[Merged by Bors] - feat: port Topology.Algebra.InfiniteSum.Order

Mathlib.lean

@@ -1243,6 +1243,7 @@ import Mathlib.Topology.Algebra.Group.Basic
 import Mathlib.Topology.Algebra.Group.Compact
 import Mathlib.Topology.Algebra.GroupWithZero
 import Mathlib.Topology.Algebra.InfiniteSum.Basic
+import Mathlib.Topology.Algebra.InfiniteSum.Order
 import Mathlib.Topology.Algebra.InfiniteSum.Ring
 import Mathlib.Topology.Algebra.Monoid
 import Mathlib.Topology.Algebra.MulAction

Mathlib/Topology/Algebra/InfiniteSum/Basic.lean

@@ -136,7 +136,7 @@ theorem hasSum_iff_hasSum {g : γ → α}
 #align has_sum_iff_has_sum hasSum_iff_hasSum
 
 theorem Function.Injective.hasSum_iff {g : γ → β} (hg : Injective g)
-    (hf : ∀ (x) (_ : x ∉ Set.range g), f x = 0) : HasSum (f ∘ g) a ↔ HasSum f a := by
+    (hf : ∀ x, x ∉ Set.range g → f x = 0) : HasSum (f ∘ g) a ↔ HasSum f a := by
   simp only [HasSum, Tendsto, comp_apply, hg.map_atTop_finset_sum_eq hf]
 #align function.injective.has_sum_iff Function.Injective.hasSum_iff
 
@@ -145,6 +145,15 @@ theorem Function.Injective.summable_iff {g : γ → β} (hg : Injective g)
   exists_congr fun _ => hg.hasSum_iff hf
 #align function.injective.summable_iff Function.Injective.summable_iff
 
+@[simp] theorem hasSum_extend_zero {g : β → γ} (hg : Injective g) :
+    HasSum (extend g f 0) a ↔ HasSum f a := by
+  rw [← hg.hasSum_iff, extend_comp hg]
+  exact extend_apply' _ _
+
+@[simp] theorem summable_extend_zero {g : β → γ} (hg : Injective g) :
+    Summable (extend g f 0) ↔ Summable f :=
+  exists_congr fun _ => hasSum_extend_zero hg
+
 theorem hasSum_subtype_iff_of_support_subset {s : Set β} (hf : support f ⊆ s) :
     HasSum (f ∘ (↑) : s → α) a ↔ HasSum f a :=
   Subtype.coe_injective.hasSum_iff <| by simpa using support_subset_iff'.1 hf

