0.04/0.13	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.13/0.14	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.14/0.35	% Computer   : n007.cluster.edu
0.14/0.35	% Model      : x86_64 x86_64
0.14/0.35	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.14/0.35	% Memory     : 8042.1875MB
0.14/0.35	% OS         : Linux 3.10.0-693.el7.x86_64
0.14/0.35	% CPULimit   : 960
0.14/0.35	% WCLimit    : 120
0.14/0.35	% DateTime   : Thu Jul  2 07:06:24 EDT 2020
0.21/0.35	% CPUTime    : 
22.63/3.24	% SZS status Theorem
22.63/3.24	
22.63/3.24	% SZS output start Proof
22.63/3.24	Take the following subset of the input axioms:
22.63/3.30	  fof(prove_distribution, conjecture, ![E]: ((![X, Y, Z, U, V, W]: ((product(X, V, W) & (product(Y, Z, V) & product(X, Y, U))) => product(U, Z, W)) & (![X]: product(X, E, X) & (![X]: product(inverse(X), X, E) & (![X]: product(X, inverse(X), E) & (![X]: product(E, X, X) & (![X, Y, Z, U, V, W]: (product(X, V, W) <= (product(X, Y, U) & (product(Y, Z, V) & product(U, Z, W)))) & ![X, Y]: ?[Z]: product(X, Y, Z))))))) => ![X, U, V, W]: ((product(V, U, X) & product(inverse(U), inverse(V), W)) => product(inverse(W), inverse(X), E)))).
22.63/3.30	
22.63/3.30	Now clausify the problem and encode Horn clauses using encoding 3 of
22.63/3.30	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
22.63/3.30	We repeatedly replace C & s=t => u=v by the two clauses:
22.63/3.30	  fresh(y, y, x1...xn) = u
22.63/3.30	  C => fresh(s, t, x1...xn) = v
22.63/3.30	where fresh is a fresh function symbol and x1..xn are the free
22.63/3.30	variables of u and v.
22.63/3.30	A predicate p(X) is encoded as p(X)=true (this is sound, because the
22.63/3.30	input problem has no model of domain size 1).
22.63/3.30	
22.63/3.30	The encoding turns the above axioms into the following unit equations and goals:
22.63/3.30	
22.63/3.30	Axiom 1 (prove_distribution_7): fresh2(X, X, Y, Z, W, V, U) = product(V, W, U).
22.63/3.30	Axiom 2 (prove_distribution_7): fresh6(X, X, Y, Z, W) = true.
22.63/3.30	Axiom 3 (prove_distribution_7): fresh5(X, X, Y, Z, W, V, U, T) = fresh6(product(Y, Z, V), true, W, V, T).
22.63/3.30	Axiom 4 (prove_distribution_8): fresh(X, X, Y, Z, W, V, U) = product(Y, V, U).
22.63/3.30	Axiom 5 (prove_distribution_8): fresh4(X, X, Y, Z, W) = true.
22.63/3.30	Axiom 6 (prove_distribution_8): fresh3(X, X, Y, Z, W, V, U, T) = fresh4(product(Y, Z, V), true, Y, U, T).
22.63/3.30	Axiom 7 (prove_distribution): product(X, sK6_prove_distribution_E, X) = true.
22.63/3.30	Axiom 8 (prove_distribution_1): product(X, inverse(X), sK6_prove_distribution_E) = true.
22.63/3.30	Axiom 9 (prove_distribution_4): product(inverse(sK3_prove_distribution_U), inverse(sK2_prove_distribution_V), sK5_prove_distribution_W) = true.
22.63/3.30	Axiom 10 (prove_distribution_5): product(sK6_prove_distribution_E, X, X) = true.
22.63/3.30	Axiom 11 (prove_distribution_6): product(sK2_prove_distribution_V, sK3_prove_distribution_U, sK4_prove_distribution_X) = true.
22.63/3.30	Axiom 12 (prove_distribution_7): fresh5(product(X, Y, Z), true, W, X, Y, V, Z, U) = fresh2(product(W, Z, U), true, W, X, Y, V, U).
22.63/3.30	Axiom 13 (prove_distribution_8): fresh3(product(X, Y, Z), true, W, V, Y, X, U, Z) = fresh(product(V, Y, U), true, W, V, X, U, Z).
22.63/3.30	
22.63/3.30	Lemma 14: fresh5(X, X, Y, Z, W, V, U, T) = fresh5(?, ?, Y, Z, W, V, ?, T).
22.63/3.30	Proof:
22.63/3.30	  fresh5(X, X, Y, Z, W, V, U, T)
22.63/3.30	= { by axiom 3 (prove_distribution_7) }
22.63/3.30	  fresh6(product(Y, Z, V), true, W, V, T)
22.63/3.30	= { by axiom 3 (prove_distribution_7) }
22.63/3.30	  fresh5(?, ?, Y, Z, W, V, ?, T)
22.63/3.30	
22.63/3.30	Lemma 15: fresh3(X, X, Y, Z, W, V, U, T) = fresh3(?, ?, Y, Z, ?, V, U, T).
22.63/3.30	Proof:
22.63/3.30	  fresh3(X, X, Y, Z, W, V, U, T)
22.63/3.30	= { by axiom 6 (prove_distribution_8) }
22.63/3.30	  fresh4(product(Y, Z, V), true, Y, U, T)
22.63/3.30	= { by axiom 6 (prove_distribution_8) }
23.24/3.36	  fresh3(?, ?, Y, Z, ?, V, U, T)
23.24/3.36	
23.24/3.36	Goal 1 (prove_distribution_9): product(inverse(sK5_prove_distribution_W), inverse(sK4_prove_distribution_X), sK6_prove_distribution_E) = true.
23.24/3.36	Proof:
23.24/3.36	  product(inverse(sK5_prove_distribution_W), inverse(sK4_prove_distribution_X), sK6_prove_distribution_E)
23.24/3.36	= { by axiom 1 (prove_distribution_7) }
23.24/3.36	  fresh2(true, true, sK4_prove_distribution_X, sK6_prove_distribution_E, inverse(sK4_prove_distribution_X), inverse(sK5_prove_distribution_W), sK6_prove_distribution_E)
23.24/3.36	= { by axiom 8 (prove_distribution_1) }
23.24/3.36	  fresh2(product(sK4_prove_distribution_X, inverse(sK4_prove_distribution_X), sK6_prove_distribution_E), true, sK4_prove_distribution_X, sK6_prove_distribution_E, inverse(sK4_prove_distribution_X), inverse(sK5_prove_distribution_W), sK6_prove_distribution_E)
23.24/3.36	= { by axiom 12 (prove_distribution_7) }
23.24/3.36	  fresh5(product(sK6_prove_distribution_E, inverse(sK4_prove_distribution_X), inverse(sK4_prove_distribution_X)), true, sK4_prove_distribution_X, sK6_prove_distribution_E, inverse(sK4_prove_distribution_X), inverse(sK5_prove_distribution_W), inverse(sK4_prove_distribution_X), sK6_prove_distribution_E)
23.24/3.36	= { by axiom 10 (prove_distribution_5) }
23.24/3.36	  fresh5(true, true, sK4_prove_distribution_X, sK6_prove_distribution_E, inverse(sK4_prove_distribution_X), inverse(sK5_prove_distribution_W), inverse(sK4_prove_distribution_X), sK6_prove_distribution_E)
23.24/3.36	= { by lemma 14 }
23.24/3.36	  fresh5(?, ?, sK4_prove_distribution_X, sK6_prove_distribution_E, inverse(sK4_prove_distribution_X), inverse(sK5_prove_distribution_W), ?, sK6_prove_distribution_E)
23.24/3.36	= { by axiom 3 (prove_distribution_7) }
23.24/3.36	  fresh6(product(sK4_prove_distribution_X, sK6_prove_distribution_E, inverse(sK5_prove_distribution_W)), true, inverse(sK4_prove_distribution_X), inverse(sK5_prove_distribution_W), sK6_prove_distribution_E)
23.24/3.36	= { by axiom 4 (prove_distribution_8) }
23.24/3.36	  fresh6(fresh(true, true, sK4_prove_distribution_X, sK5_prove_distribution_W, sK6_prove_distribution_E, sK6_prove_distribution_E, inverse(sK5_prove_distribution_W)), true, inverse(sK4_prove_distribution_X), inverse(sK5_prove_distribution_W), sK6_prove_distribution_E)
23.24/3.36	= { by axiom 8 (prove_distribution_1) }
23.24/3.36	  fresh6(fresh(product(sK5_prove_distribution_W, inverse(sK5_prove_distribution_W), sK6_prove_distribution_E), true, sK4_prove_distribution_X, sK5_prove_distribution_W, sK6_prove_distribution_E, sK6_prove_distribution_E, inverse(sK5_prove_distribution_W)), true, inverse(sK4_prove_distribution_X), inverse(sK5_prove_distribution_W), sK6_prove_distribution_E)
23.24/3.36	= { by axiom 13 (prove_distribution_8) }
23.24/3.36	  fresh6(fresh3(product(sK6_prove_distribution_E, inverse(sK5_prove_distribution_W), inverse(sK5_prove_distribution_W)), true, sK4_prove_distribution_X, sK5_prove_distribution_W, inverse(sK5_prove_distribution_W), sK6_prove_distribution_E, sK6_prove_distribution_E, inverse(sK5_prove_distribution_W)), true, inverse(sK4_prove_distribution_X), inverse(sK5_prove_distribution_W), sK6_prove_distribution_E)
23.24/3.36	= { by axiom 10 (prove_distribution_5) }
23.24/3.36	  fresh6(fresh3(true, true, sK4_prove_distribution_X, sK5_prove_distribution_W, inverse(sK5_prove_distribution_W), sK6_prove_distribution_E, sK6_prove_distribution_E, inverse(sK5_prove_distribution_W)), true, inverse(sK4_prove_distribution_X), inverse(sK5_prove_distribution_W), sK6_prove_distribution_E)
23.24/3.36	= { by lemma 15 }
23.24/3.36	  fresh6(fresh3(?, ?, sK4_prove_distribution_X, sK5_prove_distribution_W, ?, sK6_prove_distribution_E, sK6_prove_distribution_E, inverse(sK5_prove_distribution_W)), true, inverse(sK4_prove_distribution_X), inverse(sK5_prove_distribution_W), sK6_prove_distribution_E)
23.24/3.36	= { by axiom 6 (prove_distribution_8) }
23.24/3.36	  fresh6(fresh4(product(sK4_prove_distribution_X, sK5_prove_distribution_W, sK6_prove_distribution_E), true, sK4_prove_distribution_X, sK6_prove_distribution_E, inverse(sK5_prove_distribution_W)), true, inverse(sK4_prove_distribution_X), inverse(sK5_prove_distribution_W), sK6_prove_distribution_E)
23.24/3.36	= { by axiom 4 (prove_distribution_8) }
23.24/3.36	  fresh6(fresh4(fresh(true, true, sK4_prove_distribution_X, inverse(sK3_prove_distribution_U), sK2_prove_distribution_V, sK5_prove_distribution_W, sK6_prove_distribution_E), true, sK4_prove_distribution_X, sK6_prove_distribution_E, inverse(sK5_prove_distribution_W)), true, inverse(sK4_prove_distribution_X), inverse(sK5_prove_distribution_W), sK6_prove_distribution_E)
23.24/3.36	= { by axiom 9 (prove_distribution_4) }
23.24/3.36	  fresh6(fresh4(fresh(product(inverse(sK3_prove_distribution_U), inverse(sK2_prove_distribution_V), sK5_prove_distribution_W), true, sK4_prove_distribution_X, inverse(sK3_prove_distribution_U), sK2_prove_distribution_V, sK5_prove_distribution_W, sK6_prove_distribution_E), true, sK4_prove_distribution_X, sK6_prove_distribution_E, inverse(sK5_prove_distribution_W)), true, inverse(sK4_prove_distribution_X), inverse(sK5_prove_distribution_W), sK6_prove_distribution_E)
23.24/3.36	= { by axiom 13 (prove_distribution_8) }
23.24/3.36	  fresh6(fresh4(fresh3(product(sK2_prove_distribution_V, inverse(sK2_prove_distribution_V), sK6_prove_distribution_E), true, sK4_prove_distribution_X, inverse(sK3_prove_distribution_U), inverse(sK2_prove_distribution_V), sK2_prove_distribution_V, sK5_prove_distribution_W, sK6_prove_distribution_E), true, sK4_prove_distribution_X, sK6_prove_distribution_E, inverse(sK5_prove_distribution_W)), true, inverse(sK4_prove_distribution_X), inverse(sK5_prove_distribution_W), sK6_prove_distribution_E)
23.24/3.36	= { by axiom 8 (prove_distribution_1) }
23.24/3.36	  fresh6(fresh4(fresh3(true, true, sK4_prove_distribution_X, inverse(sK3_prove_distribution_U), inverse(sK2_prove_distribution_V), sK2_prove_distribution_V, sK5_prove_distribution_W, sK6_prove_distribution_E), true, sK4_prove_distribution_X, sK6_prove_distribution_E, inverse(sK5_prove_distribution_W)), true, inverse(sK4_prove_distribution_X), inverse(sK5_prove_distribution_W), sK6_prove_distribution_E)
23.24/3.36	= { by lemma 15 }
23.24/3.36	  fresh6(fresh4(fresh3(?, ?, sK4_prove_distribution_X, inverse(sK3_prove_distribution_U), ?, sK2_prove_distribution_V, sK5_prove_distribution_W, sK6_prove_distribution_E), true, sK4_prove_distribution_X, sK6_prove_distribution_E, inverse(sK5_prove_distribution_W)), true, inverse(sK4_prove_distribution_X), inverse(sK5_prove_distribution_W), sK6_prove_distribution_E)
23.24/3.36	= { by axiom 6 (prove_distribution_8) }
23.24/3.36	  fresh6(fresh4(fresh4(product(sK4_prove_distribution_X, inverse(sK3_prove_distribution_U), sK2_prove_distribution_V), true, sK4_prove_distribution_X, sK5_prove_distribution_W, sK6_prove_distribution_E), true, sK4_prove_distribution_X, sK6_prove_distribution_E, inverse(sK5_prove_distribution_W)), true, inverse(sK4_prove_distribution_X), inverse(sK5_prove_distribution_W), sK6_prove_distribution_E)
23.24/3.36	= { by axiom 1 (prove_distribution_7) }
23.24/3.36	  fresh6(fresh4(fresh4(fresh2(true, true, sK2_prove_distribution_V, sK3_prove_distribution_U, inverse(sK3_prove_distribution_U), sK4_prove_distribution_X, sK2_prove_distribution_V), true, sK4_prove_distribution_X, sK5_prove_distribution_W, sK6_prove_distribution_E), true, sK4_prove_distribution_X, sK6_prove_distribution_E, inverse(sK5_prove_distribution_W)), true, inverse(sK4_prove_distribution_X), inverse(sK5_prove_distribution_W), sK6_prove_distribution_E)
23.24/3.36	= { by axiom 7 (prove_distribution) }
23.24/3.36	  fresh6(fresh4(fresh4(fresh2(product(sK2_prove_distribution_V, sK6_prove_distribution_E, sK2_prove_distribution_V), true, sK2_prove_distribution_V, sK3_prove_distribution_U, inverse(sK3_prove_distribution_U), sK4_prove_distribution_X, sK2_prove_distribution_V), true, sK4_prove_distribution_X, sK5_prove_distribution_W, sK6_prove_distribution_E), true, sK4_prove_distribution_X, sK6_prove_distribution_E, inverse(sK5_prove_distribution_W)), true, inverse(sK4_prove_distribution_X), inverse(sK5_prove_distribution_W), sK6_prove_distribution_E)
23.24/3.36	= { by axiom 12 (prove_distribution_7) }
23.24/3.36	  fresh6(fresh4(fresh4(fresh5(product(sK3_prove_distribution_U, inverse(sK3_prove_distribution_U), sK6_prove_distribution_E), true, sK2_prove_distribution_V, sK3_prove_distribution_U, inverse(sK3_prove_distribution_U), sK4_prove_distribution_X, sK6_prove_distribution_E, sK2_prove_distribution_V), true, sK4_prove_distribution_X, sK5_prove_distribution_W, sK6_prove_distribution_E), true, sK4_prove_distribution_X, sK6_prove_distribution_E, inverse(sK5_prove_distribution_W)), true, inverse(sK4_prove_distribution_X), inverse(sK5_prove_distribution_W), sK6_prove_distribution_E)
23.24/3.36	= { by axiom 8 (prove_distribution_1) }
23.24/3.36	  fresh6(fresh4(fresh4(fresh5(true, true, sK2_prove_distribution_V, sK3_prove_distribution_U, inverse(sK3_prove_distribution_U), sK4_prove_distribution_X, sK6_prove_distribution_E, sK2_prove_distribution_V), true, sK4_prove_distribution_X, sK5_prove_distribution_W, sK6_prove_distribution_E), true, sK4_prove_distribution_X, sK6_prove_distribution_E, inverse(sK5_prove_distribution_W)), true, inverse(sK4_prove_distribution_X), inverse(sK5_prove_distribution_W), sK6_prove_distribution_E)
23.24/3.36	= { by lemma 14 }
23.24/3.36	  fresh6(fresh4(fresh4(fresh5(?, ?, sK2_prove_distribution_V, sK3_prove_distribution_U, inverse(sK3_prove_distribution_U), sK4_prove_distribution_X, ?, sK2_prove_distribution_V), true, sK4_prove_distribution_X, sK5_prove_distribution_W, sK6_prove_distribution_E), true, sK4_prove_distribution_X, sK6_prove_distribution_E, inverse(sK5_prove_distribution_W)), true, inverse(sK4_prove_distribution_X), inverse(sK5_prove_distribution_W), sK6_prove_distribution_E)
23.24/3.36	= { by axiom 3 (prove_distribution_7) }
23.24/3.36	  fresh6(fresh4(fresh4(fresh6(product(sK2_prove_distribution_V, sK3_prove_distribution_U, sK4_prove_distribution_X), true, inverse(sK3_prove_distribution_U), sK4_prove_distribution_X, sK2_prove_distribution_V), true, sK4_prove_distribution_X, sK5_prove_distribution_W, sK6_prove_distribution_E), true, sK4_prove_distribution_X, sK6_prove_distribution_E, inverse(sK5_prove_distribution_W)), true, inverse(sK4_prove_distribution_X), inverse(sK5_prove_distribution_W), sK6_prove_distribution_E)
23.24/3.36	= { by axiom 11 (prove_distribution_6) }
23.24/3.36	  fresh6(fresh4(fresh4(fresh6(true, true, inverse(sK3_prove_distribution_U), sK4_prove_distribution_X, sK2_prove_distribution_V), true, sK4_prove_distribution_X, sK5_prove_distribution_W, sK6_prove_distribution_E), true, sK4_prove_distribution_X, sK6_prove_distribution_E, inverse(sK5_prove_distribution_W)), true, inverse(sK4_prove_distribution_X), inverse(sK5_prove_distribution_W), sK6_prove_distribution_E)
23.24/3.36	= { by axiom 2 (prove_distribution_7) }
23.24/3.36	  fresh6(fresh4(fresh4(true, true, sK4_prove_distribution_X, sK5_prove_distribution_W, sK6_prove_distribution_E), true, sK4_prove_distribution_X, sK6_prove_distribution_E, inverse(sK5_prove_distribution_W)), true, inverse(sK4_prove_distribution_X), inverse(sK5_prove_distribution_W), sK6_prove_distribution_E)
23.24/3.36	= { by axiom 5 (prove_distribution_8) }
23.24/3.36	  fresh6(fresh4(true, true, sK4_prove_distribution_X, sK6_prove_distribution_E, inverse(sK5_prove_distribution_W)), true, inverse(sK4_prove_distribution_X), inverse(sK5_prove_distribution_W), sK6_prove_distribution_E)
23.24/3.36	= { by axiom 5 (prove_distribution_8) }
23.24/3.36	  fresh6(true, true, inverse(sK4_prove_distribution_X), inverse(sK5_prove_distribution_W), sK6_prove_distribution_E)
23.24/3.36	= { by axiom 2 (prove_distribution_7) }
23.24/3.36	  true
23.24/3.36	% SZS output end Proof
23.24/3.36	
23.24/3.36	RESULT: Theorem (the conjecture is true).
23.24/3.38	EOF