refactor: Lint
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jstoobysmith
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51158267d3
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be64dbb8bb
4 changed files with 121 additions and 89 deletions
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@ -25,8 +25,8 @@ open PreFeynmanRule
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inductive WickContract : {n : ℕ} → {c : Fin n → 𝓔} → (str : WickString c final) →
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{k : ℕ} → (b1 : Fin k → Fin n) → (b2 : Fin k → Fin n) → Type where
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| string {n : ℕ} {c : Fin n → 𝓔} {str : WickString c final} : WickContract str Fin.elim0 Fin.elim0
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| contr {n : ℕ} {c : Fin n → 𝓔} {str : WickString c final} {k : ℕ}
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{b1 : Fin k → Fin n} {b2 : Fin k → Fin n}: (i : Fin n) →
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| contr {n : ℕ} {c : Fin n → 𝓔} {str : WickString c final} {k : ℕ}
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{b1 : Fin k → Fin n} {b2 : Fin k → Fin n}: (i : Fin n) →
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(j : Fin n) → (h : c j = ξ (c i)) →
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(hilej : i < j) → (hb1 : ∀ r, b1 r < i) → (hb2i : ∀ r, b2 r ≠ i) → (hb2j : ∀ r, b2 r ≠ j) →
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(w : WickContract str b1 b2) →
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@ -35,7 +35,7 @@ inductive WickContract : {n : ℕ} → {c : Fin n → 𝓔} → (str : WickStrin
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namespace WickContract
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/-- The number of nodes of a Wick contraction. -/
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def size {n k : ℕ} {c : Fin n → 𝓔} {str : WickString c final} {b1 b2 : Fin k → Fin n} :
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def size {n k : ℕ} {c : Fin n → 𝓔} {str : WickString c final} {b1 b2 : Fin k → Fin n} :
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WickContract str b1 b2 → ℕ := fun
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| string => 0
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| contr _ _ _ _ _ _ _ w => w.size + 1
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@ -47,6 +47,7 @@ lemma size_eq_k {n k : ℕ} {c : Fin n → 𝓔} {str : WickString c final} {b1
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| contr _ _ _ _ _ _ _ w => by
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simpa [size] using w.size_eq_k
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/-- The map giving the vertices on the left-hand-side of a contraction. -/
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def boundFst {n k : ℕ} {c : Fin n → 𝓔} {str : WickString c final} {b1 b2 : Fin k → Fin n} :
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WickContract str b1 b2 → Fin k → Fin n := fun _ => b1
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@ -74,10 +75,10 @@ lemma boundFst_contr_last {n k : ℕ} {c : Fin n → 𝓔} {str : WickString c f
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(contr i j h hilej hb1 hb2i hb2j w).boundFst (Fin.last k) = i := by
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simp only [boundFst, Fin.snoc_last]
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lemma boundFst_strictMono {n k : ℕ} {c : Fin n → 𝓔} {str : WickString c final} {b1 b2 : Fin k → Fin n} :
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(w : WickContract str b1 b2) → StrictMono w.boundFst := fun
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lemma boundFst_strictMono {n k : ℕ} {c : Fin n → 𝓔} {str : WickString c final}
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{b1 b2 : Fin k → Fin n} : (w : WickContract str b1 b2) → StrictMono w.boundFst := fun
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| string => fun k => Fin.elim0 k
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| contr i j _ _ hb1 _ _ w => by
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| contr i j _ _ hb1 _ _ w => by
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intro r s hrs
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rcases Fin.eq_castSucc_or_eq_last r with hr | hr
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· obtain ⟨r, hr⟩ := hr
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@ -85,11 +86,11 @@ lemma boundFst_strictMono {n k : ℕ} {c : Fin n → 𝓔} {str : WickString c f
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rcases Fin.eq_castSucc_or_eq_last s with hs | hs
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· obtain ⟨s, hs⟩ := hs
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subst hs
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simp
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simp only [boundFst_contr_castSucc]
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apply w.boundFst_strictMono _
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simpa using hrs
