YES We show the termination of the TRS R: a__U11(tt(),V2) -> a__U12(a__isNat(V2)) a__U12(tt()) -> tt() a__U21(tt()) -> tt() a__U31(tt(),N) -> mark(N) a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N) a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M))) a__isNat(|0|()) -> tt() a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) a__isNat(s(V1)) -> a__U21(a__isNat(V1)) a__plus(N,|0|()) -> a__U31(a__isNat(N),N) a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N) mark(U11(X1,X2)) -> a__U11(mark(X1),X2) mark(U12(X)) -> a__U12(mark(X)) mark(isNat(X)) -> a__isNat(X) mark(U21(X)) -> a__U21(mark(X)) mark(U31(X1,X2)) -> a__U31(mark(X1),X2) mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) mark(tt()) -> tt() mark(s(X)) -> s(mark(X)) mark(|0|()) -> |0|() a__U11(X1,X2) -> U11(X1,X2) a__U12(X) -> U12(X) a__isNat(X) -> isNat(X) a__U21(X) -> U21(X) a__U31(X1,X2) -> U31(X1,X2) a__U41(X1,X2,X3) -> U41(X1,X2,X3) a__U42(X1,X2,X3) -> U42(X1,X2,X3) a__plus(X1,X2) -> plus(X1,X2) -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: a__U11#(tt(),V2) -> a__U12#(a__isNat(V2)) p2: a__U11#(tt(),V2) -> a__isNat#(V2) p3: a__U31#(tt(),N) -> mark#(N) p4: a__U41#(tt(),M,N) -> a__U42#(a__isNat(N),M,N) p5: a__U41#(tt(),M,N) -> a__isNat#(N) p6: a__U42#(tt(),M,N) -> a__plus#(mark(N),mark(M)) p7: a__U42#(tt(),M,N) -> mark#(N) p8: a__U42#(tt(),M,N) -> mark#(M) p9: a__isNat#(plus(V1,V2)) -> a__U11#(a__isNat(V1),V2) p10: a__isNat#(plus(V1,V2)) -> a__isNat#(V1) p11: a__isNat#(s(V1)) -> a__U21#(a__isNat(V1)) p12: a__isNat#(s(V1)) -> a__isNat#(V1) p13: a__plus#(N,|0|()) -> a__U31#(a__isNat(N),N) p14: a__plus#(N,|0|()) -> a__isNat#(N) p15: a__plus#(N,s(M)) -> a__U41#(a__isNat(M),M,N) p16: a__plus#(N,s(M)) -> a__isNat#(M) p17: mark#(U11(X1,X2)) -> a__U11#(mark(X1),X2) p18: mark#(U11(X1,X2)) -> mark#(X1) p19: mark#(U12(X)) -> a__U12#(mark(X)) p20: mark#(U12(X)) -> mark#(X) p21: mark#(isNat(X)) -> a__isNat#(X) p22: mark#(U21(X)) -> a__U21#(mark(X)) p23: mark#(U21(X)) -> mark#(X) p24: mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) p25: mark#(U31(X1,X2)) -> mark#(X1) p26: mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) p27: mark#(U41(X1,X2,X3)) -> mark#(X1) p28: mark#(U42(X1,X2,X3)) -> a__U42#(mark(X1),X2,X3) p29: mark#(U42(X1,X2,X3)) -> mark#(X1) p30: mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) p31: mark#(plus(X1,X2)) -> mark#(X1) p32: mark#(plus(X1,X2)) -> mark#(X2) p33: mark#(s(X)) -> mark#(X) and R consists of: r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2)) r2: a__U12(tt()) -> tt() r3: a__U21(tt()) -> tt() r4: a__U31(tt(),N) -> mark(N) r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N) r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M))) r7: a__isNat(|0|()) -> tt() r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1)) r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N) r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N) r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2) r13: mark(U12(X)) -> a__U12(mark(X)) r14: mark(isNat(X)) -> a__isNat(X) r15: mark(U21(X)) -> a__U21(mark(X)) r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2) r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) r20: mark(tt()) -> tt() r21: mark(s(X)) -> s(mark(X)) r22: mark(|0|()) -> |0|() r23: a__U11(X1,X2) -> U11(X1,X2) r24: a__U12(X) -> U12(X) r25: a__isNat(X) -> isNat(X) r26: a__U21(X) -> U21(X) r27: a__U31(X1,X2) -> U31(X1,X2) r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3) r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3) r30: a__plus(X1,X2) -> plus(X1,X2) The estimated dependency graph contains the following SCCs: {p3, p4, p6, p7, p8, p13, p15, p18, p20, p23, p24, p25, p26, p27, p28, p29, p30, p31, p32, p33} {p2, p9, p10, p12} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: a__U42#(tt(),M,N) -> mark#(M) p2: mark#(s(X)) -> mark#(X) p3: mark#(plus(X1,X2)) -> mark#(X2) p4: mark#(plus(X1,X2)) -> mark#(X1) p5: mark#(plus(X1,X2)) -> a__plus#(mark(X1),mark(X2)) p6: a__plus#(N,s(M)) -> a__U41#(a__isNat(M),M,N) p7: a__U41#(tt(),M,N) -> a__U42#(a__isNat(N),M,N) p8: a__U42#(tt(),M,N) -> mark#(N) p9: mark#(U42(X1,X2,X3)) -> mark#(X1) p10: mark#(U42(X1,X2,X3)) -> a__U42#(mark(X1),X2,X3) p11: a__U42#(tt(),M,N) -> a__plus#(mark(N),mark(M)) p12: a__plus#(N,|0|()) -> a__U31#(a__isNat(N),N) p13: a__U31#(tt(),N) -> mark#(N) p14: mark#(U41(X1,X2,X3)) -> mark#(X1) p15: mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) p16: mark#(U31(X1,X2)) -> mark#(X1) p17: mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) p18: mark#(U21(X)) -> mark#(X) p19: mark#(U12(X)) -> mark#(X) p20: mark#(U11(X1,X2)) -> mark#(X1) and R consists of: r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2)) r2: a__U12(tt()) -> tt() r3: a__U21(tt()) -> tt() r4: a__U31(tt(),N) -> mark(N) r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N) r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M))) r7: a__isNat(|0|()) -> tt() r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1)) r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N) r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N) r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2) r13: mark(U12(X)) -> a__U12(mark(X)) r14: mark(isNat(X)) -> a__isNat(X) r15: mark(U21(X)) -> a__U21(mark(X)) r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2) r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) r20: mark(tt()) -> tt() r21: mark(s(X)) -> s(mark(X)) r22: mark(|0|()) -> |0|() r23: a__U11(X1,X2) -> U11(X1,X2) r24: a__U12(X) -> U12(X) r25: a__isNat(X) -> isNat(X) r26: a__U21(X) -> U21(X) r27: a__U31(X1,X2) -> U31(X1,X2) r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3) r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3) r30: a__plus(X1,X2) -> plus(X1,X2) The set of usable rules consists of r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30 Take the reduction pair: weighted path order base order: matrix interpretations: carrier: N^1 order: lexicographic order interpretations: a__U42#_A(x1,x2,x3) = x1 + x2 + x3 + 2 tt_A() = 7 mark#_A(x1) = x1 + 8 s_A(x1) = x1 + 11 plus_A(x1,x2) = x1 + x2 + 3 a__plus#_A(x1,x2) = x1 + x2 + 2 mark_A(x1) = x1 a__U41#_A(x1,x2,x3) = x1 + x2 + x3 + 3 a__isNat_A(x1) = 7 U42_A(x1,x2,x3) = x1 + x2 + x3 + 7 |0|_A() = 15 a__U31#_A(x1,x2) = x1 + x2 + 9 U41_A(x1,x2,x3) = x1 + x2 + x3 + 7 U31_A(x1,x2) = x1 + x2 + 9 U21_A(x1) = x1 U12_A(x1) = x1 U11_A(x1,x2) = x1 a__U11_A(x1,x2) = x1 a__U12_A(x1) = x1 a__U21_A(x1) = x1 a__U31_A(x1,x2) = x1 + x2 + 9 a__U41_A(x1,x2,x3) = x1 + x2 + x3 + 7 a__U42_A(x1,x2,x3) = x1 + x2 + x3 + 7 a__plus_A(x1,x2) = x1 + x2 + 3 isNat_A(x1) = 7 precedence: mark = a__isNat = |0| = U21 = U12 = U11 = a__U11 = a__U12 = a__U21 = a__U31 > plus = a__plus = isNat > tt = U31 > s = U41 = a__U41 = a__U42 > mark# = a__plus# = a__U41# = U42 = a__U31# > a__U42# partial status: pi(a__U42#) = [] pi(tt) = [] pi(mark#) = [1] pi(s) = [] pi(plus) = [] pi(a__plus#) = [2] pi(mark) = [] pi(a__U41#) = [] pi(a__isNat) = [] pi(U42) = [2] pi(|0|) = [] pi(a__U31#) = [2] pi(U41) = [3] pi(U31) = [] pi(U21) = [] pi(U12) = [] pi(U11) = [] pi(a__U11) = [] pi(a__U12) = [] pi(a__U21) = [] pi(a__U31) = [] pi(a__U41) = [3] pi(a__U42) = [] pi(a__plus) = [] pi(isNat) = [] The next rules are strictly ordered: p5 We remove them from the problem. -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: a__U42#(tt(),M,N) -> mark#(M) p2: mark#(s(X)) -> mark#(X) p3: mark#(plus(X1,X2)) -> mark#(X2) p4: mark#(plus(X1,X2)) -> mark#(X1) p5: a__plus#(N,s(M)) -> a__U41#(a__isNat(M),M,N) p6: a__U41#(tt(),M,N) -> a__U42#(a__isNat(N),M,N) p7: a__U42#(tt(),M,N) -> mark#(N) p8: mark#(U42(X1,X2,X3)) -> mark#(X1) p9: mark#(U42(X1,X2,X3)) -> a__U42#(mark(X1),X2,X3) p10: a__U42#(tt(),M,N) -> a__plus#(mark(N),mark(M)) p11: a__plus#(N,|0|()) -> a__U31#(a__isNat(N),N) p12: a__U31#(tt(),N) -> mark#(N) p13: mark#(U41(X1,X2,X3)) -> mark#(X1) p14: mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) p15: mark#(U31(X1,X2)) -> mark#(X1) p16: mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) p17: mark#(U21(X)) -> mark#(X) p18: mark#(U12(X)) -> mark#(X) p19: mark#(U11(X1,X2)) -> mark#(X1) and R consists of: r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2)) r2: a__U12(tt()) -> tt() r3: a__U21(tt()) -> tt() r4: a__U31(tt(),N) -> mark(N) r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N) r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M))) r7: a__isNat(|0|()) -> tt() r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1)) r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N) r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N) r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2) r13: mark(U12(X)) -> a__U12(mark(X)) r14: mark(isNat(X)) -> a__isNat(X) r15: mark(U21(X)) -> a__U21(mark(X)) r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2) r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) r20: mark(tt()) -> tt() r21: mark(s(X)) -> s(mark(X)) r22: mark(|0|()) -> |0|() r23: a__U11(X1,X2) -> U11(X1,X2) r24: a__U12(X) -> U12(X) r25: a__isNat(X) -> isNat(X) r26: a__U21(X) -> U21(X) r27: a__U31(X1,X2) -> U31(X1,X2) r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3) r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3) r30: a__plus(X1,X2) -> plus(X1,X2) The estimated dependency graph contains the following SCCs: {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: a__U42#(tt(),M,N) -> mark#(M) p2: mark#(U11(X1,X2)) -> mark#(X1) p3: mark#(U12(X)) -> mark#(X) p4: mark#(U21(X)) -> mark#(X) p5: mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) p6: a__U31#(tt(),N) -> mark#(N) p7: mark#(U31(X1,X2)) -> mark#(X1) p8: mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) p9: a__U41#(tt(),M,N) -> a__U42#(a__isNat(N),M,N) p10: a__U42#(tt(),M,N) -> a__plus#(mark(N),mark(M)) p11: a__plus#(N,|0|()) -> a__U31#(a__isNat(N),N) p12: a__plus#(N,s(M)) -> a__U41#(a__isNat(M),M,N) p13: a__U42#(tt(),M,N) -> mark#(N) p14: mark#(U41(X1,X2,X3)) -> mark#(X1) p15: mark#(U42(X1,X2,X3)) -> a__U42#(mark(X1),X2,X3) p16: mark#(U42(X1,X2,X3)) -> mark#(X1) p17: mark#(plus(X1,X2)) -> mark#(X1) p18: mark#(plus(X1,X2)) -> mark#(X2) p19: mark#(s(X)) -> mark#(X) and R consists of: r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2)) r2: a__U12(tt()) -> tt() r3: a__U21(tt()) -> tt() r4: a__U31(tt(),N) -> mark(N) r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N) r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M))) r7: a__isNat(|0|()) -> tt() r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1)) r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N) r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N) r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2) r13: mark(U12(X)) -> a__U12(mark(X)) r14: mark(isNat(X)) -> a__isNat(X) r15: mark(U21(X)) -> a__U21(mark(X)) r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2) r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) r20: mark(tt()) -> tt() r21: mark(s(X)) -> s(mark(X)) r22: mark(|0|()) -> |0|() r23: a__U11(X1,X2) -> U11(X1,X2) r24: a__U12(X) -> U12(X) r25: a__isNat(X) -> isNat(X) r26: a__U21(X) -> U21(X) r27: a__U31(X1,X2) -> U31(X1,X2) r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3) r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3) r30: a__plus(X1,X2) -> plus(X1,X2) The set of usable rules consists of r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30 Take the reduction pair: weighted path order base order: matrix interpretations: carrier: N^1 order: lexicographic order interpretations: a__U42#_A(x1,x2,x3) = x1 + x2 + x3 + 1 tt_A() = 4 mark#_A(x1) = x1 + 2 U11_A(x1,x2) = x1 U12_A(x1) = x1 U21_A(x1) = x1 U31_A(x1,x2) = x1 + x2 + 2 a__U31#_A(x1,x2) = x2 + 4 mark_A(x1) = x1 U41_A(x1,x2,x3) = x1 + x2 + x3 + 6 a__U41#_A(x1,x2,x3) = x2 + x3 + 6 a__isNat_A(x1) = 4 a__plus#_A(x1,x2) = x1 + x2 + 1 |0|_A() = 5 s_A(x1) = x1 + 7 U42_A(x1,x2,x3) = x1 + x2 + x3 + 6 plus_A(x1,x2) = x1 + x2 + 3 a__U11_A(x1,x2) = x1 a__U12_A(x1) = x1 a__U21_A(x1) = x1 a__U31_A(x1,x2) = x1 + x2 + 2 a__U41_A(x1,x2,x3) = x1 + x2 + x3 + 6 a__U42_A(x1,x2,x3) = x1 + x2 + x3 + 6 a__plus_A(x1,x2) = x1 + x2 + 3 isNat_A(x1) = 4 precedence: a__isNat = isNat > U21 = a__U21 > plus = a__plus > U41 = U42 = a__U41 = a__U42 > mark > |0| > tt = U11 = a__U11 = a__U12 > mark# > U31 = a__U41# = a__U31 > a__U42# = a__plus# > a__U31# > s > U12 partial status: pi(a__U42#) = [1] pi(tt) = [] pi(mark#) = [] pi(U11) = [] pi(U12) = [] pi(U21) = [] pi(U31) = [] pi(a__U31#) = [] pi(mark) = [1] pi(U41) = [] pi(a__U41#) = [] pi(a__isNat) = [] pi(a__plus#) = [] pi(|0|) = [] pi(s) = [1] pi(U42) = [] pi(plus) = [] pi(a__U11) = [] pi(a__U12) = [] pi(a__U21) = [] pi(a__U31) = [] pi(a__U41) = [] pi(a__U42) = [] pi(a__plus) = [] pi(isNat) = [] The next rules are strictly ordered: p15 We remove them from the problem. -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: a__U42#(tt(),M,N) -> mark#(M) p2: mark#(U11(X1,X2)) -> mark#(X1) p3: mark#(U12(X)) -> mark#(X) p4: mark#(U21(X)) -> mark#(X) p5: mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) p6: a__U31#(tt(),N) -> mark#(N) p7: mark#(U31(X1,X2)) -> mark#(X1) p8: mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) p9: a__U41#(tt(),M,N) -> a__U42#(a__isNat(N),M,N) p10: a__U42#(tt(),M,N) -> a__plus#(mark(N),mark(M)) p11: a__plus#(N,|0|()) -> a__U31#(a__isNat(N),N) p12: a__plus#(N,s(M)) -> a__U41#(a__isNat(M),M,N) p13: a__U42#(tt(),M,N) -> mark#(N) p14: mark#(U41(X1,X2,X3)) -> mark#(X1) p15: mark#(U42(X1,X2,X3)) -> mark#(X1) p16: mark#(plus(X1,X2)) -> mark#(X1) p17: mark#(plus(X1,X2)) -> mark#(X2) p18: mark#(s(X)) -> mark#(X) and R consists of: r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2)) r2: a__U12(tt()) -> tt() r3: a__U21(tt()) -> tt() r4: a__U31(tt(),N) -> mark(N) r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N) r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M))) r7: a__isNat(|0|()) -> tt() r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1)) r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N) r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N) r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2) r13: mark(U12(X)) -> a__U12(mark(X)) r14: mark(isNat(X)) -> a__isNat(X) r15: mark(U21(X)) -> a__U21(mark(X)) r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2) r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) r20: mark(tt()) -> tt() r21: mark(s(X)) -> s(mark(X)) r22: mark(|0|()) -> |0|() r23: a__U11(X1,X2) -> U11(X1,X2) r24: a__U12(X) -> U12(X) r25: a__isNat(X) -> isNat(X) r26: a__U21(X) -> U21(X) r27: a__U31(X1,X2) -> U31(X1,X2) r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3) r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3) r30: a__plus(X1,X2) -> plus(X1,X2) The estimated dependency graph contains the following SCCs: {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: a__U42#(tt(),M,N) -> mark#(M) p2: mark#(s(X)) -> mark#(X) p3: mark#(plus(X1,X2)) -> mark#(X2) p4: mark#(plus(X1,X2)) -> mark#(X1) p5: mark#(U42(X1,X2,X3)) -> mark#(X1) p6: mark#(U41(X1,X2,X3)) -> mark#(X1) p7: mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) p8: a__U41#(tt(),M,N) -> a__U42#(a__isNat(N),M,N) p9: a__U42#(tt(),M,N) -> mark#(N) p10: mark#(U31(X1,X2)) -> mark#(X1) p11: mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) p12: a__U31#(tt(),N) -> mark#(N) p13: mark#(U21(X)) -> mark#(X) p14: mark#(U12(X)) -> mark#(X) p15: mark#(U11(X1,X2)) -> mark#(X1) p16: a__U42#(tt(),M,N) -> a__plus#(mark(N),mark(M)) p17: a__plus#(N,s(M)) -> a__U41#(a__isNat(M),M,N) p18: a__plus#(N,|0|()) -> a__U31#(a__isNat(N),N) and R consists of: r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2)) r2: a__U12(tt()) -> tt() r3: a__U21(tt()) -> tt() r4: a__U31(tt(),N) -> mark(N) r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N) r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M))) r7: a__isNat(|0|()) -> tt() r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1)) r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N) r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N) r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2) r13: mark(U12(X)) -> a__U12(mark(X)) r14: mark(isNat(X)) -> a__isNat(X) r15: mark(U21(X)) -> a__U21(mark(X)) r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2) r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) r20: mark(tt()) -> tt() r21: mark(s(X)) -> s(mark(X)) r22: mark(|0|()) -> |0|() r23: a__U11(X1,X2) -> U11(X1,X2) r24: a__U12(X) -> U12(X) r25: a__isNat(X) -> isNat(X) r26: a__U21(X) -> U21(X) r27: a__U31(X1,X2) -> U31(X1,X2) r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3) r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3) r30: a__plus(X1,X2) -> plus(X1,X2) The set of usable rules consists of r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30 Take the reduction pair: weighted path order base order: matrix interpretations: carrier: N^1 order: lexicographic order interpretations: a__U42#_A(x1,x2,x3) = x2 + x3 + 6 tt_A() = 0 mark#_A(x1) = x1 + 4 s_A(x1) = x1 + 3 plus_A(x1,x2) = x1 + x2 U42_A(x1,x2,x3) = x1 + x2 + x3 + 3 U41_A(x1,x2,x3) = x1 + x2 + x3 + 3 a__U41#_A(x1,x2,x3) = x2 + x3 + 6 mark_A(x1) = x1 a__isNat_A(x1) = 0 U31_A(x1,x2) = x1 + x2 a__U31#_A(x1,x2) = x2 + 4 U21_A(x1) = x1 U12_A(x1) = x1 U11_A(x1,x2) = x1 a__plus#_A(x1,x2) = x1 + x2 + 5 |0|_A() = 0 a__U11_A(x1,x2) = x1 a__U12_A(x1) = x1 a__U21_A(x1) = x1 a__U31_A(x1,x2) = x1 + x2 a__U41_A(x1,x2,x3) = x1 + x2 + x3 + 3 a__U42_A(x1,x2,x3) = x1 + x2 + x3 + 3 a__plus_A(x1,x2) = x1 + x2 isNat_A(x1) = 0 precedence: a__U42# = mark# = plus = U41 = a__U41# = mark = a__isNat = U31 = a__U31# = a__plus# = a__U11 = a__U31 = a__U41 = a__plus > isNat > U12 = |0| = a__U12 > tt = s = U42 = U21 = U11 = a__U21 = a__U42 partial status: pi(a__U42#) = [] pi(tt) = [] pi(mark#) = [] pi(s) = [] pi(plus) = [] pi(U42) = [] pi(U41) = [] pi(a__U41#) = [] pi(mark) = [] pi(a__isNat) = [] pi(U31) = [] pi(a__U31#) = [] pi(U21) = [] pi(U12) = [] pi(U11) = [] pi(a__plus#) = [] pi(|0|) = [] pi(a__U11) = [] pi(a__U12) = [] pi(a__U21) = [] pi(a__U31) = [] pi(a__U41) = [] pi(a__U42) = [] pi(a__plus) = [] pi(isNat) = [] The next rules are strictly ordered: p18 We remove them from the problem. -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: a__U42#(tt(),M,N) -> mark#(M) p2: mark#(s(X)) -> mark#(X) p3: mark#(plus(X1,X2)) -> mark#(X2) p4: mark#(plus(X1,X2)) -> mark#(X1) p5: mark#(U42(X1,X2,X3)) -> mark#(X1) p6: mark#(U41(X1,X2,X3)) -> mark#(X1) p7: mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) p8: a__U41#(tt(),M,N) -> a__U42#(a__isNat(N),M,N) p9: a__U42#(tt(),M,N) -> mark#(N) p10: mark#(U31(X1,X2)) -> mark#(X1) p11: mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) p12: a__U31#(tt(),N) -> mark#(N) p13: mark#(U21(X)) -> mark#(X) p14: mark#(U12(X)) -> mark#(X) p15: mark#(U11(X1,X2)) -> mark#(X1) p16: a__U42#(tt(),M,N) -> a__plus#(mark(N),mark(M)) p17: a__plus#(N,s(M)) -> a__U41#(a__isNat(M),M,N) and R consists of: r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2)) r2: a__U12(tt()) -> tt() r3: a__U21(tt()) -> tt() r4: a__U31(tt(),N) -> mark(N) r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N) r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M))) r7: a__isNat(|0|()) -> tt() r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1)) r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N) r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N) r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2) r13: mark(U12(X)) -> a__U12(mark(X)) r14: mark(isNat(X)) -> a__isNat(X) r15: mark(U21(X)) -> a__U21(mark(X)) r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2) r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) r20: mark(tt()) -> tt() r21: mark(s(X)) -> s(mark(X)) r22: mark(|0|()) -> |0|() r23: a__U11(X1,X2) -> U11(X1,X2) r24: a__U12(X) -> U12(X) r25: a__isNat(X) -> isNat(X) r26: a__U21(X) -> U21(X) r27: a__U31(X1,X2) -> U31(X1,X2) r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3) r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3) r30: a__plus(X1,X2) -> plus(X1,X2) The estimated dependency graph contains the following SCCs: {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: a__U42#(tt(),M,N) -> mark#(M) p2: mark#(U11(X1,X2)) -> mark#(X1) p3: mark#(U12(X)) -> mark#(X) p4: mark#(U21(X)) -> mark#(X) p5: mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) p6: a__U31#(tt(),N) -> mark#(N) p7: mark#(U31(X1,X2)) -> mark#(X1) p8: mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) p9: a__U41#(tt(),M,N) -> a__U42#(a__isNat(N),M,N) p10: a__U42#(tt(),M,N) -> a__plus#(mark(N),mark(M)) p11: a__plus#(N,s(M)) -> a__U41#(a__isNat(M),M,N) p12: a__U42#(tt(),M,N) -> mark#(N) p13: mark#(U41(X1,X2,X3)) -> mark#(X1) p14: mark#(U42(X1,X2,X3)) -> mark#(X1) p15: mark#(plus(X1,X2)) -> mark#(X1) p16: mark#(plus(X1,X2)) -> mark#(X2) p17: mark#(s(X)) -> mark#(X) and R consists of: r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2)) r2: a__U12(tt()) -> tt() r3: a__U21(tt()) -> tt() r4: a__U31(tt(),N) -> mark(N) r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N) r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M))) r7: a__isNat(|0|()) -> tt() r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1)) r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N) r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N) r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2) r13: mark(U12(X)) -> a__U12(mark(X)) r14: mark(isNat(X)) -> a__isNat(X) r15: mark(U21(X)) -> a__U21(mark(X)) r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2) r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) r20: mark(tt()) -> tt() r21: mark(s(X)) -> s(mark(X)) r22: mark(|0|()) -> |0|() r23: a__U11(X1,X2) -> U11(X1,X2) r24: a__U12(X) -> U12(X) r25: a__isNat(X) -> isNat(X) r26: a__U21(X) -> U21(X) r27: a__U31(X1,X2) -> U31(X1,X2) r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3) r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3) r30: a__plus(X1,X2) -> plus(X1,X2) The set of usable rules consists of r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30 Take the reduction pair: weighted path order base order: matrix interpretations: carrier: N^1 order: lexicographic order interpretations: a__U42#_A(x1,x2,x3) = x2 + x3 + 2 tt_A() = 0 mark#_A(x1) = x1 U11_A(x1,x2) = x1 U12_A(x1) = x1 U21_A(x1) = x1 U31_A(x1,x2) = x1 + x2 + 2 a__U31#_A(x1,x2) = x2 + 2 mark_A(x1) = x1 U41_A(x1,x2,x3) = x1 + x2 + x3 + 4 a__U41#_A(x1,x2,x3) = x2 + x3 + 3 a__isNat_A(x1) = 0 a__plus#_A(x1,x2) = x1 + x2 + 1 s_A(x1) = x1 + 2 U42_A(x1,x2,x3) = x1 + x2 + x3 + 4 plus_A(x1,x2) = x1 + x2 + 2 a__U11_A(x1,x2) = x1 a__U12_A(x1) = x1 a__U21_A(x1) = x1 a__U31_A(x1,x2) = x1 + x2 + 2 a__U41_A(x1,x2,x3) = x1 + x2 + x3 + 4 a__U42_A(x1,x2,x3) = x1 + x2 + x3 + 4 a__plus_A(x1,x2) = x1 + x2 + 2 |0|_A() = 1 isNat_A(x1) = 0 precedence: U12 = mark = a__isNat = plus = a__U11 = a__U12 = a__U21 = a__U41 = a__plus = |0| = isNat > U11 = U21 > tt = mark# = U31 = a__U31# = U41 = a__U31 > a__U42# = a__U41# = a__plus# = U42 = a__U42 > s partial status: pi(a__U42#) = [] pi(tt) = [] pi(mark#) = [] pi(U11) = [] pi(U12) = [] pi(U21) = [] pi(U31) = [] pi(a__U31#) = [] pi(mark) = [] pi(U41) = [] pi(a__U41#) = [] pi(a__isNat) = [] pi(a__plus#) = [] pi(s) = [] pi(U42) = [] pi(plus) = [] pi(a__U11) = [] pi(a__U12) = [] pi(a__U21) = [] pi(a__U31) = [] pi(a__U41) = [] pi(a__U42) = [] pi(a__plus) = [] pi(|0|) = [] pi(isNat) = [] The next rules are strictly ordered: p15 We remove them from the problem. -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: a__U42#(tt(),M,N) -> mark#(M) p2: mark#(U11(X1,X2)) -> mark#(X1) p3: mark#(U12(X)) -> mark#(X) p4: mark#(U21(X)) -> mark#(X) p5: mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) p6: a__U31#(tt(),N) -> mark#(N) p7: mark#(U31(X1,X2)) -> mark#(X1) p8: mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) p9: a__U41#(tt(),M,N) -> a__U42#(a__isNat(N),M,N) p10: a__U42#(tt(),M,N) -> a__plus#(mark(N),mark(M)) p11: a__plus#(N,s(M)) -> a__U41#(a__isNat(M),M,N) p12: a__U42#(tt(),M,N) -> mark#(N) p13: mark#(U41(X1,X2,X3)) -> mark#(X1) p14: mark#(U42(X1,X2,X3)) -> mark#(X1) p15: mark#(plus(X1,X2)) -> mark#(X2) p16: mark#(s(X)) -> mark#(X) and R consists of: r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2)) r2: a__U12(tt()) -> tt() r3: a__U21(tt()) -> tt() r4: a__U31(tt(),N) -> mark(N) r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N) r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M))) r7: a__isNat(|0|()) -> tt() r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1)) r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N) r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N) r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2) r13: mark(U12(X)) -> a__U12(mark(X)) r14: mark(isNat(X)) -> a__isNat(X) r15: mark(U21(X)) -> a__U21(mark(X)) r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2) r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) r20: mark(tt()) -> tt() r21: mark(s(X)) -> s(mark(X)) r22: mark(|0|()) -> |0|() r23: a__U11(X1,X2) -> U11(X1,X2) r24: a__U12(X) -> U12(X) r25: a__isNat(X) -> isNat(X) r26: a__U21(X) -> U21(X) r27: a__U31(X1,X2) -> U31(X1,X2) r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3) r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3) r30: a__plus(X1,X2) -> plus(X1,X2) The estimated dependency graph contains the following SCCs: {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: a__U42#(tt(),M,N) -> mark#(M) p2: mark#(s(X)) -> mark#(X) p3: mark#(plus(X1,X2)) -> mark#(X2) p4: mark#(U42(X1,X2,X3)) -> mark#(X1) p5: mark#(U41(X1,X2,X3)) -> mark#(X1) p6: mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) p7: a__U41#(tt(),M,N) -> a__U42#(a__isNat(N),M,N) p8: a__U42#(tt(),M,N) -> mark#(N) p9: mark#(U31(X1,X2)) -> mark#(X1) p10: mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) p11: a__U31#(tt(),N) -> mark#(N) p12: mark#(U21(X)) -> mark#(X) p13: mark#(U12(X)) -> mark#(X) p14: mark#(U11(X1,X2)) -> mark#(X1) p15: a__U42#(tt(),M,N) -> a__plus#(mark(N),mark(M)) p16: a__plus#(N,s(M)) -> a__U41#(a__isNat(M),M,N) and R consists of: r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2)) r2: a__U12(tt()) -> tt() r3: a__U21(tt()) -> tt() r4: a__U31(tt(),N) -> mark(N) r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N) r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M))) r7: a__isNat(|0|()) -> tt() r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1)) r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N) r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N) r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2) r13: mark(U12(X)) -> a__U12(mark(X)) r14: mark(isNat(X)) -> a__isNat(X) r15: mark(U21(X)) -> a__U21(mark(X)) r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2) r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) r20: mark(tt()) -> tt() r21: mark(s(X)) -> s(mark(X)) r22: mark(|0|()) -> |0|() r23: a__U11(X1,X2) -> U11(X1,X2) r24: a__U12(X) -> U12(X) r25: a__isNat(X) -> isNat(X) r26: a__U21(X) -> U21(X) r27: a__U31(X1,X2) -> U31(X1,X2) r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3) r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3) r30: a__plus(X1,X2) -> plus(X1,X2) The set of usable rules consists of r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30 Take the reduction pair: weighted path order base order: matrix interpretations: carrier: N^1 order: lexicographic order interpretations: a__U42#_A(x1,x2,x3) = x2 + x3 + 2 tt_A() = 4 mark#_A(x1) = x1 + 1 s_A(x1) = x1 + 5 plus_A(x1,x2) = x1 + x2 + 5 U42_A(x1,x2,x3) = x1 + x2 + x3 + 6 U41_A(x1,x2,x3) = x1 + x2 + x3 + 6 a__U41#_A(x1,x2,x3) = x2 + x3 + 3 mark_A(x1) = x1 a__isNat_A(x1) = 4 U31_A(x1,x2) = x1 + x2 + 2 a__U31#_A(x1,x2) = x1 + x2 + 1 U21_A(x1) = x1 U12_A(x1) = x1 U11_A(x1,x2) = x1 a__plus#_A(x1,x2) = x1 + x2 a__U11_A(x1,x2) = x1 a__U12_A(x1) = x1 a__U21_A(x1) = x1 a__U31_A(x1,x2) = x1 + x2 + 2 a__U41_A(x1,x2,x3) = x1 + x2 + x3 + 6 a__U42_A(x1,x2,x3) = x1 + x2 + x3 + 6 a__plus_A(x1,x2) = x1 + x2 + 5 |0|_A() = 3 isNat_A(x1) = 4 precedence: a__U42# = tt = plus = mark = a__isNat = U21 = U12 = U11 = a__plus# = a__U11 = a__U12 = a__U21 = a__U31 = a__plus = isNat > |0| > U42 = U41 = a__U41 = a__U42 > s > mark# = a__U41# = a__U31# > U31 partial status: pi(a__U42#) = [] pi(tt) = [] pi(mark#) = [1] pi(s) = [] pi(plus) = [] pi(U42) = [1, 3] pi(U41) = [] pi(a__U41#) = [] pi(mark) = [] pi(a__isNat) = [] pi(U31) = [2] pi(a__U31#) = [2] pi(U21) = [] pi(U12) = [] pi(U11) = [] pi(a__plus#) = [] pi(a__U11) = [] pi(a__U12) = [] pi(a__U21) = [] pi(a__U31) = [] pi(a__U41) = [1, 3] pi(a__U42) = [1, 3] pi(a__plus) = [] pi(|0|) = [] pi(isNat) = [] The next rules are strictly ordered: p1 We remove them from the problem. -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: mark#(s(X)) -> mark#(X) p2: mark#(plus(X1,X2)) -> mark#(X2) p3: mark#(U42(X1,X2,X3)) -> mark#(X1) p4: mark#(U41(X1,X2,X3)) -> mark#(X1) p5: mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) p6: a__U41#(tt(),M,N) -> a__U42#(a__isNat(N),M,N) p7: a__U42#(tt(),M,N) -> mark#(N) p8: mark#(U31(X1,X2)) -> mark#(X1) p9: mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) p10: a__U31#(tt(),N) -> mark#(N) p11: mark#(U21(X)) -> mark#(X) p12: mark#(U12(X)) -> mark#(X) p13: mark#(U11(X1,X2)) -> mark#(X1) p14: a__U42#(tt(),M,N) -> a__plus#(mark(N),mark(M)) p15: a__plus#(N,s(M)) -> a__U41#(a__isNat(M),M,N) and R consists of: r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2)) r2: a__U12(tt()) -> tt() r3: a__U21(tt()) -> tt() r4: a__U31(tt(),N) -> mark(N) r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N) r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M))) r7: a__isNat(|0|()) -> tt() r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1)) r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N) r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N) r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2) r13: mark(U12(X)) -> a__U12(mark(X)) r14: mark(isNat(X)) -> a__isNat(X) r15: mark(U21(X)) -> a__U21(mark(X)) r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2) r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) r20: mark(tt()) -> tt() r21: mark(s(X)) -> s(mark(X)) r22: mark(|0|()) -> |0|() r23: a__U11(X1,X2) -> U11(X1,X2) r24: a__U12(X) -> U12(X) r25: a__isNat(X) -> isNat(X) r26: a__U21(X) -> U21(X) r27: a__U31(X1,X2) -> U31(X1,X2) r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3) r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3) r30: a__plus(X1,X2) -> plus(X1,X2) The estimated dependency graph contains the following SCCs: {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: mark#(s(X)) -> mark#(X) p2: mark#(U11(X1,X2)) -> mark#(X1) p3: mark#(U12(X)) -> mark#(X) p4: mark#(U21(X)) -> mark#(X) p5: mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) p6: a__U31#(tt(),N) -> mark#(N) p7: mark#(U31(X1,X2)) -> mark#(X1) p8: mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) p9: a__U41#(tt(),M,N) -> a__U42#(a__isNat(N),M,N) p10: a__U42#(tt(),M,N) -> a__plus#(mark(N),mark(M)) p11: a__plus#(N,s(M)) -> a__U41#(a__isNat(M),M,N) p12: a__U42#(tt(),M,N) -> mark#(N) p13: mark#(U41(X1,X2,X3)) -> mark#(X1) p14: mark#(U42(X1,X2,X3)) -> mark#(X1) p15: mark#(plus(X1,X2)) -> mark#(X2) and R consists of: r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2)) r2: a__U12(tt()) -> tt() r3: a__U21(tt()) -> tt() r4: a__U31(tt(),N) -> mark(N) r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N) r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M))) r7: a__isNat(|0|()) -> tt() r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1)) r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N) r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N) r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2) r13: mark(U12(X)) -> a__U12(mark(X)) r14: mark(isNat(X)) -> a__isNat(X) r15: mark(U21(X)) -> a__U21(mark(X)) r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2) r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) r20: mark(tt()) -> tt() r21: mark(s(X)) -> s(mark(X)) r22: mark(|0|()) -> |0|() r23: a__U11(X1,X2) -> U11(X1,X2) r24: a__U12(X) -> U12(X) r25: a__isNat(X) -> isNat(X) r26: a__U21(X) -> U21(X) r27: a__U31(X1,X2) -> U31(X1,X2) r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3) r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3) r30: a__plus(X1,X2) -> plus(X1,X2) The set of usable rules consists of r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30 Take the reduction pair: weighted path order base order: matrix interpretations: carrier: N^1 order: lexicographic order interpretations: mark#_A(x1) = x1 s_A(x1) = x1 + 2 U11_A(x1,x2) = x1 U12_A(x1) = x1 U21_A(x1) = x1 U31_A(x1,x2) = x1 + x2 + 2 a__U31#_A(x1,x2) = x2 + 1 mark_A(x1) = x1 tt_A() = 0 U41_A(x1,x2,x3) = x1 + x2 + x3 + 4 a__U41#_A(x1,x2,x3) = x2 + x3 + 4 a__U42#_A(x1,x2,x3) = x2 + x3 + 3 a__isNat_A(x1) = 0 a__plus#_A(x1,x2) = x1 + x2 + 2 U42_A(x1,x2,x3) = x1 + x2 + x3 + 4 plus_A(x1,x2) = x1 + x2 + 2 a__U11_A(x1,x2) = x1 a__U12_A(x1) = x1 a__U21_A(x1) = x1 a__U31_A(x1,x2) = x1 + x2 + 2 a__U41_A(x1,x2,x3) = x1 + x2 + x3 + 4 a__U42_A(x1,x2,x3) = x1 + x2 + x3 + 4 a__plus_A(x1,x2) = x1 + x2 + 2 |0|_A() = 0 isNat_A(x1) = 0 precedence: plus = a__plus > mark = a__U31 > U11 = a__isNat = a__U11 = isNat > a__U41 > U41 > U31 > U21 = a__U21 > tt = a__U12 > U42 = a__U42 = |0| > s > mark# = U12 = a__U31# = a__U41# = a__U42# = a__plus# partial status: pi(mark#) = [] pi(s) = [1] pi(U11) = [] pi(U12) = [] pi(U21) = [] pi(U31) = [] pi(a__U31#) = [] pi(mark) = [1] pi(tt) = [] pi(U41) = [3] pi(a__U41#) = [] pi(a__U42#) = [] pi(a__isNat) = [] pi(a__plus#) = [] pi(U42) = [1, 3] pi(plus) = [] pi(a__U11) = [] pi(a__U12) = [] pi(a__U21) = [] pi(a__U31) = [2] pi(a__U41) = [] pi(a__U42) = [1, 3] pi(a__plus) = [] pi(|0|) = [] pi(isNat) = [] The next rules are strictly ordered: p1 We remove them from the problem. -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: mark#(U11(X1,X2)) -> mark#(X1) p2: mark#(U12(X)) -> mark#(X) p3: mark#(U21(X)) -> mark#(X) p4: mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) p5: a__U31#(tt(),N) -> mark#(N) p6: mark#(U31(X1,X2)) -> mark#(X1) p7: mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) p8: a__U41#(tt(),M,N) -> a__U42#(a__isNat(N),M,N) p9: a__U42#(tt(),M,N) -> a__plus#(mark(N),mark(M)) p10: a__plus#(N,s(M)) -> a__U41#(a__isNat(M),M,N) p11: a__U42#(tt(),M,N) -> mark#(N) p12: mark#(U41(X1,X2,X3)) -> mark#(X1) p13: mark#(U42(X1,X2,X3)) -> mark#(X1) p14: mark#(plus(X1,X2)) -> mark#(X2) and R consists of: r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2)) r2: a__U12(tt()) -> tt() r3: a__U21(tt()) -> tt() r4: a__U31(tt(),N) -> mark(N) r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N) r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M))) r7: a__isNat(|0|()) -> tt() r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1)) r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N) r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N) r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2) r13: mark(U12(X)) -> a__U12(mark(X)) r14: mark(isNat(X)) -> a__isNat(X) r15: mark(U21(X)) -> a__U21(mark(X)) r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2) r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) r20: mark(tt()) -> tt() r21: mark(s(X)) -> s(mark(X)) r22: mark(|0|()) -> |0|() r23: a__U11(X1,X2) -> U11(X1,X2) r24: a__U12(X) -> U12(X) r25: a__isNat(X) -> isNat(X) r26: a__U21(X) -> U21(X) r27: a__U31(X1,X2) -> U31(X1,X2) r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3) r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3) r30: a__plus(X1,X2) -> plus(X1,X2) The estimated dependency graph contains the following SCCs: {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: mark#(U11(X1,X2)) -> mark#(X1) p2: mark#(plus(X1,X2)) -> mark#(X2) p3: mark#(U42(X1,X2,X3)) -> mark#(X1) p4: mark#(U41(X1,X2,X3)) -> mark#(X1) p5: mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) p6: a__U41#(tt(),M,N) -> a__U42#(a__isNat(N),M,N) p7: a__U42#(tt(),M,N) -> mark#(N) p8: mark#(U31(X1,X2)) -> mark#(X1) p9: mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) p10: a__U31#(tt(),N) -> mark#(N) p11: mark#(U21(X)) -> mark#(X) p12: mark#(U12(X)) -> mark#(X) p13: a__U42#(tt(),M,N) -> a__plus#(mark(N),mark(M)) p14: a__plus#(N,s(M)) -> a__U41#(a__isNat(M),M,N) and R consists of: r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2)) r2: a__U12(tt()) -> tt() r3: a__U21(tt()) -> tt() r4: a__U31(tt(),N) -> mark(N) r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N) r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M))) r7: a__isNat(|0|()) -> tt() r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1)) r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N) r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N) r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2) r13: mark(U12(X)) -> a__U12(mark(X)) r14: mark(isNat(X)) -> a__isNat(X) r15: mark(U21(X)) -> a__U21(mark(X)) r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2) r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) r20: mark(tt()) -> tt() r21: mark(s(X)) -> s(mark(X)) r22: mark(|0|()) -> |0|() r23: a__U11(X1,X2) -> U11(X1,X2) r24: a__U12(X) -> U12(X) r25: a__isNat(X) -> isNat(X) r26: a__U21(X) -> U21(X) r27: a__U31(X1,X2) -> U31(X1,X2) r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3) r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3) r30: a__plus(X1,X2) -> plus(X1,X2) The set of usable rules consists of r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30 Take the reduction pair: weighted path order base order: matrix interpretations: carrier: N^1 order: lexicographic order interpretations: mark#_A(x1) = x1 U11_A(x1,x2) = x1 plus_A(x1,x2) = x1 + x2 + 3 U42_A(x1,x2,x3) = x1 + x3 + 4 U41_A(x1,x2,x3) = x1 + x3 + 5 a__U41#_A(x1,x2,x3) = x3 + 4 mark_A(x1) = x1 tt_A() = 4 a__U42#_A(x1,x2,x3) = x1 + x3 a__isNat_A(x1) = 4 U31_A(x1,x2) = x1 + x2 + 2 a__U31#_A(x1,x2) = x2 + 1 U21_A(x1) = x1 U12_A(x1) = x1 a__plus#_A(x1,x2) = x1 + 4 s_A(x1) = 7 a__U11_A(x1,x2) = x1 a__U12_A(x1) = x1 a__U21_A(x1) = x1 a__U31_A(x1,x2) = x1 + x2 + 2 a__U41_A(x1,x2,x3) = x1 + x3 + 5 a__U42_A(x1,x2,x3) = x1 + x3 + 4 a__plus_A(x1,x2) = x1 + x2 + 3 |0|_A() = 5 isNat_A(x1) = 4 precedence: U11 = a__isNat = a__U11 = isNat > U21 = a__U21 > tt = U12 = a__U12 > plus = U31 = a__U31 = a__plus > mark > U42 = a__U41 = a__U42 > mark# = U41 = a__U41# = a__U42# = a__U31# = a__plus# = s = |0| partial status: pi(mark#) = [] pi(U11) = [] pi(plus) = [] pi(U42) = [] pi(U41) = [1] pi(a__U41#) = [] pi(mark) = [1] pi(tt) = [] pi(a__U42#) = [] pi(a__isNat) = [] pi(U31) = [] pi(a__U31#) = [] pi(U21) = [] pi(U12) = [] pi(a__plus#) = [] pi(s) = [] pi(a__U11) = [] pi(a__U12) = [] pi(a__U21) = [] pi(a__U31) = [] pi(a__U41) = [1, 3] pi(a__U42) = [] pi(a__plus) = [] pi(|0|) = [] pi(isNat) = [] The next rules are strictly ordered: p7 We remove them from the problem. -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: mark#(U11(X1,X2)) -> mark#(X1) p2: mark#(plus(X1,X2)) -> mark#(X2) p3: mark#(U42(X1,X2,X3)) -> mark#(X1) p4: mark#(U41(X1,X2,X3)) -> mark#(X1) p5: mark#(U41(X1,X2,X3)) -> a__U41#(mark(X1),X2,X3) p6: a__U41#(tt(),M,N) -> a__U42#(a__isNat(N),M,N) p7: mark#(U31(X1,X2)) -> mark#(X1) p8: mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) p9: a__U31#(tt(),N) -> mark#(N) p10: mark#(U21(X)) -> mark#(X) p11: mark#(U12(X)) -> mark#(X) p12: a__U42#(tt(),M,N) -> a__plus#(mark(N),mark(M)) p13: a__plus#(N,s(M)) -> a__U41#(a__isNat(M),M,N) and R consists of: r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2)) r2: a__U12(tt()) -> tt() r3: a__U21(tt()) -> tt() r4: a__U31(tt(),N) -> mark(N) r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N) r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M))) r7: a__isNat(|0|()) -> tt() r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1)) r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N) r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N) r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2) r13: mark(U12(X)) -> a__U12(mark(X)) r14: mark(isNat(X)) -> a__isNat(X) r15: mark(U21(X)) -> a__U21(mark(X)) r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2) r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) r20: mark(tt()) -> tt() r21: mark(s(X)) -> s(mark(X)) r22: mark(|0|()) -> |0|() r23: a__U11(X1,X2) -> U11(X1,X2) r24: a__U12(X) -> U12(X) r25: a__isNat(X) -> isNat(X) r26: a__U21(X) -> U21(X) r27: a__U31(X1,X2) -> U31(X1,X2) r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3) r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3) r30: a__plus(X1,X2) -> plus(X1,X2) The estimated dependency graph contains the following SCCs: {p1, p2, p3, p4, p7, p8, p9, p10, p11} {p6, p12, p13} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: mark#(U11(X1,X2)) -> mark#(X1) p2: mark#(U12(X)) -> mark#(X) p3: mark#(U21(X)) -> mark#(X) p4: mark#(U31(X1,X2)) -> a__U31#(mark(X1),X2) p5: a__U31#(tt(),N) -> mark#(N) p6: mark#(U31(X1,X2)) -> mark#(X1) p7: mark#(U41(X1,X2,X3)) -> mark#(X1) p8: mark#(U42(X1,X2,X3)) -> mark#(X1) p9: mark#(plus(X1,X2)) -> mark#(X2) and R consists of: r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2)) r2: a__U12(tt()) -> tt() r3: a__U21(tt()) -> tt() r4: a__U31(tt(),N) -> mark(N) r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N) r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M))) r7: a__isNat(|0|()) -> tt() r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1)) r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N) r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N) r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2) r13: mark(U12(X)) -> a__U12(mark(X)) r14: mark(isNat(X)) -> a__isNat(X) r15: mark(U21(X)) -> a__U21(mark(X)) r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2) r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) r20: mark(tt()) -> tt() r21: mark(s(X)) -> s(mark(X)) r22: mark(|0|()) -> |0|() r23: a__U11(X1,X2) -> U11(X1,X2) r24: a__U12(X) -> U12(X) r25: a__isNat(X) -> isNat(X) r26: a__U21(X) -> U21(X) r27: a__U31(X1,X2) -> U31(X1,X2) r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3) r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3) r30: a__plus(X1,X2) -> plus(X1,X2) The set of usable rules consists of r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30 Take the reduction pair: weighted path order base order: matrix interpretations: carrier: N^1 order: lexicographic order interpretations: mark#_A(x1) = x1 + 1 U11_A(x1,x2) = x1 U12_A(x1) = x1 U21_A(x1) = x1 U31_A(x1,x2) = x1 + x2 + 2 a__U31#_A(x1,x2) = x2 + 1 mark_A(x1) = x1 tt_A() = 0 U41_A(x1,x2,x3) = x1 + x3 + 3 U42_A(x1,x2,x3) = x1 + 2 plus_A(x1,x2) = x1 + x2 + 3 a__U11_A(x1,x2) = x1 a__U12_A(x1) = x1 a__isNat_A(x1) = 0 a__U21_A(x1) = x1 a__U31_A(x1,x2) = x1 + x2 + 2 a__U41_A(x1,x2,x3) = x1 + x3 + 3 a__U42_A(x1,x2,x3) = x1 + 2 s_A(x1) = 1 a__plus_A(x1,x2) = x1 + x2 + 3 |0|_A() = 1 isNat_A(x1) = 0 precedence: U11 = U12 = U21 = mark = U41 = a__U11 = a__U12 = a__isNat = a__U21 = a__U31 = a__U41 = a__U42 = s = a__plus > tt > mark# = U31 = a__U31# > U42 = plus = |0| = isNat partial status: pi(mark#) = [1] pi(U11) = [] pi(U12) = [] pi(U21) = [] pi(U31) = [1, 2] pi(a__U31#) = [2] pi(mark) = [] pi(tt) = [] pi(U41) = [] pi(U42) = [] pi(plus) = [] pi(a__U11) = [] pi(a__U12) = [] pi(a__isNat) = [] pi(a__U21) = [] pi(a__U31) = [] pi(a__U41) = [] pi(a__U42) = [] pi(s) = [] pi(a__plus) = [] pi(|0|) = [] pi(isNat) = [] The next rules are strictly ordered: p4 We remove them from the problem. -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: mark#(U11(X1,X2)) -> mark#(X1) p2: mark#(U12(X)) -> mark#(X) p3: mark#(U21(X)) -> mark#(X) p4: a__U31#(tt(),N) -> mark#(N) p5: mark#(U31(X1,X2)) -> mark#(X1) p6: mark#(U41(X1,X2,X3)) -> mark#(X1) p7: mark#(U42(X1,X2,X3)) -> mark#(X1) p8: mark#(plus(X1,X2)) -> mark#(X2) and R consists of: r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2)) r2: a__U12(tt()) -> tt() r3: a__U21(tt()) -> tt() r4: a__U31(tt(),N) -> mark(N) r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N) r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M))) r7: a__isNat(|0|()) -> tt() r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1)) r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N) r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N) r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2) r13: mark(U12(X)) -> a__U12(mark(X)) r14: mark(isNat(X)) -> a__isNat(X) r15: mark(U21(X)) -> a__U21(mark(X)) r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2) r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) r20: mark(tt()) -> tt() r21: mark(s(X)) -> s(mark(X)) r22: mark(|0|()) -> |0|() r23: a__U11(X1,X2) -> U11(X1,X2) r24: a__U12(X) -> U12(X) r25: a__isNat(X) -> isNat(X) r26: a__U21(X) -> U21(X) r27: a__U31(X1,X2) -> U31(X1,X2) r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3) r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3) r30: a__plus(X1,X2) -> plus(X1,X2) The estimated dependency graph contains the following SCCs: {p1, p2, p3, p5, p6, p7, p8} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: mark#(U11(X1,X2)) -> mark#(X1) p2: mark#(plus(X1,X2)) -> mark#(X2) p3: mark#(U42(X1,X2,X3)) -> mark#(X1) p4: mark#(U41(X1,X2,X3)) -> mark#(X1) p5: mark#(U31(X1,X2)) -> mark#(X1) p6: mark#(U21(X)) -> mark#(X) p7: mark#(U12(X)) -> mark#(X) and R consists of: r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2)) r2: a__U12(tt()) -> tt() r3: a__U21(tt()) -> tt() r4: a__U31(tt(),N) -> mark(N) r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N) r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M))) r7: a__isNat(|0|()) -> tt() r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1)) r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N) r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N) r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2) r13: mark(U12(X)) -> a__U12(mark(X)) r14: mark(isNat(X)) -> a__isNat(X) r15: mark(U21(X)) -> a__U21(mark(X)) r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2) r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) r20: mark(tt()) -> tt() r21: mark(s(X)) -> s(mark(X)) r22: mark(|0|()) -> |0|() r23: a__U11(X1,X2) -> U11(X1,X2) r24: a__U12(X) -> U12(X) r25: a__isNat(X) -> isNat(X) r26: a__U21(X) -> U21(X) r27: a__U31(X1,X2) -> U31(X1,X2) r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3) r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3) r30: a__plus(X1,X2) -> plus(X1,X2) The set of usable rules consists of (no rules) Take the reduction pair: weighted path order base order: matrix interpretations: carrier: N^1 order: lexicographic order interpretations: mark#_A(x1) = x1 U11_A(x1,x2) = x1 + x2 + 1 plus_A(x1,x2) = x2 + 1 U42_A(x1,x2,x3) = x1 + x3 + 1 U41_A(x1,x2,x3) = x1 + x3 + 1 U31_A(x1,x2) = x1 + x2 + 1 U21_A(x1) = x1 + 1 U12_A(x1) = x1 + 1 precedence: mark# = U11 = plus = U41 = U31 > U42 = U21 = U12 partial status: pi(mark#) = [1] pi(U11) = [1, 2] pi(plus) = [2] pi(U42) = [1, 3] pi(U41) = [1, 3] pi(U31) = [1, 2] pi(U21) = [1] pi(U12) = [1] The next rules are strictly ordered: p1 We remove them from the problem. -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: mark#(plus(X1,X2)) -> mark#(X2) p2: mark#(U42(X1,X2,X3)) -> mark#(X1) p3: mark#(U41(X1,X2,X3)) -> mark#(X1) p4: mark#(U31(X1,X2)) -> mark#(X1) p5: mark#(U21(X)) -> mark#(X) p6: mark#(U12(X)) -> mark#(X) and R consists of: r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2)) r2: a__U12(tt()) -> tt() r3: a__U21(tt()) -> tt() r4: a__U31(tt(),N) -> mark(N) r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N) r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M))) r7: a__isNat(|0|()) -> tt() r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1)) r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N) r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N) r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2) r13: mark(U12(X)) -> a__U12(mark(X)) r14: mark(isNat(X)) -> a__isNat(X) r15: mark(U21(X)) -> a__U21(mark(X)) r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2) r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) r20: mark(tt()) -> tt() r21: mark(s(X)) -> s(mark(X)) r22: mark(|0|()) -> |0|() r23: a__U11(X1,X2) -> U11(X1,X2) r24: a__U12(X) -> U12(X) r25: a__isNat(X) -> isNat(X) r26: a__U21(X) -> U21(X) r27: a__U31(X1,X2) -> U31(X1,X2) r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3) r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3) r30: a__plus(X1,X2) -> plus(X1,X2) The estimated dependency graph contains the following SCCs: {p1, p2, p3, p4, p5, p6} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: mark#(plus(X1,X2)) -> mark#(X2) p2: mark#(U12(X)) -> mark#(X) p3: mark#(U21(X)) -> mark#(X) p4: mark#(U31(X1,X2)) -> mark#(X1) p5: mark#(U41(X1,X2,X3)) -> mark#(X1) p6: mark#(U42(X1,X2,X3)) -> mark#(X1) and R consists of: r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2)) r2: a__U12(tt()) -> tt() r3: a__U21(tt()) -> tt() r4: a__U31(tt(),N) -> mark(N) r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N) r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M))) r7: a__isNat(|0|()) -> tt() r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1)) r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N) r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N) r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2) r13: mark(U12(X)) -> a__U12(mark(X)) r14: mark(isNat(X)) -> a__isNat(X) r15: mark(U21(X)) -> a__U21(mark(X)) r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2) r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) r20: mark(tt()) -> tt() r21: mark(s(X)) -> s(mark(X)) r22: mark(|0|()) -> |0|() r23: a__U11(X1,X2) -> U11(X1,X2) r24: a__U12(X) -> U12(X) r25: a__isNat(X) -> isNat(X) r26: a__U21(X) -> U21(X) r27: a__U31(X1,X2) -> U31(X1,X2) r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3) r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3) r30: a__plus(X1,X2) -> plus(X1,X2) The set of usable rules consists of (no rules) Take the reduction pair: weighted path order base order: matrix interpretations: carrier: N^1 order: lexicographic order interpretations: mark#_A(x1) = x1 plus_A(x1,x2) = x1 + x2 + 1 U12_A(x1) = x1 + 1 U21_A(x1) = x1 + 1 U31_A(x1,x2) = x1 + x2 + 1 U41_A(x1,x2,x3) = x1 + x2 + x3 + 1 U42_A(x1,x2,x3) = x1 + x2 + x3 + 1 precedence: mark# = plus = U12 = U21 = U31 = U41 = U42 partial status: pi(mark#) = [] pi(plus) = [2] pi(U12) = [1] pi(U21) = [1] pi(U31) = [2] pi(U41) = [3] pi(U42) = [3] The next rules are strictly ordered: p1 We remove them from the problem. -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: mark#(U12(X)) -> mark#(X) p2: mark#(U21(X)) -> mark#(X) p3: mark#(U31(X1,X2)) -> mark#(X1) p4: mark#(U41(X1,X2,X3)) -> mark#(X1) p5: mark#(U42(X1,X2,X3)) -> mark#(X1) and R consists of: r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2)) r2: a__U12(tt()) -> tt() r3: a__U21(tt()) -> tt() r4: a__U31(tt(),N) -> mark(N) r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N) r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M))) r7: a__isNat(|0|()) -> tt() r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1)) r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N) r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N) r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2) r13: mark(U12(X)) -> a__U12(mark(X)) r14: mark(isNat(X)) -> a__isNat(X) r15: mark(U21(X)) -> a__U21(mark(X)) r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2) r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) r20: mark(tt()) -> tt() r21: mark(s(X)) -> s(mark(X)) r22: mark(|0|()) -> |0|() r23: a__U11(X1,X2) -> U11(X1,X2) r24: a__U12(X) -> U12(X) r25: a__isNat(X) -> isNat(X) r26: a__U21(X) -> U21(X) r27: a__U31(X1,X2) -> U31(X1,X2) r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3) r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3) r30: a__plus(X1,X2) -> plus(X1,X2) The estimated dependency graph contains the following SCCs: {p1, p2, p3, p4, p5} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: mark#(U12(X)) -> mark#(X) p2: mark#(U42(X1,X2,X3)) -> mark#(X1) p3: mark#(U41(X1,X2,X3)) -> mark#(X1) p4: mark#(U31(X1,X2)) -> mark#(X1) p5: mark#(U21(X)) -> mark#(X) and R consists of: r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2)) r2: a__U12(tt()) -> tt() r3: a__U21(tt()) -> tt() r4: a__U31(tt(),N) -> mark(N) r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N) r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M))) r7: a__isNat(|0|()) -> tt() r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1)) r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N) r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N) r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2) r13: mark(U12(X)) -> a__U12(mark(X)) r14: mark(isNat(X)) -> a__isNat(X) r15: mark(U21(X)) -> a__U21(mark(X)) r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2) r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) r20: mark(tt()) -> tt() r21: mark(s(X)) -> s(mark(X)) r22: mark(|0|()) -> |0|() r23: a__U11(X1,X2) -> U11(X1,X2) r24: a__U12(X) -> U12(X) r25: a__isNat(X) -> isNat(X) r26: a__U21(X) -> U21(X) r27: a__U31(X1,X2) -> U31(X1,X2) r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3) r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3) r30: a__plus(X1,X2) -> plus(X1,X2) The set of usable rules consists of (no rules) Take the reduction pair: weighted path order base order: matrix interpretations: carrier: N^1 order: lexicographic order interpretations: mark#_A(x1) = x1 U12_A(x1) = x1 + 1 U42_A(x1,x2,x3) = x1 + x2 + x3 + 1 U41_A(x1,x2,x3) = x1 + x3 + 1 U31_A(x1,x2) = x1 + x2 + 1 U21_A(x1) = x1 + 1 precedence: mark# > U12 = U42 > U41 = U31 = U21 partial status: pi(mark#) = [1] pi(U12) = [1] pi(U42) = [] pi(U41) = [1, 3] pi(U31) = [1, 2] pi(U21) = [1] The next rules are strictly ordered: p1 We remove them from the problem. -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: mark#(U42(X1,X2,X3)) -> mark#(X1) p2: mark#(U41(X1,X2,X3)) -> mark#(X1) p3: mark#(U31(X1,X2)) -> mark#(X1) p4: mark#(U21(X)) -> mark#(X) and R consists of: r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2)) r2: a__U12(tt()) -> tt() r3: a__U21(tt()) -> tt() r4: a__U31(tt(),N) -> mark(N) r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N) r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M))) r7: a__isNat(|0|()) -> tt() r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1)) r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N) r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N) r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2) r13: mark(U12(X)) -> a__U12(mark(X)) r14: mark(isNat(X)) -> a__isNat(X) r15: mark(U21(X)) -> a__U21(mark(X)) r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2) r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) r20: mark(tt()) -> tt() r21: mark(s(X)) -> s(mark(X)) r22: mark(|0|()) -> |0|() r23: a__U11(X1,X2) -> U11(X1,X2) r24: a__U12(X) -> U12(X) r25: a__isNat(X) -> isNat(X) r26: a__U21(X) -> U21(X) r27: a__U31(X1,X2) -> U31(X1,X2) r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3) r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3) r30: a__plus(X1,X2) -> plus(X1,X2) The estimated dependency graph contains the following SCCs: {p1, p2, p3, p4} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: mark#(U42(X1,X2,X3)) -> mark#(X1) p2: mark#(U21(X)) -> mark#(X) p3: mark#(U31(X1,X2)) -> mark#(X1) p4: mark#(U41(X1,X2,X3)) -> mark#(X1) and R consists of: r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2)) r2: a__U12(tt()) -> tt() r3: a__U21(tt()) -> tt() r4: a__U31(tt(),N) -> mark(N) r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N) r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M))) r7: a__isNat(|0|()) -> tt() r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1)) r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N) r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N) r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2) r13: mark(U12(X)) -> a__U12(mark(X)) r14: mark(isNat(X)) -> a__isNat(X) r15: mark(U21(X)) -> a__U21(mark(X)) r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2) r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) r20: mark(tt()) -> tt() r21: mark(s(X)) -> s(mark(X)) r22: mark(|0|()) -> |0|() r23: a__U11(X1,X2) -> U11(X1,X2) r24: a__U12(X) -> U12(X) r25: a__isNat(X) -> isNat(X) r26: a__U21(X) -> U21(X) r27: a__U31(X1,X2) -> U31(X1,X2) r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3) r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3) r30: a__plus(X1,X2) -> plus(X1,X2) The set of usable rules consists of (no rules) Take the reduction pair: weighted path order base order: matrix interpretations: carrier: N^1 order: lexicographic order interpretations: mark#_A(x1) = x1 U42_A(x1,x2,x3) = x1 + x2 + x3 + 1 U21_A(x1) = x1 + 1 U31_A(x1,x2) = x1 + x2 + 1 U41_A(x1,x2,x3) = x1 + x2 + x3 + 1 precedence: mark# = U42 = U21 = U31 = U41 partial status: pi(mark#) = [] pi(U42) = [3] pi(U21) = [1] pi(U31) = [2] pi(U41) = [3] The next rules are strictly ordered: p1 We remove them from the problem. -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: mark#(U21(X)) -> mark#(X) p2: mark#(U31(X1,X2)) -> mark#(X1) p3: mark#(U41(X1,X2,X3)) -> mark#(X1) and R consists of: r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2)) r2: a__U12(tt()) -> tt() r3: a__U21(tt()) -> tt() r4: a__U31(tt(),N) -> mark(N) r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N) r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M))) r7: a__isNat(|0|()) -> tt() r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1)) r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N) r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N) r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2) r13: mark(U12(X)) -> a__U12(mark(X)) r14: mark(isNat(X)) -> a__isNat(X) r15: mark(U21(X)) -> a__U21(mark(X)) r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2) r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) r20: mark(tt()) -> tt() r21: mark(s(X)) -> s(mark(X)) r22: mark(|0|()) -> |0|() r23: a__U11(X1,X2) -> U11(X1,X2) r24: a__U12(X) -> U12(X) r25: a__isNat(X) -> isNat(X) r26: a__U21(X) -> U21(X) r27: a__U31(X1,X2) -> U31(X1,X2) r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3) r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3) r30: a__plus(X1,X2) -> plus(X1,X2) The estimated dependency graph contains the following SCCs: {p1, p2, p3} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: mark#(U21(X)) -> mark#(X) p2: mark#(U41(X1,X2,X3)) -> mark#(X1) p3: mark#(U31(X1,X2)) -> mark#(X1) and R consists of: r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2)) r2: a__U12(tt()) -> tt() r3: a__U21(tt()) -> tt() r4: a__U31(tt(),N) -> mark(N) r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N) r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M))) r7: a__isNat(|0|()) -> tt() r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1)) r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N) r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N) r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2) r13: mark(U12(X)) -> a__U12(mark(X)) r14: mark(isNat(X)) -> a__isNat(X) r15: mark(U21(X)) -> a__U21(mark(X)) r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2) r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) r20: mark(tt()) -> tt() r21: mark(s(X)) -> s(mark(X)) r22: mark(|0|()) -> |0|() r23: a__U11(X1,X2) -> U11(X1,X2) r24: a__U12(X) -> U12(X) r25: a__isNat(X) -> isNat(X) r26: a__U21(X) -> U21(X) r27: a__U31(X1,X2) -> U31(X1,X2) r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3) r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3) r30: a__plus(X1,X2) -> plus(X1,X2) The set of usable rules consists of (no rules) Take the reduction pair: weighted path order base order: matrix interpretations: carrier: N^1 order: lexicographic order interpretations: mark#_A(x1) = x1 U21_A(x1) = x1 + 1 U41_A(x1,x2,x3) = x1 + x2 + x3 + 1 U31_A(x1,x2) = x1 + x2 + 1 precedence: U41 > mark# = U21 = U31 partial status: pi(mark#) = [] pi(U21) = [] pi(U41) = [3] pi(U31) = [2] The next rules are strictly ordered: p1 We remove them from the problem. -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: mark#(U41(X1,X2,X3)) -> mark#(X1) p2: mark#(U31(X1,X2)) -> mark#(X1) and R consists of: r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2)) r2: a__U12(tt()) -> tt() r3: a__U21(tt()) -> tt() r4: a__U31(tt(),N) -> mark(N) r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N) r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M))) r7: a__isNat(|0|()) -> tt() r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1)) r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N) r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N) r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2) r13: mark(U12(X)) -> a__U12(mark(X)) r14: mark(isNat(X)) -> a__isNat(X) r15: mark(U21(X)) -> a__U21(mark(X)) r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2) r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) r20: mark(tt()) -> tt() r21: mark(s(X)) -> s(mark(X)) r22: mark(|0|()) -> |0|() r23: a__U11(X1,X2) -> U11(X1,X2) r24: a__U12(X) -> U12(X) r25: a__isNat(X) -> isNat(X) r26: a__U21(X) -> U21(X) r27: a__U31(X1,X2) -> U31(X1,X2) r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3) r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3) r30: a__plus(X1,X2) -> plus(X1,X2) The estimated dependency graph contains the following SCCs: {p1, p2} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: mark#(U41(X1,X2,X3)) -> mark#(X1) p2: mark#(U31(X1,X2)) -> mark#(X1) and R consists of: r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2)) r2: a__U12(tt()) -> tt() r3: a__U21(tt()) -> tt() r4: a__U31(tt(),N) -> mark(N) r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N) r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M))) r7: a__isNat(|0|()) -> tt() r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1)) r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N) r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N) r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2) r13: mark(U12(X)) -> a__U12(mark(X)) r14: mark(isNat(X)) -> a__isNat(X) r15: mark(U21(X)) -> a__U21(mark(X)) r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2) r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) r20: mark(tt()) -> tt() r21: mark(s(X)) -> s(mark(X)) r22: mark(|0|()) -> |0|() r23: a__U11(X1,X2) -> U11(X1,X2) r24: a__U12(X) -> U12(X) r25: a__isNat(X) -> isNat(X) r26: a__U21(X) -> U21(X) r27: a__U31(X1,X2) -> U31(X1,X2) r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3) r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3) r30: a__plus(X1,X2) -> plus(X1,X2) The set of usable rules consists of (no rules) Take the reduction pair: weighted path order base order: matrix interpretations: carrier: N^1 order: lexicographic order interpretations: mark#_A(x1) = x1 U41_A(x1,x2,x3) = x1 + x2 + x3 + 1 U31_A(x1,x2) = x1 + x2 + 1 precedence: mark# = U41 = U31 partial status: pi(mark#) = [] pi(U41) = [3] pi(U31) = [2] The next rules are strictly ordered: p1 We remove them from the problem. -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: mark#(U31(X1,X2)) -> mark#(X1) and R consists of: r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2)) r2: a__U12(tt()) -> tt() r3: a__U21(tt()) -> tt() r4: a__U31(tt(),N) -> mark(N) r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N) r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M))) r7: a__isNat(|0|()) -> tt() r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1)) r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N) r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N) r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2) r13: mark(U12(X)) -> a__U12(mark(X)) r14: mark(isNat(X)) -> a__isNat(X) r15: mark(U21(X)) -> a__U21(mark(X)) r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2) r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) r20: mark(tt()) -> tt() r21: mark(s(X)) -> s(mark(X)) r22: mark(|0|()) -> |0|() r23: a__U11(X1,X2) -> U11(X1,X2) r24: a__U12(X) -> U12(X) r25: a__isNat(X) -> isNat(X) r26: a__U21(X) -> U21(X) r27: a__U31(X1,X2) -> U31(X1,X2) r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3) r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3) r30: a__plus(X1,X2) -> plus(X1,X2) The estimated dependency graph contains the following SCCs: {p1} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: mark#(U31(X1,X2)) -> mark#(X1) and R consists of: r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2)) r2: a__U12(tt()) -> tt() r3: a__U21(tt()) -> tt() r4: a__U31(tt(),N) -> mark(N) r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N) r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M))) r7: a__isNat(|0|()) -> tt() r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1)) r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N) r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N) r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2) r13: mark(U12(X)) -> a__U12(mark(X)) r14: mark(isNat(X)) -> a__isNat(X) r15: mark(U21(X)) -> a__U21(mark(X)) r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2) r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) r20: mark(tt()) -> tt() r21: mark(s(X)) -> s(mark(X)) r22: mark(|0|()) -> |0|() r23: a__U11(X1,X2) -> U11(X1,X2) r24: a__U12(X) -> U12(X) r25: a__isNat(X) -> isNat(X) r26: a__U21(X) -> U21(X) r27: a__U31(X1,X2) -> U31(X1,X2) r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3) r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3) r30: a__plus(X1,X2) -> plus(X1,X2) The set of usable rules consists of (no rules) Take the reduction pair: weighted path order base order: matrix interpretations: carrier: N^1 order: lexicographic order interpretations: mark#_A(x1) = x1 U31_A(x1,x2) = x1 + x2 + 1 precedence: mark# = U31 partial status: pi(mark#) = [] pi(U31) = [2] The next rules are strictly ordered: p1 We remove them from the problem. Then no dependency pair remains. -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: a__U41#(tt(),M,N) -> a__U42#(a__isNat(N),M,N) p2: a__U42#(tt(),M,N) -> a__plus#(mark(N),mark(M)) p3: a__plus#(N,s(M)) -> a__U41#(a__isNat(M),M,N) and R consists of: r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2)) r2: a__U12(tt()) -> tt() r3: a__U21(tt()) -> tt() r4: a__U31(tt(),N) -> mark(N) r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N) r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M))) r7: a__isNat(|0|()) -> tt() r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1)) r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N) r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N) r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2) r13: mark(U12(X)) -> a__U12(mark(X)) r14: mark(isNat(X)) -> a__isNat(X) r15: mark(U21(X)) -> a__U21(mark(X)) r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2) r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) r20: mark(tt()) -> tt() r21: mark(s(X)) -> s(mark(X)) r22: mark(|0|()) -> |0|() r23: a__U11(X1,X2) -> U11(X1,X2) r24: a__U12(X) -> U12(X) r25: a__isNat(X) -> isNat(X) r26: a__U21(X) -> U21(X) r27: a__U31(X1,X2) -> U31(X1,X2) r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3) r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3) r30: a__plus(X1,X2) -> plus(X1,X2) The set of usable rules consists of r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30 Take the reduction pair: weighted path order base order: matrix interpretations: carrier: N^1 order: lexicographic order interpretations: a__U41#_A(x1,x2,x3) = x1 + x2 + 4 tt_A() = 10 a__U42#_A(x1,x2,x3) = x2 + 13 a__isNat_A(x1) = 14 a__plus#_A(x1,x2) = x2 + 12 mark_A(x1) = x1 s_A(x1) = x1 + 7 a__U11_A(x1,x2) = 13 a__U12_A(x1) = 10 a__U21_A(x1) = 14 a__U31_A(x1,x2) = x2 + 12 a__U41_A(x1,x2,x3) = x2 + x3 + 9 a__U42_A(x1,x2,x3) = x2 + x3 + 9 a__plus_A(x1,x2) = x1 + x2 + 2 |0|_A() = 11 U11_A(x1,x2) = 13 U12_A(x1) = 10 U21_A(x1) = 14 U31_A(x1,x2) = x2 + 12 U41_A(x1,x2,x3) = x2 + x3 + 9 U42_A(x1,x2,x3) = x2 + x3 + 9 plus_A(x1,x2) = x1 + x2 + 2 isNat_A(x1) = 14 precedence: a__U41# = a__plus# > a__U42# > tt = mark = a__U12 = a__U31 = a__plus > s = a__U41 = a__U42 > a__isNat = a__U11 = U11 > |0| = U12 = U31 = U41 = U42 = plus = isNat > a__U21 = U21 partial status: pi(a__U41#) = [] pi(tt) = [] pi(a__U42#) = [2] pi(a__isNat) = [] pi(a__plus#) = [] pi(mark) = [] pi(s) = [] pi(a__U11) = [] pi(a__U12) = [] pi(a__U21) = [] pi(a__U31) = [] pi(a__U41) = [] pi(a__U42) = [] pi(a__plus) = [] pi(|0|) = [] pi(U11) = [] pi(U12) = [] pi(U21) = [] pi(U31) = [2] pi(U41) = [3] pi(U42) = [] pi(plus) = [2] pi(isNat) = [] The next rules are strictly ordered: p1 We remove them from the problem. -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: a__U42#(tt(),M,N) -> a__plus#(mark(N),mark(M)) p2: a__plus#(N,s(M)) -> a__U41#(a__isNat(M),M,N) and R consists of: r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2)) r2: a__U12(tt()) -> tt() r3: a__U21(tt()) -> tt() r4: a__U31(tt(),N) -> mark(N) r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N) r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M))) r7: a__isNat(|0|()) -> tt() r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1)) r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N) r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N) r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2) r13: mark(U12(X)) -> a__U12(mark(X)) r14: mark(isNat(X)) -> a__isNat(X) r15: mark(U21(X)) -> a__U21(mark(X)) r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2) r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) r20: mark(tt()) -> tt() r21: mark(s(X)) -> s(mark(X)) r22: mark(|0|()) -> |0|() r23: a__U11(X1,X2) -> U11(X1,X2) r24: a__U12(X) -> U12(X) r25: a__isNat(X) -> isNat(X) r26: a__U21(X) -> U21(X) r27: a__U31(X1,X2) -> U31(X1,X2) r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3) r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3) r30: a__plus(X1,X2) -> plus(X1,X2) The estimated dependency graph contains the following SCCs: (no SCCs) -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: a__U11#(tt(),V2) -> a__isNat#(V2) p2: a__isNat#(s(V1)) -> a__isNat#(V1) p3: a__isNat#(plus(V1,V2)) -> a__isNat#(V1) p4: a__isNat#(plus(V1,V2)) -> a__U11#(a__isNat(V1),V2) and R consists of: r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2)) r2: a__U12(tt()) -> tt() r3: a__U21(tt()) -> tt() r4: a__U31(tt(),N) -> mark(N) r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N) r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M))) r7: a__isNat(|0|()) -> tt() r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1)) r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N) r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N) r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2) r13: mark(U12(X)) -> a__U12(mark(X)) r14: mark(isNat(X)) -> a__isNat(X) r15: mark(U21(X)) -> a__U21(mark(X)) r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2) r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) r20: mark(tt()) -> tt() r21: mark(s(X)) -> s(mark(X)) r22: mark(|0|()) -> |0|() r23: a__U11(X1,X2) -> U11(X1,X2) r24: a__U12(X) -> U12(X) r25: a__isNat(X) -> isNat(X) r26: a__U21(X) -> U21(X) r27: a__U31(X1,X2) -> U31(X1,X2) r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3) r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3) r30: a__plus(X1,X2) -> plus(X1,X2) The set of usable rules consists of r1, r2, r3, r7, r8, r9, r23, r24, r25, r26 Take the reduction pair: weighted path order base order: matrix interpretations: carrier: N^1 order: lexicographic order interpretations: a__U11#_A(x1,x2) = x2 + 2 tt_A() = 2 a__isNat#_A(x1) = x1 + 1 s_A(x1) = x1 + 5 plus_A(x1,x2) = x1 + x2 + 5 a__isNat_A(x1) = 4 a__U12_A(x1) = 2 U12_A(x1) = 0 a__U11_A(x1,x2) = x1 a__U21_A(x1) = 3 U11_A(x1,x2) = 0 U21_A(x1) = 0 |0|_A() = 0 isNat_A(x1) = 0 precedence: s = a__isNat > a__U21 = U21 > tt = a__U12 = a__U11 = U11 > a__isNat# > a__U11# > plus = U12 = |0| = isNat partial status: pi(a__U11#) = [] pi(tt) = [] pi(a__isNat#) = [] pi(s) = [1] pi(plus) = [2] pi(a__isNat) = [] pi(a__U12) = [] pi(U12) = [] pi(a__U11) = [] pi(a__U21) = [] pi(U11) = [] pi(U21) = [] pi(|0|) = [] pi(isNat) = [] The next rules are strictly ordered: p1 We remove them from the problem. -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: a__isNat#(s(V1)) -> a__isNat#(V1) p2: a__isNat#(plus(V1,V2)) -> a__isNat#(V1) p3: a__isNat#(plus(V1,V2)) -> a__U11#(a__isNat(V1),V2) and R consists of: r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2)) r2: a__U12(tt()) -> tt() r3: a__U21(tt()) -> tt() r4: a__U31(tt(),N) -> mark(N) r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N) r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M))) r7: a__isNat(|0|()) -> tt() r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1)) r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N) r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N) r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2) r13: mark(U12(X)) -> a__U12(mark(X)) r14: mark(isNat(X)) -> a__isNat(X) r15: mark(U21(X)) -> a__U21(mark(X)) r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2) r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) r20: mark(tt()) -> tt() r21: mark(s(X)) -> s(mark(X)) r22: mark(|0|()) -> |0|() r23: a__U11(X1,X2) -> U11(X1,X2) r24: a__U12(X) -> U12(X) r25: a__isNat(X) -> isNat(X) r26: a__U21(X) -> U21(X) r27: a__U31(X1,X2) -> U31(X1,X2) r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3) r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3) r30: a__plus(X1,X2) -> plus(X1,X2) The estimated dependency graph contains the following SCCs: {p1, p2} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: a__isNat#(s(V1)) -> a__isNat#(V1) p2: a__isNat#(plus(V1,V2)) -> a__isNat#(V1) and R consists of: r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2)) r2: a__U12(tt()) -> tt() r3: a__U21(tt()) -> tt() r4: a__U31(tt(),N) -> mark(N) r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N) r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M))) r7: a__isNat(|0|()) -> tt() r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1)) r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N) r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N) r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2) r13: mark(U12(X)) -> a__U12(mark(X)) r14: mark(isNat(X)) -> a__isNat(X) r15: mark(U21(X)) -> a__U21(mark(X)) r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2) r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) r20: mark(tt()) -> tt() r21: mark(s(X)) -> s(mark(X)) r22: mark(|0|()) -> |0|() r23: a__U11(X1,X2) -> U11(X1,X2) r24: a__U12(X) -> U12(X) r25: a__isNat(X) -> isNat(X) r26: a__U21(X) -> U21(X) r27: a__U31(X1,X2) -> U31(X1,X2) r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3) r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3) r30: a__plus(X1,X2) -> plus(X1,X2) The set of usable rules consists of (no rules) Take the reduction pair: weighted path order base order: matrix interpretations: carrier: N^1 order: lexicographic order interpretations: a__isNat#_A(x1) = x1 + 1 s_A(x1) = x1 + 2 plus_A(x1,x2) = x1 + x2 + 2 precedence: a__isNat# > s = plus partial status: pi(a__isNat#) = [1] pi(s) = [1] pi(plus) = [] The next rules are strictly ordered: p2 We remove them from the problem. -- SCC decomposition. Consider the dependency pair problem (P, R), where P consists of p1: a__isNat#(s(V1)) -> a__isNat#(V1) and R consists of: r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2)) r2: a__U12(tt()) -> tt() r3: a__U21(tt()) -> tt() r4: a__U31(tt(),N) -> mark(N) r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N) r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M))) r7: a__isNat(|0|()) -> tt() r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1)) r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N) r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N) r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2) r13: mark(U12(X)) -> a__U12(mark(X)) r14: mark(isNat(X)) -> a__isNat(X) r15: mark(U21(X)) -> a__U21(mark(X)) r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2) r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) r20: mark(tt()) -> tt() r21: mark(s(X)) -> s(mark(X)) r22: mark(|0|()) -> |0|() r23: a__U11(X1,X2) -> U11(X1,X2) r24: a__U12(X) -> U12(X) r25: a__isNat(X) -> isNat(X) r26: a__U21(X) -> U21(X) r27: a__U31(X1,X2) -> U31(X1,X2) r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3) r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3) r30: a__plus(X1,X2) -> plus(X1,X2) The estimated dependency graph contains the following SCCs: {p1} -- Reduction pair. Consider the dependency pair problem (P, R), where P consists of p1: a__isNat#(s(V1)) -> a__isNat#(V1) and R consists of: r1: a__U11(tt(),V2) -> a__U12(a__isNat(V2)) r2: a__U12(tt()) -> tt() r3: a__U21(tt()) -> tt() r4: a__U31(tt(),N) -> mark(N) r5: a__U41(tt(),M,N) -> a__U42(a__isNat(N),M,N) r6: a__U42(tt(),M,N) -> s(a__plus(mark(N),mark(M))) r7: a__isNat(|0|()) -> tt() r8: a__isNat(plus(V1,V2)) -> a__U11(a__isNat(V1),V2) r9: a__isNat(s(V1)) -> a__U21(a__isNat(V1)) r10: a__plus(N,|0|()) -> a__U31(a__isNat(N),N) r11: a__plus(N,s(M)) -> a__U41(a__isNat(M),M,N) r12: mark(U11(X1,X2)) -> a__U11(mark(X1),X2) r13: mark(U12(X)) -> a__U12(mark(X)) r14: mark(isNat(X)) -> a__isNat(X) r15: mark(U21(X)) -> a__U21(mark(X)) r16: mark(U31(X1,X2)) -> a__U31(mark(X1),X2) r17: mark(U41(X1,X2,X3)) -> a__U41(mark(X1),X2,X3) r18: mark(U42(X1,X2,X3)) -> a__U42(mark(X1),X2,X3) r19: mark(plus(X1,X2)) -> a__plus(mark(X1),mark(X2)) r20: mark(tt()) -> tt() r21: mark(s(X)) -> s(mark(X)) r22: mark(|0|()) -> |0|() r23: a__U11(X1,X2) -> U11(X1,X2) r24: a__U12(X) -> U12(X) r25: a__isNat(X) -> isNat(X) r26: a__U21(X) -> U21(X) r27: a__U31(X1,X2) -> U31(X1,X2) r28: a__U41(X1,X2,X3) -> U41(X1,X2,X3) r29: a__U42(X1,X2,X3) -> U42(X1,X2,X3) r30: a__plus(X1,X2) -> plus(X1,X2) The set of usable rules consists of (no rules) Take the monotone reduction pair: weighted path order base order: matrix interpretations: carrier: N^1 order: lexicographic order interpretations: a__isNat#_A(x1) = x1 s_A(x1) = x1 + 1 precedence: a__isNat# = s partial status: pi(a__isNat#) = [1] pi(s) = [1] The next rules are strictly ordered: p1 We remove them from the problem. Then no dependency pair remains.