The rewrite relation of the following TRS is considered.
active(U11(tt,V2)) | → | mark(U12(isNat(V2))) | (1) |
active(U12(tt)) | → | mark(tt) | (2) |
active(U21(tt)) | → | mark(tt) | (3) |
active(U31(tt,N)) | → | mark(N) | (4) |
active(U41(tt,M,N)) | → | mark(U42(isNat(N),M,N)) | (5) |
active(U42(tt,M,N)) | → | mark(s(plus(N,M))) | (6) |
active(isNat(0)) | → | mark(tt) | (7) |
active(isNat(plus(V1,V2))) | → | mark(U11(isNat(V1),V2)) | (8) |
active(isNat(s(V1))) | → | mark(U21(isNat(V1))) | (9) |
active(plus(N,0)) | → | mark(U31(isNat(N),N)) | (10) |
active(plus(N,s(M))) | → | mark(U41(isNat(M),M,N)) | (11) |
mark(U11(X1,X2)) | → | active(U11(mark(X1),X2)) | (12) |
mark(tt) | → | active(tt) | (13) |
mark(U12(X)) | → | active(U12(mark(X))) | (14) |
mark(isNat(X)) | → | active(isNat(X)) | (15) |
mark(U21(X)) | → | active(U21(mark(X))) | (16) |
mark(U31(X1,X2)) | → | active(U31(mark(X1),X2)) | (17) |
mark(U41(X1,X2,X3)) | → | active(U41(mark(X1),X2,X3)) | (18) |
mark(U42(X1,X2,X3)) | → | active(U42(mark(X1),X2,X3)) | (19) |
mark(s(X)) | → | active(s(mark(X))) | (20) |
mark(plus(X1,X2)) | → | active(plus(mark(X1),mark(X2))) | (21) |
mark(0) | → | active(0) | (22) |
U11(mark(X1),X2) | → | U11(X1,X2) | (23) |
U11(X1,mark(X2)) | → | U11(X1,X2) | (24) |
U11(active(X1),X2) | → | U11(X1,X2) | (25) |
U11(X1,active(X2)) | → | U11(X1,X2) | (26) |
U12(mark(X)) | → | U12(X) | (27) |
U12(active(X)) | → | U12(X) | (28) |
isNat(mark(X)) | → | isNat(X) | (29) |
isNat(active(X)) | → | isNat(X) | (30) |
U21(mark(X)) | → | U21(X) | (31) |
U21(active(X)) | → | U21(X) | (32) |
U31(mark(X1),X2) | → | U31(X1,X2) | (33) |
U31(X1,mark(X2)) | → | U31(X1,X2) | (34) |
U31(active(X1),X2) | → | U31(X1,X2) | (35) |
U31(X1,active(X2)) | → | U31(X1,X2) | (36) |
U41(mark(X1),X2,X3) | → | U41(X1,X2,X3) | (37) |
U41(X1,mark(X2),X3) | → | U41(X1,X2,X3) | (38) |
U41(X1,X2,mark(X3)) | → | U41(X1,X2,X3) | (39) |
U41(active(X1),X2,X3) | → | U41(X1,X2,X3) | (40) |
U41(X1,active(X2),X3) | → | U41(X1,X2,X3) | (41) |
U41(X1,X2,active(X3)) | → | U41(X1,X2,X3) | (42) |
U42(mark(X1),X2,X3) | → | U42(X1,X2,X3) | (43) |
U42(X1,mark(X2),X3) | → | U42(X1,X2,X3) | (44) |
U42(X1,X2,mark(X3)) | → | U42(X1,X2,X3) | (45) |
U42(active(X1),X2,X3) | → | U42(X1,X2,X3) | (46) |
U42(X1,active(X2),X3) | → | U42(X1,X2,X3) | (47) |
U42(X1,X2,active(X3)) | → | U42(X1,X2,X3) | (48) |
s(mark(X)) | → | s(X) | (49) |
s(active(X)) | → | s(X) | (50) |
plus(mark(X1),X2) | → | plus(X1,X2) | (51) |
plus(X1,mark(X2)) | → | plus(X1,X2) | (52) |
plus(active(X1),X2) | → | plus(X1,X2) | (53) |
plus(X1,active(X2)) | → | plus(X1,X2) | (54) |
active#(U11(tt,V2)) | → | isNat#(V2) | (55) |
active#(U11(tt,V2)) | → | U12#(isNat(V2)) | (56) |
active#(U11(tt,V2)) | → | mark#(U12(isNat(V2))) | (57) |
active#(U12(tt)) | → | mark#(tt) | (58) |
active#(U21(tt)) | → | mark#(tt) | (59) |
active#(U31(tt,N)) | → | mark#(N) | (60) |
active#(U41(tt,M,N)) | → | isNat#(N) | (61) |
active#(U41(tt,M,N)) | → | U42#(isNat(N),M,N) | (62) |
active#(U41(tt,M,N)) | → | mark#(U42(isNat(N),M,N)) | (63) |
active#(U42(tt,M,N)) | → | plus#(N,M) | (64) |
active#(U42(tt,M,N)) | → | s#(plus(N,M)) | (65) |
active#(U42(tt,M,N)) | → | mark#(s(plus(N,M))) | (66) |
active#(isNat(0)) | → | mark#(tt) | (67) |
active#(isNat(plus(V1,V2))) | → | isNat#(V1) | (68) |
active#(isNat(plus(V1,V2))) | → | U11#(isNat(V1),V2) | (69) |
active#(isNat(plus(V1,V2))) | → | mark#(U11(isNat(V1),V2)) | (70) |
active#(isNat(s(V1))) | → | isNat#(V1) | (71) |
active#(isNat(s(V1))) | → | U21#(isNat(V1)) | (72) |
active#(isNat(s(V1))) | → | mark#(U21(isNat(V1))) | (73) |
active#(plus(N,0)) | → | isNat#(N) | (74) |
active#(plus(N,0)) | → | U31#(isNat(N),N) | (75) |
active#(plus(N,0)) | → | mark#(U31(isNat(N),N)) | (76) |
active#(plus(N,s(M))) | → | isNat#(M) | (77) |
active#(plus(N,s(M))) | → | U41#(isNat(M),M,N) | (78) |
active#(plus(N,s(M))) | → | mark#(U41(isNat(M),M,N)) | (79) |
mark#(U11(X1,X2)) | → | mark#(X1) | (80) |
mark#(U11(X1,X2)) | → | U11#(mark(X1),X2) | (81) |
mark#(U11(X1,X2)) | → | active#(U11(mark(X1),X2)) | (82) |
mark#(tt) | → | active#(tt) | (83) |
mark#(U12(X)) | → | mark#(X) | (84) |
mark#(U12(X)) | → | U12#(mark(X)) | (85) |
mark#(U12(X)) | → | active#(U12(mark(X))) | (86) |
mark#(isNat(X)) | → | active#(isNat(X)) | (87) |
mark#(U21(X)) | → | mark#(X) | (88) |
mark#(U21(X)) | → | U21#(mark(X)) | (89) |
mark#(U21(X)) | → | active#(U21(mark(X))) | (90) |
mark#(U31(X1,X2)) | → | mark#(X1) | (91) |
