The rewrite relation of the following TRS is considered.
There are 115 ruless (increase limit for explicit display).
The dependency pairs are split into 16
components.
-
The
1st
component contains the
pair
mark#(cons(X1,X2)) |
→ |
active#(cons(mark(X1),X2)) |
(113) |
active#(U41(tt,V2)) |
→ |
mark#(U42(isNatIList(V2))) |
(85) |
mark#(U42(X)) |
→ |
active#(U42(mark(X))) |
(130) |
active#(U51(tt,V2)) |
→ |
mark#(U52(isNatList(V2))) |
(89) |
mark#(U52(X)) |
→ |
active#(U52(mark(X))) |
(137) |
active#(U61(tt,L,N)) |
→ |
mark#(U62(isNat(N),L)) |
(93) |
mark#(U62(X1,X2)) |
→ |
active#(U62(mark(X1),X2)) |
(144) |
active#(U62(tt,L)) |
→ |
mark#(s(length(L))) |
(96) |
mark#(s(X)) |
→ |
active#(s(mark(X))) |
(148) |
active#(length(cons(N,L))) |
→ |
mark#(U61(isNatList(L),L,N)) |
(109) |
mark#(U61(X1,X2,X3)) |
→ |
active#(U61(mark(X1),X2,X3)) |
(141) |
mark#(U61(X1,X2,X3)) |
→ |
mark#(X1) |
(143) |
mark#(zeros) |
→ |
active#(zeros) |
(112) |
active#(zeros) |
→ |
mark#(cons(0,zeros)) |
(82) |
mark#(cons(X1,X2)) |
→ |
mark#(X1) |
(115) |
mark#(U11(X)) |
→ |
active#(U11(mark(X))) |
(117) |
mark#(U11(X)) |
→ |
mark#(X) |
(119) |
mark#(U21(X)) |
→ |
active#(U21(mark(X))) |
(121) |
mark#(U21(X)) |
→ |
mark#(X) |
(123) |
mark#(U31(X)) |
→ |
active#(U31(mark(X))) |
(124) |
mark#(U31(X)) |
→ |
mark#(X) |
(126) |
mark#(U41(X1,X2)) |
→ |
active#(U41(mark(X1),X2)) |
(127) |
mark#(U41(X1,X2)) |
→ |
mark#(X1) |
(129) |
mark#(U42(X)) |
→ |
mark#(X) |
(132) |
mark#(isNatIList(X)) |
→ |
active#(isNatIList(X)) |
(133) |
active#(isNatIList(cons(V1,V2))) |
→ |
mark#(U41(isNat(V1),V2)) |
(103) |
mark#(U51(X1,X2)) |
→ |
active#(U51(mark(X1),X2)) |
(134) |
mark#(U51(X1,X2)) |
→ |
mark#(X1) |
(136) |
mark#(U52(X)) |
→ |
mark#(X) |
(139) |
mark#(isNatList(X)) |
→ |
active#(isNatList(X)) |
(140) |
active#(isNatList(cons(V1,V2))) |
→ |
mark#(U51(isNat(V1),V2)) |
(106) |
mark#(U62(X1,X2)) |
→ |
mark#(X1) |
(146) |
mark#(isNat(X)) |
→ |
active#(isNat(X)) |
(147) |
active#(isNat(s(V1))) |
→ |
mark#(U21(isNat(V1))) |
(100) |
mark#(s(X)) |
→ |
mark#(X) |
(150) |
mark#(length(X)) |
→ |
active#(length(mark(X))) |
(151) |
mark#(length(X)) |
→ |
mark#(X) |
(153) |
1.1.1.1.1.1.1 Usable Rules Processor
We restrict the rewrite rules to the following usable rules of the DP problem.
mark(zeros) |
→ |
active(zeros) |
(21) |
mark(cons(X1,X2)) |
→ |
active(cons(mark(X1),X2)) |
(22) |
active(zeros) |
→ |
mark(cons(0,zeros)) |
(1) |
active(U41(tt,V2)) |
→ |
mark(U42(isNatIList(V2))) |
(5) |
mark(U42(X)) |
→ |
active(U42(mark(X))) |
(29) |
active(U51(tt,V2)) |
→ |
mark(U52(isNatList(V2))) |
(7) |
mark(U52(X)) |
→ |
active(U52(mark(X))) |
(32) |
active(U61(tt,L,N)) |
→ |
mark(U62(isNat(N),L)) |
(9) |
mark(U62(X1,X2)) |
→ |
active(U62(mark(X1),X2)) |
(35) |
active(U62(tt,L)) |
→ |
mark(s(length(L))) |
(10) |
mark(s(X)) |
→ |
active(s(mark(X))) |
(37) |
active(isNat(s(V1))) |
→ |
mark(U21(isNat(V1))) |
(13) |
mark(U21(X)) |
→ |
active(U21(mark(X))) |
(26) |
active(isNatIList(cons(V1,V2))) |
→ |
mark(U41(isNat(V1),V2)) |
(16) |
mark(U41(X1,X2)) |
→ |
active(U41(mark(X1),X2)) |
(28) |
active(isNatList(cons(V1,V2))) |
→ |
mark(U51(isNat(V1),V2)) |
(18) |
mark(U51(X1,X2)) |
→ |
active(U51(mark(X1),X2)) |
(31) |
active(length(cons(N,L))) |
→ |
mark(U61(isNatList(L),L,N)) |
(20) |
mark(U61(X1,X2,X3)) |
→ |
active(U61(mark(X1),X2,X3)) |
(34) |
mark(0) |
→ |
active(0) |
(23) |
mark(U11(X)) |
→ |
active(U11(mark(X))) |
(24) |
mark(tt) |
→ |
active(tt) |
(25) |
mark(U31(X)) |
→ |
active(U31(mark(X))) |
(27) |
mark(isNatIList(X)) |
→ |
active(isNatIList(X)) |
(30) |
mark(isNatList(X)) |
→ |
active(isNatList(X)) |
(33) |
mark(isNat(X)) |
→ |
active(isNat(X)) |
(36) |
mark(length(X)) |
→ |
active(length(mark(X))) |
(38) |
mark(nil) |
→ |
active(nil) |
(39) |
length(active(X)) |
→ |
length(X) |
(81) |
length(mark(X)) |
→ |
length(X) |
(80) |
active(U21(tt)) |
→ |
mark(tt) |
(3) |
active(U42(tt)) |
→ |
mark(tt) |
(6) |
active(U52(tt)) |
→ |
mark(tt) |
(8) |
active(isNat(0)) |
→ |
mark(tt) |
(11) |
U31(active(X)) |
→ |
U31(X) |
(49) |
U31(mark(X)) |
→ |
U31(X) |
(48) |
U11(active(X)) |
→ |
U11(X) |
(45) |
U11(mark(X)) |
→ |
U11(X) |
(44) |
cons(X1,mark(X2)) |
→ |
cons(X1,X2) |
(41) |
cons(mark(X1),X2) |
→ |
cons(X1,X2) |
(40) |
cons(active(X1),X2) |
→ |
cons(X1,X2) |
(42) |
cons(X1,active(X2)) |
→ |
cons(X1,X2) |
(43) |
U42(active(X)) |
→ |
U42(X) |
(55) |
U42(mark(X)) |
→ |
U42(X) |
(54) |
U52(active(X)) |
→ |
U52(X) |
(63) |
U52(mark(X)) |
→ |
U52(X) |
(62) |
U62(X1,mark(X2)) |
→ |
U62(X1,X2) |
(73) |
U62(mark(X1),X2) |
→ |
U62(X1,X2) |
(72) |
U62(active(X1),X2) |
→ |
U62(X1,X2) |
(74) |
U62(X1,active(X2)) |
→ |
U62(X1,X2) |
(75) |
s(active(X)) |
→ |
s(X) |
(79) |
s(mark(X)) |
→ |
s(X) |
(78) |
U21(active(X)) |
→ |
U21(X) |
(47) |
U21(mark(X)) |
→ |
U21(X) |
(46) |
U41(X1,mark(X2)) |
→ |
U41(X1,X2) |
(51) |
U41(mark(X1),X2) |
→ |
U41(X1,X2) |
(50) |
U41(active(X1),X2) |
→ |
U41(X1,X2) |
(52) |
U41(X1,active(X2)) |
→ |
U41(X1,X2) |
(53) |
U51(X1,mark(X2)) |
→ |
U51(X1,X2) |
(59) |
U51(mark(X1),X2) |
→ |
U51(X1,X2) |
(58) |
U51(active(X1),X2) |
→ |
U51(X1,X2) |
(60) |
U51(X1,active(X2)) |
→ |
U51(X1,X2) |
(61) |
U61(X1,mark(X2),X3) |
→ |
U61(X1,X2,X3) |
(67) |
U61(mark(X1),X2,X3) |
→ |
U61(X1,X2,X3) |
(66) |
U61(X1,X2,mark(X3)) |
→ |
U61(X1,X2,X3) |
(68) |
U61(active(X1),X2,X3) |
→ |
U61(X1,X2,X3) |
(69) |
U61(X1,active(X2),X3) |
→ |
U61(X1,X2,X3) |
(70) |
U61(X1,X2,active(X3)) |
→ |
U61(X1,X2,X3) |
(71) |
1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[zeros] |
= |
0 |
[active(x1)] |
= |
1 · x1
|
[cons(x1, x2)] |
= |
1 · x1 + 2 · x2
|
[0] |
= |
0 |
[U41(x1, x2)] |
= |
1 · x1 + 2 · x2
|
[tt] |
= |
0 |
[U42(x1)] |
= |
1 · x1
|
[isNatIList(x1)] |
= |
2 · x1
|
[U51(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[U52(x1)] |
= |
1 · x1
|
[isNatList(x1)] |
= |
1 · x1
|
[U61(x1, x2, x3)] |
= |
2 · x1 + 2 · x2 + 2 · x3
|
[U62(x1, x2)] |
= |
2 · x1 + 2 · x2
|
[isNat(x1)] |
= |
1 · x1
|
[s(x1)] |
= |
1 · x1
|
[length(x1)] |
= |
2 · x1
|
[U21(x1)] |
= |
1 · x1
|
[U11(x1)] |
= |
1 · x1
|
[U31(x1)] |
= |
2 + 1 · x1
|
[nil] |
= |
0 |
[mark#(x1)] |
= |
1 · x1
|
[active#(x1)] |
= |
1 · x1
|
the
pair
mark#(U31(X)) |
→ |
mark#(X) |
(126) |
and
the
rules
isNatIList(mark(X)) |
→ |
isNatIList(X) |
(56) |
isNatIList(active(X)) |
→ |
isNatIList(X) |
(57) |
isNatList(mark(X)) |
→ |
isNatList(X) |
(64) |
isNatList(active(X)) |
→ |
isNatList(X) |
(65) |
isNat(mark(X)) |
→ |
isNat(X) |
(76) |
isNat(active(X)) |
→ |
isNat(X) |
(77) |
could be deleted.
1.1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[zeros] |
= |
0 |
[active(x1)] |
= |
1 · x1
|
[cons(x1, x2)] |
= |
1 · x1 + 2 · x2
|
[0] |
= |
0 |
[U41(x1, x2)] |
= |
2 · x1 + 2 · x2
|
[tt] |
= |
0 |
[U42(x1)] |
= |
1 · x1
|
[isNatIList(x1)] |
= |
2 · x1
|
[U51(x1, x2)] |
= |
1 · x1 + 2 · x2
|
[U52(x1)] |
= |
2 · x1
|
[isNatList(x1)] |
= |
1 · x1
|
[U61(x1, x2, x3)] |
= |
2 · x1 + 2 · x2 + 2 · x3
|
[U62(x1, x2)] |
= |
2 · x1 + 2 · x2
|
[isNat(x1)] |
= |
1 · x1
|
[s(x1)] |
= |
1 · x1
|
[length(x1)] |
= |
2 · x1
|
[U21(x1)] |
= |
1 · x1
|
[U11(x1)] |
= |
2 + 1 · x1
|
[U31(x1)] |
= |
2 · x1
|
[nil] |
= |
0 |
[mark#(x1)] |
= |
2 · x1
|
[active#(x1)] |
= |
2 · x1
|
the
pair
mark#(U11(X)) |
→ |
mark#(X) |
(119) |
and
no rules
could be deleted.
1.1.1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[zeros] |
= |
0 |
[active(x1)] |
= |
1 · x1
|
[cons(x1, x2)] |
= |
2 · x1 + 2 · x2
|
[0] |
= |
0 |
[U41(x1, x2)] |
= |
1 + 1 · x1 + 2 · x2
|
[tt] |
= |
0 |
[U42(x1)] |
= |
1 · x1
|
[isNatIList(x1)] |
= |
1 + 2 · x1
|
[U51(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[U52(x1)] |
= |
1 · x1
|
[isNatList(x1)] |
= |
1 · x1
|
[U61(x1, x2, x3)] |
= |
2 · x1 + 2 · x2 + 2 · x3
|
[U62(x1, x2)] |
= |
1 · x1 + 2 · x2
|
[isNat(x1)] |
= |
2 · x1
|
[s(x1)] |
= |
1 · x1
|
[length(x1)] |
= |
2 · x1
|
[U21(x1)] |
= |
1 · x1
|
[U11(x1)] |
= |
1 · x1
|
[U31(x1)] |
= |
1 · x1
|
[nil] |
= |
0 |
[mark#(x1)] |
= |
1 · x1
|
[active#(x1)] |
= |
1 · x1
|
the
pair
mark#(U41(X1,X2)) |
→ |
mark#(X1) |
(129) |
and
no rules
could be deleted.
1.1.1.1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[zeros] |
= |
0 |
[active(x1)] |
= |
1 · x1
|
[cons(x1, x2)] |
= |
2 · x1 + 2 · x2
|
[0] |
= |
0 |
[U41(x1, x2)] |
= |
2 · x1 + 1 · x2
|
[tt] |
= |
0 |
[U42(x1)] |
= |
1 · x1
|
[isNatIList(x1)] |
= |
1 · x1
|
[U51(x1, x2)] |
= |
1 · x1 + 2 · x2
|
[U52(x1)] |
= |
1 · x1
|
[isNatList(x1)] |
= |
1 · x1
|
[U61(x1, x2, x3)] |
= |
1 + 2 · x1 + 2 · x2 + 2 · x3
|
[U62(x1, x2)] |
= |
1 + 2 · x1 + 2 · x2
|
[isNat(x1)] |
= |
1 · x1
|
[s(x1)] |
= |
1 · x1
|
[length(x1)] |
= |
1 + 2 · x1
|
[U21(x1)] |
= |
1 · x1
|
[U11(x1)] |
= |
2 · x1
|
[U31(x1)] |
= |
1 · x1
|
[nil] |
= |
0 |
[mark#(x1)] |
= |
2 · x1
|
[active#(x1)] |
= |
2 · x1
|
the
pairs
mark#(U61(X1,X2,X3)) |
→ |
mark#(X1) |
(143) |
mark#(U62(X1,X2)) |
→ |
mark#(X1) |
(146) |
mark#(length(X)) |
→ |
mark#(X) |
(153) |
and
no rules
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)] |
= |
-1 + x1
|
[U11(x1)] |
= |
-2 |
[U21(x1)] |
= |
0 |
[U31(x1)] |
= |
-2 |
[U41(x1, x2)] |
= |
2 |
[U42(x1)] |
= |
-2 |
[U51(x1, x2)] |
= |
2 |
[U52(x1)] |
= |
0 |
[U61(x1, x2, x3)] |
= |
2 |
[U62(x1, x2)] |
= |
2 |
[cons(x1, x2)] |
= |
-2 |
[length(x1)] |
= |
2 |
[s(x1)] |
= |
-2 |
[mark(x1)] |
= |
-2 |
[zeros] |
= |
2 |
[active(x1)] |
= |
2 + 2 · x1
|
[0] |
= |
2 |
[tt] |
= |
0 |
[isNatIList(x1)] |
= |
2 |
[isNatList(x1)] |
= |
2 |
[isNat(x1)] |
= |
2 |
[nil] |
= |
0 |
[mark#(x1)] |
= |
1 |
together with the usable
rules
cons(X1,mark(X2)) |
→ |
cons(X1,X2) |
(41) |
cons(mark(X1),X2) |
→ |
cons(X1,X2) |
(40) |
cons(active(X1),X2) |
→ |
cons(X1,X2) |
(42) |
cons(X1,active(X2)) |
→ |
cons(X1,X2) |
(43) |
U42(active(X)) |
→ |
U42(X) |
(55) |
U42(mark(X)) |
→ |
U42(X) |
(54) |
U52(active(X)) |
→ |
U52(X) |
(63) |
U52(mark(X)) |
→ |
U52(X) |
(62) |
U62(X1,mark(X2)) |
→ |
U62(X1,X2) |
(73) |
U62(mark(X1),X2) |
→ |
U62(X1,X2) |
(72) |
U62(active(X1),X2) |
→ |
U62(X1,X2) |
(74) |
U62(X1,active(X2)) |
→ |
U62(X1,X2) |
(75) |
s(active(X)) |
→ |
s(X) |
(79) |
s(mark(X)) |
→ |
s(X) |
(78) |
U61(X1,mark(X2),X3) |
→ |
U61(X1,X2,X3) |
(67) |
U61(mark(X1),X2,X3) |
→ |
U61(X1,X2,X3) |
(66) |
U61(X1,X2,mark(X3)) |
→ |
U61(X1,X2,X3) |
(68) |
U61(active(X1),X2,X3) |
→ |
U61(X1,X2,X3) |
(69) |
U61(X1,active(X2),X3) |
→ |
U61(X1,X2,X3) |
(70) |
U61(X1,X2,active(X3)) |
→ |
U61(X1,X2,X3) |
(71) |
U11(active(X)) |
→ |
U11(X) |
(45) |
U11(mark(X)) |
→ |
U11(X) |
(44) |
U21(active(X)) |
→ |
U21(X) |
(47) |
U21(mark(X)) |
→ |
U21(X) |
(46) |
U31(active(X)) |
→ |
U31(X) |
(49) |
U31(mark(X)) |
→ |
U31(X) |
(48) |
U41(X1,mark(X2)) |
→ |
U41(X1,X2) |
(51) |
U41(mark(X1),X2) |
→ |
U41(X1,X2) |
(50) |
U41(active(X1),X2) |
→ |
U41(X1,X2) |
(52) |
U41(X1,active(X2)) |
→ |
U41(X1,X2) |
(53) |
U51(X1,mark(X2)) |
→ |
U51(X1,X2) |
(59) |
U51(mark(X1),X2) |
→ |
U51(X1,X2) |
(58) |
U51(active(X1),X2) |
→ |
U51(X1,X2) |
(60) |
U51(X1,active(X2)) |
→ |
U51(X1,X2) |
(61) |
length(active(X)) |
→ |
length(X) |
(81) |
length(mark(X)) |
→ |
length(X) |
(80) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
mark#(cons(X1,X2)) |
→ |
active#(cons(mark(X1),X2)) |
(113) |
mark#(U42(X)) |
→ |
active#(U42(mark(X))) |
(130) |
mark#(U52(X)) |
→ |
active#(U52(mark(X))) |
(137) |
mark#(s(X)) |
→ |
active#(s(mark(X))) |
(148) |
mark#(U11(X)) |
→ |
active#(U11(mark(X))) |
(117) |
mark#(U21(X)) |
→ |
active#(U21(mark(X))) |
(121) |
mark#(U31(X)) |
→ |
active#(U31(mark(X))) |
(124) |
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)] |
= |
-2 |
[U41(x1, x2)] |
= |
-2 |
[U51(x1, x2)] |
= |
1 + 2 · x1
|
[U61(x1, x2, x3)] |
= |
-2 |
[U62(x1, x2)] |
= |
-2 |
[length(x1)] |
= |
0 |
[mark(x1)] |
= |
-2 |
[zeros] |
= |
2 |
[active(x1)] |
= |
-2 |
[cons(x1, x2)] |
= |
1 + 2 · x1
|
[0] |
= |
0 |
[tt] |
= |
0 |
[U42(x1)] |
= |
1 + 2 · x1
|
[isNatIList(x1)] |
= |
0 |
[U52(x1)] |
= |
1 + 2 · x1
|
[isNatList(x1)] |
= |
0 |
[isNat(x1)] |
= |
0 |
[s(x1)] |
= |
1 + 2 · x1
|
[U21(x1)] |
= |
1 + 2 · x1
|
[U11(x1)] |
= |
-2 |
[U31(x1)] |
= |
-2 + 2 · x1
|
[nil] |
= |
0 |
[mark#(x1)] |
= |
-1 + x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
pair
mark#(zeros) |
→ |
active#(zeros) |
(112) |
could be deleted.
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
2nd
component contains the
pair
cons#(X1,mark(X2)) |
→ |
cons#(X1,X2) |
(156) |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(155) |
cons#(active(X1),X2) |
→ |
cons#(X1,X2) |
(157) |
cons#(X1,active(X2)) |
→ |
cons#(X1,X2) |
(158) |
1.1.1.1.1.1.2 Usable Rules Processor
We restrict the rewrite rules to the following usable rules of the DP problem.
There are no rules.
1.1.1.1.1.1.2.1 Innermost Lhss Removal Processor
We restrict the innermost strategy to the following left hand sides.
active(zeros) |
active(U11(tt)) |
active(U21(tt)) |
active(U31(tt)) |
active(U41(tt,x0)) |
active(U42(tt)) |
active(U51(tt,x0)) |
active(U52(tt)) |
active(U61(tt,x0,x1)) |
active(U62(tt,x0)) |
active(isNat(0)) |
active(isNat(length(x0))) |
active(isNat(s(x0))) |
active(isNatIList(x0)) |
active(isNatList(nil)) |
active(isNatList(cons(x0,x1))) |
active(length(nil)) |
active(length(cons(x0,x1))) |
mark(zeros) |
mark(cons(x0,x1)) |
mark(0) |
mark(U11(x0)) |
mark(tt) |
mark(U21(x0)) |
mark(U31(x0)) |
mark(U41(x0,x1)) |
mark(U42(x0)) |
mark(isNatIList(x0)) |
mark(U51(x0,x1)) |
mark(U52(x0)) |
mark(isNatList(x0)) |
mark(U61(x0,x1,x2)) |
mark(U62(x0,x1)) |
mark(isNat(x0)) |
mark(s(x0)) |
mark(length(x0)) |
mark(nil) |
1.1.1.1.1.1.2.1.1 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) |
(156) |
|
1 |
≥ |
1 |
2 |
> |
2 |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(155) |
|
1 |
> |
1 |
2 |
≥ |
2 |
cons#(active(X1),X2) |
→ |
cons#(X1,X2) |
(157) |
|
1 |
> |
1 |
2 |
≥ |
2 |
cons#(X1,active(X2)) |
→ |
cons#(X1,X2) |
(158) |
|
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
U11#(active(X)) |
→ |
U11#(X) |
(160) |
U11#(mark(X)) |
→ |
U11#(X) |
(159) |
1.1.1.1.1.1.3 Usable Rules Processor
We restrict the rewrite rules to the following usable rules of the DP problem.
There are no rules.
1.1.1.1.1.1.3.1 Innermost Lhss Removal Processor
We restrict the innermost strategy to the following left hand sides.
active(zeros) |
active(U11(tt)) |
active(U21(tt)) |
active(U31(tt)) |
active(U41(tt,x0)) |
active(U42(tt)) |
active(U51(tt,x0)) |
active(U52(tt)) |
active(U61(tt,x0,x1)) |
active(U62(tt,x0)) |
active(isNat(0)) |
active(isNat(length(x0))) |
active(isNat(s(x0))) |
active(isNatIList(x0)) |
active(isNatList(nil)) |
active(isNatList(cons(x0,x1))) |
active(length(nil)) |
active(length(cons(x0,x1))) |
mark(zeros) |
mark(cons(x0,x1)) |
mark(0) |
mark(U11(x0)) |
mark(tt) |
mark(U21(x0)) |
mark(U31(x0)) |
mark(U41(x0,x1)) |
mark(U42(x0)) |
mark(isNatIList(x0)) |
mark(U51(x0,x1)) |
mark(U52(x0)) |
mark(isNatList(x0)) |
mark(U61(x0,x1,x2)) |
mark(U62(x0,x1)) |
mark(isNat(x0)) |
mark(s(x0)) |
mark(length(x0)) |
mark(nil) |
1.1.1.1.1.1.3.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
U11#(active(X)) |
→ |
U11#(X) |
(160) |
|
1 |
> |
1 |
U11#(mark(X)) |
→ |
U11#(X) |
(159) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
4th
component contains the
pair
U21#(active(X)) |
→ |
U21#(X) |
(162) |
U21#(mark(X)) |
→ |
U21#(X) |
(161) |
1.1.1.1.1.1.4 Usable Rules Processor
We restrict the rewrite rules to the following usable rules of the DP problem.
There are no rules.
1.1.1.1.1.1.4.1 Innermost Lhss Removal Processor
We restrict the innermost strategy to the following left hand sides.
active(zeros) |
active(U11(tt)) |
active(U21(tt)) |
active(U31(tt)) |
active(U41(tt,x0)) |
active(U42(tt)) |
active(U51(tt,x0)) |
active(U52(tt)) |
active(U61(tt,x0,x1)) |
active(U62(tt,x0)) |
active(isNat(0)) |
active(isNat(length(x0))) |
active(isNat(s(x0))) |
active(isNatIList(x0)) |
active(isNatList(nil)) |
active(isNatList(cons(x0,x1))) |
active(length(nil)) |
active(length(cons(x0,x1))) |
mark(zeros) |
mark(cons(x0,x1)) |
mark(0) |
mark(U11(x0)) |
mark(tt) |
mark(U21(x0)) |
mark(U31(x0)) |
mark(U41(x0,x1)) |
mark(U42(x0)) |
mark(isNatIList(x0)) |
mark(U51(x0,x1)) |
mark(U52(x0)) |
mark(isNatList(x0)) |
mark(U61(x0,x1,x2)) |
mark(U62(x0,x1)) |
mark(isNat(x0)) |
mark(s(x0)) |
mark(length(x0)) |
mark(nil) |
1.1.1.1.1.1.4.1.1 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) |
(162) |
|
1 |
> |
1 |
U21#(mark(X)) |
→ |
U21#(X) |
(161) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
5th
component contains the
pair
U31#(active(X)) |
→ |
U31#(X) |
(164) |
U31#(mark(X)) |
→ |
U31#(X) |
(163) |
1.1.1.1.1.1.5 Usable Rules Processor
We restrict the rewrite rules to the following usable rules of the DP problem.
There are no rules.
1.1.1.1.1.1.5.1 Innermost Lhss Removal Processor
We restrict the innermost strategy to the following left hand sides.
active(zeros) |
active(U11(tt)) |
active(U21(tt)) |
active(U31(tt)) |
active(U41(tt,x0)) |
active(U42(tt)) |
active(U51(tt,x0)) |
active(U52(tt)) |
active(U61(tt,x0,x1)) |
active(U62(tt,x0)) |
active(isNat(0)) |
active(isNat(length(x0))) |
active(isNat(s(x0))) |
active(isNatIList(x0)) |
active(isNatList(nil)) |
active(isNatList(cons(x0,x1))) |
active(length(nil)) |
active(length(cons(x0,x1))) |
mark(zeros) |
mark(cons(x0,x1)) |
mark(0) |
mark(U11(x0)) |
mark(tt) |
mark(U21(x0)) |
mark(U31(x0)) |
mark(U41(x0,x1)) |
mark(U42(x0)) |
mark(isNatIList(x0)) |
mark(U51(x0,x1)) |
mark(U52(x0)) |
mark(isNatList(x0)) |
mark(U61(x0,x1,x2)) |
mark(U62(x0,x1)) |
mark(isNat(x0)) |
mark(s(x0)) |
mark(length(x0)) |
mark(nil) |
1.1.1.1.1.1.5.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
U31#(active(X)) |
→ |
U31#(X) |
(164) |
|
1 |
> |
1 |
U31#(mark(X)) |
→ |
U31#(X) |
(163) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
6th
component contains the
pair
U41#(X1,mark(X2)) |
→ |
U41#(X1,X2) |
(166) |
U41#(mark(X1),X2) |
→ |
U41#(X1,X2) |
(165) |
U41#(active(X1),X2) |
→ |
U41#(X1,X2) |
(167) |
U41#(X1,active(X2)) |
→ |
U41#(X1,X2) |
(168) |
1.1.1.1.1.1.6 Usable Rules Processor
We restrict the rewrite rules to the following usable rules of the DP problem.
There are no rules.
1.1.1.1.1.1.6.1 Innermost Lhss Removal Processor
We restrict the innermost strategy to the following left hand sides.
active(zeros) |
active(U11(tt)) |
active(U21(tt)) |
active(U31(tt)) |
active(U41(tt,x0)) |
active(U42(tt)) |
active(U51(tt,x0)) |
active(U52(tt)) |
active(U61(tt,x0,x1)) |
active(U62(tt,x0)) |
active(isNat(0)) |
active(isNat(length(x0))) |
active(isNat(s(x0))) |
active(isNatIList(x0)) |
active(isNatList(nil)) |
active(isNatList(cons(x0,x1))) |
active(length(nil)) |
active(length(cons(x0,x1))) |
mark(zeros) |
mark(cons(x0,x1)) |
mark(0) |
mark(U11(x0)) |
mark(tt) |
mark(U21(x0)) |
mark(U31(x0)) |
mark(U41(x0,x1)) |
mark(U42(x0)) |
mark(isNatIList(x0)) |
mark(U51(x0,x1)) |
mark(U52(x0)) |
mark(isNatList(x0)) |
mark(U61(x0,x1,x2)) |
mark(U62(x0,x1)) |
mark(isNat(x0)) |
mark(s(x0)) |
mark(length(x0)) |
mark(nil) |
1.1.1.1.1.1.6.1.1 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)) |
→ |
U41#(X1,X2) |
(166) |
|
1 |
≥ |
1 |
2 |
> |
2 |
U41#(mark(X1),X2) |
→ |
U41#(X1,X2) |
(165) |
|
1 |
> |
1 |
2 |
≥ |
2 |
U41#(active(X1),X2) |
→ |
U41#(X1,X2) |
(167) |
|
1 |
> |
1 |
2 |
≥ |
2 |
U41#(X1,active(X2)) |
→ |
U41#(X1,X2) |
(168) |
|
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
U42#(active(X)) |
→ |
U42#(X) |
(170) |
U42#(mark(X)) |
→ |
U42#(X) |
(169) |
1.1.1.1.1.1.7 Usable Rules Processor
We restrict the rewrite rules to the following usable rules of the DP problem.
There are no rules.
1.1.1.1.1.1.7.1 Innermost Lhss Removal Processor
We restrict the innermost strategy to the following left hand sides.
active(zeros) |
active(U11(tt)) |
active(U21(tt)) |
active(U31(tt)) |
active(U41(tt,x0)) |
active(U42(tt)) |
active(U51(tt,x0)) |
active(U52(tt)) |
active(U61(tt,x0,x1)) |
active(U62(tt,x0)) |
active(isNat(0)) |
active(isNat(length(x0))) |
active(isNat(s(x0))) |
active(isNatIList(x0)) |
active(isNatList(nil)) |
active(isNatList(cons(x0,x1))) |
active(length(nil)) |
active(length(cons(x0,x1))) |
mark(zeros) |
mark(cons(x0,x1)) |
mark(0) |
mark(U11(x0)) |
mark(tt) |
mark(U21(x0)) |
mark(U31(x0)) |
mark(U41(x0,x1)) |
mark(U42(x0)) |
mark(isNatIList(x0)) |
mark(U51(x0,x1)) |
mark(U52(x0)) |
mark(isNatList(x0)) |
mark(U61(x0,x1,x2)) |
mark(U62(x0,x1)) |
mark(isNat(x0)) |
mark(s(x0)) |
mark(length(x0)) |
mark(nil) |
1.1.1.1.1.1.7.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
U42#(active(X)) |
→ |
U42#(X) |
(170) |
|
1 |
> |
1 |
U42#(mark(X)) |
→ |
U42#(X) |
(169) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
8th
component contains the
pair
isNatIList#(active(X)) |
→ |
isNatIList#(X) |
(172) |
isNatIList#(mark(X)) |
→ |
isNatIList#(X) |
(171) |
1.1.1.1.1.1.8 Usable Rules Processor
We restrict the rewrite rules to the following usable rules of the DP problem.
There are no rules.
1.1.1.1.1.1.8.1 Innermost Lhss Removal Processor
We restrict the innermost strategy to the following left hand sides.
active(zeros) |
active(U11(tt)) |
active(U21(tt)) |
active(U31(tt)) |
active(U41(tt,x0)) |
active(U42(tt)) |
active(U51(tt,x0)) |
active(U52(tt)) |
active(U61(tt,x0,x1)) |
active(U62(tt,x0)) |
active(isNat(0)) |
active(isNat(length(x0))) |
active(isNat(s(x0))) |
active(isNatIList(x0)) |
active(isNatList(nil)) |
active(isNatList(cons(x0,x1))) |
active(length(nil)) |
active(length(cons(x0,x1))) |
mark(zeros) |
mark(cons(x0,x1)) |
mark(0) |
mark(U11(x0)) |
mark(tt) |
mark(U21(x0)) |
mark(U31(x0)) |
mark(U41(x0,x1)) |
mark(U42(x0)) |
mark(isNatIList(x0)) |
mark(U51(x0,x1)) |
mark(U52(x0)) |
mark(isNatList(x0)) |
mark(U61(x0,x1,x2)) |
mark(U62(x0,x1)) |
mark(isNat(x0)) |
mark(s(x0)) |
mark(length(x0)) |
mark(nil) |
1.1.1.1.1.1.8.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
isNatIList#(active(X)) |
→ |
isNatIList#(X) |
(172) |
|
1 |
> |
1 |
isNatIList#(mark(X)) |
→ |
isNatIList#(X) |
(171) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
9th
component contains the
pair
U51#(X1,mark(X2)) |
→ |
U51#(X1,X2) |
(174) |
U51#(mark(X1),X2) |
→ |
U51#(X1,X2) |
(173) |
U51#(active(X1),X2) |
→ |
U51#(X1,X2) |
(175) |
U51#(X1,active(X2)) |
→ |
U51#(X1,X2) |
(176) |
1.1.1.1.1.1.9 Usable Rules Processor
We restrict the rewrite rules to the following usable rules of the DP problem.
There are no rules.
1.1.1.1.1.1.9.1 Innermost Lhss Removal Processor
We restrict the innermost strategy to the following left hand sides.
active(zeros) |
active(U11(tt)) |
active(U21(tt)) |
active(U31(tt)) |
active(U41(tt,x0)) |
active(U42(tt)) |
active(U51(tt,x0)) |
active(U52(tt)) |
active(U61(tt,x0,x1)) |
active(U62(tt,x0)) |
active(isNat(0)) |
active(isNat(length(x0))) |
active(isNat(s(x0))) |
active(isNatIList(x0)) |
active(isNatList(nil)) |
active(isNatList(cons(x0,x1))) |
active(length(nil)) |
active(length(cons(x0,x1))) |
mark(zeros) |
mark(cons(x0,x1)) |
mark(0) |
mark(U11(x0)) |
mark(tt) |
mark(U21(x0)) |
mark(U31(x0)) |
mark(U41(x0,x1)) |
mark(U42(x0)) |
mark(isNatIList(x0)) |
mark(U51(x0,x1)) |
mark(U52(x0)) |
mark(isNatList(x0)) |
mark(U61(x0,x1,x2)) |
mark(U62(x0,x1)) |
mark(isNat(x0)) |
mark(s(x0)) |
mark(length(x0)) |
mark(nil) |
1.1.1.1.1.1.9.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
U51#(X1,mark(X2)) |
→ |
U51#(X1,X2) |
(174) |
|
1 |
≥ |
1 |
2 |
> |
2 |
U51#(mark(X1),X2) |
→ |
U51#(X1,X2) |
(173) |
|
1 |
> |
1 |
2 |
≥ |
2 |
U51#(active(X1),X2) |
→ |
U51#(X1,X2) |
(175) |
|
1 |
> |
1 |
2 |
≥ |
2 |
U51#(X1,active(X2)) |
→ |
U51#(X1,X2) |
(176) |
|
1 |
≥ |
1 |
2 |
> |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
10th
component contains the
pair
U52#(active(X)) |
→ |
U52#(X) |
(178) |
U52#(mark(X)) |
→ |
U52#(X) |
(177) |
1.1.1.1.1.1.10 Usable Rules Processor
We restrict the rewrite rules to the following usable rules of the DP problem.
There are no rules.
1.1.1.1.1.1.10.1 Innermost Lhss Removal Processor
We restrict the innermost strategy to the following left hand sides.
active(zeros) |
active(U11(tt)) |
active(U21(tt)) |
active(U31(tt)) |
active(U41(tt,x0)) |
active(U42(tt)) |
active(U51(tt,x0)) |
active(U52(tt)) |
active(U61(tt,x0,x1)) |
active(U62(tt,x0)) |
active(isNat(0)) |
active(isNat(length(x0))) |
active(isNat(s(x0))) |
active(isNatIList(x0)) |
active(isNatList(nil)) |
active(isNatList(cons(x0,x1))) |
active(length(nil)) |
active(length(cons(x0,x1))) |
mark(zeros) |
mark(cons(x0,x1)) |
mark(0) |
mark(U11(x0)) |
mark(tt) |
mark(U21(x0)) |
mark(U31(x0)) |
mark(U41(x0,x1)) |
mark(U42(x0)) |
mark(isNatIList(x0)) |
mark(U51(x0,x1)) |
mark(U52(x0)) |
mark(isNatList(x0)) |
mark(U61(x0,x1,x2)) |
mark(U62(x0,x1)) |
mark(isNat(x0)) |
mark(s(x0)) |
mark(length(x0)) |
mark(nil) |
1.1.1.1.1.1.10.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
U52#(active(X)) |
→ |
U52#(X) |
(178) |
|
1 |
> |
1 |
U52#(mark(X)) |
→ |
U52#(X) |
(177) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
11th
component contains the
pair
isNatList#(active(X)) |
→ |
isNatList#(X) |
(180) |
isNatList#(mark(X)) |
→ |
isNatList#(X) |
(179) |
1.1.1.1.1.1.11 Usable Rules Processor
We restrict the rewrite rules to the following usable rules of the DP problem.
There are no rules.
1.1.1.1.1.1.11.1 Innermost Lhss Removal Processor
We restrict the innermost strategy to the following left hand sides.
active(zeros) |
active(U11(tt)) |
active(U21(tt)) |
active(U31(tt)) |
active(U41(tt,x0)) |
active(U42(tt)) |
active(U51(tt,x0)) |
active(U52(tt)) |
active(U61(tt,x0,x1)) |
active(U62(tt,x0)) |
active(isNat(0)) |
active(isNat(length(x0))) |
active(isNat(s(x0))) |
active(isNatIList(x0)) |
active(isNatList(nil)) |
active(isNatList(cons(x0,x1))) |
active(length(nil)) |
active(length(cons(x0,x1))) |
mark(zeros) |
mark(cons(x0,x1)) |
mark(0) |
mark(U11(x0)) |
mark(tt) |
mark(U21(x0)) |
mark(U31(x0)) |
mark(U41(x0,x1)) |
mark(U42(x0)) |
mark(isNatIList(x0)) |
mark(U51(x0,x1)) |
mark(U52(x0)) |
mark(isNatList(x0)) |
mark(U61(x0,x1,x2)) |
mark(U62(x0,x1)) |
mark(isNat(x0)) |
mark(s(x0)) |
mark(length(x0)) |
mark(nil) |
1.1.1.1.1.1.11.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
isNatList#(active(X)) |
→ |
isNatList#(X) |
(180) |
|
1 |
> |
1 |
isNatList#(mark(X)) |
→ |
isNatList#(X) |
(179) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
12th
component contains the
pair
U61#(X1,mark(X2),X3) |
→ |
U61#(X1,X2,X3) |
(182) |
U61#(mark(X1),X2,X3) |
→ |
U61#(X1,X2,X3) |
(181) |
U61#(X1,X2,mark(X3)) |
→ |
U61#(X1,X2,X3) |
(183) |
U61#(active(X1),X2,X3) |
→ |
U61#(X1,X2,X3) |
(184) |
U61#(X1,active(X2),X3) |
→ |
U61#(X1,X2,X3) |
(185) |
U61#(X1,X2,active(X3)) |
→ |
U61#(X1,X2,X3) |
(186) |
1.1.1.1.1.1.12 Usable Rules Processor
We restrict the rewrite rules to the following usable rules of the DP problem.
There are no rules.
1.1.1.1.1.1.12.1 Innermost Lhss Removal Processor
We restrict the innermost strategy to the following left hand sides.
active(zeros) |
active(U11(tt)) |
active(U21(tt)) |
active(U31(tt)) |
active(U41(tt,x0)) |
active(U42(tt)) |
active(U51(tt,x0)) |
active(U52(tt)) |
active(U61(tt,x0,x1)) |
active(U62(tt,x0)) |
active(isNat(0)) |
active(isNat(length(x0))) |
active(isNat(s(x0))) |
active(isNatIList(x0)) |
active(isNatList(nil)) |
active(isNatList(cons(x0,x1))) |
active(length(nil)) |
active(length(cons(x0,x1))) |
mark(zeros) |
mark(cons(x0,x1)) |
mark(0) |
mark(U11(x0)) |
mark(tt) |
mark(U21(x0)) |
mark(U31(x0)) |
mark(U41(x0,x1)) |
mark(U42(x0)) |
mark(isNatIList(x0)) |
mark(U51(x0,x1)) |
mark(U52(x0)) |
mark(isNatList(x0)) |
mark(U61(x0,x1,x2)) |
mark(U62(x0,x1)) |
mark(isNat(x0)) |
mark(s(x0)) |
mark(length(x0)) |
mark(nil) |
1.1.1.1.1.1.12.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
U61#(X1,mark(X2),X3) |
→ |
U61#(X1,X2,X3) |
(182) |
|
1 |
≥ |
1 |
2 |
> |
2 |
3 |
≥ |
3 |
U61#(mark(X1),X2,X3) |
→ |
U61#(X1,X2,X3) |
(181) |
|
1 |
> |
1 |
2 |
≥ |
2 |
3 |
≥ |
3 |
U61#(X1,X2,mark(X3)) |
→ |
U61#(X1,X2,X3) |
(183) |
|
1 |
≥ |
1 |
2 |
≥ |
2 |
3 |
> |
3 |
U61#(active(X1),X2,X3) |
→ |
U61#(X1,X2,X3) |
(184) |
|
1 |
> |
1 |
2 |
≥ |
2 |
3 |
≥ |
3 |
U61#(X1,active(X2),X3) |
→ |
U61#(X1,X2,X3) |
(185) |
|
1 |
≥ |
1 |
2 |
> |
2 |
3 |
≥ |
3 |
U61#(X1,X2,active(X3)) |
→ |
U61#(X1,X2,X3) |
(186) |
|
1 |
≥ |
1 |
2 |
≥ |
2 |
3 |
> |
3 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
13th
component contains the
pair
U62#(X1,mark(X2)) |
→ |
U62#(X1,X2) |
(188) |
U62#(mark(X1),X2) |
→ |
U62#(X1,X2) |
(187) |
U62#(active(X1),X2) |
→ |
U62#(X1,X2) |
(189) |
U62#(X1,active(X2)) |
→ |
U62#(X1,X2) |
(190) |
1.1.1.1.1.1.13 Usable Rules Processor
We restrict the rewrite rules to the following usable rules of the DP problem.
There are no rules.
1.1.1.1.1.1.13.1 Innermost Lhss Removal Processor
We restrict the innermost strategy to the following left hand sides.
active(zeros) |
active(U11(tt)) |
active(U21(tt)) |
active(U31(tt)) |
active(U41(tt,x0)) |
active(U42(tt)) |
active(U51(tt,x0)) |
active(U52(tt)) |
active(U61(tt,x0,x1)) |
active(U62(tt,x0)) |
active(isNat(0)) |
active(isNat(length(x0))) |
active(isNat(s(x0))) |
active(isNatIList(x0)) |
active(isNatList(nil)) |
active(isNatList(cons(x0,x1))) |
active(length(nil)) |
active(length(cons(x0,x1))) |
mark(zeros) |
mark(cons(x0,x1)) |
mark(0) |
mark(U11(x0)) |
mark(tt) |
mark(U21(x0)) |
mark(U31(x0)) |
mark(U41(x0,x1)) |
mark(U42(x0)) |
mark(isNatIList(x0)) |
mark(U51(x0,x1)) |
mark(U52(x0)) |
mark(isNatList(x0)) |
mark(U61(x0,x1,x2)) |
mark(U62(x0,x1)) |
mark(isNat(x0)) |
mark(s(x0)) |
mark(length(x0)) |
mark(nil) |
1.1.1.1.1.1.13.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
U62#(X1,mark(X2)) |
→ |
U62#(X1,X2) |
(188) |
|
1 |
≥ |
1 |
2 |
> |
2 |
U62#(mark(X1),X2) |
→ |
U62#(X1,X2) |
(187) |
|
1 |
> |
1 |
2 |
≥ |
2 |
U62#(active(X1),X2) |
→ |
U62#(X1,X2) |
(189) |
|
1 |
> |
1 |
2 |
≥ |
2 |
U62#(X1,active(X2)) |
→ |
U62#(X1,X2) |
(190) |
|
1 |
≥ |
1 |
2 |
> |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
14th
component contains the
pair
isNat#(active(X)) |
→ |
isNat#(X) |
(192) |
isNat#(mark(X)) |
→ |
isNat#(X) |
(191) |
1.1.1.1.1.1.14 Usable Rules Processor
We restrict the rewrite rules to the following usable rules of the DP problem.
There are no rules.
1.1.1.1.1.1.14.1 Innermost Lhss Removal Processor
We restrict the innermost strategy to the following left hand sides.
active(zeros) |
active(U11(tt)) |
active(U21(tt)) |
active(U31(tt)) |
active(U41(tt,x0)) |
active(U42(tt)) |
active(U51(tt,x0)) |
active(U52(tt)) |
active(U61(tt,x0,x1)) |
active(U62(tt,x0)) |
active(isNat(0)) |
active(isNat(length(x0))) |
active(isNat(s(x0))) |
active(isNatIList(x0)) |
active(isNatList(nil)) |
active(isNatList(cons(x0,x1))) |
active(length(nil)) |
active(length(cons(x0,x1))) |
mark(zeros) |
mark(cons(x0,x1)) |
mark(0) |
mark(U11(x0)) |
mark(tt) |
mark(U21(x0)) |
mark(U31(x0)) |
mark(U41(x0,x1)) |
mark(U42(x0)) |
mark(isNatIList(x0)) |
mark(U51(x0,x1)) |
mark(U52(x0)) |
mark(isNatList(x0)) |
mark(U61(x0,x1,x2)) |
mark(U62(x0,x1)) |
mark(isNat(x0)) |
mark(s(x0)) |
mark(length(x0)) |
mark(nil) |
1.1.1.1.1.1.14.1.1 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) |
(192) |
|
1 |
> |
1 |
isNat#(mark(X)) |
→ |
isNat#(X) |
(191) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
15th
component contains the
pair
s#(active(X)) |
→ |
s#(X) |
(194) |
s#(mark(X)) |
→ |
s#(X) |
(193) |
1.1.1.1.1.1.15 Usable Rules Processor
We restrict the rewrite rules to the following usable rules of the DP problem.
There are no rules.
1.1.1.1.1.1.15.1 Innermost Lhss Removal Processor
We restrict the innermost strategy to the following left hand sides.
active(zeros) |
active(U11(tt)) |
active(U21(tt)) |
active(U31(tt)) |
active(U41(tt,x0)) |
active(U42(tt)) |
active(U51(tt,x0)) |
active(U52(tt)) |
active(U61(tt,x0,x1)) |
active(U62(tt,x0)) |
active(isNat(0)) |
active(isNat(length(x0))) |
active(isNat(s(x0))) |
active(isNatIList(x0)) |
active(isNatList(nil)) |
active(isNatList(cons(x0,x1))) |
active(length(nil)) |
active(length(cons(x0,x1))) |
mark(zeros) |
mark(cons(x0,x1)) |
mark(0) |
mark(U11(x0)) |
mark(tt) |
mark(U21(x0)) |
mark(U31(x0)) |
mark(U41(x0,x1)) |
mark(U42(x0)) |
mark(isNatIList(x0)) |
mark(U51(x0,x1)) |
mark(U52(x0)) |
mark(isNatList(x0)) |
mark(U61(x0,x1,x2)) |
mark(U62(x0,x1)) |
mark(isNat(x0)) |
mark(s(x0)) |
mark(length(x0)) |
mark(nil) |
1.1.1.1.1.1.15.1.1 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) |
(194) |
|
1 |
> |
1 |
s#(mark(X)) |
→ |
s#(X) |
(193) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
16th
component contains the
pair
length#(active(X)) |
→ |
length#(X) |
(196) |
length#(mark(X)) |
→ |
length#(X) |
(195) |
1.1.1.1.1.1.16 Usable Rules Processor
We restrict the rewrite rules to the following usable rules of the DP problem.
There are no rules.
1.1.1.1.1.1.16.1 Innermost Lhss Removal Processor
We restrict the innermost strategy to the following left hand sides.
active(zeros) |
active(U11(tt)) |
active(U21(tt)) |
active(U31(tt)) |
active(U41(tt,x0)) |
active(U42(tt)) |
active(U51(tt,x0)) |
active(U52(tt)) |
active(U61(tt,x0,x1)) |
active(U62(tt,x0)) |
active(isNat(0)) |
active(isNat(length(x0))) |
active(isNat(s(x0))) |
active(isNatIList(x0)) |
active(isNatList(nil)) |
active(isNatList(cons(x0,x1))) |
active(length(nil)) |
active(length(cons(x0,x1))) |
mark(zeros) |
mark(cons(x0,x1)) |
mark(0) |
mark(U11(x0)) |
mark(tt) |
mark(U21(x0)) |
mark(U31(x0)) |
mark(U41(x0,x1)) |
mark(U42(x0)) |
mark(isNatIList(x0)) |
mark(U51(x0,x1)) |
mark(U52(x0)) |
mark(isNatList(x0)) |
mark(U61(x0,x1,x2)) |
mark(U62(x0,x1)) |
mark(isNat(x0)) |
mark(s(x0)) |
mark(length(x0)) |
mark(nil) |
1.1.1.1.1.1.16.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
length#(active(X)) |
→ |
length#(X) |
(196) |
|
1 |
> |
1 |
length#(mark(X)) |
→ |
length#(X) |
(195) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.