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) |
mark(terms(X)) |
→ |
active(terms(mark(X))) |
(10) |
mark(cons(X1,X2)) |
→ |
active(cons(mark(X1),X2)) |
(11) |
mark(recip(X)) |
→ |
active(recip(mark(X))) |
(12) |
mark(sqr(X)) |
→ |
active(sqr(mark(X))) |
(13) |
mark(s(X)) |
→ |
active(s(X)) |
(14) |
mark(0) |
→ |
active(0) |
(15) |
mark(add(X1,X2)) |
→ |
active(add(mark(X1),mark(X2))) |
(16) |
mark(dbl(X)) |
→ |
active(dbl(mark(X))) |
(17) |
mark(first(X1,X2)) |
→ |
active(first(mark(X1),mark(X2))) |
(18) |
mark(nil) |
→ |
active(nil) |
(19) |
terms(mark(X)) |
→ |
terms(X) |
(20) |
terms(active(X)) |
→ |
terms(X) |
(21) |
cons(mark(X1),X2) |
→ |
cons(X1,X2) |
(22) |
cons(X1,mark(X2)) |
→ |
cons(X1,X2) |
(23) |
cons(active(X1),X2) |
→ |
cons(X1,X2) |
(24) |
cons(X1,active(X2)) |
→ |
cons(X1,X2) |
(25) |
recip(mark(X)) |
→ |
recip(X) |
(26) |
recip(active(X)) |
→ |
recip(X) |
(27) |
sqr(mark(X)) |
→ |
sqr(X) |
(28) |
sqr(active(X)) |
→ |
sqr(X) |
(29) |
s(mark(X)) |
→ |
s(X) |
(30) |
s(active(X)) |
→ |
s(X) |
(31) |
add(mark(X1),X2) |
→ |
add(X1,X2) |
(32) |
add(X1,mark(X2)) |
→ |
add(X1,X2) |
(33) |
add(active(X1),X2) |
→ |
add(X1,X2) |
(34) |
add(X1,active(X2)) |
→ |
add(X1,X2) |
(35) |
dbl(mark(X)) |
→ |
dbl(X) |
(36) |
dbl(active(X)) |
→ |
dbl(X) |
(37) |
first(mark(X1),X2) |
→ |
first(X1,X2) |
(38) |
first(X1,mark(X2)) |
→ |
first(X1,X2) |
(39) |
first(active(X1),X2) |
→ |
first(X1,X2) |
(40) |
first(X1,active(X2)) |
→ |
first(X1,X2) |
(41) |
active#(terms(N)) |
→ |
mark#(cons(recip(sqr(N)),terms(s(N)))) |
(42) |
active#(terms(N)) |
→ |
cons#(recip(sqr(N)),terms(s(N))) |
(43) |
active#(terms(N)) |
→ |
recip#(sqr(N)) |
(44) |
active#(terms(N)) |
→ |
sqr#(N) |
(45) |
active#(terms(N)) |
→ |
terms#(s(N)) |
(46) |
active#(terms(N)) |
→ |
s#(N) |
(47) |
active#(sqr(0)) |
→ |
mark#(0) |
(48) |
active#(sqr(s(X))) |
→ |
mark#(s(add(sqr(X),dbl(X)))) |
(49) |
active#(sqr(s(X))) |
→ |
s#(add(sqr(X),dbl(X))) |
(50) |
active#(sqr(s(X))) |
→ |
add#(sqr(X),dbl(X)) |
(51) |
active#(sqr(s(X))) |
→ |
sqr#(X) |
(52) |
active#(sqr(s(X))) |
→ |
dbl#(X) |
(53) |
active#(dbl(0)) |
→ |
mark#(0) |
(54) |
active#(dbl(s(X))) |
→ |
mark#(s(s(dbl(X)))) |
(55) |
active#(dbl(s(X))) |
→ |
s#(s(dbl(X))) |
(56) |
active#(dbl(s(X))) |
→ |
s#(dbl(X)) |
(57) |
active#(dbl(s(X))) |
→ |
dbl#(X) |
(58) |
active#(add(0,X)) |
→ |
mark#(X) |
(59) |
active#(add(s(X),Y)) |
→ |
mark#(s(add(X,Y))) |
(60) |
active#(add(s(X),Y)) |
→ |
s#(add(X,Y)) |
(61) |
active#(add(s(X),Y)) |
→ |
add#(X,Y) |
(62) |
active#(first(0,X)) |
→ |
mark#(nil) |
(63) |
active#(first(s(X),cons(Y,Z))) |
→ |
mark#(cons(Y,first(X,Z))) |
(64) |
active#(first(s(X),cons(Y,Z))) |
→ |
cons#(Y,first(X,Z)) |
(65) |
active#(first(s(X),cons(Y,Z))) |
→ |
first#(X,Z) |
(66) |
mark#(terms(X)) |
→ |
active#(terms(mark(X))) |
(67) |
mark#(terms(X)) |
→ |
terms#(mark(X)) |
(68) |
mark#(terms(X)) |
→ |
mark#(X) |
(69) |
mark#(cons(X1,X2)) |
→ |
active#(cons(mark(X1),X2)) |
(70) |
mark#(cons(X1,X2)) |
→ |
cons#(mark(X1),X2) |
(71) |
mark#(cons(X1,X2)) |
→ |
mark#(X1) |
(72) |
mark#(recip(X)) |
→ |
active#(recip(mark(X))) |
(73) |
mark#(recip(X)) |
→ |
recip#(mark(X)) |
(74) |
mark#(recip(X)) |
→ |
mark#(X) |
(75) |
mark#(sqr(X)) |
→ |
active#(sqr(mark(X))) |
(76) |
mark#(sqr(X)) |
→ |
sqr#(mark(X)) |
(77) |
mark#(sqr(X)) |
→ |
mark#(X) |
(78) |
mark#(s(X)) |
→ |
active#(s(X)) |
(79) |
mark#(0) |
→ |
active#(0) |
(80) |
mark#(add(X1,X2)) |
→ |
active#(add(mark(X1),mark(X2))) |
(81) |
mark#(add(X1,X2)) |
→ |
add#(mark(X1),mark(X2)) |
(82) |
mark#(add(X1,X2)) |
→ |
mark#(X1) |
(83) |
mark#(add(X1,X2)) |
→ |
mark#(X2) |
(84) |
mark#(dbl(X)) |
→ |
active#(dbl(mark(X))) |
(85) |
mark#(dbl(X)) |
→ |
dbl#(mark(X)) |
(86) |
mark#(dbl(X)) |
→ |
mark#(X) |
(87) |
mark#(first(X1,X2)) |
→ |
active#(first(mark(X1),mark(X2))) |
(88) |
mark#(first(X1,X2)) |
→ |
first#(mark(X1),mark(X2)) |
(89) |
mark#(first(X1,X2)) |
→ |
mark#(X1) |
(90) |
mark#(first(X1,X2)) |
→ |
mark#(X2) |
(91) |
mark#(nil) |
→ |
active#(nil) |
(92) |
terms#(mark(X)) |
→ |
terms#(X) |
(93) |
terms#(active(X)) |
→ |
terms#(X) |
(94) |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(95) |
cons#(X1,mark(X2)) |
→ |
cons#(X1,X2) |
(96) |
cons#(active(X1),X2) |
→ |
cons#(X1,X2) |
(97) |
cons#(X1,active(X2)) |
→ |
cons#(X1,X2) |
(98) |
recip#(mark(X)) |
→ |
recip#(X) |
(99) |
recip#(active(X)) |
→ |
recip#(X) |
(100) |
sqr#(mark(X)) |
→ |
sqr#(X) |
(101) |
sqr#(active(X)) |
→ |
sqr#(X) |
(102) |
s#(mark(X)) |
→ |
s#(X) |
(103) |
s#(active(X)) |
→ |
s#(X) |
(104) |
add#(mark(X1),X2) |
→ |
add#(X1,X2) |
(105) |
add#(X1,mark(X2)) |
→ |
add#(X1,X2) |
(106) |
add#(active(X1),X2) |
→ |
add#(X1,X2) |
(107) |
add#(X1,active(X2)) |
→ |
add#(X1,X2) |
(108) |
dbl#(mark(X)) |
→ |
dbl#(X) |
(109) |
dbl#(active(X)) |
→ |
dbl#(X) |
(110) |
first#(mark(X1),X2) |
→ |
first#(X1,X2) |
(111) |
first#(X1,mark(X2)) |
→ |
first#(X1,X2) |
(112) |
first#(active(X1),X2) |
→ |
first#(X1,X2) |
(113) |
first#(X1,active(X2)) |
→ |
first#(X1,X2) |
(114) |
The dependency pairs are split into 9
components.
