Termination w.r.t. Q proof of /tmp/tmpZqjrhA/minsort.trs

Termination w.r.t. Q of the following Term Rewriting System could be proven:

Q restricted rewrite system:
The TRS R consists of the following rules:

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
eq(0, 0) → true
eq(0, s(y)) → false
eq(s(x), 0) → false
eq(s(x), s(y)) → eq(x, y)
minsort(nil) → nil
minsort(cons(x, xs)) → cons(min(cons(x, xs)), minsort(rm(min(cons(x, xs)), cons(x, xs))))
min(nil) → 0
min(cons(x, nil)) → x
min(cons(x, cons(y, xs))) → if1(le(x, y), x, y, xs)
if1(true, x, y, xs) → min(cons(x, xs))
if1(false, x, y, xs) → min(cons(y, xs))
rm(x, nil) → nil
rm(x, cons(y, xs)) → if2(eq(x, y), x, y, xs)
if2(true, x, y, xs) → rm(x, xs)
if2(false, x, y, xs) → cons(y, rm(x, xs))

Q is empty.

↳ QTRS

  ↳ Overlay + Local Confluence

Q restricted rewrite system:
The TRS R consists of the following rules:

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
eq(0, 0) → true
eq(0, s(y)) → false
eq(s(x), 0) → false
eq(s(x), s(y)) → eq(x, y)
minsort(nil) → nil
minsort(cons(x, xs)) → cons(min(cons(x, xs)), minsort(rm(min(cons(x, xs)), cons(x, xs))))
min(nil) → 0
min(cons(x, nil)) → x
min(cons(x, cons(y, xs))) → if1(le(x, y), x, y, xs)
if1(true, x, y, xs) → min(cons(x, xs))
if1(false, x, y, xs) → min(cons(y, xs))
rm(x, nil) → nil
rm(x, cons(y, xs)) → if2(eq(x, y), x, y, xs)
if2(true, x, y, xs) → rm(x, xs)
if2(false, x, y, xs) → cons(y, rm(x, xs))

Q is empty.

The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

Q restricted rewrite system:
The TRS R consists of the following rules:

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
eq(0, 0) → true
eq(0, s(y)) → false
eq(s(x), 0) → false
eq(s(x), s(y)) → eq(x, y)
minsort(nil) → nil
minsort(cons(x, xs)) → cons(min(cons(x, xs)), minsort(rm(min(cons(x, xs)), cons(x, xs))))
min(nil) → 0
min(cons(x, nil)) → x
min(cons(x, cons(y, xs))) → if1(le(x, y), x, y, xs)
if1(true, x, y, xs) → min(cons(x, xs))
if1(false, x, y, xs) → min(cons(y, xs))
rm(x, nil) → nil
rm(x, cons(y, xs)) → if2(eq(x, y), x, y, xs)
if2(true, x, y, xs) → rm(x, xs)
if2(false, x, y, xs) → cons(y, rm(x, xs))

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
minsort(nil)
minsort(cons(x0, x1))
min(nil)
min(cons(x0, nil))
min(cons(x0, cons(x1, x2)))
if1(true, x0, x1, x2)
if1(false, x0, x1, x2)
rm(x0, nil)
rm(x0, cons(x1, x2))
if2(true, x0, x1, x2)
if2(false, x0, x1, x2)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

LE(s(x), s(y)) → LE(x, y)
EQ(s(x), s(y)) → EQ(x, y)
MINSORT(cons(x, xs)) → MIN(cons(x, xs))
MINSORT(cons(x, xs)) → MINSORT(rm(min(cons(x, xs)), cons(x, xs)))
MINSORT(cons(x, xs)) → RM(min(cons(x, xs)), cons(x, xs))
MIN(cons(x, cons(y, xs))) → IF1(le(x, y), x, y, xs)
MIN(cons(x, cons(y, xs))) → LE(x, y)
IF1(true, x, y, xs) → MIN(cons(x, xs))
IF1(false, x, y, xs) → MIN(cons(y, xs))
RM(x, cons(y, xs)) → IF2(eq(x, y), x, y, xs)
RM(x, cons(y, xs)) → EQ(x, y)
IF2(true, x, y, xs) → RM(x, xs)
IF2(false, x, y, xs) → RM(x, xs)

The TRS R consists of the following rules:

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
eq(0, 0) → true
eq(0, s(y)) → false
eq(s(x), 0) → false
eq(s(x), s(y)) → eq(x, y)
minsort(nil) → nil
minsort(cons(x, xs)) → cons(min(cons(x, xs)), minsort(rm(min(cons(x, xs)), cons(x, xs))))
min(nil) → 0
min(cons(x, nil)) → x
min(cons(x, cons(y, xs))) → if1(le(x, y), x, y, xs)
if1(true, x, y, xs) → min(cons(x, xs))
if1(false, x, y, xs) → min(cons(y, xs))
rm(x, nil) → nil
rm(x, cons(y, xs)) → if2(eq(x, y), x, y, xs)
if2(true, x, y, xs) → rm(x, xs)
if2(false, x, y, xs) → cons(y, rm(x, xs))

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
minsort(nil)
minsort(cons(x0, x1))
min(nil)
min(cons(x0, nil))
min(cons(x0, cons(x1, x2)))
if1(true, x0, x1, x2)
if1(false, x0, x1, x2)
rm(x0, nil)
rm(x0, cons(x1, x2))
if2(true, x0, x1, x2)
if2(false, x0, x1, x2)

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

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

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

LE(s(x), s(y)) → LE(x, y)
EQ(s(x), s(y)) → EQ(x, y)
MINSORT(cons(x, xs)) → MIN(cons(x, xs))
MINSORT(cons(x, xs)) → MINSORT(rm(min(cons(x, xs)), cons(x, xs)))
MINSORT(cons(x, xs)) → RM(min(cons(x, xs)), cons(x, xs))
MIN(cons(x, cons(y, xs))) → IF1(le(x, y), x, y, xs)
MIN(cons(x, cons(y, xs))) → LE(x, y)
IF1(true, x, y, xs) → MIN(cons(x, xs))
IF1(false, x, y, xs) → MIN(cons(y, xs))
RM(x, cons(y, xs)) → IF2(eq(x, y), x, y, xs)
RM(x, cons(y, xs)) → EQ(x, y)
IF2(true, x, y, xs) → RM(x, xs)
IF2(false, x, y, xs) → RM(x, xs)

The TRS R consists of the following rules:

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
eq(0, 0) → true
eq(0, s(y)) → false
eq(s(x), 0) → false
eq(s(x), s(y)) → eq(x, y)
minsort(nil) → nil
minsort(cons(x, xs)) → cons(min(cons(x, xs)), minsort(rm(min(cons(x, xs)), cons(x, xs))))
min(nil) → 0
min(cons(x, nil)) → x
min(cons(x, cons(y, xs))) → if1(le(x, y), x, y, xs)
if1(true, x, y, xs) → min(cons(x, xs))
if1(false, x, y, xs) → min(cons(y, xs))
rm(x, nil) → nil
rm(x, cons(y, xs)) → if2(eq(x, y), x, y, xs)
if2(true, x, y, xs) → rm(x, xs)
if2(false, x, y, xs) → cons(y, rm(x, xs))

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
minsort(nil)
minsort(cons(x0, x1))
min(nil)
min(cons(x0, nil))
min(cons(x0, cons(x1, x2)))
if1(true, x0, x1, x2)
if1(false, x0, x1, x2)
rm(x0, nil)
rm(x0, cons(x1, x2))
if2(true, x0, x1, x2)
if2(false, x0, x1, x2)

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 5 SCCs with 4 less nodes.

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

EQ(s(x), s(y)) → EQ(x, y)

The TRS R consists of the following rules:

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
eq(0, 0) → true
eq(0, s(y)) → false
eq(s(x), 0) → false
eq(s(x), s(y)) → eq(x, y)
minsort(nil) → nil
minsort(cons(x, xs)) → cons(min(cons(x, xs)), minsort(rm(min(cons(x, xs)), cons(x, xs))))
min(nil) → 0
min(cons(x, nil)) → x
min(cons(x, cons(y, xs))) → if1(le(x, y), x, y, xs)
if1(true, x, y, xs) → min(cons(x, xs))
if1(false, x, y, xs) → min(cons(y, xs))
rm(x, nil) → nil
rm(x, cons(y, xs)) → if2(eq(x, y), x, y, xs)
if2(true, x, y, xs) → rm(x, xs)
if2(false, x, y, xs) → cons(y, rm(x, xs))

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
minsort(nil)
minsort(cons(x0, x1))
min(nil)
min(cons(x0, nil))
min(cons(x0, cons(x1, x2)))
if1(true, x0, x1, x2)
if1(false, x0, x1, x2)
rm(x0, nil)
rm(x0, cons(x1, x2))
if2(true, x0, x1, x2)
if2(false, x0, x1, x2)

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

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

EQ(s(x), s(y)) → EQ(x, y)

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

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
minsort(nil)
minsort(cons(x0, x1))
min(nil)
min(cons(x0, nil))
min(cons(x0, cons(x1, x2)))
if1(true, x0, x1, x2)
if1(false, x0, x1, x2)
rm(x0, nil)
rm(x0, cons(x1, x2))
if2(true, x0, x1, x2)
if2(false, x0, x1, x2)

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
minsort(nil)
minsort(cons(x0, x1))
min(nil)
min(cons(x0, nil))
min(cons(x0, cons(x1, x2)))
if1(true, x0, x1, x2)
if1(false, x0, x1, x2)
rm(x0, nil)
rm(x0, cons(x1, x2))
if2(true, x0, x1, x2)
if2(false, x0, x1, x2)

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

EQ(s(x), s(y)) → EQ(x, y)

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

EQ(s(x), s(y)) → EQ(x, y)
The graph contains the following edges 1 > 1, 2 > 2

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

RM(x, cons(y, xs)) → IF2(eq(x, y), x, y, xs)
IF2(true, x, y, xs) → RM(x, xs)
IF2(false, x, y, xs) → RM(x, xs)

The TRS R consists of the following rules:

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
eq(0, 0) → true
eq(0, s(y)) → false
eq(s(x), 0) → false
eq(s(x), s(y)) → eq(x, y)
minsort(nil) → nil
minsort(cons(x, xs)) → cons(min(cons(x, xs)), minsort(rm(min(cons(x, xs)), cons(x, xs))))
min(nil) → 0
min(cons(x, nil)) → x
min(cons(x, cons(y, xs))) → if1(le(x, y), x, y, xs)
if1(true, x, y, xs) → min(cons(x, xs))
if1(false, x, y, xs) → min(cons(y, xs))
rm(x, nil) → nil
rm(x, cons(y, xs)) → if2(eq(x, y), x, y, xs)
if2(true, x, y, xs) → rm(x, xs)
if2(false, x, y, xs) → cons(y, rm(x, xs))

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
minsort(nil)
minsort(cons(x0, x1))
min(nil)
min(cons(x0, nil))
min(cons(x0, cons(x1, x2)))
if1(true, x0, x1, x2)
if1(false, x0, x1, x2)
rm(x0, nil)
rm(x0, cons(x1, x2))
if2(true, x0, x1, x2)
if2(false, x0, x1, x2)

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

RM(x, cons(y, xs)) → IF2(eq(x, y), x, y, xs)
IF2(true, x, y, xs) → RM(x, xs)
IF2(false, x, y, xs) → RM(x, xs)

The TRS R consists of the following rules:

eq(0, 0) → true
eq(0, s(y)) → false
eq(s(x), 0) → false
eq(s(x), s(y)) → eq(x, y)

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
minsort(nil)
minsort(cons(x0, x1))
min(nil)
min(cons(x0, nil))
min(cons(x0, cons(x1, x2)))
if1(true, x0, x1, x2)
if1(false, x0, x1, x2)
rm(x0, nil)
rm(x0, cons(x1, x2))
if2(true, x0, x1, x2)
if2(false, x0, x1, x2)

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
minsort(nil)
minsort(cons(x0, x1))
min(nil)
min(cons(x0, nil))
min(cons(x0, cons(x1, x2)))
if1(true, x0, x1, x2)
if1(false, x0, x1, x2)
rm(x0, nil)
rm(x0, cons(x1, x2))
if2(true, x0, x1, x2)
if2(false, x0, x1, x2)

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ QDP

              ↳ QDP

              ↳ QDP

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

RM(x, cons(y, xs)) → IF2(eq(x, y), x, y, xs)
IF2(true, x, y, xs) → RM(x, xs)
IF2(false, x, y, xs) → RM(x, xs)

The TRS R consists of the following rules:

eq(0, 0) → true
eq(0, s(y)) → false
eq(s(x), 0) → false
eq(s(x), s(y)) → eq(x, y)

The set Q consists of the following terms:

eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))

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

RM(x, cons(y, xs)) → IF2(eq(x, y), x, y, xs)
The graph contains the following edges 1 >= 2, 2 > 3, 2 > 4
IF2(true, x, y, xs) → RM(x, xs)
The graph contains the following edges 2 >= 1, 4 >= 2
IF2(false, x, y, xs) → RM(x, xs)
The graph contains the following edges 2 >= 1, 4 >= 2

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

              ↳ QDP

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

LE(s(x), s(y)) → LE(x, y)

The TRS R consists of the following rules:

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
eq(0, 0) → true
eq(0, s(y)) → false
eq(s(x), 0) → false
eq(s(x), s(y)) → eq(x, y)
minsort(nil) → nil
minsort(cons(x, xs)) → cons(min(cons(x, xs)), minsort(rm(min(cons(x, xs)), cons(x, xs))))
min(nil) → 0
min(cons(x, nil)) → x
min(cons(x, cons(y, xs))) → if1(le(x, y), x, y, xs)
if1(true, x, y, xs) → min(cons(x, xs))
if1(false, x, y, xs) → min(cons(y, xs))
rm(x, nil) → nil
rm(x, cons(y, xs)) → if2(eq(x, y), x, y, xs)
if2(true, x, y, xs) → rm(x, xs)
if2(false, x, y, xs) → cons(y, rm(x, xs))

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
minsort(nil)
minsort(cons(x0, x1))
min(nil)
min(cons(x0, nil))
min(cons(x0, cons(x1, x2)))
if1(true, x0, x1, x2)
if1(false, x0, x1, x2)
rm(x0, nil)
rm(x0, cons(x1, x2))
if2(true, x0, x1, x2)
if2(false, x0, x1, x2)

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

              ↳ QDP

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

LE(s(x), s(y)) → LE(x, y)

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

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
minsort(nil)
minsort(cons(x0, x1))
min(nil)
min(cons(x0, nil))
min(cons(x0, cons(x1, x2)))
if1(true, x0, x1, x2)
if1(false, x0, x1, x2)
rm(x0, nil)
rm(x0, cons(x1, x2))
if2(true, x0, x1, x2)
if2(false, x0, x1, x2)

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
minsort(nil)
minsort(cons(x0, x1))
min(nil)
min(cons(x0, nil))
min(cons(x0, cons(x1, x2)))
if1(true, x0, x1, x2)
if1(false, x0, x1, x2)
rm(x0, nil)
rm(x0, cons(x1, x2))
if2(true, x0, x1, x2)
if2(false, x0, x1, x2)

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ QDP

              ↳ QDP

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

LE(s(x), s(y)) → LE(x, y)

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

LE(s(x), s(y)) → LE(x, y)
The graph contains the following edges 1 > 1, 2 > 2

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

              ↳ QDP

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

MIN(cons(x, cons(y, xs))) → IF1(le(x, y), x, y, xs)
IF1(true, x, y, xs) → MIN(cons(x, xs))
IF1(false, x, y, xs) → MIN(cons(y, xs))

The TRS R consists of the following rules:

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
eq(0, 0) → true
eq(0, s(y)) → false
eq(s(x), 0) → false
eq(s(x), s(y)) → eq(x, y)
minsort(nil) → nil
minsort(cons(x, xs)) → cons(min(cons(x, xs)), minsort(rm(min(cons(x, xs)), cons(x, xs))))
min(nil) → 0
min(cons(x, nil)) → x
min(cons(x, cons(y, xs))) → if1(le(x, y), x, y, xs)
if1(true, x, y, xs) → min(cons(x, xs))
if1(false, x, y, xs) → min(cons(y, xs))
rm(x, nil) → nil
rm(x, cons(y, xs)) → if2(eq(x, y), x, y, xs)
if2(true, x, y, xs) → rm(x, xs)
if2(false, x, y, xs) → cons(y, rm(x, xs))

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
minsort(nil)
minsort(cons(x0, x1))
min(nil)
min(cons(x0, nil))
min(cons(x0, cons(x1, x2)))
if1(true, x0, x1, x2)
if1(false, x0, x1, x2)
rm(x0, nil)
rm(x0, cons(x1, x2))
if2(true, x0, x1, x2)
if2(false, x0, x1, x2)

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

              ↳ QDP

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

MIN(cons(x, cons(y, xs))) → IF1(le(x, y), x, y, xs)
IF1(true, x, y, xs) → MIN(cons(x, xs))
IF1(false, x, y, xs) → MIN(cons(y, xs))

The TRS R consists of the following rules:

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
minsort(nil)
minsort(cons(x0, x1))
min(nil)
min(cons(x0, nil))
min(cons(x0, cons(x1, x2)))
if1(true, x0, x1, x2)
if1(false, x0, x1, x2)
rm(x0, nil)
rm(x0, cons(x1, x2))
if2(true, x0, x1, x2)
if2(false, x0, x1, x2)

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
minsort(nil)
minsort(cons(x0, x1))
min(nil)
min(cons(x0, nil))
min(cons(x0, cons(x1, x2)))
if1(true, x0, x1, x2)
if1(false, x0, x1, x2)
rm(x0, nil)
rm(x0, cons(x1, x2))
if2(true, x0, x1, x2)
if2(false, x0, x1, x2)

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ QDPOrderProof

              ↳ QDP

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

MIN(cons(x, cons(y, xs))) → IF1(le(x, y), x, y, xs)
IF1(true, x, y, xs) → MIN(cons(x, xs))
IF1(false, x, y, xs) → MIN(cons(y, xs))

The TRS R consists of the following rules:

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MIN(cons(x, cons(y, xs))) → IF1(le(x, y), x, y, xs)

The remaining pairs can at least be oriented weakly.

IF1(true, x, y, xs) → MIN(cons(x, xs))
IF1(false, x, y, xs) → MIN(cons(y, xs))

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0
POL(IF1(x₁, x₂, x₃, x₄)) = 1 + x₄
POL(MIN(x₁)) = x₁
POL(cons(x₁, x₂)) = 1 + x₂
POL(false) = 0
POL(le(x₁, x₂)) = 0
POL(s(x₁)) = 0
POL(true) = 0

The following usable rules [FROCOS05] were oriented: none

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ QDPOrderProof

                          ↳ QDP

                            ↳ DependencyGraphProof

              ↳ QDP

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

IF1(true, x, y, xs) → MIN(cons(x, xs))
IF1(false, x, y, xs) → MIN(cons(y, xs))

The TRS R consists of the following rules:

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

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

MINSORT(cons(x, xs)) → MINSORT(rm(min(cons(x, xs)), cons(x, xs)))

The TRS R consists of the following rules:

le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
eq(0, 0) → true
eq(0, s(y)) → false
eq(s(x), 0) → false
eq(s(x), s(y)) → eq(x, y)
minsort(nil) → nil
minsort(cons(x, xs)) → cons(min(cons(x, xs)), minsort(rm(min(cons(x, xs)), cons(x, xs))))
min(nil) → 0
min(cons(x, nil)) → x
min(cons(x, cons(y, xs))) → if1(le(x, y), x, y, xs)
if1(true, x, y, xs) → min(cons(x, xs))
if1(false, x, y, xs) → min(cons(y, xs))
rm(x, nil) → nil
rm(x, cons(y, xs)) → if2(eq(x, y), x, y, xs)
if2(true, x, y, xs) → rm(x, xs)
if2(false, x, y, xs) → cons(y, rm(x, xs))

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
minsort(nil)
minsort(cons(x0, x1))
min(nil)
min(cons(x0, nil))
min(cons(x0, cons(x1, x2)))
if1(true, x0, x1, x2)
if1(false, x0, x1, x2)
rm(x0, nil)
rm(x0, cons(x1, x2))
if2(true, x0, x1, x2)
if2(false, x0, x1, x2)

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

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

MINSORT(cons(x, xs)) → MINSORT(rm(min(cons(x, xs)), cons(x, xs)))

The TRS R consists of the following rules:

min(cons(x, nil)) → x
if1(true, x, y, xs) → min(cons(x, xs))
min(cons(x, cons(y, xs))) → if1(le(x, y), x, y, xs)
if1(false, x, y, xs) → min(cons(y, xs))
if2(true, x, y, xs) → rm(x, xs)
rm(x, cons(y, xs)) → if2(eq(x, y), x, y, xs)
eq(0, 0) → true
eq(0, s(y)) → false
eq(s(x), 0) → false
eq(s(x), s(y)) → eq(x, y)
if2(false, x, y, xs) → cons(y, rm(x, xs))
rm(x, nil) → nil
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
minsort(nil)
minsort(cons(x0, x1))
min(nil)
min(cons(x0, nil))
min(cons(x0, cons(x1, x2)))
if1(true, x0, x1, x2)
if1(false, x0, x1, x2)
rm(x0, nil)
rm(x0, cons(x1, x2))
if2(true, x0, x1, x2)
if2(false, x0, x1, x2)

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

minsort(nil)
minsort(cons(x0, x1))

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Induction-Processor

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

MINSORT(cons(x, xs)) → MINSORT(rm(min(cons(x, xs)), cons(x, xs)))

The TRS R consists of the following rules:

min(cons(x, nil)) → x
if1(true, x, y, xs) → min(cons(x, xs))
min(cons(x, cons(y, xs))) → if1(le(x, y), x, y, xs)
if1(false, x, y, xs) → min(cons(y, xs))
if2(true, x, y, xs) → rm(x, xs)
rm(x, cons(y, xs)) → if2(eq(x, y), x, y, xs)
eq(0, 0) → true
eq(0, s(y)) → false
eq(s(x), 0) → false
eq(s(x), s(y)) → eq(x, y)
if2(false, x, y, xs) → cons(y, rm(x, xs))
rm(x, nil) → nil
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
min(nil)
min(cons(x0, nil))
min(cons(x0, cons(x1, x2)))
if1(true, x0, x1, x2)
if1(false, x0, x1, x2)
rm(x0, nil)
rm(x0, cons(x1, x2))
if2(true, x0, x1, x2)
if2(false, x0, x1, x2)

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

This DP could be deleted by the Induction-Processor:
MINSORT(cons(x', xs')) → MINSORT(rm(min(cons(x', xs')), cons(x', xs')))

This order was computed:
Polynomial interpretation [POLO]:

POL(0) = 0
POL(MINSORT(x₁)) = 2·x₁
POL(cons(x₁, x₂)) = 1 + 2·x₁ + x₂
POL(eq(x₁, x₂)) = x₂
POL(false) = 0
POL(if1(x₁, x₂, x₃, x₄)) = 2 + 2·x₂ + 2·x₃ + x₄
POL(if2(x₁, x₂, x₃, x₄)) = 1 + 2·x₃ + x₄
POL(le(x₁, x₂)) = 1
POL(min(x₁)) = x₁
POL(nil) = 0
POL(rm(x₁, x₂)) = x₂
POL(s(x₁)) = 2·x₁
POL(true) = 0

At least one of these decreasing rules is always used after the deleted DP:
if2(true, x481, y391, xs231) → rm(x481, xs231)

The following formula is valid:
z0:sort[a34].(¬(z0 =nil)→rm'(min(z0 ), z0 )=true)

