%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SEU344+1 : TPTP v9.0.0. Released v3.3.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n008.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Sat Jun 21 05:27:39 AM UTC 2025
% Result : Theorem 119.50s 119.65s
% Output : Proof 119.80s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.04/0.13 % Problem : SEU344+1 : TPTP v9.0.0. Released v3.3.0.
% 0.04/0.13 % Command : run_E %s %d THM
% 0.12/0.35 % Computer : n008.cluster.edu
% 0.12/0.35 % Model : x86_64 x86_64
% 0.12/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.12/0.35 % Memory : 8042.1875MB
% 0.12/0.35 % OS : Linux 3.10.0-693.el7.x86_64
% 0.12/0.35 % CPULimit : 300
% 0.12/0.35 % WCLimit : 300
% 0.12/0.35 % DateTime : Fri Jun 20 06:50:03 EDT 2025
% 0.12/0.35 % CPUTime :
% 119.50/119.65 % SZS status Theorem
% 119.50/119.65 % SZS output start Proof
% 119.50/119.65 tff(below_type, type, (
% 119.50/119.65 below: ( $i * $i * $i ) > $o)).
% 119.50/119.65 tff(set_union2_type, type, (
% 119.50/119.65 set_union2: ( $i * $i ) > $i)).
% 119.50/119.65 tff(tptp_fun_C_17_type, type, (
% 119.50/119.65 tptp_fun_C_17: $i)).
% 119.50/119.65 tff(tptp_fun_B_15_type, type, (
% 119.50/119.65 tptp_fun_B_15: $i)).
% 119.50/119.65 tff(set_intersection2_type, type, (
% 119.50/119.65 set_intersection2: ( $i * $i ) > $i)).
% 119.50/119.65 tff(boole_lattice_type, type, (
% 119.50/119.65 boole_lattice: $i > $i)).
% 119.50/119.65 tff(tptp_fun_A_16_type, type, (
% 119.50/119.65 tptp_fun_A_16: $i)).
% 119.50/119.65 tff(subset_type, type, (
% 119.50/119.65 subset: ( $i * $i ) > $o)).
% 119.50/119.65 tff(join_type, type, (
% 119.50/119.65 join: ( $i * $i * $i ) > $i)).
% 119.50/119.65 tff(meet_type, type, (
% 119.50/119.65 meet: ( $i * $i * $i ) > $i)).
% 119.50/119.65 tff(element_type, type, (
% 119.50/119.65 element: ( $i * $i ) > $o)).
% 119.50/119.65 tff(the_carrier_type, type, (
% 119.50/119.65 the_carrier: $i > $i)).
% 119.50/119.65 tff(empty_carrier_type, type, (
% 119.50/119.65 empty_carrier: $i > $o)).
% 119.50/119.65 tff(strict_latt_str_type, type, (
% 119.50/119.65 strict_latt_str: $i > $o)).
% 119.50/119.65 tff(join_semilatt_str_type, type, (
% 119.50/119.65 join_semilatt_str: $i > $o)).
% 119.50/119.65 tff(latt_str_of_type, type, (
% 119.50/119.65 latt_str_of: ( $i * $i * $i ) > $i)).
% 119.50/119.65 tff(the_L_meet_type, type, (
% 119.50/119.65 the_L_meet: $i > $i)).
% 119.50/119.65 tff(the_L_join_type, type, (
% 119.50/119.65 the_L_join: $i > $i)).
% 119.50/119.65 tff(latt_str_type, type, (
% 119.50/119.65 latt_str: $i > $o)).
% 119.50/119.65 tff(meet_semilatt_str_type, type, (
% 119.50/119.65 meet_semilatt_str: $i > $o)).
% 119.50/119.65 tff(1,plain,
% 119.50/119.65 ((~![A: $i, B: $i] : ((~element(B, the_carrier(boole_lattice(A)))) | ![C: $i] : ((~element(C, the_carrier(boole_lattice(A)))) | (below(boole_lattice(A), B, C) <=> subset(B, C))))) <=> (~![A: $i, B: $i] : ((~element(B, the_carrier(boole_lattice(A)))) | ![C: $i] : ((~element(C, the_carrier(boole_lattice(A)))) | (below(boole_lattice(A), B, C) <=> subset(B, C)))))),
% 119.50/119.65 inference(rewrite,[status(thm)],[])).
% 119.50/119.65 tff(2,plain,
% 119.50/119.65 ((~![A: $i, B: $i] : (element(B, the_carrier(boole_lattice(A))) => ![C: $i] : (element(C, the_carrier(boole_lattice(A))) => (below(boole_lattice(A), B, C) <=> subset(B, C))))) <=> (~![A: $i, B: $i] : ((~element(B, the_carrier(boole_lattice(A)))) | ![C: $i] : ((~element(C, the_carrier(boole_lattice(A)))) | (below(boole_lattice(A), B, C) <=> subset(B, C)))))),
% 119.50/119.65 inference(rewrite,[status(thm)],[])).
% 119.50/119.65 tff(3,axiom,(~![A: $i, B: $i] : (element(B, the_carrier(boole_lattice(A))) => ![C: $i] : (element(C, the_carrier(boole_lattice(A))) => (below(boole_lattice(A), B, C) <=> subset(B, C))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','t2_lattice3')).
% 119.50/119.65 tff(4,plain,
% 119.50/119.65 (~![A: $i, B: $i] : ((~element(B, the_carrier(boole_lattice(A)))) | ![C: $i] : ((~element(C, the_carrier(boole_lattice(A)))) | (below(boole_lattice(A), B, C) <=> subset(B, C))))),
% 119.50/119.65 inference(modus_ponens,[status(thm)],[3, 2])).
% 119.50/119.65 tff(5,plain,
% 119.50/119.65 (~![A: $i, B: $i] : ((~element(B, the_carrier(boole_lattice(A)))) | ![C: $i] : ((~element(C, the_carrier(boole_lattice(A)))) | (below(boole_lattice(A), B, C) <=> subset(B, C))))),
% 119.50/119.65 inference(modus_ponens,[status(thm)],[4, 1])).
% 119.50/119.65 tff(6,plain,
% 119.50/119.65 (~![A: $i, B: $i] : ((~element(B, the_carrier(boole_lattice(A)))) | ![C: $i] : ((~element(C, the_carrier(boole_lattice(A)))) | (below(boole_lattice(A), B, C) <=> subset(B, C))))),
% 119.50/119.65 inference(modus_ponens,[status(thm)],[5, 1])).
% 119.50/119.65 tff(7,plain,
% 119.50/119.65 (~![A: $i, B: $i] : ((~element(B, the_carrier(boole_lattice(A)))) | ![C: $i] : ((~element(C, the_carrier(boole_lattice(A)))) | (below(boole_lattice(A), B, C) <=> subset(B, C))))),
% 119.50/119.65 inference(modus_ponens,[status(thm)],[6, 1])).
% 119.50/119.65 tff(8,plain,
% 119.50/119.65 (element(B!15, the_carrier(boole_lattice(A!16)))),
% 119.50/119.65 inference(and_elim,[status(thm)],[7])).
% 119.50/119.65 tff(9,plain,
% 119.50/119.65 (^[A: $i, B: $i] : refl(((~element(B, the_carrier(boole_lattice(A)))) | ![C: $i] : ((~element(C, the_carrier(boole_lattice(A)))) | (~((~(join(boole_lattice(A), B, C) = set_union2(B, C))) | (~(meet(boole_lattice(A), B, C) = set_intersection2(B, C))))))) <=> ((~element(B, the_carrier(boole_lattice(A)))) | ![C: $i] : ((~element(C, the_carrier(boole_lattice(A)))) | (~((~(join(boole_lattice(A), B, C) = set_union2(B, C))) | (~(meet(boole_lattice(A), B, C) = set_intersection2(B, C))))))))),
% 119.50/119.65 inference(bind,[status(th)],[])).
% 119.50/119.65 tff(10,plain,
% 119.50/119.65 (![A: $i, B: $i] : ((~element(B, the_carrier(boole_lattice(A)))) | ![C: $i] : ((~element(C, the_carrier(boole_lattice(A)))) | (~((~(join(boole_lattice(A), B, C) = set_union2(B, C))) | (~(meet(boole_lattice(A), B, C) = set_intersection2(B, C))))))) <=> ![A: $i, B: $i] : ((~element(B, the_carrier(boole_lattice(A)))) | ![C: $i] : ((~element(C, the_carrier(boole_lattice(A)))) | (~((~(join(boole_lattice(A), B, C) = set_union2(B, C))) | (~(meet(boole_lattice(A), B, C) = set_intersection2(B, C)))))))),
% 119.50/119.65 inference(quant_intro,[status(thm)],[9])).
% 119.50/119.65 tff(11,plain,
% 119.50/119.65 (^[A: $i, B: $i] : rewrite(((~element(B, the_carrier(boole_lattice(A)))) | ![C: $i] : ((~element(C, the_carrier(boole_lattice(A)))) | (~((~(join(boole_lattice(A), B, C) = set_union2(B, C))) | (~(meet(boole_lattice(A), B, C) = set_intersection2(B, C))))))) <=> ((~element(B, the_carrier(boole_lattice(A)))) | ![C: $i] : ((~element(C, the_carrier(boole_lattice(A)))) | (~((~(join(boole_lattice(A), B, C) = set_union2(B, C))) | (~(meet(boole_lattice(A), B, C) = set_intersection2(B, C))))))))),
% 119.50/119.65 inference(bind,[status(th)],[])).
% 119.50/119.65 tff(12,plain,
% 119.50/119.65 (![A: $i, B: $i] : ((~element(B, the_carrier(boole_lattice(A)))) | ![C: $i] : ((~element(C, the_carrier(boole_lattice(A)))) | (~((~(join(boole_lattice(A), B, C) = set_union2(B, C))) | (~(meet(boole_lattice(A), B, C) = set_intersection2(B, C))))))) <=> ![A: $i, B: $i] : ((~element(B, the_carrier(boole_lattice(A)))) | ![C: $i] : ((~element(C, the_carrier(boole_lattice(A)))) | (~((~(join(boole_lattice(A), B, C) = set_union2(B, C))) | (~(meet(boole_lattice(A), B, C) = set_intersection2(B, C)))))))),
% 119.50/119.65 inference(quant_intro,[status(thm)],[11])).
% 119.50/119.65 tff(13,plain,
% 119.50/119.65 (![A: $i, B: $i] : ((~element(B, the_carrier(boole_lattice(A)))) | ![C: $i] : ((~element(C, the_carrier(boole_lattice(A)))) | (~((~(join(boole_lattice(A), B, C) = set_union2(B, C))) | (~(meet(boole_lattice(A), B, C) = set_intersection2(B, C))))))) <=> ![A: $i, B: $i] : ((~element(B, the_carrier(boole_lattice(A)))) | ![C: $i] : ((~element(C, the_carrier(boole_lattice(A)))) | (~((~(join(boole_lattice(A), B, C) = set_union2(B, C))) | (~(meet(boole_lattice(A), B, C) = set_intersection2(B, C)))))))),
% 119.50/119.65 inference(transitivity,[status(thm)],[12, 10])).
% 119.50/119.65 tff(14,plain,
% 119.50/119.65 (^[A: $i, B: $i] : rewrite(((~element(B, the_carrier(boole_lattice(A)))) | ![C: $i] : ((~element(C, the_carrier(boole_lattice(A)))) | ((join(boole_lattice(A), B, C) = set_union2(B, C)) & (meet(boole_lattice(A), B, C) = set_intersection2(B, C))))) <=> ((~element(B, the_carrier(boole_lattice(A)))) | ![C: $i] : ((~element(C, the_carrier(boole_lattice(A)))) | (~((~(join(boole_lattice(A), B, C) = set_union2(B, C))) | (~(meet(boole_lattice(A), B, C) = set_intersection2(B, C))))))))),
% 119.50/119.65 inference(bind,[status(th)],[])).
% 119.50/119.65 tff(15,plain,
% 119.50/119.65 (![A: $i, B: $i] : ((~element(B, the_carrier(boole_lattice(A)))) | ![C: $i] : ((~element(C, the_carrier(boole_lattice(A)))) | ((join(boole_lattice(A), B, C) = set_union2(B, C)) & (meet(boole_lattice(A), B, C) = set_intersection2(B, C))))) <=> ![A: $i, B: $i] : ((~element(B, the_carrier(boole_lattice(A)))) | ![C: $i] : ((~element(C, the_carrier(boole_lattice(A)))) | (~((~(join(boole_lattice(A), B, C) = set_union2(B, C))) | (~(meet(boole_lattice(A), B, C) = set_intersection2(B, C)))))))),
% 119.50/119.65 inference(quant_intro,[status(thm)],[14])).
% 119.50/119.65 tff(16,plain,
% 119.50/119.65 (![A: $i, B: $i] : ((~element(B, the_carrier(boole_lattice(A)))) | ![C: $i] : ((~element(C, the_carrier(boole_lattice(A)))) | ((join(boole_lattice(A), B, C) = set_union2(B, C)) & (meet(boole_lattice(A), B, C) = set_intersection2(B, C))))) <=> ![A: $i, B: $i] : ((~element(B, the_carrier(boole_lattice(A)))) | ![C: $i] : ((~element(C, the_carrier(boole_lattice(A)))) | ((join(boole_lattice(A), B, C) = set_union2(B, C)) & (meet(boole_lattice(A), B, C) = set_intersection2(B, C)))))),
% 119.50/119.65 inference(rewrite,[status(thm)],[])).
% 119.50/119.65 tff(17,plain,
% 119.50/119.65 (^[A: $i, B: $i] : trans(monotonicity(quant_intro(proof_bind(^[C: $i] : rewrite((element(C, the_carrier(boole_lattice(A))) => ((join(boole_lattice(A), B, C) = set_union2(B, C)) & (meet(boole_lattice(A), B, C) = set_intersection2(B, C)))) <=> ((~element(C, the_carrier(boole_lattice(A)))) | ((join(boole_lattice(A), B, C) = set_union2(B, C)) & (meet(boole_lattice(A), B, C) = set_intersection2(B, C)))))), (![C: $i] : (element(C, the_carrier(boole_lattice(A))) => ((join(boole_lattice(A), B, C) = set_union2(B, C)) & (meet(boole_lattice(A), B, C) = set_intersection2(B, C)))) <=> ![C: $i] : ((~element(C, the_carrier(boole_lattice(A)))) | ((join(boole_lattice(A), B, C) = set_union2(B, C)) & (meet(boole_lattice(A), B, C) = set_intersection2(B, C)))))), ((element(B, the_carrier(boole_lattice(A))) => ![C: $i] : (element(C, the_carrier(boole_lattice(A))) => ((join(boole_lattice(A), B, C) = set_union2(B, C)) & (meet(boole_lattice(A), B, C) = set_intersection2(B, C))))) <=> (element(B, the_carrier(boole_lattice(A))) => ![C: $i] : ((~element(C, the_carrier(boole_lattice(A)))) | ((join(boole_lattice(A), B, C) = set_union2(B, C)) & (meet(boole_lattice(A), B, C) = set_intersection2(B, C))))))), rewrite((element(B, the_carrier(boole_lattice(A))) => ![C: $i] : ((~element(C, the_carrier(boole_lattice(A)))) | ((join(boole_lattice(A), B, C) = set_union2(B, C)) & (meet(boole_lattice(A), B, C) = set_intersection2(B, C))))) <=> ((~element(B, the_carrier(boole_lattice(A)))) | ![C: $i] : ((~element(C, the_carrier(boole_lattice(A)))) | ((join(boole_lattice(A), B, C) = set_union2(B, C)) & (meet(boole_lattice(A), B, C) = set_intersection2(B, C)))))), ((element(B, the_carrier(boole_lattice(A))) => ![C: $i] : (element(C, the_carrier(boole_lattice(A))) => ((join(boole_lattice(A), B, C) = set_union2(B, C)) & (meet(boole_lattice(A), B, C) = set_intersection2(B, C))))) <=> ((~element(B, the_carrier(boole_lattice(A)))) | ![C: $i] : ((~element(C, the_carrier(boole_lattice(A)))) | ((join(boole_lattice(A), B, C) = set_union2(B, C)) & (meet(boole_lattice(A), B, C) = set_intersection2(B, C)))))))),
% 119.50/119.65 inference(bind,[status(th)],[])).
% 119.50/119.65 tff(18,plain,
% 119.50/119.65 (![A: $i, B: $i] : (element(B, the_carrier(boole_lattice(A))) => ![C: $i] : (element(C, the_carrier(boole_lattice(A))) => ((join(boole_lattice(A), B, C) = set_union2(B, C)) & (meet(boole_lattice(A), B, C) = set_intersection2(B, C))))) <=> ![A: $i, B: $i] : ((~element(B, the_carrier(boole_lattice(A)))) | ![C: $i] : ((~element(C, the_carrier(boole_lattice(A)))) | ((join(boole_lattice(A), B, C) = set_union2(B, C)) & (meet(boole_lattice(A), B, C) = set_intersection2(B, C)))))),
% 119.50/119.65 inference(quant_intro,[status(thm)],[17])).
% 119.50/119.65 tff(19,axiom,(![A: $i, B: $i] : (element(B, the_carrier(boole_lattice(A))) => ![C: $i] : (element(C, the_carrier(boole_lattice(A))) => ((join(boole_lattice(A), B, C) = set_union2(B, C)) & (meet(boole_lattice(A), B, C) = set_intersection2(B, C)))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','t1_lattice3')).
% 119.50/119.65 tff(20,plain,
% 119.50/119.65 (![A: $i, B: $i] : ((~element(B, the_carrier(boole_lattice(A)))) | ![C: $i] : ((~element(C, the_carrier(boole_lattice(A)))) | ((join(boole_lattice(A), B, C) = set_union2(B, C)) & (meet(boole_lattice(A), B, C) = set_intersection2(B, C)))))),
% 119.50/119.65 inference(modus_ponens,[status(thm)],[19, 18])).
% 119.50/119.65 tff(21,plain,
% 119.50/119.65 (![A: $i, B: $i] : ((~element(B, the_carrier(boole_lattice(A)))) | ![C: $i] : ((~element(C, the_carrier(boole_lattice(A)))) | ((join(boole_lattice(A), B, C) = set_union2(B, C)) & (meet(boole_lattice(A), B, C) = set_intersection2(B, C)))))),
% 119.50/119.65 inference(modus_ponens,[status(thm)],[20, 16])).
% 119.50/119.65 tff(22,plain,(
% 119.50/119.65 ![A: $i, B: $i] : ((~element(B, the_carrier(boole_lattice(A)))) | ![C: $i] : ((~element(C, the_carrier(boole_lattice(A)))) | ((join(boole_lattice(A), B, C) = set_union2(B, C)) & (meet(boole_lattice(A), B, C) = set_intersection2(B, C)))))),
% 119.50/119.65 inference(skolemize,[status(sab)],[21])).
% 119.50/119.65 tff(23,plain,
% 119.50/119.65 (![A: $i, B: $i] : ((~element(B, the_carrier(boole_lattice(A)))) | ![C: $i] : ((~element(C, the_carrier(boole_lattice(A)))) | (~((~(join(boole_lattice(A), B, C) = set_union2(B, C))) | (~(meet(boole_lattice(A), B, C) = set_intersection2(B, C)))))))),
% 119.50/119.65 inference(modus_ponens,[status(thm)],[22, 15])).
% 119.50/119.65 tff(24,plain,
% 119.50/119.65 (![A: $i, B: $i] : ((~element(B, the_carrier(boole_lattice(A)))) | ![C: $i] : ((~element(C, the_carrier(boole_lattice(A)))) | (~((~(join(boole_lattice(A), B, C) = set_union2(B, C))) | (~(meet(boole_lattice(A), B, C) = set_intersection2(B, C)))))))),
% 119.50/119.66 inference(modus_ponens,[status(thm)],[23, 13])).
% 119.50/119.66 tff(25,plain,
% 119.50/119.66 (((~![A: $i, B: $i] : ((~element(B, the_carrier(boole_lattice(A)))) | ![C: $i] : ((~element(C, the_carrier(boole_lattice(A)))) | (~((~(join(boole_lattice(A), B, C) = set_union2(B, C))) | (~(meet(boole_lattice(A), B, C) = set_intersection2(B, C)))))))) | ((~element(B!15, the_carrier(boole_lattice(A!16)))) | ![C: $i] : ((~element(C, the_carrier(boole_lattice(A!16)))) | (~((~(join(boole_lattice(A!16), B!15, C) = set_union2(B!15, C))) | (~(meet(boole_lattice(A!16), B!15, C) = set_intersection2(B!15, C)))))))) <=> ((~![A: $i, B: $i] : ((~element(B, the_carrier(boole_lattice(A)))) | ![C: $i] : ((~element(C, the_carrier(boole_lattice(A)))) | (~((~(join(boole_lattice(A), B, C) = set_union2(B, C))) | (~(meet(boole_lattice(A), B, C) = set_intersection2(B, C)))))))) | (~element(B!15, the_carrier(boole_lattice(A!16)))) | ![C: $i] : ((~element(C, the_carrier(boole_lattice(A!16)))) | (~((~(join(boole_lattice(A!16), B!15, C) = set_union2(B!15, C))) | (~(meet(boole_lattice(A!16), B!15, C) = set_intersection2(B!15, C)))))))),
% 119.50/119.66 inference(rewrite,[status(thm)],[])).
% 119.50/119.66 tff(26,plain,
% 119.50/119.66 ((~![A: $i, B: $i] : ((~element(B, the_carrier(boole_lattice(A)))) | ![C: $i] : ((~element(C, the_carrier(boole_lattice(A)))) | (~((~(join(boole_lattice(A), B, C) = set_union2(B, C))) | (~(meet(boole_lattice(A), B, C) = set_intersection2(B, C)))))))) | ((~element(B!15, the_carrier(boole_lattice(A!16)))) | ![C: $i] : ((~element(C, the_carrier(boole_lattice(A!16)))) | (~((~(join(boole_lattice(A!16), B!15, C) = set_union2(B!15, C))) | (~(meet(boole_lattice(A!16), B!15, C) = set_intersection2(B!15, C)))))))),
% 119.50/119.66 inference(quant_inst,[status(thm)],[])).
% 119.50/119.66 tff(27,plain,
% 119.50/119.66 ((~![A: $i, B: $i] : ((~element(B, the_carrier(boole_lattice(A)))) | ![C: $i] : ((~element(C, the_carrier(boole_lattice(A)))) | (~((~(join(boole_lattice(A), B, C) = set_union2(B, C))) | (~(meet(boole_lattice(A), B, C) = set_intersection2(B, C)))))))) | (~element(B!15, the_carrier(boole_lattice(A!16)))) | ![C: $i] : ((~element(C, the_carrier(boole_lattice(A!16)))) | (~((~(join(boole_lattice(A!16), B!15, C) = set_union2(B!15, C))) | (~(meet(boole_lattice(A!16), B!15, C) = set_intersection2(B!15, C))))))),
% 119.50/119.66 inference(modus_ponens,[status(thm)],[26, 25])).
% 119.50/119.66 tff(28,plain,
% 119.50/119.66 (![C: $i] : ((~element(C, the_carrier(boole_lattice(A!16)))) | (~((~(join(boole_lattice(A!16), B!15, C) = set_union2(B!15, C))) | (~(meet(boole_lattice(A!16), B!15, C) = set_intersection2(B!15, C))))))),
% 119.50/119.66 inference(unit_resolution,[status(thm)],[27, 24, 8])).
% 119.50/119.66 tff(29,plain,
% 119.50/119.66 (~((~element(C!17, the_carrier(boole_lattice(A!16)))) | (below(boole_lattice(A!16), B!15, C!17) <=> subset(B!15, C!17)))),
% 119.50/119.66 inference(and_elim,[status(thm)],[7])).
% 119.50/119.66 tff(30,plain,
% 119.50/119.66 (element(C!17, the_carrier(boole_lattice(A!16)))),
% 119.50/119.66 inference(or_elim,[status(thm)],[29])).
% 119.50/119.66 tff(31,plain,
% 119.50/119.66 (((~![C: $i] : ((~element(C, the_carrier(boole_lattice(A!16)))) | (~((~(join(boole_lattice(A!16), B!15, C) = set_union2(B!15, C))) | (~(meet(boole_lattice(A!16), B!15, C) = set_intersection2(B!15, C))))))) | ((~element(C!17, the_carrier(boole_lattice(A!16)))) | (~((~(join(boole_lattice(A!16), B!15, C!17) = set_union2(B!15, C!17))) | (~(meet(boole_lattice(A!16), B!15, C!17) = set_intersection2(B!15, C!17))))))) <=> ((~![C: $i] : ((~element(C, the_carrier(boole_lattice(A!16)))) | (~((~(join(boole_lattice(A!16), B!15, C) = set_union2(B!15, C))) | (~(meet(boole_lattice(A!16), B!15, C) = set_intersection2(B!15, C))))))) | (~element(C!17, the_carrier(boole_lattice(A!16)))) | (~((~(join(boole_lattice(A!16), B!15, C!17) = set_union2(B!15, C!17))) | (~(meet(boole_lattice(A!16), B!15, C!17) = set_intersection2(B!15, C!17))))))),
% 119.50/119.66 inference(rewrite,[status(thm)],[])).
% 119.50/119.66 tff(32,plain,
% 119.50/119.66 ((~![C: $i] : ((~element(C, the_carrier(boole_lattice(A!16)))) | (~((~(join(boole_lattice(A!16), B!15, C) = set_union2(B!15, C))) | (~(meet(boole_lattice(A!16), B!15, C) = set_intersection2(B!15, C))))))) | ((~element(C!17, the_carrier(boole_lattice(A!16)))) | (~((~(join(boole_lattice(A!16), B!15, C!17) = set_union2(B!15, C!17))) | (~(meet(boole_lattice(A!16), B!15, C!17) = set_intersection2(B!15, C!17))))))),
% 119.50/119.66 inference(quant_inst,[status(thm)],[])).
% 119.50/119.66 tff(33,plain,
% 119.50/119.66 ((~![C: $i] : ((~element(C, the_carrier(boole_lattice(A!16)))) | (~((~(join(boole_lattice(A!16), B!15, C) = set_union2(B!15, C))) | (~(meet(boole_lattice(A!16), B!15, C) = set_intersection2(B!15, C))))))) | (~element(C!17, the_carrier(boole_lattice(A!16)))) | (~((~(join(boole_lattice(A!16), B!15, C!17) = set_union2(B!15, C!17))) | (~(meet(boole_lattice(A!16), B!15, C!17) = set_intersection2(B!15, C!17)))))),
% 119.50/119.66 inference(modus_ponens,[status(thm)],[32, 31])).
% 119.50/119.66 tff(34,plain,
% 119.50/119.66 (~((~(join(boole_lattice(A!16), B!15, C!17) = set_union2(B!15, C!17))) | (~(meet(boole_lattice(A!16), B!15, C!17) = set_intersection2(B!15, C!17))))),
% 119.50/119.66 inference(unit_resolution,[status(thm)],[33, 30, 28])).
% 119.50/119.66 tff(35,plain,
% 119.50/119.66 (((~(join(boole_lattice(A!16), B!15, C!17) = set_union2(B!15, C!17))) | (~(meet(boole_lattice(A!16), B!15, C!17) = set_intersection2(B!15, C!17)))) | (join(boole_lattice(A!16), B!15, C!17) = set_union2(B!15, C!17))),
% 119.50/119.66 inference(tautology,[status(thm)],[])).
% 119.50/119.66 tff(36,plain,
% 119.50/119.66 (join(boole_lattice(A!16), B!15, C!17) = set_union2(B!15, C!17)),
% 119.50/119.66 inference(unit_resolution,[status(thm)],[35, 34])).
% 119.50/119.66 tff(37,plain,
% 119.50/119.66 (^[A: $i] : refl((set_intersection2(A, A) = A) <=> (set_intersection2(A, A) = A))),
% 119.50/119.66 inference(bind,[status(th)],[])).
% 119.50/119.66 tff(38,plain,
% 119.50/119.66 (![A: $i] : (set_intersection2(A, A) = A) <=> ![A: $i] : (set_intersection2(A, A) = A)),
% 119.50/119.66 inference(quant_intro,[status(thm)],[37])).
% 119.50/119.66 tff(39,plain,
% 119.50/119.66 (![A: $i] : (set_intersection2(A, A) = A) <=> ![A: $i] : (set_intersection2(A, A) = A)),
% 119.50/119.66 inference(rewrite,[status(thm)],[])).
% 119.50/119.66 tff(40,plain,
% 119.50/119.66 (![A: $i, B: $i] : (set_intersection2(A, A) = A) <=> ![A: $i] : (set_intersection2(A, A) = A)),
% 119.50/119.66 inference(elim_unused_vars,[status(thm)],[])).
% 119.50/119.66 tff(41,axiom,(![A: $i, B: $i] : (set_intersection2(A, A) = A)), file('/export/starexec/sandbox/benchmark/theBenchmark.p','idempotence_k3_xboole_0')).
% 119.50/119.66 tff(42,plain,
% 119.50/119.66 (![A: $i] : (set_intersection2(A, A) = A)),
% 119.50/119.66 inference(modus_ponens,[status(thm)],[41, 40])).
% 119.50/119.66 tff(43,plain,
% 119.50/119.66 (![A: $i] : (set_intersection2(A, A) = A)),
% 119.50/119.66 inference(modus_ponens,[status(thm)],[42, 39])).
% 119.50/119.66 tff(44,plain,(
% 119.50/119.66 ![A: $i] : (set_intersection2(A, A) = A)),
% 119.50/119.66 inference(skolemize,[status(sab)],[43])).
% 119.50/119.66 tff(45,plain,
% 119.50/119.66 (![A: $i] : (set_intersection2(A, A) = A)),
% 119.50/119.66 inference(modus_ponens,[status(thm)],[44, 38])).
% 119.50/119.66 tff(46,plain,
% 119.50/119.66 ((~![A: $i] : (set_intersection2(A, A) = A)) | (set_intersection2(C!17, C!17) = C!17)),
% 119.50/119.66 inference(quant_inst,[status(thm)],[])).
% 119.50/119.66 tff(47,plain,
% 119.50/119.66 (set_intersection2(C!17, C!17) = C!17),
% 119.50/119.66 inference(unit_resolution,[status(thm)],[46, 45])).
% 119.50/119.66 tff(48,plain,
% 119.50/119.66 ((~![A: $i] : (set_intersection2(A, A) = A)) | (set_intersection2(B!15, B!15) = B!15)),
% 119.50/119.66 inference(quant_inst,[status(thm)],[])).
% 119.50/119.66 tff(49,plain,
% 119.50/119.66 (set_intersection2(B!15, B!15) = B!15),
% 119.50/119.66 inference(unit_resolution,[status(thm)],[48, 45])).
% 119.50/119.66 tff(50,plain,
% 119.50/119.66 (join(boole_lattice(A!16), set_intersection2(B!15, B!15), set_intersection2(C!17, C!17)) = join(boole_lattice(A!16), B!15, C!17)),
% 119.50/119.66 inference(monotonicity,[status(thm)],[49, 47])).
% 119.50/119.66 tff(51,plain,
% 119.50/119.66 (below(boole_lattice(A!16), set_intersection2(B!15, B!15), set_intersection2(C!17, C!17)) <=> below(boole_lattice(A!16), B!15, C!17)),
% 119.50/119.66 inference(monotonicity,[status(thm)],[49, 47])).
% 119.50/119.66 tff(52,plain,
% 119.50/119.66 (below(boole_lattice(A!16), B!15, C!17) <=> below(boole_lattice(A!16), set_intersection2(B!15, B!15), set_intersection2(C!17, C!17))),
% 119.50/119.66 inference(symmetry,[status(thm)],[51])).
% 119.50/119.66 tff(53,assumption,(~subset(B!15, C!17)), introduced(assumption)).
% 119.50/119.66 tff(54,plain,
% 119.50/119.66 (~(below(boole_lattice(A!16), B!15, C!17) <=> subset(B!15, C!17))),
% 119.50/119.66 inference(or_elim,[status(thm)],[29])).
% 119.50/119.66 tff(55,plain,
% 119.50/119.66 ((below(boole_lattice(A!16), B!15, C!17) <=> subset(B!15, C!17)) | below(boole_lattice(A!16), B!15, C!17) | subset(B!15, C!17)),
% 119.50/119.66 inference(tautology,[status(thm)],[])).
% 119.50/119.66 tff(56,plain,
% 119.50/119.66 (below(boole_lattice(A!16), B!15, C!17) | subset(B!15, C!17)),
% 119.50/119.66 inference(unit_resolution,[status(thm)],[55, 54])).
% 119.50/119.66 tff(57,plain,
% 119.50/119.66 (below(boole_lattice(A!16), B!15, C!17)),
% 119.50/119.66 inference(unit_resolution,[status(thm)],[56, 53])).
% 119.50/119.66 tff(58,plain,
% 119.50/119.66 (below(boole_lattice(A!16), set_intersection2(B!15, B!15), set_intersection2(C!17, C!17))),
% 119.50/119.66 inference(modus_ponens,[status(thm)],[57, 52])).
% 119.50/119.66 tff(59,plain,
% 119.50/119.66 (element(set_intersection2(C!17, C!17), the_carrier(boole_lattice(A!16))) <=> element(C!17, the_carrier(boole_lattice(A!16)))),
% 119.50/119.66 inference(monotonicity,[status(thm)],[47])).
% 119.50/119.66 tff(60,plain,
% 119.50/119.66 (element(C!17, the_carrier(boole_lattice(A!16))) <=> element(set_intersection2(C!17, C!17), the_carrier(boole_lattice(A!16)))),
% 119.50/119.66 inference(symmetry,[status(thm)],[59])).
% 119.50/119.66 tff(61,plain,
% 119.50/119.66 (element(set_intersection2(C!17, C!17), the_carrier(boole_lattice(A!16)))),
% 119.50/119.66 inference(modus_ponens,[status(thm)],[30, 60])).
