The rewrite relation of the following TRS is considered.
active(U11(tt,M,N)) | → | mark(U12(tt,M,N)) | (1) |
active(U12(tt,M,N)) | → | mark(s(plus(N,M))) | (2) |
active(U21(tt,M,N)) | → | mark(U22(tt,M,N)) | (3) |
active(U22(tt,M,N)) | → | mark(plus(x(N,M),N)) | (4) |
active(plus(N,0)) | → | mark(N) | (5) |
active(plus(N,s(M))) | → | mark(U11(tt,M,N)) | (6) |
active(x(N,0)) | → | mark(0) | (7) |
active(x(N,s(M))) | → | mark(U21(tt,M,N)) | (8) |
mark(U11(X1,X2,X3)) | → | active(U11(mark(X1),X2,X3)) | (9) |
mark(tt) | → | active(tt) | (10) |
mark(U12(X1,X2,X3)) | → | active(U12(mark(X1),X2,X3)) | (11) |
mark(s(X)) | → | active(s(mark(X))) | (12) |
mark(plus(X1,X2)) | → | active(plus(mark(X1),mark(X2))) | (13) |
mark(U21(X1,X2,X3)) | → | active(U21(mark(X1),X2,X3)) | (14) |
mark(U22(X1,X2,X3)) | → | active(U22(mark(X1),X2,X3)) | (15) |
mark(x(X1,X2)) | → | active(x(mark(X1),mark(X2))) | (16) |
mark(0) | → | active(0) | (17) |
U11(mark(X1),X2,X3) | → | U11(X1,X2,X3) | (18) |
U11(X1,mark(X2),X3) | → | U11(X1,X2,X3) | (19) |
U11(X1,X2,mark(X3)) | → | U11(X1,X2,X3) | (20) |
U11(active(X1),X2,X3) | → | U11(X1,X2,X3) | (21) |
U11(X1,active(X2),X3) | → | U11(X1,X2,X3) | (22) |
U11(X1,X2,active(X3)) | → | U11(X1,X2,X3) | (23) |
U12(mark(X1),X2,X3) | → | U12(X1,X2,X3) | (24) |
U12(X1,mark(X2),X3) | → | U12(X1,X2,X3) | (25) |
U12(X1,X2,mark(X3)) | → | U12(X1,X2,X3) | (26) |
U12(active(X1),X2,X3) | → | U12(X1,X2,X3) | (27) |
U12(X1,active(X2),X3) | → | U12(X1,X2,X3) | (28) |
U12(X1,X2,active(X3)) | → | U12(X1,X2,X3) | (29) |
s(mark(X)) | → | s(X) | (30) |
s(active(X)) | → | s(X) | (31) |
plus(mark(X1),X2) | → | plus(X1,X2) | (32) |
plus(X1,mark(X2)) | → | plus(X1,X2) | (33) |
plus(active(X1),X2) | → | plus(X1,X2) | (34) |
plus(X1,active(X2)) | → | plus(X1,X2) | (35) |
U21(mark(X1),X2,X3) | → | U21(X1,X2,X3) | (36) |
U21(X1,mark(X2),X3) | → | U21(X1,X2,X3) | (37) |
U21(X1,X2,mark(X3)) | → | U21(X1,X2,X3) | (38) |
U21(active(X1),X2,X3) | → | U21(X1,X2,X3) | (39) |
U21(X1,active(X2),X3) | → | U21(X1,X2,X3) | (40) |
U21(X1,X2,active(X3)) | → | U21(X1,X2,X3) | (41) |
U22(mark(X1),X2,X3) | → | U22(X1,X2,X3) | (42) |
U22(X1,mark(X2),X3) | → | U22(X1,X2,X3) | (43) |
U22(X1,X2,mark(X3)) | → | U22(X1,X2,X3) | (44) |
U22(active(X1),X2,X3) | → | U22(X1,X2,X3) | (45) |
U22(X1,active(X2),X3) | → | U22(X1,X2,X3) | (46) |
U22(X1,X2,active(X3)) | → | U22(X1,X2,X3) | (47) |
x(mark(X1),X2) | → | x(X1,X2) | (48) |
x(X1,mark(X2)) | → | x(X1,X2) | (49) |
x(active(X1),X2) | → | x(X1,X2) | (50) |
x(X1,active(X2)) | → | x(X1,X2) | (51) |
prec(U11) | = | 1 | stat(U11) | = | lex | |
prec(tt) | = | 1 | stat(tt) | = | lex | |
prec(U12) | = | 1 | stat(U12) | = | lex | |
prec(s) | = | 0 | stat(s) | = | lex | |
prec(plus) | = | 1 | stat(plus) | = | lex | |
prec(U21) | = | 2 | stat(U21) | = | lex | |
prec(U22) | = | 2 | stat(U22) | = | lex | |
prec(x) | = | 2 | stat(x) | = | lex | |
prec(0) | = | 3 | stat(0) | = | lex |
π(active) | = | 1 |
π(U11) | = | [3,2,1] |
π(tt) | = | [] |
π(mark) | = | 1 |
π(U12) | = | [3,2,1] |
π(s) | = | [1] |
π(plus) | = | [1,2] |
π(U21) | = | [2,3,1] |
π(U22) | = | [2,3,1] |
π(x) | = | [2,1] |
π(0) | = | [] |
active(U12(tt,M,N)) | → | mark(s(plus(N,M))) | (2) |
active(U22(tt,M,N)) | → | mark(plus(x(N,M),N)) | (4) |
active(plus(N,0)) | → | mark(N) | (5) |
active(plus(N,s(M))) | → | mark(U11(tt,M,N)) | (6) |
active(x(N,0)) | → | mark(0) | (7) |
active(x(N,s(M))) | → | mark(U21(tt,M,N)) | (8) |
[active(x1)] | = | 1 · x1 |
[U11(x1, x2, x3)] | = | 2 + 2 · x1 + 1 · x2 + 1 · x3 |
[tt] | = | 0 |
[mark(x1)] | = | 1 · x1 |
[U12(x1, x2, x3)] | = | 1 · x1 + 1 · x2 + 1 · x3 |
[U21(x1, x2, x3)] | = | 2 + 1 · x1 + 2 · x2 + 1 · x3 |
[U22(x1, x2, x3)] | = | 1 + 1 · x1 + 2 · x2 + 1 · x3 |
[s(x1)] | = | 1 · x1 |
[plus(x1, x2)] | = | 1 · x1 + 2 · x2 |
[x(x1, x2)] | = | 1 · x1 + 1 · x2 |
[0] | = | 0 |
active(U11(tt,M,N)) | → | mark(U12(tt,M,N)) | (1) |
active(U21(tt,M,N)) | → | mark(U22(tt,M,N)) | (3) |
[mark(x1)] | = | 2 · x1 |
[U11(x1, x2, x3)] | = | 2 · x1 + 2 · x2 + 2 · x3 |
[active(x1)] | = | 1 · x1 |
[tt] | = | 1 |
[U12(x1, x2, x3)] | = | 1 + 1 · x1 + 2 · x2 + 2 · x3 |
[s(x1)] | = | 1 + 2 · x1 |
[plus(x1, x2)] | = | 1 + 2 · x1 + 2 · x2 |
[U21(x1, x2, x3)] | = | 1 + 2 · x1 + 1 · x2 + 2 · x3 |
[U22(x1, x2, x3)] | = | 1 + 2 · x1 + 2 · x2 + 1 · x3 |
[x(x1, x2)] | = | 1 · x1 + 2 · x2 |
[0] | = | 2 |
mark(tt) | → | active(tt) | (10) |
mark(U12(X1,X2,X3)) | → | active(U12(mark(X1),X2,X3)) | (11) |
mark(s(X)) | → | active(s(mark(X))) | (12) |
mark(plus(X1,X2)) | → | active(plus(mark(X1),mark(X2))) | (13) |
mark(U21(X1,X2,X3)) | → | active(U21(mark(X1),X2,X3)) | (14) |
mark(U22(X1,X2,X3)) | → | active(U22(mark(X1),X2,X3)) | (15) |
mark(0) | → | active(0) | (17) |
[mark(x1)] | = | 2 · x1 |
[U11(x1, x2, x3)] | = | 2 + 1 · x1 + 1 · x2 + 2 · x3 |
[active(x1)] | = | 1 · x1 |
[x(x1, x2)] | = | 1 · x1 + 1 · x2 |
[U12(x1, x2, x3)] | = | 2 · x1 + 2 · x2 + 1 · x3 |
[s(x1)] | = | 2 · x1 |
[plus(x1, x2)] | = | 2 · x1 + 2 · x2 |
[U21(x1, x2, x3)] | = | 2 · x1 + 2 · x2 + 2 · x3 |
[U22(x1, x2, x3)] | = | 2 · x1 + 2 · x2 + 2 · x3 |
mark(U11(X1,X2,X3)) | → | active(U11(mark(X1),X2,X3)) | (9) |
[mark(x1)] | = | 2 · x1 |
[x(x1, x2)] | = | 2 + 2 · x1 + 2 · x2 |
[active(x1)] | = | 2 + 1 · x1 |
[U11(x1, x2, x3)] | = | 1 · x1 + 1 · x2 + 1 · x3 |
[U12(x1, x2, x3)] | = | 1 · x1 + 1 · x2 + 1 · x3 |
[s(x1)] | = | 1 · x1 |
[plus(x1, x2)] | = | 1 · x1 + 1 · x2 |
[U21(x1, x2, x3)] | = | 1 · x1 + 1 · x2 + 1 · x3 |
[U22(x1, x2, x3)] | = | 1 · x1 + 1 · x2 + 1 · x3 |
U11(active(X1),X2,X3) | → | U11(X1,X2,X3) | (21) |
U11(X1,active(X2),X3) | → | U11(X1,X2,X3) | (22) |
U11(X1,X2,active(X3)) | → | U11(X1,X2,X3) | (23) |
U12(active(X1),X2,X3) | → | U12(X1,X2,X3) | (27) |
U12(X1,active(X2),X3) | → | U12(X1,X2,X3) | (28) |
U12(X1,X2,active(X3)) | → | U12(X1,X2,X3) | (29) |
s(active(X)) | → | s(X) | (31) |
plus(active(X1),X2) | → | plus(X1,X2) | (34) |
plus(X1,active(X2)) | → | plus(X1,X2) | (35) |
U21(active(X1),X2,X3) | → | U21(X1,X2,X3) | (39) |
U21(X1,active(X2),X3) | → | U21(X1,X2,X3) | (40) |
U21(X1,X2,active(X3)) | → | U21(X1,X2,X3) | (41) |
U22(active(X1),X2,X3) | → | U22(X1,X2,X3) | (45) |
U22(X1,active(X2),X3) | → | U22(X1,X2,X3) | (46) |
U22(X1,X2,active(X3)) | → | U22(X1,X2,X3) | (47) |
x(active(X1),X2) | → | x(X1,X2) | (50) |
x(X1,active(X2)) | → | x(X1,X2) | (51) |
[mark(x1)] | = | 2 · x1 |
[x(x1, x2)] | = | 2 + 2 · x1 + 2 · x2 |
[active(x1)] | = | 1 · x1 |
[U11(x1, x2, x3)] | = | 2 · x1 + 2 · x2 + 2 · x3 |
[U12(x1, x2, x3)] | = | 2 · x1 + 2 · x2 + 2 · x3 |
[s(x1)] | = | 1 · x1 |
[plus(x1, x2)] | = | 2 · x1 + 2 · x2 |
[U21(x1, x2, x3)] | = | 1 · x1 + 2 · x2 + 2 · x3 |
[U22(x1, x2, x3)] | = | 2 · x1 + 2 · x2 + 2 · x3 |
mark(x(X1,X2)) | → | active(x(mark(X1),mark(X2))) | (16) |
prec(mark) | = | 7 | weight(mark) | = | 0 | ||||
prec(s) | = | 2 | weight(s) | = | 1 | ||||
prec(U11) | = | 0 | weight(U11) | = | 0 | ||||
prec(U12) | = | 1 | weight(U12) | = | 0 | ||||
prec(plus) | = | 3 | weight(plus) | = | 0 | ||||
prec(U21) | = | 4 | weight(U21) | = | 0 | ||||
prec(U22) | = | 5 | weight(U22) | = | 0 | ||||
prec(x) | = | 6 | weight(x) | = | 0 |
U11(mark(X1),X2,X3) | → | U11(X1,X2,X3) | (18) |
U11(X1,mark(X2),X3) | → | U11(X1,X2,X3) | (19) |
U11(X1,X2,mark(X3)) | → | U11(X1,X2,X3) | (20) |
U12(mark(X1),X2,X3) | → | U12(X1,X2,X3) | (24) |
U12(X1,mark(X2),X3) | → | U12(X1,X2,X3) | (25) |
U12(X1,X2,mark(X3)) | → | U12(X1,X2,X3) | (26) |
s(mark(X)) | → | s(X) | (30) |
plus(mark(X1),X2) | → | plus(X1,X2) | (32) |
plus(X1,mark(X2)) | → | plus(X1,X2) | (33) |
U21(mark(X1),X2,X3) | → | U21(X1,X2,X3) | (36) |
U21(X1,mark(X2),X3) | → | U21(X1,X2,X3) | (37) |
U21(X1,X2,mark(X3)) | → | U21(X1,X2,X3) | (38) |
U22(mark(X1),X2,X3) | → | U22(X1,X2,X3) | (42) |
U22(X1,mark(X2),X3) | → | U22(X1,X2,X3) | (43) |
U22(X1,X2,mark(X3)) | → | U22(X1,X2,X3) | (44) |
x(mark(X1),X2) | → | x(X1,X2) | (48) |
x(X1,mark(X2)) | → | x(X1,X2) | (49) |
There are no rules in the TRS. Hence, it is terminating.