The rewrite relation of the following TRS is considered.
a__zeros | → | cons(0,zeros) | (1) |
a__U11(tt,V1) | → | a__U12(a__isNatList(V1)) | (2) |
a__U12(tt) | → | tt | (3) |
a__U21(tt,V1) | → | a__U22(a__isNat(V1)) | (4) |
a__U22(tt) | → | tt | (5) |
a__U31(tt,V) | → | a__U32(a__isNatList(V)) | (6) |
a__U32(tt) | → | tt | (7) |
a__U41(tt,V1,V2) | → | a__U42(a__isNat(V1),V2) | (8) |
a__U42(tt,V2) | → | a__U43(a__isNatIList(V2)) | (9) |
a__U43(tt) | → | tt | (10) |
a__U51(tt,V1,V2) | → | a__U52(a__isNat(V1),V2) | (11) |
a__U52(tt,V2) | → | a__U53(a__isNatList(V2)) | (12) |
a__U53(tt) | → | tt | (13) |
a__U61(tt,L) | → | s(a__length(mark(L))) | (14) |
a__and(tt,X) | → | mark(X) | (15) |
a__isNat(0) | → | tt | (16) |
a__isNat(length(V1)) | → | a__U11(a__isNatIListKind(V1),V1) | (17) |
a__isNat(s(V1)) | → | a__U21(a__isNatKind(V1),V1) | (18) |
a__isNatIList(V) | → | a__U31(a__isNatIListKind(V),V) | (19) |
a__isNatIList(zeros) | → | tt | (20) |
a__isNatIList(cons(V1,V2)) | → | a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) | (21) |
a__isNatIListKind(nil) | → | tt | (22) |
a__isNatIListKind(zeros) | → | tt | (23) |
a__isNatIListKind(cons(V1,V2)) | → | a__and(a__isNatKind(V1),isNatIListKind(V2)) | (24) |
a__isNatKind(0) | → | tt | (25) |
a__isNatKind(length(V1)) | → | a__isNatIListKind(V1) | (26) |
a__isNatKind(s(V1)) | → | a__isNatKind(V1) | (27) |
a__isNatList(nil) | → | tt | (28) |
a__isNatList(cons(V1,V2)) | → | a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) | (29) |
a__length(nil) | → | 0 | (30) |
a__length(cons(N,L)) | → | a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) | (31) |
mark(zeros) | → | a__zeros | (32) |
mark(U11(X1,X2)) | → | a__U11(mark(X1),X2) | (33) |
mark(U12(X)) | → | a__U12(mark(X)) | (34) |
mark(isNatList(X)) | → | a__isNatList(X) | (35) |
mark(U21(X1,X2)) | → | a__U21(mark(X1),X2) | (36) |
mark(U22(X)) | → | a__U22(mark(X)) | (37) |
mark(isNat(X)) | → | a__isNat(X) | (38) |
mark(U31(X1,X2)) | → | a__U31(mark(X1),X2) | (39) |
mark(U32(X)) | → | a__U32(mark(X)) | (40) |
mark(U41(X1,X2,X3)) | → | a__U41(mark(X1),X2,X3) | (41) |
mark(U42(X1,X2)) | → | a__U42(mark(X1),X2) | (42) |
mark(U43(X)) | → | a__U43(mark(X)) | (43) |
mark(isNatIList(X)) | → | a__isNatIList(X) | (44) |
mark(U51(X1,X2,X3)) | → | a__U51(mark(X1),X2,X3) | (45) |
mark(U52(X1,X2)) | → | a__U52(mark(X1),X2) | (46) |
mark(U53(X)) | → | a__U53(mark(X)) | (47) |
mark(U61(X1,X2)) | → | a__U61(mark(X1),X2) | (48) |
mark(length(X)) | → | a__length(mark(X)) | (49) |
mark(and(X1,X2)) | → | a__and(mark(X1),X2) | (50) |
mark(isNatIListKind(X)) | → | a__isNatIListKind(X) | (51) |
mark(isNatKind(X)) | → | a__isNatKind(X) | (52) |
mark(cons(X1,X2)) | → | cons(mark(X1),X2) | (53) |
mark(0) | → | 0 | (54) |
mark(tt) | → | tt | (55) |
mark(s(X)) | → | s(mark(X)) | (56) |
mark(nil) | → | nil | (57) |
a__zeros | → | zeros | (58) |
a__U11(X1,X2) | → | U11(X1,X2) | (59) |
a__U12(X) | → | U12(X) | (60) |
a__isNatList(X) | → | isNatList(X) | (61) |
a__U21(X1,X2) | → | U21(X1,X2) | (62) |
a__U22(X) | → | U22(X) | (63) |
a__isNat(X) | → | isNat(X) | (64) |
a__U31(X1,X2) | → | U31(X1,X2) | (65) |
a__U32(X) | → | U32(X) | (66) |
a__U41(X1,X2,X3) | → | U41(X1,X2,X3) | (67) |
a__U42(X1,X2) | → | U42(X1,X2) | (68) |
a__U43(X) | → | U43(X) | (69) |
a__isNatIList(X) | → | isNatIList(X) | (70) |
a__U51(X1,X2,X3) | → | U51(X1,X2,X3) | (71) |
a__U52(X1,X2) | → | U52(X1,X2) | (72) |
a__U53(X) | → | U53(X) | (73) |
a__U61(X1,X2) | → | U61(X1,X2) | (74) |
a__length(X) | → | length(X) | (75) |
a__and(X1,X2) | → | and(X1,X2) | (76) |
a__isNatIListKind(X) | → | isNatIListKind(X) | (77) |
a__isNatKind(X) | → | isNatKind(X) | (78) |
a__U11#(tt,V1) | → | a__isNatList#(V1) | (79) |
a__U11#(tt,V1) | → | a__U12#(a__isNatList(V1)) | (80) |
a__U21#(tt,V1) | → | a__isNat#(V1) | (81) |
a__U21#(tt,V1) | → | a__U22#(a__isNat(V1)) | (82) |
a__U31#(tt,V) | → | a__isNatList#(V) | (83) |
