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

0 Prolog
  ↳1 PrologToPiTRSProof (⇐)
    ↳2 PiTRS
    ↳3 DependencyPairsProof (⇔)
    ↳4 PiDP
    ↳5 DependencyGraphProof (⇔)
    ↳6 PiDP
    ↳7 UsableRulesProof (⇔)
    ↳8 PiDP
    ↳9 PiDPToQDPProof (⇐)
    ↳10 QDP
    ↳11 QDPSizeChangeProof (⇔)
    ↳12 TRUE
  ↳13 PrologToPiTRSProof (⇐)
    ↳14 PiTRS
    ↳15 DependencyPairsProof (⇔)
    ↳16 PiDP
    ↳17 DependencyGraphProof (⇔)
    ↳18 PiDP
    ↳19 UsableRulesProof (⇔)
    ↳20 PiDP
    ↳21 PiDPToQDPProof (⇐)
    ↳22 QDP

(0) Obligation:

Clauses:

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

Queries:

p(a).

(1) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
p_in: (f)
l_in: (b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

p_in_a(.(A, [])) → U1_a(A, l_in_g(.(A, [])))
l_in_g([]) → l_out_g([])
l_in_g(.(H, T)) → U2_g(H, T, r_in_a(H))
r_in_a(1) → r_out_a(1)
U2_g(H, T, r_out_a(H)) → U3_g(H, T, l_in_g(T))
U3_g(H, T, l_out_g(T)) → l_out_g(.(H, T))
U1_a(A, l_out_g(.(A, []))) → p_out_a(.(A, []))

The argument filtering Pi contains the following mapping:
p_in_a(x1)  =  p_in_a
U1_a(x1, x2)  =  U1_a(x2)
l_in_g(x1)  =  l_in_g(x1)
.(x1, x2)  =  .(x2)
[]  =  []
l_out_g(x1)  =  l_out_g(x1)
U2_g(x1, x2, x3)  =  U2_g(x2, x3)
r_in_a(x1)  =  r_in_a
r_out_a(x1)  =  r_out_a(x1)
U3_g(x1, x2, x3)  =  U3_g(x2, x3)
p_out_a(x1)  =  p_out_a(x1)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(2) Obligation:

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

p_in_a(.(A, [])) → U1_a(A, l_in_g(.(A, [])))
l_in_g([]) → l_out_g([])
l_in_g(.(H, T)) → U2_g(H, T, r_in_a(H))
r_in_a(1) → r_out_a(1)
U2_g(H, T, r_out_a(H)) → U3_g(H, T, l_in_g(T))
U3_g(H, T, l_out_g(T)) → l_out_g(.(H, T))
U1_a(A, l_out_g(.(A, []))) → p_out_a(.(A, []))

The argument filtering Pi contains the following mapping:
p_in_a(x1)  =  p_in_a
U1_a(x1, x2)  =  U1_a(x2)
l_in_g(x1)  =  l_in_g(x1)
.(x1, x2)  =  .(x2)
[]  =  []
l_out_g(x1)  =  l_out_g(x1)
U2_g(x1, x2, x3)  =  U2_g(x2, x3)
r_in_a(x1)  =  r_in_a
r_out_a(x1)  =  r_out_a(x1)
U3_g(x1, x2, x3)  =  U3_g(x2, x3)
p_out_a(x1)  =  p_out_a(x1)

(3) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

P_IN_A(.(A, [])) → U1_A(A, l_in_g(.(A, [])))
P_IN_A(.(A, [])) → L_IN_G(.(A, []))
L_IN_G(.(H, T)) → U2_G(H, T, r_in_a(H))
L_IN_G(.(H, T)) → R_IN_A(H)
U2_G(H, T, r_out_a(H)) → U3_G(H, T, l_in_g(T))
U2_G(H, T, r_out_a(H)) → L_IN_G(T)

The TRS R consists of the following rules:

p_in_a(.(A, [])) → U1_a(A, l_in_g(.(A, [])))
l_in_g([]) → l_out_g([])
l_in_g(.(H, T)) → U2_g(H, T, r_in_a(H))
r_in_a(1) → r_out_a(1)
U2_g(H, T, r_out_a(H)) → U3_g(H, T, l_in_g(T))
U3_g(H, T, l_out_g(T)) → l_out_g(.(H, T))
U1_a(A, l_out_g(.(A, []))) → p_out_a(.(A, []))

