%------------------------------------------------------------------------------
% File : DT2H2X---1.9.5
% Problem : COM182^1 : 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 : n002.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 3.00s 1.65s
% Output : Refutation 3.00s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11 % Problem : COM182^1 : TPTP v9.2.1. Released v7.0.0.
% 0.00/0.13 % Command : /export/starexec/sandbox2/solver/bin/run_DT2H2X /export/starexec/sandbox2/benchmark/theBenchmark.p 300
% 0.16/0.33 % Computer : n002.cluster.edu
% 0.16/0.33 % Model : x86_64 x86_64
% 0.16/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.33 % Memory : 8042.1875MB
% 0.16/0.33 % 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 04:16:48 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
% 0.30/0.45 ---- Original DTF file ---
% 0.30/0.46 thf(spec,logic,$$dhol).
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 % File : COM182^1 : TPTP v9.2.1. Released v7.0.0.
% 0.30/0.46 % Domain : Computing Theory
% 0.30/0.46 % Problem : Grammars and languages 60
% 0.30/0.46 % Version : [Bla16] axioms : Especial.
% 0.30/0.46 % English :
% 0.30/0.46
% 0.30/0.46 % Refs : [BH+14] Blanchette et al. (2014), Truly Modular (Co)datatypes
% 0.30/0.46 % : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% 0.30/0.46 % : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% 0.30/0.46 % Source : [Bla16]
% 0.30/0.46 % Names : gram_lang__60.p [Bla16]
% 0.30/0.46
% 0.30/0.46 % Status : Theorem
% 0.30/0.46 % Rating : 0.33 v9.1.0, 0.00 v7.2.0, 0.25 v7.1.0
% 0.30/0.46 % Syntax : Number of formulae : 329 ( 122 unt; 53 typ; 0 def)
% 0.30/0.46 % Number of atoms : 715 ( 300 equ; 0 cnn)
% 0.30/0.46 % Maximal formula atoms : 7 ( 2 avg)
% 0.30/0.46 % Number of connectives : 4149 ( 106 ~; 11 |; 53 &;3632 @)
% 0.30/0.46 % ( 0 <=>; 347 =>; 0 <=; 0 <~>)
% 0.30/0.46 % Maximal formula depth : 21 ( 8 avg)
% 0.30/0.46 % Number of types : 4 ( 3 usr)
% 0.30/0.46 % Number of type conns : 250 ( 250 >; 0 *; 0 +; 0 <<)
% 0.30/0.46 % Number of symbols : 53 ( 50 usr; 5 con; 0-6 aty)
% 0.30/0.46 % Number of variables : 1170 ( 46 ^;1041 !; 32 ?;1170 :)
% 0.30/0.46 % ( 51 !>; 0 ?*; 0 @-; 0 @+)
% 0.30/0.46 % SPC : TH1_THM_EQU_NAR
% 0.30/0.46
% 0.30/0.46 % Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 0.30/0.46 % 2016-07-13 14:40:08.012
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 %----Could-be-implicit typings (7)
% 0.30/0.46 thf(ty_t_Sum__Type_Osum,type,
% 0.30/0.46 sum_sum: $tType > $tType > $tType ).
% 0.30/0.46
% 0.30/0.46 thf(ty_t_DTree_Odtree,type,
% 0.30/0.46 dtree: $tType ).
% 0.30/0.46
% 0.30/0.46 thf(ty_t_FSet_Ofset,type,
% 0.30/0.46 fset: $tType > $tType ).
% 0.30/0.46
% 0.30/0.46 thf(ty_t_Set_Oset,type,
% 0.30/0.46 set: $tType > $tType ).
% 0.30/0.46
% 0.30/0.46 thf(ty_t_DTree_OT,type,
% 0.30/0.46 t: $tType ).
% 0.30/0.46
% 0.30/0.46 thf(ty_t_DTree_ON,type,
% 0.30/0.46 n: $tType ).
% 0.30/0.46
% 0.30/0.46 thf(ty_t_itself,type,
% 0.30/0.46 itself: $tType > $tType ).
% 0.30/0.46
% 0.30/0.46 %----Explicit typings (46)
% 0.30/0.46 thf(sy_cl_HOL_Otype,type,
% 0.30/0.46 type:
% 0.30/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Groups_Ominus,type,
% 0.30/0.46 minus:
% 0.30/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Groups_Ogroup__add,type,
% 0.30/0.46 group_add:
% 0.30/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Finite__Set_Ofinite,type,
% 0.30/0.46 finite_finite:
% 0.30/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Orderings_Olinorder,type,
% 0.30/0.46 linorder:
% 0.30/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Lattices_Osemilattice__sup,type,
% 0.30/0.46 semilattice_sup:
% 0.30/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 0.30/0.46 ordered_ab_group_add:
% 0.30/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
% 0.30/0.46 cancel146912293up_add:
% 0.30/0.46 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Basic__BNFs_Osetl,type,
% 0.30/0.46 basic_setl:
% 0.30/0.46 !>[A: $tType,B: $tType] : ( ( sum_sum @ A @ B ) > ( set @ A ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Basic__BNFs_Osetr,type,
% 0.30/0.46 basic_setr:
% 0.30/0.46 !>[A: $tType,B: $tType] : ( ( sum_sum @ A @ B ) > ( set @ B ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_DTree_ONode,type,
% 0.30/0.46 node: n > ( set @ ( sum_sum @ t @ dtree ) ) > dtree ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_DTree_Ocont,type,
% 0.30/0.46 cont: dtree > ( set @ ( sum_sum @ t @ dtree ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_DTree_Ocorec,type,
% 0.30/0.46 corec:
% 0.30/0.46 !>[A: $tType] : ( ( A > n ) > ( A > ( set @ ( sum_sum @ t @ ( sum_sum @ dtree @ A ) ) ) ) > A > dtree ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_DTree_Odtree_ONNode,type,
% 0.30/0.46 nNode: n > ( fset @ ( sum_sum @ t @ dtree ) ) > dtree ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_DTree_Odtree_Ocase__dtree,type,
% 0.30/0.46 case_dtree:
% 0.30/0.46 !>[A: $tType] : ( ( n > ( fset @ ( sum_sum @ t @ dtree ) ) > A ) > dtree > A ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_DTree_Odtree_Occont,type,
% 0.30/0.46 ccont: dtree > ( fset @ ( sum_sum @ t @ dtree ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_DTree_Odtree_Oroot,type,
% 0.30/0.46 root: dtree > n ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_DTree_Ounfold,type,
% 0.30/0.46 unfold:
% 0.30/0.46 !>[A: $tType] : ( ( A > n ) > ( A > ( set @ ( sum_sum @ t @ A ) ) ) > A > dtree ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Finite__Set_Ofinite,type,
% 0.30/0.46 finite_finite2:
% 0.30/0.46 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Fun_Ocomp,type,
% 0.30/0.46 comp:
% 0.30/0.46 !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_OinFr,type,
% 0.30/0.46 gram_L1333338417e_inFr: ( set @ n ) > dtree > t > $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_OinFr2,type,
% 0.30/0.46 gram_L805317441_inFr2: ( set @ n ) > dtree > t > $o ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 0.30/0.46 minus_minus:
% 0.30/0.46 !>[A: $tType] : ( A > A > A ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_If,type,
% 0.30/0.46 if:
% 0.30/0.46 !>[A: $tType] : ( $o > A > A > A ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin,type,
% 0.30/0.46 lattic477160up_fin:
% 0.30/0.46 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Orderings_Obot__class_Obot,type,
% 0.30/0.46 bot_bot:
% 0.30/0.46 !>[A: $tType] : A ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Orderings_Oord__class_Oless,type,
% 0.30/0.46 ord_less:
% 0.30/0.46 !>[A: $tType] : ( A > A > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 0.30/0.46 ord_less_eq:
% 0.30/0.46 !>[A: $tType] : ( A > A > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Pure_Otype,type,
% 0.30/0.46 type2:
% 0.30/0.46 !>[A: $tType] : ( itself @ A ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Set_OCollect,type,
% 0.30/0.46 collect:
% 0.30/0.46 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Set_Oinsert,type,
% 0.30/0.46 insert:
% 0.30/0.46 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Set_Ois__empty,type,
% 0.30/0.46 is_empty:
% 0.30/0.46 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Set_Ois__singleton,type,
% 0.30/0.46 is_singleton:
% 0.30/0.46 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Set_Oremove,type,
% 0.30/0.46 remove:
% 0.30/0.46 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Set_Othe__elem,type,
% 0.30/0.46 the_elem:
% 0.30/0.46 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Sum__Type_OInl,type,
% 0.30/0.46 sum_Inl:
% 0.30/0.46 !>[A: $tType,B: $tType] : ( A > ( sum_sum @ A @ B ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Sum__Type_OInr,type,
% 0.30/0.46 sum_Inr:
% 0.30/0.46 !>[B: $tType,A: $tType] : ( B > ( sum_sum @ A @ B ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Sum__Type_OPlus,type,
% 0.30/0.46 sum_Plus:
% 0.30/0.46 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( sum_sum @ A @ B ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Sum__Type_OSuml,type,
% 0.30/0.46 sum_Suml:
% 0.30/0.46 !>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( sum_sum @ A @ B ) > C ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Sum__Type_OSumr,type,
% 0.30/0.46 sum_Sumr:
% 0.30/0.46 !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( sum_sum @ A @ B ) > C ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Sum__Type_Oold_Osum_Orec__sum,type,
% 0.30/0.46 sum_rec_sum:
% 0.30/0.46 !>[A: $tType,T: $tType,B: $tType] : ( ( A > T ) > ( B > T ) > ( sum_sum @ A @ B ) > T ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_Sum__Type_Osum_Ocase__sum,type,
% 0.30/0.46 sum_case_sum:
% 0.30/0.46 !>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( B > C ) > ( sum_sum @ A @ B ) > C ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_c_member,type,
% 0.30/0.46 member:
% 0.30/0.46 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(sy_v_nsa,type,
% 0.30/0.46 nsa: set @ n ).
% 0.30/0.46
% 0.30/0.46 thf(sy_v_ta,type,
% 0.30/0.46 ta: t ).
% 0.30/0.46
% 0.30/0.46 thf(sy_v_tra,type,
% 0.30/0.46 tra: dtree ).
% 0.30/0.46
% 0.30/0.46 %----Relevant facts (256)
% 0.30/0.46 thf(fact_0_sum_Oinject_I1_J,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,X1: A,Y1: A] :
% 0.30/0.46 ( ( ( sum_Inl @ A @ B @ X1 )
% 0.30/0.46 = ( sum_Inl @ A @ B @ Y1 ) )
% 0.30/0.46 = ( X1 = Y1 ) ) ).
% 0.30/0.46
% 0.30/0.46 % sum.inject(1)
% 0.30/0.46 thf(fact_1_old_Osum_Oinject_I1_J,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,A2: A,A3: A] :
% 0.30/0.46 ( ( ( sum_Inl @ A @ B @ A2 )
% 0.30/0.46 = ( sum_Inl @ A @ B @ A3 ) )
% 0.30/0.46 = ( A2 = A3 ) ) ).
% 0.30/0.46
% 0.30/0.46 % old.sum.inject(1)
% 0.30/0.46 thf(fact_2_dtree__cong,axiom,
% 0.30/0.46 ! [Tr: dtree,Tr2: dtree] :
% 0.30/0.46 ( ( ( root @ Tr )
% 0.30/0.46 = ( root @ Tr2 ) )
% 0.30/0.46 => ( ( ( cont @ Tr )
% 0.30/0.46 = ( cont @ Tr2 ) )
% 0.30/0.46 => ( Tr = Tr2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % dtree_cong
% 0.30/0.46 thf(fact_3_inFr2_OBase,axiom,
% 0.30/0.46 ! [Tr: dtree,Ns: set @ n,T2: t] :
% 0.30/0.46 ( ( member @ n @ ( root @ Tr ) @ Ns )
% 0.30/0.46 => ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T2 ) @ ( cont @ Tr ) )
% 0.30/0.46 => ( gram_L805317441_inFr2 @ Ns @ Tr @ T2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % inFr2.Base
% 0.30/0.46 thf(fact_4_inFr_OBase,axiom,
% 0.30/0.46 ! [Tr: dtree,Ns: set @ n,T2: t] :
% 0.30/0.46 ( ( member @ n @ ( root @ Tr ) @ Ns )
% 0.30/0.46 => ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T2 ) @ ( cont @ Tr ) )
% 0.30/0.46 => ( gram_L1333338417e_inFr @ Ns @ Tr @ T2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % inFr.Base
% 0.30/0.46 thf(fact_5_Inl__inject,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,X: A,Y: A] :
% 0.30/0.46 ( ( ( sum_Inl @ A @ B @ X )
% 0.30/0.46 = ( sum_Inl @ A @ B @ Y ) )
% 0.30/0.46 => ( X = Y ) ) ).
% 0.30/0.46
% 0.30/0.46 % Inl_inject
% 0.30/0.46 thf(fact_6_Node__root__cont,axiom,
% 0.30/0.46 ! [Tr: dtree] :
% 0.30/0.46 ( ( node @ ( root @ Tr ) @ ( cont @ Tr ) )
% 0.30/0.46 = Tr ) ).
% 0.30/0.46
% 0.30/0.46 % Node_root_cont
% 0.30/0.46 thf(fact_7_inFr_Ocases,axiom,
% 0.30/0.46 ! [A1: set @ n,A22: dtree,A32: t] :
% 0.30/0.46 ( ( gram_L1333338417e_inFr @ A1 @ A22 @ A32 )
% 0.30/0.46 => ( ( ( member @ n @ ( root @ A22 ) @ A1 )
% 0.30/0.46 => ~ ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ A32 ) @ ( cont @ A22 ) ) )
% 0.30/0.46 => ~ ( ( member @ n @ ( root @ A22 ) @ A1 )
% 0.30/0.46 => ! [Tr1: dtree] :
% 0.30/0.46 ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr1 ) @ ( cont @ A22 ) )
% 0.30/0.46 => ~ ( gram_L1333338417e_inFr @ A1 @ Tr1 @ A32 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % inFr.cases
% 0.30/0.46 thf(fact_8_inFr_Osimps,axiom,
% 0.30/0.46 ( gram_L1333338417e_inFr
% 0.30/0.46 = ( ^ [A12: set @ n,A23: dtree,A33: t] :
% 0.30/0.46 ( ? [Tr3: dtree,Ns2: set @ n,T3: t] :
% 0.30/0.46 ( ( A12 = Ns2 )
% 0.30/0.46 & ( A23 = Tr3 )
% 0.30/0.46 & ( A33 = T3 )
% 0.30/0.46 & ( member @ n @ ( root @ Tr3 ) @ Ns2 )
% 0.30/0.46 & ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T3 ) @ ( cont @ Tr3 ) ) )
% 0.30/0.46 | ? [Tr3: dtree,Ns2: set @ n,Tr12: dtree,T3: t] :
% 0.30/0.46 ( ( A12 = Ns2 )
% 0.30/0.46 & ( A23 = Tr3 )
% 0.30/0.46 & ( A33 = T3 )
% 0.30/0.46 & ( member @ n @ ( root @ Tr3 ) @ Ns2 )
% 0.30/0.46 & ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr12 ) @ ( cont @ Tr3 ) )
% 0.30/0.46 & ( gram_L1333338417e_inFr @ Ns2 @ Tr12 @ T3 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % inFr.simps
% 0.30/0.46 thf(fact_9_inFr_Oinducts,axiom,
% 0.30/0.46 ! [X1: set @ n,X2: dtree,X3: t,P: ( set @ n ) > dtree > t > $o] :
% 0.30/0.46 ( ( gram_L1333338417e_inFr @ X1 @ X2 @ X3 )
% 0.30/0.46 => ( ! [Tr4: dtree,Ns3: set @ n,T4: t] :
% 0.30/0.46 ( ( member @ n @ ( root @ Tr4 ) @ Ns3 )
% 0.30/0.46 => ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T4 ) @ ( cont @ Tr4 ) )
% 0.30/0.46 => ( P @ Ns3 @ Tr4 @ T4 ) ) )
% 0.30/0.46 => ( ! [Tr4: dtree,Ns3: set @ n,Tr1: dtree,T4: t] :
% 0.30/0.46 ( ( member @ n @ ( root @ Tr4 ) @ Ns3 )
% 0.30/0.46 => ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr1 ) @ ( cont @ Tr4 ) )
% 0.30/0.46 => ( ( gram_L1333338417e_inFr @ Ns3 @ Tr1 @ T4 )
% 0.30/0.46 => ( ( P @ Ns3 @ Tr1 @ T4 )
% 0.30/0.46 => ( P @ Ns3 @ Tr4 @ T4 ) ) ) ) )
% 0.30/0.46 => ( P @ X1 @ X2 @ X3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % inFr.inducts
% 0.30/0.46 thf(fact_10_old_Osum_Osimps_I7_J,axiom,
% 0.30/0.46 ! [B: $tType,T: $tType,A: $tType,F1: A > T,F2: B > T,A2: A] :
% 0.30/0.46 ( ( sum_rec_sum @ A @ T @ B @ F1 @ F2 @ ( sum_Inl @ A @ B @ A2 ) )
% 0.30/0.46 = ( F1 @ A2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % old.sum.simps(7)
% 0.30/0.46 thf(fact_11_inFr__root__in,axiom,
% 0.30/0.46 ! [Ns: set @ n,Tr: dtree,T2: t] :
% 0.30/0.46 ( ( gram_L1333338417e_inFr @ Ns @ Tr @ T2 )
% 0.30/0.46 => ( member @ n @ ( root @ Tr ) @ Ns ) ) ).
% 0.30/0.46
% 0.30/0.46 % inFr_root_in
% 0.30/0.46 thf(fact_12_sum_Oinject_I2_J,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,X2: B,Y2: B] :
% 0.30/0.46 ( ( ( sum_Inr @ B @ A @ X2 )
% 0.30/0.46 = ( sum_Inr @ B @ A @ Y2 ) )
% 0.30/0.46 = ( X2 = Y2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % sum.inject(2)
% 0.30/0.46 thf(fact_13_old_Osum_Oinject_I2_J,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,B2: B,B3: B] :
% 0.30/0.46 ( ( ( sum_Inr @ B @ A @ B2 )
% 0.30/0.46 = ( sum_Inr @ B @ A @ B3 ) )
% 0.30/0.46 = ( B2 = B3 ) ) ).
% 0.30/0.46
% 0.30/0.46 % old.sum.inject(2)
% 0.30/0.46 thf(fact_14_root__Node,axiom,
% 0.30/0.46 ! [N: n,As: set @ ( sum_sum @ t @ dtree )] :
% 0.30/0.46 ( ( root @ ( node @ N @ As ) )
% 0.30/0.46 = N ) ).
% 0.30/0.46
% 0.30/0.46 % root_Node
% 0.30/0.46 thf(fact_15_old_Osum_Osimps_I8_J,axiom,
% 0.30/0.46 ! [A: $tType,T: $tType,B: $tType,F1: A > T,F2: B > T,B2: B] :
% 0.30/0.46 ( ( sum_rec_sum @ A @ T @ B @ F1 @ F2 @ ( sum_Inr @ B @ A @ B2 ) )
% 0.30/0.46 = ( F2 @ B2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % old.sum.simps(8)
% 0.30/0.46 thf(fact_16_Inr__inject,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,X: B,Y: B] :
% 0.30/0.46 ( ( ( sum_Inr @ B @ A @ X )
% 0.30/0.46 = ( sum_Inr @ B @ A @ Y ) )
% 0.30/0.46 => ( X = Y ) ) ).
% 0.30/0.46
% 0.30/0.46 % Inr_inject
% 0.30/0.46 thf(fact_17_old_Osum_Oinducts,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,P: ( sum_sum @ A @ B ) > $o,Sum: sum_sum @ A @ B] :
% 0.30/0.46 ( ! [A4: A] : ( P @ ( sum_Inl @ A @ B @ A4 ) )
% 0.30/0.46 => ( ! [B4: B] : ( P @ ( sum_Inr @ B @ A @ B4 ) )
% 0.30/0.46 => ( P @ Sum ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % old.sum.inducts
% 0.30/0.46 thf(fact_18_old_Osum_Oexhaust,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,Y: sum_sum @ A @ B] :
% 0.30/0.46 ( ! [A4: A] :
% 0.30/0.46 ( Y
% 0.30/0.46 != ( sum_Inl @ A @ B @ A4 ) )
% 0.30/0.46 => ~ ! [B4: B] :
% 0.30/0.46 ( Y
% 0.30/0.46 != ( sum_Inr @ B @ A @ B4 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % old.sum.exhaust
% 0.30/0.46 thf(fact_19_split__sum__all,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( ^ [P2: ( sum_sum @ A @ B ) > $o] :
% 0.30/0.46 ! [X4: sum_sum @ A @ B] : ( P2 @ X4 ) )
% 0.30/0.46 = ( ^ [P3: ( sum_sum @ A @ B ) > $o] :
% 0.30/0.46 ( ! [X5: A] : ( P3 @ ( sum_Inl @ A @ B @ X5 ) )
% 0.30/0.46 & ! [X5: B] : ( P3 @ ( sum_Inr @ B @ A @ X5 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % split_sum_all
% 0.30/0.46 thf(fact_20_split__sum__ex,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( ^ [P2: ( sum_sum @ A @ B ) > $o] :
% 0.30/0.46 ? [X4: sum_sum @ A @ B] : ( P2 @ X4 ) )
% 0.30/0.46 = ( ^ [P3: ( sum_sum @ A @ B ) > $o] :
% 0.30/0.46 ( ? [X5: A] : ( P3 @ ( sum_Inl @ A @ B @ X5 ) )
% 0.30/0.46 | ? [X5: B] : ( P3 @ ( sum_Inr @ B @ A @ X5 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % split_sum_ex
% 0.30/0.46 thf(fact_21_Inr__not__Inl,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,B2: B,A2: A] :
% 0.30/0.46 ( ( sum_Inr @ B @ A @ B2 )
% 0.30/0.46 != ( sum_Inl @ A @ B @ A2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % Inr_not_Inl
% 0.30/0.46 thf(fact_22_sumE,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,S: sum_sum @ A @ B] :
% 0.30/0.46 ( ! [X6: A] :
% 0.30/0.46 ( S
% 0.30/0.46 != ( sum_Inl @ A @ B @ X6 ) )
% 0.30/0.46 => ~ ! [Y3: B] :
% 0.30/0.46 ( S
% 0.30/0.46 != ( sum_Inr @ B @ A @ Y3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % sumE
% 0.30/0.46 thf(fact_23_old_Osum_Odistinct_I1_J,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A2: A,B3: B] :
% 0.30/0.46 ( ( sum_Inl @ A @ B @ A2 )
% 0.30/0.46 != ( sum_Inr @ B @ A @ B3 ) ) ).
% 0.30/0.46
% 0.30/0.46 % old.sum.distinct(1)
% 0.30/0.46 thf(fact_24_old_Osum_Odistinct_I2_J,axiom,
% 0.30/0.46 ! [B5: $tType,A5: $tType,B6: B5,A6: A5] :
% 0.30/0.46 ( ( sum_Inr @ B5 @ A5 @ B6 )
% 0.30/0.46 != ( sum_Inl @ A5 @ B5 @ A6 ) ) ).
% 0.30/0.46
% 0.30/0.46 % old.sum.distinct(2)
% 0.30/0.46 thf(fact_25_sum_Odistinct_I1_J,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,X1: A,X2: B] :
% 0.30/0.46 ( ( sum_Inl @ A @ B @ X1 )
% 0.30/0.46 != ( sum_Inr @ B @ A @ X2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % sum.distinct(1)
% 0.30/0.46 thf(fact_26_inFr_OInd,axiom,
% 0.30/0.46 ! [Tr: dtree,Ns: set @ n,Tr13: dtree,T2: t] :
% 0.30/0.46 ( ( member @ n @ ( root @ Tr ) @ Ns )
% 0.30/0.46 => ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr13 ) @ ( cont @ Tr ) )
% 0.30/0.46 => ( ( gram_L1333338417e_inFr @ Ns @ Tr13 @ T2 )
% 0.30/0.46 => ( gram_L1333338417e_inFr @ Ns @ Tr @ T2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % inFr.Ind
% 0.30/0.46 thf(fact_27_Inr__Inl__False,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,X: B,Y: A] :
% 0.30/0.46 ( ( sum_Inr @ B @ A @ X )
% 0.30/0.46 != ( sum_Inl @ A @ B @ Y ) ) ).
% 0.30/0.46
% 0.30/0.46 % Inr_Inl_False
% 0.30/0.46 thf(fact_28_Inl__Inr__False,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,X: A,Y: B] :
% 0.30/0.46 ( ( sum_Inl @ A @ B @ X )
% 0.30/0.46 != ( sum_Inr @ B @ A @ Y ) ) ).
% 0.30/0.46
% 0.30/0.46 % Inl_Inr_False
% 0.30/0.46 thf(fact_29_obj__sumE,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,S: sum_sum @ A @ B] :
% 0.30/0.46 ( ! [X6: A] :
% 0.30/0.46 ( S
% 0.30/0.46 != ( sum_Inl @ A @ B @ X6 ) )
% 0.30/0.46 => ~ ! [X6: B] :
% 0.30/0.46 ( S
% 0.30/0.46 != ( sum_Inr @ B @ A @ X6 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % obj_sumE
% 0.30/0.46 thf(fact_30_obj__sumE__f,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,B: $tType,S: B,F: ( sum_sum @ A @ C ) > B,P: $o] :
% 0.30/0.46 ( ! [X6: A] :
% 0.30/0.46 ( ( S
% 0.30/0.46 = ( F @ ( sum_Inl @ A @ C @ X6 ) ) )
% 0.30/0.46 => P )
% 0.30/0.46 => ( ! [X6: C] :
% 0.30/0.46 ( ( S
% 0.30/0.46 = ( F @ ( sum_Inr @ C @ A @ X6 ) ) )
% 0.30/0.46 => P )
% 0.30/0.46 => ! [X7: sum_sum @ A @ C] :
% 0.30/0.46 ( ( S
% 0.30/0.46 = ( F @ X7 ) )
% 0.30/0.46 => P ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % obj_sumE_f
% 0.30/0.46 thf(fact_31_inFr2_Oinducts,axiom,
% 0.30/0.46 ! [X1: set @ n,X2: dtree,X3: t,P: ( set @ n ) > dtree > t > $o] :
% 0.30/0.46 ( ( gram_L805317441_inFr2 @ X1 @ X2 @ X3 )
% 0.30/0.46 => ( ! [Tr4: dtree,Ns3: set @ n,T4: t] :
% 0.30/0.46 ( ( member @ n @ ( root @ Tr4 ) @ Ns3 )
% 0.30/0.46 => ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T4 ) @ ( cont @ Tr4 ) )
% 0.30/0.46 => ( P @ Ns3 @ Tr4 @ T4 ) ) )
% 0.30/0.46 => ( ! [Tr1: dtree,Tr4: dtree,Ns1: set @ n,T4: t] :
% 0.30/0.46 ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr1 ) @ ( cont @ Tr4 ) )
% 0.30/0.46 => ( ( gram_L805317441_inFr2 @ Ns1 @ Tr1 @ T4 )
% 0.30/0.46 => ( ( P @ Ns1 @ Tr1 @ T4 )
% 0.30/0.46 => ( P @ ( insert @ n @ ( root @ Tr4 ) @ Ns1 ) @ Tr4 @ T4 ) ) ) )
% 0.30/0.46 => ( P @ X1 @ X2 @ X3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % inFr2.inducts
% 0.30/0.46 thf(fact_32_inFr2_Osimps,axiom,
% 0.30/0.46 ( gram_L805317441_inFr2
% 0.30/0.46 = ( ^ [A12: set @ n,A23: dtree,A33: t] :
% 0.30/0.46 ( ? [Tr3: dtree,Ns2: set @ n,T3: t] :
% 0.30/0.46 ( ( A12 = Ns2 )
% 0.30/0.46 & ( A23 = Tr3 )
% 0.30/0.46 & ( A33 = T3 )
% 0.30/0.46 & ( member @ n @ ( root @ Tr3 ) @ Ns2 )
% 0.30/0.46 & ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T3 ) @ ( cont @ Tr3 ) ) )
% 0.30/0.46 | ? [Tr12: dtree,Tr3: dtree,Ns12: set @ n,T3: t] :
% 0.30/0.46 ( ( A12
% 0.30/0.46 = ( insert @ n @ ( root @ Tr3 ) @ Ns12 ) )
% 0.30/0.46 & ( A23 = Tr3 )
% 0.30/0.46 & ( A33 = T3 )
% 0.30/0.46 & ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr12 ) @ ( cont @ Tr3 ) )
% 0.30/0.46 & ( gram_L805317441_inFr2 @ Ns12 @ Tr12 @ T3 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % inFr2.simps
% 0.30/0.46 thf(fact_33_inFr2_Ocases,axiom,
% 0.30/0.46 ! [A1: set @ n,A22: dtree,A32: t] :
% 0.30/0.46 ( ( gram_L805317441_inFr2 @ A1 @ A22 @ A32 )
% 0.30/0.46 => ( ( ( member @ n @ ( root @ A22 ) @ A1 )
% 0.30/0.46 => ~ ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ A32 ) @ ( cont @ A22 ) ) )
% 0.30/0.46 => ~ ! [Tr1: dtree,Tr4: dtree,Ns1: set @ n] :
% 0.30/0.46 ( ( A1
% 0.30/0.46 = ( insert @ n @ ( root @ Tr4 ) @ Ns1 ) )
% 0.30/0.46 => ( ( A22 = Tr4 )
% 0.30/0.46 => ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr1 ) @ ( cont @ Tr4 ) )
% 0.30/0.46 => ~ ( gram_L805317441_inFr2 @ Ns1 @ Tr1 @ A32 ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % inFr2.cases
% 0.30/0.46 thf(fact_34_inFr2_OInd,axiom,
% 0.30/0.46 ! [Tr13: dtree,Tr: dtree,Ns13: set @ n,T2: t] :
% 0.30/0.46 ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr13 ) @ ( cont @ Tr ) )
% 0.30/0.46 => ( ( gram_L805317441_inFr2 @ Ns13 @ Tr13 @ T2 )
% 0.30/0.46 => ( gram_L805317441_inFr2 @ ( insert @ n @ ( root @ Tr ) @ Ns13 ) @ Tr @ T2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % inFr2.Ind
% 0.30/0.46 thf(fact_35_inFr__Ind__minus,axiom,
% 0.30/0.46 ! [Ns13: set @ n,Tr13: dtree,T2: t,Tr: dtree] :
% 0.30/0.46 ( ( gram_L1333338417e_inFr @ Ns13 @ Tr13 @ T2 )
% 0.30/0.46 => ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr13 ) @ ( cont @ Tr ) )
% 0.30/0.46 => ( gram_L1333338417e_inFr @ ( insert @ n @ ( root @ Tr ) @ Ns13 ) @ Tr @ T2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % inFr_Ind_minus
% 0.30/0.46 thf(fact_36_Suml_Osimps,axiom,
% 0.30/0.46 ! [B: $tType,C: $tType,A: $tType,F: A > C,X: A] :
% 0.30/0.46 ( ( sum_Suml @ A @ C @ B @ F @ ( sum_Inl @ A @ B @ X ) )
% 0.30/0.46 = ( F @ X ) ) ).
% 0.30/0.46
% 0.30/0.46 % Suml.simps
% 0.30/0.46 thf(fact_37_Sumr_Osimps,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,B: $tType,F: B > C,X: B] :
% 0.30/0.46 ( ( sum_Sumr @ B @ C @ A @ F @ ( sum_Inr @ B @ A @ X ) )
% 0.30/0.46 = ( F @ X ) ) ).
% 0.30/0.46
% 0.30/0.46 % Sumr.simps
% 0.30/0.46 thf(fact_38_Sumr__inject,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,B: $tType,F: B > C,G: B > C] :
% 0.30/0.46 ( ( ( sum_Sumr @ B @ C @ A @ F )
% 0.30/0.46 = ( sum_Sumr @ B @ C @ A @ G ) )
% 0.30/0.46 => ( F = G ) ) ).
% 0.30/0.46
% 0.30/0.46 % Sumr_inject
% 0.30/0.46 thf(fact_39_Suml__inject,axiom,
% 0.30/0.46 ! [B: $tType,C: $tType,A: $tType,F: A > C,G: A > C] :
% 0.30/0.46 ( ( ( sum_Suml @ A @ C @ B @ F )
% 0.30/0.46 = ( sum_Suml @ A @ C @ B @ G ) )
% 0.30/0.46 => ( F = G ) ) ).
% 0.30/0.46
% 0.30/0.46 % Suml_inject
% 0.30/0.46 thf(fact_40_not__arg__cong__Inr,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,X: A,Y: A] :
% 0.30/0.46 ( ( X != Y )
% 0.30/0.46 => ( ( sum_Inr @ A @ B @ X )
% 0.30/0.46 != ( sum_Inr @ A @ B @ Y ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % not_arg_cong_Inr
% 0.30/0.46 thf(fact_41_insert__absorb2,axiom,
% 0.30/0.46 ! [A: $tType,X: A,A7: set @ A] :
% 0.30/0.46 ( ( insert @ A @ X @ ( insert @ A @ X @ A7 ) )
% 0.30/0.46 = ( insert @ A @ X @ A7 ) ) ).
% 0.30/0.46
% 0.30/0.46 % insert_absorb2
% 0.30/0.46 thf(fact_42_insert__iff,axiom,
% 0.30/0.46 ! [A: $tType,A2: A,B2: A,A7: set @ A] :
% 0.30/0.46 ( ( member @ A @ A2 @ ( insert @ A @ B2 @ A7 ) )
% 0.30/0.46 = ( ( A2 = B2 )
% 0.30/0.46 | ( member @ A @ A2 @ A7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % insert_iff
% 0.30/0.46 thf(fact_43_insertCI,axiom,
% 0.30/0.46 ! [A: $tType,A2: A,B7: set @ A,B2: A] :
% 0.30/0.46 ( ( ~ ( member @ A @ A2 @ B7 )
% 0.30/0.46 => ( A2 = B2 ) )
% 0.30/0.46 => ( member @ A @ A2 @ ( insert @ A @ B2 @ B7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % insertCI
% 0.30/0.46 thf(fact_44_corec_I1_J,axiom,
% 0.30/0.46 ! [A: $tType,Rt: A > n,Ct: A > ( set @ ( sum_sum @ t @ ( sum_sum @ dtree @ A ) ) ),B2: A] :
% 0.30/0.46 ( ( root @ ( corec @ A @ Rt @ Ct @ B2 ) )
% 0.30/0.46 = ( Rt @ B2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % corec(1)
% 0.30/0.46 thf(fact_45_mem__Collect__eq,axiom,
% 0.30/0.46 ! [A: $tType,A2: A,P: A > $o] :
% 0.30/0.46 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 0.30/0.46 = ( P @ A2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % mem_Collect_eq
% 0.30/0.46 thf(fact_46_Collect__mem__eq,axiom,
% 0.30/0.46 ! [A: $tType,A7: set @ A] :
% 0.30/0.46 ( ( collect @ A
% 0.30/0.46 @ ^ [X5: A] : ( member @ A @ X5 @ A7 ) )
% 0.30/0.46 = A7 ) ).
% 0.30/0.46
% 0.30/0.46 % Collect_mem_eq
% 0.30/0.46 thf(fact_47_Collect__cong,axiom,
% 0.30/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.30/0.46 ( ! [X6: A] :
% 0.30/0.46 ( ( P @ X6 )
% 0.30/0.46 = ( Q @ X6 ) )
% 0.30/0.46 => ( ( collect @ A @ P )
% 0.30/0.46 = ( collect @ A @ Q ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Collect_cong
% 0.30/0.46 thf(fact_48_ext,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 0.30/0.46 ( ! [X6: A] :
% 0.30/0.46 ( ( F @ X6 )
% 0.30/0.46 = ( G @ X6 ) )
% 0.30/0.46 => ( F = G ) ) ).
% 0.30/0.46
% 0.30/0.46 % ext
% 0.30/0.46 thf(fact_49_unfold_I1_J,axiom,
% 0.30/0.46 ! [A: $tType,Rt: A > n,Ct: A > ( set @ ( sum_sum @ t @ A ) ),B2: A] :
% 0.30/0.46 ( ( root @ ( unfold @ A @ Rt @ Ct @ B2 ) )
% 0.30/0.46 = ( Rt @ B2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % unfold(1)
% 0.30/0.46 thf(fact_50_cont__Node,axiom,
% 0.30/0.46 ! [As: set @ ( sum_sum @ t @ dtree ),N: n] :
% 0.30/0.46 ( ( finite_finite2 @ ( sum_sum @ t @ dtree ) @ As )
% 0.30/0.46 => ( ( cont @ ( node @ N @ As ) )
% 0.30/0.46 = As ) ) ).
% 0.30/0.46
% 0.30/0.46 % cont_Node
% 0.30/0.46 thf(fact_51_mk__disjoint__insert,axiom,
% 0.30/0.46 ! [A: $tType,A2: A,A7: set @ A] :
% 0.30/0.46 ( ( member @ A @ A2 @ A7 )
% 0.30/0.46 => ? [B8: set @ A] :
% 0.30/0.46 ( ( A7
% 0.30/0.46 = ( insert @ A @ A2 @ B8 ) )
% 0.30/0.46 & ~ ( member @ A @ A2 @ B8 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % mk_disjoint_insert
% 0.30/0.46 thf(fact_52_insert__commute,axiom,
% 0.30/0.46 ! [A: $tType,X: A,Y: A,A7: set @ A] :
% 0.30/0.46 ( ( insert @ A @ X @ ( insert @ A @ Y @ A7 ) )
% 0.30/0.46 = ( insert @ A @ Y @ ( insert @ A @ X @ A7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % insert_commute
% 0.30/0.46 thf(fact_53_insert__eq__iff,axiom,
% 0.30/0.46 ! [A: $tType,A2: A,A7: set @ A,B2: A,B7: set @ A] :
% 0.30/0.46 ( ~ ( member @ A @ A2 @ A7 )
% 0.30/0.46 => ( ~ ( member @ A @ B2 @ B7 )
% 0.30/0.46 => ( ( ( insert @ A @ A2 @ A7 )
% 0.30/0.46 = ( insert @ A @ B2 @ B7 ) )
% 0.30/0.46 = ( ( ( A2 = B2 )
% 0.30/0.46 => ( A7 = B7 ) )
% 0.30/0.46 & ( ( A2 != B2 )
% 0.30/0.46 => ? [C2: set @ A] :
% 0.30/0.46 ( ( A7
% 0.30/0.46 = ( insert @ A @ B2 @ C2 ) )
% 0.30/0.46 & ~ ( member @ A @ B2 @ C2 )
% 0.30/0.46 & ( B7
% 0.30/0.46 = ( insert @ A @ A2 @ C2 ) )
% 0.30/0.46 & ~ ( member @ A @ A2 @ C2 ) ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % insert_eq_iff
% 0.30/0.46 thf(fact_54_dtree__simps,axiom,
% 0.30/0.46 ! [As: set @ ( sum_sum @ t @ dtree ),As2: set @ ( sum_sum @ t @ dtree ),N: n,N2: n] :
% 0.30/0.46 ( ( finite_finite2 @ ( sum_sum @ t @ dtree ) @ As )
% 0.30/0.46 => ( ( finite_finite2 @ ( sum_sum @ t @ dtree ) @ As2 )
% 0.30/0.46 => ( ( ( node @ N @ As )
% 0.30/0.46 = ( node @ N2 @ As2 ) )
% 0.30/0.46 = ( ( N = N2 )
% 0.30/0.46 & ( As = As2 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % dtree_simps
% 0.30/0.46 thf(fact_55_finite__cont,axiom,
% 0.30/0.46 ! [Tr: dtree] : ( finite_finite2 @ ( sum_sum @ t @ dtree ) @ ( cont @ Tr ) ) ).
% 0.30/0.46
% 0.30/0.46 % finite_cont
% 0.30/0.46 thf(fact_56_dtree__cases,axiom,
% 0.30/0.46 ! [Tr: dtree] :
% 0.30/0.46 ~ ! [N3: n,As3: set @ ( sum_sum @ t @ dtree )] :
% 0.30/0.46 ( ( finite_finite2 @ ( sum_sum @ t @ dtree ) @ As3 )
% 0.30/0.46 => ( Tr
% 0.30/0.46 != ( node @ N3 @ As3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % dtree_cases
% 0.30/0.46 thf(fact_57_insertE,axiom,
% 0.30/0.46 ! [A: $tType,A2: A,B2: A,A7: set @ A] :
% 0.30/0.46 ( ( member @ A @ A2 @ ( insert @ A @ B2 @ A7 ) )
% 0.30/0.46 => ( ( A2 != B2 )
% 0.30/0.46 => ( member @ A @ A2 @ A7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % insertE
% 0.30/0.46 thf(fact_58_insertI1,axiom,
% 0.30/0.46 ! [A: $tType,A2: A,B7: set @ A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ B7 ) ) ).
% 0.30/0.46
% 0.30/0.46 % insertI1
% 0.30/0.46 thf(fact_59_insertI2,axiom,
% 0.30/0.46 ! [A: $tType,A2: A,B7: set @ A,B2: A] :
% 0.30/0.46 ( ( member @ A @ A2 @ B7 )
% 0.30/0.46 => ( member @ A @ A2 @ ( insert @ A @ B2 @ B7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % insertI2
% 0.30/0.46 thf(fact_60_Set_Oset__insert,axiom,
% 0.30/0.46 ! [A: $tType,X: A,A7: set @ A] :
% 0.30/0.46 ( ( member @ A @ X @ A7 )
% 0.30/0.46 => ~ ! [B8: set @ A] :
% 0.30/0.46 ( ( A7
% 0.30/0.46 = ( insert @ A @ X @ B8 ) )
% 0.30/0.46 => ( member @ A @ X @ B8 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Set.set_insert
% 0.30/0.46 thf(fact_61_insert__ident,axiom,
% 0.30/0.46 ! [A: $tType,X: A,A7: set @ A,B7: set @ A] :
% 0.30/0.46 ( ~ ( member @ A @ X @ A7 )
% 0.30/0.46 => ( ~ ( member @ A @ X @ B7 )
% 0.30/0.46 => ( ( ( insert @ A @ X @ A7 )
% 0.30/0.46 = ( insert @ A @ X @ B7 ) )
% 0.30/0.46 = ( A7 = B7 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % insert_ident
% 0.30/0.46 thf(fact_62_insert__absorb,axiom,
% 0.30/0.46 ! [A: $tType,A2: A,A7: set @ A] :
% 0.30/0.46 ( ( member @ A @ A2 @ A7 )
% 0.30/0.46 => ( ( insert @ A @ A2 @ A7 )
% 0.30/0.46 = A7 ) ) ).
% 0.30/0.46
% 0.30/0.46 % insert_absorb
% 0.30/0.46 thf(fact_63_finite__insert,axiom,
% 0.30/0.46 ! [A: $tType,A2: A,A7: set @ A] :
% 0.30/0.46 ( ( finite_finite2 @ A @ ( insert @ A @ A2 @ A7 ) )
% 0.30/0.46 = ( finite_finite2 @ A @ A7 ) ) ).
% 0.30/0.46
% 0.30/0.46 % finite_insert
% 0.30/0.46 thf(fact_64_finite__code,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( finite_finite @ A @ ( type2 @ A ) )
% 0.30/0.46 => ( ( finite_finite2 @ A )
% 0.30/0.46 = ( ^ [A8: set @ A] : $true ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % finite_code
% 0.30/0.46 thf(fact_65_finite_OinsertI,axiom,
% 0.30/0.46 ! [A: $tType,A7: set @ A,A2: A] :
% 0.30/0.46 ( ( finite_finite2 @ A @ A7 )
% 0.30/0.46 => ( finite_finite2 @ A @ ( insert @ A @ A2 @ A7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % finite.insertI
% 0.30/0.46 thf(fact_66_finite__set__choice,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,A7: set @ A,P: A > B > $o] :
% 0.30/0.46 ( ( finite_finite2 @ A @ A7 )
% 0.30/0.46 => ( ! [X6: A] :
% 0.30/0.46 ( ( member @ A @ X6 @ A7 )
% 0.30/0.46 => ? [X12: B] : ( P @ X6 @ X12 ) )
% 0.30/0.46 => ? [F3: A > B] :
% 0.30/0.46 ! [X7: A] :
% 0.30/0.46 ( ( member @ A @ X7 @ A7 )
% 0.30/0.46 => ( P @ X7 @ ( F3 @ X7 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % finite_set_choice
% 0.30/0.46 thf(fact_67_finite,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( finite_finite @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A7: set @ A] : ( finite_finite2 @ A @ A7 ) ) ).
% 0.30/0.46
% 0.30/0.46 % finite
% 0.30/0.46 thf(fact_68_case__sum__if,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,C: $tType,P4: $o,F: B > A,G: C > A,X: B,Y: C] :
% 0.30/0.46 ( ( P4
% 0.30/0.46 => ( ( sum_case_sum @ B @ A @ C @ F @ G @ ( if @ ( sum_sum @ B @ C ) @ P4 @ ( sum_Inl @ B @ C @ X ) @ ( sum_Inr @ C @ B @ Y ) ) )
% 0.30/0.46 = ( F @ X ) ) )
% 0.30/0.46 & ( ~ P4
% 0.30/0.46 => ( ( sum_case_sum @ B @ A @ C @ F @ G @ ( if @ ( sum_sum @ B @ C ) @ P4 @ ( sum_Inl @ B @ C @ X ) @ ( sum_Inr @ C @ B @ Y ) ) )
% 0.30/0.46 = ( G @ Y ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % case_sum_if
% 0.30/0.46 thf(fact_69_PlusE,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,U: sum_sum @ A @ B,A7: set @ A,B7: set @ B] :
% 0.30/0.46 ( ( member @ ( sum_sum @ A @ B ) @ U @ ( sum_Plus @ A @ B @ A7 @ B7 ) )
% 0.30/0.46 => ( ! [X6: A] :
% 0.30/0.46 ( ( member @ A @ X6 @ A7 )
% 0.30/0.46 => ( U
% 0.30/0.46 != ( sum_Inl @ A @ B @ X6 ) ) )
% 0.30/0.46 => ~ ! [Y3: B] :
% 0.30/0.46 ( ( member @ B @ Y3 @ B7 )
% 0.30/0.46 => ( U
% 0.30/0.46 != ( sum_Inr @ B @ A @ Y3 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % PlusE
% 0.30/0.46 thf(fact_70_dtree_Osel_I1_J,axiom,
% 0.30/0.46 ! [X1: n,X2: fset @ ( sum_sum @ t @ dtree )] :
% 0.30/0.46 ( ( root @ ( nNode @ X1 @ X2 ) )
% 0.30/0.46 = X1 ) ).
% 0.30/0.46
% 0.30/0.46 % dtree.sel(1)
% 0.30/0.46 thf(fact_71_dtree_Oinject,axiom,
% 0.30/0.46 ! [X1: n,X2: fset @ ( sum_sum @ t @ dtree ),Y1: n,Y2: fset @ ( sum_sum @ t @ dtree )] :
% 0.30/0.46 ( ( ( nNode @ X1 @ X2 )
% 0.30/0.46 = ( nNode @ Y1 @ Y2 ) )
% 0.30/0.46 = ( ( X1 = Y1 )
% 0.30/0.46 & ( X2 = Y2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % dtree.inject
% 0.30/0.46 thf(fact_72_InrI,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,B2: A,B7: set @ A,A7: set @ B] :
% 0.30/0.46 ( ( member @ A @ B2 @ B7 )
% 0.30/0.46 => ( member @ ( sum_sum @ B @ A ) @ ( sum_Inr @ A @ B @ B2 ) @ ( sum_Plus @ B @ A @ A7 @ B7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % InrI
% 0.30/0.46 thf(fact_73_InlI,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A2: A,A7: set @ A,B7: set @ B] :
% 0.30/0.46 ( ( member @ A @ A2 @ A7 )
% 0.30/0.46 => ( member @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B @ A2 ) @ ( sum_Plus @ A @ B @ A7 @ B7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % InlI
% 0.30/0.46 thf(fact_74_finite__Plus__iff,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A7: set @ A,B7: set @ B] :
% 0.30/0.46 ( ( finite_finite2 @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A7 @ B7 ) )
% 0.30/0.46 = ( ( finite_finite2 @ A @ A7 )
% 0.30/0.46 & ( finite_finite2 @ B @ B7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % finite_Plus_iff
% 0.30/0.46 thf(fact_75_dtree_Oexhaust,axiom,
% 0.30/0.46 ! [Y: dtree] :
% 0.30/0.46 ~ ! [X13: n,X22: fset @ ( sum_sum @ t @ dtree )] :
% 0.30/0.46 ( Y
% 0.30/0.46 != ( nNode @ X13 @ X22 ) ) ).
% 0.30/0.46
% 0.30/0.46 % dtree.exhaust
% 0.30/0.46 thf(fact_76_case__sum__inject,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,B: $tType,F1: A > C,F2: B > C,G1: A > C,G2: B > C] :
% 0.30/0.46 ( ( ( sum_case_sum @ A @ C @ B @ F1 @ F2 )
% 0.30/0.46 = ( sum_case_sum @ A @ C @ B @ G1 @ G2 ) )
% 0.30/0.46 => ~ ( ( F1 = G1 )
% 0.30/0.46 => ( F2 != G2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % case_sum_inject
% 0.30/0.46 thf(fact_77_case__sum__step_I1_J,axiom,
% 0.30/0.46 ! [D: $tType,A: $tType,C: $tType,B: $tType,F4: B > A,G3: C > A,G: D > A,P4: sum_sum @ B @ C] :
% 0.30/0.46 ( ( sum_case_sum @ ( sum_sum @ B @ C ) @ A @ D @ ( sum_case_sum @ B @ A @ C @ F4 @ G3 ) @ G @ ( sum_Inl @ ( sum_sum @ B @ C ) @ D @ P4 ) )
% 0.30/0.46 = ( sum_case_sum @ B @ A @ C @ F4 @ G3 @ P4 ) ) ).
% 0.30/0.46
% 0.30/0.46 % case_sum_step(1)
% 0.30/0.46 thf(fact_78_case__sum__step_I2_J,axiom,
% 0.30/0.46 ! [E: $tType,A: $tType,C: $tType,B: $tType,F: E > A,F4: B > A,G3: C > A,P4: sum_sum @ B @ C] :
% 0.30/0.46 ( ( sum_case_sum @ E @ A @ ( sum_sum @ B @ C ) @ F @ ( sum_case_sum @ B @ A @ C @ F4 @ G3 ) @ ( sum_Inr @ ( sum_sum @ B @ C ) @ E @ P4 ) )
% 0.30/0.46 = ( sum_case_sum @ B @ A @ C @ F4 @ G3 @ P4 ) ) ).
% 0.30/0.46
% 0.30/0.46 % case_sum_step(2)
% 0.30/0.46 thf(fact_79_old_Osum_Osimps_I6_J,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,B: $tType,F1: A > C,F2: B > C,X2: B] :
% 0.30/0.46 ( ( sum_case_sum @ A @ C @ B @ F1 @ F2 @ ( sum_Inr @ B @ A @ X2 ) )
% 0.30/0.46 = ( F2 @ X2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % old.sum.simps(6)
% 0.30/0.46 thf(fact_80_old_Osum_Osimps_I5_J,axiom,
% 0.30/0.46 ! [B: $tType,C: $tType,A: $tType,F1: A > C,F2: B > C,X1: A] :
% 0.30/0.46 ( ( sum_case_sum @ A @ C @ B @ F1 @ F2 @ ( sum_Inl @ A @ B @ X1 ) )
% 0.30/0.46 = ( F1 @ X1 ) ) ).
% 0.30/0.46
% 0.30/0.46 % old.sum.simps(5)
% 0.30/0.46 thf(fact_81_finite__Plus,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A7: set @ A,B7: set @ B] :
% 0.30/0.46 ( ( finite_finite2 @ A @ A7 )
% 0.30/0.46 => ( ( finite_finite2 @ B @ B7 )
% 0.30/0.46 => ( finite_finite2 @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A7 @ B7 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % finite_Plus
% 0.30/0.46 thf(fact_82_finite__PlusD_I1_J,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,A7: set @ A,B7: set @ B] :
% 0.30/0.46 ( ( finite_finite2 @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A7 @ B7 ) )
% 0.30/0.46 => ( finite_finite2 @ A @ A7 ) ) ).
% 0.30/0.46
% 0.30/0.46 % finite_PlusD(1)
% 0.30/0.46 thf(fact_83_finite__PlusD_I2_J,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A7: set @ A,B7: set @ B] :
% 0.30/0.46 ( ( finite_finite2 @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A7 @ B7 ) )
% 0.30/0.46 => ( finite_finite2 @ B @ B7 ) ) ).
% 0.30/0.46
% 0.30/0.46 % finite_PlusD(2)
% 0.30/0.46 thf(fact_84_dtree_Ocollapse,axiom,
% 0.30/0.46 ! [Dtree: dtree] :
% 0.30/0.46 ( ( nNode @ ( root @ Dtree ) @ ( ccont @ Dtree ) )
% 0.30/0.46 = Dtree ) ).
% 0.30/0.46
% 0.30/0.46 % dtree.collapse
% 0.30/0.46 thf(fact_85_dtree_Oexhaust__sel,axiom,
% 0.30/0.46 ! [Dtree: dtree] :
% 0.30/0.46 ( Dtree
% 0.30/0.46 = ( nNode @ ( root @ Dtree ) @ ( ccont @ Dtree ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % dtree.exhaust_sel
% 0.30/0.46 thf(fact_86_case__sum__expand__Inr__pointfree,axiom,
% 0.30/0.46 ! [B: $tType,C: $tType,A: $tType,F: ( sum_sum @ A @ C ) > B,G: A > B] :
% 0.30/0.46 ( ( ( comp @ ( sum_sum @ A @ C ) @ B @ A @ F @ ( sum_Inl @ A @ C ) )
% 0.30/0.46 = G )
% 0.30/0.46 => ( ( sum_case_sum @ A @ B @ C @ G @ ( comp @ ( sum_sum @ A @ C ) @ B @ C @ F @ ( sum_Inr @ C @ A ) ) )
% 0.30/0.46 = F ) ) ).
% 0.30/0.46
% 0.30/0.46 % case_sum_expand_Inr_pointfree
% 0.30/0.46 thf(fact_87_case__sum__expand__Inr_H,axiom,
% 0.30/0.46 ! [B: $tType,C: $tType,A: $tType,F: ( sum_sum @ A @ C ) > B,G: A > B,H: C > B] :
% 0.30/0.46 ( ( ( comp @ ( sum_sum @ A @ C ) @ B @ A @ F @ ( sum_Inl @ A @ C ) )
% 0.30/0.46 = G )
% 0.30/0.46 => ( ( H
% 0.30/0.46 = ( comp @ ( sum_sum @ A @ C ) @ B @ C @ F @ ( sum_Inr @ C @ A ) ) )
% 0.30/0.46 = ( ( sum_case_sum @ A @ B @ C @ G @ H )
% 0.30/0.46 = F ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % case_sum_expand_Inr'
% 0.30/0.46 thf(fact_88_case__sum__expand__Inr,axiom,
% 0.30/0.46 ! [B: $tType,C: $tType,A: $tType,F: ( sum_sum @ A @ C ) > B,G: A > B,X: sum_sum @ A @ C] :
% 0.30/0.46 ( ( ( comp @ ( sum_sum @ A @ C ) @ B @ A @ F @ ( sum_Inl @ A @ C ) )
% 0.30/0.46 = G )
% 0.30/0.46 => ( ( F @ X )
% 0.30/0.46 = ( sum_case_sum @ A @ B @ C @ G @ ( comp @ ( sum_sum @ A @ C ) @ B @ C @ F @ ( sum_Inr @ C @ A ) ) @ X ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % case_sum_expand_Inr
% 0.30/0.46 thf(fact_89_dtree_Ocase,axiom,
% 0.30/0.46 ! [A: $tType,F: n > ( fset @ ( sum_sum @ t @ dtree ) ) > A,X1: n,X2: fset @ ( sum_sum @ t @ dtree )] :
% 0.30/0.46 ( ( case_dtree @ A @ F @ ( nNode @ X1 @ X2 ) )
% 0.30/0.46 = ( F @ X1 @ X2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % dtree.case
% 0.30/0.46 thf(fact_90_finite__Diff__insert,axiom,
% 0.30/0.46 ! [A: $tType,A7: set @ A,A2: A,B7: set @ A] :
% 0.30/0.46 ( ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A7 @ ( insert @ A @ A2 @ B7 ) ) )
% 0.30/0.46 = ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A7 @ B7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % finite_Diff_insert
% 0.30/0.46 thf(fact_91_Diff__idemp,axiom,
% 0.30/0.46 ! [A: $tType,A7: set @ A,B7: set @ A] :
% 0.30/0.46 ( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A7 @ B7 ) @ B7 )
% 0.30/0.46 = ( minus_minus @ ( set @ A ) @ A7 @ B7 ) ) ).
% 0.30/0.46
% 0.30/0.46 % Diff_idemp
% 0.30/0.46 thf(fact_92_Diff__iff,axiom,
% 0.30/0.46 ! [A: $tType,C3: A,A7: set @ A,B7: set @ A] :
% 0.30/0.46 ( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A7 @ B7 ) )
% 0.30/0.46 = ( ( member @ A @ C3 @ A7 )
% 0.30/0.46 & ~ ( member @ A @ C3 @ B7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Diff_iff
% 0.30/0.46 thf(fact_93_DiffI,axiom,
% 0.30/0.46 ! [A: $tType,C3: A,A7: set @ A,B7: set @ A] :
% 0.30/0.46 ( ( member @ A @ C3 @ A7 )
% 0.30/0.46 => ( ~ ( member @ A @ C3 @ B7 )
% 0.30/0.46 => ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A7 @ B7 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % DiffI
% 0.30/0.46 thf(fact_94_finite__Diff2,axiom,
% 0.30/0.46 ! [A: $tType,B7: set @ A,A7: set @ A] :
% 0.30/0.46 ( ( finite_finite2 @ A @ B7 )
% 0.30/0.46 => ( ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A7 @ B7 ) )
% 0.30/0.46 = ( finite_finite2 @ A @ A7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % finite_Diff2
% 0.30/0.46 thf(fact_95_finite__Diff,axiom,
% 0.30/0.46 ! [A: $tType,A7: set @ A,B7: set @ A] :
% 0.30/0.46 ( ( finite_finite2 @ A @ A7 )
% 0.30/0.46 => ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A7 @ B7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % finite_Diff
% 0.30/0.46 thf(fact_96_Diff__insert0,axiom,
% 0.30/0.46 ! [A: $tType,X: A,A7: set @ A,B7: set @ A] :
% 0.30/0.46 ( ~ ( member @ A @ X @ A7 )
% 0.30/0.46 => ( ( minus_minus @ ( set @ A ) @ A7 @ ( insert @ A @ X @ B7 ) )
% 0.30/0.46 = ( minus_minus @ ( set @ A ) @ A7 @ B7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Diff_insert0
% 0.30/0.46 thf(fact_97_insert__Diff1,axiom,
% 0.30/0.46 ! [A: $tType,X: A,B7: set @ A,A7: set @ A] :
% 0.30/0.46 ( ( member @ A @ X @ B7 )
% 0.30/0.46 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A7 ) @ B7 )
% 0.30/0.46 = ( minus_minus @ ( set @ A ) @ A7 @ B7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % insert_Diff1
% 0.30/0.46 thf(fact_98_o__case__sum,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,D: $tType,B: $tType,H: D > C,F: A > D,G: B > D] :
% 0.30/0.46 ( ( comp @ D @ C @ ( sum_sum @ A @ B ) @ H @ ( sum_case_sum @ A @ D @ B @ F @ G ) )
% 0.30/0.46 = ( sum_case_sum @ A @ C @ B @ ( comp @ D @ C @ A @ H @ F ) @ ( comp @ D @ C @ B @ H @ G ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % o_case_sum
% 0.30/0.46 thf(fact_99_rewriteR__comp__comp2,axiom,
% 0.30/0.46 ! [C: $tType,B: $tType,E: $tType,D: $tType,A: $tType,G: C > B,H: A > C,R1: D > B,R2: A > D,F: B > E,L: D > E] :
% 0.30/0.46 ( ( ( comp @ C @ B @ A @ G @ H )
% 0.30/0.46 = ( comp @ D @ B @ A @ R1 @ R2 ) )
% 0.30/0.46 => ( ( ( comp @ B @ E @ D @ F @ R1 )
% 0.30/0.46 = L )
% 0.30/0.46 => ( ( comp @ C @ E @ A @ ( comp @ B @ E @ C @ F @ G ) @ H )
% 0.30/0.46 = ( comp @ D @ E @ A @ L @ R2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % rewriteR_comp_comp2
% 0.30/0.46 thf(fact_100_rewriteL__comp__comp2,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,B: $tType,D: $tType,E: $tType,F: C > B,G: A > C,L1: D > B,L2: A > D,H: E > A,R: E > D] :
% 0.30/0.46 ( ( ( comp @ C @ B @ A @ F @ G )
% 0.30/0.46 = ( comp @ D @ B @ A @ L1 @ L2 ) )
% 0.30/0.46 => ( ( ( comp @ A @ D @ E @ L2 @ H )
% 0.30/0.46 = R )
% 0.30/0.46 => ( ( comp @ C @ B @ E @ F @ ( comp @ A @ C @ E @ G @ H ) )
% 0.30/0.46 = ( comp @ D @ B @ E @ L1 @ R ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % rewriteL_comp_comp2
% 0.30/0.46 thf(fact_101_rewriteR__comp__comp,axiom,
% 0.30/0.46 ! [C: $tType,D: $tType,B: $tType,A: $tType,G: C > B,H: A > C,R: A > B,F: B > D] :
% 0.30/0.46 ( ( ( comp @ C @ B @ A @ G @ H )
% 0.30/0.46 = R )
% 0.30/0.46 => ( ( comp @ C @ D @ A @ ( comp @ B @ D @ C @ F @ G ) @ H )
% 0.30/0.46 = ( comp @ B @ D @ A @ F @ R ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % rewriteR_comp_comp
% 0.30/0.46 thf(fact_102_rewriteL__comp__comp,axiom,
% 0.30/0.46 ! [C: $tType,B: $tType,A: $tType,D: $tType,F: C > B,G: A > C,L: A > B,H: D > A] :
% 0.30/0.46 ( ( ( comp @ C @ B @ A @ F @ G )
% 0.30/0.46 = L )
% 0.30/0.46 => ( ( comp @ C @ B @ D @ F @ ( comp @ A @ C @ D @ G @ H ) )
% 0.30/0.46 = ( comp @ A @ B @ D @ L @ H ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % rewriteL_comp_comp
% 0.30/0.46 thf(fact_103_DiffD2,axiom,
% 0.30/0.46 ! [A: $tType,C3: A,A7: set @ A,B7: set @ A] :
% 0.30/0.46 ( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A7 @ B7 ) )
% 0.30/0.46 => ~ ( member @ A @ C3 @ B7 ) ) ).
% 0.30/0.46
% 0.30/0.46 % DiffD2
% 0.30/0.46 thf(fact_104_DiffD1,axiom,
% 0.30/0.46 ! [A: $tType,C3: A,A7: set @ A,B7: set @ A] :
% 0.30/0.46 ( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A7 @ B7 ) )
% 0.30/0.46 => ( member @ A @ C3 @ A7 ) ) ).
% 0.30/0.46
% 0.30/0.46 % DiffD1
% 0.30/0.46 thf(fact_105_DiffE,axiom,
% 0.30/0.46 ! [A: $tType,C3: A,A7: set @ A,B7: set @ A] :
% 0.30/0.46 ( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A7 @ B7 ) )
% 0.30/0.46 => ~ ( ( member @ A @ C3 @ A7 )
% 0.30/0.46 => ( member @ A @ C3 @ B7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % DiffE
% 0.30/0.46 thf(fact_106_dtree_Ocase__eq__if,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( case_dtree @ A )
% 0.30/0.46 = ( ^ [F5: n > ( fset @ ( sum_sum @ t @ dtree ) ) > A,Dtree2: dtree] : ( F5 @ ( root @ Dtree2 ) @ ( ccont @ Dtree2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % dtree.case_eq_if
% 0.30/0.46 thf(fact_107_Diff__infinite__finite,axiom,
% 0.30/0.46 ! [A: $tType,T5: set @ A,S2: set @ A] :
% 0.30/0.46 ( ( finite_finite2 @ A @ T5 )
% 0.30/0.46 => ( ~ ( finite_finite2 @ A @ S2 )
% 0.30/0.46 => ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ S2 @ T5 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Diff_infinite_finite
% 0.30/0.46 thf(fact_108_insert__Diff__if,axiom,
% 0.30/0.46 ! [A: $tType,X: A,B7: set @ A,A7: set @ A] :
% 0.30/0.46 ( ( ( member @ A @ X @ B7 )
% 0.30/0.46 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A7 ) @ B7 )
% 0.30/0.46 = ( minus_minus @ ( set @ A ) @ A7 @ B7 ) ) )
% 0.30/0.46 & ( ~ ( member @ A @ X @ B7 )
% 0.30/0.46 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A7 ) @ B7 )
% 0.30/0.46 = ( insert @ A @ X @ ( minus_minus @ ( set @ A ) @ A7 @ B7 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % insert_Diff_if
% 0.30/0.46 thf(fact_109_dtree_Oexpand,axiom,
% 0.30/0.46 ! [Dtree: dtree,Dtree3: dtree] :
% 0.30/0.46 ( ( ( ( root @ Dtree )
% 0.30/0.46 = ( root @ Dtree3 ) )
% 0.30/0.46 & ( ( ccont @ Dtree )
% 0.30/0.46 = ( ccont @ Dtree3 ) ) )
% 0.30/0.46 => ( Dtree = Dtree3 ) ) ).
% 0.30/0.46
% 0.30/0.46 % dtree.expand
% 0.30/0.46 thf(fact_110_dtree_Osplit__sel__asm,axiom,
% 0.30/0.46 ! [A: $tType,P: A > $o,F: n > ( fset @ ( sum_sum @ t @ dtree ) ) > A,Dtree: dtree] :
% 0.30/0.46 ( ( P @ ( case_dtree @ A @ F @ Dtree ) )
% 0.30/0.46 = ( ~ ( ( Dtree
% 0.30/0.46 = ( nNode @ ( root @ Dtree ) @ ( ccont @ Dtree ) ) )
% 0.30/0.46 & ~ ( P @ ( F @ ( root @ Dtree ) @ ( ccont @ Dtree ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % dtree.split_sel_asm
% 0.30/0.46 thf(fact_111_dtree_Osplit__sel,axiom,
% 0.30/0.46 ! [A: $tType,P: A > $o,F: n > ( fset @ ( sum_sum @ t @ dtree ) ) > A,Dtree: dtree] :
% 0.30/0.46 ( ( P @ ( case_dtree @ A @ F @ Dtree ) )
% 0.30/0.46 = ( ( Dtree
% 0.30/0.46 = ( nNode @ ( root @ Dtree ) @ ( ccont @ Dtree ) ) )
% 0.30/0.46 => ( P @ ( F @ ( root @ Dtree ) @ ( ccont @ Dtree ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % dtree.split_sel
% 0.30/0.46 thf(fact_112_dtree_Osel_I2_J,axiom,
% 0.30/0.46 ! [X1: n,X2: fset @ ( sum_sum @ t @ dtree )] :
% 0.30/0.46 ( ( ccont @ ( nNode @ X1 @ X2 ) )
% 0.30/0.46 = X2 ) ).
% 0.30/0.46
% 0.30/0.46 % dtree.sel(2)
% 0.30/0.46 thf(fact_113_case__sum__o__inj_I1_J,axiom,
% 0.30/0.46 ! [C: $tType,B: $tType,A: $tType,F: A > B,G: C > B] :
% 0.30/0.46 ( ( comp @ ( sum_sum @ A @ C ) @ B @ A @ ( sum_case_sum @ A @ B @ C @ F @ G ) @ ( sum_Inl @ A @ C ) )
% 0.30/0.46 = F ) ).
% 0.30/0.46
% 0.30/0.46 % case_sum_o_inj(1)
% 0.30/0.46 thf(fact_114_case__sum__o__inj_I2_J,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,C: $tType,F: A > B,G: C > B] :
% 0.30/0.46 ( ( comp @ ( sum_sum @ A @ C ) @ B @ C @ ( sum_case_sum @ A @ B @ C @ F @ G ) @ ( sum_Inr @ C @ A ) )
% 0.30/0.46 = G ) ).
% 0.30/0.46
% 0.30/0.46 % case_sum_o_inj(2)
% 0.30/0.46 thf(fact_115_comp__apply,axiom,
% 0.30/0.46 ! [C: $tType,A: $tType,B: $tType] :
% 0.30/0.46 ( ( comp @ B @ A @ C )
% 0.30/0.46 = ( ^ [F5: B > A,G4: C > B,X5: C] : ( F5 @ ( G4 @ X5 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % comp_apply
% 0.30/0.46 thf(fact_116_minus__apply,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( minus @ B @ ( type2 @ B ) )
% 0.30/0.46 => ( ( minus_minus @ ( A > B ) )
% 0.30/0.46 = ( ^ [A8: A > B,B9: A > B,X5: A] : ( minus_minus @ B @ ( A8 @ X5 ) @ ( B9 @ X5 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % minus_apply
% 0.30/0.46 thf(fact_117_comp__def,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,B: $tType] :
% 0.30/0.46 ( ( comp @ B @ C @ A )
% 0.30/0.46 = ( ^ [F5: B > C,G4: A > B,X5: A] : ( F5 @ ( G4 @ X5 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % comp_def
% 0.30/0.46 thf(fact_118_comp__assoc,axiom,
% 0.30/0.46 ! [B: $tType,D: $tType,C: $tType,A: $tType,F: D > B,G: C > D,H: A > C] :
% 0.30/0.46 ( ( comp @ C @ B @ A @ ( comp @ D @ B @ C @ F @ G ) @ H )
% 0.30/0.46 = ( comp @ D @ B @ A @ F @ ( comp @ C @ D @ A @ G @ H ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % comp_assoc
% 0.30/0.46 thf(fact_119_fun__diff__def,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType] :
% 0.30/0.46 ( ( minus @ B @ ( type2 @ B ) )
% 0.30/0.46 => ( ( minus_minus @ ( A > B ) )
% 0.30/0.46 = ( ^ [A8: A > B,B9: A > B,X5: A] : ( minus_minus @ B @ ( A8 @ X5 ) @ ( B9 @ X5 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % fun_diff_def
% 0.30/0.46 thf(fact_120_comp__apply__eq,axiom,
% 0.30/0.46 ! [B: $tType,D: $tType,A: $tType,C: $tType,F: B > A,G: C > B,X: C,H: D > A,K: C > D] :
% 0.30/0.46 ( ( ( F @ ( G @ X ) )
% 0.30/0.46 = ( H @ ( K @ X ) ) )
% 0.30/0.46 => ( ( comp @ B @ A @ C @ F @ G @ X )
% 0.30/0.46 = ( comp @ D @ A @ C @ H @ K @ X ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % comp_apply_eq
% 0.30/0.46 thf(fact_121_comp__eq__dest__lhs,axiom,
% 0.30/0.46 ! [C: $tType,B: $tType,A: $tType,A2: C > B,B2: A > C,C3: A > B,V: A] :
% 0.30/0.46 ( ( ( comp @ C @ B @ A @ A2 @ B2 )
% 0.30/0.46 = C3 )
% 0.30/0.46 => ( ( A2 @ ( B2 @ V ) )
% 0.30/0.46 = ( C3 @ V ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % comp_eq_dest_lhs
% 0.30/0.46 thf(fact_122_comp__eq__elim,axiom,
% 0.30/0.46 ! [C: $tType,B: $tType,D: $tType,A: $tType,A2: C > B,B2: A > C,C3: D > B,D2: A > D] :
% 0.30/0.46 ( ( ( comp @ C @ B @ A @ A2 @ B2 )
% 0.30/0.46 = ( comp @ D @ B @ A @ C3 @ D2 ) )
% 0.30/0.46 => ! [V2: A] :
% 0.30/0.46 ( ( A2 @ ( B2 @ V2 ) )
% 0.30/0.46 = ( C3 @ ( D2 @ V2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % comp_eq_elim
% 0.30/0.46 thf(fact_123_comp__eq__dest,axiom,
% 0.30/0.46 ! [C: $tType,B: $tType,D: $tType,A: $tType,A2: C > B,B2: A > C,C3: D > B,D2: A > D,V: A] :
% 0.30/0.46 ( ( ( comp @ C @ B @ A @ A2 @ B2 )
% 0.30/0.46 = ( comp @ D @ B @ A @ C3 @ D2 ) )
% 0.30/0.46 => ( ( A2 @ ( B2 @ V ) )
% 0.30/0.46 = ( C3 @ ( D2 @ V ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % comp_eq_dest
% 0.30/0.46 thf(fact_124_type__copy__map__cong0,axiom,
% 0.30/0.46 ! [B: $tType,D: $tType,E: $tType,A: $tType,C: $tType,M: B > A,G: C > B,X: C,N4: D > A,H: C > D,F: A > E] :
% 0.30/0.46 ( ( ( M @ ( G @ X ) )
% 0.30/0.46 = ( N4 @ ( H @ X ) ) )
% 0.30/0.46 => ( ( comp @ B @ E @ C @ ( comp @ A @ E @ B @ F @ M ) @ G @ X )
% 0.30/0.46 = ( comp @ D @ E @ C @ ( comp @ A @ E @ D @ F @ N4 ) @ H @ X ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % type_copy_map_cong0
% 0.30/0.46 thf(fact_125_fun_Omap__comp,axiom,
% 0.30/0.46 ! [B: $tType,C: $tType,A: $tType,D: $tType,G: B > C,F: A > B,V: D > A] :
% 0.30/0.46 ( ( comp @ B @ C @ D @ G @ ( comp @ A @ B @ D @ F @ V ) )
% 0.30/0.46 = ( comp @ A @ C @ D @ ( comp @ B @ C @ A @ G @ F ) @ V ) ) ).
% 0.30/0.46
% 0.30/0.46 % fun.map_comp
% 0.30/0.46 thf(fact_126_comp__cong,axiom,
% 0.30/0.46 ! [C: $tType,B: $tType,D: $tType,A: $tType,E: $tType,F: B > A,G: C > B,X: C,F4: D > A,G3: E > D,X8: E] :
% 0.30/0.46 ( ( ( F @ ( G @ X ) )
% 0.30/0.46 = ( F4 @ ( G3 @ X8 ) ) )
% 0.30/0.46 => ( ( comp @ B @ A @ C @ F @ G @ X )
% 0.30/0.46 = ( comp @ D @ A @ E @ F4 @ G3 @ X8 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % comp_cong
% 0.30/0.46 thf(fact_127_finite__empty__induct,axiom,
% 0.30/0.46 ! [A: $tType,A7: set @ A,P: ( set @ A ) > $o] :
% 0.30/0.46 ( ( finite_finite2 @ A @ A7 )
% 0.30/0.46 => ( ( P @ A7 )
% 0.30/0.46 => ( ! [A4: A,A9: set @ A] :
% 0.30/0.46 ( ( finite_finite2 @ A @ A9 )
% 0.30/0.46 => ( ( member @ A @ A4 @ A9 )
% 0.30/0.46 => ( ( P @ A9 )
% 0.30/0.46 => ( P @ ( minus_minus @ ( set @ A ) @ A9 @ ( insert @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) )
% 0.30/0.46 => ( P @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % finite_empty_induct
% 0.30/0.46 thf(fact_128_empty__Collect__eq,axiom,
% 0.30/0.46 ! [A: $tType,P: A > $o] :
% 0.30/0.46 ( ( ( bot_bot @ ( set @ A ) )
% 0.30/0.46 = ( collect @ A @ P ) )
% 0.30/0.46 = ( ! [X5: A] :
% 0.30/0.46 ~ ( P @ X5 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % empty_Collect_eq
% 0.30/0.46 thf(fact_129_Collect__empty__eq,axiom,
% 0.30/0.46 ! [A: $tType,P: A > $o] :
% 0.30/0.46 ( ( ( collect @ A @ P )
% 0.30/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.46 = ( ! [X5: A] :
% 0.30/0.46 ~ ( P @ X5 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Collect_empty_eq
% 0.30/0.46 thf(fact_130_all__not__in__conv,axiom,
% 0.30/0.46 ! [A: $tType,A7: set @ A] :
% 0.30/0.46 ( ( ! [X5: A] :
% 0.30/0.46 ~ ( member @ A @ X5 @ A7 ) )
% 0.30/0.46 = ( A7
% 0.30/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % all_not_in_conv
% 0.30/0.46 thf(fact_131_empty__iff,axiom,
% 0.30/0.46 ! [A: $tType,C3: A] :
% 0.30/0.46 ~ ( member @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % empty_iff
% 0.30/0.46 thf(fact_132_singletonI,axiom,
% 0.30/0.46 ! [A: $tType,A2: A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % singletonI
% 0.30/0.46 thf(fact_133_Diff__cancel,axiom,
% 0.30/0.46 ! [A: $tType,A7: set @ A] :
% 0.30/0.46 ( ( minus_minus @ ( set @ A ) @ A7 @ A7 )
% 0.30/0.46 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Diff_cancel
% 0.30/0.46 thf(fact_134_empty__Diff,axiom,
% 0.30/0.46 ! [A: $tType,A7: set @ A] :
% 0.30/0.46 ( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A7 )
% 0.30/0.46 = ( bot_bot @ ( set @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % empty_Diff
% 0.30/0.46 thf(fact_135_Diff__empty,axiom,
% 0.30/0.46 ! [A: $tType,A7: set @ A] :
% 0.30/0.46 ( ( minus_minus @ ( set @ A ) @ A7 @ ( bot_bot @ ( set @ A ) ) )
% 0.30/0.46 = A7 ) ).
% 0.30/0.46
% 0.30/0.46 % Diff_empty
% 0.30/0.46 thf(fact_136_Plus__eq__empty__conv,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A7: set @ A,B7: set @ B] :
% 0.30/0.46 ( ( ( sum_Plus @ A @ B @ A7 @ B7 )
% 0.30/0.46 = ( bot_bot @ ( set @ ( sum_sum @ A @ B ) ) ) )
% 0.30/0.46 = ( ( A7
% 0.30/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.46 & ( B7
% 0.30/0.46 = ( bot_bot @ ( set @ B ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Plus_eq_empty_conv
% 0.30/0.46 thf(fact_137_insert__Diff__single,axiom,
% 0.30/0.46 ! [A: $tType,A2: A,A7: set @ A] :
% 0.30/0.46 ( ( insert @ A @ A2 @ ( minus_minus @ ( set @ A ) @ A7 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.30/0.46 = ( insert @ A @ A2 @ A7 ) ) ).
% 0.30/0.46
% 0.30/0.46 % insert_Diff_single
% 0.30/0.46 thf(fact_138_finite_OemptyI,axiom,
% 0.30/0.46 ! [A: $tType] : ( finite_finite2 @ A @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % finite.emptyI
% 0.30/0.46 thf(fact_139_infinite__imp__nonempty,axiom,
% 0.30/0.46 ! [A: $tType,S2: set @ A] :
% 0.30/0.46 ( ~ ( finite_finite2 @ A @ S2 )
% 0.30/0.46 => ( S2
% 0.30/0.46 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % infinite_imp_nonempty
% 0.30/0.46 thf(fact_140_singletonD,axiom,
% 0.30/0.46 ! [A: $tType,B2: A,A2: A] :
% 0.30/0.46 ( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.46 => ( B2 = A2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % singletonD
% 0.30/0.46 thf(fact_141_singleton__iff,axiom,
% 0.30/0.46 ! [A: $tType,B2: A,A2: A] :
% 0.30/0.46 ( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.46 = ( B2 = A2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % singleton_iff
% 0.30/0.46 thf(fact_142_doubleton__eq__iff,axiom,
% 0.30/0.46 ! [A: $tType,A2: A,B2: A,C3: A,D2: A] :
% 0.30/0.46 ( ( ( insert @ A @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.46 = ( insert @ A @ C3 @ ( insert @ A @ D2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.30/0.46 = ( ( ( A2 = C3 )
% 0.30/0.46 & ( B2 = D2 ) )
% 0.30/0.46 | ( ( A2 = D2 )
% 0.30/0.46 & ( B2 = C3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % doubleton_eq_iff
% 0.30/0.46 thf(fact_143_insert__not__empty,axiom,
% 0.30/0.46 ! [A: $tType,A2: A,A7: set @ A] :
% 0.30/0.46 ( ( insert @ A @ A2 @ A7 )
% 0.30/0.46 != ( bot_bot @ ( set @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % insert_not_empty
% 0.30/0.46 thf(fact_144_singleton__inject,axiom,
% 0.30/0.46 ! [A: $tType,A2: A,B2: A] :
% 0.30/0.46 ( ( ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) )
% 0.30/0.46 = ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.46 => ( A2 = B2 ) ) ).
% 0.30/0.46
% 0.30/0.46 % singleton_inject
% 0.30/0.46 thf(fact_145_ex__in__conv,axiom,
% 0.30/0.46 ! [A: $tType,A7: set @ A] :
% 0.30/0.46 ( ( ? [X5: A] : ( member @ A @ X5 @ A7 ) )
% 0.30/0.46 = ( A7
% 0.30/0.46 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % ex_in_conv
% 0.30/0.46 thf(fact_146_equals0I,axiom,
% 0.30/0.46 ! [A: $tType,A7: set @ A] :
% 0.30/0.46 ( ! [Y3: A] :
% 0.30/0.46 ~ ( member @ A @ Y3 @ A7 )
% 0.30/0.46 => ( A7
% 0.30/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % equals0I
% 0.30/0.46 thf(fact_147_equals0D,axiom,
% 0.30/0.46 ! [A: $tType,A7: set @ A,A2: A] :
% 0.30/0.46 ( ( A7
% 0.30/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.46 => ~ ( member @ A @ A2 @ A7 ) ) ).
% 0.30/0.46
% 0.30/0.46 % equals0D
% 0.30/0.46 thf(fact_148_emptyE,axiom,
% 0.30/0.46 ! [A: $tType,A2: A] :
% 0.30/0.46 ~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % emptyE
% 0.30/0.46 thf(fact_149_infinite__finite__induct,axiom,
% 0.30/0.46 ! [A: $tType,P: ( set @ A ) > $o,A7: set @ A] :
% 0.30/0.46 ( ! [A9: set @ A] :
% 0.30/0.46 ( ~ ( finite_finite2 @ A @ A9 )
% 0.30/0.46 => ( P @ A9 ) )
% 0.30/0.46 => ( ( P @ ( bot_bot @ ( set @ A ) ) )
% 0.30/0.46 => ( ! [X6: A,F6: set @ A] :
% 0.30/0.46 ( ( finite_finite2 @ A @ F6 )
% 0.30/0.46 => ( ~ ( member @ A @ X6 @ F6 )
% 0.30/0.46 => ( ( P @ F6 )
% 0.30/0.46 => ( P @ ( insert @ A @ X6 @ F6 ) ) ) ) )
% 0.30/0.46 => ( P @ A7 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % infinite_finite_induct
% 0.30/0.46 thf(fact_150_finite__ne__induct,axiom,
% 0.30/0.46 ! [A: $tType,F7: set @ A,P: ( set @ A ) > $o] :
% 0.30/0.46 ( ( finite_finite2 @ A @ F7 )
% 0.30/0.46 => ( ( F7
% 0.30/0.46 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.46 => ( ! [X6: A] : ( P @ ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.46 => ( ! [X6: A,F6: set @ A] :
% 0.30/0.46 ( ( finite_finite2 @ A @ F6 )
% 0.30/0.46 => ( ( F6
% 0.30/0.46 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.46 => ( ~ ( member @ A @ X6 @ F6 )
% 0.30/0.46 => ( ( P @ F6 )
% 0.30/0.46 => ( P @ ( insert @ A @ X6 @ F6 ) ) ) ) ) )
% 0.30/0.46 => ( P @ F7 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % finite_ne_induct
% 0.30/0.46 thf(fact_151_finite_Oinducts,axiom,
% 0.30/0.46 ! [A: $tType,X: set @ A,P: ( set @ A ) > $o] :
% 0.30/0.46 ( ( finite_finite2 @ A @ X )
% 0.30/0.46 => ( ( P @ ( bot_bot @ ( set @ A ) ) )
% 0.30/0.46 => ( ! [A9: set @ A,A4: A] :
% 0.30/0.46 ( ( finite_finite2 @ A @ A9 )
% 0.30/0.46 => ( ( P @ A9 )
% 0.30/0.46 => ( P @ ( insert @ A @ A4 @ A9 ) ) ) )
% 0.30/0.46 => ( P @ X ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % finite.inducts
% 0.30/0.46 thf(fact_152_finite__induct,axiom,
% 0.30/0.46 ! [A: $tType,F7: set @ A,P: ( set @ A ) > $o] :
% 0.30/0.46 ( ( finite_finite2 @ A @ F7 )
% 0.30/0.46 => ( ( P @ ( bot_bot @ ( set @ A ) ) )
% 0.30/0.46 => ( ! [X6: A,F6: set @ A] :
% 0.30/0.46 ( ( finite_finite2 @ A @ F6 )
% 0.30/0.46 => ( ~ ( member @ A @ X6 @ F6 )
% 0.30/0.46 => ( ( P @ F6 )
% 0.30/0.46 => ( P @ ( insert @ A @ X6 @ F6 ) ) ) ) )
% 0.30/0.46 => ( P @ F7 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % finite_induct
% 0.30/0.46 thf(fact_153_finite_Osimps,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( finite_finite2 @ A )
% 0.30/0.46 = ( ^ [A10: set @ A] :
% 0.30/0.46 ( ( A10
% 0.30/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.46 | ? [A8: set @ A,B10: A] :
% 0.30/0.46 ( ( A10
% 0.30/0.46 = ( insert @ A @ B10 @ A8 ) )
% 0.30/0.46 & ( finite_finite2 @ A @ A8 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % finite.simps
% 0.30/0.46 thf(fact_154_finite_Ocases,axiom,
% 0.30/0.46 ! [A: $tType,A2: set @ A] :
% 0.30/0.46 ( ( finite_finite2 @ A @ A2 )
% 0.30/0.46 => ( ( A2
% 0.30/0.46 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.46 => ~ ! [A9: set @ A] :
% 0.30/0.46 ( ? [A4: A] :
% 0.30/0.46 ( A2
% 0.30/0.46 = ( insert @ A @ A4 @ A9 ) )
% 0.30/0.46 => ~ ( finite_finite2 @ A @ A9 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % finite.cases
% 0.30/0.46 thf(fact_155_Diff__insert,axiom,
% 0.30/0.46 ! [A: $tType,A7: set @ A,A2: A,B7: set @ A] :
% 0.30/0.46 ( ( minus_minus @ ( set @ A ) @ A7 @ ( insert @ A @ A2 @ B7 ) )
% 0.30/0.46 = ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A7 @ B7 ) @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Diff_insert
% 0.30/0.46 thf(fact_156_insert__Diff,axiom,
% 0.30/0.46 ! [A: $tType,A2: A,A7: set @ A] :
% 0.30/0.46 ( ( member @ A @ A2 @ A7 )
% 0.30/0.46 => ( ( insert @ A @ A2 @ ( minus_minus @ ( set @ A ) @ A7 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.30/0.46 = A7 ) ) ).
% 0.30/0.46
% 0.30/0.46 % insert_Diff
% 0.30/0.46 thf(fact_157_Diff__insert2,axiom,
% 0.30/0.46 ! [A: $tType,A7: set @ A,A2: A,B7: set @ A] :
% 0.30/0.46 ( ( minus_minus @ ( set @ A ) @ A7 @ ( insert @ A @ A2 @ B7 ) )
% 0.30/0.46 = ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A7 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) @ B7 ) ) ).
% 0.30/0.46
% 0.30/0.46 % Diff_insert2
% 0.30/0.46 thf(fact_158_Diff__insert__absorb,axiom,
% 0.30/0.46 ! [A: $tType,X: A,A7: set @ A] :
% 0.30/0.46 ( ~ ( member @ A @ X @ A7 )
% 0.30/0.46 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A7 ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.46 = A7 ) ) ).
% 0.30/0.46
% 0.30/0.46 % Diff_insert_absorb
% 0.30/0.46 thf(fact_159_infinite__remove,axiom,
% 0.30/0.46 ! [A: $tType,S2: set @ A,A2: A] :
% 0.30/0.46 ( ~ ( finite_finite2 @ A @ S2 )
% 0.30/0.46 => ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ S2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % infinite_remove
% 0.30/0.46 thf(fact_160_infinite__coinduct,axiom,
% 0.30/0.46 ! [A: $tType,X9: ( set @ A ) > $o,A7: set @ A] :
% 0.30/0.46 ( ( X9 @ A7 )
% 0.30/0.46 => ( ! [A9: set @ A] :
% 0.30/0.46 ( ( X9 @ A9 )
% 0.30/0.46 => ? [X7: A] :
% 0.30/0.46 ( ( member @ A @ X7 @ A9 )
% 0.30/0.46 & ( ( X9 @ ( minus_minus @ ( set @ A ) @ A9 @ ( insert @ A @ X7 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 0.30/0.46 | ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A9 @ ( insert @ A @ X7 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) )
% 0.30/0.46 => ~ ( finite_finite2 @ A @ A7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % infinite_coinduct
% 0.30/0.46 thf(fact_161_is__singletonI,axiom,
% 0.30/0.46 ! [A: $tType,X: A] : ( is_singleton @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % is_singletonI
% 0.30/0.46 thf(fact_162_remove__def,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( remove @ A )
% 0.30/0.46 = ( ^ [X5: A,A8: set @ A] : ( minus_minus @ ( set @ A ) @ A8 @ ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % remove_def
% 0.30/0.46 thf(fact_163_the__elem__eq,axiom,
% 0.30/0.46 ! [A: $tType,X: A] :
% 0.30/0.46 ( ( the_elem @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.46 = X ) ).
% 0.30/0.46
% 0.30/0.46 % the_elem_eq
% 0.30/0.46 thf(fact_164_sum__set__simps_I1_J,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,X: A] :
% 0.30/0.46 ( ( basic_setl @ A @ B @ ( sum_Inl @ A @ B @ X ) )
% 0.30/0.46 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % sum_set_simps(1)
% 0.30/0.46 thf(fact_165_member__remove,axiom,
% 0.30/0.46 ! [A: $tType,X: A,Y: A,A7: set @ A] :
% 0.30/0.46 ( ( member @ A @ X @ ( remove @ A @ Y @ A7 ) )
% 0.30/0.46 = ( ( member @ A @ X @ A7 )
% 0.30/0.46 & ( X != Y ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % member_remove
% 0.30/0.46 thf(fact_166_sum__set__simps_I2_J,axiom,
% 0.30/0.46 ! [A: $tType,C: $tType,X: A] :
% 0.30/0.46 ( ( basic_setl @ C @ A @ ( sum_Inr @ A @ C @ X ) )
% 0.30/0.46 = ( bot_bot @ ( set @ C ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % sum_set_simps(2)
% 0.30/0.46 thf(fact_167_setl_Ocases,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,A2: A,S: sum_sum @ A @ B] :
% 0.30/0.46 ( ( member @ A @ A2 @ ( basic_setl @ A @ B @ S ) )
% 0.30/0.46 => ( S
% 0.30/0.46 = ( sum_Inl @ A @ B @ A2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % setl.cases
% 0.30/0.46 thf(fact_168_setl_Osimps,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,A2: A,S: sum_sum @ A @ B] :
% 0.30/0.46 ( ( member @ A @ A2 @ ( basic_setl @ A @ B @ S ) )
% 0.30/0.46 = ( ? [X5: A] :
% 0.30/0.46 ( ( A2 = X5 )
% 0.30/0.46 & ( S
% 0.30/0.46 = ( sum_Inl @ A @ B @ X5 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % setl.simps
% 0.30/0.46 thf(fact_169_setl_Ointros,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,S: sum_sum @ A @ B,X: A] :
% 0.30/0.46 ( ( S
% 0.30/0.46 = ( sum_Inl @ A @ B @ X ) )
% 0.30/0.46 => ( member @ A @ X @ ( basic_setl @ A @ B @ S ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % setl.intros
% 0.30/0.46 thf(fact_170_setl_Oinducts,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,X: A,S: sum_sum @ A @ B,P: A > $o] :
% 0.30/0.46 ( ( member @ A @ X @ ( basic_setl @ A @ B @ S ) )
% 0.30/0.46 => ( ! [X6: A] :
% 0.30/0.46 ( ( S
% 0.30/0.46 = ( sum_Inl @ A @ B @ X6 ) )
% 0.30/0.46 => ( P @ X6 ) )
% 0.30/0.46 => ( P @ X ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % setl.inducts
% 0.30/0.46 thf(fact_171_is__singleton__the__elem,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( is_singleton @ A )
% 0.30/0.46 = ( ^ [A8: set @ A] :
% 0.30/0.46 ( A8
% 0.30/0.46 = ( insert @ A @ ( the_elem @ A @ A8 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % is_singleton_the_elem
% 0.30/0.46 thf(fact_172_is__singletonI_H,axiom,
% 0.30/0.46 ! [A: $tType,A7: set @ A] :
% 0.30/0.46 ( ( A7
% 0.30/0.46 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.46 => ( ! [X6: A,Y3: A] :
% 0.30/0.46 ( ( member @ A @ X6 @ A7 )
% 0.30/0.46 => ( ( member @ A @ Y3 @ A7 )
% 0.30/0.46 => ( X6 = Y3 ) ) )
% 0.30/0.46 => ( is_singleton @ A @ A7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % is_singletonI'
% 0.30/0.46 thf(fact_173_is__singleton__def,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( is_singleton @ A )
% 0.30/0.46 = ( ^ [A8: set @ A] :
% 0.30/0.46 ? [X5: A] :
% 0.30/0.46 ( A8
% 0.30/0.46 = ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % is_singleton_def
% 0.30/0.46 thf(fact_174_is__singletonE,axiom,
% 0.30/0.46 ! [A: $tType,A7: set @ A] :
% 0.30/0.46 ( ( is_singleton @ A @ A7 )
% 0.30/0.46 => ~ ! [X6: A] :
% 0.30/0.46 ( A7
% 0.30/0.46 != ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % is_singletonE
% 0.30/0.46 thf(fact_175_sum__set__simps_I4_J,axiom,
% 0.30/0.46 ! [E: $tType,A: $tType,X: A] :
% 0.30/0.46 ( ( basic_setr @ E @ A @ ( sum_Inr @ A @ E @ X ) )
% 0.30/0.46 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % sum_set_simps(4)
% 0.30/0.46 thf(fact_176_Set_Ois__empty__def,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( is_empty @ A )
% 0.30/0.46 = ( ^ [A8: set @ A] :
% 0.30/0.46 ( A8
% 0.30/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Set.is_empty_def
% 0.30/0.46 thf(fact_177_remove__induct,axiom,
% 0.30/0.46 ! [A: $tType,P: ( set @ A ) > $o,B7: set @ A] :
% 0.30/0.46 ( ( P @ ( bot_bot @ ( set @ A ) ) )
% 0.30/0.46 => ( ( ~ ( finite_finite2 @ A @ B7 )
% 0.30/0.46 => ( P @ B7 ) )
% 0.30/0.46 => ( ! [A9: set @ A] :
% 0.30/0.46 ( ( finite_finite2 @ A @ A9 )
% 0.30/0.46 => ( ( A9
% 0.30/0.46 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.46 => ( ( ord_less_eq @ ( set @ A ) @ A9 @ B7 )
% 0.30/0.46 => ( ! [X7: A] :
% 0.30/0.46 ( ( member @ A @ X7 @ A9 )
% 0.30/0.46 => ( P @ ( minus_minus @ ( set @ A ) @ A9 @ ( insert @ A @ X7 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
% 0.30/0.46 => ( P @ A9 ) ) ) ) )
% 0.30/0.46 => ( P @ B7 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % remove_induct
% 0.30/0.46 thf(fact_178_subset__antisym,axiom,
% 0.30/0.46 ! [A: $tType,A7: set @ A,B7: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
% 0.30/0.46 => ( ( ord_less_eq @ ( set @ A ) @ B7 @ A7 )
% 0.30/0.46 => ( A7 = B7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subset_antisym
% 0.30/0.46 thf(fact_179_subsetI,axiom,
% 0.30/0.46 ! [A: $tType,A7: set @ A,B7: set @ A] :
% 0.30/0.46 ( ! [X6: A] :
% 0.30/0.46 ( ( member @ A @ X6 @ A7 )
% 0.30/0.46 => ( member @ A @ X6 @ B7 ) )
% 0.30/0.46 => ( ord_less_eq @ ( set @ A ) @ A7 @ B7 ) ) ).
% 0.30/0.46
% 0.30/0.46 % subsetI
% 0.30/0.46 thf(fact_180_subset__empty,axiom,
% 0.30/0.46 ! [A: $tType,A7: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A7 @ ( bot_bot @ ( set @ A ) ) )
% 0.30/0.46 = ( A7
% 0.30/0.46 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subset_empty
% 0.30/0.46 thf(fact_181_empty__subsetI,axiom,
% 0.30/0.46 ! [A: $tType,A7: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A7 ) ).
% 0.30/0.46
% 0.30/0.46 % empty_subsetI
% 0.30/0.46 thf(fact_182_insert__subset,axiom,
% 0.30/0.46 ! [A: $tType,X: A,A7: set @ A,B7: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A7 ) @ B7 )
% 0.30/0.46 = ( ( member @ A @ X @ B7 )
% 0.30/0.46 & ( ord_less_eq @ ( set @ A ) @ A7 @ B7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % insert_subset
% 0.30/0.46 thf(fact_183_singleton__insert__inj__eq,axiom,
% 0.30/0.46 ! [A: $tType,B2: A,A2: A,A7: set @ A] :
% 0.30/0.46 ( ( ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) )
% 0.30/0.46 = ( insert @ A @ A2 @ A7 ) )
% 0.30/0.46 = ( ( A2 = B2 )
% 0.30/0.46 & ( ord_less_eq @ ( set @ A ) @ A7 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % singleton_insert_inj_eq
% 0.30/0.46 thf(fact_184_singleton__insert__inj__eq_H,axiom,
% 0.30/0.46 ! [A: $tType,A2: A,A7: set @ A,B2: A] :
% 0.30/0.46 ( ( ( insert @ A @ A2 @ A7 )
% 0.30/0.46 = ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.46 = ( ( A2 = B2 )
% 0.30/0.46 & ( ord_less_eq @ ( set @ A ) @ A7 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % singleton_insert_inj_eq'
% 0.30/0.46 thf(fact_185_Diff__eq__empty__iff,axiom,
% 0.30/0.46 ! [A: $tType,A7: set @ A,B7: set @ A] :
% 0.30/0.46 ( ( ( minus_minus @ ( set @ A ) @ A7 @ B7 )
% 0.30/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.46 = ( ord_less_eq @ ( set @ A ) @ A7 @ B7 ) ) ).
% 0.30/0.46
% 0.30/0.46 % Diff_eq_empty_iff
% 0.30/0.46 thf(fact_186_sum__set__simps_I3_J,axiom,
% 0.30/0.46 ! [A: $tType,D: $tType,X: A] :
% 0.30/0.46 ( ( basic_setr @ A @ D @ ( sum_Inl @ A @ D @ X ) )
% 0.30/0.46 = ( bot_bot @ ( set @ D ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % sum_set_simps(3)
% 0.30/0.46 thf(fact_187_inFr__mono,axiom,
% 0.30/0.46 ! [Ns: set @ n,Tr: dtree,T2: t,Ns4: set @ n] :
% 0.30/0.46 ( ( gram_L1333338417e_inFr @ Ns @ Tr @ T2 )
% 0.30/0.46 => ( ( ord_less_eq @ ( set @ n ) @ Ns @ Ns4 )
% 0.30/0.46 => ( gram_L1333338417e_inFr @ Ns4 @ Tr @ T2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % inFr_mono
% 0.30/0.46 thf(fact_188_setr_Oinducts,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,X: B,S: sum_sum @ A @ B,P: B > $o] :
% 0.30/0.46 ( ( member @ B @ X @ ( basic_setr @ A @ B @ S ) )
% 0.30/0.46 => ( ! [X6: B] :
% 0.30/0.46 ( ( S
% 0.30/0.46 = ( sum_Inr @ B @ A @ X6 ) )
% 0.30/0.46 => ( P @ X6 ) )
% 0.30/0.46 => ( P @ X ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % setr.inducts
% 0.30/0.46 thf(fact_189_setr_Ointros,axiom,
% 0.30/0.46 ! [B: $tType,A: $tType,S: sum_sum @ A @ B,X: B] :
% 0.30/0.46 ( ( S
% 0.30/0.46 = ( sum_Inr @ B @ A @ X ) )
% 0.30/0.46 => ( member @ B @ X @ ( basic_setr @ A @ B @ S ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % setr.intros
% 0.30/0.46 thf(fact_190_setr_Osimps,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A2: B,S: sum_sum @ A @ B] :
% 0.30/0.46 ( ( member @ B @ A2 @ ( basic_setr @ A @ B @ S ) )
% 0.30/0.46 = ( ? [X5: B] :
% 0.30/0.46 ( ( A2 = X5 )
% 0.30/0.46 & ( S
% 0.30/0.46 = ( sum_Inr @ B @ A @ X5 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % setr.simps
% 0.30/0.46 thf(fact_191_setr_Ocases,axiom,
% 0.30/0.46 ! [A: $tType,B: $tType,A2: B,S: sum_sum @ A @ B] :
% 0.30/0.46 ( ( member @ B @ A2 @ ( basic_setr @ A @ B @ S ) )
% 0.30/0.46 => ( S
% 0.30/0.46 = ( sum_Inr @ B @ A @ A2 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % setr.cases
% 0.30/0.46 thf(fact_192_finite__subset,axiom,
% 0.30/0.46 ! [A: $tType,A7: set @ A,B7: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
% 0.30/0.46 => ( ( finite_finite2 @ A @ B7 )
% 0.30/0.46 => ( finite_finite2 @ A @ A7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % finite_subset
% 0.30/0.46 thf(fact_193_infinite__super,axiom,
% 0.30/0.46 ! [A: $tType,S2: set @ A,T5: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ S2 @ T5 )
% 0.30/0.46 => ( ~ ( finite_finite2 @ A @ S2 )
% 0.30/0.46 => ~ ( finite_finite2 @ A @ T5 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % infinite_super
% 0.30/0.46 thf(fact_194_rev__finite__subset,axiom,
% 0.30/0.46 ! [A: $tType,B7: set @ A,A7: set @ A] :
% 0.30/0.46 ( ( finite_finite2 @ A @ B7 )
% 0.30/0.46 => ( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
% 0.30/0.46 => ( finite_finite2 @ A @ A7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % rev_finite_subset
% 0.30/0.46 thf(fact_195_insert__mono,axiom,
% 0.30/0.46 ! [A: $tType,C4: set @ A,D3: set @ A,A2: A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ C4 @ D3 )
% 0.30/0.46 => ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A2 @ C4 ) @ ( insert @ A @ A2 @ D3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % insert_mono
% 0.30/0.46 thf(fact_196_subset__insert,axiom,
% 0.30/0.46 ! [A: $tType,X: A,A7: set @ A,B7: set @ A] :
% 0.30/0.46 ( ~ ( member @ A @ X @ A7 )
% 0.30/0.46 => ( ( ord_less_eq @ ( set @ A ) @ A7 @ ( insert @ A @ X @ B7 ) )
% 0.30/0.46 = ( ord_less_eq @ ( set @ A ) @ A7 @ B7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subset_insert
% 0.30/0.46 thf(fact_197_subset__insertI,axiom,
% 0.30/0.46 ! [A: $tType,B7: set @ A,A2: A] : ( ord_less_eq @ ( set @ A ) @ B7 @ ( insert @ A @ A2 @ B7 ) ) ).
% 0.30/0.46
% 0.30/0.46 % subset_insertI
% 0.30/0.46 thf(fact_198_subset__insertI2,axiom,
% 0.30/0.46 ! [A: $tType,A7: set @ A,B7: set @ A,B2: A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
% 0.30/0.46 => ( ord_less_eq @ ( set @ A ) @ A7 @ ( insert @ A @ B2 @ B7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subset_insertI2
% 0.30/0.46 thf(fact_199_Diff__mono,axiom,
% 0.30/0.46 ! [A: $tType,A7: set @ A,C4: set @ A,D3: set @ A,B7: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A7 @ C4 )
% 0.30/0.46 => ( ( ord_less_eq @ ( set @ A ) @ D3 @ B7 )
% 0.30/0.46 => ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A7 @ B7 ) @ ( minus_minus @ ( set @ A ) @ C4 @ D3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Diff_mono
% 0.30/0.46 thf(fact_200_Diff__subset,axiom,
% 0.30/0.46 ! [A: $tType,A7: set @ A,B7: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A7 @ B7 ) @ A7 ) ).
% 0.30/0.46
% 0.30/0.46 % Diff_subset
% 0.30/0.46 thf(fact_201_double__diff,axiom,
% 0.30/0.46 ! [A: $tType,A7: set @ A,B7: set @ A,C4: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
% 0.30/0.46 => ( ( ord_less_eq @ ( set @ A ) @ B7 @ C4 )
% 0.30/0.46 => ( ( minus_minus @ ( set @ A ) @ B7 @ ( minus_minus @ ( set @ A ) @ C4 @ A7 ) )
% 0.30/0.46 = A7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % double_diff
% 0.30/0.46 thf(fact_202_Collect__mono__iff,axiom,
% 0.30/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 0.30/0.46 = ( ! [X5: A] :
% 0.30/0.46 ( ( P @ X5 )
% 0.30/0.46 => ( Q @ X5 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Collect_mono_iff
% 0.30/0.46 thf(fact_203_contra__subsetD,axiom,
% 0.30/0.46 ! [A: $tType,A7: set @ A,B7: set @ A,C3: A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
% 0.30/0.46 => ( ~ ( member @ A @ C3 @ B7 )
% 0.30/0.46 => ~ ( member @ A @ C3 @ A7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % contra_subsetD
% 0.30/0.46 thf(fact_204_set__eq__subset,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ^ [Y4: set @ A,Z: set @ A] : ( Y4 = Z ) )
% 0.30/0.46 = ( ^ [A8: set @ A,B9: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A8 @ B9 )
% 0.30/0.46 & ( ord_less_eq @ ( set @ A ) @ B9 @ A8 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % set_eq_subset
% 0.30/0.46 thf(fact_205_subset__trans,axiom,
% 0.30/0.46 ! [A: $tType,A7: set @ A,B7: set @ A,C4: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
% 0.30/0.46 => ( ( ord_less_eq @ ( set @ A ) @ B7 @ C4 )
% 0.30/0.46 => ( ord_less_eq @ ( set @ A ) @ A7 @ C4 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subset_trans
% 0.30/0.46 thf(fact_206_Collect__mono,axiom,
% 0.30/0.46 ! [A: $tType,P: A > $o,Q: A > $o] :
% 0.30/0.46 ( ! [X6: A] :
% 0.30/0.46 ( ( P @ X6 )
% 0.30/0.46 => ( Q @ X6 ) )
% 0.30/0.46 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Collect_mono
% 0.30/0.46 thf(fact_207_subset__refl,axiom,
% 0.30/0.46 ! [A: $tType,A7: set @ A] : ( ord_less_eq @ ( set @ A ) @ A7 @ A7 ) ).
% 0.30/0.46
% 0.30/0.46 % subset_refl
% 0.30/0.46 thf(fact_208_rev__subsetD,axiom,
% 0.30/0.46 ! [A: $tType,C3: A,A7: set @ A,B7: set @ A] :
% 0.30/0.46 ( ( member @ A @ C3 @ A7 )
% 0.30/0.46 => ( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
% 0.30/0.46 => ( member @ A @ C3 @ B7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % rev_subsetD
% 0.30/0.46 thf(fact_209_subset__iff,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) )
% 0.30/0.46 = ( ^ [A8: set @ A,B9: set @ A] :
% 0.30/0.46 ! [T3: A] :
% 0.30/0.46 ( ( member @ A @ T3 @ A8 )
% 0.30/0.46 => ( member @ A @ T3 @ B9 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subset_iff
% 0.30/0.46 thf(fact_210_set__rev__mp,axiom,
% 0.30/0.46 ! [A: $tType,X: A,A7: set @ A,B7: set @ A] :
% 0.30/0.46 ( ( member @ A @ X @ A7 )
% 0.30/0.46 => ( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
% 0.30/0.46 => ( member @ A @ X @ B7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % set_rev_mp
% 0.30/0.46 thf(fact_211_equalityD2,axiom,
% 0.30/0.46 ! [A: $tType,A7: set @ A,B7: set @ A] :
% 0.30/0.46 ( ( A7 = B7 )
% 0.30/0.46 => ( ord_less_eq @ ( set @ A ) @ B7 @ A7 ) ) ).
% 0.30/0.46
% 0.30/0.46 % equalityD2
% 0.30/0.46 thf(fact_212_equalityD1,axiom,
% 0.30/0.46 ! [A: $tType,A7: set @ A,B7: set @ A] :
% 0.30/0.46 ( ( A7 = B7 )
% 0.30/0.46 => ( ord_less_eq @ ( set @ A ) @ A7 @ B7 ) ) ).
% 0.30/0.46
% 0.30/0.46 % equalityD1
% 0.30/0.46 thf(fact_213_subset__eq,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) )
% 0.30/0.46 = ( ^ [A8: set @ A,B9: set @ A] :
% 0.30/0.46 ! [X5: A] :
% 0.30/0.46 ( ( member @ A @ X5 @ A8 )
% 0.30/0.46 => ( member @ A @ X5 @ B9 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subset_eq
% 0.30/0.46 thf(fact_214_equalityE,axiom,
% 0.30/0.46 ! [A: $tType,A7: set @ A,B7: set @ A] :
% 0.30/0.46 ( ( A7 = B7 )
% 0.30/0.46 => ~ ( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
% 0.30/0.46 => ~ ( ord_less_eq @ ( set @ A ) @ B7 @ A7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % equalityE
% 0.30/0.46 thf(fact_215_subsetCE,axiom,
% 0.30/0.46 ! [A: $tType,A7: set @ A,B7: set @ A,C3: A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
% 0.30/0.46 => ( ( member @ A @ C3 @ A7 )
% 0.30/0.46 => ( member @ A @ C3 @ B7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subsetCE
% 0.30/0.46 thf(fact_216_subsetD,axiom,
% 0.30/0.46 ! [A: $tType,A7: set @ A,B7: set @ A,C3: A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
% 0.30/0.46 => ( ( member @ A @ C3 @ A7 )
% 0.30/0.46 => ( member @ A @ C3 @ B7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subsetD
% 0.30/0.46 thf(fact_217_in__mono,axiom,
% 0.30/0.46 ! [A: $tType,A7: set @ A,B7: set @ A,X: A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
% 0.30/0.46 => ( ( member @ A @ X @ A7 )
% 0.30/0.46 => ( member @ A @ X @ B7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % in_mono
% 0.30/0.46 thf(fact_218_set__mp,axiom,
% 0.30/0.46 ! [A: $tType,A7: set @ A,B7: set @ A,X: A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
% 0.30/0.46 => ( ( member @ A @ X @ A7 )
% 0.30/0.46 => ( member @ A @ X @ B7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % set_mp
% 0.30/0.46 thf(fact_219_subset__singletonD,axiom,
% 0.30/0.46 ! [A: $tType,A7: set @ A,X: A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A7 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.46 => ( ( A7
% 0.30/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.46 | ( A7
% 0.30/0.46 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subset_singletonD
% 0.30/0.46 thf(fact_220_subset__singleton__iff,axiom,
% 0.30/0.46 ! [A: $tType,X9: set @ A,A2: A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ X9 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 0.30/0.46 = ( ( X9
% 0.30/0.46 = ( bot_bot @ ( set @ A ) ) )
% 0.30/0.46 | ( X9
% 0.30/0.46 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subset_singleton_iff
% 0.30/0.46 thf(fact_221_subset__Diff__insert,axiom,
% 0.30/0.46 ! [A: $tType,A7: set @ A,B7: set @ A,X: A,C4: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A7 @ ( minus_minus @ ( set @ A ) @ B7 @ ( insert @ A @ X @ C4 ) ) )
% 0.30/0.46 = ( ( ord_less_eq @ ( set @ A ) @ A7 @ ( minus_minus @ ( set @ A ) @ B7 @ C4 ) )
% 0.30/0.46 & ~ ( member @ A @ X @ A7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subset_Diff_insert
% 0.30/0.46 thf(fact_222_finite__subset__induct,axiom,
% 0.30/0.46 ! [A: $tType,F7: set @ A,A7: set @ A,P: ( set @ A ) > $o] :
% 0.30/0.46 ( ( finite_finite2 @ A @ F7 )
% 0.30/0.46 => ( ( ord_less_eq @ ( set @ A ) @ F7 @ A7 )
% 0.30/0.46 => ( ( P @ ( bot_bot @ ( set @ A ) ) )
% 0.30/0.46 => ( ! [A4: A,F6: set @ A] :
% 0.30/0.46 ( ( finite_finite2 @ A @ F6 )
% 0.30/0.46 => ( ( member @ A @ A4 @ A7 )
% 0.30/0.46 => ( ~ ( member @ A @ A4 @ F6 )
% 0.30/0.46 => ( ( P @ F6 )
% 0.30/0.46 => ( P @ ( insert @ A @ A4 @ F6 ) ) ) ) ) )
% 0.30/0.46 => ( P @ F7 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % finite_subset_induct
% 0.30/0.46 thf(fact_223_subset__insert__iff,axiom,
% 0.30/0.46 ! [A: $tType,A7: set @ A,X: A,B7: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A7 @ ( insert @ A @ X @ B7 ) )
% 0.30/0.46 = ( ( ( member @ A @ X @ A7 )
% 0.30/0.46 => ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A7 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B7 ) )
% 0.30/0.46 & ( ~ ( member @ A @ X @ A7 )
% 0.30/0.46 => ( ord_less_eq @ ( set @ A ) @ A7 @ B7 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subset_insert_iff
% 0.30/0.46 thf(fact_224_Diff__single__insert,axiom,
% 0.30/0.46 ! [A: $tType,A7: set @ A,X: A,B7: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A7 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B7 )
% 0.30/0.46 => ( ord_less_eq @ ( set @ A ) @ A7 @ ( insert @ A @ X @ B7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Diff_single_insert
% 0.30/0.46 thf(fact_225_finite__remove__induct,axiom,
% 0.30/0.46 ! [A: $tType,B7: set @ A,P: ( set @ A ) > $o] :
% 0.30/0.46 ( ( finite_finite2 @ A @ B7 )
% 0.30/0.46 => ( ( P @ ( bot_bot @ ( set @ A ) ) )
% 0.30/0.46 => ( ! [A9: set @ A] :
% 0.30/0.46 ( ( finite_finite2 @ A @ A9 )
% 0.30/0.46 => ( ( A9
% 0.30/0.46 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.46 => ( ( ord_less_eq @ ( set @ A ) @ A9 @ B7 )
% 0.30/0.46 => ( ! [X7: A] :
% 0.30/0.46 ( ( member @ A @ X7 @ A9 )
% 0.30/0.46 => ( P @ ( minus_minus @ ( set @ A ) @ A9 @ ( insert @ A @ X7 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
% 0.30/0.46 => ( P @ A9 ) ) ) ) )
% 0.30/0.46 => ( P @ B7 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % finite_remove_induct
% 0.30/0.46 thf(fact_226_insert__subsetI,axiom,
% 0.30/0.46 ! [A: $tType,X: A,A7: set @ A,X9: set @ A] :
% 0.30/0.46 ( ( member @ A @ X @ A7 )
% 0.30/0.46 => ( ( ord_less_eq @ ( set @ A ) @ X9 @ A7 )
% 0.30/0.46 => ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ X9 ) @ A7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % insert_subsetI
% 0.30/0.46 thf(fact_227_diff__mono,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A2: A,B2: A,D2: A,C3: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.46 => ( ( ord_less_eq @ A @ D2 @ C3 )
% 0.30/0.46 => ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C3 ) @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % diff_mono
% 0.30/0.46 thf(fact_228_diff__left__mono,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [B2: A,A2: A,C3: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ B2 @ A2 )
% 0.30/0.46 => ( ord_less_eq @ A @ ( minus_minus @ A @ C3 @ A2 ) @ ( minus_minus @ A @ C3 @ B2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % diff_left_mono
% 0.30/0.46 thf(fact_229_diff__right__commute,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( cancel146912293up_add @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A2: A,C3: A,B2: A] :
% 0.30/0.46 ( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C3 ) @ B2 )
% 0.30/0.46 = ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C3 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % diff_right_commute
% 0.30/0.46 thf(fact_230_diff__eq__diff__eq,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( group_add @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A2: A,B2: A,C3: A,D2: A] :
% 0.30/0.46 ( ( ( minus_minus @ A @ A2 @ B2 )
% 0.30/0.46 = ( minus_minus @ A @ C3 @ D2 ) )
% 0.30/0.46 => ( ( A2 = B2 )
% 0.30/0.46 = ( C3 = D2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % diff_eq_diff_eq
% 0.30/0.46 thf(fact_231_diff__eq__diff__less__eq,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A2: A,B2: A,C3: A,D2: A] :
% 0.30/0.46 ( ( ( minus_minus @ A @ A2 @ B2 )
% 0.30/0.46 = ( minus_minus @ A @ C3 @ D2 ) )
% 0.30/0.46 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.46 = ( ord_less_eq @ A @ C3 @ D2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % diff_eq_diff_less_eq
% 0.30/0.46 thf(fact_232_diff__right__mono,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A2: A,B2: A,C3: A] :
% 0.30/0.46 ( ( ord_less_eq @ A @ A2 @ B2 )
% 0.30/0.46 => ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C3 ) @ ( minus_minus @ A @ B2 @ C3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % diff_right_mono
% 0.30/0.46 thf(fact_233_psubset__insert__iff,axiom,
% 0.30/0.46 ! [A: $tType,A7: set @ A,X: A,B7: set @ A] :
% 0.30/0.46 ( ( ord_less @ ( set @ A ) @ A7 @ ( insert @ A @ X @ B7 ) )
% 0.30/0.46 = ( ( ( member @ A @ X @ B7 )
% 0.30/0.46 => ( ord_less @ ( set @ A ) @ A7 @ B7 ) )
% 0.30/0.46 & ( ~ ( member @ A @ X @ B7 )
% 0.30/0.46 => ( ( ( member @ A @ X @ A7 )
% 0.30/0.46 => ( ord_less @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A7 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B7 ) )
% 0.30/0.46 & ( ~ ( member @ A @ X @ A7 )
% 0.30/0.46 => ( ord_less_eq @ ( set @ A ) @ A7 @ B7 ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % psubset_insert_iff
% 0.30/0.46 thf(fact_234_finite__induct__select,axiom,
% 0.30/0.46 ! [A: $tType,S2: set @ A,P: ( set @ A ) > $o] :
% 0.30/0.46 ( ( finite_finite2 @ A @ S2 )
% 0.30/0.46 => ( ( P @ ( bot_bot @ ( set @ A ) ) )
% 0.30/0.46 => ( ! [T6: set @ A] :
% 0.30/0.46 ( ( ord_less @ ( set @ A ) @ T6 @ S2 )
% 0.30/0.46 => ( ( P @ T6 )
% 0.30/0.46 => ? [X7: A] :
% 0.30/0.46 ( ( member @ A @ X7 @ ( minus_minus @ ( set @ A ) @ S2 @ T6 ) )
% 0.30/0.46 & ( P @ ( insert @ A @ X7 @ T6 ) ) ) ) )
% 0.30/0.46 => ( P @ S2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % finite_induct_select
% 0.30/0.46 thf(fact_235_psubsetI,axiom,
% 0.30/0.46 ! [A: $tType,A7: set @ A,B7: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
% 0.30/0.46 => ( ( A7 != B7 )
% 0.30/0.46 => ( ord_less @ ( set @ A ) @ A7 @ B7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % psubsetI
% 0.30/0.46 thf(fact_236_subset__iff__psubset__eq,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) )
% 0.30/0.46 = ( ^ [A8: set @ A,B9: set @ A] :
% 0.30/0.46 ( ( ord_less @ ( set @ A ) @ A8 @ B9 )
% 0.30/0.46 | ( A8 = B9 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subset_iff_psubset_eq
% 0.30/0.46 thf(fact_237_subset__psubset__trans,axiom,
% 0.30/0.46 ! [A: $tType,A7: set @ A,B7: set @ A,C4: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
% 0.30/0.46 => ( ( ord_less @ ( set @ A ) @ B7 @ C4 )
% 0.30/0.46 => ( ord_less @ ( set @ A ) @ A7 @ C4 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subset_psubset_trans
% 0.30/0.46 thf(fact_238_subset__not__subset__eq,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ord_less @ ( set @ A ) )
% 0.30/0.46 = ( ^ [A8: set @ A,B9: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A8 @ B9 )
% 0.30/0.46 & ~ ( ord_less_eq @ ( set @ A ) @ B9 @ A8 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % subset_not_subset_eq
% 0.30/0.46 thf(fact_239_psubset__subset__trans,axiom,
% 0.30/0.46 ! [A: $tType,A7: set @ A,B7: set @ A,C4: set @ A] :
% 0.30/0.46 ( ( ord_less @ ( set @ A ) @ A7 @ B7 )
% 0.30/0.46 => ( ( ord_less_eq @ ( set @ A ) @ B7 @ C4 )
% 0.30/0.46 => ( ord_less @ ( set @ A ) @ A7 @ C4 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % psubset_subset_trans
% 0.30/0.46 thf(fact_240_psubset__imp__subset,axiom,
% 0.30/0.46 ! [A: $tType,A7: set @ A,B7: set @ A] :
% 0.30/0.46 ( ( ord_less @ ( set @ A ) @ A7 @ B7 )
% 0.30/0.46 => ( ord_less_eq @ ( set @ A ) @ A7 @ B7 ) ) ).
% 0.30/0.46
% 0.30/0.46 % psubset_imp_subset
% 0.30/0.46 thf(fact_241_psubset__eq,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ord_less @ ( set @ A ) )
% 0.30/0.46 = ( ^ [A8: set @ A,B9: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A8 @ B9 )
% 0.30/0.46 & ( A8 != B9 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % psubset_eq
% 0.30/0.46 thf(fact_242_psubsetE,axiom,
% 0.30/0.46 ! [A: $tType,A7: set @ A,B7: set @ A] :
% 0.30/0.46 ( ( ord_less @ ( set @ A ) @ A7 @ B7 )
% 0.30/0.46 => ~ ( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
% 0.30/0.46 => ( ord_less_eq @ ( set @ A ) @ B7 @ A7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % psubsetE
% 0.30/0.46 thf(fact_243_diff__strict__right__mono,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A2: A,B2: A,C3: A] :
% 0.30/0.46 ( ( ord_less @ A @ A2 @ B2 )
% 0.30/0.46 => ( ord_less @ A @ ( minus_minus @ A @ A2 @ C3 ) @ ( minus_minus @ A @ B2 @ C3 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % diff_strict_right_mono
% 0.30/0.46 thf(fact_244_diff__strict__left__mono,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [B2: A,A2: A,C3: A] :
% 0.30/0.46 ( ( ord_less @ A @ B2 @ A2 )
% 0.30/0.46 => ( ord_less @ A @ ( minus_minus @ A @ C3 @ A2 ) @ ( minus_minus @ A @ C3 @ B2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % diff_strict_left_mono
% 0.30/0.46 thf(fact_245_diff__eq__diff__less,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A2: A,B2: A,C3: A,D2: A] :
% 0.30/0.46 ( ( ( minus_minus @ A @ A2 @ B2 )
% 0.30/0.46 = ( minus_minus @ A @ C3 @ D2 ) )
% 0.30/0.46 => ( ( ord_less @ A @ A2 @ B2 )
% 0.30/0.46 = ( ord_less @ A @ C3 @ D2 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % diff_eq_diff_less
% 0.30/0.46 thf(fact_246_diff__strict__mono,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A2: A,B2: A,D2: A,C3: A] :
% 0.30/0.46 ( ( ord_less @ A @ A2 @ B2 )
% 0.30/0.46 => ( ( ord_less @ A @ D2 @ C3 )
% 0.30/0.46 => ( ord_less @ A @ ( minus_minus @ A @ A2 @ C3 ) @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % diff_strict_mono
% 0.30/0.46 thf(fact_247_not__psubset__empty,axiom,
% 0.30/0.46 ! [A: $tType,A7: set @ A] :
% 0.30/0.46 ~ ( ord_less @ ( set @ A ) @ A7 @ ( bot_bot @ ( set @ A ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % not_psubset_empty
% 0.30/0.46 thf(fact_248_finite__psubset__induct,axiom,
% 0.30/0.46 ! [A: $tType,A7: set @ A,P: ( set @ A ) > $o] :
% 0.30/0.46 ( ( finite_finite2 @ A @ A7 )
% 0.30/0.46 => ( ! [A9: set @ A] :
% 0.30/0.46 ( ( finite_finite2 @ A @ A9 )
% 0.30/0.46 => ( ! [B11: set @ A] :
% 0.30/0.46 ( ( ord_less @ ( set @ A ) @ B11 @ A9 )
% 0.30/0.46 => ( P @ B11 ) )
% 0.30/0.46 => ( P @ A9 ) ) )
% 0.30/0.46 => ( P @ A7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % finite_psubset_induct
% 0.30/0.46 thf(fact_249_psubset__imp__ex__mem,axiom,
% 0.30/0.46 ! [A: $tType,A7: set @ A,B7: set @ A] :
% 0.30/0.46 ( ( ord_less @ ( set @ A ) @ A7 @ B7 )
% 0.30/0.46 => ? [B4: A] : ( member @ A @ B4 @ ( minus_minus @ ( set @ A ) @ B7 @ A7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % psubset_imp_ex_mem
% 0.30/0.46 thf(fact_250_psubset__trans,axiom,
% 0.30/0.46 ! [A: $tType,A7: set @ A,B7: set @ A,C4: set @ A] :
% 0.30/0.46 ( ( ord_less @ ( set @ A ) @ A7 @ B7 )
% 0.30/0.46 => ( ( ord_less @ ( set @ A ) @ B7 @ C4 )
% 0.30/0.46 => ( ord_less @ ( set @ A ) @ A7 @ C4 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % psubset_trans
% 0.30/0.46 thf(fact_251_psubsetD,axiom,
% 0.30/0.46 ! [A: $tType,A7: set @ A,B7: set @ A,C3: A] :
% 0.30/0.46 ( ( ord_less @ ( set @ A ) @ A7 @ B7 )
% 0.30/0.46 => ( ( member @ A @ C3 @ A7 )
% 0.30/0.46 => ( member @ A @ C3 @ B7 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % psubsetD
% 0.30/0.46 thf(fact_252_finite__linorder__min__induct,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A7: set @ A,P: ( set @ A ) > $o] :
% 0.30/0.46 ( ( finite_finite2 @ A @ A7 )
% 0.30/0.46 => ( ( P @ ( bot_bot @ ( set @ A ) ) )
% 0.30/0.46 => ( ! [B4: A,A9: set @ A] :
% 0.30/0.46 ( ( finite_finite2 @ A @ A9 )
% 0.30/0.46 => ( ! [X7: A] :
% 0.30/0.46 ( ( member @ A @ X7 @ A9 )
% 0.30/0.46 => ( ord_less @ A @ B4 @ X7 ) )
% 0.30/0.46 => ( ( P @ A9 )
% 0.30/0.46 => ( P @ ( insert @ A @ B4 @ A9 ) ) ) ) )
% 0.30/0.46 => ( P @ A7 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % finite_linorder_min_induct
% 0.30/0.46 thf(fact_253_finite__linorder__max__induct,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A7: set @ A,P: ( set @ A ) > $o] :
% 0.30/0.46 ( ( finite_finite2 @ A @ A7 )
% 0.30/0.46 => ( ( P @ ( bot_bot @ ( set @ A ) ) )
% 0.30/0.46 => ( ! [B4: A,A9: set @ A] :
% 0.30/0.46 ( ( finite_finite2 @ A @ A9 )
% 0.30/0.46 => ( ! [X7: A] :
% 0.30/0.46 ( ( member @ A @ X7 @ A9 )
% 0.30/0.46 => ( ord_less @ A @ X7 @ B4 ) )
% 0.30/0.46 => ( ( P @ A9 )
% 0.30/0.46 => ( P @ ( insert @ A @ B4 @ A9 ) ) ) ) )
% 0.30/0.46 => ( P @ A7 ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % finite_linorder_max_induct
% 0.30/0.46 thf(fact_254_infinite__growing,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( linorder @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [X9: set @ A] :
% 0.30/0.46 ( ( X9
% 0.30/0.46 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.46 => ( ! [X6: A] :
% 0.30/0.46 ( ( member @ A @ X6 @ X9 )
% 0.30/0.46 => ? [Xa: A] :
% 0.30/0.46 ( ( member @ A @ Xa @ X9 )
% 0.30/0.46 & ( ord_less @ A @ X6 @ Xa ) ) )
% 0.30/0.46 => ~ ( finite_finite2 @ A @ X9 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % infinite_growing
% 0.30/0.46 thf(fact_255_Sup__fin_Oantimono,axiom,
% 0.30/0.46 ! [A: $tType] :
% 0.30/0.46 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 0.30/0.46 => ! [A7: set @ A,B7: set @ A] :
% 0.30/0.46 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
% 0.30/0.46 => ( ( A7
% 0.30/0.46 != ( bot_bot @ ( set @ A ) ) )
% 0.30/0.46 => ( ( finite_finite2 @ A @ B7 )
% 0.30/0.46 => ( ord_less_eq @ A @ ( lattic477160up_fin @ A @ A7 ) @ ( lattic477160up_fin @ A @ B7 ) ) ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 % Sup_fin.antimono
% 0.30/0.46
% 0.30/0.46 %----Type constructors (14)
% 0.30/0.46 thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
% 0.30/0.46 ! [A5: $tType,A11: $tType] :
% 0.30/0.46 ( ( semilattice_sup @ A11 @ ( type2 @ A11 ) )
% 0.30/0.46 => ( semilattice_sup @ ( A5 > A11 ) @ ( type2 @ ( A5 > A11 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_fun___Finite__Set_Ofinite,axiom,
% 0.30/0.46 ! [A5: $tType,A11: $tType] :
% 0.30/0.46 ( ( ( finite_finite @ A5 @ ( type2 @ A5 ) )
% 0.30/0.46 & ( finite_finite @ A11 @ ( type2 @ A11 ) ) )
% 0.30/0.46 => ( finite_finite @ ( A5 > A11 ) @ ( type2 @ ( A5 > A11 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_fun___Groups_Ominus,axiom,
% 0.30/0.46 ! [A5: $tType,A11: $tType] :
% 0.30/0.46 ( ( minus @ A11 @ ( type2 @ A11 ) )
% 0.30/0.46 => ( minus @ ( A5 > A11 ) @ ( type2 @ ( A5 > A11 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_1,axiom,
% 0.30/0.46 ! [A5: $tType] : ( semilattice_sup @ ( set @ A5 ) @ ( type2 @ ( set @ A5 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Set_Oset___Finite__Set_Ofinite_2,axiom,
% 0.30/0.46 ! [A5: $tType] :
% 0.30/0.46 ( ( finite_finite @ A5 @ ( type2 @ A5 ) )
% 0.30/0.46 => ( finite_finite @ ( set @ A5 ) @ ( type2 @ ( set @ A5 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Set_Oset___Groups_Ominus_3,axiom,
% 0.30/0.46 ! [A5: $tType] : ( minus @ ( set @ A5 ) @ ( type2 @ ( set @ A5 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_4,axiom,
% 0.30/0.46 semilattice_sup @ $o @ ( type2 @ $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 0.30/0.46 linorder @ $o @ ( type2 @ $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_HOL_Obool___Finite__Set_Ofinite_5,axiom,
% 0.30/0.46 finite_finite @ $o @ ( type2 @ $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_HOL_Obool___Groups_Ominus_6,axiom,
% 0.30/0.46 minus @ $o @ ( type2 @ $o ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_FSet_Ofset___Lattices_Osemilattice__sup_7,axiom,
% 0.30/0.46 ! [A5: $tType] : ( semilattice_sup @ ( fset @ A5 ) @ ( type2 @ ( fset @ A5 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_FSet_Ofset___Finite__Set_Ofinite_8,axiom,
% 0.30/0.46 ! [A5: $tType] :
% 0.30/0.46 ( ( finite_finite @ A5 @ ( type2 @ A5 ) )
% 0.30/0.46 => ( finite_finite @ ( fset @ A5 ) @ ( type2 @ ( fset @ A5 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_FSet_Ofset___Groups_Ominus_9,axiom,
% 0.30/0.46 ! [A5: $tType] : ( minus @ ( fset @ A5 ) @ ( type2 @ ( fset @ A5 ) ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_10,axiom,
% 0.30/0.46 ! [A5: $tType,A11: $tType] :
% 0.30/0.46 ( ( ( finite_finite @ A5 @ ( type2 @ A5 ) )
% 0.30/0.46 & ( finite_finite @ A11 @ ( type2 @ A11 ) ) )
% 0.30/0.46 => ( finite_finite @ ( sum_sum @ A5 @ A11 ) @ ( type2 @ ( sum_sum @ A5 @ A11 ) ) ) ) ).
% 0.30/0.46
% 0.30/0.46 %----Helper facts (3)
% 0.30/0.46 thf(help_If_3_1_T,axiom,
% 0.30/0.46 ! [P: $o] :
% 0.30/0.46 ( ( P = $true )
% 0.30/0.46 | ( P = $false ) ) ).
% 0.30/0.46
% 0.30/0.46 thf(help_If_2_1_T,axiom,
% 0.30/0.46 ! [A: $tType,X: A,Y: A] :
% 0.30/0.46 ( ( if @ A @ $false @ X @ Y )
% 0.30/0.46 = Y ) ).
% 0.30/0.46
% 0.30/0.46 thf(help_If_1_1_T,axiom,
% 0.30/0.46 ! [A: $tType,X: A,Y: A] :
% 0.30/0.46 ( ( if @ A @ $true @ X @ Y )
% 0.30/0.46 = X ) ).
% 0.30/0.46
% 0.30/0.46 %----Conjectures (3)
% 0.30/0.46 thf(conj_0,hypothesis,
% 0.30/0.46 member @ n @ ( root @ tra ) @ nsa ).
% 0.30/0.46
% 0.30/0.46 thf(conj_1,hypothesis,
% 0.30/0.46 member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ ta ) @ ( cont @ tra ) ).
% 0.30/0.46
% 0.30/0.46 thf(conj_2,conjecture,
% 0.30/0.46 member @ n @ ( root @ tra ) @ nsa ).
% 0.30/0.46
% 0.30/0.46 %------------------------------------------------------------------------------
% 0.30/0.46 ------------------------
% 2.92/1.52 ---- Embedded in THF ---
% 2.92/1.52 %%% This output was generated by embedproblem, version 1.9.5 (library version 1.9).
% 2.92/1.52 %%% Generated on Tue Feb 24 04:16:49 EST 2026
% 2.92/1.52 %%% using '$$dhol' embedding, version 1.3.0.
% 2.92/1.52 %%% Logic specification used:
% 2.92/1.52 %%% thf(spec, logic, $$dhol).
% 2.92/1.52
% 2.92/1.52 % SZS output start ListOfTHF for /export/starexec/sandbox2/tmp/tmp.Nuu3MkGpYK/DTF2DTF26903.p
% 2.92/1.52 thf(ty_t_Sum__Type_Osum, type, sum_sum: ($tType > ($tType > $tType))).
% 2.92/1.52 thf(ty_t_DTree_Odtree, type, dtree: $tType).
% 2.92/1.52 thf(ty_t_FSet_Ofset, type, fset: ($tType > $tType)).
% 2.92/1.52 thf(ty_t_Set_Oset, type, set: ($tType > $tType)).
% 2.92/1.52 thf(ty_t_DTree_OT, type, t: $tType).
% 2.92/1.52 thf(ty_t_DTree_ON, type, n: $tType).
% 2.92/1.52 thf(ty_t_itself, type, itself: ($tType > $tType)).
% 2.92/1.52 thf(sy_cl_HOL_Otype, type, type: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.92/1.52 thf(sy_cl_Groups_Ominus, type, minus: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.92/1.52 thf(sy_cl_Groups_Ogroup__add, type, group_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.92/1.52 thf(sy_cl_Finite__Set_Ofinite, type, finite_finite: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.92/1.52 thf(sy_cl_Orderings_Olinorder, type, linorder: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.92/1.52 thf(sy_cl_Lattices_Osemilattice__sup, type, semilattice_sup: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.92/1.52 thf(sy_cl_Groups_Oordered__ab__group__add, type, ordered_ab_group_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.92/1.52 thf(sy_cl_Groups_Ocancel__ab__semigroup__add, type, cancel146912293up_add: (!> [A:$tType]: (((itself @ A) > $o)))).
% 2.92/1.52 thf(sy_c_Basic__BNFs_Osetl, type, basic_setl: (!> [A:$tType,B:$tType]: ((((sum_sum @ A) @ B) > (set @ A))))).
% 2.92/1.52 thf(sy_c_Basic__BNFs_Osetr, type, basic_setr: (!> [A:$tType,B:$tType]: ((((sum_sum @ A) @ B) > (set @ B))))).
% 2.92/1.52 thf(sy_c_DTree_ONode, type, node: (n > ((set @ ((sum_sum @ t) @ dtree)) > dtree))).
% 2.92/1.52 thf(sy_c_DTree_Ocont, type, cont: (dtree > (set @ ((sum_sum @ t) @ dtree)))).
% 2.92/1.52 thf(sy_c_DTree_Ocorec, type, corec: (!> [A:$tType]: (((A > n) > ((A > (set @ ((sum_sum @ t) @ ((sum_sum @ dtree) @ A)))) > (A > dtree)))))).
% 2.92/1.52 thf(sy_c_DTree_Odtree_ONNode, type, nNode: (n > ((fset @ ((sum_sum @ t) @ dtree)) > dtree))).
% 2.92/1.52 thf(sy_c_DTree_Odtree_Ocase__dtree, type, case_dtree: (!> [A:$tType]: (((n > ((fset @ ((sum_sum @ t) @ dtree)) > A)) > (dtree > A))))).
% 2.92/1.52 thf(sy_c_DTree_Odtree_Occont, type, ccont: (dtree > (fset @ ((sum_sum @ t) @ dtree)))).
% 2.92/1.52 thf(sy_c_DTree_Odtree_Oroot, type, root: (dtree > n)).
% 2.92/1.52 thf(sy_c_DTree_Ounfold, type, unfold: (!> [A:$tType]: (((A > n) > ((A > (set @ ((sum_sum @ t) @ A))) > (A > dtree)))))).
% 2.92/1.52 thf(sy_c_Finite__Set_Ofinite, type, finite_finite2: (!> [A:$tType]: (((set @ A) > $o)))).
% 2.92/1.52 thf(sy_c_Fun_Ocomp, type, comp: (!> [B:$tType,C:$tType,A:$tType]: (((B > C) > ((A > B) > (A > C)))))).
% 2.92/1.52 thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_OinFr, type, gram_L1333338417e_inFr: ((set @ n) > (dtree > (t > $o)))).
% 2.92/1.52 thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_OinFr2, type, gram_L805317441_inFr2: ((set @ n) > (dtree > (t > $o)))).
% 2.92/1.52 thf(sy_c_Groups_Ominus__class_Ominus, type, minus_minus: (!> [A:$tType]: ((A > (A > A))))).
% 2.92/1.52 thf(sy_c_If, type, if: (!> [A:$tType]: (($o > (A > (A > A)))))).
% 2.92/1.52 thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin, type, lattic477160up_fin: (!> [A:$tType]: (((set @ A) > A)))).
% 2.92/1.52 thf(sy_c_Orderings_Obot__class_Obot, type, bot_bot: (!> [A:$tType]: (A))).
% 2.92/1.52 thf(sy_c_Orderings_Oord__class_Oless, type, ord_less: (!> [A:$tType]: ((A > (A > $o))))).
% 2.92/1.52 thf(sy_c_Orderings_Oord__class_Oless__eq, type, ord_less_eq: (!> [A:$tType]: ((A > (A > $o))))).
% 2.92/1.52 thf(sy_c_Pure_Otype, type, type2: (!> [A:$tType]: ((itself @ A)))).
% 2.92/1.52 thf(sy_c_Set_OCollect, type, collect: (!> [A:$tType]: (((A > $o) > (set @ A))))).
% 2.92/1.52 thf(sy_c_Set_Oinsert, type, insert: (!> [A:$tType]: ((A > ((set @ A) > (set @ A)))))).
% 2.92/1.52 thf(sy_c_Set_Ois__empty, type, is_empty: (!> [A:$tType]: (((set @ A) > $o)))).
% 2.92/1.52 thf(sy_c_Set_Ois__singleton, type, is_singleton: (!> [A:$tType]: (((set @ A) > $o)))).
% 2.92/1.52 thf(sy_c_Set_Oremove, type, remove: (!> [A:$tType]: ((A > ((set @ A) > (set @ A)))))).
% 2.92/1.52 thf(sy_c_Set_Othe__elem, type, the_elem: (!> [A:$tType]: (((set @ A) > A)))).
% 2.92/1.52 thf(sy_c_Sum__Type_OInl, type, sum_Inl: (!> [A:$tType,B:$tType]: ((A > ((sum_sum @ A) @ B))))).
% 2.92/1.52 thf(sy_c_Sum__Type_OInr, type, sum_Inr: (!> [B:$tType,A:$tType]: ((B > ((sum_sum @ A) @ B))))).
% 2.92/1.52 thf(sy_c_Sum__Type_OPlus, type, sum_Plus: (!> [A:$tType,B:$tType]: (((set @ A) > ((set @ B) > (set @ ((sum_sum @ A) @ B))))))).
% 2.92/1.52 thf(sy_c_Sum__Type_OSuml, type, sum_Suml: (!> [A:$tType,C:$tType,B:$tType]: (((A > C) > (((sum_sum @ A) @ B) > C))))).
% 2.92/1.52 thf(sy_c_Sum__Type_OSumr, type, sum_Sumr: (!> [B:$tType,C:$tType,A:$tType]: (((B > C) > (((sum_sum @ A) @ B) > C))))).
% 2.92/1.52 thf(sy_c_Sum__Type_Oold_Osum_Orec__sum, type, sum_rec_sum: (!> [A:$tType,T:$tType,B:$tType]: (((A > T) > ((B > T) > (((sum_sum @ A) @ B) > T)))))).
% 2.92/1.52 thf(sy_c_Sum__Type_Osum_Ocase__sum, type, sum_case_sum: (!> [A:$tType,C:$tType,B:$tType]: (((A > C) > ((B > C) > (((sum_sum @ A) @ B) > C)))))).
% 2.92/1.52 thf(sy_c_member, type, member: (!> [A:$tType]: ((A > ((set @ A) > $o))))).
% 2.92/1.52 thf(sy_v_nsa, type, nsa: (set @ n)).
% 2.92/1.52 thf(sy_v_ta, type, ta: t).
% 2.92/1.52 thf(sy_v_tra, type, tra: dtree).
% 2.92/1.52 thf(fact_0_sum_Oinject_I1_J, axiom, (! [B:$tType,A:$tType,X1:A,Y1:A]: ((((((sum_Inl @ A) @ B) @ X1) = (((sum_Inl @ A) @ B) @ Y1)) = (X1 = Y1))))).
% 2.92/1.52 thf(fact_1_old_Osum_Oinject_I1_J, axiom, (! [B:$tType,A:$tType,A2:A,A3:A]: ((((((sum_Inl @ A) @ B) @ A2) = (((sum_Inl @ A) @ B) @ A3)) = (A2 = A3))))).
% 2.92/1.52 thf(fact_2_dtree__cong, axiom, (! [Tr:dtree,Tr2:dtree]: ((((root @ Tr) = (root @ Tr2)) => (((cont @ Tr) = (cont @ Tr2)) => (Tr = Tr2)))))).
% 2.92/1.52 thf(fact_3_inFr2_OBase, axiom, (! [Tr:dtree,Ns:(set @ n),T2:t]: (((((member @ n) @ (root @ Tr)) @ Ns) => ((((member @ ((sum_sum @ t) @ dtree)) @ (((sum_Inl @ t) @ dtree) @ T2)) @ (cont @ Tr)) => (((gram_L805317441_inFr2 @ Ns) @ Tr) @ T2)))))).
% 2.92/1.52 thf(fact_4_inFr_OBase, axiom, (! [Tr:dtree,Ns:(set @ n),T2:t]: (((((member @ n) @ (root @ Tr)) @ Ns) => ((((member @ ((sum_sum @ t) @ dtree)) @ (((sum_Inl @ t) @ dtree) @ T2)) @ (cont @ Tr)) => (((gram_L1333338417e_inFr @ Ns) @ Tr) @ T2)))))).
% 2.92/1.52 thf(fact_5_Inl__inject, axiom, (! [B:$tType,A:$tType,X:A,Y:A]: ((((((sum_Inl @ A) @ B) @ X) = (((sum_Inl @ A) @ B) @ Y)) => (X = Y))))).
% 2.92/1.52 thf(fact_6_Node__root__cont, axiom, (! [Tr:dtree]: ((((node @ (root @ Tr)) @ (cont @ Tr)) = Tr)))).
% 2.92/1.52 thf(fact_7_inFr_Ocases, axiom, (! [A1:(set @ n),A22:dtree,A32:t]: (((((gram_L1333338417e_inFr @ A1) @ A22) @ A32) => (((((member @ n) @ (root @ A22)) @ A1) => (~ ((((member @ ((sum_sum @ t) @ dtree)) @ (((sum_Inl @ t) @ dtree) @ A32)) @ (cont @ A22))))) => (~ (((((member @ n) @ (root @ A22)) @ A1) => (! [Tr1:dtree]: (((((member @ ((sum_sum @ t) @ dtree)) @ (((sum_Inr @ dtree) @ t) @ Tr1)) @ (cont @ A22)) => (~ ((((gram_L1333338417e_inFr @ A1) @ Tr1) @ A32)))))))))))))).
% 2.92/1.52 thf(fact_8_inFr_Osimps, axiom, (gram_L1333338417e_inFr = (^ [A12:(set @ n),A23:dtree,A33:t]: (((? [Tr3:dtree,Ns2:(set @ n),T3:t]: (((A12 = Ns2) & ((A23 = Tr3) & ((A33 = T3) & ((((member @ n) @ (root @ Tr3)) @ Ns2) & (((member @ ((sum_sum @ t) @ dtree)) @ (((sum_Inl @ t) @ dtree) @ T3)) @ (cont @ Tr3)))))))) | (? [Tr3:dtree,Ns2:(set @ n),Tr12:dtree,T3:t]: (((A12 = Ns2) & ((A23 = Tr3) & ((A33 = T3) & ((((member @ n) @ (root @ Tr3)) @ Ns2) & ((((member @ ((sum_sum @ t) @ dtree)) @ (((sum_Inr @ dtree) @ t) @ Tr12)) @ (cont @ Tr3)) & (((gram_L1333338417e_inFr @ Ns2) @ Tr12) @ T3))))))))))))).
% 2.92/1.52 thf(fact_9_inFr_Oinducts, axiom, (! [X1:(set @ n),X2:dtree,X3:t,P:((set @ n) > (dtree > (t > $o)))]: (((((gram_L1333338417e_inFr @ X1) @ X2) @ X3) => ((! [Tr4:dtree,Ns3:(set @ n),T4:t]: (((((member @ n) @ (root @ Tr4)) @ Ns3) => ((((member @ ((sum_sum @ t) @ dtree)) @ (((sum_Inl @ t) @ dtree) @ T4)) @ (cont @ Tr4)) => (((P @ Ns3) @ Tr4) @ T4))))) => ((! [Tr4:dtree,Ns3:(set @ n),Tr1:dtree,T4:t]: (((((member @ n) @ (root @ Tr4)) @ Ns3) => ((((member @ ((sum_sum @ t) @ dtree)) @ (((sum_Inr @ dtree) @ t) @ Tr1)) @ (cont @ Tr4)) => ((((gram_L1333338417e_inFr @ Ns3) @ Tr1) @ T4) => ((((P @ Ns3) @ Tr1) @ T4) => (((P @ Ns3) @ Tr4) @ T4))))))) => (((P @ X1) @ X2) @ X3))))))).
% 2.92/1.52 thf(fact_10_old_Osum_Osimps_I7_J, axiom, (! [B:$tType,T:$tType,A:$tType,F1:(A > T),F2:(B > T),A2:A]: ((((((((sum_rec_sum @ A) @ T) @ B) @ F1) @ F2) @ (((sum_Inl @ A) @ B) @ A2)) = (F1 @ A2))))).
% 2.92/1.52 thf(fact_11_inFr__root__in, axiom, (! [Ns:(set @ n),Tr:dtree,T2:t]: (((((gram_L1333338417e_inFr @ Ns) @ Tr) @ T2) => (((member @ n) @ (root @ Tr)) @ Ns))))).
% 2.92/1.52 thf(fact_12_sum_Oinject_I2_J, axiom, (! [A:$tType,B:$tType,X2:B,Y2:B]: ((((((sum_Inr @ B) @ A) @ X2) = (((sum_Inr @ B) @ A) @ Y2)) = (X2 = Y2))))).
% 2.92/1.52 thf(fact_13_old_Osum_Oinject_I2_J, axiom, (! [A:$tType,B:$tType,B2:B,B3:B]: ((((((sum_Inr @ B) @ A) @ B2) = (((sum_Inr @ B) @ A) @ B3)) = (B2 = B3))))).
% 2.92/1.52 thf(fact_14_root__Node, axiom, (! [N:n,As:(set @ ((sum_sum @ t) @ dtree))]: (((root @ ((node @ N) @ As)) = N)))).
% 2.92/1.52 thf(fact_15_old_Osum_Osimps_I8_J, axiom, (! [A:$tType,T:$tType,B:$tType,F1:(A > T),F2:(B > T),B2:B]: ((((((((sum_rec_sum @ A) @ T) @ B) @ F1) @ F2) @ (((sum_Inr @ B) @ A) @ B2)) = (F2 @ B2))))).
% 2.92/1.52 thf(fact_16_Inr__inject, axiom, (! [A:$tType,B:$tType,X:B,Y:B]: ((((((sum_Inr @ B) @ A) @ X) = (((sum_Inr @ B) @ A) @ Y)) => (X = Y))))).
% 2.92/1.52 thf(fact_17_old_Osum_Oinducts, axiom, (! [B:$tType,A:$tType,P:(((sum_sum @ A) @ B) > $o),Sum:((sum_sum @ A) @ B)]: (((! [A4:A]: ((P @ (((sum_Inl @ A) @ B) @ A4)))) => ((! [B4:B]: ((P @ (((sum_Inr @ B) @ A) @ B4)))) => (P @ Sum)))))).
% 2.92/1.52 thf(fact_18_old_Osum_Oexhaust, axiom, (! [A:$tType,B:$tType,Y:((sum_sum @ A) @ B)]: (((! [A4:A]: ((Y != (((sum_Inl @ A) @ B) @ A4)))) => (~ ((! [B4:B]: ((Y != (((sum_Inr @ B) @ A) @ B4)))))))))).
% 2.92/1.52 thf(fact_19_split__sum__all, axiom, (! [B:$tType,A:$tType]: (((^ [P2:(((sum_sum @ A) @ B) > $o)]: ((! [X4:((sum_sum @ A) @ B)]: ((P2 @ X4))))) = (^ [P3:(((sum_sum @ A) @ B) > $o)]: (((! [X5:A]: ((P3 @ (((sum_Inl @ A) @ B) @ X5)))) & (! [X5:B]: ((P3 @ (((sum_Inr @ B) @ A) @ X5))))))))))).
% 2.92/1.52 thf(fact_20_split__sum__ex, axiom, (! [B:$tType,A:$tType]: (((^ [P2:(((sum_sum @ A) @ B) > $o)]: ((? [X4:((sum_sum @ A) @ B)]: ((P2 @ X4))))) = (^ [P3:(((sum_sum @ A) @ B) > $o)]: (((? [X5:A]: ((P3 @ (((sum_Inl @ A) @ B) @ X5)))) | (? [X5:B]: ((P3 @ (((sum_Inr @ B) @ A) @ X5))))))))))).
% 2.92/1.52 thf(fact_21_Inr__not__Inl, axiom, (! [B:$tType,A:$tType,B2:B,A2:A]: (((((sum_Inr @ B) @ A) @ B2) != (((sum_Inl @ A) @ B) @ A2))))).
% 2.92/1.52 thf(fact_22_sumE, axiom, (! [A:$tType,B:$tType,S:((sum_sum @ A) @ B)]: (((! [X6:A]: ((S != (((sum_Inl @ A) @ B) @ X6)))) => (~ ((! [Y3:B]: ((S != (((sum_Inr @ B) @ A) @ Y3)))))))))).
% 2.92/1.52 thf(fact_23_old_Osum_Odistinct_I1_J, axiom, (! [A:$tType,B:$tType,A2:A,B3:B]: (((((sum_Inl @ A) @ B) @ A2) != (((sum_Inr @ B) @ A) @ B3))))).
% 2.92/1.52 thf(fact_24_old_Osum_Odistinct_I2_J, axiom, (! [B5:$tType,A5:$tType,B6:B5,A6:A5]: (((((sum_Inr @ B5) @ A5) @ B6) != (((sum_Inl @ A5) @ B5) @ A6))))).
% 2.92/1.52 thf(fact_25_sum_Odistinct_I1_J, axiom, (! [A:$tType,B:$tType,X1:A,X2:B]: (((((sum_Inl @ A) @ B) @ X1) != (((sum_Inr @ B) @ A) @ X2))))).
% 2.92/1.52 thf(fact_26_inFr_OInd, axiom, (! [Tr:dtree,Ns:(set @ n),Tr13:dtree,T2:t]: (((((member @ n) @ (root @ Tr)) @ Ns) => ((((member @ ((sum_sum @ t) @ dtree)) @ (((sum_Inr @ dtree) @ t) @ Tr13)) @ (cont @ Tr)) => ((((gram_L1333338417e_inFr @ Ns) @ Tr13) @ T2) => (((gram_L1333338417e_inFr @ Ns) @ Tr) @ T2))))))).
% 2.92/1.52 thf(fact_27_Inr__Inl__False, axiom, (! [B:$tType,A:$tType,X:B,Y:A]: (((((sum_Inr @ B) @ A) @ X) != (((sum_Inl @ A) @ B) @ Y))))).
% 2.92/1.52 thf(fact_28_Inl__Inr__False, axiom, (! [A:$tType,B:$tType,X:A,Y:B]: (((((sum_Inl @ A) @ B) @ X) != (((sum_Inr @ B) @ A) @ Y))))).
% 2.92/1.52 thf(fact_29_obj__sumE, axiom, (! [A:$tType,B:$tType,S:((sum_sum @ A) @ B)]: (((! [X6:A]: ((S != (((sum_Inl @ A) @ B) @ X6)))) => (~ ((! [X6:B]: ((S != (((sum_Inr @ B) @ A) @ X6)))))))))).
% 2.92/1.52 thf(fact_30_obj__sumE__f, axiom, (! [A:$tType,C:$tType,B:$tType,S:B,F:(((sum_sum @ A) @ C) > B),P:$o]: (((! [X6:A]: (((S = (F @ (((sum_Inl @ A) @ C) @ X6))) => P))) => ((! [X6:C]: (((S = (F @ (((sum_Inr @ C) @ A) @ X6))) => P))) => (! [X7:((sum_sum @ A) @ C)]: (((S = (F @ X7)) => P)))))))).
% 2.92/1.52 thf(fact_31_inFr2_Oinducts, axiom, (! [X1:(set @ n),X2:dtree,X3:t,P:((set @ n) > (dtree > (t > $o)))]: (((((gram_L805317441_inFr2 @ X1) @ X2) @ X3) => ((! [Tr4:dtree,Ns3:(set @ n),T4:t]: (((((member @ n) @ (root @ Tr4)) @ Ns3) => ((((member @ ((sum_sum @ t) @ dtree)) @ (((sum_Inl @ t) @ dtree) @ T4)) @ (cont @ Tr4)) => (((P @ Ns3) @ Tr4) @ T4))))) => ((! [Tr1:dtree,Tr4:dtree,Ns1:(set @ n),T4:t]: (((((member @ ((sum_sum @ t) @ dtree)) @ (((sum_Inr @ dtree) @ t) @ Tr1)) @ (cont @ Tr4)) => ((((gram_L805317441_inFr2 @ Ns1) @ Tr1) @ T4) => ((((P @ Ns1) @ Tr1) @ T4) => (((P @ (((insert @ n) @ (root @ Tr4)) @ Ns1)) @ Tr4) @ T4)))))) => (((P @ X1) @ X2) @ X3))))))).
% 2.92/1.52 thf(fact_32_inFr2_Osimps, axiom, (gram_L805317441_inFr2 = (^ [A12:(set @ n),A23:dtree,A33:t]: (((? [Tr3:dtree,Ns2:(set @ n),T3:t]: (((A12 = Ns2) & ((A23 = Tr3) & ((A33 = T3) & ((((member @ n) @ (root @ Tr3)) @ Ns2) & (((member @ ((sum_sum @ t) @ dtree)) @ (((sum_Inl @ t) @ dtree) @ T3)) @ (cont @ Tr3)))))))) | (? [Tr12:dtree,Tr3:dtree,Ns12:(set @ n),T3:t]: (((A12 = (((insert @ n) @ (root @ Tr3)) @ Ns12)) & ((A23 = Tr3) & ((A33 = T3) & ((((member @ ((sum_sum @ t) @ dtree)) @ (((sum_Inr @ dtree) @ t) @ Tr12)) @ (cont @ Tr3)) & (((gram_L805317441_inFr2 @ Ns12) @ Tr12) @ T3)))))))))))).
% 2.92/1.52 thf(fact_33_inFr2_Ocases, axiom, (! [A1:(set @ n),A22:dtree,A32:t]: (((((gram_L805317441_inFr2 @ A1) @ A22) @ A32) => (((((member @ n) @ (root @ A22)) @ A1) => (~ ((((member @ ((sum_sum @ t) @ dtree)) @ (((sum_Inl @ t) @ dtree) @ A32)) @ (cont @ A22))))) => (~ ((! [Tr1:dtree,Tr4:dtree,Ns1:(set @ n)]: (((A1 = (((insert @ n) @ (root @ Tr4)) @ Ns1)) => ((A22 = Tr4) => ((((member @ ((sum_sum @ t) @ dtree)) @ (((sum_Inr @ dtree) @ t) @ Tr1)) @ (cont @ Tr4)) => (~ ((((gram_L805317441_inFr2 @ Ns1) @ Tr1) @ A32))))))))))))))).
% 2.92/1.52 thf(fact_34_inFr2_OInd, axiom, (! [Tr13:dtree,Tr:dtree,Ns13:(set @ n),T2:t]: (((((member @ ((sum_sum @ t) @ dtree)) @ (((sum_Inr @ dtree) @ t) @ Tr13)) @ (cont @ Tr)) => ((((gram_L805317441_inFr2 @ Ns13) @ Tr13) @ T2) => (((gram_L805317441_inFr2 @ (((insert @ n) @ (root @ Tr)) @ Ns13)) @ Tr) @ T2)))))).
% 2.92/1.52 thf(fact_35_inFr__Ind__minus, axiom, (! [Ns13:(set @ n),Tr13:dtree,T2:t,Tr:dtree]: (((((gram_L1333338417e_inFr @ Ns13) @ Tr13) @ T2) => ((((member @ ((sum_sum @ t) @ dtree)) @ (((sum_Inr @ dtree) @ t) @ Tr13)) @ (cont @ Tr)) => (((gram_L1333338417e_inFr @ (((insert @ n) @ (root @ Tr)) @ Ns13)) @ Tr) @ T2)))))).
% 2.92/1.52 thf(fact_36_Suml_Osimps, axiom, (! [B:$tType,C:$tType,A:$tType,F:(A > C),X:A]: (((((((sum_Suml @ A) @ C) @ B) @ F) @ (((sum_Inl @ A) @ B) @ X)) = (F @ X))))).
% 2.92/1.52 thf(fact_37_Sumr_Osimps, axiom, (! [A:$tType,C:$tType,B:$tType,F:(B > C),X:B]: (((((((sum_Sumr @ B) @ C) @ A) @ F) @ (((sum_Inr @ B) @ A) @ X)) = (F @ X))))).
% 2.92/1.52 thf(fact_38_Sumr__inject, axiom, (! [A:$tType,C:$tType,B:$tType,F:(B > C),G:(B > C)]: (((((((sum_Sumr @ B) @ C) @ A) @ F) = ((((sum_Sumr @ B) @ C) @ A) @ G)) => (F = G))))).
% 2.92/1.52 thf(fact_39_Suml__inject, axiom, (! [B:$tType,C:$tType,A:$tType,F:(A > C),G:(A > C)]: (((((((sum_Suml @ A) @ C) @ B) @ F) = ((((sum_Suml @ A) @ C) @ B) @ G)) => (F = G))))).
% 2.92/1.52 thf(fact_40_not__arg__cong__Inr, axiom, (! [B:$tType,A:$tType,X:A,Y:A]: (((X != Y) => ((((sum_Inr @ A) @ B) @ X) != (((sum_Inr @ A) @ B) @ Y)))))).
% 2.92/1.52 thf(fact_41_insert__absorb2, axiom, (! [A:$tType,X:A,A7:(set @ A)]: (((((insert @ A) @ X) @ (((insert @ A) @ X) @ A7)) = (((insert @ A) @ X) @ A7))))).
% 2.92/1.52 thf(fact_42_insert__iff, axiom, (! [A:$tType,A2:A,B2:A,A7:(set @ A)]: (((((member @ A) @ A2) @ (((insert @ A) @ B2) @ A7)) = ((A2 = B2) | (((member @ A) @ A2) @ A7)))))).
% 2.92/1.52 thf(fact_43_insertCI, axiom, (! [A:$tType,A2:A,B7:(set @ A),B2:A]: ((((~ ((((member @ A) @ A2) @ B7))) => (A2 = B2)) => (((member @ A) @ A2) @ (((insert @ A) @ B2) @ B7)))))).
% 2.92/1.52 thf(fact_44_corec_I1_J, axiom, (! [A:$tType,Rt:(A > n),Ct:(A > (set @ ((sum_sum @ t) @ ((sum_sum @ dtree) @ A)))),B2:A]: (((root @ ((((corec @ A) @ Rt) @ Ct) @ B2)) = (Rt @ B2))))).
% 2.92/1.52 thf(fact_45_mem__Collect__eq, axiom, (! [A:$tType,A2:A,P:(A > $o)]: (((((member @ A) @ A2) @ ((collect @ A) @ P)) = (P @ A2))))).
% 2.92/1.52 thf(fact_46_Collect__mem__eq, axiom, (! [A:$tType,A7:(set @ A)]: ((((collect @ A) @ (^ [X5:A]: ((((member @ A) @ X5) @ A7)))) = A7)))).
% 2.92/1.52 thf(fact_47_Collect__cong, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X6:A]: (((P @ X6) = (Q @ X6)))) => (((collect @ A) @ P) = ((collect @ A) @ Q)))))).
% 2.92/1.52 thf(fact_48_ext, axiom, (! [B:$tType,A:$tType,F:(A > B),G:(A > B)]: (((! [X6:A]: (((F @ X6) = (G @ X6)))) => (F = G))))).
% 2.92/1.52 thf(fact_49_unfold_I1_J, axiom, (! [A:$tType,Rt:(A > n),Ct:(A > (set @ ((sum_sum @ t) @ A))),B2:A]: (((root @ ((((unfold @ A) @ Rt) @ Ct) @ B2)) = (Rt @ B2))))).
% 2.92/1.52 thf(fact_50_cont__Node, axiom, (! [As:(set @ ((sum_sum @ t) @ dtree)),N:n]: ((((finite_finite2 @ ((sum_sum @ t) @ dtree)) @ As) => ((cont @ ((node @ N) @ As)) = As))))).
% 2.92/1.52 thf(fact_51_mk__disjoint__insert, axiom, (! [A:$tType,A2:A,A7:(set @ A)]: (((((member @ A) @ A2) @ A7) => (? [B8:(set @ A)]: (((A7 = (((insert @ A) @ A2) @ B8)) & (~ ((((member @ A) @ A2) @ B8)))))))))).
% 2.92/1.52 thf(fact_52_insert__commute, axiom, (! [A:$tType,X:A,Y:A,A7:(set @ A)]: (((((insert @ A) @ X) @ (((insert @ A) @ Y) @ A7)) = (((insert @ A) @ Y) @ (((insert @ A) @ X) @ A7)))))).
% 2.92/1.52 thf(fact_53_insert__eq__iff, axiom, (! [A:$tType,A2:A,A7:(set @ A),B2:A,B7:(set @ A)]: (((~ ((((member @ A) @ A2) @ A7))) => ((~ ((((member @ A) @ B2) @ B7))) => (((((insert @ A) @ A2) @ A7) = (((insert @ A) @ B2) @ B7)) = (((A2 = B2) => (A7 = B7)) & ((A2 != B2) => (? [C2:(set @ A)]: (((A7 = (((insert @ A) @ B2) @ C2)) & ((~ ((((member @ A) @ B2) @ C2))) & ((B7 = (((insert @ A) @ A2) @ C2)) & (~ ((((member @ A) @ A2) @ C2)))))))))))))))).
% 2.92/1.52 thf(fact_54_dtree__simps, axiom, (! [As:(set @ ((sum_sum @ t) @ dtree)),As2:(set @ ((sum_sum @ t) @ dtree)),N:n,N2:n]: ((((finite_finite2 @ ((sum_sum @ t) @ dtree)) @ As) => (((finite_finite2 @ ((sum_sum @ t) @ dtree)) @ As2) => ((((node @ N) @ As) = ((node @ N2) @ As2)) = ((N = N2) & (As = As2)))))))).
% 2.92/1.52 thf(fact_55_finite__cont, axiom, (! [Tr:dtree]: (((finite_finite2 @ ((sum_sum @ t) @ dtree)) @ (cont @ Tr))))).
% 2.92/1.52 thf(fact_56_dtree__cases, axiom, (! [Tr:dtree]: ((~ ((! [N3:n,As3:(set @ ((sum_sum @ t) @ dtree))]: ((((finite_finite2 @ ((sum_sum @ t) @ dtree)) @ As3) => (Tr != ((node @ N3) @ As3)))))))))).
% 2.92/1.52 thf(fact_57_insertE, axiom, (! [A:$tType,A2:A,B2:A,A7:(set @ A)]: (((((member @ A) @ A2) @ (((insert @ A) @ B2) @ A7)) => ((A2 != B2) => (((member @ A) @ A2) @ A7)))))).
% 2.92/1.52 thf(fact_58_insertI1, axiom, (! [A:$tType,A2:A,B7:(set @ A)]: ((((member @ A) @ A2) @ (((insert @ A) @ A2) @ B7))))).
% 2.92/1.52 thf(fact_59_insertI2, axiom, (! [A:$tType,A2:A,B7:(set @ A),B2:A]: (((((member @ A) @ A2) @ B7) => (((member @ A) @ A2) @ (((insert @ A) @ B2) @ B7)))))).
% 2.92/1.52 thf(fact_60_Set_Oset__insert, axiom, (! [A:$tType,X:A,A7:(set @ A)]: (((((member @ A) @ X) @ A7) => (~ ((! [B8:(set @ A)]: (((A7 = (((insert @ A) @ X) @ B8)) => (((member @ A) @ X) @ B8)))))))))).
% 2.92/1.52 thf(fact_61_insert__ident, axiom, (! [A:$tType,X:A,A7:(set @ A),B7:(set @ A)]: (((~ ((((member @ A) @ X) @ A7))) => ((~ ((((member @ A) @ X) @ B7))) => (((((insert @ A) @ X) @ A7) = (((insert @ A) @ X) @ B7)) = (A7 = B7))))))).
% 2.92/1.52 thf(fact_62_insert__absorb, axiom, (! [A:$tType,A2:A,A7:(set @ A)]: (((((member @ A) @ A2) @ A7) => ((((insert @ A) @ A2) @ A7) = A7))))).
% 2.92/1.52 thf(fact_63_finite__insert, axiom, (! [A:$tType,A2:A,A7:(set @ A)]: ((((finite_finite2 @ A) @ (((insert @ A) @ A2) @ A7)) = ((finite_finite2 @ A) @ A7))))).
% 2.92/1.52 thf(fact_64_finite__code, axiom, (! [A:$tType]: ((((finite_finite @ A) @ (type2 @ A)) => ((finite_finite2 @ A) = (^ [A8:(set @ A)]: ($true))))))).
% 2.92/1.52 thf(fact_65_finite_OinsertI, axiom, (! [A:$tType,A7:(set @ A),A2:A]: ((((finite_finite2 @ A) @ A7) => ((finite_finite2 @ A) @ (((insert @ A) @ A2) @ A7)))))).
% 2.92/1.52 thf(fact_66_finite__set__choice, axiom, (! [B:$tType,A:$tType,A7:(set @ A),P:(A > (B > $o))]: ((((finite_finite2 @ A) @ A7) => ((! [X6:A]: (((((member @ A) @ X6) @ A7) => (? [X12:B]: (((P @ X6) @ X12)))))) => (? [F3:(A > B)]: ((! [X7:A]: (((((member @ A) @ X7) @ A7) => ((P @ X7) @ (F3 @ X7)))))))))))).
% 2.92/1.52 thf(fact_67_finite, axiom, (! [A:$tType]: ((((finite_finite @ A) @ (type2 @ A)) => (! [A7:(set @ A)]: (((finite_finite2 @ A) @ A7))))))).
% 2.92/1.52 thf(fact_68_case__sum__if, axiom, (! [B:$tType,A:$tType,C:$tType,P4:$o,F:(B > A),G:(C > A),X:B,Y:C]: (((P4 => (((((((sum_case_sum @ B) @ A) @ C) @ F) @ G) @ ((((if @ ((sum_sum @ B) @ C)) @ P4) @ (((sum_Inl @ B) @ C) @ X)) @ (((sum_Inr @ C) @ B) @ Y))) = (F @ X))) & ((~ (P4)) => (((((((sum_case_sum @ B) @ A) @ C) @ F) @ G) @ ((((if @ ((sum_sum @ B) @ C)) @ P4) @ (((sum_Inl @ B) @ C) @ X)) @ (((sum_Inr @ C) @ B) @ Y))) = (G @ Y))))))).
% 2.92/1.52 thf(fact_69_PlusE, axiom, (! [A:$tType,B:$tType,U:((sum_sum @ A) @ B),A7:(set @ A),B7:(set @ B)]: (((((member @ ((sum_sum @ A) @ B)) @ U) @ ((((sum_Plus @ A) @ B) @ A7) @ B7)) => ((! [X6:A]: (((((member @ A) @ X6) @ A7) => (U != (((sum_Inl @ A) @ B) @ X6))))) => (~ ((! [Y3:B]: (((((member @ B) @ Y3) @ B7) => (U != (((sum_Inr @ B) @ A) @ Y3)))))))))))).
% 2.92/1.52 thf(fact_70_dtree_Osel_I1_J, axiom, (! [X1:n,X2:(fset @ ((sum_sum @ t) @ dtree))]: (((root @ ((nNode @ X1) @ X2)) = X1)))).
% 2.92/1.52 thf(fact_71_dtree_Oinject, axiom, (! [X1:n,X2:(fset @ ((sum_sum @ t) @ dtree)),Y1:n,Y2:(fset @ ((sum_sum @ t) @ dtree))]: (((((nNode @ X1) @ X2) = ((nNode @ Y1) @ Y2)) = ((X1 = Y1) & (X2 = Y2)))))).
% 2.92/1.52 thf(fact_72_InrI, axiom, (! [B:$tType,A:$tType,B2:A,B7:(set @ A),A7:(set @ B)]: (((((member @ A) @ B2) @ B7) => (((member @ ((sum_sum @ B) @ A)) @ (((sum_Inr @ A) @ B) @ B2)) @ ((((sum_Plus @ B) @ A) @ A7) @ B7)))))).
% 2.92/1.52 thf(fact_73_InlI, axiom, (! [A:$tType,B:$tType,A2:A,A7:(set @ A),B7:(set @ B)]: (((((member @ A) @ A2) @ A7) => (((member @ ((sum_sum @ A) @ B)) @ (((sum_Inl @ A) @ B) @ A2)) @ ((((sum_Plus @ A) @ B) @ A7) @ B7)))))).
% 2.92/1.52 thf(fact_74_finite__Plus__iff, axiom, (! [A:$tType,B:$tType,A7:(set @ A),B7:(set @ B)]: ((((finite_finite2 @ ((sum_sum @ A) @ B)) @ ((((sum_Plus @ A) @ B) @ A7) @ B7)) = (((finite_finite2 @ A) @ A7) & ((finite_finite2 @ B) @ B7)))))).
% 2.92/1.52 thf(fact_75_dtree_Oexhaust, axiom, (! [Y:dtree]: ((~ ((! [X13:n,X22:(fset @ ((sum_sum @ t) @ dtree))]: ((Y != ((nNode @ X13) @ X22))))))))).
% 2.92/1.52 thf(fact_76_case__sum__inject, axiom, (! [A:$tType,C:$tType,B:$tType,F1:(A > C),F2:(B > C),G1:(A > C),G2:(B > C)]: ((((((((sum_case_sum @ A) @ C) @ B) @ F1) @ F2) = (((((sum_case_sum @ A) @ C) @ B) @ G1) @ G2)) => (~ (((F1 = G1) => (F2 != G2)))))))).
% 2.92/1.52 thf(fact_77_case__sum__step_I1_J, axiom, (! [D:$tType,A:$tType,C:$tType,B:$tType,F4:(B > A),G3:(C > A),G:(D > A),P4:((sum_sum @ B) @ C)]: ((((((((sum_case_sum @ ((sum_sum @ B) @ C)) @ A) @ D) @ (((((sum_case_sum @ B) @ A) @ C) @ F4) @ G3)) @ G) @ (((sum_Inl @ ((sum_sum @ B) @ C)) @ D) @ P4)) = ((((((sum_case_sum @ B) @ A) @ C) @ F4) @ G3) @ P4))))).
% 2.92/1.52 thf(fact_78_case__sum__step_I2_J, axiom, (! [E:$tType,A:$tType,C:$tType,B:$tType,F:(E > A),F4:(B > A),G3:(C > A),P4:((sum_sum @ B) @ C)]: ((((((((sum_case_sum @ E) @ A) @ ((sum_sum @ B) @ C)) @ F) @ (((((sum_case_sum @ B) @ A) @ C) @ F4) @ G3)) @ (((sum_Inr @ ((sum_sum @ B) @ C)) @ E) @ P4)) = ((((((sum_case_sum @ B) @ A) @ C) @ F4) @ G3) @ P4))))).
% 2.92/1.52 thf(fact_79_old_Osum_Osimps_I6_J, axiom, (! [A:$tType,C:$tType,B:$tType,F1:(A > C),F2:(B > C),X2:B]: ((((((((sum_case_sum @ A) @ C) @ B) @ F1) @ F2) @ (((sum_Inr @ B) @ A) @ X2)) = (F2 @ X2))))).
% 2.92/1.52 thf(fact_80_old_Osum_Osimps_I5_J, axiom, (! [B:$tType,C:$tType,A:$tType,F1:(A > C),F2:(B > C),X1:A]: ((((((((sum_case_sum @ A) @ C) @ B) @ F1) @ F2) @ (((sum_Inl @ A) @ B) @ X1)) = (F1 @ X1))))).
% 2.92/1.52 thf(fact_81_finite__Plus, axiom, (! [A:$tType,B:$tType,A7:(set @ A),B7:(set @ B)]: ((((finite_finite2 @ A) @ A7) => (((finite_finite2 @ B) @ B7) => ((finite_finite2 @ ((sum_sum @ A) @ B)) @ ((((sum_Plus @ A) @ B) @ A7) @ B7))))))).
% 2.92/1.52 thf(fact_82_finite__PlusD_I1_J, axiom, (! [B:$tType,A:$tType,A7:(set @ A),B7:(set @ B)]: ((((finite_finite2 @ ((sum_sum @ A) @ B)) @ ((((sum_Plus @ A) @ B) @ A7) @ B7)) => ((finite_finite2 @ A) @ A7))))).
% 2.92/1.52 thf(fact_83_finite__PlusD_I2_J, axiom, (! [A:$tType,B:$tType,A7:(set @ A),B7:(set @ B)]: ((((finite_finite2 @ ((sum_sum @ A) @ B)) @ ((((sum_Plus @ A) @ B) @ A7) @ B7)) => ((finite_finite2 @ B) @ B7))))).
% 2.92/1.52 thf(fact_84_dtree_Ocollapse, axiom, (! [Dtree:dtree]: ((((nNode @ (root @ Dtree)) @ (ccont @ Dtree)) = Dtree)))).
% 2.92/1.52 thf(fact_85_dtree_Oexhaust__sel, axiom, (! [Dtree:dtree]: ((Dtree = ((nNode @ (root @ Dtree)) @ (ccont @ Dtree)))))).
% 2.92/1.52 thf(fact_86_case__sum__expand__Inr__pointfree, axiom, (! [B:$tType,C:$tType,A:$tType,F:(((sum_sum @ A) @ C) > B),G:(A > B)]: ((((((((comp @ ((sum_sum @ A) @ C)) @ B) @ A) @ F) @ ((sum_Inl @ A) @ C)) = G) => ((((((sum_case_sum @ A) @ B) @ C) @ G) @ (((((comp @ ((sum_sum @ A) @ C)) @ B) @ C) @ F) @ ((sum_Inr @ C) @ A))) = F))))).
% 2.92/1.52 thf(fact_87_case__sum__expand__Inr_H, axiom, (! [B:$tType,C:$tType,A:$tType,F:(((sum_sum @ A) @ C) > B),G:(A > B),H:(C > B)]: ((((((((comp @ ((sum_sum @ A) @ C)) @ B) @ A) @ F) @ ((sum_Inl @ A) @ C)) = G) => ((H = (((((comp @ ((sum_sum @ A) @ C)) @ B) @ C) @ F) @ ((sum_Inr @ C) @ A))) = ((((((sum_case_sum @ A) @ B) @ C) @ G) @ H) = F)))))).
% 2.92/1.52 thf(fact_88_case__sum__expand__Inr, axiom, (! [B:$tType,C:$tType,A:$tType,F:(((sum_sum @ A) @ C) > B),G:(A > B),X:((sum_sum @ A) @ C)]: ((((((((comp @ ((sum_sum @ A) @ C)) @ B) @ A) @ F) @ ((sum_Inl @ A) @ C)) = G) => ((F @ X) = ((((((sum_case_sum @ A) @ B) @ C) @ G) @ (((((comp @ ((sum_sum @ A) @ C)) @ B) @ C) @ F) @ ((sum_Inr @ C) @ A))) @ X)))))).
% 2.92/1.52 thf(fact_89_dtree_Ocase, axiom, (! [A:$tType,F:(n > ((fset @ ((sum_sum @ t) @ dtree)) > A)),X1:n,X2:(fset @ ((sum_sum @ t) @ dtree))]: (((((case_dtree @ A) @ F) @ ((nNode @ X1) @ X2)) = ((F @ X1) @ X2))))).
% 2.92/1.52 thf(fact_90_finite__Diff__insert, axiom, (! [A:$tType,A7:(set @ A),A2:A,B7:(set @ A)]: ((((finite_finite2 @ A) @ (((minus_minus @ (set @ A)) @ A7) @ (((insert @ A) @ A2) @ B7))) = ((finite_finite2 @ A) @ (((minus_minus @ (set @ A)) @ A7) @ B7)))))).
% 2.92/1.52 thf(fact_91_Diff__idemp, axiom, (! [A:$tType,A7:(set @ A),B7:(set @ A)]: (((((minus_minus @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A7) @ B7)) @ B7) = (((minus_minus @ (set @ A)) @ A7) @ B7))))).
% 2.92/1.52 thf(fact_92_Diff__iff, axiom, (! [A:$tType,C3:A,A7:(set @ A),B7:(set @ A)]: (((((member @ A) @ C3) @ (((minus_minus @ (set @ A)) @ A7) @ B7)) = ((((member @ A) @ C3) @ A7) & (~ ((((member @ A) @ C3) @ B7)))))))).
% 2.92/1.52 thf(fact_93_DiffI, axiom, (! [A:$tType,C3:A,A7:(set @ A),B7:(set @ A)]: (((((member @ A) @ C3) @ A7) => ((~ ((((member @ A) @ C3) @ B7))) => (((member @ A) @ C3) @ (((minus_minus @ (set @ A)) @ A7) @ B7))))))).
% 2.92/1.52 thf(fact_94_finite__Diff2, axiom, (! [A:$tType,B7:(set @ A),A7:(set @ A)]: ((((finite_finite2 @ A) @ B7) => (((finite_finite2 @ A) @ (((minus_minus @ (set @ A)) @ A7) @ B7)) = ((finite_finite2 @ A) @ A7)))))).
% 2.92/1.52 thf(fact_95_finite__Diff, axiom, (! [A:$tType,A7:(set @ A),B7:(set @ A)]: ((((finite_finite2 @ A) @ A7) => ((finite_finite2 @ A) @ (((minus_minus @ (set @ A)) @ A7) @ B7)))))).
% 2.92/1.52 thf(fact_96_Diff__insert0, axiom, (! [A:$tType,X:A,A7:(set @ A),B7:(set @ A)]: (((~ ((((member @ A) @ X) @ A7))) => ((((minus_minus @ (set @ A)) @ A7) @ (((insert @ A) @ X) @ B7)) = (((minus_minus @ (set @ A)) @ A7) @ B7)))))).
% 2.92/1.52 thf(fact_97_insert__Diff1, axiom, (! [A:$tType,X:A,B7:(set @ A),A7:(set @ A)]: (((((member @ A) @ X) @ B7) => ((((minus_minus @ (set @ A)) @ (((insert @ A) @ X) @ A7)) @ B7) = (((minus_minus @ (set @ A)) @ A7) @ B7)))))).
% 2.92/1.52 thf(fact_98_o__case__sum, axiom, (! [A:$tType,C:$tType,D:$tType,B:$tType,H:(D > C),F:(A > D),G:(B > D)]: (((((((comp @ D) @ C) @ ((sum_sum @ A) @ B)) @ H) @ (((((sum_case_sum @ A) @ D) @ B) @ F) @ G)) = (((((sum_case_sum @ A) @ C) @ B) @ (((((comp @ D) @ C) @ A) @ H) @ F)) @ (((((comp @ D) @ C) @ B) @ H) @ G)))))).
% 2.92/1.52 thf(fact_99_rewriteR__comp__comp2, axiom, (! [C:$tType,B:$tType,E:$tType,D:$tType,A:$tType,G:(C > B),H:(A > C),R1:(D > B),R2:(A > D),F:(B > E),L:(D > E)]: ((((((((comp @ C) @ B) @ A) @ G) @ H) = (((((comp @ D) @ B) @ A) @ R1) @ R2)) => (((((((comp @ B) @ E) @ D) @ F) @ R1) = L) => ((((((comp @ C) @ E) @ A) @ (((((comp @ B) @ E) @ C) @ F) @ G)) @ H) = (((((comp @ D) @ E) @ A) @ L) @ R2))))))).
% 2.92/1.52 thf(fact_100_rewriteL__comp__comp2, axiom, (! [A:$tType,C:$tType,B:$tType,D:$tType,E:$tType,F:(C > B),G:(A > C),L1:(D > B),L2:(A > D),H:(E > A),R:(E > D)]: ((((((((comp @ C) @ B) @ A) @ F) @ G) = (((((comp @ D) @ B) @ A) @ L1) @ L2)) => (((((((comp @ A) @ D) @ E) @ L2) @ H) = R) => ((((((comp @ C) @ B) @ E) @ F) @ (((((comp @ A) @ C) @ E) @ G) @ H)) = (((((comp @ D) @ B) @ E) @ L1) @ R))))))).
% 2.92/1.52 thf(fact_101_rewriteR__comp__comp, axiom, (! [C:$tType,D:$tType,B:$tType,A:$tType,G:(C > B),H:(A > C),R:(A > B),F:(B > D)]: ((((((((comp @ C) @ B) @ A) @ G) @ H) = R) => ((((((comp @ C) @ D) @ A) @ (((((comp @ B) @ D) @ C) @ F) @ G)) @ H) = (((((comp @ B) @ D) @ A) @ F) @ R)))))).
% 2.92/1.52 thf(fact_102_rewriteL__comp__comp, axiom, (! [C:$tType,B:$tType,A:$tType,D:$tType,F:(C > B),G:(A > C),L:(A > B),H:(D > A)]: ((((((((comp @ C) @ B) @ A) @ F) @ G) = L) => ((((((comp @ C) @ B) @ D) @ F) @ (((((comp @ A) @ C) @ D) @ G) @ H)) = (((((comp @ A) @ B) @ D) @ L) @ H)))))).
% 2.92/1.52 thf(fact_103_DiffD2, axiom, (! [A:$tType,C3:A,A7:(set @ A),B7:(set @ A)]: (((((member @ A) @ C3) @ (((minus_minus @ (set @ A)) @ A7) @ B7)) => (~ ((((member @ A) @ C3) @ B7))))))).
% 2.92/1.52 thf(fact_104_DiffD1, axiom, (! [A:$tType,C3:A,A7:(set @ A),B7:(set @ A)]: (((((member @ A) @ C3) @ (((minus_minus @ (set @ A)) @ A7) @ B7)) => (((member @ A) @ C3) @ A7))))).
% 2.92/1.52 thf(fact_105_DiffE, axiom, (! [A:$tType,C3:A,A7:(set @ A),B7:(set @ A)]: (((((member @ A) @ C3) @ (((minus_minus @ (set @ A)) @ A7) @ B7)) => (~ (((((member @ A) @ C3) @ A7) => (((member @ A) @ C3) @ B7)))))))).
% 2.92/1.52 thf(fact_106_dtree_Ocase__eq__if, axiom, (! [A:$tType]: (((case_dtree @ A) = (^ [F5:(n > ((fset @ ((sum_sum @ t) @ dtree)) > A)),Dtree2:dtree]: (((F5 @ (root @ Dtree2)) @ (ccont @ Dtree2)))))))).
% 2.92/1.52 thf(fact_107_Diff__infinite__finite, axiom, (! [A:$tType,T5:(set @ A),S2:(set @ A)]: ((((finite_finite2 @ A) @ T5) => ((~ (((finite_finite2 @ A) @ S2))) => (~ (((finite_finite2 @ A) @ (((minus_minus @ (set @ A)) @ S2) @ T5))))))))).
% 2.92/1.52 thf(fact_108_insert__Diff__if, axiom, (! [A:$tType,X:A,B7:(set @ A),A7:(set @ A)]: ((((((member @ A) @ X) @ B7) => ((((minus_minus @ (set @ A)) @ (((insert @ A) @ X) @ A7)) @ B7) = (((minus_minus @ (set @ A)) @ A7) @ B7))) & ((~ ((((member @ A) @ X) @ B7))) => ((((minus_minus @ (set @ A)) @ (((insert @ A) @ X) @ A7)) @ B7) = (((insert @ A) @ X) @ (((minus_minus @ (set @ A)) @ A7) @ B7)))))))).
% 2.92/1.52 thf(fact_109_dtree_Oexpand, axiom, (! [Dtree:dtree,Dtree3:dtree]: (((((root @ Dtree) = (root @ Dtree3)) & ((ccont @ Dtree) = (ccont @ Dtree3))) => (Dtree = Dtree3))))).
% 2.92/1.52 thf(fact_110_dtree_Osplit__sel__asm, axiom, (! [A:$tType,P:(A > $o),F:(n > ((fset @ ((sum_sum @ t) @ dtree)) > A)),Dtree:dtree]: (((P @ (((case_dtree @ A) @ F) @ Dtree)) = (~ (((Dtree = ((nNode @ (root @ Dtree)) @ (ccont @ Dtree))) & (~ ((P @ ((F @ (root @ Dtree)) @ (ccont @ Dtree)))))))))))).
% 2.92/1.52 thf(fact_111_dtree_Osplit__sel, axiom, (! [A:$tType,P:(A > $o),F:(n > ((fset @ ((sum_sum @ t) @ dtree)) > A)),Dtree:dtree]: (((P @ (((case_dtree @ A) @ F) @ Dtree)) = ((Dtree = ((nNode @ (root @ Dtree)) @ (ccont @ Dtree))) => (P @ ((F @ (root @ Dtree)) @ (ccont @ Dtree)))))))).
% 2.92/1.52 thf(fact_112_dtree_Osel_I2_J, axiom, (! [X1:n,X2:(fset @ ((sum_sum @ t) @ dtree))]: (((ccont @ ((nNode @ X1) @ X2)) = X2)))).
% 2.92/1.52 thf(fact_113_case__sum__o__inj_I1_J, axiom, (! [C:$tType,B:$tType,A:$tType,F:(A > B),G:(C > B)]: (((((((comp @ ((sum_sum @ A) @ C)) @ B) @ A) @ (((((sum_case_sum @ A) @ B) @ C) @ F) @ G)) @ ((sum_Inl @ A) @ C)) = F)))).
% 2.92/1.52 thf(fact_114_case__sum__o__inj_I2_J, axiom, (! [A:$tType,B:$tType,C:$tType,F:(A > B),G:(C > B)]: (((((((comp @ ((sum_sum @ A) @ C)) @ B) @ C) @ (((((sum_case_sum @ A) @ B) @ C) @ F) @ G)) @ ((sum_Inr @ C) @ A)) = G)))).
% 2.92/1.52 thf(fact_115_comp__apply, axiom, (! [C:$tType,A:$tType,B:$tType]: (((((comp @ B) @ A) @ C) = (^ [F5:(B > A),G4:(C > B),X5:C]: ((F5 @ (G4 @ X5)))))))).
% 2.92/1.52 thf(fact_116_minus__apply, axiom, (! [B:$tType,A:$tType]: ((((minus @ B) @ (type2 @ B)) => ((minus_minus @ (A > B)) = (^ [A8:(A > B),B9:(A > B),X5:A]: ((((minus_minus @ B) @ (A8 @ X5)) @ (B9 @ X5))))))))).
% 2.92/1.52 thf(fact_117_comp__def, axiom, (! [A:$tType,C:$tType,B:$tType]: (((((comp @ B) @ C) @ A) = (^ [F5:(B > C),G4:(A > B),X5:A]: ((F5 @ (G4 @ X5)))))))).
% 2.92/1.52 thf(fact_118_comp__assoc, axiom, (! [B:$tType,D:$tType,C:$tType,A:$tType,F:(D > B),G:(C > D),H:(A > C)]: (((((((comp @ C) @ B) @ A) @ (((((comp @ D) @ B) @ C) @ F) @ G)) @ H) = (((((comp @ D) @ B) @ A) @ F) @ (((((comp @ C) @ D) @ A) @ G) @ H)))))).
% 2.92/1.52 thf(fact_119_fun__diff__def, axiom, (! [B:$tType,A:$tType]: ((((minus @ B) @ (type2 @ B)) => ((minus_minus @ (A > B)) = (^ [A8:(A > B),B9:(A > B),X5:A]: ((((minus_minus @ B) @ (A8 @ X5)) @ (B9 @ X5))))))))).
% 2.92/1.52 thf(fact_120_comp__apply__eq, axiom, (! [B:$tType,D:$tType,A:$tType,C:$tType,F:(B > A),G:(C > B),X:C,H:(D > A),K:(C > D)]: ((((F @ (G @ X)) = (H @ (K @ X))) => (((((((comp @ B) @ A) @ C) @ F) @ G) @ X) = ((((((comp @ D) @ A) @ C) @ H) @ K) @ X)))))).
% 2.92/1.52 thf(fact_121_comp__eq__dest__lhs, axiom, (! [C:$tType,B:$tType,A:$tType,A2:(C > B),B2:(A > C),C3:(A > B),V:A]: ((((((((comp @ C) @ B) @ A) @ A2) @ B2) = C3) => ((A2 @ (B2 @ V)) = (C3 @ V)))))).
% 2.92/1.52 thf(fact_122_comp__eq__elim, axiom, (! [C:$tType,B:$tType,D:$tType,A:$tType,A2:(C > B),B2:(A > C),C3:(D > B),D2:(A > D)]: ((((((((comp @ C) @ B) @ A) @ A2) @ B2) = (((((comp @ D) @ B) @ A) @ C3) @ D2)) => (! [V2:A]: (((A2 @ (B2 @ V2)) = (C3 @ (D2 @ V2))))))))).
% 2.92/1.52 thf(fact_123_comp__eq__dest, axiom, (! [C:$tType,B:$tType,D:$tType,A:$tType,A2:(C > B),B2:(A > C),C3:(D > B),D2:(A > D),V:A]: ((((((((comp @ C) @ B) @ A) @ A2) @ B2) = (((((comp @ D) @ B) @ A) @ C3) @ D2)) => ((A2 @ (B2 @ V)) = (C3 @ (D2 @ V))))))).
% 2.92/1.52 thf(fact_124_type__copy__map__cong0, axiom, (! [B:$tType,D:$tType,E:$tType,A:$tType,C:$tType,M:(B > A),G:(C > B),X:C,N4:(D > A),H:(C > D),F:(A > E)]: ((((M @ (G @ X)) = (N4 @ (H @ X))) => (((((((comp @ B) @ E) @ C) @ (((((comp @ A) @ E) @ B) @ F) @ M)) @ G) @ X) = ((((((comp @ D) @ E) @ C) @ (((((comp @ A) @ E) @ D) @ F) @ N4)) @ H) @ X)))))).
% 2.92/1.52 thf(fact_125_fun_Omap__comp, axiom, (! [B:$tType,C:$tType,A:$tType,D:$tType,G:(B > C),F:(A > B),V:(D > A)]: (((((((comp @ B) @ C) @ D) @ G) @ (((((comp @ A) @ B) @ D) @ F) @ V)) = (((((comp @ A) @ C) @ D) @ (((((comp @ B) @ C) @ A) @ G) @ F)) @ V))))).
% 2.92/1.52 thf(fact_126_comp__cong, axiom, (! [C:$tType,B:$tType,D:$tType,A:$tType,E:$tType,F:(B > A),G:(C > B),X:C,F4:(D > A),G3:(E > D),X8:E]: ((((F @ (G @ X)) = (F4 @ (G3 @ X8))) => (((((((comp @ B) @ A) @ C) @ F) @ G) @ X) = ((((((comp @ D) @ A) @ E) @ F4) @ G3) @ X8)))))).
% 2.92/1.52 thf(fact_127_finite__empty__induct, axiom, (! [A:$tType,A7:(set @ A),P:((set @ A) > $o)]: ((((finite_finite2 @ A) @ A7) => ((P @ A7) => ((! [A4:A,A9:(set @ A)]: ((((finite_finite2 @ A) @ A9) => ((((member @ A) @ A4) @ A9) => ((P @ A9) => (P @ (((minus_minus @ (set @ A)) @ A9) @ (((insert @ A) @ A4) @ (bot_bot @ (set @ A)))))))))) => (P @ (bot_bot @ (set @ A))))))))).
% 2.92/1.52 thf(fact_128_empty__Collect__eq, axiom, (! [A:$tType,P:(A > $o)]: ((((bot_bot @ (set @ A)) = ((collect @ A) @ P)) = (! [X5:A]: ((~ ((P @ X5))))))))).
% 2.92/1.52 thf(fact_129_Collect__empty__eq, axiom, (! [A:$tType,P:(A > $o)]: (((((collect @ A) @ P) = (bot_bot @ (set @ A))) = (! [X5:A]: ((~ ((P @ X5))))))))).
% 2.92/1.52 thf(fact_130_all__not__in__conv, axiom, (! [A:$tType,A7:(set @ A)]: (((! [X5:A]: ((~ ((((member @ A) @ X5) @ A7))))) = (A7 = (bot_bot @ (set @ A))))))).
% 2.92/1.52 thf(fact_131_empty__iff, axiom, (! [A:$tType,C3:A]: ((~ ((((member @ A) @ C3) @ (bot_bot @ (set @ A)))))))).
% 2.92/1.52 thf(fact_132_singletonI, axiom, (! [A:$tType,A2:A]: ((((member @ A) @ A2) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))))).
% 2.92/1.52 thf(fact_133_Diff__cancel, axiom, (! [A:$tType,A7:(set @ A)]: (((((minus_minus @ (set @ A)) @ A7) @ A7) = (bot_bot @ (set @ A)))))).
% 2.92/1.52 thf(fact_134_empty__Diff, axiom, (! [A:$tType,A7:(set @ A)]: (((((minus_minus @ (set @ A)) @ (bot_bot @ (set @ A))) @ A7) = (bot_bot @ (set @ A)))))).
% 2.92/1.52 thf(fact_135_Diff__empty, axiom, (! [A:$tType,A7:(set @ A)]: (((((minus_minus @ (set @ A)) @ A7) @ (bot_bot @ (set @ A))) = A7)))).
% 2.92/1.52 thf(fact_136_Plus__eq__empty__conv, axiom, (! [A:$tType,B:$tType,A7:(set @ A),B7:(set @ B)]: (((((((sum_Plus @ A) @ B) @ A7) @ B7) = (bot_bot @ (set @ ((sum_sum @ A) @ B)))) = ((A7 = (bot_bot @ (set @ A))) & (B7 = (bot_bot @ (set @ B)))))))).
% 2.92/1.52 thf(fact_137_insert__Diff__single, axiom, (! [A:$tType,A2:A,A7:(set @ A)]: (((((insert @ A) @ A2) @ (((minus_minus @ (set @ A)) @ A7) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))) = (((insert @ A) @ A2) @ A7))))).
% 2.92/1.52 thf(fact_138_finite_OemptyI, axiom, (! [A:$tType]: (((finite_finite2 @ A) @ (bot_bot @ (set @ A)))))).
% 2.92/1.52 thf(fact_139_infinite__imp__nonempty, axiom, (! [A:$tType,S2:(set @ A)]: (((~ (((finite_finite2 @ A) @ S2))) => (S2 != (bot_bot @ (set @ A))))))).
% 2.92/1.52 thf(fact_140_singletonD, axiom, (! [A:$tType,B2:A,A2:A]: (((((member @ A) @ B2) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A)))) => (B2 = A2))))).
% 2.92/1.52 thf(fact_141_singleton__iff, axiom, (! [A:$tType,B2:A,A2:A]: (((((member @ A) @ B2) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A)))) = (B2 = A2))))).
% 2.92/1.52 thf(fact_142_doubleton__eq__iff, axiom, (! [A:$tType,A2:A,B2:A,C3:A,D2:A]: ((((((insert @ A) @ A2) @ (((insert @ A) @ B2) @ (bot_bot @ (set @ A)))) = (((insert @ A) @ C3) @ (((insert @ A) @ D2) @ (bot_bot @ (set @ A))))) = (((A2 = C3) & (B2 = D2)) | ((A2 = D2) & (B2 = C3))))))).
% 2.92/1.52 thf(fact_143_insert__not__empty, axiom, (! [A:$tType,A2:A,A7:(set @ A)]: (((((insert @ A) @ A2) @ A7) != (bot_bot @ (set @ A)))))).
% 2.92/1.52 thf(fact_144_singleton__inject, axiom, (! [A:$tType,A2:A,B2:A]: ((((((insert @ A) @ A2) @ (bot_bot @ (set @ A))) = (((insert @ A) @ B2) @ (bot_bot @ (set @ A)))) => (A2 = B2))))).
% 2.92/1.52 thf(fact_145_ex__in__conv, axiom, (! [A:$tType,A7:(set @ A)]: (((? [X5:A]: ((((member @ A) @ X5) @ A7))) = (A7 != (bot_bot @ (set @ A))))))).
% 2.92/1.52 thf(fact_146_equals0I, axiom, (! [A:$tType,A7:(set @ A)]: (((! [Y3:A]: ((~ ((((member @ A) @ Y3) @ A7))))) => (A7 = (bot_bot @ (set @ A))))))).
% 2.92/1.52 thf(fact_147_equals0D, axiom, (! [A:$tType,A7:(set @ A),A2:A]: (((A7 = (bot_bot @ (set @ A))) => (~ ((((member @ A) @ A2) @ A7))))))).
% 2.92/1.52 thf(fact_148_emptyE, axiom, (! [A:$tType,A2:A]: ((~ ((((member @ A) @ A2) @ (bot_bot @ (set @ A)))))))).
% 2.92/1.52 thf(fact_149_infinite__finite__induct, axiom, (! [A:$tType,P:((set @ A) > $o),A7:(set @ A)]: (((! [A9:(set @ A)]: (((~ (((finite_finite2 @ A) @ A9))) => (P @ A9)))) => ((P @ (bot_bot @ (set @ A))) => ((! [X6:A,F6:(set @ A)]: ((((finite_finite2 @ A) @ F6) => ((~ ((((member @ A) @ X6) @ F6))) => ((P @ F6) => (P @ (((insert @ A) @ X6) @ F6))))))) => (P @ A7))))))).
% 2.92/1.52 thf(fact_150_finite__ne__induct, axiom, (! [A:$tType,F7:(set @ A),P:((set @ A) > $o)]: ((((finite_finite2 @ A) @ F7) => ((F7 != (bot_bot @ (set @ A))) => ((! [X6:A]: ((P @ (((insert @ A) @ X6) @ (bot_bot @ (set @ A)))))) => ((! [X6:A,F6:(set @ A)]: ((((finite_finite2 @ A) @ F6) => ((F6 != (bot_bot @ (set @ A))) => ((~ ((((member @ A) @ X6) @ F6))) => ((P @ F6) => (P @ (((insert @ A) @ X6) @ F6)))))))) => (P @ F7)))))))).
% 2.92/1.52 thf(fact_151_finite_Oinducts, axiom, (! [A:$tType,X:(set @ A),P:((set @ A) > $o)]: ((((finite_finite2 @ A) @ X) => ((P @ (bot_bot @ (set @ A))) => ((! [A9:(set @ A),A4:A]: ((((finite_finite2 @ A) @ A9) => ((P @ A9) => (P @ (((insert @ A) @ A4) @ A9)))))) => (P @ X))))))).
% 2.92/1.52 thf(fact_152_finite__induct, axiom, (! [A:$tType,F7:(set @ A),P:((set @ A) > $o)]: ((((finite_finite2 @ A) @ F7) => ((P @ (bot_bot @ (set @ A))) => ((! [X6:A,F6:(set @ A)]: ((((finite_finite2 @ A) @ F6) => ((~ ((((member @ A) @ X6) @ F6))) => ((P @ F6) => (P @ (((insert @ A) @ X6) @ F6))))))) => (P @ F7))))))).
% 2.92/1.52 thf(fact_153_finite_Osimps, axiom, (! [A:$tType]: (((finite_finite2 @ A) = (^ [A10:(set @ A)]: (((A10 = (bot_bot @ (set @ A))) | (? [A8:(set @ A),B10:A]: (((A10 = (((insert @ A) @ B10) @ A8)) & ((finite_finite2 @ A) @ A8))))))))))).
% 2.92/1.52 thf(fact_154_finite_Ocases, axiom, (! [A:$tType,A2:(set @ A)]: ((((finite_finite2 @ A) @ A2) => ((A2 != (bot_bot @ (set @ A))) => (~ ((! [A9:(set @ A)]: (((? [A4:A]: ((A2 = (((insert @ A) @ A4) @ A9)))) => (~ (((finite_finite2 @ A) @ A9))))))))))))).
% 2.92/1.52 thf(fact_155_Diff__insert, axiom, (! [A:$tType,A7:(set @ A),A2:A,B7:(set @ A)]: (((((minus_minus @ (set @ A)) @ A7) @ (((insert @ A) @ A2) @ B7)) = (((minus_minus @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A7) @ B7)) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A)))))))).
% 2.92/1.52 thf(fact_156_insert__Diff, axiom, (! [A:$tType,A2:A,A7:(set @ A)]: (((((member @ A) @ A2) @ A7) => ((((insert @ A) @ A2) @ (((minus_minus @ (set @ A)) @ A7) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))) = A7))))).
% 2.92/1.52 thf(fact_157_Diff__insert2, axiom, (! [A:$tType,A7:(set @ A),A2:A,B7:(set @ A)]: (((((minus_minus @ (set @ A)) @ A7) @ (((insert @ A) @ A2) @ B7)) = (((minus_minus @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A7) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))) @ B7))))).
% 2.92/1.52 thf(fact_158_Diff__insert__absorb, axiom, (! [A:$tType,X:A,A7:(set @ A)]: (((~ ((((member @ A) @ X) @ A7))) => ((((minus_minus @ (set @ A)) @ (((insert @ A) @ X) @ A7)) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A)))) = A7))))).
% 2.92/1.52 thf(fact_159_infinite__remove, axiom, (! [A:$tType,S2:(set @ A),A2:A]: (((~ (((finite_finite2 @ A) @ S2))) => (~ (((finite_finite2 @ A) @ (((minus_minus @ (set @ A)) @ S2) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))))))))).
% 2.92/1.52 thf(fact_160_infinite__coinduct, axiom, (! [A:$tType,X9:((set @ A) > $o),A7:(set @ A)]: (((X9 @ A7) => ((! [A9:(set @ A)]: (((X9 @ A9) => (? [X7:A]: (((((member @ A) @ X7) @ A9) & ((X9 @ (((minus_minus @ (set @ A)) @ A9) @ (((insert @ A) @ X7) @ (bot_bot @ (set @ A))))) | (~ (((finite_finite2 @ A) @ (((minus_minus @ (set @ A)) @ A9) @ (((insert @ A) @ X7) @ (bot_bot @ (set @ A)))))))))))))) => (~ (((finite_finite2 @ A) @ A7)))))))).
% 2.92/1.52 thf(fact_161_is__singletonI, axiom, (! [A:$tType,X:A]: (((is_singleton @ A) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A))))))).
% 2.92/1.52 thf(fact_162_remove__def, axiom, (! [A:$tType]: (((remove @ A) = (^ [X5:A,A8:(set @ A)]: ((((minus_minus @ (set @ A)) @ A8) @ (((insert @ A) @ X5) @ (bot_bot @ (set @ A)))))))))).
% 2.92/1.52 thf(fact_163_the__elem__eq, axiom, (! [A:$tType,X:A]: ((((the_elem @ A) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A)))) = X)))).
% 2.92/1.52 thf(fact_164_sum__set__simps_I1_J, axiom, (! [B:$tType,A:$tType,X:A]: (((((basic_setl @ A) @ B) @ (((sum_Inl @ A) @ B) @ X)) = (((insert @ A) @ X) @ (bot_bot @ (set @ A))))))).
% 2.92/1.52 thf(fact_165_member__remove, axiom, (! [A:$tType,X:A,Y:A,A7:(set @ A)]: (((((member @ A) @ X) @ (((remove @ A) @ Y) @ A7)) = ((((member @ A) @ X) @ A7) & (X != Y)))))).
% 2.92/1.52 thf(fact_166_sum__set__simps_I2_J, axiom, (! [A:$tType,C:$tType,X:A]: (((((basic_setl @ C) @ A) @ (((sum_Inr @ A) @ C) @ X)) = (bot_bot @ (set @ C)))))).
% 2.92/1.52 thf(fact_167_setl_Ocases, axiom, (! [B:$tType,A:$tType,A2:A,S:((sum_sum @ A) @ B)]: (((((member @ A) @ A2) @ (((basic_setl @ A) @ B) @ S)) => (S = (((sum_Inl @ A) @ B) @ A2)))))).
% 2.92/1.52 thf(fact_168_setl_Osimps, axiom, (! [B:$tType,A:$tType,A2:A,S:((sum_sum @ A) @ B)]: (((((member @ A) @ A2) @ (((basic_setl @ A) @ B) @ S)) = (? [X5:A]: (((A2 = X5) & (S = (((sum_Inl @ A) @ B) @ X5))))))))).
% 2.92/1.52 thf(fact_169_setl_Ointros, axiom, (! [B:$tType,A:$tType,S:((sum_sum @ A) @ B),X:A]: (((S = (((sum_Inl @ A) @ B) @ X)) => (((member @ A) @ X) @ (((basic_setl @ A) @ B) @ S)))))).
% 2.92/1.52 thf(fact_170_setl_Oinducts, axiom, (! [B:$tType,A:$tType,X:A,S:((sum_sum @ A) @ B),P:(A > $o)]: (((((member @ A) @ X) @ (((basic_setl @ A) @ B) @ S)) => ((! [X6:A]: (((S = (((sum_Inl @ A) @ B) @ X6)) => (P @ X6)))) => (P @ X)))))).
% 2.92/1.52 thf(fact_171_is__singleton__the__elem, axiom, (! [A:$tType]: (((is_singleton @ A) = (^ [A8:(set @ A)]: ((A8 = (((insert @ A) @ ((the_elem @ A) @ A8)) @ (bot_bot @ (set @ A)))))))))).
% 2.92/1.52 thf(fact_172_is__singletonI_H, axiom, (! [A:$tType,A7:(set @ A)]: (((A7 != (bot_bot @ (set @ A))) => ((! [X6:A,Y3:A]: (((((member @ A) @ X6) @ A7) => ((((member @ A) @ Y3) @ A7) => (X6 = Y3))))) => ((is_singleton @ A) @ A7)))))).
% 2.92/1.52 thf(fact_173_is__singleton__def, axiom, (! [A:$tType]: (((is_singleton @ A) = (^ [A8:(set @ A)]: ((? [X5:A]: ((A8 = (((insert @ A) @ X5) @ (bot_bot @ (set @ A)))))))))))).
% 2.92/1.52 thf(fact_174_is__singletonE, axiom, (! [A:$tType,A7:(set @ A)]: ((((is_singleton @ A) @ A7) => (~ ((! [X6:A]: ((A7 != (((insert @ A) @ X6) @ (bot_bot @ (set @ A)))))))))))).
% 2.92/1.52 thf(fact_175_sum__set__simps_I4_J, axiom, (! [E:$tType,A:$tType,X:A]: (((((basic_setr @ E) @ A) @ (((sum_Inr @ A) @ E) @ X)) = (((insert @ A) @ X) @ (bot_bot @ (set @ A))))))).
% 2.92/1.52 thf(fact_176_Set_Ois__empty__def, axiom, (! [A:$tType]: (((is_empty @ A) = (^ [A8:(set @ A)]: ((A8 = (bot_bot @ (set @ A))))))))).
% 2.92/1.52 thf(fact_177_remove__induct, axiom, (! [A:$tType,P:((set @ A) > $o),B7:(set @ A)]: (((P @ (bot_bot @ (set @ A))) => (((~ (((finite_finite2 @ A) @ B7))) => (P @ B7)) => ((! [A9:(set @ A)]: ((((finite_finite2 @ A) @ A9) => ((A9 != (bot_bot @ (set @ A))) => ((((ord_less_eq @ (set @ A)) @ A9) @ B7) => ((! [X7:A]: (((((member @ A) @ X7) @ A9) => (P @ (((minus_minus @ (set @ A)) @ A9) @ (((insert @ A) @ X7) @ (bot_bot @ (set @ A)))))))) => (P @ A9))))))) => (P @ B7))))))).
% 2.92/1.52 thf(fact_178_subset__antisym, axiom, (! [A:$tType,A7:(set @ A),B7:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A7) @ B7) => ((((ord_less_eq @ (set @ A)) @ B7) @ A7) => (A7 = B7)))))).
% 2.92/1.52 thf(fact_179_subsetI, axiom, (! [A:$tType,A7:(set @ A),B7:(set @ A)]: (((! [X6:A]: (((((member @ A) @ X6) @ A7) => (((member @ A) @ X6) @ B7)))) => (((ord_less_eq @ (set @ A)) @ A7) @ B7))))).
% 2.92/1.52 thf(fact_180_subset__empty, axiom, (! [A:$tType,A7:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A7) @ (bot_bot @ (set @ A))) = (A7 = (bot_bot @ (set @ A))))))).
% 2.92/1.52 thf(fact_181_empty__subsetI, axiom, (! [A:$tType,A7:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ (bot_bot @ (set @ A))) @ A7)))).
% 2.92/1.52 thf(fact_182_insert__subset, axiom, (! [A:$tType,X:A,A7:(set @ A),B7:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ (((insert @ A) @ X) @ A7)) @ B7) = ((((member @ A) @ X) @ B7) & (((ord_less_eq @ (set @ A)) @ A7) @ B7)))))).
% 2.92/1.52 thf(fact_183_singleton__insert__inj__eq, axiom, (! [A:$tType,B2:A,A2:A,A7:(set @ A)]: ((((((insert @ A) @ B2) @ (bot_bot @ (set @ A))) = (((insert @ A) @ A2) @ A7)) = ((A2 = B2) & (((ord_less_eq @ (set @ A)) @ A7) @ (((insert @ A) @ B2) @ (bot_bot @ (set @ A))))))))).
% 2.92/1.52 thf(fact_184_singleton__insert__inj__eq_H, axiom, (! [A:$tType,A2:A,A7:(set @ A),B2:A]: ((((((insert @ A) @ A2) @ A7) = (((insert @ A) @ B2) @ (bot_bot @ (set @ A)))) = ((A2 = B2) & (((ord_less_eq @ (set @ A)) @ A7) @ (((insert @ A) @ B2) @ (bot_bot @ (set @ A))))))))).
% 2.92/1.52 thf(fact_185_Diff__eq__empty__iff, axiom, (! [A:$tType,A7:(set @ A),B7:(set @ A)]: ((((((minus_minus @ (set @ A)) @ A7) @ B7) = (bot_bot @ (set @ A))) = (((ord_less_eq @ (set @ A)) @ A7) @ B7))))).
% 2.92/1.52 thf(fact_186_sum__set__simps_I3_J, axiom, (! [A:$tType,D:$tType,X:A]: (((((basic_setr @ A) @ D) @ (((sum_Inl @ A) @ D) @ X)) = (bot_bot @ (set @ D)))))).
% 2.92/1.52 thf(fact_187_inFr__mono, axiom, (! [Ns:(set @ n),Tr:dtree,T2:t,Ns4:(set @ n)]: (((((gram_L1333338417e_inFr @ Ns) @ Tr) @ T2) => ((((ord_less_eq @ (set @ n)) @ Ns) @ Ns4) => (((gram_L1333338417e_inFr @ Ns4) @ Tr) @ T2)))))).
% 2.92/1.52 thf(fact_188_setr_Oinducts, axiom, (! [A:$tType,B:$tType,X:B,S:((sum_sum @ A) @ B),P:(B > $o)]: (((((member @ B) @ X) @ (((basic_setr @ A) @ B) @ S)) => ((! [X6:B]: (((S = (((sum_Inr @ B) @ A) @ X6)) => (P @ X6)))) => (P @ X)))))).
% 2.92/1.52 thf(fact_189_setr_Ointros, axiom, (! [B:$tType,A:$tType,S:((sum_sum @ A) @ B),X:B]: (((S = (((sum_Inr @ B) @ A) @ X)) => (((member @ B) @ X) @ (((basic_setr @ A) @ B) @ S)))))).
% 2.92/1.52 thf(fact_190_setr_Osimps, axiom, (! [A:$tType,B:$tType,A2:B,S:((sum_sum @ A) @ B)]: (((((member @ B) @ A2) @ (((basic_setr @ A) @ B) @ S)) = (? [X5:B]: (((A2 = X5) & (S = (((sum_Inr @ B) @ A) @ X5))))))))).
% 2.92/1.52 thf(fact_191_setr_Ocases, axiom, (! [A:$tType,B:$tType,A2:B,S:((sum_sum @ A) @ B)]: (((((member @ B) @ A2) @ (((basic_setr @ A) @ B) @ S)) => (S = (((sum_Inr @ B) @ A) @ A2)))))).
% 2.92/1.52 thf(fact_192_finite__subset, axiom, (! [A:$tType,A7:(set @ A),B7:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A7) @ B7) => (((finite_finite2 @ A) @ B7) => ((finite_finite2 @ A) @ A7)))))).
% 2.92/1.52 thf(fact_193_infinite__super, axiom, (! [A:$tType,S2:(set @ A),T5:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ S2) @ T5) => ((~ (((finite_finite2 @ A) @ S2))) => (~ (((finite_finite2 @ A) @ T5)))))))).
% 2.92/1.52 thf(fact_194_rev__finite__subset, axiom, (! [A:$tType,B7:(set @ A),A7:(set @ A)]: ((((finite_finite2 @ A) @ B7) => ((((ord_less_eq @ (set @ A)) @ A7) @ B7) => ((finite_finite2 @ A) @ A7)))))).
% 2.92/1.52 thf(fact_195_insert__mono, axiom, (! [A:$tType,C4:(set @ A),D3:(set @ A),A2:A]: (((((ord_less_eq @ (set @ A)) @ C4) @ D3) => (((ord_less_eq @ (set @ A)) @ (((insert @ A) @ A2) @ C4)) @ (((insert @ A) @ A2) @ D3)))))).
% 2.92/1.52 thf(fact_196_subset__insert, axiom, (! [A:$tType,X:A,A7:(set @ A),B7:(set @ A)]: (((~ ((((member @ A) @ X) @ A7))) => ((((ord_less_eq @ (set @ A)) @ A7) @ (((insert @ A) @ X) @ B7)) = (((ord_less_eq @ (set @ A)) @ A7) @ B7)))))).
% 2.92/1.52 thf(fact_197_subset__insertI, axiom, (! [A:$tType,B7:(set @ A),A2:A]: ((((ord_less_eq @ (set @ A)) @ B7) @ (((insert @ A) @ A2) @ B7))))).
% 2.92/1.52 thf(fact_198_subset__insertI2, axiom, (! [A:$tType,A7:(set @ A),B7:(set @ A),B2:A]: (((((ord_less_eq @ (set @ A)) @ A7) @ B7) => (((ord_less_eq @ (set @ A)) @ A7) @ (((insert @ A) @ B2) @ B7)))))).
% 2.92/1.52 thf(fact_199_Diff__mono, axiom, (! [A:$tType,A7:(set @ A),C4:(set @ A),D3:(set @ A),B7:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A7) @ C4) => ((((ord_less_eq @ (set @ A)) @ D3) @ B7) => (((ord_less_eq @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A7) @ B7)) @ (((minus_minus @ (set @ A)) @ C4) @ D3))))))).
% 2.92/1.52 thf(fact_200_Diff__subset, axiom, (! [A:$tType,A7:(set @ A),B7:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A7) @ B7)) @ A7)))).
% 2.92/1.52 thf(fact_201_double__diff, axiom, (! [A:$tType,A7:(set @ A),B7:(set @ A),C4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A7) @ B7) => ((((ord_less_eq @ (set @ A)) @ B7) @ C4) => ((((minus_minus @ (set @ A)) @ B7) @ (((minus_minus @ (set @ A)) @ C4) @ A7)) = A7)))))).
% 2.92/1.52 thf(fact_202_Collect__mono__iff, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)) = (! [X5:A]: (((P @ X5) => (Q @ X5)))))))).
% 2.92/1.52 thf(fact_203_contra__subsetD, axiom, (! [A:$tType,A7:(set @ A),B7:(set @ A),C3:A]: (((((ord_less_eq @ (set @ A)) @ A7) @ B7) => ((~ ((((member @ A) @ C3) @ B7))) => (~ ((((member @ A) @ C3) @ A7)))))))).
% 2.92/1.52 thf(fact_204_set__eq__subset, axiom, (! [A:$tType]: (((^ [Y4:(set @ A),Z:(set @ A)]: ((Y4 = Z))) = (^ [A8:(set @ A),B9:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A8) @ B9) & (((ord_less_eq @ (set @ A)) @ B9) @ A8)))))))).
% 2.92/1.52 thf(fact_205_subset__trans, axiom, (! [A:$tType,A7:(set @ A),B7:(set @ A),C4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A7) @ B7) => ((((ord_less_eq @ (set @ A)) @ B7) @ C4) => (((ord_less_eq @ (set @ A)) @ A7) @ C4)))))).
% 2.92/1.52 thf(fact_206_Collect__mono, axiom, (! [A:$tType,P:(A > $o),Q:(A > $o)]: (((! [X6:A]: (((P @ X6) => (Q @ X6)))) => (((ord_less_eq @ (set @ A)) @ ((collect @ A) @ P)) @ ((collect @ A) @ Q)))))).
% 2.92/1.52 thf(fact_207_subset__refl, axiom, (! [A:$tType,A7:(set @ A)]: ((((ord_less_eq @ (set @ A)) @ A7) @ A7)))).
% 2.92/1.52 thf(fact_208_rev__subsetD, axiom, (! [A:$tType,C3:A,A7:(set @ A),B7:(set @ A)]: (((((member @ A) @ C3) @ A7) => ((((ord_less_eq @ (set @ A)) @ A7) @ B7) => (((member @ A) @ C3) @ B7)))))).
% 2.92/1.52 thf(fact_209_subset__iff, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A8:(set @ A),B9:(set @ A)]: ((! [T3:A]: (((((member @ A) @ T3) @ A8) => (((member @ A) @ T3) @ B9)))))))))).
% 2.92/1.52 thf(fact_210_set__rev__mp, axiom, (! [A:$tType,X:A,A7:(set @ A),B7:(set @ A)]: (((((member @ A) @ X) @ A7) => ((((ord_less_eq @ (set @ A)) @ A7) @ B7) => (((member @ A) @ X) @ B7)))))).
% 2.92/1.52 thf(fact_211_equalityD2, axiom, (! [A:$tType,A7:(set @ A),B7:(set @ A)]: (((A7 = B7) => (((ord_less_eq @ (set @ A)) @ B7) @ A7))))).
% 2.92/1.52 thf(fact_212_equalityD1, axiom, (! [A:$tType,A7:(set @ A),B7:(set @ A)]: (((A7 = B7) => (((ord_less_eq @ (set @ A)) @ A7) @ B7))))).
% 2.92/1.52 thf(fact_213_subset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A8:(set @ A),B9:(set @ A)]: ((! [X5:A]: (((((member @ A) @ X5) @ A8) => (((member @ A) @ X5) @ B9)))))))))).
% 2.92/1.52 thf(fact_214_equalityE, axiom, (! [A:$tType,A7:(set @ A),B7:(set @ A)]: (((A7 = B7) => (~ (((((ord_less_eq @ (set @ A)) @ A7) @ B7) => (~ ((((ord_less_eq @ (set @ A)) @ B7) @ A7)))))))))).
% 2.92/1.52 thf(fact_215_subsetCE, axiom, (! [A:$tType,A7:(set @ A),B7:(set @ A),C3:A]: (((((ord_less_eq @ (set @ A)) @ A7) @ B7) => ((((member @ A) @ C3) @ A7) => (((member @ A) @ C3) @ B7)))))).
% 2.92/1.52 thf(fact_216_subsetD, axiom, (! [A:$tType,A7:(set @ A),B7:(set @ A),C3:A]: (((((ord_less_eq @ (set @ A)) @ A7) @ B7) => ((((member @ A) @ C3) @ A7) => (((member @ A) @ C3) @ B7)))))).
% 2.92/1.52 thf(fact_217_in__mono, axiom, (! [A:$tType,A7:(set @ A),B7:(set @ A),X:A]: (((((ord_less_eq @ (set @ A)) @ A7) @ B7) => ((((member @ A) @ X) @ A7) => (((member @ A) @ X) @ B7)))))).
% 2.92/1.52 thf(fact_218_set__mp, axiom, (! [A:$tType,A7:(set @ A),B7:(set @ A),X:A]: (((((ord_less_eq @ (set @ A)) @ A7) @ B7) => ((((member @ A) @ X) @ A7) => (((member @ A) @ X) @ B7)))))).
% 2.92/1.52 thf(fact_219_subset__singletonD, axiom, (! [A:$tType,A7:(set @ A),X:A]: (((((ord_less_eq @ (set @ A)) @ A7) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A)))) => ((A7 = (bot_bot @ (set @ A))) | (A7 = (((insert @ A) @ X) @ (bot_bot @ (set @ A))))))))).
% 2.92/1.52 thf(fact_220_subset__singleton__iff, axiom, (! [A:$tType,X9:(set @ A),A2:A]: (((((ord_less_eq @ (set @ A)) @ X9) @ (((insert @ A) @ A2) @ (bot_bot @ (set @ A)))) = ((X9 = (bot_bot @ (set @ A))) | (X9 = (((insert @ A) @ A2) @ (bot_bot @ (set @ A))))))))).
% 2.92/1.52 thf(fact_221_subset__Diff__insert, axiom, (! [A:$tType,A7:(set @ A),B7:(set @ A),X:A,C4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A7) @ (((minus_minus @ (set @ A)) @ B7) @ (((insert @ A) @ X) @ C4))) = ((((ord_less_eq @ (set @ A)) @ A7) @ (((minus_minus @ (set @ A)) @ B7) @ C4)) & (~ ((((member @ A) @ X) @ A7)))))))).
% 2.92/1.52 thf(fact_222_finite__subset__induct, axiom, (! [A:$tType,F7:(set @ A),A7:(set @ A),P:((set @ A) > $o)]: ((((finite_finite2 @ A) @ F7) => ((((ord_less_eq @ (set @ A)) @ F7) @ A7) => ((P @ (bot_bot @ (set @ A))) => ((! [A4:A,F6:(set @ A)]: ((((finite_finite2 @ A) @ F6) => ((((member @ A) @ A4) @ A7) => ((~ ((((member @ A) @ A4) @ F6))) => ((P @ F6) => (P @ (((insert @ A) @ A4) @ F6)))))))) => (P @ F7)))))))).
% 2.92/1.52 thf(fact_223_subset__insert__iff, axiom, (! [A:$tType,A7:(set @ A),X:A,B7:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A7) @ (((insert @ A) @ X) @ B7)) = (((((member @ A) @ X) @ A7) => (((ord_less_eq @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A7) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A))))) @ B7)) & ((~ ((((member @ A) @ X) @ A7))) => (((ord_less_eq @ (set @ A)) @ A7) @ B7))))))).
% 2.92/1.52 thf(fact_224_Diff__single__insert, axiom, (! [A:$tType,A7:(set @ A),X:A,B7:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A7) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A))))) @ B7) => (((ord_less_eq @ (set @ A)) @ A7) @ (((insert @ A) @ X) @ B7)))))).
% 2.92/1.52 thf(fact_225_finite__remove__induct, axiom, (! [A:$tType,B7:(set @ A),P:((set @ A) > $o)]: ((((finite_finite2 @ A) @ B7) => ((P @ (bot_bot @ (set @ A))) => ((! [A9:(set @ A)]: ((((finite_finite2 @ A) @ A9) => ((A9 != (bot_bot @ (set @ A))) => ((((ord_less_eq @ (set @ A)) @ A9) @ B7) => ((! [X7:A]: (((((member @ A) @ X7) @ A9) => (P @ (((minus_minus @ (set @ A)) @ A9) @ (((insert @ A) @ X7) @ (bot_bot @ (set @ A)))))))) => (P @ A9))))))) => (P @ B7))))))).
% 2.92/1.52 thf(fact_226_insert__subsetI, axiom, (! [A:$tType,X:A,A7:(set @ A),X9:(set @ A)]: (((((member @ A) @ X) @ A7) => ((((ord_less_eq @ (set @ A)) @ X9) @ A7) => (((ord_less_eq @ (set @ A)) @ (((insert @ A) @ X) @ X9)) @ A7)))))).
% 2.92/1.52 thf(fact_227_diff__mono, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [A2:A,B2:A,D2:A,C3:A]: (((((ord_less_eq @ A) @ A2) @ B2) => ((((ord_less_eq @ A) @ D2) @ C3) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ A2) @ C3)) @ (((minus_minus @ A) @ B2) @ D2)))))))))).
% 2.92/1.52 thf(fact_228_diff__left__mono, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [B2:A,A2:A,C3:A]: (((((ord_less_eq @ A) @ B2) @ A2) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ C3) @ A2)) @ (((minus_minus @ A) @ C3) @ B2))))))))).
% 2.92/1.52 thf(fact_229_diff__right__commute, axiom, (! [A:$tType]: ((((cancel146912293up_add @ A) @ (type2 @ A)) => (! [A2:A,C3:A,B2:A]: (((((minus_minus @ A) @ (((minus_minus @ A) @ A2) @ C3)) @ B2) = (((minus_minus @ A) @ (((minus_minus @ A) @ A2) @ B2)) @ C3)))))))).
% 2.92/1.52 thf(fact_230_diff__eq__diff__eq, axiom, (! [A:$tType]: ((((group_add @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C3:A,D2:A]: ((((((minus_minus @ A) @ A2) @ B2) = (((minus_minus @ A) @ C3) @ D2)) => ((A2 = B2) = (C3 = D2))))))))).
% 2.92/1.52 thf(fact_231_diff__eq__diff__less__eq, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C3:A,D2:A]: ((((((minus_minus @ A) @ A2) @ B2) = (((minus_minus @ A) @ C3) @ D2)) => ((((ord_less_eq @ A) @ A2) @ B2) = (((ord_less_eq @ A) @ C3) @ D2))))))))).
% 2.92/1.52 thf(fact_232_diff__right__mono, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C3:A]: (((((ord_less_eq @ A) @ A2) @ B2) => (((ord_less_eq @ A) @ (((minus_minus @ A) @ A2) @ C3)) @ (((minus_minus @ A) @ B2) @ C3))))))))).
% 2.92/1.52 thf(fact_233_psubset__insert__iff, axiom, (! [A:$tType,A7:(set @ A),X:A,B7:(set @ A)]: (((((ord_less @ (set @ A)) @ A7) @ (((insert @ A) @ X) @ B7)) = (((((member @ A) @ X) @ B7) => (((ord_less @ (set @ A)) @ A7) @ B7)) & ((~ ((((member @ A) @ X) @ B7))) => (((((member @ A) @ X) @ A7) => (((ord_less @ (set @ A)) @ (((minus_minus @ (set @ A)) @ A7) @ (((insert @ A) @ X) @ (bot_bot @ (set @ A))))) @ B7)) & ((~ ((((member @ A) @ X) @ A7))) => (((ord_less_eq @ (set @ A)) @ A7) @ B7))))))))).
% 2.92/1.52 thf(fact_234_finite__induct__select, axiom, (! [A:$tType,S2:(set @ A),P:((set @ A) > $o)]: ((((finite_finite2 @ A) @ S2) => ((P @ (bot_bot @ (set @ A))) => ((! [T6:(set @ A)]: (((((ord_less @ (set @ A)) @ T6) @ S2) => ((P @ T6) => (? [X7:A]: (((((member @ A) @ X7) @ (((minus_minus @ (set @ A)) @ S2) @ T6)) & (P @ (((insert @ A) @ X7) @ T6))))))))) => (P @ S2))))))).
% 2.92/1.52 thf(fact_235_psubsetI, axiom, (! [A:$tType,A7:(set @ A),B7:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A7) @ B7) => ((A7 != B7) => (((ord_less @ (set @ A)) @ A7) @ B7)))))).
% 2.92/1.52 thf(fact_236_subset__iff__psubset__eq, axiom, (! [A:$tType]: (((ord_less_eq @ (set @ A)) = (^ [A8:(set @ A),B9:(set @ A)]: (((((ord_less @ (set @ A)) @ A8) @ B9) | (A8 = B9)))))))).
% 2.92/1.52 thf(fact_237_subset__psubset__trans, axiom, (! [A:$tType,A7:(set @ A),B7:(set @ A),C4:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A7) @ B7) => ((((ord_less @ (set @ A)) @ B7) @ C4) => (((ord_less @ (set @ A)) @ A7) @ C4)))))).
% 2.92/1.52 thf(fact_238_subset__not__subset__eq, axiom, (! [A:$tType]: (((ord_less @ (set @ A)) = (^ [A8:(set @ A),B9:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A8) @ B9) & (~ ((((ord_less_eq @ (set @ A)) @ B9) @ A8)))))))))).
% 2.92/1.52 thf(fact_239_psubset__subset__trans, axiom, (! [A:$tType,A7:(set @ A),B7:(set @ A),C4:(set @ A)]: (((((ord_less @ (set @ A)) @ A7) @ B7) => ((((ord_less_eq @ (set @ A)) @ B7) @ C4) => (((ord_less @ (set @ A)) @ A7) @ C4)))))).
% 2.92/1.52 thf(fact_240_psubset__imp__subset, axiom, (! [A:$tType,A7:(set @ A),B7:(set @ A)]: (((((ord_less @ (set @ A)) @ A7) @ B7) => (((ord_less_eq @ (set @ A)) @ A7) @ B7))))).
% 2.92/1.52 thf(fact_241_psubset__eq, axiom, (! [A:$tType]: (((ord_less @ (set @ A)) = (^ [A8:(set @ A),B9:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A8) @ B9) & (A8 != B9)))))))).
% 2.92/1.52 thf(fact_242_psubsetE, axiom, (! [A:$tType,A7:(set @ A),B7:(set @ A)]: (((((ord_less @ (set @ A)) @ A7) @ B7) => (~ (((((ord_less_eq @ (set @ A)) @ A7) @ B7) => (((ord_less_eq @ (set @ A)) @ B7) @ A7)))))))).
% 2.92/1.52 thf(fact_243_diff__strict__right__mono, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C3:A]: (((((ord_less @ A) @ A2) @ B2) => (((ord_less @ A) @ (((minus_minus @ A) @ A2) @ C3)) @ (((minus_minus @ A) @ B2) @ C3))))))))).
% 2.92/1.52 thf(fact_244_diff__strict__left__mono, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [B2:A,A2:A,C3:A]: (((((ord_less @ A) @ B2) @ A2) => (((ord_less @ A) @ (((minus_minus @ A) @ C3) @ A2)) @ (((minus_minus @ A) @ C3) @ B2))))))))).
% 2.92/1.52 thf(fact_245_diff__eq__diff__less, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [A2:A,B2:A,C3:A,D2:A]: ((((((minus_minus @ A) @ A2) @ B2) = (((minus_minus @ A) @ C3) @ D2)) => ((((ord_less @ A) @ A2) @ B2) = (((ord_less @ A) @ C3) @ D2))))))))).
% 2.92/1.52 thf(fact_246_diff__strict__mono, axiom, (! [A:$tType]: ((((ordered_ab_group_add @ A) @ (type2 @ A)) => (! [A2:A,B2:A,D2:A,C3:A]: (((((ord_less @ A) @ A2) @ B2) => ((((ord_less @ A) @ D2) @ C3) => (((ord_less @ A) @ (((minus_minus @ A) @ A2) @ C3)) @ (((minus_minus @ A) @ B2) @ D2)))))))))).
% 2.92/1.52 thf(fact_247_not__psubset__empty, axiom, (! [A:$tType,A7:(set @ A)]: ((~ ((((ord_less @ (set @ A)) @ A7) @ (bot_bot @ (set @ A)))))))).
% 2.92/1.52 thf(fact_248_finite__psubset__induct, axiom, (! [A:$tType,A7:(set @ A),P:((set @ A) > $o)]: ((((finite_finite2 @ A) @ A7) => ((! [A9:(set @ A)]: ((((finite_finite2 @ A) @ A9) => ((! [B11:(set @ A)]: (((((ord_less @ (set @ A)) @ B11) @ A9) => (P @ B11)))) => (P @ A9))))) => (P @ A7)))))).
% 2.92/1.52 thf(fact_249_psubset__imp__ex__mem, axiom, (! [A:$tType,A7:(set @ A),B7:(set @ A)]: (((((ord_less @ (set @ A)) @ A7) @ B7) => (? [B4:A]: ((((member @ A) @ B4) @ (((minus_minus @ (set @ A)) @ B7) @ A7)))))))).
% 2.92/1.52 thf(fact_250_psubset__trans, axiom, (! [A:$tType,A7:(set @ A),B7:(set @ A),C4:(set @ A)]: (((((ord_less @ (set @ A)) @ A7) @ B7) => ((((ord_less @ (set @ A)) @ B7) @ C4) => (((ord_less @ (set @ A)) @ A7) @ C4)))))).
% 2.92/1.52 thf(fact_251_psubsetD, axiom, (! [A:$tType,A7:(set @ A),B7:(set @ A),C3:A]: (((((ord_less @ (set @ A)) @ A7) @ B7) => ((((member @ A) @ C3) @ A7) => (((member @ A) @ C3) @ B7)))))).
% 2.92/1.52 thf(fact_252_finite__linorder__min__induct, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [A7:(set @ A),P:((set @ A) > $o)]: ((((finite_finite2 @ A) @ A7) => ((P @ (bot_bot @ (set @ A))) => ((! [B4:A,A9:(set @ A)]: ((((finite_finite2 @ A) @ A9) => ((! [X7:A]: (((((member @ A) @ X7) @ A9) => (((ord_less @ A) @ B4) @ X7)))) => ((P @ A9) => (P @ (((insert @ A) @ B4) @ A9))))))) => (P @ A7)))))))))).
% 2.92/1.52 thf(fact_253_finite__linorder__max__induct, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [A7:(set @ A),P:((set @ A) > $o)]: ((((finite_finite2 @ A) @ A7) => ((P @ (bot_bot @ (set @ A))) => ((! [B4:A,A9:(set @ A)]: ((((finite_finite2 @ A) @ A9) => ((! [X7:A]: (((((member @ A) @ X7) @ A9) => (((ord_less @ A) @ X7) @ B4)))) => ((P @ A9) => (P @ (((insert @ A) @ B4) @ A9))))))) => (P @ A7)))))))))).
% 2.92/1.52 thf(fact_254_infinite__growing, axiom, (! [A:$tType]: ((((linorder @ A) @ (type2 @ A)) => (! [X9:(set @ A)]: (((X9 != (bot_bot @ (set @ A))) => ((! [X6:A]: (((((member @ A) @ X6) @ X9) => (? [Xa:A]: (((((member @ A) @ Xa) @ X9) & (((ord_less @ A) @ X6) @ Xa))))))) => (~ (((finite_finite2 @ A) @ X9))))))))))).
% 2.92/1.52 thf(fact_255_Sup__fin_Oantimono, axiom, (! [A:$tType]: ((((semilattice_sup @ A) @ (type2 @ A)) => (! [A7:(set @ A),B7:(set @ A)]: (((((ord_less_eq @ (set @ A)) @ A7) @ B7) => ((A7 != (bot_bot @ (set @ A))) => (((finite_finite2 @ A) @ B7) => (((ord_less_eq @ A) @ ((lattic477160up_fin @ A) @ A7)) @ ((lattic477160up_fin @ A) @ B7))))))))))).
% 2.92/1.52 thf(tcon_fun___Lattices_Osemilattice__sup, axiom, (! [A5:$tType,A11:$tType]: ((((semilattice_sup @ A11) @ (type2 @ A11)) => ((semilattice_sup @ (A5 > A11)) @ (type2 @ (A5 > A11))))))).
% 2.92/1.52 thf(tcon_fun___Finite__Set_Ofinite, axiom, (! [A5:$tType,A11:$tType]: (((((finite_finite @ A5) @ (type2 @ A5)) & ((finite_finite @ A11) @ (type2 @ A11))) => ((finite_finite @ (A5 > A11)) @ (type2 @ (A5 > A11))))))).
% 2.92/1.52 thf(tcon_fun___Groups_Ominus, axiom, (! [A5:$tType,A11:$tType]: ((((minus @ A11) @ (type2 @ A11)) => ((minus @ (A5 > A11)) @ (type2 @ (A5 > A11))))))).
% 2.92/1.52 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_1, axiom, (! [A5:$tType]: (((semilattice_sup @ (set @ A5)) @ (type2 @ (set @ A5)))))).
% 2.92/1.52 thf(tcon_Set_Oset___Finite__Set_Ofinite_2, axiom, (! [A5:$tType]: ((((finite_finite @ A5) @ (type2 @ A5)) => ((finite_finite @ (set @ A5)) @ (type2 @ (set @ A5))))))).
% 2.92/1.52 thf(tcon_Set_Oset___Groups_Ominus_3, axiom, (! [A5:$tType]: (((minus @ (set @ A5)) @ (type2 @ (set @ A5)))))).
% 2.92/1.52 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_4, axiom, ((semilattice_sup @ $o) @ (type2 @ $o))).
% 2.92/1.52 thf(tcon_HOL_Obool___Orderings_Olinorder, axiom, ((linorder @ $o) @ (type2 @ $o))).
% 2.92/1.52 thf(tcon_HOL_Obool___Finite__Set_Ofinite_5, axiom, ((finite_finite @ $o) @ (type2 @ $o))).
% 2.92/1.52 thf(tcon_HOL_Obool___Groups_Ominus_6, axiom, ((minus @ $o) @ (type2 @ $o))).
% 2.92/1.52 thf(tcon_FSet_Ofset___Lattices_Osemilattice__sup_7, axiom, (! [A5:$tType]: (((semilattice_sup @ (fset @ A5)) @ (type2 @ (fset @ A5)))))).
% 2.92/1.52 thf(tcon_FSet_Ofset___Finite__Set_Ofinite_8, axiom, (! [A5:$tType]: ((((finite_finite @ A5) @ (type2 @ A5)) => ((finite_finite @ (fset @ A5)) @ (type2 @ (fset @ A5))))))).
% 2.92/1.52 thf(tcon_FSet_Ofset___Groups_Ominus_9, axiom, (! [A5:$tType]: (((minus @ (fset @ A5)) @ (type2 @ (fset @ A5)))))).
% 2.92/1.52 thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_10, axiom, (! [A5:$tType,A11:$tType]: (((((finite_finite @ A5) @ (type2 @ A5)) & ((finite_finite @ A11) @ (type2 @ A11))) => ((finite_finite @ ((sum_sum @ A5) @ A11)) @ (type2 @ ((sum_sum @ A5) @ A11))))))).
% 2.92/1.52 thf(help_If_3_1_T, axiom, (! [P:$o]: (((P = $true) | (P = $false))))).
% 2.92/1.52 thf(help_If_2_1_T, axiom, (! [A:$tType,X:A,Y:A]: ((((((if @ A) @ $false) @ X) @ Y) = Y)))).
% 2.92/1.52 thf(help_If_1_1_T, axiom, (! [A:$tType,X:A,Y:A]: ((((((if @ A) @ $true) @ X) @ Y) = X)))).
% 2.92/1.52 thf(conj_0, hypothesis, (((member @ n) @ (root @ tra)) @ nsa)).
% 2.92/1.52 thf(conj_1, hypothesis, (((member @ ((sum_sum @ t) @ dtree)) @ (((sum_Inl @ t) @ dtree) @ ta)) @ (cont @ tra))).
% 2.92/1.52 thf(conj_2, conjecture, (((member @ n) @ (root @ tra)) @ nsa)).
% 2.92/1.52 % SZS output end ListOfTHF for /export/starexec/sandbox2/tmp/tmp.Nuu3MkGpYK/DTF2DTF26903.p
% 2.92/1.52 ------------------------
% 2.92/1.57 ---- Cleaned THF ---
% 2.92/1.57 thf(ty_t_Sum__Type_Osum,type,
% 2.92/1.57 sum_sum: $tType > $tType > $tType ).
% 2.92/1.57
% 2.92/1.57 thf(ty_t_DTree_Odtree,type,
% 2.92/1.57 dtree: $tType ).
% 2.92/1.57
% 2.92/1.57 thf(ty_t_FSet_Ofset,type,
% 2.92/1.57 fset: $tType > $tType ).
% 2.92/1.57
% 2.92/1.57 thf(ty_t_Set_Oset,type,
% 2.92/1.57 set: $tType > $tType ).
% 2.92/1.57
% 2.92/1.57 thf(ty_t_DTree_OT,type,
% 2.92/1.57 t: $tType ).
% 2.92/1.57
% 2.92/1.57 thf(ty_t_DTree_ON,type,
% 2.92/1.57 n: $tType ).
% 2.92/1.57
% 2.92/1.57 thf(ty_t_itself,type,
% 2.92/1.57 itself: $tType > $tType ).
% 2.92/1.57
% 2.92/1.57 thf(sy_cl_HOL_Otype,type,
% 2.92/1.57 type:
% 2.92/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.92/1.57
% 2.92/1.57 thf(sy_cl_Groups_Ominus,type,
% 2.92/1.57 minus:
% 2.92/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.92/1.57
% 2.92/1.57 thf(sy_cl_Groups_Ogroup__add,type,
% 2.92/1.57 group_add:
% 2.92/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.92/1.57
% 2.92/1.57 thf(sy_cl_Finite__Set_Ofinite,type,
% 2.92/1.57 finite_finite:
% 2.92/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.92/1.57
% 2.92/1.57 thf(sy_cl_Orderings_Olinorder,type,
% 2.92/1.57 linorder:
% 2.92/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.92/1.57
% 2.92/1.57 thf(sy_cl_Lattices_Osemilattice__sup,type,
% 2.92/1.57 semilattice_sup:
% 2.92/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.92/1.57
% 2.92/1.57 thf(sy_cl_Groups_Oordered__ab__group__add,type,
% 2.92/1.57 ordered_ab_group_add:
% 2.92/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.92/1.57
% 2.92/1.57 thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
% 2.92/1.57 cancel146912293up_add:
% 2.92/1.57 !>[A: $tType] : ( ( itself @ A ) > $o ) ).
% 2.92/1.57
% 2.92/1.57 thf(sy_c_Basic__BNFs_Osetl,type,
% 2.92/1.57 basic_setl:
% 2.92/1.57 !>[A: $tType,B: $tType] : ( ( sum_sum @ A @ B ) > ( set @ A ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(sy_c_Basic__BNFs_Osetr,type,
% 2.92/1.57 basic_setr:
% 2.92/1.57 !>[A: $tType,B: $tType] : ( ( sum_sum @ A @ B ) > ( set @ B ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(sy_c_DTree_ONode,type,
% 2.92/1.57 node: n > ( set @ ( sum_sum @ t @ dtree ) ) > dtree ).
% 2.92/1.57
% 2.92/1.57 thf(sy_c_DTree_Ocont,type,
% 2.92/1.57 cont: dtree > ( set @ ( sum_sum @ t @ dtree ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(sy_c_DTree_Ocorec,type,
% 2.92/1.57 corec:
% 2.92/1.57 !>[A: $tType] : ( ( A > n ) > ( A > ( set @ ( sum_sum @ t @ ( sum_sum @ dtree @ A ) ) ) ) > A > dtree ) ).
% 2.92/1.57
% 2.92/1.57 thf(sy_c_DTree_Odtree_ONNode,type,
% 2.92/1.57 nNode: n > ( fset @ ( sum_sum @ t @ dtree ) ) > dtree ).
% 2.92/1.57
% 2.92/1.57 thf(sy_c_DTree_Odtree_Ocase__dtree,type,
% 2.92/1.57 case_dtree:
% 2.92/1.57 !>[A: $tType] : ( ( n > ( fset @ ( sum_sum @ t @ dtree ) ) > A ) > dtree > A ) ).
% 2.92/1.57
% 2.92/1.57 thf(sy_c_DTree_Odtree_Occont,type,
% 2.92/1.57 ccont: dtree > ( fset @ ( sum_sum @ t @ dtree ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(sy_c_DTree_Odtree_Oroot,type,
% 2.92/1.57 root: dtree > n ).
% 2.92/1.57
% 2.92/1.57 thf(sy_c_DTree_Ounfold,type,
% 2.92/1.57 unfold:
% 2.92/1.57 !>[A: $tType] : ( ( A > n ) > ( A > ( set @ ( sum_sum @ t @ A ) ) ) > A > dtree ) ).
% 2.92/1.57
% 2.92/1.57 thf(sy_c_Finite__Set_Ofinite,type,
% 2.92/1.57 finite_finite2:
% 2.92/1.57 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 2.92/1.57
% 2.92/1.57 thf(sy_c_Fun_Ocomp,type,
% 2.92/1.57 comp:
% 2.92/1.57 !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
% 2.92/1.57
% 2.92/1.57 thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_OinFr,type,
% 2.92/1.57 gram_L1333338417e_inFr: ( set @ n ) > dtree > t > $o ).
% 2.92/1.57
% 2.92/1.57 thf(sy_c_Gram__Lang__Mirabelle__ojxrtuoybn_OinFr2,type,
% 2.92/1.57 gram_L805317441_inFr2: ( set @ n ) > dtree > t > $o ).
% 2.92/1.57
% 2.92/1.57 thf(sy_c_Groups_Ominus__class_Ominus,type,
% 2.92/1.57 minus_minus:
% 2.92/1.57 !>[A: $tType] : ( A > A > A ) ).
% 2.92/1.57
% 2.92/1.57 thf(sy_c_If,type,
% 2.92/1.57 if:
% 2.92/1.57 !>[A: $tType] : ( $o > A > A > A ) ).
% 2.92/1.57
% 2.92/1.57 thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin,type,
% 2.92/1.57 lattic477160up_fin:
% 2.92/1.57 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 2.92/1.57
% 2.92/1.57 thf(sy_c_Orderings_Obot__class_Obot,type,
% 2.92/1.57 bot_bot:
% 2.92/1.57 !>[A: $tType] : A ).
% 2.92/1.57
% 2.92/1.57 thf(sy_c_Orderings_Oord__class_Oless,type,
% 2.92/1.57 ord_less:
% 2.92/1.57 !>[A: $tType] : ( A > A > $o ) ).
% 2.92/1.57
% 2.92/1.57 thf(sy_c_Orderings_Oord__class_Oless__eq,type,
% 2.92/1.57 ord_less_eq:
% 2.92/1.57 !>[A: $tType] : ( A > A > $o ) ).
% 2.92/1.57
% 2.92/1.57 thf(sy_c_Pure_Otype,type,
% 2.92/1.57 type2:
% 2.92/1.57 !>[A: $tType] : ( itself @ A ) ).
% 2.92/1.57
% 2.92/1.57 thf(sy_c_Set_OCollect,type,
% 2.92/1.57 collect:
% 2.92/1.57 !>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(sy_c_Set_Oinsert,type,
% 2.92/1.57 insert:
% 2.92/1.57 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(sy_c_Set_Ois__empty,type,
% 2.92/1.57 is_empty:
% 2.92/1.57 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 2.92/1.57
% 2.92/1.57 thf(sy_c_Set_Ois__singleton,type,
% 2.92/1.57 is_singleton:
% 2.92/1.57 !>[A: $tType] : ( ( set @ A ) > $o ) ).
% 2.92/1.57
% 2.92/1.57 thf(sy_c_Set_Oremove,type,
% 2.92/1.57 remove:
% 2.92/1.57 !>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(sy_c_Set_Othe__elem,type,
% 2.92/1.57 the_elem:
% 2.92/1.57 !>[A: $tType] : ( ( set @ A ) > A ) ).
% 2.92/1.57
% 2.92/1.57 thf(sy_c_Sum__Type_OInl,type,
% 2.92/1.57 sum_Inl:
% 2.92/1.57 !>[A: $tType,B: $tType] : ( A > ( sum_sum @ A @ B ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(sy_c_Sum__Type_OInr,type,
% 2.92/1.57 sum_Inr:
% 2.92/1.57 !>[B: $tType,A: $tType] : ( B > ( sum_sum @ A @ B ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(sy_c_Sum__Type_OPlus,type,
% 2.92/1.57 sum_Plus:
% 2.92/1.57 !>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( sum_sum @ A @ B ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(sy_c_Sum__Type_OSuml,type,
% 2.92/1.57 sum_Suml:
% 2.92/1.57 !>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( sum_sum @ A @ B ) > C ) ).
% 2.92/1.57
% 2.92/1.57 thf(sy_c_Sum__Type_OSumr,type,
% 2.92/1.57 sum_Sumr:
% 2.92/1.57 !>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( sum_sum @ A @ B ) > C ) ).
% 2.92/1.57
% 2.92/1.57 thf(sy_c_Sum__Type_Oold_Osum_Orec__sum,type,
% 2.92/1.57 sum_rec_sum:
% 2.92/1.57 !>[A: $tType,T: $tType,B: $tType] : ( ( A > T ) > ( B > T ) > ( sum_sum @ A @ B ) > T ) ).
% 2.92/1.57
% 2.92/1.57 thf(sy_c_Sum__Type_Osum_Ocase__sum,type,
% 2.92/1.57 sum_case_sum:
% 2.92/1.57 !>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( B > C ) > ( sum_sum @ A @ B ) > C ) ).
% 2.92/1.57
% 2.92/1.57 thf(sy_c_member,type,
% 2.92/1.57 member:
% 2.92/1.57 !>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% 2.92/1.57
% 2.92/1.57 thf(sy_v_nsa,type,
% 2.92/1.57 nsa: set @ n ).
% 2.92/1.57
% 2.92/1.57 thf(sy_v_ta,type,
% 2.92/1.57 ta: t ).
% 2.92/1.57
% 2.92/1.57 thf(sy_v_tra,type,
% 2.92/1.57 tra: dtree ).
% 2.92/1.57
% 2.92/1.57 thf(fact_0_sum_Oinject_I1_J,axiom,
% 2.92/1.57 ! [B: $tType,A: $tType,X1: A,Y1: A] :
% 2.92/1.57 ( ( ( sum_Inl @ A @ B @ X1 )
% 2.92/1.57 = ( sum_Inl @ A @ B @ Y1 ) )
% 2.92/1.57 = ( X1 = Y1 ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_1_old_Osum_Oinject_I1_J,axiom,
% 2.92/1.57 ! [B: $tType,A: $tType,A2: A,A3: A] :
% 2.92/1.57 ( ( ( sum_Inl @ A @ B @ A2 )
% 2.92/1.57 = ( sum_Inl @ A @ B @ A3 ) )
% 2.92/1.57 = ( A2 = A3 ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_2_dtree__cong,axiom,
% 2.92/1.57 ! [Tr: dtree,Tr2: dtree] :
% 2.92/1.57 ( ( ( root @ Tr )
% 2.92/1.57 = ( root @ Tr2 ) )
% 2.92/1.57 => ( ( ( cont @ Tr )
% 2.92/1.57 = ( cont @ Tr2 ) )
% 2.92/1.57 => ( Tr = Tr2 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_3_inFr2_OBase,axiom,
% 2.92/1.57 ! [Tr: dtree,Ns: set @ n,T2: t] :
% 2.92/1.57 ( ( member @ n @ ( root @ Tr ) @ Ns )
% 2.92/1.57 => ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T2 ) @ ( cont @ Tr ) )
% 2.92/1.57 => ( gram_L805317441_inFr2 @ Ns @ Tr @ T2 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_4_inFr_OBase,axiom,
% 2.92/1.57 ! [Tr: dtree,Ns: set @ n,T2: t] :
% 2.92/1.57 ( ( member @ n @ ( root @ Tr ) @ Ns )
% 2.92/1.57 => ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T2 ) @ ( cont @ Tr ) )
% 2.92/1.57 => ( gram_L1333338417e_inFr @ Ns @ Tr @ T2 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_5_Inl__inject,axiom,
% 2.92/1.57 ! [B: $tType,A: $tType,X: A,Y: A] :
% 2.92/1.57 ( ( ( sum_Inl @ A @ B @ X )
% 2.92/1.57 = ( sum_Inl @ A @ B @ Y ) )
% 2.92/1.57 => ( X = Y ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_6_Node__root__cont,axiom,
% 2.92/1.57 ! [Tr: dtree] :
% 2.92/1.57 ( ( node @ ( root @ Tr ) @ ( cont @ Tr ) )
% 2.92/1.57 = Tr ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_7_inFr_Ocases,axiom,
% 2.92/1.57 ! [A1: set @ n,A22: dtree,A32: t] :
% 2.92/1.57 ( ( gram_L1333338417e_inFr @ A1 @ A22 @ A32 )
% 2.92/1.57 => ( ( ( member @ n @ ( root @ A22 ) @ A1 )
% 2.92/1.57 => ~ ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ A32 ) @ ( cont @ A22 ) ) )
% 2.92/1.57 => ~ ( ( member @ n @ ( root @ A22 ) @ A1 )
% 2.92/1.57 => ! [Tr1: dtree] :
% 2.92/1.57 ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr1 ) @ ( cont @ A22 ) )
% 2.92/1.57 => ~ ( gram_L1333338417e_inFr @ A1 @ Tr1 @ A32 ) ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_8_inFr_Osimps,axiom,
% 2.92/1.57 ( gram_L1333338417e_inFr
% 2.92/1.57 = ( ^ [A12: set @ n,A23: dtree,A33: t] :
% 2.92/1.57 ( ? [Tr3: dtree,Ns2: set @ n,T3: t] :
% 2.92/1.57 ( ( A12 = Ns2 )
% 2.92/1.57 & ( A23 = Tr3 )
% 2.92/1.57 & ( A33 = T3 )
% 2.92/1.57 & ( member @ n @ ( root @ Tr3 ) @ Ns2 )
% 2.92/1.57 & ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T3 ) @ ( cont @ Tr3 ) ) )
% 2.92/1.57 | ? [Tr3: dtree,Ns2: set @ n,Tr12: dtree,T3: t] :
% 2.92/1.57 ( ( A12 = Ns2 )
% 2.92/1.57 & ( A23 = Tr3 )
% 2.92/1.57 & ( A33 = T3 )
% 2.92/1.57 & ( member @ n @ ( root @ Tr3 ) @ Ns2 )
% 2.92/1.57 & ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr12 ) @ ( cont @ Tr3 ) )
% 2.92/1.57 & ( gram_L1333338417e_inFr @ Ns2 @ Tr12 @ T3 ) ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_9_inFr_Oinducts,axiom,
% 2.92/1.57 ! [X1: set @ n,X2: dtree,X3: t,P: ( set @ n ) > dtree > t > $o] :
% 2.92/1.57 ( ( gram_L1333338417e_inFr @ X1 @ X2 @ X3 )
% 2.92/1.57 => ( ! [Tr4: dtree,Ns3: set @ n,T4: t] :
% 2.92/1.57 ( ( member @ n @ ( root @ Tr4 ) @ Ns3 )
% 2.92/1.57 => ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T4 ) @ ( cont @ Tr4 ) )
% 2.92/1.57 => ( P @ Ns3 @ Tr4 @ T4 ) ) )
% 2.92/1.57 => ( ! [Tr4: dtree,Ns3: set @ n,Tr1: dtree,T4: t] :
% 2.92/1.57 ( ( member @ n @ ( root @ Tr4 ) @ Ns3 )
% 2.92/1.57 => ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr1 ) @ ( cont @ Tr4 ) )
% 2.92/1.57 => ( ( gram_L1333338417e_inFr @ Ns3 @ Tr1 @ T4 )
% 2.92/1.57 => ( ( P @ Ns3 @ Tr1 @ T4 )
% 2.92/1.57 => ( P @ Ns3 @ Tr4 @ T4 ) ) ) ) )
% 2.92/1.57 => ( P @ X1 @ X2 @ X3 ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_10_old_Osum_Osimps_I7_J,axiom,
% 2.92/1.57 ! [B: $tType,T: $tType,A: $tType,F1: A > T,F2: B > T,A2: A] :
% 2.92/1.57 ( ( sum_rec_sum @ A @ T @ B @ F1 @ F2 @ ( sum_Inl @ A @ B @ A2 ) )
% 2.92/1.57 = ( F1 @ A2 ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_11_inFr__root__in,axiom,
% 2.92/1.57 ! [Ns: set @ n,Tr: dtree,T2: t] :
% 2.92/1.57 ( ( gram_L1333338417e_inFr @ Ns @ Tr @ T2 )
% 2.92/1.57 => ( member @ n @ ( root @ Tr ) @ Ns ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_12_sum_Oinject_I2_J,axiom,
% 2.92/1.57 ! [A: $tType,B: $tType,X2: B,Y2: B] :
% 2.92/1.57 ( ( ( sum_Inr @ B @ A @ X2 )
% 2.92/1.57 = ( sum_Inr @ B @ A @ Y2 ) )
% 2.92/1.57 = ( X2 = Y2 ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_13_old_Osum_Oinject_I2_J,axiom,
% 2.92/1.57 ! [A: $tType,B: $tType,B2: B,B3: B] :
% 2.92/1.57 ( ( ( sum_Inr @ B @ A @ B2 )
% 2.92/1.57 = ( sum_Inr @ B @ A @ B3 ) )
% 2.92/1.57 = ( B2 = B3 ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_14_root__Node,axiom,
% 2.92/1.57 ! [N: n,As: set @ ( sum_sum @ t @ dtree )] :
% 2.92/1.57 ( ( root @ ( node @ N @ As ) )
% 2.92/1.57 = N ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_15_old_Osum_Osimps_I8_J,axiom,
% 2.92/1.57 ! [A: $tType,T: $tType,B: $tType,F1: A > T,F2: B > T,B2: B] :
% 2.92/1.57 ( ( sum_rec_sum @ A @ T @ B @ F1 @ F2 @ ( sum_Inr @ B @ A @ B2 ) )
% 2.92/1.57 = ( F2 @ B2 ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_16_Inr__inject,axiom,
% 2.92/1.57 ! [A: $tType,B: $tType,X: B,Y: B] :
% 2.92/1.57 ( ( ( sum_Inr @ B @ A @ X )
% 2.92/1.57 = ( sum_Inr @ B @ A @ Y ) )
% 2.92/1.57 => ( X = Y ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_17_old_Osum_Oinducts,axiom,
% 2.92/1.57 ! [B: $tType,A: $tType,P: ( sum_sum @ A @ B ) > $o,Sum: sum_sum @ A @ B] :
% 2.92/1.57 ( ! [A4: A] : ( P @ ( sum_Inl @ A @ B @ A4 ) )
% 2.92/1.57 => ( ! [B4: B] : ( P @ ( sum_Inr @ B @ A @ B4 ) )
% 2.92/1.57 => ( P @ Sum ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_18_old_Osum_Oexhaust,axiom,
% 2.92/1.57 ! [A: $tType,B: $tType,Y: sum_sum @ A @ B] :
% 2.92/1.57 ( ! [A4: A] :
% 2.92/1.57 ( Y
% 2.92/1.57 != ( sum_Inl @ A @ B @ A4 ) )
% 2.92/1.57 => ~ ! [B4: B] :
% 2.92/1.57 ( Y
% 2.92/1.57 != ( sum_Inr @ B @ A @ B4 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_19_split__sum__all,axiom,
% 2.92/1.57 ! [B: $tType,A: $tType] :
% 2.92/1.57 ( ( ^ [P2: ( sum_sum @ A @ B ) > $o] :
% 2.92/1.57 ! [X4: sum_sum @ A @ B] : ( P2 @ X4 ) )
% 2.92/1.57 = ( ^ [P3: ( sum_sum @ A @ B ) > $o] :
% 2.92/1.57 ( ! [X5: A] : ( P3 @ ( sum_Inl @ A @ B @ X5 ) )
% 2.92/1.57 & ! [X5: B] : ( P3 @ ( sum_Inr @ B @ A @ X5 ) ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_20_split__sum__ex,axiom,
% 2.92/1.57 ! [B: $tType,A: $tType] :
% 2.92/1.57 ( ( ^ [P2: ( sum_sum @ A @ B ) > $o] :
% 2.92/1.57 ? [X4: sum_sum @ A @ B] : ( P2 @ X4 ) )
% 2.92/1.57 = ( ^ [P3: ( sum_sum @ A @ B ) > $o] :
% 2.92/1.57 ( ? [X5: A] : ( P3 @ ( sum_Inl @ A @ B @ X5 ) )
% 2.92/1.57 | ? [X5: B] : ( P3 @ ( sum_Inr @ B @ A @ X5 ) ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_21_Inr__not__Inl,axiom,
% 2.92/1.57 ! [B: $tType,A: $tType,B2: B,A2: A] :
% 2.92/1.57 ( ( sum_Inr @ B @ A @ B2 )
% 2.92/1.57 != ( sum_Inl @ A @ B @ A2 ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_22_sumE,axiom,
% 2.92/1.57 ! [A: $tType,B: $tType,S: sum_sum @ A @ B] :
% 2.92/1.57 ( ! [X6: A] :
% 2.92/1.57 ( S
% 2.92/1.57 != ( sum_Inl @ A @ B @ X6 ) )
% 2.92/1.57 => ~ ! [Y3: B] :
% 2.92/1.57 ( S
% 2.92/1.57 != ( sum_Inr @ B @ A @ Y3 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_23_old_Osum_Odistinct_I1_J,axiom,
% 2.92/1.57 ! [A: $tType,B: $tType,A2: A,B3: B] :
% 2.92/1.57 ( ( sum_Inl @ A @ B @ A2 )
% 2.92/1.57 != ( sum_Inr @ B @ A @ B3 ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_24_old_Osum_Odistinct_I2_J,axiom,
% 2.92/1.57 ! [B5: $tType,A5: $tType,B6: B5,A6: A5] :
% 2.92/1.57 ( ( sum_Inr @ B5 @ A5 @ B6 )
% 2.92/1.57 != ( sum_Inl @ A5 @ B5 @ A6 ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_25_sum_Odistinct_I1_J,axiom,
% 2.92/1.57 ! [A: $tType,B: $tType,X1: A,X2: B] :
% 2.92/1.57 ( ( sum_Inl @ A @ B @ X1 )
% 2.92/1.57 != ( sum_Inr @ B @ A @ X2 ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_26_inFr_OInd,axiom,
% 2.92/1.57 ! [Tr: dtree,Ns: set @ n,Tr13: dtree,T2: t] :
% 2.92/1.57 ( ( member @ n @ ( root @ Tr ) @ Ns )
% 2.92/1.57 => ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr13 ) @ ( cont @ Tr ) )
% 2.92/1.57 => ( ( gram_L1333338417e_inFr @ Ns @ Tr13 @ T2 )
% 2.92/1.57 => ( gram_L1333338417e_inFr @ Ns @ Tr @ T2 ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_27_Inr__Inl__False,axiom,
% 2.92/1.57 ! [B: $tType,A: $tType,X: B,Y: A] :
% 2.92/1.57 ( ( sum_Inr @ B @ A @ X )
% 2.92/1.57 != ( sum_Inl @ A @ B @ Y ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_28_Inl__Inr__False,axiom,
% 2.92/1.57 ! [A: $tType,B: $tType,X: A,Y: B] :
% 2.92/1.57 ( ( sum_Inl @ A @ B @ X )
% 2.92/1.57 != ( sum_Inr @ B @ A @ Y ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_29_obj__sumE,axiom,
% 2.92/1.57 ! [A: $tType,B: $tType,S: sum_sum @ A @ B] :
% 2.92/1.57 ( ! [X6: A] :
% 2.92/1.57 ( S
% 2.92/1.57 != ( sum_Inl @ A @ B @ X6 ) )
% 2.92/1.57 => ~ ! [X6: B] :
% 2.92/1.57 ( S
% 2.92/1.57 != ( sum_Inr @ B @ A @ X6 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_30_obj__sumE__f,axiom,
% 2.92/1.57 ! [A: $tType,C: $tType,B: $tType,S: B,F: ( sum_sum @ A @ C ) > B,P: $o] :
% 2.92/1.57 ( ! [X6: A] :
% 2.92/1.57 ( ( S
% 2.92/1.57 = ( F @ ( sum_Inl @ A @ C @ X6 ) ) )
% 2.92/1.57 => P )
% 2.92/1.57 => ( ! [X6: C] :
% 2.92/1.57 ( ( S
% 2.92/1.57 = ( F @ ( sum_Inr @ C @ A @ X6 ) ) )
% 2.92/1.57 => P )
% 2.92/1.57 => ! [X7: sum_sum @ A @ C] :
% 2.92/1.57 ( ( S
% 2.92/1.57 = ( F @ X7 ) )
% 2.92/1.57 => P ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_31_inFr2_Oinducts,axiom,
% 2.92/1.57 ! [X1: set @ n,X2: dtree,X3: t,P: ( set @ n ) > dtree > t > $o] :
% 2.92/1.57 ( ( gram_L805317441_inFr2 @ X1 @ X2 @ X3 )
% 2.92/1.57 => ( ! [Tr4: dtree,Ns3: set @ n,T4: t] :
% 2.92/1.57 ( ( member @ n @ ( root @ Tr4 ) @ Ns3 )
% 2.92/1.57 => ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T4 ) @ ( cont @ Tr4 ) )
% 2.92/1.57 => ( P @ Ns3 @ Tr4 @ T4 ) ) )
% 2.92/1.57 => ( ! [Tr1: dtree,Tr4: dtree,Ns1: set @ n,T4: t] :
% 2.92/1.57 ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr1 ) @ ( cont @ Tr4 ) )
% 2.92/1.57 => ( ( gram_L805317441_inFr2 @ Ns1 @ Tr1 @ T4 )
% 2.92/1.57 => ( ( P @ Ns1 @ Tr1 @ T4 )
% 2.92/1.57 => ( P @ ( insert @ n @ ( root @ Tr4 ) @ Ns1 ) @ Tr4 @ T4 ) ) ) )
% 2.92/1.57 => ( P @ X1 @ X2 @ X3 ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_32_inFr2_Osimps,axiom,
% 2.92/1.57 ( gram_L805317441_inFr2
% 2.92/1.57 = ( ^ [A12: set @ n,A23: dtree,A33: t] :
% 2.92/1.57 ( ? [Tr3: dtree,Ns2: set @ n,T3: t] :
% 2.92/1.57 ( ( A12 = Ns2 )
% 2.92/1.57 & ( A23 = Tr3 )
% 2.92/1.57 & ( A33 = T3 )
% 2.92/1.57 & ( member @ n @ ( root @ Tr3 ) @ Ns2 )
% 2.92/1.57 & ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ T3 ) @ ( cont @ Tr3 ) ) )
% 2.92/1.57 | ? [Tr12: dtree,Tr3: dtree,Ns12: set @ n,T3: t] :
% 2.92/1.57 ( ( A12
% 2.92/1.57 = ( insert @ n @ ( root @ Tr3 ) @ Ns12 ) )
% 2.92/1.57 & ( A23 = Tr3 )
% 2.92/1.57 & ( A33 = T3 )
% 2.92/1.57 & ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr12 ) @ ( cont @ Tr3 ) )
% 2.92/1.57 & ( gram_L805317441_inFr2 @ Ns12 @ Tr12 @ T3 ) ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_33_inFr2_Ocases,axiom,
% 2.92/1.57 ! [A1: set @ n,A22: dtree,A32: t] :
% 2.92/1.57 ( ( gram_L805317441_inFr2 @ A1 @ A22 @ A32 )
% 2.92/1.57 => ( ( ( member @ n @ ( root @ A22 ) @ A1 )
% 2.92/1.57 => ~ ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ A32 ) @ ( cont @ A22 ) ) )
% 2.92/1.57 => ~ ! [Tr1: dtree,Tr4: dtree,Ns1: set @ n] :
% 2.92/1.57 ( ( A1
% 2.92/1.57 = ( insert @ n @ ( root @ Tr4 ) @ Ns1 ) )
% 2.92/1.57 => ( ( A22 = Tr4 )
% 2.92/1.57 => ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr1 ) @ ( cont @ Tr4 ) )
% 2.92/1.57 => ~ ( gram_L805317441_inFr2 @ Ns1 @ Tr1 @ A32 ) ) ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_34_inFr2_OInd,axiom,
% 2.92/1.57 ! [Tr13: dtree,Tr: dtree,Ns13: set @ n,T2: t] :
% 2.92/1.57 ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr13 ) @ ( cont @ Tr ) )
% 2.92/1.57 => ( ( gram_L805317441_inFr2 @ Ns13 @ Tr13 @ T2 )
% 2.92/1.57 => ( gram_L805317441_inFr2 @ ( insert @ n @ ( root @ Tr ) @ Ns13 ) @ Tr @ T2 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_35_inFr__Ind__minus,axiom,
% 2.92/1.57 ! [Ns13: set @ n,Tr13: dtree,T2: t,Tr: dtree] :
% 2.92/1.57 ( ( gram_L1333338417e_inFr @ Ns13 @ Tr13 @ T2 )
% 2.92/1.57 => ( ( member @ ( sum_sum @ t @ dtree ) @ ( sum_Inr @ dtree @ t @ Tr13 ) @ ( cont @ Tr ) )
% 2.92/1.57 => ( gram_L1333338417e_inFr @ ( insert @ n @ ( root @ Tr ) @ Ns13 ) @ Tr @ T2 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_36_Suml_Osimps,axiom,
% 2.92/1.57 ! [B: $tType,C: $tType,A: $tType,F: A > C,X: A] :
% 2.92/1.57 ( ( sum_Suml @ A @ C @ B @ F @ ( sum_Inl @ A @ B @ X ) )
% 2.92/1.57 = ( F @ X ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_37_Sumr_Osimps,axiom,
% 2.92/1.57 ! [A: $tType,C: $tType,B: $tType,F: B > C,X: B] :
% 2.92/1.57 ( ( sum_Sumr @ B @ C @ A @ F @ ( sum_Inr @ B @ A @ X ) )
% 2.92/1.57 = ( F @ X ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_38_Sumr__inject,axiom,
% 2.92/1.57 ! [A: $tType,C: $tType,B: $tType,F: B > C,G: B > C] :
% 2.92/1.57 ( ( ( sum_Sumr @ B @ C @ A @ F )
% 2.92/1.57 = ( sum_Sumr @ B @ C @ A @ G ) )
% 2.92/1.57 => ( F = G ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_39_Suml__inject,axiom,
% 2.92/1.57 ! [B: $tType,C: $tType,A: $tType,F: A > C,G: A > C] :
% 2.92/1.57 ( ( ( sum_Suml @ A @ C @ B @ F )
% 2.92/1.57 = ( sum_Suml @ A @ C @ B @ G ) )
% 2.92/1.57 => ( F = G ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_40_not__arg__cong__Inr,axiom,
% 2.92/1.57 ! [B: $tType,A: $tType,X: A,Y: A] :
% 2.92/1.57 ( ( X != Y )
% 2.92/1.57 => ( ( sum_Inr @ A @ B @ X )
% 2.92/1.57 != ( sum_Inr @ A @ B @ Y ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_41_insert__absorb2,axiom,
% 2.92/1.57 ! [A: $tType,X: A,A7: set @ A] :
% 2.92/1.57 ( ( insert @ A @ X @ ( insert @ A @ X @ A7 ) )
% 2.92/1.57 = ( insert @ A @ X @ A7 ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_42_insert__iff,axiom,
% 2.92/1.57 ! [A: $tType,A2: A,B2: A,A7: set @ A] :
% 2.92/1.57 ( ( member @ A @ A2 @ ( insert @ A @ B2 @ A7 ) )
% 2.92/1.57 = ( ( A2 = B2 )
% 2.92/1.57 | ( member @ A @ A2 @ A7 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_43_insertCI,axiom,
% 2.92/1.57 ! [A: $tType,A2: A,B7: set @ A,B2: A] :
% 2.92/1.57 ( ( ~ ( member @ A @ A2 @ B7 )
% 2.92/1.57 => ( A2 = B2 ) )
% 2.92/1.57 => ( member @ A @ A2 @ ( insert @ A @ B2 @ B7 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_44_corec_I1_J,axiom,
% 2.92/1.57 ! [A: $tType,Rt: A > n,Ct: A > ( set @ ( sum_sum @ t @ ( sum_sum @ dtree @ A ) ) ),B2: A] :
% 2.92/1.57 ( ( root @ ( corec @ A @ Rt @ Ct @ B2 ) )
% 2.92/1.57 = ( Rt @ B2 ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_45_mem__Collect__eq,axiom,
% 2.92/1.57 ! [A: $tType,A2: A,P: A > $o] :
% 2.92/1.57 ( ( member @ A @ A2 @ ( collect @ A @ P ) )
% 2.92/1.57 = ( P @ A2 ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_46_Collect__mem__eq,axiom,
% 2.92/1.57 ! [A: $tType,A7: set @ A] :
% 2.92/1.57 ( ( collect @ A
% 2.92/1.57 @ ^ [X5: A] : ( member @ A @ X5 @ A7 ) )
% 2.92/1.57 = A7 ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_47_Collect__cong,axiom,
% 2.92/1.57 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.92/1.57 ( ! [X6: A] :
% 2.92/1.57 ( ( P @ X6 )
% 2.92/1.57 = ( Q @ X6 ) )
% 2.92/1.57 => ( ( collect @ A @ P )
% 2.92/1.57 = ( collect @ A @ Q ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_48_ext,axiom,
% 2.92/1.57 ! [B: $tType,A: $tType,F: A > B,G: A > B] :
% 2.92/1.57 ( ! [X6: A] :
% 2.92/1.57 ( ( F @ X6 )
% 2.92/1.57 = ( G @ X6 ) )
% 2.92/1.57 => ( F = G ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_49_unfold_I1_J,axiom,
% 2.92/1.57 ! [A: $tType,Rt: A > n,Ct: A > ( set @ ( sum_sum @ t @ A ) ),B2: A] :
% 2.92/1.57 ( ( root @ ( unfold @ A @ Rt @ Ct @ B2 ) )
% 2.92/1.57 = ( Rt @ B2 ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_50_cont__Node,axiom,
% 2.92/1.57 ! [As: set @ ( sum_sum @ t @ dtree ),N: n] :
% 2.92/1.57 ( ( finite_finite2 @ ( sum_sum @ t @ dtree ) @ As )
% 2.92/1.57 => ( ( cont @ ( node @ N @ As ) )
% 2.92/1.57 = As ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_51_mk__disjoint__insert,axiom,
% 2.92/1.57 ! [A: $tType,A2: A,A7: set @ A] :
% 2.92/1.57 ( ( member @ A @ A2 @ A7 )
% 2.92/1.57 => ? [B8: set @ A] :
% 2.92/1.57 ( ( A7
% 2.92/1.57 = ( insert @ A @ A2 @ B8 ) )
% 2.92/1.57 & ~ ( member @ A @ A2 @ B8 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_52_insert__commute,axiom,
% 2.92/1.57 ! [A: $tType,X: A,Y: A,A7: set @ A] :
% 2.92/1.57 ( ( insert @ A @ X @ ( insert @ A @ Y @ A7 ) )
% 2.92/1.57 = ( insert @ A @ Y @ ( insert @ A @ X @ A7 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_53_insert__eq__iff,axiom,
% 2.92/1.57 ! [A: $tType,A2: A,A7: set @ A,B2: A,B7: set @ A] :
% 2.92/1.57 ( ~ ( member @ A @ A2 @ A7 )
% 2.92/1.57 => ( ~ ( member @ A @ B2 @ B7 )
% 2.92/1.57 => ( ( ( insert @ A @ A2 @ A7 )
% 2.92/1.57 = ( insert @ A @ B2 @ B7 ) )
% 2.92/1.57 = ( ( ( A2 = B2 )
% 2.92/1.57 => ( A7 = B7 ) )
% 2.92/1.57 & ( ( A2 != B2 )
% 2.92/1.57 => ? [C2: set @ A] :
% 2.92/1.57 ( ( A7
% 2.92/1.57 = ( insert @ A @ B2 @ C2 ) )
% 2.92/1.57 & ~ ( member @ A @ B2 @ C2 )
% 2.92/1.57 & ( B7
% 2.92/1.57 = ( insert @ A @ A2 @ C2 ) )
% 2.92/1.57 & ~ ( member @ A @ A2 @ C2 ) ) ) ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_54_dtree__simps,axiom,
% 2.92/1.57 ! [As: set @ ( sum_sum @ t @ dtree ),As2: set @ ( sum_sum @ t @ dtree ),N: n,N2: n] :
% 2.92/1.57 ( ( finite_finite2 @ ( sum_sum @ t @ dtree ) @ As )
% 2.92/1.57 => ( ( finite_finite2 @ ( sum_sum @ t @ dtree ) @ As2 )
% 2.92/1.57 => ( ( ( node @ N @ As )
% 2.92/1.57 = ( node @ N2 @ As2 ) )
% 2.92/1.57 = ( ( N = N2 )
% 2.92/1.57 & ( As = As2 ) ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_55_finite__cont,axiom,
% 2.92/1.57 ! [Tr: dtree] : ( finite_finite2 @ ( sum_sum @ t @ dtree ) @ ( cont @ Tr ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_56_dtree__cases,axiom,
% 2.92/1.57 ! [Tr: dtree] :
% 2.92/1.57 ~ ! [N3: n,As3: set @ ( sum_sum @ t @ dtree )] :
% 2.92/1.57 ( ( finite_finite2 @ ( sum_sum @ t @ dtree ) @ As3 )
% 2.92/1.57 => ( Tr
% 2.92/1.57 != ( node @ N3 @ As3 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_57_insertE,axiom,
% 2.92/1.57 ! [A: $tType,A2: A,B2: A,A7: set @ A] :
% 2.92/1.57 ( ( member @ A @ A2 @ ( insert @ A @ B2 @ A7 ) )
% 2.92/1.57 => ( ( A2 != B2 )
% 2.92/1.57 => ( member @ A @ A2 @ A7 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_58_insertI1,axiom,
% 2.92/1.57 ! [A: $tType,A2: A,B7: set @ A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ B7 ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_59_insertI2,axiom,
% 2.92/1.57 ! [A: $tType,A2: A,B7: set @ A,B2: A] :
% 2.92/1.57 ( ( member @ A @ A2 @ B7 )
% 2.92/1.57 => ( member @ A @ A2 @ ( insert @ A @ B2 @ B7 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_60_Set_Oset__insert,axiom,
% 2.92/1.57 ! [A: $tType,X: A,A7: set @ A] :
% 2.92/1.57 ( ( member @ A @ X @ A7 )
% 2.92/1.57 => ~ ! [B8: set @ A] :
% 2.92/1.57 ( ( A7
% 2.92/1.57 = ( insert @ A @ X @ B8 ) )
% 2.92/1.57 => ( member @ A @ X @ B8 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_61_insert__ident,axiom,
% 2.92/1.57 ! [A: $tType,X: A,A7: set @ A,B7: set @ A] :
% 2.92/1.57 ( ~ ( member @ A @ X @ A7 )
% 2.92/1.57 => ( ~ ( member @ A @ X @ B7 )
% 2.92/1.57 => ( ( ( insert @ A @ X @ A7 )
% 2.92/1.57 = ( insert @ A @ X @ B7 ) )
% 2.92/1.57 = ( A7 = B7 ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_62_insert__absorb,axiom,
% 2.92/1.57 ! [A: $tType,A2: A,A7: set @ A] :
% 2.92/1.57 ( ( member @ A @ A2 @ A7 )
% 2.92/1.57 => ( ( insert @ A @ A2 @ A7 )
% 2.92/1.57 = A7 ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_63_finite__insert,axiom,
% 2.92/1.57 ! [A: $tType,A2: A,A7: set @ A] :
% 2.92/1.57 ( ( finite_finite2 @ A @ ( insert @ A @ A2 @ A7 ) )
% 2.92/1.57 = ( finite_finite2 @ A @ A7 ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_64_finite__code,axiom,
% 2.92/1.57 ! [A: $tType] :
% 2.92/1.57 ( ( finite_finite @ A @ ( type2 @ A ) )
% 2.92/1.57 => ( ( finite_finite2 @ A )
% 2.92/1.57 = ( ^ [A8: set @ A] : $true ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_65_finite_OinsertI,axiom,
% 2.92/1.57 ! [A: $tType,A7: set @ A,A2: A] :
% 2.92/1.57 ( ( finite_finite2 @ A @ A7 )
% 2.92/1.57 => ( finite_finite2 @ A @ ( insert @ A @ A2 @ A7 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_66_finite__set__choice,axiom,
% 2.92/1.57 ! [B: $tType,A: $tType,A7: set @ A,P: A > B > $o] :
% 2.92/1.57 ( ( finite_finite2 @ A @ A7 )
% 2.92/1.57 => ( ! [X6: A] :
% 2.92/1.57 ( ( member @ A @ X6 @ A7 )
% 2.92/1.57 => ? [X12: B] : ( P @ X6 @ X12 ) )
% 2.92/1.57 => ? [F3: A > B] :
% 2.92/1.57 ! [X7: A] :
% 2.92/1.57 ( ( member @ A @ X7 @ A7 )
% 2.92/1.57 => ( P @ X7 @ ( F3 @ X7 ) ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_67_finite,axiom,
% 2.92/1.57 ! [A: $tType] :
% 2.92/1.57 ( ( finite_finite @ A @ ( type2 @ A ) )
% 2.92/1.57 => ! [A7: set @ A] : ( finite_finite2 @ A @ A7 ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_68_case__sum__if,axiom,
% 2.92/1.57 ! [B: $tType,A: $tType,C: $tType,P4: $o,F: B > A,G: C > A,X: B,Y: C] :
% 2.92/1.57 ( ( P4
% 2.92/1.57 => ( ( sum_case_sum @ B @ A @ C @ F @ G @ ( if @ ( sum_sum @ B @ C ) @ P4 @ ( sum_Inl @ B @ C @ X ) @ ( sum_Inr @ C @ B @ Y ) ) )
% 2.92/1.57 = ( F @ X ) ) )
% 2.92/1.57 & ( ~ P4
% 2.92/1.57 => ( ( sum_case_sum @ B @ A @ C @ F @ G @ ( if @ ( sum_sum @ B @ C ) @ P4 @ ( sum_Inl @ B @ C @ X ) @ ( sum_Inr @ C @ B @ Y ) ) )
% 2.92/1.57 = ( G @ Y ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_69_PlusE,axiom,
% 2.92/1.57 ! [A: $tType,B: $tType,U: sum_sum @ A @ B,A7: set @ A,B7: set @ B] :
% 2.92/1.57 ( ( member @ ( sum_sum @ A @ B ) @ U @ ( sum_Plus @ A @ B @ A7 @ B7 ) )
% 2.92/1.57 => ( ! [X6: A] :
% 2.92/1.57 ( ( member @ A @ X6 @ A7 )
% 2.92/1.57 => ( U
% 2.92/1.57 != ( sum_Inl @ A @ B @ X6 ) ) )
% 2.92/1.57 => ~ ! [Y3: B] :
% 2.92/1.57 ( ( member @ B @ Y3 @ B7 )
% 2.92/1.57 => ( U
% 2.92/1.57 != ( sum_Inr @ B @ A @ Y3 ) ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_70_dtree_Osel_I1_J,axiom,
% 2.92/1.57 ! [X1: n,X2: fset @ ( sum_sum @ t @ dtree )] :
% 2.92/1.57 ( ( root @ ( nNode @ X1 @ X2 ) )
% 2.92/1.57 = X1 ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_71_dtree_Oinject,axiom,
% 2.92/1.57 ! [X1: n,X2: fset @ ( sum_sum @ t @ dtree ),Y1: n,Y2: fset @ ( sum_sum @ t @ dtree )] :
% 2.92/1.57 ( ( ( nNode @ X1 @ X2 )
% 2.92/1.57 = ( nNode @ Y1 @ Y2 ) )
% 2.92/1.57 = ( ( X1 = Y1 )
% 2.92/1.57 & ( X2 = Y2 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_72_InrI,axiom,
% 2.92/1.57 ! [B: $tType,A: $tType,B2: A,B7: set @ A,A7: set @ B] :
% 2.92/1.57 ( ( member @ A @ B2 @ B7 )
% 2.92/1.57 => ( member @ ( sum_sum @ B @ A ) @ ( sum_Inr @ A @ B @ B2 ) @ ( sum_Plus @ B @ A @ A7 @ B7 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_73_InlI,axiom,
% 2.92/1.57 ! [A: $tType,B: $tType,A2: A,A7: set @ A,B7: set @ B] :
% 2.92/1.57 ( ( member @ A @ A2 @ A7 )
% 2.92/1.57 => ( member @ ( sum_sum @ A @ B ) @ ( sum_Inl @ A @ B @ A2 ) @ ( sum_Plus @ A @ B @ A7 @ B7 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_74_finite__Plus__iff,axiom,
% 2.92/1.57 ! [A: $tType,B: $tType,A7: set @ A,B7: set @ B] :
% 2.92/1.57 ( ( finite_finite2 @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A7 @ B7 ) )
% 2.92/1.57 = ( ( finite_finite2 @ A @ A7 )
% 2.92/1.57 & ( finite_finite2 @ B @ B7 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_75_dtree_Oexhaust,axiom,
% 2.92/1.57 ! [Y: dtree] :
% 2.92/1.57 ~ ! [X13: n,X22: fset @ ( sum_sum @ t @ dtree )] :
% 2.92/1.57 ( Y
% 2.92/1.57 != ( nNode @ X13 @ X22 ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_76_case__sum__inject,axiom,
% 2.92/1.57 ! [A: $tType,C: $tType,B: $tType,F1: A > C,F2: B > C,G1: A > C,G2: B > C] :
% 2.92/1.57 ( ( ( sum_case_sum @ A @ C @ B @ F1 @ F2 )
% 2.92/1.57 = ( sum_case_sum @ A @ C @ B @ G1 @ G2 ) )
% 2.92/1.57 => ~ ( ( F1 = G1 )
% 2.92/1.57 => ( F2 != G2 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_77_case__sum__step_I1_J,axiom,
% 2.92/1.57 ! [D: $tType,A: $tType,C: $tType,B: $tType,F4: B > A,G3: C > A,G: D > A,P4: sum_sum @ B @ C] :
% 2.92/1.57 ( ( sum_case_sum @ ( sum_sum @ B @ C ) @ A @ D @ ( sum_case_sum @ B @ A @ C @ F4 @ G3 ) @ G @ ( sum_Inl @ ( sum_sum @ B @ C ) @ D @ P4 ) )
% 2.92/1.57 = ( sum_case_sum @ B @ A @ C @ F4 @ G3 @ P4 ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_78_case__sum__step_I2_J,axiom,
% 2.92/1.57 ! [E: $tType,A: $tType,C: $tType,B: $tType,F: E > A,F4: B > A,G3: C > A,P4: sum_sum @ B @ C] :
% 2.92/1.57 ( ( sum_case_sum @ E @ A @ ( sum_sum @ B @ C ) @ F @ ( sum_case_sum @ B @ A @ C @ F4 @ G3 ) @ ( sum_Inr @ ( sum_sum @ B @ C ) @ E @ P4 ) )
% 2.92/1.57 = ( sum_case_sum @ B @ A @ C @ F4 @ G3 @ P4 ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_79_old_Osum_Osimps_I6_J,axiom,
% 2.92/1.57 ! [A: $tType,C: $tType,B: $tType,F1: A > C,F2: B > C,X2: B] :
% 2.92/1.57 ( ( sum_case_sum @ A @ C @ B @ F1 @ F2 @ ( sum_Inr @ B @ A @ X2 ) )
% 2.92/1.57 = ( F2 @ X2 ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_80_old_Osum_Osimps_I5_J,axiom,
% 2.92/1.57 ! [B: $tType,C: $tType,A: $tType,F1: A > C,F2: B > C,X1: A] :
% 2.92/1.57 ( ( sum_case_sum @ A @ C @ B @ F1 @ F2 @ ( sum_Inl @ A @ B @ X1 ) )
% 2.92/1.57 = ( F1 @ X1 ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_81_finite__Plus,axiom,
% 2.92/1.57 ! [A: $tType,B: $tType,A7: set @ A,B7: set @ B] :
% 2.92/1.57 ( ( finite_finite2 @ A @ A7 )
% 2.92/1.57 => ( ( finite_finite2 @ B @ B7 )
% 2.92/1.57 => ( finite_finite2 @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A7 @ B7 ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_82_finite__PlusD_I1_J,axiom,
% 2.92/1.57 ! [B: $tType,A: $tType,A7: set @ A,B7: set @ B] :
% 2.92/1.57 ( ( finite_finite2 @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A7 @ B7 ) )
% 2.92/1.57 => ( finite_finite2 @ A @ A7 ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_83_finite__PlusD_I2_J,axiom,
% 2.92/1.57 ! [A: $tType,B: $tType,A7: set @ A,B7: set @ B] :
% 2.92/1.57 ( ( finite_finite2 @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A7 @ B7 ) )
% 2.92/1.57 => ( finite_finite2 @ B @ B7 ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_84_dtree_Ocollapse,axiom,
% 2.92/1.57 ! [Dtree: dtree] :
% 2.92/1.57 ( ( nNode @ ( root @ Dtree ) @ ( ccont @ Dtree ) )
% 2.92/1.57 = Dtree ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_85_dtree_Oexhaust__sel,axiom,
% 2.92/1.57 ! [Dtree: dtree] :
% 2.92/1.57 ( Dtree
% 2.92/1.57 = ( nNode @ ( root @ Dtree ) @ ( ccont @ Dtree ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_86_case__sum__expand__Inr__pointfree,axiom,
% 2.92/1.57 ! [B: $tType,C: $tType,A: $tType,F: ( sum_sum @ A @ C ) > B,G: A > B] :
% 2.92/1.57 ( ( ( comp @ ( sum_sum @ A @ C ) @ B @ A @ F @ ( sum_Inl @ A @ C ) )
% 2.92/1.57 = G )
% 2.92/1.57 => ( ( sum_case_sum @ A @ B @ C @ G @ ( comp @ ( sum_sum @ A @ C ) @ B @ C @ F @ ( sum_Inr @ C @ A ) ) )
% 2.92/1.57 = F ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_87_case__sum__expand__Inr_H,axiom,
% 2.92/1.57 ! [B: $tType,C: $tType,A: $tType,F: ( sum_sum @ A @ C ) > B,G: A > B,H: C > B] :
% 2.92/1.57 ( ( ( comp @ ( sum_sum @ A @ C ) @ B @ A @ F @ ( sum_Inl @ A @ C ) )
% 2.92/1.57 = G )
% 2.92/1.57 => ( ( H
% 2.92/1.57 = ( comp @ ( sum_sum @ A @ C ) @ B @ C @ F @ ( sum_Inr @ C @ A ) ) )
% 2.92/1.57 = ( ( sum_case_sum @ A @ B @ C @ G @ H )
% 2.92/1.57 = F ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_88_case__sum__expand__Inr,axiom,
% 2.92/1.57 ! [B: $tType,C: $tType,A: $tType,F: ( sum_sum @ A @ C ) > B,G: A > B,X: sum_sum @ A @ C] :
% 2.92/1.57 ( ( ( comp @ ( sum_sum @ A @ C ) @ B @ A @ F @ ( sum_Inl @ A @ C ) )
% 2.92/1.57 = G )
% 2.92/1.57 => ( ( F @ X )
% 2.92/1.57 = ( sum_case_sum @ A @ B @ C @ G @ ( comp @ ( sum_sum @ A @ C ) @ B @ C @ F @ ( sum_Inr @ C @ A ) ) @ X ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_89_dtree_Ocase,axiom,
% 2.92/1.57 ! [A: $tType,F: n > ( fset @ ( sum_sum @ t @ dtree ) ) > A,X1: n,X2: fset @ ( sum_sum @ t @ dtree )] :
% 2.92/1.57 ( ( case_dtree @ A @ F @ ( nNode @ X1 @ X2 ) )
% 2.92/1.57 = ( F @ X1 @ X2 ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_90_finite__Diff__insert,axiom,
% 2.92/1.57 ! [A: $tType,A7: set @ A,A2: A,B7: set @ A] :
% 2.92/1.57 ( ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A7 @ ( insert @ A @ A2 @ B7 ) ) )
% 2.92/1.57 = ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A7 @ B7 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_91_Diff__idemp,axiom,
% 2.92/1.57 ! [A: $tType,A7: set @ A,B7: set @ A] :
% 2.92/1.57 ( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A7 @ B7 ) @ B7 )
% 2.92/1.57 = ( minus_minus @ ( set @ A ) @ A7 @ B7 ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_92_Diff__iff,axiom,
% 2.92/1.57 ! [A: $tType,C3: A,A7: set @ A,B7: set @ A] :
% 2.92/1.57 ( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A7 @ B7 ) )
% 2.92/1.57 = ( ( member @ A @ C3 @ A7 )
% 2.92/1.57 & ~ ( member @ A @ C3 @ B7 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_93_DiffI,axiom,
% 2.92/1.57 ! [A: $tType,C3: A,A7: set @ A,B7: set @ A] :
% 2.92/1.57 ( ( member @ A @ C3 @ A7 )
% 2.92/1.57 => ( ~ ( member @ A @ C3 @ B7 )
% 2.92/1.57 => ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A7 @ B7 ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_94_finite__Diff2,axiom,
% 2.92/1.57 ! [A: $tType,B7: set @ A,A7: set @ A] :
% 2.92/1.57 ( ( finite_finite2 @ A @ B7 )
% 2.92/1.57 => ( ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A7 @ B7 ) )
% 2.92/1.57 = ( finite_finite2 @ A @ A7 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_95_finite__Diff,axiom,
% 2.92/1.57 ! [A: $tType,A7: set @ A,B7: set @ A] :
% 2.92/1.57 ( ( finite_finite2 @ A @ A7 )
% 2.92/1.57 => ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A7 @ B7 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_96_Diff__insert0,axiom,
% 2.92/1.57 ! [A: $tType,X: A,A7: set @ A,B7: set @ A] :
% 2.92/1.57 ( ~ ( member @ A @ X @ A7 )
% 2.92/1.57 => ( ( minus_minus @ ( set @ A ) @ A7 @ ( insert @ A @ X @ B7 ) )
% 2.92/1.57 = ( minus_minus @ ( set @ A ) @ A7 @ B7 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_97_insert__Diff1,axiom,
% 2.92/1.57 ! [A: $tType,X: A,B7: set @ A,A7: set @ A] :
% 2.92/1.57 ( ( member @ A @ X @ B7 )
% 2.92/1.57 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A7 ) @ B7 )
% 2.92/1.57 = ( minus_minus @ ( set @ A ) @ A7 @ B7 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_98_o__case__sum,axiom,
% 2.92/1.57 ! [A: $tType,C: $tType,D: $tType,B: $tType,H: D > C,F: A > D,G: B > D] :
% 2.92/1.57 ( ( comp @ D @ C @ ( sum_sum @ A @ B ) @ H @ ( sum_case_sum @ A @ D @ B @ F @ G ) )
% 2.92/1.57 = ( sum_case_sum @ A @ C @ B @ ( comp @ D @ C @ A @ H @ F ) @ ( comp @ D @ C @ B @ H @ G ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_99_rewriteR__comp__comp2,axiom,
% 2.92/1.57 ! [C: $tType,B: $tType,E: $tType,D: $tType,A: $tType,G: C > B,H: A > C,R1: D > B,R2: A > D,F: B > E,L: D > E] :
% 2.92/1.57 ( ( ( comp @ C @ B @ A @ G @ H )
% 2.92/1.57 = ( comp @ D @ B @ A @ R1 @ R2 ) )
% 2.92/1.57 => ( ( ( comp @ B @ E @ D @ F @ R1 )
% 2.92/1.57 = L )
% 2.92/1.57 => ( ( comp @ C @ E @ A @ ( comp @ B @ E @ C @ F @ G ) @ H )
% 2.92/1.57 = ( comp @ D @ E @ A @ L @ R2 ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_100_rewriteL__comp__comp2,axiom,
% 2.92/1.57 ! [A: $tType,C: $tType,B: $tType,D: $tType,E: $tType,F: C > B,G: A > C,L1: D > B,L2: A > D,H: E > A,R: E > D] :
% 2.92/1.57 ( ( ( comp @ C @ B @ A @ F @ G )
% 2.92/1.57 = ( comp @ D @ B @ A @ L1 @ L2 ) )
% 2.92/1.57 => ( ( ( comp @ A @ D @ E @ L2 @ H )
% 2.92/1.57 = R )
% 2.92/1.57 => ( ( comp @ C @ B @ E @ F @ ( comp @ A @ C @ E @ G @ H ) )
% 2.92/1.57 = ( comp @ D @ B @ E @ L1 @ R ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_101_rewriteR__comp__comp,axiom,
% 2.92/1.57 ! [C: $tType,D: $tType,B: $tType,A: $tType,G: C > B,H: A > C,R: A > B,F: B > D] :
% 2.92/1.57 ( ( ( comp @ C @ B @ A @ G @ H )
% 2.92/1.57 = R )
% 2.92/1.57 => ( ( comp @ C @ D @ A @ ( comp @ B @ D @ C @ F @ G ) @ H )
% 2.92/1.57 = ( comp @ B @ D @ A @ F @ R ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_102_rewriteL__comp__comp,axiom,
% 2.92/1.57 ! [C: $tType,B: $tType,A: $tType,D: $tType,F: C > B,G: A > C,L: A > B,H: D > A] :
% 2.92/1.57 ( ( ( comp @ C @ B @ A @ F @ G )
% 2.92/1.57 = L )
% 2.92/1.57 => ( ( comp @ C @ B @ D @ F @ ( comp @ A @ C @ D @ G @ H ) )
% 2.92/1.57 = ( comp @ A @ B @ D @ L @ H ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_103_DiffD2,axiom,
% 2.92/1.57 ! [A: $tType,C3: A,A7: set @ A,B7: set @ A] :
% 2.92/1.57 ( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A7 @ B7 ) )
% 2.92/1.57 => ~ ( member @ A @ C3 @ B7 ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_104_DiffD1,axiom,
% 2.92/1.57 ! [A: $tType,C3: A,A7: set @ A,B7: set @ A] :
% 2.92/1.57 ( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A7 @ B7 ) )
% 2.92/1.57 => ( member @ A @ C3 @ A7 ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_105_DiffE,axiom,
% 2.92/1.57 ! [A: $tType,C3: A,A7: set @ A,B7: set @ A] :
% 2.92/1.57 ( ( member @ A @ C3 @ ( minus_minus @ ( set @ A ) @ A7 @ B7 ) )
% 2.92/1.57 => ~ ( ( member @ A @ C3 @ A7 )
% 2.92/1.57 => ( member @ A @ C3 @ B7 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_106_dtree_Ocase__eq__if,axiom,
% 2.92/1.57 ! [A: $tType] :
% 2.92/1.57 ( ( case_dtree @ A )
% 2.92/1.57 = ( ^ [F5: n > ( fset @ ( sum_sum @ t @ dtree ) ) > A,Dtree2: dtree] : ( F5 @ ( root @ Dtree2 ) @ ( ccont @ Dtree2 ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_107_Diff__infinite__finite,axiom,
% 2.92/1.57 ! [A: $tType,T5: set @ A,S2: set @ A] :
% 2.92/1.57 ( ( finite_finite2 @ A @ T5 )
% 2.92/1.57 => ( ~ ( finite_finite2 @ A @ S2 )
% 2.92/1.57 => ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ S2 @ T5 ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_108_insert__Diff__if,axiom,
% 2.92/1.57 ! [A: $tType,X: A,B7: set @ A,A7: set @ A] :
% 2.92/1.57 ( ( ( member @ A @ X @ B7 )
% 2.92/1.57 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A7 ) @ B7 )
% 2.92/1.57 = ( minus_minus @ ( set @ A ) @ A7 @ B7 ) ) )
% 2.92/1.57 & ( ~ ( member @ A @ X @ B7 )
% 2.92/1.57 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A7 ) @ B7 )
% 2.92/1.57 = ( insert @ A @ X @ ( minus_minus @ ( set @ A ) @ A7 @ B7 ) ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_109_dtree_Oexpand,axiom,
% 2.92/1.57 ! [Dtree: dtree,Dtree3: dtree] :
% 2.92/1.57 ( ( ( ( root @ Dtree )
% 2.92/1.57 = ( root @ Dtree3 ) )
% 2.92/1.57 & ( ( ccont @ Dtree )
% 2.92/1.57 = ( ccont @ Dtree3 ) ) )
% 2.92/1.57 => ( Dtree = Dtree3 ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_110_dtree_Osplit__sel__asm,axiom,
% 2.92/1.57 ! [A: $tType,P: A > $o,F: n > ( fset @ ( sum_sum @ t @ dtree ) ) > A,Dtree: dtree] :
% 2.92/1.57 ( ( P @ ( case_dtree @ A @ F @ Dtree ) )
% 2.92/1.57 = ( ~ ( ( Dtree
% 2.92/1.57 = ( nNode @ ( root @ Dtree ) @ ( ccont @ Dtree ) ) )
% 2.92/1.57 & ~ ( P @ ( F @ ( root @ Dtree ) @ ( ccont @ Dtree ) ) ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_111_dtree_Osplit__sel,axiom,
% 2.92/1.57 ! [A: $tType,P: A > $o,F: n > ( fset @ ( sum_sum @ t @ dtree ) ) > A,Dtree: dtree] :
% 2.92/1.57 ( ( P @ ( case_dtree @ A @ F @ Dtree ) )
% 2.92/1.57 = ( ( Dtree
% 2.92/1.57 = ( nNode @ ( root @ Dtree ) @ ( ccont @ Dtree ) ) )
% 2.92/1.57 => ( P @ ( F @ ( root @ Dtree ) @ ( ccont @ Dtree ) ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_112_dtree_Osel_I2_J,axiom,
% 2.92/1.57 ! [X1: n,X2: fset @ ( sum_sum @ t @ dtree )] :
% 2.92/1.57 ( ( ccont @ ( nNode @ X1 @ X2 ) )
% 2.92/1.57 = X2 ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_113_case__sum__o__inj_I1_J,axiom,
% 2.92/1.57 ! [C: $tType,B: $tType,A: $tType,F: A > B,G: C > B] :
% 2.92/1.57 ( ( comp @ ( sum_sum @ A @ C ) @ B @ A @ ( sum_case_sum @ A @ B @ C @ F @ G ) @ ( sum_Inl @ A @ C ) )
% 2.92/1.57 = F ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_114_case__sum__o__inj_I2_J,axiom,
% 2.92/1.57 ! [A: $tType,B: $tType,C: $tType,F: A > B,G: C > B] :
% 2.92/1.57 ( ( comp @ ( sum_sum @ A @ C ) @ B @ C @ ( sum_case_sum @ A @ B @ C @ F @ G ) @ ( sum_Inr @ C @ A ) )
% 2.92/1.57 = G ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_115_comp__apply,axiom,
% 2.92/1.57 ! [C: $tType,A: $tType,B: $tType] :
% 2.92/1.57 ( ( comp @ B @ A @ C )
% 2.92/1.57 = ( ^ [F5: B > A,G4: C > B,X5: C] : ( F5 @ ( G4 @ X5 ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_116_minus__apply,axiom,
% 2.92/1.57 ! [B: $tType,A: $tType] :
% 2.92/1.57 ( ( minus @ B @ ( type2 @ B ) )
% 2.92/1.57 => ( ( minus_minus @ ( A > B ) )
% 2.92/1.57 = ( ^ [A8: A > B,B9: A > B,X5: A] : ( minus_minus @ B @ ( A8 @ X5 ) @ ( B9 @ X5 ) ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_117_comp__def,axiom,
% 2.92/1.57 ! [A: $tType,C: $tType,B: $tType] :
% 2.92/1.57 ( ( comp @ B @ C @ A )
% 2.92/1.57 = ( ^ [F5: B > C,G4: A > B,X5: A] : ( F5 @ ( G4 @ X5 ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_118_comp__assoc,axiom,
% 2.92/1.57 ! [B: $tType,D: $tType,C: $tType,A: $tType,F: D > B,G: C > D,H: A > C] :
% 2.92/1.57 ( ( comp @ C @ B @ A @ ( comp @ D @ B @ C @ F @ G ) @ H )
% 2.92/1.57 = ( comp @ D @ B @ A @ F @ ( comp @ C @ D @ A @ G @ H ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_119_fun__diff__def,axiom,
% 2.92/1.57 ! [B: $tType,A: $tType] :
% 2.92/1.57 ( ( minus @ B @ ( type2 @ B ) )
% 2.92/1.57 => ( ( minus_minus @ ( A > B ) )
% 2.92/1.57 = ( ^ [A8: A > B,B9: A > B,X5: A] : ( minus_minus @ B @ ( A8 @ X5 ) @ ( B9 @ X5 ) ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_120_comp__apply__eq,axiom,
% 2.92/1.57 ! [B: $tType,D: $tType,A: $tType,C: $tType,F: B > A,G: C > B,X: C,H: D > A,K: C > D] :
% 2.92/1.57 ( ( ( F @ ( G @ X ) )
% 2.92/1.57 = ( H @ ( K @ X ) ) )
% 2.92/1.57 => ( ( comp @ B @ A @ C @ F @ G @ X )
% 2.92/1.57 = ( comp @ D @ A @ C @ H @ K @ X ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_121_comp__eq__dest__lhs,axiom,
% 2.92/1.57 ! [C: $tType,B: $tType,A: $tType,A2: C > B,B2: A > C,C3: A > B,V: A] :
% 2.92/1.57 ( ( ( comp @ C @ B @ A @ A2 @ B2 )
% 2.92/1.57 = C3 )
% 2.92/1.57 => ( ( A2 @ ( B2 @ V ) )
% 2.92/1.57 = ( C3 @ V ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_122_comp__eq__elim,axiom,
% 2.92/1.57 ! [C: $tType,B: $tType,D: $tType,A: $tType,A2: C > B,B2: A > C,C3: D > B,D2: A > D] :
% 2.92/1.57 ( ( ( comp @ C @ B @ A @ A2 @ B2 )
% 2.92/1.57 = ( comp @ D @ B @ A @ C3 @ D2 ) )
% 2.92/1.57 => ! [V2: A] :
% 2.92/1.57 ( ( A2 @ ( B2 @ V2 ) )
% 2.92/1.57 = ( C3 @ ( D2 @ V2 ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_123_comp__eq__dest,axiom,
% 2.92/1.57 ! [C: $tType,B: $tType,D: $tType,A: $tType,A2: C > B,B2: A > C,C3: D > B,D2: A > D,V: A] :
% 2.92/1.57 ( ( ( comp @ C @ B @ A @ A2 @ B2 )
% 2.92/1.57 = ( comp @ D @ B @ A @ C3 @ D2 ) )
% 2.92/1.57 => ( ( A2 @ ( B2 @ V ) )
% 2.92/1.57 = ( C3 @ ( D2 @ V ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_124_type__copy__map__cong0,axiom,
% 2.92/1.57 ! [B: $tType,D: $tType,E: $tType,A: $tType,C: $tType,M: B > A,G: C > B,X: C,N4: D > A,H: C > D,F: A > E] :
% 2.92/1.57 ( ( ( M @ ( G @ X ) )
% 2.92/1.57 = ( N4 @ ( H @ X ) ) )
% 2.92/1.57 => ( ( comp @ B @ E @ C @ ( comp @ A @ E @ B @ F @ M ) @ G @ X )
% 2.92/1.57 = ( comp @ D @ E @ C @ ( comp @ A @ E @ D @ F @ N4 ) @ H @ X ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_125_fun_Omap__comp,axiom,
% 2.92/1.57 ! [B: $tType,C: $tType,A: $tType,D: $tType,G: B > C,F: A > B,V: D > A] :
% 2.92/1.57 ( ( comp @ B @ C @ D @ G @ ( comp @ A @ B @ D @ F @ V ) )
% 2.92/1.57 = ( comp @ A @ C @ D @ ( comp @ B @ C @ A @ G @ F ) @ V ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_126_comp__cong,axiom,
% 2.92/1.57 ! [C: $tType,B: $tType,D: $tType,A: $tType,E: $tType,F: B > A,G: C > B,X: C,F4: D > A,G3: E > D,X8: E] :
% 2.92/1.57 ( ( ( F @ ( G @ X ) )
% 2.92/1.57 = ( F4 @ ( G3 @ X8 ) ) )
% 2.92/1.57 => ( ( comp @ B @ A @ C @ F @ G @ X )
% 2.92/1.57 = ( comp @ D @ A @ E @ F4 @ G3 @ X8 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_127_finite__empty__induct,axiom,
% 2.92/1.57 ! [A: $tType,A7: set @ A,P: ( set @ A ) > $o] :
% 2.92/1.57 ( ( finite_finite2 @ A @ A7 )
% 2.92/1.57 => ( ( P @ A7 )
% 2.92/1.57 => ( ! [A4: A,A9: set @ A] :
% 2.92/1.57 ( ( finite_finite2 @ A @ A9 )
% 2.92/1.57 => ( ( member @ A @ A4 @ A9 )
% 2.92/1.57 => ( ( P @ A9 )
% 2.92/1.57 => ( P @ ( minus_minus @ ( set @ A ) @ A9 @ ( insert @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) )
% 2.92/1.57 => ( P @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_128_empty__Collect__eq,axiom,
% 2.92/1.57 ! [A: $tType,P: A > $o] :
% 2.92/1.57 ( ( ( bot_bot @ ( set @ A ) )
% 2.92/1.57 = ( collect @ A @ P ) )
% 2.92/1.57 = ( ! [X5: A] :
% 2.92/1.57 ~ ( P @ X5 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_129_Collect__empty__eq,axiom,
% 2.92/1.57 ! [A: $tType,P: A > $o] :
% 2.92/1.57 ( ( ( collect @ A @ P )
% 2.92/1.57 = ( bot_bot @ ( set @ A ) ) )
% 2.92/1.57 = ( ! [X5: A] :
% 2.92/1.57 ~ ( P @ X5 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_130_all__not__in__conv,axiom,
% 2.92/1.57 ! [A: $tType,A7: set @ A] :
% 2.92/1.57 ( ( ! [X5: A] :
% 2.92/1.57 ~ ( member @ A @ X5 @ A7 ) )
% 2.92/1.57 = ( A7
% 2.92/1.57 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_131_empty__iff,axiom,
% 2.92/1.57 ! [A: $tType,C3: A] :
% 2.92/1.57 ~ ( member @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_132_singletonI,axiom,
% 2.92/1.57 ! [A: $tType,A2: A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_133_Diff__cancel,axiom,
% 2.92/1.57 ! [A: $tType,A7: set @ A] :
% 2.92/1.57 ( ( minus_minus @ ( set @ A ) @ A7 @ A7 )
% 2.92/1.57 = ( bot_bot @ ( set @ A ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_134_empty__Diff,axiom,
% 2.92/1.57 ! [A: $tType,A7: set @ A] :
% 2.92/1.57 ( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A7 )
% 2.92/1.57 = ( bot_bot @ ( set @ A ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_135_Diff__empty,axiom,
% 2.92/1.57 ! [A: $tType,A7: set @ A] :
% 2.92/1.57 ( ( minus_minus @ ( set @ A ) @ A7 @ ( bot_bot @ ( set @ A ) ) )
% 2.92/1.57 = A7 ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_136_Plus__eq__empty__conv,axiom,
% 2.92/1.57 ! [A: $tType,B: $tType,A7: set @ A,B7: set @ B] :
% 2.92/1.57 ( ( ( sum_Plus @ A @ B @ A7 @ B7 )
% 2.92/1.57 = ( bot_bot @ ( set @ ( sum_sum @ A @ B ) ) ) )
% 2.92/1.57 = ( ( A7
% 2.92/1.57 = ( bot_bot @ ( set @ A ) ) )
% 2.92/1.57 & ( B7
% 2.92/1.57 = ( bot_bot @ ( set @ B ) ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_137_insert__Diff__single,axiom,
% 2.92/1.57 ! [A: $tType,A2: A,A7: set @ A] :
% 2.92/1.57 ( ( insert @ A @ A2 @ ( minus_minus @ ( set @ A ) @ A7 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.92/1.57 = ( insert @ A @ A2 @ A7 ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_138_finite_OemptyI,axiom,
% 2.92/1.57 ! [A: $tType] : ( finite_finite2 @ A @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_139_infinite__imp__nonempty,axiom,
% 2.92/1.57 ! [A: $tType,S2: set @ A] :
% 2.92/1.57 ( ~ ( finite_finite2 @ A @ S2 )
% 2.92/1.57 => ( S2
% 2.92/1.57 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_140_singletonD,axiom,
% 2.92/1.57 ! [A: $tType,B2: A,A2: A] :
% 2.92/1.57 ( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.92/1.57 => ( B2 = A2 ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_141_singleton__iff,axiom,
% 2.92/1.57 ! [A: $tType,B2: A,A2: A] :
% 2.92/1.57 ( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.92/1.57 = ( B2 = A2 ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_142_doubleton__eq__iff,axiom,
% 2.92/1.57 ! [A: $tType,A2: A,B2: A,C3: A,D2: A] :
% 2.92/1.57 ( ( ( insert @ A @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.92/1.57 = ( insert @ A @ C3 @ ( insert @ A @ D2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.92/1.57 = ( ( ( A2 = C3 )
% 2.92/1.57 & ( B2 = D2 ) )
% 2.92/1.57 | ( ( A2 = D2 )
% 2.92/1.57 & ( B2 = C3 ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_143_insert__not__empty,axiom,
% 2.92/1.57 ! [A: $tType,A2: A,A7: set @ A] :
% 2.92/1.57 ( ( insert @ A @ A2 @ A7 )
% 2.92/1.57 != ( bot_bot @ ( set @ A ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_144_singleton__inject,axiom,
% 2.92/1.57 ! [A: $tType,A2: A,B2: A] :
% 2.92/1.57 ( ( ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) )
% 2.92/1.57 = ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.92/1.57 => ( A2 = B2 ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_145_ex__in__conv,axiom,
% 2.92/1.57 ! [A: $tType,A7: set @ A] :
% 2.92/1.57 ( ( ? [X5: A] : ( member @ A @ X5 @ A7 ) )
% 2.92/1.57 = ( A7
% 2.92/1.57 != ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_146_equals0I,axiom,
% 2.92/1.57 ! [A: $tType,A7: set @ A] :
% 2.92/1.57 ( ! [Y3: A] :
% 2.92/1.57 ~ ( member @ A @ Y3 @ A7 )
% 2.92/1.57 => ( A7
% 2.92/1.57 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_147_equals0D,axiom,
% 2.92/1.57 ! [A: $tType,A7: set @ A,A2: A] :
% 2.92/1.57 ( ( A7
% 2.92/1.57 = ( bot_bot @ ( set @ A ) ) )
% 2.92/1.57 => ~ ( member @ A @ A2 @ A7 ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_148_emptyE,axiom,
% 2.92/1.57 ! [A: $tType,A2: A] :
% 2.92/1.57 ~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_149_infinite__finite__induct,axiom,
% 2.92/1.57 ! [A: $tType,P: ( set @ A ) > $o,A7: set @ A] :
% 2.92/1.57 ( ! [A9: set @ A] :
% 2.92/1.57 ( ~ ( finite_finite2 @ A @ A9 )
% 2.92/1.57 => ( P @ A9 ) )
% 2.92/1.57 => ( ( P @ ( bot_bot @ ( set @ A ) ) )
% 2.92/1.57 => ( ! [X6: A,F6: set @ A] :
% 2.92/1.57 ( ( finite_finite2 @ A @ F6 )
% 2.92/1.57 => ( ~ ( member @ A @ X6 @ F6 )
% 2.92/1.57 => ( ( P @ F6 )
% 2.92/1.57 => ( P @ ( insert @ A @ X6 @ F6 ) ) ) ) )
% 2.92/1.57 => ( P @ A7 ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_150_finite__ne__induct,axiom,
% 2.92/1.57 ! [A: $tType,F7: set @ A,P: ( set @ A ) > $o] :
% 2.92/1.57 ( ( finite_finite2 @ A @ F7 )
% 2.92/1.57 => ( ( F7
% 2.92/1.57 != ( bot_bot @ ( set @ A ) ) )
% 2.92/1.57 => ( ! [X6: A] : ( P @ ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.92/1.57 => ( ! [X6: A,F6: set @ A] :
% 2.92/1.57 ( ( finite_finite2 @ A @ F6 )
% 2.92/1.57 => ( ( F6
% 2.92/1.57 != ( bot_bot @ ( set @ A ) ) )
% 2.92/1.57 => ( ~ ( member @ A @ X6 @ F6 )
% 2.92/1.57 => ( ( P @ F6 )
% 2.92/1.57 => ( P @ ( insert @ A @ X6 @ F6 ) ) ) ) ) )
% 2.92/1.57 => ( P @ F7 ) ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_151_finite_Oinducts,axiom,
% 2.92/1.57 ! [A: $tType,X: set @ A,P: ( set @ A ) > $o] :
% 2.92/1.57 ( ( finite_finite2 @ A @ X )
% 2.92/1.57 => ( ( P @ ( bot_bot @ ( set @ A ) ) )
% 2.92/1.57 => ( ! [A9: set @ A,A4: A] :
% 2.92/1.57 ( ( finite_finite2 @ A @ A9 )
% 2.92/1.57 => ( ( P @ A9 )
% 2.92/1.57 => ( P @ ( insert @ A @ A4 @ A9 ) ) ) )
% 2.92/1.57 => ( P @ X ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_152_finite__induct,axiom,
% 2.92/1.57 ! [A: $tType,F7: set @ A,P: ( set @ A ) > $o] :
% 2.92/1.57 ( ( finite_finite2 @ A @ F7 )
% 2.92/1.57 => ( ( P @ ( bot_bot @ ( set @ A ) ) )
% 2.92/1.57 => ( ! [X6: A,F6: set @ A] :
% 2.92/1.57 ( ( finite_finite2 @ A @ F6 )
% 2.92/1.57 => ( ~ ( member @ A @ X6 @ F6 )
% 2.92/1.57 => ( ( P @ F6 )
% 2.92/1.57 => ( P @ ( insert @ A @ X6 @ F6 ) ) ) ) )
% 2.92/1.57 => ( P @ F7 ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_153_finite_Osimps,axiom,
% 2.92/1.57 ! [A: $tType] :
% 2.92/1.57 ( ( finite_finite2 @ A )
% 2.92/1.57 = ( ^ [A10: set @ A] :
% 2.92/1.57 ( ( A10
% 2.92/1.57 = ( bot_bot @ ( set @ A ) ) )
% 2.92/1.57 | ? [A8: set @ A,B10: A] :
% 2.92/1.57 ( ( A10
% 2.92/1.57 = ( insert @ A @ B10 @ A8 ) )
% 2.92/1.57 & ( finite_finite2 @ A @ A8 ) ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_154_finite_Ocases,axiom,
% 2.92/1.57 ! [A: $tType,A2: set @ A] :
% 2.92/1.57 ( ( finite_finite2 @ A @ A2 )
% 2.92/1.57 => ( ( A2
% 2.92/1.57 != ( bot_bot @ ( set @ A ) ) )
% 2.92/1.57 => ~ ! [A9: set @ A] :
% 2.92/1.57 ( ? [A4: A] :
% 2.92/1.57 ( A2
% 2.92/1.57 = ( insert @ A @ A4 @ A9 ) )
% 2.92/1.57 => ~ ( finite_finite2 @ A @ A9 ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_155_Diff__insert,axiom,
% 2.92/1.57 ! [A: $tType,A7: set @ A,A2: A,B7: set @ A] :
% 2.92/1.57 ( ( minus_minus @ ( set @ A ) @ A7 @ ( insert @ A @ A2 @ B7 ) )
% 2.92/1.57 = ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A7 @ B7 ) @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_156_insert__Diff,axiom,
% 2.92/1.57 ! [A: $tType,A2: A,A7: set @ A] :
% 2.92/1.57 ( ( member @ A @ A2 @ A7 )
% 2.92/1.57 => ( ( insert @ A @ A2 @ ( minus_minus @ ( set @ A ) @ A7 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.92/1.57 = A7 ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_157_Diff__insert2,axiom,
% 2.92/1.57 ! [A: $tType,A7: set @ A,A2: A,B7: set @ A] :
% 2.92/1.57 ( ( minus_minus @ ( set @ A ) @ A7 @ ( insert @ A @ A2 @ B7 ) )
% 2.92/1.57 = ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A7 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) @ B7 ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_158_Diff__insert__absorb,axiom,
% 2.92/1.57 ! [A: $tType,X: A,A7: set @ A] :
% 2.92/1.57 ( ~ ( member @ A @ X @ A7 )
% 2.92/1.57 => ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A7 ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 2.92/1.57 = A7 ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_159_infinite__remove,axiom,
% 2.92/1.57 ! [A: $tType,S2: set @ A,A2: A] :
% 2.92/1.57 ( ~ ( finite_finite2 @ A @ S2 )
% 2.92/1.57 => ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ S2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_160_infinite__coinduct,axiom,
% 2.92/1.57 ! [A: $tType,X9: ( set @ A ) > $o,A7: set @ A] :
% 2.92/1.57 ( ( X9 @ A7 )
% 2.92/1.57 => ( ! [A9: set @ A] :
% 2.92/1.57 ( ( X9 @ A9 )
% 2.92/1.57 => ? [X7: A] :
% 2.92/1.57 ( ( member @ A @ X7 @ A9 )
% 2.92/1.57 & ( ( X9 @ ( minus_minus @ ( set @ A ) @ A9 @ ( insert @ A @ X7 @ ( bot_bot @ ( set @ A ) ) ) ) )
% 2.92/1.57 | ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A9 @ ( insert @ A @ X7 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) )
% 2.92/1.57 => ~ ( finite_finite2 @ A @ A7 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_161_is__singletonI,axiom,
% 2.92/1.57 ! [A: $tType,X: A] : ( is_singleton @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_162_remove__def,axiom,
% 2.92/1.57 ! [A: $tType] :
% 2.92/1.57 ( ( remove @ A )
% 2.92/1.57 = ( ^ [X5: A,A8: set @ A] : ( minus_minus @ ( set @ A ) @ A8 @ ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_163_the__elem__eq,axiom,
% 2.92/1.57 ! [A: $tType,X: A] :
% 2.92/1.57 ( ( the_elem @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 2.92/1.57 = X ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_164_sum__set__simps_I1_J,axiom,
% 2.92/1.57 ! [B: $tType,A: $tType,X: A] :
% 2.92/1.57 ( ( basic_setl @ A @ B @ ( sum_Inl @ A @ B @ X ) )
% 2.92/1.57 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_165_member__remove,axiom,
% 2.92/1.57 ! [A: $tType,X: A,Y: A,A7: set @ A] :
% 2.92/1.57 ( ( member @ A @ X @ ( remove @ A @ Y @ A7 ) )
% 2.92/1.57 = ( ( member @ A @ X @ A7 )
% 2.92/1.57 & ( X != Y ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_166_sum__set__simps_I2_J,axiom,
% 2.92/1.57 ! [A: $tType,C: $tType,X: A] :
% 2.92/1.57 ( ( basic_setl @ C @ A @ ( sum_Inr @ A @ C @ X ) )
% 2.92/1.57 = ( bot_bot @ ( set @ C ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_167_setl_Ocases,axiom,
% 2.92/1.57 ! [B: $tType,A: $tType,A2: A,S: sum_sum @ A @ B] :
% 2.92/1.57 ( ( member @ A @ A2 @ ( basic_setl @ A @ B @ S ) )
% 2.92/1.57 => ( S
% 2.92/1.57 = ( sum_Inl @ A @ B @ A2 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_168_setl_Osimps,axiom,
% 2.92/1.57 ! [B: $tType,A: $tType,A2: A,S: sum_sum @ A @ B] :
% 2.92/1.57 ( ( member @ A @ A2 @ ( basic_setl @ A @ B @ S ) )
% 2.92/1.57 = ( ? [X5: A] :
% 2.92/1.57 ( ( A2 = X5 )
% 2.92/1.57 & ( S
% 2.92/1.57 = ( sum_Inl @ A @ B @ X5 ) ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_169_setl_Ointros,axiom,
% 2.92/1.57 ! [B: $tType,A: $tType,S: sum_sum @ A @ B,X: A] :
% 2.92/1.57 ( ( S
% 2.92/1.57 = ( sum_Inl @ A @ B @ X ) )
% 2.92/1.57 => ( member @ A @ X @ ( basic_setl @ A @ B @ S ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_170_setl_Oinducts,axiom,
% 2.92/1.57 ! [B: $tType,A: $tType,X: A,S: sum_sum @ A @ B,P: A > $o] :
% 2.92/1.57 ( ( member @ A @ X @ ( basic_setl @ A @ B @ S ) )
% 2.92/1.57 => ( ! [X6: A] :
% 2.92/1.57 ( ( S
% 2.92/1.57 = ( sum_Inl @ A @ B @ X6 ) )
% 2.92/1.57 => ( P @ X6 ) )
% 2.92/1.57 => ( P @ X ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_171_is__singleton__the__elem,axiom,
% 2.92/1.57 ! [A: $tType] :
% 2.92/1.57 ( ( is_singleton @ A )
% 2.92/1.57 = ( ^ [A8: set @ A] :
% 2.92/1.57 ( A8
% 2.92/1.57 = ( insert @ A @ ( the_elem @ A @ A8 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_172_is__singletonI_H,axiom,
% 2.92/1.57 ! [A: $tType,A7: set @ A] :
% 2.92/1.57 ( ( A7
% 2.92/1.57 != ( bot_bot @ ( set @ A ) ) )
% 2.92/1.57 => ( ! [X6: A,Y3: A] :
% 2.92/1.57 ( ( member @ A @ X6 @ A7 )
% 2.92/1.57 => ( ( member @ A @ Y3 @ A7 )
% 2.92/1.57 => ( X6 = Y3 ) ) )
% 2.92/1.57 => ( is_singleton @ A @ A7 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_173_is__singleton__def,axiom,
% 2.92/1.57 ! [A: $tType] :
% 2.92/1.57 ( ( is_singleton @ A )
% 2.92/1.57 = ( ^ [A8: set @ A] :
% 2.92/1.57 ? [X5: A] :
% 2.92/1.57 ( A8
% 2.92/1.57 = ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_174_is__singletonE,axiom,
% 2.92/1.57 ! [A: $tType,A7: set @ A] :
% 2.92/1.57 ( ( is_singleton @ A @ A7 )
% 2.92/1.57 => ~ ! [X6: A] :
% 2.92/1.57 ( A7
% 2.92/1.57 != ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_175_sum__set__simps_I4_J,axiom,
% 2.92/1.57 ! [E: $tType,A: $tType,X: A] :
% 2.92/1.57 ( ( basic_setr @ E @ A @ ( sum_Inr @ A @ E @ X ) )
% 2.92/1.57 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_176_Set_Ois__empty__def,axiom,
% 2.92/1.57 ! [A: $tType] :
% 2.92/1.57 ( ( is_empty @ A )
% 2.92/1.57 = ( ^ [A8: set @ A] :
% 2.92/1.57 ( A8
% 2.92/1.57 = ( bot_bot @ ( set @ A ) ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_177_remove__induct,axiom,
% 2.92/1.57 ! [A: $tType,P: ( set @ A ) > $o,B7: set @ A] :
% 2.92/1.57 ( ( P @ ( bot_bot @ ( set @ A ) ) )
% 2.92/1.57 => ( ( ~ ( finite_finite2 @ A @ B7 )
% 2.92/1.57 => ( P @ B7 ) )
% 2.92/1.57 => ( ! [A9: set @ A] :
% 2.92/1.57 ( ( finite_finite2 @ A @ A9 )
% 2.92/1.57 => ( ( A9
% 2.92/1.57 != ( bot_bot @ ( set @ A ) ) )
% 2.92/1.57 => ( ( ord_less_eq @ ( set @ A ) @ A9 @ B7 )
% 2.92/1.57 => ( ! [X7: A] :
% 2.92/1.57 ( ( member @ A @ X7 @ A9 )
% 2.92/1.57 => ( P @ ( minus_minus @ ( set @ A ) @ A9 @ ( insert @ A @ X7 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
% 2.92/1.57 => ( P @ A9 ) ) ) ) )
% 2.92/1.57 => ( P @ B7 ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_178_subset__antisym,axiom,
% 2.92/1.57 ! [A: $tType,A7: set @ A,B7: set @ A] :
% 2.92/1.57 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
% 2.92/1.57 => ( ( ord_less_eq @ ( set @ A ) @ B7 @ A7 )
% 2.92/1.57 => ( A7 = B7 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_179_subsetI,axiom,
% 2.92/1.57 ! [A: $tType,A7: set @ A,B7: set @ A] :
% 2.92/1.57 ( ! [X6: A] :
% 2.92/1.57 ( ( member @ A @ X6 @ A7 )
% 2.92/1.57 => ( member @ A @ X6 @ B7 ) )
% 2.92/1.57 => ( ord_less_eq @ ( set @ A ) @ A7 @ B7 ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_180_subset__empty,axiom,
% 2.92/1.57 ! [A: $tType,A7: set @ A] :
% 2.92/1.57 ( ( ord_less_eq @ ( set @ A ) @ A7 @ ( bot_bot @ ( set @ A ) ) )
% 2.92/1.57 = ( A7
% 2.92/1.57 = ( bot_bot @ ( set @ A ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_181_empty__subsetI,axiom,
% 2.92/1.57 ! [A: $tType,A7: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A7 ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_182_insert__subset,axiom,
% 2.92/1.57 ! [A: $tType,X: A,A7: set @ A,B7: set @ A] :
% 2.92/1.57 ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A7 ) @ B7 )
% 2.92/1.57 = ( ( member @ A @ X @ B7 )
% 2.92/1.57 & ( ord_less_eq @ ( set @ A ) @ A7 @ B7 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_183_singleton__insert__inj__eq,axiom,
% 2.92/1.57 ! [A: $tType,B2: A,A2: A,A7: set @ A] :
% 2.92/1.57 ( ( ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) )
% 2.92/1.57 = ( insert @ A @ A2 @ A7 ) )
% 2.92/1.57 = ( ( A2 = B2 )
% 2.92/1.57 & ( ord_less_eq @ ( set @ A ) @ A7 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_184_singleton__insert__inj__eq_H,axiom,
% 2.92/1.57 ! [A: $tType,A2: A,A7: set @ A,B2: A] :
% 2.92/1.57 ( ( ( insert @ A @ A2 @ A7 )
% 2.92/1.57 = ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.92/1.57 = ( ( A2 = B2 )
% 2.92/1.57 & ( ord_less_eq @ ( set @ A ) @ A7 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_185_Diff__eq__empty__iff,axiom,
% 2.92/1.57 ! [A: $tType,A7: set @ A,B7: set @ A] :
% 2.92/1.57 ( ( ( minus_minus @ ( set @ A ) @ A7 @ B7 )
% 2.92/1.57 = ( bot_bot @ ( set @ A ) ) )
% 2.92/1.57 = ( ord_less_eq @ ( set @ A ) @ A7 @ B7 ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_186_sum__set__simps_I3_J,axiom,
% 2.92/1.57 ! [A: $tType,D: $tType,X: A] :
% 2.92/1.57 ( ( basic_setr @ A @ D @ ( sum_Inl @ A @ D @ X ) )
% 2.92/1.57 = ( bot_bot @ ( set @ D ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_187_inFr__mono,axiom,
% 2.92/1.57 ! [Ns: set @ n,Tr: dtree,T2: t,Ns4: set @ n] :
% 2.92/1.57 ( ( gram_L1333338417e_inFr @ Ns @ Tr @ T2 )
% 2.92/1.57 => ( ( ord_less_eq @ ( set @ n ) @ Ns @ Ns4 )
% 2.92/1.57 => ( gram_L1333338417e_inFr @ Ns4 @ Tr @ T2 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_188_setr_Oinducts,axiom,
% 2.92/1.57 ! [A: $tType,B: $tType,X: B,S: sum_sum @ A @ B,P: B > $o] :
% 2.92/1.57 ( ( member @ B @ X @ ( basic_setr @ A @ B @ S ) )
% 2.92/1.57 => ( ! [X6: B] :
% 2.92/1.57 ( ( S
% 2.92/1.57 = ( sum_Inr @ B @ A @ X6 ) )
% 2.92/1.57 => ( P @ X6 ) )
% 2.92/1.57 => ( P @ X ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_189_setr_Ointros,axiom,
% 2.92/1.57 ! [B: $tType,A: $tType,S: sum_sum @ A @ B,X: B] :
% 2.92/1.57 ( ( S
% 2.92/1.57 = ( sum_Inr @ B @ A @ X ) )
% 2.92/1.57 => ( member @ B @ X @ ( basic_setr @ A @ B @ S ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_190_setr_Osimps,axiom,
% 2.92/1.57 ! [A: $tType,B: $tType,A2: B,S: sum_sum @ A @ B] :
% 2.92/1.57 ( ( member @ B @ A2 @ ( basic_setr @ A @ B @ S ) )
% 2.92/1.57 = ( ? [X5: B] :
% 2.92/1.57 ( ( A2 = X5 )
% 2.92/1.57 & ( S
% 2.92/1.57 = ( sum_Inr @ B @ A @ X5 ) ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_191_setr_Ocases,axiom,
% 2.92/1.57 ! [A: $tType,B: $tType,A2: B,S: sum_sum @ A @ B] :
% 2.92/1.57 ( ( member @ B @ A2 @ ( basic_setr @ A @ B @ S ) )
% 2.92/1.57 => ( S
% 2.92/1.57 = ( sum_Inr @ B @ A @ A2 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_192_finite__subset,axiom,
% 2.92/1.57 ! [A: $tType,A7: set @ A,B7: set @ A] :
% 2.92/1.57 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
% 2.92/1.57 => ( ( finite_finite2 @ A @ B7 )
% 2.92/1.57 => ( finite_finite2 @ A @ A7 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_193_infinite__super,axiom,
% 2.92/1.57 ! [A: $tType,S2: set @ A,T5: set @ A] :
% 2.92/1.57 ( ( ord_less_eq @ ( set @ A ) @ S2 @ T5 )
% 2.92/1.57 => ( ~ ( finite_finite2 @ A @ S2 )
% 2.92/1.57 => ~ ( finite_finite2 @ A @ T5 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_194_rev__finite__subset,axiom,
% 2.92/1.57 ! [A: $tType,B7: set @ A,A7: set @ A] :
% 2.92/1.57 ( ( finite_finite2 @ A @ B7 )
% 2.92/1.57 => ( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
% 2.92/1.57 => ( finite_finite2 @ A @ A7 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_195_insert__mono,axiom,
% 2.92/1.57 ! [A: $tType,C4: set @ A,D3: set @ A,A2: A] :
% 2.92/1.57 ( ( ord_less_eq @ ( set @ A ) @ C4 @ D3 )
% 2.92/1.57 => ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A2 @ C4 ) @ ( insert @ A @ A2 @ D3 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_196_subset__insert,axiom,
% 2.92/1.57 ! [A: $tType,X: A,A7: set @ A,B7: set @ A] :
% 2.92/1.57 ( ~ ( member @ A @ X @ A7 )
% 2.92/1.57 => ( ( ord_less_eq @ ( set @ A ) @ A7 @ ( insert @ A @ X @ B7 ) )
% 2.92/1.57 = ( ord_less_eq @ ( set @ A ) @ A7 @ B7 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_197_subset__insertI,axiom,
% 2.92/1.57 ! [A: $tType,B7: set @ A,A2: A] : ( ord_less_eq @ ( set @ A ) @ B7 @ ( insert @ A @ A2 @ B7 ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_198_subset__insertI2,axiom,
% 2.92/1.57 ! [A: $tType,A7: set @ A,B7: set @ A,B2: A] :
% 2.92/1.57 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
% 2.92/1.57 => ( ord_less_eq @ ( set @ A ) @ A7 @ ( insert @ A @ B2 @ B7 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_199_Diff__mono,axiom,
% 2.92/1.57 ! [A: $tType,A7: set @ A,C4: set @ A,D3: set @ A,B7: set @ A] :
% 2.92/1.57 ( ( ord_less_eq @ ( set @ A ) @ A7 @ C4 )
% 2.92/1.57 => ( ( ord_less_eq @ ( set @ A ) @ D3 @ B7 )
% 2.92/1.57 => ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A7 @ B7 ) @ ( minus_minus @ ( set @ A ) @ C4 @ D3 ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_200_Diff__subset,axiom,
% 2.92/1.57 ! [A: $tType,A7: set @ A,B7: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A7 @ B7 ) @ A7 ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_201_double__diff,axiom,
% 2.92/1.57 ! [A: $tType,A7: set @ A,B7: set @ A,C4: set @ A] :
% 2.92/1.57 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
% 2.92/1.57 => ( ( ord_less_eq @ ( set @ A ) @ B7 @ C4 )
% 2.92/1.57 => ( ( minus_minus @ ( set @ A ) @ B7 @ ( minus_minus @ ( set @ A ) @ C4 @ A7 ) )
% 2.92/1.57 = A7 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_202_Collect__mono__iff,axiom,
% 2.92/1.57 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.92/1.57 ( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
% 2.92/1.57 = ( ! [X5: A] :
% 2.92/1.57 ( ( P @ X5 )
% 2.92/1.57 => ( Q @ X5 ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_203_contra__subsetD,axiom,
% 2.92/1.57 ! [A: $tType,A7: set @ A,B7: set @ A,C3: A] :
% 2.92/1.57 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
% 2.92/1.57 => ( ~ ( member @ A @ C3 @ B7 )
% 2.92/1.57 => ~ ( member @ A @ C3 @ A7 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_204_set__eq__subset,axiom,
% 2.92/1.57 ! [A: $tType] :
% 2.92/1.57 ( ( ^ [Y4: set @ A,Z: set @ A] : ( Y4 = Z ) )
% 2.92/1.57 = ( ^ [A8: set @ A,B9: set @ A] :
% 2.92/1.57 ( ( ord_less_eq @ ( set @ A ) @ A8 @ B9 )
% 2.92/1.57 & ( ord_less_eq @ ( set @ A ) @ B9 @ A8 ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_205_subset__trans,axiom,
% 2.92/1.57 ! [A: $tType,A7: set @ A,B7: set @ A,C4: set @ A] :
% 2.92/1.57 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
% 2.92/1.57 => ( ( ord_less_eq @ ( set @ A ) @ B7 @ C4 )
% 2.92/1.57 => ( ord_less_eq @ ( set @ A ) @ A7 @ C4 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_206_Collect__mono,axiom,
% 2.92/1.57 ! [A: $tType,P: A > $o,Q: A > $o] :
% 2.92/1.57 ( ! [X6: A] :
% 2.92/1.57 ( ( P @ X6 )
% 2.92/1.57 => ( Q @ X6 ) )
% 2.92/1.57 => ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_207_subset__refl,axiom,
% 2.92/1.57 ! [A: $tType,A7: set @ A] : ( ord_less_eq @ ( set @ A ) @ A7 @ A7 ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_208_rev__subsetD,axiom,
% 2.92/1.57 ! [A: $tType,C3: A,A7: set @ A,B7: set @ A] :
% 2.92/1.57 ( ( member @ A @ C3 @ A7 )
% 2.92/1.57 => ( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
% 2.92/1.57 => ( member @ A @ C3 @ B7 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_209_subset__iff,axiom,
% 2.92/1.57 ! [A: $tType] :
% 2.92/1.57 ( ( ord_less_eq @ ( set @ A ) )
% 2.92/1.57 = ( ^ [A8: set @ A,B9: set @ A] :
% 2.92/1.57 ! [T3: A] :
% 2.92/1.57 ( ( member @ A @ T3 @ A8 )
% 2.92/1.57 => ( member @ A @ T3 @ B9 ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_210_set__rev__mp,axiom,
% 2.92/1.57 ! [A: $tType,X: A,A7: set @ A,B7: set @ A] :
% 2.92/1.57 ( ( member @ A @ X @ A7 )
% 2.92/1.57 => ( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
% 2.92/1.57 => ( member @ A @ X @ B7 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_211_equalityD2,axiom,
% 2.92/1.57 ! [A: $tType,A7: set @ A,B7: set @ A] :
% 2.92/1.57 ( ( A7 = B7 )
% 2.92/1.57 => ( ord_less_eq @ ( set @ A ) @ B7 @ A7 ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_212_equalityD1,axiom,
% 2.92/1.57 ! [A: $tType,A7: set @ A,B7: set @ A] :
% 2.92/1.57 ( ( A7 = B7 )
% 2.92/1.57 => ( ord_less_eq @ ( set @ A ) @ A7 @ B7 ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_213_subset__eq,axiom,
% 2.92/1.57 ! [A: $tType] :
% 2.92/1.57 ( ( ord_less_eq @ ( set @ A ) )
% 2.92/1.57 = ( ^ [A8: set @ A,B9: set @ A] :
% 2.92/1.57 ! [X5: A] :
% 2.92/1.57 ( ( member @ A @ X5 @ A8 )
% 2.92/1.57 => ( member @ A @ X5 @ B9 ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_214_equalityE,axiom,
% 2.92/1.57 ! [A: $tType,A7: set @ A,B7: set @ A] :
% 2.92/1.57 ( ( A7 = B7 )
% 2.92/1.57 => ~ ( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
% 2.92/1.57 => ~ ( ord_less_eq @ ( set @ A ) @ B7 @ A7 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_215_subsetCE,axiom,
% 2.92/1.57 ! [A: $tType,A7: set @ A,B7: set @ A,C3: A] :
% 2.92/1.57 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
% 2.92/1.57 => ( ( member @ A @ C3 @ A7 )
% 2.92/1.57 => ( member @ A @ C3 @ B7 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_216_subsetD,axiom,
% 2.92/1.57 ! [A: $tType,A7: set @ A,B7: set @ A,C3: A] :
% 2.92/1.57 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
% 2.92/1.57 => ( ( member @ A @ C3 @ A7 )
% 2.92/1.57 => ( member @ A @ C3 @ B7 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_217_in__mono,axiom,
% 2.92/1.57 ! [A: $tType,A7: set @ A,B7: set @ A,X: A] :
% 2.92/1.57 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
% 2.92/1.57 => ( ( member @ A @ X @ A7 )
% 2.92/1.57 => ( member @ A @ X @ B7 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_218_set__mp,axiom,
% 2.92/1.57 ! [A: $tType,A7: set @ A,B7: set @ A,X: A] :
% 2.92/1.57 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
% 2.92/1.57 => ( ( member @ A @ X @ A7 )
% 2.92/1.57 => ( member @ A @ X @ B7 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_219_subset__singletonD,axiom,
% 2.92/1.57 ! [A: $tType,A7: set @ A,X: A] :
% 2.92/1.57 ( ( ord_less_eq @ ( set @ A ) @ A7 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
% 2.92/1.57 => ( ( A7
% 2.92/1.57 = ( bot_bot @ ( set @ A ) ) )
% 2.92/1.57 | ( A7
% 2.92/1.57 = ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_220_subset__singleton__iff,axiom,
% 2.92/1.57 ! [A: $tType,X9: set @ A,A2: A] :
% 2.92/1.57 ( ( ord_less_eq @ ( set @ A ) @ X9 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
% 2.92/1.57 = ( ( X9
% 2.92/1.57 = ( bot_bot @ ( set @ A ) ) )
% 2.92/1.57 | ( X9
% 2.92/1.57 = ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_221_subset__Diff__insert,axiom,
% 2.92/1.57 ! [A: $tType,A7: set @ A,B7: set @ A,X: A,C4: set @ A] :
% 2.92/1.57 ( ( ord_less_eq @ ( set @ A ) @ A7 @ ( minus_minus @ ( set @ A ) @ B7 @ ( insert @ A @ X @ C4 ) ) )
% 2.92/1.57 = ( ( ord_less_eq @ ( set @ A ) @ A7 @ ( minus_minus @ ( set @ A ) @ B7 @ C4 ) )
% 2.92/1.57 & ~ ( member @ A @ X @ A7 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_222_finite__subset__induct,axiom,
% 2.92/1.57 ! [A: $tType,F7: set @ A,A7: set @ A,P: ( set @ A ) > $o] :
% 2.92/1.57 ( ( finite_finite2 @ A @ F7 )
% 2.92/1.57 => ( ( ord_less_eq @ ( set @ A ) @ F7 @ A7 )
% 2.92/1.57 => ( ( P @ ( bot_bot @ ( set @ A ) ) )
% 2.92/1.57 => ( ! [A4: A,F6: set @ A] :
% 2.92/1.57 ( ( finite_finite2 @ A @ F6 )
% 2.92/1.57 => ( ( member @ A @ A4 @ A7 )
% 2.92/1.57 => ( ~ ( member @ A @ A4 @ F6 )
% 2.92/1.57 => ( ( P @ F6 )
% 2.92/1.57 => ( P @ ( insert @ A @ A4 @ F6 ) ) ) ) ) )
% 2.92/1.57 => ( P @ F7 ) ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_223_subset__insert__iff,axiom,
% 2.92/1.57 ! [A: $tType,A7: set @ A,X: A,B7: set @ A] :
% 2.92/1.57 ( ( ord_less_eq @ ( set @ A ) @ A7 @ ( insert @ A @ X @ B7 ) )
% 2.92/1.57 = ( ( ( member @ A @ X @ A7 )
% 2.92/1.57 => ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A7 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B7 ) )
% 2.92/1.57 & ( ~ ( member @ A @ X @ A7 )
% 2.92/1.57 => ( ord_less_eq @ ( set @ A ) @ A7 @ B7 ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_224_Diff__single__insert,axiom,
% 2.92/1.57 ! [A: $tType,A7: set @ A,X: A,B7: set @ A] :
% 2.92/1.57 ( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A7 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B7 )
% 2.92/1.57 => ( ord_less_eq @ ( set @ A ) @ A7 @ ( insert @ A @ X @ B7 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_225_finite__remove__induct,axiom,
% 2.92/1.57 ! [A: $tType,B7: set @ A,P: ( set @ A ) > $o] :
% 2.92/1.57 ( ( finite_finite2 @ A @ B7 )
% 2.92/1.57 => ( ( P @ ( bot_bot @ ( set @ A ) ) )
% 2.92/1.57 => ( ! [A9: set @ A] :
% 2.92/1.57 ( ( finite_finite2 @ A @ A9 )
% 2.92/1.57 => ( ( A9
% 2.92/1.57 != ( bot_bot @ ( set @ A ) ) )
% 2.92/1.57 => ( ( ord_less_eq @ ( set @ A ) @ A9 @ B7 )
% 2.92/1.57 => ( ! [X7: A] :
% 2.92/1.57 ( ( member @ A @ X7 @ A9 )
% 2.92/1.57 => ( P @ ( minus_minus @ ( set @ A ) @ A9 @ ( insert @ A @ X7 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
% 2.92/1.57 => ( P @ A9 ) ) ) ) )
% 2.92/1.57 => ( P @ B7 ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_226_insert__subsetI,axiom,
% 2.92/1.57 ! [A: $tType,X: A,A7: set @ A,X9: set @ A] :
% 2.92/1.57 ( ( member @ A @ X @ A7 )
% 2.92/1.57 => ( ( ord_less_eq @ ( set @ A ) @ X9 @ A7 )
% 2.92/1.57 => ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ X9 ) @ A7 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_227_diff__mono,axiom,
% 2.92/1.57 ! [A: $tType] :
% 2.92/1.57 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.92/1.57 => ! [A2: A,B2: A,D2: A,C3: A] :
% 2.92/1.57 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.92/1.57 => ( ( ord_less_eq @ A @ D2 @ C3 )
% 2.92/1.57 => ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C3 ) @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_228_diff__left__mono,axiom,
% 2.92/1.57 ! [A: $tType] :
% 2.92/1.57 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.92/1.57 => ! [B2: A,A2: A,C3: A] :
% 2.92/1.57 ( ( ord_less_eq @ A @ B2 @ A2 )
% 2.92/1.57 => ( ord_less_eq @ A @ ( minus_minus @ A @ C3 @ A2 ) @ ( minus_minus @ A @ C3 @ B2 ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_229_diff__right__commute,axiom,
% 2.92/1.57 ! [A: $tType] :
% 2.92/1.57 ( ( cancel146912293up_add @ A @ ( type2 @ A ) )
% 2.92/1.57 => ! [A2: A,C3: A,B2: A] :
% 2.92/1.57 ( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C3 ) @ B2 )
% 2.92/1.57 = ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C3 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_230_diff__eq__diff__eq,axiom,
% 2.92/1.57 ! [A: $tType] :
% 2.92/1.57 ( ( group_add @ A @ ( type2 @ A ) )
% 2.92/1.57 => ! [A2: A,B2: A,C3: A,D2: A] :
% 2.92/1.57 ( ( ( minus_minus @ A @ A2 @ B2 )
% 2.92/1.57 = ( minus_minus @ A @ C3 @ D2 ) )
% 2.92/1.57 => ( ( A2 = B2 )
% 2.92/1.57 = ( C3 = D2 ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_231_diff__eq__diff__less__eq,axiom,
% 2.92/1.57 ! [A: $tType] :
% 2.92/1.57 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.92/1.57 => ! [A2: A,B2: A,C3: A,D2: A] :
% 2.92/1.57 ( ( ( minus_minus @ A @ A2 @ B2 )
% 2.92/1.57 = ( minus_minus @ A @ C3 @ D2 ) )
% 2.92/1.57 => ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.92/1.57 = ( ord_less_eq @ A @ C3 @ D2 ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_232_diff__right__mono,axiom,
% 2.92/1.57 ! [A: $tType] :
% 2.92/1.57 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.92/1.57 => ! [A2: A,B2: A,C3: A] :
% 2.92/1.57 ( ( ord_less_eq @ A @ A2 @ B2 )
% 2.92/1.57 => ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C3 ) @ ( minus_minus @ A @ B2 @ C3 ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_233_psubset__insert__iff,axiom,
% 2.92/1.57 ! [A: $tType,A7: set @ A,X: A,B7: set @ A] :
% 2.92/1.57 ( ( ord_less @ ( set @ A ) @ A7 @ ( insert @ A @ X @ B7 ) )
% 2.92/1.57 = ( ( ( member @ A @ X @ B7 )
% 2.92/1.57 => ( ord_less @ ( set @ A ) @ A7 @ B7 ) )
% 2.92/1.57 & ( ~ ( member @ A @ X @ B7 )
% 2.92/1.57 => ( ( ( member @ A @ X @ A7 )
% 2.92/1.57 => ( ord_less @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A7 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B7 ) )
% 2.92/1.57 & ( ~ ( member @ A @ X @ A7 )
% 2.92/1.57 => ( ord_less_eq @ ( set @ A ) @ A7 @ B7 ) ) ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_234_finite__induct__select,axiom,
% 2.92/1.57 ! [A: $tType,S2: set @ A,P: ( set @ A ) > $o] :
% 2.92/1.57 ( ( finite_finite2 @ A @ S2 )
% 2.92/1.57 => ( ( P @ ( bot_bot @ ( set @ A ) ) )
% 2.92/1.57 => ( ! [T6: set @ A] :
% 2.92/1.57 ( ( ord_less @ ( set @ A ) @ T6 @ S2 )
% 2.92/1.57 => ( ( P @ T6 )
% 2.92/1.57 => ? [X7: A] :
% 2.92/1.57 ( ( member @ A @ X7 @ ( minus_minus @ ( set @ A ) @ S2 @ T6 ) )
% 2.92/1.57 & ( P @ ( insert @ A @ X7 @ T6 ) ) ) ) )
% 2.92/1.57 => ( P @ S2 ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_235_psubsetI,axiom,
% 2.92/1.57 ! [A: $tType,A7: set @ A,B7: set @ A] :
% 2.92/1.57 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
% 2.92/1.57 => ( ( A7 != B7 )
% 2.92/1.57 => ( ord_less @ ( set @ A ) @ A7 @ B7 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_236_subset__iff__psubset__eq,axiom,
% 2.92/1.57 ! [A: $tType] :
% 2.92/1.57 ( ( ord_less_eq @ ( set @ A ) )
% 2.92/1.57 = ( ^ [A8: set @ A,B9: set @ A] :
% 2.92/1.57 ( ( ord_less @ ( set @ A ) @ A8 @ B9 )
% 2.92/1.57 | ( A8 = B9 ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_237_subset__psubset__trans,axiom,
% 2.92/1.57 ! [A: $tType,A7: set @ A,B7: set @ A,C4: set @ A] :
% 2.92/1.57 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
% 2.92/1.57 => ( ( ord_less @ ( set @ A ) @ B7 @ C4 )
% 2.92/1.57 => ( ord_less @ ( set @ A ) @ A7 @ C4 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_238_subset__not__subset__eq,axiom,
% 2.92/1.57 ! [A: $tType] :
% 2.92/1.57 ( ( ord_less @ ( set @ A ) )
% 2.92/1.57 = ( ^ [A8: set @ A,B9: set @ A] :
% 2.92/1.57 ( ( ord_less_eq @ ( set @ A ) @ A8 @ B9 )
% 2.92/1.57 & ~ ( ord_less_eq @ ( set @ A ) @ B9 @ A8 ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_239_psubset__subset__trans,axiom,
% 2.92/1.57 ! [A: $tType,A7: set @ A,B7: set @ A,C4: set @ A] :
% 2.92/1.57 ( ( ord_less @ ( set @ A ) @ A7 @ B7 )
% 2.92/1.57 => ( ( ord_less_eq @ ( set @ A ) @ B7 @ C4 )
% 2.92/1.57 => ( ord_less @ ( set @ A ) @ A7 @ C4 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_240_psubset__imp__subset,axiom,
% 2.92/1.57 ! [A: $tType,A7: set @ A,B7: set @ A] :
% 2.92/1.57 ( ( ord_less @ ( set @ A ) @ A7 @ B7 )
% 2.92/1.57 => ( ord_less_eq @ ( set @ A ) @ A7 @ B7 ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_241_psubset__eq,axiom,
% 2.92/1.57 ! [A: $tType] :
% 2.92/1.57 ( ( ord_less @ ( set @ A ) )
% 2.92/1.57 = ( ^ [A8: set @ A,B9: set @ A] :
% 2.92/1.57 ( ( ord_less_eq @ ( set @ A ) @ A8 @ B9 )
% 2.92/1.57 & ( A8 != B9 ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_242_psubsetE,axiom,
% 2.92/1.57 ! [A: $tType,A7: set @ A,B7: set @ A] :
% 2.92/1.57 ( ( ord_less @ ( set @ A ) @ A7 @ B7 )
% 2.92/1.57 => ~ ( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
% 2.92/1.57 => ( ord_less_eq @ ( set @ A ) @ B7 @ A7 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_243_diff__strict__right__mono,axiom,
% 2.92/1.57 ! [A: $tType] :
% 2.92/1.57 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.92/1.57 => ! [A2: A,B2: A,C3: A] :
% 2.92/1.57 ( ( ord_less @ A @ A2 @ B2 )
% 2.92/1.57 => ( ord_less @ A @ ( minus_minus @ A @ A2 @ C3 ) @ ( minus_minus @ A @ B2 @ C3 ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_244_diff__strict__left__mono,axiom,
% 2.92/1.57 ! [A: $tType] :
% 2.92/1.57 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.92/1.57 => ! [B2: A,A2: A,C3: A] :
% 2.92/1.57 ( ( ord_less @ A @ B2 @ A2 )
% 2.92/1.57 => ( ord_less @ A @ ( minus_minus @ A @ C3 @ A2 ) @ ( minus_minus @ A @ C3 @ B2 ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_245_diff__eq__diff__less,axiom,
% 2.92/1.57 ! [A: $tType] :
% 2.92/1.57 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.92/1.57 => ! [A2: A,B2: A,C3: A,D2: A] :
% 2.92/1.57 ( ( ( minus_minus @ A @ A2 @ B2 )
% 2.92/1.57 = ( minus_minus @ A @ C3 @ D2 ) )
% 2.92/1.57 => ( ( ord_less @ A @ A2 @ B2 )
% 2.92/1.57 = ( ord_less @ A @ C3 @ D2 ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_246_diff__strict__mono,axiom,
% 2.92/1.57 ! [A: $tType] :
% 2.92/1.57 ( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
% 2.92/1.57 => ! [A2: A,B2: A,D2: A,C3: A] :
% 2.92/1.57 ( ( ord_less @ A @ A2 @ B2 )
% 2.92/1.57 => ( ( ord_less @ A @ D2 @ C3 )
% 2.92/1.57 => ( ord_less @ A @ ( minus_minus @ A @ A2 @ C3 ) @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_247_not__psubset__empty,axiom,
% 2.92/1.57 ! [A: $tType,A7: set @ A] :
% 2.92/1.57 ~ ( ord_less @ ( set @ A ) @ A7 @ ( bot_bot @ ( set @ A ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_248_finite__psubset__induct,axiom,
% 2.92/1.57 ! [A: $tType,A7: set @ A,P: ( set @ A ) > $o] :
% 2.92/1.57 ( ( finite_finite2 @ A @ A7 )
% 2.92/1.57 => ( ! [A9: set @ A] :
% 2.92/1.57 ( ( finite_finite2 @ A @ A9 )
% 2.92/1.57 => ( ! [B11: set @ A] :
% 2.92/1.57 ( ( ord_less @ ( set @ A ) @ B11 @ A9 )
% 2.92/1.57 => ( P @ B11 ) )
% 2.92/1.57 => ( P @ A9 ) ) )
% 2.92/1.57 => ( P @ A7 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_249_psubset__imp__ex__mem,axiom,
% 2.92/1.57 ! [A: $tType,A7: set @ A,B7: set @ A] :
% 2.92/1.57 ( ( ord_less @ ( set @ A ) @ A7 @ B7 )
% 2.92/1.57 => ? [B4: A] : ( member @ A @ B4 @ ( minus_minus @ ( set @ A ) @ B7 @ A7 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_250_psubset__trans,axiom,
% 2.92/1.57 ! [A: $tType,A7: set @ A,B7: set @ A,C4: set @ A] :
% 2.92/1.57 ( ( ord_less @ ( set @ A ) @ A7 @ B7 )
% 2.92/1.57 => ( ( ord_less @ ( set @ A ) @ B7 @ C4 )
% 2.92/1.57 => ( ord_less @ ( set @ A ) @ A7 @ C4 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_251_psubsetD,axiom,
% 2.92/1.57 ! [A: $tType,A7: set @ A,B7: set @ A,C3: A] :
% 2.92/1.57 ( ( ord_less @ ( set @ A ) @ A7 @ B7 )
% 2.92/1.57 => ( ( member @ A @ C3 @ A7 )
% 2.92/1.57 => ( member @ A @ C3 @ B7 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_252_finite__linorder__min__induct,axiom,
% 2.92/1.57 ! [A: $tType] :
% 2.92/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.92/1.57 => ! [A7: set @ A,P: ( set @ A ) > $o] :
% 2.92/1.57 ( ( finite_finite2 @ A @ A7 )
% 2.92/1.57 => ( ( P @ ( bot_bot @ ( set @ A ) ) )
% 2.92/1.57 => ( ! [B4: A,A9: set @ A] :
% 2.92/1.57 ( ( finite_finite2 @ A @ A9 )
% 2.92/1.57 => ( ! [X7: A] :
% 2.92/1.57 ( ( member @ A @ X7 @ A9 )
% 2.92/1.57 => ( ord_less @ A @ B4 @ X7 ) )
% 2.92/1.57 => ( ( P @ A9 )
% 2.92/1.57 => ( P @ ( insert @ A @ B4 @ A9 ) ) ) ) )
% 2.92/1.57 => ( P @ A7 ) ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_253_finite__linorder__max__induct,axiom,
% 2.92/1.57 ! [A: $tType] :
% 2.92/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.92/1.57 => ! [A7: set @ A,P: ( set @ A ) > $o] :
% 2.92/1.57 ( ( finite_finite2 @ A @ A7 )
% 2.92/1.57 => ( ( P @ ( bot_bot @ ( set @ A ) ) )
% 2.92/1.57 => ( ! [B4: A,A9: set @ A] :
% 2.92/1.57 ( ( finite_finite2 @ A @ A9 )
% 2.92/1.57 => ( ! [X7: A] :
% 2.92/1.57 ( ( member @ A @ X7 @ A9 )
% 2.92/1.57 => ( ord_less @ A @ X7 @ B4 ) )
% 2.92/1.57 => ( ( P @ A9 )
% 2.92/1.57 => ( P @ ( insert @ A @ B4 @ A9 ) ) ) ) )
% 2.92/1.57 => ( P @ A7 ) ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_254_infinite__growing,axiom,
% 2.92/1.57 ! [A: $tType] :
% 2.92/1.57 ( ( linorder @ A @ ( type2 @ A ) )
% 2.92/1.57 => ! [X9: set @ A] :
% 2.92/1.57 ( ( X9
% 2.92/1.57 != ( bot_bot @ ( set @ A ) ) )
% 2.92/1.57 => ( ! [X6: A] :
% 2.92/1.57 ( ( member @ A @ X6 @ X9 )
% 2.92/1.57 => ? [Xa: A] :
% 2.92/1.57 ( ( member @ A @ Xa @ X9 )
% 2.92/1.57 & ( ord_less @ A @ X6 @ Xa ) ) )
% 2.92/1.57 => ~ ( finite_finite2 @ A @ X9 ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(fact_255_Sup__fin_Oantimono,axiom,
% 2.92/1.57 ! [A: $tType] :
% 2.92/1.57 ( ( semilattice_sup @ A @ ( type2 @ A ) )
% 2.92/1.57 => ! [A7: set @ A,B7: set @ A] :
% 2.92/1.57 ( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
% 2.92/1.57 => ( ( A7
% 2.92/1.57 != ( bot_bot @ ( set @ A ) ) )
% 2.92/1.57 => ( ( finite_finite2 @ A @ B7 )
% 2.92/1.57 => ( ord_less_eq @ A @ ( lattic477160up_fin @ A @ A7 ) @ ( lattic477160up_fin @ A @ B7 ) ) ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
% 2.92/1.57 ! [A5: $tType,A11: $tType] :
% 2.92/1.57 ( ( semilattice_sup @ A11 @ ( type2 @ A11 ) )
% 2.92/1.57 => ( semilattice_sup @ ( A5 > A11 ) @ ( type2 @ ( A5 > A11 ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(tcon_fun___Finite__Set_Ofinite,axiom,
% 2.92/1.57 ! [A5: $tType,A11: $tType] :
% 2.92/1.57 ( ( ( finite_finite @ A5 @ ( type2 @ A5 ) )
% 2.92/1.57 & ( finite_finite @ A11 @ ( type2 @ A11 ) ) )
% 2.92/1.57 => ( finite_finite @ ( A5 > A11 ) @ ( type2 @ ( A5 > A11 ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(tcon_fun___Groups_Ominus,axiom,
% 2.92/1.57 ! [A5: $tType,A11: $tType] :
% 2.92/1.57 ( ( minus @ A11 @ ( type2 @ A11 ) )
% 2.92/1.57 => ( minus @ ( A5 > A11 ) @ ( type2 @ ( A5 > A11 ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(tcon_Set_Oset___Lattices_Osemilattice__sup_1,axiom,
% 2.92/1.57 ! [A5: $tType] : ( semilattice_sup @ ( set @ A5 ) @ ( type2 @ ( set @ A5 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(tcon_Set_Oset___Finite__Set_Ofinite_2,axiom,
% 2.92/1.57 ! [A5: $tType] :
% 2.92/1.57 ( ( finite_finite @ A5 @ ( type2 @ A5 ) )
% 2.92/1.57 => ( finite_finite @ ( set @ A5 ) @ ( type2 @ ( set @ A5 ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(tcon_Set_Oset___Groups_Ominus_3,axiom,
% 2.92/1.57 ! [A5: $tType] : ( minus @ ( set @ A5 ) @ ( type2 @ ( set @ A5 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_4,axiom,
% 2.92/1.57 semilattice_sup @ $o @ ( type2 @ $o ) ).
% 2.92/1.57
% 2.92/1.57 thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
% 2.92/1.57 linorder @ $o @ ( type2 @ $o ) ).
% 2.92/1.57
% 2.92/1.57 thf(tcon_HOL_Obool___Finite__Set_Ofinite_5,axiom,
% 2.92/1.57 finite_finite @ $o @ ( type2 @ $o ) ).
% 2.92/1.57
% 2.92/1.57 thf(tcon_HOL_Obool___Groups_Ominus_6,axiom,
% 2.92/1.57 minus @ $o @ ( type2 @ $o ) ).
% 2.92/1.57
% 2.92/1.57 thf(tcon_FSet_Ofset___Lattices_Osemilattice__sup_7,axiom,
% 2.92/1.57 ! [A5: $tType] : ( semilattice_sup @ ( fset @ A5 ) @ ( type2 @ ( fset @ A5 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(tcon_FSet_Ofset___Finite__Set_Ofinite_8,axiom,
% 2.92/1.57 ! [A5: $tType] :
% 2.92/1.57 ( ( finite_finite @ A5 @ ( type2 @ A5 ) )
% 2.92/1.57 => ( finite_finite @ ( fset @ A5 ) @ ( type2 @ ( fset @ A5 ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(tcon_FSet_Ofset___Groups_Ominus_9,axiom,
% 2.92/1.57 ! [A5: $tType] : ( minus @ ( fset @ A5 ) @ ( type2 @ ( fset @ A5 ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_10,axiom,
% 2.92/1.57 ! [A5: $tType,A11: $tType] :
% 2.92/1.57 ( ( ( finite_finite @ A5 @ ( type2 @ A5 ) )
% 2.92/1.57 & ( finite_finite @ A11 @ ( type2 @ A11 ) ) )
% 2.92/1.57 => ( finite_finite @ ( sum_sum @ A5 @ A11 ) @ ( type2 @ ( sum_sum @ A5 @ A11 ) ) ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(help_If_3_1_T,axiom,
% 2.92/1.57 ! [P: $o] :
% 2.92/1.57 ( ( P = $true )
% 2.92/1.57 | ( P = $false ) ) ).
% 2.92/1.57
% 2.92/1.57 thf(help_If_2_1_T,axiom,
% 2.92/1.57 ! [A: $tType,X: A,Y: A] :
% 2.92/1.57 ( ( if @ A @ $false @ X @ Y )
% 2.92/1.57 = Y ) ).
% 2.92/1.57
% 2.92/1.57 thf(help_If_1_1_T,axiom,
% 2.92/1.57 ! [A: $tType,X: A,Y: A] :
% 2.92/1.57 ( ( if @ A @ $true @ X @ Y )
% 2.92/1.57 = X ) ).
% 2.92/1.57
% 2.92/1.57 thf(conj_0,hypothesis,
% 2.92/1.57 member @ n @ ( root @ tra ) @ nsa ).
% 2.92/1.57
% 2.92/1.57 thf(conj_1,hypothesis,
% 2.92/1.57 member @ ( sum_sum @ t @ dtree ) @ ( sum_Inl @ t @ dtree @ ta ) @ ( cont @ tra ) ).
% 2.92/1.57
% 2.92/1.57 thf(conj_2,conjecture,
% 2.92/1.57 member @ n @ ( root @ tra ) @ nsa ).
% 2.92/1.57 ------------------------
% 3.00/1.60 % (27153)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_26903 for (2999ds/27Mi)
% 3.00/1.61 % (27151)lrs+1002_1:8_bd=off:fd=off:hud=10:tnu=1:i=183:si=on:rtra=on_0 on DTF2THF_26903 for (2999ds/183Mi)
% 3.00/1.61 % (27152)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_26903 for (2999ds/4Mi)
% 3.00/1.61 % (27154)lrs+10_1:1_au=on:inj=on:i=2:si=on:rtra=on_0 on DTF2THF_26903 for (2999ds/2Mi)
% 3.00/1.61 % (27155)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_26903 for (2999ds/2Mi)
% 3.00/1.61 % (27156)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_26903 for (2999ds/275Mi)
% 3.00/1.61 % (27157)lrs+1004_1:128_cond=on:e2e=on:sp=weighted_frequency:i=18:si=on:rtra=on_0 on DTF2THF_26903 for (2999ds/18Mi)
% 3.00/1.61 % (27153)Instruction limit reached!
% 3.00/1.61 % (27153)------------------------------
% 3.00/1.61 % (27153)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.61 % (27153)Termination reason: Unknown
% 3.00/1.61 % (27153)Termination phase: Property scanning
% 3.00/1.61
% 3.00/1.61 % (27153)Memory used [KB]: 1791
% 3.00/1.61 % (27153)Time elapsed: 0.010 s
% 3.00/1.61 % (27153)Instructions burned: 27 (million)
% 3.00/1.61 % (27153)------------------------------
% 3.00/1.61 % (27153)------------------------------
% 3.00/1.61 % (27154)Instruction limit reached!
% 3.00/1.61 % (27154)------------------------------
% 3.00/1.61 % (27154)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.61 % (27154)Termination reason: Unknown
% 3.00/1.61 % (27154)Termination phase: shuffling
% 3.00/1.61
% 3.00/1.61 % (27154)Memory used [KB]: 1407
% 3.00/1.61 % (27154)Time elapsed: 0.004 s
% 3.00/1.61 % (27154)Instructions burned: 2 (million)
% 3.00/1.61 % (27154)------------------------------
% 3.00/1.61 % (27154)------------------------------
% 3.00/1.61 % (27155)Instruction limit reached!
% 3.00/1.61 % (27155)------------------------------
% 3.00/1.61 % (27155)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.61 % (27155)Termination reason: Unknown
% 3.00/1.61 % (27155)Termination phase: shuffling
% 3.00/1.61
% 3.00/1.61 % (27155)Memory used [KB]: 1407
% 3.00/1.61 % (27155)Time elapsed: 0.004 s
% 3.00/1.61 % (27155)Instructions burned: 3 (million)
% 3.00/1.61 % (27155)------------------------------
% 3.00/1.61 % (27155)------------------------------
% 3.00/1.61 % (27152)Instruction limit reached!
% 3.00/1.61 % (27152)------------------------------
% 3.00/1.61 % (27152)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.61 % (27152)Termination reason: Unknown
% 3.00/1.61 % (27152)Termination phase: shuffling
% 3.00/1.61
% 3.00/1.61 % (27152)Memory used [KB]: 1407
% 3.00/1.61 % (27152)Time elapsed: 0.004 s
% 3.00/1.61 % (27152)Instructions burned: 4 (million)
% 3.00/1.61 % (27152)------------------------------
% 3.00/1.61 % (27152)------------------------------
% 3.00/1.62 % (27158)lrs+10_1:1_bet=on:cnfonf=off:fd=off:hud=5:inj=on:i=3:si=on:rtra=on_0 on DTF2THF_26903 for (2999ds/3Mi)
% 3.00/1.62 % (27157)Instruction limit reached!
% 3.00/1.62 % (27157)------------------------------
% 3.00/1.62 % (27157)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.62 % (27157)Termination reason: Unknown
% 3.00/1.62 % (27157)Termination phase: shuffling
% 3.00/1.62
% 3.00/1.62 % (27157)Memory used [KB]: 1663
% 3.00/1.62 % (27157)Time elapsed: 0.013 s
% 3.00/1.62 % (27157)Instructions burned: 18 (million)
% 3.00/1.62 % (27157)------------------------------
% 3.00/1.62 % (27157)------------------------------
% 3.00/1.62 % (27158)Instruction limit reached!
% 3.00/1.62 % (27158)------------------------------
% 3.00/1.62 % (27158)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.62 % (27158)Termination reason: Unknown
% 3.00/1.62 % (27158)Termination phase: shuffling
% 3.00/1.62
% 3.00/1.62 % (27158)Memory used [KB]: 1407
% 3.00/1.62 % (27158)Time elapsed: 0.003 s
% 3.00/1.62 % (27158)Instructions burned: 5 (million)
% 3.00/1.62 % (27158)------------------------------
% 3.00/1.62 % (27158)------------------------------
% 3.00/1.63 % (27159)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_26903 for (2999ds/37Mi)
% 3.00/1.63 % (27160)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_26903 for (2999ds/15Mi)
% 3.00/1.63 % (27161)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_26903 for (2999ds/3Mi)
% 3.00/1.63 % (27161)Instruction limit reached!
% 3.00/1.63 % (27161)------------------------------
% 3.00/1.63 % (27161)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.63 % (27161)Termination reason: Unknown
% 3.00/1.63 % (27161)Termination phase: shuffling
% 3.00/1.63
% 3.00/1.63 % (27161)Memory used [KB]: 1407
% 3.00/1.63 % (27161)Time elapsed: 0.004 s
% 3.00/1.63 % (27161)Instructions burned: 3 (million)
% 3.00/1.63 % (27161)------------------------------
% 3.00/1.63 % (27161)------------------------------
% 3.00/1.63 % (27163)lrs+10_1:1_av=off:chr=on:plsq=on:slsq=on:i=7:si=on:rtra=on_0 on DTF2THF_26903 for (2999ds/7Mi)
% 3.00/1.64 % (27163)Instruction limit reached!
% 3.00/1.64 % (27163)------------------------------
% 3.00/1.64 % (27163)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.64 % (27163)Termination reason: Unknown
% 3.00/1.64 % (27163)Termination phase: shuffling
% 3.00/1.64
% 3.00/1.64 % (27163)Memory used [KB]: 1535
% 3.00/1.64 % (27163)Time elapsed: 0.004 s
% 3.00/1.64 % (27163)Instructions burned: 9 (million)
% 3.00/1.64 % (27163)------------------------------
% 3.00/1.64 % (27163)------------------------------
% 3.00/1.64 % (27160)Instruction limit reached!
% 3.00/1.64 % (27160)------------------------------
% 3.00/1.64 % (27160)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.64 % (27160)Termination reason: Unknown
% 3.00/1.64 % (27160)Termination phase: shuffling
% 3.00/1.64
% 3.00/1.64 % (27160)Memory used [KB]: 1663
% 3.00/1.64 % (27160)Time elapsed: 0.012 s
% 3.00/1.64 % (27160)Instructions burned: 15 (million)
% 3.00/1.64 % (27160)------------------------------
% 3.00/1.64 % (27160)------------------------------
% 3.00/1.64 % (27162)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_26903 for (2999ds/1041Mi)
% 3.00/1.64 % (27156)First to succeed.
% 3.00/1.65 % (27156)Refutation found. Thanks to Tanya!
% 3.00/1.65 % SZS status Theorem for DTF2THF_26903
% 3.00/1.65 % SZS output start Proof for DTF2THF_26903
% 3.00/1.65 thf(type_def_5, type, itself: ($tType) > $tType).
% 3.00/1.65 thf(type_def_7, type, sum_sum: ($tType * $tType) > $tType).
% 3.00/1.65 thf(type_def_8, type, set: ($tType) > $tType).
% 3.00/1.65 thf(type_def_9, type, n: $tType).
% 3.00/1.65 thf(type_def_10, type, t: $tType).
% 3.00/1.65 thf(type_def_11, type, dtree: $tType).
% 3.00/1.65 thf(type_def_12, type, fset: ($tType) > $tType).
% 3.00/1.65 thf(func_def_0, type, sum_sum: ($tType * $tType) > $tType).
% 3.00/1.65 thf(func_def_1, type, dtree: $tType).
% 3.00/1.65 thf(func_def_2, type, fset: ($tType) > $tType).
% 3.00/1.65 thf(func_def_3, type, set: ($tType) > $tType).
% 3.00/1.65 thf(func_def_4, type, t: $tType).
% 3.00/1.65 thf(func_def_5, type, n: $tType).
% 3.00/1.65 thf(func_def_6, type, itself: ($tType) > $tType).
% 3.00/1.65 thf(func_def_7, type, type: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.00/1.65 thf(func_def_8, type, minus: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.00/1.65 thf(func_def_9, type, group_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.00/1.65 thf(func_def_10, type, finite_finite: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.00/1.65 thf(func_def_11, type, linorder: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.00/1.65 thf(func_def_12, type, semilattice_sup: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.00/1.65 thf(func_def_13, type, ordered_ab_group_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.00/1.65 thf(func_def_14, type, cancel146912293up_add: !>[X0: $tType]:(itself @ X0 > $o)).
% 3.00/1.65 thf(func_def_15, type, basic_setl: !>[X0: $tType, X1: $tType]:(sum_sum @ X0 @ X1 > set @ X0)).
% 3.00/1.65 thf(func_def_16, type, basic_setr: !>[X0: $tType, X1: $tType]:(sum_sum @ X0 @ X1 > set @ X1)).
% 3.00/1.65 thf(func_def_17, type, node: n > set @ sum_sum @ t @ dtree > dtree).
% 3.00/1.65 thf(func_def_18, type, cont: dtree > set @ sum_sum @ t @ dtree).
% 3.00/1.65 thf(func_def_19, type, corec: !>[X0: $tType]:((X0 > n) > (X0 > set @ sum_sum @ t @ sum_sum @ dtree @ X0) > X0 > dtree)).
% 3.00/1.65 thf(func_def_20, type, nNode: n > fset @ sum_sum @ t @ dtree > dtree).
% 3.00/1.65 thf(func_def_21, type, case_dtree: !>[X0: $tType]:((n > fset @ sum_sum @ t @ dtree > X0) > dtree > X0)).
% 3.00/1.65 thf(func_def_22, type, ccont: dtree > fset @ sum_sum @ t @ dtree).
% 3.00/1.65 thf(func_def_23, type, root: dtree > n).
% 3.00/1.65 thf(func_def_24, type, unfold: !>[X0: $tType]:((X0 > n) > (X0 > set @ sum_sum @ t @ X0) > X0 > dtree)).
% 3.00/1.65 thf(func_def_25, type, finite_finite2: !>[X0: $tType]:(set @ X0 > $o)).
% 3.00/1.65 thf(func_def_26, type, comp: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > (X2 > X0) > X2 > X1)).
% 3.00/1.65 thf(func_def_27, type, gram_L1333338417e_inFr: set @ n > dtree > t > $o).
% 3.00/1.65 thf(func_def_28, type, gram_L805317441_inFr2: set @ n > dtree > t > $o).
% 3.00/1.65 thf(func_def_29, type, minus_minus: !>[X0: $tType]:(X0 > X0 > X0)).
% 3.00/1.65 thf(func_def_30, type, if: !>[X0: $tType]:($o > X0 > X0 > X0)).
% 3.00/1.65 thf(func_def_31, type, lattic477160up_fin: !>[X0: $tType]:(set @ X0 > X0)).
% 3.00/1.65 thf(func_def_32, type, bot_bot: !>[X0: $tType]:(X0)).
% 3.00/1.65 thf(func_def_33, type, ord_less: !>[X0: $tType]:(X0 > X0 > $o)).
% 3.00/1.65 thf(func_def_34, type, ord_less_eq: !>[X0: $tType]:(X0 > X0 > $o)).
% 3.00/1.65 thf(func_def_35, type, type2: !>[X0: $tType]:(itself @ X0)).
% 3.00/1.65 thf(func_def_36, type, collect: !>[X0: $tType]:((X0 > $o) > set @ X0)).
% 3.00/1.65 thf(func_def_37, type, insert: !>[X0: $tType]:(X0 > set @ X0 > set @ X0)).
% 3.00/1.65 thf(func_def_38, type, is_empty: !>[X0: $tType]:(set @ X0 > $o)).
% 3.00/1.65 thf(func_def_39, type, is_singleton: !>[X0: $tType]:(set @ X0 > $o)).
% 3.00/1.65 thf(func_def_40, type, remove: !>[X0: $tType]:(X0 > set @ X0 > set @ X0)).
% 3.00/1.65 thf(func_def_41, type, the_elem: !>[X0: $tType]:(set @ X0 > X0)).
% 3.00/1.65 thf(func_def_42, type, sum_Inl: !>[X0: $tType, X1: $tType]:(X0 > sum_sum @ X0 @ X1)).
% 3.00/1.65 thf(func_def_43, type, sum_Inr: !>[X0: $tType, X1: $tType]:(X0 > sum_sum @ X1 @ X0)).
% 3.00/1.65 thf(func_def_44, type, sum_Plus: !>[X0: $tType, X1: $tType]:(set @ X0 > set @ X1 > set @ sum_sum @ X0 @ X1)).
% 3.00/1.65 thf(func_def_45, type, sum_Suml: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > sum_sum @ X0 @ X2 > X1)).
% 3.00/1.65 thf(func_def_46, type, sum_Sumr: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > sum_sum @ X2 @ X0 > X1)).
% 3.00/1.65 thf(func_def_47, type, sum_rec_sum: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > (X2 > X1) > sum_sum @ X0 @ X2 > X1)).
% 3.00/1.65 thf(func_def_48, type, sum_case_sum: !>[X0: $tType, X1: $tType, X2: $tType]:((X0 > X1) > (X2 > X1) > sum_sum @ X0 @ X2 > X1)).
% 3.00/1.65 thf(func_def_49, type, member: !>[X0: $tType]:(X0 > set @ X0 > $o)).
% 3.00/1.65 thf(func_def_50, type, nsa: set @ n).
% 3.00/1.65 thf(func_def_51, type, ta: t).
% 3.00/1.65 thf(func_def_52, type, tra: dtree).
% 3.00/1.65 thf(func_def_71, type, sP0: !>[X0: $tType]:(set @ X0 > (set @ X0 > $o) > $o)).
% 3.00/1.65 thf(func_def_72, type, sK1: !>[X0: $tType, X1: $tType, X2: $tType]:($o > (sum_sum @ X0 @ X1 > X2) > X2 > X0)).
% 3.00/1.65 thf(func_def_73, type, sK2: !>[X0: $tType, X1: $tType, X2: $tType]:($o > (sum_sum @ X0 @ X1 > X2) > X2 > X1)).
% 3.00/1.65 thf(func_def_74, type, sK3: !>[X0: $tType, X1: $tType]:(sum_sum @ X0 @ X1 > X0)).
% 3.00/1.65 thf(func_def_75, type, sK4: !>[X0: $tType, X1: $tType]:(sum_sum @ X0 @ X1 > X1)).
% 3.00/1.65 thf(func_def_76, type, sK5: !>[X0: $tType]:((set @ X0 > $o) > X0)).
% 3.00/1.65 thf(func_def_77, type, sK6: !>[X0: $tType]:((set @ X0 > $o) > X0)).
% 3.00/1.65 thf(func_def_78, type, sK7: !>[X0: $tType]:((set @ X0 > $o) > set @ X0)).
% 3.00/1.65 thf(func_def_79, type, sK8: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > set @ X1 > X1 > X0)).
% 3.00/1.65 thf(func_def_80, type, sK9: !>[X0: $tType, X1: $tType]:((X1 > X0 > $o) > set @ X1 > X1)).
% 3.00/1.65 thf(func_def_81, type, sK10: !>[X0: $tType]:((set @ X0 > $o) > set @ X0)).
% 3.00/1.65 thf(func_def_82, type, sK11: !>[X0: $tType]:((set @ X0 > $o) > X0)).
% 3.00/1.65 thf(func_def_83, type, sK12: !>[X0: $tType]:((set @ X0 > $o) > set @ X0 > set @ X0)).
% 3.00/1.65 thf(func_def_84, type, sK13: !>[X0: $tType]:((set @ X0 > $o) > X0)).
% 3.00/1.65 thf(func_def_85, type, sK14: !>[X0: $tType]:((set @ X0 > $o) > set @ X0)).
% 3.00/1.65 thf(func_def_86, type, sK15: !>[X0: $tType]:(set @ X0 > (set @ X0 > $o) > set @ X0)).
% 3.00/1.65 thf(func_def_87, type, sK16: !>[X0: $tType, X1: $tType]:(sum_sum @ X0 @ X1 > X1)).
% 3.00/1.65 thf(func_def_88, type, sK17: !>[X0: $tType, X1: $tType]:(sum_sum @ X0 @ X1 > X0)).
% 3.00/1.65 thf(func_def_89, type, sK18: !>[X0: $tType, X1: $tType]:((X1 > X0) > (X1 > X0) > X1)).
% 3.00/1.65 thf(func_def_90, type, sK19: !>[X0: $tType, X1: $tType]:((sum_sum @ X1 @ X0 > $o) > X1)).
% 3.00/1.65 thf(func_def_91, type, sK20: !>[X0: $tType, X1: $tType]:((sum_sum @ X1 @ X0 > $o) > X0)).
% 3.00/1.65 thf(func_def_92, type, sK21: !>[X0: $tType]:((set @ X0 > $o) > X0)).
% 3.00/1.65 thf(func_def_93, type, sK22: !>[X0: $tType]:((set @ X0 > $o) > set @ X0)).
% 3.00/1.65 thf(func_def_94, type, sK23: !>[X0: $tType]:((set @ X0 > $o) > set @ X0)).
% 3.00/1.65 thf(func_def_95, type, sK24: !>[X0: $tType]:((set @ X0 > $o) > X0)).
% 3.00/1.65 thf(func_def_96, type, sK25: !>[X0: $tType]:((set @ X0 > $o) > set @ X0)).
% 3.00/1.65 thf(func_def_97, type, sK26: !>[X0: $tType]:((set @ X0 > $o) > set @ X0 > X0)).
% 3.00/1.65 thf(func_def_98, type, sK27: !>[X0: $tType]:((set @ X0 > $o) > set @ X0 > set @ X0)).
% 3.00/1.65 thf(func_def_99, type, sK28: !>[X0: $tType]:((set @ X0 > $o) > set @ X0)).
% 3.00/1.65 thf(func_def_100, type, sK29: !>[X0: $tType, X1: $tType]:(sum_sum @ X0 @ X1 > X1)).
% 3.00/1.65 thf(func_def_101, type, sK30: !>[X0: $tType, X1: $tType]:(sum_sum @ X0 @ X1 > X0)).
% 3.00/1.65 thf(func_def_102, type, sK31: !>[X0: $tType]:(set @ X0 > set @ X0)).
% 3.00/1.65 thf(func_def_103, type, sK32: !>[X0: $tType]:(set @ X0 > X0)).
% 3.00/1.65 thf(func_def_105, type, ph34: !>[X0: $tType]:(X0)).
% 3.00/1.65 thf(f923,plain,(
% 3.00/1.65 $false),
% 3.00/1.65 inference(subsumption_resolution,[status(thm)],[f865,f849])).
% 3.00/1.65 thf(f849,plain,(
% 3.00/1.65 ((member @ n @ (root @ tra) @ nsa) = $true)),
% 3.00/1.65 inference(cnf_transformation,[status(thm)],[f552])).
% 3.00/1.65 thf(f552,plain,(
% 3.00/1.65 ((member @ n @ (root @ tra) @ nsa) = $true)),
% 3.00/1.65 inference(fool_elimination,[status(thm)],[f551])).
% 3.00/1.65 thf(f551,plain,(
% 3.00/1.65 (member @ n @ (root @ tra) @ nsa)),
% 3.00/1.65 inference(rectify,[status(thm)],[f274])).
% 3.00/1.65 thf(f274,axiom,(
% 3.00/1.65 (member @ n @ (root @ tra) @ nsa)),
% 3.00/1.65 file('/export/starexec/sandbox2/tmp/tmp.Nuu3MkGpYK/DTF2THF_26903.p',conj_0)).
% 3.00/1.65 thf(f865,plain,(
% 3.00/1.65 ((member @ n @ (root @ tra) @ nsa) != $true)),
% 3.00/1.65 inference(cnf_transformation,[status(thm)],[f685])).
% 3.00/1.65 thf(f685,plain,(
% 3.00/1.65 ((member @ n @ (root @ tra) @ nsa) != $true)),
% 3.00/1.65 inference(flattening,[status(thm)],[f550])).
% 3.00/1.65 thf(f550,plain,(
% 3.00/1.65 ~((member @ n @ (root @ tra) @ nsa) = $true)),
% 3.00/1.65 inference(fool_elimination,[status(thm)],[f549])).
% 3.00/1.65 thf(f549,plain,(
% 3.00/1.65 ~(member @ n @ (root @ tra) @ nsa)),
% 3.00/1.65 inference(rectify,[status(thm)],[f277])).
% 3.00/1.65 thf(f277,negated_conjecture,(
% 3.00/1.65 ~(member @ n @ (root @ tra) @ nsa)),
% 3.00/1.65 inference(negated_conjecture,[status(cth)],[f276])).
% 3.00/1.65 thf(f276,conjecture,(
% 3.00/1.65 (member @ n @ (root @ tra) @ nsa)),
% 3.00/1.65 file('/export/starexec/sandbox2/tmp/tmp.Nuu3MkGpYK/DTF2THF_26903.p',conj_2)).
% 3.00/1.65 % SZS output end Proof for DTF2THF_26903
% 3.00/1.65 % (27156)------------------------------
% 3.00/1.65 % (27156)Version: Vampire 4.8 (commit 7b44b1c2f on 2025-07-15 09:50:17 +0100)
% 3.00/1.65 % (27156)Termination reason: Refutation
% 3.00/1.65
% 3.00/1.65 % (27156)Memory used [KB]: 6524
% 3.00/1.65 % (27156)Time elapsed: 0.035 s
% 3.00/1.65 % (27156)Instructions burned: 53 (million)
% 3.00/1.65 % (27156)------------------------------
% 3.00/1.65 % (27156)------------------------------
% 3.00/1.65 % (27150)Success in time 0.065 s
%------------------------------------------------------------------------------