0.04/0.12	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.04/0.13	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.12/0.34	% Computer   : n020.cluster.edu
0.12/0.34	% Model      : x86_64 x86_64
0.12/0.34	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.12/0.34	% Memory     : 8042.1875MB
0.12/0.34	% OS         : Linux 3.10.0-693.el7.x86_64
0.12/0.34	% CPULimit   : 960
0.12/0.34	% WCLimit    : 120
0.12/0.34	% DateTime   : Thu Jul  2 07:03:17 EDT 2020
0.12/0.34	% CPUTime    : 
0.19/0.48	% SZS status Theorem
0.19/0.48	
0.19/0.48	% SZS output start Proof
0.19/0.48	Take the following subset of the input axioms:
0.19/0.48	  fof(ax56, axiom, ![U, V]: (contains_pq(U, V) <=> pi_sharp_remove(U, V))).
0.19/0.48	  fof(ax57, axiom, ![U, V]: (pi_sharp_remove(i(U), V) <=> pi_remove(U, V))).
0.19/0.48	  fof(co3, conjecture, ![U, V, W, X]: ((phi(remove_cpq(triple(U, V, W), X)) => (pi_sharp_remove(i(triple(U, V, W)), X) & i(remove_cpq(triple(U, V, W), X))=remove_pq(i(triple(U, V, W)), X))) <= pi_remove(triple(U, V, W), X))).
0.19/0.48	  fof(main3_li12, lemma, ![U, V, W, X, Y]: i(triple(U, W, X))=i(triple(V, W, Y))).
0.19/0.48	  fof(main3_li34, lemma, ![U, V, W, X]: (remove_pq(i(triple(U, V, W)), X)=i(remove_cpq(triple(U, V, W), X)) <= contains_pq(i(triple(U, V, W)), X))).
0.19/0.48	
0.19/0.48	Now clausify the problem and encode Horn clauses using encoding 3 of
0.19/0.48	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
0.19/0.48	We repeatedly replace C & s=t => u=v by the two clauses:
0.19/0.48	  fresh(y, y, x1...xn) = u
0.19/0.48	  C => fresh(s, t, x1...xn) = v
0.19/0.48	where fresh is a fresh function symbol and x1..xn are the free
0.19/0.48	variables of u and v.
0.19/0.48	A predicate p(X) is encoded as p(X)=true (this is sound, because the
0.19/0.48	input problem has no model of domain size 1).
0.19/0.48	
0.19/0.48	The encoding turns the above axioms into the following unit equations and goals:
0.19/0.48	
0.19/0.48	Axiom 1 (ax56_1): fresh27(X, X, Y, Z) = true2.
0.19/0.48	Axiom 2 (ax57_1): fresh25(X, X, Y, Z) = true2.
0.19/0.48	Axiom 3 (main3_li34): fresh7(X, X, Y, Z, W, V) = i(remove_cpq(triple(Y, Z, W), V)).
0.19/0.48	Axiom 4 (ax57_1): fresh25(pi_remove(X, Y), true2, X, Y) = pi_sharp_remove(i(X), Y).
0.19/0.48	Axiom 5 (ax56_1): fresh27(pi_sharp_remove(X, Y), true2, X, Y) = contains_pq(X, Y).
0.19/0.48	Axiom 6 (main3_li12): i(triple(X, Y, Z)) = i(triple(W, Y, V)).
0.19/0.48	Axiom 7 (main3_li34): fresh7(contains_pq(i(triple(X, Y, Z)), W), true2, X, Y, Z, W) = remove_pq(i(triple(X, Y, Z)), W).
0.19/0.48	Axiom 8 (co3): pi_remove(triple(sK4_co3_U, sK3_co3_V, sK2_co3_W), sK1_co3_X) = true2.
0.19/0.48	
0.19/0.48	Lemma 9: fresh7(X, X, Y, Z, W, V) = fresh7(?, ?, Y, Z, W, V).
0.19/0.48	Proof:
0.19/0.48	  fresh7(X, X, Y, Z, W, V)
0.19/0.48	= { by axiom 3 (main3_li34) }
0.19/0.48	  i(remove_cpq(triple(Y, Z, W), V))
0.19/0.48	= { by axiom 3 (main3_li34) }
0.19/0.48	  fresh7(?, ?, Y, Z, W, V)
0.19/0.48	
0.19/0.48	Lemma 10: i(triple(X, Y, Z)) = i(triple(?, Y, ?)).
0.19/0.48	Proof:
0.19/0.48	  i(triple(X, Y, Z))
0.19/0.48	= { by axiom 6 (main3_li12) }
0.19/0.48	  i(triple(W, Y, V))
0.19/0.48	= { by axiom 6 (main3_li12) }
0.19/0.48	  i(triple(?, Y, ?))
0.19/0.48	
0.19/0.48	Lemma 11: pi_sharp_remove(i(triple(?, sK3_co3_V, ?)), sK1_co3_X) = true2.
