Left Termination proof of /tmp/tmpVC8DZv/mul1.pl

Left Termination of the query pattern mul_in_3(g, g, a) w.r.t. the given Prolog program could not be shown:

0 Prolog

↳1 CutEliminatorProof (⇐)

↳2 Prolog

  ↳3 PrologToPiTRSProof (⇐)

    ↳4 PiTRS

    ↳5 DependencyPairsProof (⇔)

    ↳6 PiDP

    ↳7 DependencyGraphProof (⇔)

    ↳8 AND

      ↳9 PiDP

        ↳10 UsableRulesProof (⇔)

        ↳11 PiDP

        ↳12 PiDPToQDPProof (⇐)

        ↳13 QDP

        ↳14 UsableRulesReductionPairsProof (⇔)

        ↳15 QDP

        ↳16 Narrowing (⇐)

        ↳17 QDP

        ↳18 UsableRulesProof (⇔)

        ↳19 QDP

        ↳20 QReductionProof (⇔)

        ↳21 QDP

        ↳22 Instantiation (⇔)

        ↳23 QDP

        ↳24 NonTerminationProof (⇔)

        ↳25 FALSE

      ↳26 PiDP

        ↳27 UsableRulesProof (⇔)

        ↳28 PiDP

        ↳29 PiDPToQDPProof (⇐)

        ↳30 QDP

        ↳31 UsableRulesReductionPairsProof (⇔)

        ↳32 QDP

        ↳33 Narrowing (⇐)

        ↳34 QDP

        ↳35 UsableRulesProof (⇔)

        ↳36 QDP

        ↳37 QReductionProof (⇔)

        ↳38 QDP

        ↳39 Instantiation (⇔)

        ↳40 QDP

        ↳41 NonTerminationProof (⇔)

        ↳42 FALSE

  ↳43 PrologToPiTRSProof (⇐)

    ↳44 PiTRS

    ↳45 DependencyPairsProof (⇔)

    ↳46 PiDP

    ↳47 DependencyGraphProof (⇔)

    ↳48 AND

      ↳49 PiDP

        ↳50 UsableRulesProof (⇔)

        ↳51 PiDP

        ↳52 PiDPToQDPProof (⇐)

        ↳53 QDP

        ↳54 Narrowing (⇐)

        ↳55 QDP

        ↳56 UsableRulesProof (⇔)

        ↳57 QDP

        ↳58 QReductionProof (⇔)

        ↳59 QDP

        ↳60 Instantiation (⇔)

        ↳61 QDP

        ↳62 DependencyGraphProof (⇔)

        ↳63 AND

          ↳64 QDP

            ↳65 NonTerminationProof (⇔)

            ↳66 FALSE

          ↳67 QDP

            ↳68 QDPSizeChangeProof (⇔)

            ↳69 TRUE

      ↳70 PiDP

        ↳71 UsableRulesProof (⇔)

        ↳72 PiDP

        ↳73 PiDPToQDPProof (⇐)

        ↳74 QDP

        ↳75 Narrowing (⇐)

        ↳76 QDP

        ↳77 UsableRulesProof (⇔)

        ↳78 QDP

        ↳79 QReductionProof (⇔)

        ↳80 QDP

        ↳81 Instantiation (⇔)

        ↳82 QDP

        ↳83 DependencyGraphProof (⇔)

        ↳84 AND

          ↳85 QDP

            ↳86 NonTerminationProof (⇔)

            ↳87 FALSE

          ↳88 QDP

            ↳89 QDPSizeChangeProof (⇔)

            ↳90 TRUE

(0) Obligation:

Clauses:

mul(X1, 0, Z) :- ','(!, eq(Z, 0)).
mul(X, Y, Z) :- ','(p(Y, P), ','(mul(X, P, V), add(X, V, Z))).
add(X, 0, Z) :- ','(!, eq(Z, X)).
add(X, Y, Z) :- ','(p(Y, V), ','(add(X, V, W), p(Z, W))).
p(0, 0).
p(s(X), X).
eq(X, X).

Queries:

mul(g,g,a).

(1) CutEliminatorProof (SOUND transformation)

Eliminated all cuts by simply ignoring them[PROLOG].

(2) Obligation:

Clauses:

mul(X1, 0, Z) :- eq(Z, 0).
mul(X, Y, Z) :- ','(p(Y, P), ','(mul(X, P, V), add(X, V, Z))).
add(X, 0, Z) :- eq(Z, X).
add(X, Y, Z) :- ','(p(Y, V), ','(add(X, V, W), p(Z, W))).
p(0, 0).
p(s(X), X).
eq(X, X).

Queries:

mul(g,g,a).

(3) PrologToPiTRSProof (SOUND transformation)

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

mul_in_gga(X1, 0, Z) → U1_gga(X1, Z, eq_in_ag(Z, 0))
eq_in_ag(X, X) → eq_out_ag(X, X)
U1_gga(X1, Z, eq_out_ag(Z, 0)) → mul_out_gga(X1, 0, Z)
mul_in_gga(X, Y, Z) → U2_gga(X, Y, Z, p_in_ga(Y, P))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U2_gga(X, Y, Z, p_out_ga(Y, P)) → U3_gga(X, Y, Z, P, mul_in_gga(X, P, V))
U3_gga(X, Y, Z, P, mul_out_gga(X, P, V)) → U4_gga(X, Y, Z, add_in_gga(X, V, Z))
add_in_gga(X, 0, Z) → U5_gga(X, Z, eq_in_ag(Z, X))
U5_gga(X, Z, eq_out_ag(Z, X)) → add_out_gga(X, 0, Z)
add_in_gga(X, Y, Z) → U6_gga(X, Y, Z, p_in_ga(Y, V))
U6_gga(X, Y, Z, p_out_ga(Y, V)) → U7_gga(X, Y, Z, V, add_in_gga(X, V, W))
U7_gga(X, Y, Z, V, add_out_gga(X, V, W)) → U8_gga(X, Y, Z, p_in_ag(Z, W))
p_in_ag(0, 0) → p_out_ag(0, 0)
p_in_ag(s(X), X) → p_out_ag(s(X), X)
U8_gga(X, Y, Z, p_out_ag(Z, W)) → add_out_gga(X, Y, Z)
U4_gga(X, Y, Z, add_out_gga(X, V, Z)) → mul_out_gga(X, Y, Z)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(4) Obligation:

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

mul_in_gga(X1, 0, Z) → U1_gga(X1, Z, eq_in_ag(Z, 0))
eq_in_ag(X, X) → eq_out_ag(X, X)
U1_gga(X1, Z, eq_out_ag(Z, 0)) → mul_out_gga(X1, 0, Z)
mul_in_gga(X, Y, Z) → U2_gga(X, Y, Z, p_in_ga(Y, P))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U2_gga(X, Y, Z, p_out_ga(Y, P)) → U3_gga(X, Y, Z, P, mul_in_gga(X, P, V))
U3_gga(X, Y, Z, P, mul_out_gga(X, P, V)) → U4_gga(X, Y, Z, add_in_gga(X, V, Z))
add_in_gga(X, 0, Z) → U5_gga(X, Z, eq_in_ag(Z, X))
U5_gga(X, Z, eq_out_ag(Z, X)) → add_out_gga(X, 0, Z)
add_in_gga(X, Y, Z) → U6_gga(X, Y, Z, p_in_ga(Y, V))
U6_gga(X, Y, Z, p_out_ga(Y, V)) → U7_gga(X, Y, Z, V, add_in_gga(X, V, W))
U7_gga(X, Y, Z, V, add_out_gga(X, V, W)) → U8_gga(X, Y, Z, p_in_ag(Z, W))
p_in_ag(0, 0) → p_out_ag(0, 0)
p_in_ag(s(X), X) → p_out_ag(s(X), X)
U8_gga(X, Y, Z, p_out_ag(Z, W)) → add_out_gga(X, Y, Z)
U4_gga(X, Y, Z, add_out_gga(X, V, Z)) → mul_out_gga(X, Y, Z)

