order.monotone.union
⟷
Mathlib.Order.Monotone.Union
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/ce64cd319bb6b3e82f31c2d38e79080d377be451
@@ -3,7 +3,7 @@ Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel
-/
-import Mathbin.Order.Bounds.Basic
+import Order.Bounds.Basic
#align_import order.monotone.union from "leanprover-community/mathlib"@"c3291da49cfa65f0d43b094750541c0731edc932"
mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345
@@ -2,14 +2,11 @@
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel
-
-! This file was ported from Lean 3 source module order.monotone.union
-! leanprover-community/mathlib commit c3291da49cfa65f0d43b094750541c0731edc932
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathbin.Order.Bounds.Basic
+#align_import order.monotone.union from "leanprover-community/mathlib"@"c3291da49cfa65f0d43b094750541c0731edc932"
+
/-!
# Monotonicity on intervals
mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423
@@ -27,6 +27,7 @@ open Set
variable {α β : Type _} [LinearOrder α] [Preorder β] {a : α} {f : α → β}
+#print StrictMonoOn.union /-
/-- If `f` is strictly monotone both on `s` and `t`, with `s` to the left of `t` and the center
point belonging to both `s` and `t`, then `f` is strictly monotone on `s ∪ t` -/
protected theorem StrictMonoOn.union {s t : Set α} {c : α} (h₁ : StrictMonoOn f s)
@@ -54,7 +55,9 @@ protected theorem StrictMonoOn.union {s t : Set α} {c : α} (h₁ : StrictMonoO
have yt : y ∈ t := B _ hy (hcx.trans hxy.le)
exact h₂ xt yt hxy
#align strict_mono_on.union StrictMonoOn.union
+-/
+#print StrictMonoOn.Iic_union_Ici /-
/-- If `f` is strictly monotone both on `(-∞, a]` and `[a, ∞)`, then it is strictly monotone on the
whole line. -/
protected theorem StrictMonoOn.Iic_union_Ici (h₁ : StrictMonoOn f (Iic a))
@@ -63,21 +66,27 @@ protected theorem StrictMonoOn.Iic_union_Ici (h₁ : StrictMonoOn f (Iic a))
rw [← strictMonoOn_univ, ← @Iic_union_Ici _ _ a]
exact StrictMonoOn.union h₁ h₂ isGreatest_Iic isLeast_Ici
#align strict_mono_on.Iic_union_Ici StrictMonoOn.Iic_union_Ici
+-/
+#print StrictAntiOn.union /-
/-- If `f` is strictly antitone both on `s` and `t`, with `s` to the left of `t` and the center
point belonging to both `s` and `t`, then `f` is strictly antitone on `s ∪ t` -/
protected theorem StrictAntiOn.union {s t : Set α} {c : α} (h₁ : StrictAntiOn f s)
(h₂ : StrictAntiOn f t) (hs : IsGreatest s c) (ht : IsLeast t c) : StrictAntiOn f (s ∪ t) :=
(h₁.dual_right.union h₂.dual_right hs ht).dual_right
#align strict_anti_on.union StrictAntiOn.union
+-/
+#print StrictAntiOn.Iic_union_Ici /-
/-- If `f` is strictly antitone both on `(-∞, a]` and `[a, ∞)`, then it is strictly antitone on the
whole line. -/
protected theorem StrictAntiOn.Iic_union_Ici (h₁ : StrictAntiOn f (Iic a))
(h₂ : StrictAntiOn f (Ici a)) : StrictAnti f :=
(h₁.dual_right.Iic_union_Ici h₂.dual_right).dual_right
#align strict_anti_on.Iic_union_Ici StrictAntiOn.Iic_union_Ici
+-/
+#print MonotoneOn.union_right /-
/-- If `f` is monotone both on `s` and `t`, with `s` to the left of `t` and the center
point belonging to both `s` and `t`, then `f` is monotone on `s ∪ t` -/
protected theorem MonotoneOn.union_right {s t : Set α} {c : α} (h₁ : MonotoneOn f s)
@@ -105,24 +114,31 @@ protected theorem MonotoneOn.union_right {s t : Set α} {c : α} (h₁ : Monoton
