4.59/1.75	YES
4.59/1.75	
4.59/1.75	Proof:
4.59/1.76	This system is confluent.
4.59/1.76	By \cite{ALS94}, Theorem 4.1.
4.59/1.76	This system is of type 3 or smaller.
4.59/1.76	This system is strongly deterministic.
4.59/1.76	This system is quasi-decreasing.
4.59/1.76	By \cite{A14}, Theorem 11.5.9.
4.59/1.76	This system is of type 3 or smaller.
4.59/1.76	This system is deterministic.
4.59/1.76	System R transformed to V(R) + Emb.
4.59/1.76	This system is terminating.
4.59/1.76	Call external tool:
4.59/1.76	./ttt2.sh
4.59/1.76	Input:
4.59/1.76	  even(0) -> true
4.59/1.76	  even(s(x)) -> false
4.59/1.76	  even(s(x)) -> odd(x)
4.59/1.76	  even(s(x)) -> true
4.59/1.76	  odd(0) -> false
4.59/1.76	  odd(s(x)) -> false
4.59/1.76	  odd(s(x)) -> even(x)
4.59/1.76	  odd(s(x)) -> true
4.59/1.76	  even(x) -> x
4.59/1.76	  s(x) -> x
4.59/1.76	  odd(x) -> x
4.59/1.76	
4.59/1.76	 Matrix Interpretation Processor: dim=3
4.59/1.76	  
4.59/1.76	  interpretation:
4.59/1.76	               [1 0 0]  
4.59/1.76	   [odd](x0) = [1 1 0]x0
4.59/1.76	               [0 0 1]  ,
4.59/1.76	   
4.59/1.76	             [0]
4.59/1.76	   [false] = [0]
4.59/1.76	             [0],
4.59/1.76	   
4.59/1.76	             [1 0 1]     [1]
4.59/1.76	   [s](x0) = [0 1 0]x0 + [0]
4.59/1.76	             [1 0 1]     [0],
4.59/1.76	   
4.59/1.76	            [0]
4.59/1.76	   [true] = [0]
4.59/1.76	            [0],
4.59/1.76	   
4.59/1.76	                [1 0 0]  
4.59/1.76	   [even](x0) = [0 1 1]x0
4.59/1.76	                [0 0 1]  ,
4.59/1.76	   
4.59/1.76	         [1]
4.59/1.76	   [0] = [0]
4.59/1.76	         [0]
4.59/1.76	  orientation:
4.59/1.76	               [1]    [0]         
4.59/1.76	   even(0()) = [0] >= [0] = true()
4.59/1.76	               [0]    [0]         
4.59/1.76	   
4.59/1.76	                [1 0 1]    [1]    [0]          
4.59/1.76	   even(s(x)) = [1 1 1]x + [0] >= [0] = false()
4.59/1.76	                [1 0 1]    [0]    [0]          
4.59/1.76	   
4.59/1.76	                [1 0 1]    [1]    [1 0 0]          
4.59/1.76	   even(s(x)) = [1 1 1]x + [0] >= [1 1 0]x = odd(x)
4.59/1.76	                [1 0 1]    [0]    [0 0 1]          
4.59/1.76	   
4.59/1.76	                [1 0 1]    [1]    [0]         
4.59/1.76	   even(s(x)) = [1 1 1]x + [0] >= [0] = true()
4.59/1.76	                [1 0 1]    [0]    [0]         
4.59/1.76	   
4.59/1.76	              [1]    [0]          
4.59/1.76	   odd(0()) = [1] >= [0] = false()
4.59/1.76	              [0]    [0]          
4.59/1.76	   
4.59/1.76	               [1 0 1]    [1]    [0]          
4.59/1.76	   odd(s(x)) = [1 1 1]x + [1] >= [0] = false()
4.59/1.76	               [1 0 1]    [0]    [0]          
4.59/1.76	   
4.59/1.77	               [1 0 1]    [1]    [1 0 0]           
4.59/1.77	   odd(s(x)) = [1 1 1]x + [1] >= [0 1 1]x = even(x)
