The rewrite relation of the following TRS is considered.
a__zeros | → | cons(0,zeros) | (1) |
a__U11(tt) | → | tt | (2) |
a__U21(tt) | → | tt | (3) |
a__U31(tt) | → | tt | (4) |
a__U41(tt,V2) | → | a__U42(a__isNatIList(V2)) | (5) |
a__U42(tt) | → | tt | (6) |
a__U51(tt,V2) | → | a__U52(a__isNatList(V2)) | (7) |
a__U52(tt) | → | tt | (8) |
a__U61(tt,V2) | → | a__U62(a__isNatIList(V2)) | (9) |
a__U62(tt) | → | tt | (10) |
a__U71(tt,L,N) | → | a__U72(a__isNat(N),L) | (11) |
a__U72(tt,L) | → | s(a__length(mark(L))) | (12) |
a__U81(tt) | → | nil | (13) |
a__U91(tt,IL,M,N) | → | a__U92(a__isNat(M),IL,M,N) | (14) |
a__U92(tt,IL,M,N) | → | a__U93(a__isNat(N),IL,M,N) | (15) |
a__U93(tt,IL,M,N) | → | cons(mark(N),take(M,IL)) | (16) |
a__isNat(0) | → | tt | (17) |
a__isNat(length(V1)) | → | a__U11(a__isNatList(V1)) | (18) |
a__isNat(s(V1)) | → | a__U21(a__isNat(V1)) | (19) |
a__isNatIList(V) | → | a__U31(a__isNatList(V)) | (20) |
a__isNatIList(zeros) | → | tt | (21) |
a__isNatIList(cons(V1,V2)) | → | a__U41(a__isNat(V1),V2) | (22) |
a__isNatList(nil) | → | tt | (23) |
a__isNatList(cons(V1,V2)) | → | a__U51(a__isNat(V1),V2) | (24) |
a__isNatList(take(V1,V2)) | → | a__U61(a__isNat(V1),V2) | (25) |
a__length(nil) | → | 0 | (26) |
a__length(cons(N,L)) | → | a__U71(a__isNatList(L),L,N) | (27) |
a__take(0,IL) | → | a__U81(a__isNatIList(IL)) | (28) |
a__take(s(M),cons(N,IL)) | → | a__U91(a__isNatIList(IL),IL,M,N) | (29) |
mark(zeros) | → | a__zeros | (30) |
mark(U11(X)) | → | a__U11(mark(X)) | (31) |
mark(U21(X)) | → | a__U21(mark(X)) | (32) |
mark(U31(X)) | → | a__U31(mark(X)) | (33) |
mark(U41(X1,X2)) | → | a__U41(mark(X1),X2) | (34) |
mark(U42(X)) | → | a__U42(mark(X)) | (35) |
mark(isNatIList(X)) | → | a__isNatIList(X) | (36) |
mark(U51(X1,X2)) | → | a__U51(mark(X1),X2) | (37) |
mark(U52(X)) | → | a__U52(mark(X)) | (38) |
mark(isNatList(X)) | → | a__isNatList(X) | (39) |
mark(U61(X1,X2)) | → | a__U61(mark(X1),X2) | (40) |
mark(U62(X)) | → | a__U62(mark(X)) | (41) |
mark(U71(X1,X2,X3)) | → | a__U71(mark(X1),X2,X3) | (42) |
mark(U72(X1,X2)) | → | a__U72(mark(X1),X2) | (43) |
mark(isNat(X)) | → | a__isNat(X) | (44) |
mark(length(X)) | → | a__length(mark(X)) | (45) |
mark(U81(X)) | → | a__U81(mark(X)) | (46) |
mark(U91(X1,X2,X3,X4)) | → | a__U91(mark(X1),X2,X3,X4) | (47) |
mark(U92(X1,X2,X3,X4)) | → | a__U92(mark(X1),X2,X3,X4) | (48) |
mark(U93(X1,X2,X3,X4)) | → | a__U93(mark(X1),X2,X3,X4) | (49) |
mark(take(X1,X2)) | → | a__take(mark(X1),mark(X2)) | (50) |
mark(cons(X1,X2)) | → | cons(mark(X1),X2) | (51) |
mark(0) | → | 0 | (52) |
mark(tt) | → | tt | (53) |
mark(s(X)) | → | s(mark(X)) | (54) |
mark(nil) | → | nil | (55) |
a__zeros | → | zeros | (56) |
a__U11(X) | → | U11(X) | (57) |
a__U21(X) | → | U21(X) | (58) |
a__U31(X) | → | U31(X) | (59) |
a__U41(X1,X2) | → | U41(X1,X2) | (60) |
