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__U11#(tt,V1,V2) | → | a__isNat#(V1) | (42) |
a__U11#(tt,V1,V2) | → | a__U12#(a__isNat(V1),V2) | (43) |
a__U12#(tt,V2) | → | a__isNat#(V2) | (44) |
a__U12#(tt,V2) | → | a__U13#(a__isNat(V2)) | (45) |
a__U21#(tt,V1) | → | a__isNat#(V1) | (46) |
a__U21#(tt,V1) | → | a__U22#(a__isNat(V1)) | (47) |
a__U31#(tt,N) | → | mark#(N) | (48) |
a__U41#(tt,M,N) | → | mark#(M) | (49) |
a__U41#(tt,M,N) | → | mark#(N) | (50) |
a__U41#(tt,M,N) | → | a__plus#(mark(N),mark(M)) | (51) |
a__and#(tt,X) | → | mark#(X) | (52) |
a__isNat#(plus(V1,V2)) | → | a__isNatKind#(V1) | (53) |
a__isNat#(plus(V1,V2)) | → | a__and#(a__isNatKind(V1),isNatKind(V2)) | (54) |
a__isNat#(plus(V1,V2)) | → | a__U11#(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2) | (55) |
a__isNat#(s(V1)) | → | a__isNatKind#(V1) | (56) |
a__isNat#(s(V1)) | → | a__U21#(a__isNatKind(V1),V1) | (57) |
a__isNatKind#(plus(V1,V2)) | → | a__isNatKind#(V1) | (58) |
a__isNatKind#(plus(V1,V2)) | → | a__and#(a__isNatKind(V1),isNatKind(V2)) | (59) |
a__isNatKind#(s(V1)) | → | a__isNatKind#(V1) | (60) |
a__plus#(N,0) | → | a__isNat#(N) | (61) |
a__plus#(N,0) | → | a__and#(a__isNat(N),isNatKind(N)) | (62) |
a__plus#(N,0) | → | a__U31#(a__and(a__isNat(N),isNatKind(N)),N) | (63) |
a__plus#(N,s(M)) | → | a__isNat#(M) | (64) |
a__plus#(N,s(M)) | → | a__and#(a__isNat(M),isNatKind(M)) | (65) |
a__plus#(N,s(M)) | → | a__and#(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))) | (66) |
a__plus#(N,s(M)) | → | a__U41#(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N) | (67) |
mark#(U11(X1,X2,X3)) | → | mark#(X1) | (68) |
mark#(U11(X1,X2,X3)) | → | a__U11#(mark(X1),X2,X3) | (69) |
mark#(U12(X1,X2)) | → | mark#(X1) | (70) |
mark#(U12(X1,X2)) | → | a__U12#(mark(X1),X2) | (71) |
mark#(isNat(X)) | → | a__isNat#(X) | (72) |
mark#(U13(X)) | → | mark#(X) | (73) |
mark#(U13(X)) | → | a__U13#(mark(X)) | (74) |
mark#(U21(X1,X2)) | → | mark#(X1) | (75) |
mark#(U21(X1,X2)) | → | a__U21#(mark(X1),X2) | (76) |
mark#(U22(X)) | → | mark#(X) | (77) |
mark#(U22(X)) | → | a__U22#(mark(X)) | (78) |
mark#(U31(X1,X2)) | → | mark#(X1) | (79) |
mark#(U31(X1,X2)) | → | a__U31#(mark(X1),X2) | (80) |
mark#(U41(X1,X2,X3)) | → | mark#(X1) | (81) |
mark#(U41(X1,X2,X3)) | → | a__U41#(mark(X1),X2,X3) | (82) |
mark#(plus(X1,X2)) | → | mark#(X2) | (83) |
mark#(plus(X1,X2)) | → | mark#(X1) | (84) |
mark#(plus(X1,X2)) | → | a__plus#(mark(X1),mark(X2)) | (85) |
mark#(and(X1,X2)) | → | mark#(X1) | (86) |
mark#(and(X1,X2)) | → | a__and#(mark(X1),X2) | (87) |
mark#(isNatKind(X)) | → | a__isNatKind#(X) | (88) |
mark#(s(X)) | → | mark#(X) | (89) |
The dependency pairs are split into 1 component.
