MAYBE 1182.37/297.16 MAYBE 1182.37/297.16 1182.37/297.16 We are left with following problem, upon which TcT provides the 1182.37/297.16 certificate MAYBE. 1182.37/297.16 1182.37/297.16 Strict Trs: 1182.37/297.16 { a__U11(X1, X2, X3) -> U11(X1, X2, X3) 1182.37/297.16 , a__U11(tt(), V1, V2) -> a__U12(a__isNat(V1), V2) 1182.37/297.16 , a__U12(X1, X2) -> U12(X1, X2) 1182.37/297.16 , a__U12(tt(), V2) -> a__U13(a__isNat(V2)) 1182.37/297.16 , a__isNat(X) -> isNat(X) 1182.37/297.16 , a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 1182.37/297.16 , a__isNat(0()) -> tt() 1182.37/297.16 , a__isNat(plus(V1, V2)) -> 1182.37/297.16 a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2) 1182.37/297.16 , a__U13(X) -> U13(X) 1182.37/297.16 , a__U13(tt()) -> tt() 1182.37/297.16 , a__U21(X1, X2) -> U21(X1, X2) 1182.37/297.16 , a__U21(tt(), V1) -> a__U22(a__isNat(V1)) 1182.37/297.16 , a__U22(X) -> U22(X) 1182.37/297.16 , a__U22(tt()) -> tt() 1182.37/297.16 , a__U31(X1, X2) -> U31(X1, X2) 1182.37/297.16 , a__U31(tt(), N) -> mark(N) 1182.37/297.16 , mark(tt()) -> tt() 1182.37/297.16 , mark(s(X)) -> s(mark(X)) 1182.37/297.16 , mark(0()) -> 0() 1182.37/297.16 , mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 1182.37/297.16 , mark(isNatKind(X)) -> a__isNatKind(X) 1182.37/297.16 , mark(and(X1, X2)) -> a__and(mark(X1), X2) 1182.37/297.16 , mark(isNat(X)) -> a__isNat(X) 1182.37/297.16 , mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) 1182.37/297.16 , mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 1182.37/297.16 , mark(U13(X)) -> a__U13(mark(X)) 1182.37/297.16 , mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 1182.37/297.16 , mark(U22(X)) -> a__U22(mark(X)) 1182.37/297.16 , mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 1182.37/297.16 , mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 1182.37/297.16 , a__U41(X1, X2, X3) -> U41(X1, X2, X3) 1182.37/297.16 , a__U41(tt(), M, N) -> s(a__plus(mark(N), mark(M))) 1182.37/297.16 , a__plus(X1, X2) -> plus(X1, X2) 1182.37/297.16 , a__plus(N, s(M)) -> 1182.37/297.16 a__U41(a__and(a__and(a__isNat(M), isNatKind(M)), 1182.37/297.16 and(isNat(N), isNatKind(N))), 1182.37/297.16 M, 1182.37/297.16 N) 1182.37/297.16 , a__plus(N, 0()) -> a__U31(a__and(a__isNat(N), isNatKind(N)), N) 1182.37/297.16 , a__and(X1, X2) -> and(X1, X2) 1182.37/297.16 , a__and(tt(), X) -> mark(X) 1182.37/297.16 , a__isNatKind(X) -> isNatKind(X) 1182.37/297.16 , a__isNatKind(s(V1)) -> a__isNatKind(V1) 1182.37/297.16 , a__isNatKind(0()) -> tt() 1182.37/297.16 , a__isNatKind(plus(V1, V2)) -> 1182.37/297.16 a__and(a__isNatKind(V1), isNatKind(V2)) } 1182.37/297.16 Obligation: 1182.37/297.16 runtime complexity 1182.37/297.16 Answer: 1182.37/297.16 MAYBE 1182.37/297.16 1182.37/297.16 None of the processors succeeded. 