Mathlib/Topology/Algebra/InfiniteSum/Order.lean

@@ -0,0 +1,290 @@
+/-
+Copyright (c) 2017 Johannes Hölzl. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Johannes Hölzl
+
+! This file was ported from Lean 3 source module topology.algebra.infinite_sum.order
+! leanprover-community/mathlib commit 32253a1a1071173b33dc7d6a218cf722c6feb514
+! Please do not edit these lines, except to modify the commit id
+! if you have ported upstream changes.
+-/
+import Mathlib.Algebra.Order.Archimedean
+import Mathlib.Topology.Algebra.InfiniteSum.Basic
+import Mathlib.Topology.Algebra.Order.Field
+import Mathlib.Topology.Algebra.Order.MonotoneConvergence
+
+/-!
+# Infinite sum in an order
+
+This file provides lemmas about the interaction of infinite sums and order operations.
+-/
+
+
+open Finset Filter Function BigOperators
+open scoped Classical
+
+variable {ι κ α : Type _}
+
+section Preorder
+
+variable [Preorder α] [AddCommMonoid α] [TopologicalSpace α] [OrderClosedTopology α] [T2Space α]
+  {f : ℕ → α} {c : α}
+
+theorem tsum_le_of_sum_range_le (hf : Summable f) (h : ∀ n, (∑ i in range n, f i) ≤ c) :
+    (∑' n, f n) ≤ c :=
+  let ⟨_l, hl⟩ := hf
+  hl.tsum_eq.symm ▸ le_of_tendsto' hl.tendsto_sum_nat h
+#align tsum_le_of_sum_range_le tsum_le_of_sum_range_le
+
+end Preorder
+
+section OrderedAddCommMonoid
+
+variable [OrderedAddCommMonoid α] [TopologicalSpace α] [OrderClosedTopology α] {f g : ι → α}
+  {a a₁ a₂ : α}
+
+theorem hasSum_le (h : ∀ i, f i ≤ g i) (hf : HasSum f a₁) (hg : HasSum g a₂) : a₁ ≤ a₂ :=
+  le_of_tendsto_of_tendsto' hf hg fun _ => sum_le_sum fun i _ => h i
+#align has_sum_le hasSum_le
+
+@[mono]
+theorem hasSum_mono (hf : HasSum f a₁) (hg : HasSum g a₂) (h : f ≤ g) : a₁ ≤ a₂ :=
+  hasSum_le h hf hg
+#align has_sum_mono hasSum_mono
+
+theorem hasSum_le_of_sum_le (hf : HasSum f a) (h : ∀ s, (∑ i in s, f i) ≤ a₂) : a ≤ a₂ :=
+  le_of_tendsto' hf h
+#align has_sum_le_of_sum_le hasSum_le_of_sum_le
+
+theorem le_hasSum_of_le_sum (hf : HasSum f a) (h : ∀ s, a₂ ≤ ∑ i in s, f i) : a₂ ≤ a :=
+  ge_of_tendsto' hf h
+#align le_has_sum_of_le_sum le_hasSum_of_le_sum
+
+theorem hasSum_le_inj {g : κ → α} (e : ι → κ) (he : Injective e)
+    (hs : ∀ c, c ∉ Set.range e → 0 ≤ g c) (h : ∀ i, f i ≤ g (e i)) (hf : HasSum f a₁)
+    (hg : HasSum g a₂) : a₁ ≤ a₂ := by
+  rw [← hasSum_extend_zero he] at hf
+  refine hasSum_le (fun c => ?_) hf hg
+  obtain ⟨i, rfl⟩ | h := em (c ∈ Set.range e)
+  · rw [he.extend_apply]
+    exact h _
+  · rw [extend_apply' _ _ _ h]
+    exact hs _ h
+#align has_sum_le_inj hasSum_le_inj
+
+theorem tsum_le_tsum_of_inj {g : κ → α} (e : ι → κ) (he : Injective e)
+    (hs : ∀ c, c ∉ Set.range e → 0 ≤ g c) (h : ∀ i, f i ≤ g (e i)) (hf : Summable f)
+    (hg : Summable g) : tsum f ≤ tsum g :=
+  hasSum_le_inj _ he hs h hf.hasSum hg.hasSum
+#align tsum_le_tsum_of_inj tsum_le_tsum_of_inj
+
+theorem sum_le_hasSum (s : Finset ι) (hs : ∀ i, i ∉ s → 0 ≤ f i) (hf : HasSum f a) :