have yt : y ∈ t := B _ hy (hcx.trans hxy)
exact h₂ xt yt hxy
#align monotone_on.union_right MonotoneOn.union_right
+-/
+#print MonotoneOn.Iic_union_Ici /-
/-- If `f` is monotone both on `(-∞, a]` and `[a, ∞)`, then it is monotone on the whole line. -/
protected theorem MonotoneOn.Iic_union_Ici (h₁ : MonotoneOn f (Iic a)) (h₂ : MonotoneOn f (Ici a)) :
Monotone f := by
rw [← monotoneOn_univ, ← @Iic_union_Ici _ _ a]
exact MonotoneOn.union_right h₁ h₂ isGreatest_Iic isLeast_Ici
#align monotone_on.Iic_union_Ici MonotoneOn.Iic_union_Ici
+-/
+#print AntitoneOn.union_right /-
/-- If `f` is antitone both on `s` and `t`, with `s` to the left of `t` and the center
point belonging to both `s` and `t`, then `f` is antitone on `s ∪ t` -/
protected theorem AntitoneOn.union_right {s t : Set α} {c : α} (h₁ : AntitoneOn f s)
(h₂ : AntitoneOn f t) (hs : IsGreatest s c) (ht : IsLeast t c) : AntitoneOn f (s ∪ t) :=
(h₁.dual_right.union_right h₂.dual_right hs ht).dual_right
#align antitone_on.union_right AntitoneOn.union_right
+-/
+#print AntitoneOn.Iic_union_Ici /-
/-- If `f` is antitone both on `(-∞, a]` and `[a, ∞)`, then it is antitone on the whole line. -/
protected theorem AntitoneOn.Iic_union_Ici (h₁ : AntitoneOn f (Iic a)) (h₂ : AntitoneOn f (Ici a)) :
Antitone f :=
(h₁.dual_right.Iic_union_Ici h₂.dual_right).dual_right
#align antitone_on.Iic_union_Ici AntitoneOn.Iic_union_Ici
+-/
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -27,12 +27,6 @@ open Set
variable {α β : Type _} [LinearOrder α] [Preorder β] {a : α} {f : α → β}
-/- warning: strict_mono_on.union -> StrictMonoOn.union is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : LinearOrder.{u1} α] [_inst_2 : Preorder.{u2} β] {f : α -> β} {s : Set.{u1} α} {t : Set.{u1} α} {c : α}, (StrictMonoOn.{u1, u2} α β (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (LinearOrder.toLattice.{u1} α _inst_1)))) _inst_2 f s) -> (StrictMonoOn.{u1, u2} α β (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (LinearOrder.toLattice.{u1} α _inst_1)))) _inst_2 f t) -> (IsGreatest.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (LinearOrder.toLattice.{u1} α _inst_1)))) s c) -> (IsLeast.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (LinearOrder.toLattice.{u1} α _inst_1)))) t c) -> (StrictMonoOn.{u1, u2} α β (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (LinearOrder.toLattice.{u1} α _inst_1)))) _inst_2 f (Union.union.{u1} (Set.{u1} α) (Set.hasUnion.{u1} α) s t))
-but is expected to have type
- forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : LinearOrder.{u2} α] [_inst_2 : Preorder.{u1} β] {f : α -> β} {s : Set.{u2} α} {t : Set.{u2} α} {c : α}, (StrictMonoOn.{u2, u1} α β (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (DistribLattice.toLattice.{u2} α (instDistribLattice.{u2} α _inst_1))))) _inst_2 f s) -> (StrictMonoOn.{u2, u1} α β (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (DistribLattice.toLattice.{u2} α (instDistribLattice.{u2} α _inst_1))))) _inst_2 f t) -> (IsGreatest.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (DistribLattice.toLattice.{u2} α (instDistribLattice.{u2} α _inst_1))))) s c) -> (IsLeast.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (DistribLattice.toLattice.{u2} α (instDistribLattice.{u2} α _inst_1))))) t c) -> (StrictMonoOn.{u2, u1} α β (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (DistribLattice.toLattice.{u2} α (instDistribLattice.{u2} α _inst_1))))) _inst_2 f (Union.union.{u2} (Set.{u2} α) (Set.instUnionSet.{u2} α) s t))
-Case conversion may be inaccurate. Consider using '#align strict_mono_on.union StrictMonoOn.unionₓ'. -/
/-- If `f` is strictly monotone both on `s` and `t`, with `s` to the left of `t` and the center
point belonging to both `s` and `t`, then `f` is strictly monotone on `s ∪ t` -/
protected theorem StrictMonoOn.union {s t : Set α} {c : α} (h₁ : StrictMonoOn f s)
@@ -61,12 +55,6 @@ protected theorem StrictMonoOn.union {s t : Set α} {c : α} (h₁ : StrictMonoO
exact h₂ xt yt hxy
#align strict_mono_on.union StrictMonoOn.union
-/- warning: strict_mono_on.Iic_union_Ici -> StrictMonoOn.Iic_union_Ici is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : LinearOrder.{u1} α] [_inst_2 : Preorder.{u2} β] {a : α} {f : α -> β}, (StrictMonoOn.{u1, u2} α β (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (LinearOrder.toLattice.{u1} α _inst_1)))) _inst_2 f (Set.Iic.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (LinearOrder.toLattice.{u1} α _inst_1)))) a)) -> (StrictMonoOn.{u1, u2} α β (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (LinearOrder.toLattice.{u1} α _inst_1)))) _inst_2 f (Set.Ici.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (LinearOrder.toLattice.{u1} α _inst_1)))) a)) -> (StrictMono.{u1, u2} α β (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (LinearOrder.toLattice.{u1} α _inst_1)))) _inst_2 f)
-but is expected to have type
- forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : LinearOrder.{u2} α] [_inst_2 : Preorder.{u1} β] {a : α} {f : α -> β}, (StrictMonoOn.{u2, u1} α β (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (DistribLattice.toLattice.{u2} α (instDistribLattice.{u2} α _inst_1))))) _inst_2 f (Set.Iic.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (DistribLattice.toLattice.{u2} α (instDistribLattice.{u2} α _inst_1))))) a)) -> (StrictMonoOn.{u2, u1} α β (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (DistribLattice.toLattice.{u2} α (instDistribLattice.{u2} α _inst_1))))) _inst_2 f (Set.Ici.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (DistribLattice.toLattice.{u2} α (instDistribLattice.{u2} α _inst_1))))) a)) -> (StrictMono.{u2, u1} α β (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (DistribLattice.toLattice.{u2} α (instDistribLattice.{u2} α _inst_1))))) _inst_2 f)
-Case conversion may be inaccurate. Consider using '#align strict_mono_on.Iic_union_Ici StrictMonoOn.Iic_union_Iciₓ'. -/
/-- If `f` is strictly monotone both on `(-∞, a]` and `[a, ∞)`, then it is strictly monotone on the
whole line. -/
protected theorem StrictMonoOn.Iic_union_Ici (h₁ : StrictMonoOn f (Iic a))
@@ -76,12 +64,6 @@ protected theorem StrictMonoOn.Iic_union_Ici (h₁ : StrictMonoOn f (Iic a))
exact StrictMonoOn.union h₁ h₂ isGreatest_Iic isLeast_Ici
#align strict_mono_on.Iic_union_Ici StrictMonoOn.Iic_union_Ici
-/- warning: strict_anti_on.union -> StrictAntiOn.union is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : LinearOrder.{u1} α] [_inst_2 : Preorder.{u2} β] {f : α -> β} {s : Set.{u1} α} {t : Set.{u1} α} {c : α}, (StrictAntiOn.{u1, u2} α β (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (LinearOrder.toLattice.{u1} α _inst_1)))) _inst_2 f s) -> (StrictAntiOn.{u1, u2} α β (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (LinearOrder.toLattice.{u1} α _inst_1)))) _inst_2 f t) -> (IsGreatest.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (LinearOrder.toLattice.{u1} α _inst_1)))) s c) -> (IsLeast.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (LinearOrder.toLattice.{u1} α _inst_1)))) t c) -> (StrictAntiOn.{u1, u2} α β (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (LinearOrder.toLattice.{u1} α _inst_1)))) _inst_2 f (Union.union.{u1} (Set.{u1} α) (Set.hasUnion.{u1} α) s t))
-but is expected to have type
- forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : LinearOrder.{u2} α] [_inst_2 : Preorder.{u1} β] {f : α -> β} {s : Set.{u2} α} {t : Set.{u2} α} {c : α}, (StrictAntiOn.{u2, u1} α β (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (DistribLattice.toLattice.{u2} α (instDistribLattice.{u2} α _inst_1))))) _inst_2 f s) -> (StrictAntiOn.{u2, u1} α β (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (DistribLattice.toLattice.{u2} α (instDistribLattice.{u2} α _inst_1))))) _inst_2 f t) -> (IsGreatest.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (DistribLattice.toLattice.{u2} α (instDistribLattice.{u2} α _inst_1))))) s c) -> (IsLeast.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (DistribLattice.toLattice.{u2} α (instDistribLattice.{u2} α _inst_1))))) t c) -> (StrictAntiOn.{u2, u1} α β (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (DistribLattice.toLattice.{u2} α (instDistribLattice.{u2} α _inst_1))))) _inst_2 f (Union.union.{u2} (Set.{u2} α) (Set.instUnionSet.{u2} α) s t))
-Case conversion may be inaccurate. Consider using '#align strict_anti_on.union StrictAntiOn.unionₓ'. -/
/-- If `f` is strictly antitone both on `s` and `t`, with `s` to the left of `t` and the center
point belonging to both `s` and `t`, then `f` is strictly antitone on `s ∪ t` -/
protected theorem StrictAntiOn.union {s t : Set α} {c : α} (h₁ : StrictAntiOn f s)
@@ -89,12 +71,6 @@ protected theorem StrictAntiOn.union {s t : Set α} {c : α} (h₁ : StrictAntiO
(h₁.dual_right.union h₂.dual_right hs ht).dual_right
#align strict_anti_on.union StrictAntiOn.union
-/- warning: strict_anti_on.Iic_union_Ici -> StrictAntiOn.Iic_union_Ici is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : LinearOrder.{u1} α] [_inst_2 : Preorder.{u2} β] {a : α} {f : α -> β}, (StrictAntiOn.{u1, u2} α β (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (LinearOrder.toLattice.{u1} α _inst_1)))) _inst_2 f (Set.Iic.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (LinearOrder.toLattice.{u1} α _inst_1)))) a)) -> (StrictAntiOn.{u1, u2} α β (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (LinearOrder.toLattice.{u1} α _inst_1)))) _inst_2 f (Set.Ici.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (LinearOrder.toLattice.{u1} α _inst_1)))) a)) -> (StrictAnti.{u1, u2} α β (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (LinearOrder.toLattice.{u1} α _inst_1)))) _inst_2 f)
-but is expected to have type
- forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : LinearOrder.{u2} α] [_inst_2 : Preorder.{u1} β] {a : α} {f : α -> β}, (StrictAntiOn.{u2, u1} α β (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (DistribLattice.toLattice.{u2} α (instDistribLattice.{u2} α _inst_1))))) _inst_2 f (Set.Iic.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (DistribLattice.toLattice.{u2} α (instDistribLattice.{u2} α _inst_1))))) a)) -> (StrictAntiOn.{u2, u1} α β (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (DistribLattice.toLattice.{u2} α (instDistribLattice.{u2} α _inst_1))))) _inst_2 f (Set.Ici.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (DistribLattice.toLattice.{u2} α (instDistribLattice.{u2} α _inst_1))))) a)) -> (StrictAnti.{u2, u1} α β (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (DistribLattice.toLattice.{u2} α (instDistribLattice.{u2} α _inst_1))))) _inst_2 f)
-Case conversion may be inaccurate. Consider using '#align strict_anti_on.Iic_union_Ici StrictAntiOn.Iic_union_Iciₓ'. -/
/-- If `f` is strictly antitone both on `(-∞, a]` and `[a, ∞)`, then it is strictly antitone on the
whole line. -/
protected theorem StrictAntiOn.Iic_union_Ici (h₁ : StrictAntiOn f (Iic a))
@@ -102,12 +78,6 @@ protected theorem StrictAntiOn.Iic_union_Ici (h₁ : StrictAntiOn f (Iic a))
(h₁.dual_right.Iic_union_Ici h₂.dual_right).dual_right
#align strict_anti_on.Iic_union_Ici StrictAntiOn.Iic_union_Ici
-/- warning: monotone_on.union_right -> MonotoneOn.union_right is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : LinearOrder.{u1} α] [_inst_2 : Preorder.{u2} β] {f : α -> β} {s : Set.{u1} α} {t : Set.{u1} α} {c : α}, (MonotoneOn.{u1, u2} α β (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (LinearOrder.toLattice.{u1} α _inst_1)))) _inst_2 f s) -> (MonotoneOn.{u1, u2} α β (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (LinearOrder.toLattice.{u1} α _inst_1)))) _inst_2 f t) -> (IsGreatest.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (LinearOrder.toLattice.{u1} α _inst_1)))) s c) -> (IsLeast.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (LinearOrder.toLattice.{u1} α _inst_1)))) t c) -> (MonotoneOn.{u1, u2} α β (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (LinearOrder.toLattice.{u1} α _inst_1)))) _inst_2 f (Union.union.{u1} (Set.{u1} α) (Set.hasUnion.{u1} α) s t))
-but is expected to have type
- forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : LinearOrder.{u2} α] [_inst_2 : Preorder.{u1} β] {f : α -> β} {s : Set.{u2} α} {t : Set.{u2} α} {c : α}, (MonotoneOn.{u2, u1} α β (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (DistribLattice.toLattice.{u2} α (instDistribLattice.{u2} α _inst_1))))) _inst_2 f s) -> (MonotoneOn.{u2, u1} α β (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (DistribLattice.toLattice.{u2} α (instDistribLattice.{u2} α _inst_1))))) _inst_2 f t) -> (IsGreatest.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (DistribLattice.toLattice.{u2} α (instDistribLattice.{u2} α _inst_1))))) s c) -> (IsLeast.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (DistribLattice.toLattice.{u2} α (instDistribLattice.{u2} α _inst_1))))) t c) -> (MonotoneOn.{u2, u1} α β (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (DistribLattice.toLattice.{u2} α (instDistribLattice.{u2} α _inst_1))))) _inst_2 f (Union.union.{u2} (Set.{u2} α) (Set.instUnionSet.{u2} α) s t))
-Case conversion may be inaccurate. Consider using '#align monotone_on.union_right MonotoneOn.union_rightₓ'. -/
