YES(O(1),O(n^1)) 39.97/13.44 YES(O(1),O(n^1)) 39.97/13.44 39.97/13.44 We are left with following problem, upon which TcT provides the 39.97/13.44 certificate YES(O(1),O(n^1)). 39.97/13.44 39.97/13.44 Strict Trs: 39.97/13.44 { U11(tt(), N) -> activate(N) 39.97/13.44 , activate(X) -> X 39.97/13.44 , activate(n__0()) -> 0() 39.97/13.44 , activate(n__plus(X1, X2)) -> plus(X1, X2) 39.97/13.44 , activate(n__isNat(X)) -> isNat(X) 39.97/13.44 , activate(n__s(X)) -> s(X) 39.97/13.44 , activate(n__x(X1, X2)) -> x(X1, X2) 39.97/13.44 , U21(tt(), M, N) -> s(plus(activate(N), activate(M))) 39.97/13.44 , s(X) -> n__s(X) 39.97/13.44 , plus(X1, X2) -> n__plus(X1, X2) 39.97/13.44 , plus(N, s(M)) -> U21(and(isNat(M), n__isNat(N)), M, N) 39.97/13.44 , plus(N, 0()) -> U11(isNat(N), N) 39.97/13.44 , U31(tt()) -> 0() 39.97/13.44 , 0() -> n__0() 39.97/13.44 , U41(tt(), M, N) -> plus(x(activate(N), activate(M)), activate(N)) 39.97/13.44 , x(X1, X2) -> n__x(X1, X2) 39.97/13.44 , x(N, s(M)) -> U41(and(isNat(M), n__isNat(N)), M, N) 39.97/13.44 , x(N, 0()) -> U31(isNat(N)) 39.97/13.44 , and(tt(), X) -> activate(X) 39.97/13.44 , isNat(X) -> n__isNat(X) 39.97/13.44 , isNat(n__0()) -> tt() 39.97/13.44 , isNat(n__plus(V1, V2)) -> 39.97/13.44 and(isNat(activate(V1)), n__isNat(activate(V2))) 39.97/13.44 , isNat(n__s(V1)) -> isNat(activate(V1)) 39.97/13.44 , isNat(n__x(V1, V2)) -> 39.97/13.44 and(isNat(activate(V1)), n__isNat(activate(V2))) } 39.97/13.44 Obligation: 39.97/13.44 innermost runtime complexity 39.97/13.44 Answer: 39.97/13.44 YES(O(1),O(n^1)) 39.97/13.44 39.97/13.44 Arguments of following rules are not normal-forms: 39.97/13.44 39.97/13.44 { plus(N, s(M)) -> U21(and(isNat(M), n__isNat(N)), M, N) 39.97/13.44 , plus(N, 0()) -> U11(isNat(N), N) 39.97/13.44 , x(N, s(M)) -> U41(and(isNat(M), n__isNat(N)), M, N) 39.97/13.44 , x(N, 0()) -> U31(isNat(N)) } 39.97/13.44 39.97/13.44 All above mentioned rules can be savely removed. 39.97/13.44 39.97/13.44 We are left with following problem, upon which TcT provides the 39.97/13.44 certificate YES(O(1),O(n^1)). 39.97/13.44 39.97/13.44 Strict Trs: 39.97/13.44 { U11(tt(), N) -> activate(N) 39.97/13.44 , activate(X) -> X 39.97/13.44 , activate(n__0()) -> 0() 39.97/13.44 , activate(n__plus(X1, X2)) -> plus(X1, X2) 39.97/13.44 , activate(n__isNat(X)) -> isNat(X) 39.97/13.44 , activate(n__s(X)) -> s(X) 39.97/13.44 , activate(n__x(X1, X2)) -> x(X1, X2) 39.97/13.44 , U21(tt(), M, N) -> s(plus(activate(N), activate(M))) 39.97/13.44 , s(X) -> n__s(X) 39.97/13.44 , plus(X1, X2) -> n__plus(X1, X2) 39.97/13.44 , U31(tt()) -> 0() 39.97/13.44 , 0() -> n__0() 39.97/13.44 , U41(tt(), M, N) -> plus(x(activate(N), activate(M)), activate(N)) 39.97/13.44 , x(X1, X2) -> n__x(X1, X2) 39.97/13.44 , and(tt(), X) -> activate(X) 39.97/13.44 , isNat(X) -> n__isNat(X) 39.97/13.44 , isNat(n__0()) -> tt() 39.97/13.44 , isNat(n__plus(V1, V2)) -> 39.97/13.44 and(isNat(activate(V1)), n__isNat(activate(V2))) 39.97/13.44 , isNat(n__s(V1)) -> isNat(activate(V1)) 39.97/13.44 , isNat(n__x(V1, V2)) -> 39.97/13.44 and(isNat(activate(V1)), n__isNat(activate(V2))) } 39.97/13.44 Obligation: 39.97/13.44 innermost runtime complexity 39.97/13.44 Answer: 39.97/13.44 YES(O(1),O(n^1)) 39.97/13.44 39.97/13.44 We add the following dependency tuples: 39.97/13.44 39.97/13.44 Strict DPs: 39.97/13.44 { U11^#(tt(), N) -> c_1(activate^#(N)) 39.97/13.44 , activate^#(X) -> c_2() 39.97/13.44 , activate^#(n__0()) -> c_3(0^#()) 39.97/13.44 , activate^#(n__plus(X1, X2)) -> c_4(plus^#(X1, X2)) 39.97/13.44 , activate^#(n__isNat(X)) -> c_5(isNat^#(X)) 39.97/13.44 , activate^#(n__s(X)) -> c_6(s^#(X)) 39.97/13.44 , activate^#(n__x(X1, X2)) -> c_7(x^#(X1, X2)) 39.97/13.44 , 0^#() -> c_12() 39.97/13.44 , plus^#(X1, X2) -> c_10() 39.97/13.44 , isNat^#(X) -> c_16() 39.97/13.44 , isNat^#(n__0()) -> c_17() 39.97/13.44 , isNat^#(n__plus(V1, V2)) -> 39.97/13.44 c_18(and^#(isNat(activate(V1)), n__isNat(activate(V2))), 39.97/13.44 isNat^#(activate(V1)), 39.97/13.44 activate^#(V1), 39.97/13.44 activate^#(V2)) 39.97/13.44 , isNat^#(n__s(V1)) -> c_19(isNat^#(activate(V1)), activate^#(V1)) 39.97/13.44 , isNat^#(n__x(V1, V2)) -> 39.97/13.44 c_20(and^#(isNat(activate(V1)), n__isNat(activate(V2))), 39.97/13.44 isNat^#(activate(V1)), 39.97/13.44 activate^#(V1), 39.97/13.44 activate^#(V2)) 39.97/13.44 , s^#(X) -> c_9() 39.97/13.44 , x^#(X1, X2) -> c_14() 39.97/13.44 , U21^#(tt(), M, N) -> 39.97/13.44 c_8(s^#(plus(activate(N), activate(M))), 39.97/13.44 plus^#(activate(N), activate(M)), 39.97/13.44 activate^#(N), 39.97/13.44 activate^#(M)) 39.97/13.44 , U31^#(tt()) -> c_11(0^#()) 39.97/13.44 , U41^#(tt(), M, N) -> 39.97/13.44 c_13(plus^#(x(activate(N), activate(M)), activate(N)), 39.97/13.44 x^#(activate(N), activate(M)), 39.97/13.44 activate^#(N), 39.97/13.44 activate^#(M), 39.97/13.44 activate^#(N)) 39.97/13.44 , and^#(tt(), X) -> c_15(activate^#(X)) } 39.97/13.44 39.97/13.44 and mark the set of starting terms. 39.97/13.44 39.97/13.44 We are left with following problem, upon which TcT provides the 39.97/13.44 certificate YES(O(1),O(n^1)). 39.97/13.44 39.97/13.44 Strict DPs: 39.97/13.44 { U11^#(tt(), N) -> c_1(activate^#(N)) 39.97/13.44 , activate^#(X) -> c_2() 39.97/13.44 , activate^#(n__0()) -> c_3(0^#()) 39.97/13.44 , activate^#(n__plus(X1, X2)) -> c_4(plus^#(X1, X2)) 39.97/13.44 , activate^#(n__isNat(X)) -> c_5(isNat^#(X)) 39.97/13.44 , activate^#(n__s(X)) -> c_6(s^#(X)) 39.97/13.44 , activate^#(n__x(X1, X2)) -> c_7(x^#(X1, X2)) 39.97/13.44 , 0^#() -> c_12() 39.97/13.44 , plus^#(X1, X2) -> c_10() 39.97/13.44 , isNat^#(X) -> c_16() 39.97/13.44 , isNat^#(n__0()) -> c_17() 39.97/13.44 , isNat^#(n__plus(V1, V2)) -> 39.97/13.44 c_18(and^#(isNat(activate(V1)), n__isNat(activate(V2))), 39.97/13.44 isNat^#(activate(V1)), 39.97/13.44 activate^#(V1), 39.97/13.44 activate^#(V2)) 39.97/13.44 , isNat^#(n__s(V1)) -> c_19(isNat^#(activate(V1)), activate^#(V1)) 39.97/13.44 , isNat^#(n__x(V1, V2)) -> 39.97/13.44 c_20(and^#(isNat(activate(V1)), n__isNat(activate(V2))), 39.97/13.44 isNat^#(activate(V1)), 39.97/13.44 activate^#(V1), 39.97/13.44 activate^#(V2)) 39.97/13.44 , s^#(X) -> c_9() 39.97/13.44 , x^#(X1, X2) -> c_14() 39.97/13.44 , U21^#(tt(), M, N) -> 39.97/13.44 c_8(s^#(plus(activate(N), activate(M))), 39.97/13.44 plus^#(activate(N), activate(M)), 39.97/13.44 activate^#(N), 39.97/13.44 activate^#(M)) 39.97/13.44 , U31^#(tt()) -> c_11(0^#()) 39.97/13.44 , U41^#(tt(), M, N) -> 39.97/13.44 c_13(plus^#(x(activate(N), activate(M)), activate(N)), 39.97/13.44 x^#(activate(N), activate(M)), 39.97/13.44 activate^#(N), 39.97/13.44 activate^#(M), 39.97/13.44 activate^#(N)) 39.97/13.44 , and^#(tt(), X) -> c_15(activate^#(X)) } 39.97/13.44 Weak Trs: 39.97/13.44 { U11(tt(), N) -> activate(N) 39.97/13.44 , activate(X) -> X 39.97/13.44 , activate(n__0()) -> 0() 39.97/13.44 , activate(n__plus(X1, X2)) -> plus(X1, X2) 39.97/13.44 , activate(n__isNat(X)) -> isNat(X) 39.97/13.44 , activate(n__s(X)) -> s(X) 39.97/13.44 , activate(n__x(X1, X2)) -> x(X1, X2) 39.97/13.44 , U21(tt(), M, N) -> s(plus(activate(N), activate(M))) 39.97/13.44 , s(X) -> n__s(X) 39.97/13.44 , plus(X1, X2) -> n__plus(X1, X2) 39.97/13.44 , U31(tt()) -> 0() 39.97/13.44 , 0() -> n__0() 39.97/13.44 , U41(tt(), M, N) -> plus(x(activate(N), activate(M)), activate(N)) 39.97/13.44 , x(X1, X2) -> n__x(X1, X2) 39.97/13.44 , and(tt(), X) -> activate(X) 39.97/13.44 , isNat(X) -> n__isNat(X) 39.97/13.44 , isNat(n__0()) -> tt() 39.97/13.44 , isNat(n__plus(V1, V2)) -> 39.97/13.44 and(isNat(activate(V1)), n__isNat(activate(V2))) 39.97/13.44 , isNat(n__s(V1)) -> isNat(activate(V1)) 39.97/13.44 , isNat(n__x(V1, V2)) -> 39.97/13.44 and(isNat(activate(V1)), n__isNat(activate(V2))) } 39.97/13.44 Obligation: 39.97/13.44 innermost runtime complexity 39.97/13.44 Answer: 39.97/13.44 YES(O(1),O(n^1)) 39.97/13.44 39.97/13.44 We estimate the number of application of {2,8,9,10,11,15,16} by 39.97/13.44 applications of Pre({2,8,9,10,11,15,16}) = 39.97/13.44 {1,3,4,5,6,7,12,13,14,17,18,19,20}. Here rules are labeled as 39.97/13.44 follows: 39.97/13.44 39.97/13.44 DPs: 39.97/13.44 { 1: U11^#(tt(), N) -> c_1(activate^#(N)) 39.97/13.44 , 2: activate^#(X) -> c_2() 39.97/13.44 , 3: activate^#(n__0()) -> c_3(0^#()) 39.97/13.44 , 4: activate^#(n__plus(X1, X2)) -> c_4(plus^#(X1, X2)) 39.97/13.44 , 5: activate^#(n__isNat(X)) -> c_5(isNat^#(X)) 39.97/13.44 , 6: activate^#(n__s(X)) -> c_6(s^#(X)) 39.97/13.44 , 7: activate^#(n__x(X1, X2)) -> c_7(x^#(X1, X2)) 39.97/13.44 , 8: 0^#() -> c_12() 39.97/13.44 , 9: plus^#(X1, X2) -> c_10() 39.97/13.44 , 10: isNat^#(X) -> c_16() 39.97/13.44 , 11: isNat^#(n__0()) -> c_17() 39.97/13.44 , 12: isNat^#(n__plus(V1, V2)) -> 39.97/13.44 c_18(and^#(isNat(activate(V1)), n__isNat(activate(V2))), 39.97/13.44 isNat^#(activate(V1)), 39.97/13.44 activate^#(V1), 39.97/13.44 activate^#(V2)) 39.97/13.44 , 13: isNat^#(n__s(V1)) -> 39.97/13.44 c_19(isNat^#(activate(V1)), activate^#(V1)) 39.97/13.44 , 14: isNat^#(n__x(V1, V2)) -> 39.97/13.44 c_20(and^#(isNat(activate(V1)), n__isNat(activate(V2))), 39.97/13.44 isNat^#(activate(V1)), 39.97/13.44 activate^#(V1), 39.97/13.44 activate^#(V2)) 39.97/13.45 , 15: s^#(X) -> c_9() 39.97/13.45 , 16: x^#(X1, X2) -> c_14() 39.97/13.45 , 17: U21^#(tt(), M, N) -> 39.97/13.45 c_8(s^#(plus(activate(N), activate(M))), 39.97/13.45 plus^#(activate(N), activate(M)), 39.97/13.45 activate^#(N), 39.97/13.45 activate^#(M)) 39.97/13.45 , 18: U31^#(tt()) -> c_11(0^#()) 39.97/13.45 , 19: U41^#(tt(), M, N) -> 39.97/13.45 c_13(plus^#(x(activate(N), activate(M)), activate(N)), 39.97/13.45 x^#(activate(N), activate(M)), 39.97/13.45 activate^#(N), 39.97/13.45 activate^#(M), 39.97/13.45 activate^#(N)) 39.97/13.45 , 20: and^#(tt(), X) -> c_15(activate^#(X)) } 39.97/13.45 39.97/13.45 We are left with following problem, upon which TcT provides the 39.97/13.45 certificate YES(O(1),O(n^1)). 39.97/13.45 39.97/13.45 Strict DPs: 39.97/13.45 { U11^#(tt(), N) -> c_1(activate^#(N)) 39.97/13.45 , activate^#(n__0()) -> c_3(0^#()) 39.97/13.45 , activate^#(n__plus(X1, X2)) -> c_4(plus^#(X1, X2)) 39.97/13.45 , activate^#(n__isNat(X)) -> c_5(isNat^#(X)) 39.97/13.45 , activate^#(n__s(X)) -> c_6(s^#(X)) 39.97/13.45 , activate^#(n__x(X1, X2)) -> c_7(x^#(X1, X2)) 39.97/13.45 , isNat^#(n__plus(V1, V2)) -> 39.97/13.45 c_18(and^#(isNat(activate(V1)), n__isNat(activate(V2))), 39.97/13.45 isNat^#(activate(V1)), 39.97/13.45 activate^#(V1), 39.97/13.45 activate^#(V2)) 39.97/13.45 , isNat^#(n__s(V1)) -> c_19(isNat^#(activate(V1)), activate^#(V1)) 39.97/13.45 , isNat^#(n__x(V1, V2)) -> 39.97/13.45 c_20(and^#(isNat(activate(V1)), n__isNat(activate(V2))), 39.97/13.45 isNat^#(activate(V1)), 39.97/13.45 activate^#(V1), 39.97/13.45 activate^#(V2)) 39.97/13.45 , U21^#(tt(), M, N) -> 39.97/13.45 c_8(s^#(plus(activate(N), activate(M))), 39.97/13.45 plus^#(activate(N), activate(M)), 39.97/13.45 activate^#(N), 39.97/13.45 activate^#(M)) 39.97/13.45 , U31^#(tt()) -> c_11(0^#()) 39.97/13.45 , U41^#(tt(), M, N) -> 39.97/13.45 c_13(plus^#(x(activate(N), activate(M)), activate(N)), 39.97/13.45 x^#(activate(N), activate(M)), 39.97/13.45 activate^#(N), 39.97/13.45 activate^#(M), 39.97/13.45 activate^#(N)) 39.97/13.45 , and^#(tt(), X) -> c_15(activate^#(X)) } 39.97/13.45 Weak DPs: 39.97/13.45 { activate^#(X) -> c_2() 39.97/13.45 , 0^#() -> c_12() 39.97/13.45 , plus^#(X1, X2) -> c_10() 39.97/13.45 , isNat^#(X) -> c_16() 39.97/13.45 , isNat^#(n__0()) -> c_17() 39.97/13.45 , s^#(X) -> c_9() 39.97/13.45 , x^#(X1, X2) -> c_14() } 39.97/13.45 Weak Trs: 39.97/13.45 { U11(tt(), N) -> activate(N) 39.97/13.45 , activate(X) -> X 39.97/13.45 , activate(n__0()) -> 0() 39.97/13.45 , activate(n__plus(X1, X2)) -> plus(X1, X2) 39.97/13.45 , activate(n__isNat(X)) -> isNat(X) 39.97/13.45 , activate(n__s(X)) -> s(X) 39.97/13.45 , activate(n__x(X1, X2)) -> x(X1, X2) 39.97/13.45 , U21(tt(), M, N) -> s(plus(activate(N), activate(M))) 39.97/13.45 , s(X) -> n__s(X) 39.97/13.45 , plus(X1, X2) -> n__plus(X1, X2) 39.97/13.45 , U31(tt()) -> 0() 39.97/13.45 , 0() -> n__0() 39.97/13.45 , U41(tt(), M, N) -> plus(x(activate(N), activate(M)), activate(N)) 39.97/13.45 , x(X1, X2) -> n__x(X1, X2) 39.97/13.45 , and(tt(), X) -> activate(X) 39.97/13.45 , isNat(X) -> n__isNat(X) 39.97/13.45 , isNat(n__0()) -> tt() 39.97/13.45 , isNat(n__plus(V1, V2)) -> 39.97/13.45 and(isNat(activate(V1)), n__isNat(activate(V2))) 39.97/13.45 , isNat(n__s(V1)) -> isNat(activate(V1)) 39.97/13.45 , isNat(n__x(V1, V2)) -> 39.97/13.45 and(isNat(activate(V1)), n__isNat(activate(V2))) } 39.97/13.45 Obligation: 39.97/13.45 innermost runtime complexity 39.97/13.45 Answer: 39.97/13.45 YES(O(1),O(n^1)) 39.97/13.45 39.97/13.45 We estimate the number of application of {2,3,5,6,11} by 39.97/13.45 applications of Pre({2,3,5,6,11}) = {1,7,8,9,10,12,13}. Here rules 39.97/13.45 are labeled as follows: 39.97/13.45 39.97/13.45 DPs: 39.97/13.45 { 1: U11^#(tt(), N) -> c_1(activate^#(N)) 39.97/13.45 , 2: activate^#(n__0()) -> c_3(0^#()) 39.97/13.45 , 3: activate^#(n__plus(X1, X2)) -> c_4(plus^#(X1, X2)) 39.97/13.45 , 4: activate^#(n__isNat(X)) -> c_5(isNat^#(X)) 39.97/13.45 , 5: activate^#(n__s(X)) -> c_6(s^#(X)) 39.97/13.45 , 6: activate^#(n__x(X1, X2)) -> c_7(x^#(X1, X2)) 39.97/13.45 , 7: isNat^#(n__plus(V1, V2)) -> 39.97/13.45 c_18(and^#(isNat(activate(V1)), n__isNat(activate(V2))), 39.97/13.45 isNat^#(activate(V1)), 39.97/13.45 activate^#(V1), 39.97/13.45 activate^#(V2)) 39.97/13.45 , 8: isNat^#(n__s(V1)) -> 39.97/13.45 c_19(isNat^#(activate(V1)), activate^#(V1)) 39.97/13.45 , 9: isNat^#(n__x(V1, V2)) -> 39.97/13.45 c_20(and^#(isNat(activate(V1)), n__isNat(activate(V2))), 39.97/13.45 isNat^#(activate(V1)), 39.97/13.45 activate^#(V1), 39.97/13.45 activate^#(V2)) 39.97/13.45 , 10: U21^#(tt(), M, N) -> 39.97/13.45 c_8(s^#(plus(activate(N), activate(M))), 39.97/13.45 plus^#(activate(N), activate(M)), 39.97/13.45 activate^#(N), 39.97/13.45 activate^#(M)) 39.97/13.45 , 11: U31^#(tt()) -> c_11(0^#()) 39.97/13.45 , 12: U41^#(tt(), M, N) -> 39.97/13.45 c_13(plus^#(x(activate(N), activate(M)), activate(N)), 39.97/13.45 x^#(activate(N), activate(M)), 39.97/13.45 activate^#(N), 39.97/13.45 activate^#(M), 39.97/13.45 activate^#(N)) 39.97/13.45 , 13: and^#(tt(), X) -> c_15(activate^#(X)) 39.97/13.45 , 14: activate^#(X) -> c_2() 39.97/13.45 , 15: 0^#() -> c_12() 39.97/13.45 , 16: plus^#(X1, X2) -> c_10() 39.97/13.45 , 17: isNat^#(X) -> c_16() 39.97/13.45 , 18: isNat^#(n__0()) -> c_17() 39.97/13.45 , 19: s^#(X) -> c_9() 39.97/13.45 , 20: x^#(X1, X2) -> c_14() } 39.97/13.45 39.97/13.45 We are left with following problem, upon which TcT provides the 39.97/13.45 certificate YES(O(1),O(n^1)). 39.97/13.45 39.97/13.45 Strict DPs: 39.97/13.45 { U11^#(tt(), N) -> c_1(activate^#(N)) 39.97/13.45 , activate^#(n__isNat(X)) -> c_5(isNat^#(X)) 39.97/13.45 , isNat^#(n__plus(V1, V2)) -> 39.97/13.45 c_18(and^#(isNat(activate(V1)), n__isNat(activate(V2))), 39.97/13.45 isNat^#(activate(V1)), 39.97/13.45 activate^#(V1), 39.97/13.45 activate^#(V2)) 39.97/13.45 , isNat^#(n__s(V1)) -> c_19(isNat^#(activate(V1)), activate^#(V1)) 39.97/13.45 , isNat^#(n__x(V1, V2)) -> 39.97/13.45 c_20(and^#(isNat(activate(V1)), n__isNat(activate(V2))), 39.97/13.45 isNat^#(activate(V1)), 39.97/13.45 activate^#(V1), 39.97/13.45 activate^#(V2)) 39.97/13.45 , U21^#(tt(), M, N) -> 39.97/13.45 c_8(s^#(plus(activate(N), activate(M))), 39.97/13.45 plus^#(activate(N), activate(M)), 39.97/13.45 activate^#(N), 39.97/13.45 activate^#(M)) 39.97/13.45 , U41^#(tt(), M, N) -> 39.97/13.45 c_13(plus^#(x(activate(N), activate(M)), activate(N)), 39.97/13.45 x^#(activate(N), activate(M)), 39.97/13.45 activate^#(N), 39.97/13.45 activate^#(M), 39.97/13.45 activate^#(N)) 39.97/13.45 , and^#(tt(), X) -> c_15(activate^#(X)) } 39.97/13.45 Weak DPs: 39.97/13.45 { activate^#(X) -> c_2() 39.97/13.45 , activate^#(n__0()) -> c_3(0^#()) 39.97/13.45 , activate^#(n__plus(X1, X2)) -> c_4(plus^#(X1, X2)) 39.97/13.45 , activate^#(n__s(X)) -> c_6(s^#(X)) 39.97/13.45 , activate^#(n__x(X1, X2)) -> c_7(x^#(X1, X2)) 39.97/13.45 , 0^#() -> c_12() 39.97/13.45 , plus^#(X1, X2) -> c_10() 39.97/13.45 , isNat^#(X) -> c_16() 39.97/13.45 , isNat^#(n__0()) -> c_17() 39.97/13.45 , s^#(X) -> c_9() 39.97/13.45 , x^#(X1, X2) -> c_14() 39.97/13.45 , U31^#(tt()) -> c_11(0^#()) } 39.97/13.45 Weak Trs: 39.97/13.45 { U11(tt(), N) -> activate(N) 39.97/13.45 , activate(X) -> X 39.97/13.45 , activate(n__0()) -> 0() 39.97/13.45 , activate(n__plus(X1, X2)) -> plus(X1, X2) 39.97/13.45 , activate(n__isNat(X)) -> isNat(X) 39.97/13.45 , activate(n__s(X)) -> s(X) 39.97/13.45 , activate(n__x(X1, X2)) -> x(X1, X2) 39.97/13.45 , U21(tt(), M, N) -> s(plus(activate(N), activate(M))) 39.97/13.45 , s(X) -> n__s(X) 39.97/13.45 , plus(X1, X2) -> n__plus(X1, X2) 39.97/13.45 , U31(tt()) -> 0() 39.97/13.45 , 0() -> n__0() 39.97/13.45 , U41(tt(), M, N) -> plus(x(activate(N), activate(M)), activate(N)) 39.97/13.45 , x(X1, X2) -> n__x(X1, X2) 39.97/13.45 , and(tt(), X) -> activate(X) 39.97/13.45 , isNat(X) -> n__isNat(X) 39.97/13.45 , isNat(n__0()) -> tt() 39.97/13.45 , isNat(n__plus(V1, V2)) -> 39.97/13.45 and(isNat(activate(V1)), n__isNat(activate(V2))) 39.97/13.45 , isNat(n__s(V1)) -> isNat(activate(V1)) 39.97/13.45 , isNat(n__x(V1, V2)) -> 39.97/13.45 and(isNat(activate(V1)), n__isNat(activate(V2))) } 39.97/13.45 Obligation: 39.97/13.45 innermost runtime complexity 39.97/13.45 Answer: 39.97/13.45 YES(O(1),O(n^1)) 39.97/13.45 39.97/13.45 The following weak DPs constitute a sub-graph of the DG that is 39.97/13.45 closed under successors. The DPs are removed. 39.97/13.45 39.97/13.45 { activate^#(X) -> c_2() 39.97/13.45 , activate^#(n__0()) -> c_3(0^#()) 39.97/13.45 , activate^#(n__plus(X1, X2)) -> c_4(plus^#(X1, X2)) 39.97/13.45 , activate^#(n__s(X)) -> c_6(s^#(X)) 39.97/13.45 , activate^#(n__x(X1, X2)) -> c_7(x^#(X1, X2)) 39.97/13.45 , 0^#() -> c_12() 39.97/13.45 , plus^#(X1, X2) -> c_10() 39.97/13.45 , isNat^#(X) -> c_16() 39.97/13.45 , isNat^#(n__0()) -> c_17() 39.97/13.45 , s^#(X) -> c_9() 39.97/13.45 , x^#(X1, X2) -> c_14() 39.97/13.45 , U31^#(tt()) -> c_11(0^#()) } 39.97/13.45 39.97/13.45 We are left with following problem, upon which TcT provides the 39.97/13.45 certificate YES(O(1),O(n^1)). 39.97/13.45 39.97/13.45 Strict DPs: 39.97/13.45 { U11^#(tt(), N) -> c_1(activate^#(N)) 39.97/13.45 , activate^#(n__isNat(X)) -> c_5(isNat^#(X)) 39.97/13.45 , isNat^#(n__plus(V1, V2)) -> 39.97/13.45 c_18(and^#(isNat(activate(V1)), n__isNat(activate(V2))), 39.97/13.45 isNat^#(activate(V1)), 39.97/13.45 activate^#(V1), 39.97/13.45 activate^#(V2)) 39.97/13.45 , isNat^#(n__s(V1)) -> c_19(isNat^#(activate(V1)), activate^#(V1)) 39.97/13.45 , isNat^#(n__x(V1, V2)) -> 39.97/13.45 c_20(and^#(isNat(activate(V1)), n__isNat(activate(V2))), 39.97/13.45 isNat^#(activate(V1)), 39.97/13.45 activate^#(V1), 39.97/13.45 activate^#(V2)) 39.97/13.45 , U21^#(tt(), M, N) -> 39.97/13.45 c_8(s^#(plus(activate(N), activate(M))), 39.97/13.45 plus^#(activate(N), activate(M)), 39.97/13.45 activate^#(N), 39.97/13.45 activate^#(M)) 39.97/13.45 , U41^#(tt(), M, N) -> 39.97/13.45 c_13(plus^#(x(activate(N), activate(M)), activate(N)), 39.97/13.45 x^#(activate(N), activate(M)), 39.97/13.45 activate^#(N), 39.97/13.45 activate^#(M), 39.97/13.45 activate^#(N)) 39.97/13.45 , and^#(tt(), X) -> c_15(activate^#(X)) } 39.97/13.45 Weak Trs: 39.97/13.45 { U11(tt(), N) -> activate(N) 39.97/13.45 , activate(X) -> X 39.97/13.45 , activate(n__0()) -> 0() 39.97/13.45 , activate(n__plus(X1, X2)) -> plus(X1, X2) 39.97/13.45 , activate(n__isNat(X)) -> isNat(X) 39.97/13.45 , activate(n__s(X)) -> s(X) 39.97/13.45 , activate(n__x(X1, X2)) -> x(X1, X2) 39.97/13.45 , U21(tt(), M, N) -> s(plus(activate(N), activate(M))) 