The rewrite relation of the following TRS is considered.
a__app(nil,YS) | → | mark(YS) | (1) |
a__app(cons(X,XS),YS) | → | cons(mark(X),app(XS,YS)) | (2) |
a__from(X) | → | cons(mark(X),from(s(X))) | (3) |
a__zWadr(nil,YS) | → | nil | (4) |
a__zWadr(XS,nil) | → | nil | (5) |
a__zWadr(cons(X,XS),cons(Y,YS)) | → | cons(a__app(mark(Y),cons(mark(X),nil)),zWadr(XS,YS)) | (6) |
a__prefix(L) | → | cons(nil,zWadr(L,prefix(L))) | (7) |
mark(app(X1,X2)) | → | a__app(mark(X1),mark(X2)) | (8) |
mark(from(X)) | → | a__from(mark(X)) | (9) |
mark(zWadr(X1,X2)) | → | a__zWadr(mark(X1),mark(X2)) | (10) |
mark(prefix(X)) | → | a__prefix(mark(X)) | (11) |
mark(nil) | → | nil | (12) |
mark(cons(X1,X2)) | → | cons(mark(X1),X2) | (13) |
mark(s(X)) | → | s(mark(X)) | (14) |
a__app(X1,X2) | → | app(X1,X2) | (15) |
a__from(X) | → | from(X) | (16) |
a__zWadr(X1,X2) | → | zWadr(X1,X2) | (17) |
a__prefix(X) | → | prefix(X) | (18) |
a__app#(nil,YS) | → | mark#(YS) | (19) |
a__app#(cons(X,XS),YS) | → | mark#(X) | (20) |
a__from#(X) | → | mark#(X) | (21) |
a__zWadr#(cons(X,XS),cons(Y,YS)) | → | mark#(X) | (22) |
a__zWadr#(cons(X,XS),cons(Y,YS)) | → | mark#(Y) | (23) |
a__zWadr#(cons(X,XS),cons(Y,YS)) | → | a__app#(mark(Y),cons(mark(X),nil)) | (24) |
mark#(app(X1,X2)) | → | mark#(X2) | (25) |
mark#(app(X1,X2)) | → | mark#(X1) | (26) |
mark#(app(X1,X2)) | → | a__app#(mark(X1),mark(X2)) | (27) |
mark#(from(X)) | → | mark#(X) | (28) |
mark#(from(X)) | → | a__from#(mark(X)) | (29) |
mark#(zWadr(X1,X2)) | → | mark#(X2) | (30) |
mark#(zWadr(X1,X2)) | → | mark#(X1) | (31) |
mark#(zWadr(X1,X2)) | → | a__zWadr#(mark(X1),mark(X2)) | (32) |
mark#(prefix(X)) | → | mark#(X) | (33) |
mark#(prefix(X)) | → | a__prefix#(mark(X)) | (34) |
mark#(cons(X1,X2)) | → | mark#(X1) | (35) |
mark#(s(X)) | → | mark#(X) | (36) |
The dependency pairs are split into 1 component.
a__zWadr#(cons(X,XS),cons(Y,YS)) | → | mark#(Y) | (23) |
mark#(app(X1,X2)) | → | mark#(X2) | (25) |
mark#(app(X1,X2)) | → | mark#(X1) | (26) |
mark#(app(X1,X2)) | → | a__app#(mark(X1),mark(X2)) | (27) |
a__app#(nil,YS) | → | mark#(YS) | (19) |
mark#(from(X)) | → | mark#(X) | (28) |
mark#(from(X)) | → | a__from#(mark(X)) | (29) |
a__from#(X) | → | mark#(X) | (21) |
mark#(zWadr(X1,X2)) | → | mark#(X2) | (30) |
mark#(zWadr(X1,X2)) | → | mark#(X1) | (31) |
mark#(zWadr(X1,X2)) | → | a__zWadr#(mark(X1),mark(X2)) | (32) |
a__zWadr#(cons(X,XS),cons(Y,YS)) | → | mark#(X) | (22) |
mark#(prefix(X)) | → | mark#(X) | (33) |
mark#(cons(X1,X2)) | → | mark#(X1) | (35) |
mark#(s(X)) | → | mark#(X) | (36) |
a__zWadr#(cons(X,XS),cons(Y,YS)) | → | a__app#(mark(Y),cons(mark(X),nil)) | (24) |
a__app#(cons(X,XS),YS) | → | mark#(X) | (20) |
[app(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[a__zWadr#(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[a__zWadr(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[a__app(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[cons(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[a__from(x1)] | = | 0 · x1 + 0 |
[a__prefix(x1)] | = | 1 · x1 + 1 |
[nil] | = | 0 |
[from(x1)] | = | 0 · x1 + -∞ |
[a__from#(x1)] | = | 0 · x1 + 0 |
[a__app#(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[prefix(x1)] | = | 1 · x1 + 1 |
[zWadr(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[mark#(x1)] | = | 0 · x1 + 0 |
[s(x1)] | = | 0 · x1 + -∞ |
[mark(x1)] | = | 0 · x1 + 0 |
a__app(nil,YS) | → | mark(YS) | (1) |
a__app(cons(X,XS),YS) | → | cons(mark(X),app(XS,YS)) | (2) |
a__from(X) | → | cons(mark(X),from(s(X))) | (3) |
a__zWadr(nil,YS) | → | nil | (4) |
a__zWadr(XS,nil) | → | nil | (5) |
a__zWadr(cons(X,XS),cons(Y,YS)) | → | cons(a__app(mark(Y),cons(mark(X),nil)),zWadr(XS,YS)) | (6) |
a__prefix(L) | → | cons(nil,zWadr(L,prefix(L))) | (7) |
mark(app(X1,X2)) | → | a__app(mark(X1),mark(X2)) | (8) |
mark(from(X)) | → | a__from(mark(X)) | (9) |
mark(zWadr(X1,X2)) | → | a__zWadr(mark(X1),mark(X2)) | (10) |
mark(prefix(X)) | → | a__prefix(mark(X)) | (11) |
mark(nil) | → | nil | (12) |
mark(cons(X1,X2)) | → | cons(mark(X1),X2) | (13) |
mark(s(X)) | → | s(mark(X)) | (14) |
a__app(X1,X2) | → | app(X1,X2) | (15) |
a__from(X) | → | from(X) | (16) |
a__zWadr(X1,X2) | → | zWadr(X1,X2) | (17) |
a__prefix(X) | → | prefix(X) | (18) |
mark#(prefix(X)) | → | mark#(X) | (33) |
[app(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[a__zWadr#(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[a__zWadr(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[a__app(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[cons(x1, x2)] | = | 0 · x1 + -∞ · x2 + 0 |
[a__from(x1)] | = | 0 · x1 + 0 |
[a__prefix(x1)] | = | -∞ · x1 + 3 |
[nil] | = | 0 |
[from(x1)] | = | 0 · x1 + 0 |
[a__from#(x1)] | = | 0 · x1 + 0 |
[a__app#(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[prefix(x1)] | = | -∞ · x1 + 3 |
[zWadr(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[mark#(x1)] | = | 0 · x1 + 0 |
[s(x1)] | = | 3 · x1 + 4 |
[mark(x1)] | = | 0 · x1 + 0 |
a__app(nil,YS) | → | mark(YS) | (1) |
a__app(cons(X,XS),YS) | → | cons(mark(X),app(XS,YS)) | (2) |
a__from(X) | → | cons(mark(X),from(s(X))) | (3) |
a__zWadr(nil,YS) | → | nil | (4) |
a__zWadr(XS,nil) | → | nil | (5) |
a__zWadr(cons(X,XS),cons(Y,YS)) | → | cons(a__app(mark(Y),cons(mark(X),nil)),zWadr(XS,YS)) | (6) |
a__prefix(L) | → | cons(nil,zWadr(L,prefix(L))) | (7) |
mark(app(X1,X2)) | → | a__app(mark(X1),mark(X2)) | (8) |
mark(from(X)) | → | a__from(mark(X)) | (9) |
mark(zWadr(X1,X2)) | → | a__zWadr(mark(X1),mark(X2)) | (10) |
mark(prefix(X)) | → | a__prefix(mark(X)) | (11) |
mark(nil) | → | nil | (12) |
mark(cons(X1,X2)) | → | cons(mark(X1),X2) | (13) |
mark(s(X)) | → | s(mark(X)) | (14) |
a__app(X1,X2) | → | app(X1,X2) | (15) |
a__from(X) | → | from(X) | (16) |
a__zWadr(X1,X2) | → | zWadr(X1,X2) | (17) |
a__prefix(X) | → | prefix(X) | (18) |
mark#(s(X)) | → | mark#(X) | (36) |
[app(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[a__zWadr#(x1, x2)] | = | 0 · x1 + 0 · x2 + -∞ |
[a__zWadr(x1, x2)] | = | 7 · x1 + 7 · x2 + 0 |
[a__app(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[cons(x1, x2)] | = | 0 · x1 + -∞ · x2 + 0 |
[a__from(x1)] | = | 0 · x1 + 2 |
[a__prefix(x1)] | = | -∞ · x1 + 0 |
[nil] | = | 0 |
[from(x1)] | = | 0 · x1 + 2 |
[a__from#(x1)] | = | 0 · x1 + 0 |
[a__app#(x1, x2)] | = | 0 · x1 + 0 · x2 + -∞ |
[prefix(x1)] | = | -∞ · x1 + 0 |
[zWadr(x1, x2)] | = | 7 · x1 + 7 · x2 + 0 |
[mark#(x1)] | = | 0 · x1 + 0 |
[s(x1)] | = | 0 · x1 + 3 |
[mark(x1)] | = | 0 · x1 + -∞ |
a__app(nil,YS) | → | mark(YS) | (1) |
a__app(cons(X,XS),YS) | → | cons(mark(X),app(XS,YS)) | (2) |
a__from(X) | → | cons(mark(X),from(s(X))) | (3) |
a__zWadr(nil,YS) | → | nil | (4) |
a__zWadr(XS,nil) | → | nil | (5) |
a__zWadr(cons(X,XS),cons(Y,YS)) | → | cons(a__app(mark(Y),cons(mark(X),nil)),zWadr(XS,YS)) | (6) |
a__prefix(L) | → | cons(nil,zWadr(L,prefix(L))) | (7) |
mark(app(X1,X2)) | → | a__app(mark(X1),mark(X2)) | (8) |
mark(from(X)) | → | a__from(mark(X)) | (9) |
mark(zWadr(X1,X2)) | → | a__zWadr(mark(X1),mark(X2)) | (10) |
mark(prefix(X)) | → | a__prefix(mark(X)) | (11) |
mark(nil) | → | nil | (12) |
mark(cons(X1,X2)) | → | cons(mark(X1),X2) | (13) |
mark(s(X)) | → | s(mark(X)) | (14) |
a__app(X1,X2) | → | app(X1,X2) | (15) |
a__from(X) | → | from(X) | (16) |
a__zWadr(X1,X2) | → | zWadr(X1,X2) | (17) |
a__prefix(X) | → | prefix(X) | (18) |
mark#(zWadr(X1,X2)) | → | a__zWadr#(mark(X1),mark(X2)) | (32) |
The dependency pairs are split into 1 component.