4.59/1.77	               [1 0 1]    [0]    [0 0 1]           
4.59/1.77	   
4.59/1.77	               [1 0 1]    [1]    [0]         
4.59/1.77	   odd(s(x)) = [1 1 1]x + [1] >= [0] = true()
4.59/1.77	               [1 0 1]    [0]    [0]         
4.59/1.77	   
4.59/1.77	             [1 0 0]          
4.59/1.77	   even(x) = [0 1 1]x >= x = x
4.59/1.77	             [0 0 1]          
4.59/1.77	   
4.59/1.77	          [1 0 1]    [1]         
4.59/1.77	   s(x) = [0 1 0]x + [0] >= x = x
4.59/1.77	          [1 0 1]    [0]         
4.59/1.77	   
4.59/1.77	            [1 0 0]          
4.59/1.77	   odd(x) = [1 1 0]x >= x = x
4.59/1.77	            [0 0 1]          
4.59/1.77	  problem:
4.59/1.77	   even(x) -> x
4.59/1.77	   odd(x) -> x
4.59/1.77	  Matrix Interpretation Processor: dim=3
4.59/1.77	   
4.59/1.77	   interpretation:
4.59/1.77	                     [1]
4.59/1.77	    [odd](x0) = x0 + [0]
4.59/1.77	                     [0],
4.59/1.77	    
4.59/1.77	                   
4.59/1.77	    [even](x0) = x0
4.59/1.77	                   
4.59/1.77	   orientation:
4.59/1.77	                        
4.59/1.77	    even(x) = x >= x = x
4.59/1.77	                        
4.59/1.77	    
4.59/1.77	                 [1]         
4.59/1.77	    odd(x) = x + [0] >= x = x
4.59/1.77	                 [0]         
4.59/1.77	   problem:
4.59/1.77	    even(x) -> x
4.59/1.77	   Matrix Interpretation Processor: dim=3
4.59/1.77	    
4.59/1.77	    interpretation:
4.59/1.77	                       [1]
4.59/1.77	     [even](x0) = x0 + [0]
4.59/1.77	                       [0]
4.59/1.77	    orientation:
4.59/1.77	                   [1]         
4.59/1.77	     even(x) = x + [0] >= x = x
4.59/1.77	                   [0]         
4.59/1.77	    problem:
4.59/1.77	     
4.59/1.77	    Qed
4.59/1.77	All 4 critical pairs are joinable.
4.59/1.77	Overlap: (rule1: even(s(y)) -> true <= odd(y) = true, rule2: even(s(z)) -> false <= odd(z) = false, pos: ε, mgu: {(z,y)})
4.59/1.77	CP: false = true <= odd(y) = true, odd(y) = false
4.59/1.77	This critical pair is unfeasible.
4.59/1.77	Overlap: (rule1: even(s(y)) -> false <= odd(y) = false, rule2: even(s(z)) -> true <= odd(z) = true, pos: ε, mgu: {(z,y)})
4.59/1.77	CP: true = false <= odd(y) = false, odd(y) = true
4.59/1.77	This critical pair is unfeasible.
4.59/1.77	Overlap: (rule1: odd(s(y)) -> true <= even(y) = true, rule2: odd(s(z)) -> false <= even(z) = false, pos: ε, mgu: {(z,y)})
4.59/1.77	CP: false = true <= even(y) = true, even(y) = false
4.59/1.77	This critical pair is unfeasible.
4.59/1.77	Overlap: (rule1: odd(s(y)) -> false <= even(y) = false, rule2: odd(s(z)) -> true <= even(z) = true, pos: ε, mgu: {(z,y)})
4.59/1.77	CP: true = false <= even(y) = false, even(y) = true
4.59/1.77	This critical pair is unfeasible.
4.59/1.77	
4.59/1.79	EOF