+    (∑ i in s, f i) ≤ a :=
+  ge_of_tendsto hf (eventually_atTop.2
+    ⟨s, fun _t hst => sum_le_sum_of_subset_of_nonneg hst fun i _ hbs => hs i hbs⟩)
+#align sum_le_has_sum sum_le_hasSum
+
+theorem isLUB_hasSum (h : ∀ i, 0 ≤ f i) (hf : HasSum f a) :
+    IsLUB (Set.range fun s => ∑ i in s, f i) a :=
+  isLUB_of_tendsto_atTop (Finset.sum_mono_set_of_nonneg h) hf
+#align is_lub_has_sum isLUB_hasSum
+
+theorem le_hasSum (hf : HasSum f a) (i : ι) (hb : ∀ j, j ≠ i → 0 ≤ f j) : f i ≤ a :=
+  calc
+    f i = ∑ i in {i}, f i := Finset.sum_singleton.symm
+    _ ≤ a := sum_le_hasSum _ (by simpa) hf
+
+#align le_has_sum le_hasSum
+
+theorem sum_le_tsum {f : ι → α} (s : Finset ι) (hs : ∀ i, i ∉ s → 0 ≤ f i) (hf : Summable f) :
+    (∑ i in s, f i) ≤ ∑' i, f i :=
+  sum_le_hasSum s hs hf.hasSum
+#align sum_le_tsum sum_le_tsum
+
+theorem le_tsum (hf : Summable f) (i : ι) (hb : ∀ j, j ≠ i → 0 ≤ f j) : f i ≤ ∑' i, f i :=
+  le_hasSum hf.hasSum i hb
+#align le_tsum le_tsum
+
+theorem tsum_le_tsum (h : ∀ i, f i ≤ g i) (hf : Summable f) (hg : Summable g) :
+    (∑' i, f i) ≤ ∑' i, g i :=
+  hasSum_le h hf.hasSum hg.hasSum
+#align tsum_le_tsum tsum_le_tsum
+
+@[mono]
+theorem tsum_mono (hf : Summable f) (hg : Summable g) (h : f ≤ g) : (∑' n, f n) ≤ ∑' n, g n :=
+  tsum_le_tsum h hf hg
+#align tsum_mono tsum_mono
+
+theorem tsum_le_of_sum_le (hf : Summable f) (h : ∀ s, (∑ i in s, f i) ≤ a₂) : (∑' i, f i) ≤ a₂ :=
+  hasSum_le_of_sum_le hf.hasSum h
+#align tsum_le_of_sum_le tsum_le_of_sum_le
+
+theorem tsum_le_of_sum_le' (ha₂ : 0 ≤ a₂) (h : ∀ s, (∑ i in s, f i) ≤ a₂) : (∑' i, f i) ≤ a₂ := by
+  by_cases hf : Summable f
+  · exact tsum_le_of_sum_le hf h
+  · rw [tsum_eq_zero_of_not_summable hf]
+    exact ha₂
+#align tsum_le_of_sum_le' tsum_le_of_sum_le'
+
+theorem HasSum.nonneg (h : ∀ i, 0 ≤ g i) (ha : HasSum g a) : 0 ≤ a :=
+  hasSum_le h hasSum_zero ha
+#align has_sum.nonneg HasSum.nonneg
+
+theorem HasSum.nonpos (h : ∀ i, g i ≤ 0) (ha : HasSum g a) : a ≤ 0 :=
+  hasSum_le h ha hasSum_zero
+#align has_sum.nonpos HasSum.nonpos
+
+theorem tsum_nonneg (h : ∀ i, 0 ≤ g i) : 0 ≤ ∑' i, g i := by
+  by_cases hg : Summable g
+  · exact hg.hasSum.nonneg h
+  · rw [tsum_eq_zero_of_not_summable hg]
+#align tsum_nonneg tsum_nonneg
+
+theorem tsum_nonpos (h : ∀ i, f i ≤ 0) : (∑' i, f i) ≤ 0 := by
+  by_cases hf : Summable f
+  · exact hf.hasSum.nonpos h
+  · rw [tsum_eq_zero_of_not_summable hf]
+#align tsum_nonpos tsum_nonpos
+
+-- porting note: generalized from `OrderedAddCommGroup` to `OrderedAddCommMonoid`
+theorem hasSum_zero_iff_of_nonneg (hf : ∀ i, 0 ≤ f i) : HasSum f 0 ↔ f = 0 := by
+  refine' ⟨fun hf' => _, _⟩
+  · ext i
+    exact (hf i).antisymm' (le_hasSum hf' _ fun j _ => hf j)
+  · rintro rfl
+    exact hasSum_zero
+#align has_sum_zero_iff_of_nonneg hasSum_zero_iff_of_nonneg
+
+end OrderedAddCommMonoid
+
+section OrderedAddCommGroup
+
