%------------------------------------------------------------------------------
% File : Z3---4.15.1
% Problem : SEU305+1 : TPTP v9.0.0. Released v3.3.0.
% Transfm : none
% Format : tptp
% Command : run_E %s %d THM
% Computer : n007.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 05:27:33 AM UTC 2025
% Result : Theorem 0.21s 0.39s
% Output : Proof 0.21s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.12 % Problem : SEU305+1 : TPTP v9.0.0. Released v3.3.0.
% 0.03/0.12 % Command : run_E %s %d THM
% 0.13/0.34 % Computer : n007.cluster.edu
% 0.13/0.34 % Model : x86_64 x86_64
% 0.13/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.34 % Memory : 8042.1875MB
% 0.13/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.13/0.34 % CPULimit : 300
% 0.13/0.34 % WCLimit : 300
% 0.13/0.34 % DateTime : Fri Jun 20 06:42:35 EDT 2025
% 0.13/0.34 % CPUTime :
% 0.21/0.39 % SZS status Theorem
% 0.21/0.39 % SZS output start Proof
% 0.21/0.39 tff(tptp_fun_C_9_type, type, (
% 0.21/0.39 tptp_fun_C_9: $i)).
% 0.21/0.39 tff(tptp_fun_B_8_type, type, (
% 0.21/0.39 tptp_fun_B_8: $i)).
% 0.21/0.39 tff(join_type, type, (
% 0.21/0.39 join: ( $i * $i * $i ) > $i)).
% 0.21/0.39 tff(tptp_fun_A_7_type, type, (
% 0.21/0.39 tptp_fun_A_7: $i)).
% 0.21/0.39 tff(below_type, type, (
% 0.21/0.39 below: ( $i * $i * $i ) > $o)).
% 0.21/0.39 tff(element_type, type, (
% 0.21/0.39 element: ( $i * $i ) > $o)).
% 0.21/0.39 tff(the_carrier_type, type, (
% 0.21/0.39 the_carrier: $i > $i)).
% 0.21/0.39 tff(join_semilatt_str_type, type, (
% 0.21/0.39 join_semilatt_str: $i > $o)).
% 0.21/0.39 tff(join_commutative_type, type, (
% 0.21/0.39 join_commutative: $i > $o)).
% 0.21/0.39 tff(empty_carrier_type, type, (
% 0.21/0.39 empty_carrier: $i > $o)).
% 0.21/0.39 tff(apply_binary_as_element_type, type, (
% 0.21/0.39 apply_binary_as_element: ( $i * $i * $i * $i * $i * $i ) > $i)).
% 0.21/0.39 tff(the_L_join_type, type, (
% 0.21/0.39 the_L_join: $i > $i)).
% 0.21/0.39 tff(join_commut_type, type, (
% 0.21/0.39 join_commut: ( $i * $i * $i ) > $i)).
% 0.21/0.39 tff(1,plain,
% 0.21/0.39 ((((~empty_carrier(A!7)) & join_commutative(A!7) & join_semilatt_str(A!7)) & (element(B!8, the_carrier(A!7)) & (~((B!8 = C!9) | (~element(C!9, the_carrier(A!7))) | (~(below(A!7, B!8, C!9) & below(A!7, C!9, B!8))))))) <=> ((~empty_carrier(A!7)) & join_commutative(A!7) & join_semilatt_str(A!7) & element(B!8, the_carrier(A!7)) & (~((B!8 = C!9) | (~element(C!9, the_carrier(A!7))) | (~(below(A!7, B!8, C!9) & below(A!7, C!9, B!8))))))),
% 0.21/0.39 inference(rewrite,[status(thm)],[])).
% 0.21/0.39 tff(2,plain,
% 0.21/0.39 ((~((B!8 = C!9) | (~element(C!9, the_carrier(A!7))) | (~(below(A!7, B!8, C!9) & below(A!7, C!9, B!8))))) <=> (~((B!8 = C!9) | (~element(C!9, the_carrier(A!7))) | (~(below(A!7, B!8, C!9) & below(A!7, C!9, B!8)))))),
% 0.21/0.39 inference(rewrite,[status(thm)],[])).
% 0.21/0.39 tff(3,plain,
% 0.21/0.39 ((element(B!8, the_carrier(A!7)) & (~((B!8 = C!9) | (~element(C!9, the_carrier(A!7))) | (~(below(A!7, B!8, C!9) & below(A!7, C!9, B!8)))))) <=> (element(B!8, the_carrier(A!7)) & (~((B!8 = C!9) | (~element(C!9, the_carrier(A!7))) | (~(below(A!7, B!8, C!9) & below(A!7, C!9, B!8))))))),
% 0.21/0.39 inference(monotonicity,[status(thm)],[2])).
% 0.21/0.39 tff(4,plain,
% 0.21/0.39 ((((~empty_carrier(A!7)) & join_commutative(A!7) & join_semilatt_str(A!7)) & (element(B!8, the_carrier(A!7)) & (~((B!8 = C!9) | (~element(C!9, the_carrier(A!7))) | (~(below(A!7, B!8, C!9) & below(A!7, C!9, B!8))))))) <=> (((~empty_carrier(A!7)) & join_commutative(A!7) & join_semilatt_str(A!7)) & (element(B!8, the_carrier(A!7)) & (~((B!8 = C!9) | (~element(C!9, the_carrier(A!7))) | (~(below(A!7, B!8, C!9) & below(A!7, C!9, B!8)))))))),
% 0.21/0.39 inference(monotonicity,[status(thm)],[3])).
% 0.21/0.39 tff(5,plain,
% 0.21/0.39 ((((~empty_carrier(A!7)) & join_commutative(A!7) & join_semilatt_str(A!7)) & (element(B!8, the_carrier(A!7)) & (~((B!8 = C!9) | (~element(C!9, the_carrier(A!7))) | (~(below(A!7, B!8, C!9) & below(A!7, C!9, B!8))))))) <=> ((~empty_carrier(A!7)) & join_commutative(A!7) & join_semilatt_str(A!7) & element(B!8, the_carrier(A!7)) & (~((B!8 = C!9) | (~element(C!9, the_carrier(A!7))) | (~(below(A!7, B!8, C!9) & below(A!7, C!9, B!8))))))),
% 0.21/0.39 inference(transitivity,[status(thm)],[4, 1])).
% 0.21/0.39 tff(6,plain,
% 0.21/0.39 ((~![A: $i] : ((~((~empty_carrier(A)) & join_commutative(A) & join_semilatt_str(A))) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((B = C) | (~element(C, the_carrier(A))) | (~(below(A, B, C) & below(A, C, B))))))) <=> (~![A: $i] : ((~((~empty_carrier(A)) & join_commutative(A) & join_semilatt_str(A))) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((B = C) | (~element(C, the_carrier(A))) | (~(below(A, B, C) & below(A, C, B)))))))),
% 0.21/0.39 inference(rewrite,[status(thm)],[])).
% 0.21/0.39 tff(7,plain,
% 0.21/0.39 ((~![A: $i] : ((((~empty_carrier(A)) & join_commutative(A)) & join_semilatt_str(A)) => ![B: $i] : (element(B, the_carrier(A)) => ![C: $i] : (element(C, the_carrier(A)) => ((below(A, B, C) & below(A, C, B)) => (B = C)))))) <=> (~![A: $i] : ((~((~empty_carrier(A)) & join_commutative(A) & join_semilatt_str(A))) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((B = C) | (~element(C, the_carrier(A))) | (~(below(A, B, C) & below(A, C, B)))))))),
% 0.21/0.39 inference(rewrite,[status(thm)],[])).
% 0.21/0.39 tff(8,axiom,(~![A: $i] : ((((~empty_carrier(A)) & join_commutative(A)) & join_semilatt_str(A)) => ![B: $i] : (element(B, the_carrier(A)) => ![C: $i] : (element(C, the_carrier(A)) => ((below(A, B, C) & below(A, C, B)) => (B = C)))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','t26_lattices')).
% 0.21/0.39 tff(9,plain,
% 0.21/0.39 (~![A: $i] : ((~((~empty_carrier(A)) & join_commutative(A) & join_semilatt_str(A))) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((B = C) | (~element(C, the_carrier(A))) | (~(below(A, B, C) & below(A, C, B))))))),
% 0.21/0.39 inference(modus_ponens,[status(thm)],[8, 7])).
% 0.21/0.39 tff(10,plain,
% 0.21/0.39 (~![A: $i] : ((~((~empty_carrier(A)) & join_commutative(A) & join_semilatt_str(A))) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((B = C) | (~element(C, the_carrier(A))) | (~(below(A, B, C) & below(A, C, B))))))),
% 0.21/0.39 inference(modus_ponens,[status(thm)],[9, 6])).
% 0.21/0.39 tff(11,plain,
% 0.21/0.39 (~![A: $i] : ((~((~empty_carrier(A)) & join_commutative(A) & join_semilatt_str(A))) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((B = C) | (~element(C, the_carrier(A))) | (~(below(A, B, C) & below(A, C, B))))))),
% 0.21/0.39 inference(modus_ponens,[status(thm)],[10, 6])).
% 0.21/0.39 tff(12,plain,
% 0.21/0.39 (~![A: $i] : ((~((~empty_carrier(A)) & join_commutative(A) & join_semilatt_str(A))) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((B = C) | (~element(C, the_carrier(A))) | (~(below(A, B, C) & below(A, C, B))))))),
% 0.21/0.39 inference(modus_ponens,[status(thm)],[11, 6])).
% 0.21/0.39 tff(13,plain,
% 0.21/0.39 ((~empty_carrier(A!7)) & join_commutative(A!7) & join_semilatt_str(A!7) & element(B!8, the_carrier(A!7)) & (~((B!8 = C!9) | (~element(C!9, the_carrier(A!7))) | (~(below(A!7, B!8, C!9) & below(A!7, C!9, B!8)))))),
% 0.21/0.39 inference(modus_ponens,[status(thm)],[12, 5])).
% 0.21/0.39 tff(14,plain,
% 0.21/0.39 (join_semilatt_str(A!7)),
% 0.21/0.39 inference(and_elim,[status(thm)],[13])).
% 0.21/0.39 tff(15,plain,
% 0.21/0.39 (~empty_carrier(A!7)),
% 0.21/0.39 inference(and_elim,[status(thm)],[13])).
% 0.21/0.39 tff(16,plain,
% 0.21/0.39 (^[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))))))),
% 0.21/0.39 inference(bind,[status(th)],[])).
% 0.21/0.39 tff(17,plain,
% 0.21/0.39 (![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)))))),
% 0.21/0.39 inference(quant_intro,[status(thm)],[16])).
% 0.21/0.39 tff(18,plain,
% 0.21/0.39 (^[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))))),
% 0.21/0.39 inference(bind,[status(th)],[])).
% 0.21/0.39 tff(19,plain,
% 0.21/0.39 (![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)))),
% 0.21/0.39 inference(quant_intro,[status(thm)],[18])).
% 0.21/0.39 tff(20,plain,
% 0.21/0.39 (^[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))))),
% 0.21/0.39 inference(bind,[status(th)],[])).
% 0.21/0.39 tff(21,plain,
% 0.21/0.40 (![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)))),
% 0.21/0.40 inference(quant_intro,[status(thm)],[20])).
% 0.21/0.40 tff(22,plain,
% 0.21/0.40 (![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)))),
% 0.21/0.40 inference(transitivity,[status(thm)],[21, 19])).
% 0.21/0.40 tff(23,plain,
% 0.21/0.40 (^[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)))))),
% 0.21/0.40 inference(bind,[status(th)],[])).
% 0.21/0.40 tff(24,plain,
% 0.21/0.40 (![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)))),
% 0.21/0.40 inference(quant_intro,[status(thm)],[23])).