39.97/13.45 , s(X) -> n__s(X) 39.97/13.45 , plus(X1, X2) -> n__plus(X1, X2) 39.97/13.45 , U31(tt()) -> 0() 39.97/13.45 , 0() -> n__0() 39.97/13.45 , U41(tt(), M, N) -> plus(x(activate(N), activate(M)), activate(N)) 39.97/13.45 , x(X1, X2) -> n__x(X1, X2) 39.97/13.45 , and(tt(), X) -> activate(X) 39.97/13.45 , isNat(X) -> n__isNat(X) 39.97/13.45 , isNat(n__0()) -> tt() 39.97/13.45 , isNat(n__plus(V1, V2)) -> 39.97/13.45 and(isNat(activate(V1)), n__isNat(activate(V2))) 39.97/13.45 , isNat(n__s(V1)) -> isNat(activate(V1)) 39.97/13.45 , isNat(n__x(V1, V2)) -> 39.97/13.45 and(isNat(activate(V1)), n__isNat(activate(V2))) } 39.97/13.45 Obligation: 39.97/13.45 innermost runtime complexity 39.97/13.45 Answer: 39.97/13.45 YES(O(1),O(n^1)) 39.97/13.45 39.97/13.45 Due to missing edges in the dependency-graph, the right-hand sides 39.97/13.45 of following rules could be simplified: 39.97/13.45 39.97/13.45 { U21^#(tt(), M, N) -> 39.97/13.45 c_8(s^#(plus(activate(N), activate(M))), 39.97/13.45 plus^#(activate(N), activate(M)), 39.97/13.45 activate^#(N), 39.97/13.45 activate^#(M)) 39.97/13.45 , U41^#(tt(), M, N) -> 39.97/13.45 c_13(plus^#(x(activate(N), activate(M)), activate(N)), 39.97/13.45 x^#(activate(N), activate(M)), 39.97/13.45 activate^#(N), 39.97/13.45 activate^#(M), 39.97/13.45 activate^#(N)) } 39.97/13.45 39.97/13.45 We are left with following problem, upon which TcT provides the 39.97/13.45 certificate YES(O(1),O(n^1)). 39.97/13.45 39.97/13.45 Strict DPs: 39.97/13.45 { U11^#(tt(), N) -> c_1(activate^#(N)) 39.97/13.45 , activate^#(n__isNat(X)) -> c_2(isNat^#(X)) 39.97/13.45 , isNat^#(n__plus(V1, V2)) -> 39.97/13.45 c_3(and^#(isNat(activate(V1)), n__isNat(activate(V2))), 39.97/13.45 isNat^#(activate(V1)), 39.97/13.45 activate^#(V1), 39.97/13.45 activate^#(V2)) 39.97/13.45 , isNat^#(n__s(V1)) -> c_4(isNat^#(activate(V1)), activate^#(V1)) 39.97/13.45 , isNat^#(n__x(V1, V2)) -> 39.97/13.45 c_5(and^#(isNat(activate(V1)), n__isNat(activate(V2))), 39.97/13.45 isNat^#(activate(V1)), 39.97/13.45 activate^#(V1), 39.97/13.45 activate^#(V2)) 39.97/13.45 , U21^#(tt(), M, N) -> c_6(activate^#(N), activate^#(M)) 39.97/13.45 , U41^#(tt(), M, N) -> 39.97/13.45 c_7(activate^#(N), activate^#(M), activate^#(N)) 39.97/13.45 , and^#(tt(), X) -> c_8(activate^#(X)) } 39.97/13.45 Weak Trs: 39.97/13.45 { U11(tt(), N) -> activate(N) 39.97/13.45 , activate(X) -> X 39.97/13.45 , activate(n__0()) -> 0() 39.97/13.45 , activate(n__plus(X1, X2)) -> plus(X1, X2) 39.97/13.45 , activate(n__isNat(X)) -> isNat(X) 39.97/13.45 , activate(n__s(X)) -> s(X) 39.97/13.45 , activate(n__x(X1, X2)) -> x(X1, X2) 39.97/13.45 , U21(tt(), M, N) -> s(plus(activate(N), activate(M))) 39.97/13.45 , s(X) -> n__s(X) 39.97/13.45 , plus(X1, X2) -> n__plus(X1, X2) 39.97/13.45 , U31(tt()) -> 0() 39.97/13.45 , 0() -> n__0() 39.97/13.45 , U41(tt(), M, N) -> plus(x(activate(N), activate(M)), activate(N)) 39.97/13.45 , x(X1, X2) -> n__x(X1, X2) 39.97/13.45 , and(tt(), X) -> activate(X) 39.97/13.45 , isNat(X) -> n__isNat(X) 39.97/13.45 , isNat(n__0()) -> tt() 39.97/13.45 , isNat(n__plus(V1, V2)) -> 39.97/13.45 and(isNat(activate(V1)), n__isNat(activate(V2))) 39.97/13.45 , isNat(n__s(V1)) -> isNat(activate(V1)) 39.97/13.45 , isNat(n__x(V1, V2)) -> 39.97/13.45 and(isNat(activate(V1)), n__isNat(activate(V2))) } 39.97/13.45 Obligation: 39.97/13.45 innermost runtime complexity 39.97/13.45 Answer: 39.97/13.45 YES(O(1),O(n^1)) 39.97/13.45 39.97/13.45 We replace rewrite rules by usable rules: 39.97/13.45 39.97/13.45 Weak Usable Rules: 39.97/13.45 { activate(X) -> X 39.97/13.45 , activate(n__0()) -> 0() 39.97/13.45 , activate(n__plus(X1, X2)) -> plus(X1, X2) 39.97/13.45 , activate(n__isNat(X)) -> isNat(X) 39.97/13.45 , activate(n__s(X)) -> s(X) 39.97/13.45 , activate(n__x(X1, X2)) -> x(X1, X2) 39.97/13.45 , s(X) -> n__s(X) 39.97/13.45 , plus(X1, X2) -> n__plus(X1, X2) 39.97/13.45 , 0() -> n__0() 39.97/13.45 , x(X1, X2) -> n__x(X1, X2) 39.97/13.45 , and(tt(), X) -> activate(X) 39.97/13.45 , isNat(X) -> n__isNat(X) 39.97/13.45 , isNat(n__0()) -> tt() 39.97/13.45 , isNat(n__plus(V1, V2)) -> 39.97/13.45 and(isNat(activate(V1)), n__isNat(activate(V2))) 39.97/13.45 , isNat(n__s(V1)) -> isNat(activate(V1)) 39.97/13.45 , isNat(n__x(V1, V2)) -> 39.97/13.45 and(isNat(activate(V1)), n__isNat(activate(V2))) } 39.97/13.45 39.97/13.45 We are left with following problem, upon which TcT provides the 39.97/13.45 certificate YES(O(1),O(n^1)). 39.97/13.45 39.97/13.45 Strict DPs: 39.97/13.45 { U11^#(tt(), N) -> c_1(activate^#(N)) 39.97/13.45 , activate^#(n__isNat(X)) -> c_2(isNat^#(X)) 39.97/13.45 , isNat^#(n__plus(V1, V2)) -> 39.97/13.45 c_3(and^#(isNat(activate(V1)), n__isNat(activate(V2))), 39.97/13.45 isNat^#(activate(V1)), 39.97/13.45 activate^#(V1), 39.97/13.45 activate^#(V2)) 39.97/13.45 , isNat^#(n__s(V1)) -> c_4(isNat^#(activate(V1)), activate^#(V1)) 39.97/13.45 , isNat^#(n__x(V1, V2)) -> 39.97/13.45 c_5(and^#(isNat(activate(V1)), n__isNat(activate(V2))), 39.97/13.45 isNat^#(activate(V1)), 39.97/13.45 activate^#(V1), 39.97/13.45 activate^#(V2)) 39.97/13.45 , U21^#(tt(), M, N) -> c_6(activate^#(N), activate^#(M)) 39.97/13.45 , U41^#(tt(), M, N) -> 39.97/13.45 c_7(activate^#(N), activate^#(M), activate^#(N)) 39.97/13.45 , and^#(tt(), X) -> c_8(activate^#(X)) } 39.97/13.45 Weak Trs: 39.97/13.45 { activate(X) -> X 39.97/13.45 , activate(n__0()) -> 0() 39.97/13.45 , activate(n__plus(X1, X2)) -> plus(X1, X2) 39.97/13.45 , activate(n__isNat(X)) -> isNat(X) 39.97/13.45 , activate(n__s(X)) -> s(X) 39.97/13.45 , activate(n__x(X1, X2)) -> x(X1, X2) 39.97/13.45 , s(X) -> n__s(X) 39.97/13.45 , plus(X1, X2) -> n__plus(X1, X2) 39.97/13.45 , 0() -> n__0() 39.97/13.45 , x(X1, X2) -> n__x(X1, X2) 39.97/13.45 , and(tt(), X) -> activate(X) 39.97/13.45 , isNat(X) -> n__isNat(X) 39.97/13.45 , isNat(n__0()) -> tt() 39.97/13.45 , isNat(n__plus(V1, V2)) -> 39.97/13.45 and(isNat(activate(V1)), n__isNat(activate(V2))) 39.97/13.45 , isNat(n__s(V1)) -> isNat(activate(V1)) 39.97/13.45 , isNat(n__x(V1, V2)) -> 39.97/13.45 and(isNat(activate(V1)), n__isNat(activate(V2))) } 39.97/13.45 Obligation: 39.97/13.45 innermost runtime complexity 39.97/13.45 Answer: 39.97/13.45 YES(O(1),O(n^1)) 39.97/13.45 39.97/13.45 Consider the dependency graph 39.97/13.45 39.97/13.45 1: U11^#(tt(), N) -> c_1(activate^#(N)) 39.97/13.45 -->_1 activate^#(n__isNat(X)) -> c_2(isNat^#(X)) :2 39.97/13.45 39.97/13.45 2: activate^#(n__isNat(X)) -> c_2(isNat^#(X)) 39.97/13.45 -->_1 isNat^#(n__x(V1, V2)) -> 39.97/13.45 c_5(and^#(isNat(activate(V1)), n__isNat(activate(V2))), 39.97/13.45 isNat^#(activate(V1)), 39.97/13.45 activate^#(V1), 39.97/13.45 activate^#(V2)) :5 39.97/13.45 -->_1 isNat^#(n__s(V1)) -> 39.97/13.45 c_4(isNat^#(activate(V1)), activate^#(V1)) :4 39.97/13.45 -->_1 isNat^#(n__plus(V1, V2)) -> 39.97/13.45 c_3(and^#(isNat(activate(V1)), n__isNat(activate(V2))), 39.97/13.45 isNat^#(activate(V1)), 39.97/13.45 activate^#(V1), 39.97/13.45 activate^#(V2)) :3 39.97/13.45 39.97/13.45 3: isNat^#(n__plus(V1, V2)) -> 39.97/13.45 c_3(and^#(isNat(activate(V1)), n__isNat(activate(V2))), 39.97/13.45 isNat^#(activate(V1)), 39.97/13.45 activate^#(V1), 39.97/13.45 activate^#(V2)) 39.97/13.45 -->_1 and^#(tt(), X) -> c_8(activate^#(X)) :8 39.97/13.45 -->_2 isNat^#(n__x(V1, V2)) -> 39.97/13.45 c_5(and^#(isNat(activate(V1)), n__isNat(activate(V2))), 39.97/13.45 isNat^#(activate(V1)), 39.97/13.45 activate^#(V1), 39.97/13.45 activate^#(V2)) :5 39.97/13.45 -->_2 isNat^#(n__s(V1)) -> 39.97/13.45 c_4(isNat^#(activate(V1)), activate^#(V1)) :4 39.97/13.45 -->_2 isNat^#(n__plus(V1, V2)) -> 39.97/13.45 c_3(and^#(isNat(activate(V1)), n__isNat(activate(V2))), 39.97/13.45 isNat^#(activate(V1)), 39.97/13.45 activate^#(V1), 39.97/13.45 activate^#(V2)) :3 39.97/13.45 -->_4 activate^#(n__isNat(X)) -> c_2(isNat^#(X)) :2 39.97/13.45 -->_3 activate^#(n__isNat(X)) -> c_2(isNat^#(X)) :2 39.97/13.45 39.97/13.45 4: isNat^#(n__s(V1)) -> c_4(isNat^#(activate(V1)), activate^#(V1)) 39.97/13.45 -->_1 isNat^#(n__x(V1, V2)) -> 39.97/13.45 c_5(and^#(isNat(activate(V1)), n__isNat(activate(V2))), 39.97/13.45 isNat^#(activate(V1)), 39.97/13.45 activate^#(V1), 39.97/13.45 activate^#(V2)) :5 39.97/13.45 -->_1 isNat^#(n__s(V1)) -> 39.97/13.45 c_4(isNat^#(activate(V1)), activate^#(V1)) :4 39.97/13.45 -->_1 isNat^#(n__plus(V1, V2)) -> 39.97/13.45 c_3(and^#(isNat(activate(V1)), n__isNat(activate(V2))), 39.97/13.45 isNat^#(activate(V1)), 39.97/13.45 activate^#(V1), 39.97/13.45 activate^#(V2)) :3 39.97/13.45 -->_2 activate^#(n__isNat(X)) -> c_2(isNat^#(X)) :2 39.97/13.45 39.97/13.45 5: isNat^#(n__x(V1, V2)) -> 39.97/13.45 c_5(and^#(isNat(activate(V1)), n__isNat(activate(V2))), 39.97/13.45 isNat^#(activate(V1)), 39.97/13.45 activate^#(V1), 39.97/13.45 activate^#(V2)) 39.97/13.45 -->_1 and^#(tt(), X) -> c_8(activate^#(X)) :8 39.97/13.45 -->_2 isNat^#(n__x(V1, V2)) -> 39.97/13.45 c_5(and^#(isNat(activate(V1)), n__isNat(activate(V2))), 39.97/13.45 isNat^#(activate(V1)), 39.97/13.45 activate^#(V1), 39.97/13.45 activate^#(V2)) :5 39.97/13.45 -->_2 isNat^#(n__s(V1)) -> 39.97/13.45 c_4(isNat^#(activate(V1)), activate^#(V1)) :4 39.97/13.45 -->_2 isNat^#(n__plus(V1, V2)) -> 39.97/13.45 c_3(and^#(isNat(activate(V1)), n__isNat(activate(V2))), 39.97/13.45 isNat^#(activate(V1)), 39.97/13.45 activate^#(V1), 39.97/13.45 activate^#(V2)) :3 39.97/13.45 -->_4 activate^#(n__isNat(X)) -> c_2(isNat^#(X)) :2 39.97/13.45 -->_3 activate^#(n__isNat(X)) -> c_2(isNat^#(X)) :2 39.97/13.45 39.97/13.45 6: U21^#(tt(), M, N) -> c_6(activate^#(N), activate^#(M)) 39.97/13.45 -->_2 activate^#(n__isNat(X)) -> c_2(isNat^#(X)) :2 39.97/13.45 -->_1 activate^#(n__isNat(X)) -> c_2(isNat^#(X)) :2 39.97/13.45 39.97/13.45 7: U41^#(tt(), M, N) -> 39.97/13.45 c_7(activate^#(N), activate^#(M), activate^#(N)) 39.97/13.45 -->_3 activate^#(n__isNat(X)) -> c_2(isNat^#(X)) :2 39.97/13.45 -->_2 activate^#(n__isNat(X)) -> c_2(isNat^#(X)) :2 39.97/13.45 -->_1 activate^#(n__isNat(X)) -> c_2(isNat^#(X)) :2 39.97/13.45 39.97/13.45 8: and^#(tt(), X) -> c_8(activate^#(X)) 39.97/13.45 -->_1 activate^#(n__isNat(X)) -> c_2(isNat^#(X)) :2 39.97/13.45 39.97/13.45 39.97/13.45 Following roots of the dependency graph are removed, as the 39.97/13.45 considered set of starting terms is closed under reduction with 39.97/13.45 respect to these rules (modulo compound contexts). 39.97/13.46 39.97/13.46 { U11^#(tt(), N) -> c_1(activate^#(N)) 39.97/13.46 , U21^#(tt(), M, N) -> c_6(activate^#(N), activate^#(M)) 39.97/13.46 , U41^#(tt(), M, N) -> 39.97/13.46 c_7(activate^#(N), activate^#(M), activate^#(N)) } 39.97/13.46 39.97/13.46 39.97/13.46 We are left with following problem, upon which TcT provides the 39.97/13.46 certificate YES(O(1),O(n^1)). 39.97/13.46 39.97/13.46 Strict DPs: 39.97/13.46 { activate^#(n__isNat(X)) -> c_2(isNat^#(X)) 39.97/13.46 , isNat^#(n__plus(V1, V2)) -> 39.97/13.46 c_3(and^#(isNat(activate(V1)), n__isNat(activate(V2))), 39.97/13.46 isNat^#(activate(V1)), 39.97/13.46 activate^#(V1), 39.97/13.46 activate^#(V2)) 39.97/13.46 , isNat^#(n__s(V1)) -> c_4(isNat^#(activate(V1)), activate^#(V1)) 39.97/13.46 , isNat^#(n__x(V1, V2)) -> 39.97/13.46 c_5(and^#(isNat(activate(V1)), n__isNat(activate(V2))), 39.97/13.46 isNat^#(activate(V1)), 39.97/13.46 activate^#(V1), 39.97/13.46 activate^#(V2)) 39.97/13.46 , and^#(tt(), X) -> c_8(activate^#(X)) } 39.97/13.46 Weak Trs: 39.97/13.46 { activate(X) -> X 39.97/13.46 , activate(n__0()) -> 0() 39.97/13.46 , activate(n__plus(X1, X2)) -> plus(X1, X2) 39.97/13.46 , activate(n__isNat(X)) -> isNat(X) 39.97/13.46 , activate(n__s(X)) -> s(X) 39.97/13.46 , activate(n__x(X1, X2)) -> x(X1, X2) 39.97/13.46 , s(X) -> n__s(X) 39.97/13.46 , plus(X1, X2) -> n__plus(X1, X2) 39.97/13.46 , 0() -> n__0() 39.97/13.46 , x(X1, X2) -> n__x(X1, X2) 39.97/13.46 , and(tt(), X) -> activate(X) 39.97/13.46 , isNat(X) -> n__isNat(X) 39.97/13.46 , isNat(n__0()) -> tt() 39.97/13.46 , isNat(n__plus(V1, V2)) -> 39.97/13.46 and(isNat(activate(V1)), n__isNat(activate(V2))) 39.97/13.46 , isNat(n__s(V1)) -> isNat(activate(V1)) 39.97/13.46 , isNat(n__x(V1, V2)) -> 39.97/13.46 and(isNat(activate(V1)), n__isNat(activate(V2))) } 39.97/13.46 Obligation: 39.97/13.46 innermost runtime complexity 39.97/13.46 Answer: 39.97/13.46 YES(O(1),O(n^1)) 39.97/13.46 39.97/13.46 We use the processor 'matrix interpretation of dimension 2' to 39.97/13.46 orient following rules strictly. 39.97/13.46 39.97/13.46 DPs: 39.97/13.46 { 2: isNat^#(n__plus(V1, V2)) -> 39.97/13.46 c_3(and^#(isNat(activate(V1)), n__isNat(activate(V2))), 39.97/13.46 isNat^#(activate(V1)), 39.97/13.46 activate^#(V1), 39.97/13.46 activate^#(V2)) 39.97/13.46 , 3: isNat^#(n__s(V1)) -> 39.97/13.46 c_4(isNat^#(activate(V1)), activate^#(V1)) } 39.97/13.46 Trs: 39.97/13.46 { activate(n__s(X)) -> s(X) 39.97/13.46 , s(X) -> n__s(X) 39.97/13.46 , isNat(n__plus(V1, V2)) -> 39.97/13.46 and(isNat(activate(V1)), n__isNat(activate(V2))) 39.97/13.46 , isNat(n__s(V1)) -> isNat(activate(V1)) } 39.97/13.46 39.97/13.46 Sub-proof: 39.97/13.46 ---------- 39.97/13.46 The following argument positions are usable: 39.97/13.46 Uargs(c_2) = {1}, Uargs(c_3) = {1, 2, 3, 4}, Uargs(c_4) = {1, 2}, 39.97/13.46 Uargs(c_5) = {1, 2, 3, 4}, Uargs(c_8) = {1} 39.97/13.46 39.97/13.46 TcT has computed the following constructor-based matrix 39.97/13.46 interpretation satisfying not(EDA) and not(IDA(1)). 