41.59/41.73	YES
41.59/41.73	
41.59/41.73	Problem 1: 
41.59/41.73	
41.59/41.73	(VAR v_NonEmpty:S x:S y:S)
41.59/41.73	(RULES
41.59/41.73	f(k(a),k(b),x:S) -> f(x:S,x:S,x:S)
41.59/41.73	g(x:S) -> k(y:S) | h(x:S) ->* d, h(x:S) ->* c(y:S)
41.59/41.73	h(d) -> c(a)
41.59/41.73	h(d) -> c(b)
41.59/41.73	)
41.59/41.73	
41.59/41.73	Problem 1: 
41.59/41.73	Valid CTRS Processor:
41.59/41.73	-> Rules:
41.59/41.73	 f(k(a),k(b),x:S) -> f(x:S,x:S,x:S)
41.59/41.73	 g(x:S) -> k(y:S) | h(x:S) ->* d, h(x:S) ->* c(y:S)
41.59/41.73	 h(d) -> c(a)
41.59/41.73	 h(d) -> c(b)
41.59/41.73	-> The system is a deterministic 3-CTRS.
41.59/41.73	
41.59/41.73	Problem 1: 
41.59/41.73	
41.59/41.73	Dependency Pairs Processor:
41.59/41.73	
41.59/41.73	Conditional Termination Problem 1:
41.59/41.73	-> Pairs:
41.59/41.73	 F(k(a),k(b),x:S) -> F(x:S,x:S,x:S)
41.59/41.73	-> QPairs:
41.59/41.73	 Empty
41.59/41.73	-> Rules:
41.59/41.73	 f(k(a),k(b),x:S) -> f(x:S,x:S,x:S)
41.59/41.73	 g(x:S) -> k(y:S) | h(x:S) ->* d, h(x:S) ->* c(y:S)
41.59/41.73	 h(d) -> c(a)
41.59/41.73	 h(d) -> c(b)
41.59/41.73	
41.59/41.73	Conditional Termination Problem 2:
41.59/41.73	-> Pairs:
41.59/41.73	 G(x:S) -> H(x:S)
41.59/41.73	 G(x:S) -> H(x:S) | h(x:S) ->* d
41.59/41.73	-> QPairs:
41.59/41.73	 Empty
41.59/41.73	-> Rules:
41.59/41.73	 f(k(a),k(b),x:S) -> f(x:S,x:S,x:S)
41.59/41.73	 g(x:S) -> k(y:S) | h(x:S) ->* d, h(x:S) ->* c(y:S)
41.59/41.73	 h(d) -> c(a)
41.59/41.73	 h(d) -> c(b)
41.59/41.73	
41.59/41.73	
41.59/41.73	The problem is decomposed in 2 subproblems.
41.59/41.73	
41.59/41.73	Problem 1.1: 
41.59/41.73	
41.59/41.73	SCC Processor:
41.59/41.73	-> Pairs:
41.59/41.73	 F(k(a),k(b),x:S) -> F(x:S,x:S,x:S)
41.59/41.73	-> QPairs:
41.59/41.73	 Empty
41.59/41.73	-> Rules:
41.59/41.73	 f(k(a),k(b),x:S) -> f(x:S,x:S,x:S)
41.59/41.73	 g(x:S) -> k(y:S) | h(x:S) ->* d, h(x:S) ->* c(y:S)
41.59/41.73	 h(d) -> c(a)
41.59/41.73	 h(d) -> c(b)
41.59/41.73	->Strongly Connected Components:
41.59/41.73	->->Cycle:
41.59/41.73	->->-> Pairs:
41.59/41.73	 F(k(a),k(b),x:S) -> F(x:S,x:S,x:S)
41.59/41.73	-> QPairs:
41.59/41.73	 Empty
41.59/41.73	->->-> Rules:
41.59/41.73	 f(k(a),k(b),x:S) -> f(x:S,x:S,x:S)
41.59/41.73	 g(x:S) -> k(y:S) | h(x:S) ->* d, h(x:S) ->* c(y:S)
41.59/41.73	 h(d) -> c(a)
41.59/41.73	 h(d) -> c(b)
41.59/41.73	
41.59/41.73	Problem 1.1: 
41.59/41.73	
41.59/41.73	Unsatisfiable Rule Processor:
41.59/41.73	-> Pairs:
41.59/41.73	 F(k(a),k(b),x:S) -> F(x:S,x:S,x:S)
41.59/41.73	-> QPairs:
41.59/41.73	 Empty
41.59/41.73	-> Rules:
41.59/41.73	 f(k(a),k(b),x:S) -> f(x:S,x:S,x:S)
41.59/41.73	 g(x:S) -> k(y:S) | h(x:S) ->* d, h(x:S) ->* c(y:S)
41.59/41.73	 h(d) -> c(a)
41.59/41.73	 h(d) -> c(b)
41.59/41.73	->AGES Output:
41.59/41.73	 
41.59/41.73	Model Results
41.59/41.73	
41.59/41.73	System:
41.59/41.73	mod InTheory is
41.59/41.73	  sorts S Bool .
41.59/41.73	
41.59/41.73	op _->*_ : S S -> Bool .
41.59/41.73	op _->_ : S S -> Bool .
41.59/41.73	op f : S S S -> S .
41.59/41.73	op g : S -> S .
41.59/41.73	op h : S -> S .
41.59/41.73	op a :  -> S .
41.59/41.73	op b :  -> S .
41.59/41.73	op c : S -> S .
41.59/41.73	op d :  -> S .
41.59/41.73	op fSNonEmpty :  -> S .
41.59/41.73	op k : S -> S .
41.59/41.73	op gtrsim : S S -> Bool .
41.59/41.73	op sqsupset : S S -> Bool .
41.59/41.73	
41.59/41.73	endm
41.59/41.73	
41.59/41.73	
41.59/41.73	Property:
41.59/41.73	x:S ->R* x:S
41.59/41.73	x:S ->R y:S /\ y:S ->R* z:S => x:S ->R* z:S
41.59/41.73	gtrsim(x:S,y:S) /\ sqsupset(y:S,z:S) => sqsupset(x:S,z:S)
41.59/41.73	x1:S ->R y1:S => f(x1:S,x2:S,x3:S) ->R f(y1:S,x2:S,x3:S)
41.59/41.73	x2:S ->R y2:S => f(x1:S,x2:S,x3:S) ->R f(x1:S,y2:S,x3:S)
41.59/41.73	x3:S ->R y3:S => f(x1:S,x2:S,x3:S) ->R f(x1:S,x2:S,y3:S)
41.59/41.73	x1:S ->R y1:S => g(x1:S) ->R g(y1:S)
41.59/41.73	x1:S ->R y1:S => h(x1:S) ->R h(y1:S)
41.59/41.73	x1:S ->R y1:S => c(x1:S) ->R c(y1:S)
41.59/41.73	x1:S ->R y1:S => k(x1:S) ->R k(y1:S)
41.59/41.73	f(k(a),k(b),x:S) ->R f(x:S,x:S,x:S)
41.59/41.73	h(x:S) ->R* d /\ h(x:S) ->R* c(y:S) => g(x:S) ->R k(y:S)
41.59/41.73	h(d) ->R c(a)
41.59/41.73	h(d) ->R c(b)
41.59/41.73	x:S ->R y:S => gtrsim(x:S,y:S)
41.59/41.73	sqsupset(d,h(x:S))
41.59/41.73	
41.59/41.73	Results:
41.59/41.73	
41.59/41.73	
41.59/41.73	Domains:
41.59/41.73	S: |N U {-1}
41.59/41.73	
41.59/41.73	Function Interpretations:
41.59/41.73	|[f(x_1_1:S,x_2_1:S,x_3_1:S)]| = 1
41.59/41.73	|[g(x_1_1:S)]| = 1
41.59/41.73	|[h(x_1_1:S)]| = 0
41.59/41.73	|[a]| = 0
41.59/41.73	|[b]| = 0
41.59/41.73	|[c(x_1_1:S)]| = x_1_1:S
41.59/41.73	|[d]| = 1
41.59/41.73	|[fSNonEmpty]| =  - 1
41.59/41.73	|[k(x_1_1:S)]| =  - 1
41.59/41.73	
41.59/41.73	Predicate Interpretations:
41.59/41.73	 x_1_1:S ->* x_2_1:S <=> (1 + x_1_1:S >= 0)
41.59/41.73	 x_1_1:S -> x_2_1:S <=> ((1 + x_1_1:S >= 0) /\ (x_1_1:S >= x_2_1:S))
41.59/41.73	gtrsim(x_1_1:S,x_2_1:S) <=> (x_1_1:S >= x_2_1:S)
41.59/41.73	sqsupset(x_1_1:S,x_2_1:S) <=> (2.x_1_1:S >= 1+2.x_2_1:S)
41.59/41.73	
41.59/41.73	Problem 1.1: 
41.59/41.73	
41.59/41.73	SCC Processor:
41.59/41.73	-> Pairs:
41.59/41.73	 F(k(a),k(b),x:S) -> F(x:S,x:S,x:S)
41.59/41.73	-> QPairs:
41.59/41.73	 Empty
41.59/41.73	-> Rules:
41.59/41.73	 f(k(a),k(b),x:S) -> f(x:S,x:S,x:S)
41.59/41.73	 h(d) -> c(a)
41.59/41.73	 h(d) -> c(b)
41.59/41.73	->Strongly Connected Components:
41.59/41.73	->->Cycle:
41.59/41.73	->->-> Pairs:
41.59/41.73	 F(k(a),k(b),x:S) -> F(x:S,x:S,x:S)
41.59/41.73	-> QPairs:
41.59/41.73	 Empty
41.59/41.73	->->-> Rules:
41.59/41.73	 f(k(a),k(b),x:S) -> f(x:S,x:S,x:S)
41.59/41.73	 h(d) -> c(a)
41.59/41.73	 h(d) -> c(b)
41.59/41.73	
41.59/41.73	Problem 1.1: 
41.59/41.73	
41.59/41.73	Shift to Dependency Pair Processor:
41.59/41.73	-> Pairs:
41.59/41.73	 F(k(a),k(b),x:S) -> F(x:S,x:S,x:S)
41.59/41.73	-> Rules:
41.59/41.73	 f(k(a),k(b),x:S) -> f(x:S,x:S,x:S)
41.59/41.73	 h(d) -> c(a)
41.59/41.73	 h(d) -> c(b)
41.59/41.73	
41.59/41.73	Problem 1.1: 
41.59/41.73	
41.59/41.73	SCC Processor:
41.59/41.73	-> Pairs:
41.59/41.73	 F(k(a),k(b),x:S) -> F(x:S,x:S,x:S)
41.59/41.73	-> Rules:
41.59/41.73	 f(k(a),k(b),x:S) -> f(x:S,x:S,x:S)
41.59/41.73	 h(d) -> c(a)
41.59/41.73	 h(d) -> c(b)
41.59/41.73	->Strongly Connected Components:
41.59/41.73	->->Cycle:
41.59/41.73	->->-> Pairs:
41.59/41.73	 F(k(a),k(b),x:S) -> F(x:S,x:S,x:S)
41.59/41.73	->->-> Rules:
41.59/41.73	 f(k(a),k(b),x:S) -> f(x:S,x:S,x:S)
41.59/41.73	 h(d) -> c(a)
41.59/41.73	 h(d) -> c(b)
41.59/41.73	
41.59/41.73	Problem 1.1: 
41.59/41.73	
41.59/41.73	Forward Instantiation Processor:
41.59/41.73	-> Pairs:
41.59/41.73	 F(k(a),k(b),x:S) -> F(x:S,x:S,x:S)
41.59/41.73	-> Rules:
41.59/41.73	 f(k(a),k(b),x:S) -> f(x:S,x:S,x:S)
41.59/41.73	 h(d) -> c(a)
41.59/41.73	 h(d) -> c(b)
41.59/41.73	->Instantiated Pairs:
41.59/41.73	->->Original Pair:
41.59/41.73	 F(k(a),k(b),x:S) -> F(x:S,x:S,x:S)
41.59/41.73	->-> Instantiated pairs:
41.59/41.73	 Empty
41.59/41.73	
41.59/41.73	Problem 1.1: 
41.59/41.73	
41.59/41.73	SCC Processor:
41.59/41.73	-> Pairs:
41.59/41.73	 Empty
41.59/41.73	-> Rules:
41.59/41.73	 f(k(a),k(b),x:S) -> f(x:S,x:S,x:S)
41.59/41.73	 h(d) -> c(a)
41.59/41.73	 h(d) -> c(b)
41.59/41.73	->Strongly Connected Components:
41.59/41.73	 There is no strongly connected component
41.59/41.73	
41.59/41.73	The problem is finite.
41.59/41.73	
41.59/41.73	Problem 1.2: 
41.59/41.73	
41.59/41.73	SCC Processor:
41.59/41.73	-> Pairs:
41.59/41.73	 G(x:S) -> H(x:S)
41.59/41.73	 G(x:S) -> H(x:S) | h(x:S) ->* d
41.59/41.73	-> QPairs:
41.59/41.73	 Empty
41.59/41.73	-> Rules:
41.59/41.73	 f(k(a),k(b),x:S) -> f(x:S,x:S,x:S)
41.59/41.73	 g(x:S) -> k(y:S) | h(x:S) ->* d, h(x:S) ->* c(y:S)
41.59/41.73	 h(d) -> c(a)
41.59/41.73	 h(d) -> c(b)
41.59/41.73	->Strongly Connected Components:
41.59/41.73	 There is no strongly connected component
41.59/41.73	
41.59/41.73	The problem is finite.
41.59/41.73	EOF