The rewrite relation of the following TRS is considered.
a__U11(tt,V2) | → | a__U12(a__isNat(V2)) | (1) |
a__U12(tt) | → | tt | (2) |
a__U21(tt) | → | tt | (3) |
a__U31(tt,N) | → | mark(N) | (4) |
a__U41(tt,M,N) | → | a__U42(a__isNat(N),M,N) | (5) |
a__U42(tt,M,N) | → | s(a__plus(mark(N),mark(M))) | (6) |
a__isNat(0) | → | tt | (7) |
a__isNat(plus(V1,V2)) | → | a__U11(a__isNat(V1),V2) | (8) |
a__isNat(s(V1)) | → | a__U21(a__isNat(V1)) | (9) |
a__plus(N,0) | → | a__U31(a__isNat(N),N) | (10) |
a__plus(N,s(M)) | → | a__U41(a__isNat(M),M,N) | (11) |
mark(U11(X1,X2)) | → | a__U11(mark(X1),X2) | (12) |
mark(U12(X)) | → | a__U12(mark(X)) | (13) |
mark(isNat(X)) | → | a__isNat(X) | (14) |
mark(U21(X)) | → | a__U21(mark(X)) | (15) |
mark(U31(X1,X2)) | → | a__U31(mark(X1),X2) | (16) |
mark(U41(X1,X2,X3)) | → | a__U41(mark(X1),X2,X3) | (17) |
mark(U42(X1,X2,X3)) | → | a__U42(mark(X1),X2,X3) | (18) |
mark(plus(X1,X2)) | → | a__plus(mark(X1),mark(X2)) | (19) |
mark(tt) | → | tt | (20) |
mark(s(X)) | → | s(mark(X)) | (21) |
mark(0) | → | 0 | (22) |
a__U11(X1,X2) | → | U11(X1,X2) | (23) |
a__U12(X) | → | U12(X) | (24) |
a__isNat(X) | → | isNat(X) | (25) |
a__U21(X) | → | U21(X) | (26) |
a__U31(X1,X2) | → | U31(X1,X2) | (27) |
a__U41(X1,X2,X3) | → | U41(X1,X2,X3) | (28) |
a__U42(X1,X2,X3) | → | U42(X1,X2,X3) | (29) |
a__plus(X1,X2) | → | plus(X1,X2) | (30) |
a__U11#(tt,V2) | → | a__isNat#(V2) | (31) |
a__U11#(tt,V2) | → | a__U12#(a__isNat(V2)) | (32) |
a__U31#(tt,N) | → | mark#(N) | (33) |
a__U41#(tt,M,N) | → | a__isNat#(N) | (34) |
a__U41#(tt,M,N) | → | a__U42#(a__isNat(N),M,N) | (35) |
a__U42#(tt,M,N) | → | mark#(M) | (36) |
a__U42#(tt,M,N) | → | mark#(N) | (37) |
a__U42#(tt,M,N) | → | a__plus#(mark(N),mark(M)) | (38) |
a__isNat#(plus(V1,V2)) | → | a__isNat#(V1) | (39) |
a__isNat#(plus(V1,V2)) | → | a__U11#(a__isNat(V1),V2) | (40) |
a__isNat#(s(V1)) | → | a__isNat#(V1) | (41) |
a__isNat#(s(V1)) | → | a__U21#(a__isNat(V1)) | (42) |
a__plus#(N,0) | → | a__isNat#(N) | (43) |
a__plus#(N,0) | → | a__U31#(a__isNat(N),N) | (44) |
a__plus#(N,s(M)) | → | a__isNat#(M) | (45) |
a__plus#(N,s(M)) | → | a__U41#(a__isNat(M),M,N) | (46) |
mark#(U11(X1,X2)) | → | mark#(X1) | (47) |
mark#(U11(X1,X2)) | → | a__U11#(mark(X1),X2) | (48) |
mark#(U12(X)) | → | mark#(X) | (49) |
mark#(U12(X)) | → | a__U12#(mark(X)) | (50) |
mark#(isNat(X)) | → | a__isNat#(X) | (51) |
mark#(U21(X)) | → | mark#(X) | (52) |
mark#(U21(X)) | → | a__U21#(mark(X)) | (53) |
mark#(U31(X1,X2)) | → | mark#(X1) | (54) |
mark#(U31(X1,X2)) | → | a__U31#(mark(X1),X2) | (55) |
mark#(U41(X1,X2,X3)) | → | mark#(X1) | (56) |
mark#(U41(X1,X2,X3)) | → | a__U41#(mark(X1),X2,X3) | (57) |
mark#(U42(X1,X2,X3)) | → | mark#(X1) | (58) |
mark#(U42(X1,X2,X3)) | → | a__U42#(mark(X1),X2,X3) | (59) |
mark#(plus(X1,X2)) | → | mark#(X2) | (60) |
mark#(plus(X1,X2)) | → | mark#(X1) | (61) |
mark#(plus(X1,X2)) | → | a__plus#(mark(X1),mark(X2)) | (62) |
mark#(s(X)) | → | mark#(X) | (63) |
The dependency pairs are split into 2 components.
a__plus#(N,0) | → | a__U31#(a__isNat(N),N) | (44) |
a__U31#(tt,N) | → | mark#(N) | (33) |
mark#(U11(X1,X2)) | → | mark#(X1) | (47) |
mark#(U12(X)) | → | mark#(X) | (49) |
mark#(U21(X)) | → | mark#(X) | (52) |
mark#(U31(X1,X2)) | → | mark#(X1) | (54) |
mark#(U31(X1,X2)) | → | a__U31#(mark(X1),X2) | (55) |
mark#(U41(X1,X2,X3)) | → | mark#(X1) | (56) |
mark#(U41(X1,X2,X3)) | → | a__U41#(mark(X1),X2,X3) | (57) |
a__U41#(tt,M,N) | → | a__U42#(a__isNat(N),M,N) | (35) |
a__U42#(tt,M,N) | → | mark#(M) | (36) |
mark#(U42(X1,X2,X3)) | → | mark#(X1) | (58) |
mark#(U42(X1,X2,X3)) | → | a__U42#(mark(X1),X2,X3) | (59) |
a__U42#(tt,M,N) | → | mark#(N) | (37) |
mark#(plus(X1,X2)) | → | mark#(X2) | (60) |
mark#(plus(X1,X2)) | → | mark#(X1) | (61) |
mark#(plus(X1,X2)) | → | a__plus#(mark(X1),mark(X2)) | (62) |
a__plus#(N,s(M)) | → | a__U41#(a__isNat(M),M,N) | (46) |
mark#(s(X)) | → | mark#(X) | (63) |
a__U42#(tt,M,N) | → | a__plus#(mark(N),mark(M)) | (38) |
[U41(x1, x2, x3)] | = | 0 · x1 + 0 · x2 + 7 · x3 + 0 |
[a__U21(x1)] | = | 0 · x1 + 0 |
[isNat(x1)] | = | 0 · x1 + -∞ |
[a__U42(x1, x2, x3)] | = | 0 · x1 + 0 · x2 + 7 · x3 + -∞ |
[a__U11(x1, x2)] | = | 0 · x1 + 0 · x2 + -∞ |
[a__U12(x1)] | = | 0 · x1 + 0 |
[U21(x1)] | = | 0 · x1 + 0 |
[a__U41#(x1, x2, x3)] | = | -∞ · x1 + 0 · x2 + 0 · x3 + 0 |
[U42(x1, x2, x3)] | = | 0 · x1 + 0 · x2 + 7 · x3 + -∞ |
[U31(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[a__U42#(x1, x2, x3)] | = | 0 · x1 + 0 · x2 + 0 · x3 + -∞ |
[a__plus#(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[a__U31(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[s(x1)] | = | 0 · x1 + 0 |
[tt] | = | 0 |
[a__U41(x1, x2, x3)] | = | 0 · x1 + 0 · x2 + 7 · x3 + 0 |
[mark#(x1)] | = | 0 · x1 + -∞ |
[U12(x1)] | = | 0 · x1 + 0 |
[plus(x1, x2)] | = | 7 · x1 + 0 · x2 + 0 |
[a__U31#(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[0] | = | 4 |
[a__plus(x1, x2)] | = | 7 · x1 + 0 · x2 + 0 |
[U11(x1, x2)] | = | 0 · x1 + 0 · x2 + -∞ |
[mark(x1)] | = | 0 · x1 + -∞ |
[a__isNat(x1)] | = | 0 · x1 + -∞ |
a__U11(tt,V2) | → | a__U12(a__isNat(V2)) | (1) |
a__U12(tt) | → | tt | (2) |
a__U21(tt) | → | tt | (3) |
a__U31(tt,N) | → | mark(N) | (4) |
a__U41(tt,M,N) | → | a__U42(a__isNat(N),M,N) | (5) |
a__U42(tt,M,N) | → | s(a__plus(mark(N),mark(M))) | (6) |
a__isNat(0) | → | tt | (7) |
a__isNat(plus(V1,V2)) | → | a__U11(a__isNat(V1),V2) | (8) |
a__isNat(s(V1)) | → | a__U21(a__isNat(V1)) | (9) |
a__plus(N,0) | → | a__U31(a__isNat(N),N) | (10) |
a__plus(N,s(M)) | → | a__U41(a__isNat(M),M,N) | (11) |
mark(U11(X1,X2)) | → | a__U11(mark(X1),X2) | (12) |
mark(U12(X)) | → | a__U12(mark(X)) | (13) |
mark(isNat(X)) | → | a__isNat(X) | (14) |
mark(U21(X)) | → | a__U21(mark(X)) | (15) |
mark(U31(X1,X2)) | → | a__U31(mark(X1),X2) | (16) |
mark(U41(X1,X2,X3)) | → | a__U41(mark(X1),X2,X3) | (17) |
mark(U42(X1,X2,X3)) | → | a__U42(mark(X1),X2,X3) | (18) |
mark(plus(X1,X2)) | → | a__plus(mark(X1),mark(X2)) | (19) |
mark(tt) | → | tt | (20) |
mark(s(X)) | → | s(mark(X)) | (21) |
mark(0) | → | 0 | (22) |
a__U11(X1,X2) | → | U11(X1,X2) | (23) |
a__U12(X) | → | U12(X) | (24) |
a__isNat(X) | → | isNat(X) | (25) |
a__U21(X) | → | U21(X) | (26) |
a__U31(X1,X2) | → | U31(X1,X2) | (27) |
a__U41(X1,X2,X3) | → | U41(X1,X2,X3) | (28) |
a__U42(X1,X2,X3) | → | U42(X1,X2,X3) | (29) |
a__plus(X1,X2) | → | plus(X1,X2) | (30) |
mark#(plus(X1,X2)) | → | mark#(X1) | (61) |
[U41(x1, x2, x3)] | = | 0 · x1 + 4 · x2 + 7 · x3 + 0 |
[a__U21(x1)] | = | 0 · x1 + 0 |
[isNat(x1)] | = | 0 · x1 + 0 |
[a__U42(x1, x2, x3)] | = | 0 · x1 + 4 · x2 + 7 · x3 + 0 |
[a__U11(x1, x2)] | = | 0 · x1 + 1 · x2 + 0 |
[a__U12(x1)] | = | 0 · x1 + 0 |
[U21(x1)] | = | 0 · x1 + 0 |
[a__U41#(x1, x2, x3)] | = | 0 · x1 + 0 · x2 + 2 · x3 + 0 |
[U42(x1, x2, x3)] | = | 0 · x1 + 4 · x2 + 7 · x3 + 0 |
[U31(x1, x2)] | = | 2 · x1 + 7 · x2 + 0 |
[a__U42#(x1, x2, x3)] | = | 0 · x1 + 0 · x2 + 2 · x3 + 0 |
[a__plus#(x1, x2)] | = | 2 · x1 + 0 · x2 + -∞ |
[a__U31(x1, x2)] | = | 2 · x1 + 7 · x2 + 0 |
[s(x1)] | = | 0 · x1 + 0 |
[tt] | = | 0 |
[a__U41(x1, x2, x3)] | = | 0 · x1 + 4 · x2 + 7 · x3 + 0 |
[mark#(x1)] | = | 0 · x1 + -∞ |
[U12(x1)] | = | 0 · x1 + 0 |
[plus(x1, x2)] | = | 7 · x1 + 4 · x2 + -∞ |
[a__U31#(x1, x2)] | = | 2 · x1 + 2 · x2 + 0 |
[0] | = | 2 |
[a__plus(x1, x2)] | = | 7 · x1 + 4 · x2 + -∞ |
[U11(x1, x2)] | = | 0 · x1 + 1 · x2 + 0 |
[mark(x1)] | = | 0 · x1 + -∞ |
[a__isNat(x1)] | = | 0 · x1 + 0 |
a__U11(tt,V2) | → | a__U12(a__isNat(V2)) | (1) |
a__U12(tt) | → | tt | (2) |
a__U21(tt) | → | tt | (3) |
a__U31(tt,N) | → | mark(N) | (4) |
a__U41(tt,M,N) | → | a__U42(a__isNat(N),M,N) | (5) |
a__U42(tt,M,N) | → | s(a__plus(mark(N),mark(M))) | (6) |
a__isNat(0) | → | tt | (7) |
a__isNat(plus(V1,V2)) | → | a__U11(a__isNat(V1),V2) | (8) |
a__isNat(s(V1)) | → | a__U21(a__isNat(V1)) | (9) |
a__plus(N,0) | → | a__U31(a__isNat(N),N) | (10) |
a__plus(N,s(M)) | → | a__U41(a__isNat(M),M,N) | (11) |
mark(U11(X1,X2)) | → | a__U11(mark(X1),X2) | (12) |
mark(U12(X)) | → | a__U12(mark(X)) | (13) |
mark(isNat(X)) | → | a__isNat(X) | (14) |
mark(U21(X)) | → | a__U21(mark(X)) | (15) |
mark(U31(X1,X2)) | → | a__U31(mark(X1),X2) | (16) |
mark(U41(X1,X2,X3)) | → | a__U41(mark(X1),X2,X3) | (17) |
mark(U42(X1,X2,X3)) | → | a__U42(mark(X1),X2,X3) | (18) |
mark(plus(X1,X2)) | → | a__plus(mark(X1),mark(X2)) | (19) |
mark(tt) | → | tt | (20) |
mark(s(X)) | → | s(mark(X)) | (21) |
mark(0) | → | 0 | (22) |
a__U11(X1,X2) | → | U11(X1,X2) | (23) |
a__U12(X) | → | U12(X) | (24) |
a__isNat(X) | → | isNat(X) | (25) |
a__U21(X) | → | U21(X) | (26) |
a__U31(X1,X2) | → | U31(X1,X2) | (27) |
a__U41(X1,X2,X3) | → | U41(X1,X2,X3) | (28) |
a__U42(X1,X2,X3) | → | U42(X1,X2,X3) | (29) |
a__plus(X1,X2) | → | plus(X1,X2) | (30) |
a__U31#(tt,N) | → | mark#(N) | (33) |
mark#(U31(X1,X2)) | → | mark#(X1) | (54) |
a__U42#(tt,M,N) | → | mark#(N) | (37) |
mark#(plus(X1,X2)) | → | mark#(X2) | (60) |
mark#(plus(X1,X2)) | → | a__plus#(mark(X1),mark(X2)) | (62) |
The dependency pairs are split into 1 component.
a__plus#(N,s(M)) | → | a__U41#(a__isNat(M),M,N) | (46) |
a__U41#(tt,M,N) | → | a__U42#(a__isNat(N),M,N) | (35) |
a__U42#(tt,M,N) | → | mark#(M) | (36) |
mark#(U11(X1,X2)) | → | mark#(X1) | (47) |
mark#(U12(X)) | → | mark#(X) | (49) |
mark#(U21(X)) | → | mark#(X) | (52) |
mark#(U41(X1,X2,X3)) | → | mark#(X1) | (56) |
mark#(U41(X1,X2,X3)) | → | a__U41#(mark(X1),X2,X3) | (57) |
mark#(U42(X1,X2,X3)) | → | mark#(X1) | (58) |
mark#(U42(X1,X2,X3)) | → | a__U42#(mark(X1),X2,X3) | (59) |
a__U42#(tt,M,N) | → | a__plus#(mark(N),mark(M)) | (38) |
mark#(s(X)) | → | mark#(X) | (63) |
[U41(x1, x2, x3)] | = | 0 · x1 + 0 · x2 + 0 · x3 + 2 |
[a__U21(x1)] | = | 0 · x1 + -∞ |
[isNat(x1)] | = | -∞ · x1 + 0 |
[a__U42(x1, x2, x3)] | = | 1 · x1 + 0 · x2 + 0 · x3 + 2 |
[a__U11(x1, x2)] | = | 0 · x1 + -∞ · x2 + 0 |
[a__U12(x1)] | = | 0 · x1 + 0 |
[U21(x1)] | = | 0 · x1 + -∞ |
[a__U41#(x1, x2, x3)] | = | 0 · x1 + 0 · x2 + 0 · x3 + 0 |
[U42(x1, x2, x3)] | = | 1 · x1 + 0 · x2 + 0 · x3 + 2 |
[U31(x1, x2)] | = | 4 · x1 + 0 · x2 + 5 |
[a__U42#(x1, x2, x3)] | = | 0 · x1 + 0 · x2 + 0 · x3 + 0 |
[a__plus#(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[a__U31(x1, x2)] | = | 4 · x1 + 0 · x2 + 5 |
[s(x1)] | = | 0 · x1 + -∞ |
[tt] | = | 0 |
[a__U41(x1, x2, x3)] | = | 0 · x1 + 0 · x2 + 0 · x3 + 2 |
[mark#(x1)] | = | 0 · x1 + 0 |
[U12(x1)] | = | 0 · x1 + 0 |
[plus(x1, x2)] | = | 0 · x1 + 0 · x2 + 2 |
[0] | = | 6 |
[a__plus(x1, x2)] | = | 0 · x1 + 0 · x2 + 2 |
[U11(x1, x2)] | = | 0 · x1 + -∞ · x2 + 0 |
[mark(x1)] | = | 0 · x1 + -∞ |
[a__isNat(x1)] | = | -∞ · x1 + 0 |
a__U11(tt,V2) | → | a__U12(a__isNat(V2)) | (1) |
a__U12(tt) | → | tt | (2) |
a__U21(tt) | → | tt | (3) |
a__U31(tt,N) | → | mark(N) | (4) |
a__U41(tt,M,N) | → | a__U42(a__isNat(N),M,N) | (5) |
a__U42(tt,M,N) | → | s(a__plus(mark(N),mark(M))) | (6) |
a__isNat(0) | → | tt | (7) |
a__isNat(plus(V1,V2)) | → | a__U11(a__isNat(V1),V2) | (8) |
a__isNat(s(V1)) | → | a__U21(a__isNat(V1)) | (9) |
a__plus(N,0) | → | a__U31(a__isNat(N),N) | (10) |
a__plus(N,s(M)) | → | a__U41(a__isNat(M),M,N) | (11) |
mark(U11(X1,X2)) | → | a__U11(mark(X1),X2) | (12) |
mark(U12(X)) | → | a__U12(mark(X)) | (13) |
mark(isNat(X)) | → | a__isNat(X) | (14) |
mark(U21(X)) | → | a__U21(mark(X)) | (15) |
mark(U31(X1,X2)) | → | a__U31(mark(X1),X2) | (16) |
mark(U41(X1,X2,X3)) | → | a__U41(mark(X1),X2,X3) | (17) |
mark(U42(X1,X2,X3)) | → | a__U42(mark(X1),X2,X3) | (18) |
mark(plus(X1,X2)) | → | a__plus(mark(X1),mark(X2)) | (19) |
mark(tt) | → | tt | (20) |
mark(s(X)) | → | s(mark(X)) | (21) |
mark(0) | → | 0 | (22) |
a__U11(X1,X2) | → | U11(X1,X2) | (23) |
a__U12(X) | → | U12(X) | (24) |
a__isNat(X) | → | isNat(X) | (25) |
a__U21(X) | → | U21(X) | (26) |
a__U31(X1,X2) | → | U31(X1,X2) | (27) |
a__U41(X1,X2,X3) | → | U41(X1,X2,X3) | (28) |
a__U42(X1,X2,X3) | → | U42(X1,X2,X3) | (29) |
a__plus(X1,X2) | → | plus(X1,X2) | (30) |
mark#(U42(X1,X2,X3)) | → | mark#(X1) | (58) |
[U41(x1, x2, x3)] | = | 0 · x1 + 1 · x2 + 0 · x3 + -∞ |
[a__U21(x1)] | = | 0 · x1 + -∞ |
[isNat(x1)] | = | 0 · x1 + -∞ |
[a__U42(x1, x2, x3)] | = | 0 · x1 + 1 · x2 + 0 · x3 + 0 |
[a__U11(x1, x2)] | = | 0 · x1 + 1 · x2 + -∞ |
[a__U12(x1)] | = | 1 · x1 + 0 |
[U21(x1)] | = | 0 · x1 + -∞ |
[a__U41#(x1, x2, x3)] | = | 0 · x1 + 0 · x2 + 0 · x3 + -∞ |
[U42(x1, x2, x3)] | = | 0 · x1 + 1 · x2 + 0 · x3 + 0 |
[U31(x1, x2)] | = | -∞ · x1 + 0 · x2 + -∞ |
[a__U42#(x1, x2, x3)] | = | 0 · x1 + 0 · x2 + 0 · x3 + 0 |
[a__plus#(x1, x2)] | = | 0 · x1 + 0 · x2 + -∞ |
[a__U31(x1, x2)] | = | -∞ · x1 + 0 · x2 + -∞ |
[s(x1)] | = | 0 · x1 + -∞ |
[tt] | = | 0 |
[a__U41(x1, x2, x3)] | = | 0 · x1 + 1 · x2 + 0 · x3 + -∞ |
[mark#(x1)] | = | 0 · x1 + -∞ |
[U12(x1)] | = | 1 · x1 + 0 |
[plus(x1, x2)] | = | 0 · x1 + 1 · x2 + -∞ |
[0] | = | 0 |
[a__plus(x1, x2)] | = | 0 · x1 + 1 · x2 + -∞ |
[U11(x1, x2)] | = | 0 · x1 + 1 · x2 + -∞ |
[mark(x1)] | = | 0 · x1 + -∞ |
[a__isNat(x1)] | = | 0 · x1 + -∞ |
a__U11(tt,V2) | → | a__U12(a__isNat(V2)) | (1) |
a__U12(tt) | → | tt | (2) |
a__U21(tt) | → | tt | (3) |
a__U31(tt,N) | → | mark(N) | (4) |
a__U41(tt,M,N) | → | a__U42(a__isNat(N),M,N) | (5) |
a__U42(tt,M,N) | → | s(a__plus(mark(N),mark(M))) | (6) |
a__isNat(0) | → | tt | (7) |
a__isNat(plus(V1,V2)) | → | a__U11(a__isNat(V1),V2) | (8) |
a__isNat(s(V1)) | → | a__U21(a__isNat(V1)) | (9) |
a__plus(N,0) | → | a__U31(a__isNat(N),N) | (10) |
a__plus(N,s(M)) | → | a__U41(a__isNat(M),M,N) | (11) |
mark(U11(X1,X2)) | → | a__U11(mark(X1),X2) | (12) |
mark(U12(X)) | → | a__U12(mark(X)) | (13) |
mark(isNat(X)) | → | a__isNat(X) | (14) |
mark(U21(X)) | → | a__U21(mark(X)) | (15) |
mark(U31(X1,X2)) | → | a__U31(mark(X1),X2) | (16) |
mark(U41(X1,X2,X3)) | → | a__U41(mark(X1),X2,X3) | (17) |
mark(U42(X1,X2,X3)) | → | a__U42(mark(X1),X2,X3) | (18) |
mark(plus(X1,X2)) | → | a__plus(mark(X1),mark(X2)) | (19) |
mark(tt) | → | tt | (20) |
mark(s(X)) | → | s(mark(X)) | (21) |
mark(0) | → | 0 | (22) |
a__U11(X1,X2) | → | U11(X1,X2) | (23) |
a__U12(X) | → | U12(X) | (24) |
a__isNat(X) | → | isNat(X) | (25) |
a__U21(X) | → | U21(X) | (26) |
a__U31(X1,X2) | → | U31(X1,X2) | (27) |
a__U41(X1,X2,X3) | → | U41(X1,X2,X3) | (28) |
a__U42(X1,X2,X3) | → | U42(X1,X2,X3) | (29) |
a__plus(X1,X2) | → | plus(X1,X2) | (30) |
mark#(U12(X)) | → | mark#(X) | (49) |
[U41(x1, x2, x3)] | = | 0 · x1 + 4 · x2 + 2 · x3 + -∞ |
[a__U21(x1)] | = | 0 · x1 + -∞ |
[isNat(x1)] | = | 0 · x1 + -∞ |
[a__U42(x1, x2, x3)] | = | 0 · x1 + 4 · x2 + 2 · x3 + -∞ |
[a__U11(x1, x2)] | = | 0 · x1 + 4 · x2 + -∞ |
[a__U12(x1)] | = | 1 · x1 + -∞ |
[U21(x1)] | = | 0 · x1 + -∞ |
[a__U41#(x1, x2, x3)] | = | -∞ · x1 + 0 · x2 + -∞ · x3 + -∞ |
[U42(x1, x2, x3)] | = | 0 · x1 + 4 · x2 + 2 · x3 + -∞ |
[U31(x1, x2)] | = | 0 · x1 + 1 · x2 + 0 |
[a__U42#(x1, x2, x3)] | = | -∞ · x1 + 0 · x2 + -∞ · x3 + -∞ |
[a__plus#(x1, x2)] | = | -∞ · x1 + 0 · x2 + -∞ |
[a__U31(x1, x2)] | = | 0 · x1 + 1 · x2 + 0 |
[s(x1)] | = | 0 · x1 + -∞ |
[tt] | = | 0 |
[a__U41(x1, x2, x3)] | = | 0 · x1 + 4 · x2 + 2 · x3 + -∞ |
[mark#(x1)] | = | 0 · x1 + -∞ |
[U12(x1)] | = | 1 · x1 + -∞ |
[plus(x1, x2)] | = | 2 · x1 + 4 · x2 + -∞ |
[0] | = | 1 |
[a__plus(x1, x2)] | = | 2 · x1 + 4 · x2 + -∞ |
[U11(x1, x2)] | = | 0 · x1 + 4 · x2 + -∞ |
[mark(x1)] | = | 0 · x1 + -∞ |
[a__isNat(x1)] | = | 0 · x1 + -∞ |
a__U11(tt,V2) | → | a__U12(a__isNat(V2)) | (1) |
a__U12(tt) | → | tt | (2) |
a__U21(tt) | → | tt | (3) |
a__U31(tt,N) | → | mark(N) | (4) |
a__U41(tt,M,N) | → | a__U42(a__isNat(N),M,N) | (5) |
a__U42(tt,M,N) | → | s(a__plus(mark(N),mark(M))) | (6) |
a__isNat(0) | → | tt | (7) |
a__isNat(plus(V1,V2)) | → | a__U11(a__isNat(V1),V2) | (8) |
a__isNat(s(V1)) | → | a__U21(a__isNat(V1)) | (9) |
a__plus(N,0) | → | a__U31(a__isNat(N),N) | (10) |
a__plus(N,s(M)) | → | a__U41(a__isNat(M),M,N) | (11) |
mark(U11(X1,X2)) | → | a__U11(mark(X1),X2) | (12) |
mark(U12(X)) | → | a__U12(mark(X)) | (13) |
mark(isNat(X)) | → | a__isNat(X) | (14) |
mark(U21(X)) | → | a__U21(mark(X)) | (15) |
mark(U31(X1,X2)) | → | a__U31(mark(X1),X2) | (16) |
mark(U41(X1,X2,X3)) | → | a__U41(mark(X1),X2,X3) | (17) |
mark(U42(X1,X2,X3)) | → | a__U42(mark(X1),X2,X3) | (18) |
mark(plus(X1,X2)) | → | a__plus(mark(X1),mark(X2)) | (19) |
mark(tt) | → | tt | (20) |
mark(s(X)) | → | s(mark(X)) | (21) |
mark(0) | → | 0 | (22) |
a__U11(X1,X2) | → | U11(X1,X2) | (23) |
a__U12(X) | → | U12(X) | (24) |
a__isNat(X) | → | isNat(X) | (25) |
a__U21(X) | → | U21(X) | (26) |
a__U31(X1,X2) | → | U31(X1,X2) | (27) |
a__U41(X1,X2,X3) | → | U41(X1,X2,X3) | (28) |
a__U42(X1,X2,X3) | → | U42(X1,X2,X3) | (29) |
a__plus(X1,X2) | → | plus(X1,X2) | (30) |
mark#(U41(X1,X2,X3)) | → | a__U41#(mark(X1),X2,X3) | (57) |
mark#(U42(X1,X2,X3)) | → | a__U42#(mark(X1),X2,X3) | (59) |
The dependency pairs are split into 2 components.
a__plus#(N,s(M)) | → | a__U41#(a__isNat(M),M,N) | (46) |
a__U41#(tt,M,N) | → | a__U42#(a__isNat(N),M,N) | (35) |
a__U42#(tt,M,N) | → | a__plus#(mark(N),mark(M)) | (38) |
[U41(x1, x2, x3)] | = | 0 · x1 + 2 · x2 + 1 · x3 + 1 |
[a__U21(x1)] | = | 0 · x1 + 0 |
[isNat(x1)] | = | 0 · x1 + 0 |
[a__U42(x1, x2, x3)] | = | 4 · x1 + 2 · x2 + 1 · x3 + 1 |
[a__U11(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[a__U12(x1)] | = | 0 · x1 + 0 |
[U21(x1)] | = | 0 · x1 + 0 |
[a__U41#(x1, x2, x3)] | = | 0 · x1 + 1 · x2 + 0 · x3 + 0 |
[U42(x1, x2, x3)] | = | 4 · x1 + 2 · x2 + 1 · x3 + 1 |
[U31(x1, x2)] | = | 0 · x1 + 1 · x2 + 0 |
[a__U42#(x1, x2, x3)] | = | 0 · x1 + 1 · x2 + 0 · x3 + 0 |
[a__plus#(x1, x2)] | = | 0 · x1 + 1 · x2 + 0 |
[a__U31(x1, x2)] | = | 0 · x1 + 1 · x2 + 0 |
[s(x1)] | = | 1 · x1 + 1 |
[tt] | = | 0 |
[a__U41(x1, x2, x3)] | = | 0 · x1 + 2 · x2 + 1 · x3 + 1 |
[U12(x1)] | = | 0 · x1 + 0 |
[plus(x1, x2)] | = | 1 · x1 + 2 · x2 + 0 |
[0] | = | 0 |
[a__plus(x1, x2)] | = | 1 · x1 + 2 · x2 + 0 |
[U11(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[mark(x1)] | = | 1 · x1 + 0 |
[a__isNat(x1)] | = | 0 · x1 + 0 |
a__U11(tt,V2) | → | a__U12(a__isNat(V2)) | (1) |
a__U12(tt) | → | tt | (2) |
a__U21(tt) | → | tt | (3) |
a__U31(tt,N) | → | mark(N) | (4) |
a__U41(tt,M,N) | → | a__U42(a__isNat(N),M,N) | (5) |
a__U42(tt,M,N) | → | s(a__plus(mark(N),mark(M))) | (6) |
a__isNat(0) | → | tt | (7) |
a__isNat(plus(V1,V2)) | → | a__U11(a__isNat(V1),V2) | (8) |
a__isNat(s(V1)) | → | a__U21(a__isNat(V1)) | (9) |
a__plus(N,0) | → | a__U31(a__isNat(N),N) | (10) |
a__plus(N,s(M)) | → | a__U41(a__isNat(M),M,N) | (11) |
mark(U11(X1,X2)) | → | a__U11(mark(X1),X2) | (12) |
mark(U12(X)) | → | a__U12(mark(X)) | (13) |
mark(isNat(X)) | → | a__isNat(X) | (14) |
mark(U21(X)) | → | a__U21(mark(X)) | (15) |
mark(U31(X1,X2)) | → | a__U31(mark(X1),X2) | (16) |
mark(U41(X1,X2,X3)) | → | a__U41(mark(X1),X2,X3) | (17) |
mark(U42(X1,X2,X3)) | → | a__U42(mark(X1),X2,X3) | (18) |
mark(plus(X1,X2)) | → | a__plus(mark(X1),mark(X2)) | (19) |
mark(tt) | → | tt | (20) |
mark(s(X)) | → | s(mark(X)) | (21) |
mark(0) | → | 0 | (22) |
a__U11(X1,X2) | → | U11(X1,X2) | (23) |
a__U12(X) | → | U12(X) | (24) |
a__isNat(X) | → | isNat(X) | (25) |
a__U21(X) | → | U21(X) | (26) |
a__U31(X1,X2) | → | U31(X1,X2) | (27) |
a__U41(X1,X2,X3) | → | U41(X1,X2,X3) | (28) |
a__U42(X1,X2,X3) | → | U42(X1,X2,X3) | (29) |
a__plus(X1,X2) | → | plus(X1,X2) | (30) |
a__plus#(N,s(M)) | → | a__U41#(a__isNat(M),M,N) | (46) |
The dependency pairs are split into 0 components.
mark#(U11(X1,X2)) | → | mark#(X1) | (47) |
mark#(U21(X)) | → | mark#(X) | (52) |
mark#(U41(X1,X2,X3)) | → | mark#(X1) | (56) |
mark#(s(X)) | → | mark#(X) | (63) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
mark#(U11(X1,X2)) | → | mark#(X1) | (47) |
1 | > | 1 | |
mark#(U21(X)) | → | mark#(X) | (52) |
1 | > | 1 | |
mark#(U41(X1,X2,X3)) | → | mark#(X1) | (56) |
1 | > | 1 | |
mark#(s(X)) | → | mark#(X) | (63) |
1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
a__U11#(tt,V2) | → | a__isNat#(V2) | (31) |
a__isNat#(plus(V1,V2)) | → | a__isNat#(V1) | (39) |
a__isNat#(plus(V1,V2)) | → | a__U11#(a__isNat(V1),V2) | (40) |
a__isNat#(s(V1)) | → | a__isNat#(V1) | (41) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
a__U11#(tt,V2) | → | a__isNat#(V2) | (31) |
2 | ≥ | 1 | |
a__isNat#(plus(V1,V2)) | → | a__isNat#(V1) | (39) |
1 | > | 1 | |
a__isNat#(plus(V1,V2)) | → | a__U11#(a__isNat(V1),V2) | (40) |
1 | > | 2 | |
a__isNat#(s(V1)) | → | a__isNat#(V1) | (41) |
1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.