YES

Problem 1: 

(VAR h i u v x y)
(THEORY 
(AC or)
(C eq))
(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: 

Dependency Pairs Processor:
-> FAxioms:
 EQ(x6,x7) = EQ(x7,x6)
 OR(or(x6,x7),x8) = OR(x6,or(x7,x8))
 OR(x6,x7) = OR(x7,x6)
-> 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)
 OR(or(false,y),x6) -> OR(y,x6)
 OR(or(true,y),x6) -> OR(true,x6)
 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)
-> EAxioms:
 eq(x6,x7) = eq(x7,x6)
 or(or(x6,x7),x8) = or(x6,or(x7,x8))
 or(x6,x7) = or(x7,x6)
-> 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
-> SRules:
 OR(or(x6,x7),x8) -> OR(x6,x7)
 OR(x6,or(x7,x8)) -> OR(x7,x8)

Problem 1: 

SCC Processor:
-> FAxioms:
 EQ(x6,x7) = EQ(x7,x6)
 OR(or(x6,x7),x8) = OR(x6,or(x7,x8))
 OR(x6,x7) = OR(x7,x6)
-> 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)
 OR(or(false,y),x6) -> OR(y,x6)
 OR(or(true,y),x6) -> OR(true,x6)
 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)
-> EAxioms:
 eq(x6,x7) = eq(x7,x6)
 or(or(x6,x7),x8) = or(x6,or(x7,x8))
 or(x6,x7) = or(x7,x6)
-> 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
-> SRules:
 OR(or(x6,x7),x8) -> OR(x6,x7)
 OR(x6,or(x7,x8)) -> OR(x7,x8)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 UNION(edge(x,y,i),h) -> UNION(i,h)
-> FAxioms:
 eq(x6,x7) -> eq(x7,x6)
 or(or(x6,x7),x8) -> or(x6,or(x7,x8))
 or(x6,x7) -> or(x7,x6)
-> EAxioms:
 eq(x6,x7) = eq(x7,x6)
 or(or(x6,x7),x8) = or(x6,or(x7,x8))
 or(x6,x7) = or(x7,x6)
->->-> 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
-> SRules:
 Empty
->->Cycle:
->->-> Pairs:
 OR(or(false,y),x6) -> OR(y,x6)
 OR(or(true,y),x6) -> OR(true,x6)
-> FAxioms:
 eq(x6,x7) -> eq(x7,x6)
 or(or(x6,x7),x8) -> or(x6,or(x7,x8))
 or(x6,x7) -> or(x7,x6)
 OR(or(x6,x7),x8) -> OR(x6,or(x7,x8))
 OR(x6,x7) -> OR(x7,x6)
-> EAxioms:
 eq(x6,x7) = eq(x7,x6)
 or(or(x6,x7),x8) = or(x6,or(x7,x8))
 or(x6,x7) = or(x7,x6)
->->-> 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
-> SRules:
 OR(or(x6,x7),x8) -> OR(x6,x7)
 OR(x6,or(x7,x8)) -> OR(x7,x8)
->->Cycle:
->->-> Pairs:
 EQ(s(x),s(y)) -> EQ(x,y)
-> FAxioms:
 eq(x6,x7) -> eq(x7,x6)
 or(or(x6,x7),x8) -> or(x6,or(x7,x8))
 or(x6,x7) -> or(x7,x6)
 EQ(x6,x7) -> EQ(x7,x6)
-> EAxioms:
 eq(x6,x7) = eq(x7,x6)
 or(or(x6,x7),x8) = or(x6,or(x7,x8))
 or(x6,x7) = or(x7,x6)
->->-> 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
-> SRules:
 Empty
->->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)
-> FAxioms:
 eq(x6,x7) -> eq(x7,x6)
 or(or(x6,x7),x8) -> or(x6,or(x7,x8))
 or(x6,x7) -> or(x7,x6)
-> EAxioms:
 eq(x6,x7) = eq(x7,x6)
 or(or(x6,x7),x8) = or(x6,or(x7,x8))
 or(x6,x7) = or(x7,x6)
->->-> 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
-> SRules:
 Empty


The problem is decomposed in 4 subproblems.

Problem 1.1: 

Subterm Processor:
-> FAxioms:
 Empty
-> Pairs:
 UNION(edge(x,y,i),h) -> UNION(i,h)
-> EAxioms:
 eq(x6,x7) = eq(x7,x6)
 or(or(x6,x7),x8) = or(x6,or(x7,x8))
 or(x6,x7) = or(x7,x6)
-> 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
-> SRules:
 Empty
->Projection:
 pi(UNION) = [1]

Problem 1.1: 

SCC Processor:
-> FAxioms:
 Empty
-> Pairs:
 Empty
-> EAxioms:
 eq(x6,x7) = eq(x7,x6)
 or(or(x6,x7),x8) = or(x6,or(x7,x8))
 or(x6,x7) = or(x7,x6)
-> 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
-> SRules:
 Empty
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Reduction Pairs Processor:
-> FAxioms:
 OR(or(x6,x7),x8) = OR(x6,or(x7,x8))
 OR(x6,x7) = OR(x7,x6)
-> Pairs:
 OR(or(false,y),x6) -> OR(y,x6)
 OR(or(true,y),x6) -> OR(true,x6)
-> EAxioms:
 eq(x6,x7) = eq(x7,x6)
 or(or(x6,x7),x8) = or(x6,or(x7,x8))
 or(x6,x7) = or(x7,x6)
-> Usable Equations:
 or(or(x6,x7),x8) = or(x6,or(x7,x8))
 or(x6,x7) = or(x7,x6)
-> 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:
 or(false,y) -> y
 or(true,y) -> true
-> SRules:
 OR(or(x6,x7),x8) -> OR(x6,x7)
 OR(x6,or(x7,x8)) -> OR(x7,x8)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[eq](X1,X2) = 0
[if_reach_1](X1,X2,X3,X4,X5) = 0
[if_reach_2](X1,X2,X3,X4,X5) = 0
[or](X1,X2) = X1 + X2 + 1
[reach](X1,X2,X3,X4) = 0
[union](X1,X2) = 0
[0] = 0
[edge](X1,X2,X3) = 0
[empty] = 0
[false] = 1
[s](X) = 0
[true] = 1
[EQ](X1,X2) = 0
[IF_REACH_1](X1,X2,X3,X4,X5) = 0
[IF_REACH_2](X1,X2,X3,X4,X5) = 0
[OR](X1,X2) = X1 + X2
[REACH](X1,X2,X3,X4) = 0
[UNION](X1,X2) = 0

Problem 1.2: 

SCC Processor:
-> FAxioms:
 OR(or(x6,x7),x8) = OR(x6,or(x7,x8))
 OR(x6,x7) = OR(x7,x6)
-> Pairs:
 OR(or(true,y),x6) -> OR(true,x6)
-> EAxioms:
 eq(x6,x7) = eq(x7,x6)
 or(or(x6,x7),x8) = or(x6,or(x7,x8))
 or(x6,x7) = or(x7,x6)
-> 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
-> SRules:
 OR(or(x6,x7),x8) -> OR(x6,x7)
 OR(x6,or(x7,x8)) -> OR(x7,x8)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 OR(or(true,y),x6) -> OR(true,x6)
-> FAxioms:
 eq(x6,x7) -> eq(x7,x6)
 or(or(x6,x7),x8) -> or(x6,or(x7,x8))
 or(x6,x7) -> or(x7,x6)
 OR(or(x6,x7),x8) -> OR(x6,or(x7,x8))
 OR(x6,x7) -> OR(x7,x6)
-> EAxioms:
 eq(x6,x7) = eq(x7,x6)
 or(or(x6,x7),x8) = or(x6,or(x7,x8))
 or(x6,x7) = or(x7,x6)
->->-> 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
-> SRules:
 OR(or(x6,x7),x8) -> OR(x6,x7)
 OR(x6,or(x7,x8)) -> OR(x7,x8)

Problem 1.2: 

Reduction Pairs Processor:
-> FAxioms:
 OR(or(x6,x7),x8) = OR(x6,or(x7,x8))
 OR(x6,x7) = OR(x7,x6)
-> Pairs:
 OR(or(true,y),x6) -> OR(true,x6)
-> EAxioms:
 eq(x6,x7) = eq(x7,x6)
 or(or(x6,x7),x8) = or(x6,or(x7,x8))
 or(x6,x7) = or(x7,x6)
-> Usable Equations:
 or(or(x6,x7),x8) = or(x6,or(x7,x8))
 or(x6,x7) = or(x7,x6)
-> 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:
 or(false,y) -> y
 or(true,y) -> true
-> SRules:
 OR(or(x6,x7),x8) -> OR(x6,x7)
 OR(x6,or(x7,x8)) -> OR(x7,x8)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[eq](X1,X2) = 0
[if_reach_1](X1,X2,X3,X4,X5) = 0
[if_reach_2](X1,X2,X3,X4,X5) = 0
[or](X1,X2) = X1 + X2 + 1
[reach](X1,X2,X3,X4) = 0
[union](X1,X2) = 0
[0] = 0
[edge](X1,X2,X3) = 0
[empty] = 0
[false] = 0
[s](X) = 0
[true] = 2
[EQ](X1,X2) = 0
[IF_REACH_1](X1,X2,X3,X4,X5) = 0
[IF_REACH_2](X1,X2,X3,X4,X5) = 0
[OR](X1,X2) = 2.X1 + 2.X2
[REACH](X1,X2,X3,X4) = 0
[UNION](X1,X2) = 0

