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


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YES

Problem 1: 

(VAR h i u v x y)
(RULES
eq(0,0) -> true
eq(0,s(x)) -> false
eq(s(x),0) -> false
eq(s(x),s(y)) -> eq(x,y)
if_reach_1(false,x,y,edge(u,v,i),h) -> reach(x,y,i,edge(u,v,h))
if_reach_1(true,x,y,edge(u,v,i),h) -> if_reach_2(eq(y,v),x,y,edge(u,v,i),h)
if_reach_2(false,x,y,edge(u,v,i),h) -> or(reach(x,y,i,h),reach(v,y,union(i,h),empty))
if_reach_2(true,x,y,edge(u,v,i),h) -> true
or(false,y) -> y
or(true,y) -> true
reach(x,y,edge(u,v,i),h) -> if_reach_1(eq(x,u),x,y,edge(u,v,i),h)
reach(x,y,empty,h) -> false
union(edge(x,y,i),h) -> edge(x,y,union(i,h))
union(empty,h) -> h
)

Problem 1: 

Innermost Equivalent Processor:
-> Rules:
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),0) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if_reach_1(false,x,y,edge(u,v,i),h) -> reach(x,y,i,edge(u,v,h))
 if_reach_1(true,x,y,edge(u,v,i),h) -> if_reach_2(eq(y,v),x,y,edge(u,v,i),h)
 if_reach_2(false,x,y,edge(u,v,i),h) -> or(reach(x,y,i,h),reach(v,y,union(i,h),empty))
 if_reach_2(true,x,y,edge(u,v,i),h) -> true
 or(false,y) -> y
 or(true,y) -> true
 reach(x,y,edge(u,v,i),h) -> if_reach_1(eq(x,u),x,y,edge(u,v,i),h)
 reach(x,y,empty,h) -> false
 union(edge(x,y,i),h) -> edge(x,y,union(i,h))
 union(empty,h) -> h
-> The term rewriting system is non-overlaping or locally confluent overlay system. Therefore, innermost termination implies termination.
 

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 EQ(s(x),s(y)) -> EQ(x,y)
 IF_REACH_1(false,x,y,edge(u,v,i),h) -> REACH(x,y,i,edge(u,v,h))
 IF_REACH_1(true,x,y,edge(u,v,i),h) -> EQ(y,v)
 IF_REACH_1(true,x,y,edge(u,v,i),h) -> IF_REACH_2(eq(y,v),x,y,edge(u,v,i),h)
 IF_REACH_2(false,x,y,edge(u,v,i),h) -> OR(reach(x,y,i,h),reach(v,y,union(i,h),empty))
 IF_REACH_2(false,x,y,edge(u,v,i),h) -> REACH(v,y,union(i,h),empty)
 IF_REACH_2(false,x,y,edge(u,v,i),h) -> REACH(x,y,i,h)
 IF_REACH_2(false,x,y,edge(u,v,i),h) -> UNION(i,h)
 REACH(x,y,edge(u,v,i),h) -> EQ(x,u)
 REACH(x,y,edge(u,v,i),h) -> IF_REACH_1(eq(x,u),x,y,edge(u,v,i),h)
 UNION(edge(x,y,i),h) -> UNION(i,h)
-> Rules:
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),0) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if_reach_1(false,x,y,edge(u,v,i),h) -> reach(x,y,i,edge(u,v,h))
 if_reach_1(true,x,y,edge(u,v,i),h) -> if_reach_2(eq(y,v),x,y,edge(u,v,i),h)
 if_reach_2(false,x,y,edge(u,v,i),h) -> or(reach(x,y,i,h),reach(v,y,union(i,h),empty))
 if_reach_2(true,x,y,edge(u,v,i),h) -> true
 or(false,y) -> y
 or(true,y) -> true
 reach(x,y,edge(u,v,i),h) -> if_reach_1(eq(x,u),x,y,edge(u,v,i),h)
 reach(x,y,empty,h) -> false
 union(edge(x,y,i),h) -> edge(x,y,union(i,h))
 union(empty,h) -> h

