/export/starexec/sandbox2/solver/bin/starexec_run_default /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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


YES

Problem 1: 

(VAR x y)
(RULES
if_quot(false,x,y) -> 0
if_quot(true,x,y) -> s(quot(minus(x,y),y))
le(0,y) -> true
le(s(x),0) -> false
le(s(x),s(y)) -> le(x,y)
minus(s(x),s(y)) -> minus(x,y)
minus(x,0) -> x
quot(x,s(y)) -> if_quot(le(s(y),x),x,s(y))
)

Problem 1: 

Innermost Equivalent Processor:
-> Rules:
 if_quot(false,x,y) -> 0
 if_quot(true,x,y) -> s(quot(minus(x,y),y))
 le(0,y) -> true
 le(s(x),0) -> false
 le(s(x),s(y)) -> le(x,y)
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
 quot(x,s(y)) -> if_quot(le(s(y),x),x,s(y))
-> The term rewriting system is non-overlaping or locally confluent overlay system. Therefore, innermost termination implies termination.
 

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 IF_QUOT(true,x,y) -> MINUS(x,y)
 IF_QUOT(true,x,y) -> QUOT(minus(x,y),y)
 LE(s(x),s(y)) -> LE(x,y)
 MINUS(s(x),s(y)) -> MINUS(x,y)
 QUOT(x,s(y)) -> IF_QUOT(le(s(y),x),x,s(y))
 QUOT(x,s(y)) -> LE(s(y),x)
-> Rules:
 if_quot(false,x,y) -> 0
 if_quot(true,x,y) -> s(quot(minus(x,y),y))
 le(0,y) -> true
 le(s(x),0) -> false
 le(s(x),s(y)) -> le(x,y)
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
 quot(x,s(y)) -> if_quot(le(s(y),x),x,s(y))

Problem 1: 

SCC Processor:
-> Pairs:
 IF_QUOT(true,x,y) -> MINUS(x,y)
 IF_QUOT(true,x,y) -> QUOT(minus(x,y),y)
 LE(s(x),s(y)) -> LE(x,y)
 MINUS(s(x),s(y)) -> MINUS(x,y)
 QUOT(x,s(y)) -> IF_QUOT(le(s(y),x),x,s(y))
 QUOT(x,s(y)) -> LE(s(y),x)
-> Rules:
 if_quot(false,x,y) -> 0
 if_quot(true,x,y) -> s(quot(minus(x,y),y))
 le(0,y) -> true
 le(s(x),0) -> false
 le(s(x),s(y)) -> le(x,y)
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
 quot(x,s(y)) -> if_quot(le(s(y),x),x,s(y))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 MINUS(s(x),s(y)) -> MINUS(x,y)
->->-> Rules:
 if_quot(false,x,y) -> 0
 if_quot(true,x,y) -> s(quot(minus(x,y),y))
 le(0,y) -> true
 le(s(x),0) -> false
 le(s(x),s(y)) -> le(x,y)
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
 quot(x,s(y)) -> if_quot(le(s(y),x),x,s(y))
->->Cycle:
->->-> Pairs:
 LE(s(x),s(y)) -> LE(x,y)
->->-> Rules:
 if_quot(false,x,y) -> 0
 if_quot(true,x,y) -> s(quot(minus(x,y),y))
 le(0,y) -> true
 le(s(x),0) -> false
 le(s(x),s(y)) -> le(x,y)
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
 quot(x,s(y)) -> if_quot(le(s(y),x),x,s(y))
->->Cycle:
->->-> Pairs:
 IF_QUOT(true,x,y) -> QUOT(minus(x,y),y)
 QUOT(x,s(y)) -> IF_QUOT(le(s(y),x),x,s(y))
->->-> Rules:
 if_quot(false,x,y) -> 0
 if_quot(true,x,y) -> s(quot(minus(x,y),y))
 le(0,y) -> true
 le(s(x),0) -> false
 le(s(x),s(y)) -> le(x,y)
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
 quot(x,s(y)) -> if_quot(le(s(y),x),x,s(y))


