refactor: Rename superCommute for CrAnAlgebra
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jstoobysmith
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32aefb7eb7
commit
289050c829
13 changed files with 428 additions and 425 deletions
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@ -142,10 +142,10 @@ lemma timeOrder_ofState_ofState_not_ordered_eq_timeOrder {φ ψ : 𝓕.States} (
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have hx := IsTotal.total (r := timeOrderRel) ψ φ
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simp_all
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lemma timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel
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lemma timeOrder_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel
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{φ ψ : 𝓕.CrAnStates} (h : ¬ crAnTimeOrderRel φ ψ) :
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𝓣ᶠ([ofCrAnState φ, ofCrAnState ψ]ₛca) = 0 := by
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rw [superCommute_ofCrAnState_ofCrAnState]
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rw [superCommuteF_ofCrAnState_ofCrAnState]
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simp only [instCommGroup.eq_1, Algebra.smul_mul_assoc, map_sub, map_smul]
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rw [← ofCrAnList_singleton, ← ofCrAnList_singleton,
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← ofCrAnList_append, ← ofCrAnList_append, timeOrder_ofCrAnList, timeOrder_ofCrAnList]
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@ -160,51 +160,51 @@ lemma timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel
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· rw [crAnTimeOrderList_pair_ordered]
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simp_all
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lemma timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_right
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lemma timeOrder_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_right
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{φ ψ : 𝓕.CrAnStates} (h : ¬ crAnTimeOrderRel φ ψ) (a : 𝓕.CrAnAlgebra) :
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𝓣ᶠ(a * [ofCrAnState φ, ofCrAnState ψ]ₛca) = 0 := by
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rw [timeOrder_timeOrder_right,
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timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel h]
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timeOrder_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel h]
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simp
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lemma timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_left
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lemma timeOrder_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_left
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{φ ψ : 𝓕.CrAnStates} (h : ¬ crAnTimeOrderRel φ ψ) (a : 𝓕.CrAnAlgebra) :
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𝓣ᶠ([ofCrAnState φ, ofCrAnState ψ]ₛca * a) = 0 := by
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rw [timeOrder_timeOrder_left,
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timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel h]
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timeOrder_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel h]
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simp
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lemma timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_mid
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lemma timeOrder_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_mid
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{φ ψ : 𝓕.CrAnStates} (h : ¬ crAnTimeOrderRel φ ψ) (a b : 𝓕.CrAnAlgebra) :
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𝓣ᶠ(a * [ofCrAnState φ, ofCrAnState ψ]ₛca * b) = 0 := by
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rw [timeOrder_timeOrder_mid,
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timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel h]
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timeOrder_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel h]
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simp
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lemma timeOrder_superCommute_superCommute_ofCrAnState_not_crAnTimeOrderRel
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lemma timeOrder_superCommuteF_superCommuteF_ofCrAnState_not_crAnTimeOrderRel
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{φ1 φ2 : 𝓕.CrAnStates} (h : ¬ crAnTimeOrderRel φ1 φ2) (a : 𝓕.CrAnAlgebra) :
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𝓣ᶠ([a, [ofCrAnState φ1, ofCrAnState φ2]ₛca]ₛca) = 0 := by
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rw [← bosonicProj_add_fermionicProj a]
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simp only [map_add, LinearMap.add_apply]
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rw [bosonic_superCommute (Submodule.coe_mem (bosonicProj a))]
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rw [bosonic_superCommuteF (Submodule.coe_mem (bosonicProj a))]
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simp only [map_sub]
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rw [timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_left h]
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rw [timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_right h]
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rw [timeOrder_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_left h]
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rw [timeOrder_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_right h]
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simp only [sub_self, zero_add]
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rw [← ofCrAnList_singleton, ← ofCrAnList_singleton]
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rcases superCommute_ofCrAnList_ofCrAnList_bosonic_or_fermionic [φ1] [φ2] with h' | h'
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· rw [superCommute_bonsonic h']
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rcases superCommuteF_ofCrAnList_ofCrAnList_bosonic_or_fermionic [φ1] [φ2] with h' | h'
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· rw [superCommuteF_bonsonic h']
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simp only [ofCrAnList_singleton, map_sub]
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rw [timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_left h]
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rw [timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_right h]
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rw [timeOrder_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_left h]
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rw [timeOrder_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_right h]
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simp
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· rw [superCommute_fermionic_fermionic (Submodule.coe_mem (fermionicProj a)) h']
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· rw [superCommuteF_fermionic_fermionic (Submodule.coe_mem (fermionicProj a)) h']
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simp only [ofCrAnList_singleton, map_add]
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rw [timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_left h]
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rw [timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_right h]
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rw [timeOrder_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_left h]
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rw [timeOrder_superCommuteF_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel_right h]
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simp
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lemma timeOrder_superCommute_ofCrAnState_superCommute_not_crAnTimeOrderRel
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lemma timeOrder_superCommuteF_ofCrAnState_superCommuteF_not_crAnTimeOrderRel
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{φ1 φ2 φ3 : 𝓕.CrAnStates} (h12 : ¬ crAnTimeOrderRel φ1 φ2)
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(h13 : ¬ crAnTimeOrderRel φ1 φ3) :
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𝓣ᶠ([ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca) = 0 := by
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@ -213,14 +213,14 @@ lemma timeOrder_superCommute_ofCrAnState_superCommute_not_crAnTimeOrderRel
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simp only [instCommGroup.eq_1, ofList_singleton, ofCrAnList_singleton, neg_smul, map_smul,
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map_sub, map_neg, smul_eq_zero]
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right
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rw [timeOrder_superCommute_superCommute_ofCrAnState_not_crAnTimeOrderRel h12]
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rw [superCommute_ofCrAnState_ofCrAnState_symm φ3]
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rw [timeOrder_superCommuteF_superCommuteF_ofCrAnState_not_crAnTimeOrderRel h12]
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rw [superCommuteF_ofCrAnState_ofCrAnState_symm φ3]
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simp only [smul_zero, neg_zero, instCommGroup.eq_1, neg_smul, map_neg, map_smul, smul_neg,
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sub_neg_eq_add, zero_add, smul_eq_zero]
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rw [timeOrder_superCommute_superCommute_ofCrAnState_not_crAnTimeOrderRel h13]
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rw [timeOrder_superCommuteF_superCommuteF_ofCrAnState_not_crAnTimeOrderRel h13]
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simp
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lemma timeOrder_superCommute_ofCrAnState_superCommute_not_crAnTimeOrderRel'
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lemma timeOrder_superCommuteF_ofCrAnState_superCommuteF_not_crAnTimeOrderRel'
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{φ1 φ2 φ3 : 𝓕.CrAnStates} (h12 : ¬ crAnTimeOrderRel φ2 φ1)
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(h13 : ¬ crAnTimeOrderRel φ3 φ1) :
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𝓣ᶠ([ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca) = 0 := by
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@ -229,14 +229,14 @@ lemma timeOrder_superCommute_ofCrAnState_superCommute_not_crAnTimeOrderRel'
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simp only [instCommGroup.eq_1, ofList_singleton, ofCrAnList_singleton, neg_smul, map_smul,
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map_sub, map_neg, smul_eq_zero]
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right
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rw [superCommute_ofCrAnState_ofCrAnState_symm φ1]
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rw [superCommuteF_ofCrAnState_ofCrAnState_symm φ1]
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simp only [instCommGroup.eq_1, neg_smul, map_neg, map_smul, smul_neg, neg_neg]
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rw [timeOrder_superCommute_superCommute_ofCrAnState_not_crAnTimeOrderRel h12]