Problem 1: 

SCC Processor:
-> Pairs:
 EQ(s(x),s(y)) -> EQ(x,y)
 IF_REACH_1(false,x,y,edge(u,v,i),h) -> REACH(x,y,i,edge(u,v,h))
 IF_REACH_1(true,x,y,edge(u,v,i),h) -> EQ(y,v)
 IF_REACH_1(true,x,y,edge(u,v,i),h) -> IF_REACH_2(eq(y,v),x,y,edge(u,v,i),h)
 IF_REACH_2(false,x,y,edge(u,v,i),h) -> OR(reach(x,y,i,h),reach(v,y,union(i,h),empty))
 IF_REACH_2(false,x,y,edge(u,v,i),h) -> REACH(v,y,union(i,h),empty)
 IF_REACH_2(false,x,y,edge(u,v,i),h) -> REACH(x,y,i,h)
 IF_REACH_2(false,x,y,edge(u,v,i),h) -> UNION(i,h)
 REACH(x,y,edge(u,v,i),h) -> EQ(x,u)
 REACH(x,y,edge(u,v,i),h) -> IF_REACH_1(eq(x,u),x,y,edge(u,v,i),h)
 UNION(edge(x,y,i),h) -> UNION(i,h)
-> Rules:
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),0) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if_reach_1(false,x,y,edge(u,v,i),h) -> reach(x,y,i,edge(u,v,h))
 if_reach_1(true,x,y,edge(u,v,i),h) -> if_reach_2(eq(y,v),x,y,edge(u,v,i),h)
 if_reach_2(false,x,y,edge(u,v,i),h) -> or(reach(x,y,i,h),reach(v,y,union(i,h),empty))
 if_reach_2(true,x,y,edge(u,v,i),h) -> true
 or(false,y) -> y
 or(true,y) -> true
 reach(x,y,edge(u,v,i),h) -> if_reach_1(eq(x,u),x,y,edge(u,v,i),h)
 reach(x,y,empty,h) -> false
 union(edge(x,y,i),h) -> edge(x,y,union(i,h))
 union(empty,h) -> h
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 UNION(edge(x,y,i),h) -> UNION(i,h)
->->-> Rules:
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),0) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if_reach_1(false,x,y,edge(u,v,i),h) -> reach(x,y,i,edge(u,v,h))
 if_reach_1(true,x,y,edge(u,v,i),h) -> if_reach_2(eq(y,v),x,y,edge(u,v,i),h)
 if_reach_2(false,x,y,edge(u,v,i),h) -> or(reach(x,y,i,h),reach(v,y,union(i,h),empty))
 if_reach_2(true,x,y,edge(u,v,i),h) -> true
 or(false,y) -> y
 or(true,y) -> true
 reach(x,y,edge(u,v,i),h) -> if_reach_1(eq(x,u),x,y,edge(u,v,i),h)
 reach(x,y,empty,h) -> false
 union(edge(x,y,i),h) -> edge(x,y,union(i,h))
 union(empty,h) -> h
->->Cycle:
->->-> Pairs:
 EQ(s(x),s(y)) -> EQ(x,y)
->->-> Rules:
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),0) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if_reach_1(false,x,y,edge(u,v,i),h) -> reach(x,y,i,edge(u,v,h))
 if_reach_1(true,x,y,edge(u,v,i),h) -> if_reach_2(eq(y,v),x,y,edge(u,v,i),h)
 if_reach_2(false,x,y,edge(u,v,i),h) -> or(reach(x,y,i,h),reach(v,y,union(i,h),empty))
 if_reach_2(true,x,y,edge(u,v,i),h) -> true
 or(false,y) -> y
 or(true,y) -> true
 reach(x,y,edge(u,v,i),h) -> if_reach_1(eq(x,u),x,y,edge(u,v,i),h)
 reach(x,y,empty,h) -> false
 union(edge(x,y,i),h) -> edge(x,y,union(i,h))
 union(empty,h) -> h
->->Cycle:
->->-> Pairs:
 IF_REACH_1(false,x,y,edge(u,v,i),h) -> REACH(x,y,i,edge(u,v,h))
 IF_REACH_1(true,x,y,edge(u,v,i),h) -> IF_REACH_2(eq(y,v),x,y,edge(u,v,i),h)
 IF_REACH_2(false,x,y,edge(u,v,i),h) -> REACH(v,y,union(i,h),empty)
 IF_REACH_2(false,x,y,edge(u,v,i),h) -> REACH(x,y,i,h)
 REACH(x,y,edge(u,v,i),h) -> IF_REACH_1(eq(x,u),x,y,edge(u,v,i),h)
->->-> Rules:
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),0) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if_reach_1(false,x,y,edge(u,v,i),h) -> reach(x,y,i,edge(u,v,h))
 if_reach_1(true,x,y,edge(u,v,i),h) -> if_reach_2(eq(y,v),x,y,edge(u,v,i),h)
 if_reach_2(false,x,y,edge(u,v,i),h) -> or(reach(x,y,i,h),reach(v,y,union(i,h),empty))
 if_reach_2(true,x,y,edge(u,v,i),h) -> true
 or(false,y) -> y
 or(true,y) -> true
 reach(x,y,edge(u,v,i),h) -> if_reach_1(eq(x,u),x,y,edge(u,v,i),h)
 reach(x,y,empty,h) -> false
 union(edge(x,y,i),h) -> edge(x,y,union(i,h))
 union(empty,h) -> h


