TPTP Problem File: DAT229^1.p
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%------------------------------------------------------------------------------
% File : DAT229^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Red-black trees 26
% Version : [Bla16] axioms : Especial.
% English :
% Refs : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% Source : [Bla16]
% Names : rbt_impl__26.p [Bla16]
% Status : Theorem
% Rating : 0.33 v8.1.0, 0.50 v7.5.0, 0.00 v7.2.0, 0.25 v7.1.0
% Syntax : Number of formulae : 343 ( 156 unt; 59 typ; 0 def)
% Number of atoms : 674 ( 329 equ; 0 cnn)
% Maximal formula atoms : 10 ( 2 avg)
% Number of connectives : 4376 ( 63 ~; 9 |; 49 &;3989 @)
% ( 0 <=>; 266 =>; 0 <=; 0 <~>)
% Maximal formula depth : 31 ( 8 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 489 ( 489 >; 0 *; 0 +; 0 <<)
% Number of symbols : 58 ( 55 usr; 6 con; 0-10 aty)
% Number of variables : 1390 ( 51 ^;1249 !; 15 ?;1390 :)
% ( 75 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:38:44.202
%------------------------------------------------------------------------------
%----Could-be-implicit typings (7)
thf(ty_t_RBT__Impl__Mirabelle__msmaddcmtr_Ocolor,type,
rBT_Im1923302023_color: $tType ).
thf(ty_t_RBT__Impl__Mirabelle__msmaddcmtr_Orbt,type,
rBT_Im246033960le_rbt: $tType > $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
thf(ty_tf_b,type,
b: $tType ).
thf(ty_tf_a,type,
a: $tType ).
%----Explicit typings (52)
thf(sy_cl_Lattices_Obounded__lattice,type,
bounded_lattice:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Obot,type,
bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oorder__bot,type,
order_bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Obounded__lattice__bot,type,
bounded_lattice_bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Obounded__semilattice__sup__bot,type,
bounde1808546759up_bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_BNF__Composition_Oid__bnf,type,
bNF_id_bnf:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_BNF__Def_Oeq__onp,type,
bNF_eq_onp:
!>[A: $tType] : ( ( A > $o ) > A > A > $o ) ).
thf(sy_c_BNF__Def_Ovimage2p,type,
bNF_vimage2p:
!>[A: $tType,D: $tType,B: $tType,E: $tType,C: $tType] : ( ( A > D ) > ( B > E ) > ( D > E > C ) > A > B > C ) ).
thf(sy_c_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Fun_Obij__betw,type,
bij_betw:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) > $o ) ).
thf(sy_c_Fun_Ocomp,type,
comp:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
thf(sy_c_Fun_Omap__fun,type,
map_fun:
!>[C: $tType,A: $tType,B: $tType,D: $tType] : ( ( C > A ) > ( B > D ) > ( A > B ) > C > D ) ).
thf(sy_c_Fun_Ooverride__on,type,
override_on:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( A > B ) > ( set @ A ) > A > B ) ).
thf(sy_c_Groups_Ocomm__monoid,type,
comm_monoid:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Groups_Omonoid,type,
monoid:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Lattices_Oinf__class_Oinf,type,
inf_inf:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Lattices_Osemilattice__neutr,type,
semilattice_neutr:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Lattices_Osup__class_Osup,type,
sup_sup:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : ( A > nat ) ).
thf(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Ocolor_OB,type,
rBT_Impl_Mirabelle_B: rBT_Im1923302023_color ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Ocolor_OR,type,
rBT_Impl_Mirabelle_R: rBT_Im1923302023_color ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Ocolor_Ocase__color,type,
rBT_Im1975547051_color:
!>[A: $tType] : ( A > A > rBT_Im1923302023_color > A ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Ocolor_Orec__color,type,
rBT_Im1059957627_color:
!>[A: $tType] : ( A > A > rBT_Im1923302023_color > A ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Ocolor_Osize__color,type,
rBT_Im800559290_color: rBT_Im1923302023_color > nat ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Orbt_OBranch,type,
rBT_Im480247531Branch:
!>[A: $tType,B: $tType] : ( rBT_Im1923302023_color > ( rBT_Im246033960le_rbt @ A @ B ) > A > B > ( rBT_Im246033960le_rbt @ A @ B ) > ( rBT_Im246033960le_rbt @ A @ B ) ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Orbt_OEmpty,type,
rBT_Im418718756_Empty:
!>[A: $tType,B: $tType] : ( rBT_Im246033960le_rbt @ A @ B ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Orbt_Ocase__rbt,type,
rBT_Im858806507se_rbt:
!>[C: $tType,A: $tType,B: $tType] : ( C > ( rBT_Im1923302023_color > ( rBT_Im246033960le_rbt @ A @ B ) > A > B > ( rBT_Im246033960le_rbt @ A @ B ) > C ) > ( rBT_Im246033960le_rbt @ A @ B ) > C ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Orbt_Omap__rbt,type,
rBT_Im206295089ap_rbt:
!>[A: $tType,Aa: $tType,B: $tType,Ba: $tType] : ( ( A > Aa ) > ( B > Ba ) > ( rBT_Im246033960le_rbt @ A @ B ) > ( rBT_Im246033960le_rbt @ Aa @ Ba ) ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Orbt_Opred__rbt,type,
rBT_Im1931894874ed_rbt:
!>[A: $tType,B: $tType] : ( ( A > $o ) > ( B > $o ) > ( rBT_Im246033960le_rbt @ A @ B ) > $o ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Orbt_Orec__rbt,type,
rBT_Im1947144893ec_rbt:
!>[E: $tType,A: $tType,B: $tType] : ( E > ( rBT_Im1923302023_color > ( rBT_Im246033960le_rbt @ A @ B ) > A > B > ( rBT_Im246033960le_rbt @ A @ B ) > E > E > E ) > ( rBT_Im246033960le_rbt @ A @ B ) > E ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Orbt_Orel__rbt,type,
rBT_Im1000242676el_rbt:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C > $o ) > ( B > D > $o ) > ( rBT_Im246033960le_rbt @ A @ B ) > ( rBT_Im246033960le_rbt @ C @ D ) > $o ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Orbt_Oset1__rbt,type,
rBT_Im1178217900t1_rbt:
!>[A: $tType,B: $tType] : ( ( rBT_Im246033960le_rbt @ A @ B ) > ( set @ A ) ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Orbt_Oset2__rbt,type,
rBT_Im1550225131t2_rbt:
!>[A: $tType,B: $tType] : ( ( rBT_Im246033960le_rbt @ A @ B ) > ( set @ B ) ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Orbt_Osize__rbt,type,
rBT_Im1485196410ze_rbt:
!>[A: $tType,B: $tType] : ( ( A > nat ) > ( B > nat ) > ( rBT_Im246033960le_rbt @ A @ B ) > nat ) ).
thf(sy_c_Set_OBall,type,
ball:
!>[A: $tType] : ( ( set @ A ) > ( A > $o ) > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_Oinsert,type,
insert:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Set_Ois__empty,type,
is_empty:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Set_Ois__singleton,type,
is_singleton:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Set_Opairwise,type,
pairwise:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Set_Othe__elem,type,
the_elem:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Set_Ovimage,type,
vimage:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ B ) > ( set @ A ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_t,type,
t: rBT_Im246033960le_rbt @ a @ b ).
thf(sy_v_thesis,type,
thesis: $o ).
%----Relevant facts (256)
thf(fact_0_Empty,axiom,
( ( t
= ( rBT_Im418718756_Empty @ a @ b ) )
=> thesis ) ).
% Empty
thf(fact_1_rbt_Oinject,axiom,
! [B: $tType,A: $tType,X21: rBT_Im1923302023_color,X22: rBT_Im246033960le_rbt @ A @ B,X23: A,X24: B,X25: rBT_Im246033960le_rbt @ A @ B,Y21: rBT_Im1923302023_color,Y22: rBT_Im246033960le_rbt @ A @ B,Y23: A,Y24: B,Y25: rBT_Im246033960le_rbt @ A @ B] :
( ( ( rBT_Im480247531Branch @ A @ B @ X21 @ X22 @ X23 @ X24 @ X25 )
= ( rBT_Im480247531Branch @ A @ B @ Y21 @ Y22 @ Y23 @ Y24 @ Y25 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 )
& ( X23 = Y23 )
& ( X24 = Y24 )
& ( X25 = Y25 ) ) ) ).
% rbt.inject
thf(fact_2_Black,axiom,
! [L: rBT_Im246033960le_rbt @ a @ b,K: a,V: b,R: rBT_Im246033960le_rbt @ a @ b] :
( ( t
= ( rBT_Im480247531Branch @ a @ b @ rBT_Impl_Mirabelle_B @ L @ K @ V @ R ) )
=> thesis ) ).
% Black
thf(fact_3_Red,axiom,
! [L: rBT_Im246033960le_rbt @ a @ b,K: a,V: b,R: rBT_Im246033960le_rbt @ a @ b] :
( ( t
= ( rBT_Im480247531Branch @ a @ b @ rBT_Impl_Mirabelle_R @ L @ K @ V @ R ) )
=> thesis ) ).
% Red
thf(fact_4_rbt_Odistinct_I1_J,axiom,
! [B: $tType,A: $tType,X21: rBT_Im1923302023_color,X22: rBT_Im246033960le_rbt @ A @ B,X23: A,X24: B,X25: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im418718756_Empty @ A @ B )
!= ( rBT_Im480247531Branch @ A @ B @ X21 @ X22 @ X23 @ X24 @ X25 ) ) ).
% rbt.distinct(1)
thf(fact_5_color_Odistinct_I1_J,axiom,
rBT_Impl_Mirabelle_R != rBT_Impl_Mirabelle_B ).
% color.distinct(1)
thf(fact_6_rbt_Oinduct,axiom,
! [B: $tType,A: $tType,P: ( rBT_Im246033960le_rbt @ A @ B ) > $o,Rbt: rBT_Im246033960le_rbt @ A @ B] :
( ( P @ ( rBT_Im418718756_Empty @ A @ B ) )
=> ( ! [X1: rBT_Im1923302023_color,X2: rBT_Im246033960le_rbt @ A @ B,X3: A,X4: B,X5: rBT_Im246033960le_rbt @ A @ B] :
( ( P @ X2 )
=> ( ( P @ X5 )
=> ( P @ ( rBT_Im480247531Branch @ A @ B @ X1 @ X2 @ X3 @ X4 @ X5 ) ) ) )
=> ( P @ Rbt ) ) ) ).
% rbt.induct
thf(fact_7_rbt_Oexhaust,axiom,
! [B: $tType,A: $tType,Y: rBT_Im246033960le_rbt @ A @ B] :
( ( Y
!= ( rBT_Im418718756_Empty @ A @ B ) )
=> ~ ! [X212: rBT_Im1923302023_color,X222: rBT_Im246033960le_rbt @ A @ B,X232: A,X242: B,X252: rBT_Im246033960le_rbt @ A @ B] :
( Y
!= ( rBT_Im480247531Branch @ A @ B @ X212 @ X222 @ X232 @ X242 @ X252 ) ) ) ).
% rbt.exhaust
thf(fact_8_color_Oinduct,axiom,
! [P: rBT_Im1923302023_color > $o,Color: rBT_Im1923302023_color] :
( ( P @ rBT_Impl_Mirabelle_R )
=> ( ( P @ rBT_Impl_Mirabelle_B )
=> ( P @ Color ) ) ) ).
% color.induct
thf(fact_9_color_Oexhaust,axiom,
! [Y: rBT_Im1923302023_color] :
( ( Y != rBT_Impl_Mirabelle_R )
=> ( Y = rBT_Impl_Mirabelle_B ) ) ).
% color.exhaust
thf(fact_10_color_Osimps_I6_J,axiom,
! [A: $tType,F1: A,F2: A] :
( ( rBT_Im1059957627_color @ A @ F1 @ F2 @ rBT_Impl_Mirabelle_B )
= F2 ) ).
% color.simps(6)
thf(fact_11_color_Osimps_I5_J,axiom,
! [A: $tType,F1: A,F2: A] :
( ( rBT_Im1059957627_color @ A @ F1 @ F2 @ rBT_Impl_Mirabelle_R )
= F1 ) ).
% color.simps(5)
thf(fact_12_color_Osimps_I4_J,axiom,
! [A: $tType,F1: A,F2: A] :
( ( rBT_Im1975547051_color @ A @ F1 @ F2 @ rBT_Impl_Mirabelle_B )
= F2 ) ).
% color.simps(4)
thf(fact_13_color_Osimps_I3_J,axiom,
! [A: $tType,F1: A,F2: A] :
( ( rBT_Im1975547051_color @ A @ F1 @ F2 @ rBT_Impl_Mirabelle_R )
= F1 ) ).
% color.simps(3)
thf(fact_14_rbt_Osimps_I4_J,axiom,
! [A: $tType,B: $tType,C: $tType,F1: C,F2: rBT_Im1923302023_color > ( rBT_Im246033960le_rbt @ A @ B ) > A > B > ( rBT_Im246033960le_rbt @ A @ B ) > C] :
( ( rBT_Im858806507se_rbt @ C @ A @ B @ F1 @ F2 @ ( rBT_Im418718756_Empty @ A @ B ) )
= F1 ) ).
% rbt.simps(4)
thf(fact_15_rbt_Osimps_I5_J,axiom,
! [C: $tType,B: $tType,A: $tType,F1: C,F2: rBT_Im1923302023_color > ( rBT_Im246033960le_rbt @ A @ B ) > A > B > ( rBT_Im246033960le_rbt @ A @ B ) > C,X21: rBT_Im1923302023_color,X22: rBT_Im246033960le_rbt @ A @ B,X23: A,X24: B,X25: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im858806507se_rbt @ C @ A @ B @ F1 @ F2 @ ( rBT_Im480247531Branch @ A @ B @ X21 @ X22 @ X23 @ X24 @ X25 ) )
= ( F2 @ X21 @ X22 @ X23 @ X24 @ X25 ) ) ).
% rbt.simps(5)
thf(fact_16_rbt_Osimps_I6_J,axiom,
! [A: $tType,B: $tType,E: $tType,F1: E,F2: rBT_Im1923302023_color > ( rBT_Im246033960le_rbt @ A @ B ) > A > B > ( rBT_Im246033960le_rbt @ A @ B ) > E > E > E] :
( ( rBT_Im1947144893ec_rbt @ E @ A @ B @ F1 @ F2 @ ( rBT_Im418718756_Empty @ A @ B ) )
= F1 ) ).
% rbt.simps(6)
thf(fact_17_rbt_Osimps_I7_J,axiom,
! [E: $tType,B: $tType,A: $tType,F1: E,F2: rBT_Im1923302023_color > ( rBT_Im246033960le_rbt @ A @ B ) > A > B > ( rBT_Im246033960le_rbt @ A @ B ) > E > E > E,X21: rBT_Im1923302023_color,X22: rBT_Im246033960le_rbt @ A @ B,X23: A,X24: B,X25: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1947144893ec_rbt @ E @ A @ B @ F1 @ F2 @ ( rBT_Im480247531Branch @ A @ B @ X21 @ X22 @ X23 @ X24 @ X25 ) )
= ( F2 @ X21 @ X22 @ X23 @ X24 @ X25 @ ( rBT_Im1947144893ec_rbt @ E @ A @ B @ F1 @ F2 @ X22 ) @ ( rBT_Im1947144893ec_rbt @ E @ A @ B @ F1 @ F2 @ X25 ) ) ) ).
% rbt.simps(7)
thf(fact_18_rbt_Opred__inject_I2_J,axiom,
! [B: $tType,A: $tType,P1: A > $o,P2: B > $o,A2: rBT_Im1923302023_color,Aa2: rBT_Im246033960le_rbt @ A @ B,Ab: A,Ac: B,Ad: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1931894874ed_rbt @ A @ B @ P1 @ P2 @ ( rBT_Im480247531Branch @ A @ B @ A2 @ Aa2 @ Ab @ Ac @ Ad ) )
= ( ( rBT_Im1931894874ed_rbt @ A @ B @ P1 @ P2 @ Aa2 )
& ( P1 @ Ab )
& ( P2 @ Ac )
& ( rBT_Im1931894874ed_rbt @ A @ B @ P1 @ P2 @ Ad ) ) ) ).
% rbt.pred_inject(2)
thf(fact_19_rbt_Orel__distinct_I2_J,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,R1: A > C > $o,R2: B > D > $o,Y21: rBT_Im1923302023_color,Y22: rBT_Im246033960le_rbt @ A @ B,Y23: A,Y24: B,Y25: rBT_Im246033960le_rbt @ A @ B] :
~ ( rBT_Im1000242676el_rbt @ A @ C @ B @ D @ R1 @ R2 @ ( rBT_Im480247531Branch @ A @ B @ Y21 @ Y22 @ Y23 @ Y24 @ Y25 ) @ ( rBT_Im418718756_Empty @ C @ D ) ) ).
% rbt.rel_distinct(2)
thf(fact_20_rbt_Orel__distinct_I1_J,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,R1: A > C > $o,R2: B > D > $o,Y21: rBT_Im1923302023_color,Y22: rBT_Im246033960le_rbt @ C @ D,Y23: C,Y24: D,Y25: rBT_Im246033960le_rbt @ C @ D] :
~ ( rBT_Im1000242676el_rbt @ A @ C @ B @ D @ R1 @ R2 @ ( rBT_Im418718756_Empty @ A @ B ) @ ( rBT_Im480247531Branch @ C @ D @ Y21 @ Y22 @ Y23 @ Y24 @ Y25 ) ) ).
% rbt.rel_distinct(1)
thf(fact_21_rbt_Opred__inject_I1_J,axiom,
! [B: $tType,A: $tType,P1: A > $o,P2: B > $o] : ( rBT_Im1931894874ed_rbt @ A @ B @ P1 @ P2 @ ( rBT_Im418718756_Empty @ A @ B ) ) ).
% rbt.pred_inject(1)
thf(fact_22_rbt_Orel__eq,axiom,
! [B: $tType,A: $tType] :
( ( rBT_Im1000242676el_rbt @ A @ A @ B @ B
@ ^ [Y2: A,Z: A] : Y2 = Z
@ ^ [Y2: B,Z: B] : Y2 = Z )
= ( ^ [Y2: rBT_Im246033960le_rbt @ A @ B,Z: rBT_Im246033960le_rbt @ A @ B] : Y2 = Z ) ) ).
% rbt.rel_eq
thf(fact_23_rbt_Orel__refl,axiom,
! [D: $tType,C: $tType,R1a: C > C > $o,R2a: D > D > $o,X: rBT_Im246033960le_rbt @ C @ D] :
( ! [X6: C] : ( R1a @ X6 @ X6 )
=> ( ! [X6: D] : ( R2a @ X6 @ X6 )
=> ( rBT_Im1000242676el_rbt @ C @ C @ D @ D @ R1a @ R2a @ X @ X ) ) ) ).
% rbt.rel_refl
thf(fact_24_color_Ocase__distrib,axiom,
! [A: $tType,B: $tType,H: A > B,F1: A,F2: A,Color: rBT_Im1923302023_color] :
( ( H @ ( rBT_Im1975547051_color @ A @ F1 @ F2 @ Color ) )
= ( rBT_Im1975547051_color @ B @ ( H @ F1 ) @ ( H @ F2 ) @ Color ) ) ).
% color.case_distrib
thf(fact_25_rbt_Orel__intros_I2_J,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,X21: rBT_Im1923302023_color,Y21: rBT_Im1923302023_color,R1: A > C > $o,R2: B > D > $o,X22: rBT_Im246033960le_rbt @ A @ B,Y22: rBT_Im246033960le_rbt @ C @ D,X23: A,Y23: C,X24: B,Y24: D,X25: rBT_Im246033960le_rbt @ A @ B,Y25: rBT_Im246033960le_rbt @ C @ D] :
( ( X21 = Y21 )
=> ( ( rBT_Im1000242676el_rbt @ A @ C @ B @ D @ R1 @ R2 @ X22 @ Y22 )
=> ( ( R1 @ X23 @ Y23 )
=> ( ( R2 @ X24 @ Y24 )
=> ( ( rBT_Im1000242676el_rbt @ A @ C @ B @ D @ R1 @ R2 @ X25 @ Y25 )
=> ( rBT_Im1000242676el_rbt @ A @ C @ B @ D @ R1 @ R2 @ ( rBT_Im480247531Branch @ A @ B @ X21 @ X22 @ X23 @ X24 @ X25 ) @ ( rBT_Im480247531Branch @ C @ D @ Y21 @ Y22 @ Y23 @ Y24 @ Y25 ) ) ) ) ) ) ) ).
% rbt.rel_intros(2)
thf(fact_26_rbt_Orel__inject_I2_J,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,R1: A > C > $o,R2: B > D > $o,X21: rBT_Im1923302023_color,X22: rBT_Im246033960le_rbt @ A @ B,X23: A,X24: B,X25: rBT_Im246033960le_rbt @ A @ B,Y21: rBT_Im1923302023_color,Y22: rBT_Im246033960le_rbt @ C @ D,Y23: C,Y24: D,Y25: rBT_Im246033960le_rbt @ C @ D] :
( ( rBT_Im1000242676el_rbt @ A @ C @ B @ D @ R1 @ R2 @ ( rBT_Im480247531Branch @ A @ B @ X21 @ X22 @ X23 @ X24 @ X25 ) @ ( rBT_Im480247531Branch @ C @ D @ Y21 @ Y22 @ Y23 @ Y24 @ Y25 ) )
= ( ( X21 = Y21 )
& ( rBT_Im1000242676el_rbt @ A @ C @ B @ D @ R1 @ R2 @ X22 @ Y22 )
& ( R1 @ X23 @ Y23 )
& ( R2 @ X24 @ Y24 )
& ( rBT_Im1000242676el_rbt @ A @ C @ B @ D @ R1 @ R2 @ X25 @ Y25 ) ) ) ).
% rbt.rel_inject(2)
thf(fact_27_rbt_Octr__transfer_I1_J,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,R1: A > C > $o,R2: B > D > $o] : ( rBT_Im1000242676el_rbt @ A @ C @ B @ D @ R1 @ R2 @ ( rBT_Im418718756_Empty @ A @ B ) @ ( rBT_Im418718756_Empty @ C @ D ) ) ).
% rbt.ctr_transfer(1)
thf(fact_28_rbt_Opred__rel,axiom,
! [B: $tType,A: $tType] :
( ( rBT_Im1931894874ed_rbt @ A @ B )
= ( ^ [P12: A > $o,P22: B > $o,X7: rBT_Im246033960le_rbt @ A @ B] : ( rBT_Im1000242676el_rbt @ A @ A @ B @ B @ ( bNF_eq_onp @ A @ P12 ) @ ( bNF_eq_onp @ B @ P22 ) @ X7 @ X7 ) ) ) ).
% rbt.pred_rel
thf(fact_29_rbt_Oset__intros_I1_J,axiom,
! [B: $tType,A: $tType,X: A,A22: rBT_Im246033960le_rbt @ A @ B,A1: rBT_Im1923302023_color,A3: A,A4: B,A5: rBT_Im246033960le_rbt @ A @ B] :
( ( member @ A @ X @ ( rBT_Im1178217900t1_rbt @ A @ B @ A22 ) )
=> ( member @ A @ X @ ( rBT_Im1178217900t1_rbt @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ A1 @ A22 @ A3 @ A4 @ A5 ) ) ) ) ).
% rbt.set_intros(1)
thf(fact_30_rbt_Oset__intros_I2_J,axiom,
! [B: $tType,A: $tType,A3: A,A1: rBT_Im1923302023_color,A22: rBT_Im246033960le_rbt @ A @ B,A4: B,A5: rBT_Im246033960le_rbt @ A @ B] : ( member @ A @ A3 @ ( rBT_Im1178217900t1_rbt @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ A1 @ A22 @ A3 @ A4 @ A5 ) ) ) ).
% rbt.set_intros(2)
thf(fact_31_rbt_Oset__intros_I3_J,axiom,
! [B: $tType,A: $tType,Xb: A,A5: rBT_Im246033960le_rbt @ A @ B,A1: rBT_Im1923302023_color,A22: rBT_Im246033960le_rbt @ A @ B,A3: A,A4: B] :
( ( member @ A @ Xb @ ( rBT_Im1178217900t1_rbt @ A @ B @ A5 ) )
=> ( member @ A @ Xb @ ( rBT_Im1178217900t1_rbt @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ A1 @ A22 @ A3 @ A4 @ A5 ) ) ) ) ).
% rbt.set_intros(3)
thf(fact_32_rbt_Oset__intros_I4_J,axiom,
! [B: $tType,A: $tType,Xe: B,A2a: rBT_Im246033960le_rbt @ A @ B,A1a: rBT_Im1923302023_color,A3a: A,A4a: B,A5a: rBT_Im246033960le_rbt @ A @ B] :
( ( member @ B @ Xe @ ( rBT_Im1550225131t2_rbt @ A @ B @ A2a ) )
=> ( member @ B @ Xe @ ( rBT_Im1550225131t2_rbt @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ A1a @ A2a @ A3a @ A4a @ A5a ) ) ) ) ).
% rbt.set_intros(4)
thf(fact_33_rbt_Oset__intros_I5_J,axiom,
! [B: $tType,A: $tType,A4a: B,A1a: rBT_Im1923302023_color,A2a: rBT_Im246033960le_rbt @ A @ B,A3a: A,A5a: rBT_Im246033960le_rbt @ A @ B] : ( member @ B @ A4a @ ( rBT_Im1550225131t2_rbt @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ A1a @ A2a @ A3a @ A4a @ A5a ) ) ) ).
% rbt.set_intros(5)
thf(fact_34_rbt_Oset__intros_I6_J,axiom,
! [B: $tType,A: $tType,Xg: B,A5a: rBT_Im246033960le_rbt @ A @ B,A1a: rBT_Im1923302023_color,A2a: rBT_Im246033960le_rbt @ A @ B,A3a: A,A4a: B] :
( ( member @ B @ Xg @ ( rBT_Im1550225131t2_rbt @ A @ B @ A5a ) )
=> ( member @ B @ Xg @ ( rBT_Im1550225131t2_rbt @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ A1a @ A2a @ A3a @ A4a @ A5a ) ) ) ) ).
% rbt.set_intros(6)
thf(fact_35_rbt_Oset__cases_I1_J,axiom,
! [B: $tType,A: $tType,E2: A,A2: rBT_Im246033960le_rbt @ A @ B] :
( ( member @ A @ E2 @ ( rBT_Im1178217900t1_rbt @ A @ B @ A2 ) )
=> ( ! [Z1: rBT_Im1923302023_color,Z2: rBT_Im246033960le_rbt @ A @ B] :
( ? [Z3: A,Z4: B,Z5: rBT_Im246033960le_rbt @ A @ B] :
( A2
= ( rBT_Im480247531Branch @ A @ B @ Z1 @ Z2 @ Z3 @ Z4 @ Z5 ) )
=> ~ ( member @ A @ E2 @ ( rBT_Im1178217900t1_rbt @ A @ B @ Z2 ) ) )
=> ( ! [Z1: rBT_Im1923302023_color,Z2: rBT_Im246033960le_rbt @ A @ B,Z4: B,Z5: rBT_Im246033960le_rbt @ A @ B] :
( A2
!= ( rBT_Im480247531Branch @ A @ B @ Z1 @ Z2 @ E2 @ Z4 @ Z5 ) )
=> ~ ! [Z1: rBT_Im1923302023_color,Z2: rBT_Im246033960le_rbt @ A @ B,Z3: A,Z4: B,Z5: rBT_Im246033960le_rbt @ A @ B] :
( ( A2
= ( rBT_Im480247531Branch @ A @ B @ Z1 @ Z2 @ Z3 @ Z4 @ Z5 ) )
=> ~ ( member @ A @ E2 @ ( rBT_Im1178217900t1_rbt @ A @ B @ Z5 ) ) ) ) ) ) ).
% rbt.set_cases(1)
thf(fact_36_rbt_Oset__cases_I2_J,axiom,
! [B: $tType,A: $tType,E2: B,A2: rBT_Im246033960le_rbt @ A @ B] :
( ( member @ B @ E2 @ ( rBT_Im1550225131t2_rbt @ A @ B @ A2 ) )
=> ( ! [Z1: rBT_Im1923302023_color,Z2: rBT_Im246033960le_rbt @ A @ B] :
( ? [Z3: A,Z4: B,Z5: rBT_Im246033960le_rbt @ A @ B] :
( A2
= ( rBT_Im480247531Branch @ A @ B @ Z1 @ Z2 @ Z3 @ Z4 @ Z5 ) )
=> ~ ( member @ B @ E2 @ ( rBT_Im1550225131t2_rbt @ A @ B @ Z2 ) ) )
=> ( ! [Z1: rBT_Im1923302023_color,Z2: rBT_Im246033960le_rbt @ A @ B,Z3: A,Z5: rBT_Im246033960le_rbt @ A @ B] :
( A2
!= ( rBT_Im480247531Branch @ A @ B @ Z1 @ Z2 @ Z3 @ E2 @ Z5 ) )
=> ~ ! [Z1: rBT_Im1923302023_color,Z2: rBT_Im246033960le_rbt @ A @ B,Z3: A,Z4: B,Z5: rBT_Im246033960le_rbt @ A @ B] :
( ( A2
= ( rBT_Im480247531Branch @ A @ B @ Z1 @ Z2 @ Z3 @ Z4 @ Z5 ) )
=> ~ ( member @ B @ E2 @ ( rBT_Im1550225131t2_rbt @ A @ B @ Z5 ) ) ) ) ) ) ).
% rbt.set_cases(2)
thf(fact_37_rbt_Osimps_I8_J,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,F1: A > C,F2: B > D] :
( ( rBT_Im206295089ap_rbt @ A @ C @ B @ D @ F1 @ F2 @ ( rBT_Im418718756_Empty @ A @ B ) )
= ( rBT_Im418718756_Empty @ C @ D ) ) ).
% rbt.simps(8)
thf(fact_38_rbt_Oinj__map__strong,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,X: rBT_Im246033960le_rbt @ A @ B,Xa: rBT_Im246033960le_rbt @ A @ B,F1: A > C,F1a: A > C,F2: B > D,F2a: B > D] :
( ! [Z1: A,Z1a: A] :
( ( member @ A @ Z1 @ ( rBT_Im1178217900t1_rbt @ A @ B @ X ) )
=> ( ( member @ A @ Z1a @ ( rBT_Im1178217900t1_rbt @ A @ B @ Xa ) )
=> ( ( ( F1 @ Z1 )
= ( F1a @ Z1a ) )
=> ( Z1 = Z1a ) ) ) )
=> ( ! [Z2: B,Z2a: B] :
( ( member @ B @ Z2 @ ( rBT_Im1550225131t2_rbt @ A @ B @ X ) )
=> ( ( member @ B @ Z2a @ ( rBT_Im1550225131t2_rbt @ A @ B @ Xa ) )
=> ( ( ( F2 @ Z2 )
= ( F2a @ Z2a ) )
=> ( Z2 = Z2a ) ) ) )
=> ( ( ( rBT_Im206295089ap_rbt @ A @ C @ B @ D @ F1 @ F2 @ X )
= ( rBT_Im206295089ap_rbt @ A @ C @ B @ D @ F1a @ F2a @ Xa ) )
=> ( X = Xa ) ) ) ) ).
% rbt.inj_map_strong
thf(fact_39_rbt_Omap__cong0,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,X: rBT_Im246033960le_rbt @ A @ B,F1: A > C,G1: A > C,F2: B > D,G2: B > D] :
( ! [Z1: A] :
( ( member @ A @ Z1 @ ( rBT_Im1178217900t1_rbt @ A @ B @ X ) )
=> ( ( F1 @ Z1 )
= ( G1 @ Z1 ) ) )
=> ( ! [Z2: B] :
( ( member @ B @ Z2 @ ( rBT_Im1550225131t2_rbt @ A @ B @ X ) )
=> ( ( F2 @ Z2 )
= ( G2 @ Z2 ) ) )
=> ( ( rBT_Im206295089ap_rbt @ A @ C @ B @ D @ F1 @ F2 @ X )
= ( rBT_Im206295089ap_rbt @ A @ C @ B @ D @ G1 @ G2 @ X ) ) ) ) ).
% rbt.map_cong0
thf(fact_40_rbt_Omap__cong,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,X: rBT_Im246033960le_rbt @ A @ B,Ya: rBT_Im246033960le_rbt @ A @ B,F1: A > C,G1: A > C,F2: B > D,G2: B > D] :
( ( X = Ya )
=> ( ! [Z1: A] :
( ( member @ A @ Z1 @ ( rBT_Im1178217900t1_rbt @ A @ B @ Ya ) )
=> ( ( F1 @ Z1 )
= ( G1 @ Z1 ) ) )
=> ( ! [Z2: B] :
( ( member @ B @ Z2 @ ( rBT_Im1550225131t2_rbt @ A @ B @ Ya ) )
=> ( ( F2 @ Z2 )
= ( G2 @ Z2 ) ) )
=> ( ( rBT_Im206295089ap_rbt @ A @ C @ B @ D @ F1 @ F2 @ X )
= ( rBT_Im206295089ap_rbt @ A @ C @ B @ D @ G1 @ G2 @ Ya ) ) ) ) ) ).
% rbt.map_cong
thf(fact_41_rbt_Orel__refl__strong,axiom,
! [B: $tType,A: $tType,X: rBT_Im246033960le_rbt @ A @ B,R1a: A > A > $o,R2a: B > B > $o] :
( ! [Z1: A] :
( ( member @ A @ Z1 @ ( rBT_Im1178217900t1_rbt @ A @ B @ X ) )
=> ( R1a @ Z1 @ Z1 ) )
=> ( ! [Z2: B] :
( ( member @ B @ Z2 @ ( rBT_Im1550225131t2_rbt @ A @ B @ X ) )
=> ( R2a @ Z2 @ Z2 ) )
=> ( rBT_Im1000242676el_rbt @ A @ A @ B @ B @ R1a @ R2a @ X @ X ) ) ) ).
% rbt.rel_refl_strong
thf(fact_42_rbt_Orel__mono__strong,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,R1: A > C > $o,R2: B > D > $o,X: rBT_Im246033960le_rbt @ A @ B,Y: rBT_Im246033960le_rbt @ C @ D,R1a: A > C > $o,R2a: B > D > $o] :
( ( rBT_Im1000242676el_rbt @ A @ C @ B @ D @ R1 @ R2 @ X @ Y )
=> ( ! [Z1: A,Y1: C] :
( ( member @ A @ Z1 @ ( rBT_Im1178217900t1_rbt @ A @ B @ X ) )
=> ( ( member @ C @ Y1 @ ( rBT_Im1178217900t1_rbt @ C @ D @ Y ) )
=> ( ( R1 @ Z1 @ Y1 )
=> ( R1a @ Z1 @ Y1 ) ) ) )
=> ( ! [Z2: B,Y26: D] :
( ( member @ B @ Z2 @ ( rBT_Im1550225131t2_rbt @ A @ B @ X ) )
=> ( ( member @ D @ Y26 @ ( rBT_Im1550225131t2_rbt @ C @ D @ Y ) )
=> ( ( R2 @ Z2 @ Y26 )
=> ( R2a @ Z2 @ Y26 ) ) ) )
=> ( rBT_Im1000242676el_rbt @ A @ C @ B @ D @ R1a @ R2a @ X @ Y ) ) ) ) ).
% rbt.rel_mono_strong
thf(fact_43_rbt_Orel__cong,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,X: rBT_Im246033960le_rbt @ A @ B,Ya: rBT_Im246033960le_rbt @ A @ B,Y: rBT_Im246033960le_rbt @ C @ D,Xa: rBT_Im246033960le_rbt @ C @ D,R1: A > C > $o,R1a: A > C > $o,R2: B > D > $o,R2a: B > D > $o] :
( ( X = Ya )
=> ( ( Y = Xa )
=> ( ! [Z1: A,Y1: C] :
( ( member @ A @ Z1 @ ( rBT_Im1178217900t1_rbt @ A @ B @ Ya ) )
=> ( ( member @ C @ Y1 @ ( rBT_Im1178217900t1_rbt @ C @ D @ Xa ) )
=> ( ( R1 @ Z1 @ Y1 )
= ( R1a @ Z1 @ Y1 ) ) ) )
=> ( ! [Z2: B,Y26: D] :
( ( member @ B @ Z2 @ ( rBT_Im1550225131t2_rbt @ A @ B @ Ya ) )
=> ( ( member @ D @ Y26 @ ( rBT_Im1550225131t2_rbt @ C @ D @ Xa ) )
=> ( ( R2 @ Z2 @ Y26 )
= ( R2a @ Z2 @ Y26 ) ) ) )
=> ( ( rBT_Im1000242676el_rbt @ A @ C @ B @ D @ R1 @ R2 @ X @ Y )
= ( rBT_Im1000242676el_rbt @ A @ C @ B @ D @ R1a @ R2a @ Ya @ Xa ) ) ) ) ) ) ).
% rbt.rel_cong
thf(fact_44_rbt_Opred__mono__strong,axiom,
! [B: $tType,A: $tType,P1: A > $o,P2: B > $o,X: rBT_Im246033960le_rbt @ A @ B,P1a: A > $o,P2a: B > $o] :
( ( rBT_Im1931894874ed_rbt @ A @ B @ P1 @ P2 @ X )
=> ( ! [Z1: A] :
( ( member @ A @ Z1 @ ( rBT_Im1178217900t1_rbt @ A @ B @ X ) )
=> ( ( P1 @ Z1 )
=> ( P1a @ Z1 ) ) )
=> ( ! [Z2: B] :
( ( member @ B @ Z2 @ ( rBT_Im1550225131t2_rbt @ A @ B @ X ) )
=> ( ( P2 @ Z2 )
=> ( P2a @ Z2 ) ) )
=> ( rBT_Im1931894874ed_rbt @ A @ B @ P1a @ P2a @ X ) ) ) ) ).
% rbt.pred_mono_strong
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A6: set @ A] :
( ( collect @ A
@ ^ [X7: A] : ( member @ A @ X7 @ A6 ) )
= A6 ) ).
% Collect_mem_eq
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 ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X6: A] :
( ( F @ X6 )
= ( G @ X6 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_rbt_Opred__cong,axiom,
! [B: $tType,A: $tType,X: rBT_Im246033960le_rbt @ A @ B,Ya: rBT_Im246033960le_rbt @ A @ B,P1: A > $o,P1a: A > $o,P2: B > $o,P2a: B > $o] :
( ( X = Ya )
=> ( ! [Z1: A] :
( ( member @ A @ Z1 @ ( rBT_Im1178217900t1_rbt @ A @ B @ Ya ) )
=> ( ( P1 @ Z1 )
= ( P1a @ Z1 ) ) )
=> ( ! [Z2: B] :
( ( member @ B @ Z2 @ ( rBT_Im1550225131t2_rbt @ A @ B @ Ya ) )
=> ( ( P2 @ Z2 )
= ( P2a @ Z2 ) ) )
=> ( ( rBT_Im1931894874ed_rbt @ A @ B @ P1 @ P2 @ X )
= ( rBT_Im1931894874ed_rbt @ A @ B @ P1a @ P2a @ Ya ) ) ) ) ) ).
% rbt.pred_cong
thf(fact_50_rbt_Orel__eq__onp,axiom,
! [A: $tType,B: $tType,P1: A > $o,P2: B > $o] :
( ( rBT_Im1000242676el_rbt @ A @ A @ B @ B @ ( bNF_eq_onp @ A @ P1 ) @ ( bNF_eq_onp @ B @ P2 ) )
= ( bNF_eq_onp @ ( rBT_Im246033960le_rbt @ A @ B ) @ ( rBT_Im1931894874ed_rbt @ A @ B @ P1 @ P2 ) ) ) ).
% rbt.rel_eq_onp
thf(fact_51_rbt_Osimps_I9_J,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,F1: A > C,F2: B > D,X21: rBT_Im1923302023_color,X22: rBT_Im246033960le_rbt @ A @ B,X23: A,X24: B,X25: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im206295089ap_rbt @ A @ C @ B @ D @ F1 @ F2 @ ( rBT_Im480247531Branch @ A @ B @ X21 @ X22 @ X23 @ X24 @ X25 ) )
= ( rBT_Im480247531Branch @ C @ D @ X21 @ ( rBT_Im206295089ap_rbt @ A @ C @ B @ D @ F1 @ F2 @ X22 ) @ ( F1 @ X23 ) @ ( F2 @ X24 ) @ ( rBT_Im206295089ap_rbt @ A @ C @ B @ D @ F1 @ F2 @ X25 ) ) ) ).
% rbt.simps(9)
thf(fact_52_eq__onp__live__step,axiom,
! [A: $tType,X: $o,Y: $o,P: A > $o,A2: A] :
( ( X = Y )
=> ( ( ( bNF_eq_onp @ A @ P @ A2 @ A2 )
& X )
= ( ( P @ A2 )
& Y ) ) ) ).
% eq_onp_live_step
thf(fact_53_eq__onp__same__args,axiom,
! [A: $tType,P: A > $o,X: A] :
( ( bNF_eq_onp @ A @ P @ X @ X )
= ( P @ X ) ) ).
% eq_onp_same_args
thf(fact_54_eq__onp__to__eq,axiom,
! [A: $tType,P: A > $o,X: A,Y: A] :
( ( bNF_eq_onp @ A @ P @ X @ Y )
=> ( X = Y ) ) ).
% eq_onp_to_eq
thf(fact_55_eq__onp__mono0,axiom,
! [A: $tType,A6: set @ A,P: A > $o,Q: A > $o] :
( ! [X6: A] :
( ( member @ A @ X6 @ A6 )
=> ( ( P @ X6 )
=> ( Q @ X6 ) ) )
=> ! [X8: A] :
( ( member @ A @ X8 @ A6 )
=> ! [Xa2: A] :
( ( member @ A @ Xa2 @ A6 )
=> ( ( bNF_eq_onp @ A @ P @ X8 @ Xa2 )
=> ( bNF_eq_onp @ A @ Q @ X8 @ Xa2 ) ) ) ) ) ).
% eq_onp_mono0
thf(fact_56_eq__onp__eqD,axiom,
! [A: $tType,P: A > $o,Q: A > A > $o,X: A] :
( ( ( bNF_eq_onp @ A @ P )
= Q )
=> ( ( P @ X )
= ( Q @ X @ X ) ) ) ).
% eq_onp_eqD
thf(fact_57_rbt_Opred__set,axiom,
! [B: $tType,A: $tType] :
( ( rBT_Im1931894874ed_rbt @ A @ B )
= ( ^ [P12: A > $o,P22: B > $o,X7: rBT_Im246033960le_rbt @ A @ B] :
( ! [Y3: A] :
( ( member @ A @ Y3 @ ( rBT_Im1178217900t1_rbt @ A @ B @ X7 ) )
=> ( P12 @ Y3 ) )
& ! [Y3: B] :
( ( member @ B @ Y3 @ ( rBT_Im1550225131t2_rbt @ A @ B @ X7 ) )
=> ( P22 @ Y3 ) ) ) ) ) ).
% rbt.pred_set
thf(fact_58_rbt_Osimps_I16_J,axiom,
! [A: $tType,B: $tType] :
( ( rBT_Im1550225131t2_rbt @ A @ B @ ( rBT_Im418718756_Empty @ A @ B ) )
= ( bot_bot @ ( set @ B ) ) ) ).
% rbt.simps(16)
thf(fact_59_rbt_Osimps_I14_J,axiom,
! [B: $tType,A: $tType] :
( ( rBT_Im1178217900t1_rbt @ A @ B @ ( rBT_Im418718756_Empty @ A @ B ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% rbt.simps(14)
thf(fact_60_rbt_Opred__map,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,Q1: C > $o,Q2: D > $o,F1: A > C,F2: B > D,X: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1931894874ed_rbt @ C @ D @ Q1 @ Q2 @ ( rBT_Im206295089ap_rbt @ A @ C @ B @ D @ F1 @ F2 @ X ) )
= ( rBT_Im1931894874ed_rbt @ A @ B @ ( comp @ C @ $o @ A @ Q1 @ F1 ) @ ( comp @ D @ $o @ B @ Q2 @ F2 ) @ X ) ) ).
% rbt.pred_map
thf(fact_61_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 ) ) ) ).
% comp_apply_eq
thf(fact_62_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 ) ) ) ).
% rewriteL_comp_comp
thf(fact_63_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 ) ) ) ).
% rewriteR_comp_comp
thf(fact_64_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 ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_65_rewriteR__comp__comp2,axiom,
! [C: $tType,B: $tType,E: $tType,D: $tType,A: $tType,G: C > B,H: A > C,R12: D > B,R22: A > D,F: B > E,L: D > E] :
( ( ( comp @ C @ B @ A @ G @ H )
= ( comp @ D @ B @ A @ R12 @ R22 ) )
=> ( ( ( comp @ B @ E @ D @ F @ R12 )
= L )
=> ( ( comp @ C @ E @ A @ ( comp @ B @ E @ C @ F @ G ) @ H )
= ( comp @ D @ E @ A @ L @ R22 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_66_rbt_Omap__comp,axiom,
! [D: $tType,F3: $tType,E: $tType,C: $tType,B: $tType,A: $tType,G1: C > E,G2: D > F3,F1: A > C,F2: B > D,V: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im206295089ap_rbt @ C @ E @ D @ F3 @ G1 @ G2 @ ( rBT_Im206295089ap_rbt @ A @ C @ B @ D @ F1 @ F2 @ V ) )
= ( rBT_Im206295089ap_rbt @ A @ E @ B @ F3 @ ( comp @ C @ E @ A @ G1 @ F1 ) @ ( comp @ D @ F3 @ B @ G2 @ F2 ) @ V ) ) ).
% rbt.map_comp
thf(fact_67_ball__empty,axiom,
! [A: $tType,P: A > $o,X8: A] :
( ( member @ A @ X8 @ ( bot_bot @ ( set @ A ) ) )
=> ( P @ X8 ) ) ).
% ball_empty
thf(fact_68_comp__apply,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comp @ B @ A @ C )
= ( ^ [F4: B > A,G3: C > B,X7: C] : ( F4 @ ( G3 @ X7 ) ) ) ) ).
% comp_apply
thf(fact_69_empty__iff,axiom,
! [A: $tType,C2: A] :
~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_70_all__not__in__conv,axiom,
! [A: $tType,A6: set @ A] :
( ( ! [X7: A] :
~ ( member @ A @ X7 @ A6 ) )
= ( A6
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_71_Collect__empty__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X7: A] :
~ ( P @ X7 ) ) ) ).
% Collect_empty_eq
thf(fact_72_empty__Collect__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P ) )
= ( ! [X7: A] :
~ ( P @ X7 ) ) ) ).
% empty_Collect_eq
thf(fact_73_bot__apply,axiom,
! [C: $tType,D: $tType] :
( ( bot @ C @ ( type2 @ C ) )
=> ( ( bot_bot @ ( D > C ) )
= ( ^ [X7: D] : ( bot_bot @ C ) ) ) ) ).
% bot_apply
thf(fact_74_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_75_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B @ ( type2 @ B ) )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X7: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_76_ex__in__conv,axiom,
! [A: $tType,A6: set @ A] :
( ( ? [X7: A] : ( member @ A @ X7 @ A6 ) )
= ( A6
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_77_equals0I,axiom,
! [A: $tType,A6: set @ A] :
( ! [Y4: A] :
~ ( member @ A @ Y4 @ A6 )
=> ( A6
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_78_equals0D,axiom,
! [A: $tType,A6: set @ A,A2: A] :
( ( A6
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A2 @ A6 ) ) ).
% equals0D
thf(fact_79_emptyE,axiom,
! [A: $tType,A2: A] :
~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_80_comp__eq__dest__lhs,axiom,
! [C: $tType,B: $tType,A: $tType,A2: C > B,B2: A > C,C2: A > B,V: A] :
( ( ( comp @ C @ B @ A @ A2 @ B2 )
= C2 )
=> ( ( A2 @ ( B2 @ V ) )
= ( C2 @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_81_comp__eq__elim,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,A2: C > B,B2: A > C,C2: D > B,D2: A > D] :
( ( ( comp @ C @ B @ A @ A2 @ B2 )
= ( comp @ D @ B @ A @ C2 @ D2 ) )
=> ! [V2: A] :
( ( A2 @ ( B2 @ V2 ) )
= ( C2 @ ( D2 @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_82_comp__eq__dest,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,A2: C > B,B2: A > C,C2: D > B,D2: A > D,V: A] :
( ( ( comp @ C @ B @ A @ A2 @ B2 )
= ( comp @ D @ B @ A @ C2 @ D2 ) )
=> ( ( A2 @ ( B2 @ V ) )
= ( C2 @ ( D2 @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_83_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 ) ) ) ).
% comp_assoc
thf(fact_84_comp__def,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comp @ B @ C @ A )
= ( ^ [F4: B > C,G3: A > B,X7: A] : ( F4 @ ( G3 @ X7 ) ) ) ) ).
% comp_def
thf(fact_85_Ball__def,axiom,
! [A: $tType] :
( ( ball @ A )
= ( ^ [A7: set @ A,P3: A > $o] :
! [X7: A] :
( ( member @ A @ X7 @ A7 )
=> ( P3 @ X7 ) ) ) ) ).
% Ball_def
thf(fact_86_override__on__emptyset,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ( override_on @ A @ B @ F @ G @ ( bot_bot @ ( set @ A ) ) )
= F ) ).
% override_on_emptyset
thf(fact_87_Set_Ois__empty__def,axiom,
! [A: $tType] :
( ( is_empty @ A )
= ( ^ [A7: set @ A] :
( A7
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Set.is_empty_def
thf(fact_88_ball__reg,axiom,
! [A: $tType,R3: set @ A,P: A > $o,Q: A > $o] :
( ! [X6: A] :
( ( member @ A @ X6 @ R3 )
=> ( ( P @ X6 )
=> ( Q @ X6 ) ) )
=> ( ! [X6: A] :
( ( member @ A @ X6 @ R3 )
=> ( P @ X6 ) )
=> ! [X8: A] :
( ( member @ A @ X8 @ R3 )
=> ( Q @ X8 ) ) ) ) ).
% ball_reg
thf(fact_89_type__copy__map__cong0,axiom,
! [B: $tType,D: $tType,E: $tType,A: $tType,C: $tType,M: B > A,G: C > B,X: C,N: D > A,H: C > D,F: A > E] :
( ( ( M @ ( G @ X ) )
= ( N @ ( H @ X ) ) )
=> ( ( comp @ B @ E @ C @ ( comp @ A @ E @ B @ F @ M ) @ G @ X )
= ( comp @ D @ E @ C @ ( comp @ A @ E @ D @ F @ N ) @ H @ X ) ) ) ).
% type_copy_map_cong0
thf(fact_90_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 ) ) ).
% fun.map_comp
thf(fact_91_comp__cong,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,E: $tType,F: B > A,G: C > B,X: C,F5: D > A,G4: E > D,X9: E] :
( ( ( F @ ( G @ X ) )
= ( F5 @ ( G4 @ X9 ) ) )
=> ( ( comp @ B @ A @ C @ F @ G @ X )
= ( comp @ D @ A @ E @ F5 @ G4 @ X9 ) ) ) ).
% comp_cong
thf(fact_92_override__on__apply__in,axiom,
! [B: $tType,A: $tType,A2: A,A6: set @ A,F: A > B,G: A > B] :
( ( member @ A @ A2 @ A6 )
=> ( ( override_on @ A @ B @ F @ G @ A6 @ A2 )
= ( G @ A2 ) ) ) ).
% override_on_apply_in
thf(fact_93_override__on__apply__notin,axiom,
! [B: $tType,A: $tType,A2: A,A6: set @ A,F: A > B,G: A > B] :
( ~ ( member @ A @ A2 @ A6 )
=> ( ( override_on @ A @ B @ F @ G @ A6 @ A2 )
= ( F @ A2 ) ) ) ).
% override_on_apply_notin
thf(fact_94_override__on__def,axiom,
! [B: $tType,A: $tType] :
( ( override_on @ A @ B )
= ( ^ [F4: A > B,G3: A > B,A7: set @ A,A8: A] : ( if @ B @ ( member @ A @ A8 @ A7 ) @ ( G3 @ A8 ) @ ( F4 @ A8 ) ) ) ) ).
% override_on_def
thf(fact_95_Collect__empty__eq__bot,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( P
= ( bot_bot @ ( A > $o ) ) ) ) ).
% Collect_empty_eq_bot
thf(fact_96_bot__empty__eq,axiom,
! [A: $tType] :
( ( bot_bot @ ( A > $o ) )
= ( ^ [X7: A] : ( member @ A @ X7 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% bot_empty_eq
thf(fact_97_rbt_Osize__gen__o__map,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F: C > nat,Fa: D > nat,G: A > C,Ga: B > D] :
( ( comp @ ( rBT_Im246033960le_rbt @ C @ D ) @ nat @ ( rBT_Im246033960le_rbt @ A @ B ) @ ( rBT_Im1485196410ze_rbt @ C @ D @ F @ Fa ) @ ( rBT_Im206295089ap_rbt @ A @ C @ B @ D @ G @ Ga ) )
= ( rBT_Im1485196410ze_rbt @ A @ B @ ( comp @ C @ nat @ A @ F @ G ) @ ( comp @ D @ nat @ B @ Fa @ Ga ) ) ) ).
% rbt.size_gen_o_map
thf(fact_98_is__singletonI_H,axiom,
! [A: $tType,A6: set @ A] :
( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X6: A,Y4: A] :
( ( member @ A @ X6 @ A6 )
=> ( ( member @ A @ Y4 @ A6 )
=> ( X6 = Y4 ) ) )
=> ( is_singleton @ A @ A6 ) ) ) ).
% is_singletonI'
thf(fact_99_vimage2p__comp,axiom,
! [E: $tType,D: $tType,F3: $tType,A: $tType,C: $tType,B: $tType,G5: $tType,F1: F3 > A,F2: D > F3,G1: G5 > B,G2: E > G5] :
( ( bNF_vimage2p @ D @ A @ E @ B @ C @ ( comp @ F3 @ A @ D @ F1 @ F2 ) @ ( comp @ G5 @ B @ E @ G1 @ G2 ) )
= ( comp @ ( F3 > G5 > C ) @ ( D > E > C ) @ ( A > B > C ) @ ( bNF_vimage2p @ D @ F3 @ E @ G5 @ C @ F2 @ G2 ) @ ( bNF_vimage2p @ F3 @ A @ G5 @ B @ C @ F1 @ G1 ) ) ) ).
% vimage2p_comp
thf(fact_100_vimage2p__cong,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,R3: A > B > C,S: A > B > C,F: D > A,G: E > B] :
( ( R3 = S )
=> ( ( bNF_vimage2p @ D @ A @ E @ B @ C @ F @ G @ R3 )
= ( bNF_vimage2p @ D @ A @ E @ B @ C @ F @ G @ S ) ) ) ).
% vimage2p_cong
thf(fact_101_rbt_Osize__gen_I1_J,axiom,
! [A: $tType,B: $tType,Xa: A > nat,X: B > nat] :
( ( rBT_Im1485196410ze_rbt @ A @ B @ Xa @ X @ ( rBT_Im418718756_Empty @ A @ B ) )
= ( zero_zero @ nat ) ) ).
% rbt.size_gen(1)
thf(fact_102_is__singletonI,axiom,
! [A: $tType,X: A] : ( is_singleton @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% is_singletonI
thf(fact_103_map__fun_Ocomp,axiom,
! [E: $tType,C: $tType,A: $tType,F3: $tType,D: $tType,B: $tType,F: E > C,G: D > F3,H: C > A,I: B > D] :
( ( comp @ ( C > D ) @ ( E > F3 ) @ ( A > B ) @ ( map_fun @ E @ C @ D @ F3 @ F @ G ) @ ( map_fun @ C @ A @ B @ D @ H @ I ) )
= ( map_fun @ E @ A @ B @ F3 @ ( comp @ C @ A @ E @ H @ F ) @ ( comp @ D @ F3 @ B @ G @ I ) ) ) ).
% map_fun.comp
thf(fact_104_is__singletonE,axiom,
! [A: $tType,A6: set @ A] :
( ( is_singleton @ A @ A6 )
=> ~ ! [X6: A] :
( A6
!= ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% is_singletonE
thf(fact_105_is__singleton__def,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A7: set @ A] :
? [X7: A] :
( A7
= ( insert @ A @ X7 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_def
thf(fact_106_insert__absorb2,axiom,
! [A: $tType,X: A,A6: set @ A] :
( ( insert @ A @ X @ ( insert @ A @ X @ A6 ) )
= ( insert @ A @ X @ A6 ) ) ).
% insert_absorb2
thf(fact_107_insert__iff,axiom,
! [A: $tType,A2: A,B2: A,A6: set @ A] :
( ( member @ A @ A2 @ ( insert @ A @ B2 @ A6 ) )
= ( ( A2 = B2 )
| ( member @ A @ A2 @ A6 ) ) ) ).
% insert_iff
thf(fact_108_insertCI,axiom,
! [A: $tType,A2: A,B3: set @ A,B2: A] :
( ( ~ ( member @ A @ A2 @ B3 )
=> ( A2 = B2 ) )
=> ( member @ A @ A2 @ ( insert @ A @ B2 @ B3 ) ) ) ).
% insertCI
thf(fact_109_map__fun__apply,axiom,
! [D: $tType,A: $tType,C: $tType,B: $tType] :
( ( map_fun @ B @ C @ D @ A )
= ( ^ [F4: B > C,G3: D > A,H2: C > D,X7: B] : ( G3 @ ( H2 @ ( F4 @ X7 ) ) ) ) ) ).
% map_fun_apply
thf(fact_110_singletonI,axiom,
! [A: $tType,A2: A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singletonI
thf(fact_111_singletonD,axiom,
! [A: $tType,B2: A,A2: A] :
( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_112_singleton__iff,axiom,
! [A: $tType,B2: A,A2: A] :
( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_113_doubleton__eq__iff,axiom,
! [A: $tType,A2: A,B2: A,C2: A,D2: A] :
( ( ( insert @ A @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ C2 @ ( insert @ A @ D2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ( ( A2 = C2 )
& ( B2 = D2 ) )
| ( ( A2 = D2 )
& ( B2 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_114_insert__not__empty,axiom,
! [A: $tType,A2: A,A6: set @ A] :
( ( insert @ A @ A2 @ A6 )
!= ( bot_bot @ ( set @ A ) ) ) ).
% insert_not_empty
thf(fact_115_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 ) ) ).
% singleton_inject
thf(fact_116_mk__disjoint__insert,axiom,
! [A: $tType,A2: A,A6: set @ A] :
( ( member @ A @ A2 @ A6 )
=> ? [B4: set @ A] :
( ( A6
= ( insert @ A @ A2 @ B4 ) )
& ~ ( member @ A @ A2 @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_117_insert__commute,axiom,
! [A: $tType,X: A,Y: A,A6: set @ A] :
( ( insert @ A @ X @ ( insert @ A @ Y @ A6 ) )
= ( insert @ A @ Y @ ( insert @ A @ X @ A6 ) ) ) ).
% insert_commute
thf(fact_118_insert__eq__iff,axiom,
! [A: $tType,A2: A,A6: set @ A,B2: A,B3: set @ A] :
( ~ ( member @ A @ A2 @ A6 )
=> ( ~ ( member @ A @ B2 @ B3 )
=> ( ( ( insert @ A @ A2 @ A6 )
= ( insert @ A @ B2 @ B3 ) )
= ( ( ( A2 = B2 )
=> ( A6 = B3 ) )
& ( ( A2 != B2 )
=> ? [C3: set @ A] :
( ( A6
= ( insert @ A @ B2 @ C3 ) )
& ~ ( member @ A @ B2 @ C3 )
& ( B3
= ( insert @ A @ A2 @ C3 ) )
& ~ ( member @ A @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_119_insert__absorb,axiom,
! [A: $tType,A2: A,A6: set @ A] :
( ( member @ A @ A2 @ A6 )
=> ( ( insert @ A @ A2 @ A6 )
= A6 ) ) ).
% insert_absorb
thf(fact_120_insert__ident,axiom,
! [A: $tType,X: A,A6: set @ A,B3: set @ A] :
( ~ ( member @ A @ X @ A6 )
=> ( ~ ( member @ A @ X @ B3 )
=> ( ( ( insert @ A @ X @ A6 )
= ( insert @ A @ X @ B3 ) )
= ( A6 = B3 ) ) ) ) ).
% insert_ident
thf(fact_121_Set_Oset__insert,axiom,
! [A: $tType,X: A,A6: set @ A] :
( ( member @ A @ X @ A6 )
=> ~ ! [B4: set @ A] :
( ( A6
= ( insert @ A @ X @ B4 ) )
=> ( member @ A @ X @ B4 ) ) ) ).
% Set.set_insert
thf(fact_122_insertI2,axiom,
! [A: $tType,A2: A,B3: set @ A,B2: A] :
( ( member @ A @ A2 @ B3 )
=> ( member @ A @ A2 @ ( insert @ A @ B2 @ B3 ) ) ) ).
% insertI2
thf(fact_123_insertI1,axiom,
! [A: $tType,A2: A,B3: set @ A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ B3 ) ) ).
% insertI1
thf(fact_124_insertE,axiom,
! [A: $tType,A2: A,B2: A,A6: set @ A] :
( ( member @ A @ A2 @ ( insert @ A @ B2 @ A6 ) )
=> ( ( A2 != B2 )
=> ( member @ A @ A2 @ A6 ) ) ) ).
% insertE
thf(fact_125_map__fun_Ocompositionality,axiom,
! [D: $tType,F3: $tType,C: $tType,E: $tType,B: $tType,A: $tType,F: E > C,G: D > F3,H: C > A,I: B > D,Fun: A > B] :
( ( map_fun @ E @ C @ D @ F3 @ F @ G @ ( map_fun @ C @ A @ B @ D @ H @ I @ Fun ) )
= ( map_fun @ E @ A @ B @ F3 @ ( comp @ C @ A @ E @ H @ F ) @ ( comp @ D @ F3 @ B @ G @ I ) @ Fun ) ) ).
% map_fun.compositionality
thf(fact_126_map__fun__def,axiom,
! [B: $tType,D: $tType,A: $tType,C: $tType] :
( ( map_fun @ C @ A @ B @ D )
= ( ^ [F4: C > A,G3: B > D,H2: A > B] : ( comp @ A @ D @ C @ ( comp @ B @ D @ A @ G3 @ H2 ) @ F4 ) ) ) ).
% map_fun_def
thf(fact_127_color_Osize__gen_I1_J,axiom,
( ( rBT_Im800559290_color @ rBT_Impl_Mirabelle_R )
= ( zero_zero @ nat ) ) ).
% color.size_gen(1)
thf(fact_128_color_Osize__gen_I2_J,axiom,
( ( rBT_Im800559290_color @ rBT_Impl_Mirabelle_B )
= ( zero_zero @ nat ) ) ).
% color.size_gen(2)
thf(fact_129_is__singleton__the__elem,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A7: set @ A] :
( A7
= ( insert @ A @ ( the_elem @ A @ A7 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_the_elem
thf(fact_130_the__elem__eq,axiom,
! [A: $tType,X: A] :
( ( the_elem @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ).
% the_elem_eq
thf(fact_131_color_Osize_I3_J,axiom,
( ( size_size @ rBT_Im1923302023_color @ rBT_Impl_Mirabelle_R )
= ( zero_zero @ nat ) ) ).
% color.size(3)
thf(fact_132_color_Osize_I4_J,axiom,
( ( size_size @ rBT_Im1923302023_color @ rBT_Impl_Mirabelle_B )
= ( zero_zero @ nat ) ) ).
% color.size(4)
thf(fact_133_rbt_Osize_I3_J,axiom,
! [A: $tType,B: $tType] :
( ( size_size @ ( rBT_Im246033960le_rbt @ A @ B ) @ ( rBT_Im418718756_Empty @ A @ B ) )
= ( zero_zero @ nat ) ) ).
% rbt.size(3)
thf(fact_134_rbt_Osimps_I15_J,axiom,
! [B: $tType,A: $tType,X21: rBT_Im1923302023_color,X22: rBT_Im246033960le_rbt @ A @ B,X23: A,X24: B,X25: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1178217900t1_rbt @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ X21 @ X22 @ X23 @ X24 @ X25 ) )
= ( insert @ A @ X23 @ ( sup_sup @ ( set @ A ) @ ( rBT_Im1178217900t1_rbt @ A @ B @ X22 ) @ ( rBT_Im1178217900t1_rbt @ A @ B @ X25 ) ) ) ) ).
% rbt.simps(15)
thf(fact_135_rbt_Osimps_I17_J,axiom,
! [B: $tType,A: $tType,X21: rBT_Im1923302023_color,X22: rBT_Im246033960le_rbt @ A @ B,X23: A,X24: B,X25: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1550225131t2_rbt @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ X21 @ X22 @ X23 @ X24 @ X25 ) )
= ( insert @ B @ X24 @ ( sup_sup @ ( set @ B ) @ ( rBT_Im1550225131t2_rbt @ A @ B @ X22 ) @ ( rBT_Im1550225131t2_rbt @ A @ B @ X25 ) ) ) ) ).
% rbt.simps(17)
thf(fact_136_pairwise__singleton,axiom,
! [A: $tType,P: A > A > $o,A6: A] : ( pairwise @ A @ P @ ( insert @ A @ A6 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% pairwise_singleton
thf(fact_137_Un__iff,axiom,
! [A: $tType,C2: A,A6: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A6 @ B3 ) )
= ( ( member @ A @ C2 @ A6 )
| ( member @ A @ C2 @ B3 ) ) ) ).
% Un_iff
thf(fact_138_UnCI,axiom,
! [A: $tType,C2: A,B3: set @ A,A6: set @ A] :
( ( ~ ( member @ A @ C2 @ B3 )
=> ( member @ A @ C2 @ A6 ) )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A6 @ B3 ) ) ) ).
% UnCI
thf(fact_139_Un__empty,axiom,
! [A: $tType,A6: set @ A,B3: set @ A] :
( ( ( sup_sup @ ( set @ A ) @ A6 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
= ( ( A6
= ( bot_bot @ ( set @ A ) ) )
& ( B3
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Un_empty
thf(fact_140_Un__insert__left,axiom,
! [A: $tType,A2: A,B3: set @ A,C4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( insert @ A @ A2 @ B3 ) @ C4 )
= ( insert @ A @ A2 @ ( sup_sup @ ( set @ A ) @ B3 @ C4 ) ) ) ).
% Un_insert_left
thf(fact_141_Un__insert__right,axiom,
! [A: $tType,A6: set @ A,A2: A,B3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A6 @ ( insert @ A @ A2 @ B3 ) )
= ( insert @ A @ A2 @ ( sup_sup @ ( set @ A ) @ A6 @ B3 ) ) ) ).
% Un_insert_right
thf(fact_142_Un__empty__left,axiom,
! [A: $tType,B3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B3 )
= B3 ) ).
% Un_empty_left
thf(fact_143_Un__empty__right,axiom,
! [A: $tType,A6: set @ A] :
( ( sup_sup @ ( set @ A ) @ A6 @ ( bot_bot @ ( set @ A ) ) )
= A6 ) ).
% Un_empty_right
thf(fact_144_Un__left__commute,axiom,
! [A: $tType,A6: set @ A,B3: set @ A,C4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A6 @ ( sup_sup @ ( set @ A ) @ B3 @ C4 ) )
= ( sup_sup @ ( set @ A ) @ B3 @ ( sup_sup @ ( set @ A ) @ A6 @ C4 ) ) ) ).
% Un_left_commute
thf(fact_145_Un__left__absorb,axiom,
! [A: $tType,A6: set @ A,B3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A6 @ ( sup_sup @ ( set @ A ) @ A6 @ B3 ) )
= ( sup_sup @ ( set @ A ) @ A6 @ B3 ) ) ).
% Un_left_absorb
thf(fact_146_pairwise__def,axiom,
! [A: $tType] :
( ( pairwise @ A )
= ( ^ [R4: A > A > $o,S2: set @ A] :
! [X7: A] :
( ( member @ A @ X7 @ S2 )
=> ! [Y3: A] :
( ( member @ A @ Y3 @ S2 )
=> ( ( X7 != Y3 )
=> ( R4 @ X7 @ Y3 ) ) ) ) ) ) ).
% pairwise_def
thf(fact_147_Un__commute,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A7: set @ A,B5: set @ A] : ( sup_sup @ ( set @ A ) @ B5 @ A7 ) ) ) ).
% Un_commute
thf(fact_148_Un__absorb,axiom,
! [A: $tType,A6: set @ A] :
( ( sup_sup @ ( set @ A ) @ A6 @ A6 )
= A6 ) ).
% Un_absorb
thf(fact_149_Un__assoc,axiom,
! [A: $tType,A6: set @ A,B3: set @ A,C4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A6 @ B3 ) @ C4 )
= ( sup_sup @ ( set @ A ) @ A6 @ ( sup_sup @ ( set @ A ) @ B3 @ C4 ) ) ) ).
% Un_assoc
thf(fact_150_ball__Un,axiom,
! [A: $tType,A6: set @ A,B3: set @ A,P: A > $o] :
( ( ! [X7: A] :
( ( member @ A @ X7 @ ( sup_sup @ ( set @ A ) @ A6 @ B3 ) )
=> ( P @ X7 ) ) )
= ( ! [X7: A] :
( ( member @ A @ X7 @ A6 )
=> ( P @ X7 ) )
& ! [X7: A] :
( ( member @ A @ X7 @ B3 )
=> ( P @ X7 ) ) ) ) ).
% ball_Un
thf(fact_151_bex__Un,axiom,
! [A: $tType,A6: set @ A,B3: set @ A,P: A > $o] :
( ( ? [X7: A] :
( ( member @ A @ X7 @ ( sup_sup @ ( set @ A ) @ A6 @ B3 ) )
& ( P @ X7 ) ) )
= ( ? [X7: A] :
( ( member @ A @ X7 @ A6 )
& ( P @ X7 ) )
| ? [X7: A] :
( ( member @ A @ X7 @ B3 )
& ( P @ X7 ) ) ) ) ).
% bex_Un
thf(fact_152_UnI2,axiom,
! [A: $tType,C2: A,B3: set @ A,A6: set @ A] :
( ( member @ A @ C2 @ B3 )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A6 @ B3 ) ) ) ).
% UnI2
thf(fact_153_UnI1,axiom,
! [A: $tType,C2: A,A6: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ A6 )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A6 @ B3 ) ) ) ).
% UnI1
thf(fact_154_UnE,axiom,
! [A: $tType,C2: A,A6: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A6 @ B3 ) )
=> ( ~ ( member @ A @ C2 @ A6 )
=> ( member @ A @ C2 @ B3 ) ) ) ).
% UnE
thf(fact_155_singleton__Un__iff,axiom,
! [A: $tType,X: A,A6: set @ A,B3: set @ A] :
( ( ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) )
= ( sup_sup @ ( set @ A ) @ A6 @ B3 ) )
= ( ( ( A6
= ( bot_bot @ ( set @ A ) ) )
& ( B3
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
| ( ( A6
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B3
= ( bot_bot @ ( set @ A ) ) ) )
| ( ( A6
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B3
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_156_Un__singleton__iff,axiom,
! [A: $tType,A6: set @ A,B3: set @ A,X: A] :
( ( ( sup_sup @ ( set @ A ) @ A6 @ B3 )
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( ( A6
= ( bot_bot @ ( set @ A ) ) )
& ( B3
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
| ( ( A6
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B3
= ( bot_bot @ ( set @ A ) ) ) )
| ( ( A6
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B3
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_157_insert__is__Un,axiom,
! [A: $tType] :
( ( insert @ A )
= ( ^ [A8: A] : ( sup_sup @ ( set @ A ) @ ( insert @ A @ A8 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% insert_is_Un
thf(fact_158_pairwise__empty,axiom,
! [A: $tType,P: A > A > $o] : ( pairwise @ A @ P @ ( bot_bot @ ( set @ A ) ) ) ).
% pairwise_empty
thf(fact_159_pairwise__insert,axiom,
! [A: $tType,R: A > A > $o,X: A,S3: set @ A] :
( ( pairwise @ A @ R @ ( insert @ A @ X @ S3 ) )
= ( ! [Y3: A] :
( ( ( member @ A @ Y3 @ S3 )
& ( Y3 != X ) )
=> ( ( R @ X @ Y3 )
& ( R @ Y3 @ X ) ) )
& ( pairwise @ A @ R @ S3 ) ) ) ).
% pairwise_insert
thf(fact_160_sup__bot_Oright__neutral,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( sup_sup @ A @ A2 @ ( bot_bot @ A ) )
= A2 ) ) ).
% sup_bot.right_neutral
thf(fact_161_sup__bot_Oleft__neutral,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( sup_sup @ A @ ( bot_bot @ A ) @ A2 )
= A2 ) ) ).
% sup_bot.left_neutral
thf(fact_162_sup__eq__bot__iff,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ( sup_sup @ A @ X @ Y )
= ( bot_bot @ A ) )
= ( ( X
= ( bot_bot @ A ) )
& ( Y
= ( bot_bot @ A ) ) ) ) ) ).
% sup_eq_bot_iff
thf(fact_163_bot__eq__sup__iff,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ( bot_bot @ A )
= ( sup_sup @ A @ X @ Y ) )
= ( ( X
= ( bot_bot @ A ) )
& ( Y
= ( bot_bot @ A ) ) ) ) ) ).
% bot_eq_sup_iff
thf(fact_164_sup__bot__left,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( sup_sup @ A @ ( bot_bot @ A ) @ X )
= X ) ) ).
% sup_bot_left
thf(fact_165_sup__bot__right,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( sup_sup @ A @ X @ ( bot_bot @ A ) )
= X ) ) ).
% sup_bot_right
thf(fact_166_ID_Opred__set,axiom,
! [A: $tType] :
( ( bNF_id_bnf @ ( A > $o ) )
= ( ^ [P3: A > $o,X7: A] :
! [Y3: A] :
( ( member @ A @ Y3 @ ( insert @ A @ X7 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( P3 @ Y3 ) ) ) ) ).
% ID.pred_set
thf(fact_167_sup__bot_Osemilattice__neutr__axioms,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A @ ( type2 @ A ) )
=> ( semilattice_neutr @ A @ ( sup_sup @ A ) @ ( bot_bot @ A ) ) ) ).
% sup_bot.semilattice_neutr_axioms
thf(fact_168_bot__nat__def,axiom,
( ( bot_bot @ nat )
= ( zero_zero @ nat ) ) ).
% bot_nat_def
thf(fact_169_id__bnf__apply,axiom,
! [A: $tType] :
( ( bNF_id_bnf @ A )
= ( ^ [X7: A] : X7 ) ) ).
% id_bnf_apply
thf(fact_170_BNF__Composition_Oid__bnf__def,axiom,
! [A: $tType] :
( ( bNF_id_bnf @ A )
= ( ^ [X7: A] : X7 ) ) ).
% BNF_Composition.id_bnf_def
thf(fact_171_ID_Opred__map,axiom,
! [B: $tType,A: $tType,Q: B > $o,F: A > B,X: A] :
( ( bNF_id_bnf @ ( B > $o ) @ Q @ ( bNF_id_bnf @ ( A > B ) @ F @ X ) )
= ( bNF_id_bnf @ ( A > $o ) @ ( comp @ B @ $o @ A @ Q @ F ) @ X ) ) ).
% ID.pred_map
thf(fact_172_ID_Opred__mono__strong,axiom,
! [A: $tType,P: A > $o,X: A,Pa: A > $o] :
( ( bNF_id_bnf @ ( A > $o ) @ P @ X )
=> ( ! [Z6: A] :
( ( member @ A @ Z6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( P @ Z6 )
=> ( Pa @ Z6 ) ) )
=> ( bNF_id_bnf @ ( A > $o ) @ Pa @ X ) ) ) ).
% ID.pred_mono_strong
thf(fact_173_ID_Opred__cong,axiom,
! [A: $tType,X: A,Ya: A,P: A > $o,Pa: A > $o] :
( ( X = Ya )
=> ( ! [Z6: A] :
( ( member @ A @ Z6 @ ( insert @ A @ Ya @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( P @ Z6 )
= ( Pa @ Z6 ) ) )
=> ( ( bNF_id_bnf @ ( A > $o ) @ P @ X )
= ( bNF_id_bnf @ ( A > $o ) @ Pa @ Ya ) ) ) ) ).
% ID.pred_cong
thf(fact_174_sup__bot_Omonoid__axioms,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A @ ( type2 @ A ) )
=> ( monoid @ A @ ( sup_sup @ A ) @ ( bot_bot @ A ) ) ) ).
% sup_bot.monoid_axioms
thf(fact_175_ID_Orel__eq__onp,axiom,
! [A: $tType,P: A > $o] :
( ( bNF_id_bnf @ ( A > A > $o ) @ ( bNF_eq_onp @ A @ P ) )
= ( bNF_eq_onp @ A @ ( bNF_id_bnf @ ( A > $o ) @ P ) ) ) ).
% ID.rel_eq_onp
thf(fact_176_ID_Opred__rel,axiom,
! [A: $tType] :
( ( bNF_id_bnf @ ( A > $o ) )
= ( ^ [P3: A > $o,X7: A] : ( bNF_id_bnf @ ( A > A > $o ) @ ( bNF_eq_onp @ A @ P3 ) @ X7 @ X7 ) ) ) ).
% ID.pred_rel
thf(fact_177_ID_Orel__refl,axiom,
! [A: $tType,Ra: A > A > $o,X: A] :
( ! [X6: A] : ( Ra @ X6 @ X6 )
=> ( bNF_id_bnf @ ( A > A > $o ) @ Ra @ X @ X ) ) ).
% ID.rel_refl
thf(fact_178_ID_Orel__cong,axiom,
! [A: $tType,B: $tType,X: A,Ya: A,Y: B,Xa: B,R3: A > B > $o,Ra: A > B > $o] :
( ( X = Ya )
=> ( ( Y = Xa )
=> ( ! [Z6: A,Yb: B] :
( ( member @ A @ Z6 @ ( insert @ A @ Ya @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( member @ B @ Yb @ ( insert @ B @ Xa @ ( bot_bot @ ( set @ B ) ) ) )
=> ( ( R3 @ Z6 @ Yb )
= ( Ra @ Z6 @ Yb ) ) ) )
=> ( ( bNF_id_bnf @ ( A > B > $o ) @ R3 @ X @ Y )
= ( bNF_id_bnf @ ( A > B > $o ) @ Ra @ Ya @ Xa ) ) ) ) ) ).
% ID.rel_cong
thf(fact_179_ID_Orel__mono__strong,axiom,
! [A: $tType,B: $tType,R3: A > B > $o,X: A,Y: B,Ra: A > B > $o] :
( ( bNF_id_bnf @ ( A > B > $o ) @ R3 @ X @ Y )
=> ( ! [Z6: A,Yb: B] :
( ( member @ A @ Z6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( member @ B @ Yb @ ( insert @ B @ Y @ ( bot_bot @ ( set @ B ) ) ) )
=> ( ( R3 @ Z6 @ Yb )
=> ( Ra @ Z6 @ Yb ) ) ) )
=> ( bNF_id_bnf @ ( A > B > $o ) @ Ra @ X @ Y ) ) ) ).
% ID.rel_mono_strong
thf(fact_180_ID_Orel__refl__strong,axiom,
! [A: $tType,X: A,Ra: A > A > $o] :
( ! [Z6: A] :
( ( member @ A @ Z6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
=> ( Ra @ Z6 @ Z6 ) )
=> ( bNF_id_bnf @ ( A > A > $o ) @ Ra @ X @ X ) ) ).
% ID.rel_refl_strong
thf(fact_181_sup__bot_Ocomm__monoid__axioms,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A @ ( type2 @ A ) )
=> ( comm_monoid @ A @ ( sup_sup @ A ) @ ( bot_bot @ A ) ) ) ).
% sup_bot.comm_monoid_axioms
thf(fact_182_vimage__insert,axiom,
! [A: $tType,B: $tType,F: A > B,A2: B,B3: set @ B] :
( ( vimage @ A @ B @ F @ ( insert @ B @ A2 @ B3 ) )
= ( sup_sup @ ( set @ A ) @ ( vimage @ A @ B @ F @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) @ ( vimage @ A @ B @ F @ B3 ) ) ) ).
% vimage_insert
thf(fact_183_notIn__Un__bij__betw3,axiom,
! [A: $tType,B: $tType,B2: A,A6: set @ A,F: A > B,A9: set @ B] :
( ~ ( member @ A @ B2 @ A6 )
=> ( ~ ( member @ B @ ( F @ B2 ) @ A9 )
=> ( ( bij_betw @ A @ B @ F @ A6 @ A9 )
= ( bij_betw @ A @ B @ F @ ( sup_sup @ ( set @ A ) @ A6 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( sup_sup @ ( set @ B ) @ A9 @ ( insert @ B @ ( F @ B2 ) @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ).
% notIn_Un_bij_betw3
thf(fact_184_vimage__eq,axiom,
! [A: $tType,B: $tType,A2: A,F: A > B,B3: set @ B] :
( ( member @ A @ A2 @ ( vimage @ A @ B @ F @ B3 ) )
= ( member @ B @ ( F @ A2 ) @ B3 ) ) ).
% vimage_eq
thf(fact_185_vimageI,axiom,
! [B: $tType,A: $tType,F: B > A,A2: B,B2: A,B3: set @ A] :
( ( ( F @ A2 )
= B2 )
=> ( ( member @ A @ B2 @ B3 )
=> ( member @ B @ A2 @ ( vimage @ B @ A @ F @ B3 ) ) ) ) ).
% vimageI
thf(fact_186_vimage__empty,axiom,
! [B: $tType,A: $tType,F: A > B] :
( ( vimage @ A @ B @ F @ ( bot_bot @ ( set @ B ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% vimage_empty
thf(fact_187_vimage__Un,axiom,
! [A: $tType,B: $tType,F: A > B,A6: set @ B,B3: set @ B] :
( ( vimage @ A @ B @ F @ ( sup_sup @ ( set @ B ) @ A6 @ B3 ) )
= ( sup_sup @ ( set @ A ) @ ( vimage @ A @ B @ F @ A6 ) @ ( vimage @ A @ B @ F @ B3 ) ) ) ).
% vimage_Un
thf(fact_188_vimage__Collect,axiom,
! [B: $tType,A: $tType,P: B > $o,F: A > B,Q: A > $o] :
( ! [X6: A] :
( ( P @ ( F @ X6 ) )
= ( Q @ X6 ) )
=> ( ( vimage @ A @ B @ F @ ( collect @ B @ P ) )
= ( collect @ A @ Q ) ) ) ).
% vimage_Collect
thf(fact_189_bij__betw__cong,axiom,
! [A: $tType,B: $tType,A6: set @ A,F: A > B,G: A > B,A9: set @ B] :
( ! [A10: A] :
( ( member @ A @ A10 @ A6 )
=> ( ( F @ A10 )
= ( G @ A10 ) ) )
=> ( ( bij_betw @ A @ B @ F @ A6 @ A9 )
= ( bij_betw @ A @ B @ G @ A6 @ A9 ) ) ) ).
% bij_betw_cong
thf(fact_190_bij__betw__inv,axiom,
! [B: $tType,A: $tType,F: A > B,A6: set @ A,B3: set @ B] :
( ( bij_betw @ A @ B @ F @ A6 @ B3 )
=> ? [G6: B > A] : ( bij_betw @ B @ A @ G6 @ B3 @ A6 ) ) ).
% bij_betw_inv
thf(fact_191_vimageI2,axiom,
! [B: $tType,A: $tType,F: B > A,A2: B,A6: set @ A] :
( ( member @ A @ ( F @ A2 ) @ A6 )
=> ( member @ B @ A2 @ ( vimage @ B @ A @ F @ A6 ) ) ) ).
% vimageI2
thf(fact_192_vimageE,axiom,
! [A: $tType,B: $tType,A2: A,F: A > B,B3: set @ B] :
( ( member @ A @ A2 @ ( vimage @ A @ B @ F @ B3 ) )
=> ( member @ B @ ( F @ A2 ) @ B3 ) ) ).
% vimageE
thf(fact_193_vimageD,axiom,
! [A: $tType,B: $tType,A2: A,F: A > B,A6: set @ B] :
( ( member @ A @ A2 @ ( vimage @ A @ B @ F @ A6 ) )
=> ( member @ B @ ( F @ A2 ) @ A6 ) ) ).
% vimageD
thf(fact_194_set_Ocomp,axiom,
! [A: $tType,B: $tType,C: $tType,F: C > B,G: B > A] :
( ( comp @ ( set @ B ) @ ( set @ C ) @ ( set @ A ) @ ( vimage @ C @ B @ F ) @ ( vimage @ B @ A @ G ) )
= ( vimage @ C @ A @ ( comp @ B @ A @ C @ G @ F ) ) ) ).
% set.comp
thf(fact_195_vimage__comp,axiom,
! [A: $tType,B: $tType,C: $tType,F: A > B,G: B > C,X: set @ C] :
( ( vimage @ A @ B @ F @ ( vimage @ B @ C @ G @ X ) )
= ( vimage @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ X ) ) ).
% vimage_comp
thf(fact_196_set_Ocompositionality,axiom,
! [C: $tType,B: $tType,A: $tType,F: C > B,G: B > A,Set: set @ A] :
( ( vimage @ C @ B @ F @ ( vimage @ B @ A @ G @ Set ) )
= ( vimage @ C @ A @ ( comp @ B @ A @ C @ G @ F ) @ Set ) ) ).
% set.compositionality
thf(fact_197_bij__betw__trans,axiom,
! [A: $tType,B: $tType,C: $tType,F: A > B,A6: set @ A,B3: set @ B,G: B > C,C4: set @ C] :
( ( bij_betw @ A @ B @ F @ A6 @ B3 )
=> ( ( bij_betw @ B @ C @ G @ B3 @ C4 )
=> ( bij_betw @ A @ C @ ( comp @ B @ C @ A @ G @ F ) @ A6 @ C4 ) ) ) ).
% bij_betw_trans
thf(fact_198_bij__betw__comp__iff,axiom,
! [A: $tType,B: $tType,C: $tType,F: A > B,A6: set @ A,A9: set @ B,F5: B > C,A11: set @ C] :
( ( bij_betw @ A @ B @ F @ A6 @ A9 )
=> ( ( bij_betw @ B @ C @ F5 @ A9 @ A11 )
= ( bij_betw @ A @ C @ ( comp @ B @ C @ A @ F5 @ F ) @ A6 @ A11 ) ) ) ).
% bij_betw_comp_iff
thf(fact_199_vimage__singleton__eq,axiom,
! [A: $tType,B: $tType,A2: A,F: A > B,B2: B] :
( ( member @ A @ A2 @ ( vimage @ A @ B @ F @ ( insert @ B @ B2 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( ( F @ A2 )
= B2 ) ) ).
% vimage_singleton_eq
thf(fact_200_bij__betw__empty2,axiom,
! [B: $tType,A: $tType,F: A > B,A6: set @ A] :
( ( bij_betw @ A @ B @ F @ A6 @ ( bot_bot @ ( set @ B ) ) )
=> ( A6
= ( bot_bot @ ( set @ A ) ) ) ) ).
% bij_betw_empty2
thf(fact_201_bij__betw__empty1,axiom,
! [A: $tType,B: $tType,F: A > B,A6: set @ B] :
( ( bij_betw @ A @ B @ F @ ( bot_bot @ ( set @ A ) ) @ A6 )
=> ( A6
= ( bot_bot @ ( set @ B ) ) ) ) ).
% bij_betw_empty1
thf(fact_202_notIn__Un__bij__betw,axiom,
! [A: $tType,B: $tType,B2: A,A6: set @ A,F: A > B,A9: set @ B] :
( ~ ( member @ A @ B2 @ A6 )
=> ( ~ ( member @ B @ ( F @ B2 ) @ A9 )
=> ( ( bij_betw @ A @ B @ F @ A6 @ A9 )
=> ( bij_betw @ A @ B @ F @ ( sup_sup @ ( set @ A ) @ A6 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( sup_sup @ ( set @ B ) @ A9 @ ( insert @ B @ ( F @ B2 ) @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ).
% notIn_Un_bij_betw
thf(fact_203_bij__betw__combine,axiom,
! [A: $tType,B: $tType,F: A > B,A6: set @ A,B3: set @ B,C4: set @ A,D3: set @ B] :
( ( bij_betw @ A @ B @ F @ A6 @ B3 )
=> ( ( bij_betw @ A @ B @ F @ C4 @ D3 )
=> ( ( ( inf_inf @ ( set @ B ) @ B3 @ D3 )
= ( bot_bot @ ( set @ B ) ) )
=> ( bij_betw @ A @ B @ F @ ( sup_sup @ ( set @ A ) @ A6 @ C4 ) @ ( sup_sup @ ( set @ B ) @ B3 @ D3 ) ) ) ) ) ).
% bij_betw_combine
thf(fact_204_infinite__imp__bij__betw2,axiom,
! [A: $tType,A6: set @ A,A2: A] :
( ~ ( finite_finite @ A @ A6 )
=> ? [H3: A > A] : ( bij_betw @ A @ A @ H3 @ A6 @ ( sup_sup @ ( set @ A ) @ A6 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% infinite_imp_bij_betw2
thf(fact_205_singleton__insert__inj__eq_H,axiom,
! [A: $tType,A2: A,A6: set @ A,B2: A] :
( ( ( insert @ A @ A2 @ A6 )
= ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A2 = B2 )
& ( ord_less_eq @ ( set @ A ) @ A6 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_206_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_207_subsetI,axiom,
! [A: $tType,A6: set @ A,B3: set @ A] :
( ! [X6: A] :
( ( member @ A @ X6 @ A6 )
=> ( member @ A @ X6 @ B3 ) )
=> ( ord_less_eq @ ( set @ A ) @ A6 @ B3 ) ) ).
% subsetI
thf(fact_208_subset__antisym,axiom,
! [A: $tType,A6: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ A6 )
=> ( A6 = B3 ) ) ) ).
% subset_antisym
thf(fact_209_IntI,axiom,
! [A: $tType,C2: A,A6: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ A6 )
=> ( ( member @ A @ C2 @ B3 )
=> ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A6 @ B3 ) ) ) ) ).
% IntI
thf(fact_210_Int__iff,axiom,
! [A: $tType,C2: A,A6: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A6 @ B3 ) )
= ( ( member @ A @ C2 @ A6 )
& ( member @ A @ C2 @ B3 ) ) ) ).
% Int_iff
thf(fact_211_inf__bot__left,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( inf_inf @ A @ ( bot_bot @ A ) @ X )
= ( bot_bot @ A ) ) ) ).
% inf_bot_left
thf(fact_212_inf__bot__right,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( inf_inf @ A @ X @ ( bot_bot @ A ) )
= ( bot_bot @ A ) ) ) ).
% inf_bot_right
thf(fact_213_subset__empty,axiom,
! [A: $tType,A6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ ( bot_bot @ ( set @ A ) ) )
= ( A6
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_empty
thf(fact_214_empty__subsetI,axiom,
! [A: $tType,A6: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A6 ) ).
% empty_subsetI
thf(fact_215_insert__subset,axiom,
! [A: $tType,X: A,A6: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A6 ) @ B3 )
= ( ( member @ A @ X @ B3 )
& ( ord_less_eq @ ( set @ A ) @ A6 @ B3 ) ) ) ).
% insert_subset
thf(fact_216_Int__subset__iff,axiom,
! [A: $tType,C4: set @ A,A6: set @ A,B3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C4 @ ( inf_inf @ ( set @ A ) @ A6 @ B3 ) )
= ( ( ord_less_eq @ ( set @ A ) @ C4 @ A6 )
& ( ord_less_eq @ ( set @ A ) @ C4 @ B3 ) ) ) ).
% Int_subset_iff
thf(fact_217_Int__insert__right__if1,axiom,
! [A: $tType,A2: A,A6: set @ A,B3: set @ A] :
( ( member @ A @ A2 @ A6 )
=> ( ( inf_inf @ ( set @ A ) @ A6 @ ( insert @ A @ A2 @ B3 ) )
= ( insert @ A @ A2 @ ( inf_inf @ ( set @ A ) @ A6 @ B3 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_218_Int__insert__right__if0,axiom,
! [A: $tType,A2: A,A6: set @ A,B3: set @ A] :
( ~ ( member @ A @ A2 @ A6 )
=> ( ( inf_inf @ ( set @ A ) @ A6 @ ( insert @ A @ A2 @ B3 ) )
= ( inf_inf @ ( set @ A ) @ A6 @ B3 ) ) ) ).
% Int_insert_right_if0
thf(fact_219_insert__inter__insert,axiom,
! [A: $tType,A2: A,A6: set @ A,B3: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A2 @ A6 ) @ ( insert @ A @ A2 @ B3 ) )
= ( insert @ A @ A2 @ ( inf_inf @ ( set @ A ) @ A6 @ B3 ) ) ) ).
% insert_inter_insert
thf(fact_220_Int__insert__left__if1,axiom,
! [A: $tType,A2: A,C4: set @ A,B3: set @ A] :
( ( member @ A @ A2 @ C4 )
=> ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A2 @ B3 ) @ C4 )
= ( insert @ A @ A2 @ ( inf_inf @ ( set @ A ) @ B3 @ C4 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_221_Int__insert__left__if0,axiom,
! [A: $tType,A2: A,C4: set @ A,B3: set @ A] :
( ~ ( member @ A @ A2 @ C4 )
=> ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A2 @ B3 ) @ C4 )
= ( inf_inf @ ( set @ A ) @ B3 @ C4 ) ) ) ).
% Int_insert_left_if0
thf(fact_222_Un__subset__iff,axiom,
! [A: $tType,A6: set @ A,B3: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A6 @ B3 ) @ C4 )
= ( ( ord_less_eq @ ( set @ A ) @ A6 @ C4 )
& ( ord_less_eq @ ( set @ A ) @ B3 @ C4 ) ) ) ).
% Un_subset_iff
thf(fact_223_vimage__Int,axiom,
! [A: $tType,B: $tType,F: A > B,A6: set @ B,B3: set @ B] :
( ( vimage @ A @ B @ F @ ( inf_inf @ ( set @ B ) @ A6 @ B3 ) )
= ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F @ A6 ) @ ( vimage @ A @ B @ F @ B3 ) ) ) ).
% vimage_Int
thf(fact_224_singleton__insert__inj__eq,axiom,
! [A: $tType,B2: A,A2: A,A6: set @ A] :
( ( ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ A2 @ A6 ) )
= ( ( A2 = B2 )
& ( ord_less_eq @ ( set @ A ) @ A6 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_225_disjoint__insert_I2_J,axiom,
! [A: $tType,A6: set @ A,B2: A,B3: set @ A] :
( ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ A6 @ ( insert @ A @ B2 @ B3 ) ) )
= ( ~ ( member @ A @ B2 @ A6 )
& ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ A6 @ B3 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_226_disjoint__insert_I1_J,axiom,
! [A: $tType,B3: set @ A,A2: A,A6: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ B3 @ ( insert @ A @ A2 @ A6 ) )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( member @ A @ A2 @ B3 )
& ( ( inf_inf @ ( set @ A ) @ B3 @ A6 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% disjoint_insert(1)
thf(fact_227_insert__disjoint_I2_J,axiom,
! [A: $tType,A2: A,A6: set @ A,B3: set @ A] :
( ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ ( insert @ A @ A2 @ A6 ) @ B3 ) )
= ( ~ ( member @ A @ A2 @ B3 )
& ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ A6 @ B3 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_228_insert__disjoint_I1_J,axiom,
! [A: $tType,A2: A,A6: set @ A,B3: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A2 @ A6 ) @ B3 )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( member @ A @ A2 @ B3 )
& ( ( inf_inf @ ( set @ A ) @ A6 @ B3 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% insert_disjoint(1)
thf(fact_229_vimage__inter__cong,axiom,
! [B: $tType,A: $tType,S: set @ A,F: A > B,G: A > B,Y: set @ B] :
( ! [W: A] :
( ( member @ A @ W @ S )
=> ( ( F @ W )
= ( G @ W ) ) )
=> ( ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F @ Y ) @ S )
= ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ G @ Y ) @ S ) ) ) ).
% vimage_inter_cong
thf(fact_230_vimage__mono,axiom,
! [B: $tType,A: $tType,A6: set @ A,B3: set @ A,F: B > A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B3 )
=> ( ord_less_eq @ ( set @ B ) @ ( vimage @ B @ A @ F @ A6 ) @ ( vimage @ B @ A @ F @ B3 ) ) ) ).
% vimage_mono
thf(fact_231_Int__emptyI,axiom,
! [A: $tType,A6: set @ A,B3: set @ A] :
( ! [X6: A] :
( ( member @ A @ X6 @ A6 )
=> ~ ( member @ A @ X6 @ B3 ) )
=> ( ( inf_inf @ ( set @ A ) @ A6 @ B3 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% Int_emptyI
thf(fact_232_Int__empty__left,axiom,
! [A: $tType,B3: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B3 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Int_empty_left
thf(fact_233_Int__empty__right,axiom,
! [A: $tType,A6: set @ A] :
( ( inf_inf @ ( set @ A ) @ A6 @ ( bot_bot @ ( set @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Int_empty_right
thf(fact_234_disjoint__iff__not__equal,axiom,
! [A: $tType,A6: set @ A,B3: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A6 @ B3 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X7: A] :
( ( member @ A @ X7 @ A6 )
=> ! [Y3: A] :
( ( member @ A @ Y3 @ B3 )
=> ( X7 != Y3 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_235_bot_Oextremum,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A2: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A2 ) ) ).
% bot.extremum
thf(fact_236_bot_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
= ( A2
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_unique
thf(fact_237_bot_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
=> ( A2
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_uniqueI
thf(fact_238_Int__insert__right,axiom,
! [A: $tType,A2: A,A6: set @ A,B3: set @ A] :
( ( ( member @ A @ A2 @ A6 )
=> ( ( inf_inf @ ( set @ A ) @ A6 @ ( insert @ A @ A2 @ B3 ) )
= ( insert @ A @ A2 @ ( inf_inf @ ( set @ A ) @ A6 @ B3 ) ) ) )
& ( ~ ( member @ A @ A2 @ A6 )
=> ( ( inf_inf @ ( set @ A ) @ A6 @ ( insert @ A @ A2 @ B3 ) )
= ( inf_inf @ ( set @ A ) @ A6 @ B3 ) ) ) ) ).
% Int_insert_right
thf(fact_239_Int__insert__left,axiom,
! [A: $tType,A2: A,C4: set @ A,B3: set @ A] :
( ( ( member @ A @ A2 @ C4 )
=> ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A2 @ B3 ) @ C4 )
= ( insert @ A @ A2 @ ( inf_inf @ ( set @ A ) @ B3 @ C4 ) ) ) )
& ( ~ ( member @ A @ A2 @ C4 )
=> ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A2 @ B3 ) @ C4 )
= ( inf_inf @ ( set @ A ) @ B3 @ C4 ) ) ) ) ).
% Int_insert_left
thf(fact_240_IntE,axiom,
! [A: $tType,C2: A,A6: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A6 @ B3 ) )
=> ~ ( ( member @ A @ C2 @ A6 )
=> ~ ( member @ A @ C2 @ B3 ) ) ) ).
% IntE
thf(fact_241_IntD1,axiom,
! [A: $tType,C2: A,A6: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A6 @ B3 ) )
=> ( member @ A @ C2 @ A6 ) ) ).
% IntD1
thf(fact_242_IntD2,axiom,
! [A: $tType,C2: A,A6: set @ A,B3: set @ A] :
( ( member @ A @ C2 @ ( inf_inf @ ( set @ A ) @ A6 @ B3 ) )
=> ( member @ A @ C2 @ B3 ) ) ).
% IntD2
thf(fact_243_set__mp,axiom,
! [A: $tType,A6: set @ A,B3: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B3 )
=> ( ( member @ A @ X @ A6 )
=> ( member @ A @ X @ B3 ) ) ) ).
% set_mp
thf(fact_244_in__mono,axiom,
! [A: $tType,A6: set @ A,B3: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B3 )
=> ( ( member @ A @ X @ A6 )
=> ( member @ A @ X @ B3 ) ) ) ).
% in_mono
thf(fact_245_subsetD,axiom,
! [A: $tType,A6: set @ A,B3: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B3 )
=> ( ( member @ A @ C2 @ A6 )
=> ( member @ A @ C2 @ B3 ) ) ) ).
% subsetD
thf(fact_246_Int__mono,axiom,
! [A: $tType,A6: set @ A,C4: set @ A,B3: set @ A,D3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B3 @ D3 )
=> ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A6 @ B3 ) @ ( inf_inf @ ( set @ A ) @ C4 @ D3 ) ) ) ) ).
% Int_mono
thf(fact_247_subsetCE,axiom,
! [A: $tType,A6: set @ A,B3: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B3 )
=> ( ( member @ A @ C2 @ A6 )
=> ( member @ A @ C2 @ B3 ) ) ) ).
% subsetCE
thf(fact_248_Int__assoc,axiom,
! [A: $tType,A6: set @ A,B3: set @ A,C4: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A6 @ B3 ) @ C4 )
= ( inf_inf @ ( set @ A ) @ A6 @ ( inf_inf @ ( set @ A ) @ B3 @ C4 ) ) ) ).
% Int_assoc
thf(fact_249_equalityE,axiom,
! [A: $tType,A6: set @ A,B3: set @ A] :
( ( A6 = B3 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A6 @ B3 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B3 @ A6 ) ) ) ).
% equalityE
thf(fact_250_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B5: set @ A] :
! [X7: A] :
( ( member @ A @ X7 @ A7 )
=> ( member @ A @ X7 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_251_Int__absorb,axiom,
! [A: $tType,A6: set @ A] :
( ( inf_inf @ ( set @ A ) @ A6 @ A6 )
= A6 ) ).
% Int_absorb
thf(fact_252_Int__lower1,axiom,
! [A: $tType,A6: set @ A,B3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A6 @ B3 ) @ A6 ) ).
% Int_lower1
thf(fact_253_Int__lower2,axiom,
! [A: $tType,A6: set @ A,B3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A6 @ B3 ) @ B3 ) ).
% Int_lower2
thf(fact_254_equalityD1,axiom,
! [A: $tType,A6: set @ A,B3: set @ A] :
( ( A6 = B3 )
=> ( ord_less_eq @ ( set @ A ) @ A6 @ B3 ) ) ).
% equalityD1
thf(fact_255_equalityD2,axiom,
! [A: $tType,A6: set @ A,B3: set @ A] :
( ( A6 = B3 )
=> ( ord_less_eq @ ( set @ A ) @ B3 @ A6 ) ) ).
% equalityD2
%----Type constructors (21)
thf(tcon_HOL_Obool___Lattices_Obounded__lattice,axiom,
bounded_lattice @ $o @ ( type2 @ $o ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice_1,axiom,
! [A12: $tType] : ( bounded_lattice @ ( set @ A12 ) @ ( type2 @ ( set @ A12 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice_2,axiom,
! [A12: $tType,A13: $tType] :
( ( bounded_lattice @ A13 @ ( type2 @ A13 ) )
=> ( bounded_lattice @ ( A12 > A13 ) @ ( type2 @ ( A12 > A13 ) ) ) ) ).
thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
! [A12: $tType,A13: $tType] :
( ( bounded_lattice @ A13 @ ( type2 @ A13 ) )
=> ( bounde1808546759up_bot @ ( A12 > A13 ) @ ( type2 @ ( A12 > A13 ) ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice__bot,axiom,
! [A12: $tType,A13: $tType] :
( ( bounded_lattice @ A13 @ ( type2 @ A13 ) )
=> ( bounded_lattice_bot @ ( A12 > A13 ) @ ( type2 @ ( A12 > A13 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder__bot,axiom,
! [A12: $tType,A13: $tType] :
( ( order_bot @ A13 @ ( type2 @ A13 ) )
=> ( order_bot @ ( A12 > A13 ) @ ( type2 @ ( A12 > A13 ) ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A12: $tType,A13: $tType] :
( ( preorder @ A13 @ ( type2 @ A13 ) )
=> ( preorder @ ( A12 > A13 ) @ ( type2 @ ( A12 > A13 ) ) ) ) ).
thf(tcon_fun___Orderings_Obot,axiom,
! [A12: $tType,A13: $tType] :
( ( bot @ A13 @ ( type2 @ A13 ) )
=> ( bot @ ( A12 > A13 ) @ ( type2 @ ( A12 > A13 ) ) ) ) ).
thf(tcon_Nat_Onat___Orderings_Oorder__bot_3,axiom,
order_bot @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Opreorder_4,axiom,
preorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Obot_5,axiom,
bot @ nat @ ( type2 @ nat ) ).
thf(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_6,axiom,
! [A12: $tType] : ( bounde1808546759up_bot @ ( set @ A12 ) @ ( type2 @ ( set @ A12 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice__bot_7,axiom,
! [A12: $tType] : ( bounded_lattice_bot @ ( set @ A12 ) @ ( type2 @ ( set @ A12 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_8,axiom,
! [A12: $tType] : ( order_bot @ ( set @ A12 ) @ ( type2 @ ( set @ A12 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_9,axiom,
! [A12: $tType] : ( preorder @ ( set @ A12 ) @ ( type2 @ ( set @ A12 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_10,axiom,
! [A12: $tType] : ( bot @ ( set @ A12 ) @ ( type2 @ ( set @ A12 ) ) ) ).
thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_11,axiom,
bounde1808546759up_bot @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_12,axiom,
bounded_lattice_bot @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_13,axiom,
order_bot @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_14,axiom,
preorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Obot_15,axiom,
bot @ $o @ ( type2 @ $o ) ).
%----Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $true @ X @ Y )
= X ) ).
%----Conjectures (4)
thf(conj_0,hypothesis,
$true ).
thf(conj_1,hypothesis,
! [L3: rBT_Im246033960le_rbt @ a @ b,K2: a,V2: b,R5: rBT_Im246033960le_rbt @ a @ b] :
( ( t
= ( rBT_Im480247531Branch @ a @ b @ rBT_Impl_Mirabelle_R @ L3 @ K2 @ V2 @ R5 ) )
=> thesis ) ).
thf(conj_2,hypothesis,
! [L3: rBT_Im246033960le_rbt @ a @ b,K2: a,V2: b,R5: rBT_Im246033960le_rbt @ a @ b] :
( ( t
= ( rBT_Im480247531Branch @ a @ b @ rBT_Impl_Mirabelle_B @ L3 @ K2 @ V2 @ R5 ) )
=> thesis ) ).
thf(conj_3,conjecture,
thesis ).
%------------------------------------------------------------------------------