0.06/0.11	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.06/0.12	% Command    : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
0.12/0.33	% Computer   : n002.cluster.edu
0.12/0.33	% Model      : x86_64 x86_64
0.12/0.33	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.12/0.33	% Memory     : 8042.1875MB
0.12/0.33	% OS         : Linux 3.10.0-693.el7.x86_64
0.12/0.33	% CPULimit   : 1200
0.12/0.33	% WCLimit    : 120
0.12/0.33	% DateTime   : Wed Jul 14 14:20:02 EDT 2021
0.12/0.33	% CPUTime    : 
0.19/0.50	% SZS status Theorem
0.19/0.50	
0.19/0.50	% SZS output start Proof
0.19/0.50	Take the following subset of the input axioms:
0.19/0.50	  fof(ax56, axiom, ![U, V]: (pi_sharp_remove(U, V) <=> contains_pq(U, V))).
0.19/0.50	  fof(ax57, axiom, ![U, V]: (pi_remove(U, V) <=> pi_sharp_remove(i(U), V))).
0.19/0.50	  fof(co3, conjecture, ![U, V, W, X]: (pi_remove(triple(U, V, W), X) => ((i(remove_cpq(triple(U, V, W), X))=remove_pq(i(triple(U, V, W)), X) & pi_sharp_remove(i(triple(U, V, W)), X)) <= phi(remove_cpq(triple(U, V, W), X))))).
0.19/0.50	  fof(main3_li12, lemma, ![U, V, W, X, Y]: i(triple(U, W, X))=i(triple(V, W, Y))).
0.19/0.50	  fof(main3_li34, lemma, ![U, V, W, X]: (contains_pq(i(triple(U, V, W)), X) => i(remove_cpq(triple(U, V, W), X))=remove_pq(i(triple(U, V, W)), X))).
0.19/0.50	
0.19/0.50	Now clausify the problem and encode Horn clauses using encoding 3 of
0.19/0.50	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
0.19/0.50	We repeatedly replace C & s=t => u=v by the two clauses:
0.19/0.50	  fresh(y, y, x1...xn) = u
0.19/0.50	  C => fresh(s, t, x1...xn) = v
0.19/0.50	where fresh is a fresh function symbol and x1..xn are the free
0.19/0.50	variables of u and v.
0.19/0.50	A predicate p(X) is encoded as p(X)=true (this is sound, because the
0.19/0.50	input problem has no model of domain size 1).
0.19/0.50	
0.19/0.50	The encoding turns the above axioms into the following unit equations and goals:
0.19/0.50	
0.19/0.50	Axiom 1 (main3_li12): i(triple(X, Y, Z)) = i(triple(W, Y, V)).
0.19/0.50	Axiom 2 (ax56_1): fresh27(X, X, Y, Z) = true2.
0.19/0.50	Axiom 3 (ax57_1): fresh25(X, X, Y, Z) = true2.
0.19/0.50	Axiom 4 (co3): pi_remove(triple(u, v, w), x) = true2.
0.19/0.50	Axiom 5 (ax56_1): fresh27(pi_sharp_remove(X, Y), true2, X, Y) = contains_pq(X, Y).
0.19/0.50	Axiom 6 (ax57_1): fresh25(pi_remove(X, Y), true2, X, Y) = pi_sharp_remove(i(X), Y).
0.19/0.50	Axiom 7 (main3_li34): fresh7(X, X, Y, Z, W, V) = remove_pq(i(triple(Y, Z, W)), V).
0.19/0.50	Axiom 8 (main3_li34): fresh7(contains_pq(i(triple(X, Y, Z)), W), true2, X, Y, Z, W) = i(remove_cpq(triple(X, Y, Z), W)).
0.19/0.50	
0.19/0.50	Lemma 9: pi_sharp_remove(i(triple(X, v, Y)), x) = true2.
0.19/0.50	Proof:
0.19/0.50	  pi_sharp_remove(i(triple(X, v, Y)), x)
0.19/0.50	= { by axiom 1 (main3_li12) R->L }
0.19/0.50	  pi_sharp_remove(i(triple(u, v, w)), x)
0.19/0.50	= { by axiom 6 (ax57_1) R->L }
0.19/0.50	  fresh25(pi_remove(triple(u, v, w), x), true2, triple(u, v, w), x)
0.19/0.50	= { by axiom 4 (co3) }
0.19/0.50	  fresh25(true2, true2, triple(u, v, w), x)
0.19/0.50	= { by axiom 3 (ax57_1) }
0.19/0.50	  true2
0.19/0.50	
0.19/0.50	Goal 1 (co3_2): tuple2(i(remove_cpq(triple(u, v, w), x)), pi_sharp_remove(i(triple(u, v, w)), x)) = tuple2(remove_pq(i(triple(u, v, w)), x), true2).
0.19/0.50	Proof:
0.19/0.50	  tuple2(i(remove_cpq(triple(u, v, w), x)), pi_sharp_remove(i(triple(u, v, w)), x))
0.19/0.50	= { by lemma 9 }
0.19/0.50	  tuple2(i(remove_cpq(triple(u, v, w), x)), true2)
0.19/0.50	= { by axiom 8 (main3_li34) R->L }
0.19/0.50	  tuple2(fresh7(contains_pq(i(triple(u, v, w)), x), true2, u, v, w, x), true2)
0.19/0.50	= { by axiom 5 (ax56_1) R->L }
0.19/0.50	  tuple2(fresh7(fresh27(pi_sharp_remove(i(triple(u, v, w)), x), true2, i(triple(u, v, w)), x), true2, u, v, w, x), true2)
0.19/0.50	= { by lemma 9 }
0.19/0.50	  tuple2(fresh7(fresh27(true2, true2, i(triple(u, v, w)), x), true2, u, v, w, x), true2)
0.19/0.50	= { by axiom 2 (ax56_1) }
0.19/0.50	  tuple2(fresh7(true2, true2, u, v, w, x), true2)
0.19/0.50	= { by axiom 7 (main3_li34) }
0.19/0.50	  tuple2(remove_pq(i(triple(u, v, w)), x), true2)
0.19/0.50	% SZS output end Proof
0.19/0.50	
0.19/0.50	RESULT: Theorem (the conjecture is true).
0.19/0.50	EOF