The problem is decomposed in 3 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 UNION(edge(x,y,i),h) -> UNION(i,h)
-> Rules:
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),0) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if_reach_1(false,x,y,edge(u,v,i),h) -> reach(x,y,i,edge(u,v,h))
 if_reach_1(true,x,y,edge(u,v,i),h) -> if_reach_2(eq(y,v),x,y,edge(u,v,i),h)
 if_reach_2(false,x,y,edge(u,v,i),h) -> or(reach(x,y,i,h),reach(v,y,union(i,h),empty))
 if_reach_2(true,x,y,edge(u,v,i),h) -> true
 or(false,y) -> y
 or(true,y) -> true
 reach(x,y,edge(u,v,i),h) -> if_reach_1(eq(x,u),x,y,edge(u,v,i),h)
 reach(x,y,empty,h) -> false
 union(edge(x,y,i),h) -> edge(x,y,union(i,h))
 union(empty,h) -> h
->Projection:
 pi(UNION) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),0) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if_reach_1(false,x,y,edge(u,v,i),h) -> reach(x,y,i,edge(u,v,h))
 if_reach_1(true,x,y,edge(u,v,i),h) -> if_reach_2(eq(y,v),x,y,edge(u,v,i),h)
 if_reach_2(false,x,y,edge(u,v,i),h) -> or(reach(x,y,i,h),reach(v,y,union(i,h),empty))
 if_reach_2(true,x,y,edge(u,v,i),h) -> true
 or(false,y) -> y
 or(true,y) -> true
 reach(x,y,edge(u,v,i),h) -> if_reach_1(eq(x,u),x,y,edge(u,v,i),h)
 reach(x,y,empty,h) -> false
 union(edge(x,y,i),h) -> edge(x,y,union(i,h))
 union(empty,h) -> h
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 EQ(s(x),s(y)) -> EQ(x,y)
-> Rules:
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),0) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if_reach_1(false,x,y,edge(u,v,i),h) -> reach(x,y,i,edge(u,v,h))
 if_reach_1(true,x,y,edge(u,v,i),h) -> if_reach_2(eq(y,v),x,y,edge(u,v,i),h)
 if_reach_2(false,x,y,edge(u,v,i),h) -> or(reach(x,y,i,h),reach(v,y,union(i,h),empty))
 if_reach_2(true,x,y,edge(u,v,i),h) -> true
 or(false,y) -> y
 or(true,y) -> true
 reach(x,y,edge(u,v,i),h) -> if_reach_1(eq(x,u),x,y,edge(u,v,i),h)
 reach(x,y,empty,h) -> false
 union(edge(x,y,i),h) -> edge(x,y,union(i,h))
 union(empty,h) -> h
->Projection:
 pi(EQ) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),0) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if_reach_1(false,x,y,edge(u,v,i),h) -> reach(x,y,i,edge(u,v,h))
 if_reach_1(true,x,y,edge(u,v,i),h) -> if_reach_2(eq(y,v),x,y,edge(u,v,i),h)
 if_reach_2(false,x,y,edge(u,v,i),h) -> or(reach(x,y,i,h),reach(v,y,union(i,h),empty))
 if_reach_2(true,x,y,edge(u,v,i),h) -> true
 or(false,y) -> y
 or(true,y) -> true
 reach(x,y,edge(u,v,i),h) -> if_reach_1(eq(x,u),x,y,edge(u,v,i),h)
 reach(x,y,empty,h) -> false
 union(edge(x,y,i),h) -> edge(x,y,union(i,h))
 union(empty,h) -> h
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