1182.37/297.16 1182.37/297.16 Details of failed attempt(s): 1182.37/297.16 ----------------------------- 1182.37/297.16 1) 'With Problem ... (timeout of 297 seconds)' failed due to the 1182.37/297.16 following reason: 1182.37/297.16 1182.37/297.16 Computation stopped due to timeout after 297.0 seconds. 1182.37/297.16 1182.37/297.16 2) 'Best' failed due to the following reason: 1182.37/297.16 1182.37/297.16 None of the processors succeeded. 1182.37/297.16 1182.37/297.16 Details of failed attempt(s): 1182.37/297.16 ----------------------------- 1182.37/297.16 1) 'With Problem ... (timeout of 148 seconds) (timeout of 297 1182.37/297.16 seconds)' failed due to the following reason: 1182.37/297.16 1182.37/297.16 Computation stopped due to timeout after 148.0 seconds. 1182.37/297.16 1182.37/297.16 2) 'Best' failed due to the following reason: 1182.37/297.16 1182.37/297.16 None of the processors succeeded. 1182.37/297.16 1182.37/297.16 Details of failed attempt(s): 1182.37/297.16 ----------------------------- 1182.37/297.16 1) 'bsearch-popstar (timeout of 297 seconds)' failed due to the 1182.37/297.16 following reason: 1182.37/297.16 1182.37/297.16 The processor is inapplicable, reason: 1182.37/297.16 Processor only applicable for innermost runtime complexity analysis 1182.37/297.16 1182.37/297.16 2) 'Polynomial Path Order (PS) (timeout of 297 seconds)' failed due 1182.37/297.16 to the following reason: 1182.37/297.16 1182.37/297.16 The processor is inapplicable, reason: 1182.37/297.16 Processor only applicable for innermost runtime complexity analysis 1182.37/297.16 1182.37/297.16 1182.37/297.16 3) 'Fastest (timeout of 24 seconds) (timeout of 297 seconds)' 1182.37/297.16 failed due to the following reason: 1182.37/297.16 1182.37/297.16 None of the processors succeeded. 1182.37/297.16 1182.37/297.16 Details of failed attempt(s): 1182.37/297.16 ----------------------------- 1182.37/297.16 1) 'Bounds with minimal-enrichment and initial automaton 'match'' 1182.37/297.16 failed due to the following reason: 1182.37/297.16 1182.37/297.16 match-boundness of the problem could not be verified. 1182.37/297.16 1182.37/297.16 2) 'Bounds with perSymbol-enrichment and initial automaton 'match'' 1182.37/297.16 failed due to the following reason: 1182.37/297.16 1182.37/297.16 match-boundness of the problem could not be verified. 