%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : COM179^1 : TPTP v9.2.1. Released v7.0.0.
% Transfm : none
% Format : tptp:raw
% Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% Computer : n014.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 : Wed Feb 25 08:41:52 AM UTC 2026
% Result : Theorem 2.71s 1.61s
% Output : Refutation 2.71s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.12 % Problem : COM179^1 : TPTP v9.2.1. Released v7.0.0.
% 0.11/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.16/0.34 % Computer : n014.cluster.edu
% 0.16/0.34 % Model : x86_64 x86_64
% 0.16/0.34 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.34 % Memory : 8042.1875MB
% 0.16/0.34 % OS : Linux 3.10.0-693.el7.x86_64
% 0.16/0.34 % CPULimit : 300
% 0.16/0.34 % WCLimit : 300
% 0.16/0.34 % DateTime : Tue Feb 24 04:16:12 EST 2026
% 0.16/0.34 % CPUTime :
% 0.16/0.34 Running /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.19/0.46 ---- Original DTF file ---
% 0.19/0.46 thf(spec,logic,$$dhol).
% 0.19/0.46 %------------------------------------------------------------------------------
% 0.19/0.46 % File : COM179^1 : TPTP v9.2.1. Released v7.0.0.
% 0.19/0.46 % Domain : Computing Theory
% 0.19/0.46 % Problem : Koenig's lemma 231
% 0.19/0.46 % Version : [Bla16] axioms : Especial.
% 0.19/0.46 % English :
% 0.19/0.46
% 0.19/0.46 % Refs : [Loc10] Lochbihler (2010), Coinductive
% 0.19/0.46 % : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% 0.19/0.46 % : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% 0.19/0.46 % Source : [Bla16]
% 0.19/0.46 % Names : koenigslemma__231.p [Bla16]
% 0.19/0.46
% 0.19/0.46 % Status : Theorem
% 0.19/0.46 % Rating : 0.00 v7.2.0, 0.25 v7.1.0
% 0.19/0.46 % Syntax : Number of formulae : 345 ( 118 unt; 53 typ; 0 def)
% 0.19/0.46 % Number of atoms : 728 ( 227 equ; 0 cnn)
% 0.19/0.46 % Maximal formula atoms : 13 ( 2 avg)
% 0.19/0.46 % Number of connectives : 3879 ( 101 ~; 8 |; 31 &;3406 @)
% 0.19/0.46 % ( 0 <=>; 333 =>; 0 <=; 0 <~>)
% 0.19/0.46 % Maximal formula depth : 28 ( 8 avg)
% 0.19/0.46 % Number of types : 2 ( 1 usr)
% 0.19/0.46 % Number of type conns : 167 ( 167 >; 0 *; 0 +; 0 <<)
% 0.19/0.46 % Number of symbols : 54 ( 52 usr; 3 con; 0-5 aty)
% 0.19/0.46 % Number of variables : 1010 ( 35 ^; 911 !; 15 ?;1010 :)
% 0.19/0.46 % ( 49 !>; 0 ?*; 0 @-; 0 @+)
% 0.19/0.46 % SPC : TH1_THM_EQU_NAR
% 0.19/0.46
% 0.19/0.46 % Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 0.19/0.46 % 2016-07-13 14:47:31.744
% 0.19/0.46 %------------------------------------------------------------------------------
% 0.19/0.46 %----Could-be-implicit typings (7)
% 0.19/0.46 thf(ty_t_Coinductive__List_Ollist,type,
% 0.19/0.46 coinductive_llist: $tType > $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_t_Product__Type_Oprod,type,
% 0.19/0.46 product_prod: $tType > $tType > $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_t_Sum__Type_Osum,type,
% 0.19/0.46 sum_sum: $tType > $tType > $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_t_Option_Ooption,type,
% 0.19/0.46 option: $tType > $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_t_Set_Oset,type,
% 0.19/0.46 set: $tType > $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_t_itself,type,
% 0.19/0.46 itself: $tType > $tType ).
% 0.19/0.46
% 0.19/0.46 thf(ty_tf_node,type,
% 0.19/0.46 node: $tType ).
% 0.19/0.46
% 0.19/0.46 %----Explicit typings (46)
% 0.19/0.46 thf(sy_cl_HOL_Otype,type,
% 0.19/0.46 type:
% 0.19/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.19/0.46
% 0.19/0.46 thf(sy_cl_Groups_Ominus,type,
% 0.19/0.46 minus:
% 0.19/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.19/0.46
% 0.19/0.46 thf(sy_cl_Groups_Ouminus,type,
% 0.19/0.46 uminus:
% 0.19/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.19/0.46
% 0.19/0.46 thf(sy_cl_Orderings_Obot,type,
% 0.19/0.46 bot:
% 0.19/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.19/0.46
% 0.19/0.46 thf(sy_cl_Orderings_Oord,type,
% 0.19/0.46 ord:
% 0.19/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.19/0.46
% 0.19/0.46 thf(sy_cl_Orderings_Otop,type,
% 0.19/0.46 top:
% 0.19/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.19/0.46
% 0.19/0.46 thf(sy_cl_Orderings_Oorder,type,
% 0.19/0.46 order:
% 0.19/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.19/0.46
% 0.19/0.46 thf(sy_cl_Groups_Ogroup__add,type,
% 0.19/0.46 group_add:
% 0.19/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.19/0.46
% 0.19/0.46 thf(sy_cl_Finite__Set_Ofinite,type,
% 0.19/0.46 finite_finite:
% 0.19/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.19/0.46
% 0.19/0.46 thf(sy_cl_Orderings_Olinorder,type,
% 0.19/0.46 linorder:
% 0.19/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.19/0.46
% 0.19/0.46 thf(sy_cl_Orderings_Opreorder,type,
% 0.19/0.46 preorder:
% 0.19/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.19/0.46
% 0.19/0.46 thf(sy_cl_Nat_Osemiring__char__0,type,
% 0.19/0.46 semiring_char_0:
% 0.19/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.19/0.46
% 0.19/0.46 thf(sy_cl_Orderings_Oorder__bot,type,
% 0.19/0.46 order_bot:
% 0.19/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.19/0.46
% 0.19/0.46 thf(sy_cl_Orderings_Oorder__top,type,
% 0.19/0.46 order_top:
% 0.19/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.19/0.46
% 0.19/0.46 thf(sy_cl_Lattices_Oboolean__algebra,type,
% 0.19/0.46 boolean_algebra:
% 0.19/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.19/0.46
% 0.19/0.46 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 0.19/0.46 ordered_ab_group_add:
% 0.19/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.19/0.46
% 0.19/0.46 thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
% 0.19/0.46 cancel146912293up_add:
% 0.19/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.19/0.46
% 0.19/0.46 thf(sy_cl_Topological__Spaces_Operfect__space,type,
% 0.19/0.46 topolo890362671_space:
% 0.19/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Basic__BNFs_Ofsts,type,
% 0.19/0.46 basic_fsts:
% 0.19/0.46 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( set @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Basic__BNFs_Osnds,type,
% 0.19/0.46 basic_snds:
% 0.19/0.46 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( set @ B ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Coinductive__List_Ogen__lset,type,
% 0.19/0.46 coinductive_gen_lset:
% 0.19/0.46 !>[A: $tType] : ( ( set @ A ) > ( coinductive_llist @ A ) > ( set @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Coinductive__List_Olfinite,type,
% 0.19/0.46 coinductive_lfinite:
% 0.19/0.46 !>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Coinductive__List_Ollist_Olnull,type,
% 0.19/0.46 coinductive_lnull:
% 0.19/0.46 !>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Coinductive__List_Ollist_Olset,type,
% 0.19/0.46 coinductive_lset:
% 0.19/0.46 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( set @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Finite__Set_Ofinite,type,
% 0.19/0.46 finite_finite2:
% 0.19/0.46 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 0.19/0.46 minus_minus:
% 0.19/0.46 !>[A: $tType] : ( A > A > A ) ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Groups_Ouminus__class_Ouminus,type,
% 0.19/0.46 uminus_uminus:
% 0.19/0.46 !>[A: $tType] : ( A > A ) ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Koenigslemma__Mirabelle__aepjeeakgn_Oconnected,type,
% 0.19/0.46 koenig793108494nected:
% 0.19/0.46 !>[Node: $tType] : ( ( Node > Node > $o ) > $o ) ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Koenigslemma__Mirabelle__aepjeeakgn_Oreachable__via,type,
% 0.19/0.46 koenig317145564le_via:
% 0.19/0.46 !>[Node: $tType] : ( ( Node > Node > $o ) > ( set @ Node ) > Node > ( set @ Node ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Orderings_Obot__class_Obot,type,
% 0.19/0.46 bot_bot:
% 0.19/0.46 !>[A: $tType] : A ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.19/0.46 ord_less_eq:
% 0.19/0.46 !>[A: $tType] : ( A > A > $o ) ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Orderings_Otop__class_Otop,type,
% 0.19/0.46 top_top:
% 0.19/0.46 !>[A: $tType] : A ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Product__Type_OPair,type,
% 0.19/0.46 product_Pair:
% 0.19/0.46 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Product__Type_Ointernal__case__prod,type,
% 0.19/0.46 produc2004651681e_prod:
% 0.19/0.46 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 0.19/0.46 product_rec_prod:
% 0.19/0.46 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Pure_Otype,type,
% 0.19/0.46 type2:
% 0.19/0.46 !>[A: $tType] : ( itself @ A ) ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Set_OCollect,type,
% 0.19/0.46 collect:
% 0.19/0.46 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Set_Oinsert,type,
% 0.19/0.46 insert:
% 0.19/0.46 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Set_Ois__empty,type,
% 0.19/0.46 is_empty:
% 0.19/0.46 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Set_Ois__singleton,type,
% 0.19/0.46 is_singleton:
% 0.19/0.46 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_Set_Othe__elem,type,
% 0.19/0.46 the_elem:
% 0.19/0.46 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 0.19/0.46
% 0.19/0.46 thf(sy_c_member,type,
% 0.19/0.46 member:
% 0.19/0.46 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.19/0.46
% 0.19/0.46 thf(sy_v_f____,type,
% 0.19/0.46 f: ( product_prod @ node @ ( set @ node ) ) > ( coinductive_llist @ node ) ).
% 0.19/0.46
% 0.19/0.46 thf(sy_v_graph,type,
% 0.19/0.46 graph: node > node > $o ).
% 0.19/0.46
% 0.19/0.46 thf(sy_v_n,type,
% 0.19/0.46 n: node ).
% 0.19/0.46
% 0.19/0.46 thf(sy_v_ns____,type,
% 0.19/0.46 ns: set @ node ).
% 0.19/0.46
% 0.19/0.46 %----Relevant facts (254)
% 0.19/0.46 thf(fact_0_connected,axiom,
% 0.19/0.46 koenig793108494nected @ node @ graph ).
% 0.19/0.46
% 0.19/0.46 % connected
% 0.19/0.46 thf(fact_1_ns__def,axiom,
% 0.19/0.47 ( ns
% 0.19/0.47 = ( bot_bot @ ( set @ node ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % ns_def
% 0.19/0.47 thf(fact_2__092_060open_062infinite_A_Ireachable__via_Agraph_A_I_N_A_123n_125_J_An_J_092_060close_062,axiom,
% 0.19/0.47 ~ ( finite_finite2 @ node @ ( koenig317145564le_via @ node @ graph @ ( uminus_uminus @ ( set @ node ) @ ( insert @ node @ n @ ( bot_bot @ ( set @ node ) ) ) ) @ n ) ) ).
% 0.19/0.47
% 0.19/0.47 % \<open>infinite (reachable_via graph (- {n}) n)\<close>
% 0.19/0.47 thf(fact_3_infinite,axiom,
% 0.19/0.47 ~ ( finite_finite2 @ node @ ( top_top @ ( set @ node ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % infinite
% 0.19/0.47 thf(fact_4__092_060open_062_N_A_123n_125_A_092_060subseteq_062_Areachable__via_Agraph_A_I_N_A_123n_125_J_An_092_060close_062,axiom,
% 0.19/0.47 ord_less_eq @ ( set @ node ) @ ( uminus_uminus @ ( set @ node ) @ ( insert @ node @ n @ ( bot_bot @ ( set @ node ) ) ) ) @ ( koenig317145564le_via @ node @ graph @ ( uminus_uminus @ ( set @ node ) @ ( insert @ node @ n @ ( bot_bot @ ( set @ node ) ) ) ) @ n ) ).
% 0.19/0.47
% 0.19/0.47 % \<open>- {n} \<subseteq> reachable_via graph (- {n}) n\<close>
% 0.19/0.47 thf(fact_5_lset,axiom,
% 0.19/0.47 ! [X: node,Na: node,Nsa: set @ node] :
% 0.19/0.47 ( ( member @ node @ X @ ( coinductive_lset @ node @ ( f @ ( product_Pair @ node @ ( set @ node ) @ Na @ Nsa ) ) ) )
% 0.19/0.47 => ( ~ ( member @ node @ Na @ Nsa )
% 0.19/0.47 => ( ( finite_finite2 @ node @ Nsa )
% 0.19/0.47 => ( ~ ( finite_finite2 @ node @ ( koenig317145564le_via @ node @ graph @ ( uminus_uminus @ ( set @ node ) @ ( insert @ node @ Na @ Nsa ) ) @ Na ) )
% 0.19/0.47 => ~ ( member @ node @ X @ Nsa ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % lset
% 0.19/0.47 thf(fact_6_finite__insert,axiom,
% 0.19/0.47 ! [A: $tType,A2: A,A3: set @ A] :
% 0.19/0.47 ( ( finite_finite2 @ A @ ( insert @ A @ A2 @ A3 ) )
% 0.19/0.47 = ( finite_finite2 @ A @ A3 ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite_insert
% 0.19/0.47 thf(fact_7_singletonI,axiom,
% 0.19/0.47 ! [A: $tType,A2: A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % singletonI
% 0.19/0.47 thf(fact_8_finite_Ocases,axiom,
% 0.19/0.47 ! [A: $tType,A2: set @ A] :
% 0.19/0.47 ( ( finite_finite2 @ A @ A2 )
% 0.19/0.47 => ( ( A2
% 0.19/0.47 != ( bot_bot @ ( set @ A ) ) )
% 0.19/0.47 => ~ ! [A4: set @ A] :
% 0.19/0.47 ( ? [A5: A] :
% 0.19/0.47 ( A2
% 0.19/0.47 = ( insert @ A @ A5 @ A4 ) )
% 0.19/0.47 => ~ ( finite_finite2 @ A @ A4 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite.cases
% 0.19/0.47 thf(fact_9_finite_Osimps,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( finite_finite2 @ A )
% 0.19/0.47 = ( ^ [A6: set @ A] :
% 0.19/0.47 ( ( A6
% 0.19/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.19/0.47 | ? [A7: set @ A,B2: A] :
% 0.19/0.47 ( ( A6
% 0.19/0.47 = ( insert @ A @ B2 @ A7 ) )
% 0.19/0.47 & ( finite_finite2 @ A @ A7 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite.simps
% 0.19/0.47 thf(fact_10_finite__induct,axiom,
% 0.19/0.47 ! [A: $tType,F: set @ A,P: ( set @ A ) > $o] :
% 0.19/0.47 ( ( finite_finite2 @ A @ F )
% 0.19/0.47 => ( ( P @ ( bot_bot @ ( set @ A ) ) )
% 0.19/0.47 => ( ! [X2: A,F2: set @ A] :
% 0.19/0.47 ( ( finite_finite2 @ A @ F2 )
% 0.19/0.47 => ( ~ ( member @ A @ X2 @ F2 )
% 0.19/0.47 => ( ( P @ F2 )
% 0.19/0.47 => ( P @ ( insert @ A @ X2 @ F2 ) ) ) ) )
% 0.19/0.47 => ( P @ F ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite_induct
% 0.19/0.47 thf(fact_11_finite_Oinducts,axiom,
% 0.19/0.47 ! [A: $tType,X3: set @ A,P: ( set @ A ) > $o] :
% 0.19/0.47 ( ( finite_finite2 @ A @ X3 )
% 0.19/0.47 => ( ( P @ ( bot_bot @ ( set @ A ) ) )
% 0.19/0.47 => ( ! [A4: set @ A,A5: A] :
% 0.19/0.47 ( ( finite_finite2 @ A @ A4 )
% 0.19/0.47 => ( ( P @ A4 )
% 0.19/0.47 => ( P @ ( insert @ A @ A5 @ A4 ) ) ) )
% 0.19/0.47 => ( P @ X3 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite.inducts
% 0.19/0.47 thf(fact_12_finite__ne__induct,axiom,
% 0.19/0.47 ! [A: $tType,F: set @ A,P: ( set @ A ) > $o] :
% 0.19/0.47 ( ( finite_finite2 @ A @ F )
% 0.19/0.47 => ( ( F
% 0.19/0.47 != ( bot_bot @ ( set @ A ) ) )
% 0.19/0.47 => ( ! [X2: A] : ( P @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.19/0.47 => ( ! [X2: A,F2: set @ A] :
% 0.19/0.47 ( ( finite_finite2 @ A @ F2 )
% 0.19/0.47 => ( ( F2
% 0.19/0.47 != ( bot_bot @ ( set @ A ) ) )
% 0.19/0.47 => ( ~ ( member @ A @ X2 @ F2 )
% 0.19/0.47 => ( ( P @ F2 )
% 0.19/0.47 => ( P @ ( insert @ A @ X2 @ F2 ) ) ) ) ) )
% 0.19/0.47 => ( P @ F ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite_ne_induct
% 0.19/0.47 thf(fact_13_infinite__finite__induct,axiom,
% 0.19/0.47 ! [A: $tType,P: ( set @ A ) > $o,A3: set @ A] :
% 0.19/0.47 ( ! [A4: set @ A] :
% 0.19/0.47 ( ~ ( finite_finite2 @ A @ A4 )
% 0.19/0.47 => ( P @ A4 ) )
% 0.19/0.47 => ( ( P @ ( bot_bot @ ( set @ A ) ) )
% 0.19/0.47 => ( ! [X2: A,F2: set @ A] :
% 0.19/0.47 ( ( finite_finite2 @ A @ F2 )
% 0.19/0.47 => ( ~ ( member @ A @ X2 @ F2 )
% 0.19/0.47 => ( ( P @ F2 )
% 0.19/0.47 => ( P @ ( insert @ A @ X2 @ F2 ) ) ) ) )
% 0.19/0.47 => ( P @ A3 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % infinite_finite_induct
% 0.19/0.47 thf(fact_14_ComplI,axiom,
% 0.19/0.47 ! [A: $tType,C2: A,A3: set @ A] :
% 0.19/0.47 ( ~ ( member @ A @ C2 @ A3 )
% 0.19/0.47 => ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % ComplI
% 0.19/0.47 thf(fact_15_Compl__iff,axiom,
% 0.19/0.47 ! [A: $tType,C2: A,A3: set @ A] :
% 0.19/0.47 ( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A3 ) )
% 0.19/0.47 = ( ~ ( member @ A @ C2 @ A3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Compl_iff
% 0.19/0.47 thf(fact_16_empty__Collect__eq,axiom,
% 0.19/0.47 ! [A: $tType,P: A > $o] :
% 0.19/0.47 ( ( ( bot_bot @ ( set @ A ) )
% 0.19/0.47 = ( collect @ A @ P ) )
% 0.19/0.47 = ( ! [X4: A] :
% 0.19/0.47 ~ ( P @ X4 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % empty_Collect_eq
% 0.19/0.47 thf(fact_17_Collect__empty__eq,axiom,
% 0.19/0.47 ! [A: $tType,P: A > $o] :
% 0.19/0.47 ( ( ( collect @ A @ P )
% 0.19/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.19/0.47 = ( ! [X4: A] :
% 0.19/0.47 ~ ( P @ X4 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Collect_empty_eq
% 0.19/0.47 thf(fact_18_all__not__in__conv,axiom,
% 0.19/0.47 ! [A: $tType,A3: set @ A] :
% 0.19/0.47 ( ( ! [X4: A] :
% 0.19/0.47 ~ ( member @ A @ X4 @ A3 ) )
% 0.19/0.47 = ( A3
% 0.19/0.47 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % all_not_in_conv
% 0.19/0.47 thf(fact_19_empty__iff,axiom,
% 0.19/0.47 ! [A: $tType,C2: A] :
% 0.19/0.47 ~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % empty_iff
% 0.19/0.47 thf(fact_20_UNIV__I,axiom,
% 0.19/0.47 ! [A: $tType,X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % UNIV_I
% 0.19/0.47 thf(fact_21_finite__code,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( finite_finite @ A @ ( type2 @ A ) )
% 0.19/0.47 => ( ( finite_finite2 @ A )
% 0.19/0.47 = ( ^ [A7: set @ A] : $true ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite_code
% 0.19/0.47 thf(fact_22_subsetI,axiom,
% 0.19/0.47 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.19/0.47 ( ! [X2: A] :
% 0.19/0.47 ( ( member @ A @ X2 @ A3 )
% 0.19/0.47 => ( member @ A @ X2 @ B3 ) )
% 0.19/0.47 => ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% 0.19/0.47
% 0.19/0.47 % subsetI
% 0.19/0.47 thf(fact_23_subset__antisym,axiom,
% 0.19/0.47 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.19/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 )
% 0.19/0.47 => ( A3 = B3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_antisym
% 0.19/0.47 thf(fact_24_insert__absorb2,axiom,
% 0.19/0.47 ! [A: $tType,X3: A,A3: set @ A] :
% 0.19/0.47 ( ( insert @ A @ X3 @ ( insert @ A @ X3 @ A3 ) )
% 0.19/0.47 = ( insert @ A @ X3 @ A3 ) ) ).
% 0.19/0.47
% 0.19/0.47 % insert_absorb2
% 0.19/0.47 thf(fact_25_insert__iff,axiom,
% 0.19/0.47 ! [A: $tType,A2: A,B4: A,A3: set @ A] :
% 0.19/0.47 ( ( member @ A @ A2 @ ( insert @ A @ B4 @ A3 ) )
% 0.19/0.47 = ( ( A2 = B4 )
% 0.19/0.47 | ( member @ A @ A2 @ A3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % insert_iff
% 0.19/0.47 thf(fact_26_insertCI,axiom,
% 0.19/0.47 ! [A: $tType,A2: A,B3: set @ A,B4: A] :
% 0.19/0.47 ( ( ~ ( member @ A @ A2 @ B3 )
% 0.19/0.47 => ( A2 = B4 ) )
% 0.19/0.47 => ( member @ A @ A2 @ ( insert @ A @ B4 @ B3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % insertCI
% 0.19/0.47 thf(fact_27_Compl__eq__Compl__iff,axiom,
% 0.19/0.47 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.19/0.47 ( ( ( uminus_uminus @ ( set @ A ) @ A3 )
% 0.19/0.47 = ( uminus_uminus @ ( set @ A ) @ B3 ) )
% 0.19/0.47 = ( A3 = B3 ) ) ).
% 0.19/0.47
% 0.19/0.47 % Compl_eq_Compl_iff
% 0.19/0.47 thf(fact_28_finite__Plus__UNIV__iff,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType] :
% 0.19/0.47 ( ( finite_finite2 @ ( sum_sum @ A @ B ) @ ( top_top @ ( set @ ( sum_sum @ A @ B ) ) ) )
% 0.19/0.47 = ( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
% 0.19/0.47 & ( finite_finite2 @ B @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite_Plus_UNIV_iff
% 0.19/0.47 thf(fact_29_empty__subsetI,axiom,
% 0.19/0.47 ! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 ) ).
% 0.19/0.47
% 0.19/0.47 % empty_subsetI
% 0.19/0.47 thf(fact_30_subset__empty,axiom,
% 0.19/0.47 ! [A: $tType,A3: set @ A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
% 0.19/0.47 = ( A3
% 0.19/0.47 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_empty
% 0.19/0.47 thf(fact_31_insert__subset,axiom,
% 0.19/0.47 ! [A: $tType,X3: A,A3: set @ A,B3: set @ A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X3 @ A3 ) @ B3 )
% 0.19/0.47 = ( ( member @ A @ X3 @ B3 )
% 0.19/0.47 & ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % insert_subset
% 0.19/0.47 thf(fact_32_Compl__subset__Compl__iff,axiom,
% 0.19/0.47 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) @ ( uminus_uminus @ ( set @ A ) @ B3 ) )
% 0.19/0.47 = ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ).
% 0.19/0.47
% 0.19/0.47 % Compl_subset_Compl_iff
% 0.19/0.47 thf(fact_33_Compl__anti__mono,axiom,
% 0.19/0.47 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.19/0.47 => ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B3 ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Compl_anti_mono
% 0.19/0.47 thf(fact_34__092_060open_062n_A_092_060in_062_Alset_A_If_A_In_M_Ans_J_J_092_060close_062,axiom,
% 0.19/0.47 member @ node @ n @ ( coinductive_lset @ node @ ( f @ ( product_Pair @ node @ ( set @ node ) @ n @ ns ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % \<open>n \<in> lset (f (n, ns))\<close>
% 0.19/0.47 thf(fact_35_singleton__insert__inj__eq_H,axiom,
% 0.19/0.47 ! [A: $tType,A2: A,A3: set @ A,B4: A] :
% 0.19/0.47 ( ( ( insert @ A @ A2 @ A3 )
% 0.19/0.47 = ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.19/0.47 = ( ( A2 = B4 )
% 0.19/0.47 & ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % singleton_insert_inj_eq'
% 0.19/0.47 thf(fact_36_singleton__insert__inj__eq,axiom,
% 0.19/0.47 ! [A: $tType,B4: A,A2: A,A3: set @ A] :
% 0.19/0.47 ( ( ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) )
% 0.19/0.47 = ( insert @ A @ A2 @ A3 ) )
% 0.19/0.47 = ( ( A2 = B4 )
% 0.19/0.47 & ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % singleton_insert_inj_eq
% 0.19/0.47 thf(fact_37_finite__compl,axiom,
% 0.19/0.47 ! [A: $tType,A3: set @ A] :
% 0.19/0.47 ( ( finite_finite2 @ A @ A3 )
% 0.19/0.47 => ( ( finite_finite2 @ A @ ( uminus_uminus @ ( set @ A ) @ A3 ) )
% 0.19/0.47 = ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite_compl
% 0.19/0.47 thf(fact_38_subset__Compl__singleton,axiom,
% 0.19/0.47 ! [A: $tType,A3: set @ A,B4: A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.19/0.47 = ( ~ ( member @ A @ B4 @ A3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_Compl_singleton
% 0.19/0.47 thf(fact_39_set__mp,axiom,
% 0.19/0.47 ! [A: $tType,A3: set @ A,B3: set @ A,X3: A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.19/0.47 => ( ( member @ A @ X3 @ A3 )
% 0.19/0.47 => ( member @ A @ X3 @ B3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % set_mp
% 0.19/0.47 thf(fact_40_in__mono,axiom,
% 0.19/0.47 ! [A: $tType,A3: set @ A,B3: set @ A,X3: A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.19/0.47 => ( ( member @ A @ X3 @ A3 )
% 0.19/0.47 => ( member @ A @ X3 @ B3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % in_mono
% 0.19/0.47 thf(fact_41_subsetD,axiom,
% 0.19/0.47 ! [A: $tType,A3: set @ A,B3: set @ A,C2: A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.19/0.47 => ( ( member @ A @ C2 @ A3 )
% 0.19/0.47 => ( member @ A @ C2 @ B3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subsetD
% 0.19/0.47 thf(fact_42_subsetCE,axiom,
% 0.19/0.47 ! [A: $tType,A3: set @ A,B3: set @ A,C2: A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.19/0.47 => ( ( member @ A @ C2 @ A3 )
% 0.19/0.47 => ( member @ A @ C2 @ B3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subsetCE
% 0.19/0.47 thf(fact_43_UNIV__eq__I,axiom,
% 0.19/0.47 ! [A: $tType,A3: set @ A] :
% 0.19/0.47 ( ! [X2: A] : ( member @ A @ X2 @ A3 )
% 0.19/0.47 => ( ( top_top @ ( set @ A ) )
% 0.19/0.47 = A3 ) ) ).
% 0.19/0.47
% 0.19/0.47 % UNIV_eq_I
% 0.19/0.47 thf(fact_44_equalityE,axiom,
% 0.19/0.47 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.19/0.47 ( ( A3 = B3 )
% 0.19/0.47 => ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.19/0.47 => ~ ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % equalityE
% 0.19/0.47 thf(fact_45_mem__Collect__eq,axiom,
% 0.19/0.47 ! [A: $tType,A2: A,P: A > $o] :
% 0.19/0.47 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 0.19/0.47 = ( P @ A2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % mem_Collect_eq
% 0.19/0.47 thf(fact_46_Collect__mem__eq,axiom,
% 0.19/0.47 ! [A: $tType,A3: set @ A] :
% 0.19/0.47 ( ( collect @ A
% 0.19/0.47 @ ^ [X4: A] : ( member @ A @ X4 @ A3 ) )
% 0.19/0.47 = A3 ) ).
% 0.19/0.47
% 0.19/0.47 % Collect_mem_eq
% 0.19/0.47 thf(fact_47_Collect__cong,axiom,
% 0.19/0.47 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.19/0.47 ( ! [X2: A] :
% 0.19/0.47 ( ( P @ X2 )
% 0.19/0.47 = ( Q @ X2 ) )
% 0.19/0.47 => ( ( collect @ A @ P )
% 0.19/0.47 = ( collect @ A @ Q ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Collect_cong
% 0.19/0.47 thf(fact_48_ext,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,F3: A > B,G: A > B] :
% 0.19/0.47 ( ! [X2: A] :
% 0.19/0.47 ( ( F3 @ X2 )
% 0.19/0.47 = ( G @ X2 ) )
% 0.19/0.47 => ( F3 = G ) ) ).
% 0.19/0.47
% 0.19/0.47 % ext
% 0.19/0.47 thf(fact_49_subset__eq,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) )
% 0.19/0.47 = ( ^ [A7: set @ A,B5: set @ A] :
% 0.19/0.47 ! [X4: A] :
% 0.19/0.47 ( ( member @ A @ X4 @ A7 )
% 0.19/0.47 => ( member @ A @ X4 @ B5 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_eq
% 0.19/0.47 thf(fact_50_equalityD1,axiom,
% 0.19/0.47 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.19/0.47 ( ( A3 = B3 )
% 0.19/0.47 => ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% 0.19/0.47
% 0.19/0.47 % equalityD1
% 0.19/0.47 thf(fact_51_equalityD2,axiom,
% 0.19/0.47 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.19/0.47 ( ( A3 = B3 )
% 0.19/0.47 => ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ).
% 0.19/0.47
% 0.19/0.47 % equalityD2
% 0.19/0.47 thf(fact_52_set__rev__mp,axiom,
% 0.19/0.47 ! [A: $tType,X3: A,A3: set @ A,B3: set @ A] :
% 0.19/0.47 ( ( member @ A @ X3 @ A3 )
% 0.19/0.47 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.19/0.47 => ( member @ A @ X3 @ B3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % set_rev_mp
% 0.19/0.47 thf(fact_53_subset__iff,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) )
% 0.19/0.47 = ( ^ [A7: set @ A,B5: set @ A] :
% 0.19/0.47 ! [T2: A] :
% 0.19/0.47 ( ( member @ A @ T2 @ A7 )
% 0.19/0.47 => ( member @ A @ T2 @ B5 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_iff
% 0.19/0.47 thf(fact_54_rev__subsetD,axiom,
% 0.19/0.47 ! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
% 0.19/0.47 ( ( member @ A @ C2 @ A3 )
% 0.19/0.47 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.19/0.47 => ( member @ A @ C2 @ B3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % rev_subsetD
% 0.19/0.47 thf(fact_55_subset__UNIV,axiom,
% 0.19/0.47 ! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ ( top_top @ ( set @ A ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_UNIV
% 0.19/0.47 thf(fact_56_subset__refl,axiom,
% 0.19/0.47 ! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ A3 ) ).
% 0.19/0.47
% 0.19/0.47 % subset_refl
% 0.19/0.47 thf(fact_57_Collect__mono,axiom,
% 0.19/0.47 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.19/0.47 ( ! [X2: A] :
% 0.19/0.47 ( ( P @ X2 )
% 0.19/0.47 => ( Q @ X2 ) )
% 0.19/0.47 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Collect_mono
% 0.19/0.47 thf(fact_58_UNIV__witness,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ? [X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % UNIV_witness
% 0.19/0.47 thf(fact_59_subset__trans,axiom,
% 0.19/0.47 ! [A: $tType,A3: set @ A,B3: set @ A,C3: set @ A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.19/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ C3 )
% 0.19/0.47 => ( ord_less_eq @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_trans
% 0.19/0.47 thf(fact_60_set__eq__subset,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( ^ [Y: set @ A,Z: set @ A] : ( Y = Z ) )
% 0.19/0.47 = ( ^ [A7: set @ A,B5: set @ A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B5 )
% 0.19/0.47 & ( ord_less_eq @ ( set @ A ) @ B5 @ A7 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % set_eq_subset
% 0.19/0.47 thf(fact_61_contra__subsetD,axiom,
% 0.19/0.47 ! [A: $tType,A3: set @ A,B3: set @ A,C2: A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.19/0.47 => ( ~ ( member @ A @ C2 @ B3 )
% 0.19/0.47 => ~ ( member @ A @ C2 @ A3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % contra_subsetD
% 0.19/0.47 thf(fact_62_Collect__mono__iff,axiom,
% 0.19/0.47 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 0.19/0.47 = ( ! [X4: A] :
% 0.19/0.47 ( ( P @ X4 )
% 0.19/0.47 => ( Q @ X4 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Collect_mono_iff
% 0.19/0.47 thf(fact_63_Finite__Set_Ofinite__set,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( finite_finite2 @ ( set @ A ) @ ( top_top @ ( set @ ( set @ A ) ) ) )
% 0.19/0.47 = ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Finite_Set.finite_set
% 0.19/0.47 thf(fact_64_finite__prod,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType] :
% 0.19/0.47 ( ( finite_finite2 @ ( product_prod @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
% 0.19/0.47 = ( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
% 0.19/0.47 & ( finite_finite2 @ B @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite_prod
% 0.19/0.47 thf(fact_65_finite__Prod__UNIV,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType] :
% 0.19/0.47 ( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
% 0.19/0.47 => ( ( finite_finite2 @ B @ ( top_top @ ( set @ B ) ) )
% 0.19/0.47 => ( finite_finite2 @ ( product_prod @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite_Prod_UNIV
% 0.19/0.47 thf(fact_66_finite__fun__UNIVD2,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType] :
% 0.19/0.47 ( ( finite_finite2 @ ( A > B ) @ ( top_top @ ( set @ ( A > B ) ) ) )
% 0.19/0.47 => ( finite_finite2 @ B @ ( top_top @ ( set @ B ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite_fun_UNIVD2
% 0.19/0.47 thf(fact_67_rev__finite__subset,axiom,
% 0.19/0.47 ! [A: $tType,B3: set @ A,A3: set @ A] :
% 0.19/0.47 ( ( finite_finite2 @ A @ B3 )
% 0.19/0.47 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.19/0.47 => ( finite_finite2 @ A @ A3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % rev_finite_subset
% 0.19/0.47 thf(fact_68_infinite__super,axiom,
% 0.19/0.47 ! [A: $tType,S: set @ A,T3: set @ A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ S @ T3 )
% 0.19/0.47 => ( ~ ( finite_finite2 @ A @ S )
% 0.19/0.47 => ~ ( finite_finite2 @ A @ T3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % infinite_super
% 0.19/0.47 thf(fact_69_finite__subset,axiom,
% 0.19/0.47 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.19/0.47 => ( ( finite_finite2 @ A @ B3 )
% 0.19/0.47 => ( finite_finite2 @ A @ A3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite_subset
% 0.19/0.47 thf(fact_70_empty__not__UNIV,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( bot_bot @ ( set @ A ) )
% 0.19/0.47 != ( top_top @ ( set @ A ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % empty_not_UNIV
% 0.19/0.47 thf(fact_71_finite__UNIV,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( finite_finite @ A @ ( type2 @ A ) )
% 0.19/0.47 => ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite_UNIV
% 0.19/0.47 thf(fact_72_infinite__UNIV__char__0,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( semiring_char_0 @ A @ ( type2 @ A ) )
% 0.19/0.47 => ~ ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % infinite_UNIV_char_0
% 0.19/0.47 thf(fact_73_ex__new__if__finite,axiom,
% 0.19/0.47 ! [A: $tType,A3: set @ A] :
% 0.19/0.47 ( ~ ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
% 0.19/0.47 => ( ( finite_finite2 @ A @ A3 )
% 0.19/0.47 => ? [A5: A] :
% 0.19/0.47 ~ ( member @ A @ A5 @ A3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % ex_new_if_finite
% 0.19/0.47 thf(fact_74_subset__insertI2,axiom,
% 0.19/0.47 ! [A: $tType,A3: set @ A,B3: set @ A,B4: A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.19/0.47 => ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B4 @ B3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_insertI2
% 0.19/0.47 thf(fact_75_subset__insertI,axiom,
% 0.19/0.47 ! [A: $tType,B3: set @ A,A2: A] : ( ord_less_eq @ ( set @ A ) @ B3 @ ( insert @ A @ A2 @ B3 ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_insertI
% 0.19/0.47 thf(fact_76_subset__insert,axiom,
% 0.19/0.47 ! [A: $tType,X3: A,A3: set @ A,B3: set @ A] :
% 0.19/0.47 ( ~ ( member @ A @ X3 @ A3 )
% 0.19/0.47 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ B3 ) )
% 0.19/0.47 = ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_insert
% 0.19/0.47 thf(fact_77_Set_Oinsert__mono,axiom,
% 0.19/0.47 ! [A: $tType,C3: set @ A,D: set @ A,A2: A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ C3 @ D )
% 0.19/0.47 => ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A2 @ C3 ) @ ( insert @ A @ A2 @ D ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Set.insert_mono
% 0.19/0.47 thf(fact_78_insert__UNIV,axiom,
% 0.19/0.47 ! [A: $tType,X3: A] :
% 0.19/0.47 ( ( insert @ A @ X3 @ ( top_top @ ( set @ A ) ) )
% 0.19/0.47 = ( top_top @ ( set @ A ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % insert_UNIV
% 0.19/0.47 thf(fact_79_subset__singleton__iff,axiom,
% 0.19/0.47 ! [A: $tType,X5: set @ A,A2: A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ X5 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.19/0.47 = ( ( X5
% 0.19/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.19/0.47 | ( X5
% 0.19/0.47 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_singleton_iff
% 0.19/0.47 thf(fact_80_subset__singletonD,axiom,
% 0.19/0.47 ! [A: $tType,A3: set @ A,X3: A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.19/0.47 => ( ( A3
% 0.19/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.19/0.47 | ( A3
% 0.19/0.47 = ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_singletonD
% 0.19/0.47 thf(fact_81_subset__Compl__self__eq,axiom,
% 0.19/0.47 ! [A: $tType,A3: set @ A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( uminus_uminus @ ( set @ A ) @ A3 ) )
% 0.19/0.47 = ( A3
% 0.19/0.47 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_Compl_self_eq
% 0.19/0.47 thf(fact_82_Compl__empty__eq,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( uminus_uminus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) )
% 0.19/0.47 = ( top_top @ ( set @ A ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Compl_empty_eq
% 0.19/0.47 thf(fact_83_Compl__UNIV__eq,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( uminus_uminus @ ( set @ A ) @ ( top_top @ ( set @ A ) ) )
% 0.19/0.47 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Compl_UNIV_eq
% 0.19/0.47 thf(fact_84_finite__subset__induct,axiom,
% 0.19/0.47 ! [A: $tType,F: set @ A,A3: set @ A,P: ( set @ A ) > $o] :
% 0.19/0.47 ( ( finite_finite2 @ A @ F )
% 0.19/0.47 => ( ( ord_less_eq @ ( set @ A ) @ F @ A3 )
% 0.19/0.47 => ( ( P @ ( bot_bot @ ( set @ A ) ) )
% 0.19/0.47 => ( ! [A5: A,F2: set @ A] :
% 0.19/0.47 ( ( finite_finite2 @ A @ F2 )
% 0.19/0.47 => ( ( member @ A @ A5 @ A3 )
% 0.19/0.47 => ( ~ ( member @ A @ A5 @ F2 )
% 0.19/0.47 => ( ( P @ F2 )
% 0.19/0.47 => ( P @ ( insert @ A @ A5 @ F2 ) ) ) ) ) )
% 0.19/0.47 => ( P @ F ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite_subset_induct
% 0.19/0.47 thf(fact_85_ex__in__conv,axiom,
% 0.19/0.47 ! [A: $tType,A3: set @ A] :
% 0.19/0.47 ( ( ? [X4: A] : ( member @ A @ X4 @ A3 ) )
% 0.19/0.47 = ( A3
% 0.19/0.47 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % ex_in_conv
% 0.19/0.47 thf(fact_86_equals0I,axiom,
% 0.19/0.47 ! [A: $tType,A3: set @ A] :
% 0.19/0.47 ( ! [Y2: A] :
% 0.19/0.47 ~ ( member @ A @ Y2 @ A3 )
% 0.19/0.47 => ( A3
% 0.19/0.47 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % equals0I
% 0.19/0.47 thf(fact_87_equals0D,axiom,
% 0.19/0.47 ! [A: $tType,A3: set @ A,A2: A] :
% 0.19/0.47 ( ( A3
% 0.19/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.19/0.47 => ~ ( member @ A @ A2 @ A3 ) ) ).
% 0.19/0.47
% 0.19/0.47 % equals0D
% 0.19/0.47 thf(fact_88_emptyE,axiom,
% 0.19/0.47 ! [A: $tType,A2: A] :
% 0.19/0.47 ~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % emptyE
% 0.19/0.47 thf(fact_89_finite,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( finite_finite @ A @ ( type2 @ A ) )
% 0.19/0.47 => ! [A3: set @ A] : ( finite_finite2 @ A @ A3 ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite
% 0.19/0.47 thf(fact_90_finite__set__choice,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,A3: set @ A,P: A > B > $o] :
% 0.19/0.47 ( ( finite_finite2 @ A @ A3 )
% 0.19/0.47 => ( ! [X2: A] :
% 0.19/0.47 ( ( member @ A @ X2 @ A3 )
% 0.19/0.47 => ? [X1: B] : ( P @ X2 @ X1 ) )
% 0.19/0.47 => ? [F4: A > B] :
% 0.19/0.47 ! [X6: A] :
% 0.19/0.47 ( ( member @ A @ X6 @ A3 )
% 0.19/0.47 => ( P @ X6 @ ( F4 @ X6 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite_set_choice
% 0.19/0.47 thf(fact_91_mk__disjoint__insert,axiom,
% 0.19/0.47 ! [A: $tType,A2: A,A3: set @ A] :
% 0.19/0.47 ( ( member @ A @ A2 @ A3 )
% 0.19/0.47 => ? [B6: set @ A] :
% 0.19/0.47 ( ( A3
% 0.19/0.47 = ( insert @ A @ A2 @ B6 ) )
% 0.19/0.47 & ~ ( member @ A @ A2 @ B6 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % mk_disjoint_insert
% 0.19/0.47 thf(fact_92_insert__commute,axiom,
% 0.19/0.47 ! [A: $tType,X3: A,Y3: A,A3: set @ A] :
% 0.19/0.47 ( ( insert @ A @ X3 @ ( insert @ A @ Y3 @ A3 ) )
% 0.19/0.47 = ( insert @ A @ Y3 @ ( insert @ A @ X3 @ A3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % insert_commute
% 0.19/0.47 thf(fact_93_insert__eq__iff,axiom,
% 0.19/0.47 ! [A: $tType,A2: A,A3: set @ A,B4: A,B3: set @ A] :
% 0.19/0.47 ( ~ ( member @ A @ A2 @ A3 )
% 0.19/0.47 => ( ~ ( member @ A @ B4 @ B3 )
% 0.19/0.47 => ( ( ( insert @ A @ A2 @ A3 )
% 0.19/0.47 = ( insert @ A @ B4 @ B3 ) )
% 0.19/0.47 = ( ( ( A2 = B4 )
% 0.19/0.47 => ( A3 = B3 ) )
% 0.19/0.47 & ( ( A2 != B4 )
% 0.19/0.47 => ? [C4: set @ A] :
% 0.19/0.47 ( ( A3
% 0.19/0.47 = ( insert @ A @ B4 @ C4 ) )
% 0.19/0.47 & ~ ( member @ A @ B4 @ C4 )
% 0.19/0.47 & ( B3
% 0.19/0.47 = ( insert @ A @ A2 @ C4 ) )
% 0.19/0.47 & ~ ( member @ A @ A2 @ C4 ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % insert_eq_iff
% 0.19/0.47 thf(fact_94_insert__absorb,axiom,
% 0.19/0.47 ! [A: $tType,A2: A,A3: set @ A] :
% 0.19/0.47 ( ( member @ A @ A2 @ A3 )
% 0.19/0.47 => ( ( insert @ A @ A2 @ A3 )
% 0.19/0.47 = A3 ) ) ).
