feat: WickContract
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jstoobysmith
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1 changed files with 255 additions and 118 deletions
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@ -100,75 +100,55 @@ inductive WickString : {n : ℕ} → (c : FieldString n) → WickStringLast →
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WickString (Fin.cons e c) outgoing
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| endOutgoing {n : ℕ} {c : Fin n → 𝓔} (w : WickString c outgoing) : WickString c final
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inductive WickContract : {n : ℕ} → (f : FieldString n) → {m : ℕ} → (ub : Fin m → Fin n) →
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{k : ℕ} → (b1 : Fin k → Fin n) → Type where
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| string {n : ℕ} {c : Fin n → 𝓔} : WickContract c id Fin.elim0
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| contr {n : ℕ} {c : Fin n → 𝓔} {m : ℕ} {ub : Fin m.succ.succ → Fin n} {k : ℕ}
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{b1 : Fin k → Fin n} : (i : Fin m.succ.succ) →
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(j : Fin m.succ) → (h : c (ub (i.succAbove j)) = ξ (c (ub i))) →
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(hilej : i < i.succAbove j) → (hlastlej : (hk : 0 < k) → b1 ⟨k - 1,Nat.sub_one_lt_of_lt hk⟩ < ub i) →
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(w : WickContract c ub b1) → WickContract c (ub ∘ i.succAbove ∘ j.succAbove) (Fin.snoc b1 (ub i))
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inductive WickContract : {n : ℕ} → (f : FieldString n) →
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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} : WickContract c Fin.elim0 Fin.elim0
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| contr {n : ℕ} {c : Fin n → 𝓔} {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 c b1 b2) →
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WickContract c (Fin.snoc b1 i) (Fin.snoc b2 j)
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namespace WickContract
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variable {n m k : ℕ} {c : Fin n → 𝓔} {ub : Fin m → Fin n} {b1 : Fin k → Fin n}
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/-- The number of nodes of a Wick contraction. -/
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def size {n m k : ℕ} {c : Fin n → 𝓔} {ub : Fin m → Fin n} {b1 : Fin k → Fin n} :
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WickContract c ub b1 → ℕ := fun
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def size {n k : ℕ} {c : Fin n → 𝓔} {b1 b2 : Fin k → Fin n} :
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WickContract c b1 b2 → ℕ := fun
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| string => 1
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| contr _ _ _ _ _ w => w.size + 1
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| contr _ _ _ _ _ _ _ w => w.size + 1
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def unbound {n m k : ℕ} {c : Fin n → 𝓔} {ub : Fin m → Fin n} {b1 : Fin k → Fin n} :
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WickContract c ub b1 → Fin m → Fin n := fun _ => ub
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def boundFst {n k : ℕ} {c : Fin n → 𝓔} {b1 b2 : Fin k → Fin n} :
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WickContract c b1 b2 → Fin k → Fin n := fun _ => b1
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@[simp]
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lemma unbound_contr {n : ℕ} {c : Fin n → 𝓔} {m : ℕ} {ub : Fin m.succ.succ → Fin n} {k : ℕ}
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{b1 : Fin k → Fin n} (i : Fin m.succ.succ)
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(j : Fin m.succ)
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(h : c (ub (i.succAbove j)) = ξ (c (ub i)))
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(hilej : i < i.succAbove j)
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(hlastlej : (hk : 0 < k) → b1 ⟨k - 1,Nat.sub_one_lt_of_lt hk⟩ < ub i)
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(w : WickContract c ub b1) (r : Fin m) :
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(contr i j h hilej hlastlej w).unbound r = w.unbound (i.succAbove (j.succAbove r)) := rfl
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lemma unbound_strictMono {n m k : ℕ} {c : Fin n → 𝓔} {ub : Fin m → Fin n} {b1 : Fin k → Fin n} :
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(w : WickContract c ub b1) → StrictMono w.unbound := fun
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| string => by exact fun ⦃a b⦄ a => a
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| contr i j hij hilej hi w => by
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intro r s hrs
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refine w.unbound_strictMono ?_
