docs: Fix typos in docs
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27 changed files with 67 additions and 63 deletions
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@ -17,7 +17,7 @@ variable {𝓕 : FieldSpecification}
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Given a natural number `n`, which will correspond to the number of fields needing
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contracting, a Wick contraction
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is a finite set of pairs of `Fin n` (numbers `0`, ..., `n-1`), such that no
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element of `Fin n` occurs in more then one pair. The pairs are the positions of fields we
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element of `Fin n` occurs in more than one pair. The pairs are the positions of fields we
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'contract' together.
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-/
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def WickContraction (n : ℕ) : Type :=
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@ -520,8 +520,8 @@ lemma prod_finset_eq_mul_fst_snd (c : WickContraction n) (a : c.1)
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/-- For a field specification `𝓕`, `φs` a list of `𝓕.FieldOp` and a Wick contraction
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`φsΛ` of `φs`, the Wick contraction `φsΛ` is said to be `GradingCompliant` if
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for every pair in `φsΛ` the contracted fields are either both `fermionic` or both `bosonic`.
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In other words, in a `GradingCompliant` Wick contraction no contractions occur between
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`fermionic` and `bosonic` fields. -/
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In other words, in a `GradingCompliant` Wick contraction if
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no contracted pairs occur between `fermionic` and `bosonic` fields. -/
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def GradingCompliant (φs : List 𝓕.FieldOp) (φsΛ : WickContraction φs.length) :=
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∀ (a : φsΛ.1), (𝓕 |>ₛ φs[φsΛ.fstFieldOfContract a]) = (𝓕 |>ₛ φs[φsΛ.sndFieldOfContract a])
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@ -34,8 +34,8 @@ open HepLean.Fin
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of `φ` (at position `i`) with the new position of `j` after `φ` is added.
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In other words, `φsΛ.insertAndContract φ i j` is formed by adding `φ` to `φs` at position `i`,
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and contracting `φ` with the field originally at position `j` if `j` is not none.
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It is a Wick contraction of `φs.insertIdx φ i`, the list `φs` with `φ` inserted at
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and contracting `φ` with the field originally at position `j` if `j` is not `none`.
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It is a Wick contraction of the list `φs.insertIdx φ i` corresponding to `φs` with `φ` inserted at
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position `i`.
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The notation `φsΛ ↩Λ φ i j` is used to denote `φsΛ.insertAndContract φ i j`. -/
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@ -241,7 +241,7 @@ lemma signInsertNone_eq_filterset (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
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/-- For a list `φs = φ₀…φₙ` of `𝓕.FieldOp`, a graded compliant Wick contraction `φsΛ` of `φs`,
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an `i ≤ φs.length`, and a `φ` in `𝓕.FieldOp`, then
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`(φsΛ ↩Λ φ i none).sign = s * φsΛ.sign`
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where `s` is the sign got by moving `φ` through the elements of `φ₀…φᵢ₋₁` which
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where `s` is the sign arrived at by moving `φ` through the elements of `φ₀…φᵢ₋₁` which
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are contracted with some element.
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The proof of this result involves a careful consideration of the contributions of different
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@ -35,7 +35,7 @@ noncomputable def staticContract {φs : List 𝓕.FieldOp}
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`(φsΛ ↩Λ φ i none).staticContract = φsΛ.staticContract`
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The prove of this result ultimately a consequence of definitions.
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The proof of this result ultimately is a consequence of definitions.
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-/
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@[simp]
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lemma staticContract_insert_none (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
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@ -53,7 +53,7 @@ lemma staticContract_insert_none (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
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- `[anPart φ, φs[k]]ₛ` if `i ≤ k` or `[anPart φs[k], φ]ₛ` if `k < i`
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- `φsΛ.staticContract`.
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The proof of this result ultimately a consequence of definitions.
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The proof of this result ultimately is a consequence of definitions.
