The rewrite relation of the following TRS is considered.
a__U11(tt,V1,V2) | → | a__U12(a__isNatKind(V1),V1,V2) | (1) |
a__U12(tt,V1,V2) | → | a__U13(a__isNatKind(V2),V1,V2) | (2) |
a__U13(tt,V1,V2) | → | a__U14(a__isNatKind(V2),V1,V2) | (3) |
a__U14(tt,V1,V2) | → | a__U15(a__isNat(V1),V2) | (4) |
a__U15(tt,V2) | → | a__U16(a__isNat(V2)) | (5) |
a__U16(tt) | → | tt | (6) |
a__U21(tt,V1) | → | a__U22(a__isNatKind(V1),V1) | (7) |
a__U22(tt,V1) | → | a__U23(a__isNat(V1)) | (8) |
a__U23(tt) | → | tt | (9) |
a__U31(tt,V2) | → | a__U32(a__isNatKind(V2)) | (10) |
a__U32(tt) | → | tt | (11) |
a__U41(tt) | → | tt | (12) |
a__U51(tt,N) | → | a__U52(a__isNatKind(N),N) | (13) |
a__U52(tt,N) | → | mark(N) | (14) |
a__U61(tt,M,N) | → | a__U62(a__isNatKind(M),M,N) | (15) |
a__U62(tt,M,N) | → | a__U63(a__isNat(N),M,N) | (16) |
a__U63(tt,M,N) | → | a__U64(a__isNatKind(N),M,N) | (17) |
a__U64(tt,M,N) | → | s(a__plus(mark(N),mark(M))) | (18) |
a__isNat(0) | → | tt | (19) |
a__isNat(plus(V1,V2)) | → | a__U11(a__isNatKind(V1),V1,V2) | (20) |
a__isNat(s(V1)) | → | a__U21(a__isNatKind(V1),V1) | (21) |
a__isNatKind(0) | → | tt | (22) |
a__isNatKind(plus(V1,V2)) | → | a__U31(a__isNatKind(V1),V2) | (23) |
a__isNatKind(s(V1)) | → | a__U41(a__isNatKind(V1)) | (24) |
a__plus(N,0) | → | a__U51(a__isNat(N),N) | (25) |
a__plus(N,s(M)) | → | a__U61(a__isNat(M),M,N) | (26) |
mark(U11(X1,X2,X3)) | → | a__U11(mark(X1),X2,X3) | (27) |
mark(U12(X1,X2,X3)) | → | a__U12(mark(X1),X2,X3) | (28) |
mark(isNatKind(X)) | → | a__isNatKind(X) | (29) |
mark(U13(X1,X2,X3)) | → | a__U13(mark(X1),X2,X3) | (30) |
mark(U14(X1,X2,X3)) | → | a__U14(mark(X1),X2,X3) | (31) |
mark(U15(X1,X2)) | → | a__U15(mark(X1),X2) | (32) |
mark(isNat(X)) | → | a__isNat(X) | (33) |
mark(U16(X)) | → | a__U16(mark(X)) | (34) |
mark(U21(X1,X2)) | → | a__U21(mark(X1),X2) | (35) |
mark(U22(X1,X2)) | → | a__U22(mark(X1),X2) | (36) |
mark(U23(X)) | → | a__U23(mark(X)) | (37) |
mark(U31(X1,X2)) | → | a__U31(mark(X1),X2) | (38) |
mark(U32(X)) | → | a__U32(mark(X)) | (39) |
mark(U41(X)) | → | a__U41(mark(X)) | (40) |
mark(U51(X1,X2)) | → | a__U51(mark(X1),X2) | (41) |
mark(U52(X1,X2)) | → | a__U52(mark(X1),X2) | (42) |