+variable [OrderedAddCommGroup α] [TopologicalSpace α] [TopologicalAddGroup α]
+  [OrderClosedTopology α] {f g : ι → α} {a₁ a₂ : α} {i : ι}
+
+theorem hasSum_lt (h : f ≤ g) (hi : f i < g i) (hf : HasSum f a₁) (hg : HasSum g a₂) : a₁ < a₂ := by
+  have : update f i 0 ≤ update g i 0 := update_le_update_iff.mpr ⟨rfl.le, fun i _ => h i⟩
+  have : 0 - f i + a₁ ≤ 0 - g i + a₂ := hasSum_le this (hf.update i 0) (hg.update i 0)
+  simpa only [zero_sub, add_neg_cancel_left] using add_lt_add_of_lt_of_le hi this
+#align has_sum_lt hasSum_lt
+
+@[mono]
+theorem hasSum_strict_mono (hf : HasSum f a₁) (hg : HasSum g a₂) (h : f < g) : a₁ < a₂ :=
+  let ⟨hle, _i, hi⟩ := Pi.lt_def.mp h
+  hasSum_lt hle hi hf hg
+#align has_sum_strict_mono hasSum_strict_mono
+
+theorem tsum_lt_tsum (h : f ≤ g) (hi : f i < g i) (hf : Summable f) (hg : Summable g) :
+    (∑' n, f n) < ∑' n, g n :=
+  hasSum_lt h hi hf.hasSum hg.hasSum
+#align tsum_lt_tsum tsum_lt_tsum
+
+@[mono]
+theorem tsum_strict_mono (hf : Summable f) (hg : Summable g) (h : f < g) :
+    (∑' n, f n) < ∑' n, g n :=
+  let ⟨hle, _i, hi⟩ := Pi.lt_def.mp h
+  tsum_lt_tsum hle hi hf hg
+#align tsum_strict_mono tsum_strict_mono
+
+theorem tsum_pos (hsum : Summable g) (hg : ∀ i, 0 ≤ g i) (i : ι) (hi : 0 < g i) :
+    0 < ∑' i, g i := by
+  rw [← tsum_zero]
+  exact tsum_lt_tsum hg hi summable_zero hsum
+#align tsum_pos tsum_pos
+
+end OrderedAddCommGroup
+
+section CanonicallyOrderedAddMonoid
+
+variable [CanonicallyOrderedAddMonoid α] [TopologicalSpace α] [OrderClosedTopology α] {f : ι → α}
+  {a : α}
+
+theorem le_has_sum' (hf : HasSum f a) (i : ι) : f i ≤ a :=
+  le_hasSum hf i fun _ _ => zero_le _
+#align le_has_sum' le_has_sum'
+
+theorem le_tsum' (hf : Summable f) (i : ι) : f i ≤ ∑' i, f i :=
+  le_tsum hf i fun _ _ => zero_le _
+#align le_tsum' le_tsum'
+
+theorem hasSum_zero_iff : HasSum f 0 ↔ ∀ x, f x = 0 :=
+  (hasSum_zero_iff_of_nonneg fun _ => zero_le _).trans funext_iff
+#align has_sum_zero_iff hasSum_zero_iff
+
+theorem tsum_eq_zero_iff (hf : Summable f) : (∑' i, f i) = 0 ↔ ∀ x, f x = 0 := by
+  rw [← hasSum_zero_iff, hf.hasSum_iff]
+#align tsum_eq_zero_iff tsum_eq_zero_iff
+
+theorem tsum_ne_zero_iff (hf : Summable f) : (∑' i, f i) ≠ 0 ↔ ∃ x, f x ≠ 0 := by
+  rw [Ne.def, tsum_eq_zero_iff hf, not_forall]
+#align tsum_ne_zero_iff tsum_ne_zero_iff
+
+theorem isLUB_has_sum' (hf : HasSum f a) : IsLUB (Set.range fun s => ∑ i in s, f i) a :=
+  isLUB_of_tendsto_atTop (Finset.sum_mono_set f) hf
+#align is_lub_has_sum' isLUB_has_sum'
+
+end CanonicallyOrderedAddMonoid
+
+section LinearOrder
+
+/-!
+For infinite sums taking values in a linearly ordered monoid, the existence of a least upper
+bound for the finite sums is a criterion for summability.
+
+This criterion is useful when applied in a linearly ordered monoid which is also a complete or
+conditionally complete linear order, such as `ℝ`, `ℝ≥0`, `ℝ≥0∞`, because it is then easy to check
