Commit 9d6ba401 authored by MARCHE Claude's avatar MARCHE Claude

bitvetors: proof of footprint lemma

parent 20f47637
......@@ -289,13 +289,13 @@ theory BitVector
axiom to_nat_sub_zero :
forall b:bv, j i:int.
0 <= i <= j ->
0 <= i <= j < size ->
nth b j = False ->
to_nat_sub b j i = to_nat_sub b (j-1) i
axiom to_nat_sub_one :
forall b:bv, j i:int.
0 <= i <= j ->
0 <= i <= j < size ->
nth b j = True ->
to_nat_sub b j i = (pow2 (j-i)) + to_nat_sub b (j-1) i
......@@ -312,22 +312,22 @@ theory BitVector
*)
lemma to_nat_of_zero2:
forall b:bv, i j:int. j >= i >= 0 ->
forall b:bv, i j:int. size > j >= i >= 0 ->
(forall k:int. j >= k > i -> nth b k = False) ->
to_nat_sub b j 0 = to_nat_sub b i 0
lemma to_nat_of_zero:
forall b:bv, i j:int. j >= i >=0 ->
forall b:bv, i j:int. size > j >= i >= 0 ->
(forall k:int. j >= k >= i -> nth b k = False) ->
to_nat_sub b j i = 0
lemma to_nat_of_one:
forall b:bv, i j:int. j >= i >=0 ->
forall b:bv, i j:int. size > j >= i >= 0 ->
(forall k:int. j >= k >= i -> nth b k = True) ->
to_nat_sub b j i = pow2 (j-i+1) - 1
lemma to_nat_sub_footprint: forall b1 b2:bv, j i:int. j >=i>=0 ->
lemma to_nat_sub_footprint: forall b1 b2:bv, j i:int. size > j /\ i >=0 ->
(forall k:int. i <= k <= j -> nth b1 k = nth b2 k) ->
to_nat_sub b1 j i = to_nat_sub b2 j i
(*
......@@ -434,7 +434,7 @@ theory BitVector
axiom size_from_int2c: size - 1 >0
axiom nth_sign_positive:
forall n :int. n>=0 -> nth (from_int2c n) (size-1) = False
forall n :int. n>=0 -> nth (from_int2c n) (size-1) = False
axiom nth_from_int2c_high_even_positive:
forall n i:int. n>=0 /\ size-1 > i >= 0 /\ mod (div n (pow2 i)) 2 = 0 -> nth (from_int2c n) i = False
......@@ -452,7 +452,7 @@ theory BitVector
(*********************************************************************)
axiom nth_sign_negative:
forall n:int. n>=0 -> nth (from_int2c n) (size-1) = True
forall n:int. n<0 -> nth (from_int2c n) (size-1) = True
axiom nth_from_int2c_high_even_negative:
forall n i:int. n<0 /\ size-1 > i >= 0 /\ mod (div n (pow2 i)) 2 = 0 -> nth (from_int2c n) i = True
......@@ -627,7 +627,7 @@ theory TestNegAsXOR
double_of_bv64 (bw_xor x j) = -. double_of_bv64 x
lemma sign_neg:
forall x:bv. sign_value(notb(sign(x))) = -.sign_value(sign(x))
forall x:bv. sign_value(notb(sign(x))) = -.sign_value(sign(x))
lemma MainResult : forall x:bv. 0 < exp(x) < 2047 ->
double_of_bv64 (bw_xor x j) = -. double_of_bv64 x
......@@ -746,36 +746,36 @@ theory TestDoubleOfInt
lemma exp_var: forall x:int. exp(var(x)) = 1075
lemma to_nat_sub_same: forall i:BV64.bv. forall j:BV32.bv. forall m:int.
(forall k:int. 0<=k<=m -> BV64.nth i k = BV32.nth j k) ->
BV64.to_nat_sub i m 0 = BV32.to_nat_sub j m 0
(forall k:int. 0<=k<=m -> BV64.nth i k = BV32.nth j k) ->
BV64.to_nat_sub i m 0 = BV32.to_nat_sub j m 0
lemma nat_to_sub_x: forall x:int.
BV64.to_nat_sub (var x) 30 0 = BV32.to_nat_sub (BV32.from_int2c x) 30 0
lemma to_nat_sub_var:
forall x:int. BV64.to_nat_sub (var x) 30 0 = BV32.to_nat_sub (BV32.from_int2c x) 30 0
lemma x_positive:forall x:int. (BV32.nth (BV32.from_int2c x) 31) = False ->
lemma x_positive:forall x:int. (BV32.nth (BV32.from_int2c x) 31) = False ->
BV32.to_nat_sub (BV32.from_int2c x) 31 0 = BV32.to_nat_sub (BV32.from_int2c x) 30 0
lemma sign_of_x:
forall x:int. (BV32.nth (BV32.from_int2c x) 31) = False->x>0
lemma from_int2c_to_nat_sub:
forall x:int.x>0 -> BV32.to_nat_sub (BV32.from_int2c x) 31 0 = x
lemma from_int2c_to_nat_sub:
forall x:int.x>0 -> BV32.to_nat_sub (BV32.from_int2c x) 31 0 = x
lemma x_positive1: forall x:int. (BV32.nth (BV32.from_int2c x) 31) = False ->
lemma x_positive1: forall x:int. (BV32.nth (BV32.from_int2c x) 31) = False ->
BV32.to_nat_sub (BV32.from_int2c x) 30 0 = x
lemma x_positive2: forall x:int. (BV64.nth (var x) 31) = True ->
lemma x_positive2: forall x:int. (BV64.nth (var x) 31) = True ->
BV64.to_nat_sub (var x) 30 0 = x
lemma mantissa_var_x_positive:
lemma mantissa_var_x_positive:
forall x:int. (BV64.nth (var x) 31) = True ->
mantissa(var(x)) = Pow2int.pow2 31 + x
(*a verifier*)
lemma x_negative: forall x:int. (BV64.nth (var x) 31) = False ->
lemma x_negative: forall x:int. (BV64.nth (var x) 31) = False ->
BV64.to_nat_sub (var x) 30 0 = -x
(*a verifier*)
lemma mantissa_var_x_negative:
lemma mantissa_var_x_negative:
forall x:int. (BV64.nth (var x) 31) = False ->
mantissa(var(x)) = Pow2int.pow2 31 + x
......
(* This file is generated by Why3's Coq driver *)
(* Beware! Only edit allowed sections below *)
Require Import ZArith.
Require Import Rbase.
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 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).
Parameter size: Z.
Parameter bv : Type.
Axiom size_positive : (0%Z < size)%Z.
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).
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 /\
(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 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)).
(* YOU MAY EDIT THE CONTEXT BELOW *)
Open Scope Z_scope.
(* DO NOT EDIT BELOW *)
Theorem 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))).
(* YOU MAY EDIT THE PROOF BELOW *)
intros b1 b2 j i (Hij,Hipos) Hfootprint.
assert (h:(j < i \/ i <= j)) by omega.
destruct h.
do 2 (rewrite to_nat_sub_high; auto).
cut (i-1 <= j).
apply Zlt_lower_bound_ind with
(P:= fun j => to_nat_sub b1 j i = to_nat_sub b2 j i).
intros x Hind Hxi.
assert (h: x=i-1 \/ i <= x) by omega.
destruct h.
subst x.
rewrite to_nat_sub_high; auto with zarith.
rewrite to_nat_sub_high; auto with zarith.
assert (h:(nth b1 x) = true \/ (nth b1 x) = false).
destruct (nth b1 x); auto.
destruct h.
Qed.
(* DO NOT EDIT BELOW *)
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