The rewrite relation of the following TRS is considered.
a__U11(tt,V1,V2) | → | a__U12(a__isNat(V1),V2) | (1) |
a__U12(tt,V2) | → | a__U13(a__isNat(V2)) | (2) |
a__U13(tt) | → | tt | (3) |
a__U21(tt,V1) | → | a__U22(a__isNat(V1)) | (4) |
a__U22(tt) | → | tt | (5) |
a__U31(tt,N) | → | mark(N) | (6) |
a__U41(tt,M,N) | → | s(a__plus(mark(N),mark(M))) | (7) |
a__and(tt,X) | → | mark(X) | (8) |
a__isNat(0) | → | tt | (9) |
a__isNat(plus(V1,V2)) | → | a__U11(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2) | (10) |
a__isNat(s(V1)) | → | a__U21(a__isNatKind(V1),V1) | (11) |
a__isNatKind(0) | → | tt | (12) |
a__isNatKind(plus(V1,V2)) | → | a__and(a__isNatKind(V1),isNatKind(V2)) | (13) |
a__isNatKind(s(V1)) | → | a__isNatKind(V1) | (14) |
a__plus(N,0) | → | a__U31(a__and(a__isNat(N),isNatKind(N)),N) | (15) |
a__plus(N,s(M)) | → | a__U41(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N) | (16) |
mark(U11(X1,X2,X3)) | → | a__U11(mark(X1),X2,X3) | (17) |
mark(U12(X1,X2)) | → | a__U12(mark(X1),X2) | (18) |
mark(isNat(X)) | → | a__isNat(X) | (19) |
mark(U13(X)) | → | a__U13(mark(X)) | (20) |
mark(U21(X1,X2)) | → | a__U21(mark(X1),X2) | (21) |
mark(U22(X)) | → | a__U22(mark(X)) | (22) |
mark(U31(X1,X2)) | → | a__U31(mark(X1),X2) | (23) |
mark(U41(X1,X2,X3)) | → | a__U41(mark(X1),X2,X3) | (24) |
mark(plus(X1,X2)) | → | a__plus(mark(X1),mark(X2)) | (25) |
mark(and(X1,X2)) | → | a__and(mark(X1),X2) | (26) |
mark(isNatKind(X)) | → | a__isNatKind(X) | (27) |
mark(tt) | → | tt | (28) |
mark(s(X)) | → | s(mark(X)) | (29) |
mark(0) | → | 0 | (30) |
a__U11(X1,X2,X3) | → | U11(X1,X2,X3) | (31) |
a__U12(X1,X2) | → | U12(X1,X2) | (32) |
a__isNat(X) | → | isNat(X) | (33) |
a__U13(X) | → | U13(X) | (34) |
a__U21(X1,X2) | → | U21(X1,X2) | (35) |
a__U22(X) | → | U22(X) | (36) |
a__U31(X1,X2) | → | U31(X1,X2) | (37) |
a__U41(X1,X2,X3) | → | U41(X1,X2,X3) | (38) |
a__plus(X1,X2) | → | plus(X1,X2) | (39) |
a__and(X1,X2) | → | and(X1,X2) | (40) |
a__isNatKind(X) | → | isNatKind(X) | (41) |
a__U12#(tt,V2) | → | a__isNat#(V2) | (42) |
a__plus#(N,0) | → | a__and#(a__isNat(N),isNatKind(N)) | (43) |
mark#(U11(X1,X2,X3)) | → | mark#(X1) | (44) |
a__plus#(N,s(M)) | → | a__U41#(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N) | (45) |
a__U21#(tt,V1) | → | a__isNat#(V1) | (46) |
a__plus#(N,0) | → | a__U31#(a__and(a__isNat(N),isNatKind(N)),N) | (47) |
a__and#(tt,X) | → | mark#(X) | (48) |
a__plus#(N,s(M)) | → | a__and#(a__isNat(M),isNatKind(M)) | (49) |
mark#(U12(X1,X2)) | → | a__U12#(mark(X1),X2) | (50) |
mark#(U12(X1,X2)) | → | mark#(X1) | (51) |
a__plus#(N,s(M)) | → | a__isNat#(M) | (52) |
a__isNatKind#(plus(V1,V2)) | → | a__isNatKind#(V1) | (53) |
a__U11#(tt,V1,V2) | → | a__isNat#(V1) | (54) |
a__isNatKind#(s(V1)) | → | a__isNatKind#(V1) | (55) |
mark#(U41(X1,X2,X3)) | → | a__U41#(mark(X1),X2,X3) | (56) |
mark#(U31(X1,X2)) | → | a__U31#(mark(X1),X2) | (57) |
a__isNat#(plus(V1,V2)) | → | a__isNatKind#(V1) | (58) |
a__isNat#(plus(V1,V2)) | → | a__and#(a__isNatKind(V1),isNatKind(V2)) | (59) |
a__U21#(tt,V1) | → | a__U22#(a__isNat(V1)) | (60) |
mark#(U31(X1,X2)) | → | mark#(X1) | (61) |
mark#(and(X1,X2)) | → | a__and#(mark(X1),X2) | (62) |
mark#(plus(X1,X2)) | → | a__plus#(mark(X1),mark(X2)) | (63) |
mark#(isNat(X)) | → | a__isNat#(X) | (64) |
mark#(U22(X)) | → | a__U22#(mark(X)) | (65) |
a__U41#(tt,M,N) | → | mark#(N) | (66) |
a__isNat#(s(V1)) | → | a__U21#(a__isNatKind(V1),V1) | (67) |
a__U11#(tt,V1,V2) | → | a__U12#(a__isNat(V1),V2) | (68) |
mark#(U41(X1,X2,X3)) | → | mark#(X1) | (69) |
mark#(U21(X1,X2)) | → | a__U21#(mark(X1),X2) | (70) |
a__isNatKind#(plus(V1,V2)) | → | a__and#(a__isNatKind(V1),isNatKind(V2)) | (71) |
a__plus#(N,0) | → | a__isNat#(N) | (72) |
mark#(U11(X1,X2,X3)) | → | a__U11#(mark(X1),X2,X3) | (73) |
a__isNat#(plus(V1,V2)) | → | a__U11#(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2) | (74) |
mark#(U21(X1,X2)) | → | mark#(X1) | (75) |
mark#(isNatKind(X)) | → | a__isNatKind#(X) | (76) |
mark#(U13(X)) | → | a__U13#(mark(X)) | (77) |
mark#(and(X1,X2)) | → | mark#(X1) | (78) |
a__U41#(tt,M,N) | → | a__plus#(mark(N),mark(M)) | (79) |
mark#(U22(X)) | → | mark#(X) | (80) |
a__plus#(N,s(M)) | → | a__and#(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))) | (81) |
a__U41#(tt,M,N) | → | mark#(M) | (82) |
mark#(plus(X1,X2)) | → | mark#(X2) | (83) |
mark#(s(X)) | → | mark#(X) | (84) |
a__U12#(tt,V2) | → | a__U13#(a__isNat(V2)) | (85) |
a__isNat#(s(V1)) | → | a__isNatKind#(V1) | (86) |
mark#(plus(X1,X2)) | → | mark#(X1) | (87) |
a__U31#(tt,N) | → | mark#(N) | (88) |
mark#(U13(X)) | → | mark#(X) | (89) |
The dependency pairs are split into 1 component.
mark#(U13(X)) | → | mark#(X) | (89) |
a__U31#(tt,N) | → | mark#(N) | (88) |
mark#(isNat(X)) | → | a__isNat#(X) | (64) |
mark#(plus(X1,X2)) | → | a__plus#(mark(X1),mark(X2)) | (63) |
mark#(plus(X1,X2)) | → | mark#(X1) | (87) |
mark#(and(X1,X2)) | → | a__and#(mark(X1),X2) | (62) |
mark#(U31(X1,X2)) | → | mark#(X1) | (61) |
a__isNat#(s(V1)) | → | a__isNatKind#(V1) | (86) |
a__isNat#(plus(V1,V2)) | → | a__and#(a__isNatKind(V1),isNatKind(V2)) | (59) |
mark#(s(X)) | → | mark#(X) | (84) |
a__isNat#(plus(V1,V2)) | → | a__isNatKind#(V1) | (58) |
mark#(plus(X1,X2)) | → | mark#(X2) | (83) |
a__U41#(tt,M,N) | → | mark#(M) | (82) |
a__plus#(N,s(M)) | → | a__and#(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))) | (81) |
mark#(U22(X)) | → | mark#(X) | (80) |
mark#(U31(X1,X2)) | → | a__U31#(mark(X1),X2) | (57) |
mark#(U41(X1,X2,X3)) | → | a__U41#(mark(X1),X2,X3) | (56) |
a__isNatKind#(s(V1)) | → | a__isNatKind#(V1) | (55) |
mark#(and(X1,X2)) | → | mark#(X1) | (78) |
a__U41#(tt,M,N) | → | a__plus#(mark(N),mark(M)) | (79) |
a__U11#(tt,V1,V2) | → | a__isNat#(V1) | (54) |
a__isNatKind#(plus(V1,V2)) | → | a__isNatKind#(V1) | (53) |
mark#(isNatKind(X)) | → | a__isNatKind#(X) | (76) |
a__plus#(N,s(M)) | → | a__isNat#(M) | (52) |
mark#(U21(X1,X2)) | → | mark#(X1) | (75) |
mark#(U12(X1,X2)) | → | mark#(X1) | (51) |
a__isNat#(plus(V1,V2)) | → | a__U11#(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2) | (74) |
mark#(U11(X1,X2,X3)) | → | a__U11#(mark(X1),X2,X3) | (73) |
mark#(U12(X1,X2)) | → | a__U12#(mark(X1),X2) | (50) |
a__plus#(N,0) | → | a__isNat#(N) | (72) |
a__isNatKind#(plus(V1,V2)) | → | a__and#(a__isNatKind(V1),isNatKind(V2)) | (71) |
a__plus#(N,s(M)) | → | a__and#(a__isNat(M),isNatKind(M)) | (49) |
mark#(U21(X1,X2)) | → | a__U21#(mark(X1),X2) | (70) |
a__and#(tt,X) | → | mark#(X) | (48) |
a__plus#(N,0) | → | a__U31#(a__and(a__isNat(N),isNatKind(N)),N) | (47) |
mark#(U41(X1,X2,X3)) | → | mark#(X1) | (69) |
a__U21#(tt,V1) | → | a__isNat#(V1) | (46) |
a__U11#(tt,V1,V2) | → | a__U12#(a__isNat(V1),V2) | (68) |
a__isNat#(s(V1)) | → | a__U21#(a__isNatKind(V1),V1) | (67) |
a__plus#(N,s(M)) | → | a__U41#(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N) | (45) |
mark#(U11(X1,X2,X3)) | → | mark#(X1) | (44) |
a__plus#(N,0) | → | a__and#(a__isNat(N),isNatKind(N)) | (43) |
a__U12#(tt,V2) | → | a__isNat#(V2) | (42) |