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· subst hs
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simp
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simp only [boundFst_contr_castSucc, boundFst_contr_last]
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exact hb1 r
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· subst hr
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rcases Fin.eq_castSucc_or_eq_last s with hs | hs
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@ -97,12 +98,13 @@ lemma boundFst_strictMono {n k : ℕ} {c : Fin n → 𝓔} {str : WickString c f
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subst hs
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have hsp := s.prop
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rw [Fin.lt_def] at hrs
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simp at hrs
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simp only [Fin.val_last, Fin.coe_castSucc] at hrs
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omega
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· subst hs
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simp at hrs
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def boundSnd {n k : ℕ} {c : Fin n → 𝓔} {str : WickString c final} {b1 b2 : Fin k → Fin n} :
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/-- The map giving the vertices on the right-hand-side of a contraction. -/
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def boundSnd {n k : ℕ} {c : Fin n → 𝓔} {str : WickString c final} {b1 b2 : Fin k → Fin n} :
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WickContract str b1 b2 → Fin k → Fin n := fun _ => b2
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@[simp]
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@ -129,8 +131,8 @@ lemma boundSnd_contr_last {n k : ℕ} {c : Fin n → 𝓔} {str : WickString c f
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(contr i j h hilej hb1 hb2i hb2j w).boundSnd (Fin.last k) = j := by
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simp only [boundSnd, Fin.snoc_last]
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lemma boundSnd_injective {n k : ℕ} {c : Fin n → 𝓔} {str : WickString c final} {b1 b2 : Fin k → Fin n} :
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(w : WickContract str b1 b2) → Function.Injective w.boundSnd := fun
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lemma boundSnd_injective {n k : ℕ} {c : Fin n → 𝓔} {str : WickString c final}
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{b1 b2 : Fin k → Fin n} : (w : WickContract str b1 b2) → Function.Injective w.boundSnd := fun
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| string => by
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intro i j _
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exact Fin.elim0 i
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@ -142,21 +144,22 @@ lemma boundSnd_injective {n k : ℕ} {c : Fin n → 𝓔} {str : WickString c fi
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rcases Fin.eq_castSucc_or_eq_last s with hs | hs
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· obtain ⟨s, hs⟩ := hs
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subst hs
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simp at hrs
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simp only [boundSnd_contr_castSucc] at hrs
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simpa using w.boundSnd_injective hrs
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· subst hs
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simp at hrs
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simp only [boundSnd_contr_castSucc, boundSnd_contr_last] at hrs
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exact False.elim (h2j r hrs)
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· subst hr
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rcases Fin.eq_castSucc_or_eq_last s with hs | hs
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· obtain ⟨s, hs⟩ := hs
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subst hs
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simp at hrs
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simp only [boundSnd_contr_last, boundSnd_contr_castSucc] at hrs
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exact False.elim (h2j s hrs.symm)
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· subst hs
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rfl
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lemma color_boundSnd_eq_dual_boundFst {n k : ℕ} {c : Fin n → 𝓔} {str : WickString c final} {b1 b2 : Fin k → Fin n} :
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lemma color_boundSnd_eq_dual_boundFst {n k : ℕ} {c : Fin n → 𝓔} {str : WickString c final}
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{b1 b2 : Fin k → Fin n} :
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(w : WickContract str b1 b2) → (i : Fin k) → c (w.boundSnd i) = ξ (c (w.boundFst i)) := fun
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| string => fun i => Fin.elim0 i