Reduction Pairs Processor:
-> Pairs:
 IF_REACH_1(false,x,y,edge(u,v,i),h) -> REACH(x,y,i,edge(u,v,h))
 IF_REACH_1(true,x,y,edge(u,v,i),h) -> IF_REACH_2(eq(y,v),x,y,edge(u,v,i),h)
 IF_REACH_2(false,x,y,edge(u,v,i),h) -> REACH(v,y,union(i,h),empty)
 IF_REACH_2(false,x,y,edge(u,v,i),h) -> REACH(x,y,i,h)
 REACH(x,y,edge(u,v,i),h) -> IF_REACH_1(eq(x,u),x,y,edge(u,v,i),h)
-> Rules:
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),0) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if_reach_1(false,x,y,edge(u,v,i),h) -> reach(x,y,i,edge(u,v,h))
 if_reach_1(true,x,y,edge(u,v,i),h) -> if_reach_2(eq(y,v),x,y,edge(u,v,i),h)
 if_reach_2(false,x,y,edge(u,v,i),h) -> or(reach(x,y,i,h),reach(v,y,union(i,h),empty))
 if_reach_2(true,x,y,edge(u,v,i),h) -> true
 or(false,y) -> y
 or(true,y) -> true
 reach(x,y,edge(u,v,i),h) -> if_reach_1(eq(x,u),x,y,edge(u,v,i),h)
 reach(x,y,empty,h) -> false
 union(edge(x,y,i),h) -> edge(x,y,union(i,h))
 union(empty,h) -> h
-> Usable rules:
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),0) -> false
 eq(s(x),s(y)) -> eq(x,y)
 union(edge(x,y,i),h) -> edge(x,y,union(i,h))
 union(empty,h) -> h
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[eq](X1,X2) = 2
[union](X1,X2) = X1 + X2 + 1
[0] = 0
[edge](X1,X2,X3) = 2.X1 + 2.X2 + X3 + 2
[empty] = 0
[false] = 2
[s](X) = 2.X
[true] = 2
[IF_REACH_1](X1,X2,X3,X4,X5) = X1 + 2.X2 + X3 + 2.X4 + 2.X5
[IF_REACH_2](X1,X2,X3,X4,X5) = 2.X2 + X3 + 2.X4 + 2.X5
[REACH](X1,X2,X3,X4) = 2.X1 + X2 + 2.X3 + 2.X4 + 2

Problem 1.3: 