The problem is decomposed in 3 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 MINUS(s(x),s(y)) -> MINUS(x,y)
-> Rules:
 if_quot(false,x,y) -> 0
 if_quot(true,x,y) -> s(quot(minus(x,y),y))
 le(0,y) -> true
 le(s(x),0) -> false
 le(s(x),s(y)) -> le(x,y)
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
 quot(x,s(y)) -> if_quot(le(s(y),x),x,s(y))
->Projection:
 pi(MINUS) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 if_quot(false,x,y) -> 0
 if_quot(true,x,y) -> s(quot(minus(x,y),y))
 le(0,y) -> true
 le(s(x),0) -> false
 le(s(x),s(y)) -> le(x,y)
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
 quot(x,s(y)) -> if_quot(le(s(y),x),x,s(y))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 LE(s(x),s(y)) -> LE(x,y)
-> Rules:
 if_quot(false,x,y) -> 0
 if_quot(true,x,y) -> s(quot(minus(x,y),y))
 le(0,y) -> true
 le(s(x),0) -> false
 le(s(x),s(y)) -> le(x,y)
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
 quot(x,s(y)) -> if_quot(le(s(y),x),x,s(y))
->Projection:
 pi(LE) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 if_quot(false,x,y) -> 0
 if_quot(true,x,y) -> s(quot(minus(x,y),y))
 le(0,y) -> true
 le(s(x),0) -> false
 le(s(x),s(y)) -> le(x,y)
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
 quot(x,s(y)) -> if_quot(le(s(y),x),x,s(y))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

Narrowing Processor:
-> Pairs:
 IF_QUOT(true,x,y) -> QUOT(minus(x,y),y)
 QUOT(x,s(y)) -> IF_QUOT(le(s(y),x),x,s(y))
-> Rules:
 if_quot(false,x,y) -> 0
 if_quot(true,x,y) -> s(quot(minus(x,y),y))
 le(0,y) -> true
 le(s(x),0) -> false
 le(s(x),s(y)) -> le(x,y)
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
 quot(x,s(y)) -> if_quot(le(s(y),x),x,s(y))
->Narrowed Pairs:
->->Original Pair:
 QUOT(x,s(y)) -> IF_QUOT(le(s(y),x),x,s(y))
->-> Narrowed pairs:
 QUOT(0,s(x)) -> IF_QUOT(false,0,s(x))
 QUOT(s(y),s(x)) -> IF_QUOT(le(x,y),s(y),s(x))

Problem 1.3: 

SCC Processor:
-> Pairs:
 IF_QUOT(true,x,y) -> QUOT(minus(x,y),y)
 QUOT(0,s(x)) -> IF_QUOT(false,0,s(x))
 QUOT(s(y),s(x)) -> IF_QUOT(le(x,y),s(y),s(x))
-> Rules:
 if_quot(false,x,y) -> 0
 if_quot(true,x,y) -> s(quot(minus(x,y),y))
 le(0,y) -> true
 le(s(x),0) -> false
 le(s(x),s(y)) -> le(x,y)
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
 quot(x,s(y)) -> if_quot(le(s(y),x),x,s(y))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 IF_QUOT(true,x,y) -> QUOT(minus(x,y),y)
 QUOT(s(y),s(x)) -> IF_QUOT(le(x,y),s(y),s(x))
->->-> Rules:
 if_quot(false,x,y) -> 0
 if_quot(true,x,y) -> s(quot(minus(x,y),y))
 le(0,y) -> true
 le(s(x),0) -> false
 le(s(x),s(y)) -> le(x,y)
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
 quot(x,s(y)) -> if_quot(le(s(y),x),x,s(y))

Problem 1.3: 

Narrowing Processor:
-> Pairs:
 IF_QUOT(true,x,y) -> QUOT(minus(x,y),y)
 QUOT(s(y),s(x)) -> IF_QUOT(le(x,y),s(y),s(x))
-> Rules:
 if_quot(false,x,y) -> 0
 if_quot(true,x,y) -> s(quot(minus(x,y),y))
 le(0,y) -> true
 le(s(x),0) -> false
 le(s(x),s(y)) -> le(x,y)
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
 quot(x,s(y)) -> if_quot(le(s(y),x),x,s(y))
->Narrowed Pairs:
->->Original Pair:
 IF_QUOT(true,x,y) -> QUOT(minus(x,y),y)
->-> Narrowed pairs:
 IF_QUOT(true,s(x),s(y)) -> QUOT(minus(x,y),s(y))
 IF_QUOT(true,x,0) -> QUOT(x,0)
->->Original Pair:
 QUOT(s(y),s(x)) -> IF_QUOT(le(x,y),s(y),s(x))
->-> Narrowed pairs:
 QUOT(s(0),s(s(x))) -> IF_QUOT(false,s(0),s(s(x)))
 QUOT(s(s(y)),s(s(x))) -> IF_QUOT(le(x,y),s(s(y)),s(s(x)))
 QUOT(s(y),s(0)) -> IF_QUOT(true,s(y),s(0))

Problem 1.3: 

SCC Processor:
-> Pairs:
 IF_QUOT(true,s(x),s(y)) -> QUOT(minus(x,y),s(y))
 IF_QUOT(true,x,0) -> QUOT(x,0)
 QUOT(s(0),s(s(x))) -> IF_QUOT(false,s(0),s(s(x)))
 QUOT(s(s(y)),s(s(x))) -> IF_QUOT(le(x,y),s(s(y)),s(s(x)))
 QUOT(s(y),s(0)) -> IF_QUOT(true,s(y),s(0))
-> Rules:
 if_quot(false,x,y) -> 0
 if_quot(true,x,y) -> s(quot(minus(x,y),y))
 le(0,y) -> true
 le(s(x),0) -> false
 le(s(x),s(y)) -> le(x,y)
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
 quot(x,s(y)) -> if_quot(le(s(y),x),x,s(y))
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 IF_QUOT(true,s(x),s(y)) -> QUOT(minus(x,y),s(y))
 QUOT(s(s(y)),s(s(x))) -> IF_QUOT(le(x,y),s(s(y)),s(s(x)))
 QUOT(s(y),s(0)) -> IF_QUOT(true,s(y),s(0))
->->-> Rules:
 if_quot(false,x,y) -> 0
 if_quot(true,x,y) -> s(quot(minus(x,y),y))
 le(0,y) -> true
 le(s(x),0) -> false
 le(s(x),s(y)) -> le(x,y)
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
 quot(x,s(y)) -> if_quot(le(s(y),x),x,s(y))

Problem 1.3: 

Reduction Pairs Processor:
-> Pairs:
 IF_QUOT(true,s(x),s(y)) -> QUOT(minus(x,y),s(y))
 QUOT(s(s(y)),s(s(x))) -> IF_QUOT(le(x,y),s(s(y)),s(s(x)))
 QUOT(s(y),s(0)) -> IF_QUOT(true,s(y),s(0))
-> Rules:
 if_quot(false,x,y) -> 0
 if_quot(true,x,y) -> s(quot(minus(x,y),y))
 le(0,y) -> true
 le(s(x),0) -> false
 le(s(x),s(y)) -> le(x,y)
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
 quot(x,s(y)) -> if_quot(le(s(y),x),x,s(y))
-> Usable rules:
 le(0,y) -> true
 le(s(x),0) -> false
 le(s(x),s(y)) -> le(x,y)
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[le](X1,X2) = 0
[minus](X1,X2) = X1
[0] = 2
[false] = 0
[s](X) = X + 2
[true] = 0
[IF_QUOT](X1,X2,X3) = 2.X1 + 2.X2 + 2.X3 + 2
[QUOT](X1,X2) = 2.X1 + 2.X2 + 2

Problem 1.3: 

SCC Processor:
-> Pairs:
 QUOT(s(s(y)),s(s(x))) -> IF_QUOT(le(x,y),s(s(y)),s(s(x)))
 QUOT(s(y),s(0)) -> IF_QUOT(true,s(y),s(0))
-> Rules:
 if_quot(false,x,y) -> 0
 if_quot(true,x,y) -> s(quot(minus(x,y),y))
 le(0,y) -> true
 le(s(x),0) -> false
 le(s(x),s(y)) -> le(x,y)
 minus(s(x),s(y)) -> minus(x,y)
 minus(x,0) -> x
 quot(x,s(y)) -> if_quot(le(s(y),x),x,s(y))
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.