The argument filtering Pi contains the following mapping:
p_in_a(x1)  =  p_in_a
U1_a(x1, x2)  =  U1_a(x2)
l_in_g(x1)  =  l_in_g(x1)
.(x1, x2)  =  .(x2)
[]  =  []
l_out_g(x1)  =  l_out_g(x1)
U2_g(x1, x2, x3)  =  U2_g(x2, x3)
r_in_a(x1)  =  r_in_a
r_out_a(x1)  =  r_out_a(x1)
U3_g(x1, x2, x3)  =  U3_g(x2, x3)
p_out_a(x1)  =  p_out_a(x1)
P_IN_A(x1)  =  P_IN_A
U1_A(x1, x2)  =  U1_A(x2)
L_IN_G(x1)  =  L_IN_G(x1)
U2_G(x1, x2, x3)  =  U2_G(x2, x3)
R_IN_A(x1)  =  R_IN_A
U3_G(x1, x2, x3)  =  U3_G(x2, x3)

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

(4) Obligation:

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

P_IN_A(.(A, [])) → U1_A(A, l_in_g(.(A, [])))
P_IN_A(.(A, [])) → L_IN_G(.(A, []))
L_IN_G(.(H, T)) → U2_G(H, T, r_in_a(H))
L_IN_G(.(H, T)) → R_IN_A(H)
U2_G(H, T, r_out_a(H)) → U3_G(H, T, l_in_g(T))
U2_G(H, T, r_out_a(H)) → L_IN_G(T)

The TRS R consists of the following rules:

p_in_a(.(A, [])) → U1_a(A, l_in_g(.(A, [])))
l_in_g([]) → l_out_g([])
l_in_g(.(H, T)) → U2_g(H, T, r_in_a(H))
r_in_a(1) → r_out_a(1)
U2_g(H, T, r_out_a(H)) → U3_g(H, T, l_in_g(T))
U3_g(H, T, l_out_g(T)) → l_out_g(.(H, T))
U1_a(A, l_out_g(.(A, []))) → p_out_a(.(A, []))

The argument filtering Pi contains the following mapping:
p_in_a(x1)  =  p_in_a
U1_a(x1, x2)  =  U1_a(x2)
l_in_g(x1)  =  l_in_g(x1)
.(x1, x2)  =  .(x2)
[]  =  []
l_out_g(x1)  =  l_out_g(x1)
U2_g(x1, x2, x3)  =  U2_g(x2, x3)
r_in_a(x1)  =  r_in_a
r_out_a(x1)  =  r_out_a(x1)
U3_g(x1, x2, x3)  =  U3_g(x2, x3)
p_out_a(x1)  =  p_out_a(x1)
P_IN_A(x1)  =  P_IN_A
U1_A(x1, x2)  =  U1_A(x2)
L_IN_G(x1)  =  L_IN_G(x1)
U2_G(x1, x2, x3)  =  U2_G(x2, x3)
R_IN_A(x1)  =  R_IN_A
U3_G(x1, x2, x3)  =  U3_G(x2, x3)

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

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 4 less nodes.

(6) Obligation:

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

U2_G(H, T, r_out_a(H)) → L_IN_G(T)
L_IN_G(.(H, T)) → U2_G(H, T, r_in_a(H))

The TRS R consists of the following rules:

p_in_a(.(A, [])) → U1_a(A, l_in_g(.(A, [])))
l_in_g([]) → l_out_g([])
l_in_g(.(H, T)) → U2_g(H, T, r_in_a(H))
r_in_a(1) → r_out_a(1)
U2_g(H, T, r_out_a(H)) → U3_g(H, T, l_in_g(T))
U3_g(H, T, l_out_g(T)) → l_out_g(.(H, T))
U1_a(A, l_out_g(.(A, []))) → p_out_a(.(A, []))

The argument filtering Pi contains the following mapping:
p_in_a(x1)  =  p_in_a
U1_a(x1, x2)  =  U1_a(x2)
l_in_g(x1)  =  l_in_g(x1)
.(x1, x2)  =  .(x2)
[]  =  []
l_out_g(x1)  =  l_out_g(x1)
U2_g(x1, x2, x3)  =  U2_g(x2, x3)
r_in_a(x1)  =  r_in_a
r_out_a(x1)  =  r_out_a(x1)
U3_g(x1, x2, x3)  =  U3_g(x2, x3)
p_out_a(x1)  =  p_out_a(x1)
L_IN_G(x1)  =  L_IN_G(x1)
U2_G(x1, x2, x3)  =  U2_G(x2, x3)

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

(7) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(8) Obligation:

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

U2_G(H, T, r_out_a(H)) → L_IN_G(T)
L_IN_G(.(H, T)) → U2_G(H, T, r_in_a(H))

The TRS R consists of the following rules:

r_in_a(1) → r_out_a(1)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
r_in_a(x1)  =  r_in_a
r_out_a(x1)  =  r_out_a(x1)
L_IN_G(x1)  =  L_IN_G(x1)
U2_G(x1, x2, x3)  =  U2_G(x2, x3)

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

(9) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(10) Obligation:

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

U2_G(T, r_out_a(H)) → L_IN_G(T)
L_IN_G(.(T)) → U2_G(T, r_in_a)

The TRS R consists of the following rules:

r_in_a → r_out_a(1)

The set Q consists of the following terms:

r_in_a

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

(11) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] 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:

  • L_IN_G(.(T)) → U2_G(T, r_in_a)
    The graph contains the following edges 1 > 1

  • U2_G(T, r_out_a(H)) → L_IN_G(T)
    The graph contains the following edges 1 >= 1

(12) TRUE

(13) PrologToPiTRSProof (SOUND transformation)

We use the technique of [LOPSTR]. With regard to the inferred argument filtering the predicates were used in the following modes:
p_in: (f)
l_in: (b)
Transforming Prolog into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

p_in_a(.(A, [])) → U1_a(A, l_in_g(.(A, [])))
l_in_g([]) → l_out_g([])
l_in_g(.(H, T)) → U2_g(H, T, r_in_a(H))
r_in_a(1) → r_out_a(1)
U2_g(H, T, r_out_a(H)) → U3_g(H, T, l_in_g(T))
U3_g(H, T, l_out_g(T)) → l_out_g(.(H, T))
U1_a(A, l_out_g(.(A, []))) → p_out_a(.(A, []))

The argument filtering Pi contains the following mapping:
p_in_a(x1)  =  p_in_a
U1_a(x1, x2)  =  U1_a(x2)
l_in_g(x1)  =  l_in_g(x1)
.(x1, x2)  =  .(x2)
[]  =  []
l_out_g(x1)  =  l_out_g
U2_g(x1, x2, x3)  =  U2_g(x2, x3)
r_in_a(x1)  =  r_in_a
r_out_a(x1)  =  r_out_a(x1)
U3_g(x1, x2, x3)  =  U3_g(x3)
p_out_a(x1)  =  p_out_a(x1)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(14) Obligation:

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

p_in_a(.(A, [])) → U1_a(A, l_in_g(.(A, [])))
l_in_g([]) → l_out_g([])
l_in_g(.(H, T)) → U2_g(H, T, r_in_a(H))
r_in_a(1) → r_out_a(1)
U2_g(H, T, r_out_a(H)) → U3_g(H, T, l_in_g(T))
U3_g(H, T, l_out_g(T)) → l_out_g(.(H, T))
U1_a(A, l_out_g(.(A, []))) → p_out_a(.(A, []))

The argument filtering Pi contains the following mapping:
p_in_a(x1)  =  p_in_a
U1_a(x1, x2)  =  U1_a(x2)
l_in_g(x1)  =  l_in_g(x1)
.(x1, x2)  =  .(x2)
[]  =  []
l_out_g(x1)  =  l_out_g
U2_g(x1, x2, x3)  =  U2_g(x2, x3)
r_in_a(x1)  =  r_in_a
r_out_a(x1)  =  r_out_a(x1)
U3_g(x1, x2, x3)  =  U3_g(x3)
p_out_a(x1)  =  p_out_a(x1)

(15) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
Pi DP problem:
The TRS P consists of the following rules:

P_IN_A(.(A, [])) → U1_A(A, l_in_g(.(A, [])))
P_IN_A(.(A, [])) → L_IN_G(.(A, []))
L_IN_G(.(H, T)) → U2_G(H, T, r_in_a(H))
L_IN_G(.(H, T)) → R_IN_A(H)
U2_G(H, T, r_out_a(H)) → U3_G(H, T, l_in_g(T))
U2_G(H, T, r_out_a(H)) → L_IN_G(T)

The TRS R consists of the following rules:

p_in_a(.(A, [])) → U1_a(A, l_in_g(.(A, [])))
l_in_g([]) → l_out_g([])
l_in_g(.(H, T)) → U2_g(H, T, r_in_a(H))
r_in_a(1) → r_out_a(1)
U2_g(H, T, r_out_a(H)) → U3_g(H, T, l_in_g(T))
U3_g(H, T, l_out_g(T)) → l_out_g(.(H, T))
U1_a(A, l_out_g(.(A, []))) → p_out_a(.(A, []))

The argument filtering Pi contains the following mapping:
p_in_a(x1)  =  p_in_a
U1_a(x1, x2)  =  U1_a(x2)
l_in_g(x1)  =  l_in_g(x1)
.(x1, x2)  =  .(x2)
[]  =  []
l_out_g(x1)  =  l_out_g
U2_g(x1, x2, x3)  =  U2_g(x2, x3)
r_in_a(x1)  =  r_in_a
r_out_a(x1)  =  r_out_a(x1)
U3_g(x1, x2, x3)  =  U3_g(x3)
p_out_a(x1)  =  p_out_a(x1)
P_IN_A(x1)  =  P_IN_A
U1_A(x1, x2)  =  U1_A(x2)
L_IN_G(x1)  =  L_IN_G(x1)
U2_G(x1, x2, x3)  =  U2_G(x2, x3)
R_IN_A(x1)  =  R_IN_A
U3_G(x1, x2, x3)  =  U3_G(x3)

