TPTP Problem File: ITP138^2.p
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%------------------------------------------------------------------------------
% File : ITP138^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Paraconsistency problem prob_525__3267638_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : Paraconsistency/prob_525__3267638_1 [Des21]
% Status : Theorem
% Rating : 0.00 v7.5.0
% Syntax : Number of formulae : 337 ( 141 unt; 48 typ; 0 def)
% Number of atoms : 835 ( 272 equ; 0 cnn)
% Maximal formula atoms : 16 ( 2 avg)
% Number of connectives : 4144 ( 79 ~; 1 |; 69 &;3603 @)
% ( 0 <=>; 392 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 8 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 343 ( 343 >; 0 *; 0 +; 0 <<)
% Number of symbols : 47 ( 44 usr; 4 con; 0-6 aty)
% Number of variables : 1245 ( 109 ^;1073 !; 30 ?;1245 :)
% ( 33 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:21:25.150
%------------------------------------------------------------------------------
% Could-be-implicit typings (7)
thf(ty_t_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv,type,
paraco415392788lle_tv: $tType ).
thf(ty_t_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm,type,
paraco414474393lle_fm: $tType ).
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_String_Ochar,type,
char: $tType ).
thf(ty_t_List_Olist,type,
list: $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
% Explicit typings (41)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Otop,type,
top:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ono__bot,type,
no_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
thf(sy_cl_Enum_Ofinite__lattice,type,
finite_lattice:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder__top,type,
order_top:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_OSup,type,
complete_Sup:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
comple187826305attice:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_Ocomplete__linorder,type,
comple1035589618norder:
!>[A: $tType] : $o ).
thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__lattice,type,
condit378418413attice:
!>[A: $tType] : $o ).
thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
condit1037483654norder:
!>[A: $tType] : $o ).
thf(sy_c_Complete__Lattices_OSup__class_OSup,type,
complete_Sup_Sup:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Finite__Set_OFpow,type,
finite_Fpow:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
thf(sy_c_Fun_Oswap,type,
swap:
!>[A: $tType,B: $tType] : ( A > A > ( A > B ) > A > B ) ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oorder__class_OGreatest,type,
order_Greatest:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_Orderings_Oorder__class_Ostrict__mono,type,
order_strict_mono:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Orderings_Otop__class_Otop,type,
top_top:
!>[A: $tType] : A ).
thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Odomain,type,
paraco506338565domain: ( set @ nat ) > ( set @ paraco415392788lle_tv ) ).
thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Oeval,type,
paraco876059933e_eval: ( ( list @ char ) > paraco415392788lle_tv ) > paraco414474393lle_fm > paraco415392788lle_tv ).
thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OCon_H,type,
paraco2100061555le_Con: paraco414474393lle_fm > paraco414474393lle_fm > paraco414474393lle_fm ).
thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OEql,type,
paraco2084319816le_Eql: paraco414474393lle_fm > paraco414474393lle_fm > paraco414474393lle_fm ).
thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OEql_H,type,
paraco1628874225le_Eql: paraco414474393lle_fm > paraco414474393lle_fm > paraco414474393lle_fm ).
thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_ONeg_H,type,
paraco329115265le_Neg: paraco414474393lle_fm > paraco414474393lle_fm ).
thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OPro,type,
paraco27778325le_Pro: ( list @ char ) > paraco414474393lle_fm ).
thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ofm_OTruth,type,
paraco251304083_Truth: paraco414474393lle_fm ).
thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv_ODet,type,
paraco2040174112le_Det: $o > paraco415392788lle_tv ).
thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Otv_Orec__tv,type,
paraco152590079rec_tv:
!>[A: $tType] : ( ( $o > A ) > ( nat > A ) > paraco415392788lle_tv > A ) ).
thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ovalid,type,
paraco769098683_valid: paraco414474393lle_fm > $o ).
thf(sy_c_Paraconsistency__Mirabelle__pjyhsvfwkt_Ovalid__in,type,
paraco2086025920lid_in: ( set @ nat ) > paraco414474393lle_fm > $o ).
thf(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
thf(sy_c_Relation_OPowp,type,
powp:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_OPow,type,
pow:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
thf(sy_c_Set_Oimage,type,
image:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_p,type,
p: paraco414474393lle_fm ).
% Relevant facts (256)
thf(fact_0_universal__domain,axiom,
( ( paraco506338565domain
@ ( collect @ nat
@ ^ [N: nat] : $true ) )
= ( collect @ paraco415392788lle_tv
@ ^ [X: paraco415392788lle_tv] : $true ) ) ).
% universal_domain
thf(fact_1_tv_Oinject_I1_J,axiom,
! [X1: $o,Y1: $o] :
( ( ( paraco2040174112le_Det @ X1 )
= ( paraco2040174112le_Det @ Y1 ) )
= ( X1 = Y1 ) ) ).
% tv.inject(1)
thf(fact_2_valid__in__def,axiom,
( paraco2086025920lid_in
= ( ^ [U: set @ nat,P: paraco414474393lle_fm] :
! [I: ( list @ char ) > paraco415392788lle_tv] :
( ( ord_less_eq @ ( set @ paraco415392788lle_tv ) @ ( image @ ( list @ char ) @ paraco415392788lle_tv @ I @ ( top_top @ ( set @ ( list @ char ) ) ) ) @ ( paraco506338565domain @ U ) )
=> ( ( paraco876059933e_eval @ I @ P )
= ( paraco2040174112le_Det @ $true ) ) ) ) ) ).
% valid_in_def
thf(fact_3_image__ident,axiom,
! [A: $tType,Y: set @ A] :
( ( image @ A @ A
@ ^ [X: A] : X
@ Y )
= Y ) ).
% image_ident
thf(fact_4_UNIV__I,axiom,
! [A: $tType,X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_I
thf(fact_5_iso__tuple__UNIV__I,axiom,
! [A: $tType,X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% iso_tuple_UNIV_I
thf(fact_6_top__apply,axiom,
! [C: $tType,D: $tType] :
( ( top @ C )
=> ( ( top_top @ ( D > C ) )
= ( ^ [X: D] : ( top_top @ C ) ) ) ) ).
% top_apply
thf(fact_7_subsetI,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A2 )
=> ( member @ A @ X3 @ B2 ) )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% subsetI
thf(fact_8_subset__antisym,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_9_image__eqI,axiom,
! [A: $tType,B: $tType,B3: A,F: B > A,X2: B,A2: set @ B] :
( ( B3
= ( F @ X2 ) )
=> ( ( member @ B @ X2 @ A2 )
=> ( member @ A @ B3 @ ( image @ B @ A @ F @ A2 ) ) ) ) ).
% image_eqI
thf(fact_10_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% order_refl
thf(fact_11_rangeE,axiom,
! [A: $tType,B: $tType,B3: A,F: B > A] :
( ( member @ A @ B3 @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) )
=> ~ ! [X3: B] :
( B3
!= ( F @ X3 ) ) ) ).
% rangeE
thf(fact_12_range__composition,axiom,
! [A: $tType,C: $tType,B: $tType,F: C > A,G: B > C] :
( ( image @ B @ A
@ ^ [X: B] : ( F @ ( G @ X ) )
@ ( top_top @ ( set @ B ) ) )
= ( image @ C @ A @ F @ ( image @ B @ C @ G @ ( top_top @ ( set @ B ) ) ) ) ) ).
% range_composition
thf(fact_13_image__Collect__subsetI,axiom,
! [A: $tType,B: $tType,P2: A > $o,F: A > B,B2: set @ B] :
( ! [X3: A] :
( ( P2 @ X3 )
=> ( member @ B @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ ( collect @ A @ P2 ) ) @ B2 ) ) ).
% image_Collect_subsetI
thf(fact_14_top__set__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% top_set_def
thf(fact_15_less__eq__set__def,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A3: set @ A,B4: set @ A] :
( ord_less_eq @ ( A > $o )
@ ^ [X: A] : ( member @ A @ X @ A3 )
@ ^ [X: A] : ( member @ A @ X @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_16_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A4: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
=> ( ( ord_less_eq @ A @ A4 @ B3 )
=> ( A4 = B3 ) ) ) ) ).
% dual_order.antisym
thf(fact_17_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y2: A,Z: A] : Y2 = Z )
= ( ^ [A5: A,B5: A] :
( ( ord_less_eq @ A @ B5 @ A5 )
& ( ord_less_eq @ A @ A5 @ B5 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_18_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A4: A,C2: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
=> ( ( ord_less_eq @ A @ C2 @ B3 )
=> ( ord_less_eq @ A @ C2 @ A4 ) ) ) ) ).
% dual_order.trans
thf(fact_19_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P2: A > A > $o,A4: A,B3: A] :
( ! [A6: A,B6: A] :
( ( ord_less_eq @ A @ A6 @ B6 )
=> ( P2 @ A6 @ B6 ) )
=> ( ! [A6: A,B6: A] :
( ( P2 @ B6 @ A6 )
=> ( P2 @ A6 @ B6 ) )
=> ( P2 @ A4 @ B3 ) ) ) ) ).
% linorder_wlog
thf(fact_20_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: A] : ( ord_less_eq @ A @ A4 @ A4 ) ) ).
% dual_order.refl
thf(fact_21_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y3: A,Z2: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
=> ( ( ord_less_eq @ A @ Y3 @ Z2 )
=> ( ord_less_eq @ A @ X2 @ Z2 ) ) ) ) ).
% order_trans
thf(fact_22_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ A4 )
=> ( A4 = B3 ) ) ) ) ).
% order_class.order.antisym
thf(fact_23_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_24_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( A4 = B3 )
=> ( ( ord_less_eq @ A @ B3 @ C2 )
=> ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_25_order__class_Oorder_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y2: A,Z: A] : Y2 = Z )
= ( ^ [A5: A,B5: A] :
( ( ord_less_eq @ A @ A5 @ B5 )
& ( ord_less_eq @ A @ B5 @ A5 ) ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_26_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y3: A,X2: A] :
( ( ord_less_eq @ A @ Y3 @ X2 )
=> ( ( ord_less_eq @ A @ X2 @ Y3 )
= ( X2 = Y3 ) ) ) ) ).
% antisym_conv
thf(fact_27_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y3: A,Z2: A] :
( ( ( ord_less_eq @ A @ X2 @ Y3 )
=> ~ ( ord_less_eq @ A @ Y3 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ Y3 @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ X2 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ Y3 ) )
=> ( ( ( ord_less_eq @ A @ Z2 @ Y3 )
=> ~ ( ord_less_eq @ A @ Y3 @ X2 ) )
=> ( ( ( ord_less_eq @ A @ Y3 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq @ A @ Z2 @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ Y3 ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_28_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ C2 )
=> ( ord_less_eq @ A @ A4 @ C2 ) ) ) ) ).
% order.trans
thf(fact_29_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y3: A] :
( ~ ( ord_less_eq @ A @ X2 @ Y3 )
=> ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ).
% le_cases
thf(fact_30_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y3: A] :
( ( X2 = Y3 )
=> ( ord_less_eq @ A @ X2 @ Y3 ) ) ) ).
% eq_refl
thf(fact_31_linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
| ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ).
% linear
thf(fact_32_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
=> ( ( ord_less_eq @ A @ Y3 @ X2 )
=> ( X2 = Y3 ) ) ) ) ).
% antisym
thf(fact_33_eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y2: A,Z: A] : Y2 = Z )
= ( ^ [X: A,Y4: A] :
( ( ord_less_eq @ A @ X @ Y4 )
& ( ord_less_eq @ A @ Y4 @ X ) ) ) ) ) ).
% eq_iff
thf(fact_34_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A4: A,B3: A,F: A > B,C2: B] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X3: A,Y5: A] :
( ( ord_less_eq @ A @ X3 @ Y5 )
=> ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq @ B @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_35_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A4: A,F: B > A,B3: B,C2: B] :
( ( A4
= ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C2 )
=> ( ! [X3: B,Y5: B] :
( ( ord_less_eq @ B @ X3 @ Y5 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_36_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A4: A,B3: A,F: A > C,C2: C] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( ( ord_less_eq @ C @ ( F @ B3 ) @ C2 )
=> ( ! [X3: A,Y5: A] :
( ( ord_less_eq @ A @ X3 @ Y5 )
=> ( ord_less_eq @ C @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq @ C @ ( F @ A4 ) @ C2 ) ) ) ) ) ).
