120.85/121.13	NO
120.85/121.13	
120.85/121.13	Problem 1: 
120.85/121.13	
120.85/121.13	(VAR v_NonEmpty:S x:S)
120.85/121.13	(RULES
120.85/121.13	a -> b
120.85/121.13	b -> a
120.85/121.13	f(x:S,x:S) -> a
120.85/121.13	g(x:S) -> a | g(x:S) ->* b
120.85/121.13	)
120.85/121.13	
120.85/121.13	Problem 1: 
120.85/121.13	Valid CTRS Processor:
120.85/121.13	-> Rules:
120.85/121.13	 a -> b
120.85/121.13	 b -> a
120.85/121.13	 f(x:S,x:S) -> a
120.85/121.13	 g(x:S) -> a | g(x:S) ->* b
120.85/121.13	-> The system is a deterministic 3-CTRS.
120.85/121.13	
120.85/121.13	Problem 1: 
120.85/121.13	
120.85/121.13	Dependency Pairs Processor:
120.85/121.13	
120.85/121.13	Conditional Termination Problem 1:
120.85/121.13	-> Pairs:
120.85/121.13	 A -> B
120.85/121.13	 B -> A
120.85/121.13	 F(x:S,x:S) -> A
120.85/121.13	 G(x:S) -> A | g(x:S) ->* b
120.85/121.13	-> QPairs:
120.85/121.13	 Empty
120.85/121.13	-> Rules:
120.85/121.13	 a -> b
120.85/121.13	 b -> a
120.85/121.13	 f(x:S,x:S) -> a
120.85/121.13	 g(x:S) -> a | g(x:S) ->* b
120.85/121.13	
120.85/121.13	Conditional Termination Problem 2:
120.85/121.13	-> Pairs:
120.85/121.13	 G(x:S) -> G(x:S)
120.85/121.13	-> QPairs:
120.85/121.13	 A -> B
120.85/121.13	 B -> A
120.85/121.13	 G(x:S) -> A | g(x:S) ->* b
120.85/121.13	-> Rules:
120.85/121.13	 a -> b
120.85/121.13	 b -> a
120.85/121.13	 f(x:S,x:S) -> a
120.85/121.13	 g(x:S) -> a | g(x:S) ->* b
120.85/121.13	
120.85/121.13	Problem 1: 
120.85/121.13	
120.85/121.13	SCC Processor:
120.85/121.13	-> Pairs:
120.85/121.13	 A -> B
120.85/121.13	 B -> A
120.85/121.13	 F(x:S,x:S) -> A
120.85/121.13	 G(x:S) -> A | g(x:S) ->* b
120.85/121.13	-> QPairs:
120.85/121.13	 Empty
120.85/121.13	-> Rules:
120.85/121.13	 a -> b
120.85/121.13	 b -> a
120.85/121.13	 f(x:S,x:S) -> a
120.85/121.13	 g(x:S) -> a | g(x:S) ->* b
120.85/121.13	->Strongly Connected Components:
120.85/121.13	->->Cycle:
120.85/121.13	->->-> Pairs:
120.85/121.13	 A -> B
120.85/121.13	 B -> A
120.85/121.13	-> QPairs:
120.85/121.13	 Empty
120.85/121.13	->->-> Rules:
120.85/121.13	 a -> b
120.85/121.13	 b -> a
120.85/121.13	 f(x:S,x:S) -> a
120.85/121.13	 g(x:S) -> a | g(x:S) ->* b
120.85/121.13	
120.85/121.13	Problem 1: 
120.85/121.13	
120.85/121.13	Unsatisfiable Rule Processor:
120.85/121.13	-> Pairs:
120.85/121.13	 A -> B
120.85/121.13	 B -> A
120.85/121.13	-> QPairs:
120.85/121.13	 Empty
120.85/121.13	-> Rules:
120.85/121.13	 a -> b
120.85/121.13	 b -> a
120.85/121.13	 f(x:S,x:S) -> a
120.85/121.13	 g(x:S) -> a | g(x:S) ->* b
120.85/121.13	->AGES Output:
120.85/121.13	 
120.85/121.13	Model Results
120.85/121.13	
120.85/121.13	System:
120.85/121.13	mod InTheory is
120.85/121.13	  sorts S Bool .
120.85/121.13	
120.85/121.13	op _->*_ : S S -> Bool .
120.85/121.13	op _->_ : S S -> Bool .
120.85/121.13	op a :  -> S .
120.85/121.13	op b :  -> S .
120.85/121.13	op f : S S -> S .
120.85/121.13	op g : S -> S .
120.85/121.13	op fSNonEmpty :  -> S .
120.85/121.13	op gtrsim : S S -> Bool .
120.85/121.13	op sqsupset : S S -> Bool .
120.85/121.13	
120.85/121.13	endm
120.85/121.13	
120.85/121.13	
120.85/121.13	Property:
120.85/121.13	x:S ->R* x:S
120.85/121.13	x:S ->R y:S /\ y:S ->R* z:S => x:S ->R* z:S
120.85/121.13	gtrsim(x:S,y:S) /\ sqsupset(y:S,z:S) => sqsupset(x:S,z:S)
120.85/121.13	x1:S ->R y1:S => f(x1:S,x2:S) ->R f(y1:S,x2:S)
120.85/121.13	x2:S ->R y2:S => f(x1:S,x2:S) ->R f(x1:S,y2:S)
120.85/121.13	x1:S ->R y1:S => g(x1:S) ->R g(y1:S)
120.85/121.13	a ->R b
120.85/121.13	b ->R a
120.85/121.13	f(x:S,x:S) ->R a
120.85/121.13	g(x:S) ->R* b => g(x:S) ->R a
120.85/121.13	x:S ->R y:S => gtrsim(x:S,y:S)
120.85/121.13	sqsupset(b,g(x:S))
120.85/121.13	
120.85/121.13	Results:
120.85/121.13	
120.85/121.13	
120.85/121.13	Domains:
120.85/121.13	S: -|N
120.85/121.13	
120.85/121.13	Function Interpretations:
120.85/121.13	|[a]| =  - 1
120.85/121.13	|[b]| =  - 1
120.85/121.13	|[f(x_1_1:S,x_2_1:S)]| =  - 1
120.85/121.13	|[g(x_1_1:S)]| = 0
120.85/121.13	|[fSNonEmpty]| =  - 1
120.85/121.13	
120.85/121.13	Predicate Interpretations:
120.85/121.13	 x_1_1:S ->* x_2_1:S <=> (x_2_1:S >= x_1_1:S)
120.85/121.13	 x_1_1:S -> x_2_1:S <=> ((x_1_1:S >= x_2_1:S) /\ (x_2_1:S >= x_1_1:S))
120.85/121.13	gtrsim(x_1_1:S,x_2_1:S) <=> ((x_1_1:S >= x_2_1:S) /\ (x_2_1:S >= x_1_1:S))
120.85/121.13	sqsupset(x_1_1:S,x_2_1:S) <=> (x_2_1:S >= 1 + x_1_1:S)
120.85/121.13	
120.85/121.13	Problem 1: 
120.85/121.13	
120.85/121.13	SCC Processor:
120.85/121.13	-> Pairs:
120.85/121.13	 A -> B
120.85/121.13	 B -> A
120.85/121.13	-> QPairs:
120.85/121.13	 Empty
120.85/121.13	-> Rules:
120.85/121.13	 a -> b
120.85/121.13	 b -> a
120.85/121.13	 f(x:S,x:S) -> a
120.85/121.13	->Strongly Connected Components:
120.85/121.13	->->Cycle:
120.85/121.13	->->-> Pairs:
120.85/121.13	 A -> B
120.85/121.13	 B -> A
120.85/121.13	-> QPairs:
120.85/121.13	 Empty
120.85/121.13	->->-> Rules:
120.85/121.13	 a -> b
120.85/121.13	 b -> a
120.85/121.13	 f(x:S,x:S) -> a
120.85/121.13	
120.85/121.13	Problem 1: 
120.85/121.13	
120.85/121.13	Infinite Processor:
120.85/121.13	-> Pairs:
120.85/121.13	 A -> B
120.85/121.13	 B -> A
120.85/121.13	-> QPairs:
120.85/121.13	 Empty
120.85/121.13	-> Rules:
120.85/121.13	 a -> b
120.85/121.13	 b -> a
120.85/121.13	 f(x:S,x:S) -> a
120.85/121.13	-> Pairs in cycle:
120.85/121.13	 A -> B
120.85/121.13	 B -> A
120.85/121.13	
120.85/121.13	The problem is infinite.
120.85/121.13	EOF