feat: Add alternative prod_assoc lemma

HepLean/Tensors/Tree/NodeIdentities/Basic.lean

@@ -107,6 +107,14 @@ lemma perm_eq_id {n : ℕ} {c : Fin n → S.C} (σ : (OverColor.mk c) ⟶ (OverC
     (h : σ = 𝟙 _) (t : TensorTree S c) : (perm σ t).tensor = t.tensor := by
   simp [perm_tensor, h]
 
+lemma perm_eq_of_eq_perm {n m : ℕ} {c : Fin n → S.C} {c1 : Fin m → S.C}
+    (σ : (OverColor.mk c) ≅ (OverColor.mk c1))
+    {t : TensorTree S c} {t2 : TensorTree S c1} (h : (perm σ.hom t).tensor = t2.tensor) :
+    t.tensor = (perm σ.inv t2).tensor := by
+  rw [perm_tensor, ← h]
+  change _ = (S.F.map σ.hom ≫ S.F.map σ.inv).hom _
+  simp only [Iso.map_hom_inv_id, Action.id_hom, ModuleCat.id_apply]
+
 /-!
 
 ## Additive identities

HepLean/Tensors/Tree/NodeIdentities/ProdAssoc.lean

@@ -4,6 +4,7 @@ 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.Basic
 /-!
 
 # Associativity of products
@@ -104,4 +105,10 @@ theorem prod_assoc (t : TensorTree S c) (t2 : TensorTree S c2) (t3 : TensorTree
     ModuleCat.coe_comp, Function.comp_apply]
   rfl
 
+/-- The alternative version of associativity for `prod` where the permutation is on the opposite
+  side. -/
+lemma prod_assoc' (t : TensorTree S c) (t2 : TensorTree S c2) (t3 : TensorTree S c3) :
+    (prod (prod t t2) t3).tensor = (perm (assocPerm c c2 c3).inv (prod t (prod t2 t3))).tensor :=
+  perm_eq_of_eq_perm _ (prod_assoc c c2 c3 t t2 t3).symm
+
 end TensorTree