The rewrite relation of the following TRS is considered.
a__f(X) | → | cons(mark(X),f(g(X))) | (1) |
a__g(0) | → | s(0) | (2) |
a__g(s(X)) | → | s(s(a__g(mark(X)))) | (3) |
a__sel(0,cons(X,Y)) | → | mark(X) | (4) |
a__sel(s(X),cons(Y,Z)) | → | a__sel(mark(X),mark(Z)) | (5) |
mark(f(X)) | → | a__f(mark(X)) | (6) |
mark(g(X)) | → | a__g(mark(X)) | (7) |
mark(sel(X1,X2)) | → | a__sel(mark(X1),mark(X2)) | (8) |
mark(cons(X1,X2)) | → | cons(mark(X1),X2) | (9) |
mark(0) | → | 0 | (10) |
mark(s(X)) | → | s(mark(X)) | (11) |
a__f(X) | → | f(X) | (12) |
a__g(X) | → | g(X) | (13) |
a__sel(X1,X2) | → | sel(X1,X2) | (14) |
mark#(f(X)) | → | mark#(X) | (15) |
mark#(g(X)) | → | mark#(X) | (16) |
mark#(sel(X1,X2)) | → | a__sel#(mark(X1),mark(X2)) | (17) |
a__sel#(s(X),cons(Y,Z)) | → | mark#(Z) | (18) |
a__g#(s(X)) | → | mark#(X) | (19) |
a__f#(X) | → | mark#(X) | (20) |
mark#(sel(X1,X2)) | → | mark#(X1) | (21) |
a__sel#(s(X),cons(Y,Z)) | → | a__sel#(mark(X),mark(Z)) | (22) |
a__g#(s(X)) | → | a__g#(mark(X)) | (23) |
mark#(g(X)) | → | a__g#(mark(X)) | (24) |
a__sel#(0,cons(X,Y)) | → | mark#(X) | (25) |
mark#(cons(X1,X2)) | → | mark#(X1) | (26) |
mark#(f(X)) | → | a__f#(mark(X)) | (27) |
a__sel#(s(X),cons(Y,Z)) | → | mark#(X) | (28) |
mark#(sel(X1,X2)) | → | mark#(X2) | (29) |
mark#(s(X)) | → | mark#(X) | (30) |
The dependency pairs are split into 1 component.
mark#(s(X)) | → | mark#(X) | (30) |
mark#(sel(X1,X2)) | → | mark#(X1) | (21) |
mark#(sel(X1,X2)) | → | mark#(X2) | (29) |
a__f#(X) | → | mark#(X) | (20) |
a__sel#(s(X),cons(Y,Z)) | → | mark#(X) | (28) |
mark#(f(X)) | → | a__f#(mark(X)) | (27) |
mark#(cons(X1,X2)) | → | mark#(X1) | (26) |
a__sel#(0,cons(X,Y)) | → | mark#(X) | (25) |
a__g#(s(X)) | → | mark#(X) | (19) |
a__sel#(s(X),cons(Y,Z)) | → | mark#(Z) | (18) |
mark#(sel(X1,X2)) | → | a__sel#(mark(X1),mark(X2)) | (17) |
mark#(g(X)) | → | mark#(X) | (16) |
mark#(g(X)) | → | a__g#(mark(X)) | (24) |
a__g#(s(X)) | → | a__g#(mark(X)) | (23) |
a__sel#(s(X),cons(Y,Z)) | → | a__sel#(mark(X),mark(Z)) | (22) |
mark#(f(X)) | → | mark#(X) | (15) |
[a__g(x1)] | = | x1 + 0 |
[s(x1)] | = | x1 + 0 |
[a__g#(x1)] | = | x1 + 0 |
[a__f(x1)] | = | x1 + 2242 |
[f(x1)] | = | x1 + 2242 |
[mark#(x1)] | = | x1 + 0 |
[0] | = | 1 |
[sel(x1, x2)] | = | max(x1 + 28884, x2 + 1, 0) |
[a__sel#(x1, x2)] | = | max(x1 + 28884, x2 + 1, 0) |
[mark(x1)] | = | x1 + 0 |
[a__sel(x1, x2)] | = | max(x1 + 28884, x2 + 1, 0) |
[cons(x1, x2)] | = | max(x1 + 1, x2 + 0, 0) |
[a__f#(x1)] | = | x1 + 2241 |
[g(x1)] | = | x1 + 0 |
a__sel(0,cons(X,Y)) | → | mark(X) | (4) |
mark(sel(X1,X2)) | → | a__sel(mark(X1),mark(X2)) | (8) |
a__f(X) | → | cons(mark(X),f(g(X))) | (1) |
a__g(s(X)) | → | s(s(a__g(mark(X)))) | (3) |
a__sel(s(X),cons(Y,Z)) | → | a__sel(mark(X),mark(Z)) | (5) |
mark(0) | → | 0 | (10) |
mark(g(X)) | → | a__g(mark(X)) | (7) |
a__sel(X1,X2) | → | sel(X1,X2) | (14) |
a__f(X) | → | f(X) | (12) |
mark(s(X)) | → | s(mark(X)) | (11) |
mark(cons(X1,X2)) | → | cons(mark(X1),X2) | (9) |
a__g(X) | → | g(X) | (13) |
mark(f(X)) | → | a__f(mark(X)) | (6) |
a__g(0) | → | s(0) | (2) |
mark#(sel(X1,X2)) | → | mark#(X1) | (21) |
mark#(sel(X1,X2)) | → | mark#(X2) | (29) |
a__f#(X) | → | mark#(X) | (20) |
a__sel#(s(X),cons(Y,Z)) | → | mark#(X) | (28) |
mark#(f(X)) | → | a__f#(mark(X)) | (27) |
mark#(cons(X1,X2)) | → | mark#(X1) | (26) |
a__sel#(0,cons(X,Y)) | → | mark#(X) | (25) |
a__sel#(s(X),cons(Y,Z)) | → | mark#(Z) | (18) |
mark#(f(X)) | → | mark#(X) | (15) |
The dependency pairs are split into 2 components.
a__g#(s(X)) | → | mark#(X) | (19) |
a__g#(s(X)) | → | a__g#(mark(X)) | (23) |
mark#(g(X)) | → | mark#(X) | (16) |
mark#(g(X)) | → | a__g#(mark(X)) | (24) |
mark#(s(X)) | → | mark#(X) | (30) |
π(a__g#) | = | 1 |
π(mark#) | = | 1 |
π(mark) | = | 1 |
prec(a__g) | = | 4 | status(a__g) | = | [1] | list-extension(a__g) | = | Lex | ||
prec(s) | = | 3 | status(s) | = | [1] | list-extension(s) | = | Lex | ||
prec(a__f) | = | 0 | status(a__f) | = | [] | list-extension(a__f) | = | Lex | ||
prec(f) | = | 0 | status(f) | = | [] | list-extension(f) | = | Lex | ||
prec(0) | = | 2 | status(0) | = | [] | list-extension(0) | = | Lex | ||
prec(sel) | = | 0 | status(sel) | = | [] | list-extension(sel) | = | Lex | ||
prec(a__sel#) | = | 0 | status(a__sel#) | = | [1] | list-extension(a__sel#) | = | Lex | ||
prec(a__sel) | = | 0 | status(a__sel) | = | [] | list-extension(a__sel) | = | Lex | ||
prec(cons) | = | 0 | status(cons) | = | [] | list-extension(cons) | = | Lex | ||
prec(a__f#) | = | 0 | status(a__f#) | = | [] | list-extension(a__f#) | = | Lex | ||
prec(g) | = | 4 | status(g) | = | [1] | list-extension(g) | = | Lex |
[a__g(x1)] | = | x1 + 0 |
[s(x1)] | = | x1 + 0 |
[a__f(x1)] | = | x1 + 1 |
[f(x1)] | = | x1 + 1 |
[0] | = | 0 |
[sel(x1, x2)] | = | x1 + x2 + 18588 |
[a__sel#(x1, x2)] | = | x1 + 1 |
[a__sel(x1, x2)] | = | x1 + x2 + 18588 |
[cons(x1, x2)] | = | max(x1 + 0, x2 + 0, 0) |
[a__f#(x1)] | = | 1 |
[g(x1)] | = | x1 + 0 |
a__sel(0,cons(X,Y)) | → | mark(X) | (4) |
mark(sel(X1,X2)) | → | a__sel(mark(X1),mark(X2)) | (8) |
a__f(X) | → | cons(mark(X),f(g(X))) | (1) |
a__g(s(X)) | → | s(s(a__g(mark(X)))) | (3) |
a__sel(s(X),cons(Y,Z)) | → | a__sel(mark(X),mark(Z)) | (5) |
mark(0) | → | 0 | (10) |
mark(g(X)) | → | a__g(mark(X)) | (7) |
a__sel(X1,X2) | → | sel(X1,X2) | (14) |
a__f(X) | → | f(X) | (12) |
mark(s(X)) | → | s(mark(X)) | (11) |
mark(cons(X1,X2)) | → | cons(mark(X1),X2) | (9) |
a__g(X) | → | g(X) | (13) |
mark(f(X)) | → | a__f(mark(X)) | (6) |
a__g(0) | → | s(0) | (2) |
a__g#(s(X)) | → | mark#(X) | (19) |
a__g#(s(X)) | → | a__g#(mark(X)) | (23) |
mark#(g(X)) | → | mark#(X) | (16) |
mark#(g(X)) | → | a__g#(mark(X)) | (24) |
mark#(s(X)) | → | mark#(X) | (30) |
The dependency pairs are split into 0 components.
a__sel#(s(X),cons(Y,Z)) | → | a__sel#(mark(X),mark(Z)) | (22) |
π(a__g#) | = | 1 |
π(mark#) | = | 1 |
π(a__sel#) | = | 1 |
π(mark) | = | 1 |
prec(a__g) | = | 4 | status(a__g) | = | [1] | list-extension(a__g) | = | Lex | ||
prec(s) | = | 3 | status(s) | = | [1] | list-extension(s) | = | Lex | ||
prec(a__f) | = | 2 | status(a__f) | = | [] | list-extension(a__f) | = | Lex | ||
prec(f) | = | 2 | status(f) | = | [] | list-extension(f) | = | Lex | ||
prec(0) | = | 2 | status(0) | = | [] | list-extension(0) | = | Lex | ||
prec(sel) | = | 2 | status(sel) | = | [] | list-extension(sel) | = | Lex | ||
prec(a__sel) | = | 2 | status(a__sel) | = | [] | list-extension(a__sel) | = | Lex | ||
prec(cons) | = | 2 | status(cons) | = | [] | list-extension(cons) | = | Lex | ||
prec(a__f#) | = | 0 | status(a__f#) | = | [] | list-extension(a__f#) | = | Lex | ||
prec(g) | = | 4 | status(g) | = | [1] | list-extension(g) | = | Lex |
[a__g(x1)] | = | x1 + 0 |
[s(x1)] | = | x1 + 0 |
[a__f(x1)] | = | x1 + 25075 |
[f(x1)] | = | x1 + 25075 |
[0] | = | 0 |
[sel(x1, x2)] | = | x2 + 36069 |
[a__sel(x1, x2)] | = | x2 + 36069 |
[cons(x1, x2)] | = | max(x1 + 12457, x2 + 0, 0) |
[a__f#(x1)] | = | 1 |
[g(x1)] | = | x1 + 0 |
a__sel(0,cons(X,Y)) | → | mark(X) | (4) |
mark(sel(X1,X2)) | → | a__sel(mark(X1),mark(X2)) | (8) |
a__f(X) | → | cons(mark(X),f(g(X))) | (1) |
a__g(s(X)) | → | s(s(a__g(mark(X)))) | (3) |
a__sel(s(X),cons(Y,Z)) | → | a__sel(mark(X),mark(Z)) | (5) |
mark(0) | → | 0 | (10) |
mark(g(X)) | → | a__g(mark(X)) | (7) |
a__sel(X1,X2) | → | sel(X1,X2) | (14) |
a__f(X) | → | f(X) | (12) |
mark(s(X)) | → | s(mark(X)) | (11) |
mark(cons(X1,X2)) | → | cons(mark(X1),X2) | (9) |
a__g(X) | → | g(X) | (13) |
mark(f(X)) | → | a__f(mark(X)) | (6) |
a__g(0) | → | s(0) | (2) |
a__sel#(s(X),cons(Y,Z)) | → | a__sel#(mark(X),mark(Z)) | (22) |
The dependency pairs are split into 0 components.