20.02/20.02	YES
20.02/20.02	
20.02/20.02	Problem 1:
20.02/20.02	
20.02/20.02	
20.02/20.02	:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
20.02/20.02	Confluence Problem:
20.02/20.02	(VAR vNonEmpty:S x:S)
20.02/20.02	(STRATEGY CONTEXTSENSITIVE
20.02/20.02	(a)
20.02/20.02	(f 1)
20.02/20.02	(b)
20.02/20.02	(fSNonEmpty)
20.02/20.02	)
20.02/20.02	(RULES
20.02/20.02	a -> b
20.02/20.02	f(x:S) -> x:S | x:S ->* a, b ->* x:S
20.02/20.02	)
20.02/20.02	:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
20.02/20.02	
20.02/20.02	
20.02/20.02	Problem 1:
20.02/20.02	
20.02/20.02	Inlining of Conditions Processor [STERN17]:
20.02/20.02	
20.02/20.02	:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
20.02/20.02	Confluence Problem:
20.02/20.02	(VAR vNonEmpty:S x:S)
20.02/20.02	(STRATEGY CONTEXTSENSITIVE
20.02/20.02	(a)
20.02/20.02	(f 1)
20.02/20.02	(b)
20.02/20.02	(fSNonEmpty)
20.02/20.02	)
20.02/20.02	(RULES
20.02/20.02	a -> b
20.02/20.02	f(x:S) -> x:S | b ->* x:S, x:S ->* a
20.02/20.02	)
20.02/20.02	:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
20.02/20.02	
20.02/20.02	
20.02/20.02	Problem 1:
20.02/20.02	
20.02/20.02	Clean CTRS Processor:
20.02/20.02	
20.02/20.02	:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
20.02/20.02	Confluence Problem:
20.02/20.02	(STRATEGY CONTEXTSENSITIVE
20.02/20.02	(a)
20.02/20.02	(f 1)
20.02/20.02	(b)
20.02/20.02	(fSNonEmpty)
20.02/20.02	)
20.02/20.02	(RULES
20.02/20.02	a -> b
20.02/20.02	)
20.02/20.02	:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
20.02/20.02	
20.02/20.02	CRule InfChecker Info:
20.02/20.02	a -> b
20.02/20.02	Rule remains
20.02/20.02	Proof:
20.02/20.02	NO_CONDS
20.02/20.02	
20.02/20.02	CRule InfChecker Info:
20.02/20.02	f(x:S) -> x:S | b ->* x:S, x:S ->* a
20.02/20.02	Rule deleted
20.02/20.02	Proof:
20.02/20.02	YES
20.02/20.02	
20.02/20.02	Problem 1:
20.02/20.02	
20.02/20.02	Infeasibility Problem:
20.02/20.02	[(VAR vNonEmpty:S vNonEmpty:S:S x:S:S)
20.02/20.02	(STRATEGY CONTEXTSENSITIVE
20.02/20.02	(a)
20.02/20.02	(f 1)
20.02/20.02	(b)
20.02/20.02	(fSNonEmpty)
20.02/20.02	)
20.02/20.02	(RULES
20.02/20.02	a -> b
20.02/20.02	f(x:S:S) -> x:S:S | b ->* x:S:S, x:S:S ->* a
20.02/20.02	)]
20.02/20.02	
20.02/20.02	Infeasibility Conditions:
20.02/20.02	b ->* x:S:S, x:S:S ->* a
20.02/20.02	
20.02/20.02	Problem 1:
20.02/20.02	
20.02/20.02	Obtaining a model using Mace4:
20.02/20.02	
20.02/20.02	 -> Usable Rules:
20.02/20.02	a -> b
20.02/20.02	f(x:S:S) -> x:S:S | b ->* x:S:S, x:S:S ->* a
20.02/20.02	
20.02/20.02	 -> Mace4 Output:
20.02/20.02	============================== Mace4 =================================
20.02/20.02	Mace4 (64) version 2009-11A, November 2009.
20.02/20.02	Process 102631 was started by ubuntu on ubuntu,
20.02/20.02	Fri Jul 16 14:42:00 2021
20.02/20.02	The command was "./mace4 -c -f /tmp/mace4102618-2.in".
20.02/20.02	============================== end of head ===========================
20.02/20.02	
20.02/20.02	============================== INPUT =================================
20.02/20.02	
20.02/20.02	% Reading from file /tmp/mace4102618-2.in
20.02/20.02	
20.02/20.02	assign(max_seconds,10).
20.02/20.02	
20.02/20.02	formulas(assumptions).
20.02/20.02	->(x1,y) -> ->(f(x1),f(y)) # label(congruence).
20.02/20.02	->(a,b) # label(replacement).
20.02/20.02	->*(b,x2) & ->*(x2,a) -> ->(f(x2),x2) # label(replacement).
20.02/20.02	->*(x,x) # label(reflexivity).
20.02/20.02	->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity).
20.02/20.02	end_of_list.
20.02/20.02	
20.02/20.02	formulas(goals).
20.02/20.02	(exists x2 (->*(b,x2) & ->*(x2,a))) # label(goal).
20.02/20.02	end_of_list.
20.02/20.02	
20.02/20.02	============================== end of input ==========================
20.02/20.02	
20.02/20.02	============================== PROCESS NON-CLAUSAL FORMULAS ==========
20.02/20.02	
20.02/20.02	% Formulas that are not ordinary clauses:
20.02/20.02	1 ->(x1,y) -> ->(f(x1),f(y)) # label(congruence) # label(non_clause).  [assumption].
20.02/20.02	2 ->*(b,x2) & ->*(x2,a) -> ->(f(x2),x2) # label(replacement) # label(non_clause).  [assumption].
20.02/20.02	3 ->(x,y) & ->*(y,z) -> ->*(x,z) # label(transitivity) # label(non_clause).  [assumption].
20.02/20.02	4 (exists x2 (->*(b,x2) & ->*(x2,a))) # label(goal) # label(non_clause) # label(goal).  [goal].
20.02/20.02	
20.02/20.02	============================== end of process non-clausal formulas ===
20.02/20.02	
20.02/20.02	============================== CLAUSES FOR SEARCH ====================
20.02/20.02	
20.02/20.02	formulas(mace4_clauses).
20.02/20.02	-->(x,y) | ->(f(x),f(y)) # label(congruence).
20.02/20.02	->(a,b) # label(replacement).
20.02/20.02	-->*(b,x) | -->*(x,a) | ->(f(x),x) # label(replacement).
20.02/20.02	->*(x,x) # label(reflexivity).
20.02/20.02	-->(x,y) | -->*(y,z) | ->*(x,z) # label(transitivity).
20.02/20.02	-->*(b,x) | -->*(x,a) # label(goal).
20.02/20.02	end_of_list.
20.02/20.02	
20.02/20.02	============================== end of clauses for search =============
20.02/20.02	
20.02/20.02	% There are no natural numbers in the input.
20.02/20.02	
20.02/20.02	============================== DOMAIN SIZE 2 =========================
20.02/20.02	
20.02/20.02	============================== MODEL =================================
20.02/20.02	
20.02/20.02	interpretation( 2, [number=1, seconds=0], [
20.02/20.02	
20.02/20.02	        function(a, [ 0 ]),
20.02/20.02	
20.02/20.02	        function(b, [ 1 ]),
20.02/20.02	
20.02/20.02	        function(f(_), [ 0, 1 ]),
20.02/20.02	
20.02/20.02	        relation(->*(_,_), [
20.02/20.02				   1, 1,
20.02/20.02				   0, 1 ]),
20.02/20.02	
20.02/20.02	        relation(->(_,_), [
20.02/20.02				   0, 1,
20.02/20.02				   0, 0 ])
20.02/20.02	]).
20.02/20.02	
20.02/20.02	============================== end of model ==========================
20.02/20.02	
20.02/20.02	============================== STATISTICS ============================
20.02/20.02	
20.02/20.02	For domain size 2.
20.02/20.02	
20.02/20.02	Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds).
20.02/20.02	Ground clauses: seen=19, kept=15.
20.02/20.02	Selections=4, assignments=4, propagations=8, current_models=1.
20.02/20.02	Rewrite_terms=20, rewrite_bools=25, indexes=7.
20.02/20.02	Rules_from_neg_clauses=2, cross_offs=2.
20.02/20.02	
20.02/20.02	============================== end of statistics =====================
20.02/20.02	
20.02/20.02	User_CPU=0.00, System_CPU=0.00, Wall_clock=0.
20.02/20.02	
20.02/20.02	Exiting with 1 model.
20.02/20.02	
20.02/20.02	Process 102631 exit (max_models) Fri Jul 16 14:42:00 2021
20.02/20.02	The process finished Fri Jul 16 14:42:00 2021
20.02/20.02	
20.02/20.02	
20.02/20.02	Mace4 cooked interpretation:
20.02/20.02	
20.02/20.02	
20.02/20.02	
20.02/20.02	The problem is infeasible.
20.02/20.02	
20.02/20.02	
20.02/20.02	Problem 1:
20.02/20.02	
20.02/20.02	Transform No Conds CTRS Processor:
20.02/20.02	
20.02/20.02	:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
20.02/20.02	Confluence Problem:
20.02/20.02	(STRATEGY CONTEXTSENSITIVE
20.02/20.02	(a)
20.02/20.02	(f 1)
20.02/20.02	(b)
20.02/20.02	(fSNonEmpty)
20.02/20.02	)
20.02/20.02	(RULES
20.02/20.02	a -> b
20.02/20.02	)
20.02/20.02	:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
20.02/20.02	Resulting R:
20.02/20.02	(STRATEGY CONTEXTSENSITIVE
20.02/20.02	(a)
20.02/20.02	(f 1)
20.02/20.02	(b)
20.02/20.02	(fSNonEmpty)
20.02/20.02	)
20.02/20.02	(RULES
20.02/20.02	a -> b
20.02/20.02	)
20.02/20.02	
20.02/20.02	Problem 1:
20.02/20.02	
20.02/20.02	CleanTRS Processor:
20.02/20.02	
20.02/20.02	:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
20.02/20.02	Confluence Problem:
20.02/20.02	(STRATEGY CONTEXTSENSITIVE
20.02/20.02	(a)
20.02/20.02	(f 1)
20.02/20.02	(b)
20.02/20.02	(fSNonEmpty)
20.02/20.02	)
20.02/20.02	(RULES
20.02/20.02	a -> b
20.02/20.02	)
20.02/20.02	:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
20.02/20.02	
20.02/20.02	
20.02/20.02	Problem 1:
20.02/20.02	
20.02/20.02	Modular Confluence Combinations Decomposition Processor:
20.02/20.02	
20.02/20.02	No usable combinations
20.02/20.02	
20.02/20.02	Problem 1:
20.02/20.02	Linearity Processor:
20.02/20.02	Linear?
20.02/20.02	YES
20.02/20.02	Problem 1.1:
20.02/20.02	:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
20.02/20.02	Confluence Problem:
20.02/20.02	(STRATEGY CONTEXTSENSITIVE
20.02/20.02	(a)
20.02/20.02	(f 1)
20.02/20.02	(b)
20.02/20.02	(fSNonEmpty)
20.02/20.02	)
20.02/20.02	(RULES
20.02/20.02	a -> b
20.02/20.02	)
20.02/20.02	Linear:YES
20.02/20.02	:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
20.02/20.02	
20.02/20.02	Huet Levy Processor:
20.02/20.02	-> Rules:
20.02/20.02	 a -> b
20.02/20.02	-> Vars:
20.02/20.02	
20.02/20.02	-> FVars:
20.02/20.02	
20.02/20.02	-> PVars:
20.02/20.02	
20.02/20.02	
20.02/20.02	-> Rlps:
20.02/20.02	 (rule: a -> b, id: 1, possubterms: a->[])
20.02/20.02	
20.02/20.02	-> Unifications:
20.02/20.02	
20.02/20.02	
20.02/20.02	-> Critical pairs info:
20.02/20.02	
20.02/20.02	
20.02/20.02	-> Problem conclusions:
20.02/20.02	 Left linear, Right linear, Linear
20.02/20.02	 Weakly orthogonal, Almost orthogonal, Orthogonal
20.02/20.02	 Huet-Levy confluent, Not Newman confluent
20.02/20.02	 R is a TRS
20.02/20.02	
20.02/20.02	
20.02/20.02	The problem is joinable.
20.02/20.02	22.55user 0.21system 0:20.20elapsed 112%CPU (0avgtext+0avgdata 81916maxresident)k
20.02/20.02	8inputs+0outputs (0major+56290minor)pagefaults 0swaps