Jean-Christophe Filliâtre committed Aug 17, 2011 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 ``````(* This file is generated by Why3's Coq driver *) (* Beware! Only edit allowed sections below *) Require Import ZArith. Require Import Rbase. Require Import ZOdiv. Require Import Zdiv. Definition unit := unit. Parameter mark : Type. Parameter at1: forall (a:Type), a -> mark -> a. Implicit Arguments at1. Parameter old: forall (a:Type), a -> a. Implicit Arguments old. Axiom Abs_pos : forall (x:Z), (0%Z <= (Zabs x))%Z. Axiom Div_mod : forall (x:Z) (y:Z), (~ (y = 0%Z)) -> (x = ((y * (ZOdiv x y))%Z + (ZOmod x y))%Z). Axiom Div_bound : forall (x:Z) (y:Z), ((0%Z <= x)%Z /\ (0%Z < y)%Z) -> ((0%Z <= (ZOdiv x y))%Z /\ ((ZOdiv x y) <= x)%Z). Axiom Mod_bound : forall (x:Z) (y:Z), (~ (y = 0%Z)) -> (((-(Zabs y))%Z < (ZOmod x y))%Z /\ ((ZOmod x y) < (Zabs y))%Z). Axiom Div_sign_pos : forall (x:Z) (y:Z), ((0%Z <= x)%Z /\ (0%Z < y)%Z) -> (0%Z <= (ZOdiv x y))%Z. Axiom Div_sign_neg : forall (x:Z) (y:Z), ((x <= 0%Z)%Z /\ (0%Z < y)%Z) -> ((ZOdiv x y) <= 0%Z)%Z. Axiom Mod_sign_pos : forall (x:Z) (y:Z), ((0%Z <= x)%Z /\ ~ (y = 0%Z)) -> (0%Z <= (ZOmod x y))%Z. Axiom Mod_sign_neg : forall (x:Z) (y:Z), ((x <= 0%Z)%Z /\ ~ (y = 0%Z)) -> ((ZOmod x y) <= 0%Z)%Z. Axiom Rounds_toward_zero : forall (x:Z) (y:Z), (~ (y = 0%Z)) -> ((Zabs ((ZOdiv x y) * y)%Z) <= (Zabs x))%Z. Axiom Div_1 : forall (x:Z), ((ZOdiv x 1%Z) = x). Axiom Mod_1 : forall (x:Z), ((ZOmod x 1%Z) = 0%Z). Axiom Div_inf : forall (x:Z) (y:Z), ((0%Z <= x)%Z /\ (x < y)%Z) -> ((ZOdiv x y) = 0%Z). Axiom Mod_inf : forall (x:Z) (y:Z), ((0%Z <= x)%Z /\ (x < y)%Z) -> ((ZOmod x y) = x). Axiom Div_mult : forall (x:Z) (y:Z) (z:Z), ((0%Z < x)%Z /\ ((0%Z <= y)%Z /\ (0%Z <= z)%Z)) -> ((ZOdiv ((x * y)%Z + z)%Z x) = (y + (ZOdiv z x))%Z). Axiom Mod_mult : forall (x:Z) (y:Z) (z:Z), ((0%Z < x)%Z /\ ((0%Z <= y)%Z /\ (0%Z <= z)%Z)) -> ((ZOmod ((x * y)%Z + z)%Z x) = (ZOmod z x)). `````` Jean-Christophe Filliâtre committed Aug 18, 2011 61 62 63 64 65 66 67 68 ``````Definition lt_nat(x:Z) (y:Z): Prop := (0%Z <= y)%Z /\ (x < y)%Z. Inductive lex : (Z* Z)%type -> (Z* Z)%type -> Prop := | Lex_1 : forall (x1:Z) (x2:Z) (y1:Z) (y2:Z), (lt_nat x1 x2) -> (lex (x1, y1) (x2, y2)) | Lex_2 : forall (x:Z) (y1:Z) (y2:Z), (lt_nat y1 y2) -> (lex (x, y1) (x, y2)). `````` Jean-Christophe Filliâtre committed Aug 17, 2011 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 ``````Definition even(n:Z): Prop := exists k:Z, (n = (2%Z * k)%Z). Definition odd(n:Z): Prop := exists k:Z, (n = ((2%Z * k)%Z + 1%Z)%Z). Axiom even_or_odd : forall (n:Z), (even n) \/ (odd n). Axiom even_not_odd : forall (n:Z), (even n) -> ~ (odd n). Axiom odd_not_even : forall (n:Z), (odd n) -> ~ (even n). Axiom even_odd : forall (n:Z), (even n) -> (odd (n + 1%Z)%Z). Axiom odd_even : forall (n:Z), (odd n) -> (even (n + 1%Z)%Z). Axiom even_even : forall (n:Z), (even n) -> (even (n + 2%Z)%Z). Axiom odd_odd : forall (n:Z), (odd n) -> (odd (n + 2%Z)%Z). Axiom even_2k : forall (k:Z), (even (2%Z * k)%Z). Axiom odd_2k1 : forall (k:Z), (odd ((2%Z * k)%Z + 1%Z)%Z). Definition divides(d:Z) (n:Z): Prop := exists q:Z, (n = (q * d)%Z). Axiom divides_refl : forall (n:Z), (divides n n). Axiom divides_1_n : forall (n:Z), (divides 1%Z n). Axiom divides_0 : forall (n:Z), (divides n 0%Z). Axiom divides_left : forall (a:Z) (b:Z) (c:Z), (divides a b) -> (divides (c * a)%Z (c * b)%Z). Axiom divides_right : forall (a:Z) (b:Z) (c:Z), (divides a b) -> (divides (a * c)%Z (b * c)%Z). Axiom divides_oppr : forall (a:Z) (b:Z), (divides a b) -> (divides a (-b)%Z). Axiom divides_oppl : forall (a:Z) (b:Z), (divides a b) -> (divides (-a)%Z b). Axiom divides_oppr_rev : forall (a:Z) (b:Z), (divides (-a)%Z b) -> (divides a b). Axiom divides_oppl_rev : forall (a:Z) (b:Z), (divides a (-b)%Z) -> (divides a b). Axiom divides_plusr : forall (a:Z) (b:Z) (c:Z), (divides a b) -> ((divides a c) -> (divides a (b + c)%Z)). Axiom divides_minusr : forall (a:Z) (b:Z) (c:Z), (divides a b) -> ((divides a c) -> (divides a (b - c)%Z)). Axiom divides_multl : forall (a:Z) (b:Z) (c:Z), (divides a b) -> (divides a (c * b)%Z). Axiom divides_multr : forall (a:Z) (b:Z) (c:Z), (divides a b) -> (divides a (b * c)%Z). Axiom divides_factorl : forall (a:Z) (b:Z), (divides a (b * a)%Z). Axiom divides_factorr : forall (a:Z) (b:Z), (divides a (a * b)%Z). Axiom divides_n_1 : forall (n:Z), (divides n 1%Z) -> ((n = 1%Z) \/ (n = (-1%Z)%Z)). Axiom divides_antisym : forall (a:Z) (b:Z), (divides a b) -> ((divides b a) -> ((a = b) \/ (a = (-b)%Z))). Axiom divides_trans : forall (a:Z) (b:Z) (c:Z), (divides a b) -> ((divides b c) -> (divides a c)). Axiom divides_bounds : forall (a:Z) (b:Z), (divides a b) -> ((~ (b = 0%Z)) -> ((Zabs a) <= (Zabs b))%Z). Axiom Div_mod1 : forall (x:Z) (y:Z), (~ (y = 0%Z)) -> (x = ((y * (Zdiv x y))%Z + (Zmod x y))%Z). Axiom Div_bound1 : forall (x:Z) (y:Z), ((0%Z <= x)%Z /\ (0%Z < y)%Z) -> ((0%Z <= (Zdiv x y))%Z /\ ((Zdiv x y) <= x)%Z). Axiom Mod_bound1 : forall (x:Z) (y:Z), (~ (y = 0%Z)) -> ((0%Z <= (Zmod x y))%Z /\ ((Zmod x y) < (Zabs y))%Z). Axiom Mod_11 : forall (x:Z), ((Zmod x 1%Z) = 0%Z). Axiom Div_11 : forall (x:Z), ((Zdiv x 1%Z) = x). `````` Jean-Christophe Filliâtre committed Aug 18, 2011 156 ``````Axiom mod_divides_euclidean : forall (a:Z) (b:Z), (~ (b = 0%Z)) -> `````` Jean-Christophe Filliâtre committed Aug 17, 2011 157 158 `````` (((Zmod a b) = 0%Z) -> (divides b a)). `````` Jean-Christophe Filliâtre committed Aug 18, 2011 159 160 161 162 163 164 165 166 ``````Axiom divides_mod_euclidean : forall (a:Z) (b:Z), (~ (b = 0%Z)) -> ((divides b a) -> ((Zmod a b) = 0%Z)). Axiom mod_divides_computer : forall (a:Z) (b:Z), (~ (b = 0%Z)) -> (((ZOmod a b) = 0%Z) -> (divides b a)). Axiom divides_mod_computer : forall (a:Z) (b:Z), (~ (b = 0%Z)) -> ((divides b a) -> ((ZOmod a b) = 0%Z)). `````` Jean-Christophe Filliâtre committed