topology.algebra.infinite_sum.real ⟷ Mathlib.Topology.Algebra.InfiniteSum.Real

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -40,8 +40,8 @@ theorem cauchySeq_of_edist_le_of_summable [PseudoEMetricSpace α] {f : ℕ → 
   refine' (Metric.cauchySeq_iff'.1 hd ε (NNReal.coe_pos.2 εpos)).imp fun N hN n hn => _
   have hsum := hN n hn
   -- We simplify the known inequality
-  rw [dist_nndist, NNReal.nndist_eq, ← sum_range_add_sum_Ico _ hn, add_tsub_cancel_left] at hsum 
-  norm_cast at hsum 
+  rw [dist_nndist, NNReal.nndist_eq, ← sum_range_add_sum_Ico _ hn, add_tsub_cancel_left] at hsum
+  norm_cast at hsum
   replace hsum := lt_of_le_of_lt (le_max_left _ _) hsum
   rw [edist_comm]
   -- Then use `hf` to simplify the goal to the same form
@@ -64,7 +64,7 @@ theorem cauchySeq_of_dist_le_of_summable (d : ℕ → ℝ) (hf : ∀ n, dist (f
     H.tendsto_sum_nat.cauchy_seq
   refine' (Metric.cauchySeq_iff'.1 hd ε εpos).imp fun N hN n hn => _
   have hsum := hN n hn
-  rw [Real.dist_eq, ← sum_Ico_eq_sub _ hn] at hsum 
+  rw [Real.dist_eq, ← sum_Ico_eq_sub _ hn] at hsum
   calc
     dist (f n) (f N) = dist (f N) (f n) := dist_comm _ _
     _ ≤ ∑ x in Ico N n, d x := (dist_le_Ico_sum_of_dist_le hn fun k _ _ => hf k)

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451

@@ -3,9 +3,9 @@ Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel, Yury Kudryashov
 -/
-import Mathbin.Algebra.BigOperators.Intervals
-import Mathbin.Topology.Algebra.InfiniteSum.Order
-import Mathbin.Topology.Instances.Real
+import Algebra.BigOperators.Intervals
+import Topology.Algebra.InfiniteSum.Order
+import Topology.Instances.Real
 
 #align_import topology.algebra.infinite_sum.real from "leanprover-community/mathlib"@"f47581155c818e6361af4e4fda60d27d020c226b"
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345

@@ -2,16 +2,13 @@
 Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel, Yury Kudryashov
-
-! This file was ported from Lean 3 source module topology.algebra.infinite_sum.real
-! leanprover-community/mathlib commit f47581155c818e6361af4e4fda60d27d020c226b
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Algebra.BigOperators.Intervals
 import Mathbin.Topology.Algebra.InfiniteSum.Order
 import Mathbin.Topology.Instances.Real
 
+#align_import topology.algebra.infinite_sum.real from "leanprover-community/mathlib"@"f47581155c818e6361af4e4fda60d27d020c226b"
+
 /-!
 # Infinite sum in the reals
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423

@@ -28,6 +28,7 @@ open scoped BigOperators NNReal Topology
 
 variable {α : Type _}
 
+#print cauchySeq_of_edist_le_of_summable /-
 /-- If the extended distance between consecutive points of a sequence is estimated
 by a summable series of `nnreal`s, then the original sequence is a Cauchy sequence. -/
 theorem cauchySeq_of_edist_le_of_summable [PseudoEMetricSpace α] {f : ℕ → α} (d : ℕ → ℝ≥0)
@@ -50,9 +51,11 @@ theorem cauchySeq_of_edist_le_of_summable [PseudoEMetricSpace α] {f : ℕ → 
   apply lt_of_le_of_lt (edist_le_Ico_sum_of_edist_le hn fun k _ _ => hf k)
   assumption_mod_cast
 #align cauchy_seq_of_edist_le_of_summable cauchySeq_of_edist_le_of_summable
+-/
 
 variable [PseudoMetricSpace α] {f : ℕ → α} {a : α}
 
+#print cauchySeq_of_dist_le_of_summable /-
 /-- If the distance between consecutive points of a sequence is estimated by a summable series,
 then the original sequence is a Cauchy sequence. -/
 theorem cauchySeq_of_dist_le_of_summable (d : ℕ → ℝ) (hf : ∀ n, dist (f n) (f n.succ) ≤ d n)
@@ -71,6 +74,7 @@ theorem cauchySeq_of_dist_le_of_summable (d : ℕ → ℝ) (hf : ∀ n, dist (f
     _ ≤ |∑ x in Ico N n, d x| := (le_abs_self _)
     _ < ε := hsum
 #align cauchy_seq_of_dist_le_of_summable cauchySeq_of_dist_le_of_summable
+-/
 
 #print cauchySeq_of_summable_dist /-
 theorem cauchySeq_of_summable_dist (h : Summable fun n => dist (f n) (f n.succ)) : CauchySeq f :=
@@ -78,6 +82,7 @@ theorem cauchySeq_of_summable_dist (h : Summable fun n => dist (f n) (f n.succ))
 #align cauchy_seq_of_summable_dist cauchySeq_of_summable_dist
 -/
 
+#print dist_le_tsum_of_dist_le_of_tendsto /-
 theorem dist_le_tsum_of_dist_le_of_tendsto (d : ℕ → ℝ) (hf : ∀ n, dist (f n) (f n.succ) ≤ d n)
     (hd : Summable d) {a : α} (ha : Tendsto f atTop (𝓝 a)) (n : ℕ) :
     dist (f n) a ≤ ∑' m, d (n + m) :=
@@ -88,20 +93,27 @@ theorem dist_le_tsum_of_dist_le_of_tendsto (d : ℕ → ℝ) (hf : ∀ n, dist (
   refine' sum_le_tsum (range _) (fun _ _ => le_trans dist_nonneg (hf _)) _
   exact hd.comp_injective (add_right_injective n)
 #align dist_le_tsum_of_dist_le_of_tendsto dist_le_tsum_of_dist_le_of_tendsto
+-/
 
+#print dist_le_tsum_of_dist_le_of_tendsto₀ /-
 theorem dist_le_tsum_of_dist_le_of_tendsto₀ (d : ℕ → ℝ) (hf : ∀ n, dist (f n) (f n.succ) ≤ d n)
     (hd : Summable d) (ha : Tendsto f atTop (𝓝 a)) : dist (f 0) a ≤ tsum d := by
   simpa only [zero_add] using dist_le_tsum_of_dist_le_of_tendsto d hf hd ha 0
 #align dist_le_tsum_of_dist_le_of_tendsto₀ dist_le_tsum_of_dist_le_of_tendsto₀
+-/
 
+#print dist_le_tsum_dist_of_tendsto /-
 theorem dist_le_tsum_dist_of_tendsto (h : Summable fun n => dist (f n) (f n.succ))
     (ha : Tendsto f atTop (𝓝 a)) (n) : dist (f n) a ≤ ∑' m, dist (f (n + m)) (f (n + m).succ) :=
   show dist (f n) a ≤ ∑' m, (fun x => dist (f x) (f x.succ)) (n + m) from
     dist_le_tsum_of_dist_le_of_tendsto (fun n => dist (f n) (f n.succ)) (fun _ => le_rfl) h ha n
 #align dist_le_tsum_dist_of_tendsto dist_le_tsum_dist_of_tendsto
+-/
 
+#print dist_le_tsum_dist_of_tendsto₀ /-
 theorem dist_le_tsum_dist_of_tendsto₀ (h : Summable fun n => dist (f n) (f n.succ))
     (ha : Tendsto f atTop (𝓝 a)) : dist (f 0) a ≤ ∑' n, dist (f n) (f n.succ) := by
   simpa only [zero_add] using dist_le_tsum_dist_of_tendsto h ha 0
 #align dist_le_tsum_dist_of_tendsto₀ dist_le_tsum_dist_of_tendsto₀
+-/
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/7e5137f579de09a059a5ce98f364a04e221aabf0

@@ -70,7 +70,6 @@ theorem cauchySeq_of_dist_le_of_summable (d : ℕ → ℝ) (hf : ∀ n, dist (f
     _ ≤ ∑ x in Ico N n, d x := (dist_le_Ico_sum_of_dist_le hn fun k _ _ => hf k)
     _ ≤ |∑ x in Ico N n, d x| := (le_abs_self _)
     _ < ε := hsum
-    
 #align cauchy_seq_of_dist_le_of_summable cauchySeq_of_dist_le_of_summable
 
