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refactor: Simplify proofs

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jstoobysmith 2024-10-24 06:10:08 +00:00
parent 33a42c7e06
commit 95857993b5
6 changed files with 366 additions and 496 deletions
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33
HepLean/Tensors/ComplexLorentz/Basis.lean
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@ -79,11 +79,32 @@ lemma perm_basisVector {n m : ℕ} {c : Fin n → complexLorentzTensor.C}
eqToIso.hom, Functor.mapIso_inv, eqToIso.inv, LinearEquiv.ofLinear_apply]
rw [basis_eq_FDiscrete]
def contrBasisVectorMul {n : ℕ} {c : Fin n.succ.succ → complexLorentzTensor.C}
(i : Fin n.succ.succ) (j : Fin n.succ)
(b : Π k, Fin (complexLorentzTensor.repDim (c k))) : ℂ :=
(if (b i).val = (b (i.succAbove j)).val then (1 : ℂ) else 0)
lemma contrBasisVectorMul_neg {n : ℕ} {c : Fin n.succ.succ → complexLorentzTensor.C}
{i : Fin n.succ.succ} {j : Fin n.succ} {b : Π k, Fin (complexLorentzTensor.repDim (c k))}
(h : ¬ (b i).val = (b (i.succAbove j)).val := by decide) :
contrBasisVectorMul i j b = 0 := by
rw [contrBasisVectorMul]
simp
exact h
lemma contrBasisVectorMul_pos {n : ℕ} {c : Fin n.succ.succ → complexLorentzTensor.C}
{i : Fin n.succ.succ} {j : Fin n.succ} {b : Π k, Fin (complexLorentzTensor.repDim (c k))}
(h : (b i).val = (b (i.succAbove j)).val := by decide) :
contrBasisVectorMul i j b = 1 := by
rw [contrBasisVectorMul]
simp
exact h
lemma contr_basisVector {n : ℕ} {c : Fin n.succ.succ → complexLorentzTensor.C}
{i : Fin n.succ.succ} {j : Fin n.succ} {h : c (i.succAbove j) = complexLorentzTensor.τ (c i)}
(b : Π k, Fin (complexLorentzTensor.repDim (c k))) :
(contr i j h (tensorNode (basisVector c b))).tensor = (if (b i).val = (b (i.succAbove j)).val
then (1 : ℂ) else 0) • basisVector (c ∘ Fin.succAbove i ∘ Fin.succAbove j)
(contr i j h (tensorNode (basisVector c b))).tensor = (contrBasisVectorMul i j b) •
basisVector (c ∘ Fin.succAbove i ∘ Fin.succAbove j)
(fun k => b (i.succAbove (j.succAbove k))) := by
rw [contr_tensor]
simp only [Nat.succ_eq_add_one, tensorNode_tensor]
@ -102,8 +123,7 @@ lemma contr_basisVector_tree {n : ℕ} {c : Fin n.succ.succ → complexLorentzT
{i : Fin n.succ.succ} {j : Fin n.succ} {h : c (i.succAbove j) = complexLorentzTensor.τ (c i)}
(b : Π k, Fin (complexLorentzTensor.repDim (c k))) :
(contr i j h (tensorNode (basisVector c b))).tensor =
(smul ((if (b i).val = (b (i.succAbove j)).val
then (1 : ℂ) else 0)) (tensorNode ( basisVector (c ∘ Fin.succAbove i ∘ Fin.succAbove j)
(smul (contrBasisVectorMul i j b) (tensorNode ( basisVector (c ∘ Fin.succAbove i ∘ Fin.succAbove j)
(fun k => b (i.succAbove (j.succAbove k)))) )).tensor := by
exact contr_basisVector _
@ -113,7 +133,7 @@ lemma contr_basisVector_tree_pos {n : ℕ} {c : Fin n.succ.succ → complexLore
(contr i j h (tensorNode (basisVector c b))).tensor =
((tensorNode ( basisVector (c ∘ Fin.succAbove i ∘ Fin.succAbove j)
(fun k => b (i.succAbove (j.succAbove k)))))).tensor := by
rw [contr_basisVector_tree]
rw [contr_basisVector_tree, contrBasisVectorMul]
rw [if_pos hn]
simp [smul_tensor]
@ -122,10 +142,11 @@ lemma contr_basisVector_tree_neg {n : ℕ} {c : Fin n.succ.succ → complexLore
(b : Π k, Fin (complexLorentzTensor.repDim (c k))) (hn : ¬ (b i).val = (b (i.succAbove j)).val := by decide) :
(contr i j h (tensorNode (basisVector c b))).tensor =
(tensorNode 0).tensor := by
rw [contr_basisVector_tree]
rw [contr_basisVector_tree, contrBasisVectorMul]
rw [if_neg hn]
simp [smul_tensor]
def prodBasisVecEquiv {n m : ℕ} {c : Fin n → complexLorentzTensor.C}
{c1 : Fin m → complexLorentzTensor.C} (i : Fin n ⊕ Fin m) :
Sum.elim (fun i => Fin (complexLorentzTensor.repDim (c i))) (fun i => Fin (complexLorentzTensor.repDim (c1 i)))