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feat: Important fix to ContrContr

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jstoobysmith 2024-10-29 10:37:18 +00:00
parent 83ff8f5358
commit 4756466001
7 changed files with 184 additions and 2 deletions
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36
HepLean/Tensors/Tree/NodeIdentities/PermContr.lean
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@ -4,6 +4,8 @@ Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.Tensors.Tree.Basic
import HepLean.Tensors.Tree.NodeIdentities.Congr
import HepLean.Tensors.Tree.NodeIdentities.Basic
/-!
# The commutativity of Permutations and contractions.
@ -256,4 +258,38 @@ lemma perm_contr {n : ℕ} {c : Fin n.succ.succ → S.C} {c1 : Fin n.succ.succ
rw [S.contrMap_naturality σ]
rfl
lemma perm_contr_congr_mkIso_cond {n : ℕ} {c : Fin n.succ.succ → S.C} {c1 : Fin n.succ.succ → S.C}
{i : Fin n.succ.succ} {j : Fin n.succ}
{σ : (OverColor.mk c) ⟶ (OverColor.mk c1)}
{i' : Fin n.succ.succ} {j' : Fin n.succ}
(hi : i' = ((Hom.toEquiv σ).symm i))
(hj : j' = (((Hom.toEquiv (extractOne i σ))).symm j)) :
c ∘ i'.succAbove ∘ j'.succAbove =
c ∘ Fin.succAbove ((Hom.toEquiv σ).symm i) ∘ Fin.succAbove ((Hom.toEquiv (extractOne i σ)).symm j) := by
rw [hi, hj]
lemma perm_contr_congr_contr_cond {n : ℕ} {c : Fin n.succ.succ → S.C} {c1 : Fin n.succ.succ → S.C}
{i : Fin n.succ.succ} {j : Fin n.succ}
(h : c1 (i.succAbove j) = S.τ (c1 i))
{σ : (OverColor.mk c) ⟶ (OverColor.mk c1)}
{i' : Fin n.succ.succ} {j' : Fin n.succ}
(hi : i' = ((Hom.toEquiv σ).symm i))
(hj : j' = (((Hom.toEquiv (extractOne i σ))).symm j)) :
c (i'.succAbove j') = S.τ (c i') := by
rw [hi, hj]
exact S.perm_contr_cond h σ
lemma perm_contr_congr {n : ℕ} {c : Fin n.succ.succ → S.C} {c1 : Fin n.succ.succ → S.C}
{i : Fin n.succ.succ} {j : Fin n.succ}
{h : c1 (i.succAbove j) = S.τ (c1 i)} {t : TensorTree S c}
{σ : (OverColor.mk c) ⟶ (OverColor.mk c1)}
(i' : Fin n.succ.succ) (j' : Fin n.succ)
(hi : i' = ((Hom.toEquiv σ).symm i) := by decide)
(hj : j' = (((Hom.toEquiv (extractOne i σ))).symm j) := by decide) :
(contr i j h (perm σ t)).tensor = (perm ((mkIso (perm_contr_congr_mkIso_cond hi hj)).hom ≫
extractTwo i j σ) (contr i' j' (perm_contr_congr_contr_cond h hi hj) t)).tensor := by
rw [perm_contr]
rw [perm_tensor_eq <| contr_congr i' j' hi.symm hj.symm]
rw [perm_perm]
end TensorTree