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refine Fin.strictMono_succAbove _ ?_
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refine Fin.strictMono_succAbove _ hrs
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def boundFst {n m k : ℕ} {c : Fin n → 𝓔} {ub : Fin m → Fin n} {b1 : Fin k → Fin n} :
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WickContract c ub b1 → Fin k → Fin n := fun _ => b1
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@[simp]
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lemma boundFst_contr_castSucc {n : ℕ} {c : Fin n → 𝓔} {m : ℕ} {ub : Fin m.succ.succ → Fin n} {k : ℕ}
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{b1 : Fin k → Fin n} (i : Fin m.succ.succ)
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(j : Fin m.succ)
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(h : c (ub (i.succAbove j)) = ξ (c (ub i)))
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(hilej : i < i.succAbove j)
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(hlastlej : (hk : 0 < k) → b1 ⟨k - 1,Nat.sub_one_lt_of_lt hk⟩ < ub i)
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(w : WickContract c ub b1) (r : Fin k) :
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(contr i j h hilej hlastlej w).boundFst r.castSucc = w.boundFst r := by
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lemma boundFst_contr_castSucc {n k : ℕ} {c : Fin n → 𝓔}
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{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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(w : WickContract c b1 b2) (r : Fin k) :
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(contr i j h hilej hb1 hb2i hb2j w).boundFst r.castSucc = w.boundFst r := by
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simp only [boundFst, Fin.snoc_castSucc]
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@[simp]
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lemma boundFst_contr_last {n : ℕ} {c : Fin n → 𝓔} {m : ℕ} {ub : Fin m.succ.succ → Fin n} {k : ℕ}
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{b1 : Fin k → Fin n} (i : Fin m.succ.succ)
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(j : Fin m.succ)
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(h : c (ub (i.succAbove j)) = ξ (c (ub i)))
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(hilej : i < i.succAbove j)
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(hlastlej : (hk : 0 < k) → b1 ⟨k - 1,Nat.sub_one_lt_of_lt hk⟩ < ub i)
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(w : WickContract c ub b1) :
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(contr i j h hilej hlastlej w).boundFst (Fin.last k) = ub i := by
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lemma boundFst_contr_last {n k : ℕ} {c : Fin n → 𝓔}
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{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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(w : WickContract c b1 b2) :
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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 m k : ℕ} {c : Fin n → 𝓔} {ub : Fin m → Fin n} {b1 : Fin k → Fin n} :
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(w : WickContract c ub b1) → StrictMono w.boundFst := fun
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lemma boundFst_strictMono {n k : ℕ} {c : Fin n → 𝓔} {b1 b2 : Fin k → Fin n} :
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(w : WickContract c b1 b2) → StrictMono w.boundFst := fun
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| string => fun k => Fin.elim0 k
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| contr i j hij hilej hi 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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@ -181,12 +161,7 @@ lemma boundFst_strictMono {n m k : ℕ} {c : Fin n → 𝓔} {ub : Fin m → Fin
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simpa using hrs
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· subst hs
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simp
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refine Fin.lt_of_le_of_lt ?_ (hi (Nat.zero_lt_of_lt hrs))
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· refine (StrictMono.monotone w.boundFst_strictMono) ?_
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rw [Fin.le_def]
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simp
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rw [Fin.lt_def] at hrs
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omega
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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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· obtain ⟨s, hs⟩ := hs
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@ -198,61 +173,40 @@ lemma boundFst_strictMono {n m k : ℕ} {c : Fin n → 𝓔} {ub : Fin m → Fin
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· subst hs
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simp at hrs
