refactor: Change in contraction notation
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jstoobysmith
parent
6701ee7b37
commit
321e69e52e
5 changed files with 326 additions and 324 deletions
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@ -29,38 +29,38 @@ open HepLean.Fin
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`j : Option (c.uncontracted)` of `c`.
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The Wick contraction associated with `(φs.insertIdx i φ).length` formed by 'inserting' `φ`
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into `φs` after the first `i` elements and contracting it optionally with j. -/
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def insertList (φ : 𝓕.States) {φs : List 𝓕.States}
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(c : WickContraction φs.length) (i : Fin φs.length.succ) (j : Option (c.uncontracted)) :
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def insertList (φ : 𝓕.States) {φs : List 𝓕.States} (φsΛ : WickContraction φs.length)
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(i : Fin φs.length.succ) (j : Option φsΛ.uncontracted) :
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WickContraction (φs.insertIdx i φ).length :=
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congr (by simp) (c.insert i j)
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congr (by simp) (φsΛ.insert i j)
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@[simp]
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lemma insertList_fstFieldOfContract (φ : 𝓕.States) (φs : List 𝓕.States)
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(c : WickContraction φs.length) (i : Fin φs.length.succ) (j : Option (c.uncontracted))
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(a : c.1) : (c.insertList φ i j).fstFieldOfContract
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(φsΛ : WickContraction φs.length) (i : Fin φs.length.succ) (j : Option φsΛ.uncontracted)
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(a : φsΛ.1) : (φsΛ.insertList φ i j).fstFieldOfContract
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(congrLift (insertIdx_length_fin φ φs i).symm (insertLift i j a)) =
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finCongr (insertIdx_length_fin φ φs i).symm (i.succAbove (c.fstFieldOfContract a)) := by
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finCongr (insertIdx_length_fin φ φs i).symm (i.succAbove (φsΛ.fstFieldOfContract a)) := by
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simp [insertList]
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@[simp]
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lemma insertList_sndFieldOfContract (φ : 𝓕.States) (φs : List 𝓕.States)
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(c : WickContraction φs.length) (i : Fin φs.length.succ) (j : Option (c.uncontracted))
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(a : c.1) : (c.insertList φ i j).sndFieldOfContract
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(φsΛ : WickContraction φs.length) (i : Fin φs.length.succ) (j : Option (φsΛ.uncontracted))
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(a : φsΛ.1) : (φsΛ.insertList φ i j).sndFieldOfContract
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(congrLift (insertIdx_length_fin φ φs i).symm (insertLift i j a)) =
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finCongr (insertIdx_length_fin φ φs i).symm (i.succAbove (c.sndFieldOfContract a)) := by
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finCongr (insertIdx_length_fin φ φs i).symm (i.succAbove (φsΛ.sndFieldOfContract a)) := by
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simp [insertList]
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@[simp]
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lemma insertList_fstFieldOfContract_some_incl (φ : 𝓕.States) (φs : List 𝓕.States)
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(c : WickContraction φs.length) (i : Fin φs.length.succ) (j : c.uncontracted) :
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(insertList φ c i (some j)).fstFieldOfContract
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(φsΛ : WickContraction φs.length) (i : Fin φs.length.succ) (j : φsΛ.uncontracted) :
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(insertList φ φsΛ i (some j)).fstFieldOfContract
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(congrLift (insertIdx_length_fin φ φs i).symm ⟨{i, i.succAbove j}, by simp [insert]⟩) =
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if i < i.succAbove j.1 then
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finCongr (insertIdx_length_fin φ φs i).symm i else
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finCongr (insertIdx_length_fin φ φs i).symm (i.succAbove j.1) := by
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split
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· rename_i h
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refine (insertList φ c i (some j)).eq_fstFieldOfContract_of_mem
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refine (insertList φ φsΛ i (some j)).eq_fstFieldOfContract_of_mem
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(a := congrLift (insertIdx_length_fin φ φs i).symm ⟨{i, i.succAbove j}, by simp [insert]⟩)
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(i := finCongr (insertIdx_length_fin φ φs i).symm i) (j :=
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finCongr (insertIdx_length_fin φ φs i).symm (i.succAbove j)) ?_ ?_ ?_
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@ -69,7 +69,7 @@ lemma insertList_fstFieldOfContract_some_incl (φ : 𝓕.States) (φs : List
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· rw [Fin.lt_def] at h ⊢
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simp_all
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· rename_i h
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refine (insertList φ c i (some j)).eq_fstFieldOfContract_of_mem
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refine (insertList φ φsΛ i (some j)).eq_fstFieldOfContract_of_mem
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(a := congrLift (insertIdx_length_fin φ φs i).symm ⟨{i, i.succAbove j}, by simp [insert]⟩)
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(i := finCongr (insertIdx_length_fin φ φs i).symm (i.succAbove j))
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(j := finCongr (insertIdx_length_fin φ φs i).symm i) ?_ ?_ ?_
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@ -87,36 +87,36 @@ lemma insertList_fstFieldOfContract_some_incl (φ : 𝓕.States) (φs : List
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-/
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@[simp]
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lemma insertList_none_getDual?_self (φ : 𝓕.States) (φs : List 𝓕.States)
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(c : WickContraction φs.length) (i : Fin φs.length.succ) :
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(insertList φ c i none).getDual? (Fin.cast (insertIdx_length_fin φ φs i).symm i) = none := by
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(φsΛ : WickContraction φs.length) (i : Fin φs.length.succ) :
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(insertList φ φsΛ i none).getDual? (Fin.cast (insertIdx_length_fin φ φs i).symm i) = none := by
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simp only [Nat.succ_eq_add_one, insertList, getDual?_congr, finCongr_apply, Fin.cast_trans,
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Fin.cast_eq_self, Option.map_eq_none']
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have h1 := c.insert_none_getDual?_isNone i
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have h1 := φsΛ.insert_none_getDual?_isNone i
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simpa using h1
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lemma insertList_isSome_getDual?_self (φ : 𝓕.States) (φs : List 𝓕.States)
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(c : WickContraction φs.length) (i : Fin φs.length.succ) (j : c.uncontracted) :
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((insertList φ c i (some j)).getDual?
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(φsΛ : WickContraction φs.length) (i : Fin φs.length.succ) (j : φsΛ.uncontracted) :
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((insertList φ φsΛ i (some j)).getDual?
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(Fin.cast (insertIdx_length_fin φ φs i).symm i)).isSome := by
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simp [insertList, getDual?_congr]
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lemma insertList_some_getDual?_self_not_none (φ : 𝓕.States) (φs : List 𝓕.States)
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(c : WickContraction φs.length) (i : Fin φs.length.succ) (j : c.uncontracted) :
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¬ ((insertList φ c i (some j)).getDual?
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(φsΛ : WickContraction φs.length) (i : Fin φs.length.succ) (j : φsΛ.uncontracted) :
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¬ ((insertList φ φsΛ i (some j)).getDual?
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(Fin.cast (insertIdx_length_fin φ φs i).symm i)) = none := by
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simp [insertList, getDual?_congr]
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@[simp]
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lemma insertList_some_getDual?_self_eq (φ : 𝓕.States) (φs : List 𝓕.States)
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(c : WickContraction φs.length) (i : Fin φs.length.succ) (j : c.uncontracted) :
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((insertList φ c i (some j)).getDual? (Fin.cast (insertIdx_length_fin φ φs i).symm i))
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(φsΛ : WickContraction φs.length) (i : Fin φs.length.succ) (j : φsΛ.uncontracted) :
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((insertList φ φsΛ i (some j)).getDual? (Fin.cast (insertIdx_length_fin φ φs i).symm i))
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= some (Fin.cast (insertIdx_length_fin φ φs i).symm (i.succAbove j)) := by
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simp [insertList, getDual?_congr]
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@[simp]
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lemma insertList_some_getDual?_some_eq (φ : 𝓕.States) (φs : List 𝓕.States)
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(c : WickContraction φs.length) (i : Fin φs.length.succ) (j : c.uncontracted) :
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((insertList φ c i (some j)).getDual?
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(φsΛ : WickContraction φs.length) (i : Fin φs.length.succ) (j : φsΛ.uncontracted) :
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((insertList φ φsΛ i (some j)).getDual?
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(Fin.cast (insertIdx_length_fin φ φs i).symm (i.succAbove j)))
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= some (Fin.cast (insertIdx_length_fin φ φs i).symm i) := by
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rw [getDual?_eq_some_iff_mem]
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@ -126,52 +126,52 @@ lemma insertList_some_getDual?_some_eq (φ : 𝓕.States) (φs : List 𝓕.State
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@[simp]
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lemma insertList_none_succAbove_getDual?_eq_none_iff (φ : 𝓕.States) (φs : List 𝓕.States)
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(c : WickContraction φs.length) (i : Fin φs.length.succ) (j : Fin φs.length) :
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(insertList φ c i none).getDual? (Fin.cast (insertIdx_length_fin φ φs i).symm
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(i.succAbove j)) = none ↔ c.getDual? j = none := by
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(φsΛ : WickContraction φs.length) (i : Fin φs.length.succ) (j : Fin φs.length) :
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(insertList φ φsΛ i none).getDual? (Fin.cast (insertIdx_length_fin φ φs i).symm
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(i.succAbove j)) = none ↔ φsΛ.getDual? j = none := by
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simp [insertList, getDual?_congr]
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@[simp]
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lemma insertList_some_succAbove_getDual?_eq_option (φ : 𝓕.States) (φs : List 𝓕.States)
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(c : WickContraction φs.length) (i : Fin φs.length.succ) (j : Fin φs.length)
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(k : c.uncontracted) (hkj : j ≠ k.1) :
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(insertList φ c i (some k)).getDual? (Fin.cast (insertIdx_length_fin φ φs i).symm
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(φsΛ : WickContraction φs.length) (i : Fin φs.length.succ) (j : Fin φs.length)
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(k : φsΛ.uncontracted) (hkj : j ≠ k.1) :
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(insertList φ φsΛ i (some k)).getDual? (Fin.cast (insertIdx_length_fin φ φs i).symm
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(i.succAbove j)) = Option.map (Fin.cast (insertIdx_length_fin φ φs i).symm ∘ i.succAbove)
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(c.getDual? j) := by
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(φsΛ.getDual? j) := by
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simp only [Nat.succ_eq_add_one, insertList, getDual?_congr, finCongr_apply, Fin.cast_trans,
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Fin.cast_eq_self, ne_eq, hkj, not_false_eq_true, insert_some_getDual?_of_neq, Option.map_map]
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rfl
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@[simp]
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lemma insertList_none_succAbove_getDual?_isSome_iff (φ : 𝓕.States) (φs : List 𝓕.States)
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(c : WickContraction φs.length) (i : Fin φs.length.succ) (j : Fin φs.length) :
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((insertList φ c i none).getDual? (Fin.cast (insertIdx_length_fin φ φs i).symm
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(i.succAbove j))).isSome ↔ (c.getDual? j).isSome := by
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(φsΛ : WickContraction φs.length) (i : Fin φs.length.succ) (j : Fin φs.length) :
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((insertList φ φsΛ i none).getDual? (Fin.cast (insertIdx_length_fin φ φs i).symm
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(i.succAbove j))).isSome ↔ (φsΛ.getDual? j).isSome := by
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rw [← not_iff_not]
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simp
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@[simp]
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lemma insertList_none_getDual?_get_eq (φ : 𝓕.States) (φs : List 𝓕.States)
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(c : WickContraction φs.length) (i : Fin φs.length.succ) (j : Fin φs.length)
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(h : ((insertList φ c i none).getDual? (Fin.cast (insertIdx_length_fin φ φs i).symm
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(φsΛ : WickContraction φs.length) (i : Fin φs.length.succ) (j : Fin φs.length)
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(h : ((insertList φ φsΛ i none).getDual? (Fin.cast (insertIdx_length_fin φ φs i).symm
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(i.succAbove j))).isSome) :
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((insertList φ c i none).getDual? (Fin.cast (insertIdx_length_fin φ φs i).symm
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((insertList φ φsΛ i none).getDual? (Fin.cast (insertIdx_length_fin φ φs i).symm
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(i.succAbove j))).get h = Fin.cast (insertIdx_length_fin φ φs i).symm
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(i.succAbove ((c.getDual? j).get (by simpa using h))) := by
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(i.succAbove ((φsΛ.getDual? j).get (by simpa using h))) := by
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simp [insertList, getDual?_congr_get]
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/-........................................... -/
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@[simp]
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lemma insertList_sndFieldOfContract_some_incl (φ : 𝓕.States) (φs : List 𝓕.States)
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(c : WickContraction φs.length) (i : Fin φs.length.succ) (j : c.uncontracted) :
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(insertList φ c i (some j)).sndFieldOfContract
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(φsΛ : WickContraction φs.length) (i : Fin φs.length.succ) (j : φsΛ.uncontracted) :
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(insertList φ φsΛ i (some j)).sndFieldOfContract
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(congrLift (insertIdx_length_fin φ φs i).symm ⟨{i, i.succAbove j}, by simp [insert]⟩) =
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if i < i.succAbove j.1 then
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finCongr (insertIdx_length_fin φ φs i).symm (i.succAbove j.1) else
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finCongr (insertIdx_length_fin φ φs i).symm i := by
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split
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· rename_i h
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refine (insertList φ c i (some j)).eq_sndFieldOfContract_of_mem
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refine (insertList φ φsΛ i (some j)).eq_sndFieldOfContract_of_mem
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(a := congrLift (insertIdx_length_fin φ φs i).symm ⟨{i, i.succAbove j}, by simp [insert]⟩)
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(i := finCongr (insertIdx_length_fin φ φs i).symm i) (j :=
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finCongr (insertIdx_length_fin φ φs i).symm (i.succAbove j)) ?_ ?_ ?_
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@ -180,7 +180,7 @@ lemma insertList_sndFieldOfContract_some_incl (φ : 𝓕.States) (φs : List
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· rw [Fin.lt_def] at h ⊢
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simp_all
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· rename_i h
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refine (insertList φ c i (some j)).eq_sndFieldOfContract_of_mem
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refine (insertList φ φsΛ i (some j)).eq_sndFieldOfContract_of_mem
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(a := congrLift (insertIdx_length_fin φ φs i).symm ⟨{i, i.succAbove j}, by simp [insert]⟩)
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(i := finCongr (insertIdx_length_fin φ φs i).symm (i.succAbove j))
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(j := finCongr (insertIdx_length_fin φ φs i).symm i) ?_ ?_ ?_
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@ -192,27 +192,27 @@ lemma insertList_sndFieldOfContract_some_incl (φ : 𝓕.States) (φs : List
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omega
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lemma insertList_none_prod_contractions (φ : 𝓕.States) (φs : List 𝓕.States)
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(c : WickContraction φs.length) (i : Fin φs.length.succ)
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(f : (c.insertList φ i none).1 → M) [CommMonoid M] :
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∏ a, f a = ∏ (a : c.1), f (congrLift (insertIdx_length_fin φ φs i).symm
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(φsΛ : WickContraction φs.length) (i : Fin φs.length.succ)
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(f : (φsΛ.insertList φ i none).1 → M) [CommMonoid M] :
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∏ a, f a = ∏ (a : φsΛ.1), f (congrLift (insertIdx_length_fin φ φs i).symm
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(insertLift i none a)) := by
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let e1 := Equiv.ofBijective (c.insertLift i none) (insertLift_none_bijective i)
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let e1 := Equiv.ofBijective (φsΛ.insertLift i none) (insertLift_none_bijective i)
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let e2 := Equiv.ofBijective (congrLift (insertIdx_length_fin φ φs i).symm)
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((c.insert i none).congrLift_bijective ((insertIdx_length_fin φ φs i).symm))
