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Commit c20f5d7c authored by MARCHE Claude's avatar MARCHE Claude
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One more Coq proof in bitvector example

parent 6aecd3ff
(* This file is generated by Why3's Coq driver *)
(* Beware! Only edit allowed sections below *)
Require Import ZArith.
Require Import Rbase.
Require int.Int.
Require int.Abs.
Require int.EuclideanDivision.
(* Why3 assumption *)
Definition implb(x:bool) (y:bool): bool := match (x,
y) with
| (true, false) => false
| (_, _) => true
end.
Parameter pow2: Z -> Z.
Axiom Power_0 : ((pow2 0%Z) = 1%Z).
Axiom Power_s : forall (n:Z), (0%Z <= n)%Z ->
((pow2 (n + 1%Z)%Z) = (2%Z * (pow2 n))%Z).
Axiom Power_1 : ((pow2 1%Z) = 2%Z).
Axiom Power_sum : forall (n:Z) (m:Z), ((0%Z <= n)%Z /\ (0%Z <= m)%Z) ->
((pow2 (n + m)%Z) = ((pow2 n) * (pow2 m))%Z).
Axiom pow2pos : forall (i:Z), (0%Z <= i)%Z -> (0%Z < (pow2 i))%Z.
Axiom pow2_0 : ((pow2 0%Z) = 1%Z).
Axiom pow2_1 : ((pow2 1%Z) = 2%Z).
Axiom pow2_2 : ((pow2 2%Z) = 4%Z).
Axiom pow2_3 : ((pow2 3%Z) = 8%Z).
Axiom pow2_4 : ((pow2 4%Z) = 16%Z).
Axiom pow2_5 : ((pow2 5%Z) = 32%Z).
Axiom pow2_6 : ((pow2 6%Z) = 64%Z).
Axiom pow2_7 : ((pow2 7%Z) = 128%Z).
Axiom pow2_8 : ((pow2 8%Z) = 256%Z).
Axiom pow2_9 : ((pow2 9%Z) = 512%Z).
Axiom pow2_10 : ((pow2 10%Z) = 1024%Z).
Axiom pow2_11 : ((pow2 11%Z) = 2048%Z).
Axiom pow2_12 : ((pow2 12%Z) = 4096%Z).
Axiom pow2_13 : ((pow2 13%Z) = 8192%Z).
Axiom pow2_14 : ((pow2 14%Z) = 16384%Z).
Axiom pow2_15 : ((pow2 15%Z) = 32768%Z).
Axiom pow2_16 : ((pow2 16%Z) = 65536%Z).
Axiom pow2_17 : ((pow2 17%Z) = 131072%Z).
Axiom pow2_18 : ((pow2 18%Z) = 262144%Z).
Axiom pow2_19 : ((pow2 19%Z) = 524288%Z).
Axiom pow2_20 : ((pow2 20%Z) = 1048576%Z).
Axiom pow2_21 : ((pow2 21%Z) = 2097152%Z).
Axiom pow2_22 : ((pow2 22%Z) = 4194304%Z).
Axiom pow2_23 : ((pow2 23%Z) = 8388608%Z).
Axiom pow2_24 : ((pow2 24%Z) = 16777216%Z).
Axiom pow2_25 : ((pow2 25%Z) = 33554432%Z).
Axiom pow2_26 : ((pow2 26%Z) = 67108864%Z).
Axiom pow2_27 : ((pow2 27%Z) = 134217728%Z).
Axiom pow2_28 : ((pow2 28%Z) = 268435456%Z).
Axiom pow2_29 : ((pow2 29%Z) = 536870912%Z).
Axiom pow2_30 : ((pow2 30%Z) = 1073741824%Z).
Axiom pow2_31 : ((pow2 31%Z) = 2147483648%Z).
Axiom pow2_32 : ((pow2 32%Z) = 4294967296%Z).
Axiom pow2_33 : ((pow2 33%Z) = 8589934592%Z).
Axiom pow2_34 : ((pow2 34%Z) = 17179869184%Z).
Axiom pow2_35 : ((pow2 35%Z) = 34359738368%Z).
Axiom pow2_36 : ((pow2 36%Z) = 68719476736%Z).
Axiom pow2_37 : ((pow2 37%Z) = 137438953472%Z).
Axiom pow2_38 : ((pow2 38%Z) = 274877906944%Z).
Axiom pow2_39 : ((pow2 39%Z) = 549755813888%Z).
Axiom pow2_40 : ((pow2 40%Z) = 1099511627776%Z).
Axiom pow2_41 : ((pow2 41%Z) = 2199023255552%Z).
Axiom pow2_42 : ((pow2 42%Z) = 4398046511104%Z).
Axiom pow2_43 : ((pow2 43%Z) = 8796093022208%Z).
Axiom pow2_44 : ((pow2 44%Z) = 17592186044416%Z).
Axiom pow2_45 : ((pow2 45%Z) = 35184372088832%Z).
Axiom pow2_46 : ((pow2 46%Z) = 70368744177664%Z).
Axiom pow2_47 : ((pow2 47%Z) = 140737488355328%Z).
Axiom pow2_48 : ((pow2 48%Z) = 281474976710656%Z).
Axiom pow2_49 : ((pow2 49%Z) = 562949953421312%Z).
Axiom pow2_50 : ((pow2 50%Z) = 1125899906842624%Z).
Axiom pow2_51 : ((pow2 51%Z) = 2251799813685248%Z).
Axiom pow2_52 : ((pow2 52%Z) = 4503599627370496%Z).
Axiom pow2_53 : ((pow2 53%Z) = 9007199254740992%Z).
Axiom pow2_54 : ((pow2 54%Z) = 18014398509481984%Z).
Axiom pow2_55 : ((pow2 55%Z) = 36028797018963968%Z).
Axiom pow2_56 : ((pow2 56%Z) = 72057594037927936%Z).
Axiom pow2_57 : ((pow2 57%Z) = 144115188075855872%Z).
Axiom pow2_58 : ((pow2 58%Z) = 288230376151711744%Z).
Axiom pow2_59 : ((pow2 59%Z) = 576460752303423488%Z).
Axiom pow2_60 : ((pow2 60%Z) = 1152921504606846976%Z).
Axiom pow2_61 : ((pow2 61%Z) = 2305843009213693952%Z).
Axiom pow2_62 : ((pow2 62%Z) = 4611686018427387904%Z).
Axiom pow2_63 : ((pow2 63%Z) = 9223372036854775808%Z).
Axiom Div_pow : forall (x:Z) (i:Z), (((pow2 (i - 1%Z)%Z) <= x)%Z /\
(x < (pow2 i))%Z) -> ((int.EuclideanDivision.div x
(pow2 (i - 1%Z)%Z)) = 1%Z).
Axiom Div_pow2 : forall (x:Z) (i:Z), (((-(pow2 i))%Z <= x)%Z /\
(x < (-(pow2 (i - 1%Z)%Z))%Z)%Z) -> ((int.EuclideanDivision.div x
(pow2 (i - 1%Z)%Z)) = (-2%Z)%Z).
Parameter size: Z.
Axiom size_positive : (1%Z < size)%Z.
Parameter bv : Type.
Parameter nth: bv -> Z -> bool.
Parameter bvzero: bv.
Axiom Nth_zero : forall (n:Z), ((0%Z <= n)%Z /\ (n < size)%Z) -> ((nth bvzero
n) = false).
Parameter bvone: bv.
Axiom Nth_one : forall (n:Z), ((0%Z <= n)%Z /\ (n < size)%Z) -> ((nth bvone
n) = true).
(* Why3 assumption *)
Definition eq(v1:bv) (v2:bv): Prop := forall (n:Z), ((0%Z <= n)%Z /\
(n < size)%Z) -> ((nth v1 n) = (nth v2 n)).
Axiom extensionality : forall (v1:bv) (v2:bv), (eq v1 v2) -> (v1 = v2).
Parameter bw_and: bv -> bv -> bv.
Axiom Nth_bw_and : forall (v1:bv) (v2:bv) (n:Z), ((0%Z <= n)%Z /\
(n < size)%Z) -> ((nth (bw_and v1 v2) n) = (andb (nth v1 n) (nth v2 n))).
Parameter bw_or: bv -> bv -> bv.
Axiom Nth_bw_or : forall (v1:bv) (v2:bv) (n:Z), ((0%Z <= n)%Z /\
(n < size)%Z) -> ((nth (bw_or v1 v2) n) = (orb (nth v1 n) (nth v2 n))).
Parameter bw_xor: bv -> bv -> bv.
Axiom Nth_bw_xor : forall (v1:bv) (v2:bv) (n:Z), ((0%Z <= n)%Z /\
(n < size)%Z) -> ((nth (bw_xor v1 v2) n) = (xorb (nth v1 n) (nth v2 n))).
Axiom Nth_bw_xor_v1true : forall (v1:bv) (v2:bv) (n:Z), (((0%Z <= n)%Z /\
(n < size)%Z) /\ ((nth v1 n) = true)) -> ((nth (bw_xor v1 v2)
n) = (negb (nth v2 n))).
Axiom Nth_bw_xor_v1false : forall (v1:bv) (v2:bv) (n:Z), (((0%Z <= n)%Z /\
(n < size)%Z) /\ ((nth v1 n) = false)) -> ((nth (bw_xor v1 v2) n) = (nth v2
n)).
