refactor: Free simps
This commit is contained in:
jstoobysmith
parent
e5c85ac109
commit
22636db606
9 changed files with 205 additions and 171 deletions
|
|
@ -56,7 +56,7 @@ lemma ι_timeOrder_superCommute_superCommute_eq_time_ofCrAnList {φ1 φ2 φ3 :
|
|||
rw [crAnTimeOrderSign, Wick.koszulSign_perm_eq _ _ φ1 _ _ _ _ _ hp, ← crAnTimeOrderSign]
|
||||
· simp
|
||||
· intro φ4 hφ4
|
||||
simp at hφ4
|
||||
simp only [List.mem_cons, List.mem_singleton, List.not_mem_nil, or_false] at hφ4
|
||||
rcases hφ4 with hφ4 | hφ4 | hφ4
|
||||
all_goals
|
||||
subst hφ4
|
||||
|
|
@ -78,7 +78,7 @@ lemma ι_timeOrder_superCommute_superCommute_eq_time_ofCrAnList {φ1 φ2 φ3 :
|
|||
rw [crAnTimeOrderSign, Wick.koszulSign_perm_eq _ _ φ1 _ _ _ _ _ hp231, ← crAnTimeOrderSign]
|
||||
· simp
|
||||
· intro φ4 hφ4
|
||||
simp at hφ4
|
||||
simp only [List.mem_cons, List.mem_singleton, List.not_mem_nil, or_false] at hφ4
|
||||
rcases hφ4 with hφ4 | hφ4 | hφ4
|
||||
all_goals
|
||||
subst hφ4
|
||||
|
|
@ -99,24 +99,25 @@ lemma ι_timeOrder_superCommute_superCommute_eq_time_ofCrAnList {φ1 φ2 φ3 :
|
|||
rw [crAnTimeOrderSign, Wick.koszulSign_perm_eq _ _ φ1 _ _ _ _ _ hp, ← crAnTimeOrderSign]
|
||||
· simp
|
||||
· intro φ4 hφ4
|
||||
simp at hφ4
|
||||
simp only [List.mem_cons, List.mem_singleton, List.not_mem_nil, or_false] at hφ4
|
||||
rcases hφ4 with hφ4 | hφ4 | hφ4
|
||||
all_goals
|
||||
subst hφ4
|
||||
simp_all
|
||||
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, ← ofCrAnList_singleton]
|
||||
rw [superCommute_ofCrAnList_ofCrAnList]
|
||||
simp
|
||||
simp only [List.singleton_append, instCommGroup.eq_1, ofList_singleton, map_sub, map_smul]
|
||||
rw [superCommute_ofCrAnList_ofCrAnList, superCommute_ofCrAnList_ofCrAnList]
|
||||
simp [mul_sub, sub_mul, ← ofCrAnList_append]
|
||||
simp only [List.cons_append, List.nil_append, instCommGroup.eq_1, ofList_singleton, mul_sub, ←
|
||||
ofCrAnList_append, Algebra.mul_smul_comm, sub_mul, List.append_assoc, Algebra.smul_mul_assoc,
|
||||
map_sub, map_smul]
|
||||
rw [h123, h132, h231, h321]
|
||||
simp [smul_smul]
|
||||
simp only [smul_smul]
|
||||
rw [mul_comm, ← smul_smul, mul_comm, ← smul_smul]
|
||||
rw [← smul_sub, ← smul_sub, smul_smul, mul_comm, ← smul_smul, ← smul_sub]
|
||||
simp
|
||||
simp only [smul_eq_zero]
|
||||
right
|
||||
rw [← smul_mul_assoc, ← mul_smul_comm, mul_assoc]
|
||||
|
||||
rw [← smul_mul_assoc, ← mul_smul_comm]
|
||||
rw [smul_sub]
|
||||
rw [← smul_mul_assoc, ← mul_smul_comm]
|
||||
|
|
@ -129,9 +130,10 @@ lemma ι_timeOrder_superCommute_superCommute_eq_time_ofCrAnList {φ1 φ2 φ3 :
|
|||
congr
|
||||
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, ← ofCrAnList_singleton]
|
||||
rw [superCommute_ofCrAnList_ofCrAnList]
|
||||
simp
|
||||
simp only [List.singleton_append, instCommGroup.eq_1, ofList_singleton, map_sub, map_smul]
|
||||
rw [superCommute_ofCrAnList_ofCrAnList, superCommute_ofCrAnList_ofCrAnList]
|
||||
simp [smul_sub]
|
||||
simp only [List.cons_append, List.nil_append, instCommGroup.eq_1, ofList_singleton, map_sub,
|
||||
map_smul, smul_sub]
|
||||
simp_all
|
||||
|
||||
lemma ι_timeOrder_superCommute_superCommute_ofCrAnList {φ1 φ2 φ3 : 𝓕.CrAnStates}
