The rewrite relation of the following TRS is considered.
active#(U11(tt,V2)) |
→ |
mark#(U12(isNat(V2))) |
(55) |
active#(U11(tt,V2)) |
→ |
U12#(isNat(V2)) |
(56) |
active#(U11(tt,V2)) |
→ |
isNat#(V2) |
(57) |
active#(U12(tt)) |
→ |
mark#(tt) |
(58) |
active#(U21(tt)) |
→ |
mark#(tt) |
(59) |
active#(U31(tt,N)) |
→ |
mark#(N) |
(60) |
active#(U41(tt,M,N)) |
→ |
mark#(U42(isNat(N),M,N)) |
(61) |
active#(U41(tt,M,N)) |
→ |
U42#(isNat(N),M,N) |
(62) |
active#(U41(tt,M,N)) |
→ |
isNat#(N) |
(63) |
active#(U42(tt,M,N)) |
→ |
mark#(s(plus(N,M))) |
(64) |
active#(U42(tt,M,N)) |
→ |
s#(plus(N,M)) |
(65) |
active#(U42(tt,M,N)) |
→ |
plus#(N,M) |
(66) |
active#(isNat(0)) |
→ |
mark#(tt) |
(67) |
active#(isNat(plus(V1,V2))) |
→ |
mark#(U11(isNat(V1),V2)) |
(68) |
active#(isNat(plus(V1,V2))) |
→ |
U11#(isNat(V1),V2) |
(69) |
active#(isNat(plus(V1,V2))) |
→ |
isNat#(V1) |
(70) |
active#(isNat(s(V1))) |
→ |
mark#(U21(isNat(V1))) |
(71) |
active#(isNat(s(V1))) |
→ |
U21#(isNat(V1)) |
(72) |
active#(isNat(s(V1))) |
→ |
isNat#(V1) |
(73) |
active#(plus(N,0)) |
→ |
mark#(U31(isNat(N),N)) |
(74) |
active#(plus(N,0)) |
→ |
U31#(isNat(N),N) |
(75) |
active#(plus(N,0)) |
→ |
isNat#(N) |
(76) |
active#(plus(N,s(M))) |
→ |
mark#(U41(isNat(M),M,N)) |
(77) |
active#(plus(N,s(M))) |
→ |
U41#(isNat(M),M,N) |
(78) |
active#(plus(N,s(M))) |
→ |
isNat#(M) |
(79) |
mark#(U11(X1,X2)) |
→ |
active#(U11(mark(X1),X2)) |
(80) |
mark#(U11(X1,X2)) |
→ |
U11#(mark(X1),X2) |
(81) |
mark#(U11(X1,X2)) |
→ |
mark#(X1) |
(82) |
mark#(tt) |
→ |
active#(tt) |
(83) |
mark#(U12(X)) |
→ |
active#(U12(mark(X))) |
(84) |
mark#(U12(X)) |
→ |
U12#(mark(X)) |
(85) |
mark#(U12(X)) |
→ |
mark#(X) |
(86) |
mark#(isNat(X)) |
→ |
active#(isNat(X)) |
(87) |
mark#(U21(X)) |
→ |
active#(U21(mark(X))) |
(88) |
mark#(U21(X)) |
→ |
U21#(mark(X)) |
(89) |
mark#(U21(X)) |
→ |
mark#(X) |
(90) |
mark#(U31(X1,X2)) |
→ |
active#(U31(mark(X1),X2)) |
(91) |
mark#(U31(X1,X2)) |
→ |
U31#(mark(X1),X2) |
(92) |
mark#(U31(X1,X2)) |
→ |
mark#(X1) |
(93) |
mark#(U41(X1,X2,X3)) |
→ |
active#(U41(mark(X1),X2,X3)) |
(94) |
mark#(U41(X1,X2,X3)) |
→ |
U41#(mark(X1),X2,X3) |
(95) |
mark#(U41(X1,X2,X3)) |
→ |
mark#(X1) |
(96) |
mark#(U42(X1,X2,X3)) |
→ |
active#(U42(mark(X1),X2,X3)) |
(97) |
mark#(U42(X1,X2,X3)) |
→ |
U42#(mark(X1),X2,X3) |
(98) |
mark#(U42(X1,X2,X3)) |
→ |
mark#(X1) |
(99) |
mark#(s(X)) |
→ |
active#(s(mark(X))) |
(100) |
mark#(s(X)) |
→ |
s#(mark(X)) |
(101) |
mark#(s(X)) |
→ |
mark#(X) |
(102) |
mark#(plus(X1,X2)) |
→ |
active#(plus(mark(X1),mark(X2))) |
(103) |
mark#(plus(X1,X2)) |
→ |
plus#(mark(X1),mark(X2)) |
(104) |
mark#(plus(X1,X2)) |
→ |
mark#(X1) |
(105) |
mark#(plus(X1,X2)) |
→ |
mark#(X2) |
(106) |
mark#(0) |
→ |
active#(0) |
(107) |
U11#(mark(X1),X2) |
→ |
U11#(X1,X2) |
(108) |
U11#(X1,mark(X2)) |
→ |
U11#(X1,X2) |
(109) |
U11#(active(X1),X2) |
→ |
U11#(X1,X2) |
(110) |
U11#(X1,active(X2)) |
→ |
U11#(X1,X2) |
(111) |
U12#(mark(X)) |
→ |
U12#(X) |
(112) |
U12#(active(X)) |
→ |
U12#(X) |
(113) |
isNat#(mark(X)) |
→ |
isNat#(X) |
(114) |
isNat#(active(X)) |
→ |
isNat#(X) |
(115) |
U21#(mark(X)) |
→ |
U21#(X) |
(116) |
U21#(active(X)) |
→ |
U21#(X) |
(117) |
U31#(mark(X1),X2) |
→ |
U31#(X1,X2) |
(118) |
U31#(X1,mark(X2)) |
→ |
U31#(X1,X2) |
(119) |
U31#(active(X1),X2) |
→ |
U31#(X1,X2) |
(120) |
U31#(X1,active(X2)) |
→ |
U31#(X1,X2) |
(121) |
U41#(mark(X1),X2,X3) |
→ |
U41#(X1,X2,X3) |
(122) |
U41#(X1,mark(X2),X3) |
→ |
U41#(X1,X2,X3) |
(123) |
U41#(X1,X2,mark(X3)) |
→ |
U41#(X1,X2,X3) |
(124) |
U41#(active(X1),X2,X3) |
→ |
U41#(X1,X2,X3) |
(125) |
U41#(X1,active(X2),X3) |
→ |
U41#(X1,X2,X3) |
(126) |
U41#(X1,X2,active(X3)) |
→ |
U41#(X1,X2,X3) |
(127) |
U42#(mark(X1),X2,X3) |
→ |
U42#(X1,X2,X3) |
(128) |
U42#(X1,mark(X2),X3) |
→ |
U42#(X1,X2,X3) |
(129) |
U42#(X1,X2,mark(X3)) |
→ |
U42#(X1,X2,X3) |
(130) |
U42#(active(X1),X2,X3) |
→ |
U42#(X1,X2,X3) |
(131) |
U42#(X1,active(X2),X3) |
→ |
U42#(X1,X2,X3) |
(132) |
U42#(X1,X2,active(X3)) |
→ |
U42#(X1,X2,X3) |
(133) |
s#(mark(X)) |
→ |
s#(X) |
(134) |
s#(active(X)) |
→ |
s#(X) |
(135) |
plus#(mark(X1),X2) |
→ |
plus#(X1,X2) |
(136) |
plus#(X1,mark(X2)) |
→ |
plus#(X1,X2) |
(137) |
plus#(active(X1),X2) |
→ |
plus#(X1,X2) |
(138) |
plus#(X1,active(X2)) |
→ |
plus#(X1,X2) |
(139) |
The dependency pairs are split into 10
components.
-
The
1st
component contains the
pair
mark#(U11(X1,X2)) |
→ |
active#(U11(mark(X1),X2)) |
(80) |
active#(U11(tt,V2)) |
→ |
mark#(U12(isNat(V2))) |
(55) |
mark#(U11(X1,X2)) |
→ |
mark#(X1) |
(82) |
mark#(U12(X)) |
→ |
active#(U12(mark(X))) |
(84) |
active#(U31(tt,N)) |
→ |
mark#(N) |
(60) |
mark#(U12(X)) |
→ |
mark#(X) |
(86) |
mark#(isNat(X)) |
→ |
active#(isNat(X)) |
(87) |
active#(U41(tt,M,N)) |
→ |
mark#(U42(isNat(N),M,N)) |
(61) |
mark#(U21(X)) |
→ |
active#(U21(mark(X))) |
(88) |
active#(U42(tt,M,N)) |
→ |
mark#(s(plus(N,M))) |
(64) |
mark#(U21(X)) |
→ |
mark#(X) |
(90) |
mark#(U31(X1,X2)) |
→ |
active#(U31(mark(X1),X2)) |
(91) |
active#(isNat(plus(V1,V2))) |
→ |
mark#(U11(isNat(V1),V2)) |
(68) |
mark#(U31(X1,X2)) |
→ |
mark#(X1) |
(93) |
mark#(U41(X1,X2,X3)) |
→ |
active#(U41(mark(X1),X2,X3)) |
(94) |
active#(isNat(s(V1))) |
→ |
mark#(U21(isNat(V1))) |
(71) |
mark#(U41(X1,X2,X3)) |
→ |
mark#(X1) |
(96) |
mark#(U42(X1,X2,X3)) |
→ |
active#(U42(mark(X1),X2,X3)) |
(97) |
active#(plus(N,0)) |
→ |
mark#(U31(isNat(N),N)) |
(74) |
mark#(U42(X1,X2,X3)) |
→ |
mark#(X1) |
(99) |
mark#(s(X)) |
→ |
active#(s(mark(X))) |
(100) |
active#(plus(N,s(M))) |
→ |
mark#(U41(isNat(M),M,N)) |
(77) |
mark#(s(X)) |
→ |
mark#(X) |
(102) |
mark#(plus(X1,X2)) |
→ |
active#(plus(mark(X1),mark(X2))) |
(103) |
mark#(plus(X1,X2)) |
→ |
mark#(X1) |
(105) |
mark#(plus(X1,X2)) |
→ |
mark#(X2) |
(106) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active#(x1)] |
= |
x1 |
[U11(x1, x2)] |
= |
2 |
[U12(x1)] |
= |
0 |
[U21(x1)] |
= |
-2 |
[U31(x1, x2)] |
= |
2 |
[U41(x1, x2, x3)] |
= |
2 |
[U42(x1, x2, x3)] |
= |
2 |
[plus(x1, x2)] |
= |
2 |
[s(x1)] |
= |
-2 |
[mark(x1)] |
= |
-2 |
[active(x1)] |
= |
-2 |
[tt] |
= |
0 |
[isNat(x1)] |
