Plain-language statement
prop4d18₁ shows that semistability implies the best-response inequality μBstar μ ≤ μAstar μ in a linearly ordered setting.
Exact Lean statement
lemma prop4d18₁ {ℒ : Type*} [Nontrivial ℒ] [Lattice ℒ] [BoundedOrder ℒ]
{S : Type*} [CompleteLinearOrder S]
(μ : {p :ℒ × ℒ // p.1 < p.2} → S) (hμ : Semistable μ) : μBstar μ ≤ μAstar μFormal artifact
Lean source
lemma prop4d18₁ {ℒ : Type*} [Nontrivial ℒ] [Lattice ℒ] [BoundedOrder ℒ]{S : Type*} [CompleteLinearOrder S](μ : {p :ℒ × ℒ // p.1 < p.2} → S) (hμ : Semistable μ) : μBstar μ ≤ μAstar μ := by rw [semistable_iff] at hμ have : sSup {μA μ ⟨(⊥,x),hx⟩ | (x : ℒ) (hx : ⊥ < x)} ≤ μAstar μ := by apply sSup_le rintro b ⟨hb1,⟨hb2,hb3⟩⟩ have := hb3 ▸ hμ.out.choose_spec.choose_spec.1 hb1 (in_TotIntvl hb1) (Ne.symm <| bot_lt_iff_ne_bot.1 hb2) exact le_of_not_gt this refine le_trans (sSup_le_sSup_of_isCofinalFor ?_) this rintro x ⟨hx1,⟨hx2,hx3⟩⟩ use μA μ ⟨(⊥,hx1),bot_lt_iff_ne_bot.2 <| Ne.symm hx2.2⟩ rw [← hx3] constructor · use hx1, bot_lt_iff_ne_bot.2 <| Ne.symm hx2.2 · apply sInf_le_sInf_of_isCoinitialFor rintro y ⟨hy1,⟨hy2,hy3⟩⟩ use μ ⟨(hy1,hx1) , lt_of_le_of_ne hy2.1.2 hy2.2⟩ constructor · use hy1, hy2 · exact hy3 ▸ (rmk4d10₀ μ ⟨(hy1,hx1), lt_of_le_of_ne hy2.1.2 hy2.2⟩).2- Project
- Harder-Narasimhan
- License
- Apache-2.0
- Commit
- 20220e90b72c
- Source
- HarderNarasimhan/NashEquilibrium/Impl.lean:307-328
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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.