/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 x)
(RULES
A -> h(f(a),f(b))
a -> d
a -> e
b -> d
b -> e
f(x) -> x | x -> d
g(d,e) -> A
h(x,x) -> g(x,x)
)

Problem 1: 
Valid CTRS Processor:
-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)
-> The system is a deterministic 3-CTRS.

Problem 1: 

Dependency Pairs Processor:

Conditional Termination Problem 1:
-> Pairs:
 A# -> A
 A# -> B
 A# -> F(a)
 A# -> F(b)
 A# -> H(f(a),f(b))
 G(d,e) -> A#
 H(x,x) -> G(x,x)
-> QPairs:
 Empty
-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)

Conditional Termination Problem 2:
-> Pairs:
 Empty
-> QPairs:
 A# -> A
 A# -> B
 A# -> F(a)
 A# -> F(b)
 A# -> H(f(a),f(b))
 G(d,e) -> A#
 H(x,x) -> G(x,x)
-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)


The problem is decomposed in 2 subproblems.

Problem 1.1: 

SCC Processor:
-> Pairs:
 A# -> A
 A# -> B
 A# -> F(a)
 A# -> F(b)
 A# -> H(f(a),f(b))
 G(d,e) -> A#
 H(x,x) -> G(x,x)
-> QPairs:
 Empty
-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A# -> H(f(a),f(b))
 G(d,e) -> A#
 H(x,x) -> G(x,x)
->->-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)

Problem 1.1: 

Conditional Narrowing Processor:
-> Pairs:
 A# -> H(f(a),f(b))
 G(d,e) -> A#
 H(x,x) -> G(x,x)
-> QPairs:
 Empty
-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)
->Narrowed Pairs:
->->Original Pair:
 A# -> H(f(a),f(b))
->-> Narrowed pairs:
 A# -> H(a,f(b)) | a -> d
 A# -> H(f(a),b) | b -> d
 A# -> H(f(a),f(d))
 A# -> H(f(a),f(e))
 A# -> H(f(d),f(b))
 A# -> H(f(e),f(b))

Problem 1.1: 

SCC Processor:
-> Pairs:
 A# -> H(a,f(b)) | a -> d
 A# -> H(f(a),b) | b -> d
 A# -> H(f(a),f(d))
 A# -> H(f(a),f(e))
 A# -> H(f(d),f(b))
 A# -> H(f(e),f(b))
 G(d,e) -> A#
 H(x,x) -> G(x,x)
-> QPairs:
 Empty
-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A# -> H(a,f(b)) | a -> d
 A# -> H(f(a),b) | b -> d
 A# -> H(f(a),f(d))
 A# -> H(f(a),f(e))
 A# -> H(f(d),f(b))
 A# -> H(f(e),f(b))
 G(d,e) -> A#
 H(x,x) -> G(x,x)
->->-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)

Problem 1.1: 

Reduction Triple Processor:
-> Pairs:
 A# -> H(a,f(b)) | a -> d
 A# -> H(f(a),b) | b -> d
 A# -> H(f(a),f(d))
 A# -> H(f(a),f(e))
 A# -> H(f(d),f(b))
 A# -> H(f(e),f(b))
 G(d,e) -> A#
 H(x,x) -> G(x,x)
-> QPairs:
 Empty
-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)
-> Usable rules:
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
->Interpretation type:
 Linear
->Coefficients:
 Integer Numbers
->Dimension:
 1
->Convex Domain:
 D[(x_1)] = ((0 >= 0) /\ (x_1 >= 0))
->Interpretation:
 
[delta] = 1
[a] = 1
[b] = 1
[f](x_1) = x_1
[d] = 0
[e] = 1
[A#] = 3
[G](x_1,x_2) = 1+2.x_2
[H](x_1,x_2) = 1+x_1+x_2

Problem 1.1: 

SCC Processor:
-> Pairs:
 A# -> H(a,f(b)) | a -> d
 A# -> H(f(a),b) | b -> d
 A# -> H(f(a),f(e))
 A# -> H(f(d),f(b))
 A# -> H(f(e),f(b))
 G(d,e) -> A#
 H(x,x) -> G(x,x)
-> QPairs:
 Empty
-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A# -> H(a,f(b)) | a -> d
 A# -> H(f(a),b) | b -> d
 A# -> H(f(a),f(e))
 A# -> H(f(d),f(b))
 A# -> H(f(e),f(b))
 G(d,e) -> A#
 H(x,x) -> G(x,x)
->->-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)

Problem 1.1: 

Reduction Triple Processor:
-> Pairs:
 A# -> H(a,f(b)) | a -> d
 A# -> H(f(a),b) | b -> d
 A# -> H(f(a),f(e))
 A# -> H(f(d),f(b))
 A# -> H(f(e),f(b))
 G(d,e) -> A#
 H(x,x) -> G(x,x)
-> QPairs:
 Empty
-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)
-> Usable rules:
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
->Interpretation type:
 Linear
->Coefficients:
 Integer Numbers
->Dimension:
 1
->Convex Domain:
 D[(x_1)] = ((x_1 >= 0) /\ (1-x_1 >= 0))
->Interpretation:
 