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| contr i j hij hilej hi _ _ w => fun r => by
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@ -167,8 +170,9 @@ lemma color_boundSnd_eq_dual_boundFst {n k : ℕ} {c : Fin n → 𝓔} {str : Wi
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· subst hr
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simpa using hij
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lemma boundFst_lt_boundSnd {n k : ℕ} {c : Fin n → 𝓔} {str : WickString c final} {b1 b2 : Fin k → Fin n} :
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(w : WickContract str b1 b2) → (i : Fin k) → w.boundFst i < w.boundSnd i := fun
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lemma boundFst_lt_boundSnd {n k : ℕ} {c : Fin n → 𝓔} {str : WickString c final}
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{b1 b2 : Fin k → Fin n} : (w : WickContract str b1 b2) → (i : Fin k) →
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w.boundFst i < w.boundSnd i := fun
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| string => fun i => Fin.elim0 i
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| contr i j hij hilej hi _ _ w => fun r => by
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rcases Fin.eq_castSucc_or_eq_last r with hr | hr
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@ -176,11 +180,11 @@ lemma boundFst_lt_boundSnd {n k : ℕ} {c : Fin n → 𝓔} {str : WickString c
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subst hr
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simpa using w.boundFst_lt_boundSnd r
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· subst hr
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simp
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simp only [boundFst_contr_last, boundSnd_contr_last]
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exact hilej
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lemma boundFst_neq_boundSnd {n k : ℕ} {c : Fin n → 𝓔} {str : WickString c final} {b1 b2 : Fin k → Fin n} :
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(w : WickContract str b1 b2) → (r1 r2 : Fin k) → b1 r1 ≠ b2 r2 := fun
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lemma boundFst_neq_boundSnd {n k : ℕ} {c : Fin n → 𝓔} {str : WickString c final}
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{b1 b2 : Fin k → Fin n} : (w : WickContract str b1 b2) → (r1 r2 : Fin k) → b1 r1 ≠ b2 r2 := fun
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| string => fun i => Fin.elim0 i
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| contr i j _ hilej h1 h2i h2j w => fun r s => by
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rcases Fin.eq_castSucc_or_eq_last r with hr | hr
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@ -191,17 +195,19 @@ lemma boundFst_neq_boundSnd {n k : ℕ} {c : Fin n → 𝓔} {str : WickString c
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simpa using w.boundFst_neq_boundSnd r s
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· obtain ⟨r, hr⟩ := hr
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subst hr hs
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simp
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simp only [Fin.snoc_castSucc, Fin.snoc_last, ne_eq]
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have hn := h1 r
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omega
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· obtain ⟨s, hs⟩ := hs
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subst hr hs
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simp
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simp only [Fin.snoc_last, Fin.snoc_castSucc, ne_eq]
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exact (h2i s).symm
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· subst hr hs
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simp
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simp only [Fin.snoc_last, ne_eq]
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omega
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/-- Casts a Wick contraction from `WickContract str b1 b2` to `WickContract str b1' b2'` with a
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proof that `b1 = b1'` and `b2 = b2'`, and that they are defined from the same `k = k'`. -/
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def castMaps {n k k' : ℕ} {c : Fin n → 𝓔}
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{str : WickString c final} {b1 b2 : Fin k → Fin n} {b1' b2' : Fin k' → Fin n}
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(hk : k = k')
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@ -214,16 +220,16 @@ lemma castMaps_rfl {n k : ℕ} {c : Fin n → 𝓔}{str : WickString c final}
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{b1 b2 : Fin k → Fin n} (w : WickContract str b1 b2) :
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castMaps rfl rfl rfl w = w := rfl
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lemma mem_snoc' {n k : ℕ} {c : Fin n → 𝓔} {str : WickString c final} {b1' b2' : Fin k → Fin n} :
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lemma mem_snoc' {n k : ℕ} {c : Fin n → 𝓔} {str : WickString c final} {b1' b2' : Fin k → Fin n} :
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(w : WickContract str b1' b2') →
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{k' : ℕ} → (hk' : k'.succ = k ) →
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{k' : ℕ} → (hk' : k'.succ = k) →
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(b1 b2 : Fin k' → Fin n) → (i j : Fin n) → (h : c j = ξ (c i)) →
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(hilej : i < j) → (hb1 : ∀ r, b1 r < i) → (hb2i : ∀ r, b2 r ≠ i) → (hb2j : ∀ r, b2 r ≠ j) →
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(hb1' : Fin.snoc b1 i = b1' ∘ Fin.cast hk') →
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(hb2' : Fin.snoc b2 j = b2' ∘ Fin.cast hk') →
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∃ (w' : WickContract str b1 b2), w = castMaps hk' hb1' hb2' (contr i j h hilej hb1 hb2i hb2j w')
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:= fun
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| string => fun hk' => by
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(hb1' : Fin.snoc b1 i = b1' ∘ Fin.cast hk') →
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(hb2' : Fin.snoc b2 j = b2' ∘ Fin.cast hk') →
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∃ (w' : WickContract str b1 b2), w = castMaps hk' hb1' hb2' (
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contr i j h hilej hb1 hb2i hb2j w') := fun
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| string => fun hk' => by
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simp at hk'
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| contr i' j' h' hilej' hb1' hb2i' hb2j' w' => by
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intro hk b1 b2 i j h hilej hb1 hb2i hb2j hb1' hb2'
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@ -244,31 +250,27 @@ lemma mem_snoc' {n k : ℕ} {c : Fin n → 𝓔} {str : WickString c final} {b1
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· rw [hb1']
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simp
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have hi : i = i' := by
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trans (@Fin.snoc k' (fun _ => Fin n) b1 i) (Fin.last k')
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trans (@Fin.snoc k' (fun _ => Fin n) b1 i) (Fin.last k')
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· simp
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· rw [hb1']
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simp
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have hj : j = j' := by
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trans (@Fin.snoc k' (fun _ => Fin n) b2 j) (Fin.last k')
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trans (@Fin.snoc k' (fun _ => Fin n) b2 j) (Fin.last k')
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· simp
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· rw [hb2']
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simp
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subst hb1'' hb2'' hi hj
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simp
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lemma mem_snoc {n k : ℕ} {c : Fin n → 𝓔} {str : WickString c final} {b1 b2 : Fin k → Fin n} (i j : Fin n)
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(h : c j = ξ (c i))
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(hilej : i < j)
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(hb1 : ∀ r, b1 r < i)
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(hb2i : ∀ r, b2 r ≠ i)
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(hb2j : ∀ r, b2 r ≠ j)
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lemma mem_snoc {n k : ℕ} {c : Fin n → 𝓔} {str : WickString c final} {b1 b2 : Fin k → Fin n}
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(i j : Fin n) (h : c j = ξ (c i)) (hilej : i < j) (hb1 : ∀ r, b1 r < i)
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(hb2i : ∀ r, b2 r ≠ i) (hb2j : ∀ r, b2 r ≠ j)
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(w : WickContract str (Fin.snoc b1 i) (Fin.snoc b2 j)) :
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∃ (w' : WickContract str b1 b2), w = contr i j h hilej hb1 hb2i hb2j w' := by
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exact mem_snoc' w rfl b1 b2 i j h hilej hb1 hb2i hb2j rfl rfl
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lemma is_subsingleton {n k : ℕ} {c : Fin n → 𝓔} {str : WickString c final} {b1 b2 : Fin k → Fin n} :
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Subsingleton (WickContract str b1 b2) := Subsingleton.intro fun w1 w2 => by
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lemma is_subsingleton {n k : ℕ} {c : Fin n → 𝓔} {str : WickString c final} {b1 b2 : Fin k → Fin n} :
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Subsingleton (WickContract str b1 b2) := Subsingleton.intro fun w1 w2 => by