(5) 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:

MUL_IN_GGA(X1, 0, Z) → U1_GGA(X1, Z, eq_in_ag(Z, 0))
MUL_IN_GGA(X1, 0, Z) → EQ_IN_AG(Z, 0)
MUL_IN_GGA(X, Y, Z) → U2_GGA(X, Y, Z, p_in_ga(Y, P))
MUL_IN_GGA(X, Y, Z) → P_IN_GA(Y, P)
U2_GGA(X, Y, Z, p_out_ga(Y, P)) → U3_GGA(X, Y, Z, P, mul_in_gga(X, P, V))
U2_GGA(X, Y, Z, p_out_ga(Y, P)) → MUL_IN_GGA(X, P, V)
U3_GGA(X, Y, Z, P, mul_out_gga(X, P, V)) → U4_GGA(X, Y, Z, add_in_gga(X, V, Z))
U3_GGA(X, Y, Z, P, mul_out_gga(X, P, V)) → ADD_IN_GGA(X, V, Z)
ADD_IN_GGA(X, 0, Z) → U5_GGA(X, Z, eq_in_ag(Z, X))
ADD_IN_GGA(X, 0, Z) → EQ_IN_AG(Z, X)
ADD_IN_GGA(X, Y, Z) → U6_GGA(X, Y, Z, p_in_ga(Y, V))
ADD_IN_GGA(X, Y, Z) → P_IN_GA(Y, V)
U6_GGA(X, Y, Z, p_out_ga(Y, V)) → U7_GGA(X, Y, Z, V, add_in_gga(X, V, W))
U6_GGA(X, Y, Z, p_out_ga(Y, V)) → ADD_IN_GGA(X, V, W)
U7_GGA(X, Y, Z, V, add_out_gga(X, V, W)) → U8_GGA(X, Y, Z, p_in_ag(Z, W))
U7_GGA(X, Y, Z, V, add_out_gga(X, V, W)) → P_IN_AG(Z, W)

The TRS R consists of the following rules:

mul_in_gga(X1, 0, Z) → U1_gga(X1, Z, eq_in_ag(Z, 0))
eq_in_ag(X, X) → eq_out_ag(X, X)
U1_gga(X1, Z, eq_out_ag(Z, 0)) → mul_out_gga(X1, 0, Z)
mul_in_gga(X, Y, Z) → U2_gga(X, Y, Z, p_in_ga(Y, P))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U2_gga(X, Y, Z, p_out_ga(Y, P)) → U3_gga(X, Y, Z, P, mul_in_gga(X, P, V))
U3_gga(X, Y, Z, P, mul_out_gga(X, P, V)) → U4_gga(X, Y, Z, add_in_gga(X, V, Z))
add_in_gga(X, 0, Z) → U5_gga(X, Z, eq_in_ag(Z, X))
U5_gga(X, Z, eq_out_ag(Z, X)) → add_out_gga(X, 0, Z)
add_in_gga(X, Y, Z) → U6_gga(X, Y, Z, p_in_ga(Y, V))
U6_gga(X, Y, Z, p_out_ga(Y, V)) → U7_gga(X, Y, Z, V, add_in_gga(X, V, W))
U7_gga(X, Y, Z, V, add_out_gga(X, V, W)) → U8_gga(X, Y, Z, p_in_ag(Z, W))
p_in_ag(0, 0) → p_out_ag(0, 0)
p_in_ag(s(X), X) → p_out_ag(s(X), X)
U8_gga(X, Y, Z, p_out_ag(Z, W)) → add_out_gga(X, Y, Z)
U4_gga(X, Y, Z, add_out_gga(X, V, Z)) → mul_out_gga(X, Y, Z)

The argument filtering Pi contains the following mapping:
mul_in_gga(x1, x2, x3) = mul_in_gga(x1, x2)
0 = 0
U1_gga(x1, x2, x3) = U1_gga(x3)
eq_in_ag(x1, x2) = eq_in_ag(x2)
eq_out_ag(x1, x2) = eq_out_ag(x1)
mul_out_gga(x1, x2, x3) = mul_out_gga(x3)
U2_gga(x1, x2, x3, x4) = U2_gga(x1, x4)
p_in_ga(x1, x2) = p_in_ga(x1)
p_out_ga(x1, x2) = p_out_ga(x2)
s(x1) = s(x1)
U3_gga(x1, x2, x3, x4, x5) = U3_gga(x1, x5)
U4_gga(x1, x2, x3, x4) = U4_gga(x4)
add_in_gga(x1, x2, x3) = add_in_gga(x1, x2)
U5_gga(x1, x2, x3) = U5_gga(x3)
add_out_gga(x1, x2, x3) = add_out_gga(x3)
U6_gga(x1, x2, x3, x4) = U6_gga(x1, x4)
U7_gga(x1, x2, x3, x4, x5) = U7_gga(x5)
U8_gga(x1, x2, x3, x4) = U8_gga(x4)
p_in_ag(x1, x2) = p_in_ag(x2)
p_out_ag(x1, x2) = p_out_ag(x1)
MUL_IN_GGA(x1, x2, x3) = MUL_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3) = U1_GGA(x3)
EQ_IN_AG(x1, x2) = EQ_IN_AG(x2)
U2_GGA(x1, x2, x3, x4) = U2_GGA(x1, x4)
P_IN_GA(x1, x2) = P_IN_GA(x1)
U3_GGA(x1, x2, x3, x4, x5) = U3_GGA(x1, x5)
U4_GGA(x1, x2, x3, x4) = U4_GGA(x4)
ADD_IN_GGA(x1, x2, x3) = ADD_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3) = U5_GGA(x3)
U6_GGA(x1, x2, x3, x4) = U6_GGA(x1, x4)
U7_GGA(x1, x2, x3, x4, x5) = U7_GGA(x5)
U8_GGA(x1, x2, x3, x4) = U8_GGA(x4)
P_IN_AG(x1, x2) = P_IN_AG(x2)

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

(6) Obligation:

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

MUL_IN_GGA(X1, 0, Z) → U1_GGA(X1, Z, eq_in_ag(Z, 0))
MUL_IN_GGA(X1, 0, Z) → EQ_IN_AG(Z, 0)
MUL_IN_GGA(X, Y, Z) → U2_GGA(X, Y, Z, p_in_ga(Y, P))
MUL_IN_GGA(X, Y, Z) → P_IN_GA(Y, P)
U2_GGA(X, Y, Z, p_out_ga(Y, P)) → U3_GGA(X, Y, Z, P, mul_in_gga(X, P, V))
U2_GGA(X, Y, Z, p_out_ga(Y, P)) → MUL_IN_GGA(X, P, V)
U3_GGA(X, Y, Z, P, mul_out_gga(X, P, V)) → U4_GGA(X, Y, Z, add_in_gga(X, V, Z))
U3_GGA(X, Y, Z, P, mul_out_gga(X, P, V)) → ADD_IN_GGA(X, V, Z)
ADD_IN_GGA(X, 0, Z) → U5_GGA(X, Z, eq_in_ag(Z, X))
ADD_IN_GGA(X, 0, Z) → EQ_IN_AG(Z, X)
ADD_IN_GGA(X, Y, Z) → U6_GGA(X, Y, Z, p_in_ga(Y, V))
ADD_IN_GGA(X, Y, Z) → P_IN_GA(Y, V)
U6_GGA(X, Y, Z, p_out_ga(Y, V)) → U7_GGA(X, Y, Z, V, add_in_gga(X, V, W))
U6_GGA(X, Y, Z, p_out_ga(Y, V)) → ADD_IN_GGA(X, V, W)
U7_GGA(X, Y, Z, V, add_out_gga(X, V, W)) → U8_GGA(X, Y, Z, p_in_ag(Z, W))
U7_GGA(X, Y, Z, V, add_out_gga(X, V, W)) → P_IN_AG(Z, W)

The TRS R consists of the following rules:

mul_in_gga(X1, 0, Z) → U1_gga(X1, Z, eq_in_ag(Z, 0))
eq_in_ag(X, X) → eq_out_ag(X, X)
U1_gga(X1, Z, eq_out_ag(Z, 0)) → mul_out_gga(X1, 0, Z)
mul_in_gga(X, Y, Z) → U2_gga(X, Y, Z, p_in_ga(Y, P))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U2_gga(X, Y, Z, p_out_ga(Y, P)) → U3_gga(X, Y, Z, P, mul_in_gga(X, P, V))
U3_gga(X, Y, Z, P, mul_out_gga(X, P, V)) → U4_gga(X, Y, Z, add_in_gga(X, V, Z))
add_in_gga(X, 0, Z) → U5_gga(X, Z, eq_in_ag(Z, X))
U5_gga(X, Z, eq_out_ag(Z, X)) → add_out_gga(X, 0, Z)
add_in_gga(X, Y, Z) → U6_gga(X, Y, Z, p_in_ga(Y, V))
U6_gga(X, Y, Z, p_out_ga(Y, V)) → U7_gga(X, Y, Z, V, add_in_gga(X, V, W))
U7_gga(X, Y, Z, V, add_out_gga(X, V, W)) → U8_gga(X, Y, Z, p_in_ag(Z, W))
p_in_ag(0, 0) → p_out_ag(0, 0)
p_in_ag(s(X), X) → p_out_ag(s(X), X)
U8_gga(X, Y, Z, p_out_ag(Z, W)) → add_out_gga(X, Y, Z)
U4_gga(X, Y, Z, add_out_gga(X, V, Z)) → mul_out_gga(X, Y, Z)

(7) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 12 less nodes.

(8) Complex Obligation (AND)

(9) Obligation:

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

ADD_IN_GGA(X, Y, Z) → U6_GGA(X, Y, Z, p_in_ga(Y, V))
U6_GGA(X, Y, Z, p_out_ga(Y, V)) → ADD_IN_GGA(X, V, W)

The TRS R consists of the following rules:

mul_in_gga(X1, 0, Z) → U1_gga(X1, Z, eq_in_ag(Z, 0))
eq_in_ag(X, X) → eq_out_ag(X, X)
U1_gga(X1, Z, eq_out_ag(Z, 0)) → mul_out_gga(X1, 0, Z)
mul_in_gga(X, Y, Z) → U2_gga(X, Y, Z, p_in_ga(Y, P))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U2_gga(X, Y, Z, p_out_ga(Y, P)) → U3_gga(X, Y, Z, P, mul_in_gga(X, P, V))
U3_gga(X, Y, Z, P, mul_out_gga(X, P, V)) → U4_gga(X, Y, Z, add_in_gga(X, V, Z))
add_in_gga(X, 0, Z) → U5_gga(X, Z, eq_in_ag(Z, X))
U5_gga(X, Z, eq_out_ag(Z, X)) → add_out_gga(X, 0, Z)
add_in_gga(X, Y, Z) → U6_gga(X, Y, Z, p_in_ga(Y, V))
U6_gga(X, Y, Z, p_out_ga(Y, V)) → U7_gga(X, Y, Z, V, add_in_gga(X, V, W))
U7_gga(X, Y, Z, V, add_out_gga(X, V, W)) → U8_gga(X, Y, Z, p_in_ag(Z, W))
p_in_ag(0, 0) → p_out_ag(0, 0)
p_in_ag(s(X), X) → p_out_ag(s(X), X)
U8_gga(X, Y, Z, p_out_ag(Z, W)) → add_out_gga(X, Y, Z)
U4_gga(X, Y, Z, add_out_gga(X, V, Z)) → mul_out_gga(X, Y, Z)

(10) UsableRulesProof (EQUIVALENT transformation)

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

(11) Obligation:

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

ADD_IN_GGA(X, Y, Z) → U6_GGA(X, Y, Z, p_in_ga(Y, V))
U6_GGA(X, Y, Z, p_out_ga(Y, V)) → ADD_IN_GGA(X, V, W)

The TRS R consists of the following rules:

p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)

The argument filtering Pi contains the following mapping:
0 = 0
p_in_ga(x1, x2) = p_in_ga(x1)
p_out_ga(x1, x2) = p_out_ga(x2)
s(x1) = s(x1)
ADD_IN_GGA(x1, x2, x3) = ADD_IN_GGA(x1, x2)
U6_GGA(x1, x2, x3, x4) = U6_GGA(x1, x4)

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

(12) PiDPToQDPProof (SOUND transformation)

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

(13) Obligation:

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

ADD_IN_GGA(X, Y) → U6_GGA(X, p_in_ga(Y))
U6_GGA(X, p_out_ga(V)) → ADD_IN_GGA(X, V)

The TRS R consists of the following rules:

p_in_ga(0) → p_out_ga(0)
p_in_ga(s(X)) → p_out_ga(X)

The set Q consists of the following terms:

p_in_ga(x0)

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

(14) UsableRulesReductionPairsProof (EQUIVALENT transformation)

By using the usable rules with reduction pair processor [LPAR04] with a polynomial ordering [POLO], all dependency pairs and the corresponding usable rules [FROCOS05] can be oriented non-strictly. All non-usable rules are removed, and those dependency pairs and usable rules that have been oriented strictly or contain non-usable symbols in their left-hand side are removed as well.

No dependency pairs are removed.

The following rules are removed from R:

p_in_ga(s(X)) → p_out_ga(X)

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(0) = 0
POL(ADD_IN_GGA(x₁, x₂)) = x₁ + x₂
POL(U6_GGA(x₁, x₂)) = x₁ + x₂
POL(p_in_ga(x₁)) = x₁
POL(p_out_ga(x₁)) = 2·x₁
POL(s(x₁)) = 2·x₁

(15) Obligation:

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

ADD_IN_GGA(X, Y) → U6_GGA(X, p_in_ga(Y))
U6_GGA(X, p_out_ga(V)) → ADD_IN_GGA(X, V)

The TRS R consists of the following rules:

p_in_ga(0) → p_out_ga(0)

The set Q consists of the following terms:

p_in_ga(x0)

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

(16) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule ADD_IN_GGA(X, Y) → U6_GGA(X, p_in_ga(Y)) at position [1] we obtained the following new rules [LPAR04]:

ADD_IN_GGA(y0, 0) → U6_GGA(y0, p_out_ga(0))

(17) Obligation:

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

U6_GGA(X, p_out_ga(V)) → ADD_IN_GGA(X, V)
ADD_IN_GGA(y0, 0) → U6_GGA(y0, p_out_ga(0))

The TRS R consists of the following rules:

p_in_ga(0) → p_out_ga(0)

The set Q consists of the following terms:

p_in_ga(x0)

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

(18) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(19) Obligation:

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

U6_GGA(X, p_out_ga(V)) → ADD_IN_GGA(X, V)
ADD_IN_GGA(y0, 0) → U6_GGA(y0, p_out_ga(0))

R is empty.
The set Q consists of the following terms:

p_in_ga(x0)

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

(20) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

p_in_ga(x0)

(21) Obligation:

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

U6_GGA(X, p_out_ga(V)) → ADD_IN_GGA(X, V)
ADD_IN_GGA(y0, 0) → U6_GGA(y0, p_out_ga(0))

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

(22) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U6_GGA(X, p_out_ga(V)) → ADD_IN_GGA(X, V) we obtained the following new rules [LPAR04]:

U6_GGA(z0, p_out_ga(0)) → ADD_IN_GGA(z0, 0)

(23) Obligation:

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

ADD_IN_GGA(y0, 0) → U6_GGA(y0, p_out_ga(0))
U6_GGA(z0, p_out_ga(0)) → ADD_IN_GGA(z0, 0)

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

(24) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = U6_GGA(z0, p_out_ga(0)) evaluates to t =U6_GGA(z0, p_out_ga(0))

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Matcher: [ ]
Semiunifier: [ ]

Rewriting sequence

U6_GGA(z0, p_out_ga(0)) → ADD_IN_GGA(z0, 0)
with rule U6_GGA(z0', p_out_ga(0)) → ADD_IN_GGA(z0', 0) at position [] and matcher [z0' / z0]

ADD_IN_GGA(z0, 0) → U6_GGA(z0, p_out_ga(0))
with rule ADD_IN_GGA(y0, 0) → U6_GGA(y0, p_out_ga(0))

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence

All these steps are and every following step will be a correct step w.r.t to Q.

(25) FALSE

(26) Obligation:

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

MUL_IN_GGA(X, Y, Z) → U2_GGA(X, Y, Z, p_in_ga(Y, P))
U2_GGA(X, Y, Z, p_out_ga(Y, P)) → MUL_IN_GGA(X, P, V)

The TRS R consists of the following rules:

mul_in_gga(X1, 0, Z) → U1_gga(X1, Z, eq_in_ag(Z, 0))
eq_in_ag(X, X) → eq_out_ag(X, X)
U1_gga(X1, Z, eq_out_ag(Z, 0)) → mul_out_gga(X1, 0, Z)
mul_in_gga(X, Y, Z) → U2_gga(X, Y, Z, p_in_ga(Y, P))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U2_gga(X, Y, Z, p_out_ga(Y, P)) → U3_gga(X, Y, Z, P, mul_in_gga(X, P, V))
U3_gga(X, Y, Z, P, mul_out_gga(X, P, V)) → U4_gga(X, Y, Z, add_in_gga(X, V, Z))
add_in_gga(X, 0, Z) → U5_gga(X, Z, eq_in_ag(Z, X))
U5_gga(X, Z, eq_out_ag(Z, X)) → add_out_gga(X, 0, Z)
add_in_gga(X, Y, Z) → U6_gga(X, Y, Z, p_in_ga(Y, V))
U6_gga(X, Y, Z, p_out_ga(Y, V)) → U7_gga(X, Y, Z, V, add_in_gga(X, V, W))
U7_gga(X, Y, Z, V, add_out_gga(X, V, W)) → U8_gga(X, Y, Z, p_in_ag(Z, W))
p_in_ag(0, 0) → p_out_ag(0, 0)
p_in_ag(s(X), X) → p_out_ag(s(X), X)
U8_gga(X, Y, Z, p_out_ag(Z, W)) → add_out_gga(X, Y, Z)
U4_gga(X, Y, Z, add_out_gga(X, V, Z)) → mul_out_gga(X, Y, Z)

(27) UsableRulesProof (EQUIVALENT transformation)

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

(28) Obligation:

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

MUL_IN_GGA(X, Y, Z) → U2_GGA(X, Y, Z, p_in_ga(Y, P))
U2_GGA(X, Y, Z, p_out_ga(Y, P)) → MUL_IN_GGA(X, P, V)

The TRS R consists of the following rules:

p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)

The argument filtering Pi contains the following mapping:
0 = 0
p_in_ga(x1, x2) = p_in_ga(x1)
p_out_ga(x1, x2) = p_out_ga(x2)
s(x1) = s(x1)
MUL_IN_GGA(x1, x2, x3) = MUL_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4) = U2_GGA(x1, x4)

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

(29) PiDPToQDPProof (SOUND transformation)

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

(30) Obligation:

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

MUL_IN_GGA(X, Y) → U2_GGA(X, p_in_ga(Y))
U2_GGA(X, p_out_ga(P)) → MUL_IN_GGA(X, P)

The TRS R consists of the following rules:

p_in_ga(0) → p_out_ga(0)
p_in_ga(s(X)) → p_out_ga(X)

The set Q consists of the following terms:

p_in_ga(x0)

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

(31) UsableRulesReductionPairsProof (EQUIVALENT transformation)

p_in_ga(s(X)) → p_out_ga(X)

Used ordering: POLO with Polynomial interpretation [POLO]:

POL(0) = 0
POL(MUL_IN_GGA(x₁, x₂)) = x₁ + x₂
POL(U2_GGA(x₁, x₂)) = x₁ + x₂
POL(p_in_ga(x₁)) = x₁
POL(p_out_ga(x₁)) = 2·x₁
POL(s(x₁)) = 2·x₁

(32) Obligation:

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

MUL_IN_GGA(X, Y) → U2_GGA(X, p_in_ga(Y))
U2_GGA(X, p_out_ga(P)) → MUL_IN_GGA(X, P)

The TRS R consists of the following rules:

p_in_ga(0) → p_out_ga(0)

The set Q consists of the following terms:

p_in_ga(x0)

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

(33) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule MUL_IN_GGA(X, Y) → U2_GGA(X, p_in_ga(Y)) at position [1] we obtained the following new rules [LPAR04]:

MUL_IN_GGA(y0, 0) → U2_GGA(y0, p_out_ga(0))

(34) Obligation:

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

U2_GGA(X, p_out_ga(P)) → MUL_IN_GGA(X, P)
MUL_IN_GGA(y0, 0) → U2_GGA(y0, p_out_ga(0))

The TRS R consists of the following rules:

p_in_ga(0) → p_out_ga(0)

The set Q consists of the following terms:

p_in_ga(x0)

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

(35) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(36) Obligation:

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

U2_GGA(X, p_out_ga(P)) → MUL_IN_GGA(X, P)
MUL_IN_GGA(y0, 0) → U2_GGA(y0, p_out_ga(0))

R is empty.
The set Q consists of the following terms:

p_in_ga(x0)

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

(37) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

p_in_ga(x0)

(38) Obligation:

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

U2_GGA(X, p_out_ga(P)) → MUL_IN_GGA(X, P)
MUL_IN_GGA(y0, 0) → U2_GGA(y0, p_out_ga(0))

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

(39) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U2_GGA(X, p_out_ga(P)) → MUL_IN_GGA(X, P) we obtained the following new rules [LPAR04]:

U2_GGA(z0, p_out_ga(0)) → MUL_IN_GGA(z0, 0)

(40) Obligation:

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

MUL_IN_GGA(y0, 0) → U2_GGA(y0, p_out_ga(0))
U2_GGA(z0, p_out_ga(0)) → MUL_IN_GGA(z0, 0)

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

(41) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = U2_GGA(z0, p_out_ga(0)) evaluates to t =U2_GGA(z0, p_out_ga(0))

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Matcher: [ ]
Semiunifier: [ ]

Rewriting sequence

U2_GGA(z0, p_out_ga(0)) → MUL_IN_GGA(z0, 0)
with rule U2_GGA(z0', p_out_ga(0)) → MUL_IN_GGA(z0', 0) at position [] and matcher [z0' / z0]

MUL_IN_GGA(z0, 0) → U2_GGA(z0, p_out_ga(0))
with rule MUL_IN_GGA(y0, 0) → U2_GGA(y0, p_out_ga(0))

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence

All these steps are and every following step will be a correct step w.r.t to Q.

(42) FALSE

(43) PrologToPiTRSProof (SOUND transformation)

mul_in_gga(X1, 0, Z) → U1_gga(X1, Z, eq_in_ag(Z, 0))
eq_in_ag(X, X) → eq_out_ag(X, X)
U1_gga(X1, Z, eq_out_ag(Z, 0)) → mul_out_gga(X1, 0, Z)
mul_in_gga(X, Y, Z) → U2_gga(X, Y, Z, p_in_ga(Y, P))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U2_gga(X, Y, Z, p_out_ga(Y, P)) → U3_gga(X, Y, Z, P, mul_in_gga(X, P, V))
U3_gga(X, Y, Z, P, mul_out_gga(X, P, V)) → U4_gga(X, Y, Z, add_in_gga(X, V, Z))
add_in_gga(X, 0, Z) → U5_gga(X, Z, eq_in_ag(Z, X))
U5_gga(X, Z, eq_out_ag(Z, X)) → add_out_gga(X, 0, Z)
add_in_gga(X, Y, Z) → U6_gga(X, Y, Z, p_in_ga(Y, V))
U6_gga(X, Y, Z, p_out_ga(Y, V)) → U7_gga(X, Y, Z, V, add_in_gga(X, V, W))
U7_gga(X, Y, Z, V, add_out_gga(X, V, W)) → U8_gga(X, Y, Z, p_in_ag(Z, W))
p_in_ag(0, 0) → p_out_ag(0, 0)
p_in_ag(s(X), X) → p_out_ag(s(X), X)
U8_gga(X, Y, Z, p_out_ag(Z, W)) → add_out_gga(X, Y, Z)
U4_gga(X, Y, Z, add_out_gga(X, V, Z)) → mul_out_gga(X, Y, Z)

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog

(44) Obligation:

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

mul_in_gga(X1, 0, Z) → U1_gga(X1, Z, eq_in_ag(Z, 0))
eq_in_ag(X, X) → eq_out_ag(X, X)
U1_gga(X1, Z, eq_out_ag(Z, 0)) → mul_out_gga(X1, 0, Z)
mul_in_gga(X, Y, Z) → U2_gga(X, Y, Z, p_in_ga(Y, P))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U2_gga(X, Y, Z, p_out_ga(Y, P)) → U3_gga(X, Y, Z, P, mul_in_gga(X, P, V))
U3_gga(X, Y, Z, P, mul_out_gga(X, P, V)) → U4_gga(X, Y, Z, add_in_gga(X, V, Z))
add_in_gga(X, 0, Z) → U5_gga(X, Z, eq_in_ag(Z, X))
U5_gga(X, Z, eq_out_ag(Z, X)) → add_out_gga(X, 0, Z)
add_in_gga(X, Y, Z) → U6_gga(X, Y, Z, p_in_ga(Y, V))
U6_gga(X, Y, Z, p_out_ga(Y, V)) → U7_gga(X, Y, Z, V, add_in_gga(X, V, W))
U7_gga(X, Y, Z, V, add_out_gga(X, V, W)) → U8_gga(X, Y, Z, p_in_ag(Z, W))
p_in_ag(0, 0) → p_out_ag(0, 0)
p_in_ag(s(X), X) → p_out_ag(s(X), X)
U8_gga(X, Y, Z, p_out_ag(Z, W)) → add_out_gga(X, Y, Z)
U4_gga(X, Y, Z, add_out_gga(X, V, Z)) → mul_out_gga(X, Y, Z)

(45) 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:

MUL_IN_GGA(X1, 0, Z) → U1_GGA(X1, Z, eq_in_ag(Z, 0))
MUL_IN_GGA(X1, 0, Z) → EQ_IN_AG(Z, 0)
MUL_IN_GGA(X, Y, Z) → U2_GGA(X, Y, Z, p_in_ga(Y, P))
MUL_IN_GGA(X, Y, Z) → P_IN_GA(Y, P)
U2_GGA(X, Y, Z, p_out_ga(Y, P)) → U3_GGA(X, Y, Z, P, mul_in_gga(X, P, V))
U2_GGA(X, Y, Z, p_out_ga(Y, P)) → MUL_IN_GGA(X, P, V)
U3_GGA(X, Y, Z, P, mul_out_gga(X, P, V)) → U4_GGA(X, Y, Z, add_in_gga(X, V, Z))
U3_GGA(X, Y, Z, P, mul_out_gga(X, P, V)) → ADD_IN_GGA(X, V, Z)
ADD_IN_GGA(X, 0, Z) → U5_GGA(X, Z, eq_in_ag(Z, X))
ADD_IN_GGA(X, 0, Z) → EQ_IN_AG(Z, X)
ADD_IN_GGA(X, Y, Z) → U6_GGA(X, Y, Z, p_in_ga(Y, V))
ADD_IN_GGA(X, Y, Z) → P_IN_GA(Y, V)
U6_GGA(X, Y, Z, p_out_ga(Y, V)) → U7_GGA(X, Y, Z, V, add_in_gga(X, V, W))
U6_GGA(X, Y, Z, p_out_ga(Y, V)) → ADD_IN_GGA(X, V, W)
U7_GGA(X, Y, Z, V, add_out_gga(X, V, W)) → U8_GGA(X, Y, Z, p_in_ag(Z, W))
U7_GGA(X, Y, Z, V, add_out_gga(X, V, W)) → P_IN_AG(Z, W)

The TRS R consists of the following rules:

mul_in_gga(X1, 0, Z) → U1_gga(X1, Z, eq_in_ag(Z, 0))
eq_in_ag(X, X) → eq_out_ag(X, X)
U1_gga(X1, Z, eq_out_ag(Z, 0)) → mul_out_gga(X1, 0, Z)
mul_in_gga(X, Y, Z) → U2_gga(X, Y, Z, p_in_ga(Y, P))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U2_gga(X, Y, Z, p_out_ga(Y, P)) → U3_gga(X, Y, Z, P, mul_in_gga(X, P, V))
U3_gga(X, Y, Z, P, mul_out_gga(X, P, V)) → U4_gga(X, Y, Z, add_in_gga(X, V, Z))
add_in_gga(X, 0, Z) → U5_gga(X, Z, eq_in_ag(Z, X))
U5_gga(X, Z, eq_out_ag(Z, X)) → add_out_gga(X, 0, Z)
add_in_gga(X, Y, Z) → U6_gga(X, Y, Z, p_in_ga(Y, V))
U6_gga(X, Y, Z, p_out_ga(Y, V)) → U7_gga(X, Y, Z, V, add_in_gga(X, V, W))
U7_gga(X, Y, Z, V, add_out_gga(X, V, W)) → U8_gga(X, Y, Z, p_in_ag(Z, W))
p_in_ag(0, 0) → p_out_ag(0, 0)
p_in_ag(s(X), X) → p_out_ag(s(X), X)
U8_gga(X, Y, Z, p_out_ag(Z, W)) → add_out_gga(X, Y, Z)
U4_gga(X, Y, Z, add_out_gga(X, V, Z)) → mul_out_gga(X, Y, Z)

