Left Termination proof of /tmp/tmp8sofSW/sum-ioi.pl

Left Termination of the query pattern sum(b,f,b) w.r.t. the given Prolog program could successfully be proven:

↳ PROLOG

  ↳ PrologToPiTRSProof

sum₃({}₀, {}₀, {}₀).
sum₃(.₂(X1, Y1), .₂(X2, Y2), .₂(X3, Y3)) :- add₃(X1, X2, X3), sum₃(Y1, Y2, Y3).
add₃(0₀, X, X).
add₃(s₁(X), Y, s₁(Z)) :- add₃(X, Y, Z).

With regard to the inferred argument filtering the predicates were used in the following modes:
sum₃: (b,f,b)
add₃: (b,f,b)
Transforming PROLOG into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

sum_3_in_gag₃([]_0, []_0, []_0) -> sum_3_out_gag₃([]_0, []_0, []_0)
sum_3_in_gag₃(._2₂(X1, Y1), ._2₂(X2, Y2), ._2₂(X3, Y3)) -> if_sum_3_in_1_gag₇(X1, Y1, X2, Y2, X3, Y3, add_3_in_gag₃(X1, X2, X3))
add_3_in_gag₃(0_0, X, X) -> add_3_out_gag₃(0_0, X, X)
add_3_in_gag₃(s_1₁(X), Y, s_1₁(Z)) -> if_add_3_in_1_gag₄(X, Y, Z, add_3_in_gag₃(X, Y, Z))
if_add_3_in_1_gag₄(X, Y, Z, add_3_out_gag₃(X, Y, Z)) -> add_3_out_gag₃(s_1₁(X), Y, s_1₁(Z))
if_sum_3_in_1_gag₇(X1, Y1, X2, Y2, X3, Y3, add_3_out_gag₃(X1, X2, X3)) -> if_sum_3_in_2_gag₇(X1, Y1, X2, Y2, X3, Y3, sum_3_in_gag₃(Y1, Y2, Y3))
if_sum_3_in_2_gag₇(X1, Y1, X2, Y2, X3, Y3, sum_3_out_gag₃(Y1, Y2, Y3)) -> sum_3_out_gag₃(._2₂(X1, Y1), ._2₂(X2, Y2), ._2₂(X3, Y3))

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:

sum_3_in_gag₃([]_0, []_0, []_0) -> sum_3_out_gag₃([]_0, []_0, []_0)
sum_3_in_gag₃(._2₂(X1, Y1), ._2₂(X2, Y2), ._2₂(X3, Y3)) -> if_sum_3_in_1_gag₇(X1, Y1, X2, Y2, X3, Y3, add_3_in_gag₃(X1, X2, X3))
add_3_in_gag₃(0_0, X, X) -> add_3_out_gag₃(0_0, X, X)
add_3_in_gag₃(s_1₁(X), Y, s_1₁(Z)) -> if_add_3_in_1_gag₄(X, Y, Z, add_3_in_gag₃(X, Y, Z))
if_add_3_in_1_gag₄(X, Y, Z, add_3_out_gag₃(X, Y, Z)) -> add_3_out_gag₃(s_1₁(X), Y, s_1₁(Z))
if_sum_3_in_1_gag₇(X1, Y1, X2, Y2, X3, Y3, add_3_out_gag₃(X1, X2, X3)) -> if_sum_3_in_2_gag₇(X1, Y1, X2, Y2, X3, Y3, sum_3_in_gag₃(Y1, Y2, Y3))
if_sum_3_in_2_gag₇(X1, Y1, X2, Y2, X3, Y3, sum_3_out_gag₃(Y1, Y2, Y3)) -> sum_3_out_gag₃(._2₂(X1, Y1), ._2₂(X2, Y2), ._2₂(X3, Y3))