% order_subst2
thf(fact_37_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A4: A,F: B > A,B3: B,C2: B] :
( ( ord_less_eq @ A @ A4 @ ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C2 )
=> ( ! [X3: B,Y5: B] :
( ( ord_less_eq @ B @ X3 @ Y5 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y5 ) ) )
=> ( ord_less_eq @ A @ A4 @ ( F @ C2 ) ) ) ) ) ) ).
% order_subst1
thf(fact_38_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B] :
! [X: A] : ( ord_less_eq @ B @ ( F2 @ X ) @ ( G2 @ X ) ) ) ) ) ).
% le_fun_def
thf(fact_39_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B] :
( ! [X3: A] : ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) )
=> ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% le_funI
thf(fact_40_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B,X2: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% le_funE
thf(fact_41_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B,X2: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% le_funD
thf(fact_42_rev__image__eqI,axiom,
! [B: $tType,A: $tType,X2: A,A2: set @ A,B3: B,F: A > B] :
( ( member @ A @ X2 @ A2 )
=> ( ( B3
= ( F @ X2 ) )
=> ( member @ B @ B3 @ ( image @ A @ B @ F @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_43_ball__imageD,axiom,
! [A: $tType,B: $tType,F: B > A,A2: set @ B,P2: A > $o] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( image @ B @ A @ F @ A2 ) )
=> ( P2 @ X3 ) )
=> ! [X4: B] :
( ( member @ B @ X4 @ A2 )
=> ( P2 @ ( F @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_44_image__cong,axiom,
! [B: $tType,A: $tType,M: set @ A,N2: set @ A,F: A > B,G: A > B] :
( ( M = N2 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ N2 )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( image @ A @ B @ F @ M )
= ( image @ A @ B @ G @ N2 ) ) ) ) ).
% image_cong
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A4: A,P2: A > $o] :
( ( member @ A @ A4 @ ( collect @ A @ P2 ) )
= ( P2 @ A4 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A2: set @ A] :
( ( collect @ A
@ ^ [X: A] : ( member @ A @ X @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P2: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P2 @ X3 )
= ( Q @ X3 ) )
=> ( ( collect @ A @ P2 )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X3: A] :
( ( F @ X3 )
= ( G @ X3 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_bex__imageD,axiom,
! [A: $tType,B: $tType,F: B > A,A2: set @ B,P2: A > $o] :
( ? [X4: A] :
( ( member @ A @ X4 @ ( image @ B @ A @ F @ A2 ) )
& ( P2 @ X4 ) )
=> ? [X3: B] :
( ( member @ B @ X3 @ A2 )
& ( P2 @ ( F @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_50_image__iff,axiom,
! [A: $tType,B: $tType,Z2: A,F: B > A,A2: set @ B] :
( ( member @ A @ Z2 @ ( image @ B @ A @ F @ A2 ) )
= ( ? [X: B] :
( ( member @ B @ X @ A2 )
& ( Z2
= ( F @ X ) ) ) ) ) ).
% image_iff
thf(fact_51_imageI,axiom,
! [B: $tType,A: $tType,X2: A,A2: set @ A,F: A > B] :
( ( member @ A @ X2 @ A2 )
=> ( member @ B @ ( F @ X2 ) @ ( image @ A @ B @ F @ A2 ) ) ) ).
% imageI
thf(fact_52_Collect__mono__iff,axiom,
! [A: $tType,P2: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q ) )
= ( ! [X: A] :
( ( P2 @ X )
=> ( Q @ X ) ) ) ) ).
% Collect_mono_iff
thf(fact_53_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y2: set @ A,Z: set @ A] : Y2 = Z )
= ( ^ [A3: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
& ( ord_less_eq @ ( set @ A ) @ B4 @ A3 ) ) ) ) ).
% set_eq_subset
thf(fact_54_subset__trans,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_55_Collect__mono,axiom,
! [A: $tType,P2: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P2 @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_56_subset__refl,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% subset_refl
thf(fact_57_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A3: set @ A,B4: set @ A] :
! [T: A] :
( ( member @ A @ T @ A3 )
=> ( member @ A @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_58_equalityD2,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_59_equalityD1,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_60_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A3: set @ A,B4: set @ A] :
! [X: A] :
( ( member @ A @ X @ A3 )
=> ( member @ A @ X @ B4 ) ) ) ) ).
% subset_eq
thf(fact_61_equalityE,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_62_subsetD,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( member @ A @ C2 @ A2 )
=> ( member @ A @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_63_in__mono,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,X2: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( member @ A @ X2 @ A2 )
=> ( member @ A @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_64_UNIV__witness,axiom,
! [A: $tType] :
? [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_witness
thf(fact_65_UNIV__eq__I,axiom,
! [A: $tType,A2: set @ A] :
( ! [X3: A] : ( member @ A @ X3 @ A2 )
=> ( ( top_top @ ( set @ A ) )
= A2 ) ) ).
% UNIV_eq_I
thf(fact_66_Compr__image__eq,axiom,
! [A: $tType,B: $tType,F: B > A,A2: set @ B,P2: A > $o] :
( ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ ( image @ B @ A @ F @ A2 ) )
& ( P2 @ X ) ) )
= ( image @ B @ A @ F
@ ( collect @ B
@ ^ [X: B] :
( ( member @ B @ X @ A2 )
& ( P2 @ ( F @ X ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_67_image__image,axiom,
! [A: $tType,B: $tType,C: $tType,F: B > A,G: C > B,A2: set @ C] :
( ( image @ B @ A @ F @ ( image @ C @ B @ G @ A2 ) )
= ( image @ C @ A
@ ^ [X: C] : ( F @ ( G @ X ) )
@ A2 ) ) ).
% image_image
thf(fact_68_imageE,axiom,
! [A: $tType,B: $tType,B3: A,F: B > A,A2: set @ B] :
( ( member @ A @ B3 @ ( image @ B @ A @ F @ A2 ) )
=> ~ ! [X3: B] :
( ( B3
= ( F @ X3 ) )
=> ~ ( member @ B @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_69_Collect__restrict,axiom,
! [A: $tType,X5: set @ A,P2: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ X5 )
& ( P2 @ X ) ) )
@ X5 ) ).
% Collect_restrict
thf(fact_70_prop__restrict,axiom,
! [A: $tType,X2: A,Z3: set @ A,X5: set @ A,P2: A > $o] :
( ( member @ A @ X2 @ Z3 )
=> ( ( ord_less_eq @ ( set @ A ) @ Z3
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ X5 )
& ( P2 @ X ) ) ) )
=> ( P2 @ X2 ) ) ) ).
% prop_restrict
thf(fact_71_Collect__subset,axiom,
! [A: $tType,A2: set @ A,P2: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ A2 )
& ( P2 @ X ) ) )
@ A2 ) ).
% Collect_subset
thf(fact_72_UNIV__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A
@ ^ [X: A] : $true ) ) ).
% UNIV_def
thf(fact_73_top_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A4: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A4 )
=> ( A4
= ( top_top @ A ) ) ) ) ).
% top.extremum_uniqueI
thf(fact_74_top_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A4: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A4 )
= ( A4
= ( top_top @ A ) ) ) ) ).
% top.extremum_unique
thf(fact_75_top__greatest,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A4: A] : ( ord_less_eq @ A @ A4 @ ( top_top @ A ) ) ) ).
% top_greatest
thf(fact_76_subset__image__iff,axiom,
! [A: $tType,B: $tType,B2: set @ A,F: B > A,A2: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ ( image @ B @ A @ F @ A2 ) )
= ( ? [AA: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ AA @ A2 )
& ( B2
= ( image @ B @ A @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_77_image__subset__iff,axiom,
! [A: $tType,B: $tType,F: B > A,A2: set @ B,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F @ A2 ) @ B2 )
= ( ! [X: B] :
( ( member @ B @ X @ A2 )
=> ( member @ A @ ( F @ X ) @ B2 ) ) ) ) ).
% image_subset_iff
thf(fact_78_subset__imageE,axiom,
! [A: $tType,B: $tType,B2: set @ A,F: B > A,A2: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ ( image @ B @ A @ F @ A2 ) )
=> ~ ! [C4: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ C4 @ A2 )
=> ( B2
!= ( image @ B @ A @ F @ C4 ) ) ) ) ).
% subset_imageE
thf(fact_79_image__subsetI,axiom,
! [A: $tType,B: $tType,A2: set @ A,F: A > B,B2: set @ B] :
( ! [X3: A] :
( ( member @ A @ X3 @ A2 )
=> ( member @ B @ ( F @ X3 ) @ B2 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A2 ) @ B2 ) ) ).
% image_subsetI
thf(fact_80_image__mono,axiom,
! [B: $tType,A: $tType,A2: set @ A,B2: set @ A,F: A > B] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A2 ) @ ( image @ A @ B @ F @ B2 ) ) ) ).
% image_mono
thf(fact_81_range__eqI,axiom,
! [A: $tType,B: $tType,B3: A,F: B > A,X2: B] :
( ( B3
= ( F @ X2 ) )
=> ( member @ A @ B3 @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ) ).
% range_eqI
thf(fact_82_rangeI,axiom,
! [A: $tType,B: $tType,F: B > A,X2: B] : ( member @ A @ ( F @ X2 ) @ ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) ) ) ).
% rangeI
thf(fact_83_subset__UNIV,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ ( top_top @ ( set @ A ) ) ) ).
% subset_UNIV
thf(fact_84_surj__def,axiom,
! [B: $tType,A: $tType,F: B > A] :
( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
= ( ! [Y4: A] :
? [X: B] :
( Y4
= ( F @ X ) ) ) ) ).
% surj_def
thf(fact_85_surjI,axiom,
! [B: $tType,A: $tType,G: B > A,F: A > B] :
( ! [X3: A] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( ( image @ B @ A @ G @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surjI
thf(fact_86_surjE,axiom,
! [A: $tType,B: $tType,F: B > A,Y3: A] :
( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ~ ! [X3: B] :
( Y3
!= ( F @ X3 ) ) ) ).
% surjE
thf(fact_87_surjD,axiom,
! [A: $tType,B: $tType,F: B > A,Y3: A] :
( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ? [X3: B] :
( Y3
= ( F @ X3 ) ) ) ).
% surjD
thf(fact_88_all__subset__image,axiom,
! [A: $tType,B: $tType,F: B > A,A2: set @ B,P2: ( set @ A ) > $o] :
( ( ! [B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B4 @ ( image @ B @ A @ F @ A2 ) )
=> ( P2 @ B4 ) ) )
= ( ! [B4: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ B4 @ A2 )
=> ( P2 @ ( image @ B @ A @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_89_valid__def,axiom,
( paraco769098683_valid
= ( ^ [P: paraco414474393lle_fm] :
! [I: ( list @ char ) > paraco415392788lle_tv] :
( ( paraco876059933e_eval @ I @ P )
= ( paraco2040174112le_Det @ $true ) ) ) ) ).
% valid_def
thf(fact_90_pred__subset__eq,axiom,
! [A: $tType,R: set @ A,S: set @ A] :
( ( ord_less_eq @ ( A > $o )
@ ^ [X: A] : ( member @ A @ X @ R )
@ ^ [X: A] : ( member @ A @ X @ S ) )
= ( ord_less_eq @ ( set @ A ) @ R @ S ) ) ).
% pred_subset_eq
thf(fact_91_subset__Collect__iff,axiom,
! [A: $tType,B2: set @ A,A2: set @ A,P2: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ A2 )
& ( P2 @ X ) ) ) )
= ( ! [X: A] :
( ( member @ A @ X @ B2 )
=> ( P2 @ X ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_92_subset__CollectI,axiom,
! [A: $tType,B2: set @ A,A2: set @ A,Q: A > $o,P2: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ B2 )
=> ( ( Q @ X3 )
=> ( P2 @ X3 ) ) )
=> ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ B2 )
& ( Q @ X ) ) )
@ ( collect @ A
@ ^ [X: A] :
( ( member @ A @ X @ A2 )
& ( P2 @ X ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_93_conj__subset__def,axiom,
! [A: $tType,A2: set @ A,P2: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ A2
@ ( collect @ A
@ ^ [X: A] :
( ( P2 @ X )
& ( Q @ X ) ) ) )
= ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( collect @ A @ P2 ) )
& ( ord_less_eq @ ( set @ A ) @ A2 @ ( collect @ A @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_94_Sup_OSUP__identity__eq,axiom,
! [A: $tType,Sup: ( set @ A ) > A,A2: set @ A] :
( ( Sup
@ ( image @ A @ A
@ ^ [X: A] : X
@ A2 ) )
= ( Sup @ A2 ) ) ).
% Sup.SUP_identity_eq
thf(fact_95_predicate1I,axiom,
! [A: $tType,P2: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P2 @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq @ ( A > $o ) @ P2 @ Q ) ) ).
% predicate1I
thf(fact_96_top1I,axiom,
! [A: $tType,X2: A] : ( top_top @ ( A > $o ) @ X2 ) ).
% top1I
thf(fact_97_predicate1D,axiom,
! [A: $tType,P2: A > $o,Q: A > $o,X2: A] :
( ( ord_less_eq @ ( A > $o ) @ P2 @ Q )
=> ( ( P2 @ X2 )
=> ( Q @ X2 ) ) ) ).
% predicate1D
thf(fact_98_rev__predicate1D,axiom,
! [A: $tType,P2: A > $o,X2: A,Q: A > $o] :
( ( P2 @ X2 )
=> ( ( ord_less_eq @ ( A > $o ) @ P2 @ Q )
=> ( Q @ X2 ) ) ) ).
% rev_predicate1D
thf(fact_99_Inf_OINF__cong,axiom,
! [A: $tType,B: $tType,A2: set @ B,B2: set @ B,C3: B > A,D2: B > A,Inf: ( set @ A ) > A] :
( ( A2 = B2 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ B2 )
=> ( ( C3 @ X3 )
= ( D2 @ X3 ) ) )
=> ( ( Inf @ ( image @ B @ A @ C3 @ A2 ) )
= ( Inf @ ( image @ B @ A @ D2 @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_100_Sup_OSUP__cong,axiom,
! [A: $tType,B: $tType,A2: set @ B,B2: set @ B,C3: B > A,D2: B > A,Sup: ( set @ A ) > A] :
( ( A2 = B2 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ B2 )
=> ( ( C3 @ X3 )
= ( D2 @ X3 ) ) )
=> ( ( Sup @ ( image @ B @ A @ C3 @ A2 ) )
= ( Sup @ ( image @ B @ A @ D2 @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_101_top__empty__eq,axiom,
! [A: $tType] :
( ( top_top @ ( A > $o ) )
= ( ^ [X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ).
% top_empty_eq
thf(fact_102_Inf_OINF__identity__eq,axiom,
! [A: $tType,Inf: ( set @ A ) > A,A2: set @ A] :
( ( Inf
@ ( image @ A @ A
@ ^ [X: A] : X
@ A2 ) )
= ( Inf @ A2 ) ) ).
% Inf.INF_identity_eq
thf(fact_103_top__conj_I2_J,axiom,
! [A: $tType,P2: $o,X2: A] :
( ( P2
& ( top_top @ ( A > $o ) @ X2 ) )
= P2 ) ).
% top_conj(2)
thf(fact_104_top__conj_I1_J,axiom,
! [A: $tType,X2: A,P2: $o] :
( ( ( top_top @ ( A > $o ) @ X2 )
& P2 )
= P2 ) ).
% top_conj(1)
thf(fact_105_eval_Osimps_I2_J,axiom,
! [I2: ( list @ char ) > paraco415392788lle_tv] :
( ( paraco876059933e_eval @ I2 @ paraco251304083_Truth )
= ( paraco2040174112le_Det @ $true ) ) ).
% eval.simps(2)
thf(fact_106_image__Fpow__mono,axiom,
! [B: $tType,A: $tType,F: B > A,A2: set @ B,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F @ A2 ) @ B2 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( image @ ( set @ B ) @ ( set @ A ) @ ( image @ B @ A @ F ) @ ( finite_Fpow @ B @ A2 ) ) @ ( finite_Fpow @ A @ B2 ) ) ) ).
% image_Fpow_mono
thf(fact_107_eval_Osimps_I5_J,axiom,
! [I2: ( list @ char ) > paraco415392788lle_tv,P3: paraco414474393lle_fm,Q2: paraco414474393lle_fm] :
( ( ( ( paraco876059933e_eval @ I2 @ P3 )
= ( paraco876059933e_eval @ I2 @ Q2 ) )
=> ( ( paraco876059933e_eval @ I2 @ ( paraco2084319816le_Eql @ P3 @ Q2 ) )
= ( paraco2040174112le_Det @ $true ) ) )
& ( ( ( paraco876059933e_eval @ I2 @ P3 )
!= ( paraco876059933e_eval @ I2 @ Q2 ) )
=> ( ( paraco876059933e_eval @ I2 @ ( paraco2084319816le_Eql @ P3 @ Q2 ) )
= ( paraco2040174112le_Det @ $false ) ) ) ) ).
% eval.simps(5)
thf(fact_108_fm_Oinject_I4_J,axiom,
! [X51: paraco414474393lle_fm,X52: paraco414474393lle_fm,Y51: paraco414474393lle_fm,Y52: paraco414474393lle_fm] :
( ( ( paraco2084319816le_Eql @ X51 @ X52 )
= ( paraco2084319816le_Eql @ Y51 @ Y52 ) )
= ( ( X51 = Y51 )
& ( X52 = Y52 ) ) ) ).
% fm.inject(4)
thf(fact_109_fm_Odistinct_I15_J,axiom,
! [X51: paraco414474393lle_fm,X52: paraco414474393lle_fm] :
( paraco251304083_Truth
!= ( paraco2084319816le_Eql @ X51 @ X52 ) ) ).
% fm.distinct(15)
thf(fact_110_Fpow__mono,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( finite_Fpow @ A @ A2 ) @ ( finite_Fpow @ A @ B2 ) ) ) ).
% Fpow_mono
thf(fact_111_eval__equality__simplify,axiom,
! [I2: ( list @ char ) > paraco415392788lle_tv,P3: paraco414474393lle_fm,Q2: paraco414474393lle_fm] :
( ( paraco876059933e_eval @ I2 @ ( paraco2084319816le_Eql @ P3 @ Q2 ) )
= ( paraco2040174112le_Det
@ ( ( paraco876059933e_eval @ I2 @ P3 )
= ( paraco876059933e_eval @ I2 @ Q2 ) ) ) ) ).
% eval_equality_simplify
thf(fact_112_image__Pow__mono,axiom,
! [B: $tType,A: $tType,F: B > A,A2: set @ B,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F @ A2 ) @ B2 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( image @ ( set @ B ) @ ( set @ A ) @ ( image @ B @ A @ F ) @ ( pow @ B @ A2 ) ) @ ( pow @ A @ B2 ) ) ) ).
% image_Pow_mono
thf(fact_113_eval_Osimps_I4_J,axiom,
! [I2: ( list @ char ) > paraco415392788lle_tv,P3: paraco414474393lle_fm,Q2: paraco414474393lle_fm] :
( ( ( ( paraco876059933e_eval @ I2 @ P3 )
= ( paraco876059933e_eval @ I2 @ Q2 ) )
=> ( ( paraco876059933e_eval @ I2 @ ( paraco2100061555le_Con @ P3 @ Q2 ) )
= ( paraco876059933e_eval @ I2 @ P3 ) ) )
& ( ( ( paraco876059933e_eval @ I2 @ P3 )
!= ( paraco876059933e_eval @ I2 @ Q2 ) )
=> ( ( ( ( paraco876059933e_eval @ I2 @ P3 )
= ( paraco2040174112le_Det @ $true ) )
=> ( ( paraco876059933e_eval @ I2 @ ( paraco2100061555le_Con @ P3 @ Q2 ) )
= ( paraco876059933e_eval @ I2 @ Q2 ) ) )
& ( ( ( paraco876059933e_eval @ I2 @ P3 )
!= ( paraco2040174112le_Det @ $true ) )
=> ( ( ( ( paraco876059933e_eval @ I2 @ Q2 )
= ( paraco2040174112le_Det @ $true ) )
=> ( ( paraco876059933e_eval @ I2 @ ( paraco2100061555le_Con @ P3 @ Q2 ) )
= ( paraco876059933e_eval @ I2 @ P3 ) ) )
& ( ( ( paraco876059933e_eval @ I2 @ Q2 )
!= ( paraco2040174112le_Det @ $true ) )
=> ( ( paraco876059933e_eval @ I2 @ ( paraco2100061555le_Con @ P3 @ Q2 ) )
= ( paraco2040174112le_Det @ $false ) ) ) ) ) ) ) ) ).
% eval.simps(4)
thf(fact_114_Powp__mono,axiom,
! [A: $tType,A2: A > $o,B2: A > $o] :
( ( ord_less_eq @ ( A > $o ) @ A2 @ B2 )
=> ( ord_less_eq @ ( ( set @ A ) > $o ) @ ( powp @ A @ A2 ) @ ( powp @ A @ B2 ) ) ) ).
% Powp_mono
thf(fact_115_Greatest__equality,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P2: A > $o,X2: A] :
( ( P2 @ X2 )
=> ( ! [Y5: A] :
( ( P2 @ Y5 )
=> ( ord_less_eq @ A @ Y5 @ X2 ) )
=> ( ( order_Greatest @ A @ P2 )
= X2 ) ) ) ) ).
% Greatest_equality
thf(fact_116_GreatestI2__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P2: A > $o,X2: A,Q: A > $o] :
( ( P2 @ X2 )
=> ( ! [Y5: A] :
( ( P2 @ Y5 )
=> ( ord_less_eq @ A @ Y5 @ X2 ) )
=> ( ! [X3: A] :
( ( P2 @ X3 )
=> ( ! [Y6: A] :
( ( P2 @ Y6 )
=> ( ord_less_eq @ A @ Y6 @ X3 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( order_Greatest @ A @ P2 ) ) ) ) ) ) ).
% GreatestI2_order
thf(fact_117_fm_Oinject_I3_J,axiom,
! [X41: paraco414474393lle_fm,X42: paraco414474393lle_fm,Y41: paraco414474393lle_fm,Y42: paraco414474393lle_fm] :
( ( ( paraco2100061555le_Con @ X41 @ X42 )
= ( paraco2100061555le_Con @ Y41 @ Y42 ) )
= ( ( X41 = Y41 )
& ( X42 = Y42 ) ) ) ).
% fm.inject(3)
thf(fact_118_PowI,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( member @ ( set @ A ) @ A2 @ ( pow @ A @ B2 ) ) ) ).
% PowI
thf(fact_119_Pow__iff,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( member @ ( set @ A ) @ A2 @ ( pow @ A @ B2 ) )
= ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% Pow_iff
thf(fact_120_Pow__UNIV,axiom,
! [A: $tType] :
( ( pow @ A @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ ( set @ A ) ) ) ) ).
% Pow_UNIV
thf(fact_121_Pow__top,axiom,
! [A: $tType,A2: set @ A] : ( member @ ( set @ A ) @ A2 @ ( pow @ A @ A2 ) ) ).
% Pow_top
thf(fact_122_Powp__Pow__eq,axiom,
! [A: $tType,A2: set @ A] :
( ( powp @ A
@ ^ [X: A] : ( member @ A @ X @ A2 ) )
= ( ^ [X: set @ A] : ( member @ ( set @ A ) @ X @ ( pow @ A @ A2 ) ) ) ) ).
% Powp_Pow_eq
thf(fact_123_PowD,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( member @ ( set @ A ) @ A2 @ ( pow @ A @ B2 ) )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% PowD
thf(fact_124_Cantors__paradox,axiom,
! [A: $tType,A2: set @ A] :
~ ? [F3: A > ( set @ A )] :
( ( image @ A @ ( set @ A ) @ F3 @ A2 )
= ( pow @ A @ A2 ) ) ).
% Cantors_paradox
thf(fact_125_Pow__def,axiom,
! [A: $tType] :
( ( pow @ A )
= ( ^ [A3: set @ A] :
( collect @ ( set @ A )
@ ^ [B4: set @ A] : ( ord_less_eq @ ( set @ A ) @ B4 @ A3 ) ) ) ) ).
% Pow_def
thf(fact_126_fm_Odistinct_I25_J,axiom,
! [X41: paraco414474393lle_fm,X42: paraco414474393lle_fm,X51: paraco414474393lle_fm,X52: paraco414474393lle_fm] :
( ( paraco2100061555le_Con @ X41 @ X42 )
!= ( paraco2084319816le_Eql @ X51 @ X52 ) ) ).
% fm.distinct(25)
thf(fact_127_fm_Odistinct_I13_J,axiom,
! [X41: paraco414474393lle_fm,X42: paraco414474393lle_fm] :
( paraco251304083_Truth
!= ( paraco2100061555le_Con @ X41 @ X42 ) ) ).
% fm.distinct(13)
thf(fact_128_Pow__mono,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( pow @ A @ A2 ) @ ( pow @ A @ B2 ) ) ) ).
% Pow_mono
thf(fact_129_image__Pow__surj,axiom,
! [B: $tType,A: $tType,F: B > A,A2: set @ B,B2: set @ A] :
( ( ( image @ B @ A @ F @ A2 )
= B2 )
=> ( ( image @ ( set @ B ) @ ( set @ A ) @ ( image @ B @ A @ F ) @ ( pow @ B @ A2 ) )
= ( pow @ A @ B2 ) ) ) ).
% image_Pow_surj
thf(fact_130_conjunction,axiom,
! [P3: paraco414474393lle_fm,Q2: paraco414474393lle_fm] :
( ( paraco769098683_valid @ ( paraco2100061555le_Con @ P3 @ Q2 ) )
= ( ( paraco769098683_valid @ P3 )
& ( paraco769098683_valid @ Q2 ) ) ) ).
% conjunction
thf(fact_131_Fpow__subset__Pow,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( finite_Fpow @ A @ A2 ) @ ( pow @ A @ A2 ) ) ).
% Fpow_subset_Pow
thf(fact_132_tv_Osimps_I7_J,axiom,
! [A: $tType,F1: $o > A,F22: nat > A,X1: $o] :
( ( paraco152590079rec_tv @ A @ F1 @ F22 @ ( paraco2040174112le_Det @ X1 ) )
= ( F1 @ X1 ) ) ).
% tv.simps(7)
thf(fact_133_strict__mono__inv,axiom,
! [A: $tType,B: $tType] :
( ( ( linorder @ B )
& ( linorder @ A ) )
=> ! [F: A > B,G: B > A] :
( ( order_strict_mono @ A @ B @ F )
=> ( ( ( image @ A @ B @ F @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ B ) ) )
=> ( ! [X3: A] :
( ( G @ ( F @ X3 ) )
= X3 )
=> ( order_strict_mono @ B @ A @ G ) ) ) ) ) ).
% strict_mono_inv
thf(fact_134_eval__negation,axiom,
! [I2: ( list @ char ) > paraco415392788lle_tv,P3: paraco414474393lle_fm] :
( ( ( ( paraco876059933e_eval @ I2 @ P3 )
= ( paraco2040174112le_Det @ $false ) )
=> ( ( paraco876059933e_eval @ I2 @ ( paraco329115265le_Neg @ P3 ) )
= ( paraco2040174112le_Det @ $true ) ) )
& ( ( ( paraco876059933e_eval @ I2 @ P3 )
!= ( paraco2040174112le_Det @ $false ) )
=> ( ( ( ( paraco876059933e_eval @ I2 @ P3 )
= ( paraco2040174112le_Det @ $true ) )
=> ( ( paraco876059933e_eval @ I2 @ ( paraco329115265le_Neg @ P3 ) )
= ( paraco2040174112le_Det @ $false ) ) )
& ( ( ( paraco876059933e_eval @ I2 @ P3 )
!= ( paraco2040174112le_Det @ $true ) )
=> ( ( paraco876059933e_eval @ I2 @ ( paraco329115265le_Neg @ P3 ) )
= ( paraco876059933e_eval @ I2 @ P3 ) ) ) ) ) ) ).
% eval_negation
thf(fact_135_surj__swap__iff,axiom,
! [B: $tType,A: $tType,A4: B,B3: B,F: B > A] :
( ( ( image @ B @ A @ ( swap @ B @ A @ A4 @ B3 @ F ) @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
= ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surj_swap_iff
thf(fact_136_swap__nilpotent,axiom,
! [B: $tType,A: $tType,A4: A,B3: A,F: A > B] :
( ( swap @ A @ B @ A4 @ B3 @ ( swap @ A @ B @ A4 @ B3 @ F ) )
= F ) ).
% swap_nilpotent
thf(fact_137_swap__self,axiom,
! [B: $tType,A: $tType,A4: A,F: A > B] :
( ( swap @ A @ B @ A4 @ A4 @ F )
= F ) ).
% swap_self
thf(fact_138_swap__apply_I1_J,axiom,
! [A: $tType,B: $tType,A4: B,B3: B,F: B > A] :
( ( swap @ B @ A @ A4 @ B3 @ F @ A4 )
= ( F @ B3 ) ) ).
% swap_apply(1)
thf(fact_139_swap__apply_I2_J,axiom,
! [A: $tType,B: $tType,A4: B,B3: B,F: B > A] :
( ( swap @ B @ A @ A4 @ B3 @ F @ B3 )
= ( F @ A4 ) ) ).
% swap_apply(2)
thf(fact_140_swap__apply_I3_J,axiom,
! [A: $tType,B: $tType,C2: B,A4: B,B3: B,F: B > A] :
( ( C2 != A4 )
=> ( ( C2 != B3 )
=> ( ( swap @ B @ A @ A4 @ B3 @ F @ C2 )
= ( F @ C2 ) ) ) ) ).
% swap_apply(3)
thf(fact_141_fm_Oinject_I2_J,axiom,
! [X32: paraco414474393lle_fm,Y32: paraco414474393lle_fm] :
( ( ( paraco329115265le_Neg @ X32 )
= ( paraco329115265le_Neg @ Y32 ) )
= ( X32 = Y32 ) ) ).
% fm.inject(2)
thf(fact_142_swap__image__eq,axiom,
! [B: $tType,A: $tType,A4: A,A2: set @ A,B3: A,F: A > B] :
( ( member @ A @ A4 @ A2 )
=> ( ( member @ A @ B3 @ A2 )
=> ( ( image @ A @ B @ ( swap @ A @ B @ A4 @ B3 @ F ) @ A2 )
= ( image @ A @ B @ F @ A2 ) ) ) ) ).
% swap_image_eq
thf(fact_143_swap__commute,axiom,
! [B: $tType,A: $tType] :
( ( swap @ A @ B )
= ( ^ [A5: A,B5: A] : ( swap @ A @ B @ B5 @ A5 ) ) ) ).
% swap_commute
thf(fact_144_swap__triple,axiom,
! [B: $tType,A: $tType,A4: A,C2: A,B3: A,F: A > B] :
( ( A4 != C2 )
=> ( ( B3 != C2 )
=> ( ( swap @ A @ B @ A4 @ B3 @ ( swap @ A @ B @ B3 @ C2 @ ( swap @ A @ B @ A4 @ B3 @ F ) ) )
= ( swap @ A @ B @ A4 @ C2 @ F ) ) ) ) ).
% swap_triple
thf(fact_145_strict__mono__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F: A > B,X2: A,Y3: A] :
( ( order_strict_mono @ A @ B @ F )
=> ( ( ( F @ X2 )
= ( F @ Y3 ) )
= ( X2 = Y3 ) ) ) ) ).
% strict_mono_eq
thf(fact_146_double__negation,axiom,
( paraco876059933e_eval
= ( ^ [I: ( list @ char ) > paraco415392788lle_tv,P: paraco414474393lle_fm] : ( paraco876059933e_eval @ I @ ( paraco329115265le_Neg @ ( paraco329115265le_Neg @ P ) ) ) ) ) ).
% double_negation
thf(fact_147_strict__mono__less__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F: A > B,X2: A,Y3: A] :
( ( order_strict_mono @ A @ B @ F )
=> ( ( ord_less_eq @ B @ ( F @ X2 ) @ ( F @ Y3 ) )
= ( ord_less_eq @ A @ X2 @ Y3 ) ) ) ) ).
% strict_mono_less_eq
thf(fact_148_fm_Odistinct_I21_J,axiom,
! [X32: paraco414474393lle_fm,X51: paraco414474393lle_fm,X52: paraco414474393lle_fm] :
( ( paraco329115265le_Neg @ X32 )
!= ( paraco2084319816le_Eql @ X51 @ X52 ) ) ).
% fm.distinct(21)
thf(fact_149_fm_Odistinct_I19_J,axiom,
! [X32: paraco414474393lle_fm,X41: paraco414474393lle_fm,X42: paraco414474393lle_fm] :
( ( paraco329115265le_Neg @ X32 )
!= ( paraco2100061555le_Con @ X41 @ X42 ) ) ).
% fm.distinct(19)
thf(fact_150_fm_Odistinct_I11_J,axiom,
! [X32: paraco414474393lle_fm] :
( paraco251304083_Truth
!= ( paraco329115265le_Neg @ X32 ) ) ).
% fm.distinct(11)
thf(fact_151_surj__imp__surj__swap,axiom,
! [B: $tType,A: $tType,F: B > A,A4: B,B3: B] :
( ( ( image @ B @ A @ F @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( image @ B @ A @ ( swap @ B @ A @ A4 @ B3 @ F ) @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surj_imp_surj_swap
thf(fact_152_eval__equality,axiom,
! [I2: ( list @ char ) > paraco415392788lle_tv,P3: paraco414474393lle_fm,Q2: paraco414474393lle_fm] :
( ( ( ( paraco876059933e_eval @ I2 @ P3 )
= ( paraco876059933e_eval @ I2 @ Q2 ) )
=> ( ( paraco876059933e_eval @ I2 @ ( paraco1628874225le_Eql @ P3 @ Q2 ) )
= ( paraco2040174112le_Det @ $true ) ) )
& ( ( ( paraco876059933e_eval @ I2 @ P3 )
!= ( paraco876059933e_eval @ I2 @ Q2 ) )
=> ( ( ( ( paraco876059933e_eval @ I2 @ P3 )
= ( paraco2040174112le_Det @ $true ) )
=> ( ( paraco876059933e_eval @ I2 @ ( paraco1628874225le_Eql @ P3 @ Q2 ) )
= ( paraco876059933e_eval @ I2 @ Q2 ) ) )
& ( ( ( paraco876059933e_eval @ I2 @ P3 )
!= ( paraco2040174112le_Det @ $true ) )
=> ( ( ( ( paraco876059933e_eval @ I2 @ Q2 )
= ( paraco2040174112le_Det @ $true ) )
=> ( ( paraco876059933e_eval @ I2 @ ( paraco1628874225le_Eql @ P3 @ Q2 ) )
= ( paraco876059933e_eval @ I2 @ P3 ) ) )
& ( ( ( paraco876059933e_eval @ I2 @ Q2 )
!= ( paraco2040174112le_Det @ $true ) )
=> ( ( ( ( paraco876059933e_eval @ I2 @ P3 )
= ( paraco2040174112le_Det @ $false ) )
=> ( ( paraco876059933e_eval @ I2 @ ( paraco1628874225le_Eql @ P3 @ Q2 ) )
= ( paraco876059933e_eval @ I2 @ ( paraco329115265le_Neg @ Q2 ) ) ) )
& ( ( ( paraco876059933e_eval @ I2 @ P3 )
!= ( paraco2040174112le_Det @ $false ) )
=> ( ( ( ( paraco876059933e_eval @ I2 @ Q2 )
= ( paraco2040174112le_Det @ $false ) )
=> ( ( paraco876059933e_eval @ I2 @ ( paraco1628874225le_Eql @ P3 @ Q2 ) )
= ( paraco876059933e_eval @ I2 @ ( paraco329115265le_Neg @ P3 ) ) ) )
& ( ( ( paraco876059933e_eval @ I2 @ Q2 )
!= ( paraco2040174112le_Det @ $false ) )
=> ( ( paraco876059933e_eval @ I2 @ ( paraco1628874225le_Eql @ P3 @ Q2 ) )
= ( paraco2040174112le_Det @ $false ) ) ) ) ) ) ) ) ) ) ) ) ).
% eval_equality
thf(fact_153_fm_Oexhaust,axiom,
! [Y3: paraco414474393lle_fm] :
( ! [X12: list @ char] :
( Y3
!= ( paraco27778325le_Pro @ X12 ) )
=> ( ( Y3 != paraco251304083_Truth )
=> ( ! [X33: paraco414474393lle_fm] :
( Y3
!= ( paraco329115265le_Neg @ X33 ) )
=> ( ! [X412: paraco414474393lle_fm,X422: paraco414474393lle_fm] :
( Y3
!= ( paraco2100061555le_Con @ X412 @ X422 ) )
=> ( ! [X512: paraco414474393lle_fm,X522: paraco414474393lle_fm] :
( Y3
!= ( paraco2084319816le_Eql @ X512 @ X522 ) )
=> ~ ! [X61: paraco414474393lle_fm,X62: paraco414474393lle_fm] :
( Y3
!= ( paraco1628874225le_Eql @ X61 @ X62 ) ) ) ) ) ) ) ).
% fm.exhaust
thf(fact_154_fm_Oinduct,axiom,
! [P2: paraco414474393lle_fm > $o,Fm: paraco414474393lle_fm] :
( ! [X3: list @ char] : ( P2 @ ( paraco27778325le_Pro @ X3 ) )
=> ( ( P2 @ paraco251304083_Truth )
=> ( ! [X3: paraco414474393lle_fm] :
( ( P2 @ X3 )
=> ( P2 @ ( paraco329115265le_Neg @ X3 ) ) )
=> ( ! [X1a: paraco414474393lle_fm,X22: paraco414474393lle_fm] :
( ( P2 @ X1a )
=> ( ( P2 @ X22 )
=> ( P2 @ ( paraco2100061555le_Con @ X1a @ X22 ) ) ) )
=> ( ! [X1a: paraco414474393lle_fm,X22: paraco414474393lle_fm] :
( ( P2 @ X1a )
=> ( ( P2 @ X22 )
=> ( P2 @ ( paraco2084319816le_Eql @ X1a @ X22 ) ) ) )
=> ( ! [X1a: paraco414474393lle_fm,X22: paraco414474393lle_fm] :
( ( P2 @ X1a )
=> ( ( P2 @ X22 )
=> ( P2 @ ( paraco1628874225le_Eql @ X1a @ X22 ) ) ) )
=> ( P2 @ Fm ) ) ) ) ) ) ) ).
% fm.induct
thf(fact_155_SUP__subset__mono,axiom,
! [A: $tType,B: $tType] :
( ( comple187826305attice @ A )
=> ! [A2: set @ B,B2: set @ B,F: B > A,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ A2 @ B2 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A2 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A2 ) ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ G @ B2 ) ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_156_UN__ball__bex__simps_I4_J,axiom,
! [F4: $tType,E: $tType,B2: E > ( set @ F4 ),A2: set @ E,P2: F4 > $o] :
( ( ? [X: F4] :
( ( member @ F4 @ X @ ( complete_Sup_Sup @ ( set @ F4 ) @ ( image @ E @ ( set @ F4 ) @ B2 @ A2 ) ) )
& ( P2 @ X ) ) )
= ( ? [X: E] :
( ( member @ E @ X @ A2 )
& ? [Y4: F4] :
( ( member @ F4 @ Y4 @ ( B2 @ X ) )
& ( P2 @ Y4 ) ) ) ) ) ).
% UN_ball_bex_simps(4)
thf(fact_157_UN__ball__bex__simps_I2_J,axiom,
! [C: $tType,B: $tType,B2: B > ( set @ C ),A2: set @ B,P2: C > $o] :
( ( ! [X: C] :
( ( member @ C @ X @ ( complete_Sup_Sup @ ( set @ C ) @ ( image @ B @ ( set @ C ) @ B2 @ A2 ) ) )
=> ( P2 @ X ) ) )
= ( ! [X: B] :
( ( member @ B @ X @ A2 )
=> ! [Y4: C] :
( ( member @ C @ Y4 @ ( B2 @ X ) )
=> ( P2 @ Y4 ) ) ) ) ) ).
% UN_ball_bex_simps(2)
thf(fact_158_bex__UN,axiom,
! [A: $tType,B: $tType,B2: B > ( set @ A ),A2: set @ B,P2: A > $o] :
( ( ? [X: A] :
( ( member @ A @ X @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B2 @ A2 ) ) )
& ( P2 @ X ) ) )
= ( ? [X: B] :
( ( member @ B @ X @ A2 )
& ? [Y4: A] :
( ( member @ A @ Y4 @ ( B2 @ X ) )
& ( P2 @ Y4 ) ) ) ) ) ).
% bex_UN
thf(fact_159_ball__UN,axiom,
! [A: $tType,B: $tType,B2: B > ( set @ A ),A2: set @ B,P2: A > $o] :
( ( ! [X: A] :
( ( member @ A @ X @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B2 @ A2 ) ) )
=> ( P2 @ X ) ) )
= ( ! [X: B] :
( ( member @ B @ X @ A2 )
=> ! [Y4: A] :
( ( member @ A @ Y4 @ ( B2 @ X ) )
=> ( P2 @ Y4 ) ) ) ) ) ).
% ball_UN
thf(fact_160_Union__Pow__eq,axiom,
! [A: $tType,A2: set @ A] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( pow @ A @ A2 ) )
= A2 ) ).
% Union_Pow_eq
thf(fact_161_fm_Oinject_I5_J,axiom,
! [X612: paraco414474393lle_fm,X622: paraco414474393lle_fm,Y61: paraco414474393lle_fm,Y62: paraco414474393lle_fm] :
( ( ( paraco1628874225le_Eql @ X612 @ X622 )
= ( paraco1628874225le_Eql @ Y61 @ Y62 ) )
= ( ( X612 = Y61 )
& ( X622 = Y62 ) ) ) ).
% fm.inject(5)
thf(fact_162_fm_Oinject_I1_J,axiom,
! [X1: list @ char,Y1: list @ char] :
( ( ( paraco27778325le_Pro @ X1 )
= ( paraco27778325le_Pro @ Y1 ) )
= ( X1 = Y1 ) ) ).
% fm.inject(1)
thf(fact_163_Sup__apply,axiom,
! [B: $tType,A: $tType] :
( ( complete_Sup @ B )
=> ( ( complete_Sup_Sup @ ( A > B ) )
= ( ^ [A3: set @ ( A > B ),X: A] :
( complete_Sup_Sup @ B
@ ( image @ ( A > B ) @ B
@ ^ [F2: A > B] : ( F2 @ X )
@ A3 ) ) ) ) ) ).
% Sup_apply
thf(fact_164_UN__iff,axiom,
! [A: $tType,B: $tType,B3: A,B2: B > ( set @ A ),A2: set @ B] :
( ( member @ A @ B3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B2 @ A2 ) ) )
= ( ? [X: B] :
( ( member @ B @ X @ A2 )
& ( member @ A @ B3 @ ( B2 @ X ) ) ) ) ) ).
% UN_iff
thf(fact_165_UN__I,axiom,
! [B: $tType,A: $tType,A4: A,A2: set @ A,B3: B,B2: A > ( set @ B )] :
( ( member @ A @ A4 @ A2 )
=> ( ( member @ B @ B3 @ ( B2 @ A4 ) )
=> ( member @ B @ B3 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B2 @ A2 ) ) ) ) ) ).
% UN_I
thf(fact_166_SUP__apply,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( complete_Sup @ A )
=> ! [F: C > B > A,A2: set @ C,X2: B] :
( ( complete_Sup_Sup @ ( B > A ) @ ( image @ C @ ( B > A ) @ F @ A2 ) @ X2 )
= ( complete_Sup_Sup @ A
@ ( image @ C @ A
@ ^ [Y4: C] : ( F @ Y4 @ X2 )
@ A2 ) ) ) ) ).
% SUP_apply
thf(fact_167_SUP__identity__eq,axiom,
! [A: $tType] :
( ( complete_Sup @ A )
=> ! [A2: set @ A] :
( ( complete_Sup_Sup @ A
@ ( image @ A @ A
@ ^ [X: A] : X
@ A2 ) )
= ( complete_Sup_Sup @ A @ A2 ) ) ) ).
% SUP_identity_eq
thf(fact_168_Sup__UNIV,axiom,
! [A: $tType] :
( ( comple187826305attice @ A )
=> ( ( complete_Sup_Sup @ A @ ( top_top @ ( set @ A ) ) )
= ( top_top @ A ) ) ) ).
% Sup_UNIV
thf(fact_169_SUP__cong,axiom,
! [A: $tType,B: $tType] :
( ( complete_Sup @ A )
=> ! [A2: set @ B,B2: set @ B,C3: B > A,D2: B > A] :
( ( A2 = B2 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ B2 )
=> ( ( C3 @ X3 )
= ( D2 @ X3 ) ) )
=> ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ C3 @ A2 ) )
= ( complete_Sup_Sup @ A @ ( image @ B @ A @ D2 @ B2 ) ) ) ) ) ) ).
% SUP_cong
thf(fact_170_SUP__UNION,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comple187826305attice @ A )
=> ! [F: B > A,G: C > ( set @ B ),A2: set @ C] :
( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ C @ ( set @ B ) @ G @ A2 ) ) ) )
= ( complete_Sup_Sup @ A
@ ( image @ C @ A
@ ^ [Y4: C] : ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ ( G @ Y4 ) ) )
@ A2 ) ) ) ) ).
% SUP_UNION
thf(fact_171_Sup__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( complete_Sup @ B )
=> ( ( complete_Sup_Sup @ ( A > B ) )
= ( ^ [A3: set @ ( A > B ),X: A] :
( complete_Sup_Sup @ B
@ ( image @ ( A > B ) @ B
@ ^ [F2: A > B] : ( F2 @ X )
@ A3 ) ) ) ) ) ).
% Sup_fun_def
thf(fact_172_SUP__UN__eq,axiom,
! [B: $tType,A: $tType,R2: B > ( set @ A ),S: set @ B] :
( ( complete_Sup_Sup @ ( A > $o )
@ ( image @ B @ ( A > $o )
@ ^ [I: B,X: A] : ( member @ A @ X @ ( R2 @ I ) )
@ S ) )
= ( ^ [X: A] : ( member @ A @ X @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ R2 @ S ) ) ) ) ) ).
% SUP_UN_eq
thf(fact_173_SUP__commute,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comple187826305attice @ A )
=> ! [F: B > C > A,B2: set @ C,A2: set @ B] :
( ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [I: B] : ( complete_Sup_Sup @ A @ ( image @ C @ A @ ( F @ I ) @ B2 ) )
@ A2 ) )
= ( complete_Sup_Sup @ A
@ ( image @ C @ A
@ ^ [J: C] :
( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [I: B] : ( F @ I @ J )
@ A2 ) )
@ B2 ) ) ) ) ).
% SUP_commute
thf(fact_174_fm_Odistinct_I9_J,axiom,
! [X1: list @ char,X612: paraco414474393lle_fm,X622: paraco414474393lle_fm] :
( ( paraco27778325le_Pro @ X1 )
!= ( paraco1628874225le_Eql @ X612 @ X622 ) ) ).
% fm.distinct(9)
thf(fact_175_Union__least,axiom,
! [A: $tType,A2: set @ ( set @ A ),C3: set @ A] :
( ! [X6: set @ A] :
( ( member @ ( set @ A ) @ X6 @ A2 )
=> ( ord_less_eq @ ( set @ A ) @ X6 @ C3 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) @ C3 ) ) ).
% Union_least
thf(fact_176_Union__upper,axiom,
! [A: $tType,B2: set @ A,A2: set @ ( set @ A )] :
( ( member @ ( set @ A ) @ B2 @ A2 )
=> ( ord_less_eq @ ( set @ A ) @ B2 @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) ) ) ).
