/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 t x y z)
(THEORY 
(AC app plus))
(RULES
app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
app(x,app(empty,z)) -> app(empty,z)
app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
app(x,empty) -> empty
eq(0,0) -> true
eq(0,s(x)) -> false
eq(s(x),s(y)) -> eq(x,y)
if(false,x,y) -> y
if(true,x,y) -> x
plus(empty,x) -> x
)

Problem 1: 

Dependency Pairs Processor:
-> FAxioms:
 APP(app(x4,x5),x6) = APP(x4,app(x5,x6))
 APP(x4,x5) = APP(x5,x4)
 PLUS(plus(x4,x5),x6) = PLUS(x4,plus(x5,x6))
 PLUS(x4,x5) = PLUS(x5,x4)
-> Pairs:
 APP(app(singl(x),singl(y)),x4) -> APP(if(eq(x,y),singl(x),empty),x4)
 APP(app(singl(x),singl(y)),x4) -> EQ(x,y)
 APP(app(singl(x),singl(y)),x4) -> IF(eq(x,y),singl(x),empty)
 APP(app(x,app(plus(y,z),t)),x4) -> APP(app(plus(app(x,y),app(x,z)),t),x4)
 APP(app(x,app(plus(y,z),t)),x4) -> APP(plus(app(x,y),app(x,z)),t)
 APP(app(x,app(plus(y,z),t)),x4) -> APP(x,y)
 APP(app(x,app(plus(y,z),t)),x4) -> APP(x,z)
 APP(app(x,app(plus(y,z),t)),x4) -> PLUS(app(x,y),app(x,z))
 APP(app(x,app(empty,z)),x4) -> APP(app(empty,z),x4)
 APP(app(x,plus(y,z)),x4) -> APP(plus(app(x,y),app(x,z)),x4)
 APP(app(x,plus(y,z)),x4) -> APP(x,y)
 APP(app(x,plus(y,z)),x4) -> APP(x,z)
 APP(app(x,plus(y,z)),x4) -> PLUS(app(x,y),app(x,z))
 APP(app(x,empty),x4) -> APP(empty,x4)
 APP(singl(x),singl(y)) -> EQ(x,y)
 APP(singl(x),singl(y)) -> IF(eq(x,y),singl(x),empty)
 APP(x,app(plus(y,z),t)) -> APP(plus(app(x,y),app(x,z)),t)
 APP(x,app(plus(y,z),t)) -> APP(x,y)
 APP(x,app(plus(y,z),t)) -> APP(x,z)
 APP(x,app(plus(y,z),t)) -> PLUS(app(x,y),app(x,z))
 APP(x,plus(y,z)) -> APP(x,y)
 APP(x,plus(y,z)) -> APP(x,z)
 APP(x,plus(y,z)) -> PLUS(app(x,y),app(x,z))
 EQ(s(x),s(y)) -> EQ(x,y)
 PLUS(plus(empty,x),x4) -> PLUS(x,x4)
-> EAxioms:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
-> Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> SRules:
 APP(app(x4,x5),x6) -> APP(x4,x5)
 APP(x4,app(x5,x6)) -> APP(x5,x6)
 PLUS(plus(x4,x5),x6) -> PLUS(x4,x5)
 PLUS(x4,plus(x5,x6)) -> PLUS(x5,x6)

Problem 1: 

SCC Processor:
-> FAxioms:
 APP(app(x4,x5),x6) = APP(x4,app(x5,x6))
 APP(x4,x5) = APP(x5,x4)
 PLUS(plus(x4,x5),x6) = PLUS(x4,plus(x5,x6))
 PLUS(x4,x5) = PLUS(x5,x4)
-> Pairs:
 APP(app(singl(x),singl(y)),x4) -> APP(if(eq(x,y),singl(x),empty),x4)
 APP(app(singl(x),singl(y)),x4) -> EQ(x,y)
 APP(app(singl(x),singl(y)),x4) -> IF(eq(x,y),singl(x),empty)
 APP(app(x,app(plus(y,z),t)),x4) -> APP(app(plus(app(x,y),app(x,z)),t),x4)
 APP(app(x,app(plus(y,z),t)),x4) -> APP(plus(app(x,y),app(x,z)),t)
 APP(app(x,app(plus(y,z),t)),x4) -> APP(x,y)
 APP(app(x,app(plus(y,z),t)),x4) -> APP(x,z)
 APP(app(x,app(plus(y,z),t)),x4) -> PLUS(app(x,y),app(x,z))
 APP(app(x,app(empty,z)),x4) -> APP(app(empty,z),x4)
 APP(app(x,plus(y,z)),x4) -> APP(plus(app(x,y),app(x,z)),x4)
 APP(app(x,plus(y,z)),x4) -> APP(x,y)
 APP(app(x,plus(y,z)),x4) -> APP(x,z)
 APP(app(x,plus(y,z)),x4) -> PLUS(app(x,y),app(x,z))
 APP(app(x,empty),x4) -> APP(empty,x4)
 APP(singl(x),singl(y)) -> EQ(x,y)
 APP(singl(x),singl(y)) -> IF(eq(x,y),singl(x),empty)
 APP(x,app(plus(y,z),t)) -> APP(plus(app(x,y),app(x,z)),t)
 APP(x,app(plus(y,z),t)) -> APP(x,y)
 APP(x,app(plus(y,z),t)) -> APP(x,z)
 APP(x,app(plus(y,z),t)) -> PLUS(app(x,y),app(x,z))
 APP(x,plus(y,z)) -> APP(x,y)
 APP(x,plus(y,z)) -> APP(x,z)
 APP(x,plus(y,z)) -> PLUS(app(x,y),app(x,z))
 EQ(s(x),s(y)) -> EQ(x,y)
 PLUS(plus(empty,x),x4) -> PLUS(x,x4)
-> EAxioms:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
-> Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> SRules:
 APP(app(x4,x5),x6) -> APP(x4,x5)
 APP(x4,app(x5,x6)) -> APP(x5,x6)
 PLUS(plus(x4,x5),x6) -> PLUS(x4,x5)
 PLUS(x4,plus(x5,x6)) -> PLUS(x5,x6)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 PLUS(plus(empty,x),x4) -> PLUS(x,x4)
-> FAxioms:
 app(app(x4,x5),x6) -> app(x4,app(x5,x6))
 app(x4,x5) -> app(x5,x4)
 plus(plus(x4,x5),x6) -> plus(x4,plus(x5,x6))
 plus(x4,x5) -> plus(x5,x4)
 PLUS(plus(x4,x5),x6) -> PLUS(x4,plus(x5,x6))
 PLUS(x4,x5) -> PLUS(x5,x4)
-> EAxioms:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
->->-> Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> SRules:
 PLUS(plus(x4,x5),x6) -> PLUS(x4,x5)
 PLUS(x4,plus(x5,x6)) -> PLUS(x5,x6)
->->Cycle:
->->-> Pairs:
 EQ(s(x),s(y)) -> EQ(x,y)
-> FAxioms:
 app(app(x4,x5),x6) -> app(x4,app(x5,x6))
 app(x4,x5) -> app(x5,x4)
 plus(plus(x4,x5),x6) -> plus(x4,plus(x5,x6))
 plus(x4,x5) -> plus(x5,x4)
-> EAxioms:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
->->-> Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> SRules:
 Empty
->->Cycle:
->->-> Pairs:
 APP(app(singl(x),singl(y)),x4) -> APP(if(eq(x,y),singl(x),empty),x4)
 APP(app(x,app(plus(y,z),t)),x4) -> APP(app(plus(app(x,y),app(x,z)),t),x4)
 APP(app(x,app(plus(y,z),t)),x4) -> APP(plus(app(x,y),app(x,z)),t)
 APP(app(x,app(plus(y,z),t)),x4) -> APP(x,y)
 APP(app(x,app(plus(y,z),t)),x4) -> APP(x,z)
 APP(app(x,app(empty,z)),x4) -> APP(app(empty,z),x4)
 APP(app(x,plus(y,z)),x4) -> APP(plus(app(x,y),app(x,z)),x4)
 APP(app(x,plus(y,z)),x4) -> APP(x,y)
 APP(app(x,plus(y,z)),x4) -> APP(x,z)
 APP(app(x,empty),x4) -> APP(empty,x4)
 APP(x,app(plus(y,z),t)) -> APP(plus(app(x,y),app(x,z)),t)
 APP(x,app(plus(y,z),t)) -> APP(x,y)
 APP(x,app(plus(y,z),t)) -> APP(x,z)
 APP(x,plus(y,z)) -> APP(x,y)
 APP(x,plus(y,z)) -> APP(x,z)
-> FAxioms:
 app(app(x4,x5),x6) -> app(x4,app(x5,x6))
 app(x4,x5) -> app(x5,x4)
 plus(plus(x4,x5),x6) -> plus(x4,plus(x5,x6))
 plus(x4,x5) -> plus(x5,x4)
 APP(app(x4,x5),x6) -> APP(x4,app(x5,x6))
 APP(x4,x5) -> APP(x5,x4)
-> EAxioms:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
->->-> Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> SRules:
 APP(app(x4,x5),x6) -> APP(x4,x5)
 APP(x4,app(x5,x6)) -> APP(x5,x6)


The problem is decomposed in 3 subproblems.

