The rewrite relation of the following TRS is considered.
active(U11(tt,N)) |
→ |
mark(N) |
(1) |
active(U21(tt,M,N)) |
→ |
mark(s(plus(N,M))) |
(2) |
active(U31(tt)) |
→ |
mark(0) |
(3) |
active(U41(tt,M,N)) |
→ |
mark(plus(x(N,M),N)) |
(4) |
active(and(tt,X)) |
→ |
mark(X) |
(5) |
active(isNat(0)) |
→ |
mark(tt) |
(6) |
active(isNat(plus(V1,V2))) |
→ |
mark(and(isNat(V1),isNat(V2))) |
(7) |
active(isNat(s(V1))) |
→ |
mark(isNat(V1)) |
(8) |
active(isNat(x(V1,V2))) |
→ |
mark(and(isNat(V1),isNat(V2))) |
(9) |
active(plus(N,0)) |
→ |
mark(U11(isNat(N),N)) |
(10) |
active(plus(N,s(M))) |
→ |
mark(U21(and(isNat(M),isNat(N)),M,N)) |
(11) |
active(x(N,0)) |
→ |
mark(U31(isNat(N))) |
(12) |
active(x(N,s(M))) |
→ |
mark(U41(and(isNat(M),isNat(N)),M,N)) |
(13) |
active(U11(X1,X2)) |
→ |
U11(active(X1),X2) |
(14) |
active(U21(X1,X2,X3)) |
→ |
U21(active(X1),X2,X3) |
(15) |
active(s(X)) |
→ |
s(active(X)) |
(16) |
active(plus(X1,X2)) |
→ |
plus(active(X1),X2) |
(17) |
active(plus(X1,X2)) |
→ |
plus(X1,active(X2)) |
(18) |
active(U31(X)) |
→ |
U31(active(X)) |
(19) |
active(U41(X1,X2,X3)) |
→ |
U41(active(X1),X2,X3) |
(20) |
active(x(X1,X2)) |
→ |
x(active(X1),X2) |
(21) |
active(x(X1,X2)) |
→ |
x(X1,active(X2)) |
(22) |
active(and(X1,X2)) |
→ |
and(active(X1),X2) |
(23) |
U11(mark(X1),X2) |
→ |
mark(U11(X1,X2)) |
(24) |
U21(mark(X1),X2,X3) |
→ |
mark(U21(X1,X2,X3)) |
(25) |
s(mark(X)) |
→ |
mark(s(X)) |
(26) |
plus(mark(X1),X2) |
→ |
mark(plus(X1,X2)) |
(27) |
plus(X1,mark(X2)) |
→ |
mark(plus(X1,X2)) |
(28) |
U31(mark(X)) |
→ |
mark(U31(X)) |
(29) |
U41(mark(X1),X2,X3) |
→ |
mark(U41(X1,X2,X3)) |
(30) |
x(mark(X1),X2) |
→ |
mark(x(X1,X2)) |
(31) |
x(X1,mark(X2)) |
→ |
mark(x(X1,X2)) |
(32) |
and(mark(X1),X2) |
→ |
mark(and(X1,X2)) |
(33) |
proper(U11(X1,X2)) |
→ |
U11(proper(X1),proper(X2)) |
(34) |
proper(tt) |
→ |
ok(tt) |
(35) |
proper(U21(X1,X2,X3)) |
→ |
U21(proper(X1),proper(X2),proper(X3)) |
(36) |
proper(s(X)) |
→ |
s(proper(X)) |
(37) |
proper(plus(X1,X2)) |
→ |
plus(proper(X1),proper(X2)) |
(38) |
proper(U31(X)) |
→ |
U31(proper(X)) |
(39) |
proper(0) |
→ |
ok(0) |
(40) |
proper(U41(X1,X2,X3)) |
→ |
U41(proper(X1),proper(X2),proper(X3)) |
(41) |
proper(x(X1,X2)) |
→ |
x(proper(X1),proper(X2)) |
(42) |
proper(and(X1,X2)) |
→ |
and(proper(X1),proper(X2)) |
(43) |
proper(isNat(X)) |
→ |
isNat(proper(X)) |
(44) |
U11(ok(X1),ok(X2)) |
→ |
ok(U11(X1,X2)) |
(45) |
U21(ok(X1),ok(X2),ok(X3)) |
→ |
ok(U21(X1,X2,X3)) |
(46) |
s(ok(X)) |
→ |
ok(s(X)) |
(47) |
plus(ok(X1),ok(X2)) |
→ |
ok(plus(X1,X2)) |
(48) |
U31(ok(X)) |
→ |
ok(U31(X)) |
(49) |
U41(ok(X1),ok(X2),ok(X3)) |
→ |
ok(U41(X1,X2,X3)) |
(50) |
x(ok(X1),ok(X2)) |
→ |
ok(x(X1,X2)) |
(51) |
and(ok(X1),ok(X2)) |
→ |
ok(and(X1,X2)) |
(52) |
isNat(ok(X)) |
→ |
ok(isNat(X)) |
(53) |
top(mark(X)) |
→ |
top(proper(X)) |
(54) |
top(ok(X)) |
→ |
top(active(X)) |
(55) |
active#(U21(tt,M,N)) |
→ |
plus#(N,M) |
(56) |
active#(U21(X1,X2,X3)) |
→ |
U21#(active(X1),X2,X3) |
(57) |
x#(mark(X1),X2) |
→ |
x#(X1,X2) |
(58) |
active#(x(N,s(M))) |
→ |
isNat#(N) |
(59) |
active#(plus(N,s(M))) |
→ |
isNat#(M) |
(60) |
top#(mark(X)) |
→ |
top#(proper(X)) |
(61) |
proper#(U11(X1,X2)) |
→ |
proper#(X1) |
(62) |
U41#(mark(X1),X2,X3) |
→ |
U41#(X1,X2,X3) |
(63) |
active#(x(N,0)) |
→ |
isNat#(N) |
(64) |
active#(x(N,0)) |
→ |
U31#(isNat(N)) |
(65) |
U11#(mark(X1),X2) |
→ |
U11#(X1,X2) |
(66) |
active#(plus(X1,X2)) |
→ |
plus#(X1,active(X2)) |
(67) |
active#(U31(X)) |
→ |
active#(X) |
(68) |
and#(ok(X1),ok(X2)) |
→ |
and#(X1,X2) |
(69) |
active#(isNat(plus(V1,V2))) |
→ |
isNat#(V1) |
(70) |
x#(X1,mark(X2)) |
→ |
x#(X1,X2) |
(71) |
U31#(ok(X)) |
→ |
U31#(X) |
(72) |
proper#(U21(X1,X2,X3)) |
→ |
U21#(proper(X1),proper(X2),proper(X3)) |
(73) |
s#(mark(X)) |
→ |
s#(X) |
(74) |