% 0.21/0.40 tff(25,plain,
% 0.21/0.40 (![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)))))),
% 0.21/0.40 inference(rewrite,[status(thm)],[])).
% 0.21/0.40 tff(26,plain,
% 0.21/0.40 (^[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)))))))),
% 0.21/0.40 inference(bind,[status(th)],[])).
% 0.21/0.40 tff(27,plain,
% 0.21/0.40 (![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)))))),
% 0.21/0.40 inference(quant_intro,[status(thm)],[26])).
% 0.21/0.40 tff(28,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/sandbox2/benchmark/theBenchmark.p','d3_lattices')).
% 0.21/0.40 tff(29,plain,
% 0.21/0.40 (![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)))))),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[28, 27])).
% 0.21/0.40 tff(30,plain,
% 0.21/0.40 (![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)))))),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[29, 25])).
% 0.21/0.40 tff(31,plain,(
% 0.21/0.40 ![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)))))),
% 0.21/0.40 inference(skolemize,[status(sab)],[30])).
% 0.21/0.40 tff(32,plain,
% 0.21/0.40 (![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)))),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[31, 24])).
% 0.21/0.40 tff(33,plain,
% 0.21/0.40 (![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)))),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[32, 22])).
% 0.21/0.40 tff(34,plain,
% 0.21/0.40 (![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)))))),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[33, 17])).
% 0.21/0.40 tff(35,plain,
% 0.21/0.40 (((~![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(A!7) | (~join_semilatt_str(A!7)) | ![B: $i] : ((~element(B, the_carrier(A!7))) | ![C: $i] : ((~element(C, the_carrier(A!7))) | (below(A!7, B, C) <=> (join(A!7, 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(A!7) | (~join_semilatt_str(A!7)) | ![B: $i] : ((~element(B, the_carrier(A!7))) | ![C: $i] : ((~element(C, the_carrier(A!7))) | (below(A!7, B, C) <=> (join(A!7, B, C) = C)))))),
% 0.21/0.40 inference(rewrite,[status(thm)],[])).
% 0.21/0.40 tff(36,plain,
% 0.21/0.40 ((~![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(A!7) | (~join_semilatt_str(A!7)) | ![B: $i] : ((~element(B, the_carrier(A!7))) | ![C: $i] : ((~element(C, the_carrier(A!7))) | (below(A!7, B, C) <=> (join(A!7, B, C) = C)))))),
% 0.21/0.40 inference(quant_inst,[status(thm)],[])).
% 0.21/0.40 tff(37,plain,
% 0.21/0.40 ((~![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(A!7) | (~join_semilatt_str(A!7)) | ![B: $i] : ((~element(B, the_carrier(A!7))) | ![C: $i] : ((~element(C, the_carrier(A!7))) | (below(A!7, B, C) <=> (join(A!7, B, C) = C))))),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[36, 35])).
% 0.21/0.40 tff(38,plain,
% 0.21/0.40 (![B: $i] : ((~element(B, the_carrier(A!7))) | ![C: $i] : ((~element(C, the_carrier(A!7))) | (below(A!7, B, C) <=> (join(A!7, B, C) = C))))),
% 0.21/0.40 inference(unit_resolution,[status(thm)],[37, 34, 15, 14])).
% 0.21/0.40 tff(39,plain,
% 0.21/0.40 (element(B!8, the_carrier(A!7))),
% 0.21/0.40 inference(and_elim,[status(thm)],[13])).
% 0.21/0.40 tff(40,plain,
% 0.21/0.40 (((~![B: $i] : ((~element(B, the_carrier(A!7))) | ![C: $i] : ((~element(C, the_carrier(A!7))) | (below(A!7, B, C) <=> (join(A!7, B, C) = C))))) | ((~element(B!8, the_carrier(A!7))) | ![C: $i] : ((~element(C, the_carrier(A!7))) | (below(A!7, B!8, C) <=> (join(A!7, B!8, C) = C))))) <=> ((~![B: $i] : ((~element(B, the_carrier(A!7))) | ![C: $i] : ((~element(C, the_carrier(A!7))) | (below(A!7, B, C) <=> (join(A!7, B, C) = C))))) | (~element(B!8, the_carrier(A!7))) | ![C: $i] : ((~element(C, the_carrier(A!7))) | (below(A!7, B!8, C) <=> (join(A!7, B!8, C) = C))))),
% 0.21/0.40 inference(rewrite,[status(thm)],[])).
% 0.21/0.40 tff(41,plain,
% 0.21/0.40 ((~![B: $i] : ((~element(B, the_carrier(A!7))) | ![C: $i] : ((~element(C, the_carrier(A!7))) | (below(A!7, B, C) <=> (join(A!7, B, C) = C))))) | ((~element(B!8, the_carrier(A!7))) | ![C: $i] : ((~element(C, the_carrier(A!7))) | (below(A!7, B!8, C) <=> (join(A!7, B!8, C) = C))))),
% 0.21/0.40 inference(quant_inst,[status(thm)],[])).
% 0.21/0.40 tff(42,plain,
% 0.21/0.40 ((~![B: $i] : ((~element(B, the_carrier(A!7))) | ![C: $i] : ((~element(C, the_carrier(A!7))) | (below(A!7, B, C) <=> (join(A!7, B, C) = C))))) | (~element(B!8, the_carrier(A!7))) | ![C: $i] : ((~element(C, the_carrier(A!7))) | (below(A!7, B!8, C) <=> (join(A!7, B!8, C) = C)))),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[41, 40])).
% 0.21/0.40 tff(43,plain,
% 0.21/0.40 (![C: $i] : ((~element(C, the_carrier(A!7))) | (below(A!7, B!8, C) <=> (join(A!7, B!8, C) = C)))),
% 0.21/0.40 inference(unit_resolution,[status(thm)],[42, 39, 38])).
% 0.21/0.40 tff(44,plain,
% 0.21/0.40 (~((B!8 = C!9) | (~element(C!9, the_carrier(A!7))) | (~(below(A!7, B!8, C!9) & below(A!7, C!9, B!8))))),
% 0.21/0.40 inference(and_elim,[status(thm)],[13])).
% 0.21/0.40 tff(45,plain,
% 0.21/0.40 (element(C!9, the_carrier(A!7))),
% 0.21/0.40 inference(or_elim,[status(thm)],[44])).
% 0.21/0.40 tff(46,plain,
% 0.21/0.40 (((~![C: $i] : ((~element(C, the_carrier(A!7))) | (below(A!7, B!8, C) <=> (join(A!7, B!8, C) = C)))) | ((~element(C!9, the_carrier(A!7))) | (below(A!7, B!8, C!9) <=> (join(A!7, B!8, C!9) = C!9)))) <=> ((~![C: $i] : ((~element(C, the_carrier(A!7))) | (below(A!7, B!8, C) <=> (join(A!7, B!8, C) = C)))) | (~element(C!9, the_carrier(A!7))) | (below(A!7, B!8, C!9) <=> (join(A!7, B!8, C!9) = C!9)))),
% 0.21/0.40 inference(rewrite,[status(thm)],[])).
% 0.21/0.40 tff(47,plain,
% 0.21/0.40 ((~![C: $i] : ((~element(C, the_carrier(A!7))) | (below(A!7, B!8, C) <=> (join(A!7, B!8, C) = C)))) | ((~element(C!9, the_carrier(A!7))) | (below(A!7, B!8, C!9) <=> (join(A!7, B!8, C!9) = C!9)))),
% 0.21/0.40 inference(quant_inst,[status(thm)],[])).
% 0.21/0.40 tff(48,plain,
% 0.21/0.40 ((~![C: $i] : ((~element(C, the_carrier(A!7))) | (below(A!7, B!8, C) <=> (join(A!7, B!8, C) = C)))) | (~element(C!9, the_carrier(A!7))) | (below(A!7, B!8, C!9) <=> (join(A!7, B!8, C!9) = C!9))),
% 0.21/0.40 inference(modus_ponens,[status(thm)],[47, 46])).
% 0.21/0.40 tff(49,plain,
% 0.21/0.40 (below(A!7, B!8, C!9) <=> (join(A!7, B!8, C!9) = C!9)),
% 0.21/0.40 inference(unit_resolution,[status(thm)],[48, 45, 43])).
% 0.21/0.40 tff(50,plain,
% 0.21/0.40 (below(A!7, B!8, C!9) & below(A!7, C!9, B!8)),
% 0.21/0.40 inference(or_elim,[status(thm)],[44])).
% 0.21/0.40 tff(51,plain,
% 0.21/0.40 (below(A!7, B!8, C!9)),
% 0.21/0.40 inference(and_elim,[status(thm)],[50])).
% 0.21/0.40 tff(52,plain,
% 0.21/0.40 ((~(below(A!7, B!8, C!9) <=> (join(A!7, B!8, C!9) = C!9))) | (~below(A!7, B!8, C!9)) | (join(A!7, B!8, C!9) = C!9)),
% 0.21/0.40 inference(tautology,[status(thm)],[])).
% 0.21/0.40 tff(53,plain,
% 0.21/0.40 ((~(below(A!7, B!8, C!9) <=> (join(A!7, B!8, C!9) = C!9))) | (join(A!7, B!8, C!9) = C!9)),
% 0.21/0.40 inference(unit_resolution,[status(thm)],[52, 51])).
% 0.21/0.40 tff(54,plain,
% 0.21/0.40 (join(A!7, B!8, C!9) = C!9),
% 0.21/0.40 inference(unit_resolution,[status(thm)],[53, 49])).
% 0.21/0.40 tff(55,plain,
% 0.21/0.40 (^[A: $i] : rewrite((empty_carrier(A) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (join(A, B, C) = apply_binary_as_element(the_carrier(A), the_carrier(A), the_carrier(A), the_L_join(A), B, 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))) | (join(A, B, C) = apply_binary_as_element(the_carrier(A), the_carrier(A), the_carrier(A), the_L_join(A), B, C))))))),
% 0.21/0.40 inference(bind,[status(th)],[])).
% 0.21/0.40 tff(56,plain,
% 0.21/0.40 (![A: $i] : (empty_carrier(A) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (join(A, B, C) = apply_binary_as_element(the_carrier(A), the_carrier(A), the_carrier(A), the_L_join(A), B, 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))) | (join(A, B, C) = apply_binary_as_element(the_carrier(A), the_carrier(A), the_carrier(A), the_L_join(A), B, C)))))),
% 0.21/0.40 inference(quant_intro,[status(thm)],[55])).
% 0.21/0.40 tff(57,plain,
% 0.21/0.40 (^[A: $i] : refl((empty_carrier(A) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (join(A, B, C) = apply_binary_as_element(the_carrier(A), the_carrier(A), the_carrier(A), the_L_join(A), B, C)))) | (~join_semilatt_str(A))) <=> (empty_carrier(A) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (join(A, B, C) = apply_binary_as_element(the_carrier(A), the_carrier(A), the_carrier(A), the_L_join(A), B, C)))) | (~join_semilatt_str(A))))),
% 0.21/0.40 inference(bind,[status(th)],[])).
% 0.21/0.40 tff(58,plain,
% 0.21/0.40 (![A: $i] : (empty_carrier(A) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (join(A, B, C) = apply_binary_as_element(the_carrier(A), the_carrier(A), the_carrier(A), the_L_join(A), B, C)))) | (~join_semilatt_str(A))) <=> ![A: $i] : (empty_carrier(A) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (join(A, B, C) = apply_binary_as_element(the_carrier(A), the_carrier(A), the_carrier(A), the_L_join(A), B, C)))) | (~join_semilatt_str(A)))),
% 0.21/0.40 inference(quant_intro,[status(thm)],[57])).
% 0.21/0.40 tff(59,plain,
% 0.21/0.40 (^[A: $i] : rewrite((empty_carrier(A) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (join(A, B, C) = apply_binary_as_element(the_carrier(A), the_carrier(A), the_carrier(A), the_L_join(A), B, C)))) | (~join_semilatt_str(A))) <=> (empty_carrier(A) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (join(A, B, C) = apply_binary_as_element(the_carrier(A), the_carrier(A), the_carrier(A), the_L_join(A), B, C)))) | (~join_semilatt_str(A))))),
% 0.21/0.41 inference(bind,[status(th)],[])).
% 0.21/0.41 tff(60,plain,
% 0.21/0.41 (![A: $i] : (empty_carrier(A) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (join(A, B, C) = apply_binary_as_element(the_carrier(A), the_carrier(A), the_carrier(A), the_L_join(A), B, C)))) | (~join_semilatt_str(A))) <=> ![A: $i] : (empty_carrier(A) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (join(A, B, C) = apply_binary_as_element(the_carrier(A), the_carrier(A), the_carrier(A), the_L_join(A), B, C)))) | (~join_semilatt_str(A)))),
% 0.21/0.41 inference(quant_intro,[status(thm)],[59])).
% 0.21/0.41 tff(61,plain,
% 0.21/0.41 (![A: $i] : (empty_carrier(A) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (join(A, B, C) = apply_binary_as_element(the_carrier(A), the_carrier(A), the_carrier(A), the_L_join(A), B, C)))) | (~join_semilatt_str(A))) <=> ![A: $i] : (empty_carrier(A) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (join(A, B, C) = apply_binary_as_element(the_carrier(A), the_carrier(A), the_carrier(A), the_L_join(A), B, C)))) | (~join_semilatt_str(A)))),
% 0.21/0.41 inference(transitivity,[status(thm)],[60, 58])).
% 0.21/0.41 tff(62,plain,
% 0.21/0.41 (^[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))) | (join(A, B, C) = apply_binary_as_element(the_carrier(A), the_carrier(A), the_carrier(A), the_L_join(A), B, C))))) <=> ((empty_carrier(A) | (~join_semilatt_str(A))) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (join(A, B, C) = apply_binary_as_element(the_carrier(A), the_carrier(A), the_carrier(A), the_L_join(A), B, C))))))), rewrite(((empty_carrier(A) | (~join_semilatt_str(A))) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (join(A, B, C) = apply_binary_as_element(the_carrier(A), the_carrier(A), the_carrier(A), the_L_join(A), B, C))))) <=> (empty_carrier(A) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (join(A, B, C) = apply_binary_as_element(the_carrier(A), the_carrier(A), the_carrier(A), the_L_join(A), B, 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))) | (join(A, B, C) = apply_binary_as_element(the_carrier(A), the_carrier(A), the_carrier(A), the_L_join(A), B, C))))) <=> (empty_carrier(A) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (join(A, B, C) = apply_binary_as_element(the_carrier(A), the_carrier(A), the_carrier(A), the_L_join(A), B, C)))) | (~join_semilatt_str(A)))))),
% 0.21/0.41 inference(bind,[status(th)],[])).
% 0.21/0.41 tff(63,plain,
% 0.21/0.41 (![A: $i] : ((~((~empty_carrier(A)) & join_semilatt_str(A))) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (join(A, B, C) = apply_binary_as_element(the_carrier(A), the_carrier(A), the_carrier(A), the_L_join(A), B, C))))) <=> ![A: $i] : (empty_carrier(A) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (join(A, B, C) = apply_binary_as_element(the_carrier(A), the_carrier(A), the_carrier(A), the_L_join(A), B, C)))) | (~join_semilatt_str(A)))),
% 0.21/0.41 inference(quant_intro,[status(thm)],[62])).
% 0.21/0.41 tff(64,plain,
% 0.21/0.41 (![A: $i] : ((~((~empty_carrier(A)) & join_semilatt_str(A))) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (join(A, B, C) = apply_binary_as_element(the_carrier(A), the_carrier(A), the_carrier(A), the_L_join(A), B, C))))) <=> ![A: $i] : ((~((~empty_carrier(A)) & join_semilatt_str(A))) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (join(A, B, C) = apply_binary_as_element(the_carrier(A), the_carrier(A), the_carrier(A), the_L_join(A), B, C)))))),
% 0.21/0.41 inference(rewrite,[status(thm)],[])).
% 0.21/0.41 tff(65,plain,
% 0.21/0.41 (^[A: $i] : trans(monotonicity(quant_intro(proof_bind(^[B: $i] : trans(monotonicity(quant_intro(proof_bind(^[C: $i] : rewrite((element(C, the_carrier(A)) => (join(A, B, C) = apply_binary_as_element(the_carrier(A), the_carrier(A), the_carrier(A), the_L_join(A), B, C))) <=> ((~element(C, the_carrier(A))) | (join(A, B, C) = apply_binary_as_element(the_carrier(A), the_carrier(A), the_carrier(A), the_L_join(A), B, C))))), (![C: $i] : (element(C, the_carrier(A)) => (join(A, B, C) = apply_binary_as_element(the_carrier(A), the_carrier(A), the_carrier(A), the_L_join(A), B, C))) <=> ![C: $i] : ((~element(C, the_carrier(A))) | (join(A, B, C) = apply_binary_as_element(the_carrier(A), the_carrier(A), the_carrier(A), the_L_join(A), B, C))))), ((element(B, the_carrier(A)) => ![C: $i] : (element(C, the_carrier(A)) => (join(A, B, C) = apply_binary_as_element(the_carrier(A), the_carrier(A), the_carrier(A), the_L_join(A), B, C)))) <=> (element(B, the_carrier(A)) => ![C: $i] : ((~element(C, the_carrier(A))) | (join(A, B, C) = apply_binary_as_element(the_carrier(A), the_carrier(A), the_carrier(A), the_L_join(A), B, C)))))), rewrite((element(B, the_carrier(A)) => ![C: $i] : ((~element(C, the_carrier(A))) | (join(A, B, C) = apply_binary_as_element(the_carrier(A), the_carrier(A), the_carrier(A), the_L_join(A), B, C)))) <=> ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (join(A, B, C) = apply_binary_as_element(the_carrier(A), the_carrier(A), the_carrier(A), the_L_join(A), B, C))))), ((element(B, the_carrier(A)) => ![C: $i] : (element(C, the_carrier(A)) => (join(A, B, C) = apply_binary_as_element(the_carrier(A), the_carrier(A), the_carrier(A), the_L_join(A), B, C)))) <=> ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (join(A, B, C) = apply_binary_as_element(the_carrier(A), the_carrier(A), the_carrier(A), the_L_join(A), B, C))))))), (![B: $i] : (element(B, the_carrier(A)) => ![C: $i] : (element(C, the_carrier(A)) => (join(A, B, C) = apply_binary_as_element(the_carrier(A), the_carrier(A), the_carrier(A), the_L_join(A), B, C)))) <=> ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (join(A, B, C) = apply_binary_as_element(the_carrier(A), the_carrier(A), the_carrier(A), the_L_join(A), B, C)))))), ((((~empty_carrier(A)) & join_semilatt_str(A)) => ![B: $i] : (element(B, the_carrier(A)) => ![C: $i] : (element(C, the_carrier(A)) => (join(A, B, C) = apply_binary_as_element(the_carrier(A), the_carrier(A), the_carrier(A), the_L_join(A), B, C))))) <=> (((~empty_carrier(A)) & join_semilatt_str(A)) => ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (join(A, B, C) = apply_binary_as_element(the_carrier(A), the_carrier(A), the_carrier(A), the_L_join(A), B, C))))))), rewrite((((~empty_carrier(A)) & join_semilatt_str(A)) => ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (join(A, B, C) = apply_binary_as_element(the_carrier(A), the_carrier(A), the_carrier(A), the_L_join(A), B, C))))) <=> ((~((~empty_carrier(A)) & join_semilatt_str(A))) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (join(A, B, C) = apply_binary_as_element(the_carrier(A), the_carrier(A), the_carrier(A), the_L_join(A), B, C)))))), ((((~empty_carrier(A)) & join_semilatt_str(A)) => ![B: $i] : (element(B, the_carrier(A)) => ![C: $i] : (element(C, the_carrier(A)) => (join(A, B, C) = apply_binary_as_element(the_carrier(A), the_carrier(A), the_carrier(A), the_L_join(A), B, C))))) <=> ((~((~empty_carrier(A)) & join_semilatt_str(A))) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (join(A, B, C) = apply_binary_as_element(the_carrier(A), the_carrier(A), the_carrier(A), the_L_join(A), B, C)))))))),
% 0.21/0.41 inference(bind,[status(th)],[])).
% 0.21/0.41 tff(66,plain,
% 0.21/0.41 (![A: $i] : (((~empty_carrier(A)) & join_semilatt_str(A)) => ![B: $i] : (element(B, the_carrier(A)) => ![C: $i] : (element(C, the_carrier(A)) => (join(A, B, C) = apply_binary_as_element(the_carrier(A), the_carrier(A), the_carrier(A), the_L_join(A), B, C))))) <=> ![A: $i] : ((~((~empty_carrier(A)) & join_semilatt_str(A))) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (join(A, B, C) = apply_binary_as_element(the_carrier(A), the_carrier(A), the_carrier(A), the_L_join(A), B, C)))))),
% 0.21/0.41 inference(quant_intro,[status(thm)],[65])).
% 0.21/0.41 tff(67,axiom,(![A: $i] : (((~empty_carrier(A)) & join_semilatt_str(A)) => ![B: $i] : (element(B, the_carrier(A)) => ![C: $i] : (element(C, the_carrier(A)) => (join(A, B, C) = apply_binary_as_element(the_carrier(A), the_carrier(A), the_carrier(A), the_L_join(A), B, C)))))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','d1_lattices')).
% 0.21/0.41 tff(68,plain,
% 0.21/0.41 (![A: $i] : ((~((~empty_carrier(A)) & join_semilatt_str(A))) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (join(A, B, C) = apply_binary_as_element(the_carrier(A), the_carrier(A), the_carrier(A), the_L_join(A), B, C)))))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[67, 66])).
% 0.21/0.41 tff(69,plain,
% 0.21/0.41 (![A: $i] : ((~((~empty_carrier(A)) & join_semilatt_str(A))) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (join(A, B, C) = apply_binary_as_element(the_carrier(A), the_carrier(A), the_carrier(A), the_L_join(A), B, C)))))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[68, 64])).
% 0.21/0.41 tff(70,plain,(
% 0.21/0.41 ![A: $i] : ((~((~empty_carrier(A)) & join_semilatt_str(A))) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (join(A, B, C) = apply_binary_as_element(the_carrier(A), the_carrier(A), the_carrier(A), the_L_join(A), B, C)))))),
% 0.21/0.41 inference(skolemize,[status(sab)],[69])).
% 0.21/0.41 tff(71,plain,
% 0.21/0.41 (![A: $i] : (empty_carrier(A) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (join(A, B, C) = apply_binary_as_element(the_carrier(A), the_carrier(A), the_carrier(A), the_L_join(A), B, C)))) | (~join_semilatt_str(A)))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[70, 63])).
% 0.21/0.41 tff(72,plain,
% 0.21/0.41 (![A: $i] : (empty_carrier(A) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (join(A, B, C) = apply_binary_as_element(the_carrier(A), the_carrier(A), the_carrier(A), the_L_join(A), B, C)))) | (~join_semilatt_str(A)))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[71, 61])).
% 0.21/0.41 tff(73,plain,
% 0.21/0.41 (![A: $i] : (empty_carrier(A) | (~join_semilatt_str(A)) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (join(A, B, C) = apply_binary_as_element(the_carrier(A), the_carrier(A), the_carrier(A), the_L_join(A), B, C)))))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[72, 56])).
% 0.21/0.41 tff(74,plain,
% 0.21/0.41 (((~![A: $i] : (empty_carrier(A) | (~join_semilatt_str(A)) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (join(A, B, C) = apply_binary_as_element(the_carrier(A), the_carrier(A), the_carrier(A), the_L_join(A), B, C)))))) | (empty_carrier(A!7) | (~join_semilatt_str(A!7)) | ![B: $i] : ((~element(B, the_carrier(A!7))) | ![C: $i] : ((~element(C, the_carrier(A!7))) | (join(A!7, B, C) = apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B, C)))))) <=> ((~![A: $i] : (empty_carrier(A) | (~join_semilatt_str(A)) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (join(A, B, C) = apply_binary_as_element(the_carrier(A), the_carrier(A), the_carrier(A), the_L_join(A), B, C)))))) | empty_carrier(A!7) | (~join_semilatt_str(A!7)) | ![B: $i] : ((~element(B, the_carrier(A!7))) | ![C: $i] : ((~element(C, the_carrier(A!7))) | (join(A!7, B, C) = apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B, C)))))),
% 0.21/0.41 inference(rewrite,[status(thm)],[])).
% 0.21/0.41 tff(75,plain,
% 0.21/0.41 ((~![A: $i] : (empty_carrier(A) | (~join_semilatt_str(A)) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (join(A, B, C) = apply_binary_as_element(the_carrier(A), the_carrier(A), the_carrier(A), the_L_join(A), B, C)))))) | (empty_carrier(A!7) | (~join_semilatt_str(A!7)) | ![B: $i] : ((~element(B, the_carrier(A!7))) | ![C: $i] : ((~element(C, the_carrier(A!7))) | (join(A!7, B, C) = apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B, C)))))),
% 0.21/0.41 inference(quant_inst,[status(thm)],[])).
% 0.21/0.41 tff(76,plain,
% 0.21/0.41 ((~![A: $i] : (empty_carrier(A) | (~join_semilatt_str(A)) | ![B: $i] : ((~element(B, the_carrier(A))) | ![C: $i] : ((~element(C, the_carrier(A))) | (join(A, B, C) = apply_binary_as_element(the_carrier(A), the_carrier(A), the_carrier(A), the_L_join(A), B, C)))))) | empty_carrier(A!7) | (~join_semilatt_str(A!7)) | ![B: $i] : ((~element(B, the_carrier(A!7))) | ![C: $i] : ((~element(C, the_carrier(A!7))) | (join(A!7, B, C) = apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B, C))))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[75, 74])).
% 0.21/0.41 tff(77,plain,
% 0.21/0.41 (![B: $i] : ((~element(B, the_carrier(A!7))) | ![C: $i] : ((~element(C, the_carrier(A!7))) | (join(A!7, B, C) = apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B, C))))),
% 0.21/0.41 inference(unit_resolution,[status(thm)],[76, 73, 15, 14])).
% 0.21/0.41 tff(78,plain,
% 0.21/0.41 (((~![B: $i] : ((~element(B, the_carrier(A!7))) | ![C: $i] : ((~element(C, the_carrier(A!7))) | (join(A!7, B, C) = apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B, C))))) | ((~element(B!8, the_carrier(A!7))) | ![C: $i] : ((~element(C, the_carrier(A!7))) | (join(A!7, B!8, C) = apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C))))) <=> ((~![B: $i] : ((~element(B, the_carrier(A!7))) | ![C: $i] : ((~element(C, the_carrier(A!7))) | (join(A!7, B, C) = apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B, C))))) | (~element(B!8, the_carrier(A!7))) | ![C: $i] : ((~element(C, the_carrier(A!7))) | (join(A!7, B!8, C) = apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C))))),
% 0.21/0.41 inference(rewrite,[status(thm)],[])).
% 0.21/0.41 tff(79,plain,
% 0.21/0.41 ((~![B: $i] : ((~element(B, the_carrier(A!7))) | ![C: $i] : ((~element(C, the_carrier(A!7))) | (join(A!7, B, C) = apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B, C))))) | ((~element(B!8, the_carrier(A!7))) | ![C: $i] : ((~element(C, the_carrier(A!7))) | (join(A!7, B!8, C) = apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C))))),
% 0.21/0.41 inference(quant_inst,[status(thm)],[])).
% 0.21/0.41 tff(80,plain,
% 0.21/0.41 ((~![B: $i] : ((~element(B, the_carrier(A!7))) | ![C: $i] : ((~element(C, the_carrier(A!7))) | (join(A!7, B, C) = apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B, C))))) | (~element(B!8, the_carrier(A!7))) | ![C: $i] : ((~element(C, the_carrier(A!7))) | (join(A!7, B!8, C) = apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C)))),
% 0.21/0.41 inference(modus_ponens,[status(thm)],[79, 78])).
% 0.21/0.41 tff(81,plain,
% 0.21/0.41 (![C: $i] : ((~element(C, the_carrier(A!7))) | (join(A!7, B!8, C) = apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C)))),
% 0.21/0.42 inference(unit_resolution,[status(thm)],[80, 39, 77])).
% 0.21/0.42 tff(82,plain,
% 0.21/0.42 (((~![C: $i] : ((~element(C, the_carrier(A!7))) | (join(A!7, B!8, C) = apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C)))) | ((~element(C!9, the_carrier(A!7))) | (join(A!7, B!8, C!9) = apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9)))) <=> ((~![C: $i] : ((~element(C, the_carrier(A!7))) | (join(A!7, B!8, C) = apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C)))) | (~element(C!9, the_carrier(A!7))) | (join(A!7, B!8, C!9) = apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9)))),
% 0.21/0.42 inference(rewrite,[status(thm)],[])).
% 0.21/0.42 tff(83,plain,
% 0.21/0.42 ((~![C: $i] : ((~element(C, the_carrier(A!7))) | (join(A!7, B!8, C) = apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C)))) | ((~element(C!9, the_carrier(A!7))) | (join(A!7, B!8, C!9) = apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9)))),
% 0.21/0.42 inference(quant_inst,[status(thm)],[])).
% 0.21/0.42 tff(84,plain,
% 0.21/0.42 ((~![C: $i] : ((~element(C, the_carrier(A!7))) | (join(A!7, B!8, C) = apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C)))) | (~element(C!9, the_carrier(A!7))) | (join(A!7, B!8, C!9) = apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9))),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[83, 82])).
% 0.21/0.42 tff(85,plain,
% 0.21/0.42 (join(A!7, B!8, C!9) = apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9)),
% 0.21/0.42 inference(unit_resolution,[status(thm)],[84, 45, 81])).
% 0.21/0.42 tff(86,plain,
% 0.21/0.42 (apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9) = join(A!7, B!8, C!9)),
% 0.21/0.42 inference(symmetry,[status(thm)],[85])).
% 0.21/0.42 tff(87,plain,
% 0.21/0.42 (apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9) = C!9),
% 0.21/0.42 inference(transitivity,[status(thm)],[86, 54])).
% 0.21/0.42 tff(88,plain,
% 0.21/0.42 (((~![B: $i] : ((~element(B, the_carrier(A!7))) | ![C: $i] : ((~element(C, the_carrier(A!7))) | (below(A!7, B, C) <=> (join(A!7, B, C) = C))))) | ((~element(C!9, the_carrier(A!7))) | ![C: $i] : ((~element(C, the_carrier(A!7))) | (below(A!7, C!9, C) <=> (join(A!7, C!9, C) = C))))) <=> ((~![B: $i] : ((~element(B, the_carrier(A!7))) | ![C: $i] : ((~element(C, the_carrier(A!7))) | (below(A!7, B, C) <=> (join(A!7, B, C) = C))))) | (~element(C!9, the_carrier(A!7))) | ![C: $i] : ((~element(C, the_carrier(A!7))) | (below(A!7, C!9, C) <=> (join(A!7, C!9, C) = C))))),
% 0.21/0.42 inference(rewrite,[status(thm)],[])).
% 0.21/0.42 tff(89,plain,
% 0.21/0.42 ((~![B: $i] : ((~element(B, the_carrier(A!7))) | ![C: $i] : ((~element(C, the_carrier(A!7))) | (below(A!7, B, C) <=> (join(A!7, B, C) = C))))) | ((~element(C!9, the_carrier(A!7))) | ![C: $i] : ((~element(C, the_carrier(A!7))) | (below(A!7, C!9, C) <=> (join(A!7, C!9, C) = C))))),
% 0.21/0.42 inference(quant_inst,[status(thm)],[])).
% 0.21/0.42 tff(90,plain,
% 0.21/0.42 ((~![B: $i] : ((~element(B, the_carrier(A!7))) | ![C: $i] : ((~element(C, the_carrier(A!7))) | (below(A!7, B, C) <=> (join(A!7, B, C) = C))))) | (~element(C!9, the_carrier(A!7))) | ![C: $i] : ((~element(C, the_carrier(A!7))) | (below(A!7, C!9, C) <=> (join(A!7, C!9, C) = C)))),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[89, 88])).
% 0.21/0.42 tff(91,plain,
% 0.21/0.42 (![C: $i] : ((~element(C, the_carrier(A!7))) | (below(A!7, C!9, C) <=> (join(A!7, C!9, C) = C)))),
% 0.21/0.42 inference(unit_resolution,[status(thm)],[90, 45, 38])).
% 0.21/0.42 tff(92,plain,
% 0.21/0.42 (((~![C: $i] : ((~element(C, the_carrier(A!7))) | (below(A!7, C!9, C) <=> (join(A!7, C!9, C) = C)))) | ((~element(B!8, the_carrier(A!7))) | (below(A!7, C!9, B!8) <=> (join(A!7, C!9, B!8) = B!8)))) <=> ((~![C: $i] : ((~element(C, the_carrier(A!7))) | (below(A!7, C!9, C) <=> (join(A!7, C!9, C) = C)))) | (~element(B!8, the_carrier(A!7))) | (below(A!7, C!9, B!8) <=> (join(A!7, C!9, B!8) = B!8)))),
% 0.21/0.42 inference(rewrite,[status(thm)],[])).
% 0.21/0.42 tff(93,plain,
% 0.21/0.42 ((~![C: $i] : ((~element(C, the_carrier(A!7))) | (below(A!7, C!9, C) <=> (join(A!7, C!9, C) = C)))) | ((~element(B!8, the_carrier(A!7))) | (below(A!7, C!9, B!8) <=> (join(A!7, C!9, B!8) = B!8)))),
% 0.21/0.42 inference(quant_inst,[status(thm)],[])).
% 0.21/0.42 tff(94,plain,
% 0.21/0.42 ((~![C: $i] : ((~element(C, the_carrier(A!7))) | (below(A!7, C!9, C) <=> (join(A!7, C!9, C) = C)))) | (~element(B!8, the_carrier(A!7))) | (below(A!7, C!9, B!8) <=> (join(A!7, C!9, B!8) = B!8))),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[93, 92])).
% 0.21/0.42 tff(95,plain,
% 0.21/0.42 (below(A!7, C!9, B!8) <=> (join(A!7, C!9, B!8) = B!8)),
% 0.21/0.42 inference(unit_resolution,[status(thm)],[94, 39, 91])).
% 0.21/0.42 tff(96,plain,
% 0.21/0.42 (below(A!7, C!9, B!8)),
% 0.21/0.42 inference(and_elim,[status(thm)],[50])).
% 0.21/0.42 tff(97,plain,
% 0.21/0.42 ((~(below(A!7, C!9, B!8) <=> (join(A!7, C!9, B!8) = B!8))) | (~below(A!7, C!9, B!8)) | (join(A!7, C!9, B!8) = B!8)),
% 0.21/0.42 inference(tautology,[status(thm)],[])).
% 0.21/0.42 tff(98,plain,
% 0.21/0.42 ((~(below(A!7, C!9, B!8) <=> (join(A!7, C!9, B!8) = B!8))) | (join(A!7, C!9, B!8) = B!8)),
% 0.21/0.42 inference(unit_resolution,[status(thm)],[97, 96])).
% 0.21/0.42 tff(99,plain,
% 0.21/0.42 (join(A!7, C!9, B!8) = B!8),
% 0.21/0.42 inference(unit_resolution,[status(thm)],[98, 95])).
% 0.21/0.42 tff(100,plain,
% 0.21/0.42 (((~![B: $i] : ((~element(B, the_carrier(A!7))) | ![C: $i] : ((~element(C, the_carrier(A!7))) | (join(A!7, B, C) = apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B, C))))) | ((~element(C!9, the_carrier(A!7))) | ![C: $i] : ((~element(C, the_carrier(A!7))) | (join(A!7, C!9, C) = apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, C))))) <=> ((~![B: $i] : ((~element(B, the_carrier(A!7))) | ![C: $i] : ((~element(C, the_carrier(A!7))) | (join(A!7, B, C) = apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B, C))))) | (~element(C!9, the_carrier(A!7))) | ![C: $i] : ((~element(C, the_carrier(A!7))) | (join(A!7, C!9, C) = apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, C))))),
% 0.21/0.42 inference(rewrite,[status(thm)],[])).
% 0.21/0.42 tff(101,plain,
% 0.21/0.42 ((~![B: $i] : ((~element(B, the_carrier(A!7))) | ![C: $i] : ((~element(C, the_carrier(A!7))) | (join(A!7, B, C) = apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B, C))))) | ((~element(C!9, the_carrier(A!7))) | ![C: $i] : ((~element(C, the_carrier(A!7))) | (join(A!7, C!9, C) = apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, C))))),
% 0.21/0.42 inference(quant_inst,[status(thm)],[])).
% 0.21/0.42 tff(102,plain,
% 0.21/0.42 ((~![B: $i] : ((~element(B, the_carrier(A!7))) | ![C: $i] : ((~element(C, the_carrier(A!7))) | (join(A!7, B, C) = apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B, C))))) | (~element(C!9, the_carrier(A!7))) | ![C: $i] : ((~element(C, the_carrier(A!7))) | (join(A!7, C!9, C) = apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, C)))),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[101, 100])).
% 0.21/0.42 tff(103,plain,
% 0.21/0.42 (![C: $i] : ((~element(C, the_carrier(A!7))) | (join(A!7, C!9, C) = apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, C)))),
% 0.21/0.42 inference(unit_resolution,[status(thm)],[102, 45, 77])).
% 0.21/0.42 tff(104,plain,
% 0.21/0.42 (((~![C: $i] : ((~element(C, the_carrier(A!7))) | (join(A!7, C!9, C) = apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, C)))) | ((~element(B!8, the_carrier(A!7))) | (join(A!7, C!9, B!8) = apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8)))) <=> ((~![C: $i] : ((~element(C, the_carrier(A!7))) | (join(A!7, C!9, C) = apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, C)))) | (~element(B!8, the_carrier(A!7))) | (join(A!7, C!9, B!8) = apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8)))),
% 0.21/0.42 inference(rewrite,[status(thm)],[])).
% 0.21/0.42 tff(105,plain,
% 0.21/0.42 ((~![C: $i] : ((~element(C, the_carrier(A!7))) | (join(A!7, C!9, C) = apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, C)))) | ((~element(B!8, the_carrier(A!7))) | (join(A!7, C!9, B!8) = apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8)))),
% 0.21/0.42 inference(quant_inst,[status(thm)],[])).
% 0.21/0.42 tff(106,plain,
% 0.21/0.42 ((~![C: $i] : ((~element(C, the_carrier(A!7))) | (join(A!7, C!9, C) = apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, C)))) | (~element(B!8, the_carrier(A!7))) | (join(A!7, C!9, B!8) = apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8))),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[105, 104])).
% 0.21/0.42 tff(107,plain,
% 0.21/0.42 (join(A!7, C!9, B!8) = apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8)),
% 0.21/0.42 inference(unit_resolution,[status(thm)],[106, 39, 103])).
% 0.21/0.42 tff(108,plain,
% 0.21/0.42 (apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8) = join(A!7, C!9, B!8)),
% 0.21/0.42 inference(symmetry,[status(thm)],[107])).
% 0.21/0.42 tff(109,plain,
% 0.21/0.42 (apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8) = B!8),
% 0.21/0.42 inference(transitivity,[status(thm)],[108, 99])).
% 0.21/0.42 tff(110,plain,
% 0.21/0.42 (join(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9)) = join(A!7, B!8, C!9)),
% 0.21/0.42 inference(monotonicity,[status(thm)],[109, 87])).
% 0.21/0.42 tff(111,plain,
% 0.21/0.42 (element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), the_carrier(A!7)) <=> element(B!8, the_carrier(A!7))),
% 0.21/0.42 inference(monotonicity,[status(thm)],[109])).
% 0.21/0.42 tff(112,plain,
% 0.21/0.42 (element(B!8, the_carrier(A!7)) <=> element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), the_carrier(A!7))),
% 0.21/0.42 inference(symmetry,[status(thm)],[111])).
% 0.21/0.42 tff(113,plain,
% 0.21/0.42 (element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), the_carrier(A!7))),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[39, 112])).
% 0.21/0.42 tff(114,plain,
% 0.21/0.42 (element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), the_carrier(A!7)) <=> element(C!9, the_carrier(A!7))),
% 0.21/0.42 inference(monotonicity,[status(thm)],[87])).
% 0.21/0.42 tff(115,plain,
% 0.21/0.42 (element(C!9, the_carrier(A!7)) <=> element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), the_carrier(A!7))),
% 0.21/0.42 inference(symmetry,[status(thm)],[114])).
% 0.21/0.42 tff(116,plain,
% 0.21/0.42 (element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), the_carrier(A!7))),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[45, 115])).
% 0.21/0.42 tff(117,plain,
% 0.21/0.42 (join_commutative(A!7)),
% 0.21/0.42 inference(and_elim,[status(thm)],[13])).
% 0.21/0.42 tff(118,plain,
% 0.21/0.42 (^[A: $i, B: $i, C: $i] : refl((empty_carrier(A) | (join_commut(A, B, C) = join(A, B, C)) | (~element(C, the_carrier(A))) | (~join_commutative(A)) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A)))) <=> (empty_carrier(A) | (join_commut(A, B, C) = join(A, B, C)) | (~element(C, the_carrier(A))) | (~join_commutative(A)) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A)))))),
% 0.21/0.42 inference(bind,[status(th)],[])).
% 0.21/0.42 tff(119,plain,
% 0.21/0.42 (![A: $i, B: $i, C: $i] : (empty_carrier(A) | (join_commut(A, B, C) = join(A, B, C)) | (~element(C, the_carrier(A))) | (~join_commutative(A)) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A)))) <=> ![A: $i, B: $i, C: $i] : (empty_carrier(A) | (join_commut(A, B, C) = join(A, B, C)) | (~element(C, the_carrier(A))) | (~join_commutative(A)) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A))))),
% 0.21/0.42 inference(quant_intro,[status(thm)],[118])).
% 0.21/0.42 tff(120,plain,
% 0.21/0.42 (^[A: $i, B: $i, C: $i] : trans(monotonicity(trans(monotonicity(rewrite(((~empty_carrier(A)) & join_commutative(A) & join_semilatt_str(A) & element(B, the_carrier(A)) & element(C, the_carrier(A))) <=> (~(empty_carrier(A) | (~element(C, the_carrier(A))) | (~join_commutative(A)) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A)))))), ((~((~empty_carrier(A)) & join_commutative(A) & join_semilatt_str(A) & element(B, the_carrier(A)) & element(C, the_carrier(A)))) <=> (~(~(empty_carrier(A) | (~element(C, the_carrier(A))) | (~join_commutative(A)) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A)))))))), rewrite((~(~(empty_carrier(A) | (~element(C, the_carrier(A))) | (~join_commutative(A)) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A)))))) <=> (empty_carrier(A) | (~element(C, the_carrier(A))) | (~join_commutative(A)) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A))))), ((~((~empty_carrier(A)) & join_commutative(A) & join_semilatt_str(A) & element(B, the_carrier(A)) & element(C, the_carrier(A)))) <=> (empty_carrier(A) | (~element(C, the_carrier(A))) | (~join_commutative(A)) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A)))))), (((~((~empty_carrier(A)) & join_commutative(A) & join_semilatt_str(A) & element(B, the_carrier(A)) & element(C, the_carrier(A)))) | (join_commut(A, B, C) = join(A, B, C))) <=> ((empty_carrier(A) | (~element(C, the_carrier(A))) | (~join_commutative(A)) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A)))) | (join_commut(A, B, C) = join(A, B, C))))), rewrite(((empty_carrier(A) | (~element(C, the_carrier(A))) | (~join_commutative(A)) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A)))) | (join_commut(A, B, C) = join(A, B, C))) <=> (empty_carrier(A) | (join_commut(A, B, C) = join(A, B, C)) | (~element(C, the_carrier(A))) | (~join_commutative(A)) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A))))), (((~((~empty_carrier(A)) & join_commutative(A) & join_semilatt_str(A) & element(B, the_carrier(A)) & element(C, the_carrier(A)))) | (join_commut(A, B, C) = join(A, B, C))) <=> (empty_carrier(A) | (join_commut(A, B, C) = join(A, B, C)) | (~element(C, the_carrier(A))) | (~join_commutative(A)) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A))))))),
% 0.21/0.42 inference(bind,[status(th)],[])).
% 0.21/0.42 tff(121,plain,
% 0.21/0.42 (![A: $i, B: $i, C: $i] : ((~((~empty_carrier(A)) & join_commutative(A) & join_semilatt_str(A) & element(B, the_carrier(A)) & element(C, the_carrier(A)))) | (join_commut(A, B, C) = join(A, B, C))) <=> ![A: $i, B: $i, C: $i] : (empty_carrier(A) | (join_commut(A, B, C) = join(A, B, C)) | (~element(C, the_carrier(A))) | (~join_commutative(A)) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A))))),
% 0.21/0.42 inference(quant_intro,[status(thm)],[120])).
% 0.21/0.42 tff(122,plain,
% 0.21/0.42 (![A: $i, B: $i, C: $i] : ((~((~empty_carrier(A)) & join_commutative(A) & join_semilatt_str(A) & element(B, the_carrier(A)) & element(C, the_carrier(A)))) | (join_commut(A, B, C) = join(A, B, C))) <=> ![A: $i, B: $i, C: $i] : ((~((~empty_carrier(A)) & join_commutative(A) & join_semilatt_str(A) & element(B, the_carrier(A)) & element(C, the_carrier(A)))) | (join_commut(A, B, C) = join(A, B, C)))),
% 0.21/0.42 inference(rewrite,[status(thm)],[])).
% 0.21/0.42 tff(123,plain,
% 0.21/0.42 (^[A: $i, B: $i, C: $i] : trans(monotonicity(trans(monotonicity(trans(monotonicity(rewrite((((~empty_carrier(A)) & join_commutative(A)) & join_semilatt_str(A)) <=> ((~empty_carrier(A)) & join_commutative(A) & join_semilatt_str(A))), (((((~empty_carrier(A)) & join_commutative(A)) & join_semilatt_str(A)) & element(B, the_carrier(A))) <=> (((~empty_carrier(A)) & join_commutative(A) & join_semilatt_str(A)) & element(B, the_carrier(A))))), rewrite((((~empty_carrier(A)) & join_commutative(A) & join_semilatt_str(A)) & element(B, the_carrier(A))) <=> ((~empty_carrier(A)) & join_commutative(A) & join_semilatt_str(A) & element(B, the_carrier(A)))), (((((~empty_carrier(A)) & join_commutative(A)) & join_semilatt_str(A)) & element(B, the_carrier(A))) <=> ((~empty_carrier(A)) & join_commutative(A) & join_semilatt_str(A) & element(B, the_carrier(A))))), ((((((~empty_carrier(A)) & join_commutative(A)) & join_semilatt_str(A)) & element(B, the_carrier(A))) & element(C, the_carrier(A))) <=> (((~empty_carrier(A)) & join_commutative(A) & join_semilatt_str(A) & element(B, the_carrier(A))) & element(C, the_carrier(A))))), rewrite((((~empty_carrier(A)) & join_commutative(A) & join_semilatt_str(A) & element(B, the_carrier(A))) & element(C, the_carrier(A))) <=> ((~empty_carrier(A)) & join_commutative(A) & join_semilatt_str(A) & element(B, the_carrier(A)) & element(C, the_carrier(A)))), ((((((~empty_carrier(A)) & join_commutative(A)) & join_semilatt_str(A)) & element(B, the_carrier(A))) & element(C, the_carrier(A))) <=> ((~empty_carrier(A)) & join_commutative(A) & join_semilatt_str(A) & element(B, the_carrier(A)) & element(C, the_carrier(A))))), (((((((~empty_carrier(A)) & join_commutative(A)) & join_semilatt_str(A)) & element(B, the_carrier(A))) & element(C, the_carrier(A))) => (join_commut(A, B, C) = join(A, B, C))) <=> (((~empty_carrier(A)) & join_commutative(A) & join_semilatt_str(A) & element(B, the_carrier(A)) & element(C, the_carrier(A))) => (join_commut(A, B, C) = join(A, B, C))))), rewrite((((~empty_carrier(A)) & join_commutative(A) & join_semilatt_str(A) & element(B, the_carrier(A)) & element(C, the_carrier(A))) => (join_commut(A, B, C) = join(A, B, C))) <=> ((~((~empty_carrier(A)) & join_commutative(A) & join_semilatt_str(A) & element(B, the_carrier(A)) & element(C, the_carrier(A)))) | (join_commut(A, B, C) = join(A, B, C)))), (((((((~empty_carrier(A)) & join_commutative(A)) & join_semilatt_str(A)) & element(B, the_carrier(A))) & element(C, the_carrier(A))) => (join_commut(A, B, C) = join(A, B, C))) <=> ((~((~empty_carrier(A)) & join_commutative(A) & join_semilatt_str(A) & element(B, the_carrier(A)) & element(C, the_carrier(A)))) | (join_commut(A, B, C) = join(A, B, C)))))),
% 0.21/0.42 inference(bind,[status(th)],[])).
% 0.21/0.42 tff(124,plain,
% 0.21/0.42 (![A: $i, B: $i, C: $i] : ((((((~empty_carrier(A)) & join_commutative(A)) & join_semilatt_str(A)) & element(B, the_carrier(A))) & element(C, the_carrier(A))) => (join_commut(A, B, C) = join(A, B, C))) <=> ![A: $i, B: $i, C: $i] : ((~((~empty_carrier(A)) & join_commutative(A) & join_semilatt_str(A) & element(B, the_carrier(A)) & element(C, the_carrier(A)))) | (join_commut(A, B, C) = join(A, B, C)))),
% 0.21/0.42 inference(quant_intro,[status(thm)],[123])).
% 0.21/0.42 tff(125,axiom,(![A: $i, B: $i, C: $i] : ((((((~empty_carrier(A)) & join_commutative(A)) & join_semilatt_str(A)) & element(B, the_carrier(A))) & element(C, the_carrier(A))) => (join_commut(A, B, C) = join(A, B, C)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','redefinition_k3_lattices')).
% 0.21/0.42 tff(126,plain,
% 0.21/0.42 (![A: $i, B: $i, C: $i] : ((~((~empty_carrier(A)) & join_commutative(A) & join_semilatt_str(A) & element(B, the_carrier(A)) & element(C, the_carrier(A)))) | (join_commut(A, B, C) = join(A, B, C)))),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[125, 124])).
% 0.21/0.42 tff(127,plain,
% 0.21/0.42 (![A: $i, B: $i, C: $i] : ((~((~empty_carrier(A)) & join_commutative(A) & join_semilatt_str(A) & element(B, the_carrier(A)) & element(C, the_carrier(A)))) | (join_commut(A, B, C) = join(A, B, C)))),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[126, 122])).
% 0.21/0.42 tff(128,plain,(
% 0.21/0.42 ![A: $i, B: $i, C: $i] : ((~((~empty_carrier(A)) & join_commutative(A) & join_semilatt_str(A) & element(B, the_carrier(A)) & element(C, the_carrier(A)))) | (join_commut(A, B, C) = join(A, B, C)))),
% 0.21/0.42 inference(skolemize,[status(sab)],[127])).
% 0.21/0.42 tff(129,plain,
% 0.21/0.42 (![A: $i, B: $i, C: $i] : (empty_carrier(A) | (join_commut(A, B, C) = join(A, B, C)) | (~element(C, the_carrier(A))) | (~join_commutative(A)) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A))))),
% 0.21/0.42 inference(modus_ponens,[status(thm)],[128, 121])).
% 0.21/0.43 tff(130,plain,
% 0.21/0.43 (![A: $i, B: $i, C: $i] : (empty_carrier(A) | (join_commut(A, B, C) = join(A, B, C)) | (~element(C, the_carrier(A))) | (~join_commutative(A)) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A))))),
% 0.21/0.43 inference(modus_ponens,[status(thm)],[129, 119])).
% 0.21/0.43 tff(131,plain,
% 0.21/0.43 (((~![A: $i, B: $i, C: $i] : (empty_carrier(A) | (join_commut(A, B, C) = join(A, B, C)) | (~element(C, the_carrier(A))) | (~join_commutative(A)) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A))))) | (empty_carrier(A!7) | (~join_semilatt_str(A!7)) | (~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), the_carrier(A!7))) | (~join_commutative(A!7)) | (~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), the_carrier(A!7))) | (join_commut(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9)) = join(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9))))) <=> ((~![A: $i, B: $i, C: $i] : (empty_carrier(A) | (join_commut(A, B, C) = join(A, B, C)) | (~element(C, the_carrier(A))) | (~join_commutative(A)) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A))))) | empty_carrier(A!7) | (~join_semilatt_str(A!7)) | (~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), the_carrier(A!7))) | (~join_commutative(A!7)) | (~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), the_carrier(A!7))) | (join_commut(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9)) = join(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9))))),
% 0.21/0.43 inference(rewrite,[status(thm)],[])).
% 0.21/0.43 tff(132,plain,
% 0.21/0.43 ((empty_carrier(A!7) | (join_commut(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9)) = join(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9))) | (~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), the_carrier(A!7))) | (~join_commutative(A!7)) | (~join_semilatt_str(A!7)) | (~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), the_carrier(A!7)))) <=> (empty_carrier(A!7) | (~join_semilatt_str(A!7)) | (~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), the_carrier(A!7))) | (~join_commutative(A!7)) | (~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), the_carrier(A!7))) | (join_commut(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9)) = join(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9))))),
% 0.21/0.43 inference(rewrite,[status(thm)],[])).
% 0.21/0.43 tff(133,plain,
% 0.21/0.43 (((~![A: $i, B: $i, C: $i] : (empty_carrier(A) | (join_commut(A, B, C) = join(A, B, C)) | (~element(C, the_carrier(A))) | (~join_commutative(A)) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A))))) | (empty_carrier(A!7) | (join_commut(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9)) = join(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9))) | (~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), the_carrier(A!7))) | (~join_commutative(A!7)) | (~join_semilatt_str(A!7)) | (~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), the_carrier(A!7))))) <=> ((~![A: $i, B: $i, C: $i] : (empty_carrier(A) | (join_commut(A, B, C) = join(A, B, C)) | (~element(C, the_carrier(A))) | (~join_commutative(A)) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A))))) | (empty_carrier(A!7) | (~join_semilatt_str(A!7)) | (~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), the_carrier(A!7))) | (~join_commutative(A!7)) | (~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), the_carrier(A!7))) | (join_commut(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9)) = join(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9)))))),
% 0.21/0.43 inference(monotonicity,[status(thm)],[132])).
% 0.21/0.43 tff(134,plain,
% 0.21/0.43 (((~![A: $i, B: $i, C: $i] : (empty_carrier(A) | (join_commut(A, B, C) = join(A, B, C)) | (~element(C, the_carrier(A))) | (~join_commutative(A)) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A))))) | (empty_carrier(A!7) | (join_commut(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9)) = join(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9))) | (~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), the_carrier(A!7))) | (~join_commutative(A!7)) | (~join_semilatt_str(A!7)) | (~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), the_carrier(A!7))))) <=> ((~![A: $i, B: $i, C: $i] : (empty_carrier(A) | (join_commut(A, B, C) = join(A, B, C)) | (~element(C, the_carrier(A))) | (~join_commutative(A)) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A))))) | empty_carrier(A!7) | (~join_semilatt_str(A!7)) | (~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), the_carrier(A!7))) | (~join_commutative(A!7)) | (~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), the_carrier(A!7))) | (join_commut(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9)) = join(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9))))),
% 0.21/0.43 inference(transitivity,[status(thm)],[133, 131])).
% 0.21/0.43 tff(135,plain,
% 0.21/0.43 ((~![A: $i, B: $i, C: $i] : (empty_carrier(A) | (join_commut(A, B, C) = join(A, B, C)) | (~element(C, the_carrier(A))) | (~join_commutative(A)) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A))))) | (empty_carrier(A!7) | (join_commut(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9)) = join(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9))) | (~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), the_carrier(A!7))) | (~join_commutative(A!7)) | (~join_semilatt_str(A!7)) | (~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), the_carrier(A!7))))),
% 0.21/0.43 inference(quant_inst,[status(thm)],[])).
% 0.21/0.43 tff(136,plain,
% 0.21/0.43 ((~![A: $i, B: $i, C: $i] : (empty_carrier(A) | (join_commut(A, B, C) = join(A, B, C)) | (~element(C, the_carrier(A))) | (~join_commutative(A)) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A))))) | empty_carrier(A!7) | (~join_semilatt_str(A!7)) | (~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), the_carrier(A!7))) | (~join_commutative(A!7)) | (~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), the_carrier(A!7))) | (join_commut(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9)) = join(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9)))),
% 0.21/0.43 inference(modus_ponens,[status(thm)],[135, 134])).
% 0.21/0.43 tff(137,plain,
% 0.21/0.43 ((~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), the_carrier(A!7))) | (~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), the_carrier(A!7))) | (join_commut(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9)) = join(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9)))),
% 0.21/0.43 inference(unit_resolution,[status(thm)],[136, 130, 15, 117, 14])).
% 0.21/0.43 tff(138,plain,
% 0.21/0.43 (join_commut(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9)) = join(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9))),
% 0.21/0.43 inference(unit_resolution,[status(thm)],[137, 116, 113])).
% 0.21/0.43 tff(139,plain,
% 0.21/0.43 (^[A: $i, B: $i, C: $i] : refl(((join_commut(A, B, C) = join_commut(A, C, B)) | empty_carrier(A) | (~element(C, the_carrier(A))) | (~join_commutative(A)) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A)))) <=> ((join_commut(A, B, C) = join_commut(A, C, B)) | empty_carrier(A) | (~element(C, the_carrier(A))) | (~join_commutative(A)) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A)))))),
% 0.21/0.43 inference(bind,[status(th)],[])).
% 0.21/0.43 tff(140,plain,
% 0.21/0.43 (![A: $i, B: $i, C: $i] : ((join_commut(A, B, C) = join_commut(A, C, B)) | empty_carrier(A) | (~element(C, the_carrier(A))) | (~join_commutative(A)) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A)))) <=> ![A: $i, B: $i, C: $i] : ((join_commut(A, B, C) = join_commut(A, C, B)) | empty_carrier(A) | (~element(C, the_carrier(A))) | (~join_commutative(A)) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A))))),
% 0.21/0.44 inference(quant_intro,[status(thm)],[139])).
% 0.21/0.44 tff(141,plain,
% 0.21/0.44 (^[A: $i, B: $i, C: $i] : trans(monotonicity(trans(monotonicity(rewrite(((~empty_carrier(A)) & join_commutative(A) & join_semilatt_str(A) & element(B, the_carrier(A)) & element(C, the_carrier(A))) <=> (~(empty_carrier(A) | (~element(C, the_carrier(A))) | (~join_commutative(A)) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A)))))), ((~((~empty_carrier(A)) & join_commutative(A) & join_semilatt_str(A) & element(B, the_carrier(A)) & element(C, the_carrier(A)))) <=> (~(~(empty_carrier(A) | (~element(C, the_carrier(A))) | (~join_commutative(A)) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A)))))))), rewrite((~(~(empty_carrier(A) | (~element(C, the_carrier(A))) | (~join_commutative(A)) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A)))))) <=> (empty_carrier(A) | (~element(C, the_carrier(A))) | (~join_commutative(A)) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A))))), ((~((~empty_carrier(A)) & join_commutative(A) & join_semilatt_str(A) & element(B, the_carrier(A)) & element(C, the_carrier(A)))) <=> (empty_carrier(A) | (~element(C, the_carrier(A))) | (~join_commutative(A)) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A)))))), (((~((~empty_carrier(A)) & join_commutative(A) & join_semilatt_str(A) & element(B, the_carrier(A)) & element(C, the_carrier(A)))) | (join_commut(A, B, C) = join_commut(A, C, B))) <=> ((empty_carrier(A) | (~element(C, the_carrier(A))) | (~join_commutative(A)) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A)))) | (join_commut(A, B, C) = join_commut(A, C, B))))), rewrite(((empty_carrier(A) | (~element(C, the_carrier(A))) | (~join_commutative(A)) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A)))) | (join_commut(A, B, C) = join_commut(A, C, B))) <=> ((join_commut(A, B, C) = join_commut(A, C, B)) | empty_carrier(A) | (~element(C, the_carrier(A))) | (~join_commutative(A)) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A))))), (((~((~empty_carrier(A)) & join_commutative(A) & join_semilatt_str(A) & element(B, the_carrier(A)) & element(C, the_carrier(A)))) | (join_commut(A, B, C) = join_commut(A, C, B))) <=> ((join_commut(A, B, C) = join_commut(A, C, B)) | empty_carrier(A) | (~element(C, the_carrier(A))) | (~join_commutative(A)) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A))))))),
% 0.21/0.44 inference(bind,[status(th)],[])).
% 0.21/0.44 tff(142,plain,
% 0.21/0.44 (![A: $i, B: $i, C: $i] : ((~((~empty_carrier(A)) & join_commutative(A) & join_semilatt_str(A) & element(B, the_carrier(A)) & element(C, the_carrier(A)))) | (join_commut(A, B, C) = join_commut(A, C, B))) <=> ![A: $i, B: $i, C: $i] : ((join_commut(A, B, C) = join_commut(A, C, B)) | empty_carrier(A) | (~element(C, the_carrier(A))) | (~join_commutative(A)) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A))))),
% 0.21/0.44 inference(quant_intro,[status(thm)],[141])).
% 0.21/0.44 tff(143,plain,
% 0.21/0.44 (![A: $i, B: $i, C: $i] : ((~((~empty_carrier(A)) & join_commutative(A) & join_semilatt_str(A) & element(B, the_carrier(A)) & element(C, the_carrier(A)))) | (join_commut(A, B, C) = join_commut(A, C, B))) <=> ![A: $i, B: $i, C: $i] : ((~((~empty_carrier(A)) & join_commutative(A) & join_semilatt_str(A) & element(B, the_carrier(A)) & element(C, the_carrier(A)))) | (join_commut(A, B, C) = join_commut(A, C, B)))),
% 0.21/0.44 inference(rewrite,[status(thm)],[])).
% 0.21/0.44 tff(144,plain,
% 0.21/0.44 (^[A: $i, B: $i, C: $i] : trans(monotonicity(trans(monotonicity(trans(monotonicity(rewrite((((~empty_carrier(A)) & join_commutative(A)) & join_semilatt_str(A)) <=> ((~empty_carrier(A)) & join_commutative(A) & join_semilatt_str(A))), (((((~empty_carrier(A)) & join_commutative(A)) & join_semilatt_str(A)) & element(B, the_carrier(A))) <=> (((~empty_carrier(A)) & join_commutative(A) & join_semilatt_str(A)) & element(B, the_carrier(A))))), rewrite((((~empty_carrier(A)) & join_commutative(A) & join_semilatt_str(A)) & element(B, the_carrier(A))) <=> ((~empty_carrier(A)) & join_commutative(A) & join_semilatt_str(A) & element(B, the_carrier(A)))), (((((~empty_carrier(A)) & join_commutative(A)) & join_semilatt_str(A)) & element(B, the_carrier(A))) <=> ((~empty_carrier(A)) & join_commutative(A) & join_semilatt_str(A) & element(B, the_carrier(A))))), ((((((~empty_carrier(A)) & join_commutative(A)) & join_semilatt_str(A)) & element(B, the_carrier(A))) & element(C, the_carrier(A))) <=> (((~empty_carrier(A)) & join_commutative(A) & join_semilatt_str(A) & element(B, the_carrier(A))) & element(C, the_carrier(A))))), rewrite((((~empty_carrier(A)) & join_commutative(A) & join_semilatt_str(A) & element(B, the_carrier(A))) & element(C, the_carrier(A))) <=> ((~empty_carrier(A)) & join_commutative(A) & join_semilatt_str(A) & element(B, the_carrier(A)) & element(C, the_carrier(A)))), ((((((~empty_carrier(A)) & join_commutative(A)) & join_semilatt_str(A)) & element(B, the_carrier(A))) & element(C, the_carrier(A))) <=> ((~empty_carrier(A)) & join_commutative(A) & join_semilatt_str(A) & element(B, the_carrier(A)) & element(C, the_carrier(A))))), (((((((~empty_carrier(A)) & join_commutative(A)) & join_semilatt_str(A)) & element(B, the_carrier(A))) & element(C, the_carrier(A))) => (join_commut(A, B, C) = join_commut(A, C, B))) <=> (((~empty_carrier(A)) & join_commutative(A) & join_semilatt_str(A) & element(B, the_carrier(A)) & element(C, the_carrier(A))) => (join_commut(A, B, C) = join_commut(A, C, B))))), rewrite((((~empty_carrier(A)) & join_commutative(A) & join_semilatt_str(A) & element(B, the_carrier(A)) & element(C, the_carrier(A))) => (join_commut(A, B, C) = join_commut(A, C, B))) <=> ((~((~empty_carrier(A)) & join_commutative(A) & join_semilatt_str(A) & element(B, the_carrier(A)) & element(C, the_carrier(A)))) | (join_commut(A, B, C) = join_commut(A, C, B)))), (((((((~empty_carrier(A)) & join_commutative(A)) & join_semilatt_str(A)) & element(B, the_carrier(A))) & element(C, the_carrier(A))) => (join_commut(A, B, C) = join_commut(A, C, B))) <=> ((~((~empty_carrier(A)) & join_commutative(A) & join_semilatt_str(A) & element(B, the_carrier(A)) & element(C, the_carrier(A)))) | (join_commut(A, B, C) = join_commut(A, C, B)))))),
% 0.21/0.44 inference(bind,[status(th)],[])).
% 0.21/0.44 tff(145,plain,
% 0.21/0.44 (![A: $i, B: $i, C: $i] : ((((((~empty_carrier(A)) & join_commutative(A)) & join_semilatt_str(A)) & element(B, the_carrier(A))) & element(C, the_carrier(A))) => (join_commut(A, B, C) = join_commut(A, C, B))) <=> ![A: $i, B: $i, C: $i] : ((~((~empty_carrier(A)) & join_commutative(A) & join_semilatt_str(A) & element(B, the_carrier(A)) & element(C, the_carrier(A)))) | (join_commut(A, B, C) = join_commut(A, C, B)))),
% 0.21/0.44 inference(quant_intro,[status(thm)],[144])).
% 0.21/0.44 tff(146,axiom,(![A: $i, B: $i, C: $i] : ((((((~empty_carrier(A)) & join_commutative(A)) & join_semilatt_str(A)) & element(B, the_carrier(A))) & element(C, the_carrier(A))) => (join_commut(A, B, C) = join_commut(A, C, B)))), file('/export/starexec/sandbox2/benchmark/theBenchmark.p','commutativity_k3_lattices')).
% 0.21/0.44 tff(147,plain,
% 0.21/0.44 (![A: $i, B: $i, C: $i] : ((~((~empty_carrier(A)) & join_commutative(A) & join_semilatt_str(A) & element(B, the_carrier(A)) & element(C, the_carrier(A)))) | (join_commut(A, B, C) = join_commut(A, C, B)))),
% 0.21/0.44 inference(modus_ponens,[status(thm)],[146, 145])).
% 0.21/0.44 tff(148,plain,
% 0.21/0.44 (![A: $i, B: $i, C: $i] : ((~((~empty_carrier(A)) & join_commutative(A) & join_semilatt_str(A) & element(B, the_carrier(A)) & element(C, the_carrier(A)))) | (join_commut(A, B, C) = join_commut(A, C, B)))),
% 0.21/0.44 inference(modus_ponens,[status(thm)],[147, 143])).
% 0.21/0.44 tff(149,plain,(
% 0.21/0.44 ![A: $i, B: $i, C: $i] : ((~((~empty_carrier(A)) & join_commutative(A) & join_semilatt_str(A) & element(B, the_carrier(A)) & element(C, the_carrier(A)))) | (join_commut(A, B, C) = join_commut(A, C, B)))),
% 0.21/0.44 inference(skolemize,[status(sab)],[148])).
% 0.21/0.44 tff(150,plain,
% 0.21/0.44 (![A: $i, B: $i, C: $i] : ((join_commut(A, B, C) = join_commut(A, C, B)) | empty_carrier(A) | (~element(C, the_carrier(A))) | (~join_commutative(A)) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A))))),
% 0.21/0.44 inference(modus_ponens,[status(thm)],[149, 142])).
% 0.21/0.44 tff(151,plain,
% 0.21/0.44 (![A: $i, B: $i, C: $i] : ((join_commut(A, B, C) = join_commut(A, C, B)) | empty_carrier(A) | (~element(C, the_carrier(A))) | (~join_commutative(A)) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A))))),
% 0.21/0.44 inference(modus_ponens,[status(thm)],[150, 140])).
% 0.21/0.44 tff(152,plain,
% 0.21/0.44 (((~![A: $i, B: $i, C: $i] : ((join_commut(A, B, C) = join_commut(A, C, B)) | empty_carrier(A) | (~element(C, the_carrier(A))) | (~join_commutative(A)) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A))))) | (empty_carrier(A!7) | (~join_semilatt_str(A!7)) | (~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), the_carrier(A!7))) | (~join_commutative(A!7)) | (~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), the_carrier(A!7))) | (join_commut(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8)) = join_commut(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9))))) <=> ((~![A: $i, B: $i, C: $i] : ((join_commut(A, B, C) = join_commut(A, C, B)) | empty_carrier(A) | (~element(C, the_carrier(A))) | (~join_commutative(A)) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A))))) | empty_carrier(A!7) | (~join_semilatt_str(A!7)) | (~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), the_carrier(A!7))) | (~join_commutative(A!7)) | (~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), the_carrier(A!7))) | (join_commut(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8)) = join_commut(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9))))),
% 0.21/0.44 inference(rewrite,[status(thm)],[])).
