0.00/0.06	NO
0.00/0.06	
0.00/0.06	Problem 1: 
0.00/0.06	
0.00/0.06	Infeasibility Problem:
0.00/0.06	[(VAR vNonEmpty l x y vNonEmpty x2 x1)
0.00/0.06	(STRATEGY CONTEXTSENSITIVE
0.00/0.06	(le 1 2)
0.00/0.06	(min 1)
0.00/0.06	(0)
0.00/0.06	(cons 1 2)
0.00/0.06	(fSNonEmpty)
0.00/0.06	(false)
0.00/0.06	(nil)
0.00/0.06	(s 1)
0.00/0.06	(true)
0.00/0.06	)
0.00/0.06	(RULES
0.00/0.06	le(0,s(x)) -> true
0.00/0.06	le(s(x),s(y)) -> le(x,y)
0.00/0.06	le(x,0) -> false
0.00/0.06	min(cons(x,nil)) -> x
0.00/0.06	min(cons(x,l)) -> min(l) | le(x,min(l)) ->* false
0.00/0.06	min(cons(x,l)) -> min(l) | min(l) ->* x
0.00/0.06	min(cons(x,l)) -> x | le(x,min(l)) ->* true
0.00/0.06	)]
0.00/0.06	
0.00/0.06	Infeasibility Conditions:
0.00/0.06	le(x1,min(x2)) ->* false, min(x2) ->* x1
0.00/0.06	
0.00/0.06	Problem 1: 
0.00/0.06	
0.00/0.06	Obtaining a proof using Prover9:
0.00/0.06	
0.00/0.06	 -> Prover9 Output:
0.00/0.06	============================== Prover9 ===============================
0.00/0.06	Prover9 (64) version 2009-11A, November 2009.
0.00/0.06	Process 4229 was started by sandbox on z032.star.cs.uiowa.edu,
0.00/0.06	Tue Aug  1 07:38:47 2023
0.00/0.06	The command was "./prover9 -f /tmp/prover94220.in".
0.00/0.06	============================== end of head ===========================
0.00/0.06	
0.00/0.06	============================== INPUT =================================
0.00/0.06	
0.00/0.06	% Reading from file /tmp/prover94220.in
0.00/0.06	
0.00/0.06	assign(max_seconds,20).
0.00/0.06	
0.00/0.06	formulas(assumptions).
0.00/0.06	->_s0(x1,y) -> ->_s0(le(x1,x2),le(y,x2)) # label(congruence).
0.00/0.06	->_s0(x2,y) -> ->_s0(le(x1,x2),le(x1,y)) # label(congruence).
0.00/0.06	->_s0(x1,y) -> ->_s0(min(x1),min(y)) # label(congruence).
0.00/0.06	->_s0(x1,y) -> ->_s0(cons(x1,x2),cons(y,x2)) # label(congruence).
0.00/0.06	->_s0(x2,y) -> ->_s0(cons(x1,x2),cons(x1,y)) # label(congruence).
0.00/0.06	->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence).
0.00/0.06	->_s0(le(0,s(x2)),true) # label(replacement).
0.00/0.06	->_s0(le(s(x2),s(x3)),le(x2,x3)) # label(replacement).
0.00/0.06	->_s0(le(x2,0),false) # label(replacement).
0.00/0.06	->_s0(min(cons(x2,nil)),x2) # label(replacement).
0.00/0.06	->*_s0(le(x2,min(x1)),false) -> ->_s0(min(cons(x2,x1)),min(x1)) # label(replacement).
0.00/0.06	->*_s0(min(x1),x2) -> ->_s0(min(cons(x2,x1)),min(x1)) # label(replacement).
0.00/0.06	->*_s0(le(x2,min(x1)),true) -> ->_s0(min(cons(x2,x1)),x2) # label(replacement).
0.00/0.06	->*_s0(x,x) # label(reflexivity).
0.00/0.06	->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity).
0.00/0.06	end_of_list.
0.00/0.06	
0.00/0.06	formulas(goals).
0.00/0.06	(exists x5 exists x6 (->*_s0(le(x6,min(x5)),false) & ->*_s0(min(x5),x6))) # label(goal).
0.00/0.06	end_of_list.
0.00/0.06	
0.00/0.06	============================== end of input ==========================
0.00/0.06	
0.00/0.06	============================== PROCESS NON-CLAUSAL FORMULAS ==========
0.00/0.06	
0.00/0.06	% Formulas that are not ordinary clauses:
0.00/0.06	1 ->_s0(x1,y) -> ->_s0(le(x1,x2),le(y,x2)) # label(congruence) # label(non_clause).  [assumption].
0.00/0.06	2 ->_s0(x2,y) -> ->_s0(le(x1,x2),le(x1,y)) # label(congruence) # label(non_clause).  [assumption].
0.00/0.06	3 ->_s0(x1,y) -> ->_s0(min(x1),min(y)) # label(congruence) # label(non_clause).  [assumption].
0.00/0.06	4 ->_s0(x1,y) -> ->_s0(cons(x1,x2),cons(y,x2)) # label(congruence) # label(non_clause).  [assumption].
0.00/0.06	5 ->_s0(x2,y) -> ->_s0(cons(x1,x2),cons(x1,y)) # label(congruence) # label(non_clause).  [assumption].
0.00/0.06	6 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence) # label(non_clause).  [assumption].
0.00/0.06	7 ->*_s0(le(x2,min(x1)),false) -> ->_s0(min(cons(x2,x1)),min(x1)) # label(replacement) # label(non_clause).  [assumption].
0.00/0.06	8 ->*_s0(min(x1),x2) -> ->_s0(min(cons(x2,x1)),min(x1)) # label(replacement) # label(non_clause).  [assumption].
0.00/0.06	9 ->*_s0(le(x2,min(x1)),true) -> ->_s0(min(cons(x2,x1)),x2) # label(replacement) # label(non_clause).  [assumption].
0.00/0.06	10 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause).  [assumption].
0.00/0.06	11 (exists x5 exists x6 (->*_s0(le(x6,min(x5)),false) & ->*_s0(min(x5),x6))) # label(goal) # label(non_clause) # label(goal).  [goal].
0.00/0.06	
0.00/0.06	============================== end of process non-clausal formulas ===
0.00/0.06	
0.00/0.06	============================== PROCESS INITIAL CLAUSES ===============
0.00/0.06	
0.00/0.06	% Clauses before input processing:
0.00/0.06	
0.00/0.06	formulas(usable).
0.00/0.06	end_of_list.
0.00/0.06	
0.00/0.06	formulas(sos).
0.00/0.06	-->_s0(x,y) | ->_s0(le(x,z),le(y,z)) # label(congruence).  [clausify(1)].
0.00/0.06	-->_s0(x,y) | ->_s0(le(z,x),le(z,y)) # label(congruence).  [clausify(2)].
0.00/0.06	-->_s0(x,y) | ->_s0(min(x),min(y)) # label(congruence).  [clausify(3)].
0.00/0.06	-->_s0(x,y) | ->_s0(cons(x,z),cons(y,z)) # label(congruence).  [clausify(4)].
0.00/0.06	-->_s0(x,y) | ->_s0(cons(z,x),cons(z,y)) # label(congruence).  [clausify(5)].
0.00/0.06	-->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence).  [clausify(6)].
0.00/0.06	->_s0(le(0,s(x)),true) # label(replacement).  [assumption].
0.00/0.06	->_s0(le(s(x),s(y)),le(x,y)) # label(replacement).  [assumption].
0.00/0.06	->_s0(le(x,0),false) # label(replacement).  [assumption].
0.00/0.06	->_s0(min(cons(x,nil)),x) # label(replacement).  [assumption].
0.00/0.06	-->*_s0(le(x,min(y)),false) | ->_s0(min(cons(x,y)),min(y)) # label(replacement).  [clausify(7)].
0.00/0.06	-->*_s0(min(x),y) | ->_s0(min(cons(y,x)),min(x)) # label(replacement).  [clausify(8)].
0.00/0.06	-->*_s0(le(x,min(y)),true) | ->_s0(min(cons(x,y)),x) # label(replacement).  [clausify(9)].
0.00/0.06	->*_s0(x,x) # label(reflexivity).  [assumption].
0.00/0.06	-->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity).  [clausify(10)].
0.00/0.06	-->*_s0(le(x,min(y)),false) | -->*_s0(min(y),x) # label(goal).  [deny(11)].
0.00/0.06	end_of_list.
0.00/0.06	
0.00/0.06	formulas(demodulators).
0.00/0.06	end_of_list.
0.00/0.06	
0.00/0.06	============================== PREDICATE ELIMINATION =================
0.00/0.06	
0.00/0.06	No predicates eliminated.
0.00/0.06	
0.00/0.06	============================== end predicate elimination =============
0.00/0.06	
0.00/0.06	Auto_denials:
0.00/0.06	  % copying label goal to answer in negative clause
0.00/0.06	
0.00/0.06	Term ordering decisions:
0.00/0.06	Predicate symbol precedence:  predicate_order([ ->_s0, ->*_s0 ]).
