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(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) |
activate(n__0) | → | 0 | (22) |
activate(n__plus(X1,X2)) | → | plus(X1,X2) | (23) |
activate(n__isNatKind(X)) | → | isNatKind(X) | (24) |
activate(n__s(X)) | → | s(X) | (25) |
activate(n__and(X1,X2)) | → | and(X1,X2) | (26) |
activate(X) | → | X | (27) |
U11#(tt,V1,V2) | → | activate#(V2) | (28) |
U11#(tt,V1,V2) | → | activate#(V1) | (29) |
U11#(tt,V1,V2) | → | isNat#(activate(V1)) | (30) |
U11#(tt,V1,V2) | → | U12#(isNat(activate(V1)),activate(V2)) | (31) |
U12#(tt,V2) | → | activate#(V2) | (32) |
U12#(tt,V2) | → | isNat#(activate(V2)) | (33) |
U12#(tt,V2) | → | U13#(isNat(activate(V2))) | (34) |
U21#(tt,V1) | → | activate#(V1) | (35) |
U21#(tt,V1) | → | isNat#(activate(V1)) | (36) |
U21#(tt,V1) | → | U22#(isNat(activate(V1))) | (37) |
U31#(tt,N) | → | activate#(N) | (38) |
U41#(tt,M,N) | → | activate#(M) | (39) |
U41#(tt,M,N) | → | activate#(N) | (40) |
U41#(tt,M,N) | → | plus#(activate(N),activate(M)) | (41) |
U41#(tt,M,N) | → | s#(plus(activate(N),activate(M))) | (42) |
and#(tt,X) | → | activate#(X) | (43) |
isNat#(n__plus(V1,V2)) | → | activate#(V2) | (44) |
isNat#(n__plus(V1,V2)) | → | activate#(V1) | (45) |
isNat#(n__plus(V1,V2)) | → | isNatKind#(activate(V1)) | (46) |
isNat#(n__plus(V1,V2)) | → | and#(isNatKind(activate(V1)),n__isNatKind(activate(V2))) | (47) |
isNat#(n__plus(V1,V2)) | → | U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) | (48) |
isNat#(n__s(V1)) | → | activate#(V1) | (49) |
isNat#(n__s(V1)) | → | isNatKind#(activate(V1)) | (50) |
isNat#(n__s(V1)) | → | U21#(isNatKind(activate(V1)),activate(V1)) | (51) |
isNatKind#(n__plus(V1,V2)) | → | activate#(V2) | (52) |
isNatKind#(n__plus(V1,V2)) | → | activate#(V1) | (53) |
isNatKind#(n__plus(V1,V2)) | → | isNatKind#(activate(V1)) | (54) |
isNatKind#(n__plus(V1,V2)) | → | and#(isNatKind(activate(V1)),n__isNatKind(activate(V2))) | (55) |
isNatKind#(n__s(V1)) | → | activate#(V1) | (56) |
isNatKind#(n__s(V1)) | → | isNatKind#(activate(V1)) | (57) |
plus#(N,0) | → | isNat#(N) | (58) |
plus#(N,0) | → | and#(isNat(N),n__isNatKind(N)) | (59) |
plus#(N,0) | → | U31#(and(isNat(N),n__isNatKind(N)),N) | (60) |
plus#(N,s(M)) | → | isNat#(N) | (61) |
plus#(N,s(M)) | → | isNat#(M) | (62) |
plus#(N,s(M)) | → | and#(isNat(M),n__isNatKind(M)) | (63) |
plus#(N,s(M)) | → | and#(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))) | (64) |
plus#(N,s(M)) | → | U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) | (65) |
activate#(n__0) | → | 0# | (66) |
activate#(n__plus(X1,X2)) | → | plus#(X1,X2) | (67) |
activate#(n__isNatKind(X)) | → | isNatKind#(X) | (68) |
activate#(n__s(X)) | → | s#(X) | (69) |
activate#(n__and(X1,X2)) | → | and#(X1,X2) | (70) |
The dependency pairs are split into 1 component.
isNatKind#(n__s(V1)) | → | isNatKind#(activate(V1)) | (57) |
isNatKind#(n__plus(V1,V2)) | → | activate#(V2) | (52) |
activate#(n__plus(X1,X2)) | → | plus#(X1,X2) | (67) |
plus#(N,0) | → | isNat#(N) | (58) |
isNat#(n__plus(V1,V2)) | → | activate#(V2) | (44) |
activate#(n__isNatKind(X)) | → | isNatKind#(X) | (68) |
isNatKind#(n__plus(V1,V2)) | → | activate#(V1) | (53) |
activate#(n__and(X1,X2)) | → | and#(X1,X2) | (70) |
and#(tt,X) | → | activate#(X) | (43) |
isNatKind#(n__plus(V1,V2)) | → | isNatKind#(activate(V1)) | (54) |
isNatKind#(n__plus(V1,V2)) | → | and#(isNatKind(activate(V1)),n__isNatKind(activate(V2))) | (55) |
isNatKind#(n__s(V1)) | → | activate#(V1) | (56) |
isNat#(n__plus(V1,V2)) | → | activate#(V1) | (45) |
isNat#(n__plus(V1,V2)) | → | isNatKind#(activate(V1)) | (46) |
isNat#(n__plus(V1,V2)) | → | and#(isNatKind(activate(V1)),n__isNatKind(activate(V2))) | (47) |
isNat#(n__plus(V1,V2)) | → | U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) | (48) |
U11#(tt,V1,V2) | → | activate#(V2) | (28) |
U11#(tt,V1,V2) | → | activate#(V1) | (29) |
U11#(tt,V1,V2) | → | isNat#(activate(V1)) | (30) |
isNat#(n__s(V1)) | → | activate#(V1) | (49) |
isNat#(n__s(V1)) | → | isNatKind#(activate(V1)) | (50) |
isNat#(n__s(V1)) | → | U21#(isNatKind(activate(V1)),activate(V1)) | (51) |
U21#(tt,V1) | → | activate#(V1) | (35) |
U21#(tt,V1) | → | isNat#(activate(V1)) | (36) |
U11#(tt,V1,V2) | → | U12#(isNat(activate(V1)),activate(V2)) | (31) |
U12#(tt,V2) | → | activate#(V2) | (32) |
U12#(tt,V2) | → | isNat#(activate(V2)) | (33) |
plus#(N,0) | → | and#(isNat(N),n__isNatKind(N)) | (59) |
plus#(N,0) | → | U31#(and(isNat(N),n__isNatKind(N)),N) | (60) |
U31#(tt,N) | → | activate#(N) | (38) |
plus#(N,s(M)) | → | isNat#(N) | (61) |
plus#(N,s(M)) | → | isNat#(M) | (62) |
plus#(N,s(M)) | → | and#(isNat(M),n__isNatKind(M)) | (63) |
plus#(N,s(M)) | → | and#(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))) | (64) |
plus#(N,s(M)) | → | U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) | (65) |
U41#(tt,M,N) | → | activate#(M) | (39) |
U41#(tt,M,N) | → | activate#(N) | (40) |
U41#(tt,M,N) | → | plus#(activate(N),activate(M)) | (41) |
[0] | = | 2 |
[U12(x1, x2)] | = | 0 · x1 + 4 · x2 + 0 |
[isNatKind(x1)] | = | 0 · x1 + 2 |
[U31(x1, x2)] | = | 0 · x1 + 1 · x2 + 0 |
[U11(x1, x2, x3)] | = | -∞ · x1 + 2 · x2 + 6 · x3 + -∞ |
[isNat(x1)] | = | 5 · x1 + 0 |
[n__isNatKind(x1)] | = | 0 · x1 + 2 |
[activate#(x1)] | = | 0 · x1 + -∞ |
[n__and(x1, x2)] | = | -∞ · x1 + 0 · x2 + -∞ |
[isNatKind#(x1)] | = | 0 · x1 + 0 |
[n__s(x1)] | = | 0 · x1 + 2 |
[U31#(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[isNat#(x1)] | = | 0 · x1 + -∞ |
[U41#(x1, x2, x3)] | = | 0 · x1 + 0 · x2 + 0 · x3 + 0 |
[U13(x1)] | = | 0 · x1 + 0 |
[plus(x1, x2)] | = | 1 · x1 + 0 · x2 + 2 |
[tt] | = | 0 |
[U22(x1)] | = | 1 · x1 + 0 |
[U11#(x1, x2, x3)] | = | 0 · x1 + 0 · x2 + 0 · x3 + 0 |
[U21#(x1, x2)] | = | -∞ · x1 + 0 · x2 + -∞ |
[n__plus(x1, x2)] | = | 1 · x1 + 0 · x2 + 2 |
[and#(x1, x2)] | = | -∞ · x1 + 0 · x2 + -∞ |
[and(x1, x2)] | = | -∞ · x1 + 0 · x2 + -∞ |
[U12#(x1, x2)] | = | -∞ · x1 + 0 · x2 + -∞ |
[s(x1)] | = | 0 · x1 + 2 |
[U41(x1, x2, x3)] | = | 0 · x1 + 0 · x2 + 1 · x3 + 2 |
[n__0] | = | 2 |
[U21(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[plus#(x1, x2)] | = | 0 · x1 + 0 · x2 + -∞ |
[activate(x1)] | = | 0 · x1 + -∞ |
U31(tt,N) | → | activate(N) | (6) |
U41(tt,M,N) | → | s(plus(activate(N),activate(M))) | (7) |
and(tt,X) | → | activate(X) | (8) |
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(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) |
activate(n__0) | → | 0 | (22) |
activate(n__plus(X1,X2)) | → | plus(X1,X2) | (23) |
activate(n__isNatKind(X)) | → | isNatKind(X) | (24) |
activate(n__s(X)) | → | s(X) | (25) |
activate(n__and(X1,X2)) | → | and(X1,X2) | (26) |
activate(X) | → | X | (27) |
isNatKind#(n__plus(V1,V2)) | → | activate#(V1) | (53) |
isNatKind#(n__plus(V1,V2)) | → | isNatKind#(activate(V1)) | (54) |
isNat#(n__plus(V1,V2)) | → | activate#(V1) | (45) |
isNat#(n__plus(V1,V2)) | → | isNatKind#(activate(V1)) | (46) |
[0] | = | 3 |
[U12(x1, x2)] | = | -∞ · x1 + 0 · x2 + 0 |
[isNatKind(x1)] | = | 0 · x1 + -∞ |
[U31(x1, x2)] | = | -∞ · x1 + 0 · x2 + -∞ |
[U11(x1, x2, x3)] | = | -∞ · x1 + 0 · x2 + 4 · x3 + -∞ |
[isNat(x1)] | = | -∞ · x1 + 0 |
[n__isNatKind(x1)] | = | 0 · x1 + -∞ |
[activate#(x1)] | = | 0 · x1 + -∞ |
[n__and(x1, x2)] | = | -∞ · x1 + 0 · x2 + -∞ |
[isNatKind#(x1)] | = | 0 · x1 + -∞ |
[n__s(x1)] | = | 0 · x1 + 0 |
[U31#(x1, x2)] | = | 0 · x1 + 0 · x2 + -∞ |
[isNat#(x1)] | = | 0 · x1 + 0 |
[U41#(x1, x2, x3)] | = | 0 · x1 + 1 · x2 + 0 · x3 + -∞ |
[U13(x1)] | = | 5 · x1 + 6 |
[plus(x1, x2)] | = | 0 · x1 + 1 · x2 + 0 |
[tt] | = | 0 |
[U22(x1)] | = | 5 · x1 + 0 |
[U11#(x1, x2, x3)] | = | -∞ · x1 + 0 · x2 + 0 · x3 + 0 |
[U21#(x1, x2)] | = | -∞ · x1 + 0 · x2 + 0 |
[n__plus(x1, x2)] | = | 0 · x1 + 1 · x2 + 0 |
[and#(x1, x2)] | = | -∞ · x1 + 0 · x2 + -∞ |
[and(x1, x2)] | = | -∞ · x1 + 0 · x2 + -∞ |
[U12#(x1, x2)] | = | -∞ · x1 + 0 · x2 + 0 |
[s(x1)] | = | 0 · x1 + 0 |
[U41(x1, x2, x3)] | = | -∞ · x1 + 1 · x2 + 0 · x3 + 0 |
[n__0] | = | 3 |
[U21(x1, x2)] | = | 0 · x1 + 0 · x2 + -∞ |
[plus#(x1, x2)] | = | 0 · x1 + 1 · x2 + 0 |
[activate(x1)] | = | 0 · x1 + -∞ |
U31(tt,N) | → | activate(N) | (6) |
U41(tt,M,N) | → | s(plus(activate(N),activate(M))) | (7) |
and(tt,X) | → | activate(X) | (8) |
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(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) |
activate(n__0) | → | 0 | (22) |
activate(n__plus(X1,X2)) | → | plus(X1,X2) | (23) |
activate(n__isNatKind(X)) | → | isNatKind(X) | (24) |
activate(n__s(X)) | → | s(X) | (25) |
activate(n__and(X1,X2)) | → | and(X1,X2) | (26) |
activate(X) | → | X | (27) |
isNatKind#(n__plus(V1,V2)) | → | activate#(V2) | (52) |
isNat#(n__plus(V1,V2)) | → | activate#(V2) | (44) |
isNatKind#(n__plus(V1,V2)) | → | and#(isNatKind(activate(V1)),n__isNatKind(activate(V2))) | (55) |
isNat#(n__plus(V1,V2)) | → | and#(isNatKind(activate(V1)),n__isNatKind(activate(V2))) | (47) |
plus#(N,s(M)) | → | isNat#(M) | (62) |
plus#(N,s(M)) | → | and#(isNat(M),n__isNatKind(M)) | (63) |
U41#(tt,M,N) | → | activate#(M) | (39) |
[0] | = | 4 |
[U12(x1, x2)] | = | 3 · x1 + 4 · x2 + 5 |
[isNatKind(x1)] | = | 0 · x1 + 0 |
[U31(x1, x2)] | = | 0 · x1 + 2 · x2 + 5 |
[U11(x1, x2, x3)] | = | 0 · x1 + 6 · x2 + 0 · x3 + 5 |
[isNat(x1)] | = | -∞ · x1 + 0 |
[n__isNatKind(x1)] | = | 0 · x1 + 0 |
[activate#(x1)] | = | 0 · x1 + 0 |
[n__and(x1, x2)] | = | -∞ · x1 + 0 · x2 + 0 |
[isNatKind#(x1)] | = | 0 · x1 + 0 |
[n__s(x1)] | = | 0 · x1 + 0 |
[U31#(x1, x2)] | = | 3 · x1 + 0 · x2 + 0 |
[isNat#(x1)] | = | 0 · x1 + -∞ |
[U41#(x1, x2, x3)] | = | 0 · x1 + 0 · x2 + 3 · x3 + 0 |
[U13(x1)] | = | 1 · x1 + 8 |
[plus(x1, x2)] | = | 3 · x1 + 1 · x2 + 0 |
[tt] | = | 4 |
[U22(x1)] | = | -∞ · x1 + 0 |
[U11#(x1, x2, x3)] | = | 0 · x1 + 0 · x2 + 0 · x3 + 0 |
[U21#(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[n__plus(x1, x2)] | = | 3 · x1 + 1 · x2 + 0 |
[and#(x1, x2)] | = | -∞ · x1 + 0 · x2 + 0 |
[and(x1, x2)] | = | -∞ · x1 + 0 · x2 + 0 |
[U12#(x1, x2)] | = | -∞ · x1 + 0 · x2 + 0 |
[s(x1)] | = | 0 · x1 + 0 |
[U41(x1, x2, x3)] | = | 1 · x1 + 1 · x2 + 3 · x3 + 0 |
[n__0] | = | 4 |
[U21(x1, x2)] | = | 1 · x1 + 1 · x2 + 1 |
[plus#(x1, x2)] | = | 3 · x1 + 0 · x2 + -∞ |
[activate(x1)] | = | 0 · x1 + 0 |
U31(tt,N) | → | activate(N) | (6) |
U41(tt,M,N) | → | s(plus(activate(N),activate(M))) | (7) |
and(tt,X) | → | activate(X) | (8) |
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(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) |
activate(n__0) | → | 0 | (22) |
activate(n__plus(X1,X2)) | → | plus(X1,X2) | (23) |
activate(n__isNatKind(X)) | → | isNatKind(X) | (24) |
activate(n__s(X)) | → | s(X) | (25) |
activate(n__and(X1,X2)) | → | and(X1,X2) | (26) |
activate(X) | → | X | (27) |
plus#(N,0) | → | isNat#(N) | (58) |
plus#(N,0) | → | and#(isNat(N),n__isNatKind(N)) | (59) |
plus#(N,s(M)) | → | isNat#(N) | (61) |
U41#(tt,M,N) | → | activate#(N) | (40) |
The dependency pairs are split into 2 components.
