| import OAI.NumberTheory.Ostmann.Construction.FinalReassignmentInteraction |
|
|
| namespace OAI |
|
|
| /-! |
|
|
| namespace Ostmann |
|
|
| /-- The incoming anchor edge before normalization by the bulk parity. -/ |
| def anchorIncoming {n : ℕ} (α : Fin n → ℤ) (t : Fin n → Bool) |
| (j : Fin n) (b : Bool) : ℤ := |
| finalCopyParity t * anchorCodeEntry α t j b |
|
|
| theorem copyPathParity_snoc_prefix {n : ℕ} (t : Fin n → Bool) (b : Bool) |
| (j : ℕ) (hj : j ≤ n) : |
| copyPathParity (finiteCopyPath (Fin.snoc t b)) j = copyPathParity (finiteCopyPath t) j := by |
| apply copyPathParity_congr_prefix |
| intro h hh |
| have hhn : h < n := lt_of_lt_of_le hh hj |
| simp only [finiteCopyPath, hhn, Nat.lt_succ_of_lt hhn, dite_true] |
| exact @Fin.snoc_castSucc n (fun _ => Bool) b t ⟨h, hhn⟩ |
|
|
| theorem anchorCodeEntry_snoc {n : ℕ} (α : Fin (n + 1) → ℤ) |
| (t : Fin n → Bool) (u b : Bool) (j : Fin n) : |
| anchorCodeEntry α (Fin.snoc t u) j.castSucc b = |
| anchorCodeEntry (fun j => α j.castSucc) t j b := by |
| have hp := copyPathParity_snoc_prefix t u j (Nat.le_of_lt j.isLt) |
| simp only [anchorCodeEntry, finiteAnchorCode, Fin.val_castSucc, hp, |
| @Fin.snoc_castSucc n (fun _ => Bool) u t j] |
|
|
| /-- This is exactly the outside-anchor row of `transferredGraph`: every |
| later copy multiplies its incoming coefficient by that copy's sign. -/ |
| theorem anchorIncoming_snoc_old {n : ℕ} (α : Fin (n + 1) → ℤ) |
| (t : Fin n → Bool) (u b : Bool) (j : Fin n) : |
| anchorIncoming α (Fin.snoc t u) j.castSucc b = |
| transferCopySign u * anchorIncoming (fun j => α j.castSucc) t j b := by |
| unfold anchorIncoming |
| rw [finalCopyParity_snoc, anchorCodeEntry_snoc, mul_assoc] |
|
|
| theorem anchorIncoming_snoc_new {n : ℕ} (α : Fin (n + 1) → ℤ) |
| (t : Fin n → Bool) (u b : Bool) : |
| anchorIncoming α (Fin.snoc t u) (Fin.last n) b = |
| (transferCopySign u * finalCopyParity t) * |
| (if b then (anchorSignPair (α (Fin.last n)) (finalCopyParity t) u).1 |
| else (anchorSignPair (α (Fin.last n)) (finalCopyParity t) u).2) := by |
| have hp := copyPathParity_snoc_prefix t u n le_rfl |
| unfold anchorIncoming |
| rw [finalCopyParity_snoc] |
| congr 1 |
| simp only [anchorCodeEntry, finiteAnchorCode, Fin.val_last, hp, |
| @Fin.snoc_last n (fun _ => Bool) u t] |
| rfl |
|
|
| /-- At its reserved step the unnormalized coefficient is precisely the |
| edge created by the graph transfer, for either clone of the fresh anchor. -/ |
| theorem anchorIncoming_created_by_transfer {H Y : Type*} |
| (graph : Option (H ⊕ Y) → Option (H ⊕ Y) → ℤ) (a i : H) |
| {n : ℕ} (α : Fin (n + 1) → ℤ) (t : Fin n → Bool) (u b : Bool) |
| (hincoming : graph (some (.inl a)) (some (.inl i)) = finalCopyParity t) |
| (hpivot : graph (some (.inl a)) none = α (Fin.last n)) : |
| anchorIncoming α (Fin.snoc t u) (Fin.last n) b = |
| transferredGraph graph (.inl (b, a)) (.inl (u, i)) := by |
| have hs := copyPathParity_sign (finiteCopyPath t) n |
| change finalCopyParity t = 1 ∨ finalCopyParity t = -1 at hs |
| have hpair := transferred_anchor_pair graph a i (α (Fin.last n)) (finalCopyParity t) |
| hs hincoming hpivot u |
| rw [anchorIncoming_snoc_new] |
| have hsq : (transferCopySign u * finalCopyParity t) ^ 2 = 1 := by |
| rw [mul_pow, transferCopySign_sq] |
| rcases hs with hs | hs <;> simp only [hs] <;> norm_num |
| cases b |
| · have he := congrArg Prod.snd hpair |
| dsimp only at he |
| simp only [Bool.false_eq_true, ite_false] |
| rw [← he] |
| rw [← mul_assoc, ← pow_two, hsq, one_mul] |
| · have he := congrArg Prod.fst hpair |
| dsimp only at he |
| simp only [ite_true] |
| rw [← he] |
| rw [← mul_assoc, ← pow_two, hsq, one_mul] |
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|
| end Ostmann |
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| end OAI |
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