1182.37/297.16 1182.37/297.16 1182.37/297.16 1182.37/297.16 3) 'Weak Dependency Pairs (timeout of 297 seconds)' failed due to 1182.37/297.16 the following reason: 1182.37/297.16 1182.37/297.16 We add the following weak dependency pairs: 1182.37/297.16 1182.37/297.16 Strict DPs: 1182.37/297.16 { a__U11^#(X1, X2, X3) -> c_1(X1, X2, X3) 1182.37/297.16 , a__U11^#(tt(), V1, V2) -> c_2(a__U12^#(a__isNat(V1), V2)) 1182.37/297.16 , a__U12^#(X1, X2) -> c_3(X1, X2) 1182.37/297.16 , a__U12^#(tt(), V2) -> c_4(a__U13^#(a__isNat(V2))) 1182.37/297.16 , a__U13^#(X) -> c_9(X) 1182.37/297.16 , a__U13^#(tt()) -> c_10() 1182.37/297.16 , a__isNat^#(X) -> c_5(X) 1182.37/297.16 , a__isNat^#(s(V1)) -> c_6(a__U21^#(a__isNatKind(V1), V1)) 1182.37/297.16 , a__isNat^#(0()) -> c_7() 1182.37/297.16 , a__isNat^#(plus(V1, V2)) -> 1182.37/297.16 c_8(a__U11^#(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)) 1182.37/297.16 , a__U21^#(X1, X2) -> c_11(X1, X2) 1182.37/297.16 , a__U21^#(tt(), V1) -> c_12(a__U22^#(a__isNat(V1))) 1182.37/297.16 , a__U22^#(X) -> c_13(X) 1182.37/297.16 , a__U22^#(tt()) -> c_14() 1182.37/297.16 , a__U31^#(X1, X2) -> c_15(X1, X2) 1182.37/297.16 , a__U31^#(tt(), N) -> c_16(mark^#(N)) 1182.37/297.16 , mark^#(tt()) -> c_17() 1182.37/297.16 , mark^#(s(X)) -> c_18(mark^#(X)) 1182.37/297.16 , mark^#(0()) -> c_19() 1182.37/297.16 , mark^#(plus(X1, X2)) -> c_20(a__plus^#(mark(X1), mark(X2))) 1182.37/297.16 , mark^#(isNatKind(X)) -> c_21(a__isNatKind^#(X)) 1182.37/297.16 , mark^#(and(X1, X2)) -> c_22(a__and^#(mark(X1), X2)) 1182.37/297.16 , mark^#(isNat(X)) -> c_23(a__isNat^#(X)) 1182.37/297.16 , mark^#(U11(X1, X2, X3)) -> c_24(a__U11^#(mark(X1), X2, X3)) 1182.37/297.16 , mark^#(U12(X1, X2)) -> c_25(a__U12^#(mark(X1), X2)) 1182.37/297.16 , mark^#(U13(X)) -> c_26(a__U13^#(mark(X))) 1182.37/297.16 , mark^#(U21(X1, X2)) -> c_27(a__U21^#(mark(X1), X2)) 1182.37/297.16 , mark^#(U22(X)) -> c_28(a__U22^#(mark(X))) 1182.37/297.16 , mark^#(U31(X1, X2)) -> c_29(a__U31^#(mark(X1), X2)) 1182.37/297.16 , mark^#(U41(X1, X2, X3)) -> c_30(a__U41^#(mark(X1), X2, X3)) 1182.37/297.16 , a__plus^#(X1, X2) -> c_33(X1, X2) 1182.37/297.16 , a__plus^#(N, s(M)) -> 1182.37/297.16 c_34(a__U41^#(a__and(a__and(a__isNat(M), isNatKind(M)), 1182.37/297.16 and(isNat(N), isNatKind(N))), 1182.37/297.16 M, 1182.37/297.16 N)) 1182.37/297.16 , a__plus^#(N, 0()) -> 1182.37/297.16 c_35(a__U31^#(a__and(a__isNat(N), isNatKind(N)), N)) 1182.37/297.16 , a__isNatKind^#(X) -> c_38(X) 1182.37/297.16 , a__isNatKind^#(s(V1)) -> c_39(a__isNatKind^#(V1)) 1182.37/297.16 , a__isNatKind^#(0()) -> c_40() 1182.37/297.16 , a__isNatKind^#(plus(V1, V2)) -> 1182.37/297.16 c_41(a__and^#(a__isNatKind(V1), isNatKind(V2))) 1182.37/297.16 , a__and^#(X1, X2) -> c_36(X1, X2) 1182.37/297.16 , a__and^#(tt(), X) -> c_37(mark^#(X)) 1182.37/297.16 , a__U41^#(X1, X2, X3) -> c_31(X1, X2, X3) 1182.37/297.16 , a__U41^#(tt(), M, N) -> c_32(a__plus^#(mark(N), mark(M))) } 1182.37/297.16 1182.37/297.16 and mark the set of starting terms. 1182.37/297.16 1182.37/297.16 We are left with following problem, upon which TcT provides the 1182.37/297.16 certificate MAYBE. 