The rewrite relation of the following TRS is considered.
active(terms(N)) |
→ |
mark(cons(recip(sqr(N)),terms(s(N)))) |
(1) |
active(sqr(0)) |
→ |
mark(0) |
(2) |
active(sqr(s(X))) |
→ |
mark(s(add(sqr(X),dbl(X)))) |
(3) |
active(dbl(0)) |
→ |
mark(0) |
(4) |
active(dbl(s(X))) |
→ |
mark(s(s(dbl(X)))) |
(5) |
active(add(0,X)) |
→ |
mark(X) |
(6) |
active(add(s(X),Y)) |
→ |
mark(s(add(X,Y))) |
(7) |
active(first(0,X)) |
→ |
mark(nil) |
(8) |
active(first(s(X),cons(Y,Z))) |
→ |
mark(cons(Y,first(X,Z))) |
(9) |
active(terms(X)) |
→ |
terms(active(X)) |
(10) |
active(cons(X1,X2)) |
→ |
cons(active(X1),X2) |
(11) |
active(recip(X)) |
→ |
recip(active(X)) |
(12) |
active(sqr(X)) |
→ |
sqr(active(X)) |
(13) |
active(s(X)) |
→ |
s(active(X)) |
(14) |
active(add(X1,X2)) |
→ |
add(active(X1),X2) |
(15) |
active(add(X1,X2)) |
→ |
add(X1,active(X2)) |
(16) |
active(dbl(X)) |
→ |
dbl(active(X)) |
(17) |
active(first(X1,X2)) |
→ |
first(active(X1),X2) |
(18) |
active(first(X1,X2)) |
→ |
first(X1,active(X2)) |
(19) |
terms(mark(X)) |
→ |
mark(terms(X)) |
(20) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(21) |
recip(mark(X)) |
→ |
mark(recip(X)) |
(22) |
sqr(mark(X)) |
→ |
mark(sqr(X)) |
(23) |
s(mark(X)) |
→ |
mark(s(X)) |
(24) |
add(mark(X1),X2) |
→ |
mark(add(X1,X2)) |
(25) |
add(X1,mark(X2)) |
→ |
mark(add(X1,X2)) |
(26) |
dbl(mark(X)) |
→ |
mark(dbl(X)) |
(27) |
first(mark(X1),X2) |
→ |
mark(first(X1,X2)) |
(28) |
first(X1,mark(X2)) |
→ |
mark(first(X1,X2)) |
(29) |
proper(terms(X)) |
→ |
terms(proper(X)) |
(30) |
proper(cons(X1,X2)) |
→ |
cons(proper(X1),proper(X2)) |
(31) |
proper(recip(X)) |
→ |
recip(proper(X)) |
(32) |
proper(sqr(X)) |
→ |
sqr(proper(X)) |
(33) |
proper(s(X)) |
→ |
s(proper(X)) |
(34) |
proper(0) |
→ |
ok(0) |
(35) |
proper(add(X1,X2)) |
→ |
add(proper(X1),proper(X2)) |
(36) |
proper(dbl(X)) |
→ |
dbl(proper(X)) |
(37) |
proper(first(X1,X2)) |
→ |
first(proper(X1),proper(X2)) |
(38) |
proper(nil) |
→ |
ok(nil) |
(39) |
terms(ok(X)) |
→ |
ok(terms(X)) |
(40) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(41) |
recip(ok(X)) |
→ |
ok(recip(X)) |
(42) |
sqr(ok(X)) |
→ |
ok(sqr(X)) |
(43) |
s(ok(X)) |
→ |
ok(s(X)) |
(44) |
add(ok(X1),ok(X2)) |
→ |
ok(add(X1,X2)) |
(45) |
dbl(ok(X)) |
→ |
ok(dbl(X)) |
(46) |
first(ok(X1),ok(X2)) |
→ |
ok(first(X1,X2)) |
(47) |
top(mark(X)) |
→ |
top(proper(X)) |
(48) |
top(ok(X)) |
→ |
top(active(X)) |
(49) |
active#(terms(N)) |