Problem 1.2: 

SCC Processor:
-> FAxioms:
 OR(or(x6,x7),x8) = OR(x6,or(x7,x8))
 OR(x6,x7) = OR(x7,x6)
-> Pairs:
 Empty
-> EAxioms:
 eq(x6,x7) = eq(x7,x6)
 or(or(x6,x7),x8) = or(x6,or(x7,x8))
 or(x6,x7) = or(x7,x6)
-> 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
-> SRules:
 OR(or(x6,x7),x8) -> OR(x6,x7)
 OR(x6,or(x7,x8)) -> OR(x7,x8)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

Subterm Processor:
-> FAxioms:
 EQ(x6,x7) = EQ(x7,x6)
-> Pairs:
 EQ(s(x),s(y)) -> EQ(x,y)
-> EAxioms:
 eq(x6,x7) = eq(x7,x6)
 or(or(x6,x7),x8) = or(x6,or(x7,x8))
 or(x6,x7) = or(x7,x6)
-> 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
-> SRules:
 Empty
->Projection:
 pi(EQ) = [1,2]

Problem 1.3: 

SCC Processor:
-> FAxioms:
 EQ(x6,x7) = EQ(x7,x6)
-> Pairs:
 Empty
-> EAxioms:
 eq(x6,x7) = eq(x7,x6)
 or(or(x6,x7),x8) = or(x6,or(x7,x8))
 or(x6,x7) = or(x7,x6)
-> 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
-> SRules:
 Empty
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.4: 

Reduction Pairs Processor:
-> FAxioms:
 Empty
-> 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)
-> EAxioms:
 eq(x6,x7) = eq(x7,x6)
 or(or(x6,x7),x8) = or(x6,or(x7,x8))
 or(x6,x7) = or(x7,x6)
-> Usable Equations:
 eq(x6,x7) = eq(x7,x6)
-> 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
-> SRules:
 Empty
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[eq](X1,X2) = X1 + X2 + 2
[if_reach_1](X1,X2,X3,X4,X5) = 0
[if_reach_2](X1,X2,X3,X4,X5) = 0
[or](X1,X2) = 0
[reach](X1,X2,X3,X4) = 0
[union](X1,X2) = X1 + X2
[0] = 2
[edge](X1,X2,X3) = X2 + X3 + 2
[empty] = 1
[false] = 2
[s](X) = X + 2
[true] = 1
[EQ](X1,X2) = 0
[IF_REACH_1](X1,X2,X3,X4,X5) = 2.X2 + 2.X3 + 2.X4 + 2.X5 + 2
[IF_REACH_2](X1,X2,X3,X4,X5) = 2.X2 + 2.X3 + 2.X4 + 2.X5 + 1
[OR](X1,X2) = 0
[REACH](X1,X2,X3,X4) = 2.X1 + 2.X2 + 2.X3 + 2.X4 + 2
[UNION](X1,X2) = 0

Problem 1.4: 

SCC Processor:
-> FAxioms:
 Empty
-> 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)
-> EAxioms:
 eq(x6,x7) = eq(x7,x6)
 or(or(x6,x7),x8) = or(x6,or(x7,x8))
 or(x6,x7) = or(x7,x6)
-> 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
-> SRules:
 Empty
->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)
-> FAxioms:
 eq(x6,x7) -> eq(x7,x6)
 or(or(x6,x7),x8) -> or(x6,or(x7,x8))
 or(x6,x7) -> or(x7,x6)
-> EAxioms:
 eq(x6,x7) = eq(x7,x6)
 or(or(x6,x7),x8) = or(x6,or(x7,x8))
 or(x6,x7) = or(x7,x6)
->->-> 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
-> SRules:
 Empty

Problem 1.4: 

Subterm Processor:
-> FAxioms:
 Empty
-> 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)
-> EAxioms:
 eq(x6,x7) = eq(x7,x6)
 or(or(x6,x7),x8) = or(x6,or(x7,x8))
 or(x6,x7) = or(x7,x6)
-> 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
-> SRules:
 Empty
->Projection:
 pi(IF_REACH_1) = [4]
 pi(REACH) = [3]

Problem 1.4: 

SCC Processor:
-> FAxioms:
 Empty
-> Pairs:
 REACH(x,y,edge(u,v,i),h) -> IF_REACH_1(eq(x,u),x,y,edge(u,v,i),h)
-> EAxioms:
 eq(x6,x7) = eq(x7,x6)
 or(or(x6,x7),x8) = or(x6,or(x7,x8))
 or(x6,x7) = or(x7,x6)
-> 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
-> SRules:
 Empty
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.