0.08/0.13	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.08/0.14	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.14/0.35	% Computer   : n014.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   : 180
0.14/0.35	% DateTime   : Thu Aug 29 09:16:39 EDT 2019
0.14/0.35	% CPUTime    : 
1.38/1.55	% SZS status Theorem
1.38/1.55	
1.38/1.55	% SZS output start Proof
1.38/1.55	Take the following subset of the input axioms:
1.38/1.55	  fof(ax14, axiom, ![U, V]: ((issmallestelement_pq(U, V) & contains_pq(U, V)) => V=findmin_pq_res(U, V))).
1.38/1.55	  fof(ax20, axiom, ![U]: ~contains_slb(create_slb, U)).
1.38/1.55	  fof(ax22, axiom, ![U, V]: ~pair_in_list(create_slb, U, V)).
1.38/1.55	  fof(ax38, axiom, ![X, U, V, W, Y]: (strictly_less_than(X, Y) => (check_cpq(triple(U, insert_slb(V, pair(X, Y)), W)) <=> $false))).
1.38/1.55	  fof(ax40, axiom, ![U, V]: (ok(triple(U, V, bad)) <=> $false)).
1.38/1.55	  fof(ax58, axiom, ![U, V]: ((contains_pq(U, V) & issmallestelement_pq(U, V)) <=> pi_sharp_find_min(U, V))).
1.38/1.55	  fof(ax8, axiom, ![U]: ~contains_pq(create_pq, U)).
1.38/1.55	  fof(co4, conjecture, ![U, V, W]: (pi_find_min(triple(U, V, W)) => (?[X]: (findmin_cpq_res(triple(U, V, W))=findmin_pq_res(i(triple(U, V, W)), X) & pi_sharp_find_min(i(triple(U, V, W)), X)) <= phi(findmin_cpq_eff(triple(U, V, W)))))).
1.38/1.55	  fof(main4_l7, lemma, ![U, V, W]: (pi_sharp_find_min(i(triple(U, V, W)), findmin_cpq_res(triple(U, V, W))) <= phi(findmin_cpq_eff(triple(U, V, W))))).
1.38/1.55	  fof(stricly_smaller_definition, axiom, ![U, V]: (strictly_less_than(U, V) <=> (less_than(U, V) & ~less_than(V, U)))).
1.38/1.55	
1.38/1.55	Now clausify the problem and encode Horn clauses using encoding 3 of
1.38/1.55	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
1.38/1.55	We repeatedly replace C & s=t => u=v by the two clauses:
1.38/1.55	  fresh(y, y, x1...xn) = u
1.38/1.55	  C => fresh(s, t, x1...xn) = v
1.38/1.55	where fresh is a fresh function symbol and x1..xn are the free
1.38/1.55	variables of u and v.
1.38/1.55	A predicate p(X) is encoded as p(X)=true (this is sound, because the
1.38/1.55	input problem has no model of domain size 1).
1.38/1.55	
1.38/1.55	The encoding turns the above axioms into the following unit equations and goals:
1.38/1.55	
1.38/1.55	Axiom 1 (ax14): fresh(X, X, Y, Z) = Z.
1.38/1.55	Axiom 2 (ax14): fresh50(X, X, Y, Z) = findmin_pq_res(Y, Z).
1.38/1.55	Axiom 3 (ax58_1): fresh22(X, X, Y, Z) = true2.
1.38/1.55	Axiom 4 (ax58_2): fresh21(X, X, Y, Z) = true2.
1.38/1.55	Axiom 5 (main4_l7): fresh7(X, X, Y, Z, W) = true2.
1.38/1.55	Axiom 6 (ax14): fresh50(contains_pq(X, Y), true2, X, Y) = fresh(issmallestelement_pq(X, Y), true2, X, Y).
1.38/1.55	Axiom 7 (ax58_2): fresh21(pi_sharp_find_min(X, Y), true2, X, Y) = contains_pq(X, Y).
1.38/1.55	Axiom 8 (ax58_1): fresh22(pi_sharp_find_min(X, Y), true2, X, Y) = issmallestelement_pq(X, Y).
1.38/1.55	Axiom 9 (main4_l7): fresh7(phi(findmin_cpq_eff(triple(X, Y, Z))), true2, X, Y, Z) = pi_sharp_find_min(i(triple(X, Y, Z)), findmin_cpq_res(triple(X, Y, Z))).
1.38/1.55	Axiom 10 (co4_1): phi(findmin_cpq_eff(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W))) = true2.
1.38/1.55	
1.38/1.55	Lemma 11: pi_sharp_find_min(i(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W)), findmin_cpq_res(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W))) = true2.
1.38/1.55	Proof:
1.38/1.55	  pi_sharp_find_min(i(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W)), findmin_cpq_res(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W)))
1.38/1.55	= { by axiom 9 (main4_l7) }
1.38/1.55	  fresh7(phi(findmin_cpq_eff(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W))), true2, sK3_co4_U, sK2_co4_V, sK1_co4_W)
1.38/1.55	= { by axiom 10 (co4_1) }
1.38/1.55	  fresh7(true2, true2, sK3_co4_U, sK2_co4_V, sK1_co4_W)
1.38/1.55	= { by axiom 5 (main4_l7) }
1.38/1.55	  true2
1.38/1.55	
1.38/1.55	Goal 1 (co4_2): tuple2(findmin_cpq_res(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W)), pi_sharp_find_min(i(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W)), X)) = tuple2(findmin_pq_res(i(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W)), X), true2).
1.38/1.55	The goal is true when:
1.38/1.55	  X = findmin_cpq_res(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W))
1.38/1.55	
1.38/1.55	Proof:
1.38/1.55	  tuple2(findmin_cpq_res(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W)), pi_sharp_find_min(i(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W)), findmin_cpq_res(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W))))
1.38/1.55	= { by axiom 1 (ax14) }
1.38/1.55	  tuple2(fresh(true2, true2, i(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W)), findmin_cpq_res(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W))), pi_sharp_find_min(i(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W)), findmin_cpq_res(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W))))
1.38/1.55	= { by axiom 3 (ax58_1) }
1.38/1.55	  tuple2(fresh(fresh22(true2, true2, i(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W)), findmin_cpq_res(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W))), true2, i(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W)), findmin_cpq_res(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W))), pi_sharp_find_min(i(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W)), findmin_cpq_res(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W))))
1.38/1.55	= { by lemma 11 }
1.38/1.55	  tuple2(fresh(fresh22(pi_sharp_find_min(i(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W)), findmin_cpq_res(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W))), true2, i(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W)), findmin_cpq_res(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W))), true2, i(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W)), findmin_cpq_res(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W))), pi_sharp_find_min(i(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W)), findmin_cpq_res(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W))))
1.38/1.55	= { by axiom 8 (ax58_1) }
1.38/1.55	  tuple2(fresh(issmallestelement_pq(i(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W)), findmin_cpq_res(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W))), true2, i(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W)), findmin_cpq_res(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W))), pi_sharp_find_min(i(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W)), findmin_cpq_res(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W))))
1.38/1.55	= { by axiom 6 (ax14) }
1.38/1.55	  tuple2(fresh50(contains_pq(i(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W)), findmin_cpq_res(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W))), true2, i(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W)), findmin_cpq_res(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W))), pi_sharp_find_min(i(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W)), findmin_cpq_res(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W))))
1.38/1.55	= { by axiom 7 (ax58_2) }
1.38/1.55	  tuple2(fresh50(fresh21(pi_sharp_find_min(i(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W)), findmin_cpq_res(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W))), true2, i(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W)), findmin_cpq_res(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W))), true2, i(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W)), findmin_cpq_res(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W))), pi_sharp_find_min(i(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W)), findmin_cpq_res(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W))))
1.38/1.55	= { by lemma 11 }
1.38/1.55	  tuple2(fresh50(fresh21(true2, true2, i(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W)), findmin_cpq_res(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W))), true2, i(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W)), findmin_cpq_res(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W))), pi_sharp_find_min(i(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W)), findmin_cpq_res(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W))))
1.38/1.55	= { by axiom 4 (ax58_2) }
1.38/1.55	  tuple2(fresh50(true2, true2, i(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W)), findmin_cpq_res(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W))), pi_sharp_find_min(i(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W)), findmin_cpq_res(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W))))
1.38/1.55	= { by axiom 2 (ax14) }
1.38/1.55	  tuple2(findmin_pq_res(i(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W)), findmin_cpq_res(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W))), pi_sharp_find_min(i(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W)), findmin_cpq_res(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W))))
1.38/1.55	= { by lemma 11 }
1.38/1.55	  tuple2(findmin_pq_res(i(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W)), findmin_cpq_res(triple(sK3_co4_U, sK2_co4_V, sK1_co4_W))), true2)
1.38/1.55	% SZS output end Proof
1.38/1.55	
1.38/1.55	RESULT: Theorem (the conjecture is true).
1.38/1.56	EOF