= |
2 |
[0] |
= |
0 |
[mark#(x1)] |
= |
2 |
together with the usable
rules
U11(X1,mark(X2)) |
→ |
U11(X1,X2) |
(24) |
U11(mark(X1),X2) |
→ |
U11(X1,X2) |
(23) |
U11(active(X1),X2) |
→ |
U11(X1,X2) |
(25) |
U11(X1,active(X2)) |
→ |
U11(X1,X2) |
(26) |
isNat(active(X)) |
→ |
isNat(X) |
(30) |
isNat(mark(X)) |
→ |
isNat(X) |
(29) |
U12(active(X)) |
→ |
U12(X) |
(28) |
U12(mark(X)) |
→ |
U12(X) |
(27) |
U42(X1,mark(X2),X3) |
→ |
U42(X1,X2,X3) |
(44) |
U42(mark(X1),X2,X3) |
→ |
U42(X1,X2,X3) |
(43) |
U42(X1,X2,mark(X3)) |
→ |
U42(X1,X2,X3) |
(45) |
U42(active(X1),X2,X3) |
→ |
U42(X1,X2,X3) |
(46) |
U42(X1,active(X2),X3) |
→ |
U42(X1,X2,X3) |
(47) |
U42(X1,X2,active(X3)) |
→ |
U42(X1,X2,X3) |
(48) |
U21(active(X)) |
→ |
U21(X) |
(32) |
U21(mark(X)) |
→ |
U21(X) |
(31) |
plus(X1,mark(X2)) |
→ |
plus(X1,X2) |
(52) |
plus(mark(X1),X2) |
→ |
plus(X1,X2) |
(51) |
plus(active(X1),X2) |
→ |
plus(X1,X2) |
(53) |
plus(X1,active(X2)) |
→ |
plus(X1,X2) |
(54) |
s(active(X)) |
→ |
s(X) |
(50) |
s(mark(X)) |
→ |
s(X) |
(49) |
U31(X1,mark(X2)) |
→ |
U31(X1,X2) |
(34) |
U31(mark(X1),X2) |
→ |
U31(X1,X2) |
(33) |
U31(active(X1),X2) |
→ |
U31(X1,X2) |
(35) |
U31(X1,active(X2)) |
→ |
U31(X1,X2) |
(36) |
U41(X1,mark(X2),X3) |
→ |
U41(X1,X2,X3) |
(38) |
U41(mark(X1),X2,X3) |
→ |
U41(X1,X2,X3) |
(37) |
U41(X1,X2,mark(X3)) |
→ |
U41(X1,X2,X3) |
(39) |
U41(active(X1),X2,X3) |
→ |
U41(X1,X2,X3) |
(40) |
U41(X1,active(X2),X3) |
→ |
U41(X1,X2,X3) |
(41) |
U41(X1,X2,active(X3)) |
→ |
U41(X1,X2,X3) |
(42) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
mark#(U12(X)) |
→ |
active#(U12(mark(X))) |
(84) |
mark#(U21(X)) |
→ |
active#(U21(mark(X))) |
(88) |
mark#(s(X)) |
→ |
active#(s(mark(X))) |
(100) |
could be deleted.
1.1.1.1 Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
[mark#(x1)] |
= |
1 · x1
|
[U11(x1, x2)] |
= |
1 · x1
|
[active#(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[tt] |
= |
0 |
[U12(x1)] |
= |
1 · x1
|
[isNat(x1)] |
= |
0 |
[U31(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[U41(x1, x2, x3)] |
= |
1 · x1 + 1 · x2 + 1 · x3
|
[U42(x1, x2, x3)] |
= |
1 · x1 + 1 · x2 + 1 · x3
|
[s(x1)] |
= |
1 · x1
|
[plus(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[U21(x1)] |
= |
1 · x1
|
[0] |
= |
1 |
[active(x1)] |
= |
1 · x1
|
the
pair
active#(plus(N,0)) |
→ |
mark#(U31(isNat(N),N)) |
(74) |
could be deleted.
1.1.1.1.1 Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
[mark#(x1)] |
= |
1 · x1
|
[U11(x1, x2)] |
= |
1 · x1
|
[active#(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[tt] |
= |
0 |
[U12(x1)] |
= |
1 · x1
|
[isNat(x1)] |
= |
0 |
[U31(x1, x2)] |
= |
1 + 1 · x1 + 1 · x2
|
[U41(x1, x2, x3)] |
= |
1 · x1 + 1 · x2 + 1 · x3
|
[U42(x1, x2, x3)] |
= |
1 · x1 + 1 · x2 + 1 · x3
|
[s(x1)] |
= |
1 · x1
|
[plus(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[U21(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[0] |
= |
1 |
the
pairs
active#(U31(tt,N)) |
→ |
mark#(N) |
(60) |
mark#(U31(X1,X2)) |
→ |
mark#(X1) |
(93) |
could be deleted.
1.1.1.1.1.1 Reduction Pair Processor with Usable Rules
Using the
prec(mark#) |
= |
2 |
|
stat(mark#) |
= |
lex
|
prec(U11) |
= |
2 |
|
stat(U11) |
= |
lex
|
prec(tt) |
= |
3 |
|
stat(tt) |
= |
lex
|
prec(isNat) |
= |
2 |
|
stat(isNat) |
= |
lex
|
prec(U41) |
= |
2 |
|
stat(U41) |
= |
lex
|
prec(U42) |
= |
2 |
|
stat(U42) |
= |
lex
|
prec(s) |
= |
0 |
|
stat(s) |
= |
lex
|
prec(plus) |
= |
2 |
|
stat(plus) |
= |
lex
|
prec(U21) |
= |
4 |
|
stat(U21) |
= |
lex
|
prec(U31) |
= |
1 |
|
stat(U31) |
= |
lex
|
prec(0) |
= |
5 |
|
stat(0) |
= |
lex
|
π(mark#) |
= |
[] |
π(U11) |
= |
[] |
π(active#) |
= |
1 |
π(mark) |
= |
1 |
π(tt) |
= |
[] |
π(U12) |
= |
1 |
π(isNat) |
= |
[] |
π(U41) |
= |
[] |
π(U42) |
= |
[] |
π(s) |
= |
[1] |
π(plus) |
= |
[] |
π(U21) |
= |
[1] |
π(U31) |
= |
[] |
π(active) |
= |
1 |
π(0) |
= |
[] |
together with the usable
rules
U11(X1,mark(X2)) |
→ |
U11(X1,X2) |
(24) |
U11(mark(X1),X2) |
→ |
U11(X1,X2) |
(23) |
U11(active(X1),X2) |
→ |
U11(X1,X2) |
(25) |
U11(X1,active(X2)) |
→ |
U11(X1,X2) |
(26) |
isNat(active(X)) |
→ |
isNat(X) |
(30) |
isNat(mark(X)) |
→ |
isNat(X) |
(29) |
U42(X1,mark(X2),X3) |
→ |
U42(X1,X2,X3) |
(44) |
U42(mark(X1),X2,X3) |
→ |
U42(X1,X2,X3) |
(43) |
U42(X1,X2,mark(X3)) |
→ |
U42(X1,X2,X3) |
(45) |
U42(active(X1),X2,X3) |
→ |
U42(X1,X2,X3) |
(46) |
U42(X1,active(X2),X3) |
→ |
U42(X1,X2,X3) |
(47) |
U42(X1,X2,active(X3)) |
→ |
U42(X1,X2,X3) |
(48) |
plus(X1,mark(X2)) |
→ |
plus(X1,X2) |
(52) |
plus(mark(X1),X2) |
→ |
plus(X1,X2) |
(51) |
plus(active(X1),X2) |
→ |
plus(X1,X2) |
(53) |
plus(X1,active(X2)) |
→ |
plus(X1,X2) |
(54) |
U31(X1,mark(X2)) |
→ |
U31(X1,X2) |
(34) |
U31(mark(X1),X2) |
→ |
U31(X1,X2) |
(33) |
U31(active(X1),X2) |
→ |
U31(X1,X2) |
(35) |
U31(X1,active(X2)) |
→ |
U31(X1,X2) |
(36) |
U41(X1,mark(X2),X3) |
→ |
U41(X1,X2,X3) |
(38) |
U41(mark(X1),X2,X3) |
→ |
U41(X1,X2,X3) |
(37) |
U41(X1,X2,mark(X3)) |
→ |
U41(X1,X2,X3) |
(39) |
U41(active(X1),X2,X3) |
→ |
U41(X1,X2,X3) |
(40) |
U41(X1,active(X2),X3) |
→ |
U41(X1,X2,X3) |
(41) |
U41(X1,X2,active(X3)) |
→ |
U41(X1,X2,X3) |
(42) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
mark#(U31(X1,X2)) |
→ |
active#(U31(mark(X1),X2)) |
(91) |
could be deleted.
1.1.1.1.1.1.1 Reduction Pair Processor
Using the
prec(mark#) |
= |
3 |
|
stat(mark#) |
= |
mul
|
prec(active#) |
= |
3 |
|
stat(active#) |
= |
mul
|
prec(tt) |
= |
1 |
|
stat(tt) |
= |
mul
|
prec(isNat) |
= |
1 |
|
stat(isNat) |
= |
mul
|
prec(U41) |
= |
2 |
|
stat(U41) |
= |
mul
|
prec(U42) |
= |
2 |
|
stat(U42) |
= |
mul
|
prec(s) |
= |
1 |
|
stat(s) |
= |
mul
|
prec(plus) |
= |
2 |
|
stat(plus) |
= |
mul
|
prec(U31) |
= |
0 |
|
stat(U31) |
= |
lex
|
prec(0) |
= |
0 |
|
stat(0) |
= |
mul
|
π(mark#) |
= |
[1] |
π(U11) |
= |
1 |
π(active#) |
= |
[1] |
π(mark) |
= |
1 |
π(tt) |
= |
[] |
π(U12) |
= |
1 |
π(isNat) |
= |
[] |
π(U41) |
= |
[1,2,3] |
π(U42) |
= |
[1,2,3] |
π(s) |
= |
[1] |
π(plus) |
= |
[1,2] |
π(U21) |
= |
1 |
π(active) |
= |
1 |
π(U31) |
= |
[1,2] |
π(0) |
= |
[] |
the
pairs
active#(U42(tt,M,N)) |
→ |
mark#(s(plus(N,M))) |
(64) |
mark#(U41(X1,X2,X3)) |
→ |
mark#(X1) |
(96) |
mark#(U42(X1,X2,X3)) |
→ |
mark#(X1) |
(99) |
active#(plus(N,s(M))) |
→ |
mark#(U41(isNat(M),M,N)) |
(77) |
mark#(s(X)) |
→ |
mark#(X) |
(102) |
mark#(plus(X1,X2)) |
→ |
mark#(X1) |
(105) |
mark#(plus(X1,X2)) |
→ |
mark#(X2) |
(106) |
could be deleted.
1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active#(x1)] |
= |
-1 + 2 · x1
|
[U11(x1, x2)] |
= |
1 |
[U41(x1, x2, x3)] |
= |
1 |
[U42(x1, x2, x3)] |
= |
-2 |
[plus(x1, x2)] |
= |
-2 |
[mark(x1)] |
= |
-2 + 2 · x1
|
[active(x1)] |
= |
-2 + 2 · x1
|
[tt] |
= |
0 |
[U12(x1)] |
= |
2 |
[isNat(x1)] |
= |
1 |
[U31(x1, x2)] |
= |
-2 + x1
|
[U21(x1)] |
= |
-2 |
[s(x1)] |
= |
-2 + 2 · x1
|
[0] |
= |
0 |
[mark#(x1)] |
= |
1 |
together with the usable
rules
U11(X1,mark(X2)) |
→ |
U11(X1,X2) |
(24) |
U11(mark(X1),X2) |
→ |
U11(X1,X2) |
(23) |
U11(active(X1),X2) |
→ |
U11(X1,X2) |
(25) |
U11(X1,active(X2)) |
→ |
U11(X1,X2) |
(26) |
isNat(active(X)) |
→ |
isNat(X) |
(30) |
isNat(mark(X)) |
→ |
isNat(X) |
(29) |
U42(X1,mark(X2),X3) |
→ |
U42(X1,X2,X3) |
(44) |
U42(mark(X1),X2,X3) |
→ |
U42(X1,X2,X3) |
(43) |
U42(X1,X2,mark(X3)) |
→ |
U42(X1,X2,X3) |
(45) |
U42(active(X1),X2,X3) |
→ |
U42(X1,X2,X3) |
(46) |
U42(X1,active(X2),X3) |
→ |
U42(X1,X2,X3) |
(47) |
U42(X1,X2,active(X3)) |
→ |
U42(X1,X2,X3) |
(48) |
U41(X1,mark(X2),X3) |
→ |
U41(X1,X2,X3) |
(38) |
U41(mark(X1),X2,X3) |
→ |
U41(X1,X2,X3) |
(37) |
U41(X1,X2,mark(X3)) |
→ |
U41(X1,X2,X3) |
(39) |
U41(active(X1),X2,X3) |
→ |
U41(X1,X2,X3) |
(40) |
U41(X1,active(X2),X3) |
→ |
U41(X1,X2,X3) |
(41) |
U41(X1,X2,active(X3)) |
→ |
U41(X1,X2,X3) |
(42) |
plus(X1,mark(X2)) |
→ |
plus(X1,X2) |
(52) |
plus(mark(X1),X2) |
→ |
plus(X1,X2) |
(51) |
plus(active(X1),X2) |
→ |
plus(X1,X2) |
(53) |
plus(X1,active(X2)) |
→ |
plus(X1,X2) |
(54) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
mark#(U42(X1,X2,X3)) |
→ |
active#(U42(mark(X1),X2,X3)) |
(97) |
mark#(plus(X1,X2)) |
→ |
active#(plus(mark(X1),mark(X2))) |
(103) |
could be deleted.
1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active#(x1)] |
= |
0 |
[U11(x1, x2)] |
= |
1 + 2 · x1
|
[U41(x1, x2, x3)] |
= |
2 + 2 · x3
|
[mark(x1)] |
= |
2 · x1
|
[active(x1)] |
= |
2 + x1
|
[tt] |
= |
0 |
[U12(x1)] |
= |
1 + 2 · x1
|
[isNat(x1)] |
= |
0 |
[U31(x1, x2)] |
= |
2 |
[U42(x1, x2, x3)] |
= |
-2 |
[U21(x1)] |
= |
1 + 2 · x1
|
[s(x1)] |
= |
2 |
[plus(x1, x2)] |
= |
-2 + x1
|
[0] |
= |
0 |
[mark#(x1)] |
= |
-1 + x1
|
together with the usable
rules
U11(X1,mark(X2)) |
→ |
U11(X1,X2) |
(24) |
U11(mark(X1),X2) |
→ |
U11(X1,X2) |
(23) |
U11(active(X1),X2) |
→ |
U11(X1,X2) |
(25) |
U11(X1,active(X2)) |
→ |
U11(X1,X2) |
(26) |
isNat(active(X)) |
→ |
isNat(X) |
(30) |
isNat(mark(X)) |
→ |
isNat(X) |
(29) |
U12(active(X)) |
→ |
U12(X) |
(28) |
U12(mark(X)) |
→ |
U12(X) |
(27) |
U42(X1,mark(X2),X3) |
→ |
U42(X1,X2,X3) |
(44) |
U42(mark(X1),X2,X3) |
→ |
U42(X1,X2,X3) |
(43) |
U42(X1,X2,mark(X3)) |
→ |
U42(X1,X2,X3) |
(45) |
U42(active(X1),X2,X3) |
→ |
U42(X1,X2,X3) |
(46) |
U42(X1,active(X2),X3) |
→ |
U42(X1,X2,X3) |
(47) |
U42(X1,X2,active(X3)) |
→ |
U42(X1,X2,X3) |
(48) |
U21(active(X)) |
→ |
U21(X) |
(32) |
U21(mark(X)) |
→ |
U21(X) |
(31) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
mark#(U41(X1,X2,X3)) |
→ |
active#(U41(mark(X1),X2,X3)) |
(94) |
could be deleted.
1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active#(x1)] |
= |
-2 + 2 · x1
|
[U11(x1, x2)] |
= |
-2 |
[mark(x1)] |
= |
2 |
[active(x1)] |
= |
2 |
[tt] |
= |
2 |
[U12(x1)] |
= |
2 |
[isNat(x1)] |
= |
1 |
[U31(x1, x2)] |
= |
2 |
[U41(x1, x2, x3)] |
= |
2 + 2 · x1
|
[U42(x1, x2, x3)] |
= |
-2 + 2 · x3
|
[U21(x1)] |
= |
-2 + x1
|
[s(x1)] |
= |
-2 + x1
|
[plus(x1, x2)] |
= |
-2 + 2 · x1
|
[0] |
= |
0 |
[mark#(x1)] |
= |
-2 |
together with the usable
rules
U11(X1,mark(X2)) |
→ |
U11(X1,X2) |
(24) |
U11(mark(X1),X2) |
→ |
U11(X1,X2) |
(23) |
U11(active(X1),X2) |
→ |
U11(X1,X2) |
(25) |
U11(X1,active(X2)) |
→ |
U11(X1,X2) |
(26) |
isNat(active(X)) |
→ |
isNat(X) |
(30) |
isNat(mark(X)) |
→ |
isNat(X) |
(29) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
active#(U41(tt,M,N)) |
→ |
mark#(U42(isNat(N),M,N)) |
(61) |
could be deleted.
1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
[active#(x1)] |
= |
-1 + x1
|
[U11(x1, x2)] |
= |
2 · x1 + x2
|
[mark(x1)] |
= |
x1 |
[active(x1)] |
= |
x1 |
[tt] |
= |
2 |
[U12(x1)] |
= |
x1 |
[isNat(x1)] |
= |
x1 |
[U31(x1, x2)] |
= |
x2 |
[U41(x1, x2, x3)] |
= |
2 + 2 · x2 + 2 · x3
|
[U42(x1, x2, x3)] |
= |
2 + 2 · x2 + 2 · x3
|
[U21(x1)] |
= |
x1 |
[s(x1)] |
= |
1 + x1
|
[plus(x1, x2)] |
= |
1 + 2 · x1 + 2 · x2
|
[0] |
= |
2 |
[mark#(x1)] |
= |
x1 |
the
pair
active#(U11(tt,V2)) |
→ |
mark#(U12(isNat(V2))) |
(55) |
could be deleted.
1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active#(x1)] |
= |
-2 |
[U11(x1, x2)] |
= |
1 + x1
|
[mark(x1)] |
= |
2 + x1
|
[active(x1)] |
= |
2 · x1
|
[tt] |
= |
0 |
[U12(x1)] |
= |
2 + 2 · x1
|
[isNat(x1)] |
= |
0 |
[U31(x1, x2)] |
= |
2 + x2
|
[U41(x1, x2, x3)] |
= |
-2 |
[U42(x1, x2, x3)] |
= |
-1 + x1
|
[U21(x1)] |
= |
1 + x1
|
[s(x1)] |
= |
-2 + 2 · x1
|
[plus(x1, x2)] |
= |
-2 + x1
|
[0] |
= |
0 |
[mark#(x1)] |
= |
-1 + x1
|
together with the usable
rules
U11(X1,mark(X2)) |
→ |
U11(X1,X2) |
(24) |
U11(mark(X1),X2) |
→ |
U11(X1,X2) |
(23) |
U11(active(X1),X2) |
→ |
U11(X1,X2) |
(25) |
U11(X1,active(X2)) |
→ |
U11(X1,X2) |
(26) |
isNat(active(X)) |
→ |
isNat(X) |
(30) |
isNat(mark(X)) |
→ |
isNat(X) |
(29) |
U21(active(X)) |
→ |
U21(X) |
(32) |
U21(mark(X)) |
→ |
U21(X) |
(31) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
mark#(U12(X)) |
→ |
mark#(X) |
(86) |
could be deleted.
1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active#(x1)] |
= |
-1 + 2 · x1
|
[U11(x1, x2)] |
= |
-2 |
[mark(x1)] |
= |
2 |
[active(x1)] |
= |
2 + 2 · x1
|
[tt] |
= |
2 |
[U12(x1)] |
= |
0 |
[isNat(x1)] |
= |
1 |
[U31(x1, x2)] |
= |
x2 |
[U41(x1, x2, x3)] |
= |
1 + x2
|
[U42(x1, x2, x3)] |
= |
1 + x1 + x2 + x3
|
[U21(x1)] |
= |
-2 + 2 · x1
|
[s(x1)] |
= |
x1 |
[plus(x1, x2)] |
= |
-2 + 2 · x1
|
[0] |
= |
1 |
[mark#(x1)] |
= |
1 |
together with the usable
rules
U11(X1,mark(X2)) |
→ |
U11(X1,X2) |
(24) |
U11(mark(X1),X2) |
→ |
U11(X1,X2) |
(23) |
U11(active(X1),X2) |
→ |
U11(X1,X2) |
(25) |
U11(X1,active(X2)) |
→ |
U11(X1,X2) |
(26) |
isNat(active(X)) |
→ |
isNat(X) |
(30) |
isNat(mark(X)) |
→ |
isNat(X) |
(29) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
mark#(U11(X1,X2)) |
→ |
active#(U11(mark(X1),X2)) |
(80) |
could be deleted.
1.1.1.1.1.1.1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[isNat(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[U21(x1)] |
= |
1 · x1
|
[U11(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[plus(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[s(x1)] |
= |
1 · x1
|
[mark#(x1)] |
= |
1 · x1
|
[active#(x1)] |
= |
1 · x1
|
together with the usable
rules
isNat(active(X)) |
→ |
isNat(X) |
(30) |
isNat(mark(X)) |
→ |
isNat(X) |
(29) |
U21(active(X)) |
→ |
U21(X) |
(32) |
U21(mark(X)) |
→ |
U21(X) |
(31) |
U11(X1,mark(X2)) |
→ |
U11(X1,X2) |
(24) |
U11(mark(X1),X2) |
→ |
U11(X1,X2) |
(23) |
U11(active(X1),X2) |
→ |
U11(X1,X2) |
(25) |
U11(X1,active(X2)) |
→ |
U11(X1,X2) |
(26) |
(w.r.t. the implicit argument filter of the reduction pair),
the
rule
could be deleted.
1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[isNat(x1)] |
= |
2 · x1
|
[mark(x1)] |
= |
2 · x1
|
[active(x1)] |
= |
2 · x1
|
[U21(x1)] |
= |
1 · x1
|
[U11(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[mark#(x1)] |
= |
1 · x1
|
[active#(x1)] |
= |
1 · x1
|
[plus(x1, x2)] |
= |
2 + 2 · x1 + 2 · x2
|
[s(x1)] |
= |
2 · x1
|
together with the usable
rules
isNat(mark(X)) |
→ |
isNat(X) |
(29) |
isNat(active(X)) |
→ |
isNat(X) |
(30) |
U21(mark(X)) |
→ |
U21(X) |
(31) |
U21(active(X)) |
→ |
U21(X) |
(32) |
U11(mark(X1),X2) |
→ |
U11(X1,X2) |
(23) |
U11(X1,mark(X2)) |
→ |
U11(X1,X2) |
(24) |
U11(active(X1),X2) |
→ |
U11(X1,X2) |
(25) |
U11(X1,active(X2)) |
→ |
U11(X1,X2) |
(26) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
active#(isNat(plus(V1,V2))) |
→ |
mark#(U11(isNat(V1),V2)) |
(68) |
active#(isNat(s(V1))) |
→ |
mark#(U21(isNat(V1))) |
(71) |
and
the
rules
isNat(mark(X)) |
→ |
isNat(X) |
(29) |
isNat(active(X)) |
→ |
isNat(X) |
(30) |
U21(mark(X)) |
→ |
U21(X) |
(31) |
U21(active(X)) |
→ |
U21(X) |
(32) |
U11(mark(X1),X2) |
→ |
U11(X1,X2) |
(23) |
U11(X1,mark(X2)) |
→ |
U11(X1,X2) |
(24) |
U11(active(X1),X2) |
→ |
U11(X1,X2) |
(25) |
U11(X1,active(X2)) |
→ |
U11(X1,X2) |
(26) |
could be deleted.
1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
1st
component contains the
pair
mark#(U21(X)) |
→ |
mark#(X) |
(90) |
mark#(U11(X1,X2)) |
→ |
mark#(X1) |
(82) |
1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
mark#(U21(X)) |
→ |
mark#(X) |
(90) |
|
1 |
> |
1 |
mark#(U11(X1,X2)) |
→ |
mark#(X1) |
(82) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
2nd
component contains the
pair
U11#(X1,mark(X2)) |
→ |
U11#(X1,X2) |
(109) |
U11#(mark(X1),X2) |
→ |
U11#(X1,X2) |
(108) |
U11#(active(X1),X2) |
→ |
U11#(X1,X2) |
(110) |
U11#(X1,active(X2)) |
→ |
U11#(X1,X2) |
(111) |
1.1.2 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[U11#(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.
1.1.2.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
U11#(X1,mark(X2)) |
→ |
U11#(X1,X2) |
(109) |
|
1 |
≥ |
1 |
2 |
> |
2 |
U11#(mark(X1),X2) |
→ |
U11#(X1,X2) |
(108) |
|
1 |
> |
1 |
2 |
≥ |
2 |
U11#(active(X1),X2) |
→ |
U11#(X1,X2) |
(110) |
|
1 |
> |
1 |
2 |
≥ |
2 |
U11#(X1,active(X2)) |
→ |
U11#(X1,X2) |
(111) |
|
1 |
≥ |
1 |
2 |
> |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
3rd
component contains the
pair
U12#(active(X)) |
→ |
U12#(X) |
(113) |
U12#(mark(X)) |
→ |
U12#(X) |
(112) |
1.1.3 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[U12#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.3.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
U12#(active(X)) |
→ |
U12#(X) |
(113) |
|
1 |
> |
1 |
U12#(mark(X)) |
→ |
U12#(X) |
(112) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
4th
component contains the
pair
isNat#(active(X)) |
→ |
isNat#(X) |
(115) |
isNat#(mark(X)) |
→ |
isNat#(X) |
(114) |
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
|
[isNat#(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.
isNat#(active(X)) |
→ |
isNat#(X) |
(115) |
|
1 |
> |
1 |
isNat#(mark(X)) |
→ |
isNat#(X) |
(114) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
5th
component contains the
pair
U21#(active(X)) |
→ |
U21#(X) |
(117) |
U21#(mark(X)) |
→ |
U21#(X) |
(116) |
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
|
[U21#(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.
U21#(active(X)) |
→ |
U21#(X) |
(117) |
|
1 |
> |
1 |
U21#(mark(X)) |
→ |
U21#(X) |
(116) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
6th
component contains the
pair
U31#(X1,mark(X2)) |
→ |
U31#(X1,X2) |
(119) |
U31#(mark(X1),X2) |
→ |
U31#(X1,X2) |
(118) |
U31#(active(X1),X2) |
→ |
U31#(X1,X2) |
(120) |
U31#(X1,active(X2)) |
→ |
U31#(X1,X2) |
(121) |
1.1.6 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[U31#(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.
1.1.6.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
U31#(X1,mark(X2)) |
→ |
U31#(X1,X2) |
(119) |
|
1 |
≥ |
1 |
2 |
> |
2 |
U31#(mark(X1),X2) |
→ |
U31#(X1,X2) |
(118) |
|
1 |
> |
1 |
2 |
≥ |
2 |
U31#(active(X1),X2) |
→ |
U31#(X1,X2) |
(120) |
|
1 |
> |
1 |
2 |
≥ |
2 |
U31#(X1,active(X2)) |
→ |
U31#(X1,X2) |
(121) |
|
1 |
≥ |
1 |
2 |
> |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
7th
component contains the
pair
U41#(X1,mark(X2),X3) |
→ |
U41#(X1,X2,X3) |
(123) |
U41#(mark(X1),X2,X3) |
→ |
U41#(X1,X2,X3) |
(122) |
U41#(X1,X2,mark(X3)) |
→ |
U41#(X1,X2,X3) |
(124) |
U41#(active(X1),X2,X3) |
→ |
U41#(X1,X2,X3) |
(125) |
U41#(X1,active(X2),X3) |
→ |
U41#(X1,X2,X3) |
(126) |
U41#(X1,X2,active(X3)) |
→ |
U41#(X1,X2,X3) |
(127) |
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
|
[U41#(x1, x2, x3)] |
= |
1 · x1 + 1 · x2 + 1 · x3
|
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.
U41#(X1,mark(X2),X3) |
→ |
U41#(X1,X2,X3) |
(123) |
|
1 |
≥ |
1 |
2 |
> |
2 |
3 |
≥ |
3 |
U41#(mark(X1),X2,X3) |
→ |
U41#(X1,X2,X3) |
(122) |
|
1 |
> |
1 |
2 |
≥ |
2 |
3 |
≥ |
3 |
U41#(X1,X2,mark(X3)) |
→ |
U41#(X1,X2,X3) |
(124) |
|
1 |
≥ |
1 |
2 |
≥ |
2 |
3 |
> |
3 |
U41#(active(X1),X2,X3) |
→ |
U41#(X1,X2,X3) |
(125) |
|
1 |
> |
1 |
2 |
≥ |
2 |
3 |
≥ |
3 |
U41#(X1,active(X2),X3) |
→ |
U41#(X1,X2,X3) |
(126) |
|
1 |
≥ |
1 |
2 |
> |
2 |
3 |
≥ |
3 |
U41#(X1,X2,active(X3)) |
→ |
U41#(X1,X2,X3) |
(127) |
|
1 |
≥ |
1 |
2 |
≥ |
2 |
3 |
> |
3 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
8th
component contains the
pair
U42#(X1,mark(X2),X3) |
→ |
U42#(X1,X2,X3) |
(129) |
U42#(mark(X1),X2,X3) |
→ |
U42#(X1,X2,X3) |
(128) |
U42#(X1,X2,mark(X3)) |
→ |
U42#(X1,X2,X3) |
(130) |
U42#(active(X1),X2,X3) |
→ |
U42#(X1,X2,X3) |
(131) |
U42#(X1,active(X2),X3) |
→ |
U42#(X1,X2,X3) |
(132) |
U42#(X1,X2,active(X3)) |
→ |
U42#(X1,X2,X3) |
(133) |
1.1.8 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[U42#(x1, x2, x3)] |
= |
1 · x1 + 1 · x2 + 1 · x3
|
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.
U42#(X1,mark(X2),X3) |
→ |
U42#(X1,X2,X3) |
(129) |
|
1 |
≥ |
1 |
2 |
> |
2 |
3 |
≥ |
3 |
U42#(mark(X1),X2,X3) |
→ |
U42#(X1,X2,X3) |
(128) |
|
1 |
> |
1 |
2 |
≥ |
2 |
3 |
≥ |
3 |
U42#(X1,X2,mark(X3)) |
→ |
U42#(X1,X2,X3) |
(130) |
|
1 |
≥ |
1 |
2 |
≥ |
2 |
3 |
> |
3 |
U42#(active(X1),X2,X3) |
→ |
U42#(X1,X2,X3) |
(131) |
|
1 |
> |
1 |
2 |
≥ |
2 |
3 |
≥ |
3 |
U42#(X1,active(X2),X3) |
→ |
U42#(X1,X2,X3) |
(132) |
|
1 |
≥ |
1 |
2 |
> |
2 |
3 |
≥ |
3 |
U42#(X1,X2,active(X3)) |
→ |
U42#(X1,X2,X3) |
(133) |
|
1 |
≥ |
1 |
2 |
≥ |
2 |
3 |
> |
3 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
9th
component contains the
pair
s#(active(X)) |
→ |
s#(X) |
(135) |
s#(mark(X)) |
→ |
s#(X) |
(134) |
1.1.9 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.
1.1.9.1 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) |
(135) |
|
1 |
> |
1 |
s#(mark(X)) |
→ |
s#(X) |
(134) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
10th
component contains the
pair
plus#(X1,mark(X2)) |
→ |
plus#(X1,X2) |
(137) |
plus#(mark(X1),X2) |
→ |
plus#(X1,X2) |
(136) |
plus#(active(X1),X2) |
→ |
plus#(X1,X2) |
(138) |
plus#(X1,active(X2)) |
→ |
plus#(X1,X2) |
(139) |
1.1.10 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[plus#(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.
1.1.10.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
plus#(X1,mark(X2)) |
→ |
plus#(X1,X2) |
(137) |
|
1 |
≥ |
1 |
2 |
> |
2 |
plus#(mark(X1),X2) |
→ |
plus#(X1,X2) |
(136) |
|
1 |
> |
1 |
2 |
≥ |
2 |
plus#(active(X1),X2) |
→ |
plus#(X1,X2) |
(138) |
|
1 |
> |
1 |
2 |
≥ |
2 |
plus#(X1,active(X2)) |
→ |
plus#(X1,X2) |
(139) |
|
1 |
≥ |
1 |
2 |
> |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.