U21#(tt,V1) | → | isNat#(activate(V1)) | (36) |
isNat#(n__plus(V1,V2)) | → | U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) | (48) |
U11#(tt,V1,V2) | → | isNat#(activate(V1)) | (30) |
isNat#(n__s(V1)) | → | U21#(isNatKind(activate(V1)),activate(V1)) | (51) |
U11#(tt,V1,V2) | → | U12#(isNat(activate(V1)),activate(V2)) | (31) |
U12#(tt,V2) | → | isNat#(activate(V2)) | (33) |
[0] | = | 0 |
[U12(x1, x2)] | = | 0 · x1 + 0 · x2 + -∞ |
[isNatKind(x1)] | = | 1 · x1 + 0 |
[U31(x1, x2)] | = | -∞ · x1 + 0 · x2 + 1 |
[U11(x1, x2, x3)] | = | 0 · x1 + 0 · x2 + 0 · x3 + 0 |
[isNat(x1)] | = | 0 · x1 + -∞ |
[n__isNatKind(x1)] | = | 1 · x1 + 0 |
[n__and(x1, x2)] | = | -∞ · x1 + 0 · x2 + 0 |
[n__s(x1)] | = | 0 · x1 + 0 |
[isNat#(x1)] | = | 0 · x1 + -∞ |
[U13(x1)] | = | 0 · x1 + 0 |
[plus(x1, x2)] | = | 0 · x1 + 2 · x2 + 0 |
[tt] | = | 0 |
[U22(x1)] | = | -∞ · x1 + 0 |
[U11#(x1, x2, x3)] | = | 0 · x1 + 0 · x2 + 2 · x3 + 0 |
[U21#(x1, x2)] | = | -∞ · x1 + 0 · x2 + 0 |
[n__plus(x1, x2)] | = | 0 · x1 + 2 · x2 + 0 |
[and(x1, x2)] | = | -∞ · x1 + 0 · x2 + 0 |
[U12#(x1, x2)] | = | 0 · x1 + 2 · x2 + -∞ |
[s(x1)] | = | 0 · x1 + 0 |
[U41(x1, x2, x3)] | = | -∞ · x1 + 2 · x2 + 0 · x3 + 1 |
[n__0] | = | 0 |
[U21(x1, x2)] | = | -∞ · x1 + 0 · x2 + 0 |
[activate(x1)] | = | 0 · x1 + -∞ |
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(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) |
activate(n__0) | → | 0 | (22) |
activate(n__plus(X1,X2)) | → | plus(X1,X2) | (23) |
activate(n__isNatKind(X)) | → | isNatKind(X) | (24) |
activate(n__s(X)) | → | s(X) | (25) |
activate(n__and(X1,X2)) | → | and(X1,X2) | (26) |
activate(X) | → | X | (27) |
U12#(tt,V2) | → | isNat#(activate(V2)) | (33) |
The dependency pairs are split into 1 component.
U21#(tt,V1) | → | isNat#(activate(V1)) | (36) |
isNat#(n__plus(V1,V2)) | → | U11#(and(isNatKind(activate(V1)),n__isNatKind(activate(V2))),activate(V1),activate(V2)) | (48) |
U11#(tt,V1,V2) | → | isNat#(activate(V1)) | (30) |
isNat#(n__s(V1)) | → | U21#(isNatKind(activate(V1)),activate(V1)) | (51) |
[0] | = | 0 |
[U12(x1, x2)] | = | 3 · x1 + -∞ · x2 + 5 |
[isNatKind(x1)] | = | 0 · x1 + 0 |
[U31(x1, x2)] | = | -∞ · x1 + 0 · x2 + 0 |
[U11(x1, x2, x3)] | = | -∞ · x1 + -∞ · x2 + 4 · x3 + 0 |
[isNat(x1)] | = | -∞ · x1 + 1 |
[n__isNatKind(x1)] | = | 0 · x1 + -∞ |
[n__and(x1, x2)] | = | 0 · x1 + 1 · x2 + -∞ |
[n__s(x1)] | = | 0 · x1 + 0 |
[isNat#(x1)] | = | 0 · x1 + 0 |
[U13(x1)] | = | 3 · x1 + 5 |
[plus(x1, x2)] | = | 1 · x1 + 4 · x2 + 1 |
[tt] | = | 0 |
[U22(x1)] | = | 0 · x1 + 0 |
[U11#(x1, x2, x3)] | = | 0 · x1 + 1 · x2 + 1 · x3 + 1 |
[U21#(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[n__plus(x1, x2)] | = | 1 · x1 + 4 · x2 + 1 |
[and(x1, x2)] | = | 0 · x1 + 1 · x2 + 0 |
[s(x1)] | = | 0 · x1 + 0 |
[U41(x1, x2, x3)] | = | -∞ · x1 + 4 · x2 + 1 · x3 + 4 |
[n__0] | = | 0 |
[U21(x1, x2)] | = | -∞ · x1 + 2 · x2 + -∞ |
[activate(x1)] | = | 0 · x1 + 0 |
U31(tt,N) | → | activate(N) | (6) |
U41(tt,M,N) | → | s(plus(activate(N),activate(M))) | (7) |
and(tt,X) | → | activate(X) | (8) |
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(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) |
activate(n__0) | → | 0 | (22) |
activate(n__plus(X1,X2)) | → | plus(X1,X2) | (23) |
activate(n__isNatKind(X)) | → | isNatKind(X) | (24) |
activate(n__s(X)) | → | s(X) | (25) |
activate(n__and(X1,X2)) | → | and(X1,X2) | (26) |
activate(X) | → | X | (27) |
U11#(tt,V1,V2) | → | isNat#(activate(V1)) | (30) |
The dependency pairs are split into 1 component.
U21#(tt,V1) | → | isNat#(activate(V1)) | (36) |
isNat#(n__s(V1)) | → | U21#(isNatKind(activate(V1)),activate(V1)) | (51) |
[0] | = | 0 |
[U12(x1, x2)] | = | 0 · x1 + 4 · x2 + 4 |
[isNatKind(x1)] | = | 0 · x1 + 2 |
[U31(x1, x2)] | = | 0 · x1 + 1 · x2 + 0 |
[U11(x1, x2, x3)] | = | 0 · x1 + 5 · x2 + 0 · x3 + 1 |
[isNat(x1)] | = | 0 · x1 + 0 |
[n__isNatKind(x1)] | = | 0 · x1 + 2 |
[n__and(x1, x2)] | = | 0 · x1 + 1 · x2 + 0 |
[n__s(x1)] | = | 1 · x1 + 1 |
[isNat#(x1)] | = | 4 · x1 + 4 |
[U13(x1)] | = | 0 · x1 + 4 |
[plus(x1, x2)] | = | 1 · x1 + 4 · x2 + 0 |
[tt] | = | 1 |
[U22(x1)] | = | 2 · x1 + 4 |
[U21#(x1, x2)] | = | 1 · x1 + 4 · x2 + 6 |
[n__plus(x1, x2)] | = | 1 · x1 + 4 · x2 + 0 |
[and(x1, x2)] | = | 0 · x1 + 1 · x2 + 0 |
[s(x1)] | = | 1 · x1 + 1 |
[U41(x1, x2, x3)] | = | 0 · x1 + 4 · x2 + 1 · x3 + 4 |
[n__0] | = | 0 |
[U21(x1, x2)] | = | 4 · x1 + 0 · x2 + 0 |
[activate(x1)] | = | 1 · x1 + 0 |
U31(tt,N) | → | activate(N) | (6) |
U41(tt,M,N) | → | s(plus(activate(N),activate(M))) | (7) |
and(tt,X) | → | activate(X) | (8) |
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(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) |
activate(n__0) | → | 0 | (22) |
activate(n__plus(X1,X2)) | → | plus(X1,X2) | (23) |
activate(n__isNatKind(X)) | → | isNatKind(X) | (24) |
activate(n__s(X)) | → | s(X) | (25) |
activate(n__and(X1,X2)) | → | and(X1,X2) | (26) |
activate(X) | → | X | (27) |
U21#(tt,V1) | → | isNat#(activate(V1)) | (36) |
The dependency pairs are split into 0 components.
isNatKind#(n__s(V1)) | → | isNatKind#(activate(V1)) | (57) |
isNatKind#(n__s(V1)) | → | activate#(V1) | (56) |
activate#(n__plus(X1,X2)) | → | plus#(X1,X2) | (67) |
plus#(N,0) | → | U31#(and(isNat(N),n__isNatKind(N)),N) | (60) |
U31#(tt,N) | → | activate#(N) | (38) |
activate#(n__isNatKind(X)) | → | isNatKind#(X) | (68) |
activate#(n__and(X1,X2)) | → | and#(X1,X2) | (70) |
and#(tt,X) | → | activate#(X) | (43) |
plus#(N,s(M)) | → | and#(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))) | (64) |
plus#(N,s(M)) | → | U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) | (65) |
U41#(tt,M,N) | → | plus#(activate(N),activate(M)) | (41) |
[0] | = | 0 |
[U12(x1, x2)] | = | 1 · x1 + 1 · x2 + 0 |
[isNatKind(x1)] | = | 0 · x1 + -∞ |
[U31(x1, x2)] | = | -∞ · x1 + 1 · x2 + -∞ |
[U11(x1, x2, x3)] | = | 0 · x1 + -∞ · x2 + 3 · x3 + -∞ |
[isNat(x1)] | = | 2 · x1 + 5 |
[n__isNatKind(x1)] | = | 0 · x1 + -∞ |
[activate#(x1)] | = | 4 · x1 + 0 |
[n__and(x1, x2)] | = | -∞ · x1 + 0 · x2 + -∞ |
[isNatKind#(x1)] | = | 4 · x1 + 0 |
[n__s(x1)] | = | 0 · x1 + -∞ |
[U31#(x1, x2)] | = | -∞ · x1 + 4 · x2 + 0 |
[U41#(x1, x2, x3)] | = | 0 · x1 + -∞ · x2 + 4 · x3 + 0 |
[U13(x1)] | = | -∞ · x1 + 4 |
[plus(x1, x2)] | = | 1 · x1 + 2 · x2 + 1 |
[tt] | = | 0 |
[U22(x1)] | = | 2 · x1 + 0 |
[n__plus(x1, x2)] | = | 1 · x1 + 2 · x2 + 1 |
[and#(x1, x2)] | = | -∞ · x1 + 4 · x2 + 0 |
[and(x1, x2)] | = | -∞ · x1 + 0 · x2 + -∞ |
[s(x1)] | = | 0 · x1 + -∞ |
[U41(x1, x2, x3)] | = | 0 · x1 + 2 · x2 + 1 · x3 + 1 |
[n__0] | = | 0 |
[U21(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[plus#(x1, x2)] | = | 4 · x1 + -∞ · x2 + 0 |
[activate(x1)] | = | 0 · x1 + -∞ |
U31(tt,N) | → | activate(N) | (6) |
U41(tt,M,N) | → | s(plus(activate(N),activate(M))) | (7) |
and(tt,X) | → | activate(X) | (8) |
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(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) |
activate(n__0) | → | 0 | (22) |
activate(n__plus(X1,X2)) | → | plus(X1,X2) | (23) |
activate(n__isNatKind(X)) | → | isNatKind(X) | (24) |
activate(n__s(X)) | → | s(X) | (25) |
activate(n__and(X1,X2)) | → | and(X1,X2) | (26) |
activate(X) | → | X | (27) |
activate#(n__plus(X1,X2)) | → | plus#(X1,X2) | (67) |
The dependency pairs are split into 2 components.
plus#(N,s(M)) | → | U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) | (65) |
U41#(tt,M,N) | → | plus#(activate(N),activate(M)) | (41) |
[0] | = | 0 |
[U12(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[isNatKind(x1)] | = | 0 · x1 + 0 |
[U31(x1, x2)] | = | 0 · x1 + 2 · x2 + 0 |
[U11(x1, x2, x3)] | = | 0 · x1 + 0 · x2 + 0 · x3 + 1 |
[isNat(x1)] | = | 0 · x1 + 1 |
[n__isNatKind(x1)] | = | 0 · x1 + 0 |
[n__and(x1, x2)] | = | 4 · x1 + 2 · x2 + 0 |
[n__s(x1)] | = | 1 · x1 + 5 |
[U41#(x1, x2, x3)] | = | 0 · x1 + 4 · x2 + 0 · x3 + 4 |
[U13(x1)] | = | 0 · x1 + 0 |
[plus(x1, x2)] | = | 2 · x1 + 6 · x2 + 0 |
[tt] | = | 0 |
[U22(x1)] | = | 0 · x1 + 0 |
[n__plus(x1, x2)] | = | 2 · x1 + 6 · x2 + 0 |
[and(x1, x2)] | = | 4 · x1 + 2 · x2 + 0 |
[s(x1)] | = | 1 · x1 + 5 |
[U41(x1, x2, x3)] | = | 1 · x1 + 6 · x2 + 2 · x3 + 6 |
[n__0] | = | 0 |
[U21(x1, x2)] | = | 0 · x1 + 0 · x2 + 1 |
[plus#(x1, x2)] | = | 0 · x1 + 4 · x2 + 1 |
[activate(x1)] | = | 1 · x1 + 0 |
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(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) |
activate(n__0) | → | 0 | (22) |
activate(n__plus(X1,X2)) | → | plus(X1,X2) | (23) |
activate(n__isNatKind(X)) | → | isNatKind(X) | (24) |
activate(n__s(X)) | → | s(X) | (25) |
activate(n__and(X1,X2)) | → | and(X1,X2) | (26) |
activate(X) | → | X | (27) |
plus#(N,s(M)) | → | U41#(and(and(isNat(M),n__isNatKind(M)),n__and(isNat(N),n__isNatKind(N))),M,N) | (65) |
U41#(tt,M,N) | → | plus#(activate(N),activate(M)) | (41) |
There are no pairs anymore.
isNatKind#(n__s(V1)) | → | isNatKind#(activate(V1)) | (57) |
isNatKind#(n__s(V1)) | → | activate#(V1) | (56) |
activate#(n__isNatKind(X)) | → | isNatKind#(X) | (68) |
activate#(n__and(X1,X2)) | → | and#(X1,X2) | (70) |
and#(tt,X) | → | activate#(X) | (43) |
[0] | = | 2 |
[U12(x1, x2)] | = | 0 · x1 + 0 · x2 + -∞ |
[isNatKind(x1)] | = | 2 · x1 + -∞ |
[U31(x1, x2)] | = | -∞ · x1 + 1 · x2 + -∞ |
[U11(x1, x2, x3)] | = | -∞ · x1 + 0 · x2 + 2 · x3 + -∞ |
[isNat(x1)] | = | 0 · x1 + -∞ |
[n__isNatKind(x1)] | = | 2 · x1 + -∞ |
[activate#(x1)] | = | 0 · x1 + -∞ |
[n__and(x1, x2)] | = | 1 · x1 + 0 · x2 + -∞ |
[isNatKind#(x1)] | = | 0 · x1 + -∞ |
[n__s(x1)] | = | 0 · x1 + 0 |
[U13(x1)] | = | -∞ · x1 + 0 |
[plus(x1, x2)] | = | 1 · x1 + 0 · x2 + -∞ |
[tt] | = | 3 |
[U22(x1)] | = | -∞ · x1 + 0 |
[n__plus(x1, x2)] | = | 1 · x1 + 0 · x2 + -∞ |
[and#(x1, x2)] | = | 1 · x1 + 0 · x2 + -∞ |
[and(x1, x2)] | = | 1 · x1 + 0 · x2 + -∞ |
[s(x1)] | = | 0 · x1 + 0 |
[U41(x1, x2, x3)] | = | -∞ · x1 + 0 · x2 + 1 · x3 + 0 |
[n__0] | = | 2 |
[U21(x1, x2)] | = | 2 · x1 + 1 · x2 + 12 |
[activate(x1)] | = | 0 · x1 + -∞ |
U31(tt,N) | → | activate(N) | (6) |
U41(tt,M,N) | → | s(plus(activate(N),activate(M))) | (7) |
and(tt,X) | → | activate(X) | (8) |
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(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) |
activate(n__0) | → | 0 | (22) |
activate(n__plus(X1,X2)) | → | plus(X1,X2) | (23) |
activate(n__isNatKind(X)) | → | isNatKind(X) | (24) |
activate(n__s(X)) | → | s(X) | (25) |
activate(n__and(X1,X2)) | → | and(X1,X2) | (26) |
activate(X) | → | X | (27) |
activate#(n__isNatKind(X)) | → | isNatKind#(X) | (68) |
The dependency pairs are split into 2 components.
isNatKind#(n__s(V1)) | → | isNatKind#(activate(V1)) | (57) |
[0] | = | 2 |
[U12(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[isNatKind(x1)] | = | 0 · x1 + 1 |
[U31(x1, x2)] | = | 1 · x1 + 1 · x2 + 6 |
[U11(x1, x2, x3)] | = | 0 · x1 + 2 · x2 + 0 · x3 + 2 |
[isNat(x1)] | = | 0 · x1 + 0 |
[n__isNatKind(x1)] | = | 0 · x1 + 0 |
[n__and(x1, x2)] | = | 0 · x1 + 4 · x2 + 0 |
[isNatKind#(x1)] | = | 1 · x1 + 0 |
[n__s(x1)] | = | 1 · x1 + 2 |
[U13(x1)] | = | 4 · x1 + 0 |
[plus(x1, x2)] | = | 1 · x1 + 3 · x2 + 1 |
[tt] | = | 1 |
[U22(x1)] | = | 0 · x1 + 2 |
[n__plus(x1, x2)] | = | 1 · x1 + 3 · x2 + 1 |
[and(x1, x2)] | = | 0 · x1 + 4 · x2 + 1 |
[s(x1)] | = | 1 · x1 + 2 |
[U41(x1, x2, x3)] | = | 0 · x1 + 3 · x2 + 1 · x3 + 7 |
[n__0] | = | 2 |
[U21(x1, x2)] | = | 0 · x1 + 0 · x2 + 3 |
[activate(x1)] | = | 1 · x1 + 1 |
U31(tt,N) | → | activate(N) | (6) |
U41(tt,M,N) | → | s(plus(activate(N),activate(M))) | (7) |
and(tt,X) | → | activate(X) | (8) |
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(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) |
activate(n__0) | → | 0 | (22) |
activate(n__plus(X1,X2)) | → | plus(X1,X2) | (23) |
activate(n__isNatKind(X)) | → | isNatKind(X) | (24) |
activate(n__s(X)) | → | s(X) | (25) |
activate(n__and(X1,X2)) | → | and(X1,X2) | (26) |
activate(X) | → | X | (27) |
isNatKind#(n__s(V1)) | → | isNatKind#(activate(V1)) | (57) |
There are no pairs anymore.
activate#(n__and(X1,X2)) | → | and#(X1,X2) | (70) |
and#(tt,X) | → | activate#(X) | (43) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
activate#(n__and(X1,X2)) | → | and#(X1,X2) | (70) |
1 | > | 2 | |
1 | > | 1 | |
and#(tt,X) | → | activate#(X) | (43) |
2 | ≥ | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.