 #print cauchySeq_of_summable_dist /-

mathlib commit https://github.com/leanprover-community/mathlib/commit/5f25c089cb34db4db112556f23c50d12da81b297

@@ -43,7 +43,7 @@ theorem cauchySeq_of_edist_le_of_summable [PseudoEMetricSpace α] {f : ℕ → 
   have hsum := hN n hn
   -- We simplify the known inequality
   rw [dist_nndist, NNReal.nndist_eq, ← sum_range_add_sum_Ico _ hn, add_tsub_cancel_left] at hsum 
-  norm_cast  at hsum 
+  norm_cast at hsum 
   replace hsum := lt_of_le_of_lt (le_max_left _ _) hsum
   rw [edist_comm]
   -- Then use `hf` to simplify the goal to the same form

mathlib commit https://github.com/leanprover-community/mathlib/commit/cca40788df1b8755d5baf17ab2f27dacc2e17acb

@@ -42,8 +42,8 @@ theorem cauchySeq_of_edist_le_of_summable [PseudoEMetricSpace α] {f : ℕ → 
   refine' (Metric.cauchySeq_iff'.1 hd ε (NNReal.coe_pos.2 εpos)).imp fun N hN n hn => _
   have hsum := hN n hn
   -- We simplify the known inequality
-  rw [dist_nndist, NNReal.nndist_eq, ← sum_range_add_sum_Ico _ hn, add_tsub_cancel_left] at hsum
-  norm_cast  at hsum
+  rw [dist_nndist, NNReal.nndist_eq, ← sum_range_add_sum_Ico _ hn, add_tsub_cancel_left] at hsum 
+  norm_cast  at hsum 
   replace hsum := lt_of_le_of_lt (le_max_left _ _) hsum
   rw [edist_comm]
   -- Then use `hf` to simplify the goal to the same form
@@ -64,7 +64,7 @@ theorem cauchySeq_of_dist_le_of_summable (d : ℕ → ℝ) (hf : ∀ n, dist (f
     H.tendsto_sum_nat.cauchy_seq
   refine' (Metric.cauchySeq_iff'.1 hd ε εpos).imp fun N hN n hn => _
   have hsum := hN n hn
-  rw [Real.dist_eq, ← sum_Ico_eq_sub _ hn] at hsum
+  rw [Real.dist_eq, ← sum_Ico_eq_sub _ hn] at hsum 
   calc
     dist (f n) (f N) = dist (f N) (f n) := dist_comm _ _
     _ ≤ ∑ x in Ico N n, d x := (dist_le_Ico_sum_of_dist_le hn fun k _ _ => hf k)

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -24,7 +24,7 @@ This file provides lemmas about Cauchy sequences in terms of infinite sums.
 
 open Filter Finset
 
-open BigOperators NNReal Topology
+open scoped BigOperators NNReal Topology
 
 variable {α : Type _}
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -28,12 +28,6 @@ open BigOperators NNReal Topology
 
 variable {α : Type _}
 
-/- warning: cauchy_seq_of_edist_le_of_summable -> cauchySeq_of_edist_le_of_summable is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoEMetricSpace.{u1} α] {f : Nat -> α} (d : Nat -> NNReal), (forall (n : Nat), LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (EDist.edist.{u1} α (PseudoEMetricSpace.toHasEdist.{u1} α _inst_1) (f n) (f (Nat.succ n))) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal ENNReal (HasLiftT.mk.{1, 1} NNReal ENNReal (CoeTCₓ.coe.{1, 1} NNReal ENNReal (coeBase.{1, 1} NNReal ENNReal ENNReal.hasCoe))) (d n))) -> (Summable.{0, 0} NNReal Nat (OrderedCancelAddCommMonoid.toAddCommMonoid.{0} NNReal (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} NNReal NNReal.strictOrderedSemiring)) (UniformSpace.toTopologicalSpace.{0} NNReal (PseudoMetricSpace.toUniformSpace.{0} NNReal NNReal.pseudoMetricSpace)) d) -> (CauchySeq.{u1, 0} α Nat (PseudoEMetricSpace.toUniformSpace.{u1} α _inst_1) (CanonicallyLinearOrderedAddMonoid.semilatticeSup.{0} Nat Nat.canonicallyLinearOrderedAddMonoid) f)
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : PseudoEMetricSpace.{u1} α] {f : Nat -> α} (d : Nat -> NNReal), (forall (n : Nat), LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (EDist.edist.{u1} α (PseudoEMetricSpace.toEDist.{u1} α _inst_1) (f n) (f (Nat.succ n))) (ENNReal.some (d n))) -> (Summable.{0, 0} NNReal Nat (OrderedCancelAddCommMonoid.toAddCommMonoid.{0} NNReal (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} NNReal instNNRealStrictOrderedSemiring)) (UniformSpace.toTopologicalSpace.{0} NNReal (PseudoMetricSpace.toUniformSpace.{0} NNReal instPseudoMetricSpaceNNReal)) d) -> (CauchySeq.{u1, 0} α Nat (PseudoEMetricSpace.toUniformSpace.{u1} α _inst_1) (Lattice.toSemilatticeSup.{0} Nat Nat.instLatticeNat) f)
-Case conversion may be inaccurate. Consider using '#align cauchy_seq_of_edist_le_of_summable cauchySeq_of_edist_le_of_summableₓ'. -/
 /-- If the extended distance between consecutive points of a sequence is estimated
 by a summable series of `nnreal`s, then the original sequence is a Cauchy sequence. -/
 theorem cauchySeq_of_edist_le_of_summable [PseudoEMetricSpace α] {f : ℕ → α} (d : ℕ → ℝ≥0)
@@ -59,12 +53,6 @@ theorem cauchySeq_of_edist_le_of_summable [PseudoEMetricSpace α] {f : ℕ → 
 
 variable [PseudoMetricSpace α] {f : ℕ → α} {a : α}
 
-/- warning: cauchy_seq_of_dist_le_of_summable -> cauchySeq_of_dist_le_of_summable is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {f : Nat -> α} (d : Nat -> Real), (forall (n : Nat), LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (f n) (f (Nat.succ n))) (d n)) -> (Summable.{0, 0} Real Nat Real.addCommMonoid (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) d) -> (CauchySeq.{u1, 0} α Nat (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) (CanonicallyLinearOrderedAddMonoid.semilatticeSup.{0} Nat Nat.canonicallyLinearOrderedAddMonoid) f)
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {f : Nat -> α} (d : Nat -> Real), (forall (n : Nat), LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (f n) (f (Nat.succ n))) (d n)) -> (Summable.{0, 0} Real Nat Real.instAddCommMonoidReal (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) d) -> (CauchySeq.{u1, 0} α Nat (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) (Lattice.toSemilatticeSup.{0} Nat Nat.instLatticeNat) f)
-Case conversion may be inaccurate. Consider using '#align cauchy_seq_of_dist_le_of_summable cauchySeq_of_dist_le_of_summableₓ'. -/
 /-- If the distance between consecutive points of a sequence is estimated by a summable series,
 then the original sequence is a Cauchy sequence. -/
 theorem cauchySeq_of_dist_le_of_summable (d : ℕ → ℝ) (hf : ∀ n, dist (f n) (f n.succ) ≤ d n)
@@ -91,12 +79,6 @@ theorem cauchySeq_of_summable_dist (h : Summable fun n => dist (f n) (f n.succ))
 #align cauchy_seq_of_summable_dist cauchySeq_of_summable_dist
 -/
 
