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


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


YES

Problem 1: 

(VAR v_NonEmpty:S l:S x:S y:S)
(RULES
le(0,s(x:S)) -> ttrue
le(s(x:S),s(y:S)) -> le(x:S,y:S)
le(x:S,0) -> ffalse
min(cons(x:S,nil)) -> x:S
min(cons(x:S,l:S)) -> min(l:S) | le(x:S,min(l:S)) ->* ffalse
min(cons(x:S,l:S)) -> min(l:S) | min(l:S) ->* x:S
min(cons(x:S,l:S)) -> x:S | le(x:S,min(l:S)) ->* ttrue
)

Problem 1: 
Valid CTRS Processor:
-> Rules:
 le(0,s(x:S)) -> ttrue
 le(s(x:S),s(y:S)) -> le(x:S,y:S)
 le(x:S,0) -> ffalse
 min(cons(x:S,nil)) -> x:S
 min(cons(x:S,l:S)) -> min(l:S) | le(x:S,min(l:S)) ->* ffalse
 min(cons(x:S,l:S)) -> min(l:S) | min(l:S) ->* x:S
 min(cons(x:S,l:S)) -> x:S | le(x:S,min(l:S)) ->* ttrue
-> The system is a deterministic 3-CTRS.

Problem 1: 

Dependency Pairs Processor:

Conditional Termination Problem 1:
-> Pairs:
 LE(s(x:S),s(y:S)) -> LE(x:S,y:S)
 MIN(cons(x:S,l:S)) -> MIN(l:S) | le(x:S,min(l:S)) ->* ffalse
 MIN(cons(x:S,l:S)) -> MIN(l:S) | min(l:S) ->* x:S
-> QPairs:
 Empty
-> Rules:
 le(0,s(x:S)) -> ttrue
 le(s(x:S),s(y:S)) -> le(x:S,y:S)
 le(x:S,0) -> ffalse
 min(cons(x:S,nil)) -> x:S
 min(cons(x:S,l:S)) -> min(l:S) | le(x:S,min(l:S)) ->* ffalse
 min(cons(x:S,l:S)) -> min(l:S) | min(l:S) ->* x:S
 min(cons(x:S,l:S)) -> x:S | le(x:S,min(l:S)) ->* ttrue

Conditional Termination Problem 2:
-> Pairs:
 MIN(cons(x:S,l:S)) -> LE(x:S,min(l:S))
 MIN(cons(x:S,l:S)) -> MIN(l:S)
-> QPairs:
 LE(s(x:S),s(y:S)) -> LE(x:S,y:S)
 MIN(cons(x:S,l:S)) -> MIN(l:S) | le(x:S,min(l:S)) ->* ffalse
 MIN(cons(x:S,l:S)) -> MIN(l:S) | min(l:S) ->* x:S
-> Rules:
 le(0,s(x:S)) -> ttrue
 le(s(x:S),s(y:S)) -> le(x:S,y:S)
 le(x:S,0) -> ffalse
 min(cons(x:S,nil)) -> x:S
 min(cons(x:S,l:S)) -> min(l:S) | le(x:S,min(l:S)) ->* ffalse
 min(cons(x:S,l:S)) -> min(l:S) | min(l:S) ->* x:S
 min(cons(x:S,l:S)) -> x:S | le(x:S,min(l:S)) ->* ttrue


The problem is decomposed in 2 subproblems.

Problem 1.1: 