a__U41#(tt,M,N) | → | mark#(N) | (66) |
[a__isNatKind#(x1)] | = | 0 |
[isNatKind(x1)] | = | 0 |
[a__plus(x1, x2)] | = | x1 + x2 + 0 |
[U21(x1, x2)] | = | x1 + 0 |
[U11(x1, x2, x3)] | = | x1 + 0 |
[s(x1)] | = | x1 + 0 |
[a__U31#(x1, x2)] | = | x2 + 16021 |
[a__isNat#(x1)] | = | 0 |
[and(x1, x2)] | = | x1 + x2 + 0 |
[a__U13#(x1)] | = | 0 |
[a__U22(x1)] | = | x1 + 0 |
[a__U11#(x1, x2, x3)] | = | 0 |
[a__U31(x1, x2)] | = | x1 + x2 + 16022 |
[U12(x1, x2)] | = | x1 + 0 |
[a__U41(x1, x2, x3)] | = | x1 + x2 + x3 + 0 |
[a__U12#(x1, x2)] | = | 0 |
[a__U21#(x1, x2)] | = | 0 |
[a__plus#(x1, x2)] | = | x1 + x2 + 0 |
[mark#(x1)] | = | x1 + 0 |
[0] | = | 16022 |
[a__and#(x1, x2)] | = | x2 + 0 |
[a__U21(x1, x2)] | = | x1 + 0 |
[mark(x1)] | = | x1 + 0 |
[a__U11(x1, x2, x3)] | = | x1 + 0 |
[a__U12(x1, x2)] | = | x1 + 0 |
[isNat(x1)] | = | 0 |
[plus(x1, x2)] | = | x1 + x2 + 0 |
[a__U22#(x1)] | = | 0 |
[a__U13(x1)] | = | x1 + 0 |
[U31(x1, x2)] | = | x1 + x2 + 16022 |
[a__U41#(x1, x2, x3)] | = | x2 + x3 + 0 |
[tt] | = | 0 |
[a__isNat(x1)] | = | 0 |
[U13(x1)] | = | x1 + 0 |
[a__isNatKind(x1)] | = | 0 |
[U22(x1)] | = | x1 + 0 |
[a__and(x1, x2)] | = | x1 + x2 + 0 |
[U41(x1, x2, x3)] | = | x1 + x2 + x3 + 0 |
mark(U12(X1,X2)) | → | a__U12(mark(X1),X2) | (18) |
a__U21(tt,V1) | → | a__U22(a__isNat(V1)) | (4) |
a__plus(N,0) | → | a__U31(a__and(a__isNat(N),isNatKind(N)),N) | (15) |
a__and(tt,X) | → | mark(X) | (8) |
a__U11(tt,V1,V2) | → | a__U12(a__isNat(V1),V2) | (1) |
a__U13(tt) | → | tt | (3) |
a__plus(N,s(M)) | → | a__U41(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N) | (16) |
mark(U21(X1,X2)) | → | a__U21(mark(X1),X2) | (21) |
a__U22(X) | → | U22(X) | (36) |
mark(and(X1,X2)) | → | a__and(mark(X1),X2) | (26) |
mark(isNat(X)) | → | a__isNat(X) | (19) |
a__U12(X1,X2) | → | U12(X1,X2) | (32) |
mark(U11(X1,X2,X3)) | → | a__U11(mark(X1),X2,X3) | (17) |
mark(isNatKind(X)) | → | a__isNatKind(X) | (27) |
a__U13(X) | → | U13(X) | (34) |
mark(U22(X)) | → | a__U22(mark(X)) | (22) |
mark(tt) | → | tt | (28) |
a__U22(tt) | → | tt | (5) |
a__isNat(X) | → | isNat(X) | (33) |
a__isNat(plus(V1,V2)) | → | a__U11(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2) | (10) |
a__plus(X1,X2) | → | plus(X1,X2) | (39) |
a__U41(tt,M,N) | → | s(a__plus(mark(N),mark(M))) | (7) |
mark(U13(X)) | → | a__U13(mark(X)) | (20) |
mark(plus(X1,X2)) | → | a__plus(mark(X1),mark(X2)) | (25) |
mark(0) | → | 0 | (30) |
a__isNatKind(s(V1)) | → | a__isNatKind(V1) | (14) |
a__U11(X1,X2,X3) | → | U11(X1,X2,X3) | (31) |
a__isNatKind(0) | → | tt | (12) |
mark(U31(X1,X2)) | → | a__U31(mark(X1),X2) | (23) |
mark(U41(X1,X2,X3)) | → | a__U41(mark(X1),X2,X3) | (24) |
a__isNat(s(V1)) | → | a__U21(a__isNatKind(V1),V1) | (11) |
a__isNat(0) | → | tt | (9) |
a__isNatKind(plus(V1,V2)) | → | a__and(a__isNatKind(V1),isNatKind(V2)) | (13) |
a__and(X1,X2) | → | and(X1,X2) | (40) |
a__U31(tt,N) | → | mark(N) | (6) |
a__U41(X1,X2,X3) | → | U41(X1,X2,X3) | (38) |
a__U31(X1,X2) | → | U31(X1,X2) | (37) |
a__isNatKind(X) | → | isNatKind(X) | (41) |
a__U21(X1,X2) | → | U21(X1,X2) | (35) |
mark(s(X)) | → | s(mark(X)) | (29) |
a__U12(tt,V2) | → | a__U13(a__isNat(V2)) | (2) |
a__U31#(tt,N) | → | mark#(N) | (88) |
mark#(U31(X1,X2)) | → | mark#(X1) | (61) |
mark#(U31(X1,X2)) | → | a__U31#(mark(X1),X2) | (57) |
a__plus#(N,0) | → | a__isNat#(N) | (72) |
a__plus#(N,0) | → | a__U31#(a__and(a__isNat(N),isNatKind(N)),N) | (47) |
a__plus#(N,0) | → | a__and#(a__isNat(N),isNatKind(N)) | (43) |
The dependency pairs are split into 1 component.
mark#(U12(X1,X2)) | → | mark#(X1) | (51) |
mark#(U12(X1,X2)) | → | a__U12#(mark(X1),X2) | (50) |
a__U21#(tt,V1) | → | a__isNat#(V1) | (46) |
a__and#(tt,X) | → | mark#(X) | (48) |
a__U11#(tt,V1,V2) | → | a__isNat#(V1) | (54) |
a__U11#(tt,V1,V2) | → | a__U12#(a__isNat(V1),V2) | (68) |
a__plus#(N,s(M)) | → | a__isNat#(M) | (52) |
a__plus#(N,s(M)) | → | a__and#(a__isNat(M),isNatKind(M)) | (49) |
a__plus#(N,s(M)) | → | a__and#(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))) | (81) |
a__plus#(N,s(M)) | → | a__U41#(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N) | (45) |
mark#(U21(X1,X2)) | → | mark#(X1) | (75) |
mark#(U21(X1,X2)) | → | a__U21#(mark(X1),X2) | (70) |
mark#(and(X1,X2)) | → | mark#(X1) | (78) |
mark#(and(X1,X2)) | → | a__and#(mark(X1),X2) | (62) |
mark#(isNat(X)) | → | a__isNat#(X) | (64) |
mark#(U11(X1,X2,X3)) | → | mark#(X1) | (44) |
mark#(U11(X1,X2,X3)) | → | a__U11#(mark(X1),X2,X3) | (73) |
mark#(isNatKind(X)) | → | a__isNatKind#(X) | (76) |
mark#(U22(X)) | → | mark#(X) | (80) |
a__isNat#(plus(V1,V2)) | → | a__isNatKind#(V1) | (58) |
a__isNat#(plus(V1,V2)) | → | a__and#(a__isNatKind(V1),isNatKind(V2)) | (59) |
a__isNat#(plus(V1,V2)) | → | a__U11#(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2) | (74) |
a__U41#(tt,M,N) | → | mark#(M) | (82) |
a__U41#(tt,M,N) | → | mark#(N) | (66) |
a__U41#(tt,M,N) | → | a__plus#(mark(N),mark(M)) | (79) |
mark#(U13(X)) | → | mark#(X) | (89) |
mark#(plus(X1,X2)) | → | mark#(X2) | (83) |
mark#(plus(X1,X2)) | → | mark#(X1) | (87) |
mark#(plus(X1,X2)) | → | a__plus#(mark(X1),mark(X2)) | (63) |
a__isNatKind#(s(V1)) | → | a__isNatKind#(V1) | (55) |
mark#(U41(X1,X2,X3)) | → | mark#(X1) | (69) |
mark#(U41(X1,X2,X3)) | → | a__U41#(mark(X1),X2,X3) | (56) |
a__isNat#(s(V1)) | → | a__isNatKind#(V1) | (86) |
a__isNat#(s(V1)) | → | a__U21#(a__isNatKind(V1),V1) | (67) |
a__isNatKind#(plus(V1,V2)) | → | a__isNatKind#(V1) | (53) |
a__isNatKind#(plus(V1,V2)) | → | a__and#(a__isNatKind(V1),isNatKind(V2)) | (71) |
mark#(s(X)) | → | mark#(X) | (84) |
a__U12#(tt,V2) | → | a__isNat#(V2) | (42) |
[a__isNatKind#(x1)] | = | 0 |
[isNatKind(x1)] | = | 0 |
[a__plus(x1, x2)] | = | x1 + x2 + 2 |
[U21(x1, x2)] | = | x1 + 0 |
[U11(x1, x2, x3)] | = | x1 + 0 |
[s(x1)] | = | x1 + 0 |
[a__U31#(x1, x2)] | = | 16021 |
[a__isNat#(x1)] | = | 0 |
[and(x1, x2)] | = | x1 + x2 + 0 |
[a__U13#(x1)] | = | 0 |
[a__U22(x1)] | = | x1 + 0 |
[a__U11#(x1, x2, x3)] | = | 0 |
[a__U31(x1, x2)] | = | x1 + x2 + 7066 |
[U12(x1, x2)] | = | x1 + 0 |
[a__U41(x1, x2, x3)] | = | x1 + x2 + x3 + 2 |
[a__U12#(x1, x2)] | = | 0 |
[a__U21#(x1, x2)] | = | 0 |
[a__plus#(x1, x2)] | = | x1 + x2 + 1 |
[mark#(x1)] | = | x1 + 0 |
[0] | = | 16022 |
[a__and#(x1, x2)] | = | x2 + 0 |
[a__U21(x1, x2)] | = | x1 + 0 |
[mark(x1)] | = | x1 + 0 |
[a__U11(x1, x2, x3)] | = | x1 + 0 |
[a__U12(x1, x2)] | = | x1 + 0 |
[isNat(x1)] | = | 0 |
[plus(x1, x2)] | = | x1 + x2 + 2 |
[a__U22#(x1)] | = | 0 |
[a__U13(x1)] | = | x1 + 0 |
[U31(x1, x2)] | = | x1 + x2 + 7066 |
[a__U41#(x1, x2, x3)] | = | x2 + x3 + 1 |
[tt] | = | 0 |
[a__isNat(x1)] | = | 0 |
[U13(x1)] | = | x1 + 0 |
[a__isNatKind(x1)] | = | 0 |
[U22(x1)] | = | x1 + 0 |
[a__and(x1, x2)] | = | x1 + x2 + 0 |
[U41(x1, x2, x3)] | = | x1 + x2 + x3 + 2 |
mark(U12(X1,X2)) | → | a__U12(mark(X1),X2) | (18) |
a__U21(tt,V1) | → | a__U22(a__isNat(V1)) | (4) |
a__plus(N,0) | → | a__U31(a__and(a__isNat(N),isNatKind(N)),N) | (15) |
a__and(tt,X) | → | mark(X) | (8) |
a__U11(tt,V1,V2) | → | a__U12(a__isNat(V1),V2) | (1) |
a__U13(tt) | → | tt | (3) |
a__plus(N,s(M)) | → | a__U41(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N) | (16) |
mark(U21(X1,X2)) | → | a__U21(mark(X1),X2) | (21) |
a__U22(X) | → | U22(X) | (36) |
mark(and(X1,X2)) | → | a__and(mark(X1),X2) | (26) |
mark(isNat(X)) | → | a__isNat(X) | (19) |
a__U12(X1,X2) | → | U12(X1,X2) | (32) |
mark(U11(X1,X2,X3)) | → | a__U11(mark(X1),X2,X3) | (17) |
mark(isNatKind(X)) | → | a__isNatKind(X) | (27) |
a__U13(X) | → | U13(X) | (34) |
mark(U22(X)) | → | a__U22(mark(X)) | (22) |
mark(tt) | → | tt | (28) |
a__U22(tt) | → | tt | (5) |
a__isNat(X) | → | isNat(X) | (33) |
a__isNat(plus(V1,V2)) | → | a__U11(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2) | (10) |
a__plus(X1,X2) | → | plus(X1,X2) | (39) |
a__U41(tt,M,N) | → | s(a__plus(mark(N),mark(M))) | (7) |
mark(U13(X)) | → | a__U13(mark(X)) | (20) |
mark(plus(X1,X2)) | → | a__plus(mark(X1),mark(X2)) | (25) |
mark(0) | → | 0 | (30) |
a__isNatKind(s(V1)) | → | a__isNatKind(V1) | (14) |
a__U11(X1,X2,X3) | → | U11(X1,X2,X3) | (31) |
a__isNatKind(0) | → | tt | (12) |
mark(U31(X1,X2)) | → | a__U31(mark(X1),X2) | (23) |
mark(U41(X1,X2,X3)) | → | a__U41(mark(X1),X2,X3) | (24) |
a__isNat(s(V1)) | → | a__U21(a__isNatKind(V1),V1) | (11) |
a__isNat(0) | → | tt | (9) |
a__isNatKind(plus(V1,V2)) | → | a__and(a__isNatKind(V1),isNatKind(V2)) | (13) |
a__and(X1,X2) | → | and(X1,X2) | (40) |
a__U31(tt,N) | → | mark(N) | (6) |
a__U41(X1,X2,X3) | → | U41(X1,X2,X3) | (38) |
a__U31(X1,X2) | → | U31(X1,X2) | (37) |
a__isNatKind(X) | → | isNatKind(X) | (41) |
a__U21(X1,X2) | → | U21(X1,X2) | (35) |
mark(s(X)) | → | s(mark(X)) | (29) |
a__U12(tt,V2) | → | a__U13(a__isNat(V2)) | (2) |
a__plus#(N,s(M)) | → | a__isNat#(M) | (52) |
a__plus#(N,s(M)) | → | a__and#(a__isNat(M),isNatKind(M)) | (49) |
a__plus#(N,s(M)) | → | a__and#(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))) | (81) |
a__U41#(tt,M,N) | → | mark#(M) | (82) |
a__U41#(tt,M,N) | → | mark#(N) | (66) |
mark#(plus(X1,X2)) | → | mark#(X2) | (83) |
mark#(plus(X1,X2)) | → | mark#(X1) | (87) |
mark#(plus(X1,X2)) | → | a__plus#(mark(X1),mark(X2)) | (63) |
mark#(U41(X1,X2,X3)) | → | mark#(X1) | (69) |
mark#(U41(X1,X2,X3)) | → | a__U41#(mark(X1),X2,X3) | (56) |
The dependency pairs are split into 2 components.
a__plus#(N,s(M)) | → | a__U41#(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N) | (45) |
a__U41#(tt,M,N) | → | a__plus#(mark(N),mark(M)) | (79) |
[a__isNatKind#(x1)] | = | 0 |
[isNatKind(x1)] | = | 0 |
[a__plus(x1, x2)] | = | x1 + x2 + 46593 |
[U21(x1, x2)] | = | x1 + 1 |
[U11(x1, x2, x3)] | = | x1 + 0 |
[s(x1)] | = | x1 + 52062 |
[a__U31#(x1, x2)] | = | 16021 |
[a__isNat#(x1)] | = | 0 |
[and(x1, x2)] | = | x1 + x2 + 0 |
[a__U13#(x1)] | = | 0 |
[a__U22(x1)] | = | x1 + 0 |
[a__U11#(x1, x2, x3)] | = | 0 |
[a__U31(x1, x2)] | = | x2 + 32582 |
[U12(x1, x2)] | = | 0 |
[a__U41(x1, x2, x3)] | = | x2 + x3 + 98655 |
[a__U12#(x1, x2)] | = | 0 |
[a__U21#(x1, x2)] | = | 0 |
[a__plus#(x1, x2)] | = | x1 + x2 + 0 |
[mark#(x1)] | = | 0 |
[0] | = | 16022 |
[a__and#(x1, x2)] | = | 0 |
[a__U21(x1, x2)] | = | x1 + 1 |
[mark(x1)] | = | x1 + 0 |
[a__U11(x1, x2, x3)] | = | x1 + 0 |
[a__U12(x1, x2)] | = | 0 |
[isNat(x1)] | = | 1 |
[plus(x1, x2)] | = | x1 + x2 + 46593 |
[a__U22#(x1)] | = | 0 |
[a__U13(x1)] | = | 0 |
[U31(x1, x2)] | = | x2 + 32582 |
[a__U41#(x1, x2, x3)] | = | x2 + x3 + 0 |
[tt] | = | 0 |
[a__isNat(x1)] | = | 1 |
[U13(x1)] | = | 0 |
[a__isNatKind(x1)] | = | 0 |
[U22(x1)] | = | x1 + 0 |
[a__and(x1, x2)] | = | x1 + x2 + 0 |
[U41(x1, x2, x3)] | = | x2 + x3 + 98655 |
mark(U12(X1,X2)) | → | a__U12(mark(X1),X2) | (18) |
a__U21(tt,V1) | → | a__U22(a__isNat(V1)) | (4) |
a__plus(N,0) | → | a__U31(a__and(a__isNat(N),isNatKind(N)),N) | (15) |
a__and(tt,X) | → | mark(X) | (8) |
a__U11(tt,V1,V2) | → | a__U12(a__isNat(V1),V2) | (1) |
a__U13(tt) | → | tt | (3) |
a__plus(N,s(M)) | → | a__U41(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N) | (16) |
mark(U21(X1,X2)) | → | a__U21(mark(X1),X2) | (21) |
a__U22(X) | → | U22(X) | (36) |
mark(and(X1,X2)) | → | a__and(mark(X1),X2) | (26) |
mark(isNat(X)) | → | a__isNat(X) | (19) |
a__U12(X1,X2) | → | U12(X1,X2) | (32) |
mark(U11(X1,X2,X3)) | → | a__U11(mark(X1),X2,X3) | (17) |
mark(isNatKind(X)) | → | a__isNatKind(X) | (27) |
a__U13(X) | → | U13(X) | (34) |
mark(U22(X)) | → | a__U22(mark(X)) | (22) |
mark(tt) | → | tt | (28) |
a__U22(tt) | → | tt | (5) |
a__isNat(X) | → | isNat(X) | (33) |
a__isNat(plus(V1,V2)) | → | a__U11(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2) | (10) |
a__plus(X1,X2) | → | plus(X1,X2) | (39) |
a__U41(tt,M,N) | → | s(a__plus(mark(N),mark(M))) | (7) |
mark(U13(X)) | → | a__U13(mark(X)) | (20) |
mark(plus(X1,X2)) | → | a__plus(mark(X1),mark(X2)) | (25) |
mark(0) | → | 0 | (30) |
a__isNatKind(s(V1)) | → | a__isNatKind(V1) | (14) |
a__U11(X1,X2,X3) | → | U11(X1,X2,X3) | (31) |
a__isNatKind(0) | → | tt | (12) |
mark(U31(X1,X2)) | → | a__U31(mark(X1),X2) | (23) |
mark(U41(X1,X2,X3)) | → | a__U41(mark(X1),X2,X3) | (24) |
a__isNat(s(V1)) | → | a__U21(a__isNatKind(V1),V1) | (11) |
a__isNat(0) | → | tt | (9) |
a__isNatKind(plus(V1,V2)) | → | a__and(a__isNatKind(V1),isNatKind(V2)) | (13) |
a__and(X1,X2) | → | and(X1,X2) | (40) |
a__U31(tt,N) | → | mark(N) | (6) |
a__U41(X1,X2,X3) | → | U41(X1,X2,X3) | (38) |
a__U31(X1,X2) | → | U31(X1,X2) | (37) |
a__isNatKind(X) | → | isNatKind(X) | (41) |
a__U21(X1,X2) | → | U21(X1,X2) | (35) |
mark(s(X)) | → | s(mark(X)) | (29) |
a__U12(tt,V2) | → | a__U13(a__isNat(V2)) | (2) |
a__plus#(N,s(M)) | → | a__U41#(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N) | (45) |
The dependency pairs are split into 0 components.
mark#(U12(X1,X2)) | → | mark#(X1) | (51) |
mark#(U12(X1,X2)) | → | a__U12#(mark(X1),X2) | (50) |
a__U21#(tt,V1) | → | a__isNat#(V1) | (46) |
a__and#(tt,X) | → | mark#(X) | (48) |
a__U11#(tt,V1,V2) | → | a__isNat#(V1) | (54) |
a__U11#(tt,V1,V2) | → | a__U12#(a__isNat(V1),V2) | (68) |
mark#(U21(X1,X2)) | → | mark#(X1) | (75) |
mark#(U21(X1,X2)) | → | a__U21#(mark(X1),X2) | (70) |
mark#(and(X1,X2)) | → | mark#(X1) | (78) |
mark#(and(X1,X2)) | → | a__and#(mark(X1),X2) | (62) |
mark#(isNat(X)) | → | a__isNat#(X) | (64) |
mark#(U11(X1,X2,X3)) | → | mark#(X1) | (44) |
mark#(U11(X1,X2,X3)) | → | a__U11#(mark(X1),X2,X3) | (73) |
mark#(isNatKind(X)) | → | a__isNatKind#(X) | (76) |
mark#(U22(X)) | → | mark#(X) | (80) |
a__isNat#(plus(V1,V2)) | → | a__isNatKind#(V1) | (58) |
a__isNat#(plus(V1,V2)) | → | a__and#(a__isNatKind(V1),isNatKind(V2)) | (59) |
a__isNat#(plus(V1,V2)) | → | a__U11#(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2) | (74) |
mark#(U13(X)) | → | mark#(X) | (89) |
a__isNatKind#(s(V1)) | → | a__isNatKind#(V1) | (55) |
a__isNat#(s(V1)) | → | a__isNatKind#(V1) | (86) |
a__isNat#(s(V1)) | → | a__U21#(a__isNatKind(V1),V1) | (67) |
a__isNatKind#(plus(V1,V2)) | → | a__isNatKind#(V1) | (53) |
a__isNatKind#(plus(V1,V2)) | → | a__and#(a__isNatKind(V1),isNatKind(V2)) | (71) |
mark#(s(X)) | → | mark#(X) | (84) |
a__U12#(tt,V2) | → | a__isNat#(V2) | (42) |
[a__isNatKind#(x1)] | = | 1 |
[isNatKind(x1)] | = | 1 |
[a__plus(x1, x2)] | = | x1 + x2 + 10114 |
[U21(x1, x2)] | = | x1 + 3 |
[U11(x1, x2, x3)] | = | x1 + x2 + x3 + 19801 |
[s(x1)] | = | x1 + 1 |
[a__U31#(x1, x2)] | = | 16021 |
[a__isNat#(x1)] | = | 2 |
[and(x1, x2)] | = | x1 + x2 + 2 |
[a__U13#(x1)] | = | 0 |
[a__U22(x1)] | = | 2 |
[a__U11#(x1, x2, x3)] | = | 2 |
[a__U31(x1, x2)] | = | 0 |
[U12(x1, x2)] | = | x1 + 19801 |
[a__U41(x1, x2, x3)] | = | x2 + 10115 |
[a__U12#(x1, x2)] | = | 2 |
[a__U21#(x1, x2)] | = | 2 |
[a__plus#(x1, x2)] | = | 0 |
[mark#(x1)] | = | x1 + 0 |
[0] | = | 1 |
[a__and#(x1, x2)] | = | x2 + 0 |
[a__U21(x1, x2)] | = | x2 + 2 |
[mark(x1)] | = | x1 + 0 |
[a__U11(x1, x2, x3)] | = | x2 + 19800 |
[a__U12(x1, x2)] | = | 19800 |
[isNat(x1)] | = | 3 |
[plus(x1, x2)] | = | 19796 |
[a__U22#(x1)] | = | 0 |
[a__U13(x1)] | = | x1 + 10749 |
[U31(x1, x2)] | = | 15081 |
[a__U41#(x1, x2, x3)] | = | 0 |
[tt] | = | 0 |
[a__isNat(x1)] | = | x1 + 4 |
[U13(x1)] | = | x1 + 10748 |
[a__isNatKind(x1)] | = | x1 + 0 |
[U22(x1)] | = | x1 + 1 |
[a__and(x1, x2)] | = | 1 |
[U41(x1, x2, x3)] | = | x2 + x3 + 10114 |
mark#(U12(X1,X2)) | → | mark#(X1) | (51) |
mark#(U12(X1,X2)) | → | a__U12#(mark(X1),X2) | (50) |
mark#(U21(X1,X2)) | → | mark#(X1) | (75) |
mark#(U21(X1,X2)) | → | a__U21#(mark(X1),X2) | (70) |
mark#(and(X1,X2)) | → | mark#(X1) | (78) |
mark#(and(X1,X2)) | → | a__and#(mark(X1),X2) | (62) |
mark#(isNat(X)) | → | a__isNat#(X) | (64) |
mark#(U11(X1,X2,X3)) | → | mark#(X1) | (44) |
mark#(U11(X1,X2,X3)) | → | a__U11#(mark(X1),X2,X3) | (73) |
mark#(U22(X)) | → | mark#(X) | (80) |
a__isNat#(plus(V1,V2)) | → | a__isNatKind#(V1) | (58) |
a__isNat#(plus(V1,V2)) | → | a__and#(a__isNatKind(V1),isNatKind(V2)) | (59) |
mark#(U13(X)) | → | mark#(X) | (89) |
a__isNat#(s(V1)) | → | a__isNatKind#(V1) | (86) |
mark#(s(X)) | → | mark#(X) | (84) |
The dependency pairs are split into 2 components.
a__and#(tt,X) | → | mark#(X) | (48) |
mark#(isNatKind(X)) | → | a__isNatKind#(X) | (76) |
a__isNatKind#(s(V1)) | → | a__isNatKind#(V1) | (55) |
a__isNatKind#(plus(V1,V2)) | → | a__isNatKind#(V1) | (53) |
a__isNatKind#(plus(V1,V2)) | → | a__and#(a__isNatKind(V1),isNatKind(V2)) | (71) |
[a__isNatKind#(x1)] | = | x1 + 0 |
[isNatKind(x1)] | = | x1 + 1 |
[a__plus(x1, x2)] | = | x1 + x2 + 1 |
[U21(x1, x2)] | = | x1 + 30251 |
[U11(x1, x2, x3)] | = | x1 + x2 + x3 + 32161 |
[s(x1)] | = | x1 + 1 |
[a__U31#(x1, x2)] | = | 16021 |
[a__isNat#(x1)] | = | 2 |
[and(x1, x2)] | = | x1 + x2 + 2 |
[a__U13#(x1)] | = | 0 |
[a__U22(x1)] | = | 13477 |
[a__U11#(x1, x2, x3)] | = | 2 |
[a__U31(x1, x2)] | = | 0 |
[U12(x1, x2)] | = | x1 + 41703 |
[a__U41(x1, x2, x3)] | = | x2 + 2 |
[a__U12#(x1, x2)] | = | 2 |
[a__U21#(x1, x2)] | = | 2 |
[a__plus#(x1, x2)] | = | 0 |
[mark#(x1)] | = | x1 + 0 |
[0] | = | 1 |
[a__and#(x1, x2)] | = | x2 + 1 |
[a__U21(x1, x2)] | = | x2 + 30250 |
[mark(x1)] | = | x1 + 0 |
[a__U11(x1, x2, x3)] | = | x2 + 30250 |
[a__U12(x1, x2)] | = | 30250 |
[isNat(x1)] | = | 1 |
[plus(x1, x2)] | = | x1 + x2 + 3 |
[a__U22#(x1)] | = | 0 |
[a__U13(x1)] | = | x1 + 0 |
[U31(x1, x2)] | = | 15081 |
[a__U41#(x1, x2, x3)] | = | 0 |
[tt] | = | 0 |
[a__isNat(x1)] | = | 30250 |
[U13(x1)] | = | x1 + 0 |
[a__isNatKind(x1)] | = | x1 + 0 |
[U22(x1)] | = | x1 + 13476 |
[a__and(x1, x2)] | = | 1 |
[U41(x1, x2, x3)] | = | x2 + x3 + 1 |
a__and#(tt,X) | → | mark#(X) | (48) |
mark#(isNatKind(X)) | → | a__isNatKind#(X) | (76) |
a__isNatKind#(s(V1)) | → | a__isNatKind#(V1) | (55) |
a__isNatKind#(plus(V1,V2)) | → | a__isNatKind#(V1) | (53) |
a__isNatKind#(plus(V1,V2)) | → | a__and#(a__isNatKind(V1),isNatKind(V2)) | (71) |
The dependency pairs are split into 0 components.
a__U21#(tt,V1) | → | a__isNat#(V1) | (46) |
a__U11#(tt,V1,V2) | → | a__isNat#(V1) | (54) |
a__U11#(tt,V1,V2) | → | a__U12#(a__isNat(V1),V2) | (68) |
a__isNat#(plus(V1,V2)) | → | a__U11#(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2) | (74) |
a__isNat#(s(V1)) | → | a__U21#(a__isNatKind(V1),V1) | (67) |
a__U12#(tt,V2) | → | a__isNat#(V2) | (42) |
[a__isNatKind#(x1)] | = | 0 |
[isNatKind(x1)] | = | x1 + 217 |
[a__plus(x1, x2)] | = | x1 + x2 + 0 |
[U21(x1, x2)] | = | 1 |
[U11(x1, x2, x3)] | = | 3 |
[s(x1)] | = | x1 + 2 |
[a__U31#(x1, x2)] | = | 16021 |
[a__isNat#(x1)] | = | x1 + 2 |
[and(x1, x2)] | = | x1 + x2 + 15871 |
[a__U13#(x1)] | = | 0 |
[a__U22(x1)] | = | 66879 |
[a__U11#(x1, x2, x3)] | = | x2 + x3 + 4 |
[a__U31(x1, x2)] | = | x1 + x2 + 46812 |
[U12(x1, x2)] | = | x1 + x2 + 4595 |
[a__U41(x1, x2, x3)] | = | x1 + x2 + 16020 |
[a__U12#(x1, x2)] | = | x2 + 3 |
[a__U21#(x1, x2)] | = | x2 + 3 |
[a__plus#(x1, x2)] | = | 0 |
[mark#(x1)] | = | 0 |
[0] | = | 1 |
[a__and#(x1, x2)] | = | 1 |
[a__U21(x1, x2)] | = | x1 + 2 |
[mark(x1)] | = | x1 + 51409 |
[a__U11(x1, x2, x3)] | = | 2 |
[a__U12(x1, x2)] | = | 4594 |
[isNat(x1)] | = | x1 + 13458 |
[plus(x1, x2)] | = | x1 + x2 + 3 |
[a__U22#(x1)] | = | 0 |
[a__U13(x1)] | = | 4595 |
[U31(x1, x2)] | = | 46811 |
[a__U41#(x1, x2, x3)] | = | 0 |
[tt] | = | 4596 |
[a__isNat(x1)] | = | 1 |
[U13(x1)] | = | x1 + 4596 |
[a__isNatKind(x1)] | = | x1 + 213 |
[U22(x1)] | = | 15469 |
[a__and(x1, x2)] | = | x2 + 0 |
[U41(x1, x2, x3)] | = | x3 + 16019 |
a__U21#(tt,V1) | → | a__isNat#(V1) | (46) |
a__U11#(tt,V1,V2) | → | a__isNat#(V1) | (54) |
a__U11#(tt,V1,V2) | → | a__U12#(a__isNat(V1),V2) | (68) |
a__isNat#(plus(V1,V2)) | → | a__U11#(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2) | (74) |
a__isNat#(s(V1)) | → | a__U21#(a__isNatKind(V1),V1) | (67) |
a__U12#(tt,V2) | → | a__isNat#(V2) | (42) |
The dependency pairs are split into 0 components.