library list.Elements changed + new example: coincidence_count_list

parent 4e666906
(** Coincidence count
Exercise proposed by Rustan Leino at Dagstuhl seminar 16131, March 2016
You are given two sequences of integers, sorted in increasing
order and without duplicate elements, and you count the number of
elements that appear in both sequences (in linear time and constant
space).
See also coincidence_count for a version using arrays.
Authors: Jean-Christophe Filliâtre (CNRS)
*)
module CoincidenceCount
use import list.List
use import set.Fset
use import list.Elements
use list.Mem as L
use import list.Length
use import int.Int
clone export list.Sorted
with type t = int, predicate le = (<), goal Transitive.Trans
let rec coincidence_count (a b: list int) : set int
requires { sorted a }
requires { sorted b }
ensures { result == inter (elements a) (elements b) }
variant { length a + length b }
=
match a, b with
| Cons ha ta, Cons hb tb ->
if ha = hb then
add ha (coincidence_count ta tb)
else if ha < hb then
coincidence_count ta b
else
coincidence_count a tb
| _ ->
empty
end
end
<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE why3session PUBLIC "-//Why3//proof session v5//EN"
"http://why3.lri.fr/why3session.dtd">
<why3session shape_version="4">
<prover id="0" name="Alt-Ergo" version="1.01" timelimit="6" steplimit="0" memlimit="1000"/>
<prover id="1" name="CVC4" version="1.4" timelimit="6" steplimit="0" memlimit="1000"/>
<prover id="3" name="Eprover" version="1.8-001" timelimit="6" steplimit="0" memlimit="1000"/>
<file name="../coincidence_count_list.mlw" expanded="true">
<theory name="CoincidenceCount" sum="4d4c890e56bb604a55ec2e21b7ac9a92" expanded="true">
<goal name="Transitive.Trans" expanded="true">
<proof prover="0"><result status="valid" time="0.01" steps="3"/></proof>
</goal>
<goal name="WP_parameter coincidence_count" expl="VC for coincidence_count" expanded="true">
<transf name="split_goal_wp" expanded="true">
<goal name="WP_parameter coincidence_count.1" expl="1. variant decrease">
<proof prover="0"><result status="valid" time="0.01" steps="10"/></proof>
</goal>
<goal name="WP_parameter coincidence_count.2" expl="2. precondition">
<proof prover="0"><result status="valid" time="0.00" steps="18"/></proof>
</goal>
<goal name="WP_parameter coincidence_count.3" expl="3. precondition">
<proof prover="0"><result status="valid" time="0.01" steps="18"/></proof>
</goal>
<goal name="WP_parameter coincidence_count.4" expl="4. postcondition">
<proof prover="0"><result status="valid" time="1.98" steps="2625"/></proof>
<proof prover="3"><result status="timeout" time="5.99"/></proof>
</goal>
<goal name="WP_parameter coincidence_count.5" expl="5. variant decrease">
<proof prover="0"><result status="valid" time="0.01" steps="11"/></proof>
</goal>
<goal name="WP_parameter coincidence_count.6" expl="6. precondition">
<proof prover="0"><result status="valid" time="0.01" steps="19"/></proof>
</goal>
<goal name="WP_parameter coincidence_count.7" expl="7. precondition">
<proof prover="0"><result status="valid" time="0.00" steps="8"/></proof>
</goal>
<goal name="WP_parameter coincidence_count.8" expl="8. postcondition" expanded="true">
<proof prover="0"><result status="valid" time="2.36" steps="4210"/></proof>
</goal>
<goal name="WP_parameter coincidence_count.9" expl="9. variant decrease">
<proof prover="0"><result status="valid" time="0.01" steps="11"/></proof>
</goal>
<goal name="WP_parameter coincidence_count.10" expl="10. precondition">
<proof prover="0"><result status="valid" time="0.01" steps="8"/></proof>
</goal>
<goal name="WP_parameter coincidence_count.11" expl="11. precondition">
<proof prover="0"><result status="valid" time="0.01" steps="19"/></proof>
</goal>
<goal name="WP_parameter coincidence_count.12" expl="12. postcondition" expanded="true">
<proof prover="0"><result status="valid" time="3.56" steps="6215"/></proof>
</goal>
<goal name="WP_parameter coincidence_count.13" expl="13. postcondition">
<proof prover="3"><result status="valid" time="0.04"/></proof>
</goal>
<goal name="WP_parameter coincidence_count.14" expl="14. postcondition">
<proof prover="0"><result status="valid" time="0.02" steps="26"/></proof>
<proof prover="1"><result status="valid" time="0.03"/></proof>
</goal>
</transf>
</goal>
</theory>
</file>
</why3session>
......@@ -43,17 +43,17 @@ end
theory Elements
use import List
use Mem
use set.Fset as FSet
function elements (list 'a) : FSet.set 'a
use import Mem
use set.Fset as S
axiom elements_mem:
forall l:list 'a, x:'a.
Mem.mem x l <-> FSet.mem x (elements l)
function elements (l: list 'a) : S.set 'a =
match l with
| Nil -> S.empty
| Cons x r -> S.add x (elements r)
end
lemma elements_Nil:
elements (Nil : list 'a) = FSet.empty
lemma elements_mem:
forall x: 'a, l: list 'a. mem x l <-> S.mem x (elements l)
end
......
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