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-/
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lemma staticContract_insert_some
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(φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
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@ -96,8 +96,8 @@ lemma empty_mem {φs : List 𝓕.FieldOp} : empty (n := φs.length).EqTimeOnly :
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rw [eqTimeOnly_iff_forall_finset]
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simp [empty]
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/-- Let `φs` be a list of `𝓕.FieldOp` and `φsΛ` a `WickContraction` of `φs` with
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in which every contraction involves two `𝓕FieldOp`s that have the same time, then
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/-- Let `φs` be a list of `𝓕.FieldOp` and `φsΛ` a `WickContraction` of `φs` within
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which every contraction involves two `𝓕FieldOp`s that have the same time, then
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`φsΛ.staticContract = φsΛ.timeContract`. -/
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lemma staticContract_eq_timeContract_of_eqTimeOnly (h : φsΛ.EqTimeOnly) :
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φsΛ.staticContract = φsΛ.timeContract := by
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@ -193,8 +193,8 @@ lemma timeOrder_timeContract_mul_of_eqTimeOnly_mid {φs : List 𝓕.FieldOp}
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𝓣(a * φsΛ.timeContract.1 * b) = φsΛ.timeContract.1 * 𝓣(a * b) := by
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exact timeOrder_timeContract_mul_of_eqTimeOnly_mid_induction φsΛ hl a b φsΛ.1.card rfl
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/-- Let `φs` be a list of `𝓕.FieldOp`, `φsΛ` a `WickContraction` of `φs` with
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in which every contraction involves two `𝓕.FieldOp`s that have the same time and
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/-- Let `φs` be a list of `𝓕.FieldOp`, `φsΛ` a `WickContraction` of `φs` within
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which every contraction involves two `𝓕.FieldOp`s that have the same time and
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`b` a general element in `𝓕.FieldOpAlgebra`. Then
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`𝓣(φsΛ.timeContract.1 * b) = φsΛ.timeContract.1 * 𝓣(b)`.
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@ -35,7 +35,7 @@ noncomputable def timeContract {φs : List 𝓕.FieldOp}
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`(φsΛ ↩Λ φ i none).timeContract = φsΛ.timeContract`
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The prove of this result ultimately a consequence of definitions. -/
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The proof of this result ultimately is a consequence of definitions. -/
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@[simp]
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lemma timeContract_insert_none (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
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(φsΛ : WickContraction φs.length) (i : Fin φs.length.succ) :
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@ -51,7 +51,7 @@ lemma timeContract_insert_none (φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
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- `timeContract φ φs[k]` if `i ≤ k` or `timeContract φs[k] φ` if `k < i`
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- `φsΛ.timeContract`.
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The proof of this result ultimately a consequence of definitions. -/
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The proof of this result ultimately is a consequence of definitions. -/
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lemma timeContract_insertAndContract_some
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(φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
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(φsΛ : WickContraction φs.length) (i : Fin φs.length.succ) (j : φsΛ.uncontracted) :
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@ -88,7 +88,7 @@ open FieldStatistic
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- two copies of the exchange sign of `φ` with the uncontracted fields in `φ₀…φₖ₋₁`.
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These two exchange signs cancel each other out but are included for convenience.
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The proof of this result ultimately a consequence of definitions and
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The proof of this result ultimately is a consequence of definitions and
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`timeContract_of_timeOrderRel`. -/
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lemma timeContract_insert_some_of_lt
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(φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
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@ -132,7 +132,7 @@ lemma timeContract_insert_some_of_lt
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- the exchange sign of `φ` with the uncontracted fields in `φ₀…φₖ₋₁`.
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- the exchange sign of `φ` with the uncontracted fields in `φ₀…φₖ`.
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The proof of this result ultimately a consequence of definitions and
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The proof of this result ultimately is a consequence of definitions and
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`timeContract_of_not_timeOrderRel_expand`. -/
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lemma timeContract_insert_some_of_not_lt
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(φ : 𝓕.FieldOp) (φs : List 𝓕.FieldOp)
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@ -583,7 +583,7 @@ lemma uncontractedList_succAboveEmb_toFinset (c : WickContraction n) (i : Fin n.
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-/
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/-- Given a Wick contraction `c : WickContraction n` and a `Fin n.succ`, the number of elements
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of `c.uncontractedList` which are less then `i`.
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of `c.uncontractedList` which are less than `i`.
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Suppose we want to insert into `c` at position `i`, then this is the position we would
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need to insert into `c.uncontractedList`. -/
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def uncontractedListOrderPos (c : WickContraction n) (i : Fin n.succ) : ℕ :=
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