|
||||
|
|
@ -156,14 +158,14 @@ lemma ι_timeOrder_superCommute_superCommute {φ1 φ2 φ3 : 𝓕.CrAnStates} (a
|
|||
apply Submodule.span_induction
|
||||
· intro x hx
|
||||
obtain ⟨φs, rfl⟩ := hx
|
||||
simp [pb]
|
||||
simp only [ofListBasis_eq_ofList, pb]
|
||||
let pa (a : 𝓕.CrAnAlgebra) (hc : a ∈ Submodule.span ℂ (Set.range ofCrAnListBasis)) :
|
||||
Prop := ι 𝓣ᶠ(a * [ofCrAnState φ1, [ofCrAnState φ2, ofCrAnState φ3]ₛca]ₛca * ofCrAnList φs) = 0
|
||||
change pa a (Basis.mem_span _ a)
|
||||
apply Submodule.span_induction
|
||||
· intro x hx
|
||||
obtain ⟨φs', rfl⟩ := hx
|
||||
simp [pa]
|
||||
simp only [ofListBasis_eq_ofList, pa]
|
||||
exact ι_timeOrder_superCommute_superCommute_ofCrAnList φs' φs
|
||||
· simp [pa]
|
||||
· intro x y hx hy hpx hpy
|
||||
|
|
@ -187,7 +189,7 @@ lemma ι_timeOrder_superCommute_eq_time {φ ψ : 𝓕.CrAnStates}
|
|||
apply Submodule.span_induction
|
||||
· intro x hx
|
||||
obtain ⟨φs, rfl⟩ := hx
|
||||
simp [pb]
|
||||
simp only [ofListBasis_eq_ofList, map_mul, pb]
|
||||
let pa (a : 𝓕.CrAnAlgebra) (hc : a ∈ Submodule.span ℂ (Set.range ofCrAnListBasis)) :
|
||||
Prop := ι 𝓣ᶠ(a * [ofCrAnState φ, ofCrAnState ψ]ₛca * ofCrAnList φs) =
|
||||
ι ([ofCrAnState φ, ofCrAnState ψ]ₛca * 𝓣ᶠ(a* ofCrAnList φs))
|
||||
|
|
@ -195,22 +197,23 @@ lemma ι_timeOrder_superCommute_eq_time {φ ψ : 𝓕.CrAnStates}
|
|||
apply Submodule.span_induction
|
||||
· intro x hx
|
||||
obtain ⟨φs', rfl⟩ := hx
|
||||
simp [pa]
|
||||
simp only [ofListBasis_eq_ofList, map_mul, pa]
|
||||
conv_lhs =>
|
||||
rw [← ofCrAnList_singleton, ← ofCrAnList_singleton, superCommute_ofCrAnList_ofCrAnList]
|
||||
simp [mul_sub, sub_mul, ← ofCrAnList_append]
|
||||
rw [timeOrder_ofCrAnList, timeOrder_ofCrAnList]
|
||||
have h1 : crAnTimeOrderSign (φs' ++ φ :: ψ :: φs) = crAnTimeOrderSign (φs' ++ ψ :: φ :: φs) := by
|
||||
have h1 : crAnTimeOrderSign (φs' ++ φ :: ψ :: φs) =
|
||||
crAnTimeOrderSign (φs' ++ ψ :: φ :: φs) := by
|
||||
trans crAnTimeOrderSign (φs' ++ [φ, ψ] ++ φs)
|
||||
simp
|
||||
simp only [List.append_assoc, List.cons_append, List.nil_append]
|
||||
rw [crAnTimeOrderSign]
|
||||
have hp : List.Perm [φ,ψ] [ψ,φ] := by exact List.Perm.swap ψ φ []
|
||||
rw [Wick.koszulSign_perm_eq _ _ φ _ _ _ _ _ hp]
|
||||
simp
|
||||
simp only [List.append_assoc, List.cons_append, List.singleton_append]
|
||||
rfl
|
||||
simp_all
|
||||
rw [h1]
|
||||
simp
|
||||
simp only [map_smul]
|
||||
have h1 := insertionSort_of_eq_list 𝓕.crAnTimeOrderRel φ φs' [φ, ψ] φs
|
||||
(by simp_all)
|
||||
rw [crAnTimeOrderList, show φs' ++ φ :: ψ :: φs = φs' ++ [φ, ψ] ++ φs by simp, h1]
|
||||
|
|
@ -244,7 +247,8 @@ lemma ι_timeOrder_superCommute_eq_time {φ ψ : 𝓕.CrAnStates}
|
|||
rw [← map_mul, ← map_mul, ← map_mul, ← map_mul]
|
||||
rw [← ofCrAnList_append, ← ofCrAnList_append, ← ofCrAnList_append, ← ofCrAnList_append]
|
||||
have h1 := insertionSort_of_takeWhile_filter 𝓕.crAnTimeOrderRel φ φs' φs
|
||||
simp at h1 ⊢
|
||||
simp only [decide_not, Bool.decide_and, List.append_assoc, List.cons_append,
|
||||
List.singleton_append, Algebra.mul_smul_comm, map_mul] at h1 ⊢
|
||||
rw [← h1]
|
||||
rw [← crAnTimeOrderList]
|
||||
by_cases hq : (𝓕 |>ₛ φ) ≠ (𝓕 |>ₛ ψ)
|
||||
|
|
@ -252,16 +256,17 @@ lemma ι_timeOrder_superCommute_eq_time {φ ψ : 𝓕.CrAnStates}
|
|||
simp
|
||||
· rw [crAnTimeOrderSign, Wick.koszulSign_eq_rel_eq_stat _ _, ← crAnTimeOrderSign]
|
||||
rw [timeOrder_ofCrAnList]
|
||||
simp
|
||||
simp only [map_smul, Algebra.mul_smul_comm]
|
||||
simp only [List.nil_append]
|
||||
exact hψφ
|
||||
exact hφψ
|
||||
simpa using hq
|
||||
· simp [pa]
|
||||
· simp only [map_mul, zero_mul, map_zero, mul_zero, pa]
|
||||
· intro x y hx hy hpx hpy
|
||||
simp_all [pa,mul_add, add_mul]
|
||||
· intro x hx hpx
|
||||
simp_all [pa, hpx]
|
||||
· simp [pb]
|
||||
· simp only [map_mul, mul_zero, map_zero, pb]
|
||||
· intro x y hx hy hpx hpy
|
||||
simp_all [pb,mul_add, add_mul]
|
||||
· intro x hx hpx
|
||||
|
|
@ -277,12 +282,13 @@ lemma ι_timeOrder_superCommute_neq_time {φ ψ : 𝓕.CrAnStates}
|
|||
rcases hφψ with hφψ | hφψ
|
||||
· rw [timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel ]
|
||||
have ht := IsTotal.total (r := crAnTimeOrderRel) φ ψ
|
||||
simp_all
|
||||
simp_all only [false_and, not_false_eq_true, false_or, mul_zero, zero_mul, map_zero]
|
||||
simp_all
|
||||
· rw [superCommute_ofCrAnState_ofCrAnState_symm]
|
||||
simp
|
||||
simp only [instCommGroup.eq_1, neg_smul, map_neg, map_smul, mul_neg, Algebra.mul_smul_comm,
|
||||
neg_mul, Algebra.smul_mul_assoc, neg_eq_zero, smul_eq_zero]
|
||||
rw [timeOrder_superCommute_ofCrAnState_ofCrAnState_not_crAnTimeOrderRel ]
|
||||
simp
|
||||
simp only [mul_zero, zero_mul, map_zero, or_true]
|
||||
simp_all
|
||||
|
||||
|
||||
|
|
@ -306,14 +312,14 @@ lemma ι_timeOrder_zero_of_mem_ideal (a : 𝓕.CrAnAlgebra)
|
|||
match hc with
|
||||
| Or.inl hc =>
|
||||
obtain ⟨φa, φa', hφa, hφa', rfl⟩ := hc
|
||||
simp
|
||||
simp only [ι_timeOrder_superCommute_superCommute]
|
||||
| Or.inr (Or.inl hc) =>
|
||||
obtain ⟨φa, hφa, φb, hφb, rfl⟩ := hc
|
||||
by_cases heqt : (crAnTimeOrderRel φa φb ∧ crAnTimeOrderRel φb φa)
|
||||
· rw [ι_timeOrder_superCommute_eq_time]
|
||||
simp
|
||||
simp only [map_mul]
|
||||
rw [ι_superCommute_of_create_create]
|
||||
simp
|
||||
simp only [zero_mul]
|
||||
· exact hφa
|
||||
· exact hφb
|
||||
· exact heqt.1
|
||||
|
|
@ -323,9 +329,9 @@ lemma ι_timeOrder_zero_of_mem_ideal (a : 𝓕.CrAnAlgebra)
|
|||
obtain ⟨φa, hφa, φb, hφb, rfl⟩ := hc
|
||||
by_cases heqt : (crAnTimeOrderRel φa φb ∧ crAnTimeOrderRel φb φa)
|
||||
· rw [ι_timeOrder_superCommute_eq_time]
|
||||
simp
|
||||
simp only [map_mul]
|
||||
rw [ι_superCommute_of_annihilate_annihilate]
|
||||
simp
|
||||
simp only [zero_mul]
|
||||
· exact hφa
|
||||
· exact hφb
|
||||
· exact heqt.1
|
||||
|
|
@ -335,9 +341,9 @@ lemma ι_timeOrder_zero_of_mem_ideal (a : 𝓕.CrAnAlgebra)
|
|||
obtain ⟨φa, φb, hdiff, rfl⟩ := hc
|
||||
by_cases heqt : (crAnTimeOrderRel φa φb ∧ crAnTimeOrderRel φb φa)
|
||||
· rw [ι_timeOrder_superCommute_eq_time]
|
||||
simp
|
||||
simp only [map_mul]
|
||||
rw [ι_superCommute_of_diff_statistic]
|
||||
simp
|
||||
simp only [zero_mul]
|
||||
· exact hdiff
|
||||
· exact heqt.1
|
||||
· exact heqt.2
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue