Plain-language statement
prop4d12 derives the equality μmin μ TotIntvl = μmax μ TotIntvl from the stronger equality μmax μ TotIntvl = μ TotIntvl, provided a pointwise dichotomy that rules out “intermediate” points simultaneously satisfying both comparisons.
Exact Lean statement
lemma prop4d12 {ℒ : Type*} [Nontrivial ℒ] [PartialOrder ℒ] [BoundedOrder ℒ]
{S : Type*} [CompleteLattice S]
(μ : {p :ℒ × ℒ // p.1 < p.2} → S)
(h : ∀ x : ℒ, (hx : x ≠ ⊥ ∧ x ≠ ⊤) → ¬ μ ⟨(⊥,x),bot_lt_iff_ne_bot.2 hx.1⟩ ≤ μ TotIntvl ∨
μ TotIntvl ≤ μ ⟨(x,⊤),lt_top_iff_ne_top.2 hx.2⟩) :
μmax μ TotIntvl = μ TotIntvl → μmin μ TotIntvl = μmax μ TotIntvlFormal artifact
Lean source
lemma prop4d12 {ℒ : Type*} [Nontrivial ℒ] [PartialOrder ℒ] [BoundedOrder ℒ]{S : Type*} [CompleteLattice S](μ : {p :ℒ × ℒ // p.1 < p.2} → S)(h : ∀ x : ℒ, (hx : x ≠ ⊥ ∧ x ≠ ⊤) → ¬ μ ⟨(⊥,x),bot_lt_iff_ne_bot.2 hx.1⟩ ≤ μ TotIntvl ∨ μ TotIntvl ≤ μ ⟨(x,⊤),lt_top_iff_ne_top.2 hx.2⟩) :μmax μ TotIntvl = μ TotIntvl → μmin μ TotIntvl = μmax μ TotIntvl := by refine fun h' ↦ h' ▸ eq_of_le_of_ge (rmk4d10₀ μ TotIntvl).1 ?_ · apply le_sInf rintro b ⟨hb1,⟨hb2,hb3⟩⟩ rw [← hb3] by_cases hbot : hb1 = ⊥ · simp only [hbot, le_refl] refine Or.resolve_left (h hb1 <| ⟨hbot,hb2.2⟩) ?_ rw [not_not] refine h' ▸ (le_sSup ?_) use hb1, ⟨in_TotIntvl hb1, Ne.symm hbot⟩- Project
- Harder-Narasimhan
- License
- Apache-2.0
- Commit
- 20220e90b72c
- Source
- HarderNarasimhan/NashEquilibrium/Impl.lean:180-195
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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.