active#(plus(N,0)) |
→ |
isNat#(N) |
(75) |
proper#(isNat(X)) |
→ |
proper#(X) |
(76) |
active#(and(X1,X2)) |
→ |
and#(active(X1),X2) |
(77) |
U31#(mark(X)) |
→ |
U31#(X) |
(78) |
and#(mark(X1),X2) |
→ |
and#(X1,X2) |
(79) |
proper#(U31(X)) |
→ |
U31#(proper(X)) |
(80) |
proper#(U11(X1,X2)) |
→ |
proper#(X2) |
(81) |
U11#(ok(X1),ok(X2)) |
→ |
U11#(X1,X2) |
(82) |
U41#(ok(X1),ok(X2),ok(X3)) |
→ |
U41#(X1,X2,X3) |
(83) |
active#(U41(X1,X2,X3)) |
→ |
active#(X1) |
(84) |
proper#(U41(X1,X2,X3)) |
→ |
proper#(X2) |
(85) |
proper#(x(X1,X2)) |
→ |
proper#(X1) |
(86) |
proper#(U41(X1,X2,X3)) |
→ |
U41#(proper(X1),proper(X2),proper(X3)) |
(87) |
active#(plus(N,0)) |
→ |
U11#(isNat(N),N) |
(88) |
proper#(U21(X1,X2,X3)) |
→ |
proper#(X1) |
(89) |
top#(ok(X)) |
→ |
active#(X) |
(90) |
active#(plus(X1,X2)) |
→ |
active#(X1) |
(91) |
top#(ok(X)) |
→ |
top#(active(X)) |
(92) |
active#(s(X)) |
→ |
s#(active(X)) |
(93) |
proper#(U21(X1,X2,X3)) |
→ |
proper#(X3) |
(94) |
active#(U11(X1,X2)) |
→ |
active#(X1) |
(95) |
active#(plus(X1,X2)) |
→ |
plus#(active(X1),X2) |
(96) |
proper#(U41(X1,X2,X3)) |
→ |
proper#(X1) |
(97) |
active#(isNat(x(V1,V2))) |
→ |
isNat#(V1) |
(98) |
active#(x(X1,X2)) |
→ |
x#(active(X1),X2) |
(99) |
proper#(U31(X)) |
→ |
proper#(X) |
(100) |
active#(U41(tt,M,N)) |
→ |
x#(N,M) |
(101) |
active#(s(X)) |
→ |
active#(X) |
(102) |
proper#(U41(X1,X2,X3)) |
→ |
proper#(X3) |
(103) |
plus#(X1,mark(X2)) |
→ |
plus#(X1,X2) |
(104) |
active#(isNat(x(V1,V2))) |
→ |
and#(isNat(V1),isNat(V2)) |
(105) |
active#(isNat(plus(V1,V2))) |
→ |
and#(isNat(V1),isNat(V2)) |
(106) |
U21#(mark(X1),X2,X3) |
→ |
U21#(X1,X2,X3) |
(107) |
x#(ok(X1),ok(X2)) |
→ |
x#(X1,X2) |
(108) |
proper#(plus(X1,X2)) |
→ |
plus#(proper(X1),proper(X2)) |
(109) |
active#(U41(tt,M,N)) |
→ |
plus#(x(N,M),N) |
(110) |
U21#(ok(X1),ok(X2),ok(X3)) |
→ |
U21#(X1,X2,X3) |
(111) |
active#(and(X1,X2)) |
→ |
active#(X1) |
(112) |
proper#(x(X1,X2)) |
→ |
proper#(X2) |
(113) |
active#(plus(X1,X2)) |
→ |
active#(X2) |
(114) |
active#(x(X1,X2)) |
→ |
active#(X2) |
(115) |
active#(plus(N,s(M))) |
→ |
U21#(and(isNat(M),isNat(N)),M,N) |
(116) |
active#(U31(X)) |
→ |
U31#(active(X)) |
(117) |
proper#(and(X1,X2)) |
→ |
proper#(X2) |
(118) |
active#(U11(X1,X2)) |
→ |
U11#(active(X1),X2) |
(119) |
proper#(U11(X1,X2)) |
→ |
U11#(proper(X1),proper(X2)) |
(120) |
active#(x(N,s(M))) |
→ |
isNat#(M) |
(121) |
proper#(plus(X1,X2)) |
→ |
proper#(X2) |
(122) |
active#(plus(N,s(M))) |
→ |
and#(isNat(M),isNat(N)) |
(123) |
top#(mark(X)) |
→ |
proper#(X) |
(124) |
active#(U21(X1,X2,X3)) |
→ |
active#(X1) |
(125) |
active#(x(N,s(M))) |
→ |
and#(isNat(M),isNat(N)) |
(126) |
proper#(isNat(X)) |
→ |
isNat#(proper(X)) |
(127) |
s#(ok(X)) |
→ |
s#(X) |
(128) |
active#(isNat(x(V1,V2))) |
→ |
isNat#(V2) |
(129) |
plus#(ok(X1),ok(X2)) |
→ |
plus#(X1,X2) |
(130) |
active#(x(X1,X2)) |
→ |
x#(X1,active(X2)) |
(131) |
active#(x(N,s(M))) |
→ |
U41#(and(isNat(M),isNat(N)),M,N) |
(132) |
active#(isNat(s(V1))) |
→ |
isNat#(V1) |
(133) |
plus#(mark(X1),X2) |
→ |
plus#(X1,X2) |
(134) |
active#(plus(N,s(M))) |
→ |
isNat#(N) |
(135) |
proper#(plus(X1,X2)) |
→ |
proper#(X1) |
(136) |
proper#(s(X)) |
→ |
proper#(X) |
(137) |
proper#(and(X1,X2)) |
→ |
and#(proper(X1),proper(X2)) |
(138) |
proper#(U21(X1,X2,X3)) |
→ |
proper#(X2) |
(139) |
active#(U21(tt,M,N)) |
→ |
s#(plus(N,M)) |
(140) |
active#(isNat(plus(V1,V2))) |
→ |
isNat#(V2) |
(141) |
proper#(x(X1,X2)) |
→ |
x#(proper(X1),proper(X2)) |
(142) |
proper#(s(X)) |
→ |
s#(proper(X)) |
(143) |
proper#(and(X1,X2)) |
→ |
proper#(X1) |
(144) |
active#(x(X1,X2)) |
→ |
active#(X1) |
(145) |
active#(U41(X1,X2,X3)) |
→ |
U41#(active(X1),X2,X3) |
(146) |
isNat#(ok(X)) |
→ |
isNat#(X) |
(147) |
The dependency pairs are split into 12
components.
-
The
1st
component contains the
pair
top#(ok(X)) |
→ |
top#(active(X)) |
(92) |
top#(mark(X)) |
→ |
top#(proper(X)) |
(61) |
1.1.1 Reduction Pair Processor with Usable Rules
Using the argument filter
π(top) |
= |
1 |
π(top#) |
= |
1 |
π(proper) |
= |
1 |
π(ok) |
= |
1 |
π(x#) |
= |
2 |
π(isNat) |
= |
1 |
π(active) |
= |
1 |
π(U31#) |
= |
1 |
in combination with the following Weighted Path Order with the following precedence and status
prec(U21) |
= |
5 |
|
status(U21) |
= |
[3, 2, 1] |
|
list-extension(U21) |
= |
Lex |
prec(U11) |
= |
4 |
|
status(U11) |
= |
[2, 1] |
|
list-extension(U11) |
= |
Lex |
prec(s) |
= |
0 |
|
status(s) |
= |
[1] |
|
list-extension(s) |
= |
Lex |
prec(isNat#) |
= |
0 |
|
status(isNat#) |
= |
[] |
|
list-extension(isNat#) |
= |
Lex |
prec(and) |
= |
4 |
|
status(and) |
= |
[1, 2] |
|
list-extension(and) |
= |
Lex |
prec(plus#) |
= |
0 |
|
status(plus#) |
= |
[1, 2] |
|
list-extension(plus#) |
= |
Lex |
prec(x) |
= |
7 |
|
status(x) |
= |
[1, 2] |
|
list-extension(x) |
= |
Lex |
prec(0) |
= |
3 |
|
status(0) |
= |
[] |
|
list-extension(0) |
= |
Lex |
prec(s#) |
= |
0 |
|
status(s#) |
= |
[] |
|
list-extension(s#) |
= |
Lex |
prec(mark) |
= |
0 |
|
status(mark) |
= |
[1] |
|
list-extension(mark) |
= |
Lex |
prec(proper#) |
= |
0 |
|
status(proper#) |
= |
[] |
|
list-extension(proper#) |
= |
Lex |
prec(plus) |
= |
5 |
|
status(plus) |
= |
[1, 2] |
|
list-extension(plus) |
= |
Lex |
prec(U11#) |
= |
0 |
|
status(U11#) |
= |
[2, 1] |
|
list-extension(U11#) |
= |
Lex |
prec(U31) |
= |
3 |
|
status(U31) |
= |
[1] |
|
list-extension(U31) |
= |
Lex |
prec(U41#) |
= |
0 |
|
status(U41#) |
= |
[1, 3, 2] |
|
list-extension(U41#) |
= |
Lex |
prec(active#) |
= |
0 |
|
status(active#) |
= |
[] |
|
list-extension(active#) |
= |
Lex |
prec(U21#) |
= |
0 |
|
status(U21#) |
= |
[2, 1] |
|
list-extension(U21#) |
= |
Lex |
prec(tt) |
= |
2 |
|
status(tt) |
= |
[] |
|
list-extension(tt) |
= |
Lex |
prec(U41) |
= |
7 |
|
status(U41) |
= |
[3, 2, 1] |
|
list-extension(U41) |
= |
Lex |
prec(and#) |
= |
0 |
|
status(and#) |
= |
[1, 2] |
|
list-extension(and#) |
= |
Lex |
and the following
Max-polynomial interpretation
[U21(x1, x2, x3)] |
=
|
max(x1 + 0, x2 + 0, x3 + 0, 0) |
[U11(x1, x2)] |
=
|
max(x1 + 0, x2 + 0, 0) |
[s(x1)] |
=
|
x1 + 0 |
[isNat#(x1)] |
=
|
0 |
[and(x1, x2)] |
=
|
max(x1 + 0, x2 + 0, 0) |
[plus#(x1, x2)] |
=
|
max(x1 + 0, x2 + 0, 0) |
[x(x1, x2)] |
=
|
max(x1 + 0, x2 + 0, 0) |
[0] |
=
|
0 |
[s#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 0 |
[proper#(x1)] |
=
|
0 |
[plus(x1, x2)] |
=
|
max(x1 + 0, x2 + 0, 0) |
[U11#(x1, x2)] |
=
|
max(x1 + 0, x2 + 0, 0) |
[U31(x1)] |
=
|
x1 + 0 |
[U41#(x1, x2, x3)] |
=
|
max(x1 + 0, x2 + 0, x3 + 0, 0) |
[active#(x1)] |
=
|
0 |
[U21#(x1, x2, x3)] |
=
|
max(x1 + 0, x2 + 0, 0) |
[tt] |
=
|
0 |
[U41(x1, x2, x3)] |
=
|
max(x1 + 0, x2 + 0, x3 + 0, 0) |
[and#(x1, x2)] |
=
|
max(x1 + 0, x2 + 0, 0) |
together with the usable
rules
active(plus(X1,X2)) |
→ |
plus(X1,active(X2)) |
(18) |
U41(ok(X1),ok(X2),ok(X3)) |
→ |
ok(U41(X1,X2,X3)) |
(50) |
active(U41(tt,M,N)) |
→ |
mark(plus(x(N,M),N)) |
(4) |
active(U21(X1,X2,X3)) |
→ |
U21(active(X1),X2,X3) |
(15) |
active(isNat(s(V1))) |
→ |
mark(isNat(V1)) |
(8) |
active(U11(tt,N)) |
→ |
mark(N) |
(1) |
active(U31(tt)) |
→ |
mark(0) |
(3) |
active(s(X)) |
→ |
s(active(X)) |
(16) |
active(x(X1,X2)) |
→ |
x(active(X1),X2) |
(21) |
proper(U21(X1,X2,X3)) |
→ |
U21(proper(X1),proper(X2),proper(X3)) |
(36) |
s(mark(X)) |
→ |
mark(s(X)) |
(26) |
active(U31(X)) |
→ |
U31(active(X)) |
(19) |
x(X1,mark(X2)) |
→ |
mark(x(X1,X2)) |
(32) |
active(plus(X1,X2)) |
→ |
plus(active(X1),X2) |
(17) |
plus(mark(X1),X2) |
→ |
mark(plus(X1,X2)) |
(27) |
proper(U11(X1,X2)) |
→ |
U11(proper(X1),proper(X2)) |
(34) |
active(x(X1,X2)) |
→ |
x(X1,active(X2)) |
(22) |
plus(X1,mark(X2)) |
→ |
mark(plus(X1,X2)) |
(28) |
proper(isNat(X)) |
→ |
isNat(proper(X)) |
(44) |
active(and(tt,X)) |
→ |
mark(X) |
(5) |
and(mark(X1),X2) |
→ |
mark(and(X1,X2)) |
(33) |
active(plus(N,0)) |
→ |
mark(U11(isNat(N),N)) |
(10) |
proper(U31(X)) |
→ |
U31(proper(X)) |
(39) |
active(isNat(plus(V1,V2))) |
→ |
mark(and(isNat(V1),isNat(V2))) |
(7) |
active(U41(X1,X2,X3)) |
→ |
U41(active(X1),X2,X3) |
(20) |
U21(mark(X1),X2,X3) |
→ |
mark(U21(X1,X2,X3)) |
(25) |
U31(ok(X)) |
→ |
ok(U31(X)) |
(49) |
and(ok(X1),ok(X2)) |
→ |
ok(and(X1,X2)) |
(52) |
U41(mark(X1),X2,X3) |
→ |
mark(U41(X1,X2,X3)) |
(30) |
active(U11(X1,X2)) |
→ |
U11(active(X1),X2) |
(14) |
x(mark(X1),X2) |
→ |
mark(x(X1,X2)) |
(31) |
active(x(N,0)) |
→ |
mark(U31(isNat(N))) |
(12) |
U11(ok(X1),ok(X2)) |
→ |
ok(U11(X1,X2)) |
(45) |
active(and(X1,X2)) |
→ |
and(active(X1),X2) |
(23) |
U11(mark(X1),X2) |
→ |
mark(U11(X1,X2)) |
(24) |
active(plus(N,s(M))) |
→ |
mark(U21(and(isNat(M),isNat(N)),M,N)) |
(11) |
active(isNat(x(V1,V2))) |
→ |
mark(and(isNat(V1),isNat(V2))) |
(9) |
active(x(N,s(M))) |
→ |
mark(U41(and(isNat(M),isNat(N)),M,N)) |
(13) |
x(ok(X1),ok(X2)) |
→ |
ok(x(X1,X2)) |
(51) |
proper(0) |
→ |
ok(0) |
(40) |
active(isNat(0)) |
→ |
mark(tt) |
(6) |
proper(plus(X1,X2)) |
→ |
plus(proper(X1),proper(X2)) |
(38) |
plus(ok(X1),ok(X2)) |
→ |
ok(plus(X1,X2)) |
(48) |
isNat(ok(X)) |
→ |
ok(isNat(X)) |
(53) |
s(ok(X)) |
→ |
ok(s(X)) |
(47) |
proper(s(X)) |
→ |
s(proper(X)) |
(37) |
proper(U41(X1,X2,X3)) |
→ |
U41(proper(X1),proper(X2),proper(X3)) |
(41) |
proper(x(X1,X2)) |
→ |
x(proper(X1),proper(X2)) |
(42) |
U21(ok(X1),ok(X2),ok(X3)) |
→ |
ok(U21(X1,X2,X3)) |
(46) |
proper(tt) |
→ |
ok(tt) |
(35) |
U31(mark(X)) |
→ |
mark(U31(X)) |
(29) |
proper(and(X1,X2)) |
→ |
and(proper(X1),proper(X2)) |
(43) |
active(U21(tt,M,N)) |
→ |
mark(s(plus(N,M))) |
(2) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
top#(mark(X)) |
→ |
top#(proper(X)) |
(61) |
could be deleted.
1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
2nd
component contains the
pair
proper#(and(X1,X2)) |
→ |
proper#(X1) |
(144) |
proper#(U41(X1,X2,X3)) |
→ |
proper#(X3) |
(103) |
proper#(U31(X)) |
→ |
proper#(X) |
(100) |
proper#(U41(X1,X2,X3)) |
→ |
proper#(X1) |
(97) |
proper#(U21(X1,X2,X3)) |
→ |
proper#(X3) |
(94) |
proper#(U21(X1,X2,X3)) |
→ |
proper#(X2) |
(139) |
proper#(s(X)) |
→ |
proper#(X) |
(137) |
proper#(plus(X1,X2)) |
→ |
proper#(X1) |
(136) |
proper#(U21(X1,X2,X3)) |
→ |
proper#(X1) |
(89) |
proper#(x(X1,X2)) |
→ |
proper#(X1) |
(86) |
proper#(U41(X1,X2,X3)) |
→ |
proper#(X2) |
(85) |
proper#(U11(X1,X2)) |
→ |
proper#(X2) |
(81) |
proper#(isNat(X)) |
→ |
proper#(X) |
(76) |
proper#(plus(X1,X2)) |
→ |
proper#(X2) |
(122) |
proper#(and(X1,X2)) |
→ |
proper#(X2) |
(118) |
proper#(x(X1,X2)) |
→ |
proper#(X2) |
(113) |
proper#(U11(X1,X2)) |
→ |
proper#(X1) |
(62) |
1.1.2 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[U21(x1, x2, x3)] |
=
|
x1 + x2 + x3 + 1 |
[U11(x1, x2)] |
=
|
x1 + x2 + 1 |
[s(x1)] |
=
|
x1 + 1 |
[isNat#(x1)] |
=
|
0 |
[top(x1)] |
=
|
0 |
[and(x1, x2)] |
=
|
x1 + x2 + 1 |
[plus#(x1, x2)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[x(x1, x2)] |
=
|
x1 + x2 + 1 |
[proper(x1)] |
=
|
0 |
[ok(x1)] |
=
|
17891 |
[0] |
=
|
10282 |
[x#(x1, x2)] |
=
|
0 |
[s#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 0 |
[proper#(x1)] |
=
|
x1 + 0 |
[isNat(x1)] |
=
|
x1 + 1 |
[plus(x1, x2)] |
=
|
x1 + x2 + 1 |
[U11#(x1, x2)] |
=
|
0 |
[active(x1)] |
=
|
0 |
[U31(x1)] |
=
|
x1 + 6617 |
[U41#(x1, x2, x3)] |
=
|
0 |
[active#(x1)] |
=
|
0 |
[U21#(x1, x2, x3)] |
=
|
0 |
[tt] |
=
|
9123 |
[U41(x1, x2, x3)] |
=
|
x1 + x2 + x3 + 1 |
[U31#(x1)] |
=
|
0 |
[and#(x1, x2)] |
=
|
0 |
together with the usable
rules
U41(ok(X1),ok(X2),ok(X3)) |
→ |
ok(U41(X1,X2,X3)) |
(50) |
s(mark(X)) |
→ |
mark(s(X)) |
(26) |
x(X1,mark(X2)) |
→ |
mark(x(X1,X2)) |
(32) |
plus(mark(X1),X2) |
→ |
mark(plus(X1,X2)) |
(27) |
plus(X1,mark(X2)) |
→ |
mark(plus(X1,X2)) |
(28) |
and(mark(X1),X2) |
→ |
mark(and(X1,X2)) |
(33) |
U21(mark(X1),X2,X3) |
→ |
mark(U21(X1,X2,X3)) |
(25) |
U31(ok(X)) |
→ |
ok(U31(X)) |
(49) |
and(ok(X1),ok(X2)) |
→ |
ok(and(X1,X2)) |
(52) |
U41(mark(X1),X2,X3) |
→ |
mark(U41(X1,X2,X3)) |
(30) |
x(mark(X1),X2) |
→ |
mark(x(X1,X2)) |
(31) |
U11(ok(X1),ok(X2)) |
→ |
ok(U11(X1,X2)) |
(45) |
U11(mark(X1),X2) |
→ |
mark(U11(X1,X2)) |
(24) |
x(ok(X1),ok(X2)) |
→ |
ok(x(X1,X2)) |
(51) |
plus(ok(X1),ok(X2)) |
→ |
ok(plus(X1,X2)) |
(48) |
isNat(ok(X)) |
→ |
ok(isNat(X)) |
(53) |
s(ok(X)) |
→ |
ok(s(X)) |
(47) |
U21(ok(X1),ok(X2),ok(X3)) |
→ |
ok(U21(X1,X2,X3)) |
(46) |
U31(mark(X)) |
→ |
mark(U31(X)) |
(29) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
proper#(and(X1,X2)) |
→ |
proper#(X1) |
(144) |
proper#(U41(X1,X2,X3)) |
→ |
proper#(X3) |
(103) |
proper#(U31(X)) |
→ |
proper#(X) |
(100) |
proper#(U41(X1,X2,X3)) |
→ |
proper#(X1) |
(97) |
proper#(U21(X1,X2,X3)) |
→ |
proper#(X3) |
(94) |
proper#(U21(X1,X2,X3)) |
→ |
proper#(X2) |
(139) |
proper#(s(X)) |
→ |
proper#(X) |
(137) |
proper#(plus(X1,X2)) |
→ |
proper#(X1) |
(136) |
proper#(U21(X1,X2,X3)) |
→ |
proper#(X1) |
(89) |
proper#(x(X1,X2)) |
→ |
proper#(X1) |
(86) |
proper#(U41(X1,X2,X3)) |
→ |
proper#(X2) |
(85) |
proper#(U11(X1,X2)) |
→ |
proper#(X2) |
(81) |
proper#(isNat(X)) |
→ |
proper#(X) |
(76) |
proper#(plus(X1,X2)) |
→ |
proper#(X2) |
(122) |
proper#(and(X1,X2)) |
→ |
proper#(X2) |
(118) |
proper#(x(X1,X2)) |
→ |
proper#(X2) |
(113) |
proper#(U11(X1,X2)) |
→ |
proper#(X1) |
(62) |
could be deleted.
1.1.2.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
3rd
component contains the
pair
active#(x(X1,X2)) |
→ |
active#(X1) |
(145) |
active#(s(X)) |
→ |
active#(X) |
(102) |
active#(U11(X1,X2)) |
→ |
active#(X1) |
(95) |
active#(plus(X1,X2)) |
→ |
active#(X1) |
(91) |
active#(U41(X1,X2,X3)) |
→ |
active#(X1) |
(84) |
active#(U21(X1,X2,X3)) |
→ |
active#(X1) |
(125) |
active#(U31(X)) |
→ |
active#(X) |
(68) |
active#(x(X1,X2)) |
→ |
active#(X2) |
(115) |
active#(plus(X1,X2)) |
→ |
active#(X2) |
(114) |
active#(and(X1,X2)) |
→ |
active#(X1) |
(112) |
1.1.3 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[U21(x1, x2, x3)] |
=
|
x1 + x2 + x3 + 1 |
[U11(x1, x2)] |
=
|
x1 + x2 + 1 |
[s(x1)] |
=
|
x1 + 1 |
[isNat#(x1)] |
=
|
0 |
[top(x1)] |
=
|
0 |
[and(x1, x2)] |
=
|
x1 + x2 + 1 |
[plus#(x1, x2)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[x(x1, x2)] |
=
|
x1 + x2 + 1 |
[proper(x1)] |
=
|
0 |
[ok(x1)] |
=
|
1 |
[0] |
=
|
9133 |
[x#(x1, x2)] |
=
|
0 |
[s#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 0 |
[proper#(x1)] |
=
|
0 |
[isNat(x1)] |
=
|
x1 + 1 |
[plus(x1, x2)] |
=
|
x1 + x2 + 1 |
[U11#(x1, x2)] |
=
|
0 |
[active(x1)] |
=
|
0 |
[U31(x1)] |
=
|
x1 + 6617 |
[U41#(x1, x2, x3)] |
=
|
0 |
[active#(x1)] |
=
|
x1 + 0 |
[U21#(x1, x2, x3)] |
=
|
0 |
[tt] |
=
|
9123 |
[U41(x1, x2, x3)] |
=
|
x1 + x2 + x3 + 1 |
[U31#(x1)] |
=
|
0 |
[and#(x1, x2)] |
=
|
0 |
together with the usable
rules
U41(ok(X1),ok(X2),ok(X3)) |
→ |
ok(U41(X1,X2,X3)) |
(50) |
s(mark(X)) |
→ |
mark(s(X)) |
(26) |
x(X1,mark(X2)) |
→ |
mark(x(X1,X2)) |
(32) |
plus(mark(X1),X2) |
→ |
mark(plus(X1,X2)) |
(27) |
plus(X1,mark(X2)) |
→ |
mark(plus(X1,X2)) |
(28) |
and(mark(X1),X2) |
→ |
mark(and(X1,X2)) |
(33) |
U21(mark(X1),X2,X3) |
→ |
mark(U21(X1,X2,X3)) |
(25) |
U31(ok(X)) |
→ |
ok(U31(X)) |
(49) |
and(ok(X1),ok(X2)) |
→ |
ok(and(X1,X2)) |
(52) |
U41(mark(X1),X2,X3) |
→ |
mark(U41(X1,X2,X3)) |
(30) |
x(mark(X1),X2) |
→ |
mark(x(X1,X2)) |
(31) |
U11(ok(X1),ok(X2)) |
→ |
ok(U11(X1,X2)) |
(45) |
U11(mark(X1),X2) |
→ |
mark(U11(X1,X2)) |
(24) |
x(ok(X1),ok(X2)) |
→ |
ok(x(X1,X2)) |
(51) |
plus(ok(X1),ok(X2)) |
→ |
ok(plus(X1,X2)) |
(48) |
isNat(ok(X)) |
→ |
ok(isNat(X)) |
(53) |
s(ok(X)) |
→ |
ok(s(X)) |
(47) |
U21(ok(X1),ok(X2),ok(X3)) |
→ |
ok(U21(X1,X2,X3)) |
(46) |
U31(mark(X)) |
→ |
mark(U31(X)) |
(29) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
active#(x(X1,X2)) |
→ |
active#(X1) |
(145) |
active#(s(X)) |
→ |
active#(X) |
(102) |
active#(U11(X1,X2)) |
→ |
active#(X1) |
(95) |
active#(plus(X1,X2)) |
→ |
active#(X1) |
(91) |
active#(U41(X1,X2,X3)) |
→ |
active#(X1) |
(84) |
active#(U21(X1,X2,X3)) |
→ |
active#(X1) |
(125) |
active#(U31(X)) |
→ |
active#(X) |
(68) |
active#(x(X1,X2)) |
→ |
active#(X2) |
(115) |
active#(plus(X1,X2)) |
→ |
active#(X2) |
(114) |
active#(and(X1,X2)) |
→ |
active#(X1) |
(112) |
could be deleted.
1.1.3.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
4th
component contains the
pair
U41#(ok(X1),ok(X2),ok(X3)) |
→ |
U41#(X1,X2,X3) |
(83) |
U41#(mark(X1),X2,X3) |
→ |
U41#(X1,X2,X3) |
(63) |
1.1.4 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[U21(x1, x2, x3)] |
=
|
x1 + x3 + 0 |
[U11(x1, x2)] |
=
|
x1 + x2 + 1 |
[s(x1)] |
=
|
x1 + 1 |
[isNat#(x1)] |
=
|
0 |
[top(x1)] |
=
|
0 |
[and(x1, x2)] |
=
|
x1 + x2 + 1 |
[plus#(x1, x2)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[x(x1, x2)] |
=
|
x1 + x2 + 0 |
[proper(x1)] |
=
|
1 |
[ok(x1)] |
=
|
x1 + 1 |
[0] |
=
|
1 |
[x#(x1, x2)] |
=
|
0 |
[s#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 0 |
[proper#(x1)] |
=
|
0 |
[isNat(x1)] |
=
|
x1 + 1 |
[plus(x1, x2)] |
=
|
x1 + x2 + 1 |
[U11#(x1, x2)] |
=
|
0 |
[active(x1)] |
=
|
0 |
[U31(x1)] |
=
|
x1 + 1 |
[U41#(x1, x2, x3)] |
=
|
x2 + 0 |
[active#(x1)] |
=
|
0 |
[U21#(x1, x2, x3)] |
=
|
0 |
[tt] |
=
|
1 |
[U41(x1, x2, x3)] |
=
|
x1 + x2 + x3 + 0 |
[U31#(x1)] |
=
|
0 |
[and#(x1, x2)] |
=
|
0 |
together with the usable
rules
U41(ok(X1),ok(X2),ok(X3)) |
→ |
ok(U41(X1,X2,X3)) |
(50) |
s(mark(X)) |
→ |
mark(s(X)) |
(26) |
x(X1,mark(X2)) |
→ |
mark(x(X1,X2)) |
(32) |
plus(mark(X1),X2) |
→ |
mark(plus(X1,X2)) |
(27) |
plus(X1,mark(X2)) |
→ |
mark(plus(X1,X2)) |
(28) |
and(mark(X1),X2) |
→ |
mark(and(X1,X2)) |
(33) |
U21(mark(X1),X2,X3) |
→ |
mark(U21(X1,X2,X3)) |
(25) |
U31(ok(X)) |
→ |
ok(U31(X)) |
(49) |
and(ok(X1),ok(X2)) |
→ |
ok(and(X1,X2)) |
(52) |
U41(mark(X1),X2,X3) |
→ |
mark(U41(X1,X2,X3)) |
(30) |
x(mark(X1),X2) |
→ |
mark(x(X1,X2)) |
(31) |
U11(ok(X1),ok(X2)) |
→ |
ok(U11(X1,X2)) |
(45) |
U11(mark(X1),X2) |
→ |
mark(U11(X1,X2)) |
(24) |
x(ok(X1),ok(X2)) |
→ |
ok(x(X1,X2)) |
(51) |
plus(ok(X1),ok(X2)) |
→ |
ok(plus(X1,X2)) |
(48) |
isNat(ok(X)) |
→ |
ok(isNat(X)) |
(53) |
s(ok(X)) |
→ |
ok(s(X)) |
(47) |
U21(ok(X1),ok(X2),ok(X3)) |
→ |
ok(U21(X1,X2,X3)) |
(46) |
U31(mark(X)) |
→ |
mark(U31(X)) |
(29) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
U41#(ok(X1),ok(X2),ok(X3)) |
→ |
U41#(X1,X2,X3) |
(83) |
could be deleted.
1.1.4.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
5th
component contains the
pair
and#(mark(X1),X2) |
→ |
and#(X1,X2) |
(79) |
and#(ok(X1),ok(X2)) |
→ |
and#(X1,X2) |
(69) |
1.1.5 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[U21(x1, x2, x3)] |
=
|
x1 + x3 + 0 |
[U11(x1, x2)] |
=
|
x1 + x2 + 17979 |
[s(x1)] |
=
|
x1 + 1 |
[isNat#(x1)] |
=
|
0 |
[top(x1)] |
=
|
0 |
[and(x1, x2)] |
=
|
x2 + 2 |
[plus#(x1, x2)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[x(x1, x2)] |
=
|
x1 + x2 + 0 |
[proper(x1)] |
=
|
1 |
[ok(x1)] |
=
|
x1 + 1 |
[0] |
=
|
1 |
[x#(x1, x2)] |
=
|
0 |
[s#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 1 |
[proper#(x1)] |
=
|
0 |
[isNat(x1)] |
=
|
x1 + 1 |
[plus(x1, x2)] |
=
|
x1 + x2 + 1 |
[U11#(x1, x2)] |
=
|
0 |
[active(x1)] |
=
|
1 |
[U31(x1)] |
=
|
x1 + 1 |
[U41#(x1, x2, x3)] |
=
|
0 |
[active#(x1)] |
=
|
0 |
[U21#(x1, x2, x3)] |
=
|
0 |
[tt] |
=
|
1 |
[U41(x1, x2, x3)] |
=
|
x1 + x2 + x3 + 0 |
[U31#(x1)] |
=
|
0 |
[and#(x1, x2)] |
=
|
x1 + 0 |
together with the usable
rules
U41(ok(X1),ok(X2),ok(X3)) |
→ |
ok(U41(X1,X2,X3)) |
(50) |
s(mark(X)) |
→ |
mark(s(X)) |
(26) |
x(X1,mark(X2)) |
→ |
mark(x(X1,X2)) |
(32) |
plus(mark(X1),X2) |
→ |
mark(plus(X1,X2)) |
(27) |
plus(X1,mark(X2)) |
→ |
mark(plus(X1,X2)) |
(28) |
U21(mark(X1),X2,X3) |
→ |
mark(U21(X1,X2,X3)) |
(25) |
U31(ok(X)) |
→ |
ok(U31(X)) |
(49) |
U41(mark(X1),X2,X3) |
→ |
mark(U41(X1,X2,X3)) |
(30) |
x(mark(X1),X2) |
→ |
mark(x(X1,X2)) |
(31) |
U11(ok(X1),ok(X2)) |
→ |
ok(U11(X1,X2)) |
(45) |
U11(mark(X1),X2) |
→ |
mark(U11(X1,X2)) |
(24) |
x(ok(X1),ok(X2)) |
→ |
ok(x(X1,X2)) |
(51) |
plus(ok(X1),ok(X2)) |
→ |
ok(plus(X1,X2)) |
(48) |
isNat(ok(X)) |
→ |
ok(isNat(X)) |
(53) |
s(ok(X)) |
→ |
ok(s(X)) |
(47) |
U21(ok(X1),ok(X2),ok(X3)) |
→ |
ok(U21(X1,X2,X3)) |
(46) |
U31(mark(X)) |
→ |
mark(U31(X)) |
(29) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
and#(mark(X1),X2) |
→ |
and#(X1,X2) |
(79) |
and#(ok(X1),ok(X2)) |
→ |
and#(X1,X2) |
(69) |
could be deleted.
1.1.5.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
6th
component contains the
pair
U11#(ok(X1),ok(X2)) |
→ |
U11#(X1,X2) |
(82) |
U11#(mark(X1),X2) |
→ |
U11#(X1,X2) |
(66) |
1.1.6 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[U21(x1, x2, x3)] |
=
|
x1 + x3 + 0 |
[U11(x1, x2)] |
=
|
x1 + x2 + 1 |
[s(x1)] |
=
|
x1 + 1 |
[isNat#(x1)] |
=
|
0 |
[top(x1)] |
=
|
0 |
[and(x1, x2)] |
=
|
x2 + 21639 |
[plus#(x1, x2)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[x(x1, x2)] |
=
|
x1 + x2 + 0 |
[proper(x1)] |
=
|
1 |
[ok(x1)] |
=
|
x1 + 1 |
[0] |
=
|
1 |
[x#(x1, x2)] |
=
|
0 |
[s#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 1 |
[proper#(x1)] |
=
|
0 |
[isNat(x1)] |
=
|
x1 + 1 |
[plus(x1, x2)] |
=
|
x1 + x2 + 1 |
[U11#(x1, x2)] |
=
|
x2 + 0 |
[active(x1)] |
=
|
1 |
[U31(x1)] |
=
|
x1 + 32299 |
[U41#(x1, x2, x3)] |
=
|
0 |
[active#(x1)] |
=
|
0 |
[U21#(x1, x2, x3)] |
=
|
0 |
[tt] |
=
|
1 |
[U41(x1, x2, x3)] |
=
|
x1 + x2 + x3 + 0 |
[U31#(x1)] |
=
|
0 |
[and#(x1, x2)] |
=
|
0 |
together with the usable
rules
U41(ok(X1),ok(X2),ok(X3)) |
→ |
ok(U41(X1,X2,X3)) |
(50) |
s(mark(X)) |
→ |
mark(s(X)) |
(26) |
x(X1,mark(X2)) |
→ |
mark(x(X1,X2)) |
(32) |
plus(mark(X1),X2) |
→ |
mark(plus(X1,X2)) |
(27) |
plus(X1,mark(X2)) |
→ |
mark(plus(X1,X2)) |
(28) |
U21(mark(X1),X2,X3) |
→ |
mark(U21(X1,X2,X3)) |
(25) |
U31(ok(X)) |
→ |
ok(U31(X)) |
(49) |
U41(mark(X1),X2,X3) |
→ |
mark(U41(X1,X2,X3)) |
(30) |
x(mark(X1),X2) |
→ |
mark(x(X1,X2)) |
(31) |
U11(ok(X1),ok(X2)) |
→ |
ok(U11(X1,X2)) |
(45) |
U11(mark(X1),X2) |
→ |
mark(U11(X1,X2)) |
(24) |
x(ok(X1),ok(X2)) |
→ |
ok(x(X1,X2)) |
(51) |
plus(ok(X1),ok(X2)) |
→ |
ok(plus(X1,X2)) |
(48) |
isNat(ok(X)) |
→ |
ok(isNat(X)) |
(53) |
s(ok(X)) |
→ |
ok(s(X)) |
(47) |
U21(ok(X1),ok(X2),ok(X3)) |
→ |
ok(U21(X1,X2,X3)) |
(46) |
U31(mark(X)) |
→ |
mark(U31(X)) |
(29) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
U11#(ok(X1),ok(X2)) |
→ |
U11#(X1,X2) |
(82) |
could be deleted.
1.1.6.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
7th
component contains the
pair
U31#(mark(X)) |
→ |
U31#(X) |
(78) |
U31#(ok(X)) |
→ |
U31#(X) |
(72) |
1.1.7 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[U21(x1, x2, x3)] |
=
|
x1 + x3 + 0 |
[U11(x1, x2)] |
=
|
x1 + x2 + 1 |
[s(x1)] |
=
|
x1 + 1 |
[isNat#(x1)] |
=
|
0 |
[top(x1)] |
=
|
0 |
[and(x1, x2)] |
=
|
x2 + 21639 |
[plus#(x1, x2)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[x(x1, x2)] |
=
|
x1 + x2 + 0 |
[proper(x1)] |
=
|
1 |
[ok(x1)] |
=
|
x1 + 1 |
[0] |
=
|
1 |
[x#(x1, x2)] |
=
|
0 |
[s#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 1 |
[proper#(x1)] |
=
|
0 |
[isNat(x1)] |
=
|
x1 + 1 |
[plus(x1, x2)] |
=
|
x1 + x2 + 1 |
[U11#(x1, x2)] |
=
|
0 |
[active(x1)] |
=
|
1 |
[U31(x1)] |
=
|
x1 + 7362 |
[U41#(x1, x2, x3)] |
=
|
0 |
[active#(x1)] |
=
|
0 |
[U21#(x1, x2, x3)] |
=
|
0 |
[tt] |
=
|
1 |
[U41(x1, x2, x3)] |
=
|
x1 + x2 + x3 + 0 |
[U31#(x1)] |
=
|
x1 + 0 |
[and#(x1, x2)] |
=
|
0 |
together with the usable
rules
U41(ok(X1),ok(X2),ok(X3)) |
→ |
ok(U41(X1,X2,X3)) |
(50) |
s(mark(X)) |
→ |
mark(s(X)) |
(26) |
x(X1,mark(X2)) |
→ |
mark(x(X1,X2)) |
(32) |
plus(mark(X1),X2) |
→ |
mark(plus(X1,X2)) |
(27) |
plus(X1,mark(X2)) |
→ |
mark(plus(X1,X2)) |
(28) |
U21(mark(X1),X2,X3) |
→ |
mark(U21(X1,X2,X3)) |
(25) |
U31(ok(X)) |
→ |
ok(U31(X)) |
(49) |
U41(mark(X1),X2,X3) |
→ |
mark(U41(X1,X2,X3)) |
(30) |
x(mark(X1),X2) |
→ |
mark(x(X1,X2)) |
(31) |
U11(ok(X1),ok(X2)) |
→ |
ok(U11(X1,X2)) |
(45) |
U11(mark(X1),X2) |
→ |
mark(U11(X1,X2)) |
(24) |
x(ok(X1),ok(X2)) |
→ |
ok(x(X1,X2)) |
(51) |
plus(ok(X1),ok(X2)) |
→ |
ok(plus(X1,X2)) |
(48) |
isNat(ok(X)) |
→ |
ok(isNat(X)) |
(53) |
s(ok(X)) |
→ |
ok(s(X)) |
(47) |
U21(ok(X1),ok(X2),ok(X3)) |
→ |
ok(U21(X1,X2,X3)) |
(46) |
U31(mark(X)) |
→ |
mark(U31(X)) |
(29) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
U31#(mark(X)) |
→ |
U31#(X) |
(78) |
U31#(ok(X)) |
→ |
U31#(X) |
(72) |
could be deleted.
1.1.7.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
8th
component contains the
pair
s#(ok(X)) |
→ |
s#(X) |
(128) |
s#(mark(X)) |
→ |
s#(X) |
(74) |
1.1.8 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[U21(x1, x2, x3)] |
=
|
x3 + 8703 |
[U11(x1, x2)] |
=
|
21332 |
[s(x1)] |
=
|
x1 + 1 |
[isNat#(x1)] |
=
|
0 |
[top(x1)] |
=
|
0 |
[and(x1, x2)] |
=
|
28939 |
[plus#(x1, x2)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[x(x1, x2)] |
=
|
11404 |
[proper(x1)] |
=
|
1 |
[ok(x1)] |
=
|
x1 + 1 |
[0] |
=
|
1 |
[x#(x1, x2)] |
=
|
0 |
[s#(x1)] |
=
|
x1 + 0 |
[mark(x1)] |
=
|
x1 + 1 |
[proper#(x1)] |
=
|
0 |
[isNat(x1)] |
=
|
x1 + 1 |
[plus(x1, x2)] |
=
|
8137 |
[U11#(x1, x2)] |
=
|
0 |
[active(x1)] |
=
|
1 |
[U31(x1)] |
=
|
x1 + 1 |
[U41#(x1, x2, x3)] |
=
|
0 |
[active#(x1)] |
=
|
0 |
[U21#(x1, x2, x3)] |
=
|
0 |
[tt] |
=
|
1 |
[U41(x1, x2, x3)] |
=
|
27073 |
[U31#(x1)] |
=
|
0 |
[and#(x1, x2)] |
=
|
0 |
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pairs
s#(ok(X)) |
→ |
s#(X) |
(128) |
s#(mark(X)) |
→ |
s#(X) |
(74) |
could be deleted.
1.1.8.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
9th
component contains the
pair
isNat#(ok(X)) |
→ |
isNat#(X) |
(147) |
1.1.9 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[U21(x1, x2, x3)] |
=
|
2 |
[U11(x1, x2)] |
=
|
2 |
[s(x1)] |
=
|
2 |
[isNat#(x1)] |
=
|
x1 + 0 |
[top(x1)] |
=
|
0 |
[and(x1, x2)] |
=
|
2 |
[plus#(x1, x2)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[x(x1, x2)] |
=
|
2 |
[proper(x1)] |
=
|
1 |
[ok(x1)] |
=
|
x1 + 1 |
[0] |
=
|
17037 |
[x#(x1, x2)] |
=
|
0 |
[s#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
3 |
[proper#(x1)] |
=
|
0 |
[isNat(x1)] |
=
|
31607 |
[plus(x1, x2)] |
=
|
x2 + 2 |
[U11#(x1, x2)] |
=
|
0 |
[active(x1)] |
=
|
1 |
[U31(x1)] |
=
|
x1 + 1 |
[U41#(x1, x2, x3)] |
=
|
0 |
[active#(x1)] |
=
|
0 |
[U21#(x1, x2, x3)] |
=
|
0 |
[tt] |
=
|
1 |
[U41(x1, x2, x3)] |
=
|
x2 + 2 |
[U31#(x1)] |
=
|
0 |
[and#(x1, x2)] |
=
|
0 |
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pair
isNat#(ok(X)) |
→ |
isNat#(X) |
(147) |
could be deleted.
1.1.9.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
10th
component contains the
pair
U21#(mark(X1),X2,X3) |
→ |
U21#(X1,X2,X3) |
(107) |
U21#(ok(X1),ok(X2),ok(X3)) |
→ |
U21#(X1,X2,X3) |
(111) |
1.1.10 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[U21(x1, x2, x3)] |
=
|
21541 |
[U11(x1, x2)] |
=
|
27531 |
[s(x1)] |
=
|
29829 |
[isNat#(x1)] |
=
|
0 |
[top(x1)] |
=
|
0 |
[and(x1, x2)] |
=
|
15093 |
[plus#(x1, x2)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[x(x1, x2)] |
=
|
29305 |
[proper(x1)] |
=
|
1 |
[ok(x1)] |
=
|
x1 + 1 |
[0] |
=
|
17037 |
[x#(x1, x2)] |
=
|
0 |
[s#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 24228 |
[proper#(x1)] |
=
|
0 |
[isNat(x1)] |
=
|
31607 |
[plus(x1, x2)] |
=
|
x1 + 2160 |
[U11#(x1, x2)] |
=
|
0 |
[active(x1)] |
=
|
1 |
[U31(x1)] |
=
|
2 |
[U41#(x1, x2, x3)] |
=
|
0 |
[active#(x1)] |
=
|
0 |
[U21#(x1, x2, x3)] |
=
|
x2 + 0 |
[tt] |
=
|
3288 |
[U41(x1, x2, x3)] |
=
|
x2 + 2 |
[U31#(x1)] |
=
|
0 |
[and#(x1, x2)] |
=
|
0 |
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pair
U21#(ok(X1),ok(X2),ok(X3)) |
→ |
U21#(X1,X2,X3) |
(111) |
could be deleted.
1.1.10.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
11th
component contains the
pair
x#(ok(X1),ok(X2)) |
→ |
x#(X1,X2) |
(108) |
x#(X1,mark(X2)) |
→ |
x#(X1,X2) |
(71) |
x#(mark(X1),X2) |
→ |
x#(X1,X2) |
(58) |
1.1.11 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[U21(x1, x2, x3)] |
=
|
2 |
[U11(x1, x2)] |
=
|
23724 |
[s(x1)] |
=
|
19079 |
[isNat#(x1)] |
=
|
0 |
[top(x1)] |
=
|
0 |
[and(x1, x2)] |
=
|
2992 |
[plus#(x1, x2)] |
=
|
0 |
[top#(x1)] |
=
|
0 |
[x(x1, x2)] |
=
|
21966 |
[proper(x1)] |
=
|
1 |
[ok(x1)] |
=
|
x1 + 9251 |
[0] |
=
|
1 |
[x#(x1, x2)] |
=
|
x1 + x2 + 0 |
[s#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 15079 |
[proper#(x1)] |
=
|
0 |
[isNat(x1)] |
=
|
2 |
[plus(x1, x2)] |
=
|
x1 + 879 |
[U11#(x1, x2)] |
=
|
0 |
[active(x1)] |
=
|
1 |
[U31(x1)] |
=
|
13421 |
[U41#(x1, x2, x3)] |
=
|
0 |
[active#(x1)] |
=
|
0 |
[U21#(x1, x2, x3)] |
=
|
0 |
[tt] |
=
|
6753 |
[U41(x1, x2, x3)] |
=
|
21796 |
[U31#(x1)] |
=
|
0 |
[and#(x1, x2)] |
=
|
0 |
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pairs
x#(ok(X1),ok(X2)) |
→ |
x#(X1,X2) |
(108) |
x#(X1,mark(X2)) |
→ |
x#(X1,X2) |
(71) |
x#(mark(X1),X2) |
→ |
x#(X1,X2) |
(58) |
could be deleted.
1.1.11.1 Dependency Graph Processor
The dependency pairs are split into 0
components.
-
The
12th
component contains the
pair
plus#(X1,mark(X2)) |
→ |
plus#(X1,X2) |
(104) |
plus#(mark(X1),X2) |
→ |
plus#(X1,X2) |
(134) |
plus#(ok(X1),ok(X2)) |
→ |
plus#(X1,X2) |
(130) |
1.1.12 Reduction Pair Processor with Usable Rules
Using the Max-polynomial interpretation
[U21(x1, x2, x3)] |
=
|
x1 + x2 + 1 |
[U11(x1, x2)] |
=
|
2 |
[s(x1)] |
=
|
12783 |
[isNat#(x1)] |
=
|
0 |
[top(x1)] |
=
|
0 |
[and(x1, x2)] |
=
|
2 |
[plus#(x1, x2)] |
=
|
x1 + 0 |
[top#(x1)] |
=
|
0 |
[x(x1, x2)] |
=
|
21966 |
[proper(x1)] |
=
|
1 |
[ok(x1)] |
=
|
x1 + 4092 |
[0] |
=
|
20540 |
[x#(x1, x2)] |
=
|
0 |
[s#(x1)] |
=
|
0 |
[mark(x1)] |
=
|
x1 + 4331 |
[proper#(x1)] |
=
|
0 |
[isNat(x1)] |
=
|
2 |
[plus(x1, x2)] |
=
|
x1 + 9556 |
[U11#(x1, x2)] |
=
|
0 |
[active(x1)] |
=
|
1 |
[U31(x1)] |
=
|
37711 |
[U41#(x1, x2, x3)] |
=
|
0 |
[active#(x1)] |
=
|
0 |
[U21#(x1, x2, x3)] |
=
|
0 |
[tt] |
=
|
16542 |
[U41(x1, x2, x3)] |
=
|
19691 |
[U31#(x1)] |
=
|
0 |
[and#(x1, x2)] |
=
|
0 |
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pairs
plus#(mark(X1),X2) |
→ |
plus#(X1,X2) |
(134) |
plus#(ok(X1),ok(X2)) |
→ |
plus#(X1,X2) |
(130) |
could be deleted.
1.1.12.1 Dependency Graph Processor
The dependency pairs are split into 1
component.