(* This file is generated by Why3's Coq driver *)
(* Beware! Only edit allowed sections below    *)
Require Import BuiltIn.
Require BuiltIn.
Require int.Int.
Require int.Abs.
Require int.EuclideanDivision.
Require real.Real.
Require real.RealInfix.
Require real.FromInt.
Require bool.Bool.

Parameter pow2: Numbers.BinNums.Z -> Numbers.BinNums.Z.

Axiom Power_0 : ((pow2 0%Z) = 1%Z).

Axiom Power_s :
  forall (n:Numbers.BinNums.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:Numbers.BinNums.Z) (m:Numbers.BinNums.Z),
  (0%Z <= n)%Z /\ (0%Z <= m)%Z ->
  ((pow2 (n + m)%Z) = ((pow2 n) * (pow2 m))%Z).

Axiom pow2pos :
  forall (i:Numbers.BinNums.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_mult_inst :
  forall (x:Numbers.BinNums.Z) (z:Numbers.BinNums.Z), (0%Z < x)%Z ->
  ((int.EuclideanDivision.div ((x * 1%Z)%Z + z)%Z x) =
   (1%Z + (int.EuclideanDivision.div z x))%Z).

Axiom Div_double :
  forall (x:Numbers.BinNums.Z) (y:Numbers.BinNums.Z),
  (0%Z < y)%Z /\ (y <= x)%Z /\ (x < (2%Z * y)%Z)%Z ->
  ((int.EuclideanDivision.div x y) = 1%Z).

Axiom Div_pow :
  forall (x:Numbers.BinNums.Z) (i:Numbers.BinNums.Z), (0%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_double_neg :
  forall (x:Numbers.BinNums.Z) (y:Numbers.BinNums.Z),
  (((-2%Z)%Z * y)%Z <= x)%Z /\ (x < (-y)%Z)%Z /\ ((-y)%Z < 0%Z)%Z ->
  ((int.EuclideanDivision.div x y) = (-2%Z)%Z).

Axiom Div_pow2 :
  forall (x:Numbers.BinNums.Z) (i:Numbers.BinNums.Z), (0%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).

Axiom Mod_pow2_gen :
  forall (x:Numbers.BinNums.Z) (i:Numbers.BinNums.Z) (k:Numbers.BinNums.Z),
  (0%Z <= k)%Z /\ (k < i)%Z ->
  ((int.EuclideanDivision.mod1
    (int.EuclideanDivision.div (x + (pow2 i))%Z (pow2 k)) 2%Z)
   = (int.EuclideanDivision.mod1 (int.EuclideanDivision.div x (pow2 k)) 2%Z)).

Parameter pow21: Numbers.BinNums.Z -> Reals.Rdefinitions.R.

Axiom Power_01 : ((pow21 0%Z) = 1%R).

Axiom Power_s1 :
  forall (n:Numbers.BinNums.Z), (0%Z <= n)%Z ->
  ((pow21 (n + 1%Z)%Z) = (2%R * (pow21 n))%R).

Axiom Power_p :
  forall (n:Numbers.BinNums.Z), (n <= 0%Z)%Z ->
  ((pow21 (n - 1%Z)%Z) = ((1 / 2)%R * (pow21 n))%R).

Axiom Power_s_all :
  forall (n:Numbers.BinNums.Z), ((pow21 (n + 1%Z)%Z) = (2%R * (pow21 n))%R).

Axiom Power_p_all :
  forall (n:Numbers.BinNums.Z),
  ((pow21 (n - 1%Z)%Z) = ((1 / 2)%R * (pow21 n))%R).

Axiom Power_1_2 : ((1 / 2)%R = (1%R / 2%R)%R).

Axiom Power_11 : ((pow21 1%Z) = 2%R).

Axiom Power_neg1 : ((pow21 (-1%Z)%Z) = (1 / 2)%R).

Axiom Power_non_null_aux :
  forall (n:Numbers.BinNums.Z), (0%Z <= n)%Z -> ~ ((pow21 n) = 0%R).

Axiom Power_neg_aux :
  forall (n:Numbers.BinNums.Z), (0%Z <= n)%Z ->
  ((pow21 (-n)%Z) = (1%R / (pow21 n))%R).

Axiom Power_non_null : forall (n:Numbers.BinNums.Z), ~ ((pow21 n) = 0%R).

Axiom Power_neg :
  forall (n:Numbers.BinNums.Z), ((pow21 (-n)%Z) = (1%R / (pow21 n))%R).

Axiom Power_sum_aux :
  forall (m:Numbers.BinNums.Z) (n:Numbers.BinNums.Z), (0%Z <= m)%Z ->
  ((pow21 (n + m)%Z) = ((pow21 n) * (pow21 m))%R).

Axiom Power_sum1 :
  forall (n:Numbers.BinNums.Z) (m:Numbers.BinNums.Z),
  ((pow21 (n + m)%Z) = ((pow21 n) * (pow21 m))%R).

Axiom Pow2_int_real :
  forall (x:Numbers.BinNums.Z), (0%Z <= x)%Z ->
  ((pow21 x) = (BuiltIn.IZR (pow2 x))).

Axiom size_positive : (1%Z < 32%Z)%Z.

Axiom bv : Type.
Parameter bv_WhyType : WhyType bv.
Existing Instance bv_WhyType.

Parameter nth: bv -> Numbers.BinNums.Z -> Init.Datatypes.bool.

Parameter bvzero: bv.

Axiom Nth_zero :
  forall (n:Numbers.BinNums.Z), (0%Z <= n)%Z /\ (n < 32%Z)%Z ->
  ((nth bvzero n) = Init.Datatypes.false).

Parameter bvone: bv.

Axiom Nth_one :
  forall (n:Numbers.BinNums.Z), (0%Z <= n)%Z /\ (n < 32%Z)%Z ->
  ((nth bvone n) = Init.Datatypes.true).

(* Why3 assumption *)
Definition eq (v1:bv) (v2:bv) : Prop :=
  forall (n:Numbers.BinNums.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:Numbers.BinNums.Z),
  (0%Z <= n)%Z /\ (n < 32%Z)%Z ->
  ((nth (bw_and v1 v2) n) = (Init.Datatypes.andb (nth v1 n) (nth v2 n))).

Parameter bw_or: bv -> bv -> bv.

Axiom Nth_bw_or :
  forall (v1:bv) (v2:bv) (n:Numbers.BinNums.Z),
  (0%Z <= n)%Z /\ (n < 32%Z)%Z ->
  ((nth (bw_or v1 v2) n) = (Init.Datatypes.orb (nth v1 n) (nth v2 n))).

Parameter bw_xor: bv -> bv -> bv.

Axiom Nth_bw_xor :
  forall (v1:bv) (v2:bv) (n:Numbers.BinNums.Z),
  (0%Z <= n)%Z /\ (n < 32%Z)%Z ->
  ((nth (bw_xor v1 v2) n) = (Init.Datatypes.xorb (nth v1 n) (nth v2 n))).

Axiom Nth_bw_xor_v1true :
  forall (v1:bv) (v2:bv) (n:Numbers.BinNums.Z),
  ((0%Z <= n)%Z /\ (n < 32%Z)%Z) /\ ((nth v1 n) = Init.Datatypes.true) ->
  ((nth (bw_xor v1 v2) n) = (Init.Datatypes.negb (nth v2 n))).

Axiom Nth_bw_xor_v1false :
  forall (v1:bv) (v2:bv) (n:Numbers.BinNums.Z),
  ((0%Z <= n)%Z /\ (n < 32%Z)%Z) /\ ((nth v1 n) = Init.Datatypes.false) ->
  ((nth (bw_xor v1 v2) n) = (nth v2 n)).

Axiom Nth_bw_xor_v2true :
  forall (v1:bv) (v2:bv) (n:Numbers.BinNums.Z),
  ((0%Z <= n)%Z /\ (n < 32%Z)%Z) /\ ((nth v2 n) = Init.Datatypes.true) ->
  ((nth (bw_xor v1 v2) n) = (Init.Datatypes.negb (nth v1 n))).

Axiom Nth_bw_xor_v2false :
  forall (v1:bv) (v2:bv) (n:Numbers.BinNums.Z),
  ((0%Z <= n)%Z /\ (n < 32%Z)%Z) /\ ((nth v2 n) = Init.Datatypes.false) ->
  ((nth (bw_xor v1 v2) n) = (nth v1 n)).

Parameter bw_not: bv -> bv.

Axiom Nth_bw_not :
  forall (v:bv) (n:Numbers.BinNums.Z), (0%Z <= n)%Z /\ (n < 32%Z)%Z ->
  ((nth (bw_not v) n) = (Init.Datatypes.negb (nth v n))).

Parameter lsr: bv -> Numbers.BinNums.Z -> bv.

Axiom lsr_nth_low :
  forall (b:bv) (n:Numbers.BinNums.Z) (s:Numbers.BinNums.Z),
  ((0%Z <= n)%Z /\ (n < 32%Z)%Z) /\
  ((0%Z <= s)%Z /\ (s < 32%Z)%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:Numbers.BinNums.Z) (s:Numbers.BinNums.Z),
  ((0%Z <= n)%Z /\ (n < 32%Z)%Z) /\
  ((0%Z <= s)%Z /\ (s < 32%Z)%Z) /\ (32%Z <= (n + s)%Z)%Z ->
  ((nth (lsr b s) n) = Init.Datatypes.false).

Parameter asr: bv -> Numbers.BinNums.Z -> bv.

Axiom asr_nth_low :
  forall (b:bv) (n:Numbers.BinNums.Z) (s:Numbers.BinNums.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:Numbers.BinNums.Z) (s:Numbers.BinNums.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 -> Numbers.BinNums.Z -> bv.

Axiom lsl_nth_high :
  forall (b:bv) (n:Numbers.BinNums.Z) (s:Numbers.BinNums.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:Numbers.BinNums.Z) (s:Numbers.BinNums.Z),
  (0%Z <= n)%Z /\ (n < 32%Z)%Z -> (0%Z <= s)%Z -> ((n - s)%Z < 0%Z)%Z ->
  ((nth (lsl b s) n) = Init.Datatypes.false).

Parameter to_nat_sub:
  bv -> Numbers.BinNums.Z -> Numbers.BinNums.Z -> Numbers.BinNums.Z.

Axiom to_nat_sub_zero :
  forall (b:bv) (j:Numbers.BinNums.Z) (i:Numbers.BinNums.Z),
  (0%Z <= i)%Z /\ (i <= j)%Z /\ (j < 32%Z)%Z ->
  ((nth b j) = Init.Datatypes.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:Numbers.BinNums.Z) (i:Numbers.BinNums.Z),
  (0%Z <= i)%Z /\ (i <= j)%Z /\ (j < 32%Z)%Z ->
  ((nth b j) = Init.Datatypes.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:Numbers.BinNums.Z) (i:Numbers.BinNums.Z), (j < i)%Z ->
  ((to_nat_sub b j i) = 0%Z).

Axiom to_nat_of_zero2 :
  forall (b:bv) (i:Numbers.BinNums.Z) (j:Numbers.BinNums.Z),
  (j < 32%Z)%Z /\ (i <= j)%Z /\ (0%Z <= i)%Z ->
  (forall (k:Numbers.BinNums.Z), (k <= j)%Z /\ (i < k)%Z ->
   ((nth b k) = Init.Datatypes.false)) ->
  ((to_nat_sub b j 0%Z) = (to_nat_sub b i 0%Z)).

Axiom to_nat_of_zero :
  forall (b:bv) (j:Numbers.BinNums.Z) (i:Numbers.BinNums.Z),
  (j < 32%Z)%Z /\ (0%Z <= i)%Z ->
  (forall (k:Numbers.BinNums.Z), (k <= j)%Z /\ (i <= k)%Z ->
   ((nth b k) = Init.Datatypes.false)) ->
  ((to_nat_sub b j i) = 0%Z).

Axiom to_nat_of_one :
  forall (b:bv) (i:Numbers.BinNums.Z) (j:Numbers.BinNums.Z),
  (j < 32%Z)%Z /\ (i <= j)%Z /\ (0%Z <= i)%Z ->
  (forall (k:Numbers.BinNums.Z), (k <= j)%Z /\ (i <= k)%Z ->
   ((nth b k) = Init.Datatypes.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:Numbers.BinNums.Z) (i:Numbers.BinNums.Z),
  (j < 32%Z)%Z /\ (0%Z <= i)%Z ->
  (forall (k:Numbers.BinNums.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: Numbers.BinNums.Z -> bv.

Axiom nth_from_int_high_even :
  forall (n:Numbers.BinNums.Z) (i:Numbers.BinNums.Z),
  ((i < 32%Z)%Z /\ (0%Z <= i)%Z) /\
  ((int.EuclideanDivision.mod1 (int.EuclideanDivision.div n (pow2 i)) 2%Z) =
   0%Z) ->
  ((nth (from_int n) i) = Init.Datatypes.false).

Axiom nth_from_int_high_odd :
  forall (n:Numbers.BinNums.Z) (i:Numbers.BinNums.Z),
  ((i < 32%Z)%Z /\ (0%Z <= i)%Z) /\
  ~ ((int.EuclideanDivision.mod1 (int.EuclideanDivision.div n (pow2 i)) 2%Z)
     = 0%Z) ->
  ((nth (from_int n) i) = Init.Datatypes.true).

Axiom nth_from_int_low_even :
  forall (n:Numbers.BinNums.Z), ((int.EuclideanDivision.mod1 n 2%Z) = 0%Z) ->
  ((nth (from_int n) 0%Z) = Init.Datatypes.false).

Axiom nth_from_int_low_odd :
  forall (n:Numbers.BinNums.Z),
  ~ ((int.EuclideanDivision.mod1 n 2%Z) = 0%Z) ->
  ((nth (from_int n) 0%Z) = Init.Datatypes.true).

Axiom nth_from_int_0 :
  forall (i:Numbers.BinNums.Z), (i < 32%Z)%Z /\ (0%Z <= i)%Z ->
  ((nth (from_int 0%Z) i) = Init.Datatypes.false).

Parameter from_int2c: Numbers.BinNums.Z -> bv.