/-- If `f` is monotone both on `s` and `t`, with `s` to the left of `t` and the center
point belonging to both `s` and `t`, then `f` is monotone on `s ∪ t` -/
protected theorem MonotoneOn.union_right {s t : Set α} {c : α} (h₁ : MonotoneOn f s)
@@ -136,12 +106,6 @@ protected theorem MonotoneOn.union_right {s t : Set α} {c : α} (h₁ : Monoton
exact h₂ xt yt hxy
#align monotone_on.union_right MonotoneOn.union_right
-/- warning: monotone_on.Iic_union_Ici -> MonotoneOn.Iic_union_Ici is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : LinearOrder.{u1} α] [_inst_2 : Preorder.{u2} β] {a : α} {f : α -> β}, (MonotoneOn.{u1, u2} α β (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (LinearOrder.toLattice.{u1} α _inst_1)))) _inst_2 f (Set.Iic.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (LinearOrder.toLattice.{u1} α _inst_1)))) a)) -> (MonotoneOn.{u1, u2} α β (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (LinearOrder.toLattice.{u1} α _inst_1)))) _inst_2 f (Set.Ici.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (LinearOrder.toLattice.{u1} α _inst_1)))) a)) -> (Monotone.{u1, u2} α β (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (LinearOrder.toLattice.{u1} α _inst_1)))) _inst_2 f)
-but is expected to have type
- forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : LinearOrder.{u2} α] [_inst_2 : Preorder.{u1} β] {a : α} {f : α -> β}, (MonotoneOn.{u2, u1} α β (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (DistribLattice.toLattice.{u2} α (instDistribLattice.{u2} α _inst_1))))) _inst_2 f (Set.Iic.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (DistribLattice.toLattice.{u2} α (instDistribLattice.{u2} α _inst_1))))) a)) -> (MonotoneOn.{u2, u1} α β (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (DistribLattice.toLattice.{u2} α (instDistribLattice.{u2} α _inst_1))))) _inst_2 f (Set.Ici.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (DistribLattice.toLattice.{u2} α (instDistribLattice.{u2} α _inst_1))))) a)) -> (Monotone.{u2, u1} α β (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (DistribLattice.toLattice.{u2} α (instDistribLattice.{u2} α _inst_1))))) _inst_2 f)
-Case conversion may be inaccurate. Consider using '#align monotone_on.Iic_union_Ici MonotoneOn.Iic_union_Iciₓ'. -/
/-- If `f` is monotone both on `(-∞, a]` and `[a, ∞)`, then it is monotone on the whole line. -/
protected theorem MonotoneOn.Iic_union_Ici (h₁ : MonotoneOn f (Iic a)) (h₂ : MonotoneOn f (Ici a)) :
Monotone f := by
@@ -149,12 +113,6 @@ protected theorem MonotoneOn.Iic_union_Ici (h₁ : MonotoneOn f (Iic a)) (h₂ :
exact MonotoneOn.union_right h₁ h₂ isGreatest_Iic isLeast_Ici
#align monotone_on.Iic_union_Ici MonotoneOn.Iic_union_Ici
-/- warning: antitone_on.union_right -> AntitoneOn.union_right is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : LinearOrder.{u1} α] [_inst_2 : Preorder.{u2} β] {f : α -> β} {s : Set.{u1} α} {t : Set.{u1} α} {c : α}, (AntitoneOn.{u1, u2} α β (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (LinearOrder.toLattice.{u1} α _inst_1)))) _inst_2 f s) -> (AntitoneOn.{u1, u2} α β (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (LinearOrder.toLattice.{u1} α _inst_1)))) _inst_2 f t) -> (IsGreatest.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (LinearOrder.toLattice.{u1} α _inst_1)))) s c) -> (IsLeast.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (LinearOrder.toLattice.{u1} α _inst_1)))) t c) -> (AntitoneOn.{u1, u2} α β (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (LinearOrder.toLattice.{u1} α _inst_1)))) _inst_2 f (Union.union.{u1} (Set.{u1} α) (Set.hasUnion.{u1} α) s t))
-but is expected to have type
- forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : LinearOrder.{u2} α] [_inst_2 : Preorder.{u1} β] {f : α -> β} {s : Set.{u2} α} {t : Set.{u2} α} {c : α}, (AntitoneOn.{u2, u1} α β (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (DistribLattice.toLattice.{u2} α (instDistribLattice.{u2} α _inst_1))))) _inst_2 f s) -> (AntitoneOn.{u2, u1} α β (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (DistribLattice.toLattice.{u2} α (instDistribLattice.{u2} α _inst_1))))) _inst_2 f t) -> (IsGreatest.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (DistribLattice.toLattice.{u2} α (instDistribLattice.{u2} α _inst_1))))) s c) -> (IsLeast.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (DistribLattice.toLattice.{u2} α (instDistribLattice.{u2} α _inst_1))))) t c) -> (AntitoneOn.{u2, u1} α β (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (DistribLattice.toLattice.{u2} α (instDistribLattice.{u2} α _inst_1))))) _inst_2 f (Union.union.{u2} (Set.{u2} α) (Set.instUnionSet.{u2} α) s t))
-Case conversion may be inaccurate. Consider using '#align antitone_on.union_right AntitoneOn.union_rightₓ'. -/
/-- If `f` is antitone both on `s` and `t`, with `s` to the left of `t` and the center
point belonging to both `s` and `t`, then `f` is antitone on `s ∪ t` -/
protected theorem AntitoneOn.union_right {s t : Set α} {c : α} (h₁ : AntitoneOn f s)
@@ -162,12 +120,6 @@ protected theorem AntitoneOn.union_right {s t : Set α} {c : α} (h₁ : Antiton
(h₁.dual_right.union_right h₂.dual_right hs ht).dual_right
#align antitone_on.union_right AntitoneOn.union_right
-/- warning: antitone_on.Iic_union_Ici -> AntitoneOn.Iic_union_Ici is a dubious translation:
-lean 3 declaration is
- forall {α : Type.{u1}} {β : Type.{u2}} [_inst_1 : LinearOrder.{u1} α] [_inst_2 : Preorder.{u2} β] {a : α} {f : α -> β}, (AntitoneOn.{u1, u2} α β (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (LinearOrder.toLattice.{u1} α _inst_1)))) _inst_2 f (Set.Iic.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (LinearOrder.toLattice.{u1} α _inst_1)))) a)) -> (AntitoneOn.{u1, u2} α β (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (LinearOrder.toLattice.{u1} α _inst_1)))) _inst_2 f (Set.Ici.{u1} α (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (LinearOrder.toLattice.{u1} α _inst_1)))) a)) -> (Antitone.{u1, u2} α β (PartialOrder.toPreorder.{u1} α (SemilatticeInf.toPartialOrder.{u1} α (Lattice.toSemilatticeInf.{u1} α (LinearOrder.toLattice.{u1} α _inst_1)))) _inst_2 f)
-but is expected to have type
- forall {α : Type.{u2}} {β : Type.{u1}} [_inst_1 : LinearOrder.{u2} α] [_inst_2 : Preorder.{u1} β] {a : α} {f : α -> β}, (AntitoneOn.{u2, u1} α β (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (DistribLattice.toLattice.{u2} α (instDistribLattice.{u2} α _inst_1))))) _inst_2 f (Set.Iic.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (DistribLattice.toLattice.{u2} α (instDistribLattice.{u2} α _inst_1))))) a)) -> (AntitoneOn.{u2, u1} α β (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (DistribLattice.toLattice.{u2} α (instDistribLattice.{u2} α _inst_1))))) _inst_2 f (Set.Ici.{u2} α (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (DistribLattice.toLattice.{u2} α (instDistribLattice.{u2} α _inst_1))))) a)) -> (Antitone.{u2, u1} α β (PartialOrder.toPreorder.{u2} α (SemilatticeInf.toPartialOrder.{u2} α (Lattice.toSemilatticeInf.{u2} α (DistribLattice.toLattice.{u2} α (instDistribLattice.{u2} α _inst_1))))) _inst_2 f)