39.97/13.46 39.97/13.46 [tt] = [0] 39.97/13.46 [0] 39.97/13.46 39.97/13.46 [activate](x1) = [4 0] x1 + [0] 39.97/13.46 [0 1] [0] 39.97/13.46 39.97/13.46 [s](x1) = [0 0] x1 + [5] 39.97/13.46 [1 1] [2] 39.97/13.46 39.97/13.46 [plus](x1, x2) = [0 0] x1 + [0 0] x2 + [0] 39.97/13.46 [1 1] [1 1] [2] 39.97/13.46 39.97/13.46 [0] = [0] 39.97/13.46 [0] 39.97/13.46 39.97/13.46 [x](x1, x2) = [0 0] x1 + [0 0] x2 + [0] 39.97/13.46 [1 1] [1 1] [0] 39.97/13.46 39.97/13.46 [and](x1, x2) = [4 0] x2 + [0] 39.97/13.46 [0 4] [0] 39.97/13.46 39.97/13.46 [isNat](x1) = [0 4] x1 + [0] 39.97/13.46 [0 0] [0] 39.97/13.46 39.97/13.46 [n__0] = [0] 39.97/13.46 [0] 39.97/13.46 39.97/13.46 [n__plus](x1, x2) = [0 0] x1 + [0 0] x2 + [0] 39.97/13.46 [1 1] [1 1] [2] 39.97/13.46 39.97/13.46 [n__isNat](x1) = [0 1] x1 + [0] 39.97/13.46 [0 0] [0] 39.97/13.46 39.97/13.46 [n__s](x1) = [0 0] x1 + [2] 39.97/13.46 [1 1] [2] 39.97/13.46 39.97/13.46 [n__x](x1, x2) = [0 0] x1 + [0 0] x2 + [0] 39.97/13.46 [1 1] [1 1] [0] 39.97/13.46 39.97/13.46 [activate^#](x1) = [2 0] x1 + [0] 39.97/13.46 [0 0] [1] 39.97/13.46 39.97/13.46 [isNat^#](x1) = [0 2] x1 + [0] 39.97/13.46 [0 0] [4] 39.97/13.46 39.97/13.46 [and^#](x1, x2) = [2 0] x2 + [0] 39.97/13.46 [0 0] [4] 39.97/13.46 39.97/13.46 [c_2](x1) = [1 0] x1 + [0] 39.97/13.46 [0 0] [1] 39.97/13.46 39.97/13.46 [c_3](x1, x2, x3, x4) = [1 0] x1 + [1 0] x2 + [1 0] x3 + [1 39.97/13.46 0] x4 + [3] 39.97/13.46 [0 0] [0 0] [0 0] [0 39.97/13.46 0] [3] 39.97/13.46 39.97/13.46 [c_4](x1, x2) = [1 0] x1 + [1 1] x2 + [1] 39.97/13.46 [0 0] [0 0] [3] 39.97/13.46 39.97/13.46 [c_5](x1, x2, x3, x4) = [1 0] x1 + [1 0] x2 + [1 0] x3 + [1 39.97/13.46 0] x4 + [0] 39.97/13.46 [0 0] [0 0] [0 0] [0 39.97/13.46 0] [3] 39.97/13.46 39.97/13.46 [c_8](x1) = [1 0] x1 + [0] 39.97/13.46 [0 0] [3] 39.97/13.46 39.97/13.46 The order satisfies the following ordering constraints: 39.97/13.46 39.97/13.46 [activate(X)] = [4 0] X + [0] 39.97/13.46 [0 1] [0] 39.97/13.46 >= [1 0] X + [0] 39.97/13.46 [0 1] [0] 39.97/13.46 = [X] 39.97/13.46 39.97/13.46 [activate(n__0())] = [0] 39.97/13.46 [0] 39.97/13.46 >= [0] 39.97/13.46 [0] 39.97/13.46 = [0()] 39.97/13.46 39.97/13.46 [activate(n__plus(X1, X2))] = [0 0] X1 + [0 0] X2 + [0] 39.97/13.46 [1 1] [1 1] [2] 39.97/13.46 >= [0 0] X1 + [0 0] X2 + [0] 39.97/13.46 [1 1] [1 1] [2] 39.97/13.46 = [plus(X1, X2)] 39.97/13.46 39.97/13.46 [activate(n__isNat(X))] = [0 4] X + [0] 39.97/13.46 [0 0] [0] 39.97/13.46 >= [0 4] X + [0] 39.97/13.46 [0 0] [0] 39.97/13.46 = [isNat(X)] 39.97/13.46 39.97/13.46 [activate(n__s(X))] = [0 0] X + [8] 39.97/13.46 [1 1] [2] 39.97/13.46 > [0 0] X + [5] 39.97/13.46 [1 1] [2] 39.97/13.46 = [s(X)] 39.97/13.46 39.97/13.46 [activate(n__x(X1, X2))] = [0 0] X1 + [0 0] X2 + [0] 39.97/13.46 [1 1] [1 1] [0] 39.97/13.46 >= [0 0] X1 + [0 0] X2 + [0] 39.97/13.46 [1 1] [1 1] [0] 39.97/13.46 = [x(X1, X2)] 39.97/13.46 39.97/13.46 [s(X)] = [0 0] X + [5] 39.97/13.46 [1 1] [2] 39.97/13.46 > [0 0] X + [2] 39.97/13.46 [1 1] [2] 39.97/13.46 = [n__s(X)] 39.97/13.46 39.97/13.46 [plus(X1, X2)] = [0 0] X1 + [0 0] X2 + [0] 39.97/13.46 [1 1] [1 1] [2] 39.97/13.46 >= [0 0] X1 + [0 0] X2 + [0] 39.97/13.46 [1 1] [1 1] [2] 39.97/13.46 = [n__plus(X1, X2)] 39.97/13.46 39.97/13.46 [0()] = [0] 39.97/13.46 [0] 39.97/13.46 >= [0] 39.97/13.46 [0] 39.97/13.46 = [n__0()] 39.97/13.46 39.97/13.46 [x(X1, X2)] = [0 0] X1 + [0 0] X2 + [0] 39.97/13.46 [1 1] [1 1] [0] 39.97/13.46 >= [0 0] X1 + [0 0] X2 + [0] 39.97/13.46 [1 1] [1 1] [0] 39.97/13.46 = [n__x(X1, X2)] 39.97/13.46 39.97/13.46 [and(tt(), X)] = [4 0] X + [0] 39.97/13.46 [0 4] [0] 39.97/13.46 >= [4 0] X + [0] 39.97/13.46 [0 1] [0] 39.97/13.46 = [activate(X)] 39.97/13.46 39.97/13.46 [isNat(X)] = [0 4] X + [0] 39.97/13.46 [0 0] [0] 39.97/13.46 >= [0 1] X + [0] 39.97/13.46 [0 0] [0] 39.97/13.46 = [n__isNat(X)] 39.97/13.46 39.97/13.46 [isNat(n__0())] = [0] 39.97/13.46 [0] 39.97/13.46 >= [0] 39.97/13.46 [0] 39.97/13.46 = [tt()] 39.97/13.46 39.97/13.46 [isNat(n__plus(V1, V2))] = [4 4] V1 + [4 4] V2 + [8] 39.97/13.46 [0 0] [0 0] [0] 39.97/13.46 > [0 4] V2 + [0] 39.97/13.46 [0 0] [0] 39.97/13.46 = [and(isNat(activate(V1)), n__isNat(activate(V2)))] 39.97/13.46 39.97/13.46 [isNat(n__s(V1))] = [4 4] V1 + [8] 39.97/13.46 [0 0] [0] 39.97/13.46 > [0 4] V1 + [0] 39.97/13.46 [0 0] [0] 39.97/13.46 = [isNat(activate(V1))] 39.97/13.46 39.97/13.46 [isNat(n__x(V1, V2))] = [4 4] V1 + [4 4] V2 + [0] 39.97/13.46 [0 0] [0 0] [0] 39.97/13.46 >= [0 4] V2 + [0] 39.97/13.46 [0 0] [0] 39.97/13.46 = [and(isNat(activate(V1)), n__isNat(activate(V2)))] 39.97/13.46 39.97/13.46 [activate^#(n__isNat(X))] = [0 2] X + [0] 39.97/13.46 [0 0] [1] 39.97/13.46 >= [0 2] X + [0] 39.97/13.46 [0 0] [1] 39.97/13.46 = [c_2(isNat^#(X))] 39.97/13.46 39.97/13.46 [isNat^#(n__plus(V1, V2))] = [2 2] V1 + [2 2] V2 + [4] 39.97/13.46 [0 0] [0 0] [4] 39.97/13.46 > [2 2] V1 + [2 2] V2 + [3] 39.97/13.46 [0 0] [0 0] [3] 39.97/13.46 = [c_3(and^#(isNat(activate(V1)), n__isNat(activate(V2))), 39.97/13.46 isNat^#(activate(V1)), 39.97/13.46 activate^#(V1), 39.97/13.46 activate^#(V2))] 39.97/13.46 39.97/13.46 [isNat^#(n__s(V1))] = [2 2] V1 + [4] 39.97/13.46 [0 0] [4] 39.97/13.46 > [2 2] V1 + [2] 39.97/13.46 [0 0] [3] 39.97/13.46 = [c_4(isNat^#(activate(V1)), activate^#(V1))] 39.97/13.46 39.97/13.46 [isNat^#(n__x(V1, V2))] = [2 2] V1 + [2 2] V2 + [0] 39.97/13.46 [0 0] [0 0] [4] 39.97/13.46 >= [2 2] V1 + [2 2] V2 + [0] 39.97/13.46 [0 0] [0 0] [3] 39.97/13.46 = [c_5(and^#(isNat(activate(V1)), n__isNat(activate(V2))), 39.97/13.46 isNat^#(activate(V1)), 39.97/13.46 activate^#(V1), 39.97/13.46 activate^#(V2))] 39.97/13.46 39.97/13.46 [and^#(tt(), X)] = [2 0] X + [0] 39.97/13.46 [0 0] [4] 39.97/13.46 >= [2 0] X + [0] 39.97/13.46 [0 0] [3] 39.97/13.46 = [c_8(activate^#(X))] 39.97/13.46 39.97/13.46 39.97/13.46 The strictly oriented rules are moved into the weak component. 39.97/13.46 39.97/13.46 We are left with following problem, upon which TcT provides the 39.97/13.46 certificate YES(O(1),O(n^1)). 