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def boundSnd {n m k : ℕ} {c : Fin n → 𝓔} {ub : Fin m → Fin n} {b1 : Fin k → Fin n} :
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WickContract c ub b1 → Fin k → Fin n := fun
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| string => Fin.elim0
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| contr i j _ _ _ w => Fin.snoc w.boundSnd (w.unbound (i.succAbove j))
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lemma boundFst_lt_boundSnd {n m k : ℕ} {c : Fin n → 𝓔} {ub : Fin m → Fin n} {b1 : Fin k → Fin n} :
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(w : WickContract c ub b1) → (i : Fin k) → 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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simp only [boundFst, boundSnd, Nat.succ_eq_add_one]
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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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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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change w.unbound _ < _
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apply w.unbound_strictMono hilej
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lemma boundFst_dual_eq_boundSnd {n m k : ℕ} {c : Fin n → 𝓔} {ub : Fin m → Fin n} {b1 : Fin k → Fin n} :
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(w : WickContract c ub b1) → (i : Fin k) → ξ (c (w.boundFst i)) = c (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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simp only [boundFst, boundSnd, Nat.succ_eq_add_one]
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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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subst hr
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simpa using w.boundFst_dual_eq_boundSnd r
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· subst hr
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simp only [Fin.snoc_last]
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erw [hij]
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def boundSnd {n k : ℕ} {c : Fin n → 𝓔} {b1 b2 : Fin k → Fin n} :
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WickContract c b1 b2 → Fin k → Fin n := fun _ => b2
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@[simp]
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lemma boundSnd_neq_unbound {n m k : ℕ} {c : Fin n → 𝓔} {ub : Fin m → Fin n} {b1 : Fin k → Fin n} : (w : WickContract c ub b1) → (i : Fin k) → (j : Fin m) →
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w.boundSnd i ≠ ub j := fun
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| string => fun i => Fin.elim0 i
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| contr i j hij hilej hi w => fun r s => by
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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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subst hr
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simp [boundSnd]
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exact w.boundSnd_neq_unbound _ _
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· subst hr
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simp [boundSnd]
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apply (StrictMono.injective w.unbound_strictMono).eq_iff.mp.mt
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apply Fin.succAbove_right_injective.eq_iff.mp.mt
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exact Fin.ne_succAbove j s
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lemma boundSnd_contr_castSucc {n k : ℕ} {c : Fin n → 𝓔}
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{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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(w : WickContract c b1 b2) (r : Fin k) :
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(contr i j h hilej hb1 hb2i hb2j w).boundSnd r.castSucc = w.boundSnd r := by
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simp only [boundSnd, Fin.snoc_castSucc]
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lemma boundSnd_injective {n m k : ℕ} {c : Fin n → 𝓔} {ub : Fin m → Fin n} {b1 : Fin k → Fin n}: (w : WickContract c ub b1) → Function.Injective w.boundSnd := fun
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@[simp]
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lemma boundSnd_contr_last {n k : ℕ} {c : Fin n → 𝓔}
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{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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(w : WickContract c b1 b2) :
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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} {b1 b2 : Fin k → Fin n} :
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(w : WickContract c 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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| contr i j hij hilej hi w => by
