The rewrite relation of the following TRS is considered.
U11(tt,V1,V2) | → | U12(isNat(activate(V1)),activate(V2)) | (1) |
U12(tt,V2) | → | U13(isNat(activate(V2))) | (2) |
U13(tt) | → | tt | (3) |
U21(tt,V1) | → | U22(isNat(activate(V1))) | (4) |
U22(tt) | → | tt | (5) |
U31(tt,N) | → | activate(N) | (6) |
U41(tt,M,N) | → | s(plus(activate(N),activate(M))) | (7) |
and(tt,X) | → | activate(X) | (8) |
isNat(n__0) | → | tt | (9) |
isNat(n__plus(V1,V2)) | → | U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) | (10) |
isNat(n__s(V1)) | → | U21(isNatKind(activate(V1)),activate(V1)) | (11) |
isNatKind(n__0) | → | tt | (12) |
isNatKind(n__plus(V1,V2)) | → | and(isNatKind(activate(V1)),n__isNatKind(activate(V2))) | (13) |
isNatKind(n__s(V1)) | → | isNatKind(activate(V1)) | (14) |
plus(N,0) | → | U31(and(isNat(N),n__isNatKind(N)),N) | (15) |
plus(N,s(M)) | → | U41(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))),M,N) | (16) |
0 | → | n__0 | (17) |
plus(X1,X2) | → | n__plus(X1,X2) | (18) |
isNatKind(X) | → | n__isNatKind(X) | (19) |
s(X) | → | n__s(X) | (20) |
and(X1,X2) | → | n__and(X1,X2) | (21) |
isNat(X) | → | n__isNat(X) | (22) |
activate(n__0) | → | 0 | (23) |
activate(n__plus(X1,X2)) | → | plus(activate(X1),activate(X2)) | (24) |
activate(n__isNatKind(X)) | → | isNatKind(X) | (25) |
activate(n__s(X)) | → | s(activate(X)) | (26) |
activate(n__and(X1,X2)) | → | and(activate(X1),X2) | (27) |
activate(n__isNat(X)) | → | isNat(X) | (28) |
activate(X) | → | X | (29) |
U12#(tt,V2) | → | isNat#(activate(V2)) | (30) |
U11#(tt,V1,V2) | → | activate#(V2) | (31) |
isNat#(n__plus(V1,V2)) | → | activate#(V1) | (32) |
activate#(n__s(X)) | → | activate#(X) | (33) |
plus#(N,0) | → | and#(isNat(N),n__isNatKind(N)) | (34) |
U11#(tt,V1,V2) | → | activate#(V1) | (35) |
U11#(tt,V1,V2) | → | isNat#(activate(V1)) | (36) |
plus#(N,s(M)) | → | and#(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))) | (37) |
plus#(N,0) | → | isNat#(N) | (38) |
U21#(tt,V1) | → | U22#(isNat(activate(V1))) | (39) |
plus#(N,s(M)) | → | and#(isNat(M),n__isNatKind(M)) | (40) |
isNatKind#(n__plus(V1,V2)) | → | isNatKind#(activate(V1)) | (41) |
U11#(tt,V1,V2) | → | U12#(isNat(activate(V1)),activate(V2)) | (42) |
activate#(n__plus(X1,X2)) | → | plus#(activate(X1),activate(X2)) | (43) |
isNat#(n__s(V1)) | → | U21#(isNatKind(activate(V1)),activate(V1)) | (44) |
isNat#(n__s(V1)) | → | activate#(V1) | (45) |
isNat#(n__plus(V1,V2)) | → | U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) | (46) |
U41#(tt,M,N) | → | activate#(M) | (47) |
