bitvector1/add *.v

c8d03c30 · Tuyen Nguyen · e9a542f8 · c8d03c30 · c8d03c30 · c8d03c30
Commit c8d03c30 authored Nov 16, 2011 by Tuyen Nguyen
25 changed files
--- a/tests/bitvector1.why
+++ b/tests/bitvector1.why
@@ -299,7 +299,7 @@ theory BitVector
      to_nat_sub b j i = 0

  lemma to_nat_of_one:
-    forall b:bv, i j:int.
+    forall b:bv, i j:int. i >= j >=0 ->
      (forall k:int. j >= k >= i -> nth b k = True) ->
      to_nat_sub b j i = pow2 (j-i+1) - 1


--- a/tests/bitvector1/bitvector1_BitVector_lsr_to_nat_sub_1.v
+++ b/tests/bitvector1/bitvector1_BitVector_lsr_to_nat_sub_1.v
+(* This file is generated by Why3's Coq driver *)
+(* Beware! Only edit allowed sections below    *)
+Require Import ZArith.
+Require Import Rbase.
+Parameter pow_of2: Z -> Z.
+
+
+Axiom pow_of2_value : ((pow_of2 0%Z) = 1%Z) /\ (((pow_of2 1%Z) = 2%Z) /\
+  (((pow_of2 2%Z) = 4%Z) /\ (((pow_of2 3%Z) = 8%Z) /\
+  (((pow_of2 4%Z) = 16%Z) /\ (((pow_of2 5%Z) = 32%Z) /\
+  (((pow_of2 6%Z) = 64%Z) /\ (((pow_of2 7%Z) = 128%Z) /\
+  (((pow_of2 8%Z) = 256%Z) /\ (((pow_of2 9%Z) = 512%Z) /\
+  (((pow_of2 10%Z) = 1024%Z) /\ (((pow_of2 11%Z) = 2048%Z) /\
+  (((pow_of2 12%Z) = 4096%Z) /\ (((pow_of2 13%Z) = 8192%Z) /\
+  (((pow_of2 14%Z) = 16384%Z) /\ (((pow_of2 15%Z) = 32768%Z) /\
+  (((pow_of2 16%Z) = 65536%Z) /\ (((pow_of2 17%Z) = 131072%Z) /\
+  (((pow_of2 18%Z) = 262144%Z) /\ (((pow_of2 19%Z) = 524288%Z) /\
+  (((pow_of2 20%Z) = 1048576%Z) /\ (((pow_of2 21%Z) = 2097152%Z) /\
+  (((pow_of2 22%Z) = 4194304%Z) /\ (((pow_of2 23%Z) = 8388608%Z) /\
+  (((pow_of2 24%Z) = 16777216%Z) /\ (((pow_of2 25%Z) = 33554432%Z) /\
+  (((pow_of2 26%Z) = 67108864%Z) /\ (((pow_of2 27%Z) = 134217728%Z) /\
+  (((pow_of2 28%Z) = 268435456%Z) /\ (((pow_of2 29%Z) = 536870912%Z) /\
+  (((pow_of2 30%Z) = 1073741824%Z) /\ (((pow_of2 31%Z) = 2147483648%Z) /\
+  (((pow_of2 32%Z) = 4294967296%Z) /\ (((pow_of2 33%Z) = 8589934592%Z) /\
+  (((pow_of2 34%Z) = 17179869184%Z) /\ (((pow_of2 35%Z) = 34359738368%Z) /\
+  (((pow_of2 36%Z) = 68719476736%Z) /\ (((pow_of2 37%Z) = 137438953472%Z) /\
+  (((pow_of2 38%Z) = 274877906944%Z) /\ (((pow_of2 39%Z) = 549755813888%Z) /\
+  (((pow_of2 40%Z) = 1099511627776%Z) /\
+  (((pow_of2 41%Z) = 2199023255552%Z) /\
+  (((pow_of2 42%Z) = 4398046511104%Z) /\
+  (((pow_of2 43%Z) = 8796093022208%Z) /\
+  (((pow_of2 44%Z) = 17592186044416%Z) /\
+  (((pow_of2 45%Z) = 35184372088832%Z) /\
+  (((pow_of2 46%Z) = 70368744177664%Z) /\
+  (((pow_of2 47%Z) = 140737488355328%Z) /\
+  (((pow_of2 48%Z) = 281474976710656%Z) /\
+  (((pow_of2 49%Z) = 562949953421312%Z) /\
+  (((pow_of2 50%Z) = 1125899906842624%Z) /\
+  (((pow_of2 51%Z) = 2251799813685248%Z) /\
+  (((pow_of2 52%Z) = 4503599627370496%Z) /\
+  (((pow_of2 53%Z) = 9007199254740992%Z) /\
+  (((pow_of2 54%Z) = 18014398509481984%Z) /\
+  (((pow_of2 55%Z) = 36028797018963968%Z) /\
+  (((pow_of2 56%Z) = 72057594037927936%Z) /\
+  (((pow_of2 57%Z) = 144115188075855872%Z) /\
+  (((pow_of2 58%Z) = 288230376151711744%Z) /\
+  (((pow_of2 59%Z) = 576460752303423488%Z) /\
+  (((pow_of2 60%Z) = 1152921504606846976%Z) /\
+  (((pow_of2 61%Z) = 2305843009213693952%Z) /\
+  (((pow_of2 62%Z) = 4611686018427387904%Z) /\
+  ((pow_of2 63%Z) = 9223372036854775808%Z))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))).
+
+Parameter 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).
+
+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 -> (((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 -> ((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_aux: bv -> Z -> Z.
+
+
+Axiom to_nat_aux_zero : forall (b:bv) (i:Z), ((0%Z <= i)%Z /\
+  (i <  size)%Z) -> (((nth b i) = false) -> ((to_nat_aux b
+  i) = (2%Z * (to_nat_aux b (i + 1%Z)%Z))%Z)).
+
+Axiom to_nat_aux_one : forall (b:bv) (i:Z), ((0%Z <= i)%Z /\
+  (i <  size)%Z) -> (((nth b i) = true) -> ((to_nat_aux b
+  i) = (1%Z + (2%Z * (to_nat_aux b (i + 1%Z)%Z))%Z)%Z)).
+
+Axiom to_nat_aux_high : forall (b:bv) (i:Z), (size <= i)%Z -> ((to_nat_aux b
+  i) = 0%Z).
+
+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) = ((pow_of2 (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_sub_low_true : forall (b:bv) (j:Z), ((nth b j) = true) ->
+  ((to_nat_sub b j j) = 1%Z).
+
+Axiom to_nat_sub_low_false : forall (b:bv) (j:Z), ((nth b j) = false) ->
+  ((to_nat_sub b j j) = 0%Z).
+
+Axiom to_nat_of_zero : forall (b:bv) (i:Z) (j: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), (forall (k:Z),
+  ((k <= j)%Z /\ (i <= k)%Z) -> ((nth b k) = true)) -> ((to_nat_sub b j
+  i) = ((pow_of2 ((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)).
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Theorem lsr_to_nat_sub : forall (b:bv) (s:Z), (0%Z <= s)%Z ->
+  ((to_nat_sub (lsr b s) (size - 1%Z)%Z 0%Z) = (to_nat_sub b
+  ((size - 1%Z)%Z - s)%Z 0%Z)).
+(* YOU MAY EDIT THE PROOF BELOW *)
+intros.
+split.
+
+Qed.
+(* DO NOT EDIT BELOW *)
+
+
--- a/tests/bitvector1/bitvector1_BitVector_lsr_to_nat_sub_2.v
+++ b/tests/bitvector1/bitvector1_BitVector_lsr_to_nat_sub_2.v
+(* This file is generated by Why3's Coq driver *)
+(* Beware! Only edit allowed sections below    *)
+Require Import ZArith.
+Require Import Rbase.
+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.
+
+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 -> (((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 -> ((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) -> (((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 ->
+  ((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), (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)).
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Theorem lsr_to_nat_sub : forall (b:bv) (s:Z), (0%Z <= s)%Z ->
+  ((to_nat_sub (lsr b s) (size - 1%Z)%Z 0%Z) = (to_nat_sub b
+  ((size - 1%Z)%Z - s)%Z 0%Z)).
+(* YOU MAY EDIT THE PROOF BELOW *)
+intuition.
+
+Qed.
+(* DO NOT EDIT BELOW *)
+
+
--- a/tests/bitvector1/bitvector1_BitVector_nth_from_int_0_1.v
+++ b/tests/bitvector1/bitvector1_BitVector_nth_from_int_0_1.v
+(* This file is generated by Why3's Coq driver *)
+(* Beware! Only edit allowed sections below    *)
+Require Import ZArith.
+Require Import Rbase.
+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).
+