在數組上證明函數的簡單屬性

Question

假設我們有以下帶注釋的C代碼：

int a[3] = {0};
/*@ requires \valid(a+(0..2));
    assigns a[0..2];
    ensures \forall int j; 0 <= j < 3 ==> (a[j] == j); */
int main() {
    int i = 0;
    /*@ loop assigns i, a[0..(i-1)];
        loop invariant inv1: 0 <= i <= 3;
        loop invariant inv2:
        \forall int k; 0 < k <= i ==> a[k-1] == k-1; */
    while (i < L) {
      a[i] = i;
      i++;
    }
    //@ assert final_i: i == 3;
    //@ assert final_a: a[1] == 1;
}

Z3和Alt-ergo都無法證明final_a和post條件是否有效； Z3也不能證明循環不變；

Alt-Ergo的輸出：

# ~/queue $ frama-c -wp -wp-prover alt-ergo test2.c 
[kernel] Parsing FRAMAC_SHARE/libc/__fc_builtin_for_normalization.i (no preprocessing)
[kernel] Parsing test2.c (with preprocessing)
[wp] Running WP plugin...
[wp] Collecting axiomatic usage
[wp] warning: Missing RTE guards
[wp] 14 goals scheduled
[wp] [Qed] Goal typed_main_pre : Valid
[wp] [Qed] Goal typed_main_loop_inv_inv1_established : Valid
[wp] [Qed] Goal typed_main_loop_inv_inv2_established : Valid
[wp] [Qed] Goal typed_main_assert_final_i : Valid
[wp] [Alt-Ergo] Goal typed_main_loop_inv_inv1_preserved : Valid (Qed:1ms) (14ms) (15)
[wp] [Qed] Goal typed_main_loop_assign_part1 : Valid
[wp] [Alt-Ergo] Goal typed_main_loop_inv_inv2_preserved : Valid (45ms) (51)
[wp] [Qed] Goal typed_main_loop_assign_part2 : Valid
[wp] [Qed] Goal typed_main_loop_assign_part3 : Valid
[wp] [Qed] Goal typed_main_assign_part1 : Valid
[wp] [Qed] Goal typed_main_assign_part3 : Valid
[wp] [Alt-Ergo] Goal typed_main_assign_part2 : Valid (20ms) (12)
[wp] [Alt-Ergo] Goal typed_main_assert_final_a : Unknown (3.5s)
[wp] [Alt-Ergo] Goal typed_main_post : Timeout (Qed:1ms)
[wp] Proved goals:   12 / 14
     Qed:             9  (0ms-1ms)
     Alt-Ergo:        3  (14ms-45ms) (51) (interrupted: 1) (unknown: 1)

Z3的輸出：

$ frama-c -wp -wp-prover z3 test2.c                                                                                      
[kernel] Parsing FRAMAC_SHARE/libc/__fc_builtin_for_normalization.i (no preprocessing)
[kernel] Parsing test2.c (with preprocessing)
[wp] Running WP plugin...
[wp] Collecting axiomatic usage
[wp] warning: Missing RTE guards
[wp] 14 goals scheduled
[wp] [Qed] Goal typed_main_pre : Valid
[wp] [Qed] Goal typed_main_loop_inv_inv1_established : Valid
[wp] [Qed] Goal typed_main_loop_inv_inv2_established : Valid
[wp] [Qed] Goal typed_main_assert_final_i : Valid
[wp] [z3] Goal typed_main_loop_inv_inv1_preserved : Valid (20ms)
[wp] [Qed] Goal typed_main_loop_assign_part1 : Valid
[wp] [Qed] Goal typed_main_loop_assign_part2 : Valid
[wp] [Qed] Goal typed_main_loop_assign_part3 : Valid
[wp] [Qed] Goal typed_main_assign_part1 : Valid
[wp] [z3] Goal typed_main_assert_final_a : Unknown (356ms)
[wp] [Qed] Goal typed_main_assign_part3 : Valid
[wp] [z3] Goal typed_main_post : Unknown (911ms)
[wp] [z3] Goal typed_main_assign_part2 : Valid (10ms)
[wp] [z3] Goal typed_main_loop_inv_inv2_preserved : Unknown (1.3s)
[wp] Proved goals:   11 / 14
     Qed:             9 
     z3:              2  (10ms-20ms) (unknown: 3)

什么東西少了？

Answer 1

如果將inv2循環不變式更改為：

        loop invariant inv2:
        \forall int k; 0 <= k < i ==> a[k] == k;

..然后Alt-Ergo可以證明final_a斷言和main（）后置條件。