% 0.19/0.47
% 0.19/0.47 % insert_absorb
% 0.19/0.47 thf(fact_95_insert__ident,axiom,
% 0.19/0.47 ! [A: $tType,X3: A,A3: set @ A,B3: set @ A] :
% 0.19/0.47 ( ~ ( member @ A @ X3 @ A3 )
% 0.19/0.47 => ( ~ ( member @ A @ X3 @ B3 )
% 0.19/0.47 => ( ( ( insert @ A @ X3 @ A3 )
% 0.19/0.47 = ( insert @ A @ X3 @ B3 ) )
% 0.19/0.47 = ( A3 = B3 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % insert_ident
% 0.19/0.47 thf(fact_96_Set_Oset__insert,axiom,
% 0.19/0.47 ! [A: $tType,X3: A,A3: set @ A] :
% 0.19/0.47 ( ( member @ A @ X3 @ A3 )
% 0.19/0.47 => ~ ! [B6: set @ A] :
% 0.19/0.47 ( ( A3
% 0.19/0.47 = ( insert @ A @ X3 @ B6 ) )
% 0.19/0.47 => ( member @ A @ X3 @ B6 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Set.set_insert
% 0.19/0.47 thf(fact_97_insertI2,axiom,
% 0.19/0.47 ! [A: $tType,A2: A,B3: set @ A,B4: A] :
% 0.19/0.47 ( ( member @ A @ A2 @ B3 )
% 0.19/0.47 => ( member @ A @ A2 @ ( insert @ A @ B4 @ B3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % insertI2
% 0.19/0.47 thf(fact_98_insertI1,axiom,
% 0.19/0.47 ! [A: $tType,A2: A,B3: set @ A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ B3 ) ) ).
% 0.19/0.47
% 0.19/0.47 % insertI1
% 0.19/0.47 thf(fact_99_insertE,axiom,
% 0.19/0.47 ! [A: $tType,A2: A,B4: A,A3: set @ A] :
% 0.19/0.47 ( ( member @ A @ A2 @ ( insert @ A @ B4 @ A3 ) )
% 0.19/0.47 => ( ( A2 != B4 )
% 0.19/0.47 => ( member @ A @ A2 @ A3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % insertE
% 0.19/0.47 thf(fact_100_double__complement,axiom,
% 0.19/0.47 ! [A: $tType,A3: set @ A] :
% 0.19/0.47 ( ( uminus_uminus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) )
% 0.19/0.47 = A3 ) ).
% 0.19/0.47
% 0.19/0.47 % double_complement
% 0.19/0.47 thf(fact_101_ComplD,axiom,
% 0.19/0.47 ! [A: $tType,C2: A,A3: set @ A] :
% 0.19/0.47 ( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A3 ) )
% 0.19/0.47 => ~ ( member @ A @ C2 @ A3 ) ) ).
% 0.19/0.47
% 0.19/0.47 % ComplD
% 0.19/0.47 thf(fact_102_infinite__imp__nonempty,axiom,
% 0.19/0.47 ! [A: $tType,S: set @ A] :
% 0.19/0.47 ( ~ ( finite_finite2 @ A @ S )
% 0.19/0.47 => ( S
% 0.19/0.47 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % infinite_imp_nonempty
% 0.19/0.47 thf(fact_103_finite_OemptyI,axiom,
% 0.19/0.47 ! [A: $tType] : ( finite_finite2 @ A @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite.emptyI
% 0.19/0.47 thf(fact_104_singleton__inject,axiom,
% 0.19/0.47 ! [A: $tType,A2: A,B4: A] :
% 0.19/0.47 ( ( ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) )
% 0.19/0.47 = ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.19/0.47 => ( A2 = B4 ) ) ).
% 0.19/0.47
% 0.19/0.47 % singleton_inject
% 0.19/0.47 thf(fact_105_insert__not__empty,axiom,
% 0.19/0.47 ! [A: $tType,A2: A,A3: set @ A] :
% 0.19/0.47 ( ( insert @ A @ A2 @ A3 )
% 0.19/0.47 != ( bot_bot @ ( set @ A ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % insert_not_empty
% 0.19/0.47 thf(fact_106_doubleton__eq__iff,axiom,
% 0.19/0.47 ! [A: $tType,A2: A,B4: A,C2: A,D2: A] :
% 0.19/0.47 ( ( ( insert @ A @ A2 @ ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.19/0.47 = ( insert @ A @ C2 @ ( insert @ A @ D2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.19/0.47 = ( ( ( A2 = C2 )
% 0.19/0.47 & ( B4 = D2 ) )
% 0.19/0.47 | ( ( A2 = D2 )
% 0.19/0.47 & ( B4 = C2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % doubleton_eq_iff
% 0.19/0.47 thf(fact_107_singleton__iff,axiom,
% 0.19/0.47 ! [A: $tType,B4: A,A2: A] :
% 0.19/0.47 ( ( member @ A @ B4 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.19/0.47 = ( B4 = A2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % singleton_iff
% 0.19/0.47 thf(fact_108_singletonD,axiom,
% 0.19/0.47 ! [A: $tType,B4: A,A2: A] :
% 0.19/0.47 ( ( member @ A @ B4 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.19/0.47 => ( B4 = A2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % singletonD
% 0.19/0.47 thf(fact_109_finite_OinsertI,axiom,
% 0.19/0.47 ! [A: $tType,A3: set @ A,A2: A] :
% 0.19/0.47 ( ( finite_finite2 @ A @ A3 )
% 0.19/0.47 => ( finite_finite2 @ A @ ( insert @ A @ A2 @ A3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite.insertI
% 0.19/0.47 thf(fact_110_compl__top__eq,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 0.19/0.47 => ( ( uminus_uminus @ A @ ( top_top @ A ) )
% 0.19/0.47 = ( bot_bot @ A ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % compl_top_eq
% 0.19/0.47 thf(fact_111_compl__bot__eq,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 0.19/0.47 => ( ( uminus_uminus @ A @ ( bot_bot @ A ) )
% 0.19/0.47 = ( top_top @ A ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % compl_bot_eq
% 0.19/0.47 thf(fact_112__092_060open_062_092_060not_062_Alfinite_A_If_A_In_M_Ans_J_J_092_060close_062,axiom,
% 0.19/0.47 ~ ( coinductive_lfinite @ node @ ( f @ ( product_Pair @ node @ ( set @ node ) @ n @ ns ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % \<open>\<not> lfinite (f (n, ns))\<close>
% 0.19/0.47 thf(fact_113_finite__option__UNIV,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( finite_finite2 @ ( option @ A ) @ ( top_top @ ( set @ ( option @ A ) ) ) )
% 0.19/0.47 = ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite_option_UNIV
% 0.19/0.47 thf(fact_114_compl__le__compl__iff,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 0.19/0.47 => ! [X3: A,Y3: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X3 ) @ ( uminus_uminus @ A @ Y3 ) )
% 0.19/0.47 = ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % compl_le_compl_iff
% 0.19/0.47 thf(fact_115_neg__le__iff__le,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.19/0.47 => ! [B4: A,A2: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B4 ) @ ( uminus_uminus @ A @ A2 ) )
% 0.19/0.47 = ( ord_less_eq @ A @ A2 @ B4 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % neg_le_iff_le
% 0.19/0.47 thf(fact_116_Topological__Spaces_OUNIV__not__singleton,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( topolo890362671_space @ A @ ( type2 @ A ) )
% 0.19/0.47 => ! [X3: A] :
% 0.19/0.47 ( ( top_top @ ( set @ A ) )
% 0.19/0.47 != ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Topological_Spaces.UNIV_not_singleton
% 0.19/0.47 thf(fact_117_f__simps_I1_J,axiom,
% 0.19/0.47 ! [Na: node,Nsa: set @ node] :
% 0.19/0.47 ~ ( coinductive_lnull @ node @ ( f @ ( product_Pair @ node @ ( set @ node ) @ Na @ Nsa ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % f_simps(1)
% 0.19/0.47 thf(fact_118_the__elem__eq,axiom,
% 0.19/0.47 ! [A: $tType,X3: A] :
% 0.19/0.47 ( ( the_elem @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.19/0.47 = X3 ) ).
% 0.19/0.47
% 0.19/0.47 % the_elem_eq
% 0.19/0.47 thf(fact_119_iso__tuple__UNIV__I,axiom,
% 0.19/0.47 ! [A: $tType,X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % iso_tuple_UNIV_I
% 0.19/0.47 thf(fact_120_uminus__apply,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType] :
% 0.19/0.47 ( ( uminus @ B @ ( type2 @ B ) )
% 0.19/0.47 => ( ( uminus_uminus @ ( A > B ) )
% 0.19/0.47 = ( ^ [A7: A > B,X4: A] : ( uminus_uminus @ B @ ( A7 @ X4 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % uminus_apply
% 0.19/0.47 thf(fact_121_add_Oinverse__inverse,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( group_add @ A @ ( type2 @ A ) )
% 0.19/0.47 => ! [A2: A] :
% 0.19/0.47 ( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A2 ) )
% 0.19/0.47 = A2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % add.inverse_inverse
% 0.19/0.47 thf(fact_122_neg__equal__iff__equal,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( group_add @ A @ ( type2 @ A ) )
% 0.19/0.47 => ! [A2: A,B4: A] :
% 0.19/0.47 ( ( ( uminus_uminus @ A @ A2 )
% 0.19/0.47 = ( uminus_uminus @ A @ B4 ) )
% 0.19/0.47 = ( A2 = B4 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % neg_equal_iff_equal
% 0.19/0.47 thf(fact_123_double__compl,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 0.19/0.47 => ! [X3: A] :
% 0.19/0.47 ( ( uminus_uminus @ A @ ( uminus_uminus @ A @ X3 ) )
% 0.19/0.47 = X3 ) ) ).
% 0.19/0.47
% 0.19/0.47 % double_compl
% 0.19/0.47 thf(fact_124_compl__eq__compl__iff,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 0.19/0.47 => ! [X3: A,Y3: A] :
% 0.19/0.47 ( ( ( uminus_uminus @ A @ X3 )
% 0.19/0.47 = ( uminus_uminus @ A @ Y3 ) )
% 0.19/0.47 = ( X3 = Y3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % compl_eq_compl_iff
% 0.19/0.47 thf(fact_125_bot__set__def,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( bot_bot @ ( set @ A ) )
% 0.19/0.47 = ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % bot_set_def
% 0.19/0.47 thf(fact_126_top__set__def,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( top_top @ ( set @ A ) )
% 0.19/0.47 = ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % top_set_def
% 0.19/0.47 thf(fact_127_fun__Compl__def,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType] :
% 0.19/0.47 ( ( uminus @ B @ ( type2 @ B ) )
% 0.19/0.47 => ( ( uminus_uminus @ ( A > B ) )
% 0.19/0.47 = ( ^ [A7: A > B,X4: A] : ( uminus_uminus @ B @ ( A7 @ X4 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % fun_Compl_def
% 0.19/0.47 thf(fact_128_equation__minus__iff,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( group_add @ A @ ( type2 @ A ) )
% 0.19/0.47 => ! [A2: A,B4: A] :
% 0.19/0.47 ( ( A2
% 0.19/0.47 = ( uminus_uminus @ A @ B4 ) )
% 0.19/0.47 = ( B4
% 0.19/0.47 = ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % equation_minus_iff
% 0.19/0.47 thf(fact_129_minus__equation__iff,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( group_add @ A @ ( type2 @ A ) )
% 0.19/0.47 => ! [A2: A,B4: A] :
% 0.19/0.47 ( ( ( uminus_uminus @ A @ A2 )
% 0.19/0.47 = B4 )
% 0.19/0.47 = ( ( uminus_uminus @ A @ B4 )
% 0.19/0.47 = A2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % minus_equation_iff
% 0.19/0.47 thf(fact_130_compl__mono,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 0.19/0.47 => ! [X3: A,Y3: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.19/0.47 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y3 ) @ ( uminus_uminus @ A @ X3 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % compl_mono
% 0.19/0.47 thf(fact_131_compl__le__swap1,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 0.19/0.47 => ! [Y3: A,X3: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ Y3 @ ( uminus_uminus @ A @ X3 ) )
% 0.19/0.47 => ( ord_less_eq @ A @ X3 @ ( uminus_uminus @ A @ Y3 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % compl_le_swap1
% 0.19/0.47 thf(fact_132_compl__le__swap2,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 0.19/0.47 => ! [Y3: A,X3: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y3 ) @ X3 )
% 0.19/0.47 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ X3 ) @ Y3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % compl_le_swap2
% 0.19/0.47 thf(fact_133_le__minus__iff,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.19/0.47 => ! [A2: A,B4: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ B4 ) )
% 0.19/0.47 = ( ord_less_eq @ A @ B4 @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % le_minus_iff
% 0.19/0.47 thf(fact_134_minus__le__iff,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.19/0.47 => ! [A2: A,B4: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B4 )
% 0.19/0.47 = ( ord_less_eq @ A @ ( uminus_uminus @ A @ B4 ) @ A2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % minus_le_iff
% 0.19/0.47 thf(fact_135_le__imp__neg__le,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.19/0.47 => ! [A2: A,B4: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ A2 @ B4 )
% 0.19/0.47 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ B4 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % le_imp_neg_le
% 0.19/0.47 thf(fact_136_lset__eq__empty,axiom,
% 0.19/0.47 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.19/0.47 ( ( ( coinductive_lset @ A @ Xs )
% 0.19/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.19/0.47 = ( coinductive_lnull @ A @ Xs ) ) ).
% 0.19/0.47
% 0.19/0.47 % lset_eq_empty
% 0.19/0.47 thf(fact_137_lfinite__imp__finite__lset,axiom,
% 0.19/0.47 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.19/0.47 ( ( coinductive_lfinite @ A @ Xs )
% 0.19/0.47 => ( finite_finite2 @ A @ ( coinductive_lset @ A @ Xs ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % lfinite_imp_finite_lset
% 0.19/0.47 thf(fact_138_lset__lnull,axiom,
% 0.19/0.47 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.19/0.47 ( ( coinductive_lnull @ A @ Xs )
% 0.19/0.47 => ( ( coinductive_lset @ A @ Xs )
% 0.19/0.47 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % lset_lnull
% 0.19/0.47 thf(fact_139_top__apply,axiom,
% 0.19/0.47 ! [C: $tType,D3: $tType] :
% 0.19/0.47 ( ( top @ C @ ( type2 @ C ) )
% 0.19/0.47 => ( ( top_top @ ( D3 > C ) )
% 0.19/0.47 = ( ^ [X4: D3] : ( top_top @ C ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % top_apply
% 0.19/0.47 thf(fact_140_order__refl,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( preorder @ A @ ( type2 @ A ) )
% 0.19/0.47 => ! [X3: A] : ( ord_less_eq @ A @ X3 @ X3 ) ) ).
% 0.19/0.47
% 0.19/0.47 % order_refl
% 0.19/0.47 thf(fact_141_top1I,axiom,
% 0.19/0.47 ! [A: $tType,X3: A] : ( top_top @ ( A > $o ) @ X3 ) ).
% 0.19/0.47
% 0.19/0.47 % top1I
% 0.19/0.47 thf(fact_142_bot__apply,axiom,
% 0.19/0.47 ! [C: $tType,D3: $tType] :
% 0.19/0.47 ( ( bot @ C @ ( type2 @ C ) )
% 0.19/0.47 => ( ( bot_bot @ ( D3 > C ) )
% 0.19/0.47 = ( ^ [X4: D3] : ( bot_bot @ C ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % bot_apply
% 0.19/0.47 thf(fact_143_le__funD,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType] :
% 0.19/0.47 ( ( ord @ B @ ( type2 @ B ) )
% 0.19/0.47 => ! [F3: A > B,G: A > B,X3: A] :
% 0.19/0.47 ( ( ord_less_eq @ ( A > B ) @ F3 @ G )
% 0.19/0.47 => ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( G @ X3 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % le_funD
% 0.19/0.47 thf(fact_144_le__funE,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType] :
% 0.19/0.47 ( ( ord @ B @ ( type2 @ B ) )
% 0.19/0.47 => ! [F3: A > B,G: A > B,X3: A] :
% 0.19/0.47 ( ( ord_less_eq @ ( A > B ) @ F3 @ G )
% 0.19/0.47 => ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( G @ X3 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % le_funE
% 0.19/0.47 thf(fact_145_le__funI,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType] :
% 0.19/0.47 ( ( ord @ B @ ( type2 @ B ) )
% 0.19/0.47 => ! [F3: A > B,G: A > B] :
% 0.19/0.47 ( ! [X2: A] : ( ord_less_eq @ B @ ( F3 @ X2 ) @ ( G @ X2 ) )
% 0.19/0.47 => ( ord_less_eq @ ( A > B ) @ F3 @ G ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % le_funI
% 0.19/0.47 thf(fact_146_le__fun__def,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType] :
% 0.19/0.47 ( ( ord @ B @ ( type2 @ B ) )
% 0.19/0.47 => ( ( ord_less_eq @ ( A > B ) )
% 0.19/0.47 = ( ^ [F5: A > B,G2: A > B] :
% 0.19/0.47 ! [X4: A] : ( ord_less_eq @ B @ ( F5 @ X4 ) @ ( G2 @ X4 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % le_fun_def
% 0.19/0.47 thf(fact_147_order__subst1,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType] :
% 0.19/0.47 ( ( ( order @ B @ ( type2 @ B ) )
% 0.19/0.47 & ( order @ A @ ( type2 @ A ) ) )
% 0.19/0.47 => ! [A2: A,F3: B > A,B4: B,C2: B] :
% 0.19/0.47 ( ( ord_less_eq @ A @ A2 @ ( F3 @ B4 ) )
% 0.19/0.47 => ( ( ord_less_eq @ B @ B4 @ C2 )
% 0.19/0.47 => ( ! [X2: B,Y2: B] :
% 0.19/0.47 ( ( ord_less_eq @ B @ X2 @ Y2 )
% 0.19/0.47 => ( ord_less_eq @ A @ ( F3 @ X2 ) @ ( F3 @ Y2 ) ) )
% 0.19/0.47 => ( ord_less_eq @ A @ A2 @ ( F3 @ C2 ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % order_subst1
% 0.19/0.47 thf(fact_148_order__subst2,axiom,
% 0.19/0.47 ! [A: $tType,C: $tType] :
% 0.19/0.47 ( ( ( order @ C @ ( type2 @ C ) )
% 0.19/0.47 & ( order @ A @ ( type2 @ A ) ) )
% 0.19/0.47 => ! [A2: A,B4: A,F3: A > C,C2: C] :
% 0.19/0.47 ( ( ord_less_eq @ A @ A2 @ B4 )
% 0.19/0.47 => ( ( ord_less_eq @ C @ ( F3 @ B4 ) @ C2 )
% 0.19/0.47 => ( ! [X2: A,Y2: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 0.19/0.47 => ( ord_less_eq @ C @ ( F3 @ X2 ) @ ( F3 @ Y2 ) ) )
% 0.19/0.47 => ( ord_less_eq @ C @ ( F3 @ A2 ) @ C2 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % order_subst2
% 0.19/0.47 thf(fact_149_ord__eq__le__subst,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType] :
% 0.19/0.47 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.19/0.47 & ( ord @ A @ ( type2 @ A ) ) )
% 0.19/0.47 => ! [A2: A,F3: B > A,B4: B,C2: B] :
% 0.19/0.47 ( ( A2
% 0.19/0.47 = ( F3 @ B4 ) )
% 0.19/0.47 => ( ( ord_less_eq @ B @ B4 @ C2 )
% 0.19/0.47 => ( ! [X2: B,Y2: B] :
% 0.19/0.47 ( ( ord_less_eq @ B @ X2 @ Y2 )
% 0.19/0.47 => ( ord_less_eq @ A @ ( F3 @ X2 ) @ ( F3 @ Y2 ) ) )
% 0.19/0.47 => ( ord_less_eq @ A @ A2 @ ( F3 @ C2 ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % ord_eq_le_subst
% 0.19/0.47 thf(fact_150_ord__le__eq__subst,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType] :
% 0.19/0.47 ( ( ( ord @ B @ ( type2 @ B ) )
% 0.19/0.47 & ( ord @ A @ ( type2 @ A ) ) )
% 0.19/0.47 => ! [A2: A,B4: A,F3: A > B,C2: B] :
% 0.19/0.47 ( ( ord_less_eq @ A @ A2 @ B4 )
% 0.19/0.47 => ( ( ( F3 @ B4 )
% 0.19/0.47 = C2 )
% 0.19/0.47 => ( ! [X2: A,Y2: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 0.19/0.47 => ( ord_less_eq @ B @ ( F3 @ X2 ) @ ( F3 @ Y2 ) ) )
% 0.19/0.47 => ( ord_less_eq @ B @ ( F3 @ A2 ) @ C2 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % ord_le_eq_subst
% 0.19/0.47 thf(fact_151_eq__iff,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.19/0.47 => ( ( ^ [Y: A,Z: A] : ( Y = Z ) )
% 0.19/0.47 = ( ^ [X4: A,Y4: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ X4 @ Y4 )
% 0.19/0.47 & ( ord_less_eq @ A @ Y4 @ X4 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % eq_iff
% 0.19/0.47 thf(fact_152_antisym,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.19/0.47 => ! [X3: A,Y3: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.19/0.47 => ( ( ord_less_eq @ A @ Y3 @ X3 )
% 0.19/0.47 => ( X3 = Y3 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % antisym
% 0.19/0.47 thf(fact_153_linear,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.19/0.47 => ! [X3: A,Y3: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.19/0.47 | ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % linear
% 0.19/0.47 thf(fact_154_eq__refl,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( preorder @ A @ ( type2 @ A ) )
% 0.19/0.47 => ! [X3: A,Y3: A] :
% 0.19/0.47 ( ( X3 = Y3 )
% 0.19/0.47 => ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % eq_refl
% 0.19/0.47 thf(fact_155_wlog__linorder__le,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.19/0.47 => ! [P: A > A > $o,B4: A,A2: A] :
% 0.19/0.47 ( ! [A5: A,B7: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ A5 @ B7 )
% 0.19/0.47 => ( P @ A5 @ B7 ) )
% 0.19/0.47 => ( ( ( P @ B4 @ A2 )
% 0.19/0.47 => ( P @ A2 @ B4 ) )
% 0.19/0.47 => ( P @ A2 @ B4 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % wlog_linorder_le
% 0.19/0.47 thf(fact_156_le__cases,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.19/0.47 => ! [X3: A,Y3: A] :
% 0.19/0.47 ( ~ ( ord_less_eq @ A @ X3 @ Y3 )
% 0.19/0.47 => ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % le_cases
% 0.19/0.47 thf(fact_157_order_Otrans,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.19/0.47 => ! [A2: A,B4: A,C2: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ A2 @ B4 )
% 0.19/0.47 => ( ( ord_less_eq @ A @ B4 @ C2 )
% 0.19/0.47 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % order.trans
% 0.19/0.47 thf(fact_158_le__cases3,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.19/0.47 => ! [X3: A,Y3: A,Z2: A] :
% 0.19/0.47 ( ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.19/0.47 => ~ ( ord_less_eq @ A @ Y3 @ Z2 ) )
% 0.19/0.47 => ( ( ( ord_less_eq @ A @ Y3 @ X3 )
% 0.19/0.47 => ~ ( ord_less_eq @ A @ X3 @ Z2 ) )
% 0.19/0.47 => ( ( ( ord_less_eq @ A @ X3 @ Z2 )
% 0.19/0.47 => ~ ( ord_less_eq @ A @ Z2 @ Y3 ) )
% 0.19/0.47 => ( ( ( ord_less_eq @ A @ Z2 @ Y3 )
% 0.19/0.47 => ~ ( ord_less_eq @ A @ Y3 @ X3 ) )
% 0.19/0.47 => ( ( ( ord_less_eq @ A @ Y3 @ Z2 )
% 0.19/0.47 => ~ ( ord_less_eq @ A @ Z2 @ X3 ) )
% 0.19/0.47 => ~ ( ( ord_less_eq @ A @ Z2 @ X3 )
% 0.19/0.47 => ~ ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % le_cases3
% 0.19/0.47 thf(fact_159_antisym__conv,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.19/0.47 => ! [Y3: A,X3: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ Y3 @ X3 )
% 0.19/0.47 => ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.19/0.47 = ( X3 = Y3 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % antisym_conv
% 0.19/0.47 thf(fact_160_ord__eq__le__trans,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( ord @ A @ ( type2 @ A ) )
% 0.19/0.47 => ! [A2: A,B4: A,C2: A] :
% 0.19/0.47 ( ( A2 = B4 )
% 0.19/0.47 => ( ( ord_less_eq @ A @ B4 @ C2 )
% 0.19/0.47 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % ord_eq_le_trans
% 0.19/0.47 thf(fact_161_ord__le__eq__trans,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( ord @ A @ ( type2 @ A ) )
% 0.19/0.47 => ! [A2: A,B4: A,C2: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ A2 @ B4 )
% 0.19/0.47 => ( ( B4 = C2 )
% 0.19/0.47 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % ord_le_eq_trans
% 0.19/0.47 thf(fact_162_order__class_Oorder_Oantisym,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.19/0.47 => ! [A2: A,B4: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ A2 @ B4 )
% 0.19/0.47 => ( ( ord_less_eq @ A @ B4 @ A2 )
% 0.19/0.47 => ( A2 = B4 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % order_class.order.antisym
% 0.19/0.47 thf(fact_163_order__trans,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( preorder @ A @ ( type2 @ A ) )
% 0.19/0.47 => ! [X3: A,Y3: A,Z2: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 0.19/0.47 => ( ( ord_less_eq @ A @ Y3 @ Z2 )
% 0.19/0.47 => ( ord_less_eq @ A @ X3 @ Z2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % order_trans
% 0.19/0.47 thf(fact_164_dual__order_Orefl,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.19/0.47 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % dual_order.refl
% 0.19/0.47 thf(fact_165_linorder__wlog,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( linorder @ A @ ( type2 @ A ) )
% 0.19/0.47 => ! [P: A > A > $o,A2: A,B4: A] :
% 0.19/0.47 ( ! [A5: A,B7: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ A5 @ B7 )
% 0.19/0.47 => ( P @ A5 @ B7 ) )
% 0.19/0.47 => ( ! [A5: A,B7: A] :
% 0.19/0.47 ( ( P @ B7 @ A5 )
% 0.19/0.47 => ( P @ A5 @ B7 ) )
% 0.19/0.47 => ( P @ A2 @ B4 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % linorder_wlog
% 0.19/0.47 thf(fact_166_dual__order_Otrans,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.19/0.47 => ! [B4: A,A2: A,C2: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ B4 @ A2 )
% 0.19/0.47 => ( ( ord_less_eq @ A @ C2 @ B4 )
% 0.19/0.47 => ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % dual_order.trans
% 0.19/0.47 thf(fact_167_dual__order_Oantisym,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( order @ A @ ( type2 @ A ) )
% 0.19/0.47 => ! [B4: A,A2: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ B4 @ A2 )
% 0.19/0.47 => ( ( ord_less_eq @ A @ A2 @ B4 )
% 0.19/0.47 => ( A2 = B4 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % dual_order.antisym
% 0.19/0.47 thf(fact_168_bot__fun__def,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType] :
% 0.19/0.47 ( ( bot @ B @ ( type2 @ B ) )
% 0.19/0.47 => ( ( bot_bot @ ( A > B ) )
% 0.19/0.47 = ( ^ [X4: A] : ( bot_bot @ B ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % bot_fun_def
% 0.19/0.47 thf(fact_169_lzip_Oexhaust,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
% 0.19/0.47 ( ~ ( ( coinductive_lnull @ A @ Xs )
% 0.19/0.47 | ( coinductive_lnull @ B @ Ys ) )
% 0.19/0.47 => ~ ( ~ ( coinductive_lnull @ A @ Xs )
% 0.19/0.47 => ( coinductive_lnull @ B @ Ys ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % lzip.exhaust
% 0.19/0.47 thf(fact_170_lappend_Oexhaust,axiom,
% 0.19/0.47 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 0.19/0.47 ( ( ( coinductive_lnull @ A @ Xs )
% 0.19/0.47 => ~ ( coinductive_lnull @ A @ Ys ) )
% 0.19/0.47 => ( ~ ( coinductive_lnull @ A @ Xs )
% 0.19/0.47 | ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % lappend.exhaust
% 0.19/0.47 thf(fact_171_bot_Oextremum__uniqueI,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( order_bot @ A @ ( type2 @ A ) )
% 0.19/0.47 => ! [A2: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
% 0.19/0.47 => ( A2
% 0.19/0.47 = ( bot_bot @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % bot.extremum_uniqueI
% 0.19/0.47 thf(fact_172_bot_Oextremum__unique,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( order_bot @ A @ ( type2 @ A ) )
% 0.19/0.47 => ! [A2: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
% 0.19/0.47 = ( A2
% 0.19/0.47 = ( bot_bot @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % bot.extremum_unique
% 0.19/0.47 thf(fact_173_bot_Oextremum,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( order_bot @ A @ ( type2 @ A ) )
% 0.19/0.47 => ! [A2: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % bot.extremum
% 0.19/0.47 thf(fact_174_top__greatest,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( order_top @ A @ ( type2 @ A ) )
% 0.19/0.47 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ ( top_top @ A ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % top_greatest
% 0.19/0.47 thf(fact_175_top_Oextremum__unique,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( order_top @ A @ ( type2 @ A ) )
% 0.19/0.47 => ! [A2: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A2 )
% 0.19/0.47 = ( A2
% 0.19/0.47 = ( top_top @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % top.extremum_unique
% 0.19/0.47 thf(fact_176_top_Oextremum__uniqueI,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( order_top @ A @ ( type2 @ A ) )
% 0.19/0.47 => ! [A2: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A2 )
% 0.19/0.47 => ( A2
% 0.19/0.47 = ( top_top @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % top.extremum_uniqueI
% 0.19/0.47 thf(fact_177_lnull__imp__lfinite,axiom,
% 0.19/0.47 ! [A: $tType,Xs: coinductive_llist @ A] :
% 0.19/0.47 ( ( coinductive_lnull @ A @ Xs )
% 0.19/0.47 => ( coinductive_lfinite @ A @ Xs ) ) ).
% 0.19/0.47
% 0.19/0.47 % lnull_imp_lfinite
% 0.19/0.47 thf(fact_178_old_Oprod_Oinject,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,A2: A,B4: B,A8: A,B8: B] :
% 0.19/0.47 ( ( ( product_Pair @ A @ B @ A2 @ B4 )
% 0.19/0.47 = ( product_Pair @ A @ B @ A8 @ B8 ) )
% 0.19/0.47 = ( ( A2 = A8 )
% 0.19/0.47 & ( B4 = B8 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % old.prod.inject
% 0.19/0.47 thf(fact_179_prod_Oinject,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,X12: A,X22: B,Y1: A,Y22: B] :
% 0.19/0.47 ( ( ( product_Pair @ A @ B @ X12 @ X22 )
% 0.19/0.47 = ( product_Pair @ A @ B @ Y1 @ Y22 ) )
% 0.19/0.47 = ( ( X12 = Y1 )
% 0.19/0.47 & ( X22 = Y22 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % prod.inject
% 0.19/0.47 thf(fact_180_top__empty__eq,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( top_top @ ( A > $o ) )
% 0.19/0.47 = ( ^ [X4: A] : ( member @ A @ X4 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % top_empty_eq
% 0.19/0.47 thf(fact_181_Collect__empty__eq__bot,axiom,
% 0.19/0.47 ! [A: $tType,P: A > $o] :
% 0.19/0.47 ( ( ( collect @ A @ P )
% 0.19/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.19/0.47 = ( P
% 0.19/0.47 = ( bot_bot @ ( A > $o ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Collect_empty_eq_bot
% 0.19/0.47 thf(fact_182_subrelI,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
% 0.19/0.47 ( ! [X2: A,Y2: B] :
% 0.19/0.47 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ R )
% 0.19/0.47 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ S2 ) )
% 0.19/0.47 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % subrelI
% 0.19/0.47 thf(fact_183_surj__pair,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
% 0.19/0.47 ? [X2: A,Y2: B] :
% 0.19/0.47 ( P2
% 0.19/0.47 = ( product_Pair @ A @ B @ X2 @ Y2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % surj_pair
% 0.19/0.47 thf(fact_184_prod__cases,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
% 0.19/0.47 ( ! [A5: A,B7: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B7 ) )
% 0.19/0.47 => ( P @ P2 ) ) ).
% 0.19/0.47
% 0.19/0.47 % prod_cases
% 0.19/0.47 thf(fact_185_Pair__inject,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,A2: A,B4: B,A8: A,B8: B] :
% 0.19/0.47 ( ( ( product_Pair @ A @ B @ A2 @ B4 )
% 0.19/0.47 = ( product_Pair @ A @ B @ A8 @ B8 ) )
% 0.19/0.47 => ~ ( ( A2 = A8 )
% 0.19/0.47 => ( B4 != B8 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Pair_inject
% 0.19/0.47 thf(fact_186_prod__cases3,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,C: $tType,Y3: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.19/0.47 ~ ! [A5: A,B7: B,C5: C] :
% 0.19/0.47 ( Y3
% 0.19/0.47 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B7 @ C5 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % prod_cases3
% 0.19/0.47 thf(fact_187_prod__cases4,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,C: $tType,D3: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D3 ) )] :
% 0.19/0.47 ~ ! [A5: A,B7: B,C5: C,D4: D3] :
% 0.19/0.47 ( Y3
% 0.19/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D3 ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D3 ) @ B7 @ ( product_Pair @ C @ D3 @ C5 @ D4 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % prod_cases4
% 0.19/0.47 thf(fact_188_prod__cases5,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,C: $tType,D3: $tType,E: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ E ) ) )] :
% 0.19/0.47 ~ ! [A5: A,B7: B,C5: C,D4: D3,E2: E] :
% 0.19/0.47 ( Y3
% 0.19/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D3 @ E ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D3 @ E ) @ C5 @ ( product_Pair @ D3 @ E @ D4 @ E2 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % prod_cases5
% 0.19/0.47 thf(fact_189_prod__cases6,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,C: $tType,D3: $tType,E: $tType,F6: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ F6 ) ) ) )] :
% 0.19/0.47 ~ ! [A5: A,B7: B,C5: C,D4: D3,E2: E,F4: F6] :
% 0.19/0.47 ( Y3
% 0.19/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ F6 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ F6 ) ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D3 @ ( product_prod @ E @ F6 ) ) @ C5 @ ( product_Pair @ D3 @ ( product_prod @ E @ F6 ) @ D4 @ ( product_Pair @ E @ F6 @ E2 @ F4 ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % prod_cases6
% 0.19/0.47 thf(fact_190_prod__cases7,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,C: $tType,D3: $tType,E: $tType,F6: $tType,G3: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ ( product_prod @ F6 @ G3 ) ) ) ) )] :
% 0.19/0.47 ~ ! [A5: A,B7: B,C5: C,D4: D3,E2: E,F4: F6,G4: G3] :
% 0.19/0.47 ( Y3
% 0.19/0.47 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ ( product_prod @ F6 @ G3 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ ( product_prod @ F6 @ G3 ) ) ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D3 @ ( product_prod @ E @ ( product_prod @ F6 @ G3 ) ) ) @ C5 @ ( product_Pair @ D3 @ ( product_prod @ E @ ( product_prod @ F6 @ G3 ) ) @ D4 @ ( product_Pair @ E @ ( product_prod @ F6 @ G3 ) @ E2 @ ( product_Pair @ F6 @ G3 @ F4 @ G4 ) ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % prod_cases7
% 0.19/0.47 thf(fact_191_prod__induct3,axiom,
% 0.19/0.47 ! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ C )] :
% 0.19/0.47 ( ! [A5: A,B7: B,C5: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B7 @ C5 ) ) )
% 0.19/0.47 => ( P @ X3 ) ) ).
% 0.19/0.47
% 0.19/0.47 % prod_induct3
% 0.19/0.47 thf(fact_192_prod__induct4,axiom,
% 0.19/0.47 ! [D3: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D3 ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D3 ) )] :
% 0.19/0.47 ( ! [A5: A,B7: B,C5: C,D4: D3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D3 ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D3 ) @ B7 @ ( product_Pair @ C @ D3 @ C5 @ D4 ) ) ) )
% 0.19/0.47 => ( P @ X3 ) ) ).
% 0.19/0.47
% 0.19/0.47 % prod_induct4
% 0.19/0.47 thf(fact_193_prod__induct5,axiom,
% 0.19/0.47 ! [E: $tType,D3: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ E ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ E ) ) )] :
% 0.19/0.47 ( ! [A5: A,B7: B,C5: C,D4: D3,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D3 @ E ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D3 @ E ) @ C5 @ ( product_Pair @ D3 @ E @ D4 @ E2 ) ) ) ) )
% 0.19/0.47 => ( P @ X3 ) ) ).
% 0.19/0.47
% 0.19/0.47 % prod_induct5
% 0.19/0.47 thf(fact_194_prod__induct6,axiom,
% 0.19/0.47 ! [F6: $tType,E: $tType,D3: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ F6 ) ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ F6 ) ) ) )] :
% 0.19/0.47 ( ! [A5: A,B7: B,C5: C,D4: D3,E2: E,F4: F6] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ F6 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ F6 ) ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D3 @ ( product_prod @ E @ F6 ) ) @ C5 @ ( product_Pair @ D3 @ ( product_prod @ E @ F6 ) @ D4 @ ( product_Pair @ E @ F6 @ E2 @ F4 ) ) ) ) ) )
% 0.19/0.47 => ( P @ X3 ) ) ).
% 0.19/0.47
% 0.19/0.47 % prod_induct6
% 0.19/0.47 thf(fact_195_prod__induct7,axiom,
% 0.19/0.47 ! [G3: $tType,F6: $tType,E: $tType,D3: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ ( product_prod @ F6 @ G3 ) ) ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ ( product_prod @ F6 @ G3 ) ) ) ) )] :
% 0.19/0.47 ( ! [A5: A,B7: B,C5: C,D4: D3,E2: E,F4: F6,G4: G3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ ( product_prod @ F6 @ G3 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ ( product_prod @ F6 @ G3 ) ) ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D3 @ ( product_prod @ E @ ( product_prod @ F6 @ G3 ) ) ) @ C5 @ ( product_Pair @ D3 @ ( product_prod @ E @ ( product_prod @ F6 @ G3 ) ) @ D4 @ ( product_Pair @ E @ ( product_prod @ F6 @ G3 ) @ E2 @ ( product_Pair @ F6 @ G3 @ F4 @ G4 ) ) ) ) ) ) )
% 0.19/0.47 => ( P @ X3 ) ) ).
% 0.19/0.47
% 0.19/0.47 % prod_induct7
% 0.19/0.47 thf(fact_196_old_Oprod_Oexhaust,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,Y3: product_prod @ A @ B] :
% 0.19/0.47 ~ ! [A5: A,B7: B] :
% 0.19/0.47 ( Y3
% 0.19/0.47 != ( product_Pair @ A @ B @ A5 @ B7 ) ) ).
% 0.19/0.47
% 0.19/0.47 % old.prod.exhaust
% 0.19/0.47 thf(fact_197_old_Oprod_Oinducts,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 0.19/0.47 ( ! [A5: A,B7: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B7 ) )
% 0.19/0.47 => ( P @ Prod ) ) ).
% 0.19/0.47
% 0.19/0.47 % old.prod.inducts
% 0.19/0.47 thf(fact_198_bot__empty__eq,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( bot_bot @ ( A > $o ) )
% 0.19/0.47 = ( ^ [X4: A] : ( member @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % bot_empty_eq
% 0.19/0.47 thf(fact_199_old_Oprod_Orec,axiom,
% 0.19/0.47 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B4: B] :
% 0.19/0.47 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B4 ) )
% 0.19/0.47 = ( F1 @ A2 @ B4 ) ) ).
% 0.19/0.47
% 0.19/0.47 % old.prod.rec
% 0.19/0.47 thf(fact_200_is__singleton__the__elem,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( is_singleton @ A )
% 0.19/0.47 = ( ^ [A7: set @ A] :
% 0.19/0.47 ( A7
% 0.19/0.47 = ( insert @ A @ ( the_elem @ A @ A7 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % is_singleton_the_elem
% 0.19/0.47 thf(fact_201_is__singletonI,axiom,
% 0.19/0.47 ! [A: $tType,X3: A] : ( is_singleton @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % is_singletonI
% 0.19/0.47 thf(fact_202_is__singletonI_H,axiom,
% 0.19/0.47 ! [A: $tType,A3: set @ A] :
% 0.19/0.47 ( ( A3
% 0.19/0.47 != ( bot_bot @ ( set @ A ) ) )
% 0.19/0.47 => ( ! [X2: A,Y2: A] :
% 0.19/0.47 ( ( member @ A @ X2 @ A3 )
% 0.19/0.47 => ( ( member @ A @ Y2 @ A3 )
% 0.19/0.47 => ( X2 = Y2 ) ) )
% 0.19/0.47 => ( is_singleton @ A @ A3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % is_singletonI'
% 0.19/0.47 thf(fact_203_is__singleton__def,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( is_singleton @ A )
% 0.19/0.47 = ( ^ [A7: set @ A] :
% 0.19/0.47 ? [X4: A] :
% 0.19/0.47 ( A7
% 0.19/0.47 = ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % is_singleton_def
% 0.19/0.47 thf(fact_204_is__singletonE,axiom,
% 0.19/0.47 ! [A: $tType,A3: set @ A] :
% 0.19/0.47 ( ( is_singleton @ A @ A3 )
% 0.19/0.47 => ~ ! [X2: A] :
% 0.19/0.47 ( A3
% 0.19/0.47 != ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % is_singletonE
% 0.19/0.47 thf(fact_205_insert__subsetI,axiom,
% 0.19/0.47 ! [A: $tType,X3: A,A3: set @ A,X5: set @ A] :
% 0.19/0.47 ( ( member @ A @ X3 @ A3 )
% 0.19/0.47 => ( ( ord_less_eq @ ( set @ A ) @ X5 @ A3 )
% 0.19/0.47 => ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X3 @ X5 ) @ A3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % insert_subsetI
% 0.19/0.47 thf(fact_206_subset__emptyI,axiom,
% 0.19/0.47 ! [A: $tType,A3: set @ A] :
% 0.19/0.47 ( ! [X2: A] :
% 0.19/0.47 ~ ( member @ A @ X2 @ A3 )
% 0.19/0.47 => ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_emptyI
% 0.19/0.47 thf(fact_207_ssubst__Pair__rhs,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,R: A,S2: B,R2: set @ ( product_prod @ A @ B ),S3: B] :
% 0.19/0.47 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S2 ) @ R2 )
% 0.19/0.47 => ( ( S3 = S2 )
% 0.19/0.47 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S3 ) @ R2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % ssubst_Pair_rhs
% 0.19/0.47 thf(fact_208_lset__code,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( coinductive_lset @ A )
% 0.19/0.47 = ( coinductive_gen_lset @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % lset_code
% 0.19/0.47 thf(fact_209_top__conj_I1_J,axiom,
% 0.19/0.47 ! [A: $tType,X3: A,P: $o] :
% 0.19/0.47 ( ( ( top_top @ ( A > $o ) @ X3 )
% 0.19/0.47 & P )
% 0.19/0.47 = P ) ).