Aug 17, 2011 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 `````` Axiom even_divides : forall (a:Z), (even a) <-> (divides 2%Z a). Axiom odd_divides : forall (a:Z), (odd a) <-> ~ (divides 2%Z a). Definition prime(p:Z): Prop := (2%Z <= p)%Z /\ forall (n:Z), ((1%Z < n)%Z /\ (n < p)%Z) -> ~ (divides n p). Axiom not_prime_1 : ~ (prime 1%Z). Axiom prime_2 : (prime 2%Z). Axiom prime_3 : (prime 3%Z). Axiom prime_divisors : forall (p:Z), (prime p) -> forall (d:Z), (divides d p) -> ((d = 1%Z) \/ ((d = (-1%Z)%Z) \/ ((d = p) \/ (d = (-p)%Z)))). `````` Jean-Christophe Filliâtre committed Aug 18, 2011 184 185 186 187 ``````Axiom small_divisors : forall (p:Z), (2%Z <= p)%Z -> ((forall (d:Z), (2%Z <= d)%Z -> ((prime d) -> (((1%Z < (d * d)%Z)%Z /\ ((d * d)%Z <= p)%Z) -> ~ (divides d p)))) -> (prime p)). `````` Jean-Christophe Filliâtre committed Aug 17, 2011 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 ``````Axiom even_prime : forall (p:Z), (prime p) -> ((even p) -> (p = 2%Z)). Axiom odd_prime : forall (p:Z), (prime p) -> ((3%Z <= p)%Z -> (odd p)). Inductive ref (a:Type) := | mk_ref : a -> ref a. Implicit Arguments mk_ref. Definition contents (a:Type)(u:(ref a)): a := match u with | mk_ref contents1 => contents1 end. Implicit Arguments contents. Parameter map : forall (a:Type) (b:Type), Type. Parameter get: forall (a:Type) (b:Type), (map a b) -> a -> b. Implicit Arguments get. Parameter set: forall (a:Type) (b:Type), (map a b) -> a -> b -> (map a b). Implicit Arguments set. Axiom Select_eq : forall (a:Type) (b:Type), forall (m:(map a b)), forall (a1:a) (a2:a), forall (b1:b), (a1 = a2) -> ((get (set m a1 b1) a2) = b1). Axiom Select_neq : forall (a:Type) (b:Type), forall (m:(map a b)), forall (a1:a) (a2:a), forall (b1:b), (~ (a1 = a2)) -> ((get (set m a1 b1) a2) = (get m a2)). Parameter const: forall (b:Type) (a:Type), b -> (map a b). Set Contextual Implicit. Implicit Arguments const. Unset Contextual Implicit. Axiom Const : forall (b:Type) (a:Type), forall (b1:b) (a1:a), ((get (const( b1):(map a b)) a1) = b1). Inductive array (a:Type) := | mk_array : Z -> (map Z a) -> array a. Implicit Arguments mk_array. Definition elts (a:Type)(u:(array a)): (map Z a) := match u with | mk_array _ elts1 => elts1 end. Implicit Arguments elts. Definition length (a:Type)(u:(array a)): Z := match u with | mk_array length1 _ => length1 end. Implicit Arguments length. Definition get1 (a:Type)(a1:(array a)) (i:Z): a := (get (elts a1) i). Implicit Arguments get1. Definition set1 (a:Type)(a1:(array a)) (i:Z) (v:a): (array a) := match a1 with | mk_array xcl0 _ => (mk_array xcl0 (set (elts a1) i v)) end. Implicit Arguments set1. Definition sorted(p:(array Z)) (u:Z): Prop := forall (i:Z) (j:Z), (((0%Z <= i)%Z /\ (i < j)%Z) /\ (j < u)%Z) -> ((get1 p i) < (get1 p j))%Z. Definition only_primes(p:(array Z)) (u:Z): Prop := forall (i:Z), ((0%Z <= i)%Z /\ (i < u)%Z) -> (prime (get1 p i)). Definition