0.06/0.12	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.06/0.13	% Command    : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
0.13/0.33	% Computer   : n013.cluster.edu
0.13/0.33	% Model      : x86_64 x86_64
0.13/0.33	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.13/0.33	% Memory     : 8042.1875MB
0.13/0.33	% OS         : Linux 3.10.0-693.el7.x86_64
0.13/0.33	% CPULimit   : 1200
0.13/0.33	% WCLimit    : 120
0.13/0.33	% DateTime   : Tue Jul 13 16:52:03 EDT 2021
0.13/0.33	% CPUTime    : 
8.11/1.41	% SZS status Theorem
8.11/1.41	
8.11/1.41	% SZS output start Proof
8.11/1.42	Take the following subset of the input axioms:
8.11/1.42	  fof(cross_product_defn, axiom, ![X, Y, U, V]: ((member(V, Y) & member(U, X)) <=> member(ordered_pair(U, V), cross_product(X, Y)))).
8.11/1.42	  fof(first_second, axiom, ![X, Y]: ((X=first(ordered_pair(X, Y)) & Y=second(ordered_pair(X, Y))) <= (member(Y, universal_class) & member(X, universal_class)))).
8.11/1.42	  fof(infinity, axiom, ?[X]: (inductive(X) & (![Y]: (inductive(Y) => subclass(X, Y)) & member(X, universal_class)))).
8.11/1.42	  fof(ordered_pair_defn, axiom, ![X, Y]: ordered_pair(X, Y)=unordered_pair(singleton(X), unordered_pair(X, singleton(Y)))).
8.11/1.42	  fof(singleton_identified_by_element1, conjecture, ![X, Y]: ((member(X, universal_class) & singleton(Y)=singleton(X)) => Y=X)).
8.11/1.42	
8.11/1.42	Now clausify the problem and encode Horn clauses using encoding 3 of
8.11/1.42	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
8.11/1.42	We repeatedly replace C & s=t => u=v by the two clauses:
8.11/1.42	  fresh(y, y, x1...xn) = u
8.11/1.42	  C => fresh(s, t, x1...xn) = v
8.11/1.42	where fresh is a fresh function symbol and x1..xn are the free
8.11/1.42	variables of u and v.
8.11/1.42	A predicate p(X) is encoded as p(X)=true (this is sound, because the
8.11/1.42	input problem has no model of domain size 1).
8.11/1.42	
8.11/1.42	The encoding turns the above axioms into the following unit equations and goals:
8.11/1.42	
8.11/1.42	Axiom 1 (singleton_identified_by_element1): singleton(y) = singleton(x).
8.11/1.42	Axiom 2 (infinity): member(x3, universal_class) = true2.
8.11/1.42	Axiom 3 (singleton_identified_by_element1_1): member(x, universal_class) = true2.
8.11/1.42	Axiom 4 (cross_product_defn_2): fresh62(X, X, Y, Z) = true2.
8.11/1.42	Axiom 5 (first_second_1): fresh55(X, X, Y, Z) = second(ordered_pair(Y, Z)).
8.11/1.42	Axiom 6 (first_second_1): fresh7(X, X, Y, Z) = Z.
8.11/1.42	Axiom 7 (cross_product_defn): fresh64(X, X, Y, Z, W, V) = true2.
8.11/1.42	Axiom 8 (first_second_1): fresh55(member(X, universal_class), true2, Y, X) = fresh7(member(Y, universal_class), true2, Y, X).
8.11/1.42	Axiom 9 (ordered_pair_defn): ordered_pair(X, Y) = unordered_pair(singleton(X), unordered_pair(X, singleton(Y))).
8.11/1.42	Axiom 10 (cross_product_defn): fresh65(X, X, Y, Z, W, V) = member(ordered_pair(Y, Z), cross_product(W, V)).
8.11/1.42	Axiom 11 (cross_product_defn): fresh65(member(X, Y), true2, Z, X, W, Y) = fresh64(member(Z, W), true2, Z, X, W, Y).
8.11/1.42	Axiom 12 (cross_product_defn_2): fresh62(member(ordered_pair(X, Y), cross_product(Z, W)), true2, Y, W) = member(Y, W).
8.11/1.42	
8.11/1.42	Lemma 13: ordered_pair(X, x) = ordered_pair(X, y).
8.11/1.42	Proof:
8.11/1.42	  ordered_pair(X, x)
8.11/1.42	= { by axiom 9 (ordered_pair_defn) }
8.11/1.42	  unordered_pair(singleton(X), unordered_pair(X, singleton(x)))
8.11/1.42	= { by axiom 1 (singleton_identified_by_element1) R->L }
8.11/1.42	  unordered_pair(singleton(X), unordered_pair(X, singleton(y)))
8.11/1.42	= { by axiom 9 (ordered_pair_defn) R->L }
8.11/1.42	  ordered_pair(X, y)
8.11/1.42	
8.11/1.42	Lemma 14: fresh55(member(X, universal_class), true2, x3, X) = X.
8.11/1.42	Proof:
8.11/1.42	  fresh55(member(X, universal_class), true2, x3, X)
8.11/1.42	= { by axiom 8 (first_second_1) }
8.11/1.42	  fresh7(member(x3, universal_class), true2, x3, X)
8.11/1.42	= { by axiom 2 (infinity) }
8.11/1.42	  fresh7(true2, true2, x3, X)
8.11/1.42	= { by axiom 6 (first_second_1) }
8.11/1.42	  X
8.11/1.42	
8.11/1.42	Goal 1 (singleton_identified_by_element1_2): y = x.
8.11/1.42	Proof:
8.11/1.42	  y
8.11/1.42	= { by lemma 14 R->L }
8.11/1.42	  fresh55(member(y, universal_class), true2, x3, y)
8.11/1.42	= { by axiom 12 (cross_product_defn_2) R->L }
8.11/1.42	  fresh55(fresh62(member(ordered_pair(x3, y), cross_product(universal_class, universal_class)), true2, y, universal_class), true2, x3, y)
8.11/1.42	= { by lemma 13 R->L }
8.11/1.42	  fresh55(fresh62(member(ordered_pair(x3, x), cross_product(universal_class, universal_class)), true2, y, universal_class), true2, x3, y)
8.11/1.42	= { by axiom 10 (cross_product_defn) R->L }
8.11/1.42	  fresh55(fresh62(fresh65(true2, true2, x3, x, universal_class, universal_class), true2, y, universal_class), true2, x3, y)
8.11/1.42	= { by axiom 3 (singleton_identified_by_element1_1) R->L }
8.11/1.42	  fresh55(fresh62(fresh65(member(x, universal_class), true2, x3, x, universal_class, universal_class), true2, y, universal_class), true2, x3, y)
8.11/1.42	= { by axiom 11 (cross_product_defn) }
8.11/1.42	  fresh55(fresh62(fresh64(member(x3, universal_class), true2, x3, x, universal_class, universal_class), true2, y, universal_class), true2, x3, y)
8.11/1.42	= { by axiom 2 (infinity) }
8.11/1.42	  fresh55(fresh62(fresh64(true2, true2, x3, x, universal_class, universal_class), true2, y, universal_class), true2, x3, y)
8.11/1.42	= { by axiom 7 (cross_product_defn) }
8.11/1.42	  fresh55(fresh62(true2, true2, y, universal_class), true2, x3, y)
8.11/1.42	= { by axiom 4 (cross_product_defn_2) }
8.11/1.42	  fresh55(true2, true2, x3, y)
8.11/1.42	= { by axiom 5 (first_second_1) }
8.11/1.42	  second(ordered_pair(x3, y))
8.11/1.42	= { by lemma 13 R->L }
8.11/1.42	  second(ordered_pair(x3, x))
8.11/1.42	= { by axiom 5 (first_second_1) R->L }
8.11/1.42	  fresh55(true2, true2, x3, x)
8.11/1.42	= { by axiom 3 (singleton_identified_by_element1_1) R->L }
8.11/1.42	  fresh55(member(x, universal_class), true2, x3, x)
8.11/1.42	= { by lemma 14 }
8.11/1.42	  x
8.11/1.42	% SZS output end Proof
8.11/1.42	
8.11/1.42	RESULT: Theorem (the conjecture is true).
8.11/1.43	EOF