Left Termination of the query pattern reverse_in_2(g, a) w.r.t. the given Prolog program could successfully be proven:



↳ Prolog
  ↳ PrologToPiTRSProof

Clauses:

reverse(L, LR) :- revacc(L, LR, []).
revacc([], L, L).
revacc(.(EL, T), R, A) :- revacc(T, R, .(EL, A)).

Queries:

reverse(g,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:

reverse_in(L, LR) → U1(L, LR, revacc_in(L, LR, []))
revacc_in(.(EL, T), R, A) → U2(EL, T, R, A, revacc_in(T, R, .(EL, A)))
revacc_in([], L, L) → revacc_out([], L, L)
U2(EL, T, R, A, revacc_out(T, R, .(EL, A))) → revacc_out(.(EL, T), R, A)
U1(L, LR, revacc_out(L, LR, [])) → reverse_out(L, LR)

The argument filtering Pi contains the following mapping:
reverse_in(x1, x2)  =  reverse_in(x1)
U1(x1, x2, x3)  =  U1(x3)
revacc_in(x1, x2, x3)  =  revacc_in(x1, x3)
[]  =  []
.(x1, x2)  =  .(x1, x2)
U2(x1, x2, x3, x4, x5)  =  U2(x5)
revacc_out(x1, x2, x3)  =  revacc_out(x2)
reverse_out(x1, x2)  =  reverse_out(x2)

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:

reverse_in(L, LR) → U1(L, LR, revacc_in(L, LR, []))
revacc_in(.(EL, T), R, A) → U2(EL, T, R, A, revacc_in(T, R, .(EL, A)))
revacc_in([], L, L) → revacc_out([], L, L)
U2(EL, T, R, A, revacc_out(T, R, .(EL, A))) → revacc_out(.(EL, T), R, A)
U1(L, LR, revacc_out(L, LR, [])) → reverse_out(L, LR)

The argument filtering Pi contains the following mapping:
reverse_in(x1, x2)  =  reverse_in(x1)
U1(x1, x2, x3)  =  U1(x3)
revacc_in(x1, x2, x3)  =  revacc_in(x1, x3)
[]  =  []
.(x1, x2)  =  .(x1, x2)
U2(x1, x2, x3, x4, x5)  =  U2(x5)
revacc_out(x1, x2, x3)  =  revacc_out(x2)
reverse_out(x1, x2)  =  reverse_out(x2)


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

REVERSE_IN(L, LR) → U11(L, LR, revacc_in(L, LR, []))
REVERSE_IN(L, LR) → REVACC_IN(L, LR, [])
REVACC_IN(.(EL, T), R, A) → U21(EL, T, R, A, revacc_in(T, R, .(EL, A)))
REVACC_IN(.(EL, T), R, A) → REVACC_IN(T, R, .(EL, A))

The TRS R consists of the following rules:

reverse_in(L, LR) → U1(L, LR, revacc_in(L, LR, []))
revacc_in(.(EL, T), R, A) → U2(EL, T, R, A, revacc_in(T, R, .(EL, A)))
revacc_in([], L, L) → revacc_out([], L, L)
U2(EL, T, R, A, revacc_out(T, R, .(EL, A))) → revacc_out(.(EL, T), R, A)
U1(L, LR, revacc_out(L, LR, [])) → reverse_out(L, LR)

The argument filtering Pi contains the following mapping:
reverse_in(x1, x2)  =  reverse_in(x1)
U1(x1, x2, x3)  =  U1(x3)
revacc_in(x1, x2, x3)  =  revacc_in(x1, x3)
[]  =  []
.(x1, x2)  =  .(x1, x2)
U2(x1, x2, x3, x4, x5)  =  U2(x5)
revacc_out(x1, x2, x3)  =  revacc_out(x2)
reverse_out(x1, x2)  =  reverse_out(x2)
REVERSE_IN(x1, x2)  =  REVERSE_IN(x1)
REVACC_IN(x1, x2, x3)  =  REVACC_IN(x1, x3)
U21(x1, x2, x3, x4, x5)  =  U21(x5)
U11(x1, x2, x3)  =  U11(x3)

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:

REVERSE_IN(L, LR) → U11(L, LR, revacc_in(L, LR, []))
REVERSE_IN(L, LR) → REVACC_IN(L, LR, [])
REVACC_IN(.(EL, T), R, A) → U21(EL, T, R, A, revacc_in(T, R, .(EL, A)))
REVACC_IN(.(EL, T), R, A) → REVACC_IN(T, R, .(EL, A))

The TRS R consists of the following rules:

reverse_in(L, LR) → U1(L, LR, revacc_in(L, LR, []))
revacc_in(.(EL, T), R, A) → U2(EL, T, R, A, revacc_in(T, R, .(EL, A)))
revacc_in([], L, L) → revacc_out([], L, L)
U2(EL, T, R, A, revacc_out(T, R, .(EL, A))) → revacc_out(.(EL, T), R, A)
U1(L, LR, revacc_out(L, LR, [])) → reverse_out(L, LR)

The argument filtering Pi contains the following mapping:
reverse_in(x1, x2)  =  reverse_in(x1)
U1(x1, x2, x3)  =  U1(x3)
revacc_in(x1, x2, x3)  =  revacc_in(x1, x3)
[]  =  []
.(x1, x2)  =  .(x1, x2)
U2(x1, x2, x3, x4, x5)  =  U2(x5)
revacc_out(x1, x2, x3)  =  revacc_out(x2)
reverse_out(x1, x2)  =  reverse_out(x2)
REVERSE_IN(x1, x2)  =  REVERSE_IN(x1)
REVACC_IN(x1, x2, x3)  =  REVACC_IN(x1, x3)
U21(x1, x2, x3, x4, x5)  =  U21(x5)
U11(x1, x2, x3)  =  U11(x3)

We have to consider all (P,R,Pi)-chains
The approximation of the Dependency Graph [30] contains 1 SCC with 3 less nodes.

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

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

REVACC_IN(.(EL, T), R, A) → REVACC_IN(T, R, .(EL, A))

The TRS R consists of the following rules:

reverse_in(L, LR) → U1(L, LR, revacc_in(L, LR, []))
revacc_in(.(EL, T), R, A) → U2(EL, T, R, A, revacc_in(T, R, .(EL, A)))
revacc_in([], L, L) → revacc_out([], L, L)
U2(EL, T, R, A, revacc_out(T, R, .(EL, A))) → revacc_out(.(EL, T), R, A)
U1(L, LR, revacc_out(L, LR, [])) → reverse_out(L, LR)

The argument filtering Pi contains the following mapping:
reverse_in(x1, x2)  =  reverse_in(x1)
U1(x1, x2, x3)  =  U1(x3)
revacc_in(x1, x2, x3)  =  revacc_in(x1, x3)
[]  =  []
.(x1, x2)  =  .(x1, x2)
U2(x1, x2, x3, x4, x5)  =  U2(x5)
revacc_out(x1, x2, x3)  =  revacc_out(x2)
reverse_out(x1, x2)  =  reverse_out(x2)
REVACC_IN(x1, x2, x3)  =  REVACC_IN(x1, 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:

REVACC_IN(.(EL, T), R, A) → REVACC_IN(T, R, .(EL, A))

R is empty.
The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x1, x2)
REVACC_IN(x1, x2, x3)  =  REVACC_IN(x1, 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:

REVACC_IN(.(EL, T), A) → REVACC_IN(T, .(EL, A))

R is empty.
Q is empty.
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: