Termination w.r.t. Q proof of /tmp/tmpXyMu1M/aprove3.trs

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

0 QTRS

↳1 Overlay + Local Confluence (⇔)

↳2 QTRS

↳3 DependencyPairsProof (⇔)

↳4 QDP

↳5 DependencyGraphProof (⇔)

↳6 AND

  ↳7 QDP

    ↳8 UsableRulesProof (⇔)

    ↳9 QDP

    ↳10 QReductionProof (⇔)

    ↳11 QDP

    ↳12 QDPSizeChangeProof (⇔)

    ↳13 TRUE

  ↳14 QDP

    ↳15 UsableRulesProof (⇔)

    ↳16 QDP

      ↳17 Instantiation (⇔)

        ↳18 QDP

      ↳19 Rewriting (⇔)

        ↳20 QDP

        ↳21 UsableRulesProof (⇔)

        ↳22 QDP

        ↳23 Instantiation (⇔)

        ↳24 QDP

        ↳25 Narrowing (⇔)

        ↳26 QDP

        ↳27 DependencyGraphProof (⇔)

        ↳28 QDP

        ↳29 UsableRulesProof (⇔)

        ↳30 QDP

        ↳31 Narrowing (⇔)

        ↳32 QDP

        ↳33 DependencyGraphProof (⇔)

        ↳34 QDP

        ↳35 UsableRulesProof (⇔)

        ↳36 QDP

        ↳37 Instantiation (⇔)

        ↳38 QDP

        ↳39 Instantiation (⇔)

        ↳40 QDP

        ↳41 Instantiation (⇔)

        ↳42 QDP

        ↳43 ForwardInstantiation (⇔)

        ↳44 QDP

        ↳45 QDPOrderProof (⇔)

        ↳46 QDP

        ↳47 DependencyGraphProof (⇔)

        ↳48 TRUE

(0) Obligation:

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

function(iszero, 0, dummy, dummy2) → true
function(iszero, s(x), dummy, dummy2) → false
function(p, 0, dummy, dummy2) → 0
function(p, s(0), dummy, dummy2) → 0
function(p, s(s(x)), dummy, dummy2) → s(function(p, s(x), x, x))
function(plus, dummy, x, y) → function(if, function(iszero, x, x, x), x, y)
function(if, true, x, y) → y
function(if, false, x, y) → function(plus, function(third, x, y, y), function(p, x, x, y), s(y))
function(third, x, y, z) → z

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:

function(iszero, 0, dummy, dummy2) → true
function(iszero, s(x), dummy, dummy2) → false
function(p, 0, dummy, dummy2) → 0
function(p, s(0), dummy, dummy2) → 0
function(p, s(s(x)), dummy, dummy2) → s(function(p, s(x), x, x))
function(plus, dummy, x, y) → function(if, function(iszero, x, x, x), x, y)
function(if, true, x, y) → y
function(if, false, x, y) → function(plus, function(third, x, y, y), function(p, x, x, y), s(y))
function(third, x, y, z) → z

The set Q consists of the following terms:

function(iszero, 0, x0, x1)
function(iszero, s(x0), x1, x2)
function(p, 0, x0, x1)
function(p, s(0), x0, x1)
function(p, s(s(x0)), x1, x2)
function(plus, x0, x1, x2)
function(if, true, x0, x1)
function(if, false, x0, x1)
function(third, 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:

FUNCTION(p, s(s(x)), dummy, dummy2) → FUNCTION(p, s(x), x, x)
FUNCTION(plus, dummy, x, y) → FUNCTION(if, function(iszero, x, x, x), x, y)
FUNCTION(plus, dummy, x, y) → FUNCTION(iszero, x, x, x)
FUNCTION(if, false, x, y) → FUNCTION(plus, function(third, x, y, y), function(p, x, x, y), s(y))
FUNCTION(if, false, x, y) → FUNCTION(third, x, y, y)
FUNCTION(if, false, x, y) → FUNCTION(p, x, x, y)

The TRS R consists of the following rules:

function(iszero, 0, dummy, dummy2) → true
function(iszero, s(x), dummy, dummy2) → false
function(p, 0, dummy, dummy2) → 0
function(p, s(0), dummy, dummy2) → 0
function(p, s(s(x)), dummy, dummy2) → s(function(p, s(x), x, x))
function(plus, dummy, x, y) → function(if, function(iszero, x, x, x), x, y)
function(if, true, x, y) → y
function(if, false, x, y) → function(plus, function(third, x, y, y), function(p, x, x, y), s(y))
function(third, x, y, z) → z

The set Q consists of the following terms:

function(iszero, 0, x0, x1)
function(iszero, s(x0), x1, x2)
function(p, 0, x0, x1)
function(p, s(0), x0, x1)
function(p, s(s(x0)), x1, x2)
function(plus, x0, x1, x2)
function(if, true, x0, x1)
function(if, false, x0, x1)
function(third, 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 2 SCCs with 3 less nodes.

(6) Complex Obligation (AND)

(7) Obligation:

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

FUNCTION(p, s(s(x)), dummy, dummy2) → FUNCTION(p, s(x), x, x)

The TRS R consists of the following rules:

function(iszero, 0, dummy, dummy2) → true
function(iszero, s(x), dummy, dummy2) → false
function(p, 0, dummy, dummy2) → 0
function(p, s(0), dummy, dummy2) → 0
function(p, s(s(x)), dummy, dummy2) → s(function(p, s(x), x, x))
function(plus, dummy, x, y) → function(if, function(iszero, x, x, x), x, y)
function(if, true, x, y) → y
function(if, false, x, y) → function(plus, function(third, x, y, y), function(p, x, x, y), s(y))
function(third, x, y, z) → z

The set Q consists of the following terms:

function(iszero, 0, x0, x1)
function(iszero, s(x0), x1, x2)
function(p, 0, x0, x1)
function(p, s(0), x0, x1)
function(p, s(s(x0)), x1, x2)
function(plus, x0, x1, x2)
function(if, true, x0, x1)
function(if, false, x0, x1)
function(third, 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:

FUNCTION(p, s(s(x)), dummy, dummy2) → FUNCTION(p, s(x), x, x)

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

function(iszero, 0, x0, x1)
function(iszero, s(x0), x1, x2)
function(p, 0, x0, x1)
function(p, s(0), x0, x1)
function(p, s(s(x0)), x1, x2)
function(plus, x0, x1, x2)
function(if, true, x0, x1)
function(if, false, x0, x1)
function(third, 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].

function(iszero, 0, x0, x1)
function(iszero, s(x0), x1, x2)
function(p, 0, x0, x1)
function(p, s(0), x0, x1)
function(p, s(s(x0)), x1, x2)
function(plus, x0, x1, x2)
function(if, true, x0, x1)
function(if, false, x0, x1)
function(third, x0, x1, x2)

(11) Obligation:

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

FUNCTION(p, s(s(x)), dummy, dummy2) → FUNCTION(p, s(x), x, x)

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:

FUNCTION(p, s(s(x)), dummy, dummy2) → FUNCTION(p, s(x), x, x)
The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3, 2 > 4

(13) TRUE

(14) Obligation:

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

FUNCTION(if, false, x, y) → FUNCTION(plus, function(third, x, y, y), function(p, x, x, y), s(y))
FUNCTION(plus, dummy, x, y) → FUNCTION(if, function(iszero, x, x, x), x, y)

The TRS R consists of the following rules:

function(iszero, 0, dummy, dummy2) → true
function(iszero, s(x), dummy, dummy2) → false
function(p, 0, dummy, dummy2) → 0
function(p, s(0), dummy, dummy2) → 0
function(p, s(s(x)), dummy, dummy2) → s(function(p, s(x), x, x))
function(plus, dummy, x, y) → function(if, function(iszero, x, x, x), x, y)
function(if, true, x, y) → y
function(if, false, x, y) → function(plus, function(third, x, y, y), function(p, x, x, y), s(y))
function(third, x, y, z) → z

