Plain-language statement
prop4d16₂ is the main bridge: under SlopeLike μ and both chain conditions, Nash equilibrium is equivalent to the equality μmin μ TotIntvl = μmax μ TotIntvl. The proof packages the slope-like axiom into weak slope-like data on restrictions, and then combines prop4d11₁ and prop4d11₂ with the earlier characterisations.
Exact Lean statement
lemma prop4d16₂ {ℒ : Type*} [Nontrivial ℒ] [PartialOrder ℒ] [BoundedOrder ℒ]
{S : Type*} [CompleteLattice S]
(μ : {p :ℒ × ℒ // p.1 < p.2} → S) (hμ : SlopeLike μ)
(h₁ : WeakAscendingChainCondition μ) (h₂ : StrongDescendingChainCondition μ) :
μmin μ TotIntvl = μmax μ TotIntvl ↔ NashEquilibrium μFormal artifact
Lean source
lemma prop4d16₂ {ℒ : Type*} [Nontrivial ℒ] [PartialOrder ℒ] [BoundedOrder ℒ]{S : Type*} [CompleteLattice S](μ : {p :ℒ × ℒ // p.1 < p.2} → S) (hμ : SlopeLike μ)(h₁ : WeakAscendingChainCondition μ) (h₂ : StrongDescendingChainCondition μ) :μmin μ TotIntvl = μmax μ TotIntvl ↔ NashEquilibrium μ := by have : ∀ (z : { p : ℒ × ℒ // p.1 < p.2 }) (hz : z.val.2 < ⊤), μ z ≤ μ ⟨(z.val.1, ⊤), lt_trans z.prop hz⟩ ∨ μ ⟨(z.val.2, ⊤), hz⟩ ≤ μ ⟨(z.val.1, ⊤), lt_trans z.prop hz⟩ := by intro z hz rcases (hμ.slopelike z.val.1 z.val.2 ⊤ ⟨z.prop, hz⟩).1 with this | this · exact Or.inl this · exact Or.inr <| le_of_lt this refine ⟨fun h ↦ {nash_eq := eq_of_le_of_ge (impl.prop4d1₂ ℒ S μ h₁.wacc this) <| prop4d11₁ μ h}, fun h ↦ prop4d11₂ μ h₁ {wsl₁ := this} h₂ {wsl₂:=(fun z hz ↦ ?_)} h.nash_eq.symm.le⟩ rcases (hμ.slopelike ⊥ z.val.1 z.val.2 ⟨hz,z.prop⟩).2.2.1 with this | this · exact Or.inr <| le_of_lt this · exact Or.inl <| this- Project
- Harder-Narasimhan
- License
- Apache-2.0
- Commit
- 20220e90b72c
- Source
- HarderNarasimhan/NashEquilibrium/Impl.lean:284-300
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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.