plus#(N,0) | → | U31#(and(isNat(N),n__isNatKind(N)),N) | (48) |
isNat#(n__s(V1)) | → | activate#(V1) | (45) |
activate#(n__isNat(X)) | → | isNat#(X) | (49) |
activate#(n__plus(X1,X2)) | → | activate#(X1) | (50) |
isNat#(n__plus(V1,V2)) | → | activate#(V2) | (51) |
U21#(tt,V1) | → | isNat#(activate(V1)) | (52) |
isNat#(n__plus(V1,V2)) | → | and#(isNatKind(activate(V1)),n__isNatKind(activate(V2))) | (53) |
U41#(tt,M,N) | → | s#(plus(activate(N),activate(M))) | (54) |
isNatKind#(n__plus(V1,V2)) | → | activate#(V1) | (55) |
plus#(N,s(M)) | → | isNat#(M) | (56) |
isNat#(n__s(V1)) | → | isNatKind#(activate(V1)) | (57) |
activate#(n__and(X1,X2)) | → | activate#(X1) | (58) |
isNatKind#(n__plus(V1,V2)) | → | and#(isNatKind(activate(V1)),n__isNatKind(activate(V2))) | (59) |
and#(tt,X) | → | activate#(X) | (60) |
isNat#(n__plus(V1,V2)) | → | activate#(V2) | (51) |
U41#(tt,M,N) | → | activate#(N) | (61) |
activate#(n__s(X)) | → | s#(activate(X)) | (62) |
isNat#(n__plus(V1,V2)) | → | activate#(V1) | (32) |
isNatKind#(n__s(V1)) | → | isNatKind#(activate(V1)) | (63) |
activate#(n__and(X1,X2)) | → | and#(activate(X1),X2) | (64) |
activate#(n__isNatKind(X)) | → | isNatKind#(X) | (65) |
isNat#(n__plus(V1,V2)) | → | isNatKind#(activate(V1)) | (66) |
plus#(N,s(M)) | → | U41#(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))),M,N) | (67) |
U41#(tt,M,N) | → | plus#(activate(N),activate(M)) | (68) |
activate#(n__0) | → | 0# | (69) |
U12#(tt,V2) | → | activate#(V2) | (70) |
U12#(tt,V2) | → | U13#(isNat(activate(V2))) | (71) |
isNatKind#(n__plus(V1,V2)) | → | activate#(V2) | (72) |
activate#(n__plus(X1,X2)) | → | activate#(X2) | (73) |
U31#(tt,N) | → | activate#(N) | (74) |
U21#(tt,V1) | → | activate#(V1) | (75) |
isNatKind#(n__s(V1)) | → | activate#(V1) | (76) |
The dependency pairs are split into 1 component.
isNatKind#(n__s(V1)) | → | activate#(V1) | (76) |
U21#(tt,V1) | → | activate#(V1) | (75) |
isNat#(n__plus(V1,V2)) | → | and#(isNatKind(activate(V1)),n__isNatKind(activate(V2))) | (53) |
U21#(tt,V1) | → | isNat#(activate(V1)) | (52) |
U31#(tt,N) | → | activate#(N) | (74) |
isNat#(n__plus(V1,V2)) | → | activate#(V2) | (51) |
activate#(n__plus(X1,X2)) | → | activate#(X1) | (50) |
activate#(n__plus(X1,X2)) | → | activate#(X2) | (73) |
activate#(n__isNat(X)) | → | isNat#(X) | (49) |
isNat#(n__s(V1)) | → | activate#(V1) | (45) |
isNatKind#(n__plus(V1,V2)) | → | activate#(V2) | (72) |
plus#(N,0) | → | U31#(and(isNat(N),n__isNatKind(N)),N) | (48) |
U41#(tt,M,N) | → | activate#(M) | (47) |
U12#(tt,V2) | → | activate#(V2) | (70) |
isNat#(n__plus(V1,V2)) | → | U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) | (46) |
U41#(tt,M,N) | → | plus#(activate(N),activate(M)) | (68) |
plus#(N,s(M)) | → | U41#(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))),M,N) | (67) |
isNat#(n__plus(V1,V2)) | → | isNatKind#(activate(V1)) | (66) |
isNat#(n__s(V1)) | → | activate#(V1) | (45) |
isNat#(n__s(V1)) | → | U21#(isNatKind(activate(V1)),activate(V1)) | (44) |
activate#(n__plus(X1,X2)) | → | plus#(activate(X1),activate(X2)) | (43) |
activate#(n__and(X1,X2)) | → | and#(activate(X1),X2) | (64) |
activate#(n__isNatKind(X)) | → | isNatKind#(X) | (65) |
isNatKind#(n__s(V1)) | → | isNatKind#(activate(V1)) | (63) |
U11#(tt,V1,V2) | → | U12#(isNat(activate(V1)),activate(V2)) | (42) |
isNatKind#(n__plus(V1,V2)) | → | isNatKind#(activate(V1)) | (41) |
isNat#(n__plus(V1,V2)) | → | activate#(V1) | (32) |
plus#(N,s(M)) | → | and#(isNat(M),n__isNatKind(M)) | (40) |
U41#(tt,M,N) | → | activate#(N) | (61) |
isNat#(n__plus(V1,V2)) | → | activate#(V2) | (51) |
plus#(N,0) | → | isNat#(N) | (38) |
and#(tt,X) | → | activate#(X) | (60) |
isNatKind#(n__plus(V1,V2)) | → | and#(isNatKind(activate(V1)),n__isNatKind(activate(V2))) | (59) |
plus#(N,s(M)) | → | and#(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))) | (37) |
activate#(n__and(X1,X2)) | → | activate#(X1) | (58) |
U11#(tt,V1,V2) | → | isNat#(activate(V1)) | (36) |
U11#(tt,V1,V2) | → | activate#(V1) | (35) |
isNat#(n__s(V1)) | → | isNatKind#(activate(V1)) | (57) |
plus#(N,0) | → | and#(isNat(N),n__isNatKind(N)) | (34) |
plus#(N,s(M)) | → | isNat#(M) | (56) |
isNatKind#(n__plus(V1,V2)) | → | activate#(V1) | (55) |
activate#(n__s(X)) | → | activate#(X) | (33) |
isNat#(n__plus(V1,V2)) | → | activate#(V1) | (32) |
U11#(tt,V1,V2) | → | activate#(V2) | (31) |
U12#(tt,V2) | → | isNat#(activate(V2)) | (30) |
[0#] | = | 0 |
[isNatKind(x1)] | = | x1 + 2 |
[U21(x1, x2)] | = | max(x2 + 3, 0) |
[U11(x1, x2, x3)] | = | max(x2 + 3, x3 + 1664, 0) |
[s(x1)] | = | x1 + 0 |
[isNat#(x1)] | = | x1 + 3 |
[activate(x1)] | = | x1 + 0 |
[n__isNatKind(x1)] | = | x1 + 2 |
[and(x1, x2)] | = | max(x1 + 2, x2 + 3, 0) |
[plus#(x1, x2)] | = | max(x1 + 9, x2 + 8, 0) |
[activate#(x1)] | = | x1 + 1 |
[U13#(x1)] | = | 0 |
[U12(x1, x2)] | = | max(x1 + 0, x2 + 14, 0) |
[n__s(x1)] | = | x1 + 0 |
[U12#(x1, x2)] | = | max(x1 + 3, x2 + 4, 0) |
[0] | = | 7 |
[s#(x1)] | = | 0 |
[n__isNat(x1)] | = | x1 + 3 |
[n__plus(x1, x2)] | = | max(x1 + 10, x2 + 2251, 0) |
[n__0] | = | 7 |
[isNat(x1)] | = | x1 + 3 |
[plus(x1, x2)] | = | max(x1 + 10, x2 + 2251, 0) |
[U11#(x1, x2, x3)] | = | max(x1 + 0, x2 + 12, x3 + 5, 0) |
[U31(x1, x2)] | = | max(x1 + 0, x2 + 1, 0) |
[U41#(x1, x2, x3)] | = | max(x1 + 1, x2 + 8, x3 + 9, 0) |
[U21#(x1, x2)] | = | max(x1 + 0, x2 + 3, 0) |
[U22#(x1)] | = | 0 |
[tt] | = | 9 |
[n__and(x1, x2)] | = | max(x1 + 2, x2 + 3, 0) |
[U13(x1)] | = | 9 |
[U22(x1)] | = | x1 + 0 |
[isNatKind#(x1)] | = | x1 + 2 |
[U41(x1, x2, x3)] | = | max(x1 + 0, x2 + 2251, x3 + 10, 0) |
[U31#(x1, x2)] | = | max(x1 + 0, x2 + 5, 0) |
[and#(x1, x2)] | = | max(x2 + 2, 0) |
plus(X1,X2) | → | n__plus(X1,X2) | (18) |
U21(tt,V1) | → | U22(isNat(activate(V1))) | (4) |
plus(N,0) | → | U31(and(isNat(N),n__isNatKind(N)),N) | (15) |
and(tt,X) | → | activate(X) | (8) |
U11(tt,V1,V2) | → | U12(isNat(activate(V1)),activate(V2)) | (1) |
U13(tt) | → | tt | (3) |
plus(N,s(M)) | → | U41(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))),M,N) | (16) |
and(X1,X2) | → | n__and(X1,X2) | (21) |
activate(n__s(X)) | → | s(activate(X)) | (26) |
isNatKind(X) | → | n__isNatKind(X) | (19) |
0 | → | n__0 | (17) |
activate(n__and(X1,X2)) | → | and(activate(X1),X2) | (27) |
isNat(X) | → | n__isNat(X) | (22) |
activate(n__isNat(X)) | → | isNat(X) | (28) |
U22(tt) | → | tt | (5) |
isNat(n__plus(V1,V2)) | → | U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) | (10) |
U41(tt,M,N) | → | s(plus(activate(N),activate(M))) | (7) |
s(X) | → | n__s(X) | (20) |
activate(n__isNatKind(X)) | → | isNatKind(X) | (25) |
isNatKind(n__s(V1)) | → | isNatKind(activate(V1)) | (14) |
isNatKind(n__0) | → | tt | (12) |
activate(n__0) | → | 0 | (23) |
activate(n__plus(X1,X2)) | → | plus(activate(X1),activate(X2)) | (24) |
isNat(n__s(V1)) | → | U21(isNatKind(activate(V1)),activate(V1)) | (11) |
isNat(n__0) | → | tt | (9) |
isNatKind(n__plus(V1,V2)) | → | and(isNatKind(activate(V1)),n__isNatKind(activate(V2))) | (13) |
U31(tt,N) | → | activate(N) | (6) |
activate(X) | → | X | (29) |
U12(tt,V2) | → | U13(isNat(activate(V2))) | (2) |
isNatKind#(n__s(V1)) | → | activate#(V1) | (76) |
U21#(tt,V1) | → | activate#(V1) | (75) |
isNat#(n__plus(V1,V2)) | → | and#(isNatKind(activate(V1)),n__isNatKind(activate(V2))) | (53) |
U31#(tt,N) | → | activate#(N) | (74) |
isNat#(n__plus(V1,V2)) | → | activate#(V2) | (51) |
activate#(n__plus(X1,X2)) | → | activate#(X1) | (50) |
activate#(n__plus(X1,X2)) | → | activate#(X2) | (73) |
activate#(n__isNat(X)) | → | isNat#(X) | (49) |
isNat#(n__s(V1)) | → | activate#(V1) | (45) |
isNatKind#(n__plus(V1,V2)) | → | activate#(V2) | (72) |
plus#(N,0) | → | U31#(and(isNat(N),n__isNatKind(N)),N) | (48) |
U41#(tt,M,N) | → | activate#(M) | (47) |
U12#(tt,V2) | → | activate#(V2) | (70) |
isNat#(n__plus(V1,V2)) | → | U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) | (46) |
isNat#(n__plus(V1,V2)) | → | isNatKind#(activate(V1)) | (66) |
isNat#(n__s(V1)) | → | activate#(V1) | (45) |
activate#(n__plus(X1,X2)) | → | plus#(activate(X1),activate(X2)) | (43) |
activate#(n__and(X1,X2)) | → | and#(activate(X1),X2) | (64) |
activate#(n__isNatKind(X)) | → | isNatKind#(X) | (65) |
U11#(tt,V1,V2) | → | U12#(isNat(activate(V1)),activate(V2)) | (42) |
isNatKind#(n__plus(V1,V2)) | → | isNatKind#(activate(V1)) | (41) |
isNat#(n__plus(V1,V2)) | → | activate#(V1) | (32) |
plus#(N,s(M)) | → | and#(isNat(M),n__isNatKind(M)) | (40) |
U41#(tt,M,N) | → | activate#(N) | (61) |
isNat#(n__plus(V1,V2)) | → | activate#(V2) | (51) |
plus#(N,0) | → | isNat#(N) | (38) |
and#(tt,X) | → | activate#(X) | (60) |
isNatKind#(n__plus(V1,V2)) | → | and#(isNatKind(activate(V1)),n__isNatKind(activate(V2))) | (59) |
plus#(N,s(M)) | → | and#(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))) | (37) |
activate#(n__and(X1,X2)) | → | activate#(X1) | (58) |
U11#(tt,V1,V2) | → | isNat#(activate(V1)) | (36) |
U11#(tt,V1,V2) | → | activate#(V1) | (35) |
isNat#(n__s(V1)) | → | isNatKind#(activate(V1)) | (57) |
plus#(N,0) | → | and#(isNat(N),n__isNatKind(N)) | (34) |
plus#(N,s(M)) | → | isNat#(M) | (56) |
isNatKind#(n__plus(V1,V2)) | → | activate#(V1) | (55) |
isNat#(n__plus(V1,V2)) | → | activate#(V1) | (32) |
U11#(tt,V1,V2) | → | activate#(V2) | (31) |
U12#(tt,V2) | → | isNat#(activate(V2)) | (30) |
The dependency pairs are split into 4 components.
activate#(n__s(X)) | → | activate#(X) | (33) |
[0#] | = | 0 |
[isNatKind(x1)] | = | 4 |
[U21(x1, x2)] | = | x2 + 4 |
[U11(x1, x2, x3)] | = | x3 + 6 |
[s(x1)] | = | x1 + 1 |
[isNat#(x1)] | = | 4 |
[activate(x1)] | = | 3 |
[n__isNatKind(x1)] | = | 4 |
[and(x1, x2)] | = | x2 + 0 |
[plus#(x1, x2)] | = | 4 |
[activate#(x1)] | = | x1 + 4 |
[U13#(x1)] | = | 0 |
[U12(x1, x2)] | = | x1 + 0 |
[n__s(x1)] | = | x1 + 2 |
[U12#(x1, x2)] | = | 4 |
[0] | = | 4 |
[s#(x1)] | = | 0 |
[n__isNat(x1)] | = | 5 |
[n__plus(x1, x2)] | = | 2 |
[n__0] | = | 5 |
[isNat(x1)] | = | x1 + 4 |
[plus(x1, x2)] | = | x1 + 1 |
[U11#(x1, x2, x3)] | = | 4 |
[U31(x1, x2)] | = | 2 |
[U41#(x1, x2, x3)] | = | 4 |
[U21#(x1, x2)] | = | 4 |
[U22#(x1)] | = | 0 |
[tt] | = | 10 |
[n__and(x1, x2)] | = | 18005 |
[U13(x1)] | = | 11 |
[U22(x1)] | = | 9 |
[isNatKind#(x1)] | = | 4 |
[U41(x1, x2, x3)] | = | 2 |
[U31#(x1, x2)] | = | 3 |
[and#(x1, x2)] | = | 4 |
activate#(n__s(X)) | → | activate#(X) | (33) |
The dependency pairs are split into 0 components.
plus#(N,s(M)) | → | U41#(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))),M,N) | (67) |
U41#(tt,M,N) | → | plus#(activate(N),activate(M)) | (68) |
[0#] | = | 0 |
[isNatKind(x1)] | = | 1 |
[U21(x1, x2)] | = | x2 + 9 |
[U11(x1, x2, x3)] | = | x2 + x3 + 2 |
[s(x1)] | = | x1 + 2 |
[isNat#(x1)] | = | 4 |
[activate(x1)] | = | x1 + 0 |
[n__isNatKind(x1)] | = | 1 |
[and(x1, x2)] | = | x2 + 0 |
[plus#(x1, x2)] | = | x2 + 4 |
[activate#(x1)] | = | 4 |
[U13#(x1)] | = | 0 |
[U12(x1, x2)] | = | 2 |
[n__s(x1)] | = | x1 + 2 |
[U12#(x1, x2)] | = | 4 |
[0] | = | 10377 |
[s#(x1)] | = | 0 |
[n__isNat(x1)] | = | x1 + 7 |
[n__plus(x1, x2)] | = | x1 + x2 + 12063 |
[n__0] | = | 10377 |
[isNat(x1)] | = | x1 + 7 |
[plus(x1, x2)] | = | x1 + x2 + 12063 |
[U11#(x1, x2, x3)] | = | 4 |
[U31(x1, x2)] | = | x2 + 22440 |
[U41#(x1, x2, x3)] | = | x2 + 5 |
[U21#(x1, x2)] | = | 4 |
[U22#(x1)] | = | 0 |
[tt] | = | 1 |
[n__and(x1, x2)] | = | x2 + 0 |
[U13(x1)] | = | 2 |
[U22(x1)] | = | 9 |
[isNatKind#(x1)] | = | 4 |
[U41(x1, x2, x3)] | = | x2 + x3 + 12065 |
[U31#(x1, x2)] | = | 3 |
[and#(x1, x2)] | = | 4 |
plus(X1,X2) | → | n__plus(X1,X2) | (18) |
U21(tt,V1) | → | U22(isNat(activate(V1))) | (4) |
plus(N,0) | → | U31(and(isNat(N),n__isNatKind(N)),N) | (15) |
and(tt,X) | → | activate(X) | (8) |
U11(tt,V1,V2) | → | U12(isNat(activate(V1)),activate(V2)) | (1) |
U13(tt) | → | tt | (3) |
plus(N,s(M)) | → | U41(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))),M,N) | (16) |
and(X1,X2) | → | n__and(X1,X2) | (21) |
activate(n__s(X)) | → | s(activate(X)) | (26) |
isNatKind(X) | → | n__isNatKind(X) | (19) |
0 | → | n__0 | (17) |
activate(n__and(X1,X2)) | → | and(activate(X1),X2) | (27) |
isNat(X) | → | n__isNat(X) | (22) |
activate(n__isNat(X)) | → | isNat(X) | (28) |
U22(tt) | → | tt | (5) |
isNat(n__plus(V1,V2)) | → | U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) | (10) |
U41(tt,M,N) | → | s(plus(activate(N),activate(M))) | (7) |
s(X) | → | n__s(X) | (20) |
activate(n__isNatKind(X)) | → | isNatKind(X) | (25) |
isNatKind(n__s(V1)) | → | isNatKind(activate(V1)) | (14) |
isNatKind(n__0) | → | tt | (12) |
activate(n__0) | → | 0 | (23) |
activate(n__plus(X1,X2)) | → | plus(activate(X1),activate(X2)) | (24) |
isNat(n__s(V1)) | → | U21(isNatKind(activate(V1)),activate(V1)) | (11) |
isNat(n__0) | → | tt | (9) |
isNatKind(n__plus(V1,V2)) | → | and(isNatKind(activate(V1)),n__isNatKind(activate(V2))) | (13) |
U31(tt,N) | → | activate(N) | (6) |
activate(X) | → | X | (29) |
U12(tt,V2) | → | U13(isNat(activate(V2))) | (2) |
plus#(N,s(M)) | → | U41#(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))),M,N) | (67) |
U41#(tt,M,N) | → | plus#(activate(N),activate(M)) | (68) |
The dependency pairs are split into 0 components.
isNatKind#(n__s(V1)) | → | isNatKind#(activate(V1)) | (63) |
[0#] | = | 0 |
[isNatKind(x1)] | = | 1 |
[U21(x1, x2)] | = | x2 + 2 |
[U11(x1, x2, x3)] | = | x2 + x3 + 2 |
[s(x1)] | = | x1 + 1 |
[isNat#(x1)] | = | 4 |
[activate(x1)] | = | x1 + 0 |
[n__isNatKind(x1)] | = | 1 |
[and(x1, x2)] | = | x2 + 0 |
[plus#(x1, x2)] | = | x2 + 4 |
[activate#(x1)] | = | 4 |
[U13#(x1)] | = | 0 |
[U12(x1, x2)] | = | 2 |
[n__s(x1)] | = | x1 + 1 |
[U12#(x1, x2)] | = | 4 |
[0] | = | 1 |
[s#(x1)] | = | 0 |
[n__isNat(x1)] | = | x1 + 1 |
[n__plus(x1, x2)] | = | x1 + x2 + 1 |
[n__0] | = | 1 |
[isNat(x1)] | = | x1 + 1 |
[plus(x1, x2)] | = | x1 + x2 + 1 |
[U11#(x1, x2, x3)] | = | 4 |
[U31(x1, x2)] | = | x2 + 0 |
[U41#(x1, x2, x3)] | = | 5 |
[U21#(x1, x2)] | = | 4 |
[U22#(x1)] | = | 0 |
[tt] | = | 1 |
[n__and(x1, x2)] | = | x2 + 0 |
[U13(x1)] | = | 2 |
[U22(x1)] | = | 2 |
[isNatKind#(x1)] | = | x1 + 4 |
[U41(x1, x2, x3)] | = | x2 + x3 + 2 |
[U31#(x1, x2)] | = | 3 |
[and#(x1, x2)] | = | 4 |
plus(X1,X2) | → | n__plus(X1,X2) | (18) |
U21(tt,V1) | → | U22(isNat(activate(V1))) | (4) |
plus(N,0) | → | U31(and(isNat(N),n__isNatKind(N)),N) | (15) |
and(tt,X) | → | activate(X) | (8) |
U11(tt,V1,V2) | → | U12(isNat(activate(V1)),activate(V2)) | (1) |
U13(tt) | → | tt | (3) |
plus(N,s(M)) | → | U41(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))),M,N) | (16) |
and(X1,X2) | → | n__and(X1,X2) | (21) |
activate(n__s(X)) | → | s(activate(X)) | (26) |
isNatKind(X) | → | n__isNatKind(X) | (19) |
0 | → | n__0 | (17) |
activate(n__and(X1,X2)) | → | and(activate(X1),X2) | (27) |
isNat(X) | → | n__isNat(X) | (22) |
activate(n__isNat(X)) | → | isNat(X) | (28) |
U22(tt) | → | tt | (5) |
isNat(n__plus(V1,V2)) | → | U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) | (10) |
U41(tt,M,N) | → | s(plus(activate(N),activate(M))) | (7) |
s(X) | → | n__s(X) | (20) |
activate(n__isNatKind(X)) | → | isNatKind(X) | (25) |
isNatKind(n__s(V1)) | → | isNatKind(activate(V1)) | (14) |
isNatKind(n__0) | → | tt | (12) |
activate(n__0) | → | 0 | (23) |
activate(n__plus(X1,X2)) | → | plus(activate(X1),activate(X2)) | (24) |
isNat(n__s(V1)) | → | U21(isNatKind(activate(V1)),activate(V1)) | (11) |
isNat(n__0) | → | tt | (9) |
isNatKind(n__plus(V1,V2)) | → | and(isNatKind(activate(V1)),n__isNatKind(activate(V2))) | (13) |
U31(tt,N) | → | activate(N) | (6) |
activate(X) | → | X | (29) |
U12(tt,V2) | → | U13(isNat(activate(V2))) | (2) |
isNatKind#(n__s(V1)) | → | isNatKind#(activate(V1)) | (63) |
The dependency pairs are split into 0 components.
U21#(tt,V1) | → | isNat#(activate(V1)) | (52) |
isNat#(n__s(V1)) | → | U21#(isNatKind(activate(V1)),activate(V1)) | (44) |
[0#] | = | 0 |
[isNatKind(x1)] | = | 1 |
[U21(x1, x2)] | = | x2 + 2 |
[U11(x1, x2, x3)] | = | x2 + x3 + 2 |
[s(x1)] | = | x1 + 2 |
[isNat#(x1)] | = | x1 + 4 |
[activate(x1)] | = | x1 + 0 |
[n__isNatKind(x1)] | = | 1 |
[and(x1, x2)] | = | x2 + 0 |
[plus#(x1, x2)] | = | x2 + 4 |
[activate#(x1)] | = | 4 |
[U13#(x1)] | = | 0 |
[U12(x1, x2)] | = | 2 |
[n__s(x1)] | = | x1 + 2 |
[U12#(x1, x2)] | = | 4 |
[0] | = | 40990 |
[s#(x1)] | = | 0 |
[n__isNat(x1)] | = | x1 + 1 |
[n__plus(x1, x2)] | = | x1 + x2 + 1 |
[n__0] | = | 40990 |
[isNat(x1)] | = | x1 + 1 |
[plus(x1, x2)] | = | x1 + x2 + 1 |
[U11#(x1, x2, x3)] | = | 4 |
[U31(x1, x2)] | = | x2 + 0 |
[U41#(x1, x2, x3)] | = | 5 |
[U21#(x1, x2)] | = | x2 + 5 |
[U22#(x1)] | = | 0 |
[tt] | = | 1 |
[n__and(x1, x2)] | = | x2 + 0 |
[U13(x1)] | = | 2 |
[U22(x1)] | = | 2 |
[isNatKind#(x1)] | = | x1 + 4 |
[U41(x1, x2, x3)] | = | x2 + x3 + 3 |
[U31#(x1, x2)] | = | 3 |
[and#(x1, x2)] | = | 4 |
plus(X1,X2) | → | n__plus(X1,X2) | (18) |
U21(tt,V1) | → | U22(isNat(activate(V1))) | (4) |
plus(N,0) | → | U31(and(isNat(N),n__isNatKind(N)),N) | (15) |
and(tt,X) | → | activate(X) | (8) |
U11(tt,V1,V2) | → | U12(isNat(activate(V1)),activate(V2)) | (1) |
U13(tt) | → | tt | (3) |
plus(N,s(M)) | → | U41(and(and(isNat(M),n__isNatKind(M)),n__and(n__isNat(N),n__isNatKind(N))),M,N) | (16) |
and(X1,X2) | → | n__and(X1,X2) | (21) |
activate(n__s(X)) | → | s(activate(X)) | (26) |
isNatKind(X) | → | n__isNatKind(X) | (19) |
0 | → | n__0 | (17) |
activate(n__and(X1,X2)) | → | and(activate(X1),X2) | (27) |
isNat(X) | → | n__isNat(X) | (22) |
activate(n__isNat(X)) | → | isNat(X) | (28) |
U22(tt) | → | tt | (5) |
isNat(n__plus(V1,V2)) | → | U11(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) | (10) |
U41(tt,M,N) | → | s(plus(activate(N),activate(M))) | (7) |
s(X) | → | n__s(X) | (20) |
activate(n__isNatKind(X)) | → | isNatKind(X) | (25) |
isNatKind(n__s(V1)) | → | isNatKind(activate(V1)) | (14) |
isNatKind(n__0) | → | tt | (12) |
activate(n__0) | → | 0 | (23) |
activate(n__plus(X1,X2)) | → | plus(activate(X1),activate(X2)) | (24) |
isNat(n__s(V1)) | → | U21(isNatKind(activate(V1)),activate(V1)) | (11) |
isNat(n__0) | → | tt | (9) |
isNatKind(n__plus(V1,V2)) | → | and(isNatKind(activate(V1)),n__isNatKind(activate(V2))) | (13) |
U31(tt,N) | → | activate(N) | (6) |
activate(X) | → | X | (29) |
U12(tt,V2) | → | U13(isNat(activate(V2))) | (2) |
U21#(tt,V1) | → | isNat#(activate(V1)) | (52) |
isNat#(n__s(V1)) | → | U21#(isNatKind(activate(V1)),activate(V1)) | (44) |
The dependency pairs are split into 0 components.