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induction k with
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| zero =>
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have hb1 : b1 = Fin.elim0 := Subsingleton.elim _ _
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@ -284,7 +286,7 @@ lemma is_subsingleton {n k : ℕ} {c : Fin n → 𝓔} {str : WickString c final
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apply congrArg (contr i j _ _ _ _ _) (hI w w')
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lemma eq_snoc_castSucc {k n : ℕ} (b1 : Fin k.succ → Fin n) :
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b1 = Fin.snoc (b1 ∘ Fin.castSucc) (b1 (Fin.last k)) := by
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b1 = Fin.snoc (b1 ∘ Fin.castSucc) (b1 (Fin.last k)) := by
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funext i
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rcases Fin.eq_castSucc_or_eq_last i with h1 | h1
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· obtain ⟨i, rfl⟩ := h1
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@ -292,6 +294,9 @@ lemma eq_snoc_castSucc {k n : ℕ} (b1 : Fin k.succ → Fin n) :
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· subst h1
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simp
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/-- The construction of a Wick contraction from maps `b1 b2 : Fin k → Fin n`, with the former
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giving the first index to be contracted, and the latter the second index. These
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maps must satisfy a series of conditions. -/
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def fromMaps {n k : ℕ} {c : Fin n → 𝓔} {str : WickString c final} (b1 b2 : Fin k → Fin n)
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(hi : ∀ i, c (b2 i) = ξ (c (b1 i)))
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(hb1ltb2 : ∀ i, b1 i < b2 i)
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@ -312,16 +317,18 @@ def fromMaps {n k : ℕ} {c : Fin n → 𝓔} {str : WickString c final} (b1 b2
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(hb1ltb2 (Fin.last k))
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(fun r => hb1 (Fin.castSucc_lt_last r))
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(fun r a => hb1neb2 (Fin.last k) r.castSucc a.symm)
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(fun r => hb2.eq_iff.mp.mt (Fin.ne_last_of_lt (Fin.castSucc_lt_last r )))
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(fun r => hb2.eq_iff.mp.mt (Fin.ne_last_of_lt (Fin.castSucc_lt_last r)))
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(fromMaps (b1 ∘ Fin.castSucc) (b2 ∘ Fin.castSucc) (fun i => hi (Fin.castSucc i))
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(fun i => hb1ltb2 (Fin.castSucc i)) (StrictMono.comp hb1 Fin.strictMono_castSucc)
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?_ ?_))
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· exact fun r1 r2 => hb1neb2 r1.castSucc r2.castSucc
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· exact Function.Injective.comp hb2 (Fin.castSucc_injective k)
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def dropLast {n k : ℕ} {c : Fin n → 𝓔} {str : WickString c final} {b1 b2 : Fin k.succ → Fin n} (w : WickContract str b1 b2) :
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WickContract str (b1 ∘ Fin.castSucc) (b2 ∘ Fin.castSucc) :=
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fromMaps (b1 ∘ Fin.castSucc) (b2 ∘ Fin.castSucc)
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/-- Given a Wick contraction with `k.succ` contractions, returns the Wick contraction with
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`k` contractions by dropping the last contraction (defined by the first index contracted). -/
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def dropLast {n k : ℕ} {c : Fin n → 𝓔} {str : WickString c final} {b1 b2 : Fin k.succ → Fin n}
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(w : WickContract str b1 b2) : WickContract str (b1 ∘ Fin.castSucc) (b2 ∘ Fin.castSucc) :=
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fromMaps (b1 ∘ Fin.castSucc) (b2 ∘ Fin.castSucc)
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(fun i => color_boundSnd_eq_dual_boundFst w i.castSucc)
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(fun i => boundFst_lt_boundSnd w i.castSucc)
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(StrictMono.comp w.boundFst_strictMono Fin.strictMono_castSucc)
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@ -331,26 +338,30 @@ def dropLast {n k : ℕ} {c : Fin n → 𝓔} {str : WickString c final} {b1 b2
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lemma eq_from_maps {n k : ℕ} {c : Fin n → 𝓔} {str : WickString c final} {b1 b2 : Fin k → Fin n}
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(w : WickContract str b1 b2) :
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w = fromMaps w.boundFst w.boundSnd w.color_boundSnd_eq_dual_boundFst