mark(U61(X1,X2,X3)) | → | a__U61(mark(X1),X2,X3) | (43) |
mark(U62(X1,X2,X3)) | → | a__U62(mark(X1),X2,X3) | (44) |
mark(U63(X1,X2,X3)) | → | a__U63(mark(X1),X2,X3) | (45) |
mark(U64(X1,X2,X3)) | → | a__U64(mark(X1),X2,X3) | (46) |
mark(plus(X1,X2)) | → | a__plus(mark(X1),mark(X2)) | (47) |
mark(tt) | → | tt | (48) |
mark(s(X)) | → | s(mark(X)) | (49) |
mark(0) | → | 0 | (50) |
a__U11(X1,X2,X3) | → | U11(X1,X2,X3) | (51) |
a__U12(X1,X2,X3) | → | U12(X1,X2,X3) | (52) |
a__isNatKind(X) | → | isNatKind(X) | (53) |
a__U13(X1,X2,X3) | → | U13(X1,X2,X3) | (54) |
a__U14(X1,X2,X3) | → | U14(X1,X2,X3) | (55) |
a__U15(X1,X2) | → | U15(X1,X2) | (56) |
a__isNat(X) | → | isNat(X) | (57) |
a__U16(X) | → | U16(X) | (58) |
a__U21(X1,X2) | → | U21(X1,X2) | (59) |
a__U22(X1,X2) | → | U22(X1,X2) | (60) |
a__U23(X) | → | U23(X) | (61) |
a__U31(X1,X2) | → | U31(X1,X2) | (62) |
a__U32(X) | → | U32(X) | (63) |
a__U41(X) | → | U41(X) | (64) |
a__U51(X1,X2) | → | U51(X1,X2) | (65) |
a__U52(X1,X2) | → | U52(X1,X2) | (66) |
a__U61(X1,X2,X3) | → | U61(X1,X2,X3) | (67) |
a__U62(X1,X2,X3) | → | U62(X1,X2,X3) | (68) |
a__U63(X1,X2,X3) | → | U63(X1,X2,X3) | (69) |
a__U64(X1,X2,X3) | → | U64(X1,X2,X3) | (70) |
a__plus(X1,X2) | → | plus(X1,X2) | (71) |
a__U11#(tt,V1,V2) | → | a__U12#(a__isNatKind(V1),V1,V2) | (72) |
a__U11#(tt,V1,V2) | → | a__isNatKind#(V1) | (73) |
a__U12#(tt,V1,V2) | → | a__U13#(a__isNatKind(V2),V1,V2) | (74) |
a__U12#(tt,V1,V2) | → | a__isNatKind#(V2) | (75) |
a__U13#(tt,V1,V2) | → | a__U14#(a__isNatKind(V2),V1,V2) | (76) |
a__U13#(tt,V1,V2) | → | a__isNatKind#(V2) | (77) |
a__U14#(tt,V1,V2) | → | a__U15#(a__isNat(V1),V2) | (78) |
a__U14#(tt,V1,V2) | → | a__isNat#(V1) | (79) |
a__U15#(tt,V2) | → | a__U16#(a__isNat(V2)) | (80) |
a__U15#(tt,V2) | → | a__isNat#(V2) | (81) |
a__U21#(tt,V1) | → | a__U22#(a__isNatKind(V1),V1) | (82) |
a__U21#(tt,V1) | → | a__isNatKind#(V1) | (83) |
a__U22#(tt,V1) | → | a__U23#(a__isNat(V1)) | (84) |
a__U22#(tt,V1) | → | a__isNat#(V1) | (85) |
a__U31#(tt,V2) | → | a__U32#(a__isNatKind(V2)) | (86) |
a__U31#(tt,V2) | → | a__isNatKind#(V2) | (87) |
a__U51#(tt,N) | → | a__U52#(a__isNatKind(N),N) | (88) |
a__U51#(tt,N) | → | a__isNatKind#(N) | (89) |
