Commit 28683cae authored by MARCHE Claude's avatar MARCHE Claude

bitvector, var_value proved

parent 33906b7e
......@@ -294,12 +294,12 @@ theory BitVector
lemma to_nat_of_zero:
forall b:bv, i j:int. j >=i>=0 ->
forall b:bv, i j:int. 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. i >= j >=0 ->
forall b:bv, i j:int. j >= i >=0 ->
(forall k:int. j >= k >= i -> nth b k = True) ->
to_nat_sub b j i = pow2 (j-i+1) - 1
......@@ -497,8 +497,6 @@ theory BV_double
(pow2 ((exp b) - 1023)) *.
(1.0 +. (from_int (mantissa b)) *. (pow2 (-52)))
(* TODO *)
end
theory TestDouble
......@@ -556,9 +554,9 @@ theory TestNegAsXOR
lemma MainResult : forall x:bv. 0 < exp(x) < 2047 ->
double_of_bv64 (bw_xor x j) = -. double_of_bv64 x
end
theory TestDoubleOfInt
use BV32
......@@ -572,21 +570,42 @@ theory TestDoubleOfInt
use Pow2real
use import BV_double
(*********************************************************************)
(* j = 0x43300000 *)
(* j' = 0x80000000 *)
(*********************************************************************)
function j : BV32.bv = BV32.from_int 0x43300000
function j' : BV32.bv = BV32.from_int 0x80000000
(*********************************************************************)
(* definitions: *)
(* const : bv64 = concat j j' *)
(* const_as_double : real = double_of_bv64(const) *)
(*********************************************************************)
function const : BV64.bv = BV32_64.concat j j'
lemma sign_const: sign(const) = False
function const_as_double : real = double_of_bv64 const
lemma exp_const: exp(const) = 1075
(*********************************************************************)
(* next lemma: const_as_double = 2^52 + 2^31 *)
(*********************************************************************)
lemma mantissa_const_nth:
forall i:int. 0 <= i <= 30 -> BV64.nth const i = False
lemma nth_const1: forall i:int. 0 <= i <= 30 -> BV64.nth const i = False
lemma nth_const2: BV64.nth const 31 = True
lemma nth_const3: forall i:int. 32 <= i <= 51 -> BV64.nth const i = False
lemma nth_const4: forall i:int. 52 <= i <= 53 -> BV64.nth const i = True
lemma nth_const5: forall i:int. 54 <= i <= 55 -> BV64.nth const i = False
lemma nth_const6: forall i:int. 56 <= i <= 57 -> BV64.nth const i = True
lemma nth_const7: forall i:int. 58 <= i <= 61 -> BV64.nth const i = False
lemma nth_const8: BV64.nth const 62 = True
lemma nth_const9: BV64.nth const 63 = False
lemma mantissa_const_31th:
BV64.nth const 31 = True
lemma sign_const: sign(const) = False
lemma exp_const: exp(const) = 1075
lemma to_nat_mantissa_1: (BV64.to_nat_sub const 30 0) = 0
......@@ -604,18 +623,27 @@ theory TestDoubleOfInt
lemma real52_a_m52: Pow2real.pow2 (1075 - 1023) *. Pow2real.pow2 31 *. Pow2real.pow2 (-52) = Pow2real.pow2 31
function const_as_double : real = double_of_bv64 const
lemma const_value0: const_as_double =
1.0*.Pow2real.pow2 (1075 - 1023) *. (1.0 +. Pow2real.pow2 31 *. Pow2real.pow2 (-52))
lemma const_value: const_as_double = Pow2real.pow2 52 +. Pow2real.pow2 31
(*********************************************************************)
(* definitions: *)
(* var(x) : bv64 = concat j (j' xor x) *)
(* var_as_double(x) : real = double_of_bv64(var(x)) *)
(*********************************************************************)
function jpxor(i:int): BV32.bv = (BV32.bw_xor j' (BV32.from_int i))
function var(i:int): BV64.bv = (BV32_64.concat j (jpxor i))
function var_as_double(x:int) : real = double_of_bv64 (var x)
(*********************************************************************)
(* next lemma: for all integer x, var_as_double(x) = 2^52 + 2^31 + x *)
(*********************************************************************)
lemma nth30_0:
forall x:BV32.bv, j:int. 0 <= j <31 ->
(BV32.nth (BV32.bw_xor j' x) j) = (BV32.nth x j)
......@@ -642,29 +670,34 @@ theory TestDoubleOfInt
lemma mantissa_var: forall x:int. mantissa(var(x)) = Pow2int.pow2 31 + x
lemma realx_a_mx: forall x:int. Pow2real.pow2 (1075 - 1023) *. ((Pow2real.pow2 31 +. (from_int x)) *. Pow2real.pow2 (-52)) = Pow2real.pow2 31 +. (from_int x)
(*proved by Coq*)
lemma var_value0: forall x:int. var_as_double(x) =
Pow2real.pow2 (1075 - 1023) *. (1.0 +. (from_int (Pow2int.pow2 31 + x)) *. Pow2real.pow2 (-52))
function var_as_double(x:int) : real = double_of_bv64 (var x)
lemma one_mult:
forall x:real. 1.0*.x = x
lemma from_int_sum : forall x:int.
from_int (Pow2int.pow2 31 + x) = from_int (Pow2int.pow2 31) +. from_int x
lemma int_real_from_int:
forall x:int. from_int (Pow2int.pow2 31 + x) = Pow2real.pow2 31 +. (from_int x)
lemma var_value3: forall x:int. var_as_double(x) =
Pow2real.pow2 52 +. Pow2real.pow2 52 *. (from_int (Pow2int.pow2 31) +. from_int x) *. Pow2real.pow2 (-52)
(*proved by Coq*)
lemma var_value0: forall x:int. var_as_double(x) =
Pow2real.pow2 (1075 - 1023) *. (1.0 +. (from_int (Pow2int.pow2 31 + x)) *. Pow2real.pow2 (-52))
lemma distr_pow52 : forall x:real.
Pow2real.pow2 52 *. x *. Pow2real.pow2 (-52) = x
lemma var_value4: forall x:int. var_as_double(x) =
Pow2real.pow2 52 +. (from_int (Pow2int.pow2 31)) +. from_int x
lemma pow31 : from_int (Pow2int.pow2 31) = Pow2real.pow2 31
lemma var_value: forall x:int. var_as_double(x) = Pow2real.pow2 52 +. Pow2real.pow2 31 +. (from_int x)
(*********************************************************************)
(* main result *)
(*********************************************************************)
function double_of_int32 (i:int) : real = var_as_double(i) -. const_as_double
(*
function double_of_int32 (i:int) : real =
let var = BV32_64.concat j (BV32.bw_xor j' (BV32.from_int i)) in
let v = double_of_bv64 var in
v -. const_as_double
*)
lemma MainResult: forall i:int. double_of_int32 i = from_int i
end
......
......@@ -2,6 +2,12 @@
(* 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.
......
(* 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 bv : Type.
Parameter nth: bv -> Z -> bool.
Parameter bvzero: bv.
Axiom Nth_zero : forall (n:Z), ((0%Z <= n)%Z /\ (n < 32%Z)%Z) ->
((nth bvzero n) = false).
Parameter bvone: bv.
Axiom Nth_one : forall (n:Z), ((0%Z <= n)%Z /\ (n < 32%Z)%Z) -> ((nth bvone
n) = true).
Definition eq(v1:bv) (v2:bv): Prop := forall (n:Z), ((0%Z <= n)%Z /\
(n < 32%Z)%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 < 32%Z)%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 < 32%Z)%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 < 32%Z)%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 < 32%Z)%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 < 32%Z)%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 < 32%Z)%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 < 32%Z)%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 < 32%Z)%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 < 32%Z)%Z) -> ((0%Z <= s)%Z -> (((n + s)%Z < 32%Z)%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 < 32%Z)%Z) -> ((0%Z <= s)%Z -> ((32%Z <= (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 < 32%Z)%Z) -> ((0%Z <= s)%Z -> (((n + s)%Z < 32%Z)%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 < 32%Z)%Z) -> ((0%Z <= s)%Z -> ((32%Z <= (n + s)%Z)%Z -> ((nth (asr b
s) n) = (nth b (32%Z - 1%Z)%Z)))).
Parameter lsl: bv -> Z -> bv.
Axiom lsl_nth_high : forall (b:bv) (n:Z) (s:Z), ((0%Z <= n)%Z /\
(n < 32%Z)%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 < 32%Z)%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) -> (((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) -> (((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), ((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), ((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), ((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),
(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)).
Axiom lsr_to_nat_sub : forall (b:bv) (s:Z), (0%Z <= s)%Z ->
((to_nat_sub (lsr b s) (32%Z - 1%Z)%Z 0%Z) = (to_nat_sub b
((32%Z - 1%Z)%Z - s)%Z 0%Z)).
Parameter from_int: Z -> bv.
Axiom Abs_le : forall (x:Z) (y:Z), ((Zabs x) <= y)%Z <-> (((-y)%Z <= x)%Z /\
(x <= y)%Z).
Parameter div: Z -> Z -> Z.
Parameter mod1: Z -> Z -> Z.
Axiom Div_mod : forall (x:Z) (y:Z), (~ (y = 0%Z)) -> (x = ((y * (div x
y))%Z + (mod1 x y))%Z).
Axiom Div_bound : forall (x:Z) (y:Z), ((0%Z <= x)%Z /\ (0%Z < y)%Z) ->
((0%Z <= (div x y))%Z /\ ((div x y) <= x)%Z).
Axiom Mod_bound : forall (x:Z) (y:Z), (~ (y = 0%Z)) -> ((0%Z <= (mod1 x
y))%Z /\ ((mod1 x y) < (Zabs y))%Z).
Axiom Mod_1 : forall (x:Z), ((mod1 x 1%Z) = 0%Z).
Axiom Div_1 : forall (x:Z), ((div x 1%Z) = x).
Axiom nth_from_int_high_even : forall (n:Z) (i:Z), (((i < 32%Z)%Z /\
(0%Z <= i)%Z) /\ ((mod1 (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 < 32%Z)%Z /\
(0%Z <= i)%Z) /\ ~ ((mod1 (div n (pow2 i)) 2%Z) = 0%Z)) ->
((nth (from_int n) i) = true).
Axiom nth_from_int_low_even : forall (n:Z), ((mod1 n 2%Z) = 0%Z) ->
((nth (from_int n) 0%Z) = false).
Axiom nth_from_int_low_odd : forall (n:Z), (~ ((mod1 n 2%Z) = 0%Z)) ->
((nth (from_int n) 0%Z) = true).
Axiom pow2i : forall (i:Z), (0%Z <= i)%Z -> ~ ((pow2 i) = 0%Z).
Axiom nth_from_int_0 : forall (i:Z), ((i < 32%Z)%Z /\ (0%Z <= i)%Z) ->
((nth (from_int 0%Z) i) = false).
Parameter bv1 : Type.
Parameter nth1: bv1 -> Z -> bool.
Parameter bvzero1: bv1.
Axiom Nth_zero1 : forall (n:Z), ((0%Z <= n)%Z /\ (n < 64%Z)%Z) ->
((nth1 bvzero1 n) = false).
Parameter bvone1: bv1.
Axiom Nth_one1 : forall (n:Z), ((0%Z <= n)%Z /\ (n < 64%Z)%Z) ->
((nth1 bvone1 n) = true).
Definition eq1(v1:bv1) (v2:bv1): Prop := forall (n:Z), ((0%Z <= n)%Z /\
(n < 64%Z)%Z) -> ((nth1 v1 n) = (nth1 v2 n)).
Axiom extensionality1 : forall (v1:bv1) (v2:bv1), (eq1 v1 v2) -> (v1 = v2).
Parameter bw_and1: bv1 -> bv1 -> bv1.
Axiom Nth_bw_and1 : forall (v1:bv1) (v2:bv1) (n:Z), ((0%Z <= n)%Z /\
(n < 64%Z)%Z) -> ((nth1 (bw_and1 v1 v2) n) = (andb (nth1 v1 n) (nth1 v2
n))).
Parameter bw_or1: bv1 -> bv1 -> bv1.
Axiom Nth_bw_or1 : forall (v1:bv1) (v2:bv1) (n:Z), ((0%Z <= n)%Z /\
(n < 64%Z)%Z) -> ((nth1 (bw_or1 v1 v2) n) = (orb (nth1 v1 n) (nth1 v2
n))).
Parameter bw_xor1: bv1 -> bv1 -> bv1.
Axiom Nth_bw_xor1 : forall (v1:bv1) (v2:bv1) (n:Z), ((0%Z <= n)%Z /\
(n < 64%Z)%Z) -> ((nth1 (bw_xor1 v1 v2) n) = (xorb (nth1 v1 n) (nth1 v2
n))).
Axiom Nth_bw_xor_v1true1 : forall (v1:bv1) (v2:bv1) (n:Z), (((0%Z <= n)%Z /\
(n < 64%Z)%Z) /\ ((nth1 v1 n) = true)) -> ((nth1 (bw_xor1 v1 v2)
n) = (negb (nth1 v2 n))).
Axiom Nth_bw_xor_v1false1 : forall (v1:bv1) (v2:bv1) (n:Z), (((0%Z <= n)%Z /\
(n < 64%Z)%Z) /\ ((nth1 v1 n) = false)) -> ((nth1 (bw_xor1 v1 v2)
n) = (nth1 v2 n)).
Axiom Nth_bw_xor_v2true1 : forall (v1:bv1) (v2:bv1) (n:Z), (((0%Z <= n)%Z /\
(n < 64%Z)%Z) /\ ((nth1 v2 n) = true)) -> ((nth1 (bw_xor1 v1 v2)
n) = (negb (nth1 v1 n))).
Axiom Nth_bw_xor_v2false1 : forall (v1:bv1) (v2:bv1) (n:Z), (((0%Z <= n)%Z /\
(n < 64%Z)%Z) /\ ((nth1 v2 n) = false)) -> ((nth1 (bw_xor1 v1 v2)
n) = (nth1 v1 n)).
Parameter bw_not1: bv1 -> bv1.
Axiom Nth_bw_not1 : forall (v:bv1) (n:Z), ((0%Z <= n)%Z /\ (n < 64%Z)%Z) ->
((nth1 (bw_not1 v) n) = (negb (nth1 v n))).
Parameter lsr1: bv1 -> Z -> bv1.
Axiom lsr_nth_low1 : forall (b:bv1) (n:Z) (s:Z), ((0%Z <= n)%Z /\
(n < 64%Z)%Z) -> ((0%Z <= s)%Z -> (((n + s)%Z < 64%Z)%Z -> ((nth1 (lsr1 b
s) n) = (nth1 b (n + s)%Z)))).
Axiom lsr_nth_high1 : forall (b:bv1) (n:Z) (s:Z), ((0%Z <= n)%Z /\
(n < 64%Z)%Z) -> ((0%Z <= s)%Z -> ((64%Z <= (n + s)%Z)%Z -> ((nth1 (lsr1 b
s) n) = false))).
Parameter asr1: bv1 -> Z -> bv1.
Axiom asr_nth_low1 : forall (b:bv1) (n:Z) (s:Z), ((0%Z <= n)%Z /\
(n < 64%Z)%Z) -> ((0%Z <= s)%Z -> (((n + s)%Z < 64%Z)%Z -> ((nth1 (asr1 b
s) n) = (nth1 b (n + s)%Z)))).
Axiom asr_nth_high1 : forall (b:bv1) (n:Z) (s:Z), ((0%Z <= n)%Z /\
(n < 64%Z)%Z) -> ((0%Z <= s)%Z -> ((64%Z <= (n + s)%Z)%Z -> ((nth1 (asr1 b
s) n) = (nth1 b (64%Z - 1%Z)%Z)))).