Left Termination of the query pattern p_in_1(a) w.r.t. the given Prolog program could successfully be proven:



↳ Prolog
  ↳ PrologToPiTRSProof

Clauses:

p(.(A, [])) :- l(.(A, [])).
r(1).
l([]).
l(.(H, T)) :- ','(r(H), l(T)).

Queries:

p(a).

We use the technique of [30].Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

p_in(.(A, [])) → U1(A, l_in(.(A, [])))
l_in(.(H, T)) → U2(H, T, r_in(H))
r_in(1) → r_out(1)
U2(H, T, r_out(H)) → U3(H, T, l_in(T))
l_in([]) → l_out([])
U3(H, T, l_out(T)) → l_out(.(H, T))
U1(A, l_out(.(A, []))) → p_out(.(A, []))

The argument filtering Pi contains the following mapping:
p_in(x1)  =  p_in
.(x1, x2)  =  .(x2)
[]  =  []
U1(x1, x2)  =  U1(x2)
l_in(x1)  =  l_in(x1)
U2(x1, x2, x3)  =  U2(x2, x3)
r_in(x1)  =  r_in
1  =  1
r_out(x1)  =  r_out(x1)
U3(x1, x2, x3)  =  U3(x3)
l_out(x1)  =  l_out
p_out(x1)  =  p_out(x1)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog



↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof

Pi-finite rewrite system:
The TRS R consists of the following rules:

p_in(.(A, [])) → U1(A, l_in(.(A, [])))
l_in(.(H, T)) → U2(H, T, r_in(H))
r_in(1) → r_out(1)
U2(H, T, r_out(H)) → U3(H, T, l_in(T))
l_in([]) → l_out([])
U3(H, T, l_out(T)) → l_out(.(H, T))
U1(A, l_out(.(A, []))) → p_out(.(A, []))

The argument filtering Pi contains the following mapping:
p_in(x1)  =  p_in
.(x1, x2)  =  .(x2)
[]  =  []
U1(x1, x2)  =  U1(x2)
l_in(x1)  =  l_in(x1)
U2(x1, x2, x3)  =  U2(x2, x3)
r_in(x1)  =  r_in
1  =  1
r_out(x1)  =  r_out(x1)
U3(x1, x2, x3)  =  U3(x3)
l_out(x1)  =  l_out
p_out(x1)  =  p_out(x1)


Using Dependency Pairs [1,30] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

P_IN(.(A, [])) → U11(A, l_in(.(A, [])))
P_IN(.(A, [])) → L_IN(.(A, []))
L_IN(.(H, T)) → U21(H, T, r_in(H))
L_IN(.(H, T)) → R_IN(H)
U21(H, T, r_out(H)) → U31(H, T, l_in(T))
U21(H, T, r_out(H)) → L_IN(T)

The TRS R consists of the following rules:

p_in(.(A, [])) → U1(A, l_in(.(A, [])))
l_in(.(H, T)) → U2(H, T, r_in(H))
r_in(1) → r_out(1)
U2(H, T, r_out(H)) → U3(H, T, l_in(T))
l_in([]) → l_out([])
U3(H, T, l_out(T)) → l_out(.(H, T))
U1(A, l_out(.(A, []))) → p_out(.(A, []))

The argument filtering Pi contains the following mapping:
p_in(x1)  =  p_in
.(x1, x2)  =  .(x2)
[]  =  []
U1(x1, x2)  =  U1(x2)
l_in(x1)  =  l_in(x1)
U2(x1, x2, x3)  =  U2(x2, x3)
r_in(x1)  =  r_in
1  =  1
r_out(x1)  =  r_out(x1)
U3(x1, x2, x3)  =  U3(x3)
l_out(x1)  =  l_out
p_out(x1)  =  p_out(x1)
P_IN(x1)  =  P_IN
U31(x1, x2, x3)  =  U31(x3)
L_IN(x1)  =  L_IN(x1)
U11(x1, x2)  =  U11(x2)
U21(x1, x2, x3)  =  U21(x2, x3)
R_IN(x1)  =  R_IN

We have to consider all (P,R,Pi)-chains

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof

Pi DP problem:
The TRS P consists of the following rules:

P_IN(.(A, [])) → U11(A, l_in(.(A, [])))
P_IN(.(A, [])) → L_IN(.(A, []))
L_IN(.(H, T)) → U21(H, T, r_in(H))
L_IN(.(H, T)) → R_IN(H)
U21(H, T, r_out(H)) → U31(H, T, l_in(T))
U21(H, T, r_out(H)) → L_IN(T)

The TRS R consists of the following rules:

p_in(.(A, [])) → U1(A, l_in(.(A, [])))
l_in(.(H, T)) → U2(H, T, r_in(H))
r_in(1) → r_out(1)
U2(H, T, r_out(H)) → U3(H, T, l_in(T))
l_in([]) → l_out([])
U3(H, T, l_out(T)) → l_out(.(H, T))
U1(A, l_out(.(A, []))) → p_out(.(A, []))

The argument filtering Pi contains the following mapping:
p_in(x1)  =  p_in
.(x1, x2)  =  .(x2)
[]  =  []
U1(x1, x2)  =  U1(x2)
l_in(x1)  =  l_in(x1)
U2(x1, x2, x3)  =  U2(x2, x3)
r_in(x1)  =  r_in
1  =  1
r_out(x1)  =  r_out(x1)
U3(x1, x2, x3)  =  U3(x3)
l_out(x1)  =  l_out
p_out(x1)  =  p_out(x1)
P_IN(x1)  =  P_IN
U31(x1, x2, x3)  =  U31(x3)
L_IN(x1)  =  L_IN(x1)
U11(x1, x2)  =  U11(x2)
U21(x1, x2, x3)  =  U21(x2, x3)
R_IN(x1)  =  R_IN

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 1 SCC with 4 less nodes.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ PiDP
              ↳ UsableRulesProof

Pi DP problem:
The TRS P consists of the following rules:

U21(H, T, r_out(H)) → L_IN(T)
L_IN(.(H, T)) → U21(H, T, r_in(H))

The TRS R consists of the following rules:

p_in(.(A, [])) → U1(A, l_in(.(A, [])))
l_in(.(H, T)) → U2(H, T, r_in(H))
r_in(1) → r_out(1)
U2(H, T, r_out(H)) → U3(H, T, l_in(T))
l_in([]) → l_out([])
U3(H, T, l_out(T)) → l_out(.(H, T))
U1(A, l_out(.(A, []))) → p_out(.(A, []))

The argument filtering Pi contains the following mapping:
p_in(x1)  =  p_in
.(x1, x2)  =  .(x2)
[]  =  []
U1(x1, x2)  =  U1(x2)
l_in(x1)  =  l_in(x1)
U2(x1, x2, x3)  =  U2(x2, x3)
r_in(x1)  =  r_in
1  =  1
r_out(x1)  =  r_out(x1)
U3(x1, x2, x3)  =  U3(x3)
l_out(x1)  =  l_out
p_out(x1)  =  p_out(x1)
L_IN(x1)  =  L_IN(x1)
U21(x1, x2, x3)  =  U21(x2, x3)

We have to consider all (P,R,Pi)-chains
For (infinitary) constructor rewriting [30] we can delete all non-usable rules from R.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ PiDP
              ↳ UsableRulesProof
                ↳ PiDP
                  ↳ PiDPToQDPProof

Pi DP problem:
The TRS P consists of the following rules:

U21(H, T, r_out(H)) → L_IN(T)
L_IN(.(H, T)) → U21(H, T, r_in(H))

The TRS R consists of the following rules:

r_in(1) → r_out(1)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
r_in(x1)  =  r_in
1  =  1
r_out(x1)  =  r_out(x1)
L_IN(x1)  =  L_IN(x1)
U21(x1, x2, x3)  =  U21(x2, x3)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem [30] into ordinary QDP problem [15] by application of Pi.

↳ Prolog
  ↳ PrologToPiTRSProof
    ↳ PiTRS
      ↳ DependencyPairsProof
        ↳ PiDP
          ↳ DependencyGraphProof
            ↳ PiDP
              ↳ UsableRulesProof
                ↳ PiDP
                  ↳ PiDPToQDPProof
                    ↳ QDP
                      ↳ QDPSizeChangeProof

Q DP problem:
The TRS P consists of the following rules:

U21(T, r_out(H)) → L_IN(T)
L_IN(.(T)) → U21(T, r_in)

The TRS R consists of the following rules:

r_in → r_out(1)

The set Q consists of the following terms:

r_in

We have to consider all (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs: