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Why3
why3
Commits
8a6f9e75
Commit
8a6f9e75
authored
Jun 06, 2018
by
Jean-Christophe Filliâtre
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tortoise and hare example totally revamped
this is much closer now to TAOCP, vol 2, exercise 6 page 7
parent
626ca4d8
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examples/tortoise_and_hare.mlw
examples/tortoise_and_hare.mlw
+145
-71
examples/tortoise_and_hare/why3session.xml
examples/tortoise_and_hare/why3session.xml
+357
-17
examples/tortoise_and_hare/why3shapes.gz
examples/tortoise_and_hare/why3shapes.gz
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examples/tortoise_and_hare.mlw
View file @
8a6f9e75
...
...
@@ -3,88 +3,162 @@
See The Art of Computer Programming, vol 2, exercise 6 page 7. *)
module TortoiseAndHare
module TortoiseAndHare
Algorithm
use import int.Int
use int.Iter
use import int.EuclideanDivision
use import int.Iter
use import seq.Seq
use import seq.Distinct
use import ref.Refint
use import pigeon.Pigeonhole
type t
val predicate eq (x y : t)
ensures { result <-> x = y }
val function f int : int
(* let f be a function from t to t *)
val function f t : t
(* f maps 0..m-1 to 0..m-1 *)
constant m: int
axiom m_positive: m > 0
axiom f_range: forall x. 0 <= x < m -> 0 <= f x < m
(* given some initial value x0 *)
val constant x0: t
val constant x0: int
ensures { 0 <= result < m }
(* we can build the infinite sequence defined by x{i+1} = f(xi) *)
function x (i: int): t = Iter.iter f i x0
(* let use assume now that the sequence (x_i) has finitely many distinct
values (e.g. f is an integer function with values in 0..m)
it means that there exists values lambda and mu such that
(1) x0, x1, ... x{mu+lambda-1} are all distinct,
and
(2) x{n+lambda} = x_n when n >= mu. *)
(* lambda and mu are skomlemized *)
constant mu : int
constant lambda : int
(* they are axiomatized as follows *)
axiom mu_range: 0 <= mu
axiom lambda_range: 1 <= lambda
axiom distinct:
forall i j: int. 0 <= i < mu+lambda -> 0 <= j < mu+lambda ->
i <> j -> x i <> x j
axiom cycle: forall n: int. mu <= n -> x (n + lambda) = x n
lemma cycle_induction:
forall n: int. mu <= n -> forall k: int. 0 <= k -> x (n + lambda * k) = x n
(* Now comes the code.
Two references, tortoise and hare, traverses the sequence (xi)
at speed 1 and 2 respectively. We stop whenever they are equal.
The challenge is to prove termination.
Actually, we even prove that the code runs in time O(lambda + mu). *)
use import ref.Ref
use import relations.WellFounded
(* the minimal distance between x i and x j *)
function dist int int : int
(* it is defined as soon as i,j >= mu *)
axiom dist_def: forall i j: int. mu <= i -> mu <= j ->
dist i j >= 0 /\
x (i + dist i j) = x j /\
forall k: int. 0 <= k -> x (i + k) = x j -> dist i j <= k
predicate rel (t2 t1: t) =
exists i: int. t1 = x i /\ t2 = x (i+1) /\ 1 <= i <= mu + lambda /\
(i >= mu -> dist (2*i+2) (i+1) < dist (2*i) i)
axiom wfrel: well_founded rel
let tortoise_hare () =
function x (i: int): int = iter f i x0
let rec lemma x_in_range (n: int)
requires { 0 <= n } ensures { 0 <= x n < m }
variant { n }
= if n > 0 then x_in_range (n - 1)
(* First, we prove the existence of mu and lambda, with a ghost program.
Starting with x0, we repeteadly apply f, storing the elements in
a sequence, until we find a repetition. *)
let ghost periodicity () : (int, int)
returns { mu, lambda ->
0 <= mu < m /\ 1 <= lambda <= m /\ mu + lambda <= m /\
x (mu + lambda) = x mu /\
forall i j. 0 <= i < j < mu + lambda -> x i <> x j
}
= let s = ref (singleton x0) in
let cur = ref x0 in
for k = 1 to m do
invariant { 1 <= length !s = k <= m /\ !cur = !s[length !s - 1] }
invariant { forall i. 0 <= i < length !s -> !s[i] = x i }
invariant { distinct !s }
cur := f !cur;
(* look for a repetition *)
for mu = 0 to length !s - 1 do
invariant { forall i. 0 <= i < mu -> !s[i] <> !cur }
if !cur = !s[mu] then return (mu, length !s - mu)
done;
s := snoc !s !cur;
if k = m then pigeonhole (m+1) m (pure { fun i -> !s[i] })
done;
absurd
(* Then we can state the condition for two elements to be equal. *)
let lemma equality (mu lambda: int)
requires { 0 <= mu < m /\ 1 <= lambda <= m /\ mu + lambda <= m /\
x (mu + lambda) = x mu }
requires { forall i j. 0 <= i < j < mu + lambda -> x i <> x j }
ensures { forall r n. r > n >= 0 ->
x r = x n <-> n >= mu /\ exists k. k >= 1 /\ r-n = k*lambda }
= let rec lemma plus_lambda (n: int) variant { n }
requires { mu <= n }
ensures { x (n + lambda) = x n } =
if n > mu then plus_lambda (n - 1) in
let rec lemma plus_lambdas (n k: int) variant { k }
requires { mu <= n } requires { 0 <= k }
ensures { x (n + k * lambda) = x n } =
if k > 0 then begin
plus_lambdas n (k - 1); plus_lambda (n + (k - 1) * lambda) end in
let decomp (i: int) : (int, int)
requires { i >= mu }
returns { qi,mi -> i = mu + qi * lambda + mi && qi >= 0 &&
0 <= mi < lambda && x i = x (mu + mi) } =
let qi = div (i - mu) lambda in
let mi = mod (i - mu) lambda in
plus_lambdas (mu + mi) qi;
qi, mi in
let lemma equ (r n: int)
requires { r > n >= 0 } requires { x r = x n }
ensures { n >= mu /\ exists k. k >= 1 /\ r-n = k*lambda } =
if n < mu then (if r >= mu then let _ = decomp r in absurd)
else begin
let qr,mr = decomp r in let qn, mn = decomp n in
assert { mr = mn };
assert { r-n = (qr-qn) * lambda }
end in
()
(* Finally, we implement the tortoise and hare algorithm that computes
the values of mu and lambda in linear time and constant space *)
let tortoise_and_hare () : (int, int)
returns { mu, lambda ->
0 <= mu < m /\ 1 <= lambda <= m /\ mu + lambda <= m /\
x (mu + lambda) = x mu /\
forall i j. 0 <= i < j < mu + lambda -> x i <> x j
}
= let mu, lambda = periodicity () in
equality mu lambda;
(* the first loop implements the tortoise and hare,
and finds the smallest n >= 1 such that x n = x (2n) *)
let tortoise = ref (f x0) in
let hare = ref (f (f x0)) in
while not (eq !tortoise !hare) do
let n = ref 1 in
while !tortoise <> !hare do
invariant {
exists t: int.
1 <= t <= mu+lambda /\ !tortoise = x t /\ !hare = x (2 * t) /\
forall i: int. 1 <= i < t -> x i <> x (2*i) }
variant { !tortoise with rel }
1 <= !n <= mu+lambda /\ !tortoise = x !n /\ !hare = x (2 * !n) /\
forall i. 1 <= i < !n -> x i <> x (2*i) }
variant { mu + lambda - !n }
tortoise := f !tortoise;
hare := f (f !hare)
done
(* TODO: code to find out lambda and mu. See wikipedia. *)
hare := f (f !hare);
incr n;
if !n > mu + lambda then begin
let m = lambda - mod mu lambda in
let i = mu + m in
assert { 2*i - i = (div mu lambda + 1) * lambda };
absurd
end
done;
let n = !n in
assert { n >= mu };
assert { exists k. k >= 1 /\ n = k * lambda >= 1 };
assert { forall j. j >= mu -> x j = x (j + n) };
let xn = !tortoise in
(* a first loop to find mu *)
let i = ref 0 in
let xi = ref x0 in (* = x i *)
let xni = ref xn in (* = x (n+i) *)
while !xi <> !xni do
invariant { 0 <= !i <= mu }
invariant { !xi = x !i /\ !xni = x (n + !i) }
invariant { forall j. 0 <= j < !i -> x j <> x (n + j) }
variant { mu - !i }
xi := f !xi;
xni := f !xni;
incr i;
done;
let m = !i in
assert { m = mu };
(* and a second loop to find lambda
(this is slightly less efficient than the argument in TAOCP,
but since the first loop is already O(mu+lambda), using two loops
respectively O(mu) and O(lambda) is not a problem). *)
i := 1;
xni := f xn;
while !xni <> xn do
invariant { !xni = x (n + !i) }
invariant { forall j. 1 <= j < !i -> x (n + j) <> x n }
invariant { 1 <= !i <= lambda }
variant { lambda - !i }
xni := f !xni;
incr i
done;
assert { !i = lambda };
m, !i
end
examples/tortoise_and_hare/why3session.xml
View file @
8a6f9e75
...
...
@@ -2,30 +2,370 @@
<!DOCTYPE why3session PUBLIC "-//Why3//proof session v5//EN"
"http://why3.lri.fr/why3session.dtd">
<why3session
shape_version=
"4"
>
<prover
id=
"
0"
name=
"Coq"
version=
"8.7.1"
timelimit=
"5
"
steplimit=
"0"
memlimit=
"1000"
/>
<prover
id=
"
1"
name=
"Coq"
version=
"8.7.2"
timelimit=
"5"
steplimit=
"0"
memlimit=
"
0"
/>
<prover
id=
"
3"
name=
"Alt-Ergo"
version=
"1.30"
timelimit=
"5
"
steplimit=
"0"
memlimit=
"1000"
/>
<prover
id=
"
4"
name=
"Z3"
version=
"4.5.0"
timelimit=
"5
"
steplimit=
"0"
memlimit=
"1000"
/>
<prover
id=
"
2"
name=
"Alt-Ergo"
version=
"2.0.0"
timelimit=
"1
"
steplimit=
"0"
memlimit=
"1000"
/>
<prover
id=
"
5"
name=
"Z3"
version=
"4.4.0"
timelimit=
"1"
steplimit=
"0"
memlimit=
"100
0"
/>
<prover
id=
"
6"
name=
"CVC4"
version=
"1.5"
timelimit=
"1
"
steplimit=
"0"
memlimit=
"1000"
/>
<prover
id=
"
7"
name=
"Eprover"
version=
"1.8-001"
timelimit=
"16
"
steplimit=
"0"
memlimit=
"1000"
/>
<file
name=
"../tortoise_and_hare.mlw"
proved=
"true"
>
<theory
name=
"TortoiseAndHare"
proved=
"true"
>
<theory
name=
"TortoiseAndHare
Algorithm
"
proved=
"true"
>
<goal
name=
"VC x0"
expl=
"VC for x0"
proved=
"true"
>
<transf
name=
"split_goal_right"
proved=
"true"
>
<proof
prover=
"6"
><result
status=
"valid"
time=
"0.02"
/></proof>
</goal>
<goal
name=
"VC x_in_range"
expl=
"VC for x_in_range"
proved=
"true"
>
<proof
prover=
"2"
><result
status=
"valid"
time=
"0.36"
steps=
"311"
/></proof>
</goal>
<goal
name=
"VC periodicity"
expl=
"VC for periodicity"
proved=
"true"
>
<transf
name=
"split_vc"
proved=
"true"
>
<goal
name=
"VC periodicity.0"
expl=
"loop invariant init"
proved=
"true"
>
<proof
prover=
"6"
><result
status=
"valid"
time=
"0.02"
/></proof>
</goal>
<goal
name=
"VC periodicity.1"
expl=
"loop invariant init"
proved=
"true"
>
<proof
prover=
"6"
><result
status=
"valid"
time=
"0.18"
/></proof>
</goal>
<goal
name=
"VC periodicity.2"
expl=
"loop invariant init"
proved=
"true"
>
<proof
prover=
"6"
><result
status=
"valid"
time=
"0.06"
/></proof>
</goal>
<goal
name=
"VC periodicity.3"
expl=
"loop invariant init"
proved=
"true"
>
<proof
prover=
"2"
><result
status=
"valid"
time=
"0.01"
steps=
"16"
/></proof>
</goal>
<goal
name=
"VC periodicity.4"
expl=
"VC for periodicity"
proved=
"true"
>
<proof
prover=
"6"
><result
status=
"valid"
time=
"0.19"
/></proof>
</goal>
<goal
name=
"VC periodicity.5"
expl=
"loop invariant preservation"
proved=
"true"
>
<proof
prover=
"6"
><result
status=
"valid"
time=
"0.07"
/></proof>
</goal>
<goal
name=
"VC periodicity.6"
expl=
"precondition"
proved=
"true"
>
<proof
prover=
"6"
><result
status=
"valid"
time=
"0.03"
/></proof>
</goal>
<goal
name=
"VC periodicity.7"
expl=
"precondition"
proved=
"true"
>
<proof
prover=
"6"
><result
status=
"valid"
time=
"0.19"
/></proof>
</goal>
<goal
name=
"VC periodicity.8"
expl=
"loop invariant preservation"
proved=
"true"
>
<proof
prover=
"6"
><result
status=
"valid"
time=
"1.24"
/></proof>
</goal>
<goal
name=
"VC periodicity.9"
expl=
"loop invariant preservation"
proved=
"true"
>
<proof
prover=
"2"
><result
status=
"valid"
time=
"0.92"
steps=
"823"
/></proof>
</goal>
<goal
name=
"VC periodicity.10"
expl=
"loop invariant preservation"
proved=
"true"
>
<proof
prover=
"6"
><result
status=
"valid"
time=
"0.16"
/></proof>
</goal>
<goal
name=
"VC periodicity.11"
expl=
"VC for periodicity"
proved=
"true"
>
<proof
prover=
"6"
><result
status=
"valid"
time=
"0.05"
/></proof>
</goal>
<goal
name=
"VC periodicity.12"
expl=
"unreachable point"
proved=
"true"
>
<proof
prover=
"6"
><result
status=
"valid"
time=
"0.02"
/></proof>
</goal>
<goal
name=
"VC periodicity.13"
expl=
"unreachable point"
proved=
"true"
>
<proof
prover=
"6"
><result
status=
"valid"
time=
"0.06"
/></proof>
</goal>
</transf>
</goal>
<goal
name=
"cycle_induction"
proved=
"true"
>
<proof
prover=
"0"
edited=
"tortoise_and_hare_WP_TortoiseAndHare_cycle_induction_1.v"
><result
status=
"valid"
time=
"0.36"
/></proof>
<goal
name=
"VC equality"
expl=
"VC for equality"
proved=
"true"
>
<transf
name=
"split_vc"
proved=
"true"
>
<goal
name=
"VC equality.0"
expl=
"variant decrease"
proved=
"true"
>
<proof
prover=
"6"
><result
status=
"valid"
time=
"0.02"
/></proof>
</goal>
<goal
name=
"VC equality.1"
expl=
"precondition"
proved=
"true"
>
<proof
prover=
"6"
><result
status=
"valid"
time=
"0.03"
/></proof>
</goal>
<goal
name=
"VC equality.2"
expl=
"postcondition"
proved=
"true"
>
<proof
prover=
"6"
><result
status=
"valid"
time=
"0.03"
/></proof>
</goal>
<goal
name=
"VC equality.3"
expl=
"variant decrease"
proved=
"true"
>
<proof
prover=
"6"
><result
status=
"valid"
time=
"0.07"
/></proof>
</goal>
<goal
name=
"VC equality.4"
expl=
"precondition"
proved=
"true"
>
<proof
prover=
"6"
><result
status=
"valid"
time=
"0.05"
/></proof>
</goal>
<goal
name=
"VC equality.5"
expl=
"precondition"
proved=
"true"
>
<proof
prover=
"6"
><result
status=
"valid"
time=
"0.05"
/></proof>
</goal>
<goal
name=
"VC equality.6"
expl=
"precondition"
proved=
"true"
>
<transf
name=
"assert"
proved=
"true"
arg1=
"((k-1)*lambda >= 0)"
>
<goal
name=
"VC equality.6.0"
proved=
"true"
>
<proof
prover=
"7"
><result
status=
"valid"
time=
"2.45"
/></proof>
</goal>
<goal
name=
"VC equality.6.1"
expl=
"precondition"
proved=
"true"
>
<proof
prover=
"6"
><result
status=
"valid"
time=
"0.03"
/></proof>
</goal>
</transf>
</goal>
<goal
name=
"VC equality.7"
expl=
"postcondition"
proved=
"true"
>
<proof
prover=
"6"
><result
status=
"valid"
time=
"0.06"
/></proof>
</goal>
<goal
name=
"VC equality.8"
expl=
"precondition"
proved=
"true"
>
<proof
prover=
"6"
><result
status=
"valid"
time=
"0.02"
/></proof>
</goal>
<goal
name=
"VC equality.9"
expl=
"precondition"
proved=
"true"
>
<transf
name=
"split_all_full"
proved=
"true"
>
<goal
name=
"VC equality.9.0"
expl=
"precondition"
proved=
"true"
>
<proof
prover=
"6"
><result
status=
"valid"
time=
"0.09"
/></proof>
</goal>
</transf>
</goal>
<goal
name=
"VC equality.10"
expl=
"precondition"
proved=
"true"
>
<transf
name=
"split_vc"
proved=
"true"
>
<goal
name=
"VC equality.10.0"
expl=
"precondition"
proved=
"true"
>
<proof
prover=
"6"
><result
status=
"valid"
time=
"0.05"
/></proof>
</goal>
</transf>
</goal>
<goal
name=
"VC equality.11"
expl=
"precondition"
proved=
"true"
>
<proof
prover=
"5"
><result
status=
"valid"
time=
"0.02"
/></proof>
</goal>
<goal
name=
"VC equality.12"
expl=
"postcondition"
proved=
"true"
>
<proof
prover=
"5"
><result
status=
"valid"
time=
"0.18"
/></proof>
</goal>
<goal
name=
"VC equality.13"
expl=
"precondition"
proved=
"true"
>
<proof
prover=
"6"
><result
status=
"valid"
time=
"0.04"
/></proof>
</goal>
<goal
name=
"VC equality.14"
expl=
"unreachable point"
proved=
"true"
>
<transf
name=
"split_all_full"
proved=
"true"
>
<goal
name=
"VC equality.14.0"
expl=
"unreachable point"
proved=
"true"
>
<proof
prover=
"6"
><result
status=
"valid"
time=
"0.04"
/></proof>
</goal>
</transf>
</goal>
<goal
name=
"VC equality.15"
expl=
"precondition"
proved=
"true"
>
<proof
prover=
"6"
><result
status=
"valid"
time=
"0.03"
/></proof>
</goal>
<goal
name=
"VC equality.16"
expl=
"precondition"
proved=
"true"
>
<transf
name=
"split_vc"
proved=
"true"
>
<goal
name=
"VC equality.16.0"
expl=
"precondition"
proved=
"true"
>
<proof
prover=
"7"
><result
status=
"valid"
time=
"0.03"
/></proof>
</goal>
</transf>
</goal>
<goal
name=
"VC equality.17"
expl=
"assertion"
proved=
"true"
>
<proof
prover=
"6"
><result
status=
"valid"
time=
"0.03"
/></proof>
</goal>
<goal
name=
"VC equality.18"
expl=
"assertion"
proved=
"true"
>
<proof
prover=
"5"
><result
status=
"valid"
time=
"0.21"
/></proof>
</goal>
<goal
name=
"VC equality.19"
expl=
"postcondition"
proved=
"true"
>
<proof
prover=
"2"
><result
status=
"valid"
time=
"0.03"
steps=
"73"
/></proof>
</goal>
<goal
name=
"VC equality.20"
expl=
"postcondition"
proved=
"true"
>
<transf
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<goal
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<proof
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<proof
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<goal
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<transf
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"true"
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<goal
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<proof
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<goal
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<proof
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<goal
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<transf
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<goal
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<proof
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<proof
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<transf
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<goal
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<proof
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<goal
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<goal
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<proof
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<proof
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<proof
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<proof
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"2"
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<goal
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<proof
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<proof
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<goal
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<transf
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<goal
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<proof
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<proof
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<proof
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<proof
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<proof
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<proof
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<goal
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<proof
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<proof
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<proof
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<proof
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<proof
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<proof
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<proof
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>
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"case"
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"true"
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<goal
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>
<transf
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<goal
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<proof
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<proof
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<proof
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"5"
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>
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"replace"
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>
<goal
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>
<proof
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<proof
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<proof
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<proof
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<proof
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<proof
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<proof
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<proof
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<proof
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</goal>