analysis.calculus.fderiv.comp
β·
Mathlib.Analysis.Calculus.FDeriv.Comp
The following section lists changes to this file in mathlib3 and mathlib4 that occured after the initial port. Most recent changes are shown first. Hovering over a commit will show all commits associated with the same mathlib3 commit.
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mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83
@@ -3,7 +3,7 @@ Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, SΓ©bastien GouΓ«zel, Yury Kudryashov
-/
-import Analysis.Calculus.Fderiv.Basic
+import Analysis.Calculus.FDeriv.Basic
#align_import analysis.calculus.fderiv.comp from "leanprover-community/mathlib"@"38df578a6450a8c5142b3727e3ae894c2300cae0"
@@ -260,7 +260,7 @@ protected theorem HasFDerivAtFilter.iterate {f : E β E} {f' : E βL[π] E}
by
induction' n with n ihn
Β· exact hasFDerivAtFilter_id x L
- Β· rw [Function.iterate_succ, pow_succ']
+ Β· rw [Function.iterate_succ, pow_succ]
rw [β hx] at ihn
exact ihn.comp x hf hL
#align has_fderiv_at_filter.iterate HasFDerivAtFilter.iterate
@@ -293,7 +293,7 @@ protected theorem HasStrictFDerivAt.iterate {f : E β E} {f' : E βL[π] E}
by
induction' n with n ihn
Β· exact hasStrictFDerivAt_id x
- Β· rw [Function.iterate_succ, pow_succ']
+ Β· rw [Function.iterate_succ, pow_succ]
rw [β hx] at ihn
exact ihn.comp x hf
#align has_strict_fderiv_at.iterate HasStrictFDerivAt.iterate
mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83
@@ -78,7 +78,7 @@ theorem HasFDerivAtFilter.comp {g : F β G} {g' : F βL[π] G} {L' : Filter
example {g : F β G} {g' : F βL[π] G} (hg : HasFDerivAtFilter g g' (f x) (L.map f))
(hf : HasFDerivAtFilter f f' x L) : HasFDerivAtFilter (g β f) (g'.comp f') x L :=
by
- unfold HasFDerivAtFilter at hg
+ unfold HasFDerivAtFilter at hg
have :=
calc
(fun x' => g (f x') - g (f x) - g' (f x' - f x)) =o[L] fun x' => f x' - f x :=
@@ -261,7 +261,7 @@ protected theorem HasFDerivAtFilter.iterate {f : E β E} {f' : E βL[π] E}
induction' n with n ihn
Β· exact hasFDerivAtFilter_id x L
Β· rw [Function.iterate_succ, pow_succ']
- rw [β hx] at ihn
+ rw [β hx] at ihn
exact ihn.comp x hf hL
#align has_fderiv_at_filter.iterate HasFDerivAtFilter.iterate
-/
@@ -294,7 +294,7 @@ protected theorem HasStrictFDerivAt.iterate {f : E β E} {f' : E βL[π] E}
induction' n with n ihn
Β· exact hasStrictFDerivAt_id x
Β· rw [Function.iterate_succ, pow_succ']
- rw [β hx] at ihn
+ rw [β hx] at ihn
exact ihn.comp x hf
#align has_strict_fderiv_at.iterate HasStrictFDerivAt.iterate
-/
mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451
@@ -3,7 +3,7 @@ Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, SΓ©bastien GouΓ«zel, Yury Kudryashov
-/
-import Mathbin.Analysis.Calculus.Fderiv.Basic
+import Analysis.Calculus.Fderiv.Basic
#align_import analysis.calculus.fderiv.comp from "leanprover-community/mathlib"@"38df578a6450a8c5142b3727e3ae894c2300cae0"
mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345
@@ -2,14 +2,11 @@
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, SΓ©bastien GouΓ«zel, Yury Kudryashov
-
-! This file was ported from Lean 3 source module analysis.calculus.fderiv.comp
-! leanprover-community/mathlib commit 38df578a6450a8c5142b3727e3ae894c2300cae0
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathbin.Analysis.Calculus.Fderiv.Basic
+#align_import analysis.calculus.fderiv.comp from "leanprover-community/mathlib"@"38df578a6450a8c5142b3727e3ae894c2300cae0"
+
/-!
# The derivative of a composition (chain rule)
mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423
@@ -65,6 +65,7 @@ get confused since there are too many possibilities for composition -/
variable (x)
+#print HasFDerivAtFilter.comp /-
theorem HasFDerivAtFilter.comp {g : F β G} {g' : F βL[π] G} {L' : Filter F}
(hg : HasFDerivAtFilter g g' (f x) L') (hf : HasFDerivAtFilter f f' x L) (hL : Tendsto f L L') :
HasFDerivAtFilter (g β f) (g'.comp f') x L :=
@@ -73,6 +74,7 @@ theorem HasFDerivAtFilter.comp {g : F β G} {g' : F βL[π] G} {L' : Filter
let eqβ := (hg.comp_tendsto hL).trans_isBigO hf.isBigO_sub
refine' eqβ.triangle (eqβ.congr_left fun x' => _); simp
#align has_fderiv_at_filter.comp HasFDerivAtFilter.comp
+-/
/- A readable version of the previous theorem,
a general form of the chain rule. -/
@@ -93,58 +95,77 @@ example {g : F β G} {g' : F βL[π] G} (hg : HasFDerivAtFilter g g' (f x) (
_ =O[L] fun x' => f x' - f x - f' (x' - x) := (g'.is_O_comp _ _)
_ =o[L] fun x' => x' - x := hf
+#print HasFDerivWithinAt.comp /-
theorem HasFDerivWithinAt.comp {g : F β G} {g' : F βL[π] G} {t : Set F}
(hg : HasFDerivWithinAt g g' t (f x)) (hf : HasFDerivWithinAt f f' s x) (hst : MapsTo f s t) :
HasFDerivWithinAt (g β f) (g'.comp f') s x :=
hg.comp x hf <| hf.ContinuousWithinAt.tendsto_nhdsWithin hst
#align has_fderiv_within_at.comp HasFDerivWithinAt.comp
+-/
+#print HasFDerivAt.comp_hasFDerivWithinAt /-
theorem HasFDerivAt.comp_hasFDerivWithinAt {g : F β G} {g' : F βL[π] G}
(hg : HasFDerivAt g g' (f x)) (hf : HasFDerivWithinAt f f' s x) :
HasFDerivWithinAt (g β f) (g'.comp f') s x :=
hg.comp x hf hf.ContinuousWithinAt
#align has_fderiv_at.comp_has_fderiv_within_at HasFDerivAt.comp_hasFDerivWithinAt
+-/
+#print HasFDerivWithinAt.comp_of_mem /-
theorem HasFDerivWithinAt.comp_of_mem {g : F β G} {g' : F βL[π] G} {t : Set F}
(hg : HasFDerivWithinAt g g' t (f x)) (hf : HasFDerivWithinAt f f' s x)
(hst : Tendsto f (π[s] x) (π[t] f x)) : HasFDerivWithinAt (g β f) (g'.comp f') s x :=
HasFDerivAtFilter.comp x hg hf hst
#align has_fderiv_within_at.comp_of_mem HasFDerivWithinAt.comp_of_mem
+-/
+#print HasFDerivAt.comp /-
/-- The chain rule. -/
theorem HasFDerivAt.comp {g : F β G} {g' : F βL[π] G} (hg : HasFDerivAt g g' (f x))
(hf : HasFDerivAt f f' x) : HasFDerivAt (g β f) (g'.comp f') x :=
hg.comp x hf hf.ContinuousAt
#align has_fderiv_at.comp HasFDerivAt.comp
+-/
+#print DifferentiableWithinAt.comp /-
theorem DifferentiableWithinAt.comp {g : F β G} {t : Set F}
(hg : DifferentiableWithinAt π g t (f x)) (hf : DifferentiableWithinAt π f s x)
(h : MapsTo f s t) : DifferentiableWithinAt π (g β f) s x :=
(hg.HasFDerivWithinAt.comp x hf.HasFDerivWithinAt h).DifferentiableWithinAt
#align differentiable_within_at.comp DifferentiableWithinAt.comp
+-/
+#print DifferentiableWithinAt.comp' /-
theorem DifferentiableWithinAt.comp' {g : F β G} {t : Set F}
(hg : DifferentiableWithinAt π g t (f x)) (hf : DifferentiableWithinAt π f s x) :
DifferentiableWithinAt π (g β f) (s β© f β»ΒΉ' t) x :=
hg.comp x (hf.mono (inter_subset_left _ _)) (inter_subset_right _ _)
#align differentiable_within_at.comp' DifferentiableWithinAt.comp'
+-/
+#print DifferentiableAt.comp /-
theorem DifferentiableAt.comp {g : F β G} (hg : DifferentiableAt π g (f x))
(hf : DifferentiableAt π f x) : DifferentiableAt π (g β f) x :=
(hg.HasFDerivAt.comp x hf.HasFDerivAt).DifferentiableAt
#align differentiable_at.comp DifferentiableAt.comp
+-/
+#print DifferentiableAt.comp_differentiableWithinAt /-
theorem DifferentiableAt.comp_differentiableWithinAt {g : F β G} (hg : DifferentiableAt π g (f x))
(hf : DifferentiableWithinAt π f s x) : DifferentiableWithinAt π (g β f) s x :=
hg.DifferentiableWithinAt.comp x hf (mapsTo_univ _ _)
#align differentiable_at.comp_differentiable_within_at DifferentiableAt.comp_differentiableWithinAt
+-/
+#print fderivWithin.comp /-
theorem fderivWithin.comp {g : F β G} {t : Set F} (hg : DifferentiableWithinAt π g t (f x))
(hf : DifferentiableWithinAt π f s x) (h : MapsTo f s t) (hxs : UniqueDiffWithinAt π s x) :
fderivWithin π (g β f) s x = (fderivWithin π g t (f x)).comp (fderivWithin π f s x) :=
(hg.HasFDerivWithinAt.comp x hf.HasFDerivWithinAt h).fderivWithin hxs
#align fderiv_within.comp fderivWithin.comp
+-/
+#print fderivWithin_fderivWithin /-
/-- A version of `fderiv_within.comp` that is useful to rewrite the composition of two derivatives
into a single derivative. This version always applies, but creates a new side-goal `f x = y`. -/
theorem fderivWithin_fderivWithin {g : F β G} {f : E β F} {x : E} {y : F} {s : Set E} {t : Set F}
@@ -153,7 +174,9 @@ theorem fderivWithin_fderivWithin {g : F β G} {f : E β F} {x : E} {y : F} {s
fderivWithin π g t y (fderivWithin π f s x v) = fderivWithin π (g β f) s x v := by subst y;
rw [fderivWithin.comp x hg hf h hxs]; rfl
#align fderiv_within_fderiv_within fderivWithin_fderivWithin
+-/
+#print fderivWithin.compβ /-
/-- Ternary version of `fderiv_within.comp`, with equality assumptions of basepoints added, in
order to apply more easily as a rewrite from right-to-left. -/
theorem fderivWithin.compβ {g' : G β G'} {g : F β G} {t : Set F} {u : Set G} {y : F} {y' : G}
@@ -169,32 +192,44 @@ theorem fderivWithin.compβ {g' : G β G'} {g : F β G} {t : Set F} {u : Set
h2g.comp h2f).fderivWithin
hxs
#align fderiv_within.compβ fderivWithin.compβ
+-/
+#print fderiv.comp /-
theorem fderiv.comp {g : F β G} (hg : DifferentiableAt π g (f x)) (hf : DifferentiableAt π f x) :
fderiv π (g β f) x = (fderiv π g (f x)).comp (fderiv π f x) :=
(hg.HasFDerivAt.comp x hf.HasFDerivAt).fderiv
#align fderiv.comp fderiv.comp
+-/
+#print fderiv.comp_fderivWithin /-
theorem fderiv.comp_fderivWithin {g : F β G} (hg : DifferentiableAt π g (f x))
(hf : DifferentiableWithinAt π f s x) (hxs : UniqueDiffWithinAt π s x) :
fderivWithin π (g β f) s x = (fderiv π g (f x)).comp (fderivWithin π f s x) :=
(hg.HasFDerivAt.comp_hasFDerivWithinAt x hf.HasFDerivWithinAt).fderivWithin hxs
#align fderiv.comp_fderiv_within fderiv.comp_fderivWithin
+-/
+#print DifferentiableOn.comp /-
theorem DifferentiableOn.comp {g : F β G} {t : Set F} (hg : DifferentiableOn π g t)
(hf : DifferentiableOn π f s) (st : MapsTo f s t) : DifferentiableOn π (g β f) s := fun x hx =>
DifferentiableWithinAt.comp x (hg (f x) (st hx)) (hf x hx) st
#align differentiable_on.comp DifferentiableOn.comp
+-/
+#print Differentiable.comp /-
theorem Differentiable.comp {g : F β G} (hg : Differentiable π g) (hf : Differentiable π f) :
Differentiable π (g β f) := fun x => DifferentiableAt.comp x (hg (f x)) (hf x)
#align differentiable.comp Differentiable.comp
+-/
+#print Differentiable.comp_differentiableOn /-
theorem Differentiable.comp_differentiableOn {g : F β G} (hg : Differentiable π g)
(hf : DifferentiableOn π f s) : DifferentiableOn π (g β f) s :=
hg.DifferentiableOn.comp hf (mapsTo_univ _ _)
#align differentiable.comp_differentiable_on Differentiable.comp_differentiableOn
+-/
+#print HasStrictFDerivAt.comp /-
/-- The chain rule for derivatives in the sense of strict differentiability. -/
protected theorem HasStrictFDerivAt.comp {g : F β G} {g' : F βL[π] G}
(hg : HasStrictFDerivAt g g' (f x)) (hf : HasStrictFDerivAt f f' x) :
@@ -203,19 +238,25 @@ protected theorem HasStrictFDerivAt.comp {g : F β G} {g' : F βL[π] G}
hf.isBigO_sub).triangle <|
by simpa only [g'.map_sub, f'.coe_comp'] using (g'.is_O_comp _ _).trans_isLittleO hf
#align has_strict_fderiv_at.comp HasStrictFDerivAt.comp
+-/
+#print Differentiable.iterate /-
protected theorem Differentiable.iterate {f : E β E} (hf : Differentiable π f) (n : β) :
Differentiable π (f^[n]) :=
Nat.recOn n differentiable_id fun n ihn => ihn.comp hf
#align differentiable.iterate Differentiable.iterate
+-/
+#print DifferentiableOn.iterate /-
protected theorem DifferentiableOn.iterate {f : E β E} (hf : DifferentiableOn π f s)
(hs : MapsTo f s s) (n : β) : DifferentiableOn π (f^[n]) s :=
Nat.recOn n differentiableOn_id fun n ihn => ihn.comp hf hs
#align differentiable_on.iterate DifferentiableOn.iterate
+-/
variable {x}
+#print HasFDerivAtFilter.iterate /-
protected theorem HasFDerivAtFilter.iterate {f : E β E} {f' : E βL[π] E}
(hf : HasFDerivAtFilter f f' x L) (hL : Tendsto f L L) (hx : f x = x) (n : β) :
HasFDerivAtFilter (f^[n]) (f' ^ n) x L :=
@@ -226,7 +267,9 @@ protected theorem HasFDerivAtFilter.iterate {f : E β E} {f' : E βL[π] E}
rw [β hx] at ihn
exact ihn.comp x hf hL
#align has_fderiv_at_filter.iterate HasFDerivAtFilter.iterate
+-/
+#print HasFDerivAt.iterate /-
protected theorem HasFDerivAt.iterate {f : E β E} {f' : E βL[π] E} (hf : HasFDerivAt f f' x)
(hx : f x = x) (n : β) : HasFDerivAt (f^[n]) (f' ^ n) x :=
by
@@ -234,7 +277,9 @@ protected theorem HasFDerivAt.iterate {f : E β E} {f' : E βL[π] E} (hf :
convert hf.continuous_at
exact hx.symm
#align has_fderiv_at.iterate HasFDerivAt.iterate
+-/
+#print HasFDerivWithinAt.iterate /-
protected theorem HasFDerivWithinAt.iterate {f : E β E} {f' : E βL[π] E}
(hf : HasFDerivWithinAt f f' s x) (hx : f x = x) (hs : MapsTo f s s) (n : β) :
HasFDerivWithinAt (f^[n]) (f' ^ n) s x :=
@@ -243,7 +288,9 @@ protected theorem HasFDerivWithinAt.iterate {f : E β E} {f' : E βL[π] E}
convert tendsto_inf.2 β¨hf.continuous_within_at, _β©
exacts [hx.symm, (tendsto_principal_principal.2 hs).mono_left inf_le_right]
#align has_fderiv_within_at.iterate HasFDerivWithinAt.iterate
+-/
+#print HasStrictFDerivAt.iterate /-
protected theorem HasStrictFDerivAt.iterate {f : E β E} {f' : E βL[π] E}
(hf : HasStrictFDerivAt f f' x) (hx : f x = x) (n : β) : HasStrictFDerivAt (f^[n]) (f' ^ n) x :=
by
@@ -253,16 +300,21 @@ protected theorem HasStrictFDerivAt.iterate {f : E β E} {f' : E βL[π] E}
rw [β hx] at ihn
exact ihn.comp x hf
#align has_strict_fderiv_at.iterate HasStrictFDerivAt.iterate
+-/
+#print DifferentiableAt.iterate /-
protected theorem DifferentiableAt.iterate {f : E β E} (hf : DifferentiableAt π f x) (hx : f x = x)
(n : β) : DifferentiableAt π (f^[n]) x :=
(hf.HasFDerivAt.iterate hx n).DifferentiableAt
#align differentiable_at.iterate DifferentiableAt.iterate
+-/
+#print DifferentiableWithinAt.iterate /-
protected theorem DifferentiableWithinAt.iterate {f : E β E} (hf : DifferentiableWithinAt π f s x)
(hx : f x = x) (hs : MapsTo f s s) (n : β) : DifferentiableWithinAt π (f^[n]) s x :=
(hf.HasFDerivWithinAt.iterate hx hs n).DifferentiableWithinAt
#align differentiable_within_at.iterate DifferentiableWithinAt.iterate
+-/
end Composition
mathlib commit https://github.com/leanprover-community/mathlib/commit/7e5137f579de09a059a5ce98f364a04e221aabf0
@@ -85,7 +85,6 @@ example {g : F β G} {g' : F βL[π] G} (hg : HasFDerivAtFilter g g' (f x) (
(fun x' => g (f x') - g (f x) - g' (f x' - f x)) =o[L] fun x' => f x' - f x :=
hg.comp_tendsto le_rfl
_ =O[L] fun x' => x' - x := hf.is_O_sub
-
refine' this.triangle _
calc
(fun x' : E => g' (f x' - f x) - g'.comp f' (x' - x)) =αΆ [L] fun x' =>
@@ -93,7 +92,6 @@ example {g : F β G} {g' : F βL[π] G} (hg : HasFDerivAtFilter g g' (f x) (
eventually_of_forall fun x' => by simp
_ =O[L] fun x' => f x' - f x - f' (x' - x) := (g'.is_O_comp _ _)
_ =o[L] fun x' => x' - x := hf
-
theorem HasFDerivWithinAt.comp {g : F β G} {g' : F βL[π] G} {t : Set F}
(hg : HasFDerivWithinAt g g' t (f x)) (hf : HasFDerivWithinAt f f' s x) (hst : MapsTo f s t) :
mathlib commit https://github.com/leanprover-community/mathlib/commit/cca40788df1b8755d5baf17ab2f27dacc2e17acb
@@ -79,7 +79,7 @@ theorem HasFDerivAtFilter.comp {g : F β G} {g' : F βL[π] G} {L' : Filter
example {g : F β G} {g' : F βL[π] G} (hg : HasFDerivAtFilter g g' (f x) (L.map f))
(hf : HasFDerivAtFilter f f' x L) : HasFDerivAtFilter (g β f) (g'.comp f') x L :=
by
- unfold HasFDerivAtFilter at hg
+ unfold HasFDerivAtFilter at hg
have :=
calc
(fun x' => g (f x') - g (f x) - g' (f x' - f x)) =o[L] fun x' => f x' - f x :=
@@ -225,7 +225,7 @@ protected theorem HasFDerivAtFilter.iterate {f : E β E} {f' : E βL[π] E}
induction' n with n ihn
Β· exact hasFDerivAtFilter_id x L
Β· rw [Function.iterate_succ, pow_succ']
- rw [β hx] at ihn
+ rw [β hx] at ihn
exact ihn.comp x hf hL
#align has_fderiv_at_filter.iterate HasFDerivAtFilter.iterate
@@ -243,7 +243,7 @@ protected theorem HasFDerivWithinAt.iterate {f : E β E} {f' : E βL[π] E}
by
refine' hf.iterate _ hx n
convert tendsto_inf.2 β¨hf.continuous_within_at, _β©
- exacts[hx.symm, (tendsto_principal_principal.2 hs).mono_left inf_le_right]
+ exacts [hx.symm, (tendsto_principal_principal.2 hs).mono_left inf_le_right]
#align has_fderiv_within_at.iterate HasFDerivWithinAt.iterate
protected theorem HasStrictFDerivAt.iterate {f : E β E} {f' : E βL[π] E}
@@ -252,7 +252,7 @@ protected theorem HasStrictFDerivAt.iterate {f : E β E} {f' : E βL[π] E}
induction' n with n ihn
Β· exact hasStrictFDerivAt_id x
Β· rw [Function.iterate_succ, pow_succ']
- rw [β hx] at ihn
+ rw [β hx] at ihn
exact ihn.comp x hf
#align has_strict_fderiv_at.iterate HasStrictFDerivAt.iterate
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -26,7 +26,7 @@ composition of functions (the chain rule).
open Filter Asymptotics ContinuousLinearMap Set Metric
-open Topology Classical NNReal Filter Asymptotics ENNReal
+open scoped Topology Classical NNReal Filter Asymptotics ENNReal
noncomputable section
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -65,9 +65,6 @@ get confused since there are too many possibilities for composition -/
variable (x)
-/- warning: has_fderiv_at_filter.comp -> HasFDerivAtFilter.comp is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align has_fderiv_at_filter.comp HasFDerivAtFilter.compβ'. -/
theorem HasFDerivAtFilter.comp {g : F β G} {g' : F βL[π] G} {L' : Filter F}
(hg : HasFDerivAtFilter g g' (f x) L') (hf : HasFDerivAtFilter f f' x L) (hL : Tendsto f L L') :
HasFDerivAtFilter (g β f) (g'.comp f') x L :=
@@ -98,100 +95,58 @@ example {g : F β G} {g' : F βL[π] G} (hg : HasFDerivAtFilter g g' (f x) (
_ =o[L] fun x' => x' - x := hf
-/- warning: has_fderiv_within_at.comp -> HasFDerivWithinAt.comp is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align has_fderiv_within_at.comp HasFDerivWithinAt.compβ'. -/
theorem HasFDerivWithinAt.comp {g : F β G} {g' : F βL[π] G} {t : Set F}
(hg : HasFDerivWithinAt g g' t (f x)) (hf : HasFDerivWithinAt f f' s x) (hst : MapsTo f s t) :
HasFDerivWithinAt (g β f) (g'.comp f') s x :=
hg.comp x hf <| hf.ContinuousWithinAt.tendsto_nhdsWithin hst
#align has_fderiv_within_at.comp HasFDerivWithinAt.comp
-/- warning: has_fderiv_at.comp_has_fderiv_within_at -> HasFDerivAt.comp_hasFDerivWithinAt is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align has_fderiv_at.comp_has_fderiv_within_at HasFDerivAt.comp_hasFDerivWithinAtβ'. -/
theorem HasFDerivAt.comp_hasFDerivWithinAt {g : F β G} {g' : F βL[π] G}
(hg : HasFDerivAt g g' (f x)) (hf : HasFDerivWithinAt f f' s x) :
HasFDerivWithinAt (g β f) (g'.comp f') s x :=
hg.comp x hf hf.ContinuousWithinAt
#align has_fderiv_at.comp_has_fderiv_within_at HasFDerivAt.comp_hasFDerivWithinAt
-/- warning: has_fderiv_within_at.comp_of_mem -> HasFDerivWithinAt.comp_of_mem is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align has_fderiv_within_at.comp_of_mem HasFDerivWithinAt.comp_of_memβ'. -/
theorem HasFDerivWithinAt.comp_of_mem {g : F β G} {g' : F βL[π] G} {t : Set F}
(hg : HasFDerivWithinAt g g' t (f x)) (hf : HasFDerivWithinAt f f' s x)
(hst : Tendsto f (π[s] x) (π[t] f x)) : HasFDerivWithinAt (g β f) (g'.comp f') s x :=
HasFDerivAtFilter.comp x hg hf hst
#align has_fderiv_within_at.comp_of_mem HasFDerivWithinAt.comp_of_mem
-/- warning: has_fderiv_at.comp -> HasFDerivAt.comp is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align has_fderiv_at.comp HasFDerivAt.compβ'. -/
/-- The chain rule. -/
theorem HasFDerivAt.comp {g : F β G} {g' : F βL[π] G} (hg : HasFDerivAt g g' (f x))
(hf : HasFDerivAt f f' x) : HasFDerivAt (g β f) (g'.comp f') x :=
hg.comp x hf hf.ContinuousAt
#align has_fderiv_at.comp HasFDerivAt.comp
-/- warning: differentiable_within_at.comp -> DifferentiableWithinAt.comp is a dubious translation:
-lean 3 declaration is
- forall {π : Type.{u1}} [_inst_1 : NontriviallyNormedField.{u1} π] {E : Type.{u2}} [_inst_2 : NormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)] {F : Type.{u3}} [_inst_4 : NormedAddCommGroup.{u3} F] [_inst_5 : NormedSpace.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)] {G : Type.{u4}} [_inst_6 : NormedAddCommGroup.{u4} G] [_inst_7 : NormedSpace.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)] {f : E -> F} (x : E) {s : Set.{u2} E} {g : F -> G} {t : Set.{u3} F}, (DifferentiableWithinAt.{u1, u3, u4} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g t (f x)) -> (DifferentiableWithinAt.{u1, u2, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f s x) -> (Set.MapsTo.{u2, u3} E F f s t) -> (DifferentiableWithinAt.{u1, u2, u4} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (Function.comp.{succ u2, succ u3, succ u4} E F G g f) s x)
-but is expected to have type
- forall {π : Type.{u3}} [_inst_1 : NontriviallyNormedField.{u3} π] {E : Type.{u1}} [_inst_2 : NormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u3, u1} π E (NontriviallyNormedField.toNormedField.{u3} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)] {F : Type.{u4}} [_inst_4 : NormedAddCommGroup.{u4} F] [_inst_5 : NormedSpace.{u3, u4} π F (NontriviallyNormedField.toNormedField.{u3} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_4)] {G : Type.{u2}} [_inst_6 : NormedAddCommGroup.{u2} G] [_inst_7 : NormedSpace.{u3, u2} π G (NontriviallyNormedField.toNormedField.{u3} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)] {f : E -> F} (x : E) {s : Set.{u1} E} {g : F -> G} {t : Set.{u4} F}, (DifferentiableWithinAt.{u3, u4, u2} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g t (f x)) -> (DifferentiableWithinAt.{u3, u1, u4} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f s x) -> (Set.MapsTo.{u1, u4} E F f s t) -> (DifferentiableWithinAt.{u3, u1, u2} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (Function.comp.{succ u1, succ u4, succ u2} E F G g f) s x)
-Case conversion may be inaccurate. Consider using '#align differentiable_within_at.comp DifferentiableWithinAt.compβ'. -/
theorem DifferentiableWithinAt.comp {g : F β G} {t : Set F}
(hg : DifferentiableWithinAt π g t (f x)) (hf : DifferentiableWithinAt π f s x)
(h : MapsTo f s t) : DifferentiableWithinAt π (g β f) s x :=
(hg.HasFDerivWithinAt.comp x hf.HasFDerivWithinAt h).DifferentiableWithinAt
#align differentiable_within_at.comp DifferentiableWithinAt.comp
-/- warning: differentiable_within_at.comp' -> DifferentiableWithinAt.comp' is a dubious translation:
-lean 3 declaration is
- forall {π : Type.{u1}} [_inst_1 : NontriviallyNormedField.{u1} π] {E : Type.{u2}} [_inst_2 : NormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)] {F : Type.{u3}} [_inst_4 : NormedAddCommGroup.{u3} F] [_inst_5 : NormedSpace.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)] {G : Type.{u4}} [_inst_6 : NormedAddCommGroup.{u4} G] [_inst_7 : NormedSpace.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)] {f : E -> F} (x : E) {s : Set.{u2} E} {g : F -> G} {t : Set.{u3} F}, (DifferentiableWithinAt.{u1, u3, u4} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g t (f x)) -> (DifferentiableWithinAt.{u1, u2, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f s x) -> (DifferentiableWithinAt.{u1, u2, u4} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (Function.comp.{succ u2, succ u3, succ u4} E F G g f) (Inter.inter.{u2} (Set.{u2} E) (Set.hasInter.{u2} E) s (Set.preimage.{u2, u3} E F f t)) x)
-but is expected to have type
- forall {π : Type.{u3}} [_inst_1 : NontriviallyNormedField.{u3} π] {E : Type.{u1}} [_inst_2 : NormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u3, u1} π E (NontriviallyNormedField.toNormedField.{u3} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)] {F : Type.{u4}} [_inst_4 : NormedAddCommGroup.{u4} F] [_inst_5 : NormedSpace.{u3, u4} π F (NontriviallyNormedField.toNormedField.{u3} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_4)] {G : Type.{u2}} [_inst_6 : NormedAddCommGroup.{u2} G] [_inst_7 : NormedSpace.{u3, u2} π G (NontriviallyNormedField.toNormedField.{u3} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)] {f : E -> F} (x : E) {s : Set.{u1} E} {g : F -> G} {t : Set.{u4} F}, (DifferentiableWithinAt.{u3, u4, u2} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g t (f x)) -> (DifferentiableWithinAt.{u3, u1, u4} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f s x) -> (DifferentiableWithinAt.{u3, u1, u2} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (Function.comp.{succ u1, succ u4, succ u2} E F G g f) (Inter.inter.{u1} (Set.{u1} E) (Set.instInterSet.{u1} E) s (Set.preimage.{u1, u4} E F f t)) x)
-Case conversion may be inaccurate. Consider using '#align differentiable_within_at.comp' DifferentiableWithinAt.comp'β'. -/
theorem DifferentiableWithinAt.comp' {g : F β G} {t : Set F}
(hg : DifferentiableWithinAt π g t (f x)) (hf : DifferentiableWithinAt π f s x) :
DifferentiableWithinAt π (g β f) (s β© f β»ΒΉ' t) x :=
hg.comp x (hf.mono (inter_subset_left _ _)) (inter_subset_right _ _)
#align differentiable_within_at.comp' DifferentiableWithinAt.comp'
-/- warning: differentiable_at.comp -> DifferentiableAt.comp is a dubious translation:
-lean 3 declaration is
- forall {π : Type.{u1}} [_inst_1 : NontriviallyNormedField.{u1} π] {E : Type.{u2}} [_inst_2 : NormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)] {F : Type.{u3}} [_inst_4 : NormedAddCommGroup.{u3} F] [_inst_5 : NormedSpace.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)] {G : Type.{u4}} [_inst_6 : NormedAddCommGroup.{u4} G] [_inst_7 : NormedSpace.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)] {f : E -> F} (x : E) {g : F -> G}, (DifferentiableAt.{u1, u3, u4} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g (f x)) -> (DifferentiableAt.{u1, u2, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f x) -> (DifferentiableAt.{u1, u2, u4} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (Function.comp.{succ u2, succ u3, succ u4} E F G g f) x)
-but is expected to have type
- forall {π : Type.{u4}} [_inst_1 : NontriviallyNormedField.{u4} π] {E : Type.{u1}} [_inst_2 : NormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u4, u1} π E (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)] {F : Type.{u3}} [_inst_4 : NormedAddCommGroup.{u3} F] [_inst_5 : NormedSpace.{u4, u3} π F (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)] {G : Type.{u2}} [_inst_6 : NormedAddCommGroup.{u2} G] [_inst_7 : NormedSpace.{u4, u2} π G (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)] {f : E -> F} (x : E) {g : F -> G}, (DifferentiableAt.{u4, u3, u2} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g (f x)) -> (DifferentiableAt.{u4, u1, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f x) -> (DifferentiableAt.{u4, u1, u2} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (Function.comp.{succ u1, succ u3, succ u2} E F G g f) x)
-Case conversion may be inaccurate. Consider using '#align differentiable_at.comp DifferentiableAt.compβ'. -/
theorem DifferentiableAt.comp {g : F β G} (hg : DifferentiableAt π g (f x))
(hf : DifferentiableAt π f x) : DifferentiableAt π (g β f) x :=
(hg.HasFDerivAt.comp x hf.HasFDerivAt).DifferentiableAt
#align differentiable_at.comp DifferentiableAt.comp
-/- warning: differentiable_at.comp_differentiable_within_at -> DifferentiableAt.comp_differentiableWithinAt is a dubious translation:
-lean 3 declaration is
- forall {π : Type.{u1}} [_inst_1 : NontriviallyNormedField.{u1} π] {E : Type.{u2}} [_inst_2 : NormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)] {F : Type.{u3}} [_inst_4 : NormedAddCommGroup.{u3} F] [_inst_5 : NormedSpace.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)] {G : Type.{u4}} [_inst_6 : NormedAddCommGroup.{u4} G] [_inst_7 : NormedSpace.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)] {f : E -> F} (x : E) {s : Set.{u2} E} {g : F -> G}, (DifferentiableAt.{u1, u3, u4} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g (f x)) -> (DifferentiableWithinAt.{u1, u2, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f s x) -> (DifferentiableWithinAt.{u1, u2, u4} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (Function.comp.{succ u2, succ u3, succ u4} E F G g f) s x)
-but is expected to have type
- forall {π : Type.{u4}} [_inst_1 : NontriviallyNormedField.{u4} π] {E : Type.{u1}} [_inst_2 : NormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u4, u1} π E (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)] {F : Type.{u3}} [_inst_4 : NormedAddCommGroup.{u3} F] [_inst_5 : NormedSpace.{u4, u3} π F (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)] {G : Type.{u2}} [_inst_6 : NormedAddCommGroup.{u2} G] [_inst_7 : NormedSpace.{u4, u2} π G (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)] {f : E -> F} (x : E) {s : Set.{u1} E} {g : F -> G}, (DifferentiableAt.{u4, u3, u2} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g (f x)) -> (DifferentiableWithinAt.{u4, u1, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f s x) -> (DifferentiableWithinAt.{u4, u1, u2} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (Function.comp.{succ u1, succ u3, succ u2} E F G g f) s x)
-Case conversion may be inaccurate. Consider using '#align differentiable_at.comp_differentiable_within_at DifferentiableAt.comp_differentiableWithinAtβ'. -/
theorem DifferentiableAt.comp_differentiableWithinAt {g : F β G} (hg : DifferentiableAt π g (f x))
(hf : DifferentiableWithinAt π f s x) : DifferentiableWithinAt π (g β f) s x :=
hg.DifferentiableWithinAt.comp x hf (mapsTo_univ _ _)
#align differentiable_at.comp_differentiable_within_at DifferentiableAt.comp_differentiableWithinAt
-/- warning: fderiv_within.comp -> fderivWithin.comp is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align fderiv_within.comp fderivWithin.compβ'. -/
theorem fderivWithin.comp {g : F β G} {t : Set F} (hg : DifferentiableWithinAt π g t (f x))
(hf : DifferentiableWithinAt π f s x) (h : MapsTo f s t) (hxs : UniqueDiffWithinAt π s x) :
fderivWithin π (g β f) s x = (fderivWithin π g t (f x)).comp (fderivWithin π f s x) :=
(hg.HasFDerivWithinAt.comp x hf.HasFDerivWithinAt h).fderivWithin hxs
#align fderiv_within.comp fderivWithin.comp
-/- warning: fderiv_within_fderiv_within -> fderivWithin_fderivWithin is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align fderiv_within_fderiv_within fderivWithin_fderivWithinβ'. -/
/-- A version of `fderiv_within.comp` that is useful to rewrite the composition of two derivatives
into a single derivative. This version always applies, but creates a new side-goal `f x = y`. -/
theorem fderivWithin_fderivWithin {g : F β G} {f : E β F} {x : E} {y : F} {s : Set E} {t : Set F}
@@ -201,9 +156,6 @@ theorem fderivWithin_fderivWithin {g : F β G} {f : E β F} {x : E} {y : F} {s
rw [fderivWithin.comp x hg hf h hxs]; rfl
#align fderiv_within_fderiv_within fderivWithin_fderivWithin
-/- warning: fderiv_within.compβ -> fderivWithin.compβ is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align fderiv_within.compβ fderivWithin.compββ'. -/
/-- Ternary version of `fderiv_within.comp`, with equality assumptions of basepoints added, in
order to apply more easily as a rewrite from right-to-left. -/
theorem fderivWithin.compβ {g' : G β G'} {g : F β G} {t : Set F} {u : Set G} {y : F} {y' : G}
@@ -220,58 +172,31 @@ theorem fderivWithin.compβ {g' : G β G'} {g : F β G} {t : Set F} {u : Set
hxs
#align fderiv_within.compβ fderivWithin.compβ
-/- warning: fderiv.comp -> fderiv.comp is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align fderiv.comp fderiv.compβ'. -/
theorem fderiv.comp {g : F β G} (hg : DifferentiableAt π g (f x)) (hf : DifferentiableAt π f x) :
fderiv π (g β f) x = (fderiv π g (f x)).comp (fderiv π f x) :=
(hg.HasFDerivAt.comp x hf.HasFDerivAt).fderiv
#align fderiv.comp fderiv.comp
-/- warning: fderiv.comp_fderiv_within -> fderiv.comp_fderivWithin is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align fderiv.comp_fderiv_within fderiv.comp_fderivWithinβ'. -/
theorem fderiv.comp_fderivWithin {g : F β G} (hg : DifferentiableAt π g (f x))
(hf : DifferentiableWithinAt π f s x) (hxs : UniqueDiffWithinAt π s x) :
fderivWithin π (g β f) s x = (fderiv π g (f x)).comp (fderivWithin π f s x) :=
(hg.HasFDerivAt.comp_hasFDerivWithinAt x hf.HasFDerivWithinAt).fderivWithin hxs
#align fderiv.comp_fderiv_within fderiv.comp_fderivWithin
-/- warning: differentiable_on.comp -> DifferentiableOn.comp is a dubious translation:
-lean 3 declaration is
- forall {π : Type.{u1}} [_inst_1 : NontriviallyNormedField.{u1} π] {E : Type.{u2}} [_inst_2 : NormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)] {F : Type.{u3}} [_inst_4 : NormedAddCommGroup.{u3} F] [_inst_5 : NormedSpace.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)] {G : Type.{u4}} [_inst_6 : NormedAddCommGroup.{u4} G] [_inst_7 : NormedSpace.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)] {f : E -> F} {s : Set.{u2} E} {g : F -> G} {t : Set.{u3} F}, (DifferentiableOn.{u1, u3, u4} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g t) -> (DifferentiableOn.{u1, u2, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f s) -> (Set.MapsTo.{u2, u3} E F f s t) -> (DifferentiableOn.{u1, u2, u4} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (Function.comp.{succ u2, succ u3, succ u4} E F G g f) s)
-but is expected to have type
- forall {π : Type.{u3}} [_inst_1 : NontriviallyNormedField.{u3} π] {E : Type.{u1}} [_inst_2 : NormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u3, u1} π E (NontriviallyNormedField.toNormedField.{u3} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)] {F : Type.{u4}} [_inst_4 : NormedAddCommGroup.{u4} F] [_inst_5 : NormedSpace.{u3, u4} π F (NontriviallyNormedField.toNormedField.{u3} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_4)] {G : Type.{u2}} [_inst_6 : NormedAddCommGroup.{u2} G] [_inst_7 : NormedSpace.{u3, u2} π G (NontriviallyNormedField.toNormedField.{u3} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)] {f : E -> F} {s : Set.{u1} E} {g : F -> G} {t : Set.{u4} F}, (DifferentiableOn.{u3, u4, u2} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g t) -> (DifferentiableOn.{u3, u1, u4} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f s) -> (Set.MapsTo.{u1, u4} E F f s t) -> (DifferentiableOn.{u3, u1, u2} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (Function.comp.{succ u1, succ u4, succ u2} E F G g f) s)
-Case conversion may be inaccurate. Consider using '#align differentiable_on.comp DifferentiableOn.compβ'. -/
theorem DifferentiableOn.comp {g : F β G} {t : Set F} (hg : DifferentiableOn π g t)
(hf : DifferentiableOn π f s) (st : MapsTo f s t) : DifferentiableOn π (g β f) s := fun x hx =>
DifferentiableWithinAt.comp x (hg (f x) (st hx)) (hf x hx) st
#align differentiable_on.comp DifferentiableOn.comp
-/- warning: differentiable.comp -> Differentiable.comp is a dubious translation:
-lean 3 declaration is
- forall {π : Type.{u1}} [_inst_1 : NontriviallyNormedField.{u1} π] {E : Type.{u2}} [_inst_2 : NormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)] {F : Type.{u3}} [_inst_4 : NormedAddCommGroup.{u3} F] [_inst_5 : NormedSpace.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)] {G : Type.{u4}} [_inst_6 : NormedAddCommGroup.{u4} G] [_inst_7 : NormedSpace.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)] {f : E -> F} {g : F -> G}, (Differentiable.{u1, u3, u4} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g) -> (Differentiable.{u1, u2, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f) -> (Differentiable.{u1, u2, u4} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (Function.comp.{succ u2, succ u3, succ u4} E F G g f))
-but is expected to have type
- forall {π : Type.{u4}} [_inst_1 : NontriviallyNormedField.{u4} π] {E : Type.{u1}} [_inst_2 : NormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u4, u1} π E (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)] {F : Type.{u3}} [_inst_4 : NormedAddCommGroup.{u3} F] [_inst_5 : NormedSpace.{u4, u3} π F (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)] {G : Type.{u2}} [_inst_6 : NormedAddCommGroup.{u2} G] [_inst_7 : NormedSpace.{u4, u2} π G (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)] {f : E -> F} {g : F -> G}, (Differentiable.{u4, u3, u2} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g) -> (Differentiable.{u4, u1, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f) -> (Differentiable.{u4, u1, u2} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (Function.comp.{succ u1, succ u3, succ u2} E F G g f))
-Case conversion may be inaccurate. Consider using '#align differentiable.comp Differentiable.compβ'. -/
theorem Differentiable.comp {g : F β G} (hg : Differentiable π g) (hf : Differentiable π f) :
Differentiable π (g β f) := fun x => DifferentiableAt.comp x (hg (f x)) (hf x)
#align differentiable.comp Differentiable.comp
-/- warning: differentiable.comp_differentiable_on -> Differentiable.comp_differentiableOn is a dubious translation:
-lean 3 declaration is
- forall {π : Type.{u1}} [_inst_1 : NontriviallyNormedField.{u1} π] {E : Type.{u2}} [_inst_2 : NormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)] {F : Type.{u3}} [_inst_4 : NormedAddCommGroup.{u3} F] [_inst_5 : NormedSpace.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)] {G : Type.{u4}} [_inst_6 : NormedAddCommGroup.{u4} G] [_inst_7 : NormedSpace.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)] {f : E -> F} {s : Set.{u2} E} {g : F -> G}, (Differentiable.{u1, u3, u4} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g) -> (DifferentiableOn.{u1, u2, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f s) -> (DifferentiableOn.{u1, u2, u4} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (Function.comp.{succ u2, succ u3, succ u4} E F G g f) s)
-but is expected to have type
- forall {π : Type.{u4}} [_inst_1 : NontriviallyNormedField.{u4} π] {E : Type.{u1}} [_inst_2 : NormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u4, u1} π E (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)] {F : Type.{u3}} [_inst_4 : NormedAddCommGroup.{u3} F] [_inst_5 : NormedSpace.{u4, u3} π F (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)] {G : Type.{u2}} [_inst_6 : NormedAddCommGroup.{u2} G] [_inst_7 : NormedSpace.{u4, u2} π G (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)] {f : E -> F} {s : Set.{u1} E} {g : F -> G}, (Differentiable.{u4, u3, u2} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g) -> (DifferentiableOn.{u4, u1, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f s) -> (DifferentiableOn.{u4, u1, u2} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (Function.comp.{succ u1, succ u3, succ u2} E F G g f) s)
-Case conversion may be inaccurate. Consider using '#align differentiable.comp_differentiable_on Differentiable.comp_differentiableOnβ'. -/
theorem Differentiable.comp_differentiableOn {g : F β G} (hg : Differentiable π g)
(hf : DifferentiableOn π f s) : DifferentiableOn π (g β f) s :=
hg.DifferentiableOn.comp hf (mapsTo_univ _ _)
#align differentiable.comp_differentiable_on Differentiable.comp_differentiableOn
-/- warning: has_strict_fderiv_at.comp -> HasStrictFDerivAt.comp is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align has_strict_fderiv_at.comp HasStrictFDerivAt.compβ'. -/
/-- The chain rule for derivatives in the sense of strict differentiability. -/
protected theorem HasStrictFDerivAt.comp {g : F β G} {g' : F βL[π] G}
(hg : HasStrictFDerivAt g g' (f x)) (hf : HasStrictFDerivAt f f' x) :
@@ -281,23 +206,11 @@ protected theorem HasStrictFDerivAt.comp {g : F β G} {g' : F βL[π] G}
by simpa only [g'.map_sub, f'.coe_comp'] using (g'.is_O_comp _ _).trans_isLittleO hf
#align has_strict_fderiv_at.comp HasStrictFDerivAt.comp
-/- warning: differentiable.iterate -> Differentiable.iterate is a dubious translation:
-lean 3 declaration is
- forall {π : Type.{u1}} [_inst_1 : NontriviallyNormedField.{u1} π] {E : Type.{u2}} [_inst_2 : NormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)] {f : E -> E}, (Differentiable.{u1, u2, u2} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 f) -> (forall (n : Nat), Differentiable.{u1, u2, u2} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 (Nat.iterate.{succ u2} E f n))
-but is expected to have type
- forall {π : Type.{u2}} [_inst_1 : NontriviallyNormedField.{u2} π] {E : Type.{u1}} [_inst_2 : NormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)] {f : E -> E}, (Differentiable.{u2, u1, u1} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 f) -> (forall (n : Nat), Differentiable.{u2, u1, u1} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 (Nat.iterate.{succ u1} E f n))
-Case conversion may be inaccurate. Consider using '#align differentiable.iterate Differentiable.iterateβ'. -/
protected theorem Differentiable.iterate {f : E β E} (hf : Differentiable π f) (n : β) :
Differentiable π (f^[n]) :=
Nat.recOn n differentiable_id fun n ihn => ihn.comp hf
#align differentiable.iterate Differentiable.iterate
-/- warning: differentiable_on.iterate -> DifferentiableOn.iterate is a dubious translation:
-lean 3 declaration is
- forall {π : Type.{u1}} [_inst_1 : NontriviallyNormedField.{u1} π] {E : Type.{u2}} [_inst_2 : NormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)] {s : Set.{u2} E} {f : E -> E}, (DifferentiableOn.{u1, u2, u2} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 f s) -> (Set.MapsTo.{u2, u2} E E f s s) -> (forall (n : Nat), DifferentiableOn.{u1, u2, u2} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 (Nat.iterate.{succ u2} E f n) s)
-but is expected to have type
- forall {π : Type.{u2}} [_inst_1 : NontriviallyNormedField.{u2} π] {E : Type.{u1}} [_inst_2 : NormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)] {s : Set.{u1} E} {f : E -> E}, (DifferentiableOn.{u2, u1, u1} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 f s) -> (Set.MapsTo.{u1, u1} E E f s s) -> (forall (n : Nat), DifferentiableOn.{u2, u1, u1} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 (Nat.iterate.{succ u1} E f n) s)
-Case conversion may be inaccurate. Consider using '#align differentiable_on.iterate DifferentiableOn.iterateβ'. -/
protected theorem DifferentiableOn.iterate {f : E β E} (hf : DifferentiableOn π f s)
(hs : MapsTo f s s) (n : β) : DifferentiableOn π (f^[n]) s :=
Nat.recOn n differentiableOn_id fun n ihn => ihn.comp hf hs
@@ -305,9 +218,6 @@ protected theorem DifferentiableOn.iterate {f : E β E} (hf : DifferentiableOn
variable {x}
-/- warning: has_fderiv_at_filter.iterate -> HasFDerivAtFilter.iterate is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align has_fderiv_at_filter.iterate HasFDerivAtFilter.iterateβ'. -/
protected theorem HasFDerivAtFilter.iterate {f : E β E} {f' : E βL[π] E}
(hf : HasFDerivAtFilter f f' x L) (hL : Tendsto f L L) (hx : f x = x) (n : β) :
HasFDerivAtFilter (f^[n]) (f' ^ n) x L :=
@@ -319,9 +229,6 @@ protected theorem HasFDerivAtFilter.iterate {f : E β E} {f' : E βL[π] E}
exact ihn.comp x hf hL
#align has_fderiv_at_filter.iterate HasFDerivAtFilter.iterate
-/- warning: has_fderiv_at.iterate -> HasFDerivAt.iterate is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align has_fderiv_at.iterate HasFDerivAt.iterateβ'. -/
protected theorem HasFDerivAt.iterate {f : E β E} {f' : E βL[π] E} (hf : HasFDerivAt f f' x)
(hx : f x = x) (n : β) : HasFDerivAt (f^[n]) (f' ^ n) x :=
by
@@ -330,9 +237,6 @@ protected theorem HasFDerivAt.iterate {f : E β E} {f' : E βL[π] E} (hf :
exact hx.symm
#align has_fderiv_at.iterate HasFDerivAt.iterate
-/- warning: has_fderiv_within_at.iterate -> HasFDerivWithinAt.iterate is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align has_fderiv_within_at.iterate HasFDerivWithinAt.iterateβ'. -/
protected theorem HasFDerivWithinAt.iterate {f : E β E} {f' : E βL[π] E}
(hf : HasFDerivWithinAt f f' s x) (hx : f x = x) (hs : MapsTo f s s) (n : β) :
HasFDerivWithinAt (f^[n]) (f' ^ n) s x :=
@@ -342,9 +246,6 @@ protected theorem HasFDerivWithinAt.iterate {f : E β E} {f' : E βL[π] E}
exacts[hx.symm, (tendsto_principal_principal.2 hs).mono_left inf_le_right]
#align has_fderiv_within_at.iterate HasFDerivWithinAt.iterate
-/- warning: has_strict_fderiv_at.iterate -> HasStrictFDerivAt.iterate is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align has_strict_fderiv_at.iterate HasStrictFDerivAt.iterateβ'. -/
protected theorem HasStrictFDerivAt.iterate {f : E β E} {f' : E βL[π] E}
(hf : HasStrictFDerivAt f f' x) (hx : f x = x) (n : β) : HasStrictFDerivAt (f^[n]) (f' ^ n) x :=
by
@@ -355,23 +256,11 @@ protected theorem HasStrictFDerivAt.iterate {f : E β E} {f' : E βL[π] E}
exact ihn.comp x hf
#align has_strict_fderiv_at.iterate HasStrictFDerivAt.iterate
-/- warning: differentiable_at.iterate -> DifferentiableAt.iterate is a dubious translation:
-lean 3 declaration is
- forall {π : Type.{u1}} [_inst_1 : NontriviallyNormedField.{u1} π] {E : Type.{u2}} [_inst_2 : NormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)] {x : E} {f : E -> E}, (DifferentiableAt.{u1, u2, u2} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 f x) -> (Eq.{succ u2} E (f x) x) -> (forall (n : Nat), DifferentiableAt.{u1, u2, u2} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 (Nat.iterate.{succ u2} E f n) x)
-but is expected to have type
- forall {π : Type.{u2}} [_inst_1 : NontriviallyNormedField.{u2} π] {E : Type.{u1}} [_inst_2 : NormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)] {x : E} {f : E -> E}, (DifferentiableAt.{u2, u1, u1} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 f x) -> (Eq.{succ u1} E (f x) x) -> (forall (n : Nat), DifferentiableAt.{u2, u1, u1} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 (Nat.iterate.{succ u1} E f n) x)
-Case conversion may be inaccurate. Consider using '#align differentiable_at.iterate DifferentiableAt.iterateβ'. -/
protected theorem DifferentiableAt.iterate {f : E β E} (hf : DifferentiableAt π f x) (hx : f x = x)
(n : β) : DifferentiableAt π (f^[n]) x :=
(hf.HasFDerivAt.iterate hx n).DifferentiableAt
#align differentiable_at.iterate DifferentiableAt.iterate
-/- warning: differentiable_within_at.iterate -> DifferentiableWithinAt.iterate is a dubious translation:
-lean 3 declaration is
- forall {π : Type.{u1}} [_inst_1 : NontriviallyNormedField.{u1} π] {E : Type.{u2}} [_inst_2 : NormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)] {x : E} {s : Set.{u2} E} {f : E -> E}, (DifferentiableWithinAt.{u1, u2, u2} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 f s x) -> (Eq.{succ u2} E (f x) x) -> (Set.MapsTo.{u2, u2} E E f s s) -> (forall (n : Nat), DifferentiableWithinAt.{u1, u2, u2} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 (Nat.iterate.{succ u2} E f n) s x)
-but is expected to have type
- forall {π : Type.{u2}} [_inst_1 : NontriviallyNormedField.{u2} π] {E : Type.{u1}} [_inst_2 : NormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)] {x : E} {s : Set.{u1} E} {f : E -> E}, (DifferentiableWithinAt.{u2, u1, u1} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 f s x) -> (Eq.{succ u1} E (f x) x) -> (Set.MapsTo.{u1, u1} E E f s s) -> (forall (n : Nat), DifferentiableWithinAt.{u2, u1, u1} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 (Nat.iterate.{succ u1} E f n) s x)
-Case conversion may be inaccurate. Consider using '#align differentiable_within_at.iterate DifferentiableWithinAt.iterateβ'. -/
protected theorem DifferentiableWithinAt.iterate {f : E β E} (hf : DifferentiableWithinAt π f s x)
(hx : f x = x) (hs : MapsTo f s s) (n : β) : DifferentiableWithinAt π (f^[n]) s x :=
(hf.HasFDerivWithinAt.iterate hx hs n).DifferentiableWithinAt
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -74,8 +74,7 @@ theorem HasFDerivAtFilter.comp {g : F β G} {g' : F βL[π] G} {L' : Filter
by
let eqβ := (g'.isBigO_comp _ _).trans_isLittleO hf
let eqβ := (hg.comp_tendsto hL).trans_isBigO hf.isBigO_sub
- refine' eqβ.triangle (eqβ.congr_left fun x' => _)
- simp
+ refine' eqβ.triangle (eqβ.congr_left fun x' => _); simp
#align has_fderiv_at_filter.comp HasFDerivAtFilter.comp
/- A readable version of the previous theorem,
@@ -198,11 +197,8 @@ Case conversion may be inaccurate. Consider using '#align fderiv_within_fderiv_w
theorem fderivWithin_fderivWithin {g : F β G} {f : E β F} {x : E} {y : F} {s : Set E} {t : Set F}
(hg : DifferentiableWithinAt π g t y) (hf : DifferentiableWithinAt π f s x) (h : MapsTo f s t)
(hxs : UniqueDiffWithinAt π s x) (hy : f x = y) (v : E) :
- fderivWithin π g t y (fderivWithin π f s x v) = fderivWithin π (g β f) s x v :=
- by
- subst y
- rw [fderivWithin.comp x hg hf h hxs]
- rfl
+ fderivWithin π g t y (fderivWithin π f s x v) = fderivWithin π (g β f) s x v := by subst y;
+ rw [fderivWithin.comp x hg hf h hxs]; rfl
#align fderiv_within_fderiv_within fderivWithin_fderivWithin
/- warning: fderiv_within.compβ -> fderivWithin.compβ is a dubious translation:
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -66,10 +66,7 @@ get confused since there are too many possibilities for composition -/
variable (x)
/- warning: has_fderiv_at_filter.comp -> HasFDerivAtFilter.comp is a dubious translation:
-lean 3 declaration is
- forall {π : Type.{u1}} [_inst_1 : NontriviallyNormedField.{u1} π] {E : Type.{u2}} [_inst_2 : NormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)] {F : Type.{u3}} [_inst_4 : NormedAddCommGroup.{u3} F] [_inst_5 : NormedSpace.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)] {G : Type.{u4}} [_inst_6 : NormedAddCommGroup.{u4} G] [_inst_7 : NormedSpace.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)] {f : E -> F} {f' : ContinuousLinearMap.{u1, u1, u2, u3} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5)} (x : E) {L : Filter.{u2} E} {g : F -> G} {g' : ContinuousLinearMap.{u1, u1, u3, u4} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u4} G (PseudoMetricSpace.toUniformSpace.{u4} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u4} G (NormedAddCommGroup.toAddCommGroup.{u4} G _inst_6)) (NormedSpace.toModule.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6) _inst_7)} {L' : Filter.{u3} F}, (HasFDerivAtFilter.{u1, u3, u4} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g g' (f x) L') -> (HasFDerivAtFilter.{u1, u2, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f f' x L) -> (Filter.Tendsto.{u2, u3} E F f L L') -> (HasFDerivAtFilter.{u1, u2, u4} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (Function.comp.{succ u2, succ u3, succ u4} E F G g f) (ContinuousLinearMap.comp.{u1, u1, u1, u2, u3, u4} π π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u4} G (PseudoMetricSpace.toUniformSpace.{u4} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u4} G (NormedAddCommGroup.toAddCommGroup.{u4} G _inst_6)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6) _inst_7) (RingHomCompTriple.right_ids.{u1, u1} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1)))))))) g' f') x L)
-but is expected to have type
- forall {π : Type.{u4}} [_inst_1 : NontriviallyNormedField.{u4} π] {E : Type.{u1}} [_inst_2 : NormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u4, u1} π E (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)] {F : Type.{u3}} [_inst_4 : NormedAddCommGroup.{u3} F] [_inst_5 : NormedSpace.{u4, u3} π F (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)] {G : Type.{u2}} [_inst_6 : NormedAddCommGroup.{u2} G] [_inst_7 : NormedSpace.{u4, u2} π G (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)] {f : E -> F} {f' : ContinuousLinearMap.{u4, u4, u1, u3} π π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) (NormedSpace.toModule.{u4, u1} π E (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u4, u3} π F (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5)} (x : E) {L : Filter.{u1} E} {g : F -> G} {g' : ContinuousLinearMap.{u4, u4, u3, u2} π π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u2} G (PseudoMetricSpace.toUniformSpace.{u2} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u2} G (NormedAddCommGroup.toAddCommGroup.{u2} G _inst_6)) (NormedSpace.toModule.{u4, u3} π F (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u4, u2} π G (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6) _inst_7)} {L' : Filter.{u3} F}, (HasFDerivAtFilter.{u4, u3, u2} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g g' (f x) L') -> (HasFDerivAtFilter.{u4, u1, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f f' x L) -> (Filter.Tendsto.{u1, u3} E F f L L') -> (HasFDerivAtFilter.{u4, u1, u2} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (Function.comp.{succ u1, succ u3, succ u2} E F G g f) (ContinuousLinearMap.comp.{u4, u4, u4, u1, u3, u2} π π π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u2} G (PseudoMetricSpace.toUniformSpace.{u2} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u2} G (NormedAddCommGroup.toAddCommGroup.{u2} G _inst_6)) (NormedSpace.toModule.{u4, u1} π E (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u4, u3} π F (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u4, u2} π G (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6) _inst_7) (RingHomCompTriple.ids.{u4, u4} π π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1)))))))) g' f') x L)
+<too large>
Case conversion may be inaccurate. Consider using '#align has_fderiv_at_filter.comp HasFDerivAtFilter.compβ'. -/
theorem HasFDerivAtFilter.comp {g : F β G} {g' : F βL[π] G} {L' : Filter F}
(hg : HasFDerivAtFilter g g' (f x) L') (hf : HasFDerivAtFilter f f' x L) (hL : Tendsto f L L') :
@@ -103,10 +100,7 @@ example {g : F β G} {g' : F βL[π] G} (hg : HasFDerivAtFilter g g' (f x) (
/- warning: has_fderiv_within_at.comp -> HasFDerivWithinAt.comp is a dubious translation:
-lean 3 declaration is
- forall {π : Type.{u1}} [_inst_1 : NontriviallyNormedField.{u1} π] {E : Type.{u2}} [_inst_2 : NormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)] {F : Type.{u3}} [_inst_4 : NormedAddCommGroup.{u3} F] [_inst_5 : NormedSpace.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)] {G : Type.{u4}} [_inst_6 : NormedAddCommGroup.{u4} G] [_inst_7 : NormedSpace.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)] {f : E -> F} {f' : ContinuousLinearMap.{u1, u1, u2, u3} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5)} (x : E) {s : Set.{u2} E} {g : F -> G} {g' : ContinuousLinearMap.{u1, u1, u3, u4} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u4} G (PseudoMetricSpace.toUniformSpace.{u4} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u4} G (NormedAddCommGroup.toAddCommGroup.{u4} G _inst_6)) (NormedSpace.toModule.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6) _inst_7)} {t : Set.{u3} F}, (HasFDerivWithinAt.{u1, u3, u4} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g g' t (f x)) -> (HasFDerivWithinAt.{u1, u2, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f f' s x) -> (Set.MapsTo.{u2, u3} E F f s t) -> (HasFDerivWithinAt.{u1, u2, u4} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (Function.comp.{succ u2, succ u3, succ u4} E F G g f) (ContinuousLinearMap.comp.{u1, u1, u1, u2, u3, u4} π π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u4} G (PseudoMetricSpace.toUniformSpace.{u4} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u4} G (NormedAddCommGroup.toAddCommGroup.{u4} G _inst_6)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6) _inst_7) (RingHomCompTriple.right_ids.{u1, u1} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1)))))))) g' f') s x)
-but is expected to have type
- forall {π : Type.{u4}} [_inst_1 : NontriviallyNormedField.{u4} π] {E : Type.{u1}} [_inst_2 : NormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u4, u1} π E (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)] {F : Type.{u3}} [_inst_4 : NormedAddCommGroup.{u3} F] [_inst_5 : NormedSpace.{u4, u3} π F (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)] {G : Type.{u2}} [_inst_6 : NormedAddCommGroup.{u2} G] [_inst_7 : NormedSpace.{u4, u2} π G (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)] {f : E -> F} {f' : ContinuousLinearMap.{u4, u4, u1, u3} π π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) (NormedSpace.toModule.{u4, u1} π E (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u4, u3} π F (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5)} (x : E) {s : Set.{u1} E} {g : F -> G} {g' : ContinuousLinearMap.{u4, u4, u3, u2} π π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u2} G (PseudoMetricSpace.toUniformSpace.{u2} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u2} G (NormedAddCommGroup.toAddCommGroup.{u2} G _inst_6)) (NormedSpace.toModule.{u4, u3} π F (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u4, u2} π G (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6) _inst_7)} {t : Set.{u3} F}, (HasFDerivWithinAt.{u4, u3, u2} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g g' t (f x)) -> (HasFDerivWithinAt.{u4, u1, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f f' s x) -> (Set.MapsTo.{u1, u3} E F f s t) -> (HasFDerivWithinAt.{u4, u1, u2} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (Function.comp.{succ u1, succ u3, succ u2} E F G g f) (ContinuousLinearMap.comp.{u4, u4, u4, u1, u3, u2} π π π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u2} G (PseudoMetricSpace.toUniformSpace.{u2} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u2} G (NormedAddCommGroup.toAddCommGroup.{u2} G _inst_6)) (NormedSpace.toModule.{u4, u1} π E (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u4, u3} π F (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u4, u2} π G (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6) _inst_7) (RingHomCompTriple.ids.{u4, u4} π π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1)))))))) g' f') s x)
+<too large>
Case conversion may be inaccurate. Consider using '#align has_fderiv_within_at.comp HasFDerivWithinAt.compβ'. -/
theorem HasFDerivWithinAt.comp {g : F β G} {g' : F βL[π] G} {t : Set F}
(hg : HasFDerivWithinAt g g' t (f x)) (hf : HasFDerivWithinAt f f' s x) (hst : MapsTo f s t) :
@@ -115,10 +109,7 @@ theorem HasFDerivWithinAt.comp {g : F β G} {g' : F βL[π] G} {t : Set F}
#align has_fderiv_within_at.comp HasFDerivWithinAt.comp
/- warning: has_fderiv_at.comp_has_fderiv_within_at -> HasFDerivAt.comp_hasFDerivWithinAt is a dubious translation:
-lean 3 declaration is
- forall {π : Type.{u1}} [_inst_1 : NontriviallyNormedField.{u1} π] {E : Type.{u2}} [_inst_2 : NormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)] {F : Type.{u3}} [_inst_4 : NormedAddCommGroup.{u3} F] [_inst_5 : NormedSpace.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)] {G : Type.{u4}} [_inst_6 : NormedAddCommGroup.{u4} G] [_inst_7 : NormedSpace.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)] {f : E -> F} {f' : ContinuousLinearMap.{u1, u1, u2, u3} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5)} (x : E) {s : Set.{u2} E} {g : F -> G} {g' : ContinuousLinearMap.{u1, u1, u3, u4} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u4} G (PseudoMetricSpace.toUniformSpace.{u4} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u4} G (NormedAddCommGroup.toAddCommGroup.{u4} G _inst_6)) (NormedSpace.toModule.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6) _inst_7)}, (HasFDerivAt.{u1, u3, u4} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g g' (f x)) -> (HasFDerivWithinAt.{u1, u2, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f f' s x) -> (HasFDerivWithinAt.{u1, u2, u4} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (Function.comp.{succ u2, succ u3, succ u4} E F G g f) (ContinuousLinearMap.comp.{u1, u1, u1, u2, u3, u4} π π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u4} G (PseudoMetricSpace.toUniformSpace.{u4} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u4} G (NormedAddCommGroup.toAddCommGroup.{u4} G _inst_6)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6) _inst_7) (RingHomCompTriple.right_ids.{u1, u1} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1)))))))) g' f') s x)
-but is expected to have type
- forall {π : Type.{u4}} [_inst_1 : NontriviallyNormedField.{u4} π] {E : Type.{u1}} [_inst_2 : NormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u4, u1} π E (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)] {F : Type.{u3}} [_inst_4 : NormedAddCommGroup.{u3} F] [_inst_5 : NormedSpace.{u4, u3} π F (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)] {G : Type.{u2}} [_inst_6 : NormedAddCommGroup.{u2} G] [_inst_7 : NormedSpace.{u4, u2} π G (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)] {f : E -> F} {f' : ContinuousLinearMap.{u4, u4, u1, u3} π π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) (NormedSpace.toModule.{u4, u1} π E (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u4, u3} π F (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5)} (x : E) {s : Set.{u1} E} {g : F -> G} {g' : ContinuousLinearMap.{u4, u4, u3, u2} π π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u2} G (PseudoMetricSpace.toUniformSpace.{u2} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u2} G (NormedAddCommGroup.toAddCommGroup.{u2} G _inst_6)) (NormedSpace.toModule.{u4, u3} π F (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u4, u2} π G (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6) _inst_7)}, (HasFDerivAt.{u4, u3, u2} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g g' (f x)) -> (HasFDerivWithinAt.{u4, u1, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f f' s x) -> (HasFDerivWithinAt.{u4, u1, u2} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (Function.comp.{succ u1, succ u3, succ u2} E F G g f) (ContinuousLinearMap.comp.{u4, u4, u4, u1, u3, u2} π π π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u2} G (PseudoMetricSpace.toUniformSpace.{u2} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u2} G (NormedAddCommGroup.toAddCommGroup.{u2} G _inst_6)) (NormedSpace.toModule.{u4, u1} π E (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u4, u3} π F (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u4, u2} π G (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6) _inst_7) (RingHomCompTriple.ids.{u4, u4} π π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1)))))))) g' f') s x)
+<too large>
Case conversion may be inaccurate. Consider using '#align has_fderiv_at.comp_has_fderiv_within_at HasFDerivAt.comp_hasFDerivWithinAtβ'. -/
theorem HasFDerivAt.comp_hasFDerivWithinAt {g : F β G} {g' : F βL[π] G}
(hg : HasFDerivAt g g' (f x)) (hf : HasFDerivWithinAt f f' s x) :
@@ -127,10 +118,7 @@ theorem HasFDerivAt.comp_hasFDerivWithinAt {g : F β G} {g' : F βL[π] G}
#align has_fderiv_at.comp_has_fderiv_within_at HasFDerivAt.comp_hasFDerivWithinAt
/- warning: has_fderiv_within_at.comp_of_mem -> HasFDerivWithinAt.comp_of_mem is a dubious translation:
-lean 3 declaration is
- forall {π : Type.{u1}} [_inst_1 : NontriviallyNormedField.{u1} π] {E : Type.{u2}} [_inst_2 : NormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)] {F : Type.{u3}} [_inst_4 : NormedAddCommGroup.{u3} F] [_inst_5 : NormedSpace.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)] {G : Type.{u4}} [_inst_6 : NormedAddCommGroup.{u4} G] [_inst_7 : NormedSpace.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)] {f : E -> F} {f' : ContinuousLinearMap.{u1, u1, u2, u3} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5)} (x : E) {s : Set.{u2} E} {g : F -> G} {g' : ContinuousLinearMap.{u1, u1, u3, u4} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u4} G (PseudoMetricSpace.toUniformSpace.{u4} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u4} G (NormedAddCommGroup.toAddCommGroup.{u4} G _inst_6)) (NormedSpace.toModule.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6) _inst_7)} {t : Set.{u3} F}, (HasFDerivWithinAt.{u1, u3, u4} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g g' t (f x)) -> (HasFDerivWithinAt.{u1, u2, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f f' s x) -> (Filter.Tendsto.{u2, u3} E F f (nhdsWithin.{u2} E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) x s) (nhdsWithin.{u3} F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (f x) t)) -> (HasFDerivWithinAt.{u1, u2, u4} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (Function.comp.{succ u2, succ u3, succ u4} E F G g f) (ContinuousLinearMap.comp.{u1, u1, u1, u2, u3, u4} π π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u4} G (PseudoMetricSpace.toUniformSpace.{u4} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u4} G (NormedAddCommGroup.toAddCommGroup.{u4} G _inst_6)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6) _inst_7) (RingHomCompTriple.right_ids.{u1, u1} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1)))))))) g' f') s x)
-but is expected to have type
- forall {π : Type.{u4}} [_inst_1 : NontriviallyNormedField.{u4} π] {E : Type.{u1}} [_inst_2 : NormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u4, u1} π E (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)] {F : Type.{u3}} [_inst_4 : NormedAddCommGroup.{u3} F] [_inst_5 : NormedSpace.{u4, u3} π F (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)] {G : Type.{u2}} [_inst_6 : NormedAddCommGroup.{u2} G] [_inst_7 : NormedSpace.{u4, u2} π G (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)] {f : E -> F} {f' : ContinuousLinearMap.{u4, u4, u1, u3} π π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) (NormedSpace.toModule.{u4, u1} π E (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u4, u3} π F (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5)} (x : E) {s : Set.{u1} E} {g : F -> G} {g' : ContinuousLinearMap.{u4, u4, u3, u2} π π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u2} G (PseudoMetricSpace.toUniformSpace.{u2} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u2} G (NormedAddCommGroup.toAddCommGroup.{u2} G _inst_6)) (NormedSpace.toModule.{u4, u3} π F (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u4, u2} π G (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6) _inst_7)} {t : Set.{u3} F}, (HasFDerivWithinAt.{u4, u3, u2} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g g' t (f x)) -> (HasFDerivWithinAt.{u4, u1, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f f' s x) -> (Filter.Tendsto.{u1, u3} E F f (nhdsWithin.{u1} E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) x s) (nhdsWithin.{u3} F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (f x) t)) -> (HasFDerivWithinAt.{u4, u1, u2} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (Function.comp.{succ u1, succ u3, succ u2} E F G g f) (ContinuousLinearMap.comp.{u4, u4, u4, u1, u3, u2} π π π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u2} G (PseudoMetricSpace.toUniformSpace.{u2} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u2} G (NormedAddCommGroup.toAddCommGroup.{u2} G _inst_6)) (NormedSpace.toModule.{u4, u1} π E (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u4, u3} π F (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u4, u2} π G (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6) _inst_7) (RingHomCompTriple.ids.{u4, u4} π π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1)))))))) g' f') s x)
+<too large>
Case conversion may be inaccurate. Consider using '#align has_fderiv_within_at.comp_of_mem HasFDerivWithinAt.comp_of_memβ'. -/
theorem HasFDerivWithinAt.comp_of_mem {g : F β G} {g' : F βL[π] G} {t : Set F}
(hg : HasFDerivWithinAt g g' t (f x)) (hf : HasFDerivWithinAt f f' s x)
@@ -139,10 +127,7 @@ theorem HasFDerivWithinAt.comp_of_mem {g : F β G} {g' : F βL[π] G} {t : S
#align has_fderiv_within_at.comp_of_mem HasFDerivWithinAt.comp_of_mem
/- warning: has_fderiv_at.comp -> HasFDerivAt.comp is a dubious translation:
-lean 3 declaration is
- forall {π : Type.{u1}} [_inst_1 : NontriviallyNormedField.{u1} π] {E : Type.{u2}} [_inst_2 : NormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)] {F : Type.{u3}} [_inst_4 : NormedAddCommGroup.{u3} F] [_inst_5 : NormedSpace.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)] {G : Type.{u4}} [_inst_6 : NormedAddCommGroup.{u4} G] [_inst_7 : NormedSpace.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)] {f : E -> F} {f' : ContinuousLinearMap.{u1, u1, u2, u3} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5)} (x : E) {g : F -> G} {g' : ContinuousLinearMap.{u1, u1, u3, u4} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u4} G (PseudoMetricSpace.toUniformSpace.{u4} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u4} G (NormedAddCommGroup.toAddCommGroup.{u4} G _inst_6)) (NormedSpace.toModule.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6) _inst_7)}, (HasFDerivAt.{u1, u3, u4} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g g' (f x)) -> (HasFDerivAt.{u1, u2, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f f' x) -> (HasFDerivAt.{u1, u2, u4} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (Function.comp.{succ u2, succ u3, succ u4} E F G g f) (ContinuousLinearMap.comp.{u1, u1, u1, u2, u3, u4} π π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u4} G (PseudoMetricSpace.toUniformSpace.{u4} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u4} G (NormedAddCommGroup.toAddCommGroup.{u4} G _inst_6)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6) _inst_7) (RingHomCompTriple.right_ids.{u1, u1} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1)))))))) g' f') x)
-but is expected to have type
- forall {π : Type.{u4}} [_inst_1 : NontriviallyNormedField.{u4} π] {E : Type.{u1}} [_inst_2 : NormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u4, u1} π E (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)] {F : Type.{u3}} [_inst_4 : NormedAddCommGroup.{u3} F] [_inst_5 : NormedSpace.{u4, u3} π F (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)] {G : Type.{u2}} [_inst_6 : NormedAddCommGroup.{u2} G] [_inst_7 : NormedSpace.{u4, u2} π G (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)] {f : E -> F} {f' : ContinuousLinearMap.{u4, u4, u1, u3} π π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) (NormedSpace.toModule.{u4, u1} π E (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u4, u3} π F (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5)} (x : E) {g : F -> G} {g' : ContinuousLinearMap.{u4, u4, u3, u2} π π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u2} G (PseudoMetricSpace.toUniformSpace.{u2} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u2} G (NormedAddCommGroup.toAddCommGroup.{u2} G _inst_6)) (NormedSpace.toModule.{u4, u3} π F (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u4, u2} π G (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6) _inst_7)}, (HasFDerivAt.{u4, u3, u2} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g g' (f x)) -> (HasFDerivAt.{u4, u1, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f f' x) -> (HasFDerivAt.{u4, u1, u2} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (Function.comp.{succ u1, succ u3, succ u2} E F G g f) (ContinuousLinearMap.comp.{u4, u4, u4, u1, u3, u2} π π π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u2} G (PseudoMetricSpace.toUniformSpace.{u2} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u2} G (NormedAddCommGroup.toAddCommGroup.{u2} G _inst_6)) (NormedSpace.toModule.{u4, u1} π E (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u4, u3} π F (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u4, u2} π G (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6) _inst_7) (RingHomCompTriple.ids.{u4, u4} π π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1)))))))) g' f') x)
+<too large>
Case conversion may be inaccurate. Consider using '#align has_fderiv_at.comp HasFDerivAt.compβ'. -/
/-- The chain rule. -/
theorem HasFDerivAt.comp {g : F β G} {g' : F βL[π] G} (hg : HasFDerivAt g g' (f x))
@@ -197,10 +182,7 @@ theorem DifferentiableAt.comp_differentiableWithinAt {g : F β G} (hg : Differe
#align differentiable_at.comp_differentiable_within_at DifferentiableAt.comp_differentiableWithinAt
/- warning: fderiv_within.comp -> fderivWithin.comp is a dubious translation:
-lean 3 declaration is
- forall {π : Type.{u1}} [_inst_1 : NontriviallyNormedField.{u1} π] {E : Type.{u2}} [_inst_2 : NormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)] {F : Type.{u3}} [_inst_4 : NormedAddCommGroup.{u3} F] [_inst_5 : NormedSpace.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)] {G : Type.{u4}} [_inst_6 : NormedAddCommGroup.{u4} G] [_inst_7 : NormedSpace.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)] {f : E -> F} (x : E) {s : Set.{u2} E} {g : F -> G} {t : Set.{u3} F}, (DifferentiableWithinAt.{u1, u3, u4} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g t (f x)) -> (DifferentiableWithinAt.{u1, u2, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f s x) -> (Set.MapsTo.{u2, u3} E F f s t) -> (UniqueDiffWithinAt.{u1, u2} π _inst_1 E (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) s x) -> (Eq.{max (succ u2) (succ u4)} (ContinuousLinearMap.{u1, u1, u2, u4} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) G (UniformSpace.toTopologicalSpace.{u4} G (PseudoMetricSpace.toUniformSpace.{u4} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u4} G (NormedAddCommGroup.toAddCommGroup.{u4} G _inst_6)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6) _inst_7)) (fderivWithin.{u1, u2, u4} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (Function.comp.{succ u2, succ u3, succ u4} E F G g f) s x) (ContinuousLinearMap.comp.{u1, u1, u1, u2, u3, u4} π π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u4} G (PseudoMetricSpace.toUniformSpace.{u4} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u4} G (NormedAddCommGroup.toAddCommGroup.{u4} G _inst_6)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6) _inst_7) (RingHomCompTriple.right_ids.{u1, u1} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1)))))))) (fderivWithin.{u1, u3, u4} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g t (f x)) (fderivWithin.{u1, u2, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f s x)))
-but is expected to have type
- forall {π : Type.{u3}} [_inst_1 : NontriviallyNormedField.{u3} π] {E : Type.{u1}} [_inst_2 : NormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u3, u1} π E (NontriviallyNormedField.toNormedField.{u3} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)] {F : Type.{u4}} [_inst_4 : NormedAddCommGroup.{u4} F] [_inst_5 : NormedSpace.{u3, u4} π F (NontriviallyNormedField.toNormedField.{u3} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_4)] {G : Type.{u2}} [_inst_6 : NormedAddCommGroup.{u2} G] [_inst_7 : NormedSpace.{u3, u2} π G (NontriviallyNormedField.toNormedField.{u3} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)] {f : E -> F} (x : E) {s : Set.{u1} E} {g : F -> G} {t : Set.{u4} F}, (DifferentiableWithinAt.{u3, u4, u2} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g t (f x)) -> (DifferentiableWithinAt.{u3, u1, u4} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f s x) -> (Set.MapsTo.{u1, u4} E F f s t) -> (UniqueDiffWithinAt.{u3, u1} π _inst_1 E (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u3, u1} π E (NontriviallyNormedField.toNormedField.{u3} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) s x) -> (Eq.{max (succ u1) (succ u2)} (ContinuousLinearMap.{u3, u3, u1, u2} π π (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1))))) (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1))))) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) G (UniformSpace.toTopologicalSpace.{u2} G (PseudoMetricSpace.toUniformSpace.{u2} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u2} G (NormedAddCommGroup.toAddCommGroup.{u2} G _inst_6)) (NormedSpace.toModule.{u3, u1} π E (NontriviallyNormedField.toNormedField.{u3} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u3, u2} π G (NontriviallyNormedField.toNormedField.{u3} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6) _inst_7)) (fderivWithin.{u3, u1, u2} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (Function.comp.{succ u1, succ u4, succ u2} E F G g f) s x) (ContinuousLinearMap.comp.{u3, u3, u3, u1, u4, u2} π π π (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1))))) (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1))))) (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1))))) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1))))))) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1))))))) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u2} G (PseudoMetricSpace.toUniformSpace.{u2} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u2} G (NormedAddCommGroup.toAddCommGroup.{u2} G _inst_6)) (NormedSpace.toModule.{u3, u1} π E (NontriviallyNormedField.toNormedField.{u3} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u3, u4} π F (NontriviallyNormedField.toNormedField.{u3} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_4) _inst_5) (NormedSpace.toModule.{u3, u2} π G (NontriviallyNormedField.toNormedField.{u3} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6) _inst_7) (RingHomCompTriple.ids.{u3, u3} π π (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1))))) (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1))))) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1)))))))) (fderivWithin.{u3, u4, u2} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g t (f x)) (fderivWithin.{u3, u1, u4} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f s x)))
+<too large>
Case conversion may be inaccurate. Consider using '#align fderiv_within.comp fderivWithin.compβ'. -/
theorem fderivWithin.comp {g : F β G} {t : Set F} (hg : DifferentiableWithinAt π g t (f x))
(hf : DifferentiableWithinAt π f s x) (h : MapsTo f s t) (hxs : UniqueDiffWithinAt π s x) :
@@ -209,10 +191,7 @@ theorem fderivWithin.comp {g : F β G} {t : Set F} (hg : DifferentiableWithinAt
#align fderiv_within.comp fderivWithin.comp
/- warning: fderiv_within_fderiv_within -> fderivWithin_fderivWithin is a dubious translation:
-lean 3 declaration is
- forall {π : Type.{u1}} [_inst_1 : NontriviallyNormedField.{u1} π] {E : Type.{u2}} [_inst_2 : NormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)] {F : Type.{u3}} [_inst_4 : NormedAddCommGroup.{u3} F] [_inst_5 : NormedSpace.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)] {G : Type.{u4}} [_inst_6 : NormedAddCommGroup.{u4} G] [_inst_7 : NormedSpace.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)] {g : F -> G} {f : E -> F} {x : E} {y : F} {s : Set.{u2} E} {t : Set.{u3} F}, (DifferentiableWithinAt.{u1, u3, u4} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g t y) -> (DifferentiableWithinAt.{u1, u2, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f s x) -> (Set.MapsTo.{u2, u3} E F f s t) -> (UniqueDiffWithinAt.{u1, u2} π _inst_1 E (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) s x) -> (Eq.{succ u3} F (f x) y) -> (forall (v : E), Eq.{succ u4} G (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearMap.{u1, u1, u3, u4} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u4} G (PseudoMetricSpace.toUniformSpace.{u4} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u4} G (NormedAddCommGroup.toAddCommGroup.{u4} G _inst_6)) (NormedSpace.toModule.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) 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(NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π G (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} G _inst_6) _inst_7)))) (fderivWithin.{u2, u4, u1} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (Function.comp.{succ u4, succ u3, succ u1} E F G g f) s x) v))
+<too large>
Case conversion may be inaccurate. Consider using '#align fderiv_within_fderiv_within fderivWithin_fderivWithinβ'. -/
/-- A version of `fderiv_within.comp` that is useful to rewrite the composition of two derivatives
into a single derivative. This version always applies, but creates a new side-goal `f x = y`. -/
@@ -227,10 +206,7 @@ theorem fderivWithin_fderivWithin {g : F β G} {f : E β F} {x : E} {y : F} {s
#align fderiv_within_fderiv_within fderivWithin_fderivWithin
/- warning: fderiv_within.compβ -> fderivWithin.compβ is a dubious translation:
-lean 3 declaration is
- forall {π : Type.{u1}} [_inst_1 : NontriviallyNormedField.{u1} π] {E : Type.{u2}} [_inst_2 : NormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)] {F : Type.{u3}} [_inst_4 : NormedAddCommGroup.{u3} F] [_inst_5 : NormedSpace.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)] {G : Type.{u4}} [_inst_6 : NormedAddCommGroup.{u4} G] [_inst_7 : NormedSpace.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)] {G' : Type.{u5}} [_inst_8 : NormedAddCommGroup.{u5} G'] [_inst_9 : NormedSpace.{u1, u5} π G' (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G' _inst_8)] {f : E -> F} (x : E) {s : Set.{u2} E} {g' : G -> G'} {g : F -> G} {t : Set.{u3} F} {u : Set.{u4} G} {y : F} {y' : G}, (DifferentiableWithinAt.{u1, u4, u5} π _inst_1 G _inst_6 _inst_7 G' _inst_8 _inst_9 g' u y') -> (DifferentiableWithinAt.{u1, u3, u4} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g t y) -> (DifferentiableWithinAt.{u1, u2, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f s x) -> (Set.MapsTo.{u3, u4} F G g t u) -> (Set.MapsTo.{u2, u3} E F f s t) -> (Eq.{succ u4} G (g y) y') -> (Eq.{succ u3} F (f x) y) -> (UniqueDiffWithinAt.{u1, u2} π _inst_1 E (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) s x) -> (Eq.{max (succ u2) (succ u5)} (ContinuousLinearMap.{u1, u1, u2, u5} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) G' (UniformSpace.toTopologicalSpace.{u5} G' (PseudoMetricSpace.toUniformSpace.{u5} G' (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G' (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G' _inst_8)))) (AddCommGroup.toAddCommMonoid.{u5} G' (NormedAddCommGroup.toAddCommGroup.{u5} G' _inst_8)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u5} π G' (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G' _inst_8) _inst_9)) (fderivWithin.{u1, u2, u5} π _inst_1 E _inst_2 _inst_3 G' _inst_8 _inst_9 (Function.comp.{succ u2, succ u4, succ u5} E G G' g' (Function.comp.{succ u2, succ u3, succ u4} E F G g f)) s x) (ContinuousLinearMap.comp.{u1, u1, u1, u2, u4, u5} π π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) G (UniformSpace.toTopologicalSpace.{u4} G (PseudoMetricSpace.toUniformSpace.{u4} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u4} G (NormedAddCommGroup.toAddCommGroup.{u4} G _inst_6)) G' (UniformSpace.toTopologicalSpace.{u5} G' (PseudoMetricSpace.toUniformSpace.{u5} G' (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G' (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G' _inst_8)))) (AddCommGroup.toAddCommMonoid.{u5} G' (NormedAddCommGroup.toAddCommGroup.{u5} G' _inst_8)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6) _inst_7) (NormedSpace.toModule.{u1, u5} π G' (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G' _inst_8) _inst_9) (RingHomCompTriple.right_ids.{u1, u1} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1)))))))) (fderivWithin.{u1, u4, u5} π _inst_1 G _inst_6 _inst_7 G' _inst_8 _inst_9 g' u y') (ContinuousLinearMap.comp.{u1, u1, u1, u2, u3, u4} π π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u4} G (PseudoMetricSpace.toUniformSpace.{u4} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u4} G (NormedAddCommGroup.toAddCommGroup.{u4} G _inst_6)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6) _inst_7) (RingHomCompTriple.right_ids.{u1, u1} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1)))))))) (fderivWithin.{u1, u3, u4} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g t y) (fderivWithin.{u1, u2, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f s x))))
-but is expected to have type
- forall {π : Type.{u3}} [_inst_1 : NontriviallyNormedField.{u3} π] {E : Type.{u1}} [_inst_2 : NormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u3, u1} π E (NontriviallyNormedField.toNormedField.{u3} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)] {F : Type.{u5}} [_inst_4 : NormedAddCommGroup.{u5} F] [_inst_5 : NormedSpace.{u3, u5} π F (NontriviallyNormedField.toNormedField.{u3} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} F _inst_4)] {G : Type.{u4}} [_inst_6 : NormedAddCommGroup.{u4} G] [_inst_7 : NormedSpace.{u3, u4} π G (NontriviallyNormedField.toNormedField.{u3} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)] {G' : Type.{u2}} [_inst_8 : NormedAddCommGroup.{u2} G'] [_inst_9 : NormedSpace.{u3, u2} π G' (NontriviallyNormedField.toNormedField.{u3} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G' _inst_8)] {f : E -> F} (x : E) {s : Set.{u1} E} {g' : G -> G'} {g : F -> G} {t : Set.{u5} F} {u : Set.{u4} G} {y : F} {y' : G}, (DifferentiableWithinAt.{u3, u4, u2} π _inst_1 G _inst_6 _inst_7 G' _inst_8 _inst_9 g' u y') -> (DifferentiableWithinAt.{u3, u5, u4} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g t y) -> (DifferentiableWithinAt.{u3, u1, u5} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f s x) -> (Set.MapsTo.{u5, u4} F G g t u) -> (Set.MapsTo.{u1, u5} E F f s t) -> (Eq.{succ u4} G (g y) y') -> (Eq.{succ u5} F (f x) y) -> (UniqueDiffWithinAt.{u3, u1} π _inst_1 E (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u3, u1} π E (NontriviallyNormedField.toNormedField.{u3} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) s x) -> (Eq.{max (succ u1) (succ u2)} (ContinuousLinearMap.{u3, u3, u1, u2} π π (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1))))) (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1))))) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) G' (UniformSpace.toTopologicalSpace.{u2} G' (PseudoMetricSpace.toUniformSpace.{u2} G' (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} G' (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G' _inst_8)))) (AddCommGroup.toAddCommMonoid.{u2} G' (NormedAddCommGroup.toAddCommGroup.{u2} G' _inst_8)) (NormedSpace.toModule.{u3, u1} π E (NontriviallyNormedField.toNormedField.{u3} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u3, u2} π G' (NontriviallyNormedField.toNormedField.{u3} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G' _inst_8) _inst_9)) (fderivWithin.{u3, u1, u2} π _inst_1 E _inst_2 _inst_3 G' _inst_8 _inst_9 (Function.comp.{succ u1, succ u4, succ u2} E G G' g' (Function.comp.{succ u1, succ u5, succ u4} E F G g f)) s x) (ContinuousLinearMap.comp.{u3, u3, u3, u1, u4, u2} π π π (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1))))) (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1))))) (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1))))) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1))))))) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1))))))) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) G (UniformSpace.toTopologicalSpace.{u4} G (PseudoMetricSpace.toUniformSpace.{u4} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u4} G (NormedAddCommGroup.toAddCommGroup.{u4} G _inst_6)) G' (UniformSpace.toTopologicalSpace.{u2} G' (PseudoMetricSpace.toUniformSpace.{u2} G' (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} G' (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G' _inst_8)))) (AddCommGroup.toAddCommMonoid.{u2} G' (NormedAddCommGroup.toAddCommGroup.{u2} G' _inst_8)) (NormedSpace.toModule.{u3, u1} π E (NontriviallyNormedField.toNormedField.{u3} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u3, u4} π G (NontriviallyNormedField.toNormedField.{u3} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6) _inst_7) (NormedSpace.toModule.{u3, u2} π G' (NontriviallyNormedField.toNormedField.{u3} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G' _inst_8) _inst_9) (RingHomCompTriple.ids.{u3, u3} π π (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1))))) (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1))))) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1)))))))) (fderivWithin.{u3, u4, u2} π _inst_1 G _inst_6 _inst_7 G' _inst_8 _inst_9 g' u y') (ContinuousLinearMap.comp.{u3, u3, u3, u1, u5, u4} π π π (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1))))) (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1))))) (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1))))) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1))))))) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1))))))) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u5} F (PseudoMetricSpace.toUniformSpace.{u5} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u5} F (NormedAddCommGroup.toAddCommGroup.{u5} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u4} G (PseudoMetricSpace.toUniformSpace.{u4} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u4} G (NormedAddCommGroup.toAddCommGroup.{u4} G _inst_6)) (NormedSpace.toModule.{u3, u1} π E (NontriviallyNormedField.toNormedField.{u3} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u3, u5} π F (NontriviallyNormedField.toNormedField.{u3} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} F _inst_4) _inst_5) (NormedSpace.toModule.{u3, u4} π G (NontriviallyNormedField.toNormedField.{u3} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6) _inst_7) (RingHomCompTriple.ids.{u3, u3} π π (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1))))) (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1))))) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1)))))))) (fderivWithin.{u3, u5, u4} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g t y) (fderivWithin.{u3, u1, u5} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f s x))))
+<too large>
Case conversion may be inaccurate. Consider using '#align fderiv_within.compβ fderivWithin.compββ'. -/
/-- Ternary version of `fderiv_within.comp`, with equality assumptions of basepoints added, in
order to apply more easily as a rewrite from right-to-left. -/
@@ -249,10 +225,7 @@ theorem fderivWithin.compβ {g' : G β G'} {g : F β G} {t : Set F} {u : Set
#align fderiv_within.compβ fderivWithin.compβ
/- warning: fderiv.comp -> fderiv.comp is a dubious translation:
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- forall {π : Type.{u1}} [_inst_1 : NontriviallyNormedField.{u1} π] {E : Type.{u2}} [_inst_2 : NormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)] {F : Type.{u3}} [_inst_4 : NormedAddCommGroup.{u3} F] [_inst_5 : NormedSpace.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)] {G : Type.{u4}} [_inst_6 : NormedAddCommGroup.{u4} G] [_inst_7 : NormedSpace.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)] {f : E -> F} (x : E) {g : F -> G}, (DifferentiableAt.{u1, u3, u4} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g (f x)) -> (DifferentiableAt.{u1, u2, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f x) -> (Eq.{max (succ u2) (succ u4)} (ContinuousLinearMap.{u1, u1, u2, u4} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) G (UniformSpace.toTopologicalSpace.{u4} G (PseudoMetricSpace.toUniformSpace.{u4} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} G 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_inst_4 _inst_5 G _inst_6 _inst_7 g (f x)) (fderiv.{u1, u2, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f x)))
-but is expected to have type
- forall {π : Type.{u4}} [_inst_1 : NontriviallyNormedField.{u4} π] {E : Type.{u1}} [_inst_2 : NormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u4, u1} π E (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)] {F : Type.{u3}} [_inst_4 : NormedAddCommGroup.{u3} F] [_inst_5 : NormedSpace.{u4, u3} π F (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)] {G : Type.{u2}} [_inst_6 : NormedAddCommGroup.{u2} G] [_inst_7 : NormedSpace.{u4, u2} π G (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)] {f : E -> F} (x : E) {g : F -> G}, (DifferentiableAt.{u4, u3, u2} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g (f x)) -> (DifferentiableAt.{u4, u1, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f x) -> (Eq.{max (succ u1) (succ u2)} (ContinuousLinearMap.{u4, u4, u1, u2} π π 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+<too large>
Case conversion may be inaccurate. Consider using '#align fderiv.comp fderiv.compβ'. -/
theorem fderiv.comp {g : F β G} (hg : DifferentiableAt π g (f x)) (hf : DifferentiableAt π f x) :
fderiv π (g β f) x = (fderiv π g (f x)).comp (fderiv π f x) :=
@@ -260,10 +233,7 @@ theorem fderiv.comp {g : F β G} (hg : DifferentiableAt π g (f x)) (hf : Dif
#align fderiv.comp fderiv.comp
/- warning: fderiv.comp_fderiv_within -> fderiv.comp_fderivWithin is a dubious translation:
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- forall {π : Type.{u1}} [_inst_1 : NontriviallyNormedField.{u1} π] {E : Type.{u2}} [_inst_2 : NormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)] {F : Type.{u3}} [_inst_4 : NormedAddCommGroup.{u3} F] [_inst_5 : NormedSpace.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)] {G : Type.{u4}} [_inst_6 : NormedAddCommGroup.{u4} G] [_inst_7 : NormedSpace.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)] {f : E -> F} (x : E) {s : Set.{u2} E} {g : F -> G}, (DifferentiableAt.{u1, u3, u4} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g (f x)) -> (DifferentiableWithinAt.{u1, u2, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f s x) -> (UniqueDiffWithinAt.{u1, u2} π _inst_1 E (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) s x) -> (Eq.{max (succ u2) (succ u4)} (ContinuousLinearMap.{u1, u1, u2, u4} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π 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(NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u4} G (PseudoMetricSpace.toUniformSpace.{u4} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u4} G (NormedAddCommGroup.toAddCommGroup.{u4} G _inst_6)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6) _inst_7) (RingHomCompTriple.right_ids.{u1, u1} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1)))))))) (fderiv.{u1, u3, u4} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g (f x)) (fderivWithin.{u1, u2, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f s x)))
-but is expected to have type
- forall {π : Type.{u4}} [_inst_1 : NontriviallyNormedField.{u4} π] {E : Type.{u1}} [_inst_2 : NormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u4, u1} π E (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)] {F : Type.{u3}} [_inst_4 : NormedAddCommGroup.{u3} F] [_inst_5 : NormedSpace.{u4, u3} π F (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)] {G : Type.{u2}} [_inst_6 : NormedAddCommGroup.{u2} G] [_inst_7 : NormedSpace.{u4, u2} π G (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)] {f : E -> F} (x : E) {s : Set.{u1} E} {g : F -> G}, (DifferentiableAt.{u4, u3, u2} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g (f x)) -> (DifferentiableWithinAt.{u4, u1, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f s x) -> (UniqueDiffWithinAt.{u4, u1} π _inst_1 E (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u4, u1} π E (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) s x) -> (Eq.{max (succ u1) (succ u2)} (ContinuousLinearMap.{u4, u4, u1, u2} π π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) G (UniformSpace.toTopologicalSpace.{u2} G (PseudoMetricSpace.toUniformSpace.{u2} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u2} G (NormedAddCommGroup.toAddCommGroup.{u2} G _inst_6)) (NormedSpace.toModule.{u4, u1} π E (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u4, u2} π G (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6) _inst_7)) (fderivWithin.{u4, u1, u2} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (Function.comp.{succ u1, succ u3, succ u2} E F G g f) s x) (ContinuousLinearMap.comp.{u4, u4, u4, u1, u3, u2} π π π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u2} G (PseudoMetricSpace.toUniformSpace.{u2} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u2} G (NormedAddCommGroup.toAddCommGroup.{u2} G _inst_6)) (NormedSpace.toModule.{u4, u1} π E (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u4, u3} π F (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u4, u2} π G (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6) _inst_7) (RingHomCompTriple.ids.{u4, u4} π π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1)))))))) (fderiv.{u4, u3, u2} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g (f x)) (fderivWithin.{u4, u1, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f s x)))
+<too large>
Case conversion may be inaccurate. Consider using '#align fderiv.comp_fderiv_within fderiv.comp_fderivWithinβ'. -/
theorem fderiv.comp_fderivWithin {g : F β G} (hg : DifferentiableAt π g (f x))
(hf : DifferentiableWithinAt π f s x) (hxs : UniqueDiffWithinAt π s x) :
@@ -304,10 +274,7 @@ theorem Differentiable.comp_differentiableOn {g : F β G} (hg : Differentiable
#align differentiable.comp_differentiable_on Differentiable.comp_differentiableOn
/- warning: has_strict_fderiv_at.comp -> HasStrictFDerivAt.comp is a dubious translation:
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- forall {π : Type.{u1}} [_inst_1 : NontriviallyNormedField.{u1} π] {E : Type.{u2}} [_inst_2 : NormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)] {F : Type.{u3}} [_inst_4 : NormedAddCommGroup.{u3} F] [_inst_5 : NormedSpace.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)] {G : Type.{u4}} [_inst_6 : NormedAddCommGroup.{u4} G] [_inst_7 : NormedSpace.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)] {f : E -> F} {f' : ContinuousLinearMap.{u1, u1, u2, u3} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π 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(NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5)} (x : E) {g : F -> G} {g' : ContinuousLinearMap.{u1, u1, u3, u4} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u4} G (PseudoMetricSpace.toUniformSpace.{u4} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u4} G (NormedAddCommGroup.toAddCommGroup.{u4} G _inst_6)) (NormedSpace.toModule.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6) _inst_7)}, (HasStrictFDerivAt.{u1, u3, u4} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g g' (f x)) -> (HasStrictFDerivAt.{u1, u2, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f f' x) -> (HasStrictFDerivAt.{u1, u2, u4} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (fun (x : E) => g (f x)) (ContinuousLinearMap.comp.{u1, u1, u1, u2, u3, u4} π π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u4} G (PseudoMetricSpace.toUniformSpace.{u4} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u4} G (NormedAddCommGroup.toAddCommGroup.{u4} G _inst_6)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6) _inst_7) (RingHomCompTriple.right_ids.{u1, u1} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1)))))))) g' f') x)
-but is expected to have type
- forall {π : Type.{u4}} [_inst_1 : NontriviallyNormedField.{u4} π] {E : Type.{u1}} [_inst_2 : NormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u4, u1} π E (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)] {F : Type.{u3}} [_inst_4 : NormedAddCommGroup.{u3} F] [_inst_5 : NormedSpace.{u4, u3} π F (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)] {G : Type.{u2}} [_inst_6 : NormedAddCommGroup.{u2} G] [_inst_7 : NormedSpace.{u4, u2} π G (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)] {f : E -> F} {f' : ContinuousLinearMap.{u4, u4, u1, u3} π π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) (NormedSpace.toModule.{u4, u1} π E (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u4, u3} π F (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5)} (x : E) {g : F -> G} {g' : ContinuousLinearMap.{u4, u4, u3, u2} π π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u2} G (PseudoMetricSpace.toUniformSpace.{u2} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u2} G (NormedAddCommGroup.toAddCommGroup.{u2} G _inst_6)) (NormedSpace.toModule.{u4, u3} π F (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u4, u2} π G (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6) _inst_7)}, (HasStrictFDerivAt.{u4, u3, u2} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g g' (f x)) -> (HasStrictFDerivAt.{u4, u1, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f f' x) -> (HasStrictFDerivAt.{u4, u1, u2} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (fun (x : E) => g (f x)) (ContinuousLinearMap.comp.{u4, u4, u4, u1, u3, u2} π π π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u2} G (PseudoMetricSpace.toUniformSpace.{u2} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u2} G (NormedAddCommGroup.toAddCommGroup.{u2} G _inst_6)) (NormedSpace.toModule.{u4, u1} π E (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u4, u3} π F (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u4, u2} π G (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6) _inst_7) (RingHomCompTriple.ids.{u4, u4} π π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1)))))))) g' f') x)
+<too large>
Case conversion may be inaccurate. Consider using '#align has_strict_fderiv_at.comp HasStrictFDerivAt.compβ'. -/
/-- The chain rule for derivatives in the sense of strict differentiability. -/
protected theorem HasStrictFDerivAt.comp {g : F β G} {g' : F βL[π] G}
@@ -343,10 +310,7 @@ protected theorem DifferentiableOn.iterate {f : E β E} (hf : DifferentiableOn
variable {x}
/- warning: has_fderiv_at_filter.iterate -> HasFDerivAtFilter.iterate is a dubious translation:
-lean 3 declaration is
- forall {π : Type.{u1}} [_inst_1 : NontriviallyNormedField.{u1} π] {E : Type.{u2}} [_inst_2 : NormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)] {x : E} {L : Filter.{u2} E} {f : E -> E} {f' : ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)}, (HasFDerivAtFilter.{u1, u2, u2} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 f f' x L) -> (Filter.Tendsto.{u2, u2} E E f L L) -> (Eq.{succ u2} E (f x) x) -> (forall (n : Nat), HasFDerivAtFilter.{u1, u2, u2} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 (Nat.iterate.{succ u2} E f n) (HPow.hPow.{u2, 0, u2} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) Nat (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) (instHPow.{u2, 0} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) Nat (Monoid.Pow.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) (Ring.toMonoid.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) (ContinuousLinearMap.ring.{u1, u2} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1)))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (SeminormedAddCommGroup.toTopologicalAddGroup.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))))) f' n) x L)
-but is expected to have type
- forall {π : Type.{u2}} [_inst_1 : NontriviallyNormedField.{u2} π] {E : Type.{u1}} [_inst_2 : NormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)] {x : E} {L : Filter.{u1} E} {f : E -> E} {f' : ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)}, (HasFDerivAtFilter.{u2, u1, u1} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 f f' x L) -> (Filter.Tendsto.{u1, u1} E E f L L) -> (Eq.{succ u1} E (f x) x) -> (forall (n : Nat), HasFDerivAtFilter.{u2, u1, u1} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 (Nat.iterate.{succ u1} E f n) (HPow.hPow.{u1, 0, u1} (ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)) Nat (ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)) (instHPow.{u1, 0} (ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)) Nat (Monoid.Pow.{u1} (ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)) (MonoidWithZero.toMonoid.{u1} (ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)) (ContinuousLinearMap.monoidWithZero.{u2, u1} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3))))) f' n) x L)
+<too large>
Case conversion may be inaccurate. Consider using '#align has_fderiv_at_filter.iterate HasFDerivAtFilter.iterateβ'. -/
protected theorem HasFDerivAtFilter.iterate {f : E β E} {f' : E βL[π] E}
(hf : HasFDerivAtFilter f f' x L) (hL : Tendsto f L L) (hx : f x = x) (n : β) :
@@ -360,10 +324,7 @@ protected theorem HasFDerivAtFilter.iterate {f : E β E} {f' : E βL[π] E}
#align has_fderiv_at_filter.iterate HasFDerivAtFilter.iterate
/- warning: has_fderiv_at.iterate -> HasFDerivAt.iterate is a dubious translation:
-lean 3 declaration is
- forall {π : Type.{u1}} [_inst_1 : NontriviallyNormedField.{u1} π] {E : Type.{u2}} [_inst_2 : NormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)] {x : E} {f : E -> E} {f' : ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)}, (HasFDerivAt.{u1, u2, u2} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 f f' x) -> (Eq.{succ u2} E (f x) x) -> (forall (n : Nat), HasFDerivAt.{u1, u2, u2} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 (Nat.iterate.{succ u2} E f n) (HPow.hPow.{u2, 0, u2} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) Nat (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) (instHPow.{u2, 0} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) Nat (Monoid.Pow.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) (Ring.toMonoid.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) (ContinuousLinearMap.ring.{u1, u2} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1)))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (SeminormedAddCommGroup.toTopologicalAddGroup.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))))) f' n) x)
-but is expected to have type
- forall {π : Type.{u2}} [_inst_1 : NontriviallyNormedField.{u2} π] {E : Type.{u1}} [_inst_2 : NormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)] {x : E} {f : E -> E} {f' : ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)}, (HasFDerivAt.{u2, u1, u1} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 f f' x) -> (Eq.{succ u1} E (f x) x) -> (forall (n : Nat), HasFDerivAt.{u2, u1, u1} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 (Nat.iterate.{succ u1} E f n) (HPow.hPow.{u1, 0, u1} (ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)) Nat (ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)) (instHPow.{u1, 0} (ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)) Nat (Monoid.Pow.{u1} (ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)) (MonoidWithZero.toMonoid.{u1} (ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)) (ContinuousLinearMap.monoidWithZero.{u2, u1} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3))))) f' n) x)
+<too large>
Case conversion may be inaccurate. Consider using '#align has_fderiv_at.iterate HasFDerivAt.iterateβ'. -/
protected theorem HasFDerivAt.iterate {f : E β E} {f' : E βL[π] E} (hf : HasFDerivAt f f' x)
(hx : f x = x) (n : β) : HasFDerivAt (f^[n]) (f' ^ n) x :=
@@ -374,10 +335,7 @@ protected theorem HasFDerivAt.iterate {f : E β E} {f' : E βL[π] E} (hf :
#align has_fderiv_at.iterate HasFDerivAt.iterate
/- warning: has_fderiv_within_at.iterate -> HasFDerivWithinAt.iterate is a dubious translation:
-lean 3 declaration is
- forall {π : Type.{u1}} [_inst_1 : NontriviallyNormedField.{u1} π] {E : Type.{u2}} [_inst_2 : NormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)] {x : E} {s : Set.{u2} E} {f : E -> E} {f' : ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)}, (HasFDerivWithinAt.{u1, u2, u2} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 f f' s x) -> (Eq.{succ u2} E (f x) x) -> (Set.MapsTo.{u2, u2} E E f s s) -> (forall (n : Nat), HasFDerivWithinAt.{u1, u2, u2} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 (Nat.iterate.{succ u2} E f n) (HPow.hPow.{u2, 0, u2} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) Nat (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) (instHPow.{u2, 0} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) Nat (Monoid.Pow.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) (Ring.toMonoid.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) (ContinuousLinearMap.ring.{u1, u2} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1)))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (SeminormedAddCommGroup.toTopologicalAddGroup.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))))) f' n) s x)
-but is expected to have type
- forall {π : Type.{u2}} [_inst_1 : NontriviallyNormedField.{u2} π] {E : Type.{u1}} [_inst_2 : NormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)] {x : E} {s : Set.{u1} E} {f : E -> E} {f' : ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)}, (HasFDerivWithinAt.{u2, u1, u1} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 f f' s x) -> (Eq.{succ u1} E (f x) x) -> (Set.MapsTo.{u1, u1} E E f s s) -> (forall (n : Nat), HasFDerivWithinAt.{u2, u1, u1} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 (Nat.iterate.{succ u1} E f n) (HPow.hPow.{u1, 0, u1} (ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)) Nat (ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)) (instHPow.{u1, 0} (ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)) Nat (Monoid.Pow.{u1} (ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)) (MonoidWithZero.toMonoid.{u1} (ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)) (ContinuousLinearMap.monoidWithZero.{u2, u1} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3))))) f' n) s x)
+<too large>
Case conversion may be inaccurate. Consider using '#align has_fderiv_within_at.iterate HasFDerivWithinAt.iterateβ'. -/
protected theorem HasFDerivWithinAt.iterate {f : E β E} {f' : E βL[π] E}
(hf : HasFDerivWithinAt f f' s x) (hx : f x = x) (hs : MapsTo f s s) (n : β) :
@@ -389,10 +347,7 @@ protected theorem HasFDerivWithinAt.iterate {f : E β E} {f' : E βL[π] E}
#align has_fderiv_within_at.iterate HasFDerivWithinAt.iterate
/- warning: has_strict_fderiv_at.iterate -> HasStrictFDerivAt.iterate is a dubious translation:
-lean 3 declaration is
- forall {π : Type.{u1}} [_inst_1 : NontriviallyNormedField.{u1} π] {E : Type.{u2}} [_inst_2 : NormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)] {x : E} {f : E -> E} {f' : ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)}, (HasStrictFDerivAt.{u1, u2, u2} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 f f' x) -> (Eq.{succ u2} E (f x) x) -> (forall (n : Nat), HasStrictFDerivAt.{u1, u2, u2} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 (Nat.iterate.{succ u2} E f n) (HPow.hPow.{u2, 0, u2} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) Nat (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) (instHPow.{u2, 0} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) Nat (Monoid.Pow.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) (Ring.toMonoid.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) (ContinuousLinearMap.ring.{u1, u2} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1)))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (SeminormedAddCommGroup.toTopologicalAddGroup.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))))) f' n) x)
-but is expected to have type
- forall {π : Type.{u2}} [_inst_1 : NontriviallyNormedField.{u2} π] {E : Type.{u1}} [_inst_2 : NormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)] {x : E} {f : E -> E} {f' : ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)}, (HasStrictFDerivAt.{u2, u1, u1} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 f f' x) -> (Eq.{succ u1} E (f x) x) -> (forall (n : Nat), HasStrictFDerivAt.{u2, u1, u1} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 (Nat.iterate.{succ u1} E f n) (HPow.hPow.{u1, 0, u1} (ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)) Nat (ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)) (instHPow.{u1, 0} (ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)) Nat (Monoid.Pow.{u1} (ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)) (MonoidWithZero.toMonoid.{u1} (ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)) (ContinuousLinearMap.monoidWithZero.{u2, u1} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3))))) f' n) x)
+<too large>
Case conversion may be inaccurate. Consider using '#align has_strict_fderiv_at.iterate HasStrictFDerivAt.iterateβ'. -/
protected theorem HasStrictFDerivAt.iterate {f : E β E} {f' : E βL[π] E}
(hf : HasStrictFDerivAt f f' x) (hx : f x = x) (n : β) : HasStrictFDerivAt (f^[n]) (f' ^ n) x :=
mathlib commit https://github.com/leanprover-community/mathlib/commit/8d33f09cd7089ecf074b4791907588245aec5d1b
@@ -344,7 +344,7 @@ variable {x}
/- warning: has_fderiv_at_filter.iterate -> HasFDerivAtFilter.iterate is a dubious translation:
lean 3 declaration is
- forall {π : Type.{u1}} [_inst_1 : NontriviallyNormedField.{u1} π] {E : Type.{u2}} [_inst_2 : NormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)] {x : E} {L : Filter.{u2} E} {f : E -> E} {f' : ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)}, (HasFDerivAtFilter.{u1, u2, u2} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 f f' x L) -> (Filter.Tendsto.{u2, u2} E E f L L) -> (Eq.{succ u2} E (f x) x) -> (forall (n : Nat), HasFDerivAtFilter.{u1, u2, u2} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 (Nat.iterate.{succ u2} E f n) (HPow.hPow.{u2, 0, u2} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) Nat (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) (instHPow.{u2, 0} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) Nat (Monoid.Pow.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) (Ring.toMonoid.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) (ContinuousLinearMap.ring.{u1, u2} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1)))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (SeminormedAddCommGroup.to_topologicalAddGroup.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))))) f' n) x L)
+ forall {π : Type.{u1}} [_inst_1 : NontriviallyNormedField.{u1} π] {E : Type.{u2}} [_inst_2 : NormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)] {x : E} {L : Filter.{u2} E} {f : E -> E} {f' : ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)}, (HasFDerivAtFilter.{u1, u2, u2} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 f f' x L) -> (Filter.Tendsto.{u2, u2} E E f L L) -> (Eq.{succ u2} E (f x) x) -> (forall (n : Nat), HasFDerivAtFilter.{u1, u2, u2} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 (Nat.iterate.{succ u2} E f n) (HPow.hPow.{u2, 0, u2} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) Nat (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) (instHPow.{u2, 0} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) Nat (Monoid.Pow.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) (Ring.toMonoid.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) (ContinuousLinearMap.ring.{u1, u2} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1)))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (SeminormedAddCommGroup.toTopologicalAddGroup.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))))) f' n) x L)
but is expected to have type
forall {π : Type.{u2}} [_inst_1 : NontriviallyNormedField.{u2} π] {E : Type.{u1}} [_inst_2 : NormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)] {x : E} {L : Filter.{u1} E} {f : E -> E} {f' : ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)}, (HasFDerivAtFilter.{u2, u1, u1} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 f f' x L) -> (Filter.Tendsto.{u1, u1} E E f L L) -> (Eq.{succ u1} E (f x) x) -> (forall (n : Nat), HasFDerivAtFilter.{u2, u1, u1} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 (Nat.iterate.{succ u1} E f n) (HPow.hPow.{u1, 0, u1} (ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)) Nat (ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)) (instHPow.{u1, 0} (ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)) Nat (Monoid.Pow.{u1} (ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)) (MonoidWithZero.toMonoid.{u1} (ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)) (ContinuousLinearMap.monoidWithZero.{u2, u1} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3))))) f' n) x L)
Case conversion may be inaccurate. Consider using '#align has_fderiv_at_filter.iterate HasFDerivAtFilter.iterateβ'. -/
@@ -361,7 +361,7 @@ protected theorem HasFDerivAtFilter.iterate {f : E β E} {f' : E βL[π] E}
/- warning: has_fderiv_at.iterate -> HasFDerivAt.iterate is a dubious translation:
lean 3 declaration is
- forall {π : Type.{u1}} [_inst_1 : NontriviallyNormedField.{u1} π] {E : Type.{u2}} [_inst_2 : NormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)] {x : E} {f : E -> E} {f' : ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)}, (HasFDerivAt.{u1, u2, u2} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 f f' x) -> (Eq.{succ u2} E (f x) x) -> (forall (n : Nat), HasFDerivAt.{u1, u2, u2} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 (Nat.iterate.{succ u2} E f n) (HPow.hPow.{u2, 0, u2} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) Nat (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) (instHPow.{u2, 0} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) Nat (Monoid.Pow.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) (Ring.toMonoid.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) (ContinuousLinearMap.ring.{u1, u2} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1)))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (SeminormedAddCommGroup.to_topologicalAddGroup.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))))) f' n) x)
+ forall {π : Type.{u1}} [_inst_1 : NontriviallyNormedField.{u1} π] {E : Type.{u2}} [_inst_2 : NormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)] {x : E} {f : E -> E} {f' : ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)}, (HasFDerivAt.{u1, u2, u2} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 f f' x) -> (Eq.{succ u2} E (f x) x) -> (forall (n : Nat), HasFDerivAt.{u1, u2, u2} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 (Nat.iterate.{succ u2} E f n) (HPow.hPow.{u2, 0, u2} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) Nat (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) (instHPow.{u2, 0} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) Nat (Monoid.Pow.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) (Ring.toMonoid.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) (ContinuousLinearMap.ring.{u1, u2} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1)))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (SeminormedAddCommGroup.toTopologicalAddGroup.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))))) f' n) x)
but is expected to have type
forall {π : Type.{u2}} [_inst_1 : NontriviallyNormedField.{u2} π] {E : Type.{u1}} [_inst_2 : NormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)] {x : E} {f : E -> E} {f' : ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)}, (HasFDerivAt.{u2, u1, u1} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 f f' x) -> (Eq.{succ u1} E (f x) x) -> (forall (n : Nat), HasFDerivAt.{u2, u1, u1} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 (Nat.iterate.{succ u1} E f n) (HPow.hPow.{u1, 0, u1} (ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)) Nat (ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)) (instHPow.{u1, 0} (ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)) Nat (Monoid.Pow.{u1} (ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)) (MonoidWithZero.toMonoid.{u1} (ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)) (ContinuousLinearMap.monoidWithZero.{u2, u1} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3))))) f' n) x)
Case conversion may be inaccurate. Consider using '#align has_fderiv_at.iterate HasFDerivAt.iterateβ'. -/
@@ -375,7 +375,7 @@ protected theorem HasFDerivAt.iterate {f : E β E} {f' : E βL[π] E} (hf :
/- warning: has_fderiv_within_at.iterate -> HasFDerivWithinAt.iterate is a dubious translation:
lean 3 declaration is
- forall {π : Type.{u1}} [_inst_1 : NontriviallyNormedField.{u1} π] {E : Type.{u2}} [_inst_2 : NormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)] {x : E} {s : Set.{u2} E} {f : E -> E} {f' : ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)}, (HasFDerivWithinAt.{u1, u2, u2} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 f f' s x) -> (Eq.{succ u2} E (f x) x) -> (Set.MapsTo.{u2, u2} E E f s s) -> (forall (n : Nat), HasFDerivWithinAt.{u1, u2, u2} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 (Nat.iterate.{succ u2} E f n) (HPow.hPow.{u2, 0, u2} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) Nat (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) (instHPow.{u2, 0} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) Nat (Monoid.Pow.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) (Ring.toMonoid.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) (ContinuousLinearMap.ring.{u1, u2} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1)))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (SeminormedAddCommGroup.to_topologicalAddGroup.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))))) f' n) s x)
+ forall {π : Type.{u1}} [_inst_1 : NontriviallyNormedField.{u1} π] {E : Type.{u2}} [_inst_2 : NormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)] {x : E} {s : Set.{u2} E} {f : E -> E} {f' : ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)}, (HasFDerivWithinAt.{u1, u2, u2} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 f f' s x) -> (Eq.{succ u2} E (f x) x) -> (Set.MapsTo.{u2, u2} E E f s s) -> (forall (n : Nat), HasFDerivWithinAt.{u1, u2, u2} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 (Nat.iterate.{succ u2} E f n) (HPow.hPow.{u2, 0, u2} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) Nat (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) (instHPow.{u2, 0} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) Nat (Monoid.Pow.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) (Ring.toMonoid.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) (ContinuousLinearMap.ring.{u1, u2} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1)))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (SeminormedAddCommGroup.toTopologicalAddGroup.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))))) f' n) s x)
but is expected to have type
forall {π : Type.{u2}} [_inst_1 : NontriviallyNormedField.{u2} π] {E : Type.{u1}} [_inst_2 : NormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)] {x : E} {s : Set.{u1} E} {f : E -> E} {f' : ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)}, (HasFDerivWithinAt.{u2, u1, u1} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 f f' s x) -> (Eq.{succ u1} E (f x) x) -> (Set.MapsTo.{u1, u1} E E f s s) -> (forall (n : Nat), HasFDerivWithinAt.{u2, u1, u1} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 (Nat.iterate.{succ u1} E f n) (HPow.hPow.{u1, 0, u1} (ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)) Nat (ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)) (instHPow.{u1, 0} (ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)) Nat (Monoid.Pow.{u1} (ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)) (MonoidWithZero.toMonoid.{u1} (ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)) (ContinuousLinearMap.monoidWithZero.{u2, u1} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3))))) f' n) s x)
Case conversion may be inaccurate. Consider using '#align has_fderiv_within_at.iterate HasFDerivWithinAt.iterateβ'. -/
@@ -390,7 +390,7 @@ protected theorem HasFDerivWithinAt.iterate {f : E β E} {f' : E βL[π] E}
/- warning: has_strict_fderiv_at.iterate -> HasStrictFDerivAt.iterate is a dubious translation:
lean 3 declaration is
- forall {π : Type.{u1}} [_inst_1 : NontriviallyNormedField.{u1} π] {E : Type.{u2}} [_inst_2 : NormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)] {x : E} {f : E -> E} {f' : ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)}, (HasStrictFDerivAt.{u1, u2, u2} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 f f' x) -> (Eq.{succ u2} E (f x) x) -> (forall (n : Nat), HasStrictFDerivAt.{u1, u2, u2} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 (Nat.iterate.{succ u2} E f n) (HPow.hPow.{u2, 0, u2} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) Nat (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) (instHPow.{u2, 0} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) Nat (Monoid.Pow.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) (Ring.toMonoid.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) (ContinuousLinearMap.ring.{u1, u2} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1)))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (SeminormedAddCommGroup.to_topologicalAddGroup.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))))) f' n) x)
+ forall {π : Type.{u1}} [_inst_1 : NontriviallyNormedField.{u1} π] {E : Type.{u2}} [_inst_2 : NormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)] {x : E} {f : E -> E} {f' : ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)}, (HasStrictFDerivAt.{u1, u2, u2} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 f f' x) -> (Eq.{succ u2} E (f x) x) -> (forall (n : Nat), HasStrictFDerivAt.{u1, u2, u2} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 (Nat.iterate.{succ u2} E f n) (HPow.hPow.{u2, 0, u2} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) Nat (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) (instHPow.{u2, 0} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) Nat (Monoid.Pow.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) (Ring.toMonoid.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) (ContinuousLinearMap.ring.{u1, u2} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1)))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (SeminormedAddCommGroup.toTopologicalAddGroup.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))))) f' n) x)
but is expected to have type
forall {π : Type.{u2}} [_inst_1 : NontriviallyNormedField.{u2} π] {E : Type.{u1}} [_inst_2 : NormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)] {x : E} {f : E -> E} {f' : ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)}, (HasStrictFDerivAt.{u2, u1, u1} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 f f' x) -> (Eq.{succ u1} E (f x) x) -> (forall (n : Nat), HasStrictFDerivAt.{u2, u1, u1} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 (Nat.iterate.{succ u1} E f n) (HPow.hPow.{u1, 0, u1} (ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)) Nat (ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)) (instHPow.{u1, 0} (ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)) Nat (Monoid.Pow.{u1} (ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)) (MonoidWithZero.toMonoid.{u1} (ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)) (ContinuousLinearMap.monoidWithZero.{u2, u1} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3))))) f' n) x)
Case conversion may be inaccurate. Consider using '#align has_strict_fderiv_at.iterate HasStrictFDerivAt.iterateβ'. -/
mathlib commit https://github.com/leanprover-community/mathlib/commit/8d33f09cd7089ecf074b4791907588245aec5d1b
@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, SΓ©bastien GouΓ«zel, Yury Kudryashov
! This file was ported from Lean 3 source module analysis.calculus.fderiv.comp
-! leanprover-community/mathlib commit e3fb84046afd187b710170887195d50bada934ee
+! leanprover-community/mathlib commit 38df578a6450a8c5142b3727e3ae894c2300cae0
! Please do not edit these lines, except to modify the commit id
! if you have ported upstream changes.
-/
@@ -13,6 +13,9 @@ import Mathbin.Analysis.Calculus.Fderiv.Basic
/-!
# The derivative of a composition (chain rule)
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
For detailed documentation of the FrΓ©chet derivative,
see the module docstring of `analysis/calculus/fderiv/basic.lean`.
mathlib commit https://github.com/leanprover-community/mathlib/commit/75e7fca56381d056096ce5d05e938f63a6567828
@@ -62,6 +62,12 @@ get confused since there are too many possibilities for composition -/
variable (x)
+/- warning: has_fderiv_at_filter.comp -> HasFDerivAtFilter.comp is a dubious translation:
+lean 3 declaration is
+ forall {π : Type.{u1}} [_inst_1 : NontriviallyNormedField.{u1} π] {E : Type.{u2}} [_inst_2 : NormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)] {F : Type.{u3}} [_inst_4 : NormedAddCommGroup.{u3} F] [_inst_5 : NormedSpace.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)] {G : Type.{u4}} [_inst_6 : NormedAddCommGroup.{u4} G] [_inst_7 : NormedSpace.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)] {f : E -> F} {f' : ContinuousLinearMap.{u1, u1, u2, u3} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5)} (x : E) {L : Filter.{u2} E} {g : F -> G} {g' : ContinuousLinearMap.{u1, u1, u3, u4} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u4} G (PseudoMetricSpace.toUniformSpace.{u4} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u4} G (NormedAddCommGroup.toAddCommGroup.{u4} G _inst_6)) (NormedSpace.toModule.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6) _inst_7)} {L' : Filter.{u3} F}, (HasFDerivAtFilter.{u1, u3, u4} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g g' (f x) L') -> (HasFDerivAtFilter.{u1, u2, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f f' x L) -> (Filter.Tendsto.{u2, u3} E F f L L') -> (HasFDerivAtFilter.{u1, u2, u4} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (Function.comp.{succ u2, succ u3, succ u4} E F G g f) (ContinuousLinearMap.comp.{u1, u1, u1, u2, u3, u4} π π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u4} G (PseudoMetricSpace.toUniformSpace.{u4} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u4} G (NormedAddCommGroup.toAddCommGroup.{u4} G _inst_6)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6) _inst_7) (RingHomCompTriple.right_ids.{u1, u1} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1)))))))) g' f') x L)
+but is expected to have type
+ forall {π : Type.{u4}} [_inst_1 : NontriviallyNormedField.{u4} π] {E : Type.{u1}} [_inst_2 : NormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u4, u1} π E (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)] {F : Type.{u3}} [_inst_4 : NormedAddCommGroup.{u3} F] [_inst_5 : NormedSpace.{u4, u3} π F (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)] {G : Type.{u2}} [_inst_6 : NormedAddCommGroup.{u2} G] [_inst_7 : NormedSpace.{u4, u2} π G (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)] {f : E -> F} {f' : ContinuousLinearMap.{u4, u4, u1, u3} π π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) (NormedSpace.toModule.{u4, u1} π E (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u4, u3} π F (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5)} (x : E) {L : Filter.{u1} E} {g : F -> G} {g' : ContinuousLinearMap.{u4, u4, u3, u2} π π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u2} G (PseudoMetricSpace.toUniformSpace.{u2} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u2} G (NormedAddCommGroup.toAddCommGroup.{u2} G _inst_6)) (NormedSpace.toModule.{u4, u3} π F (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u4, u2} π G (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6) _inst_7)} {L' : Filter.{u3} F}, (HasFDerivAtFilter.{u4, u3, u2} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g g' (f x) L') -> (HasFDerivAtFilter.{u4, u1, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f f' x L) -> (Filter.Tendsto.{u1, u3} E F f L L') -> (HasFDerivAtFilter.{u4, u1, u2} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (Function.comp.{succ u1, succ u3, succ u2} E F G g f) (ContinuousLinearMap.comp.{u4, u4, u4, u1, u3, u2} π π π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u2} G (PseudoMetricSpace.toUniformSpace.{u2} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u2} G (NormedAddCommGroup.toAddCommGroup.{u2} G _inst_6)) (NormedSpace.toModule.{u4, u1} π E (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u4, u3} π F (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u4, u2} π G (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6) _inst_7) (RingHomCompTriple.ids.{u4, u4} π π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1)))))))) g' f') x L)
+Case conversion may be inaccurate. Consider using '#align has_fderiv_at_filter.comp HasFDerivAtFilter.compβ'. -/
theorem HasFDerivAtFilter.comp {g : F β G} {g' : F βL[π] G} {L' : Filter F}
(hg : HasFDerivAtFilter g g' (f x) L') (hf : HasFDerivAtFilter f f' x L) (hL : Tendsto f L L') :
HasFDerivAtFilter (g β f) (g'.comp f') x L :=
@@ -93,58 +99,118 @@ example {g : F β G} {g' : F βL[π] G} (hg : HasFDerivAtFilter g g' (f x) (
_ =o[L] fun x' => x' - x := hf
+/- warning: has_fderiv_within_at.comp -> HasFDerivWithinAt.comp is a dubious translation:
+lean 3 declaration is
+ forall {π : Type.{u1}} [_inst_1 : NontriviallyNormedField.{u1} π] {E : Type.{u2}} [_inst_2 : NormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)] {F : Type.{u3}} [_inst_4 : NormedAddCommGroup.{u3} F] [_inst_5 : NormedSpace.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)] {G : Type.{u4}} [_inst_6 : NormedAddCommGroup.{u4} G] [_inst_7 : NormedSpace.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)] {f : E -> F} {f' : ContinuousLinearMap.{u1, u1, u2, u3} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5)} (x : E) {s : Set.{u2} E} {g : F -> G} {g' : ContinuousLinearMap.{u1, u1, u3, u4} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u4} G (PseudoMetricSpace.toUniformSpace.{u4} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u4} G (NormedAddCommGroup.toAddCommGroup.{u4} G _inst_6)) (NormedSpace.toModule.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6) _inst_7)} {t : Set.{u3} F}, (HasFDerivWithinAt.{u1, u3, u4} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g g' t (f x)) -> (HasFDerivWithinAt.{u1, u2, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f f' s x) -> (Set.MapsTo.{u2, u3} E F f s t) -> (HasFDerivWithinAt.{u1, u2, u4} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (Function.comp.{succ u2, succ u3, succ u4} E F G g f) (ContinuousLinearMap.comp.{u1, u1, u1, u2, u3, u4} π π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u4} G (PseudoMetricSpace.toUniformSpace.{u4} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u4} G (NormedAddCommGroup.toAddCommGroup.{u4} G _inst_6)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6) _inst_7) (RingHomCompTriple.right_ids.{u1, u1} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1)))))))) g' f') s x)
+but is expected to have type
+ forall {π : Type.{u4}} [_inst_1 : NontriviallyNormedField.{u4} π] {E : Type.{u1}} [_inst_2 : NormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u4, u1} π E (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)] {F : Type.{u3}} [_inst_4 : NormedAddCommGroup.{u3} F] [_inst_5 : NormedSpace.{u4, u3} π F (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)] {G : Type.{u2}} [_inst_6 : NormedAddCommGroup.{u2} G] [_inst_7 : NormedSpace.{u4, u2} π G (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)] {f : E -> F} {f' : ContinuousLinearMap.{u4, u4, u1, u3} π π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) (NormedSpace.toModule.{u4, u1} π E (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u4, u3} π F (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5)} (x : E) {s : Set.{u1} E} {g : F -> G} {g' : ContinuousLinearMap.{u4, u4, u3, u2} π π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u2} G (PseudoMetricSpace.toUniformSpace.{u2} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u2} G (NormedAddCommGroup.toAddCommGroup.{u2} G _inst_6)) (NormedSpace.toModule.{u4, u3} π F (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u4, u2} π G (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6) _inst_7)} {t : Set.{u3} F}, (HasFDerivWithinAt.{u4, u3, u2} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g g' t (f x)) -> (HasFDerivWithinAt.{u4, u1, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f f' s x) -> (Set.MapsTo.{u1, u3} E F f s t) -> (HasFDerivWithinAt.{u4, u1, u2} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (Function.comp.{succ u1, succ u3, succ u2} E F G g f) (ContinuousLinearMap.comp.{u4, u4, u4, u1, u3, u2} π π π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u2} G (PseudoMetricSpace.toUniformSpace.{u2} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u2} G (NormedAddCommGroup.toAddCommGroup.{u2} G _inst_6)) (NormedSpace.toModule.{u4, u1} π E (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u4, u3} π F (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u4, u2} π G (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6) _inst_7) (RingHomCompTriple.ids.{u4, u4} π π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1)))))))) g' f') s x)
+Case conversion may be inaccurate. Consider using '#align has_fderiv_within_at.comp HasFDerivWithinAt.compβ'. -/
theorem HasFDerivWithinAt.comp {g : F β G} {g' : F βL[π] G} {t : Set F}
(hg : HasFDerivWithinAt g g' t (f x)) (hf : HasFDerivWithinAt f f' s x) (hst : MapsTo f s t) :
HasFDerivWithinAt (g β f) (g'.comp f') s x :=
hg.comp x hf <| hf.ContinuousWithinAt.tendsto_nhdsWithin hst
#align has_fderiv_within_at.comp HasFDerivWithinAt.comp
+/- warning: has_fderiv_at.comp_has_fderiv_within_at -> HasFDerivAt.comp_hasFDerivWithinAt is a dubious translation:
+lean 3 declaration is
+ forall {π : Type.{u1}} [_inst_1 : NontriviallyNormedField.{u1} π] {E : Type.{u2}} [_inst_2 : NormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)] {F : Type.{u3}} [_inst_4 : NormedAddCommGroup.{u3} F] [_inst_5 : NormedSpace.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)] {G : Type.{u4}} [_inst_6 : NormedAddCommGroup.{u4} G] [_inst_7 : NormedSpace.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)] {f : E -> F} {f' : ContinuousLinearMap.{u1, u1, u2, u3} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5)} (x : E) {s : Set.{u2} E} {g : F -> G} {g' : ContinuousLinearMap.{u1, u1, u3, u4} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u4} G (PseudoMetricSpace.toUniformSpace.{u4} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u4} G (NormedAddCommGroup.toAddCommGroup.{u4} G _inst_6)) (NormedSpace.toModule.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6) _inst_7)}, (HasFDerivAt.{u1, u3, u4} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g g' (f x)) -> (HasFDerivWithinAt.{u1, u2, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f f' s x) -> (HasFDerivWithinAt.{u1, u2, u4} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (Function.comp.{succ u2, succ u3, succ u4} E F G g f) (ContinuousLinearMap.comp.{u1, u1, u1, u2, u3, u4} π π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u4} G (PseudoMetricSpace.toUniformSpace.{u4} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u4} G (NormedAddCommGroup.toAddCommGroup.{u4} G _inst_6)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6) _inst_7) (RingHomCompTriple.right_ids.{u1, u1} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1)))))))) g' f') s x)
+but is expected to have type
+ forall {π : Type.{u4}} [_inst_1 : NontriviallyNormedField.{u4} π] {E : Type.{u1}} [_inst_2 : NormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u4, u1} π E (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)] {F : Type.{u3}} [_inst_4 : NormedAddCommGroup.{u3} F] [_inst_5 : NormedSpace.{u4, u3} π F (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)] {G : Type.{u2}} [_inst_6 : NormedAddCommGroup.{u2} G] [_inst_7 : NormedSpace.{u4, u2} π G (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)] {f : E -> F} {f' : ContinuousLinearMap.{u4, u4, u1, u3} π π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) (NormedSpace.toModule.{u4, u1} π E (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u4, u3} π F (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5)} (x : E) {s : Set.{u1} E} {g : F -> G} {g' : ContinuousLinearMap.{u4, u4, u3, u2} π π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u2} G (PseudoMetricSpace.toUniformSpace.{u2} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u2} G (NormedAddCommGroup.toAddCommGroup.{u2} G _inst_6)) (NormedSpace.toModule.{u4, u3} π F (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u4, u2} π G (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6) _inst_7)}, (HasFDerivAt.{u4, u3, u2} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g g' (f x)) -> (HasFDerivWithinAt.{u4, u1, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f f' s x) -> (HasFDerivWithinAt.{u4, u1, u2} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (Function.comp.{succ u1, succ u3, succ u2} E F G g f) (ContinuousLinearMap.comp.{u4, u4, u4, u1, u3, u2} π π π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u2} G (PseudoMetricSpace.toUniformSpace.{u2} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u2} G (NormedAddCommGroup.toAddCommGroup.{u2} G _inst_6)) (NormedSpace.toModule.{u4, u1} π E (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u4, u3} π F (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u4, u2} π G (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6) _inst_7) (RingHomCompTriple.ids.{u4, u4} π π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1)))))))) g' f') s x)
+Case conversion may be inaccurate. Consider using '#align has_fderiv_at.comp_has_fderiv_within_at HasFDerivAt.comp_hasFDerivWithinAtβ'. -/
theorem HasFDerivAt.comp_hasFDerivWithinAt {g : F β G} {g' : F βL[π] G}
(hg : HasFDerivAt g g' (f x)) (hf : HasFDerivWithinAt f f' s x) :
HasFDerivWithinAt (g β f) (g'.comp f') s x :=
hg.comp x hf hf.ContinuousWithinAt
#align has_fderiv_at.comp_has_fderiv_within_at HasFDerivAt.comp_hasFDerivWithinAt
+/- warning: has_fderiv_within_at.comp_of_mem -> HasFDerivWithinAt.comp_of_mem is a dubious translation:
+lean 3 declaration is
+ forall {π : Type.{u1}} [_inst_1 : NontriviallyNormedField.{u1} π] {E : Type.{u2}} [_inst_2 : NormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)] {F : Type.{u3}} [_inst_4 : NormedAddCommGroup.{u3} F] [_inst_5 : NormedSpace.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)] {G : Type.{u4}} [_inst_6 : NormedAddCommGroup.{u4} G] [_inst_7 : NormedSpace.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)] {f : E -> F} {f' : ContinuousLinearMap.{u1, u1, u2, u3} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5)} (x : E) {s : Set.{u2} E} {g : F -> G} {g' : ContinuousLinearMap.{u1, u1, u3, u4} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u4} G (PseudoMetricSpace.toUniformSpace.{u4} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u4} G (NormedAddCommGroup.toAddCommGroup.{u4} G _inst_6)) (NormedSpace.toModule.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6) _inst_7)} {t : Set.{u3} F}, (HasFDerivWithinAt.{u1, u3, u4} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g g' t (f x)) -> (HasFDerivWithinAt.{u1, u2, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f f' s x) -> (Filter.Tendsto.{u2, u3} E F f (nhdsWithin.{u2} E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) x s) (nhdsWithin.{u3} F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (f x) t)) -> (HasFDerivWithinAt.{u1, u2, u4} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (Function.comp.{succ u2, succ u3, succ u4} E F G g f) (ContinuousLinearMap.comp.{u1, u1, u1, u2, u3, u4} π π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u4} G (PseudoMetricSpace.toUniformSpace.{u4} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u4} G (NormedAddCommGroup.toAddCommGroup.{u4} G _inst_6)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6) _inst_7) (RingHomCompTriple.right_ids.{u1, u1} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1)))))))) g' f') s x)
+but is expected to have type
+ forall {π : Type.{u4}} [_inst_1 : NontriviallyNormedField.{u4} π] {E : Type.{u1}} [_inst_2 : NormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u4, u1} π E (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)] {F : Type.{u3}} [_inst_4 : NormedAddCommGroup.{u3} F] [_inst_5 : NormedSpace.{u4, u3} π F (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)] {G : Type.{u2}} [_inst_6 : NormedAddCommGroup.{u2} G] [_inst_7 : NormedSpace.{u4, u2} π G (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)] {f : E -> F} {f' : ContinuousLinearMap.{u4, u4, u1, u3} π π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) (NormedSpace.toModule.{u4, u1} π E (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u4, u3} π F (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5)} (x : E) {s : Set.{u1} E} {g : F -> G} {g' : ContinuousLinearMap.{u4, u4, u3, u2} π π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u2} G (PseudoMetricSpace.toUniformSpace.{u2} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u2} G (NormedAddCommGroup.toAddCommGroup.{u2} G _inst_6)) (NormedSpace.toModule.{u4, u3} π F (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u4, u2} π G (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6) _inst_7)} {t : Set.{u3} F}, (HasFDerivWithinAt.{u4, u3, u2} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g g' t (f x)) -> (HasFDerivWithinAt.{u4, u1, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f f' s x) -> (Filter.Tendsto.{u1, u3} E F f (nhdsWithin.{u1} E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) x s) (nhdsWithin.{u3} F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (f x) t)) -> (HasFDerivWithinAt.{u4, u1, u2} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (Function.comp.{succ u1, succ u3, succ u2} E F G g f) (ContinuousLinearMap.comp.{u4, u4, u4, u1, u3, u2} π π π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u2} G (PseudoMetricSpace.toUniformSpace.{u2} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u2} G (NormedAddCommGroup.toAddCommGroup.{u2} G _inst_6)) (NormedSpace.toModule.{u4, u1} π E (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u4, u3} π F (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u4, u2} π G (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6) _inst_7) (RingHomCompTriple.ids.{u4, u4} π π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1)))))))) g' f') s x)
+Case conversion may be inaccurate. Consider using '#align has_fderiv_within_at.comp_of_mem HasFDerivWithinAt.comp_of_memβ'. -/
theorem HasFDerivWithinAt.comp_of_mem {g : F β G} {g' : F βL[π] G} {t : Set F}
(hg : HasFDerivWithinAt g g' t (f x)) (hf : HasFDerivWithinAt f f' s x)
(hst : Tendsto f (π[s] x) (π[t] f x)) : HasFDerivWithinAt (g β f) (g'.comp f') s x :=
HasFDerivAtFilter.comp x hg hf hst
#align has_fderiv_within_at.comp_of_mem HasFDerivWithinAt.comp_of_mem
+/- warning: has_fderiv_at.comp -> HasFDerivAt.comp is a dubious translation:
+lean 3 declaration is
+ forall {π : Type.{u1}} [_inst_1 : NontriviallyNormedField.{u1} π] {E : Type.{u2}} [_inst_2 : NormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)] {F : Type.{u3}} [_inst_4 : NormedAddCommGroup.{u3} F] [_inst_5 : NormedSpace.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)] {G : Type.{u4}} [_inst_6 : NormedAddCommGroup.{u4} G] [_inst_7 : NormedSpace.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)] {f : E -> F} {f' : ContinuousLinearMap.{u1, u1, u2, u3} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5)} (x : E) {g : F -> G} {g' : ContinuousLinearMap.{u1, u1, u3, u4} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u4} G (PseudoMetricSpace.toUniformSpace.{u4} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u4} G (NormedAddCommGroup.toAddCommGroup.{u4} G _inst_6)) (NormedSpace.toModule.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6) _inst_7)}, (HasFDerivAt.{u1, u3, u4} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g g' (f x)) -> (HasFDerivAt.{u1, u2, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f f' x) -> (HasFDerivAt.{u1, u2, u4} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (Function.comp.{succ u2, succ u3, succ u4} E F G g f) (ContinuousLinearMap.comp.{u1, u1, u1, u2, u3, u4} π π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u4} G (PseudoMetricSpace.toUniformSpace.{u4} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u4} G (NormedAddCommGroup.toAddCommGroup.{u4} G _inst_6)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6) _inst_7) (RingHomCompTriple.right_ids.{u1, u1} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1)))))))) g' f') x)
+but is expected to have type
+ forall {π : Type.{u4}} [_inst_1 : NontriviallyNormedField.{u4} π] {E : Type.{u1}} [_inst_2 : NormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u4, u1} π E (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)] {F : Type.{u3}} [_inst_4 : NormedAddCommGroup.{u3} F] [_inst_5 : NormedSpace.{u4, u3} π F (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)] {G : Type.{u2}} [_inst_6 : NormedAddCommGroup.{u2} G] [_inst_7 : NormedSpace.{u4, u2} π G (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)] {f : E -> F} {f' : ContinuousLinearMap.{u4, u4, u1, u3} π π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) (NormedSpace.toModule.{u4, u1} π E (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u4, u3} π F (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5)} (x : E) {g : F -> G} {g' : ContinuousLinearMap.{u4, u4, u3, u2} π π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u2} G (PseudoMetricSpace.toUniformSpace.{u2} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u2} G (NormedAddCommGroup.toAddCommGroup.{u2} G _inst_6)) (NormedSpace.toModule.{u4, u3} π F (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u4, u2} π G (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6) _inst_7)}, (HasFDerivAt.{u4, u3, u2} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g g' (f x)) -> (HasFDerivAt.{u4, u1, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f f' x) -> (HasFDerivAt.{u4, u1, u2} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (Function.comp.{succ u1, succ u3, succ u2} E F G g f) (ContinuousLinearMap.comp.{u4, u4, u4, u1, u3, u2} π π π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u2} G (PseudoMetricSpace.toUniformSpace.{u2} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u2} G (NormedAddCommGroup.toAddCommGroup.{u2} G _inst_6)) (NormedSpace.toModule.{u4, u1} π E (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u4, u3} π F (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u4, u2} π G (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6) _inst_7) (RingHomCompTriple.ids.{u4, u4} π π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1)))))))) g' f') x)
+Case conversion may be inaccurate. Consider using '#align has_fderiv_at.comp HasFDerivAt.compβ'. -/
/-- The chain rule. -/
theorem HasFDerivAt.comp {g : F β G} {g' : F βL[π] G} (hg : HasFDerivAt g g' (f x))
(hf : HasFDerivAt f f' x) : HasFDerivAt (g β f) (g'.comp f') x :=
hg.comp x hf hf.ContinuousAt
#align has_fderiv_at.comp HasFDerivAt.comp
+/- warning: differentiable_within_at.comp -> DifferentiableWithinAt.comp is a dubious translation:
+lean 3 declaration is
+ forall {π : Type.{u1}} [_inst_1 : NontriviallyNormedField.{u1} π] {E : Type.{u2}} [_inst_2 : NormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)] {F : Type.{u3}} [_inst_4 : NormedAddCommGroup.{u3} F] [_inst_5 : NormedSpace.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)] {G : Type.{u4}} [_inst_6 : NormedAddCommGroup.{u4} G] [_inst_7 : NormedSpace.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)] {f : E -> F} (x : E) {s : Set.{u2} E} {g : F -> G} {t : Set.{u3} F}, (DifferentiableWithinAt.{u1, u3, u4} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g t (f x)) -> (DifferentiableWithinAt.{u1, u2, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f s x) -> (Set.MapsTo.{u2, u3} E F f s t) -> (DifferentiableWithinAt.{u1, u2, u4} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (Function.comp.{succ u2, succ u3, succ u4} E F G g f) s x)
+but is expected to have type
+ forall {π : Type.{u3}} [_inst_1 : NontriviallyNormedField.{u3} π] {E : Type.{u1}} [_inst_2 : NormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u3, u1} π E (NontriviallyNormedField.toNormedField.{u3} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)] {F : Type.{u4}} [_inst_4 : NormedAddCommGroup.{u4} F] [_inst_5 : NormedSpace.{u3, u4} π F (NontriviallyNormedField.toNormedField.{u3} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_4)] {G : Type.{u2}} [_inst_6 : NormedAddCommGroup.{u2} G] [_inst_7 : NormedSpace.{u3, u2} π G (NontriviallyNormedField.toNormedField.{u3} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)] {f : E -> F} (x : E) {s : Set.{u1} E} {g : F -> G} {t : Set.{u4} F}, (DifferentiableWithinAt.{u3, u4, u2} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g t (f x)) -> (DifferentiableWithinAt.{u3, u1, u4} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f s x) -> (Set.MapsTo.{u1, u4} E F f s t) -> (DifferentiableWithinAt.{u3, u1, u2} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (Function.comp.{succ u1, succ u4, succ u2} E F G g f) s x)
+Case conversion may be inaccurate. Consider using '#align differentiable_within_at.comp DifferentiableWithinAt.compβ'. -/
theorem DifferentiableWithinAt.comp {g : F β G} {t : Set F}
(hg : DifferentiableWithinAt π g t (f x)) (hf : DifferentiableWithinAt π f s x)
(h : MapsTo f s t) : DifferentiableWithinAt π (g β f) s x :=
(hg.HasFDerivWithinAt.comp x hf.HasFDerivWithinAt h).DifferentiableWithinAt
#align differentiable_within_at.comp DifferentiableWithinAt.comp
+/- warning: differentiable_within_at.comp' -> DifferentiableWithinAt.comp' is a dubious translation:
+lean 3 declaration is
+ forall {π : Type.{u1}} [_inst_1 : NontriviallyNormedField.{u1} π] {E : Type.{u2}} [_inst_2 : NormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)] {F : Type.{u3}} [_inst_4 : NormedAddCommGroup.{u3} F] [_inst_5 : NormedSpace.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)] {G : Type.{u4}} [_inst_6 : NormedAddCommGroup.{u4} G] [_inst_7 : NormedSpace.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)] {f : E -> F} (x : E) {s : Set.{u2} E} {g : F -> G} {t : Set.{u3} F}, (DifferentiableWithinAt.{u1, u3, u4} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g t (f x)) -> (DifferentiableWithinAt.{u1, u2, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f s x) -> (DifferentiableWithinAt.{u1, u2, u4} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (Function.comp.{succ u2, succ u3, succ u4} E F G g f) (Inter.inter.{u2} (Set.{u2} E) (Set.hasInter.{u2} E) s (Set.preimage.{u2, u3} E F f t)) x)
+but is expected to have type
+ forall {π : Type.{u3}} [_inst_1 : NontriviallyNormedField.{u3} π] {E : Type.{u1}} [_inst_2 : NormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u3, u1} π E (NontriviallyNormedField.toNormedField.{u3} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)] {F : Type.{u4}} [_inst_4 : NormedAddCommGroup.{u4} F] [_inst_5 : NormedSpace.{u3, u4} π F (NontriviallyNormedField.toNormedField.{u3} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_4)] {G : Type.{u2}} [_inst_6 : NormedAddCommGroup.{u2} G] [_inst_7 : NormedSpace.{u3, u2} π G (NontriviallyNormedField.toNormedField.{u3} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)] {f : E -> F} (x : E) {s : Set.{u1} E} {g : F -> G} {t : Set.{u4} F}, (DifferentiableWithinAt.{u3, u4, u2} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g t (f x)) -> (DifferentiableWithinAt.{u3, u1, u4} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f s x) -> (DifferentiableWithinAt.{u3, u1, u2} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (Function.comp.{succ u1, succ u4, succ u2} E F G g f) (Inter.inter.{u1} (Set.{u1} E) (Set.instInterSet.{u1} E) s (Set.preimage.{u1, u4} E F f t)) x)
+Case conversion may be inaccurate. Consider using '#align differentiable_within_at.comp' DifferentiableWithinAt.comp'β'. -/
theorem DifferentiableWithinAt.comp' {g : F β G} {t : Set F}
(hg : DifferentiableWithinAt π g t (f x)) (hf : DifferentiableWithinAt π f s x) :
DifferentiableWithinAt π (g β f) (s β© f β»ΒΉ' t) x :=
hg.comp x (hf.mono (inter_subset_left _ _)) (inter_subset_right _ _)
#align differentiable_within_at.comp' DifferentiableWithinAt.comp'
+/- warning: differentiable_at.comp -> DifferentiableAt.comp is a dubious translation:
+lean 3 declaration is
+ forall {π : Type.{u1}} [_inst_1 : NontriviallyNormedField.{u1} π] {E : Type.{u2}} [_inst_2 : NormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)] {F : Type.{u3}} [_inst_4 : NormedAddCommGroup.{u3} F] [_inst_5 : NormedSpace.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)] {G : Type.{u4}} [_inst_6 : NormedAddCommGroup.{u4} G] [_inst_7 : NormedSpace.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)] {f : E -> F} (x : E) {g : F -> G}, (DifferentiableAt.{u1, u3, u4} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g (f x)) -> (DifferentiableAt.{u1, u2, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f x) -> (DifferentiableAt.{u1, u2, u4} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (Function.comp.{succ u2, succ u3, succ u4} E F G g f) x)
+but is expected to have type
+ forall {π : Type.{u4}} [_inst_1 : NontriviallyNormedField.{u4} π] {E : Type.{u1}} [_inst_2 : NormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u4, u1} π E (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)] {F : Type.{u3}} [_inst_4 : NormedAddCommGroup.{u3} F] [_inst_5 : NormedSpace.{u4, u3} π F (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)] {G : Type.{u2}} [_inst_6 : NormedAddCommGroup.{u2} G] [_inst_7 : NormedSpace.{u4, u2} π G (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)] {f : E -> F} (x : E) {g : F -> G}, (DifferentiableAt.{u4, u3, u2} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g (f x)) -> (DifferentiableAt.{u4, u1, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f x) -> (DifferentiableAt.{u4, u1, u2} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (Function.comp.{succ u1, succ u3, succ u2} E F G g f) x)
+Case conversion may be inaccurate. Consider using '#align differentiable_at.comp DifferentiableAt.compβ'. -/
theorem DifferentiableAt.comp {g : F β G} (hg : DifferentiableAt π g (f x))
(hf : DifferentiableAt π f x) : DifferentiableAt π (g β f) x :=
(hg.HasFDerivAt.comp x hf.HasFDerivAt).DifferentiableAt
#align differentiable_at.comp DifferentiableAt.comp
+/- warning: differentiable_at.comp_differentiable_within_at -> DifferentiableAt.comp_differentiableWithinAt is a dubious translation:
+lean 3 declaration is
+ forall {π : Type.{u1}} [_inst_1 : NontriviallyNormedField.{u1} π] {E : Type.{u2}} [_inst_2 : NormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)] {F : Type.{u3}} [_inst_4 : NormedAddCommGroup.{u3} F] [_inst_5 : NormedSpace.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)] {G : Type.{u4}} [_inst_6 : NormedAddCommGroup.{u4} G] [_inst_7 : NormedSpace.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)] {f : E -> F} (x : E) {s : Set.{u2} E} {g : F -> G}, (DifferentiableAt.{u1, u3, u4} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g (f x)) -> (DifferentiableWithinAt.{u1, u2, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f s x) -> (DifferentiableWithinAt.{u1, u2, u4} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (Function.comp.{succ u2, succ u3, succ u4} E F G g f) s x)
+but is expected to have type
+ forall {π : Type.{u4}} [_inst_1 : NontriviallyNormedField.{u4} π] {E : Type.{u1}} [_inst_2 : NormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u4, u1} π E (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)] {F : Type.{u3}} [_inst_4 : NormedAddCommGroup.{u3} F] [_inst_5 : NormedSpace.{u4, u3} π F (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)] {G : Type.{u2}} [_inst_6 : NormedAddCommGroup.{u2} G] [_inst_7 : NormedSpace.{u4, u2} π G (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)] {f : E -> F} (x : E) {s : Set.{u1} E} {g : F -> G}, (DifferentiableAt.{u4, u3, u2} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g (f x)) -> (DifferentiableWithinAt.{u4, u1, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f s x) -> (DifferentiableWithinAt.{u4, u1, u2} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (Function.comp.{succ u1, succ u3, succ u2} E F G g f) s x)
+Case conversion may be inaccurate. Consider using '#align differentiable_at.comp_differentiable_within_at DifferentiableAt.comp_differentiableWithinAtβ'. -/
theorem DifferentiableAt.comp_differentiableWithinAt {g : F β G} (hg : DifferentiableAt π g (f x))
(hf : DifferentiableWithinAt π f s x) : DifferentiableWithinAt π (g β f) s x :=
hg.DifferentiableWithinAt.comp x hf (mapsTo_univ _ _)
#align differentiable_at.comp_differentiable_within_at DifferentiableAt.comp_differentiableWithinAt
+/- warning: fderiv_within.comp -> fderivWithin.comp is a dubious translation:
+lean 3 declaration is
+ forall {π : Type.{u1}} [_inst_1 : NontriviallyNormedField.{u1} π] {E : Type.{u2}} [_inst_2 : NormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)] {F : Type.{u3}} [_inst_4 : NormedAddCommGroup.{u3} F] [_inst_5 : NormedSpace.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)] {G : Type.{u4}} [_inst_6 : NormedAddCommGroup.{u4} G] [_inst_7 : NormedSpace.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)] {f : E -> F} (x : E) {s : Set.{u2} E} {g : F -> G} {t : Set.{u3} F}, (DifferentiableWithinAt.{u1, u3, u4} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g t (f x)) -> (DifferentiableWithinAt.{u1, u2, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f s x) -> (Set.MapsTo.{u2, u3} E F f s t) -> (UniqueDiffWithinAt.{u1, u2} π _inst_1 E (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) s x) -> (Eq.{max (succ u2) (succ u4)} (ContinuousLinearMap.{u1, u1, u2, u4} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) G (UniformSpace.toTopologicalSpace.{u4} G (PseudoMetricSpace.toUniformSpace.{u4} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u4} G (NormedAddCommGroup.toAddCommGroup.{u4} G _inst_6)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6) _inst_7)) (fderivWithin.{u1, u2, u4} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (Function.comp.{succ u2, succ u3, succ u4} E F G g f) s x) (ContinuousLinearMap.comp.{u1, u1, u1, u2, u3, u4} π π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u4} G (PseudoMetricSpace.toUniformSpace.{u4} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u4} G (NormedAddCommGroup.toAddCommGroup.{u4} G _inst_6)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6) _inst_7) (RingHomCompTriple.right_ids.{u1, u1} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1)))))))) (fderivWithin.{u1, u3, u4} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g t (f x)) (fderivWithin.{u1, u2, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f s x)))
+but is expected to have type
+ forall {π : Type.{u3}} [_inst_1 : NontriviallyNormedField.{u3} π] {E : Type.{u1}} [_inst_2 : NormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u3, u1} π E (NontriviallyNormedField.toNormedField.{u3} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)] {F : Type.{u4}} [_inst_4 : NormedAddCommGroup.{u4} F] [_inst_5 : NormedSpace.{u3, u4} π F (NontriviallyNormedField.toNormedField.{u3} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_4)] {G : Type.{u2}} [_inst_6 : NormedAddCommGroup.{u2} G] [_inst_7 : NormedSpace.{u3, u2} π G (NontriviallyNormedField.toNormedField.{u3} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)] {f : E -> F} (x : E) {s : Set.{u1} E} {g : F -> G} {t : Set.{u4} F}, (DifferentiableWithinAt.{u3, u4, u2} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g t (f x)) -> (DifferentiableWithinAt.{u3, u1, u4} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f s x) -> (Set.MapsTo.{u1, u4} E F f s t) -> (UniqueDiffWithinAt.{u3, u1} π _inst_1 E (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u3, u1} π E (NontriviallyNormedField.toNormedField.{u3} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) s x) -> (Eq.{max (succ u1) (succ u2)} (ContinuousLinearMap.{u3, u3, u1, u2} π π (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1))))) (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1))))) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) G (UniformSpace.toTopologicalSpace.{u2} G (PseudoMetricSpace.toUniformSpace.{u2} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u2} G (NormedAddCommGroup.toAddCommGroup.{u2} G _inst_6)) (NormedSpace.toModule.{u3, u1} π E (NontriviallyNormedField.toNormedField.{u3} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u3, u2} π G (NontriviallyNormedField.toNormedField.{u3} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6) _inst_7)) (fderivWithin.{u3, u1, u2} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (Function.comp.{succ u1, succ u4, succ u2} E F G g f) s x) (ContinuousLinearMap.comp.{u3, u3, u3, u1, u4, u2} π π π (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1))))) (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1))))) (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1))))) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1))))))) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1))))))) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u4} F (PseudoMetricSpace.toUniformSpace.{u4} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u4} F (NormedAddCommGroup.toAddCommGroup.{u4} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u2} G (PseudoMetricSpace.toUniformSpace.{u2} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u2} G (NormedAddCommGroup.toAddCommGroup.{u2} G _inst_6)) (NormedSpace.toModule.{u3, u1} π E (NontriviallyNormedField.toNormedField.{u3} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u3, u4} π F (NontriviallyNormedField.toNormedField.{u3} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_4) _inst_5) (NormedSpace.toModule.{u3, u2} π G (NontriviallyNormedField.toNormedField.{u3} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6) _inst_7) (RingHomCompTriple.ids.{u3, u3} π π (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1))))) (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1))))) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1)))))))) (fderivWithin.{u3, u4, u2} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g t (f x)) (fderivWithin.{u3, u1, u4} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f s x)))
+Case conversion may be inaccurate. Consider using '#align fderiv_within.comp fderivWithin.compβ'. -/
theorem fderivWithin.comp {g : F β G} {t : Set F} (hg : DifferentiableWithinAt π g t (f x))
(hf : DifferentiableWithinAt π f s x) (h : MapsTo f s t) (hxs : UniqueDiffWithinAt π s x) :
fderivWithin π (g β f) s x = (fderivWithin π g t (f x)).comp (fderivWithin π f s x) :=
(hg.HasFDerivWithinAt.comp x hf.HasFDerivWithinAt h).fderivWithin hxs
#align fderiv_within.comp fderivWithin.comp
+/- warning: fderiv_within_fderiv_within -> fderivWithin_fderivWithin is a dubious translation:
+lean 3 declaration is
+ forall {π : Type.{u1}} [_inst_1 : NontriviallyNormedField.{u1} π] {E : Type.{u2}} [_inst_2 : NormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)] {F : Type.{u3}} [_inst_4 : NormedAddCommGroup.{u3} F] [_inst_5 : NormedSpace.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)] {G : Type.{u4}} [_inst_6 : NormedAddCommGroup.{u4} G] [_inst_7 : NormedSpace.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)] {g : F -> G} {f : E -> F} {x : E} {y : F} {s : Set.{u2} E} {t : Set.{u3} F}, (DifferentiableWithinAt.{u1, u3, u4} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g t y) -> (DifferentiableWithinAt.{u1, u2, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f s x) -> (Set.MapsTo.{u2, u3} E F f s t) -> (UniqueDiffWithinAt.{u1, u2} π _inst_1 E (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) s x) -> (Eq.{succ u3} F (f x) y) -> (forall (v : E), Eq.{succ u4} G (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (ContinuousLinearMap.{u1, u1, u3, u4} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u4} G (PseudoMetricSpace.toUniformSpace.{u4} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u4} G (NormedAddCommGroup.toAddCommGroup.{u4} G _inst_6)) (NormedSpace.toModule.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6) _inst_7)) (fun (_x : ContinuousLinearMap.{u1, u1, u3, u4} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u4} G (PseudoMetricSpace.toUniformSpace.{u4} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u4} G (NormedAddCommGroup.toAddCommGroup.{u4} G _inst_6)) (NormedSpace.toModule.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6) _inst_7)) => F -> G) (ContinuousLinearMap.toFun.{u1, u1, u3, u4} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u4} G (PseudoMetricSpace.toUniformSpace.{u4} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u4} G (NormedAddCommGroup.toAddCommGroup.{u4} G _inst_6)) (NormedSpace.toModule.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6) _inst_7)) (fderivWithin.{u1, u3, u4} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g t y) (coeFn.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (ContinuousLinearMap.{u1, u1, u2, u3} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5)) (fun (_x : ContinuousLinearMap.{u1, u1, u2, u3} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5)) => E -> F) (ContinuousLinearMap.toFun.{u1, u1, u2, u3} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5)) (fderivWithin.{u1, u2, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f s x) v)) (coeFn.{max (succ u2) (succ u4), max (succ u2) (succ u4)} (ContinuousLinearMap.{u1, u1, u2, u4} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) G (UniformSpace.toTopologicalSpace.{u4} G (PseudoMetricSpace.toUniformSpace.{u4} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u4} G (NormedAddCommGroup.toAddCommGroup.{u4} G _inst_6)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6) _inst_7)) (fun (_x : ContinuousLinearMap.{u1, u1, u2, u4} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) G (UniformSpace.toTopologicalSpace.{u4} G (PseudoMetricSpace.toUniformSpace.{u4} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u4} G (NormedAddCommGroup.toAddCommGroup.{u4} G _inst_6)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6) _inst_7)) => E -> G) (ContinuousLinearMap.toFun.{u1, u1, u2, u4} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) G (UniformSpace.toTopologicalSpace.{u4} G (PseudoMetricSpace.toUniformSpace.{u4} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u4} G (NormedAddCommGroup.toAddCommGroup.{u4} G _inst_6)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6) _inst_7)) (fderivWithin.{u1, u2, u4} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (Function.comp.{succ u2, succ u3, succ u4} E F G g f) s x) v))
+but is expected to have type
+ forall {π : Type.{u2}} [_inst_1 : NontriviallyNormedField.{u2} π] {E : Type.{u4}} [_inst_2 : NormedAddCommGroup.{u4} E] [_inst_3 : NormedSpace.{u2, u4} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_2)] {F : Type.{u3}} [_inst_4 : NormedAddCommGroup.{u3} F] [_inst_5 : NormedSpace.{u2, u3} π F (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)] {G : Type.{u1}} [_inst_6 : NormedAddCommGroup.{u1} G] [_inst_7 : NormedSpace.{u2, u1} π G (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} G _inst_6)] {g : F -> G} {f : E -> F} {x : E} {y : F} {s : Set.{u4} E} {t : Set.{u3} F}, (DifferentiableWithinAt.{u2, u3, u1} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g t y) -> (DifferentiableWithinAt.{u2, u4, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f s x) -> (Set.MapsTo.{u4, u3} E F f s t) -> (UniqueDiffWithinAt.{u2, u4} π _inst_1 E (AddCommGroup.toAddCommMonoid.{u4} E (NormedAddCommGroup.toAddCommGroup.{u4} E _inst_2)) (NormedSpace.toModule.{u2, u4} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_2) _inst_3) (UniformSpace.toTopologicalSpace.{u4} E (PseudoMetricSpace.toUniformSpace.{u4} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_2)))) s x) -> (Eq.{succ u3} F (f x) y) -> (forall (v : E), Eq.{succ u1} ((fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : F) => G) (FunLike.coe.{max (succ u4) (succ u3), succ u4, succ u3} (ContinuousLinearMap.{u2, u2, u4, u3} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u4} E (PseudoMetricSpace.toUniformSpace.{u4} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u4} E (NormedAddCommGroup.toAddCommGroup.{u4} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) (NormedSpace.toModule.{u2, u4} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u3} π F (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5)) E (fun (a : E) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : E) => F) a) (ContinuousMapClass.toFunLike.{max u4 u3, u4, u3} (ContinuousLinearMap.{u2, u2, u4, u3} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u4} E (PseudoMetricSpace.toUniformSpace.{u4} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u4} E (NormedAddCommGroup.toAddCommGroup.{u4} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) (NormedSpace.toModule.{u2, u4} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u3} π F (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5)) E F (UniformSpace.toTopologicalSpace.{u4} E (PseudoMetricSpace.toUniformSpace.{u4} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_2)))) (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (ContinuousSemilinearMapClass.toContinuousMapClass.{max u4 u3, u2, u2, u4, u3} (ContinuousLinearMap.{u2, u2, u4, u3} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u4} E (PseudoMetricSpace.toUniformSpace.{u4} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u4} E (NormedAddCommGroup.toAddCommGroup.{u4} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) (NormedSpace.toModule.{u2, u4} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u3} π F (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5)) π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u4} E (PseudoMetricSpace.toUniformSpace.{u4} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u4} E (NormedAddCommGroup.toAddCommGroup.{u4} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) (NormedSpace.toModule.{u2, u4} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u3} π F (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (ContinuousLinearMap.continuousSemilinearMapClass.{u2, u2, u4, u3} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u4} E (PseudoMetricSpace.toUniformSpace.{u4} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u4} E (NormedAddCommGroup.toAddCommGroup.{u4} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) (NormedSpace.toModule.{u2, u4} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u3} π F (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5)))) (fderivWithin.{u2, u4, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f s x) v)) (FunLike.coe.{max (succ u3) (succ u1), succ u3, succ u1} (ContinuousLinearMap.{u2, u2, u3, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u1} G (PseudoMetricSpace.toUniformSpace.{u1} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u1} G (NormedAddCommGroup.toAddCommGroup.{u1} G _inst_6)) (NormedSpace.toModule.{u2, u3} π F (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u2, u1} π G (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} G _inst_6) _inst_7)) F (fun (_x : F) => (fun (x._@.Mathlib.Topology.ContinuousFunction.Basic._hyg.699 : F) => G) _x) (ContinuousMapClass.toFunLike.{max u3 u1, u3, u1} (ContinuousLinearMap.{u2, u2, u3, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u1} G (PseudoMetricSpace.toUniformSpace.{u1} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u1} G (NormedAddCommGroup.toAddCommGroup.{u1} G _inst_6)) (NormedSpace.toModule.{u2, u3} π F (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u2, u1} π G (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} G _inst_6) _inst_7)) F G (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (UniformSpace.toTopologicalSpace.{u1} G (PseudoMetricSpace.toUniformSpace.{u1} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} G _inst_6)))) (ContinuousSemilinearMapClass.toContinuousMapClass.{max u3 u1, u2, u2, u3, u1} (ContinuousLinearMap.{u2, u2, u3, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u1} G (PseudoMetricSpace.toUniformSpace.{u1} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u1} G (NormedAddCommGroup.toAddCommGroup.{u1} G _inst_6)) (NormedSpace.toModule.{u2, u3} π F (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u2, u1} π G (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} G _inst_6) _inst_7)) π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u1} G (PseudoMetricSpace.toUniformSpace.{u1} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u1} G (NormedAddCommGroup.toAddCommGroup.{u1} G _inst_6)) (NormedSpace.toModule.{u2, u3} π 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(Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u4} E (PseudoMetricSpace.toUniformSpace.{u4} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u4} E (NormedAddCommGroup.toAddCommGroup.{u4} E _inst_2)) G (UniformSpace.toTopologicalSpace.{u1} G (PseudoMetricSpace.toUniformSpace.{u1} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u1} G (NormedAddCommGroup.toAddCommGroup.{u1} G _inst_6)) (NormedSpace.toModule.{u2, u4} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π G (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} G _inst_6) _inst_7)))) (fderivWithin.{u2, u4, u1} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (Function.comp.{succ u4, succ u3, succ u1} E F G g f) s x) v))
+Case conversion may be inaccurate. Consider using '#align fderiv_within_fderiv_within fderivWithin_fderivWithinβ'. -/
/-- A version of `fderiv_within.comp` that is useful to rewrite the composition of two derivatives
into a single derivative. This version always applies, but creates a new side-goal `f x = y`. -/
theorem fderivWithin_fderivWithin {g : F β G} {f : E β F} {x : E} {y : F} {s : Set E} {t : Set F}
@@ -157,6 +223,12 @@ theorem fderivWithin_fderivWithin {g : F β G} {f : E β F} {x : E} {y : F} {s
rfl
#align fderiv_within_fderiv_within fderivWithin_fderivWithin
+/- warning: fderiv_within.compβ -> fderivWithin.compβ is a dubious translation:
+lean 3 declaration is
+ forall {π : Type.{u1}} [_inst_1 : NontriviallyNormedField.{u1} π] {E : Type.{u2}} [_inst_2 : NormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)] {F : Type.{u3}} [_inst_4 : NormedAddCommGroup.{u3} F] [_inst_5 : NormedSpace.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)] {G : Type.{u4}} [_inst_6 : NormedAddCommGroup.{u4} G] [_inst_7 : NormedSpace.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)] {G' : Type.{u5}} [_inst_8 : NormedAddCommGroup.{u5} G'] [_inst_9 : NormedSpace.{u1, u5} π G' (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G' _inst_8)] {f : E -> F} (x : E) {s : Set.{u2} E} {g' : G -> G'} {g : F -> G} {t : Set.{u3} F} {u : Set.{u4} G} {y : F} {y' : G}, (DifferentiableWithinAt.{u1, u4, u5} π _inst_1 G _inst_6 _inst_7 G' _inst_8 _inst_9 g' u y') -> (DifferentiableWithinAt.{u1, u3, u4} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g t y) -> (DifferentiableWithinAt.{u1, u2, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f s x) -> (Set.MapsTo.{u3, u4} F G g t u) -> (Set.MapsTo.{u2, u3} E F f s t) -> (Eq.{succ u4} G (g y) y') -> (Eq.{succ u3} F (f x) y) -> (UniqueDiffWithinAt.{u1, u2} π _inst_1 E (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) s x) -> (Eq.{max (succ u2) (succ u5)} (ContinuousLinearMap.{u1, u1, u2, u5} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) G' (UniformSpace.toTopologicalSpace.{u5} G' (PseudoMetricSpace.toUniformSpace.{u5} G' (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G' (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G' _inst_8)))) (AddCommGroup.toAddCommMonoid.{u5} G' (NormedAddCommGroup.toAddCommGroup.{u5} G' _inst_8)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u5} π G' (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G' _inst_8) _inst_9)) (fderivWithin.{u1, u2, u5} π _inst_1 E _inst_2 _inst_3 G' _inst_8 _inst_9 (Function.comp.{succ u2, succ u4, succ u5} E G G' g' (Function.comp.{succ u2, succ u3, succ u4} E F G g f)) s x) (ContinuousLinearMap.comp.{u1, u1, u1, u2, u4, u5} π π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) G (UniformSpace.toTopologicalSpace.{u4} G (PseudoMetricSpace.toUniformSpace.{u4} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u4} G (NormedAddCommGroup.toAddCommGroup.{u4} G _inst_6)) G' (UniformSpace.toTopologicalSpace.{u5} G' (PseudoMetricSpace.toUniformSpace.{u5} G' (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} G' (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G' _inst_8)))) (AddCommGroup.toAddCommMonoid.{u5} G' (NormedAddCommGroup.toAddCommGroup.{u5} G' _inst_8)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6) _inst_7) (NormedSpace.toModule.{u1, u5} π G' (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} G' _inst_8) _inst_9) (RingHomCompTriple.right_ids.{u1, u1} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1)))))))) (fderivWithin.{u1, u4, u5} π _inst_1 G _inst_6 _inst_7 G' _inst_8 _inst_9 g' u y') (ContinuousLinearMap.comp.{u1, u1, u1, u2, u3, u4} π π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u4} G (PseudoMetricSpace.toUniformSpace.{u4} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u4} G (NormedAddCommGroup.toAddCommGroup.{u4} G _inst_6)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6) _inst_7) (RingHomCompTriple.right_ids.{u1, u1} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1)))))))) (fderivWithin.{u1, u3, u4} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g t y) (fderivWithin.{u1, u2, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f s x))))
+but is expected to have type
+ forall {π : Type.{u3}} [_inst_1 : NontriviallyNormedField.{u3} π] {E : Type.{u1}} [_inst_2 : NormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u3, u1} π E (NontriviallyNormedField.toNormedField.{u3} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)] {F : Type.{u5}} [_inst_4 : NormedAddCommGroup.{u5} F] [_inst_5 : NormedSpace.{u3, u5} π F (NontriviallyNormedField.toNormedField.{u3} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} F _inst_4)] {G : Type.{u4}} [_inst_6 : NormedAddCommGroup.{u4} G] [_inst_7 : NormedSpace.{u3, u4} π G (NontriviallyNormedField.toNormedField.{u3} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)] {G' : Type.{u2}} [_inst_8 : NormedAddCommGroup.{u2} G'] [_inst_9 : NormedSpace.{u3, u2} π G' (NontriviallyNormedField.toNormedField.{u3} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G' _inst_8)] {f : E -> F} (x : E) {s : Set.{u1} E} {g' : G -> G'} {g : F -> G} {t : Set.{u5} F} {u : Set.{u4} G} {y : F} {y' : G}, (DifferentiableWithinAt.{u3, u4, u2} π _inst_1 G _inst_6 _inst_7 G' _inst_8 _inst_9 g' u y') -> (DifferentiableWithinAt.{u3, u5, u4} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g t y) -> (DifferentiableWithinAt.{u3, u1, u5} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f s x) -> (Set.MapsTo.{u5, u4} F G g t u) -> (Set.MapsTo.{u1, u5} E F f s t) -> (Eq.{succ u4} G (g y) y') -> (Eq.{succ u5} F (f x) y) -> (UniqueDiffWithinAt.{u3, u1} π _inst_1 E (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u3, u1} π E (NontriviallyNormedField.toNormedField.{u3} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) s x) -> (Eq.{max (succ u1) (succ u2)} (ContinuousLinearMap.{u3, u3, u1, u2} π π (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1))))) (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1))))) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) G' (UniformSpace.toTopologicalSpace.{u2} G' (PseudoMetricSpace.toUniformSpace.{u2} G' (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} G' (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G' _inst_8)))) (AddCommGroup.toAddCommMonoid.{u2} G' (NormedAddCommGroup.toAddCommGroup.{u2} G' _inst_8)) (NormedSpace.toModule.{u3, u1} π E (NontriviallyNormedField.toNormedField.{u3} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u3, u2} π G' (NontriviallyNormedField.toNormedField.{u3} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G' _inst_8) _inst_9)) (fderivWithin.{u3, u1, u2} π _inst_1 E _inst_2 _inst_3 G' _inst_8 _inst_9 (Function.comp.{succ u1, succ u4, succ u2} E G G' g' (Function.comp.{succ u1, succ u5, succ u4} E F G g f)) s x) (ContinuousLinearMap.comp.{u3, u3, u3, u1, u4, u2} π π π (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1))))) (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1))))) (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1))))) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1))))))) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1))))))) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) G (UniformSpace.toTopologicalSpace.{u4} G (PseudoMetricSpace.toUniformSpace.{u4} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u4} G (NormedAddCommGroup.toAddCommGroup.{u4} G _inst_6)) G' (UniformSpace.toTopologicalSpace.{u2} G' (PseudoMetricSpace.toUniformSpace.{u2} G' (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} G' (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G' _inst_8)))) (AddCommGroup.toAddCommMonoid.{u2} G' (NormedAddCommGroup.toAddCommGroup.{u2} G' _inst_8)) (NormedSpace.toModule.{u3, u1} π E (NontriviallyNormedField.toNormedField.{u3} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u3, u4} π G (NontriviallyNormedField.toNormedField.{u3} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6) _inst_7) (NormedSpace.toModule.{u3, u2} π G' (NontriviallyNormedField.toNormedField.{u3} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G' _inst_8) _inst_9) (RingHomCompTriple.ids.{u3, u3} π π (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1))))) (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1))))) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1)))))))) (fderivWithin.{u3, u4, u2} π _inst_1 G _inst_6 _inst_7 G' _inst_8 _inst_9 g' u y') (ContinuousLinearMap.comp.{u3, u3, u3, u1, u5, u4} π π π (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1))))) (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1))))) (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1))))) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1))))))) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1))))))) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u5} F (PseudoMetricSpace.toUniformSpace.{u5} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u5} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u5} F (NormedAddCommGroup.toAddCommGroup.{u5} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u4} G (PseudoMetricSpace.toUniformSpace.{u4} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u4} G (NormedAddCommGroup.toAddCommGroup.{u4} G _inst_6)) (NormedSpace.toModule.{u3, u1} π E (NontriviallyNormedField.toNormedField.{u3} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u3, u5} π F (NontriviallyNormedField.toNormedField.{u3} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u5} F _inst_4) _inst_5) (NormedSpace.toModule.{u3, u4} π G (NontriviallyNormedField.toNormedField.{u3} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6) _inst_7) (RingHomCompTriple.ids.{u3, u3} π π (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1))))) (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1))))) (RingHom.id.{u3} π (Semiring.toNonAssocSemiring.{u3} π (DivisionSemiring.toSemiring.{u3} π (Semifield.toDivisionSemiring.{u3} π (Field.toSemifield.{u3} π (NormedField.toField.{u3} π (NontriviallyNormedField.toNormedField.{u3} π _inst_1)))))))) (fderivWithin.{u3, u5, u4} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g t y) (fderivWithin.{u3, u1, u5} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f s x))))
+Case conversion may be inaccurate. Consider using '#align fderiv_within.compβ fderivWithin.compββ'. -/
/-- Ternary version of `fderiv_within.comp`, with equality assumptions of basepoints added, in
order to apply more easily as a rewrite from right-to-left. -/
theorem fderivWithin.compβ {g' : G β G'} {g : F β G} {t : Set F} {u : Set G} {y : F} {y' : G}
@@ -173,31 +245,67 @@ theorem fderivWithin.compβ {g' : G β G'} {g : F β G} {t : Set F} {u : Set
hxs
#align fderiv_within.compβ fderivWithin.compβ
+/- warning: fderiv.comp -> fderiv.comp is a dubious translation:
+lean 3 declaration is
+ forall {π : Type.{u1}} [_inst_1 : NontriviallyNormedField.{u1} π] {E : Type.{u2}} [_inst_2 : NormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)] {F : Type.{u3}} [_inst_4 : NormedAddCommGroup.{u3} F] [_inst_5 : NormedSpace.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)] {G : Type.{u4}} [_inst_6 : NormedAddCommGroup.{u4} G] [_inst_7 : NormedSpace.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)] {f : E -> F} (x : E) {g : F -> G}, (DifferentiableAt.{u1, u3, u4} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g (f x)) -> (DifferentiableAt.{u1, u2, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f x) -> (Eq.{max (succ u2) (succ u4)} (ContinuousLinearMap.{u1, u1, u2, u4} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) G (UniformSpace.toTopologicalSpace.{u4} G (PseudoMetricSpace.toUniformSpace.{u4} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u4} G (NormedAddCommGroup.toAddCommGroup.{u4} G _inst_6)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6) _inst_7)) (fderiv.{u1, u2, u4} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (Function.comp.{succ u2, succ u3, succ u4} E F G g f) x) (ContinuousLinearMap.comp.{u1, u1, u1, u2, u3, u4} π π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u4} G (PseudoMetricSpace.toUniformSpace.{u4} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u4} G (NormedAddCommGroup.toAddCommGroup.{u4} G _inst_6)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6) _inst_7) (RingHomCompTriple.right_ids.{u1, u1} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1)))))))) (fderiv.{u1, u3, u4} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g (f x)) (fderiv.{u1, u2, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f x)))
+but is expected to have type
+ forall {π : Type.{u4}} [_inst_1 : NontriviallyNormedField.{u4} π] {E : Type.{u1}} [_inst_2 : NormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u4, u1} π E (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)] {F : Type.{u3}} [_inst_4 : NormedAddCommGroup.{u3} F] [_inst_5 : NormedSpace.{u4, u3} π F (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)] {G : Type.{u2}} [_inst_6 : NormedAddCommGroup.{u2} G] [_inst_7 : NormedSpace.{u4, u2} π G (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)] {f : E -> F} (x : E) {g : F -> G}, (DifferentiableAt.{u4, u3, u2} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g (f x)) -> (DifferentiableAt.{u4, u1, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f x) -> (Eq.{max (succ u1) (succ u2)} (ContinuousLinearMap.{u4, u4, u1, u2} π π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) G (UniformSpace.toTopologicalSpace.{u2} G (PseudoMetricSpace.toUniformSpace.{u2} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u2} G (NormedAddCommGroup.toAddCommGroup.{u2} G _inst_6)) (NormedSpace.toModule.{u4, u1} π E (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u4, u2} π G (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6) _inst_7)) (fderiv.{u4, u1, u2} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (Function.comp.{succ u1, succ u3, succ u2} E F G g f) x) (ContinuousLinearMap.comp.{u4, u4, u4, u1, u3, u2} π π π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u2} G (PseudoMetricSpace.toUniformSpace.{u2} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u2} G (NormedAddCommGroup.toAddCommGroup.{u2} G _inst_6)) (NormedSpace.toModule.{u4, u1} π E (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u4, u3} π F (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u4, u2} π G (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6) _inst_7) (RingHomCompTriple.ids.{u4, u4} π π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1)))))))) (fderiv.{u4, u3, u2} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g (f x)) (fderiv.{u4, u1, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f x)))
+Case conversion may be inaccurate. Consider using '#align fderiv.comp fderiv.compβ'. -/
theorem fderiv.comp {g : F β G} (hg : DifferentiableAt π g (f x)) (hf : DifferentiableAt π f x) :
fderiv π (g β f) x = (fderiv π g (f x)).comp (fderiv π f x) :=
(hg.HasFDerivAt.comp x hf.HasFDerivAt).fderiv
#align fderiv.comp fderiv.comp
+/- warning: fderiv.comp_fderiv_within -> fderiv.comp_fderivWithin is a dubious translation:
+lean 3 declaration is
+ forall {π : Type.{u1}} [_inst_1 : NontriviallyNormedField.{u1} π] {E : Type.{u2}} [_inst_2 : NormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)] {F : Type.{u3}} [_inst_4 : NormedAddCommGroup.{u3} F] [_inst_5 : NormedSpace.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)] {G : Type.{u4}} [_inst_6 : NormedAddCommGroup.{u4} G] [_inst_7 : NormedSpace.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)] {f : E -> F} (x : E) {s : Set.{u2} E} {g : F -> G}, (DifferentiableAt.{u1, u3, u4} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g (f x)) -> (DifferentiableWithinAt.{u1, u2, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f s x) -> (UniqueDiffWithinAt.{u1, u2} π _inst_1 E (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) s x) -> (Eq.{max (succ u2) (succ u4)} (ContinuousLinearMap.{u1, u1, u2, u4} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) G (UniformSpace.toTopologicalSpace.{u4} G (PseudoMetricSpace.toUniformSpace.{u4} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u4} G (NormedAddCommGroup.toAddCommGroup.{u4} G _inst_6)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6) _inst_7)) (fderivWithin.{u1, u2, u4} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (Function.comp.{succ u2, succ u3, succ u4} E F G g f) s x) (ContinuousLinearMap.comp.{u1, u1, u1, u2, u3, u4} π π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u4} G (PseudoMetricSpace.toUniformSpace.{u4} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u4} G (NormedAddCommGroup.toAddCommGroup.{u4} G _inst_6)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6) _inst_7) (RingHomCompTriple.right_ids.{u1, u1} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1)))))))) (fderiv.{u1, u3, u4} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g (f x)) (fderivWithin.{u1, u2, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f s x)))
+but is expected to have type
+ forall {π : Type.{u4}} [_inst_1 : NontriviallyNormedField.{u4} π] {E : Type.{u1}} [_inst_2 : NormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u4, u1} π E (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)] {F : Type.{u3}} [_inst_4 : NormedAddCommGroup.{u3} F] [_inst_5 : NormedSpace.{u4, u3} π F (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)] {G : Type.{u2}} [_inst_6 : NormedAddCommGroup.{u2} G] [_inst_7 : NormedSpace.{u4, u2} π G (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)] {f : E -> F} (x : E) {s : Set.{u1} E} {g : F -> G}, (DifferentiableAt.{u4, u3, u2} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g (f x)) -> (DifferentiableWithinAt.{u4, u1, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f s x) -> (UniqueDiffWithinAt.{u4, u1} π _inst_1 E (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u4, u1} π E (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) s x) -> (Eq.{max (succ u1) (succ u2)} (ContinuousLinearMap.{u4, u4, u1, u2} π π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) G (UniformSpace.toTopologicalSpace.{u2} G (PseudoMetricSpace.toUniformSpace.{u2} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u2} G (NormedAddCommGroup.toAddCommGroup.{u2} G _inst_6)) (NormedSpace.toModule.{u4, u1} π E (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u4, u2} π G (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6) _inst_7)) (fderivWithin.{u4, u1, u2} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (Function.comp.{succ u1, succ u3, succ u2} E F G g f) s x) (ContinuousLinearMap.comp.{u4, u4, u4, u1, u3, u2} π π π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u2} G (PseudoMetricSpace.toUniformSpace.{u2} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u2} G (NormedAddCommGroup.toAddCommGroup.{u2} G _inst_6)) (NormedSpace.toModule.{u4, u1} π E (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u4, u3} π F (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u4, u2} π G (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6) _inst_7) (RingHomCompTriple.ids.{u4, u4} π π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1)))))))) (fderiv.{u4, u3, u2} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g (f x)) (fderivWithin.{u4, u1, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f s x)))
+Case conversion may be inaccurate. Consider using '#align fderiv.comp_fderiv_within fderiv.comp_fderivWithinβ'. -/
theorem fderiv.comp_fderivWithin {g : F β G} (hg : DifferentiableAt π g (f x))
(hf : DifferentiableWithinAt π f s x) (hxs : UniqueDiffWithinAt π s x) :
fderivWithin π (g β f) s x = (fderiv π g (f x)).comp (fderivWithin π f s x) :=
(hg.HasFDerivAt.comp_hasFDerivWithinAt x hf.HasFDerivWithinAt).fderivWithin hxs
#align fderiv.comp_fderiv_within fderiv.comp_fderivWithin
+/- warning: differentiable_on.comp -> DifferentiableOn.comp is a dubious translation:
+lean 3 declaration is
+ forall {π : Type.{u1}} [_inst_1 : NontriviallyNormedField.{u1} π] {E : Type.{u2}} [_inst_2 : NormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)] {F : Type.{u3}} [_inst_4 : NormedAddCommGroup.{u3} F] [_inst_5 : NormedSpace.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)] {G : Type.{u4}} [_inst_6 : NormedAddCommGroup.{u4} G] [_inst_7 : NormedSpace.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)] {f : E -> F} {s : Set.{u2} E} {g : F -> G} {t : Set.{u3} F}, (DifferentiableOn.{u1, u3, u4} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g t) -> (DifferentiableOn.{u1, u2, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f s) -> (Set.MapsTo.{u2, u3} E F f s t) -> (DifferentiableOn.{u1, u2, u4} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (Function.comp.{succ u2, succ u3, succ u4} E F G g f) s)
+but is expected to have type
+ forall {π : Type.{u3}} [_inst_1 : NontriviallyNormedField.{u3} π] {E : Type.{u1}} [_inst_2 : NormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u3, u1} π E (NontriviallyNormedField.toNormedField.{u3} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)] {F : Type.{u4}} [_inst_4 : NormedAddCommGroup.{u4} F] [_inst_5 : NormedSpace.{u3, u4} π F (NontriviallyNormedField.toNormedField.{u3} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} F _inst_4)] {G : Type.{u2}} [_inst_6 : NormedAddCommGroup.{u2} G] [_inst_7 : NormedSpace.{u3, u2} π G (NontriviallyNormedField.toNormedField.{u3} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)] {f : E -> F} {s : Set.{u1} E} {g : F -> G} {t : Set.{u4} F}, (DifferentiableOn.{u3, u4, u2} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g t) -> (DifferentiableOn.{u3, u1, u4} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f s) -> (Set.MapsTo.{u1, u4} E F f s t) -> (DifferentiableOn.{u3, u1, u2} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (Function.comp.{succ u1, succ u4, succ u2} E F G g f) s)
+Case conversion may be inaccurate. Consider using '#align differentiable_on.comp DifferentiableOn.compβ'. -/
theorem DifferentiableOn.comp {g : F β G} {t : Set F} (hg : DifferentiableOn π g t)
(hf : DifferentiableOn π f s) (st : MapsTo f s t) : DifferentiableOn π (g β f) s := fun x hx =>
DifferentiableWithinAt.comp x (hg (f x) (st hx)) (hf x hx) st
#align differentiable_on.comp DifferentiableOn.comp
+/- warning: differentiable.comp -> Differentiable.comp is a dubious translation:
+lean 3 declaration is
+ forall {π : Type.{u1}} [_inst_1 : NontriviallyNormedField.{u1} π] {E : Type.{u2}} [_inst_2 : NormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)] {F : Type.{u3}} [_inst_4 : NormedAddCommGroup.{u3} F] [_inst_5 : NormedSpace.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)] {G : Type.{u4}} [_inst_6 : NormedAddCommGroup.{u4} G] [_inst_7 : NormedSpace.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)] {f : E -> F} {g : F -> G}, (Differentiable.{u1, u3, u4} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g) -> (Differentiable.{u1, u2, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f) -> (Differentiable.{u1, u2, u4} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (Function.comp.{succ u2, succ u3, succ u4} E F G g f))
+but is expected to have type
+ forall {π : Type.{u4}} [_inst_1 : NontriviallyNormedField.{u4} π] {E : Type.{u1}} [_inst_2 : NormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u4, u1} π E (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)] {F : Type.{u3}} [_inst_4 : NormedAddCommGroup.{u3} F] [_inst_5 : NormedSpace.{u4, u3} π F (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)] {G : Type.{u2}} [_inst_6 : NormedAddCommGroup.{u2} G] [_inst_7 : NormedSpace.{u4, u2} π G (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)] {f : E -> F} {g : F -> G}, (Differentiable.{u4, u3, u2} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g) -> (Differentiable.{u4, u1, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f) -> (Differentiable.{u4, u1, u2} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (Function.comp.{succ u1, succ u3, succ u2} E F G g f))
+Case conversion may be inaccurate. Consider using '#align differentiable.comp Differentiable.compβ'. -/
theorem Differentiable.comp {g : F β G} (hg : Differentiable π g) (hf : Differentiable π f) :
Differentiable π (g β f) := fun x => DifferentiableAt.comp x (hg (f x)) (hf x)
#align differentiable.comp Differentiable.comp
+/- warning: differentiable.comp_differentiable_on -> Differentiable.comp_differentiableOn is a dubious translation:
+lean 3 declaration is
+ forall {π : Type.{u1}} [_inst_1 : NontriviallyNormedField.{u1} π] {E : Type.{u2}} [_inst_2 : NormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)] {F : Type.{u3}} [_inst_4 : NormedAddCommGroup.{u3} F] [_inst_5 : NormedSpace.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)] {G : Type.{u4}} [_inst_6 : NormedAddCommGroup.{u4} G] [_inst_7 : NormedSpace.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)] {f : E -> F} {s : Set.{u2} E} {g : F -> G}, (Differentiable.{u1, u3, u4} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g) -> (DifferentiableOn.{u1, u2, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f s) -> (DifferentiableOn.{u1, u2, u4} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (Function.comp.{succ u2, succ u3, succ u4} E F G g f) s)
+but is expected to have type
+ forall {π : Type.{u4}} [_inst_1 : NontriviallyNormedField.{u4} π] {E : Type.{u1}} [_inst_2 : NormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u4, u1} π E (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)] {F : Type.{u3}} [_inst_4 : NormedAddCommGroup.{u3} F] [_inst_5 : NormedSpace.{u4, u3} π F (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)] {G : Type.{u2}} [_inst_6 : NormedAddCommGroup.{u2} G] [_inst_7 : NormedSpace.{u4, u2} π G (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)] {f : E -> F} {s : Set.{u1} E} {g : F -> G}, (Differentiable.{u4, u3, u2} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g) -> (DifferentiableOn.{u4, u1, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f s) -> (DifferentiableOn.{u4, u1, u2} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (Function.comp.{succ u1, succ u3, succ u2} E F G g f) s)
+Case conversion may be inaccurate. Consider using '#align differentiable.comp_differentiable_on Differentiable.comp_differentiableOnβ'. -/
theorem Differentiable.comp_differentiableOn {g : F β G} (hg : Differentiable π g)
(hf : DifferentiableOn π f s) : DifferentiableOn π (g β f) s :=
hg.DifferentiableOn.comp hf (mapsTo_univ _ _)
#align differentiable.comp_differentiable_on Differentiable.comp_differentiableOn
+/- warning: has_strict_fderiv_at.comp -> HasStrictFDerivAt.comp is a dubious translation:
+lean 3 declaration is
+ forall {π : Type.{u1}} [_inst_1 : NontriviallyNormedField.{u1} π] {E : Type.{u2}} [_inst_2 : NormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)] {F : Type.{u3}} [_inst_4 : NormedAddCommGroup.{u3} F] [_inst_5 : NormedSpace.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)] {G : Type.{u4}} [_inst_6 : NormedAddCommGroup.{u4} G] [_inst_7 : NormedSpace.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)] {f : E -> F} {f' : ContinuousLinearMap.{u1, u1, u2, u3} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5)} (x : E) {g : F -> G} {g' : ContinuousLinearMap.{u1, u1, u3, u4} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u4} G (PseudoMetricSpace.toUniformSpace.{u4} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u4} G (NormedAddCommGroup.toAddCommGroup.{u4} G _inst_6)) (NormedSpace.toModule.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6) _inst_7)}, (HasStrictFDerivAt.{u1, u3, u4} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g g' (f x)) -> (HasStrictFDerivAt.{u1, u2, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f f' x) -> (HasStrictFDerivAt.{u1, u2, u4} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (fun (x : E) => g (f x)) (ContinuousLinearMap.comp.{u1, u1, u1, u2, u3, u4} π π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u4} G (PseudoMetricSpace.toUniformSpace.{u4} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u4} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u4} G (NormedAddCommGroup.toAddCommGroup.{u4} G _inst_6)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u3} π F (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u1, u4} π G (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u4} G _inst_6) _inst_7) (RingHomCompTriple.right_ids.{u1, u1} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1)))))))) g' f') x)
+but is expected to have type
+ forall {π : Type.{u4}} [_inst_1 : NontriviallyNormedField.{u4} π] {E : Type.{u1}} [_inst_2 : NormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u4, u1} π E (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)] {F : Type.{u3}} [_inst_4 : NormedAddCommGroup.{u3} F] [_inst_5 : NormedSpace.{u4, u3} π F (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)] {G : Type.{u2}} [_inst_6 : NormedAddCommGroup.{u2} G] [_inst_7 : NormedSpace.{u4, u2} π G (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)] {f : E -> F} {f' : ContinuousLinearMap.{u4, u4, u1, u3} π π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) (NormedSpace.toModule.{u4, u1} π E (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u4, u3} π F (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5)} (x : E) {g : F -> G} {g' : ContinuousLinearMap.{u4, u4, u3, u2} π π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u2} G (PseudoMetricSpace.toUniformSpace.{u2} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u2} G (NormedAddCommGroup.toAddCommGroup.{u2} G _inst_6)) (NormedSpace.toModule.{u4, u3} π F (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u4, u2} π G (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6) _inst_7)}, (HasStrictFDerivAt.{u4, u3, u2} π _inst_1 F _inst_4 _inst_5 G _inst_6 _inst_7 g g' (f x)) -> (HasStrictFDerivAt.{u4, u1, u3} π _inst_1 E _inst_2 _inst_3 F _inst_4 _inst_5 f f' x) -> (HasStrictFDerivAt.{u4, u1, u2} π _inst_1 E _inst_2 _inst_3 G _inst_6 _inst_7 (fun (x : E) => g (f x)) (ContinuousLinearMap.comp.{u4, u4, u4, u1, u3, u2} π π π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) F (UniformSpace.toTopologicalSpace.{u3} F (PseudoMetricSpace.toUniformSpace.{u3} F (SeminormedAddCommGroup.toPseudoMetricSpace.{u3} F (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4)))) (AddCommGroup.toAddCommMonoid.{u3} F (NormedAddCommGroup.toAddCommGroup.{u3} F _inst_4)) G (UniformSpace.toTopologicalSpace.{u2} G (PseudoMetricSpace.toUniformSpace.{u2} G (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} G (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6)))) (AddCommGroup.toAddCommMonoid.{u2} G (NormedAddCommGroup.toAddCommGroup.{u2} G _inst_6)) (NormedSpace.toModule.{u4, u1} π E (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u4, u3} π F (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u3} F _inst_4) _inst_5) (NormedSpace.toModule.{u4, u2} π G (NontriviallyNormedField.toNormedField.{u4} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} G _inst_6) _inst_7) (RingHomCompTriple.ids.{u4, u4} π π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1))))) (RingHom.id.{u4} π (Semiring.toNonAssocSemiring.{u4} π (DivisionSemiring.toSemiring.{u4} π (Semifield.toDivisionSemiring.{u4} π (Field.toSemifield.{u4} π (NormedField.toField.{u4} π (NontriviallyNormedField.toNormedField.{u4} π _inst_1)))))))) g' f') x)
+Case conversion may be inaccurate. Consider using '#align has_strict_fderiv_at.comp HasStrictFDerivAt.compβ'. -/
/-- The chain rule for derivatives in the sense of strict differentiability. -/
protected theorem HasStrictFDerivAt.comp {g : F β G} {g' : F βL[π] G}
(hg : HasStrictFDerivAt g g' (f x)) (hf : HasStrictFDerivAt f f' x) :
@@ -207,11 +315,23 @@ protected theorem HasStrictFDerivAt.comp {g : F β G} {g' : F βL[π] G}
by simpa only [g'.map_sub, f'.coe_comp'] using (g'.is_O_comp _ _).trans_isLittleO hf
#align has_strict_fderiv_at.comp HasStrictFDerivAt.comp
+/- warning: differentiable.iterate -> Differentiable.iterate is a dubious translation:
+lean 3 declaration is
+ forall {π : Type.{u1}} [_inst_1 : NontriviallyNormedField.{u1} π] {E : Type.{u2}} [_inst_2 : NormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)] {f : E -> E}, (Differentiable.{u1, u2, u2} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 f) -> (forall (n : Nat), Differentiable.{u1, u2, u2} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 (Nat.iterate.{succ u2} E f n))
+but is expected to have type
+ forall {π : Type.{u2}} [_inst_1 : NontriviallyNormedField.{u2} π] {E : Type.{u1}} [_inst_2 : NormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)] {f : E -> E}, (Differentiable.{u2, u1, u1} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 f) -> (forall (n : Nat), Differentiable.{u2, u1, u1} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 (Nat.iterate.{succ u1} E f n))
+Case conversion may be inaccurate. Consider using '#align differentiable.iterate Differentiable.iterateβ'. -/
protected theorem Differentiable.iterate {f : E β E} (hf : Differentiable π f) (n : β) :
Differentiable π (f^[n]) :=
Nat.recOn n differentiable_id fun n ihn => ihn.comp hf
#align differentiable.iterate Differentiable.iterate
+/- warning: differentiable_on.iterate -> DifferentiableOn.iterate is a dubious translation:
+lean 3 declaration is
+ forall {π : Type.{u1}} [_inst_1 : NontriviallyNormedField.{u1} π] {E : Type.{u2}} [_inst_2 : NormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)] {s : Set.{u2} E} {f : E -> E}, (DifferentiableOn.{u1, u2, u2} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 f s) -> (Set.MapsTo.{u2, u2} E E f s s) -> (forall (n : Nat), DifferentiableOn.{u1, u2, u2} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 (Nat.iterate.{succ u2} E f n) s)
+but is expected to have type
+ forall {π : Type.{u2}} [_inst_1 : NontriviallyNormedField.{u2} π] {E : Type.{u1}} [_inst_2 : NormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)] {s : Set.{u1} E} {f : E -> E}, (DifferentiableOn.{u2, u1, u1} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 f s) -> (Set.MapsTo.{u1, u1} E E f s s) -> (forall (n : Nat), DifferentiableOn.{u2, u1, u1} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 (Nat.iterate.{succ u1} E f n) s)
+Case conversion may be inaccurate. Consider using '#align differentiable_on.iterate DifferentiableOn.iterateβ'. -/
protected theorem DifferentiableOn.iterate {f : E β E} (hf : DifferentiableOn π f s)
(hs : MapsTo f s s) (n : β) : DifferentiableOn π (f^[n]) s :=
Nat.recOn n differentiableOn_id fun n ihn => ihn.comp hf hs
@@ -219,6 +339,12 @@ protected theorem DifferentiableOn.iterate {f : E β E} (hf : DifferentiableOn
variable {x}
+/- warning: has_fderiv_at_filter.iterate -> HasFDerivAtFilter.iterate is a dubious translation:
+lean 3 declaration is
+ forall {π : Type.{u1}} [_inst_1 : NontriviallyNormedField.{u1} π] {E : Type.{u2}} [_inst_2 : NormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)] {x : E} {L : Filter.{u2} E} {f : E -> E} {f' : ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)}, (HasFDerivAtFilter.{u1, u2, u2} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 f f' x L) -> (Filter.Tendsto.{u2, u2} E E f L L) -> (Eq.{succ u2} E (f x) x) -> (forall (n : Nat), HasFDerivAtFilter.{u1, u2, u2} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 (Nat.iterate.{succ u2} E f n) (HPow.hPow.{u2, 0, u2} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) Nat (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) (instHPow.{u2, 0} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) Nat (Monoid.Pow.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) (Ring.toMonoid.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) (ContinuousLinearMap.ring.{u1, u2} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1)))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (SeminormedAddCommGroup.to_topologicalAddGroup.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))))) f' n) x L)
+but is expected to have type
+ forall {π : Type.{u2}} [_inst_1 : NontriviallyNormedField.{u2} π] {E : Type.{u1}} [_inst_2 : NormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)] {x : E} {L : Filter.{u1} E} {f : E -> E} {f' : ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)}, (HasFDerivAtFilter.{u2, u1, u1} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 f f' x L) -> (Filter.Tendsto.{u1, u1} E E f L L) -> (Eq.{succ u1} E (f x) x) -> (forall (n : Nat), HasFDerivAtFilter.{u2, u1, u1} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 (Nat.iterate.{succ u1} E f n) (HPow.hPow.{u1, 0, u1} (ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)) Nat (ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)) (instHPow.{u1, 0} (ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)) Nat (Monoid.Pow.{u1} (ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)) (MonoidWithZero.toMonoid.{u1} (ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)) (ContinuousLinearMap.monoidWithZero.{u2, u1} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3))))) f' n) x L)
+Case conversion may be inaccurate. Consider using '#align has_fderiv_at_filter.iterate HasFDerivAtFilter.iterateβ'. -/
protected theorem HasFDerivAtFilter.iterate {f : E β E} {f' : E βL[π] E}
(hf : HasFDerivAtFilter f f' x L) (hL : Tendsto f L L) (hx : f x = x) (n : β) :
HasFDerivAtFilter (f^[n]) (f' ^ n) x L :=
@@ -230,6 +356,12 @@ protected theorem HasFDerivAtFilter.iterate {f : E β E} {f' : E βL[π] E}
exact ihn.comp x hf hL
#align has_fderiv_at_filter.iterate HasFDerivAtFilter.iterate
+/- warning: has_fderiv_at.iterate -> HasFDerivAt.iterate is a dubious translation:
+lean 3 declaration is
+ forall {π : Type.{u1}} [_inst_1 : NontriviallyNormedField.{u1} π] {E : Type.{u2}} [_inst_2 : NormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)] {x : E} {f : E -> E} {f' : ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)}, (HasFDerivAt.{u1, u2, u2} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 f f' x) -> (Eq.{succ u2} E (f x) x) -> (forall (n : Nat), HasFDerivAt.{u1, u2, u2} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 (Nat.iterate.{succ u2} E f n) (HPow.hPow.{u2, 0, u2} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) Nat (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) (instHPow.{u2, 0} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) Nat (Monoid.Pow.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) (Ring.toMonoid.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) (ContinuousLinearMap.ring.{u1, u2} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1)))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (SeminormedAddCommGroup.to_topologicalAddGroup.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))))) f' n) x)
+but is expected to have type
+ forall {π : Type.{u2}} [_inst_1 : NontriviallyNormedField.{u2} π] {E : Type.{u1}} [_inst_2 : NormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)] {x : E} {f : E -> E} {f' : ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)}, (HasFDerivAt.{u2, u1, u1} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 f f' x) -> (Eq.{succ u1} E (f x) x) -> (forall (n : Nat), HasFDerivAt.{u2, u1, u1} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 (Nat.iterate.{succ u1} E f n) (HPow.hPow.{u1, 0, u1} (ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)) Nat (ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)) (instHPow.{u1, 0} (ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)) Nat (Monoid.Pow.{u1} (ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)) (MonoidWithZero.toMonoid.{u1} (ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)) (ContinuousLinearMap.monoidWithZero.{u2, u1} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3))))) f' n) x)
+Case conversion may be inaccurate. Consider using '#align has_fderiv_at.iterate HasFDerivAt.iterateβ'. -/
protected theorem HasFDerivAt.iterate {f : E β E} {f' : E βL[π] E} (hf : HasFDerivAt f f' x)
(hx : f x = x) (n : β) : HasFDerivAt (f^[n]) (f' ^ n) x :=
by
@@ -238,6 +370,12 @@ protected theorem HasFDerivAt.iterate {f : E β E} {f' : E βL[π] E} (hf :
exact hx.symm
#align has_fderiv_at.iterate HasFDerivAt.iterate
+/- warning: has_fderiv_within_at.iterate -> HasFDerivWithinAt.iterate is a dubious translation:
+lean 3 declaration is
+ forall {π : Type.{u1}} [_inst_1 : NontriviallyNormedField.{u1} π] {E : Type.{u2}} [_inst_2 : NormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)] {x : E} {s : Set.{u2} E} {f : E -> E} {f' : ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)}, (HasFDerivWithinAt.{u1, u2, u2} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 f f' s x) -> (Eq.{succ u2} E (f x) x) -> (Set.MapsTo.{u2, u2} E E f s s) -> (forall (n : Nat), HasFDerivWithinAt.{u1, u2, u2} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 (Nat.iterate.{succ u2} E f n) (HPow.hPow.{u2, 0, u2} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) Nat (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) (instHPow.{u2, 0} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) Nat (Monoid.Pow.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) (Ring.toMonoid.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) (ContinuousLinearMap.ring.{u1, u2} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1)))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (SeminormedAddCommGroup.to_topologicalAddGroup.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))))) f' n) s x)
+but is expected to have type
+ forall {π : Type.{u2}} [_inst_1 : NontriviallyNormedField.{u2} π] {E : Type.{u1}} [_inst_2 : NormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)] {x : E} {s : Set.{u1} E} {f : E -> E} {f' : ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)}, (HasFDerivWithinAt.{u2, u1, u1} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 f f' s x) -> (Eq.{succ u1} E (f x) x) -> (Set.MapsTo.{u1, u1} E E f s s) -> (forall (n : Nat), HasFDerivWithinAt.{u2, u1, u1} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 (Nat.iterate.{succ u1} E f n) (HPow.hPow.{u1, 0, u1} (ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)) Nat (ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)) (instHPow.{u1, 0} (ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)) Nat (Monoid.Pow.{u1} (ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)) (MonoidWithZero.toMonoid.{u1} (ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)) (ContinuousLinearMap.monoidWithZero.{u2, u1} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3))))) f' n) s x)
+Case conversion may be inaccurate. Consider using '#align has_fderiv_within_at.iterate HasFDerivWithinAt.iterateβ'. -/
protected theorem HasFDerivWithinAt.iterate {f : E β E} {f' : E βL[π] E}
(hf : HasFDerivWithinAt f f' s x) (hx : f x = x) (hs : MapsTo f s s) (n : β) :
HasFDerivWithinAt (f^[n]) (f' ^ n) s x :=
@@ -247,6 +385,12 @@ protected theorem HasFDerivWithinAt.iterate {f : E β E} {f' : E βL[π] E}
exacts[hx.symm, (tendsto_principal_principal.2 hs).mono_left inf_le_right]
#align has_fderiv_within_at.iterate HasFDerivWithinAt.iterate
+/- warning: has_strict_fderiv_at.iterate -> HasStrictFDerivAt.iterate is a dubious translation:
+lean 3 declaration is
+ forall {π : Type.{u1}} [_inst_1 : NontriviallyNormedField.{u1} π] {E : Type.{u2}} [_inst_2 : NormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)] {x : E} {f : E -> E} {f' : ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)}, (HasStrictFDerivAt.{u1, u2, u2} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 f f' x) -> (Eq.{succ u2} E (f x) x) -> (forall (n : Nat), HasStrictFDerivAt.{u1, u2, u2} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 (Nat.iterate.{succ u2} E f n) (HPow.hPow.{u2, 0, u2} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) Nat (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) (instHPow.{u2, 0} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) Nat (Monoid.Pow.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) (Ring.toMonoid.{u2} (ContinuousLinearMap.{u1, u1, u2, u2} π π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))) (RingHom.id.{u1} π (Semiring.toNonAssocSemiring.{u1} π (Ring.toSemiring.{u1} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u2} E (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2)) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3)) (ContinuousLinearMap.ring.{u1, u2} π (NormedRing.toRing.{u1} π (NormedCommRing.toNormedRing.{u1} π (NormedField.toNormedCommRing.{u1} π (NontriviallyNormedField.toNormedField.{u1} π _inst_1)))) E (UniformSpace.toTopologicalSpace.{u2} E (PseudoMetricSpace.toUniformSpace.{u2} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))) (NormedAddCommGroup.toAddCommGroup.{u2} E _inst_2) (NormedSpace.toModule.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2) _inst_3) (SeminormedAddCommGroup.to_topologicalAddGroup.{u2} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)))))) f' n) x)
+but is expected to have type
+ forall {π : Type.{u2}} [_inst_1 : NontriviallyNormedField.{u2} π] {E : Type.{u1}} [_inst_2 : NormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)] {x : E} {f : E -> E} {f' : ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)}, (HasStrictFDerivAt.{u2, u1, u1} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 f f' x) -> (Eq.{succ u1} E (f x) x) -> (forall (n : Nat), HasStrictFDerivAt.{u2, u1, u1} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 (Nat.iterate.{succ u1} E f n) (HPow.hPow.{u1, 0, u1} (ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)) Nat (ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)) (instHPow.{u1, 0} (ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)) Nat (Monoid.Pow.{u1} (ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)) (MonoidWithZero.toMonoid.{u1} (ContinuousLinearMap.{u2, u2, u1, u1} π π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) (RingHom.id.{u2} π (Semiring.toNonAssocSemiring.{u2} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3)) (ContinuousLinearMap.monoidWithZero.{u2, u1} π (DivisionSemiring.toSemiring.{u2} π (Semifield.toDivisionSemiring.{u2} π (Field.toSemifield.{u2} π (NormedField.toField.{u2} π (NontriviallyNormedField.toNormedField.{u2} π _inst_1))))) E (UniformSpace.toTopologicalSpace.{u1} E (PseudoMetricSpace.toUniformSpace.{u1} E (SeminormedAddCommGroup.toPseudoMetricSpace.{u1} E (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)))) (AddCommGroup.toAddCommMonoid.{u1} E (NormedAddCommGroup.toAddCommGroup.{u1} E _inst_2)) (NormedSpace.toModule.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2) _inst_3))))) f' n) x)
+Case conversion may be inaccurate. Consider using '#align has_strict_fderiv_at.iterate HasStrictFDerivAt.iterateβ'. -/
protected theorem HasStrictFDerivAt.iterate {f : E β E} {f' : E βL[π] E}
(hf : HasStrictFDerivAt f f' x) (hx : f x = x) (n : β) : HasStrictFDerivAt (f^[n]) (f' ^ n) x :=
by
@@ -257,11 +401,23 @@ protected theorem HasStrictFDerivAt.iterate {f : E β E} {f' : E βL[π] E}
exact ihn.comp x hf
#align has_strict_fderiv_at.iterate HasStrictFDerivAt.iterate
+/- warning: differentiable_at.iterate -> DifferentiableAt.iterate is a dubious translation:
+lean 3 declaration is
+ forall {π : Type.{u1}} [_inst_1 : NontriviallyNormedField.{u1} π] {E : Type.{u2}} [_inst_2 : NormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)] {x : E} {f : E -> E}, (DifferentiableAt.{u1, u2, u2} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 f x) -> (Eq.{succ u2} E (f x) x) -> (forall (n : Nat), DifferentiableAt.{u1, u2, u2} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 (Nat.iterate.{succ u2} E f n) x)
+but is expected to have type
+ forall {π : Type.{u2}} [_inst_1 : NontriviallyNormedField.{u2} π] {E : Type.{u1}} [_inst_2 : NormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)] {x : E} {f : E -> E}, (DifferentiableAt.{u2, u1, u1} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 f x) -> (Eq.{succ u1} E (f x) x) -> (forall (n : Nat), DifferentiableAt.{u2, u1, u1} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 (Nat.iterate.{succ u1} E f n) x)
+Case conversion may be inaccurate. Consider using '#align differentiable_at.iterate DifferentiableAt.iterateβ'. -/
protected theorem DifferentiableAt.iterate {f : E β E} (hf : DifferentiableAt π f x) (hx : f x = x)
(n : β) : DifferentiableAt π (f^[n]) x :=
(hf.HasFDerivAt.iterate hx n).DifferentiableAt
#align differentiable_at.iterate DifferentiableAt.iterate
+/- warning: differentiable_within_at.iterate -> DifferentiableWithinAt.iterate is a dubious translation:
+lean 3 declaration is
+ forall {π : Type.{u1}} [_inst_1 : NontriviallyNormedField.{u1} π] {E : Type.{u2}} [_inst_2 : NormedAddCommGroup.{u2} E] [_inst_3 : NormedSpace.{u1, u2} π E (NontriviallyNormedField.toNormedField.{u1} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u2} E _inst_2)] {x : E} {s : Set.{u2} E} {f : E -> E}, (DifferentiableWithinAt.{u1, u2, u2} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 f s x) -> (Eq.{succ u2} E (f x) x) -> (Set.MapsTo.{u2, u2} E E f s s) -> (forall (n : Nat), DifferentiableWithinAt.{u1, u2, u2} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 (Nat.iterate.{succ u2} E f n) s x)
+but is expected to have type
+ forall {π : Type.{u2}} [_inst_1 : NontriviallyNormedField.{u2} π] {E : Type.{u1}} [_inst_2 : NormedAddCommGroup.{u1} E] [_inst_3 : NormedSpace.{u2, u1} π E (NontriviallyNormedField.toNormedField.{u2} π _inst_1) (NormedAddCommGroup.toSeminormedAddCommGroup.{u1} E _inst_2)] {x : E} {s : Set.{u1} E} {f : E -> E}, (DifferentiableWithinAt.{u2, u1, u1} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 f s x) -> (Eq.{succ u1} E (f x) x) -> (Set.MapsTo.{u1, u1} E E f s s) -> (forall (n : Nat), DifferentiableWithinAt.{u2, u1, u1} π _inst_1 E _inst_2 _inst_3 E _inst_2 _inst_3 (Nat.iterate.{succ u1} E f n) s x)
+Case conversion may be inaccurate. Consider using '#align differentiable_within_at.iterate DifferentiableWithinAt.iterateβ'. -/
protected theorem DifferentiableWithinAt.iterate {f : E β E} (hf : DifferentiableWithinAt π f s x)
(hx : f x = x) (hs : MapsTo f s s) (n : β) : DifferentiableWithinAt π (f^[n]) s x :=
(hf.HasFDerivWithinAt.iterate hx hs n).DifferentiableWithinAt
mathlib commit https://github.com/leanprover-community/mathlib/commit/33c67ae661dd8988516ff7f247b0be3018cdd952
@@ -62,22 +62,22 @@ get confused since there are too many possibilities for composition -/
variable (x)
-theorem HasFderivAtFilter.comp {g : F β G} {g' : F βL[π] G} {L' : Filter F}
- (hg : HasFderivAtFilter g g' (f x) L') (hf : HasFderivAtFilter f f' x L) (hL : Tendsto f L L') :
- HasFderivAtFilter (g β f) (g'.comp f') x L :=
+theorem HasFDerivAtFilter.comp {g : F β G} {g' : F βL[π] G} {L' : Filter F}
+ (hg : HasFDerivAtFilter g g' (f x) L') (hf : HasFDerivAtFilter f f' x L) (hL : Tendsto f L L') :
+ HasFDerivAtFilter (g β f) (g'.comp f') x L :=
by
let eqβ := (g'.isBigO_comp _ _).trans_isLittleO hf
let eqβ := (hg.comp_tendsto hL).trans_isBigO hf.isBigO_sub
refine' eqβ.triangle (eqβ.congr_left fun x' => _)
simp
-#align has_fderiv_at_filter.comp HasFderivAtFilter.comp
+#align has_fderiv_at_filter.comp HasFDerivAtFilter.comp
/- A readable version of the previous theorem,
a general form of the chain rule. -/
-example {g : F β G} {g' : F βL[π] G} (hg : HasFderivAtFilter g g' (f x) (L.map f))
- (hf : HasFderivAtFilter f f' x L) : HasFderivAtFilter (g β f) (g'.comp f') x L :=
+example {g : F β G} {g' : F βL[π] G} (hg : HasFDerivAtFilter g g' (f x) (L.map f))
+ (hf : HasFDerivAtFilter f f' x L) : HasFDerivAtFilter (g β f) (g'.comp f') x L :=
by
- unfold HasFderivAtFilter at hg
+ unfold HasFDerivAtFilter at hg
have :=
calc
(fun x' => g (f x') - g (f x) - g' (f x' - f x)) =o[L] fun x' => f x' - f x :=
@@ -93,34 +93,34 @@ example {g : F β G} {g' : F βL[π] G} (hg : HasFderivAtFilter g g' (f x) (
_ =o[L] fun x' => x' - x := hf
-theorem HasFderivWithinAt.comp {g : F β G} {g' : F βL[π] G} {t : Set F}
- (hg : HasFderivWithinAt g g' t (f x)) (hf : HasFderivWithinAt f f' s x) (hst : MapsTo f s t) :
- HasFderivWithinAt (g β f) (g'.comp f') s x :=
+theorem HasFDerivWithinAt.comp {g : F β G} {g' : F βL[π] G} {t : Set F}
+ (hg : HasFDerivWithinAt g g' t (f x)) (hf : HasFDerivWithinAt f f' s x) (hst : MapsTo f s t) :
+ HasFDerivWithinAt (g β f) (g'.comp f') s x :=
hg.comp x hf <| hf.ContinuousWithinAt.tendsto_nhdsWithin hst
-#align has_fderiv_within_at.comp HasFderivWithinAt.comp
+#align has_fderiv_within_at.comp HasFDerivWithinAt.comp
-theorem HasFderivAt.comp_hasFderivWithinAt {g : F β G} {g' : F βL[π] G}
- (hg : HasFderivAt g g' (f x)) (hf : HasFderivWithinAt f f' s x) :
- HasFderivWithinAt (g β f) (g'.comp f') s x :=
+theorem HasFDerivAt.comp_hasFDerivWithinAt {g : F β G} {g' : F βL[π] G}
+ (hg : HasFDerivAt g g' (f x)) (hf : HasFDerivWithinAt f f' s x) :
+ HasFDerivWithinAt (g β f) (g'.comp f') s x :=
hg.comp x hf hf.ContinuousWithinAt
-#align has_fderiv_at.comp_has_fderiv_within_at HasFderivAt.comp_hasFderivWithinAt
+#align has_fderiv_at.comp_has_fderiv_within_at HasFDerivAt.comp_hasFDerivWithinAt
-theorem HasFderivWithinAt.comp_of_mem {g : F β G} {g' : F βL[π] G} {t : Set F}
- (hg : HasFderivWithinAt g g' t (f x)) (hf : HasFderivWithinAt f f' s x)
- (hst : Tendsto f (π[s] x) (π[t] f x)) : HasFderivWithinAt (g β f) (g'.comp f') s x :=
- HasFderivAtFilter.comp x hg hf hst
-#align has_fderiv_within_at.comp_of_mem HasFderivWithinAt.comp_of_mem
+theorem HasFDerivWithinAt.comp_of_mem {g : F β G} {g' : F βL[π] G} {t : Set F}
+ (hg : HasFDerivWithinAt g g' t (f x)) (hf : HasFDerivWithinAt f f' s x)
+ (hst : Tendsto f (π[s] x) (π[t] f x)) : HasFDerivWithinAt (g β f) (g'.comp f') s x :=
+ HasFDerivAtFilter.comp x hg hf hst
+#align has_fderiv_within_at.comp_of_mem HasFDerivWithinAt.comp_of_mem
/-- The chain rule. -/
-theorem HasFderivAt.comp {g : F β G} {g' : F βL[π] G} (hg : HasFderivAt g g' (f x))
- (hf : HasFderivAt f f' x) : HasFderivAt (g β f) (g'.comp f') x :=
+theorem HasFDerivAt.comp {g : F β G} {g' : F βL[π] G} (hg : HasFDerivAt g g' (f x))
+ (hf : HasFDerivAt f f' x) : HasFDerivAt (g β f) (g'.comp f') x :=
hg.comp x hf hf.ContinuousAt
-#align has_fderiv_at.comp HasFderivAt.comp
+#align has_fderiv_at.comp HasFDerivAt.comp
theorem DifferentiableWithinAt.comp {g : F β G} {t : Set F}
(hg : DifferentiableWithinAt π g t (f x)) (hf : DifferentiableWithinAt π f s x)
(h : MapsTo f s t) : DifferentiableWithinAt π (g β f) s x :=
- (hg.HasFderivWithinAt.comp x hf.HasFderivWithinAt h).DifferentiableWithinAt
+ (hg.HasFDerivWithinAt.comp x hf.HasFDerivWithinAt h).DifferentiableWithinAt
#align differentiable_within_at.comp DifferentiableWithinAt.comp
theorem DifferentiableWithinAt.comp' {g : F β G} {t : Set F}
@@ -131,7 +131,7 @@ theorem DifferentiableWithinAt.comp' {g : F β G} {t : Set F}
theorem DifferentiableAt.comp {g : F β G} (hg : DifferentiableAt π g (f x))
(hf : DifferentiableAt π f x) : DifferentiableAt π (g β f) x :=
- (hg.HasFderivAt.comp x hf.HasFderivAt).DifferentiableAt
+ (hg.HasFDerivAt.comp x hf.HasFDerivAt).DifferentiableAt
#align differentiable_at.comp DifferentiableAt.comp
theorem DifferentiableAt.comp_differentiableWithinAt {g : F β G} (hg : DifferentiableAt π g (f x))
@@ -142,7 +142,7 @@ theorem DifferentiableAt.comp_differentiableWithinAt {g : F β G} (hg : Differe
theorem fderivWithin.comp {g : F β G} {t : Set F} (hg : DifferentiableWithinAt π g t (f x))
(hf : DifferentiableWithinAt π f s x) (h : MapsTo f s t) (hxs : UniqueDiffWithinAt π s x) :
fderivWithin π (g β f) s x = (fderivWithin π g t (f x)).comp (fderivWithin π f s x) :=
- (hg.HasFderivWithinAt.comp x hf.HasFderivWithinAt h).fderivWithin hxs
+ (hg.HasFDerivWithinAt.comp x hf.HasFDerivWithinAt h).fderivWithin hxs
#align fderiv_within.comp fderivWithin.comp
/-- A version of `fderiv_within.comp` that is useful to rewrite the composition of two derivatives
@@ -175,13 +175,13 @@ theorem fderivWithin.compβ {g' : G β G'} {g : F β G} {t : Set F} {u : Set
theorem fderiv.comp {g : F β G} (hg : DifferentiableAt π g (f x)) (hf : DifferentiableAt π f x) :
fderiv π (g β f) x = (fderiv π g (f x)).comp (fderiv π f x) :=
- (hg.HasFderivAt.comp x hf.HasFderivAt).fderiv
+ (hg.HasFDerivAt.comp x hf.HasFDerivAt).fderiv
#align fderiv.comp fderiv.comp
theorem fderiv.comp_fderivWithin {g : F β G} (hg : DifferentiableAt π g (f x))
(hf : DifferentiableWithinAt π f s x) (hxs : UniqueDiffWithinAt π s x) :
fderivWithin π (g β f) s x = (fderiv π g (f x)).comp (fderivWithin π f s x) :=
- (hg.HasFderivAt.comp_hasFderivWithinAt x hf.HasFderivWithinAt).fderivWithin hxs
+ (hg.HasFDerivAt.comp_hasFDerivWithinAt x hf.HasFDerivWithinAt).fderivWithin hxs
#align fderiv.comp_fderiv_within fderiv.comp_fderivWithin
theorem DifferentiableOn.comp {g : F β G} {t : Set F} (hg : DifferentiableOn π g t)
@@ -199,13 +199,13 @@ theorem Differentiable.comp_differentiableOn {g : F β G} (hg : Differentiable
#align differentiable.comp_differentiable_on Differentiable.comp_differentiableOn
/-- The chain rule for derivatives in the sense of strict differentiability. -/
-protected theorem HasStrictFderivAt.comp {g : F β G} {g' : F βL[π] G}
- (hg : HasStrictFderivAt g g' (f x)) (hf : HasStrictFderivAt f f' x) :
- HasStrictFderivAt (fun x => g (f x)) (g'.comp f') x :=
+protected theorem HasStrictFDerivAt.comp {g : F β G} {g' : F βL[π] G}
+ (hg : HasStrictFDerivAt g g' (f x)) (hf : HasStrictFDerivAt f f' x) :
+ HasStrictFDerivAt (fun x => g (f x)) (g'.comp f') x :=
((hg.comp_tendsto (hf.ContinuousAt.prod_map' hf.ContinuousAt)).trans_isBigO
hf.isBigO_sub).triangle <|
by simpa only [g'.map_sub, f'.coe_comp'] using (g'.is_O_comp _ _).trans_isLittleO hf
-#align has_strict_fderiv_at.comp HasStrictFderivAt.comp
+#align has_strict_fderiv_at.comp HasStrictFDerivAt.comp
protected theorem Differentiable.iterate {f : E β E} (hf : Differentiable π f) (n : β) :
Differentiable π (f^[n]) :=
@@ -219,52 +219,52 @@ protected theorem DifferentiableOn.iterate {f : E β E} (hf : DifferentiableOn
variable {x}
-protected theorem HasFderivAtFilter.iterate {f : E β E} {f' : E βL[π] E}
- (hf : HasFderivAtFilter f f' x L) (hL : Tendsto f L L) (hx : f x = x) (n : β) :
- HasFderivAtFilter (f^[n]) (f' ^ n) x L :=
+protected theorem HasFDerivAtFilter.iterate {f : E β E} {f' : E βL[π] E}
+ (hf : HasFDerivAtFilter f f' x L) (hL : Tendsto f L L) (hx : f x = x) (n : β) :
+ HasFDerivAtFilter (f^[n]) (f' ^ n) x L :=
by
induction' n with n ihn
- Β· exact hasFderivAtFilter_id x L
+ Β· exact hasFDerivAtFilter_id x L
Β· rw [Function.iterate_succ, pow_succ']
rw [β hx] at ihn
exact ihn.comp x hf hL
-#align has_fderiv_at_filter.iterate HasFderivAtFilter.iterate
+#align has_fderiv_at_filter.iterate HasFDerivAtFilter.iterate
-protected theorem HasFderivAt.iterate {f : E β E} {f' : E βL[π] E} (hf : HasFderivAt f f' x)
- (hx : f x = x) (n : β) : HasFderivAt (f^[n]) (f' ^ n) x :=
+protected theorem HasFDerivAt.iterate {f : E β E} {f' : E βL[π] E} (hf : HasFDerivAt f f' x)
+ (hx : f x = x) (n : β) : HasFDerivAt (f^[n]) (f' ^ n) x :=
by
refine' hf.iterate _ hx n
convert hf.continuous_at
exact hx.symm
-#align has_fderiv_at.iterate HasFderivAt.iterate
+#align has_fderiv_at.iterate HasFDerivAt.iterate
-protected theorem HasFderivWithinAt.iterate {f : E β E} {f' : E βL[π] E}
- (hf : HasFderivWithinAt f f' s x) (hx : f x = x) (hs : MapsTo f s s) (n : β) :
- HasFderivWithinAt (f^[n]) (f' ^ n) s x :=
+protected theorem HasFDerivWithinAt.iterate {f : E β E} {f' : E βL[π] E}
+ (hf : HasFDerivWithinAt f f' s x) (hx : f x = x) (hs : MapsTo f s s) (n : β) :
+ HasFDerivWithinAt (f^[n]) (f' ^ n) s x :=
by
refine' hf.iterate _ hx n
convert tendsto_inf.2 β¨hf.continuous_within_at, _β©
exacts[hx.symm, (tendsto_principal_principal.2 hs).mono_left inf_le_right]
-#align has_fderiv_within_at.iterate HasFderivWithinAt.iterate
+#align has_fderiv_within_at.iterate HasFDerivWithinAt.iterate
-protected theorem HasStrictFderivAt.iterate {f : E β E} {f' : E βL[π] E}
- (hf : HasStrictFderivAt f f' x) (hx : f x = x) (n : β) : HasStrictFderivAt (f^[n]) (f' ^ n) x :=
+protected theorem HasStrictFDerivAt.iterate {f : E β E} {f' : E βL[π] E}
+ (hf : HasStrictFDerivAt f f' x) (hx : f x = x) (n : β) : HasStrictFDerivAt (f^[n]) (f' ^ n) x :=
by
induction' n with n ihn
- Β· exact hasStrictFderivAt_id x
+ Β· exact hasStrictFDerivAt_id x
Β· rw [Function.iterate_succ, pow_succ']
rw [β hx] at ihn
exact ihn.comp x hf
-#align has_strict_fderiv_at.iterate HasStrictFderivAt.iterate
+#align has_strict_fderiv_at.iterate HasStrictFDerivAt.iterate
protected theorem DifferentiableAt.iterate {f : E β E} (hf : DifferentiableAt π f x) (hx : f x = x)
(n : β) : DifferentiableAt π (f^[n]) x :=
- (hf.HasFderivAt.iterate hx n).DifferentiableAt
+ (hf.HasFDerivAt.iterate hx n).DifferentiableAt
#align differentiable_at.iterate DifferentiableAt.iterate
protected theorem DifferentiableWithinAt.iterate {f : E β E} (hf : DifferentiableWithinAt π f s x)
(hx : f x = x) (hs : MapsTo f s s) (n : β) : DifferentiableWithinAt π (f^[n]) s x :=
- (hf.HasFderivWithinAt.iterate hx hs n).DifferentiableWithinAt
+ (hf.HasFDerivWithinAt.iterate hx hs n).DifferentiableWithinAt
#align differentiable_within_at.iterate DifferentiableWithinAt.iterate
end Composition
mathlib commit https://github.com/leanprover-community/mathlib/commit/e3fb84046afd187b710170887195d50bada934ee
@@ -71,7 +71,7 @@ example {g : F β G} {g' : F βL[π] G} (hg : HasFDerivAtFilter g g' (f x) (
calc
(fun x' : E => g' (f x' - f x) - g'.comp f' (x' - x))
_ =αΆ [L] fun x' => g' (f x' - f x - f' (x' - x)) := eventually_of_forall fun x' => by simp
- _ =O[L] fun x' => f x' - f x - f' (x' - x) := (g'.isBigO_comp _ _)
+ _ =O[L] fun x' => f x' - f x - f' (x' - x) := g'.isBigO_comp _ _
_ =o[L] fun x' => x' - x := hf.isLittleO
@[fun_prop]
We change the following field in the definition of an additive commutative monoid:
nsmul_succ : β (n : β) (x : G),
- AddMonoid.nsmul (n + 1) x = x + AddMonoid.nsmul n x
+ AddMonoid.nsmul (n + 1) x = AddMonoid.nsmul n x + x
where the latter is more natural
We adjust the definitions of ^
in monoids, groups, etc.