no_prime_in(l:Z) (u:Z): Prop := forall (x:Z), ((l < x)%Z /\ (x < u)%Z) -> ~ (prime x). Definition all_primes(p:(array Z)) (u:Z): Prop := forall (i:Z), ((0%Z <= i)%Z /\ (i < (u - 1%Z)%Z)%Z) -> (no_prime_in (get1 p i) (get1 p (i + 1%Z)%Z)). `````` Jean-Christophe Filliâtre committed Aug 18, 2011 268 269 270 271 272 273 274 275 276 ``````Axiom exists_prime : forall (p:(array Z)) (u:Z), (1%Z <= u)%Z -> (((get1 p 0%Z) = 2%Z) -> ((sorted p u) -> ((only_primes p u) -> ((all_primes p u) -> forall (d:Z), ((2%Z <= d)%Z /\ (d <= (get1 p (u - 1%Z)%Z))%Z) -> ((prime d) -> exists i:Z, ((0%Z <= i)%Z /\ (i < u)%Z) /\ (d = (get1 p i))))))). Axiom Bertrand_postulate : forall (p:Z), (prime p) -> ~ (no_prime_in p (2%Z * p)%Z). `````` Jean-Christophe Filliâtre committed Aug 17, 2011 277 278 279 280 281 282 283 284 285 286 287 ``````(* YOU MAY EDIT THE CONTEXT BELOW *) (* DO NOT EDIT BELOW *) Theorem WP_parameter_prime_numbers : forall (m:Z), (2%Z <= m)%Z -> ((0%Z <= m)%Z -> (((0%Z <= 0%Z)%Z /\ (0%Z < m)%Z) -> forall (p:(map Z Z)), (p = (set (const(0%Z):(map Z Z)) 0%Z 2%Z)) -> (((0%Z <= 1%Z)%Z /\ (1%Z < m)%Z) -> forall (p1:(map Z Z)), (p1 = (set p 1%Z 3%Z)) -> ((2%Z <= (m - 1%Z)%Z)%Z -> forall (n:Z), forall (p2:(map Z Z)), let p3 := (mk_array m p2) in forall (j:Z), ((2%Z <= j)%Z /\ (j <= (m - 1%Z)%Z)%Z) -> ((((get p2 0%Z) = 2%Z) /\ ((sorted p3 j) /\ ((only_primes p3 j) /\ `````` Jean-Christophe Filliâtre committed Aug 18, 2011 288 289 `````` ((all_primes p3 j) /\ ((((get p2 (j - 1%Z)%Z) < n)%Z /\ (n < (2%Z * (get p2 (j - 1%Z)%Z))%Z)%Z) /\ ((odd n) /\ `````` Jean-Christophe Filliâtre committed Aug 17, 2011 290 `````` (no_prime_in (get p2 (j - 1%Z)%Z) n))))))) -> ((((1%Z <= 1%Z)%Z /\ `````` Jean-Christophe Filliâtre committed Aug 18, 2011 291 292 293 294 295 296 297 298 299 `````` (1%Z < j)%Z) /\ (((get p2 0%Z) = 2%Z) /\ ((sorted p3 j) /\ ((only_primes p3 j) /\ ((all_primes p3 j) /\ ((((get p2 (j - 1%Z)%Z) < n)%Z /\ (n < (2%Z * (get p2 (j - 1%Z)%Z))%Z)%Z) /\ ((odd n) /\ ((no_prime_in (get p2 (j - 1%Z)%Z) n) /\ forall (i:Z), ((0%Z <= i)%Z /\ (i < 1%Z)%Z) -> ~ (divides (get p2 i) n))))))))) -> forall (n1:Z), (((get p2 (j - 1%Z)%Z) < n1)%Z /\ ((prime n1) /\ (no_prime_in (get p2 (j - 1%Z)%Z) n1))) -> (((0%Z <= j)%Z /\ (j < m)%Z) -> forall (p4:(map Z Z)), (p4 = (set p2 j n1)) -> forall (n2:Z), (n2 = (n1 + 2%Z)%Z) -> (sorted (mk_array m p4) (j + 1%Z)%Z)))))))). `````` Jean-Christophe Filliâtre committed Aug 17, 2011 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 ``````(* YOU MAY EDIT THE PROOF BELOW *) unfold sorted, get1; simpl; intuition. assert (case: (j0 < j \/ j0 = j)%Z) by omega. destruct case. subst p4. do 2 (rewrite Select_neq; try omega). apply H10; omega. subst p4. rewrite Select_neq; try omega. subst j0; rewrite Select_eq; auto. assert (case: (i < j-1 \/ i = j-1)%Z) by omega. destruct case. assert (get p2 i < get p2 (j-1)); intuition. subst i; auto. Qed. (* DO NOT EDIT BELOW *) ``````