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

(16) Obligation:

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

P_IN_A(.(A, [])) → U1_A(A, l_in_g(.(A, [])))
P_IN_A(.(A, [])) → L_IN_G(.(A, []))
L_IN_G(.(H, T)) → U2_G(H, T, r_in_a(H))
L_IN_G(.(H, T)) → R_IN_A(H)
U2_G(H, T, r_out_a(H)) → U3_G(H, T, l_in_g(T))
U2_G(H, T, r_out_a(H)) → L_IN_G(T)

The TRS R consists of the following rules:

p_in_a(.(A, [])) → U1_a(A, l_in_g(.(A, [])))
l_in_g([]) → l_out_g([])
l_in_g(.(H, T)) → U2_g(H, T, r_in_a(H))
r_in_a(1) → r_out_a(1)
U2_g(H, T, r_out_a(H)) → U3_g(H, T, l_in_g(T))
U3_g(H, T, l_out_g(T)) → l_out_g(.(H, T))
U1_a(A, l_out_g(.(A, []))) → p_out_a(.(A, []))

The argument filtering Pi contains the following mapping:
p_in_a(x1)  =  p_in_a
U1_a(x1, x2)  =  U1_a(x2)
l_in_g(x1)  =  l_in_g(x1)
.(x1, x2)  =  .(x2)
[]  =  []
l_out_g(x1)  =  l_out_g
U2_g(x1, x2, x3)  =  U2_g(x2, x3)
r_in_a(x1)  =  r_in_a
r_out_a(x1)  =  r_out_a(x1)
U3_g(x1, x2, x3)  =  U3_g(x3)
p_out_a(x1)  =  p_out_a(x1)
P_IN_A(x1)  =  P_IN_A
U1_A(x1, x2)  =  U1_A(x2)
L_IN_G(x1)  =  L_IN_G(x1)
U2_G(x1, x2, x3)  =  U2_G(x2, x3)
R_IN_A(x1)  =  R_IN_A
U3_G(x1, x2, x3)  =  U3_G(x3)

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

(17) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 1 SCC with 4 less nodes.

(18) Obligation:

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

U2_G(H, T, r_out_a(H)) → L_IN_G(T)
L_IN_G(.(H, T)) → U2_G(H, T, r_in_a(H))

The TRS R consists of the following rules:

p_in_a(.(A, [])) → U1_a(A, l_in_g(.(A, [])))
l_in_g([]) → l_out_g([])
l_in_g(.(H, T)) → U2_g(H, T, r_in_a(H))
r_in_a(1) → r_out_a(1)
U2_g(H, T, r_out_a(H)) → U3_g(H, T, l_in_g(T))
U3_g(H, T, l_out_g(T)) → l_out_g(.(H, T))
U1_a(A, l_out_g(.(A, []))) → p_out_a(.(A, []))

The argument filtering Pi contains the following mapping:
p_in_a(x1)  =  p_in_a
U1_a(x1, x2)  =  U1_a(x2)
l_in_g(x1)  =  l_in_g(x1)
.(x1, x2)  =  .(x2)
[]  =  []
l_out_g(x1)  =  l_out_g
U2_g(x1, x2, x3)  =  U2_g(x2, x3)
r_in_a(x1)  =  r_in_a
r_out_a(x1)  =  r_out_a(x1)
U3_g(x1, x2, x3)  =  U3_g(x3)
p_out_a(x1)  =  p_out_a(x1)
L_IN_G(x1)  =  L_IN_G(x1)
U2_G(x1, x2, x3)  =  U2_G(x2, x3)

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

(19) UsableRulesProof (EQUIVALENT transformation)

For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.

(20) Obligation:

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

U2_G(H, T, r_out_a(H)) → L_IN_G(T)
L_IN_G(.(H, T)) → U2_G(H, T, r_in_a(H))

The TRS R consists of the following rules:

r_in_a(1) → r_out_a(1)

The argument filtering Pi contains the following mapping:
.(x1, x2)  =  .(x2)
r_in_a(x1)  =  r_in_a
r_out_a(x1)  =  r_out_a(x1)
L_IN_G(x1)  =  L_IN_G(x1)
U2_G(x1, x2, x3)  =  U2_G(x2, x3)

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

(21) PiDPToQDPProof (SOUND transformation)

Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.

(22) Obligation:

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

U2_G(T, r_out_a(H)) → L_IN_G(T)
L_IN_G(.(T)) → U2_G(T, r_in_a)

The TRS R consists of the following rules:

r_in_a → r_out_a(1)

The set Q consists of the following terms:

r_in_a

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