→ |
s#(N) |
(50) |
active#(terms(N)) |
→ |
terms#(s(N)) |
(51) |
active#(terms(N)) |
→ |
sqr#(N) |
(52) |
active#(terms(N)) |
→ |
recip#(sqr(N)) |
(53) |
active#(terms(N)) |
→ |
cons#(recip(sqr(N)),terms(s(N))) |
(54) |
active#(sqr(s(X))) |
→ |
dbl#(X) |
(55) |
active#(sqr(s(X))) |
→ |
sqr#(X) |
(56) |
active#(sqr(s(X))) |
→ |
add#(sqr(X),dbl(X)) |
(57) |
active#(sqr(s(X))) |
→ |
s#(add(sqr(X),dbl(X))) |
(58) |
active#(dbl(s(X))) |
→ |
dbl#(X) |
(59) |
active#(dbl(s(X))) |
→ |
s#(dbl(X)) |
(60) |
active#(dbl(s(X))) |
→ |
s#(s(dbl(X))) |
(61) |
active#(add(s(X),Y)) |
→ |
add#(X,Y) |
(62) |
active#(add(s(X),Y)) |
→ |
s#(add(X,Y)) |
(63) |
active#(first(s(X),cons(Y,Z))) |
→ |
first#(X,Z) |
(64) |
active#(first(s(X),cons(Y,Z))) |
→ |
cons#(Y,first(X,Z)) |
(65) |
active#(terms(X)) |
→ |
active#(X) |
(66) |
active#(terms(X)) |
→ |
terms#(active(X)) |
(67) |
active#(cons(X1,X2)) |
→ |
active#(X1) |
(68) |
active#(cons(X1,X2)) |
→ |
cons#(active(X1),X2) |
(69) |
active#(recip(X)) |
→ |
active#(X) |
(70) |
active#(recip(X)) |
→ |
recip#(active(X)) |
(71) |
active#(sqr(X)) |
→ |
active#(X) |
(72) |
active#(sqr(X)) |
→ |
sqr#(active(X)) |
(73) |
active#(s(X)) |
→ |
active#(X) |
(74) |
active#(s(X)) |
→ |
s#(active(X)) |
(75) |
active#(add(X1,X2)) |
→ |
active#(X1) |
(76) |
active#(add(X1,X2)) |
→ |
add#(active(X1),X2) |
(77) |
active#(add(X1,X2)) |
→ |
active#(X2) |
(78) |
active#(add(X1,X2)) |
→ |
add#(X1,active(X2)) |
(79) |
active#(dbl(X)) |
→ |
active#(X) |
(80) |
active#(dbl(X)) |
→ |
dbl#(active(X)) |
(81) |
active#(first(X1,X2)) |
→ |
active#(X1) |
(82) |
active#(first(X1,X2)) |
→ |
first#(active(X1),X2) |
(83) |
active#(first(X1,X2)) |
→ |
active#(X2) |
(84) |
active#(first(X1,X2)) |
→ |
first#(X1,active(X2)) |
(85) |
terms#(mark(X)) |
→ |
terms#(X) |
(86) |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(87) |
recip#(mark(X)) |
→ |
recip#(X) |
(88) |
sqr#(mark(X)) |
→ |
sqr#(X) |
(89) |
s#(mark(X)) |
→ |
s#(X) |
(90) |
add#(mark(X1),X2) |
→ |
add#(X1,X2) |
(91) |
add#(X1,mark(X2)) |
→ |
add#(X1,X2) |
(92) |
dbl#(mark(X)) |
→ |
dbl#(X) |
(93) |
first#(mark(X1),X2) |
→ |
first#(X1,X2) |
(94) |
first#(X1,mark(X2)) |
→ |
first#(X1,X2) |
(95) |
proper#(terms(X)) |
→ |
proper#(X) |
(96) |
proper#(terms(X)) |
→ |
terms#(proper(X)) |
(97) |
proper#(cons(X1,X2)) |
→ |
proper#(X2) |
(98) |
proper#(cons(X1,X2)) |
→ |
proper#(X1) |
(99) |
proper#(cons(X1,X2)) |
→ |
cons#(proper(X1),proper(X2)) |
(100) |
proper#(recip(X)) |
→ |
proper#(X) |
(101) |
proper#(recip(X)) |
→ |
recip#(proper(X)) |
(102) |
proper#(sqr(X)) |
→ |
proper#(X) |
(103) |
proper#(sqr(X)) |
→ |
sqr#(proper(X)) |
(104) |
proper#(s(X)) |
→ |
proper#(X) |
(105) |
proper#(s(X)) |
→ |
s#(proper(X)) |
(106) |
proper#(add(X1,X2)) |
→ |
proper#(X2) |
(107) |
proper#(add(X1,X2)) |
→ |
proper#(X1) |
(108) |
proper#(add(X1,X2)) |
→ |
add#(proper(X1),proper(X2)) |
(109) |
proper#(dbl(X)) |
→ |
proper#(X) |
(110) |
proper#(dbl(X)) |
→ |
dbl#(proper(X)) |
(111) |
proper#(first(X1,X2)) |
→ |
proper#(X2) |
(112) |
proper#(first(X1,X2)) |
→ |
proper#(X1) |
(113) |
proper#(first(X1,X2)) |
→ |
first#(proper(X1),proper(X2)) |
(114) |
terms#(ok(X)) |
→ |
terms#(X) |
(115) |
cons#(ok(X1),ok(X2)) |
→ |
cons#(X1,X2) |
(116) |
recip#(ok(X)) |
→ |
recip#(X) |
(117) |
sqr#(ok(X)) |
→ |
sqr#(X) |
(118) |
s#(ok(X)) |
→ |
s#(X) |
(119) |
add#(ok(X1),ok(X2)) |
→ |
add#(X1,X2) |
(120) |
dbl#(ok(X)) |
→ |
dbl#(X) |
(121) |
first#(ok(X1),ok(X2)) |
→ |
first#(X1,X2) |
(122) |
top#(mark(X)) |
→ |
proper#(X) |
(123) |
top#(mark(X)) |
→ |
top#(proper(X)) |
(124) |
top#(ok(X)) |
→ |
active#(X) |
(125) |
top#(ok(X)) |
→ |
top#(active(X)) |
(126) |
The dependency pairs are split into 11
components.
-
The
1st
component contains the
pair
top#(ok(X)) |
→ |
top#(active(X)) |
(126) |
top#(mark(X)) |
→ |
top#(proper(X)) |
(124) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the
prec(top#) |
= |
0 |
|
stat(top#) |
= |
lex
|
prec(ok) |
= |
0 |
|
stat(ok) |
= |
lex
|
prec(proper) |
= |
0 |
|
stat(proper) |
= |
lex
|
prec(nil) |
= |
0 |
|
stat(nil) |
= |
lex
|
prec(first) |
= |
1 |
|
stat(first) |
= |
lex
|
prec(add) |
= |
2 |
|
stat(add) |
= |
lex
|
prec(dbl) |
= |
2 |
|
stat(dbl) |
= |
lex
|
prec(0) |
= |
0 |
|
stat(0) |
= |
lex
|
prec(mark) |
= |
0 |
|
stat(mark) |
= |
lex
|
prec(cons) |
= |
8 |
|
stat(cons) |
= |
lex
|
prec(s) |
= |
1 |
|
stat(s) |
= |
lex
|
prec(recip) |
= |
1 |
|
stat(recip) |
= |
lex
|
prec(sqr) |
= |
3 |
|
stat(sqr) |
= |
lex
|
prec(active) |
= |
0 |
|
stat(active) |
= |
lex
|
prec(terms) |
= |
9 |
|
stat(terms) |
= |
lex
|
π(top#) |
= |
1 |
π(ok) |
= |
1 |
π(proper) |
= |
1 |
π(nil) |
= |
[] |
π(first) |
= |
[1,2] |
π(add) |
= |
[1,2] |
π(dbl) |
= |
[1] |
π(0) |
= |
[] |
π(mark) |
= |
[1] |
π(cons) |
= |
[1] |
π(s) |
= |
[1] |
π(recip) |
= |
[1] |
π(sqr) |
= |
[1] |
π(active) |
= |
1 |
π(terms) |