mark#(app(X1,X2)) | → | mark#(X2) | (25) |
mark#(app(X1,X2)) | → | mark#(X1) | (26) |
mark#(app(X1,X2)) | → | a__app#(mark(X1),mark(X2)) | (27) |
a__app#(nil,YS) | → | mark#(YS) | (19) |
mark#(from(X)) | → | mark#(X) | (28) |
mark#(from(X)) | → | a__from#(mark(X)) | (29) |
a__from#(X) | → | mark#(X) | (21) |
mark#(zWadr(X1,X2)) | → | mark#(X2) | (30) |
mark#(zWadr(X1,X2)) | → | mark#(X1) | (31) |
mark#(cons(X1,X2)) | → | mark#(X1) | (35) |
a__app#(cons(X,XS),YS) | → | mark#(X) | (20) |
[app(x1, x2)] | = | 6 · x1 + 0 · x2 + -∞ |
[a__zWadr(x1, x2)] | = | 0 · x1 + 6 · x2 + -∞ |
[a__app(x1, x2)] | = | 6 · x1 + 0 · x2 + -∞ |
[cons(x1, x2)] | = | 0 · x1 + -∞ · x2 + 0 |
[a__from(x1)] | = | 0 · x1 + 0 |
[a__prefix(x1)] | = | 0 · x1 + 0 |
[nil] | = | 0 |
[from(x1)] | = | 0 · x1 + 0 |
[a__from#(x1)] | = | 0 · x1 + -∞ |
[a__app#(x1, x2)] | = | 6 · x1 + 0 · x2 + -∞ |
[prefix(x1)] | = | 0 · x1 + 0 |
[zWadr(x1, x2)] | = | 0 · x1 + 6 · x2 + -∞ |
[mark#(x1)] | = | 0 · x1 + -∞ |
[s(x1)] | = | -∞ · x1 + 1 |
[mark(x1)] | = | 0 · x1 + -∞ |
a__app(nil,YS) | → | mark(YS) | (1) |
a__app(cons(X,XS),YS) | → | cons(mark(X),app(XS,YS)) | (2) |
a__from(X) | → | cons(mark(X),from(s(X))) | (3) |
a__zWadr(nil,YS) | → | nil | (4) |
a__zWadr(XS,nil) | → | nil | (5) |
a__zWadr(cons(X,XS),cons(Y,YS)) | → | cons(a__app(mark(Y),cons(mark(X),nil)),zWadr(XS,YS)) | (6) |
a__prefix(L) | → | cons(nil,zWadr(L,prefix(L))) | (7) |
mark(app(X1,X2)) | → | a__app(mark(X1),mark(X2)) | (8) |
mark(from(X)) | → | a__from(mark(X)) | (9) |
mark(zWadr(X1,X2)) | → | a__zWadr(mark(X1),mark(X2)) | (10) |
mark(prefix(X)) | → | a__prefix(mark(X)) | (11) |
mark(nil) | → | nil | (12) |
mark(cons(X1,X2)) | → | cons(mark(X1),X2) | (13) |
mark(s(X)) | → | s(mark(X)) | (14) |
a__app(X1,X2) | → | app(X1,X2) | (15) |
a__from(X) | → | from(X) | (16) |
a__zWadr(X1,X2) | → | zWadr(X1,X2) | (17) |
a__prefix(X) | → | prefix(X) | (18) |
mark#(app(X1,X2)) | → | mark#(X1) | (26) |
mark#(zWadr(X1,X2)) | → | mark#(X2) | (30) |
a__app#(cons(X,XS),YS) | → | mark#(X) | (20) |
[app(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[a__zWadr(x1, x2)] | = | 1 · x1 + 0 · x2 + 0 |
[a__app(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[cons(x1, x2)] | = | 0 · x1 + -∞ · x2 + -∞ |
[a__from(x1)] | = | 0 · x1 + -∞ |
[a__prefix(x1)] | = | -∞ · x1 + 0 |
[nil] | = | 0 |
[from(x1)] | = | 0 · x1 + -∞ |
[a__from#(x1)] | = | 0 · x1 + -∞ |
[a__app#(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[prefix(x1)] | = | -∞ · x1 + 0 |
[zWadr(x1, x2)] | = | 1 · x1 + 0 · x2 + 0 |
[mark#(x1)] | = | 0 · x1 + -∞ |
[s(x1)] | = | 1 · x1 + -∞ |
[mark(x1)] | = | 0 · x1 + -∞ |
a__app(nil,YS) | → | mark(YS) | (1) |
a__app(cons(X,XS),YS) | → | cons(mark(X),app(XS,YS)) | (2) |
a__from(X) | → | cons(mark(X),from(s(X))) | (3) |
a__zWadr(nil,YS) | → | nil | (4) |
a__zWadr(XS,nil) | → | nil | (5) |
a__zWadr(cons(X,XS),cons(Y,YS)) | → | cons(a__app(mark(Y),cons(mark(X),nil)),zWadr(XS,YS)) | (6) |
a__prefix(L) | → | cons(nil,zWadr(L,prefix(L))) | (7) |
mark(app(X1,X2)) | → | a__app(mark(X1),mark(X2)) | (8) |
mark(from(X)) | → | a__from(mark(X)) | (9) |
mark(zWadr(X1,X2)) | → | a__zWadr(mark(X1),mark(X2)) | (10) |
mark(prefix(X)) | → | a__prefix(mark(X)) | (11) |
mark(nil) | → | nil | (12) |
mark(cons(X1,X2)) | → | cons(mark(X1),X2) | (13) |
mark(s(X)) | → | s(mark(X)) | (14) |
a__app(X1,X2) | → | app(X1,X2) | (15) |
a__from(X) | → | from(X) | (16) |
a__zWadr(X1,X2) | → | zWadr(X1,X2) | (17) |
a__prefix(X) | → | prefix(X) | (18) |
mark#(zWadr(X1,X2)) | → | mark#(X1) | (31) |
[app(x1, x2)] | = | 1 · x1 + 5 · x2 + -∞ |
[a__zWadr(x1, x2)] | = | 6 · x1 + 2 · x2 + 0 |
[a__app(x1, x2)] | = | 1 · x1 + 5 · x2 + -∞ |
[cons(x1, x2)] | = | 1 · x1 + -∞ · x2 + 0 |
[a__from(x1)] | = | 1 · x1 + 0 |
[a__prefix(x1)] | = | 0 · x1 + 1 |
[nil] | = | 0 |
[from(x1)] | = | 1 · x1 + 0 |
[a__from#(x1)] | = | 0 · x1 + 0 |
[a__app#(x1, x2)] | = | 0 · x1 + 4 · x2 + -∞ |
[prefix(x1)] | = | 0 · x1 + 1 |
[zWadr(x1, x2)] | = | 6 · x1 + 2 · x2 + 0 |
[mark#(x1)] | = | 0 · x1 + 0 |
[s(x1)] | = | 0 · x1 + -∞ |
[mark(x1)] | = | 0 · x1 + -∞ |
a__app(nil,YS) | → | mark(YS) | (1) |
a__app(cons(X,XS),YS) | → | cons(mark(X),app(XS,YS)) | (2) |
a__from(X) | → | cons(mark(X),from(s(X))) | (3) |
a__zWadr(nil,YS) | → | nil | (4) |
a__zWadr(XS,nil) | → | nil | (5) |
a__zWadr(cons(X,XS),cons(Y,YS)) | → | cons(a__app(mark(Y),cons(mark(X),nil)),zWadr(XS,YS)) | (6) |
a__prefix(L) | → | cons(nil,zWadr(L,prefix(L))) | (7) |
mark(app(X1,X2)) | → | a__app(mark(X1),mark(X2)) | (8) |
mark(from(X)) | → | a__from(mark(X)) | (9) |
mark(zWadr(X1,X2)) | → | a__zWadr(mark(X1),mark(X2)) | (10) |
mark(prefix(X)) | → | a__prefix(mark(X)) | (11) |
mark(nil) | → | nil | (12) |
mark(cons(X1,X2)) | → | cons(mark(X1),X2) | (13) |
mark(s(X)) | → | s(mark(X)) | (14) |
a__app(X1,X2) | → | app(X1,X2) | (15) |
a__from(X) | → | from(X) | (16) |
a__zWadr(X1,X2) | → | zWadr(X1,X2) | (17) |
a__prefix(X) | → | prefix(X) | (18) |
mark#(app(X1,X2)) | → | a__app#(mark(X1),mark(X2)) | (27) |
The dependency pairs are split into 1 component.
a__from#(X) | → | mark#(X) | (21) |
mark#(app(X1,X2)) | → | mark#(X2) | (25) |
mark#(from(X)) | → | mark#(X) | (28) |
mark#(from(X)) | → | a__from#(mark(X)) | (29) |
mark#(cons(X1,X2)) | → | mark#(X1) | (35) |
[app(x1, x2)] | = | 0 · x1 + 0 · x2 + -∞ |
[a__zWadr(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[a__app(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[cons(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[a__from(x1)] | = | 4 · x1 + 6 |
[a__prefix(x1)] | = | 0 · x1 + 0 |
[nil] | = | 0 |
[from(x1)] | = | 4 · x1 + 6 |
[a__from#(x1)] | = | 4 · x1 + 6 |
[prefix(x1)] | = | 0 · x1 + -∞ |
[zWadr(x1, x2)] | = | 0 · x1 + 0 · x2 + 0 |
[mark#(x1)] | = | 0 · x1 + 0 |
[s(x1)] | = | 0 · x1 + 0 |
[mark(x1)] | = | 0 · x1 + 0 |
a__app(nil,YS) | → | mark(YS) | (1) |
a__app(cons(X,XS),YS) | → | cons(mark(X),app(XS,YS)) | (2) |
a__from(X) | → | cons(mark(X),from(s(X))) | (3) |
a__zWadr(nil,YS) | → | nil | (4) |
a__zWadr(XS,nil) | → | nil | (5) |
a__zWadr(cons(X,XS),cons(Y,YS)) | → | cons(a__app(mark(Y),cons(mark(X),nil)),zWadr(XS,YS)) | (6) |
a__prefix(L) | → | cons(nil,zWadr(L,prefix(L))) | (7) |
mark(app(X1,X2)) | → | a__app(mark(X1),mark(X2)) | (8) |
mark(from(X)) | → | a__from(mark(X)) | (9) |
mark(zWadr(X1,X2)) | → | a__zWadr(mark(X1),mark(X2)) | (10) |
mark(prefix(X)) | → | a__prefix(mark(X)) | (11) |
mark(nil) | → | nil | (12) |
mark(cons(X1,X2)) | → | cons(mark(X1),X2) | (13) |
mark(s(X)) | → | s(mark(X)) | (14) |
a__app(X1,X2) | → | app(X1,X2) | (15) |
a__from(X) | → | from(X) | (16) |
a__zWadr(X1,X2) | → | zWadr(X1,X2) | (17) |
a__prefix(X) | → | prefix(X) | (18) |
a__from#(X) | → | mark#(X) | (21) |
mark#(from(X)) | → | mark#(X) | (28) |
The dependency pairs are split into 1 component.
mark#(app(X1,X2)) | → | mark#(X2) | (25) |
mark#(cons(X1,X2)) | → | mark#(X1) | (35) |
Using size-change termination in combination with the subterm criterion one obtains the following initial size-change graphs.
mark#(app(X1,X2)) | → | mark#(X2) | (25) |
1 | > | 1 | |
mark#(cons(X1,X2)) | → | mark#(X1) | (35) |
1 | > | 1 |
As there is no critical graph in the transitive closure, there are no infinite chains.