0.00/0.03	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.03/0.04	% Command    : twee %s --tstp --casc --quiet --conditional-encoding if --smaller --drop-non-horn
0.03/0.24	% Computer   : n005.star.cs.uiowa.edu
0.03/0.24	% Model      : x86_64 x86_64
0.03/0.24	% CPU        : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
0.03/0.24	% Memory     : 32218.625MB
0.03/0.24	% OS         : Linux 3.10.0-693.2.2.el7.x86_64
0.03/0.24	% CPULimit   : 300
0.03/0.24	% DateTime   : Sat Jul 14 04:21:55 CDT 2018
0.03/0.24	% CPUTime    : 
0.07/0.38	% SZS status Theorem
0.07/0.38	
0.07/0.38	% SZS output start Proof
0.07/0.38	Take the following subset of the input axioms:
0.07/0.38	  fof(first_second, axiom,
0.07/0.38	      ![X, Y]:
0.07/0.38	        ((member(Y, universal_class) & member(X, universal_class))
0.07/0.38	           => (first(ordered_pair(X, Y))=X & Y=second(ordered_pair(X, Y))))).
0.07/0.38	  fof(unique_1st_and_2nd_in_pair_of_sets1, conjecture,
0.07/0.38	      ![X, U, V]:
0.07/0.38	        ((member(V, universal_class)
0.07/0.38	            & (ordered_pair(U, V)=X & member(U, universal_class)))
0.07/0.38	           => (V=second(X) & first(X)=U))).
0.07/0.38	
0.07/0.38	Now clausify the problem and encode Horn clauses using encoding 3 of
0.07/0.38	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
0.07/0.38	We repeatedly replace C & s=t => u=v by the two clauses:
0.07/0.38	  $$fresh(y, y, x1...xn) = u
0.07/0.38	  C => $$fresh(s, t, x1...xn) = v
0.07/0.38	where $$fresh is a fresh function symbol and x1..xn are the free
0.07/0.38	variables of u and v.
0.07/0.38	A predicate p(X) is encoded as p(X)=$$true (this is sound, because the
0.07/0.38	input problem has no model of domain size 1).
0.07/0.38	
0.07/0.38	The encoding turns the above axioms into the following unit equations and goals:
0.07/0.38	
0.07/0.38	Axiom 18 (first_second): $$fresh4(X, X, Y, Z) = Y.
0.07/0.38	Axiom 19 (first_second): $$fresh56(X, X, Y, Z) = first(ordered_pair(Y, Z)).
0.07/0.38	Axiom 20 (first_second_1): $$fresh5(X, X, Y, Z) = Z.
0.07/0.38	Axiom 21 (first_second_1): $$fresh55(X, X, Y, Z) = second(ordered_pair(Y, Z)).
0.07/0.38	Axiom 116 (first_second_1): $$fresh55(member(X, universal_class), $$true2, Y, X) = $$fresh5(member(Y, universal_class), $$true2, Y, X).
0.07/0.38	Axiom 117 (first_second): $$fresh56(member(X, universal_class), $$true2, Y, X) = $$fresh4(member(Y, universal_class), $$true2, Y, X).
0.07/0.38	Axiom 146 (unique_1st_and_2nd_in_pair_of_sets1): ordered_pair(sK2_unique_1st_and_2nd_in_pair_of_sets1_U, sK3_unique_1st_and_2nd_in_pair_of_sets1_V) = sK1_unique_1st_and_2nd_in_pair_of_sets1_X.
0.07/0.38	Axiom 147 (unique_1st_and_2nd_in_pair_of_sets1_1): member(sK3_unique_1st_and_2nd_in_pair_of_sets1_V, universal_class) = $$true2.
0.07/0.38	Axiom 148 (unique_1st_and_2nd_in_pair_of_sets1_2): member(sK2_unique_1st_and_2nd_in_pair_of_sets1_U, universal_class) = $$true2.
0.07/0.38	
0.07/0.38	Goal 1 (unique_1st_and_2nd_in_pair_of_sets1_3): tuple4(first(sK1_unique_1st_and_2nd_in_pair_of_sets1_X), sK3_unique_1st_and_2nd_in_pair_of_sets1_V) = tuple4(sK2_unique_1st_and_2nd_in_pair_of_sets1_U, second(sK1_unique_1st_and_2nd_in_pair_of_sets1_X)).
