TPTP Problem File: SLH0821^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Prefix_Free_Code_Combinators/0000_Prefix_Free_Code_Combinators/prob_00417_014203__11921386_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1654 ( 583 unt; 371 typ; 0 def)
% Number of atoms : 3787 (1808 equ; 0 cnn)
% Maximal formula atoms : 16 ( 2 avg)
% Number of connectives : 10765 ( 501 ~; 56 |; 228 &;8346 @)
% ( 0 <=>;1634 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Number of types : 40 ( 39 usr)
% Number of type conns : 1686 (1686 >; 0 *; 0 +; 0 <<)
% Number of symbols : 335 ( 332 usr; 25 con; 0-4 aty)
% Number of variables : 3529 ( 200 ^;3223 !; 106 ?;3529 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 09:58:19.421
%------------------------------------------------------------------------------
% Could-be-implicit typings (39)
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thf(ty_n_t__Set__Oset_It__Option__Ooption_It__Int__Oint_J_J,type,
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thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
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% Explicit typings (332)
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thf(sy_c_Fun_Ofun__upd_001t__Int__Oint_001t__Option__Ooption_It__List__Olist_I_Eo_J_J,type,
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thf(sy_c_Fun_Ofun__upd_001t__Int__Oint_001t__Option__Ooption_It__Nat__Onat_J,type,
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thf(sy_c_Fun_Ofun__upd_001t__Option__Ooption_It__Nat__Onat_J_001t__Option__Ooption_It__Int__Oint_J,type,
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thf(sy_c_Fun_Ofun__upd_001t__Set__Oset_It__Nat__Onat_J_001t__Option__Ooption_It__Int__Oint_J,type,
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thf(sy_c_Fun_Oinj__on_001t__Int__Oint_001t__Int__Oint,type,
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thf(sy_c_Fun_Oinj__on_001t__Int__Oint_001t__Nat__Onat,type,
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thf(sy_c_Fun_Oinj__on_001t__Int__Oint_001t__Option__Ooption_It__Int__Oint_J,type,
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thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Int__Oint,type,
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thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Option__Ooption_It__List__Olist_I_Eo_J_J,type,
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thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Option__Ooption_It__Nat__Onat_J,type,
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thf(sy_c_Fun_Oinj__on_001t__Option__Ooption_It__List__Olist_I_Eo_J_J_001t__Int__Oint,type,
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thf(sy_c_Fun_Oinj__on_001t__Option__Ooption_It__List__Olist_I_Eo_J_J_001t__Nat__Onat,type,
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thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Int__Oint_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Option__Ooption_It__List__Olist_I_Eo_J_J,type,
inj_on4526473911404666215list_o: ( produc1062882101152166346at_nat > option_list_o ) > set_Pr6620829329807565824at_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J_001t__Option__Ooption_It__List__Olist_I_Eo_J_J,type,
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thf(sy_c_Fun_Omonotone__on_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Fun_Othe__inv__into_001t__Int__Oint_001t__Int__Oint,type,
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thf(sy_c_Fun_Othe__inv__into_001t__Int__Oint_001t__Nat__Onat,type,
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thf(sy_c_Fun_Othe__inv__into_001t__Int__Oint_001t__Option__Ooption_It__List__Olist_I_Eo_J_J,type,
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thf(sy_c_Fun_Othe__inv__into_001t__Nat__Onat_001t__Int__Oint,type,
the_inv_into_nat_int: set_nat > ( nat > int ) > int > nat ).
thf(sy_c_Fun_Othe__inv__into_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Fun_Othe__inv__into_001t__Nat__Onat_001t__Option__Ooption_It__List__Olist_I_Eo_J_J,type,
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thf(sy_c_Fun_Othe__inv__into_001t__Option__Ooption_It__List__Olist_I_Eo_J_J_001t__Int__Oint,type,
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thf(sy_c_Fun_Othe__inv__into_001t__Option__Ooption_It__List__Olist_I_Eo_J_J_001t__Nat__Onat,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Int__Oint_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__List__Olist_I_Eo_J_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Option__Ooption_It__List__Olist_I_Eo_J_J_J,type,
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thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Int__Oint_J,type,
uminus1532241313380277803et_int: set_int > set_int ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Option__Ooption_It__List__Olist_I_Eo_J_J_J,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_If_001t__Option__Ooption_It__List__Olist_I_Eo_J_J,type,
if_option_list_o: $o > option_list_o > option_list_o > option_list_o ).
thf(sy_c_If_001t__Option__Ooption_It__Nat__Onat_J,type,
if_option_nat: $o > option_nat > option_nat > option_nat ).
thf(sy_c_If_001t__Option__Ooption_It__Option__Ooption_It__List__Olist_I_Eo_J_J_J,type,
if_opt4852585628301394032list_o: $o > option_option_list_o > option_option_list_o > option_option_list_o ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Int__Oint,type,
inf_inf_int: int > int > int ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
inf_inf_nat: nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Int__Oint_J,type,
inf_inf_set_int: set_int > set_int > set_int ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
inf_inf_set_nat: set_nat > set_nat > set_nat ).
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collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Option__Ooption_It__Int__Oint_J,type,
collect_option_int: ( option_int > $o ) > set_option_int ).
thf(sy_c_Set_OCollect_001t__Option__Ooption_It__Nat__Onat_J,type,
collect_option_nat: ( option_nat > $o ) > set_option_nat ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Int__Oint_J,type,
collect_set_int: ( set_int > $o ) > set_set_int ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Nat__Onat_J,type,
collect_set_nat: ( set_nat > $o ) > set_set_nat ).
thf(sy_c_Set_Odisjnt_001t__Int__Oint,type,
disjnt_int: set_int > set_int > $o ).
thf(sy_c_Set_Odisjnt_001t__Nat__Onat,type,
disjnt_nat: set_nat > set_nat > $o ).
thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Int__Oint,type,
image_int_int: ( int > int ) > set_int > set_int ).
thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Nat__Onat,type,
image_int_nat: ( int > nat ) > set_int > set_nat ).
thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Option__Ooption_It__List__Olist_I_Eo_J_J,type,
image_3148138840367899121list_o: ( int > option_list_o ) > set_int > set_option_list_o ).
thf(sy_c_Set_Oimage_001t__List__Olist_I_Eo_J_001t__Option__Ooption_It__List__Olist_I_Eo_J_J,type,
image_3961372166222858059list_o: ( list_o > option_list_o ) > set_list_o > set_option_list_o ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Int__Oint,type,
image_nat_int: ( nat > int ) > set_nat > set_int ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Option__Ooption_It__List__Olist_I_Eo_J_J,type,
image_4575287668734308173list_o: ( nat > option_list_o ) > set_nat > set_option_list_o ).
thf(sy_c_Set_Oimage_001t__Option__Ooption_It__List__Olist_I_Eo_J_J_001t__Int__Oint,type,
image_5399048050737105611_o_int: ( option_list_o > int ) > set_option_list_o > set_int ).
thf(sy_c_Set_Oimage_001t__Option__Ooption_It__List__Olist_I_Eo_J_J_001t__Nat__Onat,type,
image_5401538521246155887_o_nat: ( option_list_o > nat ) > set_option_list_o > set_nat ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Int__Oint_J_001t__Set__Oset_It__Int__Oint_J,type,
image_524474410958335435et_int: ( set_int > set_int ) > set_set_int > set_set_int ).
thf(sy_c_Set_Oimage_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
image_7916887816326733075et_nat: ( set_nat > set_nat ) > set_set_nat > set_set_nat ).
thf(sy_c_Set_Oinsert_001t__Int__Oint,type,
insert_int: int > set_int > set_int ).
thf(sy_c_Set_Oinsert_001t__List__Olist_I_Eo_J,type,
insert_list_o: list_o > set_list_o > set_list_o ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Option__Ooption_It__List__Olist_I_Eo_J_J,type,
insert_option_list_o: option_list_o > set_option_list_o > set_option_list_o ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
insert8211810215607154385at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat ).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Option__Ooption_It__List__Olist_I_Eo_J_J_Mt__Option__Ooption_It__List__Olist_I_Eo_J_J_J,type,
insert7989621605229100023list_o: produc4882884732533091879list_o > set_Pr7497825696620840711list_o > set_Pr7497825696620840711list_o ).
thf(sy_c_Set_Ois__empty_001t__Int__Oint,type,
is_empty_int: set_int > $o ).
thf(sy_c_Set_Ois__empty_001t__List__Olist_I_Eo_J,type,
is_empty_list_o: set_list_o > $o ).
thf(sy_c_Set_Ois__empty_001t__Nat__Onat,type,
is_empty_nat: set_nat > $o ).
thf(sy_c_Set_Ois__singleton_001t__Int__Oint,type,
is_singleton_int: set_int > $o ).
thf(sy_c_Set_Ois__singleton_001t__Nat__Onat,type,
is_singleton_nat: set_nat > $o ).
thf(sy_c_Set_Opairwise_001t__Int__Oint,type,
pairwise_int: ( int > int > $o ) > set_int > $o ).
thf(sy_c_Set_Opairwise_001t__Nat__Onat,type,
pairwise_nat: ( nat > nat > $o ) > set_nat > $o ).
thf(sy_c_Set_Oremove_001t__Int__Oint,type,
remove_int: int > set_int > set_int ).
thf(sy_c_Set_Oremove_001t__Nat__Onat,type,
remove_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Ovimage_001t__Int__Oint_001t__Int__Oint,type,
vimage_int_int: ( int > int ) > set_int > set_int ).
thf(sy_c_Set_Ovimage_001t__Int__Oint_001t__Nat__Onat,type,
vimage_int_nat: ( int > nat ) > set_nat > set_int ).
thf(sy_c_Set_Ovimage_001t__Int__Oint_001t__Option__Ooption_It__List__Olist_I_Eo_J_J,type,
vimage8119226661570972603list_o: ( int > option_list_o ) > set_option_list_o > set_int ).
thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Int__Oint,type,
vimage_nat_int: ( nat > int ) > set_int > set_nat ).
thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Nat__Onat,type,
vimage_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Option__Ooption_It__List__Olist_I_Eo_J_J,type,
vimage323003453082605847list_o: ( nat > option_list_o ) > set_option_list_o > set_nat ).
thf(sy_c_Sublist_Oprefix_001_Eo,type,
prefix_o: list_o > list_o > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Option__Ooption_It__List__Olist_I_Eo_J_J_Mt__Option__Ooption_It__List__Olist_I_Eo_J_J_J,type,
accp_P4987914878654715440list_o: ( produc4882884732533091879list_o > produc4882884732533091879list_o > $o ) > produc4882884732533091879list_o > $o ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__List__Olist_I_Eo_J,type,
member_list_o: list_o > set_list_o > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Option__Ooption_It__Int__Oint_J,type,
member_option_int: option_int > set_option_int > $o ).
thf(sy_c_member_001t__Option__Ooption_It__List__Olist_I_Eo_J_J,type,
member_option_list_o: option_list_o > set_option_list_o > $o ).
thf(sy_c_member_001t__Option__Ooption_It__Nat__Onat_J,type,
member_option_nat: option_nat > set_option_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
member5262025264175285858nt_int: product_prod_int_int > set_Pr958786334691620121nt_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Nat__Onat_J,type,
member216504246829706758nt_nat: product_prod_int_nat > set_Pr3448869479623346877nt_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J,type,
member4262671552274231302at_int: product_prod_nat_int > set_Pr7995236796853374141at_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Option__Ooption_It__List__Olist_I_Eo_J_J_Mt__Option__Ooption_It__List__Olist_I_Eo_J_J_J,type,
member1589324699396745552list_o: produc4882884732533091879list_o > set_Pr7497825696620840711list_o > $o ).
thf(sy_c_member_001t__Set__Oset_It__Int__Oint_J,type,
member_set_int: set_int > set_set_int > $o ).
thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
member_set_nat: set_nat > set_set_nat > $o ).
% Relevant facts (1275)
thf(fact_0_is__encodingD,axiom,
! [E: int > option_list_o,X: int,Y: int] :
( ( prefix3555694663680348106ng_int @ E )
=> ( ( prefix454693708527911765comp_o @ ( E @ X ) @ ( E @ Y ) )
=> ( X = Y ) ) ) ).
% is_encodingD
thf(fact_1_is__encodingD,axiom,
! [E: nat > option_list_o,X: nat,Y: nat] :
( ( prefix3558185134189398382ng_nat @ E )
=> ( ( prefix454693708527911765comp_o @ ( E @ X ) @ ( E @ Y ) )
=> ( X = Y ) ) ) ).
% is_encodingD
thf(fact_2_is__encodingD,axiom,
! [E: product_prod_nat_nat > option_list_o,X: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( prefix5049516368958873059at_nat @ E )
=> ( ( prefix454693708527911765comp_o @ ( E @ X ) @ ( E @ Y ) )
=> ( X = Y ) ) ) ).
% is_encodingD
thf(fact_3_is__encodingD,axiom,
! [E: product_prod_nat_int > option_list_o,X: product_prod_nat_int,Y: product_prod_nat_int] :
( ( prefix871665349449676351at_int @ E )
=> ( ( prefix454693708527911765comp_o @ ( E @ X ) @ ( E @ Y ) )
=> ( X = Y ) ) ) ).
% is_encodingD
thf(fact_4_is__encodingD,axiom,
! [E: product_prod_int_nat > option_list_o,X: product_prod_int_nat,Y: product_prod_int_nat] :
( ( prefix6048870080859927615nt_nat @ E )
=> ( ( prefix454693708527911765comp_o @ ( E @ X ) @ ( E @ Y ) )
=> ( X = Y ) ) ) ).
% is_encodingD
thf(fact_5_is__encodingD,axiom,
! [E: product_prod_int_int > option_list_o,X: product_prod_int_int,Y: product_prod_int_int] :
( ( prefix1871019061350730907nt_int @ E )
=> ( ( prefix454693708527911765comp_o @ ( E @ X ) @ ( E @ Y ) )
=> ( X = Y ) ) ) ).
% is_encodingD
thf(fact_6_is__encodingD,axiom,
! [E: produc6945250483304103390nt_int > option_list_o,X: produc6945250483304103390nt_int,Y: produc6945250483304103390nt_int] :
( ( prefix422932351439244924nt_int @ E )
=> ( ( prefix454693708527911765comp_o @ ( E @ X ) @ ( E @ Y ) )
=> ( X = Y ) ) ) ).
% is_encodingD
thf(fact_7_is__encodingD,axiom,
! [E: produc1062882101152166346at_nat > option_list_o,X: produc1062882101152166346at_nat,Y: produc1062882101152166346at_nat] :
( ( prefix3763936006142083688at_nat @ E )
=> ( ( prefix454693708527911765comp_o @ ( E @ X ) @ ( E @ Y ) )
=> ( X = Y ) ) ) ).
% is_encodingD
thf(fact_8_is__encodingD,axiom,
! [E: produc7796170993075215398at_int > option_list_o,X: produc7796170993075215398at_int,Y: produc7796170993075215398at_int] :
( ( prefix1273852861210356932at_int @ E )
=> ( ( prefix454693708527911765comp_o @ ( E @ X ) @ ( E @ Y ) )
=> ( X = Y ) ) ) ).
% is_encodingD
thf(fact_9_is__encodingD,axiom,
! [E: produc3249803675845188134nt_nat > option_list_o,X: produc3249803675845188134nt_nat,Y: produc3249803675845188134nt_nat] :
( ( prefix5950857580835105476nt_nat @ E )
=> ( ( prefix454693708527911765comp_o @ ( E @ X ) @ ( E @ Y ) )
=> ( X = Y ) ) ) ).
% is_encodingD
thf(fact_10_is__encodingI__2,axiom,
! [E: int > option_list_o] :
( ! [X2: int,Y2: int] :
( ( prefix454693708527911765comp_o @ ( E @ X2 ) @ ( E @ Y2 ) )
=> ( X2 = Y2 ) )
=> ( prefix3555694663680348106ng_int @ E ) ) ).
% is_encodingI_2
thf(fact_11_is__encodingI__2,axiom,
! [E: nat > option_list_o] :
( ! [X2: nat,Y2: nat] :
( ( prefix454693708527911765comp_o @ ( E @ X2 ) @ ( E @ Y2 ) )
=> ( X2 = Y2 ) )
=> ( prefix3558185134189398382ng_nat @ E ) ) ).
% is_encodingI_2
thf(fact_12_is__encodingI__2,axiom,
! [E: product_prod_nat_nat > option_list_o] :
( ! [X2: product_prod_nat_nat,Y2: product_prod_nat_nat] :
( ( prefix454693708527911765comp_o @ ( E @ X2 ) @ ( E @ Y2 ) )
=> ( X2 = Y2 ) )
=> ( prefix5049516368958873059at_nat @ E ) ) ).
% is_encodingI_2
thf(fact_13_is__encodingI__2,axiom,
! [E: product_prod_nat_int > option_list_o] :
( ! [X2: product_prod_nat_int,Y2: product_prod_nat_int] :
( ( prefix454693708527911765comp_o @ ( E @ X2 ) @ ( E @ Y2 ) )
=> ( X2 = Y2 ) )
=> ( prefix871665349449676351at_int @ E ) ) ).
% is_encodingI_2
thf(fact_14_is__encodingI__2,axiom,
! [E: product_prod_int_nat > option_list_o] :
( ! [X2: product_prod_int_nat,Y2: product_prod_int_nat] :
( ( prefix454693708527911765comp_o @ ( E @ X2 ) @ ( E @ Y2 ) )
=> ( X2 = Y2 ) )
=> ( prefix6048870080859927615nt_nat @ E ) ) ).
% is_encodingI_2
thf(fact_15_is__encodingI__2,axiom,
! [E: product_prod_int_int > option_list_o] :
( ! [X2: product_prod_int_int,Y2: product_prod_int_int] :
( ( prefix454693708527911765comp_o @ ( E @ X2 ) @ ( E @ Y2 ) )
=> ( X2 = Y2 ) )
=> ( prefix1871019061350730907nt_int @ E ) ) ).
% is_encodingI_2
thf(fact_16_is__encodingI__2,axiom,
! [E: produc6945250483304103390nt_int > option_list_o] :
( ! [X2: produc6945250483304103390nt_int,Y2: produc6945250483304103390nt_int] :
( ( prefix454693708527911765comp_o @ ( E @ X2 ) @ ( E @ Y2 ) )
=> ( X2 = Y2 ) )
=> ( prefix422932351439244924nt_int @ E ) ) ).
% is_encodingI_2
thf(fact_17_is__encodingI__2,axiom,
! [E: produc1062882101152166346at_nat > option_list_o] :
( ! [X2: produc1062882101152166346at_nat,Y2: produc1062882101152166346at_nat] :
( ( prefix454693708527911765comp_o @ ( E @ X2 ) @ ( E @ Y2 ) )
=> ( X2 = Y2 ) )
=> ( prefix3763936006142083688at_nat @ E ) ) ).
% is_encodingI_2
thf(fact_18_is__encodingI__2,axiom,
! [E: produc7796170993075215398at_int > option_list_o] :
( ! [X2: produc7796170993075215398at_int,Y2: produc7796170993075215398at_int] :
( ( prefix454693708527911765comp_o @ ( E @ X2 ) @ ( E @ Y2 ) )
=> ( X2 = Y2 ) )
=> ( prefix1273852861210356932at_int @ E ) ) ).
% is_encodingI_2
thf(fact_19_is__encodingI__2,axiom,
! [E: produc3249803675845188134nt_nat > option_list_o] :
( ! [X2: produc3249803675845188134nt_nat,Y2: produc3249803675845188134nt_nat] :
( ( prefix454693708527911765comp_o @ ( E @ X2 ) @ ( E @ Y2 ) )
=> ( X2 = Y2 ) )
=> ( prefix5950857580835105476nt_nat @ E ) ) ).
% is_encodingI_2
thf(fact_20_is__encoding__def,axiom,
( prefix3555694663680348106ng_int
= ( ^ [F: int > option_list_o] :
! [X3: int,Y3: int] :
( ( prefix8824957607401505554efix_o @ ( F @ X3 ) @ ( F @ Y3 ) )
=> ( X3 = Y3 ) ) ) ) ).
% is_encoding_def
thf(fact_21_is__encoding__def,axiom,
( prefix3558185134189398382ng_nat
= ( ^ [F: nat > option_list_o] :
! [X3: nat,Y3: nat] :
( ( prefix8824957607401505554efix_o @ ( F @ X3 ) @ ( F @ Y3 ) )
=> ( X3 = Y3 ) ) ) ) ).
% is_encoding_def
thf(fact_22_is__encoding__def,axiom,
( prefix5049516368958873059at_nat
= ( ^ [F: product_prod_nat_nat > option_list_o] :
! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
( ( prefix8824957607401505554efix_o @ ( F @ X3 ) @ ( F @ Y3 ) )
=> ( X3 = Y3 ) ) ) ) ).
% is_encoding_def
thf(fact_23_is__encoding__def,axiom,
( prefix871665349449676351at_int
= ( ^ [F: product_prod_nat_int > option_list_o] :
! [X3: product_prod_nat_int,Y3: product_prod_nat_int] :
( ( prefix8824957607401505554efix_o @ ( F @ X3 ) @ ( F @ Y3 ) )
=> ( X3 = Y3 ) ) ) ) ).
% is_encoding_def
thf(fact_24_is__encoding__def,axiom,
( prefix6048870080859927615nt_nat
= ( ^ [F: product_prod_int_nat > option_list_o] :
! [X3: product_prod_int_nat,Y3: product_prod_int_nat] :
( ( prefix8824957607401505554efix_o @ ( F @ X3 ) @ ( F @ Y3 ) )
=> ( X3 = Y3 ) ) ) ) ).
% is_encoding_def
thf(fact_25_is__encoding__def,axiom,
( prefix1871019061350730907nt_int
= ( ^ [F: product_prod_int_int > option_list_o] :
! [X3: product_prod_int_int,Y3: product_prod_int_int] :
( ( prefix8824957607401505554efix_o @ ( F @ X3 ) @ ( F @ Y3 ) )
=> ( X3 = Y3 ) ) ) ) ).
% is_encoding_def
thf(fact_26_is__encoding__def,axiom,
( prefix422932351439244924nt_int
= ( ^ [F: produc6945250483304103390nt_int > option_list_o] :
! [X3: produc6945250483304103390nt_int,Y3: produc6945250483304103390nt_int] :
( ( prefix8824957607401505554efix_o @ ( F @ X3 ) @ ( F @ Y3 ) )
=> ( X3 = Y3 ) ) ) ) ).
% is_encoding_def
thf(fact_27_is__encoding__def,axiom,
( prefix3763936006142083688at_nat
= ( ^ [F: produc1062882101152166346at_nat > option_list_o] :
! [X3: produc1062882101152166346at_nat,Y3: produc1062882101152166346at_nat] :
( ( prefix8824957607401505554efix_o @ ( F @ X3 ) @ ( F @ Y3 ) )
=> ( X3 = Y3 ) ) ) ) ).
% is_encoding_def
thf(fact_28_is__encoding__def,axiom,
( prefix1273852861210356932at_int
= ( ^ [F: produc7796170993075215398at_int > option_list_o] :
! [X3: produc7796170993075215398at_int,Y3: produc7796170993075215398at_int] :
( ( prefix8824957607401505554efix_o @ ( F @ X3 ) @ ( F @ Y3 ) )
=> ( X3 = Y3 ) ) ) ) ).
% is_encoding_def
thf(fact_29_is__encoding__def,axiom,
( prefix5950857580835105476nt_nat
= ( ^ [F: produc3249803675845188134nt_nat > option_list_o] :
! [X3: produc3249803675845188134nt_nat,Y3: produc3249803675845188134nt_nat] :
( ( prefix8824957607401505554efix_o @ ( F @ X3 ) @ ( F @ Y3 ) )
=> ( X3 = Y3 ) ) ) ) ).
% is_encoding_def
thf(fact_30_opt__comp__sym,axiom,
( prefix454693708527911765comp_o
= ( ^ [X3: option_list_o,Y3: option_list_o] : ( prefix454693708527911765comp_o @ Y3 @ X3 ) ) ) ).
% opt_comp_sym
thf(fact_31_dependent__encoding,axiom,
! [E1: int > option_list_o,E2: int > int > option_list_o] :
( ( prefix3555694663680348106ng_int @ E1 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ ( dom_int_list_o @ E1 ) )
=> ( prefix3555694663680348106ng_int @ ( E2 @ X2 ) ) )
=> ( prefix1871019061350730907nt_int @ ( prefix2353203243457709914nt_int @ E1 @ E2 ) ) ) ) ).
% dependent_encoding
thf(fact_32_dependent__encoding,axiom,
! [E1: int > option_list_o,E2: int > nat > option_list_o] :
( ( prefix3555694663680348106ng_int @ E1 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ ( dom_int_list_o @ E1 ) )
=> ( prefix3558185134189398382ng_nat @ ( E2 @ X2 ) ) )
=> ( prefix6048870080859927615nt_nat @ ( prefix2355693713966760190nt_nat @ E1 @ E2 ) ) ) ) ).
% dependent_encoding
thf(fact_33_dependent__encoding,axiom,
! [E1: nat > option_list_o,E2: nat > int > option_list_o] :
( ( prefix3558185134189398382ng_nat @ E1 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( dom_nat_list_o @ E1 ) )
=> ( prefix3555694663680348106ng_int @ ( E2 @ X2 ) ) )
=> ( prefix871665349449676351at_int @ ( prefix1353849531556655358at_int @ E1 @ E2 ) ) ) ) ).
% dependent_encoding
thf(fact_34_dependent__encoding,axiom,
! [E1: nat > option_list_o,E2: nat > nat > option_list_o] :
( ( prefix3558185134189398382ng_nat @ E1 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( dom_nat_list_o @ E1 ) )
=> ( prefix3558185134189398382ng_nat @ ( E2 @ X2 ) ) )
=> ( prefix5049516368958873059at_nat @ ( prefix1356340002065705634at_nat @ E1 @ E2 ) ) ) ) ).
% dependent_encoding
thf(fact_35_dependent__encoding,axiom,
! [E1: int > option_list_o,E2: int > product_prod_nat_nat > option_list_o] :
( ( prefix3555694663680348106ng_int @ E1 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ ( dom_int_list_o @ E1 ) )
=> ( prefix5049516368958873059at_nat @ ( E2 @ X2 ) ) )
=> ( prefix3763936006142083688at_nat @ ( prefix8798362131150533203at_nat @ E1 @ E2 ) ) ) ) ).
% dependent_encoding
thf(fact_36_dependent__encoding,axiom,
! [E1: int > option_list_o,E2: int > product_prod_nat_int > option_list_o] :
( ( prefix3555694663680348106ng_int @ E1 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ ( dom_int_list_o @ E1 ) )
=> ( prefix871665349449676351at_int @ ( E2 @ X2 ) ) )
=> ( prefix1273852861210356932at_int @ ( prefix4620511111641336495at_int @ E1 @ E2 ) ) ) ) ).
% dependent_encoding
thf(fact_37_dependent__encoding,axiom,
! [E1: int > option_list_o,E2: int > product_prod_int_nat > option_list_o] :
( ( prefix3555694663680348106ng_int @ E1 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ ( dom_int_list_o @ E1 ) )
=> ( prefix6048870080859927615nt_nat @ ( E2 @ X2 ) ) )
=> ( prefix5950857580835105476nt_nat @ ( prefix574343806196811951nt_nat @ E1 @ E2 ) ) ) ) ).
% dependent_encoding
thf(fact_38_dependent__encoding,axiom,
! [E1: int > option_list_o,E2: int > product_prod_int_int > option_list_o] :
( ( prefix3555694663680348106ng_int @ E1 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ ( dom_int_list_o @ E1 ) )
=> ( prefix1871019061350730907nt_int @ ( E2 @ X2 ) ) )
=> ( prefix3460774435903378720nt_int @ ( prefix5619864823542391051nt_int @ E1 @ E2 ) ) ) ) ).
% dependent_encoding
thf(fact_39_dependent__encoding,axiom,
! [E1: nat > option_list_o,E2: nat > product_prod_nat_nat > option_list_o] :
( ( prefix3558185134189398382ng_nat @ E1 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( dom_nat_list_o @ E1 ) )
=> ( prefix5049516368958873059at_nat @ ( E2 @ X2 ) ) )
=> ( prefix726093921677949892at_nat @ ( prefix5760520046686399407at_nat @ E1 @ E2 ) ) ) ) ).
% dependent_encoding
thf(fact_40_dependent__encoding,axiom,
! [E1: nat > option_list_o,E2: nat > product_prod_nat_int > option_list_o] :
( ( prefix3558185134189398382ng_nat @ E1 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ ( dom_nat_list_o @ E1 ) )
=> ( prefix871665349449676351at_int @ ( E2 @ X2 ) ) )
=> ( prefix7459382813600998944at_int @ ( prefix1582669027177202699at_int @ E1 @ E2 ) ) ) ) ).
% dependent_encoding
thf(fact_41_is__encodingI,axiom,
! [E: int > option_list_o] :
( ! [X2: int,X4: list_o,Y2: int,Y4: list_o] :
( ( ( E @ X2 )
= ( some_list_o @ X4 ) )
=> ( ( ( E @ Y2 )
= ( some_list_o @ Y4 ) )
=> ( ( prefix_o @ X4 @ Y4 )
=> ( X2 = Y2 ) ) ) )
=> ( prefix3555694663680348106ng_int @ E ) ) ).
% is_encodingI
thf(fact_42_is__encodingI,axiom,
! [E: nat > option_list_o] :
( ! [X2: nat,X4: list_o,Y2: nat,Y4: list_o] :
( ( ( E @ X2 )
= ( some_list_o @ X4 ) )
=> ( ( ( E @ Y2 )
= ( some_list_o @ Y4 ) )
=> ( ( prefix_o @ X4 @ Y4 )
=> ( X2 = Y2 ) ) ) )
=> ( prefix3558185134189398382ng_nat @ E ) ) ).
% is_encodingI
thf(fact_43_is__encodingI,axiom,
! [E: product_prod_nat_nat > option_list_o] :
( ! [X2: product_prod_nat_nat,X4: list_o,Y2: product_prod_nat_nat,Y4: list_o] :
( ( ( E @ X2 )
= ( some_list_o @ X4 ) )
=> ( ( ( E @ Y2 )
= ( some_list_o @ Y4 ) )
=> ( ( prefix_o @ X4 @ Y4 )
=> ( X2 = Y2 ) ) ) )
=> ( prefix5049516368958873059at_nat @ E ) ) ).
% is_encodingI
thf(fact_44_is__encodingI,axiom,
! [E: product_prod_nat_int > option_list_o] :
( ! [X2: product_prod_nat_int,X4: list_o,Y2: product_prod_nat_int,Y4: list_o] :
( ( ( E @ X2 )
= ( some_list_o @ X4 ) )
=> ( ( ( E @ Y2 )
= ( some_list_o @ Y4 ) )
=> ( ( prefix_o @ X4 @ Y4 )
=> ( X2 = Y2 ) ) ) )
=> ( prefix871665349449676351at_int @ E ) ) ).
% is_encodingI
thf(fact_45_is__encodingI,axiom,
! [E: product_prod_int_nat > option_list_o] :
( ! [X2: product_prod_int_nat,X4: list_o,Y2: product_prod_int_nat,Y4: list_o] :
( ( ( E @ X2 )
= ( some_list_o @ X4 ) )
=> ( ( ( E @ Y2 )
= ( some_list_o @ Y4 ) )
=> ( ( prefix_o @ X4 @ Y4 )
=> ( X2 = Y2 ) ) ) )
=> ( prefix6048870080859927615nt_nat @ E ) ) ).
% is_encodingI
thf(fact_46_is__encodingI,axiom,
! [E: product_prod_int_int > option_list_o] :
( ! [X2: product_prod_int_int,X4: list_o,Y2: product_prod_int_int,Y4: list_o] :
( ( ( E @ X2 )
= ( some_list_o @ X4 ) )
=> ( ( ( E @ Y2 )
= ( some_list_o @ Y4 ) )
=> ( ( prefix_o @ X4 @ Y4 )
=> ( X2 = Y2 ) ) ) )
=> ( prefix1871019061350730907nt_int @ E ) ) ).
% is_encodingI
thf(fact_47_is__encodingI,axiom,
! [E: produc6945250483304103390nt_int > option_list_o] :
( ! [X2: produc6945250483304103390nt_int,X4: list_o,Y2: produc6945250483304103390nt_int,Y4: list_o] :
( ( ( E @ X2 )
= ( some_list_o @ X4 ) )
=> ( ( ( E @ Y2 )
= ( some_list_o @ Y4 ) )
=> ( ( prefix_o @ X4 @ Y4 )
=> ( X2 = Y2 ) ) ) )
=> ( prefix422932351439244924nt_int @ E ) ) ).
% is_encodingI
thf(fact_48_is__encodingI,axiom,
! [E: produc1062882101152166346at_nat > option_list_o] :
( ! [X2: produc1062882101152166346at_nat,X4: list_o,Y2: produc1062882101152166346at_nat,Y4: list_o] :
( ( ( E @ X2 )
= ( some_list_o @ X4 ) )
=> ( ( ( E @ Y2 )
= ( some_list_o @ Y4 ) )
=> ( ( prefix_o @ X4 @ Y4 )
=> ( X2 = Y2 ) ) ) )
=> ( prefix3763936006142083688at_nat @ E ) ) ).
% is_encodingI
thf(fact_49_is__encodingI,axiom,
! [E: produc7796170993075215398at_int > option_list_o] :
( ! [X2: produc7796170993075215398at_int,X4: list_o,Y2: produc7796170993075215398at_int,Y4: list_o] :
( ( ( E @ X2 )
= ( some_list_o @ X4 ) )
=> ( ( ( E @ Y2 )
= ( some_list_o @ Y4 ) )
=> ( ( prefix_o @ X4 @ Y4 )
=> ( X2 = Y2 ) ) ) )
=> ( prefix1273852861210356932at_int @ E ) ) ).
% is_encodingI
thf(fact_50_is__encodingI,axiom,
! [E: produc3249803675845188134nt_nat > option_list_o] :
( ! [X2: produc3249803675845188134nt_nat,X4: list_o,Y2: produc3249803675845188134nt_nat,Y4: list_o] :
( ( ( E @ X2 )
= ( some_list_o @ X4 ) )
=> ( ( ( E @ Y2 )
= ( some_list_o @ Y4 ) )
=> ( ( prefix_o @ X4 @ Y4 )
=> ( X2 = Y2 ) ) ) )
=> ( prefix5950857580835105476nt_nat @ E ) ) ).
% is_encodingI
thf(fact_51_encoding__imp__inj,axiom,
! [F2: int > option_list_o] :
( ( prefix3555694663680348106ng_int @ F2 )
=> ( inj_on203032006962319749list_o @ F2 @ ( dom_int_list_o @ F2 ) ) ) ).
% encoding_imp_inj
thf(fact_52_encoding__imp__inj,axiom,
! [F2: nat > option_list_o] :
( ( prefix3558185134189398382ng_nat @ F2 )
=> ( inj_on1630180835328728801list_o @ F2 @ ( dom_nat_list_o @ F2 ) ) ) ).
% encoding_imp_inj
thf(fact_53_encoding__imp__inj,axiom,
! [F2: product_prod_nat_nat > option_list_o] :
( ( prefix5049516368958873059at_nat @ F2 )
=> ( inj_on8168085447206903444list_o @ F2 @ ( dom_Pr5202128127435093295list_o @ F2 ) ) ) ).
% encoding_imp_inj
thf(fact_54_encoding__imp__inj,axiom,
! [F2: product_prod_nat_int > option_list_o] :
( ( prefix871665349449676351at_int @ F2 )
=> ( inj_on7977379531974639416list_o @ F2 @ ( dom_Pr7669787946167753683list_o @ F2 ) ) ) ).
% encoding_imp_inj
thf(fact_55_encoding__imp__inj,axiom,
! [F2: product_prod_int_nat > option_list_o] :
( ( prefix6048870080859927615nt_nat @ F2 )
=> ( inj_on622226642133877048list_o @ F2 @ ( dom_Pr5309327336824135123list_o @ F2 ) ) ) ).
% encoding_imp_inj
thf(fact_56_encoding__imp__inj,axiom,
! [F2: product_prod_int_int > option_list_o] :
( ( prefix1871019061350730907nt_int @ F2 )
=> ( inj_on431520726901613020list_o @ F2 @ ( dom_Pr7776987155556795511list_o @ F2 ) ) ) ).
% encoding_imp_inj
thf(fact_57_encoding__imp__inj,axiom,
! [F2: list_o > option_list_o] :
( ( prefix1117678025990076032list_o @ F2 )
=> ( inj_on2239224958054982839list_o @ F2 @ ( dom_list_o_list_o @ F2 ) ) ) ).
% encoding_imp_inj
thf(fact_58_encoding__imp__inj,axiom,
! [F2: produc6945250483304103390nt_int > option_list_o] :
( ( prefix422932351439244924nt_int @ F2 )
=> ( inj_on7585825339795045459list_o @ F2 @ ( dom_Pr5162500942570483308list_o @ F2 ) ) ) ).
% encoding_imp_inj
thf(fact_59_encoding__imp__inj,axiom,
! [F2: produc1062882101152166346at_nat > option_list_o] :
( ( prefix3763936006142083688at_nat @ F2 )
=> ( inj_on4526473911404666215list_o @ F2 @ ( dom_Pr6680471568419110272list_o @ F2 ) ) ) ).
% encoding_imp_inj
thf(fact_60_encoding__imp__inj,axiom,
! [F2: produc7796170993075215398at_int > option_list_o] :
( ( prefix1273852861210356932at_int @ F2 )
=> ( inj_on4764051053055241227list_o @ F2 @ ( dom_Pr3796893515621290532list_o @ F2 ) ) ) ).
% encoding_imp_inj
thf(fact_61_opt__prefix_Oelims_I2_J,axiom,
! [X: option_list_o,Xa: option_list_o] :
( ( prefix8824957607401505554efix_o @ X @ Xa )
=> ~ ! [X2: list_o] :
( ( X
= ( some_list_o @ X2 ) )
=> ! [Y2: list_o] :
( ( Xa
= ( some_list_o @ Y2 ) )
=> ~ ( prefix_o @ X2 @ Y2 ) ) ) ) ).
% opt_prefix.elims(2)
thf(fact_62_opt__prefix_Osimps_I1_J,axiom,
! [X: list_o,Y: list_o] :
( ( prefix8824957607401505554efix_o @ ( some_list_o @ X ) @ ( some_list_o @ Y ) )
= ( prefix_o @ X @ Y ) ) ).
% opt_prefix.simps(1)
thf(fact_63_opt__comp__def,axiom,
( prefix454693708527911765comp_o
= ( ^ [X3: option_list_o,Y3: option_list_o] :
( ( prefix8824957607401505554efix_o @ X3 @ Y3 )
| ( prefix8824957607401505554efix_o @ Y3 @ X3 ) ) ) ) ).
% opt_comp_def
thf(fact_64_option_Oinject,axiom,
! [X22: int,Y22: int] :
( ( ( some_int @ X22 )
= ( some_int @ Y22 ) )
= ( X22 = Y22 ) ) ).
% option.inject
thf(fact_65_option_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( some_nat @ X22 )
= ( some_nat @ Y22 ) )
= ( X22 = Y22 ) ) ).
% option.inject
thf(fact_66_option_Oinject,axiom,
! [X22: option_list_o,Y22: option_list_o] :
( ( ( some_option_list_o @ X22 )
= ( some_option_list_o @ Y22 ) )
= ( X22 = Y22 ) ) ).
% option.inject
thf(fact_67_option_Oinject,axiom,
! [X22: list_o,Y22: list_o] :
( ( ( some_list_o @ X22 )
= ( some_list_o @ Y22 ) )
= ( X22 = Y22 ) ) ).
% option.inject
thf(fact_68_inj__Some,axiom,
! [A: set_int] : ( inj_on8482933962677695643on_int @ some_int @ A ) ).
% inj_Some
thf(fact_69_inj__Some,axiom,
! [A: set_nat] : ( inj_on1310046215330054115on_nat @ some_nat @ A ) ).
% inj_Some
thf(fact_70_inj__Some,axiom,
! [A: set_option_list_o] : ( inj_on1889704059425077559list_o @ some_option_list_o @ A ) ).
% inj_Some
thf(fact_71_inj__Some,axiom,
! [A: set_list_o] : ( inj_on2239224958054982839list_o @ some_list_o @ A ) ).
% inj_Some
thf(fact_72_prefix__order_Oorder__refl,axiom,
! [X: list_o] : ( prefix_o @ X @ X ) ).
% prefix_order.order_refl
thf(fact_73_prefix__order_Odual__order_Orefl,axiom,
! [A2: list_o] : ( prefix_o @ A2 @ A2 ) ).
% prefix_order.dual_order.refl
thf(fact_74_domD,axiom,
! [A2: nat,M: nat > option_list_o] :
( ( member_nat @ A2 @ ( dom_nat_list_o @ M ) )
=> ? [B: list_o] :
( ( M @ A2 )
= ( some_list_o @ B ) ) ) ).
% domD
thf(fact_75_domD,axiom,
! [A2: int,M: int > option_list_o] :
( ( member_int @ A2 @ ( dom_int_list_o @ M ) )
=> ? [B: list_o] :
( ( M @ A2 )
= ( some_list_o @ B ) ) ) ).
% domD
thf(fact_76_domD,axiom,
! [A2: nat,M: nat > option_int] :
( ( member_nat @ A2 @ ( dom_nat_int @ M ) )
=> ? [B: int] :
( ( M @ A2 )
= ( some_int @ B ) ) ) ).
% domD
thf(fact_77_domD,axiom,
! [A2: int,M: int > option_int] :
( ( member_int @ A2 @ ( dom_int_int @ M ) )
=> ? [B: int] :
( ( M @ A2 )
= ( some_int @ B ) ) ) ).
% domD
thf(fact_78_domD,axiom,
! [A2: nat,M: nat > option_nat] :
( ( member_nat @ A2 @ ( dom_nat_nat @ M ) )
=> ? [B: nat] :
( ( M @ A2 )
= ( some_nat @ B ) ) ) ).
% domD
thf(fact_79_domD,axiom,
! [A2: int,M: int > option_nat] :
( ( member_int @ A2 @ ( dom_int_nat @ M ) )
=> ? [B: nat] :
( ( M @ A2 )
= ( some_nat @ B ) ) ) ).
% domD
thf(fact_80_domD,axiom,
! [A2: option_nat,M: option_nat > option_int] :
( ( member_option_nat @ A2 @ ( dom_option_nat_int @ M ) )
=> ? [B: int] :
( ( M @ A2 )
= ( some_int @ B ) ) ) ).
% domD
thf(fact_81_domD,axiom,
! [A2: option_int,M: option_int > option_int] :
( ( member_option_int @ A2 @ ( dom_option_int_int @ M ) )
=> ? [B: int] :
( ( M @ A2 )
= ( some_int @ B ) ) ) ).
% domD
thf(fact_82_domD,axiom,
! [A2: list_o,M: list_o > option_int] :
( ( member_list_o @ A2 @ ( dom_list_o_int @ M ) )
=> ? [B: int] :
( ( M @ A2 )
= ( some_int @ B ) ) ) ).
% domD
thf(fact_83_domD,axiom,
! [A2: set_nat,M: set_nat > option_int] :
( ( member_set_nat @ A2 @ ( dom_set_nat_int @ M ) )
=> ? [B: int] :
( ( M @ A2 )
= ( some_int @ B ) ) ) ).
% domD
thf(fact_84_domI,axiom,
! [M: nat > option_list_o,A2: nat,B2: list_o] :
( ( ( M @ A2 )
= ( some_list_o @ B2 ) )
=> ( member_nat @ A2 @ ( dom_nat_list_o @ M ) ) ) ).
% domI
thf(fact_85_domI,axiom,
! [M: int > option_list_o,A2: int,B2: list_o] :
( ( ( M @ A2 )
= ( some_list_o @ B2 ) )
=> ( member_int @ A2 @ ( dom_int_list_o @ M ) ) ) ).
% domI
thf(fact_86_domI,axiom,
! [M: nat > option_int,A2: nat,B2: int] :
( ( ( M @ A2 )
= ( some_int @ B2 ) )
=> ( member_nat @ A2 @ ( dom_nat_int @ M ) ) ) ).
% domI
thf(fact_87_domI,axiom,
! [M: int > option_int,A2: int,B2: int] :
( ( ( M @ A2 )
= ( some_int @ B2 ) )
=> ( member_int @ A2 @ ( dom_int_int @ M ) ) ) ).
% domI
thf(fact_88_domI,axiom,
! [M: nat > option_nat,A2: nat,B2: nat] :
( ( ( M @ A2 )
= ( some_nat @ B2 ) )
=> ( member_nat @ A2 @ ( dom_nat_nat @ M ) ) ) ).
% domI
thf(fact_89_domI,axiom,
! [M: int > option_nat,A2: int,B2: nat] :
( ( ( M @ A2 )
= ( some_nat @ B2 ) )
=> ( member_int @ A2 @ ( dom_int_nat @ M ) ) ) ).
% domI
thf(fact_90_domI,axiom,
! [M: option_nat > option_int,A2: option_nat,B2: int] :
( ( ( M @ A2 )
= ( some_int @ B2 ) )
=> ( member_option_nat @ A2 @ ( dom_option_nat_int @ M ) ) ) ).
% domI
thf(fact_91_domI,axiom,
! [M: option_int > option_int,A2: option_int,B2: int] :
( ( ( M @ A2 )
= ( some_int @ B2 ) )
=> ( member_option_int @ A2 @ ( dom_option_int_int @ M ) ) ) ).
% domI
thf(fact_92_domI,axiom,
! [M: list_o > option_int,A2: list_o,B2: int] :
( ( ( M @ A2 )
= ( some_int @ B2 ) )
=> ( member_list_o @ A2 @ ( dom_list_o_int @ M ) ) ) ).
% domI
thf(fact_93_domI,axiom,
! [M: set_nat > option_int,A2: set_nat,B2: int] :
( ( ( M @ A2 )
= ( some_int @ B2 ) )
=> ( member_set_nat @ A2 @ ( dom_set_nat_int @ M ) ) ) ).
% domI
thf(fact_94_encode__unary__nat__prefix__free,axiom,
! [U: nat,V: nat] :
( ( prefix_o @ ( prefix979155009907353096ry_nat @ U ) @ ( prefix979155009907353096ry_nat @ V ) )
=> ( U = V ) ) ).
% encode_unary_nat_prefix_free
thf(fact_95_opt__prefix_Oelims_I3_J,axiom,
! [X: option_list_o,Xa: option_list_o] :
( ~ ( prefix8824957607401505554efix_o @ X @ Xa )
=> ( ! [X2: list_o] :
( ( X
= ( some_list_o @ X2 ) )
=> ! [Y2: list_o] :
( ( Xa
= ( some_list_o @ Y2 ) )
=> ( prefix_o @ X2 @ Y2 ) ) )
=> ( ( X != none_list_o )
=> ( Xa = none_list_o ) ) ) ) ).
% opt_prefix.elims(3)
thf(fact_96_opt__prefix_Oelims_I1_J,axiom,
! [X: option_list_o,Xa: option_list_o,Y: $o] :
( ( ( prefix8824957607401505554efix_o @ X @ Xa )
= Y )
=> ( ! [X2: list_o] :
( ( X
= ( some_list_o @ X2 ) )
=> ! [Y2: list_o] :
( ( Xa
= ( some_list_o @ Y2 ) )
=> ( Y
= ( ~ ( prefix_o @ X2 @ Y2 ) ) ) ) )
=> ( ( ( X = none_list_o )
=> Y )
=> ~ ( ( Xa = none_list_o )
=> Y ) ) ) ) ).
% opt_prefix.elims(1)
thf(fact_97_prefix__order_Otrans,axiom,
! [A2: list_o,B2: list_o,C: list_o] :
( ( prefix_o @ A2 @ B2 )
=> ( ( prefix_o @ B2 @ C )
=> ( prefix_o @ A2 @ C ) ) ) ).
% prefix_order.trans
thf(fact_98_prefix__order_Oeq__iff,axiom,
( ( ^ [Y5: list_o,Z: list_o] : ( Y5 = Z ) )
= ( ^ [A3: list_o,B3: list_o] :
( ( prefix_o @ A3 @ B3 )
& ( prefix_o @ B3 @ A3 ) ) ) ) ).
% prefix_order.eq_iff
thf(fact_99_prefix__order_Oantisym,axiom,
! [A2: list_o,B2: list_o] :
( ( prefix_o @ A2 @ B2 )
=> ( ( prefix_o @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% prefix_order.antisym
thf(fact_100_not__Some__eq,axiom,
! [X: option_int] :
( ( ! [Y3: int] :
( X
!= ( some_int @ Y3 ) ) )
= ( X = none_int ) ) ).
% not_Some_eq
thf(fact_101_not__Some__eq,axiom,
! [X: option_nat] :
( ( ! [Y3: nat] :
( X
!= ( some_nat @ Y3 ) ) )
= ( X = none_nat ) ) ).
% not_Some_eq
thf(fact_102_not__Some__eq,axiom,
! [X: option_option_list_o] :
( ( ! [Y3: option_list_o] :
( X
!= ( some_option_list_o @ Y3 ) ) )
= ( X = none_option_list_o ) ) ).
% not_Some_eq
thf(fact_103_not__Some__eq,axiom,
! [X: option_list_o] :
( ( ! [Y3: list_o] :
( X
!= ( some_list_o @ Y3 ) ) )
= ( X = none_list_o ) ) ).
% not_Some_eq
thf(fact_104_not__None__eq,axiom,
! [X: option_int] :
( ( X != none_int )
= ( ? [Y3: int] :
( X
= ( some_int @ Y3 ) ) ) ) ).
% not_None_eq
thf(fact_105_not__None__eq,axiom,
! [X: option_nat] :
( ( X != none_nat )
= ( ? [Y3: nat] :
( X
= ( some_nat @ Y3 ) ) ) ) ).
% not_None_eq
thf(fact_106_not__None__eq,axiom,
! [X: option_option_list_o] :
( ( X != none_option_list_o )
= ( ? [Y3: option_list_o] :
( X
= ( some_option_list_o @ Y3 ) ) ) ) ).
% not_None_eq
thf(fact_107_not__None__eq,axiom,
! [X: option_list_o] :
( ( X != none_list_o )
= ( ? [Y3: list_o] :
( X
= ( some_list_o @ Y3 ) ) ) ) ).
% not_None_eq
thf(fact_108_combine__options__cases,axiom,
! [X: option_list_o,P: option_list_o > option_list_o > $o,Y: option_list_o] :
( ( ( X = none_list_o )
=> ( P @ X @ Y ) )
=> ( ( ( Y = none_list_o )
=> ( P @ X @ Y ) )
=> ( ! [A4: list_o,B: list_o] :
( ( X
= ( some_list_o @ A4 ) )
=> ( ( Y
= ( some_list_o @ B ) )
=> ( P @ X @ Y ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% combine_options_cases
thf(fact_109_combine__options__cases,axiom,
! [X: option_int,P: option_int > option_int > $o,Y: option_int] :
( ( ( X = none_int )
=> ( P @ X @ Y ) )
=> ( ( ( Y = none_int )
=> ( P @ X @ Y ) )
=> ( ! [A4: int,B: int] :
( ( X
= ( some_int @ A4 ) )
=> ( ( Y
= ( some_int @ B ) )
=> ( P @ X @ Y ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% combine_options_cases
thf(fact_110_combine__options__cases,axiom,
! [X: option_int,P: option_int > option_nat > $o,Y: option_nat] :
( ( ( X = none_int )
=> ( P @ X @ Y ) )
=> ( ( ( Y = none_nat )
=> ( P @ X @ Y ) )
=> ( ! [A4: int,B: nat] :
( ( X
= ( some_int @ A4 ) )
=> ( ( Y
= ( some_nat @ B ) )
=> ( P @ X @ Y ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% combine_options_cases
thf(fact_111_combine__options__cases,axiom,
! [X: option_nat,P: option_nat > option_int > $o,Y: option_int] :
( ( ( X = none_nat )
=> ( P @ X @ Y ) )
=> ( ( ( Y = none_int )
=> ( P @ X @ Y ) )
=> ( ! [A4: nat,B: int] :
( ( X
= ( some_nat @ A4 ) )
=> ( ( Y
= ( some_int @ B ) )
=> ( P @ X @ Y ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% combine_options_cases
thf(fact_112_combine__options__cases,axiom,
! [X: option_nat,P: option_nat > option_nat > $o,Y: option_nat] :
( ( ( X = none_nat )
=> ( P @ X @ Y ) )
=> ( ( ( Y = none_nat )
=> ( P @ X @ Y ) )
=> ( ! [A4: nat,B: nat] :
( ( X
= ( some_nat @ A4 ) )
=> ( ( Y
= ( some_nat @ B ) )
=> ( P @ X @ Y ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% combine_options_cases
thf(fact_113_combine__options__cases,axiom,
! [X: option_list_o,P: option_list_o > option_int > $o,Y: option_int] :
( ( ( X = none_list_o )
=> ( P @ X @ Y ) )
=> ( ( ( Y = none_int )
=> ( P @ X @ Y ) )
=> ( ! [A4: list_o,B: int] :
( ( X
= ( some_list_o @ A4 ) )
=> ( ( Y
= ( some_int @ B ) )
=> ( P @ X @ Y ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% combine_options_cases
thf(fact_114_combine__options__cases,axiom,
! [X: option_list_o,P: option_list_o > option_nat > $o,Y: option_nat] :
( ( ( X = none_list_o )
=> ( P @ X @ Y ) )
=> ( ( ( Y = none_nat )
=> ( P @ X @ Y ) )
=> ( ! [A4: list_o,B: nat] :
( ( X
= ( some_list_o @ A4 ) )
=> ( ( Y
= ( some_nat @ B ) )
=> ( P @ X @ Y ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% combine_options_cases
thf(fact_115_combine__options__cases,axiom,
! [X: option_int,P: option_int > option_list_o > $o,Y: option_list_o] :
( ( ( X = none_int )
=> ( P @ X @ Y ) )
=> ( ( ( Y = none_list_o )
=> ( P @ X @ Y ) )
=> ( ! [A4: int,B: list_o] :
( ( X
= ( some_int @ A4 ) )
=> ( ( Y
= ( some_list_o @ B ) )
=> ( P @ X @ Y ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% combine_options_cases
thf(fact_116_combine__options__cases,axiom,
! [X: option_nat,P: option_nat > option_list_o > $o,Y: option_list_o] :
( ( ( X = none_nat )
=> ( P @ X @ Y ) )
=> ( ( ( Y = none_list_o )
=> ( P @ X @ Y ) )
=> ( ! [A4: nat,B: list_o] :
( ( X
= ( some_nat @ A4 ) )
=> ( ( Y
= ( some_list_o @ B ) )
=> ( P @ X @ Y ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% combine_options_cases
thf(fact_117_combine__options__cases,axiom,
! [X: option_int,P: option_int > option_option_list_o > $o,Y: option_option_list_o] :
( ( ( X = none_int )
=> ( P @ X @ Y ) )
=> ( ( ( Y = none_option_list_o )
=> ( P @ X @ Y ) )
=> ( ! [A4: int,B: option_list_o] :
( ( X
= ( some_int @ A4 ) )
=> ( ( Y
= ( some_option_list_o @ B ) )
=> ( P @ X @ Y ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% combine_options_cases
thf(fact_118_split__option__all,axiom,
( ( ^ [P2: option_int > $o] :
! [X5: option_int] : ( P2 @ X5 ) )
= ( ^ [P3: option_int > $o] :
( ( P3 @ none_int )
& ! [X3: int] : ( P3 @ ( some_int @ X3 ) ) ) ) ) ).
% split_option_all
thf(fact_119_split__option__all,axiom,
( ( ^ [P2: option_nat > $o] :
! [X5: option_nat] : ( P2 @ X5 ) )
= ( ^ [P3: option_nat > $o] :
( ( P3 @ none_nat )
& ! [X3: nat] : ( P3 @ ( some_nat @ X3 ) ) ) ) ) ).
% split_option_all
thf(fact_120_split__option__all,axiom,
( ( ^ [P2: option_option_list_o > $o] :
! [X5: option_option_list_o] : ( P2 @ X5 ) )
= ( ^ [P3: option_option_list_o > $o] :
( ( P3 @ none_option_list_o )
& ! [X3: option_list_o] : ( P3 @ ( some_option_list_o @ X3 ) ) ) ) ) ).
% split_option_all
thf(fact_121_split__option__all,axiom,
( ( ^ [P2: option_list_o > $o] :
! [X5: option_list_o] : ( P2 @ X5 ) )
= ( ^ [P3: option_list_o > $o] :
( ( P3 @ none_list_o )
& ! [X3: list_o] : ( P3 @ ( some_list_o @ X3 ) ) ) ) ) ).
% split_option_all
thf(fact_122_split__option__ex,axiom,
( ( ^ [P2: option_int > $o] :
? [X5: option_int] : ( P2 @ X5 ) )
= ( ^ [P3: option_int > $o] :
( ( P3 @ none_int )
| ? [X3: int] : ( P3 @ ( some_int @ X3 ) ) ) ) ) ).
% split_option_ex
thf(fact_123_split__option__ex,axiom,
( ( ^ [P2: option_nat > $o] :
? [X5: option_nat] : ( P2 @ X5 ) )
= ( ^ [P3: option_nat > $o] :
( ( P3 @ none_nat )
| ? [X3: nat] : ( P3 @ ( some_nat @ X3 ) ) ) ) ) ).
% split_option_ex
thf(fact_124_split__option__ex,axiom,
( ( ^ [P2: option_option_list_o > $o] :
? [X5: option_option_list_o] : ( P2 @ X5 ) )
= ( ^ [P3: option_option_list_o > $o] :
( ( P3 @ none_option_list_o )
| ? [X3: option_list_o] : ( P3 @ ( some_option_list_o @ X3 ) ) ) ) ) ).
% split_option_ex
thf(fact_125_split__option__ex,axiom,
( ( ^ [P2: option_list_o > $o] :
? [X5: option_list_o] : ( P2 @ X5 ) )
= ( ^ [P3: option_list_o > $o] :
( ( P3 @ none_list_o )
| ? [X3: list_o] : ( P3 @ ( some_list_o @ X3 ) ) ) ) ) ).
% split_option_ex
thf(fact_126_option_Oexhaust,axiom,
! [Y: option_int] :
( ( Y != none_int )
=> ~ ! [X23: int] :
( Y
!= ( some_int @ X23 ) ) ) ).
% option.exhaust
thf(fact_127_option_Oexhaust,axiom,
! [Y: option_nat] :
( ( Y != none_nat )
=> ~ ! [X23: nat] :
( Y
!= ( some_nat @ X23 ) ) ) ).
% option.exhaust
thf(fact_128_option_Oexhaust,axiom,
! [Y: option_option_list_o] :
( ( Y != none_option_list_o )
=> ~ ! [X23: option_list_o] :
( Y
!= ( some_option_list_o @ X23 ) ) ) ).
% option.exhaust
thf(fact_129_option_Oexhaust,axiom,
! [Y: option_list_o] :
( ( Y != none_list_o )
=> ~ ! [X23: list_o] :
( Y
!= ( some_list_o @ X23 ) ) ) ).
% option.exhaust
thf(fact_130_option_OdiscI,axiom,
! [Option: option_int,X22: int] :
( ( Option
= ( some_int @ X22 ) )
=> ( Option != none_int ) ) ).
% option.discI
thf(fact_131_option_OdiscI,axiom,
! [Option: option_nat,X22: nat] :
( ( Option
= ( some_nat @ X22 ) )
=> ( Option != none_nat ) ) ).
% option.discI
thf(fact_132_option_OdiscI,axiom,
! [Option: option_option_list_o,X22: option_list_o] :
( ( Option
= ( some_option_list_o @ X22 ) )
=> ( Option != none_option_list_o ) ) ).
% option.discI
thf(fact_133_option_OdiscI,axiom,
! [Option: option_list_o,X22: list_o] :
( ( Option
= ( some_list_o @ X22 ) )
=> ( Option != none_list_o ) ) ).
% option.discI
thf(fact_134_option_Odistinct_I1_J,axiom,
! [X22: int] :
( none_int
!= ( some_int @ X22 ) ) ).
% option.distinct(1)
thf(fact_135_option_Odistinct_I1_J,axiom,
! [X22: nat] :
( none_nat
!= ( some_nat @ X22 ) ) ).
% option.distinct(1)
thf(fact_136_option_Odistinct_I1_J,axiom,
! [X22: option_list_o] :
( none_option_list_o
!= ( some_option_list_o @ X22 ) ) ).
% option.distinct(1)
thf(fact_137_option_Odistinct_I1_J,axiom,
! [X22: list_o] :
( none_list_o
!= ( some_list_o @ X22 ) ) ).
% option.distinct(1)
thf(fact_138_domIff,axiom,
! [A2: nat,M: nat > option_list_o] :
( ( member_nat @ A2 @ ( dom_nat_list_o @ M ) )
= ( ( M @ A2 )
!= none_list_o ) ) ).
% domIff
thf(fact_139_domIff,axiom,
! [A2: int,M: int > option_list_o] :
( ( member_int @ A2 @ ( dom_int_list_o @ M ) )
= ( ( M @ A2 )
!= none_list_o ) ) ).
% domIff
thf(fact_140_domIff,axiom,
! [A2: nat,M: nat > option_int] :
( ( member_nat @ A2 @ ( dom_nat_int @ M ) )
= ( ( M @ A2 )
!= none_int ) ) ).
% domIff
thf(fact_141_domIff,axiom,
! [A2: int,M: int > option_int] :
( ( member_int @ A2 @ ( dom_int_int @ M ) )
= ( ( M @ A2 )
!= none_int ) ) ).
% domIff
thf(fact_142_domIff,axiom,
! [A2: nat,M: nat > option_nat] :
( ( member_nat @ A2 @ ( dom_nat_nat @ M ) )
= ( ( M @ A2 )
!= none_nat ) ) ).
% domIff
thf(fact_143_domIff,axiom,
! [A2: int,M: int > option_nat] :
( ( member_int @ A2 @ ( dom_int_nat @ M ) )
= ( ( M @ A2 )
!= none_nat ) ) ).
% domIff
thf(fact_144_domIff,axiom,
! [A2: option_nat,M: option_nat > option_int] :
( ( member_option_nat @ A2 @ ( dom_option_nat_int @ M ) )
= ( ( M @ A2 )
!= none_int ) ) ).
% domIff
thf(fact_145_domIff,axiom,
! [A2: option_int,M: option_int > option_int] :
( ( member_option_int @ A2 @ ( dom_option_int_int @ M ) )
= ( ( M @ A2 )
!= none_int ) ) ).
% domIff
thf(fact_146_domIff,axiom,
! [A2: list_o,M: list_o > option_int] :
( ( member_list_o @ A2 @ ( dom_list_o_int @ M ) )
= ( ( M @ A2 )
!= none_int ) ) ).
% domIff
thf(fact_147_domIff,axiom,
! [A2: set_nat,M: set_nat > option_int] :
( ( member_set_nat @ A2 @ ( dom_set_nat_int @ M ) )
= ( ( M @ A2 )
!= none_int ) ) ).
% domIff
thf(fact_148_bit__count_Ocases,axiom,
! [X: option_list_o] :
( ( X != none_list_o )
=> ~ ! [X2: list_o] :
( X
!= ( some_list_o @ X2 ) ) ) ).
% bit_count.cases
thf(fact_149_opt__prefix_Osimps_I2_J,axiom,
! [Uv: option_list_o] :
~ ( prefix8824957607401505554efix_o @ none_list_o @ Uv ) ).
% opt_prefix.simps(2)
thf(fact_150_opt__prefix_Osimps_I3_J,axiom,
! [Uu: option_list_o] :
~ ( prefix8824957607401505554efix_o @ Uu @ none_list_o ) ).
% opt_prefix.simps(3)
thf(fact_151_prefix__same__cases,axiom,
! [Xs_1: list_o,Ys: list_o,Xs_2: list_o] :
( ( prefix_o @ Xs_1 @ Ys )
=> ( ( prefix_o @ Xs_2 @ Ys )
=> ( ( prefix_o @ Xs_1 @ Xs_2 )
| ( prefix_o @ Xs_2 @ Xs_1 ) ) ) ) ).
% prefix_same_cases
thf(fact_152_prefix__order_Odual__order_Oantisym,axiom,
! [B2: list_o,A2: list_o] :
( ( prefix_o @ B2 @ A2 )
=> ( ( prefix_o @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% prefix_order.dual_order.antisym
thf(fact_153_prefix__order_Odual__order_Oeq__iff,axiom,
( ( ^ [Y5: list_o,Z: list_o] : ( Y5 = Z ) )
= ( ^ [A3: list_o,B3: list_o] :
( ( prefix_o @ B3 @ A3 )
& ( prefix_o @ A3 @ B3 ) ) ) ) ).
% prefix_order.dual_order.eq_iff
thf(fact_154_prefix__order_Odual__order_Otrans,axiom,
! [B2: list_o,A2: list_o,C: list_o] :
( ( prefix_o @ B2 @ A2 )
=> ( ( prefix_o @ C @ B2 )
=> ( prefix_o @ C @ A2 ) ) ) ).
% prefix_order.dual_order.trans
thf(fact_155_prefix__order_Oord__le__eq__trans,axiom,
! [A2: list_o,B2: list_o,C: list_o] :
( ( prefix_o @ A2 @ B2 )
=> ( ( B2 = C )
=> ( prefix_o @ A2 @ C ) ) ) ).
% prefix_order.ord_le_eq_trans
thf(fact_156_prefix__order_Oord__eq__le__trans,axiom,
! [A2: list_o,B2: list_o,C: list_o] :
( ( A2 = B2 )
=> ( ( prefix_o @ B2 @ C )
=> ( prefix_o @ A2 @ C ) ) ) ).
% prefix_order.ord_eq_le_trans
thf(fact_157_mem__Collect__eq,axiom,
! [A2: option_nat,P: option_nat > $o] :
( ( member_option_nat @ A2 @ ( collect_option_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_158_mem__Collect__eq,axiom,
! [A2: option_int,P: option_int > $o] :
( ( member_option_int @ A2 @ ( collect_option_int @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_159_mem__Collect__eq,axiom,
! [A2: list_o,P: list_o > $o] :
( ( member_list_o @ A2 @ ( collect_list_o @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_160_mem__Collect__eq,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( member_set_nat @ A2 @ ( collect_set_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_161_mem__Collect__eq,axiom,
! [A2: set_int,P: set_int > $o] :
( ( member_set_int @ A2 @ ( collect_set_int @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_162_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_163_mem__Collect__eq,axiom,
! [A2: int,P: int > $o] :
( ( member_int @ A2 @ ( collect_int @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_164_Collect__mem__eq,axiom,
! [A: set_option_nat] :
( ( collect_option_nat
@ ^ [X3: option_nat] : ( member_option_nat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_165_Collect__mem__eq,axiom,
! [A: set_option_int] :
( ( collect_option_int
@ ^ [X3: option_int] : ( member_option_int @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_166_Collect__mem__eq,axiom,
! [A: set_list_o] :
( ( collect_list_o
@ ^ [X3: list_o] : ( member_list_o @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_167_Collect__mem__eq,axiom,
! [A: set_set_nat] :
( ( collect_set_nat
@ ^ [X3: set_nat] : ( member_set_nat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_168_Collect__mem__eq,axiom,
! [A: set_set_int] :
( ( collect_set_int
@ ^ [X3: set_int] : ( member_set_int @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_169_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X3: nat] : ( member_nat @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_170_Collect__mem__eq,axiom,
! [A: set_int] :
( ( collect_int
@ ^ [X3: int] : ( member_int @ X3 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_171_Collect__cong,axiom,
! [P: int > $o,Q: int > $o] :
( ! [X2: int] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_int @ P )
= ( collect_int @ Q ) ) ) ).
% Collect_cong
thf(fact_172_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X2: nat] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_173_prefix__order_Oorder__antisym,axiom,
! [X: list_o,Y: list_o] :
( ( prefix_o @ X @ Y )
=> ( ( prefix_o @ Y @ X )
=> ( X = Y ) ) ) ).
% prefix_order.order_antisym
thf(fact_174_prefix__order_Oorder__eq__iff,axiom,
( ( ^ [Y5: list_o,Z: list_o] : ( Y5 = Z ) )
= ( ^ [X3: list_o,Y3: list_o] :
( ( prefix_o @ X3 @ Y3 )
& ( prefix_o @ Y3 @ X3 ) ) ) ) ).
% prefix_order.order_eq_iff
thf(fact_175_prefix__order_Oantisym__conv,axiom,
! [Y: list_o,X: list_o] :
( ( prefix_o @ Y @ X )
=> ( ( prefix_o @ X @ Y )
= ( X = Y ) ) ) ).
% prefix_order.antisym_conv
thf(fact_176_prefix__order_Oorder__trans,axiom,
! [X: list_o,Y: list_o,Z2: list_o] :
( ( prefix_o @ X @ Y )
=> ( ( prefix_o @ Y @ Z2 )
=> ( prefix_o @ X @ Z2 ) ) ) ).
% prefix_order.order_trans
thf(fact_177_prefix__order_Oeq__refl,axiom,
! [X: list_o,Y: list_o] :
( ( X = Y )
=> ( prefix_o @ X @ Y ) ) ).
% prefix_order.eq_refl
thf(fact_178_Nu_092_060_094sub_062e__def,axiom,
( prefix8864127203703499552e_Nu_e
= ( ^ [N: nat] : ( some_list_o @ ( prefix979155009907353096ry_nat @ N ) ) ) ) ).
% Nu\<^sub>e_def
thf(fact_179_inj__on__map__add__dom,axiom,
! [M: product_prod_int_int > option_list_o,M2: product_prod_int_int > option_list_o] :
( ( inj_on431520726901613020list_o @ ( map_ad4066630804642346330list_o @ M @ M2 ) @ ( dom_Pr7776987155556795511list_o @ M2 ) )
= ( inj_on431520726901613020list_o @ M2 @ ( dom_Pr7776987155556795511list_o @ M2 ) ) ) ).
% inj_on_map_add_dom
thf(fact_180_inj__on__map__add__dom,axiom,
! [M: product_prod_int_nat > option_list_o,M2: product_prod_int_nat > option_list_o] :
( ( inj_on622226642133877048list_o @ ( map_ad1598970985909685942list_o @ M @ M2 ) @ ( dom_Pr5309327336824135123list_o @ M2 ) )
= ( inj_on622226642133877048list_o @ M2 @ ( dom_Pr5309327336824135123list_o @ M2 ) ) ) ).
% inj_on_map_add_dom
thf(fact_181_inj__on__map__add__dom,axiom,
! [M: product_prod_nat_int > option_list_o,M2: product_prod_nat_int > option_list_o] :
( ( inj_on7977379531974639416list_o @ ( map_ad3959431595253304502list_o @ M @ M2 ) @ ( dom_Pr7669787946167753683list_o @ M2 ) )
= ( inj_on7977379531974639416list_o @ M2 @ ( dom_Pr7669787946167753683list_o @ M2 ) ) ) ).
% inj_on_map_add_dom
thf(fact_182_inj__on__map__add__dom,axiom,
! [M: product_prod_nat_nat > option_list_o,M2: product_prod_nat_nat > option_list_o] :
( ( inj_on8168085447206903444list_o @ ( map_ad1491771776520644114list_o @ M @ M2 ) @ ( dom_Pr5202128127435093295list_o @ M2 ) )
= ( inj_on8168085447206903444list_o @ M2 @ ( dom_Pr5202128127435093295list_o @ M2 ) ) ) ).
% inj_on_map_add_dom
thf(fact_183_inj__on__map__add__dom,axiom,
! [M: list_o > option_list_o,M2: list_o > option_list_o] :
( ( inj_on2239224958054982839list_o @ ( map_ad7290972094422621877list_o @ M @ M2 ) @ ( dom_list_o_list_o @ M2 ) )
= ( inj_on2239224958054982839list_o @ M2 @ ( dom_list_o_list_o @ M2 ) ) ) ).
% inj_on_map_add_dom
thf(fact_184_inj__on__map__add__dom,axiom,
! [M: nat > option_list_o,M2: nat > option_list_o] :
( ( inj_on1630180835328728801list_o @ ( map_add_nat_list_o @ M @ M2 ) @ ( dom_nat_list_o @ M2 ) )
= ( inj_on1630180835328728801list_o @ M2 @ ( dom_nat_list_o @ M2 ) ) ) ).
% inj_on_map_add_dom
thf(fact_185_inj__on__map__add__dom,axiom,
! [M: int > option_list_o,M2: int > option_list_o] :
( ( inj_on203032006962319749list_o @ ( map_add_int_list_o @ M @ M2 ) @ ( dom_int_list_o @ M2 ) )
= ( inj_on203032006962319749list_o @ M2 @ ( dom_int_list_o @ M2 ) ) ) ).
% inj_on_map_add_dom
thf(fact_186_inj__on__inverseI,axiom,
! [A: set_nat,G: nat > nat,F2: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( G @ ( F2 @ X2 ) )
= X2 ) )
=> ( inj_on_nat_nat @ F2 @ A ) ) ).
% inj_on_inverseI
thf(fact_187_inj__on__inverseI,axiom,
! [A: set_nat,G: int > nat,F2: nat > int] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( G @ ( F2 @ X2 ) )
= X2 ) )
=> ( inj_on_nat_int @ F2 @ A ) ) ).
% inj_on_inverseI
thf(fact_188_inj__on__inverseI,axiom,
! [A: set_int,G: nat > int,F2: int > nat] :
( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ( ( G @ ( F2 @ X2 ) )
= X2 ) )
=> ( inj_on_int_nat @ F2 @ A ) ) ).
% inj_on_inverseI
thf(fact_189_inj__on__inverseI,axiom,
! [A: set_int,G: int > int,F2: int > int] :
( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ( ( G @ ( F2 @ X2 ) )
= X2 ) )
=> ( inj_on_int_int @ F2 @ A ) ) ).
% inj_on_inverseI
thf(fact_190_inj__on__inverseI,axiom,
! [A: set_list_o,G: option_list_o > list_o,F2: list_o > option_list_o] :
( ! [X2: list_o] :
( ( member_list_o @ X2 @ A )
=> ( ( G @ ( F2 @ X2 ) )
= X2 ) )
=> ( inj_on2239224958054982839list_o @ F2 @ A ) ) ).
% inj_on_inverseI
thf(fact_191_inj__on__inverseI,axiom,
! [A: set_nat,G: option_list_o > nat,F2: nat > option_list_o] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( G @ ( F2 @ X2 ) )
= X2 ) )
=> ( inj_on1630180835328728801list_o @ F2 @ A ) ) ).
% inj_on_inverseI
thf(fact_192_inj__on__inverseI,axiom,
! [A: set_int,G: option_list_o > int,F2: int > option_list_o] :
( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ( ( G @ ( F2 @ X2 ) )
= X2 ) )
=> ( inj_on203032006962319749list_o @ F2 @ A ) ) ).
% inj_on_inverseI
thf(fact_193_inj__on__contraD,axiom,
! [F2: nat > nat,A: set_nat,X: nat,Y: nat] :
( ( inj_on_nat_nat @ F2 @ A )
=> ( ( X != Y )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y @ A )
=> ( ( F2 @ X )
!= ( F2 @ Y ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_194_inj__on__contraD,axiom,
! [F2: nat > int,A: set_nat,X: nat,Y: nat] :
( ( inj_on_nat_int @ F2 @ A )
=> ( ( X != Y )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y @ A )
=> ( ( F2 @ X )
!= ( F2 @ Y ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_195_inj__on__contraD,axiom,
! [F2: int > nat,A: set_int,X: int,Y: int] :
( ( inj_on_int_nat @ F2 @ A )
=> ( ( X != Y )
=> ( ( member_int @ X @ A )
=> ( ( member_int @ Y @ A )
=> ( ( F2 @ X )
!= ( F2 @ Y ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_196_inj__on__contraD,axiom,
! [F2: int > int,A: set_int,X: int,Y: int] :
( ( inj_on_int_int @ F2 @ A )
=> ( ( X != Y )
=> ( ( member_int @ X @ A )
=> ( ( member_int @ Y @ A )
=> ( ( F2 @ X )
!= ( F2 @ Y ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_197_inj__on__contraD,axiom,
! [F2: list_o > option_list_o,A: set_list_o,X: list_o,Y: list_o] :
( ( inj_on2239224958054982839list_o @ F2 @ A )
=> ( ( X != Y )
=> ( ( member_list_o @ X @ A )
=> ( ( member_list_o @ Y @ A )
=> ( ( F2 @ X )
!= ( F2 @ Y ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_198_inj__on__contraD,axiom,
! [F2: nat > option_list_o,A: set_nat,X: nat,Y: nat] :
( ( inj_on1630180835328728801list_o @ F2 @ A )
=> ( ( X != Y )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y @ A )
=> ( ( F2 @ X )
!= ( F2 @ Y ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_199_inj__on__contraD,axiom,
! [F2: int > option_list_o,A: set_int,X: int,Y: int] :
( ( inj_on203032006962319749list_o @ F2 @ A )
=> ( ( X != Y )
=> ( ( member_int @ X @ A )
=> ( ( member_int @ Y @ A )
=> ( ( F2 @ X )
!= ( F2 @ Y ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_200_inj__on__eq__iff,axiom,
! [F2: nat > nat,A: set_nat,X: nat,Y: nat] :
( ( inj_on_nat_nat @ F2 @ A )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y @ A )
=> ( ( ( F2 @ X )
= ( F2 @ Y ) )
= ( X = Y ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_201_inj__on__eq__iff,axiom,
! [F2: nat > int,A: set_nat,X: nat,Y: nat] :
( ( inj_on_nat_int @ F2 @ A )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y @ A )
=> ( ( ( F2 @ X )
= ( F2 @ Y ) )
= ( X = Y ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_202_inj__on__eq__iff,axiom,
! [F2: int > nat,A: set_int,X: int,Y: int] :
( ( inj_on_int_nat @ F2 @ A )
=> ( ( member_int @ X @ A )
=> ( ( member_int @ Y @ A )
=> ( ( ( F2 @ X )
= ( F2 @ Y ) )
= ( X = Y ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_203_inj__on__eq__iff,axiom,
! [F2: int > int,A: set_int,X: int,Y: int] :
( ( inj_on_int_int @ F2 @ A )
=> ( ( member_int @ X @ A )
=> ( ( member_int @ Y @ A )
=> ( ( ( F2 @ X )
= ( F2 @ Y ) )
= ( X = Y ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_204_inj__on__eq__iff,axiom,
! [F2: list_o > option_list_o,A: set_list_o,X: list_o,Y: list_o] :
( ( inj_on2239224958054982839list_o @ F2 @ A )
=> ( ( member_list_o @ X @ A )
=> ( ( member_list_o @ Y @ A )
=> ( ( ( F2 @ X )
= ( F2 @ Y ) )
= ( X = Y ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_205_inj__on__eq__iff,axiom,
! [F2: nat > option_list_o,A: set_nat,X: nat,Y: nat] :
( ( inj_on1630180835328728801list_o @ F2 @ A )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y @ A )
=> ( ( ( F2 @ X )
= ( F2 @ Y ) )
= ( X = Y ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_206_inj__on__eq__iff,axiom,
! [F2: int > option_list_o,A: set_int,X: int,Y: int] :
( ( inj_on203032006962319749list_o @ F2 @ A )
=> ( ( member_int @ X @ A )
=> ( ( member_int @ Y @ A )
=> ( ( ( F2 @ X )
= ( F2 @ Y ) )
= ( X = Y ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_207_inj__on__cong,axiom,
! [A: set_nat,F2: nat > nat,G: nat > nat] :
( ! [A4: nat] :
( ( member_nat @ A4 @ A )
=> ( ( F2 @ A4 )
= ( G @ A4 ) ) )
=> ( ( inj_on_nat_nat @ F2 @ A )
= ( inj_on_nat_nat @ G @ A ) ) ) ).
% inj_on_cong
thf(fact_208_inj__on__cong,axiom,
! [A: set_nat,F2: nat > int,G: nat > int] :
( ! [A4: nat] :
( ( member_nat @ A4 @ A )
=> ( ( F2 @ A4 )
= ( G @ A4 ) ) )
=> ( ( inj_on_nat_int @ F2 @ A )
= ( inj_on_nat_int @ G @ A ) ) ) ).
% inj_on_cong
thf(fact_209_inj__on__cong,axiom,
! [A: set_int,F2: int > nat,G: int > nat] :
( ! [A4: int] :
( ( member_int @ A4 @ A )
=> ( ( F2 @ A4 )
= ( G @ A4 ) ) )
=> ( ( inj_on_int_nat @ F2 @ A )
= ( inj_on_int_nat @ G @ A ) ) ) ).
% inj_on_cong
thf(fact_210_inj__on__cong,axiom,
! [A: set_int,F2: int > int,G: int > int] :
( ! [A4: int] :
( ( member_int @ A4 @ A )
=> ( ( F2 @ A4 )
= ( G @ A4 ) ) )
=> ( ( inj_on_int_int @ F2 @ A )
= ( inj_on_int_int @ G @ A ) ) ) ).
% inj_on_cong
thf(fact_211_inj__on__cong,axiom,
! [A: set_list_o,F2: list_o > option_list_o,G: list_o > option_list_o] :
( ! [A4: list_o] :
( ( member_list_o @ A4 @ A )
=> ( ( F2 @ A4 )
= ( G @ A4 ) ) )
=> ( ( inj_on2239224958054982839list_o @ F2 @ A )
= ( inj_on2239224958054982839list_o @ G @ A ) ) ) ).
% inj_on_cong
thf(fact_212_inj__on__cong,axiom,
! [A: set_nat,F2: nat > option_list_o,G: nat > option_list_o] :
( ! [A4: nat] :
( ( member_nat @ A4 @ A )
=> ( ( F2 @ A4 )
= ( G @ A4 ) ) )
=> ( ( inj_on1630180835328728801list_o @ F2 @ A )
= ( inj_on1630180835328728801list_o @ G @ A ) ) ) ).
% inj_on_cong
thf(fact_213_inj__on__cong,axiom,
! [A: set_int,F2: int > option_list_o,G: int > option_list_o] :
( ! [A4: int] :
( ( member_int @ A4 @ A )
=> ( ( F2 @ A4 )
= ( G @ A4 ) ) )
=> ( ( inj_on203032006962319749list_o @ F2 @ A )
= ( inj_on203032006962319749list_o @ G @ A ) ) ) ).
% inj_on_cong
thf(fact_214_inj__on__def,axiom,
( inj_on2239224958054982839list_o
= ( ^ [F: list_o > option_list_o,A5: set_list_o] :
! [X3: list_o] :
( ( member_list_o @ X3 @ A5 )
=> ! [Y3: list_o] :
( ( member_list_o @ Y3 @ A5 )
=> ( ( ( F @ X3 )
= ( F @ Y3 ) )
=> ( X3 = Y3 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_215_inj__on__def,axiom,
( inj_on_nat_nat
= ( ^ [F: nat > nat,A5: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A5 )
=> ! [Y3: nat] :
( ( member_nat @ Y3 @ A5 )
=> ( ( ( F @ X3 )
= ( F @ Y3 ) )
=> ( X3 = Y3 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_216_inj__on__def,axiom,
( inj_on_nat_int
= ( ^ [F: nat > int,A5: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A5 )
=> ! [Y3: nat] :
( ( member_nat @ Y3 @ A5 )
=> ( ( ( F @ X3 )
= ( F @ Y3 ) )
=> ( X3 = Y3 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_217_inj__on__def,axiom,
( inj_on_int_nat
= ( ^ [F: int > nat,A5: set_int] :
! [X3: int] :
( ( member_int @ X3 @ A5 )
=> ! [Y3: int] :
( ( member_int @ Y3 @ A5 )
=> ( ( ( F @ X3 )
= ( F @ Y3 ) )
=> ( X3 = Y3 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_218_inj__on__def,axiom,
( inj_on_int_int
= ( ^ [F: int > int,A5: set_int] :
! [X3: int] :
( ( member_int @ X3 @ A5 )
=> ! [Y3: int] :
( ( member_int @ Y3 @ A5 )
=> ( ( ( F @ X3 )
= ( F @ Y3 ) )
=> ( X3 = Y3 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_219_inj__on__def,axiom,
( inj_on1630180835328728801list_o
= ( ^ [F: nat > option_list_o,A5: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A5 )
=> ! [Y3: nat] :
( ( member_nat @ Y3 @ A5 )
=> ( ( ( F @ X3 )
= ( F @ Y3 ) )
=> ( X3 = Y3 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_220_inj__on__def,axiom,
( inj_on203032006962319749list_o
= ( ^ [F: int > option_list_o,A5: set_int] :
! [X3: int] :
( ( member_int @ X3 @ A5 )
=> ! [Y3: int] :
( ( member_int @ Y3 @ A5 )
=> ( ( ( F @ X3 )
= ( F @ Y3 ) )
=> ( X3 = Y3 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_221_inj__onI,axiom,
! [A: set_nat,F2: nat > nat] :
( ! [X2: nat,Y2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_nat @ Y2 @ A )
=> ( ( ( F2 @ X2 )
= ( F2 @ Y2 ) )
=> ( X2 = Y2 ) ) ) )
=> ( inj_on_nat_nat @ F2 @ A ) ) ).
% inj_onI
thf(fact_222_inj__onI,axiom,
! [A: set_nat,F2: nat > int] :
( ! [X2: nat,Y2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_nat @ Y2 @ A )
=> ( ( ( F2 @ X2 )
= ( F2 @ Y2 ) )
=> ( X2 = Y2 ) ) ) )
=> ( inj_on_nat_int @ F2 @ A ) ) ).
% inj_onI
thf(fact_223_inj__onI,axiom,
! [A: set_int,F2: int > nat] :
( ! [X2: int,Y2: int] :
( ( member_int @ X2 @ A )
=> ( ( member_int @ Y2 @ A )
=> ( ( ( F2 @ X2 )
= ( F2 @ Y2 ) )
=> ( X2 = Y2 ) ) ) )
=> ( inj_on_int_nat @ F2 @ A ) ) ).
% inj_onI
thf(fact_224_inj__onI,axiom,
! [A: set_int,F2: int > int] :
( ! [X2: int,Y2: int] :
( ( member_int @ X2 @ A )
=> ( ( member_int @ Y2 @ A )
=> ( ( ( F2 @ X2 )
= ( F2 @ Y2 ) )
=> ( X2 = Y2 ) ) ) )
=> ( inj_on_int_int @ F2 @ A ) ) ).
% inj_onI
thf(fact_225_inj__onI,axiom,
! [A: set_list_o,F2: list_o > option_list_o] :
( ! [X2: list_o,Y2: list_o] :
( ( member_list_o @ X2 @ A )
=> ( ( member_list_o @ Y2 @ A )
=> ( ( ( F2 @ X2 )
= ( F2 @ Y2 ) )
=> ( X2 = Y2 ) ) ) )
=> ( inj_on2239224958054982839list_o @ F2 @ A ) ) ).
% inj_onI
thf(fact_226_inj__onI,axiom,
! [A: set_nat,F2: nat > option_list_o] :
( ! [X2: nat,Y2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_nat @ Y2 @ A )
=> ( ( ( F2 @ X2 )
= ( F2 @ Y2 ) )
=> ( X2 = Y2 ) ) ) )
=> ( inj_on1630180835328728801list_o @ F2 @ A ) ) ).
% inj_onI
thf(fact_227_inj__onI,axiom,
! [A: set_int,F2: int > option_list_o] :
( ! [X2: int,Y2: int] :
( ( member_int @ X2 @ A )
=> ( ( member_int @ Y2 @ A )
=> ( ( ( F2 @ X2 )
= ( F2 @ Y2 ) )
=> ( X2 = Y2 ) ) ) )
=> ( inj_on203032006962319749list_o @ F2 @ A ) ) ).
% inj_onI
thf(fact_228_inj__onD,axiom,
! [F2: nat > nat,A: set_nat,X: nat,Y: nat] :
( ( inj_on_nat_nat @ F2 @ A )
=> ( ( ( F2 @ X )
= ( F2 @ Y ) )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y @ A )
=> ( X = Y ) ) ) ) ) ).
% inj_onD
thf(fact_229_inj__onD,axiom,
! [F2: nat > int,A: set_nat,X: nat,Y: nat] :
( ( inj_on_nat_int @ F2 @ A )
=> ( ( ( F2 @ X )
= ( F2 @ Y ) )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y @ A )
=> ( X = Y ) ) ) ) ) ).
% inj_onD
thf(fact_230_inj__onD,axiom,
! [F2: int > nat,A: set_int,X: int,Y: int] :
( ( inj_on_int_nat @ F2 @ A )
=> ( ( ( F2 @ X )
= ( F2 @ Y ) )
=> ( ( member_int @ X @ A )
=> ( ( member_int @ Y @ A )
=> ( X = Y ) ) ) ) ) ).
% inj_onD
thf(fact_231_inj__onD,axiom,
! [F2: int > int,A: set_int,X: int,Y: int] :
( ( inj_on_int_int @ F2 @ A )
=> ( ( ( F2 @ X )
= ( F2 @ Y ) )
=> ( ( member_int @ X @ A )
=> ( ( member_int @ Y @ A )
=> ( X = Y ) ) ) ) ) ).
% inj_onD
thf(fact_232_inj__onD,axiom,
! [F2: list_o > option_list_o,A: set_list_o,X: list_o,Y: list_o] :
( ( inj_on2239224958054982839list_o @ F2 @ A )
=> ( ( ( F2 @ X )
= ( F2 @ Y ) )
=> ( ( member_list_o @ X @ A )
=> ( ( member_list_o @ Y @ A )
=> ( X = Y ) ) ) ) ) ).
% inj_onD
thf(fact_233_inj__onD,axiom,
! [F2: nat > option_list_o,A: set_nat,X: nat,Y: nat] :
( ( inj_on1630180835328728801list_o @ F2 @ A )
=> ( ( ( F2 @ X )
= ( F2 @ Y ) )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y @ A )
=> ( X = Y ) ) ) ) ) ).
% inj_onD
thf(fact_234_inj__onD,axiom,
! [F2: int > option_list_o,A: set_int,X: int,Y: int] :
( ( inj_on203032006962319749list_o @ F2 @ A )
=> ( ( ( F2 @ X )
= ( F2 @ Y ) )
=> ( ( member_int @ X @ A )
=> ( ( member_int @ Y @ A )
=> ( X = Y ) ) ) ) ) ).
% inj_onD
thf(fact_235_map__add__assoc,axiom,
! [M1: int > option_list_o,M22: int > option_list_o,M3: int > option_list_o] :
( ( map_add_int_list_o @ M1 @ ( map_add_int_list_o @ M22 @ M3 ) )
= ( map_add_int_list_o @ ( map_add_int_list_o @ M1 @ M22 ) @ M3 ) ) ).