-/- warning: dist_le_tsum_of_dist_le_of_tendsto -> dist_le_tsum_of_dist_le_of_tendsto is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {f : Nat -> α} (d : Nat -> Real), (forall (n : Nat), LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (f n) (f (Nat.succ n))) (d n)) -> (Summable.{0, 0} Real Nat Real.addCommMonoid (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) d) -> (forall {a : α}, (Filter.Tendsto.{0, u1} Nat α f (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring)))) (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) a)) -> (forall (n : Nat), LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (f n) a) (tsum.{0, 0} Real Real.addCommMonoid (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) Nat (fun (m : Nat) => d (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n m)))))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {f : Nat -> α} (d : Nat -> Real), (forall (n : Nat), LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (f n) (f (Nat.succ n))) (d n)) -> (Summable.{0, 0} Real Nat Real.instAddCommMonoidReal (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) d) -> (forall {a : α}, (Filter.Tendsto.{0, u1} Nat α f (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring))) (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) a)) -> (forall (n : Nat), LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (f n) a) (tsum.{0, 0} Real Real.instAddCommMonoidReal (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) Nat (fun (m : Nat) => d (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n m)))))
-Case conversion may be inaccurate. Consider using '#align dist_le_tsum_of_dist_le_of_tendsto dist_le_tsum_of_dist_le_of_tendstoₓ'. -/
 theorem dist_le_tsum_of_dist_le_of_tendsto (d : ℕ → ℝ) (hf : ∀ n, dist (f n) (f n.succ) ≤ d n)
     (hd : Summable d) {a : α} (ha : Tendsto f atTop (𝓝 a)) (n : ℕ) :
     dist (f n) a ≤ ∑' m, d (n + m) :=
@@ -108,35 +90,17 @@ theorem dist_le_tsum_of_dist_le_of_tendsto (d : ℕ → ℝ) (hf : ∀ n, dist (
   exact hd.comp_injective (add_right_injective n)
 #align dist_le_tsum_of_dist_le_of_tendsto dist_le_tsum_of_dist_le_of_tendsto
 
-/- warning: dist_le_tsum_of_dist_le_of_tendsto₀ -> dist_le_tsum_of_dist_le_of_tendsto₀ is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {f : Nat -> α} {a : α} (d : Nat -> Real), (forall (n : Nat), LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (f n) (f (Nat.succ n))) (d n)) -> (Summable.{0, 0} Real Nat Real.addCommMonoid (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) d) -> (Filter.Tendsto.{0, u1} Nat α f (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring)))) (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) a)) -> (LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (f (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero)))) a) (tsum.{0, 0} Real Real.addCommMonoid (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) Nat d))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {f : Nat -> α} {a : α} (d : Nat -> Real), (forall (n : Nat), LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (f n) (f (Nat.succ n))) (d n)) -> (Summable.{0, 0} Real Nat Real.instAddCommMonoidReal (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) d) -> (Filter.Tendsto.{0, u1} Nat α f (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring))) (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) a)) -> (LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (f (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) a) (tsum.{0, 0} Real Real.instAddCommMonoidReal (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) Nat d))
-Case conversion may be inaccurate. Consider using '#align dist_le_tsum_of_dist_le_of_tendsto₀ dist_le_tsum_of_dist_le_of_tendsto₀ₓ'. -/
 theorem dist_le_tsum_of_dist_le_of_tendsto₀ (d : ℕ → ℝ) (hf : ∀ n, dist (f n) (f n.succ) ≤ d n)
     (hd : Summable d) (ha : Tendsto f atTop (𝓝 a)) : dist (f 0) a ≤ tsum d := by
   simpa only [zero_add] using dist_le_tsum_of_dist_le_of_tendsto d hf hd ha 0
 #align dist_le_tsum_of_dist_le_of_tendsto₀ dist_le_tsum_of_dist_le_of_tendsto₀
 
-/- warning: dist_le_tsum_dist_of_tendsto -> dist_le_tsum_dist_of_tendsto is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {f : Nat -> α} {a : α}, (Summable.{0, 0} Real Nat Real.addCommMonoid (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (fun (n : Nat) => Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (f n) (f (Nat.succ n)))) -> (Filter.Tendsto.{0, u1} Nat α f (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring)))) (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) a)) -> (forall (n : Nat), LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (f n) a) (tsum.{0, 0} Real Real.addCommMonoid (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) Nat (fun (m : Nat) => Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (f (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n m)) (f (Nat.succ (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n m))))))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {f : Nat -> α} {a : α}, (Summable.{0, 0} Real Nat Real.instAddCommMonoidReal (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (fun (n : Nat) => Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (f n) (f (Nat.succ n)))) -> (Filter.Tendsto.{0, u1} Nat α f (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring))) (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) a)) -> (forall (n : Nat), LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (f n) a) (tsum.{0, 0} Real Real.instAddCommMonoidReal (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) Nat (fun (m : Nat) => Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (f (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n m)) (f (Nat.succ (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n m))))))
-Case conversion may be inaccurate. Consider using '#align dist_le_tsum_dist_of_tendsto dist_le_tsum_dist_of_tendstoₓ'. -/
 theorem dist_le_tsum_dist_of_tendsto (h : Summable fun n => dist (f n) (f n.succ))
     (ha : Tendsto f atTop (𝓝 a)) (n) : dist (f n) a ≤ ∑' m, dist (f (n + m)) (f (n + m).succ) :=
   show dist (f n) a ≤ ∑' m, (fun x => dist (f x) (f x.succ)) (n + m) from
     dist_le_tsum_of_dist_le_of_tendsto (fun n => dist (f n) (f n.succ)) (fun _ => le_rfl) h ha n
 #align dist_le_tsum_dist_of_tendsto dist_le_tsum_dist_of_tendsto
 
-/- warning: dist_le_tsum_dist_of_tendsto₀ -> dist_le_tsum_dist_of_tendsto₀ is a dubious translation:
-lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {f : Nat -> α} {a : α}, (Summable.{0, 0} Real Nat Real.addCommMonoid (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (fun (n : Nat) => Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (f n) (f (Nat.succ n)))) -> (Filter.Tendsto.{0, u1} Nat α f (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring)))) (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) a)) -> (LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (f (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero)))) a) (tsum.{0, 0} Real Real.addCommMonoid (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) Nat (fun (n : Nat) => Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (f n) (f (Nat.succ n)))))
-but is expected to have type
-  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {f : Nat -> α} {a : α}, (Summable.{0, 0} Real Nat Real.instAddCommMonoidReal (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (fun (n : Nat) => Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (f n) (f (Nat.succ n)))) -> (Filter.Tendsto.{0, u1} Nat α f (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring))) (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) a)) -> (LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (f (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) a) (tsum.{0, 0} Real Real.instAddCommMonoidReal (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) Nat (fun (n : Nat) => Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (f n) (f (Nat.succ n)))))
-Case conversion may be inaccurate. Consider using '#align dist_le_tsum_dist_of_tendsto₀ dist_le_tsum_dist_of_tendsto₀ₓ'. -/
 theorem dist_le_tsum_dist_of_tendsto₀ (h : Summable fun n => dist (f n) (f n.succ))
     (ha : Tendsto f atTop (𝓝 a)) : dist (f 0) a ≤ ∑' n, dist (f n) (f n.succ) := by
   simpa only [zero_add] using dist_le_tsum_dist_of_tendsto h ha 0

mathlib commit https://github.com/leanprover-community/mathlib/commit/0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8

@@ -30,7 +30,7 @@ variable {α : Type _}
 
 /- warning: cauchy_seq_of_edist_le_of_summable -> cauchySeq_of_edist_le_of_summable is a dubious translation:
 lean 3 declaration is
-  forall {α : Type.{u1}} [_inst_1 : PseudoEMetricSpace.{u1} α] {f : Nat -> α} (d : Nat -> NNReal), (forall (n : Nat), LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (EDist.edist.{u1} α (PseudoEMetricSpace.toHasEdist.{u1} α _inst_1) (f n) (f (Nat.succ n))) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal ENNReal (HasLiftT.mk.{1, 1} NNReal ENNReal (CoeTCₓ.coe.{1, 1} NNReal ENNReal (coeBase.{1, 1} NNReal ENNReal ENNReal.hasCoe))) (d n))) -> (Summable.{0, 0} NNReal Nat (OrderedCancelAddCommMonoid.toAddCommMonoid.{0} NNReal (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} NNReal NNReal.strictOrderedSemiring)) (UniformSpace.toTopologicalSpace.{0} NNReal (PseudoMetricSpace.toUniformSpace.{0} NNReal NNReal.pseudoMetricSpace)) d) -> (CauchySeq.{u1, 0} α Nat (PseudoEMetricSpace.toUniformSpace.{u1} α _inst_1) (CanonicallyLinearOrderedAddMonoid.semilatticeSup.{0} Nat Nat.canonicallyLinearOrderedAddMonoid) f)
+  forall {α : Type.{u1}} [_inst_1 : PseudoEMetricSpace.{u1} α] {f : Nat -> α} (d : Nat -> NNReal), (forall (n : Nat), LE.le.{0} ENNReal (Preorder.toHasLe.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (EDist.edist.{u1} α (PseudoEMetricSpace.toHasEdist.{u1} α _inst_1) (f n) (f (Nat.succ n))) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal ENNReal (HasLiftT.mk.{1, 1} NNReal ENNReal (CoeTCₓ.coe.{1, 1} NNReal ENNReal (coeBase.{1, 1} NNReal ENNReal ENNReal.hasCoe))) (d n))) -> (Summable.{0, 0} NNReal Nat (OrderedCancelAddCommMonoid.toAddCommMonoid.{0} NNReal (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} NNReal NNReal.strictOrderedSemiring)) (UniformSpace.toTopologicalSpace.{0} NNReal (PseudoMetricSpace.toUniformSpace.{0} NNReal NNReal.pseudoMetricSpace)) d) -> (CauchySeq.{u1, 0} α Nat (PseudoEMetricSpace.toUniformSpace.{u1} α _inst_1) (CanonicallyLinearOrderedAddMonoid.semilatticeSup.{0} Nat Nat.canonicallyLinearOrderedAddMonoid) f)
 but is expected to have type
   forall {α : Type.{u1}} [_inst_1 : PseudoEMetricSpace.{u1} α] {f : Nat -> α} (d : Nat -> NNReal), (forall (n : Nat), LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (EDist.edist.{u1} α (PseudoEMetricSpace.toEDist.{u1} α _inst_1) (f n) (f (Nat.succ n))) (ENNReal.some (d n))) -> (Summable.{0, 0} NNReal Nat (OrderedCancelAddCommMonoid.toAddCommMonoid.{0} NNReal (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} NNReal instNNRealStrictOrderedSemiring)) (UniformSpace.toTopologicalSpace.{0} NNReal (PseudoMetricSpace.toUniformSpace.{0} NNReal instPseudoMetricSpaceNNReal)) d) -> (CauchySeq.{u1, 0} α Nat (PseudoEMetricSpace.toUniformSpace.{u1} α _inst_1) (Lattice.toSemilatticeSup.{0} Nat Nat.instLatticeNat) f)
 Case conversion may be inaccurate. Consider using '#align cauchy_seq_of_edist_le_of_summable cauchySeq_of_edist_le_of_summableₓ'. -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/21e3562c5e12d846c7def5eff8cdbc520d7d4936

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel, Yury Kudryashov
 
 ! This file was ported from Lean 3 source module topology.algebra.infinite_sum.real
-! leanprover-community/mathlib commit 9a59dcb7a2d06bf55da57b9030169219980660cd
+! leanprover-community/mathlib commit f47581155c818e6361af4e4fda60d27d020c226b
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -15,6 +15,9 @@ import Mathbin.Topology.Instances.Real
 /-!
 # Infinite sum in the reals
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 This file provides lemmas about Cauchy sequences in terms of infinite sums.
 -/
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/3b267e70a936eebb21ab546f49a8df34dd300b25

@@ -25,6 +25,12 @@ open BigOperators NNReal Topology
 
 variable {α : Type _}
 
+/- warning: cauchy_seq_of_edist_le_of_summable -> cauchySeq_of_edist_le_of_summable is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoEMetricSpace.{u1} α] {f : Nat -> α} (d : Nat -> NNReal), (forall (n : Nat), LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.completeLinearOrder))))) (EDist.edist.{u1} α (PseudoEMetricSpace.toHasEdist.{u1} α _inst_1) (f n) (f (Nat.succ n))) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) NNReal ENNReal (HasLiftT.mk.{1, 1} NNReal ENNReal (CoeTCₓ.coe.{1, 1} NNReal ENNReal (coeBase.{1, 1} NNReal ENNReal ENNReal.hasCoe))) (d n))) -> (Summable.{0, 0} NNReal Nat (OrderedCancelAddCommMonoid.toAddCommMonoid.{0} NNReal (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} NNReal NNReal.strictOrderedSemiring)) (UniformSpace.toTopologicalSpace.{0} NNReal (PseudoMetricSpace.toUniformSpace.{0} NNReal NNReal.pseudoMetricSpace)) d) -> (CauchySeq.{u1, 0} α Nat (PseudoEMetricSpace.toUniformSpace.{u1} α _inst_1) (CanonicallyLinearOrderedAddMonoid.semilatticeSup.{0} Nat Nat.canonicallyLinearOrderedAddMonoid) f)
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoEMetricSpace.{u1} α] {f : Nat -> α} (d : Nat -> NNReal), (forall (n : Nat), LE.le.{0} ENNReal (Preorder.toLE.{0} ENNReal (PartialOrder.toPreorder.{0} ENNReal (CompleteSemilatticeInf.toPartialOrder.{0} ENNReal (CompleteLattice.toCompleteSemilatticeInf.{0} ENNReal (CompleteLinearOrder.toCompleteLattice.{0} ENNReal ENNReal.instCompleteLinearOrderENNReal))))) (EDist.edist.{u1} α (PseudoEMetricSpace.toEDist.{u1} α _inst_1) (f n) (f (Nat.succ n))) (ENNReal.some (d n))) -> (Summable.{0, 0} NNReal Nat (OrderedCancelAddCommMonoid.toAddCommMonoid.{0} NNReal (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} NNReal instNNRealStrictOrderedSemiring)) (UniformSpace.toTopologicalSpace.{0} NNReal (PseudoMetricSpace.toUniformSpace.{0} NNReal instPseudoMetricSpaceNNReal)) d) -> (CauchySeq.{u1, 0} α Nat (PseudoEMetricSpace.toUniformSpace.{u1} α _inst_1) (Lattice.toSemilatticeSup.{0} Nat Nat.instLatticeNat) f)
+Case conversion may be inaccurate. Consider using '#align cauchy_seq_of_edist_le_of_summable cauchySeq_of_edist_le_of_summableₓ'. -/
 /-- If the extended distance between consecutive points of a sequence is estimated
 by a summable series of `nnreal`s, then the original sequence is a Cauchy sequence. -/
 theorem cauchySeq_of_edist_le_of_summable [PseudoEMetricSpace α] {f : ℕ → α} (d : ℕ → ℝ≥0)
@@ -50,6 +56,12 @@ theorem cauchySeq_of_edist_le_of_summable [PseudoEMetricSpace α] {f : ℕ → 
 
 variable [PseudoMetricSpace α] {f : ℕ → α} {a : α}
 
+/- warning: cauchy_seq_of_dist_le_of_summable -> cauchySeq_of_dist_le_of_summable is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {f : Nat -> α} (d : Nat -> Real), (forall (n : Nat), LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (f n) (f (Nat.succ n))) (d n)) -> (Summable.{0, 0} Real Nat Real.addCommMonoid (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) d) -> (CauchySeq.{u1, 0} α Nat (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) (CanonicallyLinearOrderedAddMonoid.semilatticeSup.{0} Nat Nat.canonicallyLinearOrderedAddMonoid) f)
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {f : Nat -> α} (d : Nat -> Real), (forall (n : Nat), LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (f n) (f (Nat.succ n))) (d n)) -> (Summable.{0, 0} Real Nat Real.instAddCommMonoidReal (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) d) -> (CauchySeq.{u1, 0} α Nat (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1) (Lattice.toSemilatticeSup.{0} Nat Nat.instLatticeNat) f)
+Case conversion may be inaccurate. Consider using '#align cauchy_seq_of_dist_le_of_summable cauchySeq_of_dist_le_of_summableₓ'. -/
 /-- If the distance between consecutive points of a sequence is estimated by a summable series,
 then the original sequence is a Cauchy sequence. -/
 theorem cauchySeq_of_dist_le_of_summable (d : ℕ → ℝ) (hf : ∀ n, dist (f n) (f n.succ) ≤ d n)
@@ -70,10 +82,18 @@ theorem cauchySeq_of_dist_le_of_summable (d : ℕ → ℝ) (hf : ∀ n, dist (f
     
 #align cauchy_seq_of_dist_le_of_summable cauchySeq_of_dist_le_of_summable
 
+#print cauchySeq_of_summable_dist /-
 theorem cauchySeq_of_summable_dist (h : Summable fun n => dist (f n) (f n.succ)) : CauchySeq f :=
   cauchySeq_of_dist_le_of_summable _ (fun _ => le_rfl) h
 #align cauchy_seq_of_summable_dist cauchySeq_of_summable_dist
+-/
 
+/- warning: dist_le_tsum_of_dist_le_of_tendsto -> dist_le_tsum_of_dist_le_of_tendsto is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {f : Nat -> α} (d : Nat -> Real), (forall (n : Nat), LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (f n) (f (Nat.succ n))) (d n)) -> (Summable.{0, 0} Real Nat Real.addCommMonoid (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) d) -> (forall {a : α}, (Filter.Tendsto.{0, u1} Nat α f (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring)))) (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) a)) -> (forall (n : Nat), LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (f n) a) (tsum.{0, 0} Real Real.addCommMonoid (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) Nat (fun (m : Nat) => d (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n m)))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {f : Nat -> α} (d : Nat -> Real), (forall (n : Nat), LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (f n) (f (Nat.succ n))) (d n)) -> (Summable.{0, 0} Real Nat Real.instAddCommMonoidReal (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) d) -> (forall {a : α}, (Filter.Tendsto.{0, u1} Nat α f (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring))) (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) a)) -> (forall (n : Nat), LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (f n) a) (tsum.{0, 0} Real Real.instAddCommMonoidReal (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) Nat (fun (m : Nat) => d (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n m)))))
+Case conversion may be inaccurate. Consider using '#align dist_le_tsum_of_dist_le_of_tendsto dist_le_tsum_of_dist_le_of_tendstoₓ'. -/
 theorem dist_le_tsum_of_dist_le_of_tendsto (d : ℕ → ℝ) (hf : ∀ n, dist (f n) (f n.succ) ≤ d n)
     (hd : Summable d) {a : α} (ha : Tendsto f atTop (𝓝 a)) (n : ℕ) :
     dist (f n) a ≤ ∑' m, d (n + m) :=
@@ -85,17 +105,35 @@ theorem dist_le_tsum_of_dist_le_of_tendsto (d : ℕ → ℝ) (hf : ∀ n, dist (
   exact hd.comp_injective (add_right_injective n)
 #align dist_le_tsum_of_dist_le_of_tendsto dist_le_tsum_of_dist_le_of_tendsto
 
+/- warning: dist_le_tsum_of_dist_le_of_tendsto₀ -> dist_le_tsum_of_dist_le_of_tendsto₀ is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {f : Nat -> α} {a : α} (d : Nat -> Real), (forall (n : Nat), LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (f n) (f (Nat.succ n))) (d n)) -> (Summable.{0, 0} Real Nat Real.addCommMonoid (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) d) -> (Filter.Tendsto.{0, u1} Nat α f (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring)))) (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) a)) -> (LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (f (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero)))) a) (tsum.{0, 0} Real Real.addCommMonoid (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) Nat d))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {f : Nat -> α} {a : α} (d : Nat -> Real), (forall (n : Nat), LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (f n) (f (Nat.succ n))) (d n)) -> (Summable.{0, 0} Real Nat Real.instAddCommMonoidReal (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) d) -> (Filter.Tendsto.{0, u1} Nat α f (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring))) (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) a)) -> (LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (f (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) a) (tsum.{0, 0} Real Real.instAddCommMonoidReal (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) Nat d))
+Case conversion may be inaccurate. Consider using '#align dist_le_tsum_of_dist_le_of_tendsto₀ dist_le_tsum_of_dist_le_of_tendsto₀ₓ'. -/
 theorem dist_le_tsum_of_dist_le_of_tendsto₀ (d : ℕ → ℝ) (hf : ∀ n, dist (f n) (f n.succ) ≤ d n)
     (hd : Summable d) (ha : Tendsto f atTop (𝓝 a)) : dist (f 0) a ≤ tsum d := by
   simpa only [zero_add] using dist_le_tsum_of_dist_le_of_tendsto d hf hd ha 0
 #align dist_le_tsum_of_dist_le_of_tendsto₀ dist_le_tsum_of_dist_le_of_tendsto₀
 
+/- warning: dist_le_tsum_dist_of_tendsto -> dist_le_tsum_dist_of_tendsto is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {f : Nat -> α} {a : α}, (Summable.{0, 0} Real Nat Real.addCommMonoid (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (fun (n : Nat) => Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (f n) (f (Nat.succ n)))) -> (Filter.Tendsto.{0, u1} Nat α f (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring)))) (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) a)) -> (forall (n : Nat), LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (f n) a) (tsum.{0, 0} Real Real.addCommMonoid (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) Nat (fun (m : Nat) => Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (f (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n m)) (f (Nat.succ (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n m))))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {f : Nat -> α} {a : α}, (Summable.{0, 0} Real Nat Real.instAddCommMonoidReal (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (fun (n : Nat) => Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (f n) (f (Nat.succ n)))) -> (Filter.Tendsto.{0, u1} Nat α f (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring))) (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) a)) -> (forall (n : Nat), LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (f n) a) (tsum.{0, 0} Real Real.instAddCommMonoidReal (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) Nat (fun (m : Nat) => Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (f (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n m)) (f (Nat.succ (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n m))))))
+Case conversion may be inaccurate. Consider using '#align dist_le_tsum_dist_of_tendsto dist_le_tsum_dist_of_tendstoₓ'. -/
 theorem dist_le_tsum_dist_of_tendsto (h : Summable fun n => dist (f n) (f n.succ))
     (ha : Tendsto f atTop (𝓝 a)) (n) : dist (f n) a ≤ ∑' m, dist (f (n + m)) (f (n + m).succ) :=
   show dist (f n) a ≤ ∑' m, (fun x => dist (f x) (f x.succ)) (n + m) from
     dist_le_tsum_of_dist_le_of_tendsto (fun n => dist (f n) (f n.succ)) (fun _ => le_rfl) h ha n
 #align dist_le_tsum_dist_of_tendsto dist_le_tsum_dist_of_tendsto
 
+/- warning: dist_le_tsum_dist_of_tendsto₀ -> dist_le_tsum_dist_of_tendsto₀ is a dubious translation:
+lean 3 declaration is
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {f : Nat -> α} {a : α}, (Summable.{0, 0} Real Nat Real.addCommMonoid (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (fun (n : Nat) => Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (f n) (f (Nat.succ n)))) -> (Filter.Tendsto.{0, u1} Nat α f (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (OrderedCancelAddCommMonoid.toPartialOrder.{0} Nat (StrictOrderedSemiring.toOrderedCancelAddCommMonoid.{0} Nat Nat.strictOrderedSemiring)))) (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) a)) -> (LE.le.{0} Real Real.hasLe (Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (f (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero)))) a) (tsum.{0, 0} Real Real.addCommMonoid (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) Nat (fun (n : Nat) => Dist.dist.{u1} α (PseudoMetricSpace.toHasDist.{u1} α _inst_1) (f n) (f (Nat.succ n)))))
+but is expected to have type
+  forall {α : Type.{u1}} [_inst_1 : PseudoMetricSpace.{u1} α] {f : Nat -> α} {a : α}, (Summable.{0, 0} Real Nat Real.instAddCommMonoidReal (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) (fun (n : Nat) => Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (f n) (f (Nat.succ n)))) -> (Filter.Tendsto.{0, u1} Nat α f (Filter.atTop.{0} Nat (PartialOrder.toPreorder.{0} Nat (StrictOrderedSemiring.toPartialOrder.{0} Nat Nat.strictOrderedSemiring))) (nhds.{u1} α (UniformSpace.toTopologicalSpace.{u1} α (PseudoMetricSpace.toUniformSpace.{u1} α _inst_1)) a)) -> (LE.le.{0} Real Real.instLEReal (Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (f (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) a) (tsum.{0, 0} Real Real.instAddCommMonoidReal (UniformSpace.toTopologicalSpace.{0} Real (PseudoMetricSpace.toUniformSpace.{0} Real Real.pseudoMetricSpace)) Nat (fun (n : Nat) => Dist.dist.{u1} α (PseudoMetricSpace.toDist.{u1} α _inst_1) (f n) (f (Nat.succ n)))))
+Case conversion may be inaccurate. Consider using '#align dist_le_tsum_dist_of_tendsto₀ dist_le_tsum_dist_of_tendsto₀ₓ'. -/
 theorem dist_le_tsum_dist_of_tendsto₀ (h : Summable fun n => dist (f n) (f n.succ))
     (ha : Tendsto f atTop (𝓝 a)) : dist (f 0) a ≤ ∑' n, dist (f n) (f n.succ) := by
   simpa only [zero_add] using dist_le_tsum_dist_of_tendsto h ha 0

mathlib commit https://github.com/leanprover-community/mathlib/commit/62e8311c791f02c47451bf14aa2501048e7c2f33