SCC Processor:
-> Pairs:
 LE(s(x:S),s(y:S)) -> LE(x:S,y:S)
 MIN(cons(x:S,l:S)) -> MIN(l:S) | le(x:S,min(l:S)) ->* ffalse
 MIN(cons(x:S,l:S)) -> MIN(l:S) | min(l:S) ->* x:S
-> QPairs:
 Empty
-> Rules:
 le(0,s(x:S)) -> ttrue
 le(s(x:S),s(y:S)) -> le(x:S,y:S)
 le(x:S,0) -> ffalse
 min(cons(x:S,nil)) -> x:S
 min(cons(x:S,l:S)) -> min(l:S) | le(x:S,min(l:S)) ->* ffalse
 min(cons(x:S,l:S)) -> min(l:S) | min(l:S) ->* x:S
 min(cons(x:S,l:S)) -> x:S | le(x:S,min(l:S)) ->* ttrue
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 MIN(cons(x:S,l:S)) -> MIN(l:S) | le(x:S,min(l:S)) ->* ffalse
 MIN(cons(x:S,l:S)) -> MIN(l:S) | min(l:S) ->* x:S
-> QPairs:
 Empty
->->-> Rules:
 le(0,s(x:S)) -> ttrue
 le(s(x:S),s(y:S)) -> le(x:S,y:S)
 le(x:S,0) -> ffalse
 min(cons(x:S,nil)) -> x:S
 min(cons(x:S,l:S)) -> min(l:S) | le(x:S,min(l:S)) ->* ffalse
 min(cons(x:S,l:S)) -> min(l:S) | min(l:S) ->* x:S
 min(cons(x:S,l:S)) -> x:S | le(x:S,min(l:S)) ->* ttrue
->->Cycle:
->->-> Pairs:
 LE(s(x:S),s(y:S)) -> LE(x:S,y:S)
-> QPairs:
 Empty
->->-> Rules:
 le(0,s(x:S)) -> ttrue
 le(s(x:S),s(y:S)) -> le(x:S,y:S)
 le(x:S,0) -> ffalse
 min(cons(x:S,nil)) -> x:S
 min(cons(x:S,l:S)) -> min(l:S) | le(x:S,min(l:S)) ->* ffalse
 min(cons(x:S,l:S)) -> min(l:S) | min(l:S) ->* x:S
 min(cons(x:S,l:S)) -> x:S | le(x:S,min(l:S)) ->* ttrue


The problem is decomposed in 2 subproblems.

Problem 1.1.1: 

Conditional Subterm Processor:
-> Pairs:
 MIN(cons(x:S,l:S)) -> MIN(l:S) | le(x:S,min(l:S)) ->* ffalse
 MIN(cons(x:S,l:S)) -> MIN(l:S) | min(l:S) ->* x:S
-> QPairs:
 Empty
-> Rules:
 le(0,s(x:S)) -> ttrue
 le(s(x:S),s(y:S)) -> le(x:S,y:S)
 le(x:S,0) -> ffalse
 min(cons(x:S,nil)) -> x:S
 min(cons(x:S,l:S)) -> min(l:S) | le(x:S,min(l:S)) ->* ffalse
 min(cons(x:S,l:S)) -> min(l:S) | min(l:S) ->* x:S
 min(cons(x:S,l:S)) -> x:S | le(x:S,min(l:S)) ->* ttrue
->Projection:
 pi(MIN) = 1

Problem 1.1.1: 

SCC Processor:
-> Pairs:
 Empty
-> QPairs:
 Empty
-> Rules:
 le(0,s(x:S)) -> ttrue
 le(s(x:S),s(y:S)) -> le(x:S,y:S)
 le(x:S,0) -> ffalse
 min(cons(x:S,nil)) -> x:S
 min(cons(x:S,l:S)) -> min(l:S) | le(x:S,min(l:S)) ->* ffalse
 min(cons(x:S,l:S)) -> min(l:S) | min(l:S) ->* x:S
 min(cons(x:S,l:S)) -> x:S | le(x:S,min(l:S)) ->* ttrue
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.1.2: 

Conditional Subterm Processor:
-> Pairs:
 LE(s(x:S),s(y:S)) -> LE(x:S,y:S)
-> QPairs:
 Empty
-> Rules:
 le(0,s(x:S)) -> ttrue
 le(s(x:S),s(y:S)) -> le(x:S,y:S)
 le(x:S,0) -> ffalse
 min(cons(x:S,nil)) -> x:S
 min(cons(x:S,l:S)) -> min(l:S) | le(x:S,min(l:S)) ->* ffalse
 min(cons(x:S,l:S)) -> min(l:S) | min(l:S) ->* x:S
 min(cons(x:S,l:S)) -> x:S | le(x:S,min(l:S)) ->* ttrue
->Projection:
 pi(LE) = 1

Problem 1.1.2: 

