NO Non-Confluence Proof

Non-Confluence Proof

by Hakusan

Input

The rewrite relation of the following TRS is considered.

0(2(1(x)))	→	0(4(0(1(4(5(1(5(0(1(x))))))))))
1(1(3(x)))	→	1(2(1(2(2(2(5(1(4(2(x))))))))))
0(0(0(3(x))))	→	4(2(4(0(2(5(3(3(4(5(x))))))))))
1(0(2(3(x))))	→	1(1(2(5(4(1(2(4(3(2(x))))))))))
1(3(5(3(x))))	→	3(5(4(5(2(4(3(2(5(4(x))))))))))
0(2(3(1(3(x)))))	→	1(0(1(2(1(3(1(3(1(2(x))))))))))
0(3(4(5(3(x)))))	→	2(0(2(5(1(2(4(4(5(5(x))))))))))
0(5(2(1(3(x)))))	→	2(4(2(5(2(4(3(0(2(4(x))))))))))
2(1(3(1(0(x)))))	→	0(1(4(5(1(5(5(2(3(0(x))))))))))
0(5(2(2(2(0(x))))))	→	2(5(4(3(0(2(5(1(2(1(x))))))))))
2(0(0(5(2(0(x))))))	→	4(0(4(2(1(4(4(4(0(1(x))))))))))
2(0(5(3(0(2(x))))))	→	2(5(3(5(1(4(5(0(0(2(x))))))))))
2(1(0(2(1(5(x))))))	→	2(5(4(1(3(2(2(5(4(5(x))))))))))
5(1(5(1(0(2(x))))))	→	4(5(0(0(4(3(1(1(0(4(x))))))))))
0(5(2(2(2(1(0(x)))))))	→	0(3(2(3(1(4(1(0(1(0(x))))))))))
0(5(3(5(3(1(5(x)))))))	→	0(1(3(4(0(1(4(5(1(5(x))))))))))
1(1(5(1(4(4(3(x)))))))	→	1(0(3(4(4(1(0(2(5(5(x))))))))))
1(3(2(3(0(5(3(x)))))))	→	1(4(0(1(5(4(0(3(2(5(x))))))))))
1(5(2(4(2(1(1(x)))))))	→	4(4(1(4(1(4(3(1(0(3(x))))))))))
2(0(2(0(2(1(0(x)))))))	→	2(0(2(3(4(2(4(4(4(0(x))))))))))
2(4(5(5(1(3(5(x)))))))	→	2(1(2(1(4(4(4(3(4(4(x))))))))))
3(0(0(5(5(2(1(x)))))))	→	0(2(4(3(2(3(2(1(0(3(x))))))))))
3(1(5(2(3(0(5(x)))))))	→	5(3(4(0(4(5(2(0(0(4(x))))))))))

Proof

1 Non-Joinable Fork

The system is not confluent due to the following forking derivations.

t₀ = 0(2(1(3(1(0(x1))))))

→₁ 0(0(1(4(5(1(5(5(2(3(0(x1)))))))))))

= t₁

t₀ = 0(2(1(3(1(0(x1))))))

→_ε 0(4(0(1(4(5(1(5(0(1(3(1(0(x1)))))))))))))

= t₁

The two resulting terms cannot be joined for the following reason:

When applying the cap-function on both terms (where variables may be treated like constants) then the resulting terms do not unify.

Tool configuration

Hakusan

version: 0.10