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rw [timeOrder_superCommuteF_superCommuteF_ofCrAnState_not_crAnTimeOrderRel h12]
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simp only [smul_zero, zero_sub, neg_eq_zero, smul_eq_zero]
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rw [timeOrder_superCommute_superCommute_ofCrAnState_not_crAnTimeOrderRel h13]
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rw [timeOrder_superCommuteF_superCommuteF_ofCrAnState_not_crAnTimeOrderRel h13]
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simp
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lemma timeOrder_superCommute_ofCrAnState_superCommute_all_not_crAnTimeOrderRel
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lemma timeOrder_superCommuteF_ofCrAnState_superCommuteF_all_not_crAnTimeOrderRel
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(φ1 φ2 φ3 : 𝓕.CrAnStates) (h : ¬
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(crAnTimeOrderRel φ1 φ2 ∧ crAnTimeOrderRel φ1 φ3 ∧
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crAnTimeOrderRel φ2 φ1 ∧ crAnTimeOrderRel φ2 φ3 ∧
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@ -245,13 +245,13 @@ lemma timeOrder_superCommute_ofCrAnState_superCommute_all_not_crAnTimeOrderRel
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simp only [not_and] at h
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by_cases h23 : ¬ crAnTimeOrderRel φ2 φ3
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· simp_all only [IsEmpty.forall_iff, implies_true]
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rw [timeOrder_superCommute_superCommute_ofCrAnState_not_crAnTimeOrderRel h23]
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rw [timeOrder_superCommuteF_superCommuteF_ofCrAnState_not_crAnTimeOrderRel h23]
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simp_all only [Decidable.not_not, forall_const]
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by_cases h32 : ¬ crAnTimeOrderRel φ3 φ2
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· simp_all only [not_false_eq_true, implies_true]
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rw [superCommute_ofCrAnState_ofCrAnState_symm]
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rw [superCommuteF_ofCrAnState_ofCrAnState_symm]
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simp only [instCommGroup.eq_1, neg_smul, map_neg, map_smul, neg_eq_zero, smul_eq_zero]
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rw [timeOrder_superCommute_superCommute_ofCrAnState_not_crAnTimeOrderRel h32]
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rw [timeOrder_superCommuteF_superCommuteF_ofCrAnState_not_crAnTimeOrderRel h32]
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simp
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simp_all only [imp_false, Decidable.not_not]
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by_cases h12 : ¬ crAnTimeOrderRel φ1 φ2
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@ -259,7 +259,7 @@ lemma timeOrder_superCommute_ofCrAnState_superCommute_all_not_crAnTimeOrderRel
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intro h13
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apply h12
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exact IsTrans.trans φ1 φ3 φ2 h13 h32
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rw [timeOrder_superCommute_ofCrAnState_superCommute_not_crAnTimeOrderRel h12 h13]
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rw [timeOrder_superCommuteF_ofCrAnState_superCommuteF_not_crAnTimeOrderRel h12 h13]
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simp_all only [Decidable.not_not, forall_const]
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have h13 : crAnTimeOrderRel φ1 φ3 := IsTrans.trans φ1 φ2 φ3 h12 h23
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simp_all only [forall_const]
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@ -269,15 +269,15 @@ lemma timeOrder_superCommute_ofCrAnState_superCommute_all_not_crAnTimeOrderRel
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intro h31
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apply h21
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exact IsTrans.trans φ2 φ3 φ1 h23 h31
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rw [timeOrder_superCommute_ofCrAnState_superCommute_not_crAnTimeOrderRel' h21 h31]
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rw [timeOrder_superCommuteF_ofCrAnState_superCommuteF_not_crAnTimeOrderRel' h21 h31]
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simp_all only [Decidable.not_not, forall_const]
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refine False.elim (h ?_)
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exact IsTrans.trans φ3 φ2 φ1 h32 h21
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lemma timeOrder_superCommute_ofCrAnState_ofCrAnState_eq_time
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lemma timeOrder_superCommuteF_ofCrAnState_ofCrAnState_eq_time
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{φ ψ : 𝓕.CrAnStates} (h1 : crAnTimeOrderRel φ ψ) (h2 : crAnTimeOrderRel ψ φ) :
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𝓣ᶠ([ofCrAnState φ, ofCrAnState ψ]ₛca) = [ofCrAnState φ, ofCrAnState ψ]ₛca := by
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rw [superCommute_ofCrAnState_ofCrAnState]
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rw [superCommuteF_ofCrAnState_ofCrAnState]
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simp only [instCommGroup.eq_1, Algebra.smul_mul_assoc, map_sub, map_smul]
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rw [← ofCrAnList_singleton, ← ofCrAnList_singleton,
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← ofCrAnList_append, ← ofCrAnList_append, timeOrder_ofCrAnList, timeOrder_ofCrAnList]
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