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Copyright 2025 The Formal Conjectures Authors.
Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
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-/
module
public import Mathlib.Algebra.BigOperators.Group.Finset.Basic
public import Mathlib.Data.Set.Finite.Lattice
public import Mathlib.Order.Filter.AtTopBot.Defs@[expose] public sectionvariable {M : Type*} [AddCommMonoid M]open scoped List
The set of subset sums of a set A ⊆ M.
def subsetSums (A : Set M) : Set M :=
{n | ∃ B : Finset M, ↑B ⊆ A ∧ n = ∑ i ∈ B, i}
If A ⊆ B, then subsetSums A ⊆ subsetSums B.
@[gcongr]
theorem subsetSums_mono {A B : Set M} (h : A ⊆ B) : subsetSums A ⊆ subsetSums B :=
fun _ ⟨C, hC⟩ => ⟨C, hC.1.trans h, hC.2⟩
The set of subset sums of a sequence ℕ → M, where repetition is allowed.
def subseqSums' (A : ℕ → M) : Set M :=
{n | ∃ B : Finset ℕ, n = ∑ i ∈ B, A i}variable [Preorder M]
A set A ⊆ M is complete if every sufficiently large element of M is a subset sum of A.
def IsAddComplete (A : Set M) : Prop :=
∀ᶠ k in Filter.atTop, k ∈ subsetSums A
If A ⊆ B and A is complete, then B is also complete.
@[gcongr]
theorem IsAddComplete.mono {A B : Set M} (h : A ⊆ B) (ha : IsAddComplete A) : IsAddComplete B := M:Type u_1inst✝¹:AddCommMonoid Minst✝:Preorder MA:Set MB:Set Mh:A ⊆ Bha:IsAddComplete A⊢ IsAddComplete B
filter_upwards [ha] with x M:Type u_1inst✝¹:AddCommMonoid Minst✝:Preorder MA:Set MB:Set Mh:A ⊆ Bha:IsAddComplete Ax:Mhx:x ∈ subsetSums A⊢ x ∈ subsetSums B
All goals completed! 🐙
A set A ⊆ M is complete if every sufficiently large element of M is a subset sum of A.
def IsAddStronglyComplete (A : Set M) : Prop :=
∀ ⦃B : Set M⦄, B.Finite → IsAddComplete (A \ B)A strongly complete set is complete.
theorem IsAddStronglyComplete.isAddComplete {A : Set M} (hA : IsAddStronglyComplete A) :
IsAddComplete A := M:Type u_1inst✝¹:AddCommMonoid Minst✝:Preorder MA:Set MhA:IsAddStronglyComplete A⊢ IsAddComplete A All goals completed! 🐙
If A ⊆ B and A is strongly complete, then B is also strongly complete.
theorem IsAddStronglyComplete.mono {A B : Set M} (h : A ⊆ B) (ha : IsAddStronglyComplete A) :
IsAddStronglyComplete B := fun C hC => (ha hC).mono (M:Type u_1inst✝¹:AddCommMonoid Minst✝:Preorder MA:Set MB:Set Mh:A ⊆ Bha:IsAddStronglyComplete AC:Set MhC:C.Finite⊢ A \ C ⊆ B \ C All goals completed! 🐙)
A sequence A is strongly complete if fun m => A (n + m) is still complete for all n.
def IsAddStronglyCompleteNatSeq (A : ℕ → M) : Prop :=
∀ n, IsAddComplete (Set.range (fun m => A (n + m)))A strongly complete sequence is complete.
theorem IsAddStronglyCompleteNatSeq.isAddComplete {A : ℕ → M}
(hA : IsAddStronglyCompleteNatSeq A) :
IsAddComplete (Set.range A) := M:Type u_1inst✝¹:AddCommMonoid Minst✝:Preorder MA:ℕ → MhA:IsAddStronglyCompleteNatSeq A⊢ IsAddComplete (Set.range A) All goals completed! 🐙open scoped Classical in
If the range of a sequence A is strongly complete, then A is strongly complete.
theorem IsAddStronglyCompleteNatSeq.of_isAddStronglyComplete {A : ℕ → M}
(h : IsAddStronglyComplete (.range A)) : IsAddStronglyCompleteNatSeq A :=
fun n => (h (Finset.finite_toSet _)).mono (A := .range A \ ((Finset.range n).image A))
(fun _ ⟨⟨y, hy⟩, q⟩ => ⟨y - n, M:Type u_1inst✝¹:AddCommMonoid Minst✝:Preorder MA:ℕ → Mh:IsAddStronglyComplete (Set.range A)n:ℕx✝¹:Mx✝:x✝¹ ∈ Set.range A \ ↑(Finset.image A (Finset.range n))y:ℕhy:A y = x✝¹q:x✝¹ ∉ ↑(Finset.image A (Finset.range n))⊢ (fun m ↦ A (n + m)) (y - n) = x✝¹ All goals completed! 🐙⟩)
If A is strongly complete and the preimage of each element is finite, then the range of A
is strongly complete.
M:Type u_1inst✝¹:AddCommMonoid Minst✝:Preorder MA:ℕ → Mh:IsAddStronglyCompleteNatSeq AhA:∀ (m : M), (A ⁻¹' {m}).FiniteB:Set MhB:B.Finiten:ℕhn:A ⁻¹' B ⊆ Set.Iio nx:Mx✝:x ∈ Set.range fun m ↦ A (n + 1 + m)y:ℕhy:(fun m ↦ A (n + 1 + m)) y = xhx:x ∈ Bthis:n + 1 + y ∈ Set.Iio n⊢ False
All goals completed! 🐙
A sequence A is complete if every sufficiently large element of M is a sum of
(not necessarily distinct) terms of A.
def IsAddCompleteNatSeq' (A : ℕ → M) : Prop :=
∀ᶠ k in Filter.atTop, k ∈ subseqSums' A