[delta] = 1
[a] = 1
[b] = 1
[f](x_1) = x_1
[d] = 1
[e] = 0
[A#] = 1
[G](x_1,x_2) = x_1
[H](x_1,x_2) = x_2

Problem 1.1: 

SCC Processor:
-> Pairs:
 A# -> H(a,f(b)) | a -> d
 A# -> H(f(a),b) | b -> d
 A# -> H(f(d),f(b))
 A# -> H(f(e),f(b))
 G(d,e) -> A#
 H(x,x) -> G(x,x)
-> QPairs:
 Empty
-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A# -> H(a,f(b)) | a -> d
 A# -> H(f(a),b) | b -> d
 A# -> H(f(d),f(b))
 A# -> H(f(e),f(b))
 G(d,e) -> A#
 H(x,x) -> G(x,x)
->->-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)

Problem 1.1: 

Reduction Triple Processor:
-> Pairs:
 A# -> H(a,f(b)) | a -> d
 A# -> H(f(a),b) | b -> d
 A# -> H(f(d),f(b))
 A# -> H(f(e),f(b))
 G(d,e) -> A#
 H(x,x) -> G(x,x)
-> QPairs:
 Empty
-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)
-> Usable rules:
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
->Interpretation type:
 Linear
->Coefficients:
 Integer Numbers
->Dimension:
 1
->Convex Domain:
 D[(x_1)] = ((0 >= 0) /\ (x_1 >= 0))
->Interpretation:
 
[delta] = 1
[a] = 1
[b] = 1
[f](x_1) = x_1
[d] = 0
[e] = 1
[A#] = 1
[G](x_1,x_2) = x_2
[H](x_1,x_2) = x_1

Problem 1.1: 

SCC Processor:
-> Pairs:
 A# -> H(a,f(b)) | a -> d
 A# -> H(f(a),b) | b -> d
 A# -> H(f(e),f(b))
 G(d,e) -> A#
 H(x,x) -> G(x,x)
-> QPairs:
 Empty
-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A# -> H(a,f(b)) | a -> d
 A# -> H(f(a),b) | b -> d
 A# -> H(f(e),f(b))
 G(d,e) -> A#
 H(x,x) -> G(x,x)
->->-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)

Problem 1.1: 

Reduction Triple Processor:
-> Pairs:
 A# -> H(a,f(b)) | a -> d
 A# -> H(f(a),b) | b -> d
 A# -> H(f(e),f(b))
 G(d,e) -> A#
 H(x,x) -> G(x,x)
-> QPairs:
 Empty
-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)
-> Usable rules:
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
->Interpretation type:
 Linear
->Coefficients:
 Integer Numbers
->Dimension:
 1
->Convex Domain:
 D[(x_1)] = ((x_1 >= 0) /\ (0 >= 0))
->Interpretation:
 
[delta] = 1
[a] = 1
[b] = 1
[f](x_1) = x_1
[d] = 1
[e] = 0
[A#] = 2
[G](x_1,x_2) = 2.x_1
[H](x_1,x_2) = x_1+x_2

Problem 1.1: 

SCC Processor:
-> Pairs:
 A# -> H(a,f(b)) | a -> d
 A# -> H(f(a),b) | b -> d
 G(d,e) -> A#
 H(x,x) -> G(x,x)
-> QPairs:
 Empty
-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A# -> H(a,f(b)) | a -> d
 A# -> H(f(a),b) | b -> d
 G(d,e) -> A#
 H(x,x) -> G(x,x)
->->-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)

Problem 1.1: 

Conditional Narrowing Processor:
-> Pairs:
 A# -> H(a,f(b)) | a -> d
 A# -> H(f(a),b) | b -> d
 G(d,e) -> A#
 H(x,x) -> G(x,x)
-> QPairs:
 Empty
-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)
->Narrowed Pairs:
->->Original Pair:
 A# -> H(a,f(b)) | a -> d
->-> Narrowed pairs:
 A# -> H(a,b) | b -> d, a -> d
 A# -> H(a,f(d)) | a -> d
 A# -> H(a,f(e)) | a -> d
 A# -> H(d,f(b)) | a -> d
 A# -> H(e,f(b)) | a -> d
->->Original Pair:
 A# -> H(f(a),b) | b -> d
->-> Narrowed pairs:
 A# -> H(a,b) | a -> d, b -> d
 A# -> H(f(a),d) | b -> d
 A# -> H(f(a),e) | b -> d
 A# -> H(f(d),b) | b -> d
 A# -> H(f(e),b) | b -> d

Problem 1.1: 

