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example decrease1: Coq proof and recursive version
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Jean-Christophe Filliatre
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(* This file is generated by Why3's Coq driver *)
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(* Beware! Only edit allowed sections below *)
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Require Import ZArith.
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Require Import Rbase.
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Definition unit := unit.
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Parameter ignore: forall (a:Type), a -> unit.
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Implicit Arguments ignore.
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Parameter label_ : Type.
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Parameter at1: forall (a:Type), a -> label_ -> a.
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Implicit Arguments at1.
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Parameter old: forall (a:Type), a -> a.
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Implicit Arguments old.
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Definition ref (a:Type) := a.
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Parameter map : forall (a:Type) (b:Type), Type.
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Parameter get: forall (a:Type) (b:Type), (map a b) -> a -> b.
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Implicit Arguments get.
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Parameter set: forall (a:Type) (b:Type), (map a b) -> a -> b -> (map a b).
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Implicit Arguments set.
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Axiom Select_eq : forall (a:Type) (b:Type), forall (m:(map a b)),
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forall (a1:a) (a2:a), forall (b1:b), (a1 = a2) -> ((get (set m a1 b1)
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a2) = b1).
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Axiom Select_neq : forall (a:Type) (b:Type), forall (m:(map a b)),
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forall (a1:a) (a2:a), forall (b1:b), (~ (a1 = a2)) -> ((get (set m a1 b1)
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a2) = (get m a2)).
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Parameter create_const: forall (b:Type) (a:Type), b -> (map a b).
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Set Contextual Implicit.
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Implicit Arguments create_const.
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Unset Contextual Implicit.
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Axiom Const : forall (b:Type) (a:Type), forall (b1:b) (a1:a),
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((get (create_const(b1):(map a b)) a1) = b1).
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Inductive array (a:Type) :=
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| mk_array : Z -> (map Z a) -> array a.
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Implicit Arguments mk_array.
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Definition elts (a:Type)(u:(array a)): (map Z a) :=
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match u with
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| mk_array _ elts1 => elts1
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end.
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Implicit Arguments elts.
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Definition length (a:Type)(u:(array a)): Z :=
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match u with
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| mk_array length1 _ => length1
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end.
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Implicit Arguments length.
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Definition mixfix_lbrb (a:Type)(a1:(array a)) (i:Z): a := (get (elts a1) i).
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Implicit Arguments mixfix_lbrb.
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Definition decrease1(a:(array Z)): Prop := forall (i:Z), ((0%Z <= i)%Z /\
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(i < ((length a) - 1%Z)%Z)%Z) -> (((mixfix_lbrb a
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i) - 1%Z)%Z <= (mixfix_lbrb a (i + 1%Z)%Z))%Z.
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Theorem decrease1_induction : forall (a:(array Z)), (decrease1 a) ->
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forall (i:Z) (j:Z), (((0%Z <= i)%Z /\ (i <= j)%Z) /\
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(j < (length a))%Z) -> ((((mixfix_lbrb a
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i) + i)%Z - j)%Z <= (mixfix_lbrb a j))%Z.
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(* YOU MAY EDIT THE PROOF BELOW *)
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unfold decrease1.
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intros a Ha i j Hij.
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generalize Hij; pattern j.
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apply (Zlt_lower_bound_ind _ i).
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2: omega.
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intuition.
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assert (x = i \/ i < x)%Z by omega.
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destruct H4.
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subst x.
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ring_simplify.
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omega.
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apply Zle_trans with (mixfix_lbrb a (x-1) - 1)%Z.
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assert (i <= x-1 < x)%Z by omega.
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assert (0 <= i <= x-1 /\ x-1 < length a)%Z by omega.
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generalize (H (x-1)%Z H8 H9); clear H; intuition.
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apply Zle_trans with (mixfix_lbrb a (x-1+1))%Z.
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apply (Ha (x-1)%Z); omega.
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ring_simplify (x-1+1)%Z.
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omega.
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Qed.