% map_add_assoc
thf(fact_236_map__add__assoc,axiom,
! [M1: nat > option_list_o,M22: nat > option_list_o,M3: nat > option_list_o] :
( ( map_add_nat_list_o @ M1 @ ( map_add_nat_list_o @ M22 @ M3 ) )
= ( map_add_nat_list_o @ ( map_add_nat_list_o @ M1 @ M22 ) @ M3 ) ) ).
% map_add_assoc
thf(fact_237_map__add__find__right,axiom,
! [N2: int > option_list_o,K: int,Xx: list_o,M: int > option_list_o] :
( ( ( N2 @ K )
= ( some_list_o @ Xx ) )
=> ( ( map_add_int_list_o @ M @ N2 @ K )
= ( some_list_o @ Xx ) ) ) ).
% map_add_find_right
thf(fact_238_map__add__find__right,axiom,
! [N2: nat > option_list_o,K: nat,Xx: list_o,M: nat > option_list_o] :
( ( ( N2 @ K )
= ( some_list_o @ Xx ) )
=> ( ( map_add_nat_list_o @ M @ N2 @ K )
= ( some_list_o @ Xx ) ) ) ).
% map_add_find_right
thf(fact_239_map__add__None,axiom,
! [M: int > option_list_o,N2: int > option_list_o,K: int] :
( ( ( map_add_int_list_o @ M @ N2 @ K )
= none_list_o )
= ( ( ( N2 @ K )
= none_list_o )
& ( ( M @ K )
= none_list_o ) ) ) ).
% map_add_None
thf(fact_240_map__add__None,axiom,
! [M: nat > option_list_o,N2: nat > option_list_o,K: nat] :
( ( ( map_add_nat_list_o @ M @ N2 @ K )
= none_list_o )
= ( ( ( N2 @ K )
= none_list_o )
& ( ( M @ K )
= none_list_o ) ) ) ).
% map_add_None
thf(fact_241_map__add__eq__empty__iff,axiom,
! [F2: int > option_list_o,G: int > option_list_o] :
( ( ( map_add_int_list_o @ F2 @ G )
= ( ^ [X3: int] : none_list_o ) )
= ( ( F2
= ( ^ [X3: int] : none_list_o ) )
& ( G
= ( ^ [X3: int] : none_list_o ) ) ) ) ).
% map_add_eq_empty_iff
thf(fact_242_map__add__eq__empty__iff,axiom,
! [F2: nat > option_list_o,G: nat > option_list_o] :
( ( ( map_add_nat_list_o @ F2 @ G )
= ( ^ [X3: nat] : none_list_o ) )
= ( ( F2
= ( ^ [X3: nat] : none_list_o ) )
& ( G
= ( ^ [X3: nat] : none_list_o ) ) ) ) ).
% map_add_eq_empty_iff
thf(fact_243_unary__encoding,axiom,
prefix3558185134189398382ng_nat @ prefix8864127203703499552e_Nu_e ).
% unary_encoding
thf(fact_244_map__add__dom__app__simps_I3_J,axiom,
! [M: product_prod_int_int,L2: product_prod_int_int > option_list_o,L1: product_prod_int_int > option_list_o] :
( ~ ( member5262025264175285858nt_int @ M @ ( dom_Pr7776987155556795511list_o @ L2 ) )
=> ( ( map_ad4066630804642346330list_o @ L1 @ L2 @ M )
= ( L1 @ M ) ) ) ).
% map_add_dom_app_simps(3)
thf(fact_245_map__add__dom__app__simps_I3_J,axiom,
! [M: product_prod_int_nat,L2: product_prod_int_nat > option_list_o,L1: product_prod_int_nat > option_list_o] :
( ~ ( member216504246829706758nt_nat @ M @ ( dom_Pr5309327336824135123list_o @ L2 ) )
=> ( ( map_ad1598970985909685942list_o @ L1 @ L2 @ M )
= ( L1 @ M ) ) ) ).
% map_add_dom_app_simps(3)
thf(fact_246_map__add__dom__app__simps_I3_J,axiom,
! [M: product_prod_nat_int,L2: product_prod_nat_int > option_list_o,L1: product_prod_nat_int > option_list_o] :
( ~ ( member4262671552274231302at_int @ M @ ( dom_Pr7669787946167753683list_o @ L2 ) )
=> ( ( map_ad3959431595253304502list_o @ L1 @ L2 @ M )
= ( L1 @ M ) ) ) ).
% map_add_dom_app_simps(3)
thf(fact_247_map__add__dom__app__simps_I3_J,axiom,
! [M: product_prod_nat_nat,L2: product_prod_nat_nat > option_list_o,L1: product_prod_nat_nat > option_list_o] :
( ~ ( member8440522571783428010at_nat @ M @ ( dom_Pr5202128127435093295list_o @ L2 ) )
=> ( ( map_ad1491771776520644114list_o @ L1 @ L2 @ M )
= ( L1 @ M ) ) ) ).
% map_add_dom_app_simps(3)
thf(fact_248_map__add__dom__app__simps_I3_J,axiom,
! [M: nat,L2: nat > option_list_o,L1: nat > option_list_o] :
( ~ ( member_nat @ M @ ( dom_nat_list_o @ L2 ) )
=> ( ( map_add_nat_list_o @ L1 @ L2 @ M )
= ( L1 @ M ) ) ) ).
% map_add_dom_app_simps(3)
thf(fact_249_map__add__dom__app__simps_I3_J,axiom,
! [M: int,L2: int > option_list_o,L1: int > option_list_o] :
( ~ ( member_int @ M @ ( dom_int_list_o @ L2 ) )
=> ( ( map_add_int_list_o @ L1 @ L2 @ M )
= ( L1 @ M ) ) ) ).
% map_add_dom_app_simps(3)
thf(fact_250_map__add__dom__app__simps_I2_J,axiom,
! [M: product_prod_int_int,L1: product_prod_int_int > option_list_o,L2: product_prod_int_int > option_list_o] :
( ~ ( member5262025264175285858nt_int @ M @ ( dom_Pr7776987155556795511list_o @ L1 ) )
=> ( ( map_ad4066630804642346330list_o @ L1 @ L2 @ M )
= ( L2 @ M ) ) ) ).
% map_add_dom_app_simps(2)
thf(fact_251_map__add__dom__app__simps_I2_J,axiom,
! [M: product_prod_int_nat,L1: product_prod_int_nat > option_list_o,L2: product_prod_int_nat > option_list_o] :
( ~ ( member216504246829706758nt_nat @ M @ ( dom_Pr5309327336824135123list_o @ L1 ) )
=> ( ( map_ad1598970985909685942list_o @ L1 @ L2 @ M )
= ( L2 @ M ) ) ) ).
% map_add_dom_app_simps(2)
thf(fact_252_map__add__dom__app__simps_I2_J,axiom,
! [M: product_prod_nat_int,L1: product_prod_nat_int > option_list_o,L2: product_prod_nat_int > option_list_o] :
( ~ ( member4262671552274231302at_int @ M @ ( dom_Pr7669787946167753683list_o @ L1 ) )
=> ( ( map_ad3959431595253304502list_o @ L1 @ L2 @ M )
= ( L2 @ M ) ) ) ).
% map_add_dom_app_simps(2)
thf(fact_253_map__add__dom__app__simps_I2_J,axiom,
! [M: product_prod_nat_nat,L1: product_prod_nat_nat > option_list_o,L2: product_prod_nat_nat > option_list_o] :
( ~ ( member8440522571783428010at_nat @ M @ ( dom_Pr5202128127435093295list_o @ L1 ) )
=> ( ( map_ad1491771776520644114list_o @ L1 @ L2 @ M )
= ( L2 @ M ) ) ) ).
% map_add_dom_app_simps(2)
thf(fact_254_map__add__dom__app__simps_I2_J,axiom,
! [M: nat,L1: nat > option_list_o,L2: nat > option_list_o] :
( ~ ( member_nat @ M @ ( dom_nat_list_o @ L1 ) )
=> ( ( map_add_nat_list_o @ L1 @ L2 @ M )
= ( L2 @ M ) ) ) ).
% map_add_dom_app_simps(2)
thf(fact_255_map__add__dom__app__simps_I2_J,axiom,
! [M: int,L1: int > option_list_o,L2: int > option_list_o] :
( ~ ( member_int @ M @ ( dom_int_list_o @ L1 ) )
=> ( ( map_add_int_list_o @ L1 @ L2 @ M )
= ( L2 @ M ) ) ) ).
% map_add_dom_app_simps(2)
thf(fact_256_map__add__dom__app__simps_I1_J,axiom,
! [M: product_prod_int_int,L2: product_prod_int_int > option_list_o,L1: product_prod_int_int > option_list_o] :
( ( member5262025264175285858nt_int @ M @ ( dom_Pr7776987155556795511list_o @ L2 ) )
=> ( ( map_ad4066630804642346330list_o @ L1 @ L2 @ M )
= ( L2 @ M ) ) ) ).
% map_add_dom_app_simps(1)
thf(fact_257_map__add__dom__app__simps_I1_J,axiom,
! [M: product_prod_int_nat,L2: product_prod_int_nat > option_list_o,L1: product_prod_int_nat > option_list_o] :
( ( member216504246829706758nt_nat @ M @ ( dom_Pr5309327336824135123list_o @ L2 ) )
=> ( ( map_ad1598970985909685942list_o @ L1 @ L2 @ M )
= ( L2 @ M ) ) ) ).
% map_add_dom_app_simps(1)
thf(fact_258_map__add__dom__app__simps_I1_J,axiom,
! [M: product_prod_nat_int,L2: product_prod_nat_int > option_list_o,L1: product_prod_nat_int > option_list_o] :
( ( member4262671552274231302at_int @ M @ ( dom_Pr7669787946167753683list_o @ L2 ) )
=> ( ( map_ad3959431595253304502list_o @ L1 @ L2 @ M )
= ( L2 @ M ) ) ) ).
% map_add_dom_app_simps(1)
thf(fact_259_map__add__dom__app__simps_I1_J,axiom,
! [M: product_prod_nat_nat,L2: product_prod_nat_nat > option_list_o,L1: product_prod_nat_nat > option_list_o] :
( ( member8440522571783428010at_nat @ M @ ( dom_Pr5202128127435093295list_o @ L2 ) )
=> ( ( map_ad1491771776520644114list_o @ L1 @ L2 @ M )
= ( L2 @ M ) ) ) ).
% map_add_dom_app_simps(1)
thf(fact_260_map__add__dom__app__simps_I1_J,axiom,
! [M: nat,L2: nat > option_list_o,L1: nat > option_list_o] :
( ( member_nat @ M @ ( dom_nat_list_o @ L2 ) )
=> ( ( map_add_nat_list_o @ L1 @ L2 @ M )
= ( L2 @ M ) ) ) ).
% map_add_dom_app_simps(1)
thf(fact_261_map__add__dom__app__simps_I1_J,axiom,
! [M: int,L2: int > option_list_o,L1: int > option_list_o] :
( ( member_int @ M @ ( dom_int_list_o @ L2 ) )
=> ( ( map_add_int_list_o @ L1 @ L2 @ M )
= ( L2 @ M ) ) ) ).
% map_add_dom_app_simps(1)
thf(fact_262_map__add__Some__iff,axiom,
! [M: int > option_list_o,N2: int > option_list_o,K: int,X: list_o] :
( ( ( map_add_int_list_o @ M @ N2 @ K )
= ( some_list_o @ X ) )
= ( ( ( N2 @ K )
= ( some_list_o @ X ) )
| ( ( ( N2 @ K )
= none_list_o )
& ( ( M @ K )
= ( some_list_o @ X ) ) ) ) ) ).
% map_add_Some_iff
thf(fact_263_map__add__Some__iff,axiom,
! [M: nat > option_list_o,N2: nat > option_list_o,K: nat,X: list_o] :
( ( ( map_add_nat_list_o @ M @ N2 @ K )
= ( some_list_o @ X ) )
= ( ( ( N2 @ K )
= ( some_list_o @ X ) )
| ( ( ( N2 @ K )
= none_list_o )
& ( ( M @ K )
= ( some_list_o @ X ) ) ) ) ) ).
% map_add_Some_iff
thf(fact_264_map__add__SomeD,axiom,
! [M: int > option_list_o,N2: int > option_list_o,K: int,X: list_o] :
( ( ( map_add_int_list_o @ M @ N2 @ K )
= ( some_list_o @ X ) )
=> ( ( ( N2 @ K )
= ( some_list_o @ X ) )
| ( ( ( N2 @ K )
= none_list_o )
& ( ( M @ K )
= ( some_list_o @ X ) ) ) ) ) ).
% map_add_SomeD
thf(fact_265_map__add__SomeD,axiom,
! [M: nat > option_list_o,N2: nat > option_list_o,K: nat,X: list_o] :
( ( ( map_add_nat_list_o @ M @ N2 @ K )
= ( some_list_o @ X ) )
=> ( ( ( N2 @ K )
= ( some_list_o @ X ) )
| ( ( ( N2 @ K )
= none_list_o )
& ( ( M @ K )
= ( some_list_o @ X ) ) ) ) ) ).
% map_add_SomeD
thf(fact_266_option_Ocollapse,axiom,
! [Option: option_int] :
( ( Option != none_int )
=> ( ( some_int @ ( the_int @ Option ) )
= Option ) ) ).
% option.collapse
thf(fact_267_option_Ocollapse,axiom,
! [Option: option_nat] :
( ( Option != none_nat )
=> ( ( some_nat @ ( the_nat @ Option ) )
= Option ) ) ).
% option.collapse
thf(fact_268_option_Ocollapse,axiom,
! [Option: option_option_list_o] :
( ( Option != none_option_list_o )
=> ( ( some_option_list_o @ ( the_option_list_o @ Option ) )
= Option ) ) ).
% option.collapse
thf(fact_269_option_Ocollapse,axiom,
! [Option: option_list_o] :
( ( Option != none_list_o )
=> ( ( some_list_o @ ( the_list_o @ Option ) )
= Option ) ) ).
% option.collapse
thf(fact_270_dom__eq__empty__conv,axiom,
! [F2: nat > option_list_o] :
( ( ( dom_nat_list_o @ F2 )
= bot_bot_set_nat )
= ( F2
= ( ^ [X3: nat] : none_list_o ) ) ) ).
% dom_eq_empty_conv
thf(fact_271_dom__eq__empty__conv,axiom,
! [F2: int > option_list_o] :
( ( ( dom_int_list_o @ F2 )
= bot_bot_set_int )
= ( F2
= ( ^ [X3: int] : none_list_o ) ) ) ).
% dom_eq_empty_conv
thf(fact_272_dom__eq__empty__conv,axiom,
! [F2: nat > option_int] :
( ( ( dom_nat_int @ F2 )
= bot_bot_set_nat )
= ( F2
= ( ^ [X3: nat] : none_int ) ) ) ).
% dom_eq_empty_conv
thf(fact_273_dom__eq__empty__conv,axiom,
! [F2: nat > option_nat] :
( ( ( dom_nat_nat @ F2 )
= bot_bot_set_nat )
= ( F2
= ( ^ [X3: nat] : none_nat ) ) ) ).
% dom_eq_empty_conv
thf(fact_274_dom__eq__empty__conv,axiom,
! [F2: int > option_int] :
( ( ( dom_int_int @ F2 )
= bot_bot_set_int )
= ( F2
= ( ^ [X3: int] : none_int ) ) ) ).
% dom_eq_empty_conv
thf(fact_275_dom__eq__empty__conv,axiom,
! [F2: int > option_nat] :
( ( ( dom_int_nat @ F2 )
= bot_bot_set_int )
= ( F2
= ( ^ [X3: int] : none_nat ) ) ) ).
% dom_eq_empty_conv
thf(fact_276_dom__eq__empty__conv,axiom,
! [F2: list_o > option_int] :
( ( ( dom_list_o_int @ F2 )
= bot_bot_set_list_o )
= ( F2
= ( ^ [X3: list_o] : none_int ) ) ) ).
% dom_eq_empty_conv
thf(fact_277_dom__eq__empty__conv,axiom,
! [F2: list_o > option_nat] :
( ( ( dom_list_o_nat @ F2 )
= bot_bot_set_list_o )
= ( F2
= ( ^ [X3: list_o] : none_nat ) ) ) ).
% dom_eq_empty_conv
thf(fact_278_dom__eq__empty__conv,axiom,
! [F2: list_o > option_list_o] :
( ( ( dom_list_o_list_o @ F2 )
= bot_bot_set_list_o )
= ( F2
= ( ^ [X3: list_o] : none_list_o ) ) ) ).
% dom_eq_empty_conv
thf(fact_279_dom__eq__empty__conv,axiom,
! [F2: nat > option_option_list_o] :
( ( ( dom_na4357407098760616128list_o @ F2 )
= bot_bot_set_nat )
= ( F2
= ( ^ [X3: nat] : none_option_list_o ) ) ) ).
% dom_eq_empty_conv
thf(fact_280_map__add__upd__left,axiom,
! [M: nat,E2: nat > option_list_o,E1: nat > option_list_o,U1: list_o] :
( ~ ( member_nat @ M @ ( dom_nat_list_o @ E2 ) )
=> ( ( map_add_nat_list_o @ ( fun_up6685928411843432661list_o @ E1 @ M @ ( some_list_o @ U1 ) ) @ E2 )
= ( fun_up6685928411843432661list_o @ ( map_add_nat_list_o @ E1 @ E2 ) @ M @ ( some_list_o @ U1 ) ) ) ) ).
% map_add_upd_left
thf(fact_281_map__add__upd__left,axiom,
! [M: int,E2: int > option_list_o,E1: int > option_list_o,U1: list_o] :
( ~ ( member_int @ M @ ( dom_int_list_o @ E2 ) )
=> ( ( map_add_int_list_o @ ( fun_up5258779583477023609list_o @ E1 @ M @ ( some_list_o @ U1 ) ) @ E2 )
= ( fun_up5258779583477023609list_o @ ( map_add_int_list_o @ E1 @ E2 ) @ M @ ( some_list_o @ U1 ) ) ) ) ).
% map_add_upd_left
thf(fact_282_map__add__upd__left,axiom,
! [M: nat,E2: nat > option_int,E1: nat > option_int,U1: int] :
( ~ ( member_nat @ M @ ( dom_nat_int @ E2 ) )
=> ( ( map_add_nat_int @ ( fun_up6538678405303910731on_int @ E1 @ M @ ( some_int @ U1 ) ) @ E2 )
= ( fun_up6538678405303910731on_int @ ( map_add_nat_int @ E1 @ E2 ) @ M @ ( some_int @ U1 ) ) ) ) ).
% map_add_upd_left
thf(fact_283_map__add__upd__left,axiom,
! [M: int,E2: int > option_int,E1: int > option_int,U1: int] :
( ~ ( member_int @ M @ ( dom_int_int @ E2 ) )
=> ( ( map_add_int_int @ ( fun_up8666045135305973159on_int @ E1 @ M @ ( some_int @ U1 ) ) @ E2 )
= ( fun_up8666045135305973159on_int @ ( map_add_int_int @ E1 @ E2 ) @ M @ ( some_int @ U1 ) ) ) ) ).
% map_add_upd_left
thf(fact_284_map__add__upd__left,axiom,
! [M: int,E2: int > option_nat,E1: int > option_nat,U1: nat] :
( ~ ( member_int @ M @ ( dom_int_nat @ E2 ) )
=> ( ( map_add_int_nat @ ( fun_up3620524117960394059on_nat @ E1 @ M @ ( some_nat @ U1 ) ) @ E2 )
= ( fun_up3620524117960394059on_nat @ ( map_add_int_nat @ E1 @ E2 ) @ M @ ( some_nat @ U1 ) ) ) ) ).
% map_add_upd_left
thf(fact_285_map__add__upd__left,axiom,
! [M: nat,E2: nat > option_nat,E1: nat > option_nat,U1: nat] :
( ~ ( member_nat @ M @ ( dom_nat_nat @ E2 ) )
=> ( ( map_add_nat_nat @ ( fun_up1493157387958331631on_nat @ E1 @ M @ ( some_nat @ U1 ) ) @ E2 )
= ( fun_up1493157387958331631on_nat @ ( map_add_nat_nat @ E1 @ E2 ) @ M @ ( some_nat @ U1 ) ) ) ) ).
% map_add_upd_left
thf(fact_286_map__add__upd__left,axiom,
! [M: option_nat,E2: option_nat > option_int,E1: option_nat > option_int,U1: int] :
( ~ ( member_option_nat @ M @ ( dom_option_nat_int @ E2 ) )
=> ( ( map_ad6030038764658420611at_int @ ( fun_up1794774578788926875on_int @ E1 @ M @ ( some_int @ U1 ) ) @ E2 )
= ( fun_up1794774578788926875on_int @ ( map_ad6030038764658420611at_int @ E1 @ E2 ) @ M @ ( some_int @ U1 ) ) ) ) ).
% map_add_upd_left
thf(fact_287_map__add__upd__left,axiom,
! [M: option_int,E2: option_int > option_int,E1: option_int > option_int,U1: int] :
( ~ ( member_option_int @ M @ ( dom_option_int_int @ E2 ) )
=> ( ( map_ad8216960339351442399nt_int @ ( fun_up7205856760492389367on_int @ E1 @ M @ ( some_int @ U1 ) ) @ E2 )
= ( fun_up7205856760492389367on_int @ ( map_ad8216960339351442399nt_int @ E1 @ E2 ) @ M @ ( some_int @ U1 ) ) ) ) ).
% map_add_upd_left
thf(fact_288_map__add__upd__left,axiom,
! [M: list_o,E2: list_o > option_int,E1: list_o > option_int,U1: int] :
( ~ ( member_list_o @ M @ ( dom_list_o_int @ E2 ) )
=> ( ( map_add_list_o_int @ ( fun_up5495103757224608477on_int @ E1 @ M @ ( some_int @ U1 ) ) @ E2 )
= ( fun_up5495103757224608477on_int @ ( map_add_list_o_int @ E1 @ E2 ) @ M @ ( some_int @ U1 ) ) ) ) ).
% map_add_upd_left
thf(fact_289_map__add__upd__left,axiom,
! [M: set_nat,E2: set_nat > option_int,E1: set_nat > option_int,U1: int] :
( ~ ( member_set_nat @ M @ ( dom_set_nat_int @ E2 ) )
=> ( ( map_add_set_nat_int @ ( fun_up1329515020090725377on_int @ E1 @ M @ ( some_int @ U1 ) ) @ E2 )
= ( fun_up1329515020090725377on_int @ ( map_add_set_nat_int @ E1 @ E2 ) @ M @ ( some_int @ U1 ) ) ) ) ).
% map_add_upd_left
thf(fact_290_dom__map__add,axiom,
! [M: product_prod_int_int > option_list_o,N2: product_prod_int_int > option_list_o] :
( ( dom_Pr7776987155556795511list_o @ ( map_ad4066630804642346330list_o @ M @ N2 ) )
= ( sup_su6024340866399070445nt_int @ ( dom_Pr7776987155556795511list_o @ N2 ) @ ( dom_Pr7776987155556795511list_o @ M ) ) ) ).
% dom_map_add
thf(fact_291_dom__map__add,axiom,
! [M: product_prod_int_nat > option_list_o,N2: product_prod_int_nat > option_list_o] :
( ( dom_Pr5309327336824135123list_o @ ( map_ad1598970985909685942list_o @ M @ N2 ) )
= ( sup_su8514424011330797201nt_nat @ ( dom_Pr5309327336824135123list_o @ N2 ) @ ( dom_Pr5309327336824135123list_o @ M ) ) ) ).
% dom_map_add
thf(fact_292_dom__map__add,axiom,
! [M: product_prod_nat_int > option_list_o,N2: product_prod_nat_int > option_list_o] :
( ( dom_Pr7669787946167753683list_o @ ( map_ad3959431595253304502list_o @ M @ N2 ) )
= ( sup_su3837419291706048657at_int @ ( dom_Pr7669787946167753683list_o @ N2 ) @ ( dom_Pr7669787946167753683list_o @ M ) ) ) ).
% dom_map_add
thf(fact_293_dom__map__add,axiom,
! [M: product_prod_nat_nat > option_list_o,N2: product_prod_nat_nat > option_list_o] :
( ( dom_Pr5202128127435093295list_o @ ( map_ad1491771776520644114list_o @ M @ N2 ) )
= ( sup_su6327502436637775413at_nat @ ( dom_Pr5202128127435093295list_o @ N2 ) @ ( dom_Pr5202128127435093295list_o @ M ) ) ) ).
% dom_map_add
thf(fact_294_dom__map__add,axiom,
! [M: nat > option_list_o,N2: nat > option_list_o] :
( ( dom_nat_list_o @ ( map_add_nat_list_o @ M @ N2 ) )
= ( sup_sup_set_nat @ ( dom_nat_list_o @ N2 ) @ ( dom_nat_list_o @ M ) ) ) ).
% dom_map_add
thf(fact_295_dom__map__add,axiom,
! [M: int > option_list_o,N2: int > option_list_o] :
( ( dom_int_list_o @ ( map_add_int_list_o @ M @ N2 ) )
= ( sup_sup_set_int @ ( dom_int_list_o @ N2 ) @ ( dom_int_list_o @ M ) ) ) ).
% dom_map_add
thf(fact_296_opt__comp__append,axiom,
! [X: option_list_o,Y: option_list_o,Z2: option_list_o] :
( ( prefix454693708527911765comp_o @ ( prefix5314359684614007693append @ X @ Y ) @ Z2 )
=> ( prefix454693708527911765comp_o @ X @ Z2 ) ) ).
% opt_comp_append
thf(fact_297_opt__comp__append__2,axiom,
! [X: option_list_o,Y: option_list_o,Z2: option_list_o] :
( ( prefix454693708527911765comp_o @ X @ ( prefix5314359684614007693append @ Y @ Z2 ) )
=> ( prefix454693708527911765comp_o @ X @ Y ) ) ).
% opt_comp_append_2
thf(fact_298_opt__comp__append__3,axiom,
! [X: option_list_o,Y: option_list_o,Z2: option_list_o] :
( ( prefix454693708527911765comp_o @ ( prefix5314359684614007693append @ X @ Y ) @ ( prefix5314359684614007693append @ X @ Z2 ) )
=> ( prefix454693708527911765comp_o @ Y @ Z2 ) ) ).
% opt_comp_append_3
thf(fact_299_bind__split__asm,axiom,
! [P: option_list_o > $o,M: option_list_o,F2: list_o > option_list_o] :
( ( P @ ( bind_list_o_list_o @ M @ F2 ) )
= ( ~ ( ( ( M = none_list_o )
& ~ ( P @ none_list_o ) )
| ? [X3: list_o] :
( ( M
= ( some_list_o @ X3 ) )
& ~ ( P @ ( F2 @ X3 ) ) ) ) ) ) ).
% bind_split_asm
thf(fact_300_bind__split__asm,axiom,
! [P: option_int > $o,M: option_int,F2: int > option_int] :
( ( P @ ( bind_int_int @ M @ F2 ) )
= ( ~ ( ( ( M = none_int )
& ~ ( P @ none_int ) )
| ? [X3: int] :
( ( M
= ( some_int @ X3 ) )
& ~ ( P @ ( F2 @ X3 ) ) ) ) ) ) ).
% bind_split_asm
thf(fact_301_bind__split__asm,axiom,
! [P: option_nat > $o,M: option_int,F2: int > option_nat] :
( ( P @ ( bind_int_nat @ M @ F2 ) )
= ( ~ ( ( ( M = none_int )
& ~ ( P @ none_nat ) )
| ? [X3: int] :
( ( M
= ( some_int @ X3 ) )
& ~ ( P @ ( F2 @ X3 ) ) ) ) ) ) ).
% bind_split_asm
thf(fact_302_bind__split__asm,axiom,
! [P: option_int > $o,M: option_nat,F2: nat > option_int] :
( ( P @ ( bind_nat_int @ M @ F2 ) )
= ( ~ ( ( ( M = none_nat )
& ~ ( P @ none_int ) )
| ? [X3: nat] :
( ( M
= ( some_nat @ X3 ) )
& ~ ( P @ ( F2 @ X3 ) ) ) ) ) ) ).
% bind_split_asm
thf(fact_303_bind__split__asm,axiom,
! [P: option_nat > $o,M: option_nat,F2: nat > option_nat] :
( ( P @ ( bind_nat_nat @ M @ F2 ) )
= ( ~ ( ( ( M = none_nat )
& ~ ( P @ none_nat ) )
| ? [X3: nat] :
( ( M
= ( some_nat @ X3 ) )
& ~ ( P @ ( F2 @ X3 ) ) ) ) ) ) ).
% bind_split_asm
thf(fact_304_bind__split__asm,axiom,
! [P: option_int > $o,M: option_list_o,F2: list_o > option_int] :
( ( P @ ( bind_list_o_int @ M @ F2 ) )
= ( ~ ( ( ( M = none_list_o )
& ~ ( P @ none_int ) )
| ? [X3: list_o] :
( ( M
= ( some_list_o @ X3 ) )
& ~ ( P @ ( F2 @ X3 ) ) ) ) ) ) ).
% bind_split_asm
thf(fact_305_bind__split__asm,axiom,
! [P: option_nat > $o,M: option_list_o,F2: list_o > option_nat] :
( ( P @ ( bind_list_o_nat @ M @ F2 ) )
= ( ~ ( ( ( M = none_list_o )
& ~ ( P @ none_nat ) )
| ? [X3: list_o] :
( ( M
= ( some_list_o @ X3 ) )
& ~ ( P @ ( F2 @ X3 ) ) ) ) ) ) ).
% bind_split_asm
thf(fact_306_bind__split__asm,axiom,
! [P: option_list_o > $o,M: option_int,F2: int > option_list_o] :
( ( P @ ( bind_int_list_o @ M @ F2 ) )
= ( ~ ( ( ( M = none_int )
& ~ ( P @ none_list_o ) )
| ? [X3: int] :
( ( M
= ( some_int @ X3 ) )
& ~ ( P @ ( F2 @ X3 ) ) ) ) ) ) ).
% bind_split_asm
thf(fact_307_bind__split__asm,axiom,
! [P: option_list_o > $o,M: option_nat,F2: nat > option_list_o] :
( ( P @ ( bind_nat_list_o @ M @ F2 ) )
= ( ~ ( ( ( M = none_nat )
& ~ ( P @ none_list_o ) )
| ? [X3: nat] :
( ( M
= ( some_nat @ X3 ) )
& ~ ( P @ ( F2 @ X3 ) ) ) ) ) ) ).
% bind_split_asm
thf(fact_308_bind__split__asm,axiom,
! [P: option_option_list_o > $o,M: option_int,F2: int > option_option_list_o] :
( ( P @ ( bind_i874725573076888056list_o @ M @ F2 ) )
= ( ~ ( ( ( M = none_int )
& ~ ( P @ none_option_list_o ) )
| ? [X3: int] :
( ( M
= ( some_int @ X3 ) )
& ~ ( P @ ( F2 @ X3 ) ) ) ) ) ) ).
% bind_split_asm
thf(fact_309_fun__upd__upd,axiom,
! [F2: int > option_list_o,X: int,Y: option_list_o,Z2: option_list_o] :
( ( fun_up5258779583477023609list_o @ ( fun_up5258779583477023609list_o @ F2 @ X @ Y ) @ X @ Z2 )
= ( fun_up5258779583477023609list_o @ F2 @ X @ Z2 ) ) ).
% fun_upd_upd
thf(fact_310_fun__upd__upd,axiom,
! [F2: nat > option_list_o,X: nat,Y: option_list_o,Z2: option_list_o] :
( ( fun_up6685928411843432661list_o @ ( fun_up6685928411843432661list_o @ F2 @ X @ Y ) @ X @ Z2 )
= ( fun_up6685928411843432661list_o @ F2 @ X @ Z2 ) ) ).
% fun_upd_upd
thf(fact_311_fun__upd__upd,axiom,
! [F2: nat > option_nat,X: nat,Y: option_nat,Z2: option_nat] :
( ( fun_up1493157387958331631on_nat @ ( fun_up1493157387958331631on_nat @ F2 @ X @ Y ) @ X @ Z2 )
= ( fun_up1493157387958331631on_nat @ F2 @ X @ Z2 ) ) ).
% fun_upd_upd
thf(fact_312_fun__upd__upd,axiom,
! [F2: option_list_o > option_option_list_o,X: option_list_o,Y: option_option_list_o,Z2: option_option_list_o] :
( ( fun_up4915027116007172163list_o @ ( fun_up4915027116007172163list_o @ F2 @ X @ Y ) @ X @ Z2 )
= ( fun_up4915027116007172163list_o @ F2 @ X @ Z2 ) ) ).
% fun_upd_upd
thf(fact_313_fun__upd__triv,axiom,
! [F2: int > option_list_o,X: int] :
( ( fun_up5258779583477023609list_o @ F2 @ X @ ( F2 @ X ) )
= F2 ) ).
% fun_upd_triv
thf(fact_314_fun__upd__triv,axiom,
! [F2: nat > option_list_o,X: nat] :
( ( fun_up6685928411843432661list_o @ F2 @ X @ ( F2 @ X ) )
= F2 ) ).
% fun_upd_triv
thf(fact_315_fun__upd__triv,axiom,
! [F2: nat > option_nat,X: nat] :
( ( fun_up1493157387958331631on_nat @ F2 @ X @ ( F2 @ X ) )
= F2 ) ).
% fun_upd_triv
thf(fact_316_fun__upd__triv,axiom,
! [F2: option_list_o > option_option_list_o,X: option_list_o] :
( ( fun_up4915027116007172163list_o @ F2 @ X @ ( F2 @ X ) )
= F2 ) ).
% fun_upd_triv
thf(fact_317_fun__upd__apply,axiom,
( fun_up5258779583477023609list_o
= ( ^ [F: int > option_list_o,X3: int,Y3: option_list_o,Z3: int] : ( if_option_list_o @ ( Z3 = X3 ) @ Y3 @ ( F @ Z3 ) ) ) ) ).
% fun_upd_apply
thf(fact_318_fun__upd__apply,axiom,
( fun_up6685928411843432661list_o
= ( ^ [F: nat > option_list_o,X3: nat,Y3: option_list_o,Z3: nat] : ( if_option_list_o @ ( Z3 = X3 ) @ Y3 @ ( F @ Z3 ) ) ) ) ).
% fun_upd_apply
thf(fact_319_fun__upd__apply,axiom,
( fun_up1493157387958331631on_nat
= ( ^ [F: nat > option_nat,X3: nat,Y3: option_nat,Z3: nat] : ( if_option_nat @ ( Z3 = X3 ) @ Y3 @ ( F @ Z3 ) ) ) ) ).
% fun_upd_apply
thf(fact_320_fun__upd__apply,axiom,
( fun_up4915027116007172163list_o
= ( ^ [F: option_list_o > option_option_list_o,X3: option_list_o,Y3: option_option_list_o,Z3: option_list_o] : ( if_opt4852585628301394032list_o @ ( Z3 = X3 ) @ Y3 @ ( F @ Z3 ) ) ) ) ).
% fun_upd_apply
thf(fact_321_inj__on__empty,axiom,
! [F2: list_o > option_list_o] : ( inj_on2239224958054982839list_o @ F2 @ bot_bot_set_list_o ) ).
% inj_on_empty
thf(fact_322_inj__on__empty,axiom,
! [F2: nat > nat] : ( inj_on_nat_nat @ F2 @ bot_bot_set_nat ) ).
% inj_on_empty
thf(fact_323_inj__on__empty,axiom,
! [F2: nat > int] : ( inj_on_nat_int @ F2 @ bot_bot_set_nat ) ).
% inj_on_empty
thf(fact_324_inj__on__empty,axiom,
! [F2: int > nat] : ( inj_on_int_nat @ F2 @ bot_bot_set_int ) ).
% inj_on_empty
thf(fact_325_inj__on__empty,axiom,
! [F2: int > int] : ( inj_on_int_int @ F2 @ bot_bot_set_int ) ).
% inj_on_empty
thf(fact_326_inj__on__empty,axiom,
! [F2: nat > option_list_o] : ( inj_on1630180835328728801list_o @ F2 @ bot_bot_set_nat ) ).
% inj_on_empty
thf(fact_327_inj__on__empty,axiom,
! [F2: int > option_list_o] : ( inj_on203032006962319749list_o @ F2 @ bot_bot_set_int ) ).
% inj_on_empty
thf(fact_328_bind__runit,axiom,
! [X: option_int] :
( ( bind_int_int @ X @ some_int )
= X ) ).
% bind_runit
thf(fact_329_bind__runit,axiom,
! [X: option_nat] :
( ( bind_nat_nat @ X @ some_nat )
= X ) ).
% bind_runit
thf(fact_330_bind__runit,axiom,
! [X: option_option_list_o] :
( ( bind_o5440579883579728062list_o @ X @ some_option_list_o )
= X ) ).
% bind_runit
thf(fact_331_bind__runit,axiom,
! [X: option_list_o] :
( ( bind_list_o_list_o @ X @ some_list_o )
= X ) ).
% bind_runit
thf(fact_332_fun__upd__None__if__notin__dom,axiom,
! [K: nat,M: nat > option_list_o] :
( ~ ( member_nat @ K @ ( dom_nat_list_o @ M ) )
=> ( ( fun_up6685928411843432661list_o @ M @ K @ none_list_o )
= M ) ) ).
% fun_upd_None_if_notin_dom
thf(fact_333_fun__upd__None__if__notin__dom,axiom,
! [K: int,M: int > option_list_o] :
( ~ ( member_int @ K @ ( dom_int_list_o @ M ) )
=> ( ( fun_up5258779583477023609list_o @ M @ K @ none_list_o )
= M ) ) ).
% fun_upd_None_if_notin_dom
thf(fact_334_fun__upd__None__if__notin__dom,axiom,
! [K: nat,M: nat > option_int] :
( ~ ( member_nat @ K @ ( dom_nat_int @ M ) )
=> ( ( fun_up6538678405303910731on_int @ M @ K @ none_int )
= M ) ) ).
% fun_upd_None_if_notin_dom
thf(fact_335_fun__upd__None__if__notin__dom,axiom,
! [K: int,M: int > option_int] :
( ~ ( member_int @ K @ ( dom_int_int @ M ) )
=> ( ( fun_up8666045135305973159on_int @ M @ K @ none_int )
= M ) ) ).
% fun_upd_None_if_notin_dom
thf(fact_336_fun__upd__None__if__notin__dom,axiom,
! [K: int,M: int > option_nat] :
( ~ ( member_int @ K @ ( dom_int_nat @ M ) )
=> ( ( fun_up3620524117960394059on_nat @ M @ K @ none_nat )
= M ) ) ).
% fun_upd_None_if_notin_dom
thf(fact_337_fun__upd__None__if__notin__dom,axiom,
! [K: nat,M: nat > option_nat] :
( ~ ( member_nat @ K @ ( dom_nat_nat @ M ) )
=> ( ( fun_up1493157387958331631on_nat @ M @ K @ none_nat )
= M ) ) ).
% fun_upd_None_if_notin_dom
thf(fact_338_fun__upd__None__if__notin__dom,axiom,
! [K: option_nat,M: option_nat > option_int] :
( ~ ( member_option_nat @ K @ ( dom_option_nat_int @ M ) )
=> ( ( fun_up1794774578788926875on_int @ M @ K @ none_int )
= M ) ) ).
% fun_upd_None_if_notin_dom
thf(fact_339_fun__upd__None__if__notin__dom,axiom,
! [K: option_int,M: option_int > option_int] :
( ~ ( member_option_int @ K @ ( dom_option_int_int @ M ) )
=> ( ( fun_up7205856760492389367on_int @ M @ K @ none_int )
= M ) ) ).
% fun_upd_None_if_notin_dom
thf(fact_340_fun__upd__None__if__notin__dom,axiom,
! [K: list_o,M: list_o > option_int] :
( ~ ( member_list_o @ K @ ( dom_list_o_int @ M ) )
=> ( ( fun_up5495103757224608477on_int @ M @ K @ none_int )
= M ) ) ).
% fun_upd_None_if_notin_dom
thf(fact_341_fun__upd__None__if__notin__dom,axiom,
! [K: set_nat,M: set_nat > option_int] :
( ~ ( member_set_nat @ K @ ( dom_set_nat_int @ M ) )
=> ( ( fun_up1329515020090725377on_int @ M @ K @ none_int )
= M ) ) ).
% fun_upd_None_if_notin_dom
thf(fact_342_map__add__upd,axiom,
! [F2: int > option_list_o,G: int > option_list_o,X: int,Y: list_o] :
( ( map_add_int_list_o @ F2 @ ( fun_up5258779583477023609list_o @ G @ X @ ( some_list_o @ Y ) ) )
= ( fun_up5258779583477023609list_o @ ( map_add_int_list_o @ F2 @ G ) @ X @ ( some_list_o @ Y ) ) ) ).
% map_add_upd
thf(fact_343_map__add__upd,axiom,
! [F2: nat > option_list_o,G: nat > option_list_o,X: nat,Y: list_o] :
( ( map_add_nat_list_o @ F2 @ ( fun_up6685928411843432661list_o @ G @ X @ ( some_list_o @ Y ) ) )
= ( fun_up6685928411843432661list_o @ ( map_add_nat_list_o @ F2 @ G ) @ X @ ( some_list_o @ Y ) ) ) ).
% map_add_upd
thf(fact_344_map__add__upd,axiom,
! [F2: nat > option_nat,G: nat > option_nat,X: nat,Y: nat] :
( ( map_add_nat_nat @ F2 @ ( fun_up1493157387958331631on_nat @ G @ X @ ( some_nat @ Y ) ) )
= ( fun_up1493157387958331631on_nat @ ( map_add_nat_nat @ F2 @ G ) @ X @ ( some_nat @ Y ) ) ) ).
% map_add_upd
thf(fact_345_map__add__upd,axiom,
! [F2: option_list_o > option_option_list_o,G: option_list_o > option_option_list_o,X: option_list_o,Y: option_list_o] :
( ( map_ad7444082643596406197list_o @ F2 @ ( fun_up4915027116007172163list_o @ G @ X @ ( some_option_list_o @ Y ) ) )
= ( fun_up4915027116007172163list_o @ ( map_ad7444082643596406197list_o @ F2 @ G ) @ X @ ( some_option_list_o @ Y ) ) ) ).
% map_add_upd
thf(fact_346_fun__upd__def,axiom,
( fun_up5258779583477023609list_o
= ( ^ [F: int > option_list_o,A3: int,B3: option_list_o,X3: int] : ( if_option_list_o @ ( X3 = A3 ) @ B3 @ ( F @ X3 ) ) ) ) ).
% fun_upd_def
thf(fact_347_fun__upd__def,axiom,
( fun_up6685928411843432661list_o
= ( ^ [F: nat > option_list_o,A3: nat,B3: option_list_o,X3: nat] : ( if_option_list_o @ ( X3 = A3 ) @ B3 @ ( F @ X3 ) ) ) ) ).
% fun_upd_def
thf(fact_348_fun__upd__def,axiom,
( fun_up1493157387958331631on_nat
= ( ^ [F: nat > option_nat,A3: nat,B3: option_nat,X3: nat] : ( if_option_nat @ ( X3 = A3 ) @ B3 @ ( F @ X3 ) ) ) ) ).
% fun_upd_def
thf(fact_349_fun__upd__def,axiom,
( fun_up4915027116007172163list_o
= ( ^ [F: option_list_o > option_option_list_o,A3: option_list_o,B3: option_option_list_o,X3: option_list_o] : ( if_opt4852585628301394032list_o @ ( X3 = A3 ) @ B3 @ ( F @ X3 ) ) ) ) ).
% fun_upd_def
thf(fact_350_fun__upd__eqD,axiom,
! [F2: int > option_list_o,X: int,Y: option_list_o,G: int > option_list_o,Z2: option_list_o] :
( ( ( fun_up5258779583477023609list_o @ F2 @ X @ Y )
= ( fun_up5258779583477023609list_o @ G @ X @ Z2 ) )
=> ( Y = Z2 ) ) ).
% fun_upd_eqD
thf(fact_351_fun__upd__eqD,axiom,
! [F2: nat > option_list_o,X: nat,Y: option_list_o,G: nat > option_list_o,Z2: option_list_o] :
( ( ( fun_up6685928411843432661list_o @ F2 @ X @ Y )
= ( fun_up6685928411843432661list_o @ G @ X @ Z2 ) )
=> ( Y = Z2 ) ) ).
% fun_upd_eqD
thf(fact_352_fun__upd__eqD,axiom,
! [F2: nat > option_nat,X: nat,Y: option_nat,G: nat > option_nat,Z2: option_nat] :
( ( ( fun_up1493157387958331631on_nat @ F2 @ X @ Y )
= ( fun_up1493157387958331631on_nat @ G @ X @ Z2 ) )
=> ( Y = Z2 ) ) ).
% fun_upd_eqD
thf(fact_353_fun__upd__eqD,axiom,
! [F2: option_list_o > option_option_list_o,X: option_list_o,Y: option_option_list_o,G: option_list_o > option_option_list_o,Z2: option_option_list_o] :
( ( ( fun_up4915027116007172163list_o @ F2 @ X @ Y )
= ( fun_up4915027116007172163list_o @ G @ X @ Z2 ) )
=> ( Y = Z2 ) ) ).
% fun_upd_eqD
thf(fact_354_fun__upd__idem,axiom,
! [F2: int > option_list_o,X: int,Y: option_list_o] :
( ( ( F2 @ X )
= Y )
=> ( ( fun_up5258779583477023609list_o @ F2 @ X @ Y )
= F2 ) ) ).
% fun_upd_idem
thf(fact_355_fun__upd__idem,axiom,
! [F2: nat > option_list_o,X: nat,Y: option_list_o] :
( ( ( F2 @ X )
= Y )
=> ( ( fun_up6685928411843432661list_o @ F2 @ X @ Y )
= F2 ) ) ).
% fun_upd_idem
thf(fact_356_fun__upd__idem,axiom,
! [F2: nat > option_nat,X: nat,Y: option_nat] :
( ( ( F2 @ X )
= Y )
=> ( ( fun_up1493157387958331631on_nat @ F2 @ X @ Y )
= F2 ) ) ).
% fun_upd_idem
thf(fact_357_fun__upd__idem,axiom,
! [F2: option_list_o > option_option_list_o,X: option_list_o,Y: option_option_list_o] :
( ( ( F2 @ X )
= Y )
=> ( ( fun_up4915027116007172163list_o @ F2 @ X @ Y )
= F2 ) ) ).
% fun_upd_idem
thf(fact_358_fun__upd__same,axiom,
! [F2: int > option_list_o,X: int,Y: option_list_o] :
( ( fun_up5258779583477023609list_o @ F2 @ X @ Y @ X )
= Y ) ).
% fun_upd_same
thf(fact_359_fun__upd__same,axiom,
! [F2: nat > option_list_o,X: nat,Y: option_list_o] :
( ( fun_up6685928411843432661list_o @ F2 @ X @ Y @ X )
= Y ) ).
% fun_upd_same
thf(fact_360_fun__upd__same,axiom,
! [F2: nat > option_nat,X: nat,Y: option_nat] :
( ( fun_up1493157387958331631on_nat @ F2 @ X @ Y @ X )
= Y ) ).
% fun_upd_same
thf(fact_361_fun__upd__same,axiom,
! [F2: option_list_o > option_option_list_o,X: option_list_o,Y: option_option_list_o] :
( ( fun_up4915027116007172163list_o @ F2 @ X @ Y @ X )
= Y ) ).
% fun_upd_same
thf(fact_362_fun__upd__other,axiom,
! [Z2: int,X: int,F2: int > option_list_o,Y: option_list_o] :
( ( Z2 != X )
=> ( ( fun_up5258779583477023609list_o @ F2 @ X @ Y @ Z2 )
= ( F2 @ Z2 ) ) ) ).
% fun_upd_other
thf(fact_363_fun__upd__other,axiom,
! [Z2: nat,X: nat,F2: nat > option_list_o,Y: option_list_o] :
( ( Z2 != X )
=> ( ( fun_up6685928411843432661list_o @ F2 @ X @ Y @ Z2 )
= ( F2 @ Z2 ) ) ) ).
% fun_upd_other
thf(fact_364_fun__upd__other,axiom,
! [Z2: nat,X: nat,F2: nat > option_nat,Y: option_nat] :
( ( Z2 != X )
=> ( ( fun_up1493157387958331631on_nat @ F2 @ X @ Y @ Z2 )
= ( F2 @ Z2 ) ) ) ).
% fun_upd_other
thf(fact_365_fun__upd__other,axiom,
! [Z2: option_list_o,X: option_list_o,F2: option_list_o > option_option_list_o,Y: option_option_list_o] :
( ( Z2 != X )
=> ( ( fun_up4915027116007172163list_o @ F2 @ X @ Y @ Z2 )
= ( F2 @ Z2 ) ) ) ).
% fun_upd_other
thf(fact_366_fun__upd__twist,axiom,
! [A2: int,C: int,M: int > option_list_o,B2: option_list_o,D: option_list_o] :
( ( A2 != C )
=> ( ( fun_up5258779583477023609list_o @ ( fun_up5258779583477023609list_o @ M @ A2 @ B2 ) @ C @ D )
= ( fun_up5258779583477023609list_o @ ( fun_up5258779583477023609list_o @ M @ C @ D ) @ A2 @ B2 ) ) ) ).
% fun_upd_twist
thf(fact_367_fun__upd__twist,axiom,
! [A2: nat,C: nat,M: nat > option_list_o,B2: option_list_o,D: option_list_o] :
( ( A2 != C )
=> ( ( fun_up6685928411843432661list_o @ ( fun_up6685928411843432661list_o @ M @ A2 @ B2 ) @ C @ D )
= ( fun_up6685928411843432661list_o @ ( fun_up6685928411843432661list_o @ M @ C @ D ) @ A2 @ B2 ) ) ) ).
% fun_upd_twist
thf(fact_368_fun__upd__twist,axiom,
! [A2: nat,C: nat,M: nat > option_nat,B2: option_nat,D: option_nat] :
( ( A2 != C )
=> ( ( fun_up1493157387958331631on_nat @ ( fun_up1493157387958331631on_nat @ M @ A2 @ B2 ) @ C @ D )
= ( fun_up1493157387958331631on_nat @ ( fun_up1493157387958331631on_nat @ M @ C @ D ) @ A2 @ B2 ) ) ) ).
% fun_upd_twist
thf(fact_369_fun__upd__twist,axiom,
! [A2: option_list_o,C: option_list_o,M: option_list_o > option_option_list_o,B2: option_option_list_o,D: option_option_list_o] :
( ( A2 != C )
=> ( ( fun_up4915027116007172163list_o @ ( fun_up4915027116007172163list_o @ M @ A2 @ B2 ) @ C @ D )
= ( fun_up4915027116007172163list_o @ ( fun_up4915027116007172163list_o @ M @ C @ D ) @ A2 @ B2 ) ) ) ).
% fun_upd_twist
thf(fact_370_fun__upd__idem__iff,axiom,
! [F2: int > option_list_o,X: int,Y: option_list_o] :
( ( ( fun_up5258779583477023609list_o @ F2 @ X @ Y )
= F2 )
= ( ( F2 @ X )
= Y ) ) ).
% fun_upd_idem_iff
thf(fact_371_fun__upd__idem__iff,axiom,
! [F2: nat > option_list_o,X: nat,Y: option_list_o] :
( ( ( fun_up6685928411843432661list_o @ F2 @ X @ Y )
= F2 )
= ( ( F2 @ X )
= Y ) ) ).
% fun_upd_idem_iff
thf(fact_372_fun__upd__idem__iff,axiom,
! [F2: nat > option_nat,X: nat,Y: option_nat] :
( ( ( fun_up1493157387958331631on_nat @ F2 @ X @ Y )
= F2 )
= ( ( F2 @ X )
= Y ) ) ).
% fun_upd_idem_iff
thf(fact_373_fun__upd__idem__iff,axiom,
! [F2: option_list_o > option_option_list_o,X: option_list_o,Y: option_option_list_o] :
( ( ( fun_up4915027116007172163list_o @ F2 @ X @ Y )
= F2 )
= ( ( F2 @ X )
= Y ) ) ).
% fun_upd_idem_iff
thf(fact_374_bind__option__cong__code,axiom,
! [X: option_list_o,Y: option_list_o,F2: list_o > option_list_o] :
( ( X = Y )
=> ( ( bind_list_o_list_o @ X @ F2 )
= ( bind_list_o_list_o @ Y @ F2 ) ) ) ).
% bind_option_cong_code
thf(fact_375_bind__eq__None__conv,axiom,
! [A2: option_list_o,F2: list_o > option_list_o] :
( ( ( bind_list_o_list_o @ A2 @ F2 )
= none_list_o )
= ( ( A2 = none_list_o )
| ( ( F2 @ ( the_list_o @ A2 ) )
= none_list_o ) ) ) ).
% bind_eq_None_conv
thf(fact_376_bind__eq__None__conv,axiom,
! [A2: option_int,F2: int > option_int] :
( ( ( bind_int_int @ A2 @ F2 )
= none_int )
= ( ( A2 = none_int )
| ( ( F2 @ ( the_int @ A2 ) )
= none_int ) ) ) ).
% bind_eq_None_conv
thf(fact_377_bind__eq__None__conv,axiom,
! [A2: option_nat,F2: nat > option_int] :
( ( ( bind_nat_int @ A2 @ F2 )
= none_int )
= ( ( A2 = none_nat )
| ( ( F2 @ ( the_nat @ A2 ) )
= none_int ) ) ) ).
% bind_eq_None_conv
thf(fact_378_bind__eq__None__conv,axiom,
! [A2: option_int,F2: int > option_nat] :
( ( ( bind_int_nat @ A2 @ F2 )
= none_nat )
= ( ( A2 = none_int )
| ( ( F2 @ ( the_int @ A2 ) )
= none_nat ) ) ) ).
% bind_eq_None_conv
thf(fact_379_bind__eq__None__conv,axiom,
! [A2: option_nat,F2: nat > option_nat] :
( ( ( bind_nat_nat @ A2 @ F2 )
= none_nat )
= ( ( A2 = none_nat )
| ( ( F2 @ ( the_nat @ A2 ) )
= none_nat ) ) ) ).
% bind_eq_None_conv
thf(fact_380_bind__eq__None__conv,axiom,
! [A2: option_int,F2: int > option_list_o] :
( ( ( bind_int_list_o @ A2 @ F2 )
= none_list_o )
= ( ( A2 = none_int )
| ( ( F2 @ ( the_int @ A2 ) )
= none_list_o ) ) ) ).
% bind_eq_None_conv
thf(fact_381_bind__eq__None__conv,axiom,
! [A2: option_nat,F2: nat > option_list_o] :
( ( ( bind_nat_list_o @ A2 @ F2 )
= none_list_o )
= ( ( A2 = none_nat )
| ( ( F2 @ ( the_nat @ A2 ) )
= none_list_o ) ) ) ).
% bind_eq_None_conv
thf(fact_382_bind__eq__None__conv,axiom,
! [A2: option_list_o,F2: list_o > option_int] :
( ( ( bind_list_o_int @ A2 @ F2 )
= none_int )
= ( ( A2 = none_list_o )
| ( ( F2 @ ( the_list_o @ A2 ) )
= none_int ) ) ) ).
% bind_eq_None_conv
thf(fact_383_bind__eq__None__conv,axiom,
! [A2: option_list_o,F2: list_o > option_nat] :
( ( ( bind_list_o_nat @ A2 @ F2 )
= none_nat )
= ( ( A2 = none_list_o )
| ( ( F2 @ ( the_list_o @ A2 ) )
= none_nat ) ) ) ).
% bind_eq_None_conv
thf(fact_384_bind__eq__None__conv,axiom,
! [A2: option_option_list_o,F2: option_list_o > option_int] :
( ( ( bind_o3125634783446094546_o_int @ A2 @ F2 )
= none_int )
= ( ( A2 = none_option_list_o )
| ( ( F2 @ ( the_option_list_o @ A2 ) )
= none_int ) ) ) ).
% bind_eq_None_conv
thf(fact_385_map__upd__Some__unfold,axiom,
! [M: int > option_list_o,A2: int,B2: list_o,X: int,Y: list_o] :
( ( ( fun_up5258779583477023609list_o @ M @ A2 @ ( some_list_o @ B2 ) @ X )
= ( some_list_o @ Y ) )
= ( ( ( X = A2 )
& ( B2 = Y ) )
| ( ( X != A2 )
& ( ( M @ X )
= ( some_list_o @ Y ) ) ) ) ) ).
% map_upd_Some_unfold
thf(fact_386_map__upd__Some__unfold,axiom,
! [M: nat > option_list_o,A2: nat,B2: list_o,X: nat,Y: list_o] :
( ( ( fun_up6685928411843432661list_o @ M @ A2 @ ( some_list_o @ B2 ) @ X )
= ( some_list_o @ Y ) )
= ( ( ( X = A2 )
& ( B2 = Y ) )
| ( ( X != A2 )
& ( ( M @ X )
= ( some_list_o @ Y ) ) ) ) ) ).
% map_upd_Some_unfold
thf(fact_387_map__upd__Some__unfold,axiom,
! [M: nat > option_nat,A2: nat,B2: nat,X: nat,Y: nat] :
( ( ( fun_up1493157387958331631on_nat @ M @ A2 @ ( some_nat @ B2 ) @ X )
= ( some_nat @ Y ) )
= ( ( ( X = A2 )
& ( B2 = Y ) )
| ( ( X != A2 )
& ( ( M @ X )
= ( some_nat @ Y ) ) ) ) ) ).
% map_upd_Some_unfold
thf(fact_388_map__upd__Some__unfold,axiom,
! [M: option_list_o > option_option_list_o,A2: option_list_o,B2: option_list_o,X: option_list_o,Y: option_list_o] :
( ( ( fun_up4915027116007172163list_o @ M @ A2 @ ( some_option_list_o @ B2 ) @ X )
= ( some_option_list_o @ Y ) )
= ( ( ( X = A2 )
& ( B2 = Y ) )
| ( ( X != A2 )
& ( ( M @ X )
= ( some_option_list_o @ Y ) ) ) ) ) ).
% map_upd_Some_unfold
thf(fact_389_map__upd__triv,axiom,
! [T: int > option_list_o,K: int,X: list_o] :
( ( ( T @ K )
= ( some_list_o @ X ) )
=> ( ( fun_up5258779583477023609list_o @ T @ K @ ( some_list_o @ X ) )
= T ) ) ).
% map_upd_triv
thf(fact_390_map__upd__triv,axiom,
! [T: nat > option_list_o,K: nat,X: list_o] :
( ( ( T @ K )
= ( some_list_o @ X ) )
=> ( ( fun_up6685928411843432661list_o @ T @ K @ ( some_list_o @ X ) )
= T ) ) ).
% map_upd_triv
thf(fact_391_map__upd__triv,axiom,
! [T: nat > option_nat,K: nat,X: nat] :
( ( ( T @ K )
= ( some_nat @ X ) )
=> ( ( fun_up1493157387958331631on_nat @ T @ K @ ( some_nat @ X ) )
= T ) ) ).
% map_upd_triv
thf(fact_392_map__upd__triv,axiom,
! [T: option_list_o > option_option_list_o,K: option_list_o,X: option_list_o] :
( ( ( T @ K )
= ( some_option_list_o @ X ) )
=> ( ( fun_up4915027116007172163list_o @ T @ K @ ( some_option_list_o @ X ) )
= T ) ) ).
% map_upd_triv
thf(fact_393_map__upd__eqD1,axiom,
! [M: int > option_list_o,A2: int,X: list_o,N2: int > option_list_o,Y: list_o] :
( ( ( fun_up5258779583477023609list_o @ M @ A2 @ ( some_list_o @ X ) )
= ( fun_up5258779583477023609list_o @ N2 @ A2 @ ( some_list_o @ Y ) ) )
=> ( X = Y ) ) ).
% map_upd_eqD1
thf(fact_394_map__upd__eqD1,axiom,
! [M: nat > option_list_o,A2: nat,X: list_o,N2: nat > option_list_o,Y: list_o] :
( ( ( fun_up6685928411843432661list_o @ M @ A2 @ ( some_list_o @ X ) )
= ( fun_up6685928411843432661list_o @ N2 @ A2 @ ( some_list_o @ Y ) ) )
=> ( X = Y ) ) ).
% map_upd_eqD1
thf(fact_395_map__upd__eqD1,axiom,
! [M: nat > option_nat,A2: nat,X: nat,N2: nat > option_nat,Y: nat] :
( ( ( fun_up1493157387958331631on_nat @ M @ A2 @ ( some_nat @ X ) )
= ( fun_up1493157387958331631on_nat @ N2 @ A2 @ ( some_nat @ Y ) ) )
=> ( X = Y ) ) ).
% map_upd_eqD1
thf(fact_396_map__upd__eqD1,axiom,
! [M: option_list_o > option_option_list_o,A2: option_list_o,X: option_list_o,N2: option_list_o > option_option_list_o,Y: option_list_o] :
( ( ( fun_up4915027116007172163list_o @ M @ A2 @ ( some_option_list_o @ X ) )
= ( fun_up4915027116007172163list_o @ N2 @ A2 @ ( some_option_list_o @ Y ) ) )
=> ( X = Y ) ) ).
% map_upd_eqD1
thf(fact_397_option_Oexpand,axiom,
! [Option: option_int,Option2: option_int] :
( ( ( Option = none_int )
= ( Option2 = none_int ) )
=> ( ( ( Option != none_int )
=> ( ( Option2 != none_int )
=> ( ( the_int @ Option )
= ( the_int @ Option2 ) ) ) )
=> ( Option = Option2 ) ) ) ).
% option.expand
thf(fact_398_option_Oexpand,axiom,
! [Option: option_nat,Option2: option_nat] :
( ( ( Option = none_nat )
= ( Option2 = none_nat ) )
=> ( ( ( Option != none_nat )
=> ( ( Option2 != none_nat )
=> ( ( the_nat @ Option )
= ( the_nat @ Option2 ) ) ) )
=> ( Option = Option2 ) ) ) ).
% option.expand
thf(fact_399_option_Oexpand,axiom,
! [Option: option_option_list_o,Option2: option_option_list_o] :
( ( ( Option = none_option_list_o )
= ( Option2 = none_option_list_o ) )
=> ( ( ( Option != none_option_list_o )
=> ( ( Option2 != none_option_list_o )
=> ( ( the_option_list_o @ Option )
= ( the_option_list_o @ Option2 ) ) ) )
=> ( Option = Option2 ) ) ) ).
% option.expand
thf(fact_400_option_Oexpand,axiom,
! [Option: option_list_o,Option2: option_list_o] :
( ( ( Option = none_list_o )
= ( Option2 = none_list_o ) )
=> ( ( ( Option != none_list_o )
=> ( ( Option2 != none_list_o )
=> ( ( the_list_o @ Option )
= ( the_list_o @ Option2 ) ) ) )
=> ( Option = Option2 ) ) ) ).
% option.expand
thf(fact_401_option_Osel,axiom,
! [X22: int] :
( ( the_int @ ( some_int @ X22 ) )
= X22 ) ).
% option.sel
thf(fact_402_option_Osel,axiom,
! [X22: nat] :
( ( the_nat @ ( some_nat @ X22 ) )
= X22 ) ).
% option.sel
thf(fact_403_option_Osel,axiom,
! [X22: option_list_o] :
( ( the_option_list_o @ ( some_option_list_o @ X22 ) )
= X22 ) ).
% option.sel
thf(fact_404_option_Osel,axiom,
! [X22: list_o] :
( ( the_list_o @ ( some_list_o @ X22 ) )
= X22 ) ).
% option.sel
thf(fact_405_bind_Obind__lzero,axiom,
! [F2: list_o > option_list_o] :
( ( bind_list_o_list_o @ none_list_o @ F2 )
= none_list_o ) ).
% bind.bind_lzero
thf(fact_406_bind_Obind__lzero,axiom,
! [F2: int > option_int] :
( ( bind_int_int @ none_int @ F2 )
= none_int ) ).
% bind.bind_lzero
thf(fact_407_bind_Obind__lzero,axiom,
! [F2: int > option_nat] :
( ( bind_int_nat @ none_int @ F2 )
= none_nat ) ).
% bind.bind_lzero
thf(fact_408_bind_Obind__lzero,axiom,
! [F2: nat > option_int] :
( ( bind_nat_int @ none_nat @ F2 )
= none_int ) ).
% bind.bind_lzero
thf(fact_409_bind_Obind__lzero,axiom,
! [F2: nat > option_nat] :
( ( bind_nat_nat @ none_nat @ F2 )
= none_nat ) ).
% bind.bind_lzero
thf(fact_410_bind_Obind__lzero,axiom,
! [F2: list_o > option_int] :
( ( bind_list_o_int @ none_list_o @ F2 )
= none_int ) ).
% bind.bind_lzero
thf(fact_411_bind_Obind__lzero,axiom,
! [F2: list_o > option_nat] :
( ( bind_list_o_nat @ none_list_o @ F2 )
= none_nat ) ).
% bind.bind_lzero
thf(fact_412_bind_Obind__lzero,axiom,
! [F2: int > option_list_o] :
( ( bind_int_list_o @ none_int @ F2 )
= none_list_o ) ).
% bind.bind_lzero
thf(fact_413_bind_Obind__lzero,axiom,
! [F2: nat > option_list_o] :
( ( bind_nat_list_o @ none_nat @ F2 )
= none_list_o ) ).
% bind.bind_lzero
thf(fact_414_bind_Obind__lzero,axiom,
! [F2: int > option_option_list_o] :
( ( bind_i874725573076888056list_o @ none_int @ F2 )
= none_option_list_o ) ).
% bind.bind_lzero
thf(fact_415_bind__eq__Some__conv,axiom,
! [F2: option_list_o,G: list_o > option_list_o,X: list_o] :
( ( ( bind_list_o_list_o @ F2 @ G )
= ( some_list_o @ X ) )
= ( ? [Y3: list_o] :
( ( F2
= ( some_list_o @ Y3 ) )
& ( ( G @ Y3 )
= ( some_list_o @ X ) ) ) ) ) ).
% bind_eq_Some_conv
thf(fact_416_bind__eq__Some__conv,axiom,
! [F2: option_int,G: int > option_int,X: int] :
( ( ( bind_int_int @ F2 @ G )
= ( some_int @ X ) )
= ( ? [Y3: int] :
( ( F2
= ( some_int @ Y3 ) )
& ( ( G @ Y3 )
= ( some_int @ X ) ) ) ) ) ).
% bind_eq_Some_conv
thf(fact_417_bind__eq__Some__conv,axiom,
! [F2: option_nat,G: nat > option_int,X: int] :
( ( ( bind_nat_int @ F2 @ G )
= ( some_int @ X ) )
= ( ? [Y3: nat] :
( ( F2
= ( some_nat @ Y3 ) )
& ( ( G @ Y3 )
= ( some_int @ X ) ) ) ) ) ).
% bind_eq_Some_conv
thf(fact_418_bind__eq__Some__conv,axiom,
! [F2: option_int,G: int > option_nat,X: nat] :
( ( ( bind_int_nat @ F2 @ G )
= ( some_nat @ X ) )
= ( ? [Y3: int] :
( ( F2
= ( some_int @ Y3 ) )
& ( ( G @ Y3 )
= ( some_nat @ X ) ) ) ) ) ).
% bind_eq_Some_conv
thf(fact_419_bind__eq__Some__conv,axiom,
! [F2: option_nat,G: nat > option_nat,X: nat] :
( ( ( bind_nat_nat @ F2 @ G )
= ( some_nat @ X ) )
= ( ? [Y3: nat] :
( ( F2
= ( some_nat @ Y3 ) )
& ( ( G @ Y3 )
= ( some_nat @ X ) ) ) ) ) ).
% bind_eq_Some_conv
thf(fact_420_bind__eq__Some__conv,axiom,
! [F2: option_int,G: int > option_list_o,X: list_o] :
( ( ( bind_int_list_o @ F2 @ G )
= ( some_list_o @ X ) )
= ( ? [Y3: int] :
( ( F2
= ( some_int @ Y3 ) )
& ( ( G @ Y3 )
= ( some_list_o @ X ) ) ) ) ) ).
% bind_eq_Some_conv
thf(fact_421_bind__eq__Some__conv,axiom,
! [F2: option_nat,G: nat > option_list_o,X: list_o] :
( ( ( bind_nat_list_o @ F2 @ G )
= ( some_list_o @ X ) )
= ( ? [Y3: nat] :
( ( F2
= ( some_nat @ Y3 ) )
& ( ( G @ Y3 )
= ( some_list_o @ X ) ) ) ) ) ).
% bind_eq_Some_conv
thf(fact_422_bind__eq__Some__conv,axiom,
! [F2: option_list_o,G: list_o > option_int,X: int] :
( ( ( bind_list_o_int @ F2 @ G )
= ( some_int @ X ) )
= ( ? [Y3: list_o] :
( ( F2
= ( some_list_o @ Y3 ) )
& ( ( G @ Y3 )
= ( some_int @ X ) ) ) ) ) ).
% bind_eq_Some_conv
thf(fact_423_bind__eq__Some__conv,axiom,
! [F2: option_list_o,G: list_o > option_nat,X: nat] :
( ( ( bind_list_o_nat @ F2 @ G )
= ( some_nat @ X ) )
= ( ? [Y3: list_o] :
( ( F2
= ( some_list_o @ Y3 ) )
& ( ( G @ Y3 )
= ( some_nat @ X ) ) ) ) ) ).
% bind_eq_Some_conv
thf(fact_424_bind__eq__Some__conv,axiom,
! [F2: option_option_list_o,G: option_list_o > option_int,X: int] :
( ( ( bind_o3125634783446094546_o_int @ F2 @ G )
= ( some_int @ X ) )
= ( ? [Y3: option_list_o] :
( ( F2
= ( some_option_list_o @ Y3 ) )
& ( ( G @ Y3 )
= ( some_int @ X ) ) ) ) ) ).
% bind_eq_Some_conv
thf(fact_425_Option_Obind__cong,axiom,
! [X: option_list_o,Y: option_list_o,F2: list_o > option_list_o,G: list_o > option_list_o] :
( ( X = Y )
=> ( ! [A4: list_o] :
( ( Y
= ( some_list_o @ A4 ) )
=> ( ( F2 @ A4 )
= ( G @ A4 ) ) )
=> ( ( bind_list_o_list_o @ X @ F2 )
= ( bind_list_o_list_o @ Y @ G ) ) ) ) ).
% Option.bind_cong
thf(fact_426_bind_Obind__lunit,axiom,
! [X: list_o,F2: list_o > option_list_o] :
( ( bind_list_o_list_o @ ( some_list_o @ X ) @ F2 )
= ( F2 @ X ) ) ).
% bind.bind_lunit
thf(fact_427_map__upd__nonempty,axiom,
! [T: int > option_list_o,K: int,X: list_o] :
( ( fun_up5258779583477023609list_o @ T @ K @ ( some_list_o @ X ) )
!= ( ^ [X3: int] : none_list_o ) ) ).
% map_upd_nonempty
thf(fact_428_map__upd__nonempty,axiom,
! [T: nat > option_list_o,K: nat,X: list_o] :
( ( fun_up6685928411843432661list_o @ T @ K @ ( some_list_o @ X ) )
!= ( ^ [X3: nat] : none_list_o ) ) ).
% map_upd_nonempty
thf(fact_429_map__upd__nonempty,axiom,
! [T: nat > option_nat,K: nat,X: nat] :
( ( fun_up1493157387958331631on_nat @ T @ K @ ( some_nat @ X ) )
!= ( ^ [X3: nat] : none_nat ) ) ).
% map_upd_nonempty
thf(fact_430_map__upd__nonempty,axiom,
! [T: option_list_o > option_option_list_o,K: option_list_o,X: option_list_o] :
( ( fun_up4915027116007172163list_o @ T @ K @ ( some_option_list_o @ X ) )
!= ( ^ [X3: option_list_o] : none_option_list_o ) ) ).
% map_upd_nonempty
thf(fact_431_opt__append_Osimps_I2_J,axiom,
! [Uv: option_list_o] :
( ( prefix5314359684614007693append @ none_list_o @ Uv )
= none_list_o ) ).
% opt_append.simps(2)
thf(fact_432_opt__append_Osimps_I3_J,axiom,
! [Uu: option_list_o] :
( ( prefix5314359684614007693append @ Uu @ none_list_o )
= none_list_o ) ).
% opt_append.simps(3)
thf(fact_433_option_Oexhaust__sel,axiom,
! [Option: option_int] :
( ( Option != none_int )
=> ( Option
= ( some_int @ ( the_int @ Option ) ) ) ) ).
% option.exhaust_sel
thf(fact_434_option_Oexhaust__sel,axiom,
! [Option: option_nat] :
( ( Option != none_nat )
=> ( Option
= ( some_nat @ ( the_nat @ Option ) ) ) ) ).
% option.exhaust_sel
thf(fact_435_option_Oexhaust__sel,axiom,
! [Option: option_option_list_o] :
( ( Option != none_option_list_o )
=> ( Option
= ( some_option_list_o @ ( the_option_list_o @ Option ) ) ) ) ).
% option.exhaust_sel
thf(fact_436_option_Oexhaust__sel,axiom,
! [Option: option_list_o] :
( ( Option != none_list_o )
=> ( Option
= ( some_list_o @ ( the_list_o @ Option ) ) ) ) ).
% option.exhaust_sel
thf(fact_437_bind__split,axiom,
! [P: option_list_o > $o,M: option_list_o,F2: list_o > option_list_o] :
( ( P @ ( bind_list_o_list_o @ M @ F2 ) )
= ( ( ( M = none_list_o )
=> ( P @ none_list_o ) )
& ! [V2: list_o] :
( ( M
= ( some_list_o @ V2 ) )
=> ( P @ ( F2 @ V2 ) ) ) ) ) ).
% bind_split
thf(fact_438_bind__split,axiom,
! [P: option_int > $o,M: option_int,F2: int > option_int] :
( ( P @ ( bind_int_int @ M @ F2 ) )
= ( ( ( M = none_int )
=> ( P @ none_int ) )
& ! [V2: int] :
( ( M
= ( some_int @ V2 ) )
=> ( P @ ( F2 @ V2 ) ) ) ) ) ).
% bind_split
thf(fact_439_bind__split,axiom,
! [P: option_nat > $o,M: option_int,F2: int > option_nat] :
( ( P @ ( bind_int_nat @ M @ F2 ) )
= ( ( ( M = none_int )
=> ( P @ none_nat ) )
& ! [V2: int] :
( ( M
= ( some_int @ V2 ) )
=> ( P @ ( F2 @ V2 ) ) ) ) ) ).
% bind_split
thf(fact_440_bind__split,axiom,
! [P: option_int > $o,M: option_nat,F2: nat > option_int] :
( ( P @ ( bind_nat_int @ M @ F2 ) )
= ( ( ( M = none_nat )
=> ( P @ none_int ) )
& ! [V2: nat] :
( ( M
= ( some_nat @ V2 ) )
=> ( P @ ( F2 @ V2 ) ) ) ) ) ).
% bind_split
thf(fact_441_bind__split,axiom,
! [P: option_nat > $o,M: option_nat,F2: nat > option_nat] :
( ( P @ ( bind_nat_nat @ M @ F2 ) )
= ( ( ( M = none_nat )
=> ( P @ none_nat ) )
& ! [V2: nat] :
( ( M
= ( some_nat @ V2 ) )
=> ( P @ ( F2 @ V2 ) ) ) ) ) ).
% bind_split
thf(fact_442_bind__split,axiom,
! [P: option_int > $o,M: option_list_o,F2: list_o > option_int] :
( ( P @ ( bind_list_o_int @ M @ F2 ) )
= ( ( ( M = none_list_o )
=> ( P @ none_int ) )
& ! [V2: list_o] :
( ( M
= ( some_list_o @ V2 ) )
=> ( P @ ( F2 @ V2 ) ) ) ) ) ).
% bind_split
thf(fact_443_bind__split,axiom,
! [P: option_nat > $o,M: option_list_o,F2: list_o > option_nat] :
( ( P @ ( bind_list_o_nat @ M @ F2 ) )
= ( ( ( M = none_list_o )
=> ( P @ none_nat ) )
& ! [V2: list_o] :
( ( M
= ( some_list_o @ V2 ) )
=> ( P @ ( F2 @ V2 ) ) ) ) ) ).
% bind_split
thf(fact_444_bind__split,axiom,
! [P: option_list_o > $o,M: option_int,F2: int > option_list_o] :
( ( P @ ( bind_int_list_o @ M @ F2 ) )
= ( ( ( M = none_int )
=> ( P @ none_list_o ) )
& ! [V2: int] :
( ( M
= ( some_int @ V2 ) )
=> ( P @ ( F2 @ V2 ) ) ) ) ) ).
% bind_split
thf(fact_445_bind__split,axiom,
! [P: option_list_o > $o,M: option_nat,F2: nat > option_list_o] :
( ( P @ ( bind_nat_list_o @ M @ F2 ) )
= ( ( ( M = none_nat )
=> ( P @ none_list_o ) )
& ! [V2: nat] :
( ( M
= ( some_nat @ V2 ) )
=> ( P @ ( F2 @ V2 ) ) ) ) ) ).
% bind_split
thf(fact_446_bind__split,axiom,
! [P: option_option_list_o > $o,M: option_int,F2: int > option_option_list_o] :
( ( P @ ( bind_i874725573076888056list_o @ M @ F2 ) )
= ( ( ( M = none_int )
=> ( P @ none_option_list_o ) )
& ! [V2: int] :
( ( M
= ( some_int @ V2 ) )
=> ( P @ ( F2 @ V2 ) ) ) ) ) ).
% bind_split
thf(fact_447_Un__empty,axiom,
! [A: set_list_o,B4: set_list_o] :
( ( ( sup_sup_set_list_o @ A @ B4 )
= bot_bot_set_list_o )
= ( ( A = bot_bot_set_list_o )
& ( B4 = bot_bot_set_list_o ) ) ) ).
% Un_empty
thf(fact_448_Un__empty,axiom,
! [A: set_nat,B4: set_nat] :
( ( ( sup_sup_set_nat @ A @ B4 )
= bot_bot_set_nat )
= ( ( A = bot_bot_set_nat )
& ( B4 = bot_bot_set_nat ) ) ) ).
% Un_empty
thf(fact_449_Un__empty,axiom,
! [A: set_int,B4: set_int] :
( ( ( sup_sup_set_int @ A @ B4 )
= bot_bot_set_int )
= ( ( A = bot_bot_set_int )
& ( B4 = bot_bot_set_int ) ) ) ).
% Un_empty
thf(fact_450_sup__bot_Oright__neutral,axiom,
! [A2: set_list_o] :
( ( sup_sup_set_list_o @ A2 @ bot_bot_set_list_o )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_451_sup__bot_Oright__neutral,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ bot_bot_set_nat )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_452_sup__bot_Oright__neutral,axiom,
! [A2: set_int] :
( ( sup_sup_set_int @ A2 @ bot_bot_set_int )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_453_sup__bot_Oright__neutral,axiom,
! [A2: nat > $o] :
( ( sup_sup_nat_o @ A2 @ bot_bot_nat_o )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_454_sup__bot_Oright__neutral,axiom,
! [A2: int > $o] :
( ( sup_sup_int_o @ A2 @ bot_bot_int_o )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_455_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_list_o,B2: set_list_o] :
( ( bot_bot_set_list_o
= ( sup_sup_set_list_o @ A2 @ B2 ) )
= ( ( A2 = bot_bot_set_list_o )
& ( B2 = bot_bot_set_list_o ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_456_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ A2 @ B2 ) )
= ( ( A2 = bot_bot_set_nat )
& ( B2 = bot_bot_set_nat ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_457_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_int,B2: set_int] :
( ( bot_bot_set_int
= ( sup_sup_set_int @ A2 @ B2 ) )
= ( ( A2 = bot_bot_set_int )
& ( B2 = bot_bot_set_int ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_458_sup__bot_Oneutr__eq__iff,axiom,
! [A2: nat > $o,B2: nat > $o] :
( ( bot_bot_nat_o
= ( sup_sup_nat_o @ A2 @ B2 ) )
= ( ( A2 = bot_bot_nat_o )
& ( B2 = bot_bot_nat_o ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_459_sup__bot_Oneutr__eq__iff,axiom,
! [A2: int > $o,B2: int > $o] :
( ( bot_bot_int_o
= ( sup_sup_int_o @ A2 @ B2 ) )
= ( ( A2 = bot_bot_int_o )
& ( B2 = bot_bot_int_o ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_460_sup__bot_Oleft__neutral,axiom,
! [A2: set_list_o] :
( ( sup_sup_set_list_o @ bot_bot_set_list_o @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_461_sup__bot_Oleft__neutral,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_462_sup__bot_Oleft__neutral,axiom,
! [A2: set_int] :
( ( sup_sup_set_int @ bot_bot_set_int @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_463_sup__bot_Oleft__neutral,axiom,
! [A2: nat > $o] :
( ( sup_sup_nat_o @ bot_bot_nat_o @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_464_sup__bot_Oleft__neutral,axiom,
! [A2: int > $o] :
( ( sup_sup_int_o @ bot_bot_int_o @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_465_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_list_o,B2: set_list_o] :
( ( ( sup_sup_set_list_o @ A2 @ B2 )
= bot_bot_set_list_o )
= ( ( A2 = bot_bot_set_list_o )
& ( B2 = bot_bot_set_list_o ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_466_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( sup_sup_set_nat @ A2 @ B2 )
= bot_bot_set_nat )
= ( ( A2 = bot_bot_set_nat )
& ( B2 = bot_bot_set_nat ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_467_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_int,B2: set_int] :
( ( ( sup_sup_set_int @ A2 @ B2 )
= bot_bot_set_int )
= ( ( A2 = bot_bot_set_int )
& ( B2 = bot_bot_set_int ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_468_sup__bot_Oeq__neutr__iff,axiom,
! [A2: nat > $o,B2: nat > $o] :
( ( ( sup_sup_nat_o @ A2 @ B2 )
= bot_bot_nat_o )
= ( ( A2 = bot_bot_nat_o )
& ( B2 = bot_bot_nat_o ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_469_sup__bot_Oeq__neutr__iff,axiom,
! [A2: int > $o,B2: int > $o] :
( ( ( sup_sup_int_o @ A2 @ B2 )
= bot_bot_int_o )
= ( ( A2 = bot_bot_int_o )
& ( B2 = bot_bot_int_o ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_470_sup__eq__bot__iff,axiom,
! [X: set_list_o,Y: set_list_o] :
( ( ( sup_sup_set_list_o @ X @ Y )
= bot_bot_set_list_o )
= ( ( X = bot_bot_set_list_o )
& ( Y = bot_bot_set_list_o ) ) ) ).
% sup_eq_bot_iff
thf(fact_471_sup__eq__bot__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( sup_sup_set_nat @ X @ Y )
= bot_bot_set_nat )
= ( ( X = bot_bot_set_nat )
& ( Y = bot_bot_set_nat ) ) ) ).
% sup_eq_bot_iff
thf(fact_472_sup__eq__bot__iff,axiom,
! [X: set_int,Y: set_int] :
( ( ( sup_sup_set_int @ X @ Y )
= bot_bot_set_int )
= ( ( X = bot_bot_set_int )
& ( Y = bot_bot_set_int ) ) ) ).
% sup_eq_bot_iff
thf(fact_473_sup__eq__bot__iff,axiom,
! [X: nat > $o,Y: nat > $o] :
( ( ( sup_sup_nat_o @ X @ Y )
= bot_bot_nat_o )
= ( ( X = bot_bot_nat_o )
& ( Y = bot_bot_nat_o ) ) ) ).
% sup_eq_bot_iff
thf(fact_474_sup__eq__bot__iff,axiom,
! [X: int > $o,Y: int > $o] :
( ( ( sup_sup_int_o @ X @ Y )
= bot_bot_int_o )
= ( ( X = bot_bot_int_o )
& ( Y = bot_bot_int_o ) ) ) ).
% sup_eq_bot_iff
thf(fact_475_bot__eq__sup__iff,axiom,
! [X: set_list_o,Y: set_list_o] :
( ( bot_bot_set_list_o
= ( sup_sup_set_list_o @ X @ Y ) )
= ( ( X = bot_bot_set_list_o )
& ( Y = bot_bot_set_list_o ) ) ) ).
% bot_eq_sup_iff
thf(fact_476_bot__eq__sup__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( bot_bot_set_nat
= ( sup_sup_set_nat @ X @ Y ) )
= ( ( X = bot_bot_set_nat )
& ( Y = bot_bot_set_nat ) ) ) ).
% bot_eq_sup_iff
thf(fact_477_bot__eq__sup__iff,axiom,
! [X: set_int,Y: set_int] :
( ( bot_bot_set_int
= ( sup_sup_set_int @ X @ Y ) )
= ( ( X = bot_bot_set_int )
& ( Y = bot_bot_set_int ) ) ) ).
% bot_eq_sup_iff
thf(fact_478_bot__eq__sup__iff,axiom,
! [X: nat > $o,Y: nat > $o] :
( ( bot_bot_nat_o
= ( sup_sup_nat_o @ X @ Y ) )
= ( ( X = bot_bot_nat_o )
& ( Y = bot_bot_nat_o ) ) ) ).
% bot_eq_sup_iff
thf(fact_479_bot__eq__sup__iff,axiom,
! [X: int > $o,Y: int > $o] :
( ( bot_bot_int_o
= ( sup_sup_int_o @ X @ Y ) )
= ( ( X = bot_bot_int_o )
& ( Y = bot_bot_int_o ) ) ) ).
% bot_eq_sup_iff
thf(fact_480_sup__bot__right,axiom,
! [X: set_list_o] :
( ( sup_sup_set_list_o @ X @ bot_bot_set_list_o )
= X ) ).
% sup_bot_right
thf(fact_481_sup__bot__right,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ bot_bot_set_nat )
= X ) ).
% sup_bot_right
thf(fact_482_sup__bot__right,axiom,
! [X: set_int] :
( ( sup_sup_set_int @ X @ bot_bot_set_int )
= X ) ).
% sup_bot_right
thf(fact_483_sup__bot__right,axiom,
! [X: nat > $o] :
( ( sup_sup_nat_o @ X @ bot_bot_nat_o )
= X ) ).
% sup_bot_right
thf(fact_484_sup__bot__right,axiom,
! [X: int > $o] :
( ( sup_sup_int_o @ X @ bot_bot_int_o )
= X ) ).
% sup_bot_right
thf(fact_485_sup__bot__left,axiom,
! [X: set_list_o] :
( ( sup_sup_set_list_o @ bot_bot_set_list_o @ X )
= X ) ).
% sup_bot_left
thf(fact_486_sup__bot__left,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ X )
= X ) ).
% sup_bot_left
thf(fact_487_sup__bot__left,axiom,
! [X: set_int] :
( ( sup_sup_set_int @ bot_bot_set_int @ X )
= X ) ).
% sup_bot_left
thf(fact_488_sup__bot__left,axiom,
! [X: nat > $o] :
( ( sup_sup_nat_o @ bot_bot_nat_o @ X )
= X ) ).
% sup_bot_left
thf(fact_489_sup__bot__left,axiom,
! [X: int > $o] :
( ( sup_sup_int_o @ bot_bot_int_o @ X )
= X ) ).
% sup_bot_left
thf(fact_490_Un__iff,axiom,
! [C: option_nat,A: set_option_nat,B4: set_option_nat] :
( ( member_option_nat @ C @ ( sup_su7692501479792361346on_nat @ A @ B4 ) )
= ( ( member_option_nat @ C @ A )
| ( member_option_nat @ C @ B4 ) ) ) ).
% Un_iff
thf(fact_491_Un__iff,axiom,
! [C: option_int,A: set_option_int,B4: set_option_int] :
( ( member_option_int @ C @ ( sup_su5202418334860634590on_int @ A @ B4 ) )
= ( ( member_option_int @ C @ A )
| ( member_option_int @ C @ B4 ) ) ) ).
% Un_iff
thf(fact_492_Un__iff,axiom,
! [C: set_nat,A: set_set_nat,B4: set_set_nat] :
( ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A @ B4 ) )
= ( ( member_set_nat @ C @ A )
| ( member_set_nat @ C @ B4 ) ) ) ).
% Un_iff
thf(fact_493_Un__iff,axiom,
! [C: set_int,A: set_set_int,B4: set_set_int] :
( ( member_set_int @ C @ ( sup_sup_set_set_int @ A @ B4 ) )
= ( ( member_set_int @ C @ A )
| ( member_set_int @ C @ B4 ) ) ) ).
% Un_iff
thf(fact_494_Un__iff,axiom,
! [C: list_o,A: set_list_o,B4: set_list_o] :
( ( member_list_o @ C @ ( sup_sup_set_list_o @ A @ B4 ) )
= ( ( member_list_o @ C @ A )
| ( member_list_o @ C @ B4 ) ) ) ).
% Un_iff
thf(fact_495_Un__iff,axiom,
! [C: nat,A: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B4 ) )
= ( ( member_nat @ C @ A )
| ( member_nat @ C @ B4 ) ) ) ).
% Un_iff
thf(fact_496_Un__iff,axiom,
! [C: int,A: set_int,B4: set_int] :
( ( member_int @ C @ ( sup_sup_set_int @ A @ B4 ) )
= ( ( member_int @ C @ A )
| ( member_int @ C @ B4 ) ) ) ).
% Un_iff
thf(fact_497_empty__iff,axiom,
! [C: option_nat] :
~ ( member_option_nat @ C @ bot_bo5009843511495006442on_nat ) ).
% empty_iff
thf(fact_498_empty__iff,axiom,
! [C: option_int] :
~ ( member_option_int @ C @ bot_bo2519760366563279686on_int ) ).
% empty_iff
thf(fact_499_empty__iff,axiom,
! [C: set_nat] :
~ ( member_set_nat @ C @ bot_bot_set_set_nat ) ).
% empty_iff
thf(fact_500_empty__iff,axiom,
! [C: set_int] :
~ ( member_set_int @ C @ bot_bot_set_set_int ) ).
% empty_iff
thf(fact_501_empty__iff,axiom,
! [C: list_o] :
~ ( member_list_o @ C @ bot_bot_set_list_o ) ).
% empty_iff
thf(fact_502_empty__iff,axiom,
! [C: nat] :
~ ( member_nat @ C @ bot_bot_set_nat ) ).
% empty_iff
thf(fact_503_empty__iff,axiom,
! [C: int] :
~ ( member_int @ C @ bot_bot_set_int ) ).
% empty_iff
thf(fact_504_all__not__in__conv,axiom,
! [A: set_option_nat] :
( ( ! [X3: option_nat] :
~ ( member_option_nat @ X3 @ A ) )
= ( A = bot_bo5009843511495006442on_nat ) ) ).