a__U52#(tt,N) | → | mark#(N) | (90) |
a__U61#(tt,M,N) | → | a__U62#(a__isNatKind(M),M,N) | (91) |
a__U61#(tt,M,N) | → | a__isNatKind#(M) | (92) |
a__U62#(tt,M,N) | → | a__U63#(a__isNat(N),M,N) | (93) |
a__U62#(tt,M,N) | → | a__isNat#(N) | (94) |
a__U63#(tt,M,N) | → | a__U64#(a__isNatKind(N),M,N) | (95) |
a__U63#(tt,M,N) | → | a__isNatKind#(N) | (96) |
a__U64#(tt,M,N) | → | a__plus#(mark(N),mark(M)) | (97) |
a__U64#(tt,M,N) | → | mark#(N) | (98) |
a__U64#(tt,M,N) | → | mark#(M) | (99) |
a__isNat#(plus(V1,V2)) | → | a__U11#(a__isNatKind(V1),V1,V2) | (100) |
a__isNat#(plus(V1,V2)) | → | a__isNatKind#(V1) | (101) |
a__isNat#(s(V1)) | → | a__U21#(a__isNatKind(V1),V1) | (102) |
a__isNat#(s(V1)) | → | a__isNatKind#(V1) | (103) |
a__isNatKind#(plus(V1,V2)) | → | a__U31#(a__isNatKind(V1),V2) | (104) |
a__isNatKind#(plus(V1,V2)) | → | a__isNatKind#(V1) | (105) |
a__isNatKind#(s(V1)) | → | a__U41#(a__isNatKind(V1)) | (106) |
a__isNatKind#(s(V1)) | → | a__isNatKind#(V1) | (107) |
a__plus#(N,0) | → | a__U51#(a__isNat(N),N) | (108) |
a__plus#(N,0) | → | a__isNat#(N) | (109) |
a__plus#(N,s(M)) | → | a__U61#(a__isNat(M),M,N) | (110) |
a__plus#(N,s(M)) | → | a__isNat#(M) | (111) |
mark#(U11(X1,X2,X3)) | → | a__U11#(mark(X1),X2,X3) | (112) |
mark#(U11(X1,X2,X3)) | → | mark#(X1) | (113) |
mark#(U12(X1,X2,X3)) | → | a__U12#(mark(X1),X2,X3) | (114) |
mark#(U12(X1,X2,X3)) | → | mark#(X1) | (115) |
mark#(isNatKind(X)) | → | a__isNatKind#(X) | (116) |
mark#(U13(X1,X2,X3)) | → | a__U13#(mark(X1),X2,X3) | (117) |
mark#(U13(X1,X2,X3)) | → | mark#(X1) | (118) |
mark#(U14(X1,X2,X3)) | → | a__U14#(mark(X1),X2,X3) | (119) |
mark#(U14(X1,X2,X3)) | → | mark#(X1) | (120) |
mark#(U15(X1,X2)) | → | a__U15#(mark(X1),X2) | (121) |
mark#(U15(X1,X2)) | → | mark#(X1) | (122) |
mark#(isNat(X)) | → | a__isNat#(X) | (123) |
mark#(U16(X)) | → | a__U16#(mark(X)) | (124) |
mark#(U16(X)) | → | mark#(X) | (125) |
mark#(U21(X1,X2)) | → | a__U21#(mark(X1),X2) | (126) |
mark#(U21(X1,X2)) | → | mark#(X1) | (127) |
mark#(U22(X1,X2)) | → | a__U22#(mark(X1),X2) | (128) |
mark#(U22(X1,X2)) | → | mark#(X1) | (129) |
mark#(U23(X)) | → | a__U23#(mark(X)) | (130) |
mark#(U23(X)) | → | mark#(X) | (131) |
mark#(U31(X1,X2)) | → | a__U31#(mark(X1),X2) | (132) |
mark#(U31(X1,X2)) | → | mark#(X1) | (133) |
mark#(U32(X)) | → | a__U32#(mark(X)) | (134) |
mark#(U32(X)) | → | mark#(X) | (135) |
mark#(U41(X)) | → | a__U41#(mark(X)) | (136) |
mark#(U41(X)) | → | mark#(X) | (137) |
mark#(U51(X1,X2)) | → | a__U51#(mark(X1),X2) | (138) |
mark#(U51(X1,X2)) | → | mark#(X1) | (139) |
mark#(U52(X1,X2)) | → | a__U52#(mark(X1),X2) | (140) |
mark#(U52(X1,X2)) | → | mark#(X1) | (141) |
mark#(U61(X1,X2,X3)) | → | a__U61#(mark(X1),X2,X3) | (142) |
mark#(U61(X1,X2,X3)) | → | mark#(X1) | (143) |
mark#(U62(X1,X2,X3)) | → | a__U62#(mark(X1),X2,X3) | (144) |
mark#(U62(X1,X2,X3)) | → | mark#(X1) | (145) |
mark#(U63(X1,X2,X3)) | → | a__U63#(mark(X1),X2,X3) | (146) |
mark#(U63(X1,X2,X3)) | → | mark#(X1) | (147) |
mark#(U64(X1,X2,X3)) | → | a__U64#(mark(X1),X2,X3) | (148) |
mark#(U64(X1,X2,X3)) | → | mark#(X1) | (149) |
mark#(plus(X1,X2)) | → | a__plus#(mark(X1),mark(X2)) | (150) |
mark#(plus(X1,X2)) | → | mark#(X1) | (151) |
mark#(plus(X1,X2)) | → | mark#(X2) | (152) |
mark#(s(X)) | → | mark#(X) | (153) |
The dependency pairs are split into 3 components.
mark#(U11(X1,X2,X3)) | → | mark#(X1) | (113) |
mark#(U12(X1,X2,X3)) | → | mark#(X1) | (115) |
mark#(U13(X1,X2,X3)) | → | mark#(X1) | (118) |
mark#(U14(X1,X2,X3)) | → | mark#(X1) | (120) |
mark#(U15(X1,X2)) | → | mark#(X1) | (122) |
mark#(U16(X)) | → | mark#(X) | (125) |
mark#(U21(X1,X2)) | → | mark#(X1) | (127) |
mark#(U22(X1,X2)) | → | mark#(X1) | (129) |
mark#(U23(X)) | → | mark#(X) | (131) |
mark#(U31(X1,X2)) | → | mark#(X1) | (133) |
mark#(U32(X)) | → | mark#(X) | (135) |
mark#(U41(X)) | → | mark#(X) | (137) |
mark#(U51(X1,X2)) | → | a__U51#(mark(X1),X2) | (138) |
a__U51#(tt,N) | → | a__U52#(a__isNatKind(N),N) | (88) |
a__U52#(tt,N) | → | mark#(N) | (90) |
mark#(U51(X1,X2)) | → | mark#(X1) | (139) |
mark#(U52(X1,X2)) | → | a__U52#(mark(X1),X2) | (140) |
mark#(U52(X1,X2)) | → | mark#(X1) | (141) |
mark#(U61(X1,X2,X3)) | → | a__U61#(mark(X1),X2,X3) | (142) |
a__U61#(tt,M,N) | → | a__U62#(a__isNatKind(M),M,N) | (91) |
a__U62#(tt,M,N) | → | a__U63#(a__isNat(N),M,N) | (93) |
a__U63#(tt,M,N) | → | a__U64#(a__isNatKind(N),M,N) | (95) |
a__U64#(tt,M,N) | → | a__plus#(mark(N),mark(M)) | (97) |
a__plus#(N,0) | → | a__U51#(a__isNat(N),N) | (108) |
a__plus#(N,s(M)) | → | a__U61#(a__isNat(M),M,N) | (110) |
a__U64#(tt,M,N) | → | mark#(N) | (98) |
mark#(U61(X1,X2,X3)) | → | mark#(X1) | (143) |
mark#(U62(X1,X2,X3)) | → | a__U62#(mark(X1),X2,X3) | (144) |
mark#(U62(X1,X2,X3)) | → | mark#(X1) | (145) |
mark#(U63(X1,X2,X3)) | → | a__U63#(mark(X1),X2,X3) | (146) |
mark#(U63(X1,X2,X3)) | → | mark#(X1) | (147) |
mark#(U64(X1,X2,X3)) | → | a__U64#(mark(X1),X2,X3) | (148) |
a__U64#(tt,M,N) | → | mark#(M) | (99) |
mark#(U64(X1,X2,X3)) | → | mark#(X1) | (149) |
mark#(plus(X1,X2)) | → | a__plus#(mark(X1),mark(X2)) | (150) |
mark#(plus(X1,X2)) | → | mark#(X1) | (151) |
mark#(plus(X1,X2)) | → | mark#(X2) | (152) |
mark#(s(X)) | → | mark#(X) | (153) |
[a__plus#(x1, x2)] | = | 2 · x1 + 2 · x2 |
[a__U51#(x1, x2)] | = | 1 + 2 · x2 |
[a__U52#(x1, x2)] | = | 1 + 2 · x2 |
[a__U61#(x1, x2, x3)] | = | 1 + 2 · x2 + 2 · x3 |
[a__U62#(x1, x2, x3)] | = | 1 + 2 · x2 + 2 · x3 |
[a__U63#(x1, x2, x3)] | = | 1 + 2 · x2 + 2 · x3 |
[a__U64#(x1, x2, x3)] | = | 2 · x2 + 2 · x3 |
[mark(x1)] | = | x1 |
[U11(x1, x2, x3)] | = | 2 · x1 |
[a__U11(x1, x2, x3)] | = | 2 · x1 |
[U12(x1, x2, x3)] | = | 2 · x1 |
[a__U12(x1, x2, x3)] | = | 2 · x1 |
[isNatKind(x1)] | = | 0 |
[a__isNatKind(x1)] | = | 0 |
[U13(x1, x2, x3)] | = | x1 |
[a__U13(x1, x2, x3)] | = | x1 |
[U14(x1, x2, x3)] | = | x1 |
[a__U14(x1, x2, x3)] | = | x1 |
[U15(x1, x2)] | = | 2 · x1 |
[a__U15(x1, x2)] | = | 2 · x1 |
[isNat(x1)] | = | 0 |
[a__isNat(x1)] | = | 0 |
[U16(x1)] | = | 2 · x1 |
[a__U16(x1)] | = | 2 · x1 |
[U21(x1, x2)] | = | x1 |
[a__U21(x1, x2)] | = | x1 |
[U22(x1, x2)] | = | 2 · x1 |
[a__U22(x1, x2)] | = | 2 · x1 |
[U23(x1)] | = | x1 |
[a__U23(x1)] | = | x1 |
[U31(x1, x2)] | = | x1 |
[a__U31(x1, x2)] | = | x1 |
[U32(x1)] | = | x1 |
[a__U32(x1)] | = | x1 |
[U41(x1)] | = | x1 |
[a__U41(x1)] | = | x1 |
[U51(x1, x2)] | = | 1 + x1 + x2 |
[a__U51(x1, x2)] | = | 1 + x1 + x2 |
[tt] | = | 0 |
[a__U52(x1, x2)] | = | 1 + x1 + x2 |
[U52(x1, x2)] | = | 1 + x1 + x2 |
[plus(x1, x2)] | = | x1 + 2 · x2 |
[a__plus(x1, x2)] | = | x1 + 2 · x2 |
[0] | = | 1 |
[U61(x1, x2, x3)] | = | 1 + x1 + 2 · x2 + x3 |
[a__U61(x1, x2, x3)] | = | 1 + x1 + 2 · x2 + x3 |
[U62(x1, x2, x3)] | = | 1 + x1 + 2 · x2 + x3 |
[a__U62(x1, x2, x3)] | = | 1 + x1 + 2 · x2 + x3 |
[U63(x1, x2, x3)] | = | 1 + x1 + 2 · x2 + x3 |
[a__U63(x1, x2, x3)] | = | 1 + x1 + 2 · x2 + x3 |
[U64(x1, x2, x3)] | = | 1 + 2 · x1 + 2 · x2 + x3 |
[a__U64(x1, x2, x3)] | = | 1 + 2 · x1 + 2 · x2 + x3 |
[s(x1)] | = | 1 + x1 |
[mark#(x1)] | = | 2 · x1 |
mark#(U51(X1,X2)) | → | a__U51#(mark(X1),X2) | (138) |
a__U52#(tt,N) | → | mark#(N) | (90) |
mark#(U51(X1,X2)) | → | mark#(X1) | (139) |
mark#(U52(X1,X2)) | → | a__U52#(mark(X1),X2) | (140) |
mark#(U52(X1,X2)) | → | mark#(X1) | (141) |
mark#(U61(X1,X2,X3)) | → | a__U61#(mark(X1),X2,X3) | (142) |
a__U63#(tt,M,N) | → | a__U64#(a__isNatKind(N),M,N) | (95) |
a__plus#(N,0) | → | a__U51#(a__isNat(N),N) | (108) |
a__plus#(N,s(M)) | → | a__U61#(a__isNat(M),M,N) | (110) |
mark#(U61(X1,X2,X3)) | → | mark#(X1) | (143) |
mark#(U62(X1,X2,X3)) | → | a__U62#(mark(X1),X2,X3) | (144) |
mark#(U62(X1,X2,X3)) | → | mark#(X1) | (145) |
mark#(U63(X1,X2,X3)) | → | a__U63#(mark(X1),X2,X3) | (146) |
mark#(U63(X1,X2,X3)) | → | mark#(X1) | (147) |
mark#(U64(X1,X2,X3)) | → | a__U64#(mark(X1),X2,X3) | (148) |
mark#(U64(X1,X2,X3)) | → | mark#(X1) | (149) |
mark#(s(X)) | → | mark#(X) | (153) |
The dependency pairs are split into 1 component.
mark#(U12(X1,X2,X3)) | → | mark#(X1) | (115) |
mark#(U11(X1,X2,X3)) | → | mark#(X1) | (113) |
mark#(U13(X1,X2,X3)) | → | mark#(X1) | (118) |
mark#(U14(X1,X2,X3)) | → | mark#(X1) | (120) |
mark#(U15(X1,X2)) | → | mark#(X1) | (122) |
mark#(U16(X)) | → | mark#(X) | (125) |
mark#(U21(X1,X2)) | → | mark#(X1) | (127) |
mark#(U22(X1,X2)) | → | mark#(X1) | (129) |
mark#(U23(X)) | → | mark#(X) | (131) |
mark#(U31(X1,X2)) | → | mark#(X1) | (133) |
mark#(U32(X)) | → | mark#(X) | (135) |
mark#(U41(X)) | → | mark#(X) | (137) |
mark#(plus(X1,X2)) | → | mark#(X1) | (151) |
mark#(plus(X1,X2)) | → | mark#(X2) | (152) |
We restrict the rewrite rules to the following usable rules of the DP problem.
There are no rules.
We restrict the innermost strategy to the following left hand sides.
There are no lhss.
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
mark#(U12(X1,X2,X3)) | → | mark#(X1) | (115) |
1 | > | 1 | |
mark#(U11(X1,X2,X3)) | → | mark#(X1) | (113) |
1 | > | 1 | |
mark#(U13(X1,X2,X3)) | → | mark#(X1) | (118) |
1 | > | 1 | |
mark#(U14(X1,X2,X3)) | → | mark#(X1) | (120) |
1 | > | 1 | |
mark#(U15(X1,X2)) | → | mark#(X1) | (122) |
1 | > | 1 | |
mark#(U16(X)) | → | mark#(X) | (125) |
1 | > | 1 | |
mark#(U21(X1,X2)) | → | mark#(X1) | (127) |
1 | > | 1 | |
mark#(U22(X1,X2)) | → | mark#(X1) | (129) |
1 | > | 1 | |
mark#(U23(X)) | → | mark#(X) | (131) |
1 | > | 1 | |
mark#(U31(X1,X2)) | → | mark#(X1) | (133) |
1 | > | 1 | |
mark#(U32(X)) | → | mark#(X) | (135) |
1 | > | 1 | |
mark#(U41(X)) | → | mark#(X) | (137) |
1 | > | 1 | |
mark#(plus(X1,X2)) | → | mark#(X1) | (151) |
1 | > | 1 | |
mark#(plus(X1,X2)) | → | mark#(X2) | (152) |
1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
a__U12#(tt,V1,V2) | → | a__U13#(a__isNatKind(V2),V1,V2) | (74) |
a__U13#(tt,V1,V2) | → | a__U14#(a__isNatKind(V2),V1,V2) | (76) |
a__U14#(tt,V1,V2) | → | a__U15#(a__isNat(V1),V2) | (78) |
a__U15#(tt,V2) | → | a__isNat#(V2) | (81) |
a__isNat#(plus(V1,V2)) | → | a__U11#(a__isNatKind(V1),V1,V2) | (100) |
a__U11#(tt,V1,V2) | → | a__U12#(a__isNatKind(V1),V1,V2) | (72) |
a__isNat#(s(V1)) | → | a__U21#(a__isNatKind(V1),V1) | (102) |
a__U21#(tt,V1) | → | a__U22#(a__isNatKind(V1),V1) | (82) |
a__U22#(tt,V1) | → | a__isNat#(V1) | (85) |
a__U14#(tt,V1,V2) | → | a__isNat#(V1) | (79) |
We restrict the rewrite rules to the following usable rules of the DP problem.
a__isNatKind(0) | → | tt | (22) |
a__isNatKind(plus(V1,V2)) | → | a__U31(a__isNatKind(V1),V2) | (23) |
a__isNatKind(s(V1)) | → | a__U41(a__isNatKind(V1)) | (24) |
a__isNatKind(X) | → | isNatKind(X) | (53) |
a__U41(tt) | → | tt | (12) |
a__U41(X) | → | U41(X) | (64) |
a__U31(tt,V2) | → | a__U32(a__isNatKind(V2)) | (10) |
a__U31(X1,X2) | → | U31(X1,X2) | (62) |
a__U32(tt) | → | tt | (11) |
a__U32(X) | → | U32(X) | (63) |
a__isNat(0) | → | tt | (19) |
a__isNat(plus(V1,V2)) | → | a__U11(a__isNatKind(V1),V1,V2) | (20) |
a__isNat(s(V1)) | → | a__U21(a__isNatKind(V1),V1) | (21) |
a__isNat(X) | → | isNat(X) | (57) |
a__U21(tt,V1) | → | a__U22(a__isNatKind(V1),V1) | (7) |
a__U21(X1,X2) | → | U21(X1,X2) | (59) |
a__U22(tt,V1) | → | a__U23(a__isNat(V1)) | (8) |
a__U22(X1,X2) | → | U22(X1,X2) | (60) |
a__U23(tt) | → | tt | (9) |
a__U23(X) | → | U23(X) | (61) |
a__U11(tt,V1,V2) | → | a__U12(a__isNatKind(V1),V1,V2) | (1) |
a__U11(X1,X2,X3) | → | U11(X1,X2,X3) | (51) |
a__U12(tt,V1,V2) | → | a__U13(a__isNatKind(V2),V1,V2) | (2) |
a__U12(X1,X2,X3) | → | U12(X1,X2,X3) | (52) |
a__U13(tt,V1,V2) | → | a__U14(a__isNatKind(V2),V1,V2) | (3) |
a__U13(X1,X2,X3) | → | U13(X1,X2,X3) | (54) |
a__U14(tt,V1,V2) | → | a__U15(a__isNat(V1),V2) | (4) |
a__U14(X1,X2,X3) | → | U14(X1,X2,X3) | (55) |
a__U15(tt,V2) | → | a__U16(a__isNat(V2)) | (5) |
a__U15(X1,X2) | → | U15(X1,X2) | (56) |
a__U16(tt) | → | tt | (6) |
a__U16(X) | → | U16(X) | (58) |
We restrict the innermost strategy to the following left hand sides.
a__U11(x0,x1,x2) |
a__U12(x0,x1,x2) |
a__isNatKind(x0) |
a__U13(x0,x1,x2) |
a__U14(x0,x1,x2) |
a__U15(x0,x1) |
a__isNat(x0) |
a__U16(x0) |
a__U21(x0,x1) |
a__U22(x0,x1) |
a__U23(x0) |
a__U31(x0,x1) |
a__U32(x0) |
a__U41(x0) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
a__U13#(tt,V1,V2) | → | a__U14#(a__isNatKind(V2),V1,V2) | (76) |
2 | ≥ | 2 | |
3 | ≥ | 3 | |
a__U11#(tt,V1,V2) | → | a__U12#(a__isNatKind(V1),V1,V2) | (72) |
2 | ≥ | 2 | |
3 | ≥ | 3 | |
a__U12#(tt,V1,V2) | → | a__U13#(a__isNatKind(V2),V1,V2) | (74) |
2 | ≥ | 2 | |
3 | ≥ | 3 | |
a__U15#(tt,V2) | → | a__isNat#(V2) | (81) |
2 | ≥ | 1 | |
a__U14#(tt,V1,V2) | → | a__U15#(a__isNat(V1),V2) | (78) |
3 | ≥ | 2 | |
a__U14#(tt,V1,V2) | → | a__isNat#(V1) | (79) |
2 | ≥ | 1 | |
a__U22#(tt,V1) | → | a__isNat#(V1) | (85) |
2 | ≥ | 1 | |
a__U21#(tt,V1) | → | a__U22#(a__isNatKind(V1),V1) | (82) |
2 | ≥ | 2 | |
a__isNat#(plus(V1,V2)) | → | a__U11#(a__isNatKind(V1),V1,V2) | (100) |
1 | > | 2 | |
1 | > | 3 | |
a__isNat#(s(V1)) | → | a__U21#(a__isNatKind(V1),V1) | (102) |
1 | > | 2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
a__U31#(tt,V2) | → | a__isNatKind#(V2) | (87) |
a__isNatKind#(plus(V1,V2)) | → | a__U31#(a__isNatKind(V1),V2) | (104) |
a__isNatKind#(plus(V1,V2)) | → | a__isNatKind#(V1) | (105) |
a__isNatKind#(s(V1)) | → | a__isNatKind#(V1) | (107) |
We restrict the rewrite rules to the following usable rules of the DP problem.
a__isNatKind(0) | → | tt | (22) |
a__isNatKind(plus(V1,V2)) | → | a__U31(a__isNatKind(V1),V2) | (23) |
a__isNatKind(s(V1)) | → | a__U41(a__isNatKind(V1)) | (24) |
a__isNatKind(X) | → | isNatKind(X) | (53) |
a__U41(tt) | → | tt | (12) |
a__U41(X) | → | U41(X) | (64) |
a__U31(tt,V2) | → | a__U32(a__isNatKind(V2)) | (10) |
a__U31(X1,X2) | → | U31(X1,X2) | (62) |
a__U32(tt) | → | tt | (11) |
a__U32(X) | → | U32(X) | (63) |
We restrict the innermost strategy to the following left hand sides.
a__isNatKind(x0) |
a__U31(x0,x1) |
a__U32(x0) |
a__U41(x0) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
a__isNatKind#(plus(V1,V2)) | → | a__U31#(a__isNatKind(V1),V2) | (104) |
1 | > | 2 | |
a__U31#(tt,V2) | → | a__isNatKind#(V2) | (87) |
2 | ≥ | 1 | |
a__isNatKind#(plus(V1,V2)) | → | a__isNatKind#(V1) | (105) |
1 | > | 1 | |
a__isNatKind#(s(V1)) | → | a__isNatKind#(V1) | (107) |
1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.