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refactor: Docs

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jstoobysmith 2024-10-19 10:50:38 +00:00
parent 3e1cd363bd
commit ae7f8dea1e
4 changed files with 26 additions and 7 deletions
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21
HepLean/Tensors/Tree/NodeIdentities/ContrContr.lean
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@ -22,27 +22,43 @@ namespace TensorTree
variable {S : TensorStruct}
/-- A structure containing two pairs of indices (i, j) and (k, l) to be sequentially
contracted in a tensor.-/
structure ContrQuartet {n : ℕ} (c : Fin n.succ.succ.succ.succ → S.C) where
/-- The first index of the first pair to be contracted. -/
i : Fin n.succ.succ.succ.succ
/-- The second index of the first pair to be contracted. -/
j : Fin n.succ.succ.succ
/-- The first index of the second pair to be contracted. -/
k : Fin n.succ.succ
/-- The second index of the second pair to be contracted. -/
l : Fin n.succ
/-- The condition on the first pair of indices permitting their contraction. -/
hij : c (i.succAbove j) = S.τ (c i)
/-- The condition on the second pair of indices permitting their contraction. -/
hkl : (c ∘ i.succAbove ∘ j.succAbove) (k.succAbove l) = S.τ ((c ∘ i.succAbove ∘ j.succAbove) k)
namespace ContrQuartet
variable {n : ℕ} {c : Fin n.succ.succ.succ.succ → S.C} (q : ContrQuartet c)
/-- On swapping the order of contraction (notionally `(i, j) - (k, l)` vs `(k, l) - (i, j)`), this
is the new `i` index. -/
def swapI : Fin n.succ.succ.succ.succ := q.i.succAbove (q.j.succAbove q.k)
/-- On swapping the order of contraction (notionally `(i, j) - (k, l)` vs `(k, l) - (i, j)`), this
is the new `j` index. -/
def swapJ : Fin n.succ.succ.succ := (predAboveI (q.i.succAbove (q.j.succAbove q.k)) q.i).succAbove
((predAboveI (q.j.succAbove q.k) q.j).succAbove q.l)
/-- On swapping the order of contraction (notionally `(i, j) - (k, l)` vs `(k, l) - (i, j)`), this
is the new `k` index. -/
def swapK : Fin n.succ.succ := predAboveI
((predAboveI (q.i.succAbove (q.j.succAbove q.k)) q.i).succAbove
((predAboveI (q.j.succAbove q.k) q.j).succAbove q.l))
(predAboveI (q.i.succAbove (q.j.succAbove q.k)) q.i)
/-- On swapping the order of contraction (notionally `(i, j) - (k, l)` vs `(k, l) - (i, j)`), this
is the new `l` index. -/
def swapL : Fin n.succ := predAboveI ((predAboveI (q.j.succAbove q.k) q.j).succAbove q.l)
(predAboveI (q.j.succAbove q.k) q.j)
@ -113,6 +129,8 @@ lemma swapL_swapK_swapJ_swapI_succAbove :
exact Fin.succAbove_ne q.j q.k
exact Fin.succAbove_ne (predAboveI (q.j.succAbove q.k) q.j) q.l
/-- The `ContrQuartet` corresponding to swapping the order of contraction
(notionally `(i, j) - (k, l)` vs `(k, l) - (i, j)`). -/
def swap : ContrQuartet c where
i := q.swapI
j := q.swapJ
@ -126,10 +144,13 @@ def swap : ContrQuartet c where
noncomputable section
/-- The contraction map for the first pair of indices. -/
def contrMapFst := S.contrMap c q.i q.j q.hij
/-- The contractoin map for the second pair of indices. -/
def contrMapSnd := S.contrMap (c ∘ q.i.succAbove ∘ q.j.succAbove) q.k q.l q.hkl
/-- The homomorphism one must apply on swapping the order of contractions. -/
def contrSwapHom : (OverColor.mk ((c ∘ q.swap.i.succAbove ∘ q.swap.j.succAbove) ∘
q.swap.k.succAbove ∘ q.swap.l.succAbove)) ⟶
(OverColor.mk fun x => c (q.i.succAbove (q.j.succAbove (q.k.succAbove (q.l.succAbove x))))) :=