a__U31#(tt,V) | → | a__U32#(a__isNatList(V)) | (84) |
a__U41#(tt,V1,V2) | → | a__isNat#(V1) | (85) |
a__U41#(tt,V1,V2) | → | a__U42#(a__isNat(V1),V2) | (86) |
a__U42#(tt,V2) | → | a__isNatIList#(V2) | (87) |
a__U42#(tt,V2) | → | a__U43#(a__isNatIList(V2)) | (88) |
a__U51#(tt,V1,V2) | → | a__isNat#(V1) | (89) |
a__U51#(tt,V1,V2) | → | a__U52#(a__isNat(V1),V2) | (90) |
a__U52#(tt,V2) | → | a__isNatList#(V2) | (91) |
a__U52#(tt,V2) | → | a__U53#(a__isNatList(V2)) | (92) |
a__U61#(tt,L) | → | mark#(L) | (93) |
a__U61#(tt,L) | → | a__length#(mark(L)) | (94) |
a__and#(tt,X) | → | mark#(X) | (95) |
a__isNat#(length(V1)) | → | a__isNatIListKind#(V1) | (96) |
a__isNat#(length(V1)) | → | a__U11#(a__isNatIListKind(V1),V1) | (97) |
a__isNat#(s(V1)) | → | a__isNatKind#(V1) | (98) |
a__isNat#(s(V1)) | → | a__U21#(a__isNatKind(V1),V1) | (99) |
a__isNatIList#(V) | → | a__isNatIListKind#(V) | (100) |
a__isNatIList#(V) | → | a__U31#(a__isNatIListKind(V),V) | (101) |
a__isNatIList#(cons(V1,V2)) | → | a__isNatKind#(V1) | (102) |
a__isNatIList#(cons(V1,V2)) | → | a__and#(a__isNatKind(V1),isNatIListKind(V2)) | (103) |
a__isNatIList#(cons(V1,V2)) | → | a__U41#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) | (104) |
a__isNatIListKind#(cons(V1,V2)) | → | a__isNatKind#(V1) | (105) |
a__isNatIListKind#(cons(V1,V2)) | → | a__and#(a__isNatKind(V1),isNatIListKind(V2)) | (106) |
a__isNatKind#(length(V1)) | → | a__isNatIListKind#(V1) | (107) |
a__isNatKind#(s(V1)) | → | a__isNatKind#(V1) | (108) |
a__isNatList#(cons(V1,V2)) | → | a__isNatKind#(V1) | (109) |
a__isNatList#(cons(V1,V2)) | → | a__and#(a__isNatKind(V1),isNatIListKind(V2)) | (110) |
a__isNatList#(cons(V1,V2)) | → | a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) | (111) |
a__length#(cons(N,L)) | → | a__isNatList#(L) | (112) |
a__length#(cons(N,L)) | → | a__and#(a__isNatList(L),isNatIListKind(L)) | (113) |
a__length#(cons(N,L)) | → | a__and#(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) | (114) |
a__length#(cons(N,L)) | → | a__U61#(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) | (115) |
mark#(zeros) | → | a__zeros# | (116) |
mark#(U11(X1,X2)) | → | mark#(X1) | (117) |
mark#(U11(X1,X2)) | → | a__U11#(mark(X1),X2) | (118) |
mark#(U12(X)) | → | mark#(X) | (119) |
mark#(U12(X)) | → | a__U12#(mark(X)) | (120) |
mark#(isNatList(X)) | → | a__isNatList#(X) | (121) |
mark#(U21(X1,X2)) | → | mark#(X1) | (122) |
mark#(U21(X1,X2)) | → | a__U21#(mark(X1),X2) | (123) |
mark#(U22(X)) | → | mark#(X) | (124) |
mark#(U22(X)) | → | a__U22#(mark(X)) | (125) |
mark#(isNat(X)) | → | a__isNat#(X) | (126) |
mark#(U31(X1,X2)) | → | mark#(X1) | (127) |
mark#(U31(X1,X2)) | → | a__U31#(mark(X1),X2) | (128) |
mark#(U32(X)) | → | mark#(X) | (129) |
mark#(U32(X)) | → | a__U32#(mark(X)) | (130) |
mark#(U41(X1,X2,X3)) | → | mark#(X1) | (131) |
mark#(U41(X1,X2,X3)) | → | a__U41#(mark(X1),X2,X3) | (132) |
mark#(U42(X1,X2)) | → | mark#(X1) | (133) |
mark#(U42(X1,X2)) | → | a__U42#(mark(X1),X2) | (134) |
mark#(U43(X)) | → | mark#(X) | (135) |
mark#(U43(X)) | → | a__U43#(mark(X)) | (136) |
mark#(isNatIList(X)) | → | a__isNatIList#(X) | (137) |
mark#(U51(X1,X2,X3)) | → | mark#(X1) | (138) |
mark#(U51(X1,X2,X3)) | → | a__U51#(mark(X1),X2,X3) | (139) |
mark#(U52(X1,X2)) | → | mark#(X1) | (140) |
mark#(U52(X1,X2)) | → | a__U52#(mark(X1),X2) | (141) |
mark#(U53(X)) | → | mark#(X) | (142) |
mark#(U53(X)) | → | a__U53#(mark(X)) | (143) |
mark#(U61(X1,X2)) | → | mark#(X1) | (144) |
mark#(U61(X1,X2)) | → | a__U61#(mark(X1),X2) | (145) |
mark#(length(X)) | → | mark#(X) | (146) |
mark#(length(X)) | → | a__length#(mark(X)) | (147) |
mark#(and(X1,X2)) | → | mark#(X1) | (148) |
mark#(and(X1,X2)) | → | a__and#(mark(X1),X2) | (149) |
mark#(isNatIListKind(X)) | → | a__isNatIListKind#(X) | (150) |
mark#(isNatKind(X)) | → | a__isNatKind#(X) | (151) |
mark#(cons(X1,X2)) | → | mark#(X1) | (152) |
mark#(s(X)) | → | mark#(X) | (153) |
The dependency pairs are split into 1 component.