SCC Processor:
-> Pairs:
 IF_REACH_1(false,x,y,edge(u,v,i),h) -> REACH(x,y,i,edge(u,v,h))
 IF_REACH_2(false,x,y,edge(u,v,i),h) -> REACH(v,y,union(i,h),empty)
 IF_REACH_2(false,x,y,edge(u,v,i),h) -> REACH(x,y,i,h)
 REACH(x,y,edge(u,v,i),h) -> IF_REACH_1(eq(x,u),x,y,edge(u,v,i),h)
-> Rules:
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),0) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if_reach_1(false,x,y,edge(u,v,i),h) -> reach(x,y,i,edge(u,v,h))
 if_reach_1(true,x,y,edge(u,v,i),h) -> if_reach_2(eq(y,v),x,y,edge(u,v,i),h)
 if_reach_2(false,x,y,edge(u,v,i),h) -> or(reach(x,y,i,h),reach(v,y,union(i,h),empty))
 if_reach_2(true,x,y,edge(u,v,i),h) -> true
 or(false,y) -> y
 or(true,y) -> true
 reach(x,y,edge(u,v,i),h) -> if_reach_1(eq(x,u),x,y,edge(u,v,i),h)
 reach(x,y,empty,h) -> false
 union(edge(x,y,i),h) -> edge(x,y,union(i,h))
 union(empty,h) -> h
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 IF_REACH_1(false,x,y,edge(u,v,i),h) -> REACH(x,y,i,edge(u,v,h))
 REACH(x,y,edge(u,v,i),h) -> IF_REACH_1(eq(x,u),x,y,edge(u,v,i),h)
->->-> Rules:
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),0) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if_reach_1(false,x,y,edge(u,v,i),h) -> reach(x,y,i,edge(u,v,h))
 if_reach_1(true,x,y,edge(u,v,i),h) -> if_reach_2(eq(y,v),x,y,edge(u,v,i),h)
 if_reach_2(false,x,y,edge(u,v,i),h) -> or(reach(x,y,i,h),reach(v,y,union(i,h),empty))
 if_reach_2(true,x,y,edge(u,v,i),h) -> true
 or(false,y) -> y
 or(true,y) -> true
 reach(x,y,edge(u,v,i),h) -> if_reach_1(eq(x,u),x,y,edge(u,v,i),h)
 reach(x,y,empty,h) -> false
 union(edge(x,y,i),h) -> edge(x,y,union(i,h))
 union(empty,h) -> h

Problem 1.3: 

Subterm Processor:
-> Pairs:
 IF_REACH_1(false,x,y,edge(u,v,i),h) -> REACH(x,y,i,edge(u,v,h))
 REACH(x,y,edge(u,v,i),h) -> IF_REACH_1(eq(x,u),x,y,edge(u,v,i),h)
-> Rules:
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),0) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if_reach_1(false,x,y,edge(u,v,i),h) -> reach(x,y,i,edge(u,v,h))
 if_reach_1(true,x,y,edge(u,v,i),h) -> if_reach_2(eq(y,v),x,y,edge(u,v,i),h)
 if_reach_2(false,x,y,edge(u,v,i),h) -> or(reach(x,y,i,h),reach(v,y,union(i,h),empty))
 if_reach_2(true,x,y,edge(u,v,i),h) -> true
 or(false,y) -> y
 or(true,y) -> true
 reach(x,y,edge(u,v,i),h) -> if_reach_1(eq(x,u),x,y,edge(u,v,i),h)
 reach(x,y,empty,h) -> false
 union(edge(x,y,i),h) -> edge(x,y,union(i,h))
 union(empty,h) -> h
->Projection:
 pi(IF_REACH_1) = 4
 pi(REACH) = 3

Problem 1.3: 

SCC Processor:
-> Pairs:
 REACH(x,y,edge(u,v,i),h) -> IF_REACH_1(eq(x,u),x,y,edge(u,v,i),h)
-> Rules:
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),0) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if_reach_1(false,x,y,edge(u,v,i),h) -> reach(x,y,i,edge(u,v,h))
 if_reach_1(true,x,y,edge(u,v,i),h) -> if_reach_2(eq(y,v),x,y,edge(u,v,i),h)
 if_reach_2(false,x,y,edge(u,v,i),h) -> or(reach(x,y,i,h),reach(v,y,union(i,h),empty))
 if_reach_2(true,x,y,edge(u,v,i),h) -> true
 or(false,y) -> y
 or(true,y) -> true
 reach(x,y,edge(u,v,i),h) -> if_reach_1(eq(x,u),x,y,edge(u,v,i),h)
 reach(x,y,empty,h) -> false
 union(edge(x,y,i),h) -> edge(x,y,union(i,h))
 union(empty,h) -> h
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.