The set Q consists of the following terms:

function(iszero, 0, x0, x1)
function(iszero, s(x0), x1, x2)
function(p, 0, x0, x1)
function(p, s(0), x0, x1)
function(p, s(s(x0)), x1, x2)
function(plus, x0, x1, x2)
function(if, true, x0, x1)
function(if, false, x0, x1)
function(third, 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:

FUNCTION(if, false, x, y) → FUNCTION(plus, function(third, x, y, y), function(p, x, x, y), s(y))
FUNCTION(plus, dummy, x, y) → FUNCTION(if, function(iszero, x, x, x), x, y)

The TRS R consists of the following rules:

function(iszero, 0, dummy, dummy2) → true
function(iszero, s(x), dummy, dummy2) → false
function(third, x, y, z) → z
function(p, 0, dummy, dummy2) → 0
function(p, s(0), dummy, dummy2) → 0
function(p, s(s(x)), dummy, dummy2) → s(function(p, s(x), x, x))

The set Q consists of the following terms:

function(iszero, 0, x0, x1)
function(iszero, s(x0), x1, x2)
function(p, 0, x0, x1)
function(p, s(0), x0, x1)
function(p, s(s(x0)), x1, x2)
function(plus, x0, x1, x2)
function(if, true, x0, x1)
function(if, false, x0, x1)
function(third, x0, x1, x2)

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

(17) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule FUNCTION(plus, dummy, x, y) → FUNCTION(if, function(iszero, x, x, x), x, y) we obtained the following new rules [LPAR04]:

FUNCTION(plus, y_0, y_1, s(z1)) → FUNCTION(if, function(iszero, y_1, y_1, y_1), y_1, s(z1))

(18) Obligation:

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

FUNCTION(if, false, x, y) → FUNCTION(plus, function(third, x, y, y), function(p, x, x, y), s(y))
FUNCTION(plus, y_0, y_1, s(z1)) → FUNCTION(if, function(iszero, y_1, y_1, y_1), y_1, s(z1))

The TRS R consists of the following rules:

function(iszero, 0, dummy, dummy2) → true
function(iszero, s(x), dummy, dummy2) → false
function(third, x, y, z) → z
function(p, 0, dummy, dummy2) → 0
function(p, s(0), dummy, dummy2) → 0
function(p, s(s(x)), dummy, dummy2) → s(function(p, s(x), x, x))

The set Q consists of the following terms:

function(iszero, 0, x0, x1)
function(iszero, s(x0), x1, x2)
function(p, 0, x0, x1)
function(p, s(0), x0, x1)
function(p, s(s(x0)), x1, x2)
function(plus, x0, x1, x2)
function(if, true, x0, x1)
function(if, false, x0, x1)
function(third, x0, x1, x2)

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

(19) Rewriting (EQUIVALENT transformation)

By rewriting [LPAR04] the rule FUNCTION(if, false, x, y) → FUNCTION(plus, function(third, x, y, y), function(p, x, x, y), s(y)) at position [1] we obtained the following new rules [LPAR04]:

FUNCTION(if, false, x, y) → FUNCTION(plus, y, function(p, x, x, y), s(y))

(20) Obligation:

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

FUNCTION(plus, dummy, x, y) → FUNCTION(if, function(iszero, x, x, x), x, y)
FUNCTION(if, false, x, y) → FUNCTION(plus, y, function(p, x, x, y), s(y))

The TRS R consists of the following rules:

function(iszero, 0, dummy, dummy2) → true
function(iszero, s(x), dummy, dummy2) → false
function(third, x, y, z) → z
function(p, 0, dummy, dummy2) → 0
function(p, s(0), dummy, dummy2) → 0
function(p, s(s(x)), dummy, dummy2) → s(function(p, s(x), x, x))

The set Q consists of the following terms:

function(iszero, 0, x0, x1)
function(iszero, s(x0), x1, x2)
function(p, 0, x0, x1)
function(p, s(0), x0, x1)
function(p, s(s(x0)), x1, x2)
function(plus, x0, x1, x2)
function(if, true, x0, x1)
function(if, false, x0, x1)
function(third, x0, x1, x2)

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

(21) 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.

(22) Obligation:

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

FUNCTION(plus, dummy, x, y) → FUNCTION(if, function(iszero, x, x, x), x, y)
FUNCTION(if, false, x, y) → FUNCTION(plus, y, function(p, x, x, y), s(y))

The TRS R consists of the following rules:

function(p, 0, dummy, dummy2) → 0
function(p, s(0), dummy, dummy2) → 0
function(p, s(s(x)), dummy, dummy2) → s(function(p, s(x), x, x))
function(iszero, 0, dummy, dummy2) → true
function(iszero, s(x), dummy, dummy2) → false

The set Q consists of the following terms:

function(iszero, 0, x0, x1)
function(iszero, s(x0), x1, x2)
function(p, 0, x0, x1)
function(p, s(0), x0, x1)
function(p, s(s(x0)), x1, x2)
function(plus, x0, x1, x2)
function(if, true, x0, x1)
function(if, false, x0, x1)
function(third, x0, x1, x2)

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

(23) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule FUNCTION(plus, dummy, x, y) → FUNCTION(if, function(iszero, x, x, x), x, y) we obtained the following new rules [LPAR04]:

FUNCTION(plus, z1, y_0, s(z1)) → FUNCTION(if, function(iszero, y_0, y_0, y_0), y_0, s(z1))

(24) Obligation:

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

FUNCTION(if, false, x, y) → FUNCTION(plus, y, function(p, x, x, y), s(y))
FUNCTION(plus, z1, y_0, s(z1)) → FUNCTION(if, function(iszero, y_0, y_0, y_0), y_0, s(z1))

The TRS R consists of the following rules:

function(p, 0, dummy, dummy2) → 0
function(p, s(0), dummy, dummy2) → 0
function(p, s(s(x)), dummy, dummy2) → s(function(p, s(x), x, x))
function(iszero, 0, dummy, dummy2) → true
function(iszero, s(x), dummy, dummy2) → false

The set Q consists of the following terms:

function(iszero, 0, x0, x1)
function(iszero, s(x0), x1, x2)
function(p, 0, x0, x1)
function(p, s(0), x0, x1)
function(p, s(s(x0)), x1, x2)
function(plus, x0, x1, x2)
function(if, true, x0, x1)
function(if, false, x0, x1)
function(third, x0, x1, x2)

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

(25) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule FUNCTION(plus, z1, y_0, s(z1)) → FUNCTION(if, function(iszero, y_0, y_0, y_0), y_0, s(z1)) at position [1] we obtained the following new rules [LPAR04]:

FUNCTION(plus, y0, 0, s(y0)) → FUNCTION(if, true, 0, s(y0))
FUNCTION(plus, y0, s(x0), s(y0)) → FUNCTION(if, false, s(x0), s(y0))

(26) Obligation:

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

FUNCTION(if, false, x, y) → FUNCTION(plus, y, function(p, x, x, y), s(y))
FUNCTION(plus, y0, 0, s(y0)) → FUNCTION(if, true, 0, s(y0))
FUNCTION(plus, y0, s(x0), s(y0)) → FUNCTION(if, false, s(x0), s(y0))

The TRS R consists of the following rules:

function(p, 0, dummy, dummy2) → 0
function(p, s(0), dummy, dummy2) → 0
function(p, s(s(x)), dummy, dummy2) → s(function(p, s(x), x, x))
function(iszero, 0, dummy, dummy2) → true
function(iszero, s(x), dummy, dummy2) → false

The set Q consists of the following terms:

function(iszero, 0, x0, x1)
function(iszero, s(x0), x1, x2)
function(p, 0, x0, x1)
function(p, s(0), x0, x1)
function(p, s(s(x0)), x1, x2)
function(plus, x0, x1, x2)
function(if, true, x0, x1)
function(if, false, x0, x1)
function(third, x0, x1, x2)

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

(27) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.

(28) Obligation:

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

FUNCTION(plus, y0, s(x0), s(y0)) → FUNCTION(if, false, s(x0), s(y0))
FUNCTION(if, false, x, y) → FUNCTION(plus, y, function(p, x, x, y), s(y))

The TRS R consists of the following rules:

function(p, 0, dummy, dummy2) → 0
function(p, s(0), dummy, dummy2) → 0
function(p, s(s(x)), dummy, dummy2) → s(function(p, s(x), x, x))
function(iszero, 0, dummy, dummy2) → true
function(iszero, s(x), dummy, dummy2) → false

The set Q consists of the following terms:

function(iszero, 0, x0, x1)
function(iszero, s(x0), x1, x2)
function(p, 0, x0, x1)
function(p, s(0), x0, x1)
function(p, s(s(x0)), x1, x2)
function(plus, x0, x1, x2)
function(if, true, x0, x1)
function(if, false, x0, x1)
function(third, 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:

FUNCTION(plus, y0, s(x0), s(y0)) → FUNCTION(if, false, s(x0), s(y0))
FUNCTION(if, false, x, y) → FUNCTION(plus, y, function(p, x, x, y), s(y))

The TRS R consists of the following rules:

function(p, 0, dummy, dummy2) → 0
function(p, s(0), dummy, dummy2) → 0
function(p, s(s(x)), dummy, dummy2) → s(function(p, s(x), x, x))

The set Q consists of the following terms:

function(iszero, 0, x0, x1)
function(iszero, s(x0), x1, x2)
function(p, 0, x0, x1)
function(p, s(0), x0, x1)
function(p, s(s(x0)), x1, x2)
function(plus, x0, x1, x2)
function(if, true, x0, x1)
function(if, false, x0, x1)
function(third, x0, x1, x2)

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

(31) Narrowing (EQUIVALENT transformation)

By narrowing [LPAR04] the rule FUNCTION(if, false, x, y) → FUNCTION(plus, y, function(p, x, x, y), s(y)) at position [2] we obtained the following new rules [LPAR04]:

FUNCTION(if, false, 0, x1) → FUNCTION(plus, x1, 0, s(x1))
FUNCTION(if, false, s(0), x1) → FUNCTION(plus, x1, 0, s(x1))
FUNCTION(if, false, s(s(x0)), x2) → FUNCTION(plus, x2, s(function(p, s(x0), x0, x0)), s(x2))

(32) Obligation:

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

FUNCTION(plus, y0, s(x0), s(y0)) → FUNCTION(if, false, s(x0), s(y0))
FUNCTION(if, false, 0, x1) → FUNCTION(plus, x1, 0, s(x1))
FUNCTION(if, false, s(0), x1) → FUNCTION(plus, x1, 0, s(x1))
FUNCTION(if, false, s(s(x0)), x2) → FUNCTION(plus, x2, s(function(p, s(x0), x0, x0)), s(x2))

The TRS R consists of the following rules:

function(p, 0, dummy, dummy2) → 0
function(p, s(0), dummy, dummy2) → 0
function(p, s(s(x)), dummy, dummy2) → s(function(p, s(x), x, x))

The set Q consists of the following terms:

function(iszero, 0, x0, x1)
function(iszero, s(x0), x1, x2)
function(p, 0, x0, x1)
function(p, s(0), x0, x1)
function(p, s(s(x0)), x1, x2)
function(plus, x0, x1, x2)
function(if, true, x0, x1)
function(if, false, x0, x1)
function(third, x0, x1, x2)

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

(33) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.

(34) Obligation:

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

FUNCTION(if, false, s(s(x0)), x2) → FUNCTION(plus, x2, s(function(p, s(x0), x0, x0)), s(x2))
FUNCTION(plus, y0, s(x0), s(y0)) → FUNCTION(if, false, s(x0), s(y0))

The TRS R consists of the following rules:

function(p, 0, dummy, dummy2) → 0
function(p, s(0), dummy, dummy2) → 0
function(p, s(s(x)), dummy, dummy2) → s(function(p, s(x), x, x))

The set Q consists of the following terms:

function(iszero, 0, x0, x1)
function(iszero, s(x0), x1, x2)
function(p, 0, x0, x1)
function(p, s(0), x0, x1)
function(p, s(s(x0)), x1, x2)
function(plus, x0, x1, x2)
function(if, true, x0, x1)
function(if, false, x0, x1)
function(third, x0, x1, x2)

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

(35) UsableRulesProof (EQUIVALENT transformation)

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

(36) Obligation:

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

FUNCTION(if, false, s(s(x0)), x2) → FUNCTION(plus, x2, s(function(p, s(x0), x0, x0)), s(x2))
FUNCTION(plus, y0, s(x0), s(y0)) → FUNCTION(if, false, s(x0), s(y0))

The TRS R consists of the following rules:

function(p, s(0), dummy, dummy2) → 0
function(p, s(s(x)), dummy, dummy2) → s(function(p, s(x), x, x))

The set Q consists of the following terms:

function(iszero, 0, x0, x1)
function(iszero, s(x0), x1, x2)
function(p, 0, x0, x1)
function(p, s(0), x0, x1)
function(p, s(s(x0)), x1, x2)
function(plus, x0, x1, x2)
function(if, true, x0, x1)
function(if, false, x0, x1)
function(third, x0, x1, x2)

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

(37) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule FUNCTION(if, false, s(s(x0)), x2) → FUNCTION(plus, x2, s(function(p, s(x0), x0, x0)), s(x2)) we obtained the following new rules [LPAR04]:

FUNCTION(if, false, s(s(x0)), s(z0)) → FUNCTION(plus, s(z0), s(function(p, s(x0), x0, x0)), s(s(z0)))

(38) Obligation:

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

FUNCTION(plus, y0, s(x0), s(y0)) → FUNCTION(if, false, s(x0), s(y0))
FUNCTION(if, false, s(s(x0)), s(z0)) → FUNCTION(plus, s(z0), s(function(p, s(x0), x0, x0)), s(s(z0)))

The TRS R consists of the following rules:

function(p, s(0), dummy, dummy2) → 0
function(p, s(s(x)), dummy, dummy2) → s(function(p, s(x), x, x))

The set Q consists of the following terms:

function(iszero, 0, x0, x1)
function(iszero, s(x0), x1, x2)
function(p, 0, x0, x1)
function(p, s(0), x0, x1)
function(p, s(s(x0)), x1, x2)
function(plus, x0, x1, x2)
function(if, true, x0, x1)
function(if, false, x0, x1)
function(third, x0, x1, x2)

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

(39) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule FUNCTION(plus, y0, s(x0), s(y0)) → FUNCTION(if, false, s(x0), s(y0)) we obtained the following new rules [LPAR04]:

FUNCTION(plus, s(z1), s(y_0), s(s(z1))) → FUNCTION(if, false, s(y_0), s(s(z1)))

(40) Obligation:

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

FUNCTION(if, false, s(s(x0)), s(z0)) → FUNCTION(plus, s(z0), s(function(p, s(x0), x0, x0)), s(s(z0)))
FUNCTION(plus, s(z1), s(y_0), s(s(z1))) → FUNCTION(if, false, s(y_0), s(s(z1)))

