20.02/20.46	NO
20.02/20.46	
20.02/20.46	Problem 1: 
20.02/20.46	
20.02/20.46	(VAR v_NonEmpty:S)
20.02/20.46	(RULES
20.02/20.46	a -> b | f(a) ->* b
20.02/20.46	)
20.02/20.46	
20.02/20.46	Problem 1: 
20.02/20.46	Valid CTRS Processor:
20.02/20.46	-> Rules:
20.02/20.46	 a -> b | f(a) ->* b
20.02/20.46	-> The system is a deterministic 3-CTRS.
20.02/20.46	
20.02/20.46	Problem 1: 
20.02/20.46	
20.02/20.46	Dependency Pairs Processor:
20.02/20.46	
20.02/20.46	Conditional Termination Problem 1:
20.02/20.46	-> Pairs:
20.02/20.46	 Empty
20.02/20.46	-> QPairs:
20.02/20.46	 Empty
20.02/20.46	-> Rules:
20.02/20.46	 a -> b | f(a) ->* b
20.02/20.46	
20.02/20.46	Conditional Termination Problem 2:
20.02/20.46	-> Pairs:
20.02/20.46	 A -> A
20.02/20.46	-> QPairs:
20.02/20.46	 Empty
20.02/20.46	-> Rules:
20.02/20.46	 a -> b | f(a) ->* b
20.02/20.46	
20.02/20.46	Problem 1: 
20.02/20.46	
20.02/20.46	SCC Processor:
20.02/20.46	-> Pairs:
20.02/20.46	 A -> A
20.02/20.46	-> QPairs:
20.02/20.46	 Empty
20.02/20.46	-> Rules:
20.02/20.46	 a -> b | f(a) ->* b
20.02/20.46	->Strongly Connected Components:
20.02/20.46	->->Cycle:
20.02/20.46	->->-> Pairs:
20.02/20.46	 A -> A
20.02/20.46	-> QPairs:
20.02/20.46	 Empty
20.02/20.46	->->-> Rules:
20.02/20.46	 a -> b | f(a) ->* b
20.02/20.46	
20.02/20.46	Problem 1: 
20.02/20.46	
20.02/20.46	Unsatisfiable Rule Processor:
20.02/20.46	-> Pairs:
20.02/20.46	 A -> A
20.02/20.46	-> QPairs:
20.02/20.46	 Empty
20.02/20.46	-> Rules:
20.02/20.46	 a -> b | f(a) ->* b
20.02/20.46	->AGES Output:
20.02/20.46	 
20.02/20.46	Model Results
20.02/20.46	
20.02/20.46	System:
20.02/20.46	mod InTheory is
20.02/20.46	  sorts S Bool .
20.02/20.46	
20.02/20.46	op _->*_ : S S -> Bool .
20.02/20.46	op _->_ : S S -> Bool .
20.02/20.46	op a :  -> S .
20.02/20.46	op b :  -> S .
20.02/20.46	op f : S -> S .
20.02/20.46	op fSNonEmpty :  -> S .
20.02/20.46	op gtrsim : S S -> Bool .
20.02/20.46	op sqsupset : S S -> Bool .
20.02/20.46	
20.02/20.46	endm
20.02/20.46	
20.02/20.46	
20.02/20.46	Property:
20.02/20.46	x:S ->R* x:S
20.02/20.46	x:S ->R y:S /\ y:S ->R* z:S => x:S ->R* z:S
20.02/20.46	gtrsim(x:S,y:S) /\ sqsupset(y:S,z:S) => sqsupset(x:S,z:S)
20.02/20.46	x1:S ->R y1:S => f(x1:S) ->R f(y1:S)
20.02/20.46	f(a) ->R* b => a ->R b
20.02/20.46	x:S ->R y:S => gtrsim(x:S,y:S)
20.02/20.46	sqsupset(b,f(a))
20.02/20.46	
20.02/20.46	Results:
20.02/20.46	
20.02/20.46	
20.02/20.46	Domains:
20.02/20.46	S: |N
20.02/20.46	
20.02/20.46	Function Interpretations:
20.02/20.46	|[a]| = 0
20.02/20.46	|[b]| = 1
20.02/20.46	|[f(x_1_1:S)]| = 0
20.02/20.46	|[fSNonEmpty]| = 0
20.02/20.46	
20.02/20.46	Predicate Interpretations:
20.02/20.46	 x_1_1:S ->* x_2_1:S <=> (x_1_1:S >= x_2_1:S)
20.02/20.46	 x_1_1:S -> x_2_1:S <=> ((0 >= 1 + x_1_1:S + x_2_1:S) /\ (1 >= x_2_1:S))
20.02/20.46	gtrsim(x_1_1:S,x_2_1:S) <=> ((0 >= 1+2.x_1_1:S) /\ (2+3.x_1_1:S >= 2.x_2_1:S))
20.02/20.46	sqsupset(x_1_1:S,x_2_1:S) <=> (2.x_1_1:S >= 2+2.x_2_1:S)
20.02/20.46	
20.02/20.46	Problem 1: 
20.02/20.46	
20.02/20.46	Infinite Processor:
20.02/20.46	-> Pairs:
20.02/20.46	 A -> A
20.02/20.46	-> QPairs:
20.02/20.46	 Empty
20.02/20.46	-> Rules:
20.02/20.46	 Empty
20.02/20.46	-> Pairs in cycle:
20.02/20.46	 A -> A
20.02/20.46	
20.02/20.46	The problem is infinite.
20.02/20.46	EOF