0.03/0.13	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.14/0.13	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.14/0.35	% Computer   : n023.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 13:36:52 EDT 2019
0.14/0.35	% CPUTime    : 
179.37/179.68	% SZS status Theorem
179.37/179.68	
179.37/179.68	% SZS output start Proof
179.37/179.68	Take the following subset of the input axioms:
179.37/179.68	  fof(conj_0, conjecture, c_Hoare__Mirabelle_Ohoare__derivs(t_a, hAPP(hAPP(c_Set_Oinsert(tc_Hoare__Mirabelle_Otriple(t_a)), v_t), c_Orderings_Obot__class_Obot(tc_fun(tc_Hoare__Mirabelle_Otriple(t_a), tc_HOL_Obool))), hAPP(hAPP(c_Set_Oinsert(tc_Hoare__Mirabelle_Otriple(t_a)), v_t), c_Orderings_Obot__class_Obot(tc_fun(tc_Hoare__Mirabelle_Otriple(t_a), tc_HOL_Obool))))).
179.37/179.68	  fof(fact_Collect__def, axiom, ![V_P_2, T_b]: hAPP(c_Set_OCollect(T_b), V_P_2)=V_P_2).
179.37/179.68	  fof(fact_asm, axiom, ![T_b, V_ts_2, V_G_2]: (c_Hoare__Mirabelle_Ohoare__derivs(T_b, V_G_2, V_ts_2) <= hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_Hoare__Mirabelle_Otriple(T_b), tc_HOL_Obool)), V_ts_2), V_G_2)))).
179.37/179.68	  fof(fact_equalityD2, axiom, ![T_b, V_A_2, V_B_2]: (V_B_2=V_A_2 => hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_b, tc_HOL_Obool)), V_B_2), V_A_2)))).
179.37/179.68	  fof(fact_singleton__conv2, axiom, ![T_b, V_a_2]: hAPP(c_Set_OCollect(T_b), hAPP(c_fequal, V_a_2))=hAPP(hAPP(c_Set_Oinsert(T_b), V_a_2), c_Orderings_Obot__class_Obot(tc_fun(T_b, tc_HOL_Obool)))).
179.37/179.68	
179.37/179.68	Now clausify the problem and encode Horn clauses using encoding 3 of
179.37/179.68	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
179.37/179.68	We repeatedly replace C & s=t => u=v by the two clauses:
179.37/179.68	  fresh(y, y, x1...xn) = u
179.37/179.68	  C => fresh(s, t, x1...xn) = v
179.37/179.68	where fresh is a fresh function symbol and x1..xn are the free
179.37/179.68	variables of u and v.
179.37/179.68	A predicate p(X) is encoded as p(X)=true (this is sound, because the
179.37/179.68	input problem has no model of domain size 1).
179.37/179.68	
179.37/179.68	The encoding turns the above axioms into the following unit equations and goals:
179.37/179.68	
179.37/179.68	Axiom 1 (fact_Collect__conv__if2_1): fresh4462(X, X, Y, Z, W) = hAPP(hAPP(c_Set_Oinsert(Y), Z), c_Orderings_Obot__class_Obot(tc_fun(Y, tc_HOL_Obool))).
179.37/179.68	Axiom 2 (fact_abs__le__D1): fresh4036(X, X, Y, Z, W) = hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(W), Z), Y)).
179.37/179.68	Axiom 3 (fact_asm): fresh3835(X, X, Y, Z, W) = true2.
179.37/179.68	Axiom 4 (fact_atMost__Int__atLeast): fresh3812(X, X, Y, Z) = hAPP(hAPP(c_Set_Oinsert(Z), Y), c_Orderings_Obot__class_Obot(tc_fun(Z, tc_HOL_Obool))).
179.37/179.68	Axiom 5 (fact_le__fun__def): fresh2679(X, X, Y, Z, W, V) = hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(W, V)), Z), Y)).
179.37/179.68	Axiom 6 (fact_order__fun_I1_J_1): fresh5427(X, X, Y, Z, W, V) = hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(W, V)), Z), Y)).
179.37/179.68	Axiom 7 (fact_order__fun_I2_J): fresh6743(X, X, Y, Z, W, V) = hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(W, V)), Z), Y)).
179.37/179.68	Axiom 8 (fact_Collect__def): hAPP(c_Set_OCollect(X), Y) = Y.
179.37/179.68	Axiom 9 (fact_equalityD2): hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(X, tc_HOL_Obool)), Y), Y)) = true2.
179.37/179.68	Axiom 10 (fact_asm): fresh3835(hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_Hoare__Mirabelle_Otriple(X), tc_HOL_Obool)), Y), Z)), true2, Z, Y, X) = c_Hoare__Mirabelle_Ohoare__derivs(X, Z, Y).
179.37/179.68	Axiom 11 (fact_singleton__conv2): hAPP(c_Set_OCollect(X), hAPP(c_fequal, Y)) = hAPP(hAPP(c_Set_Oinsert(X), Y), c_Orderings_Obot__class_Obot(tc_fun(X, tc_HOL_Obool))).
179.37/179.68	
179.37/179.68	Lemma 12: fresh4462(?, ?, Z, Y, ?) = fresh3812(?, ?, Y, Z).
179.37/179.68	Proof:
179.37/179.68	  fresh4462(?, ?, Z, Y, ?)
179.37/179.68	= { by axiom 1 (fact_Collect__conv__if2_1) }
179.37/179.68	  hAPP(hAPP(c_Set_Oinsert(Z), Y), c_Orderings_Obot__class_Obot(tc_fun(Z, tc_HOL_Obool)))
179.37/179.68	= { by axiom 4 (fact_atMost__Int__atLeast) }
179.37/179.68	  fresh3812(?, ?, Y, Z)
179.37/179.68	
179.37/179.68	Lemma 13: hAPP(hAPP(c_Set_Oinsert(Y), Z), c_Orderings_Obot__class_Obot(tc_fun(Y, tc_HOL_Obool))) = hAPP(c_fequal, Z).
179.37/179.68	Proof:
179.37/179.68	  hAPP(hAPP(c_Set_Oinsert(Y), Z), c_Orderings_Obot__class_Obot(tc_fun(Y, tc_HOL_Obool)))
179.37/179.68	= { by axiom 1 (fact_Collect__conv__if2_1) }
179.37/179.68	  fresh4462(?, ?, Y, Z, ?)
179.37/179.68	= { by axiom 1 (fact_Collect__conv__if2_1) }
179.37/179.68	  hAPP(hAPP(c_Set_Oinsert(Y), Z), c_Orderings_Obot__class_Obot(tc_fun(Y, tc_HOL_Obool)))
179.37/179.68	= { by axiom 11 (fact_singleton__conv2) }
179.37/179.68	  hAPP(c_Set_OCollect(Y), hAPP(c_fequal, Z))
179.37/179.68	= { by axiom 8 (fact_Collect__def) }
179.37/179.68	  hAPP(c_fequal, Z)
179.37/179.68	
179.37/179.68	Lemma 14: fresh4036(?, ?, Y, Z, tc_fun(W, V)) = fresh2679(?, ?, Y, Z, W, V).
179.37/179.68	Proof:
179.37/179.68	  fresh4036(?, ?, Y, Z, tc_fun(W, V))
179.37/179.68	= { by axiom 2 (fact_abs__le__D1) }
179.37/179.68	  hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(W, V)), Z), Y))
179.37/179.68	= { by axiom 5 (fact_le__fun__def) }
179.37/179.68	  fresh2679(?, ?, Y, Z, W, V)
179.37/179.68	
179.37/179.68	Lemma 15: fresh2679(?, ?, Y, Z, W, V) = fresh5427(?, ?, Y, Z, W, V).
179.37/179.68	Proof:
179.37/179.68	  fresh2679(?, ?, Y, Z, W, V)
179.37/179.68	= { by lemma 14 }
179.37/179.68	  fresh4036(?, ?, Y, Z, tc_fun(W, V))
179.37/179.68	= { by axiom 2 (fact_abs__le__D1) }
179.37/179.68	  hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(W, V)), Z), Y))
179.37/179.68	= { by axiom 6 (fact_order__fun_I1_J_1) }
179.37/179.68	  fresh5427(?, ?, Y, Z, W, V)
179.37/179.68	
179.37/179.68	Lemma 16: fresh5427(?, ?, Y, Z, W, V) = fresh6743(?, ?, Y, Z, W, V).
179.37/179.68	Proof:
179.37/179.68	  fresh5427(?, ?, Y, Z, W, V)
179.37/179.68	= { by lemma 15 }
179.37/179.68	  fresh2679(?, ?, Y, Z, W, V)
179.37/179.68	= { by lemma 14 }
179.37/179.68	  fresh4036(?, ?, Y, Z, tc_fun(W, V))
179.37/179.68	= { by axiom 2 (fact_abs__le__D1) }
179.37/179.68	  hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(W, V)), Z), Y))
179.37/179.68	= { by axiom 7 (fact_order__fun_I2_J) }
179.37/179.68	  fresh6743(?, ?, Y, Z, W, V)
179.37/179.68	
179.37/179.68	Goal 1 (conj_0): c_Hoare__Mirabelle_Ohoare__derivs(t_a, hAPP(hAPP(c_Set_Oinsert(tc_Hoare__Mirabelle_Otriple(t_a)), v_t), c_Orderings_Obot__class_Obot(tc_fun(tc_Hoare__Mirabelle_Otriple(t_a), tc_HOL_Obool))), hAPP(hAPP(c_Set_Oinsert(tc_Hoare__Mirabelle_Otriple(t_a)), v_t), c_Orderings_Obot__class_Obot(tc_fun(tc_Hoare__Mirabelle_Otriple(t_a), tc_HOL_Obool)))) = true2.
179.37/179.68	Proof:
179.37/179.68	  c_Hoare__Mirabelle_Ohoare__derivs(t_a, hAPP(hAPP(c_Set_Oinsert(tc_Hoare__Mirabelle_Otriple(t_a)), v_t), c_Orderings_Obot__class_Obot(tc_fun(tc_Hoare__Mirabelle_Otriple(t_a), tc_HOL_Obool))), hAPP(hAPP(c_Set_Oinsert(tc_Hoare__Mirabelle_Otriple(t_a)), v_t), c_Orderings_Obot__class_Obot(tc_fun(tc_Hoare__Mirabelle_Otriple(t_a), tc_HOL_Obool))))
179.37/179.68	= { by lemma 13 }
179.37/179.68	  c_Hoare__Mirabelle_Ohoare__derivs(t_a, hAPP(c_fequal, v_t), hAPP(hAPP(c_Set_Oinsert(tc_Hoare__Mirabelle_Otriple(t_a)), v_t), c_Orderings_Obot__class_Obot(tc_fun(tc_Hoare__Mirabelle_Otriple(t_a), tc_HOL_Obool))))
179.37/179.68	= { by lemma 13 }
179.37/179.68	  c_Hoare__Mirabelle_Ohoare__derivs(t_a, hAPP(c_fequal, v_t), hAPP(c_fequal, v_t))
179.37/179.68	= { by axiom 10 (fact_asm) }
179.37/179.68	  fresh3835(hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_Hoare__Mirabelle_Otriple(t_a), tc_HOL_Obool)), hAPP(c_fequal, v_t)), hAPP(c_fequal, v_t))), true2, hAPP(c_fequal, v_t), hAPP(c_fequal, v_t), t_a)
179.37/179.68	= { by axiom 9 (fact_equalityD2) }
179.37/179.68	  fresh3835(true2, true2, hAPP(c_fequal, v_t), hAPP(c_fequal, v_t), t_a)
179.37/179.68	= { by axiom 3 (fact_asm) }
179.37/179.68	  true2
179.37/179.68	% SZS output end Proof
179.37/179.68	
179.37/179.68	RESULT: Theorem (the conjecture is true).
179.62/179.90	EOF