example decrease1: Coq proof and recursive version

parent b7c7d6f9
......@@ -18,21 +18,38 @@ module Decrease1
forall a: array int. decrease1 a ->
forall i j: int. 0 <= i <= j < length a -> a[j] >= a[i] + i - j
exception Found int
exception Found
let search (a: array int) =
{ decrease1 a }
let i = ref 0 in
while !i < length a do
invariant { 0 <= i and
forall j: int. 0 <= j < i -> j < length a -> a[j] <> 0 }
variant { length a - i }
if get a !i = 0 then raise (Found !i);
if get a !i > 0 then i := !i + get a !i else i := !i + 1
done
{ forall j: int. 0 <= j < length a -> a[j] <> 0 }
| Found -> { 0 <= result < length a and a[result] = 0 and
forall j: int. 0 <= j < result -> a[j] <> 0 }
try
while !i < length a do
invariant { 0 <= i and
forall j: int. 0 <= j < i -> j < length a -> a[j] <> 0 }
variant { length a - i }
if get a !i = 0 then raise Found;
if get a !i > 0 then i := !i + get a !i else i := !i + 1
done;
-1
with Found ->
!i
end
{ (result = -1 and forall j: int. 0 <= j < length a -> a[j] <> 0)
or (0 <= result < length a and a[result] = 0 and
forall j: int. 0 <= j < result -> a[j] <> 0) }
let rec search_rec (a: array int) (i : int) =
{ decrease1 a and 0 <= i }
if i < length a then
if get a i = 0 then i
else if get a i > 0 then search_rec a (i + get a i)
else search_rec a (i + 1)
else
-1
{ (result = -1 and forall j: int. i <= j < length a -> a[j] <> 0)
or (i <= result < length a and a[result] = 0 and
forall j: int. i <= j < result -> a[j] <> 0) }
end
......
(* 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 ignore: forall (a:Type), a -> unit.
Implicit Arguments ignore.
Parameter label_ : Type.
Parameter at1: forall (a:Type), a -> label_ -> a.
Implicit Arguments at1.
Parameter old: forall (a:Type), a -> a.
Implicit Arguments old.
Definition ref (a:Type) := a.
Parameter map : forall (a:Type) (b:Type), Type.
Parameter get: forall (a:Type) (b:Type), (map a b) -> a -> b.
Implicit Arguments get.
Parameter set: forall (a:Type) (b:Type), (map a b) -> a -> b -> (map a b).
Implicit Arguments set.
Axiom Select_eq : forall (a:Type) (b:Type), forall (m:(map a b)),
forall (a1:a) (a2:a), forall (b1:b), (a1 = a2) -> ((get (set 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 (set m a1 b1)
a2) = (get m a2)).
Parameter create_const: forall (b:Type) (a:Type), b -> (map a b).
Set Contextual Implicit.
Implicit Arguments create_const.
Unset Contextual Implicit.
Axiom Const : forall (b:Type) (a:Type), forall (b1:b) (a1:a),
((get (create_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 mixfix_lbrb (a:Type)(a1:(array a)) (i:Z): a := (get (elts a1) i).
Implicit Arguments mixfix_lbrb.
Definition decrease1(a:(array Z)): Prop := forall (i:Z), ((0%Z <= i)%Z /\
(i < ((length a) - 1%Z)%Z)%Z) -> (((mixfix_lbrb a
i) - 1%Z)%Z <= (mixfix_lbrb a (i + 1%Z)%Z))%Z.
Theorem decrease1_induction : forall (a:(array Z)), (decrease1 a) ->
forall (i:Z) (j:Z), (((0%Z <= i)%Z /\ (i <= j)%Z) /\
(j < (length a))%Z) -> ((((mixfix_lbrb a
i) + i)%Z - j)%Z <= (mixfix_lbrb a j))%Z.
(* YOU MAY EDIT THE PROOF BELOW *)
unfold decrease1.
intros a Ha i j Hij.
generalize Hij; pattern j.
apply (Zlt_lower_bound_ind _ i).
2: omega.
intuition.
assert (x = i \/ i < x)%Z by omega.
destruct H4.
subst x.
ring_simplify.
omega.
apply Zle_trans with (mixfix_lbrb a (x-1) - 1)%Z.
assert (i <= x-1 < x)%Z by omega.
assert (0 <= i <= x-1 /\ x-1 < length a)%Z by omega.
generalize (H (x-1)%Z H8 H9); clear H; intuition.
apply Zle_trans with (mixfix_lbrb a (x-1+1))%Z.
apply (Ha (x-1)%Z); omega.
ring_simplify (x-1+1)%Z.
omega.
Qed.
(* DO NOT EDIT BELOW *)
<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE why3session SYSTEM "why3session.dtd">
<why3session name="examples/programs/decrease1/why3session.xml">
<file name="../decrease1.mlw" verified="true" expanded="true">
<theory name="Decrease1" verified="true" expanded="true">
<goal name="decrease1_induction" sum="8319c3299f6c5b391aec5943ca151429" proved="true" expanded="true">
<proof prover="coq" timelimit="10" edited="examples/programs/decrease1/decrease1.mlw_Decrease1_decrease1_induction_2.v" obsolete="false">
<result status="valid" time="0.75"/>
</proof>
<proof prover="alt-ergo" timelimit="10" edited="" obsolete="false">
<result status="unknown" time="0.43"/>
</proof>
</goal>
<goal name="WP_search" expl="correctness of search" sum="5f9e29fcb501d5550c29bfbd1c8fd584" proved="true" expanded="true">
<proof prover="alt-ergo" timelimit="10" edited="" obsolete="true">
<result status="timeout" time="10.12"/>
</proof>
<transf name="split_goal" proved="true" expanded="true">
<goal name="WP_search.1" expl="loop invariant init" sum="5d4fd28ee3c69be535f8ee535e6cb802" proved="true" expanded="true">
<proof prover="alt-ergo" timelimit="10" edited="" obsolete="false">
<result status="valid" time="0.02"/>
</proof>
</goal>
<goal name="WP_search.2" expl="precondition" sum="d315642af07745fcac77676c671ffee3" proved="true" expanded="true">
<proof prover="alt-ergo" timelimit="10" edited="" obsolete="false">
<result status="valid" time="0.02"/>
</proof>
</goal>
<goal name="WP_search.3" expl="normal postcondition" sum="fa8f0f929008e376cd5e388239a0320a" proved="true" expanded="true">
<proof prover="alt-ergo" timelimit="10" edited="" obsolete="false">
<result status="valid" time="0.05"/>
</proof>
</goal>
<goal name="WP_search.4" expl="precondition" sum="026ddab308389db6e2db01c19192eb40" proved="true" expanded="true">
<proof prover="alt-ergo" timelimit="10" edited="" obsolete="false">
<result status="valid" time="0.02"/>
</proof>
</goal>
<goal name="WP_search.5" expl="precondition" sum="3f6bcaf82657cc2a967502da8329005b" proved="true" expanded="true">
<proof prover="alt-ergo" timelimit="10" edited="" obsolete="false">
<result status="valid" time="0.02"/>
</proof>
</goal>
<goal name="WP_search.6" expl="loop invariant preservation" sum="5bce229f4e8d1fc1fc6e04c42e6e408b" proved="true" expanded="true">
<proof prover="cvc3" timelimit="10" edited="" obsolete="false">
<result status="unknown" time="3.17"/>
</proof>
<proof prover="alt-ergo" timelimit="10" edited="" obsolete="false">
<result status="timeout" time="10.19"/>
</proof>
<proof prover="z3" timelimit="10" edited="" obsolete="false">
<result status="valid" time="0.10"/>
</proof>
</goal>
<goal name="WP_search.7" expl="loop variant decreases" sum="996216bff5e7480e0a1fe0e3968b8ffb" proved="true" expanded="true">
<proof prover="cvc3" timelimit="10" edited="" obsolete="false">
<result status="valid" time="0.02"/>
</proof>
<proof prover="alt-ergo" timelimit="10" edited="" obsolete="false">
<result status="valid" time="0.02"/>
</proof>
<proof prover="z3" timelimit="10" edited="" obsolete="false">
<result status="valid" time="0.04"/>
</proof>
<transf name="split_goal" proved="true" expanded="true">
<goal name="WP_search.7.1" expl="correctness of search" sum="d97f71408572fc6c730314dd5bebfe22" proved="true" expanded="true">
<proof prover="alt-ergo" timelimit="10" edited="" obsolete="false">
<result status="valid" time="0.02"/>
</proof>
</goal>
<goal name="WP_search.7.2" expl="correctness of search" sum="1f8eb20d029c95d115aa48530508032c" proved="true" expanded="true">
<proof prover="alt-ergo" timelimit="10" edited="" obsolete="false">
<result status="valid" time="0.02"/>
</proof>
<proof prover="z3" timelimit="10" edited="" obsolete="false">
<result status="valid" time="0.03"/>
</proof>
<proof prover="yices" timelimit="10" edited="" obsolete="false">
<result status="valid" time="0.03"/>
</proof>
</goal>
</transf>
</goal>
<goal name="WP_search.8" expl="loop invariant preservation" sum="fc6718c43629f048e462fa6063b11410" proved="true" expanded="true">
<proof prover="alt-ergo" timelimit="10" edited="" obsolete="false">
<result status="valid" time="0.02"/>
</proof>
</goal>
<goal name="WP_search.9" expl="loop variant decreases" sum="14c9dd5d9133e195e8e688d19a5d7e19" proved="true" expanded="true">
<proof prover="alt-ergo" timelimit="10" edited="" obsolete="false">
<result status="valid" time="0.02"/>
</proof>
</goal>
<goal name="WP_search.10" expl="normal postcondition" sum="ff98d91de211d86ceddf8f37215c787c" proved="true" expanded="true">
<proof prover="alt-ergo" timelimit="10" edited="" obsolete="false">
<result status="valid" time="0.03"/>
</proof>
</goal>
</transf>
</goal>
<goal name="WP_search_rec" expl="correctness of search_rec" sum="359800e7e063b55fb2e4665ffe6c7a62" proved="true" expanded="true">
<proof prover="cvc3" timelimit="10" edited="" obsolete="true">
<result status="unknown" time="0.21"/>
</proof>
<proof prover="alt-ergo" timelimit="10" edited="" obsolete="true">
<result status="timeout" time="10.22"/>
</proof>
<proof prover="z3" timelimit="10" edited="" obsolete="true">
<result status="timeout" time="10.50"/>
</proof>
<transf name="split_goal" proved="true" expanded="true">
<goal name="WP_search_rec.1" expl="precondition" sum="b862eaa14642db1bca4cfd29e71dc85a" proved="true" expanded="true">
<proof prover="alt-ergo" timelimit="10" edited="" obsolete="false">
<result status="valid" time="0.02"/>
</proof>
</goal>
<goal name="WP_search_rec.2" expl="normal postcondition" sum="028a83456b4c241f325751fe75ff7cfe" proved="true" expanded="true">
<proof prover="alt-ergo" timelimit="10" edited="" obsolete="true">
<result status="valid" time="0.03"/>
</proof>
</goal>
<goal name="WP_search_rec.3" expl="precondition" sum="58493f837aaa39a3890ba4c802e46109" proved="true" expanded="true">
<proof prover="alt-ergo" timelimit="10" edited="" obsolete="true">
<result status="valid" time="0.02"/>
</proof>
</goal>
<goal name="WP_search_rec.4" expl="precondition" sum="99a18ac863afa1e6b4acfb99b4d0dae6" proved="true" expanded="true">
<proof prover="alt-ergo" timelimit="10" edited="" obsolete="true">
<result status="valid" time="0.02"/>
</proof>
</goal>
<goal name="WP_search_rec.5" expl="precondition" sum="44a5c5d945aad7cfe2bcc7d4adf728f2" proved="true" expanded="true">
<proof prover="alt-ergo" timelimit="10" edited="" obsolete="true">
<result status="valid" time="0.02"/>
</proof>
</goal>
<goal name="WP_search_rec.6" expl="normal postcondition" sum="699e81654d2a303c491cfab413eb3501" proved="true" expanded="true">
<proof prover="alt-ergo" timelimit="10" edited="" obsolete="false">
<result status="timeout" time="10.08"/>
</proof>
<proof prover="z3" timelimit="10" edited="" obsolete="false">
<result status="valid" time="0.24"/>
</proof>
</goal>
<goal name="WP_search_rec.7" expl="precondition" sum="9453d56ae3a433905973837d2fab5f11" proved="true" expanded="true">
<proof prover="alt-ergo" timelimit="10" edited="" obsolete="true">
<result status="valid" time="0.03"/>
</proof>
</goal>
<goal name="WP_search_rec.8" expl="normal postcondition" sum="179d0569c8256c2c7e5e5730ab534329" proved="true" expanded="true">
<proof prover="alt-ergo" timelimit="10" edited="" obsolete="true">
<result status="valid" time="0.05"/>
</proof>
</goal>
<goal name="WP_search_rec.9" expl="normal postcondition" sum="a5e3a9553c1dbb830d1064e70335f3a9" proved="true" expanded="true">
<proof prover="alt-ergo" timelimit="10" edited="" obsolete="true">
<result status="valid" time="0.02"/>
</proof>
</goal>
</transf>
</goal>
</theory>
</file>
</why3session>
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