Originally there was a warning comment about why this natural order was preferred
use
x * npowRec n x
and notnpowRec n x * x
in the definition to make sure that definitional unfolding ofnpowRec
is blocked, to avoid deep recursion issues.
but it seems to no longer apply.
Remarks on the PR :
pow_succ
and pow_succ'
have switched their meanings.Ideal.IsPrime.mul_mem_pow
which is defined in [Mathlib/RingTheory/DedekindDomain/Ideal.lean]. Changing the order of operation forced me to add the symmetric lemma Ideal.IsPrime.mem_pow_mul
.@@ -215,7 +215,7 @@ protected theorem HasFDerivAtFilter.iterate {f : E β E} {f' : E βL[π] E}
HasFDerivAtFilter f^[n] (f' ^ n) x L := by
induction' n with n ihn
Β· exact hasFDerivAtFilter_id x L
- Β· rw [Function.iterate_succ, pow_succ']
+ Β· rw [Function.iterate_succ, pow_succ]
rw [β hx] at ihn
exact ihn.comp x hf hL
#align has_fderiv_at_filter.iterate HasFDerivAtFilter.iterate
@@ -245,7 +245,7 @@ protected theorem HasStrictFDerivAt.iterate {f : E β E} {f' : E βL[π] E}
HasStrictFDerivAt f^[n] (f' ^ n) x := by
induction' n with n ihn
Β· exact hasStrictFDerivAt_id x
- Β· rw [Function.iterate_succ, pow_succ']
+ Β· rw [Function.iterate_succ, pow_succ]
rw [β hx] at ihn
exact ihn.comp x hf
#align has_strict_fderiv_at.iterate HasStrictFDerivAt.iterate
Empty lines were removed by executing the following Python script twice
import os
import re
# Loop through each file in the repository
for dir_path, dirs, files in os.walk('.'):
for filename in files:
if filename.endswith('.lean'):
file_path = os.path.join(dir_path, filename)
# Open the file and read its contents
with open(file_path, 'r') as file:
content = file.read()
# Use a regular expression to replace sequences of "variable" lines separated by empty lines
# with sequences without empty lines
modified_content = re.sub(r'(variable.*\n)\n(variable(?! .* in))', r'\1\2', content)
# Write the modified content back to the file
with open(file_path, 'w') as file:
file.write(modified_content)
@@ -28,25 +28,15 @@ noncomputable section
section
variable {π : Type*} [NontriviallyNormedField π]
-
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace π E]
-
variable {F : Type*} [NormedAddCommGroup F] [NormedSpace π F]
-
variable {G : Type*} [NormedAddCommGroup G] [NormedSpace π G]
-
variable {G' : Type*} [NormedAddCommGroup G'] [NormedSpace π G']
-
variable {f fβ fβ g : E β F}
-
variable {f' fβ' fβ' g' : E βL[π] F}
-
variable (e : E βL[π] F)
-
variable {x : E}
-
variable {s t : Set E}
-
variable {L Lβ Lβ : Filter E}
section Composition
Basic setup for fun_prop
for Differentiable(At/On/Within) and HasFDeriv(At/Within/Strict).
Mainly consists of marking theorems with fun_prop
attribute but I had to formulate appropriate _pi
and _apply
theorems. Proofs of _apply
theorems can probably be golfed into neater form.
@@ -84,18 +84,21 @@ example {g : F β G} {g' : F βL[π] G} (hg : HasFDerivAtFilter g g' (f x) (
_ =O[L] fun x' => f x' - f x - f' (x' - x) := (g'.isBigO_comp _ _)
_ =o[L] fun x' => x' - x := hf.isLittleO
+@[fun_prop]
theorem HasFDerivWithinAt.comp {g : F β G} {g' : F βL[π] G} {t : Set F}
(hg : HasFDerivWithinAt g g' t (f x)) (hf : HasFDerivWithinAt f f' s x) (hst : MapsTo f s t) :
HasFDerivWithinAt (g β f) (g'.comp f') s x :=
HasFDerivAtFilter.comp x hg hf <| hf.continuousWithinAt.tendsto_nhdsWithin hst
#align has_fderiv_within_at.comp HasFDerivWithinAt.comp
+@[fun_prop]
theorem HasFDerivAt.comp_hasFDerivWithinAt {g : F β G} {g' : F βL[π] G}
(hg : HasFDerivAt g g' (f x)) (hf : HasFDerivWithinAt f f' s x) :
HasFDerivWithinAt (g β f) (g'.comp f') s x :=
hg.comp x hf hf.continuousWithinAt
#align has_fderiv_at.comp_has_fderiv_within_at HasFDerivAt.comp_hasFDerivWithinAt
+@[fun_prop]
theorem HasFDerivWithinAt.comp_of_mem {g : F β G} {g' : F βL[π] G} {t : Set F}
(hg : HasFDerivWithinAt g g' t (f x)) (hf : HasFDerivWithinAt f f' s x)
(hst : Tendsto f (π[s] x) (π[t] f x)) : HasFDerivWithinAt (g β f) (g'.comp f') s x :=
@@ -103,28 +106,33 @@ theorem HasFDerivWithinAt.comp_of_mem {g : F β G} {g' : F βL[π] G} {t : S
#align has_fderiv_within_at.comp_of_mem HasFDerivWithinAt.comp_of_mem
/-- The chain rule. -/
+@[fun_prop]
theorem HasFDerivAt.comp {g : F β G} {g' : F βL[π] G} (hg : HasFDerivAt g g' (f x))
(hf : HasFDerivAt f f' x) : HasFDerivAt (g β f) (g'.comp f') x :=
HasFDerivAtFilter.comp x hg hf hf.continuousAt
#align has_fderiv_at.comp HasFDerivAt.comp
+@[fun_prop]
theorem DifferentiableWithinAt.comp {g : F β G} {t : Set F}
(hg : DifferentiableWithinAt π g t (f x)) (hf : DifferentiableWithinAt π f s x)
(h : MapsTo f s t) : DifferentiableWithinAt π (g β f) s x :=
(hg.hasFDerivWithinAt.comp x hf.hasFDerivWithinAt h).differentiableWithinAt
#align differentiable_within_at.comp DifferentiableWithinAt.comp
+@[fun_prop]
theorem DifferentiableWithinAt.comp' {g : F β G} {t : Set F}
(hg : DifferentiableWithinAt π g t (f x)) (hf : DifferentiableWithinAt π f s x) :
DifferentiableWithinAt π (g β f) (s β© f β»ΒΉ' t) x :=
hg.comp x (hf.mono (inter_subset_left _ _)) (inter_subset_right _ _)
#align differentiable_within_at.comp' DifferentiableWithinAt.comp'
+@[fun_prop]
theorem DifferentiableAt.comp {g : F β G} (hg : DifferentiableAt π g (f x))
(hf : DifferentiableAt π f x) : DifferentiableAt π (g β f) x :=
(hg.hasFDerivAt.comp x hf.hasFDerivAt).differentiableAt
#align differentiable_at.comp DifferentiableAt.comp
+@[fun_prop]
theorem DifferentiableAt.comp_differentiableWithinAt {g : F β G} (hg : DifferentiableAt π g (f x))
(hf : DifferentiableWithinAt π f s x) : DifferentiableWithinAt π (g β f) s x :=
hg.differentiableWithinAt.comp x hf (mapsTo_univ _ _)
@@ -170,22 +178,26 @@ theorem fderiv.comp_fderivWithin {g : F β G} (hg : DifferentiableAt π g (f
(hg.hasFDerivAt.comp_hasFDerivWithinAt x hf.hasFDerivWithinAt).fderivWithin hxs
#align fderiv.comp_fderiv_within fderiv.comp_fderivWithin
+@[fun_prop]
theorem DifferentiableOn.comp {g : F β G} {t : Set F} (hg : DifferentiableOn π g t)
(hf : DifferentiableOn π f s) (st : MapsTo f s t) : DifferentiableOn π (g β f) s :=
fun x hx => DifferentiableWithinAt.comp x (hg (f x) (st hx)) (hf x hx) st
#align differentiable_on.comp DifferentiableOn.comp
+@[fun_prop]
theorem Differentiable.comp {g : F β G} (hg : Differentiable π g) (hf : Differentiable π f) :
Differentiable π (g β f) :=
fun x => DifferentiableAt.comp x (hg (f x)) (hf x)
#align differentiable.comp Differentiable.comp
+@[fun_prop]
theorem Differentiable.comp_differentiableOn {g : F β G} (hg : Differentiable π g)
(hf : DifferentiableOn π f s) : DifferentiableOn π (g β f) s :=
hg.differentiableOn.comp hf (mapsTo_univ _ _)
#align differentiable.comp_differentiable_on Differentiable.comp_differentiableOn
/-- The chain rule for derivatives in the sense of strict differentiability. -/
+@[fun_prop]
protected theorem HasStrictFDerivAt.comp {g : F β G} {g' : F βL[π] G}
(hg : HasStrictFDerivAt g g' (f x)) (hf : HasStrictFDerivAt f f' x) :
HasStrictFDerivAt (fun x => g (f x)) (g'.comp f') x :=
@@ -194,11 +206,13 @@ protected theorem HasStrictFDerivAt.comp {g : F β G} {g' : F βL[π] G}
simpa only [g'.map_sub, f'.coe_comp'] using (g'.isBigO_comp _ _).trans_isLittleO hf
#align has_strict_fderiv_at.comp HasStrictFDerivAt.comp
+@[fun_prop]
protected theorem Differentiable.iterate {f : E β E} (hf : Differentiable π f) (n : β) :
Differentiable π f^[n] :=
Nat.recOn n differentiable_id fun _ ihn => ihn.comp hf
#align differentiable.iterate Differentiable.iterate
+@[fun_prop]
protected theorem DifferentiableOn.iterate {f : E β E} (hf : DifferentiableOn π f s)
(hs : MapsTo f s s) (n : β) : DifferentiableOn π f^[n] s :=
Nat.recOn n differentiableOn_id fun _ ihn => ihn.comp hf hs
@@ -216,6 +230,7 @@ protected theorem HasFDerivAtFilter.iterate {f : E β E} {f' : E βL[π] E}
exact ihn.comp x hf hL
#align has_fderiv_at_filter.iterate HasFDerivAtFilter.iterate
+@[fun_prop]
protected theorem HasFDerivAt.iterate {f : E β E} {f' : E βL[π] E} (hf : HasFDerivAt f f' x)
(hx : f x = x) (n : β) : HasFDerivAt f^[n] (f' ^ n) x := by
refine' HasFDerivAtFilter.iterate hf _ hx n
@@ -224,6 +239,7 @@ protected theorem HasFDerivAt.iterate {f : E β E} {f' : E βL[π] E} (hf :
exact hx.symm
#align has_fderiv_at.iterate HasFDerivAt.iterate
+@[fun_prop]
protected theorem HasFDerivWithinAt.iterate {f : E β E} {f' : E βL[π] E}
(hf : HasFDerivWithinAt f f' s x) (hx : f x = x) (hs : MapsTo f s s) (n : β) :
HasFDerivWithinAt f^[n] (f' ^ n) s x := by
@@ -233,6 +249,7 @@ protected theorem HasFDerivWithinAt.iterate {f : E β E} {f' : E βL[π] E}
exacts [hx.symm, (tendsto_principal_principal.2 hs).mono_left inf_le_right]
#align has_fderiv_within_at.iterate HasFDerivWithinAt.iterate
+@[fun_prop]
protected theorem HasStrictFDerivAt.iterate {f : E β E} {f' : E βL[π] E}
(hf : HasStrictFDerivAt f f' x) (hx : f x = x) (n : β) :
HasStrictFDerivAt f^[n] (f' ^ n) x := by
@@ -243,11 +260,13 @@ protected theorem HasStrictFDerivAt.iterate {f : E β E} {f' : E βL[π] E}
exact ihn.comp x hf
#align has_strict_fderiv_at.iterate HasStrictFDerivAt.iterate
+@[fun_prop]
protected theorem DifferentiableAt.iterate {f : E β E} (hf : DifferentiableAt π f x) (hx : f x = x)
(n : β) : DifferentiableAt π f^[n] x :=
(hf.hasFDerivAt.iterate hx n).differentiableAt
#align differentiable_at.iterate DifferentiableAt.iterate
+@[fun_prop]
protected theorem DifferentiableWithinAt.iterate {f : E β E} (hf : DifferentiableWithinAt π f s x)
(hx : f x = x) (hs : MapsTo f s s) (n : β) : DifferentiableWithinAt π f^[n] s x :=
(hf.hasFDerivWithinAt.iterate hx hs n).differentiableWithinAt
open Classical
(#11199)
We remove all but one open Classical
s, instead preferring to use open scoped Classical
. The only real side-effect this led to is moving a couple declarations to use Exists.choose
instead of Classical.choose
.
The first few commits are explicitly labelled regex replaces for ease of review.
@@ -20,7 +20,8 @@ composition of functions (the chain rule).
open Filter Asymptotics ContinuousLinearMap Set Metric
-open Topology Classical NNReal Filter Asymptotics ENNReal
+open scoped Classical
+open Topology NNReal Filter Asymptotics ENNReal
noncomputable section
structure
(#8907)
This way we can easily change the definition so that it works for topological vector spaces without generalizing any of the theorems right away.
@@ -62,27 +62,26 @@ variable (x)
theorem HasFDerivAtFilter.comp {g : F β G} {g' : F βL[π] G} {L' : Filter F}
(hg : HasFDerivAtFilter g g' (f x) L') (hf : HasFDerivAtFilter f f' x L) (hL : Tendsto f L L') :
HasFDerivAtFilter (g β f) (g'.comp f') x L := by
- let eqβ := (g'.isBigO_comp _ _).trans_isLittleO hf
- let eqβ := (hg.comp_tendsto hL).trans_isBigO hf.isBigO_sub
- refine' eqβ.triangle (eqβ.congr_left fun x' => _)
+ let eqβ := (g'.isBigO_comp _ _).trans_isLittleO hf.isLittleO
+ let eqβ := (hg.isLittleO.comp_tendsto hL).trans_isBigO hf.isBigO_sub
+ refine .of_isLittleO <| eqβ.triangle <| eqβ.congr_left fun x' => ?_
simp
#align has_fderiv_at_filter.comp HasFDerivAtFilter.comp
/- A readable version of the previous theorem, a general form of the chain rule. -/
example {g : F β G} {g' : F βL[π] G} (hg : HasFDerivAtFilter g g' (f x) (L.map f))
(hf : HasFDerivAtFilter f f' x L) : HasFDerivAtFilter (g β f) (g'.comp f') x L := by
- unfold HasFDerivAtFilter at hg
have :=
calc
(fun x' => g (f x') - g (f x) - g' (f x' - f x)) =o[L] fun x' => f x' - f x :=
- hg.comp_tendsto le_rfl
+ hg.isLittleO.comp_tendsto le_rfl
_ =O[L] fun x' => x' - x := hf.isBigO_sub
- refine' this.triangle _
+ refine' .of_isLittleO <| this.triangle _
calc
(fun x' : E => g' (f x' - f x) - g'.comp f' (x' - x))
_ =αΆ [L] fun x' => g' (f x' - f x - f' (x' - x)) := eventually_of_forall fun x' => by simp
_ =O[L] fun x' => f x' - f x - f' (x' - x) := (g'.isBigO_comp _ _)
- _ =o[L] fun x' => x' - x := hf
+ _ =o[L] fun x' => x' - x := hf.isLittleO
theorem HasFDerivWithinAt.comp {g : F β G} {g' : F βL[π] G} {t : Set F}
(hg : HasFDerivWithinAt g g' t (f x)) (hf : HasFDerivWithinAt f f' s x) (hst : MapsTo f s t) :
@@ -69,8 +69,6 @@ theorem HasFDerivAtFilter.comp {g : F β G} {g' : F βL[π] G} {L' : Filter
#align has_fderiv_at_filter.comp HasFDerivAtFilter.comp
/- A readable version of the previous theorem, a general form of the chain rule. -/
-/- porting note: todo: restore the example
-Compile fails because `calc` fails to generate a `Trans` instance
example {g : F β G} {g' : F βL[π] G} (hg : HasFDerivAtFilter g g' (f x) (L.map f))
(hf : HasFDerivAtFilter f f' x L) : HasFDerivAtFilter (g β f) (g'.comp f') x L := by
unfold HasFDerivAtFilter at hg
@@ -81,12 +79,10 @@ example {g : F β G} {g' : F βL[π] G} (hg : HasFDerivAtFilter g g' (f x) (
_ =O[L] fun x' => x' - x := hf.isBigO_sub
refine' this.triangle _
calc
- (fun x' : E => g' (f x' - f x) - g'.comp f' (x' - x)) =αΆ [L] fun x' =>
- g' (f x' - f x - f' (x' - x)) :=
- eventually_of_forall fun x' => by simp
+ (fun x' : E => g' (f x' - f x) - g'.comp f' (x' - x))
+ _ =αΆ [L] fun x' => g' (f x' - f x - f' (x' - x)) := eventually_of_forall fun x' => by simp
_ =O[L] fun x' => f x' - f x - f' (x' - x) := (g'.isBigO_comp _ _)
_ =o[L] fun x' => x' - x := hf
--/
theorem HasFDerivWithinAt.comp {g : F β G} {g' : F βL[π] G} {t : Set F}
(hg : HasFDerivWithinAt g g' t (f x)) (hf : HasFDerivWithinAt f f' s x) (hst : MapsTo f s t) :
@@ -147,8 +143,7 @@ theorem fderivWithin_fderivWithin {g : F β G} {f : E β F} {x : E} {y : F} {s
(hxs : UniqueDiffWithinAt π s x) (hy : f x = y) (v : E) :
fderivWithin π g t y (fderivWithin π f s x v) = fderivWithin π (g β f) s x v := by
subst y
- rw [fderivWithin.comp x hg hf h hxs]
- rfl
+ rw [fderivWithin.comp x hg hf h hxs, coe_comp', Function.comp_apply]
#align fderiv_within_fderiv_within fderivWithin_fderivWithin
/-- Ternary version of `fderivWithin.comp`, with equality assumptions of basepoints added, in
@@ -225,9 +220,7 @@ protected theorem HasFDerivAt.iterate {f : E β E} {f' : E βL[π] E} (hf :
(hx : f x = x) (n : β) : HasFDerivAt f^[n] (f' ^ n) x := by
refine' HasFDerivAtFilter.iterate hf _ hx n
-- Porting note: was `convert hf.continuousAt`
- have := hf.continuousAt
- unfold ContinuousAt at this
- convert this
+ convert hf.continuousAt.tendsto
exact hx.symm
#align has_fderiv_at.iterate HasFDerivAt.iterate
Type _
and Sort _
(#6499)
We remove all possible occurences of Type _
and Sort _
in favor of Type*
and Sort*
.
This has nice performance benefits.
@@ -26,15 +26,15 @@ noncomputable section
section
-variable {π : Type _} [NontriviallyNormedField π]
+variable {π : Type*} [NontriviallyNormedField π]
-variable {E : Type _} [NormedAddCommGroup E] [NormedSpace π E]
+variable {E : Type*} [NormedAddCommGroup E] [NormedSpace π E]
-variable {F : Type _} [NormedAddCommGroup F] [NormedSpace π F]
+variable {F : Type*} [NormedAddCommGroup F] [NormedSpace π F]
-variable {G : Type _} [NormedAddCommGroup G] [NormedSpace π G]
+variable {G : Type*} [NormedAddCommGroup G] [NormedSpace π G]
-variable {G' : Type _} [NormedAddCommGroup G'] [NormedSpace π G']
+variable {G' : Type*} [NormedAddCommGroup G'] [NormedSpace π G']
variable {f fβ fβ g : E β F}
@@ -2,14 +2,11 @@
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, SΓ©bastien GouΓ«zel, Yury Kudryashov
-
-! This file was ported from Lean 3 source module analysis.calculus.fderiv.comp
-! leanprover-community/mathlib commit e3fb84046afd187b710170887195d50bada934ee
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathlib.Analysis.Calculus.FDeriv.Basic
+#align_import analysis.calculus.fderiv.comp from "leanprover-community/mathlib"@"e3fb84046afd187b710170887195d50bada934ee"
+
/-!
# The derivative of a composition (chain rule)
@@ -203,12 +203,12 @@ protected theorem HasStrictFDerivAt.comp {g : F β G} {g' : F βL[π] G}
#align has_strict_fderiv_at.comp HasStrictFDerivAt.comp
protected theorem Differentiable.iterate {f : E β E} (hf : Differentiable π f) (n : β) :
- Differentiable π (f^[n]) :=
+ Differentiable π f^[n] :=
Nat.recOn n differentiable_id fun _ ihn => ihn.comp hf
#align differentiable.iterate Differentiable.iterate
protected theorem DifferentiableOn.iterate {f : E β E} (hf : DifferentiableOn π f s)
- (hs : MapsTo f s s) (n : β) : DifferentiableOn π (f^[n]) s :=
+ (hs : MapsTo f s s) (n : β) : DifferentiableOn π f^[n] s :=
Nat.recOn n differentiableOn_id fun _ ihn => ihn.comp hf hs
#align differentiable_on.iterate DifferentiableOn.iterate
@@ -216,7 +216,7 @@ variable {x}
protected theorem HasFDerivAtFilter.iterate {f : E β E} {f' : E βL[π] E}
(hf : HasFDerivAtFilter f f' x L) (hL : Tendsto f L L) (hx : f x = x) (n : β) :
- HasFDerivAtFilter (f^[n]) (f' ^ n) x L := by
+ HasFDerivAtFilter f^[n] (f' ^ n) x L := by
induction' n with n ihn
Β· exact hasFDerivAtFilter_id x L
Β· rw [Function.iterate_succ, pow_succ']
@@ -225,7 +225,7 @@ protected theorem HasFDerivAtFilter.iterate {f : E β E} {f' : E βL[π] E}
#align has_fderiv_at_filter.iterate HasFDerivAtFilter.iterate
protected theorem HasFDerivAt.iterate {f : E β E} {f' : E βL[π] E} (hf : HasFDerivAt f f' x)
- (hx : f x = x) (n : β) : HasFDerivAt (f^[n]) (f' ^ n) x := by
+ (hx : f x = x) (n : β) : HasFDerivAt f^[n] (f' ^ n) x := by
refine' HasFDerivAtFilter.iterate hf _ hx n
-- Porting note: was `convert hf.continuousAt`
have := hf.continuousAt
@@ -236,7 +236,7 @@ protected theorem HasFDerivAt.iterate {f : E β E} {f' : E βL[π] E} (hf :
protected theorem HasFDerivWithinAt.iterate {f : E β E} {f' : E βL[π] E}
(hf : HasFDerivWithinAt f f' s x) (hx : f x = x) (hs : MapsTo f s s) (n : β) :
- HasFDerivWithinAt (f^[n]) (f' ^ n) s x := by
+ HasFDerivWithinAt f^[n] (f' ^ n) s x := by
refine' HasFDerivAtFilter.iterate hf _ hx n
rw [_root_.nhdsWithin] -- Porting note: Added `rw` to get rid of an error
convert tendsto_inf.2 β¨hf.continuousWithinAt, _β©
@@ -245,7 +245,7 @@ protected theorem HasFDerivWithinAt.iterate {f : E β E} {f' : E βL[π] E}
protected theorem HasStrictFDerivAt.iterate {f : E β E} {f' : E βL[π] E}
(hf : HasStrictFDerivAt f f' x) (hx : f x = x) (n : β) :
- HasStrictFDerivAt (f^[n]) (f' ^ n) x := by
+ HasStrictFDerivAt f^[n] (f' ^ n) x := by
induction' n with n ihn
Β· exact hasStrictFDerivAt_id x
Β· rw [Function.iterate_succ, pow_succ']
@@ -254,12 +254,12 @@ protected theorem HasStrictFDerivAt.iterate {f : E β E} {f' : E βL[π] E}
#align has_strict_fderiv_at.iterate HasStrictFDerivAt.iterate
protected theorem DifferentiableAt.iterate {f : E β E} (hf : DifferentiableAt π f x) (hx : f x = x)
- (n : β) : DifferentiableAt π (f^[n]) x :=
+ (n : β) : DifferentiableAt π f^[n] x :=
(hf.hasFDerivAt.iterate hx n).differentiableAt
#align differentiable_at.iterate DifferentiableAt.iterate
protected theorem DifferentiableWithinAt.iterate {f : E β E} (hf : DifferentiableWithinAt π f s x)
- (hx : f x = x) (hs : MapsTo f s s) (n : β) : DifferentiableWithinAt π (f^[n]) s x :=
+ (hx : f x = x) (hs : MapsTo f s s) (n : β) : DifferentiableWithinAt π f^[n] s x :=
(hf.hasFDerivWithinAt.iterate hx hs n).differentiableWithinAt
#align differentiable_within_at.iterate DifferentiableWithinAt.iterate
The unported dependencies are
algebra.order.module
init.core
algebra.order.monoid.cancel.defs
algebra.abs
algebra.group_power.lemmas
init.data.list.basic
algebra.order.monoid.cancel.basic
init.data.list.default
topology.subset_properties
init.logic
The following 1 dependencies have changed in mathlib3 since they were ported, which may complicate porting this file