Axiom Nth_bw_xor_v2true : forall (v1:bv) (v2:bv) (n:Z), (((0%Z <= n)%Z /\
(n < size)%Z) /\ ((nth v2 n) = true)) -> ((nth (bw_xor v1 v2)
n) = (negb (nth v1 n))).
Axiom Nth_bw_xor_v2false : forall (v1:bv) (v2:bv) (n:Z), (((0%Z <= n)%Z /\
(n < size)%Z) /\ ((nth v2 n) = false)) -> ((nth (bw_xor v1 v2) n) = (nth v1
n)).
Parameter bw_not: bv -> bv.
Axiom Nth_bw_not : forall (v:bv) (n:Z), ((0%Z <= n)%Z /\ (n < size)%Z) ->
((nth (bw_not v) n) = (negb (nth v n))).
Parameter lsr: bv -> Z -> bv.
Axiom lsr_nth_low : forall (b:bv) (n:Z) (s:Z), (((0%Z <= n)%Z /\
(n < size)%Z) /\ (((0%Z <= s)%Z /\ (s < size)%Z) /\
((n + s)%Z < size)%Z)) -> ((nth (lsr b s) n) = (nth b (n + s)%Z)).
Axiom lsr_nth_high : forall (b:bv) (n:Z) (s:Z), (((0%Z <= n)%Z /\
(n < size)%Z) /\ (((0%Z <= s)%Z /\ (s < size)%Z) /\
(size <= (n + s)%Z)%Z)) -> ((nth (lsr b s) n) = false).
Parameter asr: bv -> Z -> bv.
Axiom asr_nth_low : forall (b:bv) (n:Z) (s:Z), ((0%Z <= n)%Z /\
(n < size)%Z) -> ((0%Z <= s)%Z -> (((n + s)%Z < size)%Z -> ((nth (asr b s)
n) = (nth b (n + s)%Z)))).
Axiom asr_nth_high : forall (b:bv) (n:Z) (s:Z), ((0%Z <= n)%Z /\
(n < size)%Z) -> ((0%Z <= s)%Z -> ((size <= (n + s)%Z)%Z -> ((nth (asr b s)
n) = (nth b (size - 1%Z)%Z)))).
Parameter lsl: bv -> Z -> bv.
Axiom lsl_nth_high : forall (b:bv) (n:Z) (s:Z), ((0%Z <= n)%Z /\
(n < size)%Z) -> ((0%Z <= s)%Z -> ((0%Z <= (n - s)%Z)%Z -> ((nth (lsl b s)
n) = (nth b (n - s)%Z)))).
Axiom lsl_nth_low : forall (b:bv) (n:Z) (s:Z), ((0%Z <= n)%Z /\
(n < size)%Z) -> ((0%Z <= s)%Z -> (((n - s)%Z < 0%Z)%Z -> ((nth (lsl b s)
n) = false))).
Parameter to_nat_sub: bv -> Z -> Z -> Z.
Axiom to_nat_sub_zero : forall (b:bv) (j:Z) (i:Z), (((0%Z <= i)%Z /\
(i <= j)%Z) /\ (j < size)%Z) -> (((nth b j) = false) -> ((to_nat_sub b j
i) = (to_nat_sub b (j - 1%Z)%Z i))).
Axiom to_nat_sub_one : forall (b:bv) (j:Z) (i:Z), (((0%Z <= i)%Z /\
(i <= j)%Z) /\ (j < size)%Z) -> (((nth b j) = true) -> ((to_nat_sub b j
i) = ((pow2 (j - i)%Z) + (to_nat_sub b (j - 1%Z)%Z i))%Z)).
Axiom to_nat_sub_high : forall (b:bv) (j:Z) (i:Z), (j < i)%Z ->
((to_nat_sub b j i) = 0%Z).
Axiom to_nat_of_zero2 : forall (b:bv) (i:Z) (j:Z), (((j < size)%Z /\
(i <= j)%Z) /\ (0%Z <= i)%Z) -> ((forall (k:Z), ((k <= j)%Z /\
(i < k)%Z) -> ((nth b k) = false)) -> ((to_nat_sub b j 0%Z) = (to_nat_sub b
i 0%Z))).
Axiom to_nat_of_zero : forall (b:bv) (i:Z) (j:Z), ((j < size)%Z /\
(0%Z <= i)%Z) -> ((forall (k:Z), ((k <= j)%Z /\ (i <= k)%Z) -> ((nth b
k) = false)) -> ((to_nat_sub b j i) = 0%Z)).
Axiom to_nat_of_one : forall (b:bv) (i:Z) (j:Z), (((j < size)%Z /\
(i <= j)%Z) /\ (0%Z <= i)%Z) -> ((forall (k:Z), ((k <= j)%Z /\
(i <= k)%Z) -> ((nth b k) = true)) -> ((to_nat_sub b j
i) = ((pow2 ((j - i)%Z + 1%Z)%Z) - 1%Z)%Z)).
Axiom to_nat_sub_footprint : forall (b1:bv) (b2:bv) (j:Z) (i:Z),
((j < size)%Z /\ (0%Z <= i)%Z) -> ((forall (k:Z), ((i <= k)%Z /\
(k <= j)%Z) -> ((nth b1 k) = (nth b2 k))) -> ((to_nat_sub b1 j
i) = (to_nat_sub b2 j i))).
Parameter from_int: Z -> bv.
Axiom nth_from_int_high_even : forall (n:Z) (i:Z), (((i < size)%Z /\
(0%Z <= i)%Z) /\ ((int.EuclideanDivision.mod1 (int.EuclideanDivision.div n
(pow2 i)) 2%Z) = 0%Z)) -> ((nth (from_int n) i) = false).
Axiom nth_from_int_high_odd : forall (n:Z) (i:Z), (((i < size)%Z /\
(0%Z <= i)%Z) /\
~ ((int.EuclideanDivision.mod1 (int.EuclideanDivision.div n (pow2 i))
2%Z) = 0%Z)) -> ((nth (from_int n) i) = true).
Axiom nth_from_int_low_even : forall (n:Z), ((int.EuclideanDivision.mod1 n
2%Z) = 0%Z) -> ((nth (from_int n) 0%Z) = false).
Axiom nth_from_int_low_odd : forall (n:Z), (~ ((int.EuclideanDivision.mod1 n
2%Z) = 0%Z)) -> ((nth (from_int n) 0%Z) = true).
Axiom nth_from_int_0 : forall (i:Z), ((i < size)%Z /\ (0%Z <= i)%Z) ->
((nth (from_int 0%Z) i) = false).
Parameter from_int2c: Z -> bv.
Axiom nth_sign_positive : forall (n:Z), (0%Z <= n)%Z -> ((nth (from_int2c n)
(size - 1%Z)%Z) = false).
Axiom nth_sign_negative : forall (n:Z), (n < 0%Z)%Z -> ((nth (from_int2c n)
(size - 1%Z)%Z) = true).
Axiom nth_from_int2c_high_even : forall (n:Z) (i:Z),
(((i < (size - 1%Z)%Z)%Z /\ (0%Z <= i)%Z) /\
((int.EuclideanDivision.mod1 (int.EuclideanDivision.div n (pow2 i))
2%Z) = 0%Z)) -> ((nth (from_int2c n) i) = false).
Axiom nth_from_int2c_high_odd : forall (n:Z) (i:Z),
(((i < (size - 1%Z)%Z)%Z /\ (0%Z <= i)%Z) /\
~ ((int.EuclideanDivision.mod1 (int.EuclideanDivision.div n (pow2 i))
2%Z) = 0%Z)) -> ((nth (from_int2c n) i) = true).
Axiom nth_from_int2c_low_even : forall (n:Z), ((int.EuclideanDivision.mod1 n
2%Z) = 0%Z) -> ((nth (from_int2c n) 0%Z) = false).
Axiom nth_from_int2c_low_odd : forall (n:Z),
(~ ((int.EuclideanDivision.mod1 n 2%Z) = 0%Z)) -> ((nth (from_int2c n)
0%Z) = true).
Axiom nth_from_int2c_0 : forall (i:Z), ((i < size)%Z /\ (0%Z <= i)%Z) ->
((nth (from_int2c 0%Z) i) = false).
Open Scope Z_scope.
Require Import Why3.
Ltac ae := why3 "alt-ergo" timelimit 3.
(* Why3 goal *)
Theorem nth_from_int2c_plus_pow2 : forall (x:Z) (k:Z) (i:Z), ((0%Z <= k)%Z /\
(k < i)%Z) -> ((nth (from_int2c (x + (pow2 i))%Z) k) = (nth (from_int2c x)
k)).
intros x k i (h1 & h2).
generalize i h1 h2.
pattern k; apply Z_lt_induction; auto.
clear k i h1 h2.
intros k Hind i Hk Hki.
assert (h: k = 0 \/ k > 0) by omega.
destruct h.
(*case k = 0*)
subst k.
assert (h: nth (from_int2c (x)) 0 = true
\/ nth (from_int2c (x)) 0 = false).
destruct (nth (from_int2c (x)) 0);auto.
destruct h.
rewrite H.
apply nth_from_int2c_low_odd.
rewrite Mod_pow2.
(*case k >0*)
Qed.
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