7.22/2.81	YES
7.22/2.81	
7.22/2.81	Proof:
7.22/2.82	This system is confluent.
7.22/2.82	By \cite{ALS94}, Theorem 4.1.
7.22/2.82	This system is of type 3 or smaller.
7.22/2.82	This system is strongly deterministic.
7.22/2.82	This system is quasi-decreasing.
7.22/2.82	By \cite{A14}, Theorem 11.5.9.
7.22/2.82	This system is of type 3 or smaller.
7.22/2.82	This system is deterministic.
7.22/2.82	System R transformed to V(R) + Emb.
7.22/2.82	This system is terminating.
7.22/2.82	Call external tool:
7.22/2.82	./ttt2.sh
7.22/2.82	Input:
7.22/2.82	(VAR x)
7.22/2.82	(RULES
7.22/2.82	  even(0) -> true
7.22/2.82	  even(s(x)) -> false
7.22/2.82	  even(s(x)) -> odd(x)
7.22/2.82	  even(s(x)) -> true
7.22/2.82	  odd(0) -> false
7.22/2.82	  odd(s(x)) -> false
7.22/2.82	  odd(s(x)) -> even(x)
7.22/2.82	  odd(s(x)) -> true
7.22/2.82	  even(x) -> x
7.22/2.82	  s(x) -> x
7.22/2.82	  odd(x) -> x
7.22/2.82	)
7.22/2.82	
7.22/2.82	 Matrix Interpretation Processor: dim=3
7.22/2.82	  
7.22/2.82	  interpretation:
7.22/2.82	               [1 0 0]  
7.22/2.82	   [odd](x0) = [1 1 0]x0
7.22/2.82	               [0 0 1]  ,
7.22/2.82	   
7.22/2.82	             [0]
7.22/2.82	   [false] = [0]
7.22/2.82	             [0],
7.22/2.82	   
7.22/2.82	             [1 0 1]     [1]
7.22/2.82	   [s](x0) = [0 1 0]x0 + [0]
7.22/2.82	             [1 0 1]     [0],
7.22/2.82	   
7.22/2.82	            [0]
7.22/2.82	   [true] = [0]
7.22/2.82	            [0],
7.22/2.82	   
7.22/2.82	                [1 0 0]  
7.22/2.82	   [even](x0) = [0 1 1]x0
7.22/2.82	                [0 0 1]  ,
7.22/2.82	   
7.22/2.82	         [1]
7.22/2.82	   [0] = [0]
7.22/2.82	         [0]
7.22/2.82	  orientation:
7.22/2.82	               [1]    [0]         
7.22/2.82	   even(0()) = [0] >= [0] = true()
7.22/2.82	               [0]    [0]         
7.22/2.82	   
7.22/2.82	                [1 0 1]    [1]    [0]          
7.22/2.82	   even(s(x)) = [1 1 1]x + [0] >= [0] = false()
7.22/2.82	                [1 0 1]    [0]    [0]          
7.22/2.82	   
7.22/2.82	                [1 0 1]    [1]    [1 0 0]          
7.22/2.82	   even(s(x)) = [1 1 1]x + [0] >= [1 1 0]x = odd(x)
7.22/2.82	                [1 0 1]    [0]    [0 0 1]          
7.22/2.82	   
7.22/2.82	                [1 0 1]    [1]    [0]         
7.22/2.82	   even(s(x)) = [1 1 1]x + [0] >= [0] = true()
7.22/2.82	                [1 0 1]    [0]    [0]         
7.22/2.82	   
7.22/2.82	              [1]    [0]          
7.22/2.82	   odd(0()) = [1] >= [0] = false()
7.22/2.82	              [0]    [0]          
7.22/2.82	   
7.22/2.82	               [1 0 1]    [1]    [0]          
7.22/2.82	   odd(s(x)) = [1 1 1]x + [1] >= [0] = false()
7.22/2.82	               [1 0 1]    [0]    [0]          
7.22/2.82	   
7.22/2.82	               [1 0 1]    [1]    [1 0 0]           
7.22/2.82	   odd(s(x)) = [1 1 1]x + [1] >= [0 1 1]x = even(x)
7.22/2.82	               [1 0 1]    [0]    [0 0 1]           
7.22/2.82	   
7.22/2.82	               [1 0 1]    [1]    [0]         
7.22/2.82	   odd(s(x)) = [1 1 1]x + [1] >= [0] = true()
7.22/2.82	               [1 0 1]    [0]    [0]         
7.22/2.82	   
7.22/2.82	             [1 0 0]          
7.22/2.82	   even(x) = [0 1 1]x >= x = x
7.22/2.82	             [0 0 1]          
7.22/2.82	   
7.22/2.82	          [1 0 1]    [1]         
7.22/2.83	   s(x) = [0 1 0]x + [0] >= x = x
7.22/2.83	          [1 0 1]    [0]         
7.22/2.83	   
7.22/2.83	            [1 0 0]          
7.22/2.83	   odd(x) = [1 1 0]x >= x = x
7.22/2.83	            [0 0 1]          
7.22/2.83	  problem:
7.22/2.83	   even(x) -> x
7.22/2.83	   odd(x) -> x
7.22/2.83	  Matrix Interpretation Processor: dim=3
7.22/2.83	   
7.22/2.83	   interpretation:
7.22/2.83	                     [1]
7.22/2.83	    [odd](x0) = x0 + [0]
7.22/2.83	                     [0],
7.22/2.83	    
7.22/2.83	                   
7.22/2.83	    [even](x0) = x0
7.22/2.83	                   
7.22/2.83	   orientation:
7.22/2.83	                        
7.22/2.83	    even(x) = x >= x = x
7.22/2.83	                        
7.22/2.83	    
7.22/2.83	                 [1]         
7.22/2.83	    odd(x) = x + [0] >= x = x
7.22/2.83	                 [0]         
7.22/2.83	   problem:
7.22/2.83	    even(x) -> x
7.22/2.83	   Matrix Interpretation Processor: dim=3
7.22/2.83	    
7.22/2.83	    interpretation:
7.22/2.83	                       [1]
7.22/2.83	     [even](x0) = x0 + [0]
7.22/2.83	                       [0]
7.22/2.83	    orientation:
7.22/2.83	                   [1]         
7.22/2.83	     even(x) = x + [0] >= x = x
7.22/2.83	                   [0]         
7.22/2.83	    problem:
7.22/2.83	     
7.22/2.83	    Qed
7.22/2.83	All 4 critical pairs are joinable.
7.22/2.83	Overlap: (rule1: even(s(y)) -> true <= odd(y) = true, rule2: even(s(z)) -> false <= odd(z) = false, pos: ε, mgu: {(y,z)})
7.22/2.83	CP: false = true <= odd(z) = true, odd(z) = false
7.22/2.83	This critical pair is unfeasible.
7.22/2.83	Overlap: (rule1: even(s(y)) -> false <= odd(y) = false, rule2: even(s(z)) -> true <= odd(z) = true, pos: ε, mgu: {(y,z)})
7.22/2.83	CP: true = false <= odd(z) = false, odd(z) = true
7.22/2.83	This critical pair is unfeasible.
7.22/2.83	Overlap: (rule1: odd(s(y)) -> true <= even(y) = true, rule2: odd(s(z)) -> false <= even(z) = false, pos: ε, mgu: {(y,z)})
7.22/2.83	CP: false = true <= even(z) = true, even(z) = false
7.22/2.83	This critical pair is unfeasible.
7.22/2.83	Overlap: (rule1: odd(s(y)) -> false <= even(y) = false, rule2: odd(s(z)) -> true <= even(z) = true, pos: ε, mgu: {(y,z)})
7.22/2.83	CP: true = false <= even(z) = false, even(z) = true
7.22/2.83	This critical pair is unfeasible.
7.22/2.83	
7.57/2.89	EOF