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w.boundFst_lt_boundSnd w.boundFst_strictMono w.boundFst_neq_boundSnd w.boundSnd_injective := by
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exact is_subsingleton.allEq w _
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w.boundFst_lt_boundSnd w.boundFst_strictMono w.boundFst_neq_boundSnd
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w.boundSnd_injective := is_subsingleton.allEq w _
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lemma eq_dropLast_contr {n k : ℕ} {c : Fin n → 𝓔} {str : WickString c final} {b1 b2 : Fin k.succ → Fin n}
|
||||
(w : WickContract str b1 b2) :
|
||||
lemma eq_dropLast_contr {n k : ℕ} {c : Fin n → 𝓔} {str : WickString c final}
|
||||
{b1 b2 : Fin k.succ → Fin n} (w : WickContract str b1 b2) :
|
||||
w = castMaps rfl (eq_snoc_castSucc b1).symm (eq_snoc_castSucc b2).symm
|
||||
(contr (b1 (Fin.last k)) (b2 (Fin.last k))
|
||||
(w.color_boundSnd_eq_dual_boundFst (Fin.last k))
|
||||
(w.boundFst_lt_boundSnd (Fin.last k))
|
||||
(fun r => w.boundFst_strictMono (Fin.castSucc_lt_last r))
|
||||
(fun r a => w.boundFst_neq_boundSnd (Fin.last k) r.castSucc a.symm)
|
||||
(fun r => w.boundSnd_injective.eq_iff.mp.mt (Fin.ne_last_of_lt (Fin.castSucc_lt_last r )))
|
||||
(fun r => w.boundSnd_injective.eq_iff.mp.mt (Fin.ne_last_of_lt (Fin.castSucc_lt_last r)))
|
||||
(dropLast w)) := by
|
||||
rw [eq_from_maps w]
|
||||
rfl
|
||||
|
||||
/-- Wick contractions of a given Wick string with `k` different contractions. -/
|
||||
def Level {n : ℕ} {c : Fin n → 𝓔} (str : WickString c final) (k : ℕ) : Type :=
|
||||
Σ (b1 : Fin k → Fin n) (b2 : Fin k → Fin n), WickContract str b1 b2
|
||||
|
||||
instance levelZeroFintype {n : ℕ} {c : Fin n → 𝓔} (str : WickString c final) : Fintype (Level str 0) where
|
||||
/-- There is a finite number of Wick contractions with no contractions. In particular,
|
||||
this is just the original Wick string. -/
|
||||
instance levelZeroFintype {n : ℕ} {c : Fin n → 𝓔} (str : WickString c final) :
|
||||
Fintype (Level str 0) where
|
||||
elems := {⟨Fin.elim0, Fin.elim0, WickContract.string⟩}
|
||||
complete := by
|
||||
intro x
|
||||
|
|
@ -362,8 +373,12 @@ instance levelZeroFintype {n : ℕ} {c : Fin n → 𝓔} (str : WickString c fin
|
|||
simp only [Finset.mem_singleton]
|
||||
rw [is_subsingleton.allEq w string]
|
||||
|
||||
structure ContrPair {n : ℕ} {c : Fin n → 𝓔} {str : WickString c final} {b1 b2 : Fin k → Fin n} (w : WickContract str b1 b2) where
|
||||
/-- The pairs of additional indices which can be contracted given a Wick contraction. -/
|
||||
structure ContrPair {n : ℕ} {c : Fin n → 𝓔} {str : WickString c final} {b1 b2 : Fin k → Fin n}
|
||||
(w : WickContract str b1 b2) where
|
||||
/-- The first index in the contraction pair. -/
|
||||
i : Fin n
|
||||
/-- The second index in the contraction pair. -/
|
||||
j : Fin n
|
||||
h : c j = ξ (c i)
|
||||
hilej : i < j
|
||||
|
|
@ -371,9 +386,13 @@ structure ContrPair {n : ℕ} {c : Fin n → 𝓔} {str : WickString c final} {
|
|||
hb2i : ∀ r, b2 r ≠ i
|
||||
hb2j : ∀ r, b2 r ≠ j
|
||||
|
||||
def contrPairEquivSubtype {n : ℕ} {c : Fin n → 𝓔} {str : WickString c final} {b1 b2 : Fin k → Fin n} (w : WickContract str b1 b2) :
|
||||
/-- The pairs of additional indices which can be contracted, given an existing wick contraction,
|
||||
is equivalent to the a subtype of `Fin n × Fin n` defined by certain conditions equivalent
|
||||
to the conditions appearing in `ContrPair`. -/
|
||||
def contrPairEquivSubtype {n : ℕ} {c : Fin n → 𝓔} {str : WickString c final}
|
||||
{b1 b2 : Fin k → Fin n} (w : WickContract str b1 b2) :
|
||||
ContrPair w ≃ {x : Fin n × Fin n // c x.2 = ξ (c x.1) ∧ x.1 < x.2 ∧
|
||||
(∀ r, b1 r < x.1) ∧ (∀ r, b2 r ≠ x.1) ∧ (∀ r, b2 r ≠ x.2)} where
|
||||
(∀ r, b1 r < x.1) ∧ (∀ r, b2 r ≠ x.1) ∧ (∀ r, b2 r ≠ x.2)} where
|
||||
toFun cp := ⟨⟨cp.i, cp.j⟩, ⟨cp.h, cp.hilej, cp.hb1, cp.hb2i, cp.hb2j⟩⟩
|
||||
invFun x :=
|
||||
match x with
|
||||
|
|
@ -393,16 +412,18 @@ lemma heq_eq {n : ℕ} {c : Fin n → 𝓔} {b1 b2 b1' b2' : Fin k → Fin n} {s
|
|||
(w : WickContract str b1 b2)
|
||||
(w' : WickContract str b1' b2') (h1 : b1 = b1') (h2 : b2 = b2') : HEq w w':= by
|
||||
subst h1 h2
|
||||
simp
|
||||
simp only [heq_eq_eq]
|
||||
exact is_subsingleton.allEq w w'
|
||||
|
||||
/-- The equivalence between Wick contractions consisting of `k.succ` contractions and
|
||||
those with `k` contractions paired with a suitable contraction pair. -/
|
||||
def levelSuccEquiv {n : ℕ} {c : Fin n → 𝓔} (str : WickString c final) (k : ℕ) :
|
||||
Level str k.succ ≃ (w : Level str k) × ContrPair w.2.2 where
|
||||
toFun w :=
|
||||
match w with
|
||||
| ⟨b1, b2, w⟩ =>
|
||||
⟨⟨b1 ∘ Fin.castSucc, b2 ∘ Fin.castSucc, dropLast w⟩,
|
||||
⟨b1 (Fin.last k), b2 (Fin.last k),
|
||||
⟨b1 (Fin.last k), b2 (Fin.last k),
|
||||
w.color_boundSnd_eq_dual_boundFst (Fin.last k),
|
||||
w.boundFst_lt_boundSnd (Fin.last k),
|
||||
fun r => w.boundFst_strictMono (Fin.castSucc_lt_last r),
|
||||
|
|
@ -415,7 +436,7 @@ def levelSuccEquiv {n : ℕ} {c : Fin n → 𝓔} (str : WickString c final) (k
|
|||
left_inv w := by
|
||||
match w with
