 ### new example: distance (in progress)

parent 0ec3c954
 (* Author: Jean-Christophe Filliatre (CNRS) Tool: Why3 (see http://why3.lri.fr/) The following problem was suggested to me by Ernie Cohen (Microsoft Research) We are given an integer N>0 and a function f such that 0 <= f(i) < i for all i in 0..N-1. We define a reachability as follows: each integer i in 0..N-1 can be reached from any integer in f(i)..i-1 in one step. The problem is then to compute the distance from 0 to N-1 in O(N). Even better, we want to compute this distance, say d, for all i in 0..N-1 and to build a predecessor function g such that i <-- g(i) <-- g(g(i)) <-- ... <-- 0 is the path of length d[i] from 0 to i. *) module Distance use import int.Int use import module ref.Refint use import module array.Array (* parameters [N] and [f] are introduced as logic symbols *) constant n: int axiom n_nonneg: 0 < n function f int: int axiom f_prop: forall k: int. 0 <= k < n -> 0 <= f k < k (* path from 0 to i of distance d *) inductive path int int = | path0: path 0 0 | paths: forall i: int. 0 <= i < n -> forall d j: int. path d j -> f i <= j < i -> path (d+1) i predicate distance (d i: int) = path d i /\ forall d': int. path d' i -> d <= d' (* function [g] is built in an array and then returned *) let distance () = { } let g = make n 0 in g <- -1; (* sentinel *) let d = make n 0 in d <- 0; (* redundant *) let count = ref 0 in (* ghost *) for i = 1 to n-1 do invariant { g = -1 /\ (forall k: int. 0 < k < i -> f k <= g[k] < k) /\ (forall k: int. 0 < k < i -> d[k] = d[g[k]] + 1) (* /\ (forall k: int. 0 <= k < i -> path d[k] k) *) } let j = ref (i-1) in while g[!j] >= f i do invariant { f i <= !j < i } variant { !j } j := g[!j]; incr count done; d[i] <- 1 + d[!j]; g[i] <- !j done; g { forall k: int. 0 < k < n -> f k <= result[k] < k } end (* Local Variables: compile-command: "why3ide distance.mlw" End: *)
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