% 0.21/0.44 tff(153,plain,
% 0.21/0.44 (((join_commut(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8)) = join_commut(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9))) | empty_carrier(A!7) | (~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), the_carrier(A!7))) | (~join_commutative(A!7)) | (~join_semilatt_str(A!7)) | (~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), the_carrier(A!7)))) <=> (empty_carrier(A!7) | (~join_semilatt_str(A!7)) | (~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), the_carrier(A!7))) | (~join_commutative(A!7)) | (~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), the_carrier(A!7))) | (join_commut(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8)) = join_commut(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9))))),
% 0.21/0.44 inference(rewrite,[status(thm)],[])).
% 0.21/0.44 tff(154,plain,
% 0.21/0.44 (((~![A: $i, B: $i, C: $i] : ((join_commut(A, B, C) = join_commut(A, C, B)) | empty_carrier(A) | (~element(C, the_carrier(A))) | (~join_commutative(A)) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A))))) | ((join_commut(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8)) = join_commut(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9))) | empty_carrier(A!7) | (~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), the_carrier(A!7))) | (~join_commutative(A!7)) | (~join_semilatt_str(A!7)) | (~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), the_carrier(A!7))))) <=> ((~![A: $i, B: $i, C: $i] : ((join_commut(A, B, C) = join_commut(A, C, B)) | empty_carrier(A) | (~element(C, the_carrier(A))) | (~join_commutative(A)) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A))))) | (empty_carrier(A!7) | (~join_semilatt_str(A!7)) | (~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), the_carrier(A!7))) | (~join_commutative(A!7)) | (~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), the_carrier(A!7))) | (join_commut(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8)) = join_commut(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9)))))),
% 0.21/0.44 inference(monotonicity,[status(thm)],[153])).
% 0.21/0.44 tff(155,plain,
% 0.21/0.44 (((~![A: $i, B: $i, C: $i] : ((join_commut(A, B, C) = join_commut(A, C, B)) | empty_carrier(A) | (~element(C, the_carrier(A))) | (~join_commutative(A)) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A))))) | ((join_commut(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8)) = join_commut(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9))) | empty_carrier(A!7) | (~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), the_carrier(A!7))) | (~join_commutative(A!7)) | (~join_semilatt_str(A!7)) | (~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), the_carrier(A!7))))) <=> ((~![A: $i, B: $i, C: $i] : ((join_commut(A, B, C) = join_commut(A, C, B)) | empty_carrier(A) | (~element(C, the_carrier(A))) | (~join_commutative(A)) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A))))) | empty_carrier(A!7) | (~join_semilatt_str(A!7)) | (~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), the_carrier(A!7))) | (~join_commutative(A!7)) | (~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), the_carrier(A!7))) | (join_commut(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8)) = join_commut(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9))))),
% 0.21/0.45 inference(transitivity,[status(thm)],[154, 152])).
% 0.21/0.45 tff(156,plain,
% 0.21/0.45 ((~![A: $i, B: $i, C: $i] : ((join_commut(A, B, C) = join_commut(A, C, B)) | empty_carrier(A) | (~element(C, the_carrier(A))) | (~join_commutative(A)) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A))))) | ((join_commut(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8)) = join_commut(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9))) | empty_carrier(A!7) | (~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), the_carrier(A!7))) | (~join_commutative(A!7)) | (~join_semilatt_str(A!7)) | (~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), the_carrier(A!7))))),
% 0.21/0.45 inference(quant_inst,[status(thm)],[])).
% 0.21/0.45 tff(157,plain,
% 0.21/0.45 ((~![A: $i, B: $i, C: $i] : ((join_commut(A, B, C) = join_commut(A, C, B)) | empty_carrier(A) | (~element(C, the_carrier(A))) | (~join_commutative(A)) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A))))) | empty_carrier(A!7) | (~join_semilatt_str(A!7)) | (~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), the_carrier(A!7))) | (~join_commutative(A!7)) | (~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), the_carrier(A!7))) | (join_commut(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8)) = join_commut(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9)))),
% 0.21/0.45 inference(modus_ponens,[status(thm)],[156, 155])).
% 0.21/0.45 tff(158,plain,
% 0.21/0.45 (join_commut(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8)) = join_commut(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9))),
% 0.21/0.45 inference(unit_resolution,[status(thm)],[157, 151, 15, 117, 14, 113, 116])).
% 0.21/0.45 tff(159,plain,
% 0.21/0.45 (((~![A: $i, B: $i, C: $i] : (empty_carrier(A) | (join_commut(A, B, C) = join(A, B, C)) | (~element(C, the_carrier(A))) | (~join_commutative(A)) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A))))) | (empty_carrier(A!7) | (~join_semilatt_str(A!7)) | (join_commut(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8)) = join(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8))) | (~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), the_carrier(A!7))) | (~join_commutative(A!7)) | (~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), the_carrier(A!7))))) <=> ((~![A: $i, B: $i, C: $i] : (empty_carrier(A) | (join_commut(A, B, C) = join(A, B, C)) | (~element(C, the_carrier(A))) | (~join_commutative(A)) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A))))) | empty_carrier(A!7) | (~join_semilatt_str(A!7)) | (join_commut(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8)) = join(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8))) | (~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), the_carrier(A!7))) | (~join_commutative(A!7)) | (~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), the_carrier(A!7))))),
% 0.21/0.45 inference(rewrite,[status(thm)],[])).
% 0.21/0.45 tff(160,plain,
% 0.21/0.45 ((empty_carrier(A!7) | (join_commut(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8)) = join(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8))) | (~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), the_carrier(A!7))) | (~join_commutative(A!7)) | (~join_semilatt_str(A!7)) | (~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), the_carrier(A!7)))) <=> (empty_carrier(A!7) | (~join_semilatt_str(A!7)) | (join_commut(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8)) = join(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8))) | (~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), the_carrier(A!7))) | (~join_commutative(A!7)) | (~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), the_carrier(A!7))))),
% 0.21/0.45 inference(rewrite,[status(thm)],[])).
% 0.21/0.45 tff(161,plain,
% 0.21/0.45 (((~![A: $i, B: $i, C: $i] : (empty_carrier(A) | (join_commut(A, B, C) = join(A, B, C)) | (~element(C, the_carrier(A))) | (~join_commutative(A)) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A))))) | (empty_carrier(A!7) | (join_commut(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8)) = join(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8))) | (~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), the_carrier(A!7))) | (~join_commutative(A!7)) | (~join_semilatt_str(A!7)) | (~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), the_carrier(A!7))))) <=> ((~![A: $i, B: $i, C: $i] : (empty_carrier(A) | (join_commut(A, B, C) = join(A, B, C)) | (~element(C, the_carrier(A))) | (~join_commutative(A)) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A))))) | (empty_carrier(A!7) | (~join_semilatt_str(A!7)) | (join_commut(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8)) = join(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8))) | (~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), the_carrier(A!7))) | (~join_commutative(A!7)) | (~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), the_carrier(A!7)))))),
% 0.21/0.45 inference(monotonicity,[status(thm)],[160])).
% 0.21/0.45 tff(162,plain,
% 0.21/0.45 (((~![A: $i, B: $i, C: $i] : (empty_carrier(A) | (join_commut(A, B, C) = join(A, B, C)) | (~element(C, the_carrier(A))) | (~join_commutative(A)) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A))))) | (empty_carrier(A!7) | (join_commut(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8)) = join(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8))) | (~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), the_carrier(A!7))) | (~join_commutative(A!7)) | (~join_semilatt_str(A!7)) | (~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), the_carrier(A!7))))) <=> ((~![A: $i, B: $i, C: $i] : (empty_carrier(A) | (join_commut(A, B, C) = join(A, B, C)) | (~element(C, the_carrier(A))) | (~join_commutative(A)) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A))))) | empty_carrier(A!7) | (~join_semilatt_str(A!7)) | (join_commut(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8)) = join(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8))) | (~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), the_carrier(A!7))) | (~join_commutative(A!7)) | (~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), the_carrier(A!7))))),
% 0.21/0.45 inference(transitivity,[status(thm)],[161, 159])).
% 0.21/0.45 tff(163,plain,
% 0.21/0.45 ((~![A: $i, B: $i, C: $i] : (empty_carrier(A) | (join_commut(A, B, C) = join(A, B, C)) | (~element(C, the_carrier(A))) | (~join_commutative(A)) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A))))) | (empty_carrier(A!7) | (join_commut(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8)) = join(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8))) | (~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), the_carrier(A!7))) | (~join_commutative(A!7)) | (~join_semilatt_str(A!7)) | (~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), the_carrier(A!7))))),
% 0.21/0.45 inference(quant_inst,[status(thm)],[])).
% 0.21/0.46 tff(164,plain,
% 0.21/0.46 ((~![A: $i, B: $i, C: $i] : (empty_carrier(A) | (join_commut(A, B, C) = join(A, B, C)) | (~element(C, the_carrier(A))) | (~join_commutative(A)) | (~join_semilatt_str(A)) | (~element(B, the_carrier(A))))) | empty_carrier(A!7) | (~join_semilatt_str(A!7)) | (join_commut(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8)) = join(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8))) | (~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), the_carrier(A!7))) | (~join_commutative(A!7)) | (~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), the_carrier(A!7)))),
% 0.21/0.46 inference(modus_ponens,[status(thm)],[163, 162])).
% 0.21/0.46 tff(165,plain,
% 0.21/0.46 ((join_commut(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8)) = join(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8))) | (~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8), the_carrier(A!7))) | (~element(apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), the_carrier(A!7)))),
% 0.21/0.46 inference(unit_resolution,[status(thm)],[164, 130, 15, 117, 14])).
% 0.21/0.46 tff(166,plain,
% 0.21/0.46 (join_commut(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8)) = join(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8))),
% 0.21/0.46 inference(unit_resolution,[status(thm)],[165, 113, 116])).
% 0.21/0.46 tff(167,plain,
% 0.21/0.46 (join(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8)) = join_commut(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8))),
% 0.21/0.46 inference(symmetry,[status(thm)],[166])).
% 0.21/0.46 tff(168,plain,
% 0.21/0.46 (B!8 = join(A!7, C!9, B!8)),
% 0.21/0.46 inference(symmetry,[status(thm)],[99])).
% 0.21/0.46 tff(169,plain,
% 0.21/0.46 (B!8 = apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8)),
% 0.21/0.46 inference(transitivity,[status(thm)],[168, 107])).
% 0.21/0.46 tff(170,plain,
% 0.21/0.46 (C!9 = join(A!7, B!8, C!9)),
% 0.21/0.46 inference(symmetry,[status(thm)],[54])).
% 0.21/0.46 tff(171,plain,
% 0.21/0.46 (C!9 = apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9)),
% 0.21/0.46 inference(transitivity,[status(thm)],[170, 85])).
% 0.21/0.46 tff(172,plain,
% 0.21/0.46 (join(A!7, C!9, B!8) = join(A!7, apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), B!8, C!9), apply_binary_as_element(the_carrier(A!7), the_carrier(A!7), the_carrier(A!7), the_L_join(A!7), C!9, B!8))),
% 0.21/0.46 inference(monotonicity,[status(thm)],[171, 169])).
% 0.21/0.46 tff(173,plain,
% 0.21/0.46 (B!8 = C!9),
% 0.21/0.46 inference(transitivity,[status(thm)],[168, 172, 167, 158, 138, 110, 54])).
% 0.21/0.46 tff(174,plain,
% 0.21/0.46 (~(B!8 = C!9)),
% 0.21/0.46 inference(or_elim,[status(thm)],[44])).
% 0.21/0.46 tff(175,plain,
% 0.21/0.46 ($false),
% 0.21/0.46 inference(unit_resolution,[status(thm)],[174, 173])).
% 0.21/0.46 % SZS output end Proof
% 0.21/0.46 % E exiting
%------------------------------------------------------------------------------