The rewrite relation of the following TRS is considered.
active(zeros) |
→ |
mark(cons(0,zeros)) |
(1) |
active(U11(tt,L)) |
→ |
mark(s(length(L))) |
(2) |
active(and(tt,X)) |
→ |
mark(X) |
(3) |
active(isNat(0)) |
→ |
mark(tt) |
(4) |
active(isNat(length(V1))) |
→ |
mark(isNatList(V1)) |
(5) |
active(isNat(s(V1))) |
→ |
mark(isNat(V1)) |
(6) |
active(isNatIList(V)) |
→ |
mark(isNatList(V)) |
(7) |
active(isNatIList(zeros)) |
→ |
mark(tt) |
(8) |
active(isNatIList(cons(V1,V2))) |
→ |
mark(and(isNat(V1),isNatIList(V2))) |
(9) |
active(isNatList(nil)) |
→ |
mark(tt) |
(10) |
active(isNatList(cons(V1,V2))) |
→ |
mark(and(isNat(V1),isNatList(V2))) |
(11) |
active(length(nil)) |
→ |
mark(0) |
(12) |
active(length(cons(N,L))) |
→ |
mark(U11(and(isNatList(L),isNat(N)),L)) |
(13) |
mark(zeros) |
→ |
active(zeros) |
(14) |
mark(cons(X1,X2)) |
→ |
active(cons(mark(X1),X2)) |
(15) |
mark(0) |
→ |
active(0) |
(16) |
mark(U11(X1,X2)) |
→ |
active(U11(mark(X1),X2)) |
(17) |
mark(tt) |
→ |
active(tt) |
(18) |
mark(s(X)) |
→ |
active(s(mark(X))) |
(19) |
mark(length(X)) |
→ |
active(length(mark(X))) |
(20) |
mark(and(X1,X2)) |
→ |
active(and(mark(X1),X2)) |
(21) |
mark(isNat(X)) |
→ |
active(isNat(X)) |
(22) |
mark(isNatList(X)) |
→ |
active(isNatList(X)) |
(23) |
mark(isNatIList(X)) |
→ |
active(isNatIList(X)) |
(24) |
mark(nil) |
→ |
active(nil) |
(25) |
cons(mark(X1),X2) |
→ |
cons(X1,X2) |
(26) |
cons(X1,mark(X2)) |
→ |
cons(X1,X2) |
(27) |
cons(active(X1),X2) |
→ |
cons(X1,X2) |
(28) |
cons(X1,active(X2)) |
→ |
cons(X1,X2) |
(29) |
U11(mark(X1),X2) |
→ |
U11(X1,X2) |
(30) |
U11(X1,mark(X2)) |
→ |
U11(X1,X2) |
(31) |
U11(active(X1),X2) |
→ |
U11(X1,X2) |
(32) |
U11(X1,active(X2)) |
→ |
U11(X1,X2) |
(33) |
s(mark(X)) |
→ |
s(X) |
(34) |
s(active(X)) |
→ |
s(X) |
(35) |
length(mark(X)) |
→ |
length(X) |
(36) |
length(active(X)) |
→ |
length(X) |
(37) |
and(mark(X1),X2) |
→ |
and(X1,X2) |
(38) |
and(X1,mark(X2)) |
→ |
and(X1,X2) |
(39) |
and(active(X1),X2) |
→ |
and(X1,X2) |
(40) |
and(X1,active(X2)) |
→ |
and(X1,X2) |
(41) |
isNat(mark(X)) |
→ |
isNat(X) |
(42) |
isNat(active(X)) |
→ |
isNat(X) |
(43) |
isNatList(mark(X)) |
→ |
isNatList(X) |
(44) |
isNatList(active(X)) |
→ |
isNatList(X) |
(45) |
isNatIList(mark(X)) |
→ |
isNatIList(X) |
(46) |
isNatIList(active(X)) |
→ |
isNatIList(X) |
(47) |
active#(zeros) |
→ |
mark#(cons(0,zeros)) |
(48) |
active#(zeros) |
→ |
cons#(0,zeros) |
(49) |
active#(U11(tt,L)) |
→ |
mark#(s(length(L))) |
(50) |
active#(U11(tt,L)) |
→ |
s#(length(L)) |
(51) |
active#(U11(tt,L)) |
→ |
length#(L) |
(52) |
active#(and(tt,X)) |
→ |
mark#(X) |
(53) |
active#(isNat(0)) |
→ |
mark#(tt) |
(54) |
active#(isNat(s(V1))) |
→ |
mark#(isNat(V1)) |
(55) |
active#(isNat(s(V1))) |
→ |
isNat#(V1) |
(56) |
active#(isNatIList(cons(V1,V2))) |
→ |
mark#(and(isNat(V1),isNatIList(V2))) |
(57) |
active#(isNatIList(cons(V1,V2))) |
→ |
and#(isNat(V1),isNatIList(V2)) |
(58) |
active#(isNatIList(cons(V1,V2))) |
→ |
isNat#(V1) |
(59) |
active#(isNatIList(cons(V1,V2))) |
→ |
isNatIList#(V2) |
(60) |
active#(isNatList(cons(V1,V2))) |
→ |
mark#(and(isNat(V1),isNatList(V2))) |
(61) |
active#(isNatList(cons(V1,V2))) |
→ |
and#(isNat(V1),isNatList(V2)) |
(62) |
active#(isNatList(cons(V1,V2))) |
→ |
isNat#(V1) |
(63) |
active#(isNatList(cons(V1,V2))) |
→ |
isNatList#(V2) |
(64) |
active#(length(cons(N,L))) |
→ |
mark#(U11(and(isNatList(L),isNat(N)),L)) |
(65) |
active#(length(cons(N,L))) |
→ |
U11#(and(isNatList(L),isNat(N)),L) |
(66) |
active#(length(cons(N,L))) |
→ |
and#(isNatList(L),isNat(N)) |
(67) |
active#(length(cons(N,L))) |
→ |
isNatList#(L) |
(68) |
active#(length(cons(N,L))) |
→ |
isNat#(N) |
(69) |
mark#(zeros) |
→ |
active#(zeros) |
(70) |
mark#(cons(X1,X2)) |
→ |
active#(cons(mark(X1),X2)) |
(71) |
mark#(cons(X1,X2)) |
→ |
cons#(mark(X1),X2) |
(72) |
mark#(cons(X1,X2)) |
→ |
mark#(X1) |
(73) |
mark#(0) |
→ |
active#(0) |
(74) |
mark#(U11(X1,X2)) |
→ |
active#(U11(mark(X1),X2)) |
(75) |
mark#(U11(X1,X2)) |
→ |
U11#(mark(X1),X2) |
(76) |
mark#(U11(X1,X2)) |
→ |
mark#(X1) |
(77) |
mark#(tt) |
→ |
active#(tt) |
(78) |
mark#(s(X)) |
→ |
active#(s(mark(X))) |
(79) |
mark#(s(X)) |
→ |
s#(mark(X)) |
(80) |
mark#(s(X)) |
→ |
mark#(X) |
(81) |
mark#(length(X)) |
→ |
active#(length(mark(X))) |
(82) |
mark#(length(X)) |
→ |
length#(mark(X)) |
(83) |
mark#(length(X)) |
→ |
mark#(X) |
(84) |
mark#(and(X1,X2)) |
→ |
active#(and(mark(X1),X2)) |
(85) |
mark#(and(X1,X2)) |
→ |
and#(mark(X1),X2) |
(86) |
mark#(and(X1,X2)) |
→ |
mark#(X1) |
(87) |
mark#(isNat(X)) |
→ |
active#(isNat(X)) |
(88) |
mark#(isNatList(X)) |
→ |
active#(isNatList(X)) |
(89) |
mark#(isNatIList(X)) |
→ |
active#(isNatIList(X)) |
(90) |
mark#(nil) |
→ |
active#(nil) |
(91) |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(92) |
cons#(X1,mark(X2)) |
→ |
cons#(X1,X2) |
(93) |
cons#(active(X1),X2) |
→ |
cons#(X1,X2) |
(94) |
cons#(X1,active(X2)) |
→ |
cons#(X1,X2) |
(95) |
U11#(mark(X1),X2) |
→ |
U11#(X1,X2) |
(96) |
U11#(X1,mark(X2)) |
→ |
U11#(X1,X2) |
(97) |
U11#(active(X1),X2) |
→ |
U11#(X1,X2) |
(98) |
U11#(X1,active(X2)) |
→ |
U11#(X1,X2) |
(99) |
s#(mark(X)) |
→ |
s#(X) |
(100) |
s#(active(X)) |
→ |
s#(X) |
(101) |
length#(mark(X)) |
→ |
length#(X) |
(102) |
length#(active(X)) |
→ |
length#(X) |
(103) |
and#(mark(X1),X2) |
→ |
and#(X1,X2) |
(104) |
and#(X1,mark(X2)) |
→ |
and#(X1,X2) |
(105) |
and#(active(X1),X2) |
→ |
and#(X1,X2) |
(106) |
and#(X1,active(X2)) |
→ |
and#(X1,X2) |
(107) |
isNat#(mark(X)) |
→ |
isNat#(X) |
(108) |
isNat#(active(X)) |
→ |
isNat#(X) |
(109) |
isNatList#(mark(X)) |
→ |
isNatList#(X) |
(110) |
isNatList#(active(X)) |
→ |
isNatList#(X) |
(111) |
isNatIList#(mark(X)) |
→ |
isNatIList#(X) |
(112) |
isNatIList#(active(X)) |
→ |
isNatIList#(X) |
(113) |
The dependency pairs are split into 9
components.
-
The
1st
component contains the
pair
mark#(cons(X1,X2)) |
→ |
active#(cons(mark(X1),X2)) |
(71) |
active#(U11(tt,L)) |
→ |
mark#(s(length(L))) |
(50) |
mark#(cons(X1,X2)) |
→ |
mark#(X1) |
(73) |
mark#(zeros) |
→ |
active#(zeros) |
(70) |
active#(zeros) |
→ |
mark#(cons(0,zeros)) |
(48) |
mark#(U11(X1,X2)) |
→ |
active#(U11(mark(X1),X2)) |
(75) |
active#(and(tt,X)) |
→ |
mark#(X) |
(53) |
mark#(U11(X1,X2)) |
→ |
mark#(X1) |
(77) |
mark#(s(X)) |
→ |
active#(s(mark(X))) |
(79) |
active#(isNat(s(V1))) |
→ |
mark#(isNat(V1)) |
(55) |
mark#(s(X)) |
→ |
mark#(X) |
(81) |
mark#(length(X)) |
→ |
active#(length(mark(X))) |
(82) |
active#(isNatIList(cons(V1,V2))) |
→ |
mark#(and(isNat(V1),isNatIList(V2))) |
(57) |
mark#(length(X)) |
→ |
mark#(X) |
(84) |
mark#(and(X1,X2)) |
→ |
active#(and(mark(X1),X2)) |
(85) |
active#(isNatList(cons(V1,V2))) |
→ |
mark#(and(isNat(V1),isNatList(V2))) |
(61) |
mark#(and(X1,X2)) |
→ |
mark#(X1) |
(87) |
mark#(isNat(X)) |
→ |
active#(isNat(X)) |
(88) |
active#(length(cons(N,L))) |
→ |
mark#(U11(and(isNatList(L),isNat(N)),L)) |
(65) |
mark#(isNatList(X)) |
→ |
active#(isNatList(X)) |
(89) |
mark#(isNatIList(X)) |
→ |
active#(isNatIList(X)) |
(90) |
1.1.1.1.1.1 Monotonic Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[zeros] |
= |
0 |
[mark(x1)] |
= |
1 · x1
|
[cons(x1, x2)] |
= |
2 · x1 + 2 · x2
|
[0] |
= |
0 |
[U11(x1, x2)] |
= |
1 + 1 · x1 + 2 · x2
|
[tt] |
= |
0 |
[s(x1)] |
= |
1 · x1
|
[length(x1)] |
= |
1 + 2 · x1
|
[and(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[isNat(x1)] |
= |
2 · x1
|
[isNatIList(x1)] |
= |
1 · x1
|
[isNatList(x1)] |
= |
2 · x1
|
[nil] |
= |
0 |
[mark#(x1)] |
= |
2 · x1
|
[active#(x1)] |
= |
2 · x1
|
the
pairs
mark#(U11(X1,X2)) |
→ |
mark#(X1) |
(77) |
mark#(length(X)) |
→ |
mark#(X) |
(84) |
and
no rules
could be deleted.
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, x2)] |
= |
2 |
[and(x1, x2)] |
= |
2 |
[cons(x1, x2)] |
= |
-2 |
[length(x1)] |
= |
2 |
[s(x1)] |
= |
0 |
[mark(x1)] |
= |
0 |
[active(x1)] |
= |
2 |
[zeros] |
= |
2 |
[0] |
= |
0 |
[tt] |
= |
0 |
[isNat(x1)] |
= |
2 |
[isNatIList(x1)] |
= |
2 |
[isNatList(x1)] |
= |
2 |
[nil] |
= |
0 |
[mark#(x1)] |
= |
1 |
together with the usable
rules
cons(X1,mark(X2)) |
→ |
cons(X1,X2) |
(27) |
cons(mark(X1),X2) |
→ |
cons(X1,X2) |
(26) |
cons(active(X1),X2) |
→ |
cons(X1,X2) |
(28) |
cons(X1,active(X2)) |
→ |
cons(X1,X2) |
(29) |
length(active(X)) |
→ |
length(X) |
(37) |
length(mark(X)) |
→ |
length(X) |
(36) |
s(active(X)) |
→ |
s(X) |
(35) |
s(mark(X)) |
→ |
s(X) |
(34) |
U11(X1,mark(X2)) |
→ |
U11(X1,X2) |
(31) |
U11(mark(X1),X2) |
→ |
U11(X1,X2) |
(30) |
U11(active(X1),X2) |
→ |
U11(X1,X2) |
(32) |
U11(X1,active(X2)) |
→ |
U11(X1,X2) |
(33) |
isNat(active(X)) |
→ |
isNat(X) |
(43) |
isNat(mark(X)) |
→ |
isNat(X) |
(42) |
isNatIList(active(X)) |
→ |
isNatIList(X) |
(47) |
isNatIList(mark(X)) |
→ |
isNatIList(X) |
(46) |
and(X1,mark(X2)) |
→ |
and(X1,X2) |
(39) |
and(mark(X1),X2) |
→ |
and(X1,X2) |
(38) |
and(active(X1),X2) |
→ |
and(X1,X2) |
(40) |
and(X1,active(X2)) |
→ |
and(X1,X2) |
(41) |
isNatList(active(X)) |
→ |
isNatList(X) |
(45) |
isNatList(mark(X)) |
→ |
isNatList(X) |
(44) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
mark#(cons(X1,X2)) |
→ |
active#(cons(mark(X1),X2)) |
(71) |
mark#(s(X)) |
→ |
active#(s(mark(X))) |
(79) |
could be deleted.
1.1.1.1.1.1.1.1 Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
[active#(x1)] |
= |
2 + x1
|
[mark#(x1)] |
= |
2 + 2 · x1
|
[s(x1)] |
= |
2 · x1
|
[length(x1)] |
= |
-2 |
[active(x1)] |
= |
x1 |
[mark(x1)] |
= |
2 · x1
|
[U11(x1, x2)] |
= |
-2 |
[and(x1, x2)] |
= |
x1 + 2 · x2
|
[cons(x1, x2)] |
= |
2 · x1
|
[zeros] |
= |
2 |
[0] |
= |
0 |
[tt] |
= |
0 |
[isNat(x1)] |
= |
0 |
[isNatIList(x1)] |
= |
0 |
[isNatList(x1)] |
= |
0 |
[nil] |
= |
2 |
the
pairs
mark#(zeros) |
→ |
active#(zeros) |
(70) |
active#(zeros) |
→ |
mark#(cons(0,zeros)) |
(48) |
could be deleted.
1.1.1.1.1.1.1.1.1 Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
[active#(x1)] |
= |
x1 |
[mark#(x1)] |
= |
x1 |
[s(x1)] |
= |
x1 |
[length(x1)] |
= |
2 |
[active(x1)] |
= |
x1 |
[mark(x1)] |
= |
x1 |
[U11(x1, x2)] |
= |
2 |
[and(x1, x2)] |
= |
x1 + x2
|
[cons(x1, x2)] |
= |
2 + x1
|
[zeros] |
= |
2 |
[0] |
= |
0 |
[tt] |
= |
0 |
[isNat(x1)] |
= |
0 |
[isNatIList(x1)] |
= |
0 |
[isNatList(x1)] |
= |
0 |
[nil] |
= |
0 |
the
pair
mark#(cons(X1,X2)) |
→ |
mark#(X1) |
(73) |
could be deleted.
1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor
Using the matrix interpretations of dimension 1 with strict dimension 1 over the arctic semiring over the naturals
[active#(x1)] |
= |
+ · x1
|
[U11(x1, x2)] |
= |
+ · x1 + · x2
|
[tt] |
= |
|
[mark#(x1)] |
= |
+ · x1
|
[s(x1)] |
= |
+ · x1
|
[length(x1)] |
= |
+ · x1
|
[mark(x1)] |
= |
+ · x1
|
[and(x1, x2)] |
= |
+ · x1 + · x2
|
[isNat(x1)] |
= |
+ · x1
|
[isNatIList(x1)] |
= |
+ · x1
|
[cons(x1, x2)] |
= |
+ · x1 + · x2
|
[isNatList(x1)] |
= |
+ · x1
|
[active(x1)] |
= |
+ · x1
|
[zeros] |
= |
|
[0] |
= |
|
[nil] |
= |
|
the
pair
mark#(and(X1,X2)) |
→ |
mark#(X1) |
(87) |
could be deleted.
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 + 2 · x1
|
[mark#(x1)] |
= |
-2 + 2 · x1
|
[s(x1)] |
= |
1 + x1
|
[length(x1)] |
= |
-2 |
[active(x1)] |
= |
x1 |
[mark(x1)] |
= |
2 + x1
|
[U11(x1, x2)] |
= |
-2 |
[and(x1, x2)] |
= |
1 + x2
|
[cons(x1, x2)] |
= |
-2 |
[zeros] |
= |
1 |
[0] |
= |
0 |
[tt] |
= |
0 |
[isNat(x1)] |
= |
1 + 2 · x1
|
[isNatIList(x1)] |
= |
0 |
[isNatList(x1)] |
= |
0 |
[nil] |
= |
2 |
together with the usable
rules
length(active(X)) |
→ |
length(X) |
(37) |
length(mark(X)) |
→ |
length(X) |
(36) |
s(active(X)) |
→ |
s(X) |
(35) |
s(mark(X)) |
→ |
s(X) |
(34) |
U11(X1,mark(X2)) |
→ |
U11(X1,X2) |
(31) |
U11(mark(X1),X2) |
→ |
U11(X1,X2) |
(30) |
U11(active(X1),X2) |
→ |
U11(X1,X2) |
(32) |
U11(X1,active(X2)) |
→ |
U11(X1,X2) |
(33) |
isNat(active(X)) |
→ |
isNat(X) |
(43) |
isNat(mark(X)) |
→ |
isNat(X) |
(42) |
isNatIList(active(X)) |
→ |
isNatIList(X) |
(47) |
isNatIList(mark(X)) |
→ |
isNatIList(X) |
(46) |
and(X1,mark(X2)) |
→ |
and(X1,X2) |
(39) |
and(mark(X1),X2) |
→ |
and(X1,X2) |
(38) |
and(active(X1),X2) |
→ |
and(X1,X2) |
(40) |
and(X1,active(X2)) |
→ |
and(X1,X2) |
(41) |
isNatList(active(X)) |
→ |
isNatList(X) |
(45) |
isNatList(mark(X)) |
→ |
isNatList(X) |
(44) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
active#(isNat(s(V1))) |
→ |
mark#(isNat(V1)) |
(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 + 2 · x1
|
[mark#(x1)] |
= |
2 |
[s(x1)] |
= |
-2 |
[length(x1)] |
= |
2 |
[active(x1)] |
= |
-2 |
[mark(x1)] |
= |
-2 |
[U11(x1, x2)] |
= |
2 |
[and(x1, x2)] |
= |
2 |
[cons(x1, x2)] |
= |
-2 + x1
|
[zeros] |
= |
0 |
[0] |
= |
0 |
[tt] |
= |
0 |
[isNat(x1)] |
= |
0 |
[isNatIList(x1)] |
= |
2 |
[isNatList(x1)] |
= |
2 |
[nil] |
= |
0 |
together with the usable
rules
length(active(X)) |
→ |
length(X) |
(37) |
length(mark(X)) |
→ |
length(X) |
(36) |
U11(X1,mark(X2)) |
→ |
U11(X1,X2) |
(31) |
U11(mark(X1),X2) |
→ |
U11(X1,X2) |
(30) |
U11(active(X1),X2) |
→ |
U11(X1,X2) |
(32) |
U11(X1,active(X2)) |
→ |
U11(X1,X2) |
(33) |
isNat(active(X)) |
→ |
isNat(X) |
(43) |
isNat(mark(X)) |
→ |
isNat(X) |
(42) |
isNatIList(active(X)) |
→ |
isNatIList(X) |
(47) |
isNatIList(mark(X)) |
→ |
isNatIList(X) |
(46) |
and(X1,mark(X2)) |
→ |
and(X1,X2) |
(39) |
and(mark(X1),X2) |
→ |
and(X1,X2) |
(38) |
and(active(X1),X2) |
→ |
and(X1,X2) |
(40) |
and(X1,active(X2)) |
→ |
and(X1,X2) |
(41) |
isNatList(active(X)) |
→ |
isNatList(X) |
(45) |
isNatList(mark(X)) |
→ |
isNatList(X) |
(44) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
mark#(isNat(X)) |
→ |
active#(isNat(X)) |
(88) |
could be deleted.
1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules
Using the
prec(active#) |
= |
2 |
|
stat(active#) |
= |
mul
|
prec(U11) |
= |
4 |
|
stat(U11) |
= |
mul
|
prec(tt) |
= |
4 |
|
stat(tt) |
= |
mul
|
prec(mark#) |
= |
2 |
|
stat(mark#) |
= |
mul
|
prec(length) |
= |
4 |
|
stat(length) |
= |
mul
|
prec(mark) |
= |
1 |
|
stat(mark) |
= |
mul
|
prec(isNatIList) |
= |
1 |
|
stat(isNatIList) |
= |
mul
|
prec(cons) |
= |
1 |
|
stat(cons) |
= |
mul
|
prec(isNat) |
= |
2 |
|
stat(isNat) |
= |
mul
|
prec(isNatList) |
= |
3 |
|
stat(isNatList) |
= |
mul
|
prec(zeros) |
= |
5 |
|
stat(zeros) |
= |
mul
|
prec(0) |
= |
0 |
|
stat(0) |
= |
mul
|
prec(nil) |
= |
6 |
|
stat(nil) |
= |
mul
|
π(active#) |
= |
[1] |
π(U11) |
= |
[] |
π(tt) |
= |
[] |
π(mark#) |
= |
[1] |
π(s) |
= |
1 |
π(length) |
= |
[] |
π(mark) |
= |
[1] |
π(and) |
= |
2 |
π(isNatIList) |
= |
[1] |
π(cons) |
= |
[1,2] |
π(isNat) |
= |
[] |
π(isNatList) |
= |
[] |
π(active) |
= |
1 |
π(zeros) |
= |
[] |
π(0) |
= |
[] |
π(nil) |
= |
[] |
together with the usable
rules
length(active(X)) |
→ |
length(X) |
(37) |
length(mark(X)) |
→ |
length(X) |
(36) |
s(active(X)) |
→ |
s(X) |
(35) |
s(mark(X)) |
→ |
s(X) |
(34) |
U11(X1,mark(X2)) |
→ |
U11(X1,X2) |
(31) |
U11(mark(X1),X2) |
→ |
U11(X1,X2) |
(30) |
U11(active(X1),X2) |
→ |
U11(X1,X2) |
(32) |
U11(X1,active(X2)) |
→ |
U11(X1,X2) |
(33) |
isNatIList(active(X)) |
→ |
isNatIList(X) |
(47) |
isNatIList(mark(X)) |
→ |
isNatIList(X) |
(46) |
and(X1,mark(X2)) |
→ |
and(X1,X2) |
(39) |
and(mark(X1),X2) |
→ |
and(X1,X2) |
(38) |
and(active(X1),X2) |
→ |
and(X1,X2) |
(40) |
and(X1,active(X2)) |
→ |
and(X1,X2) |
(41) |
isNatList(active(X)) |
→ |
isNatList(X) |
(45) |
isNatList(mark(X)) |
→ |
isNatList(X) |
(44) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
active#(isNatIList(cons(V1,V2))) |
→ |
mark#(and(isNat(V1),isNatIList(V2))) |
(57) |
could be deleted.
1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active#(x1)] |
= |
-1 + 2 · x1
|
[mark#(x1)] |
= |
1 |
[s(x1)] |
= |
-2 |
[length(x1)] |
= |
1 |
[active(x1)] |
= |
2 |
[mark(x1)] |
= |
-2 |
[U11(x1, x2)] |
= |
1 |
[and(x1, x2)] |
= |
1 |
[cons(x1, x2)] |
= |
-2 + 2 · x2
|
[zeros] |
= |
0 |
[0] |
= |
0 |
[tt] |
= |
0 |
[isNat(x1)] |
= |
0 |
[isNatIList(x1)] |
= |
0 |
[isNatList(x1)] |
= |
1 |
[nil] |
= |
0 |
together with the usable
rules
length(active(X)) |
→ |
length(X) |
(37) |
length(mark(X)) |
→ |
length(X) |
(36) |
U11(X1,mark(X2)) |
→ |
U11(X1,X2) |
(31) |
U11(mark(X1),X2) |
→ |
U11(X1,X2) |
(30) |
U11(active(X1),X2) |
→ |
U11(X1,X2) |
(32) |
U11(X1,active(X2)) |
→ |
U11(X1,X2) |
(33) |
and(X1,mark(X2)) |
→ |
and(X1,X2) |
(39) |
and(mark(X1),X2) |
→ |
and(X1,X2) |
(38) |
and(active(X1),X2) |
→ |
and(X1,X2) |
(40) |
and(X1,active(X2)) |
→ |
and(X1,X2) |
(41) |
isNatList(active(X)) |
→ |
isNatList(X) |
(45) |
isNatList(mark(X)) |
→ |
isNatList(X) |
(44) |
isNatIList(active(X)) |
→ |
isNatIList(X) |
(47) |
isNatIList(mark(X)) |
→ |
isNatIList(X) |
(46) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
mark#(isNatIList(X)) |
→ |
active#(isNatIList(X)) |
(90) |
could be deleted.
1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active#(x1)] |
= |
x1 |
[mark#(x1)] |
= |
x1 |
[s(x1)] |
= |
x1 |
[length(x1)] |
= |
-2 |
[active(x1)] |
= |
x1 |
[mark(x1)] |
= |
x1 |
[U11(x1, x2)] |
= |
-2 |
[and(x1, x2)] |
= |
2 · x2
|
[cons(x1, x2)] |
= |
2 + 2 · x1 + 2 · x2
|
[zeros] |
= |
0 |
[0] |
= |
2 |
[tt] |
= |
0 |
[isNat(x1)] |
= |
0 |
[isNatIList(x1)] |
= |
2 |
[isNatList(x1)] |
= |
x1 |
[nil] |
= |
2 |
together with the usable
rules
length(active(X)) |
→ |
length(X) |
(37) |
length(mark(X)) |
→ |
length(X) |
(36) |
s(active(X)) |
→ |
s(X) |
(35) |
s(mark(X)) |
→ |
s(X) |
(34) |
U11(X1,mark(X2)) |
→ |
U11(X1,X2) |
(31) |
U11(mark(X1),X2) |
→ |
U11(X1,X2) |
(30) |
U11(active(X1),X2) |
→ |
U11(X1,X2) |
(32) |
U11(X1,active(X2)) |
→ |
U11(X1,X2) |
(33) |
and(X1,mark(X2)) |
→ |
and(X1,X2) |
(39) |
and(mark(X1),X2) |
→ |
and(X1,X2) |
(38) |
and(active(X1),X2) |
→ |
and(X1,X2) |
(40) |
and(X1,active(X2)) |
→ |
and(X1,X2) |
(41) |
isNatList(active(X)) |
→ |
isNatList(X) |
(45) |
isNatList(mark(X)) |
→ |
isNatList(X) |
(44) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
active#(isNatList(cons(V1,V2))) |
→ |
mark#(and(isNat(V1),isNatList(V2))) |
(61) |
could be deleted.
1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active#(x1)] |
= |
-1 + x1
|
[mark#(x1)] |
= |
1 |
[s(x1)] |
= |
-2 |
[length(x1)] |
= |
2 |
[active(x1)] |
= |
-2 |
[mark(x1)] |
= |
2 |
[U11(x1, x2)] |
= |
2 |
[and(x1, x2)] |
= |
2 |
[cons(x1, x2)] |
= |
-2 + x2
|
[zeros] |
= |
0 |
[0] |
= |
0 |
[tt] |
= |
0 |
[isNat(x1)] |
= |
0 |
[isNatIList(x1)] |
= |
0 |
[isNatList(x1)] |
= |
0 |
[nil] |
= |
0 |
together with the usable
rules
length(active(X)) |
→ |
length(X) |
(37) |
length(mark(X)) |
→ |
length(X) |
(36) |
U11(X1,mark(X2)) |
→ |
U11(X1,X2) |
(31) |
U11(mark(X1),X2) |
→ |
U11(X1,X2) |
(30) |
U11(active(X1),X2) |
→ |
U11(X1,X2) |
(32) |
U11(X1,active(X2)) |
→ |
U11(X1,X2) |
(33) |
and(X1,mark(X2)) |
→ |
and(X1,X2) |
(39) |
and(mark(X1),X2) |
→ |
and(X1,X2) |
(38) |
and(active(X1),X2) |
→ |
and(X1,X2) |
(40) |
and(X1,active(X2)) |
→ |
and(X1,X2) |
(41) |
isNatList(active(X)) |
→ |
isNatList(X) |
(45) |
isNatList(mark(X)) |
→ |
isNatList(X) |
(44) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
mark#(isNatList(X)) |
→ |
active#(isNatList(X)) |
(89) |
could be deleted.
1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules
Using the
prec(active#) |
= |
3 |
|
stat(active#) |
= |
lex
|
prec(U11) |
= |
2 |
|
stat(U11) |
= |
lex
|
prec(tt) |
= |
4 |
|
stat(tt) |
= |
lex
|
prec(mark#) |
= |
3 |
|
stat(mark#) |
= |
lex
|
prec(length) |
= |
2 |
|
stat(length) |
= |
lex
|
prec(and) |
= |
3 |
|
stat(and) |
= |
lex
|
prec(isNatList) |
= |
1 |
|
stat(isNatList) |
= |
lex
|
prec(isNat) |
= |
0 |
|
stat(isNat) |
= |
lex
|
prec(zeros) |
= |
5 |
|
stat(zeros) |
= |
lex
|
prec(0) |
= |
6 |
|
stat(0) |
= |
lex
|
prec(isNatIList) |
= |
7 |
|
stat(isNatIList) |
= |
lex
|
prec(nil) |
= |
8 |
|
stat(nil) |
= |
lex
|
π(active#) |
= |
[1] |
π(U11) |
= |
[] |
π(tt) |
= |
[] |
π(mark#) |
= |
[1] |
π(s) |
= |
1 |
π(length) |
= |
[] |
π(mark) |
= |
1 |
π(and) |
= |
[2] |
π(cons) |
= |
1 |
π(isNatList) |
= |
[1] |
π(isNat) |
= |
[] |
π(active) |
= |
1 |
π(zeros) |
= |
[] |
π(0) |
= |
[] |
π(isNatIList) |
= |
[] |
π(nil) |
= |
[] |
together with the usable
rules
length(active(X)) |
→ |
length(X) |
(37) |
length(mark(X)) |
→ |
length(X) |
(36) |
s(active(X)) |
→ |
s(X) |
(35) |
s(mark(X)) |
→ |
s(X) |
(34) |
U11(X1,mark(X2)) |
→ |
U11(X1,X2) |
(31) |
U11(mark(X1),X2) |
→ |
U11(X1,X2) |
(30) |
U11(active(X1),X2) |
→ |
U11(X1,X2) |
(32) |
U11(X1,active(X2)) |
→ |
U11(X1,X2) |
(33) |
and(X1,mark(X2)) |
→ |
and(X1,X2) |
(39) |
and(mark(X1),X2) |
→ |
and(X1,X2) |
(38) |
and(active(X1),X2) |
→ |
and(X1,X2) |
(40) |
and(X1,active(X2)) |
→ |
and(X1,X2) |
(41) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
active#(and(tt,X)) |
→ |
mark#(X) |
(53) |
could be deleted.
1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active#(x1)] |
= |
-1 + 2 · x1
|
[mark#(x1)] |
= |
1 |
[s(x1)] |
= |
-2 |
[length(x1)] |
= |
1 |
[active(x1)] |
= |
-2 |
[mark(x1)] |
= |
0 |
[U11(x1, x2)] |
= |
1 |
[and(x1, x2)] |
= |
-2 |
[cons(x1, x2)] |
= |
2 |
[zeros] |
= |
0 |
[0] |
= |
0 |
[tt] |
= |
0 |
[isNat(x1)] |
= |
0 |
[isNatIList(x1)] |
= |
0 |
[isNatList(x1)] |
= |
0 |
[nil] |
= |
0 |
together with the usable
rules
length(active(X)) |
→ |
length(X) |
(37) |
length(mark(X)) |
→ |
length(X) |
(36) |
U11(X1,mark(X2)) |
→ |
U11(X1,X2) |
(31) |
U11(mark(X1),X2) |
→ |
U11(X1,X2) |
(30) |
U11(active(X1),X2) |
→ |
U11(X1,X2) |
(32) |
U11(X1,active(X2)) |
→ |
U11(X1,X2) |
(33) |
and(X1,mark(X2)) |
→ |
and(X1,X2) |
(39) |
and(mark(X1),X2) |
→ |
and(X1,X2) |
(38) |
and(active(X1),X2) |
→ |
and(X1,X2) |
(40) |
and(X1,active(X2)) |
→ |
and(X1,X2) |
(41) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
mark#(and(X1,X2)) |
→ |
active#(and(mark(X1),X2)) |
(85) |
could be deleted.
1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Narrowing Processor
We consider all narrowings of the pair
below position
1
to get the following set of pairs
mark#(U11(x0,x1)) |
→ |
active#(U11(x0,x1)) |
(114) |
mark#(U11(y0,mark(x1))) |
→ |
active#(U11(mark(y0),x1)) |
(115) |
mark#(U11(y0,active(x1))) |
→ |
active#(U11(mark(y0),x1)) |
(116) |
mark#(U11(zeros,y1)) |
→ |
active#(U11(active(zeros),y1)) |
(117) |
mark#(U11(cons(x0,x1),y1)) |
→ |
active#(U11(active(cons(mark(x0),x1)),y1)) |
(118) |
mark#(U11(0,y1)) |
→ |
active#(U11(active(0),y1)) |
(119) |
mark#(U11(U11(x0,x1),y1)) |
→ |
active#(U11(active(U11(mark(x0),x1)),y1)) |
(120) |
mark#(U11(tt,y1)) |
→ |
active#(U11(active(tt),y1)) |
(121) |
mark#(U11(s(x0),y1)) |
→ |
active#(U11(active(s(mark(x0))),y1)) |
(122) |
mark#(U11(length(x0),y1)) |
→ |
active#(U11(active(length(mark(x0))),y1)) |
(123) |
mark#(U11(and(x0,x1),y1)) |
→ |
active#(U11(active(and(mark(x0),x1)),y1)) |
(124) |
mark#(U11(isNat(x0),y1)) |
→ |
active#(U11(active(isNat(x0)),y1)) |
(125) |
mark#(U11(isNatList(x0),y1)) |
→ |
active#(U11(active(isNatList(x0)),y1)) |
(126) |
mark#(U11(isNatIList(x0),y1)) |
→ |
active#(U11(active(isNatIList(x0)),y1)) |
(127) |
mark#(U11(nil,y1)) |
→ |
active#(U11(active(nil),y1)) |
(128) |
1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Narrowing Processor
We consider all narrowings of the pair
below position
1
to get the following set of pairs
mark#(length(x0)) |
→ |
active#(length(x0)) |
(129) |
mark#(length(zeros)) |
→ |
active#(length(active(zeros))) |
(130) |
mark#(length(cons(x0,x1))) |
→ |
active#(length(active(cons(mark(x0),x1)))) |
(131) |
mark#(length(0)) |
→ |
active#(length(active(0))) |
(132) |
mark#(length(U11(x0,x1))) |
→ |
active#(length(active(U11(mark(x0),x1)))) |
(133) |
mark#(length(tt)) |
→ |
active#(length(active(tt))) |
(134) |
mark#(length(s(x0))) |
→ |
active#(length(active(s(mark(x0))))) |
(135) |
mark#(length(length(x0))) |
→ |
active#(length(active(length(mark(x0))))) |
(136) |
mark#(length(and(x0,x1))) |
→ |
active#(length(active(and(mark(x0),x1)))) |
(137) |
mark#(length(isNat(x0))) |
→ |
active#(length(active(isNat(x0)))) |
(138) |
mark#(length(isNatList(x0))) |
→ |
active#(length(active(isNatList(x0)))) |
(139) |
mark#(length(isNatIList(x0))) |
→ |
active#(length(active(isNatIList(x0)))) |
(140) |
mark#(length(nil)) |
→ |
active#(length(active(nil))) |
(141) |
1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active#(x1)] |
= |
-2 |
[mark#(x1)] |
= |
-2 + 2 · x1
|
[s(x1)] |
= |
1 + x1
|
[length(x1)] |
= |
0 |
[active(x1)] |
= |
2 + x1
|
[mark(x1)] |
= |
2 + x1
|
[U11(x1, x2)] |
= |
1 + 2 · x1
|
[and(x1, x2)] |
= |
-2 |
[isNatList(x1)] |
= |
0 |
[isNat(x1)] |
= |
0 |
[cons(x1, x2)] |
= |
0 |
[zeros] |
= |
0 |
[0] |
= |
0 |
[tt] |
= |
1 |
[isNatIList(x1)] |
= |
0 |
[nil] |
= |
0 |
together with the usable
rules
length(active(X)) |
→ |
length(X) |
(37) |
length(mark(X)) |
→ |
length(X) |
(36) |
s(active(X)) |
→ |
s(X) |
(35) |
s(mark(X)) |
→ |
s(X) |
(34) |
and(X1,mark(X2)) |
→ |
and(X1,X2) |
(39) |
and(mark(X1),X2) |
→ |
and(X1,X2) |
(38) |
and(active(X1),X2) |
→ |
and(X1,X2) |
(40) |
and(X1,active(X2)) |
→ |
and(X1,X2) |
(41) |
U11(X1,mark(X2)) |
→ |
U11(X1,X2) |
(31) |
U11(mark(X1),X2) |
→ |
U11(X1,X2) |
(30) |
U11(active(X1),X2) |
→ |
U11(X1,X2) |
(32) |
U11(X1,active(X2)) |
→ |
U11(X1,X2) |
(33) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
mark#(U11(U11(x0,x1),y1)) |
→ |
active#(U11(active(U11(mark(x0),x1)),y1)) |
(120) |
mark#(U11(tt,y1)) |
→ |
active#(U11(active(tt),y1)) |
(121) |
mark#(U11(s(x0),y1)) |
→ |
active#(U11(active(s(mark(x0))),y1)) |
(122) |
could be deleted.
1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active#(x1)] |
= |
-2 |
[mark#(x1)] |
= |
-2 + 2 · x1
|
[s(x1)] |
= |
1 + 2 · x1
|
[length(x1)] |
= |
-2 |
[active(x1)] |
= |
x1 |
[mark(x1)] |
= |
x1 |
[U11(x1, x2)] |
= |
1 + 2 · x1
|
[and(x1, x2)] |
= |
-2 |
[isNatList(x1)] |
= |
0 |
[isNat(x1)] |
= |
0 |
[cons(x1, x2)] |
= |
1 |
[zeros] |
= |
0 |
[0] |
= |
0 |
[tt] |
= |
0 |
[isNatIList(x1)] |
= |
0 |
[nil] |
= |
0 |
together with the usable
rules
length(active(X)) |
→ |
length(X) |
(37) |
length(mark(X)) |
→ |
length(X) |
(36) |
s(active(X)) |
→ |
s(X) |
(35) |
s(mark(X)) |
→ |
s(X) |
(34) |
and(X1,mark(X2)) |
→ |
and(X1,X2) |
(39) |
and(mark(X1),X2) |
→ |
and(X1,X2) |
(38) |
and(active(X1),X2) |
→ |
and(X1,X2) |
(40) |
and(X1,active(X2)) |
→ |
and(X1,X2) |
(41) |
U11(X1,mark(X2)) |
→ |
U11(X1,X2) |
(31) |
U11(mark(X1),X2) |
→ |
U11(X1,X2) |
(30) |
U11(active(X1),X2) |
→ |
U11(X1,X2) |
(32) |
U11(X1,active(X2)) |
→ |
U11(X1,X2) |
(33) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
mark#(U11(cons(x0,x1),y1)) |
→ |
active#(U11(active(cons(mark(x0),x1)),y1)) |
(118) |
could be deleted.
1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active#(x1)] |
= |
-2 |
[mark#(x1)] |
= |
-1 + x1
|
[s(x1)] |
= |
1 + x1
|
[length(x1)] |
= |
-2 |
[active(x1)] |
= |
2 + 2 · x1
|
[mark(x1)] |
= |
2 + x1
|
[U11(x1, x2)] |
= |
1 + 2 · x1
|
[and(x1, x2)] |
= |
-2 |
[isNatList(x1)] |
= |
1 |
[isNat(x1)] |
= |
2 |
[cons(x1, x2)] |
= |
2 + 2 · x1
|
[zeros] |
= |
0 |
[0] |
= |
0 |
[tt] |
= |
0 |
[isNatIList(x1)] |
= |
1 |
[nil] |
= |
0 |
together with the usable
rules
length(active(X)) |
→ |
length(X) |
(37) |
length(mark(X)) |
→ |
length(X) |
(36) |
s(active(X)) |
→ |
s(X) |
(35) |
s(mark(X)) |
→ |
s(X) |
(34) |
and(X1,mark(X2)) |
→ |
and(X1,X2) |
(39) |
and(mark(X1),X2) |
→ |
and(X1,X2) |
(38) |
and(active(X1),X2) |
→ |
and(X1,X2) |
(40) |
and(X1,active(X2)) |
→ |
and(X1,X2) |
(41) |
U11(X1,mark(X2)) |
→ |
U11(X1,X2) |
(31) |
U11(mark(X1),X2) |
→ |
U11(X1,X2) |
(30) |
U11(active(X1),X2) |
→ |
U11(X1,X2) |
(32) |
U11(X1,active(X2)) |
→ |
U11(X1,X2) |
(33) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
mark#(U11(isNat(x0),y1)) |
→ |
active#(U11(active(isNat(x0)),y1)) |
(125) |
mark#(U11(isNatList(x0),y1)) |
→ |
active#(U11(active(isNatList(x0)),y1)) |
(126) |
mark#(U11(isNatIList(x0),y1)) |
→ |
active#(U11(active(isNatIList(x0)),y1)) |
(127) |
could be deleted.
1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active#(x1)] |
= |
-2 |
[mark#(x1)] |
= |
-2 + x1
|
[s(x1)] |
= |
2 + x1
|
[length(x1)] |
= |
-2 |
[active(x1)] |
= |
2 · x1
|
[mark(x1)] |
= |
2 + x1
|
[U11(x1, x2)] |
= |
2 + 2 · x1
|
[and(x1, x2)] |
= |
-2 |
[isNatList(x1)] |
= |
0 |
[isNat(x1)] |
= |
0 |
[cons(x1, x2)] |
= |
-2 + x1
|
[zeros] |
= |
2 |
[0] |
= |
0 |
[tt] |
= |
0 |
[isNatIList(x1)] |
= |
x1 |
[nil] |
= |
2 |
together with the usable
rules
length(active(X)) |
→ |
length(X) |
(37) |
length(mark(X)) |
→ |
length(X) |
(36) |
s(active(X)) |
→ |
s(X) |
(35) |
s(mark(X)) |
→ |
s(X) |
(34) |
and(X1,mark(X2)) |
→ |
and(X1,X2) |
(39) |
and(mark(X1),X2) |
→ |
and(X1,X2) |
(38) |
and(active(X1),X2) |
→ |
and(X1,X2) |
(40) |
and(X1,active(X2)) |
→ |
and(X1,X2) |
(41) |
U11(X1,mark(X2)) |
→ |
U11(X1,X2) |
(31) |
U11(mark(X1),X2) |
→ |
U11(X1,X2) |
(30) |
U11(active(X1),X2) |
→ |
U11(X1,X2) |
(32) |
U11(X1,active(X2)) |
→ |
U11(X1,X2) |
(33) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
mark#(U11(zeros,y1)) |
→ |
active#(U11(active(zeros),y1)) |
(117) |
mark#(U11(nil,y1)) |
→ |
active#(U11(active(nil),y1)) |
(128) |
could be deleted.
1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active#(x1)] |
= |
-2 |
[mark#(x1)] |
= |
-2 + x1
|
[s(x1)] |
= |
2 + 2 · x1
|
[length(x1)] |
= |
-2 |
[active(x1)] |
= |
2 + 2 · x1
|
[mark(x1)] |
= |
2 · x1
|
[U11(x1, x2)] |
= |
2 + 2 · x1
|
[and(x1, x2)] |
= |
-2 |
[isNatList(x1)] |
= |
0 |
[isNat(x1)] |
= |
0 |
[cons(x1, x2)] |
= |
-2 + x2
|
[zeros] |
= |
0 |
[0] |
= |
1 |
[tt] |
= |
0 |
[isNatIList(x1)] |
= |
1 + x1
|
[nil] |
= |
0 |
together with the usable
rules
length(active(X)) |
→ |
length(X) |
(37) |
length(mark(X)) |
→ |
length(X) |
(36) |
s(active(X)) |
→ |
s(X) |
(35) |
s(mark(X)) |
→ |
s(X) |
(34) |
and(X1,mark(X2)) |
→ |
and(X1,X2) |
(39) |
and(mark(X1),X2) |
→ |
and(X1,X2) |
(38) |
and(active(X1),X2) |
→ |
and(X1,X2) |
(40) |
and(X1,active(X2)) |
→ |
and(X1,X2) |
(41) |
U11(X1,mark(X2)) |
→ |
U11(X1,X2) |
(31) |
U11(mark(X1),X2) |
→ |
U11(X1,X2) |
(30) |
U11(active(X1),X2) |
→ |
U11(X1,X2) |
(32) |
U11(X1,active(X2)) |
→ |
U11(X1,X2) |
(33) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
mark#(U11(0,y1)) |
→ |
active#(U11(active(0),y1)) |
(119) |
could be deleted.
1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active#(x1)] |
= |
2 |
[mark#(x1)] |
= |
x1 |
[s(x1)] |
= |
x1 |
[length(x1)] |
= |
2 |
[active(x1)] |
= |
1 + x1
|
[mark(x1)] |
= |
2 + x1
|
[U11(x1, x2)] |
= |
2 + 2 · x1
|
[and(x1, x2)] |
= |
-2 |
[isNatList(x1)] |
= |
0 |
[isNat(x1)] |
= |
0 |
[cons(x1, x2)] |
= |
-2 + 2 · x2
|
[zeros] |
= |
1 |
[0] |
= |
0 |
[tt] |
= |
0 |
[isNatIList(x1)] |
= |
0 |
[nil] |
= |
0 |
together with the usable
rules
length(active(X)) |
→ |
length(X) |
(37) |
length(mark(X)) |
→ |
length(X) |
(36) |
s(active(X)) |
→ |
s(X) |
(35) |
s(mark(X)) |
→ |
s(X) |
(34) |
and(X1,mark(X2)) |
→ |
and(X1,X2) |
(39) |
and(mark(X1),X2) |
→ |
and(X1,X2) |
(38) |
and(active(X1),X2) |
→ |
and(X1,X2) |
(40) |
and(X1,active(X2)) |
→ |
and(X1,X2) |
(41) |
U11(X1,mark(X2)) |
→ |
U11(X1,X2) |
(31) |
U11(mark(X1),X2) |
→ |
U11(X1,X2) |
(30) |
U11(active(X1),X2) |
→ |
U11(X1,X2) |
(32) |
U11(X1,active(X2)) |
→ |
U11(X1,X2) |
(33) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
mark#(U11(length(x0),y1)) |
→ |
active#(U11(active(length(mark(x0))),y1)) |
(123) |
could be deleted.
1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor
Using the matrix interpretations of dimension 1 with strict dimension 1 over the arctic semiring over the naturals
[active#(x1)] |
= |
+ · x1
|
[U11(x1, x2)] |
= |
+ · x1 + · x2
|
[tt] |
= |
|
[mark#(x1)] |
= |
+ · x1
|
[s(x1)] |
= |
+ · x1
|
[length(x1)] |
= |
+ · x1
|
[cons(x1, x2)] |
= |
+ · x1 + · x2
|
[and(x1, x2)] |
= |
+ · x1 + · x2
|
[isNatList(x1)] |
= |
+ · x1
|
[isNat(x1)] |
= |
+ · x1
|
[mark(x1)] |
= |
+ · x1
|
[active(x1)] |
= |
+ · x1
|
[zeros] |
= |
|
[0] |
= |
|
[isNatIList(x1)] |
= |
+ · x1
|
[nil] |
= |
|
the
pair
mark#(length(nil)) |
→ |
active#(length(active(nil))) |
(141) |
could be deleted.
1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor
Using the matrix interpretations of dimension 1 with strict dimension 1 over the arctic semiring over the naturals
[active#(x1)] |
= |
+ · x1
|
[U11(x1, x2)] |
= |
+ · x1 + · x2
|
[tt] |
= |
|
[mark#(x1)] |
= |
+ · x1
|
[s(x1)] |
= |
+ · x1
|
[length(x1)] |
= |
+ · x1
|
[cons(x1, x2)] |
= |
+ · x1 + · x2
|
[and(x1, x2)] |
= |
+ · x1 + · x2
|
[isNatList(x1)] |
= |
+ · x1
|
[isNat(x1)] |
= |
+ · x1
|
[mark(x1)] |
= |
+ · x1
|
[active(x1)] |
= |
+ · x1
|
[zeros] |
= |
|
[0] |
= |
|
[isNatIList(x1)] |
= |
+ · x1
|
[nil] |
= |
|
the
pair
mark#(length(isNatIList(x0))) |
→ |
active#(length(active(isNatIList(x0)))) |
(140) |
could be deleted.
1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor
Using the matrix interpretations of dimension 1 with strict dimension 1 over the arctic semiring over the naturals
[active#(x1)] |
= |
+ · x1
|
[U11(x1, x2)] |
= |
+ · x1 + · x2
|
[tt] |
= |
|
[mark#(x1)] |
= |
+ · x1
|
[s(x1)] |
= |
+ · x1
|
[length(x1)] |
= |
+ · x1
|
[cons(x1, x2)] |
= |
+ · x1 + · x2
|
[and(x1, x2)] |
= |
+ · x1 + · x2
|
[isNatList(x1)] |
= |
+ · x1
|
[isNat(x1)] |
= |
+ · x1
|
[mark(x1)] |
= |
+ · x1
|
[active(x1)] |
= |
+ · x1
|
[zeros] |
= |
|
[0] |
= |
|
[isNatIList(x1)] |
= |
+ · x1
|
[nil] |
= |
|
the
pair
mark#(length(0)) |
→ |
active#(length(active(0))) |
(132) |
could be deleted.
1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor
Using the matrix interpretations of dimension 1 with strict dimension 1 over the arctic semiring over the naturals
[active#(x1)] |
= |
+ · x1
|
[U11(x1, x2)] |
= |
+ · x1 + · x2
|
[tt] |
= |
|
[mark#(x1)] |
= |
+ · x1
|
[s(x1)] |
= |
+ · x1
|
[length(x1)] |
= |
+ · x1
|
[cons(x1, x2)] |
= |
+ · x1 + · x2
|
[and(x1, x2)] |
= |
+ · x1 + · x2
|
[isNatList(x1)] |
= |
+ · x1
|
[isNat(x1)] |
= |
+ · x1
|
[mark(x1)] |
= |
+ · x1
|
[active(x1)] |
= |
+ · x1
|
[zeros] |
= |
|
[0] |
= |
|
[isNatIList(x1)] |
= |
+ · x1
|
[nil] |
= |
|
the
pair
mark#(length(tt)) |
→ |
active#(length(active(tt))) |
(134) |
could be deleted.
1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor
Using the matrix interpretations of dimension 1 with strict dimension 1 over the arctic semiring over the naturals
[active#(x1)] |
= |
+ · x1
|
[U11(x1, x2)] |
= |
+ · x1 + · x2
|
[tt] |
= |
|
[mark#(x1)] |
= |
+ · x1
|
[s(x1)] |
= |
+ · x1
|
[length(x1)] |
= |
+ · x1
|
[cons(x1, x2)] |
= |
+ · x1 + · x2
|
[and(x1, x2)] |
= |
+ · x1 + · x2
|
[isNatList(x1)] |
= |
+ · x1
|
[isNat(x1)] |
= |
+ · x1
|
[mark(x1)] |
= |
+ · x1
|
[active(x1)] |
= |
+ · x1
|
[zeros] |
= |
|
[0] |
= |
|
[isNatIList(x1)] |
= |
+ · x1
|
[nil] |
= |
|
the
pair
mark#(length(isNat(x0))) |
→ |
active#(length(active(isNat(x0)))) |
(138) |
could be deleted.
1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor
Using the matrix interpretations of dimension 1 with strict dimension 1 over the arctic semiring over the naturals
[active#(x1)] |
= |
+ · x1
|
[U11(x1, x2)] |
= |
+ · x1 + · x2
|
[tt] |
= |
|
[mark#(x1)] |
= |
+ · x1
|
[s(x1)] |
= |
+ · x1
|
[length(x1)] |
= |
+ · x1
|
[cons(x1, x2)] |
= |
+ · x1 + · x2
|
[and(x1, x2)] |
= |
+ · x1 + · x2
|
[isNatList(x1)] |
= |
+ · x1
|
[isNat(x1)] |
= |
+ · x1
|
[mark(x1)] |
= |
+ · x1
|
[active(x1)] |
= |
+ · x1
|
[zeros] |
= |
|
[0] |
= |
|
[isNatIList(x1)] |
= |
+ · x1
|
[nil] |
= |
|
the
pair
mark#(length(isNatList(x0))) |
→ |
active#(length(active(isNatList(x0)))) |
(139) |
could be deleted.
1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor
Using the matrix interpretations of dimension 1 with strict dimension 1 over the arctic semiring over the naturals
[active#(x1)] |
= |
+ · x1
|
[U11(x1, x2)] |
= |
+ · x1 + · x2
|
[tt] |
= |
|
[mark#(x1)] |
= |
+ · x1
|
[s(x1)] |
= |
+ · x1
|
[length(x1)] |
= |
+ · x1
|
[cons(x1, x2)] |
= |
+ · x1 + · x2
|
[and(x1, x2)] |
= |
+ · x1 + · x2
|
[isNatList(x1)] |
= |
+ · x1
|
[isNat(x1)] |
= |
+ · x1
|
[mark(x1)] |
= |
+ · x1
|
[active(x1)] |
= |
+ · x1
|
[zeros] |
= |
|
[0] |
= |
|
[isNatIList(x1)] |
= |
+ · x1
|
[nil] |
= |
|
the
pairs
mark#(U11(y0,mark(x1))) |
→ |
active#(U11(mark(y0),x1)) |
(115) |
mark#(U11(y0,active(x1))) |
→ |
active#(U11(mark(y0),x1)) |
(116) |
could be deleted.
1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor
Using the matrix interpretations of dimension 1 with strict dimension 1 over the arctic semiring over the naturals
[active#(x1)] |
= |
+ · x1
|
[U11(x1, x2)] |
= |
+ · x1 + · x2
|
[tt] |
= |
|
[mark#(x1)] |
= |
+ · x1
|
[s(x1)] |
= |
+ · x1
|
[length(x1)] |
= |
+ · x1
|
[cons(x1, x2)] |
= |
+ · x1 + · x2
|
[and(x1, x2)] |
= |
+ · x1 + · x2
|
[isNatList(x1)] |
= |
+ · x1
|
[isNat(x1)] |
= |
+ · x1
|
[active(x1)] |
= |
+ · x1
|
[mark(x1)] |
= |
+ · x1
|
[zeros] |
= |
|
[0] |
= |
|
[isNatIList(x1)] |
= |
+ · x1
|
[nil] |
= |
|
the
pair
mark#(length(cons(x0,x1))) |
→ |
active#(length(active(cons(mark(x0),x1)))) |
(131) |
could be deleted.
1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor
Using the matrix interpretations of dimension 1 with strict dimension 1 over the arctic semiring over the naturals
[active#(x1)] |
= |
+ · x1
|
[U11(x1, x2)] |
= |
+ · x1 + · x2
|
[tt] |
= |
|
[mark#(x1)] |
= |
+ · x1
|
[s(x1)] |
= |
+ · x1
|
[length(x1)] |
= |
+ · x1
|
[cons(x1, x2)] |
= |
+ · x1 + · x2
|
[and(x1, x2)] |
= |
+ · x1 + · x2
|
[isNatList(x1)] |
= |
+ · x1
|
[isNat(x1)] |
= |
+ · x1
|
[active(x1)] |
= |
+ · x1
|
[mark(x1)] |
= |
+ · x1
|
[zeros] |
= |
|
[0] |
= |
|
[isNatIList(x1)] |
= |
+ · x1
|
[nil] |
= |
|
the
pairs
mark#(U11(x0,x1)) |
→ |
active#(U11(x0,x1)) |
(114) |
mark#(length(x0)) |
→ |
active#(length(x0)) |
(129) |
could be deleted.
1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor
Using the matrix interpretations of dimension 1 with strict dimension 1 over the arctic semiring over the naturals
[active#(x1)] |
= |
+ · x1
|
[U11(x1, x2)] |
= |
+ · x1 + · x2
|
[tt] |
= |
|
[mark#(x1)] |
= |
+ · x1
|
[s(x1)] |
= |
+ · x1
|
[length(x1)] |
= |
+ · x1
|
[cons(x1, x2)] |
= |
+ · x1 + · x2
|
[and(x1, x2)] |
= |
+ · x1 + · x2
|
[isNatList(x1)] |
= |
+ · x1
|
[isNat(x1)] |
= |
+ · x1
|
[active(x1)] |
= |
+ · x1
|
[mark(x1)] |
= |
+ · x1
|
[zeros] |
= |
|
[0] |
= |
|
[isNatIList(x1)] |
= |
+ · x1
|
[nil] |
= |
|
the
pairs
mark#(length(length(x0))) |
→ |
active#(length(active(length(mark(x0))))) |
(136) |
mark#(length(and(x0,x1))) |
→ |
active#(length(active(and(mark(x0),x1)))) |
(137) |
could be deleted.
1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor
Using the linear polynomial interpretation over the naturals
[active#(x1)] |
= |
-1 + 2 · x1
|
[mark#(x1)] |
= |
1 |
[s(x1)] |
= |
0 |
[length(x1)] |
= |
x1 |
[active(x1)] |
= |
x1 |
[mark(x1)] |
= |
x1 |
[U11(x1, x2)] |
= |
1 |
[and(x1, x2)] |
= |
x2 |
[isNatList(x1)] |
= |
0 |
[isNat(x1)] |
= |
0 |
[cons(x1, x2)] |
= |
1 |
[zeros] |
= |
1 |
[0] |
= |
0 |
[tt] |
= |
0 |
[isNatIList(x1)] |
= |
0 |
[nil] |
= |
0 |
the
pair
mark#(length(s(x0))) |
→ |
active#(length(active(s(mark(x0))))) |
(135) |
could be deleted.
1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor
Using the matrix interpretations of dimension 1 with strict dimension 1 over the arctic semiring over the naturals
[active#(x1)] |
= |
+ · x1
|
[U11(x1, x2)] |
= |
+ · x1 + · x2
|
[tt] |
= |
|
[mark#(x1)] |
= |
+ · x1
|
[s(x1)] |
= |
+ · x1
|
[length(x1)] |
= |
+ · x1
|
[cons(x1, x2)] |
= |
+ · x1 + · x2
|
[and(x1, x2)] |
= |
+ · x1 + · x2
|
[isNatList(x1)] |
= |
+ · x1
|
[isNat(x1)] |
= |
+ · x1
|
[active(x1)] |
= |
+ · x1
|
[mark(x1)] |
= |
+ · x1
|
[zeros] |
= |
|
[0] |
= |
|
[isNatIList(x1)] |
= |
+ · x1
|
[nil] |
= |
|
the
pair
mark#(length(U11(x0,x1))) |
→ |
active#(length(active(U11(mark(x0),x1)))) |
(133) |
could be deleted.
1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor
Using the non-linear polynomial interpretation over the naturals
[active#(x1)] |
= |
1 · x1
|
[U11(x1, x2)] |
= |
1 · x1 · x2
|
[tt] |
= |
1 |
[mark#(x1)] |
= |
1 · x1
|
[s(x1)] |
= |
1 · x1
|
[length(x1)] |
= |
1 · x1
|
[cons(x1, x2)] |
= |
1 |
[and(x1, x2)] |
= |
1 · x1 · x2
|
[isNatList(x1)] |
= |
0 |
[isNat(x1)] |
= |
1 · x1 · x1
|
[active(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[zeros] |
= |
1 |
[0] |
= |
1 |
[isNatIList(x1)] |
= |
0 |
[nil] |
= |
0 |
the
pair
active#(length(cons(N,L))) |
→ |
mark#(U11(and(isNatList(L),isNat(N)),L)) |
(65) |
could be deleted.
1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules
Using the Knuth Bendix order with w0 = 1 and the following precedence and weight functions
prec(U11) |
= |
1 |
|
weight(U11) |
= |
2 |
|
|
|
prec(length) |
= |
0 |
|
weight(length) |
= |
1 |
|
|
|
in combination with the following argument filter
π(active#) |
= |
1 |
π(U11) |
= |
[] |
π(mark#) |
= |
1 |
π(s) |
= |
1 |
π(length) |
= |
[] |
π(active) |
= |
1 |
π(mark) |
= |
1 |
together with the usable
rules
length(active(X)) |
→ |
length(X) |
(37) |
length(mark(X)) |
→ |
length(X) |
(36) |
s(active(X)) |
→ |
s(X) |
(35) |
s(mark(X)) |
→ |
s(X) |
(34) |
U11(X1,mark(X2)) |
→ |
U11(X1,X2) |
(31) |
U11(mark(X1),X2) |
→ |
U11(X1,X2) |
(30) |
U11(active(X1),X2) |
→ |
U11(X1,X2) |
(32) |
U11(X1,active(X2)) |
→ |
U11(X1,X2) |
(33) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
active#(U11(tt,L)) |
→ |
mark#(s(length(L))) |
(50) |
could be deleted.
1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Dependency Graph Processor
The dependency pairs are split into 1
component.
-
The
1st
component contains the
pair
mark#(s(X)) |
→ |
mark#(X) |
(81) |
1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[s(x1)] |
= |
1 · x1
|
[mark#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
mark#(s(X)) |
→ |
mark#(X) |
(81) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
2nd
component contains the
pair
cons#(X1,mark(X2)) |
→ |
cons#(X1,X2) |
(93) |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(92) |
cons#(active(X1),X2) |
→ |
cons#(X1,X2) |
(94) |
cons#(X1,active(X2)) |
→ |
cons#(X1,X2) |
(95) |
1.1.1.1.1.2 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[cons#(x1, x2)] |
= |
1 · x1 + 1 · x2
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.1.2.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
cons#(X1,mark(X2)) |
→ |
cons#(X1,X2) |
(93) |
|
1 |
≥ |
1 |
2 |
> |
2 |
cons#(mark(X1),X2) |
→ |
cons#(X1,X2) |
(92) |
|
1 |
> |
1 |
2 |
≥ |
2 |
cons#(active(X1),X2) |
→ |
cons#(X1,X2) |
(94) |
|
1 |
> |
1 |
2 |
≥ |
2 |
cons#(X1,active(X2)) |
→ |
cons#(X1,X2) |
(95) |
|
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#(X1,mark(X2)) |
→ |
U11#(X1,X2) |
(97) |
U11#(mark(X1),X2) |
→ |
U11#(X1,X2) |
(96) |
U11#(active(X1),X2) |
→ |
U11#(X1,X2) |
(98) |
U11#(X1,active(X2)) |
→ |
U11#(X1,X2) |
(99) |
1.1.1.1.1.3 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[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.
1.1.1.1.1.3.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
U11#(X1,mark(X2)) |
→ |
U11#(X1,X2) |
(97) |
|
1 |
≥ |
1 |
2 |
> |
2 |
U11#(mark(X1),X2) |
→ |
U11#(X1,X2) |
(96) |
|
1 |
> |
1 |
2 |
≥ |
2 |
U11#(active(X1),X2) |
→ |
U11#(X1,X2) |
(98) |
|
1 |
> |
1 |
2 |
≥ |
2 |
U11#(X1,active(X2)) |
→ |
U11#(X1,X2) |
(99) |
|
1 |
≥ |
1 |
2 |
> |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
4th
component contains the
pair
s#(active(X)) |
→ |
s#(X) |
(101) |
s#(mark(X)) |
→ |
s#(X) |
(100) |
1.1.1.1.1.4 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[s#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.1.4.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
s#(active(X)) |
→ |
s#(X) |
(101) |
|
1 |
> |
1 |
s#(mark(X)) |
→ |
s#(X) |
(100) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
5th
component contains the
pair
length#(active(X)) |
→ |
length#(X) |
(103) |
length#(mark(X)) |
→ |
length#(X) |
(102) |
1.1.1.1.1.5 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[length#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.1.5.1 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) |
(103) |
|
1 |
> |
1 |
length#(mark(X)) |
→ |
length#(X) |
(102) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
6th
component contains the
pair
and#(X1,mark(X2)) |
→ |
and#(X1,X2) |
(105) |
and#(mark(X1),X2) |
→ |
and#(X1,X2) |
(104) |
and#(active(X1),X2) |
→ |
and#(X1,X2) |
(106) |
and#(X1,active(X2)) |
→ |
and#(X1,X2) |
(107) |
1.1.1.1.1.6 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[and#(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.
1.1.1.1.1.6.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
and#(X1,mark(X2)) |
→ |
and#(X1,X2) |
(105) |
|
1 |
≥ |
1 |
2 |
> |
2 |
and#(mark(X1),X2) |
→ |
and#(X1,X2) |
(104) |
|
1 |
> |
1 |
2 |
≥ |
2 |
and#(active(X1),X2) |
→ |
and#(X1,X2) |
(106) |
|
1 |
> |
1 |
2 |
≥ |
2 |
and#(X1,active(X2)) |
→ |
and#(X1,X2) |
(107) |
|
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
isNat#(active(X)) |
→ |
isNat#(X) |
(109) |
isNat#(mark(X)) |
→ |
isNat#(X) |
(108) |
1.1.1.1.1.7 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.
1.1.1.1.1.7.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
isNat#(active(X)) |
→ |
isNat#(X) |
(109) |
|
1 |
> |
1 |
isNat#(mark(X)) |
→ |
isNat#(X) |
(108) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
8th
component contains the
pair
isNatList#(active(X)) |
→ |
isNatList#(X) |
(111) |
isNatList#(mark(X)) |
→ |
isNatList#(X) |
(110) |
1.1.1.1.1.8 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[isNatList#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.1.8.1 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) |
(111) |
|
1 |
> |
1 |
isNatList#(mark(X)) |
→ |
isNatList#(X) |
(110) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
9th
component contains the
pair
isNatIList#(active(X)) |
→ |
isNatIList#(X) |
(113) |
isNatIList#(mark(X)) |
→ |
isNatIList#(X) |
(112) |
1.1.1.1.1.9 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[isNatIList#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.1.1.1.9.1 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) |
(113) |
|
1 |
> |
1 |
isNatIList#(mark(X)) |
→ |
isNatIList#(X) |
(112) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.