= |
[1] |
together with the usable
rules
active(terms(N)) |
→ |
mark(cons(recip(sqr(N)),terms(s(N)))) |
(1) |
active(sqr(0)) |
→ |
mark(0) |
(2) |
active(sqr(s(X))) |
→ |
mark(s(add(sqr(X),dbl(X)))) |
(3) |
active(dbl(0)) |
→ |
mark(0) |
(4) |
active(dbl(s(X))) |
→ |
mark(s(s(dbl(X)))) |
(5) |
active(add(0,X)) |
→ |
mark(X) |
(6) |
active(add(s(X),Y)) |
→ |
mark(s(add(X,Y))) |
(7) |
active(first(0,X)) |
→ |
mark(nil) |
(8) |
active(first(s(X),cons(Y,Z))) |
→ |
mark(cons(Y,first(X,Z))) |
(9) |
active(terms(X)) |
→ |
terms(active(X)) |
(10) |
active(cons(X1,X2)) |
→ |
cons(active(X1),X2) |
(11) |
active(recip(X)) |
→ |
recip(active(X)) |
(12) |
active(sqr(X)) |
→ |
sqr(active(X)) |
(13) |
active(s(X)) |
→ |
s(active(X)) |
(14) |
active(add(X1,X2)) |
→ |
add(active(X1),X2) |
(15) |
active(add(X1,X2)) |
→ |
add(X1,active(X2)) |
(16) |
active(dbl(X)) |
→ |
dbl(active(X)) |
(17) |
active(first(X1,X2)) |
→ |
first(active(X1),X2) |
(18) |
active(first(X1,X2)) |
→ |
first(X1,active(X2)) |
(19) |
s(mark(X)) |
→ |
mark(s(X)) |
(24) |
s(ok(X)) |
→ |
ok(s(X)) |
(44) |
terms(mark(X)) |
→ |
mark(terms(X)) |
(20) |
terms(ok(X)) |
→ |
ok(terms(X)) |
(40) |
sqr(mark(X)) |
→ |
mark(sqr(X)) |
(23) |
sqr(ok(X)) |
→ |
ok(sqr(X)) |
(43) |
recip(mark(X)) |
→ |
mark(recip(X)) |
(22) |
recip(ok(X)) |
→ |
ok(recip(X)) |
(42) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(21) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(41) |
dbl(mark(X)) |
→ |
mark(dbl(X)) |
(27) |
dbl(ok(X)) |
→ |
ok(dbl(X)) |
(46) |
add(mark(X1),X2) |
→ |
mark(add(X1,X2)) |
(25) |
add(X1,mark(X2)) |
→ |
mark(add(X1,X2)) |
(26) |
add(ok(X1),ok(X2)) |
→ |
ok(add(X1,X2)) |
(45) |
first(mark(X1),X2) |
→ |
mark(first(X1,X2)) |
(28) |
first(X1,mark(X2)) |
→ |
mark(first(X1,X2)) |
(29) |
first(ok(X1),ok(X2)) |
→ |
ok(first(X1,X2)) |
(47) |
proper(terms(X)) |
→ |
terms(proper(X)) |
(30) |
proper(cons(X1,X2)) |
→ |
cons(proper(X1),proper(X2)) |
(31) |
proper(recip(X)) |
→ |
recip(proper(X)) |
(32) |
proper(sqr(X)) |
→ |
sqr(proper(X)) |
(33) |
proper(s(X)) |
→ |
s(proper(X)) |
(34) |
proper(0) |
→ |
ok(0) |
(35) |
proper(add(X1,X2)) |
→ |
add(proper(X1),proper(X2)) |
(36) |
proper(dbl(X)) |
→ |
dbl(proper(X)) |
(37) |
proper(first(X1,X2)) |
→ |
first(proper(X1),proper(X2)) |
(38) |
proper(nil) |
→ |
ok(nil) |
(39) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
top#(mark(X)) |
→ |
top#(proper(X)) |
(124) |
could be deleted.
1.1.1.1 Reduction Pair Processor with Usable Rules
Using the
prec(top#) |
= |
0 |
|
stat(top#) |
= |
lex
|
prec(ok) |
= |
0 |
|
stat(ok) |
= |
lex
|
prec(nil) |
= |
0 |
|
stat(nil) |
= |
lex
|
prec(first) |
= |
5 |
|
stat(first) |
= |
lex
|
prec(add) |
= |
0 |
|
stat(add) |
= |
lex
|
prec(dbl) |
= |
0 |
|
stat(dbl) |
= |
lex
|
prec(0) |
= |
0 |
|
stat(0) |
= |
lex
|
prec(mark) |
= |
0 |
|
stat(mark) |
= |
lex
|
prec(cons) |
= |
0 |
|
stat(cons) |
= |
lex
|
prec(s) |
= |
0 |
|
stat(s) |
= |
lex
|
prec(recip) |
= |
0 |
|
stat(recip) |
= |
lex
|
prec(sqr) |
= |
0 |
|
stat(sqr) |
= |
lex
|
prec(active) |
= |
0 |
|
stat(active) |
= |
lex
|
prec(terms) |
= |
0 |
|
stat(terms) |
= |
lex
|
π(top#) |
= |
1 |
π(ok) |
= |
[1] |
π(nil) |
= |
[] |
π(first) |
= |
[2] |
π(add) |
= |
2 |
π(dbl) |
= |
1 |
π(0) |
= |
[] |
π(mark) |
= |
1 |
π(cons) |
= |
2 |
π(s) |
= |
1 |
π(recip) |
= |
1 |
π(sqr) |
= |
1 |
π(active) |
= |
1 |
π(terms) |
= |
1 |
together with the usable
rules
active(terms(N)) |
→ |
mark(cons(recip(sqr(N)),terms(s(N)))) |
(1) |
active(sqr(0)) |
→ |
mark(0) |
(2) |
active(sqr(s(X))) |
→ |
mark(s(add(sqr(X),dbl(X)))) |
(3) |
active(dbl(0)) |
→ |
mark(0) |
(4) |
active(dbl(s(X))) |
→ |
mark(s(s(dbl(X)))) |
(5) |
active(add(0,X)) |
→ |
mark(X) |
(6) |
active(add(s(X),Y)) |
→ |
mark(s(add(X,Y))) |
(7) |
active(first(0,X)) |
→ |
mark(nil) |
(8) |
active(first(s(X),cons(Y,Z))) |
→ |
mark(cons(Y,first(X,Z))) |
(9) |
active(terms(X)) |
→ |
terms(active(X)) |
(10) |
active(cons(X1,X2)) |
→ |
cons(active(X1),X2) |
(11) |
active(recip(X)) |
→ |
recip(active(X)) |
(12) |
active(sqr(X)) |
→ |
sqr(active(X)) |
(13) |
active(s(X)) |
→ |
s(active(X)) |
(14) |
active(add(X1,X2)) |
→ |
add(active(X1),X2) |
(15) |
active(add(X1,X2)) |
→ |
add(X1,active(X2)) |
(16) |
active(dbl(X)) |
→ |
dbl(active(X)) |
(17) |
active(first(X1,X2)) |
→ |
first(active(X1),X2) |
(18) |
active(first(X1,X2)) |
→ |
first(X1,active(X2)) |
(19) |
s(mark(X)) |
→ |
mark(s(X)) |
(24) |
s(ok(X)) |
→ |
ok(s(X)) |
(44) |
terms(mark(X)) |
→ |
mark(terms(X)) |
(20) |
terms(ok(X)) |
→ |
ok(terms(X)) |
(40) |
sqr(mark(X)) |
→ |
mark(sqr(X)) |
(23) |
sqr(ok(X)) |
→ |
ok(sqr(X)) |
(43) |
recip(mark(X)) |
→ |
mark(recip(X)) |
(22) |
recip(ok(X)) |
→ |
ok(recip(X)) |
(42) |
cons(mark(X1),X2) |
→ |
mark(cons(X1,X2)) |
(21) |
cons(ok(X1),ok(X2)) |
→ |
ok(cons(X1,X2)) |
(41) |
dbl(mark(X)) |
→ |
mark(dbl(X)) |
(27) |
dbl(ok(X)) |
→ |
ok(dbl(X)) |
(46) |
add(mark(X1),X2) |
→ |
mark(add(X1,X2)) |
(25) |
add(X1,mark(X2)) |
→ |
mark(add(X1,X2)) |
(26) |
add(ok(X1),ok(X2)) |
→ |
ok(add(X1,X2)) |
(45) |
first(mark(X1),X2) |
→ |
mark(first(X1,X2)) |
(28) |
first(X1,mark(X2)) |
→ |
mark(first(X1,X2)) |
(29) |
first(ok(X1),ok(X2)) |
→ |
ok(first(X1,X2)) |
(47) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
top#(ok(X)) |
→ |
top#(active(X)) |
(126) |
could be deleted.
1.1.1.1.1 P is empty
There are no pairs anymore.
-
The
2nd
component contains the
pair
active#(terms(X)) |
→ |
active#(X) |
(66) |
active#(cons(X1,X2)) |
→ |
active#(X1) |
(68) |
active#(recip(X)) |
→ |
active#(X) |
(70) |
active#(sqr(X)) |
→ |
active#(X) |
(72) |
active#(s(X)) |
→ |
active#(X) |
(74) |
active#(add(X1,X2)) |
→ |
active#(X1) |
(76) |
active#(add(X1,X2)) |
→ |
active#(X2) |
(78) |
active#(dbl(X)) |
→ |
active#(X) |
(80) |
active#(first(X1,X2)) |
→ |
active#(X1) |
(82) |
active#(first(X1,X2)) |
→ |
active#(X2) |
(84) |
1.1.2 Subterm Criterion Processor
We use the projection
and remove the pairs:
active#(terms(X)) |
→ |
active#(X) |
(66) |
active#(cons(X1,X2)) |
→ |
active#(X1) |
(68) |
active#(recip(X)) |
→ |
active#(X) |
(70) |
active#(sqr(X)) |
→ |
active#(X) |
(72) |
active#(s(X)) |
→ |
active#(X) |
(74) |
active#(add(X1,X2)) |
→ |
active#(X1) |
(76) |
active#(add(X1,X2)) |
→ |
active#(X2) |
(78) |
active#(dbl(X)) |
→ |
active#(X) |
(80) |
active#(first(X1,X2)) |
→ |
active#(X1) |
(82) |
active#(first(X1,X2)) |
→ |
active#(X2) |
(84) |
1.1.2.1 P is empty
There are no pairs anymore.
-
The
3rd
component contains the
pair
proper#(terms(X)) |
→ |
proper#(X) |
(96) |
proper#(cons(X1,X2)) |
→ |
proper#(X2) |
(98) |
proper#(cons(X1,X2)) |
→ |
proper#(X1) |
(99) |
proper#(recip(X)) |
→ |
proper#(X) |
(101) |
proper#(sqr(X)) |
→ |
proper#(X) |
(103) |
proper#(s(X)) |
→ |
proper#(X) |
(105) |
proper#(add(X1,X2)) |
→ |
proper#(X2) |
(107) |
proper#(add(X1,X2)) |
→ |
proper#(X1) |
(108) |
proper#(dbl(X)) |
→ |
proper#(X) |
(110) |
proper#(first(X1,X2)) |
→ |
proper#(X2) |
(112) |
proper#(first(X1,X2)) |
→ |
proper#(X1) |
(113) |
1.1.3 Subterm Criterion Processor
We use the projection
and remove the pairs:
proper#(terms(X)) |
→ |
proper#(X) |
(96) |
proper#(cons(X1,X2)) |
→ |
proper#(X2) |
(98) |
proper#(cons(X1,X2)) |
→ |
proper#(X1) |
(99) |
proper#(recip(X)) |
→ |
proper#(X) |
(101) |
proper#(sqr(X)) |
→ |
proper#(X) |
(103) |
proper#(s(X)) |
→ |
proper#(X) |
(105) |
proper#(add(X1,X2)) |
→ |
proper#(X2) |
(107) |
proper#(add(X1,X2)) |
→ |
proper#(X1) |
(108) |
proper#(dbl(X)) |
→ |
proper#(X) |
(110) |
proper#(first(X1,X2)) |
→ |
proper#(X2) |
(112) |
proper#(first(X1,X2)) |
→ |
proper#(X1) |
(113) |
1.1.3.1 P is empty
There are no pairs anymore.
-
The
4th
component contains the
pair
first#(mark(X1),X2) |
→ |
first#(X1,X2) |
(94) |
first#(X1,mark(X2)) |
→ |
first#(X1,X2) |
(95) |
first#(ok(X1),ok(X2)) |
→ |
first#(X1,X2) |
(122) |
1.1.4 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
first#(mark(X1),X2) |
→ |
first#(X1,X2) |
(94) |
|
2 |
≥ |
2 |
1 |
> |
1 |
first#(X1,mark(X2)) |
→ |
first#(X1,X2) |
(95) |
|
2 |
> |
2 |
1 |
≥ |
1 |
first#(ok(X1),ok(X2)) |
→ |
first#(X1,X2) |
(122) |
|
2 |
> |
2 |
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
5th
component contains the
pair
dbl#(mark(X)) |
→ |
dbl#(X) |
(93) |
dbl#(ok(X)) |
→ |
dbl#(X) |
(121) |
1.1.5 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
dbl#(mark(X)) |
→ |
dbl#(X) |
(93) |
|
1 |
> |
1 |
dbl#(ok(X)) |
→ |
dbl#(X) |
(121) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
6th
component contains the
pair
add#(mark(X1),X2) |
→ |
add#(X1,X2) |
(91) |
add#(X1,mark(X2)) |
→ |
add#(X1,X2) |
(92) |
add#(ok(X1),ok(X2)) |
→ |
add#(X1,X2) |
(120) |
1.1.6 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
add#(mark(X1),X2) |
→ |
add#(X1,X2) |
(91) |
|
2 |
≥ |
2 |
1 |
> |
1 |
add#(X1,mark(X2)) |
→ |
add#(X1,X2) |
(92) |
|
2 |
> |
2 |
1 |
≥ |
1 |
add#(ok(X1),ok(X2)) |
→ |
add#(X1,X2) |
(120) |
|
2 |
> |
2 |
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
7th
component contains the
pair
s#(mark(X)) |
→ |
s#(X) |
(90) |
s#(ok(X)) |
→ |
s#(X) |
(119) |
1.1.7 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
s#(mark(X)) |
→ |
s#(X) |
(90) |
|
1 |
> |
1 |
s#(ok(X)) |
→ |
s#(X) |
(119) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
8th
component contains the
pair
sqr#(mark(X)) |
→ |
sqr#(X) |
(89) |
sqr#(ok(X)) |
→ |
sqr#(X) |
(118) |
1.1.8 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
sqr#(mark(X)) |
→ |
sqr#(X) |
(89) |
|
1 |
> |
1 |
sqr#(ok(X)) |
→ |
sqr#(X) |
(118) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
9th
component contains the
pair
recip#(mark(X)) |
→ |
recip#(X) |
(88) |
recip#(ok(X)) |
→ |
recip#(X) |
(117) |
1.1.9 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
recip#(mark(X)) |
→ |
recip#(X) |
(88) |
|
1 |
> |
1 |
recip#(ok(X)) |
→ |
recip#(X) |
(117) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
10th
component contains the
pair
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(87) |
cons#(ok(X1),ok(X2)) |
→ |
cons#(X1,X2) |
(116) |
1.1.10 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(87) |
|
2 |
≥ |
2 |
1 |
> |
1 |
cons#(ok(X1),ok(X2)) |
→ |
cons#(X1,X2) |
(116) |
|
2 |
> |
2 |
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
11th
component contains the
pair
terms#(mark(X)) |
→ |
terms#(X) |
(86) |
terms#(ok(X)) |
→ |
terms#(X) |
(115) |
1.1.11 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
terms#(mark(X)) |
→ |
terms#(X) |
(86) |
|
1 |
> |
1 |
terms#(ok(X)) |
→ |
terms#(X) |
(115) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.