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Project-declaredLean 4.32.0 ยท mathlib@81a5d257c8e4

Koszul Sign perm eq append

Wick.koszulSign_perm_eq_append

physicsquantum field theoryrelativitykoszul sign

Project documentation

Gives a factor of - 1 for every fermion-fermion (q is 1) crossing that occurs when sorting a list of based on r. -/ def koszulSign (q : ๐“• โ†’ FieldStatistic) (le : ๐“• โ†’ ๐“• โ†’ Prop) [DecidableRel le] : List ๐“• โ†’ โ„‚ | [] => 1 | a :: l => koszulSignInsert q le a l * koszulSign q le l @[simp] lemma koszulSign_singleton (q : ๐“• โ†’ FieldStatistic) (le : ๐“•...

Exact Lean statement

lemma koszulSign_perm_eq_append [IsTrans ๐“• le] (ฯ† : ๐“•) (ฯ†s ฯ†s' ฯ†s2 : List ๐“•)
    (hp : ฯ†s.Perm ฯ†s') : (h : โˆ€ ฯ†' โˆˆ ฯ†s, le ฯ† ฯ†' โˆง le ฯ†' ฯ†) โ†’
    koszulSign q le (ฯ†s ++ ฯ†s2) = koszulSign q le (ฯ†s' ++ ฯ†s2)

Formal artifact

Lean source

Canonical source
Full Lean sourceLean 4
lemma koszulSign_perm_eq_append [IsTrans ๐“• le] (ฯ† : ๐“•) (ฯ†s ฯ†s' ฯ†s2 : List ๐“•)    (hp : ฯ†s.Perm ฯ†s') : (h : โˆ€ ฯ†' โˆˆ ฯ†s, le ฯ† ฯ†' โˆง le ฯ†' ฯ†) โ†’    koszulSign q le (ฯ†s ++ ฯ†s2) = koszulSign q le (ฯ†s' ++ ฯ†s2) := by  let motive (ฯ†s ฯ†s' : List ๐“•) (hp : ฯ†s.Perm ฯ†s') : Prop :=    (h : โˆ€ ฯ†' โˆˆ ฯ†s, le ฯ† ฯ†' โˆง le ฯ†' ฯ†) โ†’    koszulSign q le (ฯ†s ++ ฯ†s2) = koszulSign q le (ฯ†s' ++ ฯ†s2)  change motive ฯ†s ฯ†s' hp  apply List.Perm.recOn  ยท simp [motive]  ยท intro x l1 l2 h ih hxฯ†    simp_all only [List.mem_cons, or_true, and_self, implies_true, nonempty_prop, forall_const,      forall_eq_or_imp, List.cons_append, motive]    simp only [koszulSign, ih, mul_eq_mul_right_iff]    left    apply koszulSignInsert_eq_perm    exact (List.perm_append_right_iff ฯ†s2).mpr h  ยท intro x y l h    simp_all only [List.mem_cons, forall_eq_or_imp, List.cons_append]    apply Wick.koszulSign_swap_eq_rel_cons    ยท exact IsTrans.trans y ฯ† x h.1.2 h.2.1.1    ยท exact IsTrans.trans x ฯ† y h.2.1.2 h.1.1  ยท intro l1 l2 l3 h1 h2 ih1 ih2 h    simp_all only [and_self, implies_true, nonempty_prop, forall_const, motive]    refine (ih2 ?_)    intro ฯ†' hฯ†    refine h ฯ†' ?_    exact (List.Perm.mem_iff (id (List.Perm.symm h1))).mp hฯ†
Project
Physlib
License
Apache-2.0
Commit
dd43e9e65791
Source
Physlib/QFT/PerturbationTheory/Koszul/KoszulSign.lean:393-419

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Person-level attribution pending.

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