feat: Some simple extensions of lemmas

HepLean/Tensors/TensorSpecies/MetricTensor.lean

@@ -5,6 +5,10 @@ Authors: Joseph Tooby-Smith
 -/
 import HepLean.Tensors.TensorSpecies.UnitTensor
 import HepLean.Tensors.TensorSpecies.ContractLemmas
+import HepLean.Tensors.Tree.NodeIdentities.ProdComm
+import HepLean.Tensors.Tree.NodeIdentities.PermProd
+import HepLean.Tensors.Tree.NodeIdentities.ContrSwap
+import HepLean.Tensors.Tree.NodeIdentities.PermContr
 /-!
 
 ## Metrics in tensor trees
@@ -28,6 +32,14 @@ def metricTensor (S : TensorSpecies) (c : S.C) : S.F.obj (OverColor.mk ![c, c])
 
 variable {S : TensorSpecies}
 
+lemma metricTensor_congr {c c' : S.C} (h : c = c') : {S.metricTensor c | μ ν}ᵀ.tensor =
+    (perm  (OverColor.equivToHomEq (Equiv.refl _) (fun x => by subst h; fin_cases x <;> rfl ))
+    {S.metricTensor c' | μ ν}ᵀ).tensor := by
+  subst h
+  change _ = (S.F.map (𝟙 _)).hom (S.metricTensor c)
+  simp
+
+
 lemma pairIsoSep_inv_metricTensor (c : S.C) :
     (Discrete.pairIsoSep S.FD).inv.hom (S.metricTensor c) =
     (S.metric.app (Discrete.mk c)).hom (1 : S.k) := by
@@ -48,12 +60,9 @@ lemma contr_metric_braid_unit (c : S.C) :  (((S.FD.obj (Discrete.mk c)) ◁
     (OverColor.Discrete.pairIsoSep S.FD).inv.hom (S.metricTensor (S.τ c)))))))) =
     (β_ (S.FD.obj (Discrete.mk (S.τ c))) (S.FD.obj (Discrete.mk c))).hom.hom
       ((S.unit.app (Discrete.mk c)).hom (1 : S.k)) := by
-  have hx : Function.Injective (β_ (S.FD.obj (Discrete.mk c)) (S.FD.obj (Discrete.mk (S.τ c))) ).hom.hom := by
-    change  Function.Injective (β_ (S.FD.obj (Discrete.mk c)).V (S.FD.obj (Discrete.mk (S.τ c))).V ).hom
-    exact (β_ (S.FD.obj (Discrete.mk c)).V (S.FD.obj (Discrete.mk (S.τ c))).V ).toLinearEquiv.toEquiv.injective
-  apply hx
+  apply (β_ _ _).toLinearEquiv.toEquiv.injective
   rw [pairIsoSep_inv_metricTensor, pairIsoSep_inv_metricTensor]
-  rw [S.contr_metric c]
+  erw [S.contr_metric c]
   change  _ = (β_  (S.FD.obj { as := S.τ c }) (S.FD.obj { as := c })).inv.hom
     ((β_ (S.FD.obj { as := S.τ c }) (S.FD.obj { as := c })).hom.hom _)
   rw [Discrete.rep_iso_inv_hom_apply]
@@ -66,16 +75,53 @@ lemma metricTensor_contr_dual_metricTensor_perm_cond (c : S.C) : ∀ (x : Fin (N
   · rfl
   · rfl
 