-
The
1st
component contains the
pair
mark#(terms(X)) |
→ |
active#(terms(mark(X))) |
(67) |
active#(terms(N)) |
→ |
mark#(cons(recip(sqr(N)),terms(s(N)))) |
(42) |
mark#(terms(X)) |
→ |
mark#(X) |
(69) |
mark#(cons(X1,X2)) |
→ |
active#(cons(mark(X1),X2)) |
(70) |
active#(sqr(s(X))) |
→ |
mark#(s(add(sqr(X),dbl(X)))) |
(49) |
mark#(cons(X1,X2)) |
→ |
mark#(X1) |
(72) |
mark#(recip(X)) |
→ |
active#(recip(mark(X))) |
(73) |
active#(dbl(s(X))) |
→ |
mark#(s(s(dbl(X)))) |
(55) |
mark#(recip(X)) |
→ |
mark#(X) |
(75) |
mark#(sqr(X)) |
→ |
active#(sqr(mark(X))) |
(76) |
active#(add(0,X)) |
→ |
mark#(X) |
(59) |
mark#(sqr(X)) |
→ |
mark#(X) |
(78) |
mark#(s(X)) |
→ |
active#(s(X)) |
(79) |
active#(add(s(X),Y)) |
→ |
mark#(s(add(X,Y))) |
(60) |
mark#(add(X1,X2)) |
→ |
active#(add(mark(X1),mark(X2))) |
(81) |
active#(first(s(X),cons(Y,Z))) |
→ |
mark#(cons(Y,first(X,Z))) |
(64) |
mark#(add(X1,X2)) |
→ |
mark#(X1) |
(83) |
mark#(add(X1,X2)) |
→ |
mark#(X2) |
(84) |
mark#(dbl(X)) |
→ |
active#(dbl(mark(X))) |
(85) |
mark#(dbl(X)) |
→ |
mark#(X) |
(87) |
mark#(first(X1,X2)) |
→ |
active#(first(mark(X1),mark(X2))) |
(88) |
mark#(first(X1,X2)) |
→ |
mark#(X1) |
(90) |
mark#(first(X1,X2)) |
→ |
mark#(X2) |
(91) |
1.1.1 Reduction Pair Processor
Using the Knuth Bendix order with w0 = 2 and the following precedence and weight functions
prec(s) |
= |
0 |
|
weight(s) |
= |
2 |
|
|
|
prec(add) |
= |
1 |
|
weight(add) |
= |
1 |
|
|
|
prec(0) |
= |
2 |
|
weight(0) |
= |
4 |
|
|
|
prec(first) |
= |
3 |
|
weight(first) |
= |
2 |
|
|
|
prec(nil) |
= |
4 |
|
weight(nil) |
= |
5 |
|
|
|
in combination with the following argument filter
π(mark#) |
= |
1 |
π(terms) |
= |
1 |
π(active#) |
= |
1 |
π(mark) |
= |
1 |
π(cons) |
= |
1 |
π(recip) |
= |
1 |
π(sqr) |
= |
1 |
π(s) |
= |
[] |
π(dbl) |
= |
1 |
π(add) |
= |
[1,2] |
π(0) |
= |
[] |
π(first) |
= |
[1,2] |
π(active) |
= |
1 |
π(nil) |
= |
[] |
the
pairs
active#(add(0,X)) |
→ |
mark#(X) |
(59) |
active#(add(s(X),Y)) |
→ |
mark#(s(add(X,Y))) |
(60) |
active#(first(s(X),cons(Y,Z))) |
→ |
mark#(cons(Y,first(X,Z))) |
(64) |
mark#(add(X1,X2)) |
→ |
mark#(X1) |
(83) |
mark#(add(X1,X2)) |
→ |
mark#(X2) |
(84) |
mark#(first(X1,X2)) |
→ |
mark#(X1) |
(90) |
mark#(first(X1,X2)) |
→ |
mark#(X2) |
(91) |
could be deleted.
1.1.1.1 Reduction Pair Processor
Using the Knuth Bendix order with w0 = 1 and the following precedence and weight functions
prec(s) |
= |
0 |
|
weight(s) |
= |
1 |
|
|
|
prec(dbl) |
= |
1 |
|
weight(dbl) |
= |
1 |
|
|
|
prec(first) |
= |
3 |
|
weight(first) |
= |
2 |
|
|
|
prec(0) |
= |
2 |
|
weight(0) |
= |
2 |
|
|
|
prec(nil) |
= |
4 |
|
weight(nil) |
= |
2 |
|
|
|
in combination with the following argument filter
π(mark#) |
= |
1 |
π(terms) |
= |
1 |
π(active#) |
= |
1 |
π(mark) |
= |
1 |
π(cons) |
= |
1 |
π(recip) |
= |
1 |
π(sqr) |
= |
1 |
π(s) |
= |
[] |
π(dbl) |
= |
[1] |
π(add) |
= |
2 |
π(first) |
= |
[2] |
π(active) |
= |
1 |
π(0) |
= |
[] |
π(nil) |
= |
[] |
the
pairs
active#(dbl(s(X))) |
→ |
mark#(s(s(dbl(X)))) |
(55) |
mark#(dbl(X)) |
→ |
mark#(X) |
(87) |
could be deleted.
1.1.1.1.1 Reduction Pair Processor with Usable Rules
Using the
prec(mark#) |
= |
3 |
|
stat(mark#) |
= |
lex
|
prec(terms) |
= |
3 |
|
stat(terms) |
= |
lex
|
prec(cons) |
= |
3 |
|
stat(cons) |
= |
lex
|
prec(recip) |
= |
3 |
|
stat(recip) |
= |
lex
|
prec(sqr) |
= |
3 |
|
stat(sqr) |
= |
lex
|
prec(s) |
= |
0 |
|
stat(s) |
= |
lex
|
prec(add) |
= |
3 |
|
stat(add) |
= |
lex
|
prec(dbl) |
= |
1 |
|
stat(dbl) |
= |
lex
|
prec(first) |
= |
2 |
|
stat(first) |
= |
lex
|
prec(0) |
= |
1 |
|
stat(0) |
= |
lex
|
prec(nil) |
= |
4 |
|
stat(nil) |
= |
lex
|
π(mark#) |
= |
[] |
π(terms) |
= |
[] |
π(active#) |
= |
1 |
π(mark) |
= |
1 |
π(cons) |
= |
[] |
π(recip) |
= |
[] |
π(sqr) |
= |
[] |
π(s) |
= |
[] |
π(add) |
= |
[] |
π(dbl) |
= |
[] |
π(first) |
= |
[] |
π(active) |
= |
1 |
π(0) |
= |
[] |
π(nil) |
= |
[] |
together with the usable
rules
terms(active(X)) |
→ |
terms(X) |
(21) |
terms(mark(X)) |
→ |
terms(X) |
(20) |
sqr(active(X)) |
→ |
sqr(X) |
(29) |
sqr(mark(X)) |
→ |
sqr(X) |
(28) |
recip(active(X)) |
→ |
recip(X) |
(27) |
recip(mark(X)) |
→ |
recip(X) |
(26) |
s(active(X)) |
→ |
s(X) |
(31) |
s(mark(X)) |
→ |
s(X) |
(30) |
cons(X1,mark(X2)) |
→ |
cons(X1,X2) |
(23) |
cons(mark(X1),X2) |
→ |
cons(X1,X2) |
(22) |
cons(active(X1),X2) |
→ |
cons(X1,X2) |
(24) |
cons(X1,active(X2)) |
→ |
cons(X1,X2) |
(25) |
dbl(active(X)) |
→ |
dbl(X) |
(37) |
dbl(mark(X)) |
→ |
dbl(X) |
(36) |
add(X1,mark(X2)) |
→ |
add(X1,X2) |
(33) |
add(mark(X1),X2) |
→ |
add(X1,X2) |
(32) |
add(active(X1),X2) |
→ |
add(X1,X2) |
(34) |
add(X1,active(X2)) |
→ |
add(X1,X2) |
(35) |
first(X1,mark(X2)) |
→ |
first(X1,X2) |
(39) |
first(mark(X1),X2) |
→ |
first(X1,X2) |
(38) |
first(active(X1),X2) |
→ |
first(X1,X2) |
(40) |
first(X1,active(X2)) |
→ |
first(X1,X2) |
(41) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
mark#(s(X)) |
→ |
active#(s(X)) |
(79) |
mark#(dbl(X)) |
→ |
active#(dbl(mark(X))) |
(85) |
mark#(first(X1,X2)) |
→ |
active#(first(mark(X1),mark(X2))) |
(88) |
could be deleted.
1.1.1.1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active#(x1)] |
= |
-1 + x1
|
[add(x1, x2)] |
= |
-2 |
[cons(x1, x2)] |
= |
-2 |
[recip(x1)] |
= |
0 |
[sqr(x1)] |
= |
2 |
[terms(x1)] |
= |
2 |
[mark(x1)] |
= |
0 |
[active(x1)] |
= |
-2 |
[s(x1)] |
= |
-2 + x1
|
[dbl(x1)] |
= |
1 |
[0] |
= |
0 |
[first(x1, x2)] |
= |
-2 + 2 · x1
|
[nil] |
= |
0 |
[mark#(x1)] |
= |
1 |
together with the usable
rules
terms(active(X)) |
→ |
terms(X) |
(21) |
terms(mark(X)) |
→ |
terms(X) |
(20) |
sqr(active(X)) |
→ |
sqr(X) |
(29) |
sqr(mark(X)) |
→ |
sqr(X) |
(28) |
recip(active(X)) |
→ |
recip(X) |
(27) |
recip(mark(X)) |
→ |
recip(X) |
(26) |
cons(X1,mark(X2)) |
→ |
cons(X1,X2) |
(23) |
cons(mark(X1),X2) |
→ |
cons(X1,X2) |
(22) |
cons(active(X1),X2) |
→ |
cons(X1,X2) |
(24) |
cons(X1,active(X2)) |
→ |
cons(X1,X2) |
(25) |
add(X1,mark(X2)) |
→ |
add(X1,X2) |
(33) |
add(mark(X1),X2) |
→ |
add(X1,X2) |
(32) |
add(active(X1),X2) |
→ |
add(X1,X2) |
(34) |
add(X1,active(X2)) |
→ |
add(X1,X2) |
(35) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
mark#(cons(X1,X2)) |
→ |
active#(cons(mark(X1),X2)) |
(70) |
mark#(recip(X)) |
→ |
active#(recip(mark(X))) |
(73) |
mark#(add(X1,X2)) |
→ |
active#(add(mark(X1),mark(X2))) |
(81) |
could be deleted.
1.1.1.1.1.1.1 Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
[active#(x1)] |
= |
1 + x1
|
[sqr(x1)] |
= |
x1 |
[terms(x1)] |
= |
2 + 2 · x1
|
[mark(x1)] |
= |
2 · x1
|
[active(x1)] |
= |
x1 |
[cons(x1, x2)] |
= |
x1 |
[recip(x1)] |
= |
x1 |
[s(x1)] |
= |
0 |
[add(x1, x2)] |
= |
1 + 2 · x2
|
[dbl(x1)] |
= |
-2 |
[0] |
= |
0 |
[first(x1, x2)] |
= |
1 + 2 · x2
|
[nil] |
= |
0 |
[mark#(x1)] |
= |
1 + 2 · x1
|
the
pairs
mark#(terms(X)) |
→ |
active#(terms(mark(X))) |
(67) |
active#(terms(N)) |
→ |
mark#(cons(recip(sqr(N)),terms(s(N)))) |
(42) |
mark#(terms(X)) |
→ |
mark#(X) |
(69) |
could be deleted.
1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules
Using the Knuth Bendix order with w0 = 1 and the following precedence and weight functions
prec(active#) |
= |
2 |
|
weight(active#) |
= |
2 |
|
|
|
prec(s) |
= |
0 |
|
weight(s) |
= |
1 |
|
|
|
prec(sqr) |
= |
1 |
|
weight(sqr) |
= |
2 |
|
|
|
in combination with the following argument filter
π(active#) |
= |
[] |
π(mark#) |
= |
1 |
π(s) |
= |
[] |
π(cons) |
= |
1 |
π(recip) |
= |
1 |
π(sqr) |
= |
[1] |
together with the usable
rules
s(active(X)) |
→ |
s(X) |
(31) |
s(mark(X)) |
→ |
s(X) |
(30) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
active#(sqr(s(X))) |
→ |
mark#(s(add(sqr(X),dbl(X)))) |
(49) |
mark#(sqr(X)) |
→ |
active#(sqr(mark(X))) |
(76) |
mark#(sqr(X)) |
→ |
mark#(X) |
(78) |
could be deleted.
1.1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[cons(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[recip(x1)] |
= |
1 · x1
|
[mark#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.1.1.1.1.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
mark#(cons(X1,X2)) |
→ |
mark#(X1) |
(72) |
|
1 |
> |
1 |
mark#(recip(X)) |
→ |
mark#(X) |
(75) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
2nd
component contains the
pair
terms#(active(X)) |
→ |
terms#(X) |
(94) |
terms#(mark(X)) |
→ |
terms#(X) |
(93) |
1.1.2 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[terms#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.2.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
terms#(active(X)) |
→ |
terms#(X) |
(94) |
|
1 |
> |
1 |
terms#(mark(X)) |
→ |
terms#(X) |
(93) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
3rd
component contains the
pair
cons#(X1,mark(X2)) |
→ |
cons#(X1,X2) |
(96) |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(95) |
cons#(active(X1),X2) |
→ |
cons#(X1,X2) |
(97) |
cons#(X1,active(X2)) |
→ |
cons#(X1,X2) |
(98) |
1.1.3 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[cons#(x1, x2)] |
= |
1 · x1 + 1 · x2
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.3.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
cons#(X1,mark(X2)) |
→ |
cons#(X1,X2) |
(96) |
|
1 |
≥ |
1 |
2 |
> |
2 |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(95) |
|
1 |
> |
1 |
2 |
≥ |
2 |
cons#(active(X1),X2) |
→ |
cons#(X1,X2) |
(97) |
|
1 |
> |
1 |
2 |
≥ |
2 |
cons#(X1,active(X2)) |
→ |
cons#(X1,X2) |
(98) |
|
1 |
≥ |
1 |
2 |
> |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
4th
component contains the
pair
recip#(active(X)) |
→ |
recip#(X) |
(100) |
recip#(mark(X)) |
→ |
recip#(X) |
(99) |
1.1.4 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[recip#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.4.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
recip#(active(X)) |
→ |
recip#(X) |
(100) |
|
1 |
> |
1 |
recip#(mark(X)) |
→ |
recip#(X) |
(99) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
5th
component contains the
pair
sqr#(active(X)) |
→ |
sqr#(X) |
(102) |
sqr#(mark(X)) |
→ |
sqr#(X) |
(101) |
1.1.5 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[sqr#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.5.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
sqr#(active(X)) |
→ |
sqr#(X) |
(102) |
|
1 |
> |
1 |
sqr#(mark(X)) |
→ |
sqr#(X) |
(101) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
6th
component contains the
pair
s#(active(X)) |
→ |
s#(X) |
(104) |
s#(mark(X)) |
→ |
s#(X) |
(103) |
1.1.6 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[s#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.6.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
s#(active(X)) |
→ |
s#(X) |
(104) |
|
1 |
> |
1 |
s#(mark(X)) |
→ |
s#(X) |
(103) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
7th
component contains the
pair
add#(X1,mark(X2)) |
→ |
add#(X1,X2) |
(106) |
add#(mark(X1),X2) |
→ |
add#(X1,X2) |
(105) |
add#(active(X1),X2) |
→ |
add#(X1,X2) |
(107) |
add#(X1,active(X2)) |
→ |
add#(X1,X2) |
(108) |
1.1.7 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[add#(x1, x2)] |
= |
1 · x1 + 1 · x2
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.7.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
add#(X1,mark(X2)) |
→ |
add#(X1,X2) |
(106) |
|
1 |
≥ |
1 |
2 |
> |
2 |
add#(mark(X1),X2) |
→ |
add#(X1,X2) |
(105) |
|
1 |
> |
1 |
2 |
≥ |
2 |
add#(active(X1),X2) |
→ |
add#(X1,X2) |
(107) |
|
1 |
> |
1 |
2 |
≥ |
2 |
add#(X1,active(X2)) |
→ |
add#(X1,X2) |
(108) |
|
1 |
≥ |
1 |
2 |
> |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
8th
component contains the
pair
dbl#(active(X)) |
→ |
dbl#(X) |
(110) |
dbl#(mark(X)) |
→ |
dbl#(X) |
(109) |
1.1.8 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[dbl#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.8.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
dbl#(active(X)) |
→ |
dbl#(X) |
(110) |
|
1 |
> |
1 |
dbl#(mark(X)) |
→ |
dbl#(X) |
(109) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
9th
component contains the
pair
first#(X1,mark(X2)) |
→ |
first#(X1,X2) |
(112) |
first#(mark(X1),X2) |
→ |
first#(X1,X2) |
(111) |
first#(active(X1),X2) |
→ |
first#(X1,X2) |
(113) |
first#(X1,active(X2)) |
→ |
first#(X1,X2) |
(114) |
1.1.9 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[first#(x1, x2)] |
= |
1 · x1 + 1 · x2
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.9.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
first#(X1,mark(X2)) |
→ |
first#(X1,X2) |
(112) |
|
1 |
≥ |
1 |
2 |
> |
2 |
first#(mark(X1),X2) |
→ |
first#(X1,X2) |
(111) |
|
1 |
> |
1 |
2 |
≥ |
2 |
first#(active(X1),X2) |
→ |
first#(X1,X2) |
(113) |
|
1 |
> |
1 |
2 |
≥ |
2 |
first#(X1,active(X2)) |
→ |
first#(X1,X2) |
(114) |
|
1 |
≥ |
1 |
2 |
> |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.