% 0.19/0.47
% 0.19/0.47 % top_conj(1)
% 0.19/0.47 thf(fact_210_top__conj_I2_J,axiom,
% 0.19/0.47 ! [A: $tType,P: $o,X3: A] :
% 0.19/0.47 ( ( P
% 0.19/0.47 & ( top_top @ ( A > $o ) @ X3 ) )
% 0.19/0.47 = P ) ).
% 0.19/0.47
% 0.19/0.47 % top_conj(2)
% 0.19/0.47 thf(fact_211_prod__set__simps_I1_J,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,X3: A,Y3: B] :
% 0.19/0.47 ( ( basic_fsts @ A @ B @ ( product_Pair @ A @ B @ X3 @ Y3 ) )
% 0.19/0.47 = ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % prod_set_simps(1)
% 0.19/0.47 thf(fact_212_prod__set__simps_I2_J,axiom,
% 0.19/0.47 ! [A: $tType,B: $tType,X3: A,Y3: B] :
% 0.19/0.47 ( ( basic_snds @ A @ B @ ( product_Pair @ A @ B @ X3 @ Y3 ) )
% 0.19/0.47 = ( insert @ B @ Y3 @ ( bot_bot @ ( set @ B ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % prod_set_simps(2)
% 0.19/0.47 thf(fact_213_Set_Ois__empty__def,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( is_empty @ A )
% 0.19/0.47 = ( ^ [A7: set @ A] :
% 0.19/0.47 ( A7
% 0.19/0.47 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Set.is_empty_def
% 0.19/0.47 thf(fact_214_internal__case__prod__conv,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType,C: $tType,C2: B > C > A,A2: B,B4: C] :
% 0.19/0.47 ( ( produc2004651681e_prod @ B @ C @ A @ C2 @ ( product_Pair @ B @ C @ A2 @ B4 ) )
% 0.19/0.47 = ( C2 @ A2 @ B4 ) ) ).
% 0.19/0.47
% 0.19/0.47 % internal_case_prod_conv
% 0.19/0.47 thf(fact_215_finite__remove__induct,axiom,
% 0.19/0.47 ! [A: $tType,B3: set @ A,P: ( set @ A ) > $o] :
% 0.19/0.47 ( ( finite_finite2 @ A @ B3 )
% 0.19/0.47 => ( ( P @ ( bot_bot @ ( set @ A ) ) )
% 0.19/0.47 => ( ! [A4: set @ A] :
% 0.19/0.47 ( ( finite_finite2 @ A @ A4 )
% 0.19/0.47 => ( ( A4
% 0.19/0.47 != ( bot_bot @ ( set @ A ) ) )
% 0.19/0.47 => ( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
% 0.19/0.47 => ( ! [X6: A] :
% 0.19/0.47 ( ( member @ A @ X6 @ A4 )
% 0.19/0.47 => ( P @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
% 0.19/0.47 => ( P @ A4 ) ) ) ) )
% 0.19/0.47 => ( P @ B3 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite_remove_induct
% 0.19/0.47 thf(fact_216_remove__induct,axiom,
% 0.19/0.47 ! [A: $tType,P: ( set @ A ) > $o,B3: set @ A] :
% 0.19/0.47 ( ( P @ ( bot_bot @ ( set @ A ) ) )
% 0.19/0.47 => ( ( ~ ( finite_finite2 @ A @ B3 )
% 0.19/0.47 => ( P @ B3 ) )
% 0.19/0.47 => ( ! [A4: set @ A] :
% 0.19/0.47 ( ( finite_finite2 @ A @ A4 )
% 0.19/0.47 => ( ( A4
% 0.19/0.47 != ( bot_bot @ ( set @ A ) ) )
% 0.19/0.47 => ( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
% 0.19/0.47 => ( ! [X6: A] :
% 0.19/0.47 ( ( member @ A @ X6 @ A4 )
% 0.19/0.47 => ( P @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
% 0.19/0.47 => ( P @ A4 ) ) ) ) )
% 0.19/0.47 => ( P @ B3 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % remove_induct
% 0.19/0.47 thf(fact_217_minus__apply,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType] :
% 0.19/0.47 ( ( minus @ B @ ( type2 @ B ) )
% 0.19/0.47 => ( ( minus_minus @ ( A > B ) )
% 0.19/0.47 = ( ^ [A7: A > B,B5: A > B,X4: A] : ( minus_minus @ B @ ( A7 @ X4 ) @ ( B5 @ X4 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % minus_apply
% 0.19/0.47 thf(fact_218_DiffI,axiom,
% 0.19/0.47 ! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
% 0.19/0.47 ( ( member @ A @ C2 @ A3 )
% 0.19/0.47 => ( ~ ( member @ A @ C2 @ B3 )
% 0.19/0.47 => ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % DiffI
% 0.19/0.47 thf(fact_219_Diff__iff,axiom,
% 0.19/0.47 ! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
% 0.19/0.47 ( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
% 0.19/0.47 = ( ( member @ A @ C2 @ A3 )
% 0.19/0.47 & ~ ( member @ A @ C2 @ B3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Diff_iff
% 0.19/0.47 thf(fact_220_Diff__idemp,axiom,
% 0.19/0.47 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.19/0.47 ( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) @ B3 )
% 0.19/0.47 = ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ).
% 0.19/0.47
% 0.19/0.47 % Diff_idemp
% 0.19/0.47 thf(fact_221_minus__diff__eq,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( group_add @ A @ ( type2 @ A ) )
% 0.19/0.47 => ! [A2: A,B4: A] :
% 0.19/0.47 ( ( uminus_uminus @ A @ ( minus_minus @ A @ A2 @ B4 ) )
% 0.19/0.47 = ( minus_minus @ A @ B4 @ A2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % minus_diff_eq
% 0.19/0.47 thf(fact_222_Diff__empty,axiom,
% 0.19/0.47 ! [A: $tType,A3: set @ A] :
% 0.19/0.47 ( ( minus_minus @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
% 0.19/0.47 = A3 ) ).
% 0.19/0.47
% 0.19/0.47 % Diff_empty
% 0.19/0.47 thf(fact_223_empty__Diff,axiom,
% 0.19/0.47 ! [A: $tType,A3: set @ A] :
% 0.19/0.47 ( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 )
% 0.19/0.47 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % empty_Diff
% 0.19/0.47 thf(fact_224_Diff__cancel,axiom,
% 0.19/0.47 ! [A: $tType,A3: set @ A] :
% 0.19/0.47 ( ( minus_minus @ ( set @ A ) @ A3 @ A3 )
% 0.19/0.47 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Diff_cancel
% 0.19/0.47 thf(fact_225_finite__Diff2,axiom,
% 0.19/0.47 ! [A: $tType,B3: set @ A,A3: set @ A] :
% 0.19/0.47 ( ( finite_finite2 @ A @ B3 )
% 0.19/0.47 => ( ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
% 0.19/0.47 = ( finite_finite2 @ A @ A3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite_Diff2
% 0.19/0.47 thf(fact_226_finite__Diff,axiom,
% 0.19/0.47 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.19/0.47 ( ( finite_finite2 @ A @ A3 )
% 0.19/0.47 => ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite_Diff
% 0.19/0.47 thf(fact_227_insert__Diff1,axiom,
% 0.19/0.47 ! [A: $tType,X3: A,B3: set @ A,A3: set @ A] :
% 0.19/0.47 ( ( member @ A @ X3 @ B3 )
% 0.19/0.47 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X3 @ A3 ) @ B3 )
% 0.19/0.47 = ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % insert_Diff1
% 0.19/0.47 thf(fact_228_Diff__insert0,axiom,
% 0.19/0.47 ! [A: $tType,X3: A,A3: set @ A,B3: set @ A] :
% 0.19/0.47 ( ~ ( member @ A @ X3 @ A3 )
% 0.19/0.47 => ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ B3 ) )
% 0.19/0.47 = ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Diff_insert0
% 0.19/0.47 thf(fact_229_Diff__UNIV,axiom,
% 0.19/0.47 ! [A: $tType,A3: set @ A] :
% 0.19/0.47 ( ( minus_minus @ ( set @ A ) @ A3 @ ( top_top @ ( set @ A ) ) )
% 0.19/0.47 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Diff_UNIV
% 0.19/0.47 thf(fact_230_Diff__eq__empty__iff,axiom,
% 0.19/0.47 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 0.19/0.47 ( ( ( minus_minus @ ( set @ A ) @ A3 @ B3 )
% 0.19/0.47 = ( bot_bot @ ( set @ A ) ) )
% 0.19/0.47 = ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% 0.19/0.47
% 0.19/0.47 % Diff_eq_empty_iff
% 0.19/0.47 thf(fact_231_insert__Diff__single,axiom,
% 0.19/0.47 ! [A: $tType,A2: A,A3: set @ A] :
% 0.19/0.47 ( ( insert @ A @ A2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.19/0.47 = ( insert @ A @ A2 @ A3 ) ) ).
% 0.19/0.47
% 0.19/0.47 % insert_Diff_single
% 0.19/0.47 thf(fact_232_finite__Diff__insert,axiom,
% 0.19/0.47 ! [A: $tType,A3: set @ A,A2: A,B3: set @ A] :
% 0.19/0.47 ( ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B3 ) ) )
% 0.19/0.47 = ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % finite_Diff_insert
% 0.19/0.47 thf(fact_233_Diff__mono,axiom,
% 0.19/0.47 ! [A: $tType,A3: set @ A,C3: set @ A,D: set @ A,B3: set @ A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ A3 @ C3 )
% 0.19/0.47 => ( ( ord_less_eq @ ( set @ A ) @ D @ B3 )
% 0.19/0.47 => ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) @ ( minus_minus @ ( set @ A ) @ C3 @ D ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Diff_mono
% 0.19/0.47 thf(fact_234_Diff__subset,axiom,
% 0.19/0.47 ! [A: $tType,A3: set @ A,B3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) @ A3 ) ).
% 0.19/0.47
% 0.19/0.47 % Diff_subset
% 0.19/0.47 thf(fact_235_double__diff,axiom,
% 0.19/0.47 ! [A: $tType,A3: set @ A,B3: set @ A,C3: set @ A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 0.19/0.47 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ C3 )
% 0.19/0.47 => ( ( minus_minus @ ( set @ A ) @ B3 @ ( minus_minus @ ( set @ A ) @ C3 @ A3 ) )
% 0.19/0.47 = A3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % double_diff
% 0.19/0.47 thf(fact_236_diff__mono,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.19/0.47 => ! [A2: A,B4: A,D2: A,C2: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ A2 @ B4 )
% 0.19/0.47 => ( ( ord_less_eq @ A @ D2 @ C2 )
% 0.19/0.47 => ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B4 @ D2 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % diff_mono
% 0.19/0.47 thf(fact_237_diff__left__mono,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.19/0.47 => ! [B4: A,A2: A,C2: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ B4 @ A2 )
% 0.19/0.47 => ( ord_less_eq @ A @ ( minus_minus @ A @ C2 @ A2 ) @ ( minus_minus @ A @ C2 @ B4 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % diff_left_mono
% 0.19/0.47 thf(fact_238_diff__right__mono,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.19/0.47 => ! [A2: A,B4: A,C2: A] :
% 0.19/0.47 ( ( ord_less_eq @ A @ A2 @ B4 )
% 0.19/0.47 => ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B4 @ C2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % diff_right_mono
% 0.19/0.47 thf(fact_239_diff__eq__diff__less__eq,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.19/0.47 => ! [A2: A,B4: A,C2: A,D2: A] :
% 0.19/0.47 ( ( ( minus_minus @ A @ A2 @ B4 )
% 0.19/0.47 = ( minus_minus @ A @ C2 @ D2 ) )
% 0.19/0.47 => ( ( ord_less_eq @ A @ A2 @ B4 )
% 0.19/0.47 = ( ord_less_eq @ A @ C2 @ D2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % diff_eq_diff_less_eq
% 0.19/0.47 thf(fact_240_subset__Diff__insert,axiom,
% 0.19/0.47 ! [A: $tType,A3: set @ A,B3: set @ A,X3: A,C3: set @ A] :
% 0.19/0.47 ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ B3 @ ( insert @ A @ X3 @ C3 ) ) )
% 0.19/0.47 = ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ B3 @ C3 ) )
% 0.19/0.47 & ~ ( member @ A @ X3 @ A3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % subset_Diff_insert
% 0.19/0.47 thf(fact_241_Compl__eq__Diff__UNIV,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( uminus_uminus @ ( set @ A ) )
% 0.19/0.47 = ( minus_minus @ ( set @ A ) @ ( top_top @ ( set @ A ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Compl_eq_Diff_UNIV
% 0.19/0.47 thf(fact_242_DiffE,axiom,
% 0.19/0.47 ! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
% 0.19/0.47 ( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
% 0.19/0.47 => ~ ( ( member @ A @ C2 @ A3 )
% 0.19/0.47 => ( member @ A @ C2 @ B3 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % DiffE
% 0.19/0.47 thf(fact_243_DiffD1,axiom,
% 0.19/0.47 ! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
% 0.19/0.47 ( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
% 0.19/0.47 => ( member @ A @ C2 @ A3 ) ) ).
% 0.19/0.47
% 0.19/0.47 % DiffD1
% 0.19/0.47 thf(fact_244_DiffD2,axiom,
% 0.19/0.47 ! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
% 0.19/0.47 ( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
% 0.19/0.47 => ~ ( member @ A @ C2 @ B3 ) ) ).
% 0.19/0.47
% 0.19/0.47 % DiffD2
% 0.19/0.47 thf(fact_245_insert__Diff__if,axiom,
% 0.19/0.47 ! [A: $tType,X3: A,B3: set @ A,A3: set @ A] :
% 0.19/0.47 ( ( ( member @ A @ X3 @ B3 )
% 0.19/0.47 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X3 @ A3 ) @ B3 )
% 0.19/0.47 = ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) )
% 0.19/0.47 & ( ~ ( member @ A @ X3 @ B3 )
% 0.19/0.47 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X3 @ A3 ) @ B3 )
% 0.19/0.47 = ( insert @ A @ X3 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % insert_Diff_if
% 0.19/0.47 thf(fact_246_Diff__infinite__finite,axiom,
% 0.19/0.47 ! [A: $tType,T3: set @ A,S: set @ A] :
% 0.19/0.47 ( ( finite_finite2 @ A @ T3 )
% 0.19/0.47 => ( ~ ( finite_finite2 @ A @ S )
% 0.19/0.47 => ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ S @ T3 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Diff_infinite_finite
% 0.19/0.47 thf(fact_247_fun__diff__def,axiom,
% 0.19/0.47 ! [B: $tType,A: $tType] :
% 0.19/0.47 ( ( minus @ B @ ( type2 @ B ) )
% 0.19/0.47 => ( ( minus_minus @ ( A > B ) )
% 0.19/0.47 = ( ^ [A7: A > B,B5: A > B,X4: A] : ( minus_minus @ B @ ( A7 @ X4 ) @ ( B5 @ X4 ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % fun_diff_def
% 0.19/0.47 thf(fact_248_diff__eq__diff__eq,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( group_add @ A @ ( type2 @ A ) )
% 0.19/0.47 => ! [A2: A,B4: A,C2: A,D2: A] :
% 0.19/0.47 ( ( ( minus_minus @ A @ A2 @ B4 )
% 0.19/0.47 = ( minus_minus @ A @ C2 @ D2 ) )
% 0.19/0.47 => ( ( A2 = B4 )
% 0.19/0.47 = ( C2 = D2 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % diff_eq_diff_eq
% 0.19/0.47 thf(fact_249_diff__right__commute,axiom,
% 0.19/0.47 ! [A: $tType] :
% 0.19/0.47 ( ( cancel146912293up_add @ A @ ( type2 @ A ) )
% 0.19/0.47 => ! [A2: A,C2: A,B4: A] :
% 0.19/0.47 ( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C2 ) @ B4 )
% 0.19/0.47 = ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B4 ) @ C2 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % diff_right_commute
% 0.19/0.47 thf(fact_250_Diff__insert__absorb,axiom,
% 0.19/0.47 ! [A: $tType,X3: A,A3: set @ A] :
% 0.19/0.47 ( ~ ( member @ A @ X3 @ A3 )
% 0.19/0.47 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X3 @ A3 ) @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.19/0.47 = A3 ) ) ).
% 0.19/0.47
% 0.19/0.47 % Diff_insert_absorb
% 0.19/0.47 thf(fact_251_Diff__insert2,axiom,
% 0.19/0.47 ! [A: $tType,A3: set @ A,A2: A,B3: set @ A] :
% 0.19/0.47 ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B3 ) )
% 0.19/0.47 = ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) @ B3 ) ) ).
% 0.19/0.47
% 0.19/0.47 % Diff_insert2
% 0.19/0.47 thf(fact_252_insert__Diff,axiom,
% 0.19/0.47 ! [A: $tType,A2: A,A3: set @ A] :
% 0.19/0.47 ( ( member @ A @ A2 @ A3 )
% 0.19/0.47 => ( ( insert @ A @ A2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.19/0.47 = A3 ) ) ).
% 0.19/0.47
% 0.19/0.47 % insert_Diff
% 0.19/0.47 thf(fact_253_Diff__insert,axiom,
% 0.19/0.47 ! [A: $tType,A3: set @ A,A2: A,B3: set @ A] :
% 0.19/0.47 ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B3 ) )
% 0.19/0.47 = ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 % Diff_insert
% 0.19/0.47
% 0.19/0.47 %----Type constructors (37)
% 0.19/0.47 thf(tcon_fun___Lattices_Oboolean__algebra,axiom,
% 0.19/0.47 ! [A9: $tType,A10: $tType] :
% 0.19/0.47 ( ( boolean_algebra @ A10 @ ( type2 @ A10 ) )
% 0.19/0.47 => ( boolean_algebra @ ( A9 > A10 ) @ ( type2 @ ( A9 > A10 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_fun___Orderings_Oorder__top,axiom,
% 0.19/0.47 ! [A9: $tType,A10: $tType] :
% 0.19/0.47 ( ( order_top @ A10 @ ( type2 @ A10 ) )
% 0.19/0.47 => ( order_top @ ( A9 > A10 ) @ ( type2 @ ( A9 > A10 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_fun___Orderings_Oorder__bot,axiom,
% 0.19/0.47 ! [A9: $tType,A10: $tType] :
% 0.19/0.47 ( ( order_bot @ A10 @ ( type2 @ A10 ) )
% 0.19/0.47 => ( order_bot @ ( A9 > A10 ) @ ( type2 @ ( A9 > A10 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_fun___Orderings_Opreorder,axiom,
% 0.19/0.47 ! [A9: $tType,A10: $tType] :
% 0.19/0.47 ( ( preorder @ A10 @ ( type2 @ A10 ) )
% 0.19/0.47 => ( preorder @ ( A9 > A10 ) @ ( type2 @ ( A9 > A10 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_fun___Finite__Set_Ofinite,axiom,
% 0.19/0.47 ! [A9: $tType,A10: $tType] :
% 0.19/0.47 ( ( ( finite_finite @ A9 @ ( type2 @ A9 ) )
% 0.19/0.47 & ( finite_finite @ A10 @ ( type2 @ A10 ) ) )
% 0.19/0.47 => ( finite_finite @ ( A9 > A10 ) @ ( type2 @ ( A9 > A10 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_fun___Orderings_Oorder,axiom,
% 0.19/0.47 ! [A9: $tType,A10: $tType] :
% 0.19/0.47 ( ( order @ A10 @ ( type2 @ A10 ) )
% 0.19/0.47 => ( order @ ( A9 > A10 ) @ ( type2 @ ( A9 > A10 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_fun___Orderings_Otop,axiom,
% 0.19/0.47 ! [A9: $tType,A10: $tType] :
% 0.19/0.47 ( ( top @ A10 @ ( type2 @ A10 ) )
% 0.19/0.47 => ( top @ ( A9 > A10 ) @ ( type2 @ ( A9 > A10 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_fun___Orderings_Oord,axiom,
% 0.19/0.47 ! [A9: $tType,A10: $tType] :
% 0.19/0.47 ( ( ord @ A10 @ ( type2 @ A10 ) )
% 0.19/0.47 => ( ord @ ( A9 > A10 ) @ ( type2 @ ( A9 > A10 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_fun___Orderings_Obot,axiom,
% 0.19/0.47 ! [A9: $tType,A10: $tType] :
% 0.19/0.47 ( ( bot @ A10 @ ( type2 @ A10 ) )
% 0.19/0.47 => ( bot @ ( A9 > A10 ) @ ( type2 @ ( A9 > A10 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_fun___Groups_Ouminus,axiom,
% 0.19/0.47 ! [A9: $tType,A10: $tType] :
% 0.19/0.47 ( ( uminus @ A10 @ ( type2 @ A10 ) )
% 0.19/0.47 => ( uminus @ ( A9 > A10 ) @ ( type2 @ ( A9 > A10 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_fun___Groups_Ominus,axiom,
% 0.19/0.47 ! [A9: $tType,A10: $tType] :
% 0.19/0.47 ( ( minus @ A10 @ ( type2 @ A10 ) )
% 0.19/0.47 => ( minus @ ( A9 > A10 ) @ ( type2 @ ( A9 > A10 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_Set_Oset___Lattices_Oboolean__algebra_1,axiom,
% 0.19/0.47 ! [A9: $tType] : ( boolean_algebra @ ( set @ A9 ) @ ( type2 @ ( set @ A9 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_Set_Oset___Orderings_Oorder__top_2,axiom,
% 0.19/0.47 ! [A9: $tType] : ( order_top @ ( set @ A9 ) @ ( type2 @ ( set @ A9 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_Set_Oset___Orderings_Oorder__bot_3,axiom,
% 0.19/0.47 ! [A9: $tType] : ( order_bot @ ( set @ A9 ) @ ( type2 @ ( set @ A9 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_Set_Oset___Orderings_Opreorder_4,axiom,
% 0.19/0.47 ! [A9: $tType] : ( preorder @ ( set @ A9 ) @ ( type2 @ ( set @ A9 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_Set_Oset___Finite__Set_Ofinite_5,axiom,
% 0.19/0.47 ! [A9: $tType] :
% 0.19/0.47 ( ( finite_finite @ A9 @ ( type2 @ A9 ) )
% 0.19/0.47 => ( finite_finite @ ( set @ A9 ) @ ( type2 @ ( set @ A9 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_Set_Oset___Orderings_Oorder_6,axiom,
% 0.19/0.47 ! [A9: $tType] : ( order @ ( set @ A9 ) @ ( type2 @ ( set @ A9 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_Set_Oset___Orderings_Otop_7,axiom,
% 0.19/0.47 ! [A9: $tType] : ( top @ ( set @ A9 ) @ ( type2 @ ( set @ A9 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_Set_Oset___Orderings_Oord_8,axiom,
% 0.19/0.47 ! [A9: $tType] : ( ord @ ( set @ A9 ) @ ( type2 @ ( set @ A9 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_Set_Oset___Orderings_Obot_9,axiom,
% 0.19/0.47 ! [A9: $tType] : ( bot @ ( set @ A9 ) @ ( type2 @ ( set @ A9 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_Set_Oset___Groups_Ouminus_10,axiom,
% 0.19/0.47 ! [A9: $tType] : ( uminus @ ( set @ A9 ) @ ( type2 @ ( set @ A9 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_Set_Oset___Groups_Ominus_11,axiom,
% 0.19/0.47 ! [A9: $tType] : ( minus @ ( set @ A9 ) @ ( type2 @ ( set @ A9 ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_12,axiom,
% 0.19/0.47 boolean_algebra @ $o @ ( type2 @ $o ) ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_HOL_Obool___Orderings_Oorder__top_13,axiom,
% 0.19/0.47 order_top @ $o @ ( type2 @ $o ) ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_HOL_Obool___Orderings_Oorder__bot_14,axiom,
% 0.19/0.47 order_bot @ $o @ ( type2 @ $o ) ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_HOL_Obool___Orderings_Opreorder_15,axiom,
% 0.19/0.47 preorder @ $o @ ( type2 @ $o ) ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 0.19/0.47 linorder @ $o @ ( type2 @ $o ) ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_HOL_Obool___Finite__Set_Ofinite_16,axiom,
% 0.19/0.47 finite_finite @ $o @ ( type2 @ $o ) ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_HOL_Obool___Orderings_Oorder_17,axiom,
% 0.19/0.47 order @ $o @ ( type2 @ $o ) ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_HOL_Obool___Orderings_Otop_18,axiom,
% 0.19/0.47 top @ $o @ ( type2 @ $o ) ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_HOL_Obool___Orderings_Oord_19,axiom,
% 0.19/0.47 ord @ $o @ ( type2 @ $o ) ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_HOL_Obool___Orderings_Obot_20,axiom,
% 0.19/0.47 bot @ $o @ ( type2 @ $o ) ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_HOL_Obool___Groups_Ouminus_21,axiom,
% 0.19/0.47 uminus @ $o @ ( type2 @ $o ) ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_HOL_Obool___Groups_Ominus_22,axiom,
% 0.19/0.47 minus @ $o @ ( type2 @ $o ) ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_23,axiom,
% 0.19/0.47 ! [A9: $tType,A10: $tType] :
% 0.19/0.47 ( ( ( finite_finite @ A9 @ ( type2 @ A9 ) )
% 0.19/0.47 & ( finite_finite @ A10 @ ( type2 @ A10 ) ) )
% 0.19/0.47 => ( finite_finite @ ( sum_sum @ A9 @ A10 ) @ ( type2 @ ( sum_sum @ A9 @ A10 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_Option_Ooption___Finite__Set_Ofinite_24,axiom,
% 0.19/0.47 ! [A9: $tType] :
% 0.19/0.47 ( ( finite_finite @ A9 @ ( type2 @ A9 ) )
% 0.19/0.47 => ( finite_finite @ ( option @ A9 ) @ ( type2 @ ( option @ A9 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_25,axiom,
% 0.19/0.47 ! [A9: $tType,A10: $tType] :
% 0.19/0.47 ( ( ( finite_finite @ A9 @ ( type2 @ A9 ) )
% 0.19/0.47 & ( finite_finite @ A10 @ ( type2 @ A10 ) ) )
% 0.19/0.47 => ( finite_finite @ ( product_prod @ A9 @ A10 ) @ ( type2 @ ( product_prod @ A9 @ A10 ) ) ) ) ).
% 0.19/0.47
% 0.19/0.47 %----Conjectures (1)
% 0.19/0.47 thf(conj_0,conjecture,
% 0.19/0.47 ? [N: node,Ns: set @ node] :
% 0.19/0.47 ( ( ( f @ ( product_Pair @ node @ ( set @ node ) @ n @ ( bot_bot @ ( set @ node ) ) ) )
% 0.19/0.47 = ( f @ ( product_Pair @ node @ ( set @ node ) @ N @ Ns ) ) )
% 0.19/0.47 & ( finite_finite2 @ node @ Ns )
% 0.19/0.47 & ~ ( finite_finite2 @ node @ ( koenig317145564le_via @ node @ graph @ ( uminus_uminus @ ( set @ node ) @ ( insert @ node @ N @ Ns ) ) @ N ) ) ) ).
% 0.19/0.47
% 0.19/0.47 %------------------------------------------------------------------------------
% 0.19/0.47 ------------------------
% 2.48/1.49 ---- Embedded in THF ---
% 2.48/1.49 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.48/1.49 %%% Generated on Tue Feb 24 04:16:13 EST 2026
% 2.48/1.49 %%% using '$$dhol' embedding, version 1.3.0.
% 2.48/1.49 %%% Logic specification used:
% 2.48/1.49 %%% thf(spec, logic, $$dhol).
% 2.48/1.49
% 2.48/1.49 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.sjZ2TfJLxv/DTF2DTF22201.p
% 2.48/1.49 thf(ty_t_Coinductive__List_Ollist, type, coinductive_llist: ($tType > $tType)).
% 2.48/1.49 thf(ty_t_Product__Type_Oprod, type, product_prod: ($tType > ($tType > $tType))).
% 2.48/1.49 thf(ty_t_Sum__Type_Osum, type, sum_sum: ($tType > ($tType > $tType))).
% 2.48/1.49 thf(ty_t_Option_Ooption, type, option: ($tType > $tType)).
% 2.48/1.49 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.48/1.49 thf(ty_t_itself, type, itself: ($tType > $tType)).
% 2.48/1.49 thf(ty_tf_node, type, node: $tType).
% 2.48/1.49 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.48/1.49 thf(sy_cl_Groups_Ominus, type, minus: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.48/1.49 thf(sy_cl_Groups_Ouminus, type, uminus: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.48/1.49 thf(sy_cl_Orderings_Obot, type, bot: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.48/1.49 thf(sy_cl_Orderings_Oord, type, ord: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.48/1.49 thf(sy_cl_Orderings_Otop, type, top: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.48/1.49 thf(sy_cl_Orderings_Oorder, type, order: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.48/1.49 thf(sy_cl_Groups_Ogroup__add, type, group_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.48/1.49 thf(sy_cl_Finite__Set_Ofinite, type, finite_finite: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.48/1.49 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.48/1.49 thf(sy_cl_Orderings_Opreorder, type, preorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.48/1.49 thf(sy_cl_Nat_Osemiring__char__0, type, semiring_char_0: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.48/1.49 thf(sy_cl_Orderings_Oorder__bot, type, order_bot: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.48/1.49 thf(sy_cl_Orderings_Oorder__top, type, order_top: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.48/1.49 thf(sy_cl_Lattices_Oboolean__algebra, type, boolean_algebra: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.48/1.49 thf(sy_cl_Groups_Oordered__ab__group__add, type, ordered_ab_group_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.48/1.49 thf(sy_cl_Groups_Ocancel__ab__semigroup__add, type, cancel146912293up_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.48/1.49 thf(sy_cl_Topological__Spaces_Operfect__space, type, topolo890362671_space: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.48/1.49 thf(sy_c_Basic__BNFs_Ofsts, type, basic_fsts: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > (set @ A))))).
% 2.48/1.49 thf(sy_c_Basic__BNFs_Osnds, type, basic_snds: (!> [A:$tType,B:$tType]: ((((product_prod @ A) @ B) > (set @ B))))).
% 2.48/1.49 thf(sy_c_Coinductive__List_Ogen__lset, type, coinductive_gen_lset: (!> [A:$tType]: (((set @ A) > ((coinductive_llist @ A) > (set @ A)))))).
% 2.48/1.49 thf(sy_c_Coinductive__List_Olfinite, type, coinductive_lfinite: (!> [A:$tType]: (((coinductive_llist @ A) > $o)))).
% 2.48/1.49 thf(sy_c_Coinductive__List_Ollist_Olnull, type, coinductive_lnull: (!> [A:$tType]: (((coinductive_llist @ A) > $o)))).
% 2.48/1.49 thf(sy_c_Coinductive__List_Ollist_Olset, type, coinductive_lset: (!> [A:$tType]: (((coinductive_llist @ A) > (set @ A))))).
% 2.48/1.49 thf(sy_c_Finite__Set_Ofinite, type, finite_finite2: (!> [A:$tType]: (((set @ A) > $o)))).
% 2.48/1.49 thf(sy_c_Groups_Ominus__class_Ominus, type, minus_minus: (!> [A:$tType]: ((A > (A > A))))).
% 2.48/1.49 thf(sy_c_Groups_Ouminus__class_Ouminus, type, uminus_uminus: (!> [A:$tType]: ((A > A)))).
% 2.48/1.49 thf(sy_c_Koenigslemma__Mirabelle__aepjeeakgn_Oconnected, type, koenig793108494nected: (!> [Node:$tType]: (((Node > (Node > $o)) > $o)))).
% 2.48/1.49 thf(sy_c_Koenigslemma__Mirabelle__aepjeeakgn_Oreachable__via, type, koenig317145564le_via: (!> [Node:$tType]: (((Node > (Node > $o)) > ((set @ Node) > (Node > (set @ Node))))))).
% 2.48/1.49 thf(sy_c_Orderings_Obot__class_Obot, type, bot_bot: (!> [A:$tType]: (A))).
% 2.48/1.49 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.48/1.49 thf(sy_c_Orderings_Otop__class_Otop, type, top_top: (!> [A:$tType]: (A))).
% 2.48/1.49 thf(sy_c_Product__Type_OPair, type, product_Pair: (!> [A:$tType,B:$tType]: ((A > (B > ((product_prod @ A) @ B)))))).
% 2.48/1.49 thf(sy_c_Product__Type_Ointernal__case__prod, type, produc2004651681e_prod: (!> [A:$tType,B:$tType,C:$tType]: (((A > (B > C)) > (((product_prod @ A) @ B) > C))))).
% 2.48/1.49 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod, type, product_rec_prod: (!> [A:$tType,B:$tType,T:$tType]: (((A > (B > T)) > (((product_prod @ A) @ B) > T))))).
% 2.48/1.49 thf(sy_c_Pure_Otype, type, type2: (!> [A:$tType]: ((itself @ A)))).
% 2.48/1.49 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.48/1.49 thf(sy_c_Set_Oinsert, type, insert: (!> [A:$tType]: ((A > ((set @ A) > (set @ A)))))).
% 2.48/1.49 thf(sy_c_Set_Ois__empty, type, is_empty: (!> [A:$tType]: (((set @ A) > $o)))).
% 2.48/1.49 thf(sy_c_Set_Ois__singleton, type, is_singleton: (!> [A:$tType]: (((set @ A) > $o)))).
% 2.48/1.49 thf(sy_c_Set_Othe__elem, type, the_elem: (!> [A:$tType]: (((set @ A) > A)))).
% 2.48/1.49 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.48/1.49 thf(sy_v_f____, type, f: (((product_prod @ node) @ (set @ node)) > (coinductive_llist @ node))).
% 2.48/1.49 thf(sy_v_graph, type, graph: (node > (node > $o))).
% 2.48/1.49 thf(sy_v_n, type, n: node).
% 2.48/1.49 thf(sy_v_ns____, type, ns: (set @ node)).
% 2.48/1.49 thf(fact_0_connected, axiom, ((koenig793108494nected @ node) @ graph)).
% 2.48/1.49 thf(fact_1_ns__def, axiom, (ns = (bot_bot @ (set @ node)))).
% 2.48/1.49 thf(fact_2__092_060open_062infinite_A_Ireachable__via_Agraph_A_I_N_A_123n_125_J_An_J_092_060close_062, axiom, (~ (((finite_finite2 @ node) @ ((((koenig317145564le_via @ node) @ graph) @ ((uminus_uminus @ (set @ node)) @ (((insert @ node) @ n) @ (bot_bot @ (set @ node))))) @ n))))).
% 2.48/1.49 thf(fact_3_infinite, axiom, (~ (((finite_finite2 @ node) @ (top_top @ (set @ node)))))).
% 2.48/1.49 thf(fact_4__092_060open_062_N_A_123n_125_A_092_060subseteq_062_Areachable__via_Agraph_A_I_N_A_123n_125_J_An_092_060close_062, axiom, (((ord_less_eq @ (set @ node)) @ ((uminus_uminus @ (set @ node)) @ (((insert @ node) @ n) @ (bot_bot @ (set @ node))))) @ ((((koenig317145564le_via @ node) @ graph) @ ((uminus_uminus @ (set @ node)) @ (((insert @ node) @ n) @ (bot_bot @ (set @ node))))) @ n))).
% 2.48/1.49 thf(fact_5_lset, axiom, (! [X:node,Na:node,Nsa:(set @ node)]: (((((member @ node) @ X) @ ((coinductive_lset @ node) @ (f @ ((((product_Pair @ node) @ (set @ node)) @ Na) @ Nsa)))) => ((~ ((((member @ node) @ Na) @ Nsa))) => (((finite_finite2 @ node) @ Nsa) => ((~ (((finite_finite2 @ node) @ ((((koenig317145564le_via @ node) @ graph) @ ((uminus_uminus @ (set @ node)) @ (((insert @ node) @ Na) @ Nsa))) @ Na)))) => (~ ((((member @ node) @ X) @ Nsa)))))))))).
% 2.48/1.49 thf(fact_6_finite__insert, axiom, (! [A:$tType,A2:A,A3:(set @ A)]: ((((finite_finite2 @ A) @ (((insert @ A) @ A2) @ A3)) = ((finite_finite2 @ A) @ A3))))).
% 2.48/1.49 thf(fact_7_singletonI, axiom, (! [A:$tType,A2:A]: ((((member @ A) @ A2) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))))).
% 2.48/1.49 thf(fact_8_finite_Ocases, axiom, (! [A:$tType,A2:(set @ A)]: ((((finite_finite2 @ A) @ A2) => ((A2 != (bot_bot @ (set @ A))) => (~ ((! [A4:(set @ A)]: (((? [A5:A]: ((A2 = (((insert @ A) @ A5) @ A4)))) => (~ (((finite_finite2 @ A) @ A4))))))))))))).
% 2.48/1.49 thf(fact_9_finite_Osimps, axiom, (! [A:$tType]: (((finite_finite2 @ A) = (^ [A6:(set @ A)]: (((A6 = (bot_bot @ (set @ A))) | (? [A7:(set @ A),B2:A]: (((A6 = (((insert @ A) @ B2) @ A7)) & ((finite_finite2 @ A) @ A7))))))))))).
% 2.48/1.49 thf(fact_10_finite__induct, axiom, (! [A:$tType,F:(set @ A),P:((set @ A) > $o)]: ((((finite_finite2 @ A) @ F) => ((P @ (bot_bot @ (set @ A))) => ((! [X2:A,F2:(set @ A)]: ((((finite_finite2 @ A) @ F2) => ((~ ((((member @ A) @ X2) @ F2))) => ((P @ F2) => (P @ (((insert @ A) @ X2) @ F2))))))) => (P @ F))))))).
% 2.48/1.49 thf(fact_11_finite_Oinducts, axiom, (! [A:$tType,X3:(set @ A),P:((set @ A) > $o)]: ((((finite_finite2 @ A) @ X3) => ((P @ (bot_bot @ (set @ A))) => ((! [A4:(set @ A),A5:A]: ((((finite_finite2 @ A) @ A4) => ((P @ A4) => (P @ (((insert @ A) @ A5) @ A4)))))) => (P @ X3))))))).
% 2.48/1.49 thf(fact_12_finite__ne__induct, axiom, (! [A:$tType,F:(set @ A),P:((set @ A) > $o)]: ((((finite_finite2 @ A) @ F) => ((F != (bot_bot @ (set @ A))) => ((! [X2:A]: ((P @ (((insert @ A) @ X2) @ (bot_bot @ (set @ A)))))) => ((! [X2:A,F2:(set @ A)]: ((((finite_finite2 @ A) @ F2) => ((F2 != (bot_bot @ (set @ A))) => ((~ ((((member @ A) @ X2) @ F2))) => ((P @ F2) => (P @ (((insert @ A) @ X2) @ F2)))))))) => (P @ F)))))))).
% 2.48/1.49 thf(fact_13_infinite__finite__induct, axiom, (! [A:$tType,P:((set @ A) > $o),A3:(set @ A)]: (((! [A4:(set @ A)]: (((~ (((finite_finite2 @ A) @ A4))) => (P @ A4)))) => ((P @ (bot_bot @ (set @ A))) => ((! [X2:A,F2:(set @ A)]: ((((finite_finite2 @ A) @ F2) => ((~ ((((member @ A) @ X2) @ F2))) => ((P @ F2) => (P @ (((insert @ A) @ X2) @ F2))))))) => (P @ A3))))))).
% 2.48/1.49 thf(fact_14_ComplI, axiom, (! [A:$tType,C2:A,A3:(set @ A)]: (((~ ((((member @ A) @ C2) @ A3))) => (((member @ A) @ C2) @ ((uminus_uminus @ (set @ A)) @ A3)))))).
% 2.48/1.49 thf(fact_15_Compl__iff, axiom, (! [A:$tType,C2:A,A3:(set @ A)]: (((((member @ A) @ C2) @ ((uminus_uminus @ (set @ A)) @ A3)) = (~ ((((member @ A) @ C2) @ A3))))))).
% 2.48/1.49 thf(fact_16_empty__Collect__eq, axiom, (! [A:$tType,P:(A > $o)]: ((((bot_bot @ (set @ A)) = ((collect @ A) @ P)) = (! [X4:A]: ((~ ((P @ X4))))))))).
% 2.48/1.49 thf(fact_17_Collect__empty__eq, axiom, (! [A:$tType,P:(A > $o)]: (((((collect @ A) @ P) = (bot_bot @ (set @ A))) = (! [X4:A]: ((~ ((P @ X4))))))))).
% 2.48/1.49 thf(fact_18_all__not__in__conv, axiom, (! [A:$tType,A3:(set @ A)]: (((! [X4:A]: ((~ ((((member @ A) @ X4) @ A3))))) = (A3 = (bot_bot @ (set @ A))))))).
% 2.48/1.49 thf(fact_19_empty__iff, axiom, (! [A:$tType,C2:A]: ((~ ((((member @ A) @ C2) @ (bot_bot @ (set @ A)))))))).
% 2.48/1.49 thf(fact_20_UNIV__I, axiom, (! [A:$tType,X3:A]: ((((member @ A) @ X3) @ (top_top @ (set @ A)))))).
% 2.48/1.49 thf(fact_21_finite__code, axiom, (! [A:$tType]: ((((finite_finite @ A) @ (type2 @ A)) => ((finite_finite2 @ A) = (^ [A7:(set @ A)]: ($true))))))).
% 2.48/1.49 thf(fact_22_subsetI, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((! [X2:A]: (((((member @ A) @ X2) @ A3) => (((member @ A) @ X2) @ B3)))) => (((ord_less_eq @ (set @ A)) @ A3) @ B3))))).
% 2.48/1.49 thf(fact_23_subset__antisym, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((((ord_less_eq @ (set @ A)) @ B3) @ A3) => (A3 = B3)))))).
% 2.48/1.49 thf(fact_24_insert__absorb2, axiom, (! [A:$tType,X3:A,A3:(set @ A)]: (((((insert @ A) @ X3) @ (((insert @ A) @ X3) @ A3)) = (((insert @ A) @ X3) @ A3))))).
% 2.48/1.49 thf(fact_25_insert__iff, axiom, (! [A:$tType,A2:A,B4:A,A3:(set @ A)]: (((((member @ A) @ A2) @ (((insert @ A) @ B4) @ A3)) = ((A2 = B4) | (((member @ A) @ A2) @ A3)))))).
% 2.48/1.49 thf(fact_26_insertCI, axiom, (! [A:$tType,A2:A,B3:(set @ A),B4:A]: ((((~ ((((member @ A) @ A2) @ B3))) => (A2 = B4)) => (((member @ A) @ A2) @ (((insert @ A) @ B4) @ B3)))))).
% 2.48/1.49 thf(fact_27_Compl__eq__Compl__iff, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((uminus_uminus @ (set @ A)) @ A3) = ((uminus_uminus @ (set @ A)) @ B3)) = (A3 = B3))))).
% 2.48/1.49 thf(fact_28_finite__Plus__UNIV__iff, axiom, (! [A:$tType,B:$tType]: ((((finite_finite2 @ ((sum_sum @ A) @ B)) @ (top_top @ (set @ ((sum_sum @ A) @ B)))) = (((finite_finite2 @ A) @ (top_top @ (set @ A))) & ((finite_finite2 @ B) @ (top_top @ (set @ B)))))))).
% 2.48/1.49 thf(fact_29_empty__subsetI, axiom, (! [A:$tType,A3:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ (bot_bot @ (set @ A))) @ A3)))).
% 2.48/1.49 thf(fact_30_subset__empty, axiom, (! [A:$tType,A3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ (bot_bot @ (set @ A))) = (A3 = (bot_bot @ (set @ A))))))).
% 2.48/1.49 thf(fact_31_insert__subset, axiom, (! [A:$tType,X3:A,A3:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ (((insert @ A) @ X3) @ A3)) @ B3) = ((((member @ A) @ X3) @ B3) & (((ord_less_eq @ (set @ A)) @ A3) @ B3)))))).
% 2.48/1.49 thf(fact_32_Compl__subset__Compl__iff, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ A3)) @ ((uminus_uminus @ (set @ A)) @ B3)) = (((ord_less_eq @ (set @ A)) @ B3) @ A3))))).
% 2.48/1.49 thf(fact_33_Compl__anti__mono, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => (((ord_less_eq @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ B3)) @ ((uminus_uminus @ (set @ A)) @ A3)))))).
% 2.48/1.49 thf(fact_34__092_060open_062n_A_092_060in_062_Alset_A_If_A_In_M_Ans_J_J_092_060close_062, axiom, (((member @ node) @ n) @ ((coinductive_lset @ node) @ (f @ ((((product_Pair @ node) @ (set @ node)) @ n) @ ns))))).
% 2.48/1.49 thf(fact_35_singleton__insert__inj__eq_H, axiom, (! [A:$tType,A2:A,A3:(set @ A),B4:A]: ((((((insert @ A) @ A2) @ A3) = (((insert @ A) @ B4) @ (bot_bot @ (set @ A)))) = ((A2 = B4) & (((ord_less_eq @ (set @ A)) @ A3) @ (((insert @ A) @ B4) @ (bot_bot @ (set @ A))))))))).
% 2.48/1.49 thf(fact_36_singleton__insert__inj__eq, axiom, (! [A:$tType,B4:A,A2:A,A3:(set @ A)]: ((((((insert @ A) @ B4) @ (bot_bot @ (set @ A))) = (((insert @ A) @ A2) @ A3)) = ((A2 = B4) & (((ord_less_eq @ (set @ A)) @ A3) @ (((insert @ A) @ B4) @ (bot_bot @ (set @ A))))))))).
% 2.48/1.49 thf(fact_37_finite__compl, axiom, (! [A:$tType,A3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => (((finite_finite2 @ A) @ ((uminus_uminus @ (set @ A)) @ A3)) = ((finite_finite2 @ A) @ (top_top @ (set @ A)))))))).
% 2.48/1.49 thf(fact_38_subset__Compl__singleton, axiom, (! [A:$tType,A3:(set @ A),B4:A]: (((((ord_less_eq @ (set @ A)) @ A3) @ ((uminus_uminus @ (set @ A)) @ (((insert @ A) @ B4) @ (bot_bot @ (set @ A))))) = (~ ((((member @ A) @ B4) @ A3))))))).
% 2.48/1.49 thf(fact_39_set__mp, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),X3:A]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((((member @ A) @ X3) @ A3) => (((member @ A) @ X3) @ B3)))))).
% 2.48/1.49 thf(fact_40_in__mono, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),X3:A]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((((member @ A) @ X3) @ A3) => (((member @ A) @ X3) @ B3)))))).
% 2.48/1.49 thf(fact_41_subsetD, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),C2:A]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((((member @ A) @ C2) @ A3) => (((member @ A) @ C2) @ B3)))))).
% 2.48/1.49 thf(fact_42_subsetCE, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),C2:A]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((((member @ A) @ C2) @ A3) => (((member @ A) @ C2) @ B3)))))).
% 2.48/1.49 thf(fact_43_UNIV__eq__I, axiom, (! [A:$tType,A3:(set @ A)]: (((! [X2:A]: ((((member @ A) @ X2) @ A3))) => ((top_top @ (set @ A)) = A3))))).
% 2.48/1.49 thf(fact_44_equalityE, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((A3 = B3) => (~ (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => (~ ((((ord_less_eq @ (set @ A)) @ B3) @ A3)))))))))).
% 2.48/1.49 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P)) = (P @ A2))))).
% 2.48/1.49 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A3:(set @ A)]: ((((collect @ A) @ (^ [X4:A]: ((((member @ A) @ X4) @ A3)))) = A3)))).
% 2.48/1.49 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X2:A]: (((P @ X2) = (Q @ X2)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 2.48/1.49 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F3:(A > B),G:(A > B)]: (((! [X2:A]: (((F3 @ X2) = (G @ X2)))) => (F3 = G))))).
% 2.48/1.49 thf(fact_49_subset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A7:(set @ A),B5:(set @ A)]: ((! [X4:A]: (((((member @ A) @ X4) @ A7) => (((member @ A) @ X4) @ B5)))))))))).
% 2.48/1.49 thf(fact_50_equalityD1, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((A3 = B3) => (((ord_less_eq @ (set @ A)) @ A3) @ B3))))).
% 2.48/1.49 thf(fact_51_equalityD2, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((A3 = B3) => (((ord_less_eq @ (set @ A)) @ B3) @ A3))))).
% 2.48/1.49 thf(fact_52_set__rev__mp, axiom, (! [A:$tType,X3:A,A3:(set @ A),B3:(set @ A)]: (((((member @ A) @ X3) @ A3) => ((((ord_less_eq @ (set @ A)) @ A3) @ B3) => (((member @ A) @ X3) @ B3)))))).
% 2.48/1.49 thf(fact_53_subset__iff, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A7:(set @ A),B5:(set @ A)]: ((! [T2:A]: (((((member @ A) @ T2) @ A7) => (((member @ A) @ T2) @ B5)))))))))).
% 2.48/1.49 thf(fact_54_rev__subsetD, axiom, (! [A:$tType,C2:A,A3:(set @ A),B3:(set @ A)]: (((((member @ A) @ C2) @ A3) => ((((ord_less_eq @ (set @ A)) @ A3) @ B3) => (((member @ A) @ C2) @ B3)))))).
% 2.48/1.49 thf(fact_55_subset__UNIV, axiom, (! [A:$tType,A3:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A3) @ (top_top @ (set @ A)))))).
% 2.48/1.49 thf(fact_56_subset__refl, axiom, (! [A:$tType,A3:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A3) @ A3)))).
% 2.48/1.49 thf(fact_57_Collect__mono, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X2:A]: (((P @ X2) => (Q @ X2)))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)))))).
% 2.48/1.49 thf(fact_58_UNIV__witness, axiom, (! [A:$tType]: ((? [X2:A]: ((((member @ A) @ X2) @ (top_top @ (set @ A)))))))).
% 2.48/1.49 thf(fact_59_subset__trans, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),C3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((((ord_less_eq @ (set @ A)) @ B3) @ C3) => (((ord_less_eq @ (set @ A)) @ A3) @ C3)))))).
% 2.48/1.49 thf(fact_60_set__eq__subset, axiom, (! [A:$tType]: (((^ [Y:(set @ A),Z:(set @ A)]: ((Y = Z))) = (^ [A7:(set @ A),B5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A7) @ B5) & (((ord_less_eq @ (set @ A)) @ B5) @ A7)))))))).
% 2.48/1.49 thf(fact_61_contra__subsetD, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),C2:A]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((~ ((((member @ A) @ C2) @ B3))) => (~ ((((member @ A) @ C2) @ A3)))))))).
% 2.48/1.49 thf(fact_62_Collect__mono__iff, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)) = (! [X4:A]: (((P @ X4) => (Q @ X4)))))))).
% 2.48/1.49 thf(fact_63_Finite__Set_Ofinite__set, axiom, (! [A:$tType]: ((((finite_finite2 @ (set @ A)) @ (top_top @ (set @ (set @ A)))) = ((finite_finite2 @ A) @ (top_top @ (set @ A))))))).
% 2.48/1.49 thf(fact_64_finite__prod, axiom, (! [A:$tType,B:$tType]: ((((finite_finite2 @ ((product_prod @ A) @ B)) @ (top_top @ (set @ ((product_prod @ A) @ B)))) = (((finite_finite2 @ A) @ (top_top @ (set @ A))) & ((finite_finite2 @ B) @ (top_top @ (set @ B)))))))).
% 2.48/1.49 thf(fact_65_finite__Prod__UNIV, axiom, (! [B:$tType,A:$tType]: ((((finite_finite2 @ A) @ (top_top @ (set @ A))) => (((finite_finite2 @ B) @ (top_top @ (set @ B))) => ((finite_finite2 @ ((product_prod @ A) @ B)) @ (top_top @ (set @ ((product_prod @ A) @ B))))))))).
% 2.48/1.49 thf(fact_66_finite__fun__UNIVD2, axiom, (! [A:$tType,B:$tType]: ((((finite_finite2 @ (A > B)) @ (top_top @ (set @ (A > B)))) => ((finite_finite2 @ B) @ (top_top @ (set @ B))))))).
% 2.48/1.49 thf(fact_67_rev__finite__subset, axiom, (! [A:$tType,B3:(set @ A),A3:(set @ A)]: ((((finite_finite2 @ A) @ B3) => ((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((finite_finite2 @ A) @ A3)))))).
% 2.48/1.49 thf(fact_68_infinite__super, axiom, (! [A:$tType,S:(set @ A),T3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ S) @ T3) => ((~ (((finite_finite2 @ A) @ S))) => (~ (((finite_finite2 @ A) @ T3)))))))).
% 2.48/1.49 thf(fact_69_finite__subset, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => (((finite_finite2 @ A) @ B3) => ((finite_finite2 @ A) @ A3)))))).
% 2.48/1.49 thf(fact_70_empty__not__UNIV, axiom, (! [A:$tType]: (((bot_bot @ (set @ A)) != (top_top @ (set @ A)))))).
% 2.48/1.49 thf(fact_71_finite__UNIV, axiom, (! [A:$tType]: ((((finite_finite @ A) @ (type2 @ A)) => ((finite_finite2 @ A) @ (top_top @ (set @ A))))))).
% 2.48/1.49 thf(fact_72_infinite__UNIV__char__0, axiom, (! [A:$tType]: ((((semiring_char_0 @ A) @ (type2 @ A)) => (~ (((finite_finite2 @ A) @ (top_top @ (set @ A))))))))).
% 2.48/1.49 thf(fact_73_ex__new__if__finite, axiom, (! [A:$tType,A3:(set @ A)]: (((~ (((finite_finite2 @ A) @ (top_top @ (set @ A))))) => (((finite_finite2 @ A) @ A3) => (? [A5:A]: ((~ ((((member @ A) @ A5) @ A3)))))))))).
% 2.48/1.49 thf(fact_74_subset__insertI2, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),B4:A]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => (((ord_less_eq @ (set @ A)) @ A3) @ (((insert @ A) @ B4) @ B3)))))).
% 2.48/1.49 thf(fact_75_subset__insertI, axiom, (! [A:$tType,B3:(set @ A),A2:A]: ((((ord_less_eq @ (set @ A)) @ B3) @ (((insert @ A) @ A2) @ B3))))).
% 2.48/1.49 thf(fact_76_subset__insert, axiom, (! [A:$tType,X3:A,A3:(set @ A),B3:(set @ A)]: (((~ ((((member @ A) @ X3) @ A3))) => ((((ord_less_eq @ (set @ A)) @ A3) @ (((insert @ A) @ X3) @ B3)) = (((ord_less_eq @ (set @ A)) @ A3) @ B3)))))).
% 2.48/1.49 thf(fact_77_Set_Oinsert__mono, axiom, (! [A:$tType,C3:(set @ A),D:(set @ A),A2:A]: (((((ord_less_eq @ (set @ A)) @ C3) @ D) => (((ord_less_eq @ (set @ A)) @ (((insert @ A) @ A2) @ C3)) @ (((insert @ A) @ A2) @ D)))))).
% 2.48/1.49 thf(fact_78_insert__UNIV, axiom, (! [A:$tType,X3:A]: (((((insert @ A) @ X3) @ (top_top @ (set @ A))) = (top_top @ (set @ A)))))).
% 2.48/1.49 thf(fact_79_subset__singleton__iff, axiom, (! [A:$tType,X5:(set @ A),A2:A]: (((((ord_less_eq @ (set @ A)) @ X5) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A)))) = ((X5 = (bot_bot @ (set @ A))) | (X5 = (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))))))).
% 2.48/1.49 thf(fact_80_subset__singletonD, axiom, (! [A:$tType,A3:(set @ A),X3:A]: (((((ord_less_eq @ (set @ A)) @ A3) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) => ((A3 = (bot_bot @ (set @ A))) | (A3 = (((insert @ A) @ X3) @ (bot_bot @ (set @ A))))))))).
% 2.48/1.49 thf(fact_81_subset__Compl__self__eq, axiom, (! [A:$tType,A3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ ((uminus_uminus @ (set @ A)) @ A3)) = (A3 = (bot_bot @ (set @ A))))))).
% 2.48/1.49 thf(fact_82_Compl__empty__eq, axiom, (! [A:$tType]: ((((uminus_uminus @ (set @ A)) @ (bot_bot @ (set @ A))) = (top_top @ (set @ A)))))).
% 2.48/1.49 thf(fact_83_Compl__UNIV__eq, axiom, (! [A:$tType]: ((((uminus_uminus @ (set @ A)) @ (top_top @ (set @ A))) = (bot_bot @ (set @ A)))))).
% 2.48/1.49 thf(fact_84_finite__subset__induct, axiom, (! [A:$tType,F:(set @ A),A3:(set @ A),P:((set @ A) > $o)]: ((((finite_finite2 @ A) @ F) => ((((ord_less_eq @ (set @ A)) @ F) @ A3) => ((P @ (bot_bot @ (set @ A))) => ((! [A5:A,F2:(set @ A)]: ((((finite_finite2 @ A) @ F2) => ((((member @ A) @ A5) @ A3) => ((~ ((((member @ A) @ A5) @ F2))) => ((P @ F2) => (P @ (((insert @ A) @ A5) @ F2)))))))) => (P @ F)))))))).
% 2.48/1.49 thf(fact_85_ex__in__conv, axiom, (! [A:$tType,A3:(set @ A)]: (((? [X4:A]: ((((member @ A) @ X4) @ A3))) = (A3 != (bot_bot @ (set @ A))))))).
% 2.48/1.49 thf(fact_86_equals0I, axiom, (! [A:$tType,A3:(set @ A)]: (((! [Y2:A]: ((~ ((((member @ A) @ Y2) @ A3))))) => (A3 = (bot_bot @ (set @ A))))))).
% 2.48/1.49 thf(fact_87_equals0D, axiom, (! [A:$tType,A3:(set @ A),A2:A]: (((A3 = (bot_bot @ (set @ A))) => (~ ((((member @ A) @ A2) @ A3))))))).
% 2.48/1.49 thf(fact_88_emptyE, axiom, (! [A:$tType,A2:A]: ((~ ((((member @ A) @ A2) @ (bot_bot @ (set @ A)))))))).
% 2.48/1.49 thf(fact_89_finite, axiom, (! [A:$tType]: ((((finite_finite @ A) @ (type2 @ A)) => (! [A3:(set @ A)]: (((finite_finite2 @ A) @ A3))))))).
% 2.48/1.49 thf(fact_90_finite__set__choice, axiom, (! [B:$tType,A:$tType,A3:(set @ A),P:(A > (B > $o))]: ((((finite_finite2 @ A) @ A3) => ((! [X2:A]: (((((member @ A) @ X2) @ A3) => (? [X1:B]: (((P @ X2) @ X1)))))) => (? [F4:(A > B)]: ((! [X6:A]: (((((member @ A) @ X6) @ A3) => ((P @ X6) @ (F4 @ X6)))))))))))).
% 2.48/1.49 thf(fact_91_mk__disjoint__insert, axiom, (! [A:$tType,A2:A,A3:(set @ A)]: (((((member @ A) @ A2) @ A3) => (? [B6:(set @ A)]: (((A3 = (((insert @ A) @ A2) @ B6)) & (~ ((((member @ A) @ A2) @ B6)))))))))).
% 2.48/1.49 thf(fact_92_insert__commute, axiom, (! [A:$tType,X3:A,Y3:A,A3:(set @ A)]: (((((insert @ A) @ X3) @ (((insert @ A) @ Y3) @ A3)) = (((insert @ A) @ Y3) @ (((insert @ A) @ X3) @ A3)))))).
% 2.48/1.49 thf(fact_93_insert__eq__iff, axiom, (! [A:$tType,A2:A,A3:(set @ A),B4:A,B3:(set @ A)]: (((~ ((((member @ A) @ A2) @ A3))) => ((~ ((((member @ A) @ B4) @ B3))) => (((((insert @ A) @ A2) @ A3) = (((insert @ A) @ B4) @ B3)) = (((A2 = B4) => (A3 = B3)) & ((A2 != B4) => (? [C4:(set @ A)]: (((A3 = (((insert @ A) @ B4) @ C4)) & ((~ ((((member @ A) @ B4) @ C4))) & ((B3 = (((insert @ A) @ A2) @ C4)) & (~ ((((member @ A) @ A2) @ C4)))))))))))))))).
% 2.48/1.49 thf(fact_94_insert__absorb, axiom, (! [A:$tType,A2:A,A3:(set @ A)]: (((((member @ A) @ A2) @ A3) => ((((insert @ A) @ A2) @ A3) = A3))))).
% 2.48/1.49 thf(fact_95_insert__ident, axiom, (! [A:$tType,X3:A,A3:(set @ A),B3:(set @ A)]: (((~ ((((member @ A) @ X3) @ A3))) => ((~ ((((member @ A) @ X3) @ B3))) => (((((insert @ A) @ X3) @ A3) = (((insert @ A) @ X3) @ B3)) = (A3 = B3))))))).
% 2.48/1.49 thf(fact_96_Set_Oset__insert, axiom, (! [A:$tType,X3:A,A3:(set @ A)]: (((((member @ A) @ X3) @ A3) => (~ ((! [B6:(set @ A)]: (((A3 = (((insert @ A) @ X3) @ B6)) => (((member @ A) @ X3) @ B6)))))))))).
% 2.48/1.49 thf(fact_97_insertI2, axiom, (! [A:$tType,A2:A,B3:(set @ A),B4:A]: (((((member @ A) @ A2) @ B3) => (((member @ A) @ A2) @ (((insert @ A) @ B4) @ B3)))))).
% 2.48/1.49 thf(fact_98_insertI1, axiom, (! [A:$tType,A2:A,B3:(set @ A)]: ((((member @ A) @ A2) @ (((insert @ A) @ A2) @ B3))))).
% 2.48/1.49 thf(fact_99_insertE, axiom, (! [A:$tType,A2:A,B4:A,A3:(set @ A)]: (((((member @ A) @ A2) @ (((insert @ A) @ B4) @ A3)) => ((A2 != B4) => (((member @ A) @ A2) @ A3)))))).
% 2.48/1.49 thf(fact_100_double__complement, axiom, (! [A:$tType,A3:(set @ A)]: ((((uminus_uminus @ (set @ A)) @ ((uminus_uminus @ (set @ A)) @ A3)) = A3)))).
% 2.48/1.49 thf(fact_101_ComplD, axiom, (! [A:$tType,C2:A,A3:(set @ A)]: (((((member @ A) @ C2) @ ((uminus_uminus @ (set @ A)) @ A3)) => (~ ((((member @ A) @ C2) @ A3))))))).
% 2.48/1.49 thf(fact_102_infinite__imp__nonempty, axiom, (! [A:$tType,S:(set @ A)]: (((~ (((finite_finite2 @ A) @ S))) => (S != (bot_bot @ (set @ A))))))).
% 2.48/1.49 thf(fact_103_finite_OemptyI, axiom, (! [A:$tType]: (((finite_finite2 @ A) @ (bot_bot @ (set @ A)))))).
% 2.48/1.49 thf(fact_104_singleton__inject, axiom, (! [A:$tType,A2:A,B4:A]: ((((((insert @ A) @ A2) @ (bot_bot @ (set @ A))) = (((insert @ A) @ B4) @ (bot_bot @ (set @ A)))) => (A2 = B4))))).
% 2.48/1.49 thf(fact_105_insert__not__empty, axiom, (! [A:$tType,A2:A,A3:(set @ A)]: (((((insert @ A) @ A2) @ A3) != (bot_bot @ (set @ A)))))).
% 2.48/1.49 thf(fact_106_doubleton__eq__iff, axiom, (! [A:$tType,A2:A,B4:A,C2:A,D2:A]: ((((((insert @ A) @ A2) @ (((insert @ A) @ B4) @ (bot_bot @ (set @ A)))) = (((insert @ A) @ C2) @ (((insert @ A) @ D2) @ (bot_bot @ (set @ A))))) = (((A2 = C2) & (B4 = D2)) | ((A2 = D2) & (B4 = C2))))))).
% 2.48/1.49 thf(fact_107_singleton__iff, axiom, (! [A:$tType,B4:A,A2:A]: (((((member @ A) @ B4) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A)))) = (B4 = A2))))).
% 2.48/1.49 thf(fact_108_singletonD, axiom, (! [A:$tType,B4:A,A2:A]: (((((member @ A) @ B4) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A)))) => (B4 = A2))))).
% 2.48/1.49 thf(fact_109_finite_OinsertI, axiom, (! [A:$tType,A3:(set @ A),A2:A]: ((((finite_finite2 @ A) @ A3) => ((finite_finite2 @ A) @ (((insert @ A) @ A2) @ A3)))))).
% 2.48/1.49 thf(fact_110_compl__top__eq, axiom, (! [A:$tType]: ((((boolean_algebra @ A) @ (type2 @ A)) => (((uminus_uminus @ A) @ (top_top @ A)) = (bot_bot @ A)))))).
% 2.48/1.49 thf(fact_111_compl__bot__eq, axiom, (! [A:$tType]: ((((boolean_algebra @ A) @ (type2 @ A)) => (((uminus_uminus @ A) @ (bot_bot @ A)) = (top_top @ A)))))).
% 2.48/1.49 thf(fact_112__092_060open_062_092_060not_062_Alfinite_A_If_A_In_M_Ans_J_J_092_060close_062, axiom, (~ (((coinductive_lfinite @ node) @ (f @ ((((product_Pair @ node) @ (set @ node)) @ n) @ ns)))))).
% 2.48/1.49 thf(fact_113_finite__option__UNIV, axiom, (! [A:$tType]: ((((finite_finite2 @ (option @ A)) @ (top_top @ (set @ (option @ A)))) = ((finite_finite2 @ A) @ (top_top @ (set @ A))))))).
% 2.48/1.49 thf(fact_114_compl__le__compl__iff, axiom, (! [A:$tType]: ((((boolean_algebra @ A) @ (type2 @ A)) => (! [X3:A,Y3:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ X3)) @ ((uminus_uminus @ A) @ Y3)) = (((ord_less_eq @ A) @ Y3) @ X3)))))))).
% 2.48/1.49 thf(fact_115_neg__le__iff__le, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [B4:A,A2:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ B4)) @ ((uminus_uminus @ A) @ A2)) = (((ord_less_eq @ A) @ A2) @ B4)))))))).
% 2.48/1.49 thf(fact_116_Topological__Spaces_OUNIV__not__singleton, axiom, (! [A:$tType]: ((((topolo890362671_space @ A) @ (type2 @ A)) => (! [X3:A]: (((top_top @ (set @ A)) != (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))))))))).
% 2.48/1.49 thf(fact_117_f__simps_I1_J, axiom, (! [Na:node,Nsa:(set @ node)]: ((~ (((coinductive_lnull @ node) @ (f @ ((((product_Pair @ node) @ (set @ node)) @ Na) @ Nsa)))))))).
% 2.48/1.49 thf(fact_118_the__elem__eq, axiom, (! [A:$tType,X3:A]: ((((the_elem @ A) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) = X3)))).
% 2.48/1.49 thf(fact_119_iso__tuple__UNIV__I, axiom, (! [A:$tType,X3:A]: ((((member @ A) @ X3) @ (top_top @ (set @ A)))))).
% 2.48/1.49 thf(fact_120_uminus__apply, axiom, (! [B:$tType,A:$tType]: ((((uminus @ B) @ (type2 @ B)) => ((uminus_uminus @ (A > B)) = (^ [A7:(A > B),X4:A]: (((uminus_uminus @ B) @ (A7 @ X4))))))))).
% 2.48/1.49 thf(fact_121_add_Oinverse__inverse, axiom, (! [A:$tType]: ((((group_add @ A) @ (type2 @ A)) => (! [A2:A]: ((((uminus_uminus @ A) @ ((uminus_uminus @ A) @ A2)) = A2))))))).
% 2.48/1.49 thf(fact_122_neg__equal__iff__equal, axiom, (! [A:$tType]: ((((group_add @ A) @ (type2 @ A)) => (! [A2:A,B4:A]: (((((uminus_uminus @ A) @ A2) = ((uminus_uminus @ A) @ B4)) = (A2 = B4)))))))).
% 2.48/1.49 thf(fact_123_double__compl, axiom, (! [A:$tType]: ((((boolean_algebra @ A) @ (type2 @ A)) => (! [X3:A]: ((((uminus_uminus @ A) @ ((uminus_uminus @ A) @ X3)) = X3))))))).
% 2.48/1.49 thf(fact_124_compl__eq__compl__iff, axiom, (! [A:$tType]: ((((boolean_algebra @ A) @ (type2 @ A)) => (! [X3:A,Y3:A]: (((((uminus_uminus @ A) @ X3) = ((uminus_uminus @ A) @ Y3)) = (X3 = Y3)))))))).
% 2.48/1.49 thf(fact_125_bot__set__def, axiom, (! [A:$tType]: (((bot_bot @ (set @ A)) = ((collect @ A) @ (bot_bot @ (A > $o))))))).
% 2.48/1.49 thf(fact_126_top__set__def, axiom, (! [A:$tType]: (((top_top @ (set @ A)) = ((collect @ A) @ (top_top @ (A > $o))))))).
% 2.48/1.49 thf(fact_127_fun__Compl__def, axiom, (! [B:$tType,A:$tType]: ((((uminus @ B) @ (type2 @ B)) => ((uminus_uminus @ (A > B)) = (^ [A7:(A > B),X4:A]: (((uminus_uminus @ B) @ (A7 @ X4))))))))).
% 2.48/1.49 thf(fact_128_equation__minus__iff, axiom, (! [A:$tType]: ((((group_add @ A) @ (type2 @ A)) => (! [A2:A,B4:A]: (((A2 = ((uminus_uminus @ A) @ B4)) = (B4 = ((uminus_uminus @ A) @ A2))))))))).
% 2.48/1.49 thf(fact_129_minus__equation__iff, axiom, (! [A:$tType]: ((((group_add @ A) @ (type2 @ A)) => (! [A2:A,B4:A]: (((((uminus_uminus @ A) @ A2) = B4) = (((uminus_uminus @ A) @ B4) = A2)))))))).
% 2.48/1.49 thf(fact_130_compl__mono, axiom, (! [A:$tType]: ((((boolean_algebra @ A) @ (type2 @ A)) => (! [X3:A,Y3:A]: (((((ord_less_eq @ A) @ X3) @ Y3) => (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ Y3)) @ ((uminus_uminus @ A) @ X3))))))))).
% 2.48/1.49 thf(fact_131_compl__le__swap1, axiom, (! [A:$tType]: ((((boolean_algebra @ A) @ (type2 @ A)) => (! [Y3:A,X3:A]: (((((ord_less_eq @ A) @ Y3) @ ((uminus_uminus @ A) @ X3)) => (((ord_less_eq @ A) @ X3) @ ((uminus_uminus @ A) @ Y3))))))))).
% 2.48/1.49 thf(fact_132_compl__le__swap2, axiom, (! [A:$tType]: ((((boolean_algebra @ A) @ (type2 @ A)) => (! [Y3:A,X3:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ Y3)) @ X3) => (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ X3)) @ Y3)))))))).
% 2.48/1.49 thf(fact_133_le__minus__iff, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [A2:A,B4:A]: (((((ord_less_eq @ A) @ A2) @ ((uminus_uminus @ A) @ B4)) = (((ord_less_eq @ A) @ B4) @ ((uminus_uminus @ A) @ A2))))))))).
% 2.48/1.49 thf(fact_134_minus__le__iff, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [A2:A,B4:A]: (((((ord_less_eq @ A) @ ((uminus_uminus @ A) @ A2)) @ B4) = (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ B4)) @ A2)))))))).
% 2.48/1.49 thf(fact_135_le__imp__neg__le, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [A2:A,B4:A]: (((((ord_less_eq @ A) @ A2) @ B4) => (((ord_less_eq @ A) @ ((uminus_uminus @ A) @ B4)) @ ((uminus_uminus @ A) @ A2))))))))).
% 2.48/1.49 thf(fact_136_lset__eq__empty, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: (((((coinductive_lset @ A) @ Xs) = (bot_bot @ (set @ A))) = ((coinductive_lnull @ A) @ Xs))))).
% 2.48/1.49 thf(fact_137_lfinite__imp__finite__lset, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: ((((coinductive_lfinite @ A) @ Xs) => ((finite_finite2 @ A) @ ((coinductive_lset @ A) @ Xs)))))).
% 2.48/1.49 thf(fact_138_lset__lnull, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: ((((coinductive_lnull @ A) @ Xs) => (((coinductive_lset @ A) @ Xs) = (bot_bot @ (set @ A))))))).
% 2.48/1.49 thf(fact_139_top__apply, axiom, (! [C:$tType,D3:$tType]: ((((top @ C) @ (type2 @ C)) => ((top_top @ (D3 > C)) = (^ [X4:D3]: ((top_top @ C)))))))).
% 2.48/1.49 thf(fact_140_order__refl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X3:A]: ((((ord_less_eq @ A) @ X3) @ X3))))))).
% 2.48/1.49 thf(fact_141_top1I, axiom, (! [A:$tType,X3:A]: (((top_top @ (A > $o)) @ X3)))).
% 2.48/1.49 thf(fact_142_bot__apply, axiom, (! [C:$tType,D3:$tType]: ((((bot @ C) @ (type2 @ C)) => ((bot_bot @ (D3 > C)) = (^ [X4:D3]: ((bot_bot @ C)))))))).
% 2.48/1.49 thf(fact_143_le__funD, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [F3:(A > B),G:(A > B),X3:A]: (((((ord_less_eq @ (A > B)) @ F3) @ G) => (((ord_less_eq @ B) @ (F3 @ X3)) @ (G @ X3))))))))).
% 2.48/1.49 thf(fact_144_le__funE, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [F3:(A > B),G:(A > B),X3:A]: (((((ord_less_eq @ (A > B)) @ F3) @ G) => (((ord_less_eq @ B) @ (F3 @ X3)) @ (G @ X3))))))))).
% 2.48/1.49 thf(fact_145_le__funI, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => (! [F3:(A > B),G:(A > B)]: (((! [X2:A]: ((((ord_less_eq @ B) @ (F3 @ X2)) @ (G @ X2)))) => (((ord_less_eq @ (A > B)) @ F3) @ G)))))))).
% 2.48/1.49 thf(fact_146_le__fun__def, axiom, (! [B:$tType,A:$tType]: ((((ord @ B) @ (type2 @ B)) => ((ord_less_eq @ (A > B)) = (^ [F5:(A > B),G2:(A > B)]: ((! [X4:A]: ((((ord_less_eq @ B) @ (F5 @ X4)) @ (G2 @ X4))))))))))).
% 2.48/1.49 thf(fact_147_order__subst1, axiom, (! [A:$tType,B:$tType]: (((((order @ B) @ (type2 @ B)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,F3:(B > A),B4:B,C2:B]: (((((ord_less_eq @ A) @ A2) @ (F3 @ B4)) => ((((ord_less_eq @ B) @ B4) @ C2) => ((! [X2:B,Y2:B]: (((((ord_less_eq @ B) @ X2) @ Y2) => (((ord_less_eq @ A) @ (F3 @ X2)) @ (F3 @ Y2))))) => (((ord_less_eq @ A) @ A2) @ (F3 @ C2))))))))))).
% 2.48/1.49 thf(fact_148_order__subst2, axiom, (! [A:$tType,C:$tType]: (((((order @ C) @ (type2 @ C)) & ((order @ A) @ (type2 @ A))) => (! [A2:A,B4:A,F3:(A > C),C2:C]: (((((ord_less_eq @ A) @ A2) @ B4) => ((((ord_less_eq @ C) @ (F3 @ B4)) @ C2) => ((! [X2:A,Y2:A]: (((((ord_less_eq @ A) @ X2) @ Y2) => (((ord_less_eq @ C) @ (F3 @ X2)) @ (F3 @ Y2))))) => (((ord_less_eq @ C) @ (F3 @ A2)) @ C2)))))))))).
% 2.48/1.49 thf(fact_149_ord__eq__le__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A2:A,F3:(B > A),B4:B,C2:B]: (((A2 = (F3 @ B4)) => ((((ord_less_eq @ B) @ B4) @ C2) => ((! [X2:B,Y2:B]: (((((ord_less_eq @ B) @ X2) @ Y2) => (((ord_less_eq @ A) @ (F3 @ X2)) @ (F3 @ Y2))))) => (((ord_less_eq @ A) @ A2) @ (F3 @ C2))))))))))).
% 2.48/1.49 thf(fact_150_ord__le__eq__subst, axiom, (! [A:$tType,B:$tType]: (((((ord @ B) @ (type2 @ B)) & ((ord @ A) @ (type2 @ A))) => (! [A2:A,B4:A,F3:(A > B),C2:B]: (((((ord_less_eq @ A) @ A2) @ B4) => (((F3 @ B4) = C2) => ((! [X2:A,Y2:A]: (((((ord_less_eq @ A) @ X2) @ Y2) => (((ord_less_eq @ B) @ (F3 @ X2)) @ (F3 @ Y2))))) => (((ord_less_eq @ B) @ (F3 @ A2)) @ C2)))))))))).
% 2.48/1.49 thf(fact_151_eq__iff, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => ((^ [Y:A,Z:A]: ((Y = Z))) = (^ [X4:A,Y4:A]: (((((ord_less_eq @ A) @ X4) @ Y4) & (((ord_less_eq @ A) @ Y4) @ X4))))))))).
% 2.48/1.49 thf(fact_152_antisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [X3:A,Y3:A]: (((((ord_less_eq @ A) @ X3) @ Y3) => ((((ord_less_eq @ A) @ Y3) @ X3) => (X3 = Y3))))))))).
% 2.48/1.49 thf(fact_153_linear, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X3:A,Y3:A]: (((((ord_less_eq @ A) @ X3) @ Y3) | (((ord_less_eq @ A) @ Y3) @ X3)))))))).
% 2.48/1.49 thf(fact_154_eq__refl, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X3:A,Y3:A]: (((X3 = Y3) => (((ord_less_eq @ A) @ X3) @ Y3)))))))).
% 2.48/1.49 thf(fact_155_wlog__linorder__le, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > (A > $o)),B4:A,A2:A]: (((! [A5:A,B7:A]: (((((ord_less_eq @ A) @ A5) @ B7) => ((P @ A5) @ B7)))) => ((((P @ B4) @ A2) => ((P @ A2) @ B4)) => ((P @ A2) @ B4))))))))).
% 2.48/1.49 thf(fact_156_le__cases, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X3:A,Y3:A]: (((~ ((((ord_less_eq @ A) @ X3) @ Y3))) => (((ord_less_eq @ A) @ Y3) @ X3)))))))).
% 2.48/1.49 thf(fact_157_order_Otrans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B4:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B4) => ((((ord_less_eq @ A) @ B4) @ C2) => (((ord_less_eq @ A) @ A2) @ C2))))))))).
% 2.48/1.49 thf(fact_158_le__cases3, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X3:A,Y3:A,Z2:A]: ((((((ord_less_eq @ A) @ X3) @ Y3) => (~ ((((ord_less_eq @ A) @ Y3) @ Z2)))) => (((((ord_less_eq @ A) @ Y3) @ X3) => (~ ((((ord_less_eq @ A) @ X3) @ Z2)))) => (((((ord_less_eq @ A) @ X3) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ Y3)))) => (((((ord_less_eq @ A) @ Z2) @ Y3) => (~ ((((ord_less_eq @ A) @ Y3) @ X3)))) => (((((ord_less_eq @ A) @ Y3) @ Z2) => (~ ((((ord_less_eq @ A) @ Z2) @ X3)))) => (~ (((((ord_less_eq @ A) @ Z2) @ X3) => (~ ((((ord_less_eq @ A) @ X3) @ Y3))))))))))))))))).
% 2.48/1.49 thf(fact_159_antisym__conv, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [Y3:A,X3:A]: (((((ord_less_eq @ A) @ Y3) @ X3) => ((((ord_less_eq @ A) @ X3) @ Y3) = (X3 = Y3))))))))).
% 2.48/1.49 thf(fact_160_ord__eq__le__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A2:A,B4:A,C2:A]: (((A2 = B4) => ((((ord_less_eq @ A) @ B4) @ C2) => (((ord_less_eq @ A) @ A2) @ C2))))))))).
% 2.48/1.49 thf(fact_161_ord__le__eq__trans, axiom, (! [A:$tType]: ((((ord @ A) @ (type2 @ A)) => (! [A2:A,B4:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B4) => ((B4 = C2) => (((ord_less_eq @ A) @ A2) @ C2))))))))).
% 2.48/1.49 thf(fact_162_order__class_Oorder_Oantisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A,B4:A]: (((((ord_less_eq @ A) @ A2) @ B4) => ((((ord_less_eq @ A) @ B4) @ A2) => (A2 = B4))))))))).
% 2.48/1.49 thf(fact_163_order__trans, axiom, (! [A:$tType]: ((((preorder @ A) @ (type2 @ A)) => (! [X3:A,Y3:A,Z2:A]: (((((ord_less_eq @ A) @ X3) @ Y3) => ((((ord_less_eq @ A) @ Y3) @ Z2) => (((ord_less_eq @ A) @ X3) @ Z2))))))))).
% 2.48/1.49 thf(fact_164_dual__order_Orefl, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [A2:A]: ((((ord_less_eq @ A) @ A2) @ A2))))))).
% 2.48/1.49 thf(fact_165_linorder__wlog, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [P:(A > (A > $o)),A2:A,B4:A]: (((! [A5:A,B7:A]: (((((ord_less_eq @ A) @ A5) @ B7) => ((P @ A5) @ B7)))) => ((! [A5:A,B7:A]: ((((P @ B7) @ A5) => ((P @ A5) @ B7)))) => ((P @ A2) @ B4))))))))).
% 2.48/1.49 thf(fact_166_dual__order_Otrans, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B4:A,A2:A,C2:A]: (((((ord_less_eq @ A) @ B4) @ A2) => ((((ord_less_eq @ A) @ C2) @ B4) => (((ord_less_eq @ A) @ C2) @ A2))))))))).
% 2.48/1.49 thf(fact_167_dual__order_Oantisym, axiom, (! [A:$tType]: ((((order @ A) @ (type2 @ A)) => (! [B4:A,A2:A]: (((((ord_less_eq @ A) @ B4) @ A2) => ((((ord_less_eq @ A) @ A2) @ B4) => (A2 = B4))))))))).
% 2.48/1.49 thf(fact_168_bot__fun__def, axiom, (! [B:$tType,A:$tType]: ((((bot @ B) @ (type2 @ B)) => ((bot_bot @ (A > B)) = (^ [X4:A]: ((bot_bot @ B)))))))).
% 2.48/1.49 thf(fact_169_lzip_Oexhaust, axiom, (! [A:$tType,B:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ B)]: (((~ ((((coinductive_lnull @ A) @ Xs) | ((coinductive_lnull @ B) @ Ys)))) => (~ (((~ (((coinductive_lnull @ A) @ Xs))) => ((coinductive_lnull @ B) @ Ys)))))))).
% 2.48/1.49 thf(fact_170_lappend_Oexhaust, axiom, (! [A:$tType,Xs:(coinductive_llist @ A),Ys:(coinductive_llist @ A)]: (((((coinductive_lnull @ A) @ Xs) => (~ (((coinductive_lnull @ A) @ Ys)))) => ((~ (((coinductive_lnull @ A) @ Xs))) | (~ (((coinductive_lnull @ A) @ Ys)))))))).
% 2.48/1.49 thf(fact_171_bot_Oextremum__uniqueI, axiom, (! [A:$tType]: ((((order_bot @ A) @ (type2 @ A)) => (! [A2:A]: (((((ord_less_eq @ A) @ A2) @ (bot_bot @ A)) => (A2 = (bot_bot @ A))))))))).
% 2.48/1.49 thf(fact_172_bot_Oextremum__unique, axiom, (! [A:$tType]: ((((order_bot @ A) @ (type2 @ A)) => (! [A2:A]: (((((ord_less_eq @ A) @ A2) @ (bot_bot @ A)) = (A2 = (bot_bot @ A))))))))).
% 2.48/1.49 thf(fact_173_bot_Oextremum, axiom, (! [A:$tType]: ((((order_bot @ A) @ (type2 @ A)) => (! [A2:A]: ((((ord_less_eq @ A) @ (bot_bot @ A)) @ A2))))))).
% 2.48/1.49 thf(fact_174_top__greatest, axiom, (! [A:$tType]: ((((order_top @ A) @ (type2 @ A)) => (! [A2:A]: ((((ord_less_eq @ A) @ A2) @ (top_top @ A)))))))).
% 2.48/1.49 thf(fact_175_top_Oextremum__unique, axiom, (! [A:$tType]: ((((order_top @ A) @ (type2 @ A)) => (! [A2:A]: (((((ord_less_eq @ A) @ (top_top @ A)) @ A2) = (A2 = (top_top @ A))))))))).
% 2.48/1.49 thf(fact_176_top_Oextremum__uniqueI, axiom, (! [A:$tType]: ((((order_top @ A) @ (type2 @ A)) => (! [A2:A]: (((((ord_less_eq @ A) @ (top_top @ A)) @ A2) => (A2 = (top_top @ A))))))))).
% 2.48/1.49 thf(fact_177_lnull__imp__lfinite, axiom, (! [A:$tType,Xs:(coinductive_llist @ A)]: ((((coinductive_lnull @ A) @ Xs) => ((coinductive_lfinite @ A) @ Xs))))).
% 2.48/1.49 thf(fact_178_old_Oprod_Oinject, axiom, (! [A:$tType,B:$tType,A2:A,B4:B,A8:A,B8:B]: (((((((product_Pair @ A) @ B) @ A2) @ B4) = ((((product_Pair @ A) @ B) @ A8) @ B8)) = ((A2 = A8) & (B4 = B8)))))).
% 2.48/1.49 thf(fact_179_prod_Oinject, axiom, (! [A:$tType,B:$tType,X12:A,X22:B,Y1:A,Y22:B]: (((((((product_Pair @ A) @ B) @ X12) @ X22) = ((((product_Pair @ A) @ B) @ Y1) @ Y22)) = ((X12 = Y1) & (X22 = Y22)))))).
% 2.48/1.49 thf(fact_180_top__empty__eq, axiom, (! [A:$tType]: (((top_top @ (A > $o)) = (^ [X4:A]: ((((member @ A) @ X4) @ (top_top @ (set @ A))))))))).
% 2.48/1.49 thf(fact_181_Collect__empty__eq__bot, axiom, (! [A:$tType,P:(A > $o)]: (((((collect @ A) @ P) = (bot_bot @ (set @ A))) = (P = (bot_bot @ (A > $o))))))).
% 2.48/1.49 thf(fact_182_subrelI, axiom, (! [B:$tType,A:$tType,R:(set @ ((product_prod @ A) @ B)),S2:(set @ ((product_prod @ A) @ B))]: (((! [X2:A,Y2:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X2) @ Y2)) @ R) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ X2) @ Y2)) @ S2)))) => (((ord_less_eq @ (set @ ((product_prod @ A) @ B))) @ R) @ S2))))).
% 2.48/1.49 thf(fact_183_surj__pair, axiom, (! [A:$tType,B:$tType,P2:((product_prod @ A) @ B)]: ((? [X2:A,Y2:B]: ((P2 = ((((product_Pair @ A) @ B) @ X2) @ Y2))))))).
% 2.48/1.49 thf(fact_184_prod__cases, axiom, (! [B:$tType,A:$tType,P:(((product_prod @ A) @ B) > $o),P2:((product_prod @ A) @ B)]: (((! [A5:A,B7:B]: ((P @ ((((product_Pair @ A) @ B) @ A5) @ B7)))) => (P @ P2))))).
% 2.48/1.49 thf(fact_185_Pair__inject, axiom, (! [A:$tType,B:$tType,A2:A,B4:B,A8:A,B8:B]: (((((((product_Pair @ A) @ B) @ A2) @ B4) = ((((product_Pair @ A) @ B) @ A8) @ B8)) => (~ (((A2 = A8) => (B4 != B8)))))))).
% 2.48/1.49 thf(fact_186_prod__cases3, axiom, (! [A:$tType,B:$tType,C:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ C))]: ((~ ((! [A5:A,B7:B,C5:C]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A5) @ ((((product_Pair @ B) @ C) @ B7) @ C5)))))))))).
% 2.48/1.49 thf(fact_187_prod__cases4, axiom, (! [A:$tType,B:$tType,C:$tType,D3:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D3)))]: ((~ ((! [A5:A,B7:B,C5:C,D4:D3]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D3))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D3)) @ B7) @ ((((product_Pair @ C) @ D3) @ C5) @ D4))))))))))).
