fibonacci: fixed proof (using Coq)

examples/programs/fibonacci/fibonacci_WP_FibonacciLogarithmic_WP_parameter_logfib_1.v

+(* This file is generated by Why3's Coq driver *)
+(* Beware! Only edit allowed sections below    *)
+Require Import ZArith.
+Require Import Rbase.
+Definition unit  := unit.
+
+Parameter qtmark : Type.
+
+Parameter at1: forall (a:Type), a -> qtmark -> a.
+
+Implicit Arguments at1.
+
+Parameter old: forall (a:Type), a -> a.
+
+Implicit Arguments old.
+
+Parameter fib: Z -> Z.
+
+
+Axiom fib0 : ((fib 0%Z) = 0%Z).
+
+Axiom fib1 : ((fib 1%Z) = 1%Z).
+
+Axiom fibn : forall (n:Z), (2%Z <= n)%Z ->
+  ((fib n) = ((fib (n - 1%Z)%Z) + (fib (n - 2%Z)%Z))%Z).
+
+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).
+
+Inductive t  :=
+  | mk_t : Z -> Z -> Z -> Z -> t .
+
+Definition a11(u:t): Z := match u with
+  | (mk_t a111 _ _ _) => a111
+  end.
+
+Definition a12(u:t): Z := match u with
+  | (mk_t _ a121 _ _) => a121
+  end.
+
+Definition a21(u:t): Z := match u with
+  | (mk_t _ _ a211 _) => a211
+  end.
+
+Definition a22(u:t): Z := match u with
+  | (mk_t _ _ _ a221) => a221
+  end.
+
+Definition mult(x:t) (y:t): t :=
+  (mk_t (((a11 x) * (a11 y))%Z + ((a12 x) * (a21 y))%Z)%Z
+  (((a11 x) * (a12 y))%Z + ((a12 x) * (a22 y))%Z)%Z
+  (((a21 x) * (a11 y))%Z + ((a22 x) * (a21 y))%Z)%Z
+  (((a21 x) * (a12 y))%Z + ((a22 x) * (a22 y))%Z)%Z).
+
+Parameter power: t -> Z -> t.
+
+
+Axiom Power_0 : forall (x:t), ((power x 0%Z) = (mk_t 1%Z 0%Z 0%Z 1%Z)).
+
+Axiom Power_s : forall (x:t) (n:Z), (0%Z <= n)%Z -> ((power x
+  (n + 1%Z)%Z) = (mult x (power x n))).
+
+Axiom Power_1 : forall (x:t), ((power x 1%Z) = x).
+
+Axiom Power_sum : forall (x:t) (n:Z) (m:Z), ((0%Z <= n)%Z /\ (0%Z <= m)%Z) ->
+  ((power x (n + m)%Z) = (mult (power x n) (power x m))).
+
+Axiom Power_mult : forall (x:t) (n:Z) (m:Z), (0%Z <= n)%Z -> ((0%Z <= m)%Z ->
+  ((power x (n * m)%Z) = (power (power x n) m))).
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Theorem WP_parameter_logfib : forall (n:Z), (0%Z <= n)%Z -> ((~ (n = 0%Z)) ->
+  ((((0%Z <= n)%Z /\ ((div n 2%Z) <  n)%Z) /\ (0%Z <= (div n 2%Z))%Z) ->
+  forall (result:Z) (result1:Z), ((power (mk_t 1%Z 1%Z 1%Z 0%Z) (div n
+  2%Z)) = (mk_t (result + result1)%Z result1 result1 result)) -> ((((mod1 n
+  2%Z) = 0%Z) -> match (((result * result)%Z + (result1 * result1)%Z)%Z,
+  (result1 * (result + (result + result1)%Z)%Z)%Z) with
+  | (a, b) => ((power (mk_t 1%Z 1%Z 1%Z 0%Z) n) = (mk_t (a + b)%Z b b a))
+  end) /\ ((~ ((mod1 n 2%Z) = 0%Z)) -> match (
+  (result1 * (result + (result + result1)%Z)%Z)%Z,
+  (((result + result1)%Z * (result + result1)%Z)%Z + (result1 * result1)%Z)%Z) with
+  | (a, b) => ((power (mk_t 1%Z 1%Z 1%Z 0%Z) n) = (mk_t (a + b)%Z b b a))
+  end)))).
+(* YOU MAY EDIT THE PROOF BELOW *)
+intros.
+assert (h: (2 <> 0)%Z) by omega.
+generalize (Div_mod n 2 h)%Z.
+intuition.
+rewrite H4 in H3.
+rewrite H3.
+replace (2 * div n 2 + 0)%Z with (div n 2 + div n 2)%Z by omega.
+rewrite Power_sum.
+rewrite H2; simpl.
+unfold mult; simpl.
+apply f_equal4; ring.
+intuition.
+
+generalize (Mod_bound n 2 h)%Z.
+simpl.
+intro.
+assert (h': (mod1 n 2 = 1)%Z) by omega.
+rewrite h' in H3.
+rewrite H3.
+replace (2 * div n 2 + 1)%Z with ((div n 2 + div n 2) + 1)%Z by omega.
+rewrite Power_s.
+rewrite Power_sum.
+rewrite H2; simpl.
+unfold mult at 2; simpl.
+unfold mult.
+apply f_equal4; unfold a11, a12, a21, a22; try ring.
+intuition.
+omega.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+

examples/programs/fibonacci/fibonacci_WP_FibonacciLogarithmic_fib_m_1.v

+(* This file is generated by Why3's Coq driver *)
+(* Beware! Only edit allowed sections below    *)
+Require Import ZArith.
+Require Import Rbase.
+Definition unit  := unit.
+
+Parameter qtmark : Type.
+
+Parameter at1: forall (a:Type), a -> qtmark -> a.
+
+Implicit Arguments at1.
+
+Parameter old: forall (a:Type), a -> a.
+
+Implicit Arguments old.
+
+Parameter fib: Z -> Z.
+
+
+Axiom fib0 : ((fib 0%Z) = 0%Z).
+
+Axiom fib1 : ((fib 1%Z) = 1%Z).
+
+Axiom fibn : forall (n:Z), (2%Z <= n)%Z ->
+  ((fib n) = ((fib (n - 1%Z)%Z) + (fib (n - 2%Z)%Z))%Z).
+
+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).
+
+Inductive t  :=
+  | mk_t : Z -> Z -> Z -> Z -> t .
+
+Definition a11(u:t): Z := match u with
+  | (mk_t a111 _ _ _) => a111
+  end.
+
+Definition a12(u:t): Z := match u with
+  | (mk_t _ a121 _ _) => a121
+  end.
+
+Definition a21(u:t): Z := match u with
+  | (mk_t _ _ a211 _) => a211
+  end.
+
+Definition a22(u:t): Z := match u with
+  | (mk_t _ _ _ a221) => a221
+  end.
+
+Definition mult(x:t) (y:t): t :=
+  (mk_t (((a11 x) * (a11 y))%Z + ((a12 x) * (a21 y))%Z)%Z
+  (((a11 x) * (a12 y))%Z + ((a12 x) * (a22 y))%Z)%Z
+  (((a21 x) * (a11 y))%Z + ((a22 x) * (a21 y))%Z)%Z
+  (((a21 x) * (a12 y))%Z + ((a22 x) * (a22 y))%Z)%Z).
+
+Parameter power: t -> Z -> t.
+
+
+Axiom Power_0 : forall (x:t), ((power x 0%Z) = (mk_t 1%Z 0%Z 0%Z 1%Z)).
+
+Axiom Power_s : forall (x:t) (n:Z), (0%Z <= n)%Z -> ((power x
+  (n + 1%Z)%Z) = (mult x (power x n))).
+
+Axiom Power_1 : forall (x:t), ((power x 1%Z) = x).
+
+Axiom Power_sum : forall (x:t) (n:Z) (m:Z), ((0%Z <= n)%Z /\ (0%Z <= m)%Z) ->
+  ((power x (n + m)%Z) = (mult (power x n) (power x m))).
+
+Axiom Power_mult : forall (x:t) (n:Z) (m:Z), (0%Z <= n)%Z -> ((0%Z <= m)%Z ->
+  ((power x (n * m)%Z) = (power (power x n) m))).
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+Hint Resolve fib0 fib1.
+(* DO NOT EDIT BELOW *)
+
+Theorem fib_m : forall (n:Z), (0%Z <= n)%Z -> let p := (power (mk_t 1%Z 1%Z
+  1%Z 0%Z) n) in (((fib (n + 1%Z)%Z) = (a11 p)) /\ ((fib n) = (a21 p))).
+(* YOU MAY EDIT THE PROOF BELOW *)
+intros n hn.
+pattern n; apply natlike_ind; intuition.
+rewrite Power_0.
+unfold a11, a21; simpl; auto.
+replace (Zsucc  x) with (x+1)%Z by omega.
+rewrite Power_s; auto.
+destruct H0 as (h1,h2).
+split.
+rewrite fibn; try omega.
+ring_simplify (x+1+1-1)%Z.
+ring_simplify (x+1+1-2)%Z.
+unfold a11, mult.
+rewrite <- h1. rewrite <- h2.
+unfold a11, a12. ring.
+unfold a21, mult.
+rewrite <- h1. rewrite <- h2.
+unfold a21, a22. ring.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+

examples/programs/fibonacci/why3session.xml

@@ -5,7 +5,7 @@
  <prover
   id="alt-ergo"
   name="Alt-Ergo"
-  version="0.93"/>
+  version="0.93.1"/>
  <prover
   id="coq"
   name="Coq"
@@ -17,11 +17,11 @@
  <prover
   id="eprover"
   name="Eprover"
-  version="1.4 Namring"/>
+  version="0.8 Steinthal"/>
  <prover
   id="gappa"
   name="Gappa"
-  version="0.15.1"/>
+  version="0.15.0"/>
  <prover
   id="simplify"
   name="Simplify"
@@ -34,21 +34,17 @@
   id="vampire"
   name="Vampire"
   version="0.6"/>
- <prover
-  id="verit"
-  name="veriT"
-  version="dev"/>
  <prover
   id="yices"
   name="Yices"
-  version="1.0.25"/>
+  version="1.0.27"/>
  <prover
   id="z3"
   name="Z3"
   version="2.19"/>
  <file
   name="../fibonacci.mlw"
-  verified="false"
+  verified="true"
   expanded="true">
   <theory
    name="Fibonacci"
@@ -58,12 +54,12 @@
   <theory
    name="FibonacciTest"
    verified="true"
-   expanded="true">
+   expanded="false">
    <goal
     name="isfib_2_1"
     sum="94f544153b551bfc1648c9451a1f66dd"
     proved="true"
-    expanded="true"
+    expanded="false"
     shape="ainfix =afibc2c1">
     <proof
      prover="cvc3"
@@ -77,7 +73,7 @@
     name="isfib_6_8"
     sum="b684c14a9915d623d32b9339ae110d94"
     proved="true"
-    expanded="true"
+    expanded="false"
     shape="ainfix =afibc6c8">
     <proof
      prover="cvc3"
@@ -91,7 +87,7 @@
     name="not_isfib_2_2"
     sum="1eabfb3d0f770ce04a4dfa55f318bda3"
     proved="true"
-    expanded="true"
+    expanded="false"
     shape="ainfix =afibc2c2N">
     <proof
      prover="cvc3"
@@ -133,13 +129,13 @@
   <theory
    name="WP FibonacciLinear"
    verified="true"
-   expanded="true">
+   expanded="false">
    <goal
     name="WP_parameter fib"
     expl="parameter fib"
     sum="124b5d5590957581d602b9d1b81cc12e"
     proved="true"
-    expanded="true"
+    expanded="false"
     shape="ainfix =afibV0V2Iainfix =afibainfix +ainfix -V0c1c1V2Aainfix =afibainfix +ainfix +ainfix -V0c1c1c1V1Aainfix <=ainfix +ainfix -V0c1c1V0Aainfix <=c0ainfix +ainfix -V0c1c1Aainfix =afibainfix +V3c1V4Aainfix =afibainfix +ainfix +V3c1c1V5Aainfix <=ainfix +V3c1V0Aainfix <=c0ainfix +V3c1Iainfix =V5ainfix +V1V2FIainfix =V4V1FIainfix =afibV3V2Aainfix =afibainfix +V3c1V1Aainfix <=V3V0Aainfix <=c0V3Iainfix <=V3ainfix -V0c1Aainfix <=c0V3FFFAainfix =afibc0c0Aainfix =afibainfix +c0c1c1Aainfix <=c0V0Aainfix <=c0c0Iainfix <=c0ainfix -V0c1Aainfix =afibV0c0Iainfix >c0ainfix -V0c1Iainfix >=V0c0F">
     <proof
      prover="cvc3"
@@ -157,25 +153,25 @@
   </theory>
   <theory
    name="WP FibonacciLogarithmic"
-   verified="false"
+   verified="true"
    expanded="true">
    <goal
     name="WP_parameter logfib"
     expl="parameter logfib"
     sum="fe911abf17f3353090572ac6fa13f505"
-    proved="false"
-    expanded="true"
+    proved="true"
+    expanded="false"
     shape="iainfix =V0c0ainfix =apoweramk tc1c1c1c0V0amk tainfix +c1c0c0c0c1Ciainfix =amodV0c2c0aTuple2ainfix +ainfix *V1V1ainfix *V2V2ainfix *V2ainfix +V1ainfix +V1V2aTuple2ainfix *V2ainfix +V1ainfix +V1V2ainfix +ainfix *ainfix +V1V2ainfix +V1V2ainfix *V2V2aTuple2VVainfix =apoweramk tc1c1c1c0V0amk tainfix +V3V4V4V4V3Iainfix =apoweramk tc1c1c1c0adivV0c2amk tainfix +V1V2V2V2V1FAainfix >=adivV0c2c0Aainfix <adivV0c2V0Aainfix <=c0V0Iainfix >=V0c0F">
     <transf
      name="split_goal"
-     proved="false"
-     expanded="true">
+     proved="true"
+     expanded="false">
      <goal
       name="WP_parameter logfib.1"
       expl="normal postcondition"
       sum="97a33180db2d5fd938a5be052ce9455b"
       proved="true"
-      expanded="true"
+      expanded="false"
       shape="ainfix =apoweramk tc1c1c1c0V0amk tainfix +c1c0c0c0c1Iainfix =V0c0Iainfix >=V0c0F">
       <proof
        prover="cvc3"
@@ -211,15 +207,8 @@
       expl="precondition"
       sum="4d8a4a68e7db8bd86a6cc356c477d12c"
       proved="true"
-      expanded="true"
+      expanded="false"
       shape="ainfix >=adivV0c2c0Aainfix <adivV0c2V0Aainfix <=c0V0Iainfix =V0c0NIainfix >=V0c0F">
-      <proof
-       prover="alt-ergo"
-       timelimit="3"
-       edited=""
-       obsolete="false">
-       <result status="unknown" time="0.02"/>
-      </proof>
       <proof
        prover="cvc3"
        timelimit="5"
@@ -234,48 +223,20 @@
        obsolete="false">
        <result status="valid" time="0.01"/>
       </proof>
-      <proof
-       prover="vampire"
-       timelimit="3"
-       edited=""
-       obsolete="false">
-       <result status="timeout" time="2.99"/>
-      </proof>
      </goal>
      <goal
       name="WP_parameter logfib.3"
       expl="normal postcondition"
       sum="f94510de22468298510f134ac259cb4e"
-      proved="false"
-      expanded="true"
+      proved="true"
+      expanded="false"
       shape="Ciainfix =amodV0c2c0aTuple2ainfix +ainfix *V1V1ainfix *V2V2ainfix *V2ainfix +V1ainfix +V1V2aTuple2ainfix *V2ainfix +V1ainfix +V1V2ainfix +ainfix *ainfix +V1V2ainfix +V1V2ainfix *V2V2aTuple2VVainfix =apoweramk tc1c1c1c0V0amk tainfix +V3V4V4V4V3Iainfix =apoweramk tc1c1c1c0adivV0c2amk tainfix +V1V2V2V2V1FIainfix >=adivV0c2c0Aainfix <adivV0c2V0Aainfix <=c0V0Iainfix =V0c0NIainfix >=V0c0F">
       <proof
-       prover="cvc3"
-       timelimit="5"
-       edited=""
-       obsolete="false">
-       <result status="timeout" time="5.02"/>
-      </proof>
-      <proof
-       prover="alt-ergo"
-       timelimit="5"
-       edited=""
-       obsolete="false">
-       <result status="timeout" time="3.02"/>
-      </proof>
-      <proof
-       prover="z3"
-       timelimit="5"
-       edited=""
-       obsolete="false">
-       <result status="timeout" time="5.03"/>
-      </proof>
-      <proof
-       prover="vampire"
-       timelimit="3"
-       edited=""
+       prover="coq"
+       timelimit="10"
+       edited="fibonacci_WP_FibonacciLogarithmic_WP_parameter_logfib_1.v"
        obsolete="false">
-       <result status="timeout" time="2.99"/>
+       <result status="valid" time="0.69"/>
       </proof>
      </goal>
     </transf>
@@ -283,91 +244,23 @@
    <goal
     name="fib_m"
     sum="ebf63c237f713cc9fd86defe979892af"
-    proved="false"
+    proved="true"
     expanded="true"
     shape="Lapoweram1110V0ainfix =afibV0aa21V1Aainfix =afibainfix +V0c1aa11V1Iainfix >=V0c0F">
-    <transf
-     name="split_goal"
-     proved="false"
-     expanded="true">
-     <goal
-      name="fib_m.1"
-      sum="db01e70460edcdd6ffcb21494335a666"
-      proved="false"
-      expanded="true"
-      shape="Lapoweram1110V0ainfix =afibainfix +V0c1aa11V1Iainfix >=V0c0F">
-      <proof
-       prover="cvc3"
-       timelimit="5"
-       edited=""
-       obsolete="false">
-       <result status="timeout" time="5.05"/>
-      </proof>
-      <proof
-       prover="alt-ergo"
-       timelimit="5"
-       edited=""
-       obsolete="false">
-       <result status="timeout" time="5.03"/>
-      </proof>
-      <proof
-       prover="z3"
-       timelimit="5"
-       edited=""
-       obsolete="false">
-       <result status="timeout" time="5.03"/>
-      </proof>
-      <proof
-       prover="vampire"
-       timelimit="3"
-       edited=""
-       obsolete="false">
-       <result status="timeout" time="2.99"/>
-      </proof>
-     </goal>
-     <goal
-      name="fib_m.2"
-      sum="bafc463e470a81f635d7d28f18788abb"
-      proved="false"
-      expanded="true"
-      shape="Lapoweram1110V0ainfix =afibV0aa21V1Iainfix >=V0c0F">
-      <proof
-       prover="cvc3"
-       timelimit="5"
-       edited=""
-       obsolete="false">
-       <result status="timeout" time="5.04"/>
-      </proof>
-      <proof
-       prover="alt-ergo"
-       timelimit="5"
-       edited=""
-       obsolete="false">
-       <result status="timeout" time="5.03"/>
-      </proof>
-      <proof
-       prover="z3"
-       timelimit="5"
-       edited=""
-       obsolete="false">
-       <result status="timeout" time="5.03"/>
-      </proof>
-      <proof
-       prover="vampire"
-       timelimit="3"
-       edited=""
-       obsolete="false">
-       <result status="timeout" time="2.98"/>
-      </proof>
-     </goal>
-    </transf>
+    <proof
+     prover="coq"
+     timelimit="10"
+     edited="fibonacci_WP_FibonacciLogarithmic_fib_m_1.v"
+     obsolete="false">
+     <result status="valid" time="0.53"/>
+    </proof>
    </goal>
    <goal
     name="WP_parameter fibo"
     expl="parameter fibo"
     sum="ca4814a90b164bca8a0fb53063fc4b8b"
     proved="true"
-    expanded="true"
+    expanded="false"
     shape="ainfix =V2afibV0Iainfix =apoweramk tc1c1c1c0V0amk tainfix +V1V2V2V2V1FAainfix >=V0c0Iainfix >=V0c0F">
     <proof
      prover="cvc3"
@@ -381,14 +274,7 @@
      timelimit="5"
      edited=""
      obsolete="false">
-     <result status="valid" time="0.02"/>
-    </proof>
-    <proof
-     prover="vampire"
-     timelimit="3"
-     edited=""
-     obsolete="false">
-     <result status="timeout" time="2.99"/>
+     <result status="valid" time="0.01"/>
     </proof>
    </goal>
   </theory>