The transformed set:
if2'(true, x48, y39, xs23) → true
rm'(x61, cons(y50, xs30)) → if2'(eq(x61, y50), x61, y50, xs30)
if2'(false, x124, y103, xs61) → rm'(x124, xs61)
rm'(x137, nil) → false
min(cons(x', nil)) → x'
if1(true, x9, y6, xs2) → min(cons(x9, xs2))
min(cons(x22, cons(y17, xs9))) → if1(le(x22, y17), x22, y17, xs9)
if1(false, x35, y28, xs16) → min(cons(y28, xs16))
if2(true, x48, y39, xs23) → rm(x48, xs23)
rm(x61, cons(y50, xs30)) → if2(eq(x61, y50), x61, y50, xs30)
eq(0, 0) → true
eq(0, s(y71)) → false
eq(s(x98), 0) → false
eq(s(x111), s(y92)) → eq(x111, y92)
if2(false, x124, y103, xs61) → cons(y103, rm(x124, xs61))
rm(x137, nil) → nil
le(0, y124) → true
le(s(x162), 0) → false
le(s(x175), s(y145)) → le(x175, y145)
min(nil) → 0
equal_bool(true, false) → false
equal_bool(false, true) → false
equal_bool(true, true) → true
equal_bool(false, false) → true
and(true, x) → x
and(false, x) → false
or(true, x) → true
or(false, x) → x
not(false) → true
not(true) → false
isa_true(true) → true
isa_true(false) → false
isa_false(true) → false
isa_false(false) → true
equal_sort[a0](0, 0) → true
equal_sort[a0](0, s(x0)) → false
equal_sort[a0](s(x0), 0) → false
equal_sort[a0](s(x0), s(x1)) → equal_sort[a0](x0, x1)
equal_sort[a34](nil, nil) → true
equal_sort[a34](nil, cons(x0, x1)) → false
equal_sort[a34](cons(x0, x1), nil) → false
equal_sort[a34](cons(x0, x1), cons(x2, x3)) → and(equal_sort[a34](x0, x2), equal_sort[a34](x1, x3))
equal_sort[a60](witness_sort[a60], witness_sort[a60]) → true

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Induction-Processor

                          ↳ AND

                            ↳ QDP

                              ↳ PisEmptyProof

                            ↳ QTRS

Q DP problem:
P is empty.
The TRS R consists of the following rules:

min(cons(x, nil)) → x
if1(true, x, y, xs) → min(cons(x, xs))
min(cons(x, cons(y, xs))) → if1(le(x, y), x, y, xs)
if1(false, x, y, xs) → min(cons(y, xs))
if2(true, x, y, xs) → rm(x, xs)
rm(x, cons(y, xs)) → if2(eq(x, y), x, y, xs)
eq(0, 0) → true
eq(0, s(y)) → false
eq(s(x), 0) → false
eq(s(x), s(y)) → eq(x, y)
if2(false, x, y, xs) → cons(y, rm(x, xs))
rm(x, nil) → nil
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
min(nil)
min(cons(x0, nil))
min(cons(x0, cons(x1, x2)))
if1(true, x0, x1, x2)
if1(false, x0, x1, x2)
rm(x0, nil)
rm(x0, cons(x1, x2))
if2(true, x0, x1, x2)
if2(false, x0, x1, x2)

We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Induction-Processor

                          ↳ AND

                            ↳ QDP

                            ↳ QTRS

                              ↳ Overlay + Local Confluence

Q restricted rewrite system:
The TRS R consists of the following rules:

if2'(true, x48, y39, xs23) → true
rm'(x61, cons(y50, xs30)) → if2'(eq(x61, y50), x61, y50, xs30)
if2'(false, x124, y103, xs61) → rm'(x124, xs61)
rm'(x137, nil) → false
min(cons(x', nil)) → x'
if1(true, x9, y6, xs2) → min(cons(x9, xs2))
min(cons(x22, cons(y17, xs9))) → if1(le(x22, y17), x22, y17, xs9)
if1(false, x35, y28, xs16) → min(cons(y28, xs16))
if2(true, x48, y39, xs23) → rm(x48, xs23)
rm(x61, cons(y50, xs30)) → if2(eq(x61, y50), x61, y50, xs30)
eq(0, 0) → true
eq(0, s(y71)) → false
eq(s(x98), 0) → false
eq(s(x111), s(y92)) → eq(x111, y92)
if2(false, x124, y103, xs61) → cons(y103, rm(x124, xs61))
rm(x137, nil) → nil
le(0, y124) → true
le(s(x162), 0) → false
le(s(x175), s(y145)) → le(x175, y145)
min(nil) → 0
equal_bool(true, false) → false
equal_bool(false, true) → false
equal_bool(true, true) → true
equal_bool(false, false) → true
and(true, x) → x
and(false, x) → false
or(true, x) → true
or(false, x) → x
not(false) → true
not(true) → false
isa_true(true) → true
isa_true(false) → false
isa_false(true) → false
isa_false(false) → true
equal_sort[a0](0, 0) → true
equal_sort[a0](0, s(x0)) → false
equal_sort[a0](s(x0), 0) → false
equal_sort[a0](s(x0), s(x1)) → equal_sort[a0](x0, x1)
equal_sort[a34](nil, nil) → true
equal_sort[a34](nil, cons(x0, x1)) → false
equal_sort[a34](cons(x0, x1), nil) → false
equal_sort[a34](cons(x0, x1), cons(x2, x3)) → and(equal_sort[a34](x0, x2), equal_sort[a34](x1, x3))
equal_sort[a60](witness_sort[a60], witness_sort[a60]) → true

Q is empty.

The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Induction-Processor

                          ↳ AND

                            ↳ QDP

                            ↳ QTRS

                              ↳ Overlay + Local Confluence

                                ↳ QTRS

                                  ↳ DependencyPairsProof

Q restricted rewrite system:
The TRS R consists of the following rules:

if2'(true, x48, y39, xs23) → true
rm'(x61, cons(y50, xs30)) → if2'(eq(x61, y50), x61, y50, xs30)
if2'(false, x124, y103, xs61) → rm'(x124, xs61)
rm'(x137, nil) → false
min(cons(x', nil)) → x'
if1(true, x9, y6, xs2) → min(cons(x9, xs2))
min(cons(x22, cons(y17, xs9))) → if1(le(x22, y17), x22, y17, xs9)
if1(false, x35, y28, xs16) → min(cons(y28, xs16))
if2(true, x48, y39, xs23) → rm(x48, xs23)
rm(x61, cons(y50, xs30)) → if2(eq(x61, y50), x61, y50, xs30)
eq(0, 0) → true
eq(0, s(y71)) → false
eq(s(x98), 0) → false
eq(s(x111), s(y92)) → eq(x111, y92)
if2(false, x124, y103, xs61) → cons(y103, rm(x124, xs61))
rm(x137, nil) → nil
le(0, y124) → true
le(s(x162), 0) → false
le(s(x175), s(y145)) → le(x175, y145)
min(nil) → 0
equal_bool(true, false) → false
equal_bool(false, true) → false
equal_bool(true, true) → true
equal_bool(false, false) → true
and(true, x) → x
and(false, x) → false
or(true, x) → true
or(false, x) → x
not(false) → true
not(true) → false
isa_true(true) → true
isa_true(false) → false
isa_false(true) → false
isa_false(false) → true
equal_sort[a0](0, 0) → true
equal_sort[a0](0, s(x0)) → false
equal_sort[a0](s(x0), 0) → false
equal_sort[a0](s(x0), s(x1)) → equal_sort[a0](x0, x1)
equal_sort[a34](nil, nil) → true
equal_sort[a34](nil, cons(x0, x1)) → false
equal_sort[a34](cons(x0, x1), nil) → false
equal_sort[a34](cons(x0, x1), cons(x2, x3)) → and(equal_sort[a34](x0, x2), equal_sort[a34](x1, x3))
equal_sort[a60](witness_sort[a60], witness_sort[a60]) → true

The set Q consists of the following terms:

if2'(true, x0, x1, x2)
rm'(x0, cons(x1, x2))
if2'(false, x0, x1, x2)
rm'(x0, nil)
min(cons(x0, nil))
if1(true, x0, x1, x2)
min(cons(x0, cons(x1, x2)))
if1(false, x0, x1, x2)
if2(true, x0, x1, x2)
rm(x0, cons(x1, x2))
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
if2(false, x0, x1, x2)
rm(x0, nil)
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
min(nil)
equal_bool(true, false)
equal_bool(false, true)
equal_bool(true, true)
equal_bool(false, false)
and(true, x0)
and(false, x0)
or(true, x0)
or(false, x0)
not(false)
not(true)
isa_true(true)
isa_true(false)
isa_false(true)
isa_false(false)
equal_sort[a0](0, 0)
equal_sort[a0](0, s(x0))
equal_sort[a0](s(x0), 0)
equal_sort[a0](s(x0), s(x1))
equal_sort[a34](nil, nil)
equal_sort[a34](nil, cons(x0, x1))
equal_sort[a34](cons(x0, x1), nil)
equal_sort[a34](cons(x0, x1), cons(x2, x3))
equal_sort[a60](witness_sort[a60], witness_sort[a60])

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:

RM'(x61, cons(y50, xs30)) → IF2'(eq(x61, y50), x61, y50, xs30)
RM'(x61, cons(y50, xs30)) → EQ(x61, y50)
IF2'(false, x124, y103, xs61) → RM'(x124, xs61)
IF1(true, x9, y6, xs2) → MIN(cons(x9, xs2))
MIN(cons(x22, cons(y17, xs9))) → IF1(le(x22, y17), x22, y17, xs9)
MIN(cons(x22, cons(y17, xs9))) → LE(x22, y17)
IF1(false, x35, y28, xs16) → MIN(cons(y28, xs16))
IF2(true, x48, y39, xs23) → RM(x48, xs23)
RM(x61, cons(y50, xs30)) → IF2(eq(x61, y50), x61, y50, xs30)
RM(x61, cons(y50, xs30)) → EQ(x61, y50)
EQ(s(x111), s(y92)) → EQ(x111, y92)
IF2(false, x124, y103, xs61) → RM(x124, xs61)
LE(s(x175), s(y145)) → LE(x175, y145)
EQUAL_SORT[A0](s(x0), s(x1)) → EQUAL_SORT[A0](x0, x1)
EQUAL_SORT[A34](cons(x0, x1), cons(x2, x3)) → AND(equal_sort[a34](x0, x2), equal_sort[a34](x1, x3))
EQUAL_SORT[A34](cons(x0, x1), cons(x2, x3)) → EQUAL_SORT[A34](x0, x2)
EQUAL_SORT[A34](cons(x0, x1), cons(x2, x3)) → EQUAL_SORT[A34](x1, x3)

The TRS R consists of the following rules:

if2'(true, x48, y39, xs23) → true
rm'(x61, cons(y50, xs30)) → if2'(eq(x61, y50), x61, y50, xs30)
if2'(false, x124, y103, xs61) → rm'(x124, xs61)
rm'(x137, nil) → false
min(cons(x', nil)) → x'
if1(true, x9, y6, xs2) → min(cons(x9, xs2))
min(cons(x22, cons(y17, xs9))) → if1(le(x22, y17), x22, y17, xs9)
if1(false, x35, y28, xs16) → min(cons(y28, xs16))
if2(true, x48, y39, xs23) → rm(x48, xs23)
rm(x61, cons(y50, xs30)) → if2(eq(x61, y50), x61, y50, xs30)
eq(0, 0) → true
eq(0, s(y71)) → false
eq(s(x98), 0) → false
eq(s(x111), s(y92)) → eq(x111, y92)
if2(false, x124, y103, xs61) → cons(y103, rm(x124, xs61))
rm(x137, nil) → nil
le(0, y124) → true
le(s(x162), 0) → false
le(s(x175), s(y145)) → le(x175, y145)
min(nil) → 0
equal_bool(true, false) → false
equal_bool(false, true) → false
equal_bool(true, true) → true
equal_bool(false, false) → true
and(true, x) → x
and(false, x) → false
or(true, x) → true
or(false, x) → x
not(false) → true
not(true) → false
isa_true(true) → true
isa_true(false) → false
isa_false(true) → false
isa_false(false) → true
equal_sort[a0](0, 0) → true
equal_sort[a0](0, s(x0)) → false
equal_sort[a0](s(x0), 0) → false
equal_sort[a0](s(x0), s(x1)) → equal_sort[a0](x0, x1)
equal_sort[a34](nil, nil) → true
equal_sort[a34](nil, cons(x0, x1)) → false
equal_sort[a34](cons(x0, x1), nil) → false
equal_sort[a34](cons(x0, x1), cons(x2, x3)) → and(equal_sort[a34](x0, x2), equal_sort[a34](x1, x3))
equal_sort[a60](witness_sort[a60], witness_sort[a60]) → true

The set Q consists of the following terms:

if2'(true, x0, x1, x2)
rm'(x0, cons(x1, x2))
if2'(false, x0, x1, x2)
rm'(x0, nil)
min(cons(x0, nil))
if1(true, x0, x1, x2)
min(cons(x0, cons(x1, x2)))
if1(false, x0, x1, x2)
if2(true, x0, x1, x2)
rm(x0, cons(x1, x2))
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
if2(false, x0, x1, x2)
rm(x0, nil)
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
min(nil)
equal_bool(true, false)
equal_bool(false, true)
equal_bool(true, true)
equal_bool(false, false)
and(true, x0)
and(false, x0)
or(true, x0)
or(false, x0)
not(false)
not(true)
isa_true(true)
isa_true(false)
isa_false(true)
isa_false(false)
equal_sort[a0](0, 0)
equal_sort[a0](0, s(x0))
equal_sort[a0](s(x0), 0)
equal_sort[a0](s(x0), s(x1))
equal_sort[a34](nil, nil)
equal_sort[a34](nil, cons(x0, x1))
equal_sort[a34](cons(x0, x1), nil)
equal_sort[a34](cons(x0, x1), cons(x2, x3))
equal_sort[a60](witness_sort[a60], witness_sort[a60])

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

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Induction-Processor

                          ↳ AND

                            ↳ QDP

                            ↳ QTRS

                              ↳ Overlay + Local Confluence

                                ↳ QTRS

                                  ↳ DependencyPairsProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

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

RM'(x61, cons(y50, xs30)) → IF2'(eq(x61, y50), x61, y50, xs30)
RM'(x61, cons(y50, xs30)) → EQ(x61, y50)
IF2'(false, x124, y103, xs61) → RM'(x124, xs61)
IF1(true, x9, y6, xs2) → MIN(cons(x9, xs2))
MIN(cons(x22, cons(y17, xs9))) → IF1(le(x22, y17), x22, y17, xs9)
MIN(cons(x22, cons(y17, xs9))) → LE(x22, y17)
IF1(false, x35, y28, xs16) → MIN(cons(y28, xs16))
IF2(true, x48, y39, xs23) → RM(x48, xs23)
RM(x61, cons(y50, xs30)) → IF2(eq(x61, y50), x61, y50, xs30)
RM(x61, cons(y50, xs30)) → EQ(x61, y50)
EQ(s(x111), s(y92)) → EQ(x111, y92)
IF2(false, x124, y103, xs61) → RM(x124, xs61)
LE(s(x175), s(y145)) → LE(x175, y145)
EQUAL_SORT[A0](s(x0), s(x1)) → EQUAL_SORT[A0](x0, x1)
EQUAL_SORT[A34](cons(x0, x1), cons(x2, x3)) → AND(equal_sort[a34](x0, x2), equal_sort[a34](x1, x3))
EQUAL_SORT[A34](cons(x0, x1), cons(x2, x3)) → EQUAL_SORT[A34](x0, x2)
EQUAL_SORT[A34](cons(x0, x1), cons(x2, x3)) → EQUAL_SORT[A34](x1, x3)

The TRS R consists of the following rules:

if2'(true, x48, y39, xs23) → true
rm'(x61, cons(y50, xs30)) → if2'(eq(x61, y50), x61, y50, xs30)
if2'(false, x124, y103, xs61) → rm'(x124, xs61)
rm'(x137, nil) → false
min(cons(x', nil)) → x'
if1(true, x9, y6, xs2) → min(cons(x9, xs2))
min(cons(x22, cons(y17, xs9))) → if1(le(x22, y17), x22, y17, xs9)
if1(false, x35, y28, xs16) → min(cons(y28, xs16))
if2(true, x48, y39, xs23) → rm(x48, xs23)
rm(x61, cons(y50, xs30)) → if2(eq(x61, y50), x61, y50, xs30)
eq(0, 0) → true
eq(0, s(y71)) → false
eq(s(x98), 0) → false
eq(s(x111), s(y92)) → eq(x111, y92)
if2(false, x124, y103, xs61) → cons(y103, rm(x124, xs61))
rm(x137, nil) → nil
le(0, y124) → true
le(s(x162), 0) → false
le(s(x175), s(y145)) → le(x175, y145)
min(nil) → 0
equal_bool(true, false) → false
equal_bool(false, true) → false
equal_bool(true, true) → true
equal_bool(false, false) → true
and(true, x) → x
and(false, x) → false
or(true, x) → true
or(false, x) → x
not(false) → true
not(true) → false
isa_true(true) → true
isa_true(false) → false
isa_false(true) → false
isa_false(false) → true
equal_sort[a0](0, 0) → true
equal_sort[a0](0, s(x0)) → false
equal_sort[a0](s(x0), 0) → false
equal_sort[a0](s(x0), s(x1)) → equal_sort[a0](x0, x1)
equal_sort[a34](nil, nil) → true
equal_sort[a34](nil, cons(x0, x1)) → false
equal_sort[a34](cons(x0, x1), nil) → false
equal_sort[a34](cons(x0, x1), cons(x2, x3)) → and(equal_sort[a34](x0, x2), equal_sort[a34](x1, x3))
equal_sort[a60](witness_sort[a60], witness_sort[a60]) → true

The set Q consists of the following terms:

if2'(true, x0, x1, x2)
rm'(x0, cons(x1, x2))
if2'(false, x0, x1, x2)
rm'(x0, nil)
min(cons(x0, nil))
if1(true, x0, x1, x2)
min(cons(x0, cons(x1, x2)))
if1(false, x0, x1, x2)
if2(true, x0, x1, x2)
rm(x0, cons(x1, x2))
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
if2(false, x0, x1, x2)
rm(x0, nil)
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
min(nil)
equal_bool(true, false)
equal_bool(false, true)
equal_bool(true, true)
equal_bool(false, false)
and(true, x0)
and(false, x0)
or(true, x0)
or(false, x0)
not(false)
not(true)
isa_true(true)
isa_true(false)
isa_false(true)
isa_false(false)
equal_sort[a0](0, 0)
equal_sort[a0](0, s(x0))
equal_sort[a0](s(x0), 0)
equal_sort[a0](s(x0), s(x1))
equal_sort[a34](nil, nil)
equal_sort[a34](nil, cons(x0, x1))
equal_sort[a34](cons(x0, x1), nil)
equal_sort[a34](cons(x0, x1), cons(x2, x3))
equal_sort[a60](witness_sort[a60], witness_sort[a60])

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 7 SCCs with 4 less nodes.

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Induction-Processor

                          ↳ AND

                            ↳ QDP

                            ↳ QTRS

                              ↳ Overlay + Local Confluence

                                ↳ QTRS

                                  ↳ DependencyPairsProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ AND

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

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

EQUAL_SORT[A34](cons(x0, x1), cons(x2, x3)) → EQUAL_SORT[A34](x1, x3)
EQUAL_SORT[A34](cons(x0, x1), cons(x2, x3)) → EQUAL_SORT[A34](x0, x2)

The TRS R consists of the following rules:

if2'(true, x48, y39, xs23) → true
rm'(x61, cons(y50, xs30)) → if2'(eq(x61, y50), x61, y50, xs30)
if2'(false, x124, y103, xs61) → rm'(x124, xs61)
rm'(x137, nil) → false
min(cons(x', nil)) → x'
if1(true, x9, y6, xs2) → min(cons(x9, xs2))
min(cons(x22, cons(y17, xs9))) → if1(le(x22, y17), x22, y17, xs9)
if1(false, x35, y28, xs16) → min(cons(y28, xs16))
if2(true, x48, y39, xs23) → rm(x48, xs23)
rm(x61, cons(y50, xs30)) → if2(eq(x61, y50), x61, y50, xs30)
eq(0, 0) → true
eq(0, s(y71)) → false
eq(s(x98), 0) → false
eq(s(x111), s(y92)) → eq(x111, y92)
if2(false, x124, y103, xs61) → cons(y103, rm(x124, xs61))
rm(x137, nil) → nil
le(0, y124) → true
le(s(x162), 0) → false
le(s(x175), s(y145)) → le(x175, y145)
min(nil) → 0
equal_bool(true, false) → false
equal_bool(false, true) → false
equal_bool(true, true) → true
equal_bool(false, false) → true
and(true, x) → x
and(false, x) → false
or(true, x) → true
or(false, x) → x
not(false) → true
not(true) → false
isa_true(true) → true
isa_true(false) → false
isa_false(true) → false
isa_false(false) → true
equal_sort[a0](0, 0) → true
equal_sort[a0](0, s(x0)) → false
equal_sort[a0](s(x0), 0) → false
equal_sort[a0](s(x0), s(x1)) → equal_sort[a0](x0, x1)
equal_sort[a34](nil, nil) → true
equal_sort[a34](nil, cons(x0, x1)) → false
equal_sort[a34](cons(x0, x1), nil) → false
equal_sort[a34](cons(x0, x1), cons(x2, x3)) → and(equal_sort[a34](x0, x2), equal_sort[a34](x1, x3))
equal_sort[a60](witness_sort[a60], witness_sort[a60]) → true

The set Q consists of the following terms:

if2'(true, x0, x1, x2)
rm'(x0, cons(x1, x2))
if2'(false, x0, x1, x2)
rm'(x0, nil)
min(cons(x0, nil))
if1(true, x0, x1, x2)
min(cons(x0, cons(x1, x2)))
if1(false, x0, x1, x2)
if2(true, x0, x1, x2)
rm(x0, cons(x1, x2))
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
if2(false, x0, x1, x2)
rm(x0, nil)
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
min(nil)
equal_bool(true, false)
equal_bool(false, true)
equal_bool(true, true)
equal_bool(false, false)
and(true, x0)
and(false, x0)
or(true, x0)
or(false, x0)
not(false)
not(true)
isa_true(true)
isa_true(false)
isa_false(true)
isa_false(false)
equal_sort[a0](0, 0)
equal_sort[a0](0, s(x0))
equal_sort[a0](s(x0), 0)
equal_sort[a0](s(x0), s(x1))
equal_sort[a34](nil, nil)
equal_sort[a34](nil, cons(x0, x1))
equal_sort[a34](cons(x0, x1), nil)
equal_sort[a34](cons(x0, x1), cons(x2, x3))
equal_sort[a60](witness_sort[a60], witness_sort[a60])

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Induction-Processor

                          ↳ AND

                            ↳ QDP

                            ↳ QTRS

                              ↳ Overlay + Local Confluence

                                ↳ QTRS

                                  ↳ DependencyPairsProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ AND

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ QReductionProof

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

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

EQUAL_SORT[A34](cons(x0, x1), cons(x2, x3)) → EQUAL_SORT[A34](x1, x3)
EQUAL_SORT[A34](cons(x0, x1), cons(x2, x3)) → EQUAL_SORT[A34](x0, x2)

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

if2'(true, x0, x1, x2)
rm'(x0, cons(x1, x2))
if2'(false, x0, x1, x2)
rm'(x0, nil)
min(cons(x0, nil))
if1(true, x0, x1, x2)
min(cons(x0, cons(x1, x2)))
if1(false, x0, x1, x2)
if2(true, x0, x1, x2)
rm(x0, cons(x1, x2))
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
if2(false, x0, x1, x2)
rm(x0, nil)
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
min(nil)
equal_bool(true, false)
equal_bool(false, true)
equal_bool(true, true)
equal_bool(false, false)
and(true, x0)
and(false, x0)
or(true, x0)
or(false, x0)
not(false)
not(true)
isa_true(true)
isa_true(false)
isa_false(true)
isa_false(false)
equal_sort[a0](0, 0)
equal_sort[a0](0, s(x0))
equal_sort[a0](s(x0), 0)
equal_sort[a0](s(x0), s(x1))
equal_sort[a34](nil, nil)
equal_sort[a34](nil, cons(x0, x1))
equal_sort[a34](cons(x0, x1), nil)
equal_sort[a34](cons(x0, x1), cons(x2, x3))
equal_sort[a60](witness_sort[a60], witness_sort[a60])

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

if2'(true, x0, x1, x2)
rm'(x0, cons(x1, x2))
if2'(false, x0, x1, x2)
rm'(x0, nil)
min(cons(x0, nil))
if1(true, x0, x1, x2)
min(cons(x0, cons(x1, x2)))
if1(false, x0, x1, x2)
if2(true, x0, x1, x2)
rm(x0, cons(x1, x2))
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
if2(false, x0, x1, x2)
rm(x0, nil)
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
min(nil)
equal_bool(true, false)
equal_bool(false, true)
equal_bool(true, true)
equal_bool(false, false)
and(true, x0)
and(false, x0)
or(true, x0)
or(false, x0)
not(false)
not(true)
isa_true(true)
isa_true(false)
isa_false(true)
isa_false(false)
equal_sort[a0](0, 0)
equal_sort[a0](0, s(x0))
equal_sort[a0](s(x0), 0)
equal_sort[a0](s(x0), s(x1))
equal_sort[a34](nil, nil)
equal_sort[a34](nil, cons(x0, x1))
equal_sort[a34](cons(x0, x1), nil)
equal_sort[a34](cons(x0, x1), cons(x2, x3))
equal_sort[a60](witness_sort[a60], witness_sort[a60])

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Induction-Processor

                          ↳ AND

                            ↳ QDP

                            ↳ QTRS

                              ↳ Overlay + Local Confluence

                                ↳ QTRS

                                  ↳ DependencyPairsProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ AND

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ QReductionProof

                                                  ↳ QDP

                                                    ↳ QDPSizeChangeProof

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

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

EQUAL_SORT[A34](cons(x0, x1), cons(x2, x3)) → EQUAL_SORT[A34](x1, x3)
EQUAL_SORT[A34](cons(x0, x1), cons(x2, x3)) → EQUAL_SORT[A34](x0, x2)

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

EQUAL_SORT[A34](cons(x0, x1), cons(x2, x3)) → EQUAL_SORT[A34](x1, x3)
The graph contains the following edges 1 > 1, 2 > 2
EQUAL_SORT[A34](cons(x0, x1), cons(x2, x3)) → EQUAL_SORT[A34](x0, x2)
The graph contains the following edges 1 > 1, 2 > 2

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Induction-Processor

                          ↳ AND

                            ↳ QDP

                            ↳ QTRS

                              ↳ Overlay + Local Confluence

                                ↳ QTRS

                                  ↳ DependencyPairsProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ AND

                                          ↳ QDP

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

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

EQUAL_SORT[A0](s(x0), s(x1)) → EQUAL_SORT[A0](x0, x1)

The TRS R consists of the following rules:

if2'(true, x48, y39, xs23) → true
rm'(x61, cons(y50, xs30)) → if2'(eq(x61, y50), x61, y50, xs30)
if2'(false, x124, y103, xs61) → rm'(x124, xs61)
rm'(x137, nil) → false
min(cons(x', nil)) → x'
if1(true, x9, y6, xs2) → min(cons(x9, xs2))
min(cons(x22, cons(y17, xs9))) → if1(le(x22, y17), x22, y17, xs9)
if1(false, x35, y28, xs16) → min(cons(y28, xs16))
if2(true, x48, y39, xs23) → rm(x48, xs23)
rm(x61, cons(y50, xs30)) → if2(eq(x61, y50), x61, y50, xs30)
eq(0, 0) → true
eq(0, s(y71)) → false
eq(s(x98), 0) → false
eq(s(x111), s(y92)) → eq(x111, y92)
if2(false, x124, y103, xs61) → cons(y103, rm(x124, xs61))
rm(x137, nil) → nil
le(0, y124) → true
le(s(x162), 0) → false
le(s(x175), s(y145)) → le(x175, y145)
min(nil) → 0
equal_bool(true, false) → false
equal_bool(false, true) → false
equal_bool(true, true) → true
equal_bool(false, false) → true
and(true, x) → x
and(false, x) → false
or(true, x) → true
or(false, x) → x
not(false) → true
not(true) → false
isa_true(true) → true
isa_true(false) → false
isa_false(true) → false
isa_false(false) → true
equal_sort[a0](0, 0) → true
equal_sort[a0](0, s(x0)) → false
equal_sort[a0](s(x0), 0) → false
equal_sort[a0](s(x0), s(x1)) → equal_sort[a0](x0, x1)
equal_sort[a34](nil, nil) → true
equal_sort[a34](nil, cons(x0, x1)) → false
equal_sort[a34](cons(x0, x1), nil) → false
equal_sort[a34](cons(x0, x1), cons(x2, x3)) → and(equal_sort[a34](x0, x2), equal_sort[a34](x1, x3))
equal_sort[a60](witness_sort[a60], witness_sort[a60]) → true

The set Q consists of the following terms:

if2'(true, x0, x1, x2)
rm'(x0, cons(x1, x2))
if2'(false, x0, x1, x2)
rm'(x0, nil)
min(cons(x0, nil))
if1(true, x0, x1, x2)
min(cons(x0, cons(x1, x2)))
if1(false, x0, x1, x2)
if2(true, x0, x1, x2)
rm(x0, cons(x1, x2))
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
if2(false, x0, x1, x2)
rm(x0, nil)
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
min(nil)
equal_bool(true, false)
equal_bool(false, true)
equal_bool(true, true)
equal_bool(false, false)
and(true, x0)
and(false, x0)
or(true, x0)
or(false, x0)
not(false)
not(true)
isa_true(true)
isa_true(false)
isa_false(true)
isa_false(false)
equal_sort[a0](0, 0)
equal_sort[a0](0, s(x0))
equal_sort[a0](s(x0), 0)
equal_sort[a0](s(x0), s(x1))
equal_sort[a34](nil, nil)
equal_sort[a34](nil, cons(x0, x1))
equal_sort[a34](cons(x0, x1), nil)
equal_sort[a34](cons(x0, x1), cons(x2, x3))
equal_sort[a60](witness_sort[a60], witness_sort[a60])

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Induction-Processor

                          ↳ AND

                            ↳ QDP

                            ↳ QTRS

                              ↳ Overlay + Local Confluence

                                ↳ QTRS

                                  ↳ DependencyPairsProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ AND

                                          ↳ QDP

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ QReductionProof

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

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

EQUAL_SORT[A0](s(x0), s(x1)) → EQUAL_SORT[A0](x0, x1)

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

if2'(true, x0, x1, x2)
rm'(x0, cons(x1, x2))
if2'(false, x0, x1, x2)
rm'(x0, nil)
min(cons(x0, nil))
if1(true, x0, x1, x2)
min(cons(x0, cons(x1, x2)))
if1(false, x0, x1, x2)
if2(true, x0, x1, x2)
rm(x0, cons(x1, x2))
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
if2(false, x0, x1, x2)
rm(x0, nil)
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
min(nil)
equal_bool(true, false)
equal_bool(false, true)
equal_bool(true, true)
equal_bool(false, false)
and(true, x0)
and(false, x0)
or(true, x0)
or(false, x0)
not(false)
not(true)
isa_true(true)
isa_true(false)
isa_false(true)
isa_false(false)
equal_sort[a0](0, 0)
equal_sort[a0](0, s(x0))
equal_sort[a0](s(x0), 0)
equal_sort[a0](s(x0), s(x1))
equal_sort[a34](nil, nil)
equal_sort[a34](nil, cons(x0, x1))
equal_sort[a34](cons(x0, x1), nil)
equal_sort[a34](cons(x0, x1), cons(x2, x3))
equal_sort[a60](witness_sort[a60], witness_sort[a60])

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

if2'(true, x0, x1, x2)
rm'(x0, cons(x1, x2))
if2'(false, x0, x1, x2)
rm'(x0, nil)
min(cons(x0, nil))
if1(true, x0, x1, x2)
min(cons(x0, cons(x1, x2)))
if1(false, x0, x1, x2)
if2(true, x0, x1, x2)
rm(x0, cons(x1, x2))
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
if2(false, x0, x1, x2)
rm(x0, nil)
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
min(nil)
equal_bool(true, false)
equal_bool(false, true)
equal_bool(true, true)
equal_bool(false, false)
and(true, x0)
and(false, x0)
or(true, x0)
or(false, x0)
not(false)
not(true)
isa_true(true)
isa_true(false)
isa_false(true)
isa_false(false)
equal_sort[a0](0, 0)
equal_sort[a0](0, s(x0))
equal_sort[a0](s(x0), 0)
equal_sort[a0](s(x0), s(x1))
equal_sort[a34](nil, nil)
equal_sort[a34](nil, cons(x0, x1))
equal_sort[a34](cons(x0, x1), nil)
equal_sort[a34](cons(x0, x1), cons(x2, x3))
equal_sort[a60](witness_sort[a60], witness_sort[a60])

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Induction-Processor

                          ↳ AND

                            ↳ QDP

                            ↳ QTRS

                              ↳ Overlay + Local Confluence

                                ↳ QTRS

                                  ↳ DependencyPairsProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ AND

                                          ↳ QDP

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ QReductionProof

                                                  ↳ QDP

                                                    ↳ QDPSizeChangeProof

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

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

EQUAL_SORT[A0](s(x0), s(x1)) → EQUAL_SORT[A0](x0, x1)

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

EQUAL_SORT[A0](s(x0), s(x1)) → EQUAL_SORT[A0](x0, x1)
The graph contains the following edges 1 > 1, 2 > 2

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Induction-Processor

                          ↳ AND

                            ↳ QDP

                            ↳ QTRS

                              ↳ Overlay + Local Confluence

                                ↳ QTRS

                                  ↳ DependencyPairsProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ AND

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

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

LE(s(x175), s(y145)) → LE(x175, y145)

The TRS R consists of the following rules:

if2'(true, x48, y39, xs23) → true
rm'(x61, cons(y50, xs30)) → if2'(eq(x61, y50), x61, y50, xs30)
if2'(false, x124, y103, xs61) → rm'(x124, xs61)
rm'(x137, nil) → false
min(cons(x', nil)) → x'
if1(true, x9, y6, xs2) → min(cons(x9, xs2))
min(cons(x22, cons(y17, xs9))) → if1(le(x22, y17), x22, y17, xs9)
if1(false, x35, y28, xs16) → min(cons(y28, xs16))
if2(true, x48, y39, xs23) → rm(x48, xs23)
rm(x61, cons(y50, xs30)) → if2(eq(x61, y50), x61, y50, xs30)
eq(0, 0) → true
eq(0, s(y71)) → false
eq(s(x98), 0) → false
eq(s(x111), s(y92)) → eq(x111, y92)
if2(false, x124, y103, xs61) → cons(y103, rm(x124, xs61))
rm(x137, nil) → nil
le(0, y124) → true
le(s(x162), 0) → false
le(s(x175), s(y145)) → le(x175, y145)
min(nil) → 0
equal_bool(true, false) → false
equal_bool(false, true) → false
equal_bool(true, true) → true
equal_bool(false, false) → true
and(true, x) → x
and(false, x) → false
or(true, x) → true
or(false, x) → x
not(false) → true
not(true) → false
isa_true(true) → true
isa_true(false) → false
isa_false(true) → false
isa_false(false) → true
equal_sort[a0](0, 0) → true
equal_sort[a0](0, s(x0)) → false
equal_sort[a0](s(x0), 0) → false
equal_sort[a0](s(x0), s(x1)) → equal_sort[a0](x0, x1)
equal_sort[a34](nil, nil) → true
equal_sort[a34](nil, cons(x0, x1)) → false
equal_sort[a34](cons(x0, x1), nil) → false
equal_sort[a34](cons(x0, x1), cons(x2, x3)) → and(equal_sort[a34](x0, x2), equal_sort[a34](x1, x3))
equal_sort[a60](witness_sort[a60], witness_sort[a60]) → true

The set Q consists of the following terms:

if2'(true, x0, x1, x2)
rm'(x0, cons(x1, x2))
if2'(false, x0, x1, x2)
rm'(x0, nil)
min(cons(x0, nil))
if1(true, x0, x1, x2)
min(cons(x0, cons(x1, x2)))
if1(false, x0, x1, x2)
if2(true, x0, x1, x2)
rm(x0, cons(x1, x2))
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
if2(false, x0, x1, x2)
rm(x0, nil)
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
min(nil)
equal_bool(true, false)
equal_bool(false, true)
equal_bool(true, true)
equal_bool(false, false)
and(true, x0)
and(false, x0)
or(true, x0)
or(false, x0)
not(false)
not(true)
isa_true(true)
isa_true(false)
isa_false(true)
isa_false(false)
equal_sort[a0](0, 0)
equal_sort[a0](0, s(x0))
equal_sort[a0](s(x0), 0)
equal_sort[a0](s(x0), s(x1))
equal_sort[a34](nil, nil)
equal_sort[a34](nil, cons(x0, x1))
equal_sort[a34](cons(x0, x1), nil)
equal_sort[a34](cons(x0, x1), cons(x2, x3))
equal_sort[a60](witness_sort[a60], witness_sort[a60])

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Induction-Processor

                          ↳ AND

                            ↳ QDP

                            ↳ QTRS

                              ↳ Overlay + Local Confluence

                                ↳ QTRS

                                  ↳ DependencyPairsProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ AND

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ QReductionProof

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

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

LE(s(x175), s(y145)) → LE(x175, y145)

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

if2'(true, x0, x1, x2)
rm'(x0, cons(x1, x2))
if2'(false, x0, x1, x2)
rm'(x0, nil)
min(cons(x0, nil))
if1(true, x0, x1, x2)
min(cons(x0, cons(x1, x2)))
if1(false, x0, x1, x2)
if2(true, x0, x1, x2)
rm(x0, cons(x1, x2))
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
if2(false, x0, x1, x2)
rm(x0, nil)
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
min(nil)
equal_bool(true, false)
equal_bool(false, true)
equal_bool(true, true)
equal_bool(false, false)
and(true, x0)
and(false, x0)
or(true, x0)
or(false, x0)
not(false)
not(true)
isa_true(true)
isa_true(false)
isa_false(true)
isa_false(false)
equal_sort[a0](0, 0)
equal_sort[a0](0, s(x0))
equal_sort[a0](s(x0), 0)
equal_sort[a0](s(x0), s(x1))
equal_sort[a34](nil, nil)
equal_sort[a34](nil, cons(x0, x1))
equal_sort[a34](cons(x0, x1), nil)
equal_sort[a34](cons(x0, x1), cons(x2, x3))
equal_sort[a60](witness_sort[a60], witness_sort[a60])

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

if2'(true, x0, x1, x2)
rm'(x0, cons(x1, x2))
if2'(false, x0, x1, x2)
rm'(x0, nil)
min(cons(x0, nil))
if1(true, x0, x1, x2)
min(cons(x0, cons(x1, x2)))
if1(false, x0, x1, x2)
if2(true, x0, x1, x2)
rm(x0, cons(x1, x2))
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
if2(false, x0, x1, x2)
rm(x0, nil)
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
min(nil)
equal_bool(true, false)
equal_bool(false, true)
equal_bool(true, true)
equal_bool(false, false)
and(true, x0)
and(false, x0)
or(true, x0)
or(false, x0)
not(false)
not(true)
isa_true(true)
isa_true(false)
isa_false(true)
isa_false(false)
equal_sort[a0](0, 0)
equal_sort[a0](0, s(x0))
equal_sort[a0](s(x0), 0)
equal_sort[a0](s(x0), s(x1))
equal_sort[a34](nil, nil)
equal_sort[a34](nil, cons(x0, x1))
equal_sort[a34](cons(x0, x1), nil)
equal_sort[a34](cons(x0, x1), cons(x2, x3))
equal_sort[a60](witness_sort[a60], witness_sort[a60])

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Induction-Processor

                          ↳ AND

                            ↳ QDP

                            ↳ QTRS

                              ↳ Overlay + Local Confluence

                                ↳ QTRS

                                  ↳ DependencyPairsProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ AND

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ QReductionProof

                                                  ↳ QDP

                                                    ↳ QDPSizeChangeProof

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

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

LE(s(x175), s(y145)) → LE(x175, y145)

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

LE(s(x175), s(y145)) → LE(x175, y145)
The graph contains the following edges 1 > 1, 2 > 2

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Induction-Processor

                          ↳ AND

                            ↳ QDP

                            ↳ QTRS

                              ↳ Overlay + Local Confluence

                                ↳ QTRS

                                  ↳ DependencyPairsProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ AND

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

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

EQ(s(x111), s(y92)) → EQ(x111, y92)

The TRS R consists of the following rules:

if2'(true, x48, y39, xs23) → true
rm'(x61, cons(y50, xs30)) → if2'(eq(x61, y50), x61, y50, xs30)
if2'(false, x124, y103, xs61) → rm'(x124, xs61)
rm'(x137, nil) → false
min(cons(x', nil)) → x'
if1(true, x9, y6, xs2) → min(cons(x9, xs2))
min(cons(x22, cons(y17, xs9))) → if1(le(x22, y17), x22, y17, xs9)
if1(false, x35, y28, xs16) → min(cons(y28, xs16))
if2(true, x48, y39, xs23) → rm(x48, xs23)
rm(x61, cons(y50, xs30)) → if2(eq(x61, y50), x61, y50, xs30)
eq(0, 0) → true
eq(0, s(y71)) → false
eq(s(x98), 0) → false
eq(s(x111), s(y92)) → eq(x111, y92)
if2(false, x124, y103, xs61) → cons(y103, rm(x124, xs61))
rm(x137, nil) → nil
le(0, y124) → true
le(s(x162), 0) → false
le(s(x175), s(y145)) → le(x175, y145)
min(nil) → 0
equal_bool(true, false) → false
equal_bool(false, true) → false
equal_bool(true, true) → true
equal_bool(false, false) → true
and(true, x) → x
and(false, x) → false
or(true, x) → true
or(false, x) → x
not(false) → true
not(true) → false
isa_true(true) → true
isa_true(false) → false
isa_false(true) → false
isa_false(false) → true
equal_sort[a0](0, 0) → true
equal_sort[a0](0, s(x0)) → false
equal_sort[a0](s(x0), 0) → false
equal_sort[a0](s(x0), s(x1)) → equal_sort[a0](x0, x1)
equal_sort[a34](nil, nil) → true
equal_sort[a34](nil, cons(x0, x1)) → false
equal_sort[a34](cons(x0, x1), nil) → false
equal_sort[a34](cons(x0, x1), cons(x2, x3)) → and(equal_sort[a34](x0, x2), equal_sort[a34](x1, x3))
equal_sort[a60](witness_sort[a60], witness_sort[a60]) → true

The set Q consists of the following terms:

if2'(true, x0, x1, x2)
rm'(x0, cons(x1, x2))
if2'(false, x0, x1, x2)
rm'(x0, nil)
min(cons(x0, nil))
if1(true, x0, x1, x2)
min(cons(x0, cons(x1, x2)))
if1(false, x0, x1, x2)
if2(true, x0, x1, x2)
rm(x0, cons(x1, x2))
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
if2(false, x0, x1, x2)
rm(x0, nil)
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
min(nil)
equal_bool(true, false)
equal_bool(false, true)
equal_bool(true, true)
equal_bool(false, false)
and(true, x0)
and(false, x0)
or(true, x0)
or(false, x0)
not(false)
not(true)
isa_true(true)
isa_true(false)
isa_false(true)
isa_false(false)
equal_sort[a0](0, 0)
equal_sort[a0](0, s(x0))
equal_sort[a0](s(x0), 0)
equal_sort[a0](s(x0), s(x1))
equal_sort[a34](nil, nil)
equal_sort[a34](nil, cons(x0, x1))
equal_sort[a34](cons(x0, x1), nil)
equal_sort[a34](cons(x0, x1), cons(x2, x3))
equal_sort[a60](witness_sort[a60], witness_sort[a60])

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Induction-Processor

                          ↳ AND

                            ↳ QDP

                            ↳ QTRS

                              ↳ Overlay + Local Confluence

                                ↳ QTRS

                                  ↳ DependencyPairsProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ AND

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ QReductionProof

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

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

EQ(s(x111), s(y92)) → EQ(x111, y92)

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

if2'(true, x0, x1, x2)
rm'(x0, cons(x1, x2))
if2'(false, x0, x1, x2)
rm'(x0, nil)
min(cons(x0, nil))
if1(true, x0, x1, x2)
min(cons(x0, cons(x1, x2)))
if1(false, x0, x1, x2)
if2(true, x0, x1, x2)
rm(x0, cons(x1, x2))
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
if2(false, x0, x1, x2)
rm(x0, nil)
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
min(nil)
equal_bool(true, false)
equal_bool(false, true)
equal_bool(true, true)
equal_bool(false, false)
and(true, x0)
and(false, x0)
or(true, x0)
or(false, x0)
not(false)
not(true)
isa_true(true)
isa_true(false)
isa_false(true)
isa_false(false)
equal_sort[a0](0, 0)
equal_sort[a0](0, s(x0))
equal_sort[a0](s(x0), 0)
equal_sort[a0](s(x0), s(x1))
equal_sort[a34](nil, nil)
equal_sort[a34](nil, cons(x0, x1))
equal_sort[a34](cons(x0, x1), nil)
equal_sort[a34](cons(x0, x1), cons(x2, x3))
equal_sort[a60](witness_sort[a60], witness_sort[a60])

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

if2'(true, x0, x1, x2)
rm'(x0, cons(x1, x2))
if2'(false, x0, x1, x2)
rm'(x0, nil)
min(cons(x0, nil))
if1(true, x0, x1, x2)
min(cons(x0, cons(x1, x2)))
if1(false, x0, x1, x2)
if2(true, x0, x1, x2)
rm(x0, cons(x1, x2))
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
if2(false, x0, x1, x2)
rm(x0, nil)
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
min(nil)
equal_bool(true, false)
equal_bool(false, true)
equal_bool(true, true)
equal_bool(false, false)
and(true, x0)
and(false, x0)
or(true, x0)
or(false, x0)
not(false)
not(true)
isa_true(true)
isa_true(false)
isa_false(true)
isa_false(false)
equal_sort[a0](0, 0)
equal_sort[a0](0, s(x0))
equal_sort[a0](s(x0), 0)
equal_sort[a0](s(x0), s(x1))
equal_sort[a34](nil, nil)
equal_sort[a34](nil, cons(x0, x1))
equal_sort[a34](cons(x0, x1), nil)
equal_sort[a34](cons(x0, x1), cons(x2, x3))
equal_sort[a60](witness_sort[a60], witness_sort[a60])

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Induction-Processor

                          ↳ AND

                            ↳ QDP

                            ↳ QTRS

                              ↳ Overlay + Local Confluence

                                ↳ QTRS

                                  ↳ DependencyPairsProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ AND

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ QReductionProof

                                                  ↳ QDP

                                                    ↳ QDPSizeChangeProof

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

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

EQ(s(x111), s(y92)) → EQ(x111, y92)

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

EQ(s(x111), s(y92)) → EQ(x111, y92)
The graph contains the following edges 1 > 1, 2 > 2

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Induction-Processor

                          ↳ AND

                            ↳ QDP

                            ↳ QTRS

                              ↳ Overlay + Local Confluence

                                ↳ QTRS

                                  ↳ DependencyPairsProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ AND

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                          ↳ QDP

                                          ↳ QDP

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

RM(x61, cons(y50, xs30)) → IF2(eq(x61, y50), x61, y50, xs30)
IF2(true, x48, y39, xs23) → RM(x48, xs23)
IF2(false, x124, y103, xs61) → RM(x124, xs61)

The TRS R consists of the following rules:

if2'(true, x48, y39, xs23) → true
rm'(x61, cons(y50, xs30)) → if2'(eq(x61, y50), x61, y50, xs30)
if2'(false, x124, y103, xs61) → rm'(x124, xs61)
rm'(x137, nil) → false
min(cons(x', nil)) → x'
if1(true, x9, y6, xs2) → min(cons(x9, xs2))
min(cons(x22, cons(y17, xs9))) → if1(le(x22, y17), x22, y17, xs9)
if1(false, x35, y28, xs16) → min(cons(y28, xs16))
if2(true, x48, y39, xs23) → rm(x48, xs23)
rm(x61, cons(y50, xs30)) → if2(eq(x61, y50), x61, y50, xs30)
eq(0, 0) → true
eq(0, s(y71)) → false
eq(s(x98), 0) → false
eq(s(x111), s(y92)) → eq(x111, y92)
if2(false, x124, y103, xs61) → cons(y103, rm(x124, xs61))
rm(x137, nil) → nil
le(0, y124) → true
le(s(x162), 0) → false
le(s(x175), s(y145)) → le(x175, y145)
min(nil) → 0
equal_bool(true, false) → false
equal_bool(false, true) → false
equal_bool(true, true) → true
equal_bool(false, false) → true
and(true, x) → x
and(false, x) → false
or(true, x) → true
or(false, x) → x
not(false) → true
not(true) → false
isa_true(true) → true
isa_true(false) → false
isa_false(true) → false
isa_false(false) → true
equal_sort[a0](0, 0) → true
equal_sort[a0](0, s(x0)) → false
equal_sort[a0](s(x0), 0) → false
equal_sort[a0](s(x0), s(x1)) → equal_sort[a0](x0, x1)
equal_sort[a34](nil, nil) → true
equal_sort[a34](nil, cons(x0, x1)) → false
equal_sort[a34](cons(x0, x1), nil) → false
equal_sort[a34](cons(x0, x1), cons(x2, x3)) → and(equal_sort[a34](x0, x2), equal_sort[a34](x1, x3))
equal_sort[a60](witness_sort[a60], witness_sort[a60]) → true

The set Q consists of the following terms:

if2'(true, x0, x1, x2)
rm'(x0, cons(x1, x2))
if2'(false, x0, x1, x2)
rm'(x0, nil)
min(cons(x0, nil))
if1(true, x0, x1, x2)
min(cons(x0, cons(x1, x2)))
if1(false, x0, x1, x2)
if2(true, x0, x1, x2)
rm(x0, cons(x1, x2))
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
if2(false, x0, x1, x2)
rm(x0, nil)
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
min(nil)
equal_bool(true, false)
equal_bool(false, true)
equal_bool(true, true)
equal_bool(false, false)
and(true, x0)
and(false, x0)
or(true, x0)
or(false, x0)
not(false)
not(true)
isa_true(true)
isa_true(false)
isa_false(true)
isa_false(false)
equal_sort[a0](0, 0)
equal_sort[a0](0, s(x0))
equal_sort[a0](s(x0), 0)
equal_sort[a0](s(x0), s(x1))
equal_sort[a34](nil, nil)
equal_sort[a34](nil, cons(x0, x1))
equal_sort[a34](cons(x0, x1), nil)
equal_sort[a34](cons(x0, x1), cons(x2, x3))
equal_sort[a60](witness_sort[a60], witness_sort[a60])

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Induction-Processor

                          ↳ AND

                            ↳ QDP

                            ↳ QTRS

                              ↳ Overlay + Local Confluence

                                ↳ QTRS

                                  ↳ DependencyPairsProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ AND

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ QReductionProof

                                          ↳ QDP

                                          ↳ QDP

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

RM(x61, cons(y50, xs30)) → IF2(eq(x61, y50), x61, y50, xs30)
IF2(true, x48, y39, xs23) → RM(x48, xs23)
IF2(false, x124, y103, xs61) → RM(x124, xs61)

The TRS R consists of the following rules:

eq(0, 0) → true
eq(0, s(y71)) → false
eq(s(x98), 0) → false
eq(s(x111), s(y92)) → eq(x111, y92)

The set Q consists of the following terms:

if2'(true, x0, x1, x2)
rm'(x0, cons(x1, x2))
if2'(false, x0, x1, x2)
rm'(x0, nil)
min(cons(x0, nil))
if1(true, x0, x1, x2)
min(cons(x0, cons(x1, x2)))
if1(false, x0, x1, x2)
if2(true, x0, x1, x2)
rm(x0, cons(x1, x2))
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
if2(false, x0, x1, x2)
rm(x0, nil)
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
min(nil)
equal_bool(true, false)
equal_bool(false, true)
equal_bool(true, true)
equal_bool(false, false)
and(true, x0)
and(false, x0)
or(true, x0)
or(false, x0)
not(false)
not(true)
isa_true(true)
isa_true(false)
isa_false(true)
isa_false(false)
equal_sort[a0](0, 0)
equal_sort[a0](0, s(x0))
equal_sort[a0](s(x0), 0)
equal_sort[a0](s(x0), s(x1))
equal_sort[a34](nil, nil)
equal_sort[a34](nil, cons(x0, x1))
equal_sort[a34](cons(x0, x1), nil)
equal_sort[a34](cons(x0, x1), cons(x2, x3))
equal_sort[a60](witness_sort[a60], witness_sort[a60])

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

if2'(true, x0, x1, x2)
rm'(x0, cons(x1, x2))
if2'(false, x0, x1, x2)
rm'(x0, nil)
min(cons(x0, nil))
if1(true, x0, x1, x2)
min(cons(x0, cons(x1, x2)))
if1(false, x0, x1, x2)
if2(true, x0, x1, x2)
rm(x0, cons(x1, x2))
if2(false, x0, x1, x2)
rm(x0, nil)
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
min(nil)
equal_bool(true, false)
equal_bool(false, true)
equal_bool(true, true)
equal_bool(false, false)
and(true, x0)
and(false, x0)
or(true, x0)
or(false, x0)
not(false)
not(true)
isa_true(true)
isa_true(false)
isa_false(true)
isa_false(false)
equal_sort[a0](0, 0)
equal_sort[a0](0, s(x0))
equal_sort[a0](s(x0), 0)
equal_sort[a0](s(x0), s(x1))
equal_sort[a34](nil, nil)
equal_sort[a34](nil, cons(x0, x1))
equal_sort[a34](cons(x0, x1), nil)
equal_sort[a34](cons(x0, x1), cons(x2, x3))
equal_sort[a60](witness_sort[a60], witness_sort[a60])

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Induction-Processor

                          ↳ AND

                            ↳ QDP

                            ↳ QTRS

                              ↳ Overlay + Local Confluence

                                ↳ QTRS

                                  ↳ DependencyPairsProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ AND

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ QReductionProof

                                                  ↳ QDP

                                                    ↳ QDPSizeChangeProof

                                          ↳ QDP

                                          ↳ QDP

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

RM(x61, cons(y50, xs30)) → IF2(eq(x61, y50), x61, y50, xs30)
IF2(true, x48, y39, xs23) → RM(x48, xs23)
IF2(false, x124, y103, xs61) → RM(x124, xs61)

The TRS R consists of the following rules:

eq(0, 0) → true
eq(0, s(y71)) → false
eq(s(x98), 0) → false
eq(s(x111), s(y92)) → eq(x111, y92)

The set Q consists of the following terms:

eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))

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

RM(x61, cons(y50, xs30)) → IF2(eq(x61, y50), x61, y50, xs30)
The graph contains the following edges 1 >= 2, 2 > 3, 2 > 4
IF2(true, x48, y39, xs23) → RM(x48, xs23)
The graph contains the following edges 2 >= 1, 4 >= 2
IF2(false, x124, y103, xs61) → RM(x124, xs61)
The graph contains the following edges 2 >= 1, 4 >= 2

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Induction-Processor

                          ↳ AND

                            ↳ QDP

                            ↳ QTRS

                              ↳ Overlay + Local Confluence

                                ↳ QTRS

                                  ↳ DependencyPairsProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ AND

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                          ↳ QDP

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

MIN(cons(x22, cons(y17, xs9))) → IF1(le(x22, y17), x22, y17, xs9)
IF1(true, x9, y6, xs2) → MIN(cons(x9, xs2))
IF1(false, x35, y28, xs16) → MIN(cons(y28, xs16))

The TRS R consists of the following rules:

if2'(true, x48, y39, xs23) → true
rm'(x61, cons(y50, xs30)) → if2'(eq(x61, y50), x61, y50, xs30)
if2'(false, x124, y103, xs61) → rm'(x124, xs61)
rm'(x137, nil) → false
min(cons(x', nil)) → x'
if1(true, x9, y6, xs2) → min(cons(x9, xs2))
min(cons(x22, cons(y17, xs9))) → if1(le(x22, y17), x22, y17, xs9)
if1(false, x35, y28, xs16) → min(cons(y28, xs16))
if2(true, x48, y39, xs23) → rm(x48, xs23)
rm(x61, cons(y50, xs30)) → if2(eq(x61, y50), x61, y50, xs30)
eq(0, 0) → true
eq(0, s(y71)) → false
eq(s(x98), 0) → false
eq(s(x111), s(y92)) → eq(x111, y92)
if2(false, x124, y103, xs61) → cons(y103, rm(x124, xs61))
rm(x137, nil) → nil
le(0, y124) → true
le(s(x162), 0) → false
le(s(x175), s(y145)) → le(x175, y145)
min(nil) → 0
equal_bool(true, false) → false
equal_bool(false, true) → false
equal_bool(true, true) → true
equal_bool(false, false) → true
and(true, x) → x
and(false, x) → false
or(true, x) → true
or(false, x) → x
not(false) → true
not(true) → false
isa_true(true) → true
isa_true(false) → false
isa_false(true) → false
isa_false(false) → true
equal_sort[a0](0, 0) → true
equal_sort[a0](0, s(x0)) → false
equal_sort[a0](s(x0), 0) → false
equal_sort[a0](s(x0), s(x1)) → equal_sort[a0](x0, x1)
equal_sort[a34](nil, nil) → true
equal_sort[a34](nil, cons(x0, x1)) → false
equal_sort[a34](cons(x0, x1), nil) → false
equal_sort[a34](cons(x0, x1), cons(x2, x3)) → and(equal_sort[a34](x0, x2), equal_sort[a34](x1, x3))
equal_sort[a60](witness_sort[a60], witness_sort[a60]) → true

The set Q consists of the following terms:

if2'(true, x0, x1, x2)
rm'(x0, cons(x1, x2))
if2'(false, x0, x1, x2)
rm'(x0, nil)
min(cons(x0, nil))
if1(true, x0, x1, x2)
min(cons(x0, cons(x1, x2)))
if1(false, x0, x1, x2)
if2(true, x0, x1, x2)
rm(x0, cons(x1, x2))
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
if2(false, x0, x1, x2)
rm(x0, nil)
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
min(nil)
equal_bool(true, false)
equal_bool(false, true)
equal_bool(true, true)
equal_bool(false, false)
and(true, x0)
and(false, x0)
or(true, x0)
or(false, x0)
not(false)
not(true)
isa_true(true)
isa_true(false)
isa_false(true)
isa_false(false)
equal_sort[a0](0, 0)
equal_sort[a0](0, s(x0))
equal_sort[a0](s(x0), 0)
equal_sort[a0](s(x0), s(x1))
equal_sort[a34](nil, nil)
equal_sort[a34](nil, cons(x0, x1))
equal_sort[a34](cons(x0, x1), nil)
equal_sort[a34](cons(x0, x1), cons(x2, x3))
equal_sort[a60](witness_sort[a60], witness_sort[a60])

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Induction-Processor

                          ↳ AND

                            ↳ QDP

                            ↳ QTRS

                              ↳ Overlay + Local Confluence

                                ↳ QTRS

                                  ↳ DependencyPairsProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ AND

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ QReductionProof

                                          ↳ QDP

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

MIN(cons(x22, cons(y17, xs9))) → IF1(le(x22, y17), x22, y17, xs9)
IF1(true, x9, y6, xs2) → MIN(cons(x9, xs2))
IF1(false, x35, y28, xs16) → MIN(cons(y28, xs16))

The TRS R consists of the following rules:

le(0, y124) → true
le(s(x162), 0) → false
le(s(x175), s(y145)) → le(x175, y145)

The set Q consists of the following terms:

if2'(true, x0, x1, x2)
rm'(x0, cons(x1, x2))
if2'(false, x0, x1, x2)
rm'(x0, nil)
min(cons(x0, nil))
if1(true, x0, x1, x2)
min(cons(x0, cons(x1, x2)))
if1(false, x0, x1, x2)
if2(true, x0, x1, x2)
rm(x0, cons(x1, x2))
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
if2(false, x0, x1, x2)
rm(x0, nil)
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
min(nil)
equal_bool(true, false)
equal_bool(false, true)
equal_bool(true, true)
equal_bool(false, false)
and(true, x0)
and(false, x0)
or(true, x0)
or(false, x0)
not(false)
not(true)
isa_true(true)
isa_true(false)
isa_false(true)
isa_false(false)
equal_sort[a0](0, 0)
equal_sort[a0](0, s(x0))
equal_sort[a0](s(x0), 0)
equal_sort[a0](s(x0), s(x1))
equal_sort[a34](nil, nil)
equal_sort[a34](nil, cons(x0, x1))
equal_sort[a34](cons(x0, x1), nil)
equal_sort[a34](cons(x0, x1), cons(x2, x3))
equal_sort[a60](witness_sort[a60], witness_sort[a60])

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

if2'(true, x0, x1, x2)
rm'(x0, cons(x1, x2))
if2'(false, x0, x1, x2)
rm'(x0, nil)
min(cons(x0, nil))
if1(true, x0, x1, x2)
min(cons(x0, cons(x1, x2)))
if1(false, x0, x1, x2)
if2(true, x0, x1, x2)
rm(x0, cons(x1, x2))
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
if2(false, x0, x1, x2)
rm(x0, nil)
min(nil)
equal_bool(true, false)
equal_bool(false, true)
equal_bool(true, true)
equal_bool(false, false)
and(true, x0)
and(false, x0)
or(true, x0)
or(false, x0)
not(false)
not(true)
isa_true(true)
isa_true(false)
isa_false(true)
isa_false(false)
equal_sort[a0](0, 0)
equal_sort[a0](0, s(x0))
equal_sort[a0](s(x0), 0)
equal_sort[a0](s(x0), s(x1))
equal_sort[a34](nil, nil)
equal_sort[a34](nil, cons(x0, x1))
equal_sort[a34](cons(x0, x1), nil)
equal_sort[a34](cons(x0, x1), cons(x2, x3))
equal_sort[a60](witness_sort[a60], witness_sort[a60])

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Induction-Processor

                          ↳ AND

                            ↳ QDP

                            ↳ QTRS

                              ↳ Overlay + Local Confluence

                                ↳ QTRS

                                  ↳ DependencyPairsProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ AND

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ QReductionProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                          ↳ QDP

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

MIN(cons(x22, cons(y17, xs9))) → IF1(le(x22, y17), x22, y17, xs9)
IF1(true, x9, y6, xs2) → MIN(cons(x9, xs2))
IF1(false, x35, y28, xs16) → MIN(cons(y28, xs16))

The TRS R consists of the following rules:

le(0, y124) → true
le(s(x162), 0) → false
le(s(x175), s(y145)) → le(x175, y145)

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

MIN(cons(x22, cons(y17, xs9))) → IF1(le(x22, y17), x22, y17, xs9)

The remaining pairs can at least be oriented weakly.

IF1(true, x9, y6, xs2) → MIN(cons(x9, xs2))
IF1(false, x35, y28, xs16) → MIN(cons(y28, xs16))

Used ordering: Polynomial interpretation [POLO]:

POL(0) = 0
POL(IF1(x₁, x₂, x₃, x₄)) = 1 + x₄
POL(MIN(x₁)) = x₁
POL(cons(x₁, x₂)) = 1 + x₂
POL(false) = 0
POL(le(x₁, x₂)) = 0
POL(s(x₁)) = 0
POL(true) = 0

The following usable rules [FROCOS05] were oriented: none

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Induction-Processor

                          ↳ AND

                            ↳ QDP

                            ↳ QTRS

                              ↳ Overlay + Local Confluence

                                ↳ QTRS

                                  ↳ DependencyPairsProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ AND

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ QReductionProof

                                                  ↳ QDP

                                                    ↳ QDPOrderProof

                                                      ↳ QDP

                                                        ↳ DependencyGraphProof

                                          ↳ QDP

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

IF1(true, x9, y6, xs2) → MIN(cons(x9, xs2))
IF1(false, x35, y28, xs16) → MIN(cons(y28, xs16))

The TRS R consists of the following rules:

le(0, y124) → true
le(s(x162), 0) → false
le(s(x175), s(y145)) → le(x175, y145)

The set Q consists of the following terms:

le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))

We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Induction-Processor

                          ↳ AND

                            ↳ QDP

                            ↳ QTRS

                              ↳ Overlay + Local Confluence

                                ↳ QTRS

                                  ↳ DependencyPairsProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ AND

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                            ↳ UsableRulesProof

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

IF2'(false, x124, y103, xs61) → RM'(x124, xs61)
RM'(x61, cons(y50, xs30)) → IF2'(eq(x61, y50), x61, y50, xs30)

The TRS R consists of the following rules:

if2'(true, x48, y39, xs23) → true
rm'(x61, cons(y50, xs30)) → if2'(eq(x61, y50), x61, y50, xs30)
if2'(false, x124, y103, xs61) → rm'(x124, xs61)
rm'(x137, nil) → false
min(cons(x', nil)) → x'
if1(true, x9, y6, xs2) → min(cons(x9, xs2))
min(cons(x22, cons(y17, xs9))) → if1(le(x22, y17), x22, y17, xs9)
if1(false, x35, y28, xs16) → min(cons(y28, xs16))
if2(true, x48, y39, xs23) → rm(x48, xs23)
rm(x61, cons(y50, xs30)) → if2(eq(x61, y50), x61, y50, xs30)
eq(0, 0) → true
eq(0, s(y71)) → false
eq(s(x98), 0) → false
eq(s(x111), s(y92)) → eq(x111, y92)
if2(false, x124, y103, xs61) → cons(y103, rm(x124, xs61))
rm(x137, nil) → nil
le(0, y124) → true
le(s(x162), 0) → false
le(s(x175), s(y145)) → le(x175, y145)
min(nil) → 0
equal_bool(true, false) → false
equal_bool(false, true) → false
equal_bool(true, true) → true
equal_bool(false, false) → true
and(true, x) → x
and(false, x) → false
or(true, x) → true
or(false, x) → x
not(false) → true
not(true) → false
isa_true(true) → true
isa_true(false) → false
isa_false(true) → false
isa_false(false) → true
equal_sort[a0](0, 0) → true
equal_sort[a0](0, s(x0)) → false
equal_sort[a0](s(x0), 0) → false
equal_sort[a0](s(x0), s(x1)) → equal_sort[a0](x0, x1)
equal_sort[a34](nil, nil) → true
equal_sort[a34](nil, cons(x0, x1)) → false
equal_sort[a34](cons(x0, x1), nil) → false
equal_sort[a34](cons(x0, x1), cons(x2, x3)) → and(equal_sort[a34](x0, x2), equal_sort[a34](x1, x3))
equal_sort[a60](witness_sort[a60], witness_sort[a60]) → true

The set Q consists of the following terms:

if2'(true, x0, x1, x2)
rm'(x0, cons(x1, x2))
if2'(false, x0, x1, x2)
rm'(x0, nil)
min(cons(x0, nil))
if1(true, x0, x1, x2)
min(cons(x0, cons(x1, x2)))
if1(false, x0, x1, x2)
if2(true, x0, x1, x2)
rm(x0, cons(x1, x2))
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
if2(false, x0, x1, x2)
rm(x0, nil)
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
min(nil)
equal_bool(true, false)
equal_bool(false, true)
equal_bool(true, true)
equal_bool(false, false)
and(true, x0)
and(false, x0)
or(true, x0)
or(false, x0)
not(false)
not(true)
isa_true(true)
isa_true(false)
isa_false(true)
isa_false(false)
equal_sort[a0](0, 0)
equal_sort[a0](0, s(x0))
equal_sort[a0](s(x0), 0)
equal_sort[a0](s(x0), s(x1))
equal_sort[a34](nil, nil)
equal_sort[a34](nil, cons(x0, x1))
equal_sort[a34](cons(x0, x1), nil)
equal_sort[a34](cons(x0, x1), cons(x2, x3))
equal_sort[a60](witness_sort[a60], witness_sort[a60])

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Induction-Processor

                          ↳ AND

                            ↳ QDP

                            ↳ QTRS

                              ↳ Overlay + Local Confluence

                                ↳ QTRS

                                  ↳ DependencyPairsProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ AND

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ QReductionProof

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

IF2'(false, x124, y103, xs61) → RM'(x124, xs61)
RM'(x61, cons(y50, xs30)) → IF2'(eq(x61, y50), x61, y50, xs30)

The TRS R consists of the following rules:

eq(0, 0) → true
eq(0, s(y71)) → false
eq(s(x98), 0) → false
eq(s(x111), s(y92)) → eq(x111, y92)

The set Q consists of the following terms:

if2'(true, x0, x1, x2)
rm'(x0, cons(x1, x2))
if2'(false, x0, x1, x2)
rm'(x0, nil)
min(cons(x0, nil))
if1(true, x0, x1, x2)
min(cons(x0, cons(x1, x2)))
if1(false, x0, x1, x2)
if2(true, x0, x1, x2)
rm(x0, cons(x1, x2))
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
if2(false, x0, x1, x2)
rm(x0, nil)
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
min(nil)
equal_bool(true, false)
equal_bool(false, true)
equal_bool(true, true)
equal_bool(false, false)
and(true, x0)
and(false, x0)
or(true, x0)
or(false, x0)
not(false)
not(true)
isa_true(true)
isa_true(false)
isa_false(true)
isa_false(false)
equal_sort[a0](0, 0)
equal_sort[a0](0, s(x0))
equal_sort[a0](s(x0), 0)
equal_sort[a0](s(x0), s(x1))
equal_sort[a34](nil, nil)
equal_sort[a34](nil, cons(x0, x1))
equal_sort[a34](cons(x0, x1), nil)
equal_sort[a34](cons(x0, x1), cons(x2, x3))
equal_sort[a60](witness_sort[a60], witness_sort[a60])

We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].

if2'(true, x0, x1, x2)
rm'(x0, cons(x1, x2))
if2'(false, x0, x1, x2)
rm'(x0, nil)
min(cons(x0, nil))
if1(true, x0, x1, x2)
min(cons(x0, cons(x1, x2)))
if1(false, x0, x1, x2)
if2(true, x0, x1, x2)
rm(x0, cons(x1, x2))
if2(false, x0, x1, x2)
rm(x0, nil)
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
min(nil)
equal_bool(true, false)
equal_bool(false, true)
equal_bool(true, true)
equal_bool(false, false)
and(true, x0)
and(false, x0)
or(true, x0)
or(false, x0)
not(false)
not(true)
isa_true(true)
isa_true(false)
isa_false(true)
isa_false(false)
equal_sort[a0](0, 0)
equal_sort[a0](0, s(x0))
equal_sort[a0](s(x0), 0)
equal_sort[a0](s(x0), s(x1))
equal_sort[a34](nil, nil)
equal_sort[a34](nil, cons(x0, x1))
equal_sort[a34](cons(x0, x1), nil)
equal_sort[a34](cons(x0, x1), cons(x2, x3))
equal_sort[a60](witness_sort[a60], witness_sort[a60])

↳ QTRS

  ↳ Overlay + Local Confluence

    ↳ QTRS

      ↳ DependencyPairsProof

        ↳ QDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

              ↳ QDP

                ↳ UsableRulesProof

                  ↳ QDP

                    ↳ QReductionProof

                      ↳ QDP

                        ↳ Induction-Processor

                          ↳ AND

                            ↳ QDP

                            ↳ QTRS

                              ↳ Overlay + Local Confluence

                                ↳ QTRS

                                  ↳ DependencyPairsProof

                                    ↳ QDP

                                      ↳ DependencyGraphProof

                                        ↳ AND

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                          ↳ QDP

                                            ↳ UsableRulesProof

                                              ↳ QDP

                                                ↳ QReductionProof

                                                  ↳ QDP

                                                    ↳ QDPSizeChangeProof

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

IF2'(false, x124, y103, xs61) → RM'(x124, xs61)
RM'(x61, cons(y50, xs30)) → IF2'(eq(x61, y50), x61, y50, xs30)

The TRS R consists of the following rules:

eq(0, 0) → true
eq(0, s(y71)) → false
eq(s(x98), 0) → false
eq(s(x111), s(y92)) → eq(x111, y92)

The set Q consists of the following terms:

eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))

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

RM'(x61, cons(y50, xs30)) → IF2'(eq(x61, y50), x61, y50, xs30)
The graph contains the following edges 1 >= 2, 2 > 3, 2 > 4
IF2'(false, x124, y103, xs61) → RM'(x124, xs61)
The graph contains the following edges 2 >= 1, 4 >= 2