0.00/0.06	Function symbol precedence:  function_order([ true, 0, false, nil, le, cons, min, s ]).
0.00/0.06	After inverse_order:  (no changes).
0.00/0.06	Unfolding symbols: (none).
0.00/0.06	
0.00/0.06	Auto_inference settings:
0.00/0.06	  % set(neg_binary_resolution).  % (HNE depth_diff=-7)
0.00/0.06	  % clear(ordered_res).  % (HNE depth_diff=-7)
0.00/0.06	  % set(ur_resolution).  % (HNE depth_diff=-7)
0.00/0.06	    % set(ur_resolution) -> set(pos_ur_resolution).
0.00/0.06	    % set(ur_resolution) -> set(neg_ur_resolution).
0.00/0.06	
0.00/0.06	Auto_process settings:
0.00/0.06	  % set(unit_deletion).  % (Horn set with negative nonunits)
0.00/0.06	
0.00/0.06	kept:      12 -->_s0(x,y) | ->_s0(le(x,z),le(y,z)) # label(congruence).  [clausify(1)].
0.00/0.06	kept:      13 -->_s0(x,y) | ->_s0(le(z,x),le(z,y)) # label(congruence).  [clausify(2)].
0.00/0.06	kept:      14 -->_s0(x,y) | ->_s0(min(x),min(y)) # label(congruence).  [clausify(3)].
0.00/0.06	kept:      15 -->_s0(x,y) | ->_s0(cons(x,z),cons(y,z)) # label(congruence).  [clausify(4)].
0.00/0.06	kept:      16 -->_s0(x,y) | ->_s0(cons(z,x),cons(z,y)) # label(congruence).  [clausify(5)].
0.00/0.06	kept:      17 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence).  [clausify(6)].
0.00/0.06	kept:      18 ->_s0(le(0,s(x)),true) # label(replacement).  [assumption].
0.00/0.06	kept:      19 ->_s0(le(s(x),s(y)),le(x,y)) # label(replacement).  [assumption].
0.00/0.06	kept:      20 ->_s0(le(x,0),false) # label(replacement).  [assumption].
0.00/0.06	kept:      21 ->_s0(min(cons(x,nil)),x) # label(replacement).  [assumption].
0.00/0.06	kept:      22 -->*_s0(le(x,min(y)),false) | ->_s0(min(cons(x,y)),min(y)) # label(replacement).  [clausify(7)].
0.00/0.06	kept:      23 -->*_s0(min(x),y) | ->_s0(min(cons(y,x)),min(x)) # label(replacement).  [clausify(8)].
0.00/0.06	kept:      24 -->*_s0(le(x,min(y)),true) | ->_s0(min(cons(x,y)),x) # label(replacement).  [clausify(9)].
0.00/0.06	kept:      25 ->*_s0(x,x) # label(reflexivity).  [assumption].
0.00/0.06	kept:      26 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity).  [clausify(10)].
0.00/0.06	kept:      27 -->*_s0(le(x,min(y)),false) | -->*_s0(min(y),x) # label(goal) # answer(goal).  [deny(11)].
0.00/0.06	
0.00/0.06	============================== end of process initial clauses ========
0.00/0.06	
0.00/0.06	============================== CLAUSES FOR SEARCH ====================
0.00/0.06	
0.00/0.06	% Clauses after input processing:
0.00/0.06	
0.00/0.06	formulas(usable).
0.00/0.06	end_of_list.
0.00/0.06	
0.00/0.06	formulas(sos).
0.00/0.06	12 -->_s0(x,y) | ->_s0(le(x,z),le(y,z)) # label(congruence).  [clausify(1)].
0.00/0.06	13 -->_s0(x,y) | ->_s0(le(z,x),le(z,y)) # label(congruence).  [clausify(2)].
0.00/0.06	14 -->_s0(x,y) | ->_s0(min(x),min(y)) # label(congruence).  [clausify(3)].
0.00/0.06	15 -->_s0(x,y) | ->_s0(cons(x,z),cons(y,z)) # label(congruence).  [clausify(4)].
0.00/0.06	16 -->_s0(x,y) | ->_s0(cons(z,x),cons(z,y)) # label(congruence).  [clausify(5)].
0.00/0.06	17 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence).  [clausify(6)].
0.00/0.06	18 ->_s0(le(0,s(x)),true) # label(replacement).  [assumption].
0.00/0.06	19 ->_s0(le(s(x),s(y)),le(x,y)) # label(replacement).  [assumption].
