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Commit 26d05648 authored by Jean-Christophe Filliâtre's avatar Jean-Christophe Filliâtre
Browse files

new example euler290

parent 4850f8fb
...@@ -10,10 +10,12 @@ ...@@ -10,10 +10,12 @@
In the following, we prove the correctness of a recursive function f In the following, we prove the correctness of a recursive function f
which takes m, a, and b as arguments and returns the number of such n. which takes m, a, and b as arguments and returns the number of such n.
Memoization must be added if one wants to solve the initial problem. Memoization must be added if one wants to solve the initial problem
in a reasonable amount of time.
(See for instance fib_memo.mlw for an example of memoized function.) *) (See for instance fib_memo.mlw for an example of memoized function.) *)
module M module Euler290
use import int.Int use import int.Int
use import ref.Ref use import ref.Ref
use import int.EuclideanDivision use import int.EuclideanDivision
...@@ -21,68 +23,70 @@ module M ...@@ -21,68 +23,70 @@ module M
function sum_digits int : int function sum_digits int : int
axiom Sum_digits_def : forall n : int. sum_digits n = axiom Sum_digits_def: forall n : int. sum_digits n =
if n <= 0 then 0 else sum_digits (div n 10) + mod n 10 if n <= 0 then 0 else sum_digits (div n 10) + mod n 10
(* the number of n st 0 <= n mod 10 < c and sd(n) = sd(137n+a)+b *) (* the number of n st 0 <= n mod 10 < c and sd(n) = sd(137n+a)+b *)
type int3 = (int,int,int) type int3 = (int,int,int)
predicate p (d:int3) (n:int) = predicate p (d: int3) (n: int) =
let (a,b,c) = d in let (a,b,c) = d in
0 <= mod n 10 < c /\ sum_digits n = sum_digits (137 * n + a) + b 0 <= mod n 10 < c /\ sum_digits n = sum_digits (137 * n + a) + b
clone int.NumOfParam as P with type param = int3, predicate pr = p clone int.NumOfParam as P with type param = int3, predicate pr = p
function solution(a b m : int) : int = P.num_of (a,b,10) 0 (power 10 m) function solution(a b m: int) : int = P.num_of (a,b,10) 0 (power 10 m)
(* short cut for the number of n st n mod 10 = c and ... *) (* short cut for the number of n st n mod 10 = c and ... *)
function num_of_modc (d:int3) (x y:int) : int = function num_of_modc (d: int3) (x y: int) : int =
let (a,b,c) = d in let (a,b,c) = d in
P.num_of (a,b,c+1) x y - P.num_of (a,b,c) x y P.num_of (a,b,c+1) x y - P.num_of (a,b,c) x y
(* helper lemmas *) (* helper lemmas *)
lemma Base: lemma Base:
forall a b : int. 0 <= a -> sum_digits a + b = 0 -> p (a,b,10) 0 forall a b: int. 0 <= a -> sum_digits a + b = 0 -> p (a,b,10) 0
lemma Empty: lemma Empty:
forall a b x y : int. P.num_of (a,b,0) x y = 0 forall a b x y: int. P.num_of (a,b,0) x y = 0
lemma Induc: lemma Induc:
forall a b c : int. 0 <= a -> 0 <= c < 10 -> forall a b c: int. 0 <= a -> 0 <= c < 10 ->
let x = 137 * c + a in let x = 137 * c + a in
let a' = div x 10 in let a' = div x 10 in
let b' = mod x 10 + b - c in let b' = mod x 10 + b - c in
forall m : int. m > 0 -> forall m: int. m > 0 ->
solution a' b' (m-1) = num_of_modc (a,b,c) 0 (power 10 m) solution a' b' (m-1) = num_of_modc (a,b,c) 0 (power 10 m)
(* code *)
use import int.ComputerDivision use import int.ComputerDivision
let rec sd n let rec sd (n: int) : int
requires { n >= 0 } ensures { result = sum_digits n } requires { n >= 0 }
ensures { result = sum_digits n }
variant { n }
= if n = 0 then 0 else sd (div n 10) + mod n 10 = if n = 0 then 0 else sd (div n 10) + mod n 10
(* f(m,a,b) = the number of 0 <= n < 10^m such that (* f(m,a,b) = the number of 0 <= n < 10^m such that
digitsum(137n+a) + b = digitsum(n). *) digitsum(137n + a) + b = digitsum(n). *)
let rec f m a b let rec f (m a b: int) : int
requires { 0 <= m /\ 0 <= a } requires { 0 <= m /\ 0 <= a }
ensures { result = solution a b m } ensures { result = solution a b m }
variant { m }
= if m = 0 then begin = if m = 0 then begin
(* only n = 0 is possible *) (* only n = 0 is possible *)
if sd a + b = 0 then 1 else 0 if sd a + b = 0 then 1 else 0
end else begin end else begin
let sum = ref 0 in let sum = ref 0 in
let c = ref 0 in for c = 0 to 9 do (* count the n st n mod 10 = c *)
while !c <= 9 do (* n = 10n' + c *) invariant { !sum = P.num_of (a,b,c) 0 (power 10 m) }
invariant { 0 <= !c <= 10 /\ !sum = P.num_of (a,b,!c) 0 (power 10 m) } let x = 137 * c + a in
variant { 10 - !c }
let x = 137 * !c + a in
let q = div x 10 in let q = div x 10 in
let r = mod x 10 in let r = mod x 10 in
sum := !sum + f (m-1) q (r + b - !c); sum := !sum + f (m-1) q (r + b - c)
c := !c + 1
done; done;
!sum !sum
end end
......
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<!DOCTYPE why3session PUBLIC "-//Why3//proof session v2//EN" "http://why3.lri.fr/why3session.dtd">
<why3session shape_version="4">
<prover
id="0"
name="Alt-Ergo"
version="0.95.2"/>
<prover
id="1"
name="CVC3"
version="2.4.1"/>
<file
name="../euler290.mlw"
verified="false"
expanded="true">
<theory
name="Euler290"
locfile="../euler290.mlw"
loclnum="17" loccnumb="7" loccnume="15"
verified="false"
expanded="true">
<goal
name="Base"
locfile="../euler290.mlw"
loclnum="49" loccnumb="8" loccnume="12"
sum="a800be5522f47c3fbb0d82c9e60e4aa0"
proved="true"
expanded="false"
shape="apaTuple3V0V1c10c0Iainfix =ainfix +asum_digitsV0V1c0Iainfix &lt;=c0V0F">
<proof
prover="0"
timelimit="10"
memlimit="1000"
obsolete="false"
archived="false">
<result status="valid" time="0.01"/>
</proof>
</goal>
<goal
name="Empty"
locfile="../euler290.mlw"
loclnum="52" loccnumb="8" loccnume="13"
sum="ec43ace9c5ab9d4d7394fe6c1c604544"
proved="true"
expanded="false"
shape="ainfix =anum_ofaTuple3V0V1c0V2V3c0F">
<proof
prover="0"
timelimit="10"
memlimit="1000"
obsolete="false"
archived="false">
<result status="valid" time="0.06"/>
</proof>
</goal>
<goal
name="Induc"
locfile="../euler290.mlw"
loclnum="55" loccnumb="8" loccnume="13"
sum="cf43691d3fcaec2492b10c5ac20df569"
proved="false"
expanded="true"
shape="ainfix =asolutionV4V5ainfix -V6c1anum_of_modcaTuple3V0V1V2c0apowerc10V6Iainfix &gt;V6c0FLainfix -ainfix +amodV3c10V1V2LadivV3c10Lainfix +ainfix *c137V2V0Iainfix &lt;V2c10Aainfix &lt;=c0V2Iainfix &lt;=c0V0F">
</goal>
<goal
name="WP_parameter sd"
locfile="../euler290.mlw"
loclnum="67" loccnumb="10" loccnume="12"
expl="VC for sd"
sum="2d339bcd0e2d3a0a64e418be1b8470c6"
proved="true"
expanded="false"
shape="iainfix =ainfix +asum_digitsV1amodV0c10asum_digitsV0Aainfix &gt;=V1c0Aainfix &lt;V1V0Aainfix &lt;=c0V0LadivV0c10ainfix =c0asum_digitsV0ainfix =V0c0Iainfix &gt;=V0c0F">
<label
name="expl:VC for sd"/>
<proof
prover="0"
timelimit="10"
memlimit="1000"
obsolete="false"
archived="false">
<result status="valid" time="0.22"/>
</proof>
</goal>
<goal
name="WP_parameter f"
locfile="../euler290.mlw"
loclnum="75" loccnumb="10" loccnume="11"
expl="VC for f"
sum="747029b96bc67621d69f2403c2b42456"
proved="true"
expanded="false"
shape="iainfix =V3asolutionV1V2V0Iainfix =V3anum_ofaTuple3V1V2ainfix +c9c1c0apowerc10V0Aainfix =V8anum_ofaTuple3V1V2ainfix +V4c1c0apowerc10V0Iainfix =V8ainfix +V3asolutionV6ainfix -ainfix +amodV5c10V2V4V7FAainfix &lt;=c0V6Aainfix &lt;=c0V7Aainfix &lt;V7V0Aainfix &lt;=c0V0Lainfix -V0c1LadivV5c10Lainfix +ainfix *c137V4V1Iainfix =V3anum_ofaTuple3V1V2V4c0apowerc10V0Iainfix &lt;=V4c9Aainfix &lt;=c0V4FFAainfix =c0anum_ofaTuple3V1V2c0c0apowerc10V0Iainfix &lt;=c0c9Aainfix =c0asolutionV1V2V0Iainfix &gt;c0c9ainfix =ic0c1ainfix =ainfix +asum_digitsV1V2c0asolutionV1V2V0Aainfix &gt;=V1c0ainfix =V0c0Iainfix &lt;=c0V1Aainfix &lt;=c0V0F">
<label
name="expl:VC for f"/>
<transf
name="split_goal_wp"
proved="true"
expanded="false">
<goal
name="WP_parameter f.1"
locfile="../euler290.mlw"
loclnum="75" loccnumb="10" loccnume="11"
expl="1. precondition"
sum="33549ab3d57be34931226ccd0b65225e"
proved="true"
expanded="false"
shape="preconditionainfix &gt;=V1c0Iainfix =V0c0Iainfix &lt;=c0V1Aainfix &lt;=c0V0F">
<label
name="expl:VC for f"/>
<proof
prover="0"
timelimit="10"
memlimit="1000"
obsolete="false"
archived="false">
<result status="valid" time="0.03"/>
</proof>
</goal>
<goal
name="WP_parameter f.2"
locfile="../euler290.mlw"
loclnum="75" loccnumb="10" loccnume="11"
expl="2. postcondition"
sum="7755175ab539038173c1ea75868f2bd8"
proved="true"
expanded="false"
shape="postconditionainfix =ic0c1ainfix =ainfix +asum_digitsV1V2c0asolutionV1V2V0Iainfix &gt;=V1c0Iainfix =V0c0Iainfix &lt;=c0V1Aainfix &lt;=c0V0F">
<label
name="expl:VC for f"/>
<proof
prover="0"
timelimit="10"
memlimit="1000"
obsolete="false"
archived="false">
<result status="valid" time="1.70"/>
</proof>
</goal>
<goal
name="WP_parameter f.3"
locfile="../euler290.mlw"
loclnum="75" loccnumb="10" loccnume="11"
expl="3. postcondition"
sum="0e636ef68fb8ecb78a3fc9776fed83e4"
proved="true"
expanded="false"
shape="postconditionainfix =c0asolutionV1V2V0Iainfix &gt;c0c9INainfix =V0c0Iainfix &lt;=c0V1Aainfix &lt;=c0V0F">
<label
name="expl:VC for f"/>
<proof
prover="0"
timelimit="10"
memlimit="1000"
obsolete="false"
archived="false">
<result status="valid" time="0.02"/>
</proof>
</goal>
<goal
name="WP_parameter f.4"
locfile="../euler290.mlw"
loclnum="75" loccnumb="10" loccnume="11"
expl="4. loop invariant init"
sum="d6b312ab912bcac4e24806c370c84aa3"
proved="true"
expanded="false"
shape="loop invariant initainfix =c0anum_ofaTuple3V1V2c0c0apowerc10V0Iainfix &lt;=c0c9INainfix =V0c0Iainfix &lt;=c0V1Aainfix &lt;=c0V0F">
<label
name="expl:VC for f"/>
<proof
prover="0"
timelimit="10"
memlimit="1000"
obsolete="false"
archived="false">
<result status="valid" time="0.03"/>
</proof>
</goal>
<goal
name="WP_parameter f.5"
locfile="../euler290.mlw"
loclnum="75" loccnumb="10" loccnume="11"
expl="5. variant decrease"
sum="03bcd47b917f7102016280a9d25451f3"
proved="true"
expanded="false"
shape="variant decreaseainfix &lt;V7V0Aainfix &lt;=c0V0Lainfix -V0c1LadivV5c10Lainfix +ainfix *c137V4V1Iainfix =V3anum_ofaTuple3V1V2V4c0apowerc10V0Iainfix &lt;=V4c9Aainfix &lt;=c0V4FFIainfix &lt;=c0c9INainfix =V0c0Iainfix &lt;=c0V1Aainfix &lt;=c0V0F">
<label
name="expl:VC for f"/>
<proof
prover="0"
timelimit="10"
memlimit="1000"
obsolete="false"
archived="false">
<result status="valid" time="0.02"/>
</proof>
</goal>
<goal
name="WP_parameter f.6"
locfile="../euler290.mlw"
loclnum="75" loccnumb="10" loccnume="11"
expl="6. precondition"
sum="8c4279522a5afd47f25feddb2bf6da50"
proved="true"
expanded="false"
shape="preconditionainfix &lt;=c0V6Aainfix &lt;=c0V7Lainfix -V0c1LadivV5c10Lainfix +ainfix *c137V4V1Iainfix =V3anum_ofaTuple3V1V2V4c0apowerc10V0Iainfix &lt;=V4c9Aainfix &lt;=c0V4FFIainfix &lt;=c0c9INainfix =V0c0Iainfix &lt;=c0V1Aainfix &lt;=c0V0F">
<label
name="expl:VC for f"/>
<proof
prover="0"
timelimit="10"
memlimit="1000"
obsolete="false"
archived="false">
<result status="valid" time="0.02"/>
</proof>
</goal>
<goal
name="WP_parameter f.7"
locfile="../euler290.mlw"
loclnum="75" loccnumb="10" loccnume="11"
expl="7. loop invariant preservation"
sum="bfa28bd4703499893c6377c9d72f5924"
proved="true"
expanded="false"
shape="loop invariant preservationainfix =V8anum_ofaTuple3V1V2ainfix +V4c1c0apowerc10V0Iainfix =V8ainfix +V3asolutionV6ainfix -ainfix +amodV5c10V2V4V7FIainfix &lt;=c0V6Aainfix &lt;=c0V7Lainfix -V0c1LadivV5c10Lainfix +ainfix *c137V4V1Iainfix =V3anum_ofaTuple3V1V2V4c0apowerc10V0Iainfix &lt;=V4c9Aainfix &lt;=c0V4FFIainfix &lt;=c0c9INainfix =V0c0Iainfix &lt;=c0V1Aainfix &lt;=c0V0F">
<label
name="expl:VC for f"/>
<transf
name="inline_goal"
proved="true"
expanded="false">
<goal
name="WP_parameter f.7.1"
locfile="../euler290.mlw"
loclnum="75" loccnumb="10" loccnume="11"
expl="1. loop invariant preservation"
sum="140d7d31a2c129f7f00367a760487594"
proved="true"
expanded="false"
shape="loop invariant preservationainfix =V8anum_ofaTuple3V1V2ainfix +V4c1c0apowerc10V0Iainfix =V8ainfix +V3asolutionV6ainfix -ainfix +amodV5c10V2V4V7FIainfix =c0V6Oainfix &lt;c0V6Aainfix =c0V7Oainfix &lt;c0V7Lainfix -V0c1LadivV5c10Lainfix +ainfix *c137V4V1Iainfix =V3anum_ofaTuple3V1V2V4c0apowerc10V0Iainfix =V4c9Oainfix &lt;V4c9Aainfix =c0V4Oainfix &lt;c0V4FFIainfix =c0c9Oainfix &lt;c0c9INainfix =V0c0Iainfix =c0V1Oainfix &lt;c0V1Aainfix =c0V0Oainfix &lt;c0V0F">
<label
name="expl:VC for f"/>
<proof
prover="1"
timelimit="60"
memlimit="1000"
obsolete="false"
archived="false">
<result status="valid" time="47.04"/>
</proof>
</goal>
</transf>
</goal>
<goal
name="WP_parameter f.8"
locfile="../euler290.mlw"
loclnum="75" loccnumb="10" loccnume="11"
expl="8. postcondition"
sum="ca337390e8505769e64d34de5d331c65"
proved="true"
expanded="false"
shape="postconditionainfix =V3asolutionV1V2V0Iainfix =V3anum_ofaTuple3V1V2ainfix +c9c1c0apowerc10V0FIainfix &lt;=c0c9INainfix =V0c0Iainfix &lt;=c0V1Aainfix &lt;=c0V0F">
<label
name="expl:VC for f"/>
<proof
prover="0"
timelimit="10"
memlimit="1000"
obsolete="false"
archived="false">
<result status="valid" time="0.01"/>
</proof>
</goal>
</transf>
</goal>
</theory>
</file>
</why3session>
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