0.07/0.38	Proof:
0.07/0.38	  tuple4(first(sK1_unique_1st_and_2nd_in_pair_of_sets1_X), sK3_unique_1st_and_2nd_in_pair_of_sets1_V)
0.07/0.38	= { by axiom 146 (unique_1st_and_2nd_in_pair_of_sets1) }
0.07/0.38	  tuple4(first(ordered_pair(sK2_unique_1st_and_2nd_in_pair_of_sets1_U, sK3_unique_1st_and_2nd_in_pair_of_sets1_V)), sK3_unique_1st_and_2nd_in_pair_of_sets1_V)
0.07/0.38	= { by axiom 19 (first_second) }
0.07/0.38	  tuple4($$fresh56($$true2, $$true2, sK2_unique_1st_and_2nd_in_pair_of_sets1_U, sK3_unique_1st_and_2nd_in_pair_of_sets1_V), sK3_unique_1st_and_2nd_in_pair_of_sets1_V)
0.07/0.38	= { by axiom 147 (unique_1st_and_2nd_in_pair_of_sets1_1) }
0.07/0.38	  tuple4($$fresh56(member(sK3_unique_1st_and_2nd_in_pair_of_sets1_V, universal_class), $$true2, sK2_unique_1st_and_2nd_in_pair_of_sets1_U, sK3_unique_1st_and_2nd_in_pair_of_sets1_V), sK3_unique_1st_and_2nd_in_pair_of_sets1_V)
0.07/0.38	= { by axiom 117 (first_second) }
0.07/0.38	  tuple4($$fresh4(member(sK2_unique_1st_and_2nd_in_pair_of_sets1_U, universal_class), $$true2, sK2_unique_1st_and_2nd_in_pair_of_sets1_U, sK3_unique_1st_and_2nd_in_pair_of_sets1_V), sK3_unique_1st_and_2nd_in_pair_of_sets1_V)
0.07/0.38	= { by axiom 148 (unique_1st_and_2nd_in_pair_of_sets1_2) }
0.07/0.38	  tuple4($$fresh4($$true2, $$true2, sK2_unique_1st_and_2nd_in_pair_of_sets1_U, sK3_unique_1st_and_2nd_in_pair_of_sets1_V), sK3_unique_1st_and_2nd_in_pair_of_sets1_V)
0.07/0.38	= { by axiom 18 (first_second) }
0.07/0.38	  tuple4(sK2_unique_1st_and_2nd_in_pair_of_sets1_U, sK3_unique_1st_and_2nd_in_pair_of_sets1_V)
0.07/0.38	= { by axiom 20 (first_second_1) }
0.07/0.38	  tuple4(sK2_unique_1st_and_2nd_in_pair_of_sets1_U, $$fresh5($$true2, $$true2, sK2_unique_1st_and_2nd_in_pair_of_sets1_U, sK3_unique_1st_and_2nd_in_pair_of_sets1_V))
0.07/0.38	= { by axiom 148 (unique_1st_and_2nd_in_pair_of_sets1_2) }
0.07/0.38	  tuple4(sK2_unique_1st_and_2nd_in_pair_of_sets1_U, $$fresh5(member(sK2_unique_1st_and_2nd_in_pair_of_sets1_U, universal_class), $$true2, sK2_unique_1st_and_2nd_in_pair_of_sets1_U, sK3_unique_1st_and_2nd_in_pair_of_sets1_V))
0.07/0.38	= { by axiom 116 (first_second_1) }
0.07/0.38	  tuple4(sK2_unique_1st_and_2nd_in_pair_of_sets1_U, $$fresh55(member(sK3_unique_1st_and_2nd_in_pair_of_sets1_V, universal_class), $$true2, sK2_unique_1st_and_2nd_in_pair_of_sets1_U, sK3_unique_1st_and_2nd_in_pair_of_sets1_V))
0.07/0.38	= { by axiom 147 (unique_1st_and_2nd_in_pair_of_sets1_1) }
0.07/0.38	  tuple4(sK2_unique_1st_and_2nd_in_pair_of_sets1_U, $$fresh55($$true2, $$true2, sK2_unique_1st_and_2nd_in_pair_of_sets1_U, sK3_unique_1st_and_2nd_in_pair_of_sets1_V))
0.07/0.38	= { by axiom 21 (first_second_1) }
0.07/0.38	  tuple4(sK2_unique_1st_and_2nd_in_pair_of_sets1_U, second(ordered_pair(sK2_unique_1st_and_2nd_in_pair_of_sets1_U, sK3_unique_1st_and_2nd_in_pair_of_sets1_V)))
0.07/0.38	= { by axiom 146 (unique_1st_and_2nd_in_pair_of_sets1) }
0.07/0.38	  tuple4(sK2_unique_1st_and_2nd_in_pair_of_sets1_U, second(sK1_unique_1st_and_2nd_in_pair_of_sets1_X))
0.07/0.38	% SZS output end Proof
0.07/0.38	
0.07/0.38	RESULT: Theorem (the conjecture is true).
0.07/0.38	EOF