mark#(U31(X1,X2)) | → | U31#(mark(X1),X2) | (92) |
mark#(U31(X1,X2)) | → | active#(U31(mark(X1),X2)) | (93) |
mark#(U41(X1,X2,X3)) | → | mark#(X1) | (94) |
mark#(U41(X1,X2,X3)) | → | U41#(mark(X1),X2,X3) | (95) |
mark#(U41(X1,X2,X3)) | → | active#(U41(mark(X1),X2,X3)) | (96) |
mark#(U42(X1,X2,X3)) | → | mark#(X1) | (97) |
mark#(U42(X1,X2,X3)) | → | U42#(mark(X1),X2,X3) | (98) |
mark#(U42(X1,X2,X3)) | → | active#(U42(mark(X1),X2,X3)) | (99) |
mark#(s(X)) | → | mark#(X) | (100) |
mark#(s(X)) | → | s#(mark(X)) | (101) |
mark#(s(X)) | → | active#(s(mark(X))) | (102) |
mark#(plus(X1,X2)) | → | mark#(X2) | (103) |
mark#(plus(X1,X2)) | → | mark#(X1) | (104) |
mark#(plus(X1,X2)) | → | plus#(mark(X1),mark(X2)) | (105) |
mark#(plus(X1,X2)) | → | active#(plus(mark(X1),mark(X2))) | (106) |
mark#(0) | → | active#(0) | (107) |
U11#(mark(X1),X2) | → | U11#(X1,X2) | (108) |
U11#(X1,mark(X2)) | → | U11#(X1,X2) | (109) |
U11#(active(X1),X2) | → | U11#(X1,X2) | (110) |
U11#(X1,active(X2)) | → | U11#(X1,X2) | (111) |
U12#(mark(X)) | → | U12#(X) | (112) |
U12#(active(X)) | → | U12#(X) | (113) |
isNat#(mark(X)) | → | isNat#(X) | (114) |
isNat#(active(X)) | → | isNat#(X) | (115) |
U21#(mark(X)) | → | U21#(X) | (116) |
U21#(active(X)) | → | U21#(X) | (117) |
U31#(mark(X1),X2) | → | U31#(X1,X2) | (118) |
U31#(X1,mark(X2)) | → | U31#(X1,X2) | (119) |
U31#(active(X1),X2) | → | U31#(X1,X2) | (120) |
U31#(X1,active(X2)) | → | U31#(X1,X2) | (121) |
U41#(mark(X1),X2,X3) | → | U41#(X1,X2,X3) | (122) |
U41#(X1,mark(X2),X3) | → | U41#(X1,X2,X3) | (123) |
U41#(X1,X2,mark(X3)) | → | U41#(X1,X2,X3) | (124) |
U41#(active(X1),X2,X3) | → | U41#(X1,X2,X3) | (125) |
U41#(X1,active(X2),X3) | → | U41#(X1,X2,X3) | (126) |
U41#(X1,X2,active(X3)) | → | U41#(X1,X2,X3) | (127) |
U42#(mark(X1),X2,X3) | → | U42#(X1,X2,X3) | (128) |
U42#(X1,mark(X2),X3) | → | U42#(X1,X2,X3) | (129) |
U42#(X1,X2,mark(X3)) | → | U42#(X1,X2,X3) | (130) |
U42#(active(X1),X2,X3) | → | U42#(X1,X2,X3) | (131) |
U42#(X1,active(X2),X3) | → | U42#(X1,X2,X3) | (132) |
U42#(X1,X2,active(X3)) | → | U42#(X1,X2,X3) | (133) |
s#(mark(X)) | → | s#(X) | (134) |
s#(active(X)) | → | s#(X) | (135) |
plus#(mark(X1),X2) | → | plus#(X1,X2) | (136) |
plus#(X1,mark(X2)) | → | plus#(X1,X2) | (137) |
plus#(active(X1),X2) | → | plus#(X1,X2) | (138) |
plus#(X1,active(X2)) | → | plus#(X1,X2) | (139) |
The dependency pairs are split into 10 components.
mark#(s(X)) | → | mark#(X) | (100) |
mark#(plus(X1,X2)) | → | active#(plus(mark(X1),mark(X2))) | (106) |
active#(plus(N,s(M))) | → | mark#(U41(isNat(M),M,N)) | (79) |
mark#(U41(X1,X2,X3)) | → | active#(U41(mark(X1),X2,X3)) | (96) |
active#(U41(tt,M,N)) | → | mark#(U42(isNat(N),M,N)) | (63) |
mark#(U42(X1,X2,X3)) | → | active#(U42(mark(X1),X2,X3)) | (99) |
active#(U42(tt,M,N)) | → | mark#(s(plus(N,M))) | (66) |
mark#(U42(X1,X2,X3)) | → | mark#(X1) | (97) |
mark#(plus(X1,X2)) | → | mark#(X1) | (104) |
mark#(plus(X1,X2)) | → | mark#(X2) | (103) |
mark#(U41(X1,X2,X3)) | → | mark#(X1) | (94) |
mark#(U31(X1,X2)) | → | active#(U31(mark(X1),X2)) | (93) |
active#(U31(tt,N)) | → | mark#(N) | (60) |
mark#(U31(X1,X2)) | → | mark#(X1) | (91) |
mark#(U21(X)) | → | mark#(X) | (88) |
mark#(isNat(X)) | → | active#(isNat(X)) | (87) |
active#(isNat(s(V1))) | → | mark#(U21(isNat(V1))) | (73) |
active#(isNat(plus(V1,V2))) | → | mark#(U11(isNat(V1),V2)) | (70) |
mark#(U11(X1,X2)) | → | active#(U11(mark(X1),X2)) | (82) |
active#(U11(tt,V2)) | → | mark#(U12(isNat(V2))) | (57) |
mark#(U12(X)) | → | mark#(X) | (84) |
mark#(U11(X1,X2)) | → | mark#(X1) | (80) |
active#(plus(N,0)) | → | mark#(U31(isNat(N),N)) | (76) |
[U12(x1)] | = | 0 · x1 + 0 |
[U41(x1, x2, x3)] | = | 4 · x1 + 6 · x2 + 1 · x3 + 0 |
[U11(x1, x2)] | = | 0 · x1 + -∞ · x2 + 0 |
[isNat(x1)] | = | -∞ · x1 + 0 |
[mark#(x1)] | = | 0 · x1 + 0 |
[mark(x1)] | = | 0 · x1 + -∞ |
[plus(x1, x2)] | = | 1 · x1 + 6 · x2 + 0 |
[tt] | = | 0 |
[U31(x1, x2)] | = | 4 · x1 + 1 · x2 + 4 |
[0] | = | 0 |
[s(x1)] | = | 0 · x1 + 0 |
[U42(x1, x2, x3)] | = | 4 · x1 + 6 · x2 + 1 · x3 + 0 |
[active#(x1)] | = | 0 · x1 + 0 |
[U21(x1)] | = | 0 · x1 + 0 |
[active(x1)] | = | 0 · x1 + 0 |
active(U11(tt,V2)) | → | mark(U12(isNat(V2))) | (1) |
active(U12(tt)) | → | mark(tt) | (2) |
active(U21(tt)) | → | mark(tt) | (3) |
active(U31(tt,N)) | → | mark(N) | (4) |
active(U41(tt,M,N)) | → | mark(U42(isNat(N),M,N)) | (5) |
active(U42(tt,M,N)) | → | mark(s(plus(N,M))) | (6) |
active(isNat(0)) | → | mark(tt) | (7) |
active(isNat(plus(V1,V2))) | → | mark(U11(isNat(V1),V2)) | (8) |
active(isNat(s(V1))) | → | mark(U21(isNat(V1))) | (9) |
active(plus(N,0)) | → | mark(U31(isNat(N),N)) | (10) |
active(plus(N,s(M))) | → | mark(U41(isNat(M),M,N)) | (11) |
mark(U11(X1,X2)) | → | active(U11(mark(X1),X2)) | (12) |
mark(tt) | → | active(tt) | (13) |
mark(U12(X)) | → | active(U12(mark(X))) | (14) |
mark(isNat(X)) | → | active(isNat(X)) | (15) |
mark(U21(X)) | → | active(U21(mark(X))) | (16) |
mark(U31(X1,X2)) | → | active(U31(mark(X1),X2)) | (17) |
mark(U41(X1,X2,X3)) | → | active(U41(mark(X1),X2,X3)) | (18) |
mark(U42(X1,X2,X3)) | → | active(U42(mark(X1),X2,X3)) | (19) |
mark(s(X)) | → | active(s(mark(X))) | (20) |
mark(plus(X1,X2)) | → | active(plus(mark(X1),mark(X2))) | (21) |
mark(0) | → | active(0) | (22) |
U11(mark(X1),X2) | → | U11(X1,X2) | (23) |
U11(X1,mark(X2)) | → | U11(X1,X2) | (24) |
U11(active(X1),X2) | → | U11(X1,X2) | (25) |
U11(X1,active(X2)) | → | U11(X1,X2) | (26) |
U12(mark(X)) | → | U12(X) | (27) |
U12(active(X)) | → | U12(X) | (28) |
isNat(mark(X)) | → | isNat(X) | (29) |
isNat(active(X)) | → | isNat(X) | (30) |
U21(mark(X)) | → | U21(X) | (31) |
U21(active(X)) | → | U21(X) | (32) |
U31(mark(X1),X2) | → | U31(X1,X2) | (33) |
U31(X1,mark(X2)) | → | U31(X1,X2) | (34) |
U31(active(X1),X2) | → | U31(X1,X2) | (35) |
U31(X1,active(X2)) | → | U31(X1,X2) | (36) |
U41(mark(X1),X2,X3) | → | U41(X1,X2,X3) | (37) |
U41(X1,mark(X2),X3) | → | U41(X1,X2,X3) | (38) |
U41(X1,X2,mark(X3)) | → | U41(X1,X2,X3) | (39) |
U41(active(X1),X2,X3) | → | U41(X1,X2,X3) | (40) |
U41(X1,active(X2),X3) | → | U41(X1,X2,X3) | (41) |
U41(X1,X2,active(X3)) | → | U41(X1,X2,X3) | (42) |
U42(mark(X1),X2,X3) | → | U42(X1,X2,X3) | (43) |
U42(X1,mark(X2),X3) | → | U42(X1,X2,X3) | (44) |
U42(X1,X2,mark(X3)) | → | U42(X1,X2,X3) | (45) |
U42(active(X1),X2,X3) | → | U42(X1,X2,X3) | (46) |
U42(X1,active(X2),X3) | → | U42(X1,X2,X3) | (47) |
U42(X1,X2,active(X3)) | → | U42(X1,X2,X3) | (48) |
s(mark(X)) | → | s(X) | (49) |
s(active(X)) | → | s(X) | (50) |
plus(mark(X1),X2) | → | plus(X1,X2) | (51) |
plus(X1,mark(X2)) | → | plus(X1,X2) | (52) |
plus(active(X1),X2) | → | plus(X1,X2) | (53) |
plus(X1,active(X2)) | → | plus(X1,X2) | (54) |
active#(U31(tt,N)) | → | mark#(N) | (60) |
mark#(U31(X1,X2)) | → | mark#(X1) | (91) |
The dependency pairs are split into 1 component.
mark#(s(X)) | → | mark#(X) | (100) |
mark#(plus(X1,X2)) | → | active#(plus(mark(X1),mark(X2))) | (106) |
active#(plus(N,s(M))) | → | mark#(U41(isNat(M),M,N)) | (79) |
mark#(U41(X1,X2,X3)) | → | active#(U41(mark(X1),X2,X3)) | (96) |
active#(U41(tt,M,N)) | → | mark#(U42(isNat(N),M,N)) | (63) |
mark#(U42(X1,X2,X3)) | → | active#(U42(mark(X1),X2,X3)) | (99) |
active#(U42(tt,M,N)) | → | mark#(s(plus(N,M))) | (66) |
mark#(U42(X1,X2,X3)) | → | mark#(X1) | (97) |
mark#(plus(X1,X2)) | → | mark#(X1) | (104) |
mark#(plus(X1,X2)) | → | mark#(X2) | (103) |
mark#(U41(X1,X2,X3)) | → | mark#(X1) | (94) |
mark#(U21(X)) | → | mark#(X) | (88) |
mark#(isNat(X)) | → | active#(isNat(X)) | (87) |
active#(isNat(s(V1))) | → | mark#(U21(isNat(V1))) | (73) |
active#(isNat(plus(V1,V2))) | → | mark#(U11(isNat(V1),V2)) | (70) |
mark#(U11(X1,X2)) | → | active#(U11(mark(X1),X2)) | (82) |
active#(U11(tt,V2)) | → | mark#(U12(isNat(V2))) | (57) |
mark#(U12(X)) | → | mark#(X) | (84) |
mark#(U11(X1,X2)) | → | mark#(X1) | (80) |
[U12(x1)] | = | 0 · x1 + 0 |
[U41(x1, x2, x3)] | = | 0 · x1 + 0 · x2 + 2 · x3 + 4 |
[U11(x1, x2)] | = | 0 · x1 + 0 · x2 + -∞ |
[isNat(x1)] | = | 0 · x1 + 4 |
[mark#(x1)] | = | 0 · x1 + -∞ |
[mark(x1)] | = | 0 · x1 + -∞ |
[plus(x1, x2)] | = | 2 · x1 + 0 · x2 + -∞ |
[tt] | = | 6 |
[U31(x1, x2)] | = | 0 · x1 + 2 · x2 + 0 |
[0] | = | 6 |
[s(x1)] | = | 0 · x1 + 4 |
[U42(x1, x2, x3)] | = | 0 · x1 + 0 · x2 + 2 · x3 + 0 |
[active#(x1)] | = | 0 · x1 + -∞ |
[U21(x1)] | = | 0 · x1 + -∞ |
[active(x1)] | = | 0 · x1 + -∞ |
active(U11(tt,V2)) | → | mark(U12(isNat(V2))) | (1) |
active(U12(tt)) | → | mark(tt) | (2) |
active(U21(tt)) | → | mark(tt) | (3) |
active(U31(tt,N)) | → | mark(N) | (4) |
active(U41(tt,M,N)) | → | mark(U42(isNat(N),M,N)) | (5) |
active(U42(tt,M,N)) | → | mark(s(plus(N,M))) | (6) |
active(isNat(0)) | → | mark(tt) | (7) |
active(isNat(plus(V1,V2))) | → | mark(U11(isNat(V1),V2)) | (8) |
active(isNat(s(V1))) | → | mark(U21(isNat(V1))) | (9) |
active(plus(N,0)) | → | mark(U31(isNat(N),N)) | (10) |
active(plus(N,s(M))) | → | mark(U41(isNat(M),M,N)) | (11) |
mark(U11(X1,X2)) | → | active(U11(mark(X1),X2)) | (12) |
mark(tt) | → | active(tt) | (13) |
mark(U12(X)) | → | active(U12(mark(X))) | (14) |
mark(isNat(X)) | → | active(isNat(X)) | (15) |
mark(U21(X)) | → | active(U21(mark(X))) | (16) |
mark(U31(X1,X2)) | → | active(U31(mark(X1),X2)) | (17) |
mark(U41(X1,X2,X3)) | → | active(U41(mark(X1),X2,X3)) | (18) |
mark(U42(X1,X2,X3)) | → | active(U42(mark(X1),X2,X3)) | (19) |
mark(s(X)) | → | active(s(mark(X))) | (20) |
mark(plus(X1,X2)) | → | active(plus(mark(X1),mark(X2))) | (21) |
mark(0) | → | active(0) | (22) |
U11(mark(X1),X2) | → | U11(X1,X2) | (23) |
U11(X1,mark(X2)) | → | U11(X1,X2) | (24) |
U11(active(X1),X2) | → | U11(X1,X2) | (25) |
U11(X1,active(X2)) | → | U11(X1,X2) | (26) |
U12(mark(X)) | → | U12(X) | (27) |
U12(active(X)) | → | U12(X) | (28) |
isNat(mark(X)) | → | isNat(X) | (29) |
isNat(active(X)) | → | isNat(X) | (30) |
U21(mark(X)) | → | U21(X) | (31) |
U21(active(X)) | → | U21(X) | (32) |
U31(mark(X1),X2) | → | U31(X1,X2) | (33) |
U31(X1,mark(X2)) | → | U31(X1,X2) | (34) |
U31(active(X1),X2) | → | U31(X1,X2) | (35) |
U31(X1,active(X2)) | → | U31(X1,X2) | (36) |
U41(mark(X1),X2,X3) | → | U41(X1,X2,X3) | (37) |
U41(X1,mark(X2),X3) | → | U41(X1,X2,X3) | (38) |
U41(X1,X2,mark(X3)) | → | U41(X1,X2,X3) | (39) |
U41(active(X1),X2,X3) | → | U41(X1,X2,X3) | (40) |
U41(X1,active(X2),X3) | → | U41(X1,X2,X3) | (41) |
U41(X1,X2,active(X3)) | → | U41(X1,X2,X3) | (42) |
U42(mark(X1),X2,X3) | → | U42(X1,X2,X3) | (43) |
U42(X1,mark(X2),X3) | → | U42(X1,X2,X3) | (44) |
U42(X1,X2,mark(X3)) | → | U42(X1,X2,X3) | (45) |
U42(active(X1),X2,X3) | → | U42(X1,X2,X3) | (46) |
U42(X1,active(X2),X3) | → | U42(X1,X2,X3) | (47) |
U42(X1,X2,active(X3)) | → | U42(X1,X2,X3) | (48) |
s(mark(X)) | → | s(X) | (49) |
s(active(X)) | → | s(X) | (50) |
plus(mark(X1),X2) | → | plus(X1,X2) | (51) |
plus(X1,mark(X2)) | → | plus(X1,X2) | (52) |
plus(active(X1),X2) | → | plus(X1,X2) | (53) |
plus(X1,active(X2)) | → | plus(X1,X2) | (54) |
mark#(plus(X1,X2)) | → | mark#(X1) | (104) |
[U12(x1)] | = | 0 · x1 + -∞ |
[U41(x1, x2, x3)] | = | 1 · x1 + 1 · x2 + 1 · x3 + 0 |
[U11(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[isNat(x1)] | = | 0 · x1 + 0 |
[mark#(x1)] | = | 0 · x1 + 0 |
[mark(x1)] | = | 0 · x1 + -∞ |
[plus(x1, x2)] | = | 1 · x1 + 1 · x2 + 1 |
[tt] | = | 0 |
[U31(x1, x2)] | = | -∞ · x1 + 1 · x2 + -∞ |
[0] | = | 2 |
[s(x1)] | = | 0 · x1 + -∞ |
[U42(x1, x2, x3)] | = | 1 · x1 + 1 · x2 + 1 · x3 + -∞ |
[active#(x1)] | = | 0 · x1 + 0 |
[U21(x1)] | = | 0 · x1 + 0 |
[active(x1)] | = | 0 · x1 + -∞ |
active(U11(tt,V2)) | → | mark(U12(isNat(V2))) | (1) |
active(U12(tt)) | → | mark(tt) | (2) |
active(U21(tt)) | → | mark(tt) | (3) |
active(U31(tt,N)) | → | mark(N) | (4) |
active(U41(tt,M,N)) | → | mark(U42(isNat(N),M,N)) | (5) |
active(U42(tt,M,N)) | → | mark(s(plus(N,M))) | (6) |
active(isNat(0)) | → | mark(tt) | (7) |
active(isNat(plus(V1,V2))) | → | mark(U11(isNat(V1),V2)) | (8) |
active(isNat(s(V1))) | → | mark(U21(isNat(V1))) | (9) |
active(plus(N,0)) | → | mark(U31(isNat(N),N)) | (10) |
active(plus(N,s(M))) | → | mark(U41(isNat(M),M,N)) | (11) |
mark(U11(X1,X2)) | → | active(U11(mark(X1),X2)) | (12) |
mark(tt) | → | active(tt) | (13) |
mark(U12(X)) | → | active(U12(mark(X))) | (14) |
mark(isNat(X)) | → | active(isNat(X)) | (15) |
mark(U21(X)) | → | active(U21(mark(X))) | (16) |
mark(U31(X1,X2)) | → | active(U31(mark(X1),X2)) | (17) |
mark(U41(X1,X2,X3)) | → | active(U41(mark(X1),X2,X3)) | (18) |
mark(U42(X1,X2,X3)) | → | active(U42(mark(X1),X2,X3)) | (19) |
mark(s(X)) | → | active(s(mark(X))) | (20) |
mark(plus(X1,X2)) | → | active(plus(mark(X1),mark(X2))) | (21) |
mark(0) | → | active(0) | (22) |
U11(mark(X1),X2) | → | U11(X1,X2) | (23) |
U11(X1,mark(X2)) | → | U11(X1,X2) | (24) |
U11(active(X1),X2) | → | U11(X1,X2) | (25) |
U11(X1,active(X2)) | → | U11(X1,X2) | (26) |
U12(mark(X)) | → | U12(X) | (27) |
U12(active(X)) | → | U12(X) | (28) |
isNat(mark(X)) | → | isNat(X) | (29) |
isNat(active(X)) | → | isNat(X) | (30) |
U21(mark(X)) | → | U21(X) | (31) |
U21(active(X)) | → | U21(X) | (32) |
U31(mark(X1),X2) | → | U31(X1,X2) | (33) |
U31(X1,mark(X2)) | → | U31(X1,X2) | (34) |
U31(active(X1),X2) | → | U31(X1,X2) | (35) |
U31(X1,active(X2)) | → | U31(X1,X2) | (36) |
U41(mark(X1),X2,X3) | → | U41(X1,X2,X3) | (37) |
U41(X1,mark(X2),X3) | → | U41(X1,X2,X3) | (38) |
U41(X1,X2,mark(X3)) | → | U41(X1,X2,X3) | (39) |
U41(active(X1),X2,X3) | → | U41(X1,X2,X3) | (40) |
U41(X1,active(X2),X3) | → | U41(X1,X2,X3) | (41) |
U41(X1,X2,active(X3)) | → | U41(X1,X2,X3) | (42) |
U42(mark(X1),X2,X3) | → | U42(X1,X2,X3) | (43) |
U42(X1,mark(X2),X3) | → | U42(X1,X2,X3) | (44) |
U42(X1,X2,mark(X3)) | → | U42(X1,X2,X3) | (45) |
U42(active(X1),X2,X3) | → | U42(X1,X2,X3) | (46) |
U42(X1,active(X2),X3) | → | U42(X1,X2,X3) | (47) |
U42(X1,X2,active(X3)) | → | U42(X1,X2,X3) | (48) |
s(mark(X)) | → | s(X) | (49) |
s(active(X)) | → | s(X) | (50) |
plus(mark(X1),X2) | → | plus(X1,X2) | (51) |
plus(X1,mark(X2)) | → | plus(X1,X2) | (52) |
plus(active(X1),X2) | → | plus(X1,X2) | (53) |
plus(X1,active(X2)) | → | plus(X1,X2) | (54) |
mark#(plus(X1,X2)) | → | mark#(X2) | (103) |
active#(isNat(plus(V1,V2))) | → | mark#(U11(isNat(V1),V2)) | (70) |
[U12(x1)] | = | 1 · x1 + 0 |
[U41(x1, x2, x3)] | = | 2 · x1 + 0 · x2 + 0 · x3 + 0 |
[U11(x1, x2)] | = | 1 · x1 + 4 · x2 + 3 |
[isNat(x1)] | = | 0 · x1 + 0 |
[mark#(x1)] | = | 4 · x1 + 0 |
[mark(x1)] | = | 2 · x1 + 1 |
[plus(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[tt] | = | 3 |
[U31(x1, x2)] | = | 0 · x1 + 0 · x2 + 1 |
[0] | = | 1 |
[s(x1)] | = | 2 · x1 + 0 |
[U42(x1, x2, x3)] | = | 1 · x1 + 0 · x2 + 0 · x3 + 0 |
[active#(x1)] | = | 0 · x1 + 0 |
[U21(x1)] | = | 2 · x1 + 0 |
[active(x1)] | = | 4 · x1 + 5 |
U12(mark(X)) | → | U12(X) | (27) |
U12(active(X)) | → | U12(X) | (28) |
isNat(mark(X)) | → | isNat(X) | (29) |
isNat(active(X)) | → | isNat(X) | (30) |
U21(mark(X)) | → | U21(X) | (31) |
U21(active(X)) | → | U21(X) | (32) |
U41(mark(X1),X2,X3) | → | U41(X1,X2,X3) | (37) |
U41(X1,mark(X2),X3) | → | U41(X1,X2,X3) | (38) |
U41(X1,X2,mark(X3)) | → | U41(X1,X2,X3) | (39) |
U41(active(X1),X2,X3) | → | U41(X1,X2,X3) | (40) |
U41(X1,active(X2),X3) | → | U41(X1,X2,X3) | (41) |
U41(X1,X2,active(X3)) | → | U41(X1,X2,X3) | (42) |
U42(mark(X1),X2,X3) | → | U42(X1,X2,X3) | (43) |
U42(X1,mark(X2),X3) | → | U42(X1,X2,X3) | (44) |
U42(X1,X2,mark(X3)) | → | U42(X1,X2,X3) | (45) |
U42(active(X1),X2,X3) | → | U42(X1,X2,X3) | (46) |
U42(X1,active(X2),X3) | → | U42(X1,X2,X3) | (47) |
U42(X1,X2,active(X3)) | → | U42(X1,X2,X3) | (48) |
s(mark(X)) | → | s(X) | (49) |
s(active(X)) | → | s(X) | (50) |
plus(mark(X1),X2) | → | plus(X1,X2) | (51) |
plus(X1,mark(X2)) | → | plus(X1,X2) | (52) |
plus(active(X1),X2) | → | plus(X1,X2) | (53) |
plus(X1,active(X2)) | → | plus(X1,X2) | (54) |
mark#(U11(X1,X2)) | → | active#(U11(mark(X1),X2)) | (82) |
mark#(U11(X1,X2)) | → | mark#(X1) | (80) |
The dependency pairs are split into 1 component.
mark#(s(X)) | → | mark#(X) | (100) |
mark#(plus(X1,X2)) | → | active#(plus(mark(X1),mark(X2))) | (106) |
active#(plus(N,s(M))) | → | mark#(U41(isNat(M),M,N)) | (79) |
mark#(U41(X1,X2,X3)) | → | active#(U41(mark(X1),X2,X3)) | (96) |
active#(U41(tt,M,N)) | → | mark#(U42(isNat(N),M,N)) | (63) |
mark#(U42(X1,X2,X3)) | → | active#(U42(mark(X1),X2,X3)) | (99) |
active#(U42(tt,M,N)) | → | mark#(s(plus(N,M))) | (66) |
mark#(U42(X1,X2,X3)) | → | mark#(X1) | (97) |
mark#(U41(X1,X2,X3)) | → | mark#(X1) | (94) |
mark#(U21(X)) | → | mark#(X) | (88) |
mark#(isNat(X)) | → | active#(isNat(X)) | (87) |
active#(isNat(s(V1))) | → | mark#(U21(isNat(V1))) | (73) |
mark#(U12(X)) | → | mark#(X) | (84) |
[U12(x1)] | = | 0 · x1 + 0 |
[U41(x1, x2, x3)] | = | 0 · x1 + 0 · x2 + 1 · x3 + 0 |
[U11(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[isNat(x1)] | = | 0 · x1 + 0 |
[mark#(x1)] | = | 0 · x1 + 0 |
[mark(x1)] | = | 0 · x1 + -∞ |
[plus(x1, x2)] | = | 1 · x1 + 0 · x2 + -∞ |
[tt] | = | 4 |
[U31(x1, x2)] | = | 0 · x1 + 0 · x2 + -∞ |
[0] | = | 4 |
[s(x1)] | = | 0 · x1 + 0 |
[U42(x1, x2, x3)] | = | 1 · x1 + 0 · x2 + 1 · x3 + 4 |
[active#(x1)] | = | 0 · x1 + 0 |
[U21(x1)] | = | 0 · x1 + 0 |
[active(x1)] | = | 0 · x1 + -∞ |
active(U11(tt,V2)) | → | mark(U12(isNat(V2))) | (1) |
active(U12(tt)) | → | mark(tt) | (2) |
active(U21(tt)) | → | mark(tt) | (3) |
active(U31(tt,N)) | → | mark(N) | (4) |
active(U41(tt,M,N)) | → | mark(U42(isNat(N),M,N)) | (5) |
active(U42(tt,M,N)) | → | mark(s(plus(N,M))) | (6) |
active(isNat(0)) | → | mark(tt) | (7) |
active(isNat(plus(V1,V2))) | → | mark(U11(isNat(V1),V2)) | (8) |
active(isNat(s(V1))) | → | mark(U21(isNat(V1))) | (9) |
active(plus(N,0)) | → | mark(U31(isNat(N),N)) | (10) |
active(plus(N,s(M))) | → | mark(U41(isNat(M),M,N)) | (11) |
mark(U11(X1,X2)) | → | active(U11(mark(X1),X2)) | (12) |
mark(tt) | → | active(tt) | (13) |
mark(U12(X)) | → | active(U12(mark(X))) | (14) |
mark(isNat(X)) | → | active(isNat(X)) | (15) |
mark(U21(X)) | → | active(U21(mark(X))) | (16) |
mark(U31(X1,X2)) | → | active(U31(mark(X1),X2)) | (17) |
mark(U41(X1,X2,X3)) | → | active(U41(mark(X1),X2,X3)) | (18) |
mark(U42(X1,X2,X3)) | → | active(U42(mark(X1),X2,X3)) | (19) |
mark(s(X)) | → | active(s(mark(X))) | (20) |
mark(plus(X1,X2)) | → | active(plus(mark(X1),mark(X2))) | (21) |
mark(0) | → | active(0) | (22) |
U11(mark(X1),X2) | → | U11(X1,X2) | (23) |
U11(X1,mark(X2)) | → | U11(X1,X2) | (24) |
U11(active(X1),X2) | → | U11(X1,X2) | (25) |
U11(X1,active(X2)) | → | U11(X1,X2) | (26) |
U12(mark(X)) | → | U12(X) | (27) |
U12(active(X)) | → | U12(X) | (28) |
isNat(mark(X)) | → | isNat(X) | (29) |
isNat(active(X)) | → | isNat(X) | (30) |
U21(mark(X)) | → | U21(X) | (31) |
U21(active(X)) | → | U21(X) | (32) |
U31(mark(X1),X2) | → | U31(X1,X2) | (33) |
U31(X1,mark(X2)) | → | U31(X1,X2) | (34) |
U31(active(X1),X2) | → | U31(X1,X2) | (35) |
U31(X1,active(X2)) | → | U31(X1,X2) | (36) |
U41(mark(X1),X2,X3) | → | U41(X1,X2,X3) | (37) |
U41(X1,mark(X2),X3) | → | U41(X1,X2,X3) | (38) |
U41(X1,X2,mark(X3)) | → | U41(X1,X2,X3) | (39) |
U41(active(X1),X2,X3) | → | U41(X1,X2,X3) | (40) |
U41(X1,active(X2),X3) | → | U41(X1,X2,X3) | (41) |
U41(X1,X2,active(X3)) | → | U41(X1,X2,X3) | (42) |
U42(mark(X1),X2,X3) | → | U42(X1,X2,X3) | (43) |
U42(X1,mark(X2),X3) | → | U42(X1,X2,X3) | (44) |
U42(X1,X2,mark(X3)) | → | U42(X1,X2,X3) | (45) |
U42(active(X1),X2,X3) | → | U42(X1,X2,X3) | (46) |
U42(X1,active(X2),X3) | → | U42(X1,X2,X3) | (47) |
U42(X1,X2,active(X3)) | → | U42(X1,X2,X3) | (48) |
s(mark(X)) | → | s(X) | (49) |
s(active(X)) | → | s(X) | (50) |
plus(mark(X1),X2) | → | plus(X1,X2) | (51) |
plus(X1,mark(X2)) | → | plus(X1,X2) | (52) |
plus(active(X1),X2) | → | plus(X1,X2) | (53) |
plus(X1,active(X2)) | → | plus(X1,X2) | (54) |
mark#(U42(X1,X2,X3)) | → | mark#(X1) | (97) |
[U12(x1)] | = | 5 · x1 + 5 |
[U41(x1, x2, x3)] | = | 0 · x1 + -∞ · x2 + -∞ · x3 + 0 |
[U11(x1, x2)] | = | -∞ · x1 + 0 · x2 + -∞ |
[isNat(x1)] | = | -∞ · x1 + 0 |
[mark#(x1)] | = | 1 · x1 + 4 |
[mark(x1)] | = | 0 · x1 + 0 |
[plus(x1, x2)] | = | -∞ · x1 + -∞ · x2 + 3 |
[tt] | = | 0 |
[U31(x1, x2)] | = | 1 · x1 + 4 · x2 + 0 |
[0] | = | 1 |
[s(x1)] | = | 0 · x1 + -∞ |
[U42(x1, x2, x3)] | = | 3 · x1 + -∞ · x2 + -∞ · x3 + 3 |
[active#(x1)] | = | -∞ · x1 + 4 |
[U21(x1)] | = | 0 · x1 + 3 |
[active(x1)] | = | 0 · x1 + 0 |
isNat(mark(X)) | → | isNat(X) | (29) |
isNat(active(X)) | → | isNat(X) | (30) |
U21(mark(X)) | → | U21(X) | (31) |
U21(active(X)) | → | U21(X) | (32) |
U41(mark(X1),X2,X3) | → | U41(X1,X2,X3) | (37) |
U41(X1,mark(X2),X3) | → | U41(X1,X2,X3) | (38) |
U41(X1,X2,mark(X3)) | → | U41(X1,X2,X3) | (39) |
U41(active(X1),X2,X3) | → | U41(X1,X2,X3) | (40) |
U41(X1,active(X2),X3) | → | U41(X1,X2,X3) | (41) |
U41(X1,X2,active(X3)) | → | U41(X1,X2,X3) | (42) |
U42(mark(X1),X2,X3) | → | U42(X1,X2,X3) | (43) |
U42(X1,mark(X2),X3) | → | U42(X1,X2,X3) | (44) |
U42(X1,X2,mark(X3)) | → | U42(X1,X2,X3) | (45) |
U42(active(X1),X2,X3) | → | U42(X1,X2,X3) | (46) |
U42(X1,active(X2),X3) | → | U42(X1,X2,X3) | (47) |
U42(X1,X2,active(X3)) | → | U42(X1,X2,X3) | (48) |
s(mark(X)) | → | s(X) | (49) |
s(active(X)) | → | s(X) | (50) |
plus(mark(X1),X2) | → | plus(X1,X2) | (51) |
plus(X1,mark(X2)) | → | plus(X1,X2) | (52) |
plus(active(X1),X2) | → | plus(X1,X2) | (53) |
plus(X1,active(X2)) | → | plus(X1,X2) | (54) |
mark#(U12(X)) | → | mark#(X) | (84) |
[U12(x1)] | = | 0 · x1 + 1 |
[U41(x1, x2, x3)] | = | 2 · x1 + 4 · x2 + 1 · x3 + 0 |
[U11(x1, x2)] | = | 0 · x1 + 0 · x2 + 1 |
[isNat(x1)] | = | 0 · x1 + 4 |
[mark#(x1)] | = | 2 · x1 + 0 |
[mark(x1)] | = | 1 · x1 + 0 |
[plus(x1, x2)] | = | 1 · x1 + 4 · x2 + 0 |
[tt] | = | 1 |
[U31(x1, x2)] | = | 0 · x1 + 1 · x2 + 0 |
[0] | = | 0 |
[s(x1)] | = | 1 · x1 + 2 |
[U42(x1, x2, x3)] | = | 0 · x1 + 4 · x2 + 1 · x3 + 2 |
[active#(x1)] | = | 2 · x1 + 0 |
[U21(x1)] | = | 1 · x1 + 0 |
[active(x1)] | = | 1 · x1 + 0 |
active(U11(tt,V2)) | → | mark(U12(isNat(V2))) | (1) |
active(U12(tt)) | → | mark(tt) | (2) |
active(U21(tt)) | → | mark(tt) | (3) |
active(U31(tt,N)) | → | mark(N) | (4) |
active(U41(tt,M,N)) | → | mark(U42(isNat(N),M,N)) | (5) |
active(U42(tt,M,N)) | → | mark(s(plus(N,M))) | (6) |
active(isNat(0)) | → | mark(tt) | (7) |
active(isNat(plus(V1,V2))) | → | mark(U11(isNat(V1),V2)) | (8) |
active(isNat(s(V1))) | → | mark(U21(isNat(V1))) | (9) |
active(plus(N,0)) | → | mark(U31(isNat(N),N)) | (10) |
active(plus(N,s(M))) | → | mark(U41(isNat(M),M,N)) | (11) |
mark(U11(X1,X2)) | → | active(U11(mark(X1),X2)) | (12) |
mark(tt) | → | active(tt) | (13) |
mark(U12(X)) | → | active(U12(mark(X))) | (14) |
mark(isNat(X)) | → | active(isNat(X)) | (15) |
mark(U21(X)) | → | active(U21(mark(X))) | (16) |
mark(U31(X1,X2)) | → | active(U31(mark(X1),X2)) | (17) |
mark(U41(X1,X2,X3)) | → | active(U41(mark(X1),X2,X3)) | (18) |
mark(U42(X1,X2,X3)) | → | active(U42(mark(X1),X2,X3)) | (19) |
mark(s(X)) | → | active(s(mark(X))) | (20) |
mark(plus(X1,X2)) | → | active(plus(mark(X1),mark(X2))) | (21) |
mark(0) | → | active(0) | (22) |
U11(mark(X1),X2) | → | U11(X1,X2) | (23) |
U11(X1,mark(X2)) | → | U11(X1,X2) | (24) |
U11(active(X1),X2) | → | U11(X1,X2) | (25) |
U11(X1,active(X2)) | → | U11(X1,X2) | (26) |
U12(mark(X)) | → | U12(X) | (27) |
U12(active(X)) | → | U12(X) | (28) |
isNat(mark(X)) | → | isNat(X) | (29) |
isNat(active(X)) | → | isNat(X) | (30) |
U21(mark(X)) | → | U21(X) | (31) |
U21(active(X)) | → | U21(X) | (32) |
U31(mark(X1),X2) | → | U31(X1,X2) | (33) |
U31(X1,mark(X2)) | → | U31(X1,X2) | (34) |
U31(active(X1),X2) | → | U31(X1,X2) | (35) |
U31(X1,active(X2)) | → | U31(X1,X2) | (36) |
U41(mark(X1),X2,X3) | → | U41(X1,X2,X3) | (37) |
U41(X1,mark(X2),X3) | → | U41(X1,X2,X3) | (38) |
U41(X1,X2,mark(X3)) | → | U41(X1,X2,X3) | (39) |
U41(active(X1),X2,X3) | → | U41(X1,X2,X3) | (40) |
U41(X1,active(X2),X3) | → | U41(X1,X2,X3) | (41) |
U41(X1,X2,active(X3)) | → | U41(X1,X2,X3) | (42) |
U42(mark(X1),X2,X3) | → | U42(X1,X2,X3) | (43) |
U42(X1,mark(X2),X3) | → | U42(X1,X2,X3) | (44) |
U42(X1,X2,mark(X3)) | → | U42(X1,X2,X3) | (45) |
U42(active(X1),X2,X3) | → | U42(X1,X2,X3) | (46) |
U42(X1,active(X2),X3) | → | U42(X1,X2,X3) | (47) |
U42(X1,X2,active(X3)) | → | U42(X1,X2,X3) | (48) |
s(mark(X)) | → | s(X) | (49) |
s(active(X)) | → | s(X) | (50) |
plus(mark(X1),X2) | → | plus(X1,X2) | (51) |
plus(X1,mark(X2)) | → | plus(X1,X2) | (52) |
plus(active(X1),X2) | → | plus(X1,X2) | (53) |
plus(X1,active(X2)) | → | plus(X1,X2) | (54) |
mark#(s(X)) | → | mark#(X) | (100) |
The dependency pairs are split into 1 component.
mark#(plus(X1,X2)) | → | active#(plus(mark(X1),mark(X2))) | (106) |
active#(plus(N,s(M))) | → | mark#(U41(isNat(M),M,N)) | (79) |
mark#(U41(X1,X2,X3)) | → | mark#(X1) | (94) |
mark#(U21(X)) | → | mark#(X) | (88) |
mark#(isNat(X)) | → | active#(isNat(X)) | (87) |
active#(isNat(s(V1))) | → | mark#(U21(isNat(V1))) | (73) |
prec(mark#) | = | 0 | stat(mark#) | = | lex | |
prec(active#) | = | 0 | stat(active#) | = | lex | |
prec(0) | = | 0 | stat(0) | = | lex | |
prec(s) | = | 0 | stat(s) | = | lex | |
prec(plus) | = | 8 | stat(plus) | = | lex | |
prec(U42) | = | 0 | stat(U42) | = | lex | |
prec(U41) | = | 0 | stat(U41) | = | lex | |
prec(U31) | = | 0 | stat(U31) | = | lex | |
prec(U21) | = | 0 | stat(U21) | = | lex | |
prec(mark) | = | 4 | stat(mark) | = | lex | |
prec(U12) | = | 0 | stat(U12) | = | lex | |
prec(isNat) | = | 0 | stat(isNat) | = | lex | |
prec(active) | = | 0 | stat(active) | = | lex | |
prec(U11) | = | 0 | stat(U11) | = | lex | |
prec(tt) | = | 0 | stat(tt) | = | lex |
π(mark#) | = | 1 |
π(active#) | = | 1 |
π(0) | = | [] |
π(s) | = | 1 |
π(plus) | = | [] |
π(U42) | = | 1 |
π(U41) | = | [1] |
π(U31) | = | [] |
π(U21) | = | 1 |
π(mark) | = | [1] |
π(U12) | = | [] |
π(isNat) | = | [] |
π(active) | = | 1 |
π(U11) | = | 2 |
π(tt) | = | [] |
isNat(mark(X)) | → | isNat(X) | (29) |
isNat(active(X)) | → | isNat(X) | (30) |
U21(mark(X)) | → | U21(X) | (31) |
U21(active(X)) | → | U21(X) | (32) |
U41(mark(X1),X2,X3) | → | U41(X1,X2,X3) | (37) |
U41(X1,mark(X2),X3) | → | U41(X1,X2,X3) | (38) |
U41(X1,X2,mark(X3)) | → | U41(X1,X2,X3) | (39) |
U41(active(X1),X2,X3) | → | U41(X1,X2,X3) | (40) |
U41(X1,active(X2),X3) | → | U41(X1,X2,X3) | (41) |
U41(X1,X2,active(X3)) | → | U41(X1,X2,X3) | (42) |
plus(mark(X1),X2) | → | plus(X1,X2) | (51) |
plus(X1,mark(X2)) | → | plus(X1,X2) | (52) |
plus(active(X1),X2) | → | plus(X1,X2) | (53) |
plus(X1,active(X2)) | → | plus(X1,X2) | (54) |
active#(plus(N,s(M))) | → | mark#(U41(isNat(M),M,N)) | (79) |
mark#(U41(X1,X2,X3)) | → | mark#(X1) | (94) |
The dependency pairs are split into 1 component.
mark#(U21(X)) | → | mark#(X) | (88) |
mark#(isNat(X)) | → | active#(isNat(X)) | (87) |
active#(isNat(s(V1))) | → | mark#(U21(isNat(V1))) | (73) |
π(mark#) | = | { 1 } |
π(active#) | = | { 1 } |
π(U21) | = | { 1 } |
π(isNat) | = | { 1 } |
active#(isNat(s(V1))) | → | mark#(U21(isNat(V1))) | (73) |
The dependency pairs are split into 1 component.
mark#(U21(X)) | → | mark#(X) | (88) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
mark#(U21(X)) | → | mark#(X) | (88) |
1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
U12#(active(X)) | → | U12#(X) | (113) |
U12#(mark(X)) | → | U12#(X) | (112) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
U12#(active(X)) | → | U12#(X) | (113) |
1 | > | 1 | |
U12#(mark(X)) | → | U12#(X) | (112) |
1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
isNat#(active(X)) | → | isNat#(X) | (115) |
isNat#(mark(X)) | → | isNat#(X) | (114) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
isNat#(active(X)) | → | isNat#(X) | (115) |
1 | > | 1 | |
isNat#(mark(X)) | → | isNat#(X) | (114) |
1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
U42#(mark(X1),X2,X3) | → | U42#(X1,X2,X3) | (128) |
U42#(X1,X2,active(X3)) | → | U42#(X1,X2,X3) | (133) |
U42#(X1,active(X2),X3) | → | U42#(X1,X2,X3) | (132) |
U42#(active(X1),X2,X3) | → | U42#(X1,X2,X3) | (131) |
U42#(X1,X2,mark(X3)) | → | U42#(X1,X2,X3) | (130) |
U42#(X1,mark(X2),X3) | → | U42#(X1,X2,X3) | (129) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
U42#(mark(X1),X2,X3) | → | U42#(X1,X2,X3) | (128) |
3 | ≥ | 3 | |
2 | ≥ | 2 | |
1 | > | 1 | |
U42#(X1,X2,active(X3)) | → | U42#(X1,X2,X3) | (133) |
3 | > | 3 | |
2 | ≥ | 2 | |
1 | ≥ | 1 | |
U42#(X1,active(X2),X3) | → | U42#(X1,X2,X3) | (132) |
3 | ≥ | 3 | |
2 | > | 2 | |
1 | ≥ | 1 | |
U42#(active(X1),X2,X3) | → | U42#(X1,X2,X3) | (131) |
3 | ≥ | 3 | |
2 | ≥ | 2 | |
1 | > | 1 | |
U42#(X1,X2,mark(X3)) | → | U42#(X1,X2,X3) | (130) |
3 | > | 3 | |
2 | ≥ | 2 | |
1 | ≥ | 1 | |
U42#(X1,mark(X2),X3) | → | U42#(X1,X2,X3) | (129) |
3 | ≥ | 3 | |
2 | > | 2 | |
1 | ≥ | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
plus#(mark(X1),X2) | → | plus#(X1,X2) | (136) |
plus#(X1,active(X2)) | → | plus#(X1,X2) | (139) |
plus#(active(X1),X2) | → | plus#(X1,X2) | (138) |
plus#(X1,mark(X2)) | → | plus#(X1,X2) | (137) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
plus#(mark(X1),X2) | → | plus#(X1,X2) | (136) |
2 | ≥ | 2 | |
1 | > | 1 | |
plus#(X1,active(X2)) | → | plus#(X1,X2) | (139) |
2 | > | 2 | |
1 | ≥ | 1 | |
plus#(active(X1),X2) | → | plus#(X1,X2) | (138) |
2 | ≥ | 2 | |
1 | > | 1 | |
plus#(X1,mark(X2)) | → | plus#(X1,X2) | (137) |
2 | > | 2 | |
1 | ≥ | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
s#(mark(X)) | → | s#(X) | (134) |
s#(active(X)) | → | s#(X) | (135) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
s#(mark(X)) | → | s#(X) | (134) |
1 | > | 1 | |
s#(active(X)) | → | s#(X) | (135) |
1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
U11#(mark(X1),X2) | → | U11#(X1,X2) | (108) |
U11#(X1,active(X2)) | → | U11#(X1,X2) | (111) |
U11#(active(X1),X2) | → | U11#(X1,X2) | (110) |
U11#(X1,mark(X2)) | → | U11#(X1,X2) | (109) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
U11#(mark(X1),X2) | → | U11#(X1,X2) | (108) |
2 | ≥ | 2 | |
1 | > | 1 | |
U11#(X1,active(X2)) | → | U11#(X1,X2) | (111) |
2 | > | 2 | |
1 | ≥ | 1 | |
U11#(active(X1),X2) | → | U11#(X1,X2) | (110) |
2 | ≥ | 2 | |
1 | > | 1 | |
U11#(X1,mark(X2)) | → | U11#(X1,X2) | (109) |
2 | > | 2 | |
1 | ≥ | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
U21#(mark(X)) | → | U21#(X) | (116) |
U21#(active(X)) | → | U21#(X) | (117) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
U21#(mark(X)) | → | U21#(X) | (116) |
1 | > | 1 | |
U21#(active(X)) | → | U21#(X) | (117) |
1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
U31#(mark(X1),X2) | → | U31#(X1,X2) | (118) |
U31#(X1,active(X2)) | → | U31#(X1,X2) | (121) |
U31#(active(X1),X2) | → | U31#(X1,X2) | (120) |
U31#(X1,mark(X2)) | → | U31#(X1,X2) | (119) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
U31#(mark(X1),X2) | → | U31#(X1,X2) | (118) |
2 | ≥ | 2 | |
1 | > | 1 | |
U31#(X1,active(X2)) | → | U31#(X1,X2) | (121) |
2 | > | 2 | |
1 | ≥ | 1 | |
U31#(active(X1),X2) | → | U31#(X1,X2) | (120) |
2 | ≥ | 2 | |
1 | > | 1 | |
U31#(X1,mark(X2)) | → | U31#(X1,X2) | (119) |
2 | > | 2 | |
1 | ≥ | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
U41#(mark(X1),X2,X3) | → | U41#(X1,X2,X3) | (122) |
U41#(X1,X2,active(X3)) | → | U41#(X1,X2,X3) | (127) |
U41#(X1,active(X2),X3) | → | U41#(X1,X2,X3) | (126) |
U41#(active(X1),X2,X3) | → | U41#(X1,X2,X3) | (125) |
U41#(X1,X2,mark(X3)) | → | U41#(X1,X2,X3) | (124) |
U41#(X1,mark(X2),X3) | → | U41#(X1,X2,X3) | (123) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
U41#(mark(X1),X2,X3) | → | U41#(X1,X2,X3) | (122) |
3 | ≥ | 3 | |
2 | ≥ | 2 | |
1 | > | 1 | |
U41#(X1,X2,active(X3)) | → | U41#(X1,X2,X3) | (127) |
3 | > | 3 | |
2 | ≥ | 2 | |
1 | ≥ | 1 | |
U41#(X1,active(X2),X3) | → | U41#(X1,X2,X3) | (126) |
3 | ≥ | 3 | |
2 | > | 2 | |
1 | ≥ | 1 | |
U41#(active(X1),X2,X3) | → | U41#(X1,X2,X3) | (125) |
3 | ≥ | 3 | |
2 | ≥ | 2 | |
1 | > | 1 | |
U41#(X1,X2,mark(X3)) | → | U41#(X1,X2,X3) | (124) |
3 | > | 3 | |
2 | ≥ | 2 | |
1 | ≥ | 1 | |
U41#(X1,mark(X2),X3) | → | U41#(X1,X2,X3) | (123) |
3 | ≥ | 3 | |
2 | > | 2 | |
1 | ≥ | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.