Termination w.r.t. Q of the following Term Rewriting System could be proven:
Q restricted rewrite system:
The TRS R consists of the following rules:
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
div(x, y) → ify(ge(y, s(0)), x, y)
ify(false, x, y) → divByZeroError
ify(true, x, y) → if(ge(x, y), x, y)
if(false, x, y) → 0
if(true, x, y) → s(div(minus(x, y), y))
Q is empty.
↳ QTRS
↳ Overlay + Local Confluence
Q restricted rewrite system:
The TRS R consists of the following rules:
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
div(x, y) → ify(ge(y, s(0)), x, y)
ify(false, x, y) → divByZeroError
ify(true, x, y) → if(ge(x, y), x, y)
if(false, x, y) → 0
if(true, x, y) → s(div(minus(x, y), y))
Q is empty.
The TRS is overlay and locally confluent. By [19] we can switch to innermost.
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
Q restricted rewrite system:
The TRS R consists of the following rules:
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
div(x, y) → ify(ge(y, s(0)), x, y)
ify(false, x, y) → divByZeroError
ify(true, x, y) → if(ge(x, y), x, y)
if(false, x, y) → 0
if(true, x, y) → s(div(minus(x, y), y))
The set Q consists of the following terms:
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
minus(x0, 0)
minus(s(x0), s(x1))
div(x0, x1)
ify(false, x0, x1)
ify(true, x0, x1)
if(false, x0, x1)
if(true, x0, x1)
Using Dependency Pairs [1,15] we result in the following initial DP problem:
Q DP problem:
The TRS P consists of the following rules:
MINUS(s(x), s(y)) → MINUS(x, y)
IF(true, x, y) → DIV(minus(x, y), y)
IF(true, x, y) → MINUS(x, y)
DIV(x, y) → IFY(ge(y, s(0)), x, y)
IFY(true, x, y) → IF(ge(x, y), x, y)
DIV(x, y) → GE(y, s(0))
GE(s(x), s(y)) → GE(x, y)
IFY(true, x, y) → GE(x, y)
The TRS R consists of the following rules:
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
div(x, y) → ify(ge(y, s(0)), x, y)
ify(false, x, y) → divByZeroError
ify(true, x, y) → if(ge(x, y), x, y)
if(false, x, y) → 0
if(true, x, y) → s(div(minus(x, y), y))
The set Q consists of the following terms:
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
minus(x0, 0)
minus(s(x0), s(x1))
div(x0, x1)
ify(false, x0, x1)
ify(true, x0, x1)
if(false, x0, x1)
if(true, x0, x1)
We have to consider all minimal (P,Q,R)-chains.
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
MINUS(s(x), s(y)) → MINUS(x, y)
IF(true, x, y) → DIV(minus(x, y), y)
IF(true, x, y) → MINUS(x, y)
DIV(x, y) → IFY(ge(y, s(0)), x, y)
IFY(true, x, y) → IF(ge(x, y), x, y)
DIV(x, y) → GE(y, s(0))
GE(s(x), s(y)) → GE(x, y)
IFY(true, x, y) → GE(x, y)
The TRS R consists of the following rules:
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
div(x, y) → ify(ge(y, s(0)), x, y)
ify(false, x, y) → divByZeroError
ify(true, x, y) → if(ge(x, y), x, y)
if(false, x, y) → 0
if(true, x, y) → s(div(minus(x, y), y))
The set Q consists of the following terms:
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
minus(x0, 0)
minus(s(x0), s(x1))
div(x0, x1)
ify(false, x0, x1)
ify(true, x0, x1)
if(false, x0, x1)
if(true, x0, x1)
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 3 SCCs with 3 less nodes.
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
MINUS(s(x), s(y)) → MINUS(x, y)
The TRS R consists of the following rules:
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
div(x, y) → ify(ge(y, s(0)), x, y)
ify(false, x, y) → divByZeroError
ify(true, x, y) → if(ge(x, y), x, y)
if(false, x, y) → 0
if(true, x, y) → s(div(minus(x, y), y))
The set Q consists of the following terms:
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
minus(x0, 0)
minus(s(x0), s(x1))
div(x0, x1)
ify(false, x0, x1)
ify(true, x0, x1)
if(false, x0, x1)
if(true, x0, x1)
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
MINUS(s(x), s(y)) → MINUS(x, y)
R is empty.
The set Q consists of the following terms:
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
minus(x0, 0)
minus(s(x0), s(x1))
div(x0, x1)
ify(false, x0, x1)
ify(true, x0, x1)
if(false, x0, x1)
if(true, x0, x1)
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
minus(x0, 0)
minus(s(x0), s(x1))
div(x0, x1)
ify(false, x0, x1)
ify(true, x0, x1)
if(false, x0, x1)
if(true, x0, x1)
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
MINUS(s(x), s(y)) → MINUS(x, y)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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:
- MINUS(s(x), s(y)) → MINUS(x, y)
The graph contains the following edges 1 > 1, 2 > 2
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
GE(s(x), s(y)) → GE(x, y)
The TRS R consists of the following rules:
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
div(x, y) → ify(ge(y, s(0)), x, y)
ify(false, x, y) → divByZeroError
ify(true, x, y) → if(ge(x, y), x, y)
if(false, x, y) → 0
if(true, x, y) → s(div(minus(x, y), y))
The set Q consists of the following terms:
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
minus(x0, 0)
minus(s(x0), s(x1))
div(x0, x1)
ify(false, x0, x1)
ify(true, x0, x1)
if(false, x0, x1)
if(true, x0, x1)
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
GE(s(x), s(y)) → GE(x, y)
R is empty.
The set Q consists of the following terms:
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
minus(x0, 0)
minus(s(x0), s(x1))
div(x0, x1)
ify(false, x0, x1)
ify(true, x0, x1)
if(false, x0, x1)
if(true, x0, x1)
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
minus(x0, 0)
minus(s(x0), s(x1))
div(x0, x1)
ify(false, x0, x1)
ify(true, x0, x1)
if(false, x0, x1)
if(true, x0, x1)
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
GE(s(x), s(y)) → GE(x, y)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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:
- GE(s(x), s(y)) → GE(x, y)
The graph contains the following edges 1 > 1, 2 > 2
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
Q DP problem:
The TRS P consists of the following rules:
IF(true, x, y) → DIV(minus(x, y), y)
DIV(x, y) → IFY(ge(y, s(0)), x, y)
IFY(true, x, y) → IF(ge(x, y), x, y)
The TRS R consists of the following rules:
ge(x, 0) → true
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
div(x, y) → ify(ge(y, s(0)), x, y)
ify(false, x, y) → divByZeroError
ify(true, x, y) → if(ge(x, y), x, y)
if(false, x, y) → 0
if(true, x, y) → s(div(minus(x, y), y))
The set Q consists of the following terms:
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
minus(x0, 0)
minus(s(x0), s(x1))
div(x0, x1)
ify(false, x0, x1)
ify(true, x0, x1)
if(false, x0, x1)
if(true, x0, x1)
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
Q DP problem:
The TRS P consists of the following rules:
IF(true, x, y) → DIV(minus(x, y), y)
DIV(x, y) → IFY(ge(y, s(0)), x, y)
IFY(true, x, y) → IF(ge(x, y), x, y)
The TRS R consists of the following rules:
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
ge(x, 0) → true
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
The set Q consists of the following terms:
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
minus(x0, 0)
minus(s(x0), s(x1))
div(x0, x1)
ify(false, x0, x1)
ify(true, x0, x1)
if(false, x0, x1)
if(true, x0, x1)
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
div(x0, x1)
ify(false, x0, x1)
ify(true, x0, x1)
if(false, x0, x1)
if(true, x0, x1)
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
IF(true, x, y) → DIV(minus(x, y), y)
DIV(x, y) → IFY(ge(y, s(0)), x, y)
IFY(true, x, y) → IF(ge(x, y), x, y)
The TRS R consists of the following rules:
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
ge(x, 0) → true
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
The set Q consists of the following terms:
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
minus(x0, 0)
minus(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule DIV(x, y) → IFY(ge(y, s(0)), x, y) at position [0] we obtained the following new rules:
DIV(y0, s(x0)) → IFY(ge(x0, 0), y0, s(x0))
DIV(y0, 0) → IFY(false, y0, 0)
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
DIV(y0, s(x0)) → IFY(ge(x0, 0), y0, s(x0))
DIV(y0, 0) → IFY(false, y0, 0)
IF(true, x, y) → DIV(minus(x, y), y)
IFY(true, x, y) → IF(ge(x, y), x, y)
The TRS R consists of the following rules:
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
ge(x, 0) → true
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
The set Q consists of the following terms:
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
minus(x0, 0)
minus(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 1 less node.
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
DIV(y0, s(x0)) → IFY(ge(x0, 0), y0, s(x0))
IF(true, x, y) → DIV(minus(x, y), y)
IFY(true, x, y) → IF(ge(x, y), x, y)
The TRS R consists of the following rules:
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
ge(x, 0) → true
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
The set Q consists of the following terms:
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
minus(x0, 0)
minus(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule DIV(y0, s(x0)) → IFY(ge(x0, 0), y0, s(x0)) at position [0] we obtained the following new rules:
DIV(y0, s(x0)) → IFY(true, y0, s(x0))
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
DIV(y0, s(x0)) → IFY(true, y0, s(x0))
IF(true, x, y) → DIV(minus(x, y), y)
IFY(true, x, y) → IF(ge(x, y), x, y)
The TRS R consists of the following rules:
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
ge(x, 0) → true
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
The set Q consists of the following terms:
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
minus(x0, 0)
minus(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule IFY(true, x, y) → IF(ge(x, y), x, y) at position [0] we obtained the following new rules:
IFY(true, s(x0), s(x1)) → IF(ge(x0, x1), s(x0), s(x1))
IFY(true, x0, 0) → IF(true, x0, 0)
IFY(true, 0, s(x0)) → IF(false, 0, s(x0))
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
IFY(true, s(x0), s(x1)) → IF(ge(x0, x1), s(x0), s(x1))
DIV(y0, s(x0)) → IFY(true, y0, s(x0))
IFY(true, x0, 0) → IF(true, x0, 0)
IF(true, x, y) → DIV(minus(x, y), y)
IFY(true, 0, s(x0)) → IF(false, 0, s(x0))
The TRS R consists of the following rules:
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
ge(x, 0) → true
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
The set Q consists of the following terms:
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
minus(x0, 0)
minus(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 1 SCC with 2 less nodes.
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
Q DP problem:
The TRS P consists of the following rules:
IFY(true, s(x0), s(x1)) → IF(ge(x0, x1), s(x0), s(x1))
DIV(y0, s(x0)) → IFY(true, y0, s(x0))
IF(true, x, y) → DIV(minus(x, y), y)
The TRS R consists of the following rules:
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
ge(x, 0) → true
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
The set Q consists of the following terms:
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
minus(x0, 0)
minus(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule IF(true, x, y) → DIV(minus(x, y), y) we obtained the following new rules:
IF(true, s(z0), s(z1)) → DIV(minus(s(z0), s(z1)), s(z1))
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
IFY(true, s(x0), s(x1)) → IF(ge(x0, x1), s(x0), s(x1))
DIV(y0, s(x0)) → IFY(true, y0, s(x0))
IF(true, s(z0), s(z1)) → DIV(minus(s(z0), s(z1)), s(z1))
The TRS R consists of the following rules:
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
ge(x, 0) → true
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
The set Q consists of the following terms:
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
minus(x0, 0)
minus(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule IF(true, s(z0), s(z1)) → DIV(minus(s(z0), s(z1)), s(z1)) at position [0] we obtained the following new rules:
IF(true, s(z0), s(z1)) → DIV(minus(z0, z1), s(z1))
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ ForwardInstantiation
Q DP problem:
The TRS P consists of the following rules:
IFY(true, s(x0), s(x1)) → IF(ge(x0, x1), s(x0), s(x1))
DIV(y0, s(x0)) → IFY(true, y0, s(x0))
IF(true, s(z0), s(z1)) → DIV(minus(z0, z1), s(z1))
The TRS R consists of the following rules:
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
ge(x, 0) → true
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
The set Q consists of the following terms:
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
minus(x0, 0)
minus(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule DIV(y0, s(x0)) → IFY(true, y0, s(x0)) we obtained the following new rules:
DIV(s(y_0), s(x1)) → IFY(true, s(y_0), s(x1))
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ Narrowing
Q DP problem:
The TRS P consists of the following rules:
IFY(true, s(x0), s(x1)) → IF(ge(x0, x1), s(x0), s(x1))
IF(true, s(z0), s(z1)) → DIV(minus(z0, z1), s(z1))
DIV(s(y_0), s(x1)) → IFY(true, s(y_0), s(x1))
The TRS R consists of the following rules:
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
ge(x, 0) → true
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
The set Q consists of the following terms:
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
minus(x0, 0)
minus(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
By narrowing [15] the rule IF(true, s(z0), s(z1)) → DIV(minus(z0, z1), s(z1)) at position [0] we obtained the following new rules:
IF(true, s(s(x0)), s(s(x1))) → DIV(minus(x0, x1), s(s(x1)))
IF(true, s(x0), s(0)) → DIV(x0, s(0))
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ Instantiation
Q DP problem:
The TRS P consists of the following rules:
IFY(true, s(x0), s(x1)) → IF(ge(x0, x1), s(x0), s(x1))
DIV(s(y_0), s(x1)) → IFY(true, s(y_0), s(x1))
IF(true, s(x0), s(0)) → DIV(x0, s(0))
IF(true, s(s(x0)), s(s(x1))) → DIV(minus(x0, x1), s(s(x1)))
The TRS R consists of the following rules:
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
ge(x, 0) → true
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
The set Q consists of the following terms:
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
minus(x0, 0)
minus(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule DIV(s(y_0), s(x1)) → IFY(true, s(y_0), s(x1)) we obtained the following new rules:
DIV(s(x0), s(0)) → IFY(true, s(x0), s(0))
DIV(s(x0), s(s(z1))) → IFY(true, s(x0), s(s(z1)))
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
Q DP problem:
The TRS P consists of the following rules:
IFY(true, s(x0), s(x1)) → IF(ge(x0, x1), s(x0), s(x1))
DIV(s(x0), s(0)) → IFY(true, s(x0), s(0))
DIV(s(x0), s(s(z1))) → IFY(true, s(x0), s(s(z1)))
IF(true, s(s(x0)), s(s(x1))) → DIV(minus(x0, x1), s(s(x1)))
IF(true, s(x0), s(0)) → DIV(x0, s(0))
The TRS R consists of the following rules:
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
ge(x, 0) → true
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
The set Q consists of the following terms:
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
minus(x0, 0)
minus(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
By instantiating [15] the rule IFY(true, s(x0), s(x1)) → IF(ge(x0, x1), s(x0), s(x1)) we obtained the following new rules:
IFY(true, s(z0), s(s(z1))) → IF(ge(z0, s(z1)), s(z0), s(s(z1)))
IFY(true, s(z0), s(0)) → IF(ge(z0, 0), s(z0), s(0))
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
Q DP problem:
The TRS P consists of the following rules:
IFY(true, s(z0), s(s(z1))) → IF(ge(z0, s(z1)), s(z0), s(s(z1)))
DIV(s(x0), s(0)) → IFY(true, s(x0), s(0))
IF(true, s(x0), s(0)) → DIV(x0, s(0))
IF(true, s(s(x0)), s(s(x1))) → DIV(minus(x0, x1), s(s(x1)))
DIV(s(x0), s(s(z1))) → IFY(true, s(x0), s(s(z1)))
IFY(true, s(z0), s(0)) → IF(ge(z0, 0), s(z0), s(0))
The TRS R consists of the following rules:
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
ge(x, 0) → true
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
The set Q consists of the following terms:
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
minus(x0, 0)
minus(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 2 SCCs.