a__isNatKind#(length(V1)) | → | a__isNatIListKind#(V1) | (107) |
a__isNatIListKind#(cons(V1,V2)) | → | a__isNatKind#(V1) | (105) |
a__isNatKind#(s(V1)) | → | a__isNatKind#(V1) | (108) |
a__isNatIListKind#(cons(V1,V2)) | → | a__and#(a__isNatKind(V1),isNatIListKind(V2)) | (106) |
a__and#(tt,X) | → | mark#(X) | (95) |
mark#(U11(X1,X2)) | → | mark#(X1) | (117) |
mark#(U11(X1,X2)) | → | a__U11#(mark(X1),X2) | (118) |
a__U11#(tt,V1) | → | a__isNatList#(V1) | (79) |
a__isNatList#(cons(V1,V2)) | → | a__isNatKind#(V1) | (109) |
a__isNatList#(cons(V1,V2)) | → | a__and#(a__isNatKind(V1),isNatIListKind(V2)) | (110) |
a__isNatList#(cons(V1,V2)) | → | a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) | (111) |
a__U51#(tt,V1,V2) | → | a__isNat#(V1) | (89) |
a__isNat#(length(V1)) | → | a__isNatIListKind#(V1) | (96) |
a__isNat#(length(V1)) | → | a__U11#(a__isNatIListKind(V1),V1) | (97) |
a__isNat#(s(V1)) | → | a__isNatKind#(V1) | (98) |
a__isNat#(s(V1)) | → | a__U21#(a__isNatKind(V1),V1) | (99) |
a__U21#(tt,V1) | → | a__isNat#(V1) | (81) |
a__U51#(tt,V1,V2) | → | a__U52#(a__isNat(V1),V2) | (90) |
a__U52#(tt,V2) | → | a__isNatList#(V2) | (91) |
mark#(U12(X)) | → | mark#(X) | (119) |
mark#(isNatList(X)) | → | a__isNatList#(X) | (121) |
mark#(U21(X1,X2)) | → | mark#(X1) | (122) |
mark#(U21(X1,X2)) | → | a__U21#(mark(X1),X2) | (123) |
mark#(U22(X)) | → | mark#(X) | (124) |
mark#(isNat(X)) | → | a__isNat#(X) | (126) |
mark#(U31(X1,X2)) | → | mark#(X1) | (127) |
mark#(U31(X1,X2)) | → | a__U31#(mark(X1),X2) | (128) |
a__U31#(tt,V) | → | a__isNatList#(V) | (83) |
mark#(U32(X)) | → | mark#(X) | (129) |
mark#(U41(X1,X2,X3)) | → | mark#(X1) | (131) |
mark#(U41(X1,X2,X3)) | → | a__U41#(mark(X1),X2,X3) | (132) |
a__U41#(tt,V1,V2) | → | a__isNat#(V1) | (85) |
a__U41#(tt,V1,V2) | → | a__U42#(a__isNat(V1),V2) | (86) |
a__U42#(tt,V2) | → | a__isNatIList#(V2) | (87) |
a__isNatIList#(V) | → | a__isNatIListKind#(V) | (100) |
a__isNatIList#(V) | → | a__U31#(a__isNatIListKind(V),V) | (101) |
a__isNatIList#(cons(V1,V2)) | → | a__isNatKind#(V1) | (102) |
a__isNatIList#(cons(V1,V2)) | → | a__and#(a__isNatKind(V1),isNatIListKind(V2)) | (103) |
a__isNatIList#(cons(V1,V2)) | → | a__U41#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) | (104) |
mark#(U42(X1,X2)) | → | mark#(X1) | (133) |
mark#(U42(X1,X2)) | → | a__U42#(mark(X1),X2) | (134) |
mark#(U43(X)) | → | mark#(X) | (135) |
mark#(isNatIList(X)) | → | a__isNatIList#(X) | (137) |
mark#(U51(X1,X2,X3)) | → | mark#(X1) | (138) |
mark#(U51(X1,X2,X3)) | → | a__U51#(mark(X1),X2,X3) | (139) |
mark#(U52(X1,X2)) | → | mark#(X1) | (140) |
mark#(U52(X1,X2)) | → | a__U52#(mark(X1),X2) | (141) |
mark#(U53(X)) | → | mark#(X) | (142) |
mark#(U61(X1,X2)) | → | mark#(X1) | (144) |
mark#(U61(X1,X2)) | → | a__U61#(mark(X1),X2) | (145) |
a__U61#(tt,L) | → | mark#(L) | (93) |
mark#(length(X)) | → | mark#(X) | (146) |
mark#(length(X)) | → | a__length#(mark(X)) | (147) |
a__length#(cons(N,L)) | → | a__isNatList#(L) | (112) |
a__length#(cons(N,L)) | → | a__and#(a__isNatList(L),isNatIListKind(L)) | (113) |
a__length#(cons(N,L)) | → | a__and#(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) | (114) |
a__length#(cons(N,L)) | → | a__U61#(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) | (115) |
a__U61#(tt,L) | → | a__length#(mark(L)) | (94) |
mark#(and(X1,X2)) | → | mark#(X1) | (148) |
mark#(and(X1,X2)) | → | a__and#(mark(X1),X2) | (149) |
mark#(isNatIListKind(X)) | → | a__isNatIListKind#(X) | (150) |
mark#(isNatKind(X)) | → | a__isNatKind#(X) | (151) |
mark#(cons(X1,X2)) | → | mark#(X1) | (152) |
mark#(s(X)) | → | mark#(X) | (153) |
[a__U52(x1, x2)] | = | 0 · x1 + 0 · x2 + -∞ |
[a__isNatList#(x1)] | = | 0 · x1 + -∞ |
[tt] | = | 0 |
[a__isNatIList(x1)] | = | 4 · x1 + -∞ |
[a__U21(x1, x2)] | = | 0 · x1 + 0 · x2 + -∞ |
[0] | = | 0 |
[cons(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[a__U43(x1)] | = | 0 · x1 + -∞ |
[mark(x1)] | = | 0 · x1 + -∞ |
[U21(x1, x2)] | = | 0 · x1 + 0 · x2 + -∞ |
[a__isNatIListKind#(x1)] | = | 0 · x1 + -∞ |
[a__isNatIListKind(x1)] | = | 0 · x1 + -∞ |
[a__U53(x1)] | = | 0 · x1 + 0 |
[and(x1, x2)] | = | 0 · x1 + 0 · x2 + -∞ |
[a__U51(x1, x2, x3)] | = | 0 · x1 + 0 · x2 + 0 · x3 + 0 |
[a__isNatKind(x1)] | = | 0 · x1 + -∞ |
[U12(x1)] | = | 0 · x1 + -∞ |
[a__length(x1)] | = | 6 · x1 + -∞ |
[isNatIListKind(x1)] | = | 0 · x1 + -∞ |
[mark#(x1)] | = | 0 · x1 + -∞ |
[U51(x1, x2, x3)] | = | 0 · x1 + 0 · x2 + 0 · x3 + 0 |
[a__U11(x1, x2)] | = | 0 · x1 + 4 · x2 + -∞ |
[a__U61#(x1, x2)] | = | -∞ · x1 + 4 · x2 + -∞ |
[U11(x1, x2)] | = | 0 · x1 + 4 · x2 + -∞ |
[a__U22(x1)] | = | 0 · x1 + -∞ |
[a__zeros] | = | 0 |
[U32(x1)] | = | 0 · x1 + 0 |
[a__U12(x1)] | = | 0 · x1 + -∞ |
[a__U61(x1, x2)] | = | 0 · x1 + 6 · x2 + 0 |
[length(x1)] | = | 6 · x1 + -∞ |
[a__U21#(x1, x2)] | = | 0 · x1 + 0 · x2 + -∞ |
[a__U11#(x1, x2)] | = | 0 · x1 + 4 · x2 + -∞ |
[isNat(x1)] | = | 0 · x1 + -∞ |
[a__U52#(x1, x2)] | = | 0 · x1 + 0 · x2 + -∞ |
[a__U42(x1, x2)] | = | 0 · x1 + 4 · x2 + 0 |
[isNatKind(x1)] | = | 0 · x1 + -∞ |
[a__U32(x1)] | = | 0 · x1 + 0 |
[a__isNatKind#(x1)] | = | 0 · x1 + -∞ |
[U52(x1, x2)] | = | 0 · x1 + 0 · x2 + -∞ |
[s(x1)] | = | 0 · x1 + -∞ |
[a__and(x1, x2)] | = | 0 · x1 + 0 · x2 + -∞ |
[a__U41#(x1, x2, x3)] | = | 0 · x1 + 0 · x2 + 0 · x3 + -∞ |
[a__U31(x1, x2)] | = | 0 · x1 + 4 · x2 + -∞ |
[a__isNat(x1)] | = | 0 · x1 + -∞ |
[a__U41(x1, x2, x3)] | = | 0 · x1 + 4 · x2 + 4 · x3 + 0 |
[U41(x1, x2, x3)] | = | 0 · x1 + 4 · x2 + 4 · x3 + 0 |
[a__and#(x1, x2)] | = | 0 · x1 + 0 · x2 + -∞ |
[a__U31#(x1, x2)] | = | -∞ · x1 + 0 · x2 + -∞ |
[a__isNatIList#(x1)] | = | 0 · x1 + -∞ |
[a__isNatList(x1)] | = | 0 · x1 + 0 |
[U31(x1, x2)] | = | 0 · x1 + 4 · x2 + -∞ |
[a__U51#(x1, x2, x3)] | = | 0 · x1 + 0 · x2 + 0 · x3 + -∞ |
[a__U42#(x1, x2)] | = | -∞ · x1 + 0 · x2 + -∞ |
[a__isNat#(x1)] | = | 0 · x1 + -∞ |
[U61(x1, x2)] | = | 0 · x1 + 6 · x2 + 0 |
[U22(x1)] | = | 0 · x1 + -∞ |
[U42(x1, x2)] | = | 0 · x1 + 4 · x2 + 0 |
[U43(x1)] | = | 0 · x1 + -∞ |
[U53(x1)] | = | 0 · x1 + 0 |
[a__length#(x1)] | = | 4 · x1 + -∞ |
[isNatList(x1)] | = | 0 · x1 + 0 |
[nil] | = | 0 |
[isNatIList(x1)] | = | 4 · x1 + -∞ |
[zeros] | = | 0 |
a__zeros | → | cons(0,zeros) | (1) |
a__U11(tt,V1) | → | a__U12(a__isNatList(V1)) | (2) |
a__U12(tt) | → | tt | (3) |
a__U21(tt,V1) | → | a__U22(a__isNat(V1)) | (4) |
a__U22(tt) | → | tt | (5) |
a__U31(tt,V) | → | a__U32(a__isNatList(V)) | (6) |
a__U32(tt) | → | tt | (7) |
a__U41(tt,V1,V2) | → | a__U42(a__isNat(V1),V2) | (8) |
a__U42(tt,V2) | → | a__U43(a__isNatIList(V2)) | (9) |
a__U43(tt) | → | tt | (10) |
a__U51(tt,V1,V2) | → | a__U52(a__isNat(V1),V2) | (11) |
a__U52(tt,V2) | → | a__U53(a__isNatList(V2)) | (12) |
a__U53(tt) | → | tt | (13) |
a__U61(tt,L) | → | s(a__length(mark(L))) | (14) |
a__and(tt,X) | → | mark(X) | (15) |
a__isNat(0) | → | tt | (16) |
a__isNat(length(V1)) | → | a__U11(a__isNatIListKind(V1),V1) | (17) |
a__isNat(s(V1)) | → | a__U21(a__isNatKind(V1),V1) | (18) |
a__isNatIList(V) | → | a__U31(a__isNatIListKind(V),V) | (19) |
a__isNatIList(zeros) | → | tt | (20) |
a__isNatIList(cons(V1,V2)) | → | a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) | (21) |
a__isNatIListKind(nil) | → | tt | (22) |
a__isNatIListKind(zeros) | → | tt | (23) |
a__isNatIListKind(cons(V1,V2)) | → | a__and(a__isNatKind(V1),isNatIListKind(V2)) | (24) |
a__isNatKind(0) | → | tt | (25) |
a__isNatKind(length(V1)) | → | a__isNatIListKind(V1) | (26) |
a__isNatKind(s(V1)) | → | a__isNatKind(V1) | (27) |
a__isNatList(nil) | → | tt | (28) |
a__isNatList(cons(V1,V2)) | → | a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) | (29) |
a__length(nil) | → | 0 | (30) |
a__length(cons(N,L)) | → | a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) | (31) |
mark(zeros) | → | a__zeros | (32) |
mark(U11(X1,X2)) | → | a__U11(mark(X1),X2) | (33) |
mark(U12(X)) | → | a__U12(mark(X)) | (34) |
mark(isNatList(X)) | → | a__isNatList(X) | (35) |
mark(U21(X1,X2)) | → | a__U21(mark(X1),X2) | (36) |
mark(U22(X)) | → | a__U22(mark(X)) | (37) |
mark(isNat(X)) | → | a__isNat(X) | (38) |
mark(U31(X1,X2)) | → | a__U31(mark(X1),X2) | (39) |
mark(U32(X)) | → | a__U32(mark(X)) | (40) |
mark(U41(X1,X2,X3)) | → | a__U41(mark(X1),X2,X3) | (41) |
mark(U42(X1,X2)) | → | a__U42(mark(X1),X2) | (42) |
mark(U43(X)) | → | a__U43(mark(X)) | (43) |
mark(isNatIList(X)) | → | a__isNatIList(X) | (44) |
mark(U51(X1,X2,X3)) | → | a__U51(mark(X1),X2,X3) | (45) |
mark(U52(X1,X2)) | → | a__U52(mark(X1),X2) | (46) |
mark(U53(X)) | → | a__U53(mark(X)) | (47) |
mark(U61(X1,X2)) | → | a__U61(mark(X1),X2) | (48) |
mark(length(X)) | → | a__length(mark(X)) | (49) |
mark(and(X1,X2)) | → | a__and(mark(X1),X2) | (50) |
mark(isNatIListKind(X)) | → | a__isNatIListKind(X) | (51) |
mark(isNatKind(X)) | → | a__isNatKind(X) | (52) |
mark(cons(X1,X2)) | → | cons(mark(X1),X2) | (53) |
mark(0) | → | 0 | (54) |
mark(tt) | → | tt | (55) |
mark(s(X)) | → | s(mark(X)) | (56) |
mark(nil) | → | nil | (57) |
a__zeros | → | zeros | (58) |
a__U11(X1,X2) | → | U11(X1,X2) | (59) |
a__U12(X) | → | U12(X) | (60) |
a__isNatList(X) | → | isNatList(X) | (61) |
a__U21(X1,X2) | → | U21(X1,X2) | (62) |
a__U22(X) | → | U22(X) | (63) |
a__isNat(X) | → | isNat(X) | (64) |
a__U31(X1,X2) | → | U31(X1,X2) | (65) |
a__U32(X) | → | U32(X) | (66) |
a__U41(X1,X2,X3) | → | U41(X1,X2,X3) | (67) |
a__U42(X1,X2) | → | U42(X1,X2) | (68) |
a__U43(X) | → | U43(X) | (69) |
a__isNatIList(X) | → | isNatIList(X) | (70) |
a__U51(X1,X2,X3) | → | U51(X1,X2,X3) | (71) |
a__U52(X1,X2) | → | U52(X1,X2) | (72) |
a__U53(X) | → | U53(X) | (73) |
a__U61(X1,X2) | → | U61(X1,X2) | (74) |
a__length(X) | → | length(X) | (75) |
a__and(X1,X2) | → | and(X1,X2) | (76) |
a__isNatIListKind(X) | → | isNatIListKind(X) | (77) |
a__isNatKind(X) | → | isNatKind(X) | (78) |
a__isNatKind#(length(V1)) | → | a__isNatIListKind#(V1) | (107) |
a__U11#(tt,V1) | → | a__isNatList#(V1) | (79) |
a__isNat#(length(V1)) | → | a__isNatIListKind#(V1) | (96) |
a__isNat#(length(V1)) | → | a__U11#(a__isNatIListKind(V1),V1) | (97) |
mark#(U31(X1,X2)) | → | a__U31#(mark(X1),X2) | (128) |
mark#(U42(X1,X2)) | → | a__U42#(mark(X1),X2) | (134) |
mark#(isNatIList(X)) | → | a__isNatIList#(X) | (137) |
mark#(U61(X1,X2)) | → | a__U61#(mark(X1),X2) | (145) |
a__U61#(tt,L) | → | mark#(L) | (93) |
mark#(length(X)) | → | mark#(X) | (146) |
mark#(length(X)) | → | a__length#(mark(X)) | (147) |
a__length#(cons(N,L)) | → | a__isNatList#(L) | (112) |
a__length#(cons(N,L)) | → | a__and#(a__isNatList(L),isNatIListKind(L)) | (113) |
a__length#(cons(N,L)) | → | a__and#(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))) | (114) |
The dependency pairs are split into 4 components.
a__length#(cons(N,L)) | → | a__U61#(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) | (115) |
a__U61#(tt,L) | → | a__length#(mark(L)) | (94) |
[a__U52(x1, x2)] | = | 0 · x1 + 1/2 · x2 + 0 |
[tt] | = | 1/2 |
[a__isNatIList(x1)] | = | 0 · x1 + 2 |
[a__U21(x1, x2)] | = | 0 · x1 + 0 · x2 + 1/2 |
[0] | = | 0 |
[cons(x1, x2)] | = | 0 · x1 + 3 · x2 + 0 |
[a__U43(x1)] | = | 0 · x1 + 1/2 |
[mark(x1)] | = | 2 · x1 + 1/2 |
[U21(x1, x2)] | = | 0 · x1 + 0 · x2 + 1/2 |
[a__isNatIListKind(x1)] | = | 0 · x1 + 1/2 |
[a__U53(x1)] | = | 1 · x1 + 0 |
[and(x1, x2)] | = | 1 · x1 + 2 · x2 + 0 |
[a__U51(x1, x2, x3)] | = | 0 · x1 + 0 · x2 + 1/2 · x3 + 0 |
[a__isNatKind(x1)] | = | 0 · x1 + 1/2 |
[U12(x1)] | = | 0 · x1 + 0 |
[a__length(x1)] | = | 0 · x1 + 0 |
[isNatIListKind(x1)] | = | 0 · x1 + 0 |
[U51(x1, x2, x3)] | = | 0 · x1 + 0 · x2 + 1/2 · x3 + 0 |
[a__U11(x1, x2)] | = | 1 · x1 + 0 · x2 + 0 |
[U11(x1, x2)] | = | 1 · x1 + 0 · x2 + 0 |
[a__U61#(x1, x2)] | = | 2 · x1 + 2 · x2 + 0 |
[a__U22(x1)] | = | 1 · x1 + 0 |
[a__zeros] | = | 1/2 |
[U32(x1)] | = | 0 · x1 + 1/2 |
[a__U12(x1)] | = | 0 · x1 + 1/2 |
[a__U61(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[length(x1)] | = | 0 · x1 + 0 |
[isNat(x1)] | = | 0 · x1 + 0 |
[a__U42(x1, x2)] | = | 1 · x1 + 0 · x2 + 0 |
[isNatKind(x1)] | = | 0 · x1 + 0 |
[a__U32(x1)] | = | 0 · x1 + 1 |
[U52(x1, x2)] | = | 0 · x1 + 1/2 · x2 + 0 |
[s(x1)] | = | 0 · x1 + 0 |
[a__and(x1, x2)] | = | 1 · x1 + 2 · x2 + 0 |
[a__U31(x1, x2)] | = | 1 · x1 + 0 · x2 + 1/2 |
[a__isNat(x1)] | = | 0 · x1 + 1/2 |
[a__U41(x1, x2, x3)] | = | 0 · x1 + 0 · x2 + 0 · x3 + 1/2 |
[U41(x1, x2, x3)] | = | 0 · x1 + 0 · x2 + 0 · x3 + 1/2 |
[a__isNatList(x1)] | = | 1/2 · x1 + 0 |
[U31(x1, x2)] | = | 1 · x1 + 0 · x2 + 1/2 |
[U61(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[U22(x1)] | = | 1 · x1 + 0 |
[U42(x1, x2)] | = | 1 · x1 + 0 · x2 + 0 |
[U43(x1)] | = | 0 · x1 + 1/2 |
[U53(x1)] | = | 1 · x1 + 0 |
[a__length#(x1)] | = | 1 · x1 + 0 |
[isNatList(x1)] | = | 1/2 · x1 + 0 |
[nil] | = | 1 |
[isNatIList(x1)] | = | 0 · x1 + 2 |
[zeros] | = | 0 |
a__zeros | → | cons(0,zeros) | (1) |
a__U11(tt,V1) | → | a__U12(a__isNatList(V1)) | (2) |
a__U12(tt) | → | tt | (3) |
a__U21(tt,V1) | → | a__U22(a__isNat(V1)) | (4) |
a__U22(tt) | → | tt | (5) |
a__U31(tt,V) | → | a__U32(a__isNatList(V)) | (6) |
a__U32(tt) | → | tt | (7) |
a__U41(tt,V1,V2) | → | a__U42(a__isNat(V1),V2) | (8) |
a__U42(tt,V2) | → | a__U43(a__isNatIList(V2)) | (9) |
a__U43(tt) | → | tt | (10) |
a__U51(tt,V1,V2) | → | a__U52(a__isNat(V1),V2) | (11) |
a__U52(tt,V2) | → | a__U53(a__isNatList(V2)) | (12) |
a__U53(tt) | → | tt | (13) |
a__U61(tt,L) | → | s(a__length(mark(L))) | (14) |
a__and(tt,X) | → | mark(X) | (15) |
a__isNat(0) | → | tt | (16) |
a__isNat(length(V1)) | → | a__U11(a__isNatIListKind(V1),V1) | (17) |
a__isNat(s(V1)) | → | a__U21(a__isNatKind(V1),V1) | (18) |
a__isNatIList(V) | → | a__U31(a__isNatIListKind(V),V) | (19) |
a__isNatIList(zeros) | → | tt | (20) |
a__isNatIList(cons(V1,V2)) | → | a__U41(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) | (21) |
a__isNatIListKind(nil) | → | tt | (22) |
a__isNatIListKind(zeros) | → | tt | (23) |
a__isNatIListKind(cons(V1,V2)) | → | a__and(a__isNatKind(V1),isNatIListKind(V2)) | (24) |
a__isNatKind(0) | → | tt | (25) |
a__isNatKind(length(V1)) | → | a__isNatIListKind(V1) | (26) |
a__isNatKind(s(V1)) | → | a__isNatKind(V1) | (27) |
a__isNatList(nil) | → | tt | (28) |
a__isNatList(cons(V1,V2)) | → | a__U51(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) | (29) |
a__length(nil) | → | 0 | (30) |
a__length(cons(N,L)) | → | a__U61(a__and(a__and(a__isNatList(L),isNatIListKind(L)),and(isNat(N),isNatKind(N))),L) | (31) |
mark(zeros) | → | a__zeros | (32) |
mark(U11(X1,X2)) | → | a__U11(mark(X1),X2) | (33) |
mark(U12(X)) | → | a__U12(mark(X)) | (34) |
mark(isNatList(X)) | → | a__isNatList(X) | (35) |
mark(U21(X1,X2)) | → | a__U21(mark(X1),X2) | (36) |
mark(U22(X)) | → | a__U22(mark(X)) | (37) |
mark(isNat(X)) | → | a__isNat(X) | (38) |
mark(U31(X1,X2)) | → | a__U31(mark(X1),X2) | (39) |
mark(U32(X)) | → | a__U32(mark(X)) | (40) |
mark(U41(X1,X2,X3)) | → | a__U41(mark(X1),X2,X3) | (41) |
mark(U42(X1,X2)) | → | a__U42(mark(X1),X2) | (42) |
mark(U43(X)) | → | a__U43(mark(X)) | (43) |
mark(isNatIList(X)) | → | a__isNatIList(X) | (44) |
mark(U51(X1,X2,X3)) | → | a__U51(mark(X1),X2,X3) | (45) |
mark(U52(X1,X2)) | → | a__U52(mark(X1),X2) | (46) |
mark(U53(X)) | → | a__U53(mark(X)) | (47) |
mark(U61(X1,X2)) | → | a__U61(mark(X1),X2) | (48) |
mark(length(X)) | → | a__length(mark(X)) | (49) |
mark(and(X1,X2)) | → | a__and(mark(X1),X2) | (50) |
mark(isNatIListKind(X)) | → | a__isNatIListKind(X) | (51) |
mark(isNatKind(X)) | → | a__isNatKind(X) | (52) |
mark(cons(X1,X2)) | → | cons(mark(X1),X2) | (53) |
mark(0) | → | 0 | (54) |
mark(tt) | → | tt | (55) |
mark(s(X)) | → | s(mark(X)) | (56) |
mark(nil) | → | nil | (57) |
a__zeros | → | zeros | (58) |
a__U11(X1,X2) | → | U11(X1,X2) | (59) |
a__U12(X) | → | U12(X) | (60) |
a__isNatList(X) | → | isNatList(X) | (61) |
a__U21(X1,X2) | → | U21(X1,X2) | (62) |
a__U22(X) | → | U22(X) | (63) |
a__isNat(X) | → | isNat(X) | (64) |
a__U31(X1,X2) | → | U31(X1,X2) | (65) |
a__U32(X) | → | U32(X) | (66) |
a__U41(X1,X2,X3) | → | U41(X1,X2,X3) | (67) |
a__U42(X1,X2) | → | U42(X1,X2) | (68) |
a__U43(X) | → | U43(X) | (69) |
a__isNatIList(X) | → | isNatIList(X) | (70) |
a__U51(X1,X2,X3) | → | U51(X1,X2,X3) | (71) |
a__U52(X1,X2) | → | U52(X1,X2) | (72) |
a__U53(X) | → | U53(X) | (73) |
a__U61(X1,X2) | → | U61(X1,X2) | (74) |
a__length(X) | → | length(X) | (75) |
a__and(X1,X2) | → | and(X1,X2) | (76) |
a__isNatIListKind(X) | → | isNatIListKind(X) | (77) |
a__isNatKind(X) | → | isNatKind(X) | (78) |
a__U61#(tt,L) | → | a__length#(mark(L)) | (94) |
The dependency pairs are split into 0 components.
a__isNatIListKind#(cons(V1,V2)) | → | a__and#(a__isNatKind(V1),isNatIListKind(V2)) | (106) |
a__and#(tt,X) | → | mark#(X) | (95) |
mark#(U11(X1,X2)) | → | mark#(X1) | (117) |
mark#(U12(X)) | → | mark#(X) | (119) |
mark#(isNatList(X)) | → | a__isNatList#(X) | (121) |
a__isNatList#(cons(V1,V2)) | → | a__and#(a__isNatKind(V1),isNatIListKind(V2)) | (110) |
a__isNatList#(cons(V1,V2)) | → | a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) | (111) |
a__U51#(tt,V1,V2) | → | a__U52#(a__isNat(V1),V2) | (90) |
a__U52#(tt,V2) | → | a__isNatList#(V2) | (91) |
mark#(U21(X1,X2)) | → | mark#(X1) | (122) |
mark#(U22(X)) | → | mark#(X) | (124) |
mark#(U31(X1,X2)) | → | mark#(X1) | (127) |
mark#(U32(X)) | → | mark#(X) | (129) |
mark#(U41(X1,X2,X3)) | → | mark#(X1) | (131) |
mark#(U41(X1,X2,X3)) | → | a__U41#(mark(X1),X2,X3) | (132) |
a__U41#(tt,V1,V2) | → | a__U42#(a__isNat(V1),V2) | (86) |
a__U42#(tt,V2) | → | a__isNatIList#(V2) | (87) |
a__isNatIList#(V) | → | a__isNatIListKind#(V) | (100) |
a__isNatIList#(V) | → | a__U31#(a__isNatIListKind(V),V) | (101) |
a__U31#(tt,V) | → | a__isNatList#(V) | (83) |
a__isNatIList#(cons(V1,V2)) | → | a__and#(a__isNatKind(V1),isNatIListKind(V2)) | (103) |
a__isNatIList#(cons(V1,V2)) | → | a__U41#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) | (104) |
mark#(U42(X1,X2)) | → | mark#(X1) | (133) |
mark#(U43(X)) | → | mark#(X) | (135) |
mark#(U51(X1,X2,X3)) | → | mark#(X1) | (138) |
mark#(U51(X1,X2,X3)) | → | a__U51#(mark(X1),X2,X3) | (139) |
mark#(U52(X1,X2)) | → | mark#(X1) | (140) |
mark#(U52(X1,X2)) | → | a__U52#(mark(X1),X2) | (141) |
mark#(U53(X)) | → | mark#(X) | (142) |
mark#(U61(X1,X2)) | → | mark#(X1) | (144) |
mark#(and(X1,X2)) | → | mark#(X1) | (148) |
mark#(and(X1,X2)) | → | a__and#(mark(X1),X2) | (149) |
mark#(isNatIListKind(X)) | → | a__isNatIListKind#(X) | (150) |
mark#(cons(X1,X2)) | → | mark#(X1) | (152) |
mark#(s(X)) | → | mark#(X) | (153) |
π(a__isNatIListKind#) | = | { 1 } |
π(a__and#) | = | { 2 } |
π(mark#) | = | { 1 } |
π(a__U52#) | = | { 2 } |
π(a__U51#) | = | { 3 } |
π(a__isNatIList#) | = | { 1 } |
π(a__U42#) | = | { 2 } |
π(a__U41#) | = | { 3 } |
π(a__U31#) | = | { 2 } |
π(a__isNatList#) | = | { 1 } |
π(U42) | = | { 1 } |
π(U31) | = | { 1 } |
π(U21) | = | { 1 } |
π(isNatIListKind) | = | { 1 } |
a__isNatIListKind#(cons(V1,V2)) | → | a__and#(a__isNatKind(V1),isNatIListKind(V2)) | (106) |
mark#(U11(X1,X2)) | → | mark#(X1) | (117) |
mark#(U12(X)) | → | mark#(X) | (119) |
mark#(isNatList(X)) | → | a__isNatList#(X) | (121) |
a__isNatList#(cons(V1,V2)) | → | a__and#(a__isNatKind(V1),isNatIListKind(V2)) | (110) |
a__isNatList#(cons(V1,V2)) | → | a__U51#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) | (111) |
mark#(U22(X)) | → | mark#(X) | (124) |
mark#(U32(X)) | → | mark#(X) | (129) |
mark#(U41(X1,X2,X3)) | → | mark#(X1) | (131) |
mark#(U41(X1,X2,X3)) | → | a__U41#(mark(X1),X2,X3) | (132) |
a__isNatIList#(cons(V1,V2)) | → | a__and#(a__isNatKind(V1),isNatIListKind(V2)) | (103) |
a__isNatIList#(cons(V1,V2)) | → | a__U41#(a__and(a__isNatKind(V1),isNatIListKind(V2)),V1,V2) | (104) |
mark#(U43(X)) | → | mark#(X) | (135) |
mark#(U51(X1,X2,X3)) | → | mark#(X1) | (138) |
mark#(U51(X1,X2,X3)) | → | a__U51#(mark(X1),X2,X3) | (139) |
mark#(U52(X1,X2)) | → | mark#(X1) | (140) |
mark#(U52(X1,X2)) | → | a__U52#(mark(X1),X2) | (141) |
mark#(U53(X)) | → | mark#(X) | (142) |
mark#(U61(X1,X2)) | → | mark#(X1) | (144) |
mark#(and(X1,X2)) | → | mark#(X1) | (148) |
mark#(and(X1,X2)) | → | a__and#(mark(X1),X2) | (149) |
mark#(cons(X1,X2)) | → | mark#(X1) | (152) |
mark#(s(X)) | → | mark#(X) | (153) |
The dependency pairs are split into 1 component.
mark#(U21(X1,X2)) | → | mark#(X1) | (122) |
mark#(U31(X1,X2)) | → | mark#(X1) | (127) |
mark#(U42(X1,X2)) | → | mark#(X1) | (133) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
mark#(U21(X1,X2)) | → | mark#(X1) | (122) |
1 | > | 1 | |
mark#(U31(X1,X2)) | → | mark#(X1) | (127) |
1 | > | 1 | |
mark#(U42(X1,X2)) | → | mark#(X1) | (133) |
1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
a__isNat#(s(V1)) | → | a__U21#(a__isNatKind(V1),V1) | (99) |
a__U21#(tt,V1) | → | a__isNat#(V1) | (81) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
a__isNat#(s(V1)) | → | a__U21#(a__isNatKind(V1),V1) | (99) |
1 | > | 2 | |
a__U21#(tt,V1) | → | a__isNat#(V1) | (81) |
2 | ≥ | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
a__isNatKind#(s(V1)) | → | a__isNatKind#(V1) | (108) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
a__isNatKind#(s(V1)) | → | a__isNatKind#(V1) | (108) |
1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.