SCC Processor:
-> Pairs:
 A# -> H(a,b) | a -> d, b -> d
 A# -> H(a,b) | b -> d, a -> d
 A# -> H(a,f(d)) | a -> d
 A# -> H(a,f(e)) | a -> d
 A# -> H(f(a),d) | b -> d
 A# -> H(f(a),e) | b -> d
 A# -> H(f(d),b) | b -> d
 A# -> H(f(e),b) | b -> d
 A# -> H(d,f(b)) | a -> d
 A# -> H(e,f(b)) | a -> d
 G(d,e) -> A#
 H(x,x) -> G(x,x)
-> QPairs:
 Empty
-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A# -> H(a,b) | a -> d, b -> d
 A# -> H(a,b) | b -> d, a -> d
 A# -> H(a,f(d)) | a -> d
 A# -> H(a,f(e)) | a -> d
 A# -> H(f(a),d) | b -> d
 A# -> H(f(a),e) | b -> d
 A# -> H(f(d),b) | b -> d
 A# -> H(f(e),b) | b -> d
 A# -> H(d,f(b)) | a -> d
 A# -> H(e,f(b)) | a -> d
 G(d,e) -> A#
 H(x,x) -> G(x,x)
->->-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)

Problem 1.1: 

Reduction Triple Processor:
-> Pairs:
 A# -> H(a,b) | a -> d, b -> d
 A# -> H(a,b) | b -> d, a -> d
 A# -> H(a,f(d)) | a -> d
 A# -> H(a,f(e)) | a -> d
 A# -> H(f(a),d) | b -> d
 A# -> H(f(a),e) | b -> d
 A# -> H(f(d),b) | b -> d
 A# -> H(f(e),b) | b -> d
 A# -> H(d,f(b)) | a -> d
 A# -> H(e,f(b)) | a -> d
 G(d,e) -> A#
 H(x,x) -> G(x,x)
-> QPairs:
 Empty
-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)
-> Usable rules:
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
->Interpretation type:
 Linear
->Coefficients:
 Integer Numbers
->Dimension:
 1
->Convex Domain:
 D[(x_1)] = ((x_1 >= 0) /\ (1-x_1 >= 0))
->Interpretation:
 
[delta] = 1
[a] = 1
[b] = 1
[f](x_1) = x_1
[d] = 0
[e] = 1
[A#] = 1
[G](x_1,x_2) = x_2
[H](x_1,x_2) = x_2

Problem 1.1: 

SCC Processor:
-> Pairs:
 A# -> H(a,b) | a -> d, b -> d
 A# -> H(a,b) | b -> d, a -> d
 A# -> H(a,f(e)) | a -> d
 A# -> H(f(a),d) | b -> d
 A# -> H(f(a),e) | b -> d
 A# -> H(f(d),b) | b -> d
 A# -> H(f(e),b) | b -> d
 A# -> H(d,f(b)) | a -> d
 A# -> H(e,f(b)) | a -> d
 G(d,e) -> A#
 H(x,x) -> G(x,x)
-> QPairs:
 Empty
-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A# -> H(a,b) | a -> d, b -> d
 A# -> H(a,b) | b -> d, a -> d
 A# -> H(a,f(e)) | a -> d
 A# -> H(f(a),d) | b -> d
 A# -> H(f(a),e) | b -> d
 A# -> H(f(d),b) | b -> d
 A# -> H(f(e),b) | b -> d
 A# -> H(d,f(b)) | a -> d
 A# -> H(e,f(b)) | a -> d
 G(d,e) -> A#
 H(x,x) -> G(x,x)
->->-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)

Problem 1.1: 

Reduction Triple Processor:
-> Pairs:
 A# -> H(a,b) | a -> d, b -> d
 A# -> H(a,b) | b -> d, a -> d
 A# -> H(a,f(e)) | a -> d
 A# -> H(f(a),d) | b -> d
 A# -> H(f(a),e) | b -> d
 A# -> H(f(d),b) | b -> d
 A# -> H(f(e),b) | b -> d
 A# -> H(d,f(b)) | a -> d
 A# -> H(e,f(b)) | a -> d
 G(d,e) -> A#
 H(x,x) -> G(x,x)
-> QPairs:
 Empty
-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)
-> Usable rules:
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
->Interpretation type:
 Linear
->Coefficients:
 Integer Numbers
->Dimension:
 1
->Convex Domain:
 D[(x_1)] = ((x_1 >= 0) /\ (x_1 >= 0))
->Interpretation:
 
[delta] = 1
[a] = 1
[b] = 1
[f](x_1) = x_1
[d] = 1
[e] = 0
[A#] = 1
[G](x_1,x_2) = x_1
[H](x_1,x_2) = x_2

Problem 1.1: 

SCC Processor:
-> Pairs:
 A# -> H(a,b) | a -> d, b -> d
 A# -> H(a,b) | b -> d, a -> d
 A# -> H(f(a),d) | b -> d
 A# -> H(f(a),e) | b -> d
 A# -> H(f(d),b) | b -> d
 A# -> H(f(e),b) | b -> d
 A# -> H(d,f(b)) | a -> d
 A# -> H(e,f(b)) | a -> d
 G(d,e) -> A#
 H(x,x) -> G(x,x)
-> QPairs:
 Empty
-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A# -> H(a,b) | a -> d, b -> d
 A# -> H(a,b) | b -> d, a -> d
 A# -> H(f(a),d) | b -> d
 A# -> H(f(a),e) | b -> d
 A# -> H(f(d),b) | b -> d
 A# -> H(f(e),b) | b -> d
 A# -> H(d,f(b)) | a -> d
 A# -> H(e,f(b)) | a -> d
 G(d,e) -> A#
 H(x,x) -> G(x,x)
->->-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)