Axiom nth_sign_positive :
  forall (n:Numbers.BinNums.Z), (0%Z <= n)%Z ->
  ((nth (from_int2c n) (32%Z - 1%Z)%Z) = Init.Datatypes.false).

Axiom nth_sign_negative :
  forall (n:Numbers.BinNums.Z), (n < 0%Z)%Z ->
  ((nth (from_int2c n) (32%Z - 1%Z)%Z) = Init.Datatypes.true).

Axiom nth_from_int2c_high_even :
  forall (n:Numbers.BinNums.Z) (i:Numbers.BinNums.Z),
  ((i < (32%Z - 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) = Init.Datatypes.false).

Axiom nth_from_int2c_high_odd :
  forall (n:Numbers.BinNums.Z) (i:Numbers.BinNums.Z),
  ((i < (32%Z - 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) = Init.Datatypes.true).

Axiom nth_from_int2c_low_even :
  forall (n:Numbers.BinNums.Z), ((int.EuclideanDivision.mod1 n 2%Z) = 0%Z) ->
  ((nth (from_int2c n) 0%Z) = Init.Datatypes.false).

Axiom nth_from_int2c_low_odd :
  forall (n:Numbers.BinNums.Z),
  ~ ((int.EuclideanDivision.mod1 n 2%Z) = 0%Z) ->
  ((nth (from_int2c n) 0%Z) = Init.Datatypes.true).

Axiom nth_from_int2c_0 :
  forall (i:Numbers.BinNums.Z), (i < 32%Z)%Z /\ (0%Z <= i)%Z ->
  ((nth (from_int2c 0%Z) i) = Init.Datatypes.false).

Axiom nth_from_int2c_plus_pow2 :
  forall (x:Numbers.BinNums.Z) (k:Numbers.BinNums.Z) (i:Numbers.BinNums.Z),
  ((0%Z <= k)%Z /\ (k < i)%Z) /\ (k < (32%Z - 1%Z)%Z)%Z ->
  ((nth (from_int2c (x + (pow2 i))%Z) k) = (nth (from_int2c x) k)).

Axiom size_positive1 : (1%Z < 64%Z)%Z.

Axiom bv1 : Type.
Parameter bv1_WhyType : WhyType bv1.
Existing Instance bv1_WhyType.

Parameter nth1: bv1 -> Numbers.BinNums.Z -> Init.Datatypes.bool.

Parameter bvzero1: bv1.

Axiom Nth_zero1 :
  forall (n:Numbers.BinNums.Z), (0%Z <= n)%Z /\ (n < 64%Z)%Z ->
  ((nth1 bvzero1 n) = Init.Datatypes.false).

Parameter bvone1: bv1.

Axiom Nth_one1 :
  forall (n:Numbers.BinNums.Z), (0%Z <= n)%Z /\ (n < 64%Z)%Z ->
  ((nth1 bvone1 n) = Init.Datatypes.true).

(* Why3 assumption *)
Definition eq1 (v1:bv1) (v2:bv1) : Prop :=
  forall (n:Numbers.BinNums.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:Numbers.BinNums.Z),
  (0%Z <= n)%Z /\ (n < 64%Z)%Z ->
  ((nth1 (bw_and1 v1 v2) n) = (Init.Datatypes.andb (nth1 v1 n) (nth1 v2 n))).

Parameter bw_or1: bv1 -> bv1 -> bv1.

Axiom Nth_bw_or1 :
  forall (v1:bv1) (v2:bv1) (n:Numbers.BinNums.Z),
  (0%Z <= n)%Z /\ (n < 64%Z)%Z ->
  ((nth1 (bw_or1 v1 v2) n) = (Init.Datatypes.orb (nth1 v1 n) (nth1 v2 n))).

Parameter bw_xor1: bv1 -> bv1 -> bv1.

Axiom Nth_bw_xor1 :
  forall (v1:bv1) (v2:bv1) (n:Numbers.BinNums.Z),
  (0%Z <= n)%Z /\ (n < 64%Z)%Z ->
  ((nth1 (bw_xor1 v1 v2) n) = (Init.Datatypes.xorb (nth1 v1 n) (nth1 v2 n))).

Axiom Nth_bw_xor_v1true1 :
  forall (v1:bv1) (v2:bv1) (n:Numbers.BinNums.Z),
  ((0%Z <= n)%Z /\ (n < 64%Z)%Z) /\ ((nth1 v1 n) = Init.Datatypes.true) ->
  ((nth1 (bw_xor1 v1 v2) n) = (Init.Datatypes.negb (nth1 v2 n))).

Axiom Nth_bw_xor_v1false1 :
  forall (v1:bv1) (v2:bv1) (n:Numbers.BinNums.Z),
  ((0%Z <= n)%Z /\ (n < 64%Z)%Z) /\ ((nth1 v1 n) = Init.Datatypes.false) ->
  ((nth1 (bw_xor1 v1 v2) n) = (nth1 v2 n)).

Axiom Nth_bw_xor_v2true1 :
  forall (v1:bv1) (v2:bv1) (n:Numbers.BinNums.Z),
  ((0%Z <= n)%Z /\ (n < 64%Z)%Z) /\ ((nth1 v2 n) = Init.Datatypes.true) ->
  ((nth1 (bw_xor1 v1 v2) n) = (Init.Datatypes.negb (nth1 v1 n))).

Axiom Nth_bw_xor_v2false1 :
  forall (v1:bv1) (v2:bv1) (n:Numbers.BinNums.Z),
  ((0%Z <= n)%Z /\ (n < 64%Z)%Z) /\ ((nth1 v2 n) = Init.Datatypes.false) ->
  ((nth1 (bw_xor1 v1 v2) n) = (nth1 v1 n)).

Parameter bw_not1: bv1 -> bv1.

Axiom Nth_bw_not1 :
  forall (v:bv1) (n:Numbers.BinNums.Z), (0%Z <= n)%Z /\ (n < 64%Z)%Z ->
  ((nth1 (bw_not1 v) n) = (Init.Datatypes.negb (nth1 v n))).

Parameter lsr1: bv1 -> Numbers.BinNums.Z -> bv1.

Axiom lsr_nth_low1 :
  forall (b:bv1) (n:Numbers.BinNums.Z) (s:Numbers.BinNums.Z),
  ((0%Z <= n)%Z /\ (n < 64%Z)%Z) /\
  ((0%Z <= s)%Z /\ (s < 64%Z)%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:Numbers.BinNums.Z) (s:Numbers.BinNums.Z),
  ((0%Z <= n)%Z /\ (n < 64%Z)%Z) /\
  ((0%Z <= s)%Z /\ (s < 64%Z)%Z) /\ (64%Z <= (n + s)%Z)%Z ->
  ((nth1 (lsr1 b s) n) = Init.Datatypes.false).

Parameter asr1: bv1 -> Numbers.BinNums.Z -> bv1.

Axiom asr_nth_low1 :
  forall (b:bv1) (n:Numbers.BinNums.Z) (s:Numbers.BinNums.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:Numbers.BinNums.Z) (s:Numbers.BinNums.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)).

Parameter lsl1: bv1 -> Numbers.BinNums.Z -> bv1.

Axiom lsl_nth_high1 :
  forall (b:bv1) (n:Numbers.BinNums.Z) (s:Numbers.BinNums.Z),
  (0%Z <= n)%Z /\ (n < 64%Z)%Z -> (0%Z <= s)%Z -> (0%Z <= (n - s)%Z)%Z ->
  ((nth1 (lsl1 b s) n) = (nth1 b (n - s)%Z)).

Axiom lsl_nth_low1 :
  forall (b:bv1) (n:Numbers.BinNums.Z) (s:Numbers.BinNums.Z),
  (0%Z <= n)%Z /\ (n < 64%Z)%Z -> (0%Z <= s)%Z -> ((n - s)%Z < 0%Z)%Z ->
  ((nth1 (lsl1 b s) n) = Init.Datatypes.false).

Parameter to_nat_sub1:
  bv1 -> Numbers.BinNums.Z -> Numbers.BinNums.Z -> Numbers.BinNums.Z.

Axiom to_nat_sub_zero1 :
  forall (b:bv1) (j:Numbers.BinNums.Z) (i:Numbers.BinNums.Z),
  (0%Z <= i)%Z /\ (i <= j)%Z /\ (j < 64%Z)%Z ->
  ((nth1 b j) = Init.Datatypes.false) ->
  ((to_nat_sub1 b j i) = (to_nat_sub1 b (j - 1%Z)%Z i)).

Axiom to_nat_sub_one1 :
  forall (b:bv1) (j:Numbers.BinNums.Z) (i:Numbers.BinNums.Z),
  (0%Z <= i)%Z /\ (i <= j)%Z /\ (j < 64%Z)%Z ->
  ((nth1 b j) = Init.Datatypes.true) ->
  ((to_nat_sub1 b j i) =
   ((pow2 (j - i)%Z) + (to_nat_sub1 b (j - 1%Z)%Z i))%Z).

Axiom to_nat_sub_high1 :
  forall (b:bv1) (j:Numbers.BinNums.Z) (i:Numbers.BinNums.Z), (j < i)%Z ->
  ((to_nat_sub1 b j i) = 0%Z).

Axiom to_nat_of_zero21 :
  forall (b:bv1) (i:Numbers.BinNums.Z) (j:Numbers.BinNums.Z),
  (j < 64%Z)%Z /\ (i <= j)%Z /\ (0%Z <= i)%Z ->
  (forall (k:Numbers.BinNums.Z), (k <= j)%Z /\ (i < k)%Z ->
   ((nth1 b k) = Init.Datatypes.false)) ->
  ((to_nat_sub1 b j 0%Z) = (to_nat_sub1 b i 0%Z)).

Axiom to_nat_of_zero1 :
  forall (b:bv1) (j:Numbers.BinNums.Z) (i:Numbers.BinNums.Z),
  (j < 64%Z)%Z /\ (0%Z <= i)%Z ->
  (forall (k:Numbers.BinNums.Z), (k <= j)%Z /\ (i <= k)%Z ->
   ((nth1 b k) = Init.Datatypes.false)) ->
  ((to_nat_sub1 b j i) = 0%Z).

Axiom to_nat_of_one1 :
  forall (b:bv1) (i:Numbers.BinNums.Z) (j:Numbers.BinNums.Z),
  (j < 64%Z)%Z /\ (i <= j)%Z /\ (0%Z <= i)%Z ->
  (forall (k:Numbers.BinNums.Z), (k <= j)%Z /\ (i <= k)%Z ->
   ((nth1 b k) = Init.Datatypes.true)) ->
  ((to_nat_sub1 b j i) = ((pow2 ((j - i)%Z + 1%Z)%Z) - 1%Z)%Z).

Axiom to_nat_sub_footprint1 :
  forall (b1:bv1) (b2:bv1) (j:Numbers.BinNums.Z) (i:Numbers.BinNums.Z),
  (j < 64%Z)%Z /\ (0%Z <= i)%Z ->
  (forall (k:Numbers.BinNums.Z), (i <= k)%Z /\ (k <= j)%Z ->
   ((nth1 b1 k) = (nth1 b2 k))) ->
  ((to_nat_sub1 b1 j i) = (to_nat_sub1 b2 j i)).

Parameter from_int1: Numbers.BinNums.Z -> bv1.

Axiom nth_from_int_high_even1 :
  forall (n:Numbers.BinNums.Z) (i:Numbers.BinNums.Z),
  ((i < 64%Z)%Z /\ (0%Z <= i)%Z) /\
  ((int.EuclideanDivision.mod1 (int.EuclideanDivision.div n (pow2 i)) 2%Z) =
   0%Z) ->
  ((nth1 (from_int1 n) i) = Init.Datatypes.false).

Axiom nth_from_int_high_odd1 :
  forall (n:Numbers.BinNums.Z) (i:Numbers.BinNums.Z),
  ((i < 64%Z)%Z /\ (0%Z <= i)%Z) /\
  ~ ((int.EuclideanDivision.mod1 (int.EuclideanDivision.div n (pow2 i)) 2%Z)
     = 0%Z) ->
  ((nth1 (from_int1 n) i) = Init.Datatypes.true).

Axiom nth_from_int_low_even1 :
  forall (n:Numbers.BinNums.Z), ((int.EuclideanDivision.mod1 n 2%Z) = 0%Z) ->
  ((nth1 (from_int1 n) 0%Z) = Init.Datatypes.false).

Axiom nth_from_int_low_odd1 :
  forall (n:Numbers.BinNums.Z),
  ~ ((int.EuclideanDivision.mod1 n 2%Z) = 0%Z) ->
  ((nth1 (from_int1 n) 0%Z) = Init.Datatypes.true).

Axiom nth_from_int_01 :
  forall (i:Numbers.BinNums.Z), (i < 64%Z)%Z /\ (0%Z <= i)%Z ->
  ((nth1 (from_int1 0%Z) i) = Init.Datatypes.false).

Parameter from_int2c1: Numbers.BinNums.Z -> bv1.

Axiom nth_sign_positive1 :
  forall (n:Numbers.BinNums.Z), (0%Z <= n)%Z ->
  ((nth1 (from_int2c1 n) (64%Z - 1%Z)%Z) = Init.Datatypes.false).

Axiom nth_sign_negative1 :
  forall (n:Numbers.BinNums.Z), (n < 0%Z)%Z ->
  ((nth1 (from_int2c1 n) (64%Z - 1%Z)%Z) = Init.Datatypes.true).

Axiom nth_from_int2c_high_even1 :
  forall (n:Numbers.BinNums.Z) (i:Numbers.BinNums.Z),
  ((i < (64%Z - 1%Z)%Z)%Z /\ (0%Z <= i)%Z) /\
  ((int.EuclideanDivision.mod1 (int.EuclideanDivision.div n (pow2 i)) 2%Z) =
   0%Z) ->
  ((nth1 (from_int2c1 n) i) = Init.Datatypes.false).

Axiom nth_from_int2c_high_odd1 :
  forall (n:Numbers.BinNums.Z) (i:Numbers.BinNums.Z),
  ((i < (64%Z - 1%Z)%Z)%Z /\ (0%Z <= i)%Z) /\
  ~ ((int.EuclideanDivision.mod1 (int.EuclideanDivision.div n (pow2 i)) 2%Z)
     = 0%Z) ->
  ((nth1 (from_int2c1 n) i) = Init.Datatypes.true).

Axiom nth_from_int2c_low_even1 :
  forall (n:Numbers.BinNums.Z), ((int.EuclideanDivision.mod1 n 2%Z) = 0%Z) ->
  ((nth1 (from_int2c1 n) 0%Z) = Init.Datatypes.false).

Axiom nth_from_int2c_low_odd1 :
  forall (n:Numbers.BinNums.Z),
  ~ ((int.EuclideanDivision.mod1 n 2%Z) = 0%Z) ->
  ((nth1 (from_int2c1 n) 0%Z) = Init.Datatypes.true).

Axiom nth_from_int2c_01 :
  forall (i:Numbers.BinNums.Z), (i < 64%Z)%Z /\ (0%Z <= i)%Z ->
  ((nth1 (from_int2c1 0%Z) i) = Init.Datatypes.false).

Axiom nth_from_int2c_plus_pow21 :
  forall (x:Numbers.BinNums.Z) (k:Numbers.BinNums.Z) (i:Numbers.BinNums.Z),
  ((0%Z <= k)%Z /\ (k < i)%Z) /\ (k < (64%Z - 1%Z)%Z)%Z ->
  ((nth1 (from_int2c1 (x + (pow2 i))%Z) k) = (nth1 (from_int2c1 x) k)).

Parameter concat: bv -> bv -> bv1.

Axiom concat_low :
  forall (b1:bv) (b2:bv), forall (i:Numbers.BinNums.Z),
  (0%Z <= i)%Z /\ (i < 32%Z)%Z -> ((nth1 (concat b1 b2) i) = (nth b2 i)).

Axiom concat_high :
  forall (b1:bv) (b2:bv), forall (i:Numbers.BinNums.Z),
  (32%Z <= i)%Z /\ (i < 64%Z)%Z ->
  ((nth1 (concat b1 b2) i) = (nth b1 (i - 32%Z)%Z)).

Parameter double_of_bv64: bv1 -> Reals.Rdefinitions.R.

Parameter sign_value: Init.Datatypes.bool -> Reals.Rdefinitions.R.

Axiom sign_value_false : ((sign_value Init.Datatypes.false) = 1%R).

Axiom sign_value_true : ((sign_value Init.Datatypes.true) = (-1%R)%R).

Axiom zero :
  forall (b:bv1),
  ((to_nat_sub1 b 62%Z 52%Z) = 0%Z) /\ ((to_nat_sub1 b 51%Z 0%Z) = 0%Z) ->
  ((double_of_bv64 b) = 0%R).

Axiom sign_of_double_positive :
  forall (b:bv1), ((nth1 b 63%Z) = Init.Datatypes.false) ->
  (0%R <= (double_of_bv64 b))%R.

Axiom sign_of_double_negative :
  forall (b:bv1), ((nth1 b 63%Z) = Init.Datatypes.true) ->
  ((double_of_bv64 b) <= 0%R)%R.

Axiom double_of_bv64_value :
  forall (b:bv1),
  (0%Z < (to_nat_sub1 b 62%Z 52%Z))%Z /\
  ((to_nat_sub1 b 62%Z 52%Z) < 2047%Z)%Z ->
  ((double_of_bv64 b) =
   (((sign_value (nth1 b 63%Z)) *
     (pow21 ((to_nat_sub1 b 62%Z 52%Z) - 1023%Z)%Z))%R
    *
    (1%R + ((BuiltIn.IZR (to_nat_sub1 b 51%Z 0%Z)) * (pow21 (-52%Z)%Z))%R)%R)%R).

Axiom nth_j1 :
  forall (i:Numbers.BinNums.Z), (0%Z <= i)%Z /\ (i <= 19%Z)%Z ->
  ((nth (from_int 1127219200%Z) i) = Init.Datatypes.false).

Axiom nth_j2 :
  forall (i:Numbers.BinNums.Z), (20%Z <= i)%Z /\ (i <= 21%Z)%Z ->
  ((nth (from_int 1127219200%Z) i) = Init.Datatypes.true).

Axiom nth_j3 :
  forall (i:Numbers.BinNums.Z), (22%Z <= i)%Z /\ (i <= 23%Z)%Z ->
  ((nth (from_int 1127219200%Z) i) = Init.Datatypes.false).

Axiom nth_j4 :
  forall (i:Numbers.BinNums.Z), (24%Z <= i)%Z /\ (i <= 25%Z)%Z ->
  ((nth (from_int 1127219200%Z) i) = Init.Datatypes.true).

Axiom nth_j5 :
  forall (i:Numbers.BinNums.Z), (26%Z <= i)%Z /\ (i <= 29%Z)%Z ->
  ((nth (from_int 1127219200%Z) i) = Init.Datatypes.false).

Axiom nth_j6 : ((nth (from_int 1127219200%Z) 30%Z) = Init.Datatypes.true).

Axiom nth_j7 : ((nth (from_int 1127219200%Z) 31%Z) = Init.Datatypes.false).

Axiom jp0_30 :
  forall (i:Numbers.BinNums.Z), (0%Z <= i)%Z /\ (i < 30%Z)%Z ->
  ((nth (from_int 2147483648%Z) i) = Init.Datatypes.false).

Axiom nth_const1 :
  forall (i:Numbers.BinNums.Z), (0%Z <= i)%Z /\ (i <= 30%Z)%Z ->
  ((nth1 (concat (from_int 1127219200%Z) (from_int 2147483648%Z)) i) =
   Init.Datatypes.false).

Axiom nth_const2 :
  ((nth1 (concat (from_int 1127219200%Z) (from_int 2147483648%Z)) 31%Z) =
   Init.Datatypes.true).

Axiom nth_const3 :
  forall (i:Numbers.BinNums.Z), (32%Z <= i)%Z /\ (i <= 51%Z)%Z ->
  ((nth1 (concat (from_int 1127219200%Z) (from_int 2147483648%Z)) i) =
   Init.Datatypes.false).

Axiom nth_const4 :
  forall (i:Numbers.BinNums.Z), (52%Z <= i)%Z /\ (i <= 53%Z)%Z ->
  ((nth1 (concat (from_int 1127219200%Z) (from_int 2147483648%Z)) i) =
   Init.Datatypes.true).

Axiom nth_const5 :
  forall (i:Numbers.BinNums.Z), (54%Z <= i)%Z /\ (i <= 55%Z)%Z ->
  ((nth1 (concat (from_int 1127219200%Z) (from_int 2147483648%Z)) i) =
   Init.Datatypes.false).

Axiom nth_const6 :
  forall (i:Numbers.BinNums.Z), (56%Z <= i)%Z /\ (i <= 57%Z)%Z ->
  ((nth1 (concat (from_int 1127219200%Z) (from_int 2147483648%Z)) i) =
   Init.Datatypes.true).

Axiom nth_const7 :
  forall (i:Numbers.BinNums.Z), (58%Z <= i)%Z /\ (i <= 61%Z)%Z ->
  ((nth1 (concat (from_int 1127219200%Z) (from_int 2147483648%Z)) i) =
   Init.Datatypes.false).

Axiom nth_const8 :
  ((nth1 (concat (from_int 1127219200%Z) (from_int 2147483648%Z)) 62%Z) =
   Init.Datatypes.true).

Axiom nth_const9 :
  ((nth1 (concat (from_int 1127219200%Z) (from_int 2147483648%Z)) 63%Z) =
   Init.Datatypes.false).

Axiom sign_const :
  ((nth1 (concat (from_int 1127219200%Z) (from_int 2147483648%Z)) 63%Z) =
   Init.Datatypes.false).

Open Scope Z_scope.

Require Import Lia.

(* Why3 goal *)
Theorem exp_const :
  ((to_nat_sub1 (concat (from_int 1127219200%Z) (from_int 2147483648%Z)) 62%Z
    52%Z)
   = 1075%Z).
rewrite to_nat_sub_one1; auto with zarith.
  2: apply nth_const8.
replace (62 - 52) with 10 by lia.
 rewrite pow2_10.
rewrite to_nat_sub_zero1; auto with zarith.
  2: apply nth_const7; auto with zarith.
rewrite to_nat_sub_zero1; auto with zarith.
  2: apply nth_const7; auto with zarith.
rewrite to_nat_sub_zero1; auto with zarith.
  2: apply nth_const7; auto with zarith.
rewrite to_nat_sub_zero1; auto with zarith.
  2: apply nth_const7; auto with zarith.
rewrite to_nat_sub_one1; auto with zarith.
  2: apply nth_const6; auto with zarith.
replace (62 - 1 - 1 - 1 - 1 - 1 - 52) with 5 by lia.
 rewrite pow2_5.
rewrite to_nat_sub_one1; auto with zarith.
  2: apply nth_const6; auto with zarith.
replace (62 - 1 - 1 - 1 - 1 - 1 - 1 -  52) with 4 by lia.
 rewrite pow2_4.
rewrite to_nat_sub_zero1; auto with zarith.
  2: apply nth_const5; auto with zarith.
rewrite to_nat_sub_zero1; auto with zarith.
  2: apply nth_const5; auto with zarith.
rewrite to_nat_of_one1; auto with zarith.
replace (62 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 52) with 1 by lia.
replace (1+1) with 2 by lia.
 rewrite pow2_2; auto.
intros.
apply nth_const4; auto with zarith.
Qed.