+the existence of a least upper bound.
+-/
+
+theorem hasSum_of_isLUB_of_nonneg [LinearOrderedAddCommMonoid α] [TopologicalSpace α]
+    [OrderTopology α] {f : ι → α} (i : α) (h : ∀ i, 0 ≤ f i)
+    (hf : IsLUB (Set.range fun s => ∑ i in s, f i) i) : HasSum f i :=
+  tendsto_atTop_isLUB (Finset.sum_mono_set_of_nonneg h) hf
+#align has_sum_of_is_lub_of_nonneg hasSum_of_isLUB_of_nonneg
+
+theorem hasSum_of_isLUB [CanonicallyLinearOrderedAddMonoid α] [TopologicalSpace α] [OrderTopology α]
+    {f : ι → α} (b : α) (hf : IsLUB (Set.range fun s => ∑ i in s, f i) b) : HasSum f b :=
+  tendsto_atTop_isLUB (Finset.sum_mono_set f) hf
+#align has_sum_of_is_lub hasSum_of_isLUB
+
+theorem summable_abs_iff [LinearOrderedAddCommGroup α] [UniformSpace α] [UniformAddGroup α]
+    [CompleteSpace α] {f : ι → α} : (Summable fun x => |f x|) ↔ Summable f :=
+  let s := { x | 0 ≤ f x }
+  have h1 : ∀ x : s, |f x| = f x := fun x => abs_of_nonneg x.2
+  have h2 : ∀ x : ↑(sᶜ), |f x| = -f x := fun x => abs_of_neg (not_le.1 x.2)
+  calc (Summable fun x => |f x|) ↔
+      (Summable fun x : s => |f x|) ∧ Summable fun x : ↑(sᶜ) => |f x| :=
+        summable_subtype_and_compl.symm
+  _ ↔ (Summable fun x : s => f x) ∧ Summable fun x : ↑(sᶜ) => -f x := by simp only [h1, h2]
+  _ ↔ Summable f := by simp only [summable_neg_iff, summable_subtype_and_compl]
+#align summable_abs_iff summable_abs_iff
+
+alias summable_abs_iff ↔ Summable.of_abs Summable.abs
+#align summable.of_abs Summable.of_abs
+#align summable.abs Summable.abs
+
+theorem Finite.of_summable_const [LinearOrderedAddCommGroup α] [TopologicalSpace α] [Archimedean α]
+    [OrderClosedTopology α] {b : α} (hb : 0 < b) (hf : Summable fun _ : ι => b) :
+    Finite ι := by
+  have H : ∀ s : Finset ι, s.card • b ≤ ∑' _i : ι, b := fun s => by
+    simpa using sum_le_hasSum s (fun a _ => hb.le) hf.hasSum
+  obtain ⟨n, hn⟩ := Archimedean.arch (∑' _i : ι, b) hb
+  have : ∀ s : Finset ι, s.card ≤ n := fun s => by
+    simpa [nsmul_le_nsmul_iff hb] using (H s).trans hn
+  have : Fintype ι := fintypeOfFinsetCardLe n this
+  infer_instance
+
+theorem Set.Finite.of_summable_const [LinearOrderedAddCommGroup α] [TopologicalSpace α]
+    [Archimedean α] [OrderClosedTopology α] {b : α} (hb : 0 < b) (hf : Summable fun _ : ι => b) :
+    (Set.univ : Set ι).Finite :=
+  finite_univ_iff.2 <| .of_summable_const hb hf
+#align finite_of_summable_const Set.Finite.of_summable_const
+
+end LinearOrder
+
+theorem Summable.tendsto_atTop_of_pos [LinearOrderedField α] [TopologicalSpace α] [OrderTopology α]
+    {f : ℕ → α} (hf : Summable f⁻¹) (hf' : ∀ n, 0 < f n) : Tendsto f atTop atTop :=
+  inv_inv f ▸ Filter.Tendsto.inv_tendsto_zero <|
+    tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within _ hf.tendsto_atTop_zero <|
+      eventually_of_forall fun _ => inv_pos.2 (hf' _)
+#align summable.tendsto_top_of_pos Summable.tendsto_atTop_of_pos