Plain-language statement
prop4d11₁ shows that if the global extremal values on TotIntvl coincide, then the best-response inequality μBstar μ ≤ μAstar μ holds.
Exact Lean statement
lemma prop4d11₁ {ℒ : Type*} [Nontrivial ℒ] [PartialOrder ℒ] [BoundedOrder ℒ]
{S : Type*} [CompleteLattice S]
(μ : {p :ℒ × ℒ // p.1 < p.2} → S) :
μmin μ TotIntvl = μmax μ TotIntvl → μBstar μ ≤ μAstar μFormal artifact
Lean source
lemma prop4d11₁ {ℒ : Type*} [Nontrivial ℒ] [PartialOrder ℒ] [BoundedOrder ℒ]{S : Type*} [CompleteLattice S](μ : {p :ℒ × ℒ // p.1 < p.2} → S) :μmin μ TotIntvl = μmax μ TotIntvl → μBstar μ ≤ μAstar μ := by have h₁ : μBstar μ ≤ μmax μ TotIntvl := by unfold μBstar μB μmax TotIntvl apply sSup_le rintro b ⟨hb1,⟨hb2,hb3⟩⟩ exact hb3 ▸ le_trans (rmk4d10₀ μ ⟨(⊥,hb1), bot_lt_iff_ne_bot.2 <| Ne.symm hb2.2⟩).1 <| le_sSup ⟨hb1, ⟨in_TotIntvl hb1, hb2.2⟩, rfl⟩ have h₂ : μmin μ TotIntvl ≤ μAstar μ := by unfold μAstar μA μmin TotIntvl apply le_sInf rintro b ⟨hb1,⟨hb2,hb3⟩⟩ exact hb3 ▸ le_trans (sInf_le ⟨hb1, ⟨in_TotIntvl hb1, hb2.2⟩, rfl⟩) (rmk4d10₀ μ ⟨(hb1,⊤), lt_top_iff_ne_top.2 <| hb2.2⟩).2 exact fun h ↦ le_trans h₁ (h ▸ h₂)- Project
- Harder-Narasimhan
- License
- Apache-2.0
- Commit
- 20220e90b72c
- Source
- HarderNarasimhan/NashEquilibrium/Impl.lean:140-157
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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.