1182.37/297.16 1182.37/297.16 Strict DPs: 1182.37/297.16 { a__U11^#(X1, X2, X3) -> c_1(X1, X2, X3) 1182.37/297.16 , a__U11^#(tt(), V1, V2) -> c_2(a__U12^#(a__isNat(V1), V2)) 1182.37/297.16 , a__U12^#(X1, X2) -> c_3(X1, X2) 1182.37/297.16 , a__U12^#(tt(), V2) -> c_4(a__U13^#(a__isNat(V2))) 1182.37/297.16 , a__U13^#(X) -> c_9(X) 1182.37/297.16 , a__U13^#(tt()) -> c_10() 1182.37/297.16 , a__isNat^#(X) -> c_5(X) 1182.37/297.16 , a__isNat^#(s(V1)) -> c_6(a__U21^#(a__isNatKind(V1), V1)) 1182.37/297.16 , a__isNat^#(0()) -> c_7() 1182.37/297.16 , a__isNat^#(plus(V1, V2)) -> 1182.37/297.16 c_8(a__U11^#(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)) 1182.37/297.16 , a__U21^#(X1, X2) -> c_11(X1, X2) 1182.37/297.16 , a__U21^#(tt(), V1) -> c_12(a__U22^#(a__isNat(V1))) 1182.37/297.16 , a__U22^#(X) -> c_13(X) 1182.37/297.16 , a__U22^#(tt()) -> c_14() 1182.37/297.16 , a__U31^#(X1, X2) -> c_15(X1, X2) 1182.37/297.16 , a__U31^#(tt(), N) -> c_16(mark^#(N)) 1182.37/297.16 , mark^#(tt()) -> c_17() 1182.37/297.16 , mark^#(s(X)) -> c_18(mark^#(X)) 1182.37/297.16 , mark^#(0()) -> c_19() 1182.37/297.16 , mark^#(plus(X1, X2)) -> c_20(a__plus^#(mark(X1), mark(X2))) 1182.37/297.16 , mark^#(isNatKind(X)) -> c_21(a__isNatKind^#(X)) 1182.37/297.16 , mark^#(and(X1, X2)) -> c_22(a__and^#(mark(X1), X2)) 1182.37/297.16 , mark^#(isNat(X)) -> c_23(a__isNat^#(X)) 1182.37/297.16 , mark^#(U11(X1, X2, X3)) -> c_24(a__U11^#(mark(X1), X2, X3)) 1182.37/297.16 , mark^#(U12(X1, X2)) -> c_25(a__U12^#(mark(X1), X2)) 1182.37/297.16 , mark^#(U13(X)) -> c_26(a__U13^#(mark(X))) 1182.37/297.16 , mark^#(U21(X1, X2)) -> c_27(a__U21^#(mark(X1), X2)) 1182.37/297.16 , mark^#(U22(X)) -> c_28(a__U22^#(mark(X))) 1182.37/297.16 , mark^#(U31(X1, X2)) -> c_29(a__U31^#(mark(X1), X2)) 1182.37/297.16 , mark^#(U41(X1, X2, X3)) -> c_30(a__U41^#(mark(X1), X2, X3)) 1182.37/297.16 , a__plus^#(X1, X2) -> c_33(X1, X2) 1182.37/297.16 , a__plus^#(N, s(M)) -> 1182.37/297.16 c_34(a__U41^#(a__and(a__and(a__isNat(M), isNatKind(M)), 1182.37/297.16 and(isNat(N), isNatKind(N))), 1182.37/297.16 M, 1182.37/297.16 N)) 1182.37/297.17 , a__plus^#(N, 0()) -> 1182.37/297.17 c_35(a__U31^#(a__and(a__isNat(N), isNatKind(N)), N)) 1182.37/297.17 , a__isNatKind^#(X) -> c_38(X) 1182.37/297.17 , a__isNatKind^#(s(V1)) -> c_39(a__isNatKind^#(V1)) 1182.37/297.17 , a__isNatKind^#(0()) -> c_40() 1182.37/297.17 , a__isNatKind^#(plus(V1, V2)) -> 1182.37/297.17 c_41(a__and^#(a__isNatKind(V1), isNatKind(V2))) 1182.37/297.17 , a__and^#(X1, X2) -> c_36(X1, X2) 1182.37/297.17 , a__and^#(tt(), X) -> c_37(mark^#(X)) 1182.37/297.17 , a__U41^#(X1, X2, X3) -> c_31(X1, X2, X3) 1182.37/297.17 , a__U41^#(tt(), M, N) -> c_32(a__plus^#(mark(N), mark(M))) } 1182.37/297.17 Strict Trs: 1182.37/297.17 { a__U11(X1, X2, X3) -> U11(X1, X2, X3) 1182.37/297.17 , a__U11(tt(), V1, V2) -> a__U12(a__isNat(V1), V2) 1182.37/297.17 , a__U12(X1, X2) -> U12(X1, X2) 1182.37/297.17 , a__U12(tt(), V2) -> a__U13(a__isNat(V2)) 1182.37/297.17 , a__isNat(X) -> isNat(X) 1182.37/297.17 , a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 1182.37/297.17 , a__isNat(0()) -> tt() 1182.37/297.17 , a__isNat(plus(V1, V2)) -> 1182.37/297.17 a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2) 1182.37/297.17 , a__U13(X) -> U13(X) 1182.37/297.17 , a__U13(tt()) -> tt() 1182.37/297.17 , a__U21(X1, X2) -> U21(X1, X2) 1182.37/297.17 , a__U21(tt(), V1) -> a__U22(a__isNat(V1)) 1182.37/297.17 , a__U22(X) -> U22(X) 1182.37/297.17 , a__U22(tt()) -> tt() 1182.37/297.17 , a__U31(X1, X2) -> U31(X1, X2) 1182.37/297.17 , a__U31(tt(), N) -> mark(N) 1182.37/297.17 , mark(tt()) -> tt() 1182.37/297.17 , mark(s(X)) -> s(mark(X)) 1182.37/297.17 , mark(0()) -> 0() 1182.37/297.17 , mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 1182.37/297.17 , mark(isNatKind(X)) -> a__isNatKind(X) 1182.37/297.17 , mark(and(X1, X2)) -> a__and(mark(X1), X2) 1182.37/297.17 , mark(isNat(X)) -> a__isNat(X) 1182.37/297.17 , mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) 1182.37/297.17 , mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 1182.37/297.17 , mark(U13(X)) -> a__U13(mark(X)) 1182.37/297.17 , mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 1182.37/297.17 , mark(U22(X)) -> a__U22(mark(X)) 1182.37/297.17 , mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 1182.37/297.17 , mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 1182.37/297.17 , a__U41(X1, X2, X3) -> U41(X1, X2, X3) 1182.37/297.17 , a__U41(tt(), M, N) -> s(a__plus(mark(N), mark(M))) 1182.37/297.17 , a__plus(X1, X2) -> plus(X1, X2) 1182.37/297.17 , a__plus(N, s(M)) -> 1182.37/297.17 a__U41(a__and(a__and(a__isNat(M), isNatKind(M)), 1182.37/297.17 and(isNat(N), isNatKind(N))), 1182.37/297.17 M, 1182.37/297.17 N) 1182.37/297.17 , a__plus(N, 0()) -> a__U31(a__and(a__isNat(N), isNatKind(N)), N) 1182.37/297.17 , a__and(X1, X2) -> and(X1, X2) 1182.37/297.17 , a__and(tt(), X) -> mark(X) 1182.37/297.17 , a__isNatKind(X) -> isNatKind(X) 1182.37/297.17 , a__isNatKind(s(V1)) -> a__isNatKind(V1) 1182.37/297.17 , a__isNatKind(0()) -> tt() 1182.37/297.17 , a__isNatKind(plus(V1, V2)) -> 1182.37/297.17 a__and(a__isNatKind(V1), isNatKind(V2)) } 1182.37/297.17 Obligation: 1182.37/297.17 runtime complexity 1182.37/297.17 Answer: 1182.37/297.17 MAYBE 1182.37/297.17 1182.37/297.17 We estimate the number of application of {6,9,14,17,19,36} by 1182.37/297.17 applications of Pre({6,9,14,17,19,36}) = 1182.37/297.17 {1,3,4,5,7,11,12,13,15,16,18,21,23,26,28,31,34,35,38,39,40}. Here 1182.37/297.17 rules are labeled as follows: 1182.37/297.17 1182.37/297.17 DPs: 