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| contr i j hij hilej hi h2i h2j w => by
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intro r s hrs
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simp [boundSnd] at 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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subst hr
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@ -263,23 +217,206 @@ lemma boundSnd_injective {n m k : ℕ} {c : Fin n → 𝓔} {ub : Fin m → Fin
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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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exact False.elim (w.boundSnd_neq_unbound r (i.succAbove j) hrs)
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exact False.elim (h2j r hrs)
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· subst hr
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simp at hrs
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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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exact False.elim (w.boundSnd_neq_unbound s (i.succAbove j) hrs.symm)
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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 no_fields_eq_unbound_plus_two_bound {n m k : ℕ} {c : Fin n → 𝓔} {ub : Fin m → Fin n} {b1 : Fin k → Fin n} :
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(w : WickContract c ub b1) → n = m + 2 * k := fun
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| string => rfl
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| contr i j hij hilej hi w => by
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rw [w.no_fields_eq_unbound_plus_two_bound]
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omega
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lemma color_boundSnd_eq_dual_boundFst {n k : ℕ} {c : Fin n → 𝓔} {b1 b2 : Fin k → Fin n} :
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(w : WickContract c 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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rcases Fin.eq_castSucc_or_eq_last r with hr | hr
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· obtain ⟨r, hr⟩ := hr
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subst hr
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simpa using w.color_boundSnd_eq_dual_boundFst r
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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} {b1 b2 : Fin k → Fin n} :
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(w : WickContract c b1 b2) → (i : Fin k) → 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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· obtain ⟨r, hr⟩ := hr
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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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exact hilej
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lemma boundFst_neq_boundSnd {n k : ℕ} {c : Fin n → 𝓔} {b1 b2 : Fin k → Fin n} :
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(w : WickContract c 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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<;> rcases Fin.eq_castSucc_or_eq_last s with hs | hs
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· obtain ⟨r, hr⟩ := hr
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obtain ⟨s, hs⟩ := hs
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subst hr hs
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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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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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exact (h2i s).symm
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· subst hr hs
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simp
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omega
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def castMaps {n k k' : ℕ} {c : Fin n → 𝓔} {b1 b2 : Fin k → Fin n} {b1' b2' : Fin k' → Fin n}
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(hk : k = k')
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(hb1 : b1 = b1' ∘ Fin.cast hk) (hb2 : b2 = b2' ∘ Fin.cast hk) (w : WickContract c b1 b2) :
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WickContract c b1' b2' :=
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cast (by subst hk; rfl) (hb2 ▸ hb1 ▸ w)
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@[simp]
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lemma castMaps_rfl {n k : ℕ} {c : Fin n → 𝓔} {b1 b2 : Fin k → Fin n} (w : WickContract c 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} {b1' b2' : Fin k → Fin n} :
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(w : WickContract c b1' b2') →
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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 c 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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simp at hk'
|
||||
| contr i' j' h' hilej' hb1' hb2i' hb2j' w' => by
|
||||
intro hk b1 b2 i j h hilej hb1 hb2i hb2j hb1' hb2'
|
||||
rename_i k' k b1' b2' f
|
||||
have hk2 : k' = k := Nat.succ_inj'.mp hk
|
||||
subst hk2
|
||||
simp_all
|
||||
have hb2'' : b2 = b2' := by
|
||||
funext k
|
||||
trans (@Fin.snoc k' (fun _ => Fin n) b2 j) (Fin.castSucc k)
|
||||
· simp
|
||||
· rw [hb2']
|
||||
simp
|
||||
have hb1'' : b1 = b1' := by
|
||||
funext k
|
||||
trans (@Fin.snoc k' (fun _ => Fin n) b1 i) (Fin.castSucc k)
|
||||
· simp
|
||||
· rw [hb1']
|
||||
simp
|
||||
have hi : i = i' := by
|
||||
trans (@Fin.snoc k' (fun _ => Fin n) b1 i) (Fin.last k')
|
||||
· simp
|
||||
· rw [hb1']
|
||||
simp
|
||||
have hj : j = j' := by
|
||||
trans (@Fin.snoc k' (fun _ => Fin n) b2 j) (Fin.last k')
|
||||
· simp
|
||||
· rw [hb2']
|
||||
simp
|
||||
subst hb1'' hb2'' hi hj
|
||||
simp
|
||||
|
||||
|
||||
lemma mem_snoc {n k : ℕ} {c : Fin n → 𝓔} {b1 b2 : Fin k → Fin n} (i j : Fin n)
|
||||
(h : c j = ξ (c i))
|
||||
(hilej : i < j)
|
||||
(hb1 : ∀ r, b1 r < i)
|
||||
(hb2i : ∀ r, b2 r ≠ i)
|
||||
(hb2j : ∀ r, b2 r ≠ j)
|
||||
(w : WickContract c (Fin.snoc b1 i) (Fin.snoc b2 j)) :
|
||||
∃ (w' : WickContract c b1 b2), w = contr i j h hilej hb1 hb2i hb2j w' := by
|
||||
exact mem_snoc' w rfl b1 b2 i j h hilej hb1 hb2i hb2j rfl rfl
|
||||
|
||||
lemma is_subsingleton {n k : ℕ} {c : Fin n → 𝓔} {b1 b2 : Fin k → Fin n} :
|
||||
Subsingleton (WickContract c b1 b2) := Subsingleton.intro fun w1 w2 => by
|
||||
induction k with
|
||||
| zero =>
|
||||
have hb1 : b1 = Fin.elim0 := Subsingleton.elim _ _
|
||||
have hb2 : b2 = Fin.elim0 := Subsingleton.elim _ _
|
||||
subst hb1 hb2
|
||||
match w1, w2 with
|
||||
| string, string => rfl
|
||||
| succ k hI =>
|
||||
match w1, w2 with
|
||||
| contr i j h hilej hb1 hb2i hb2j w, w2 =>
|
||||
let ⟨w', hw'⟩ := mem_snoc i j h hilej hb1 hb2i hb2j w2
|
||||
rw [hw']
|
||||
apply congrArg (contr i j _ _ _ _ _) (hI w w')
|
||||
|
||||
lemma eq_snoc_castSucc {k n : ℕ} (b1 : Fin k.succ → Fin n) :
|
||||
b1 = Fin.snoc (b1 ∘ Fin.castSucc) (b1 (Fin.last k)) := by
|
||||
funext i
|
||||
rcases Fin.eq_castSucc_or_eq_last i with h1 | h1
|
||||
· obtain ⟨i, rfl⟩ := h1
|
||||
simp
|
||||
· subst h1
|
||||
simp
|
||||
|
||||
def fromMaps {n k : ℕ} (c : Fin n → 𝓔) (b1 b2 : Fin k → Fin n)
|
||||
(hi : ∀ i, c (b2 i) = ξ (c (b1 i)))
|
||||
(hb1ltb2 : ∀ i, b1 i < b2 i)
|
||||
(hb1 : StrictMono b1)
|
||||
(hb1neb2 : ∀ r1 r2, b1 r1 ≠ b2 r2)
|
||||
(hb2 : Function.Injective b2) :
|
||||
WickContract c b1 b2 := by
|
||||
match k with
|
||||
| 0 =>
|
||||
refine castMaps ?_ ?_ ?_ string
|
||||
· rfl
|
||||
· exact funext (fun i => Fin.elim0 i)
|
||||
· exact funext (fun i => Fin.elim0 i)
|
||||
| Nat.succ k =>
|
||||
refine castMaps rfl (eq_snoc_castSucc b1).symm (eq_snoc_castSucc b2).symm
|
||||
(contr (b1 (Fin.last k)) (b2 (Fin.last k)) (hi (Fin.last k)) (hb1ltb2 (Fin.last k)) (fun r => hb1 (Fin.castSucc_lt_last r)) ?_ ?_
|
||||
(fromMaps c (b1 ∘ Fin.castSucc) (b2 ∘ Fin.castSucc) (fun i => hi (Fin.castSucc i))
|
||||
(fun i => hb1ltb2 (Fin.castSucc i)) (StrictMono.comp hb1 Fin.strictMono_castSucc)
|
||||
?_ ?_
|
||||
))
|
||||
· exact fun r a => hb1neb2 (Fin.last k) r.castSucc a.symm
|
||||
· exact fun r => hb2.eq_iff.mp.mt (Fin.ne_last_of_lt (Fin.castSucc_lt_last r ))
|
||||
· exact fun r1 r2 => hb1neb2 r1.castSucc r2.castSucc
|
||||
· exact Function.Injective.comp hb2 (Fin.castSucc_injective k)
|
||||
|
||||
lemma eq_from_maps {n k : ℕ} {c : Fin n → 𝓔} {b1 b2 : Fin k → Fin n}
|
||||
(w : WickContract c b1 b2) :
|
||||
w = fromMaps c w.boundFst w.boundSnd w.color_boundSnd_eq_dual_boundFst
|
||||
w.boundFst_lt_boundSnd w.boundFst_strictMono w.boundFst_neq_boundSnd w.boundSnd_injective := by
|
||||
exact is_subsingleton.allEq w _
|
||||
|
||||
structure struc {n : ℕ} (c : Fin n → 𝓔) where
|
||||
k : ℕ
|
||||
b1 : Fin k ↪o Fin n
|
||||
b2 : Fin k ↪ Fin n
|
||||
b2_color_eq_dual_b1 : ∀ i, c (b2 i) = ξ (c (b1 i))
|
||||
b1_lt_b2 : ∀ i, b1 i < b2 i
|
||||
b1_neq_b2 : ∀ r1 r2, b1 r1 ≠ b2 r2
|
||||
|
||||
def strucEquivSigma {n : ℕ} (c : Fin n → 𝓔) :
|
||||
struc c ≃ Σ (k : ℕ) (b1 : Fin k → Fin n) (b2 : Fin k → Fin n), WickContract c b1 b2 where
|
||||
toFun s := ⟨s.k, s.b1, s.b2, fromMaps c s.b1 s.b2 s.b2_color_eq_dual_b1
|
||||
s.b1_lt_b2 s.b1.strictMono s.b1_neq_b2 s.b2.inj'⟩
|
||||
invFun x :=
|
||||
match x with
|
||||
| ⟨k, b1, b2, w⟩ => ⟨k, OrderEmbedding.ofStrictMono b1 w.boundFst_strictMono,
|
||||
⟨b2, w.boundSnd_injective⟩,
|
||||
w.color_boundSnd_eq_dual_boundFst, w.boundFst_lt_boundSnd, w.boundFst_neq_boundSnd⟩
|
||||
left_inv s := rfl
|
||||
right_inv w := by
|
||||
match w with
|
||||
| ⟨k, b1, b2, w⟩ =>
|
||||
simp only [OrderEmbedding.coe_ofStrictMono, Function.Embedding.coeFn_mk, Sigma.mk.inj_iff,
|
||||
heq_eq_eq, true_and]
|
||||
exact (eq_from_maps w).symm
|
||||
|
||||
|
||||
end WickContract
|
||||
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue