Termination w.r.t. Q proof of AProVE_09_Inductive

Termination w.r.t. Q of the given QTRS could be proven:

0 QTRS

↳1 Overlay + Local Confluence (⇔, 0 ms)

↳2 QTRS

↳3 DependencyPairsProof (⇔, 0 ms)

↳4 QDP

↳5 DependencyGraphProof (⇔, 0 ms)

↳6 AND

    ↳7 QDP

        ↳8 UsableRulesProof (⇔, 0 ms)

        ↳9 QDP

        ↳10 QReductionProof (⇔, 0 ms)

        ↳11 QDP

        ↳12 QDPSizeChangeProof (⇔, 0 ms)

        ↳13 YES

    ↳14 QDP

        ↳15 UsableRulesProof (⇔, 0 ms)

        ↳16 QDP

        ↳17 QReductionProof (⇔, 0 ms)

        ↳18 QDP

        ↳19 QDPSizeChangeProof (⇔, 0 ms)

        ↳20 YES

    ↳21 QDP

        ↳22 UsableRulesProof (⇔, 0 ms)

        ↳23 QDP

        ↳24 QReductionProof (⇔, 0 ms)

        ↳25 QDP

        ↳26 QDPSizeChangeProof (⇔, 0 ms)

        ↳27 YES

    ↳28 QDP

        ↳29 UsableRulesProof (⇔, 0 ms)

        ↳30 QDP

        ↳31 QReductionProof (⇔, 0 ms)

        ↳32 QDP

        ↳33 QDPOrderProof (⇔, 0 ms)

        ↳34 QDP

        ↳35 PisEmptyProof (⇔, 0 ms)

        ↳36 YES

    ↳37 QDP

        ↳38 UsableRulesProof (⇔, 0 ms)

        ↳39 QDP

        ↳40 QReductionProof (⇔, 0 ms)

        ↳41 QDP

        ↳42 Induction-Processor (⇒, 59 ms)

        ↳43 AND

            ↳44 QDP

                ↳45 PisEmptyProof (⇔, 0 ms)

                ↳46 YES

            ↳47 QTRS

                ↳48 Overlay + Local Confluence (⇔, 0 ms)

                ↳49 QTRS

                ↳50 DependencyPairsProof (⇔, 0 ms)

                ↳51 QDP

                ↳52 DependencyGraphProof (⇔, 0 ms)

                ↳53 AND

                    ↳54 QDP

                        ↳55 UsableRulesProof (⇔, 0 ms)

                        ↳56 QDP

                        ↳57 QReductionProof (⇔, 0 ms)

                        ↳58 QDP

                        ↳59 QDPSizeChangeProof (⇔, 0 ms)

                        ↳60 YES

                    ↳61 QDP

                        ↳62 UsableRulesProof (⇔, 0 ms)

                        ↳63 QDP

                        ↳64 QReductionProof (⇔, 0 ms)

                        ↳65 QDP

                        ↳66 QDPSizeChangeProof (⇔, 0 ms)

                        ↳67 YES

                    ↳68 QDP

                        ↳69 UsableRulesProof (⇔, 0 ms)

                        ↳70 QDP

                        ↳71 QReductionProof (⇔, 0 ms)

                        ↳72 QDP

                        ↳73 QDPSizeChangeProof (⇔, 0 ms)

                        ↳74 YES

                    ↳75 QDP

                        ↳76 UsableRulesProof (⇔, 0 ms)

                        ↳77 QDP

                        ↳78 QReductionProof (⇔, 0 ms)

                        ↳79 QDP

                        ↳80 QDPSizeChangeProof (⇔, 0 ms)

                        ↳81 YES

                    ↳82 QDP

                        ↳83 UsableRulesProof (⇔, 0 ms)

                        ↳84 QDP

                        ↳85 QReductionProof (⇔, 0 ms)

                        ↳86 QDP

                        ↳87 QDPSizeChangeProof (⇔, 0 ms)

                        ↳88 YES

                    ↳89 QDP

                        ↳90 UsableRulesProof (⇔, 0 ms)

                        ↳91 QDP

                        ↳92 QReductionProof (⇔, 0 ms)

                        ↳93 QDP

                        ↳94 QDPOrderProof (⇔, 0 ms)

                        ↳95 QDP

                        ↳96 PisEmptyProof (⇔, 0 ms)

                        ↳97 YES

                    ↳98 QDP

                        ↳99 UsableRulesProof (⇔, 0 ms)

                        ↳100 QDP

                        ↳101 QReductionProof (⇔, 0 ms)

                        ↳102 QDP

                        ↳103 QDPSizeChangeProof (⇔, 0 ms)

                        ↳104 YES

(0) Obligation:

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.

(1) Overlay + Local Confluence (EQUIVALENT transformation)

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

(2) Obligation:

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)

(3) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(4) Obligation:

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.

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 5 SCCs with 4 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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.

(8) UsableRulesProof (EQUIVALENT transformation)

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

(9) Obligation:

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.

(10) QReductionProof (EQUIVALENT transformation)

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

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)

(11) Obligation:

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.

(12) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

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

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

(13) YES

(14) Obligation:

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)

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

(15) UsableRulesProof (EQUIVALENT transformation)

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

(16) Obligation:

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.

(17) QReductionProof (EQUIVALENT transformation)

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

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)

(18) Obligation:

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.

(19) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

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

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

(20) YES

(21) Obligation:

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)

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

(22) UsableRulesProof (EQUIVALENT transformation)

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

(23) Obligation:

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.

(24) QReductionProof (EQUIVALENT transformation)

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

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)

(25) Obligation:

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

LE(s(x), s(y)) → LE(x, y)

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

(26) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

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

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

(27) YES

(28) Obligation:

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)

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

(29) UsableRulesProof (EQUIVALENT transformation)

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

(30) Obligation:

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.

(31) QReductionProof (EQUIVALENT transformation)

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

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)

(32) Obligation:

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.

(33) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].

The following pairs can be oriented strictly and are deleted.

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 remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
MIN(x1) = x1
cons(x1, x2) = cons(x2)
IF1(x1, x2, x3, x4) = IF1(x4)

Knuth-Bendix order [KBO] with precedence:

trivial

and weight map:

dummyConstant=1
IF1_1=4
cons_1=3

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
none

(34) Obligation:

Q DP problem:
P is empty.
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.

(35) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(36) YES

(37) Obligation:

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)

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

(38) UsableRulesProof (EQUIVALENT transformation)

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

(39) Obligation:

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
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, cons(y, xs)) → if2(eq(x, y), x, y, xs)
if2(true, x, y, xs) → rm(x, 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.

(40) QReductionProof (EQUIVALENT transformation)

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

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

(41) Obligation:

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
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, cons(y, xs)) → if2(eq(x, y), x, y, xs)
if2(true, x, y, xs) → rm(x, 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.

(42) Induction-Processor (SOUND transformation)

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)) = x₁
POL(cons(x₁, x₂)) = 1 + x₁ + x₂
POL(eq(x₁, x₂)) = 3·x₁ + 2·x₂
POL(false_renamed) = 1
POL(if1(x₁, x₂, x₃, x₄)) = 1 + x₂ + x₃ + x₄
POL(if2(x₁, x₂, x₃, x₄)) = 1 + x₃ + x₄
POL(le(x₁, x₂)) = 1
POL(min(x₁)) = x₁
POL(nil) = 0
POL(rm(x₁, x₂)) = x₂
POL(s(x₁)) = 1 + 3·x₁
POL(true_renamed) = 0

At least one of these decreasing rules is always used after the deleted DP:
if2(true_renamed, x4, y2, xs3) → rm(x4, xs3)

The following formula is valid:
z0:sort[a35].(¬(z0 =nil)→rm'(min(z0 ), z0 )=true)

The transformed set:
rm'(x3, cons(y1, xs2)) → if2'(eq(x3, y1), x3, y1, xs2)
if2'(true_renamed, x4, y2, xs3) → true
if2'(false_renamed, x7, y5, xs4) → rm'(x7, xs4)
rm'(x8, nil) → false
min(cons(x', nil)) → x'
min(cons(x'', cons(y, xs'))) → if1(le(x'', y), x'', y, xs')
if1(true_renamed, x1, y', xs'') → min(cons(x1, xs''))
if1(false_renamed, x2, y'', xs1) → min(cons(y'', xs1))
rm(x3, cons(y1, xs2)) → if2(eq(x3, y1), x3, y1, xs2)
if2(true_renamed, x4, y2, xs3) → rm(x4, xs3)
eq(0, 0) → true_renamed
eq(0, s(y3)) → false_renamed
eq(s(x5), 0) → false_renamed
eq(s(x6), s(y4)) → eq(x6, y4)
if2(false_renamed, x7, y5, xs4) → cons(y5, rm(x7, xs4))
rm(x8, nil) → nil
le(0, y6) → true_renamed
le(s(x9), 0) → false_renamed
le(s(x10), s(y7)) → le(x10, y7)
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(v26)) → false
equal_sort[a0](s(v27), 0) → false
equal_sort[a0](s(v27), s(v28)) → equal_sort[a0](v27, v28)
equal_sort[a35](cons(v29, v30), cons(v31, v32)) → and(equal_sort[a0](v29, v31), equal_sort[a35](v30, v32))
equal_sort[a35](cons(v29, v30), nil) → false
equal_sort[a35](nil, cons(v33, v34)) → false
equal_sort[a35](nil, nil) → true
equal_sort[a43](true_renamed, true_renamed) → true
equal_sort[a43](true_renamed, false_renamed) → false
equal_sort[a43](false_renamed, true_renamed) → false
equal_sort[a43](false_renamed, false_renamed) → true
equal_sort[a61](witness_sort[a61], witness_sort[a61]) → true

The proof given by the theorem prover:
The following output was given by the internal theorem prover:

proof of internal
# AProVE Commit ID: 3a20a6ef7432c3f292db1a8838479c42bf5e3b22 root 20240618 unpublished


Partial correctness of the following Program

   [x, v26, v27, v28, v29, v30, v31, v32, v33, v34, x4, y2, xs3, x7, y5, y1, xs2, x8, x3, x', x'', y, xs', y3, x5, x6, y4, y6, x9, x10, y7, x1, y', x2, y'', xs'', xs1]
   equal_bool(true, false) -> false
   equal_bool(false, true) -> false
   equal_bool(true, true) -> true
   equal_bool(false, false) -> true
   true and x -> x
   false and x -> false
   true or x -> true
   false or 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(v26)) -> false
   equal_sort[a0](s(v27), 0) -> false
   equal_sort[a0](s(v27), s(v28)) -> equal_sort[a0](v27, v28)
   equal_sort[a35](cons(v29, v30), cons(v31, v32)) -> equal_sort[a0](v29, v31) and equal_sort[a35](v30, v32)
   equal_sort[a35](cons(v29, v30), nil) -> false
   equal_sort[a35](nil, cons(v33, v34)) -> false
   equal_sort[a35](nil, nil) -> true
   equal_sort[a43](true_renamed, true_renamed) -> true
   equal_sort[a43](true_renamed, false_renamed) -> false
   equal_sort[a43](false_renamed, true_renamed) -> false
   equal_sort[a43](false_renamed, false_renamed) -> true
   equal_sort[a61](witness_sort[a61], witness_sort[a61]) -> true
   if2'(true_renamed, x4, y2, xs3) -> true
   if2'(false_renamed, x7, y5, cons(y1, xs2)) -> if2'(eq(x7, y1), x7, y1, xs2)
   if2'(false_renamed, x7, y5, nil) -> false
   rm'(x8, nil) -> false
   equal_sort[a43](eq(x3, y1), true_renamed) -> true | rm'(x3, cons(y1, xs2)) -> true
   equal_sort[a43](eq(x3, y1), true_renamed) -> false | rm'(x3, cons(y1, xs2)) -> rm'(x3, xs2)
   min(cons(x', nil)) -> x'
   min(nil) -> 0
   equal_sort[a43](le(x'', y), true_renamed) -> true | min(cons(x'', cons(y, xs'))) -> min(cons(x'', xs'))
   equal_sort[a43](le(x'', y), true_renamed) -> false | min(cons(x'', cons(y, xs'))) -> min(cons(y, xs'))
   eq(0, 0) -> true_renamed
   eq(0, s(y3)) -> false_renamed
   eq(s(x5), 0) -> false_renamed
   eq(s(x6), s(y4)) -> eq(x6, y4)
   rm(x8, nil) -> nil
   equal_sort[a43](eq(x3, y1), true_renamed) -> true | rm(x3, cons(y1, xs2)) -> rm(x3, xs2)
   equal_sort[a43](eq(x3, y1), true_renamed) -> false | rm(x3, cons(y1, xs2)) -> cons(y1, rm(x3, xs2))
   le(0, y6) -> true_renamed
   le(s(x9), 0) -> false_renamed
   le(s(x10), s(y7)) -> le(x10, y7)
   if1(true_renamed, x1, y', nil) -> x1
   if1(true_renamed, x1, y', cons(y, xs')) -> if1(le(x1, y), x1, y, xs')
   if1(false_renamed, x2, y'', nil) -> y''
   if1(false_renamed, x2, y'', cons(y, xs')) -> if1(le(y'', y), y'', y, xs')
   if1(true_renamed, x1, y', xs'') -> 0
   if1(false_renamed, x2, y'', xs1) -> 0
   if2(true_renamed, x4, y2, cons(y1, xs2)) -> if2(eq(x4, y1), x4, y1, xs2)
   if2(true_renamed, x4, y2, nil) -> nil
   if2(false_renamed, x7, y5, cons(y1, xs2)) -> cons(y5, if2(eq(x7, y1), x7, y1, xs2))
   if2(false_renamed, x7, y5, nil) -> cons(y5, nil)

using the following formula:
z0:sort[a35].(~(z0=nil)->rm'(min(z0), z0)=true)

could be successfully shown:
(0) Formula
(1) Induction by algorithm [EQUIVALENT, 0 ms]
(2) AND
    (3) Formula
        (4) Symbolic evaluation [EQUIVALENT, 0 ms]
        (5) Formula
        (6) Induction by data structure [EQUIVALENT, 0 ms]
        (7) AND
            (8) Formula
                (9) Symbolic evaluation [EQUIVALENT, 0 ms]
                (10) YES
            (11) Formula
                (12) Conditional Evaluation [EQUIVALENT, 0 ms]
                (13) AND
                    (14) Formula
                        (15) Symbolic evaluation [EQUIVALENT, 0 ms]
                        (16) YES
                    (17) Formula
                        (18) Symbolic evaluation [EQUIVALENT, 0 ms]
                        (19) Formula
                        (20) Hypothesis Lifting [EQUIVALENT, 0 ms]
                        (21) Formula
                        (22) Symbolic evaluation under hypothesis [SOUND, 0 ms]
                        (23) Formula
                        (24) Hypothesis Lifting [EQUIVALENT, 0 ms]
                        (25) Formula
                        (26) Hypothesis Lifting [EQUIVALENT, 0 ms]
                        (27) Formula
                        (28) Conditional Evaluation [EQUIVALENT, 0 ms]
                        (29) AND
                            (30) Formula
                                (31) Symbolic evaluation under hypothesis [EQUIVALENT, 0 ms]
                                (32) YES
                            (33) Formula
                                (34) Symbolic evaluation [EQUIVALENT, 0 ms]
                                (35) YES
    (36) Formula
        (37) Symbolic evaluation [EQUIVALENT, 0 ms]
        (38) YES
    (39) Formula
        (40) Symbolic evaluation [EQUIVALENT, 0 ms]
        (41) Formula
        (42) Conditional Evaluation [EQUIVALENT, 0 ms]
        (43) Formula
        (44) Conditional Evaluation [EQUIVALENT, 0 ms]
        (45) AND
            (46) Formula
                (47) Symbolic evaluation [EQUIVALENT, 0 ms]
                (48) YES
            (49) Formula
                (50) Conditional Evaluation [EQUIVALENT, 0 ms]
                (51) AND
                    (52) Formula
                        (53) Symbolic evaluation [EQUIVALENT, 0 ms]
                        (54) YES
                    (55) Formula
                        (56) Hypothesis Lifting [EQUIVALENT, 0 ms]
                        (57) Formula
                        (58) Conditional Evaluation [EQUIVALENT, 0 ms]
                        (59) Formula
                        (60) Symbolic evaluation [EQUIVALENT, 0 ms]
                        (61) YES
    (62) Formula
        (63) Symbolic evaluation [EQUIVALENT, 0 ms]
        (64) Formula
        (65) Conditional Evaluation [EQUIVALENT, 0 ms]
        (66) Formula
        (67) Conditional Evaluation [EQUIVALENT, 0 ms]
        (68) AND
            (69) Formula
                (70) Symbolic evaluation [EQUIVALENT, 0 ms]
                (71) YES
            (72) Formula
                (73) Symbolic evaluation under hypothesis [EQUIVALENT, 0 ms]
                (74) YES


----------------------------------------

(0)
Obligation:
Formula:
z0:sort[a35].(~(z0=nil)->rm'(min(z0), z0)=true)

There are no hypotheses.




----------------------------------------

(1) Induction by algorithm (EQUIVALENT)
Induction by algorithm min(z0) generates the following cases:

1. Base Case:
Formula:
x':sort[a0].(~(cons(x', nil)=nil)->rm'(min(cons(x', nil)), cons(x', nil))=true)

There are no hypotheses.





2. Base Case:
Formula:
(~(nil=nil)->rm'(min(nil), nil)=true)

There are no hypotheses.





1. Step Case:
Formula:
x'':sort[a0],y:sort[a0],xs':sort[a35].(~(cons(x'', cons(y, xs'))=nil)->rm'(min(cons(x'', cons(y, xs'))), cons(x'', cons(y, xs')))=true)

Hypotheses:
x'':sort[a0],xs':sort[a35].rm'(min(cons(x'', xs')), cons(x'', xs'))=true
x'':sort[a0],y:sort[a0].equal_sort[a43](le(x'', y), true_renamed)=true





2. Step Case:
Formula:
x'':sort[a0],y:sort[a0],xs':sort[a35].(~(cons(x'', cons(y, xs'))=nil)->rm'(min(cons(x'', cons(y, xs'))), cons(x'', cons(y, xs')))=true)

Hypotheses:
y:sort[a0],xs':sort[a35].rm'(min(cons(y, xs')), cons(y, xs'))=true
x'':sort[a0],y:sort[a0].equal_sort[a43](le(x'', y), true_renamed)=false






----------------------------------------

(2)
Complex Obligation (AND)

----------------------------------------

(3)
Obligation:
Formula:
x':sort[a0].(~(cons(x', nil)=nil)->rm'(min(cons(x', nil)), cons(x', nil))=true)

There are no hypotheses.




----------------------------------------

(4) Symbolic evaluation (EQUIVALENT)
Could be shown by simple symbolic evaluation.
----------------------------------------

(5)
Obligation:
Formula:
x':sort[a0].rm'(x', cons(x', nil))=true

There are no hypotheses.




----------------------------------------

(6) Induction by data structure (EQUIVALENT)
Induction by data structure sort[a0] generates the following cases:



1. Base Case:
Formula:
rm'(0, cons(0, nil))=true

There are no hypotheses.





1. Step Case:
Formula:
n:sort[a0].rm'(s(n), cons(s(n), nil))=true

Hypotheses:
n:sort[a0].rm'(n, cons(n, nil))=true






----------------------------------------

(7)
Complex Obligation (AND)

----------------------------------------

(8)
Obligation:
Formula:
rm'(0, cons(0, nil))=true

There are no hypotheses.




----------------------------------------

(9) Symbolic evaluation (EQUIVALENT)
Could be reduced to the following new obligation by simple symbolic evaluation:
True
----------------------------------------

(10)
YES

----------------------------------------

(11)
Obligation:
Formula:
n:sort[a0].rm'(s(n), cons(s(n), nil))=true

Hypotheses:
n:sort[a0].rm'(n, cons(n, nil))=true




----------------------------------------

(12) Conditional Evaluation (EQUIVALENT)
The formula could be reduced to the following new obligations by conditional evaluation:
Formula:
true=true

Hypotheses:
n:sort[a0].rm'(n, cons(n, nil))=true
n:sort[a0].equal_sort[a43](eq(s(n), s(n)), true_renamed)=true





Formula:
n:sort[a0].rm'(s(n), nil)=true

Hypotheses:
n:sort[a0].rm'(n, cons(n, nil))=true
n:sort[a0].equal_sort[a43](eq(s(n), s(n)), true_renamed)=false






----------------------------------------

(13)
Complex Obligation (AND)

----------------------------------------

(14)
Obligation:
Formula:
true=true

Hypotheses:
n:sort[a0].rm'(n, cons(n, nil))=true
n:sort[a0].equal_sort[a43](eq(s(n), s(n)), true_renamed)=true




----------------------------------------

(15) Symbolic evaluation (EQUIVALENT)
Could be reduced to the following new obligation by simple symbolic evaluation:
True
----------------------------------------

(16)
YES

----------------------------------------

(17)
Obligation:
Formula:
n:sort[a0].rm'(s(n), nil)=true

Hypotheses:
n:sort[a0].rm'(n, cons(n, nil))=true
n:sort[a0].equal_sort[a43](eq(s(n), s(n)), true_renamed)=false




----------------------------------------

(18) Symbolic evaluation (EQUIVALENT)
Could be shown by simple symbolic evaluation.
----------------------------------------

(19)
Obligation:
Formula:
False

Hypotheses:
n:sort[a0].rm'(n, cons(n, nil))=true
n:sort[a0].equal_sort[a43](eq(s(n), s(n)), true_renamed)=false




----------------------------------------

(20) Hypothesis Lifting (EQUIVALENT)
Formula could be generalised by hypothesis lifting to the following new obligation:
Formula:
n:sort[a0].((rm'(n, cons(n, nil))=true/\equal_sort[a43](eq(s(n), s(n)), true_renamed)=false)->False)

Hypotheses:
n:sort[a0].rm'(n, cons(n, nil))=true
n:sort[a0].equal_sort[a43](eq(s(n), s(n)), true_renamed)=false




----------------------------------------

(21)
Obligation:
Formula:
n:sort[a0].((rm'(n, cons(n, nil))=true/\equal_sort[a43](eq(s(n), s(n)), true_renamed)=false)->False)

Hypotheses:
n:sort[a0].rm'(n, cons(n, nil))=true
n:sort[a0].equal_sort[a43](eq(s(n), s(n)), true_renamed)=false




----------------------------------------

(22) Symbolic evaluation under hypothesis (SOUND)
Could be reduced by symbolic evaluation under hypothesis to:
n:sort[a0].~(equal_sort[a43](eq(n, n), true_renamed)=false)

By using the following hypotheses:
n:sort[a0].rm'(n, cons(n, nil))=true

----------------------------------------

(23)
Obligation:
Formula:
n:sort[a0].~(equal_sort[a43](eq(n, n), true_renamed)=false)

Hypotheses:
n:sort[a0].rm'(n, cons(n, nil))=true
n:sort[a0].equal_sort[a43](eq(s(n), s(n)), true_renamed)=false




----------------------------------------

(24) Hypothesis Lifting (EQUIVALENT)
Formula could be generalised by hypothesis lifting to the following new obligation:
Formula:
n:sort[a0].(equal_sort[a43](eq(n, n), true_renamed)=false->~(equal_sort[a43](eq(n, n), true_renamed)=false))

Hypotheses:
n:sort[a0].rm'(n, cons(n, nil))=true




----------------------------------------

(25)
Obligation:
Formula:
n:sort[a0].(equal_sort[a43](eq(n, n), true_renamed)=false->~(equal_sort[a43](eq(n, n), true_renamed)=false))

Hypotheses:
n:sort[a0].rm'(n, cons(n, nil))=true




----------------------------------------

(26) Hypothesis Lifting (EQUIVALENT)
Formula could be generalised by hypothesis lifting to the following new obligation:
Formula:
n:sort[a0].(rm'(n, cons(n, nil))=true->(equal_sort[a43](eq(n, n), true_renamed)=false->~(equal_sort[a43](eq(n, n), true_renamed)=false)))

There are no hypotheses.




----------------------------------------

(27)
Obligation:
Formula:
n:sort[a0].(rm'(n, cons(n, nil))=true->(equal_sort[a43](eq(n, n), true_renamed)=false->~(equal_sort[a43](eq(n, n), true_renamed)=false)))

There are no hypotheses.




----------------------------------------

(28) Conditional Evaluation (EQUIVALENT)
The formula could be reduced to the following new obligations by conditional evaluation:
Formula:
n:sort[a0].(true=true->(equal_sort[a43](eq(n, n), true_renamed)=false->~(equal_sort[a43](eq(n, n), true_renamed)=false)))

Hypotheses:
n:sort[a0].equal_sort[a43](eq(n, n), true_renamed)=true





Formula:
n:sort[a0].(rm'(n, nil)=true->(equal_sort[a43](eq(n, n), true_renamed)=false->~(equal_sort[a43](eq(n, n), true_renamed)=false)))

Hypotheses:
n:sort[a0].equal_sort[a43](eq(n, n), true_renamed)=false






----------------------------------------

(29)
Complex Obligation (AND)

----------------------------------------

(30)
Obligation:
Formula:
n:sort[a0].(true=true->(equal_sort[a43](eq(n, n), true_renamed)=false->~(equal_sort[a43](eq(n, n), true_renamed)=false)))

Hypotheses:
n:sort[a0].equal_sort[a43](eq(n, n), true_renamed)=true




----------------------------------------

(31) Symbolic evaluation under hypothesis (EQUIVALENT)
Could be shown using symbolic evaluation under hypothesis, by using the following hypotheses:

n:sort[a0].equal_sort[a43](eq(n, n), true_renamed)=true

----------------------------------------

(32)
YES

----------------------------------------

(33)
Obligation:
Formula:
n:sort[a0].(rm'(n, nil)=true->(equal_sort[a43](eq(n, n), true_renamed)=false->~(equal_sort[a43](eq(n, n), true_renamed)=false)))

Hypotheses:
n:sort[a0].equal_sort[a43](eq(n, n), true_renamed)=false




----------------------------------------

(34) Symbolic evaluation (EQUIVALENT)
Could be reduced to the following new obligation by simple symbolic evaluation:
True
----------------------------------------

(35)
YES

----------------------------------------

(36)
Obligation:
Formula:
(~(nil=nil)->rm'(min(nil), nil)=true)

There are no hypotheses.




----------------------------------------

(37) Symbolic evaluation (EQUIVALENT)
Could be reduced to the following new obligation by simple symbolic evaluation:
True
----------------------------------------

(38)
YES

----------------------------------------

(39)
Obligation:
Formula:
x'':sort[a0],y:sort[a0],xs':sort[a35].(~(cons(x'', cons(y, xs'))=nil)->rm'(min(cons(x'', cons(y, xs'))), cons(x'', cons(y, xs')))=true)

Hypotheses:
x'':sort[a0],xs':sort[a35].rm'(min(cons(x'', xs')), cons(x'', xs'))=true
x'':sort[a0],y:sort[a0].equal_sort[a43](le(x'', y), true_renamed)=true




----------------------------------------

(40) Symbolic evaluation (EQUIVALENT)
Could be shown by simple symbolic evaluation.
----------------------------------------

(41)
Obligation:
Formula:
x'':sort[a0],y:sort[a0],xs':sort[a35].rm'(min(cons(x'', cons(y, xs'))), cons(x'', cons(y, xs')))=true

Hypotheses:
x'':sort[a0],xs':sort[a35].rm'(min(cons(x'', xs')), cons(x'', xs'))=true
x'':sort[a0],y:sort[a0].equal_sort[a43](le(x'', y), true_renamed)=true




----------------------------------------

(42) Conditional Evaluation (EQUIVALENT)
The formula could be reduced to the following new obligations by conditional evaluation:
Formula:
x'':sort[a0],xs':sort[a35],y:sort[a0].rm'(min(cons(x'', xs')), cons(x'', cons(y, xs')))=true

Hypotheses:
x'':sort[a0],xs':sort[a35].rm'(min(cons(x'', xs')), cons(x'', xs'))=true
x'':sort[a0],y:sort[a0].equal_sort[a43](le(x'', y), true_renamed)=true






----------------------------------------

(43)
Obligation:
Formula:
x'':sort[a0],xs':sort[a35],y:sort[a0].rm'(min(cons(x'', xs')), cons(x'', cons(y, xs')))=true

Hypotheses:
x'':sort[a0],xs':sort[a35].rm'(min(cons(x'', xs')), cons(x'', xs'))=true
x'':sort[a0],y:sort[a0].equal_sort[a43](le(x'', y), true_renamed)=true




----------------------------------------

(44) Conditional Evaluation (EQUIVALENT)
The formula could be reduced to the following new obligations by conditional evaluation:
Formula:
true=true

Hypotheses:
x'':sort[a0],xs':sort[a35].rm'(min(cons(x'', xs')), cons(x'', xs'))=true
x'':sort[a0],y:sort[a0].equal_sort[a43](le(x'', y), true_renamed)=true
x'':sort[a0],xs':sort[a35].equal_sort[a43](eq(min(cons(x'', xs')), x''), true_renamed)=true





Formula:
x'':sort[a0],xs':sort[a35],y:sort[a0].rm'(min(cons(x'', xs')), cons(y, xs'))=true

Hypotheses:
x'':sort[a0],xs':sort[a35].rm'(min(cons(x'', xs')), cons(x'', xs'))=true
x'':sort[a0],y:sort[a0].equal_sort[a43](le(x'', y), true_renamed)=true
x'':sort[a0],xs':sort[a35].equal_sort[a43](eq(min(cons(x'', xs')), x''), true_renamed)=false






----------------------------------------

(45)
Complex Obligation (AND)

----------------------------------------

(46)
Obligation:
Formula:
true=true

Hypotheses:
x'':sort[a0],xs':sort[a35].rm'(min(cons(x'', xs')), cons(x'', xs'))=true
x'':sort[a0],y:sort[a0].equal_sort[a43](le(x'', y), true_renamed)=true
x'':sort[a0],xs':sort[a35].equal_sort[a43](eq(min(cons(x'', xs')), x''), true_renamed)=true




----------------------------------------

(47) Symbolic evaluation (EQUIVALENT)
Could be reduced to the following new obligation by simple symbolic evaluation:
True
----------------------------------------

(48)
YES

----------------------------------------

(49)
Obligation:
Formula:
x'':sort[a0],xs':sort[a35],y:sort[a0].rm'(min(cons(x'', xs')), cons(y, xs'))=true

Hypotheses:
x'':sort[a0],xs':sort[a35].rm'(min(cons(x'', xs')), cons(x'', xs'))=true
x'':sort[a0],y:sort[a0].equal_sort[a43](le(x'', y), true_renamed)=true
x'':sort[a0],xs':sort[a35].equal_sort[a43](eq(min(cons(x'', xs')), x''), true_renamed)=false




----------------------------------------

(50) Conditional Evaluation (EQUIVALENT)
The formula could be reduced to the following new obligations by conditional evaluation:
Formula:
true=true

Hypotheses:
x'':sort[a0],xs':sort[a35].rm'(min(cons(x'', xs')), cons(x'', xs'))=true
x'':sort[a0],y:sort[a0].equal_sort[a43](le(x'', y), true_renamed)=true
x'':sort[a0],xs':sort[a35].equal_sort[a43](eq(min(cons(x'', xs')), x''), true_renamed)=false
x'':sort[a0],xs':sort[a35],y:sort[a0].equal_sort[a43](eq(min(cons(x'', xs')), y), true_renamed)=true





Formula:
x'':sort[a0],xs':sort[a35].rm'(min(cons(x'', xs')), xs')=true

Hypotheses:
x'':sort[a0],xs':sort[a35].rm'(min(cons(x'', xs')), cons(x'', xs'))=true
x'':sort[a0],y:sort[a0].equal_sort[a43](le(x'', y), true_renamed)=true
x'':sort[a0],xs':sort[a35].equal_sort[a43](eq(min(cons(x'', xs')), x''), true_renamed)=false
x'':sort[a0],xs':sort[a35],y:sort[a0].equal_sort[a43](eq(min(cons(x'', xs')), y), true_renamed)=false






----------------------------------------

(51)
Complex Obligation (AND)

----------------------------------------

(52)
Obligation:
Formula:
true=true

Hypotheses:
x'':sort[a0],xs':sort[a35].rm'(min(cons(x'', xs')), cons(x'', xs'))=true
x'':sort[a0],y:sort[a0].equal_sort[a43](le(x'', y), true_renamed)=true
x'':sort[a0],xs':sort[a35].equal_sort[a43](eq(min(cons(x'', xs')), x''), true_renamed)=false
x'':sort[a0],xs':sort[a35],y:sort[a0].equal_sort[a43](eq(min(cons(x'', xs')), y), true_renamed)=true




----------------------------------------

(53) Symbolic evaluation (EQUIVALENT)
Could be reduced to the following new obligation by simple symbolic evaluation:
True
----------------------------------------

(54)
YES

----------------------------------------

(55)
Obligation:
Formula:
x'':sort[a0],xs':sort[a35].rm'(min(cons(x'', xs')), xs')=true

Hypotheses:
x'':sort[a0],xs':sort[a35].rm'(min(cons(x'', xs')), cons(x'', xs'))=true
x'':sort[a0],y:sort[a0].equal_sort[a43](le(x'', y), true_renamed)=true
x'':sort[a0],xs':sort[a35].equal_sort[a43](eq(min(cons(x'', xs')), x''), true_renamed)=false
x'':sort[a0],xs':sort[a35],y:sort[a0].equal_sort[a43](eq(min(cons(x'', xs')), y), true_renamed)=false




----------------------------------------

(56) Hypothesis Lifting (EQUIVALENT)
Formula could be generalised by hypothesis lifting to the following new obligation:
Formula:
x'':sort[a0],xs':sort[a35].(rm'(min(cons(x'', xs')), cons(x'', xs'))=true->rm'(min(cons(x'', xs')), xs')=true)

Hypotheses:
x'':sort[a0],y:sort[a0].equal_sort[a43](le(x'', y), true_renamed)=true
x'':sort[a0],xs':sort[a35].equal_sort[a43](eq(min(cons(x'', xs')), x''), true_renamed)=false
x'':sort[a0],xs':sort[a35],y:sort[a0].equal_sort[a43](eq(min(cons(x'', xs')), y), true_renamed)=false




----------------------------------------

(57)
Obligation:
Formula:
x'':sort[a0],xs':sort[a35].(rm'(min(cons(x'', xs')), cons(x'', xs'))=true->rm'(min(cons(x'', xs')), xs')=true)

Hypotheses:
x'':sort[a0],y:sort[a0].equal_sort[a43](le(x'', y), true_renamed)=true
x'':sort[a0],xs':sort[a35].equal_sort[a43](eq(min(cons(x'', xs')), x''), true_renamed)=false
x'':sort[a0],xs':sort[a35],y:sort[a0].equal_sort[a43](eq(min(cons(x'', xs')), y), true_renamed)=false




----------------------------------------

(58) Conditional Evaluation (EQUIVALENT)
The formula could be reduced to the following new obligations by conditional evaluation:
Formula:
x'':sort[a0],xs':sort[a35].(rm'(min(cons(x'', xs')), xs')=true->rm'(min(cons(x'', xs')), xs')=true)

Hypotheses:
x'':sort[a0],y:sort[a0].equal_sort[a43](le(x'', y), true_renamed)=true
x'':sort[a0],xs':sort[a35].equal_sort[a43](eq(min(cons(x'', xs')), x''), true_renamed)=false
x'':sort[a0],xs':sort[a35],y:sort[a0].equal_sort[a43](eq(min(cons(x'', xs')), y), true_renamed)=false






----------------------------------------

(59)
Obligation:
Formula:
x'':sort[a0],xs':sort[a35].(rm'(min(cons(x'', xs')), xs')=true->rm'(min(cons(x'', xs')), xs')=true)

Hypotheses:
x'':sort[a0],y:sort[a0].equal_sort[a43](le(x'', y), true_renamed)=true
x'':sort[a0],xs':sort[a35].equal_sort[a43](eq(min(cons(x'', xs')), x''), true_renamed)=false
x'':sort[a0],xs':sort[a35],y:sort[a0].equal_sort[a43](eq(min(cons(x'', xs')), y), true_renamed)=false




----------------------------------------

(60) Symbolic evaluation (EQUIVALENT)
Could be reduced to the following new obligation by simple symbolic evaluation:
True
----------------------------------------

(61)
YES

----------------------------------------

(62)
Obligation:
Formula:
x'':sort[a0],y:sort[a0],xs':sort[a35].(~(cons(x'', cons(y, xs'))=nil)->rm'(min(cons(x'', cons(y, xs'))), cons(x'', cons(y, xs')))=true)

Hypotheses:
y:sort[a0],xs':sort[a35].rm'(min(cons(y, xs')), cons(y, xs'))=true
x'':sort[a0],y:sort[a0].equal_sort[a43](le(x'', y), true_renamed)=false




----------------------------------------

(63) Symbolic evaluation (EQUIVALENT)
Could be shown by simple symbolic evaluation.
----------------------------------------

(64)
Obligation:
Formula:
x'':sort[a0],y:sort[a0],xs':sort[a35].rm'(min(cons(x'', cons(y, xs'))), cons(x'', cons(y, xs')))=true

Hypotheses:
y:sort[a0],xs':sort[a35].rm'(min(cons(y, xs')), cons(y, xs'))=true
x'':sort[a0],y:sort[a0].equal_sort[a43](le(x'', y), true_renamed)=false




----------------------------------------

(65) Conditional Evaluation (EQUIVALENT)
The formula could be reduced to the following new obligations by conditional evaluation:
Formula:
y:sort[a0],xs':sort[a35],x'':sort[a0].rm'(min(cons(y, xs')), cons(x'', cons(y, xs')))=true

Hypotheses:
y:sort[a0],xs':sort[a35].rm'(min(cons(y, xs')), cons(y, xs'))=true
x'':sort[a0],y:sort[a0].equal_sort[a43](le(x'', y), true_renamed)=false






----------------------------------------

(66)
Obligation:
Formula:
y:sort[a0],xs':sort[a35],x'':sort[a0].rm'(min(cons(y, xs')), cons(x'', cons(y, xs')))=true

Hypotheses:
y:sort[a0],xs':sort[a35].rm'(min(cons(y, xs')), cons(y, xs'))=true
x'':sort[a0],y:sort[a0].equal_sort[a43](le(x'', y), true_renamed)=false




----------------------------------------

(67) Conditional Evaluation (EQUIVALENT)
The formula could be reduced to the following new obligations by conditional evaluation:
Formula:
true=true

Hypotheses:
y:sort[a0],xs':sort[a35].rm'(min(cons(y, xs')), cons(y, xs'))=true
x'':sort[a0],y:sort[a0].equal_sort[a43](le(x'', y), true_renamed)=false
y:sort[a0],xs':sort[a35],x'':sort[a0].equal_sort[a43](eq(min(cons(y, xs')), x''), true_renamed)=true





Formula:
y:sort[a0],xs':sort[a35].rm'(min(cons(y, xs')), cons(y, xs'))=true

Hypotheses:
y:sort[a0],xs':sort[a35].rm'(min(cons(y, xs')), cons(y, xs'))=true
x'':sort[a0],y:sort[a0].equal_sort[a43](le(x'', y), true_renamed)=false
y:sort[a0],xs':sort[a35],x'':sort[a0].equal_sort[a43](eq(min(cons(y, xs')), x''), true_renamed)=false






----------------------------------------

(68)
Complex Obligation (AND)

----------------------------------------

(69)
Obligation:
Formula:
true=true

Hypotheses:
y:sort[a0],xs':sort[a35].rm'(min(cons(y, xs')), cons(y, xs'))=true
x'':sort[a0],y:sort[a0].equal_sort[a43](le(x'', y), true_renamed)=false
y:sort[a0],xs':sort[a35],x'':sort[a0].equal_sort[a43](eq(min(cons(y, xs')), x''), true_renamed)=true




----------------------------------------

(70) Symbolic evaluation (EQUIVALENT)
Could be reduced to the following new obligation by simple symbolic evaluation:
True
----------------------------------------

(71)
YES

----------------------------------------

(72)
Obligation:
Formula:
y:sort[a0],xs':sort[a35].rm'(min(cons(y, xs')), cons(y, xs'))=true

Hypotheses:
y:sort[a0],xs':sort[a35].rm'(min(cons(y, xs')), cons(y, xs'))=true
x'':sort[a0],y:sort[a0].equal_sort[a43](le(x'', y), true_renamed)=false
y:sort[a0],xs':sort[a35],x'':sort[a0].equal_sort[a43](eq(min(cons(y, xs')), x''), true_renamed)=false




----------------------------------------

(73) Symbolic evaluation under hypothesis (EQUIVALENT)
Could be shown using symbolic evaluation under hypothesis, by using the following hypotheses:

y:sort[a0],xs':sort[a35].rm'(min(cons(y, xs')), cons(y, xs'))=true

----------------------------------------

(74)
YES

(43) Complex Obligation (AND)

(44) Obligation:

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

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, cons(y, xs)) → if2(eq(x, y), x, y, xs)
if2(true, x, y, xs) → rm(x, 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.

(45) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(46) YES

(47) Obligation:

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

rm'(x3, cons(y1, xs2)) → if2'(eq(x3, y1), x3, y1, xs2)
if2'(true_renamed, x4, y2, xs3) → true
if2'(false_renamed, x7, y5, xs4) → rm'(x7, xs4)
rm'(x8, nil) → false
min(cons(x', nil)) → x'
min(cons(x'', cons(y, xs'))) → if1(le(x'', y), x'', y, xs')
if1(true_renamed, x1, y', xs'') → min(cons(x1, xs''))
if1(false_renamed, x2, y'', xs1) → min(cons(y'', xs1))
rm(x3, cons(y1, xs2)) → if2(eq(x3, y1), x3, y1, xs2)
if2(true_renamed, x4, y2, xs3) → rm(x4, xs3)
eq(0, 0) → true_renamed
eq(0, s(y3)) → false_renamed
eq(s(x5), 0) → false_renamed
eq(s(x6), s(y4)) → eq(x6, y4)
if2(false_renamed, x7, y5, xs4) → cons(y5, rm(x7, xs4))
rm(x8, nil) → nil
le(0, y6) → true_renamed
le(s(x9), 0) → false_renamed
le(s(x10), s(y7)) → le(x10, y7)
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(v26)) → false
equal_sort[a0](s(v27), 0) → false
equal_sort[a0](s(v27), s(v28)) → equal_sort[a0](v27, v28)
equal_sort[a35](cons(v29, v30), cons(v31, v32)) → and(equal_sort[a0](v29, v31), equal_sort[a35](v30, v32))
equal_sort[a35](cons(v29, v30), nil) → false
equal_sort[a35](nil, cons(v33, v34)) → false
equal_sort[a35](nil, nil) → true
equal_sort[a43](true_renamed, true_renamed) → true
equal_sort[a43](true_renamed, false_renamed) → false
equal_sort[a43](false_renamed, true_renamed) → false
equal_sort[a43](false_renamed, false_renamed) → true
equal_sort[a61](witness_sort[a61], witness_sort[a61]) → true

Q is empty.

(48) Overlay + Local Confluence (EQUIVALENT transformation)

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

(49) Obligation:

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

rm'(x3, cons(y1, xs2)) → if2'(eq(x3, y1), x3, y1, xs2)
if2'(true_renamed, x4, y2, xs3) → true
if2'(false_renamed, x7, y5, xs4) → rm'(x7, xs4)
rm'(x8, nil) → false
min(cons(x', nil)) → x'
min(cons(x'', cons(y, xs'))) → if1(le(x'', y), x'', y, xs')
if1(true_renamed, x1, y', xs'') → min(cons(x1, xs''))
if1(false_renamed, x2, y'', xs1) → min(cons(y'', xs1))
rm(x3, cons(y1, xs2)) → if2(eq(x3, y1), x3, y1, xs2)
if2(true_renamed, x4, y2, xs3) → rm(x4, xs3)
eq(0, 0) → true_renamed
eq(0, s(y3)) → false_renamed
eq(s(x5), 0) → false_renamed
eq(s(x6), s(y4)) → eq(x6, y4)
if2(false_renamed, x7, y5, xs4) → cons(y5, rm(x7, xs4))
rm(x8, nil) → nil
le(0, y6) → true_renamed
le(s(x9), 0) → false_renamed
le(s(x10), s(y7)) → le(x10, y7)
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(v26)) → false
equal_sort[a0](s(v27), 0) → false
equal_sort[a0](s(v27), s(v28)) → equal_sort[a0](v27, v28)
equal_sort[a35](cons(v29, v30), cons(v31, v32)) → and(equal_sort[a0](v29, v31), equal_sort[a35](v30, v32))
equal_sort[a35](cons(v29, v30), nil) → false
equal_sort[a35](nil, cons(v33, v34)) → false
equal_sort[a35](nil, nil) → true
equal_sort[a43](true_renamed, true_renamed) → true
equal_sort[a43](true_renamed, false_renamed) → false
equal_sort[a43](false_renamed, true_renamed) → false
equal_sort[a43](false_renamed, false_renamed) → true
equal_sort[a61](witness_sort[a61], witness_sort[a61]) → true

The set Q consists of the following terms:

rm'(x0, cons(x1, x2))
if2'(true_renamed, x0, x1, x2)
if2'(false_renamed, x0, x1, x2)
rm'(x0, nil)
min(cons(x0, nil))
min(cons(x0, cons(x1, x2)))
if1(true_renamed, x0, x1, x2)
if1(false_renamed, x0, x1, x2)
rm(x0, cons(x1, x2))
if2(true_renamed, x0, x1, x2)
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
if2(false_renamed, 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[a35](cons(x0, x1), cons(x2, x3))
equal_sort[a35](cons(x0, x1), nil)
equal_sort[a35](nil, cons(x0, x1))
equal_sort[a35](nil, nil)
equal_sort[a43](true_renamed, true_renamed)
equal_sort[a43](true_renamed, false_renamed)
equal_sort[a43](false_renamed, true_renamed)
equal_sort[a43](false_renamed, false_renamed)
equal_sort[a61](witness_sort[a61], witness_sort[a61])

(50) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(51) Obligation:

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

RM'(x3, cons(y1, xs2)) → IF2'(eq(x3, y1), x3, y1, xs2)
RM'(x3, cons(y1, xs2)) → EQ(x3, y1)
IF2'(false_renamed, x7, y5, xs4) → RM'(x7, xs4)
MIN(cons(x'', cons(y, xs'))) → IF1(le(x'', y), x'', y, xs')
MIN(cons(x'', cons(y, xs'))) → LE(x'', y)
IF1(true_renamed, x1, y', xs'') → MIN(cons(x1, xs''))
IF1(false_renamed, x2, y'', xs1) → MIN(cons(y'', xs1))
RM(x3, cons(y1, xs2)) → IF2(eq(x3, y1), x3, y1, xs2)
RM(x3, cons(y1, xs2)) → EQ(x3, y1)
IF2(true_renamed, x4, y2, xs3) → RM(x4, xs3)
EQ(s(x6), s(y4)) → EQ(x6, y4)
IF2(false_renamed, x7, y5, xs4) → RM(x7, xs4)
LE(s(x10), s(y7)) → LE(x10, y7)
EQUAL_SORT[A0](s(v27), s(v28)) → EQUAL_SORT[A0](v27, v28)
EQUAL_SORT[A35](cons(v29, v30), cons(v31, v32)) → AND(equal_sort[a0](v29, v31), equal_sort[a35](v30, v32))
EQUAL_SORT[A35](cons(v29, v30), cons(v31, v32)) → EQUAL_SORT[A0](v29, v31)
EQUAL_SORT[A35](cons(v29, v30), cons(v31, v32)) → EQUAL_SORT[A35](v30, v32)

The TRS R consists of the following rules:

rm'(x3, cons(y1, xs2)) → if2'(eq(x3, y1), x3, y1, xs2)
if2'(true_renamed, x4, y2, xs3) → true
if2'(false_renamed, x7, y5, xs4) → rm'(x7, xs4)
rm'(x8, nil) → false
min(cons(x', nil)) → x'
min(cons(x'', cons(y, xs'))) → if1(le(x'', y), x'', y, xs')
if1(true_renamed, x1, y', xs'') → min(cons(x1, xs''))
if1(false_renamed, x2, y'', xs1) → min(cons(y'', xs1))
rm(x3, cons(y1, xs2)) → if2(eq(x3, y1), x3, y1, xs2)
if2(true_renamed, x4, y2, xs3) → rm(x4, xs3)
eq(0, 0) → true_renamed
eq(0, s(y3)) → false_renamed
eq(s(x5), 0) → false_renamed
eq(s(x6), s(y4)) → eq(x6, y4)
if2(false_renamed, x7, y5, xs4) → cons(y5, rm(x7, xs4))
rm(x8, nil) → nil
le(0, y6) → true_renamed
le(s(x9), 0) → false_renamed
le(s(x10), s(y7)) → le(x10, y7)
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(v26)) → false
equal_sort[a0](s(v27), 0) → false
equal_sort[a0](s(v27), s(v28)) → equal_sort[a0](v27, v28)
equal_sort[a35](cons(v29, v30), cons(v31, v32)) → and(equal_sort[a0](v29, v31), equal_sort[a35](v30, v32))
equal_sort[a35](cons(v29, v30), nil) → false
equal_sort[a35](nil, cons(v33, v34)) → false
equal_sort[a35](nil, nil) → true
equal_sort[a43](true_renamed, true_renamed) → true
equal_sort[a43](true_renamed, false_renamed) → false
equal_sort[a43](false_renamed, true_renamed) → false
equal_sort[a43](false_renamed, false_renamed) → true
equal_sort[a61](witness_sort[a61], witness_sort[a61]) → true

The set Q consists of the following terms:

rm'(x0, cons(x1, x2))
if2'(true_renamed, x0, x1, x2)
if2'(false_renamed, x0, x1, x2)
rm'(x0, nil)
min(cons(x0, nil))
min(cons(x0, cons(x1, x2)))
if1(true_renamed, x0, x1, x2)
if1(false_renamed, x0, x1, x2)
rm(x0, cons(x1, x2))
if2(true_renamed, x0, x1, x2)
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
if2(false_renamed, 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[a35](cons(x0, x1), cons(x2, x3))
equal_sort[a35](cons(x0, x1), nil)
equal_sort[a35](nil, cons(x0, x1))
equal_sort[a35](nil, nil)
equal_sort[a43](true_renamed, true_renamed)
equal_sort[a43](true_renamed, false_renamed)
equal_sort[a43](false_renamed, true_renamed)
equal_sort[a43](false_renamed, false_renamed)
equal_sort[a61](witness_sort[a61], witness_sort[a61])

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

(52) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 7 SCCs with 5 less nodes.

(53) Complex Obligation (AND)

(54) Obligation:

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

EQUAL_SORT[A0](s(v27), s(v28)) → EQUAL_SORT[A0](v27, v28)

The TRS R consists of the following rules:

rm'(x3, cons(y1, xs2)) → if2'(eq(x3, y1), x3, y1, xs2)
if2'(true_renamed, x4, y2, xs3) → true
if2'(false_renamed, x7, y5, xs4) → rm'(x7, xs4)
rm'(x8, nil) → false
min(cons(x', nil)) → x'
min(cons(x'', cons(y, xs'))) → if1(le(x'', y), x'', y, xs')
if1(true_renamed, x1, y', xs'') → min(cons(x1, xs''))
if1(false_renamed, x2, y'', xs1) → min(cons(y'', xs1))
rm(x3, cons(y1, xs2)) → if2(eq(x3, y1), x3, y1, xs2)
if2(true_renamed, x4, y2, xs3) → rm(x4, xs3)
eq(0, 0) → true_renamed
eq(0, s(y3)) → false_renamed
eq(s(x5), 0) → false_renamed
eq(s(x6), s(y4)) → eq(x6, y4)
if2(false_renamed, x7, y5, xs4) → cons(y5, rm(x7, xs4))
rm(x8, nil) → nil
le(0, y6) → true_renamed
le(s(x9), 0) → false_renamed
le(s(x10), s(y7)) → le(x10, y7)
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(v26)) → false
equal_sort[a0](s(v27), 0) → false
equal_sort[a0](s(v27), s(v28)) → equal_sort[a0](v27, v28)
equal_sort[a35](cons(v29, v30), cons(v31, v32)) → and(equal_sort[a0](v29, v31), equal_sort[a35](v30, v32))
equal_sort[a35](cons(v29, v30), nil) → false
equal_sort[a35](nil, cons(v33, v34)) → false
equal_sort[a35](nil, nil) → true
equal_sort[a43](true_renamed, true_renamed) → true
equal_sort[a43](true_renamed, false_renamed) → false
equal_sort[a43](false_renamed, true_renamed) → false
equal_sort[a43](false_renamed, false_renamed) → true
equal_sort[a61](witness_sort[a61], witness_sort[a61]) → true

The set Q consists of the following terms:

rm'(x0, cons(x1, x2))
if2'(true_renamed, x0, x1, x2)
if2'(false_renamed, x0, x1, x2)
rm'(x0, nil)
min(cons(x0, nil))
min(cons(x0, cons(x1, x2)))
if1(true_renamed, x0, x1, x2)
if1(false_renamed, x0, x1, x2)
rm(x0, cons(x1, x2))
if2(true_renamed, x0, x1, x2)
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
if2(false_renamed, 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[a35](cons(x0, x1), cons(x2, x3))
equal_sort[a35](cons(x0, x1), nil)
equal_sort[a35](nil, cons(x0, x1))
equal_sort[a35](nil, nil)
equal_sort[a43](true_renamed, true_renamed)
equal_sort[a43](true_renamed, false_renamed)
equal_sort[a43](false_renamed, true_renamed)
equal_sort[a43](false_renamed, false_renamed)
equal_sort[a61](witness_sort[a61], witness_sort[a61])

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

(55) UsableRulesProof (EQUIVALENT transformation)

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

(56) Obligation:

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

EQUAL_SORT[A0](s(v27), s(v28)) → EQUAL_SORT[A0](v27, v28)

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

rm'(x0, cons(x1, x2))
if2'(true_renamed, x0, x1, x2)
if2'(false_renamed, x0, x1, x2)
rm'(x0, nil)
min(cons(x0, nil))
min(cons(x0, cons(x1, x2)))
if1(true_renamed, x0, x1, x2)
if1(false_renamed, x0, x1, x2)
rm(x0, cons(x1, x2))
if2(true_renamed, x0, x1, x2)
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
if2(false_renamed, 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[a35](cons(x0, x1), cons(x2, x3))
equal_sort[a35](cons(x0, x1), nil)
equal_sort[a35](nil, cons(x0, x1))
equal_sort[a35](nil, nil)
equal_sort[a43](true_renamed, true_renamed)
equal_sort[a43](true_renamed, false_renamed)
equal_sort[a43](false_renamed, true_renamed)
equal_sort[a43](false_renamed, false_renamed)
equal_sort[a61](witness_sort[a61], witness_sort[a61])

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

(57) QReductionProof (EQUIVALENT transformation)

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

rm'(x0, cons(x1, x2))
if2'(true_renamed, x0, x1, x2)
if2'(false_renamed, x0, x1, x2)
rm'(x0, nil)
min(cons(x0, nil))
min(cons(x0, cons(x1, x2)))
if1(true_renamed, x0, x1, x2)
if1(false_renamed, x0, x1, x2)
rm(x0, cons(x1, x2))
if2(true_renamed, x0, x1, x2)
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
if2(false_renamed, 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[a35](cons(x0, x1), cons(x2, x3))
equal_sort[a35](cons(x0, x1), nil)
equal_sort[a35](nil, cons(x0, x1))
equal_sort[a35](nil, nil)
equal_sort[a43](true_renamed, true_renamed)
equal_sort[a43](true_renamed, false_renamed)
equal_sort[a43](false_renamed, true_renamed)
equal_sort[a43](false_renamed, false_renamed)
equal_sort[a61](witness_sort[a61], witness_sort[a61])

(58) Obligation:

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

EQUAL_SORT[A0](s(v27), s(v28)) → EQUAL_SORT[A0](v27, v28)

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

(59) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

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

EQUAL_SORT[A0](s(v27), s(v28)) → EQUAL_SORT[A0](v27, v28)
The graph contains the following edges 1 > 1, 2 > 2

(60) YES

(61) Obligation:

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

EQUAL_SORT[A35](cons(v29, v30), cons(v31, v32)) → EQUAL_SORT[A35](v30, v32)

The TRS R consists of the following rules:

rm'(x3, cons(y1, xs2)) → if2'(eq(x3, y1), x3, y1, xs2)
if2'(true_renamed, x4, y2, xs3) → true
if2'(false_renamed, x7, y5, xs4) → rm'(x7, xs4)
rm'(x8, nil) → false
min(cons(x', nil)) → x'
min(cons(x'', cons(y, xs'))) → if1(le(x'', y), x'', y, xs')
if1(true_renamed, x1, y', xs'') → min(cons(x1, xs''))
if1(false_renamed, x2, y'', xs1) → min(cons(y'', xs1))
rm(x3, cons(y1, xs2)) → if2(eq(x3, y1), x3, y1, xs2)
if2(true_renamed, x4, y2, xs3) → rm(x4, xs3)
eq(0, 0) → true_renamed
eq(0, s(y3)) → false_renamed
eq(s(x5), 0) → false_renamed
eq(s(x6), s(y4)) → eq(x6, y4)
if2(false_renamed, x7, y5, xs4) → cons(y5, rm(x7, xs4))
rm(x8, nil) → nil
le(0, y6) → true_renamed
le(s(x9), 0) → false_renamed
le(s(x10), s(y7)) → le(x10, y7)
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(v26)) → false
equal_sort[a0](s(v27), 0) → false
equal_sort[a0](s(v27), s(v28)) → equal_sort[a0](v27, v28)
equal_sort[a35](cons(v29, v30), cons(v31, v32)) → and(equal_sort[a0](v29, v31), equal_sort[a35](v30, v32))
equal_sort[a35](cons(v29, v30), nil) → false
equal_sort[a35](nil, cons(v33, v34)) → false
equal_sort[a35](nil, nil) → true
equal_sort[a43](true_renamed, true_renamed) → true
equal_sort[a43](true_renamed, false_renamed) → false
equal_sort[a43](false_renamed, true_renamed) → false
equal_sort[a43](false_renamed, false_renamed) → true
equal_sort[a61](witness_sort[a61], witness_sort[a61]) → true

The set Q consists of the following terms:

rm'(x0, cons(x1, x2))
if2'(true_renamed, x0, x1, x2)
if2'(false_renamed, x0, x1, x2)
rm'(x0, nil)
min(cons(x0, nil))
min(cons(x0, cons(x1, x2)))
if1(true_renamed, x0, x1, x2)
if1(false_renamed, x0, x1, x2)
rm(x0, cons(x1, x2))
if2(true_renamed, x0, x1, x2)
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
if2(false_renamed, 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[a35](cons(x0, x1), cons(x2, x3))
equal_sort[a35](cons(x0, x1), nil)
equal_sort[a35](nil, cons(x0, x1))
equal_sort[a35](nil, nil)
equal_sort[a43](true_renamed, true_renamed)
equal_sort[a43](true_renamed, false_renamed)
equal_sort[a43](false_renamed, true_renamed)
equal_sort[a43](false_renamed, false_renamed)
equal_sort[a61](witness_sort[a61], witness_sort[a61])

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

(62) UsableRulesProof (EQUIVALENT transformation)

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

(63) Obligation:

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

EQUAL_SORT[A35](cons(v29, v30), cons(v31, v32)) → EQUAL_SORT[A35](v30, v32)

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

rm'(x0, cons(x1, x2))
if2'(true_renamed, x0, x1, x2)
if2'(false_renamed, x0, x1, x2)
rm'(x0, nil)
min(cons(x0, nil))
min(cons(x0, cons(x1, x2)))
if1(true_renamed, x0, x1, x2)
if1(false_renamed, x0, x1, x2)
rm(x0, cons(x1, x2))
if2(true_renamed, x0, x1, x2)
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
if2(false_renamed, 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[a35](cons(x0, x1), cons(x2, x3))
equal_sort[a35](cons(x0, x1), nil)
equal_sort[a35](nil, cons(x0, x1))
equal_sort[a35](nil, nil)
equal_sort[a43](true_renamed, true_renamed)
equal_sort[a43](true_renamed, false_renamed)
equal_sort[a43](false_renamed, true_renamed)
equal_sort[a43](false_renamed, false_renamed)
equal_sort[a61](witness_sort[a61], witness_sort[a61])

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

(64) QReductionProof (EQUIVALENT transformation)

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

rm'(x0, cons(x1, x2))
if2'(true_renamed, x0, x1, x2)
if2'(false_renamed, x0, x1, x2)
rm'(x0, nil)
min(cons(x0, nil))
min(cons(x0, cons(x1, x2)))
if1(true_renamed, x0, x1, x2)
if1(false_renamed, x0, x1, x2)
rm(x0, cons(x1, x2))
if2(true_renamed, x0, x1, x2)
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
if2(false_renamed, 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[a35](cons(x0, x1), cons(x2, x3))
equal_sort[a35](cons(x0, x1), nil)
equal_sort[a35](nil, cons(x0, x1))
equal_sort[a35](nil, nil)
equal_sort[a43](true_renamed, true_renamed)
equal_sort[a43](true_renamed, false_renamed)
equal_sort[a43](false_renamed, true_renamed)
equal_sort[a43](false_renamed, false_renamed)
equal_sort[a61](witness_sort[a61], witness_sort[a61])

(65) Obligation:

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

EQUAL_SORT[A35](cons(v29, v30), cons(v31, v32)) → EQUAL_SORT[A35](v30, v32)

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

(66) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

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

EQUAL_SORT[A35](cons(v29, v30), cons(v31, v32)) → EQUAL_SORT[A35](v30, v32)
The graph contains the following edges 1 > 1, 2 > 2

(67) YES

(68) Obligation:

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

LE(s(x10), s(y7)) → LE(x10, y7)

The TRS R consists of the following rules:

rm'(x3, cons(y1, xs2)) → if2'(eq(x3, y1), x3, y1, xs2)
if2'(true_renamed, x4, y2, xs3) → true
if2'(false_renamed, x7, y5, xs4) → rm'(x7, xs4)
rm'(x8, nil) → false
min(cons(x', nil)) → x'
min(cons(x'', cons(y, xs'))) → if1(le(x'', y), x'', y, xs')
if1(true_renamed, x1, y', xs'') → min(cons(x1, xs''))
if1(false_renamed, x2, y'', xs1) → min(cons(y'', xs1))
rm(x3, cons(y1, xs2)) → if2(eq(x3, y1), x3, y1, xs2)
if2(true_renamed, x4, y2, xs3) → rm(x4, xs3)
eq(0, 0) → true_renamed
eq(0, s(y3)) → false_renamed
eq(s(x5), 0) → false_renamed
eq(s(x6), s(y4)) → eq(x6, y4)
if2(false_renamed, x7, y5, xs4) → cons(y5, rm(x7, xs4))
rm(x8, nil) → nil
le(0, y6) → true_renamed
le(s(x9), 0) → false_renamed
le(s(x10), s(y7)) → le(x10, y7)
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(v26)) → false
equal_sort[a0](s(v27), 0) → false
equal_sort[a0](s(v27), s(v28)) → equal_sort[a0](v27, v28)
equal_sort[a35](cons(v29, v30), cons(v31, v32)) → and(equal_sort[a0](v29, v31), equal_sort[a35](v30, v32))
equal_sort[a35](cons(v29, v30), nil) → false
equal_sort[a35](nil, cons(v33, v34)) → false
equal_sort[a35](nil, nil) → true
equal_sort[a43](true_renamed, true_renamed) → true
equal_sort[a43](true_renamed, false_renamed) → false
equal_sort[a43](false_renamed, true_renamed) → false
equal_sort[a43](false_renamed, false_renamed) → true
equal_sort[a61](witness_sort[a61], witness_sort[a61]) → true

The set Q consists of the following terms:

rm'(x0, cons(x1, x2))
if2'(true_renamed, x0, x1, x2)
if2'(false_renamed, x0, x1, x2)
rm'(x0, nil)
min(cons(x0, nil))
min(cons(x0, cons(x1, x2)))
if1(true_renamed, x0, x1, x2)
if1(false_renamed, x0, x1, x2)
rm(x0, cons(x1, x2))
if2(true_renamed, x0, x1, x2)
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
if2(false_renamed, 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[a35](cons(x0, x1), cons(x2, x3))
equal_sort[a35](cons(x0, x1), nil)
equal_sort[a35](nil, cons(x0, x1))
equal_sort[a35](nil, nil)
equal_sort[a43](true_renamed, true_renamed)
equal_sort[a43](true_renamed, false_renamed)
equal_sort[a43](false_renamed, true_renamed)
equal_sort[a43](false_renamed, false_renamed)
equal_sort[a61](witness_sort[a61], witness_sort[a61])

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

(69) UsableRulesProof (EQUIVALENT transformation)

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

(70) Obligation:

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

LE(s(x10), s(y7)) → LE(x10, y7)

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

rm'(x0, cons(x1, x2))
if2'(true_renamed, x0, x1, x2)
if2'(false_renamed, x0, x1, x2)
rm'(x0, nil)
min(cons(x0, nil))
min(cons(x0, cons(x1, x2)))
if1(true_renamed, x0, x1, x2)
if1(false_renamed, x0, x1, x2)
rm(x0, cons(x1, x2))
if2(true_renamed, x0, x1, x2)
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
if2(false_renamed, 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[a35](cons(x0, x1), cons(x2, x3))
equal_sort[a35](cons(x0, x1), nil)
equal_sort[a35](nil, cons(x0, x1))
equal_sort[a35](nil, nil)
equal_sort[a43](true_renamed, true_renamed)
equal_sort[a43](true_renamed, false_renamed)
equal_sort[a43](false_renamed, true_renamed)
equal_sort[a43](false_renamed, false_renamed)
equal_sort[a61](witness_sort[a61], witness_sort[a61])

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

(71) QReductionProof (EQUIVALENT transformation)

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

rm'(x0, cons(x1, x2))
if2'(true_renamed, x0, x1, x2)
if2'(false_renamed, x0, x1, x2)
rm'(x0, nil)
min(cons(x0, nil))
min(cons(x0, cons(x1, x2)))
if1(true_renamed, x0, x1, x2)
if1(false_renamed, x0, x1, x2)
rm(x0, cons(x1, x2))
if2(true_renamed, x0, x1, x2)
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
if2(false_renamed, 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[a35](cons(x0, x1), cons(x2, x3))
equal_sort[a35](cons(x0, x1), nil)
equal_sort[a35](nil, cons(x0, x1))
equal_sort[a35](nil, nil)
equal_sort[a43](true_renamed, true_renamed)
equal_sort[a43](true_renamed, false_renamed)
equal_sort[a43](false_renamed, true_renamed)
equal_sort[a43](false_renamed, false_renamed)
equal_sort[a61](witness_sort[a61], witness_sort[a61])

(72) Obligation:

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

LE(s(x10), s(y7)) → LE(x10, y7)

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

(73) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

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

LE(s(x10), s(y7)) → LE(x10, y7)
The graph contains the following edges 1 > 1, 2 > 2

(74) YES

(75) Obligation:

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

EQ(s(x6), s(y4)) → EQ(x6, y4)

The TRS R consists of the following rules:

rm'(x3, cons(y1, xs2)) → if2'(eq(x3, y1), x3, y1, xs2)
if2'(true_renamed, x4, y2, xs3) → true
if2'(false_renamed, x7, y5, xs4) → rm'(x7, xs4)
rm'(x8, nil) → false
min(cons(x', nil)) → x'
min(cons(x'', cons(y, xs'))) → if1(le(x'', y), x'', y, xs')
if1(true_renamed, x1, y', xs'') → min(cons(x1, xs''))
if1(false_renamed, x2, y'', xs1) → min(cons(y'', xs1))
rm(x3, cons(y1, xs2)) → if2(eq(x3, y1), x3, y1, xs2)
if2(true_renamed, x4, y2, xs3) → rm(x4, xs3)
eq(0, 0) → true_renamed
eq(0, s(y3)) → false_renamed
eq(s(x5), 0) → false_renamed
eq(s(x6), s(y4)) → eq(x6, y4)
if2(false_renamed, x7, y5, xs4) → cons(y5, rm(x7, xs4))
rm(x8, nil) → nil
le(0, y6) → true_renamed
le(s(x9), 0) → false_renamed
le(s(x10), s(y7)) → le(x10, y7)
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(v26)) → false
equal_sort[a0](s(v27), 0) → false
equal_sort[a0](s(v27), s(v28)) → equal_sort[a0](v27, v28)
equal_sort[a35](cons(v29, v30), cons(v31, v32)) → and(equal_sort[a0](v29, v31), equal_sort[a35](v30, v32))
equal_sort[a35](cons(v29, v30), nil) → false
equal_sort[a35](nil, cons(v33, v34)) → false
equal_sort[a35](nil, nil) → true
equal_sort[a43](true_renamed, true_renamed) → true
equal_sort[a43](true_renamed, false_renamed) → false
equal_sort[a43](false_renamed, true_renamed) → false
equal_sort[a43](false_renamed, false_renamed) → true
equal_sort[a61](witness_sort[a61], witness_sort[a61]) → true

The set Q consists of the following terms:

rm'(x0, cons(x1, x2))
if2'(true_renamed, x0, x1, x2)
if2'(false_renamed, x0, x1, x2)
rm'(x0, nil)
min(cons(x0, nil))
min(cons(x0, cons(x1, x2)))
if1(true_renamed, x0, x1, x2)
if1(false_renamed, x0, x1, x2)
rm(x0, cons(x1, x2))
if2(true_renamed, x0, x1, x2)
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
if2(false_renamed, 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[a35](cons(x0, x1), cons(x2, x3))
equal_sort[a35](cons(x0, x1), nil)
equal_sort[a35](nil, cons(x0, x1))
equal_sort[a35](nil, nil)
equal_sort[a43](true_renamed, true_renamed)
equal_sort[a43](true_renamed, false_renamed)
equal_sort[a43](false_renamed, true_renamed)
equal_sort[a43](false_renamed, false_renamed)
equal_sort[a61](witness_sort[a61], witness_sort[a61])

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

(76) UsableRulesProof (EQUIVALENT transformation)

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

(77) Obligation:

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

EQ(s(x6), s(y4)) → EQ(x6, y4)

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

rm'(x0, cons(x1, x2))
if2'(true_renamed, x0, x1, x2)
if2'(false_renamed, x0, x1, x2)
rm'(x0, nil)
min(cons(x0, nil))
min(cons(x0, cons(x1, x2)))
if1(true_renamed, x0, x1, x2)
if1(false_renamed, x0, x1, x2)
rm(x0, cons(x1, x2))
if2(true_renamed, x0, x1, x2)
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
if2(false_renamed, 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[a35](cons(x0, x1), cons(x2, x3))
equal_sort[a35](cons(x0, x1), nil)
equal_sort[a35](nil, cons(x0, x1))
equal_sort[a35](nil, nil)
equal_sort[a43](true_renamed, true_renamed)
equal_sort[a43](true_renamed, false_renamed)
equal_sort[a43](false_renamed, true_renamed)
equal_sort[a43](false_renamed, false_renamed)
equal_sort[a61](witness_sort[a61], witness_sort[a61])

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

(78) QReductionProof (EQUIVALENT transformation)

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

rm'(x0, cons(x1, x2))
if2'(true_renamed, x0, x1, x2)
if2'(false_renamed, x0, x1, x2)
rm'(x0, nil)
min(cons(x0, nil))
min(cons(x0, cons(x1, x2)))
if1(true_renamed, x0, x1, x2)
if1(false_renamed, x0, x1, x2)
rm(x0, cons(x1, x2))
if2(true_renamed, x0, x1, x2)
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
if2(false_renamed, 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[a35](cons(x0, x1), cons(x2, x3))
equal_sort[a35](cons(x0, x1), nil)
equal_sort[a35](nil, cons(x0, x1))
equal_sort[a35](nil, nil)
equal_sort[a43](true_renamed, true_renamed)
equal_sort[a43](true_renamed, false_renamed)
equal_sort[a43](false_renamed, true_renamed)
equal_sort[a43](false_renamed, false_renamed)
equal_sort[a61](witness_sort[a61], witness_sort[a61])

(79) Obligation:

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

EQ(s(x6), s(y4)) → EQ(x6, y4)

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

(80) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

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

EQ(s(x6), s(y4)) → EQ(x6, y4)
The graph contains the following edges 1 > 1, 2 > 2

(81) YES

(82) Obligation:

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

IF2(true_renamed, x4, y2, xs3) → RM(x4, xs3)
RM(x3, cons(y1, xs2)) → IF2(eq(x3, y1), x3, y1, xs2)
IF2(false_renamed, x7, y5, xs4) → RM(x7, xs4)

The TRS R consists of the following rules:

rm'(x3, cons(y1, xs2)) → if2'(eq(x3, y1), x3, y1, xs2)
if2'(true_renamed, x4, y2, xs3) → true
if2'(false_renamed, x7, y5, xs4) → rm'(x7, xs4)
rm'(x8, nil) → false
min(cons(x', nil)) → x'
min(cons(x'', cons(y, xs'))) → if1(le(x'', y), x'', y, xs')
if1(true_renamed, x1, y', xs'') → min(cons(x1, xs''))
if1(false_renamed, x2, y'', xs1) → min(cons(y'', xs1))
rm(x3, cons(y1, xs2)) → if2(eq(x3, y1), x3, y1, xs2)
if2(true_renamed, x4, y2, xs3) → rm(x4, xs3)
eq(0, 0) → true_renamed
eq(0, s(y3)) → false_renamed
eq(s(x5), 0) → false_renamed
eq(s(x6), s(y4)) → eq(x6, y4)
if2(false_renamed, x7, y5, xs4) → cons(y5, rm(x7, xs4))
rm(x8, nil) → nil
le(0, y6) → true_renamed
le(s(x9), 0) → false_renamed
le(s(x10), s(y7)) → le(x10, y7)
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(v26)) → false
equal_sort[a0](s(v27), 0) → false
equal_sort[a0](s(v27), s(v28)) → equal_sort[a0](v27, v28)
equal_sort[a35](cons(v29, v30), cons(v31, v32)) → and(equal_sort[a0](v29, v31), equal_sort[a35](v30, v32))
equal_sort[a35](cons(v29, v30), nil) → false
equal_sort[a35](nil, cons(v33, v34)) → false
equal_sort[a35](nil, nil) → true
equal_sort[a43](true_renamed, true_renamed) → true
equal_sort[a43](true_renamed, false_renamed) → false
equal_sort[a43](false_renamed, true_renamed) → false
equal_sort[a43](false_renamed, false_renamed) → true
equal_sort[a61](witness_sort[a61], witness_sort[a61]) → true

The set Q consists of the following terms:

rm'(x0, cons(x1, x2))
if2'(true_renamed, x0, x1, x2)
if2'(false_renamed, x0, x1, x2)
rm'(x0, nil)
min(cons(x0, nil))
min(cons(x0, cons(x1, x2)))
if1(true_renamed, x0, x1, x2)
if1(false_renamed, x0, x1, x2)
rm(x0, cons(x1, x2))
if2(true_renamed, x0, x1, x2)
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
if2(false_renamed, 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[a35](cons(x0, x1), cons(x2, x3))
equal_sort[a35](cons(x0, x1), nil)
equal_sort[a35](nil, cons(x0, x1))
equal_sort[a35](nil, nil)
equal_sort[a43](true_renamed, true_renamed)
equal_sort[a43](true_renamed, false_renamed)
equal_sort[a43](false_renamed, true_renamed)
equal_sort[a43](false_renamed, false_renamed)
equal_sort[a61](witness_sort[a61], witness_sort[a61])

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

(83) UsableRulesProof (EQUIVALENT transformation)

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

(84) Obligation:

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

IF2(true_renamed, x4, y2, xs3) → RM(x4, xs3)
RM(x3, cons(y1, xs2)) → IF2(eq(x3, y1), x3, y1, xs2)
IF2(false_renamed, x7, y5, xs4) → RM(x7, xs4)

The TRS R consists of the following rules:

eq(0, 0) → true_renamed
eq(0, s(y3)) → false_renamed
eq(s(x5), 0) → false_renamed
eq(s(x6), s(y4)) → eq(x6, y4)

The set Q consists of the following terms:

rm'(x0, cons(x1, x2))
if2'(true_renamed, x0, x1, x2)
if2'(false_renamed, x0, x1, x2)
rm'(x0, nil)
min(cons(x0, nil))
min(cons(x0, cons(x1, x2)))
if1(true_renamed, x0, x1, x2)
if1(false_renamed, x0, x1, x2)
rm(x0, cons(x1, x2))
if2(true_renamed, x0, x1, x2)
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
if2(false_renamed, 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[a35](cons(x0, x1), cons(x2, x3))
equal_sort[a35](cons(x0, x1), nil)
equal_sort[a35](nil, cons(x0, x1))
equal_sort[a35](nil, nil)
equal_sort[a43](true_renamed, true_renamed)
equal_sort[a43](true_renamed, false_renamed)
equal_sort[a43](false_renamed, true_renamed)
equal_sort[a43](false_renamed, false_renamed)
equal_sort[a61](witness_sort[a61], witness_sort[a61])

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

(85) QReductionProof (EQUIVALENT transformation)

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

rm'(x0, cons(x1, x2))
if2'(true_renamed, x0, x1, x2)
if2'(false_renamed, x0, x1, x2)
rm'(x0, nil)
min(cons(x0, nil))
min(cons(x0, cons(x1, x2)))
if1(true_renamed, x0, x1, x2)
if1(false_renamed, x0, x1, x2)
rm(x0, cons(x1, x2))
if2(true_renamed, x0, x1, x2)
if2(false_renamed, 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[a35](cons(x0, x1), cons(x2, x3))
equal_sort[a35](cons(x0, x1), nil)
equal_sort[a35](nil, cons(x0, x1))
equal_sort[a35](nil, nil)
equal_sort[a43](true_renamed, true_renamed)
equal_sort[a43](true_renamed, false_renamed)
equal_sort[a43](false_renamed, true_renamed)
equal_sort[a43](false_renamed, false_renamed)
equal_sort[a61](witness_sort[a61], witness_sort[a61])

(86) Obligation:

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

IF2(true_renamed, x4, y2, xs3) → RM(x4, xs3)
RM(x3, cons(y1, xs2)) → IF2(eq(x3, y1), x3, y1, xs2)
IF2(false_renamed, x7, y5, xs4) → RM(x7, xs4)

The TRS R consists of the following rules:

eq(0, 0) → true_renamed
eq(0, s(y3)) → false_renamed
eq(s(x5), 0) → false_renamed
eq(s(x6), s(y4)) → eq(x6, y4)

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.

(87) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

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

RM(x3, cons(y1, xs2)) → IF2(eq(x3, y1), x3, y1, xs2)
The graph contains the following edges 1 >= 2, 2 > 3, 2 > 4
IF2(true_renamed, x4, y2, xs3) → RM(x4, xs3)
The graph contains the following edges 2 >= 1, 4 >= 2
IF2(false_renamed, x7, y5, xs4) → RM(x7, xs4)
The graph contains the following edges 2 >= 1, 4 >= 2

(88) YES

(89) Obligation:

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

IF1(true_renamed, x1, y', xs'') → MIN(cons(x1, xs''))
MIN(cons(x'', cons(y, xs'))) → IF1(le(x'', y), x'', y, xs')
IF1(false_renamed, x2, y'', xs1) → MIN(cons(y'', xs1))

The TRS R consists of the following rules:

rm'(x3, cons(y1, xs2)) → if2'(eq(x3, y1), x3, y1, xs2)
if2'(true_renamed, x4, y2, xs3) → true
if2'(false_renamed, x7, y5, xs4) → rm'(x7, xs4)
rm'(x8, nil) → false
min(cons(x', nil)) → x'
min(cons(x'', cons(y, xs'))) → if1(le(x'', y), x'', y, xs')
if1(true_renamed, x1, y', xs'') → min(cons(x1, xs''))
if1(false_renamed, x2, y'', xs1) → min(cons(y'', xs1))
rm(x3, cons(y1, xs2)) → if2(eq(x3, y1), x3, y1, xs2)
if2(true_renamed, x4, y2, xs3) → rm(x4, xs3)
eq(0, 0) → true_renamed
eq(0, s(y3)) → false_renamed
eq(s(x5), 0) → false_renamed
eq(s(x6), s(y4)) → eq(x6, y4)
if2(false_renamed, x7, y5, xs4) → cons(y5, rm(x7, xs4))
rm(x8, nil) → nil
le(0, y6) → true_renamed
le(s(x9), 0) → false_renamed
le(s(x10), s(y7)) → le(x10, y7)
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(v26)) → false
equal_sort[a0](s(v27), 0) → false
equal_sort[a0](s(v27), s(v28)) → equal_sort[a0](v27, v28)
equal_sort[a35](cons(v29, v30), cons(v31, v32)) → and(equal_sort[a0](v29, v31), equal_sort[a35](v30, v32))
equal_sort[a35](cons(v29, v30), nil) → false
equal_sort[a35](nil, cons(v33, v34)) → false
equal_sort[a35](nil, nil) → true
equal_sort[a43](true_renamed, true_renamed) → true
equal_sort[a43](true_renamed, false_renamed) → false
equal_sort[a43](false_renamed, true_renamed) → false
equal_sort[a43](false_renamed, false_renamed) → true
equal_sort[a61](witness_sort[a61], witness_sort[a61]) → true

The set Q consists of the following terms:

rm'(x0, cons(x1, x2))
if2'(true_renamed, x0, x1, x2)
if2'(false_renamed, x0, x1, x2)
rm'(x0, nil)
min(cons(x0, nil))
min(cons(x0, cons(x1, x2)))
if1(true_renamed, x0, x1, x2)
if1(false_renamed, x0, x1, x2)
rm(x0, cons(x1, x2))
if2(true_renamed, x0, x1, x2)
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
if2(false_renamed, 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[a35](cons(x0, x1), cons(x2, x3))
equal_sort[a35](cons(x0, x1), nil)
equal_sort[a35](nil, cons(x0, x1))
equal_sort[a35](nil, nil)
equal_sort[a43](true_renamed, true_renamed)
equal_sort[a43](true_renamed, false_renamed)
equal_sort[a43](false_renamed, true_renamed)
equal_sort[a43](false_renamed, false_renamed)
equal_sort[a61](witness_sort[a61], witness_sort[a61])

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

(90) UsableRulesProof (EQUIVALENT transformation)

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

(91) Obligation:

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

IF1(true_renamed, x1, y', xs'') → MIN(cons(x1, xs''))
MIN(cons(x'', cons(y, xs'))) → IF1(le(x'', y), x'', y, xs')
IF1(false_renamed, x2, y'', xs1) → MIN(cons(y'', xs1))

The TRS R consists of the following rules:

le(0, y6) → true_renamed
le(s(x9), 0) → false_renamed
le(s(x10), s(y7)) → le(x10, y7)

The set Q consists of the following terms:

rm'(x0, cons(x1, x2))
if2'(true_renamed, x0, x1, x2)
if2'(false_renamed, x0, x1, x2)
rm'(x0, nil)
min(cons(x0, nil))
min(cons(x0, cons(x1, x2)))
if1(true_renamed, x0, x1, x2)
if1(false_renamed, x0, x1, x2)
rm(x0, cons(x1, x2))
if2(true_renamed, x0, x1, x2)
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
if2(false_renamed, 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[a35](cons(x0, x1), cons(x2, x3))
equal_sort[a35](cons(x0, x1), nil)
equal_sort[a35](nil, cons(x0, x1))
equal_sort[a35](nil, nil)
equal_sort[a43](true_renamed, true_renamed)
equal_sort[a43](true_renamed, false_renamed)
equal_sort[a43](false_renamed, true_renamed)
equal_sort[a43](false_renamed, false_renamed)
equal_sort[a61](witness_sort[a61], witness_sort[a61])

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

(92) QReductionProof (EQUIVALENT transformation)

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

rm'(x0, cons(x1, x2))
if2'(true_renamed, x0, x1, x2)
if2'(false_renamed, x0, x1, x2)
rm'(x0, nil)
min(cons(x0, nil))
min(cons(x0, cons(x1, x2)))
if1(true_renamed, x0, x1, x2)
if1(false_renamed, x0, x1, x2)
rm(x0, cons(x1, x2))
if2(true_renamed, x0, x1, x2)
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
if2(false_renamed, 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[a35](cons(x0, x1), cons(x2, x3))
equal_sort[a35](cons(x0, x1), nil)
equal_sort[a35](nil, cons(x0, x1))
equal_sort[a35](nil, nil)
equal_sort[a43](true_renamed, true_renamed)
equal_sort[a43](true_renamed, false_renamed)
equal_sort[a43](false_renamed, true_renamed)
equal_sort[a43](false_renamed, false_renamed)
equal_sort[a61](witness_sort[a61], witness_sort[a61])

(93) Obligation:

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

IF1(true_renamed, x1, y', xs'') → MIN(cons(x1, xs''))
MIN(cons(x'', cons(y, xs'))) → IF1(le(x'', y), x'', y, xs')
IF1(false_renamed, x2, y'', xs1) → MIN(cons(y'', xs1))

The TRS R consists of the following rules:

le(0, y6) → true_renamed
le(s(x9), 0) → false_renamed
le(s(x10), s(y7)) → le(x10, y7)

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.

(94) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04,JAR06].

The following pairs can be oriented strictly and are deleted.

IF1(true_renamed, x1, y', xs'') → MIN(cons(x1, xs''))
MIN(cons(x'', cons(y, xs'))) → IF1(le(x'', y), x'', y, xs')
IF1(false_renamed, x2, y'', xs1) → MIN(cons(y'', xs1))

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
IF1(x1, x2, x3, x4) = IF1(x4)
MIN(x1) = x1
cons(x1, x2) = cons(x2)

Knuth-Bendix order [KBO] with precedence:

trivial

and weight map:

dummyConstant=1
IF1_1=3
cons_1=2

The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
none

(95) Obligation:

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

le(0, y6) → true_renamed
le(s(x9), 0) → false_renamed
le(s(x10), s(y7)) → le(x10, y7)

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.

(96) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(97) YES

(98) Obligation:

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

IF2'(false_renamed, x7, y5, xs4) → RM'(x7, xs4)
RM'(x3, cons(y1, xs2)) → IF2'(eq(x3, y1), x3, y1, xs2)

The TRS R consists of the following rules:

rm'(x3, cons(y1, xs2)) → if2'(eq(x3, y1), x3, y1, xs2)
if2'(true_renamed, x4, y2, xs3) → true
if2'(false_renamed, x7, y5, xs4) → rm'(x7, xs4)
rm'(x8, nil) → false
min(cons(x', nil)) → x'
min(cons(x'', cons(y, xs'))) → if1(le(x'', y), x'', y, xs')
if1(true_renamed, x1, y', xs'') → min(cons(x1, xs''))
if1(false_renamed, x2, y'', xs1) → min(cons(y'', xs1))
rm(x3, cons(y1, xs2)) → if2(eq(x3, y1), x3, y1, xs2)
if2(true_renamed, x4, y2, xs3) → rm(x4, xs3)
eq(0, 0) → true_renamed
eq(0, s(y3)) → false_renamed
eq(s(x5), 0) → false_renamed
eq(s(x6), s(y4)) → eq(x6, y4)
if2(false_renamed, x7, y5, xs4) → cons(y5, rm(x7, xs4))
rm(x8, nil) → nil
le(0, y6) → true_renamed
le(s(x9), 0) → false_renamed
le(s(x10), s(y7)) → le(x10, y7)
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(v26)) → false
equal_sort[a0](s(v27), 0) → false
equal_sort[a0](s(v27), s(v28)) → equal_sort[a0](v27, v28)
equal_sort[a35](cons(v29, v30), cons(v31, v32)) → and(equal_sort[a0](v29, v31), equal_sort[a35](v30, v32))
equal_sort[a35](cons(v29, v30), nil) → false
equal_sort[a35](nil, cons(v33, v34)) → false
equal_sort[a35](nil, nil) → true
equal_sort[a43](true_renamed, true_renamed) → true
equal_sort[a43](true_renamed, false_renamed) → false
equal_sort[a43](false_renamed, true_renamed) → false
equal_sort[a43](false_renamed, false_renamed) → true
equal_sort[a61](witness_sort[a61], witness_sort[a61]) → true

The set Q consists of the following terms:

rm'(x0, cons(x1, x2))
if2'(true_renamed, x0, x1, x2)
if2'(false_renamed, x0, x1, x2)
rm'(x0, nil)
min(cons(x0, nil))
min(cons(x0, cons(x1, x2)))
if1(true_renamed, x0, x1, x2)
if1(false_renamed, x0, x1, x2)
rm(x0, cons(x1, x2))
if2(true_renamed, x0, x1, x2)
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
if2(false_renamed, 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[a35](cons(x0, x1), cons(x2, x3))
equal_sort[a35](cons(x0, x1), nil)
equal_sort[a35](nil, cons(x0, x1))
equal_sort[a35](nil, nil)
equal_sort[a43](true_renamed, true_renamed)
equal_sort[a43](true_renamed, false_renamed)
equal_sort[a43](false_renamed, true_renamed)
equal_sort[a43](false_renamed, false_renamed)
equal_sort[a61](witness_sort[a61], witness_sort[a61])

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

(99) UsableRulesProof (EQUIVALENT transformation)

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

(100) Obligation:

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

IF2'(false_renamed, x7, y5, xs4) → RM'(x7, xs4)
RM'(x3, cons(y1, xs2)) → IF2'(eq(x3, y1), x3, y1, xs2)

The TRS R consists of the following rules:

eq(0, 0) → true_renamed
eq(0, s(y3)) → false_renamed
eq(s(x5), 0) → false_renamed
eq(s(x6), s(y4)) → eq(x6, y4)

The set Q consists of the following terms:

rm'(x0, cons(x1, x2))
if2'(true_renamed, x0, x1, x2)
if2'(false_renamed, x0, x1, x2)
rm'(x0, nil)
min(cons(x0, nil))
min(cons(x0, cons(x1, x2)))
if1(true_renamed, x0, x1, x2)
if1(false_renamed, x0, x1, x2)
rm(x0, cons(x1, x2))
if2(true_renamed, x0, x1, x2)
eq(0, 0)
eq(0, s(x0))
eq(s(x0), 0)
eq(s(x0), s(x1))
if2(false_renamed, 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[a35](cons(x0, x1), cons(x2, x3))
equal_sort[a35](cons(x0, x1), nil)
equal_sort[a35](nil, cons(x0, x1))
equal_sort[a35](nil, nil)
equal_sort[a43](true_renamed, true_renamed)
equal_sort[a43](true_renamed, false_renamed)
equal_sort[a43](false_renamed, true_renamed)
equal_sort[a43](false_renamed, false_renamed)
equal_sort[a61](witness_sort[a61], witness_sort[a61])

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

(101) QReductionProof (EQUIVALENT transformation)

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

rm'(x0, cons(x1, x2))
if2'(true_renamed, x0, x1, x2)
if2'(false_renamed, x0, x1, x2)
rm'(x0, nil)
min(cons(x0, nil))
min(cons(x0, cons(x1, x2)))
if1(true_renamed, x0, x1, x2)
if1(false_renamed, x0, x1, x2)
rm(x0, cons(x1, x2))
if2(true_renamed, x0, x1, x2)
if2(false_renamed, 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[a35](cons(x0, x1), cons(x2, x3))
equal_sort[a35](cons(x0, x1), nil)
equal_sort[a35](nil, cons(x0, x1))
equal_sort[a35](nil, nil)
equal_sort[a43](true_renamed, true_renamed)
equal_sort[a43](true_renamed, false_renamed)
equal_sort[a43](false_renamed, true_renamed)
equal_sort[a43](false_renamed, false_renamed)
equal_sort[a61](witness_sort[a61], witness_sort[a61])

(102) Obligation:

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

IF2'(false_renamed, x7, y5, xs4) → RM'(x7, xs4)
RM'(x3, cons(y1, xs2)) → IF2'(eq(x3, y1), x3, y1, xs2)

The TRS R consists of the following rules:

eq(0, 0) → true_renamed
eq(0, s(y3)) → false_renamed
eq(s(x5), 0) → false_renamed
eq(s(x6), s(y4)) → eq(x6, y4)

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.

(103) QDPSizeChangeProof (EQUIVALENT transformation)

By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.

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

RM'(x3, cons(y1, xs2)) → IF2'(eq(x3, y1), x3, y1, xs2)
The graph contains the following edges 1 >= 2, 2 > 3, 2 > 4
IF2'(false_renamed, x7, y5, xs4) → RM'(x7, xs4)
The graph contains the following edges 2 >= 1, 4 >= 2