Problem 1.1: 

Reduction Pairs Processor:
-> FAxioms:
 PLUS(plus(x4,x5),x6) = PLUS(x4,plus(x5,x6))
 PLUS(x4,x5) = PLUS(x5,x4)
-> Pairs:
 PLUS(plus(empty,x),x4) -> PLUS(x,x4)
-> EAxioms:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
-> Usable Equations:
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
-> Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> Usable Rules:
 plus(empty,x) -> x
-> SRules:
 PLUS(plus(x4,x5),x6) -> PLUS(x4,x5)
 PLUS(x4,plus(x5,x6)) -> PLUS(x5,x6)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[app](X1,X2) = 0
[eq](X1,X2) = 0
[if](X1,X2,X3) = 0
[plus](X1,X2) = X1 + X2
[0] = 0
[empty] = 2
[false] = 0
[s](X) = 0
[singl](X) = 0
[true] = 0
[APP](X1,X2) = 0
[EQ](X1,X2) = 0
[IF](X1,X2,X3) = 0
[PLUS](X1,X2) = 2.X1 + 2.X2

Problem 1.1: 

SCC Processor:
-> FAxioms:
 PLUS(plus(x4,x5),x6) = PLUS(x4,plus(x5,x6))
 PLUS(x4,x5) = PLUS(x5,x4)
-> Pairs:
 Empty
-> EAxioms:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
-> Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> SRules:
 PLUS(plus(x4,x5),x6) -> PLUS(x4,x5)
 PLUS(x4,plus(x5,x6)) -> PLUS(x5,x6)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> FAxioms:
 Empty
-> Pairs:
 EQ(s(x),s(y)) -> EQ(x,y)
-> EAxioms:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
-> Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> SRules:
 Empty
->Projection:
 pi(EQ) = [1]

Problem 1.2: 

SCC Processor:
-> FAxioms:
 Empty
-> Pairs:
 Empty
-> EAxioms:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
-> Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> SRules:
 Empty
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.3: 

Reduction Pairs Processor:
-> FAxioms:
 APP(app(x4,x5),x6) = APP(x4,app(x5,x6))
 APP(x4,x5) = APP(x5,x4)
-> Pairs:
 APP(app(singl(x),singl(y)),x4) -> APP(if(eq(x,y),singl(x),empty),x4)
 APP(app(x,app(plus(y,z),t)),x4) -> APP(app(plus(app(x,y),app(x,z)),t),x4)
 APP(app(x,app(plus(y,z),t)),x4) -> APP(plus(app(x,y),app(x,z)),t)
 APP(app(x,app(plus(y,z),t)),x4) -> APP(x,y)
 APP(app(x,app(plus(y,z),t)),x4) -> APP(x,z)
 APP(app(x,app(empty,z)),x4) -> APP(app(empty,z),x4)
 APP(app(x,plus(y,z)),x4) -> APP(plus(app(x,y),app(x,z)),x4)
 APP(app(x,plus(y,z)),x4) -> APP(x,y)
 APP(app(x,plus(y,z)),x4) -> APP(x,z)
 APP(app(x,empty),x4) -> APP(empty,x4)
 APP(x,app(plus(y,z),t)) -> APP(plus(app(x,y),app(x,z)),t)
 APP(x,app(plus(y,z),t)) -> APP(x,y)
 APP(x,app(plus(y,z),t)) -> APP(x,z)
 APP(x,plus(y,z)) -> APP(x,y)
 APP(x,plus(y,z)) -> APP(x,z)
-> EAxioms:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
-> Usable Equations:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
-> Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> Usable Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> SRules:
 APP(app(x4,x5),x6) -> APP(x4,x5)
 APP(x4,app(x5,x6)) -> APP(x5,x6)
->Interpretation type:
 Simple mixed
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 1
->Interpretation:
 
[app](X1,X2) = X1.X2 + X1 + X2
[eq](X1,X2) = X2
[if](X1,X2,X3) = X1.X2 + X1.X3 + X2.X3 + X2
[plus](X1,X2) = X1 + X2 + 1
[0] = 1
[empty] = 1
[false] = 1
[s](X) = X + 1
[singl](X) = X.X + X + 1
[true] = 1
[APP](X1,X2) = X1.X2 + X1 + X2
[EQ](X1,X2) = 0
[IF](X1,X2,X3) = 0
[PLUS](X1,X2) = 0

Problem 1.3: 

SCC Processor:
-> FAxioms:
 APP(app(x4,x5),x6) = APP(x4,app(x5,x6))
 APP(x4,x5) = APP(x5,x4)
-> Pairs:
 APP(app(x,app(plus(y,z),t)),x4) -> APP(app(plus(app(x,y),app(x,z)),t),x4)
 APP(app(x,app(plus(y,z),t)),x4) -> APP(plus(app(x,y),app(x,z)),t)
 APP(app(x,app(plus(y,z),t)),x4) -> APP(x,y)
 APP(app(x,app(plus(y,z),t)),x4) -> APP(x,z)
 APP(app(x,app(empty,z)),x4) -> APP(app(empty,z),x4)
 APP(app(x,plus(y,z)),x4) -> APP(plus(app(x,y),app(x,z)),x4)
 APP(app(x,plus(y,z)),x4) -> APP(x,y)
 APP(app(x,plus(y,z)),x4) -> APP(x,z)
 APP(app(x,empty),x4) -> APP(empty,x4)
 APP(x,app(plus(y,z),t)) -> APP(plus(app(x,y),app(x,z)),t)
 APP(x,app(plus(y,z),t)) -> APP(x,y)
 APP(x,app(plus(y,z),t)) -> APP(x,z)
 APP(x,plus(y,z)) -> APP(x,y)
 APP(x,plus(y,z)) -> APP(x,z)
-> EAxioms:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
-> Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> SRules:
 APP(app(x4,x5),x6) -> APP(x4,x5)
 APP(x4,app(x5,x6)) -> APP(x5,x6)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 APP(app(x,app(plus(y,z),t)),x4) -> APP(app(plus(app(x,y),app(x,z)),t),x4)
 APP(app(x,app(plus(y,z),t)),x4) -> APP(plus(app(x,y),app(x,z)),t)
 APP(app(x,app(plus(y,z),t)),x4) -> APP(x,y)
 APP(app(x,app(plus(y,z),t)),x4) -> APP(x,z)
 APP(app(x,app(empty,z)),x4) -> APP(app(empty,z),x4)
 APP(app(x,plus(y,z)),x4) -> APP(plus(app(x,y),app(x,z)),x4)
 APP(app(x,plus(y,z)),x4) -> APP(x,y)
 APP(app(x,plus(y,z)),x4) -> APP(x,z)
 APP(app(x,empty),x4) -> APP(empty,x4)
 APP(x,app(plus(y,z),t)) -> APP(plus(app(x,y),app(x,z)),t)
 APP(x,app(plus(y,z),t)) -> APP(x,y)
 APP(x,app(plus(y,z),t)) -> APP(x,z)
 APP(x,plus(y,z)) -> APP(x,y)
 APP(x,plus(y,z)) -> APP(x,z)
-> FAxioms:
 app(app(x4,x5),x6) -> app(x4,app(x5,x6))
 app(x4,x5) -> app(x5,x4)
 plus(plus(x4,x5),x6) -> plus(x4,plus(x5,x6))
 plus(x4,x5) -> plus(x5,x4)
 APP(app(x4,x5),x6) -> APP(x4,app(x5,x6))
 APP(x4,x5) -> APP(x5,x4)
-> EAxioms:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
->->-> Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> SRules:
 APP(app(x4,x5),x6) -> APP(x4,x5)
 APP(x4,app(x5,x6)) -> APP(x5,x6)

Problem 1.3: 

Reduction Pairs Processor:
-> FAxioms:
 APP(app(x4,x5),x6) = APP(x4,app(x5,x6))
 APP(x4,x5) = APP(x5,x4)
-> Pairs:
 APP(app(x,app(plus(y,z),t)),x4) -> APP(app(plus(app(x,y),app(x,z)),t),x4)
 APP(app(x,app(plus(y,z),t)),x4) -> APP(plus(app(x,y),app(x,z)),t)
 APP(app(x,app(plus(y,z),t)),x4) -> APP(x,y)
 APP(app(x,app(plus(y,z),t)),x4) -> APP(x,z)
 APP(app(x,app(empty,z)),x4) -> APP(app(empty,z),x4)
 APP(app(x,plus(y,z)),x4) -> APP(plus(app(x,y),app(x,z)),x4)
 APP(app(x,plus(y,z)),x4) -> APP(x,y)
 APP(app(x,plus(y,z)),x4) -> APP(x,z)
 APP(app(x,empty),x4) -> APP(empty,x4)
 APP(x,app(plus(y,z),t)) -> APP(plus(app(x,y),app(x,z)),t)
 APP(x,app(plus(y,z),t)) -> APP(x,y)
 APP(x,app(plus(y,z),t)) -> APP(x,z)
 APP(x,plus(y,z)) -> APP(x,y)
 APP(x,plus(y,z)) -> APP(x,z)
-> EAxioms:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
-> Usable Equations:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
-> Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> Usable Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> SRules:
 APP(app(x4,x5),x6) -> APP(x4,x5)
 APP(x4,app(x5,x6)) -> APP(x5,x6)
->Interpretation type:
 Simple mixed
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 1
->Interpretation:
 
[app](X1,X2) = X1.X2 + X1 + X2
[eq](X1,X2) = X1.X2 + X1 + X2 + 1
[if](X1,X2,X3) = X1.X3 + X2 + X3
[plus](X1,X2) = X1 + X2 + 1
[0] = 1
[empty] = 1
[false] = 1
[s](X) = X + 1
[singl](X) = X.X + X + 1
[true] = 1
[APP](X1,X2) = X1.X2 + X1 + X2
[EQ](X1,X2) = 0
[IF](X1,X2,X3) = 0
[PLUS](X1,X2) = 0

Problem 1.3: 

SCC Processor:
-> FAxioms:
 APP(app(x4,x5),x6) = APP(x4,app(x5,x6))
 APP(x4,x5) = APP(x5,x4)
-> Pairs:
 APP(app(x,app(plus(y,z),t)),x4) -> APP(app(plus(app(x,y),app(x,z)),t),x4)
 APP(app(x,app(plus(y,z),t)),x4) -> APP(plus(app(x,y),app(x,z)),t)
 APP(app(x,app(plus(y,z),t)),x4) -> APP(x,z)
 APP(app(x,app(empty,z)),x4) -> APP(app(empty,z),x4)
 APP(app(x,plus(y,z)),x4) -> APP(plus(app(x,y),app(x,z)),x4)
 APP(app(x,plus(y,z)),x4) -> APP(x,y)
 APP(app(x,plus(y,z)),x4) -> APP(x,z)
 APP(app(x,empty),x4) -> APP(empty,x4)
 APP(x,app(plus(y,z),t)) -> APP(plus(app(x,y),app(x,z)),t)
 APP(x,app(plus(y,z),t)) -> APP(x,y)
 APP(x,app(plus(y,z),t)) -> APP(x,z)
 APP(x,plus(y,z)) -> APP(x,y)
 APP(x,plus(y,z)) -> APP(x,z)
-> EAxioms:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
-> Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> SRules:
 APP(app(x4,x5),x6) -> APP(x4,x5)
 APP(x4,app(x5,x6)) -> APP(x5,x6)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 APP(app(x,app(plus(y,z),t)),x4) -> APP(app(plus(app(x,y),app(x,z)),t),x4)
 APP(app(x,app(plus(y,z),t)),x4) -> APP(plus(app(x,y),app(x,z)),t)
 APP(app(x,app(plus(y,z),t)),x4) -> APP(x,z)
 APP(app(x,app(empty,z)),x4) -> APP(app(empty,z),x4)
 APP(app(x,plus(y,z)),x4) -> APP(plus(app(x,y),app(x,z)),x4)
 APP(app(x,plus(y,z)),x4) -> APP(x,y)
 APP(app(x,plus(y,z)),x4) -> APP(x,z)
 APP(app(x,empty),x4) -> APP(empty,x4)
 APP(x,app(plus(y,z),t)) -> APP(plus(app(x,y),app(x,z)),t)
 APP(x,app(plus(y,z),t)) -> APP(x,y)
 APP(x,app(plus(y,z),t)) -> APP(x,z)
 APP(x,plus(y,z)) -> APP(x,y)
 APP(x,plus(y,z)) -> APP(x,z)
-> FAxioms:
 app(app(x4,x5),x6) -> app(x4,app(x5,x6))
 app(x4,x5) -> app(x5,x4)
 plus(plus(x4,x5),x6) -> plus(x4,plus(x5,x6))
 plus(x4,x5) -> plus(x5,x4)
 APP(app(x4,x5),x6) -> APP(x4,app(x5,x6))
 APP(x4,x5) -> APP(x5,x4)
-> EAxioms:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
->->-> Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> SRules:
 APP(app(x4,x5),x6) -> APP(x4,x5)
 APP(x4,app(x5,x6)) -> APP(x5,x6)

Problem 1.3: 

Reduction Pairs Processor:
-> FAxioms:
 APP(app(x4,x5),x6) = APP(x4,app(x5,x6))
 APP(x4,x5) = APP(x5,x4)
-> Pairs:
 APP(app(x,app(plus(y,z),t)),x4) -> APP(app(plus(app(x,y),app(x,z)),t),x4)
 APP(app(x,app(plus(y,z),t)),x4) -> APP(plus(app(x,y),app(x,z)),t)
 APP(app(x,app(plus(y,z),t)),x4) -> APP(x,z)
 APP(app(x,app(empty,z)),x4) -> APP(app(empty,z),x4)
 APP(app(x,plus(y,z)),x4) -> APP(plus(app(x,y),app(x,z)),x4)
 APP(app(x,plus(y,z)),x4) -> APP(x,y)
 APP(app(x,plus(y,z)),x4) -> APP(x,z)
 APP(app(x,empty),x4) -> APP(empty,x4)
 APP(x,app(plus(y,z),t)) -> APP(plus(app(x,y),app(x,z)),t)
 APP(x,app(plus(y,z),t)) -> APP(x,y)
 APP(x,app(plus(y,z),t)) -> APP(x,z)
 APP(x,plus(y,z)) -> APP(x,y)
 APP(x,plus(y,z)) -> APP(x,z)
-> EAxioms:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
-> Usable Equations:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
-> Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> Usable Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> SRules:
 APP(app(x4,x5),x6) -> APP(x4,x5)
 APP(x4,app(x5,x6)) -> APP(x5,x6)
->Interpretation type:
 Simple mixed
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 1
->Interpretation:
 
[app](X1,X2) = X1.X2 + X1 + X2
[eq](X1,X2) = X1.X2 + X1 + X2 + 1
[if](X1,X2,X3) = X1.X3 + X2 + 1
[plus](X1,X2) = X1 + X2 + 1
[0] = 1
[empty] = 1
[false] = 1
[s](X) = X + 1
[singl](X) = X + 1
[true] = 1
[APP](X1,X2) = X1.X2 + X1 + X2
[EQ](X1,X2) = 0
[IF](X1,X2,X3) = 0
[PLUS](X1,X2) = 0

Problem 1.3: 

SCC Processor:
-> FAxioms:
 APP(app(x4,x5),x6) = APP(x4,app(x5,x6))
 APP(x4,x5) = APP(x5,x4)
-> Pairs:
 APP(app(x,app(plus(y,z),t)),x4) -> APP(app(plus(app(x,y),app(x,z)),t),x4)
 APP(app(x,app(plus(y,z),t)),x4) -> APP(plus(app(x,y),app(x,z)),t)
 APP(app(x,app(empty,z)),x4) -> APP(app(empty,z),x4)
 APP(app(x,plus(y,z)),x4) -> APP(plus(app(x,y),app(x,z)),x4)
 APP(app(x,plus(y,z)),x4) -> APP(x,y)
 APP(app(x,plus(y,z)),x4) -> APP(x,z)
 APP(app(x,empty),x4) -> APP(empty,x4)
 APP(x,app(plus(y,z),t)) -> APP(plus(app(x,y),app(x,z)),t)
 APP(x,app(plus(y,z),t)) -> APP(x,y)
 APP(x,app(plus(y,z),t)) -> APP(x,z)
 APP(x,plus(y,z)) -> APP(x,y)
 APP(x,plus(y,z)) -> APP(x,z)
-> EAxioms:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
-> Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> SRules:
 APP(app(x4,x5),x6) -> APP(x4,x5)
 APP(x4,app(x5,x6)) -> APP(x5,x6)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 APP(app(x,app(plus(y,z),t)),x4) -> APP(app(plus(app(x,y),app(x,z)),t),x4)
 APP(app(x,app(plus(y,z),t)),x4) -> APP(plus(app(x,y),app(x,z)),t)
 APP(app(x,app(empty,z)),x4) -> APP(app(empty,z),x4)
 APP(app(x,plus(y,z)),x4) -> APP(plus(app(x,y),app(x,z)),x4)
 APP(app(x,plus(y,z)),x4) -> APP(x,y)
 APP(app(x,plus(y,z)),x4) -> APP(x,z)
 APP(app(x,empty),x4) -> APP(empty,x4)
 APP(x,app(plus(y,z),t)) -> APP(plus(app(x,y),app(x,z)),t)
 APP(x,app(plus(y,z),t)) -> APP(x,y)
 APP(x,app(plus(y,z),t)) -> APP(x,z)
 APP(x,plus(y,z)) -> APP(x,y)
 APP(x,plus(y,z)) -> APP(x,z)
-> FAxioms:
 app(app(x4,x5),x6) -> app(x4,app(x5,x6))
 app(x4,x5) -> app(x5,x4)
 plus(plus(x4,x5),x6) -> plus(x4,plus(x5,x6))
 plus(x4,x5) -> plus(x5,x4)
 APP(app(x4,x5),x6) -> APP(x4,app(x5,x6))
 APP(x4,x5) -> APP(x5,x4)
-> EAxioms:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
->->-> Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> SRules:
 APP(app(x4,x5),x6) -> APP(x4,x5)
 APP(x4,app(x5,x6)) -> APP(x5,x6)

Problem 1.3: 

Reduction Pairs Processor:
-> FAxioms:
 APP(app(x4,x5),x6) = APP(x4,app(x5,x6))
 APP(x4,x5) = APP(x5,x4)
-> Pairs:
 APP(app(x,app(plus(y,z),t)),x4) -> APP(app(plus(app(x,y),app(x,z)),t),x4)
 APP(app(x,app(plus(y,z),t)),x4) -> APP(plus(app(x,y),app(x,z)),t)
 APP(app(x,app(empty,z)),x4) -> APP(app(empty,z),x4)
 APP(app(x,plus(y,z)),x4) -> APP(plus(app(x,y),app(x,z)),x4)
 APP(app(x,plus(y,z)),x4) -> APP(x,y)
 APP(app(x,plus(y,z)),x4) -> APP(x,z)
 APP(app(x,empty),x4) -> APP(empty,x4)
 APP(x,app(plus(y,z),t)) -> APP(plus(app(x,y),app(x,z)),t)
 APP(x,app(plus(y,z),t)) -> APP(x,y)
 APP(x,app(plus(y,z),t)) -> APP(x,z)
 APP(x,plus(y,z)) -> APP(x,y)
 APP(x,plus(y,z)) -> APP(x,z)
-> EAxioms:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
-> Usable Equations:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
-> Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> Usable Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> SRules:
 APP(app(x4,x5),x6) -> APP(x4,x5)
 APP(x4,app(x5,x6)) -> APP(x5,x6)
->Interpretation type:
 Simple mixed
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 1
->Interpretation:
 
[app](X1,X2) = X1.X2 + X1 + X2
[eq](X1,X2) = 1
[if](X1,X2,X3) = X1.X2.X3 + X2 + X3
[plus](X1,X2) = X1 + X2 + 1
[0] = 0
[empty] = 1
[false] = 1
[s](X) = 1
[singl](X) = X.X + X + 1
[true] = 1
[APP](X1,X2) = X1.X2 + X1 + X2
[EQ](X1,X2) = 0
[IF](X1,X2,X3) = 0
[PLUS](X1,X2) = 0

Problem 1.3: 

SCC Processor:
-> FAxioms:
 APP(app(x4,x5),x6) = APP(x4,app(x5,x6))
 APP(x4,x5) = APP(x5,x4)
-> Pairs:
 APP(app(x,app(plus(y,z),t)),x4) -> APP(app(plus(app(x,y),app(x,z)),t),x4)
 APP(app(x,app(plus(y,z),t)),x4) -> APP(plus(app(x,y),app(x,z)),t)
 APP(app(x,app(empty,z)),x4) -> APP(app(empty,z),x4)
 APP(app(x,plus(y,z)),x4) -> APP(plus(app(x,y),app(x,z)),x4)
 APP(app(x,plus(y,z)),x4) -> APP(x,z)
 APP(app(x,empty),x4) -> APP(empty,x4)
 APP(x,app(plus(y,z),t)) -> APP(plus(app(x,y),app(x,z)),t)
 APP(x,app(plus(y,z),t)) -> APP(x,y)
 APP(x,app(plus(y,z),t)) -> APP(x,z)
 APP(x,plus(y,z)) -> APP(x,y)
 APP(x,plus(y,z)) -> APP(x,z)
-> EAxioms:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
-> Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> SRules:
 APP(app(x4,x5),x6) -> APP(x4,x5)
 APP(x4,app(x5,x6)) -> APP(x5,x6)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 APP(app(x,app(plus(y,z),t)),x4) -> APP(app(plus(app(x,y),app(x,z)),t),x4)
 APP(app(x,app(plus(y,z),t)),x4) -> APP(plus(app(x,y),app(x,z)),t)
 APP(app(x,app(empty,z)),x4) -> APP(app(empty,z),x4)
 APP(app(x,plus(y,z)),x4) -> APP(plus(app(x,y),app(x,z)),x4)
 APP(app(x,plus(y,z)),x4) -> APP(x,z)
 APP(app(x,empty),x4) -> APP(empty,x4)
 APP(x,app(plus(y,z),t)) -> APP(plus(app(x,y),app(x,z)),t)
 APP(x,app(plus(y,z),t)) -> APP(x,y)
 APP(x,app(plus(y,z),t)) -> APP(x,z)
 APP(x,plus(y,z)) -> APP(x,y)
 APP(x,plus(y,z)) -> APP(x,z)
-> FAxioms:
 app(app(x4,x5),x6) -> app(x4,app(x5,x6))
 app(x4,x5) -> app(x5,x4)
 plus(plus(x4,x5),x6) -> plus(x4,plus(x5,x6))
 plus(x4,x5) -> plus(x5,x4)
 APP(app(x4,x5),x6) -> APP(x4,app(x5,x6))
 APP(x4,x5) -> APP(x5,x4)
-> EAxioms:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
->->-> Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> SRules:
 APP(app(x4,x5),x6) -> APP(x4,x5)
 APP(x4,app(x5,x6)) -> APP(x5,x6)

Problem 1.3: 

Reduction Pairs Processor:
-> FAxioms:
 APP(app(x4,x5),x6) = APP(x4,app(x5,x6))
 APP(x4,x5) = APP(x5,x4)
-> Pairs:
 APP(app(x,app(plus(y,z),t)),x4) -> APP(app(plus(app(x,y),app(x,z)),t),x4)
 APP(app(x,app(plus(y,z),t)),x4) -> APP(plus(app(x,y),app(x,z)),t)
 APP(app(x,app(empty,z)),x4) -> APP(app(empty,z),x4)
 APP(app(x,plus(y,z)),x4) -> APP(plus(app(x,y),app(x,z)),x4)
 APP(app(x,plus(y,z)),x4) -> APP(x,z)
 APP(app(x,empty),x4) -> APP(empty,x4)
 APP(x,app(plus(y,z),t)) -> APP(plus(app(x,y),app(x,z)),t)
 APP(x,app(plus(y,z),t)) -> APP(x,y)
 APP(x,app(plus(y,z),t)) -> APP(x,z)
 APP(x,plus(y,z)) -> APP(x,y)
 APP(x,plus(y,z)) -> APP(x,z)
-> EAxioms:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
-> Usable Equations:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
-> Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> Usable Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> SRules:
 APP(app(x4,x5),x6) -> APP(x4,x5)
 APP(x4,app(x5,x6)) -> APP(x5,x6)
->Interpretation type:
 Simple mixed
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 1
->Interpretation:
 
[app](X1,X2) = X1.X2 + X1 + X2
[eq](X1,X2) = X1 + X2 + 1
[if](X1,X2,X3) = X1.X3 + X2.X3 + X1 + X2 + X3 + 1
[plus](X1,X2) = X1 + X2 + 1
[0] = 0
[empty] = 0
[false] = 1
[s](X) = X + 1
[singl](X) = X.X + X + 1
[true] = 1
[APP](X1,X2) = X1.X2 + X1 + X2
[EQ](X1,X2) = 0
[IF](X1,X2,X3) = 0
[PLUS](X1,X2) = 0

Problem 1.3: 

SCC Processor:
-> FAxioms:
 APP(app(x4,x5),x6) = APP(x4,app(x5,x6))
 APP(x4,x5) = APP(x5,x4)
-> Pairs:
 APP(app(x,app(plus(y,z),t)),x4) -> APP(app(plus(app(x,y),app(x,z)),t),x4)
 APP(app(x,app(plus(y,z),t)),x4) -> APP(plus(app(x,y),app(x,z)),t)
 APP(app(x,app(empty,z)),x4) -> APP(app(empty,z),x4)
 APP(app(x,plus(y,z)),x4) -> APP(plus(app(x,y),app(x,z)),x4)
 APP(app(x,empty),x4) -> APP(empty,x4)
 APP(x,app(plus(y,z),t)) -> APP(plus(app(x,y),app(x,z)),t)
 APP(x,app(plus(y,z),t)) -> APP(x,y)
 APP(x,app(plus(y,z),t)) -> APP(x,z)
 APP(x,plus(y,z)) -> APP(x,y)
 APP(x,plus(y,z)) -> APP(x,z)
-> EAxioms:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
-> Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> SRules:
 APP(app(x4,x5),x6) -> APP(x4,x5)
 APP(x4,app(x5,x6)) -> APP(x5,x6)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 APP(app(x,app(plus(y,z),t)),x4) -> APP(app(plus(app(x,y),app(x,z)),t),x4)
 APP(app(x,app(plus(y,z),t)),x4) -> APP(plus(app(x,y),app(x,z)),t)
 APP(app(x,app(empty,z)),x4) -> APP(app(empty,z),x4)
 APP(app(x,plus(y,z)),x4) -> APP(plus(app(x,y),app(x,z)),x4)
 APP(app(x,empty),x4) -> APP(empty,x4)
 APP(x,app(plus(y,z),t)) -> APP(plus(app(x,y),app(x,z)),t)
 APP(x,app(plus(y,z),t)) -> APP(x,y)
 APP(x,app(plus(y,z),t)) -> APP(x,z)
 APP(x,plus(y,z)) -> APP(x,y)
 APP(x,plus(y,z)) -> APP(x,z)
-> FAxioms:
 app(app(x4,x5),x6) -> app(x4,app(x5,x6))
 app(x4,x5) -> app(x5,x4)
 plus(plus(x4,x5),x6) -> plus(x4,plus(x5,x6))
 plus(x4,x5) -> plus(x5,x4)
 APP(app(x4,x5),x6) -> APP(x4,app(x5,x6))
 APP(x4,x5) -> APP(x5,x4)
-> EAxioms:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
->->-> Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> SRules:
 APP(app(x4,x5),x6) -> APP(x4,x5)
 APP(x4,app(x5,x6)) -> APP(x5,x6)

Problem 1.3: 

Reduction Pairs Processor:
-> FAxioms:
 APP(app(x4,x5),x6) = APP(x4,app(x5,x6))
 APP(x4,x5) = APP(x5,x4)
-> Pairs:
 APP(app(x,app(plus(y,z),t)),x4) -> APP(app(plus(app(x,y),app(x,z)),t),x4)
 APP(app(x,app(plus(y,z),t)),x4) -> APP(plus(app(x,y),app(x,z)),t)
 APP(app(x,app(empty,z)),x4) -> APP(app(empty,z),x4)
 APP(app(x,plus(y,z)),x4) -> APP(plus(app(x,y),app(x,z)),x4)
 APP(app(x,empty),x4) -> APP(empty,x4)
 APP(x,app(plus(y,z),t)) -> APP(plus(app(x,y),app(x,z)),t)
 APP(x,app(plus(y,z),t)) -> APP(x,y)
 APP(x,app(plus(y,z),t)) -> APP(x,z)
 APP(x,plus(y,z)) -> APP(x,y)
 APP(x,plus(y,z)) -> APP(x,z)
-> EAxioms:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
-> Usable Equations:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
-> Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> Usable Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> SRules:
 APP(app(x4,x5),x6) -> APP(x4,x5)
 APP(x4,app(x5,x6)) -> APP(x5,x6)
->Interpretation type:
 Simple mixed
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 1
->Interpretation:
 
[app](X1,X2) = X1.X2 + X1 + X2
[eq](X1,X2) = X1.X2 + X1 + X2
[if](X1,X2,X3) = X1.X3 + X2 + X3 + 1
[plus](X1,X2) = X1 + X2 + 1
[0] = 1
[empty] = 1
[false] = 1
[s](X) = X + 1
[singl](X) = X + 1
[true] = 1
[APP](X1,X2) = X1.X2 + X1 + X2
[EQ](X1,X2) = 0
[IF](X1,X2,X3) = 0
[PLUS](X1,X2) = 0

Problem 1.3: 

SCC Processor:
-> FAxioms:
 APP(app(x4,x5),x6) = APP(x4,app(x5,x6))
 APP(x4,x5) = APP(x5,x4)
-> Pairs:
 APP(app(x,app(plus(y,z),t)),x4) -> APP(app(plus(app(x,y),app(x,z)),t),x4)
 APP(app(x,app(plus(y,z),t)),x4) -> APP(plus(app(x,y),app(x,z)),t)
 APP(app(x,app(empty,z)),x4) -> APP(app(empty,z),x4)
 APP(app(x,plus(y,z)),x4) -> APP(plus(app(x,y),app(x,z)),x4)
 APP(app(x,empty),x4) -> APP(empty,x4)
 APP(x,app(plus(y,z),t)) -> APP(plus(app(x,y),app(x,z)),t)
 APP(x,app(plus(y,z),t)) -> APP(x,z)
 APP(x,plus(y,z)) -> APP(x,y)
 APP(x,plus(y,z)) -> APP(x,z)
-> EAxioms:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
-> Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> SRules:
 APP(app(x4,x5),x6) -> APP(x4,x5)
 APP(x4,app(x5,x6)) -> APP(x5,x6)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 APP(app(x,app(plus(y,z),t)),x4) -> APP(app(plus(app(x,y),app(x,z)),t),x4)
 APP(app(x,app(plus(y,z),t)),x4) -> APP(plus(app(x,y),app(x,z)),t)
 APP(app(x,app(empty,z)),x4) -> APP(app(empty,z),x4)
 APP(app(x,plus(y,z)),x4) -> APP(plus(app(x,y),app(x,z)),x4)
 APP(app(x,empty),x4) -> APP(empty,x4)
 APP(x,app(plus(y,z),t)) -> APP(plus(app(x,y),app(x,z)),t)
 APP(x,app(plus(y,z),t)) -> APP(x,z)
 APP(x,plus(y,z)) -> APP(x,y)
 APP(x,plus(y,z)) -> APP(x,z)
-> FAxioms:
 app(app(x4,x5),x6) -> app(x4,app(x5,x6))
 app(x4,x5) -> app(x5,x4)
 plus(plus(x4,x5),x6) -> plus(x4,plus(x5,x6))
 plus(x4,x5) -> plus(x5,x4)
 APP(app(x4,x5),x6) -> APP(x4,app(x5,x6))
 APP(x4,x5) -> APP(x5,x4)
-> EAxioms:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
->->-> Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> SRules:
 APP(app(x4,x5),x6) -> APP(x4,x5)
 APP(x4,app(x5,x6)) -> APP(x5,x6)

Problem 1.3: 

Reduction Pairs Processor:
-> FAxioms:
 APP(app(x4,x5),x6) = APP(x4,app(x5,x6))
 APP(x4,x5) = APP(x5,x4)
-> Pairs:
 APP(app(x,app(plus(y,z),t)),x4) -> APP(app(plus(app(x,y),app(x,z)),t),x4)
 APP(app(x,app(plus(y,z),t)),x4) -> APP(plus(app(x,y),app(x,z)),t)
 APP(app(x,app(empty,z)),x4) -> APP(app(empty,z),x4)
 APP(app(x,plus(y,z)),x4) -> APP(plus(app(x,y),app(x,z)),x4)
 APP(app(x,empty),x4) -> APP(empty,x4)
 APP(x,app(plus(y,z),t)) -> APP(plus(app(x,y),app(x,z)),t)
 APP(x,app(plus(y,z),t)) -> APP(x,z)
 APP(x,plus(y,z)) -> APP(x,y)
 APP(x,plus(y,z)) -> APP(x,z)
-> EAxioms:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
-> Usable Equations:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
-> Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> Usable Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> SRules:
 APP(app(x4,x5),x6) -> APP(x4,x5)
 APP(x4,app(x5,x6)) -> APP(x5,x6)
->Interpretation type:
 Simple mixed
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 1
->Interpretation:
 
[app](X1,X2) = X1.X2 + X1 + X2
[eq](X1,X2) = X1 + X2
[if](X1,X2,X3) = X1.X3 + X2 + X3 + 1
[plus](X1,X2) = X1 + X2 + 1
[0] = 1
[empty] = 1
[false] = 1
[s](X) = X + 1
[singl](X) = X + 1
[true] = 1
[APP](X1,X2) = X1.X2 + X1 + X2
[EQ](X1,X2) = 0
[IF](X1,X2,X3) = 0
[PLUS](X1,X2) = 0

Problem 1.3: 

SCC Processor:
-> FAxioms:
 APP(app(x4,x5),x6) = APP(x4,app(x5,x6))
 APP(x4,x5) = APP(x5,x4)
-> Pairs:
 APP(app(x,app(plus(y,z),t)),x4) -> APP(app(plus(app(x,y),app(x,z)),t),x4)
 APP(app(x,app(plus(y,z),t)),x4) -> APP(plus(app(x,y),app(x,z)),t)
 APP(app(x,app(empty,z)),x4) -> APP(app(empty,z),x4)
 APP(app(x,plus(y,z)),x4) -> APP(plus(app(x,y),app(x,z)),x4)
 APP(app(x,empty),x4) -> APP(empty,x4)
 APP(x,app(plus(y,z),t)) -> APP(plus(app(x,y),app(x,z)),t)
 APP(x,plus(y,z)) -> APP(x,y)
 APP(x,plus(y,z)) -> APP(x,z)
-> EAxioms:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
-> Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> SRules:
 APP(app(x4,x5),x6) -> APP(x4,x5)
 APP(x4,app(x5,x6)) -> APP(x5,x6)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 APP(app(x,app(plus(y,z),t)),x4) -> APP(app(plus(app(x,y),app(x,z)),t),x4)
 APP(app(x,app(plus(y,z),t)),x4) -> APP(plus(app(x,y),app(x,z)),t)
 APP(app(x,app(empty,z)),x4) -> APP(app(empty,z),x4)
 APP(app(x,plus(y,z)),x4) -> APP(plus(app(x,y),app(x,z)),x4)
 APP(app(x,empty),x4) -> APP(empty,x4)
 APP(x,app(plus(y,z),t)) -> APP(plus(app(x,y),app(x,z)),t)
 APP(x,plus(y,z)) -> APP(x,y)
 APP(x,plus(y,z)) -> APP(x,z)
-> FAxioms:
 app(app(x4,x5),x6) -> app(x4,app(x5,x6))
 app(x4,x5) -> app(x5,x4)
 plus(plus(x4,x5),x6) -> plus(x4,plus(x5,x6))
 plus(x4,x5) -> plus(x5,x4)
 APP(app(x4,x5),x6) -> APP(x4,app(x5,x6))
 APP(x4,x5) -> APP(x5,x4)
-> EAxioms:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
->->-> Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> SRules:
 APP(app(x4,x5),x6) -> APP(x4,x5)
 APP(x4,app(x5,x6)) -> APP(x5,x6)

Problem 1.3: 

Reduction Pairs Processor:
-> FAxioms:
 APP(app(x4,x5),x6) = APP(x4,app(x5,x6))
 APP(x4,x5) = APP(x5,x4)
-> Pairs:
 APP(app(x,app(plus(y,z),t)),x4) -> APP(app(plus(app(x,y),app(x,z)),t),x4)
 APP(app(x,app(plus(y,z),t)),x4) -> APP(plus(app(x,y),app(x,z)),t)
 APP(app(x,app(empty,z)),x4) -> APP(app(empty,z),x4)
 APP(app(x,plus(y,z)),x4) -> APP(plus(app(x,y),app(x,z)),x4)
 APP(app(x,empty),x4) -> APP(empty,x4)
 APP(x,app(plus(y,z),t)) -> APP(plus(app(x,y),app(x,z)),t)
 APP(x,plus(y,z)) -> APP(x,y)
 APP(x,plus(y,z)) -> APP(x,z)
-> EAxioms:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
-> Usable Equations:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
-> Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> Usable Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> SRules:
 APP(app(x4,x5),x6) -> APP(x4,x5)
 APP(x4,app(x5,x6)) -> APP(x5,x6)
->Interpretation type:
 Simple mixed
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 1
->Interpretation:
 
[app](X1,X2) = X1.X2 + X1 + X2
[eq](X1,X2) = X2 + 1
[if](X1,X2,X3) = X1.X2 + X1.X3 + X2.X3 + X1 + X2 + X3
[plus](X1,X2) = X1 + X2 + 1
[0] = 1
[empty] = 0
[false] = 1
[s](X) = X + 1
[singl](X) = X + 1
[true] = 0
[APP](X1,X2) = X1.X2 + X1 + X2
[EQ](X1,X2) = 0
[IF](X1,X2,X3) = 0
[PLUS](X1,X2) = 0

Problem 1.3: 

SCC Processor:
-> FAxioms:
 APP(app(x4,x5),x6) = APP(x4,app(x5,x6))
 APP(x4,x5) = APP(x5,x4)
-> Pairs:
 APP(app(x,app(plus(y,z),t)),x4) -> APP(app(plus(app(x,y),app(x,z)),t),x4)
 APP(app(x,app(plus(y,z),t)),x4) -> APP(plus(app(x,y),app(x,z)),t)
 APP(app(x,app(empty,z)),x4) -> APP(app(empty,z),x4)
 APP(app(x,plus(y,z)),x4) -> APP(plus(app(x,y),app(x,z)),x4)
 APP(app(x,empty),x4) -> APP(empty,x4)
 APP(x,app(plus(y,z),t)) -> APP(plus(app(x,y),app(x,z)),t)
 APP(x,plus(y,z)) -> APP(x,z)
-> EAxioms:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
-> Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> SRules:
 APP(app(x4,x5),x6) -> APP(x4,x5)
 APP(x4,app(x5,x6)) -> APP(x5,x6)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 APP(app(x,app(plus(y,z),t)),x4) -> APP(app(plus(app(x,y),app(x,z)),t),x4)
 APP(app(x,app(plus(y,z),t)),x4) -> APP(plus(app(x,y),app(x,z)),t)
 APP(app(x,app(empty,z)),x4) -> APP(app(empty,z),x4)
 APP(app(x,plus(y,z)),x4) -> APP(plus(app(x,y),app(x,z)),x4)
 APP(app(x,empty),x4) -> APP(empty,x4)
 APP(x,app(plus(y,z),t)) -> APP(plus(app(x,y),app(x,z)),t)
 APP(x,plus(y,z)) -> APP(x,z)
-> FAxioms:
 app(app(x4,x5),x6) -> app(x4,app(x5,x6))
 app(x4,x5) -> app(x5,x4)
 plus(plus(x4,x5),x6) -> plus(x4,plus(x5,x6))
 plus(x4,x5) -> plus(x5,x4)
 APP(app(x4,x5),x6) -> APP(x4,app(x5,x6))
 APP(x4,x5) -> APP(x5,x4)
-> EAxioms:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
->->-> Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> SRules:
 APP(app(x4,x5),x6) -> APP(x4,x5)
 APP(x4,app(x5,x6)) -> APP(x5,x6)

Problem 1.3: 

Reduction Pairs Processor:
-> FAxioms:
 APP(app(x4,x5),x6) = APP(x4,app(x5,x6))
 APP(x4,x5) = APP(x5,x4)
-> Pairs:
 APP(app(x,app(plus(y,z),t)),x4) -> APP(app(plus(app(x,y),app(x,z)),t),x4)
 APP(app(x,app(plus(y,z),t)),x4) -> APP(plus(app(x,y),app(x,z)),t)
 APP(app(x,app(empty,z)),x4) -> APP(app(empty,z),x4)
 APP(app(x,plus(y,z)),x4) -> APP(plus(app(x,y),app(x,z)),x4)
 APP(app(x,empty),x4) -> APP(empty,x4)
 APP(x,app(plus(y,z),t)) -> APP(plus(app(x,y),app(x,z)),t)
 APP(x,plus(y,z)) -> APP(x,z)
-> EAxioms:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
-> Usable Equations:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
-> Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> Usable Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> SRules:
 APP(app(x4,x5),x6) -> APP(x4,x5)
 APP(x4,app(x5,x6)) -> APP(x5,x6)
->Interpretation type:
 Simple mixed
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 1
->Interpretation:
 
[app](X1,X2) = X1.X2 + X1 + X2
[eq](X1,X2) = X1 + X2 + 1
[if](X1,X2,X3) = X1.X2.X3 + X1.X3 + X2.X3 + X1 + X2 + X3 + 1
[plus](X1,X2) = X1 + X2 + 1
[0] = 1
[empty] = 0
[false] = 1
[s](X) = X + 1
[singl](X) = X + 1
[true] = 1
[APP](X1,X2) = X1.X2 + X1 + X2
[EQ](X1,X2) = 0
[IF](X1,X2,X3) = 0
[PLUS](X1,X2) = 0

Problem 1.3: 

SCC Processor:
-> FAxioms:
 APP(app(x4,x5),x6) = APP(x4,app(x5,x6))
 APP(x4,x5) = APP(x5,x4)
-> Pairs:
 APP(app(x,app(plus(y,z),t)),x4) -> APP(app(plus(app(x,y),app(x,z)),t),x4)
 APP(app(x,app(plus(y,z),t)),x4) -> APP(plus(app(x,y),app(x,z)),t)
 APP(app(x,app(empty,z)),x4) -> APP(app(empty,z),x4)
 APP(app(x,plus(y,z)),x4) -> APP(plus(app(x,y),app(x,z)),x4)
 APP(app(x,empty),x4) -> APP(empty,x4)
 APP(x,app(plus(y,z),t)) -> APP(plus(app(x,y),app(x,z)),t)
-> EAxioms:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
-> Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> SRules:
 APP(app(x4,x5),x6) -> APP(x4,x5)
 APP(x4,app(x5,x6)) -> APP(x5,x6)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 APP(app(x,app(plus(y,z),t)),x4) -> APP(app(plus(app(x,y),app(x,z)),t),x4)
 APP(app(x,app(plus(y,z),t)),x4) -> APP(plus(app(x,y),app(x,z)),t)
 APP(app(x,app(empty,z)),x4) -> APP(app(empty,z),x4)
 APP(app(x,plus(y,z)),x4) -> APP(plus(app(x,y),app(x,z)),x4)
 APP(app(x,empty),x4) -> APP(empty,x4)
 APP(x,app(plus(y,z),t)) -> APP(plus(app(x,y),app(x,z)),t)
-> FAxioms:
 app(app(x4,x5),x6) -> app(x4,app(x5,x6))
 app(x4,x5) -> app(x5,x4)
 plus(plus(x4,x5),x6) -> plus(x4,plus(x5,x6))
 plus(x4,x5) -> plus(x5,x4)
 APP(app(x4,x5),x6) -> APP(x4,app(x5,x6))
 APP(x4,x5) -> APP(x5,x4)
-> EAxioms:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
->->-> Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> SRules:
 APP(app(x4,x5),x6) -> APP(x4,x5)
 APP(x4,app(x5,x6)) -> APP(x5,x6)

Problem 1.3: 

Reduction Pairs Processor:
-> FAxioms:
 APP(app(x4,x5),x6) = APP(x4,app(x5,x6))
 APP(x4,x5) = APP(x5,x4)
-> Pairs:
 APP(app(x,app(plus(y,z),t)),x4) -> APP(app(plus(app(x,y),app(x,z)),t),x4)
 APP(app(x,app(plus(y,z),t)),x4) -> APP(plus(app(x,y),app(x,z)),t)
 APP(app(x,app(empty,z)),x4) -> APP(app(empty,z),x4)
 APP(app(x,plus(y,z)),x4) -> APP(plus(app(x,y),app(x,z)),x4)
 APP(app(x,empty),x4) -> APP(empty,x4)
 APP(x,app(plus(y,z),t)) -> APP(plus(app(x,y),app(x,z)),t)
-> EAxioms:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
-> Usable Equations:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
-> Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> Usable Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> SRules:
 APP(app(x4,x5),x6) -> APP(x4,x5)
 APP(x4,app(x5,x6)) -> APP(x5,x6)
->Interpretation type:
 Simple mixed
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 3
->Interpretation:
 
[app](X1,X2) = 3.X1.X2 + 3.X1 + 3.X2 + 2
[eq](X1,X2) = 2
[if](X1,X2,X3) = 2/3.X1.X2 + 3.X2.X3 + 2/3.X1 + 2/3.X2 + X3
[plus](X1,X2) = X1 + X2 + 3
[0] = 1/3
[empty] = 0
[false] = 2
[s](X) = 2/3.X + 1/2
[singl](X) = 2/3.X + 2
[true] = 1/2
[APP](X1,X2) = 1/3.X1.X2 + 1/3.X1 + 1/3.X2
[EQ](X1,X2) = 0
[IF](X1,X2,X3) = 0
[PLUS](X1,X2) = 0

Problem 1.3: 

SCC Processor:
-> FAxioms:
 APP(app(x4,x5),x6) = APP(x4,app(x5,x6))
 APP(x4,x5) = APP(x5,x4)
-> Pairs:
 APP(app(x,app(plus(y,z),t)),x4) -> APP(plus(app(x,y),app(x,z)),t)
 APP(app(x,app(empty,z)),x4) -> APP(app(empty,z),x4)
 APP(app(x,plus(y,z)),x4) -> APP(plus(app(x,y),app(x,z)),x4)
 APP(app(x,empty),x4) -> APP(empty,x4)
 APP(x,app(plus(y,z),t)) -> APP(plus(app(x,y),app(x,z)),t)
-> EAxioms:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
-> Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> SRules:
 APP(app(x4,x5),x6) -> APP(x4,x5)
 APP(x4,app(x5,x6)) -> APP(x5,x6)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 APP(app(x,app(plus(y,z),t)),x4) -> APP(plus(app(x,y),app(x,z)),t)
 APP(app(x,app(empty,z)),x4) -> APP(app(empty,z),x4)
 APP(app(x,plus(y,z)),x4) -> APP(plus(app(x,y),app(x,z)),x4)
 APP(app(x,empty),x4) -> APP(empty,x4)
 APP(x,app(plus(y,z),t)) -> APP(plus(app(x,y),app(x,z)),t)
-> FAxioms:
 app(app(x4,x5),x6) -> app(x4,app(x5,x6))
 app(x4,x5) -> app(x5,x4)
 plus(plus(x4,x5),x6) -> plus(x4,plus(x5,x6))
 plus(x4,x5) -> plus(x5,x4)
 APP(app(x4,x5),x6) -> APP(x4,app(x5,x6))
 APP(x4,x5) -> APP(x5,x4)
-> EAxioms:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
->->-> Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> SRules:
 APP(app(x4,x5),x6) -> APP(x4,x5)
 APP(x4,app(x5,x6)) -> APP(x5,x6)

Problem 1.3: 

Reduction Pairs Processor:
-> FAxioms:
 APP(app(x4,x5),x6) = APP(x4,app(x5,x6))
 APP(x4,x5) = APP(x5,x4)
-> Pairs:
 APP(app(x,app(plus(y,z),t)),x4) -> APP(plus(app(x,y),app(x,z)),t)
 APP(app(x,app(empty,z)),x4) -> APP(app(empty,z),x4)
 APP(app(x,plus(y,z)),x4) -> APP(plus(app(x,y),app(x,z)),x4)
 APP(app(x,empty),x4) -> APP(empty,x4)
 APP(x,app(plus(y,z),t)) -> APP(plus(app(x,y),app(x,z)),t)
-> EAxioms:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
-> Usable Equations:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
-> Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> Usable Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> SRules:
 APP(app(x4,x5),x6) -> APP(x4,x5)
 APP(x4,app(x5,x6)) -> APP(x5,x6)
->Interpretation type:
 Simple mixed
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 3
->Interpretation:
 
[app](X1,X2) = 3.X1.X2 + 3.X1 + 3.X2 + 2
[eq](X1,X2) = 2
[if](X1,X2,X3) = 3.X1.X2.X3 + 3/2.X1.X3 + 1/3.X2.X3 + 3.X2 + 2/3.X3 + 1/2
[plus](X1,X2) = X1 + X2 + 3
[0] = 1
[empty] = 0
[false] = 1/3
[s](X) = 0
[singl](X) = 3/2.X.X + 1/3.X + 2
[true] = 1
[APP](X1,X2) = 1/3.X1.X2 + 1/3.X1 + 1/3.X2
[EQ](X1,X2) = 0
[IF](X1,X2,X3) = 0
[PLUS](X1,X2) = 0

Problem 1.3: 

SCC Processor:
-> FAxioms:
 APP(app(x4,x5),x6) = APP(x4,app(x5,x6))
 APP(x4,x5) = APP(x5,x4)
-> Pairs:
 APP(app(x,app(empty,z)),x4) -> APP(app(empty,z),x4)
 APP(app(x,plus(y,z)),x4) -> APP(plus(app(x,y),app(x,z)),x4)
 APP(app(x,empty),x4) -> APP(empty,x4)
 APP(x,app(plus(y,z),t)) -> APP(plus(app(x,y),app(x,z)),t)
-> EAxioms:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
-> Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> SRules:
 APP(app(x4,x5),x6) -> APP(x4,x5)
 APP(x4,app(x5,x6)) -> APP(x5,x6)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 APP(app(x,app(empty,z)),x4) -> APP(app(empty,z),x4)
 APP(app(x,plus(y,z)),x4) -> APP(plus(app(x,y),app(x,z)),x4)
 APP(app(x,empty),x4) -> APP(empty,x4)
 APP(x,app(plus(y,z),t)) -> APP(plus(app(x,y),app(x,z)),t)
-> FAxioms:
 app(app(x4,x5),x6) -> app(x4,app(x5,x6))
 app(x4,x5) -> app(x5,x4)
 plus(plus(x4,x5),x6) -> plus(x4,plus(x5,x6))
 plus(x4,x5) -> plus(x5,x4)
 APP(app(x4,x5),x6) -> APP(x4,app(x5,x6))
 APP(x4,x5) -> APP(x5,x4)
-> EAxioms:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
->->-> Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> SRules:
 APP(app(x4,x5),x6) -> APP(x4,x5)
 APP(x4,app(x5,x6)) -> APP(x5,x6)

Problem 1.3: 

Reduction Pairs Processor:
-> FAxioms:
 APP(app(x4,x5),x6) = APP(x4,app(x5,x6))
 APP(x4,x5) = APP(x5,x4)
-> Pairs:
 APP(app(x,app(empty,z)),x4) -> APP(app(empty,z),x4)
 APP(app(x,plus(y,z)),x4) -> APP(plus(app(x,y),app(x,z)),x4)
 APP(app(x,empty),x4) -> APP(empty,x4)
 APP(x,app(plus(y,z),t)) -> APP(plus(app(x,y),app(x,z)),t)
-> EAxioms:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
-> Usable Equations:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
-> Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> Usable Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> SRules:
 APP(app(x4,x5),x6) -> APP(x4,x5)
 APP(x4,app(x5,x6)) -> APP(x5,x6)
->Interpretation type:
 Simple mixed
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 3
->Interpretation:
 
[app](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 1
[eq](X1,X2) = 2.X1.X2 + X2 + 1/2
[if](X1,X2,X3) = 2/3.X1 + 3/2.X2 + 2.X3 + 3
[plus](X1,X2) = X1 + X2 + 2
[0] = 3
[empty] = 0
[false] = 0
[s](X) = 3.X
[singl](X) = X + 3
[true] = 1/2
[APP](X1,X2) = 1/2.X1.X2 + 1/2.X1 + 1/2.X2
[EQ](X1,X2) = 0
[IF](X1,X2,X3) = 0
[PLUS](X1,X2) = 0

Problem 1.3: 

SCC Processor:
-> FAxioms:
 APP(app(x4,x5),x6) = APP(x4,app(x5,x6))
 APP(x4,x5) = APP(x5,x4)
-> Pairs:
 APP(app(x,plus(y,z)),x4) -> APP(plus(app(x,y),app(x,z)),x4)
 APP(app(x,empty),x4) -> APP(empty,x4)
 APP(x,app(plus(y,z),t)) -> APP(plus(app(x,y),app(x,z)),t)
-> EAxioms:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
-> Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> SRules:
 APP(app(x4,x5),x6) -> APP(x4,x5)
 APP(x4,app(x5,x6)) -> APP(x5,x6)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 APP(app(x,plus(y,z)),x4) -> APP(plus(app(x,y),app(x,z)),x4)
 APP(app(x,empty),x4) -> APP(empty,x4)
 APP(x,app(plus(y,z),t)) -> APP(plus(app(x,y),app(x,z)),t)
-> FAxioms:
 app(app(x4,x5),x6) -> app(x4,app(x5,x6))
 app(x4,x5) -> app(x5,x4)
 plus(plus(x4,x5),x6) -> plus(x4,plus(x5,x6))
 plus(x4,x5) -> plus(x5,x4)
 APP(app(x4,x5),x6) -> APP(x4,app(x5,x6))
 APP(x4,x5) -> APP(x5,x4)
-> EAxioms:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
->->-> Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> SRules:
 APP(app(x4,x5),x6) -> APP(x4,x5)
 APP(x4,app(x5,x6)) -> APP(x5,x6)

Problem 1.3: 

Reduction Pairs Processor:
-> FAxioms:
 APP(app(x4,x5),x6) = APP(x4,app(x5,x6))
 APP(x4,x5) = APP(x5,x4)
-> Pairs:
 APP(app(x,plus(y,z)),x4) -> APP(plus(app(x,y),app(x,z)),x4)
 APP(app(x,empty),x4) -> APP(empty,x4)
 APP(x,app(plus(y,z),t)) -> APP(plus(app(x,y),app(x,z)),t)
-> EAxioms:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
-> Usable Equations:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
-> Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> Usable Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> SRules:
 APP(app(x4,x5),x6) -> APP(x4,x5)
 APP(x4,app(x5,x6)) -> APP(x5,x6)
->Interpretation type:
 Simple mixed
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 3
->Interpretation:
 
[app](X1,X2) = 2.X1.X2 + 2.X1 + 2.X2 + 1
[eq](X1,X2) = 2
[if](X1,X2,X3) = 3.X1.X2 + 2/3.X1.X3 + X2.X3 + 1/2.X2 + 3/2.X3 + 2/3
[plus](X1,X2) = X1 + X2 + 3
[0] = 1
[empty] = 0
[false] = 0
[s](X) = 1/3.X + 3/2
[singl](X) = 2.X + 3
[true] = 2
[APP](X1,X2) = 1/2.X1.X2 + 1/2.X1 + 1/2.X2
[EQ](X1,X2) = 0
[IF](X1,X2,X3) = 0
[PLUS](X1,X2) = 0

Problem 1.3: 

SCC Processor:
-> FAxioms:
 APP(app(x4,x5),x6) = APP(x4,app(x5,x6))
 APP(x4,x5) = APP(x5,x4)
-> Pairs:
 APP(app(x,empty),x4) -> APP(empty,x4)
 APP(x,app(plus(y,z),t)) -> APP(plus(app(x,y),app(x,z)),t)
-> EAxioms:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
-> Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> SRules:
 APP(app(x4,x5),x6) -> APP(x4,x5)
 APP(x4,app(x5,x6)) -> APP(x5,x6)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 APP(app(x,empty),x4) -> APP(empty,x4)
 APP(x,app(plus(y,z),t)) -> APP(plus(app(x,y),app(x,z)),t)
-> FAxioms:
 app(app(x4,x5),x6) -> app(x4,app(x5,x6))
 app(x4,x5) -> app(x5,x4)
 plus(plus(x4,x5),x6) -> plus(x4,plus(x5,x6))
 plus(x4,x5) -> plus(x5,x4)
 APP(app(x4,x5),x6) -> APP(x4,app(x5,x6))
 APP(x4,x5) -> APP(x5,x4)
-> EAxioms:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
->->-> Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> SRules:
 APP(app(x4,x5),x6) -> APP(x4,x5)
 APP(x4,app(x5,x6)) -> APP(x5,x6)

Problem 1.3: 

Reduction Pairs Processor:
-> FAxioms:
 APP(app(x4,x5),x6) = APP(x4,app(x5,x6))
 APP(x4,x5) = APP(x5,x4)
-> Pairs:
 APP(app(x,empty),x4) -> APP(empty,x4)
 APP(x,app(plus(y,z),t)) -> APP(plus(app(x,y),app(x,z)),t)
-> EAxioms:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
-> Usable Equations:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
-> Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> Usable Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> SRules:
 APP(app(x4,x5),x6) -> APP(x4,x5)
 APP(x4,app(x5,x6)) -> APP(x5,x6)
->Interpretation type:
 Simple mixed
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 3
->Interpretation:
 
[app](X1,X2) = 3/2.X1.X2 + 3/2.X1 + 3/2.X2 + 1/2
[eq](X1,X2) = 2
[if](X1,X2,X3) = 2.X1.X2 + 2.X1.X3 + 1/3.X2.X3 + 2/3.X1 + 1/2.X2 + 3.X3
[plus](X1,X2) = X1 + X2 + 3/2
[0] = 2
[empty] = 1/2
[false] = 2/3
[s](X) = 1/2.X
[singl](X) = 3.X + 3
[true] = 1/2
[APP](X1,X2) = 1/3.X1.X2 + 1/3.X1 + 1/3.X2
[EQ](X1,X2) = 0
[IF](X1,X2,X3) = 0
[PLUS](X1,X2) = 0

Problem 1.3: 

SCC Processor:
-> FAxioms:
 APP(app(x4,x5),x6) = APP(x4,app(x5,x6))
 APP(x4,x5) = APP(x5,x4)
-> Pairs:
 APP(x,app(plus(y,z),t)) -> APP(plus(app(x,y),app(x,z)),t)
-> EAxioms:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
-> Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> SRules:
 APP(app(x4,x5),x6) -> APP(x4,x5)
 APP(x4,app(x5,x6)) -> APP(x5,x6)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 APP(x,app(plus(y,z),t)) -> APP(plus(app(x,y),app(x,z)),t)
-> FAxioms:
 app(app(x4,x5),x6) -> app(x4,app(x5,x6))
 app(x4,x5) -> app(x5,x4)
 plus(plus(x4,x5),x6) -> plus(x4,plus(x5,x6))
 plus(x4,x5) -> plus(x5,x4)
 APP(app(x4,x5),x6) -> APP(x4,app(x5,x6))
 APP(x4,x5) -> APP(x5,x4)
-> EAxioms:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
->->-> Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> SRules:
 APP(app(x4,x5),x6) -> APP(x4,x5)
 APP(x4,app(x5,x6)) -> APP(x5,x6)

Problem 1.3: 

Reduction Pairs Processor:
-> FAxioms:
 APP(app(x4,x5),x6) = APP(x4,app(x5,x6))
 APP(x4,x5) = APP(x5,x4)
-> Pairs:
 APP(x,app(plus(y,z),t)) -> APP(plus(app(x,y),app(x,z)),t)
-> EAxioms:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
-> Usable Equations:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
-> Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> Usable Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> SRules:
 APP(app(x4,x5),x6) -> APP(x4,x5)
 APP(x4,app(x5,x6)) -> APP(x5,x6)
->Interpretation type:
 Simple mixed
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 3
->Interpretation:
 
[app](X1,X2) = 2/3.X1.X2 + 2.X1 + 2.X2 + 3
[eq](X1,X2) = 3
[if](X1,X2,X3) = 3.X1.X2.X3 + 1/3.X1.X3 + 3/2.X2.X3 + 2/3.X1 + 3/2.X2 + 3.X3
[plus](X1,X2) = X1 + X2 + 3
[0] = 3
[empty] = 0
[false] = 3/2
[s](X) = 2/3
[singl](X) = 1/3.X + 3
[true] = 3
[APP](X1,X2) = 1/3.X1.X2 + X1 + X2
[EQ](X1,X2) = 0
[IF](X1,X2,X3) = 0
[PLUS](X1,X2) = 0

Problem 1.3: 

SCC Processor:
-> FAxioms:
 APP(app(x4,x5),x6) = APP(x4,app(x5,x6))
 APP(x4,x5) = APP(x5,x4)
-> Pairs:
 APP(x,app(plus(y,z),t)) -> APP(plus(app(x,y),app(x,z)),t)
-> EAxioms:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
-> Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> SRules:
 APP(x4,app(x5,x6)) -> APP(x5,x6)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 APP(x,app(plus(y,z),t)) -> APP(plus(app(x,y),app(x,z)),t)
-> FAxioms:
 app(app(x4,x5),x6) -> app(x4,app(x5,x6))
 app(x4,x5) -> app(x5,x4)
 plus(plus(x4,x5),x6) -> plus(x4,plus(x5,x6))
 plus(x4,x5) -> plus(x5,x4)
 APP(app(x4,x5),x6) -> APP(x4,app(x5,x6))
 APP(x4,x5) -> APP(x5,x4)
-> EAxioms:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
->->-> Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> SRules:
 APP(x4,app(x5,x6)) -> APP(x5,x6)

Problem 1.3: 

Reduction Pairs Processor:
-> FAxioms:
 APP(app(x4,x5),x6) = APP(x4,app(x5,x6))
 APP(x4,x5) = APP(x5,x4)
-> Pairs:
 APP(x,app(plus(y,z),t)) -> APP(plus(app(x,y),app(x,z)),t)
-> EAxioms:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
-> Usable Equations:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
-> Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> Usable Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> SRules:
 APP(x4,app(x5,x6)) -> APP(x5,x6)
->Interpretation type:
 Simple mixed
->Coefficients:
 All rationals
->Dimension:
 1
->Bound:
 3
->Interpretation:
 
[app](X1,X2) = 3.X1.X2 + 2.X1 + 2.X2 + 2/3
[eq](X1,X2) = X2 + 1/3
[if](X1,X2,X3) = 1/3.X1.X2 + 3.X2.X3 + 1/3.X2 + X3
[plus](X1,X2) = X1 + X2 + 3
[0] = 2
[empty] = 1/2
[false] = 0
[s](X) = 3.X
[singl](X) = 2.X.X + 3.X
[true] = 2
[APP](X1,X2) = X1.X2 + 2/3.X1 + 2/3.X2
[EQ](X1,X2) = 0
[IF](X1,X2,X3) = 0
[PLUS](X1,X2) = 0

Problem 1.3: 

SCC Processor:
-> FAxioms:
 APP(app(x4,x5),x6) = APP(x4,app(x5,x6))
 APP(x4,x5) = APP(x5,x4)
-> Pairs:
 Empty
-> EAxioms:
 app(app(x4,x5),x6) = app(x4,app(x5,x6))
 app(x4,x5) = app(x5,x4)
 plus(plus(x4,x5),x6) = plus(x4,plus(x5,x6))
 plus(x4,x5) = plus(x5,x4)
-> Rules:
 app(singl(x),singl(y)) -> if(eq(x,y),singl(x),empty)
 app(x,app(plus(y,z),t)) -> app(plus(app(x,y),app(x,z)),t)
 app(x,app(empty,z)) -> app(empty,z)
 app(x,plus(y,z)) -> plus(app(x,y),app(x,z))
 app(x,empty) -> empty
 eq(0,0) -> true
 eq(0,s(x)) -> false
 eq(s(x),s(y)) -> eq(x,y)
 if(false,x,y) -> y
 if(true,x,y) -> x
 plus(empty,x) -> x
-> SRules:
 APP(x4,app(x5,x6)) -> APP(x5,x6)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.