% 119.50/119.66 tff(62,plain,
% 119.50/119.66 (element(set_intersection2(B!15, B!15), the_carrier(boole_lattice(A!16))) <=> element(B!15, the_carrier(boole_lattice(A!16)))),
% 119.50/119.66 inference(monotonicity,[status(thm)],[49])).
% 119.50/119.66 tff(63,plain,
% 119.50/119.66 (element(B!15, the_carrier(boole_lattice(A!16))) <=> element(set_intersection2(B!15, B!15), the_carrier(boole_lattice(A!16)))),
% 119.50/119.66 inference(symmetry,[status(thm)],[62])).
% 119.50/119.66 tff(64,plain,
% 119.50/119.66 (element(set_intersection2(B!15, B!15), the_carrier(boole_lattice(A!16)))),
% 119.50/119.66 inference(modus_ponens,[status(thm)],[8, 63])).
% 119.50/119.66 tff(65,plain,
% 119.50/119.66 (^[A: $i] : refl((~(empty_carrier(boole_lattice(A)) | (~strict_latt_str(boole_lattice(A))))) <=> (~(empty_carrier(boole_lattice(A)) | (~strict_latt_str(boole_lattice(A))))))),
% 119.50/119.66 inference(bind,[status(th)],[])).
% 119.50/119.66 tff(66,plain,
% 119.50/119.66 (![A: $i] : (~(empty_carrier(boole_lattice(A)) | (~strict_latt_str(boole_lattice(A))))) <=> ![A: $i] : (~(empty_carrier(boole_lattice(A)) | (~strict_latt_str(boole_lattice(A)))))),
% 119.50/119.66 inference(quant_intro,[status(thm)],[65])).
% 119.50/119.66 tff(67,plain,
% 119.50/119.66 (^[A: $i] : rewrite(((~empty_carrier(boole_lattice(A))) & strict_latt_str(boole_lattice(A))) <=> (~(empty_carrier(boole_lattice(A)) | (~strict_latt_str(boole_lattice(A))))))),
% 119.50/119.66 inference(bind,[status(th)],[])).
% 119.50/119.66 tff(68,plain,
% 119.50/119.66 (![A: $i] : ((~empty_carrier(boole_lattice(A))) & strict_latt_str(boole_lattice(A))) <=> ![A: $i] : (~(empty_carrier(boole_lattice(A)) | (~strict_latt_str(boole_lattice(A)))))),
% 119.50/119.66 inference(quant_intro,[status(thm)],[67])).
% 119.50/119.66 tff(69,plain,
% 119.50/119.66 (![A: $i] : ((~empty_carrier(boole_lattice(A))) & strict_latt_str(boole_lattice(A))) <=> ![A: $i] : ((~empty_carrier(boole_lattice(A))) & strict_latt_str(boole_lattice(A)))),
% 119.50/119.66 inference(rewrite,[status(thm)],[])).
% 119.50/119.66 tff(70,axiom,(![A: $i] : ((~empty_carrier(boole_lattice(A))) & strict_latt_str(boole_lattice(A)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','fc1_lattice3')).
% 119.50/119.66 tff(71,plain,
% 119.50/119.66 (![A: $i] : ((~empty_carrier(boole_lattice(A))) & strict_latt_str(boole_lattice(A)))),
% 119.50/119.66 inference(modus_ponens,[status(thm)],[70, 69])).
% 119.50/119.66 tff(72,plain,(
% 119.50/119.66 ![A: $i] : ((~empty_carrier(boole_lattice(A))) & strict_latt_str(boole_lattice(A)))),
% 119.50/119.66 inference(skolemize,[status(sab)],[71])).
% 119.50/119.66 tff(73,plain,
% 119.50/119.66 (![A: $i] : (~(empty_carrier(boole_lattice(A)) | (~strict_latt_str(boole_lattice(A)))))),
% 119.50/119.66 inference(modus_ponens,[status(thm)],[72, 68])).
% 119.50/119.66 tff(74,plain,
% 119.50/119.66 (![A: $i] : (~(empty_carrier(boole_lattice(A)) | (~strict_latt_str(boole_lattice(A)))))),
% 119.50/119.66 inference(modus_ponens,[status(thm)],[73, 66])).
% 119.50/119.66 tff(75,plain,
% 119.50/119.66 ((~![A: $i] : (~(empty_carrier(boole_lattice(A)) | (~strict_latt_str(boole_lattice(A)))))) | (~(empty_carrier(boole_lattice(A!16)) | (~strict_latt_str(boole_lattice(A!16)))))),
% 119.50/119.66 inference(quant_inst,[status(thm)],[])).
% 119.50/119.66 tff(76,plain,
% 119.50/119.66 (~(empty_carrier(boole_lattice(A!16)) | (~strict_latt_str(boole_lattice(A!16))))),
% 119.50/119.66 inference(unit_resolution,[status(thm)],[75, 74])).
% 119.50/119.66 tff(77,plain,
% 119.50/119.66 ((empty_carrier(boole_lattice(A!16)) | (~strict_latt_str(boole_lattice(A!16)))) | (~empty_carrier(boole_lattice(A!16)))),
% 119.50/119.66 inference(tautology,[status(thm)],[])).
% 119.50/119.66 tff(78,plain,
% 119.50/119.66 (~empty_carrier(boole_lattice(A!16))),
% 119.50/119.66 inference(unit_resolution,[status(thm)],[77, 76])).
% 119.50/119.66 tff(79,plain,
% 119.50/119.66 (^[A: $i] : refl((~((~strict_latt_str(boole_lattice(A))) | (~latt_str(boole_lattice(A))))) <=> (~((~strict_latt_str(boole_lattice(A))) | (~latt_str(boole_lattice(A))))))),
% 119.50/119.66 inference(bind,[status(th)],[])).
% 119.50/119.66 tff(80,plain,
% 119.50/119.66 (![A: $i] : (~((~strict_latt_str(boole_lattice(A))) | (~latt_str(boole_lattice(A))))) <=> ![A: $i] : (~((~strict_latt_str(boole_lattice(A))) | (~latt_str(boole_lattice(A)))))),
% 119.50/119.66 inference(quant_intro,[status(thm)],[79])).
% 119.50/119.66 tff(81,plain,
% 119.50/119.66 (^[A: $i] : rewrite((strict_latt_str(boole_lattice(A)) & latt_str(boole_lattice(A))) <=> (~((~strict_latt_str(boole_lattice(A))) | (~latt_str(boole_lattice(A))))))),
% 119.50/119.66 inference(bind,[status(th)],[])).
% 119.50/119.66 tff(82,plain,
% 119.50/119.66 (![A: $i] : (strict_latt_str(boole_lattice(A)) & latt_str(boole_lattice(A))) <=> ![A: $i] : (~((~strict_latt_str(boole_lattice(A))) | (~latt_str(boole_lattice(A)))))),
% 119.50/119.66 inference(quant_intro,[status(thm)],[81])).
% 119.50/119.66 tff(83,plain,
% 119.50/119.66 (![A: $i] : (strict_latt_str(boole_lattice(A)) & latt_str(boole_lattice(A))) <=> ![A: $i] : (strict_latt_str(boole_lattice(A)) & latt_str(boole_lattice(A)))),
% 119.50/119.66 inference(rewrite,[status(thm)],[])).
% 119.50/119.66 tff(84,axiom,(![A: $i] : (strict_latt_str(boole_lattice(A)) & latt_str(boole_lattice(A)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','dt_k1_lattice3')).
% 119.50/119.66 tff(85,plain,
% 119.50/119.66 (![A: $i] : (strict_latt_str(boole_lattice(A)) & latt_str(boole_lattice(A)))),
% 119.50/119.66 inference(modus_ponens,[status(thm)],[84, 83])).
% 119.50/119.66 tff(86,plain,(
% 119.50/119.66 ![A: $i] : (strict_latt_str(boole_lattice(A)) & latt_str(boole_lattice(A)))),
% 119.50/119.66 inference(skolemize,[status(sab)],[85])).
% 119.50/119.66 tff(87,plain,
% 119.50/119.66 (![A: $i] : (~((~strict_latt_str(boole_lattice(A))) | (~latt_str(boole_lattice(A)))))),
% 119.50/119.66 inference(modus_ponens,[status(thm)],[86, 82])).
% 119.50/119.66 tff(88,plain,
% 119.50/119.66 (![A: $i] : (~((~strict_latt_str(boole_lattice(A))) | (~latt_str(boole_lattice(A)))))),
% 119.50/119.66 inference(modus_ponens,[status(thm)],[87, 80])).
% 119.50/119.66 tff(89,plain,
% 119.50/119.66 ((~![A: $i] : (~((~strict_latt_str(boole_lattice(A))) | (~latt_str(boole_lattice(A)))))) | (~((~strict_latt_str(boole_lattice(A!16))) | (~latt_str(boole_lattice(A!16)))))),
% 119.50/119.66 inference(quant_inst,[status(thm)],[])).
% 119.50/119.66 tff(90,plain,
% 119.50/119.66 (~((~strict_latt_str(boole_lattice(A!16))) | (~latt_str(boole_lattice(A!16))))),
% 119.50/119.66 inference(unit_resolution,[status(thm)],[89, 88])).
% 119.50/119.66 tff(91,plain,
% 119.50/119.66 (((~strict_latt_str(boole_lattice(A!16))) | (~latt_str(boole_lattice(A!16)))) | latt_str(boole_lattice(A!16))),
% 119.50/119.66 inference(tautology,[status(thm)],[])).
% 119.50/119.66 tff(92,plain,
% 119.50/119.66 (latt_str(boole_lattice(A!16))),
% 119.50/119.66 inference(unit_resolution,[status(thm)],[91, 90])).
% 119.50/119.66 tff(93,plain,
% 119.50/119.66 (((~strict_latt_str(boole_lattice(A!16))) | (~latt_str(boole_lattice(A!16)))) | strict_latt_str(boole_lattice(A!16))),
% 119.50/119.66 inference(tautology,[status(thm)],[])).
% 119.50/119.66 tff(94,plain,
% 119.50/119.66 (strict_latt_str(boole_lattice(A!16))),
% 119.50/119.66 inference(unit_resolution,[status(thm)],[93, 90])).
% 119.50/119.66 tff(95,plain,
% 119.50/119.66 (^[A: $i] : refl(((A = latt_str_of(the_carrier(A), the_L_join(A), the_L_meet(A))) | (~strict_latt_str(A)) | (~latt_str(A))) <=> ((A = latt_str_of(the_carrier(A), the_L_join(A), the_L_meet(A))) | (~strict_latt_str(A)) | (~latt_str(A))))),
% 119.50/119.66 inference(bind,[status(th)],[])).
% 119.50/119.66 tff(96,plain,
% 119.50/119.66 (![A: $i] : ((A = latt_str_of(the_carrier(A), the_L_join(A), the_L_meet(A))) | (~strict_latt_str(A)) | (~latt_str(A))) <=> ![A: $i] : ((A = latt_str_of(the_carrier(A), the_L_join(A), the_L_meet(A))) | (~strict_latt_str(A)) | (~latt_str(A)))),
% 119.50/119.66 inference(quant_intro,[status(thm)],[95])).
% 119.50/119.66 tff(97,plain,
% 119.50/119.66 (![A: $i] : ((A = latt_str_of(the_carrier(A), the_L_join(A), the_L_meet(A))) | (~strict_latt_str(A)) | (~latt_str(A))) <=> ![A: $i] : ((A = latt_str_of(the_carrier(A), the_L_join(A), the_L_meet(A))) | (~strict_latt_str(A)) | (~latt_str(A)))),
% 119.50/119.66 inference(rewrite,[status(thm)],[])).
% 119.50/119.66 tff(98,plain,
% 119.50/119.66 (^[A: $i] : trans(monotonicity(rewrite((strict_latt_str(A) => (A = latt_str_of(the_carrier(A), the_L_join(A), the_L_meet(A)))) <=> ((~strict_latt_str(A)) | (A = latt_str_of(the_carrier(A), the_L_join(A), the_L_meet(A))))), ((latt_str(A) => (strict_latt_str(A) => (A = latt_str_of(the_carrier(A), the_L_join(A), the_L_meet(A))))) <=> (latt_str(A) => ((~strict_latt_str(A)) | (A = latt_str_of(the_carrier(A), the_L_join(A), the_L_meet(A))))))), rewrite((latt_str(A) => ((~strict_latt_str(A)) | (A = latt_str_of(the_carrier(A), the_L_join(A), the_L_meet(A))))) <=> ((A = latt_str_of(the_carrier(A), the_L_join(A), the_L_meet(A))) | (~strict_latt_str(A)) | (~latt_str(A)))), ((latt_str(A) => (strict_latt_str(A) => (A = latt_str_of(the_carrier(A), the_L_join(A), the_L_meet(A))))) <=> ((A = latt_str_of(the_carrier(A), the_L_join(A), the_L_meet(A))) | (~strict_latt_str(A)) | (~latt_str(A)))))),
% 119.50/119.66 inference(bind,[status(th)],[])).
% 119.50/119.66 tff(99,plain,
% 119.50/119.66 (![A: $i] : (latt_str(A) => (strict_latt_str(A) => (A = latt_str_of(the_carrier(A), the_L_join(A), the_L_meet(A))))) <=> ![A: $i] : ((A = latt_str_of(the_carrier(A), the_L_join(A), the_L_meet(A))) | (~strict_latt_str(A)) | (~latt_str(A)))),
% 119.50/119.66 inference(quant_intro,[status(thm)],[98])).
% 119.50/119.66 tff(100,axiom,(![A: $i] : (latt_str(A) => (strict_latt_str(A) => (A = latt_str_of(the_carrier(A), the_L_join(A), the_L_meet(A)))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','abstractness_v3_lattices')).
% 119.50/119.66 tff(101,plain,
% 119.50/119.66 (![A: $i] : ((A = latt_str_of(the_carrier(A), the_L_join(A), the_L_meet(A))) | (~strict_latt_str(A)) | (~latt_str(A)))),
% 119.50/119.66 inference(modus_ponens,[status(thm)],[100, 99])).
% 119.50/119.66 tff(102,plain,
% 119.50/119.66 (![A: $i] : ((A = latt_str_of(the_carrier(A), the_L_join(A), the_L_meet(A))) | (~strict_latt_str(A)) | (~latt_str(A)))),
% 119.50/119.66 inference(modus_ponens,[status(thm)],[101, 97])).
% 119.50/119.66 tff(103,plain,(
% 119.50/119.66 ![A: $i] : ((A = latt_str_of(the_carrier(A), the_L_join(A), the_L_meet(A))) | (~strict_latt_str(A)) | (~latt_str(A)))),
% 119.50/119.66 inference(skolemize,[status(sab)],[102])).
% 119.50/119.66 tff(104,plain,
% 119.50/119.66 (![A: $i] : ((A = latt_str_of(the_carrier(A), the_L_join(A), the_L_meet(A))) | (~strict_latt_str(A)) | (~latt_str(A)))),
% 119.50/119.66 inference(modus_ponens,[status(thm)],[103, 96])).
% 119.50/119.66 tff(105,plain,
% 119.50/119.66 (((~![A: $i] : ((A = latt_str_of(the_carrier(A), the_L_join(A), the_L_meet(A))) | (~strict_latt_str(A)) | (~latt_str(A)))) | ((boole_lattice(A!16) = latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16)))) | (~strict_latt_str(boole_lattice(A!16))) | (~latt_str(boole_lattice(A!16))))) <=> ((~![A: $i] : ((A = latt_str_of(the_carrier(A), the_L_join(A), the_L_meet(A))) | (~strict_latt_str(A)) | (~latt_str(A)))) | (boole_lattice(A!16) = latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16)))) | (~strict_latt_str(boole_lattice(A!16))) | (~latt_str(boole_lattice(A!16))))),
% 119.50/119.66 inference(rewrite,[status(thm)],[])).
% 119.50/119.66 tff(106,plain,
% 119.50/119.66 ((~![A: $i] : ((A = latt_str_of(the_carrier(A), the_L_join(A), the_L_meet(A))) | (~strict_latt_str(A)) | (~latt_str(A)))) | ((boole_lattice(A!16) = latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16)))) | (~strict_latt_str(boole_lattice(A!16))) | (~latt_str(boole_lattice(A!16))))),
% 119.50/119.66 inference(quant_inst,[status(thm)],[])).
% 119.50/119.66 tff(107,plain,
% 119.50/119.66 ((~![A: $i] : ((A = latt_str_of(the_carrier(A), the_L_join(A), the_L_meet(A))) | (~strict_latt_str(A)) | (~latt_str(A)))) | (boole_lattice(A!16) = latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16)))) | (~strict_latt_str(boole_lattice(A!16))) | (~latt_str(boole_lattice(A!16)))),
% 119.50/119.66 inference(modus_ponens,[status(thm)],[106, 105])).