The argument filtering Pi contains the following mapping:
mul_in_gga(x1, x2, x3) = mul_in_gga(x1, x2)
0 = 0
U1_gga(x1, x2, x3) = U1_gga(x1, x3)
eq_in_ag(x1, x2) = eq_in_ag(x2)
eq_out_ag(x1, x2) = eq_out_ag(x1, x2)
mul_out_gga(x1, x2, x3) = mul_out_gga(x1, x2, x3)
U2_gga(x1, x2, x3, x4) = U2_gga(x1, x2, x4)
p_in_ga(x1, x2) = p_in_ga(x1)
p_out_ga(x1, x2) = p_out_ga(x1, x2)
s(x1) = s(x1)
U3_gga(x1, x2, x3, x4, x5) = U3_gga(x1, x2, x5)
U4_gga(x1, x2, x3, x4) = U4_gga(x1, x2, x4)
add_in_gga(x1, x2, x3) = add_in_gga(x1, x2)
U5_gga(x1, x2, x3) = U5_gga(x1, x3)
add_out_gga(x1, x2, x3) = add_out_gga(x1, x2, x3)
U6_gga(x1, x2, x3, x4) = U6_gga(x1, x2, x4)
U7_gga(x1, x2, x3, x4, x5) = U7_gga(x1, x2, x5)
U8_gga(x1, x2, x3, x4) = U8_gga(x1, x2, x4)
p_in_ag(x1, x2) = p_in_ag(x2)
p_out_ag(x1, x2) = p_out_ag(x1, x2)
MUL_IN_GGA(x1, x2, x3) = MUL_IN_GGA(x1, x2)
U1_GGA(x1, x2, x3) = U1_GGA(x1, x3)
EQ_IN_AG(x1, x2) = EQ_IN_AG(x2)
U2_GGA(x1, x2, x3, x4) = U2_GGA(x1, x2, x4)
P_IN_GA(x1, x2) = P_IN_GA(x1)
U3_GGA(x1, x2, x3, x4, x5) = U3_GGA(x1, x2, x5)
U4_GGA(x1, x2, x3, x4) = U4_GGA(x1, x2, x4)
ADD_IN_GGA(x1, x2, x3) = ADD_IN_GGA(x1, x2)
U5_GGA(x1, x2, x3) = U5_GGA(x1, x3)
U6_GGA(x1, x2, x3, x4) = U6_GGA(x1, x2, x4)
U7_GGA(x1, x2, x3, x4, x5) = U7_GGA(x1, x2, x5)
U8_GGA(x1, x2, x3, x4) = U8_GGA(x1, x2, x4)
P_IN_AG(x1, x2) = P_IN_AG(x2)

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

(46) Obligation:

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

MUL_IN_GGA(X1, 0, Z) → U1_GGA(X1, Z, eq_in_ag(Z, 0))
MUL_IN_GGA(X1, 0, Z) → EQ_IN_AG(Z, 0)
MUL_IN_GGA(X, Y, Z) → U2_GGA(X, Y, Z, p_in_ga(Y, P))
MUL_IN_GGA(X, Y, Z) → P_IN_GA(Y, P)
U2_GGA(X, Y, Z, p_out_ga(Y, P)) → U3_GGA(X, Y, Z, P, mul_in_gga(X, P, V))
U2_GGA(X, Y, Z, p_out_ga(Y, P)) → MUL_IN_GGA(X, P, V)
U3_GGA(X, Y, Z, P, mul_out_gga(X, P, V)) → U4_GGA(X, Y, Z, add_in_gga(X, V, Z))
U3_GGA(X, Y, Z, P, mul_out_gga(X, P, V)) → ADD_IN_GGA(X, V, Z)
ADD_IN_GGA(X, 0, Z) → U5_GGA(X, Z, eq_in_ag(Z, X))
ADD_IN_GGA(X, 0, Z) → EQ_IN_AG(Z, X)
ADD_IN_GGA(X, Y, Z) → U6_GGA(X, Y, Z, p_in_ga(Y, V))
ADD_IN_GGA(X, Y, Z) → P_IN_GA(Y, V)
U6_GGA(X, Y, Z, p_out_ga(Y, V)) → U7_GGA(X, Y, Z, V, add_in_gga(X, V, W))
U6_GGA(X, Y, Z, p_out_ga(Y, V)) → ADD_IN_GGA(X, V, W)
U7_GGA(X, Y, Z, V, add_out_gga(X, V, W)) → U8_GGA(X, Y, Z, p_in_ag(Z, W))
U7_GGA(X, Y, Z, V, add_out_gga(X, V, W)) → P_IN_AG(Z, W)

The TRS R consists of the following rules:

mul_in_gga(X1, 0, Z) → U1_gga(X1, Z, eq_in_ag(Z, 0))
eq_in_ag(X, X) → eq_out_ag(X, X)
U1_gga(X1, Z, eq_out_ag(Z, 0)) → mul_out_gga(X1, 0, Z)
mul_in_gga(X, Y, Z) → U2_gga(X, Y, Z, p_in_ga(Y, P))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U2_gga(X, Y, Z, p_out_ga(Y, P)) → U3_gga(X, Y, Z, P, mul_in_gga(X, P, V))
U3_gga(X, Y, Z, P, mul_out_gga(X, P, V)) → U4_gga(X, Y, Z, add_in_gga(X, V, Z))
add_in_gga(X, 0, Z) → U5_gga(X, Z, eq_in_ag(Z, X))
U5_gga(X, Z, eq_out_ag(Z, X)) → add_out_gga(X, 0, Z)
add_in_gga(X, Y, Z) → U6_gga(X, Y, Z, p_in_ga(Y, V))
U6_gga(X, Y, Z, p_out_ga(Y, V)) → U7_gga(X, Y, Z, V, add_in_gga(X, V, W))
U7_gga(X, Y, Z, V, add_out_gga(X, V, W)) → U8_gga(X, Y, Z, p_in_ag(Z, W))
p_in_ag(0, 0) → p_out_ag(0, 0)
p_in_ag(s(X), X) → p_out_ag(s(X), X)
U8_gga(X, Y, Z, p_out_ag(Z, W)) → add_out_gga(X, Y, Z)
U4_gga(X, Y, Z, add_out_gga(X, V, Z)) → mul_out_gga(X, Y, Z)

(47) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LOPSTR] contains 2 SCCs with 12 less nodes.

(48) Complex Obligation (AND)

(49) Obligation:

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

ADD_IN_GGA(X, Y, Z) → U6_GGA(X, Y, Z, p_in_ga(Y, V))
U6_GGA(X, Y, Z, p_out_ga(Y, V)) → ADD_IN_GGA(X, V, W)

The TRS R consists of the following rules:

mul_in_gga(X1, 0, Z) → U1_gga(X1, Z, eq_in_ag(Z, 0))
eq_in_ag(X, X) → eq_out_ag(X, X)
U1_gga(X1, Z, eq_out_ag(Z, 0)) → mul_out_gga(X1, 0, Z)
mul_in_gga(X, Y, Z) → U2_gga(X, Y, Z, p_in_ga(Y, P))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U2_gga(X, Y, Z, p_out_ga(Y, P)) → U3_gga(X, Y, Z, P, mul_in_gga(X, P, V))
U3_gga(X, Y, Z, P, mul_out_gga(X, P, V)) → U4_gga(X, Y, Z, add_in_gga(X, V, Z))
add_in_gga(X, 0, Z) → U5_gga(X, Z, eq_in_ag(Z, X))
U5_gga(X, Z, eq_out_ag(Z, X)) → add_out_gga(X, 0, Z)
add_in_gga(X, Y, Z) → U6_gga(X, Y, Z, p_in_ga(Y, V))
U6_gga(X, Y, Z, p_out_ga(Y, V)) → U7_gga(X, Y, Z, V, add_in_gga(X, V, W))
U7_gga(X, Y, Z, V, add_out_gga(X, V, W)) → U8_gga(X, Y, Z, p_in_ag(Z, W))
p_in_ag(0, 0) → p_out_ag(0, 0)
p_in_ag(s(X), X) → p_out_ag(s(X), X)
U8_gga(X, Y, Z, p_out_ag(Z, W)) → add_out_gga(X, Y, Z)
U4_gga(X, Y, Z, add_out_gga(X, V, Z)) → mul_out_gga(X, Y, Z)

(50) UsableRulesProof (EQUIVALENT transformation)

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

(51) Obligation:

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

ADD_IN_GGA(X, Y, Z) → U6_GGA(X, Y, Z, p_in_ga(Y, V))
U6_GGA(X, Y, Z, p_out_ga(Y, V)) → ADD_IN_GGA(X, V, W)

The TRS R consists of the following rules:

p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)

The argument filtering Pi contains the following mapping:
0 = 0
p_in_ga(x1, x2) = p_in_ga(x1)
p_out_ga(x1, x2) = p_out_ga(x1, x2)
s(x1) = s(x1)
ADD_IN_GGA(x1, x2, x3) = ADD_IN_GGA(x1, x2)
U6_GGA(x1, x2, x3, x4) = U6_GGA(x1, x2, x4)

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

(52) PiDPToQDPProof (SOUND transformation)

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

(53) Obligation:

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

ADD_IN_GGA(X, Y) → U6_GGA(X, Y, p_in_ga(Y))
U6_GGA(X, Y, p_out_ga(Y, V)) → ADD_IN_GGA(X, V)

The TRS R consists of the following rules:

p_in_ga(0) → p_out_ga(0, 0)
p_in_ga(s(X)) → p_out_ga(s(X), X)

The set Q consists of the following terms:

p_in_ga(x0)

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

(54) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule ADD_IN_GGA(X, Y) → U6_GGA(X, Y, p_in_ga(Y)) at position [2] we obtained the following new rules [LPAR04]:

ADD_IN_GGA(y0, 0) → U6_GGA(y0, 0, p_out_ga(0, 0))
ADD_IN_GGA(y0, s(x0)) → U6_GGA(y0, s(x0), p_out_ga(s(x0), x0))

(55) Obligation:

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

U6_GGA(X, Y, p_out_ga(Y, V)) → ADD_IN_GGA(X, V)
ADD_IN_GGA(y0, 0) → U6_GGA(y0, 0, p_out_ga(0, 0))
ADD_IN_GGA(y0, s(x0)) → U6_GGA(y0, s(x0), p_out_ga(s(x0), x0))

The TRS R consists of the following rules:

p_in_ga(0) → p_out_ga(0, 0)
p_in_ga(s(X)) → p_out_ga(s(X), X)

The set Q consists of the following terms:

p_in_ga(x0)

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

(56) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(57) Obligation:

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

U6_GGA(X, Y, p_out_ga(Y, V)) → ADD_IN_GGA(X, V)
ADD_IN_GGA(y0, 0) → U6_GGA(y0, 0, p_out_ga(0, 0))
ADD_IN_GGA(y0, s(x0)) → U6_GGA(y0, s(x0), p_out_ga(s(x0), x0))

R is empty.
The set Q consists of the following terms:

p_in_ga(x0)

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

(58) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

p_in_ga(x0)

(59) Obligation:

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

U6_GGA(X, Y, p_out_ga(Y, V)) → ADD_IN_GGA(X, V)
ADD_IN_GGA(y0, 0) → U6_GGA(y0, 0, p_out_ga(0, 0))
ADD_IN_GGA(y0, s(x0)) → U6_GGA(y0, s(x0), p_out_ga(s(x0), x0))

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

(60) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U6_GGA(X, Y, p_out_ga(Y, V)) → ADD_IN_GGA(X, V) we obtained the following new rules [LPAR04]:

U6_GGA(z0, 0, p_out_ga(0, 0)) → ADD_IN_GGA(z0, 0)
U6_GGA(z0, s(z1), p_out_ga(s(z1), z1)) → ADD_IN_GGA(z0, z1)

(61) Obligation:

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

ADD_IN_GGA(y0, 0) → U6_GGA(y0, 0, p_out_ga(0, 0))
ADD_IN_GGA(y0, s(x0)) → U6_GGA(y0, s(x0), p_out_ga(s(x0), x0))
U6_GGA(z0, 0, p_out_ga(0, 0)) → ADD_IN_GGA(z0, 0)
U6_GGA(z0, s(z1), p_out_ga(s(z1), z1)) → ADD_IN_GGA(z0, z1)

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

(62) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs.

(63) Complex Obligation (AND)

(64) Obligation:

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

U6_GGA(z0, 0, p_out_ga(0, 0)) → ADD_IN_GGA(z0, 0)
ADD_IN_GGA(y0, 0) → U6_GGA(y0, 0, p_out_ga(0, 0))

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

(65) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = ADD_IN_GGA(y0, 0) evaluates to t =ADD_IN_GGA(y0, 0)

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Matcher: [ ]
Semiunifier: [ ]

Rewriting sequence

ADD_IN_GGA(y0, 0) → U6_GGA(y0, 0, p_out_ga(0, 0))
with rule ADD_IN_GGA(y0', 0) → U6_GGA(y0', 0, p_out_ga(0, 0)) at position [] and matcher [y0' / y0]

U6_GGA(y0, 0, p_out_ga(0, 0)) → ADD_IN_GGA(y0, 0)
with rule U6_GGA(z0, 0, p_out_ga(0, 0)) → ADD_IN_GGA(z0, 0)

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence

All these steps are and every following step will be a correct step w.r.t to Q.

(66) FALSE

(67) Obligation:

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

ADD_IN_GGA(y0, s(x0)) → U6_GGA(y0, s(x0), p_out_ga(s(x0), x0))
U6_GGA(z0, s(z1), p_out_ga(s(z1), z1)) → ADD_IN_GGA(z0, z1)

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

(68) 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:

U6_GGA(z0, s(z1), p_out_ga(s(z1), z1)) → ADD_IN_GGA(z0, z1)
The graph contains the following edges 1 >= 1, 2 > 2, 3 > 2
ADD_IN_GGA(y0, s(x0)) → U6_GGA(y0, s(x0), p_out_ga(s(x0), x0))
The graph contains the following edges 1 >= 1, 2 >= 2

(69) TRUE

(70) Obligation:

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

MUL_IN_GGA(X, Y, Z) → U2_GGA(X, Y, Z, p_in_ga(Y, P))
U2_GGA(X, Y, Z, p_out_ga(Y, P)) → MUL_IN_GGA(X, P, V)

The TRS R consists of the following rules:

mul_in_gga(X1, 0, Z) → U1_gga(X1, Z, eq_in_ag(Z, 0))
eq_in_ag(X, X) → eq_out_ag(X, X)
U1_gga(X1, Z, eq_out_ag(Z, 0)) → mul_out_gga(X1, 0, Z)
mul_in_gga(X, Y, Z) → U2_gga(X, Y, Z, p_in_ga(Y, P))
p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)
U2_gga(X, Y, Z, p_out_ga(Y, P)) → U3_gga(X, Y, Z, P, mul_in_gga(X, P, V))
U3_gga(X, Y, Z, P, mul_out_gga(X, P, V)) → U4_gga(X, Y, Z, add_in_gga(X, V, Z))
add_in_gga(X, 0, Z) → U5_gga(X, Z, eq_in_ag(Z, X))
U5_gga(X, Z, eq_out_ag(Z, X)) → add_out_gga(X, 0, Z)
add_in_gga(X, Y, Z) → U6_gga(X, Y, Z, p_in_ga(Y, V))
U6_gga(X, Y, Z, p_out_ga(Y, V)) → U7_gga(X, Y, Z, V, add_in_gga(X, V, W))
U7_gga(X, Y, Z, V, add_out_gga(X, V, W)) → U8_gga(X, Y, Z, p_in_ag(Z, W))
p_in_ag(0, 0) → p_out_ag(0, 0)
p_in_ag(s(X), X) → p_out_ag(s(X), X)
U8_gga(X, Y, Z, p_out_ag(Z, W)) → add_out_gga(X, Y, Z)
U4_gga(X, Y, Z, add_out_gga(X, V, Z)) → mul_out_gga(X, Y, Z)

(71) UsableRulesProof (EQUIVALENT transformation)

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

(72) Obligation:

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

MUL_IN_GGA(X, Y, Z) → U2_GGA(X, Y, Z, p_in_ga(Y, P))
U2_GGA(X, Y, Z, p_out_ga(Y, P)) → MUL_IN_GGA(X, P, V)

The TRS R consists of the following rules:

p_in_ga(0, 0) → p_out_ga(0, 0)
p_in_ga(s(X), X) → p_out_ga(s(X), X)

The argument filtering Pi contains the following mapping:
0 = 0
p_in_ga(x1, x2) = p_in_ga(x1)
p_out_ga(x1, x2) = p_out_ga(x1, x2)
s(x1) = s(x1)
MUL_IN_GGA(x1, x2, x3) = MUL_IN_GGA(x1, x2)
U2_GGA(x1, x2, x3, x4) = U2_GGA(x1, x2, x4)

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

(73) PiDPToQDPProof (SOUND transformation)

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

(74) Obligation:

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

MUL_IN_GGA(X, Y) → U2_GGA(X, Y, p_in_ga(Y))
U2_GGA(X, Y, p_out_ga(Y, P)) → MUL_IN_GGA(X, P)

The TRS R consists of the following rules:

p_in_ga(0) → p_out_ga(0, 0)
p_in_ga(s(X)) → p_out_ga(s(X), X)

The set Q consists of the following terms:

p_in_ga(x0)

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

(75) Narrowing (SOUND transformation)

By narrowing [LPAR04] the rule MUL_IN_GGA(X, Y) → U2_GGA(X, Y, p_in_ga(Y)) at position [2] we obtained the following new rules [LPAR04]:

MUL_IN_GGA(y0, 0) → U2_GGA(y0, 0, p_out_ga(0, 0))
MUL_IN_GGA(y0, s(x0)) → U2_GGA(y0, s(x0), p_out_ga(s(x0), x0))

(76) Obligation:

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

U2_GGA(X, Y, p_out_ga(Y, P)) → MUL_IN_GGA(X, P)
MUL_IN_GGA(y0, 0) → U2_GGA(y0, 0, p_out_ga(0, 0))
MUL_IN_GGA(y0, s(x0)) → U2_GGA(y0, s(x0), p_out_ga(s(x0), x0))

The TRS R consists of the following rules:

p_in_ga(0) → p_out_ga(0, 0)
p_in_ga(s(X)) → p_out_ga(s(X), X)

The set Q consists of the following terms:

p_in_ga(x0)

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

(77) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(78) Obligation:

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

U2_GGA(X, Y, p_out_ga(Y, P)) → MUL_IN_GGA(X, P)
MUL_IN_GGA(y0, 0) → U2_GGA(y0, 0, p_out_ga(0, 0))
MUL_IN_GGA(y0, s(x0)) → U2_GGA(y0, s(x0), p_out_ga(s(x0), x0))

R is empty.
The set Q consists of the following terms:

p_in_ga(x0)

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

(79) QReductionProof (EQUIVALENT transformation)

We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

p_in_ga(x0)

(80) Obligation:

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

U2_GGA(X, Y, p_out_ga(Y, P)) → MUL_IN_GGA(X, P)
MUL_IN_GGA(y0, 0) → U2_GGA(y0, 0, p_out_ga(0, 0))
MUL_IN_GGA(y0, s(x0)) → U2_GGA(y0, s(x0), p_out_ga(s(x0), x0))

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

(81) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule U2_GGA(X, Y, p_out_ga(Y, P)) → MUL_IN_GGA(X, P) we obtained the following new rules [LPAR04]:

U2_GGA(z0, 0, p_out_ga(0, 0)) → MUL_IN_GGA(z0, 0)
U2_GGA(z0, s(z1), p_out_ga(s(z1), z1)) → MUL_IN_GGA(z0, z1)

(82) Obligation:

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

MUL_IN_GGA(y0, 0) → U2_GGA(y0, 0, p_out_ga(0, 0))
MUL_IN_GGA(y0, s(x0)) → U2_GGA(y0, s(x0), p_out_ga(s(x0), x0))
U2_GGA(z0, 0, p_out_ga(0, 0)) → MUL_IN_GGA(z0, 0)
U2_GGA(z0, s(z1), p_out_ga(s(z1), z1)) → MUL_IN_GGA(z0, z1)

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

(83) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs.

(84) Complex Obligation (AND)

(85) Obligation:

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

U2_GGA(z0, 0, p_out_ga(0, 0)) → MUL_IN_GGA(z0, 0)
MUL_IN_GGA(y0, 0) → U2_GGA(y0, 0, p_out_ga(0, 0))

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

(86) NonTerminationProof (EQUIVALENT transformation)

We used the non-termination processor [FROCOS05] to show that the DP problem is infinite.
Found a loop by narrowing to the left:

s = MUL_IN_GGA(y0, 0) evaluates to t =MUL_IN_GGA(y0, 0)

Thus s starts an infinite chain as s semiunifies with t with the following substitutions:

Semiunifier: [ ]
Matcher: [ ]

Rewriting sequence

MUL_IN_GGA(y0, 0) → U2_GGA(y0, 0, p_out_ga(0, 0))
with rule MUL_IN_GGA(y0', 0) → U2_GGA(y0', 0, p_out_ga(0, 0)) at position [] and matcher [y0' / y0]

U2_GGA(y0, 0, p_out_ga(0, 0)) → MUL_IN_GGA(y0, 0)
with rule U2_GGA(z0, 0, p_out_ga(0, 0)) → MUL_IN_GGA(z0, 0)

Now applying the matcher to the start term leads to a term which is equal to the last term in the rewriting sequence

All these steps are and every following step will be a correct step w.r.t to Q.

(87) FALSE

(88) Obligation:

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

MUL_IN_GGA(y0, s(x0)) → U2_GGA(y0, s(x0), p_out_ga(s(x0), x0))
U2_GGA(z0, s(z1), p_out_ga(s(z1), z1)) → MUL_IN_GGA(z0, z1)

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

(89) 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:

U2_GGA(z0, s(z1), p_out_ga(s(z1), z1)) → MUL_IN_GGA(z0, z1)
The graph contains the following edges 1 >= 1, 2 > 2, 3 > 2
MUL_IN_GGA(y0, s(x0)) → U2_GGA(y0, s(x0), p_out_ga(s(x0), x0))
The graph contains the following edges 1 >= 1, 2 >= 2