@@ -4,10 +4,11 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel, Yury Kudryashov
 
 ! This file was ported from Lean 3 source module topology.algebra.infinite_sum.real
-! leanprover-community/mathlib commit 32253a1a1071173b33dc7d6a218cf722c6feb514
+! leanprover-community/mathlib commit 9a59dcb7a2d06bf55da57b9030169219980660cd
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
+import Mathbin.Algebra.BigOperators.Intervals
 import Mathbin.Topology.Algebra.InfiniteSum.Order
 import Mathbin.Topology.Instances.Real
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/195fcd60ff2bfe392543bceb0ec2adcdb472db4c

@@ -26,10 +26,10 @@ variable {α : Type _}
 
 /-- If the extended distance between consecutive points of a sequence is estimated
 by a summable series of `nnreal`s, then the original sequence is a Cauchy sequence. -/
-theorem cauchySeq_of_edist_le_of_summable [PseudoEmetricSpace α] {f : ℕ → α} (d : ℕ → ℝ≥0)
+theorem cauchySeq_of_edist_le_of_summable [PseudoEMetricSpace α] {f : ℕ → α} (d : ℕ → ℝ≥0)
     (hf : ∀ n, edist (f n) (f n.succ) ≤ d n) (hd : Summable d) : CauchySeq f :=
   by
-  refine' Emetric.cauchySeq_iff_nNReal.2 fun ε εpos => _
+  refine' EMetric.cauchySeq_iff_NNReal.2 fun ε εpos => _
   -- Actually we need partial sums of `d` to be a Cauchy sequence
   replace hd : CauchySeq fun n : ℕ => ∑ x in range n, d x :=
     let ⟨_, H⟩ := hd

mathlib commit https://github.com/leanprover-community/mathlib/commit/4c586d291f189eecb9d00581aeb3dd998ac34442

@@ -63,8 +63,8 @@ theorem cauchySeq_of_dist_le_of_summable (d : ℕ → ℝ) (hf : ∀ n, dist (f
   rw [Real.dist_eq, ← sum_Ico_eq_sub _ hn] at hsum
   calc
     dist (f n) (f N) = dist (f N) (f n) := dist_comm _ _
-    _ ≤ ∑ x in Ico N n, d x := dist_le_Ico_sum_of_dist_le hn fun k _ _ => hf k
-    _ ≤ |∑ x in Ico N n, d x| := le_abs_self _
+    _ ≤ ∑ x in Ico N n, d x := (dist_le_Ico_sum_of_dist_le hn fun k _ _ => hf k)
+    _ ≤ |∑ x in Ico N n, d x| := (le_abs_self _)
     _ < ε := hsum
     
 #align cauchy_seq_of_dist_le_of_summable cauchySeq_of_dist_le_of_summable

mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc

We move some lemmas out of Topology/Instances/ENNReal into Topology/Algebra/InfiniteSum/Real. Also use this to address a porting TODO.

This was originally part of #12446

Co-authored-by: Chris Birkbeck <c.birkbeck@uea.ac.uk>

@@ -6,63 +6,30 @@ Authors: Sébastien Gouëzel, Yury Kudryashov
 import Mathlib.Algebra.BigOperators.Intervals
 import Mathlib.Topology.Algebra.InfiniteSum.Order
 import Mathlib.Topology.Instances.Real
+import Mathlib.Topology.Instances.ENNReal
 
 #align_import topology.algebra.infinite_sum.real from "leanprover-community/mathlib"@"9a59dcb7a2d06bf55da57b9030169219980660cd"
 
 /-!
 # Infinite sum in the reals
 
-This file provides lemmas about Cauchy sequences in terms of infinite sums.
+This file provides lemmas about Cauchy sequences in terms of infinite sums and infinite sums valued
+in the reals.
 -/
 
 open Filter Finset BigOperators NNReal Topology
 
-variable {α : Type*}
-
-/-- If the extended distance between consecutive points of a sequence is estimated
-by a summable series of `NNReal`s, then the original sequence is a Cauchy sequence. -/
-theorem cauchySeq_of_edist_le_of_summable [PseudoEMetricSpace α] {f : ℕ → α} (d : ℕ → ℝ≥0)
-    (hf : ∀ n, edist (f n) (f n.succ) ≤ d n) (hd : Summable d) : CauchySeq f := by
-  refine EMetric.cauchySeq_iff_NNReal.2 fun ε εpos ↦ ?_
-  -- Actually we need partial sums of `d` to be a Cauchy sequence
-  replace hd : CauchySeq fun n : ℕ ↦ ∑ x in range n, d x :=
-    let ⟨_, H⟩ := hd
-    H.tendsto_sum_nat.cauchySeq
-  -- Now we take the same `N` as in one of the definitions of a Cauchy sequence
-  refine (Metric.cauchySeq_iff'.1 hd ε (NNReal.coe_pos.2 εpos)).imp fun N hN n hn ↦ ?_
-  specialize hN n hn
-  -- We simplify the known inequality
-  rw [dist_nndist, NNReal.nndist_eq, ← sum_range_add_sum_Ico _ hn, add_tsub_cancel_left,
-    NNReal.coe_lt_coe, max_lt_iff] at hN
-  rw [edist_comm]
-  -- Then use `hf` to simplify the goal to the same form
-  refine lt_of_le_of_lt (edist_le_Ico_sum_of_edist_le hn fun _ _ ↦ hf _) ?_
-  exact mod_cast hN.1
-#align cauchy_seq_of_edist_le_of_summable cauchySeq_of_edist_le_of_summable
-
-variable [PseudoMetricSpace α] {f : ℕ → α} {a : α}
+variable {α β : Type*} [PseudoMetricSpace α] {f : ℕ → α} {a : α}
 
 /-- If the distance between consecutive points of a sequence is estimated by a summable series,
 then the original sequence is a Cauchy sequence. -/
 theorem cauchySeq_of_dist_le_of_summable (d : ℕ → ℝ) (hf : ∀ n, dist (f n) (f n.succ) ≤ d n)
     (hd : Summable d) : CauchySeq f := by
-  -- Porting note (#11215): TODO: with `Topology.Instances.NNReal` we can use this:
-  -- lift d to ℕ → ℝ≥0 using fun n ↦ dist_nonneg.trans (hf n)
-  -- refine cauchySeq_of_edist_le_of_summable d ?_ ?_
-  -- · exact_mod_cast hf
-  -- · exact_mod_cast hd
-  refine' Metric.cauchySeq_iff'.2 fun ε εpos ↦ _
-  replace hd : CauchySeq fun n : ℕ ↦ ∑ x in range n, d x :=
-    let ⟨_, H⟩ := hd
-    H.tendsto_sum_nat.cauchySeq
-  refine' (Metric.cauchySeq_iff'.1 hd ε εpos).imp fun N hN n hn ↦ _
-  have hsum := hN n hn
-  rw [Real.dist_eq, ← sum_Ico_eq_sub _ hn] at hsum
-  calc
-    dist (f n) (f N) = dist (f N) (f n) := dist_comm _ _
-    _ ≤ ∑ x in Ico N n, d x := dist_le_Ico_sum_of_dist_le hn fun _ _ ↦ hf _
-    _ ≤ |∑ x in Ico N n, d x| := le_abs_self _
-    _ < ε := hsum
+  lift d to ℕ → ℝ≥0 using fun n ↦ dist_nonneg.trans (hf n)
+  apply cauchySeq_of_edist_le_of_summable d (α := α) (f := f)
+  · exact_mod_cast hf
+  · exact_mod_cast hd
+
 #align cauchy_seq_of_dist_le_of_summable cauchySeq_of_dist_le_of_summable
 