% 119.50/119.66 tff(108,plain,
% 119.50/119.66 ((boole_lattice(A!16) = latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16)))) | (~strict_latt_str(boole_lattice(A!16))) | (~latt_str(boole_lattice(A!16)))),
% 119.50/119.66 inference(unit_resolution,[status(thm)],[107, 104])).
% 119.50/119.66 tff(109,plain,
% 119.50/119.66 (boole_lattice(A!16) = latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16)))),
% 119.50/119.66 inference(unit_resolution,[status(thm)],[108, 94, 92])).
% 119.50/119.66 tff(110,plain,
% 119.50/119.66 (latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16))) = boole_lattice(A!16)),
% 119.50/119.66 inference(symmetry,[status(thm)],[109])).
% 119.50/119.66 tff(111,plain,
% 119.50/119.66 (join_semilatt_str(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16)))) <=> join_semilatt_str(boole_lattice(A!16))),
% 119.50/119.66 inference(monotonicity,[status(thm)],[110])).
% 119.50/119.66 tff(112,plain,
% 119.50/119.66 (latt_str(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16)))) <=> latt_str(boole_lattice(A!16))),
% 119.50/119.66 inference(monotonicity,[status(thm)],[110])).
% 119.50/119.66 tff(113,plain,
% 119.50/119.66 (latt_str(boole_lattice(A!16)) <=> latt_str(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16))))),
% 119.50/119.66 inference(symmetry,[status(thm)],[112])).
% 119.50/119.66 tff(114,plain,
% 119.50/119.66 (latt_str(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16))))),
% 119.50/119.66 inference(modus_ponens,[status(thm)],[92, 113])).
% 119.50/119.66 tff(115,plain,
% 119.50/119.66 (^[A: $i] : refl(((~latt_str(A)) | (~((~meet_semilatt_str(A)) | (~join_semilatt_str(A))))) <=> ((~latt_str(A)) | (~((~meet_semilatt_str(A)) | (~join_semilatt_str(A))))))),
% 119.50/119.66 inference(bind,[status(th)],[])).
% 119.50/119.66 tff(116,plain,
% 119.50/119.66 (![A: $i] : ((~latt_str(A)) | (~((~meet_semilatt_str(A)) | (~join_semilatt_str(A))))) <=> ![A: $i] : ((~latt_str(A)) | (~((~meet_semilatt_str(A)) | (~join_semilatt_str(A)))))),
% 119.50/119.66 inference(quant_intro,[status(thm)],[115])).
% 119.50/119.66 tff(117,plain,
% 119.50/119.66 (^[A: $i] : rewrite(((~latt_str(A)) | (meet_semilatt_str(A) & join_semilatt_str(A))) <=> ((~latt_str(A)) | (~((~meet_semilatt_str(A)) | (~join_semilatt_str(A))))))),
% 119.50/119.66 inference(bind,[status(th)],[])).
% 119.50/119.66 tff(118,plain,
% 119.50/119.66 (![A: $i] : ((~latt_str(A)) | (meet_semilatt_str(A) & join_semilatt_str(A))) <=> ![A: $i] : ((~latt_str(A)) | (~((~meet_semilatt_str(A)) | (~join_semilatt_str(A)))))),
% 119.50/119.66 inference(quant_intro,[status(thm)],[117])).
% 119.50/119.66 tff(119,plain,
% 119.50/119.66 (![A: $i] : ((~latt_str(A)) | (meet_semilatt_str(A) & join_semilatt_str(A))) <=> ![A: $i] : ((~latt_str(A)) | (meet_semilatt_str(A) & join_semilatt_str(A)))),
% 119.50/119.66 inference(rewrite,[status(thm)],[])).
% 119.50/119.66 tff(120,plain,
% 119.50/119.66 (^[A: $i] : rewrite((latt_str(A) => (meet_semilatt_str(A) & join_semilatt_str(A))) <=> ((~latt_str(A)) | (meet_semilatt_str(A) & join_semilatt_str(A))))),
% 119.50/119.66 inference(bind,[status(th)],[])).
% 119.50/119.66 tff(121,plain,
% 119.50/119.66 (![A: $i] : (latt_str(A) => (meet_semilatt_str(A) & join_semilatt_str(A))) <=> ![A: $i] : ((~latt_str(A)) | (meet_semilatt_str(A) & join_semilatt_str(A)))),
% 119.50/119.66 inference(quant_intro,[status(thm)],[120])).
% 119.50/119.66 tff(122,axiom,(![A: $i] : (latt_str(A) => (meet_semilatt_str(A) & join_semilatt_str(A)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','dt_l3_lattices')).
% 119.50/119.66 tff(123,plain,
% 119.50/119.66 (![A: $i] : ((~latt_str(A)) | (meet_semilatt_str(A) & join_semilatt_str(A)))),
% 119.50/119.66 inference(modus_ponens,[status(thm)],[122, 121])).
% 119.50/119.66 tff(124,plain,
% 119.50/119.66 (![A: $i] : ((~latt_str(A)) | (meet_semilatt_str(A) & join_semilatt_str(A)))),
% 119.50/119.66 inference(modus_ponens,[status(thm)],[123, 119])).
% 119.50/119.66 tff(125,plain,(
% 119.50/119.66 ![A: $i] : ((~latt_str(A)) | (meet_semilatt_str(A) & join_semilatt_str(A)))),
% 119.50/119.66 inference(skolemize,[status(sab)],[124])).
% 119.50/119.66 tff(126,plain,
% 119.50/119.66 (![A: $i] : ((~latt_str(A)) | (~((~meet_semilatt_str(A)) | (~join_semilatt_str(A)))))),
% 119.50/119.66 inference(modus_ponens,[status(thm)],[125, 118])).
% 119.50/119.66 tff(127,plain,
% 119.50/119.66 (![A: $i] : ((~latt_str(A)) | (~((~meet_semilatt_str(A)) | (~join_semilatt_str(A)))))),
% 119.50/119.66 inference(modus_ponens,[status(thm)],[126, 116])).
% 119.50/119.66 tff(128,plain,
% 119.50/119.66 (((~![A: $i] : ((~latt_str(A)) | (~((~meet_semilatt_str(A)) | (~join_semilatt_str(A)))))) | ((~latt_str(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16))))) | (~((~meet_semilatt_str(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16))))) | (~join_semilatt_str(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16))))))))) <=> ((~![A: $i] : ((~latt_str(A)) | (~((~meet_semilatt_str(A)) | (~join_semilatt_str(A)))))) | (~latt_str(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16))))) | (~((~meet_semilatt_str(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16))))) | (~join_semilatt_str(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16))))))))),
% 119.50/119.66 inference(rewrite,[status(thm)],[])).
% 119.50/119.66 tff(129,plain,
% 119.50/119.66 ((~![A: $i] : ((~latt_str(A)) | (~((~meet_semilatt_str(A)) | (~join_semilatt_str(A)))))) | ((~latt_str(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16))))) | (~((~meet_semilatt_str(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16))))) | (~join_semilatt_str(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16))))))))),
% 119.50/119.66 inference(quant_inst,[status(thm)],[])).
% 119.50/119.66 tff(130,plain,
% 119.50/119.66 ((~![A: $i] : ((~latt_str(A)) | (~((~meet_semilatt_str(A)) | (~join_semilatt_str(A)))))) | (~latt_str(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16))))) | (~((~meet_semilatt_str(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16))))) | (~join_semilatt_str(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16)))))))),
% 119.50/119.66 inference(modus_ponens,[status(thm)],[129, 128])).
% 119.50/119.66 tff(131,plain,
% 119.50/119.66 ((~latt_str(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16))))) | (~((~meet_semilatt_str(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16))))) | (~join_semilatt_str(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16)))))))),
% 119.50/119.66 inference(unit_resolution,[status(thm)],[130, 127])).
% 119.50/119.66 tff(132,plain,
% 119.50/119.66 (~((~meet_semilatt_str(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16))))) | (~join_semilatt_str(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16))))))),
% 119.50/119.66 inference(unit_resolution,[status(thm)],[131, 114])).
% 119.50/119.66 tff(133,plain,
% 119.50/119.66 (((~meet_semilatt_str(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16))))) | (~join_semilatt_str(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16)))))) | join_semilatt_str(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16))))),
% 119.50/119.66 inference(tautology,[status(thm)],[])).
% 119.50/119.66 tff(134,plain,
% 119.50/119.66 (join_semilatt_str(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16))))),
% 119.50/119.66 inference(unit_resolution,[status(thm)],[133, 132])).
% 119.50/119.66 tff(135,plain,
% 119.50/119.66 (join_semilatt_str(boole_lattice(A!16))),
% 119.50/119.66 inference(modus_ponens,[status(thm)],[134, 111])).
% 119.50/119.66 tff(136,plain,
% 119.50/119.66 (^[A: $i] : rewrite((empty_carrier(A) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (below(A, B, C) <=> (join(A, B, C) = C)))) | (~join_semilatt_str(A))) <=> (empty_carrier(A) | (~join_semilatt_str(A)) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (below(A, B, C) <=> (join(A, B, C) = C))))))),
% 119.50/119.66 inference(bind,[status(th)],[])).
% 119.50/119.66 tff(137,plain,
% 119.50/119.66 (![A: $i] : (empty_carrier(A) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (below(A, B, C) <=> (join(A, B, C) = C)))) | (~join_semilatt_str(A))) <=> ![A: $i] : (empty_carrier(A) | (~join_semilatt_str(A)) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (below(A, B, C) <=> (join(A, B, C) = C)))))),
% 119.50/119.66 inference(quant_intro,[status(thm)],[136])).
% 119.50/119.66 tff(138,plain,
% 119.50/119.66 (^[A: $i] : refl((empty_carrier(A) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (below(A, B, C) <=> (join(A, B, C) = C)))) | (~join_semilatt_str(A))) <=> (empty_carrier(A) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (below(A, B, C) <=> (join(A, B, C) = C)))) | (~join_semilatt_str(A))))),
% 119.50/119.66 inference(bind,[status(th)],[])).
% 119.50/119.66 tff(139,plain,
% 119.50/119.66 (![A: $i] : (empty_carrier(A) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (below(A, B, C) <=> (join(A, B, C) = C)))) | (~join_semilatt_str(A))) <=> ![A: $i] : (empty_carrier(A) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (below(A, B, C) <=> (join(A, B, C) = C)))) | (~join_semilatt_str(A)))),
% 119.50/119.66 inference(quant_intro,[status(thm)],[138])).
% 119.50/119.66 tff(140,plain,
% 119.50/119.66 (^[A: $i] : rewrite((empty_carrier(A) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (below(A, B, C) <=> (join(A, B, C) = C)))) | (~join_semilatt_str(A))) <=> (empty_carrier(A) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (below(A, B, C) <=> (join(A, B, C) = C)))) | (~join_semilatt_str(A))))),
% 119.50/119.66 inference(bind,[status(th)],[])).
% 119.50/119.66 tff(141,plain,
% 119.50/119.66 (![A: $i] : (empty_carrier(A) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (below(A, B, C) <=> (join(A, B, C) = C)))) | (~join_semilatt_str(A))) <=> ![A: $i] : (empty_carrier(A) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (below(A, B, C) <=> (join(A, B, C) = C)))) | (~join_semilatt_str(A)))),
% 119.50/119.66 inference(quant_intro,[status(thm)],[140])).
% 119.50/119.66 tff(142,plain,
% 119.50/119.66 (![A: $i] : (empty_carrier(A) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (below(A, B, C) <=> (join(A, B, C) = C)))) | (~join_semilatt_str(A))) <=> ![A: $i] : (empty_carrier(A) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (below(A, B, C) <=> (join(A, B, C) = C)))) | (~join_semilatt_str(A)))),
% 119.50/119.66 inference(transitivity,[status(thm)],[141, 139])).
% 119.50/119.66 tff(143,plain,
% 119.50/119.66 (^[A: $i] : trans(monotonicity(trans(monotonicity(rewrite(((~empty_carrier(A)) & join_semilatt_str(A)) <=> (~(empty_carrier(A) | (~join_semilatt_str(A))))), ((~((~empty_carrier(A)) & join_semilatt_str(A))) <=> (~(~(empty_carrier(A) | (~join_semilatt_str(A))))))), rewrite((~(~(empty_carrier(A) | (~join_semilatt_str(A))))) <=> (empty_carrier(A) | (~join_semilatt_str(A)))), ((~((~empty_carrier(A)) & join_semilatt_str(A))) <=> (empty_carrier(A) | (~join_semilatt_str(A))))), (((~((~empty_carrier(A)) & join_semilatt_str(A))) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (below(A, B, C) <=> (join(A, B, C) = C))))) <=> ((empty_carrier(A) | (~join_semilatt_str(A))) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (below(A, B, C) <=> (join(A, B, C) = C))))))), rewrite(((empty_carrier(A) | (~join_semilatt_str(A))) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (below(A, B, C) <=> (join(A, B, C) = C))))) <=> (empty_carrier(A) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (below(A, B, C) <=> (join(A, B, C) = C)))) | (~join_semilatt_str(A)))), (((~((~empty_carrier(A)) & join_semilatt_str(A))) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (below(A, B, C) <=> (join(A, B, C) = C))))) <=> (empty_carrier(A) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (below(A, B, C) <=> (join(A, B, C) = C)))) | (~join_semilatt_str(A)))))),
% 119.50/119.66 inference(bind,[status(th)],[])).
% 119.50/119.66 tff(144,plain,
% 119.50/119.66 (![A: $i] : ((~((~empty_carrier(A)) & join_semilatt_str(A))) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (below(A, B, C) <=> (join(A, B, C) = C))))) <=> ![A: $i] : (empty_carrier(A) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (below(A, B, C) <=> (join(A, B, C) = C)))) | (~join_semilatt_str(A)))),
% 119.50/119.66 inference(quant_intro,[status(thm)],[143])).
% 119.50/119.66 tff(145,plain,
% 119.50/119.66 (![A: $i] : ((~((~empty_carrier(A)) & join_semilatt_str(A))) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (below(A, B, C) <=> (join(A, B, C) = C))))) <=> ![A: $i] : ((~((~empty_carrier(A)) & join_semilatt_str(A))) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (below(A, B, C) <=> (join(A, B, C) = C)))))),
% 119.50/119.66 inference(rewrite,[status(thm)],[])).
% 119.50/119.66 tff(146,plain,
% 119.50/119.66 (^[A: $i] : trans(monotonicity(quant_intro(proof_bind(^[B: $i] : trans(monotonicity(quant_intro(proof_bind(^[C: $i] : rewrite((element(C, the_carrier(A)) => (below(A, B, C) <=> (join(A, B, C) = C))) <=> ((~element(C, the_carrier(A))) | (below(A, B, C) <=> (join(A, B, C) = C))))), (![C: $i] : (element(C, the_carrier(A)) => (below(A, B, C) <=> (join(A, B, C) = C))) <=> ![C: $i] : ((~element(C, the_carrier(A))) | (below(A, B, C) <=> (join(A, B, C) = C))))), ((element(B, the_carrier(A)) => ![C: $i] : (element(C, the_carrier(A)) => (below(A, B, C) <=> (join(A, B, C) = C)))) <=> (element(B, the_carrier(A)) => ![C: $i] : ((~element(C, the_carrier(A))) | (below(A, B, C) <=> (join(A, B, C) = C)))))), rewrite((element(B, the_carrier(A)) => ![C: $i] : ((~element(C, the_carrier(A))) | (below(A, B, C) <=> (join(A, B, C) = C)))) <=> ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (below(A, B, C) <=> (join(A, B, C) = C))))), ((element(B, the_carrier(A)) => ![C: $i] : (element(C, the_carrier(A)) => (below(A, B, C) <=> (join(A, B, C) = C)))) <=> ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (below(A, B, C) <=> (join(A, B, C) = C))))))), (![B: $i] : (element(B, the_carrier(A)) => ![C: $i] : (element(C, the_carrier(A)) => (below(A, B, C) <=> (join(A, B, C) = C)))) <=> ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (below(A, B, C) <=> (join(A, B, C) = C)))))), ((((~empty_carrier(A)) & join_semilatt_str(A)) => ![B: $i] : (element(B, the_carrier(A)) => ![C: $i] : (element(C, the_carrier(A)) => (below(A, B, C) <=> (join(A, B, C) = C))))) <=> (((~empty_carrier(A)) & join_semilatt_str(A)) => ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (below(A, B, C) <=> (join(A, B, C) = C))))))), rewrite((((~empty_carrier(A)) & join_semilatt_str(A)) => ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (below(A, B, C) <=> (join(A, B, C) = C))))) <=> ((~((~empty_carrier(A)) & join_semilatt_str(A))) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (below(A, B, C) <=> (join(A, B, C) = C)))))), ((((~empty_carrier(A)) & join_semilatt_str(A)) => ![B: $i] : (element(B, the_carrier(A)) => ![C: $i] : (element(C, the_carrier(A)) => (below(A, B, C) <=> (join(A, B, C) = C))))) <=> ((~((~empty_carrier(A)) & join_semilatt_str(A))) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (below(A, B, C) <=> (join(A, B, C) = C)))))))),
% 119.50/119.66 inference(bind,[status(th)],[])).
% 119.50/119.66 tff(147,plain,
% 119.50/119.66 (![A: $i] : (((~empty_carrier(A)) & join_semilatt_str(A)) => ![B: $i] : (element(B, the_carrier(A)) => ![C: $i] : (element(C, the_carrier(A)) => (below(A, B, C) <=> (join(A, B, C) = C))))) <=> ![A: $i] : ((~((~empty_carrier(A)) & join_semilatt_str(A))) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (below(A, B, C) <=> (join(A, B, C) = C)))))),
% 119.50/119.66 inference(quant_intro,[status(thm)],[146])).
% 119.50/119.66 tff(148,axiom,(![A: $i] : (((~empty_carrier(A)) & join_semilatt_str(A)) => ![B: $i] : (element(B, the_carrier(A)) => ![C: $i] : (element(C, the_carrier(A)) => (below(A, B, C) <=> (join(A, B, C) = C)))))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','d3_lattices')).
% 119.50/119.66 tff(149,plain,
% 119.50/119.66 (![A: $i] : ((~((~empty_carrier(A)) & join_semilatt_str(A))) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (below(A, B, C) <=> (join(A, B, C) = C)))))),
% 119.50/119.66 inference(modus_ponens,[status(thm)],[148, 147])).
% 119.50/119.66 tff(150,plain,
% 119.50/119.66 (![A: $i] : ((~((~empty_carrier(A)) & join_semilatt_str(A))) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (below(A, B, C) <=> (join(A, B, C) = C)))))),
% 119.50/119.66 inference(modus_ponens,[status(thm)],[149, 145])).
% 119.50/119.66 tff(151,plain,(
% 119.50/119.66 ![A: $i] : ((~((~empty_carrier(A)) & join_semilatt_str(A))) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (below(A, B, C) <=> (join(A, B, C) = C)))))),
% 119.50/119.66 inference(skolemize,[status(sab)],[150])).
% 119.50/119.66 tff(152,plain,
% 119.50/119.66 (![A: $i] : (empty_carrier(A) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (below(A, B, C) <=> (join(A, B, C) = C)))) | (~join_semilatt_str(A)))),
% 119.50/119.66 inference(modus_ponens,[status(thm)],[151, 144])).
% 119.50/119.66 tff(153,plain,
% 119.50/119.66 (![A: $i] : (empty_carrier(A) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (below(A, B, C) <=> (join(A, B, C) = C)))) | (~join_semilatt_str(A)))),
% 119.50/119.66 inference(modus_ponens,[status(thm)],[152, 142])).
% 119.50/119.66 tff(154,plain,
% 119.50/119.66 (![A: $i] : (empty_carrier(A) | (~join_semilatt_str(A)) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (below(A, B, C) <=> (join(A, B, C) = C)))))),
% 119.50/119.66 inference(modus_ponens,[status(thm)],[153, 137])).
% 119.50/119.66 tff(155,plain,
% 119.50/119.66 (((~![A: $i] : (empty_carrier(A) | (~join_semilatt_str(A)) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (below(A, B, C) <=> (join(A, B, C) = C)))))) | (empty_carrier(boole_lattice(A!16)) | (~join_semilatt_str(boole_lattice(A!16))) | ![B: $i] : ((~element(B, the_carrier(boole_lattice(A!16)))) | ![C: $i] : ((~element(C, the_carrier(boole_lattice(A!16)))) | (below(boole_lattice(A!16), B, C) <=> (join(boole_lattice(A!16), B, C) = C)))))) <=> ((~![A: $i] : (empty_carrier(A) | (~join_semilatt_str(A)) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (below(A, B, C) <=> (join(A, B, C) = C)))))) | empty_carrier(boole_lattice(A!16)) | (~join_semilatt_str(boole_lattice(A!16))) | ![B: $i] : ((~element(B, the_carrier(boole_lattice(A!16)))) | ![C: $i] : ((~element(C, the_carrier(boole_lattice(A!16)))) | (below(boole_lattice(A!16), B, C) <=> (join(boole_lattice(A!16), B, C) = C)))))),
% 119.50/119.66 inference(rewrite,[status(thm)],[])).
% 119.50/119.66 tff(156,plain,
% 119.50/119.66 ((~![A: $i] : (empty_carrier(A) | (~join_semilatt_str(A)) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (below(A, B, C) <=> (join(A, B, C) = C)))))) | (empty_carrier(boole_lattice(A!16)) | (~join_semilatt_str(boole_lattice(A!16))) | ![B: $i] : ((~element(B, the_carrier(boole_lattice(A!16)))) | ![C: $i] : ((~element(C, the_carrier(boole_lattice(A!16)))) | (below(boole_lattice(A!16), B, C) <=> (join(boole_lattice(A!16), B, C) = C)))))),
% 119.50/119.66 inference(quant_inst,[status(thm)],[])).
% 119.50/119.66 tff(157,plain,
% 119.50/119.66 ((~![A: $i] : (empty_carrier(A) | (~join_semilatt_str(A)) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (below(A, B, C) <=> (join(A, B, C) = C)))))) | empty_carrier(boole_lattice(A!16)) | (~join_semilatt_str(boole_lattice(A!16))) | ![B: $i] : ((~element(B, the_carrier(boole_lattice(A!16)))) | ![C: $i] : ((~element(C, the_carrier(boole_lattice(A!16)))) | (below(boole_lattice(A!16), B, C) <=> (join(boole_lattice(A!16), B, C) = C))))),
% 119.50/119.66 inference(modus_ponens,[status(thm)],[156, 155])).
% 119.50/119.66 tff(158,plain,
% 119.50/119.66 (empty_carrier(boole_lattice(A!16)) | (~join_semilatt_str(boole_lattice(A!16))) | ![B: $i] : ((~element(B, the_carrier(boole_lattice(A!16)))) | ![C: $i] : ((~element(C, the_carrier(boole_lattice(A!16)))) | (below(boole_lattice(A!16), B, C) <=> (join(boole_lattice(A!16), B, C) = C))))),
% 119.50/119.66 inference(unit_resolution,[status(thm)],[157, 154])).
% 119.50/119.66 tff(159,plain,
% 119.50/119.66 (![B: $i] : ((~element(B, the_carrier(boole_lattice(A!16)))) | ![C: $i] : ((~element(C, the_carrier(boole_lattice(A!16)))) | (below(boole_lattice(A!16), B, C) <=> (join(boole_lattice(A!16), B, C) = C))))),
% 119.50/119.66 inference(unit_resolution,[status(thm)],[158, 135, 78])).
% 119.50/119.66 tff(160,plain,
% 119.50/119.66 (((~![B: $i] : ((~element(B, the_carrier(boole_lattice(A!16)))) | ![C: $i] : ((~element(C, the_carrier(boole_lattice(A!16)))) | (below(boole_lattice(A!16), B, C) <=> (join(boole_lattice(A!16), B, C) = C))))) | ((~element(set_intersection2(B!15, B!15), the_carrier(boole_lattice(A!16)))) | ![C: $i] : ((~element(C, the_carrier(boole_lattice(A!16)))) | (below(boole_lattice(A!16), set_intersection2(B!15, B!15), C) <=> (join(boole_lattice(A!16), set_intersection2(B!15, B!15), C) = C))))) <=> ((~![B: $i] : ((~element(B, the_carrier(boole_lattice(A!16)))) | ![C: $i] : ((~element(C, the_carrier(boole_lattice(A!16)))) | (below(boole_lattice(A!16), B, C) <=> (join(boole_lattice(A!16), B, C) = C))))) | (~element(set_intersection2(B!15, B!15), the_carrier(boole_lattice(A!16)))) | ![C: $i] : ((~element(C, the_carrier(boole_lattice(A!16)))) | (below(boole_lattice(A!16), set_intersection2(B!15, B!15), C) <=> (join(boole_lattice(A!16), set_intersection2(B!15, B!15), C) = C))))),
% 119.50/119.66 inference(rewrite,[status(thm)],[])).
% 119.50/119.66 tff(161,plain,
% 119.50/119.66 ((~![B: $i] : ((~element(B, the_carrier(boole_lattice(A!16)))) | ![C: $i] : ((~element(C, the_carrier(boole_lattice(A!16)))) | (below(boole_lattice(A!16), B, C) <=> (join(boole_lattice(A!16), B, C) = C))))) | ((~element(set_intersection2(B!15, B!15), the_carrier(boole_lattice(A!16)))) | ![C: $i] : ((~element(C, the_carrier(boole_lattice(A!16)))) | (below(boole_lattice(A!16), set_intersection2(B!15, B!15), C) <=> (join(boole_lattice(A!16), set_intersection2(B!15, B!15), C) = C))))),
% 119.50/119.66 inference(quant_inst,[status(thm)],[])).
% 119.50/119.66 tff(162,plain,
% 119.50/119.66 ((~![B: $i] : ((~element(B, the_carrier(boole_lattice(A!16)))) | ![C: $i] : ((~element(C, the_carrier(boole_lattice(A!16)))) | (below(boole_lattice(A!16), B, C) <=> (join(boole_lattice(A!16), B, C) = C))))) | (~element(set_intersection2(B!15, B!15), the_carrier(boole_lattice(A!16)))) | ![C: $i] : ((~element(C, the_carrier(boole_lattice(A!16)))) | (below(boole_lattice(A!16), set_intersection2(B!15, B!15), C) <=> (join(boole_lattice(A!16), set_intersection2(B!15, B!15), C) = C)))),
% 119.50/119.66 inference(modus_ponens,[status(thm)],[161, 160])).
% 119.50/119.66 tff(163,plain,
% 119.50/119.66 ((~element(set_intersection2(B!15, B!15), the_carrier(boole_lattice(A!16)))) | ![C: $i] : ((~element(C, the_carrier(boole_lattice(A!16)))) | (below(boole_lattice(A!16), set_intersection2(B!15, B!15), C) <=> (join(boole_lattice(A!16), set_intersection2(B!15, B!15), C) = C)))),
% 119.50/119.66 inference(unit_resolution,[status(thm)],[162, 159])).
% 119.50/119.66 tff(164,plain,
% 119.50/119.66 (![C: $i] : ((~element(C, the_carrier(boole_lattice(A!16)))) | (below(boole_lattice(A!16), set_intersection2(B!15, B!15), C) <=> (join(boole_lattice(A!16), set_intersection2(B!15, B!15), C) = C)))),
% 119.50/119.66 inference(unit_resolution,[status(thm)],[163, 64])).
% 119.50/119.66 tff(165,plain,
% 119.50/119.66 (((~![C: $i] : ((~element(C, the_carrier(boole_lattice(A!16)))) | (below(boole_lattice(A!16), set_intersection2(B!15, B!15), C) <=> (join(boole_lattice(A!16), set_intersection2(B!15, B!15), C) = C)))) | ((~element(set_intersection2(C!17, C!17), the_carrier(boole_lattice(A!16)))) | (below(boole_lattice(A!16), set_intersection2(B!15, B!15), set_intersection2(C!17, C!17)) <=> (join(boole_lattice(A!16), set_intersection2(B!15, B!15), set_intersection2(C!17, C!17)) = set_intersection2(C!17, C!17))))) <=> ((~![C: $i] : ((~element(C, the_carrier(boole_lattice(A!16)))) | (below(boole_lattice(A!16), set_intersection2(B!15, B!15), C) <=> (join(boole_lattice(A!16), set_intersection2(B!15, B!15), C) = C)))) | (~element(set_intersection2(C!17, C!17), the_carrier(boole_lattice(A!16)))) | (below(boole_lattice(A!16), set_intersection2(B!15, B!15), set_intersection2(C!17, C!17)) <=> (join(boole_lattice(A!16), set_intersection2(B!15, B!15), set_intersection2(C!17, C!17)) = set_intersection2(C!17, C!17))))),
% 119.50/119.67 inference(rewrite,[status(thm)],[])).
% 119.50/119.67 tff(166,plain,
% 119.50/119.67 ((~![C: $i] : ((~element(C, the_carrier(boole_lattice(A!16)))) | (below(boole_lattice(A!16), set_intersection2(B!15, B!15), C) <=> (join(boole_lattice(A!16), set_intersection2(B!15, B!15), C) = C)))) | ((~element(set_intersection2(C!17, C!17), the_carrier(boole_lattice(A!16)))) | (below(boole_lattice(A!16), set_intersection2(B!15, B!15), set_intersection2(C!17, C!17)) <=> (join(boole_lattice(A!16), set_intersection2(B!15, B!15), set_intersection2(C!17, C!17)) = set_intersection2(C!17, C!17))))),
% 119.50/119.67 inference(quant_inst,[status(thm)],[])).
% 119.50/119.67 tff(167,plain,
% 119.50/119.67 ((~![C: $i] : ((~element(C, the_carrier(boole_lattice(A!16)))) | (below(boole_lattice(A!16), set_intersection2(B!15, B!15), C) <=> (join(boole_lattice(A!16), set_intersection2(B!15, B!15), C) = C)))) | (~element(set_intersection2(C!17, C!17), the_carrier(boole_lattice(A!16)))) | (below(boole_lattice(A!16), set_intersection2(B!15, B!15), set_intersection2(C!17, C!17)) <=> (join(boole_lattice(A!16), set_intersection2(B!15, B!15), set_intersection2(C!17, C!17)) = set_intersection2(C!17, C!17)))),
% 119.50/119.67 inference(modus_ponens,[status(thm)],[166, 165])).
% 119.50/119.67 tff(168,plain,
% 119.50/119.67 (below(boole_lattice(A!16), set_intersection2(B!15, B!15), set_intersection2(C!17, C!17)) <=> (join(boole_lattice(A!16), set_intersection2(B!15, B!15), set_intersection2(C!17, C!17)) = set_intersection2(C!17, C!17))),
% 119.50/119.67 inference(unit_resolution,[status(thm)],[167, 164, 61])).
% 119.50/119.67 tff(169,plain,
% 119.50/119.67 ((~(below(boole_lattice(A!16), set_intersection2(B!15, B!15), set_intersection2(C!17, C!17)) <=> (join(boole_lattice(A!16), set_intersection2(B!15, B!15), set_intersection2(C!17, C!17)) = set_intersection2(C!17, C!17)))) | (~below(boole_lattice(A!16), set_intersection2(B!15, B!15), set_intersection2(C!17, C!17))) | (join(boole_lattice(A!16), set_intersection2(B!15, B!15), set_intersection2(C!17, C!17)) = set_intersection2(C!17, C!17))),
% 119.50/119.67 inference(tautology,[status(thm)],[])).
% 119.50/119.67 tff(170,plain,
% 119.50/119.67 ((~below(boole_lattice(A!16), set_intersection2(B!15, B!15), set_intersection2(C!17, C!17))) | (join(boole_lattice(A!16), set_intersection2(B!15, B!15), set_intersection2(C!17, C!17)) = set_intersection2(C!17, C!17))),
% 119.50/119.67 inference(unit_resolution,[status(thm)],[169, 168])).
% 119.50/119.67 tff(171,plain,
% 119.50/119.67 (join(boole_lattice(A!16), set_intersection2(B!15, B!15), set_intersection2(C!17, C!17)) = set_intersection2(C!17, C!17)),
% 119.50/119.67 inference(unit_resolution,[status(thm)],[170, 58])).
% 119.50/119.67 tff(172,plain,
% 119.50/119.67 (set_intersection2(C!17, C!17) = join(boole_lattice(A!16), set_intersection2(B!15, B!15), set_intersection2(C!17, C!17))),
% 119.50/119.67 inference(symmetry,[status(thm)],[171])).
% 119.50/119.67 tff(173,plain,
% 119.50/119.67 (C!17 = set_intersection2(C!17, C!17)),
% 119.50/119.67 inference(symmetry,[status(thm)],[47])).
% 119.50/119.67 tff(174,plain,
% 119.50/119.67 (C!17 = set_union2(B!15, C!17)),
% 119.50/119.67 inference(transitivity,[status(thm)],[173, 172, 50, 36])).
% 119.50/119.67 tff(175,plain,
% 119.50/119.67 (subset(B!15, C!17) <=> subset(B!15, set_union2(B!15, C!17))),
% 119.50/119.67 inference(monotonicity,[status(thm)],[174])).
% 119.50/119.67 tff(176,plain,
% 119.50/119.67 (subset(B!15, set_union2(B!15, C!17)) <=> subset(B!15, C!17)),
% 119.50/119.67 inference(symmetry,[status(thm)],[175])).
% 119.50/119.67 tff(177,plain,
% 119.50/119.67 (^[A: $i, B: $i] : refl(subset(A, set_union2(A, B)) <=> subset(A, set_union2(A, B)))),
% 119.50/119.67 inference(bind,[status(th)],[])).
% 119.50/119.67 tff(178,plain,
% 119.50/119.67 (![A: $i, B: $i] : subset(A, set_union2(A, B)) <=> ![A: $i, B: $i] : subset(A, set_union2(A, B))),
% 119.50/119.67 inference(quant_intro,[status(thm)],[177])).
% 119.50/119.67 tff(179,plain,
% 119.50/119.67 (![A: $i, B: $i] : subset(A, set_union2(A, B)) <=> ![A: $i, B: $i] : subset(A, set_union2(A, B))),
% 119.50/119.67 inference(rewrite,[status(thm)],[])).
% 119.50/119.67 tff(180,axiom,(![A: $i, B: $i] : subset(A, set_union2(A, B))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','t7_xboole_1')).
% 119.50/119.67 tff(181,plain,
% 119.50/119.67 (![A: $i, B: $i] : subset(A, set_union2(A, B))),
% 119.50/119.67 inference(modus_ponens,[status(thm)],[180, 179])).
% 119.50/119.67 tff(182,plain,(
% 119.50/119.67 ![A: $i, B: $i] : subset(A, set_union2(A, B))),
% 119.50/119.67 inference(skolemize,[status(sab)],[181])).
% 119.50/119.67 tff(183,plain,
% 119.50/119.67 (![A: $i, B: $i] : subset(A, set_union2(A, B))),
% 119.50/119.67 inference(modus_ponens,[status(thm)],[182, 178])).
% 119.50/119.67 tff(184,plain,
% 119.50/119.67 ((~![A: $i, B: $i] : subset(A, set_union2(A, B))) | subset(B!15, set_union2(B!15, C!17))),
% 119.50/119.67 inference(quant_inst,[status(thm)],[])).
% 119.50/119.67 tff(185,plain,
% 119.50/119.67 (subset(B!15, set_union2(B!15, C!17))),
% 119.50/119.67 inference(unit_resolution,[status(thm)],[184, 183])).
% 119.50/119.67 tff(186,plain,
% 119.50/119.67 (subset(B!15, C!17)),
% 119.50/119.67 inference(modus_ponens,[status(thm)],[185, 176])).
% 119.50/119.67 tff(187,plain,
% 119.50/119.67 ($false),
% 119.50/119.67 inference(unit_resolution,[status(thm)],[53, 186])).
% 119.50/119.67 tff(188,plain,(subset(B!15, C!17)), inference(lemma,lemma(discharge,[]))).
% 119.50/119.67 tff(189,plain,
% 119.50/119.67 (^[A: $i, B: $i] : refl(((~subset(A, B)) | (set_union2(A, B) = B)) <=> ((~subset(A, B)) | (set_union2(A, B) = B)))),
% 119.50/119.67 inference(bind,[status(th)],[])).
% 119.50/119.67 tff(190,plain,
% 119.50/119.67 (![A: $i, B: $i] : ((~subset(A, B)) | (set_union2(A, B) = B)) <=> ![A: $i, B: $i] : ((~subset(A, B)) | (set_union2(A, B) = B))),
% 119.50/119.67 inference(quant_intro,[status(thm)],[189])).
% 119.50/119.67 tff(191,plain,
% 119.50/119.67 (![A: $i, B: $i] : ((~subset(A, B)) | (set_union2(A, B) = B)) <=> ![A: $i, B: $i] : ((~subset(A, B)) | (set_union2(A, B) = B))),
% 119.50/119.67 inference(rewrite,[status(thm)],[])).
% 119.50/119.67 tff(192,plain,
% 119.50/119.67 (^[A: $i, B: $i] : rewrite((subset(A, B) => (set_union2(A, B) = B)) <=> ((~subset(A, B)) | (set_union2(A, B) = B)))),
% 119.50/119.67 inference(bind,[status(th)],[])).
% 119.50/119.67 tff(193,plain,
% 119.50/119.67 (![A: $i, B: $i] : (subset(A, B) => (set_union2(A, B) = B)) <=> ![A: $i, B: $i] : ((~subset(A, B)) | (set_union2(A, B) = B))),
% 119.50/119.67 inference(quant_intro,[status(thm)],[192])).
% 119.50/119.67 tff(194,axiom,(![A: $i, B: $i] : (subset(A, B) => (set_union2(A, B) = B))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','t12_xboole_1')).
% 119.50/119.67 tff(195,plain,
% 119.50/119.67 (![A: $i, B: $i] : ((~subset(A, B)) | (set_union2(A, B) = B))),
% 119.50/119.67 inference(modus_ponens,[status(thm)],[194, 193])).
% 119.50/119.67 tff(196,plain,
% 119.50/119.67 (![A: $i, B: $i] : ((~subset(A, B)) | (set_union2(A, B) = B))),
% 119.50/119.67 inference(modus_ponens,[status(thm)],[195, 191])).
% 119.50/119.67 tff(197,plain,(
% 119.50/119.67 ![A: $i, B: $i] : ((~subset(A, B)) | (set_union2(A, B) = B))),
% 119.50/119.67 inference(skolemize,[status(sab)],[196])).
% 119.50/119.67 tff(198,plain,
% 119.50/119.67 (![A: $i, B: $i] : ((~subset(A, B)) | (set_union2(A, B) = B))),
% 119.50/119.67 inference(modus_ponens,[status(thm)],[197, 190])).
% 119.50/119.67 tff(199,plain,
% 119.50/119.67 (((~![A: $i, B: $i] : ((~subset(A, B)) | (set_union2(A, B) = B))) | ((~subset(B!15, C!17)) | (set_union2(B!15, C!17) = C!17))) <=> ((~![A: $i, B: $i] : ((~subset(A, B)) | (set_union2(A, B) = B))) | (~subset(B!15, C!17)) | (set_union2(B!15, C!17) = C!17))),
% 119.50/119.67 inference(rewrite,[status(thm)],[])).
% 119.50/119.67 tff(200,plain,
% 119.50/119.67 ((~![A: $i, B: $i] : ((~subset(A, B)) | (set_union2(A, B) = B))) | ((~subset(B!15, C!17)) | (set_union2(B!15, C!17) = C!17))),
% 119.50/119.67 inference(quant_inst,[status(thm)],[])).
% 119.50/119.67 tff(201,plain,
% 119.50/119.67 ((~![A: $i, B: $i] : ((~subset(A, B)) | (set_union2(A, B) = B))) | (~subset(B!15, C!17)) | (set_union2(B!15, C!17) = C!17)),
% 119.50/119.67 inference(modus_ponens,[status(thm)],[200, 199])).
% 119.50/119.67 tff(202,plain,
% 119.50/119.67 ((~subset(B!15, C!17)) | (set_union2(B!15, C!17) = C!17)),
% 119.50/119.67 inference(unit_resolution,[status(thm)],[201, 198])).
% 119.50/119.67 tff(203,plain,
% 119.50/119.67 (set_union2(B!15, C!17) = C!17),
% 119.50/119.67 inference(unit_resolution,[status(thm)],[202, 188])).
% 119.50/119.67 tff(204,plain,
% 119.50/119.67 (below(boole_lattice(A!16), set_intersection2(B!15, B!15), set_union2(B!15, C!17)) <=> below(boole_lattice(A!16), B!15, C!17)),
% 119.50/119.67 inference(monotonicity,[status(thm)],[49, 203])).
% 119.50/119.67 tff(205,plain,
% 119.50/119.67 (below(boole_lattice(A!16), B!15, C!17) <=> below(boole_lattice(A!16), set_intersection2(B!15, B!15), set_union2(B!15, C!17))),
% 119.50/119.67 inference(symmetry,[status(thm)],[204])).
% 119.50/119.67 tff(206,plain,
% 119.50/119.67 ((~below(boole_lattice(A!16), B!15, C!17)) <=> (~below(boole_lattice(A!16), set_intersection2(B!15, B!15), set_union2(B!15, C!17)))),
% 119.50/119.67 inference(monotonicity,[status(thm)],[205])).
% 119.50/119.67 tff(207,plain,
% 119.50/119.67 ((below(boole_lattice(A!16), B!15, C!17) <=> subset(B!15, C!17)) | (~below(boole_lattice(A!16), B!15, C!17)) | (~subset(B!15, C!17))),
% 119.50/119.67 inference(tautology,[status(thm)],[])).
% 119.50/119.67 tff(208,plain,
% 119.50/119.67 ((~below(boole_lattice(A!16), B!15, C!17)) | (~subset(B!15, C!17))),
% 119.50/119.67 inference(unit_resolution,[status(thm)],[207, 54])).
% 119.50/119.67 tff(209,plain,
% 119.50/119.67 (~below(boole_lattice(A!16), B!15, C!17)),
% 119.50/119.67 inference(unit_resolution,[status(thm)],[208, 188])).
% 119.50/119.67 tff(210,plain,
% 119.50/119.67 (~below(boole_lattice(A!16), set_intersection2(B!15, B!15), set_union2(B!15, C!17))),
% 119.50/119.67 inference(modus_ponens,[status(thm)],[209, 206])).
% 119.50/119.67 tff(211,plain,
% 119.50/119.67 (the_carrier(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16)))) = the_carrier(boole_lattice(A!16))),
% 119.50/119.67 inference(monotonicity,[status(thm)],[110])).
% 119.50/119.67 tff(212,plain,
% 119.50/119.67 (the_carrier(boole_lattice(A!16)) = the_carrier(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16))))),
% 119.50/119.67 inference(symmetry,[status(thm)],[211])).
% 119.50/119.67 tff(213,plain,
% 119.50/119.67 (join(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16))), C!17, B!15) = join(boole_lattice(A!16), C!17, B!15)),
% 119.50/119.67 inference(monotonicity,[status(thm)],[110])).
% 119.50/119.67 tff(214,plain,
% 119.50/119.67 (join(boole_lattice(A!16), C!17, B!15) = join(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16))), C!17, B!15)),
% 119.50/119.67 inference(symmetry,[status(thm)],[213])).
% 119.50/119.67 tff(215,plain,
% 119.50/119.67 (((~![A: $i, B: $i] : ((~element(B, the_carrier(boole_lattice(A)))) | ![C: $i] : ((~element(C, the_carrier(boole_lattice(A)))) | (~((~(join(boole_lattice(A), B, C) = set_union2(B, C))) | (~(meet(boole_lattice(A), B, C) = set_intersection2(B, C)))))))) | ((~element(C!17, the_carrier(boole_lattice(A!16)))) | ![C: $i] : ((~element(C, the_carrier(boole_lattice(A!16)))) | (~((~(join(boole_lattice(A!16), C!17, C) = set_union2(C!17, C))) | (~(meet(boole_lattice(A!16), C!17, C) = set_intersection2(C!17, C)))))))) <=> ((~![A: $i, B: $i] : ((~element(B, the_carrier(boole_lattice(A)))) | ![C: $i] : ((~element(C, the_carrier(boole_lattice(A)))) | (~((~(join(boole_lattice(A), B, C) = set_union2(B, C))) | (~(meet(boole_lattice(A), B, C) = set_intersection2(B, C)))))))) | (~element(C!17, the_carrier(boole_lattice(A!16)))) | ![C: $i] : ((~element(C, the_carrier(boole_lattice(A!16)))) | (~((~(join(boole_lattice(A!16), C!17, C) = set_union2(C!17, C))) | (~(meet(boole_lattice(A!16), C!17, C) = set_intersection2(C!17, C)))))))),
% 119.50/119.67 inference(rewrite,[status(thm)],[])).
% 119.50/119.67 tff(216,plain,
% 119.50/119.67 ((~![A: $i, B: $i] : ((~element(B, the_carrier(boole_lattice(A)))) | ![C: $i] : ((~element(C, the_carrier(boole_lattice(A)))) | (~((~(join(boole_lattice(A), B, C) = set_union2(B, C))) | (~(meet(boole_lattice(A), B, C) = set_intersection2(B, C)))))))) | ((~element(C!17, the_carrier(boole_lattice(A!16)))) | ![C: $i] : ((~element(C, the_carrier(boole_lattice(A!16)))) | (~((~(join(boole_lattice(A!16), C!17, C) = set_union2(C!17, C))) | (~(meet(boole_lattice(A!16), C!17, C) = set_intersection2(C!17, C)))))))),
% 119.50/119.67 inference(quant_inst,[status(thm)],[])).
% 119.50/119.67 tff(217,plain,
% 119.50/119.67 ((~![A: $i, B: $i] : ((~element(B, the_carrier(boole_lattice(A)))) | ![C: $i] : ((~element(C, the_carrier(boole_lattice(A)))) | (~((~(join(boole_lattice(A), B, C) = set_union2(B, C))) | (~(meet(boole_lattice(A), B, C) = set_intersection2(B, C)))))))) | (~element(C!17, the_carrier(boole_lattice(A!16)))) | ![C: $i] : ((~element(C, the_carrier(boole_lattice(A!16)))) | (~((~(join(boole_lattice(A!16), C!17, C) = set_union2(C!17, C))) | (~(meet(boole_lattice(A!16), C!17, C) = set_intersection2(C!17, C))))))),
% 119.50/119.67 inference(modus_ponens,[status(thm)],[216, 215])).
% 119.50/119.67 tff(218,plain,
% 119.50/119.67 (![C: $i] : ((~element(C, the_carrier(boole_lattice(A!16)))) | (~((~(join(boole_lattice(A!16), C!17, C) = set_union2(C!17, C))) | (~(meet(boole_lattice(A!16), C!17, C) = set_intersection2(C!17, C))))))),
% 119.50/119.67 inference(unit_resolution,[status(thm)],[217, 24, 30])).
% 119.50/119.67 tff(219,plain,
% 119.50/119.67 (((~![C: $i] : ((~element(C, the_carrier(boole_lattice(A!16)))) | (~((~(join(boole_lattice(A!16), C!17, C) = set_union2(C!17, C))) | (~(meet(boole_lattice(A!16), C!17, C) = set_intersection2(C!17, C))))))) | ((~element(B!15, the_carrier(boole_lattice(A!16)))) | (~((~(join(boole_lattice(A!16), C!17, B!15) = set_union2(C!17, B!15))) | (~(meet(boole_lattice(A!16), C!17, B!15) = set_intersection2(C!17, B!15))))))) <=> ((~![C: $i] : ((~element(C, the_carrier(boole_lattice(A!16)))) | (~((~(join(boole_lattice(A!16), C!17, C) = set_union2(C!17, C))) | (~(meet(boole_lattice(A!16), C!17, C) = set_intersection2(C!17, C))))))) | (~element(B!15, the_carrier(boole_lattice(A!16)))) | (~((~(join(boole_lattice(A!16), C!17, B!15) = set_union2(C!17, B!15))) | (~(meet(boole_lattice(A!16), C!17, B!15) = set_intersection2(C!17, B!15))))))),
% 119.50/119.67 inference(rewrite,[status(thm)],[])).
% 119.50/119.67 tff(220,plain,
% 119.50/119.67 ((~![C: $i] : ((~element(C, the_carrier(boole_lattice(A!16)))) | (~((~(join(boole_lattice(A!16), C!17, C) = set_union2(C!17, C))) | (~(meet(boole_lattice(A!16), C!17, C) = set_intersection2(C!17, C))))))) | ((~element(B!15, the_carrier(boole_lattice(A!16)))) | (~((~(join(boole_lattice(A!16), C!17, B!15) = set_union2(C!17, B!15))) | (~(meet(boole_lattice(A!16), C!17, B!15) = set_intersection2(C!17, B!15))))))),
% 119.50/119.67 inference(quant_inst,[status(thm)],[])).
% 119.50/119.67 tff(221,plain,
% 119.50/119.67 ((~![C: $i] : ((~element(C, the_carrier(boole_lattice(A!16)))) | (~((~(join(boole_lattice(A!16), C!17, C) = set_union2(C!17, C))) | (~(meet(boole_lattice(A!16), C!17, C) = set_intersection2(C!17, C))))))) | (~element(B!15, the_carrier(boole_lattice(A!16)))) | (~((~(join(boole_lattice(A!16), C!17, B!15) = set_union2(C!17, B!15))) | (~(meet(boole_lattice(A!16), C!17, B!15) = set_intersection2(C!17, B!15)))))),
% 119.50/119.67 inference(modus_ponens,[status(thm)],[220, 219])).
% 119.50/119.67 tff(222,plain,
% 119.50/119.67 (~((~(join(boole_lattice(A!16), C!17, B!15) = set_union2(C!17, B!15))) | (~(meet(boole_lattice(A!16), C!17, B!15) = set_intersection2(C!17, B!15))))),
% 119.50/119.67 inference(unit_resolution,[status(thm)],[221, 8, 218])).
% 119.50/119.67 tff(223,plain,
% 119.50/119.67 (((~(join(boole_lattice(A!16), C!17, B!15) = set_union2(C!17, B!15))) | (~(meet(boole_lattice(A!16), C!17, B!15) = set_intersection2(C!17, B!15)))) | (join(boole_lattice(A!16), C!17, B!15) = set_union2(C!17, B!15))),
% 119.50/119.67 inference(tautology,[status(thm)],[])).
% 119.50/119.67 tff(224,plain,
% 119.50/119.67 (join(boole_lattice(A!16), C!17, B!15) = set_union2(C!17, B!15)),
% 119.50/119.67 inference(unit_resolution,[status(thm)],[223, 222])).
% 119.50/119.67 tff(225,plain,
% 119.50/119.67 (set_union2(C!17, B!15) = join(boole_lattice(A!16), C!17, B!15)),
% 119.50/119.67 inference(symmetry,[status(thm)],[224])).
% 119.50/119.67 tff(226,plain,
% 119.50/119.67 (^[A: $i, B: $i] : refl((set_union2(A, B) = set_union2(B, A)) <=> (set_union2(A, B) = set_union2(B, A)))),
% 119.50/119.67 inference(bind,[status(th)],[])).
% 119.50/119.67 tff(227,plain,
% 119.50/119.67 (![A: $i, B: $i] : (set_union2(A, B) = set_union2(B, A)) <=> ![A: $i, B: $i] : (set_union2(A, B) = set_union2(B, A))),
% 119.50/119.67 inference(quant_intro,[status(thm)],[226])).
% 119.50/119.67 tff(228,plain,
% 119.50/119.67 (![A: $i, B: $i] : (set_union2(A, B) = set_union2(B, A)) <=> ![A: $i, B: $i] : (set_union2(A, B) = set_union2(B, A))),
% 119.50/119.67 inference(rewrite,[status(thm)],[])).
% 119.50/119.67 tff(229,axiom,(![A: $i, B: $i] : (set_union2(A, B) = set_union2(B, A))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','commutativity_k2_xboole_0')).
% 119.50/119.67 tff(230,plain,
% 119.50/119.67 (![A: $i, B: $i] : (set_union2(A, B) = set_union2(B, A))),
% 119.50/119.67 inference(modus_ponens,[status(thm)],[229, 228])).
% 119.50/119.67 tff(231,plain,(
% 119.50/119.67 ![A: $i, B: $i] : (set_union2(A, B) = set_union2(B, A))),
% 119.50/119.67 inference(skolemize,[status(sab)],[230])).
% 119.50/119.67 tff(232,plain,
% 119.50/119.67 (![A: $i, B: $i] : (set_union2(A, B) = set_union2(B, A))),
% 119.50/119.67 inference(modus_ponens,[status(thm)],[231, 227])).
% 119.50/119.67 tff(233,plain,
% 119.50/119.67 ((~![A: $i, B: $i] : (set_union2(A, B) = set_union2(B, A))) | (set_union2(C!17, B!15) = set_union2(B!15, C!17))),
% 119.50/119.67 inference(quant_inst,[status(thm)],[])).
% 119.50/119.67 tff(234,plain,
% 119.50/119.67 (set_union2(C!17, B!15) = set_union2(B!15, C!17)),
% 119.50/119.67 inference(unit_resolution,[status(thm)],[233, 232])).
% 119.50/119.67 tff(235,plain,
% 119.50/119.67 (set_union2(B!15, C!17) = set_union2(C!17, B!15)),
% 119.50/119.67 inference(symmetry,[status(thm)],[234])).
% 119.50/119.67 tff(236,plain,
% 119.50/119.67 (set_union2(B!15, C!17) = join(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16))), C!17, B!15)),
% 119.50/119.67 inference(transitivity,[status(thm)],[235, 225, 214])).
% 119.50/119.67 tff(237,plain,
% 119.50/119.67 (element(set_union2(B!15, C!17), the_carrier(boole_lattice(A!16))) <=> element(join(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16))), C!17, B!15), the_carrier(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16)))))),
% 119.50/119.67 inference(monotonicity,[status(thm)],[236, 212])).
% 119.50/119.67 tff(238,plain,
% 119.50/119.67 (element(join(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16))), C!17, B!15), the_carrier(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16))))) <=> element(set_union2(B!15, C!17), the_carrier(boole_lattice(A!16)))),
% 119.50/119.67 inference(symmetry,[status(thm)],[237])).
% 119.50/119.67 tff(239,plain,
% 119.50/119.67 (empty_carrier(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16)))) <=> empty_carrier(boole_lattice(A!16))),
% 119.50/119.67 inference(monotonicity,[status(thm)],[110])).
% 119.50/119.67 tff(240,plain,
% 119.50/119.67 (empty_carrier(boole_lattice(A!16)) <=> empty_carrier(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16))))),
% 119.50/119.67 inference(symmetry,[status(thm)],[239])).
% 119.50/119.67 tff(241,plain,
% 119.50/119.67 ((~empty_carrier(boole_lattice(A!16))) <=> (~empty_carrier(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16)))))),
% 119.50/119.67 inference(monotonicity,[status(thm)],[240])).
% 119.50/119.67 tff(242,plain,
% 119.50/119.67 (~empty_carrier(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16))))),
% 119.50/119.67 inference(modus_ponens,[status(thm)],[78, 241])).
% 119.50/119.67 tff(243,plain,
% 119.50/119.67 (element(C!17, the_carrier(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16))))) <=> element(C!17, the_carrier(boole_lattice(A!16)))),
% 119.50/119.67 inference(monotonicity,[status(thm)],[211])).
% 119.50/119.67 tff(244,plain,
% 119.50/119.67 (element(C!17, the_carrier(boole_lattice(A!16))) <=> element(C!17, the_carrier(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16)))))),
% 119.50/119.67 inference(symmetry,[status(thm)],[243])).
% 119.50/119.67 tff(245,plain,
% 119.50/119.67 (element(C!17, the_carrier(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16)))))),
% 119.50/119.67 inference(modus_ponens,[status(thm)],[30, 244])).
% 119.50/119.67 tff(246,plain,
% 119.50/119.67 (element(B!15, the_carrier(boole_lattice(A!16))) <=> element(B!15, the_carrier(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16)))))),
% 119.50/119.67 inference(monotonicity,[status(thm)],[212])).
% 119.50/119.67 tff(247,plain,
% 119.50/119.67 (element(B!15, the_carrier(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16)))))),
% 119.50/119.67 inference(modus_ponens,[status(thm)],[8, 246])).
% 119.50/119.67 tff(248,plain,
% 119.50/119.67 (^[A: $i, B: $i, C: $i] : refl((element(join(A, B, C), the_carrier(A)) | empty_carrier(A) | (~element(C, the_carrier(A))) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A)))) <=> (element(join(A, B, C), the_carrier(A)) | empty_carrier(A) | (~element(C, the_carrier(A))) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A)))))),
% 119.50/119.67 inference(bind,[status(th)],[])).
% 119.50/119.67 tff(249,plain,
% 119.50/119.67 (![A: $i, B: $i, C: $i] : (element(join(A, B, C), the_carrier(A)) | empty_carrier(A) | (~element(C, the_carrier(A))) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A)))) <=> ![A: $i, B: $i, C: $i] : (element(join(A, B, C), the_carrier(A)) | empty_carrier(A) | (~element(C, the_carrier(A))) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A))))),
% 119.50/119.67 inference(quant_intro,[status(thm)],[248])).
% 119.50/119.67 tff(250,plain,
% 119.50/119.67 (^[A: $i, B: $i, C: $i] : trans(monotonicity(trans(monotonicity(rewrite(((~empty_carrier(A)) & join_semilatt_str(A) & element(B, the_carrier(A)) & element(C, the_carrier(A))) <=> (~(empty_carrier(A) | (~element(C, the_carrier(A))) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A)))))), ((~((~empty_carrier(A)) & join_semilatt_str(A) & element(B, the_carrier(A)) & element(C, the_carrier(A)))) <=> (~(~(empty_carrier(A) | (~element(C, the_carrier(A))) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A)))))))), rewrite((~(~(empty_carrier(A) | (~element(C, the_carrier(A))) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A)))))) <=> (empty_carrier(A) | (~element(C, the_carrier(A))) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A))))), ((~((~empty_carrier(A)) & join_semilatt_str(A) & element(B, the_carrier(A)) & element(C, the_carrier(A)))) <=> (empty_carrier(A) | (~element(C, the_carrier(A))) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A)))))), (((~((~empty_carrier(A)) & join_semilatt_str(A) & element(B, the_carrier(A)) & element(C, the_carrier(A)))) | element(join(A, B, C), the_carrier(A))) <=> ((empty_carrier(A) | (~element(C, the_carrier(A))) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A)))) | element(join(A, B, C), the_carrier(A))))), rewrite(((empty_carrier(A) | (~element(C, the_carrier(A))) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A)))) | element(join(A, B, C), the_carrier(A))) <=> (element(join(A, B, C), the_carrier(A)) | empty_carrier(A) | (~element(C, the_carrier(A))) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A))))), (((~((~empty_carrier(A)) & join_semilatt_str(A) & element(B, the_carrier(A)) & element(C, the_carrier(A)))) | element(join(A, B, C), the_carrier(A))) <=> (element(join(A, B, C), the_carrier(A)) | empty_carrier(A) | (~element(C, the_carrier(A))) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A))))))),
% 119.50/119.67 inference(bind,[status(th)],[])).
% 119.50/119.67 tff(251,plain,
% 119.50/119.67 (![A: $i, B: $i, C: $i] : ((~((~empty_carrier(A)) & join_semilatt_str(A) & element(B, the_carrier(A)) & element(C, the_carrier(A)))) | element(join(A, B, C), the_carrier(A))) <=> ![A: $i, B: $i, C: $i] : (element(join(A, B, C), the_carrier(A)) | empty_carrier(A) | (~element(C, the_carrier(A))) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A))))),
% 119.50/119.67 inference(quant_intro,[status(thm)],[250])).
% 119.50/119.67 tff(252,plain,
% 119.50/119.67 (![A: $i, B: $i, C: $i] : ((~((~empty_carrier(A)) & join_semilatt_str(A) & element(B, the_carrier(A)) & element(C, the_carrier(A)))) | element(join(A, B, C), the_carrier(A))) <=> ![A: $i, B: $i, C: $i] : ((~((~empty_carrier(A)) & join_semilatt_str(A) & element(B, the_carrier(A)) & element(C, the_carrier(A)))) | element(join(A, B, C), the_carrier(A)))),
% 119.50/119.67 inference(rewrite,[status(thm)],[])).
% 119.50/119.67 tff(253,plain,
% 119.50/119.67 (^[A: $i, B: $i, C: $i] : trans(monotonicity(trans(monotonicity(rewrite((((~empty_carrier(A)) & join_semilatt_str(A)) & element(B, the_carrier(A))) <=> ((~empty_carrier(A)) & join_semilatt_str(A) & element(B, the_carrier(A)))), (((((~empty_carrier(A)) & join_semilatt_str(A)) & element(B, the_carrier(A))) & element(C, the_carrier(A))) <=> (((~empty_carrier(A)) & join_semilatt_str(A) & element(B, the_carrier(A))) & element(C, the_carrier(A))))), rewrite((((~empty_carrier(A)) & join_semilatt_str(A) & element(B, the_carrier(A))) & element(C, the_carrier(A))) <=> ((~empty_carrier(A)) & join_semilatt_str(A) & element(B, the_carrier(A)) & element(C, the_carrier(A)))), (((((~empty_carrier(A)) & join_semilatt_str(A)) & element(B, the_carrier(A))) & element(C, the_carrier(A))) <=> ((~empty_carrier(A)) & join_semilatt_str(A) & element(B, the_carrier(A)) & element(C, the_carrier(A))))), ((((((~empty_carrier(A)) & join_semilatt_str(A)) & element(B, the_carrier(A))) & element(C, the_carrier(A))) => element(join(A, B, C), the_carrier(A))) <=> (((~empty_carrier(A)) & join_semilatt_str(A) & element(B, the_carrier(A)) & element(C, the_carrier(A))) => element(join(A, B, C), the_carrier(A))))), rewrite((((~empty_carrier(A)) & join_semilatt_str(A) & element(B, the_carrier(A)) & element(C, the_carrier(A))) => element(join(A, B, C), the_carrier(A))) <=> ((~((~empty_carrier(A)) & join_semilatt_str(A) & element(B, the_carrier(A)) & element(C, the_carrier(A)))) | element(join(A, B, C), the_carrier(A)))), ((((((~empty_carrier(A)) & join_semilatt_str(A)) & element(B, the_carrier(A))) & element(C, the_carrier(A))) => element(join(A, B, C), the_carrier(A))) <=> ((~((~empty_carrier(A)) & join_semilatt_str(A) & element(B, the_carrier(A)) & element(C, the_carrier(A)))) | element(join(A, B, C), the_carrier(A)))))),
% 119.50/119.67 inference(bind,[status(th)],[])).
% 119.50/119.67 tff(254,plain,
% 119.50/119.67 (![A: $i, B: $i, C: $i] : (((((~empty_carrier(A)) & join_semilatt_str(A)) & element(B, the_carrier(A))) & element(C, the_carrier(A))) => element(join(A, B, C), the_carrier(A))) <=> ![A: $i, B: $i, C: $i] : ((~((~empty_carrier(A)) & join_semilatt_str(A) & element(B, the_carrier(A)) & element(C, the_carrier(A)))) | element(join(A, B, C), the_carrier(A)))),
% 119.50/119.67 inference(quant_intro,[status(thm)],[253])).
% 119.50/119.67 tff(255,axiom,(![A: $i, B: $i, C: $i] : (((((~empty_carrier(A)) & join_semilatt_str(A)) & element(B, the_carrier(A))) & element(C, the_carrier(A))) => element(join(A, B, C), the_carrier(A)))), file('/export/starexec/sandbox/benchmark/theBenchmark.p','dt_k1_lattices')).
% 119.50/119.67 tff(256,plain,
% 119.50/119.67 (![A: $i, B: $i, C: $i] : ((~((~empty_carrier(A)) & join_semilatt_str(A) & element(B, the_carrier(A)) & element(C, the_carrier(A)))) | element(join(A, B, C), the_carrier(A)))),
% 119.50/119.67 inference(modus_ponens,[status(thm)],[255, 254])).
% 119.50/119.67 tff(257,plain,
% 119.50/119.67 (![A: $i, B: $i, C: $i] : ((~((~empty_carrier(A)) & join_semilatt_str(A) & element(B, the_carrier(A)) & element(C, the_carrier(A)))) | element(join(A, B, C), the_carrier(A)))),
% 119.50/119.67 inference(modus_ponens,[status(thm)],[256, 252])).
% 119.50/119.67 tff(258,plain,(
% 119.50/119.67 ![A: $i, B: $i, C: $i] : ((~((~empty_carrier(A)) & join_semilatt_str(A) & element(B, the_carrier(A)) & element(C, the_carrier(A)))) | element(join(A, B, C), the_carrier(A)))),
% 119.50/119.67 inference(skolemize,[status(sab)],[257])).
% 119.50/119.67 tff(259,plain,
% 119.50/119.67 (![A: $i, B: $i, C: $i] : (element(join(A, B, C), the_carrier(A)) | empty_carrier(A) | (~element(C, the_carrier(A))) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A))))),
% 119.50/119.67 inference(modus_ponens,[status(thm)],[258, 251])).
% 119.50/119.67 tff(260,plain,
% 119.50/119.67 (![A: $i, B: $i, C: $i] : (element(join(A, B, C), the_carrier(A)) | empty_carrier(A) | (~element(C, the_carrier(A))) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A))))),
% 119.50/119.67 inference(modus_ponens,[status(thm)],[259, 249])).
% 119.50/119.67 tff(261,plain,
% 119.50/119.67 (((~![A: $i, B: $i, C: $i] : (element(join(A, B, C), the_carrier(A)) | empty_carrier(A) | (~element(C, the_carrier(A))) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A))))) | ((~join_semilatt_str(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16))))) | empty_carrier(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16)))) | (~element(B!15, the_carrier(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16)))))) | (~element(C!17, the_carrier(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16)))))) | element(join(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16))), C!17, B!15), the_carrier(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16))))))) <=> ((~![A: $i, B: $i, C: $i] : (element(join(A, B, C), the_carrier(A)) | empty_carrier(A) | (~element(C, the_carrier(A))) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A))))) | (~join_semilatt_str(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16))))) | empty_carrier(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16)))) | (~element(B!15, the_carrier(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16)))))) | (~element(C!17, the_carrier(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16)))))) | element(join(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16))), C!17, B!15), the_carrier(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16))))))),
% 119.50/119.67 inference(rewrite,[status(thm)],[])).
% 119.50/119.67 tff(262,plain,
% 119.50/119.67 ((element(join(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16))), C!17, B!15), the_carrier(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16))))) | empty_carrier(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16)))) | (~element(B!15, the_carrier(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16)))))) | (~join_semilatt_str(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16))))) | (~element(C!17, the_carrier(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16))))))) <=> ((~join_semilatt_str(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16))))) | empty_carrier(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16)))) | (~element(B!15, the_carrier(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16)))))) | (~element(C!17, the_carrier(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16)))))) | element(join(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16))), C!17, B!15), the_carrier(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16))))))),
% 119.50/119.67 inference(rewrite,[status(thm)],[])).
% 119.50/119.67 tff(263,plain,
% 119.50/119.67 (((~![A: $i, B: $i, C: $i] : (element(join(A, B, C), the_carrier(A)) | empty_carrier(A) | (~element(C, the_carrier(A))) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A))))) | (element(join(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16))), C!17, B!15), the_carrier(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16))))) | empty_carrier(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16)))) | (~element(B!15, the_carrier(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16)))))) | (~join_semilatt_str(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16))))) | (~element(C!17, the_carrier(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16)))))))) <=> ((~![A: $i, B: $i, C: $i] : (element(join(A, B, C), the_carrier(A)) | empty_carrier(A) | (~element(C, the_carrier(A))) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A))))) | ((~join_semilatt_str(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16))))) | empty_carrier(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16)))) | (~element(B!15, the_carrier(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16)))))) | (~element(C!17, the_carrier(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16)))))) | element(join(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16))), C!17, B!15), the_carrier(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16)))))))),
% 119.50/119.67 inference(monotonicity,[status(thm)],[262])).