% Union_upper
thf(fact_177_Union__subsetI,axiom,
! [A: $tType,A2: set @ ( set @ A ),B2: set @ ( set @ A )] :
( ! [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ A2 )
=> ? [Y6: set @ A] :
( ( member @ ( set @ A ) @ Y6 @ B2 )
& ( ord_less_eq @ ( set @ A ) @ X3 @ Y6 ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) @ ( complete_Sup_Sup @ ( set @ A ) @ B2 ) ) ) ).
% Union_subsetI
thf(fact_178_Sup__upper2,axiom,
! [A: $tType] :
( ( comple187826305attice @ A )
=> ! [U2: A,A2: set @ A,V: A] :
( ( member @ A @ U2 @ A2 )
=> ( ( ord_less_eq @ A @ V @ U2 )
=> ( ord_less_eq @ A @ V @ ( complete_Sup_Sup @ A @ A2 ) ) ) ) ) ).
% Sup_upper2
thf(fact_179_Sup__le__iff,axiom,
! [A: $tType] :
( ( comple187826305attice @ A )
=> ! [A2: set @ A,B3: A] :
( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A2 ) @ B3 )
= ( ! [X: A] :
( ( member @ A @ X @ A2 )
=> ( ord_less_eq @ A @ X @ B3 ) ) ) ) ) ).
% Sup_le_iff
thf(fact_180_Sup__upper,axiom,
! [A: $tType] :
( ( comple187826305attice @ A )
=> ! [X2: A,A2: set @ A] :
( ( member @ A @ X2 @ A2 )
=> ( ord_less_eq @ A @ X2 @ ( complete_Sup_Sup @ A @ A2 ) ) ) ) ).
% Sup_upper
thf(fact_181_Sup__least,axiom,
! [A: $tType] :
( ( comple187826305attice @ A )
=> ! [A2: set @ A,Z2: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A2 )
=> ( ord_less_eq @ A @ X3 @ Z2 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A2 ) @ Z2 ) ) ) ).
% Sup_least
thf(fact_182_Sup__mono,axiom,
! [A: $tType] :
( ( comple187826305attice @ A )
=> ! [A2: set @ A,B2: set @ A] :
( ! [A6: A] :
( ( member @ A @ A6 @ A2 )
=> ? [X4: A] :
( ( member @ A @ X4 @ B2 )
& ( ord_less_eq @ A @ A6 @ X4 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A2 ) @ ( complete_Sup_Sup @ A @ B2 ) ) ) ) ).
% Sup_mono
thf(fact_183_Sup__eqI,axiom,
! [A: $tType] :
( ( comple187826305attice @ A )
=> ! [A2: set @ A,X2: A] :
( ! [Y5: A] :
( ( member @ A @ Y5 @ A2 )
=> ( ord_less_eq @ A @ Y5 @ X2 ) )
=> ( ! [Y5: A] :
( ! [Z4: A] :
( ( member @ A @ Z4 @ A2 )
=> ( ord_less_eq @ A @ Z4 @ Y5 ) )
=> ( ord_less_eq @ A @ X2 @ Y5 ) )
=> ( ( complete_Sup_Sup @ A @ A2 )
= X2 ) ) ) ) ).
% Sup_eqI
thf(fact_184_UN__UN__flatten,axiom,
! [A: $tType,B: $tType,C: $tType,C3: B > ( set @ A ),B2: C > ( set @ B ),A2: set @ C] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ C3 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ C @ ( set @ B ) @ B2 @ A2 ) ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image @ C @ ( set @ A )
@ ^ [Y4: C] : ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ C3 @ ( B2 @ Y4 ) ) )
@ A2 ) ) ) ).
% UN_UN_flatten
thf(fact_185_UN__E,axiom,
! [A: $tType,B: $tType,B3: A,B2: B > ( set @ A ),A2: set @ B] :
( ( member @ A @ B3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B2 @ A2 ) ) )
=> ~ ! [X3: B] :
( ( member @ B @ X3 @ A2 )
=> ~ ( member @ A @ B3 @ ( B2 @ X3 ) ) ) ) ).
% UN_E
thf(fact_186_UN__extend__simps_I9_J,axiom,
! [S2: $tType,R3: $tType,Q3: $tType,C3: R3 > ( set @ S2 ),B2: Q3 > ( set @ R3 ),A2: set @ Q3] :
( ( complete_Sup_Sup @ ( set @ S2 )
@ ( image @ Q3 @ ( set @ S2 )
@ ^ [X: Q3] : ( complete_Sup_Sup @ ( set @ S2 ) @ ( image @ R3 @ ( set @ S2 ) @ C3 @ ( B2 @ X ) ) )
@ A2 ) )
= ( complete_Sup_Sup @ ( set @ S2 ) @ ( image @ R3 @ ( set @ S2 ) @ C3 @ ( complete_Sup_Sup @ ( set @ R3 ) @ ( image @ Q3 @ ( set @ R3 ) @ B2 @ A2 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_187_UN__Pow__subset,axiom,
! [A: $tType,B: $tType,B2: B > ( set @ A ),A2: set @ B] :
( ord_less_eq @ ( set @ ( set @ A ) )
@ ( complete_Sup_Sup @ ( set @ ( set @ A ) )
@ ( image @ B @ ( set @ ( set @ A ) )
@ ^ [X: B] : ( pow @ A @ ( B2 @ X ) )
@ A2 ) )
@ ( pow @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B2 @ A2 ) ) ) ) ).
% UN_Pow_subset
thf(fact_188_SUP__eq,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comple187826305attice @ A )
=> ! [A2: set @ B,B2: set @ C,F: B > A,G: C > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A2 )
=> ? [X4: C] :
( ( member @ C @ X4 @ B2 )
& ( ord_less_eq @ A @ ( F @ I3 ) @ ( G @ X4 ) ) ) )
=> ( ! [J2: C] :
( ( member @ C @ J2 @ B2 )
=> ? [X4: B] :
( ( member @ B @ X4 @ A2 )
& ( ord_less_eq @ A @ ( G @ J2 ) @ ( F @ X4 ) ) ) )
=> ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A2 ) )
= ( complete_Sup_Sup @ A @ ( image @ C @ A @ G @ B2 ) ) ) ) ) ) ).
% SUP_eq
thf(fact_189_Sup__subset__mono,axiom,
! [A: $tType] :
( ( comple187826305attice @ A )
=> ! [A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A2 ) @ ( complete_Sup_Sup @ A @ B2 ) ) ) ) ).
% Sup_subset_mono
thf(fact_190_eval_Osimps_I1_J,axiom,
! [I2: ( list @ char ) > paraco415392788lle_tv,S3: list @ char] :
( ( paraco876059933e_eval @ I2 @ ( paraco27778325le_Pro @ S3 ) )
= ( I2 @ S3 ) ) ).
% eval.simps(1)
thf(fact_191_fm_Odistinct_I23_J,axiom,
! [X32: paraco414474393lle_fm,X612: paraco414474393lle_fm,X622: paraco414474393lle_fm] :
( ( paraco329115265le_Neg @ X32 )
!= ( paraco1628874225le_Eql @ X612 @ X622 ) ) ).
% fm.distinct(23)
thf(fact_192_fm_Odistinct_I3_J,axiom,
! [X1: list @ char,X32: paraco414474393lle_fm] :
( ( paraco27778325le_Pro @ X1 )
!= ( paraco329115265le_Neg @ X32 ) ) ).
% fm.distinct(3)
thf(fact_193_fm_Odistinct_I27_J,axiom,
! [X41: paraco414474393lle_fm,X42: paraco414474393lle_fm,X612: paraco414474393lle_fm,X622: paraco414474393lle_fm] :
( ( paraco2100061555le_Con @ X41 @ X42 )
!= ( paraco1628874225le_Eql @ X612 @ X622 ) ) ).
% fm.distinct(27)
thf(fact_194_fm_Odistinct_I29_J,axiom,
! [X51: paraco414474393lle_fm,X52: paraco414474393lle_fm,X612: paraco414474393lle_fm,X622: paraco414474393lle_fm] :
( ( paraco2084319816le_Eql @ X51 @ X52 )
!= ( paraco1628874225le_Eql @ X612 @ X622 ) ) ).
% fm.distinct(29)
thf(fact_195_Union__mono,axiom,
! [A: $tType,A2: set @ ( set @ A ),B2: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ A2 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) @ ( complete_Sup_Sup @ ( set @ A ) @ B2 ) ) ) ).
% Union_mono
thf(fact_196_fm_Odistinct_I5_J,axiom,
! [X1: list @ char,X41: paraco414474393lle_fm,X42: paraco414474393lle_fm] :
( ( paraco27778325le_Pro @ X1 )
!= ( paraco2100061555le_Con @ X41 @ X42 ) ) ).
% fm.distinct(5)
thf(fact_197_fm_Odistinct_I7_J,axiom,
! [X1: list @ char,X51: paraco414474393lle_fm,X52: paraco414474393lle_fm] :
( ( paraco27778325le_Pro @ X1 )
!= ( paraco2084319816le_Eql @ X51 @ X52 ) ) ).
% fm.distinct(7)
thf(fact_198_fm_Odistinct_I17_J,axiom,
! [X612: paraco414474393lle_fm,X622: paraco414474393lle_fm] :
( paraco251304083_Truth
!= ( paraco1628874225le_Eql @ X612 @ X622 ) ) ).
% fm.distinct(17)
thf(fact_199_SUP__eqI,axiom,
! [B: $tType,A: $tType] :
( ( comple187826305attice @ A )
=> ! [A2: set @ B,F: B > A,X2: A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A2 )
=> ( ord_less_eq @ A @ ( F @ I3 ) @ X2 ) )
=> ( ! [Y5: A] :
( ! [I4: B] :
( ( member @ B @ I4 @ A2 )
=> ( ord_less_eq @ A @ ( F @ I4 ) @ Y5 ) )
=> ( ord_less_eq @ A @ X2 @ Y5 ) )
=> ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A2 ) )
= X2 ) ) ) ) ).
% SUP_eqI
thf(fact_200_SUP__mono,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comple187826305attice @ A )
=> ! [A2: set @ B,B2: set @ C,F: B > A,G: C > A] :
( ! [N3: B] :
( ( member @ B @ N3 @ A2 )
=> ? [X4: C] :
( ( member @ C @ X4 @ B2 )
& ( ord_less_eq @ A @ ( F @ N3 ) @ ( G @ X4 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A2 ) ) @ ( complete_Sup_Sup @ A @ ( image @ C @ A @ G @ B2 ) ) ) ) ) ).
% SUP_mono
thf(fact_201_SUP__least,axiom,
! [B: $tType,A: $tType] :
( ( comple187826305attice @ A )
=> ! [A2: set @ B,F: B > A,U2: A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A2 )
=> ( ord_less_eq @ A @ ( F @ I3 ) @ U2 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A2 ) ) @ U2 ) ) ) ).
% SUP_least
thf(fact_202_SUP__mono_H,axiom,
! [A: $tType,B: $tType] :
( ( comple187826305attice @ A )
=> ! [F: B > A,G: B > A,A2: set @ B] :
( ! [X3: B] : ( ord_less_eq @ A @ ( F @ X3 ) @ ( G @ X3 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A2 ) ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ G @ A2 ) ) ) ) ) ).
% SUP_mono'
thf(fact_203_SUP__upper,axiom,
! [A: $tType,B: $tType] :
( ( comple187826305attice @ A )
=> ! [I2: B,A2: set @ B,F: B > A] :
( ( member @ B @ I2 @ A2 )
=> ( ord_less_eq @ A @ ( F @ I2 ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A2 ) ) ) ) ) ).
% SUP_upper
thf(fact_204_SUP__le__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple187826305attice @ A )
=> ! [F: B > A,A2: set @ B,U2: A] :
( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A2 ) ) @ U2 )
= ( ! [X: B] :
( ( member @ B @ X @ A2 )
=> ( ord_less_eq @ A @ ( F @ X ) @ U2 ) ) ) ) ) ).
