/*
Author: Yihao Yan

See challenge 1 of: http://etaps2015.verifythis.org/challenges

-----------------
Problem description:

Verify a function isRelaxedPrefix determining if a list _pat_ (for
pattern) is a relaxed prefix of another list _a_.

The relaxed prefix property holds iff _pat_ is a prefix of _a_ after
removing at most one element from _pat_.

examples:
pat = {1,3}   is a relaxed prefix of a = {1,3,2,3} (standard prefix)
pat = {1,2,3} is a relaxed prefix of a = {1,3,2,3} (remove 2 from pat)
pat = {1,2,4} is not a relaxed prefix of a = {1,3,2,3}.

-----------------
Verification Task:

Implement the isRelaxedPrefix function which takes two arrays and their length
as input and verify that it behaves as described.

-----------------
Result:

For any array X1 with length less than 5 and any array X2 with length less than 4, 
function isRelaxedPrefix will tell if X1 can become a prefix of X2 by removing at 
most one element. Therefore, the isRelaxedPrefix behaves correctly.

-----------------
command: civl verify relaxedPrefix.c

*/

#include <stdio.h>
#include <stdbool.h>
#include <civlc.cvh>

$input int N1_BOUND = 4;
$input int N2_BOUND = 3;
$input int n1;
$input int n2;
$assume (n2>0 && n2 < N2_BOUND);
$assume (n1>0 && n1 < N1_BOUND);
$input int X2[n2];
$input int X1[n1];

bool isRelaxedPrefix(int* pat, int patLen, int* a, int aLen) {
  int shift = 0;
  int i;

  if(patLen > aLen+1) return false;

  if(aLen == 0) return true;

  for(i=0; i<patLen; i++) {
     if(pat[i] != a[i-shift]) {
      if(shift == 0)
        shift = 1;
      else
        return false;
     }
     if(i == aLen - 1 && shift == 0) return true;
   }
  return true;
}

void main() {
	bool result = isRelaxedPrefix(X1, n1, X2, n2);

  if(n1 > n2+1) {
    $assert(!result);
  } else if(n1 == n2+1) {
    $assert( result ==
	     ($exists (int k: 0 .. n1-1)
        (
	 ($forall (int i: 0 .. k-1) X1[i] == X2[i]) &&
	 ($forall (int i: k+1 .. n1-1) X1[i] == X2[i-1])
        )
      )
    );
  } else {
    $assert(result ==
	    ($exists (int k: 0 .. n1)
        (
	 ($forall (int i: 0 .. k-1) X1[i] == X2[i]) &&
	 ($forall (int i: k+1 .. n1-1) X1[i] == X2[i-1])
        )
      )
    );
  }
}