The rewrite relation of the following TRS is considered.
sel(s(X),cons(Y,Z)) |
→ |
sel(X,activate(Z)) |
(1) |
sel(0,cons(X,Z)) |
→ |
X |
(2) |
first(0,Z) |
→ |
nil |
(3) |
first(s(X),cons(Y,Z)) |
→ |
cons(Y,n__first(X,activate(Z))) |
(4) |
from(X) |
→ |
cons(X,n__from(s(X))) |
(5) |
sel1(s(X),cons(Y,Z)) |
→ |
sel1(X,activate(Z)) |
(6) |
sel1(0,cons(X,Z)) |
→ |
quote(X) |
(7) |
first1(0,Z) |
→ |
nil1 |
(8) |
first1(s(X),cons(Y,Z)) |
→ |
cons1(quote(Y),first1(X,activate(Z))) |
(9) |
quote(n__0) |
→ |
01 |
(10) |
quote1(n__cons(X,Z)) |
→ |
cons1(quote(activate(X)),quote1(activate(Z))) |
(11) |
quote1(n__nil) |
→ |
nil1 |
(12) |
quote(n__s(X)) |
→ |
s1(quote(activate(X))) |
(13) |
quote(n__sel(X,Z)) |
→ |
sel1(activate(X),activate(Z)) |
(14) |
quote1(n__first(X,Z)) |
→ |
first1(activate(X),activate(Z)) |
(15) |
unquote(01) |
→ |
0 |
(16) |
unquote(s1(X)) |
→ |
s(unquote(X)) |
(17) |
unquote1(nil1) |
→ |
nil |
(18) |
unquote1(cons1(X,Z)) |
→ |
fcons(unquote(X),unquote1(Z)) |
(19) |
fcons(X,Z) |
→ |
cons(X,Z) |
(20) |
first(X1,X2) |
→ |
n__first(X1,X2) |
(21) |
from(X) |
→ |
n__from(X) |
(22) |
0 |
→ |
n__0 |
(23) |
cons(X1,X2) |
→ |
n__cons(X1,X2) |
(24) |
nil |
→ |
n__nil |
(25) |
s(X) |
→ |
n__s(X) |
(26) |
sel(X1,X2) |
→ |
n__sel(X1,X2) |
(27) |
activate(n__first(X1,X2)) |
→ |
first(X1,X2) |
(28) |
activate(n__from(X)) |
→ |
from(X) |
(29) |
activate(n__0) |
→ |
0 |
(30) |
activate(n__cons(X1,X2)) |
→ |
cons(X1,X2) |
(31) |
activate(n__nil) |
→ |
nil |
(32) |
activate(n__s(X)) |
→ |
s(X) |
(33) |
activate(n__sel(X1,X2)) |
→ |
sel(X1,X2) |
(34) |
activate(X) |
→ |
X |
(35) |
sel#(s(X),cons(Y,Z)) |
→ |
sel#(X,activate(Z)) |
(36) |
sel#(s(X),cons(Y,Z)) |
→ |
activate#(Z) |
(37) |
first#(0,Z) |
→ |
nil# |
(38) |
first#(s(X),cons(Y,Z)) |
→ |
cons#(Y,n__first(X,activate(Z))) |
(39) |
first#(s(X),cons(Y,Z)) |
→ |
activate#(Z) |
(40) |
from#(X) |
→ |
cons#(X,n__from(s(X))) |
(41) |
from#(X) |
→ |
s#(X) |
(42) |
sel1#(s(X),cons(Y,Z)) |
→ |
sel1#(X,activate(Z)) |
(43) |
sel1#(s(X),cons(Y,Z)) |
→ |
activate#(Z) |
(44) |
sel1#(0,cons(X,Z)) |
→ |
quote#(X) |
(45) |
first1#(s(X),cons(Y,Z)) |
→ |
quote#(Y) |
(46) |
first1#(s(X),cons(Y,Z)) |
→ |
first1#(X,activate(Z)) |
(47) |
first1#(s(X),cons(Y,Z)) |
→ |
activate#(Z) |
(48) |
quote1#(n__cons(X,Z)) |
→ |
quote#(activate(X)) |
(49) |
quote1#(n__cons(X,Z)) |
→ |
activate#(X) |
(50) |
quote1#(n__cons(X,Z)) |
→ |
quote1#(activate(Z)) |
(51) |
quote1#(n__cons(X,Z)) |
→ |
activate#(Z) |
(52) |
quote#(n__s(X)) |
→ |
quote#(activate(X)) |
(53) |
quote#(n__s(X)) |
→ |
activate#(X) |
(54) |
quote#(n__sel(X,Z)) |
→ |
sel1#(activate(X),activate(Z)) |
(55) |
quote#(n__sel(X,Z)) |
→ |
activate#(X) |
(56) |
quote#(n__sel(X,Z)) |
→ |
activate#(Z) |
(57) |
quote1#(n__first(X,Z)) |
→ |
first1#(activate(X),activate(Z)) |
(58) |
quote1#(n__first(X,Z)) |
→ |
activate#(X) |
(59) |
quote1#(n__first(X,Z)) |
→ |
activate#(Z) |
(60) |
unquote#(01) |
→ |
0# |
(61) |
unquote#(s1(X)) |
→ |
s#(unquote(X)) |
(62) |
unquote#(s1(X)) |
→ |
unquote#(X) |
(63) |
unquote1#(nil1) |
→ |
nil# |
(64) |
unquote1#(cons1(X,Z)) |
→ |
fcons#(unquote(X),unquote1(Z)) |
(65) |
unquote1#(cons1(X,Z)) |
→ |
unquote#(X) |
(66) |
unquote1#(cons1(X,Z)) |
→ |
unquote1#(Z) |
(67) |
fcons#(X,Z) |
→ |
cons#(X,Z) |
(68) |
activate#(n__first(X1,X2)) |
→ |
first#(X1,X2) |
(69) |
activate#(n__from(X)) |
→ |
from#(X) |
(70) |
activate#(n__0) |
→ |
0# |
(71) |
activate#(n__cons(X1,X2)) |
→ |
cons#(X1,X2) |
(72) |
activate#(n__nil) |
→ |
nil# |
(73) |
activate#(n__s(X)) |
→ |
s#(X) |
(74) |
activate#(n__sel(X1,X2)) |
→ |
sel#(X1,X2) |
(75) |
It remains to prove infiniteness of the resulting DP problem.
sel#(s(X),cons(Y,Z)) |
→ |
sel#(X,activate(Z)) |
(36) |
sel#(s(X),cons(Y,Z)) |
→ |
activate#(Z) |
(37) |
first#(0,Z) |
→ |
nil# |
(38) |
first#(s(X),cons(Y,Z)) |
→ |
cons#(Y,n__first(X,activate(Z))) |
(39) |
first#(s(X),cons(Y,Z)) |
→ |
activate#(Z) |
(40) |
from#(X) |
→ |
cons#(X,n__from(s(X))) |
(41) |
from#(X) |
→ |
s#(X) |
(42) |
sel1#(s(X),cons(Y,Z)) |
→ |
sel1#(X,activate(Z)) |
(43) |
sel1#(s(X),cons(Y,Z)) |
→ |
activate#(Z) |
(44) |
sel1#(0,cons(X,Z)) |
→ |
quote#(X) |
(45) |
first1#(s(X),cons(Y,Z)) |
→ |
quote#(Y) |
(46) |
first1#(s(X),cons(Y,Z)) |
→ |
first1#(X,activate(Z)) |
(47) |
first1#(s(X),cons(Y,Z)) |
→ |
activate#(Z) |
(48) |
quote1#(n__cons(X,Z)) |
→ |
quote#(activate(X)) |
(49) |
quote1#(n__cons(X,Z)) |
→ |
activate#(X) |
(50) |
quote1#(n__cons(X,Z)) |
→ |
activate#(Z) |
(52) |
quote#(n__s(X)) |
→ |
quote#(activate(X)) |
(53) |
quote#(n__s(X)) |
→ |
activate#(X) |
(54) |
quote#(n__sel(X,Z)) |
→ |
sel1#(activate(X),activate(Z)) |
(55) |
quote#(n__sel(X,Z)) |
→ |
activate#(X) |
(56) |
quote#(n__sel(X,Z)) |
→ |
activate#(Z) |
(57) |
quote1#(n__first(X,Z)) |
→ |
first1#(activate(X),activate(Z)) |
(58) |
quote1#(n__first(X,Z)) |
→ |
activate#(X) |
(59) |
quote1#(n__first(X,Z)) |
→ |
activate#(Z) |
(60) |
unquote#(01) |
→ |
0# |
(61) |
unquote#(s1(X)) |
→ |
s#(unquote(X)) |
(62) |
unquote#(s1(X)) |
→ |
unquote#(X) |
(63) |
unquote1#(nil1) |
→ |
nil# |
(64) |
unquote1#(cons1(X,Z)) |
→ |
fcons#(unquote(X),unquote1(Z)) |
(65) |
unquote1#(cons1(X,Z)) |
→ |
unquote#(X) |
(66) |
unquote1#(cons1(X,Z)) |
→ |
unquote1#(Z) |
(67) |
fcons#(X,Z) |
→ |
cons#(X,Z) |
(68) |
activate#(n__first(X1,X2)) |
→ |
first#(X1,X2) |
(69) |
activate#(n__from(X)) |
→ |
from#(X) |
(70) |
activate#(n__0) |
→ |
0# |
(71) |
activate#(n__cons(X1,X2)) |
→ |
cons#(X1,X2) |
(72) |
activate#(n__nil) |
→ |
nil# |
(73) |
activate#(n__s(X)) |
→ |
s#(X) |
(74) |
activate#(n__sel(X1,X2)) |
→ |
sel#(X1,X2) |
(75) |
and the following rules have been deleted.
quote1#(n__cons(y0,n__first(x0,x1))) |
→ |
quote1#(first(x0,x1)) |
(76) |
quote1#(n__cons(y0,n__from(x0))) |
→ |
quote1#(from(x0)) |
(77) |
quote1#(n__cons(y0,n__0)) |
→ |
quote1#(0) |
(78) |
quote1#(n__cons(y0,n__cons(x0,x1))) |
→ |
quote1#(cons(x0,x1)) |
(79) |
quote1#(n__cons(y0,n__nil)) |
→ |
quote1#(nil) |
(80) |
quote1#(n__cons(y0,n__s(x0))) |
→ |
quote1#(s(x0)) |
(81) |
quote1#(n__cons(y0,n__sel(x0,x1))) |
→ |
quote1#(sel(x0,x1)) |
(82) |
quote1#(n__cons(y0,x0)) |
→ |
quote1#(x0) |
(83) |