% 119.50/119.67 tff(264,plain,
% 119.50/119.67 (((~![A: $i, B: $i, C: $i] : (element(join(A, B, C), the_carrier(A)) | empty_carrier(A) | (~element(C, the_carrier(A))) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A))))) | (element(join(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16))), C!17, B!15), the_carrier(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16))))) | empty_carrier(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16)))) | (~element(B!15, the_carrier(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16)))))) | (~join_semilatt_str(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16))))) | (~element(C!17, the_carrier(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16)))))))) <=> ((~![A: $i, B: $i, C: $i] : (element(join(A, B, C), the_carrier(A)) | empty_carrier(A) | (~element(C, the_carrier(A))) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A))))) | (~join_semilatt_str(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16))))) | empty_carrier(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16)))) | (~element(B!15, the_carrier(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16)))))) | (~element(C!17, the_carrier(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16)))))) | element(join(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16))), C!17, B!15), the_carrier(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16))))))),
% 119.50/119.67 inference(transitivity,[status(thm)],[263, 261])).
% 119.50/119.67 tff(265,plain,
% 119.50/119.67 ((~![A: $i, B: $i, C: $i] : (element(join(A, B, C), the_carrier(A)) | empty_carrier(A) | (~element(C, the_carrier(A))) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A))))) | (element(join(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16))), C!17, B!15), the_carrier(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16))))) | empty_carrier(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16)))) | (~element(B!15, the_carrier(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16)))))) | (~join_semilatt_str(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16))))) | (~element(C!17, the_carrier(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16)))))))),
% 119.50/119.67 inference(quant_inst,[status(thm)],[])).
% 119.50/119.67 tff(266,plain,
% 119.50/119.67 ((~![A: $i, B: $i, C: $i] : (element(join(A, B, C), the_carrier(A)) | empty_carrier(A) | (~element(C, the_carrier(A))) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A))))) | (~join_semilatt_str(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16))))) | empty_carrier(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16)))) | (~element(B!15, the_carrier(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16)))))) | (~element(C!17, the_carrier(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16)))))) | element(join(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16))), C!17, B!15), the_carrier(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16)))))),
% 119.50/119.68 inference(modus_ponens,[status(thm)],[265, 264])).
% 119.50/119.68 tff(267,plain,
% 119.50/119.68 ((~join_semilatt_str(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16))))) | empty_carrier(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16)))) | (~element(B!15, the_carrier(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16)))))) | (~element(C!17, the_carrier(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16)))))) | element(join(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16))), C!17, B!15), the_carrier(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16)))))),
% 119.50/119.68 inference(unit_resolution,[status(thm)],[266, 260])).
% 119.50/119.68 tff(268,plain,
% 119.50/119.68 (element(join(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16))), C!17, B!15), the_carrier(latt_str_of(the_carrier(boole_lattice(A!16)), the_L_join(boole_lattice(A!16)), the_L_meet(boole_lattice(A!16)))))),
% 119.50/119.68 inference(unit_resolution,[status(thm)],[267, 247, 134, 245, 242])).
% 119.50/119.68 tff(269,plain,
% 119.50/119.68 (element(set_union2(B!15, C!17), the_carrier(boole_lattice(A!16)))),
% 119.50/119.68 inference(modus_ponens,[status(thm)],[268, 238])).
% 119.50/119.68 tff(270,plain,
% 119.50/119.68 (((~![C: $i] : ((~element(C, the_carrier(boole_lattice(A!16)))) | (below(boole_lattice(A!16), set_intersection2(B!15, B!15), C) <=> (join(boole_lattice(A!16), set_intersection2(B!15, B!15), C) = C)))) | ((~element(set_union2(B!15, C!17), the_carrier(boole_lattice(A!16)))) | (below(boole_lattice(A!16), set_intersection2(B!15, B!15), set_union2(B!15, C!17)) <=> (join(boole_lattice(A!16), set_intersection2(B!15, B!15), set_union2(B!15, C!17)) = set_union2(B!15, C!17))))) <=> ((~![C: $i] : ((~element(C, the_carrier(boole_lattice(A!16)))) | (below(boole_lattice(A!16), set_intersection2(B!15, B!15), C) <=> (join(boole_lattice(A!16), set_intersection2(B!15, B!15), C) = C)))) | (~element(set_union2(B!15, C!17), the_carrier(boole_lattice(A!16)))) | (below(boole_lattice(A!16), set_intersection2(B!15, B!15), set_union2(B!15, C!17)) <=> (join(boole_lattice(A!16), set_intersection2(B!15, B!15), set_union2(B!15, C!17)) = set_union2(B!15, C!17))))),
% 119.50/119.68 inference(rewrite,[status(thm)],[])).
% 119.50/119.68 tff(271,plain,
% 119.50/119.68 ((~![C: $i] : ((~element(C, the_carrier(boole_lattice(A!16)))) | (below(boole_lattice(A!16), set_intersection2(B!15, B!15), C) <=> (join(boole_lattice(A!16), set_intersection2(B!15, B!15), C) = C)))) | ((~element(set_union2(B!15, C!17), the_carrier(boole_lattice(A!16)))) | (below(boole_lattice(A!16), set_intersection2(B!15, B!15), set_union2(B!15, C!17)) <=> (join(boole_lattice(A!16), set_intersection2(B!15, B!15), set_union2(B!15, C!17)) = set_union2(B!15, C!17))))),
% 119.50/119.68 inference(quant_inst,[status(thm)],[])).
% 119.50/119.68 tff(272,plain,
% 119.50/119.68 ((~![C: $i] : ((~element(C, the_carrier(boole_lattice(A!16)))) | (below(boole_lattice(A!16), set_intersection2(B!15, B!15), C) <=> (join(boole_lattice(A!16), set_intersection2(B!15, B!15), C) = C)))) | (~element(set_union2(B!15, C!17), the_carrier(boole_lattice(A!16)))) | (below(boole_lattice(A!16), set_intersection2(B!15, B!15), set_union2(B!15, C!17)) <=> (join(boole_lattice(A!16), set_intersection2(B!15, B!15), set_union2(B!15, C!17)) = set_union2(B!15, C!17)))),
% 119.50/119.68 inference(modus_ponens,[status(thm)],[271, 270])).
% 119.50/119.68 tff(273,plain,
% 119.50/119.68 (below(boole_lattice(A!16), set_intersection2(B!15, B!15), set_union2(B!15, C!17)) <=> (join(boole_lattice(A!16), set_intersection2(B!15, B!15), set_union2(B!15, C!17)) = set_union2(B!15, C!17))),
% 119.50/119.68 inference(unit_resolution,[status(thm)],[272, 269, 164])).
% 119.50/119.68 tff(274,plain,
% 119.50/119.68 (subset(set_intersection2(B!15, B!15), set_union2(B!15, C!17)) <=> subset(B!15, set_union2(B!15, C!17))),
% 119.50/119.68 inference(monotonicity,[status(thm)],[49])).
% 119.50/119.68 tff(275,plain,
% 119.50/119.68 (subset(B!15, set_union2(B!15, C!17)) <=> subset(set_intersection2(B!15, B!15), set_union2(B!15, C!17))),
% 119.50/119.68 inference(symmetry,[status(thm)],[274])).
% 119.50/119.68 tff(276,plain,
% 119.50/119.68 (subset(set_intersection2(B!15, B!15), set_union2(B!15, C!17))),
% 119.50/119.68 inference(modus_ponens,[status(thm)],[185, 275])).
% 119.50/119.68 tff(277,plain,
% 119.50/119.68 (((~![A: $i, B: $i] : ((~subset(A, B)) | (set_union2(A, B) = B))) | ((~subset(set_intersection2(B!15, B!15), set_union2(B!15, C!17))) | (set_union2(set_intersection2(B!15, B!15), set_union2(B!15, C!17)) = set_union2(B!15, C!17)))) <=> ((~![A: $i, B: $i] : ((~subset(A, B)) | (set_union2(A, B) = B))) | (~subset(set_intersection2(B!15, B!15), set_union2(B!15, C!17))) | (set_union2(set_intersection2(B!15, B!15), set_union2(B!15, C!17)) = set_union2(B!15, C!17)))),
% 119.50/119.68 inference(rewrite,[status(thm)],[])).
% 119.50/119.68 tff(278,plain,
% 119.50/119.68 ((~![A: $i, B: $i] : ((~subset(A, B)) | (set_union2(A, B) = B))) | ((~subset(set_intersection2(B!15, B!15), set_union2(B!15, C!17))) | (set_union2(set_intersection2(B!15, B!15), set_union2(B!15, C!17)) = set_union2(B!15, C!17)))),
% 119.50/119.68 inference(quant_inst,[status(thm)],[])).
% 119.50/119.68 tff(279,plain,
% 119.50/119.68 ((~![A: $i, B: $i] : ((~subset(A, B)) | (set_union2(A, B) = B))) | (~subset(set_intersection2(B!15, B!15), set_union2(B!15, C!17))) | (set_union2(set_intersection2(B!15, B!15), set_union2(B!15, C!17)) = set_union2(B!15, C!17))),
% 119.50/119.68 inference(modus_ponens,[status(thm)],[278, 277])).
% 119.50/119.68 tff(280,plain,
% 119.50/119.68 ((~subset(set_intersection2(B!15, B!15), set_union2(B!15, C!17))) | (set_union2(set_intersection2(B!15, B!15), set_union2(B!15, C!17)) = set_union2(B!15, C!17))),
% 119.50/119.68 inference(unit_resolution,[status(thm)],[279, 198])).
% 119.50/119.68 tff(281,plain,
% 119.50/119.68 (set_union2(set_intersection2(B!15, B!15), set_union2(B!15, C!17)) = set_union2(B!15, C!17)),
% 119.50/119.68 inference(unit_resolution,[status(thm)],[280, 276])).
% 119.50/119.68 tff(282,plain,
% 119.50/119.68 (B!15 = set_intersection2(B!15, B!15)),
% 119.50/119.68 inference(symmetry,[status(thm)],[49])).
% 119.50/119.68 tff(283,plain,
% 119.50/119.68 (set_union2(B!15, set_union2(B!15, C!17)) = set_union2(set_intersection2(B!15, B!15), set_union2(B!15, C!17))),
% 119.50/119.68 inference(monotonicity,[status(thm)],[282])).
% 119.50/119.68 tff(284,plain,
% 119.50/119.68 (((~![C: $i] : ((~element(C, the_carrier(boole_lattice(A!16)))) | (~((~(join(boole_lattice(A!16), B!15, C) = set_union2(B!15, C))) | (~(meet(boole_lattice(A!16), B!15, C) = set_intersection2(B!15, C))))))) | ((~element(set_union2(B!15, C!17), the_carrier(boole_lattice(A!16)))) | (~((~(join(boole_lattice(A!16), B!15, set_union2(B!15, C!17)) = set_union2(B!15, set_union2(B!15, C!17)))) | (~(meet(boole_lattice(A!16), B!15, set_union2(B!15, C!17)) = set_intersection2(B!15, set_union2(B!15, C!17)))))))) <=> ((~![C: $i] : ((~element(C, the_carrier(boole_lattice(A!16)))) | (~((~(join(boole_lattice(A!16), B!15, C) = set_union2(B!15, C))) | (~(meet(boole_lattice(A!16), B!15, C) = set_intersection2(B!15, C))))))) | (~element(set_union2(B!15, C!17), the_carrier(boole_lattice(A!16)))) | (~((~(join(boole_lattice(A!16), B!15, set_union2(B!15, C!17)) = set_union2(B!15, set_union2(B!15, C!17)))) | (~(meet(boole_lattice(A!16), B!15, set_union2(B!15, C!17)) = set_intersection2(B!15, set_union2(B!15, C!17)))))))),
% 119.50/119.68 inference(rewrite,[status(thm)],[])).
% 119.50/119.68 tff(285,plain,
% 119.50/119.68 ((~![C: $i] : ((~element(C, the_carrier(boole_lattice(A!16)))) | (~((~(join(boole_lattice(A!16), B!15, C) = set_union2(B!15, C))) | (~(meet(boole_lattice(A!16), B!15, C) = set_intersection2(B!15, C))))))) | ((~element(set_union2(B!15, C!17), the_carrier(boole_lattice(A!16)))) | (~((~(join(boole_lattice(A!16), B!15, set_union2(B!15, C!17)) = set_union2(B!15, set_union2(B!15, C!17)))) | (~(meet(boole_lattice(A!16), B!15, set_union2(B!15, C!17)) = set_intersection2(B!15, set_union2(B!15, C!17)))))))),
% 119.50/119.68 inference(quant_inst,[status(thm)],[])).
% 119.50/119.68 tff(286,plain,
% 119.50/119.68 ((~![C: $i] : ((~element(C, the_carrier(boole_lattice(A!16)))) | (~((~(join(boole_lattice(A!16), B!15, C) = set_union2(B!15, C))) | (~(meet(boole_lattice(A!16), B!15, C) = set_intersection2(B!15, C))))))) | (~element(set_union2(B!15, C!17), the_carrier(boole_lattice(A!16)))) | (~((~(join(boole_lattice(A!16), B!15, set_union2(B!15, C!17)) = set_union2(B!15, set_union2(B!15, C!17)))) | (~(meet(boole_lattice(A!16), B!15, set_union2(B!15, C!17)) = set_intersection2(B!15, set_union2(B!15, C!17))))))),
% 119.80/119.97 inference(modus_ponens,[status(thm)],[285, 284])).
% 119.80/119.97 tff(287,plain,
% 119.80/119.97 (~((~(join(boole_lattice(A!16), B!15, set_union2(B!15, C!17)) = set_union2(B!15, set_union2(B!15, C!17)))) | (~(meet(boole_lattice(A!16), B!15, set_union2(B!15, C!17)) = set_intersection2(B!15, set_union2(B!15, C!17)))))),
% 119.80/119.97 inference(unit_resolution,[status(thm)],[286, 28, 269])).
% 119.80/119.97 tff(288,plain,
% 119.80/119.97 (((~(join(boole_lattice(A!16), B!15, set_union2(B!15, C!17)) = set_union2(B!15, set_union2(B!15, C!17)))) | (~(meet(boole_lattice(A!16), B!15, set_union2(B!15, C!17)) = set_intersection2(B!15, set_union2(B!15, C!17))))) | (join(boole_lattice(A!16), B!15, set_union2(B!15, C!17)) = set_union2(B!15, set_union2(B!15, C!17)))),
% 119.80/119.97 inference(tautology,[status(thm)],[])).
% 119.80/119.97 tff(289,plain,
% 119.80/119.97 (join(boole_lattice(A!16), B!15, set_union2(B!15, C!17)) = set_union2(B!15, set_union2(B!15, C!17))),
% 119.80/119.97 inference(unit_resolution,[status(thm)],[288, 287])).
% 119.80/119.97 tff(290,plain,
% 119.80/119.97 (join(boole_lattice(A!16), set_intersection2(B!15, B!15), set_union2(B!15, C!17)) = join(boole_lattice(A!16), B!15, set_union2(B!15, C!17))),
% 119.80/119.97 inference(monotonicity,[status(thm)],[49])).
% 119.80/119.97 tff(291,plain,
% 119.80/119.97 (join(boole_lattice(A!16), set_intersection2(B!15, B!15), set_union2(B!15, C!17)) = set_union2(B!15, C!17)),
% 119.80/119.97 inference(transitivity,[status(thm)],[290, 289, 283, 281])).
% 119.80/119.97 tff(292,plain,
% 119.80/119.97 ((~(below(boole_lattice(A!16), set_intersection2(B!15, B!15), set_union2(B!15, C!17)) <=> (join(boole_lattice(A!16), set_intersection2(B!15, B!15), set_union2(B!15, C!17)) = set_union2(B!15, C!17)))) | below(boole_lattice(A!16), set_intersection2(B!15, B!15), set_union2(B!15, C!17)) | (~(join(boole_lattice(A!16), set_intersection2(B!15, B!15), set_union2(B!15, C!17)) = set_union2(B!15, C!17)))),
% 119.80/119.97 inference(tautology,[status(thm)],[])).
% 119.80/119.97 tff(293,plain,
% 119.80/119.97 (below(boole_lattice(A!16), set_intersection2(B!15, B!15), set_union2(B!15, C!17))),
% 119.80/119.97 inference(unit_resolution,[status(thm)],[292, 291, 273])).
% 119.80/119.97 tff(294,plain,
% 119.80/119.97 ($false),
% 119.80/119.97 inference(unit_resolution,[status(thm)],[293, 210])).
% 119.80/119.97 % SZS output end Proof
% 119.80/120.03 % E exiting
%------------------------------------------------------------------------------