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((φsΛ.insert i none).congrLift_bijective ((insertIdx_length_fin φ φs i).symm))
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erw [← e2.prod_comp]
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erw [← e1.prod_comp]
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rfl
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lemma insertList_some_prod_contractions (φ : 𝓕.States) (φs : List 𝓕.States)
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(c : WickContraction φs.length) (i : Fin φs.length.succ) (j : c.uncontracted)
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(f : (c.insertList φ i (some j)).1 → M) [CommMonoid M] :
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(φsΛ : WickContraction φs.length) (i : Fin φs.length.succ) (j : φsΛ.uncontracted)
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(f : (φsΛ.insertList φ i (some j)).1 → M) [CommMonoid M] :
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∏ a, f a = f (congrLift (insertIdx_length_fin φ φs i).symm
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⟨{i, i.succAbove j}, by simp [insert]⟩) *
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∏ (a : c.1), f (congrLift (insertIdx_length_fin φ φs i).symm (insertLift i (some j) a)) := by
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∏ (a : φsΛ.1), f (congrLift (insertIdx_length_fin φ φs i).symm (insertLift i (some j) a)) := by
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let e2 := Equiv.ofBijective (congrLift (insertIdx_length_fin φ φs i).symm)
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((c.insert i (some j)).congrLift_bijective ((insertIdx_length_fin φ φs i).symm))
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((φsΛ.insert i (some j)).congrLift_bijective ((insertIdx_length_fin φ φs i).symm))
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erw [← e2.prod_comp]
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let e1 := Equiv.ofBijective (c.insertLiftSome i j) (insertLiftSome_bijective i j)
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let e1 := Equiv.ofBijective (φsΛ.insertLiftSome i j) (insertLiftSome_bijective i j)
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erw [← e1.prod_comp]
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rw [Fintype.prod_sum_type]
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simp only [Finset.univ_unique, PUnit.default_eq_unit, Nat.succ_eq_add_one, Finset.prod_singleton,
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@ -267,10 +267,9 @@ lemma insert_fin_eq_self (φ : 𝓕.States) {φs : List 𝓕.States}
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rfl
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lemma insertList_uncontractedList_none_map (φ : 𝓕.States) {φs : List 𝓕.States}
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(c : WickContraction φs.length) (i : Fin φs.length.succ) :
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List.map (List.insertIdx (↑i) φ φs).get (insertList φ c i none).uncontractedList =
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List.insertIdx (c.uncontractedListOrderPos i) φ (List.map φs.get c.uncontractedList) := by
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simp only [Nat.succ_eq_add_one, insertList]
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(φsΛ : WickContraction φs.length) (i : Fin φs.length.succ) :
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[φsΛ.insertList φ i none]ᵘᶜ = List.insertIdx (φsΛ.uncontractedListOrderPos i) φ [φsΛ]ᵘᶜ := by
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simp only [Nat.succ_eq_add_one, insertList, uncontractedListGet]
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rw [congr_uncontractedList]
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erw [uncontractedList_extractEquiv_symm_none]
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rw [orderedInsert_succAboveEmb_uncontractedList_eq_insertIdx]
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@ -284,8 +283,8 @@ lemma insertList_uncontractedList_none_map (φ : 𝓕.States) {φs : List 𝓕.S
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lemma insertLift_sum (φ : 𝓕.States) {φs : List 𝓕.States}
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(i : Fin φs.length.succ) [AddCommMonoid M] (f : WickContraction (φs.insertIdx i φ).length → M) :
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∑ c, f c = ∑ (c : WickContraction φs.length), ∑ (k : Option (c.uncontracted)),
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f (insertList φ c i k) := by
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∑ c, f c = ∑ (φsΛ : WickContraction φs.length), ∑ (k : Option φsΛ.uncontracted),
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f (φsΛ.insertList φ i k) := by
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rw [sum_extractEquiv_congr (finCongr (insertIdx_length_fin φ φs i).symm i) f
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(insertIdx_length_fin φ φs i)]
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rfl
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Reference in a new issue