Problem 1.1: 

Reduction Triple Processor:
-> Pairs:
 A# -> H(a,b) | a -> d, b -> d
 A# -> H(a,b) | b -> d, a -> d
 A# -> H(f(a),d) | b -> d
 A# -> H(f(a),e) | b -> d
 A# -> H(f(d),b) | b -> d
 A# -> H(f(e),b) | b -> d
 A# -> H(d,f(b)) | a -> d
 A# -> H(e,f(b)) | a -> d
 G(d,e) -> A#
 H(x,x) -> G(x,x)
-> QPairs:
 Empty
-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)
-> Usable rules:
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
->Interpretation type:
 Linear
->Coefficients:
 Integer Numbers
->Dimension:
 1
->Convex Domain:
 D[(x_1)] = ((x_1 >= 0) /\ (0 >= 0))
->Interpretation:
 
[delta] = 1
[a] = 1
[b] = 1
[f](x_1) = x_1
[d] = 0
[e] = 1
[A#] = 2
[G](x_1,x_2) = 2.x_2
[H](x_1,x_2) = x_1+x_2

Problem 1.1: 

SCC Processor:
-> Pairs:
 A# -> H(a,b) | a -> d, b -> d
 A# -> H(a,b) | b -> d, a -> d
 A# -> H(f(a),e) | b -> d
 A# -> H(f(d),b) | b -> d
 A# -> H(f(e),b) | b -> d
 A# -> H(d,f(b)) | a -> d
 A# -> H(e,f(b)) | a -> d
 G(d,e) -> A#
 H(x,x) -> G(x,x)
-> QPairs:
 Empty
-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A# -> H(a,b) | a -> d, b -> d
 A# -> H(a,b) | b -> d, a -> d
 A# -> H(f(a),e) | b -> d
 A# -> H(f(d),b) | b -> d
 A# -> H(f(e),b) | b -> d
 A# -> H(d,f(b)) | a -> d
 A# -> H(e,f(b)) | a -> d
 G(d,e) -> A#
 H(x,x) -> G(x,x)
->->-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)

Problem 1.1: 

Reduction Triple Processor:
-> Pairs:
 A# -> H(a,b) | a -> d, b -> d
 A# -> H(a,b) | b -> d, a -> d
 A# -> H(f(a),e) | b -> d
 A# -> H(f(d),b) | b -> d
 A# -> H(f(e),b) | b -> d
 A# -> H(d,f(b)) | a -> d
 A# -> H(e,f(b)) | a -> d
 G(d,e) -> A#
 H(x,x) -> G(x,x)
-> QPairs:
 Empty
-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)
-> Usable rules:
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
->Interpretation type:
 Linear
->Coefficients:
 Integer Numbers
->Dimension:
 1
->Convex Domain:
 D[(x_1)] = ((1-x_1 >= 0) /\ (1+x_1 >= 0))
->Interpretation:
 
[delta] = 2
[a] = 1
[b] = 1
[f](x_1) = 1
[d] = 1
[e] = -1
[A#] = 1
[G](x_1,x_2) = x_1
[H](x_1,x_2) = x_2

Problem 1.1: 

SCC Processor:
-> Pairs:
 A# -> H(a,b) | a -> d, b -> d
 A# -> H(a,b) | b -> d, a -> d
 A# -> H(f(d),b) | b -> d
 A# -> H(f(e),b) | b -> d
 A# -> H(d,f(b)) | a -> d
 A# -> H(e,f(b)) | a -> d
 G(d,e) -> A#
 H(x,x) -> G(x,x)
-> QPairs:
 Empty
-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A# -> H(a,b) | a -> d, b -> d
 A# -> H(a,b) | b -> d, a -> d
 A# -> H(f(d),b) | b -> d
 A# -> H(f(e),b) | b -> d
 A# -> H(d,f(b)) | a -> d
 A# -> H(e,f(b)) | a -> d
 G(d,e) -> A#
 H(x,x) -> G(x,x)
->->-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)

Problem 1.1: 

Reduction Triple Processor:
-> Pairs:
 A# -> H(a,b) | a -> d, b -> d
 A# -> H(a,b) | b -> d, a -> d
 A# -> H(f(d),b) | b -> d
 A# -> H(f(e),b) | b -> d
 A# -> H(d,f(b)) | a -> d
 A# -> H(e,f(b)) | a -> d
 G(d,e) -> A#
 H(x,x) -> G(x,x)
-> QPairs:
 Empty
-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)
-> Usable rules:
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
->Interpretation type:
 Linear
->Coefficients:
 Integer Numbers
->Dimension:
 1
->Convex Domain:
 D[(x_1)] = ((x_1 >= 0) /\ (1-x_1 >= 0))
->Interpretation:
 
[delta] = 1
[a] = 1
[b] = 1
[f](x_1) = x_1
[d] = 0
[e] = 1
[A#] = 1
[G](x_1,x_2) = x_2
[H](x_1,x_2) = x_1

Problem 1.1: 

SCC Processor:
-> Pairs:
 A# -> H(a,b) | a -> d, b -> d
 A# -> H(a,b) | b -> d, a -> d
 A# -> H(f(e),b) | b -> d
 A# -> H(d,f(b)) | a -> d
 A# -> H(e,f(b)) | a -> d
 G(d,e) -> A#
 H(x,x) -> G(x,x)
-> QPairs:
 Empty
-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A# -> H(a,b) | a -> d, b -> d
 A# -> H(a,b) | b -> d, a -> d
 A# -> H(f(e),b) | b -> d
 A# -> H(d,f(b)) | a -> d
 A# -> H(e,f(b)) | a -> d
 G(d,e) -> A#
 H(x,x) -> G(x,x)
->->-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)

Problem 1.1: 

Reduction Triple Processor:
-> Pairs:
 A# -> H(a,b) | a -> d, b -> d
 A# -> H(a,b) | b -> d, a -> d
 A# -> H(f(e),b) | b -> d
 A# -> H(d,f(b)) | a -> d
 A# -> H(e,f(b)) | a -> d
 G(d,e) -> A#
 H(x,x) -> G(x,x)
-> QPairs:
 Empty
-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)
-> Usable rules:
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
->Interpretation type:
 Linear
->Coefficients:
 Integer Numbers
->Dimension:
 1
->Convex Domain:
 D[(x_1)] = ((1+x_1 >= 0) /\ (1-x_1 >= 0))
->Interpretation:
 
[delta] = 1
[a] = 0
[b] = 0
[f](x_1) = x_1
[d] = 0
[e] = -1
[A#] = 0
[G](x_1,x_2) = x_1
[H](x_1,x_2) = x_1

Problem 1.1: 

SCC Processor:
-> Pairs:
 A# -> H(a,b) | a -> d, b -> d
 A# -> H(a,b) | b -> d, a -> d
 A# -> H(d,f(b)) | a -> d
 A# -> H(e,f(b)) | a -> d
 G(d,e) -> A#
 H(x,x) -> G(x,x)
-> QPairs:
 Empty
-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A# -> H(a,b) | a -> d, b -> d
 A# -> H(a,b) | b -> d, a -> d
 A# -> H(d,f(b)) | a -> d
 A# -> H(e,f(b)) | a -> d
 G(d,e) -> A#
 H(x,x) -> G(x,x)
->->-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)

Problem 1.1: 

Reduction Triple Processor:
-> Pairs:
 A# -> H(a,b) | a -> d, b -> d
 A# -> H(a,b) | b -> d, a -> d
 A# -> H(d,f(b)) | a -> d
 A# -> H(e,f(b)) | a -> d
 G(d,e) -> A#
 H(x,x) -> G(x,x)
-> QPairs:
 Empty
-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)
-> Usable rules:
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
->Interpretation type:
 Linear
->Coefficients:
 Integer Numbers
->Dimension:
 1
->Convex Domain:
 D[(x_1)] = ((1-x_1 >= 0) /\ (1+x_1 >= 0))
->Interpretation:
 
[delta] = 1
[a] = 0
[b] = 0
[f](x_1) = x_1
[d] = -1
[e] = 0
[A#] = 0
[G](x_1,x_2) = x_2
[H](x_1,x_2) = x_1

Problem 1.1: 

SCC Processor:
-> Pairs:
 A# -> H(a,b) | a -> d, b -> d
 A# -> H(a,b) | b -> d, a -> d
 A# -> H(e,f(b)) | a -> d
 G(d,e) -> A#
 H(x,x) -> G(x,x)
-> QPairs:
 Empty
-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A# -> H(a,b) | a -> d, b -> d
 A# -> H(a,b) | b -> d, a -> d
 A# -> H(e,f(b)) | a -> d
 G(d,e) -> A#
 H(x,x) -> G(x,x)
->->-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)

Problem 1.1: 

Reduction Triple Processor:
-> Pairs:
 A# -> H(a,b) | a -> d, b -> d
 A# -> H(a,b) | b -> d, a -> d
 A# -> H(e,f(b)) | a -> d
 G(d,e) -> A#
 H(x,x) -> G(x,x)
-> QPairs:
 Empty
-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)
-> Usable rules:
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
->Interpretation type:
 Linear
->Coefficients:
 Integer Numbers
->Dimension:
 1
->Convex Domain:
 D[(x_1)] = ((1+x_1 >= 0) /\ (1-x_1 >= 0))
->Interpretation:
 
[delta] = 1
[a] = 1
[b] = 1
[f](x_1) = x_1
[d] = 1
[e] = 0
[A#] = 1
[G](x_1,x_2) = x_1
[H](x_1,x_2) = x_1

Problem 1.1: 

SCC Processor:
-> Pairs:
 A# -> H(a,b) | a -> d, b -> d
 A# -> H(a,b) | b -> d, a -> d
 G(d,e) -> A#
 H(x,x) -> G(x,x)
-> QPairs:
 Empty
-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A# -> H(a,b) | a -> d, b -> d
 A# -> H(a,b) | b -> d, a -> d
 G(d,e) -> A#
 H(x,x) -> G(x,x)
->->-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)

Problem 1.1: 

Conditional Narrowing Processor:
-> Pairs:
 A# -> H(a,b) | a -> d, b -> d
 A# -> H(a,b) | b -> d, a -> d
 G(d,e) -> A#
 H(x,x) -> G(x,x)
-> QPairs:
 Empty
-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)
->Narrowed Pairs:
->->Original Pair:
 A# -> H(a,b) | a -> d, b -> d
->-> Narrowed pairs:
 A# -> H(a,d) | a -> d, b -> d
 A# -> H(a,e) | a -> d, b -> d
 A# -> H(d,b) | a -> d, b -> d
 A# -> H(e,b) | a -> d, b -> d
->->Original Pair:
 A# -> H(a,b) | b -> d, a -> d
->-> Narrowed pairs:
 A# -> H(a,d) | b -> d, a -> d
 A# -> H(a,e) | b -> d, a -> d
 A# -> H(d,b) | b -> d, a -> d
 A# -> H(e,b) | b -> d, a -> d

Problem 1.1: 

SCC Processor:
-> Pairs:
 A# -> H(a,d) | a -> d, b -> d
 A# -> H(a,d) | b -> d, a -> d
 A# -> H(a,e) | a -> d, b -> d
 A# -> H(a,e) | b -> d, a -> d
 A# -> H(d,b) | a -> d, b -> d
 A# -> H(d,b) | b -> d, a -> d
 A# -> H(e,b) | a -> d, b -> d
 A# -> H(e,b) | b -> d, a -> d
 G(d,e) -> A#
 H(x,x) -> G(x,x)
-> QPairs:
 Empty
-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A# -> H(a,d) | a -> d, b -> d
 A# -> H(a,d) | b -> d, a -> d
 A# -> H(a,e) | a -> d, b -> d
 A# -> H(a,e) | b -> d, a -> d
 A# -> H(d,b) | a -> d, b -> d
 A# -> H(d,b) | b -> d, a -> d
 A# -> H(e,b) | a -> d, b -> d
 A# -> H(e,b) | b -> d, a -> d
 G(d,e) -> A#
 H(x,x) -> G(x,x)
->->-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)

Problem 1.1: 

Reduction Triple Processor:
-> Pairs:
 A# -> H(a,d) | a -> d, b -> d
 A# -> H(a,d) | b -> d, a -> d
 A# -> H(a,e) | a -> d, b -> d
 A# -> H(a,e) | b -> d, a -> d
 A# -> H(d,b) | a -> d, b -> d
 A# -> H(d,b) | b -> d, a -> d
 A# -> H(e,b) | a -> d, b -> d
 A# -> H(e,b) | b -> d, a -> d
 G(d,e) -> A#
 H(x,x) -> G(x,x)
-> QPairs:
 Empty
-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)
-> Usable rules:
 a -> d
 a -> e
 b -> d
 b -> e
->Interpretation type:
 Linear
->Coefficients:
 Integer Numbers
->Dimension:
 1
->Convex Domain:
 D[(x_1)] = ((-x_1 >= 0) /\ (1+x_1 >= 0))
->Interpretation:
 
[delta] = 1
[a] = 0
[b] = 0
[d] = -1
[e] = 0
[A#] = 0
[G](x_1,x_2) = x_2
[H](x_1,x_2) = x_2

Problem 1.1: 

SCC Processor:
-> Pairs:
 A# -> H(a,d) | b -> d, a -> d
 A# -> H(a,e) | a -> d, b -> d
 A# -> H(a,e) | b -> d, a -> d
 A# -> H(d,b) | a -> d, b -> d
 A# -> H(d,b) | b -> d, a -> d
 A# -> H(e,b) | a -> d, b -> d
 A# -> H(e,b) | b -> d, a -> d
 G(d,e) -> A#
 H(x,x) -> G(x,x)
-> QPairs:
 Empty
-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A# -> H(a,d) | b -> d, a -> d
 A# -> H(a,e) | a -> d, b -> d
 A# -> H(a,e) | b -> d, a -> d
 A# -> H(d,b) | a -> d, b -> d
 A# -> H(d,b) | b -> d, a -> d
 A# -> H(e,b) | a -> d, b -> d
 A# -> H(e,b) | b -> d, a -> d
 G(d,e) -> A#
 H(x,x) -> G(x,x)
->->-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)

Problem 1.1: 

Reduction Triple Processor:
-> Pairs:
 A# -> H(a,d) | b -> d, a -> d
 A# -> H(a,e) | a -> d, b -> d
 A# -> H(a,e) | b -> d, a -> d
 A# -> H(d,b) | a -> d, b -> d
 A# -> H(d,b) | b -> d, a -> d
 A# -> H(e,b) | a -> d, b -> d
 A# -> H(e,b) | b -> d, a -> d
 G(d,e) -> A#
 H(x,x) -> G(x,x)
-> QPairs:
 Empty
-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)
-> Usable rules:
 a -> d
 a -> e
 b -> d
 b -> e
->Interpretation type:
 Linear
->Coefficients:
 Integer Numbers
->Dimension:
 1
->Convex Domain:
 D[(x_1)] = ((-x_1 >= 0) /\ (1+x_1 >= 0))
->Interpretation:
 
[delta] = 1
[a] = 0
[b] = 0
[d] = -1
[e] = 0
[A#] = 0
[G](x_1,x_2) = x_2
[H](x_1,x_2) = x_2

Problem 1.1: 

SCC Processor:
-> Pairs:
 A# -> H(a,e) | a -> d, b -> d
 A# -> H(a,e) | b -> d, a -> d
 A# -> H(d,b) | a -> d, b -> d
 A# -> H(d,b) | b -> d, a -> d
 A# -> H(e,b) | a -> d, b -> d
 A# -> H(e,b) | b -> d, a -> d
 G(d,e) -> A#
 H(x,x) -> G(x,x)
-> QPairs:
 Empty
-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A# -> H(a,e) | a -> d, b -> d
 A# -> H(a,e) | b -> d, a -> d
 A# -> H(d,b) | a -> d, b -> d
 A# -> H(d,b) | b -> d, a -> d
 A# -> H(e,b) | a -> d, b -> d
 A# -> H(e,b) | b -> d, a -> d
 G(d,e) -> A#
 H(x,x) -> G(x,x)
->->-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)

Problem 1.1: 

Reduction Triple Processor:
-> Pairs:
 A# -> H(a,e) | a -> d, b -> d
 A# -> H(a,e) | b -> d, a -> d
 A# -> H(d,b) | a -> d, b -> d
 A# -> H(d,b) | b -> d, a -> d
 A# -> H(e,b) | a -> d, b -> d
 A# -> H(e,b) | b -> d, a -> d
 G(d,e) -> A#
 H(x,x) -> G(x,x)
-> QPairs:
 Empty
-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)
-> Usable rules:
 a -> d
 a -> e
 b -> d
 b -> e
->Interpretation type:
 Linear
->Coefficients:
 Integer Numbers
->Dimension:
 1
->Convex Domain:
 D[(x_1)] = ((-x_1 >= 0) /\ (1+x_1 >= 0))
->Interpretation:
 
[delta] = 1
[a] = 0
[b] = 0
[d] = 0
[e] = -1
[A#] = 0
[G](x_1,x_2) = x_1
[H](x_1,x_2) = x_2

Problem 1.1: 

SCC Processor:
-> Pairs:
 A# -> H(a,e) | b -> d, a -> d
 A# -> H(d,b) | a -> d, b -> d
 A# -> H(d,b) | b -> d, a -> d
 A# -> H(e,b) | a -> d, b -> d
 A# -> H(e,b) | b -> d, a -> d
 G(d,e) -> A#
 H(x,x) -> G(x,x)
-> QPairs:
 Empty
-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A# -> H(a,e) | b -> d, a -> d
 A# -> H(d,b) | a -> d, b -> d
 A# -> H(d,b) | b -> d, a -> d
 A# -> H(e,b) | a -> d, b -> d
 A# -> H(e,b) | b -> d, a -> d
 G(d,e) -> A#
 H(x,x) -> G(x,x)
->->-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)

Problem 1.1: 

Reduction Triple Processor:
-> Pairs:
 A# -> H(a,e) | b -> d, a -> d
 A# -> H(d,b) | a -> d, b -> d
 A# -> H(d,b) | b -> d, a -> d
 A# -> H(e,b) | a -> d, b -> d
 A# -> H(e,b) | b -> d, a -> d
 G(d,e) -> A#
 H(x,x) -> G(x,x)
-> QPairs:
 Empty
-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)
-> Usable rules:
 a -> d
 a -> e
 b -> d
 b -> e
->Interpretation type:
 Linear
->Coefficients:
 Integer Numbers
->Dimension:
 1
->Convex Domain:
 D[(x_1)] = ((-x_1 >= 0) /\ (1+x_1 >= 0))
->Interpretation:
 
[delta] = 1
[a] = 0
[b] = 0
[d] = 0
[e] = -1
[A#] = 0
[G](x_1,x_2) = x_1
[H](x_1,x_2) = x_2

Problem 1.1: 

SCC Processor:
-> Pairs:
 A# -> H(d,b) | a -> d, b -> d
 A# -> H(d,b) | b -> d, a -> d
 A# -> H(e,b) | a -> d, b -> d
 A# -> H(e,b) | b -> d, a -> d
 G(d,e) -> A#
 H(x,x) -> G(x,x)
-> QPairs:
 Empty
-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A# -> H(d,b) | a -> d, b -> d
 A# -> H(d,b) | b -> d, a -> d
 A# -> H(e,b) | a -> d, b -> d
 A# -> H(e,b) | b -> d, a -> d
 G(d,e) -> A#
 H(x,x) -> G(x,x)
->->-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)

Problem 1.1: 

