N-queens: proof in progress

437ed722 · Jean-Christophe Filliâtre · 18d45331 · 437ed722 · 437ed722 · 437ed722
Commit 437ed722 authored Aug 18, 2011 by Jean-Christophe Filliâtre
4 changed files
--- a/examples/programs/queens.mlw
+++ b/examples/programs/queens.mlw
+
+(* Verification of the following 2-lines C program solving the N-queens puzzle:
+
+   t(a,b,c){int d=0,e=a&~b&~c,f=1;if(a)for(f=0;d=(e-=d)&-e;f+=t(a-d,(b+d)*2,(
+   c+d)/2));return f;}main(q){scanf("%d",&q);printf("%d\n",t(~(~0<<q),0,0));}
+*)
+
+theory BitwiseArithmetic
+
+  use export int.Int
+
+  (* logical and *)
+  function (&) int int : int
+
+  (* logical shift left *)
+  function (<<) int int : int
+
+  (* logical negation *)
+  function (~_) int : int
+
+end
+
+theory Bits "the bits of an integer, as a set of integers"
+
+  use export set.Fsetint
+
+  function bits int : set int
+
+  axiom bits_0:
+    forall x: int. is_empty (bits x) <-> x = 0
+
+  axiom bits_remove_singleton:
+    forall x a b: int. bits b = singleton x -> mem x (bits a) ->
+    bits (a - b) = remove x (bits a)
+
+  use export BitwiseArithmetic
+
+  axiom bits_diff:
+    forall a b: int. bits (a & ~b) = diff (bits a) (bits b)
+
+  axiom rightmost_bit_trick:
+    forall x: int. x <> 0 -> bits (x & -x) = singleton (min_elt (bits x))
+
+end
+
+module NQueens
+
+  use import Bits
+  use import module arith.Int
+  use import module ref.Refint
+  use import module array.Array
+
+  (* warmup 1: termination of the loop *)
+  let rec t1 (a b c : int) =
+    if a <> 0 then begin
+      let e = ref (a & ~b & ~c) in
+      let f = ref 0 in
+      while !e <> 0 do variant { cardinal (bits !e) }
+        let d = !e & (- !e) in
+        f += t1 (a - d) ((b+d) * 2) ((c+d)/2);
+        e -= d
+      done;
+      !f
+    end else
+      1
+
+  (* warmup 2: termination of the recursive function *)
+  let rec t2 (a b c : int) variant { cardinal (bits a) } =
+    if a <> 0 then begin
+      let e = ref (a & ~b & ~c) in
+      let f = ref 0 in
+      while !e <> 0 do invariant { subset (bits !e) (bits a) }
+        let d = !e & (- !e) in
+        assert { bits d = singleton (min_elt (bits !e)) };
+        f += t2 (a - d) ((b+d) * 2) ((c+d)/2);
+        e -= d
+      done;
+      !f
+    end else
+      1
+
+end
+
+(*
+Local Variables:
+compile-command: "unset LANG; make -C ../.. examples/programs/queens.gui"
+End:
+*)
--- a/examples/programs/queens/queens_WP_NQueens_WP_parameter_t2_1.v
+++ b/examples/programs/queens/queens_WP_NQueens_WP_parameter_t2_1.v
+(* This file is generated by Why3's Coq driver *)
+(* Beware! Only edit allowed sections below    *)
+Require Import ZArith.
+Require Import Rbase.
+Require Import ZOdiv.
+Definition unit  := unit.
+
+Parameter mark : Type.
+
+Parameter at1: forall (a:Type), a -> mark  -> a.
+
+Implicit Arguments at1.
+
+Parameter old: forall (a:Type), a  -> a.
+
+Implicit Arguments old.
+
+Parameter set : forall (a:Type), Type.
+
+Parameter mem: forall (a:Type), a -> (set a)  -> Prop.
+
+Implicit Arguments mem.
+
+Definition infix_eqeq (a:Type)(s1:(set a)) (s2:(set a)): Prop :=
+  forall (x:a), (mem x s1) <-> (mem x s2).
+Implicit Arguments infix_eqeq.
+
+Axiom extensionality : forall (a:Type), forall (s1:(set a)) (s2:(set a)),
+  (infix_eqeq s1 s2) -> (s1 = s2).
+
+Definition subset (a:Type)(s1:(set a)) (s2:(set a)): Prop := forall (x:a),
+  (mem x s1) -> (mem x s2).
+Implicit Arguments subset.
+
+Axiom subset_trans : forall (a:Type), forall (s1:(set a)) (s2:(set a))
+  (s3:(set a)), (subset s1 s2) -> ((subset s2 s3) -> (subset s1 s3)).
+
+Parameter empty: forall (a:Type),  (set a).
+
+Set Contextual Implicit.
+Implicit Arguments empty.
+Unset Contextual Implicit.
+
+Definition is_empty (a:Type)(s:(set a)): Prop := forall (x:a), ~ (mem x s).
+Implicit Arguments is_empty.
+
+Axiom empty_def1 : forall (a:Type), (is_empty (empty:(set a))).
+
+Parameter add: forall (a:Type), a -> (set a)  -> (set a).
+
+Implicit Arguments add.
+
+Axiom add_def1 : forall (a:Type), forall (x:a) (y:a), forall (s:(set a)),
+  (mem x (add y s)) <-> ((x = y) \/ (mem x s)).
+
+Parameter remove: forall (a:Type), a -> (set a)  -> (set a).
+
+Implicit Arguments remove.
+
+Axiom remove_def1 : forall (a:Type), forall (x:a) (y:a) (s:(set a)), (mem x
+  (remove y s)) <-> ((~ (x = y)) /\ (mem x s)).
+
+Axiom subset_remove : forall (a:Type), forall (x:a) (s:(set a)),
+  (subset (remove x s) s).
+
+Parameter union: forall (a:Type), (set a) -> (set a)  -> (set a).
+
+Implicit Arguments union.
+
+Axiom union_def1 : forall (a:Type), forall (s1:(set a)) (s2:(set a)) (x:a),
+  (mem x (union s1 s2)) <-> ((mem x s1) \/ (mem x s2)).
+
+Parameter inter: forall (a:Type), (set a) -> (set a)  -> (set a).
+
+Implicit Arguments inter.
+
+Axiom inter_def1 : forall (a:Type), forall (s1:(set a)) (s2:(set a)) (x:a),
+  (mem x (inter s1 s2)) <-> ((mem x s1) /\ (mem x s2)).
+
+Parameter diff: forall (a:Type), (set a) -> (set a)  -> (set a).
+
+Implicit Arguments diff.
+
+Axiom diff_def1 : forall (a:Type), forall (s1:(set a)) (s2:(set a)) (x:a),
+  (mem x (diff s1 s2)) <-> ((mem x s1) /\ ~ (mem x s2)).
+
+Axiom subset_diff : forall (a:Type), forall (s1:(set a)) (s2:(set a)),
+  (subset (diff s1 s2) s1).
+
+Parameter cardinal: forall (a:Type), (set a)  -> Z.
+
+Implicit Arguments cardinal.
+
+Axiom cardinal_nonneg : forall (a:Type), forall (s:(set a)),
+  (0%Z <= (cardinal s))%Z.
+
+Axiom cardinal_empty : forall (a:Type), forall (s:(set a)),
+  ((cardinal s) = 0%Z) <-> (is_empty s).
+
+Axiom cardinal_add : forall (a:Type), forall (x:a), forall (s:(set a)),
+  (~ (mem x s)) -> ((cardinal (add x s)) = (1%Z + (cardinal s))%Z).
+