0.00/0.06	20 ->_s0(le(x,0),false) # label(replacement).  [assumption].
0.00/0.06	21 ->_s0(min(cons(x,nil)),x) # label(replacement).  [assumption].
0.00/0.06	22 -->*_s0(le(x,min(y)),false) | ->_s0(min(cons(x,y)),min(y)) # label(replacement).  [clausify(7)].
0.00/0.06	23 -->*_s0(min(x),y) | ->_s0(min(cons(y,x)),min(x)) # label(replacement).  [clausify(8)].
0.00/0.06	24 -->*_s0(le(x,min(y)),true) | ->_s0(min(cons(x,y)),x) # label(replacement).  [clausify(9)].
0.00/0.06	25 ->*_s0(x,x) # label(reflexivity).  [assumption].
0.00/0.06	26 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity).  [clausify(10)].
0.00/0.06	27 -->*_s0(le(x,min(y)),false) | -->*_s0(min(y),x) # label(goal) # answer(goal).  [deny(11)].
0.00/0.06	end_of_list.
0.00/0.06	
0.00/0.06	formulas(demodulators).
0.00/0.06	end_of_list.
0.00/0.06	
0.00/0.06	============================== end of clauses for search =============
0.00/0.06	
0.00/0.06	============================== SEARCH ================================
0.00/0.06	
0.00/0.06	% Starting search at 0.01 seconds.
0.00/0.06	
0.00/0.06	given #1 (I,wt=10): 12 -->_s0(x,y) | ->_s0(le(x,z),le(y,z)) # label(congruence).  [clausify(1)].
0.00/0.06	
0.00/0.06	given #2 (I,wt=10): 13 -->_s0(x,y) | ->_s0(le(z,x),le(z,y)) # label(congruence).  [clausify(2)].
0.00/0.06	
0.00/0.06	given #3 (I,wt=8): 14 -->_s0(x,y) | ->_s0(min(x),min(y)) # label(congruence).  [clausify(3)].
0.00/0.06	
0.00/0.06	given #4 (I,wt=10): 15 -->_s0(x,y) | ->_s0(cons(x,z),cons(y,z)) # label(congruence).  [clausify(4)].
0.00/0.06	
0.00/0.06	given #5 (I,wt=10): 16 -->_s0(x,y) | ->_s0(cons(z,x),cons(z,y)) # label(congruence).  [clausify(5)].
0.00/0.06	
0.00/0.06	given #6 (I,wt=8): 17 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence).  [clausify(6)].
0.00/0.06	
0.00/0.06	given #7 (I,wt=6): 18 ->_s0(le(0,s(x)),true) # label(replacement).  [assumption].
0.00/0.06	
0.00/0.06	given #8 (I,wt=9): 19 ->_s0(le(s(x),s(y)),le(x,y)) # label(replacement).  [assumption].
0.00/0.06	
0.00/0.06	given #9 (I,wt=5): 20 ->_s0(le(x,0),false) # label(replacement).  [assumption].
0.00/0.06	
0.00/0.06	given #10 (I,wt=6): 21 ->_s0(min(cons(x,nil)),x) # label(replacement).  [assumption].
0.00/0.06	
0.00/0.06	given #11 (I,wt=13): 22 -->*_s0(le(x,min(y)),false) | ->_s0(min(cons(x,y)),min(y)) # label(replacement).  [clausify(7)].
0.00/0.06	
0.00/0.06	given #12 (I,wt=11): 23 -->*_s0(min(x),y) | ->_s0(min(cons(y,x)),min(x)) # label(replacement).  [clausify(8)].
0.00/0.06	
0.00/0.06	given #13 (I,wt=12): 24 -->*_s0(le(x,min(y)),true) | ->_s0(min(cons(x,y)),x) # label(replacement).  [clausify(9)].
0.00/0.06	
0.00/0.06	given #14 (I,wt=3): 25 ->*_s0(x,x) # label(reflexivity).  [assumption].
0.00/0.06	
0.00/0.06	given #15 (I,wt=9): 26 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity).  [clausify(10)].
0.00/0.06	
0.00/0.06	given #16 (I,wt=10): 27 -->*_s0(le(x,min(y)),false) | -->*_s0(min(y),x) # label(goal) # answer(goal).  [deny(11)].
0.00/0.06	
0.00/0.06	given #17 (A,wt=8): 28 ->_s0(s(le(0,s(x))),s(true)).  [ur(17,a,18,a)].