% SUP_le_iff
thf(fact_205_SUP__upper2,axiom,
! [A: $tType,B: $tType] :
( ( comple187826305attice @ A )
=> ! [I2: B,A2: set @ B,U2: A,F: B > A] :
( ( member @ B @ I2 @ A2 )
=> ( ( ord_less_eq @ A @ U2 @ ( F @ I2 ) )
=> ( ord_less_eq @ A @ U2 @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A2 ) ) ) ) ) ) ).
% SUP_upper2
thf(fact_206_fm_Odistinct_I1_J,axiom,
! [X1: list @ char] :
( ( paraco27778325le_Pro @ X1 )
!= paraco251304083_Truth ) ).
% fm.distinct(1)
thf(fact_207_subset__Pow__Union,axiom,
! [A: $tType,A2: set @ ( set @ A )] : ( ord_less_eq @ ( set @ ( set @ A ) ) @ A2 @ ( pow @ A @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) ) ) ).
% subset_Pow_Union
thf(fact_208_UN__extend__simps_I10_J,axiom,
! [V2: $tType,U3: $tType,T2: $tType,B2: U3 > ( set @ V2 ),F: T2 > U3,A2: set @ T2] :
( ( complete_Sup_Sup @ ( set @ V2 )
@ ( image @ T2 @ ( set @ V2 )
@ ^ [A5: T2] : ( B2 @ ( F @ A5 ) )
@ A2 ) )
= ( complete_Sup_Sup @ ( set @ V2 ) @ ( image @ U3 @ ( set @ V2 ) @ B2 @ ( image @ T2 @ U3 @ F @ A2 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_209_image__UN,axiom,
! [A: $tType,B: $tType,C: $tType,F: B > A,B2: C > ( set @ B ),A2: set @ C] :
( ( image @ B @ A @ F @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ C @ ( set @ B ) @ B2 @ A2 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image @ C @ ( set @ A )
@ ^ [X: C] : ( image @ B @ A @ F @ ( B2 @ X ) )
@ A2 ) ) ) ).
% image_UN
thf(fact_210_UN__mono,axiom,
! [B: $tType,A: $tType,A2: set @ A,B2: set @ A,F: A > ( set @ B ),G: A > ( set @ B )] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A2 )
=> ( ord_less_eq @ ( set @ B ) @ ( F @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less_eq @ ( set @ B ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ F @ A2 ) ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ G @ B2 ) ) ) ) ) ).
% UN_mono
thf(fact_211_UN__least,axiom,
! [A: $tType,B: $tType,A2: set @ A,B2: A > ( set @ B ),C3: set @ B] :
( ! [X3: A] :
( ( member @ A @ X3 @ A2 )
=> ( ord_less_eq @ ( set @ B ) @ ( B2 @ X3 ) @ C3 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B2 @ A2 ) ) @ C3 ) ) ).
% UN_least
thf(fact_212_UN__upper,axiom,
! [B: $tType,A: $tType,A4: A,A2: set @ A,B2: A > ( set @ B )] :
( ( member @ A @ A4 @ A2 )
=> ( ord_less_eq @ ( set @ B ) @ ( B2 @ A4 ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B2 @ A2 ) ) ) ) ).
% UN_upper
thf(fact_213_UN__image__subset,axiom,
! [C: $tType,A: $tType,B: $tType,F: B > ( set @ A ),G: C > ( set @ B ),X2: C,X5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ F @ ( G @ X2 ) ) ) @ X5 )
= ( ord_less_eq @ ( set @ B ) @ ( G @ X2 )
@ ( collect @ B
@ ^ [X: B] : ( ord_less_eq @ ( set @ A ) @ ( F @ X ) @ X5 ) ) ) ) ).
% UN_image_subset
thf(fact_214_UN__subset__iff,axiom,
! [A: $tType,B: $tType,A2: B > ( set @ A ),I5: set @ B,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ A2 @ I5 ) ) @ B2 )
= ( ! [X: B] :
( ( member @ B @ X @ I5 )
=> ( ord_less_eq @ ( set @ A ) @ ( A2 @ X ) @ B2 ) ) ) ) ).
% UN_subset_iff
thf(fact_215_Union__UNIV,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( top_top @ ( set @ ( set @ A ) ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% Union_UNIV
thf(fact_216_image__Union,axiom,
! [A: $tType,B: $tType,F: B > A,S: set @ ( set @ B )] :
( ( image @ B @ A @ F @ ( complete_Sup_Sup @ ( set @ B ) @ S ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ ( set @ B ) @ ( set @ A ) @ ( image @ B @ A @ F ) @ S ) ) ) ).
% image_Union
thf(fact_217_UN__extend__simps_I8_J,axiom,
! [P4: $tType,O: $tType,B2: O > ( set @ P4 ),A2: set @ ( set @ O )] :
( ( complete_Sup_Sup @ ( set @ P4 )
@ ( image @ ( set @ O ) @ ( set @ P4 )
@ ^ [Y4: set @ O] : ( complete_Sup_Sup @ ( set @ P4 ) @ ( image @ O @ ( set @ P4 ) @ B2 @ Y4 ) )
@ A2 ) )
= ( complete_Sup_Sup @ ( set @ P4 ) @ ( image @ O @ ( set @ P4 ) @ B2 @ ( complete_Sup_Sup @ ( set @ O ) @ A2 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_218_top__finite__def,axiom,
! [A: $tType] :
( ( finite_lattice @ A )
=> ( ( top_top @ A )
= ( complete_Sup_Sup @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% top_finite_def
thf(fact_219_eval_Ocases,axiom,
! [X2: product_prod @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm] :
( ! [I3: ( list @ char ) > paraco415392788lle_tv,S4: list @ char] :
( X2
!= ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ ( paraco27778325le_Pro @ S4 ) ) )
=> ( ! [I3: ( list @ char ) > paraco415392788lle_tv] :
( X2
!= ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ paraco251304083_Truth ) )
=> ( ! [I3: ( list @ char ) > paraco415392788lle_tv,P5: paraco414474393lle_fm] :
( X2
!= ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ ( paraco329115265le_Neg @ P5 ) ) )
=> ( ! [I3: ( list @ char ) > paraco415392788lle_tv,P5: paraco414474393lle_fm,Q4: paraco414474393lle_fm] :
( X2
!= ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ ( paraco2100061555le_Con @ P5 @ Q4 ) ) )
=> ( ! [I3: ( list @ char ) > paraco415392788lle_tv,P5: paraco414474393lle_fm,Q4: paraco414474393lle_fm] :
( X2
!= ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ ( paraco2084319816le_Eql @ P5 @ Q4 ) ) )
=> ~ ! [I3: ( list @ char ) > paraco415392788lle_tv,P5: paraco414474393lle_fm,Q4: paraco414474393lle_fm] :
( X2
!= ( product_Pair @ ( ( list @ char ) > paraco415392788lle_tv ) @ paraco414474393lle_fm @ I3 @ ( paraco1628874225le_Eql @ P5 @ Q4 ) ) ) ) ) ) ) ) ).
% eval.cases
thf(fact_220_cSup__eq__maximum,axiom,
! [A: $tType] :
( ( condit378418413attice @ A )
=> ! [Z2: A,X5: set @ A] :
( ( member @ A @ Z2 @ X5 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X5 )
=> ( ord_less_eq @ A @ X3 @ Z2 ) )
=> ( ( complete_Sup_Sup @ A @ X5 )
= Z2 ) ) ) ) ).
% cSup_eq_maximum
thf(fact_221_UN__ball__bex__simps_I3_J,axiom,
! [D: $tType,A2: set @ ( set @ D ),P2: D > $o] :
( ( ? [X: D] :
( ( member @ D @ X @ ( complete_Sup_Sup @ ( set @ D ) @ A2 ) )
& ( P2 @ X ) ) )
= ( ? [X: set @ D] :
( ( member @ ( set @ D ) @ X @ A2 )
& ? [Y4: D] :
( ( member @ D @ Y4 @ X )
& ( P2 @ Y4 ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_222_UN__ball__bex__simps_I1_J,axiom,
! [A: $tType,A2: set @ ( set @ A ),P2: A > $o] :
( ( ! [X: A] :
( ( member @ A @ X @ ( complete_Sup_Sup @ ( set @ A ) @ A2 ) )
=> ( P2 @ X ) ) )
= ( ! [X: set @ A] :
( ( member @ ( set @ A ) @ X @ A2 )
=> ! [Y4: A] :
( ( member @ A @ Y4 @ X )
=> ( P2 @ Y4 ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_223_UnionI,axiom,
! [A: $tType,X5: set @ A,C3: set @ ( set @ A ),A2: A] :
( ( member @ ( set @ A ) @ X5 @ C3 )
=> ( ( member @ A @ A2 @ X5 )
=> ( member @ A @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ C3 ) ) ) ) ).
% UnionI
thf(fact_224_Union__iff,axiom,
! [A: $tType,A2: A,C3: set @ ( set @ A )] :
( ( member @ A @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ C3 ) )
= ( ? [X: set @ A] :
( ( member @ ( set @ A ) @ X @ C3 )
& ( member @ A @ A2 @ X ) ) ) ) ).
% Union_iff
thf(fact_225_Sup1__I,axiom,
! [A: $tType,P2: A > $o,A2: set @ ( A > $o ),A4: A] :
( ( member @ ( A > $o ) @ P2 @ A2 )
=> ( ( P2 @ A4 )
=> ( complete_Sup_Sup @ ( A > $o ) @ A2 @ A4 ) ) ) ).
% Sup1_I
thf(fact_226_SUP1__I,axiom,
! [A: $tType,B: $tType,A4: A,A2: set @ A,B2: A > B > $o,B3: B] :
( ( member @ A @ A4 @ A2 )
=> ( ( B2 @ A4 @ B3 )
=> ( complete_Sup_Sup @ ( B > $o ) @ ( image @ A @ ( B > $o ) @ B2 @ A2 ) @ B3 ) ) ) ).
% SUP1_I
thf(fact_227_Sup__set__def,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( set @ A ) )
= ( ^ [A3: set @ ( set @ A )] :
( collect @ A
@ ^ [X: A] : ( complete_Sup_Sup @ $o @ ( image @ ( set @ A ) @ $o @ ( member @ A @ X ) @ A3 ) ) ) ) ) ).
% Sup_set_def
thf(fact_228_SUP1__E,axiom,
! [B: $tType,A: $tType,B2: B > A > $o,A2: set @ B,B3: A] :
( ( complete_Sup_Sup @ ( A > $o ) @ ( image @ B @ ( A > $o ) @ B2 @ A2 ) @ B3 )
=> ~ ! [X3: B] :
( ( member @ B @ X3 @ A2 )
=> ~ ( B2 @ X3 @ B3 ) ) ) ).
% SUP1_E
thf(fact_229_SUP__Sup__eq2,axiom,
! [B: $tType,A: $tType,S: set @ ( set @ ( product_prod @ A @ B ) )] :
( ( complete_Sup_Sup @ ( A > B > $o )
@ ( image @ ( set @ ( product_prod @ A @ B ) ) @ ( A > B > $o )
@ ^ [I: set @ ( product_prod @ A @ B ),X: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ I )
@ S ) )
= ( ^ [X: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ S ) ) ) ) ).
% SUP_Sup_eq2
thf(fact_230_Sup__SUP__eq,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( A > $o ) )
= ( ^ [S5: set @ ( A > $o ),X: A] : ( member @ A @ X @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ ( A > $o ) @ ( set @ A ) @ ( collect @ A ) @ S5 ) ) ) ) ) ).
% Sup_SUP_eq
thf(fact_231_SUP__UN__eq2,axiom,
! [B: $tType,C: $tType,A: $tType,R2: C > ( set @ ( product_prod @ A @ B ) ),S: set @ C] :
( ( complete_Sup_Sup @ ( A > B > $o )
@ ( image @ C @ ( A > B > $o )
@ ^ [I: C,X: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ ( R2 @ I ) )
@ S ) )
= ( ^ [X: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ ( image @ C @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S ) ) ) ) ) ).
% SUP_UN_eq2
thf(fact_232_SUP__Sup__eq,axiom,
! [A: $tType,S: set @ ( set @ A )] :
( ( complete_Sup_Sup @ ( A > $o )
@ ( image @ ( set @ A ) @ ( A > $o )
@ ^ [I: set @ A,X: A] : ( member @ A @ X @ I )
@ S ) )
= ( ^ [X: A] : ( member @ A @ X @ ( complete_Sup_Sup @ ( set @ A ) @ S ) ) ) ) ).
% SUP_Sup_eq
thf(fact_233_Sup1__E,axiom,
! [A: $tType,A2: set @ ( A > $o ),A4: A] :
( ( complete_Sup_Sup @ ( A > $o ) @ A2 @ A4 )
=> ~ ! [P6: A > $o] :
( ( member @ ( A > $o ) @ P6 @ A2 )
=> ~ ( P6 @ A4 ) ) ) ).
% Sup1_E
thf(fact_234_UnionE,axiom,
! [A: $tType,A2: A,C3: set @ ( set @ A )] :
( ( member @ A @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ C3 ) )
=> ~ ! [X6: set @ A] :
( ( member @ A @ A2 @ X6 )
=> ~ ( member @ ( set @ A ) @ X6 @ C3 ) ) ) ).
% UnionE
thf(fact_235_subrelI,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
( ! [X3: A,Y5: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y5 ) @ R2 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y5 ) @ S3 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S3 ) ) ).
% subrelI
thf(fact_236_pred__subset__eq2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( A > B > $o )
@ ^ [X: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ R )
@ ^ [X: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ S ) )
= ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S ) ) ).
% pred_subset_eq2
thf(fact_237_pred__equals__eq2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ( ( ^ [X: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ R ) )
= ( ^ [X: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ S ) ) )
= ( R = S ) ) ).
% pred_equals_eq2
thf(fact_238_ssubst__Pair__rhs,axiom,
! [B: $tType,A: $tType,R2: A,S3: B,R: set @ ( product_prod @ A @ B ),S6: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S3 ) @ R )
=> ( ( S6 = S3 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S6 ) @ R ) ) ) ).
% ssubst_Pair_rhs
thf(fact_239_top__empty__eq2,axiom,
! [B: $tType,A: $tType] :
( ( top_top @ ( A > B > $o ) )
= ( ^ [X: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y4 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% top_empty_eq2
thf(fact_240_cSup__eq,axiom,
! [A: $tType] :
( ( ( condit378418413attice @ A )
& ( no_bot @ A ) )
=> ! [X5: set @ A,A4: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ X5 )
=> ( ord_less_eq @ A @ X3 @ A4 ) )
=> ( ! [Y5: A] :
( ! [X4: A] :
( ( member @ A @ X4 @ X5 )
=> ( ord_less_eq @ A @ X4 @ Y5 ) )
=> ( ord_less_eq @ A @ A4 @ Y5 ) )
=> ( ( complete_Sup_Sup @ A @ X5 )
= A4 ) ) ) ) ).
% cSup_eq
thf(fact_241_UN__constant__eq,axiom,
! [A: $tType,B: $tType,A4: A,A2: set @ A,F: A > ( set @ B ),C2: set @ B] :
( ( member @ A @ A4 @ A2 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A2 )
=> ( ( F @ X3 )
= C2 ) )
=> ( ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ F @ A2 ) )
= C2 ) ) ) ).
% UN_constant_eq
thf(fact_242_SUP__eq__top__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple1035589618norder @ A )
=> ! [F: B > A,A2: set @ B] :
( ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A2 ) )
= ( top_top @ A ) )
= ( ! [X: A] :
( ( ord_less @ A @ X @ ( top_top @ A ) )
=> ? [Y4: B] :
( ( member @ B @ Y4 @ A2 )
& ( ord_less @ A @ X @ ( F @ Y4 ) ) ) ) ) ) ) ).
% SUP_eq_top_iff
thf(fact_243_psubsetI,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% psubsetI
thf(fact_244_Sup2__I,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,A2: set @ ( A > B > $o ),A4: A,B3: B] :
( ( member @ ( A > B > $o ) @ R2 @ A2 )
=> ( ( R2 @ A4 @ B3 )
=> ( complete_Sup_Sup @ ( A > B > $o ) @ A2 @ A4 @ B3 ) ) ) ).
% Sup2_I
thf(fact_245_top2I,axiom,
! [A: $tType,B: $tType,X2: A,Y3: B] : ( top_top @ ( A > B > $o ) @ X2 @ Y3 ) ).
% top2I
thf(fact_246_Sup__eq__top__iff,axiom,
! [A: $tType] :
( ( comple1035589618norder @ A )
=> ! [A2: set @ A] :
( ( ( complete_Sup_Sup @ A @ A2 )
= ( top_top @ A ) )
= ( ! [X: A] :
( ( ord_less @ A @ X @ ( top_top @ A ) )
=> ? [Y4: A] :
( ( member @ A @ Y4 @ A2 )
& ( ord_less @ A @ X @ Y4 ) ) ) ) ) ) ).
% Sup_eq_top_iff
thf(fact_247_SUP2__I,axiom,
! [B: $tType,A: $tType,C: $tType,A4: A,A2: set @ A,B2: A > B > C > $o,B3: B,C2: C] :
( ( member @ A @ A4 @ A2 )
=> ( ( B2 @ A4 @ B3 @ C2 )
=> ( complete_Sup_Sup @ ( B > C > $o ) @ ( image @ A @ ( B > C > $o ) @ B2 @ A2 ) @ B3 @ C2 ) ) ) ).
% SUP2_I
thf(fact_248_SUP2__E,axiom,
! [A: $tType,C: $tType,B: $tType,B2: C > A > B > $o,A2: set @ C,B3: A,C2: B] :
( ( complete_Sup_Sup @ ( A > B > $o ) @ ( image @ C @ ( A > B > $o ) @ B2 @ A2 ) @ B3 @ C2 )
=> ~ ! [X3: C] :
( ( member @ C @ X3 @ A2 )
=> ~ ( B2 @ X3 @ B3 @ C2 ) ) ) ).
% SUP2_E
thf(fact_249_Sup2__E,axiom,
! [A: $tType,B: $tType,A2: set @ ( A > B > $o ),A4: A,B3: B] :
( ( complete_Sup_Sup @ ( A > B > $o ) @ A2 @ A4 @ B3 )
=> ~ ! [R4: A > B > $o] :
( ( member @ ( A > B > $o ) @ R4 @ A2 )
=> ~ ( R4 @ A4 @ B3 ) ) ) ).
% Sup2_E
thf(fact_250_Sup__bool__def,axiom,
( ( complete_Sup_Sup @ $o )
= ( member @ $o @ $true ) ) ).
% Sup_bool_def
thf(fact_251_less__Sup__iff,axiom,
! [A: $tType] :
( ( comple1035589618norder @ A )
=> ! [A4: A,S: set @ A] :
( ( ord_less @ A @ A4 @ ( complete_Sup_Sup @ A @ S ) )
= ( ? [X: A] :
( ( member @ A @ X @ S )
& ( ord_less @ A @ A4 @ X ) ) ) ) ) ).
% less_Sup_iff
thf(fact_252_complete__interval,axiom,
! [A: $tType] :
( ( condit1037483654norder @ A )
=> ! [A4: A,B3: A,P2: A > $o] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( P2 @ A4 )
=> ( ~ ( P2 @ B3 )
=> ? [C5: A] :
( ( ord_less_eq @ A @ A4 @ C5 )
& ( ord_less_eq @ A @ C5 @ B3 )
& ! [X4: A] :
( ( ( ord_less_eq @ A @ A4 @ X4 )
& ( ord_less @ A @ X4 @ C5 ) )
=> ( P2 @ X4 ) )
& ! [D3: A] :
( ! [X3: A] :
( ( ( ord_less_eq @ A @ A4 @ X3 )
& ( ord_less @ A @ X3 @ D3 ) )
=> ( P2 @ X3 ) )
=> ( ord_less_eq @ A @ D3 @ C5 ) ) ) ) ) ) ) ).
% complete_interval
thf(fact_253_le__Sup__iff,axiom,
! [A: $tType] :
( ( comple1035589618norder @ A )
=> ! [X2: A,A2: set @ A] :
( ( ord_less_eq @ A @ X2 @ ( complete_Sup_Sup @ A @ A2 ) )
= ( ! [Y4: A] :
( ( ord_less @ A @ Y4 @ X2 )
=> ? [X: A] :
( ( member @ A @ X @ A2 )
& ( ord_less @ A @ Y4 @ X ) ) ) ) ) ) ).
% le_Sup_iff
thf(fact_254_less__SUP__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple1035589618norder @ A )
=> ! [A4: A,F: B > A,A2: set @ B] :
( ( ord_less @ A @ A4 @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A2 ) ) )
= ( ? [X: B] :
( ( member @ B @ X @ A2 )
& ( ord_less @ A @ A4 @ ( F @ X ) ) ) ) ) ) ).
% less_SUP_iff
thf(fact_255_SUP__lessD,axiom,
! [B: $tType,A: $tType] :
( ( comple187826305attice @ A )
=> ! [F: B > A,A2: set @ B,Y3: A,I2: B] :
( ( ord_less @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F @ A2 ) ) @ Y3 )
=> ( ( member @ B @ I2 @ A2 )
=> ( ord_less @ A @ ( F @ I2 ) @ Y3 ) ) ) ) ).
% SUP_lessD
% Type constructors (32)
thf(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
! [A7: $tType,A8: $tType] :
( ( comple187826305attice @ A8 )
=> ( condit378418413attice @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
! [A7: $tType,A8: $tType] :
( ( comple187826305attice @ A8 )
=> ( comple187826305attice @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Complete__Lattices_OSup,axiom,
! [A7: $tType,A8: $tType] :
( ( complete_Sup @ A8 )
=> ( complete_Sup @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__top,axiom,
! [A7: $tType,A8: $tType] :
( ( order_top @ A8 )
=> ( order_top @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A7: $tType,A8: $tType] :
( ( preorder @ A8 )
=> ( preorder @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A7: $tType,A8: $tType] :
( ( order @ A8 )
=> ( order @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Otop,axiom,
! [A7: $tType,A8: $tType] :
( ( top @ A8 )
=> ( top @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A7: $tType,A8: $tType] :
( ( ord @ A8 )
=> ( ord @ ( A7 > A8 ) ) ) ).
thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
condit1037483654norder @ nat ).
thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_1,axiom,
condit378418413attice @ nat ).
thf(tcon_Nat_Onat___Complete__Lattices_OSup_2,axiom,
complete_Sup @ nat ).
thf(tcon_Nat_Onat___Orderings_Opreorder_3,axiom,
preorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
linorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder_4,axiom,
order @ nat ).
thf(tcon_Nat_Onat___Orderings_Oord_5,axiom,
ord @ nat ).
thf(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_6,axiom,
! [A7: $tType] : ( condit378418413attice @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_7,axiom,
! [A7: $tType] : ( comple187826305attice @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_OSup_8,axiom,
! [A7: $tType] : ( complete_Sup @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__top_9,axiom,
! [A7: $tType] : ( order_top @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_10,axiom,
! [A7: $tType] : ( preorder @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_11,axiom,
! [A7: $tType] : ( order @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Otop_12,axiom,
! [A7: $tType] : ( top @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_13,axiom,
! [A7: $tType] : ( ord @ ( set @ A7 ) ) ).
thf(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_14,axiom,
condit378418413attice @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_15,axiom,
comple187826305attice @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_OSup_16,axiom,
complete_Sup @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__top_17,axiom,
order_top @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_18,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder_19,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_20,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Otop_21,axiom,
top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_22,axiom,
ord @ $o ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( ! [I: ( list @ char ) > paraco415392788lle_tv] :
( ( paraco876059933e_eval @ I @ p )
= ( paraco2040174112le_Det @ $true ) ) )
!= ( ~ ! [I: ( list @ char ) > paraco415392788lle_tv] :
( ( ord_less_eq @ ( set @ paraco415392788lle_tv ) @ ( image @ ( list @ char ) @ paraco415392788lle_tv @ I @ ( top_top @ ( set @ ( list @ char ) ) ) )
@ ( paraco506338565domain
@ ( collect @ nat
@ ^ [N: nat] : $true ) ) )
=> ( ( paraco876059933e_eval @ I @ p )
= ( paraco2040174112le_Det @ $true ) ) ) ) ) ).
%------------------------------------------------------------------------------