+Axiom cardinal_remove : forall (a:Type), forall (x:a), forall (s:(set a)),
+  (mem x s) -> ((cardinal s) = (1%Z + (cardinal (remove x s)))%Z).
+
+Axiom cardinal_subset : forall (a:Type), forall (s1:(set a)) (s2:(set a)),
+  (subset s1 s2) -> ((cardinal s1) <= (cardinal s2))%Z.
+
+Parameter min_elt: (set Z)  -> Z.
+
+
+Axiom min_elt_def1 : forall (s:(set Z)), (~ (is_empty s)) -> (mem (min_elt s)
+  s).
+
+Axiom min_elt_def2 : forall (s:(set Z)), (~ (is_empty s)) -> forall (x:Z),
+  (mem x s) -> ((min_elt s) <= x)%Z.
+
+Parameter max_elt: (set Z)  -> Z.
+
+
+Axiom max_elt_def1 : forall (s:(set Z)), (~ (is_empty s)) -> (mem (max_elt s)
+  s).
+
+Axiom max_elt_def2 : forall (s:(set Z)), (~ (is_empty s)) -> forall (x:Z),
+  (mem x s) -> (x <= (max_elt s))%Z.
+
+Parameter below: Z  -> (set Z).
+
+
+Axiom below_def : forall (x:Z) (n:Z), (mem x (below n)) <-> ((0%Z <= x)%Z /\
+  (x <  n)%Z).
+
+Parameter bits: Z  -> (set Z).
+
+
+Axiom bits_0 : forall (x:Z), (is_empty (bits x)) <-> (x = 0%Z).
+
+Axiom bits_remove_singleton : forall (x:Z) (a:Z) (b:Z), ((bits b) = (add x
+  (empty:(set Z)))) -> ((mem x (bits a)) -> ((bits (a - b)%Z) = (remove x
+  (bits a)))).
+
+Parameter infix_et: Z -> Z  -> Z.
+
+
+Parameter infix_lsls: Z -> Z  -> Z.
+
+
+Parameter prefix_tl: Z  -> Z.
+
+
+Axiom bits_diff : forall (a:Z) (b:Z), ((bits (infix_et a
+  (prefix_tl b))) = (diff (bits a) (bits b))).
+
+Axiom rightmost_bit_trick : forall (x:Z), (~ (x = 0%Z)) -> ((bits (infix_et x
+  (-x)%Z)) = (add (min_elt (bits x)) (empty:(set Z)))).
+
+Axiom Abs_pos : forall (x:Z), (0%Z <= (Zabs x))%Z.
+
+Axiom Div_mod : forall (x:Z) (y:Z), (~ (y = 0%Z)) ->
+  (x = ((y * (ZOdiv x y))%Z + (ZOmod x y))%Z).
+
+Axiom Div_bound : forall (x:Z) (y:Z), ((0%Z <= x)%Z /\ (0%Z <  y)%Z) ->
+  ((0%Z <= (ZOdiv x y))%Z /\ ((ZOdiv x y) <= x)%Z).
+
+Axiom Mod_bound : forall (x:Z) (y:Z), (~ (y = 0%Z)) ->
+  (((-(Zabs y))%Z <  (ZOmod x y))%Z /\ ((ZOmod x y) <  (Zabs y))%Z).
+
+Axiom Div_sign_pos : forall (x:Z) (y:Z), ((0%Z <= x)%Z /\ (0%Z <  y)%Z) ->
+  (0%Z <= (ZOdiv x y))%Z.
+
+Axiom Div_sign_neg : forall (x:Z) (y:Z), ((x <= 0%Z)%Z /\ (0%Z <  y)%Z) ->
+  ((ZOdiv x y) <= 0%Z)%Z.
+
+Axiom Mod_sign_pos : forall (x:Z) (y:Z), ((0%Z <= x)%Z /\ ~ (y = 0%Z)) ->
+  (0%Z <= (ZOmod x y))%Z.
+
+Axiom Mod_sign_neg : forall (x:Z) (y:Z), ((x <= 0%Z)%Z /\ ~ (y = 0%Z)) ->
+  ((ZOmod x y) <= 0%Z)%Z.
+
+Axiom Rounds_toward_zero : forall (x:Z) (y:Z), (~ (y = 0%Z)) ->
+  ((Zabs ((ZOdiv x y) * y)%Z) <= (Zabs x))%Z.
+
+Axiom Div_1 : forall (x:Z), ((ZOdiv x 1%Z) = x).
+
+Axiom Mod_1 : forall (x:Z), ((ZOmod x 1%Z) = 0%Z).
+
+Axiom Div_inf : forall (x:Z) (y:Z), ((0%Z <= x)%Z /\ (x <  y)%Z) ->
+  ((ZOdiv x y) = 0%Z).
+
+Axiom Mod_inf : forall (x:Z) (y:Z), ((0%Z <= x)%Z /\ (x <  y)%Z) ->
+  ((ZOmod x y) = x).
+
+Axiom Div_mult : forall (x:Z) (y:Z) (z:Z), ((0%Z <  x)%Z /\ ((0%Z <= y)%Z /\
