YES
0 RelTRS
↳1 RelTRStoRelADPProof (⇔, 0 ms)
↳2 RelADPP
↳3 RelADPDepGraphProof (⇔, 0 ms)
↳4 RelADPP
↳5 RelADPCleverAfsProof (⇒, 64 ms)
↳6 QDP
↳7 MRRProof (⇔, 0 ms)
↳8 QDP
↳9 MRRProof (⇔, 0 ms)
↳10 QDP
↳11 QDPOrderProof (⇔, 0 ms)
↳12 QDP
↳13 PisEmptyProof (⇔, 0 ms)
↳14 YES
g(s(x)) → f(x)
f(0) → s(0)
f(s(x)) → s(s(g(x)))
g(0) → 0
rand(x) → rand(s(x))
rand(x) → x
We upgrade the RelTRS problem to an equivalent Relative ADP Problem [IJCAR24].
g(s(x)) → F(x)
f(0) → s(0)
f(s(x)) → s(s(G(x)))
g(0) → 0
rand(x) → RAND(s(x))
rand(x) → x
We use the relative dependency graph processor [IJCAR24].
The approximation of the Relative Dependency Graph contains:
1 SCC with nodes from P_abs,
0 Lassos,
Result: This relative DT problem is equivalent to 1 subproblem.
f(0) → s(0)
f(s(x)) → s(s(G(x)))
g(s(x)) → F(x)
g(0) → 0
rand(x) → rand(s(x))
rand(x) → x
Furthermore, We use an argument filter [LPAR04].
Filtering:s_1 =
G_1 =
f_1 =
0 =
F_1 =
rand_1 =
g_1 =
Found this filtering by looking at the following order that orders at least one DP strictly:Combined order from the following AFS and order.
G(x1) = G(x1)
s(x1) = s(x1)
F(x1) = F(x1)
f(x1) = f(x1)
0 = 0
g(x1) = x1
Recursive path order with status [RPO].
Quasi-Precedence:
[G1, s1, f1] > F1
[G1, s1, f1] > 0
G1: multiset
s1: multiset
F1: multiset
f1: multiset
0: multiset
G0(s0(x)) → F0(x)
F0(s0(x)) → G0(x)
f0(00) → s0(00)
g0(00) → 00
f0(s0(x)) → s0(s0(g0(x)))
rand0(x) → rand0(s0(x))
g0(s0(x)) → f0(x)
rand0(x) → x
rand0(x) → x
POL(00) = 0
POL(F0(x1)) = x1
POL(G0(x1)) = x1
POL(f0(x1)) = 2·x1
POL(g0(x1)) = 2·x1
POL(rand0(x1)) = 2 + x1
POL(s0(x1)) = x1
G0(s0(x)) → F0(x)
F0(s0(x)) → G0(x)
f0(00) → s0(00)
g0(00) → 00
f0(s0(x)) → s0(s0(g0(x)))
rand0(x) → rand0(s0(x))
g0(s0(x)) → f0(x)
f0(00) → s0(00)
g0(00) → 00
POL(00) = 2
POL(F0(x1)) = x1
POL(G0(x1)) = x1
POL(f0(x1)) = 1 + 2·x1
POL(g0(x1)) = 1 + 2·x1
POL(rand0(x1)) = x1
POL(s0(x1)) = x1
G0(s0(x)) → F0(x)
F0(s0(x)) → G0(x)
f0(s0(x)) → s0(s0(g0(x)))
rand0(x) → rand0(s0(x))
g0(s0(x)) → f0(x)
The following pairs can be oriented strictly and are deleted.
The remaining pairs can at least be oriented weakly.
G0(s0(x)) → F0(x)
F0(s0(x)) → G0(x)
rand0 > [s01, f01] > F01 > G01
G01: multiset
s01: multiset
F01: multiset
f01: multiset
rand0: multiset
f0(s0(x)) → s0(s0(g0(x)))
rand0(x) → rand0(s0(x))
g0(s0(x)) → f0(x)
f0(s0(x)) → s0(s0(g0(x)))
rand0(x) → rand0(s0(x))
g0(s0(x)) → f0(x)