SCC Processor:
-> Pairs:
 Empty
-> QPairs:
 Empty
-> Rules:
 le(0,s(x:S)) -> ttrue
 le(s(x:S),s(y:S)) -> le(x:S,y:S)
 le(x:S,0) -> ffalse
 min(cons(x:S,nil)) -> x:S
 min(cons(x:S,l:S)) -> min(l:S) | le(x:S,min(l:S)) ->* ffalse
 min(cons(x:S,l:S)) -> min(l:S) | min(l:S) ->* x:S
 min(cons(x:S,l:S)) -> x:S | le(x:S,min(l:S)) ->* ttrue
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

SCC Processor:
-> Pairs:
 MIN(cons(x:S,l:S)) -> LE(x:S,min(l:S))
 MIN(cons(x:S,l:S)) -> MIN(l:S)
-> QPairs:
 LE(s(x:S),s(y:S)) -> LE(x:S,y:S)
 MIN(cons(x:S,l:S)) -> MIN(l:S) | le(x:S,min(l:S)) ->* ffalse
 MIN(cons(x:S,l:S)) -> MIN(l:S) | min(l:S) ->* x:S
-> Rules:
 le(0,s(x:S)) -> ttrue
 le(s(x:S),s(y:S)) -> le(x:S,y:S)
 le(x:S,0) -> ffalse
 min(cons(x:S,nil)) -> x:S
 min(cons(x:S,l:S)) -> min(l:S) | le(x:S,min(l:S)) ->* ffalse
 min(cons(x:S,l:S)) -> min(l:S) | min(l:S) ->* x:S
 min(cons(x:S,l:S)) -> x:S | le(x:S,min(l:S)) ->* ttrue
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 MIN(cons(x:S,l:S)) -> MIN(l:S)
-> QPairs:
 MIN(cons(x:S,l:S)) -> MIN(l:S) | le(x:S,min(l:S)) ->* ffalse
 MIN(cons(x:S,l:S)) -> MIN(l:S) | min(l:S) ->* x:S
->->-> Rules:
 le(0,s(x:S)) -> ttrue
 le(s(x:S),s(y:S)) -> le(x:S,y:S)
 le(x:S,0) -> ffalse
 min(cons(x:S,nil)) -> x:S
 min(cons(x:S,l:S)) -> min(l:S) | le(x:S,min(l:S)) ->* ffalse
 min(cons(x:S,l:S)) -> min(l:S) | min(l:S) ->* x:S
 min(cons(x:S,l:S)) -> x:S | le(x:S,min(l:S)) ->* ttrue

Problem 1.2: 

Conditional Subterm Processor:
-> Pairs:
 MIN(cons(x:S,l:S)) -> MIN(l:S)
-> QPairs:
 MIN(cons(x:S,l:S)) -> MIN(l:S) | le(x:S,min(l:S)) ->* ffalse
 MIN(cons(x:S,l:S)) -> MIN(l:S) | min(l:S) ->* x:S
-> Rules:
 le(0,s(x:S)) -> ttrue
 le(s(x:S),s(y:S)) -> le(x:S,y:S)
 le(x:S,0) -> ffalse
 min(cons(x:S,nil)) -> x:S
 min(cons(x:S,l:S)) -> min(l:S) | le(x:S,min(l:S)) ->* ffalse
 min(cons(x:S,l:S)) -> min(l:S) | min(l:S) ->* x:S
 min(cons(x:S,l:S)) -> x:S | le(x:S,min(l:S)) ->* ttrue
->Projection:
 pi(MIN) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> QPairs:
 Empty
-> Rules:
 le(0,s(x:S)) -> ttrue
 le(s(x:S),s(y:S)) -> le(x:S,y:S)
 le(x:S,0) -> ffalse
 min(cons(x:S,nil)) -> x:S
 min(cons(x:S,l:S)) -> min(l:S) | le(x:S,min(l:S)) ->* ffalse
 min(cons(x:S,l:S)) -> min(l:S) | min(l:S) ->* x:S
 min(cons(x:S,l:S)) -> x:S | le(x:S,min(l:S)) ->* ttrue
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.