The rewrite relation of the following TRS is considered.
active(U11(tt,V2)) |
→ |
mark(U12(isNat(V2))) |
(1) |
active(U12(tt)) |
→ |
mark(tt) |
(2) |
active(U21(tt)) |
→ |
mark(tt) |
(3) |
active(U31(tt,V2)) |
→ |
mark(U32(isNat(V2))) |
(4) |
active(U32(tt)) |
→ |
mark(tt) |
(5) |
active(U41(tt,N)) |
→ |
mark(N) |
(6) |
active(U51(tt,M,N)) |
→ |
mark(U52(isNat(N),M,N)) |
(7) |
active(U52(tt,M,N)) |
→ |
mark(s(plus(N,M))) |
(8) |
active(U61(tt)) |
→ |
mark(0) |
(9) |
active(U71(tt,M,N)) |
→ |
mark(U72(isNat(N),M,N)) |
(10) |
active(U72(tt,M,N)) |
→ |
mark(plus(x(N,M),N)) |
(11) |
active(isNat(0)) |
→ |
mark(tt) |
(12) |
active(isNat(plus(V1,V2))) |
→ |
mark(U11(isNat(V1),V2)) |
(13) |
active(isNat(s(V1))) |
→ |
mark(U21(isNat(V1))) |
(14) |
active(isNat(x(V1,V2))) |
→ |
mark(U31(isNat(V1),V2)) |
(15) |
active(plus(N,0)) |
→ |
mark(U41(isNat(N),N)) |
(16) |
active(plus(N,s(M))) |
→ |
mark(U51(isNat(M),M,N)) |
(17) |
active(x(N,0)) |
→ |
mark(U61(isNat(N))) |
(18) |
active(x(N,s(M))) |
→ |
mark(U71(isNat(M),M,N)) |
(19) |
mark(U11(X1,X2)) |
→ |
active(U11(mark(X1),X2)) |
(20) |
mark(tt) |
→ |
active(tt) |
(21) |
mark(U12(X)) |
→ |
active(U12(mark(X))) |
(22) |
mark(isNat(X)) |
→ |
active(isNat(X)) |
(23) |
mark(U21(X)) |
→ |
active(U21(mark(X))) |
(24) |
mark(U31(X1,X2)) |
→ |
active(U31(mark(X1),X2)) |
(25) |
mark(U32(X)) |
→ |
active(U32(mark(X))) |
(26) |
mark(U41(X1,X2)) |
→ |
active(U41(mark(X1),X2)) |
(27) |
mark(U51(X1,X2,X3)) |
→ |
active(U51(mark(X1),X2,X3)) |
(28) |
mark(U52(X1,X2,X3)) |
→ |
active(U52(mark(X1),X2,X3)) |
(29) |
mark(s(X)) |
→ |
active(s(mark(X))) |
(30) |
mark(plus(X1,X2)) |
→ |
active(plus(mark(X1),mark(X2))) |
(31) |
mark(U61(X)) |
→ |
active(U61(mark(X))) |
(32) |
mark(0) |
→ |
active(0) |
(33) |
mark(U71(X1,X2,X3)) |
→ |
active(U71(mark(X1),X2,X3)) |
(34) |
mark(U72(X1,X2,X3)) |
→ |
active(U72(mark(X1),X2,X3)) |
(35) |
mark(x(X1,X2)) |
→ |
active(x(mark(X1),mark(X2))) |
(36) |
U11(mark(X1),X2) |
→ |
U11(X1,X2) |
(37) |
U11(X1,mark(X2)) |
→ |
U11(X1,X2) |
(38) |
U11(active(X1),X2) |
→ |
U11(X1,X2) |
(39) |
U11(X1,active(X2)) |
→ |
U11(X1,X2) |
(40) |
U12(mark(X)) |
→ |
U12(X) |
(41) |
U12(active(X)) |
→ |
U12(X) |
(42) |
isNat(mark(X)) |
→ |
isNat(X) |
(43) |
isNat(active(X)) |
→ |
isNat(X) |
(44) |
U21(mark(X)) |
→ |
U21(X) |
(45) |
U21(active(X)) |
→ |
U21(X) |
(46) |
U31(mark(X1),X2) |
→ |
U31(X1,X2) |
(47) |
U31(X1,mark(X2)) |
→ |
U31(X1,X2) |
(48) |
U31(active(X1),X2) |
→ |
U31(X1,X2) |
(49) |
U31(X1,active(X2)) |
→ |
U31(X1,X2) |
(50) |
U32(mark(X)) |
→ |
U32(X) |
(51) |
U32(active(X)) |
→ |
U32(X) |
(52) |
U41(mark(X1),X2) |
→ |
U41(X1,X2) |
(53) |
U41(X1,mark(X2)) |
→ |
U41(X1,X2) |
(54) |
U41(active(X1),X2) |
→ |
U41(X1,X2) |
(55) |
U41(X1,active(X2)) |
→ |
U41(X1,X2) |
(56) |
U51(mark(X1),X2,X3) |
→ |
U51(X1,X2,X3) |
(57) |
U51(X1,mark(X2),X3) |
→ |
U51(X1,X2,X3) |
(58) |
U51(X1,X2,mark(X3)) |
→ |
U51(X1,X2,X3) |
(59) |
U51(active(X1),X2,X3) |
→ |
U51(X1,X2,X3) |
(60) |
U51(X1,active(X2),X3) |
→ |
U51(X1,X2,X3) |
(61) |
U51(X1,X2,active(X3)) |
→ |
U51(X1,X2,X3) |
(62) |
U52(mark(X1),X2,X3) |
→ |
U52(X1,X2,X3) |
(63) |
U52(X1,mark(X2),X3) |
→ |
U52(X1,X2,X3) |
(64) |
U52(X1,X2,mark(X3)) |
→ |
U52(X1,X2,X3) |
(65) |
U52(active(X1),X2,X3) |
→ |
U52(X1,X2,X3) |
(66) |
U52(X1,active(X2),X3) |
→ |
U52(X1,X2,X3) |
(67) |
U52(X1,X2,active(X3)) |
→ |
U52(X1,X2,X3) |
(68) |
s(mark(X)) |
→ |
s(X) |
(69) |
s(active(X)) |
→ |
s(X) |
(70) |
plus(mark(X1),X2) |
→ |
plus(X1,X2) |
(71) |
plus(X1,mark(X2)) |
→ |
plus(X1,X2) |
(72) |
plus(active(X1),X2) |
→ |
plus(X1,X2) |
(73) |
plus(X1,active(X2)) |
→ |
plus(X1,X2) |
(74) |
U61(mark(X)) |
→ |
U61(X) |
(75) |
U61(active(X)) |
→ |
U61(X) |
(76) |
U71(mark(X1),X2,X3) |
→ |
U71(X1,X2,X3) |
(77) |
U71(X1,mark(X2),X3) |
→ |
U71(X1,X2,X3) |
(78) |
U71(X1,X2,mark(X3)) |
→ |
U71(X1,X2,X3) |
(79) |
U71(active(X1),X2,X3) |
→ |
U71(X1,X2,X3) |
(80) |
U71(X1,active(X2),X3) |
→ |
U71(X1,X2,X3) |
(81) |
U71(X1,X2,active(X3)) |
→ |
U71(X1,X2,X3) |
(82) |
U72(mark(X1),X2,X3) |
→ |
U72(X1,X2,X3) |
(83) |
U72(X1,mark(X2),X3) |
→ |
U72(X1,X2,X3) |
(84) |
U72(X1,X2,mark(X3)) |
→ |
U72(X1,X2,X3) |
(85) |
U72(active(X1),X2,X3) |
→ |
U72(X1,X2,X3) |
(86) |
U72(X1,active(X2),X3) |
→ |
U72(X1,X2,X3) |
(87) |
U72(X1,X2,active(X3)) |
→ |
U72(X1,X2,X3) |
(88) |
x(mark(X1),X2) |
→ |
x(X1,X2) |
(89) |
x(X1,mark(X2)) |
→ |
x(X1,X2) |
(90) |
x(active(X1),X2) |
→ |
x(X1,X2) |
(91) |
x(X1,active(X2)) |
→ |
x(X1,X2) |
(92) |
There are 148 ruless (increase limit for explicit display).
The dependency pairs are split into 16
components.
-
The
1st
component contains the
pair
mark#(U11(X1,X2)) |
→ |
active#(U11(mark(X1),X2)) |
(138) |
active#(U11(tt,V2)) |
→ |
mark#(U12(isNat(V2))) |
(93) |
mark#(U11(X1,X2)) |
→ |
mark#(X1) |
(140) |
mark#(U12(X)) |
→ |
active#(U12(mark(X))) |
(142) |
active#(U31(tt,V2)) |
→ |
mark#(U32(isNat(V2))) |
(98) |
mark#(U12(X)) |
→ |
mark#(X) |
(144) |
mark#(isNat(X)) |
→ |
active#(isNat(X)) |
(145) |
active#(U41(tt,N)) |
→ |
mark#(N) |
(102) |
mark#(U21(X)) |
→ |
active#(U21(mark(X))) |
(146) |
active#(U51(tt,M,N)) |
→ |
mark#(U52(isNat(N),M,N)) |
(103) |
mark#(U21(X)) |
→ |
mark#(X) |
(148) |
mark#(U31(X1,X2)) |
→ |
active#(U31(mark(X1),X2)) |
(149) |
active#(U52(tt,M,N)) |
→ |
mark#(s(plus(N,M))) |
(106) |
mark#(U31(X1,X2)) |
→ |
mark#(X1) |
(151) |
mark#(U32(X)) |
→ |
active#(U32(mark(X))) |
(152) |
active#(U71(tt,M,N)) |
→ |
mark#(U72(isNat(N),M,N)) |
(110) |
mark#(U32(X)) |
→ |
mark#(X) |
(154) |
mark#(U41(X1,X2)) |
→ |
active#(U41(mark(X1),X2)) |
(155) |
active#(U72(tt,M,N)) |
→ |
mark#(plus(x(N,M),N)) |
(113) |
mark#(U41(X1,X2)) |
→ |
mark#(X1) |
(157) |
mark#(U51(X1,X2,X3)) |
→ |
active#(U51(mark(X1),X2,X3)) |
(158) |
active#(isNat(plus(V1,V2))) |
→ |
mark#(U11(isNat(V1),V2)) |
(117) |
mark#(U51(X1,X2,X3)) |
→ |
mark#(X1) |
(160) |
mark#(U52(X1,X2,X3)) |
→ |
active#(U52(mark(X1),X2,X3)) |
(161) |
active#(isNat(s(V1))) |
→ |
mark#(U21(isNat(V1))) |
(120) |
mark#(U52(X1,X2,X3)) |
→ |
mark#(X1) |
(163) |
mark#(s(X)) |
→ |
active#(s(mark(X))) |
(164) |
active#(isNat(x(V1,V2))) |
→ |
mark#(U31(isNat(V1),V2)) |
(123) |
mark#(s(X)) |
→ |
mark#(X) |
(166) |
mark#(plus(X1,X2)) |
→ |
active#(plus(mark(X1),mark(X2))) |
(167) |
active#(plus(N,0)) |
→ |
mark#(U41(isNat(N),N)) |
(126) |
mark#(plus(X1,X2)) |
→ |
mark#(X1) |
(169) |
mark#(plus(X1,X2)) |
→ |
mark#(X2) |
(170) |
mark#(U61(X)) |
→ |
active#(U61(mark(X))) |
(171) |
active#(plus(N,s(M))) |
→ |
mark#(U51(isNat(M),M,N)) |
(129) |
mark#(U61(X)) |
→ |
mark#(X) |
(173) |
mark#(U71(X1,X2,X3)) |
→ |
active#(U71(mark(X1),X2,X3)) |
(175) |
active#(x(N,0)) |
→ |
mark#(U61(isNat(N))) |
(132) |
mark#(U71(X1,X2,X3)) |
→ |
mark#(X1) |
(177) |
mark#(U72(X1,X2,X3)) |
→ |
active#(U72(mark(X1),X2,X3)) |
(178) |
active#(x(N,s(M))) |
→ |
mark#(U71(isNat(M),M,N)) |
(135) |
mark#(U72(X1,X2,X3)) |
→ |
mark#(X1) |
(180) |
mark#(x(X1,X2)) |
→ |
active#(x(mark(X1),mark(X2))) |
(181) |
mark#(x(X1,X2)) |
→ |
mark#(X1) |
(183) |
mark#(x(X1,X2)) |
→ |
mark#(X2) |
(184) |
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 |
[U12(x1)] |
= |
-2 |
[U21(x1)] |
= |
0 |
[U31(x1, x2)] |
= |
2 |
[U32(x1)] |
= |
-2 |
[U41(x1, x2)] |
= |
2 |
[U51(x1, x2, x3)] |
= |
2 |
[U52(x1, x2, x3)] |
= |
2 |
[U61(x1)] |
= |
2 |
[U71(x1, x2, x3)] |
= |
2 |
[U72(x1, x2, x3)] |
= |
2 |
[plus(x1, x2)] |
= |
2 |
[s(x1)] |
= |
-2 |
[x(x1, x2)] |
= |
2 |
[mark(x1)] |
= |
-2 |
[active(x1)] |
= |
-2 |
[tt] |
= |
0 |
[isNat(x1)] |
= |
2 |
[0] |
= |
0 |
[mark#(x1)] |
= |
1 |
together with the usable
rules
U11(X1,mark(X2)) |
→ |
U11(X1,X2) |
(38) |
U11(mark(X1),X2) |
→ |
U11(X1,X2) |
(37) |
U11(active(X1),X2) |
→ |
U11(X1,X2) |
(39) |
U11(X1,active(X2)) |
→ |
U11(X1,X2) |
(40) |
isNat(active(X)) |
→ |
isNat(X) |
(44) |
isNat(mark(X)) |
→ |
isNat(X) |
(43) |
U12(active(X)) |
→ |
U12(X) |
(42) |
U12(mark(X)) |
→ |
U12(X) |
(41) |
U32(active(X)) |
→ |
U32(X) |
(52) |
U32(mark(X)) |
→ |
U32(X) |
(51) |
U21(active(X)) |
→ |
U21(X) |
(46) |
U21(mark(X)) |
→ |
U21(X) |
(45) |
U52(X1,mark(X2),X3) |
→ |
U52(X1,X2,X3) |
(64) |
U52(mark(X1),X2,X3) |
→ |
U52(X1,X2,X3) |
(63) |
U52(X1,X2,mark(X3)) |
→ |
U52(X1,X2,X3) |
(65) |
U52(active(X1),X2,X3) |
→ |
U52(X1,X2,X3) |
(66) |
U52(X1,active(X2),X3) |
→ |
U52(X1,X2,X3) |
(67) |
U52(X1,X2,active(X3)) |
→ |
U52(X1,X2,X3) |
(68) |
U31(X1,mark(X2)) |
→ |
U31(X1,X2) |
(48) |
U31(mark(X1),X2) |
→ |
U31(X1,X2) |
(47) |
U31(active(X1),X2) |
→ |
U31(X1,X2) |
(49) |
U31(X1,active(X2)) |
→ |
U31(X1,X2) |
(50) |
plus(X1,mark(X2)) |
→ |
plus(X1,X2) |
(72) |
plus(mark(X1),X2) |
→ |
plus(X1,X2) |
(71) |
plus(active(X1),X2) |
→ |
plus(X1,X2) |
(73) |
plus(X1,active(X2)) |
→ |
plus(X1,X2) |
(74) |
s(active(X)) |
→ |
s(X) |
(70) |
s(mark(X)) |
→ |
s(X) |
(69) |
U72(X1,mark(X2),X3) |
→ |
U72(X1,X2,X3) |
(84) |
U72(mark(X1),X2,X3) |
→ |
U72(X1,X2,X3) |
(83) |
U72(X1,X2,mark(X3)) |
→ |
U72(X1,X2,X3) |
(85) |
U72(active(X1),X2,X3) |
→ |
U72(X1,X2,X3) |
(86) |
U72(X1,active(X2),X3) |
→ |
U72(X1,X2,X3) |
(87) |
U72(X1,X2,active(X3)) |
→ |
U72(X1,X2,X3) |
(88) |
U41(X1,mark(X2)) |
→ |
U41(X1,X2) |
(54) |
U41(mark(X1),X2) |
→ |
U41(X1,X2) |
(53) |
U41(active(X1),X2) |
→ |
U41(X1,X2) |
(55) |
U41(X1,active(X2)) |
→ |
U41(X1,X2) |
(56) |
x(X1,mark(X2)) |
→ |
x(X1,X2) |
(90) |
x(mark(X1),X2) |
→ |
x(X1,X2) |
(89) |
x(active(X1),X2) |
→ |
x(X1,X2) |
(91) |
x(X1,active(X2)) |
→ |
x(X1,X2) |
(92) |
U51(X1,mark(X2),X3) |
→ |
U51(X1,X2,X3) |
(58) |
U51(mark(X1),X2,X3) |
→ |
U51(X1,X2,X3) |
(57) |
U51(X1,X2,mark(X3)) |
→ |
U51(X1,X2,X3) |
(59) |
U51(active(X1),X2,X3) |
→ |
U51(X1,X2,X3) |
(60) |
U51(X1,active(X2),X3) |
→ |
U51(X1,X2,X3) |
(61) |
U51(X1,X2,active(X3)) |
→ |
U51(X1,X2,X3) |
(62) |
U61(active(X)) |
→ |
U61(X) |
(76) |
U61(mark(X)) |
→ |
U61(X) |
(75) |
U71(X1,mark(X2),X3) |
→ |
U71(X1,X2,X3) |
(78) |
U71(mark(X1),X2,X3) |
→ |
U71(X1,X2,X3) |
(77) |
U71(X1,X2,mark(X3)) |
→ |
U71(X1,X2,X3) |
(79) |
U71(active(X1),X2,X3) |
→ |
U71(X1,X2,X3) |
(80) |
U71(X1,active(X2),X3) |
→ |
U71(X1,X2,X3) |
(81) |
U71(X1,X2,active(X3)) |
→ |
U71(X1,X2,X3) |
(82) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
mark#(U12(X)) |
→ |
active#(U12(mark(X))) |
(142) |
mark#(U21(X)) |
→ |
active#(U21(mark(X))) |
(146) |
mark#(U32(X)) |
→ |
active#(U32(mark(X))) |
(152) |
mark#(s(X)) |
→ |
active#(s(mark(X))) |
(164) |
could be deleted.
1.1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark#(x1)] |
= |
1 |
[U11(x1, x2)] |
= |
1 |
[active#(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
0 |
[tt] |
= |
0 |
[U12(x1)] |
= |
0 |
[isNat(x1)] |
= |
1 |
[U31(x1, x2)] |
= |
1 |
[U32(x1)] |
= |
0 |
[U41(x1, x2)] |
= |
1 |
[U51(x1, x2, x3)] |
= |
1 |
[U52(x1, x2, x3)] |
= |
1 |
[U21(x1)] |
= |
0 |
[s(x1)] |
= |
0 |
[plus(x1, x2)] |
= |
1 |
[U71(x1, x2, x3)] |
= |
1 |
[U72(x1, x2, x3)] |
= |
1 |
[x(x1, x2)] |
= |
1 |
[0] |
= |
0 |
[U61(x1)] |
= |
0 |
[active(x1)] |
= |
0 |
together with the usable
rules
U11(X1,mark(X2)) |
→ |
U11(X1,X2) |
(38) |
U11(mark(X1),X2) |
→ |
U11(X1,X2) |
(37) |
U11(active(X1),X2) |
→ |
U11(X1,X2) |
(39) |
U11(X1,active(X2)) |
→ |
U11(X1,X2) |
(40) |
isNat(active(X)) |
→ |
isNat(X) |
(44) |
isNat(mark(X)) |
→ |
isNat(X) |
(43) |
U52(X1,mark(X2),X3) |
→ |
U52(X1,X2,X3) |
(64) |
U52(mark(X1),X2,X3) |
→ |
U52(X1,X2,X3) |
(63) |
U52(X1,X2,mark(X3)) |
→ |
U52(X1,X2,X3) |
(65) |
U52(active(X1),X2,X3) |
→ |
U52(X1,X2,X3) |
(66) |
U52(X1,active(X2),X3) |
→ |
U52(X1,X2,X3) |
(67) |
U52(X1,X2,active(X3)) |
→ |
U52(X1,X2,X3) |
(68) |
U31(X1,mark(X2)) |
→ |
U31(X1,X2) |
(48) |
U31(mark(X1),X2) |
→ |
U31(X1,X2) |
(47) |
U31(active(X1),X2) |
→ |
U31(X1,X2) |
(49) |
U31(X1,active(X2)) |
→ |
U31(X1,X2) |
(50) |
plus(X1,mark(X2)) |
→ |
plus(X1,X2) |
(72) |
plus(mark(X1),X2) |
→ |
plus(X1,X2) |
(71) |
plus(active(X1),X2) |
→ |
plus(X1,X2) |
(73) |
plus(X1,active(X2)) |
→ |
plus(X1,X2) |
(74) |
U72(X1,mark(X2),X3) |
→ |
U72(X1,X2,X3) |
(84) |
U72(mark(X1),X2,X3) |
→ |
U72(X1,X2,X3) |
(83) |
U72(X1,X2,mark(X3)) |
→ |
U72(X1,X2,X3) |
(85) |
U72(active(X1),X2,X3) |
→ |
U72(X1,X2,X3) |
(86) |
U72(X1,active(X2),X3) |
→ |
U72(X1,X2,X3) |
(87) |
U72(X1,X2,active(X3)) |
→ |
U72(X1,X2,X3) |
(88) |
U41(X1,mark(X2)) |
→ |
U41(X1,X2) |
(54) |
U41(mark(X1),X2) |
→ |
U41(X1,X2) |
(53) |
U41(active(X1),X2) |
→ |
U41(X1,X2) |
(55) |
U41(X1,active(X2)) |
→ |
U41(X1,X2) |
(56) |
x(X1,mark(X2)) |
→ |
x(X1,X2) |
(90) |
x(mark(X1),X2) |
→ |
x(X1,X2) |
(89) |
x(active(X1),X2) |
→ |
x(X1,X2) |
(91) |
x(X1,active(X2)) |
→ |
x(X1,X2) |
(92) |
U51(X1,mark(X2),X3) |
→ |
U51(X1,X2,X3) |
(58) |
U51(mark(X1),X2,X3) |
→ |
U51(X1,X2,X3) |
(57) |
U51(X1,X2,mark(X3)) |
→ |
U51(X1,X2,X3) |
(59) |
U51(active(X1),X2,X3) |
→ |
U51(X1,X2,X3) |
(60) |
U51(X1,active(X2),X3) |
→ |
U51(X1,X2,X3) |
(61) |
U51(X1,X2,active(X3)) |
→ |
U51(X1,X2,X3) |
(62) |
U61(active(X)) |
→ |
U61(X) |
(76) |
U61(mark(X)) |
→ |
U61(X) |
(75) |
U71(X1,mark(X2),X3) |
→ |
U71(X1,X2,X3) |
(78) |
U71(mark(X1),X2,X3) |
→ |
U71(X1,X2,X3) |
(77) |
U71(X1,X2,mark(X3)) |
→ |
U71(X1,X2,X3) |
(79) |
U71(active(X1),X2,X3) |
→ |
U71(X1,X2,X3) |
(80) |
U71(X1,active(X2),X3) |
→ |
U71(X1,X2,X3) |
(81) |
U71(X1,X2,active(X3)) |
→ |
U71(X1,X2,X3) |
(82) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
mark#(U61(X)) |
→ |
active#(U61(mark(X))) |
(171) |
could be deleted.
1.1.1.1.1 Reduction Pair Processor
Using the
prec(mark#) |
= |
2 |
|
stat(mark#) |
= |
mul
|
prec(active#) |
= |
2 |
|
stat(active#) |
= |
mul
|
prec(tt) |
= |
0 |
|
stat(tt) |
= |
mul
|
prec(isNat) |
= |
0 |
|
stat(isNat) |
= |
mul
|
prec(U41) |
= |
1 |
|
stat(U41) |
= |
mul
|
prec(U51) |
= |
1 |
|
stat(U51) |
= |
mul
|
prec(U52) |
= |
1 |
|
stat(U52) |
= |
mul
|
prec(s) |
= |
0 |
|
stat(s) |
= |
mul
|
prec(plus) |
= |
1 |
|
stat(plus) |
= |
mul
|
prec(U71) |
= |
2 |
|
stat(U71) |
= |
mul
|
prec(U72) |
= |
2 |
|
stat(U72) |
= |
mul
|
prec(x) |
= |
2 |
|
stat(x) |
= |
mul
|
prec(0) |
= |
2 |
|
stat(0) |
= |
mul
|
prec(U61) |
= |
2 |
|
stat(U61) |
= |
mul
|
π(mark#) |
= |
[1] |
π(U11) |
= |
1 |
π(active#) |
= |
[1] |
π(mark) |
= |
1 |
π(tt) |
= |
[] |
π(U12) |
= |
1 |
π(isNat) |
= |
[] |
π(U31) |
= |
1 |
π(U32) |
= |
1 |
π(U41) |
= |
[1,2] |
π(U51) |
= |
[1,2,3] |
π(U52) |
= |
[1,2,3] |
π(U21) |
= |
1 |
π(s) |
= |
[1] |
π(plus) |
= |
[1,2] |
π(U71) |
= |
[1,2,3] |
π(U72) |
= |
[1,2,3] |
π(x) |
= |
[1,2] |
π(0) |
= |
[] |
π(U61) |
= |
[1] |
π(active) |
= |
1 |
the
pairs
active#(U41(tt,N)) |
→ |
mark#(N) |
(102) |
active#(U52(tt,M,N)) |
→ |
mark#(s(plus(N,M))) |
(106) |
active#(U72(tt,M,N)) |
→ |
mark#(plus(x(N,M),N)) |
(113) |
mark#(U41(X1,X2)) |
→ |
mark#(X1) |
(157) |
mark#(U51(X1,X2,X3)) |
→ |
mark#(X1) |
(160) |
mark#(U52(X1,X2,X3)) |
→ |
mark#(X1) |
(163) |
mark#(s(X)) |
→ |
mark#(X) |
(166) |
active#(plus(N,0)) |
→ |
mark#(U41(isNat(N),N)) |
(126) |
mark#(plus(X1,X2)) |
→ |
mark#(X1) |
(169) |
mark#(plus(X1,X2)) |
→ |
mark#(X2) |
(170) |
active#(plus(N,s(M))) |
→ |
mark#(U51(isNat(M),M,N)) |
(129) |
mark#(U61(X)) |
→ |
mark#(X) |
(173) |
active#(x(N,0)) |
→ |
mark#(U61(isNat(N))) |
(132) |
mark#(U71(X1,X2,X3)) |
→ |
mark#(X1) |
(177) |
active#(x(N,s(M))) |
→ |
mark#(U71(isNat(M),M,N)) |
(135) |
mark#(U72(X1,X2,X3)) |
→ |
mark#(X1) |
(180) |
mark#(x(X1,X2)) |
→ |
mark#(X1) |
(183) |
mark#(x(X1,X2)) |
→ |
mark#(X2) |
(184) |
could be deleted.
1.1.1.1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active#(x1)] |
= |
-2 |
[U11(x1, x2)] |
= |
1 + 2 · x1
|
[U31(x1, x2)] |
= |
1 + 2 · x1
|
[U41(x1, x2)] |
= |
2 |
[U51(x1, x2, x3)] |
= |
2 + x2 + 2 · x3
|
[U52(x1, x2, x3)] |
= |
-2 |
[U71(x1, x2, x3)] |
= |
2 + x2 + x3
|
[U72(x1, x2, x3)] |
= |
1 |
[plus(x1, x2)] |
= |
2 |
[x(x1, x2)] |
= |
2 |
[mark(x1)] |
= |
2 · x1
|
[active(x1)] |
= |
2 + 2 · x1
|
[tt] |
= |
0 |
[U12(x1)] |
= |
1 + 2 · x1
|
[isNat(x1)] |
= |
0 |
[U32(x1)] |
= |
1 + x1
|
[U21(x1)] |
= |
1 + 2 · x1
|
[s(x1)] |
= |
-2 |
[0] |
= |
0 |
[U61(x1)] |
= |
2 + x1
|
[mark#(x1)] |
= |
-2 + 2 · x1
|
together with the usable
rules
U11(X1,mark(X2)) |
→ |
U11(X1,X2) |
(38) |
U11(mark(X1),X2) |
→ |
U11(X1,X2) |
(37) |
U11(active(X1),X2) |
→ |
U11(X1,X2) |
(39) |
U11(X1,active(X2)) |
→ |
U11(X1,X2) |
(40) |
isNat(active(X)) |
→ |
isNat(X) |
(44) |
isNat(mark(X)) |
→ |
isNat(X) |
(43) |
U12(active(X)) |
→ |
U12(X) |
(42) |
U12(mark(X)) |
→ |
U12(X) |
(41) |
U32(active(X)) |
→ |
U32(X) |
(52) |
U32(mark(X)) |
→ |
U32(X) |
(51) |
U52(X1,mark(X2),X3) |
→ |
U52(X1,X2,X3) |
(64) |
U52(mark(X1),X2,X3) |
→ |
U52(X1,X2,X3) |
(63) |
U52(X1,X2,mark(X3)) |
→ |
U52(X1,X2,X3) |
(65) |
U52(active(X1),X2,X3) |
→ |
U52(X1,X2,X3) |
(66) |
U52(X1,active(X2),X3) |
→ |
U52(X1,X2,X3) |
(67) |
U52(X1,X2,active(X3)) |
→ |
U52(X1,X2,X3) |
(68) |
U31(X1,mark(X2)) |
→ |
U31(X1,X2) |
(48) |
U31(mark(X1),X2) |
→ |
U31(X1,X2) |
(47) |
U31(active(X1),X2) |
→ |
U31(X1,X2) |
(49) |
U31(X1,active(X2)) |
→ |
U31(X1,X2) |
(50) |
U72(X1,mark(X2),X3) |
→ |
U72(X1,X2,X3) |
(84) |
U72(mark(X1),X2,X3) |
→ |
U72(X1,X2,X3) |
(83) |
U72(X1,X2,mark(X3)) |
→ |
U72(X1,X2,X3) |
(85) |
U72(active(X1),X2,X3) |
→ |
U72(X1,X2,X3) |
(86) |
U72(X1,active(X2),X3) |
→ |
U72(X1,X2,X3) |
(87) |
U72(X1,X2,active(X3)) |
→ |
U72(X1,X2,X3) |
(88) |
U21(active(X)) |
→ |
U21(X) |
(46) |
U21(mark(X)) |
→ |
U21(X) |
(45) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
mark#(U41(X1,X2)) |
→ |
active#(U41(mark(X1),X2)) |
(155) |
mark#(U51(X1,X2,X3)) |
→ |
active#(U51(mark(X1),X2,X3)) |
(158) |
mark#(plus(X1,X2)) |
→ |
active#(plus(mark(X1),mark(X2))) |
(167) |
mark#(U71(X1,X2,X3)) |
→ |
active#(U71(mark(X1),X2,X3)) |
(175) |
mark#(x(X1,X2)) |
→ |
active#(x(mark(X1),mark(X2))) |
(181) |
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)] |
= |
2 + x1
|
[U11(x1, x2)] |
= |
0 |
[U31(x1, x2)] |
= |
-2 |
[U52(x1, x2, x3)] |
= |
-2 |
[U72(x1, x2, x3)] |
= |
-2 |
[mark(x1)] |
= |
2 |
[active(x1)] |
= |
-2 + x1
|
[tt] |
= |
2 |
[U12(x1)] |
= |
2 |
[isNat(x1)] |
= |
0 |
[U32(x1)] |
= |
2 |
[U41(x1, x2)] |
= |
-2 + 2 · x2
|
[U21(x1)] |
= |
2 |
[U51(x1, x2, x3)] |
= |
-2 + x1
|
[s(x1)] |
= |
2 + x1
|
[plus(x1, x2)] |
= |
-2 |
[U71(x1, x2, x3)] |
= |
2 + 2 · x1
|
[x(x1, x2)] |
= |
-2 + x1 + x2
|
[0] |
= |
0 |
[U61(x1)] |
= |
-2 |
[mark#(x1)] |
= |
2 |
together with the usable
rules
U11(X1,mark(X2)) |
→ |
U11(X1,X2) |
(38) |
U11(mark(X1),X2) |
→ |
U11(X1,X2) |
(37) |
U11(active(X1),X2) |
→ |
U11(X1,X2) |
(39) |
U11(X1,active(X2)) |
→ |
U11(X1,X2) |
(40) |
isNat(active(X)) |
→ |
isNat(X) |
(44) |
isNat(mark(X)) |
→ |
isNat(X) |
(43) |
U52(X1,mark(X2),X3) |
→ |
U52(X1,X2,X3) |
(64) |
U52(mark(X1),X2,X3) |
→ |
U52(X1,X2,X3) |
(63) |
U52(X1,X2,mark(X3)) |
→ |
U52(X1,X2,X3) |
(65) |
U52(active(X1),X2,X3) |
→ |
U52(X1,X2,X3) |
(66) |
U52(X1,active(X2),X3) |
→ |
U52(X1,X2,X3) |
(67) |
U52(X1,X2,active(X3)) |
→ |
U52(X1,X2,X3) |
(68) |
U31(X1,mark(X2)) |
→ |
U31(X1,X2) |
(48) |
U31(mark(X1),X2) |
→ |
U31(X1,X2) |
(47) |
U31(active(X1),X2) |
→ |
U31(X1,X2) |
(49) |
U31(X1,active(X2)) |
→ |
U31(X1,X2) |
(50) |
U72(X1,mark(X2),X3) |
→ |
U72(X1,X2,X3) |
(84) |
U72(mark(X1),X2,X3) |
→ |
U72(X1,X2,X3) |
(83) |
U72(X1,X2,mark(X3)) |
→ |
U72(X1,X2,X3) |
(85) |
U72(active(X1),X2,X3) |
→ |
U72(X1,X2,X3) |
(86) |
U72(X1,active(X2),X3) |
→ |
U72(X1,X2,X3) |
(87) |
U72(X1,X2,active(X3)) |
→ |
U72(X1,X2,X3) |
(88) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
active#(U71(tt,M,N)) |
→ |
mark#(U72(isNat(N),M,N)) |
(110) |
could be deleted.
1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active#(x1)] |
= |
x1 |
[U11(x1, x2)] |
= |
-2 |
[U31(x1, x2)] |
= |
-2 |
[U52(x1, x2, x3)] |
= |
-2 |
[U72(x1, x2, x3)] |
= |
0 |
[mark(x1)] |
= |
1 |
[active(x1)] |
= |
-2 + x1
|
[tt] |
= |
1 |
[U12(x1)] |
= |
0 |
[isNat(x1)] |
= |
0 |
[U32(x1)] |
= |
2 |
[U41(x1, x2)] |
= |
-1 + 2 · x1 + 2 · x2
|
[U21(x1)] |
= |
-2 |
[U51(x1, x2, x3)] |
= |
1 + 2 · x1 + x2
|
[s(x1)] |
= |
2 + x1
|
[plus(x1, x2)] |
= |
2 + x2
|
[U71(x1, x2, x3)] |
= |
-2 + x2 + x3
|
[x(x1, x2)] |
= |
2 + 2 · x2
|
[0] |
= |
0 |
[U61(x1)] |
= |
-2 |
[mark#(x1)] |
= |
-2 |
together with the usable
rules
U11(X1,mark(X2)) |
→ |
U11(X1,X2) |
(38) |
U11(mark(X1),X2) |
→ |
U11(X1,X2) |
(37) |
U11(active(X1),X2) |
→ |
U11(X1,X2) |
(39) |
U11(X1,active(X2)) |
→ |
U11(X1,X2) |
(40) |
isNat(active(X)) |
→ |
isNat(X) |
(44) |
isNat(mark(X)) |
→ |
isNat(X) |
(43) |
U52(X1,mark(X2),X3) |
→ |
U52(X1,X2,X3) |
(64) |
U52(mark(X1),X2,X3) |
→ |
U52(X1,X2,X3) |
(63) |
U52(X1,X2,mark(X3)) |
→ |
U52(X1,X2,X3) |
(65) |
U52(active(X1),X2,X3) |
→ |
U52(X1,X2,X3) |
(66) |
U52(X1,active(X2),X3) |
→ |
U52(X1,X2,X3) |
(67) |
U52(X1,X2,active(X3)) |
→ |
U52(X1,X2,X3) |
(68) |
U31(X1,mark(X2)) |
→ |
U31(X1,X2) |
(48) |
U31(mark(X1),X2) |
→ |
U31(X1,X2) |
(47) |
U31(active(X1),X2) |
→ |
U31(X1,X2) |
(49) |
U31(X1,active(X2)) |
→ |
U31(X1,X2) |
(50) |
U72(X1,mark(X2),X3) |
→ |
U72(X1,X2,X3) |
(84) |
U72(mark(X1),X2,X3) |
→ |
U72(X1,X2,X3) |
(83) |
U72(X1,X2,mark(X3)) |
→ |
U72(X1,X2,X3) |
(85) |
U72(active(X1),X2,X3) |
→ |
U72(X1,X2,X3) |
(86) |
U72(X1,active(X2),X3) |
→ |
U72(X1,X2,X3) |
(87) |
U72(X1,X2,active(X3)) |
→ |
U72(X1,X2,X3) |
(88) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
active#(U51(tt,M,N)) |
→ |
mark#(U52(isNat(N),M,N)) |
(103) |
could be deleted.
1.1.1.1.1.1.1.1.1 Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active#(x1)] |
= |
x1 |
[U11(x1, x2)] |
= |
2 |
[U31(x1, x2)] |
= |
2 |
[U52(x1, x2, x3)] |
= |
1 |
[U72(x1, x2, x3)] |
= |
1 |
[mark(x1)] |
= |
-2 |
[active(x1)] |
= |
-2 + x1
|
[tt] |
= |
0 |
[U12(x1)] |
= |
2 |
[isNat(x1)] |
= |
2 |
[U32(x1)] |
= |
-2 |
[U41(x1, x2)] |
= |
-2 + 2 · x2
|
[U21(x1)] |
= |
-2 |
[U51(x1, x2, x3)] |
= |
-2 + 2 · x2
|
[s(x1)] |
= |
-2 + 2 · x1
|
[plus(x1, x2)] |
= |
-2 + x1
|
[U71(x1, x2, x3)] |
= |
-2 + 2 · x2 + 2 · x3
|
[x(x1, x2)] |
= |
-2 + 2 · x1 + 2 · x2
|
[0] |
= |
0 |
[U61(x1)] |
= |
2 |
[mark#(x1)] |
= |
2 |
together with the usable
rules
U11(X1,mark(X2)) |
→ |
U11(X1,X2) |
(38) |
U11(mark(X1),X2) |
→ |
U11(X1,X2) |
(37) |
U11(active(X1),X2) |
→ |
U11(X1,X2) |
(39) |
U11(X1,active(X2)) |
→ |
U11(X1,X2) |
(40) |
isNat(active(X)) |
→ |
isNat(X) |
(44) |
isNat(mark(X)) |
→ |
isNat(X) |
(43) |
U31(X1,mark(X2)) |
→ |
U31(X1,X2) |
(48) |
U31(mark(X1),X2) |
→ |
U31(X1,X2) |
(47) |
U31(active(X1),X2) |
→ |
U31(X1,X2) |
(49) |
U31(X1,active(X2)) |
→ |
U31(X1,X2) |
(50) |
U52(X1,mark(X2),X3) |
→ |
U52(X1,X2,X3) |
(64) |
U52(mark(X1),X2,X3) |
→ |
U52(X1,X2,X3) |
(63) |
U52(X1,X2,mark(X3)) |
→ |
U52(X1,X2,X3) |
(65) |
U52(active(X1),X2,X3) |
→ |
U52(X1,X2,X3) |
(66) |
U52(X1,active(X2),X3) |
→ |
U52(X1,X2,X3) |
(67) |
U52(X1,X2,active(X3)) |
→ |
U52(X1,X2,X3) |
(68) |
U72(X1,mark(X2),X3) |
→ |
U72(X1,X2,X3) |
(84) |
U72(mark(X1),X2,X3) |
→ |
U72(X1,X2,X3) |
(83) |
U72(X1,X2,mark(X3)) |
→ |
U72(X1,X2,X3) |
(85) |
U72(active(X1),X2,X3) |
→ |
U72(X1,X2,X3) |
(86) |
U72(X1,active(X2),X3) |
→ |
U72(X1,X2,X3) |
(87) |
U72(X1,X2,active(X3)) |
→ |
U72(X1,X2,X3) |
(88) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
mark#(U52(X1,X2,X3)) |
→ |
active#(U52(mark(X1),X2,X3)) |
(161) |
mark#(U72(X1,X2,X3)) |
→ |
active#(U72(mark(X1),X2,X3)) |
(178) |
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
[mark#(x1)] |
= |
+ · x1
|
[U11(x1, x2)] |
= |
+ · x1 + · x2
|
[active#(x1)] |
= |
+ · x1
|
[mark(x1)] |
= |
+ · x1
|
[tt] |
= |
|
[U12(x1)] |
= |
+ · x1
|
[isNat(x1)] |
= |
+ · x1
|
[U31(x1, x2)] |
= |
+ · x1 + · x2
|
[U32(x1)] |
= |
+ · x1
|
[U21(x1)] |
= |
+ · x1
|
[plus(x1, x2)] |
= |
+ · x1 + · x2
|
[s(x1)] |
= |
+ · x1
|
[x(x1, x2)] |
= |
+ · x1 + · x2
|
[active(x1)] |
= |
+ · x1
|
[U41(x1, x2)] |
= |
+ · x1 + · x2
|
[U51(x1, x2, x3)] |
= |
+ · x1 + · x2 + · x3
|
[U52(x1, x2, x3)] |
= |
+ · x1 + · x2 + · x3
|
[U71(x1, x2, x3)] |
= |
+ · x1 + · x2 + · x3
|
[U72(x1, x2, x3)] |
= |
+ · x1 + · x2 + · x3
|
[0] |
= |
|
[U61(x1)] |
= |
+ · x1
|
the
pair
active#(U31(tt,V2)) |
→ |
mark#(U32(isNat(V2))) |
(98) |
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)] |
= |
x1 |
[U11(x1, x2)] |
= |
2 |
[U31(x1, x2)] |
= |
-2 |
[mark(x1)] |
= |
-2 + x1
|
[active(x1)] |
= |
-2 + x1
|
[tt] |
= |
2 |
[U12(x1)] |
= |
-2 |
[isNat(x1)] |
= |
2 |
[U32(x1)] |
= |
0 |
[U41(x1, x2)] |
= |
2 |
[U21(x1)] |
= |
2 · x1
|
[U51(x1, x2, x3)] |
= |
2 |
[U52(x1, x2, x3)] |
= |
-2 + x1 + x3
|
[s(x1)] |
= |
-2 + x1
|
[plus(x1, x2)] |
= |
-2 + 2 · x2
|
[U71(x1, x2, x3)] |
= |
-2 + x1 + 2 · x2
|
[U72(x1, x2, x3)] |
= |
2 + 2 · x2
|
[x(x1, x2)] |
= |
-2 + 2 · x1
|
[0] |
= |
0 |
[U61(x1)] |
= |
-2 |
[mark#(x1)] |
= |
2 |
together with the usable
rules
U11(X1,mark(X2)) |
→ |
U11(X1,X2) |
(38) |
U11(mark(X1),X2) |
→ |
U11(X1,X2) |
(37) |
U11(active(X1),X2) |
→ |
U11(X1,X2) |
(39) |
U11(X1,active(X2)) |
→ |
U11(X1,X2) |
(40) |
isNat(active(X)) |
→ |
isNat(X) |
(44) |
isNat(mark(X)) |
→ |
isNat(X) |
(43) |
U31(X1,mark(X2)) |
→ |
U31(X1,X2) |
(48) |
U31(mark(X1),X2) |
→ |
U31(X1,X2) |
(47) |
U31(active(X1),X2) |
→ |
U31(X1,X2) |
(49) |
U31(X1,active(X2)) |
→ |
U31(X1,X2) |
(50) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
mark#(U31(X1,X2)) |
→ |
active#(U31(mark(X1),X2)) |
(149) |
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)] |
= |
0 |
[U11(x1, x2)] |
= |
1 + x1
|
[mark(x1)] |
= |
2 + 2 · x1
|
[active(x1)] |
= |
1 + x1
|
[tt] |
= |
1 |
[U12(x1)] |
= |
1 + 2 · x1
|
[isNat(x1)] |
= |
0 |
[U31(x1, x2)] |
= |
1 + 2 · x1
|
[U32(x1)] |
= |
2 + x1
|
[U41(x1, x2)] |
= |
1 + x1 + x2
|
[U21(x1)] |
= |
1 + x1
|
[U51(x1, x2, x3)] |
= |
-2 + 2 · x2
|
[U52(x1, x2, x3)] |
= |
-1 + 2 · x1
|
[s(x1)] |
= |
x1 |
[plus(x1, x2)] |
= |
-2 + x2
|
[U71(x1, x2, x3)] |
= |
2 + 2 · x1 + x3
|
[U72(x1, x2, x3)] |
= |
-2 + 2 · x2 + 2 · x3
|
[x(x1, x2)] |
= |
2 + 2 · x1
|
[0] |
= |
2 |
[U61(x1)] |
= |
2 |
[mark#(x1)] |
= |
-2 + 2 · x1
|
together with the usable
rules
U11(X1,mark(X2)) |
→ |
U11(X1,X2) |
(38) |
U11(mark(X1),X2) |
→ |
U11(X1,X2) |
(37) |
U11(active(X1),X2) |
→ |
U11(X1,X2) |
(39) |
U11(X1,active(X2)) |
→ |
U11(X1,X2) |
(40) |
isNat(active(X)) |
→ |
isNat(X) |
(44) |
isNat(mark(X)) |
→ |
isNat(X) |
(43) |
U12(active(X)) |
→ |
U12(X) |
(42) |
U12(mark(X)) |
→ |
U12(X) |
(41) |
U21(active(X)) |
→ |
U21(X) |
(46) |
U21(mark(X)) |
→ |
U21(X) |
(45) |
U31(X1,mark(X2)) |
→ |
U31(X1,X2) |
(48) |
U31(mark(X1),X2) |
→ |
U31(X1,X2) |
(47) |
U31(active(X1),X2) |
→ |
U31(X1,X2) |
(49) |
U31(X1,active(X2)) |
→ |
U31(X1,X2) |
(50) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
mark#(U32(X)) |
→ |
mark#(X) |
(154) |
could be deleted.
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
[mark#(x1)] |
= |
+ · x1
|
[U11(x1, x2)] |
= |
+ · x1 + · x2
|
[active#(x1)] |
= |
+ · x1
|
[mark(x1)] |
= |
+ · x1
|
[tt] |
= |
|
[U12(x1)] |
= |
+ · x1
|
[isNat(x1)] |
= |
+ · x1
|
[U21(x1)] |
= |
+ · x1
|
[U31(x1, x2)] |
= |
+ · x1 + · x2
|
[plus(x1, x2)] |
= |
+ · x1 + · x2
|
[s(x1)] |
= |
+ · x1
|
[x(x1, x2)] |
= |
+ · x1 + · x2
|
[active(x1)] |
= |
+ · x1
|
[U32(x1)] |
= |
+ · x1
|
[U41(x1, x2)] |
= |
+ · x1 + · x2
|
[U51(x1, x2, x3)] |
= |
+ · x1 + · x2 + · x3
|
[U52(x1, x2, x3)] |
= |
+ · x1 + · x2 + · x3
|
[U71(x1, x2, x3)] |
= |
+ · x1 + · x2 + · x3
|
[U72(x1, x2, x3)] |
= |
+ · x1 + · x2 + · x3
|
[0] |
= |
|
[U61(x1)] |
= |
+ · x1
|
the
pair
active#(U11(tt,V2)) |
→ |
mark#(U12(isNat(V2))) |
(93) |
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)] |
= |
-2 |
[U11(x1, x2)] |
= |
1 + x1
|
[mark(x1)] |
= |
1 + 2 · x1
|
[active(x1)] |
= |
2 + x1
|
[tt] |
= |
0 |
[U12(x1)] |
= |
2 + x1
|
[isNat(x1)] |
= |
0 |
[U31(x1, x2)] |
= |
1 + x1
|
[U32(x1)] |
= |
2 |
[U41(x1, x2)] |
= |
-2 + 2 · x1
|
[U21(x1)] |
= |
1 + x1
|
[U51(x1, x2, x3)] |
= |
2 + x2 + 2 · x3
|
[U52(x1, x2, x3)] |
= |
-1 + 2 · x1
|
[s(x1)] |
= |
1 + x1
|
[plus(x1, x2)] |
= |
-2 |
[U71(x1, x2, x3)] |
= |
-2 + 2 · x1 + x2 + x3
|
[U72(x1, x2, x3)] |
= |
2 + 2 · x3
|
[x(x1, x2)] |
= |
0 |
[0] |
= |
0 |
[U61(x1)] |
= |
-2 + x1
|
[mark#(x1)] |
= |
-1 + x1
|
together with the usable
rules
U11(X1,mark(X2)) |
→ |
U11(X1,X2) |
(38) |
U11(mark(X1),X2) |
→ |
U11(X1,X2) |
(37) |
U11(active(X1),X2) |
→ |
U11(X1,X2) |
(39) |
U11(X1,active(X2)) |
→ |
U11(X1,X2) |
(40) |
isNat(active(X)) |
→ |
isNat(X) |
(44) |
isNat(mark(X)) |
→ |
isNat(X) |
(43) |
U21(active(X)) |
→ |
U21(X) |
(46) |
U21(mark(X)) |
→ |
U21(X) |
(45) |
U31(X1,mark(X2)) |
→ |
U31(X1,X2) |
(48) |
U31(mark(X1),X2) |
→ |
U31(X1,X2) |
(47) |
U31(active(X1),X2) |
→ |
U31(X1,X2) |
(49) |
U31(X1,active(X2)) |
→ |
U31(X1,X2) |
(50) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
mark#(U12(X)) |
→ |
mark#(X) |
(144) |
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)] |
= |
-2 + 2 · x1
|
[U11(x1, x2)] |
= |
-2 |
[mark(x1)] |
= |
2 |
[active(x1)] |
= |
-2 |
[tt] |
= |
0 |
[U12(x1)] |
= |
0 |
[isNat(x1)] |
= |
2 |
[U31(x1, x2)] |
= |
-2 |
[U32(x1)] |
= |
-2 + 2 · x1
|
[U41(x1, x2)] |
= |
-2 |
[U21(x1)] |
= |
-2 |
[U51(x1, x2, x3)] |
= |
2 + 2 · x2 + 2 · x3
|
[U52(x1, x2, x3)] |
= |
-2 + x1 + 2 · x2
|
[s(x1)] |
= |
-2 + x1
|
[plus(x1, x2)] |
= |
-2 + 2 · x1 + x2
|
[U71(x1, x2, x3)] |
= |
2 |
[U72(x1, x2, x3)] |
= |
1 + x1 + x3
|
[x(x1, x2)] |
= |
-2 + x1
|
[0] |
= |
0 |
[U61(x1)] |
= |
2 |
[mark#(x1)] |
= |
2 |
together with the usable
rules
U11(X1,mark(X2)) |
→ |
U11(X1,X2) |
(38) |
U11(mark(X1),X2) |
→ |
U11(X1,X2) |
(37) |
U11(active(X1),X2) |
→ |
U11(X1,X2) |
(39) |
U11(X1,active(X2)) |
→ |
U11(X1,X2) |
(40) |
isNat(active(X)) |
→ |
isNat(X) |
(44) |
isNat(mark(X)) |
→ |
isNat(X) |
(43) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pair
mark#(U11(X1,X2)) |
→ |
active#(U11(mark(X1),X2)) |
(138) |
could be deleted.
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
[isNat(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[U31(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[U21(x1)] |
= |
1 · x1
|
[U11(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[plus(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[s(x1)] |
= |
1 · x1
|
[x(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[mark#(x1)] |
= |
1 · x1
|
[active#(x1)] |
= |
1 · x1
|
together with the usable
rules
isNat(active(X)) |
→ |
isNat(X) |
(44) |
isNat(mark(X)) |
→ |
isNat(X) |
(43) |
U31(X1,mark(X2)) |
→ |
U31(X1,X2) |
(48) |
U31(mark(X1),X2) |
→ |
U31(X1,X2) |
(47) |
U31(active(X1),X2) |
→ |
U31(X1,X2) |
(49) |
U31(X1,active(X2)) |
→ |
U31(X1,X2) |
(50) |
U21(active(X)) |
→ |
U21(X) |
(46) |
U21(mark(X)) |
→ |
U21(X) |
(45) |
U11(X1,mark(X2)) |
→ |
U11(X1,X2) |
(38) |
U11(mark(X1),X2) |
→ |
U11(X1,X2) |
(37) |
U11(active(X1),X2) |
→ |
U11(X1,X2) |
(39) |
U11(X1,active(X2)) |
→ |
U11(X1,X2) |
(40) |
(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 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[isNat(x1)] |
= |
2 · x1
|
[mark(x1)] |
= |
2 · x1
|
[active(x1)] |
= |
2 · x1
|
[U31(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[U21(x1)] |
= |
1 · x1
|
[U11(x1, x2)] |
= |
1 · x1 + 1 · x2
|
[mark#(x1)] |
= |
2 · x1
|
[active#(x1)] |
= |
1 · x1
|
[plus(x1, x2)] |
= |
1 + 2 · x1 + 1 · x2
|
[s(x1)] |
= |
2 · x1
|
[x(x1, x2)] |
= |
1 + 2 · x1 + 2 · x2
|
together with the usable
rules
isNat(mark(X)) |
→ |
isNat(X) |
(43) |
isNat(active(X)) |
→ |
isNat(X) |
(44) |
U31(mark(X1),X2) |
→ |
U31(X1,X2) |
(47) |
U31(X1,mark(X2)) |
→ |
U31(X1,X2) |
(48) |
U31(active(X1),X2) |
→ |
U31(X1,X2) |
(49) |
U31(X1,active(X2)) |
→ |
U31(X1,X2) |
(50) |
U21(mark(X)) |
→ |
U21(X) |
(45) |
U21(active(X)) |
→ |
U21(X) |
(46) |
U11(mark(X1),X2) |
→ |
U11(X1,X2) |
(37) |
U11(X1,mark(X2)) |
→ |
U11(X1,X2) |
(38) |
U11(active(X1),X2) |
→ |
U11(X1,X2) |
(39) |
U11(X1,active(X2)) |
→ |
U11(X1,X2) |
(40) |
(w.r.t. the implicit argument filter of the reduction pair),
the
pairs
active#(isNat(plus(V1,V2))) |
→ |
mark#(U11(isNat(V1),V2)) |
(117) |
active#(isNat(s(V1))) |
→ |
mark#(U21(isNat(V1))) |
(120) |
active#(isNat(x(V1,V2))) |
→ |
mark#(U31(isNat(V1),V2)) |
(123) |
and
the
rules
isNat(mark(X)) |
→ |
isNat(X) |
(43) |
isNat(active(X)) |
→ |
isNat(X) |
(44) |
U31(mark(X1),X2) |
→ |
U31(X1,X2) |
(47) |
U31(X1,mark(X2)) |
→ |
U31(X1,X2) |
(48) |
U31(active(X1),X2) |
→ |
U31(X1,X2) |
(49) |
U31(X1,active(X2)) |
→ |
U31(X1,X2) |
(50) |
U21(mark(X)) |
→ |
U21(X) |
(45) |
U21(active(X)) |
→ |
U21(X) |
(46) |
U11(mark(X1),X2) |
→ |
U11(X1,X2) |
(37) |
U11(X1,mark(X2)) |
→ |
U11(X1,X2) |
(38) |
U11(active(X1),X2) |
→ |
U11(X1,X2) |
(39) |
U11(X1,active(X2)) |
→ |
U11(X1,X2) |
(40) |
could be deleted.
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#(U21(X)) |
→ |
mark#(X) |
(148) |
mark#(U11(X1,X2)) |
→ |
mark#(X1) |
(140) |
mark#(U31(X1,X2)) |
→ |
mark#(X1) |
(151) |
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#(U21(X)) |
→ |
mark#(X) |
(148) |
|
1 |
> |
1 |
mark#(U11(X1,X2)) |
→ |
mark#(X1) |
(140) |
|
1 |
> |
1 |
mark#(U31(X1,X2)) |
→ |
mark#(X1) |
(151) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
2nd
component contains the
pair
U11#(X1,mark(X2)) |
→ |
U11#(X1,X2) |
(186) |
U11#(mark(X1),X2) |
→ |
U11#(X1,X2) |
(185) |
U11#(active(X1),X2) |
→ |
U11#(X1,X2) |
(187) |
U11#(X1,active(X2)) |
→ |
U11#(X1,X2) |
(188) |
1.1.2 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[U11#(x1, x2)] |
= |
1 · x1 + 1 · x2
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.2.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
U11#(X1,mark(X2)) |
→ |
U11#(X1,X2) |
(186) |
|
1 |
≥ |
1 |
2 |
> |
2 |
U11#(mark(X1),X2) |
→ |
U11#(X1,X2) |
(185) |
|
1 |
> |
1 |
2 |
≥ |
2 |
U11#(active(X1),X2) |
→ |
U11#(X1,X2) |
(187) |
|
1 |
> |
1 |
2 |
≥ |
2 |
U11#(X1,active(X2)) |
→ |
U11#(X1,X2) |
(188) |
|
1 |
≥ |
1 |
2 |
> |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
3rd
component contains the
pair
U12#(active(X)) |
→ |
U12#(X) |
(190) |
U12#(mark(X)) |
→ |
U12#(X) |
(189) |
1.1.3 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[U12#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.3.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
U12#(active(X)) |
→ |
U12#(X) |
(190) |
|
1 |
> |
1 |
U12#(mark(X)) |
→ |
U12#(X) |
(189) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
4th
component contains the
pair
isNat#(active(X)) |
→ |
isNat#(X) |
(192) |
isNat#(mark(X)) |
→ |
isNat#(X) |
(191) |
1.1.4 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[isNat#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.4.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
isNat#(active(X)) |
→ |
isNat#(X) |
(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
5th
component contains the
pair
U21#(active(X)) |
→ |
U21#(X) |
(194) |
U21#(mark(X)) |
→ |
U21#(X) |
(193) |
1.1.5 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[U21#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.5.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
U21#(active(X)) |
→ |
U21#(X) |
(194) |
|
1 |
> |
1 |
U21#(mark(X)) |
→ |
U21#(X) |
(193) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
6th
component contains the
pair
U31#(X1,mark(X2)) |
→ |
U31#(X1,X2) |
(196) |
U31#(mark(X1),X2) |
→ |
U31#(X1,X2) |
(195) |
U31#(active(X1),X2) |
→ |
U31#(X1,X2) |
(197) |
U31#(X1,active(X2)) |
→ |
U31#(X1,X2) |
(198) |
1.1.6 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[U31#(x1, x2)] |
= |
1 · x1 + 1 · x2
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.6.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
U31#(X1,mark(X2)) |
→ |
U31#(X1,X2) |
(196) |
|
1 |
≥ |
1 |
2 |
> |
2 |
U31#(mark(X1),X2) |
→ |
U31#(X1,X2) |
(195) |
|
1 |
> |
1 |
2 |
≥ |
2 |
U31#(active(X1),X2) |
→ |
U31#(X1,X2) |
(197) |
|
1 |
> |
1 |
2 |
≥ |
2 |
U31#(X1,active(X2)) |
→ |
U31#(X1,X2) |
(198) |
|
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
U32#(active(X)) |
→ |
U32#(X) |
(200) |
U32#(mark(X)) |
→ |
U32#(X) |
(199) |
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
|
[U32#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.7.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
U32#(active(X)) |
→ |
U32#(X) |
(200) |
|
1 |
> |
1 |
U32#(mark(X)) |
→ |
U32#(X) |
(199) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
8th
component contains the
pair
U41#(X1,mark(X2)) |
→ |
U41#(X1,X2) |
(202) |
U41#(mark(X1),X2) |
→ |
U41#(X1,X2) |
(201) |
U41#(active(X1),X2) |
→ |
U41#(X1,X2) |
(203) |
U41#(X1,active(X2)) |
→ |
U41#(X1,X2) |
(204) |
1.1.8 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[U41#(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.
1.1.8.1 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) |
(202) |
|
1 |
≥ |
1 |
2 |
> |
2 |
U41#(mark(X1),X2) |
→ |
U41#(X1,X2) |
(201) |
|
1 |
> |
1 |
2 |
≥ |
2 |
U41#(active(X1),X2) |
→ |
U41#(X1,X2) |
(203) |
|
1 |
> |
1 |
2 |
≥ |
2 |
U41#(X1,active(X2)) |
→ |
U41#(X1,X2) |
(204) |
|
1 |
≥ |
1 |
2 |
> |
2 |
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),X3) |
→ |
U51#(X1,X2,X3) |
(206) |
U51#(mark(X1),X2,X3) |
→ |
U51#(X1,X2,X3) |
(205) |
U51#(X1,X2,mark(X3)) |
→ |
U51#(X1,X2,X3) |
(207) |
U51#(active(X1),X2,X3) |
→ |
U51#(X1,X2,X3) |
(208) |
U51#(X1,active(X2),X3) |
→ |
U51#(X1,X2,X3) |
(209) |
U51#(X1,X2,active(X3)) |
→ |
U51#(X1,X2,X3) |
(210) |
1.1.9 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[U51#(x1, x2, x3)] |
= |
1 · x1 + 1 · x2 + 1 · x3
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.9.1 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),X3) |
→ |
U51#(X1,X2,X3) |
(206) |
|
1 |
≥ |
1 |
2 |
> |
2 |
3 |
≥ |
3 |
U51#(mark(X1),X2,X3) |
→ |
U51#(X1,X2,X3) |
(205) |
|
1 |
> |
1 |
2 |
≥ |
2 |
3 |
≥ |
3 |
U51#(X1,X2,mark(X3)) |
→ |
U51#(X1,X2,X3) |
(207) |
|
1 |
≥ |
1 |
2 |
≥ |
2 |
3 |
> |
3 |
U51#(active(X1),X2,X3) |
→ |
U51#(X1,X2,X3) |
(208) |
|
1 |
> |
1 |
2 |
≥ |
2 |
3 |
≥ |
3 |
U51#(X1,active(X2),X3) |
→ |
U51#(X1,X2,X3) |
(209) |
|
1 |
≥ |
1 |
2 |
> |
2 |
3 |
≥ |
3 |
U51#(X1,X2,active(X3)) |
→ |
U51#(X1,X2,X3) |
(210) |
|
1 |
≥ |
1 |
2 |
≥ |
2 |
3 |
> |
3 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
10th
component contains the
pair
U52#(X1,mark(X2),X3) |
→ |
U52#(X1,X2,X3) |
(212) |
U52#(mark(X1),X2,X3) |
→ |
U52#(X1,X2,X3) |
(211) |
U52#(X1,X2,mark(X3)) |
→ |
U52#(X1,X2,X3) |
(213) |
U52#(active(X1),X2,X3) |
→ |
U52#(X1,X2,X3) |
(214) |
U52#(X1,active(X2),X3) |
→ |
U52#(X1,X2,X3) |
(215) |
U52#(X1,X2,active(X3)) |
→ |
U52#(X1,X2,X3) |
(216) |
1.1.10 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[U52#(x1, x2, x3)] |
= |
1 · x1 + 1 · x2 + 1 · x3
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.10.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
U52#(X1,mark(X2),X3) |
→ |
U52#(X1,X2,X3) |
(212) |
|
1 |
≥ |
1 |
2 |
> |
2 |
3 |
≥ |
3 |
U52#(mark(X1),X2,X3) |
→ |
U52#(X1,X2,X3) |
(211) |
|
1 |
> |
1 |
2 |
≥ |
2 |
3 |
≥ |
3 |
U52#(X1,X2,mark(X3)) |
→ |
U52#(X1,X2,X3) |
(213) |
|
1 |
≥ |
1 |
2 |
≥ |
2 |
3 |
> |
3 |
U52#(active(X1),X2,X3) |
→ |
U52#(X1,X2,X3) |
(214) |
|
1 |
> |
1 |
2 |
≥ |
2 |
3 |
≥ |
3 |
U52#(X1,active(X2),X3) |
→ |
U52#(X1,X2,X3) |
(215) |
|
1 |
≥ |
1 |
2 |
> |
2 |
3 |
≥ |
3 |
U52#(X1,X2,active(X3)) |
→ |
U52#(X1,X2,X3) |
(216) |
|
1 |
≥ |
1 |
2 |
≥ |
2 |
3 |
> |
3 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
11th
component contains the
pair
s#(active(X)) |
→ |
s#(X) |
(218) |
s#(mark(X)) |
→ |
s#(X) |
(217) |
1.1.11 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.
1.1.11.1 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) |
(218) |
|
1 |
> |
1 |
s#(mark(X)) |
→ |
s#(X) |
(217) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
12th
component contains the
pair
plus#(X1,mark(X2)) |
→ |
plus#(X1,X2) |
(220) |
plus#(mark(X1),X2) |
→ |
plus#(X1,X2) |
(219) |
plus#(active(X1),X2) |
→ |
plus#(X1,X2) |
(221) |
plus#(X1,active(X2)) |
→ |
plus#(X1,X2) |
(222) |
1.1.12 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[plus#(x1, x2)] |
= |
1 · x1 + 1 · x2
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.12.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
plus#(X1,mark(X2)) |
→ |
plus#(X1,X2) |
(220) |
|
1 |
≥ |
1 |
2 |
> |
2 |
plus#(mark(X1),X2) |
→ |
plus#(X1,X2) |
(219) |
|
1 |
> |
1 |
2 |
≥ |
2 |
plus#(active(X1),X2) |
→ |
plus#(X1,X2) |
(221) |
|
1 |
> |
1 |
2 |
≥ |
2 |
plus#(X1,active(X2)) |
→ |
plus#(X1,X2) |
(222) |
|
1 |
≥ |
1 |
2 |
> |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
13th
component contains the
pair
U61#(active(X)) |
→ |
U61#(X) |
(224) |
U61#(mark(X)) |
→ |
U61#(X) |
(223) |
1.1.13 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[active(x1)] |
= |
1 · x1
|
[mark(x1)] |
= |
1 · x1
|
[U61#(x1)] |
= |
1 · x1
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.13.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
U61#(active(X)) |
→ |
U61#(X) |
(224) |
|
1 |
> |
1 |
U61#(mark(X)) |
→ |
U61#(X) |
(223) |
|
1 |
> |
1 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
14th
component contains the
pair
U71#(X1,mark(X2),X3) |
→ |
U71#(X1,X2,X3) |
(226) |
U71#(mark(X1),X2,X3) |
→ |
U71#(X1,X2,X3) |
(225) |
U71#(X1,X2,mark(X3)) |
→ |
U71#(X1,X2,X3) |
(227) |
U71#(active(X1),X2,X3) |
→ |
U71#(X1,X2,X3) |
(228) |
U71#(X1,active(X2),X3) |
→ |
U71#(X1,X2,X3) |
(229) |
U71#(X1,X2,active(X3)) |
→ |
U71#(X1,X2,X3) |
(230) |
1.1.14 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[U71#(x1, x2, x3)] |
= |
1 · x1 + 1 · x2 + 1 · x3
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.14.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
U71#(X1,mark(X2),X3) |
→ |
U71#(X1,X2,X3) |
(226) |
|
1 |
≥ |
1 |
2 |
> |
2 |
3 |
≥ |
3 |
U71#(mark(X1),X2,X3) |
→ |
U71#(X1,X2,X3) |
(225) |
|
1 |
> |
1 |
2 |
≥ |
2 |
3 |
≥ |
3 |
U71#(X1,X2,mark(X3)) |
→ |
U71#(X1,X2,X3) |
(227) |
|
1 |
≥ |
1 |
2 |
≥ |
2 |
3 |
> |
3 |
U71#(active(X1),X2,X3) |
→ |
U71#(X1,X2,X3) |
(228) |
|
1 |
> |
1 |
2 |
≥ |
2 |
3 |
≥ |
3 |
U71#(X1,active(X2),X3) |
→ |
U71#(X1,X2,X3) |
(229) |
|
1 |
≥ |
1 |
2 |
> |
2 |
3 |
≥ |
3 |
U71#(X1,X2,active(X3)) |
→ |
U71#(X1,X2,X3) |
(230) |
|
1 |
≥ |
1 |
2 |
≥ |
2 |
3 |
> |
3 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
15th
component contains the
pair
U72#(X1,mark(X2),X3) |
→ |
U72#(X1,X2,X3) |
(232) |
U72#(mark(X1),X2,X3) |
→ |
U72#(X1,X2,X3) |
(231) |
U72#(X1,X2,mark(X3)) |
→ |
U72#(X1,X2,X3) |
(233) |
U72#(active(X1),X2,X3) |
→ |
U72#(X1,X2,X3) |
(234) |
U72#(X1,active(X2),X3) |
→ |
U72#(X1,X2,X3) |
(235) |
U72#(X1,X2,active(X3)) |
→ |
U72#(X1,X2,X3) |
(236) |
1.1.15 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[U72#(x1, x2, x3)] |
= |
1 · x1 + 1 · x2 + 1 · x3
|
having no usable rules (w.r.t. the implicit argument filter of the
reduction pair),
the
rule
could be deleted.
1.1.15.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
U72#(X1,mark(X2),X3) |
→ |
U72#(X1,X2,X3) |
(232) |
|
1 |
≥ |
1 |
2 |
> |
2 |
3 |
≥ |
3 |
U72#(mark(X1),X2,X3) |
→ |
U72#(X1,X2,X3) |
(231) |
|
1 |
> |
1 |
2 |
≥ |
2 |
3 |
≥ |
3 |
U72#(X1,X2,mark(X3)) |
→ |
U72#(X1,X2,X3) |
(233) |
|
1 |
≥ |
1 |
2 |
≥ |
2 |
3 |
> |
3 |
U72#(active(X1),X2,X3) |
→ |
U72#(X1,X2,X3) |
(234) |
|
1 |
> |
1 |
2 |
≥ |
2 |
3 |
≥ |
3 |
U72#(X1,active(X2),X3) |
→ |
U72#(X1,X2,X3) |
(235) |
|
1 |
≥ |
1 |
2 |
> |
2 |
3 |
≥ |
3 |
U72#(X1,X2,active(X3)) |
→ |
U72#(X1,X2,X3) |
(236) |
|
1 |
≥ |
1 |
2 |
≥ |
2 |
3 |
> |
3 |
As there is no critical graph in the transitive closure, there are no infinite chains.
-
The
16th
component contains the
pair
x#(X1,mark(X2)) |
→ |
x#(X1,X2) |
(238) |
x#(mark(X1),X2) |
→ |
x#(X1,X2) |
(237) |
x#(active(X1),X2) |
→ |
x#(X1,X2) |
(239) |
x#(X1,active(X2)) |
→ |
x#(X1,X2) |
(240) |
1.1.16 Monotonic Reduction Pair Processor with Usable Rules
Using the linear polynomial interpretation over the naturals
[mark(x1)] |
= |
1 · x1
|
[active(x1)] |
= |
1 · x1
|
[x#(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.
1.1.16.1 Size-Change Termination
Using size-change termination in combination with
the subterm criterion
one obtains the following initial size-change graphs.
x#(X1,mark(X2)) |
→ |
x#(X1,X2) |
(238) |
|
1 |
≥ |
1 |
2 |
> |
2 |
x#(mark(X1),X2) |
→ |
x#(X1,X2) |
(237) |
|
1 |
> |
1 |
2 |
≥ |
2 |
x#(active(X1),X2) |
→ |
x#(X1,X2) |
(239) |
|
1 |
> |
1 |
2 |
≥ |
2 |
x#(X1,active(X2)) |
→ |
x#(X1,X2) |
(240) |
|
1 |
≥ |
1 |
2 |
> |
2 |
As there is no critical graph in the transitive closure, there are no infinite chains.