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2b3d0c0 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 | import OAI.NumberTheory.Ostmann.Construction.FinalReassignmentInteraction
namespace OAI
/-! # The anchor coefficients through their actual creation and later copies -/
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]
end Ostmann
end OAI
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