4.80/2.12	YES
4.80/2.12	
4.80/2.12	Proof:
4.80/2.12	This system is confluent.
4.80/2.12	By \cite{ALS94}, Theorem 4.1.
4.80/2.12	This system is of type 3 or smaller.
4.80/2.12	This system is strongly deterministic.
4.80/2.12	This system is quasi-decreasing.
4.80/2.12	By \cite{O02}, p. 214, Proposition 7.2.50.
4.80/2.12	This system is of type 3 or smaller.
5.11/2.15	This system is deterministic.
5.11/2.15	System R transformed to U(R).
5.11/2.15	This system is terminating.
5.11/2.15	Call external tool:
5.11/2.15	./ttt2.sh
5.11/2.15	Input:
5.11/2.15	(VAR x y)
5.11/2.15	(RULES
5.11/2.15	  ?1(c(a), x, y) -> g(s(x))
5.11/2.15	  f(x, y) -> ?1(c(g(x)), x, y)
5.11/2.15	  ?2(c(a), x, y) -> h(s(x))
5.11/2.15	  f(x, y) -> ?2(c(h(x)), x, y)
5.11/2.15	  g(s(x)) -> x
5.11/2.15	  h(s(x)) -> x
5.11/2.15	)
5.11/2.15	
5.11/2.15	 Matrix Interpretation Processor: dim=1
5.11/2.15	  
5.11/2.15	  interpretation:
5.11/2.15	   [h](x0) = 2x0 + 1,
5.11/2.15	   
5.11/2.15	   [?2](x0, x1, x2) = x0 + 2x1 + 2x2 + 3,
5.11/2.15	   
5.11/2.15	   [f](x0, x1) = 6x0 + 2x1 + 7,
5.11/2.15	   
5.11/2.15	   [g](x0) = 2x0 + 2,
5.11/2.15	   
5.11/2.15	   [s](x0) = x0,
5.11/2.15	   
5.11/2.15	   [?1](x0, x1, x2) = x0 + 2x1 + 2x2,
5.11/2.15	   
5.11/2.15	   [c](x0) = 2x0,
5.11/2.15	   
5.11/2.15	   [a] = 1
5.11/2.15	  orientation:
5.11/2.15	   ?1(c(a()),x,y) = 2x + 2y + 2 >= 2x + 2 = g(s(x))
5.11/2.15	   
5.11/2.15	   f(x,y) = 6x + 2y + 7 >= 6x + 2y + 4 = ?1(c(g(x)),x,y)
5.11/2.15	   
5.11/2.15	   ?2(c(a()),x,y) = 2x + 2y + 5 >= 2x + 1 = h(s(x))
5.11/2.15	   
5.11/2.15	   f(x,y) = 6x + 2y + 7 >= 6x + 2y + 5 = ?2(c(h(x)),x,y)
5.11/2.15	   
5.11/2.15	   g(s(x)) = 2x + 2 >= x = x
5.11/2.15	   
5.11/2.15	   h(s(x)) = 2x + 1 >= x = x
5.11/2.15	  problem:
5.11/2.15	   ?1(c(a()),x,y) -> g(s(x))
5.11/2.15	  Matrix Interpretation Processor: dim=3
5.11/2.15	   
5.11/2.15	   interpretation:
5.11/2.15	              [1 0 0]  
5.11/2.15	    [g](x0) = [0 0 0]x0
5.11/2.15	              [0 0 0]  ,
5.11/2.15	    
5.11/2.15	              [1 0 0]  
5.11/2.15	    [s](x0) = [0 0 0]x0
5.11/2.15	              [0 0 0]  ,
5.11/2.15	    
5.11/2.15	                       [1 1 0]     [1 0 0]     [1 0 0]     [0]
5.11/2.15	    [?1](x0, x1, x2) = [0 0 0]x0 + [0 0 0]x1 + [0 0 0]x2 + [1]
5.11/2.15	                       [0 0 0]     [0 0 0]     [0 0 0]     [0],
5.11/2.15	    
5.11/2.15	              [1 0 0]     [0]
5.11/2.15	    [c](x0) = [0 0 0]x0 + [1]
5.11/2.15	              [0 0 0]     [0],
5.11/2.15	    
5.11/2.15	          [0]
5.11/2.15	    [a] = [0]
5.11/2.15	          [0]
5.11/2.15	   orientation:
5.11/2.15	                     [1 0 0]    [1 0 0]    [1]    [1 0 0]           
5.11/2.15	    ?1(c(a()),x,y) = [0 0 0]x + [0 0 0]y + [1] >= [0 0 0]x = g(s(x))
5.11/2.15	                     [0 0 0]    [0 0 0]    [0]    [0 0 0]           
5.11/2.15	   problem:
5.11/2.15	    
5.11/2.15	   Qed
5.11/2.15	All 2 critical pairs are joinable.
5.11/2.15	Overlap: (rule1: f(z, x') -> h(s(z)) <= c(h(z)) = c(a), rule2: f(y', z') -> g(s(y')) <= c(g(y')) = c(a), pos: ε, mgu: {(z,y'), (x',z')})
5.11/2.15	CP: g(s(y')) = h(s(y')) <= c(h(y')) = c(a), c(g(y')) = c(a)
5.11/2.15	This critical pair is context-joinable.
5.11/2.15	Overlap: (rule1: f(z, x') -> g(s(z)) <= c(g(z)) = c(a), rule2: f(y', z') -> h(s(y')) <= c(h(y')) = c(a), pos: ε, mgu: {(z,y'), (x',z')})
5.11/2.15	CP: h(s(y')) = g(s(y')) <= c(g(y')) = c(a), c(h(y')) = c(a)
5.11/2.15	This critical pair is context-joinable.
5.11/2.15	
5.23/2.21	EOF