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
DIV(s(x0), s(0)) → IFY(true, s(x0), s(0))
IF(true, s(x0), s(0)) → DIV(x0, s(0))
IFY(true, s(z0), s(0)) → IF(ge(z0, 0), s(z0), s(0))
The TRS R consists of the following rules:
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
ge(x, 0) → true
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
The set Q consists of the following terms:
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
minus(x0, 0)
minus(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
DIV(s(x0), s(0)) → IFY(true, s(x0), s(0))
IF(true, s(x0), s(0)) → DIV(x0, s(0))
IFY(true, s(z0), s(0)) → IF(ge(z0, 0), s(z0), s(0))
The TRS R consists of the following rules:
ge(x, 0) → true
The set Q consists of the following terms:
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
minus(x0, 0)
minus(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
minus(x0, 0)
minus(s(x0), s(x1))
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
DIV(s(x0), s(0)) → IFY(true, s(x0), s(0))
IF(true, s(x0), s(0)) → DIV(x0, s(0))
IFY(true, s(z0), s(0)) → IF(ge(z0, 0), s(z0), s(0))
The TRS R consists of the following rules:
ge(x, 0) → true
The set Q consists of the following terms:
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule IFY(true, s(z0), s(0)) → IF(ge(z0, 0), s(z0), s(0)) at position [0] we obtained the following new rules:
IFY(true, s(z0), s(0)) → IF(true, s(z0), s(0))
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
IFY(true, s(z0), s(0)) → IF(true, s(z0), s(0))
DIV(s(x0), s(0)) → IFY(true, s(x0), s(0))
IF(true, s(x0), s(0)) → DIV(x0, s(0))
The TRS R consists of the following rules:
ge(x, 0) → true
The set Q consists of the following terms:
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [15] we can delete all non-usable rules [17] from R.
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
IFY(true, s(z0), s(0)) → IF(true, s(z0), s(0))
DIV(s(x0), s(0)) → IFY(true, s(x0), s(0))
IF(true, s(x0), s(0)) → DIV(x0, s(0))
R is empty.
The set Q consists of the following terms:
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
IFY(true, s(z0), s(0)) → IF(true, s(z0), s(0))
DIV(s(x0), s(0)) → IFY(true, s(x0), s(0))
IF(true, s(x0), s(0)) → DIV(x0, s(0))
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule IF(true, s(x0), s(0)) → DIV(x0, s(0)) we obtained the following new rules:
IF(true, s(s(y_0)), s(0)) → DIV(s(y_0), s(0))
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
IFY(true, s(z0), s(0)) → IF(true, s(z0), s(0))
IF(true, s(s(y_0)), s(0)) → DIV(s(y_0), s(0))
DIV(s(x0), s(0)) → IFY(true, s(x0), s(0))
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule IFY(true, s(z0), s(0)) → IF(true, s(z0), s(0)) we obtained the following new rules:
IFY(true, s(s(y_0)), s(0)) → IF(true, s(s(y_0)), s(0))
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Rewriting
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ QDPSizeChangeProof
↳ QDP
Q DP problem:
The TRS P consists of the following rules:
IFY(true, s(s(y_0)), s(0)) → IF(true, s(s(y_0)), s(0))
IF(true, s(s(y_0)), s(0)) → DIV(s(y_0), s(0))
DIV(s(x0), s(0)) → IFY(true, s(x0), s(0))
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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:
- IFY(true, s(s(y_0)), s(0)) → IF(true, s(s(y_0)), s(0))
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3
- IF(true, s(s(y_0)), s(0)) → DIV(s(y_0), s(0))
The graph contains the following edges 2 > 1, 3 >= 2
- DIV(s(x0), s(0)) → IFY(true, s(x0), s(0))
The graph contains the following edges 1 >= 2, 2 >= 3
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ ForwardInstantiation
Q DP problem:
The TRS P consists of the following rules:
IFY(true, s(z0), s(s(z1))) → IF(ge(z0, s(z1)), s(z0), s(s(z1)))
IF(true, s(s(x0)), s(s(x1))) → DIV(minus(x0, x1), s(s(x1)))
DIV(s(x0), s(s(z1))) → IFY(true, s(x0), s(s(z1)))
The TRS R consists of the following rules:
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
ge(x, 0) → true
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
The set Q consists of the following terms:
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
minus(x0, 0)
minus(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
By forward instantiating [14] the rule IFY(true, s(z0), s(s(z1))) → IF(ge(z0, s(z1)), s(z0), s(s(z1))) we obtained the following new rules:
IFY(true, s(s(y_1)), s(s(x1))) → IF(ge(s(y_1), s(x1)), s(s(y_1)), s(s(x1)))
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ Rewriting
Q DP problem:
The TRS P consists of the following rules:
IFY(true, s(s(y_1)), s(s(x1))) → IF(ge(s(y_1), s(x1)), s(s(y_1)), s(s(x1)))
DIV(s(x0), s(s(z1))) → IFY(true, s(x0), s(s(z1)))
IF(true, s(s(x0)), s(s(x1))) → DIV(minus(x0, x1), s(s(x1)))
The TRS R consists of the following rules:
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
ge(x, 0) → true
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
The set Q consists of the following terms:
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
minus(x0, 0)
minus(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
By rewriting [15] the rule IFY(true, s(s(y_1)), s(s(x1))) → IF(ge(s(y_1), s(x1)), s(s(y_1)), s(s(x1))) at position [0] we obtained the following new rules:
IFY(true, s(s(y_1)), s(s(x1))) → IF(ge(y_1, x1), s(s(y_1)), s(s(x1)))
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
Q DP problem:
The TRS P consists of the following rules:
IFY(true, s(s(y_1)), s(s(x1))) → IF(ge(y_1, x1), s(s(y_1)), s(s(x1)))
IF(true, s(s(x0)), s(s(x1))) → DIV(minus(x0, x1), s(s(x1)))
DIV(s(x0), s(s(z1))) → IFY(true, s(x0), s(s(z1)))
The TRS R consists of the following rules:
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
ge(x, 0) → true
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
The set Q consists of the following terms:
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
minus(x0, 0)
minus(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
We use the reduction pair processor [15].
The following pairs can be oriented strictly and are deleted.
IFY(true, s(s(y_1)), s(s(x1))) → IF(ge(y_1, x1), s(s(y_1)), s(s(x1)))
IF(true, s(s(x0)), s(s(x1))) → DIV(minus(x0, x1), s(s(x1)))
DIV(s(x0), s(s(z1))) → IFY(true, s(x0), s(s(z1)))
The remaining pairs can at least be oriented weakly.
none
Used ordering: Matrix interpretation [3]:
Non-tuple symbols:
| M( minus(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
| M( ge(x1, x2) ) = | | + | | · | x1 | + | | · | x2 |
Tuple symbols:
| M( DIV(x1, x2) ) = | 1 | + | | · | x1 | + | | · | x2 |
| M( IFY(x1, ..., x3) ) = | 0 | + | | · | x1 | + | | · | x2 | + | | · | x3 |
| M( IF(x1, ..., x3) ) = | 1 | + | | · | x1 | + | | · | x2 | + | | · | x3 |
Matrix type:
We used a basic matrix type which is not further parametrizeable.
As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
The following usable rules [17] were oriented:
minus(s(x), s(y)) → minus(x, y)
minus(x, 0) → x
↳ QTRS
↳ Overlay + Local Confluence
↳ QTRS
↳ DependencyPairsProof
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ QDP
↳ UsableRulesProof
↳ QDP
↳ QReductionProof
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Rewriting
↳ QDP
↳ Narrowing
↳ QDP
↳ DependencyGraphProof
↳ QDP
↳ Instantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ Narrowing
↳ QDP
↳ Instantiation
↳ QDP
↳ Instantiation
↳ QDP
↳ DependencyGraphProof
↳ AND
↳ QDP
↳ QDP
↳ ForwardInstantiation
↳ QDP
↳ Rewriting
↳ QDP
↳ QDPOrderProof
↳ QDP
↳ PisEmptyProof
Q DP problem:
P is empty.
The TRS R consists of the following rules:
ge(0, s(x)) → false
ge(s(x), s(y)) → ge(x, y)
ge(x, 0) → true
minus(x, 0) → x
minus(s(x), s(y)) → minus(x, y)
The set Q consists of the following terms:
ge(x0, 0)
ge(0, s(x0))
ge(s(x0), s(x1))
minus(x0, 0)
minus(s(x0), s(x1))
We have to consider all minimal (P,Q,R)-chains.
The TRS P is empty. Hence, there is no (P,Q,R) chain.