Reduction Triple Processor:
-> Pairs:
 A# -> H(d,b) | a -> d, b -> d
 A# -> H(d,b) | b -> d, a -> d
 A# -> H(e,b) | a -> d, b -> d
 A# -> H(e,b) | b -> d, a -> d
 G(d,e) -> A#
 H(x,x) -> G(x,x)
-> QPairs:
 Empty
-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)
-> Usable rules:
 a -> d
 a -> e
 b -> d
 b -> e
->Interpretation type:
 Linear
->Coefficients:
 Integer Numbers
->Dimension:
 1
->Convex Domain:
 D[(x_1)] = ((x_1 >= 0) /\ (1 >= 0))
->Interpretation:
 
[delta] = 1
[a] = 1
[b] = 1
[d] = 0
[e] = 1
[A#] = 1
[G](x_1,x_2) = x_2
[H](x_1,x_2) = x_1

Problem 1.1: 

SCC Processor:
-> Pairs:
 A# -> H(d,b) | b -> d, a -> d
 A# -> H(e,b) | a -> d, b -> d
 A# -> H(e,b) | b -> d, a -> d
 G(d,e) -> A#
 H(x,x) -> G(x,x)
-> QPairs:
 Empty
-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A# -> H(d,b) | b -> d, a -> d
 A# -> H(e,b) | a -> d, b -> d
 A# -> H(e,b) | b -> d, a -> d
 G(d,e) -> A#
 H(x,x) -> G(x,x)
->->-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)

Problem 1.1: 

Reduction Triple Processor:
-> Pairs:
 A# -> H(d,b) | b -> d, a -> d
 A# -> H(e,b) | a -> d, b -> d
 A# -> H(e,b) | b -> d, a -> d
 G(d,e) -> A#
 H(x,x) -> G(x,x)
-> QPairs:
 Empty
-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)
-> Usable rules:
 a -> d
 a -> e
 b -> d
 b -> e
->Interpretation type:
 Linear
->Coefficients:
 Integer Numbers
->Dimension:
 1
->Convex Domain:
 D[(x_1)] = ((x_1 >= 0) /\ (1 >= 0))
->Interpretation:
 
[delta] = 1
[a] = 1
[b] = 1
[d] = 0
[e] = 1
[A#] = 1
[G](x_1,x_2) = x_2
[H](x_1,x_2) = x_1

Problem 1.1: 

SCC Processor:
-> Pairs:
 A# -> H(e,b) | a -> d, b -> d
 A# -> H(e,b) | b -> d, a -> d
 G(d,e) -> A#
 H(x,x) -> G(x,x)
-> QPairs:
 Empty
-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A# -> H(e,b) | a -> d, b -> d
 A# -> H(e,b) | b -> d, a -> d
 G(d,e) -> A#
 H(x,x) -> G(x,x)
->->-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)

Problem 1.1: 

Reduction Triple Processor:
-> Pairs:
 A# -> H(e,b) | a -> d, b -> d
 A# -> H(e,b) | b -> d, a -> d
 G(d,e) -> A#
 H(x,x) -> G(x,x)
-> QPairs:
 Empty
-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)
-> Usable rules:
 a -> d
 a -> e
 b -> d
 b -> e
->Interpretation type:
 Linear
->Coefficients:
 Integer Numbers
->Dimension:
 1
->Convex Domain:
 D[(x_1)] = ((0 >= 0) /\ (x_1 >= 0))
->Interpretation:
 
[delta] = 1
[a] = 1
[b] = 1
[d] = 1
[e] = 0
[A#] = 2
[G](x_1,x_2) = 2.x_1
[H](x_1,x_2) = 2.x_1+x_2

Problem 1.1: 

SCC Processor:
-> Pairs:
 A# -> H(e,b) | b -> d, a -> d
 G(d,e) -> A#
 H(x,x) -> G(x,x)
-> QPairs:
 Empty
-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 A# -> H(e,b) | b -> d, a -> d
 G(d,e) -> A#
 H(x,x) -> G(x,x)
->->-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)

Problem 1.1: 

Reduction Triple Processor:
-> Pairs:
 A# -> H(e,b) | b -> d, a -> d
 G(d,e) -> A#
 H(x,x) -> G(x,x)
-> QPairs:
 Empty
-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)
-> Usable rules:
 a -> d
 a -> e
 b -> d
 b -> e
->Interpretation type:
 Linear
->Coefficients:
 Integer Numbers
->Dimension:
 1
->Convex Domain:
 D[(x_1)] = ((0 >= 0) /\ (1+x_1 >= 0))
->Interpretation:
 
[delta] = 1
[a] = 0
[b] = 0
[d] = 0
[e] = -1
[A#] = 0
[G](x_1,x_2) = x_1
[H](x_1,x_2) = 1+2.x_1

Problem 1.1: 

SCC Processor:
-> Pairs:
 G(d,e) -> A#
 H(x,x) -> G(x,x)
-> QPairs:
 Empty
-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> QPairs:
 A# -> A
 A# -> B
 A# -> F(a)
 A# -> F(b)
 A# -> H(f(a),f(b))
 G(d,e) -> A#
 H(x,x) -> G(x,x)
-> Rules:
 A -> h(f(a),f(b))
 a -> d
 a -> e
 b -> d
 b -> e
 f(x) -> x | x -> d
 g(d,e) -> A
 h(x,x) -> g(x,x)
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.