 theorem cauchySeq_of_summable_dist (h : Summable fun n ↦ dist (f n) (f n.succ)) : CauchySeq f :=
@@ -94,3 +61,54 @@ theorem dist_le_tsum_dist_of_tendsto₀ (h : Summable fun n ↦ dist (f n) (f n.
     (ha : Tendsto f atTop (𝓝 a)) : dist (f 0) a ≤ ∑' n, dist (f n) (f n.succ) := by
   simpa only [zero_add] using dist_le_tsum_dist_of_tendsto h ha 0
 #align dist_le_tsum_dist_of_tendsto₀ dist_le_tsum_dist_of_tendsto₀
+
+section summable
+
+theorem not_summable_iff_tendsto_nat_atTop_of_nonneg {f : ℕ → ℝ} (hf : ∀ n, 0 ≤ f n) :
+    ¬Summable f ↔ Tendsto (fun n : ℕ => ∑ i in Finset.range n, f i) atTop atTop := by
+  lift f to ℕ → ℝ≥0 using hf
+  exact mod_cast NNReal.not_summable_iff_tendsto_nat_atTop
+#align not_summable_iff_tendsto_nat_at_top_of_nonneg not_summable_iff_tendsto_nat_atTop_of_nonneg
+
+theorem summable_iff_not_tendsto_nat_atTop_of_nonneg {f : ℕ → ℝ} (hf : ∀ n, 0 ≤ f n) :
+    Summable f ↔ ¬Tendsto (fun n : ℕ => ∑ i in Finset.range n, f i) atTop atTop := by
+  rw [← not_iff_not, Classical.not_not, not_summable_iff_tendsto_nat_atTop_of_nonneg hf]
+#align summable_iff_not_tendsto_nat_at_top_of_nonneg summable_iff_not_tendsto_nat_atTop_of_nonneg
+
+theorem summable_sigma_of_nonneg {β : α → Type*} {f : (Σ x, β x) → ℝ} (hf : ∀ x, 0 ≤ f x) :
+    Summable f ↔ (∀ x, Summable fun y => f ⟨x, y⟩) ∧ Summable fun x => ∑' y, f ⟨x, y⟩ := by
+  lift f to (Σx, β x) → ℝ≥0 using hf
+  exact mod_cast NNReal.summable_sigma
+#align summable_sigma_of_nonneg summable_sigma_of_nonneg
+
+theorem summable_prod_of_nonneg {f : (α × β) → ℝ} (hf : 0 ≤ f) :
+    Summable f ↔ (∀ x, Summable fun y ↦ f (x, y)) ∧ Summable fun x ↦ ∑' y, f (x, y) :=
+  (Equiv.sigmaEquivProd _ _).summable_iff.symm.trans <| summable_sigma_of_nonneg fun _ ↦ hf _
+
+theorem summable_of_sum_le {ι : Type*} {f : ι → ℝ} {c : ℝ} (hf : 0 ≤ f)
+    (h : ∀ u : Finset ι, ∑ x in u, f x ≤ c) : Summable f :=
+  ⟨⨆ u : Finset ι, ∑ x in u, f x,
+    tendsto_atTop_ciSup (Finset.sum_mono_set_of_nonneg hf) ⟨c, fun _ ⟨u, hu⟩ => hu ▸ h u⟩⟩
+#align summable_of_sum_le summable_of_sum_le
+
+theorem summable_of_sum_range_le {f : ℕ → ℝ} {c : ℝ} (hf : ∀ n, 0 ≤ f n)
+    (h : ∀ n, ∑ i in Finset.range n, f i ≤ c) : Summable f := by
+  refine (summable_iff_not_tendsto_nat_atTop_of_nonneg hf).2 fun H => ?_
+  rcases exists_lt_of_tendsto_atTop H 0 c with ⟨n, -, hn⟩
+  exact lt_irrefl _ (hn.trans_le (h n))
+#align summable_of_sum_range_le summable_of_sum_range_le
+
+theorem Real.tsum_le_of_sum_range_le {f : ℕ → ℝ} {c : ℝ} (hf : ∀ n, 0 ≤ f n)
+    (h : ∀ n, ∑ i in Finset.range n, f i ≤ c) : ∑' n, f n ≤ c :=
+  _root_.tsum_le_of_sum_range_le (summable_of_sum_range_le hf h) h
+#align real.tsum_le_of_sum_range_le Real.tsum_le_of_sum_range_le
+
+/-- If a sequence `f` with non-negative terms is dominated by a sequence `g` with summable
+series and at least one term of `f` is strictly smaller than the corresponding term in `g`,
+then the series of `f` is strictly smaller than the series of `g`. -/
+theorem tsum_lt_tsum_of_nonneg {i : ℕ} {f g : ℕ → ℝ} (h0 : ∀ b : ℕ, 0 ≤ f b)
+    (h : ∀ b : ℕ, f b ≤ g b) (hi : f i < g i) (hg : Summable g) : ∑' n, f n < ∑' n, g n :=
+  tsum_lt_tsum h hi (.of_nonneg_of_le h0 h hg) hg
+#align tsum_lt_tsum_of_nonneg tsum_lt_tsum_of_nonneg
+
+end summable

@@ -46,7 +46,7 @@ variable [PseudoMetricSpace α] {f : ℕ → α} {a : α}
 then the original sequence is a Cauchy sequence. -/
 theorem cauchySeq_of_dist_le_of_summable (d : ℕ → ℝ) (hf : ∀ n, dist (f n) (f n.succ) ≤ d n)
     (hd : Summable d) : CauchySeq f := by
-  -- Porting note (#11215): TODO: with `Topology/Instances/NNReal` we can use this:
+  -- Porting note (#11215): TODO: with `Topology.Instances.NNReal` we can use this:
   -- lift d to ℕ → ℝ≥0 using fun n ↦ dist_nonneg.trans (hf n)
   -- refine cauchySeq_of_edist_le_of_summable d ?_ ?_
   -- · exact_mod_cast hf

Classifies by adding issue number #11215 to porting notes claiming "TODO".

@@ -46,7 +46,7 @@ variable [PseudoMetricSpace α] {f : ℕ → α} {a : α}
 then the original sequence is a Cauchy sequence. -/
 theorem cauchySeq_of_dist_le_of_summable (d : ℕ → ℝ) (hf : ∀ n, dist (f n) (f n.succ) ≤ d n)
     (hd : Summable d) : CauchySeq f := by
-  -- Porting note: todo: with `Topology/Instances/NNReal` we can use this:
+  -- Porting note (#11215): TODO: with `Topology/Instances/NNReal` we can use this:
   -- lift d to ℕ → ℝ≥0 using fun n ↦ dist_nonneg.trans (hf n)
   -- refine cauchySeq_of_edist_le_of_summable d ?_ ?_
   -- · exact_mod_cast hf

Split up Mathlib.Topology.Algebra.InfiniteSum.Basic (1600 lines) into smaller files.

@@ -23,20 +23,20 @@ variable {α : Type*}
 by a summable series of `NNReal`s, then the original sequence is a Cauchy sequence. -/
 theorem cauchySeq_of_edist_le_of_summable [PseudoEMetricSpace α] {f : ℕ → α} (d : ℕ → ℝ≥0)
     (hf : ∀ n, edist (f n) (f n.succ) ≤ d n) (hd : Summable d) : CauchySeq f := by
-  refine EMetric.cauchySeq_iff_NNReal.2 fun ε εpos => ?_
+  refine EMetric.cauchySeq_iff_NNReal.2 fun ε εpos ↦ ?_
   -- Actually we need partial sums of `d` to be a Cauchy sequence
-  replace hd : CauchySeq fun n : ℕ => ∑ x in range n, d x :=
+  replace hd : CauchySeq fun n : ℕ ↦ ∑ x in range n, d x :=
     let ⟨_, H⟩ := hd
     H.tendsto_sum_nat.cauchySeq
   -- Now we take the same `N` as in one of the definitions of a Cauchy sequence
-  refine (Metric.cauchySeq_iff'.1 hd ε (NNReal.coe_pos.2 εpos)).imp fun N hN n hn => ?_
+  refine (Metric.cauchySeq_iff'.1 hd ε (NNReal.coe_pos.2 εpos)).imp fun N hN n hn ↦ ?_
   specialize hN n hn
   -- We simplify the known inequality
   rw [dist_nndist, NNReal.nndist_eq, ← sum_range_add_sum_Ico _ hn, add_tsub_cancel_left,
     NNReal.coe_lt_coe, max_lt_iff] at hN
   rw [edist_comm]
   -- Then use `hf` to simplify the goal to the same form
-  refine lt_of_le_of_lt (edist_le_Ico_sum_of_edist_le hn fun _ _ => hf _) ?_
+  refine lt_of_le_of_lt (edist_le_Ico_sum_of_edist_le hn fun _ _ ↦ hf _) ?_
   exact mod_cast hN.1
 #align cauchy_seq_of_edist_le_of_summable cauchySeq_of_edist_le_of_summable
 
@@ -47,35 +47,35 @@ then the original sequence is a Cauchy sequence. -/
 theorem cauchySeq_of_dist_le_of_summable (d : ℕ → ℝ) (hf : ∀ n, dist (f n) (f n.succ) ≤ d n)
     (hd : Summable d) : CauchySeq f := by
   -- Porting note: todo: with `Topology/Instances/NNReal` we can use this:
-  -- lift d to ℕ → ℝ≥0 using fun n => dist_nonneg.trans (hf n)
+  -- lift d to ℕ → ℝ≥0 using fun n ↦ dist_nonneg.trans (hf n)
   -- refine cauchySeq_of_edist_le_of_summable d ?_ ?_
   -- · exact_mod_cast hf
   -- · exact_mod_cast hd
-  refine' Metric.cauchySeq_iff'.2 fun ε εpos => _
-  replace hd : CauchySeq fun n : ℕ => ∑ x in range n, d x :=
+  refine' Metric.cauchySeq_iff'.2 fun ε εpos ↦ _
+  replace hd : CauchySeq fun n : ℕ ↦ ∑ x in range n, d x :=
     let ⟨_, H⟩ := hd
     H.tendsto_sum_nat.cauchySeq
-  refine' (Metric.cauchySeq_iff'.1 hd ε εpos).imp fun N hN n hn => _
+  refine' (Metric.cauchySeq_iff'.1 hd ε εpos).imp fun N hN n hn ↦ _
   have hsum := hN n hn
   rw [Real.dist_eq, ← sum_Ico_eq_sub _ hn] at hsum
   calc
     dist (f n) (f N) = dist (f N) (f n) := dist_comm _ _
-    _ ≤ ∑ x in Ico N n, d x := dist_le_Ico_sum_of_dist_le hn fun _ _ => hf _
+    _ ≤ ∑ x in Ico N n, d x := dist_le_Ico_sum_of_dist_le hn fun _ _ ↦ hf _
     _ ≤ |∑ x in Ico N n, d x| := le_abs_self _
     _ < ε := hsum
 #align cauchy_seq_of_dist_le_of_summable cauchySeq_of_dist_le_of_summable
 
-theorem cauchySeq_of_summable_dist (h : Summable fun n => dist (f n) (f n.succ)) : CauchySeq f :=
-  cauchySeq_of_dist_le_of_summable _ (fun _ => le_rfl) h
+theorem cauchySeq_of_summable_dist (h : Summable fun n ↦ dist (f n) (f n.succ)) : CauchySeq f :=
+  cauchySeq_of_dist_le_of_summable _ (fun _ ↦ le_rfl) h
 #align cauchy_seq_of_summable_dist cauchySeq_of_summable_dist
 
 theorem dist_le_tsum_of_dist_le_of_tendsto (d : ℕ → ℝ) (hf : ∀ n, dist (f n) (f n.succ) ≤ d n)
     (hd : Summable d) {a : α} (ha : Tendsto f atTop (𝓝 a)) (n : ℕ) :
     dist (f n) a ≤ ∑' m, d (n + m) := by
-  refine' le_of_tendsto (tendsto_const_nhds.dist ha) (eventually_atTop.2 ⟨n, fun m hnm => _⟩)
-  refine' le_trans (dist_le_Ico_sum_of_dist_le hnm fun _ _ => hf _) _
+  refine' le_of_tendsto (tendsto_const_nhds.dist ha) (eventually_atTop.2 ⟨n, fun m hnm ↦ _⟩)
+  refine' le_trans (dist_le_Ico_sum_of_dist_le hnm fun _ _ ↦ hf _) _
   rw [sum_Ico_eq_sum_range]
-  refine' sum_le_tsum (range _) (fun _ _ => le_trans dist_nonneg (hf _)) _
+  refine' sum_le_tsum (range _) (fun _ _ ↦ le_trans dist_nonneg (hf _)) _
   exact hd.comp_injective (add_right_injective n)
 #align dist_le_tsum_of_dist_le_of_tendsto dist_le_tsum_of_dist_le_of_tendsto
 
@@ -84,13 +84,13 @@ theorem dist_le_tsum_of_dist_le_of_tendsto₀ (d : ℕ → ℝ) (hf : ∀ n, dis
   simpa only [zero_add] using dist_le_tsum_of_dist_le_of_tendsto d hf hd ha 0
 #align dist_le_tsum_of_dist_le_of_tendsto₀ dist_le_tsum_of_dist_le_of_tendsto₀
 
-theorem dist_le_tsum_dist_of_tendsto (h : Summable fun n => dist (f n) (f n.succ))
+theorem dist_le_tsum_dist_of_tendsto (h : Summable fun n ↦ dist (f n) (f n.succ))
     (ha : Tendsto f atTop (𝓝 a)) (n) : dist (f n) a ≤ ∑' m, dist (f (n + m)) (f (n + m).succ) :=
-  show dist (f n) a ≤ ∑' m, (fun x => dist (f x) (f x.succ)) (n + m) from
-    dist_le_tsum_of_dist_le_of_tendsto (fun n => dist (f n) (f n.succ)) (fun _ => le_rfl) h ha n
+  show dist (f n) a ≤ ∑' m, (fun x ↦ dist (f x) (f x.succ)) (n + m) from
+    dist_le_tsum_of_dist_le_of_tendsto (fun n ↦ dist (f n) (f n.succ)) (fun _ ↦ le_rfl) h ha n
 #align dist_le_tsum_dist_of_tendsto dist_le_tsum_dist_of_tendsto
 
-theorem dist_le_tsum_dist_of_tendsto₀ (h : Summable fun n => dist (f n) (f n.succ))
+theorem dist_le_tsum_dist_of_tendsto₀ (h : Summable fun n ↦ dist (f n) (f n.succ))
     (ha : Tendsto f atTop (𝓝 a)) : dist (f 0) a ≤ ∑' n, dist (f n) (f n.succ) := by
   simpa only [zero_add] using dist_le_tsum_dist_of_tendsto h ha 0
 #align dist_le_tsum_dist_of_tendsto₀ dist_le_tsum_dist_of_tendsto₀

We still have the exact_mod_cast tactic, used in a few places, which somehow (?) works a little bit harder to prevent the expected type influencing the elaboration of the term. I would like to get to the bottom of this, and it will be easier once the only usages of exact_mod_cast are the ones that don't work using the term elaborator by itself.

Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -37,7 +37,7 @@ theorem cauchySeq_of_edist_le_of_summable [PseudoEMetricSpace α] {f : ℕ → 
   rw [edist_comm]
   -- Then use `hf` to simplify the goal to the same form
   refine lt_of_le_of_lt (edist_le_Ico_sum_of_edist_le hn fun _ _ => hf _) ?_
-  exact_mod_cast hN.1
+  exact mod_cast hN.1
 #align cauchy_seq_of_edist_le_of_summable cauchySeq_of_edist_le_of_summable
 
 variable [PseudoMetricSpace α] {f : ℕ → α} {a : α}

We remove all possible occurences of Type _ and Sort _ in favor of Type* and Sort*.

This has nice performance benefits.

@@ -17,7 +17,7 @@ This file provides lemmas about Cauchy sequences in terms of infinite sums.
 
 open Filter Finset BigOperators NNReal Topology
 
-variable {α : Type _}
+variable {α : Type*}
 
 /-- If the extended distance between consecutive points of a sequence is estimated
 by a summable series of `NNReal`s, then the original sequence is a Cauchy sequence. -/

• depends on: #5966

Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -2,16 +2,13 @@
 Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Sébastien Gouëzel, Yury Kudryashov
-
-! This file was ported from Lean 3 source module topology.algebra.infinite_sum.real
-! leanprover-community/mathlib commit 9a59dcb7a2d06bf55da57b9030169219980660cd
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Algebra.BigOperators.Intervals
 import Mathlib.Topology.Algebra.InfiniteSum.Order
 import Mathlib.Topology.Instances.Real
 
+#align_import topology.algebra.infinite_sum.real from "leanprover-community/mathlib"@"9a59dcb7a2d06bf55da57b9030169219980660cd"
+
 /-!
 # Infinite sum in the reals