0.19/0.48	Proof:
0.19/0.48	  pi_sharp_remove(i(triple(?, sK3_co3_V, ?)), sK1_co3_X)
0.19/0.48	= { by lemma 10 }
0.19/0.48	  pi_sharp_remove(i(triple(sK4_co3_U, sK3_co3_V, sK2_co3_W)), sK1_co3_X)
0.19/0.48	= { by axiom 4 (ax57_1) }
0.19/0.48	  fresh25(pi_remove(triple(sK4_co3_U, sK3_co3_V, sK2_co3_W), sK1_co3_X), true2, triple(sK4_co3_U, sK3_co3_V, sK2_co3_W), sK1_co3_X)
0.19/0.48	= { by axiom 8 (co3) }
0.19/0.48	  fresh25(true2, true2, triple(sK4_co3_U, sK3_co3_V, sK2_co3_W), sK1_co3_X)
0.19/0.48	= { by axiom 2 (ax57_1) }
0.19/0.48	  true2
0.19/0.48	
0.19/0.48	Lemma 12: contains_pq(i(triple(?, sK3_co3_V, ?)), sK1_co3_X) = true2.
0.19/0.48	Proof:
0.19/0.48	  contains_pq(i(triple(?, sK3_co3_V, ?)), sK1_co3_X)
0.19/0.48	= { by axiom 5 (ax56_1) }
0.19/0.48	  fresh27(pi_sharp_remove(i(triple(?, sK3_co3_V, ?)), sK1_co3_X), true2, i(triple(?, sK3_co3_V, ?)), sK1_co3_X)
0.19/0.48	= { by lemma 11 }
0.19/0.48	  fresh27(true2, true2, i(triple(?, sK3_co3_V, ?)), sK1_co3_X)
0.19/0.48	= { by axiom 1 (ax56_1) }
0.19/0.48	  true2
0.19/0.48	
0.19/0.48	Lemma 13: fresh7(contains_pq(i(triple(?, Y, ?)), W), true2, X, Y, Z, W) = remove_pq(i(triple(?, Y, ?)), W).
0.19/0.48	Proof:
0.19/0.48	  fresh7(contains_pq(i(triple(?, Y, ?)), W), true2, X, Y, Z, W)
0.19/0.48	= { by lemma 10 }
0.19/0.48	  fresh7(contains_pq(i(triple(X, Y, Z)), W), true2, X, Y, Z, W)
0.19/0.48	= { by axiom 7 (main3_li34) }
0.19/0.48	  remove_pq(i(triple(X, Y, Z)), W)
0.19/0.48	= { by lemma 10 }
0.19/0.48	  remove_pq(i(triple(?, Y, ?)), W)
0.19/0.48	
0.19/0.48	Goal 1 (co3_2): tuple2(i(remove_cpq(triple(sK4_co3_U, sK3_co3_V, sK2_co3_W), sK1_co3_X)), pi_sharp_remove(i(triple(sK4_co3_U, sK3_co3_V, sK2_co3_W)), sK1_co3_X)) = tuple2(remove_pq(i(triple(sK4_co3_U, sK3_co3_V, sK2_co3_W)), sK1_co3_X), true2).
0.19/0.48	Proof:
0.19/0.48	  tuple2(i(remove_cpq(triple(sK4_co3_U, sK3_co3_V, sK2_co3_W), sK1_co3_X)), pi_sharp_remove(i(triple(sK4_co3_U, sK3_co3_V, sK2_co3_W)), sK1_co3_X))
0.19/0.48	= { by axiom 3 (main3_li34) }
0.19/0.48	  tuple2(fresh7(?, ?, sK4_co3_U, sK3_co3_V, sK2_co3_W, sK1_co3_X), pi_sharp_remove(i(triple(sK4_co3_U, sK3_co3_V, sK2_co3_W)), sK1_co3_X))
0.19/0.48	= { by lemma 9 }
0.19/0.48	  tuple2(fresh7(true2, true2, sK4_co3_U, sK3_co3_V, sK2_co3_W, sK1_co3_X), pi_sharp_remove(i(triple(sK4_co3_U, sK3_co3_V, sK2_co3_W)), sK1_co3_X))
0.19/0.48	= { by lemma 12 }
0.19/0.48	  tuple2(fresh7(contains_pq(i(triple(?, sK3_co3_V, ?)), sK1_co3_X), true2, sK4_co3_U, sK3_co3_V, sK2_co3_W, sK1_co3_X), pi_sharp_remove(i(triple(sK4_co3_U, sK3_co3_V, sK2_co3_W)), sK1_co3_X))
0.19/0.48	= { by lemma 13 }
0.19/0.48	  tuple2(remove_pq(i(triple(?, sK3_co3_V, ?)), sK1_co3_X), pi_sharp_remove(i(triple(sK4_co3_U, sK3_co3_V, sK2_co3_W)), sK1_co3_X))
0.19/0.48	= { by lemma 10 }
0.19/0.48	  tuple2(remove_pq(i(triple(sK4_co3_U, sK3_co3_V, sK2_co3_W)), sK1_co3_X), pi_sharp_remove(i(triple(sK4_co3_U, sK3_co3_V, sK2_co3_W)), sK1_co3_X))
0.19/0.48	= { by lemma 10 }
0.19/0.48	  tuple2(remove_pq(i(triple(sK4_co3_U, sK3_co3_V, sK2_co3_W)), sK1_co3_X), pi_sharp_remove(i(triple(?, sK3_co3_V, ?)), sK1_co3_X))
0.19/0.48	= { by lemma 11 }
0.19/0.48	  tuple2(remove_pq(i(triple(sK4_co3_U, sK3_co3_V, sK2_co3_W)), sK1_co3_X), true2)
0.19/0.48	% SZS output end Proof
0.19/0.48	
0.19/0.48	RESULT: Theorem (the conjecture is true).
0.19/0.48	EOF