The TRS R consists of the following rules:

function(p, s(0), dummy, dummy2) → 0
function(p, s(s(x)), dummy, dummy2) → s(function(p, s(x), x, x))

The set Q consists of the following terms:

function(iszero, 0, x0, x1)
function(iszero, s(x0), x1, x2)
function(p, 0, x0, x1)
function(p, s(0), x0, x1)
function(p, s(s(x0)), x1, x2)
function(plus, x0, x1, x2)
function(if, true, x0, x1)
function(if, false, x0, x1)
function(third, x0, x1, x2)

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

(41) Instantiation (EQUIVALENT transformation)

By instantiating [LPAR04] the rule FUNCTION(if, false, s(s(x0)), s(z0)) → FUNCTION(plus, s(z0), s(function(p, s(x0), x0, x0)), s(s(z0))) we obtained the following new rules [LPAR04]:

FUNCTION(if, false, s(s(x0)), s(s(z0))) → FUNCTION(plus, s(s(z0)), s(function(p, s(x0), x0, x0)), s(s(s(z0))))

(42) Obligation:

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

FUNCTION(plus, s(z1), s(y_0), s(s(z1))) → FUNCTION(if, false, s(y_0), s(s(z1)))
FUNCTION(if, false, s(s(x0)), s(s(z0))) → FUNCTION(plus, s(s(z0)), s(function(p, s(x0), x0, x0)), s(s(s(z0))))

The TRS R consists of the following rules:

function(p, s(0), dummy, dummy2) → 0
function(p, s(s(x)), dummy, dummy2) → s(function(p, s(x), x, x))

The set Q consists of the following terms:

function(iszero, 0, x0, x1)
function(iszero, s(x0), x1, x2)
function(p, 0, x0, x1)
function(p, s(0), x0, x1)
function(p, s(s(x0)), x1, x2)
function(plus, x0, x1, x2)
function(if, true, x0, x1)
function(if, false, x0, x1)
function(third, x0, x1, x2)

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

(43) ForwardInstantiation (EQUIVALENT transformation)

By forward instantiating [JAR06] the rule FUNCTION(plus, s(z1), s(y_0), s(s(z1))) → FUNCTION(if, false, s(y_0), s(s(z1))) we obtained the following new rules [LPAR04]:

FUNCTION(plus, s(x0), s(s(y_0)), s(s(x0))) → FUNCTION(if, false, s(s(y_0)), s(s(x0)))

(44) Obligation:

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

FUNCTION(if, false, s(s(x0)), s(s(z0))) → FUNCTION(plus, s(s(z0)), s(function(p, s(x0), x0, x0)), s(s(s(z0))))
FUNCTION(plus, s(x0), s(s(y_0)), s(s(x0))) → FUNCTION(if, false, s(s(y_0)), s(s(x0)))

The TRS R consists of the following rules:

function(p, s(0), dummy, dummy2) → 0
function(p, s(s(x)), dummy, dummy2) → s(function(p, s(x), x, x))

The set Q consists of the following terms:

function(iszero, 0, x0, x1)
function(iszero, s(x0), x1, x2)
function(p, 0, x0, x1)
function(p, s(0), x0, x1)
function(p, s(s(x0)), x1, x2)
function(plus, x0, x1, x2)
function(if, true, x0, x1)
function(if, false, x0, x1)
function(third, x0, x1, x2)

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

(45) QDPOrderProof (EQUIVALENT transformation)

We use the reduction pair processor [LPAR04].

The following pairs can be oriented strictly and are deleted.

FUNCTION(if, false, s(s(x0)), s(s(z0))) → FUNCTION(plus, s(s(z0)), s(function(p, s(x0), x0, x0)), s(s(s(z0))))

The remaining pairs can at least be oriented weakly.
Used ordering: Matrix interpretation [MATRO]:

POL(FUNCTION(x₁, x₂, x₃, x₄)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 0 \

\ 0 0 /

· x₂ +
/ 0 1 \

\ 0 0 /

· x₃ +
/ 0 0 \

\ 0 0 /

· x₄

POL(if) =
/ 0 \

\ 0 /

POL(false) =
/ 0 \

\ 0 /

POL(s(x₁)) =
/ 1 \

\ 0 /

+
/ 1 1 \

\ 1 1 /

· x₁

POL(plus) =
/ 0 \

\ 0 /

POL(function(x₁, x₂, x₃, x₄)) =
/ 0 \

\ 0 /

+
/ 0 0 \

\ 0 0 /

· x₁ +
/ 0 1 \

\ 0 1 /

· x₂ +
/ 0 0 \

\ 0 0 /

· x₃ +
/ 0 0 \

\ 0 0 /

· x₄

POL(p) =
/ 0 \

\ 0 /

POL(0) =
/ 0 \

\ 0 /

The following usable rules [FROCOS05] were oriented:

function(p, s(0), dummy, dummy2) → 0
function(p, s(s(x)), dummy, dummy2) → s(function(p, s(x), x, x))

(46) Obligation:

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

FUNCTION(plus, s(x0), s(s(y_0)), s(s(x0))) → FUNCTION(if, false, s(s(y_0)), s(s(x0)))

The TRS R consists of the following rules:

function(p, s(0), dummy, dummy2) → 0
function(p, s(s(x)), dummy, dummy2) → s(function(p, s(x), x, x))

The set Q consists of the following terms:

function(iszero, 0, x0, x1)
function(iszero, s(x0), x1, x2)
function(p, 0, x0, x1)
function(p, s(0), x0, x1)
function(p, s(s(x0)), x1, x2)
function(plus, x0, x1, x2)
function(if, true, x0, x1)
function(if, false, x0, x1)
function(third, x0, x1, x2)

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

(47) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node.

(0) Obligation:

(1) Overlay + Local Confluence (EQUIVALENT transformation)

(2) Obligation:

(3) DependencyPairsProof (EQUIVALENT transformation)

(4) Obligation:

(5) DependencyGraphProof (EQUIVALENT transformation)

(6) Complex Obligation (AND)

(7) Obligation:

(8) UsableRulesProof (EQUIVALENT transformation)

(9) Obligation:

(10) QReductionProof (EQUIVALENT transformation)

(11) Obligation:

(12) QDPSizeChangeProof (EQUIVALENT transformation)

(13) TRUE

(14) Obligation:

(15) UsableRulesProof (EQUIVALENT transformation)

(16) Obligation:

(17) Instantiation (EQUIVALENT transformation)

(18) Obligation:

(19) Rewriting (EQUIVALENT transformation)

(20) Obligation:

(21) UsableRulesProof (EQUIVALENT transformation)

(22) Obligation:

(23) Instantiation (EQUIVALENT transformation)

(24) Obligation:

(25) Narrowing (EQUIVALENT transformation)

(26) Obligation:

(27) DependencyGraphProof (EQUIVALENT transformation)

(28) Obligation:

(29) UsableRulesProof (EQUIVALENT transformation)

(30) Obligation:

(31) Narrowing (EQUIVALENT transformation)

(32) Obligation:

(33) DependencyGraphProof (EQUIVALENT transformation)

(34) Obligation:

(35) UsableRulesProof (EQUIVALENT transformation)

(36) Obligation:

(37) Instantiation (EQUIVALENT transformation)

(38) Obligation:

(39) Instantiation (EQUIVALENT transformation)

(40) Obligation:

(41) Instantiation (EQUIVALENT transformation)

(42) Obligation:

(43) ForwardInstantiation (EQUIVALENT transformation)

(44) Obligation:

(45) QDPOrderProof (EQUIVALENT transformation)

(46) Obligation:

(47) DependencyGraphProof (EQUIVALENT transformation)

(48) TRUE