a__isNatKind#(plus(V1,V2)) | → | a__isNatKind#(V1) | (58) |
a__isNatKind#(plus(V1,V2)) | → | a__and#(a__isNatKind(V1),isNatKind(V2)) | (59) |
a__and#(tt,X) | → | mark#(X) | (52) |
mark#(U11(X1,X2,X3)) | → | mark#(X1) | (68) |
mark#(U11(X1,X2,X3)) | → | a__U11#(mark(X1),X2,X3) | (69) |
a__U11#(tt,V1,V2) | → | a__isNat#(V1) | (42) |
a__isNat#(plus(V1,V2)) | → | a__isNatKind#(V1) | (53) |
a__isNatKind#(s(V1)) | → | a__isNatKind#(V1) | (60) |
a__isNat#(plus(V1,V2)) | → | a__and#(a__isNatKind(V1),isNatKind(V2)) | (54) |
a__isNat#(plus(V1,V2)) | → | a__U11#(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2) | (55) |
a__U11#(tt,V1,V2) | → | a__U12#(a__isNat(V1),V2) | (43) |
a__U12#(tt,V2) | → | a__isNat#(V2) | (44) |
a__isNat#(s(V1)) | → | a__isNatKind#(V1) | (56) |
a__isNat#(s(V1)) | → | a__U21#(a__isNatKind(V1),V1) | (57) |
a__U21#(tt,V1) | → | a__isNat#(V1) | (46) |
mark#(U12(X1,X2)) | → | mark#(X1) | (70) |
mark#(U12(X1,X2)) | → | a__U12#(mark(X1),X2) | (71) |
mark#(isNat(X)) | → | a__isNat#(X) | (72) |
mark#(U13(X)) | → | mark#(X) | (73) |
mark#(U21(X1,X2)) | → | mark#(X1) | (75) |
mark#(U21(X1,X2)) | → | a__U21#(mark(X1),X2) | (76) |
mark#(U22(X)) | → | mark#(X) | (77) |
mark#(U31(X1,X2)) | → | mark#(X1) | (79) |
mark#(U31(X1,X2)) | → | a__U31#(mark(X1),X2) | (80) |
a__U31#(tt,N) | → | mark#(N) | (48) |
mark#(U41(X1,X2,X3)) | → | mark#(X1) | (81) |
mark#(U41(X1,X2,X3)) | → | a__U41#(mark(X1),X2,X3) | (82) |
a__U41#(tt,M,N) | → | mark#(M) | (49) |
mark#(plus(X1,X2)) | → | mark#(X2) | (83) |
mark#(plus(X1,X2)) | → | mark#(X1) | (84) |
mark#(plus(X1,X2)) | → | a__plus#(mark(X1),mark(X2)) | (85) |
a__plus#(N,0) | → | a__isNat#(N) | (61) |
a__plus#(N,0) | → | a__and#(a__isNat(N),isNatKind(N)) | (62) |
a__plus#(N,0) | → | a__U31#(a__and(a__isNat(N),isNatKind(N)),N) | (63) |
a__plus#(N,s(M)) | → | a__isNat#(M) | (64) |
a__plus#(N,s(M)) | → | a__and#(a__isNat(M),isNatKind(M)) | (65) |
a__plus#(N,s(M)) | → | a__and#(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))) | (66) |
a__plus#(N,s(M)) | → | a__U41#(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N) | (67) |
a__U41#(tt,M,N) | → | mark#(N) | (50) |
mark#(and(X1,X2)) | → | mark#(X1) | (86) |