|
||||
| ⟨b1, b2, w⟩ =>
|
||||
simp
|
||||
simp only [Nat.succ_eq_add_one, Function.comp_apply]
|
||||
congr
|
||||
· exact Eq.symm (eq_snoc_castSucc b1)
|
||||
· funext b2
|
||||
|
|
@ -429,7 +450,7 @@ def levelSuccEquiv {n : ℕ} {c : Fin n → 𝓔} (str : WickString c final) (k
|
|||
right_inv w := by
|
||||
match w with
|
||||
| ⟨⟨b1, b2, w⟩, cp⟩ =>
|
||||
simp
|
||||
simp only [Nat.succ_eq_add_one, Fin.snoc_last, Sigma.mk.inj_iff]
|
||||
apply And.intro
|
||||
· congr
|
||||
· exact Fin.snoc_comp_castSucc
|
||||
|
|
@ -446,38 +467,40 @@ def levelSuccEquiv {n : ℕ} {c : Fin n → 𝓔} (str : WickString c final) (k
|
|||
· simp
|
||||
· simp
|
||||
|
||||
|
||||
def bound {n k : ℕ} {c : Fin n → 𝓔} {str : WickString c final} {b1 b2 : Fin k → Fin n}
|
||||
(w : WickContract str b1 b2) : Fin k ⊕ Fin k → Fin n :=
|
||||
/-- The sum of `boundFst` and `boundSnd`, giving on `Sum.inl k` the first index
|
||||
in the `k`th contraction, and on `Sum.inr k` the second index in the `k`th contraction. -/
|
||||
def bound {n k : ℕ} {c : Fin n → 𝓔} {str : WickString c final} {b1 b2 : Fin k → Fin n}
|
||||
(w : WickContract str b1 b2) : Fin k ⊕ Fin k → Fin n :=
|
||||
Sum.elim w.boundFst w.boundSnd
|
||||
|
||||
/-- On `Sum.inl k` the map `bound` acts via `boundFst`. -/
|
||||
@[simp]
|
||||
lemma bound_inl {n k : ℕ} {c : Fin n → 𝓔} {str : WickString c final} {b1 b2 : Fin k → Fin n}
|
||||
lemma bound_inl {n k : ℕ} {c : Fin n → 𝓔} {str : WickString c final} {b1 b2 : Fin k → Fin n}
|
||||
(w : WickContract str b1 b2) (i : Fin k) : w.bound (Sum.inl i) = w.boundFst i := rfl
|
||||
|
||||
/-- On `Sum.inr k` the map `bound` acts via `boundSnd`. -/
|
||||
@[simp]
|
||||
lemma bound_inr {n k : ℕ} {c : Fin n → 𝓔} {str : WickString c final} {b1 b2 : Fin k → Fin n}
|
||||
lemma bound_inr {n k : ℕ} {c : Fin n → 𝓔} {str : WickString c final} {b1 b2 : Fin k → Fin n}
|
||||
(w : WickContract str b1 b2) (i : Fin k) : w.bound (Sum.inr i) = w.boundSnd i := rfl
|
||||
|
||||
lemma bound_injection {n k : ℕ} {c : Fin n → 𝓔} {str : WickString c final} {b1 b2 : Fin k → Fin n}
|
||||
lemma bound_injection {n k : ℕ} {c : Fin n → 𝓔} {str : WickString c final} {b1 b2 : Fin k → Fin n}
|
||||
(w : WickContract str b1 b2) : Function.Injective w.bound := by
|
||||
intro x y h
|
||||
match x, y with
|
||||
| Sum.inl x, Sum.inl y =>
|
||||
simp at h
|
||||
simp only [bound_inl] at h
|
||||
simpa using (StrictMono.injective w.boundFst_strictMono).eq_iff.mp h
|
||||
| Sum.inr x, Sum.inr y =>
|
||||
simp at h
|
||||
simp only [bound_inr] at h
|
||||
simpa using w.boundSnd_injective h
|
||||
| Sum.inl x, Sum.inr y =>
|
||||
simp at h
|
||||
simp only [bound_inl, bound_inr] at h
|
||||
exact False.elim (w.boundFst_neq_boundSnd x y h)
|
||||
| Sum.inr x, Sum.inl y =>
|
||||
simp at h
|
||||
simp only [bound_inr, bound_inl] at h
|
||||
exact False.elim (w.boundFst_neq_boundSnd y x h.symm)
|
||||
|
||||
|
||||
lemma bound_le_total {n k : ℕ} {c : Fin n → 𝓔} {str : WickString c final} {b1 b2 : Fin k → Fin n}
|
||||
lemma bound_le_total {n k : ℕ} {c : Fin n → 𝓔} {str : WickString c final} {b1 b2 : Fin k → Fin n}
|
||||
(w : WickContract str b1 b2) : 2 * k ≤ n := by
|
||||
refine Fin.nonempty_embedding_iff.mp ⟨w.bound ∘ finSumFinEquiv.symm ∘ Fin.cast (Nat.two_mul k),
|
||||
?_⟩
|
||||
|
|
@ -485,39 +508,49 @@ lemma bound_le_total {n k : ℕ} {c : Fin n → 𝓔} {str : WickString c final}
|
|||
· exact Fin.cast_injective (Nat.two_mul k)
|
||||
· exact bound_injection w
|
||||
|
||||
/-- The list of fields (indexed by `Fin n`) in a Wick contraction which are not bound,
|
||||
i.e. which do not appear in any contraction. -/
|
||||
def unboundList {n k : ℕ} {c : Fin n → 𝓔} {str : WickString c final} {b1 b2 : Fin k → Fin n}
|
||||
(w : WickContract str b1 b2) : List (Fin n) :=
|
||||
List.filter (fun i => decide (∀ r, w.bound r ≠ i)) (List.finRange n)
|
||||
|
||||
lemma unboundList_nodup {n k : ℕ} {c : Fin n → 𝓔} {str : WickString c final} {b1 b2 : Fin k → Fin n}
|
||||
(w : WickContract str b1 b2) : (w.unboundList).Nodup := List.Nodup.filter _ (List.nodup_finRange n)
|
||||
(w : WickContract str b1 b2) : (w.unboundList).Nodup :=
|
||||
List.Nodup.filter _ (List.nodup_finRange n)
|
||||
|
||||
lemma unboundList_length {n k : ℕ} {c : Fin n → 𝓔} {str : WickString c final} {b1 b2 : Fin k → Fin n}
|
||||
(w : WickContract str b1 b2) : w.unboundList.length = n - 2 * k := by
|
||||
lemma unboundList_length {n k : ℕ} {c : Fin n → 𝓔} {str : WickString c final}
|
||||
{b1 b2 : Fin k → Fin n} (w : WickContract str b1 b2) :
|
||||
w.unboundList.length = n - 2 * k := by
|
||||
rw [← List.Nodup.dedup w.unboundList_nodup]
|
||||
rw [← List.card_toFinset, unboundList]
|
||||
rw [List.toFinset_filter, List.toFinset_finRange]
|
||||
have hn := Finset.filter_card_add_filter_neg_card_eq_card (s := Finset.univ) (fun (i : Fin n) => i ∈ Finset.image w.bound Finset.univ)
|
||||
have hn := Finset.filter_card_add_filter_neg_card_eq_card (s := Finset.univ)
|
||||
(fun (i : Fin n) => i ∈ Finset.image w.bound Finset.univ)
|
||||
have hn' :(Finset.filter (fun i => i ∈ Finset.image w.bound Finset.univ) Finset.univ).card =
|
||||