% 2.48/1.49 thf(fact_188_prod__cases5, axiom, (! [A:$tType,B:$tType,C:$tType,D3:$tType,E:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D3) @ E))))]: ((~ ((! [A5:A,B7:B,C5:C,D4:D3,E2:E]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D3) @ E)))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D3) @ E))) @ B7) @ ((((product_Pair @ C) @ ((product_prod @ D3) @ E)) @ C5) @ ((((product_Pair @ D3) @ E) @ D4) @ E2)))))))))))).
% 2.48/1.49 thf(fact_189_prod__cases6, axiom, (! [A:$tType,B:$tType,C:$tType,D3:$tType,E:$tType,F6:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D3) @ ((product_prod @ E) @ F6)))))]: ((~ ((! [A5:A,B7:B,C5:C,D4:D3,E2:E,F4:F6]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D3) @ ((product_prod @ E) @ F6))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D3) @ ((product_prod @ E) @ F6)))) @ B7) @ ((((product_Pair @ C) @ ((product_prod @ D3) @ ((product_prod @ E) @ F6))) @ C5) @ ((((product_Pair @ D3) @ ((product_prod @ E) @ F6)) @ D4) @ ((((product_Pair @ E) @ F6) @ E2) @ F4))))))))))))).
% 2.48/1.49 thf(fact_190_prod__cases7, axiom, (! [A:$tType,B:$tType,C:$tType,D3:$tType,E:$tType,F6:$tType,G3:$tType,Y3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D3) @ ((product_prod @ E) @ ((product_prod @ F6) @ G3))))))]: ((~ ((! [A5:A,B7:B,C5:C,D4:D3,E2:E,F4:F6,G4:G3]: ((Y3 != ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D3) @ ((product_prod @ E) @ ((product_prod @ F6) @ G3)))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D3) @ ((product_prod @ E) @ ((product_prod @ F6) @ G3))))) @ B7) @ ((((product_Pair @ C) @ ((product_prod @ D3) @ ((product_prod @ E) @ ((product_prod @ F6) @ G3)))) @ C5) @ ((((product_Pair @ D3) @ ((product_prod @ E) @ ((product_prod @ F6) @ G3))) @ D4) @ ((((product_Pair @ E) @ ((product_prod @ F6) @ G3)) @ E2) @ ((((product_Pair @ F6) @ G3) @ F4) @ G4)))))))))))))).
% 2.48/1.49 thf(fact_191_prod__induct3, axiom, (! [C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ C)) > $o),X3:((product_prod @ A) @ ((product_prod @ B) @ C))]: (((! [A5:A,B7:B,C5:C]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ C)) @ A5) @ ((((product_Pair @ B) @ C) @ B7) @ C5))))) => (P @ X3))))).
% 2.48/1.49 thf(fact_192_prod__induct4, axiom, (! [D3:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D3))) > $o),X3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D3)))]: (((! [A5:A,B7:B,C5:C,D4:D3]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ D3))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ D3)) @ B7) @ ((((product_Pair @ C) @ D3) @ C5) @ D4)))))) => (P @ X3))))).
% 2.48/1.49 thf(fact_193_prod__induct5, axiom, (! [E:$tType,D3:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D3) @ E)))) > $o),X3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D3) @ E))))]: (((! [A5:A,B7:B,C5:C,D4:D3,E2:E]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D3) @ E)))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D3) @ E))) @ B7) @ ((((product_Pair @ C) @ ((product_prod @ D3) @ E)) @ C5) @ ((((product_Pair @ D3) @ E) @ D4) @ E2))))))) => (P @ X3))))).
% 2.48/1.49 thf(fact_194_prod__induct6, axiom, (! [F6:$tType,E:$tType,D3:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D3) @ ((product_prod @ E) @ F6))))) > $o),X3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D3) @ ((product_prod @ E) @ F6)))))]: (((! [A5:A,B7:B,C5:C,D4:D3,E2:E,F4:F6]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D3) @ ((product_prod @ E) @ F6))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D3) @ ((product_prod @ E) @ F6)))) @ B7) @ ((((product_Pair @ C) @ ((product_prod @ D3) @ ((product_prod @ E) @ F6))) @ C5) @ ((((product_Pair @ D3) @ ((product_prod @ E) @ F6)) @ D4) @ ((((product_Pair @ E) @ F6) @ E2) @ F4)))))))) => (P @ X3))))).
% 2.48/1.49 thf(fact_195_prod__induct7, axiom, (! [G3:$tType,F6:$tType,E:$tType,D3:$tType,C:$tType,B:$tType,A:$tType,P:(((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D3) @ ((product_prod @ E) @ ((product_prod @ F6) @ G3)))))) > $o),X3:((product_prod @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D3) @ ((product_prod @ E) @ ((product_prod @ F6) @ G3))))))]: (((! [A5:A,B7:B,C5:C,D4:D3,E2:E,F4:F6,G4:G3]: ((P @ ((((product_Pair @ A) @ ((product_prod @ B) @ ((product_prod @ C) @ ((product_prod @ D3) @ ((product_prod @ E) @ ((product_prod @ F6) @ G3)))))) @ A5) @ ((((product_Pair @ B) @ ((product_prod @ C) @ ((product_prod @ D3) @ ((product_prod @ E) @ ((product_prod @ F6) @ G3))))) @ B7) @ ((((product_Pair @ C) @ ((product_prod @ D3) @ ((product_prod @ E) @ ((product_prod @ F6) @ G3)))) @ C5) @ ((((product_Pair @ D3) @ ((product_prod @ E) @ ((product_prod @ F6) @ G3))) @ D4) @ ((((product_Pair @ E) @ ((product_prod @ F6) @ G3)) @ E2) @ ((((product_Pair @ F6) @ G3) @ F4) @ G4))))))))) => (P @ X3))))).
% 2.48/1.49 thf(fact_196_old_Oprod_Oexhaust, axiom, (! [A:$tType,B:$tType,Y3:((product_prod @ A) @ B)]: ((~ ((! [A5:A,B7:B]: ((Y3 != ((((product_Pair @ A) @ B) @ A5) @ B7))))))))).
% 2.48/1.49 thf(fact_197_old_Oprod_Oinducts, axiom, (! [B:$tType,A:$tType,P:(((product_prod @ A) @ B) > $o),Prod:((product_prod @ A) @ B)]: (((! [A5:A,B7:B]: ((P @ ((((product_Pair @ A) @ B) @ A5) @ B7)))) => (P @ Prod))))).
% 2.48/1.49 thf(fact_198_bot__empty__eq, axiom, (! [A:$tType]: (((bot_bot @ (A > $o)) = (^ [X4:A]: ((((member @ A) @ X4) @ (bot_bot @ (set @ A))))))))).
% 2.48/1.49 thf(fact_199_old_Oprod_Orec, axiom, (! [A:$tType,T:$tType,B:$tType,F1:(A > (B > T)),A2:A,B4:B]: (((((((product_rec_prod @ A) @ B) @ T) @ F1) @ ((((product_Pair @ A) @ B) @ A2) @ B4)) = ((F1 @ A2) @ B4))))).
% 2.48/1.49 thf(fact_200_is__singleton__the__elem, axiom, (! [A:$tType]: (((is_singleton @ A) = (^ [A7:(set @ A)]: ((A7 = (((insert @ A) @ ((the_elem @ A) @ A7)) @ (bot_bot @ (set @ A)))))))))).
% 2.48/1.49 thf(fact_201_is__singletonI, axiom, (! [A:$tType,X3:A]: (((is_singleton @ A) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A))))))).
% 2.48/1.49 thf(fact_202_is__singletonI_H, axiom, (! [A:$tType,A3:(set @ A)]: (((A3 != (bot_bot @ (set @ A))) => ((! [X2:A,Y2:A]: (((((member @ A) @ X2) @ A3) => ((((member @ A) @ Y2) @ A3) => (X2 = Y2))))) => ((is_singleton @ A) @ A3)))))).
% 2.48/1.49 thf(fact_203_is__singleton__def, axiom, (! [A:$tType]: (((is_singleton @ A) = (^ [A7:(set @ A)]: ((? [X4:A]: ((A7 = (((insert @ A) @ X4) @ (bot_bot @ (set @ A)))))))))))).
% 2.48/1.49 thf(fact_204_is__singletonE, axiom, (! [A:$tType,A3:(set @ A)]: ((((is_singleton @ A) @ A3) => (~ ((! [X2:A]: ((A3 != (((insert @ A) @ X2) @ (bot_bot @ (set @ A)))))))))))).
% 2.48/1.49 thf(fact_205_insert__subsetI, axiom, (! [A:$tType,X3:A,A3:(set @ A),X5:(set @ A)]: (((((member @ A) @ X3) @ A3) => ((((ord_less_eq @ (set @ A)) @ X5) @ A3) => (((ord_less_eq @ (set @ A)) @ (((insert @ A) @ X3) @ X5)) @ A3)))))).
% 2.48/1.49 thf(fact_206_subset__emptyI, axiom, (! [A:$tType,A3:(set @ A)]: (((! [X2:A]: ((~ ((((member @ A) @ X2) @ A3))))) => (((ord_less_eq @ (set @ A)) @ A3) @ (bot_bot @ (set @ A))))))).
% 2.48/1.49 thf(fact_207_ssubst__Pair__rhs, axiom, (! [B:$tType,A:$tType,R:A,S2:B,R2:(set @ ((product_prod @ A) @ B)),S3:B]: (((((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ R) @ S2)) @ R2) => ((S3 = S2) => (((member @ ((product_prod @ A) @ B)) @ ((((product_Pair @ A) @ B) @ R) @ S3)) @ R2)))))).
% 2.48/1.49 thf(fact_208_lset__code, axiom, (! [A:$tType]: (((coinductive_lset @ A) = ((coinductive_gen_lset @ A) @ (bot_bot @ (set @ A))))))).
% 2.48/1.49 thf(fact_209_top__conj_I1_J, axiom, (! [A:$tType,X3:A,P:$o]: (((((top_top @ (A > $o)) @ X3) & P) = P)))).
% 2.48/1.49 thf(fact_210_top__conj_I2_J, axiom, (! [A:$tType,P:$o,X3:A]: (((P & ((top_top @ (A > $o)) @ X3)) = P)))).
% 2.48/1.49 thf(fact_211_prod__set__simps_I1_J, axiom, (! [B:$tType,A:$tType,X3:A,Y3:B]: (((((basic_fsts @ A) @ B) @ ((((product_Pair @ A) @ B) @ X3) @ Y3)) = (((insert @ A) @ X3) @ (bot_bot @ (set @ A))))))).
% 2.48/1.49 thf(fact_212_prod__set__simps_I2_J, axiom, (! [A:$tType,B:$tType,X3:A,Y3:B]: (((((basic_snds @ A) @ B) @ ((((product_Pair @ A) @ B) @ X3) @ Y3)) = (((insert @ B) @ Y3) @ (bot_bot @ (set @ B))))))).
% 2.48/1.49 thf(fact_213_Set_Ois__empty__def, axiom, (! [A:$tType]: (((is_empty @ A) = (^ [A7:(set @ A)]: ((A7 = (bot_bot @ (set @ A))))))))).
% 2.48/1.49 thf(fact_214_internal__case__prod__conv, axiom, (! [B:$tType,A:$tType,C:$tType,C2:(B > (C > A)),A2:B,B4:C]: (((((((produc2004651681e_prod @ B) @ C) @ A) @ C2) @ ((((product_Pair @ B) @ C) @ A2) @ B4)) = ((C2 @ A2) @ B4))))).
% 2.48/1.49 thf(fact_215_finite__remove__induct, axiom, (! [A:$tType,B3:(set @ A),P:((set @ A) > $o)]: ((((finite_finite2 @ A) @ B3) => ((P @ (bot_bot @ (set @ A))) => ((! [A4:(set @ A)]: ((((finite_finite2 @ A) @ A4) => ((A4 != (bot_bot @ (set @ A))) => ((((ord_less_eq @ (set @ A)) @ A4) @ B3) => ((! [X6:A]: (((((member @ A) @ X6) @ A4) => (P @ (((minus_minus @ (set @ A)) @ A4) @ (((insert @ A) @ X6) @ (bot_bot @ (set @ A)))))))) => (P @ A4))))))) => (P @ B3))))))).
% 2.48/1.49 thf(fact_216_remove__induct, axiom, (! [A:$tType,P:((set @ A) > $o),B3:(set @ A)]: (((P @ (bot_bot @ (set @ A))) => (((~ (((finite_finite2 @ A) @ B3))) => (P @ B3)) => ((! [A4:(set @ A)]: ((((finite_finite2 @ A) @ A4) => ((A4 != (bot_bot @ (set @ A))) => ((((ord_less_eq @ (set @ A)) @ A4) @ B3) => ((! [X6:A]: (((((member @ A) @ X6) @ A4) => (P @ (((minus_minus @ (set @ A)) @ A4) @ (((insert @ A) @ X6) @ (bot_bot @ (set @ A)))))))) => (P @ A4))))))) => (P @ B3))))))).
% 2.48/1.49 thf(fact_217_minus__apply, axiom, (! [B:$tType,A:$tType]: ((((minus @ B) @ (type2 @ B)) => ((minus_minus @ (A > B)) = (^ [A7:(A > B),B5:(A > B),X4:A]: ((((minus_minus @ B) @ (A7 @ X4)) @ (B5 @ X4))))))))).
% 2.48/1.49 thf(fact_218_DiffI, axiom, (! [A:$tType,C2:A,A3:(set @ A),B3:(set @ A)]: (((((member @ A) @ C2) @ A3) => ((~ ((((member @ A) @ C2) @ B3))) => (((member @ A) @ C2) @ (((minus_minus @ (set @ A)) @ A3) @ B3))))))).
% 2.48/1.49 thf(fact_219_Diff__iff, axiom, (! [A:$tType,C2:A,A3:(set @ A),B3:(set @ A)]: (((((member @ A) @ C2) @ (((minus_minus @ (set @ A)) @ A3) @ B3)) = ((((member @ A) @ C2) @ A3) & (~ ((((member @ A) @ C2) @ B3)))))))).
% 2.48/1.49 thf(fact_220_Diff__idemp, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: (((((minus_minus @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A3) @ B3)) @ B3) = (((minus_minus @ (set @ A)) @ A3) @ B3))))).
% 2.48/1.49 thf(fact_221_minus__diff__eq, axiom, (! [A:$tType]: ((((group_add @ A) @ (type2 @ A)) => (! [A2:A,B4:A]: ((((uminus_uminus @ A) @ (((minus_minus @ A) @ A2) @ B4)) = (((minus_minus @ A) @ B4) @ A2)))))))).
% 2.48/1.49 thf(fact_222_Diff__empty, axiom, (! [A:$tType,A3:(set @ A)]: (((((minus_minus @ (set @ A)) @ A3) @ (bot_bot @ (set @ A))) = A3)))).
% 2.48/1.49 thf(fact_223_empty__Diff, axiom, (! [A:$tType,A3:(set @ A)]: (((((minus_minus @ (set @ A)) @ (bot_bot @ (set @ A))) @ A3) = (bot_bot @ (set @ A)))))).
% 2.48/1.49 thf(fact_224_Diff__cancel, axiom, (! [A:$tType,A3:(set @ A)]: (((((minus_minus @ (set @ A)) @ A3) @ A3) = (bot_bot @ (set @ A)))))).
% 2.48/1.49 thf(fact_225_finite__Diff2, axiom, (! [A:$tType,B3:(set @ A),A3:(set @ A)]: ((((finite_finite2 @ A) @ B3) => (((finite_finite2 @ A) @ (((minus_minus @ (set @ A)) @ A3) @ B3)) = ((finite_finite2 @ A) @ A3)))))).
% 2.48/1.49 thf(fact_226_finite__Diff, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: ((((finite_finite2 @ A) @ A3) => ((finite_finite2 @ A) @ (((minus_minus @ (set @ A)) @ A3) @ B3)))))).
% 2.48/1.49 thf(fact_227_insert__Diff1, axiom, (! [A:$tType,X3:A,B3:(set @ A),A3:(set @ A)]: (((((member @ A) @ X3) @ B3) => ((((minus_minus @ (set @ A)) @ (((insert @ A) @ X3) @ A3)) @ B3) = (((minus_minus @ (set @ A)) @ A3) @ B3)))))).
% 2.48/1.49 thf(fact_228_Diff__insert0, axiom, (! [A:$tType,X3:A,A3:(set @ A),B3:(set @ A)]: (((~ ((((member @ A) @ X3) @ A3))) => ((((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ X3) @ B3)) = (((minus_minus @ (set @ A)) @ A3) @ B3)))))).
% 2.48/1.49 thf(fact_229_Diff__UNIV, axiom, (! [A:$tType,A3:(set @ A)]: (((((minus_minus @ (set @ A)) @ A3) @ (top_top @ (set @ A))) = (bot_bot @ (set @ A)))))).
% 2.48/1.49 thf(fact_230_Diff__eq__empty__iff, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: ((((((minus_minus @ (set @ A)) @ A3) @ B3) = (bot_bot @ (set @ A))) = (((ord_less_eq @ (set @ A)) @ A3) @ B3))))).
% 2.48/1.49 thf(fact_231_insert__Diff__single, axiom, (! [A:$tType,A2:A,A3:(set @ A)]: (((((insert @ A) @ A2) @ (((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))) = (((insert @ A) @ A2) @ A3))))).
% 2.48/1.49 thf(fact_232_finite__Diff__insert, axiom, (! [A:$tType,A3:(set @ A),A2:A,B3:(set @ A)]: ((((finite_finite2 @ A) @ (((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ A2) @ B3))) = ((finite_finite2 @ A) @ (((minus_minus @ (set @ A)) @ A3) @ B3)))))).
% 2.48/1.49 thf(fact_233_Diff__mono, axiom, (! [A:$tType,A3:(set @ A),C3:(set @ A),D:(set @ A),B3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ C3) => ((((ord_less_eq @ (set @ A)) @ D) @ B3) => (((ord_less_eq @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A3) @ B3)) @ (((minus_minus @ (set @ A)) @ C3) @ D))))))).
% 2.48/1.49 thf(fact_234_Diff__subset, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A3) @ B3)) @ A3)))).
% 2.48/1.49 thf(fact_235_double__diff, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),C3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ B3) => ((((ord_less_eq @ (set @ A)) @ B3) @ C3) => ((((minus_minus @ (set @ A)) @ B3) @ (((minus_minus @ (set @ A)) @ C3) @ A3)) = A3)))))).
% 2.48/1.49 thf(fact_236_diff__mono, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [A2:A,B4:A,D2:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B4) => ((((ord_less_eq @ A) @ D2) @ C2) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ A2) @ C2)) @ (((minus_minus @ A) @ B4) @ D2)))))))))).
% 2.48/1.49 thf(fact_237_diff__left__mono, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [B4:A,A2:A,C2:A]: (((((ord_less_eq @ A) @ B4) @ A2) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ C2) @ A2)) @ (((minus_minus @ A) @ C2) @ B4))))))))).
% 2.48/1.49 thf(fact_238_diff__right__mono, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [A2:A,B4:A,C2:A]: (((((ord_less_eq @ A) @ A2) @ B4) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ A2) @ C2)) @ (((minus_minus @ A) @ B4) @ C2))))))))).
% 2.48/1.49 thf(fact_239_diff__eq__diff__less__eq, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [A2:A,B4:A,C2:A,D2:A]: ((((((minus_minus @ A) @ A2) @ B4) = (((minus_minus @ A) @ C2) @ D2)) => ((((ord_less_eq @ A) @ A2) @ B4) = (((ord_less_eq @ A) @ C2) @ D2))))))))).
% 2.48/1.49 thf(fact_240_subset__Diff__insert, axiom, (! [A:$tType,A3:(set @ A),B3:(set @ A),X3:A,C3:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A3) @ (((minus_minus @ (set @ A)) @ B3) @ (((insert @ A) @ X3) @ C3))) = ((((ord_less_eq @ (set @ A)) @ A3) @ (((minus_minus @ (set @ A)) @ B3) @ C3)) & (~ ((((member @ A) @ X3) @ A3)))))))).
% 2.48/1.49 thf(fact_241_Compl__eq__Diff__UNIV, axiom, (! [A:$tType]: (((uminus_uminus @ (set @ A)) = ((minus_minus @ (set @ A)) @ (top_top @ (set @ A))))))).
% 2.48/1.49 thf(fact_242_DiffE, axiom, (! [A:$tType,C2:A,A3:(set @ A),B3:(set @ A)]: (((((member @ A) @ C2) @ (((minus_minus @ (set @ A)) @ A3) @ B3)) => (~ (((((member @ A) @ C2) @ A3) => (((member @ A) @ C2) @ B3)))))))).
% 2.48/1.49 thf(fact_243_DiffD1, axiom, (! [A:$tType,C2:A,A3:(set @ A),B3:(set @ A)]: (((((member @ A) @ C2) @ (((minus_minus @ (set @ A)) @ A3) @ B3)) => (((member @ A) @ C2) @ A3))))).
% 2.48/1.49 thf(fact_244_DiffD2, axiom, (! [A:$tType,C2:A,A3:(set @ A),B3:(set @ A)]: (((((member @ A) @ C2) @ (((minus_minus @ (set @ A)) @ A3) @ B3)) => (~ ((((member @ A) @ C2) @ B3))))))).
% 2.48/1.49 thf(fact_245_insert__Diff__if, axiom, (! [A:$tType,X3:A,B3:(set @ A),A3:(set @ A)]: ((((((member @ A) @ X3) @ B3) => ((((minus_minus @ (set @ A)) @ (((insert @ A) @ X3) @ A3)) @ B3) = (((minus_minus @ (set @ A)) @ A3) @ B3))) & ((~ ((((member @ A) @ X3) @ B3))) => ((((minus_minus @ (set @ A)) @ (((insert @ A) @ X3) @ A3)) @ B3) = (((insert @ A) @ X3) @ (((minus_minus @ (set @ A)) @ A3) @ B3)))))))).
% 2.48/1.49 thf(fact_246_Diff__infinite__finite, axiom, (! [A:$tType,T3:(set @ A),S:(set @ A)]: ((((finite_finite2 @ A) @ T3) => ((~ (((finite_finite2 @ A) @ S))) => (~ (((finite_finite2 @ A) @ (((minus_minus @ (set @ A)) @ S) @ T3))))))))).
% 2.48/1.49 thf(fact_247_fun__diff__def, axiom, (! [B:$tType,A:$tType]: ((((minus @ B) @ (type2 @ B)) => ((minus_minus @ (A > B)) = (^ [A7:(A > B),B5:(A > B),X4:A]: ((((minus_minus @ B) @ (A7 @ X4)) @ (B5 @ X4))))))))).
% 2.48/1.49 thf(fact_248_diff__eq__diff__eq, axiom, (! [A:$tType]: ((((group_add @ A) @ (type2 @ A)) => (! [A2:A,B4:A,C2:A,D2:A]: ((((((minus_minus @ A) @ A2) @ B4) = (((minus_minus @ A) @ C2) @ D2)) => ((A2 = B4) = (C2 = D2))))))))).
% 2.48/1.49 thf(fact_249_diff__right__commute, axiom, (! [A:$tType]: ((((cancel146912293up_add @ A) @ (type2 @ A)) => (! [A2:A,C2:A,B4:A]: (((((minus_minus @ A) @ (((minus_minus @ A) @ A2) @ C2)) @ B4) = (((minus_minus @ A) @ (((minus_minus @ A) @ A2) @ B4)) @ C2)))))))).
% 2.48/1.49 thf(fact_250_Diff__insert__absorb, axiom, (! [A:$tType,X3:A,A3:(set @ A)]: (((~ ((((member @ A) @ X3) @ A3))) => ((((minus_minus @ (set @ A)) @ (((insert @ A) @ X3) @ A3)) @ (((insert @ A) @ X3) @ (bot_bot @ (set @ A)))) = A3))))).
% 2.48/1.49 thf(fact_251_Diff__insert2, axiom, (! [A:$tType,A3:(set @ A),A2:A,B3:(set @ A)]: (((((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ A2) @ B3)) = (((minus_minus @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))) @ B3))))).
% 2.48/1.49 thf(fact_252_insert__Diff, axiom, (! [A:$tType,A2:A,A3:(set @ A)]: (((((member @ A) @ A2) @ A3) => ((((insert @ A) @ A2) @ (((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))) = A3))))).
% 2.48/1.49 thf(fact_253_Diff__insert, axiom, (! [A:$tType,A3:(set @ A),A2:A,B3:(set @ A)]: (((((minus_minus @ (set @ A)) @ A3) @ (((insert @ A) @ A2) @ B3)) = (((minus_minus @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A3) @ B3)) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A)))))))).
% 2.48/1.49 thf(tcon_fun___Lattices_Oboolean__algebra, axiom, (! [A9:$tType,A10:$tType]: ((((boolean_algebra @ A10) @ (type2 @ A10)) => ((boolean_algebra @ (A9 > A10)) @ (type2 @ (A9 > A10))))))).
% 2.48/1.49 thf(tcon_fun___Orderings_Oorder__top, axiom, (! [A9:$tType,A10:$tType]: ((((order_top @ A10) @ (type2 @ A10)) => ((order_top @ (A9 > A10)) @ (type2 @ (A9 > A10))))))).
% 2.48/1.49 thf(tcon_fun___Orderings_Oorder__bot, axiom, (! [A9:$tType,A10:$tType]: ((((order_bot @ A10) @ (type2 @ A10)) => ((order_bot @ (A9 > A10)) @ (type2 @ (A9 > A10))))))).
% 2.48/1.49 thf(tcon_fun___Orderings_Opreorder, axiom, (! [A9:$tType,A10:$tType]: ((((preorder @ A10) @ (type2 @ A10)) => ((preorder @ (A9 > A10)) @ (type2 @ (A9 > A10))))))).
% 2.48/1.49 thf(tcon_fun___Finite__Set_Ofinite, axiom, (! [A9:$tType,A10:$tType]: (((((finite_finite @ A9) @ (type2 @ A9)) & ((finite_finite @ A10) @ (type2 @ A10))) => ((finite_finite @ (A9 > A10)) @ (type2 @ (A9 > A10))))))).
% 2.48/1.49 thf(tcon_fun___Orderings_Oorder, axiom, (! [A9:$tType,A10:$tType]: ((((order @ A10) @ (type2 @ A10)) => ((order @ (A9 > A10)) @ (type2 @ (A9 > A10))))))).
% 2.48/1.49 thf(tcon_fun___Orderings_Otop, axiom, (! [A9:$tType,A10:$tType]: ((((top @ A10) @ (type2 @ A10)) => ((top @ (A9 > A10)) @ (type2 @ (A9 > A10))))))).
% 2.48/1.49 thf(tcon_fun___Orderings_Oord, axiom, (! [A9:$tType,A10:$tType]: ((((ord @ A10) @ (type2 @ A10)) => ((ord @ (A9 > A10)) @ (type2 @ (A9 > A10))))))).
% 2.48/1.49 thf(tcon_fun___Orderings_Obot, axiom, (! [A9:$tType,A10:$tType]: ((((bot @ A10) @ (type2 @ A10)) => ((bot @ (A9 > A10)) @ (type2 @ (A9 > A10))))))).
% 2.48/1.49 thf(tcon_fun___Groups_Ouminus, axiom, (! [A9:$tType,A10:$tType]: ((((uminus @ A10) @ (type2 @ A10)) => ((uminus @ (A9 > A10)) @ (type2 @ (A9 > A10))))))).
% 2.48/1.49 thf(tcon_fun___Groups_Ominus, axiom, (! [A9:$tType,A10:$tType]: ((((minus @ A10) @ (type2 @ A10)) => ((minus @ (A9 > A10)) @ (type2 @ (A9 > A10))))))).
% 2.48/1.49 thf(tcon_Set_Oset___Lattices_Oboolean__algebra_1, axiom, (! [A9:$tType]: (((boolean_algebra @ (set @ A9)) @ (type2 @ (set @ A9)))))).
% 2.48/1.49 thf(tcon_Set_Oset___Orderings_Oorder__top_2, axiom, (! [A9:$tType]: (((order_top @ (set @ A9)) @ (type2 @ (set @ A9)))))).
% 2.48/1.49 thf(tcon_Set_Oset___Orderings_Oorder__bot_3, axiom, (! [A9:$tType]: (((order_bot @ (set @ A9)) @ (type2 @ (set @ A9)))))).
% 2.48/1.49 thf(tcon_Set_Oset___Orderings_Opreorder_4, axiom, (! [A9:$tType]: (((preorder @ (set @ A9)) @ (type2 @ (set @ A9)))))).
% 2.48/1.49 thf(tcon_Set_Oset___Finite__Set_Ofinite_5, axiom, (! [A9:$tType]: ((((finite_finite @ A9) @ (type2 @ A9)) => ((finite_finite @ (set @ A9)) @ (type2 @ (set @ A9))))))).
% 2.48/1.49 thf(tcon_Set_Oset___Orderings_Oorder_6, axiom, (! [A9:$tType]: (((order @ (set @ A9)) @ (type2 @ (set @ A9)))))).
% 2.48/1.49 thf(tcon_Set_Oset___Orderings_Otop_7, axiom, (! [A9:$tType]: (((top @ (set @ A9)) @ (type2 @ (set @ A9)))))).
% 2.48/1.49 thf(tcon_Set_Oset___Orderings_Oord_8, axiom, (! [A9:$tType]: (((ord @ (set @ A9)) @ (type2 @ (set @ A9)))))).
% 2.48/1.49 thf(tcon_Set_Oset___Orderings_Obot_9, axiom, (! [A9:$tType]: (((bot @ (set @ A9)) @ (type2 @ (set @ A9)))))).
% 2.48/1.49 thf(tcon_Set_Oset___Groups_Ouminus_10, axiom, (! [A9:$tType]: (((uminus @ (set @ A9)) @ (type2 @ (set @ A9)))))).
% 2.48/1.49 thf(tcon_Set_Oset___Groups_Ominus_11, axiom, (! [A9:$tType]: (((minus @ (set @ A9)) @ (type2 @ (set @ A9)))))).
% 2.48/1.49 thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_12, axiom, ((boolean_algebra @ $o) @ (type2 @ $o))).
% 2.48/1.49 thf(tcon_HOL_Obool___Orderings_Oorder__top_13, axiom, ((order_top @ $o) @ (type2 @ $o))).
% 2.48/1.49 thf(tcon_HOL_Obool___Orderings_Oorder__bot_14, axiom, ((order_bot @ $o) @ (type2 @ $o))).
% 2.48/1.49 thf(tcon_HOL_Obool___Orderings_Opreorder_15, axiom, ((preorder @ $o) @ (type2 @ $o))).
% 2.48/1.49 thf(tcon_HOL_Obool___Orderings_Olinorder, axiom, ((linorder @ $o) @ (type2 @ $o))).
% 2.48/1.49 thf(tcon_HOL_Obool___Finite__Set_Ofinite_16, axiom, ((finite_finite @ $o) @ (type2 @ $o))).
% 2.48/1.49 thf(tcon_HOL_Obool___Orderings_Oorder_17, axiom, ((order @ $o) @ (type2 @ $o))).
% 2.48/1.49 thf(tcon_HOL_Obool___Orderings_Otop_18, axiom, ((top @ $o) @ (type2 @ $o))).
% 2.48/1.49 thf(tcon_HOL_Obool___Orderings_Oord_19, axiom, ((ord @ $o) @ (type2 @ $o))).
% 2.48/1.49 thf(tcon_HOL_Obool___Orderings_Obot_20, axiom, ((bot @ $o) @ (type2 @ $o))).
% 2.48/1.49 thf(tcon_HOL_Obool___Groups_Ouminus_21, axiom, ((uminus @ $o) @ (type2 @ $o))).
% 2.48/1.49 thf(tcon_HOL_Obool___Groups_Ominus_22, axiom, ((minus @ $o) @ (type2 @ $o))).
% 2.48/1.49 thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_23, axiom, (! [A9:$tType,A10:$tType]: (((((finite_finite @ A9) @ (type2 @ A9)) & ((finite_finite @ A10) @ (type2 @ A10))) => ((finite_finite @ ((sum_sum @ A9) @ A10)) @ (type2 @ ((sum_sum @ A9) @ A10))))))).
% 2.48/1.49 thf(tcon_Option_Ooption___Finite__Set_Ofinite_24, axiom, (! [A9:$tType]: ((((finite_finite @ A9) @ (type2 @ A9)) => ((finite_finite @ (option @ A9)) @ (type2 @ (option @ A9))))))).
% 2.48/1.49 thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_25, axiom, (! [A9:$tType,A10:$tType]: (((((finite_finite @ A9) @ (type2 @ A9)) & ((finite_finite @ A10) @ (type2 @ A10))) => ((finite_finite @ ((product_prod @ A9) @ A10)) @ (type2 @ ((product_prod @ A9) @ A10))))))).
% 2.48/1.49 thf(conj_0, conjecture, (? [N:node,Ns:(set @ node)]: ((((f @ ((((product_Pair @ node) @ (set @ node)) @ n) @ (bot_bot @ (set @ node)))) = (f @ ((((product_Pair @ node) @ (set @ node)) @ N) @ Ns))) & (((finite_finite2 @ node) @ Ns) & (~ (((finite_finite2 @ node) @ ((((koenig317145564le_via @ node) @ graph) @ ((uminus_uminus @ (set @ node)) @ (((insert @ node) @ N) @ Ns))) @ N))))))))).
% 2.48/1.49 % SZS output end ListOfTHF for /export/starexec/sandbox2/tmp/tmp.sjZ2TfJLxv/DTF2DTF22201.p
% 2.48/1.49 ------------------------
% 2.69/1.51 ---- Cleaned THF ---
% 2.69/1.52 thf(ty_t_Coinductive__List_Ollist,type,
% 2.69/1.52 coinductive_llist: $tType > $tType ).
% 2.69/1.52
% 2.69/1.52 thf(ty_t_Product__Type_Oprod,type,
% 2.69/1.52 product_prod: $tType > $tType > $tType ).
% 2.69/1.52
% 2.69/1.52 thf(ty_t_Sum__Type_Osum,type,
% 2.69/1.52 sum_sum: $tType > $tType > $tType ).
% 2.69/1.52
% 2.69/1.52 thf(ty_t_Option_Ooption,type,
% 2.69/1.52 option: $tType > $tType ).
% 2.69/1.52
% 2.69/1.52 thf(ty_t_Set_Oset,type,
% 2.69/1.52 set: $tType > $tType ).
% 2.69/1.52
% 2.69/1.52 thf(ty_t_itself,type,
% 2.69/1.52 itself: $tType > $tType ).
% 2.69/1.52
% 2.69/1.52 thf(ty_tf_node,type,
% 2.69/1.52 node: $tType ).
% 2.69/1.52
% 2.69/1.52 thf(sy_cl_HOL_Otype,type,
% 2.69/1.52 type:
% 2.69/1.52 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.69/1.52
% 2.69/1.52 thf(sy_cl_Groups_Ominus,type,
% 2.69/1.52 minus:
% 2.69/1.52 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.69/1.52
% 2.69/1.52 thf(sy_cl_Groups_Ouminus,type,
% 2.69/1.52 uminus:
% 2.69/1.52 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.69/1.52
% 2.69/1.52 thf(sy_cl_Orderings_Obot,type,
% 2.69/1.52 bot:
% 2.69/1.52 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.69/1.52
% 2.69/1.52 thf(sy_cl_Orderings_Oord,type,
% 2.69/1.52 ord:
% 2.69/1.52 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.69/1.52
% 2.69/1.52 thf(sy_cl_Orderings_Otop,type,
% 2.69/1.52 top:
% 2.69/1.52 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.69/1.52
% 2.69/1.52 thf(sy_cl_Orderings_Oorder,type,
% 2.69/1.52 order:
% 2.69/1.52 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.69/1.52
% 2.69/1.52 thf(sy_cl_Groups_Ogroup__add,type,
% 2.69/1.52 group_add:
% 2.69/1.52 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.69/1.52
% 2.69/1.52 thf(sy_cl_Finite__Set_Ofinite,type,
% 2.69/1.52 finite_finite:
% 2.69/1.52 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.69/1.52
% 2.69/1.52 thf(sy_cl_Orderings_Olinorder,type,
% 2.69/1.52 linorder:
% 2.69/1.52 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.69/1.52
% 2.69/1.52 thf(sy_cl_Orderings_Opreorder,type,
% 2.69/1.52 preorder:
% 2.69/1.52 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.69/1.52
% 2.69/1.52 thf(sy_cl_Nat_Osemiring__char__0,type,
% 2.69/1.52 semiring_char_0:
% 2.69/1.52 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.69/1.52
% 2.69/1.52 thf(sy_cl_Orderings_Oorder__bot,type,
% 2.69/1.52 order_bot:
% 2.69/1.52 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.69/1.52
% 2.69/1.52 thf(sy_cl_Orderings_Oorder__top,type,
% 2.69/1.52 order_top:
% 2.69/1.52 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.69/1.52
% 2.69/1.52 thf(sy_cl_Lattices_Oboolean__algebra,type,
% 2.69/1.52 boolean_algebra:
% 2.69/1.52 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.69/1.52
% 2.69/1.52 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 2.69/1.52 ordered_ab_group_add:
% 2.69/1.52 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.69/1.52
% 2.69/1.52 thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
% 2.69/1.52 cancel146912293up_add:
% 2.69/1.52 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.69/1.52
% 2.69/1.52 thf(sy_cl_Topological__Spaces_Operfect__space,type,
% 2.69/1.52 topolo890362671_space:
% 2.69/1.52 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.69/1.52
% 2.69/1.52 thf(sy_c_Basic__BNFs_Ofsts,type,
% 2.69/1.52 basic_fsts:
% 2.69/1.52 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( set @ A ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(sy_c_Basic__BNFs_Osnds,type,
% 2.69/1.52 basic_snds:
% 2.69/1.52 !>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( set @ B ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(sy_c_Coinductive__List_Ogen__lset,type,
% 2.69/1.52 coinductive_gen_lset:
% 2.69/1.52 !>[A: $tType] : ( ( set @ A ) > ( coinductive_llist @ A ) > ( set @ A ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(sy_c_Coinductive__List_Olfinite,type,
% 2.69/1.52 coinductive_lfinite:
% 2.69/1.52 !>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
% 2.69/1.52
% 2.69/1.52 thf(sy_c_Coinductive__List_Ollist_Olnull,type,
% 2.69/1.52 coinductive_lnull:
% 2.69/1.52 !>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
% 2.69/1.52
% 2.69/1.52 thf(sy_c_Coinductive__List_Ollist_Olset,type,
% 2.69/1.52 coinductive_lset:
% 2.69/1.52 !>[A: $tType] : ( ( coinductive_llist @ A ) > ( set @ A ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(sy_c_Finite__Set_Ofinite,type,
% 2.69/1.52 finite_finite2:
% 2.69/1.52 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 2.69/1.52
% 2.69/1.52 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 2.69/1.52 minus_minus:
% 2.69/1.52 !>[A: $tType] : ( A > A > A ) ).
% 2.69/1.52
% 2.69/1.52 thf(sy_c_Groups_Ouminus__class_Ouminus,type,
% 2.69/1.52 uminus_uminus:
% 2.69/1.52 !>[A: $tType] : ( A > A ) ).
% 2.69/1.52
% 2.69/1.52 thf(sy_c_Koenigslemma__Mirabelle__aepjeeakgn_Oconnected,type,
% 2.69/1.52 koenig793108494nected:
% 2.69/1.52 !>[Node: $tType] : ( ( Node > Node > $o ) > $o ) ).
% 2.69/1.52
% 2.69/1.52 thf(sy_c_Koenigslemma__Mirabelle__aepjeeakgn_Oreachable__via,type,
% 2.69/1.52 koenig317145564le_via:
% 2.69/1.52 !>[Node: $tType] : ( ( Node > Node > $o ) > ( set @ Node ) > Node > ( set @ Node ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(sy_c_Orderings_Obot__class_Obot,type,
% 2.69/1.52 bot_bot:
% 2.69/1.52 !>[A: $tType] : A ).
% 2.69/1.52
% 2.69/1.52 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.69/1.52 ord_less_eq:
% 2.69/1.52 !>[A: $tType] : ( A > A > $o ) ).
% 2.69/1.52
% 2.69/1.52 thf(sy_c_Orderings_Otop__class_Otop,type,
% 2.69/1.52 top_top:
% 2.69/1.52 !>[A: $tType] : A ).
% 2.69/1.52
% 2.69/1.52 thf(sy_c_Product__Type_OPair,type,
% 2.69/1.52 product_Pair:
% 2.69/1.52 !>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(sy_c_Product__Type_Ointernal__case__prod,type,
% 2.69/1.52 produc2004651681e_prod:
% 2.69/1.52 !>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
% 2.69/1.52
% 2.69/1.52 thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
% 2.69/1.52 product_rec_prod:
% 2.69/1.52 !>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
% 2.69/1.52
% 2.69/1.52 thf(sy_c_Pure_Otype,type,
% 2.69/1.52 type2:
% 2.69/1.52 !>[A: $tType] : ( itself @ A ) ).
% 2.69/1.52
% 2.69/1.52 thf(sy_c_Set_OCollect,type,
% 2.69/1.52 collect:
% 2.69/1.52 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(sy_c_Set_Oinsert,type,
% 2.69/1.52 insert:
% 2.69/1.52 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(sy_c_Set_Ois__empty,type,
% 2.69/1.52 is_empty:
% 2.69/1.52 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 2.69/1.52
% 2.69/1.52 thf(sy_c_Set_Ois__singleton,type,
% 2.69/1.52 is_singleton:
% 2.69/1.52 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 2.69/1.52
% 2.69/1.52 thf(sy_c_Set_Othe__elem,type,
% 2.69/1.52 the_elem:
% 2.69/1.52 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 2.69/1.52
% 2.69/1.52 thf(sy_c_member,type,
% 2.69/1.52 member:
% 2.69/1.52 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.69/1.52
% 2.69/1.52 thf(sy_v_f____,type,
% 2.69/1.52 f: ( product_prod @ node @ ( set @ node ) ) > ( coinductive_llist @ node ) ).
% 2.69/1.52
% 2.69/1.52 thf(sy_v_graph,type,
% 2.69/1.52 graph: node > node > $o ).
% 2.69/1.52
% 2.69/1.52 thf(sy_v_n,type,
% 2.69/1.52 n: node ).
% 2.69/1.52
% 2.69/1.52 thf(sy_v_ns____,type,
% 2.69/1.52 ns: set @ node ).
% 2.69/1.52
% 2.69/1.52 thf(fact_0_connected,axiom,
% 2.69/1.52 koenig793108494nected @ node @ graph ).
% 2.69/1.52
% 2.69/1.52 thf(fact_1_ns__def,axiom,
% 2.69/1.52 ( ns
% 2.69/1.52 = ( bot_bot @ ( set @ node ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_2__092_060open_062infinite_A_Ireachable__via_Agraph_A_I_N_A_123n_125_J_An_J_092_060close_062,axiom,
% 2.69/1.52 ~ ( finite_finite2 @ node @ ( koenig317145564le_via @ node @ graph @ ( uminus_uminus @ ( set @ node ) @ ( insert @ node @ n @ ( bot_bot @ ( set @ node ) ) ) ) @ n ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_3_infinite,axiom,
% 2.69/1.52 ~ ( finite_finite2 @ node @ ( top_top @ ( set @ node ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_4__092_060open_062_N_A_123n_125_A_092_060subseteq_062_Areachable__via_Agraph_A_I_N_A_123n_125_J_An_092_060close_062,axiom,
% 2.69/1.52 ord_less_eq @ ( set @ node ) @ ( uminus_uminus @ ( set @ node ) @ ( insert @ node @ n @ ( bot_bot @ ( set @ node ) ) ) ) @ ( koenig317145564le_via @ node @ graph @ ( uminus_uminus @ ( set @ node ) @ ( insert @ node @ n @ ( bot_bot @ ( set @ node ) ) ) ) @ n ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_5_lset,axiom,
% 2.69/1.52 ! [X: node,Na: node,Nsa: set @ node] :
% 2.69/1.52 ( ( member @ node @ X @ ( coinductive_lset @ node @ ( f @ ( product_Pair @ node @ ( set @ node ) @ Na @ Nsa ) ) ) )
% 2.69/1.52 => ( ~ ( member @ node @ Na @ Nsa )
% 2.69/1.52 => ( ( finite_finite2 @ node @ Nsa )
% 2.69/1.52 => ( ~ ( finite_finite2 @ node @ ( koenig317145564le_via @ node @ graph @ ( uminus_uminus @ ( set @ node ) @ ( insert @ node @ Na @ Nsa ) ) @ Na ) )
% 2.69/1.52 => ~ ( member @ node @ X @ Nsa ) ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_6_finite__insert,axiom,
% 2.69/1.52 ! [A: $tType,A2: A,A3: set @ A] :
% 2.69/1.52 ( ( finite_finite2 @ A @ ( insert @ A @ A2 @ A3 ) )
% 2.69/1.52 = ( finite_finite2 @ A @ A3 ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_7_singletonI,axiom,
% 2.69/1.52 ! [A: $tType,A2: A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_8_finite_Ocases,axiom,
% 2.69/1.52 ! [A: $tType,A2: set @ A] :
% 2.69/1.52 ( ( finite_finite2 @ A @ A2 )
% 2.69/1.52 => ( ( A2
% 2.69/1.52 != ( bot_bot @ ( set @ A ) ) )
% 2.69/1.52 => ~ ! [A4: set @ A] :
% 2.69/1.52 ( ? [A5: A] :
% 2.69/1.52 ( A2
% 2.69/1.52 = ( insert @ A @ A5 @ A4 ) )
% 2.69/1.52 => ~ ( finite_finite2 @ A @ A4 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_9_finite_Osimps,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( finite_finite2 @ A )
% 2.69/1.52 = ( ^ [A6: set @ A] :
% 2.69/1.52 ( ( A6
% 2.69/1.52 = ( bot_bot @ ( set @ A ) ) )
% 2.69/1.52 | ? [A7: set @ A,B2: A] :
% 2.69/1.52 ( ( A6
% 2.69/1.52 = ( insert @ A @ B2 @ A7 ) )
% 2.69/1.52 & ( finite_finite2 @ A @ A7 ) ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_10_finite__induct,axiom,
% 2.69/1.52 ! [A: $tType,F: set @ A,P: ( set @ A ) > $o] :
% 2.69/1.52 ( ( finite_finite2 @ A @ F )
% 2.69/1.52 => ( ( P @ ( bot_bot @ ( set @ A ) ) )
% 2.69/1.52 => ( ! [X2: A,F2: set @ A] :
% 2.69/1.52 ( ( finite_finite2 @ A @ F2 )
% 2.69/1.52 => ( ~ ( member @ A @ X2 @ F2 )
% 2.69/1.52 => ( ( P @ F2 )
% 2.69/1.52 => ( P @ ( insert @ A @ X2 @ F2 ) ) ) ) )
% 2.69/1.52 => ( P @ F ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_11_finite_Oinducts,axiom,
% 2.69/1.52 ! [A: $tType,X3: set @ A,P: ( set @ A ) > $o] :
% 2.69/1.52 ( ( finite_finite2 @ A @ X3 )
% 2.69/1.52 => ( ( P @ ( bot_bot @ ( set @ A ) ) )
% 2.69/1.52 => ( ! [A4: set @ A,A5: A] :
% 2.69/1.52 ( ( finite_finite2 @ A @ A4 )
% 2.69/1.52 => ( ( P @ A4 )
% 2.69/1.52 => ( P @ ( insert @ A @ A5 @ A4 ) ) ) )
% 2.69/1.52 => ( P @ X3 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_12_finite__ne__induct,axiom,
% 2.69/1.52 ! [A: $tType,F: set @ A,P: ( set @ A ) > $o] :
% 2.69/1.52 ( ( finite_finite2 @ A @ F )
% 2.69/1.52 => ( ( F
% 2.69/1.52 != ( bot_bot @ ( set @ A ) ) )
% 2.69/1.52 => ( ! [X2: A] : ( P @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.69/1.52 => ( ! [X2: A,F2: set @ A] :
% 2.69/1.52 ( ( finite_finite2 @ A @ F2 )
% 2.69/1.52 => ( ( F2
% 2.69/1.52 != ( bot_bot @ ( set @ A ) ) )
% 2.69/1.52 => ( ~ ( member @ A @ X2 @ F2 )
% 2.69/1.52 => ( ( P @ F2 )
% 2.69/1.52 => ( P @ ( insert @ A @ X2 @ F2 ) ) ) ) ) )
% 2.69/1.52 => ( P @ F ) ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_13_infinite__finite__induct,axiom,
% 2.69/1.52 ! [A: $tType,P: ( set @ A ) > $o,A3: set @ A] :
% 2.69/1.52 ( ! [A4: set @ A] :
% 2.69/1.52 ( ~ ( finite_finite2 @ A @ A4 )
% 2.69/1.52 => ( P @ A4 ) )
% 2.69/1.52 => ( ( P @ ( bot_bot @ ( set @ A ) ) )
% 2.69/1.52 => ( ! [X2: A,F2: set @ A] :
% 2.69/1.52 ( ( finite_finite2 @ A @ F2 )
% 2.69/1.52 => ( ~ ( member @ A @ X2 @ F2 )
% 2.69/1.52 => ( ( P @ F2 )
% 2.69/1.52 => ( P @ ( insert @ A @ X2 @ F2 ) ) ) ) )
% 2.69/1.52 => ( P @ A3 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_14_ComplI,axiom,
% 2.69/1.52 ! [A: $tType,C2: A,A3: set @ A] :
% 2.69/1.52 ( ~ ( member @ A @ C2 @ A3 )
% 2.69/1.52 => ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A3 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_15_Compl__iff,axiom,
% 2.69/1.52 ! [A: $tType,C2: A,A3: set @ A] :
% 2.69/1.52 ( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A3 ) )
% 2.69/1.52 = ( ~ ( member @ A @ C2 @ A3 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_16_empty__Collect__eq,axiom,
% 2.69/1.52 ! [A: $tType,P: A > $o] :
% 2.69/1.52 ( ( ( bot_bot @ ( set @ A ) )
% 2.69/1.52 = ( collect @ A @ P ) )
% 2.69/1.52 = ( ! [X4: A] :
% 2.69/1.52 ~ ( P @ X4 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_17_Collect__empty__eq,axiom,
% 2.69/1.52 ! [A: $tType,P: A > $o] :
% 2.69/1.52 ( ( ( collect @ A @ P )
% 2.69/1.52 = ( bot_bot @ ( set @ A ) ) )
% 2.69/1.52 = ( ! [X4: A] :
% 2.69/1.52 ~ ( P @ X4 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_18_all__not__in__conv,axiom,
% 2.69/1.52 ! [A: $tType,A3: set @ A] :
% 2.69/1.52 ( ( ! [X4: A] :
% 2.69/1.52 ~ ( member @ A @ X4 @ A3 ) )
% 2.69/1.52 = ( A3
% 2.69/1.52 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_19_empty__iff,axiom,
% 2.69/1.52 ! [A: $tType,C2: A] :
% 2.69/1.52 ~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_20_UNIV__I,axiom,
% 2.69/1.52 ! [A: $tType,X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_21_finite__code,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( finite_finite @ A @ ( type2 @ A ) )
% 2.69/1.52 => ( ( finite_finite2 @ A )
% 2.69/1.52 = ( ^ [A7: set @ A] : $true ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_22_subsetI,axiom,
% 2.69/1.52 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 2.69/1.52 ( ! [X2: A] :
% 2.69/1.52 ( ( member @ A @ X2 @ A3 )
% 2.69/1.52 => ( member @ A @ X2 @ B3 ) )
% 2.69/1.52 => ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_23_subset__antisym,axiom,
% 2.69/1.52 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 2.69/1.52 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 2.69/1.52 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ A3 )
% 2.69/1.52 => ( A3 = B3 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_24_insert__absorb2,axiom,
% 2.69/1.52 ! [A: $tType,X3: A,A3: set @ A] :
% 2.69/1.52 ( ( insert @ A @ X3 @ ( insert @ A @ X3 @ A3 ) )
% 2.69/1.52 = ( insert @ A @ X3 @ A3 ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_25_insert__iff,axiom,
% 2.69/1.52 ! [A: $tType,A2: A,B4: A,A3: set @ A] :
% 2.69/1.52 ( ( member @ A @ A2 @ ( insert @ A @ B4 @ A3 ) )
% 2.69/1.52 = ( ( A2 = B4 )
% 2.69/1.52 | ( member @ A @ A2 @ A3 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_26_insertCI,axiom,
% 2.69/1.52 ! [A: $tType,A2: A,B3: set @ A,B4: A] :
% 2.69/1.52 ( ( ~ ( member @ A @ A2 @ B3 )
% 2.69/1.52 => ( A2 = B4 ) )
% 2.69/1.52 => ( member @ A @ A2 @ ( insert @ A @ B4 @ B3 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_27_Compl__eq__Compl__iff,axiom,
% 2.69/1.52 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 2.69/1.52 ( ( ( uminus_uminus @ ( set @ A ) @ A3 )
% 2.69/1.52 = ( uminus_uminus @ ( set @ A ) @ B3 ) )
% 2.69/1.52 = ( A3 = B3 ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_28_finite__Plus__UNIV__iff,axiom,
% 2.69/1.52 ! [A: $tType,B: $tType] :
% 2.69/1.52 ( ( finite_finite2 @ ( sum_sum @ A @ B ) @ ( top_top @ ( set @ ( sum_sum @ A @ B ) ) ) )
% 2.69/1.52 = ( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
% 2.69/1.52 & ( finite_finite2 @ B @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_29_empty__subsetI,axiom,
% 2.69/1.52 ! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_30_subset__empty,axiom,
% 2.69/1.52 ! [A: $tType,A3: set @ A] :
% 2.69/1.52 ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
% 2.69/1.52 = ( A3
% 2.69/1.52 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_31_insert__subset,axiom,
% 2.69/1.52 ! [A: $tType,X3: A,A3: set @ A,B3: set @ A] :
% 2.69/1.52 ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X3 @ A3 ) @ B3 )
% 2.69/1.52 = ( ( member @ A @ X3 @ B3 )
% 2.69/1.52 & ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_32_Compl__subset__Compl__iff,axiom,
% 2.69/1.52 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 2.69/1.52 ( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) @ ( uminus_uminus @ ( set @ A ) @ B3 ) )
% 2.69/1.52 = ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_33_Compl__anti__mono,axiom,
% 2.69/1.52 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 2.69/1.52 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 2.69/1.52 => ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B3 ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_34__092_060open_062n_A_092_060in_062_Alset_A_If_A_In_M_Ans_J_J_092_060close_062,axiom,
% 2.69/1.52 member @ node @ n @ ( coinductive_lset @ node @ ( f @ ( product_Pair @ node @ ( set @ node ) @ n @ ns ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_35_singleton__insert__inj__eq_H,axiom,
% 2.69/1.52 ! [A: $tType,A2: A,A3: set @ A,B4: A] :
% 2.69/1.52 ( ( ( insert @ A @ A2 @ A3 )
% 2.69/1.52 = ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.69/1.52 = ( ( A2 = B4 )
% 2.69/1.52 & ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_36_singleton__insert__inj__eq,axiom,
% 2.69/1.52 ! [A: $tType,B4: A,A2: A,A3: set @ A] :
% 2.69/1.52 ( ( ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) )
% 2.69/1.52 = ( insert @ A @ A2 @ A3 ) )
% 2.69/1.52 = ( ( A2 = B4 )
% 2.69/1.52 & ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_37_finite__compl,axiom,
% 2.69/1.52 ! [A: $tType,A3: set @ A] :
% 2.69/1.52 ( ( finite_finite2 @ A @ A3 )
% 2.69/1.52 => ( ( finite_finite2 @ A @ ( uminus_uminus @ ( set @ A ) @ A3 ) )
% 2.69/1.52 = ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_38_subset__Compl__singleton,axiom,
% 2.69/1.52 ! [A: $tType,A3: set @ A,B4: A] :
% 2.69/1.52 ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.69/1.52 = ( ~ ( member @ A @ B4 @ A3 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_39_set__mp,axiom,
% 2.69/1.52 ! [A: $tType,A3: set @ A,B3: set @ A,X3: A] :
% 2.69/1.52 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 2.69/1.52 => ( ( member @ A @ X3 @ A3 )
% 2.69/1.52 => ( member @ A @ X3 @ B3 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_40_in__mono,axiom,
% 2.69/1.52 ! [A: $tType,A3: set @ A,B3: set @ A,X3: A] :
% 2.69/1.52 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 2.69/1.52 => ( ( member @ A @ X3 @ A3 )
% 2.69/1.52 => ( member @ A @ X3 @ B3 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_41_subsetD,axiom,
% 2.69/1.52 ! [A: $tType,A3: set @ A,B3: set @ A,C2: A] :
% 2.69/1.52 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 2.69/1.52 => ( ( member @ A @ C2 @ A3 )
% 2.69/1.52 => ( member @ A @ C2 @ B3 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_42_subsetCE,axiom,
% 2.69/1.52 ! [A: $tType,A3: set @ A,B3: set @ A,C2: A] :
% 2.69/1.52 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 2.69/1.52 => ( ( member @ A @ C2 @ A3 )
% 2.69/1.52 => ( member @ A @ C2 @ B3 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_43_UNIV__eq__I,axiom,
% 2.69/1.52 ! [A: $tType,A3: set @ A] :
% 2.69/1.52 ( ! [X2: A] : ( member @ A @ X2 @ A3 )
% 2.69/1.52 => ( ( top_top @ ( set @ A ) )
% 2.69/1.52 = A3 ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_44_equalityE,axiom,
% 2.69/1.52 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 2.69/1.52 ( ( A3 = B3 )
% 2.69/1.52 => ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 2.69/1.52 => ~ ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_45_mem__Collect__eq,axiom,
% 2.69/1.52 ! [A: $tType,A2: A,P: A > $o] :
% 2.69/1.52 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 2.69/1.52 = ( P @ A2 ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_46_Collect__mem__eq,axiom,
% 2.69/1.52 ! [A: $tType,A3: set @ A] :
% 2.69/1.52 ( ( collect @ A
% 2.69/1.52 @ ^ [X4: A] : ( member @ A @ X4 @ A3 ) )
% 2.69/1.52 = A3 ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_47_Collect__cong,axiom,
% 2.69/1.52 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.69/1.52 ( ! [X2: A] :
% 2.69/1.52 ( ( P @ X2 )
% 2.69/1.52 = ( Q @ X2 ) )
% 2.69/1.52 => ( ( collect @ A @ P )
% 2.69/1.52 = ( collect @ A @ Q ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_48_ext,axiom,
% 2.69/1.52 ! [B: $tType,A: $tType,F3: A > B,G: A > B] :
% 2.69/1.52 ( ! [X2: A] :
% 2.69/1.52 ( ( F3 @ X2 )
% 2.69/1.52 = ( G @ X2 ) )
% 2.69/1.52 => ( F3 = G ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_49_subset__eq,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( ord_less_eq @ ( set @ A ) )
% 2.69/1.52 = ( ^ [A7: set @ A,B5: set @ A] :
% 2.69/1.52 ! [X4: A] :
% 2.69/1.52 ( ( member @ A @ X4 @ A7 )
% 2.69/1.52 => ( member @ A @ X4 @ B5 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_50_equalityD1,axiom,
% 2.69/1.52 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 2.69/1.52 ( ( A3 = B3 )
% 2.69/1.52 => ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_51_equalityD2,axiom,
% 2.69/1.52 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 2.69/1.52 ( ( A3 = B3 )
% 2.69/1.52 => ( ord_less_eq @ ( set @ A ) @ B3 @ A3 ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_52_set__rev__mp,axiom,
% 2.69/1.52 ! [A: $tType,X3: A,A3: set @ A,B3: set @ A] :
% 2.69/1.52 ( ( member @ A @ X3 @ A3 )
% 2.69/1.52 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 2.69/1.52 => ( member @ A @ X3 @ B3 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_53_subset__iff,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( ord_less_eq @ ( set @ A ) )
% 2.69/1.52 = ( ^ [A7: set @ A,B5: set @ A] :
% 2.69/1.52 ! [T2: A] :
% 2.69/1.52 ( ( member @ A @ T2 @ A7 )
% 2.69/1.52 => ( member @ A @ T2 @ B5 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_54_rev__subsetD,axiom,
% 2.69/1.52 ! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
% 2.69/1.52 ( ( member @ A @ C2 @ A3 )
% 2.69/1.52 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 2.69/1.52 => ( member @ A @ C2 @ B3 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_55_subset__UNIV,axiom,
% 2.69/1.52 ! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ ( top_top @ ( set @ A ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_56_subset__refl,axiom,
% 2.69/1.52 ! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ A3 ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_57_Collect__mono,axiom,
% 2.69/1.52 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.69/1.52 ( ! [X2: A] :
% 2.69/1.52 ( ( P @ X2 )
% 2.69/1.52 => ( Q @ X2 ) )
% 2.69/1.52 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_58_UNIV__witness,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ? [X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_59_subset__trans,axiom,
% 2.69/1.52 ! [A: $tType,A3: set @ A,B3: set @ A,C3: set @ A] :
% 2.69/1.52 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 2.69/1.52 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ C3 )
% 2.69/1.52 => ( ord_less_eq @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_60_set__eq__subset,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( ^ [Y: set @ A,Z: set @ A] : ( Y = Z ) )
% 2.69/1.52 = ( ^ [A7: set @ A,B5: set @ A] :
% 2.69/1.52 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B5 )
% 2.69/1.52 & ( ord_less_eq @ ( set @ A ) @ B5 @ A7 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_61_contra__subsetD,axiom,
% 2.69/1.52 ! [A: $tType,A3: set @ A,B3: set @ A,C2: A] :
% 2.69/1.52 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 2.69/1.52 => ( ~ ( member @ A @ C2 @ B3 )
% 2.69/1.52 => ~ ( member @ A @ C2 @ A3 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_62_Collect__mono__iff,axiom,
% 2.69/1.52 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.69/1.52 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 2.69/1.52 = ( ! [X4: A] :
% 2.69/1.52 ( ( P @ X4 )
% 2.69/1.52 => ( Q @ X4 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_63_Finite__Set_Ofinite__set,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( finite_finite2 @ ( set @ A ) @ ( top_top @ ( set @ ( set @ A ) ) ) )
% 2.69/1.52 = ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_64_finite__prod,axiom,
% 2.69/1.52 ! [A: $tType,B: $tType] :
% 2.69/1.52 ( ( finite_finite2 @ ( product_prod @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
% 2.69/1.52 = ( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
% 2.69/1.52 & ( finite_finite2 @ B @ ( top_top @ ( set @ B ) ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_65_finite__Prod__UNIV,axiom,
% 2.69/1.52 ! [B: $tType,A: $tType] :
% 2.69/1.52 ( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
% 2.69/1.52 => ( ( finite_finite2 @ B @ ( top_top @ ( set @ B ) ) )
% 2.69/1.52 => ( finite_finite2 @ ( product_prod @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_66_finite__fun__UNIVD2,axiom,
% 2.69/1.52 ! [A: $tType,B: $tType] :
% 2.69/1.52 ( ( finite_finite2 @ ( A > B ) @ ( top_top @ ( set @ ( A > B ) ) ) )
% 2.69/1.52 => ( finite_finite2 @ B @ ( top_top @ ( set @ B ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_67_rev__finite__subset,axiom,
% 2.69/1.52 ! [A: $tType,B3: set @ A,A3: set @ A] :
% 2.69/1.52 ( ( finite_finite2 @ A @ B3 )
% 2.69/1.52 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 2.69/1.52 => ( finite_finite2 @ A @ A3 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_68_infinite__super,axiom,
% 2.69/1.52 ! [A: $tType,S: set @ A,T3: set @ A] :
% 2.69/1.52 ( ( ord_less_eq @ ( set @ A ) @ S @ T3 )
% 2.69/1.52 => ( ~ ( finite_finite2 @ A @ S )
% 2.69/1.52 => ~ ( finite_finite2 @ A @ T3 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_69_finite__subset,axiom,
% 2.69/1.52 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 2.69/1.52 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 2.69/1.52 => ( ( finite_finite2 @ A @ B3 )
% 2.69/1.52 => ( finite_finite2 @ A @ A3 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_70_empty__not__UNIV,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( bot_bot @ ( set @ A ) )
% 2.69/1.52 != ( top_top @ ( set @ A ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_71_finite__UNIV,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( finite_finite @ A @ ( type2 @ A ) )
% 2.69/1.52 => ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_72_infinite__UNIV__char__0,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( semiring_char_0 @ A @ ( type2 @ A ) )
% 2.69/1.52 => ~ ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_73_ex__new__if__finite,axiom,
% 2.69/1.52 ! [A: $tType,A3: set @ A] :
% 2.69/1.52 ( ~ ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
% 2.69/1.52 => ( ( finite_finite2 @ A @ A3 )
% 2.69/1.52 => ? [A5: A] :
% 2.69/1.52 ~ ( member @ A @ A5 @ A3 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_74_subset__insertI2,axiom,
% 2.69/1.52 ! [A: $tType,A3: set @ A,B3: set @ A,B4: A] :
% 2.69/1.52 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 2.69/1.52 => ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B4 @ B3 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_75_subset__insertI,axiom,
% 2.69/1.52 ! [A: $tType,B3: set @ A,A2: A] : ( ord_less_eq @ ( set @ A ) @ B3 @ ( insert @ A @ A2 @ B3 ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_76_subset__insert,axiom,
% 2.69/1.52 ! [A: $tType,X3: A,A3: set @ A,B3: set @ A] :
% 2.69/1.52 ( ~ ( member @ A @ X3 @ A3 )
% 2.69/1.52 => ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ B3 ) )
% 2.69/1.52 = ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_77_Set_Oinsert__mono,axiom,
% 2.69/1.52 ! [A: $tType,C3: set @ A,D: set @ A,A2: A] :
% 2.69/1.52 ( ( ord_less_eq @ ( set @ A ) @ C3 @ D )
% 2.69/1.52 => ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A2 @ C3 ) @ ( insert @ A @ A2 @ D ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_78_insert__UNIV,axiom,
% 2.69/1.52 ! [A: $tType,X3: A] :
% 2.69/1.52 ( ( insert @ A @ X3 @ ( top_top @ ( set @ A ) ) )
% 2.69/1.52 = ( top_top @ ( set @ A ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_79_subset__singleton__iff,axiom,
% 2.69/1.52 ! [A: $tType,X5: set @ A,A2: A] :
% 2.69/1.52 ( ( ord_less_eq @ ( set @ A ) @ X5 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.69/1.52 = ( ( X5
% 2.69/1.52 = ( bot_bot @ ( set @ A ) ) )
% 2.69/1.52 | ( X5
% 2.69/1.52 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_80_subset__singletonD,axiom,
% 2.69/1.52 ! [A: $tType,A3: set @ A,X3: A] :
% 2.69/1.52 ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.69/1.52 => ( ( A3
% 2.69/1.52 = ( bot_bot @ ( set @ A ) ) )
% 2.69/1.52 | ( A3
% 2.69/1.52 = ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_81_subset__Compl__self__eq,axiom,
% 2.69/1.52 ! [A: $tType,A3: set @ A] :
% 2.69/1.52 ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( uminus_uminus @ ( set @ A ) @ A3 ) )
% 2.69/1.52 = ( A3
% 2.69/1.52 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_82_Compl__empty__eq,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( uminus_uminus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) )
% 2.69/1.52 = ( top_top @ ( set @ A ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_83_Compl__UNIV__eq,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( uminus_uminus @ ( set @ A ) @ ( top_top @ ( set @ A ) ) )
% 2.69/1.52 = ( bot_bot @ ( set @ A ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_84_finite__subset__induct,axiom,
% 2.69/1.52 ! [A: $tType,F: set @ A,A3: set @ A,P: ( set @ A ) > $o] :
% 2.69/1.52 ( ( finite_finite2 @ A @ F )
% 2.69/1.52 => ( ( ord_less_eq @ ( set @ A ) @ F @ A3 )
% 2.69/1.52 => ( ( P @ ( bot_bot @ ( set @ A ) ) )
% 2.69/1.52 => ( ! [A5: A,F2: set @ A] :
% 2.69/1.52 ( ( finite_finite2 @ A @ F2 )
% 2.69/1.52 => ( ( member @ A @ A5 @ A3 )
% 2.69/1.52 => ( ~ ( member @ A @ A5 @ F2 )
% 2.69/1.52 => ( ( P @ F2 )
% 2.69/1.52 => ( P @ ( insert @ A @ A5 @ F2 ) ) ) ) ) )
% 2.69/1.52 => ( P @ F ) ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_85_ex__in__conv,axiom,
% 2.69/1.52 ! [A: $tType,A3: set @ A] :
% 2.69/1.52 ( ( ? [X4: A] : ( member @ A @ X4 @ A3 ) )
% 2.69/1.52 = ( A3
% 2.69/1.52 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_86_equals0I,axiom,
% 2.69/1.52 ! [A: $tType,A3: set @ A] :
% 2.69/1.52 ( ! [Y2: A] :
% 2.69/1.52 ~ ( member @ A @ Y2 @ A3 )
% 2.69/1.52 => ( A3
% 2.69/1.52 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_87_equals0D,axiom,
% 2.69/1.52 ! [A: $tType,A3: set @ A,A2: A] :
% 2.69/1.52 ( ( A3
% 2.69/1.52 = ( bot_bot @ ( set @ A ) ) )
% 2.69/1.52 => ~ ( member @ A @ A2 @ A3 ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_88_emptyE,axiom,
% 2.69/1.52 ! [A: $tType,A2: A] :
% 2.69/1.52 ~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_89_finite,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( finite_finite @ A @ ( type2 @ A ) )
% 2.69/1.52 => ! [A3: set @ A] : ( finite_finite2 @ A @ A3 ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_90_finite__set__choice,axiom,
% 2.69/1.52 ! [B: $tType,A: $tType,A3: set @ A,P: A > B > $o] :
% 2.69/1.52 ( ( finite_finite2 @ A @ A3 )
% 2.69/1.52 => ( ! [X2: A] :
% 2.69/1.52 ( ( member @ A @ X2 @ A3 )
% 2.69/1.52 => ? [X1: B] : ( P @ X2 @ X1 ) )
% 2.69/1.52 => ? [F4: A > B] :
% 2.69/1.52 ! [X6: A] :
% 2.69/1.52 ( ( member @ A @ X6 @ A3 )
% 2.69/1.52 => ( P @ X6 @ ( F4 @ X6 ) ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_91_mk__disjoint__insert,axiom,
% 2.69/1.52 ! [A: $tType,A2: A,A3: set @ A] :
% 2.69/1.52 ( ( member @ A @ A2 @ A3 )
% 2.69/1.52 => ? [B6: set @ A] :
% 2.69/1.52 ( ( A3
% 2.69/1.52 = ( insert @ A @ A2 @ B6 ) )
% 2.69/1.52 & ~ ( member @ A @ A2 @ B6 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_92_insert__commute,axiom,
% 2.69/1.52 ! [A: $tType,X3: A,Y3: A,A3: set @ A] :
% 2.69/1.52 ( ( insert @ A @ X3 @ ( insert @ A @ Y3 @ A3 ) )
% 2.69/1.52 = ( insert @ A @ Y3 @ ( insert @ A @ X3 @ A3 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_93_insert__eq__iff,axiom,
% 2.69/1.52 ! [A: $tType,A2: A,A3: set @ A,B4: A,B3: set @ A] :
% 2.69/1.52 ( ~ ( member @ A @ A2 @ A3 )
% 2.69/1.52 => ( ~ ( member @ A @ B4 @ B3 )
% 2.69/1.52 => ( ( ( insert @ A @ A2 @ A3 )
% 2.69/1.52 = ( insert @ A @ B4 @ B3 ) )
% 2.69/1.52 = ( ( ( A2 = B4 )
% 2.69/1.52 => ( A3 = B3 ) )
% 2.69/1.52 & ( ( A2 != B4 )
% 2.69/1.52 => ? [C4: set @ A] :
% 2.69/1.52 ( ( A3
% 2.69/1.52 = ( insert @ A @ B4 @ C4 ) )
% 2.69/1.52 & ~ ( member @ A @ B4 @ C4 )
% 2.69/1.52 & ( B3
% 2.69/1.52 = ( insert @ A @ A2 @ C4 ) )
% 2.69/1.52 & ~ ( member @ A @ A2 @ C4 ) ) ) ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_94_insert__absorb,axiom,
% 2.69/1.52 ! [A: $tType,A2: A,A3: set @ A] :
% 2.69/1.52 ( ( member @ A @ A2 @ A3 )
% 2.69/1.52 => ( ( insert @ A @ A2 @ A3 )
% 2.69/1.52 = A3 ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_95_insert__ident,axiom,
% 2.69/1.52 ! [A: $tType,X3: A,A3: set @ A,B3: set @ A] :
% 2.69/1.52 ( ~ ( member @ A @ X3 @ A3 )
% 2.69/1.52 => ( ~ ( member @ A @ X3 @ B3 )
% 2.69/1.52 => ( ( ( insert @ A @ X3 @ A3 )
% 2.69/1.52 = ( insert @ A @ X3 @ B3 ) )
% 2.69/1.52 = ( A3 = B3 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_96_Set_Oset__insert,axiom,
% 2.69/1.52 ! [A: $tType,X3: A,A3: set @ A] :
% 2.69/1.52 ( ( member @ A @ X3 @ A3 )
% 2.69/1.52 => ~ ! [B6: set @ A] :
% 2.69/1.52 ( ( A3
% 2.69/1.52 = ( insert @ A @ X3 @ B6 ) )
% 2.69/1.52 => ( member @ A @ X3 @ B6 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_97_insertI2,axiom,
% 2.69/1.52 ! [A: $tType,A2: A,B3: set @ A,B4: A] :
% 2.69/1.52 ( ( member @ A @ A2 @ B3 )
% 2.69/1.52 => ( member @ A @ A2 @ ( insert @ A @ B4 @ B3 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_98_insertI1,axiom,
% 2.69/1.52 ! [A: $tType,A2: A,B3: set @ A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ B3 ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_99_insertE,axiom,
% 2.69/1.52 ! [A: $tType,A2: A,B4: A,A3: set @ A] :
% 2.69/1.52 ( ( member @ A @ A2 @ ( insert @ A @ B4 @ A3 ) )
% 2.69/1.52 => ( ( A2 != B4 )
% 2.69/1.52 => ( member @ A @ A2 @ A3 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_100_double__complement,axiom,
% 2.69/1.52 ! [A: $tType,A3: set @ A] :
% 2.69/1.52 ( ( uminus_uminus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) )
% 2.69/1.52 = A3 ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_101_ComplD,axiom,
% 2.69/1.52 ! [A: $tType,C2: A,A3: set @ A] :
% 2.69/1.52 ( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A3 ) )
% 2.69/1.52 => ~ ( member @ A @ C2 @ A3 ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_102_infinite__imp__nonempty,axiom,
% 2.69/1.52 ! [A: $tType,S: set @ A] :
% 2.69/1.52 ( ~ ( finite_finite2 @ A @ S )
% 2.69/1.52 => ( S
% 2.69/1.52 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_103_finite_OemptyI,axiom,
% 2.69/1.52 ! [A: $tType] : ( finite_finite2 @ A @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_104_singleton__inject,axiom,
% 2.69/1.52 ! [A: $tType,A2: A,B4: A] :
% 2.69/1.52 ( ( ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) )
% 2.69/1.52 = ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.69/1.52 => ( A2 = B4 ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_105_insert__not__empty,axiom,
% 2.69/1.52 ! [A: $tType,A2: A,A3: set @ A] :
% 2.69/1.52 ( ( insert @ A @ A2 @ A3 )
% 2.69/1.52 != ( bot_bot @ ( set @ A ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_106_doubleton__eq__iff,axiom,
% 2.69/1.52 ! [A: $tType,A2: A,B4: A,C2: A,D2: A] :
% 2.69/1.52 ( ( ( insert @ A @ A2 @ ( insert @ A @ B4 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.69/1.52 = ( insert @ A @ C2 @ ( insert @ A @ D2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.69/1.52 = ( ( ( A2 = C2 )
% 2.69/1.52 & ( B4 = D2 ) )
% 2.69/1.52 | ( ( A2 = D2 )
% 2.69/1.52 & ( B4 = C2 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_107_singleton__iff,axiom,
% 2.69/1.52 ! [A: $tType,B4: A,A2: A] :
% 2.69/1.52 ( ( member @ A @ B4 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.69/1.52 = ( B4 = A2 ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_108_singletonD,axiom,
% 2.69/1.52 ! [A: $tType,B4: A,A2: A] :
% 2.69/1.52 ( ( member @ A @ B4 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.69/1.52 => ( B4 = A2 ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_109_finite_OinsertI,axiom,
% 2.69/1.52 ! [A: $tType,A3: set @ A,A2: A] :
% 2.69/1.52 ( ( finite_finite2 @ A @ A3 )
% 2.69/1.52 => ( finite_finite2 @ A @ ( insert @ A @ A2 @ A3 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_110_compl__top__eq,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 2.69/1.52 => ( ( uminus_uminus @ A @ ( top_top @ A ) )
% 2.69/1.52 = ( bot_bot @ A ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_111_compl__bot__eq,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 2.69/1.52 => ( ( uminus_uminus @ A @ ( bot_bot @ A ) )
% 2.69/1.52 = ( top_top @ A ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_112__092_060open_062_092_060not_062_Alfinite_A_If_A_In_M_Ans_J_J_092_060close_062,axiom,
% 2.69/1.52 ~ ( coinductive_lfinite @ node @ ( f @ ( product_Pair @ node @ ( set @ node ) @ n @ ns ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_113_finite__option__UNIV,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( finite_finite2 @ ( option @ A ) @ ( top_top @ ( set @ ( option @ A ) ) ) )
% 2.69/1.52 = ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_114_compl__le__compl__iff,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 2.69/1.52 => ! [X3: A,Y3: A] :
% 2.69/1.52 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X3 ) @ ( uminus_uminus @ A @ Y3 ) )
% 2.69/1.52 = ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_115_neg__le__iff__le,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.69/1.52 => ! [B4: A,A2: A] :
% 2.69/1.52 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B4 ) @ ( uminus_uminus @ A @ A2 ) )
% 2.69/1.52 = ( ord_less_eq @ A @ A2 @ B4 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_116_Topological__Spaces_OUNIV__not__singleton,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( topolo890362671_space @ A @ ( type2 @ A ) )
% 2.69/1.52 => ! [X3: A] :
% 2.69/1.52 ( ( top_top @ ( set @ A ) )
% 2.69/1.52 != ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_117_f__simps_I1_J,axiom,
% 2.69/1.52 ! [Na: node,Nsa: set @ node] :
% 2.69/1.52 ~ ( coinductive_lnull @ node @ ( f @ ( product_Pair @ node @ ( set @ node ) @ Na @ Nsa ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_118_the__elem__eq,axiom,
% 2.69/1.52 ! [A: $tType,X3: A] :
% 2.69/1.52 ( ( the_elem @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.69/1.52 = X3 ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_119_iso__tuple__UNIV__I,axiom,
% 2.69/1.52 ! [A: $tType,X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_120_uminus__apply,axiom,
% 2.69/1.52 ! [B: $tType,A: $tType] :
% 2.69/1.52 ( ( uminus @ B @ ( type2 @ B ) )
% 2.69/1.52 => ( ( uminus_uminus @ ( A > B ) )
% 2.69/1.52 = ( ^ [A7: A > B,X4: A] : ( uminus_uminus @ B @ ( A7 @ X4 ) ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_121_add_Oinverse__inverse,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( group_add @ A @ ( type2 @ A ) )
% 2.69/1.52 => ! [A2: A] :
% 2.69/1.52 ( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A2 ) )
% 2.69/1.52 = A2 ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_122_neg__equal__iff__equal,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( group_add @ A @ ( type2 @ A ) )
% 2.69/1.52 => ! [A2: A,B4: A] :
% 2.69/1.52 ( ( ( uminus_uminus @ A @ A2 )
% 2.69/1.52 = ( uminus_uminus @ A @ B4 ) )
% 2.69/1.52 = ( A2 = B4 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_123_double__compl,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 2.69/1.52 => ! [X3: A] :
% 2.69/1.52 ( ( uminus_uminus @ A @ ( uminus_uminus @ A @ X3 ) )
% 2.69/1.52 = X3 ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_124_compl__eq__compl__iff,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 2.69/1.52 => ! [X3: A,Y3: A] :
% 2.69/1.52 ( ( ( uminus_uminus @ A @ X3 )
% 2.69/1.52 = ( uminus_uminus @ A @ Y3 ) )
% 2.69/1.52 = ( X3 = Y3 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_125_bot__set__def,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( bot_bot @ ( set @ A ) )
% 2.69/1.52 = ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_126_top__set__def,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( top_top @ ( set @ A ) )
% 2.69/1.52 = ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_127_fun__Compl__def,axiom,
% 2.69/1.52 ! [B: $tType,A: $tType] :
% 2.69/1.52 ( ( uminus @ B @ ( type2 @ B ) )
% 2.69/1.52 => ( ( uminus_uminus @ ( A > B ) )
% 2.69/1.52 = ( ^ [A7: A > B,X4: A] : ( uminus_uminus @ B @ ( A7 @ X4 ) ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_128_equation__minus__iff,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( group_add @ A @ ( type2 @ A ) )
% 2.69/1.52 => ! [A2: A,B4: A] :
% 2.69/1.52 ( ( A2
% 2.69/1.52 = ( uminus_uminus @ A @ B4 ) )
% 2.69/1.52 = ( B4
% 2.69/1.52 = ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_129_minus__equation__iff,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( group_add @ A @ ( type2 @ A ) )
% 2.69/1.52 => ! [A2: A,B4: A] :
% 2.69/1.52 ( ( ( uminus_uminus @ A @ A2 )
% 2.69/1.52 = B4 )
% 2.69/1.52 = ( ( uminus_uminus @ A @ B4 )
% 2.69/1.52 = A2 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_130_compl__mono,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 2.69/1.52 => ! [X3: A,Y3: A] :
% 2.69/1.52 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 2.69/1.52 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y3 ) @ ( uminus_uminus @ A @ X3 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_131_compl__le__swap1,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 2.69/1.52 => ! [Y3: A,X3: A] :
% 2.69/1.52 ( ( ord_less_eq @ A @ Y3 @ ( uminus_uminus @ A @ X3 ) )
% 2.69/1.52 => ( ord_less_eq @ A @ X3 @ ( uminus_uminus @ A @ Y3 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_132_compl__le__swap2,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( boolean_algebra @ A @ ( type2 @ A ) )
% 2.69/1.52 => ! [Y3: A,X3: A] :
% 2.69/1.52 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y3 ) @ X3 )
% 2.69/1.52 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ X3 ) @ Y3 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_133_le__minus__iff,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.69/1.52 => ! [A2: A,B4: A] :
% 2.69/1.52 ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ B4 ) )
% 2.69/1.52 = ( ord_less_eq @ A @ B4 @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_134_minus__le__iff,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.69/1.52 => ! [A2: A,B4: A] :
% 2.69/1.52 ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B4 )
% 2.69/1.52 = ( ord_less_eq @ A @ ( uminus_uminus @ A @ B4 ) @ A2 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_135_le__imp__neg__le,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.69/1.52 => ! [A2: A,B4: A] :
% 2.69/1.52 ( ( ord_less_eq @ A @ A2 @ B4 )
% 2.69/1.52 => ( ord_less_eq @ A @ ( uminus_uminus @ A @ B4 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_136_lset__eq__empty,axiom,
% 2.69/1.52 ! [A: $tType,Xs: coinductive_llist @ A] :
% 2.69/1.52 ( ( ( coinductive_lset @ A @ Xs )
% 2.69/1.52 = ( bot_bot @ ( set @ A ) ) )
% 2.69/1.52 = ( coinductive_lnull @ A @ Xs ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_137_lfinite__imp__finite__lset,axiom,
% 2.69/1.52 ! [A: $tType,Xs: coinductive_llist @ A] :
% 2.69/1.52 ( ( coinductive_lfinite @ A @ Xs )
% 2.69/1.52 => ( finite_finite2 @ A @ ( coinductive_lset @ A @ Xs ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_138_lset__lnull,axiom,
% 2.69/1.52 ! [A: $tType,Xs: coinductive_llist @ A] :
% 2.69/1.52 ( ( coinductive_lnull @ A @ Xs )
% 2.69/1.52 => ( ( coinductive_lset @ A @ Xs )
% 2.69/1.52 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_139_top__apply,axiom,
% 2.69/1.52 ! [C: $tType,D3: $tType] :
% 2.69/1.52 ( ( top @ C @ ( type2 @ C ) )
% 2.69/1.52 => ( ( top_top @ ( D3 > C ) )
% 2.69/1.52 = ( ^ [X4: D3] : ( top_top @ C ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_140_order__refl,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( preorder @ A @ ( type2 @ A ) )
% 2.69/1.52 => ! [X3: A] : ( ord_less_eq @ A @ X3 @ X3 ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_141_top1I,axiom,
% 2.69/1.52 ! [A: $tType,X3: A] : ( top_top @ ( A > $o ) @ X3 ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_142_bot__apply,axiom,
% 2.69/1.52 ! [C: $tType,D3: $tType] :
% 2.69/1.52 ( ( bot @ C @ ( type2 @ C ) )
% 2.69/1.52 => ( ( bot_bot @ ( D3 > C ) )
% 2.69/1.52 = ( ^ [X4: D3] : ( bot_bot @ C ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_143_le__funD,axiom,
% 2.69/1.52 ! [B: $tType,A: $tType] :
% 2.69/1.52 ( ( ord @ B @ ( type2 @ B ) )
% 2.69/1.52 => ! [F3: A > B,G: A > B,X3: A] :
% 2.69/1.52 ( ( ord_less_eq @ ( A > B ) @ F3 @ G )
% 2.69/1.52 => ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( G @ X3 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_144_le__funE,axiom,
% 2.69/1.52 ! [B: $tType,A: $tType] :
% 2.69/1.52 ( ( ord @ B @ ( type2 @ B ) )
% 2.69/1.52 => ! [F3: A > B,G: A > B,X3: A] :
% 2.69/1.52 ( ( ord_less_eq @ ( A > B ) @ F3 @ G )
% 2.69/1.52 => ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( G @ X3 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_145_le__funI,axiom,
% 2.69/1.52 ! [B: $tType,A: $tType] :
% 2.69/1.52 ( ( ord @ B @ ( type2 @ B ) )
% 2.69/1.52 => ! [F3: A > B,G: A > B] :
% 2.69/1.52 ( ! [X2: A] : ( ord_less_eq @ B @ ( F3 @ X2 ) @ ( G @ X2 ) )
% 2.69/1.52 => ( ord_less_eq @ ( A > B ) @ F3 @ G ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_146_le__fun__def,axiom,
% 2.69/1.52 ! [B: $tType,A: $tType] :
% 2.69/1.52 ( ( ord @ B @ ( type2 @ B ) )
% 2.69/1.52 => ( ( ord_less_eq @ ( A > B ) )
% 2.69/1.52 = ( ^ [F5: A > B,G2: A > B] :
% 2.69/1.52 ! [X4: A] : ( ord_less_eq @ B @ ( F5 @ X4 ) @ ( G2 @ X4 ) ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_147_order__subst1,axiom,
% 2.69/1.52 ! [A: $tType,B: $tType] :
% 2.69/1.52 ( ( ( order @ B @ ( type2 @ B ) )
% 2.69/1.52 & ( order @ A @ ( type2 @ A ) ) )
% 2.69/1.52 => ! [A2: A,F3: B > A,B4: B,C2: B] :
% 2.69/1.52 ( ( ord_less_eq @ A @ A2 @ ( F3 @ B4 ) )
% 2.69/1.52 => ( ( ord_less_eq @ B @ B4 @ C2 )
% 2.69/1.52 => ( ! [X2: B,Y2: B] :
% 2.69/1.52 ( ( ord_less_eq @ B @ X2 @ Y2 )