-Case conversion may be inaccurate. Consider using '#align antitone_on.Iic_union_Ici AntitoneOn.Iic_union_Iciₓ'. -/
/-- If `f` is antitone both on `(-∞, a]` and `[a, ∞)`, then it is antitone on the whole line. -/
protected theorem AntitoneOn.Iic_union_Ici (h₁ : AntitoneOn f (Iic a)) (h₂ : AntitoneOn f (Ici a)) :
Antitone f :=
mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb
@@ -41,19 +41,14 @@ protected theorem StrictMonoOn.union {s t : Set α} {c : α} (h₁ : StrictMonoO
have A : ∀ x, x ∈ s ∪ t → x ≤ c → x ∈ s :=
by
intro x hx hxc
- cases hx
- · exact hx
- rcases eq_or_lt_of_le hxc with (rfl | h'x)
- · exact hs.1
+ cases hx; · exact hx
+ rcases eq_or_lt_of_le hxc with (rfl | h'x); · exact hs.1
exact (lt_irrefl _ (h'x.trans_le (ht.2 hx))).elim
have B : ∀ x, x ∈ s ∪ t → c ≤ x → x ∈ t :=
by
intro x hx hxc
- cases hx
- swap
- · exact hx
- rcases eq_or_lt_of_le hxc with (rfl | h'x)
- · exact ht.1
+ cases hx; swap; · exact hx
+ rcases eq_or_lt_of_le hxc with (rfl | h'x); · exact ht.1
exact (lt_irrefl _ (h'x.trans_le (hs.2 hx))).elim
intro x hx y hy hxy
rcases lt_or_le x c with (hxc | hcx)
@@ -121,19 +116,14 @@ protected theorem MonotoneOn.union_right {s t : Set α} {c : α} (h₁ : Monoton
have A : ∀ x, x ∈ s ∪ t → x ≤ c → x ∈ s :=
by
intro x hx hxc
- cases hx
- · exact hx
- rcases eq_or_lt_of_le hxc with (rfl | h'x)
- · exact hs.1
+ cases hx; · exact hx
+ rcases eq_or_lt_of_le hxc with (rfl | h'x); · exact hs.1
exact (lt_irrefl _ (h'x.trans_le (ht.2 hx))).elim
have B : ∀ x, x ∈ s ∪ t → c ≤ x → x ∈ t :=
by
intro x hx hxc
- cases hx
- swap
- · exact hx
- rcases eq_or_lt_of_le hxc with (rfl | h'x)
- · exact ht.1
+ cases hx; swap; · exact hx
+ rcases eq_or_lt_of_le hxc with (rfl | h'x); · exact ht.1
exact (lt_irrefl _ (h'x.trans_le (hs.2 hx))).elim
intro x hx y hy hxy
rcases lt_or_le x c with (hxc | hcx)
mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc
Type _
and Sort _
(#6499)
We remove all possible occurences of Type _
and Sort _
in favor of Type*
and Sort*
.
This has nice performance benefits.
@@ -19,7 +19,7 @@ set.
open Set
-variable {α β : Type _} [LinearOrder α] [Preorder β] {a : α} {f : α → β}
+variable {α β : Type*} [LinearOrder α] [Preorder β] {a : α} {f : α → β}
/-- If `f` is strictly monotone both on `s` and `t`, with `s` to the left of `t` and the center
point belonging to both `s` and `t`, then `f` is strictly monotone on `s ∪ t` -/
@@ -2,14 +2,11 @@
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel
-
-! This file was ported from Lean 3 source module order.monotone.union
-! leanprover-community/mathlib commit d012cd09a9b256d870751284dd6a29882b0be105
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
-/
import Mathlib.Order.Bounds.Basic
+#align_import order.monotone.union from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
+
/-!
# Monotonicity on intervals
All dependencies are ported!