SUM_3_IN_GAG₃(._2₂(X1, Y1), ._2₂(X2, Y2), ._2₂(X3, Y3)) -> IF_SUM_3_IN_1_GAG₇(X1, Y1, X2, Y2, X3, Y3, add_3_in_gag₃(X1, X2, X3))
SUM_3_IN_GAG₃(._2₂(X1, Y1), ._2₂(X2, Y2), ._2₂(X3, Y3)) -> ADD_3_IN_GAG₃(X1, X2, X3)
ADD_3_IN_GAG₃(s_1₁(X), Y, s_1₁(Z)) -> IF_ADD_3_IN_1_GAG₄(X, Y, Z, add_3_in_gag₃(X, Y, Z))
ADD_3_IN_GAG₃(s_1₁(X), Y, s_1₁(Z)) -> ADD_3_IN_GAG₃(X, Y, Z)
IF_SUM_3_IN_1_GAG₇(X1, Y1, X2, Y2, X3, Y3, add_3_out_gag₃(X1, X2, X3)) -> IF_SUM_3_IN_2_GAG₇(X1, Y1, X2, Y2, X3, Y3, sum_3_in_gag₃(Y1, Y2, Y3))
IF_SUM_3_IN_1_GAG₇(X1, Y1, X2, Y2, X3, Y3, add_3_out_gag₃(X1, X2, X3)) -> SUM_3_IN_GAG₃(Y1, Y2, Y3)

The TRS R consists of the following rules:

sum_3_in_gag₃([]_0, []_0, []_0) -> sum_3_out_gag₃([]_0, []_0, []_0)
sum_3_in_gag₃(._2₂(X1, Y1), ._2₂(X2, Y2), ._2₂(X3, Y3)) -> if_sum_3_in_1_gag₇(X1, Y1, X2, Y2, X3, Y3, add_3_in_gag₃(X1, X2, X3))
add_3_in_gag₃(0_0, X, X) -> add_3_out_gag₃(0_0, X, X)
add_3_in_gag₃(s_1₁(X), Y, s_1₁(Z)) -> if_add_3_in_1_gag₄(X, Y, Z, add_3_in_gag₃(X, Y, Z))
if_add_3_in_1_gag₄(X, Y, Z, add_3_out_gag₃(X, Y, Z)) -> add_3_out_gag₃(s_1₁(X), Y, s_1₁(Z))
if_sum_3_in_1_gag₇(X1, Y1, X2, Y2, X3, Y3, add_3_out_gag₃(X1, X2, X3)) -> if_sum_3_in_2_gag₇(X1, Y1, X2, Y2, X3, Y3, sum_3_in_gag₃(Y1, Y2, Y3))
if_sum_3_in_2_gag₇(X1, Y1, X2, Y2, X3, Y3, sum_3_out_gag₃(Y1, Y2, Y3)) -> sum_3_out_gag₃(._2₂(X1, Y1), ._2₂(X2, Y2), ._2₂(X3, Y3))

The argument filtering Pi contains the following mapping:
sum_3_in_gag₃(x1, x2, x3) = sum_3_in_gag₂(x1, x3)
[]_0 = []_0
._2₂(x1, x2) = ._2₂(x1, x2)
0_0 = 0_0
s_1₁(x1) = s_1₁(x1)
sum_3_out_gag₃(x1, x2, x3) = sum_3_out_gag₁(x2)
if_sum_3_in_1_gag₇(x1, x2, x3, x4, x5, x6, x7) = if_sum_3_in_1_gag₃(x2, x6, x7)
add_3_in_gag₃(x1, x2, x3) = add_3_in_gag₂(x1, x3)
add_3_out_gag₃(x1, x2, x3) = add_3_out_gag₁(x2)
if_add_3_in_1_gag₄(x1, x2, x3, x4) = if_add_3_in_1_gag₁(x4)
if_sum_3_in_2_gag₇(x1, x2, x3, x4, x5, x6, x7) = if_sum_3_in_2_gag₂(x3, x7)
IF_SUM_3_IN_2_GAG₇(x1, x2, x3, x4, x5, x6, x7) = IF_SUM_3_IN_2_GAG₂(x3, x7)
IF_ADD_3_IN_1_GAG₄(x1, x2, x3, x4) = IF_ADD_3_IN_1_GAG₁(x4)
SUM_3_IN_GAG₃(x1, x2, x3) = SUM_3_IN_GAG₂(x1, x3)
IF_SUM_3_IN_1_GAG₇(x1, x2, x3, x4, x5, x6, x7) = IF_SUM_3_IN_1_GAG₃(x2, x6, x7)
ADD_3_IN_GAG₃(x1, x2, x3) = ADD_3_IN_GAG₂(x1, 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:

SUM_3_IN_GAG₃(._2₂(X1, Y1), ._2₂(X2, Y2), ._2₂(X3, Y3)) -> IF_SUM_3_IN_1_GAG₇(X1, Y1, X2, Y2, X3, Y3, add_3_in_gag₃(X1, X2, X3))
SUM_3_IN_GAG₃(._2₂(X1, Y1), ._2₂(X2, Y2), ._2₂(X3, Y3)) -> ADD_3_IN_GAG₃(X1, X2, X3)
ADD_3_IN_GAG₃(s_1₁(X), Y, s_1₁(Z)) -> IF_ADD_3_IN_1_GAG₄(X, Y, Z, add_3_in_gag₃(X, Y, Z))
ADD_3_IN_GAG₃(s_1₁(X), Y, s_1₁(Z)) -> ADD_3_IN_GAG₃(X, Y, Z)
IF_SUM_3_IN_1_GAG₇(X1, Y1, X2, Y2, X3, Y3, add_3_out_gag₃(X1, X2, X3)) -> IF_SUM_3_IN_2_GAG₇(X1, Y1, X2, Y2, X3, Y3, sum_3_in_gag₃(Y1, Y2, Y3))
IF_SUM_3_IN_1_GAG₇(X1, Y1, X2, Y2, X3, Y3, add_3_out_gag₃(X1, X2, X3)) -> SUM_3_IN_GAG₃(Y1, Y2, Y3)

The TRS R consists of the following rules:

sum_3_in_gag₃([]_0, []_0, []_0) -> sum_3_out_gag₃([]_0, []_0, []_0)
sum_3_in_gag₃(._2₂(X1, Y1), ._2₂(X2, Y2), ._2₂(X3, Y3)) -> if_sum_3_in_1_gag₇(X1, Y1, X2, Y2, X3, Y3, add_3_in_gag₃(X1, X2, X3))
add_3_in_gag₃(0_0, X, X) -> add_3_out_gag₃(0_0, X, X)
add_3_in_gag₃(s_1₁(X), Y, s_1₁(Z)) -> if_add_3_in_1_gag₄(X, Y, Z, add_3_in_gag₃(X, Y, Z))
if_add_3_in_1_gag₄(X, Y, Z, add_3_out_gag₃(X, Y, Z)) -> add_3_out_gag₃(s_1₁(X), Y, s_1₁(Z))
if_sum_3_in_1_gag₇(X1, Y1, X2, Y2, X3, Y3, add_3_out_gag₃(X1, X2, X3)) -> if_sum_3_in_2_gag₇(X1, Y1, X2, Y2, X3, Y3, sum_3_in_gag₃(Y1, Y2, Y3))
if_sum_3_in_2_gag₇(X1, Y1, X2, Y2, X3, Y3, sum_3_out_gag₃(Y1, Y2, Y3)) -> sum_3_out_gag₃(._2₂(X1, Y1), ._2₂(X2, Y2), ._2₂(X3, Y3))

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

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

ADD_3_IN_GAG₃(s_1₁(X), Y, s_1₁(Z)) -> ADD_3_IN_GAG₃(X, Y, Z)

The TRS R consists of the following rules:

sum_3_in_gag₃([]_0, []_0, []_0) -> sum_3_out_gag₃([]_0, []_0, []_0)
sum_3_in_gag₃(._2₂(X1, Y1), ._2₂(X2, Y2), ._2₂(X3, Y3)) -> if_sum_3_in_1_gag₇(X1, Y1, X2, Y2, X3, Y3, add_3_in_gag₃(X1, X2, X3))
add_3_in_gag₃(0_0, X, X) -> add_3_out_gag₃(0_0, X, X)
add_3_in_gag₃(s_1₁(X), Y, s_1₁(Z)) -> if_add_3_in_1_gag₄(X, Y, Z, add_3_in_gag₃(X, Y, Z))
if_add_3_in_1_gag₄(X, Y, Z, add_3_out_gag₃(X, Y, Z)) -> add_3_out_gag₃(s_1₁(X), Y, s_1₁(Z))
if_sum_3_in_1_gag₇(X1, Y1, X2, Y2, X3, Y3, add_3_out_gag₃(X1, X2, X3)) -> if_sum_3_in_2_gag₇(X1, Y1, X2, Y2, X3, Y3, sum_3_in_gag₃(Y1, Y2, Y3))
if_sum_3_in_2_gag₇(X1, Y1, X2, Y2, X3, Y3, sum_3_out_gag₃(Y1, Y2, Y3)) -> sum_3_out_gag₃(._2₂(X1, Y1), ._2₂(X2, Y2), ._2₂(X3, Y3))

The argument filtering Pi contains the following mapping:
sum_3_in_gag₃(x1, x2, x3) = sum_3_in_gag₂(x1, x3)
[]_0 = []_0
._2₂(x1, x2) = ._2₂(x1, x2)
0_0 = 0_0
s_1₁(x1) = s_1₁(x1)
sum_3_out_gag₃(x1, x2, x3) = sum_3_out_gag₁(x2)
if_sum_3_in_1_gag₇(x1, x2, x3, x4, x5, x6, x7) = if_sum_3_in_1_gag₃(x2, x6, x7)
add_3_in_gag₃(x1, x2, x3) = add_3_in_gag₂(x1, x3)
add_3_out_gag₃(x1, x2, x3) = add_3_out_gag₁(x2)
if_add_3_in_1_gag₄(x1, x2, x3, x4) = if_add_3_in_1_gag₁(x4)
if_sum_3_in_2_gag₇(x1, x2, x3, x4, x5, x6, x7) = if_sum_3_in_2_gag₂(x3, x7)
ADD_3_IN_GAG₃(x1, x2, x3) = ADD_3_IN_GAG₂(x1, x3)

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

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

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

ADD_3_IN_GAG₃(s_1₁(X), Y, s_1₁(Z)) -> ADD_3_IN_GAG₃(X, Y, Z)

R is empty.
The argument filtering Pi contains the following mapping:
s_1₁(x1) = s_1₁(x1)
ADD_3_IN_GAG₃(x1, x2, x3) = ADD_3_IN_GAG₂(x1, x3)

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

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

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

ADD_3_IN_GAG₂(s_1₁(X), s_1₁(Z)) -> ADD_3_IN_GAG₂(X, Z)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {ADD_3_IN_GAG₂}.
By using the subterm criterion together with the size-change analysis 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:

ADD_3_IN_GAG₂(s_1₁(X), s_1₁(Z)) -> ADD_3_IN_GAG₂(X, Z)
The graph contains the following edges 1 > 1, 2 > 2

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

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

SUM_3_IN_GAG₃(._2₂(X1, Y1), ._2₂(X2, Y2), ._2₂(X3, Y3)) -> IF_SUM_3_IN_1_GAG₇(X1, Y1, X2, Y2, X3, Y3, add_3_in_gag₃(X1, X2, X3))
IF_SUM_3_IN_1_GAG₇(X1, Y1, X2, Y2, X3, Y3, add_3_out_gag₃(X1, X2, X3)) -> SUM_3_IN_GAG₃(Y1, Y2, Y3)

The TRS R consists of the following rules:

sum_3_in_gag₃([]_0, []_0, []_0) -> sum_3_out_gag₃([]_0, []_0, []_0)
sum_3_in_gag₃(._2₂(X1, Y1), ._2₂(X2, Y2), ._2₂(X3, Y3)) -> if_sum_3_in_1_gag₇(X1, Y1, X2, Y2, X3, Y3, add_3_in_gag₃(X1, X2, X3))
add_3_in_gag₃(0_0, X, X) -> add_3_out_gag₃(0_0, X, X)
add_3_in_gag₃(s_1₁(X), Y, s_1₁(Z)) -> if_add_3_in_1_gag₄(X, Y, Z, add_3_in_gag₃(X, Y, Z))
if_add_3_in_1_gag₄(X, Y, Z, add_3_out_gag₃(X, Y, Z)) -> add_3_out_gag₃(s_1₁(X), Y, s_1₁(Z))
if_sum_3_in_1_gag₇(X1, Y1, X2, Y2, X3, Y3, add_3_out_gag₃(X1, X2, X3)) -> if_sum_3_in_2_gag₇(X1, Y1, X2, Y2, X3, Y3, sum_3_in_gag₃(Y1, Y2, Y3))
if_sum_3_in_2_gag₇(X1, Y1, X2, Y2, X3, Y3, sum_3_out_gag₃(Y1, Y2, Y3)) -> sum_3_out_gag₃(._2₂(X1, Y1), ._2₂(X2, Y2), ._2₂(X3, Y3))

The argument filtering Pi contains the following mapping:
sum_3_in_gag₃(x1, x2, x3) = sum_3_in_gag₂(x1, x3)
[]_0 = []_0
._2₂(x1, x2) = ._2₂(x1, x2)
0_0 = 0_0
s_1₁(x1) = s_1₁(x1)
sum_3_out_gag₃(x1, x2, x3) = sum_3_out_gag₁(x2)
if_sum_3_in_1_gag₇(x1, x2, x3, x4, x5, x6, x7) = if_sum_3_in_1_gag₃(x2, x6, x7)
add_3_in_gag₃(x1, x2, x3) = add_3_in_gag₂(x1, x3)
add_3_out_gag₃(x1, x2, x3) = add_3_out_gag₁(x2)
if_add_3_in_1_gag₄(x1, x2, x3, x4) = if_add_3_in_1_gag₁(x4)
if_sum_3_in_2_gag₇(x1, x2, x3, x4, x5, x6, x7) = if_sum_3_in_2_gag₂(x3, x7)
SUM_3_IN_GAG₃(x1, x2, x3) = SUM_3_IN_GAG₂(x1, x3)
IF_SUM_3_IN_1_GAG₇(x1, x2, x3, x4, x5, x6, x7) = IF_SUM_3_IN_1_GAG₃(x2, x6, x7)

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

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

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

SUM_3_IN_GAG₃(._2₂(X1, Y1), ._2₂(X2, Y2), ._2₂(X3, Y3)) -> IF_SUM_3_IN_1_GAG₇(X1, Y1, X2, Y2, X3, Y3, add_3_in_gag₃(X1, X2, X3))
IF_SUM_3_IN_1_GAG₇(X1, Y1, X2, Y2, X3, Y3, add_3_out_gag₃(X1, X2, X3)) -> SUM_3_IN_GAG₃(Y1, Y2, Y3)

The TRS R consists of the following rules:

add_3_in_gag₃(0_0, X, X) -> add_3_out_gag₃(0_0, X, X)
add_3_in_gag₃(s_1₁(X), Y, s_1₁(Z)) -> if_add_3_in_1_gag₄(X, Y, Z, add_3_in_gag₃(X, Y, Z))
if_add_3_in_1_gag₄(X, Y, Z, add_3_out_gag₃(X, Y, Z)) -> add_3_out_gag₃(s_1₁(X), Y, s_1₁(Z))

The argument filtering Pi contains the following mapping:
._2₂(x1, x2) = ._2₂(x1, x2)
0_0 = 0_0
s_1₁(x1) = s_1₁(x1)
add_3_in_gag₃(x1, x2, x3) = add_3_in_gag₂(x1, x3)
add_3_out_gag₃(x1, x2, x3) = add_3_out_gag₁(x2)
if_add_3_in_1_gag₄(x1, x2, x3, x4) = if_add_3_in_1_gag₁(x4)
SUM_3_IN_GAG₃(x1, x2, x3) = SUM_3_IN_GAG₂(x1, x3)
IF_SUM_3_IN_1_GAG₇(x1, x2, x3, x4, x5, x6, x7) = IF_SUM_3_IN_1_GAG₃(x2, x6, x7)

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

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

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

SUM_3_IN_GAG₂(._2₂(X1, Y1), ._2₂(X3, Y3)) -> IF_SUM_3_IN_1_GAG₃(Y1, Y3, add_3_in_gag₂(X1, X3))
IF_SUM_3_IN_1_GAG₃(Y1, Y3, add_3_out_gag₁(X2)) -> SUM_3_IN_GAG₂(Y1, Y3)

The TRS R consists of the following rules:

add_3_in_gag₂(0_0, X) -> add_3_out_gag₁(X)
add_3_in_gag₂(s_1₁(X), s_1₁(Z)) -> if_add_3_in_1_gag₁(add_3_in_gag₂(X, Z))
if_add_3_in_1_gag₁(add_3_out_gag₁(Y)) -> add_3_out_gag₁(Y)

The set Q consists of the following terms:

add_3_in_gag₂(x0, x1)
if_add_3_in_1_gag₁(x0)

We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {IF_SUM_3_IN_1_GAG₃, SUM_3_IN_GAG₂}.
By using the subterm criterion together with the size-change analysis 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:

IF_SUM_3_IN_1_GAG₃(Y1, Y3, add_3_out_gag₁(X2)) -> SUM_3_IN_GAG₂(Y1, Y3)
The graph contains the following edges 1 >= 1, 2 >= 2
SUM_3_IN_GAG₂(._2₂(X1, Y1), ._2₂(X3, Y3)) -> IF_SUM_3_IN_1_GAG₃(Y1, Y3, add_3_in_gag₂(X1, X3))
The graph contains the following edges 1 > 1, 2 > 2