1182.37/297.17 { 1: a__U11^#(X1, X2, X3) -> c_1(X1, X2, X3) 1182.37/297.17 , 2: a__U11^#(tt(), V1, V2) -> c_2(a__U12^#(a__isNat(V1), V2)) 1182.37/297.17 , 3: a__U12^#(X1, X2) -> c_3(X1, X2) 1182.37/297.17 , 4: a__U12^#(tt(), V2) -> c_4(a__U13^#(a__isNat(V2))) 1182.37/297.17 , 5: a__U13^#(X) -> c_9(X) 1182.37/297.17 , 6: a__U13^#(tt()) -> c_10() 1182.37/297.17 , 7: a__isNat^#(X) -> c_5(X) 1182.37/297.17 , 8: a__isNat^#(s(V1)) -> c_6(a__U21^#(a__isNatKind(V1), V1)) 1182.37/297.17 , 9: a__isNat^#(0()) -> c_7() 1182.37/297.17 , 10: a__isNat^#(plus(V1, V2)) -> 1182.37/297.17 c_8(a__U11^#(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)) 1182.37/297.17 , 11: a__U21^#(X1, X2) -> c_11(X1, X2) 1182.37/297.17 , 12: a__U21^#(tt(), V1) -> c_12(a__U22^#(a__isNat(V1))) 1182.37/297.17 , 13: a__U22^#(X) -> c_13(X) 1182.37/297.17 , 14: a__U22^#(tt()) -> c_14() 1182.37/297.17 , 15: a__U31^#(X1, X2) -> c_15(X1, X2) 1182.37/297.17 , 16: a__U31^#(tt(), N) -> c_16(mark^#(N)) 1182.37/297.17 , 17: mark^#(tt()) -> c_17() 1182.37/297.17 , 18: mark^#(s(X)) -> c_18(mark^#(X)) 1182.37/297.17 , 19: mark^#(0()) -> c_19() 1182.37/297.17 , 20: mark^#(plus(X1, X2)) -> c_20(a__plus^#(mark(X1), mark(X2))) 1182.37/297.17 , 21: mark^#(isNatKind(X)) -> c_21(a__isNatKind^#(X)) 1182.37/297.17 , 22: mark^#(and(X1, X2)) -> c_22(a__and^#(mark(X1), X2)) 1182.37/297.17 , 23: mark^#(isNat(X)) -> c_23(a__isNat^#(X)) 1182.37/297.17 , 24: mark^#(U11(X1, X2, X3)) -> c_24(a__U11^#(mark(X1), X2, X3)) 1182.37/297.17 , 25: mark^#(U12(X1, X2)) -> c_25(a__U12^#(mark(X1), X2)) 1182.37/297.17 , 26: mark^#(U13(X)) -> c_26(a__U13^#(mark(X))) 1182.37/297.17 , 27: mark^#(U21(X1, X2)) -> c_27(a__U21^#(mark(X1), X2)) 1182.37/297.17 , 28: mark^#(U22(X)) -> c_28(a__U22^#(mark(X))) 1182.37/297.17 , 29: mark^#(U31(X1, X2)) -> c_29(a__U31^#(mark(X1), X2)) 1182.37/297.17 , 30: mark^#(U41(X1, X2, X3)) -> c_30(a__U41^#(mark(X1), X2, X3)) 1182.37/297.17 , 31: a__plus^#(X1, X2) -> c_33(X1, X2) 1182.37/297.17 , 32: a__plus^#(N, s(M)) -> 1182.37/297.17 c_34(a__U41^#(a__and(a__and(a__isNat(M), isNatKind(M)), 1182.37/297.17 and(isNat(N), isNatKind(N))), 1182.37/297.17 M, 1182.37/297.17 N)) 1182.37/297.17 , 33: a__plus^#(N, 0()) -> 1182.37/297.17 c_35(a__U31^#(a__and(a__isNat(N), isNatKind(N)), N)) 1182.37/297.17 , 34: a__isNatKind^#(X) -> c_38(X) 1182.37/297.17 , 35: a__isNatKind^#(s(V1)) -> c_39(a__isNatKind^#(V1)) 1182.37/297.17 , 36: a__isNatKind^#(0()) -> c_40() 1182.37/297.17 , 37: a__isNatKind^#(plus(V1, V2)) -> 1182.37/297.17 c_41(a__and^#(a__isNatKind(V1), isNatKind(V2))) 1182.37/297.17 , 38: a__and^#(X1, X2) -> c_36(X1, X2) 1182.37/297.17 , 39: a__and^#(tt(), X) -> c_37(mark^#(X)) 1182.37/297.17 , 40: a__U41^#(X1, X2, X3) -> c_31(X1, X2, X3) 1182.37/297.17 , 41: a__U41^#(tt(), M, N) -> c_32(a__plus^#(mark(N), mark(M))) } 1182.37/297.17 1182.37/297.17 We are left with following problem, upon which TcT provides the 1182.37/297.17 certificate MAYBE. 1182.37/297.17 1182.37/297.17 Strict DPs: 1182.37/297.17 { a__U11^#(X1, X2, X3) -> c_1(X1, X2, X3) 1182.37/297.17 , a__U11^#(tt(), V1, V2) -> c_2(a__U12^#(a__isNat(V1), V2)) 1182.37/297.17 , a__U12^#(X1, X2) -> c_3(X1, X2) 1182.37/297.17 , a__U12^#(tt(), V2) -> c_4(a__U13^#(a__isNat(V2))) 1182.37/297.17 , a__U13^#(X) -> c_9(X) 1182.37/297.17 , a__isNat^#(X) -> c_5(X) 1182.37/297.17 , a__isNat^#(s(V1)) -> c_6(a__U21^#(a__isNatKind(V1), V1)) 1182.37/297.17 , a__isNat^#(plus(V1, V2)) -> 1182.37/297.17 c_8(a__U11^#(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2)) 1182.37/297.17 , a__U21^#(X1, X2) -> c_11(X1, X2) 1182.37/297.17 , a__U21^#(tt(), V1) -> c_12(a__U22^#(a__isNat(V1))) 1182.37/297.17 , a__U22^#(X) -> c_13(X) 1182.37/297.17 , a__U31^#(X1, X2) -> c_15(X1, X2) 1182.37/297.17 , a__U31^#(tt(), N) -> c_16(mark^#(N)) 1182.37/297.17 , mark^#(s(X)) -> c_18(mark^#(X)) 1182.37/297.17 , mark^#(plus(X1, X2)) -> c_20(a__plus^#(mark(X1), mark(X2))) 1182.37/297.17 , mark^#(isNatKind(X)) -> c_21(a__isNatKind^#(X)) 1182.37/297.17 , mark^#(and(X1, X2)) -> c_22(a__and^#(mark(X1), X2)) 1182.37/297.17 , mark^#(isNat(X)) -> c_23(a__isNat^#(X)) 1182.37/297.17 , mark^#(U11(X1, X2, X3)) -> c_24(a__U11^#(mark(X1), X2, X3)) 1182.37/297.17 , mark^#(U12(X1, X2)) -> c_25(a__U12^#(mark(X1), X2)) 1182.37/297.17 , mark^#(U13(X)) -> c_26(a__U13^#(mark(X))) 1182.37/297.17 , mark^#(U21(X1, X2)) -> c_27(a__U21^#(mark(X1), X2)) 1182.37/297.17 , mark^#(U22(X)) -> c_28(a__U22^#(mark(X))) 1182.37/297.17 , mark^#(U31(X1, X2)) -> c_29(a__U31^#(mark(X1), X2)) 1182.37/297.17 , mark^#(U41(X1, X2, X3)) -> c_30(a__U41^#(mark(X1), X2, X3)) 1182.37/297.17 , a__plus^#(X1, X2) -> c_33(X1, X2) 1182.37/297.17 , a__plus^#(N, s(M)) -> 1182.37/297.17 c_34(a__U41^#(a__and(a__and(a__isNat(M), isNatKind(M)), 1182.37/297.17 and(isNat(N), isNatKind(N))), 1182.37/297.17 M, 1182.37/297.17 N)) 1182.37/297.17 , a__plus^#(N, 0()) -> 1182.37/297.17 c_35(a__U31^#(a__and(a__isNat(N), isNatKind(N)), N)) 1182.37/297.17 , a__isNatKind^#(X) -> c_38(X) 1182.37/297.17 , a__isNatKind^#(s(V1)) -> c_39(a__isNatKind^#(V1)) 1182.37/297.17 , a__isNatKind^#(plus(V1, V2)) -> 1182.37/297.17 c_41(a__and^#(a__isNatKind(V1), isNatKind(V2))) 1182.37/297.17 , a__and^#(X1, X2) -> c_36(X1, X2) 1182.37/297.17 , a__and^#(tt(), X) -> c_37(mark^#(X)) 1182.37/297.17 , a__U41^#(X1, X2, X3) -> c_31(X1, X2, X3) 1182.37/297.17 , a__U41^#(tt(), M, N) -> c_32(a__plus^#(mark(N), mark(M))) } 1182.37/297.17 Strict Trs: 1182.37/297.17 { a__U11(X1, X2, X3) -> U11(X1, X2, X3) 1182.37/297.17 , a__U11(tt(), V1, V2) -> a__U12(a__isNat(V1), V2) 1182.37/297.17 , a__U12(X1, X2) -> U12(X1, X2) 1182.37/297.17 , a__U12(tt(), V2) -> a__U13(a__isNat(V2)) 1182.37/297.17 , a__isNat(X) -> isNat(X) 1182.37/297.17 , a__isNat(s(V1)) -> a__U21(a__isNatKind(V1), V1) 1182.37/297.17 , a__isNat(0()) -> tt() 1182.37/297.17 , a__isNat(plus(V1, V2)) -> 1182.37/297.17 a__U11(a__and(a__isNatKind(V1), isNatKind(V2)), V1, V2) 1182.37/297.17 , a__U13(X) -> U13(X) 1182.37/297.17 , a__U13(tt()) -> tt() 1182.37/297.17 , a__U21(X1, X2) -> U21(X1, X2) 1182.37/297.17 , a__U21(tt(), V1) -> a__U22(a__isNat(V1)) 1182.37/297.17 , a__U22(X) -> U22(X) 1182.37/297.17 , a__U22(tt()) -> tt() 1182.37/297.17 , a__U31(X1, X2) -> U31(X1, X2) 1182.37/297.17 , a__U31(tt(), N) -> mark(N) 1182.37/297.17 , mark(tt()) -> tt() 1182.77/297.23 , mark(s(X)) -> s(mark(X)) 1182.77/297.23 , mark(0()) -> 0() 1182.77/297.23 , mark(plus(X1, X2)) -> a__plus(mark(X1), mark(X2)) 1182.77/297.23 , mark(isNatKind(X)) -> a__isNatKind(X) 1182.77/297.23 , mark(and(X1, X2)) -> a__and(mark(X1), X2) 1182.77/297.23 , mark(isNat(X)) -> a__isNat(X) 1182.77/297.23 , mark(U11(X1, X2, X3)) -> a__U11(mark(X1), X2, X3) 1182.77/297.23 , mark(U12(X1, X2)) -> a__U12(mark(X1), X2) 1182.77/297.23 , mark(U13(X)) -> a__U13(mark(X)) 1182.77/297.23 , mark(U21(X1, X2)) -> a__U21(mark(X1), X2) 1182.77/297.23 , mark(U22(X)) -> a__U22(mark(X)) 1182.77/297.23 , mark(U31(X1, X2)) -> a__U31(mark(X1), X2) 1182.77/297.23 , mark(U41(X1, X2, X3)) -> a__U41(mark(X1), X2, X3) 1182.77/297.23 , a__U41(X1, X2, X3) -> U41(X1, X2, X3) 1182.77/297.23 , a__U41(tt(), M, N) -> s(a__plus(mark(N), mark(M))) 1182.77/297.23 , a__plus(X1, X2) -> plus(X1, X2) 1182.77/297.23 , a__plus(N, s(M)) -> 1182.77/297.23 a__U41(a__and(a__and(a__isNat(M), isNatKind(M)), 1182.77/297.23 and(isNat(N), isNatKind(N))), 1182.77/297.23 M, 1182.77/297.23 N) 1182.77/297.23 , a__plus(N, 0()) -> a__U31(a__and(a__isNat(N), isNatKind(N)), N) 1182.77/297.23 , a__and(X1, X2) -> and(X1, X2) 1182.77/297.24 , a__and(tt(), X) -> mark(X) 1182.77/297.24 , a__isNatKind(X) -> isNatKind(X) 1182.77/297.24 , a__isNatKind(s(V1)) -> a__isNatKind(V1) 1182.77/297.24 , a__isNatKind(0()) -> tt() 1182.77/297.24 , a__isNatKind(plus(V1, V2)) -> 1182.77/297.24 a__and(a__isNatKind(V1), isNatKind(V2)) } 1182.77/297.24 Weak DPs: 1182.77/297.24 { a__U13^#(tt()) -> c_10() 1182.77/297.24 , a__isNat^#(0()) -> c_7() 1182.77/297.24 , a__U22^#(tt()) -> c_14() 1182.77/297.24 , mark^#(tt()) -> c_17() 1182.77/297.24 , mark^#(0()) -> c_19() 1182.77/297.24 , a__isNatKind^#(0()) -> c_40() } 1182.77/297.24 Obligation: 1182.77/297.24 runtime complexity 1182.77/297.24 Answer: 1182.77/297.24 MAYBE 1182.77/297.24 1182.77/297.24 Empty strict component of the problem is NOT empty. 1182.77/297.24 1182.77/297.24 1182.77/297.24 Arrrr.. 1183.88/298.01 EOF