% 2.69/1.52 => ( ord_less_eq @ A @ ( F3 @ X2 ) @ ( F3 @ Y2 ) ) )
% 2.69/1.52 => ( ord_less_eq @ A @ A2 @ ( F3 @ C2 ) ) ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_148_order__subst2,axiom,
% 2.69/1.52 ! [A: $tType,C: $tType] :
% 2.69/1.52 ( ( ( order @ C @ ( type2 @ C ) )
% 2.69/1.52 & ( order @ A @ ( type2 @ A ) ) )
% 2.69/1.52 => ! [A2: A,B4: A,F3: A > C,C2: C] :
% 2.69/1.52 ( ( ord_less_eq @ A @ A2 @ B4 )
% 2.69/1.52 => ( ( ord_less_eq @ C @ ( F3 @ B4 ) @ C2 )
% 2.69/1.52 => ( ! [X2: A,Y2: A] :
% 2.69/1.52 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 2.69/1.52 => ( ord_less_eq @ C @ ( F3 @ X2 ) @ ( F3 @ Y2 ) ) )
% 2.69/1.52 => ( ord_less_eq @ C @ ( F3 @ A2 ) @ C2 ) ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_149_ord__eq__le__subst,axiom,
% 2.69/1.52 ! [A: $tType,B: $tType] :
% 2.69/1.52 ( ( ( ord @ B @ ( type2 @ B ) )
% 2.69/1.52 & ( ord @ A @ ( type2 @ A ) ) )
% 2.69/1.52 => ! [A2: A,F3: B > A,B4: B,C2: B] :
% 2.69/1.52 ( ( A2
% 2.69/1.52 = ( F3 @ B4 ) )
% 2.69/1.52 => ( ( ord_less_eq @ B @ B4 @ C2 )
% 2.69/1.52 => ( ! [X2: B,Y2: B] :
% 2.69/1.52 ( ( ord_less_eq @ B @ X2 @ Y2 )
% 2.69/1.52 => ( ord_less_eq @ A @ ( F3 @ X2 ) @ ( F3 @ Y2 ) ) )
% 2.69/1.52 => ( ord_less_eq @ A @ A2 @ ( F3 @ C2 ) ) ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_150_ord__le__eq__subst,axiom,
% 2.69/1.52 ! [A: $tType,B: $tType] :
% 2.69/1.52 ( ( ( ord @ B @ ( type2 @ B ) )
% 2.69/1.52 & ( ord @ A @ ( type2 @ A ) ) )
% 2.69/1.52 => ! [A2: A,B4: A,F3: A > B,C2: B] :
% 2.69/1.52 ( ( ord_less_eq @ A @ A2 @ B4 )
% 2.69/1.52 => ( ( ( F3 @ B4 )
% 2.69/1.52 = C2 )
% 2.69/1.52 => ( ! [X2: A,Y2: A] :
% 2.69/1.52 ( ( ord_less_eq @ A @ X2 @ Y2 )
% 2.69/1.52 => ( ord_less_eq @ B @ ( F3 @ X2 ) @ ( F3 @ Y2 ) ) )
% 2.69/1.52 => ( ord_less_eq @ B @ ( F3 @ A2 ) @ C2 ) ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_151_eq__iff,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( order @ A @ ( type2 @ A ) )
% 2.69/1.52 => ( ( ^ [Y: A,Z: A] : ( Y = Z ) )
% 2.69/1.52 = ( ^ [X4: A,Y4: A] :
% 2.69/1.52 ( ( ord_less_eq @ A @ X4 @ Y4 )
% 2.69/1.52 & ( ord_less_eq @ A @ Y4 @ X4 ) ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_152_antisym,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( order @ A @ ( type2 @ A ) )
% 2.69/1.52 => ! [X3: A,Y3: A] :
% 2.69/1.52 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 2.69/1.52 => ( ( ord_less_eq @ A @ Y3 @ X3 )
% 2.69/1.52 => ( X3 = Y3 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_153_linear,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( linorder @ A @ ( type2 @ A ) )
% 2.69/1.52 => ! [X3: A,Y3: A] :
% 2.69/1.52 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 2.69/1.52 | ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_154_eq__refl,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( preorder @ A @ ( type2 @ A ) )
% 2.69/1.52 => ! [X3: A,Y3: A] :
% 2.69/1.52 ( ( X3 = Y3 )
% 2.69/1.52 => ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_155_wlog__linorder__le,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( linorder @ A @ ( type2 @ A ) )
% 2.69/1.52 => ! [P: A > A > $o,B4: A,A2: A] :
% 2.69/1.52 ( ! [A5: A,B7: A] :
% 2.69/1.52 ( ( ord_less_eq @ A @ A5 @ B7 )
% 2.69/1.52 => ( P @ A5 @ B7 ) )
% 2.69/1.52 => ( ( ( P @ B4 @ A2 )
% 2.69/1.52 => ( P @ A2 @ B4 ) )
% 2.69/1.52 => ( P @ A2 @ B4 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_156_le__cases,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( linorder @ A @ ( type2 @ A ) )
% 2.69/1.52 => ! [X3: A,Y3: A] :
% 2.69/1.52 ( ~ ( ord_less_eq @ A @ X3 @ Y3 )
% 2.69/1.52 => ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_157_order_Otrans,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( order @ A @ ( type2 @ A ) )
% 2.69/1.52 => ! [A2: A,B4: A,C2: A] :
% 2.69/1.52 ( ( ord_less_eq @ A @ A2 @ B4 )
% 2.69/1.52 => ( ( ord_less_eq @ A @ B4 @ C2 )
% 2.69/1.52 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_158_le__cases3,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( linorder @ A @ ( type2 @ A ) )
% 2.69/1.52 => ! [X3: A,Y3: A,Z2: A] :
% 2.69/1.52 ( ( ( ord_less_eq @ A @ X3 @ Y3 )
% 2.69/1.52 => ~ ( ord_less_eq @ A @ Y3 @ Z2 ) )
% 2.69/1.52 => ( ( ( ord_less_eq @ A @ Y3 @ X3 )
% 2.69/1.52 => ~ ( ord_less_eq @ A @ X3 @ Z2 ) )
% 2.69/1.52 => ( ( ( ord_less_eq @ A @ X3 @ Z2 )
% 2.69/1.52 => ~ ( ord_less_eq @ A @ Z2 @ Y3 ) )
% 2.69/1.52 => ( ( ( ord_less_eq @ A @ Z2 @ Y3 )
% 2.69/1.52 => ~ ( ord_less_eq @ A @ Y3 @ X3 ) )
% 2.69/1.52 => ( ( ( ord_less_eq @ A @ Y3 @ Z2 )
% 2.69/1.52 => ~ ( ord_less_eq @ A @ Z2 @ X3 ) )
% 2.69/1.52 => ~ ( ( ord_less_eq @ A @ Z2 @ X3 )
% 2.69/1.52 => ~ ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ) ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_159_antisym__conv,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( order @ A @ ( type2 @ A ) )
% 2.69/1.52 => ! [Y3: A,X3: A] :
% 2.69/1.52 ( ( ord_less_eq @ A @ Y3 @ X3 )
% 2.69/1.52 => ( ( ord_less_eq @ A @ X3 @ Y3 )
% 2.69/1.52 = ( X3 = Y3 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_160_ord__eq__le__trans,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( ord @ A @ ( type2 @ A ) )
% 2.69/1.52 => ! [A2: A,B4: A,C2: A] :
% 2.69/1.52 ( ( A2 = B4 )
% 2.69/1.52 => ( ( ord_less_eq @ A @ B4 @ C2 )
% 2.69/1.52 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_161_ord__le__eq__trans,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( ord @ A @ ( type2 @ A ) )
% 2.69/1.52 => ! [A2: A,B4: A,C2: A] :
% 2.69/1.52 ( ( ord_less_eq @ A @ A2 @ B4 )
% 2.69/1.52 => ( ( B4 = C2 )
% 2.69/1.52 => ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_162_order__class_Oorder_Oantisym,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( order @ A @ ( type2 @ A ) )
% 2.69/1.52 => ! [A2: A,B4: A] :
% 2.69/1.52 ( ( ord_less_eq @ A @ A2 @ B4 )
% 2.69/1.52 => ( ( ord_less_eq @ A @ B4 @ A2 )
% 2.69/1.52 => ( A2 = B4 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_163_order__trans,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( preorder @ A @ ( type2 @ A ) )
% 2.69/1.52 => ! [X3: A,Y3: A,Z2: A] :
% 2.69/1.52 ( ( ord_less_eq @ A @ X3 @ Y3 )
% 2.69/1.52 => ( ( ord_less_eq @ A @ Y3 @ Z2 )
% 2.69/1.52 => ( ord_less_eq @ A @ X3 @ Z2 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_164_dual__order_Orefl,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( order @ A @ ( type2 @ A ) )
% 2.69/1.52 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_165_linorder__wlog,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( linorder @ A @ ( type2 @ A ) )
% 2.69/1.52 => ! [P: A > A > $o,A2: A,B4: A] :
% 2.69/1.52 ( ! [A5: A,B7: A] :
% 2.69/1.52 ( ( ord_less_eq @ A @ A5 @ B7 )
% 2.69/1.52 => ( P @ A5 @ B7 ) )
% 2.69/1.52 => ( ! [A5: A,B7: A] :
% 2.69/1.52 ( ( P @ B7 @ A5 )
% 2.69/1.52 => ( P @ A5 @ B7 ) )
% 2.69/1.52 => ( P @ A2 @ B4 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_166_dual__order_Otrans,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( order @ A @ ( type2 @ A ) )
% 2.69/1.52 => ! [B4: A,A2: A,C2: A] :
% 2.69/1.52 ( ( ord_less_eq @ A @ B4 @ A2 )
% 2.69/1.52 => ( ( ord_less_eq @ A @ C2 @ B4 )
% 2.69/1.52 => ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_167_dual__order_Oantisym,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( order @ A @ ( type2 @ A ) )
% 2.69/1.52 => ! [B4: A,A2: A] :
% 2.69/1.52 ( ( ord_less_eq @ A @ B4 @ A2 )
% 2.69/1.52 => ( ( ord_less_eq @ A @ A2 @ B4 )
% 2.69/1.52 => ( A2 = B4 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_168_bot__fun__def,axiom,
% 2.69/1.52 ! [B: $tType,A: $tType] :
% 2.69/1.52 ( ( bot @ B @ ( type2 @ B ) )
% 2.69/1.52 => ( ( bot_bot @ ( A > B ) )
% 2.69/1.52 = ( ^ [X4: A] : ( bot_bot @ B ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_169_lzip_Oexhaust,axiom,
% 2.69/1.52 ! [A: $tType,B: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
% 2.69/1.52 ( ~ ( ( coinductive_lnull @ A @ Xs )
% 2.69/1.52 | ( coinductive_lnull @ B @ Ys ) )
% 2.69/1.52 => ~ ( ~ ( coinductive_lnull @ A @ Xs )
% 2.69/1.52 => ( coinductive_lnull @ B @ Ys ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_170_lappend_Oexhaust,axiom,
% 2.69/1.52 ! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
% 2.69/1.52 ( ( ( coinductive_lnull @ A @ Xs )
% 2.69/1.52 => ~ ( coinductive_lnull @ A @ Ys ) )
% 2.69/1.52 => ( ~ ( coinductive_lnull @ A @ Xs )
% 2.69/1.52 | ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_171_bot_Oextremum__uniqueI,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( order_bot @ A @ ( type2 @ A ) )
% 2.69/1.52 => ! [A2: A] :
% 2.69/1.52 ( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
% 2.69/1.52 => ( A2
% 2.69/1.52 = ( bot_bot @ A ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_172_bot_Oextremum__unique,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( order_bot @ A @ ( type2 @ A ) )
% 2.69/1.52 => ! [A2: A] :
% 2.69/1.52 ( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
% 2.69/1.52 = ( A2
% 2.69/1.52 = ( bot_bot @ A ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_173_bot_Oextremum,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( order_bot @ A @ ( type2 @ A ) )
% 2.69/1.52 => ! [A2: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A2 ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_174_top__greatest,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( order_top @ A @ ( type2 @ A ) )
% 2.69/1.52 => ! [A2: A] : ( ord_less_eq @ A @ A2 @ ( top_top @ A ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_175_top_Oextremum__unique,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( order_top @ A @ ( type2 @ A ) )
% 2.69/1.52 => ! [A2: A] :
% 2.69/1.52 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A2 )
% 2.69/1.52 = ( A2
% 2.69/1.52 = ( top_top @ A ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_176_top_Oextremum__uniqueI,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( order_top @ A @ ( type2 @ A ) )
% 2.69/1.52 => ! [A2: A] :
% 2.69/1.52 ( ( ord_less_eq @ A @ ( top_top @ A ) @ A2 )
% 2.69/1.52 => ( A2
% 2.69/1.52 = ( top_top @ A ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_177_lnull__imp__lfinite,axiom,
% 2.69/1.52 ! [A: $tType,Xs: coinductive_llist @ A] :
% 2.69/1.52 ( ( coinductive_lnull @ A @ Xs )
% 2.69/1.52 => ( coinductive_lfinite @ A @ Xs ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_178_old_Oprod_Oinject,axiom,
% 2.69/1.52 ! [A: $tType,B: $tType,A2: A,B4: B,A8: A,B8: B] :
% 2.69/1.52 ( ( ( product_Pair @ A @ B @ A2 @ B4 )
% 2.69/1.52 = ( product_Pair @ A @ B @ A8 @ B8 ) )
% 2.69/1.52 = ( ( A2 = A8 )
% 2.69/1.52 & ( B4 = B8 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_179_prod_Oinject,axiom,
% 2.69/1.52 ! [A: $tType,B: $tType,X12: A,X22: B,Y1: A,Y22: B] :
% 2.69/1.52 ( ( ( product_Pair @ A @ B @ X12 @ X22 )
% 2.69/1.52 = ( product_Pair @ A @ B @ Y1 @ Y22 ) )
% 2.69/1.52 = ( ( X12 = Y1 )
% 2.69/1.52 & ( X22 = Y22 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_180_top__empty__eq,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( top_top @ ( A > $o ) )
% 2.69/1.52 = ( ^ [X4: A] : ( member @ A @ X4 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_181_Collect__empty__eq__bot,axiom,
% 2.69/1.52 ! [A: $tType,P: A > $o] :
% 2.69/1.52 ( ( ( collect @ A @ P )
% 2.69/1.52 = ( bot_bot @ ( set @ A ) ) )
% 2.69/1.52 = ( P
% 2.69/1.52 = ( bot_bot @ ( A > $o ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_182_subrelI,axiom,
% 2.69/1.52 ! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
% 2.69/1.52 ( ! [X2: A,Y2: B] :
% 2.69/1.52 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ R )
% 2.69/1.52 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) @ S2 ) )
% 2.69/1.52 => ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S2 ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_183_surj__pair,axiom,
% 2.69/1.52 ! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
% 2.69/1.52 ? [X2: A,Y2: B] :
% 2.69/1.52 ( P2
% 2.69/1.52 = ( product_Pair @ A @ B @ X2 @ Y2 ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_184_prod__cases,axiom,
% 2.69/1.52 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
% 2.69/1.52 ( ! [A5: A,B7: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B7 ) )
% 2.69/1.52 => ( P @ P2 ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_185_Pair__inject,axiom,
% 2.69/1.52 ! [A: $tType,B: $tType,A2: A,B4: B,A8: A,B8: B] :
% 2.69/1.52 ( ( ( product_Pair @ A @ B @ A2 @ B4 )
% 2.69/1.52 = ( product_Pair @ A @ B @ A8 @ B8 ) )
% 2.69/1.52 => ~ ( ( A2 = A8 )
% 2.69/1.52 => ( B4 != B8 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_186_prod__cases3,axiom,
% 2.69/1.52 ! [A: $tType,B: $tType,C: $tType,Y3: product_prod @ A @ ( product_prod @ B @ C )] :
% 2.69/1.52 ~ ! [A5: A,B7: B,C5: C] :
% 2.69/1.52 ( Y3
% 2.69/1.52 != ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B7 @ C5 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_187_prod__cases4,axiom,
% 2.69/1.52 ! [A: $tType,B: $tType,C: $tType,D3: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D3 ) )] :
% 2.69/1.52 ~ ! [A5: A,B7: B,C5: C,D4: D3] :
% 2.69/1.52 ( Y3
% 2.69/1.52 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D3 ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D3 ) @ B7 @ ( product_Pair @ C @ D3 @ C5 @ D4 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_188_prod__cases5,axiom,
% 2.69/1.52 ! [A: $tType,B: $tType,C: $tType,D3: $tType,E: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ E ) ) )] :
% 2.69/1.52 ~ ! [A5: A,B7: B,C5: C,D4: D3,E2: E] :
% 2.69/1.52 ( Y3
% 2.69/1.52 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D3 @ E ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D3 @ E ) @ C5 @ ( product_Pair @ D3 @ E @ D4 @ E2 ) ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_189_prod__cases6,axiom,
% 2.69/1.52 ! [A: $tType,B: $tType,C: $tType,D3: $tType,E: $tType,F6: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ F6 ) ) ) )] :
% 2.69/1.52 ~ ! [A5: A,B7: B,C5: C,D4: D3,E2: E,F4: F6] :
% 2.69/1.52 ( Y3
% 2.69/1.52 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ F6 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ F6 ) ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D3 @ ( product_prod @ E @ F6 ) ) @ C5 @ ( product_Pair @ D3 @ ( product_prod @ E @ F6 ) @ D4 @ ( product_Pair @ E @ F6 @ E2 @ F4 ) ) ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_190_prod__cases7,axiom,
% 2.69/1.52 ! [A: $tType,B: $tType,C: $tType,D3: $tType,E: $tType,F6: $tType,G3: $tType,Y3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ ( product_prod @ F6 @ G3 ) ) ) ) )] :
% 2.69/1.52 ~ ! [A5: A,B7: B,C5: C,D4: D3,E2: E,F4: F6,G4: G3] :
% 2.69/1.52 ( Y3
% 2.69/1.52 != ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ ( product_prod @ F6 @ G3 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ ( product_prod @ F6 @ G3 ) ) ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D3 @ ( product_prod @ E @ ( product_prod @ F6 @ G3 ) ) ) @ C5 @ ( product_Pair @ D3 @ ( product_prod @ E @ ( product_prod @ F6 @ G3 ) ) @ D4 @ ( product_Pair @ E @ ( product_prod @ F6 @ G3 ) @ E2 @ ( product_Pair @ F6 @ G3 @ F4 @ G4 ) ) ) ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_191_prod__induct3,axiom,
% 2.69/1.52 ! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ C )] :
% 2.69/1.52 ( ! [A5: A,B7: B,C5: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A5 @ ( product_Pair @ B @ C @ B7 @ C5 ) ) )
% 2.69/1.52 => ( P @ X3 ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_192_prod__induct4,axiom,
% 2.69/1.52 ! [D3: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D3 ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D3 ) )] :
% 2.69/1.52 ( ! [A5: A,B7: B,C5: C,D4: D3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D3 ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ D3 ) @ B7 @ ( product_Pair @ C @ D3 @ C5 @ D4 ) ) ) )
% 2.69/1.52 => ( P @ X3 ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_193_prod__induct5,axiom,
% 2.69/1.52 ! [E: $tType,D3: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ E ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ E ) ) )] :
% 2.69/1.52 ( ! [A5: A,B7: B,C5: C,D4: D3,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ E ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D3 @ E ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D3 @ E ) @ C5 @ ( product_Pair @ D3 @ E @ D4 @ E2 ) ) ) ) )
% 2.69/1.52 => ( P @ X3 ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_194_prod__induct6,axiom,
% 2.69/1.52 ! [F6: $tType,E: $tType,D3: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ F6 ) ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ F6 ) ) ) )] :
% 2.69/1.52 ( ! [A5: A,B7: B,C5: C,D4: D3,E2: E,F4: F6] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ F6 ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ F6 ) ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D3 @ ( product_prod @ E @ F6 ) ) @ C5 @ ( product_Pair @ D3 @ ( product_prod @ E @ F6 ) @ D4 @ ( product_Pair @ E @ F6 @ E2 @ F4 ) ) ) ) ) )
% 2.69/1.52 => ( P @ X3 ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_195_prod__induct7,axiom,
% 2.69/1.52 ! [G3: $tType,F6: $tType,E: $tType,D3: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ ( product_prod @ F6 @ G3 ) ) ) ) ) ) > $o,X3: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ ( product_prod @ F6 @ G3 ) ) ) ) )] :
% 2.69/1.52 ( ! [A5: A,B7: B,C5: C,D4: D3,E2: E,F4: F6,G4: G3] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ ( product_prod @ F6 @ G3 ) ) ) ) ) @ A5 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D3 @ ( product_prod @ E @ ( product_prod @ F6 @ G3 ) ) ) ) @ B7 @ ( product_Pair @ C @ ( product_prod @ D3 @ ( product_prod @ E @ ( product_prod @ F6 @ G3 ) ) ) @ C5 @ ( product_Pair @ D3 @ ( product_prod @ E @ ( product_prod @ F6 @ G3 ) ) @ D4 @ ( product_Pair @ E @ ( product_prod @ F6 @ G3 ) @ E2 @ ( product_Pair @ F6 @ G3 @ F4 @ G4 ) ) ) ) ) ) )
% 2.69/1.52 => ( P @ X3 ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_196_old_Oprod_Oexhaust,axiom,
% 2.69/1.52 ! [A: $tType,B: $tType,Y3: product_prod @ A @ B] :
% 2.69/1.52 ~ ! [A5: A,B7: B] :
% 2.69/1.52 ( Y3
% 2.69/1.52 != ( product_Pair @ A @ B @ A5 @ B7 ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_197_old_Oprod_Oinducts,axiom,
% 2.69/1.52 ! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
% 2.69/1.52 ( ! [A5: A,B7: B] : ( P @ ( product_Pair @ A @ B @ A5 @ B7 ) )
% 2.69/1.52 => ( P @ Prod ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_198_bot__empty__eq,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( bot_bot @ ( A > $o ) )
% 2.69/1.52 = ( ^ [X4: A] : ( member @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_199_old_Oprod_Orec,axiom,
% 2.69/1.52 ! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B4: B] :
% 2.69/1.52 ( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B4 ) )
% 2.69/1.52 = ( F1 @ A2 @ B4 ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_200_is__singleton__the__elem,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( is_singleton @ A )
% 2.69/1.52 = ( ^ [A7: set @ A] :
% 2.69/1.52 ( A7
% 2.69/1.52 = ( insert @ A @ ( the_elem @ A @ A7 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_201_is__singletonI,axiom,
% 2.69/1.52 ! [A: $tType,X3: A] : ( is_singleton @ A @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_202_is__singletonI_H,axiom,
% 2.69/1.52 ! [A: $tType,A3: set @ A] :
% 2.69/1.52 ( ( A3
% 2.69/1.52 != ( bot_bot @ ( set @ A ) ) )
% 2.69/1.52 => ( ! [X2: A,Y2: A] :
% 2.69/1.52 ( ( member @ A @ X2 @ A3 )
% 2.69/1.52 => ( ( member @ A @ Y2 @ A3 )
% 2.69/1.52 => ( X2 = Y2 ) ) )
% 2.69/1.52 => ( is_singleton @ A @ A3 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_203_is__singleton__def,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( is_singleton @ A )
% 2.69/1.52 = ( ^ [A7: set @ A] :
% 2.69/1.52 ? [X4: A] :
% 2.69/1.52 ( A7
% 2.69/1.52 = ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_204_is__singletonE,axiom,
% 2.69/1.52 ! [A: $tType,A3: set @ A] :
% 2.69/1.52 ( ( is_singleton @ A @ A3 )
% 2.69/1.52 => ~ ! [X2: A] :
% 2.69/1.52 ( A3
% 2.69/1.52 != ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_205_insert__subsetI,axiom,
% 2.69/1.52 ! [A: $tType,X3: A,A3: set @ A,X5: set @ A] :
% 2.69/1.52 ( ( member @ A @ X3 @ A3 )
% 2.69/1.52 => ( ( ord_less_eq @ ( set @ A ) @ X5 @ A3 )
% 2.69/1.52 => ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X3 @ X5 ) @ A3 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_206_subset__emptyI,axiom,
% 2.69/1.52 ! [A: $tType,A3: set @ A] :
% 2.69/1.52 ( ! [X2: A] :
% 2.69/1.52 ~ ( member @ A @ X2 @ A3 )
% 2.69/1.52 => ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_207_ssubst__Pair__rhs,axiom,
% 2.69/1.52 ! [B: $tType,A: $tType,R: A,S2: B,R2: set @ ( product_prod @ A @ B ),S3: B] :
% 2.69/1.52 ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S2 ) @ R2 )
% 2.69/1.52 => ( ( S3 = S2 )
% 2.69/1.52 => ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R @ S3 ) @ R2 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_208_lset__code,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( coinductive_lset @ A )
% 2.69/1.52 = ( coinductive_gen_lset @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_209_top__conj_I1_J,axiom,
% 2.69/1.52 ! [A: $tType,X3: A,P: $o] :
% 2.69/1.52 ( ( ( top_top @ ( A > $o ) @ X3 )
% 2.69/1.52 & P )
% 2.69/1.52 = P ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_210_top__conj_I2_J,axiom,
% 2.69/1.52 ! [A: $tType,P: $o,X3: A] :
% 2.69/1.52 ( ( P
% 2.69/1.52 & ( top_top @ ( A > $o ) @ X3 ) )
% 2.69/1.52 = P ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_211_prod__set__simps_I1_J,axiom,
% 2.69/1.52 ! [B: $tType,A: $tType,X3: A,Y3: B] :
% 2.69/1.52 ( ( basic_fsts @ A @ B @ ( product_Pair @ A @ B @ X3 @ Y3 ) )
% 2.69/1.52 = ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_212_prod__set__simps_I2_J,axiom,
% 2.69/1.52 ! [A: $tType,B: $tType,X3: A,Y3: B] :
% 2.69/1.52 ( ( basic_snds @ A @ B @ ( product_Pair @ A @ B @ X3 @ Y3 ) )
% 2.69/1.52 = ( insert @ B @ Y3 @ ( bot_bot @ ( set @ B ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_213_Set_Ois__empty__def,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( is_empty @ A )
% 2.69/1.52 = ( ^ [A7: set @ A] :
% 2.69/1.52 ( A7
% 2.69/1.52 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_214_internal__case__prod__conv,axiom,
% 2.69/1.52 ! [B: $tType,A: $tType,C: $tType,C2: B > C > A,A2: B,B4: C] :
% 2.69/1.52 ( ( produc2004651681e_prod @ B @ C @ A @ C2 @ ( product_Pair @ B @ C @ A2 @ B4 ) )
% 2.69/1.52 = ( C2 @ A2 @ B4 ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_215_finite__remove__induct,axiom,
% 2.69/1.52 ! [A: $tType,B3: set @ A,P: ( set @ A ) > $o] :
% 2.69/1.52 ( ( finite_finite2 @ A @ B3 )
% 2.69/1.52 => ( ( P @ ( bot_bot @ ( set @ A ) ) )
% 2.69/1.52 => ( ! [A4: set @ A] :
% 2.69/1.52 ( ( finite_finite2 @ A @ A4 )
% 2.69/1.52 => ( ( A4
% 2.69/1.52 != ( bot_bot @ ( set @ A ) ) )
% 2.69/1.52 => ( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
% 2.69/1.52 => ( ! [X6: A] :
% 2.69/1.52 ( ( member @ A @ X6 @ A4 )
% 2.69/1.52 => ( P @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
% 2.69/1.52 => ( P @ A4 ) ) ) ) )
% 2.69/1.52 => ( P @ B3 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_216_remove__induct,axiom,
% 2.69/1.52 ! [A: $tType,P: ( set @ A ) > $o,B3: set @ A] :
% 2.69/1.52 ( ( P @ ( bot_bot @ ( set @ A ) ) )
% 2.69/1.52 => ( ( ~ ( finite_finite2 @ A @ B3 )
% 2.69/1.52 => ( P @ B3 ) )
% 2.69/1.52 => ( ! [A4: set @ A] :
% 2.69/1.52 ( ( finite_finite2 @ A @ A4 )
% 2.69/1.52 => ( ( A4
% 2.69/1.52 != ( bot_bot @ ( set @ A ) ) )
% 2.69/1.52 => ( ( ord_less_eq @ ( set @ A ) @ A4 @ B3 )
% 2.69/1.52 => ( ! [X6: A] :
% 2.69/1.52 ( ( member @ A @ X6 @ A4 )
% 2.69/1.52 => ( P @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
% 2.69/1.52 => ( P @ A4 ) ) ) ) )
% 2.69/1.52 => ( P @ B3 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_217_minus__apply,axiom,
% 2.69/1.52 ! [B: $tType,A: $tType] :
% 2.69/1.52 ( ( minus @ B @ ( type2 @ B ) )
% 2.69/1.52 => ( ( minus_minus @ ( A > B ) )
% 2.69/1.52 = ( ^ [A7: A > B,B5: A > B,X4: A] : ( minus_minus @ B @ ( A7 @ X4 ) @ ( B5 @ X4 ) ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_218_DiffI,axiom,
% 2.69/1.52 ! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
% 2.69/1.52 ( ( member @ A @ C2 @ A3 )
% 2.69/1.52 => ( ~ ( member @ A @ C2 @ B3 )
% 2.69/1.52 => ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_219_Diff__iff,axiom,
% 2.69/1.52 ! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
% 2.69/1.52 ( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
% 2.69/1.52 = ( ( member @ A @ C2 @ A3 )
% 2.69/1.52 & ~ ( member @ A @ C2 @ B3 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_220_Diff__idemp,axiom,
% 2.69/1.52 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 2.69/1.52 ( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) @ B3 )
% 2.69/1.52 = ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_221_minus__diff__eq,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( group_add @ A @ ( type2 @ A ) )
% 2.69/1.52 => ! [A2: A,B4: A] :
% 2.69/1.52 ( ( uminus_uminus @ A @ ( minus_minus @ A @ A2 @ B4 ) )
% 2.69/1.52 = ( minus_minus @ A @ B4 @ A2 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_222_Diff__empty,axiom,
% 2.69/1.52 ! [A: $tType,A3: set @ A] :
% 2.69/1.52 ( ( minus_minus @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
% 2.69/1.52 = A3 ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_223_empty__Diff,axiom,
% 2.69/1.52 ! [A: $tType,A3: set @ A] :
% 2.69/1.52 ( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 )
% 2.69/1.52 = ( bot_bot @ ( set @ A ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_224_Diff__cancel,axiom,
% 2.69/1.52 ! [A: $tType,A3: set @ A] :
% 2.69/1.52 ( ( minus_minus @ ( set @ A ) @ A3 @ A3 )
% 2.69/1.52 = ( bot_bot @ ( set @ A ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_225_finite__Diff2,axiom,
% 2.69/1.52 ! [A: $tType,B3: set @ A,A3: set @ A] :
% 2.69/1.52 ( ( finite_finite2 @ A @ B3 )
% 2.69/1.52 => ( ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
% 2.69/1.52 = ( finite_finite2 @ A @ A3 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_226_finite__Diff,axiom,
% 2.69/1.52 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 2.69/1.52 ( ( finite_finite2 @ A @ A3 )
% 2.69/1.52 => ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_227_insert__Diff1,axiom,
% 2.69/1.52 ! [A: $tType,X3: A,B3: set @ A,A3: set @ A] :
% 2.69/1.52 ( ( member @ A @ X3 @ B3 )
% 2.69/1.52 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X3 @ A3 ) @ B3 )
% 2.69/1.52 = ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_228_Diff__insert0,axiom,
% 2.69/1.52 ! [A: $tType,X3: A,A3: set @ A,B3: set @ A] :
% 2.69/1.52 ( ~ ( member @ A @ X3 @ A3 )
% 2.69/1.52 => ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X3 @ B3 ) )
% 2.69/1.52 = ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_229_Diff__UNIV,axiom,
% 2.69/1.52 ! [A: $tType,A3: set @ A] :
% 2.69/1.52 ( ( minus_minus @ ( set @ A ) @ A3 @ ( top_top @ ( set @ A ) ) )
% 2.69/1.52 = ( bot_bot @ ( set @ A ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_230_Diff__eq__empty__iff,axiom,
% 2.69/1.52 ! [A: $tType,A3: set @ A,B3: set @ A] :
% 2.69/1.52 ( ( ( minus_minus @ ( set @ A ) @ A3 @ B3 )
% 2.69/1.52 = ( bot_bot @ ( set @ A ) ) )
% 2.69/1.52 = ( ord_less_eq @ ( set @ A ) @ A3 @ B3 ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_231_insert__Diff__single,axiom,
% 2.69/1.52 ! [A: $tType,A2: A,A3: set @ A] :
% 2.69/1.52 ( ( insert @ A @ A2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.69/1.52 = ( insert @ A @ A2 @ A3 ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_232_finite__Diff__insert,axiom,
% 2.69/1.52 ! [A: $tType,A3: set @ A,A2: A,B3: set @ A] :
% 2.69/1.52 ( ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B3 ) ) )
% 2.69/1.52 = ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_233_Diff__mono,axiom,
% 2.69/1.52 ! [A: $tType,A3: set @ A,C3: set @ A,D: set @ A,B3: set @ A] :
% 2.69/1.52 ( ( ord_less_eq @ ( set @ A ) @ A3 @ C3 )
% 2.69/1.52 => ( ( ord_less_eq @ ( set @ A ) @ D @ B3 )
% 2.69/1.52 => ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) @ ( minus_minus @ ( set @ A ) @ C3 @ D ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_234_Diff__subset,axiom,
% 2.69/1.52 ! [A: $tType,A3: set @ A,B3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) @ A3 ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_235_double__diff,axiom,
% 2.69/1.52 ! [A: $tType,A3: set @ A,B3: set @ A,C3: set @ A] :
% 2.69/1.52 ( ( ord_less_eq @ ( set @ A ) @ A3 @ B3 )
% 2.69/1.52 => ( ( ord_less_eq @ ( set @ A ) @ B3 @ C3 )
% 2.69/1.52 => ( ( minus_minus @ ( set @ A ) @ B3 @ ( minus_minus @ ( set @ A ) @ C3 @ A3 ) )
% 2.69/1.52 = A3 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_236_diff__mono,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.69/1.52 => ! [A2: A,B4: A,D2: A,C2: A] :
% 2.69/1.52 ( ( ord_less_eq @ A @ A2 @ B4 )
% 2.69/1.52 => ( ( ord_less_eq @ A @ D2 @ C2 )
% 2.69/1.52 => ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B4 @ D2 ) ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_237_diff__left__mono,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.69/1.52 => ! [B4: A,A2: A,C2: A] :
% 2.69/1.52 ( ( ord_less_eq @ A @ B4 @ A2 )
% 2.69/1.52 => ( ord_less_eq @ A @ ( minus_minus @ A @ C2 @ A2 ) @ ( minus_minus @ A @ C2 @ B4 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_238_diff__right__mono,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.69/1.52 => ! [A2: A,B4: A,C2: A] :
% 2.69/1.52 ( ( ord_less_eq @ A @ A2 @ B4 )
% 2.69/1.52 => ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B4 @ C2 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_239_diff__eq__diff__less__eq,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.69/1.52 => ! [A2: A,B4: A,C2: A,D2: A] :
% 2.69/1.52 ( ( ( minus_minus @ A @ A2 @ B4 )
% 2.69/1.52 = ( minus_minus @ A @ C2 @ D2 ) )
% 2.69/1.52 => ( ( ord_less_eq @ A @ A2 @ B4 )
% 2.69/1.52 = ( ord_less_eq @ A @ C2 @ D2 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_240_subset__Diff__insert,axiom,
% 2.69/1.52 ! [A: $tType,A3: set @ A,B3: set @ A,X3: A,C3: set @ A] :
% 2.69/1.52 ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ B3 @ ( insert @ A @ X3 @ C3 ) ) )
% 2.69/1.52 = ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ B3 @ C3 ) )
% 2.69/1.52 & ~ ( member @ A @ X3 @ A3 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_241_Compl__eq__Diff__UNIV,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( uminus_uminus @ ( set @ A ) )
% 2.69/1.52 = ( minus_minus @ ( set @ A ) @ ( top_top @ ( set @ A ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_242_DiffE,axiom,
% 2.69/1.52 ! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
% 2.69/1.52 ( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
% 2.69/1.52 => ~ ( ( member @ A @ C2 @ A3 )
% 2.69/1.52 => ( member @ A @ C2 @ B3 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_243_DiffD1,axiom,
% 2.69/1.52 ! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
% 2.69/1.52 ( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
% 2.69/1.52 => ( member @ A @ C2 @ A3 ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_244_DiffD2,axiom,
% 2.69/1.52 ! [A: $tType,C2: A,A3: set @ A,B3: set @ A] :
% 2.69/1.52 ( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) )
% 2.69/1.52 => ~ ( member @ A @ C2 @ B3 ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_245_insert__Diff__if,axiom,
% 2.69/1.52 ! [A: $tType,X3: A,B3: set @ A,A3: set @ A] :
% 2.69/1.52 ( ( ( member @ A @ X3 @ B3 )
% 2.69/1.52 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X3 @ A3 ) @ B3 )
% 2.69/1.52 = ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) )
% 2.69/1.52 & ( ~ ( member @ A @ X3 @ B3 )
% 2.69/1.52 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X3 @ A3 ) @ B3 )
% 2.69/1.52 = ( insert @ A @ X3 @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_246_Diff__infinite__finite,axiom,
% 2.69/1.52 ! [A: $tType,T3: set @ A,S: set @ A] :
% 2.69/1.52 ( ( finite_finite2 @ A @ T3 )
% 2.69/1.52 => ( ~ ( finite_finite2 @ A @ S )
% 2.69/1.52 => ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ S @ T3 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_247_fun__diff__def,axiom,
% 2.69/1.52 ! [B: $tType,A: $tType] :
% 2.69/1.52 ( ( minus @ B @ ( type2 @ B ) )
% 2.69/1.52 => ( ( minus_minus @ ( A > B ) )
% 2.69/1.52 = ( ^ [A7: A > B,B5: A > B,X4: A] : ( minus_minus @ B @ ( A7 @ X4 ) @ ( B5 @ X4 ) ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_248_diff__eq__diff__eq,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( group_add @ A @ ( type2 @ A ) )
% 2.69/1.52 => ! [A2: A,B4: A,C2: A,D2: A] :
% 2.69/1.52 ( ( ( minus_minus @ A @ A2 @ B4 )
% 2.69/1.52 = ( minus_minus @ A @ C2 @ D2 ) )
% 2.69/1.52 => ( ( A2 = B4 )
% 2.69/1.52 = ( C2 = D2 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_249_diff__right__commute,axiom,
% 2.69/1.52 ! [A: $tType] :
% 2.69/1.52 ( ( cancel146912293up_add @ A @ ( type2 @ A ) )
% 2.69/1.52 => ! [A2: A,C2: A,B4: A] :
% 2.69/1.52 ( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C2 ) @ B4 )
% 2.69/1.52 = ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B4 ) @ C2 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_250_Diff__insert__absorb,axiom,
% 2.69/1.52 ! [A: $tType,X3: A,A3: set @ A] :
% 2.69/1.52 ( ~ ( member @ A @ X3 @ A3 )
% 2.69/1.52 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X3 @ A3 ) @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.69/1.52 = A3 ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_251_Diff__insert2,axiom,
% 2.69/1.52 ! [A: $tType,A3: set @ A,A2: A,B3: set @ A] :
% 2.69/1.52 ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B3 ) )
% 2.69/1.52 = ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) @ B3 ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_252_insert__Diff,axiom,
% 2.69/1.52 ! [A: $tType,A2: A,A3: set @ A] :
% 2.69/1.52 ( ( member @ A @ A2 @ A3 )
% 2.69/1.52 => ( ( insert @ A @ A2 @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.69/1.52 = A3 ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(fact_253_Diff__insert,axiom,
% 2.69/1.52 ! [A: $tType,A3: set @ A,A2: A,B3: set @ A] :
% 2.69/1.52 ( ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B3 ) )
% 2.69/1.52 = ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B3 ) @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(tcon_fun___Lattices_Oboolean__algebra,axiom,
% 2.69/1.52 ! [A9: $tType,A10: $tType] :
% 2.69/1.52 ( ( boolean_algebra @ A10 @ ( type2 @ A10 ) )
% 2.69/1.52 => ( boolean_algebra @ ( A9 > A10 ) @ ( type2 @ ( A9 > A10 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(tcon_fun___Orderings_Oorder__top,axiom,
% 2.69/1.52 ! [A9: $tType,A10: $tType] :
% 2.69/1.52 ( ( order_top @ A10 @ ( type2 @ A10 ) )
% 2.69/1.52 => ( order_top @ ( A9 > A10 ) @ ( type2 @ ( A9 > A10 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(tcon_fun___Orderings_Oorder__bot,axiom,
% 2.69/1.52 ! [A9: $tType,A10: $tType] :
% 2.69/1.52 ( ( order_bot @ A10 @ ( type2 @ A10 ) )
% 2.69/1.52 => ( order_bot @ ( A9 > A10 ) @ ( type2 @ ( A9 > A10 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(tcon_fun___Orderings_Opreorder,axiom,
% 2.69/1.52 ! [A9: $tType,A10: $tType] :
% 2.69/1.52 ( ( preorder @ A10 @ ( type2 @ A10 ) )
% 2.69/1.52 => ( preorder @ ( A9 > A10 ) @ ( type2 @ ( A9 > A10 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(tcon_fun___Finite__Set_Ofinite,axiom,
% 2.69/1.52 ! [A9: $tType,A10: $tType] :
% 2.69/1.52 ( ( ( finite_finite @ A9 @ ( type2 @ A9 ) )
% 2.69/1.52 & ( finite_finite @ A10 @ ( type2 @ A10 ) ) )
% 2.69/1.52 => ( finite_finite @ ( A9 > A10 ) @ ( type2 @ ( A9 > A10 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(tcon_fun___Orderings_Oorder,axiom,
% 2.69/1.52 ! [A9: $tType,A10: $tType] :
% 2.69/1.52 ( ( order @ A10 @ ( type2 @ A10 ) )
% 2.69/1.52 => ( order @ ( A9 > A10 ) @ ( type2 @ ( A9 > A10 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(tcon_fun___Orderings_Otop,axiom,
% 2.69/1.52 ! [A9: $tType,A10: $tType] :
% 2.69/1.52 ( ( top @ A10 @ ( type2 @ A10 ) )
% 2.69/1.52 => ( top @ ( A9 > A10 ) @ ( type2 @ ( A9 > A10 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(tcon_fun___Orderings_Oord,axiom,
% 2.69/1.52 ! [A9: $tType,A10: $tType] :
% 2.69/1.52 ( ( ord @ A10 @ ( type2 @ A10 ) )
% 2.69/1.52 => ( ord @ ( A9 > A10 ) @ ( type2 @ ( A9 > A10 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(tcon_fun___Orderings_Obot,axiom,
% 2.69/1.52 ! [A9: $tType,A10: $tType] :
% 2.69/1.52 ( ( bot @ A10 @ ( type2 @ A10 ) )
% 2.69/1.52 => ( bot @ ( A9 > A10 ) @ ( type2 @ ( A9 > A10 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(tcon_fun___Groups_Ouminus,axiom,
% 2.69/1.52 ! [A9: $tType,A10: $tType] :
% 2.69/1.52 ( ( uminus @ A10 @ ( type2 @ A10 ) )
% 2.69/1.52 => ( uminus @ ( A9 > A10 ) @ ( type2 @ ( A9 > A10 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(tcon_fun___Groups_Ominus,axiom,
% 2.69/1.52 ! [A9: $tType,A10: $tType] :
% 2.69/1.52 ( ( minus @ A10 @ ( type2 @ A10 ) )
% 2.69/1.52 => ( minus @ ( A9 > A10 ) @ ( type2 @ ( A9 > A10 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(tcon_Set_Oset___Lattices_Oboolean__algebra_1,axiom,
% 2.69/1.52 ! [A9: $tType] : ( boolean_algebra @ ( set @ A9 ) @ ( type2 @ ( set @ A9 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(tcon_Set_Oset___Orderings_Oorder__top_2,axiom,
% 2.69/1.52 ! [A9: $tType] : ( order_top @ ( set @ A9 ) @ ( type2 @ ( set @ A9 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(tcon_Set_Oset___Orderings_Oorder__bot_3,axiom,
% 2.69/1.52 ! [A9: $tType] : ( order_bot @ ( set @ A9 ) @ ( type2 @ ( set @ A9 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(tcon_Set_Oset___Orderings_Opreorder_4,axiom,
% 2.69/1.52 ! [A9: $tType] : ( preorder @ ( set @ A9 ) @ ( type2 @ ( set @ A9 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(tcon_Set_Oset___Finite__Set_Ofinite_5,axiom,
% 2.69/1.52 ! [A9: $tType] :
% 2.69/1.52 ( ( finite_finite @ A9 @ ( type2 @ A9 ) )
% 2.69/1.52 => ( finite_finite @ ( set @ A9 ) @ ( type2 @ ( set @ A9 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(tcon_Set_Oset___Orderings_Oorder_6,axiom,
% 2.69/1.52 ! [A9: $tType] : ( order @ ( set @ A9 ) @ ( type2 @ ( set @ A9 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(tcon_Set_Oset___Orderings_Otop_7,axiom,
% 2.69/1.52 ! [A9: $tType] : ( top @ ( set @ A9 ) @ ( type2 @ ( set @ A9 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(tcon_Set_Oset___Orderings_Oord_8,axiom,
% 2.69/1.52 ! [A9: $tType] : ( ord @ ( set @ A9 ) @ ( type2 @ ( set @ A9 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(tcon_Set_Oset___Orderings_Obot_9,axiom,
% 2.69/1.52 ! [A9: $tType] : ( bot @ ( set @ A9 ) @ ( type2 @ ( set @ A9 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(tcon_Set_Oset___Groups_Ouminus_10,axiom,
% 2.69/1.52 ! [A9: $tType] : ( uminus @ ( set @ A9 ) @ ( type2 @ ( set @ A9 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(tcon_Set_Oset___Groups_Ominus_11,axiom,
% 2.69/1.52 ! [A9: $tType] : ( minus @ ( set @ A9 ) @ ( type2 @ ( set @ A9 ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_12,axiom,
% 2.69/1.52 boolean_algebra @ $o @ ( type2 @ $o ) ).
% 2.69/1.52
% 2.69/1.52 thf(tcon_HOL_Obool___Orderings_Oorder__top_13,axiom,
% 2.69/1.52 order_top @ $o @ ( type2 @ $o ) ).
% 2.69/1.52
% 2.69/1.52 thf(tcon_HOL_Obool___Orderings_Oorder__bot_14,axiom,
% 2.69/1.52 order_bot @ $o @ ( type2 @ $o ) ).
% 2.69/1.52
% 2.69/1.52 thf(tcon_HOL_Obool___Orderings_Opreorder_15,axiom,
% 2.69/1.52 preorder @ $o @ ( type2 @ $o ) ).
% 2.69/1.52
% 2.69/1.52 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 2.69/1.52 linorder @ $o @ ( type2 @ $o ) ).
% 2.69/1.52
% 2.69/1.52 thf(tcon_HOL_Obool___Finite__Set_Ofinite_16,axiom,
% 2.69/1.52 finite_finite @ $o @ ( type2 @ $o ) ).
% 2.69/1.52
% 2.69/1.52 thf(tcon_HOL_Obool___Orderings_Oorder_17,axiom,
% 2.69/1.52 order @ $o @ ( type2 @ $o ) ).
% 2.69/1.52
% 2.69/1.52 thf(tcon_HOL_Obool___Orderings_Otop_18,axiom,
% 2.69/1.52 top @ $o @ ( type2 @ $o ) ).
% 2.69/1.52
% 2.69/1.52 thf(tcon_HOL_Obool___Orderings_Oord_19,axiom,
% 2.69/1.52 ord @ $o @ ( type2 @ $o ) ).
% 2.69/1.52
% 2.69/1.52 thf(tcon_HOL_Obool___Orderings_Obot_20,axiom,
% 2.69/1.52 bot @ $o @ ( type2 @ $o ) ).
% 2.69/1.52
% 2.69/1.52 thf(tcon_HOL_Obool___Groups_Ouminus_21,axiom,
% 2.69/1.52 uminus @ $o @ ( type2 @ $o ) ).
% 2.69/1.52
% 2.69/1.52 thf(tcon_HOL_Obool___Groups_Ominus_22,axiom,
% 2.69/1.52 minus @ $o @ ( type2 @ $o ) ).
% 2.69/1.52
% 2.69/1.52 thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_23,axiom,
% 2.69/1.52 ! [A9: $tType,A10: $tType] :
% 2.69/1.52 ( ( ( finite_finite @ A9 @ ( type2 @ A9 ) )
% 2.69/1.52 & ( finite_finite @ A10 @ ( type2 @ A10 ) ) )
% 2.69/1.52 => ( finite_finite @ ( sum_sum @ A9 @ A10 ) @ ( type2 @ ( sum_sum @ A9 @ A10 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(tcon_Option_Ooption___Finite__Set_Ofinite_24,axiom,
% 2.69/1.52 ! [A9: $tType] :
% 2.69/1.52 ( ( finite_finite @ A9 @ ( type2 @ A9 ) )
% 2.69/1.52 => ( finite_finite @ ( option @ A9 ) @ ( type2 @ ( option @ A9 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_25,axiom,
% 2.69/1.52 ! [A9: $tType,A10: $tType] :
% 2.69/1.52 ( ( ( finite_finite @ A9 @ ( type2 @ A9 ) )
% 2.69/1.52 & ( finite_finite @ A10 @ ( type2 @ A10 ) ) )
% 2.69/1.52 => ( finite_finite @ ( product_prod @ A9 @ A10 ) @ ( type2 @ ( product_prod @ A9 @ A10 ) ) ) ) ).
% 2.69/1.52
% 2.69/1.52 thf(conj_0,conjecture,
% 2.69/1.52 ? [N: node,Ns: set @ node] :
% 2.69/1.52 ( ( ( f @ ( product_Pair @ node @ ( set @ node ) @ n @ ( bot_bot @ ( set @ node ) ) ) )
% 2.69/1.52 = ( f @ ( product_Pair @ node @ ( set @ node ) @ N @ Ns ) ) )
% 2.69/1.52 & ( finite_finite2 @ node @ Ns )
% 2.69/1.52 & ~ ( finite_finite2 @ node @ ( koenig317145564le_via @ node @ graph @ ( uminus_uminus @ ( set @ node ) @ ( insert @ node @ N @ Ns ) ) @ N ) ) ) ).
% 2.69/1.52 ------------------------
% 2.69/1.55 % (22378)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_22201 for (2999ds/183Mi)
% 2.69/1.55 % (22379)lrs+10_1:1_c=on:cnfonf=conj_eager:fd=off:fe=off:kws=frequency:spb=intro:i=4:si=on:rtra=on_0 on DTF2THF_22201 for (2999ds/4Mi)
% 2.69/1.55 % (22379)Instruction limit reached!
% 2.69/1.55 % (22379)------------------------------
% 2.69/1.55 % (22379)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.55 % (22379)Termination reason: Unknown
% 2.69/1.55 % (22379)Termination phase: shuffling
% 2.69/1.55
% 2.69/1.55 % (22379)Memory used [KB]: 1407
% 2.69/1.55 % (22379)Time elapsed: 0.003 s
% 2.69/1.55 % (22379)Instructions burned: 5 (million)
% 2.69/1.55 % (22379)------------------------------
% 2.69/1.55 % (22379)------------------------------
% 2.69/1.55 % (22384)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_22201 for (2999ds/18Mi)
% 2.69/1.55 % (22380)dis+1010_1:1_au=on:cbe=off:chr=on:fsr=off:hfsq=on:nm=64:sos=theory:sp=weighted_frequency:i=27:si=on:rtra=on_0 on DTF2THF_22201 for (2999ds/27Mi)
% 2.69/1.56 % (22381)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_22201 for (2999ds/2Mi)
% 2.69/1.56 % (22382)lrs+1002_1:128_aac=none:au=on:cnfonf=lazy_not_gen_be_off:sos=all:i=2:si=on:rtra=on_0 on DTF2THF_22201 for (2999ds/2Mi)
% 2.69/1.56 % (22383)lrs+1002_1:1_au=on:bd=off:e2e=on:sd=2:sos=on:ss=axioms:i=275:si=on:rtra=on_0 on DTF2THF_22201 for (2999ds/275Mi)
% 2.69/1.56 % (22384)Instruction limit reached!
% 2.69/1.56 % (22384)------------------------------
% 2.69/1.56 % (22384)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.56 % (22384)Termination reason: Unknown
% 2.69/1.56 % (22384)Termination phase: shuffling
% 2.69/1.56
% 2.69/1.56 % (22384)Memory used [KB]: 1663
% 2.69/1.56 % (22384)Time elapsed: 0.009 s
% 2.69/1.56 % (22384)Instructions burned: 18 (million)
% 2.69/1.56 % (22384)------------------------------
% 2.69/1.56 % (22384)------------------------------
% 2.69/1.56 % (22381)Instruction limit reached!
% 2.69/1.56 % (22381)------------------------------
% 2.69/1.56 % (22381)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.56 % (22381)Termination reason: Unknown
% 2.69/1.56 % (22381)Termination phase: shuffling
% 2.69/1.56
% 2.69/1.56 % (22381)Memory used [KB]: 1279
% 2.69/1.56 % (22381)Time elapsed: 0.004 s
% 2.69/1.56 % (22381)Instructions burned: 3 (million)
% 2.69/1.56 % (22381)------------------------------
% 2.69/1.56 % (22381)------------------------------
% 2.69/1.56 % (22382)Instruction limit reached!
% 2.69/1.56 % (22382)------------------------------
% 2.69/1.56 % (22382)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.56 % (22382)Termination reason: Unknown
% 2.69/1.56 % (22382)Termination phase: shuffling
% 2.69/1.56
% 2.69/1.56 % (22382)Memory used [KB]: 1407
% 2.69/1.56 % (22382)Time elapsed: 0.004 s
% 2.69/1.56 % (22382)Instructions burned: 3 (million)
% 2.69/1.56 % (22382)------------------------------
% 2.69/1.56 % (22382)------------------------------
% 2.69/1.56 % (22385)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_22201 for (2999ds/3Mi)
% 2.69/1.56 % (22385)Instruction limit reached!
% 2.69/1.56 % (22385)------------------------------
% 2.69/1.56 % (22385)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.56 % (22385)Termination reason: Unknown
% 2.69/1.56 % (22385)Termination phase: shuffling
% 2.69/1.56
% 2.69/1.56 % (22385)Memory used [KB]: 1407
% 2.69/1.56 % (22385)Time elapsed: 0.002 s
% 2.69/1.56 % (22385)Instructions burned: 3 (million)
% 2.69/1.56 % (22385)------------------------------
% 2.69/1.56 % (22385)------------------------------
% 2.69/1.57 % (22392)lrs+1002_1:1_cnfonf=lazy_not_be_gen:hud=14:prag=on:sp=weighted_frequency:tnu=1:i=37:si=on:rtra=on_0 on DTF2THF_22201 for (2999ds/37Mi)
% 2.69/1.57 % (22399)lrs+1002_1:1_aac=none:au=on:cnfonf=lazy_gen:plsq=on:plsqc=1:plsqr=4203469,65536:i=1041:si=on:rtra=on_0 on DTF2THF_22201 for (2999ds/1041Mi)
% 2.69/1.57 % (22380)Instruction limit reached!
% 2.69/1.57 % (22380)------------------------------
% 2.69/1.57 % (22380)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.57 % (22380)Termination reason: Unknown
% 2.69/1.57 % (22380)Termination phase: Property scanning
% 2.69/1.57
% 2.69/1.57 % (22380)Memory used [KB]: 1663
% 2.69/1.57 % (22380)Time elapsed: 0.019 s
% 2.69/1.57 % (22380)Instructions burned: 28 (million)
% 2.69/1.57 % (22380)------------------------------
% 2.69/1.57 % (22380)------------------------------
% 2.69/1.57 % (22395)lrs+2_16:1_acc=model:au=on:bd=off:c=on:e2e=on:nm=2:sos=all:i=15:si=on:rtra=on_0 on DTF2THF_22201 for (2999ds/15Mi)
% 2.69/1.57 % (22396)dis+21_1:1_cbe=off:cnfonf=off:fs=off:fsr=off:hud=1:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_22201 for (2999ds/3Mi)
% 2.69/1.58 % (22396)Instruction limit reached!
% 2.69/1.58 % (22396)------------------------------
% 2.69/1.58 % (22396)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.58 % (22396)Termination reason: Unknown
% 2.69/1.58 % (22396)Termination phase: shuffling
% 2.69/1.58
% 2.69/1.58 % (22396)Memory used [KB]: 1407
% 2.69/1.58 % (22396)Time elapsed: 0.004 s
% 2.69/1.58 % (22396)Instructions burned: 3 (million)
% 2.69/1.58 % (22396)------------------------------
% 2.69/1.58 % (22396)------------------------------
% 2.69/1.58 % (22392)Instruction limit reached!
% 2.69/1.58 % (22392)------------------------------
% 2.69/1.58 % (22392)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.58 % (22392)Termination reason: Unknown
% 2.69/1.58 % (22392)Termination phase: Property scanning
% 2.69/1.58
% 2.69/1.58 % (22392)Memory used [KB]: 2046
% 2.69/1.58 % (22392)Time elapsed: 0.013 s
% 2.69/1.58 % (22392)Instructions burned: 37 (million)
% 2.69/1.58 % (22392)------------------------------
% 2.69/1.58 % (22392)------------------------------
% 2.69/1.58 % (22395)Instruction limit reached!
% 2.69/1.58 % (22395)------------------------------
% 2.69/1.58 % (22395)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.58 % (22395)Termination reason: Unknown
% 2.69/1.58 % (22395)Termination phase: shuffling
% 2.69/1.58
% 2.69/1.58 % (22395)Memory used [KB]: 1663
% 2.69/1.58 % (22395)Time elapsed: 0.009 s
% 2.69/1.58 % (22395)Instructions burned: 15 (million)
% 2.69/1.58 % (22395)------------------------------
% 2.69/1.58 % (22395)------------------------------
% 2.69/1.59 % (22412)lrs+21_1:1_au=on:cnfonf=off:fd=preordered:fe=off:fsr=off:hud=11:inj=on:kws=precedence:s2pl=no:sp=weighted_frequency:tgt=full:i=3:si=on:rtra=on_0 on DTF2THF_22201 for (2999ds/3Mi)
% 2.69/1.59 % (22412)Instruction limit reached!
% 2.69/1.59 % (22412)------------------------------
% 2.69/1.59 % (22412)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.59 % (22412)Termination reason: Unknown
% 2.69/1.59 % (22412)Termination phase: shuffling
% 2.69/1.59
% 2.69/1.59 % (22412)Memory used [KB]: 1407
% 2.69/1.59 % (22412)Time elapsed: 0.003 s
% 2.69/1.59 % (22412)Instructions burned: 3 (million)
% 2.69/1.59 % (22412)------------------------------
% 2.69/1.59 % (22412)------------------------------
% 2.69/1.59 % (22405)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_22201 for (2999ds/7Mi)
% 2.69/1.59 % (22413)lrs+2_1:1_apa=on:au=on:bd=preordered:cnfonf=off:cs=on:ixr=off:sos=on:i=3:si=on:rtra=on_0 on DTF2THF_22201 for (2999ds/3Mi)
% 2.69/1.59 % (22413)Instruction limit reached!
% 2.69/1.59 % (22413)------------------------------
% 2.69/1.59 % (22413)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.59 % (22413)Termination reason: Unknown
% 2.69/1.59 % (22413)Termination phase: shuffling
% 2.69/1.59
% 2.69/1.59 % (22413)Memory used [KB]: 1407
% 2.69/1.59 % (22413)Time elapsed: 0.003 s
% 2.69/1.59 % (22413)Instructions burned: 4 (million)
% 2.69/1.59 % (22413)------------------------------
% 2.69/1.59 % (22413)------------------------------
% 2.69/1.59 % (22409)lrs+10_1:1_acc=on:amm=sco:cs=on:tgt=full:i=16:si=on:rtra=on_0 on DTF2THF_22201 for (2999ds/16Mi)
% 2.69/1.59 % (22405)Instruction limit reached!
% 2.69/1.59 % (22405)------------------------------
% 2.69/1.59 % (22405)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.69/1.59 % (22405)Termination reason: Unknown
% 2.69/1.59 % (22405)Termination phase: shuffling
% 2.69/1.59
% 2.69/1.59 % (22405)Memory used [KB]: 1407
% 2.69/1.59 % (22405)Time elapsed: 0.007 s
% 2.69/1.59 % (22405)Instructions burned: 8 (million)
% 2.69/1.59 % (22405)------------------------------
% 2.69/1.59 % (22405)------------------------------
% 2.71/1.60 % (22414)lrs+10_1:1_cnfonf=off:cs=on:hud=3:prag=on:sup=off:i=7:si=on:rtra=on_0 on DTF2THF_22201 for (2999ds/7Mi)
% 2.71/1.60 % (22414)Instruction limit reached!
% 2.71/1.60 % (22414)------------------------------
% 2.71/1.60 % (22414)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.60 % (22414)Termination reason: Unknown
% 2.71/1.60 % (22414)Termination phase: shuffling
% 2.71/1.60
% 2.71/1.60 % (22414)Memory used [KB]: 1535
% 2.71/1.60 % (22414)Time elapsed: 0.005 s
% 2.71/1.60 % (22414)Instructions burned: 9 (million)
% 2.71/1.60 % (22414)------------------------------
% 2.71/1.60 % (22414)------------------------------
% 2.71/1.60 % (22378)Instruction limit reached!
% 2.71/1.60 % (22378)------------------------------
% 2.71/1.60 % (22378)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.60 % (22378)Termination reason: Unknown
% 2.71/1.60 % (22378)Termination phase: Saturation
% 2.71/1.60
% 2.71/1.60 % (22378)Memory used [KB]: 7164
% 2.71/1.60 % (22378)Time elapsed: 0.058 s
% 2.71/1.60 % (22378)Instructions burned: 183 (million)
% 2.71/1.60 % (22378)------------------------------
% 2.71/1.60 % (22378)------------------------------
% 2.71/1.60 % (22415)dis+1002_1:1_add=large:cnfonf=lazy_pi_sigma_gen:fe=off:prag=on:i=3:si=on:rtra=on_0 on DTF2THF_22201 for (2999ds/3Mi)
% 2.71/1.60 % (22409)Instruction limit reached!
% 2.71/1.60 % (22409)------------------------------
% 2.71/1.60 % (22409)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.60 % (22409)Termination reason: Unknown
% 2.71/1.60 % (22409)Termination phase: shuffling
% 2.71/1.60
% 2.71/1.60 % (22409)Memory used [KB]: 1663
% 2.71/1.60 % (22409)Time elapsed: 0.013 s
% 2.71/1.60 % (22409)Instructions burned: 17 (million)
% 2.71/1.60 % (22409)------------------------------
% 2.71/1.60 % (22409)------------------------------
% 2.71/1.60 % (22415)Instruction limit reached!
% 2.71/1.60 % (22415)------------------------------
% 2.71/1.60 % (22415)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.60 % (22415)Termination reason: Unknown
% 2.71/1.60 % (22415)Termination phase: shuffling
% 2.71/1.60
% 2.71/1.60 % (22415)Memory used [KB]: 1407
% 2.71/1.60 % (22415)Time elapsed: 0.003 s
% 2.71/1.60 % (22415)Instructions burned: 5 (million)
% 2.71/1.60 % (22415)------------------------------
% 2.71/1.60 % (22415)------------------------------
% 2.71/1.61 % (22383)First to succeed.
% 2.71/1.61 % (22416)dis+1004_1:1_cha=on:cs=on:fe=off:hud=1:i=4:si=on:rtra=on_0 on DTF2THF_22201 for (2999ds/4Mi)
% 2.71/1.61 % (22416)Instruction limit reached!
% 2.71/1.61 % (22416)------------------------------
% 2.71/1.61 % (22416)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.61 % (22416)Termination reason: Unknown
% 2.71/1.61 % (22416)Termination phase: shuffling
% 2.71/1.61
% 2.71/1.61 % (22416)Memory used [KB]: 1407
% 2.71/1.61 % (22416)Time elapsed: 0.004 s
% 2.71/1.61 % (22416)Instructions burned: 6 (million)
% 2.71/1.61 % (22416)------------------------------
% 2.71/1.61 % (22416)------------------------------
% 2.71/1.61 % (22383)Refutation found. Thanks to Tanya!
% 2.71/1.61 % SZS status Theorem for DTF2THF_22201
% 2.71/1.61 % SZS output start Proof for DTF2THF_22201
% 2.71/1.61 thf(type_def_5, type, itself: ($tType) > $tType).
% 2.71/1.61 thf(type_def_7, type, product_prod: ($tType * $tType) > $tType).
% 2.71/1.61 thf(type_def_8, type, set: ($tType) > $tType).
% 2.71/1.61 thf(type_def_9, type, coinductive_llist: ($tType) > $tType).
% 2.71/1.61 thf(type_def_10, type, node: $tType).
% 2.71/1.61 thf(type_def_11, type, sum_sum: ($tType * $tType) > $tType).
% 2.71/1.61 thf(type_def_12, type, option: ($tType) > $tType).
% 2.71/1.61 thf(func_def_0, type, coinductive_llist: ($tType) > $tType).
% 2.71/1.61 thf(func_def_1, type, product_prod: ($tType * $tType) > $tType).
% 2.71/1.61 thf(func_def_2, type, sum_sum: ($tType * $tType) > $tType).
% 2.71/1.61 thf(func_def_3, type, option: ($tType) > $tType).
% 2.71/1.61 thf(func_def_4, type, set: ($tType) > $tType).
% 2.71/1.61 thf(func_def_5, type, itself: ($tType) > $tType).
% 2.71/1.61 thf(func_def_6, type, node: $tType).
% 2.71/1.61 thf(func_def_7, type, type: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.71/1.61 thf(func_def_8, type, minus: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.71/1.61 thf(func_def_9, type, uminus: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.71/1.61 thf(func_def_10, type, bot: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.71/1.61 thf(func_def_11, type, ord: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.71/1.61 thf(func_def_12, type, top: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.71/1.61 thf(func_def_13, type, order: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.71/1.61 thf(func_def_14, type, group_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.71/1.61 thf(func_def_15, type, finite_finite: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.71/1.61 thf(func_def_16, type, linorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.71/1.61 thf(func_def_17, type, preorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.71/1.61 thf(func_def_18, type, semiring_char_0: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.71/1.61 thf(func_def_19, type, order_bot: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.71/1.61 thf(func_def_20, type, order_top: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.71/1.61 thf(func_def_21, type, boolean_algebra: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.71/1.61 thf(func_def_22, type, ordered_ab_group_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.71/1.61 thf(func_def_23, type, cancel146912293up_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.71/1.61 thf(func_def_24, type, topolo890362671_space: !>[X0: $tType]:(itself @ X0 > $o)).
% 2.71/1.61 thf(func_def_25, type, basic_fsts: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > set @ X0)).
% 2.71/1.61 thf(func_def_26, type, basic_snds: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > set @ X1)).
% 2.71/1.61 thf(func_def_27, type, coinductive_gen_lset: !>[X0: $tType]:(set @ X0 > coinductive_llist @ X0 > set @ X0)).
% 2.71/1.61 thf(func_def_28, type, coinductive_lfinite: !>[X0: $tType]:(coinductive_llist @ X0 > $o)).
% 2.71/1.61 thf(func_def_29, type, coinductive_lnull: !>[X0: $tType]:(coinductive_llist @ X0 > $o)).
% 2.71/1.61 thf(func_def_30, type, coinductive_lset: !>[X0: $tType]:(coinductive_llist @ X0 > set @ X0)).
% 2.71/1.61 thf(func_def_31, type, finite_finite2: !>[X0: $tType]:(set @ X0 > $o)).
% 2.71/1.61 thf(func_def_32, type, minus_minus: !>[X0: $tType]:(X0 > X0 > X0)).
% 2.71/1.61 thf(func_def_33, type, uminus_uminus: !>[X0: $tType]:(X0 > X0)).
% 2.71/1.61 thf(func_def_34, type, koenig793108494nected: !>[X0: $tType]:((X0 > X0 > $o) > $o)).
% 2.71/1.61 thf(func_def_35, type, koenig317145564le_via: !>[X0: $tType]:((X0 > X0 > $o) > set @ X0 > X0 > set @ X0)).
% 2.71/1.61 thf(func_def_36, type, bot_bot: !>[X0: $tType]:(X0)).
% 2.71/1.61 thf(func_def_37, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 2.71/1.61 thf(func_def_38, type, top_top: !>[X0: $tType]:(X0)).
% 2.71/1.61 thf(func_def_39, type, product_Pair: !>[X0: $tType, X1: $tType]:(X0 > X1 > product_prod @ X0 @ X1)).
% 2.71/1.61 thf(func_def_40, type, produc2004651681e_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 2.71/1.61 thf(func_def_41, type, product_rec_prod: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1 > X2) > product_prod @ X0 @ X1 > X2)).
% 2.71/1.61 thf(func_def_42, type, type2: !>[X0: $tType]:(itself @ X0)).
% 2.71/1.61 thf(func_def_43, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 2.71/1.61 thf(func_def_44, type, insert: !>[X0: $tType]:(X0 > set @ X0 > set @ X0)).
% 2.71/1.61 thf(func_def_45, type, is_empty: !>[X0: $tType]:(set @ X0 > $o)).
% 2.71/1.61 thf(func_def_46, type, is_singleton: !>[X0: $tType]:(set @ X0 > $o)).
% 2.71/1.61 thf(func_def_47, type, the_elem: !>[X0: $tType]:(set @ X0 > X0)).
% 2.71/1.61 thf(func_def_48, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 2.71/1.61 thf(func_def_49, type, f: product_prod @ node @ set @ node > coinductive_llist @ node).
% 2.71/1.61 thf(func_def_50, type, graph: node > node > $o).
% 2.71/1.61 thf(func_def_51, type, n: node).
% 2.71/1.61 thf(func_def_52, type, ns: set @ node).
% 2.71/1.61 thf(func_def_66, type, sP0: !>[X0: $tType]:(X0 > X0 > set @ X0 > set @ X0 > $o)).
% 2.71/1.61 thf(func_def_67, type, sP1: !>[X0: $tType]:(X0 > X0 > set @ X0 > set @ X0 > $o)).
% 2.71/1.61 thf(func_def_68, type, sP2: !>[X0: $tType]:((set @ X0 > $o) > set @ X0 > $o)).
% 2.71/1.61 thf(func_def_69, type, sK3: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X0)).
% 2.71/1.61 thf(func_def_70, type, sK4: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X1)).
% 2.71/1.61 thf(func_def_71, type, sK5: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X2)).
% 2.71/1.61 thf(func_def_72, type, sK6: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X3)).
% 2.71/1.61 thf(func_def_73, type, sK7: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X4)).
% 2.71/1.61 thf(func_def_74, type, sK8: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X5)).
% 2.71/1.61 thf(func_def_75, type, sK9: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ product_prod @ X5 @ X6 > X6)).
% 2.71/1.61 thf(func_def_76, type, sK10: !>[X0: $tType]:(set @ X0 > set @ X0)).
% 2.71/1.61 thf(func_def_77, type, sK11: !>[X0: $tType]:(set @ X0 > X0)).
% 2.71/1.61 thf(func_def_78, type, sK12: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 2.71/1.61 thf(func_def_79, type, sK13: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 2.71/1.61 thf(func_def_80, type, sK14: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X4)).
% 2.71/1.61 thf(func_def_81, type, sK15: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 2.71/1.61 thf(func_def_82, type, sK16: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 2.71/1.61 thf(func_def_83, type, sK17: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 2.71/1.61 thf(func_def_84, type, sK18: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:((product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 2.71/1.61 thf(func_def_85, type, sK19: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X0)).
% 2.71/1.61 thf(func_def_86, type, sK20: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X1)).
% 2.71/1.61 thf(func_def_87, type, sK21: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X2)).
% 2.71/1.61 thf(func_def_88, type, sK22: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ X3 > X3)).
% 2.71/1.61 thf(func_def_89, type, sK23: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X5)).
% 2.71/1.61 thf(func_def_90, type, sK24: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X4)).
% 2.71/1.61 thf(func_def_91, type, sK25: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 2.71/1.61 thf(func_def_92, type, sK26: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 2.71/1.61 thf(func_def_93, type, sK27: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 2.71/1.61 thf(func_def_94, type, sK28: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:((product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 2.71/1.61 thf(func_def_95, type, sK29: !>[X0: $tType]:(X0)).
% 2.71/1.61 thf(func_def_96, type, sK30: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X1 @ X0 > set @ product_prod @ X1 @ X0 > X1)).
% 2.71/1.61 thf(func_def_97, type, sK31: !>[X0: $tType, X1: $tType]:(set @ product_prod @ X1 @ X0 > set @ product_prod @ X1 @ X0 > X0)).
% 2.71/1.61 thf(func_def_98, type, sK32: !>[X0: $tType]:(set @ X0 > X0 > set @ X0)).
% 2.71/1.61 thf(func_def_99, type, sK33: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 2.71/1.61 thf(func_def_100, type, sK34: !>[X0: $tType]:((set @ X0 > $o) > X0)).
% 2.71/1.61 thf(func_def_101, type, sK35: !>[X0: $tType]:((set @ X0 > $o) > X0)).
% 2.71/1.61 thf(func_def_102, type, sK36: !>[X0: $tType]:((set @ X0 > $o) > set @ X0)).
% 2.71/1.61 thf(func_def_103, type, sK37: !>[X0: $tType]:(set @ X0 > X0)).
% 2.71/1.61 thf(func_def_104, type, sK38: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 2.71/1.61 thf(func_def_105, type, sK39: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 2.71/1.61 thf(func_def_106, type, sK40: !>[X0: $tType, X1: $tType, X2: $tType]:((product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 2.71/1.61 thf(func_def_107, type, sK41: !>[X0: $tType]:(set @ X0 > X0)).
% 2.71/1.61 thf(func_def_108, type, sK42: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X0)).
% 2.71/1.61 thf(func_def_109, type, sK43: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X1)).
% 2.71/1.61 thf(func_def_110, type, sK44: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X2)).
% 2.71/1.61 thf(func_def_111, type, sK45: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X3)).
% 2.71/1.61 thf(func_def_112, type, sK46: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X4)).
% 2.71/1.61 thf(func_def_113, type, sK47: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ product_prod @ X4 @ X5 > X5)).
% 2.71/1.61 thf(func_def_114, type, sK48: !>[X0: $tType]:((set @ X0 > $o) > set @ X0 > X0)).
% 2.71/1.61 thf(func_def_115, type, sK49: !>[X0: $tType]:((set @ X0 > $o) > set @ X0 > set @ X0)).
% 2.71/1.61 thf(func_def_116, type, sK50: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 2.71/1.61 thf(func_def_117, type, sK51: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 2.71/1.61 thf(func_def_118, type, sK52: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 2.71/1.61 thf(func_def_119, type, sK53: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType]:((product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 2.71/1.61 thf(func_def_120, type, sK54: !>[X0: $tType]:(set @ X0 > set @ X0 > X0)).
% 2.71/1.61 thf(func_def_121, type, sK55: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X0)).
% 2.71/1.61 thf(func_def_122, type, sK56: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X1)).
% 2.71/1.61 thf(func_def_123, type, sK57: !>[X0: $tType, X1: $tType, X2: $tType]:(product_prod @ X0 @ product_prod @ X1 @ X2 > X2)).
% 2.71/1.61 thf(func_def_124, type, sK58: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X0)).
% 2.71/1.61 thf(func_def_125, type, sK59: !>[X0: $tType, X1: $tType]:(product_prod @ X0 @ X1 > X1)).
% 2.71/1.61 thf(func_def_126, type, sK60: !>[X0: $tType]:(X0 > X0 > set @ X0 > set @ X0 > set @ X0)).
% 2.71/1.61 thf(func_def_127, type, sK61: !>[X0: $tType]:((set @ X0 > $o) > X0)).
% 2.71/1.61 thf(func_def_128, type, sK62: !>[X0: $tType]:((set @ X0 > $o) > set @ X0)).
% 2.71/1.61 thf(func_def_129, type, sK63: !>[X0: $tType]:((set @ X0 > $o) > set @ X0)).
% 2.71/1.61 thf(func_def_130, type, sK64: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X6)).
% 2.71/1.61 thf(func_def_131, type, sK65: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X5)).
% 2.71/1.61 thf(func_def_132, type, sK66: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X4)).
% 2.71/1.61 thf(func_def_133, type, sK67: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X3)).
% 2.71/1.61 thf(func_def_134, type, sK68: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X2)).
% 2.71/1.61 thf(func_def_135, type, sK69: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X1)).
% 2.71/1.61 thf(func_def_136, type, sK70: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType, X5: $tType, X6: $tType]:((product_prod @ X6 @ product_prod @ X5 @ product_prod @ X4 @ product_prod @ X3 @ product_prod @ X2 @ product_prod @ X1 @ X0 > $o) > X0)).
% 2.71/1.61 thf(func_def_137, type, sK71: !>[X0: $tType]:(set @ X0 > X0)).
% 2.71/1.61 thf(func_def_138, type, sK72: !>[X0: $tType]:(set @ X0 > X0 > set @ X0)).
% 2.71/1.61 thf(func_def_139, type, sK73: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X0)).
% 2.71/1.61 thf(func_def_140, type, sK74: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X1)).
% 2.71/1.61 thf(func_def_141, type, sK75: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X2)).
% 2.71/1.61 thf(func_def_142, type, sK76: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X3)).
% 2.71/1.61 thf(func_def_143, type, sK77: !>[X0: $tType, X1: $tType, X2: $tType, X3: $tType, X4: $tType]:(product_prod @ X0 @ product_prod @ X1 @ product_prod @ X2 @ product_prod @ X3 @ X4 > X4)).
% 2.71/1.61 thf(func_def_144, type, sK78: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > set @ X1 > X1)).
% 2.71/1.61 thf(func_def_145, type, sK79: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > set @ X1 > X1 > X0)).
% 2.71/1.61 thf(func_def_146, type, sK80: !>[X0: $tType]:(set @ X0 > X0)).
% 2.71/1.61 thf(func_def_147, type, sK81: !>[X0: $tType]:((set @ X0 > $o) > set @ X0 > set @ X0)).
% 2.71/1.61 thf(func_def_148, type, sK82: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X1)).
% 2.71/1.61 thf(func_def_149, type, sK83: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X0)).
% 2.71/1.61 thf(func_def_150, type, sK84: !>[X0: $tType]:((set @ X0 > $o) > set @ X0 > set @ X0)).
% 2.71/1.61 thf(func_def_151, type, sK85: !>[X0: $tType]:(set @ X0 > X0)).
% 2.71/1.61 thf(func_def_152, type, sK86: !>[X0: $tType]:((set @ X0 > $o) > X0)).
% 2.71/1.61 thf(func_def_153, type, sK87: !>[X0: $tType]:((set @ X0 > $o) > set @ X0)).
% 2.71/1.61 thf(func_def_154, type, sK88: !>[X0: $tType]:(set @ X0 > X0)).
% 2.71/1.61 thf(func_def_155, type, sK89: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X1)).
% 2.71/1.61 thf(func_def_156, type, sK90: !>[X0: $tType, X1: $tType]:((product_prod @ X1 @ X0 > $o) > X0)).
% 2.71/1.61 thf(func_def_157, type, sK91: !>[X0: $tType]:((set @ X0 > $o) > set @ X0)).
% 2.71/1.61 thf(func_def_158, type, sK92: !>[X0: $tType]:((set @ X0 > $o) > X0)).
% 2.71/1.61 thf(func_def_160, type, ph94: !>[X0: $tType]:(X0)).
% 2.71/1.61 thf(f1406,plain,(
% 2.71/1.61 $false),
% 2.71/1.61 inference(subsumption_resolution,[status(thm)],[f1405,f1180])).
% 2.71/1.61 thf(f1180,plain,(
% 2.71/1.61 ( ! [X0 : $tType] : (((finite_finite2 @ X0 @ bot_bot @ set @ X0) = $true)) )),
% 2.71/1.61 inference(cnf_transformation,[status(thm)],[f384])).
% 2.71/1.61 thf(f384,plain,(
% 2.71/1.61 ! [X0 : $tType] : ((finite_finite2 @ X0 @ bot_bot @ set @ X0) = $true)),
% 2.71/1.61 inference(fool_elimination,[status(thm)],[f383])).
% 2.71/1.61 thf(f383,plain,(
% 2.71/1.61 ! [X0 : $tType] : (finite_finite2 @ X0 @ bot_bot @ set @ X0)),
% 2.71/1.61 inference(rectify,[status(thm)],[f104])).
% 2.71/1.61 thf(f104,axiom,(
% 2.71/1.61 ! [X0 : $tType] : (finite_finite2 @ X0 @ bot_bot @ set @ X0)),
% 2.71/1.61 file('/export/starexec/sandbox2/tmp/tmp.sjZ2TfJLxv/DTF2THF_22201.p',fact_103_finite_OemptyI)).
% 2.71/1.61 thf(f1405,plain,(
% 2.71/1.61 ($true != (finite_finite2 @ node @ bot_bot @ set @ node))),
% 2.71/1.61 inference(subsumption_resolution,[status(thm)],[f1404,f1119])).
% 2.71/1.61 thf(f1119,plain,(
% 2.71/1.61 ((finite_finite2 @ node @ (koenig317145564le_via @ node @ graph @ (uminus_uminus @ set @ node @ (insert @ node @ n @ bot_bot @ set @ node)) @ n)) != $true)),
% 2.71/1.61 inference(cnf_transformation,[status(thm)],[f813])).
% 2.71/1.61 thf(f813,plain,(
% 2.71/1.61 ((finite_finite2 @ node @ (koenig317145564le_via @ node @ graph @ (uminus_uminus @ set @ node @ (insert @ node @ n @ bot_bot @ set @ node)) @ n)) != $true)),
% 2.71/1.61 inference(flattening,[status(thm)],[f576])).
% 2.71/1.61 thf(f576,plain,(
% 2.71/1.61 ~((finite_finite2 @ node @ (koenig317145564le_via @ node @ graph @ (uminus_uminus @ set @ node @ (insert @ node @ n @ bot_bot @ set @ node)) @ n)) = $true)),
% 2.71/1.61 inference(fool_elimination,[status(thm)],[f575])).
% 2.71/1.61 thf(f575,plain,(
% 2.71/1.61 ~(finite_finite2 @ node @ (koenig317145564le_via @ node @ graph @ (uminus_uminus @ set @ node @ (insert @ node @ n @ bot_bot @ set @ node)) @ n))),
% 2.71/1.61 inference(rectify,[status(thm)],[f3])).
% 2.71/1.61 thf(f3,axiom,(
% 2.71/1.61 ~(finite_finite2 @ node @ (koenig317145564le_via @ node @ graph @ (uminus_uminus @ set @ node @ (insert @ node @ n @ bot_bot @ set @ node)) @ n))),
% 2.71/1.61 file('/export/starexec/sandbox2/tmp/tmp.sjZ2TfJLxv/DTF2THF_22201.p',fact_2__092_060open_062infinite_A_Ireachable__via_Agraph_A_I_N_A_123n_125_J_An_J_092_060close_062)).
% 2.71/1.61 thf(f1404,plain,(
% 2.71/1.61 ((finite_finite2 @ node @ (koenig317145564le_via @ node @ graph @ (uminus_uminus @ set @ node @ (insert @ node @ n @ bot_bot @ set @ node)) @ n)) = $true) | ($true != (finite_finite2 @ node @ bot_bot @ set @ node))),
% 2.71/1.61 inference(equality_resolution,[status(thm)],[f1135])).
% 2.71/1.61 thf(f1135,plain,(
% 2.71/1.61 ( ! [X0 : node,X1 : set @ node] : (((f @ (product_Pair @ node @ set @ node @ n @ bot_bot @ set @ node)) != (f @ (product_Pair @ node @ set @ node @ X0 @ X1))) | ((finite_finite2 @ node @ (koenig317145564le_via @ node @ graph @ (uminus_uminus @ set @ node @ (insert @ node @ X0 @ X1)) @ X0)) = $true) | ($true != (finite_finite2 @ node @ X1))) )),
% 2.71/1.61 inference(cnf_transformation,[status(thm)],[f955])).
% 2.71/1.61 thf(f955,plain,(
% 2.71/1.61 ! [X0 : node,X1 : set @ node] : (((f @ (product_Pair @ node @ set @ node @ n @ bot_bot @ set @ node)) != (f @ (product_Pair @ node @ set @ node @ X0 @ X1))) | ($true != (finite_finite2 @ node @ X1)) | ((finite_finite2 @ node @ (koenig317145564le_via @ node @ graph @ (uminus_uminus @ set @ node @ (insert @ node @ X0 @ X1)) @ X0)) = $true))),
% 2.71/1.61 inference(ennf_transformation,[status(thm)],[f793])).
% 2.71/1.61 thf(f793,plain,(
% 2.71/1.61 ~? [X0 : node,X1 : set @ node] : (($true = (finite_finite2 @ node @ X1)) & ((finite_finite2 @ node @ (koenig317145564le_via @ node @ graph @ (uminus_uminus @ set @ node @ (insert @ node @ X0 @ X1)) @ X0)) != $true) & ((f @ (product_Pair @ node @ set @ node @ n @ bot_bot @ set @ node)) = (f @ (product_Pair @ node @ set @ node @ X0 @ X1))))),
% 2.71/1.61 inference(flattening,[status(thm)],[f400])).
% 2.71/1.61 thf(f400,plain,(
% 2.71/1.61 ~? [X0 : node,X1 : set @ node] : (($true = (finite_finite2 @ node @ X1)) & ((f @ (product_Pair @ node @ set @ node @ n @ bot_bot @ set @ node)) = (f @ (product_Pair @ node @ set @ node @ X0 @ X1))) & ~((finite_finite2 @ node @ (koenig317145564le_via @ node @ graph @ (uminus_uminus @ set @ node @ (insert @ node @ X0 @ X1)) @ X0)) = $true))),
% 2.71/1.61 inference(fool_elimination,[status(thm)],[f399])).
% 2.71/1.61 thf(f399,plain,(
% 2.71/1.61 ~? [X0 : node,X1 : set @ node] : ((finite_finite2 @ node @ X1) & ((f @ (product_Pair @ node @ set @ node @ n @ bot_bot @ set @ node)) = (f @ (product_Pair @ node @ set @ node @ X0 @ X1))) & ~(finite_finite2 @ node @ (koenig317145564le_via @ node @ graph @ (uminus_uminus @ set @ node @ (insert @ node @ X0 @ X1)) @ X0)))),
% 2.71/1.61 inference(rectify,[status(thm)],[f293])).
% 2.71/1.61 thf(f293,negated_conjecture,(
% 2.71/1.61 ~? [X74 : node,X75 : set @ node] : ((finite_finite2 @ node @ X75) & ((f @ (product_Pair @ node @ set @ node @ n @ bot_bot @ set @ node)) = (f @ (product_Pair @ node @ set @ node @ X74 @ X75))) & ~(finite_finite2 @ node @ (koenig317145564le_via @ node @ graph @ (uminus_uminus @ set @ node @ (insert @ node @ X74 @ X75)) @ X74)))),
% 2.71/1.61 inference(negated_conjecture,[status(cth)],[f292])).
% 2.71/1.61 thf(f292,conjecture,(
% 2.71/1.61 ? [X74 : node,X75 : set @ node] : ((finite_finite2 @ node @ X75) & ((f @ (product_Pair @ node @ set @ node @ n @ bot_bot @ set @ node)) = (f @ (product_Pair @ node @ set @ node @ X74 @ X75))) & ~(finite_finite2 @ node @ (koenig317145564le_via @ node @ graph @ (uminus_uminus @ set @ node @ (insert @ node @ X74 @ X75)) @ X74)))),
% 2.71/1.61 file('/export/starexec/sandbox2/tmp/tmp.sjZ2TfJLxv/DTF2THF_22201.p',conj_0)).
% 2.71/1.61 % SZS output end Proof for DTF2THF_22201
% 2.71/1.61 % (22383)------------------------------
% 2.71/1.61 % (22383)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 2.71/1.61 % (22383)Termination reason: Refutation
% 2.71/1.61
% 2.71/1.61 % (22383)Memory used [KB]: 6908
% 2.71/1.61 % (22383)Time elapsed: 0.058 s
% 2.71/1.61 % (22383)Instructions burned: 91 (million)
% 2.71/1.61 % (22383)------------------------------
% 2.71/1.61 % (22383)------------------------------
% 2.71/1.61 % (22368)Success in time 0.081 s
%------------------------------------------------------------------------------