a__U42(X) | → | U42(X) | (61) |
a__isNatIList(X) | → | isNatIList(X) | (62) |
a__U51(X1,X2) | → | U51(X1,X2) | (63) |
a__U52(X) | → | U52(X) | (64) |
a__isNatList(X) | → | isNatList(X) | (65) |
a__U61(X1,X2) | → | U61(X1,X2) | (66) |
a__U62(X) | → | U62(X) | (67) |
a__U71(X1,X2,X3) | → | U71(X1,X2,X3) | (68) |
a__U72(X1,X2) | → | U72(X1,X2) | (69) |
a__isNat(X) | → | isNat(X) | (70) |
a__length(X) | → | length(X) | (71) |
a__U81(X) | → | U81(X) | (72) |
a__U91(X1,X2,X3,X4) | → | U91(X1,X2,X3,X4) | (73) |
a__U92(X1,X2,X3,X4) | → | U92(X1,X2,X3,X4) | (74) |
a__U93(X1,X2,X3,X4) | → | U93(X1,X2,X3,X4) | (75) |
a__take(X1,X2) | → | take(X1,X2) | (76) |
a__U41#(tt,V2) | → | a__U42#(a__isNatIList(V2)) | (77) |
a__U41#(tt,V2) | → | a__isNatIList#(V2) | (78) |
a__U51#(tt,V2) | → | a__U52#(a__isNatList(V2)) | (79) |
a__U51#(tt,V2) | → | a__isNatList#(V2) | (80) |
a__U61#(tt,V2) | → | a__U62#(a__isNatIList(V2)) | (81) |
a__U61#(tt,V2) | → | a__isNatIList#(V2) | (82) |
a__U71#(tt,L,N) | → | a__U72#(a__isNat(N),L) | (83) |
a__U71#(tt,L,N) | → | a__isNat#(N) | (84) |
a__U72#(tt,L) | → | a__length#(mark(L)) | (85) |
a__U72#(tt,L) | → | mark#(L) | (86) |
a__U91#(tt,IL,M,N) | → | a__U92#(a__isNat(M),IL,M,N) | (87) |
a__U91#(tt,IL,M,N) | → | a__isNat#(M) | (88) |
a__U92#(tt,IL,M,N) | → | a__U93#(a__isNat(N),IL,M,N) | (89) |
a__U92#(tt,IL,M,N) | → | a__isNat#(N) | (90) |
a__U93#(tt,IL,M,N) | → | mark#(N) | (91) |
a__isNat#(length(V1)) | → | a__U11#(a__isNatList(V1)) | (92) |
a__isNat#(length(V1)) | → | a__isNatList#(V1) | (93) |
a__isNat#(s(V1)) | → | a__U21#(a__isNat(V1)) | (94) |
a__isNat#(s(V1)) | → | a__isNat#(V1) | (95) |
a__isNatIList#(V) | → | a__U31#(a__isNatList(V)) | (96) |
a__isNatIList#(V) | → | a__isNatList#(V) | (97) |
a__isNatIList#(cons(V1,V2)) | → | a__U41#(a__isNat(V1),V2) | (98) |
a__isNatIList#(cons(V1,V2)) | → | a__isNat#(V1) | (99) |
a__isNatList#(cons(V1,V2)) | → | a__U51#(a__isNat(V1),V2) | (100) |
a__isNatList#(cons(V1,V2)) | → | a__isNat#(V1) | (101) |
a__isNatList#(take(V1,V2)) | → | a__U61#(a__isNat(V1),V2) | (102) |
a__isNatList#(take(V1,V2)) | → | a__isNat#(V1) | (103) |
a__length#(cons(N,L)) | → | a__U71#(a__isNatList(L),L,N) | (104) |
a__length#(cons(N,L)) | → | a__isNatList#(L) | (105) |
a__take#(0,IL) | → | a__U81#(a__isNatIList(IL)) | (106) |
a__take#(0,IL) | → | a__isNatIList#(IL) | (107) |
a__take#(s(M),cons(N,IL)) | → | a__U91#(a__isNatIList(IL),IL,M,N) | (108) |
a__take#(s(M),cons(N,IL)) | → | a__isNatIList#(IL) | (109) |
mark#(zeros) | → | a__zeros# | (110) |
mark#(U11(X)) | → | a__U11#(mark(X)) | (111) |
mark#(U11(X)) | → | mark#(X) | (112) |
mark#(U21(X)) | → | a__U21#(mark(X)) | (113) |
mark#(U21(X)) | → | mark#(X) | (114) |
mark#(U31(X)) | → | a__U31#(mark(X)) | (115) |
mark#(U31(X)) | → | mark#(X) | (116) |
mark#(U41(X1,X2)) | → | a__U41#(mark(X1),X2) | (117) |
mark#(U41(X1,X2)) | → | mark#(X1) | (118) |
mark#(U42(X)) | → | a__U42#(mark(X)) | (119) |
mark#(U42(X)) | → | mark#(X) | (120) |
mark#(isNatIList(X)) | → | a__isNatIList#(X) | (121) |
mark#(U51(X1,X2)) | → | a__U51#(mark(X1),X2) | (122) |
mark#(U51(X1,X2)) | → | mark#(X1) | (123) |
mark#(U52(X)) | → | a__U52#(mark(X)) | (124) |
mark#(U52(X)) | → | mark#(X) | (125) |
mark#(isNatList(X)) | → | a__isNatList#(X) | (126) |
mark#(U61(X1,X2)) | → | a__U61#(mark(X1),X2) | (127) |
mark#(U61(X1,X2)) | → | mark#(X1) | (128) |
mark#(U62(X)) | → | a__U62#(mark(X)) | (129) |
mark#(U62(X)) | → | mark#(X) | (130) |
mark#(U71(X1,X2,X3)) | → | a__U71#(mark(X1),X2,X3) | (131) |
mark#(U71(X1,X2,X3)) | → | mark#(X1) | (132) |
mark#(U72(X1,X2)) | → | a__U72#(mark(X1),X2) | (133) |
mark#(U72(X1,X2)) | → | mark#(X1) | (134) |
mark#(isNat(X)) | → | a__isNat#(X) | (135) |
mark#(length(X)) | → | a__length#(mark(X)) | (136) |
mark#(length(X)) | → | mark#(X) | (137) |
mark#(U81(X)) | → | a__U81#(mark(X)) | (138) |
mark#(U81(X)) | → | mark#(X) | (139) |
mark#(U91(X1,X2,X3,X4)) | → | a__U91#(mark(X1),X2,X3,X4) | (140) |
mark#(U91(X1,X2,X3,X4)) | → | mark#(X1) | (141) |
mark#(U92(X1,X2,X3,X4)) | → | a__U92#(mark(X1),X2,X3,X4) | (142) |
mark#(U92(X1,X2,X3,X4)) | → | mark#(X1) | (143) |
mark#(U93(X1,X2,X3,X4)) | → | a__U93#(mark(X1),X2,X3,X4) | (144) |
mark#(U93(X1,X2,X3,X4)) | → | mark#(X1) | (145) |
mark#(take(X1,X2)) | → | a__take#(mark(X1),mark(X2)) | (146) |
mark#(take(X1,X2)) | → | mark#(X1) | (147) |
mark#(take(X1,X2)) | → | mark#(X2) | (148) |
mark#(cons(X1,X2)) | → | mark#(X1) | (149) |
mark#(s(X)) | → | mark#(X) | (150) |
The dependency pairs are split into 2 components.
mark#(U11(X)) | → | mark#(X) | (112) |
mark#(U21(X)) | → | mark#(X) | (114) |
mark#(U31(X)) | → | mark#(X) | (116) |
mark#(U41(X1,X2)) | → | mark#(X1) | (118) |
mark#(U42(X)) | → | mark#(X) | (120) |
mark#(U51(X1,X2)) | → | mark#(X1) | (123) |
mark#(U52(X)) | → | mark#(X) | (125) |
mark#(U61(X1,X2)) | → | mark#(X1) | (128) |
mark#(U62(X)) | → | mark#(X) | (130) |
mark#(U71(X1,X2,X3)) | → | a__U71#(mark(X1),X2,X3) | (131) |
a__U71#(tt,L,N) | → | a__U72#(a__isNat(N),L) | (83) |
a__U72#(tt,L) | → | a__length#(mark(L)) | (85) |
a__length#(cons(N,L)) | → | a__U71#(a__isNatList(L),L,N) | (104) |
a__U72#(tt,L) | → | mark#(L) | (86) |
mark#(U71(X1,X2,X3)) | → | mark#(X1) | (132) |
mark#(U72(X1,X2)) | → | a__U72#(mark(X1),X2) | (133) |
mark#(U72(X1,X2)) | → | mark#(X1) | (134) |
mark#(length(X)) | → | a__length#(mark(X)) | (136) |
mark#(length(X)) | → | mark#(X) | (137) |
mark#(U81(X)) | → | mark#(X) | (139) |
mark#(U91(X1,X2,X3,X4)) | → | a__U91#(mark(X1),X2,X3,X4) | (140) |
a__U91#(tt,IL,M,N) | → | a__U92#(a__isNat(M),IL,M,N) | (87) |
a__U92#(tt,IL,M,N) | → | a__U93#(a__isNat(N),IL,M,N) | (89) |
a__U93#(tt,IL,M,N) | → | mark#(N) | (91) |
mark#(U91(X1,X2,X3,X4)) | → | mark#(X1) | (141) |
mark#(U92(X1,X2,X3,X4)) | → | a__U92#(mark(X1),X2,X3,X4) | (142) |
mark#(U92(X1,X2,X3,X4)) | → | mark#(X1) | (143) |
mark#(U93(X1,X2,X3,X4)) | → | a__U93#(mark(X1),X2,X3,X4) | (144) |
mark#(U93(X1,X2,X3,X4)) | → | mark#(X1) | (145) |
mark#(take(X1,X2)) | → | a__take#(mark(X1),mark(X2)) | (146) |
a__take#(s(M),cons(N,IL)) | → | a__U91#(a__isNatIList(IL),IL,M,N) | (108) |
mark#(take(X1,X2)) | → | mark#(X1) | (147) |
mark#(take(X1,X2)) | → | mark#(X2) | (148) |
mark#(cons(X1,X2)) | → | mark#(X1) | (149) |
mark#(s(X)) | → | mark#(X) | (150) |
[a__length#(x1)] | = | x1 |
[a__take#(x1, x2)] | = | 2 · x2 |
[a__U71#(x1, x2, x3)] | = | x2 + x3 |
[a__U72#(x1, x2)] | = | x2 |
[a__U91#(x1,...,x4)] | = | x2 + 2 · x4 |
[a__U92#(x1,...,x4)] | = | x4 |
[a__U93#(x1,...,x4)] | = | x4 |
[mark(x1)] | = | x1 |
[zeros] | = | 0 |
[a__zeros] | = | 0 |
[U11(x1)] | = | 2 · x1 |
[a__U11(x1)] | = | 2 · x1 |
[U21(x1)] | = | 2 · x1 |
[a__U21(x1)] | = | 2 · x1 |
[U31(x1)] | = | x1 |
[a__U31(x1)] | = | x1 |
[U41(x1, x2)] | = | 2 · x1 |
[a__U41(x1, x2)] | = | 2 · x1 |
[U42(x1)] | = | x1 |
[a__U42(x1)] | = | x1 |
[isNatIList(x1)] | = | 0 |
[a__isNatIList(x1)] | = | 0 |
[U51(x1, x2)] | = | x1 |
[a__U51(x1, x2)] | = | x1 |
[U52(x1)] | = | 2 · x1 |
[a__U52(x1)] | = | 2 · x1 |
[isNatList(x1)] | = | 0 |
[a__isNatList(x1)] | = | 0 |
[U61(x1, x2)] | = | x1 |
[a__U61(x1, x2)] | = | x1 |
[U62(x1)] | = | x1 |
[a__U62(x1)] | = | x1 |
[U71(x1, x2, x3)] | = | 2 + 2 · x1 + 2 · x2 + x3 |
[a__U71(x1, x2, x3)] | = | 2 + 2 · x1 + 2 · x2 + x3 |
[U72(x1, x2)] | = | 2 + 2 · x1 + 2 · x2 |
[a__U72(x1, x2)] | = | 2 + 2 · x1 + 2 · x2 |
[isNat(x1)] | = | 0 |
[a__isNat(x1)] | = | 0 |
[length(x1)] | = | 2 + 2 · x1 |
[a__length(x1)] | = | 2 + 2 · x1 |
[U81(x1)] | = | 2 · x1 |
[a__U81(x1)] | = | 2 · x1 |
[U91(x1,...,x4)] | = | 2 · x1 + 2 · x2 + 2 · x3 + 2 · x4 |
[a__U91(x1,...,x4)] | = | 2 · x1 + 2 · x2 + 2 · x3 + 2 · x4 |
[U92(x1,...,x4)] | = | 2 · x1 + 2 · x2 + 2 · x3 + 2 · x4 |
[a__U92(x1,...,x4)] | = | 2 · x1 + 2 · x2 + 2 · x3 + 2 · x4 |
[U93(x1,...,x4)] | = | x1 + 2 · x2 + 2 · x3 + 2 · x4 |
[a__U93(x1,...,x4)] | = | x1 + 2 · x2 + 2 · x3 + 2 · x4 |
[take(x1, x2)] | = | 2 · x1 + 2 · x2 |
[a__take(x1, x2)] | = | 2 · x1 + 2 · x2 |
[cons(x1, x2)] | = | 2 · x1 + x2 |
[0] | = | 0 |
[tt] | = | 0 |
[s(x1)] | = | x1 |
[nil] | = | 0 |
[mark#(x1)] | = | x1 |
mark#(U71(X1,X2,X3)) | → | a__U71#(mark(X1),X2,X3) | (131) |
mark#(U71(X1,X2,X3)) | → | mark#(X1) | (132) |
mark#(U72(X1,X2)) | → | a__U72#(mark(X1),X2) | (133) |
mark#(U72(X1,X2)) | → | mark#(X1) | (134) |
mark#(length(X)) | → | a__length#(mark(X)) | (136) |
mark#(length(X)) | → | mark#(X) | (137) |
The dependency pairs are split into 2 components.
a__U72#(tt,L) | → | a__length#(mark(L)) | (85) |
a__length#(cons(N,L)) | → | a__U71#(a__isNatList(L),L,N) | (104) |
a__U71#(tt,L,N) | → | a__U72#(a__isNat(N),L) | (83) |
[a__length#(x1)] | = | 2 · x1 |
[mark(x1)] | = | x1 |
[zeros] | = | 0 |
[a__zeros] | = | 0 |
[U11(x1)] | = | x1 |
[a__U11(x1)] | = | x1 |
[U21(x1)] | = | x1 |
[a__U21(x1)] | = | x1 |
[U31(x1)] | = | x1 |
[a__U31(x1)] | = | x1 |
[U41(x1, x2)] | = | 2 + 2 · x2 |
[a__U41(x1, x2)] | = | 2 + 2 · x2 |
[U42(x1)] | = | x1 |
[a__U42(x1)] | = | x1 |
[isNatIList(x1)] | = | 2 + 2 · x1 |
[a__isNatIList(x1)] | = | 2 + 2 · x1 |
[U51(x1, x2)] | = | 1 + 2 · x2 |
[a__U51(x1, x2)] | = | 1 + 2 · x2 |
[U52(x1)] | = | x1 |
[a__U52(x1)] | = | x1 |
[isNatList(x1)] | = | 1 + x1 |
[a__isNatList(x1)] | = | 1 + x1 |
[U61(x1, x2)] | = | x1 |
[a__U61(x1, x2)] | = | x1 |
[U62(x1)] | = | 2 |
[a__U62(x1)] | = | 2 |
[U71(x1, x2, x3)] | = | 2 · x2 |
[a__U71(x1, x2, x3)] | = | 2 · x2 |
[U72(x1, x2)] | = | 2 · x2 |
[a__U72(x1, x2)] | = | 2 · x2 |
[isNat(x1)] | = | 1 + x1 |
[a__isNat(x1)] | = | 1 + x1 |
[length(x1)] | = | x1 |
[a__length(x1)] | = | x1 |
[U81(x1)] | = | 1 |
[a__U81(x1)] | = | 1 |
[U91(x1,...,x4)] | = | 2 · x3 |
[a__U91(x1,...,x4)] | = | 2 · x3 |
[U92(x1,...,x4)] | = | 2 · x3 |
[a__U92(x1,...,x4)] | = | 2 · x3 |
[U93(x1,...,x4)] | = | 2 · x3 |
[a__U93(x1,...,x4)] | = | 2 · x3 |
[take(x1, x2)] | = | x1 |
[a__take(x1, x2)] | = | x1 |
[cons(x1, x2)] | = | 2 · x2 |
[0] | = | 1 |
[tt] | = | 2 |
[s(x1)] | = | 2 · x1 |
[nil] | = | 1 |
[a__U71#(x1, x2, x3)] | = | -2 + 2 · x1 + 2 · x2 |
[a__U72#(x1, x2)] | = | 2 · x2 |
a__U71#(tt,L,N) | → | a__U72#(a__isNat(N),L) | (83) |
The dependency pairs are split into 0 components.
mark#(U21(X)) | → | mark#(X) | (114) |
mark#(U11(X)) | → | mark#(X) | (112) |
mark#(U31(X)) | → | mark#(X) | (116) |
mark#(U41(X1,X2)) | → | mark#(X1) | (118) |
mark#(U42(X)) | → | mark#(X) | (120) |
mark#(U51(X1,X2)) | → | mark#(X1) | (123) |
mark#(U52(X)) | → | mark#(X) | (125) |
mark#(U61(X1,X2)) | → | mark#(X1) | (128) |
mark#(U62(X)) | → | mark#(X) | (130) |
mark#(U81(X)) | → | mark#(X) | (139) |
mark#(U91(X1,X2,X3,X4)) | → | a__U91#(mark(X1),X2,X3,X4) | (140) |
a__U91#(tt,IL,M,N) | → | a__U92#(a__isNat(M),IL,M,N) | (87) |
a__U92#(tt,IL,M,N) | → | a__U93#(a__isNat(N),IL,M,N) | (89) |
a__U93#(tt,IL,M,N) | → | mark#(N) | (91) |
mark#(U91(X1,X2,X3,X4)) | → | mark#(X1) | (141) |
mark#(U92(X1,X2,X3,X4)) | → | a__U92#(mark(X1),X2,X3,X4) | (142) |
mark#(U92(X1,X2,X3,X4)) | → | mark#(X1) | (143) |
mark#(U93(X1,X2,X3,X4)) | → | a__U93#(mark(X1),X2,X3,X4) | (144) |
mark#(U93(X1,X2,X3,X4)) | → | mark#(X1) | (145) |
mark#(take(X1,X2)) | → | a__take#(mark(X1),mark(X2)) | (146) |
a__take#(s(M),cons(N,IL)) | → | a__U91#(a__isNatIList(IL),IL,M,N) | (108) |
mark#(take(X1,X2)) | → | mark#(X1) | (147) |
mark#(take(X1,X2)) | → | mark#(X2) | (148) |
mark#(cons(X1,X2)) | → | mark#(X1) | (149) |
mark#(s(X)) | → | mark#(X) | (150) |
[a__take#(x1, x2)] | = | -2 + 2 · x2 |
[a__U91#(x1,...,x4)] | = | 2 + 2 · x4 |
[a__U92#(x1,...,x4)] | = | 2 + 2 · x4 |
[a__U93#(x1,...,x4)] | = | 2 + 2 · x4 |
[mark(x1)] | = | 2 · x1 |
[zeros] | = | 2 |
[a__zeros] | = | 2 |
[U11(x1)] | = | x1 |
[a__U11(x1)] | = | x1 |
[U21(x1)] | = | 2 · x1 |
[a__U21(x1)] | = | 2 · x1 |
[U31(x1)] | = | 2 · x1 |
[a__U31(x1)] | = | 2 · x1 |
[U41(x1, x2)] | = | 2 · x1 |
[a__U41(x1, x2)] | = | 2 · x1 |
[U42(x1)] | = | 2 · x1 |
[a__U42(x1)] | = | 2 · x1 |
[isNatIList(x1)] | = | 0 |
[a__isNatIList(x1)] | = | 0 |
[U51(x1, x2)] | = | x1 |
[a__U51(x1, x2)] | = | x1 |
[U52(x1)] | = | 2 · x1 |
[a__U52(x1)] | = | 2 · x1 |
[isNatList(x1)] | = | 0 |
[a__isNatList(x1)] | = | 0 |
[U61(x1, x2)] | = | x1 |
[a__U61(x1, x2)] | = | x1 |
[U62(x1)] | = | x1 |
[a__U62(x1)] | = | x1 |
[U71(x1, x2, x3)] | = | 0 |
[a__U71(x1, x2, x3)] | = | -2 |
[U72(x1, x2)] | = | 0 |
[a__U72(x1, x2)] | = | -2 |
[isNat(x1)] | = | 0 |
[a__isNat(x1)] | = | 0 |
[length(x1)] | = | 0 |
[a__length(x1)] | = | -2 |
[U81(x1)] | = | 2 · x1 |
[a__U81(x1)] | = | 2 · x1 |
[U91(x1,...,x4)] | = | 2 + x1 + x3 + 2 · x4 |
[a__U91(x1,...,x4)] | = | 2 + x1 + x3 + 2 · x4 |
[U92(x1,...,x4)] | = | 1 + x1 + x3 + x4 |
[a__U92(x1,...,x4)] | = | 2 + x1 + x3 + 2 · x4 |
[U93(x1,...,x4)] | = | 1 + x1 + x3 + 2 · x4 |
[a__U93(x1,...,x4)] | = | 2 + x1 + x3 + 2 · x4 |
[take(x1, x2)] | = | 2 + x1 + 2 · x2 |
[a__take(x1, x2)] | = | 2 + x1 + 2 · x2 |
[cons(x1, x2)] | = | 2 + x1 |
[0] | = | 0 |
[tt] | = | 0 |
[s(x1)] | = | x1 |
[nil] | = | 0 |
[mark#(x1)] | = | 2 + 2 · x1 |
mark#(U91(X1,X2,X3,X4)) | → | a__U91#(mark(X1),X2,X3,X4) | (140) |
mark#(U91(X1,X2,X3,X4)) | → | mark#(X1) | (141) |
mark#(U92(X1,X2,X3,X4)) | → | a__U92#(mark(X1),X2,X3,X4) | (142) |
mark#(U92(X1,X2,X3,X4)) | → | mark#(X1) | (143) |
mark#(U93(X1,X2,X3,X4)) | → | a__U93#(mark(X1),X2,X3,X4) | (144) |
mark#(U93(X1,X2,X3,X4)) | → | mark#(X1) | (145) |
mark#(take(X1,X2)) | → | a__take#(mark(X1),mark(X2)) | (146) |
mark#(take(X1,X2)) | → | mark#(X1) | (147) |
mark#(take(X1,X2)) | → | mark#(X2) | (148) |
mark#(cons(X1,X2)) | → | mark#(X1) | (149) |
The dependency pairs are split into 1 component.
mark#(U11(X)) | → | mark#(X) | (112) |
mark#(U21(X)) | → | mark#(X) | (114) |
mark#(U31(X)) | → | mark#(X) | (116) |
mark#(U41(X1,X2)) | → | mark#(X1) | (118) |
mark#(U42(X)) | → | mark#(X) | (120) |
mark#(U51(X1,X2)) | → | mark#(X1) | (123) |
mark#(U52(X)) | → | mark#(X) | (125) |
mark#(U61(X1,X2)) | → | mark#(X1) | (128) |
mark#(U62(X)) | → | mark#(X) | (130) |
mark#(U81(X)) | → | mark#(X) | (139) |
mark#(s(X)) | → | mark#(X) | (150) |
[U11(x1)] | = | 1 · x1 |
[U21(x1)] | = | 1 · x1 |
[U31(x1)] | = | 1 · x1 |
[U41(x1, x2)] | = | 1 · x1 + 1 · x2 |
[U42(x1)] | = | 1 · x1 |
[U51(x1, x2)] | = | 1 · x1 + 1 · x2 |
[U52(x1)] | = | 1 · x1 |
[U61(x1, x2)] | = | 1 · x1 + 1 · x2 |
[U62(x1)] | = | 1 · x1 |
[U81(x1)] | = | 1 · x1 |
[s(x1)] | = | 1 · x1 |
[mark#(x1)] | = | 1 · x1 |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
mark#(U11(X)) | → | mark#(X) | (112) |
1 | > | 1 | |
mark#(U21(X)) | → | mark#(X) | (114) |
1 | > | 1 | |
mark#(U31(X)) | → | mark#(X) | (116) |
1 | > | 1 | |
mark#(U41(X1,X2)) | → | mark#(X1) | (118) |
1 | > | 1 | |
mark#(U42(X)) | → | mark#(X) | (120) |
1 | > | 1 | |
mark#(U51(X1,X2)) | → | mark#(X1) | (123) |
1 | > | 1 | |
mark#(U52(X)) | → | mark#(X) | (125) |
1 | > | 1 | |
mark#(U61(X1,X2)) | → | mark#(X1) | (128) |
1 | > | 1 | |
mark#(U62(X)) | → | mark#(X) | (130) |
1 | > | 1 | |
mark#(U81(X)) | → | mark#(X) | (139) |
1 | > | 1 | |
mark#(s(X)) | → | mark#(X) | (150) |
1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
a__U41#(tt,V2) | → | a__isNatIList#(V2) | (78) |
a__isNatIList#(V) | → | a__isNatList#(V) | (97) |
a__isNatList#(cons(V1,V2)) | → | a__U51#(a__isNat(V1),V2) | (100) |
a__U51#(tt,V2) | → | a__isNatList#(V2) | (80) |
a__isNatList#(cons(V1,V2)) | → | a__isNat#(V1) | (101) |
a__isNat#(length(V1)) | → | a__isNatList#(V1) | (93) |
a__isNatList#(take(V1,V2)) | → | a__U61#(a__isNat(V1),V2) | (102) |
a__U61#(tt,V2) | → | a__isNatIList#(V2) | (82) |
a__isNatIList#(cons(V1,V2)) | → | a__U41#(a__isNat(V1),V2) | (98) |
a__isNatIList#(cons(V1,V2)) | → | a__isNat#(V1) | (99) |
a__isNat#(s(V1)) | → | a__isNat#(V1) | (95) |
a__isNatList#(take(V1,V2)) | → | a__isNat#(V1) | (103) |
[a__isNat(x1)] | = | 1 · x1 |
[0] | = | 0 |
[tt] | = | 0 |
[length(x1)] | = | 1 · x1 |
[a__U11(x1)] | = | 1 · x1 |
[a__isNatList(x1)] | = | 1 · x1 |
[s(x1)] | = | 1 · x1 |
[a__U21(x1)] | = | 1 · x1 |
[isNat(x1)] | = | 1 · x1 |
[U21(x1)] | = | 1 · x1 |
[nil] | = | 0 |
[cons(x1, x2)] | = | 1 · x1 + 1 · x2 |
[a__U51(x1, x2)] | = | 1 · x1 + 1 · x2 |
[take(x1, x2)] | = | 1 · x1 + 1 · x2 |
[a__U61(x1, x2)] | = | 1 · x1 + 1 · x2 |
[isNatList(x1)] | = | 1 · x1 |
[U11(x1)] | = | 1 · x1 |
[a__U62(x1)] | = | 1 · x1 |
[a__isNatIList(x1)] | = | 1 · x1 |
[U61(x1, x2)] | = | 1 · x1 + 1 · x2 |
[a__U31(x1)] | = | 1 · x1 |
[zeros] | = | 0 |
[a__U41(x1, x2)] | = | 1 · x1 + 1 · x2 |
[isNatIList(x1)] | = | 1 · x1 |
[U62(x1)] | = | 1 · x1 |
[a__U42(x1)] | = | 1 · x1 |
[U41(x1, x2)] | = | 1 · x1 + 1 · x2 |
[U42(x1)] | = | 1 · x1 |
[U31(x1)] | = | 1 · x1 |
[a__U52(x1)] | = | 1 · x1 |
[U51(x1, x2)] | = | 1 · x1 + 1 · x2 |
[U52(x1)] | = | 1 · x1 |
[a__isNatIList#(x1)] | = | 1 · x1 |
[a__U41#(x1, x2)] | = | 1 · x1 + 1 · x2 |
[a__isNatList#(x1)] | = | 1 · x1 |
[a__U51#(x1, x2)] | = | 1 · x1 + 1 · x2 |
[a__isNat#(x1)] | = | 1 · x1 |
[a__U61#(x1, x2)] | = | 1 · x1 + 1 · x2 |
a__isNat(0) | → | tt | (17) |
a__isNat(length(V1)) | → | a__U11(a__isNatList(V1)) | (18) |
a__isNat(s(V1)) | → | a__U21(a__isNat(V1)) | (19) |
a__isNat(X) | → | isNat(X) | (70) |
a__U21(tt) | → | tt | (3) |
a__U21(X) | → | U21(X) | (58) |
a__isNatList(nil) | → | tt | (23) |
a__isNatList(cons(V1,V2)) | → | a__U51(a__isNat(V1),V2) | (24) |
a__isNatList(take(V1,V2)) | → | a__U61(a__isNat(V1),V2) | (25) |
a__isNatList(X) | → | isNatList(X) | (65) |
a__U11(tt) | → | tt | (2) |
a__U11(X) | → | U11(X) | (57) |
a__U61(tt,V2) | → | a__U62(a__isNatIList(V2)) | (9) |
a__U61(X1,X2) | → | U61(X1,X2) | (66) |
a__isNatIList(V) | → | a__U31(a__isNatList(V)) | (20) |
a__isNatIList(zeros) | → | tt | (21) |
a__isNatIList(cons(V1,V2)) | → | a__U41(a__isNat(V1),V2) | (22) |
a__isNatIList(X) | → | isNatIList(X) | (62) |
a__U62(tt) | → | tt | (10) |
a__U62(X) | → | U62(X) | (67) |
a__U41(tt,V2) | → | a__U42(a__isNatIList(V2)) | (5) |
a__U41(X1,X2) | → | U41(X1,X2) | (60) |
a__U42(tt) | → | tt | (6) |
a__U42(X) | → | U42(X) | (61) |
a__U31(tt) | → | tt | (4) |
a__U31(X) | → | U31(X) | (59) |
a__U51(tt,V2) | → | a__U52(a__isNatList(V2)) | (7) |
a__U51(X1,X2) | → | U51(X1,X2) | (63) |
a__U52(tt) | → | tt | (8) |
a__U52(X) | → | U52(X) | (64) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
a__isNatIList#(cons(V1,V2)) | → | a__U41#(a__isNat(V1),V2) | (98) |
1 | > | 2 | |
a__isNatIList#(V) | → | a__isNatList#(V) | (97) |
1 | ≥ | 1 | |
a__isNatIList#(cons(V1,V2)) | → | a__isNat#(V1) | (99) |
1 | > | 1 | |
a__U51#(tt,V2) | → | a__isNatList#(V2) | (80) |
2 | ≥ | 1 | |
a__isNat#(length(V1)) | → | a__isNatList#(V1) | (93) |
1 | > | 1 | |
a__isNat#(s(V1)) | → | a__isNat#(V1) | (95) |
1 | > | 1 | |
a__isNatList#(cons(V1,V2)) | → | a__U51#(a__isNat(V1),V2) | (100) |
1 | > | 2 | |
a__isNatList#(take(V1,V2)) | → | a__U61#(a__isNat(V1),V2) | (102) |
1 | > | 2 | |
a__U61#(tt,V2) | → | a__isNatIList#(V2) | (82) |
2 | ≥ | 1 | |
a__U41#(tt,V2) | → | a__isNatIList#(V2) | (78) |
2 | ≥ | 1 | |
a__isNatList#(cons(V1,V2)) | → | a__isNat#(V1) | (101) |
1 | > | 1 | |
a__isNatList#(take(V1,V2)) | → | a__isNat#(V1) | (103) |
1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.