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(* DO NOT EDIT BELOW *)
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164
examples/programs/decrease1/why3session.xml
Normal file
164
examples/programs/decrease1/why3session.xml
Normal file
@@ -0,0 +1,164 @@
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<?xml version="1.0" encoding="UTF-8"?>
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<!DOCTYPE why3session SYSTEM "why3session.dtd">
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<why3session name="examples/programs/decrease1/why3session.xml">
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<file name="../decrease1.mlw" verified="true" expanded="true">
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<theory name="Decrease1" verified="true" expanded="true">
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<goal name="decrease1_induction" sum="8319c3299f6c5b391aec5943ca151429" proved="true" expanded="true">
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<proof prover="coq" timelimit="10" edited="examples/programs/decrease1/decrease1.mlw_Decrease1_decrease1_induction_2.v" obsolete="false">
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<result status="valid" time="0.75"/>
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</proof>
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<proof prover="alt-ergo" timelimit="10" edited="" obsolete="false">
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<result status="unknown" time="0.43"/>
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</proof>
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</goal>
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<goal name="WP_search" expl="correctness of search" sum="5f9e29fcb501d5550c29bfbd1c8fd584" proved="true" expanded="true">
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<proof prover="alt-ergo" timelimit="10" edited="" obsolete="true">
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<result status="timeout" time="10.12"/>
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</proof>
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<transf name="split_goal" proved="true" expanded="true">
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<goal name="WP_search.1" expl="loop invariant init" sum="5d4fd28ee3c69be535f8ee535e6cb802" proved="true" expanded="true">
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<proof prover="alt-ergo" timelimit="10" edited="" obsolete="false">
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<result status="valid" time="0.02"/>
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</proof>
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</goal>
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<goal name="WP_search.2" expl="precondition" sum="d315642af07745fcac77676c671ffee3" proved="true" expanded="true">
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<proof prover="alt-ergo" timelimit="10" edited="" obsolete="false">
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<result status="valid" time="0.02"/>
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</proof>
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</goal>
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<goal name="WP_search.3" expl="normal postcondition" sum="fa8f0f929008e376cd5e388239a0320a" proved="true" expanded="true">
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<proof prover="alt-ergo" timelimit="10" edited="" obsolete="false">
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<result status="valid" time="0.05"/>
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</proof>
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</goal>
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<goal name="WP_search.4" expl="precondition" sum="026ddab308389db6e2db01c19192eb40" proved="true" expanded="true">
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<proof prover="alt-ergo" timelimit="10" edited="" obsolete="false">
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<result status="valid" time="0.02"/>
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</proof>
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</goal>
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<goal name="WP_search.5" expl="precondition" sum="3f6bcaf82657cc2a967502da8329005b" proved="true" expanded="true">
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<proof prover="alt-ergo" timelimit="10" edited="" obsolete="false">
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<result status="valid" time="0.02"/>
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</proof>
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</goal>
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<goal name="WP_search.6" expl="loop invariant preservation" sum="5bce229f4e8d1fc1fc6e04c42e6e408b" proved="true" expanded="true">
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<proof prover="cvc3" timelimit="10" edited="" obsolete="false">
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<result status="unknown" time="3.17"/>
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</proof>
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<proof prover="alt-ergo" timelimit="10" edited="" obsolete="false">
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<result status="timeout" time="10.19"/>
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</proof>
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<proof prover="z3" timelimit="10" edited="" obsolete="false">
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<result status="valid" time="0.10"/>
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</proof>
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</goal>
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<goal name="WP_search.7" expl="loop variant decreases" sum="996216bff5e7480e0a1fe0e3968b8ffb" proved="true" expanded="true">
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<proof prover="cvc3" timelimit="10" edited="" obsolete="false">
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<result status="valid" time="0.02"/>
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</proof>
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<proof prover="alt-ergo" timelimit="10" edited="" obsolete="false">
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<result status="valid" time="0.02"/>
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</proof>
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<proof prover="z3" timelimit="10" edited="" obsolete="false">
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<result status="valid" time="0.04"/>
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</proof>
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<transf name="split_goal" proved="true" expanded="true">
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<goal name="WP_search.7.1" expl="correctness of search" sum="d97f71408572fc6c730314dd5bebfe22" proved="true" expanded="true">
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<proof prover="alt-ergo" timelimit="10" edited="" obsolete="false">
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<result status="valid" time="0.02"/>
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</proof>
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</goal>
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<goal name="WP_search.7.2" expl="correctness of search" sum="1f8eb20d029c95d115aa48530508032c" proved="true" expanded="true">
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<proof prover="alt-ergo" timelimit="10" edited="" obsolete="false">
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<result status="valid" time="0.02"/>
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</proof>
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<proof prover="z3" timelimit="10" edited="" obsolete="false">
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<result status="valid" time="0.03"/>
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</proof>
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<proof prover="yices" timelimit="10" edited="" obsolete="false">
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<result status="valid" time="0.03"/>
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</proof>
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</goal>
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</transf>
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</goal>
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<goal name="WP_search.8" expl="loop invariant preservation" sum="fc6718c43629f048e462fa6063b11410" proved="true" expanded="true">
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<proof prover="alt-ergo" timelimit="10" edited="" obsolete="false">
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<result status="valid" time="0.02"/>
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</proof>
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</goal>
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<goal name="WP_search.9" expl="loop variant decreases" sum="14c9dd5d9133e195e8e688d19a5d7e19" proved="true" expanded="true">
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<proof prover="alt-ergo" timelimit="10" edited="" obsolete="false">
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<result status="valid" time="0.02"/>
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</proof>
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</goal>
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<goal name="WP_search.10" expl="normal postcondition" sum="ff98d91de211d86ceddf8f37215c787c" proved="true" expanded="true">
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<proof prover="alt-ergo" timelimit="10" edited="" obsolete="false">
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<result status="valid" time="0.03"/>
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</proof>
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</goal>
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</transf>
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</goal>
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<goal name="WP_search_rec" expl="correctness of search_rec" sum="359800e7e063b55fb2e4665ffe6c7a62" proved="true" expanded="true">
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<proof prover="cvc3" timelimit="10" edited="" obsolete="true">
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<result status="unknown" time="0.21"/>
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</proof>
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<proof prover="alt-ergo" timelimit="10" edited="" obsolete="true">
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<result status="timeout" time="10.22"/>
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</proof>
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<proof prover="z3" timelimit="10" edited="" obsolete="true">
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<result status="timeout" time="10.50"/>
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</proof>
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<transf name="split_goal" proved="true" expanded="true">
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<goal name="WP_search_rec.1" expl="precondition" sum="b862eaa14642db1bca4cfd29e71dc85a" proved="true" expanded="true">
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<proof prover="alt-ergo" timelimit="10" edited="" obsolete="false">
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<result status="valid" time="0.02"/>
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</proof>
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</goal>
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<goal name="WP_search_rec.2" expl="normal postcondition" sum="028a83456b4c241f325751fe75ff7cfe" proved="true" expanded="true">
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<proof prover="alt-ergo" timelimit="10" edited="" obsolete="true">
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<result status="valid" time="0.03"/>
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</proof>
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</goal>
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<goal name="WP_search_rec.3" expl="precondition" sum="58493f837aaa39a3890ba4c802e46109" proved="true" expanded="true">
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<proof prover="alt-ergo" timelimit="10" edited="" obsolete="true">
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<result status="valid" time="0.02"/>
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</proof>
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</goal>
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<goal name="WP_search_rec.4" expl="precondition" sum="99a18ac863afa1e6b4acfb99b4d0dae6" proved="true" expanded="true">
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<proof prover="alt-ergo" timelimit="10" edited="" obsolete="true">
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<result status="valid" time="0.02"/>
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</proof>
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</goal>
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<goal name="WP_search_rec.5" expl="precondition" sum="44a5c5d945aad7cfe2bcc7d4adf728f2" proved="true" expanded="true">
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<proof prover="alt-ergo" timelimit="10" edited="" obsolete="true">
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<result status="valid" time="0.02"/>
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</proof>
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</goal>
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<goal name="WP_search_rec.6" expl="normal postcondition" sum="699e81654d2a303c491cfab413eb3501" proved="true" expanded="true">
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<proof prover="alt-ergo" timelimit="10" edited="" obsolete="false">
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<result status="timeout" time="10.08"/>
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</proof>
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<proof prover="z3" timelimit="10" edited="" obsolete="false">
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<result status="valid" time="0.24"/>
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</proof>
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</goal>
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<goal name="WP_search_rec.7" expl="precondition" sum="9453d56ae3a433905973837d2fab5f11" proved="true" expanded="true">
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<proof prover="alt-ergo" timelimit="10" edited="" obsolete="true">
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<result status="valid" time="0.03"/>
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</proof>
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</goal>
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<goal name="WP_search_rec.8" expl="normal postcondition" sum="179d0569c8256c2c7e5e5730ab534329" proved="true" expanded="true">
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<proof prover="alt-ergo" timelimit="10" edited="" obsolete="true">
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<result status="valid" time="0.05"/>
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</proof>
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</goal>
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<goal name="WP_search_rec.9" expl="normal postcondition" sum="a5e3a9553c1dbb830d1064e70335f3a9" proved="true" expanded="true">
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<proof prover="alt-ergo" timelimit="10" edited="" obsolete="true">
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<result status="valid" time="0.02"/>
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</proof>
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</goal>
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</transf>
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</goal>
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</theory>
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</file>
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</why3session>
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