bitvectors: removed some Coq proofs

parent 5f0df6ba
......@@ -10,12 +10,13 @@ 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_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) ->
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.
......@@ -148,24 +149,29 @@ Axiom pow2_62 : ((pow2 62%Z) = 4611686018427387904%Z).
Axiom pow2_63 : ((pow2 63%Z) = 9223372036854775808%Z).
Axiom Div_mult_inst : forall (x:Z) (z: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_mult_inst :
forall (x:Z) (z: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:Z) (y:Z), ((0%Z < y)%Z /\ ((y <= x)%Z /\
(x < (2%Z * y)%Z)%Z)) -> ((int.EuclideanDivision.div x y) = 1%Z).
Axiom Div_double :
forall (x:Z) (y: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:Z) (i: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_pow :
forall (x:Z) (i: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:Z) (y: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_double_neg :
forall (x:Z) (y: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:Z) (i: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 Div_pow2 :
forall (x:Z) (i: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).
Open Scope Z_scope.
......@@ -174,11 +180,11 @@ Ltac ae := why3 "alt-ergo" timelimit 3; admit.
(* Why3 goal *)
Theorem Mod_pow2_gen : forall (x:Z) (i:Z) (k: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)).
Theorem Mod_pow2_gen :
forall (x:Z) (i:Z) (k: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)).
(* Why3 intros x i k (h1,h2). *)
(* intros x i k (h1,h2). *)
intros x i k (h1,h2).
......
......@@ -8,8 +8,8 @@ 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_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).
......@@ -17,7 +17,8 @@ Open Scope Z_scope.
Require Import Why3.
(* Why3 goal *)
Theorem Power_sum : forall (n:Z) (m:Z), ((0%Z <= n)%Z /\ (0%Z <= m)%Z) ->
Theorem Power_sum :
forall (n:Z) (m:Z), ((0%Z <= n)%Z /\ (0%Z <= m)%Z) ->
((pow2 (n + m)%Z) = ((pow2 n) * (pow2 m))%Z).
(* Why3 intros n m (h1,h2). *)
intros n m (Hn & Hm).
......
(* This file is generated by Why3's Coq driver *)
(* Beware! Only edit allowed sections below *)
Require Import ZArith.
Require Import Rbase.
Require Import BuiltIn.
Require BuiltIn.
Require int.Int.
Require int.Abs.
Require int.EuclideanDivision.
Require real.Real.
Require real.RealInfix.
Require real.FromInt.
......@@ -11,18 +13,19 @@ Parameter pow2: Z -> R.
Axiom Power_0 : ((pow2 0%Z) = 1%R).
Axiom Power_s : forall (n:Z), (0%Z <= n)%Z ->
((pow2 (n + 1%Z)%Z) = (2%R * (pow2 n))%R).
Axiom Power_s :
forall (n:Z), (0%Z <= n)%Z -> ((pow2 (n + 1%Z)%Z) = (2%R * (pow2 n))%R).
Axiom Power_p : forall (n:Z), (n <= 0%Z)%Z ->
Axiom Power_p :
forall (n:Z), (n <= 0%Z)%Z ->
((pow2 (n - 1%Z)%Z) = ((05 / 10)%R * (pow2 n))%R).
Axiom Power_s_all : forall (n:Z), ((pow2 (n + 1%Z)%Z) = (2%R * (pow2 n))%R).
Axiom Power_p_all : forall (n:Z),
((pow2 (n - 1%Z)%Z) = ((05 / 10)%R * (pow2 n))%R).
Axiom Power_p_all :
forall (n:Z), ((pow2 (n - 1%Z)%Z) = ((05 / 10)%R * (pow2 n))%R).
Axiom Power_1_2 : ((05 / 10)%R = (Rdiv 1%R 2%R)%R).
Axiom Power_1_2 : ((05 / 10)%R = (1%R / 2%R)%R).
Axiom Power_1 : ((pow2 1%Z) = 2%R).
......@@ -30,31 +33,35 @@ Axiom Power_neg1 : ((pow2 (-1%Z)%Z) = (05 / 10)%R).
Axiom Power_non_null_aux : forall (n:Z), (0%Z <= n)%Z -> ~ ((pow2 n) = 0%R).
Axiom Power_neg_aux : forall (n:Z), (0%Z <= n)%Z ->
((pow2 (-n)%Z) = (Rdiv 1%R (pow2 n))%R).
Axiom Power_neg_aux :
forall (n:Z), (0%Z <= n)%Z -> ((pow2 (-n)%Z) = (1%R / (pow2 n))%R).
Axiom Power_non_null : forall (n:Z), ~ ((pow2 n) = 0%R).
Axiom Power_neg : forall (n:Z), ((pow2 (-n)%Z) = (Rdiv 1%R (pow2 n))%R).
Axiom Power_neg : forall (n:Z), ((pow2 (-n)%Z) = (1%R / (pow2 n))%R).
Axiom Power_sum_aux : forall (n:Z) (m:Z), (0%Z <= m)%Z ->
Axiom Power_sum_aux :
forall (n:Z) (m:Z), (0%Z <= m)%Z ->
((pow2 (n + m)%Z) = ((pow2 n) * (pow2 m))%R).
Axiom Power_sum : forall (n:Z) (m:Z),
((pow2 (n + m)%Z) = ((pow2 n) * (pow2 m))%R).
Axiom Power_sum :
forall (n:Z) (m:Z), ((pow2 (n + m)%Z) = ((pow2 n) * (pow2 m))%R).
Parameter pow21: Z -> Z.
Axiom Power_01 : ((pow21 0%Z) = 1%Z).
Axiom Power_s1 : forall (n:Z), (0%Z <= n)%Z ->
((pow21 (n + 1%Z)%Z) = (2%Z * (pow21 n))%Z).
Axiom Power_s1 :
forall (n:Z), (0%Z <= n)%Z -> ((pow21 (n + 1%Z)%Z) = (2%Z * (pow21 n))%Z).
Axiom Power_11 : ((pow21 1%Z) = 2%Z).
Axiom Power_sum1 : forall (n:Z) (m:Z), ((0%Z <= n)%Z /\ (0%Z <= m)%Z) ->
Axiom Power_sum1 :
forall (n:Z) (m:Z), ((0%Z <= n)%Z /\ (0%Z <= m)%Z) ->
((pow21 (n + m)%Z) = ((pow21 n) * (pow21 m))%Z).
Axiom pow2pos : forall (i:Z), (0%Z <= i)%Z -> (0%Z < (pow21 i))%Z.
Axiom pow2_0 : ((pow21 0%Z) = 1%Z).
Axiom pow2_1 : ((pow21 1%Z) = 2%Z).
......@@ -183,13 +190,45 @@ Axiom pow2_62 : ((pow21 62%Z) = 4611686018427387904%Z).
Axiom pow2_63 : ((pow21 63%Z) = 9223372036854775808%Z).
Axiom Div_mult_inst :
forall (x:Z) (z: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:Z) (y: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:Z) (i:Z), (0%Z < i)%Z ->
(((pow21 (i - 1%Z)%Z) <= x)%Z /\ (x < (pow21 i))%Z) ->
((int.EuclideanDivision.div x (pow21 (i - 1%Z)%Z)) = 1%Z).
Axiom Div_double_neg :
forall (x:Z) (y: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:Z) (i:Z), (0%Z < i)%Z ->
(((-(pow21 i))%Z <= x)%Z /\ (x < (-(pow21 (i - 1%Z)%Z))%Z)%Z) ->
((int.EuclideanDivision.div x (pow21 (i - 1%Z)%Z)) = (-2%Z)%Z).
Axiom Mod_pow2_gen :
forall (x:Z) (i:Z) (k:Z), ((0%Z <= k)%Z /\ (k < i)%Z) ->
((int.EuclideanDivision.mod1
(int.EuclideanDivision.div (x + (pow21 i))%Z (pow21 k)) 2%Z)
=
(int.EuclideanDivision.mod1 (int.EuclideanDivision.div x (pow21 k)) 2%Z)).
Require Import Why3.
Ltac ae := why3 "alt-ergo" timelimit 2.
Open Scope Z_scope.
(* Why3 goal *)
Theorem Pow2_int_real : forall (x:Z), (0%Z <= x)%Z ->
((pow2 x) = (IZR (pow21 x))).
Theorem Pow2_int_real :
forall (x:Z), (0%Z <= x)%Z -> ((pow2 x) = (BuiltIn.IZR (pow21 x))).
(* Why3 intros x h1. *)
intros x Hx.
generalize Hx.
pattern x; apply Z_lt_induction; auto.
......@@ -209,4 +248,3 @@ rewrite mult_IZR.
rewrite Hind; auto with zarith.
Qed.
......@@ -10,16 +10,17 @@ Parameter pow2: Z -> R.
Axiom Power_0 : ((pow2 0%Z) = 1%R).
Axiom Power_s : forall (n:Z), (0%Z <= n)%Z ->
((pow2 (n + 1%Z)%Z) = (2%R * (pow2 n))%R).
Axiom Power_s :
forall (n:Z), (0%Z <= n)%Z -> ((pow2 (n + 1%Z)%Z) = (2%R * (pow2 n))%R).
Axiom Power_p : forall (n:Z), (n <= 0%Z)%Z ->
Axiom Power_p :
forall (n:Z), (n <= 0%Z)%Z ->
((pow2 (n - 1%Z)%Z) = ((05 / 10)%R * (pow2 n))%R).
Axiom Power_s_all : forall (n:Z), ((pow2 (n + 1%Z)%Z) = (2%R * (pow2 n))%R).
Axiom Power_p_all : forall (n:Z),
((pow2 (n - 1%Z)%Z) = ((05 / 10)%R * (pow2 n))%R).
Axiom Power_p_all :
forall (n:Z), ((pow2 (n - 1%Z)%Z) = ((05 / 10)%R * (pow2 n))%R).
Axiom Power_1_2 : ((05 / 10)%R = (1%R / 2%R)%R).
......@@ -34,8 +35,8 @@ Ltac ae := why3 "alt-ergo" timelimit 2; admit.
Open Scope Z_scope.
(* Why3 goal *)
Theorem Power_neg_aux : forall (n:Z), (0%Z <= n)%Z ->
((pow2 (-n)%Z) = (1%R / (pow2 n))%R).
Theorem Power_neg_aux :
forall (n:Z), (0%Z <= n)%Z -> ((pow2 (-n)%Z) = (1%R / (pow2 n))%R).
(* Why3 intros n h1. *)
intros n Hn.
generalize Hn.
......
(* This file is generated by Why3's Coq driver *)
(* Beware! Only edit allowed sections below *)
Require Import BuiltIn.
Require BuiltIn.
Require int.Int.
Require real.Real.
Require real.RealInfix.
Parameter pow2: Z -> R.
Axiom Power_0 : ((pow2 0%Z) = 1%R).
Axiom Power_s : forall (n:Z), (0%Z <= n)%Z ->
((pow2 (n + 1%Z)%Z) = (2%R * (pow2 n))%R).
Axiom Power_p : forall (n:Z), (n <= 0%Z)%Z ->
((pow2 (n - 1%Z)%Z) = ((05 / 10)%R * (pow2 n))%R).
Axiom Power_s_all : forall (n:Z), ((pow2 (n + 1%Z)%Z) = (2%R * (pow2 n))%R).
Axiom Power_p_all : forall (n:Z),
((pow2 (n - 1%Z)%Z) = ((05 / 10)%R * (pow2 n))%R).
Axiom Power_1_2 : ((05 / 10)%R = (1%R / 2%R)%R).
Axiom Power_1 : ((pow2 1%Z) = 2%R).
Axiom Power_neg1 : ((pow2 (-1%Z)%Z) = (05 / 10)%R).
Axiom Power_non_null_aux : forall (n:Z), (0%Z <= n)%Z -> ~ ((pow2 n) = 0%R).
Axiom Power_neg_aux : forall (n:Z), (0%Z <= n)%Z ->
((pow2 (-n)%Z) = (1%R / (pow2 n))%R).
Require Import Why3.
Ltac ae := why3 "alt-ergo" timelimit 2; admit.
Open Scope Z_scope.
(* Why3 goal *)
Theorem Power_non_null : forall (n:Z), ~ ((pow2 n) = 0%R).
(* Why3 intros n. *)
intro n.
assert (h:n>=0 \/ n<0) by omega.
destruct h.
ae.
pose (n':=-n).
assert (Hn': n' > 0) by (subst n'; omega).
replace n with (-n') by (subst n'; omega).
rewrite Power_neg_aux; auto with zarith.
unfold Rdiv.
rewrite Rmult_1_l.
apply Rinv_neq_0_compat.
ae.
Admitted.
(* This file is generated by Why3's Coq driver *)
(* Beware! Only edit allowed sections below *)
Require Import BuiltIn.
Require BuiltIn.
Require int.Int.
Require real.Real.
Require real.RealInfix.
Parameter pow2: Z -> R.
Axiom Power_0 : ((pow2 0%Z) = 1%R).
Axiom Power_s : forall (n:Z), (0%Z <= n)%Z ->
((pow2 (n + 1%Z)%Z) = (2%R * (pow2 n))%R).
Axiom Power_p : forall (n:Z), (n <= 0%Z)%Z ->
((pow2 (n - 1%Z)%Z) = ((05 / 10)%R * (pow2 n))%R).
Axiom Power_s_all : forall (n:Z), ((pow2 (n + 1%Z)%Z) = (2%R * (pow2 n))%R).
Axiom Power_p_all : forall (n:Z),
((pow2 (n - 1%Z)%Z) = ((05 / 10)%R * (pow2 n))%R).
Axiom Power_1_2 : ((05 / 10)%R = (1%R / 2%R)%R).
Axiom Power_1 : ((pow2 1%Z) = 2%R).
Axiom Power_neg1 : ((pow2 (-1%Z)%Z) = (05 / 10)%R).
Require Import Why3.
Ltac ae := why3 "alt-ergo" timelimit 2; admit.
(* Why3 goal *)
Theorem Power_non_null_aux : forall (n:Z), (0%Z <= n)%Z ->
~ ((pow2 n) = 0%R).
(* Why3 intros n h1. *)
intros n Hn.
generalize Hn.
pattern n; apply Z_lt_induction; auto.
ae.
Admitted.
(* This file is generated by Why3's Coq driver *)
(* Beware! Only edit allowed sections below *)
Require Import BuiltIn.
Require BuiltIn.
Require int.Int.
Require real.Real.
Require real.RealInfix.
Parameter pow2: Z -> R.
Axiom Power_0 : ((pow2 0%Z) = 1%R).
Axiom Power_s : forall (n:Z), (0%Z <= n)%Z ->
((pow2 (n + 1%Z)%Z) = (2%R * (pow2 n))%R).
Axiom Power_p : forall (n:Z), (n <= 0%Z)%Z ->
((pow2 (n - 1%Z)%Z) = ((05 / 10)%R * (pow2 n))%R).
Axiom Power_s_all : forall (n:Z), ((pow2 (n + 1%Z)%Z) = (2%R * (pow2 n))%R).
Axiom Power_p_all : forall (n:Z),
((pow2 (n - 1%Z)%Z) = ((05 / 10)%R * (pow2 n))%R).
Axiom Power_1_2 : ((05 / 10)%R = (1%R / 2%R)%R).
Axiom Power_1 : ((pow2 1%Z) = 2%R).
Axiom Power_neg1 : ((pow2 (-1%Z)%Z) = (05 / 10)%R).
Axiom Power_non_null_aux : forall (n:Z), (0%Z <= n)%Z -> ~ ((pow2 n) = 0%R).
Axiom Power_neg_aux : forall (n:Z), (0%Z <= n)%Z ->
((pow2 (-n)%Z) = (1%R / (pow2 n))%R).
Axiom Power_non_null : forall (n:Z), ~ ((pow2 n) = 0%R).
Axiom Power_neg : forall (n:Z), ((pow2 (-n)%Z) = (1%R / (pow2 n))%R).
Axiom Power_sum_aux : forall (n:Z) (m:Z), (0%Z <= m)%Z ->
((pow2 (n + m)%Z) = ((pow2 n) * (pow2 m))%R).
Require Import Why3.
Ltac ae := why3 "alt-ergo" timelimit 2; admit.
Open Scope Z_scope.
(* Why3 goal *)
Theorem Power_sum : forall (n:Z) (m:Z),
((pow2 (n + m)%Z) = ((pow2 n) * (pow2 m))%R).
(* Why3 intros n m. *)
intros n m.
assert (h:m>=0 \/ m <=0) by omega; destruct h.
apply Power_sum_aux; auto with zarith.
pose (m' := -m).
assert (0 <= m') by (subst m'; omega).
replace m with (-m') by (subst m'; omega).
replace (n+ - m') with (- ((-n) + m')) by omega.
repeat rewrite Power_neg; auto.
rewrite Power_sum_aux; auto.
ae.
Admitted.
......@@ -3,9 +3,11 @@
"http://why3.lri.fr/why3session.dtd">
<why3session shape_version="4">
<prover id="0" name="Gappa" version="1.3.0" timelimit="5" steplimit="0" memlimit="1000"/>
<prover id="1" name="Eprover" version="2.0" timelimit="5" steplimit="0" memlimit="2000"/>
<prover id="2" name="CVC3" version="2.4.1" timelimit="5" steplimit="0" memlimit="1000"/>
<prover id="3" name="CVC4" version="1.4" timelimit="5" steplimit="0" memlimit="1000"/>
<prover id="4" name="Coq" version="8.7.1" timelimit="5" steplimit="0" memlimit="1000"/>
<prover id="5" name="CVC4" version="1.5" timelimit="1" steplimit="0" memlimit="1000"/>
<prover id="6" name="Z3" version="3.2" timelimit="5" steplimit="0" memlimit="1000"/>
<prover id="9" name="Z3" version="4.3.2" timelimit="5" steplimit="0" memlimit="1000"/>
<prover id="10" name="Alt-Ergo" version="0.99.1" timelimit="5" steplimit="0" memlimit="1000"/>
......@@ -244,7 +246,7 @@
<goal name="pow2_36" proved="true">
<proof prover="2"><result status="valid" time="0.02"/></proof>
<proof prover="6"><result status="valid" time="0.72"/></proof>
<proof prover="9"><result status="valid" time="0.87"/></proof>
<proof prover="9"><result status="valid" time="0.67"/></proof>
<proof prover="10"><result status="valid" time="0.02" steps="39"/></proof>
</goal>
<goal name="pow2_37" proved="true">
......@@ -303,7 +305,7 @@
</goal>
<goal name="pow2_46" proved="true">
<proof prover="2"><result status="valid" time="0.04"/></proof>
<proof prover="6"><result status="valid" time="1.42"/></proof>
<proof prover="6"><result status="valid" time="1.17"/></proof>
<proof prover="9"><result status="valid" time="1.44"/></proof>
<proof prover="10"><result status="valid" time="0.02" steps="49"/></proof>
</goal>
......@@ -467,13 +469,24 @@
<proof prover="10"><result status="valid" time="0.00" steps="5"/></proof>
</goal>
<goal name="Power_non_null_aux" proved="true">
<proof prover="4" edited="power2_Pow2real_Power_non_null_aux_1.v"><result status="valid" time="0.65"/></proof>
<transf name="introduce_premises" proved="true" >
<goal name="Power_non_null_aux.0" proved="true">
<transf name="induction" proved="true" arg1="n">
<goal name="Power_non_null_aux.0.0" expl="base case" proved="true">
<proof prover="5"><result status="valid" time="0.01"/></proof>
</goal>
<goal name="Power_non_null_aux.0.1" expl="recursive case" proved="true">
<proof prover="5"><result status="valid" time="0.01"/></proof>
</goal>
</transf>
</goal>
</transf>
</goal>
<goal name="Power_neg_aux" proved="true">
<proof prover="4" edited="power2_Pow2real_Power_neg_aux_1.v"><result status="valid" time="0.85"/></proof>
</goal>
<goal name="Power_non_null" proved="true">
<proof prover="4" edited="power2_Pow2real_Power_non_null_1.v"><result status="valid" time="0.84"/></proof>
<proof prover="5"><result status="valid" time="0.02"/></proof>
</goal>
<goal name="Power_neg" proved="true">
<proof prover="10"><result status="valid" time="0.02" steps="42"/></proof>
......@@ -482,10 +495,10 @@
<proof prover="4" edited="power2_Pow2real_Power_sum_aux_1.v"><result status="valid" time="0.82"/></proof>
</goal>
<goal name="Power_sum" proved="true">
<proof prover="4" edited="power2_Pow2real_Power_sum_1.v"><result status="valid" time="0.62"/></proof>
<proof prover="1"><result status="valid" time="3.43"/></proof>
</goal>
<goal name="Pow2_int_real" proved="true">
<proof prover="4" edited="power2_Pow2real_Pow2_int_real_1.v"><result status="valid" time="0.51"/></proof>
<proof prover="4" edited="power2_Pow2real_Pow2_int_real_1.v"><result status="valid" time="0.68"/></proof>
</goal>
</theory>
</file>
......
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