-/-- The contraction of a metric tensor with its dual gives the unit. -/
+/-- The contraction of a metric tensor with its dual via the inner indices gives the unit. -/
 lemma metricTensor_contr_dual_metricTensor_eq_unit (c : S.C) :
-    {S.metricTensor c | μ ν ⊗ S.metricTensor (S.τ c) | ν ρ}ᵀ.tensor =
-    (perm (OverColor.equivToHomEq (finMapToEquiv ![1, 0] ![1, 0])
-      (metricTensor_contr_dual_metricTensor_perm_cond c)) {S.unitTensor c | μ ρ}ᵀ).tensor := by
+    {S.metricTensor c | μ ν ⊗ S.metricTensor (S.τ c) | ν ρ}ᵀ.tensor = ({S.unitTensor c | μ ρ}ᵀ |>
+    perm (OverColor.equivToHomEq (finMapToEquiv ![1, 0] ![1, 0])
+      (metricTensor_contr_dual_metricTensor_perm_cond c))).tensor := by
   rw [contr_two_two_inner, contr_metric_braid_unit, Discrete.pairIsoSep_β]
   change (S.F.map _ ≫ S.F.map _ ).hom _ = _
   rw [← S.F.map_comp]
   rfl
 
+/-- The contraction of a metric tensor with its dual via the outer indices gives the unit. -/
+lemma metricTensor_contr_dual_metricTensor_outer_eq_unit  (c : S.C) :
+    {S.metricTensor c | ν μ ⊗ S.metricTensor (S.τ c) | ρ ν}ᵀ.tensor = ({S.unitTensor c | μ ρ}ᵀ |>
+    perm (OverColor.equivToHomEq
+      (finMapToEquiv ![1, 0] ![1, 0]) (fun x => by fin_cases x <;> rfl))).tensor := by
+  conv_lhs =>
+    rw [contr_tensor_eq <| prod_tensor_eq_fst <| metricTensor_congr (S.τ_involution c).symm]
+    rw [contr_tensor_eq <| prod_comm _ _ _ _]
+    rw [perm_contr_congr 2 1 (by rfl) (by rfl)]
+    rw [perm_tensor_eq <| contr_tensor_eq <| prod_perm_right _ _ _ _]
+    rw [perm_tensor_eq <| perm_contr_congr 2 1 (by rfl) (by rfl)]
+    rw [perm_perm]
+    rw [perm_tensor_eq <| contr_swap _ _]
+    rw [perm_perm]
+    erw [perm_tensor_eq <| metricTensor_contr_dual_metricTensor_eq_unit _]
+    rw [perm_perm]
+    rw [perm_tensor_eq <| dual_unitTensor_eq_perm _]
+    rw [perm_perm]
+  apply perm_congr _ rfl
+  apply OverColor.Hom.fin_ext
+  intro i
+  simp only [Functor.id_obj, mk_hom, Function.comp_apply, Equiv.refl_symm, Equiv.coe_refl, id_eq,
+    Fin.zero_eta, Matrix.cons_val_zero, List.pmap.eq_1, ContrPair.contrSwapHom,
+    extractOne_homToEquiv, Category.assoc, Hom.hom_comp, Over.comp_left, equivToHomEq_hom_left,
+    Equiv.toFun_as_coe, types_comp_apply, finMapToEquiv_apply, mkIso_hom_hom_left_apply]
+  rw [extractTwo_hom_left_apply]
+  simp only [mk_left, braidPerm_toEquiv, Equiv.symm_trans_apply, Equiv.symm_symm,
+    Equiv.sumComm_symm, Equiv.sumComm_apply, finExtractOnePerm_symm_apply, Equiv.trans_apply,
+    Equiv.symm_apply_apply, Sum.swap_swap, Equiv.apply_symm_apply, finExtractOne_symm_inr_apply,
+    Fin.zero_succAbove, List.pmap.eq_2, Fin.mk_one, List.pmap.eq_1, Matrix.cons_val_one,
+    Matrix.head_cons, extractTwo_hom_left_apply, permProdRight_toEquiv, equivToHomEq_toEquiv,
+    Equiv.sumCongr_refl, Equiv.refl_trans, Equiv.symm_trans_self, Equiv.refl_symm, Equiv.refl_apply,
+    predAboveI_succAbove, finExtractOnePerm_apply]
+  fin_cases i
+  · decide
+  · decide
+
 end TensorSpecies
 
 end