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


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YES

Problem 1: 

(VAR v_NonEmpty:S x:S y:S z:S)
(RULES
add(x:S,0) -> x:S
add(x:S,s(y:S)) -> s(add(x:S,y:S))
quad(x:S) -> z:S | add(x:S,x:S) ->* y:S, add(y:S,y:S) ->* z:S
)

Problem 1: 
Valid CTRS Processor:
-> Rules:
 add(x:S,0) -> x:S
 add(x:S,s(y:S)) -> s(add(x:S,y:S))
 quad(x:S) -> z:S | add(x:S,x:S) ->* y:S, add(y:S,y:S) ->* z:S
-> The system is a deterministic 3-CTRS.

Problem 1: 

Dependency Pairs Processor:

Conditional Termination Problem 1:
-> Pairs:
 ADD(x:S,s(y:S)) -> ADD(x:S,y:S)
-> QPairs:
 Empty
-> Rules:
 add(x:S,0) -> x:S
 add(x:S,s(y:S)) -> s(add(x:S,y:S))
 quad(x:S) -> z:S | add(x:S,x:S) ->* y:S, add(y:S,y:S) ->* z:S

Conditional Termination Problem 2:
-> Pairs:
 QUAD(x:S) -> ADD(x:S,x:S)
 QUAD(x:S) -> ADD(y:S,y:S) | add(x:S,x:S) ->* y:S
-> QPairs:
 ADD(x:S,s(y:S)) -> ADD(x:S,y:S)
-> Rules:
 add(x:S,0) -> x:S
 add(x:S,s(y:S)) -> s(add(x:S,y:S))
 quad(x:S) -> z:S | add(x:S,x:S) ->* y:S, add(y:S,y:S) ->* z:S


The problem is decomposed in 2 subproblems.

Problem 1.1: 

SCC Processor:
-> Pairs:
 ADD(x:S,s(y:S)) -> ADD(x:S,y:S)
-> QPairs:
 Empty
-> Rules:
 add(x:S,0) -> x:S
 add(x:S,s(y:S)) -> s(add(x:S,y:S))
 quad(x:S) -> z:S | add(x:S,x:S) ->* y:S, add(y:S,y:S) ->* z:S
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 ADD(x:S,s(y:S)) -> ADD(x:S,y:S)
-> QPairs:
 Empty
->->-> Rules:
 add(x:S,0) -> x:S
 add(x:S,s(y:S)) -> s(add(x:S,y:S))
 quad(x:S) -> z:S | add(x:S,x:S) ->* y:S, add(y:S,y:S) ->* z:S

Problem 1.1: 

Conditional Subterm Processor:
-> Pairs:
 ADD(x:S,s(y:S)) -> ADD(x:S,y:S)
-> QPairs:
 Empty
-> Rules:
 add(x:S,0) -> x:S
 add(x:S,s(y:S)) -> s(add(x:S,y:S))
 quad(x:S) -> z:S | add(x:S,x:S) ->* y:S, add(y:S,y:S) ->* z:S
->Projection:
 pi(ADD) = 2

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> QPairs:
 Empty
-> Rules:
 add(x:S,0) -> x:S
 add(x:S,s(y:S)) -> s(add(x:S,y:S))
 quad(x:S) -> z:S | add(x:S,x:S) ->* y:S, add(y:S,y:S) ->* z:S
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

SCC Processor:
-> Pairs:
 QUAD(x:S) -> ADD(x:S,x:S)
 QUAD(x:S) -> ADD(y:S,y:S) | add(x:S,x:S) ->* y:S
-> QPairs:
 ADD(x:S,s(y:S)) -> ADD(x:S,y:S)
-> Rules:
 add(x:S,0) -> x:S
 add(x:S,s(y:S)) -> s(add(x:S,y:S))
 quad(x:S) -> z:S | add(x:S,x:S) ->* y:S, add(y:S,y:S) ->* z:S
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.