(Finset.image w.bound Finset.univ).card := by
|
||||
refine Finset.card_equiv (Equiv.refl _) fun i => ?_
|
||||
simp
|
||||
rw [hn'] at hn
|
||||
rw [Finset.card_image_of_injective] at hn
|
||||
simp only [Finset.card_univ, Fintype.card_sum, Fintype.card_fin,
|
||||
simp only [Finset.card_univ, Fintype.card_sum, Fintype.card_fin,
|
||||
Finset.mem_univ, true_and, Sum.exists, bound_inl, bound_inr, not_or, not_exists] at hn
|
||||
have hn'' : (Finset.filter (fun a => a ∉ Finset.image w.bound Finset.univ) Finset.univ).card = n - 2 * k := by
|
||||
have hn'' : (Finset.filter (fun a => a ∉ Finset.image w.bound Finset.univ) Finset.univ).card =
|
||||
n - 2 * k := by
|
||||
omega
|
||||
rw [← hn'']
|
||||
congr
|
||||
funext x
|
||||
simp
|
||||
simp only [ne_eq, Sum.forall, bound_inl, bound_inr, Bool.decide_and, Bool.and_eq_true,
|
||||
decide_eq_true_eq, Finset.mem_image, Finset.mem_univ, true_and, Sum.exists, not_or, not_exists]
|
||||
exact bound_injection w
|
||||
|
||||
lemma unboundList_sorted {n k : ℕ} {c : Fin n → 𝓔} {str : WickString c final} {b1 b2 : Fin k → Fin n}
|
||||
(w : WickContract str b1 b2) : List.Sorted (fun i j => i < j) w.unboundList :=
|
||||
lemma unboundList_sorted {n k : ℕ} {c : Fin n → 𝓔} {str : WickString c final}
|
||||
{b1 b2 : Fin k → Fin n} (w : WickContract str b1 b2) :
|
||||
List.Sorted (fun i j => i < j) w.unboundList :=
|
||||
List.Pairwise.sublist (List.filter_sublist (List.finRange n)) (List.pairwise_lt_finRange n)
|
||||
|
||||
/-- The map giving the fields which are not bound in a contraction. These
|
||||
are the fields that will appear in a normal operator in Wick's theorem. -/
|
||||
def unbound {n k : ℕ} {c : Fin n → 𝓔} {str : WickString c final} {b1 b2 : Fin k → Fin n}
|
||||
(w : WickContract str b1 b2) : Fin (n - 2 * k) → Fin n :=
|
||||
w.unboundList.get ∘ Fin.cast w.unboundList_length.symm
|
||||
|
|
@ -529,8 +562,8 @@ lemma unbound_injective {n k : ℕ} {c : Fin n → 𝓔} {str : WickString c fin
|
|||
exact w.unboundList_nodup
|
||||
· exact Fin.cast_injective _
|
||||
|
||||
lemma unbound_strictMono {n k : ℕ} {c : Fin n → 𝓔} {str : WickString c final} {b1 b2 : Fin k → Fin n}
|
||||
(w : WickContract str b1 b2) : StrictMono w.unbound := by
|
||||
lemma unbound_strictMono {n k : ℕ} {c : Fin n → 𝓔} {str : WickString c final}
|
||||
{b1 b2 : Fin k → Fin n} (w : WickContract str b1 b2) : StrictMono w.unbound := by
|
||||
apply StrictMono.comp
|
||||
· refine List.Sorted.get_strictMono w.unboundList_sorted
|
||||
· exact fun ⦃a b⦄ a => a
|
||||
|
|
|
|||
|
|
@ -42,10 +42,10 @@ scoped[TensorSpecies] notation "⟪" ψ "," φ "⟫ₜₛ" => contractSelfField
|
|||
|
||||
/-- The map `contractSelfField` is equivariant with respect to the group action. -/
|
||||
@[simp]
|
||||
lemma contractSelfField_equivariant {S : TensorSpecies} {c : S.C} {g : S.G}
|
||||
lemma contractSelfField_equivariant {S : TensorSpecies} {c : S.C} {g : S.G}
|
||||
(ψ : S.FD.obj (Discrete.mk c)) (φ : S.FD.obj (Discrete.mk c)) :
|
||||
⟪(S.FD.obj (Discrete.mk c)).ρ g ψ, (S.FD.obj (Discrete.mk c)).ρ g φ⟫ₜₛ = ⟪ψ, φ⟫ₜₛ := by
|
||||
simpa using congrFun (congrArg (fun x => x.toFun) ((S.contractSelfHom c).comm g )) (ψ ⊗ₜ[S.k] φ)
|
||||
simpa using congrFun (congrArg (fun x => x.toFun) ((S.contractSelfHom c).comm g)) (ψ ⊗ₜ[S.k] φ)
|
||||
|
||||
informal_lemma contractSelfField_non_degenerate where
|
||||
math :≈ "The contraction of two vectors of the same color is non-degenerate.
|
||||
|
|
@ -85,12 +85,12 @@ def IsNormZero {c : S.C} (ψ : S.FD.obj (Discrete.mk c)) : Prop := ⟪ψ, ψ⟫
|
|||
|
||||
/-- The zero vector has norm equal to zero. -/
|
||||
@[simp]
|
||||
lemma zero_isNormZero {c : S.C} : @IsNormZero S c 0 := by
|
||||
lemma zero_isNormZero {c : S.C} : @IsNormZero S c 0 := by
|
||||
simp only [IsNormZero, tmul_zero, map_zero]
|
||||
|
||||
/-- If a vector is norm-zero, then any scalar multiple of that vector is also norm-zero. -/
|
||||
lemma smul_isNormZero_of_isNormZero {c : S.C} {ψ : S.FD.obj (Discrete.mk c)}
|
||||
(h : S.IsNormZero ψ ) (a : S.k) : S.IsNormZero (a • ψ) := by
|
||||
(h : S.IsNormZero ψ) (a : S.k) : S.IsNormZero (a • ψ) := by
|
||||
simp only [IsNormZero, tmul_smul, map_smul, smul_tmul]
|
||||
rw [h]
|
||||
simp only [smul_eq_mul, mul_zero]
|
||||
|
|
|
|||
|
|
@ -22,7 +22,6 @@ namespace TensorSpecies
|
|||
|
||||
variable (S : TensorSpecies)
|
||||
|
||||
|
||||
/-- The isomorphism between the image of a map `Fin 1 ⊕ Fin 1 → S.C` contructed by `finExtractTwo`
|
||||
under `S.F.obj`, and an object in the image of `OverColor.Discrete.pairτ S.FD`. -/
|
||||
def contrFin1Fin1 {n : ℕ} (c : Fin n.succ.succ → S.C)
|
||||
|
|
|
|||
|
|
@ -48,7 +48,7 @@ lemma tprod_equivariant (g : S.G) (p : Pure S c) : (ρ g p).tprod = (S.F.obj c).
|
|||
|
||||
end Pure
|
||||
|
||||
/-- We say a tensor is pure if it is `⨂[S.k] i, p i` for some `p : Pure c`. -/
|
||||
/-- A tensor is pure if it is `⨂[S.k] i, p i` for some `p : Pure c`. -/
|
||||
def IsPure {c : OverColor S.C} (t : S.F.obj c) : Prop := ∃ p : Pure S c, t = p.tprod
|
||||
|
||||
/-- As long as we are dealing with tensors with at least one index, then the zero
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue