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module TriEx1 | |
use import int.Int | |
use import ref.Ref | |
use import array.Array | |
use import array.IntArraySorted | |
use import array.ArraySwap | |
use import array.ArrayPermut | |
use import array.ArrayEq | |
(* Prédicats et sous-programme utiles (et vus au cours 3) *) | |
(* | |
predicate sorted (a : array int) | |
predicate sorted_sub (a : array int) (i j : int) | |
predicate permut_all (a1 a2: array 'a) = | |
permut a1 a2 | |
let swap (a: array int) (i: int) (j: int) | |
requires { 0 <= i < length a /\ 0 <= j < length a } | |
ensures { exchange (old a) a i j } | |
= | |
let v = a[i] in | |
a[i] <- a[j]; | |
a[j] <- v | |
*) | |
let minimum (t:array int) (m n:int) : int | |
requires { 0 <= m < t.length && 0 <= n < t.length } | |
= | |
let indiceMinimum = ref m in | |
for i = m to n-1 do | |
invariant { m <= i <= n } | |
invariant { m <= !indiceMinimum <= n } | |
if t[!indiceMinimum] > t[i] then | |
indiceMinimum := i; | |
done; | |
!indiceMinimum | |
let tri_selection (t: array int) : unit | |
ensures { permut_all t (old (t))} | |
ensures { sorted t } | |
= | |
for i = 0 to t.length - 1 do | |
invariant { 0 <= i <= t.length } | |
invariant { sorted_sub t 0 i } | |
swap t i (minimum t i t.length) | |
done | |
let test1 () = | |
let t = make 3 0 in | |
t[0] <- 7; t[1] <- 3; t[2] <- 1; | |
tri_selection t; | |
assert { t [0] = 1 }; | |
assert { t [1] = 3 }; | |
assert { t [2] = 7 }; | |
t | |
let test2 () ensures { result.length = 6 } = | |
let t = make 6 0 in | |
t[0] <- 53; t[1] <- 91; t[2] <- 17; t[3] <- -5; | |
t[4] <- 413; t[5] <- 42; | |
tri_selection t; | |
assert { t[0] = -5 }; | |
assert { t[1] = 17 }; | |
assert { t[2] = 42 }; | |
assert { t[3] = 53 }; | |
assert { t[4] = 91 }; | |
assert { t[5] = 413 }; | |
t | |
end | |
module TriEx2 | |
use import int.Int | |
use import ref.Ref | |
use import array.Array | |
use import array.IntArraySorted | |
use import array.ArrayPermut | |
use import array.ArrayEq | |
(* | |
let tri_insertion (t: array int) : unit | |
ensures { permut_all t (old t)} | |
ensures { sorted t } | |
= | |
for i = 0 to t.length -1 do | |
invariant { 0 <= i <= t.length -1 } | |
invariant { sorted_sub t 0 i } | |
let x = t[i]; | |
let j = i; | |
while j > 0 && t[j-1] > x do | |
invariant { 0 < j <= i } | |
t[j] <- t[j-1]; | |
j := !j-1; | |
done; | |
t[j] <- x; | |
done; | |
*) | |
end |
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