+  (0%Z <= z)%Z)) -> ((ZOdiv ((x * y)%Z + z)%Z x) = (y + (ZOdiv z x))%Z).
+
+Axiom Mod_mult : forall (x:Z) (y:Z) (z:Z), ((0%Z <  x)%Z /\ ((0%Z <= y)%Z /\
+  (0%Z <= z)%Z)) -> ((ZOmod ((x * y)%Z + z)%Z x) = (ZOmod z x)).
+
+Inductive ref (a:Type) :=
+  | mk_ref : a -> ref a.
+Implicit Arguments mk_ref.
+
+Definition contents (a:Type)(u:(ref a)): a :=
+  match u with
+  | mk_ref contents1 => contents1
+  end.
+Implicit Arguments contents.
+
+Parameter map : forall (a:Type) (b:Type), Type.
+
+Parameter get: forall (a:Type) (b:Type), (map a b) -> a  -> b.
+
+Implicit Arguments get.
+
+Parameter set1: forall (a:Type) (b:Type), (map a b) -> a -> b  -> (map a b).
+
+Implicit Arguments set1.
+
+Axiom Select_eq : forall (a:Type) (b:Type), forall (m:(map a b)),
+  forall (a1:a) (a2:a), forall (b1:b), (a1 = a2) -> ((get (set1 m a1 b1)
+  a2) = b1).
+
+Axiom Select_neq : forall (a:Type) (b:Type), forall (m:(map a b)),
+  forall (a1:a) (a2:a), forall (b1:b), (~ (a1 = a2)) -> ((get (set1 m a1 b1)
+  a2) = (get m a2)).
+
+Parameter const: forall (b:Type) (a:Type), b  -> (map a b).
+
+Set Contextual Implicit.
+Implicit Arguments const.
+Unset Contextual Implicit.
+
+Axiom Const : forall (b:Type) (a:Type), forall (b1:b) (a1:a), ((get (const(
+  b1):(map a b)) a1) = b1).
+
+Inductive array (a:Type) :=
+  | mk_array : Z -> (map Z a) -> array a.
+Implicit Arguments mk_array.
+
+Definition elts (a:Type)(u:(array a)): (map Z a) :=
+  match u with
+  | mk_array _ elts1 => elts1
+  end.
+Implicit Arguments elts.
+
+Definition length (a:Type)(u:(array a)): Z :=
+  match u with
+  | mk_array length1 _ => length1
+  end.
+Implicit Arguments length.
+
+Definition get1 (a:Type)(a1:(array a)) (i:Z): a := (get (elts a1) i).
+Implicit Arguments get1.
+
+Definition set2 (a:Type)(a1:(array a)) (i:Z) (v:a): (array a) :=
+  match a1 with
+  | mk_array xcl0 _ => (mk_array xcl0 (set1 (elts a1) i v))
+  end.
+Implicit Arguments set2.
+
+(* YOU MAY EDIT THE CONTEXT BELOW *)
+
+(* DO NOT EDIT BELOW *)
+
+Theorem WP_parameter_t2 : forall (a:Z), (~ (a = 0%Z)) -> forall (f:Z),
+  forall (e:Z), (subset (bits e) (bits a)) -> ((~ (e = 0%Z)) ->
+  (((bits (infix_et e (-e)%Z)) = (add (min_elt (bits e)) (empty:(set Z)))) ->
+  ((~ (2%Z = 0%Z)) -> (((0%Z <= (cardinal (bits a)))%Z /\
+  ((cardinal (bits (a - (infix_et e
+  (-e)%Z))%Z)) <  (cardinal (bits a)))%Z) -> forall (result:Z),
+  forall (f1:Z), (f1 = (f + result)%Z) -> forall (e1:Z),
+  (e1 = (e - (infix_et e (-e)%Z))%Z) -> (subset (bits e1) (bits a)))))).
+(* YOU MAY EDIT THE PROOF BELOW *)
+intuition.
+assert (bits e1 = remove (min_elt (bits e)) (bits e)).
+subst e1.
+apply bits_remove_singleton.
+apply rightmost_bit_trick.
+omega.
+apply min_elt_def1.
+generalize (bits_0 e); intuition.
+apply subset_trans with (bits e); auto.
+rewrite H8.
+apply subset_remove; auto.
+Qed.
+(* DO NOT EDIT BELOW *)
+
+
--- a/examples/programs/queens/why3session.xml
+++ b/examples/programs/queens/why3session.xml
+<?xml version="1.0" encoding="UTF-8"?>
+<!DOCTYPE why3session SYSTEM "why3session.dtd">
+<why3session name="examples/programs/queens/why3session.xml">
+ <prover id="alt-ergo" name="Alt-Ergo" version="0.93"/>
+ <prover id="coq" name="Coq" version="8.2pl1"/>
+ <prover id="cvc3" name="CVC3" version="2.2"/>
+ <prover id="eprover" name="Eprover" version="1.0-004 Temi"/>
+ <prover id="gappa" name="Gappa" version="0.14.0"/>
+ <prover id="simplify" name="Simplify" version="1.5.4"/>
+ <prover id="spass" name="Spass" version="3.5"/>
+ <prover id="yices" name="Yices" version="1.0.27"/>
+ <prover id="z3" name="Z3" version="2.19"/>
+ <file name="../queens.mlw" verified="true" expanded="true">
+  <theory name="BitwiseArithmetic" verified="true" expanded="true">
+  </theory>
+  <theory name="Bits" verified="true" expanded="true">
+  </theory>
+  <theory name="WP NQueens" verified="true" expanded="true">
+   <goal name="WP_parameter t1" expl="correctness of parameter t1" sum="9844cd8c7f55016fd45ae33278e8d93f" proved="true" expanded="true" shape="iainfix =V0c0Niainfix =V2c0Nainfix <acardinalabitsV5acardinalabitsV2Aainfix <=c0acardinalabitsV2Iainfix =V5ainfix -V2ainfix &V2aprefix -V2FIainfix =V4ainfix +V1V3FFAainfix =c2c0NtFFtF">
+    <transf name="split_goal" proved="true" expanded="true">
+     <goal name="WP_parameter t1.1" expl="precondition" sum="7d8a8971a03ce57a28f3e4a1fddda896" proved="true" expanded="true" shape="ainfix =c2c0NIainfix =V2c0NFFIainfix =V0c0NF">
+      <proof prover="alt-ergo" timelimit="10" edited="" obsolete="false">
+       <result status="valid" time="0.02"/>
+      </proof>
+     </goal>
+     <goal name="WP_parameter t1.2" expl="loop variant decreases" sum="098c3a0d333c7dfc3bc02c6d833e7a60" proved="true" expanded="true" shape="ainfix <acardinalabitsV5acardinalabitsV2Aainfix <=c0acardinalabitsV2Iainfix =V5ainfix -V2ainfix &V2aprefix -V2FIainfix =V4ainfix +V1V3FFIainfix =c2c0NIainfix =V2c0NFFIainfix =V0c0NF">
+      <transf name="split_goal" proved="true" expanded="true">
+       <goal name="WP_parameter t1.2.1" expl="correctness of parameter t1" sum="7c572289b8be1ace3edf8a30024b9aaa" proved="true" expanded="true" shape="ainfix <=c0acardinalabitsV2Iainfix =V5ainfix -V2ainfix &V2aprefix -V2FIainfix =V4ainfix +V1V3FFIainfix =c2c0NIainfix =V2c0NFFIainfix =V0c0NF">
+        <proof prover="alt-ergo" timelimit="10" edited="" obsolete="false">
+         <result status="valid" time="0.03"/>
+        </proof>
+       </goal>
+       <goal name="WP_parameter t1.2.2" expl="correctness of parameter t1" sum="7754973086bdc451d97dc36dee15942c" proved="true" expanded="true" shape="ainfix <acardinalabitsV5acardinalabitsV2Iainfix =V5ainfix -V2ainfix &V2aprefix -V2FIainfix =V4ainfix +V1V3FFIainfix =c2c0NIainfix =V2c0NFFIainfix =V0c0NF">
+        <proof prover="z3" timelimit="10" edited="" obsolete="false">
+         <result status="valid" time="0.08"/>
+        </proof>
+       </goal>
+      </transf>
+     </goal>
+     <goal name="WP_parameter t1.3" expl="normal postcondition" sum="08539b7da399a7e2c76a2cfbd846123b" proved="true" expanded="true" shape="tIainfix =V2c0NNFFIainfix =V0c0NF">
+      <proof prover="alt-ergo" timelimit="10" edited="" obsolete="false">
+       <result status="valid" time="0.02"/>
+      </proof>
+     </goal>
+     <goal name="WP_parameter t1.4" expl="normal postcondition" sum="9aef56a481bc1b058d23bb09449d3f22" proved="true" expanded="true" shape="tIainfix =V0c0NNF">
+      <proof prover="alt-ergo" timelimit="10" edited="" obsolete="false">
+       <result status="valid" time="0.03"/>
+      </proof>
+     </goal>
+    </transf>
+   </goal>
+   <goal name="WP_parameter t2" expl="correctness of parameter t2" sum="ae1c369ef635350b8f5302d7f17894c6" proved="true" expanded="true" shape="iainfix =V0c0Niainfix =V4c0NasubsetabitsV7abitsV0Iainfix =V7ainfix -V4ainfix &V4aprefix -V4FIainfix =V6ainfix +V3V5FFAtAainfix <acardinalabitsainfix -V0ainfix &V4aprefix -V4acardinalabitsV0Aainfix <=c0acardinalabitsV0Aainfix =c2c0NAainfix =abitsainfix &V4aprefix -V4asingletonamin_eltabitsV4tIasubsetabitsV4abitsV0FFAasubsetabitsainfix &ainfix &V0aprefix ~V1aprefix ~V2abitsV0tFFF">
+    <transf name="split_goal" proved="true" expanded="true">
+     <goal name="WP_parameter t2.1" expl="loop invariant init" sum="84fd7f60ba18cd77c2a67d40008804bd" proved="true" expanded="true" shape="asubsetabitsainfix &ainfix &V0aprefix ~V1aprefix ~V2abitsV0Iainfix =V0c0NFFF">
+      <proof prover="alt-ergo" timelimit="10" edited="" obsolete="false">
+       <result status="valid" time="0.05"/>
+      </proof>
+     </goal>
+     <goal name="WP_parameter t2.2" expl="assertion" sum="2a54a03011dd0ce17358573a065d9a09" proved="true" expanded="true" shape="ainfix =abitsainfix &V4aprefix -V4asingletonamin_eltabitsV4Iainfix =V4c0NIasubsetabitsV4abitsV0FFIainfix =V0c0NFFF">
+      <proof prover="alt-ergo" timelimit="10" edited="" obsolete="false">
+       <result status="valid" time="0.03"/>
+      </proof>
+     </goal>
+     <goal name="WP_parameter t2.3" expl="precondition" sum="4f12ace363f89ac270a1ba8a478b1011" proved="true" expanded="true" shape="ainfix =c2c0NIainfix =abitsainfix &V4aprefix -V4asingletonamin_eltabitsV4Iainfix =V4c0NIasubsetabitsV4abitsV0FFIainfix =V0c0NFFF">
+      <proof prover="alt-ergo" timelimit="10" edited="" obsolete="false">
+       <result status="valid" time="0.03"/>
+      </proof>
+     </goal>
+     <goal name="WP_parameter t2.4" expl="precondition" sum="0a21712328585890a3798a6e17b698c7" proved="true" expanded="true" shape="tAainfix <acardinalabitsainfix -V0ainfix &V4aprefix -V4acardinalabitsV0Aainfix <=c0acardinalabitsV0Iainfix =c2c0NIainfix =abitsainfix &V4aprefix -V4asingletonamin_eltabitsV4Iainfix =V4c0NIasubsetabitsV4abitsV0FFIainfix =V0c0NFFF">
+      <transf name="split_goal" proved="true" expanded="true">
+       <goal name="WP_parameter t2.4.1" expl="correctness of parameter t2" sum="9ad392c07fcdd8f70b7e8ee3841ef5ef" proved="true" expanded="true" shape="ainfix <=c0acardinalabitsV0Iainfix =c2c0NIainfix =abitsainfix &V4aprefix -V4asingletonamin_eltabitsV4Iainfix =V4c0NIasubsetabitsV4abitsV0FFIainfix =V0c0NFFF">
+        <proof prover="alt-ergo" timelimit="10" edited="" obsolete="false">
+         <result status="valid" time="0.02"/>
+        </proof>
+       </goal>
+       <goal name="WP_parameter t2.4.2" expl="correctness of parameter t2" sum="9adbc8e82df5a12fef6630e780b3861c" proved="true" expanded="true" shape="ainfix <acardinalabitsainfix -V0ainfix &V4aprefix -V4acardinalabitsV0Iainfix =c2c0NIainfix =abitsainfix &V4aprefix -V4asingletonamin_eltabitsV4Iainfix =V4c0NIasubsetabitsV4abitsV0FFIainfix =V0c0NFFF">
+        <proof prover="z3" timelimit="10" edited="" obsolete="false">
+         <result status="valid" time="7.86"/>
+        </proof>
+       </goal>
+      </transf>
+     </goal>
+     <goal name="WP_parameter t2.5" expl="loop invariant preservation" sum="638feec51a96015020e77cb80f359a29" proved="true" expanded="true" shape="asubsetabitsV7abitsV0Iainfix =V7ainfix -V4ainfix &V4aprefix -V4FIainfix =V6ainfix +V3V5FFItAainfix <acardinalabitsainfix -V0ainfix &V4aprefix -V4acardinalabitsV0Aainfix <=c0acardinalabitsV0Iainfix =c2c0NIainfix =abitsainfix &V4aprefix -V4asingletonamin_eltabitsV4Iainfix =V4c0NIasubsetabitsV4abitsV0FFIainfix =V0c0NFFF">
+      <proof prover="coq" timelimit="10" edited="queens_WP_NQueens_WP_parameter_t2_1.v" obsolete="false">
+       <result status="valid" time="0.68"/>
+      </proof>
+     </goal>
+     <goal name="WP_parameter t2.6" expl="normal postcondition" sum="4903c9b24c965c6cefaddad3cd9820ec" proved="true" expanded="true" shape="tIainfix =V4c0NNIasubsetabitsV4abitsV0FFIainfix =V0c0NFFF">
+      <proof prover="alt-ergo" timelimit="10" edited="" obsolete="false">
+       <result status="valid" time="0.02"/>
+      </proof>
+     </goal>
+     <goal name="WP_parameter t2.7" expl="normal postcondition" sum="a17d91fc3e9e97cdfbc4dbce099c0c9a" proved="true" expanded="true" shape="tIainfix =V0c0NNFFF">
+      <proof prover="alt-ergo" timelimit="10" edited="" obsolete="false">
+       <result status="valid" time="0.02"/>
+      </proof>
+     </goal>
+    </transf>
+   </goal>
+  </theory>
+ </file>
+</why3session>
--- a/theories/set.why
+++ b/theories/set.why
@@ -3,47 +3,70 @@ theory Set

  type set 'a

+  (* membership *)
  predicate mem 'a (set 'a)

+  (* equality *)
+  predicate (==) (s1 s2: set 'a) = forall x : 'a. mem x s1 <-> mem x s2
+
+  axiom extensionality:
+    forall s1 s2: set 'a. s1 == s2 -> s1 = s2
+
+  (* inclusion *)
+  predicate subset (s1 s2: set 'a) = forall x : 'a. mem x s1 -> mem x s2
+
+  lemma subset_trans:
+    forall s1 s2 s3: set 'a. subset s1 s2 -> subset s2 s3 -> subset s1 s3
+
+  (* empty set *)
  function empty : set 'a

-  predicate is_empty (s : set 'a) = forall x : 'a. not mem x s
+  predicate is_empty (s: set 'a) = forall x: 'a. not (mem x s)

-  axiom Empty_def1 : is_empty(empty : set 'a)
+  axiom empty_def1: is_empty (empty : set 'a)

+  (* addition *)
  function add 'a (set 'a) : set 'a

-  axiom Add_def1 :
-    forall x y : 'a. forall s : set 'a.
+  axiom add_def1:
+    forall x y: 'a. forall s: set 'a.
    mem x (add y s) <-> x = y \/ mem x s

+  function singleton (x: 'a) : set 'a = add x empty
+
+  (* removal *)
  function remove 'a (set 'a) : set 'a

-  axiom Remove_def1 :
-    forall x y : 'a. forall s : set 'a.
+  axiom remove_def1:
+    forall x y : 'a, s : set 'a.
    mem x (remove y s) <-> x <> y /\ mem x s

+  lemma subset_remove:
+    forall x: 'a, s: set 'a. subset (remove x s) s
+
+  (* union *)
  function union (set 'a) (set 'a) : set 'a

-  axiom Union_def1 :
-    forall s1 s2 : set 'a. forall x : 'a.
+  axiom union_def1:
+    forall s1 s2: set 'a, x: 'a.
    mem x (union s1 s2) <-> mem x s1 \/ mem x s2

+  (* intersection *)
  function inter (set 'a) (set 'a) : set 'a

-  axiom Inter_def1 :
-    forall s1 s2 : set 'a. forall x : 'a.
+  axiom inter_def1:
+    forall s1 s2: set 'a, x: 'a.
    mem x (inter s1 s2) <-> mem x s1 /\ mem x s2

+  (* difference *)
  function diff (set 'a) (set 'a) : set 'a

-  axiom Diff_def1 :
-    forall s1 s2 : set 'a. forall x : 'a.
-    mem x (diff s1 s2) <-> mem x s1 /\ not mem x s2
-
-  predicate equal(s1 s2 : set 'a) = forall x : 'a. mem x s1 <-> mem x s2
+  axiom diff_def1:
+    forall s1 s2: set 'a, x: 'a.
+    mem x (diff s1 s2) <-> mem x s1 /\ not (mem x s2)

-  predicate subset(s1 s2 : set 'a) = forall x : 'a. mem x s1 -> mem x s2
+  lemma subset_diff:
+    forall s1 s2: set 'a. subset (diff s1 s2) s1

 end

@@ -54,23 +77,53 @@ theory Fset

  function cardinal (set 'a) : int

-  axiom Cardinal_nonneg : forall s : set 'a. cardinal s >= 0
+  axiom cardinal_nonneg: forall s: set 'a. cardinal s >= 0

-  axiom Cardinal_empty : cardinal(empty : set 'a) = 0
+  axiom cardinal_empty:
+    forall s: set 'a. cardinal s = 0 <-> is_empty s

-  axiom Cardinal_add :
+  axiom cardinal_add:
    forall x : 'a. forall s : set 'a.
-    not mem x s -> cardinal (add x s) = 1 + cardinal s
+    not (mem x s) -> cardinal (add x s) = 1 + cardinal s