Plain-language statement
prop4d3₁ is the dual form of Proposition 4.1: under hypotheses h₁ and h₂ phrased for strict anti-chains and bottom-anchored alternatives, the best-response value μBstar μ coincides with the global supremum μmax μ TotIntvl. The proof reduces to prop4d1₁ on the order dual, and then translates the result back via the duality lemmas.
Exact Lean statement
lemma prop4d3₁ {ℒ : Type*} [Nontrivial ℒ] [PartialOrder ℒ] [BoundedOrder ℒ]
{S : Type*} [CompleteLattice S]
(μ : {p :ℒ × ℒ // p.1 < p.2} → S)
(h₁ : ∀ x : ℕ → ℒ, (saf : StrictAnti x) →
∃ N : ℕ, μ ⟨(⊥ , x N), lt_of_le_of_lt bot_le <| saf <| Nat.lt_add_one N⟩ ≤
μ ⟨(x (N+1), x N), saf <| Nat.lt_add_one N⟩)
(h₂ : ∀ z : {p :ℒ × ℒ // p.1 < p.2}, (hz : ⊥ < z.val.1) →
μ ⟨(⊥,z.val.2),lt_trans hz z.prop⟩ ≤ μ z ∨ μ ⟨(⊥,z.val.2),lt_trans hz z.prop⟩ ≤
μ ⟨(⊥,z.val.1),hz⟩) :
μBstar μ = μmax μ TotIntvlFormal artifact
Lean source
lemma prop4d3₁ {ℒ : Type*} [Nontrivial ℒ] [PartialOrder ℒ] [BoundedOrder ℒ]{S : Type*} [CompleteLattice S](μ : {p :ℒ × ℒ // p.1 < p.2} → S)(h₁ : ∀ x : ℕ → ℒ, (saf : StrictAnti x) → ∃ N : ℕ, μ ⟨(⊥ , x N), lt_of_le_of_lt bot_le <| saf <| Nat.lt_add_one N⟩ ≤ μ ⟨(x (N+1), x N), saf <| Nat.lt_add_one N⟩)(h₂ : ∀ z : {p :ℒ × ℒ // p.1 < p.2}, (hz : ⊥ < z.val.1) → μ ⟨(⊥,z.val.2),lt_trans hz z.prop⟩ ≤ μ z ∨ μ ⟨(⊥,z.val.2),lt_trans hz z.prop⟩ ≤ μ ⟨(⊥,z.val.1),hz⟩) :μBstar μ = μmax μ TotIntvl := by have := prop4d1₁ ℒᵒᵈ Sᵒᵈ (fun (p : {p : ℒᵒᵈ × ℒᵒᵈ // p.1 < p.2}) ↦ OrderDual.toDual <| μ ⟨(p.val.2, p.val.1), p.prop⟩) (h₁_dual_of_h₁ h₁) (h₂_dual_of_h₂ h₂) rw [← prop4d1_helper] at this rw [← prop4d3_helper] simp only [OrderDual.exists] at this rw [← dualμAstar_eq_μBstar, this] refine congrArg sSup <| Set.ext fun r ↦ ?_ exact ⟨fun ⟨a, ha, ha'⟩ ↦ ⟨a, ha, ha'⟩, fun ⟨a, ha, ha'⟩ ↦ ⟨a, ha, ha'⟩⟩- Project
- Harder-Narasimhan
- License
- Apache-2.0
- Commit
- 20220e90b72c
- Source
- HarderNarasimhan/FirstMoverAdvantage/Impl.lean:310-327
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Related declarations
Associated Primes ker mk Linear Map eq
HarderNarasimhan.CommutativeAlgebra.associatedPrimes_ker_mkLinearMap_eq
Project documentation
Associated primes of the kernel of the localization map. This identifies the associated primes of ker (LocalizedModule.mkLinearMap S M) : Submodule R M with the associated primes of M that do meet the multiplicative set S. Equivalently, these are the associated primes of M after removing those disjoint from S. This lemma is used as the “kernel...
Source project: Harder-Narasimhan
Person-level attribution pending.
Associated Primes ker mk Linear Map subset
HarderNarasimhan.CommutativeAlgebra.associatedPrimes_ker_mkLinearMap_subset
Plain-language statement
One-sided inclusion for associated primes of the kernel of the localization map. If p ∈ associatedPrimes R (ker (mkLinearMap S M)), then p is an associated prime of M and p is not disjoint from the multiplicative set S. This is one direction of associatedPrimes_ker_mkLinearMap_eq.
Source project: Harder-Narasimhan
Person-level attribution pending.
Associated Primes localized Module subset disjoint
HarderNarasimhan.CommutativeAlgebra.associatedPrimes_localizedModule_subset_disjoint
Plain-language statement
Associated primes of a localized module are disjoint from the multiplicative set. More precisely, if p ∈ associatedPrimes R (LocalizedModule S M), then p.carrier ∩ S = ∅. This is a standard fact in commutative algebra: an element of S becomes a unit after localization, so no associated prime of the localized module can contain an element of S.
Source project: Harder-Narasimhan
Person-level attribution pending.