39.97/13.46 39.97/13.46 Strict DPs: 39.97/13.46 { activate^#(n__isNat(X)) -> c_2(isNat^#(X)) 39.97/13.46 , isNat^#(n__x(V1, V2)) -> 39.97/13.46 c_5(and^#(isNat(activate(V1)), n__isNat(activate(V2))), 39.97/13.46 isNat^#(activate(V1)), 39.97/13.46 activate^#(V1), 39.97/13.46 activate^#(V2)) 39.97/13.46 , and^#(tt(), X) -> c_8(activate^#(X)) } 39.97/13.46 Weak DPs: 39.97/13.46 { isNat^#(n__plus(V1, V2)) -> 39.97/13.46 c_3(and^#(isNat(activate(V1)), n__isNat(activate(V2))), 39.97/13.46 isNat^#(activate(V1)), 39.97/13.46 activate^#(V1), 39.97/13.46 activate^#(V2)) 39.97/13.46 , isNat^#(n__s(V1)) -> c_4(isNat^#(activate(V1)), activate^#(V1)) } 39.97/13.46 Weak Trs: 39.97/13.46 { activate(X) -> X 39.97/13.46 , activate(n__0()) -> 0() 39.97/13.46 , activate(n__plus(X1, X2)) -> plus(X1, X2) 39.97/13.46 , activate(n__isNat(X)) -> isNat(X) 39.97/13.46 , activate(n__s(X)) -> s(X) 39.97/13.46 , activate(n__x(X1, X2)) -> x(X1, X2) 39.97/13.46 , s(X) -> n__s(X) 39.97/13.46 , plus(X1, X2) -> n__plus(X1, X2) 39.97/13.46 , 0() -> n__0() 39.97/13.46 , x(X1, X2) -> n__x(X1, X2) 39.97/13.46 , and(tt(), X) -> activate(X) 39.97/13.46 , isNat(X) -> n__isNat(X) 39.97/13.46 , isNat(n__0()) -> tt() 39.97/13.46 , isNat(n__plus(V1, V2)) -> 39.97/13.46 and(isNat(activate(V1)), n__isNat(activate(V2))) 39.97/13.46 , isNat(n__s(V1)) -> isNat(activate(V1)) 39.97/13.46 , isNat(n__x(V1, V2)) -> 39.97/13.46 and(isNat(activate(V1)), n__isNat(activate(V2))) } 39.97/13.46 Obligation: 39.97/13.46 innermost runtime complexity 39.97/13.46 Answer: 39.97/13.46 YES(O(1),O(n^1)) 39.97/13.46 39.97/13.46 We use the processor 'matrix interpretation of dimension 2' to 39.97/13.46 orient following rules strictly. 39.97/13.46 39.97/13.46 DPs: 39.97/13.46 { 2: isNat^#(n__x(V1, V2)) -> 39.97/13.46 c_5(and^#(isNat(activate(V1)), n__isNat(activate(V2))), 39.97/13.46 isNat^#(activate(V1)), 39.97/13.46 activate^#(V1), 39.97/13.46 activate^#(V2)) 39.97/13.46 , 4: isNat^#(n__plus(V1, V2)) -> 39.97/13.46 c_3(and^#(isNat(activate(V1)), n__isNat(activate(V2))), 39.97/13.46 isNat^#(activate(V1)), 39.97/13.46 activate^#(V1), 39.97/13.46 activate^#(V2)) 39.97/13.46 , 5: isNat^#(n__s(V1)) -> 39.97/13.46 c_4(isNat^#(activate(V1)), activate^#(V1)) } 39.97/13.46 39.97/13.46 Sub-proof: 39.97/13.46 ---------- 39.97/13.46 The following argument positions are usable: 39.97/13.46 Uargs(c_2) = {1}, Uargs(c_3) = {1, 2, 3, 4}, Uargs(c_4) = {1, 2}, 39.97/13.46 Uargs(c_5) = {1, 2, 3, 4}, Uargs(c_8) = {1} 39.97/13.46 39.97/13.46 TcT has computed the following constructor-based matrix 39.97/13.46 interpretation satisfying not(EDA) and not(IDA(1)). 39.97/13.46 39.97/13.46 [tt] = [0] 39.97/13.46 [0] 39.97/13.46 39.97/13.46 [activate](x1) = [1 0] x1 + [0] 39.97/13.46 [0 2] [0] 39.97/13.46 39.97/13.46 [s](x1) = [1 1] x1 + [2] 39.97/13.46 [0 0] [0] 39.97/13.46 39.97/13.46 [plus](x1, x2) = [1 1] x1 + [1 1] x2 + [4] 39.97/13.46 [0 0] [0 0] [0] 39.97/13.46 39.97/13.46 [0] = [0] 39.97/13.46 [0] 39.97/13.46 39.97/13.46 [x](x1, x2) = [1 1] x1 + [1 1] x2 + [6] 39.97/13.46 [0 0] [0 0] [0] 39.97/13.46 39.97/13.46 [and](x1, x2) = [1 0] x2 + [0] 39.97/13.46 [0 2] [4] 39.97/13.46 39.97/13.46 [isNat](x1) = [0 0] x1 + [0] 39.97/13.46 [2 0] [0] 39.97/13.46 39.97/13.46 [n__0] = [0] 39.97/13.46 [0] 39.97/13.46 39.97/13.46 [n__plus](x1, x2) = [1 1] x1 + [1 1] x2 + [4] 39.97/13.46 [0 0] [0 0] [0] 39.97/13.46 39.97/13.46 [n__isNat](x1) = [0 0] x1 + [0] 39.97/13.46 [1 0] [0] 39.97/13.46 39.97/13.46 [n__s](x1) = [1 1] x1 + [2] 39.97/13.46 [0 0] [0] 39.97/13.46 39.97/13.46 [n__x](x1, x2) = [1 1] x1 + [1 1] x2 + [6] 39.97/13.46 [0 0] [0 0] [0] 39.97/13.46 39.97/13.46 [activate^#](x1) = [0 2] x1 + [1] 39.97/13.46 [0 0] [1] 39.97/13.46 39.97/13.46 [isNat^#](x1) = [2 0] x1 + [0] 39.97/13.46 [0 0] [4] 39.97/13.46 39.97/13.46 [and^#](x1, x2) = [0 2] x2 + [1] 39.97/13.46 [0 0] [4] 39.97/13.46 39.97/13.46 [c_2](x1) = [1 0] x1 + [1] 39.97/13.46 [0 0] [1] 39.97/13.46 39.97/13.46 [c_3](x1, x2, x3, x4) = [1 0] x1 + [1 0] x2 + [1 0] x3 + [1 39.97/13.46 0] x4 + [4] 39.97/13.46 [0 0] [0 0] [0 0] [0 39.97/13.46 0] [3] 39.97/13.46 39.97/13.46 [c_4](x1, x2) = [1 0] x1 + [1 0] x2 + [1] 39.97/13.46 [0 0] [0 0] [3] 39.97/13.46 39.97/13.46 [c_5](x1, x2, x3, x4) = [1 0] x1 + [1 0] x2 + [1 4] x3 + [1 39.97/13.46 1] x4 + [1] 39.97/13.46 [0 0] [0 0] [0 0] [0 39.97/13.46 0] [3] 39.97/13.46 39.97/13.46 [c_8](x1) = [1 0] x1 + [0] 39.97/13.46 [0 0] [3] 39.97/13.46 39.97/13.46 The order satisfies the following ordering constraints: 39.97/13.46 39.97/13.46 [activate(X)] = [1 0] X + [0] 39.97/13.46 [0 2] [0] 39.97/13.46 >= [1 0] X + [0] 39.97/13.46 [0 1] [0] 39.97/13.46 = [X] 39.97/13.46 39.97/13.46 [activate(n__0())] = [0] 39.97/13.46 [0] 39.97/13.46 >= [0] 39.97/13.46 [0] 39.97/13.46 = [0()] 39.97/13.46 39.97/13.46 [activate(n__plus(X1, X2))] = [1 1] X1 + [1 1] X2 + [4] 39.97/13.46 [0 0] [0 0] [0] 39.97/13.46 >= [1 1] X1 + [1 1] X2 + [4] 39.97/13.46 [0 0] [0 0] [0] 39.97/13.46 = [plus(X1, X2)] 39.97/13.46 39.97/13.46 [activate(n__isNat(X))] = [0 0] X + [0] 39.97/13.46 [2 0] [0] 39.97/13.46 >= [0 0] X + [0] 39.97/13.46 [2 0] [0] 39.97/13.46 = [isNat(X)] 39.97/13.46 39.97/13.46 [activate(n__s(X))] = [1 1] X + [2] 39.97/13.46 [0 0] [0] 39.97/13.46 >= [1 1] X + [2] 39.97/13.46 [0 0] [0] 39.97/13.46 = [s(X)] 39.97/13.46 39.97/13.46 [activate(n__x(X1, X2))] = [1 1] X1 + [1 1] X2 + [6] 39.97/13.46 [0 0] [0 0] [0] 39.97/13.46 >= [1 1] X1 + [1 1] X2 + [6] 39.97/13.46 [0 0] [0 0] [0] 39.97/13.46 = [x(X1, X2)] 39.97/13.46 39.97/13.46 [s(X)] = [1 1] X + [2] 39.97/13.46 [0 0] [0] 39.97/13.46 >= [1 1] X + [2] 39.97/13.46 [0 0] [0] 39.97/13.46 = [n__s(X)] 39.97/13.46 39.97/13.46 [plus(X1, X2)] = [1 1] X1 + [1 1] X2 + [4] 39.97/13.46 [0 0] [0 0] [0] 39.97/13.46 >= [1 1] X1 + [1 1] X2 + [4] 39.97/13.46 [0 0] [0 0] [0] 39.97/13.46 = [n__plus(X1, X2)] 39.97/13.46 39.97/13.46 [0()] = [0] 39.97/13.46 [0] 39.97/13.46 >= [0] 39.97/13.46 [0] 39.97/13.46 = [n__0()] 39.97/13.46 39.97/13.46 [x(X1, X2)] = [1 1] X1 + [1 1] X2 + [6] 39.97/13.46 [0 0] [0 0] [0] 39.97/13.46 >= [1 1] X1 + [1 1] X2 + [6] 39.97/13.46 [0 0] [0 0] [0] 39.97/13.46 = [n__x(X1, X2)] 39.97/13.46 39.97/13.46 [and(tt(), X)] = [1 0] X + [0] 39.97/13.46 [0 2] [4] 39.97/13.46 >= [1 0] X + [0] 39.97/13.46 [0 2] [0] 39.97/13.46 = [activate(X)] 39.97/13.46 39.97/13.46 [isNat(X)] = [0 0] X + [0] 39.97/13.46 [2 0] [0] 39.97/13.46 >= [0 0] X + [0] 39.97/13.46 [1 0] [0] 39.97/13.46 = [n__isNat(X)] 39.97/13.46 39.97/13.46 [isNat(n__0())] = [0] 39.97/13.46 [0] 39.97/13.46 >= [0] 39.97/13.46 [0] 39.97/13.46 = [tt()] 39.97/13.46 39.97/13.46 [isNat(n__plus(V1, V2))] = [0 0] V1 + [0 0] V2 + [0] 39.97/13.46 [2 2] [2 2] [8] 39.97/13.46 >= [0 0] V2 + [0] 39.97/13.46 [2 0] [4] 39.97/13.46 = [and(isNat(activate(V1)), n__isNat(activate(V2)))] 39.97/13.46 39.97/13.46 [isNat(n__s(V1))] = [0 0] V1 + [0] 39.97/13.46 [2 2] [4] 39.97/13.46 >= [0 0] V1 + [0] 39.97/13.46 [2 0] [0] 39.97/13.46 = [isNat(activate(V1))] 39.97/13.46 39.97/13.46 [isNat(n__x(V1, V2))] = [0 0] V1 + [0 0] V2 + [0] 39.97/13.46 [2 2] [2 2] [12] 39.97/13.46 >= [0 0] V2 + [0] 39.97/13.46 [2 0] [4] 39.97/13.46 = [and(isNat(activate(V1)), n__isNat(activate(V2)))] 39.97/13.46 39.97/13.46 [activate^#(n__isNat(X))] = [2 0] X + [1] 39.97/13.46 [0 0] [1] 39.97/13.46 >= [2 0] X + [1] 39.97/13.46 [0 0] [1] 39.97/13.46 = [c_2(isNat^#(X))] 39.97/13.46 39.97/13.46 [isNat^#(n__plus(V1, V2))] = [2 2] V1 + [2 2] V2 + [8] 39.97/13.46 [0 0] [0 0] [4] 39.97/13.46 > [2 2] V1 + [2 2] V2 + [7] 39.97/13.46 [0 0] [0 0] [3] 39.97/13.46 = [c_3(and^#(isNat(activate(V1)), n__isNat(activate(V2))), 39.97/13.46 isNat^#(activate(V1)), 39.97/13.46 activate^#(V1), 39.97/13.46 activate^#(V2))] 39.97/13.46 39.97/13.46 [isNat^#(n__s(V1))] = [2 2] V1 + [4] 39.97/13.46 [0 0] [4] 39.97/13.46 > [2 2] V1 + [2] 39.97/13.46 [0 0] [3] 39.97/13.46 = [c_4(isNat^#(activate(V1)), activate^#(V1))] 39.97/13.46 39.97/13.46 [isNat^#(n__x(V1, V2))] = [2 2] V1 + [2 2] V2 + [12] 39.97/13.46 [0 0] [0 0] [4] 39.97/13.46 > [2 2] V1 + [2 2] V2 + [9] 39.97/13.46 [0 0] [0 0] [3] 39.97/13.46 = [c_5(and^#(isNat(activate(V1)), n__isNat(activate(V2))), 39.97/13.46 isNat^#(activate(V1)), 39.97/13.46 activate^#(V1), 39.97/13.46 activate^#(V2))] 39.97/13.46 39.97/13.46 [and^#(tt(), X)] = [0 2] X + [1] 39.97/13.46 [0 0] [4] 39.97/13.46 >= [0 2] X + [1] 39.97/13.46 [0 0] [3] 39.97/13.46 = [c_8(activate^#(X))] 39.97/13.46 39.97/13.46 39.97/13.46 We return to the main proof. Consider the set of all dependency 39.97/13.46 pairs 39.97/13.46 39.97/13.46 : 39.97/13.46 { 1: activate^#(n__isNat(X)) -> c_2(isNat^#(X)) 39.97/13.46 , 2: isNat^#(n__x(V1, V2)) -> 39.97/13.46 c_5(and^#(isNat(activate(V1)), n__isNat(activate(V2))), 39.97/13.46 isNat^#(activate(V1)), 39.97/13.46 activate^#(V1), 39.97/13.46 activate^#(V2)) 39.97/13.46 , 3: and^#(tt(), X) -> c_8(activate^#(X)) 39.97/13.46 , 4: isNat^#(n__plus(V1, V2)) -> 39.97/13.46 c_3(and^#(isNat(activate(V1)), n__isNat(activate(V2))), 39.97/13.46 isNat^#(activate(V1)), 39.97/13.46 activate^#(V1), 39.97/13.46 activate^#(V2)) 39.97/13.46 , 5: isNat^#(n__s(V1)) -> 39.97/13.46 c_4(isNat^#(activate(V1)), activate^#(V1)) } 39.97/13.46 39.97/13.46 Processor 'matrix interpretation of dimension 2' induces the 39.97/13.46 complexity certificate YES(?,O(n^1)) on application of dependency 39.97/13.46 pairs {2,4,5}. These cover all (indirect) predecessors of 39.97/13.46 dependency pairs {1,2,3,4,5}, their number of application is 39.97/13.46 equally bounded. The dependency pairs are shifted into the weak 39.97/13.46 component. 39.97/13.46 39.97/13.46 We are left with following problem, upon which TcT provides the 39.97/13.46 certificate YES(O(1),O(1)). 39.97/13.46 39.97/13.46 Weak DPs: 39.97/13.46 { activate^#(n__isNat(X)) -> c_2(isNat^#(X)) 39.97/13.46 , isNat^#(n__plus(V1, V2)) -> 39.97/13.46 c_3(and^#(isNat(activate(V1)), n__isNat(activate(V2))), 39.97/13.46 isNat^#(activate(V1)), 39.97/13.46 activate^#(V1), 39.97/13.46 activate^#(V2)) 39.97/13.46 , isNat^#(n__s(V1)) -> c_4(isNat^#(activate(V1)), activate^#(V1)) 39.97/13.46 , isNat^#(n__x(V1, V2)) -> 39.97/13.46 c_5(and^#(isNat(activate(V1)), n__isNat(activate(V2))), 39.97/13.46 isNat^#(activate(V1)), 39.97/13.46 activate^#(V1), 39.97/13.46 activate^#(V2)) 39.97/13.46 , and^#(tt(), X) -> c_8(activate^#(X)) } 39.97/13.46 Weak Trs: 39.97/13.46 { activate(X) -> X 39.97/13.46 , activate(n__0()) -> 0() 39.97/13.46 , activate(n__plus(X1, X2)) -> plus(X1, X2) 39.97/13.46 , activate(n__isNat(X)) -> isNat(X) 39.97/13.46 , activate(n__s(X)) -> s(X) 39.97/13.46 , activate(n__x(X1, X2)) -> x(X1, X2) 39.97/13.46 , s(X) -> n__s(X) 39.97/13.46 , plus(X1, X2) -> n__plus(X1, X2) 39.97/13.46 , 0() -> n__0() 39.97/13.46 , x(X1, X2) -> n__x(X1, X2) 39.97/13.46 , and(tt(), X) -> activate(X) 39.97/13.46 , isNat(X) -> n__isNat(X) 39.97/13.46 , isNat(n__0()) -> tt() 39.97/13.46 , isNat(n__plus(V1, V2)) -> 39.97/13.46 and(isNat(activate(V1)), n__isNat(activate(V2))) 39.97/13.46 , isNat(n__s(V1)) -> isNat(activate(V1)) 39.97/13.46 , isNat(n__x(V1, V2)) -> 39.97/13.46 and(isNat(activate(V1)), n__isNat(activate(V2))) } 39.97/13.46 Obligation: 39.97/13.46 innermost runtime complexity 39.97/13.46 Answer: 39.97/13.46 YES(O(1),O(1)) 39.97/13.46 39.97/13.46 The following weak DPs constitute a sub-graph of the DG that is 39.97/13.46 closed under successors. The DPs are removed. 39.97/13.46 39.97/13.46 { activate^#(n__isNat(X)) -> c_2(isNat^#(X)) 39.97/13.46 , isNat^#(n__plus(V1, V2)) -> 39.97/13.46 c_3(and^#(isNat(activate(V1)), n__isNat(activate(V2))), 39.97/13.46 isNat^#(activate(V1)), 39.97/13.46 activate^#(V1), 39.97/13.46 activate^#(V2)) 39.97/13.46 , isNat^#(n__s(V1)) -> c_4(isNat^#(activate(V1)), activate^#(V1)) 39.97/13.46 , isNat^#(n__x(V1, V2)) -> 39.97/13.46 c_5(and^#(isNat(activate(V1)), n__isNat(activate(V2))), 39.97/13.46 isNat^#(activate(V1)), 39.97/13.46 activate^#(V1), 39.97/13.46 activate^#(V2)) 39.97/13.46 , and^#(tt(), X) -> c_8(activate^#(X)) } 39.97/13.46 39.97/13.46 We are left with following problem, upon which TcT provides the 39.97/13.46 certificate YES(O(1),O(1)). 39.97/13.46 39.97/13.46 Weak Trs: 39.97/13.46 { activate(X) -> X 39.97/13.46 , activate(n__0()) -> 0() 39.97/13.46 , activate(n__plus(X1, X2)) -> plus(X1, X2) 39.97/13.46 , activate(n__isNat(X)) -> isNat(X) 39.97/13.46 , activate(n__s(X)) -> s(X) 39.97/13.46 , activate(n__x(X1, X2)) -> x(X1, X2) 39.97/13.46 , s(X) -> n__s(X) 39.97/13.46 , plus(X1, X2) -> n__plus(X1, X2) 39.97/13.46 , 0() -> n__0() 39.97/13.46 , x(X1, X2) -> n__x(X1, X2) 39.97/13.46 , and(tt(), X) -> activate(X) 39.97/13.46 , isNat(X) -> n__isNat(X) 39.97/13.46 , isNat(n__0()) -> tt() 39.97/13.46 , isNat(n__plus(V1, V2)) -> 39.97/13.46 and(isNat(activate(V1)), n__isNat(activate(V2))) 39.97/13.46 , isNat(n__s(V1)) -> isNat(activate(V1)) 39.97/13.46 , isNat(n__x(V1, V2)) -> 39.97/13.46 and(isNat(activate(V1)), n__isNat(activate(V2))) } 39.97/13.46 Obligation: 39.97/13.46 innermost runtime complexity 39.97/13.46 Answer: 39.97/13.46 YES(O(1),O(1)) 39.97/13.46 39.97/13.46 No rule is usable, rules are removed from the input problem. 39.97/13.46 39.97/13.46 We are left with following problem, upon which TcT provides the 39.97/13.46 certificate YES(O(1),O(1)). 39.97/13.46 39.97/13.46 Rules: Empty 39.97/13.46 Obligation: 39.97/13.46 innermost runtime complexity 39.97/13.46 Answer: 39.97/13.46 YES(O(1),O(1)) 39.97/13.46 39.97/13.46 Empty rules are trivially bounded 39.97/13.46 39.97/13.46 Hurray, we answered YES(O(1),O(n^1)) 39.97/13.49 EOF