mark#(and(X1,X2)) | → | a__and#(mark(X1),X2) | (87) |
mark#(isNatKind(X)) | → | a__isNatKind#(X) | (88) |
mark#(s(X)) | → | mark#(X) | (89) |
a__U41#(tt,M,N) | → | a__plus#(mark(N),mark(M)) | (51) |
[and(x1, x2)] | = | 0 · x1 + 0 · x2 + -∞ |
[a__U13(x1)] | = | 0 · x1 + 0 |
[a__isNatKind(x1)] | = | 0 · x1 + 0 |
[mark(x1)] | = | 0 · x1 + 0 |
[a__U11(x1, x2, x3)] | = | 0 · x1 + 4 · x2 + 0 · x3 + 0 |
[a__U12(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[isNatKind(x1)] | = | 0 · x1 + -∞ |
[U13(x1)] | = | 0 · x1 + -∞ |
[a__and#(x1, x2)] | = | -∞ · x1 + 0 · x2 + -∞ |
[a__U11#(x1, x2, x3)] | = | 0 · x1 + 4 · x2 + 0 · x3 + 0 |
[a__U31#(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[U11(x1, x2, x3)] | = | 0 · x1 + 4 · x2 + 0 · x3 + 0 |
[isNat(x1)] | = | 0 · x1 + -∞ |
[a__isNat#(x1)] | = | 0 · x1 + -∞ |
[a__U41#(x1, x2, x3)] | = | 0 · x1 + 0 · x2 + 7 · x3 + 0 |
[U21(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[a__U12#(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[a__U21(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[mark#(x1)] | = | 0 · x1 + -∞ |
[a__plus(x1, x2)] | = | 7 · x1 + 0 · x2 + 7 |
[tt] | = | 7 |
[a__U31(x1, x2)] | = | 0 · x1 + 7 · x2 + 3 |
[U12(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[U41(x1, x2, x3)] | = | 0 · x1 + 0 · x2 + 7 · x3 + 2 |
[a__U21#(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[plus(x1, x2)] | = | 7 · x1 + 0 · x2 + 7 |
[a__and(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[U22(x1)] | = | 0 · x1 + -∞ |
[U31(x1, x2)] | = | 0 · x1 + 7 · x2 + 3 |
[s(x1)] | = | 0 · x1 + 0 |
[a__U41(x1, x2, x3)] | = | 0 · x1 + 0 · x2 + 7 · x3 + 2 |
[0] | = | 7 |
[a__U22(x1)] | = | 0 · x1 + 0 |
[a__isNatKind#(x1)] | = | 0 · x1 + -∞ |
[a__plus#(x1, x2)] | = | 7 · x1 + 0 · x2 + 0 |
[a__isNat(x1)] | = | 0 · x1 + 0 |
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__isNatKind#(plus(V1,V2)) | → | a__isNatKind#(V1) | (58) |
a__U11#(tt,V1,V2) | → | a__isNat#(V1) | (42) |
a__isNat#(plus(V1,V2)) | → | a__isNatKind#(V1) | (53) |
mark#(plus(X1,X2)) | → | mark#(X1) | (84) |
a__plus#(N,0) | → | a__isNat#(N) | (61) |
a__plus#(N,0) | → | a__and#(a__isNat(N),isNatKind(N)) | (62) |
a__plus#(N,0) | → | a__U31#(a__and(a__isNat(N),isNatKind(N)),N) | (63) |
a__plus#(N,s(M)) | → | a__and#(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))) | (66) |
a__U41#(tt,M,N) | → | mark#(N) | (50) |
[and(x1, x2)] | = | 2 · x1 + 1 · x2 + 0 |
[a__U13(x1)] | = | 1 · x1 + 0 |
[a__isNatKind(x1)] | = | 0 · x1 + 0 |
[mark(x1)] | = | 1 · x1 + 0 |
[a__U11(x1, x2, x3)] | = | 1 · x1 + 0 · x2 + 0 · x3 + 0 |
[a__U12(x1, x2)] | = | 4 · x1 + 0 · x2 + 0 |
[isNatKind(x1)] | = | 0 · x1 + 0 |
[U13(x1)] | = | 1 · x1 + 0 |
[a__and#(x1, x2)] | = | 0 · x1 + 4 · x2 + 0 |
[a__U11#(x1, x2, x3)] | = | 0 · x1 + 0 · x2 + 0 · x3 + 0 |
[a__U31#(x1, x2)] | = | 0 · x1 + 4 · x2 + 0 |
[U11(x1, x2, x3)] | = | 1 · x1 + 0 · x2 + 0 · x3 + 0 |
[isNat(x1)] | = | 0 · x1 + 0 |
[a__isNat#(x1)] | = | 0 · x1 + 0 |
[a__U41#(x1, x2, x3)] | = | 0 · x1 + 4 · x2 + 0 · x3 + 0 |
[U21(x1, x2)] | = | 1 · x1 + 0 · x2 + 0 |
[a__U12#(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[a__U21(x1, x2)] | = | 1 · x1 + 0 · x2 + 0 |
[mark#(x1)] | = | 4 · x1 + 0 |
[a__plus(x1, x2)] | = | 1 · x1 + 1 · x2 + 0 |
[tt] | = | 0 |
[a__U31(x1, x2)] | = | 4 · x1 + 1 · x2 + 0 |
[U12(x1, x2)] | = | 4 · x1 + 0 · x2 + 0 |
[U41(x1, x2, x3)] | = | 2 · x1 + 1 · x2 + 1 · x3 + 1 |
[a__U21#(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[plus(x1, x2)] | = | 1 · x1 + 1 · x2 + 0 |
[a__and(x1, x2)] | = | 2 · x1 + 1 · x2 + 0 |
[U22(x1)] | = | 4 · x1 + 0 |
[U31(x1, x2)] | = | 4 · x1 + 1 · x2 + 0 |
[s(x1)] | = | 1 · x1 + 1 |
[a__U41(x1, x2, x3)] | = | 2 · x1 + 1 · x2 + 1 · x3 + 1 |
[0] | = | 0 |
[a__U22(x1)] | = | 4 · x1 + 0 |
[a__isNatKind#(x1)] | = | 0 · x1 + 0 |
[a__plus#(x1, x2)] | = | 0 · x1 + 4 · x2 + 0 |
[a__isNat(x1)] | = | 0 · x1 + 0 |
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) |
mark#(U41(X1,X2,X3)) | → | mark#(X1) | (81) |
mark#(U41(X1,X2,X3)) | → | a__U41#(mark(X1),X2,X3) | (82) |
a__plus#(N,s(M)) | → | a__isNat#(M) | (64) |
a__plus#(N,s(M)) | → | a__and#(a__isNat(M),isNatKind(M)) | (65) |
a__plus#(N,s(M)) | → | a__U41#(a__and(a__and(a__isNat(M),isNatKind(M)),and(isNat(N),isNatKind(N))),M,N) | (67) |
mark#(s(X)) | → | mark#(X) | (89) |
The dependency pairs are split into 1 component.
a__isNatKind#(plus(V1,V2)) | → | a__and#(a__isNatKind(V1),isNatKind(V2)) | (59) |
a__and#(tt,X) | → | mark#(X) | (52) |
mark#(U11(X1,X2,X3)) | → | mark#(X1) | (68) |
mark#(U11(X1,X2,X3)) | → | a__U11#(mark(X1),X2,X3) | (69) |
a__U11#(tt,V1,V2) | → | a__U12#(a__isNat(V1),V2) | (43) |
a__U12#(tt,V2) | → | a__isNat#(V2) | (44) |
a__isNat#(plus(V1,V2)) | → | a__and#(a__isNatKind(V1),isNatKind(V2)) | (54) |
a__isNat#(plus(V1,V2)) | → | a__U11#(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2) | (55) |
a__isNat#(s(V1)) | → | a__isNatKind#(V1) | (56) |
a__isNatKind#(s(V1)) | → | a__isNatKind#(V1) | (60) |
a__isNat#(s(V1)) | → | a__U21#(a__isNatKind(V1),V1) | (57) |
a__U21#(tt,V1) | → | a__isNat#(V1) | (46) |
mark#(U12(X1,X2)) | → | mark#(X1) | (70) |
mark#(U12(X1,X2)) | → | a__U12#(mark(X1),X2) | (71) |
mark#(isNat(X)) | → | a__isNat#(X) | (72) |
mark#(U13(X)) | → | mark#(X) | (73) |
mark#(U21(X1,X2)) | → | mark#(X1) | (75) |
mark#(U21(X1,X2)) | → | a__U21#(mark(X1),X2) | (76) |
mark#(U22(X)) | → | mark#(X) | (77) |
mark#(U31(X1,X2)) | → | mark#(X1) | (79) |
mark#(U31(X1,X2)) | → | a__U31#(mark(X1),X2) | (80) |
a__U31#(tt,N) | → | mark#(N) | (48) |
mark#(plus(X1,X2)) | → | mark#(X2) | (83) |
mark#(and(X1,X2)) | → | mark#(X1) | (86) |
mark#(and(X1,X2)) | → | a__and#(mark(X1),X2) | (87) |
mark#(isNatKind(X)) | → | a__isNatKind#(X) | (88) |
π(a__isNatKind#) | = | { 1 } |
π(a__and#) | = | { 2 } |
π(mark#) | = | { 1 } |
π(a__U31#) | = | { 2 } |
π(a__U21#) | = | { 2 } |
π(a__U12#) | = | { 2 } |
π(a__isNat#) | = | { 1 } |
π(a__U11#) | = | { 3 } |
π(isNatKind) | = | { 1 } |
a__isNatKind#(plus(V1,V2)) | → | a__and#(a__isNatKind(V1),isNatKind(V2)) | (59) |
mark#(U11(X1,X2,X3)) | → | mark#(X1) | (68) |
mark#(U11(X1,X2,X3)) | → | a__U11#(mark(X1),X2,X3) | (69) |
a__isNat#(plus(V1,V2)) | → | a__and#(a__isNatKind(V1),isNatKind(V2)) | (54) |
a__isNat#(plus(V1,V2)) | → | a__U11#(a__and(a__isNatKind(V1),isNatKind(V2)),V1,V2) | (55) |
a__isNat#(s(V1)) | → | a__isNatKind#(V1) | (56) |
a__isNatKind#(s(V1)) | → | a__isNatKind#(V1) | (60) |
a__isNat#(s(V1)) | → | a__U21#(a__isNatKind(V1),V1) | (57) |
mark#(U12(X1,X2)) | → | mark#(X1) | (70) |
mark#(U12(X1,X2)) | → | a__U12#(mark(X1),X2) | (71) |
mark#(isNat(X)) | → | a__isNat#(X) | (72) |
mark#(U13(X)) | → | mark#(X) | (73) |
mark#(U21(X1,X2)) | → | mark#(X1) | (75) |
mark#(U21(X1,X2)) | → | a__U21#(mark(X1),X2) | (76) |
mark#(U22(X)) | → | mark#(X) | (77) |
mark#(U31(X1,X2)) | → | mark#(X1) | (79) |
mark#(U31(X1,X2)) | → | a__U31#(mark(X1),X2) | (80) |
mark#(plus(X1,X2)) | → | mark#(X2) | (83) |
mark#(and(X1,X2)) | → | mark#(X1) | (86) |
mark#(and(X1,X2)) | → | a__and#(mark(X1),X2) | (87) |
The dependency pairs are split into 0 components.