0.00/0.06	
0.00/0.06	given #18 (F,wt=7): 59 -->*_s0(le(min(x),min(x)),false) # answer(goal).  [resolve(27,b,25,a)].
0.00/0.06	
0.00/0.06	given #19 (F,wt=7): 68 -->_s0(le(min(x),min(x)),false) # answer(goal).  [ur(26,b,25,a,c,59,a)].
0.00/0.06	
0.00/0.06	given #20 (F,wt=10): 67 -->_s0(le(min(x),min(x)),y) | -->*_s0(y,false) # answer(goal).  [resolve(59,a,26,c)].
0.00/0.06	
0.00/0.06	given #21 (F,wt=10): 69 -->*_s0(le(min(x),y),false) | -->_s0(min(x),y) # answer(goal).  [resolve(67,a,13,b)].
0.00/0.06	
0.00/0.06	============================== PROOF =================================
0.00/0.06	
0.00/0.06	% Proof 1 at 0.01 (+ 0.00) seconds: goal.
0.00/0.06	% Length of proof is 15.
0.00/0.06	% Level of proof is 6.
0.00/0.06	% Maximum clause weight is 10.000.
0.00/0.06	% Given clauses 21.
0.00/0.06	
0.00/0.06	2 ->_s0(x2,y) -> ->_s0(le(x1,x2),le(x1,y)) # label(congruence) # label(non_clause).  [assumption].
0.00/0.06	10 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause).  [assumption].
0.00/0.06	11 (exists x5 exists x6 (->*_s0(le(x6,min(x5)),false) & ->*_s0(min(x5),x6))) # label(goal) # label(non_clause) # label(goal).  [goal].
0.00/0.06	13 -->_s0(x,y) | ->_s0(le(z,x),le(z,y)) # label(congruence).  [clausify(2)].
0.00/0.06	20 ->_s0(le(x,0),false) # label(replacement).  [assumption].
0.00/0.06	21 ->_s0(min(cons(x,nil)),x) # label(replacement).  [assumption].
0.00/0.06	25 ->*_s0(x,x) # label(reflexivity).  [assumption].
0.00/0.06	26 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity).  [clausify(10)].
0.00/0.06	27 -->*_s0(le(x,min(y)),false) | -->*_s0(min(y),x) # label(goal) # answer(goal).  [deny(11)].
0.00/0.06	54 ->*_s0(le(x,0),false).  [ur(26,a,20,a,b,25,a)].
0.00/0.06	59 -->*_s0(le(min(x),min(x)),false) # answer(goal).  [resolve(27,b,25,a)].
0.00/0.06	67 -->_s0(le(min(x),min(x)),y) | -->*_s0(y,false) # answer(goal).  [resolve(59,a,26,c)].
0.00/0.06	69 -->*_s0(le(min(x),y),false) | -->_s0(min(x),y) # answer(goal).  [resolve(67,a,13,b)].
0.00/0.06	76 -->*_s0(le(min(cons(x,nil)),x),false) # answer(goal).  [resolve(69,b,21,a)].
0.00/0.06	77 $F # answer(goal).  [resolve(76,a,54,a)].
0.00/0.06	
0.00/0.06	============================== end of proof ==========================
0.00/0.06	
0.00/0.06	============================== STATISTICS ============================
0.00/0.06	
0.00/0.06	Given=21. Generated=68. Kept=65. proofs=1.
0.00/0.06	Usable=21. Sos=39. Demods=0. Limbo=4, Disabled=16. Hints=0.
0.00/0.06	Kept_by_rule=0, Deleted_by_rule=0.
0.00/0.06	Forward_subsumed=3. Back_subsumed=0.
0.00/0.06	Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0.
0.00/0.06	New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0.
0.00/0.06	Demod_attempts=0. Demod_rewrites=0.
0.00/0.06	Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0.
0.00/0.06	Nonunit_fsub_feature_tests=19. Nonunit_bsub_feature_tests=43.
0.00/0.06	Megabytes=0.18.
0.00/0.06	User_CPU=0.01, System_CPU=0.00, Wall_clock=0.
0.00/0.06	
0.00/0.06	============================== end of statistics =====================
0.00/0.06	
0.00/0.06	============================== end of search =========================
0.00/0.06	
0.00/0.06	THEOREM PROVED
0.00/0.06	
0.00/0.06	Exiting with 1 proof.
0.00/0.06	
0.00/0.06	Process 4229 exit (max_proofs) Tue Aug  1 07:38:47 2023
0.00/0.06	
0.00/0.06	
0.00/0.06	The problem is feasible.
0.00/100.51	EOF