% all_not_in_conv
thf(fact_505_all__not__in__conv,axiom,
! [A: set_option_int] :
( ( ! [X3: option_int] :
~ ( member_option_int @ X3 @ A ) )
= ( A = bot_bo2519760366563279686on_int ) ) ).
% all_not_in_conv
thf(fact_506_all__not__in__conv,axiom,
! [A: set_set_nat] :
( ( ! [X3: set_nat] :
~ ( member_set_nat @ X3 @ A ) )
= ( A = bot_bot_set_set_nat ) ) ).
% all_not_in_conv
thf(fact_507_all__not__in__conv,axiom,
! [A: set_set_int] :
( ( ! [X3: set_int] :
~ ( member_set_int @ X3 @ A ) )
= ( A = bot_bot_set_set_int ) ) ).
% all_not_in_conv
thf(fact_508_all__not__in__conv,axiom,
! [A: set_list_o] :
( ( ! [X3: list_o] :
~ ( member_list_o @ X3 @ A ) )
= ( A = bot_bot_set_list_o ) ) ).
% all_not_in_conv
thf(fact_509_all__not__in__conv,axiom,
! [A: set_nat] :
( ( ! [X3: nat] :
~ ( member_nat @ X3 @ A ) )
= ( A = bot_bot_set_nat ) ) ).
% all_not_in_conv
thf(fact_510_all__not__in__conv,axiom,
! [A: set_int] :
( ( ! [X3: int] :
~ ( member_int @ X3 @ A ) )
= ( A = bot_bot_set_int ) ) ).
% all_not_in_conv
thf(fact_511_Collect__empty__eq,axiom,
! [P: list_o > $o] :
( ( ( collect_list_o @ P )
= bot_bot_set_list_o )
= ( ! [X3: list_o] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_512_Collect__empty__eq,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_513_Collect__empty__eq,axiom,
! [P: int > $o] :
( ( ( collect_int @ P )
= bot_bot_set_int )
= ( ! [X3: int] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_514_empty__Collect__eq,axiom,
! [P: list_o > $o] :
( ( bot_bot_set_list_o
= ( collect_list_o @ P ) )
= ( ! [X3: list_o] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_515_empty__Collect__eq,axiom,
! [P: nat > $o] :
( ( bot_bot_set_nat
= ( collect_nat @ P ) )
= ( ! [X3: nat] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_516_empty__Collect__eq,axiom,
! [P: int > $o] :
( ( bot_bot_set_int
= ( collect_int @ P ) )
= ( ! [X3: int] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_517_sup_Oidem,axiom,
! [A2: set_list_o] :
( ( sup_sup_set_list_o @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_518_sup_Oidem,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_519_sup_Oidem,axiom,
! [A2: set_int] :
( ( sup_sup_set_int @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_520_sup_Oidem,axiom,
! [A2: nat] :
( ( sup_sup_nat @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_521_sup_Oidem,axiom,
! [A2: int] :
( ( sup_sup_int @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_522_sup__idem,axiom,
! [X: set_list_o] :
( ( sup_sup_set_list_o @ X @ X )
= X ) ).
% sup_idem
thf(fact_523_sup__idem,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ X )
= X ) ).
% sup_idem
thf(fact_524_sup__idem,axiom,
! [X: set_int] :
( ( sup_sup_set_int @ X @ X )
= X ) ).
% sup_idem
thf(fact_525_sup__idem,axiom,
! [X: nat] :
( ( sup_sup_nat @ X @ X )
= X ) ).
% sup_idem
thf(fact_526_sup__idem,axiom,
! [X: int] :
( ( sup_sup_int @ X @ X )
= X ) ).
% sup_idem
thf(fact_527_sup_Oleft__idem,axiom,
! [A2: set_list_o,B2: set_list_o] :
( ( sup_sup_set_list_o @ A2 @ ( sup_sup_set_list_o @ A2 @ B2 ) )
= ( sup_sup_set_list_o @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_528_sup_Oleft__idem,axiom,
! [A2: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_529_sup_Oleft__idem,axiom,
! [A2: set_int,B2: set_int] :
( ( sup_sup_set_int @ A2 @ ( sup_sup_set_int @ A2 @ B2 ) )
= ( sup_sup_set_int @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_530_sup_Oleft__idem,axiom,
! [A2: nat,B2: nat] :
( ( sup_sup_nat @ A2 @ ( sup_sup_nat @ A2 @ B2 ) )
= ( sup_sup_nat @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_531_sup_Oleft__idem,axiom,
! [A2: int,B2: int] :
( ( sup_sup_int @ A2 @ ( sup_sup_int @ A2 @ B2 ) )
= ( sup_sup_int @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_532_sup__left__idem,axiom,
! [X: set_list_o,Y: set_list_o] :
( ( sup_sup_set_list_o @ X @ ( sup_sup_set_list_o @ X @ Y ) )
= ( sup_sup_set_list_o @ X @ Y ) ) ).
% sup_left_idem
thf(fact_533_sup__left__idem,axiom,
! [X: set_nat,Y: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) )
= ( sup_sup_set_nat @ X @ Y ) ) ).
% sup_left_idem
thf(fact_534_sup__left__idem,axiom,
! [X: set_int,Y: set_int] :
( ( sup_sup_set_int @ X @ ( sup_sup_set_int @ X @ Y ) )
= ( sup_sup_set_int @ X @ Y ) ) ).
% sup_left_idem
thf(fact_535_sup__left__idem,axiom,
! [X: nat,Y: nat] :
( ( sup_sup_nat @ X @ ( sup_sup_nat @ X @ Y ) )
= ( sup_sup_nat @ X @ Y ) ) ).
% sup_left_idem
thf(fact_536_sup__left__idem,axiom,
! [X: int,Y: int] :
( ( sup_sup_int @ X @ ( sup_sup_int @ X @ Y ) )
= ( sup_sup_int @ X @ Y ) ) ).
% sup_left_idem
thf(fact_537_sup_Oright__idem,axiom,
! [A2: set_list_o,B2: set_list_o] :
( ( sup_sup_set_list_o @ ( sup_sup_set_list_o @ A2 @ B2 ) @ B2 )
= ( sup_sup_set_list_o @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_538_sup_Oright__idem,axiom,
! [A2: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ B2 )
= ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_539_sup_Oright__idem,axiom,
! [A2: set_int,B2: set_int] :
( ( sup_sup_set_int @ ( sup_sup_set_int @ A2 @ B2 ) @ B2 )
= ( sup_sup_set_int @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_540_sup_Oright__idem,axiom,
! [A2: nat,B2: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ A2 @ B2 ) @ B2 )
= ( sup_sup_nat @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_541_sup_Oright__idem,axiom,
! [A2: int,B2: int] :
( ( sup_sup_int @ ( sup_sup_int @ A2 @ B2 ) @ B2 )
= ( sup_sup_int @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_542_UnCI,axiom,
! [C: option_nat,B4: set_option_nat,A: set_option_nat] :
( ( ~ ( member_option_nat @ C @ B4 )
=> ( member_option_nat @ C @ A ) )
=> ( member_option_nat @ C @ ( sup_su7692501479792361346on_nat @ A @ B4 ) ) ) ).
% UnCI
thf(fact_543_UnCI,axiom,
! [C: option_int,B4: set_option_int,A: set_option_int] :
( ( ~ ( member_option_int @ C @ B4 )
=> ( member_option_int @ C @ A ) )
=> ( member_option_int @ C @ ( sup_su5202418334860634590on_int @ A @ B4 ) ) ) ).
% UnCI
thf(fact_544_UnCI,axiom,
! [C: set_nat,B4: set_set_nat,A: set_set_nat] :
( ( ~ ( member_set_nat @ C @ B4 )
=> ( member_set_nat @ C @ A ) )
=> ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A @ B4 ) ) ) ).
% UnCI
thf(fact_545_UnCI,axiom,
! [C: set_int,B4: set_set_int,A: set_set_int] :
( ( ~ ( member_set_int @ C @ B4 )
=> ( member_set_int @ C @ A ) )
=> ( member_set_int @ C @ ( sup_sup_set_set_int @ A @ B4 ) ) ) ).
% UnCI
thf(fact_546_UnCI,axiom,
! [C: list_o,B4: set_list_o,A: set_list_o] :
( ( ~ ( member_list_o @ C @ B4 )
=> ( member_list_o @ C @ A ) )
=> ( member_list_o @ C @ ( sup_sup_set_list_o @ A @ B4 ) ) ) ).
% UnCI
thf(fact_547_UnCI,axiom,
! [C: nat,B4: set_nat,A: set_nat] :
( ( ~ ( member_nat @ C @ B4 )
=> ( member_nat @ C @ A ) )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B4 ) ) ) ).
% UnCI
thf(fact_548_UnCI,axiom,
! [C: int,B4: set_int,A: set_int] :
( ( ~ ( member_int @ C @ B4 )
=> ( member_int @ C @ A ) )
=> ( member_int @ C @ ( sup_sup_set_int @ A @ B4 ) ) ) ).
% UnCI
thf(fact_549_bot__set__def,axiom,
( bot_bot_set_list_o
= ( collect_list_o @ bot_bot_list_o_o ) ) ).
% bot_set_def
thf(fact_550_bot__set__def,axiom,
( bot_bot_set_nat
= ( collect_nat @ bot_bot_nat_o ) ) ).
% bot_set_def
thf(fact_551_bot__set__def,axiom,
( bot_bot_set_int
= ( collect_int @ bot_bot_int_o ) ) ).
% bot_set_def
thf(fact_552_emptyE,axiom,
! [A2: option_nat] :
~ ( member_option_nat @ A2 @ bot_bo5009843511495006442on_nat ) ).
% emptyE
thf(fact_553_emptyE,axiom,
! [A2: option_int] :
~ ( member_option_int @ A2 @ bot_bo2519760366563279686on_int ) ).
% emptyE
thf(fact_554_emptyE,axiom,
! [A2: set_nat] :
~ ( member_set_nat @ A2 @ bot_bot_set_set_nat ) ).
% emptyE
thf(fact_555_emptyE,axiom,
! [A2: set_int] :
~ ( member_set_int @ A2 @ bot_bot_set_set_int ) ).
% emptyE
thf(fact_556_emptyE,axiom,
! [A2: list_o] :
~ ( member_list_o @ A2 @ bot_bot_set_list_o ) ).
% emptyE
thf(fact_557_emptyE,axiom,
! [A2: nat] :
~ ( member_nat @ A2 @ bot_bot_set_nat ) ).
% emptyE
thf(fact_558_emptyE,axiom,
! [A2: int] :
~ ( member_int @ A2 @ bot_bot_set_int ) ).
% emptyE
thf(fact_559_equals0D,axiom,
! [A: set_option_nat,A2: option_nat] :
( ( A = bot_bo5009843511495006442on_nat )
=> ~ ( member_option_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_560_equals0D,axiom,
! [A: set_option_int,A2: option_int] :
( ( A = bot_bo2519760366563279686on_int )
=> ~ ( member_option_int @ A2 @ A ) ) ).
% equals0D
thf(fact_561_equals0D,axiom,
! [A: set_set_nat,A2: set_nat] :
( ( A = bot_bot_set_set_nat )
=> ~ ( member_set_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_562_equals0D,axiom,
! [A: set_set_int,A2: set_int] :
( ( A = bot_bot_set_set_int )
=> ~ ( member_set_int @ A2 @ A ) ) ).
% equals0D
thf(fact_563_equals0D,axiom,
! [A: set_list_o,A2: list_o] :
( ( A = bot_bot_set_list_o )
=> ~ ( member_list_o @ A2 @ A ) ) ).
% equals0D
thf(fact_564_equals0D,axiom,
! [A: set_nat,A2: nat] :
( ( A = bot_bot_set_nat )
=> ~ ( member_nat @ A2 @ A ) ) ).
% equals0D
thf(fact_565_equals0D,axiom,
! [A: set_int,A2: int] :
( ( A = bot_bot_set_int )
=> ~ ( member_int @ A2 @ A ) ) ).
% equals0D
thf(fact_566_equals0I,axiom,
! [A: set_option_nat] :
( ! [Y2: option_nat] :
~ ( member_option_nat @ Y2 @ A )
=> ( A = bot_bo5009843511495006442on_nat ) ) ).
% equals0I
thf(fact_567_equals0I,axiom,
! [A: set_option_int] :
( ! [Y2: option_int] :
~ ( member_option_int @ Y2 @ A )
=> ( A = bot_bo2519760366563279686on_int ) ) ).
% equals0I
thf(fact_568_equals0I,axiom,
! [A: set_set_nat] :
( ! [Y2: set_nat] :
~ ( member_set_nat @ Y2 @ A )
=> ( A = bot_bot_set_set_nat ) ) ).
% equals0I
thf(fact_569_equals0I,axiom,
! [A: set_set_int] :
( ! [Y2: set_int] :
~ ( member_set_int @ Y2 @ A )
=> ( A = bot_bot_set_set_int ) ) ).
% equals0I
thf(fact_570_equals0I,axiom,
! [A: set_list_o] :
( ! [Y2: list_o] :
~ ( member_list_o @ Y2 @ A )
=> ( A = bot_bot_set_list_o ) ) ).
% equals0I
thf(fact_571_equals0I,axiom,
! [A: set_nat] :
( ! [Y2: nat] :
~ ( member_nat @ Y2 @ A )
=> ( A = bot_bot_set_nat ) ) ).
% equals0I
thf(fact_572_equals0I,axiom,
! [A: set_int] :
( ! [Y2: int] :
~ ( member_int @ Y2 @ A )
=> ( A = bot_bot_set_int ) ) ).
% equals0I
thf(fact_573_ex__in__conv,axiom,
! [A: set_option_nat] :
( ( ? [X3: option_nat] : ( member_option_nat @ X3 @ A ) )
= ( A != bot_bo5009843511495006442on_nat ) ) ).
% ex_in_conv
thf(fact_574_ex__in__conv,axiom,
! [A: set_option_int] :
( ( ? [X3: option_int] : ( member_option_int @ X3 @ A ) )
= ( A != bot_bo2519760366563279686on_int ) ) ).
% ex_in_conv
thf(fact_575_ex__in__conv,axiom,
! [A: set_set_nat] :
( ( ? [X3: set_nat] : ( member_set_nat @ X3 @ A ) )
= ( A != bot_bot_set_set_nat ) ) ).
% ex_in_conv
thf(fact_576_ex__in__conv,axiom,
! [A: set_set_int] :
( ( ? [X3: set_int] : ( member_set_int @ X3 @ A ) )
= ( A != bot_bot_set_set_int ) ) ).
% ex_in_conv
thf(fact_577_ex__in__conv,axiom,
! [A: set_list_o] :
( ( ? [X3: list_o] : ( member_list_o @ X3 @ A ) )
= ( A != bot_bot_set_list_o ) ) ).
% ex_in_conv
thf(fact_578_ex__in__conv,axiom,
! [A: set_nat] :
( ( ? [X3: nat] : ( member_nat @ X3 @ A ) )
= ( A != bot_bot_set_nat ) ) ).
% ex_in_conv
thf(fact_579_ex__in__conv,axiom,
! [A: set_int] :
( ( ? [X3: int] : ( member_int @ X3 @ A ) )
= ( A != bot_bot_set_int ) ) ).
% ex_in_conv
thf(fact_580_inf__sup__aci_I8_J,axiom,
! [X: set_list_o,Y: set_list_o] :
( ( sup_sup_set_list_o @ X @ ( sup_sup_set_list_o @ X @ Y ) )
= ( sup_sup_set_list_o @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_581_inf__sup__aci_I8_J,axiom,
! [X: set_nat,Y: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) )
= ( sup_sup_set_nat @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_582_inf__sup__aci_I8_J,axiom,
! [X: set_int,Y: set_int] :
( ( sup_sup_set_int @ X @ ( sup_sup_set_int @ X @ Y ) )
= ( sup_sup_set_int @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_583_inf__sup__aci_I8_J,axiom,
! [X: nat,Y: nat] :
( ( sup_sup_nat @ X @ ( sup_sup_nat @ X @ Y ) )
= ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_584_inf__sup__aci_I8_J,axiom,
! [X: int,Y: int] :
( ( sup_sup_int @ X @ ( sup_sup_int @ X @ Y ) )
= ( sup_sup_int @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_585_inf__sup__aci_I7_J,axiom,
! [X: set_list_o,Y: set_list_o,Z2: set_list_o] :
( ( sup_sup_set_list_o @ X @ ( sup_sup_set_list_o @ Y @ Z2 ) )
= ( sup_sup_set_list_o @ Y @ ( sup_sup_set_list_o @ X @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_586_inf__sup__aci_I7_J,axiom,
! [X: set_nat,Y: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z2 ) )
= ( sup_sup_set_nat @ Y @ ( sup_sup_set_nat @ X @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_587_inf__sup__aci_I7_J,axiom,
! [X: set_int,Y: set_int,Z2: set_int] :
( ( sup_sup_set_int @ X @ ( sup_sup_set_int @ Y @ Z2 ) )
= ( sup_sup_set_int @ Y @ ( sup_sup_set_int @ X @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_588_inf__sup__aci_I7_J,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z2 ) )
= ( sup_sup_nat @ Y @ ( sup_sup_nat @ X @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_589_inf__sup__aci_I7_J,axiom,
! [X: int,Y: int,Z2: int] :
( ( sup_sup_int @ X @ ( sup_sup_int @ Y @ Z2 ) )
= ( sup_sup_int @ Y @ ( sup_sup_int @ X @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_590_inf__sup__aci_I6_J,axiom,
! [X: set_list_o,Y: set_list_o,Z2: set_list_o] :
( ( sup_sup_set_list_o @ ( sup_sup_set_list_o @ X @ Y ) @ Z2 )
= ( sup_sup_set_list_o @ X @ ( sup_sup_set_list_o @ Y @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_591_inf__sup__aci_I6_J,axiom,
! [X: set_nat,Y: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ Z2 )
= ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_592_inf__sup__aci_I6_J,axiom,
! [X: set_int,Y: set_int,Z2: set_int] :
( ( sup_sup_set_int @ ( sup_sup_set_int @ X @ Y ) @ Z2 )
= ( sup_sup_set_int @ X @ ( sup_sup_set_int @ Y @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_593_inf__sup__aci_I6_J,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ X @ Y ) @ Z2 )
= ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_594_inf__sup__aci_I6_J,axiom,
! [X: int,Y: int,Z2: int] :
( ( sup_sup_int @ ( sup_sup_int @ X @ Y ) @ Z2 )
= ( sup_sup_int @ X @ ( sup_sup_int @ Y @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_595_inf__sup__aci_I5_J,axiom,
( sup_sup_set_list_o
= ( ^ [X3: set_list_o,Y3: set_list_o] : ( sup_sup_set_list_o @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_596_inf__sup__aci_I5_J,axiom,
( sup_sup_set_nat
= ( ^ [X3: set_nat,Y3: set_nat] : ( sup_sup_set_nat @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_597_inf__sup__aci_I5_J,axiom,
( sup_sup_set_int
= ( ^ [X3: set_int,Y3: set_int] : ( sup_sup_set_int @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_598_inf__sup__aci_I5_J,axiom,
( sup_sup_nat
= ( ^ [X3: nat,Y3: nat] : ( sup_sup_nat @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_599_inf__sup__aci_I5_J,axiom,
( sup_sup_int
= ( ^ [X3: int,Y3: int] : ( sup_sup_int @ Y3 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_600_sup_Oassoc,axiom,
! [A2: set_list_o,B2: set_list_o,C: set_list_o] :
( ( sup_sup_set_list_o @ ( sup_sup_set_list_o @ A2 @ B2 ) @ C )
= ( sup_sup_set_list_o @ A2 @ ( sup_sup_set_list_o @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_601_sup_Oassoc,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ C )
= ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_602_sup_Oassoc,axiom,
! [A2: set_int,B2: set_int,C: set_int] :
( ( sup_sup_set_int @ ( sup_sup_set_int @ A2 @ B2 ) @ C )
= ( sup_sup_set_int @ A2 @ ( sup_sup_set_int @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_603_sup_Oassoc,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ A2 @ B2 ) @ C )
= ( sup_sup_nat @ A2 @ ( sup_sup_nat @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_604_sup_Oassoc,axiom,
! [A2: int,B2: int,C: int] :
( ( sup_sup_int @ ( sup_sup_int @ A2 @ B2 ) @ C )
= ( sup_sup_int @ A2 @ ( sup_sup_int @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_605_sup__assoc,axiom,
! [X: set_list_o,Y: set_list_o,Z2: set_list_o] :
( ( sup_sup_set_list_o @ ( sup_sup_set_list_o @ X @ Y ) @ Z2 )
= ( sup_sup_set_list_o @ X @ ( sup_sup_set_list_o @ Y @ Z2 ) ) ) ).
% sup_assoc
thf(fact_606_sup__assoc,axiom,
! [X: set_nat,Y: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ Z2 )
= ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z2 ) ) ) ).
% sup_assoc
thf(fact_607_sup__assoc,axiom,
! [X: set_int,Y: set_int,Z2: set_int] :
( ( sup_sup_set_int @ ( sup_sup_set_int @ X @ Y ) @ Z2 )
= ( sup_sup_set_int @ X @ ( sup_sup_set_int @ Y @ Z2 ) ) ) ).
% sup_assoc
thf(fact_608_sup__assoc,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ X @ Y ) @ Z2 )
= ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z2 ) ) ) ).
% sup_assoc
thf(fact_609_sup__assoc,axiom,
! [X: int,Y: int,Z2: int] :
( ( sup_sup_int @ ( sup_sup_int @ X @ Y ) @ Z2 )
= ( sup_sup_int @ X @ ( sup_sup_int @ Y @ Z2 ) ) ) ).
% sup_assoc
thf(fact_610_sup_Ocommute,axiom,
( sup_sup_set_list_o
= ( ^ [A3: set_list_o,B3: set_list_o] : ( sup_sup_set_list_o @ B3 @ A3 ) ) ) ).
% sup.commute
thf(fact_611_sup_Ocommute,axiom,
( sup_sup_set_nat
= ( ^ [A3: set_nat,B3: set_nat] : ( sup_sup_set_nat @ B3 @ A3 ) ) ) ).
% sup.commute
thf(fact_612_sup_Ocommute,axiom,
( sup_sup_set_int
= ( ^ [A3: set_int,B3: set_int] : ( sup_sup_set_int @ B3 @ A3 ) ) ) ).
% sup.commute
thf(fact_613_sup_Ocommute,axiom,
( sup_sup_nat
= ( ^ [A3: nat,B3: nat] : ( sup_sup_nat @ B3 @ A3 ) ) ) ).
% sup.commute
thf(fact_614_sup_Ocommute,axiom,
( sup_sup_int
= ( ^ [A3: int,B3: int] : ( sup_sup_int @ B3 @ A3 ) ) ) ).
% sup.commute
thf(fact_615_sup__commute,axiom,
( sup_sup_set_list_o
= ( ^ [X3: set_list_o,Y3: set_list_o] : ( sup_sup_set_list_o @ Y3 @ X3 ) ) ) ).
% sup_commute
thf(fact_616_sup__commute,axiom,
( sup_sup_set_nat
= ( ^ [X3: set_nat,Y3: set_nat] : ( sup_sup_set_nat @ Y3 @ X3 ) ) ) ).
% sup_commute
thf(fact_617_sup__commute,axiom,
( sup_sup_set_int
= ( ^ [X3: set_int,Y3: set_int] : ( sup_sup_set_int @ Y3 @ X3 ) ) ) ).
% sup_commute
thf(fact_618_sup__commute,axiom,
( sup_sup_nat
= ( ^ [X3: nat,Y3: nat] : ( sup_sup_nat @ Y3 @ X3 ) ) ) ).
% sup_commute
thf(fact_619_sup__commute,axiom,
( sup_sup_int
= ( ^ [X3: int,Y3: int] : ( sup_sup_int @ Y3 @ X3 ) ) ) ).
% sup_commute
thf(fact_620_sup_Oleft__commute,axiom,
! [B2: set_list_o,A2: set_list_o,C: set_list_o] :
( ( sup_sup_set_list_o @ B2 @ ( sup_sup_set_list_o @ A2 @ C ) )
= ( sup_sup_set_list_o @ A2 @ ( sup_sup_set_list_o @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_621_sup_Oleft__commute,axiom,
! [B2: set_nat,A2: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ B2 @ ( sup_sup_set_nat @ A2 @ C ) )
= ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_622_sup_Oleft__commute,axiom,
! [B2: set_int,A2: set_int,C: set_int] :
( ( sup_sup_set_int @ B2 @ ( sup_sup_set_int @ A2 @ C ) )
= ( sup_sup_set_int @ A2 @ ( sup_sup_set_int @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_623_sup_Oleft__commute,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( sup_sup_nat @ B2 @ ( sup_sup_nat @ A2 @ C ) )
= ( sup_sup_nat @ A2 @ ( sup_sup_nat @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_624_sup_Oleft__commute,axiom,
! [B2: int,A2: int,C: int] :
( ( sup_sup_int @ B2 @ ( sup_sup_int @ A2 @ C ) )
= ( sup_sup_int @ A2 @ ( sup_sup_int @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_625_sup__left__commute,axiom,
! [X: set_list_o,Y: set_list_o,Z2: set_list_o] :
( ( sup_sup_set_list_o @ X @ ( sup_sup_set_list_o @ Y @ Z2 ) )
= ( sup_sup_set_list_o @ Y @ ( sup_sup_set_list_o @ X @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_626_sup__left__commute,axiom,
! [X: set_nat,Y: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z2 ) )
= ( sup_sup_set_nat @ Y @ ( sup_sup_set_nat @ X @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_627_sup__left__commute,axiom,
! [X: set_int,Y: set_int,Z2: set_int] :
( ( sup_sup_set_int @ X @ ( sup_sup_set_int @ Y @ Z2 ) )
= ( sup_sup_set_int @ Y @ ( sup_sup_set_int @ X @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_628_sup__left__commute,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( sup_sup_nat @ X @ ( sup_sup_nat @ Y @ Z2 ) )
= ( sup_sup_nat @ Y @ ( sup_sup_nat @ X @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_629_sup__left__commute,axiom,
! [X: int,Y: int,Z2: int] :
( ( sup_sup_int @ X @ ( sup_sup_int @ Y @ Z2 ) )
= ( sup_sup_int @ Y @ ( sup_sup_int @ X @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_630_UnE,axiom,
! [C: option_nat,A: set_option_nat,B4: set_option_nat] :
( ( member_option_nat @ C @ ( sup_su7692501479792361346on_nat @ A @ B4 ) )
=> ( ~ ( member_option_nat @ C @ A )
=> ( member_option_nat @ C @ B4 ) ) ) ).
% UnE
thf(fact_631_UnE,axiom,
! [C: option_int,A: set_option_int,B4: set_option_int] :
( ( member_option_int @ C @ ( sup_su5202418334860634590on_int @ A @ B4 ) )
=> ( ~ ( member_option_int @ C @ A )
=> ( member_option_int @ C @ B4 ) ) ) ).
% UnE
thf(fact_632_UnE,axiom,
! [C: set_nat,A: set_set_nat,B4: set_set_nat] :
( ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A @ B4 ) )
=> ( ~ ( member_set_nat @ C @ A )
=> ( member_set_nat @ C @ B4 ) ) ) ).
% UnE
thf(fact_633_UnE,axiom,
! [C: set_int,A: set_set_int,B4: set_set_int] :
( ( member_set_int @ C @ ( sup_sup_set_set_int @ A @ B4 ) )
=> ( ~ ( member_set_int @ C @ A )
=> ( member_set_int @ C @ B4 ) ) ) ).
% UnE
thf(fact_634_UnE,axiom,
! [C: list_o,A: set_list_o,B4: set_list_o] :
( ( member_list_o @ C @ ( sup_sup_set_list_o @ A @ B4 ) )
=> ( ~ ( member_list_o @ C @ A )
=> ( member_list_o @ C @ B4 ) ) ) ).
% UnE
thf(fact_635_UnE,axiom,
! [C: nat,A: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B4 ) )
=> ( ~ ( member_nat @ C @ A )
=> ( member_nat @ C @ B4 ) ) ) ).
% UnE
thf(fact_636_UnE,axiom,
! [C: int,A: set_int,B4: set_int] :
( ( member_int @ C @ ( sup_sup_set_int @ A @ B4 ) )
=> ( ~ ( member_int @ C @ A )
=> ( member_int @ C @ B4 ) ) ) ).
% UnE
thf(fact_637_UnI1,axiom,
! [C: option_nat,A: set_option_nat,B4: set_option_nat] :
( ( member_option_nat @ C @ A )
=> ( member_option_nat @ C @ ( sup_su7692501479792361346on_nat @ A @ B4 ) ) ) ).
% UnI1
thf(fact_638_UnI1,axiom,
! [C: option_int,A: set_option_int,B4: set_option_int] :
( ( member_option_int @ C @ A )
=> ( member_option_int @ C @ ( sup_su5202418334860634590on_int @ A @ B4 ) ) ) ).
% UnI1
thf(fact_639_UnI1,axiom,
! [C: set_nat,A: set_set_nat,B4: set_set_nat] :
( ( member_set_nat @ C @ A )
=> ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A @ B4 ) ) ) ).
% UnI1
thf(fact_640_UnI1,axiom,
! [C: set_int,A: set_set_int,B4: set_set_int] :
( ( member_set_int @ C @ A )
=> ( member_set_int @ C @ ( sup_sup_set_set_int @ A @ B4 ) ) ) ).
% UnI1
thf(fact_641_UnI1,axiom,
! [C: list_o,A: set_list_o,B4: set_list_o] :
( ( member_list_o @ C @ A )
=> ( member_list_o @ C @ ( sup_sup_set_list_o @ A @ B4 ) ) ) ).
% UnI1
thf(fact_642_UnI1,axiom,
! [C: nat,A: set_nat,B4: set_nat] :
( ( member_nat @ C @ A )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B4 ) ) ) ).
% UnI1
thf(fact_643_UnI1,axiom,
! [C: int,A: set_int,B4: set_int] :
( ( member_int @ C @ A )
=> ( member_int @ C @ ( sup_sup_set_int @ A @ B4 ) ) ) ).
% UnI1
thf(fact_644_UnI2,axiom,
! [C: option_nat,B4: set_option_nat,A: set_option_nat] :
( ( member_option_nat @ C @ B4 )
=> ( member_option_nat @ C @ ( sup_su7692501479792361346on_nat @ A @ B4 ) ) ) ).
% UnI2
thf(fact_645_UnI2,axiom,
! [C: option_int,B4: set_option_int,A: set_option_int] :
( ( member_option_int @ C @ B4 )
=> ( member_option_int @ C @ ( sup_su5202418334860634590on_int @ A @ B4 ) ) ) ).
% UnI2
thf(fact_646_UnI2,axiom,
! [C: set_nat,B4: set_set_nat,A: set_set_nat] :
( ( member_set_nat @ C @ B4 )
=> ( member_set_nat @ C @ ( sup_sup_set_set_nat @ A @ B4 ) ) ) ).
% UnI2
thf(fact_647_UnI2,axiom,
! [C: set_int,B4: set_set_int,A: set_set_int] :
( ( member_set_int @ C @ B4 )
=> ( member_set_int @ C @ ( sup_sup_set_set_int @ A @ B4 ) ) ) ).
% UnI2
thf(fact_648_UnI2,axiom,
! [C: list_o,B4: set_list_o,A: set_list_o] :
( ( member_list_o @ C @ B4 )
=> ( member_list_o @ C @ ( sup_sup_set_list_o @ A @ B4 ) ) ) ).
% UnI2
thf(fact_649_UnI2,axiom,
! [C: nat,B4: set_nat,A: set_nat] :
( ( member_nat @ C @ B4 )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B4 ) ) ) ).
% UnI2
thf(fact_650_UnI2,axiom,
! [C: int,B4: set_int,A: set_int] :
( ( member_int @ C @ B4 )
=> ( member_int @ C @ ( sup_sup_set_int @ A @ B4 ) ) ) ).
% UnI2
thf(fact_651_bex__Un,axiom,
! [A: set_list_o,B4: set_list_o,P: list_o > $o] :
( ( ? [X3: list_o] :
( ( member_list_o @ X3 @ ( sup_sup_set_list_o @ A @ B4 ) )
& ( P @ X3 ) ) )
= ( ? [X3: list_o] :
( ( member_list_o @ X3 @ A )
& ( P @ X3 ) )
| ? [X3: list_o] :
( ( member_list_o @ X3 @ B4 )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_652_bex__Un,axiom,
! [A: set_nat,B4: set_nat,P: nat > $o] :
( ( ? [X3: nat] :
( ( member_nat @ X3 @ ( sup_sup_set_nat @ A @ B4 ) )
& ( P @ X3 ) ) )
= ( ? [X3: nat] :
( ( member_nat @ X3 @ A )
& ( P @ X3 ) )
| ? [X3: nat] :
( ( member_nat @ X3 @ B4 )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_653_bex__Un,axiom,
! [A: set_int,B4: set_int,P: int > $o] :
( ( ? [X3: int] :
( ( member_int @ X3 @ ( sup_sup_set_int @ A @ B4 ) )
& ( P @ X3 ) ) )
= ( ? [X3: int] :
( ( member_int @ X3 @ A )
& ( P @ X3 ) )
| ? [X3: int] :
( ( member_int @ X3 @ B4 )
& ( P @ X3 ) ) ) ) ).
% bex_Un
thf(fact_654_ball__Un,axiom,
! [A: set_list_o,B4: set_list_o,P: list_o > $o] :
( ( ! [X3: list_o] :
( ( member_list_o @ X3 @ ( sup_sup_set_list_o @ A @ B4 ) )
=> ( P @ X3 ) ) )
= ( ! [X3: list_o] :
( ( member_list_o @ X3 @ A )
=> ( P @ X3 ) )
& ! [X3: list_o] :
( ( member_list_o @ X3 @ B4 )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_655_ball__Un,axiom,
! [A: set_nat,B4: set_nat,P: nat > $o] :
( ( ! [X3: nat] :
( ( member_nat @ X3 @ ( sup_sup_set_nat @ A @ B4 ) )
=> ( P @ X3 ) ) )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( P @ X3 ) )
& ! [X3: nat] :
( ( member_nat @ X3 @ B4 )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_656_ball__Un,axiom,
! [A: set_int,B4: set_int,P: int > $o] :
( ( ! [X3: int] :
( ( member_int @ X3 @ ( sup_sup_set_int @ A @ B4 ) )
=> ( P @ X3 ) ) )
= ( ! [X3: int] :
( ( member_int @ X3 @ A )
=> ( P @ X3 ) )
& ! [X3: int] :
( ( member_int @ X3 @ B4 )
=> ( P @ X3 ) ) ) ) ).
% ball_Un
thf(fact_657_Un__assoc,axiom,
! [A: set_list_o,B4: set_list_o,C2: set_list_o] :
( ( sup_sup_set_list_o @ ( sup_sup_set_list_o @ A @ B4 ) @ C2 )
= ( sup_sup_set_list_o @ A @ ( sup_sup_set_list_o @ B4 @ C2 ) ) ) ).
% Un_assoc
thf(fact_658_Un__assoc,axiom,
! [A: set_nat,B4: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A @ B4 ) @ C2 )
= ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B4 @ C2 ) ) ) ).
% Un_assoc
thf(fact_659_Un__assoc,axiom,
! [A: set_int,B4: set_int,C2: set_int] :
( ( sup_sup_set_int @ ( sup_sup_set_int @ A @ B4 ) @ C2 )
= ( sup_sup_set_int @ A @ ( sup_sup_set_int @ B4 @ C2 ) ) ) ).
% Un_assoc
thf(fact_660_Un__absorb,axiom,
! [A: set_list_o] :
( ( sup_sup_set_list_o @ A @ A )
= A ) ).
% Un_absorb
thf(fact_661_Un__absorb,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ A )
= A ) ).
% Un_absorb
thf(fact_662_Un__absorb,axiom,
! [A: set_int] :
( ( sup_sup_set_int @ A @ A )
= A ) ).
% Un_absorb
thf(fact_663_Un__commute,axiom,
( sup_sup_set_list_o
= ( ^ [A5: set_list_o,B5: set_list_o] : ( sup_sup_set_list_o @ B5 @ A5 ) ) ) ).
% Un_commute
thf(fact_664_Un__commute,axiom,
( sup_sup_set_nat
= ( ^ [A5: set_nat,B5: set_nat] : ( sup_sup_set_nat @ B5 @ A5 ) ) ) ).
% Un_commute
thf(fact_665_Un__commute,axiom,
( sup_sup_set_int
= ( ^ [A5: set_int,B5: set_int] : ( sup_sup_set_int @ B5 @ A5 ) ) ) ).
% Un_commute
thf(fact_666_Un__left__absorb,axiom,
! [A: set_list_o,B4: set_list_o] :
( ( sup_sup_set_list_o @ A @ ( sup_sup_set_list_o @ A @ B4 ) )
= ( sup_sup_set_list_o @ A @ B4 ) ) ).
% Un_left_absorb
thf(fact_667_Un__left__absorb,axiom,
! [A: set_nat,B4: set_nat] :
( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ A @ B4 ) )
= ( sup_sup_set_nat @ A @ B4 ) ) ).
% Un_left_absorb
thf(fact_668_Un__left__absorb,axiom,
! [A: set_int,B4: set_int] :
( ( sup_sup_set_int @ A @ ( sup_sup_set_int @ A @ B4 ) )
= ( sup_sup_set_int @ A @ B4 ) ) ).
% Un_left_absorb
thf(fact_669_Un__left__commute,axiom,
! [A: set_list_o,B4: set_list_o,C2: set_list_o] :
( ( sup_sup_set_list_o @ A @ ( sup_sup_set_list_o @ B4 @ C2 ) )
= ( sup_sup_set_list_o @ B4 @ ( sup_sup_set_list_o @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_670_Un__left__commute,axiom,
! [A: set_nat,B4: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B4 @ C2 ) )
= ( sup_sup_set_nat @ B4 @ ( sup_sup_set_nat @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_671_Un__left__commute,axiom,
! [A: set_int,B4: set_int,C2: set_int] :
( ( sup_sup_set_int @ A @ ( sup_sup_set_int @ B4 @ C2 ) )
= ( sup_sup_set_int @ B4 @ ( sup_sup_set_int @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_672_Un__empty__left,axiom,
! [B4: set_list_o] :
( ( sup_sup_set_list_o @ bot_bot_set_list_o @ B4 )
= B4 ) ).
% Un_empty_left
thf(fact_673_Un__empty__left,axiom,
! [B4: set_nat] :
( ( sup_sup_set_nat @ bot_bot_set_nat @ B4 )
= B4 ) ).
% Un_empty_left
thf(fact_674_Un__empty__left,axiom,
! [B4: set_int] :
( ( sup_sup_set_int @ bot_bot_set_int @ B4 )
= B4 ) ).
% Un_empty_left
thf(fact_675_Un__empty__right,axiom,
! [A: set_list_o] :
( ( sup_sup_set_list_o @ A @ bot_bot_set_list_o )
= A ) ).
% Un_empty_right
thf(fact_676_Un__empty__right,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ bot_bot_set_nat )
= A ) ).
% Un_empty_right
thf(fact_677_Un__empty__right,axiom,
! [A: set_int] :
( ( sup_sup_set_int @ A @ bot_bot_set_int )
= A ) ).
% Un_empty_right
thf(fact_678_bot__apply,axiom,
( bot_bot_nat_o
= ( ^ [X3: nat] : bot_bot_o ) ) ).
% bot_apply
thf(fact_679_bot__apply,axiom,
( bot_bot_int_o
= ( ^ [X3: int] : bot_bot_o ) ) ).
% bot_apply
thf(fact_680_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_list_o] :
( ( sup_sup_set_list_o @ X @ bot_bot_set_list_o )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_681_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ bot_bot_set_nat )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_682_boolean__algebra_Odisj__zero__right,axiom,
! [X: set_int] :
( ( sup_sup_set_int @ X @ bot_bot_set_int )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_683_boolean__algebra_Odisj__zero__right,axiom,
! [X: nat > $o] :
( ( sup_sup_nat_o @ X @ bot_bot_nat_o )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_684_boolean__algebra_Odisj__zero__right,axiom,
! [X: int > $o] :
( ( sup_sup_int_o @ X @ bot_bot_int_o )
= X ) ).
% boolean_algebra.disj_zero_right
thf(fact_685_elias__gamma__encoding,axiom,
prefix3558185134189398382ng_nat @ prefix1649127329469935890e_Ng_e ).
% elias_gamma_encoding
thf(fact_686_Set_Ois__empty__def,axiom,
( is_empty_list_o
= ( ^ [A5: set_list_o] : ( A5 = bot_bot_set_list_o ) ) ) ).
% Set.is_empty_def
thf(fact_687_Set_Ois__empty__def,axiom,
( is_empty_nat
= ( ^ [A5: set_nat] : ( A5 = bot_bot_set_nat ) ) ) ).
% Set.is_empty_def
thf(fact_688_Set_Ois__empty__def,axiom,
( is_empty_int
= ( ^ [A5: set_int] : ( A5 = bot_bot_set_int ) ) ) ).
% Set.is_empty_def
thf(fact_689_bounded__nat__encoding,axiom,
! [L: nat] : ( prefix3558185134189398382ng_nat @ ( prefix6319276831915272717e_Nb_e @ L ) ) ).
% bounded_nat_encoding
thf(fact_690_exp__golomb__encoding,axiom,
prefix3558185134189398382ng_nat @ prefix_Free_Code_N_e ).
% exp_golomb_encoding
thf(fact_691_map__le__imp__upd__le,axiom,
! [M1: int > option_list_o,M22: int > option_list_o,X: int,Y: list_o] :
( ( map_le_int_list_o @ M1 @ M22 )
=> ( map_le_int_list_o @ ( fun_up5258779583477023609list_o @ M1 @ X @ none_list_o ) @ ( fun_up5258779583477023609list_o @ M22 @ X @ ( some_list_o @ Y ) ) ) ) ).
% map_le_imp_upd_le
thf(fact_692_map__le__imp__upd__le,axiom,
! [M1: nat > option_list_o,M22: nat > option_list_o,X: nat,Y: list_o] :
( ( map_le_nat_list_o @ M1 @ M22 )
=> ( map_le_nat_list_o @ ( fun_up6685928411843432661list_o @ M1 @ X @ none_list_o ) @ ( fun_up6685928411843432661list_o @ M22 @ X @ ( some_list_o @ Y ) ) ) ) ).
% map_le_imp_upd_le
thf(fact_693_map__le__imp__upd__le,axiom,
! [M1: nat > option_nat,M22: nat > option_nat,X: nat,Y: nat] :
( ( map_le_nat_nat @ M1 @ M22 )
=> ( map_le_nat_nat @ ( fun_up1493157387958331631on_nat @ M1 @ X @ none_nat ) @ ( fun_up1493157387958331631on_nat @ M22 @ X @ ( some_nat @ Y ) ) ) ) ).
% map_le_imp_upd_le
thf(fact_694_map__le__imp__upd__le,axiom,
! [M1: option_list_o > option_option_list_o,M22: option_list_o > option_option_list_o,X: option_list_o,Y: option_list_o] :
( ( map_le539069141309449381list_o @ M1 @ M22 )
=> ( map_le539069141309449381list_o @ ( fun_up4915027116007172163list_o @ M1 @ X @ none_option_list_o ) @ ( fun_up4915027116007172163list_o @ M22 @ X @ ( some_option_list_o @ Y ) ) ) ) ).
% map_le_imp_upd_le
thf(fact_695_map__le__refl,axiom,
! [F2: int > option_list_o] : ( map_le_int_list_o @ F2 @ F2 ) ).
% map_le_refl
thf(fact_696_map__le__refl,axiom,
! [F2: nat > option_list_o] : ( map_le_nat_list_o @ F2 @ F2 ) ).
% map_le_refl
thf(fact_697_map__le__trans,axiom,
! [M1: int > option_list_o,M22: int > option_list_o,M3: int > option_list_o] :
( ( map_le_int_list_o @ M1 @ M22 )
=> ( ( map_le_int_list_o @ M22 @ M3 )
=> ( map_le_int_list_o @ M1 @ M3 ) ) ) ).
% map_le_trans
thf(fact_698_map__le__trans,axiom,
! [M1: nat > option_list_o,M22: nat > option_list_o,M3: nat > option_list_o] :
( ( map_le_nat_list_o @ M1 @ M22 )
=> ( ( map_le_nat_list_o @ M22 @ M3 )
=> ( map_le_nat_list_o @ M1 @ M3 ) ) ) ).
% map_le_trans
thf(fact_699_map__le__antisym,axiom,
! [F2: int > option_list_o,G: int > option_list_o] :
( ( map_le_int_list_o @ F2 @ G )
=> ( ( map_le_int_list_o @ G @ F2 )
=> ( F2 = G ) ) ) ).
% map_le_antisym
thf(fact_700_map__le__antisym,axiom,
! [F2: nat > option_list_o,G: nat > option_list_o] :
( ( map_le_nat_list_o @ F2 @ G )
=> ( ( map_le_nat_list_o @ G @ F2 )
=> ( F2 = G ) ) ) ).
% map_le_antisym
thf(fact_701_map__le__def,axiom,
( map_le8466019216010085450list_o
= ( ^ [M_1: product_prod_int_int > option_list_o,M_2: product_prod_int_int > option_list_o] :
! [X3: product_prod_int_int] :
( ( member5262025264175285858nt_int @ X3 @ ( dom_Pr7776987155556795511list_o @ M_1 ) )
=> ( ( M_1 @ X3 )
= ( M_2 @ X3 ) ) ) ) ) ).
% map_le_def
thf(fact_702_map__le__def,axiom,
( map_le5998359397277425062list_o
= ( ^ [M_1: product_prod_int_nat > option_list_o,M_2: product_prod_int_nat > option_list_o] :
! [X3: product_prod_int_nat] :
( ( member216504246829706758nt_nat @ X3 @ ( dom_Pr5309327336824135123list_o @ M_1 ) )
=> ( ( M_1 @ X3 )
= ( M_2 @ X3 ) ) ) ) ) ).
% map_le_def
thf(fact_703_map__le__def,axiom,
( map_le8358820006621043622list_o
= ( ^ [M_1: product_prod_nat_int > option_list_o,M_2: product_prod_nat_int > option_list_o] :
! [X3: product_prod_nat_int] :
( ( member4262671552274231302at_int @ X3 @ ( dom_Pr7669787946167753683list_o @ M_1 ) )
=> ( ( M_1 @ X3 )
= ( M_2 @ X3 ) ) ) ) ) ).
% map_le_def
thf(fact_704_map__le__def,axiom,
( map_le5891160187888383234list_o
= ( ^ [M_1: product_prod_nat_nat > option_list_o,M_2: product_prod_nat_nat > option_list_o] :
! [X3: product_prod_nat_nat] :
( ( member8440522571783428010at_nat @ X3 @ ( dom_Pr5202128127435093295list_o @ M_1 ) )
=> ( ( M_1 @ X3 )
= ( M_2 @ X3 ) ) ) ) ) ).
% map_le_def
thf(fact_705_map__le__def,axiom,
( map_le_nat_list_o
= ( ^ [M_1: nat > option_list_o,M_2: nat > option_list_o] :
! [X3: nat] :
( ( member_nat @ X3 @ ( dom_nat_list_o @ M_1 ) )
=> ( ( M_1 @ X3 )
= ( M_2 @ X3 ) ) ) ) ) ).
% map_le_def
thf(fact_706_map__le__def,axiom,
( map_le_int_list_o
= ( ^ [M_1: int > option_list_o,M_2: int > option_list_o] :
! [X3: int] :
( ( member_int @ X3 @ ( dom_int_list_o @ M_1 ) )
=> ( ( M_1 @ X3 )
= ( M_2 @ X3 ) ) ) ) ) ).
% map_le_def
thf(fact_707_map__le__upd,axiom,
! [F2: int > option_list_o,G: int > option_list_o,A2: int,B2: option_list_o] :
( ( map_le_int_list_o @ F2 @ G )
=> ( map_le_int_list_o @ ( fun_up5258779583477023609list_o @ F2 @ A2 @ B2 ) @ ( fun_up5258779583477023609list_o @ G @ A2 @ B2 ) ) ) ).
% map_le_upd
thf(fact_708_map__le__upd,axiom,
! [F2: nat > option_list_o,G: nat > option_list_o,A2: nat,B2: option_list_o] :
( ( map_le_nat_list_o @ F2 @ G )
=> ( map_le_nat_list_o @ ( fun_up6685928411843432661list_o @ F2 @ A2 @ B2 ) @ ( fun_up6685928411843432661list_o @ G @ A2 @ B2 ) ) ) ).
% map_le_upd
thf(fact_709_map__le__upd,axiom,
! [F2: nat > option_nat,G: nat > option_nat,A2: nat,B2: option_nat] :
( ( map_le_nat_nat @ F2 @ G )
=> ( map_le_nat_nat @ ( fun_up1493157387958331631on_nat @ F2 @ A2 @ B2 ) @ ( fun_up1493157387958331631on_nat @ G @ A2 @ B2 ) ) ) ).
% map_le_upd
thf(fact_710_map__le__upd,axiom,
! [F2: option_list_o > option_option_list_o,G: option_list_o > option_option_list_o,A2: option_list_o,B2: option_option_list_o] :
( ( map_le539069141309449381list_o @ F2 @ G )
=> ( map_le539069141309449381list_o @ ( fun_up4915027116007172163list_o @ F2 @ A2 @ B2 ) @ ( fun_up4915027116007172163list_o @ G @ A2 @ B2 ) ) ) ).
% map_le_upd
thf(fact_711_map__le__map__add,axiom,
! [F2: int > option_list_o,G: int > option_list_o] : ( map_le_int_list_o @ F2 @ ( map_add_int_list_o @ G @ F2 ) ) ).
% map_le_map_add
thf(fact_712_map__le__map__add,axiom,
! [F2: nat > option_list_o,G: nat > option_list_o] : ( map_le_nat_list_o @ F2 @ ( map_add_nat_list_o @ G @ F2 ) ) ).
% map_le_map_add
thf(fact_713_map__add__le__mapE,axiom,
! [F2: int > option_list_o,G: int > option_list_o,H: int > option_list_o] :
( ( map_le_int_list_o @ ( map_add_int_list_o @ F2 @ G ) @ H )
=> ( map_le_int_list_o @ G @ H ) ) ).
% map_add_le_mapE
thf(fact_714_map__add__le__mapE,axiom,
! [F2: nat > option_list_o,G: nat > option_list_o,H: nat > option_list_o] :
( ( map_le_nat_list_o @ ( map_add_nat_list_o @ F2 @ G ) @ H )
=> ( map_le_nat_list_o @ G @ H ) ) ).
% map_add_le_mapE
thf(fact_715_map__add__le__mapI,axiom,
! [F2: int > option_list_o,H: int > option_list_o,G: int > option_list_o] :
( ( map_le_int_list_o @ F2 @ H )
=> ( ( map_le_int_list_o @ G @ H )
=> ( map_le_int_list_o @ ( map_add_int_list_o @ F2 @ G ) @ H ) ) ) ).
% map_add_le_mapI
thf(fact_716_map__add__le__mapI,axiom,
! [F2: nat > option_list_o,H: nat > option_list_o,G: nat > option_list_o] :
( ( map_le_nat_list_o @ F2 @ H )
=> ( ( map_le_nat_list_o @ G @ H )
=> ( map_le_nat_list_o @ ( map_add_nat_list_o @ F2 @ G ) @ H ) ) ) ).
% map_add_le_mapI
thf(fact_717_map__add__subsumed1,axiom,
! [F2: int > option_list_o,G: int > option_list_o] :
( ( map_le_int_list_o @ F2 @ G )
=> ( ( map_add_int_list_o @ F2 @ G )
= G ) ) ).
% map_add_subsumed1
thf(fact_718_map__add__subsumed1,axiom,
! [F2: nat > option_list_o,G: nat > option_list_o] :
( ( map_le_nat_list_o @ F2 @ G )
=> ( ( map_add_nat_list_o @ F2 @ G )
= G ) ) ).
% map_add_subsumed1
thf(fact_719_map__add__subsumed2,axiom,
! [F2: int > option_list_o,G: int > option_list_o] :
( ( map_le_int_list_o @ F2 @ G )
=> ( ( map_add_int_list_o @ G @ F2 )
= G ) ) ).
% map_add_subsumed2
thf(fact_720_map__add__subsumed2,axiom,
! [F2: nat > option_list_o,G: nat > option_list_o] :
( ( map_le_nat_list_o @ F2 @ G )
=> ( ( map_add_nat_list_o @ G @ F2 )
= G ) ) ).
% map_add_subsumed2
thf(fact_721_map__le__iff__map__add__commute,axiom,
! [F2: int > option_list_o,G: int > option_list_o] :
( ( map_le_int_list_o @ F2 @ ( map_add_int_list_o @ F2 @ G ) )
= ( ( map_add_int_list_o @ F2 @ G )
= ( map_add_int_list_o @ G @ F2 ) ) ) ).
% map_le_iff_map_add_commute
thf(fact_722_map__le__iff__map__add__commute,axiom,
! [F2: nat > option_list_o,G: nat > option_list_o] :
( ( map_le_nat_list_o @ F2 @ ( map_add_nat_list_o @ F2 @ G ) )
= ( ( map_add_nat_list_o @ F2 @ G )
= ( map_add_nat_list_o @ G @ F2 ) ) ) ).
% map_le_iff_map_add_commute
thf(fact_723_upd__None__map__le,axiom,
! [F2: int > option_list_o,X: int] : ( map_le_int_list_o @ ( fun_up5258779583477023609list_o @ F2 @ X @ none_list_o ) @ F2 ) ).
% upd_None_map_le
thf(fact_724_upd__None__map__le,axiom,
! [F2: nat > option_list_o,X: nat] : ( map_le_nat_list_o @ ( fun_up6685928411843432661list_o @ F2 @ X @ none_list_o ) @ F2 ) ).
% upd_None_map_le
thf(fact_725_upd__None__map__le,axiom,
! [F2: nat > option_nat,X: nat] : ( map_le_nat_nat @ ( fun_up1493157387958331631on_nat @ F2 @ X @ none_nat ) @ F2 ) ).
% upd_None_map_le
thf(fact_726_upd__None__map__le,axiom,
! [F2: option_list_o > option_option_list_o,X: option_list_o] : ( map_le539069141309449381list_o @ ( fun_up4915027116007172163list_o @ F2 @ X @ none_option_list_o ) @ F2 ) ).
% upd_None_map_le
thf(fact_727_bot__fun__def,axiom,
( bot_bot_nat_o
= ( ^ [X3: nat] : bot_bot_o ) ) ).
% bot_fun_def
thf(fact_728_bot__fun__def,axiom,
( bot_bot_int_o
= ( ^ [X3: int] : bot_bot_o ) ) ).
% bot_fun_def
thf(fact_729_boolean__algebra__cancel_Osup2,axiom,
! [B4: set_list_o,K: set_list_o,B2: set_list_o,A2: set_list_o] :
( ( B4
= ( sup_sup_set_list_o @ K @ B2 ) )
=> ( ( sup_sup_set_list_o @ A2 @ B4 )
= ( sup_sup_set_list_o @ K @ ( sup_sup_set_list_o @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_730_boolean__algebra__cancel_Osup2,axiom,
! [B4: set_nat,K: set_nat,B2: set_nat,A2: set_nat] :
( ( B4
= ( sup_sup_set_nat @ K @ B2 ) )
=> ( ( sup_sup_set_nat @ A2 @ B4 )
= ( sup_sup_set_nat @ K @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_731_boolean__algebra__cancel_Osup2,axiom,
! [B4: set_int,K: set_int,B2: set_int,A2: set_int] :
( ( B4
= ( sup_sup_set_int @ K @ B2 ) )
=> ( ( sup_sup_set_int @ A2 @ B4 )
= ( sup_sup_set_int @ K @ ( sup_sup_set_int @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_732_boolean__algebra__cancel_Osup2,axiom,
! [B4: nat,K: nat,B2: nat,A2: nat] :
( ( B4
= ( sup_sup_nat @ K @ B2 ) )
=> ( ( sup_sup_nat @ A2 @ B4 )
= ( sup_sup_nat @ K @ ( sup_sup_nat @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_733_boolean__algebra__cancel_Osup2,axiom,
! [B4: int,K: int,B2: int,A2: int] :
( ( B4
= ( sup_sup_int @ K @ B2 ) )
=> ( ( sup_sup_int @ A2 @ B4 )
= ( sup_sup_int @ K @ ( sup_sup_int @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_734_boolean__algebra__cancel_Osup1,axiom,
! [A: set_list_o,K: set_list_o,A2: set_list_o,B2: set_list_o] :
( ( A
= ( sup_sup_set_list_o @ K @ A2 ) )
=> ( ( sup_sup_set_list_o @ A @ B2 )
= ( sup_sup_set_list_o @ K @ ( sup_sup_set_list_o @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_735_boolean__algebra__cancel_Osup1,axiom,
! [A: set_nat,K: set_nat,A2: set_nat,B2: set_nat] :
( ( A
= ( sup_sup_set_nat @ K @ A2 ) )
=> ( ( sup_sup_set_nat @ A @ B2 )
= ( sup_sup_set_nat @ K @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_736_boolean__algebra__cancel_Osup1,axiom,
! [A: set_int,K: set_int,A2: set_int,B2: set_int] :
( ( A
= ( sup_sup_set_int @ K @ A2 ) )
=> ( ( sup_sup_set_int @ A @ B2 )
= ( sup_sup_set_int @ K @ ( sup_sup_set_int @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_737_boolean__algebra__cancel_Osup1,axiom,
! [A: nat,K: nat,A2: nat,B2: nat] :
( ( A
= ( sup_sup_nat @ K @ A2 ) )
=> ( ( sup_sup_nat @ A @ B2 )
= ( sup_sup_nat @ K @ ( sup_sup_nat @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_738_boolean__algebra__cancel_Osup1,axiom,
! [A: int,K: int,A2: int,B2: int] :
( ( A
= ( sup_sup_int @ K @ A2 ) )
=> ( ( sup_sup_int @ A @ B2 )
= ( sup_sup_int @ K @ ( sup_sup_int @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_739_Collect__empty__eq__bot,axiom,
! [P: list_o > $o] :
( ( ( collect_list_o @ P )
= bot_bot_set_list_o )
= ( P = bot_bot_list_o_o ) ) ).
% Collect_empty_eq_bot
thf(fact_740_Collect__empty__eq__bot,axiom,
! [P: nat > $o] :
( ( ( collect_nat @ P )
= bot_bot_set_nat )
= ( P = bot_bot_nat_o ) ) ).
% Collect_empty_eq_bot
thf(fact_741_Collect__empty__eq__bot,axiom,
! [P: int > $o] :
( ( ( collect_int @ P )
= bot_bot_set_int )
= ( P = bot_bot_int_o ) ) ).
% Collect_empty_eq_bot
thf(fact_742_bot__empty__eq,axiom,
( bot_bot_option_nat_o
= ( ^ [X3: option_nat] : ( member_option_nat @ X3 @ bot_bo5009843511495006442on_nat ) ) ) ).
% bot_empty_eq
thf(fact_743_bot__empty__eq,axiom,
( bot_bot_option_int_o
= ( ^ [X3: option_int] : ( member_option_int @ X3 @ bot_bo2519760366563279686on_int ) ) ) ).
% bot_empty_eq
thf(fact_744_bot__empty__eq,axiom,
( bot_bot_set_nat_o
= ( ^ [X3: set_nat] : ( member_set_nat @ X3 @ bot_bot_set_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_745_bot__empty__eq,axiom,
( bot_bot_set_int_o
= ( ^ [X3: set_int] : ( member_set_int @ X3 @ bot_bot_set_set_int ) ) ) ).
% bot_empty_eq
thf(fact_746_bot__empty__eq,axiom,
( bot_bot_list_o_o
= ( ^ [X3: list_o] : ( member_list_o @ X3 @ bot_bot_set_list_o ) ) ) ).
% bot_empty_eq
thf(fact_747_bot__empty__eq,axiom,
( bot_bot_nat_o
= ( ^ [X3: nat] : ( member_nat @ X3 @ bot_bot_set_nat ) ) ) ).
% bot_empty_eq
thf(fact_748_bot__empty__eq,axiom,
( bot_bot_int_o
= ( ^ [X3: int] : ( member_int @ X3 @ bot_bot_set_int ) ) ) ).
% bot_empty_eq
thf(fact_749_opt__append_Oelims,axiom,
! [X: option_list_o,Xa: option_list_o,Y: option_list_o] :
( ( ( prefix5314359684614007693append @ X @ Xa )
= Y )
=> ( ! [X2: list_o] :
( ( X
= ( some_list_o @ X2 ) )
=> ! [Y2: list_o] :
( ( Xa
= ( some_list_o @ Y2 ) )
=> ( Y
!= ( some_list_o @ ( append_o @ X2 @ Y2 ) ) ) ) )
=> ( ( ( X = none_list_o )
=> ( Y != none_list_o ) )
=> ~ ( ( Xa = none_list_o )
=> ( Y != none_list_o ) ) ) ) ) ).
% opt_append.elims
thf(fact_750_is__none__bind,axiom,
! [F2: option_list_o,G: list_o > option_list_o] :
( ( is_none_list_o @ ( bind_list_o_list_o @ F2 @ G ) )
= ( ( is_none_list_o @ F2 )
| ( is_none_list_o @ ( G @ ( the_list_o @ F2 ) ) ) ) ) ).
% is_none_bind
thf(fact_751_is__none__bind,axiom,
! [F2: option_int,G: int > option_list_o] :
( ( is_none_list_o @ ( bind_int_list_o @ F2 @ G ) )
= ( ( is_none_int @ F2 )
| ( is_none_list_o @ ( G @ ( the_int @ F2 ) ) ) ) ) ).
% is_none_bind
thf(fact_752_is__none__bind,axiom,
! [F2: option_nat,G: nat > option_list_o] :
( ( is_none_list_o @ ( bind_nat_list_o @ F2 @ G ) )
= ( ( is_none_nat @ F2 )
| ( is_none_list_o @ ( G @ ( the_nat @ F2 ) ) ) ) ) ).
% is_none_bind
thf(fact_753_map__add__comm,axiom,
! [M1: product_prod_int_nat > option_list_o,M22: product_prod_int_nat > option_list_o] :
( ( ( inf_in4759246646417213867nt_nat @ ( dom_Pr5309327336824135123list_o @ M1 ) @ ( dom_Pr5309327336824135123list_o @ M22 ) )
= bot_bo4286715327455315753nt_nat )
=> ( ( map_ad1598970985909685942list_o @ M1 @ M22 )
= ( map_ad1598970985909685942list_o @ M22 @ M1 ) ) ) ).
% map_add_comm
thf(fact_754_map__add__comm,axiom,
! [M1: product_prod_nat_int > option_list_o,M22: product_prod_nat_int > option_list_o] :
( ( ( inf_in82241926792465323at_int @ ( dom_Pr7669787946167753683list_o @ M1 ) @ ( dom_Pr7669787946167753683list_o @ M22 ) )
= bot_bo8833082644685343017at_int )
=> ( ( map_ad3959431595253304502list_o @ M1 @ M22 )
= ( map_ad3959431595253304502list_o @ M22 @ M1 ) ) ) ).
% map_add_comm
thf(fact_755_map__add__comm,axiom,
! [M1: product_prod_nat_nat > option_list_o,M22: product_prod_nat_nat > option_list_o] :
( ( ( inf_in2572325071724192079at_nat @ ( dom_Pr5202128127435093295list_o @ M1 ) @ ( dom_Pr5202128127435093295list_o @ M22 ) )
= bot_bo2099793752762293965at_nat )
=> ( ( map_ad1491771776520644114list_o @ M1 @ M22 )
= ( map_ad1491771776520644114list_o @ M22 @ M1 ) ) ) ).
% map_add_comm
thf(fact_756_map__add__comm,axiom,
! [M1: nat > option_list_o,M22: nat > option_list_o] :
( ( ( inf_inf_set_nat @ ( dom_nat_list_o @ M1 ) @ ( dom_nat_list_o @ M22 ) )
= bot_bot_set_nat )
=> ( ( map_add_nat_list_o @ M1 @ M22 )
= ( map_add_nat_list_o @ M22 @ M1 ) ) ) ).
% map_add_comm
thf(fact_757_map__add__comm,axiom,
! [M1: int > option_list_o,M22: int > option_list_o] :
( ( ( inf_inf_set_int @ ( dom_int_list_o @ M1 ) @ ( dom_int_list_o @ M22 ) )
= bot_bot_set_int )
=> ( ( map_add_int_list_o @ M1 @ M22 )
= ( map_add_int_list_o @ M22 @ M1 ) ) ) ).
% map_add_comm
thf(fact_758_set__empty__eq,axiom,
! [Xo: option_list_o] :
( ( ( set_option_list_o2 @ Xo )
= bot_bot_set_list_o )
= ( Xo = none_list_o ) ) ).
% set_empty_eq
thf(fact_759_Int__iff,axiom,
! [C: nat,A: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B4 ) )
= ( ( member_nat @ C @ A )
& ( member_nat @ C @ B4 ) ) ) ).
% Int_iff
thf(fact_760_Int__iff,axiom,
! [C: int,A: set_int,B4: set_int] :
( ( member_int @ C @ ( inf_inf_set_int @ A @ B4 ) )
= ( ( member_int @ C @ A )
& ( member_int @ C @ B4 ) ) ) ).
% Int_iff
thf(fact_761_IntI,axiom,
! [C: nat,A: set_nat,B4: set_nat] :
( ( member_nat @ C @ A )
=> ( ( member_nat @ C @ B4 )
=> ( member_nat @ C @ ( inf_inf_set_nat @ A @ B4 ) ) ) ) ).
% IntI
thf(fact_762_IntI,axiom,
! [C: int,A: set_int,B4: set_int] :
( ( member_int @ C @ A )
=> ( ( member_int @ C @ B4 )
=> ( member_int @ C @ ( inf_inf_set_int @ A @ B4 ) ) ) ) ).
% IntI
thf(fact_763_same__prefix__prefix,axiom,
! [Xs: list_o,Ys: list_o,Zs: list_o] :
( ( prefix_o @ ( append_o @ Xs @ Ys ) @ ( append_o @ Xs @ Zs ) )
= ( prefix_o @ Ys @ Zs ) ) ).
% same_prefix_prefix
thf(fact_764_elem__set,axiom,
! [X: nat,Xo: option_nat] :
( ( member_nat @ X @ ( set_option_nat2 @ Xo ) )
= ( Xo
= ( some_nat @ X ) ) ) ).
% elem_set
thf(fact_765_elem__set,axiom,
! [X: int,Xo: option_int] :
( ( member_int @ X @ ( set_option_int2 @ Xo ) )
= ( Xo
= ( some_int @ X ) ) ) ).
% elem_set
thf(fact_766_elem__set,axiom,
! [X: list_o,Xo: option_list_o] :
( ( member_list_o @ X @ ( set_option_list_o2 @ Xo ) )
= ( Xo
= ( some_list_o @ X ) ) ) ).
% elem_set
thf(fact_767_is__none__code_I2_J,axiom,
! [X: list_o] :
~ ( is_none_list_o @ ( some_list_o @ X ) ) ).
% is_none_code(2)
thf(fact_768_is__none__code_I1_J,axiom,
is_none_list_o @ none_list_o ).
% is_none_code(1)
thf(fact_769_IntD2,axiom,
! [C: nat,A: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B4 ) )
=> ( member_nat @ C @ B4 ) ) ).
% IntD2
thf(fact_770_IntD2,axiom,
! [C: int,A: set_int,B4: set_int] :
( ( member_int @ C @ ( inf_inf_set_int @ A @ B4 ) )
=> ( member_int @ C @ B4 ) ) ).
% IntD2
thf(fact_771_IntD1,axiom,
! [C: nat,A: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B4 ) )
=> ( member_nat @ C @ A ) ) ).
% IntD1
thf(fact_772_IntD1,axiom,
! [C: int,A: set_int,B4: set_int] :
( ( member_int @ C @ ( inf_inf_set_int @ A @ B4 ) )
=> ( member_int @ C @ A ) ) ).
% IntD1
thf(fact_773_IntE,axiom,
! [C: nat,A: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B4 ) )
=> ~ ( ( member_nat @ C @ A )
=> ~ ( member_nat @ C @ B4 ) ) ) ).
% IntE
thf(fact_774_IntE,axiom,
! [C: int,A: set_int,B4: set_int] :
( ( member_int @ C @ ( inf_inf_set_int @ A @ B4 ) )
=> ~ ( ( member_int @ C @ A )
=> ~ ( member_int @ C @ B4 ) ) ) ).
% IntE
thf(fact_775_Int__emptyI,axiom,
! [A: set_nat,B4: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ~ ( member_nat @ X2 @ B4 ) )
=> ( ( inf_inf_set_nat @ A @ B4 )
= bot_bot_set_nat ) ) ).
% Int_emptyI
thf(fact_776_Int__emptyI,axiom,
! [A: set_int,B4: set_int] :
( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ~ ( member_int @ X2 @ B4 ) )
=> ( ( inf_inf_set_int @ A @ B4 )
= bot_bot_set_int ) ) ).
% Int_emptyI
thf(fact_777_disjoint__iff,axiom,
! [A: set_nat,B4: set_nat] :
( ( ( inf_inf_set_nat @ A @ B4 )
= bot_bot_set_nat )
= ( ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ~ ( member_nat @ X3 @ B4 ) ) ) ) ).
% disjoint_iff
thf(fact_778_disjoint__iff,axiom,
! [A: set_int,B4: set_int] :
( ( ( inf_inf_set_int @ A @ B4 )
= bot_bot_set_int )
= ( ! [X3: int] :
( ( member_int @ X3 @ A )
=> ~ ( member_int @ X3 @ B4 ) ) ) ) ).
% disjoint_iff
thf(fact_779_prefixE,axiom,
! [Xs: list_o,Ys: list_o] :
( ( prefix_o @ Xs @ Ys )
=> ~ ! [Zs2: list_o] :
( Ys
!= ( append_o @ Xs @ Zs2 ) ) ) ).
% prefixE
thf(fact_780_prefixI,axiom,
! [Ys: list_o,Xs: list_o,Zs: list_o] :
( ( Ys
= ( append_o @ Xs @ Zs ) )
=> ( prefix_o @ Xs @ Ys ) ) ).
% prefixI
thf(fact_781_prefix__def,axiom,
( prefix_o
= ( ^ [Xs2: list_o,Ys2: list_o] :
? [Zs3: list_o] :
( Ys2
= ( append_o @ Xs2 @ Zs3 ) ) ) ) ).
% prefix_def
thf(fact_782_prefix__append,axiom,
! [Xs: list_o,Ys: list_o,Zs: list_o] :
( ( prefix_o @ Xs @ ( append_o @ Ys @ Zs ) )
= ( ( prefix_o @ Xs @ Ys )
| ? [Us: list_o] :
( ( Xs
= ( append_o @ Ys @ Us ) )
& ( prefix_o @ Us @ Zs ) ) ) ) ).
% prefix_append
thf(fact_783_prefix__prefix,axiom,
! [Xs: list_o,Ys: list_o,Zs: list_o] :
( ( prefix_o @ Xs @ Ys )
=> ( prefix_o @ Xs @ ( append_o @ Ys @ Zs ) ) ) ).
% prefix_prefix
thf(fact_784_append__prefixD,axiom,
! [Xs: list_o,Ys: list_o,Zs: list_o] :
( ( prefix_o @ ( append_o @ Xs @ Ys ) @ Zs )
=> ( prefix_o @ Xs @ Zs ) ) ).
% append_prefixD
thf(fact_785_inj__on__Int,axiom,
! [F2: nat > option_list_o,A: set_nat,B4: set_nat] :
( ( ( inj_on1630180835328728801list_o @ F2 @ A )
| ( inj_on1630180835328728801list_o @ F2 @ B4 ) )
=> ( inj_on1630180835328728801list_o @ F2 @ ( inf_inf_set_nat @ A @ B4 ) ) ) ).
% inj_on_Int
thf(fact_786_inj__on__Int,axiom,
! [F2: int > option_list_o,A: set_int,B4: set_int] :
( ( ( inj_on203032006962319749list_o @ F2 @ A )
| ( inj_on203032006962319749list_o @ F2 @ B4 ) )
=> ( inj_on203032006962319749list_o @ F2 @ ( inf_inf_set_int @ A @ B4 ) ) ) ).
% inj_on_Int
thf(fact_787_option_Oset__cases,axiom,
! [E: nat,A2: option_nat] :
( ( member_nat @ E @ ( set_option_nat2 @ A2 ) )
=> ( A2
= ( some_nat @ E ) ) ) ).
% option.set_cases
thf(fact_788_option_Oset__cases,axiom,
! [E: int,A2: option_int] :
( ( member_int @ E @ ( set_option_int2 @ A2 ) )
=> ( A2
= ( some_int @ E ) ) ) ).
% option.set_cases
thf(fact_789_option_Oset__cases,axiom,
! [E: list_o,A2: option_list_o] :
( ( member_list_o @ E @ ( set_option_list_o2 @ A2 ) )
=> ( A2
= ( some_list_o @ E ) ) ) ).
% option.set_cases
thf(fact_790_option_Oset__intros,axiom,
! [X22: nat] : ( member_nat @ X22 @ ( set_option_nat2 @ ( some_nat @ X22 ) ) ) ).
% option.set_intros
thf(fact_791_option_Oset__intros,axiom,
! [X22: int] : ( member_int @ X22 @ ( set_option_int2 @ ( some_int @ X22 ) ) ) ).
% option.set_intros
thf(fact_792_option_Oset__intros,axiom,
! [X22: list_o] : ( member_list_o @ X22 @ ( set_option_list_o2 @ ( some_list_o @ X22 ) ) ) ).
% option.set_intros
thf(fact_793_ospec,axiom,
! [A: option_list_o,P: list_o > $o,X: list_o] :
( ! [X2: list_o] :
( ( member_list_o @ X2 @ ( set_option_list_o2 @ A ) )
=> ( P @ X2 ) )
=> ( ( A
= ( some_list_o @ X ) )
=> ( P @ X ) ) ) ).
% ospec
thf(fact_794_is__none__simps_I2_J,axiom,
! [X: list_o] :
~ ( is_none_list_o @ ( some_list_o @ X ) ) ).
% is_none_simps(2)
thf(fact_795_is__none__simps_I1_J,axiom,
is_none_list_o @ none_list_o ).
% is_none_simps(1)
thf(fact_796_Option_Ois__none__def,axiom,
( is_none_list_o
= ( ^ [X3: option_list_o] : ( X3 = none_list_o ) ) ) ).
% Option.is_none_def
thf(fact_797_option_Osimps_I14_J,axiom,
( ( set_option_list_o2 @ none_list_o )
= bot_bot_set_list_o ) ).
% option.simps(14)
thf(fact_798_opt__append_Osimps_I1_J,axiom,
! [X: list_o,Y: list_o] :
( ( prefix5314359684614007693append @ ( some_list_o @ X ) @ ( some_list_o @ Y ) )
= ( some_list_o @ ( append_o @ X @ Y ) ) ) ).
% opt_append.simps(1)
thf(fact_799_option_Oset__sel,axiom,
! [A2: option_nat] :
( ( A2 != none_nat )
=> ( member_nat @ ( the_nat @ A2 ) @ ( set_option_nat2 @ A2 ) ) ) ).
% option.set_sel
thf(fact_800_option_Oset__sel,axiom,
! [A2: option_int] :
( ( A2 != none_int )
=> ( member_int @ ( the_int @ A2 ) @ ( set_option_int2 @ A2 ) ) ) ).
% option.set_sel
thf(fact_801_option_Oset__sel,axiom,
! [A2: option_list_o] :
( ( A2 != none_list_o )
=> ( member_list_o @ ( the_list_o @ A2 ) @ ( set_option_list_o2 @ A2 ) ) ) ).
% option.set_sel
thf(fact_802_ran__map__add,axiom,
! [M1: nat > option_list_o,M22: nat > option_list_o] :
( ( ( inf_inf_set_nat @ ( dom_nat_list_o @ M1 ) @ ( dom_nat_list_o @ M22 ) )
= bot_bot_set_nat )
=> ( ( ran_nat_list_o @ ( map_add_nat_list_o @ M1 @ M22 ) )
= ( sup_sup_set_list_o @ ( ran_nat_list_o @ M1 ) @ ( ran_nat_list_o @ M22 ) ) ) ) ).
% ran_map_add
thf(fact_803_ran__map__add,axiom,
! [M1: int > option_list_o,M22: int > option_list_o] :
( ( ( inf_inf_set_int @ ( dom_int_list_o @ M1 ) @ ( dom_int_list_o @ M22 ) )
= bot_bot_set_int )
=> ( ( ran_int_list_o @ ( map_add_int_list_o @ M1 @ M22 ) )
= ( sup_sup_set_list_o @ ( ran_int_list_o @ M1 ) @ ( ran_int_list_o @ M22 ) ) ) ) ).
% ran_map_add
thf(fact_804_graph__map__add,axiom,
! [M1: nat > option_list_o,M22: nat > option_list_o] :
( ( ( inf_inf_set_nat @ ( dom_nat_list_o @ M1 ) @ ( dom_nat_list_o @ M22 ) )
= bot_bot_set_nat )
=> ( ( graph_nat_list_o @ ( map_add_nat_list_o @ M1 @ M22 ) )
= ( sup_su4620230350752640421list_o @ ( graph_nat_list_o @ M1 ) @ ( graph_nat_list_o @ M22 ) ) ) ) ).
% graph_map_add
thf(fact_805_graph__map__add,axiom,
! [M1: int > option_list_o,M22: int > option_list_o] :
( ( ( inf_inf_set_int @ ( dom_int_list_o @ M1 ) @ ( dom_int_list_o @ M22 ) )
= bot_bot_set_int )
=> ( ( graph_int_list_o @ ( map_add_int_list_o @ M1 @ M22 ) )
= ( sup_su7087890169485300809list_o @ ( graph_int_list_o @ M1 ) @ ( graph_int_list_o @ M22 ) ) ) ) ).
% graph_map_add
thf(fact_806_dom__restrict,axiom,
! [M: nat > option_list_o,A: set_nat] :
( ( dom_nat_list_o @ ( restri60617331834859036list_o @ M @ A ) )
= ( inf_inf_set_nat @ ( dom_nat_list_o @ M ) @ A ) ) ).
% dom_restrict
thf(fact_807_dom__restrict,axiom,
! [M: int > option_list_o,A: set_int] :
( ( dom_int_list_o @ ( restri2456407693429728960list_o @ M @ A ) )
= ( inf_inf_set_int @ ( dom_int_list_o @ M ) @ A ) ) ).
% dom_restrict
thf(fact_808_option_Osimps_I15_J,axiom,
! [X22: list_o] :
( ( set_option_list_o2 @ ( some_list_o @ X22 ) )
= ( insert_list_o @ X22 @ bot_bot_set_list_o ) ) ).
% option.simps(15)
thf(fact_809_insertCI,axiom,
! [A2: nat,B4: set_nat,B2: nat] :
( ( ~ ( member_nat @ A2 @ B4 )
=> ( A2 = B2 ) )
=> ( member_nat @ A2 @ ( insert_nat @ B2 @ B4 ) ) ) ).
% insertCI
thf(fact_810_insertCI,axiom,
! [A2: int,B4: set_int,B2: int] :
( ( ~ ( member_int @ A2 @ B4 )
=> ( A2 = B2 ) )
=> ( member_int @ A2 @ ( insert_int @ B2 @ B4 ) ) ) ).
% insertCI
thf(fact_811_insert__iff,axiom,
! [A2: nat,B2: nat,A: set_nat] :
( ( member_nat @ A2 @ ( insert_nat @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_nat @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_812_insert__iff,axiom,
! [A2: int,B2: int,A: set_int] :
( ( member_int @ A2 @ ( insert_int @ B2 @ A ) )
= ( ( A2 = B2 )
| ( member_int @ A2 @ A ) ) ) ).
% insert_iff
thf(fact_813_singletonI,axiom,
! [A2: nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ).
% singletonI
thf(fact_814_singletonI,axiom,
! [A2: int] : ( member_int @ A2 @ ( insert_int @ A2 @ bot_bot_set_int ) ) ).
% singletonI
thf(fact_815_Int__insert__left__if0,axiom,
! [A2: nat,C2: set_nat,B4: set_nat] :
( ~ ( member_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B4 ) @ C2 )
= ( inf_inf_set_nat @ B4 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_816_Int__insert__left__if0,axiom,
! [A2: int,C2: set_int,B4: set_int] :
( ~ ( member_int @ A2 @ C2 )
=> ( ( inf_inf_set_int @ ( insert_int @ A2 @ B4 ) @ C2 )
= ( inf_inf_set_int @ B4 @ C2 ) ) ) ).
% Int_insert_left_if0
thf(fact_817_Int__insert__left__if1,axiom,
! [A2: nat,C2: set_nat,B4: set_nat] :
( ( member_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B4 ) @ C2 )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ B4 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_818_Int__insert__left__if1,axiom,
! [A2: int,C2: set_int,B4: set_int] :
( ( member_int @ A2 @ C2 )
=> ( ( inf_inf_set_int @ ( insert_int @ A2 @ B4 ) @ C2 )
= ( insert_int @ A2 @ ( inf_inf_set_int @ B4 @ C2 ) ) ) ) ).
% Int_insert_left_if1
thf(fact_819_Int__insert__right__if0,axiom,
! [A2: nat,A: set_nat,B4: set_nat] :
( ~ ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B4 ) )
= ( inf_inf_set_nat @ A @ B4 ) ) ) ).
% Int_insert_right_if0
thf(fact_820_Int__insert__right__if0,axiom,
! [A2: int,A: set_int,B4: set_int] :
( ~ ( member_int @ A2 @ A )
=> ( ( inf_inf_set_int @ A @ ( insert_int @ A2 @ B4 ) )
= ( inf_inf_set_int @ A @ B4 ) ) ) ).
% Int_insert_right_if0
thf(fact_821_Int__insert__right__if1,axiom,
! [A2: nat,A: set_nat,B4: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B4 ) )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ A @ B4 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_822_Int__insert__right__if1,axiom,
! [A2: int,A: set_int,B4: set_int] :
( ( member_int @ A2 @ A )
=> ( ( inf_inf_set_int @ A @ ( insert_int @ A2 @ B4 ) )
= ( insert_int @ A2 @ ( inf_inf_set_int @ A @ B4 ) ) ) ) ).
% Int_insert_right_if1
thf(fact_823_restrict__out,axiom,
! [X: nat,A: set_nat,M: nat > option_list_o] :
( ~ ( member_nat @ X @ A )
=> ( ( restri60617331834859036list_o @ M @ A @ X )
= none_list_o ) ) ).
% restrict_out
thf(fact_824_restrict__out,axiom,
! [X: int,A: set_int,M: int > option_list_o] :
( ~ ( member_int @ X @ A )
=> ( ( restri2456407693429728960list_o @ M @ A @ X )
= none_list_o ) ) ).
% restrict_out
thf(fact_825_these__insert__None,axiom,
! [A: set_option_list_o] :
( ( these_list_o @ ( insert_option_list_o @ none_list_o @ A ) )
= ( these_list_o @ A ) ) ).
% these_insert_None
thf(fact_826_disjoint__insert_I2_J,axiom,
! [A: set_nat,B2: nat,B4: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ ( insert_nat @ B2 @ B4 ) ) )
= ( ~ ( member_nat @ B2 @ A )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ B4 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_827_disjoint__insert_I2_J,axiom,
! [A: set_int,B2: int,B4: set_int] :
( ( bot_bot_set_int
= ( inf_inf_set_int @ A @ ( insert_int @ B2 @ B4 ) ) )
= ( ~ ( member_int @ B2 @ A )
& ( bot_bot_set_int
= ( inf_inf_set_int @ A @ B4 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_828_disjoint__insert_I1_J,axiom,
! [B4: set_nat,A2: nat,A: set_nat] :
( ( ( inf_inf_set_nat @ B4 @ ( insert_nat @ A2 @ A ) )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A2 @ B4 )
& ( ( inf_inf_set_nat @ B4 @ A )
= bot_bot_set_nat ) ) ) ).
% disjoint_insert(1)
thf(fact_829_disjoint__insert_I1_J,axiom,
! [B4: set_int,A2: int,A: set_int] :
( ( ( inf_inf_set_int @ B4 @ ( insert_int @ A2 @ A ) )
= bot_bot_set_int )
= ( ~ ( member_int @ A2 @ B4 )
& ( ( inf_inf_set_int @ B4 @ A )
= bot_bot_set_int ) ) ) ).
% disjoint_insert(1)
thf(fact_830_insert__disjoint_I2_J,axiom,
! [A2: nat,A: set_nat,B4: set_nat] :
( ( bot_bot_set_nat
= ( inf_inf_set_nat @ ( insert_nat @ A2 @ A ) @ B4 ) )
= ( ~ ( member_nat @ A2 @ B4 )
& ( bot_bot_set_nat
= ( inf_inf_set_nat @ A @ B4 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_831_insert__disjoint_I2_J,axiom,
! [A2: int,A: set_int,B4: set_int] :
( ( bot_bot_set_int
= ( inf_inf_set_int @ ( insert_int @ A2 @ A ) @ B4 ) )
= ( ~ ( member_int @ A2 @ B4 )
& ( bot_bot_set_int
= ( inf_inf_set_int @ A @ B4 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_832_insert__disjoint_I1_J,axiom,
! [A2: nat,A: set_nat,B4: set_nat] :
( ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ A ) @ B4 )
= bot_bot_set_nat )
= ( ~ ( member_nat @ A2 @ B4 )
& ( ( inf_inf_set_nat @ A @ B4 )
= bot_bot_set_nat ) ) ) ).
% insert_disjoint(1)
thf(fact_833_insert__disjoint_I1_J,axiom,
! [A2: int,A: set_int,B4: set_int] :
( ( ( inf_inf_set_int @ ( insert_int @ A2 @ A ) @ B4 )
= bot_bot_set_int )
= ( ~ ( member_int @ A2 @ B4 )
& ( ( inf_inf_set_int @ A @ B4 )
= bot_bot_set_int ) ) ) ).
% insert_disjoint(1)
thf(fact_834_these__insert__Some,axiom,
! [X: list_o,A: set_option_list_o] :
( ( these_list_o @ ( insert_option_list_o @ ( some_list_o @ X ) @ A ) )
= ( insert_list_o @ X @ ( these_list_o @ A ) ) ) ).
% these_insert_Some
thf(fact_835_insertE,axiom,
! [A2: nat,B2: nat,A: set_nat] :
( ( member_nat @ A2 @ ( insert_nat @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_nat @ A2 @ A ) ) ) ).
% insertE
thf(fact_836_insertE,axiom,
! [A2: int,B2: int,A: set_int] :
( ( member_int @ A2 @ ( insert_int @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( member_int @ A2 @ A ) ) ) ).
% insertE
thf(fact_837_insertI1,axiom,
! [A2: nat,B4: set_nat] : ( member_nat @ A2 @ ( insert_nat @ A2 @ B4 ) ) ).
% insertI1
thf(fact_838_insertI1,axiom,
! [A2: int,B4: set_int] : ( member_int @ A2 @ ( insert_int @ A2 @ B4 ) ) ).
% insertI1
thf(fact_839_insertI2,axiom,
! [A2: nat,B4: set_nat,B2: nat] :
( ( member_nat @ A2 @ B4 )
=> ( member_nat @ A2 @ ( insert_nat @ B2 @ B4 ) ) ) ).
% insertI2
thf(fact_840_insertI2,axiom,
! [A2: int,B4: set_int,B2: int] :
( ( member_int @ A2 @ B4 )
=> ( member_int @ A2 @ ( insert_int @ B2 @ B4 ) ) ) ).
% insertI2
thf(fact_841_Set_Oset__insert,axiom,
! [X: nat,A: set_nat] :
( ( member_nat @ X @ A )
=> ~ ! [B6: set_nat] :
( ( A
= ( insert_nat @ X @ B6 ) )
=> ( member_nat @ X @ B6 ) ) ) ).
% Set.set_insert
thf(fact_842_Set_Oset__insert,axiom,
! [X: int,A: set_int] :
( ( member_int @ X @ A )
=> ~ ! [B6: set_int] :
( ( A
= ( insert_int @ X @ B6 ) )
=> ( member_int @ X @ B6 ) ) ) ).
% Set.set_insert
thf(fact_843_insert__ident,axiom,
! [X: nat,A: set_nat,B4: set_nat] :
( ~ ( member_nat @ X @ A )
=> ( ~ ( member_nat @ X @ B4 )
=> ( ( ( insert_nat @ X @ A )
= ( insert_nat @ X @ B4 ) )
= ( A = B4 ) ) ) ) ).
% insert_ident
thf(fact_844_insert__ident,axiom,
! [X: int,A: set_int,B4: set_int] :
( ~ ( member_int @ X @ A )
=> ( ~ ( member_int @ X @ B4 )
=> ( ( ( insert_int @ X @ A )
= ( insert_int @ X @ B4 ) )
= ( A = B4 ) ) ) ) ).
% insert_ident
thf(fact_845_insert__absorb,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( insert_nat @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_846_insert__absorb,axiom,
! [A2: int,A: set_int] :
( ( member_int @ A2 @ A )
=> ( ( insert_int @ A2 @ A )
= A ) ) ).
% insert_absorb
thf(fact_847_insert__eq__iff,axiom,
! [A2: nat,A: set_nat,B2: nat,B4: set_nat] :
( ~ ( member_nat @ A2 @ A )
=> ( ~ ( member_nat @ B2 @ B4 )
=> ( ( ( insert_nat @ A2 @ A )
= ( insert_nat @ B2 @ B4 ) )
= ( ( ( A2 = B2 )
=> ( A = B4 ) )
& ( ( A2 != B2 )
=> ? [C3: set_nat] :
( ( A
= ( insert_nat @ B2 @ C3 ) )
& ~ ( member_nat @ B2 @ C3 )
& ( B4
= ( insert_nat @ A2 @ C3 ) )
& ~ ( member_nat @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_848_insert__eq__iff,axiom,
! [A2: int,A: set_int,B2: int,B4: set_int] :
( ~ ( member_int @ A2 @ A )
=> ( ~ ( member_int @ B2 @ B4 )
=> ( ( ( insert_int @ A2 @ A )
= ( insert_int @ B2 @ B4 ) )
= ( ( ( A2 = B2 )
=> ( A = B4 ) )
& ( ( A2 != B2 )
=> ? [C3: set_int] :
( ( A
= ( insert_int @ B2 @ C3 ) )
& ~ ( member_int @ B2 @ C3 )
& ( B4
= ( insert_int @ A2 @ C3 ) )
& ~ ( member_int @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_849_mk__disjoint__insert,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ? [B6: set_nat] :
( ( A
= ( insert_nat @ A2 @ B6 ) )
& ~ ( member_nat @ A2 @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_850_mk__disjoint__insert,axiom,
! [A2: int,A: set_int] :
( ( member_int @ A2 @ A )
=> ? [B6: set_int] :
( ( A
= ( insert_int @ A2 @ B6 ) )
& ~ ( member_int @ A2 @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_851_singleton__iff,axiom,
! [B2: nat,A2: nat] :
( ( member_nat @ B2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_852_singleton__iff,axiom,
! [B2: int,A2: int] :
( ( member_int @ B2 @ ( insert_int @ A2 @ bot_bot_set_int ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_853_singletonD,axiom,
! [B2: nat,A2: nat] :
( ( member_nat @ B2 @ ( insert_nat @ A2 @ bot_bot_set_nat ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_854_singletonD,axiom,
! [B2: int,A2: int] :
( ( member_int @ B2 @ ( insert_int @ A2 @ bot_bot_set_int ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_855_Int__insert__left,axiom,
! [A2: nat,C2: set_nat,B4: set_nat] :
( ( ( member_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B4 ) @ C2 )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ B4 @ C2 ) ) ) )
& ( ~ ( member_nat @ A2 @ C2 )
=> ( ( inf_inf_set_nat @ ( insert_nat @ A2 @ B4 ) @ C2 )
= ( inf_inf_set_nat @ B4 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_856_Int__insert__left,axiom,
! [A2: int,C2: set_int,B4: set_int] :
( ( ( member_int @ A2 @ C2 )
=> ( ( inf_inf_set_int @ ( insert_int @ A2 @ B4 ) @ C2 )
= ( insert_int @ A2 @ ( inf_inf_set_int @ B4 @ C2 ) ) ) )
& ( ~ ( member_int @ A2 @ C2 )
=> ( ( inf_inf_set_int @ ( insert_int @ A2 @ B4 ) @ C2 )
= ( inf_inf_set_int @ B4 @ C2 ) ) ) ) ).
% Int_insert_left
thf(fact_857_Int__insert__right,axiom,
! [A2: nat,A: set_nat,B4: set_nat] :
( ( ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B4 ) )
= ( insert_nat @ A2 @ ( inf_inf_set_nat @ A @ B4 ) ) ) )
& ( ~ ( member_nat @ A2 @ A )
=> ( ( inf_inf_set_nat @ A @ ( insert_nat @ A2 @ B4 ) )
= ( inf_inf_set_nat @ A @ B4 ) ) ) ) ).
% Int_insert_right
thf(fact_858_Int__insert__right,axiom,
! [A2: int,A: set_int,B4: set_int] :
( ( ( member_int @ A2 @ A )
=> ( ( inf_inf_set_int @ A @ ( insert_int @ A2 @ B4 ) )
= ( insert_int @ A2 @ ( inf_inf_set_int @ A @ B4 ) ) ) )
& ( ~ ( member_int @ A2 @ A )
=> ( ( inf_inf_set_int @ A @ ( insert_int @ A2 @ B4 ) )
= ( inf_inf_set_int @ A @ B4 ) ) ) ) ).
% Int_insert_right
thf(fact_859_these__not__empty__eq,axiom,
! [B4: set_option_list_o] :
( ( ( these_list_o @ B4 )
!= bot_bot_set_list_o )
= ( ( B4 != bot_bo3275880064859030064list_o )
& ( B4
!= ( insert_option_list_o @ none_list_o @ bot_bo3275880064859030064list_o ) ) ) ) ).
% these_not_empty_eq
thf(fact_860_these__empty__eq,axiom,
! [B4: set_option_list_o] :
( ( ( these_list_o @ B4 )
= bot_bot_set_list_o )
= ( ( B4 = bot_bo3275880064859030064list_o )
| ( B4
= ( insert_option_list_o @ none_list_o @ bot_bo3275880064859030064list_o ) ) ) ) ).
% these_empty_eq
thf(fact_861_inj__def,axiom,
! [F2: nat > option_list_o] :
( ( inj_on1630180835328728801list_o @ F2 @ top_top_set_nat )
= ( ! [X3: nat,Y3: nat] :
( ( ( F2 @ X3 )
= ( F2 @ Y3 ) )
=> ( X3 = Y3 ) ) ) ) ).
% inj_def
thf(fact_862_inj__def,axiom,
! [F2: int > option_list_o] :
( ( inj_on203032006962319749list_o @ F2 @ top_top_set_int )
= ( ! [X3: int,Y3: int] :
( ( ( F2 @ X3 )
= ( F2 @ Y3 ) )
=> ( X3 = Y3 ) ) ) ) ).
% inj_def
thf(fact_863_inj__eq,axiom,
! [F2: nat > option_list_o,X: nat,Y: nat] :
( ( inj_on1630180835328728801list_o @ F2 @ top_top_set_nat )
=> ( ( ( F2 @ X )
= ( F2 @ Y ) )
= ( X = Y ) ) ) ).
% inj_eq
thf(fact_864_inj__eq,axiom,
! [F2: int > option_list_o,X: int,Y: int] :
( ( inj_on203032006962319749list_o @ F2 @ top_top_set_int )
=> ( ( ( F2 @ X )
= ( F2 @ Y ) )
= ( X = Y ) ) ) ).
% inj_eq
thf(fact_865_injI,axiom,
! [F2: nat > option_list_o] :
( ! [X2: nat,Y2: nat] :
( ( ( F2 @ X2 )
= ( F2 @ Y2 ) )
=> ( X2 = Y2 ) )
=> ( inj_on1630180835328728801list_o @ F2 @ top_top_set_nat ) ) ).
% injI
thf(fact_866_injI,axiom,
! [F2: int > option_list_o] :
( ! [X2: int,Y2: int] :
( ( ( F2 @ X2 )
= ( F2 @ Y2 ) )
=> ( X2 = Y2 ) )
=> ( inj_on203032006962319749list_o @ F2 @ top_top_set_int ) ) ).
% injI
thf(fact_867_injD,axiom,
! [F2: nat > option_list_o,X: nat,Y: nat] :
( ( inj_on1630180835328728801list_o @ F2 @ top_top_set_nat )
=> ( ( ( F2 @ X )
= ( F2 @ Y ) )
=> ( X = Y ) ) ) ).
% injD
thf(fact_868_injD,axiom,
! [F2: int > option_list_o,X: int,Y: int] :
( ( inj_on203032006962319749list_o @ F2 @ top_top_set_int )
=> ( ( ( F2 @ X )
= ( F2 @ Y ) )
=> ( X = Y ) ) ) ).
% injD
thf(fact_869_restrict__map__def,axiom,
( restri60617331834859036list_o
= ( ^ [M4: nat > option_list_o,A5: set_nat,X3: nat] : ( if_option_list_o @ ( member_nat @ X3 @ A5 ) @ ( M4 @ X3 ) @ none_list_o ) ) ) ).
% restrict_map_def
thf(fact_870_restrict__map__def,axiom,
( restri2456407693429728960list_o
= ( ^ [M4: int > option_list_o,A5: set_int,X3: int] : ( if_option_list_o @ ( member_int @ X3 @ A5 ) @ ( M4 @ X3 ) @ none_list_o ) ) ) ).
% restrict_map_def
thf(fact_871_in__these__eq,axiom,
! [X: nat,A: set_option_nat] :
( ( member_nat @ X @ ( these_nat @ A ) )
= ( member_option_nat @ ( some_nat @ X ) @ A ) ) ).
% in_these_eq
thf(fact_872_in__these__eq,axiom,
! [X: int,A: set_option_int] :
( ( member_int @ X @ ( these_int @ A ) )
= ( member_option_int @ ( some_int @ X ) @ A ) ) ).
% in_these_eq
thf(fact_873_in__these__eq,axiom,
! [X: list_o,A: set_option_list_o] :
( ( member_list_o @ X @ ( these_list_o @ A ) )
= ( member_option_list_o @ ( some_list_o @ X ) @ A ) ) ).
% in_these_eq
thf(fact_874_insert__dom,axiom,
! [F2: nat > option_list_o,X: nat,Y: list_o] :
( ( ( F2 @ X )
= ( some_list_o @ Y ) )
=> ( ( insert_nat @ X @ ( dom_nat_list_o @ F2 ) )
= ( dom_nat_list_o @ F2 ) ) ) ).
% insert_dom
thf(fact_875_insert__dom,axiom,
! [F2: int > option_list_o,X: int,Y: list_o] :
( ( ( F2 @ X )
= ( some_list_o @ Y ) )
=> ( ( insert_int @ X @ ( dom_int_list_o @ F2 ) )
= ( dom_int_list_o @ F2 ) ) ) ).
% insert_dom
thf(fact_876_ran__map__upd__Some,axiom,
! [M: nat > option_list_o,X: nat,Y: list_o,Z2: list_o] :
( ( ( M @ X )
= ( some_list_o @ Y ) )
=> ( ( inj_on1630180835328728801list_o @ M @ ( dom_nat_list_o @ M ) )
=> ( ~ ( member_list_o @ Z2 @ ( ran_nat_list_o @ M ) )
=> ( ( ran_nat_list_o @ ( fun_up6685928411843432661list_o @ M @ X @ ( some_list_o @ Z2 ) ) )
= ( sup_sup_set_list_o @ ( minus_8912710245716896613list_o @ ( ran_nat_list_o @ M ) @ ( insert_list_o @ Y @ bot_bot_set_list_o ) ) @ ( insert_list_o @ Z2 @ bot_bot_set_list_o ) ) ) ) ) ) ).
% ran_map_upd_Some
thf(fact_877_ran__map__upd__Some,axiom,
! [M: int > option_list_o,X: int,Y: list_o,Z2: list_o] :
( ( ( M @ X )
= ( some_list_o @ Y ) )
=> ( ( inj_on203032006962319749list_o @ M @ ( dom_int_list_o @ M ) )
=> ( ~ ( member_list_o @ Z2 @ ( ran_int_list_o @ M ) )
=> ( ( ran_int_list_o @ ( fun_up5258779583477023609list_o @ M @ X @ ( some_list_o @ Z2 ) ) )
= ( sup_sup_set_list_o @ ( minus_8912710245716896613list_o @ ( ran_int_list_o @ M ) @ ( insert_list_o @ Y @ bot_bot_set_list_o ) ) @ ( insert_list_o @ Z2 @ bot_bot_set_list_o ) ) ) ) ) ) ).
% ran_map_upd_Some
thf(fact_878_fun__upd__None__restrict,axiom,
! [X: nat,D2: set_nat,M: nat > option_list_o] :
( ( ( member_nat @ X @ D2 )
=> ( ( fun_up6685928411843432661list_o @ ( restri60617331834859036list_o @ M @ D2 ) @ X @ none_list_o )
= ( restri60617331834859036list_o @ M @ ( minus_minus_set_nat @ D2 @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) )
& ( ~ ( member_nat @ X @ D2 )
=> ( ( fun_up6685928411843432661list_o @ ( restri60617331834859036list_o @ M @ D2 ) @ X @ none_list_o )
= ( restri60617331834859036list_o @ M @ D2 ) ) ) ) ).
% fun_upd_None_restrict
thf(fact_879_fun__upd__None__restrict,axiom,
! [X: int,D2: set_int,M: int > option_list_o] :
( ( ( member_int @ X @ D2 )
=> ( ( fun_up5258779583477023609list_o @ ( restri2456407693429728960list_o @ M @ D2 ) @ X @ none_list_o )
= ( restri2456407693429728960list_o @ M @ ( minus_minus_set_int @ D2 @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) )
& ( ~ ( member_int @ X @ D2 )
=> ( ( fun_up5258779583477023609list_o @ ( restri2456407693429728960list_o @ M @ D2 ) @ X @ none_list_o )
= ( restri2456407693429728960list_o @ M @ D2 ) ) ) ) ).
% fun_upd_None_restrict
thf(fact_880_UNIV__I,axiom,
! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% UNIV_I
thf(fact_881_UNIV__I,axiom,
! [X: int] : ( member_int @ X @ top_top_set_int ) ).
% UNIV_I
thf(fact_882_Diff__iff,axiom,
! [C: nat,A: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B4 ) )
= ( ( member_nat @ C @ A )
& ~ ( member_nat @ C @ B4 ) ) ) ).
% Diff_iff
thf(fact_883_Diff__iff,axiom,
! [C: int,A: set_int,B4: set_int] :
( ( member_int @ C @ ( minus_minus_set_int @ A @ B4 ) )
= ( ( member_int @ C @ A )
& ~ ( member_int @ C @ B4 ) ) ) ).
% Diff_iff
thf(fact_884_DiffI,axiom,
! [C: nat,A: set_nat,B4: set_nat] :
( ( member_nat @ C @ A )
=> ( ~ ( member_nat @ C @ B4 )
=> ( member_nat @ C @ ( minus_minus_set_nat @ A @ B4 ) ) ) ) ).
% DiffI
thf(fact_885_DiffI,axiom,
! [C: int,A: set_int,B4: set_int] :
( ( member_int @ C @ A )
=> ( ~ ( member_int @ C @ B4 )
=> ( member_int @ C @ ( minus_minus_set_int @ A @ B4 ) ) ) ) ).
% DiffI
thf(fact_886_Compl__iff,axiom,
! [C: nat,A: set_nat] :
( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A ) )
= ( ~ ( member_nat @ C @ A ) ) ) ).
% Compl_iff
thf(fact_887_Compl__iff,axiom,
! [C: int,A: set_int] :
( ( member_int @ C @ ( uminus1532241313380277803et_int @ A ) )
= ( ~ ( member_int @ C @ A ) ) ) ).
% Compl_iff
thf(fact_888_ComplI,axiom,
! [C: nat,A: set_nat] :
( ~ ( member_nat @ C @ A )
=> ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A ) ) ) ).
% ComplI
thf(fact_889_ComplI,axiom,
! [C: int,A: set_int] :
( ~ ( member_int @ C @ A )
=> ( member_int @ C @ ( uminus1532241313380277803et_int @ A ) ) ) ).
% ComplI
thf(fact_890_insert__Diff1,axiom,
! [X: nat,B4: set_nat,A: set_nat] :
( ( member_nat @ X @ B4 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A ) @ B4 )
= ( minus_minus_set_nat @ A @ B4 ) ) ) ).
% insert_Diff1
thf(fact_891_insert__Diff1,axiom,
! [X: int,B4: set_int,A: set_int] :
( ( member_int @ X @ B4 )
=> ( ( minus_minus_set_int @ ( insert_int @ X @ A ) @ B4 )
= ( minus_minus_set_int @ A @ B4 ) ) ) ).
% insert_Diff1
thf(fact_892_Diff__insert0,axiom,
! [X: nat,A: set_nat,B4: set_nat] :
( ~ ( member_nat @ X @ A )
=> ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ B4 ) )
= ( minus_minus_set_nat @ A @ B4 ) ) ) ).
% Diff_insert0
thf(fact_893_Diff__insert0,axiom,
! [X: int,A: set_int,B4: set_int] :
( ~ ( member_int @ X @ A )
=> ( ( minus_minus_set_int @ A @ ( insert_int @ X @ B4 ) )
= ( minus_minus_set_int @ A @ B4 ) ) ) ).
% Diff_insert0
thf(fact_894_dom__fun__upd,axiom,
! [Y: option_list_o,F2: nat > option_list_o,X: nat] :
( ( ( Y = none_list_o )
=> ( ( dom_nat_list_o @ ( fun_up6685928411843432661list_o @ F2 @ X @ Y ) )
= ( minus_minus_set_nat @ ( dom_nat_list_o @ F2 ) @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) )
& ( ( Y != none_list_o )
=> ( ( dom_nat_list_o @ ( fun_up6685928411843432661list_o @ F2 @ X @ Y ) )
= ( insert_nat @ X @ ( dom_nat_list_o @ F2 ) ) ) ) ) ).
% dom_fun_upd
thf(fact_895_dom__fun__upd,axiom,
! [Y: option_list_o,F2: int > option_list_o,X: int] :
( ( ( Y = none_list_o )
=> ( ( dom_int_list_o @ ( fun_up5258779583477023609list_o @ F2 @ X @ Y ) )
= ( minus_minus_set_int @ ( dom_int_list_o @ F2 ) @ ( insert_int @ X @ bot_bot_set_int ) ) ) )
& ( ( Y != none_list_o )
=> ( ( dom_int_list_o @ ( fun_up5258779583477023609list_o @ F2 @ X @ Y ) )
= ( insert_int @ X @ ( dom_int_list_o @ F2 ) ) ) ) ) ).
% dom_fun_upd
thf(fact_896_UNIV__eq__I,axiom,
! [A: set_nat] :
( ! [X2: nat] : ( member_nat @ X2 @ A )
=> ( top_top_set_nat = A ) ) ).
% UNIV_eq_I
thf(fact_897_UNIV__eq__I,axiom,
! [A: set_int] :
( ! [X2: int] : ( member_int @ X2 @ A )
=> ( top_top_set_int = A ) ) ).
% UNIV_eq_I
thf(fact_898_UNIV__witness,axiom,
? [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ).
% UNIV_witness
thf(fact_899_UNIV__witness,axiom,
? [X2: int] : ( member_int @ X2 @ top_top_set_int ) ).
% UNIV_witness
thf(fact_900_DiffD2,axiom,
! [C: nat,A: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B4 ) )
=> ~ ( member_nat @ C @ B4 ) ) ).
% DiffD2
thf(fact_901_DiffD2,axiom,
! [C: int,A: set_int,B4: set_int] :
( ( member_int @ C @ ( minus_minus_set_int @ A @ B4 ) )
=> ~ ( member_int @ C @ B4 ) ) ).
% DiffD2
thf(fact_902_DiffD1,axiom,
! [C: nat,A: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B4 ) )
=> ( member_nat @ C @ A ) ) ).
% DiffD1
thf(fact_903_DiffD1,axiom,
! [C: int,A: set_int,B4: set_int] :
( ( member_int @ C @ ( minus_minus_set_int @ A @ B4 ) )
=> ( member_int @ C @ A ) ) ).
% DiffD1
thf(fact_904_ComplD,axiom,
! [C: nat,A: set_nat] :
( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A ) )
=> ~ ( member_nat @ C @ A ) ) ).
% ComplD
thf(fact_905_ComplD,axiom,
! [C: int,A: set_int] :
( ( member_int @ C @ ( uminus1532241313380277803et_int @ A ) )
=> ~ ( member_int @ C @ A ) ) ).
% ComplD
thf(fact_906_DiffE,axiom,
! [C: nat,A: set_nat,B4: set_nat] :
( ( member_nat @ C @ ( minus_minus_set_nat @ A @ B4 ) )
=> ~ ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B4 ) ) ) ).
% DiffE
thf(fact_907_DiffE,axiom,
! [C: int,A: set_int,B4: set_int] :
( ( member_int @ C @ ( minus_minus_set_int @ A @ B4 ) )
=> ~ ( ( member_int @ C @ A )
=> ( member_int @ C @ B4 ) ) ) ).
% DiffE
thf(fact_908_insert__Diff__if,axiom,
! [X: nat,B4: set_nat,A: set_nat] :
( ( ( member_nat @ X @ B4 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A ) @ B4 )
= ( minus_minus_set_nat @ A @ B4 ) ) )
& ( ~ ( member_nat @ X @ B4 )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A ) @ B4 )
= ( insert_nat @ X @ ( minus_minus_set_nat @ A @ B4 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_909_insert__Diff__if,axiom,
! [X: int,B4: set_int,A: set_int] :
( ( ( member_int @ X @ B4 )
=> ( ( minus_minus_set_int @ ( insert_int @ X @ A ) @ B4 )
= ( minus_minus_set_int @ A @ B4 ) ) )
& ( ~ ( member_int @ X @ B4 )
=> ( ( minus_minus_set_int @ ( insert_int @ X @ A ) @ B4 )
= ( insert_int @ X @ ( minus_minus_set_int @ A @ B4 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_910_inj__on__diff,axiom,
! [F2: nat > option_list_o,A: set_nat,B4: set_nat] :
( ( inj_on1630180835328728801list_o @ F2 @ A )
=> ( inj_on1630180835328728801list_o @ F2 @ ( minus_minus_set_nat @ A @ B4 ) ) ) ).
% inj_on_diff
thf(fact_911_inj__on__diff,axiom,
! [F2: int > option_list_o,A: set_int,B4: set_int] :
( ( inj_on203032006962319749list_o @ F2 @ A )
=> ( inj_on203032006962319749list_o @ F2 @ ( minus_minus_set_int @ A @ B4 ) ) ) ).
% inj_on_diff
thf(fact_912_Diff__insert__absorb,axiom,
! [X: nat,A: set_nat] :
( ~ ( member_nat @ X @ A )
=> ( ( minus_minus_set_nat @ ( insert_nat @ X @ A ) @ ( insert_nat @ X @ bot_bot_set_nat ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_913_Diff__insert__absorb,axiom,
! [X: int,A: set_int] :
( ~ ( member_int @ X @ A )
=> ( ( minus_minus_set_int @ ( insert_int @ X @ A ) @ ( insert_int @ X @ bot_bot_set_int ) )
= A ) ) ).
% Diff_insert_absorb
thf(fact_914_insert__Diff,axiom,
! [A2: nat,A: set_nat] :
( ( member_nat @ A2 @ A )
=> ( ( insert_nat @ A2 @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) )
= A ) ) ).
% insert_Diff
thf(fact_915_insert__Diff,axiom,
! [A2: int,A: set_int] :
( ( member_int @ A2 @ A )
=> ( ( insert_int @ A2 @ ( minus_minus_set_int @ A @ ( insert_int @ A2 @ bot_bot_set_int ) ) )
= A ) ) ).
% insert_Diff
thf(fact_916_is__singletonI_H,axiom,
! [A: set_nat] :
( ( A != bot_bot_set_nat )
=> ( ! [X2: nat,Y2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_nat @ Y2 @ A )
=> ( X2 = Y2 ) ) )
=> ( is_singleton_nat @ A ) ) ) ).
% is_singletonI'
thf(fact_917_is__singletonI_H,axiom,
! [A: set_int] :
( ( A != bot_bot_set_int )
=> ( ! [X2: int,Y2: int] :
( ( member_int @ X2 @ A )
=> ( ( member_int @ Y2 @ A )
=> ( X2 = Y2 ) ) )
=> ( is_singleton_int @ A ) ) ) ).
% is_singletonI'
thf(fact_918_dom__minus,axiom,
! [F2: nat > option_list_o,X: nat,A: set_nat] :
( ( ( F2 @ X )
= none_list_o )
=> ( ( minus_minus_set_nat @ ( dom_nat_list_o @ F2 ) @ ( insert_nat @ X @ A ) )
= ( minus_minus_set_nat @ ( dom_nat_list_o @ F2 ) @ A ) ) ) ).
% dom_minus
thf(fact_919_dom__minus,axiom,
! [F2: int > option_list_o,X: int,A: set_int] :
( ( ( F2 @ X )
= none_list_o )
=> ( ( minus_minus_set_int @ ( dom_int_list_o @ F2 ) @ ( insert_int @ X @ A ) )
= ( minus_minus_set_int @ ( dom_int_list_o @ F2 ) @ A ) ) ) ).
% dom_minus
thf(fact_920_member__remove,axiom,
! [X: nat,Y: nat,A: set_nat] :
( ( member_nat @ X @ ( remove_nat @ Y @ A ) )
= ( ( member_nat @ X @ A )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_921_member__remove,axiom,
! [X: int,Y: int,A: set_int] :
( ( member_int @ X @ ( remove_int @ Y @ A ) )
= ( ( member_int @ X @ A )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_922_graph__map__upd,axiom,
! [M: option_list_o > option_option_list_o,K: option_list_o,V: option_list_o] :
( ( graph_1138927715465675390list_o @ ( fun_up4915027116007172163list_o @ M @ K @ ( some_option_list_o @ V ) ) )
= ( insert7989621605229100023list_o @ ( produc5745850523778858007list_o @ K @ V ) @ ( graph_1138927715465675390list_o @ ( fun_up4915027116007172163list_o @ M @ K @ none_option_list_o ) ) ) ) ).
% graph_map_upd
thf(fact_923_graph__map__upd,axiom,
! [M: nat > option_nat,K: nat,V: nat] :
( ( graph_nat_nat @ ( fun_up1493157387958331631on_nat @ M @ K @ ( some_nat @ V ) ) )
= ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ K @ V ) @ ( graph_nat_nat @ ( fun_up1493157387958331631on_nat @ M @ K @ none_nat ) ) ) ) ).
% graph_map_upd
thf(fact_924_inj__on__add,axiom,
! [A2: nat,A: set_nat] : ( inj_on_nat_nat @ ( plus_plus_nat @ A2 ) @ A ) ).
% inj_on_add
thf(fact_925_in__graphD,axiom,
! [K: option_list_o,V: option_list_o,M: option_list_o > option_option_list_o] :
( ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ K @ V ) @ ( graph_1138927715465675390list_o @ M ) )
=> ( ( M @ K )
= ( some_option_list_o @ V ) ) ) ).
% in_graphD
thf(fact_926_in__graphD,axiom,
! [K: nat,V: nat,M: nat > option_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ K @ V ) @ ( graph_nat_nat @ M ) )
=> ( ( M @ K )
= ( some_nat @ V ) ) ) ).
% in_graphD
thf(fact_927_in__graphI,axiom,
! [M: option_list_o > option_option_list_o,K: option_list_o,V: option_list_o] :
( ( ( M @ K )
= ( some_option_list_o @ V ) )
=> ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ K @ V ) @ ( graph_1138927715465675390list_o @ M ) ) ) ).
% in_graphI
thf(fact_928_in__graphI,axiom,
! [M: nat > option_nat,K: nat,V: nat] :
( ( ( M @ K )
= ( some_nat @ V ) )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ K @ V ) @ ( graph_nat_nat @ M ) ) ) ).
% in_graphI
thf(fact_929_opt__prefix_Ocases,axiom,
! [X: produc4882884732533091879list_o] :
( ! [X2: list_o,Y2: list_o] :
( X
!= ( produc5745850523778858007list_o @ ( some_list_o @ X2 ) @ ( some_list_o @ Y2 ) ) )
=> ( ! [Uv2: option_list_o] :
( X
!= ( produc5745850523778858007list_o @ none_list_o @ Uv2 ) )
=> ~ ! [Uu2: option_list_o] :
( X
!= ( produc5745850523778858007list_o @ Uu2 @ none_list_o ) ) ) ) ).
% opt_prefix.cases
thf(fact_930_graph__restrictD_I1_J,axiom,
! [K: option_list_o,V: option_list_o,M: option_list_o > option_option_list_o,A: set_option_list_o] :
( ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ K @ V ) @ ( graph_1138927715465675390list_o @ ( restri411196559532073264list_o @ M @ A ) ) )
=> ( member_option_list_o @ K @ A ) ) ).
% graph_restrictD(1)
thf(fact_931_graph__restrictD_I1_J,axiom,
! [K: nat,V: nat,M: nat > option_nat,A: set_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ K @ V ) @ ( graph_nat_nat @ ( restrict_map_nat_nat @ M @ A ) ) )
=> ( member_nat @ K @ A ) ) ).
% graph_restrictD(1)
thf(fact_932_opt__append_Ocases,axiom,
! [X: produc4882884732533091879list_o] :
( ! [X2: list_o,Y2: list_o] :
( X
!= ( produc5745850523778858007list_o @ ( some_list_o @ X2 ) @ ( some_list_o @ Y2 ) ) )
=> ( ! [Uv2: option_list_o] :
( X
!= ( produc5745850523778858007list_o @ none_list_o @ Uv2 ) )
=> ~ ! [Uu2: option_list_o] :
( X
!= ( produc5745850523778858007list_o @ Uu2 @ none_list_o ) ) ) ) ).
% opt_append.cases
thf(fact_933_graph__restrictD_I2_J,axiom,
! [K: option_list_o,V: option_list_o,M: option_list_o > option_option_list_o,A: set_option_list_o] :
( ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ K @ V ) @ ( graph_1138927715465675390list_o @ ( restri411196559532073264list_o @ M @ A ) ) )
=> ( ( M @ K )
= ( some_option_list_o @ V ) ) ) ).
% graph_restrictD(2)
thf(fact_934_graph__restrictD_I2_J,axiom,
! [K: nat,V: nat,M: nat > option_nat,A: set_nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ K @ V ) @ ( graph_nat_nat @ ( restrict_map_nat_nat @ M @ A ) ) )
=> ( ( M @ K )
= ( some_nat @ V ) ) ) ).
% graph_restrictD(2)
thf(fact_935_Field__insert,axiom,
! [A2: option_list_o,B2: option_list_o,R: set_Pr7497825696620840711list_o] :
( ( field_option_list_o @ ( insert7989621605229100023list_o @ ( produc5745850523778858007list_o @ A2 @ B2 ) @ R ) )
= ( sup_su6641994874487927192list_o @ ( insert_option_list_o @ A2 @ ( insert_option_list_o @ B2 @ bot_bo3275880064859030064list_o ) ) @ ( field_option_list_o @ R ) ) ) ).
% Field_insert
thf(fact_936_Field__insert,axiom,
! [A2: nat,B2: nat,R: set_Pr1261947904930325089at_nat] :
( ( field_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ R ) )
= ( sup_sup_set_nat @ ( insert_nat @ A2 @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) @ ( field_nat @ R ) ) ) ).
% Field_insert
thf(fact_937_inj__on__Un,axiom,
! [F2: nat > option_list_o,A: set_nat,B4: set_nat] :
( ( inj_on1630180835328728801list_o @ F2 @ ( sup_sup_set_nat @ A @ B4 ) )
= ( ( inj_on1630180835328728801list_o @ F2 @ A )
& ( inj_on1630180835328728801list_o @ F2 @ B4 )
& ( ( inf_in6922378751903173298list_o @ ( image_4575287668734308173list_o @ F2 @ ( minus_minus_set_nat @ A @ B4 ) ) @ ( image_4575287668734308173list_o @ F2 @ ( minus_minus_set_nat @ B4 @ A ) ) )
= bot_bo3275880064859030064list_o ) ) ) ).
% inj_on_Un
thf(fact_938_inj__on__Un,axiom,
! [F2: int > option_list_o,A: set_int,B4: set_int] :
( ( inj_on203032006962319749list_o @ F2 @ ( sup_sup_set_int @ A @ B4 ) )
= ( ( inj_on203032006962319749list_o @ F2 @ A )
& ( inj_on203032006962319749list_o @ F2 @ B4 )
& ( ( inf_in6922378751903173298list_o @ ( image_3148138840367899121list_o @ F2 @ ( minus_minus_set_int @ A @ B4 ) ) @ ( image_3148138840367899121list_o @ F2 @ ( minus_minus_set_int @ B4 @ A ) ) )
= bot_bo3275880064859030064list_o ) ) ) ).
% inj_on_Un
thf(fact_939_inj__on__insert,axiom,
! [F2: nat > option_list_o,A2: nat,A: set_nat] :
( ( inj_on1630180835328728801list_o @ F2 @ ( insert_nat @ A2 @ A ) )
= ( ( inj_on1630180835328728801list_o @ F2 @ A )
& ~ ( member_option_list_o @ ( F2 @ A2 ) @ ( image_4575287668734308173list_o @ F2 @ ( minus_minus_set_nat @ A @ ( insert_nat @ A2 @ bot_bot_set_nat ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_940_inj__on__insert,axiom,
! [F2: int > option_list_o,A2: int,A: set_int] :
( ( inj_on203032006962319749list_o @ F2 @ ( insert_int @ A2 @ A ) )
= ( ( inj_on203032006962319749list_o @ F2 @ A )
& ~ ( member_option_list_o @ ( F2 @ A2 ) @ ( image_3148138840367899121list_o @ F2 @ ( minus_minus_set_int @ A @ ( insert_int @ A2 @ bot_bot_set_int ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_941_UNIV__option__conv,axiom,
( top_to633166595683317524list_o
= ( insert_option_list_o @ none_list_o @ ( image_3961372166222858059list_o @ some_list_o @ top_top_set_list_o ) ) ) ).
% UNIV_option_conv
thf(fact_942_image__eqI,axiom,
! [B2: nat,F2: nat > nat,X: nat,A: set_nat] :
( ( B2
= ( F2 @ X ) )
=> ( ( member_nat @ X @ A )
=> ( member_nat @ B2 @ ( image_nat_nat @ F2 @ A ) ) ) ) ).
% image_eqI
thf(fact_943_image__eqI,axiom,
! [B2: int,F2: nat > int,X: nat,A: set_nat] :
( ( B2
= ( F2 @ X ) )
=> ( ( member_nat @ X @ A )
=> ( member_int @ B2 @ ( image_nat_int @ F2 @ A ) ) ) ) ).
% image_eqI
thf(fact_944_image__eqI,axiom,
! [B2: nat,F2: int > nat,X: int,A: set_int] :
( ( B2
= ( F2 @ X ) )
=> ( ( member_int @ X @ A )
=> ( member_nat @ B2 @ ( image_int_nat @ F2 @ A ) ) ) ) ).
% image_eqI
thf(fact_945_image__eqI,axiom,
! [B2: int,F2: int > int,X: int,A: set_int] :
( ( B2
= ( F2 @ X ) )
=> ( ( member_int @ X @ A )
=> ( member_int @ B2 @ ( image_int_int @ F2 @ A ) ) ) ) ).
% image_eqI
thf(fact_946_image__map__upd,axiom,
! [X: nat,A: set_nat,M: nat > option_list_o,Y: list_o] :
( ~ ( member_nat @ X @ A )
=> ( ( image_4575287668734308173list_o @ ( fun_up6685928411843432661list_o @ M @ X @ ( some_list_o @ Y ) ) @ A )
= ( image_4575287668734308173list_o @ M @ A ) ) ) ).
% image_map_upd
thf(fact_947_image__map__upd,axiom,
! [X: int,A: set_int,M: int > option_list_o,Y: list_o] :
( ~ ( member_int @ X @ A )
=> ( ( image_3148138840367899121list_o @ ( fun_up5258779583477023609list_o @ M @ X @ ( some_list_o @ Y ) ) @ A )
= ( image_3148138840367899121list_o @ M @ A ) ) ) ).
% image_map_upd
thf(fact_948_these__image__Some__eq,axiom,
! [A: set_list_o] :
( ( these_list_o @ ( image_3961372166222858059list_o @ some_list_o @ A ) )
= A ) ).
% these_image_Some_eq
thf(fact_949_imageI,axiom,
! [X: nat,A: set_nat,F2: nat > nat] :
( ( member_nat @ X @ A )
=> ( member_nat @ ( F2 @ X ) @ ( image_nat_nat @ F2 @ A ) ) ) ).
% imageI
thf(fact_950_imageI,axiom,
! [X: nat,A: set_nat,F2: nat > int] :
( ( member_nat @ X @ A )
=> ( member_int @ ( F2 @ X ) @ ( image_nat_int @ F2 @ A ) ) ) ).
% imageI
thf(fact_951_imageI,axiom,
! [X: int,A: set_int,F2: int > nat] :
( ( member_int @ X @ A )
=> ( member_nat @ ( F2 @ X ) @ ( image_int_nat @ F2 @ A ) ) ) ).
% imageI
thf(fact_952_imageI,axiom,
! [X: int,A: set_int,F2: int > int] :
( ( member_int @ X @ A )
=> ( member_int @ ( F2 @ X ) @ ( image_int_int @ F2 @ A ) ) ) ).
% imageI
thf(fact_953_rev__image__eqI,axiom,
! [X: nat,A: set_nat,B2: nat,F2: nat > nat] :
( ( member_nat @ X @ A )
=> ( ( B2
= ( F2 @ X ) )
=> ( member_nat @ B2 @ ( image_nat_nat @ F2 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_954_rev__image__eqI,axiom,
! [X: nat,A: set_nat,B2: int,F2: nat > int] :
( ( member_nat @ X @ A )
=> ( ( B2
= ( F2 @ X ) )
=> ( member_int @ B2 @ ( image_nat_int @ F2 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_955_rev__image__eqI,axiom,
! [X: int,A: set_int,B2: nat,F2: int > nat] :
( ( member_int @ X @ A )
=> ( ( B2
= ( F2 @ X ) )
=> ( member_nat @ B2 @ ( image_int_nat @ F2 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_956_rev__image__eqI,axiom,
! [X: int,A: set_int,B2: int,F2: int > int] :
( ( member_int @ X @ A )
=> ( ( B2
= ( F2 @ X ) )
=> ( member_int @ B2 @ ( image_int_int @ F2 @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_957_inj__on__image__iff,axiom,
! [A: set_nat,G: nat > option_list_o,F2: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A )
=> ( ( ( G @ ( F2 @ X2 ) )
= ( G @ ( F2 @ Xa2 ) ) )
= ( ( G @ X2 )
= ( G @ Xa2 ) ) ) ) )
=> ( ( inj_on_nat_nat @ F2 @ A )
=> ( ( inj_on1630180835328728801list_o @ G @ ( image_nat_nat @ F2 @ A ) )
= ( inj_on1630180835328728801list_o @ G @ A ) ) ) ) ).
% inj_on_image_iff
thf(fact_958_inj__on__image__iff,axiom,
! [A: set_int,G: int > option_list_o,F2: int > int] :
( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ! [Xa2: int] :
( ( member_int @ Xa2 @ A )
=> ( ( ( G @ ( F2 @ X2 ) )
= ( G @ ( F2 @ Xa2 ) ) )
= ( ( G @ X2 )
= ( G @ Xa2 ) ) ) ) )
=> ( ( inj_on_int_int @ F2 @ A )
=> ( ( inj_on203032006962319749list_o @ G @ ( image_int_int @ F2 @ A ) )
= ( inj_on203032006962319749list_o @ G @ A ) ) ) ) ).
% inj_on_image_iff
thf(fact_959_range__ex1__eq,axiom,
! [F2: nat > option_list_o,B2: option_list_o] :
( ( inj_on1630180835328728801list_o @ F2 @ top_top_set_nat )
=> ( ( member_option_list_o @ B2 @ ( image_4575287668734308173list_o @ F2 @ top_top_set_nat ) )
= ( ? [X3: nat] :
( ( B2
= ( F2 @ X3 ) )
& ! [Y3: nat] :
( ( B2
= ( F2 @ Y3 ) )
=> ( Y3 = X3 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_960_range__ex1__eq,axiom,
! [F2: int > option_list_o,B2: option_list_o] :
( ( inj_on203032006962319749list_o @ F2 @ top_top_set_int )
=> ( ( member_option_list_o @ B2 @ ( image_3148138840367899121list_o @ F2 @ top_top_set_int ) )
= ( ? [X3: int] :
( ( B2
= ( F2 @ X3 ) )
& ! [Y3: int] :
( ( B2
= ( F2 @ Y3 ) )
=> ( Y3 = X3 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_961_inj__image__eq__iff,axiom,
! [F2: nat > option_list_o,A: set_nat,B4: set_nat] :
( ( inj_on1630180835328728801list_o @ F2 @ top_top_set_nat )
=> ( ( ( image_4575287668734308173list_o @ F2 @ A )
= ( image_4575287668734308173list_o @ F2 @ B4 ) )
= ( A = B4 ) ) ) ).
% inj_image_eq_iff
thf(fact_962_inj__image__eq__iff,axiom,
! [F2: int > option_list_o,A: set_int,B4: set_int] :
( ( inj_on203032006962319749list_o @ F2 @ top_top_set_int )
=> ( ( ( image_3148138840367899121list_o @ F2 @ A )
= ( image_3148138840367899121list_o @ F2 @ B4 ) )
= ( A = B4 ) ) ) ).
% inj_image_eq_iff
thf(fact_963_inj__image__mem__iff,axiom,
! [F2: nat > option_list_o,A2: nat,A: set_nat] :
( ( inj_on1630180835328728801list_o @ F2 @ top_top_set_nat )
=> ( ( member_option_list_o @ ( F2 @ A2 ) @ ( image_4575287668734308173list_o @ F2 @ A ) )
= ( member_nat @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_964_inj__image__mem__iff,axiom,
! [F2: int > option_list_o,A2: int,A: set_int] :
( ( inj_on203032006962319749list_o @ F2 @ top_top_set_int )
=> ( ( member_option_list_o @ ( F2 @ A2 ) @ ( image_3148138840367899121list_o @ F2 @ A ) )
= ( member_int @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_965_inj__image__mem__iff,axiom,
! [F2: nat > nat,A2: nat,A: set_nat] :
( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
=> ( ( member_nat @ ( F2 @ A2 ) @ ( image_nat_nat @ F2 @ A ) )
= ( member_nat @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_966_inj__image__mem__iff,axiom,
! [F2: int > nat,A2: int,A: set_int] :
( ( inj_on_int_nat @ F2 @ top_top_set_int )
=> ( ( member_nat @ ( F2 @ A2 ) @ ( image_int_nat @ F2 @ A ) )
= ( member_int @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_967_inj__image__mem__iff,axiom,
! [F2: nat > int,A2: nat,A: set_nat] :
( ( inj_on_nat_int @ F2 @ top_top_set_nat )
=> ( ( member_int @ ( F2 @ A2 ) @ ( image_nat_int @ F2 @ A ) )
= ( member_nat @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_968_inj__image__mem__iff,axiom,
! [F2: int > int,A2: int,A: set_int] :
( ( inj_on_int_int @ F2 @ top_top_set_int )
=> ( ( member_int @ ( F2 @ A2 ) @ ( image_int_int @ F2 @ A ) )
= ( member_int @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_969_inj__img__insertE,axiom,
! [F2: nat > option_list_o,A: set_nat,X: option_list_o,B4: set_option_list_o] :
( ( inj_on1630180835328728801list_o @ F2 @ A )
=> ( ~ ( member_option_list_o @ X @ B4 )
=> ( ( ( insert_option_list_o @ X @ B4 )
= ( image_4575287668734308173list_o @ F2 @ A ) )
=> ~ ! [X4: nat,A6: set_nat] :
( ~ ( member_nat @ X4 @ A6 )
=> ( ( A
= ( insert_nat @ X4 @ A6 ) )
=> ( ( X
= ( F2 @ X4 ) )
=> ( B4
!= ( image_4575287668734308173list_o @ F2 @ A6 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_970_inj__img__insertE,axiom,
! [F2: int > option_list_o,A: set_int,X: option_list_o,B4: set_option_list_o] :
( ( inj_on203032006962319749list_o @ F2 @ A )
=> ( ~ ( member_option_list_o @ X @ B4 )
=> ( ( ( insert_option_list_o @ X @ B4 )
= ( image_3148138840367899121list_o @ F2 @ A ) )
=> ~ ! [X4: int,A6: set_int] :
( ~ ( member_int @ X4 @ A6 )
=> ( ( A
= ( insert_int @ X4 @ A6 ) )
=> ( ( X
= ( F2 @ X4 ) )
=> ( B4
!= ( image_3148138840367899121list_o @ F2 @ A6 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_971_inj__img__insertE,axiom,
! [F2: nat > nat,A: set_nat,X: nat,B4: set_nat] :
( ( inj_on_nat_nat @ F2 @ A )
=> ( ~ ( member_nat @ X @ B4 )
=> ( ( ( insert_nat @ X @ B4 )
= ( image_nat_nat @ F2 @ A ) )
=> ~ ! [X4: nat,A6: set_nat] :
( ~ ( member_nat @ X4 @ A6 )
=> ( ( A
= ( insert_nat @ X4 @ A6 ) )
=> ( ( X
= ( F2 @ X4 ) )
=> ( B4
!= ( image_nat_nat @ F2 @ A6 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_972_inj__img__insertE,axiom,
! [F2: int > nat,A: set_int,X: nat,B4: set_nat] :
( ( inj_on_int_nat @ F2 @ A )
=> ( ~ ( member_nat @ X @ B4 )
=> ( ( ( insert_nat @ X @ B4 )
= ( image_int_nat @ F2 @ A ) )
=> ~ ! [X4: int,A6: set_int] :
( ~ ( member_int @ X4 @ A6 )
=> ( ( A
= ( insert_int @ X4 @ A6 ) )
=> ( ( X
= ( F2 @ X4 ) )
=> ( B4
!= ( image_int_nat @ F2 @ A6 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_973_inj__img__insertE,axiom,
! [F2: nat > int,A: set_nat,X: int,B4: set_int] :
( ( inj_on_nat_int @ F2 @ A )
=> ( ~ ( member_int @ X @ B4 )
=> ( ( ( insert_int @ X @ B4 )
= ( image_nat_int @ F2 @ A ) )
=> ~ ! [X4: nat,A6: set_nat] :
( ~ ( member_nat @ X4 @ A6 )
=> ( ( A
= ( insert_nat @ X4 @ A6 ) )
=> ( ( X
= ( F2 @ X4 ) )
=> ( B4
!= ( image_nat_int @ F2 @ A6 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_974_inj__img__insertE,axiom,
! [F2: int > int,A: set_int,X: int,B4: set_int] :
( ( inj_on_int_int @ F2 @ A )
=> ( ~ ( member_int @ X @ B4 )
=> ( ( ( insert_int @ X @ B4 )
= ( image_int_int @ F2 @ A ) )
=> ~ ! [X4: int,A6: set_int] :
( ~ ( member_int @ X4 @ A6 )
=> ( ( A
= ( insert_int @ X4 @ A6 ) )
=> ( ( X
= ( F2 @ X4 ) )
=> ( B4
!= ( image_int_int @ F2 @ A6 ) ) ) ) ) ) ) ) ).
% inj_img_insertE
thf(fact_975_inj__on__Un__image__eq__iff,axiom,
! [F2: nat > option_list_o,A: set_nat,B4: set_nat] :
( ( inj_on1630180835328728801list_o @ F2 @ ( sup_sup_set_nat @ A @ B4 ) )
=> ( ( ( image_4575287668734308173list_o @ F2 @ A )
= ( image_4575287668734308173list_o @ F2 @ B4 ) )
= ( A = B4 ) ) ) ).
% inj_on_Un_image_eq_iff
thf(fact_976_inj__on__Un__image__eq__iff,axiom,
! [F2: int > option_list_o,A: set_int,B4: set_int] :
( ( inj_on203032006962319749list_o @ F2 @ ( sup_sup_set_int @ A @ B4 ) )
=> ( ( ( image_3148138840367899121list_o @ F2 @ A )
= ( image_3148138840367899121list_o @ F2 @ B4 ) )
= ( A = B4 ) ) ) ).
% inj_on_Un_image_eq_iff
thf(fact_977_None__notin__image__Some,axiom,
! [A: set_list_o] :
~ ( member_option_list_o @ none_list_o @ ( image_3961372166222858059list_o @ some_list_o @ A ) ) ).
% None_notin_image_Some
thf(fact_978_inj__on__fun__updI,axiom,
! [F2: nat > option_list_o,A: set_nat,Y: option_list_o,X: nat] :
( ( inj_on1630180835328728801list_o @ F2 @ A )
=> ( ~ ( member_option_list_o @ Y @ ( image_4575287668734308173list_o @ F2 @ A ) )
=> ( inj_on1630180835328728801list_o @ ( fun_up6685928411843432661list_o @ F2 @ X @ Y ) @ A ) ) ) ).
% inj_on_fun_updI
thf(fact_979_inj__on__fun__updI,axiom,
! [F2: int > option_list_o,A: set_int,Y: option_list_o,X: int] :
( ( inj_on203032006962319749list_o @ F2 @ A )
=> ( ~ ( member_option_list_o @ Y @ ( image_3148138840367899121list_o @ F2 @ A ) )
=> ( inj_on203032006962319749list_o @ ( fun_up5258779583477023609list_o @ F2 @ X @ Y ) @ A ) ) ) ).
% inj_on_fun_updI
thf(fact_980_image__Int,axiom,
! [F2: nat > option_list_o,A: set_nat,B4: set_nat] :
( ( inj_on1630180835328728801list_o @ F2 @ top_top_set_nat )
=> ( ( image_4575287668734308173list_o @ F2 @ ( inf_inf_set_nat @ A @ B4 ) )
= ( inf_in6922378751903173298list_o @ ( image_4575287668734308173list_o @ F2 @ A ) @ ( image_4575287668734308173list_o @ F2 @ B4 ) ) ) ) ).
% image_Int
thf(fact_981_image__Int,axiom,
! [F2: int > option_list_o,A: set_int,B4: set_int] :
( ( inj_on203032006962319749list_o @ F2 @ top_top_set_int )
=> ( ( image_3148138840367899121list_o @ F2 @ ( inf_inf_set_int @ A @ B4 ) )
= ( inf_in6922378751903173298list_o @ ( image_3148138840367899121list_o @ F2 @ A ) @ ( image_3148138840367899121list_o @ F2 @ B4 ) ) ) ) ).
% image_Int
thf(fact_982_image__set__diff,axiom,
! [F2: nat > option_list_o,A: set_nat,B4: set_nat] :
( ( inj_on1630180835328728801list_o @ F2 @ top_top_set_nat )
=> ( ( image_4575287668734308173list_o @ F2 @ ( minus_minus_set_nat @ A @ B4 ) )
= ( minus_972179126788192427list_o @ ( image_4575287668734308173list_o @ F2 @ A ) @ ( image_4575287668734308173list_o @ F2 @ B4 ) ) ) ) ).
% image_set_diff
thf(fact_983_image__set__diff,axiom,
! [F2: int > option_list_o,A: set_int,B4: set_int] :
( ( inj_on203032006962319749list_o @ F2 @ top_top_set_int )
=> ( ( image_3148138840367899121list_o @ F2 @ ( minus_minus_set_int @ A @ B4 ) )
= ( minus_972179126788192427list_o @ ( image_3148138840367899121list_o @ F2 @ A ) @ ( image_3148138840367899121list_o @ F2 @ B4 ) ) ) ) ).
% image_set_diff
thf(fact_984_in__image__insert__iff,axiom,
! [B4: set_set_nat,X: nat,A: set_nat] :
( ! [C4: set_nat] :
( ( member_set_nat @ C4 @ B4 )
=> ~ ( member_nat @ X @ C4 ) )
=> ( ( member_set_nat @ A @ ( image_7916887816326733075et_nat @ ( insert_nat @ X ) @ B4 ) )
= ( ( member_nat @ X @ A )
& ( member_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B4 ) ) ) ) ).
% in_image_insert_iff
thf(fact_985_in__image__insert__iff,axiom,
! [B4: set_set_int,X: int,A: set_int] :
( ! [C4: set_int] :
( ( member_set_int @ C4 @ B4 )
=> ~ ( member_int @ X @ C4 ) )
=> ( ( member_set_int @ A @ ( image_524474410958335435et_int @ ( insert_int @ X ) @ B4 ) )
= ( ( member_int @ X @ A )
& ( member_set_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) @ B4 ) ) ) ) ).
% in_image_insert_iff
thf(fact_986_notin__range__Some,axiom,
! [X: option_list_o] :
( ( ~ ( member_option_list_o @ X @ ( image_3961372166222858059list_o @ some_list_o @ top_top_set_list_o ) ) )
= ( X = none_list_o ) ) ).
% notin_range_Some
thf(fact_987_opt__append_Opelims,axiom,
! [X: option_list_o,Xa: option_list_o,Y: option_list_o] :
( ( ( prefix5314359684614007693append @ X @ Xa )
= Y )
=> ( ( accp_P4987914878654715440list_o @ prefix6152564574919515564nd_rel @ ( produc5745850523778858007list_o @ X @ Xa ) )
=> ( ! [X2: list_o] :
( ( X
= ( some_list_o @ X2 ) )
=> ! [Y2: list_o] :
( ( Xa
= ( some_list_o @ Y2 ) )
=> ( ( Y
= ( some_list_o @ ( append_o @ X2 @ Y2 ) ) )
=> ~ ( accp_P4987914878654715440list_o @ prefix6152564574919515564nd_rel @ ( produc5745850523778858007list_o @ ( some_list_o @ X2 ) @ ( some_list_o @ Y2 ) ) ) ) ) )
=> ( ( ( X = none_list_o )
=> ( ( Y = none_list_o )
=> ~ ( accp_P4987914878654715440list_o @ prefix6152564574919515564nd_rel @ ( produc5745850523778858007list_o @ none_list_o @ Xa ) ) ) )
=> ~ ( ( Xa = none_list_o )
=> ( ( Y = none_list_o )
=> ~ ( accp_P4987914878654715440list_o @ prefix6152564574919515564nd_rel @ ( produc5745850523778858007list_o @ X @ none_list_o ) ) ) ) ) ) ) ) ).
% opt_append.pelims
thf(fact_988_refl__on__singleton,axiom,
! [X: option_list_o] : ( refl_o7334115775069118051list_o @ ( insert_option_list_o @ X @ bot_bo3275880064859030064list_o ) @ ( insert7989621605229100023list_o @ ( produc5745850523778858007list_o @ X @ X ) @ bot_bo6949037146550090099list_o ) ) ).
% refl_on_singleton
thf(fact_989_refl__on__singleton,axiom,
! [X: nat] : ( refl_on_nat @ ( insert_nat @ X @ bot_bot_set_nat ) @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X @ X ) @ bot_bo2099793752762293965at_nat ) ) ).
% refl_on_singleton
thf(fact_990_inj__image__Compl__subset,axiom,
! [F2: nat > option_list_o,A: set_nat] :
( ( inj_on1630180835328728801list_o @ F2 @ top_top_set_nat )
=> ( ord_le1162937763994921316list_o @ ( image_4575287668734308173list_o @ F2 @ ( uminus5710092332889474511et_nat @ A ) ) @ ( uminus2228965239982383419list_o @ ( image_4575287668734308173list_o @ F2 @ A ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_991_inj__image__Compl__subset,axiom,
! [F2: int > option_list_o,A: set_int] :
( ( inj_on203032006962319749list_o @ F2 @ top_top_set_int )
=> ( ord_le1162937763994921316list_o @ ( image_3148138840367899121list_o @ F2 @ ( uminus1532241313380277803et_int @ A ) ) @ ( uminus2228965239982383419list_o @ ( image_3148138840367899121list_o @ F2 @ A ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_992_subsetI,axiom,
! [A: set_nat,B4: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat @ X2 @ B4 ) )
=> ( ord_less_eq_set_nat @ A @ B4 ) ) ).
% subsetI
thf(fact_993_subsetI,axiom,
! [A: set_int,B4: set_int] :
( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ( member_int @ X2 @ B4 ) )
=> ( ord_less_eq_set_int @ A @ B4 ) ) ).
% subsetI
thf(fact_994_insert__subset,axiom,
! [X: nat,A: set_nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ ( insert_nat @ X @ A ) @ B4 )
= ( ( member_nat @ X @ B4 )
& ( ord_less_eq_set_nat @ A @ B4 ) ) ) ).
% insert_subset
thf(fact_995_insert__subset,axiom,
! [X: int,A: set_int,B4: set_int] :
( ( ord_less_eq_set_int @ ( insert_int @ X @ A ) @ B4 )
= ( ( member_int @ X @ B4 )
& ( ord_less_eq_set_int @ A @ B4 ) ) ) ).
% insert_subset
thf(fact_996_finite__graph__iff__finite__dom,axiom,
! [M: nat > option_list_o] :
( ( finite1686248346847189978list_o @ ( graph_nat_list_o @ M ) )
= ( finite_finite_nat @ ( dom_nat_list_o @ M ) ) ) ).
% finite_graph_iff_finite_dom
thf(fact_997_finite__graph__iff__finite__dom,axiom,
! [M: int > option_list_o] :
( ( finite4082038708442059902list_o @ ( graph_int_list_o @ M ) )
= ( finite_finite_int @ ( dom_int_list_o @ M ) ) ) ).
% finite_graph_iff_finite_dom
thf(fact_998_subset__Compl__singleton,axiom,
! [A: set_nat,B2: nat] :
( ( ord_less_eq_set_nat @ A @ ( uminus5710092332889474511et_nat @ ( insert_nat @ B2 @ bot_bot_set_nat ) ) )
= ( ~ ( member_nat @ B2 @ A ) ) ) ).
% subset_Compl_singleton
thf(fact_999_subset__Compl__singleton,axiom,
! [A: set_int,B2: int] :
( ( ord_less_eq_set_int @ A @ ( uminus1532241313380277803et_int @ ( insert_int @ B2 @ bot_bot_set_int ) ) )
= ( ~ ( member_int @ B2 @ A ) ) ) ).
% subset_Compl_singleton
thf(fact_1000_image__subsetI,axiom,
! [A: set_nat,F2: nat > nat,B4: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat @ ( F2 @ X2 ) @ B4 ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F2 @ A ) @ B4 ) ) ).
% image_subsetI
thf(fact_1001_image__subsetI,axiom,
! [A: set_nat,F2: nat > int,B4: set_int] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_int @ ( F2 @ X2 ) @ B4 ) )
=> ( ord_less_eq_set_int @ ( image_nat_int @ F2 @ A ) @ B4 ) ) ).
% image_subsetI
thf(fact_1002_image__subsetI,axiom,
! [A: set_int,F2: int > nat,B4: set_nat] :
( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ( member_nat @ ( F2 @ X2 ) @ B4 ) )
=> ( ord_less_eq_set_nat @ ( image_int_nat @ F2 @ A ) @ B4 ) ) ).
% image_subsetI
thf(fact_1003_image__subsetI,axiom,
! [A: set_int,F2: int > int,B4: set_int] :
( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ( member_int @ ( F2 @ X2 ) @ B4 ) )
=> ( ord_less_eq_set_int @ ( image_int_int @ F2 @ A ) @ B4 ) ) ).
% image_subsetI
thf(fact_1004_Int__Collect__mono,axiom,
! [A: set_nat,B4: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B4 @ ( collect_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1005_Int__Collect__mono,axiom,
! [A: set_int,B4: set_int,P: int > $o,Q: int > $o] :
( ( ord_less_eq_set_int @ A @ B4 )
=> ( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_less_eq_set_int @ ( inf_inf_set_int @ A @ ( collect_int @ P ) ) @ ( inf_inf_set_int @ B4 @ ( collect_int @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_1006_subset__inj__on,axiom,
! [F2: nat > option_list_o,B4: set_nat,A: set_nat] :
( ( inj_on1630180835328728801list_o @ F2 @ B4 )
=> ( ( ord_less_eq_set_nat @ A @ B4 )
=> ( inj_on1630180835328728801list_o @ F2 @ A ) ) ) ).
% subset_inj_on
thf(fact_1007_subset__inj__on,axiom,
! [F2: int > option_list_o,B4: set_int,A: set_int] :
( ( inj_on203032006962319749list_o @ F2 @ B4 )
=> ( ( ord_less_eq_set_int @ A @ B4 )
=> ( inj_on203032006962319749list_o @ F2 @ A ) ) ) ).
% subset_inj_on
thf(fact_1008_inj__on__subset,axiom,
! [F2: nat > option_list_o,A: set_nat,B4: set_nat] :
( ( inj_on1630180835328728801list_o @ F2 @ A )
=> ( ( ord_less_eq_set_nat @ B4 @ A )
=> ( inj_on1630180835328728801list_o @ F2 @ B4 ) ) ) ).
% inj_on_subset
thf(fact_1009_inj__on__subset,axiom,
! [F2: int > option_list_o,A: set_int,B4: set_int] :
( ( inj_on203032006962319749list_o @ F2 @ A )
=> ( ( ord_less_eq_set_int @ B4 @ A )
=> ( inj_on203032006962319749list_o @ F2 @ B4 ) ) ) ).
% inj_on_subset
thf(fact_1010_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A5 )
=> ( member_nat @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_1011_subset__iff,axiom,
( ord_less_eq_set_int
= ( ^ [A5: set_int,B5: set_int] :
! [T2: int] :
( ( member_int @ T2 @ A5 )
=> ( member_int @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_1012_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
! [X3: nat] :
( ( member_nat @ X3 @ A5 )
=> ( member_nat @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_1013_subset__eq,axiom,
( ord_less_eq_set_int
= ( ^ [A5: set_int,B5: set_int] :
! [X3: int] :
( ( member_int @ X3 @ A5 )
=> ( member_int @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_1014_subsetD,axiom,
! [A: set_nat,B4: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B4 ) ) ) ).
% subsetD
thf(fact_1015_subsetD,axiom,
! [A: set_int,B4: set_int,C: int] :
( ( ord_less_eq_set_int @ A @ B4 )
=> ( ( member_int @ C @ A )
=> ( member_int @ C @ B4 ) ) ) ).
% subsetD
thf(fact_1016_in__mono,axiom,
! [A: set_nat,B4: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ( member_nat @ X @ A )
=> ( member_nat @ X @ B4 ) ) ) ).
% in_mono
thf(fact_1017_in__mono,axiom,
! [A: set_int,B4: set_int,X: int] :
( ( ord_less_eq_set_int @ A @ B4 )
=> ( ( member_int @ X @ A )
=> ( member_int @ X @ B4 ) ) ) ).
% in_mono
thf(fact_1018_subset__insert,axiom,
! [X: nat,A: set_nat,B4: set_nat] :
( ~ ( member_nat @ X @ A )
=> ( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X @ B4 ) )
= ( ord_less_eq_set_nat @ A @ B4 ) ) ) ).
% subset_insert
thf(fact_1019_subset__insert,axiom,
! [X: int,A: set_int,B4: set_int] :
( ~ ( member_int @ X @ A )
=> ( ( ord_less_eq_set_int @ A @ ( insert_int @ X @ B4 ) )
= ( ord_less_eq_set_int @ A @ B4 ) ) ) ).
% subset_insert
thf(fact_1020_finite__range__Some,axiom,
( ( finite7007496012504252301list_o @ ( image_3961372166222858059list_o @ some_list_o @ top_top_set_list_o ) )
= ( finite_finite_list_o @ top_top_set_list_o ) ) ).
% finite_range_Some
thf(fact_1021_inj__on__image__mem__iff,axiom,
! [F2: nat > option_list_o,B4: set_nat,A2: nat,A: set_nat] :
( ( inj_on1630180835328728801list_o @ F2 @ B4 )
=> ( ( member_nat @ A2 @ B4 )
=> ( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ( member_option_list_o @ ( F2 @ A2 ) @ ( image_4575287668734308173list_o @ F2 @ A ) )
= ( member_nat @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1022_inj__on__image__mem__iff,axiom,
! [F2: nat > nat,B4: set_nat,A2: nat,A: set_nat] :
( ( inj_on_nat_nat @ F2 @ B4 )
=> ( ( member_nat @ A2 @ B4 )
=> ( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ( member_nat @ ( F2 @ A2 ) @ ( image_nat_nat @ F2 @ A ) )
= ( member_nat @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1023_inj__on__image__mem__iff,axiom,
! [F2: nat > int,B4: set_nat,A2: nat,A: set_nat] :
( ( inj_on_nat_int @ F2 @ B4 )
=> ( ( member_nat @ A2 @ B4 )
=> ( ( ord_less_eq_set_nat @ A @ B4 )
=> ( ( member_int @ ( F2 @ A2 ) @ ( image_nat_int @ F2 @ A ) )
= ( member_nat @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1024_inj__on__image__mem__iff,axiom,
! [F2: int > option_list_o,B4: set_int,A2: int,A: set_int] :
( ( inj_on203032006962319749list_o @ F2 @ B4 )
=> ( ( member_int @ A2 @ B4 )
=> ( ( ord_less_eq_set_int @ A @ B4 )
=> ( ( member_option_list_o @ ( F2 @ A2 ) @ ( image_3148138840367899121list_o @ F2 @ A ) )
= ( member_int @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1025_inj__on__image__mem__iff,axiom,
! [F2: int > nat,B4: set_int,A2: int,A: set_int] :
( ( inj_on_int_nat @ F2 @ B4 )
=> ( ( member_int @ A2 @ B4 )
=> ( ( ord_less_eq_set_int @ A @ B4 )
=> ( ( member_nat @ ( F2 @ A2 ) @ ( image_int_nat @ F2 @ A ) )
= ( member_int @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1026_inj__on__image__mem__iff,axiom,
! [F2: int > int,B4: set_int,A2: int,A: set_int] :
( ( inj_on_int_int @ F2 @ B4 )
=> ( ( member_int @ A2 @ B4 )
=> ( ( ord_less_eq_set_int @ A @ B4 )
=> ( ( member_int @ ( F2 @ A2 ) @ ( image_int_int @ F2 @ A ) )
= ( member_int @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1027_inj__on__image__eq__iff,axiom,
! [F2: nat > option_list_o,C2: set_nat,A: set_nat,B4: set_nat] :
( ( inj_on1630180835328728801list_o @ F2 @ C2 )
=> ( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B4 @ C2 )
=> ( ( ( image_4575287668734308173list_o @ F2 @ A )
= ( image_4575287668734308173list_o @ F2 @ B4 ) )
= ( A = B4 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_1028_inj__on__image__eq__iff,axiom,
! [F2: int > option_list_o,C2: set_int,A: set_int,B4: set_int] :
( ( inj_on203032006962319749list_o @ F2 @ C2 )
=> ( ( ord_less_eq_set_int @ A @ C2 )
=> ( ( ord_less_eq_set_int @ B4 @ C2 )
=> ( ( ( image_3148138840367899121list_o @ F2 @ A )
= ( image_3148138840367899121list_o @ F2 @ B4 ) )
= ( A = B4 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_1029_subset__Diff__insert,axiom,
! [A: set_nat,B4: set_nat,X: nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( minus_minus_set_nat @ B4 @ ( insert_nat @ X @ C2 ) ) )
= ( ( ord_less_eq_set_nat @ A @ ( minus_minus_set_nat @ B4 @ C2 ) )
& ~ ( member_nat @ X @ A ) ) ) ).
% subset_Diff_insert
thf(fact_1030_subset__Diff__insert,axiom,
! [A: set_int,B4: set_int,X: int,C2: set_int] :
( ( ord_less_eq_set_int @ A @ ( minus_minus_set_int @ B4 @ ( insert_int @ X @ C2 ) ) )
= ( ( ord_less_eq_set_int @ A @ ( minus_minus_set_int @ B4 @ C2 ) )
& ~ ( member_int @ X @ A ) ) ) ).
% subset_Diff_insert
thf(fact_1031_finite__ran,axiom,
! [P4: nat > option_list_o] :
( ( finite_finite_nat @ ( dom_nat_list_o @ P4 ) )
=> ( finite_finite_list_o @ ( ran_nat_list_o @ P4 ) ) ) ).
% finite_ran
thf(fact_1032_finite__ran,axiom,
! [P4: int > option_list_o] :
( ( finite_finite_int @ ( dom_int_list_o @ P4 ) )
=> ( finite_finite_list_o @ ( ran_int_list_o @ P4 ) ) ) ).
% finite_ran
thf(fact_1033_map__le__implies__dom__le,axiom,
! [F2: nat > option_list_o,G: nat > option_list_o] :
( ( map_le_nat_list_o @ F2 @ G )
=> ( ord_less_eq_set_nat @ ( dom_nat_list_o @ F2 ) @ ( dom_nat_list_o @ G ) ) ) ).
% map_le_implies_dom_le
thf(fact_1034_map__le__implies__dom__le,axiom,
! [F2: int > option_list_o,G: int > option_list_o] :
( ( map_le_int_list_o @ F2 @ G )
=> ( ord_less_eq_set_int @ ( dom_int_list_o @ F2 ) @ ( dom_int_list_o @ G ) ) ) ).
% map_le_implies_dom_le
thf(fact_1035_inj__image__subset__iff,axiom,
! [F2: nat > option_list_o,A: set_nat,B4: set_nat] :
( ( inj_on1630180835328728801list_o @ F2 @ top_top_set_nat )
=> ( ( ord_le1162937763994921316list_o @ ( image_4575287668734308173list_o @ F2 @ A ) @ ( image_4575287668734308173list_o @ F2 @ B4 ) )
= ( ord_less_eq_set_nat @ A @ B4 ) ) ) ).
% inj_image_subset_iff
thf(fact_1036_inj__image__subset__iff,axiom,
! [F2: int > option_list_o,A: set_int,B4: set_int] :
( ( inj_on203032006962319749list_o @ F2 @ top_top_set_int )
=> ( ( ord_le1162937763994921316list_o @ ( image_3148138840367899121list_o @ F2 @ A ) @ ( image_3148138840367899121list_o @ F2 @ B4 ) )
= ( ord_less_eq_set_int @ A @ B4 ) ) ) ).
% inj_image_subset_iff
thf(fact_1037_subset__insert__iff,axiom,
! [A: set_nat,X: nat,B4: set_nat] :
( ( ord_less_eq_set_nat @ A @ ( insert_nat @ X @ B4 ) )
= ( ( ( member_nat @ X @ A )
=> ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B4 ) )
& ( ~ ( member_nat @ X @ A )
=> ( ord_less_eq_set_nat @ A @ B4 ) ) ) ) ).
% subset_insert_iff
thf(fact_1038_subset__insert__iff,axiom,
! [A: set_int,X: int,B4: set_int] :
( ( ord_less_eq_set_int @ A @ ( insert_int @ X @ B4 ) )
= ( ( ( member_int @ X @ A )
=> ( ord_less_eq_set_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) @ B4 ) )
& ( ~ ( member_int @ X @ A )
=> ( ord_less_eq_set_int @ A @ B4 ) ) ) ) ).
% subset_insert_iff
thf(fact_1039_inj__on__image__Int,axiom,
! [F2: nat > option_list_o,C2: set_nat,A: set_nat,B4: set_nat] :
( ( inj_on1630180835328728801list_o @ F2 @ C2 )
=> ( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B4 @ C2 )
=> ( ( image_4575287668734308173list_o @ F2 @ ( inf_inf_set_nat @ A @ B4 ) )
= ( inf_in6922378751903173298list_o @ ( image_4575287668734308173list_o @ F2 @ A ) @ ( image_4575287668734308173list_o @ F2 @ B4 ) ) ) ) ) ) ).
% inj_on_image_Int
thf(fact_1040_inj__on__image__Int,axiom,
! [F2: int > option_list_o,C2: set_int,A: set_int,B4: set_int] :
( ( inj_on203032006962319749list_o @ F2 @ C2 )
=> ( ( ord_less_eq_set_int @ A @ C2 )
=> ( ( ord_less_eq_set_int @ B4 @ C2 )
=> ( ( image_3148138840367899121list_o @ F2 @ ( inf_inf_set_int @ A @ B4 ) )
= ( inf_in6922378751903173298list_o @ ( image_3148138840367899121list_o @ F2 @ A ) @ ( image_3148138840367899121list_o @ F2 @ B4 ) ) ) ) ) ) ).
% inj_on_image_Int
thf(fact_1041_inj__on__image__set__diff,axiom,
! [F2: nat > option_list_o,C2: set_nat,A: set_nat,B4: set_nat] :
( ( inj_on1630180835328728801list_o @ F2 @ C2 )
=> ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A @ B4 ) @ C2 )
=> ( ( ord_less_eq_set_nat @ B4 @ C2 )
=> ( ( image_4575287668734308173list_o @ F2 @ ( minus_minus_set_nat @ A @ B4 ) )
= ( minus_972179126788192427list_o @ ( image_4575287668734308173list_o @ F2 @ A ) @ ( image_4575287668734308173list_o @ F2 @ B4 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_1042_inj__on__image__set__diff,axiom,
! [F2: int > option_list_o,C2: set_int,A: set_int,B4: set_int] :
( ( inj_on203032006962319749list_o @ F2 @ C2 )
=> ( ( ord_less_eq_set_int @ ( minus_minus_set_int @ A @ B4 ) @ C2 )
=> ( ( ord_less_eq_set_int @ B4 @ C2 )
=> ( ( image_3148138840367899121list_o @ F2 @ ( minus_minus_set_int @ A @ B4 ) )
= ( minus_972179126788192427list_o @ ( image_3148138840367899121list_o @ F2 @ A ) @ ( image_3148138840367899121list_o @ F2 @ B4 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_1043_finite__map__freshness,axiom,
! [F2: nat > option_list_o] :
( ( finite_finite_nat @ ( dom_nat_list_o @ F2 ) )
=> ( ~ ( finite_finite_nat @ top_top_set_nat )
=> ? [X2: nat] :
( ( F2 @ X2 )
= none_list_o ) ) ) ).
% finite_map_freshness
thf(fact_1044_finite__map__freshness,axiom,
! [F2: int > option_list_o] :
( ( finite_finite_int @ ( dom_int_list_o @ F2 ) )
=> ( ~ ( finite_finite_int @ top_top_set_int )
=> ? [X2: int] :
( ( F2 @ X2 )
= none_list_o ) ) ) ).
% finite_map_freshness
thf(fact_1045_remove__induct,axiom,
! [P: set_nat > $o,B4: set_nat] :
( ( P @ bot_bot_set_nat )
=> ( ( ~ ( finite_finite_nat @ B4 )
=> ( P @ B4 ) )
=> ( ! [A7: set_nat] :
( ( finite_finite_nat @ A7 )
=> ( ( A7 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A7 @ B4 )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ A7 )
=> ( P @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B4 ) ) ) ) ).
% remove_induct
thf(fact_1046_remove__induct,axiom,
! [P: set_int > $o,B4: set_int] :
( ( P @ bot_bot_set_int )
=> ( ( ~ ( finite_finite_int @ B4 )
=> ( P @ B4 ) )
=> ( ! [A7: set_int] :
( ( finite_finite_int @ A7 )
=> ( ( A7 != bot_bot_set_int )
=> ( ( ord_less_eq_set_int @ A7 @ B4 )
=> ( ! [X6: int] :
( ( member_int @ X6 @ A7 )
=> ( P @ ( minus_minus_set_int @ A7 @ ( insert_int @ X6 @ bot_bot_set_int ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B4 ) ) ) ) ).
% remove_induct
thf(fact_1047_finite__remove__induct,axiom,
! [B4: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ B4 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A7: set_nat] :
( ( finite_finite_nat @ A7 )
=> ( ( A7 != bot_bot_set_nat )
=> ( ( ord_less_eq_set_nat @ A7 @ B4 )
=> ( ! [X6: nat] :
( ( member_nat @ X6 @ A7 )
=> ( P @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X6 @ bot_bot_set_nat ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B4 ) ) ) ) ).
% finite_remove_induct
thf(fact_1048_finite__remove__induct,axiom,
! [B4: set_int,P: set_int > $o] :
( ( finite_finite_int @ B4 )
=> ( ( P @ bot_bot_set_int )
=> ( ! [A7: set_int] :
( ( finite_finite_int @ A7 )
=> ( ( A7 != bot_bot_set_int )
=> ( ( ord_less_eq_set_int @ A7 @ B4 )
=> ( ! [X6: int] :
( ( member_int @ X6 @ A7 )
=> ( P @ ( minus_minus_set_int @ A7 @ ( insert_int @ X6 @ bot_bot_set_int ) ) ) )
=> ( P @ A7 ) ) ) ) )
=> ( P @ B4 ) ) ) ) ).
% finite_remove_induct
thf(fact_1049_prefix__order_Ofinite__has__maximal,axiom,
! [A: set_list_o] :
( ( finite_finite_list_o @ A )
=> ( ( A != bot_bot_set_list_o )
=> ? [X2: list_o] :
( ( member_list_o @ X2 @ A )
& ! [Xa3: list_o] :
( ( member_list_o @ Xa3 @ A )
=> ( ( prefix_o @ X2 @ Xa3 )
=> ( X2 = Xa3 ) ) ) ) ) ) ).
% prefix_order.finite_has_maximal
thf(fact_1050_prefix__order_Ofinite__has__minimal,axiom,
! [A: set_list_o] :
( ( finite_finite_list_o @ A )
=> ( ( A != bot_bot_set_list_o )
=> ? [X2: list_o] :
( ( member_list_o @ X2 @ A )
& ! [Xa3: list_o] :
( ( member_list_o @ Xa3 @ A )
=> ( ( prefix_o @ Xa3 @ X2 )
=> ( X2 = Xa3 ) ) ) ) ) ) ).
% prefix_order.finite_has_minimal
thf(fact_1051_prefix__order_Ofinite__has__minimal2,axiom,
! [A: set_list_o,A2: list_o] :
( ( finite_finite_list_o @ A )
=> ( ( member_list_o @ A2 @ A )
=> ? [X2: list_o] :
( ( member_list_o @ X2 @ A )
& ( prefix_o @ X2 @ A2 )
& ! [Xa3: list_o] :
( ( member_list_o @ Xa3 @ A )
=> ( ( prefix_o @ Xa3 @ X2 )
=> ( X2 = Xa3 ) ) ) ) ) ) ).
% prefix_order.finite_has_minimal2
thf(fact_1052_prefix__order_Ofinite__has__maximal2,axiom,
! [A: set_list_o,A2: list_o] :
( ( finite_finite_list_o @ A )
=> ( ( member_list_o @ A2 @ A )
=> ? [X2: list_o] :
( ( member_list_o @ X2 @ A )
& ( prefix_o @ A2 @ X2 )
& ! [Xa3: list_o] :
( ( member_list_o @ Xa3 @ A )
=> ( ( prefix_o @ X2 @ Xa3 )
=> ( X2 = Xa3 ) ) ) ) ) ) ).
% prefix_order.finite_has_maximal2
thf(fact_1053_finite__induct,axiom,
! [F3: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F3 )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X2: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ~ ( member_nat @ X2 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_nat @ X2 @ F4 ) ) ) ) )
=> ( P @ F3 ) ) ) ) ).
% finite_induct
thf(fact_1054_finite__induct,axiom,
! [F3: set_int,P: set_int > $o] :
( ( finite_finite_int @ F3 )
=> ( ( P @ bot_bot_set_int )
=> ( ! [X2: int,F4: set_int] :
( ( finite_finite_int @ F4 )
=> ( ~ ( member_int @ X2 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_int @ X2 @ F4 ) ) ) ) )
=> ( P @ F3 ) ) ) ) ).
% finite_induct
thf(fact_1055_finite__ne__induct,axiom,
! [F3: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F3 )
=> ( ( F3 != bot_bot_set_nat )
=> ( ! [X2: nat] : ( P @ ( insert_nat @ X2 @ bot_bot_set_nat ) )
=> ( ! [X2: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ( F4 != bot_bot_set_nat )
=> ( ~ ( member_nat @ X2 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_nat @ X2 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_1056_finite__ne__induct,axiom,
! [F3: set_int,P: set_int > $o] :
( ( finite_finite_int @ F3 )
=> ( ( F3 != bot_bot_set_int )
=> ( ! [X2: int] : ( P @ ( insert_int @ X2 @ bot_bot_set_int ) )
=> ( ! [X2: int,F4: set_int] :
( ( finite_finite_int @ F4 )
=> ( ( F4 != bot_bot_set_int )
=> ( ~ ( member_int @ X2 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_int @ X2 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_1057_infinite__finite__induct,axiom,
! [P: set_nat > $o,A: set_nat] :
( ! [A7: set_nat] :
( ~ ( finite_finite_nat @ A7 )
=> ( P @ A7 ) )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [X2: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ~ ( member_nat @ X2 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_nat @ X2 @ F4 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_1058_infinite__finite__induct,axiom,
! [P: set_int > $o,A: set_int] :
( ! [A7: set_int] :
( ~ ( finite_finite_int @ A7 )
=> ( P @ A7 ) )
=> ( ( P @ bot_bot_set_int )
=> ( ! [X2: int,F4: set_int] :
( ( finite_finite_int @ F4 )
=> ( ~ ( member_int @ X2 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_int @ X2 @ F4 ) ) ) ) )
=> ( P @ A ) ) ) ) ).
% infinite_finite_induct
thf(fact_1059_finite__image__iff,axiom,
! [F2: nat > option_list_o,A: set_nat] :
( ( inj_on1630180835328728801list_o @ F2 @ A )
=> ( ( finite7007496012504252301list_o @ ( image_4575287668734308173list_o @ F2 @ A ) )
= ( finite_finite_nat @ A ) ) ) ).
% finite_image_iff
thf(fact_1060_finite__image__iff,axiom,
! [F2: int > option_list_o,A: set_int] :
( ( inj_on203032006962319749list_o @ F2 @ A )
=> ( ( finite7007496012504252301list_o @ ( image_3148138840367899121list_o @ F2 @ A ) )
= ( finite_finite_int @ A ) ) ) ).
% finite_image_iff
thf(fact_1061_finite__imageD,axiom,
! [F2: nat > option_list_o,A: set_nat] :
( ( finite7007496012504252301list_o @ ( image_4575287668734308173list_o @ F2 @ A ) )
=> ( ( inj_on1630180835328728801list_o @ F2 @ A )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_imageD
thf(fact_1062_finite__imageD,axiom,
! [F2: int > option_list_o,A: set_int] :
( ( finite7007496012504252301list_o @ ( image_3148138840367899121list_o @ F2 @ A ) )
=> ( ( inj_on203032006962319749list_o @ F2 @ A )
=> ( finite_finite_int @ A ) ) ) ).
% finite_imageD
thf(fact_1063_finite__subset__induct_H,axiom,
! [F3: set_nat,A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F3 )
=> ( ( ord_less_eq_set_nat @ F3 @ A )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A4: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ( member_nat @ A4 @ A )
=> ( ( ord_less_eq_set_nat @ F4 @ A )
=> ( ~ ( member_nat @ A4 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_nat @ A4 @ F4 ) ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1064_finite__subset__induct_H,axiom,
! [F3: set_int,A: set_int,P: set_int > $o] :
( ( finite_finite_int @ F3 )
=> ( ( ord_less_eq_set_int @ F3 @ A )
=> ( ( P @ bot_bot_set_int )
=> ( ! [A4: int,F4: set_int] :
( ( finite_finite_int @ F4 )
=> ( ( member_int @ A4 @ A )
=> ( ( ord_less_eq_set_int @ F4 @ A )
=> ( ~ ( member_int @ A4 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_int @ A4 @ F4 ) ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_1065_finite__subset__induct,axiom,
! [F3: set_nat,A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ F3 )
=> ( ( ord_less_eq_set_nat @ F3 @ A )
=> ( ( P @ bot_bot_set_nat )
=> ( ! [A4: nat,F4: set_nat] :
( ( finite_finite_nat @ F4 )
=> ( ( member_nat @ A4 @ A )
=> ( ~ ( member_nat @ A4 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_nat @ A4 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1066_finite__subset__induct,axiom,
! [F3: set_int,A: set_int,P: set_int > $o] :
( ( finite_finite_int @ F3 )
=> ( ( ord_less_eq_set_int @ F3 @ A )
=> ( ( P @ bot_bot_set_int )
=> ( ! [A4: int,F4: set_int] :
( ( finite_finite_int @ F4 )
=> ( ( member_int @ A4 @ A )
=> ( ~ ( member_int @ A4 @ F4 )
=> ( ( P @ F4 )
=> ( P @ ( insert_int @ A4 @ F4 ) ) ) ) ) )
=> ( P @ F3 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_1067_inj__on__finite,axiom,
! [F2: nat > option_list_o,A: set_nat,B4: set_option_list_o] :
( ( inj_on1630180835328728801list_o @ F2 @ A )
=> ( ( ord_le1162937763994921316list_o @ ( image_4575287668734308173list_o @ F2 @ A ) @ B4 )
=> ( ( finite7007496012504252301list_o @ B4 )
=> ( finite_finite_nat @ A ) ) ) ) ).
% inj_on_finite
thf(fact_1068_inj__on__finite,axiom,
! [F2: int > option_list_o,A: set_int,B4: set_option_list_o] :
( ( inj_on203032006962319749list_o @ F2 @ A )
=> ( ( ord_le1162937763994921316list_o @ ( image_3148138840367899121list_o @ F2 @ A ) @ B4 )
=> ( ( finite7007496012504252301list_o @ B4 )
=> ( finite_finite_int @ A ) ) ) ) ).
% inj_on_finite
thf(fact_1069_finite__empty__induct,axiom,
! [A: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A )
=> ( ( P @ A )
=> ( ! [A4: nat,A7: set_nat] :
( ( finite_finite_nat @ A7 )
=> ( ( member_nat @ A4 @ A7 )
=> ( ( P @ A7 )
=> ( P @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ A4 @ bot_bot_set_nat ) ) ) ) ) )
=> ( P @ bot_bot_set_nat ) ) ) ) ).
% finite_empty_induct
thf(fact_1070_finite__empty__induct,axiom,
! [A: set_int,P: set_int > $o] :
( ( finite_finite_int @ A )
=> ( ( P @ A )
=> ( ! [A4: int,A7: set_int] :
( ( finite_finite_int @ A7 )
=> ( ( member_int @ A4 @ A7 )
=> ( ( P @ A7 )
=> ( P @ ( minus_minus_set_int @ A7 @ ( insert_int @ A4 @ bot_bot_set_int ) ) ) ) ) )
=> ( P @ bot_bot_set_int ) ) ) ) ).
% finite_empty_induct
thf(fact_1071_inj__on__iff__surj,axiom,
! [A: set_nat,A8: set_option_list_o] :
( ( A != bot_bot_set_nat )
=> ( ( ? [F: nat > option_list_o] :
( ( inj_on1630180835328728801list_o @ F @ A )
& ( ord_le1162937763994921316list_o @ ( image_4575287668734308173list_o @ F @ A ) @ A8 ) ) )
= ( ? [G2: option_list_o > nat] :
( ( image_5401538521246155887_o_nat @ G2 @ A8 )
= A ) ) ) ) ).
% inj_on_iff_surj
thf(fact_1072_inj__on__iff__surj,axiom,
! [A: set_int,A8: set_option_list_o] :
( ( A != bot_bot_set_int )
=> ( ( ? [F: int > option_list_o] :
( ( inj_on203032006962319749list_o @ F @ A )
& ( ord_le1162937763994921316list_o @ ( image_3148138840367899121list_o @ F @ A ) @ A8 ) ) )
= ( ? [G2: option_list_o > int] :
( ( image_5399048050737105611_o_int @ G2 @ A8 )
= A ) ) ) ) ).
% inj_on_iff_surj
thf(fact_1073_opt__prefix_Opelims_I1_J,axiom,
! [X: option_list_o,Xa: option_list_o,Y: $o] :
( ( ( prefix8824957607401505554efix_o @ X @ Xa )
= Y )
=> ( ( accp_P4987914878654715440list_o @ prefix2776050468385694435_rel_o @ ( produc5745850523778858007list_o @ X @ Xa ) )
=> ( ! [X2: list_o] :
( ( X
= ( some_list_o @ X2 ) )
=> ! [Y2: list_o] :
( ( Xa
= ( some_list_o @ Y2 ) )
=> ( ( Y
= ( prefix_o @ X2 @ Y2 ) )
=> ~ ( accp_P4987914878654715440list_o @ prefix2776050468385694435_rel_o @ ( produc5745850523778858007list_o @ ( some_list_o @ X2 ) @ ( some_list_o @ Y2 ) ) ) ) ) )
=> ( ( ( X = none_list_o )
=> ( ~ Y
=> ~ ( accp_P4987914878654715440list_o @ prefix2776050468385694435_rel_o @ ( produc5745850523778858007list_o @ none_list_o @ Xa ) ) ) )
=> ~ ( ( Xa = none_list_o )
=> ( ~ Y
=> ~ ( accp_P4987914878654715440list_o @ prefix2776050468385694435_rel_o @ ( produc5745850523778858007list_o @ X @ none_list_o ) ) ) ) ) ) ) ) ).
% opt_prefix.pelims(1)
thf(fact_1074_subset__image__inj,axiom,
! [S: set_option_list_o,F2: nat > option_list_o,T3: set_nat] :
( ( ord_le1162937763994921316list_o @ S @ ( image_4575287668734308173list_o @ F2 @ T3 ) )
= ( ? [U2: set_nat] :
( ( ord_less_eq_set_nat @ U2 @ T3 )
& ( inj_on1630180835328728801list_o @ F2 @ U2 )
& ( S
= ( image_4575287668734308173list_o @ F2 @ U2 ) ) ) ) ) ).
% subset_image_inj
thf(fact_1075_subset__image__inj,axiom,
! [S: set_option_list_o,F2: int > option_list_o,T3: set_int] :
( ( ord_le1162937763994921316list_o @ S @ ( image_3148138840367899121list_o @ F2 @ T3 ) )
= ( ? [U2: set_int] :
( ( ord_less_eq_set_int @ U2 @ T3 )
& ( inj_on203032006962319749list_o @ F2 @ U2 )
& ( S
= ( image_3148138840367899121list_o @ F2 @ U2 ) ) ) ) ) ).
% subset_image_inj
thf(fact_1076_opt__prefix_Opelims_I2_J,axiom,
! [X: option_list_o,Xa: option_list_o] :
( ( prefix8824957607401505554efix_o @ X @ Xa )
=> ( ( accp_P4987914878654715440list_o @ prefix2776050468385694435_rel_o @ ( produc5745850523778858007list_o @ X @ Xa ) )
=> ~ ! [X2: list_o] :
( ( X
= ( some_list_o @ X2 ) )
=> ! [Y2: list_o] :
( ( Xa
= ( some_list_o @ Y2 ) )
=> ( ( accp_P4987914878654715440list_o @ prefix2776050468385694435_rel_o @ ( produc5745850523778858007list_o @ ( some_list_o @ X2 ) @ ( some_list_o @ Y2 ) ) )
=> ~ ( prefix_o @ X2 @ Y2 ) ) ) ) ) ) ).
% opt_prefix.pelims(2)
thf(fact_1077_opt__prefix_Opelims_I3_J,axiom,
! [X: option_list_o,Xa: option_list_o] :
( ~ ( prefix8824957607401505554efix_o @ X @ Xa )
=> ( ( accp_P4987914878654715440list_o @ prefix2776050468385694435_rel_o @ ( produc5745850523778858007list_o @ X @ Xa ) )
=> ( ! [X2: list_o] :
( ( X
= ( some_list_o @ X2 ) )
=> ! [Y2: list_o] :
( ( Xa
= ( some_list_o @ Y2 ) )
=> ( ( accp_P4987914878654715440list_o @ prefix2776050468385694435_rel_o @ ( produc5745850523778858007list_o @ ( some_list_o @ X2 ) @ ( some_list_o @ Y2 ) ) )
=> ( prefix_o @ X2 @ Y2 ) ) ) )
=> ( ( ( X = none_list_o )
=> ~ ( accp_P4987914878654715440list_o @ prefix2776050468385694435_rel_o @ ( produc5745850523778858007list_o @ none_list_o @ Xa ) ) )
=> ~ ( ( Xa = none_list_o )
=> ~ ( accp_P4987914878654715440list_o @ prefix2776050468385694435_rel_o @ ( produc5745850523778858007list_o @ X @ none_list_o ) ) ) ) ) ) ) ).
% opt_prefix.pelims(3)
thf(fact_1078_ex__subset__image__inj,axiom,
! [F2: nat > option_list_o,S: set_nat,P: set_option_list_o > $o] :
( ( ? [T4: set_option_list_o] :
( ( ord_le1162937763994921316list_o @ T4 @ ( image_4575287668734308173list_o @ F2 @ S ) )
& ( P @ T4 ) ) )
= ( ? [T4: set_nat] :
( ( ord_less_eq_set_nat @ T4 @ S )
& ( inj_on1630180835328728801list_o @ F2 @ T4 )
& ( P @ ( image_4575287668734308173list_o @ F2 @ T4 ) ) ) ) ) ).
% ex_subset_image_inj
thf(fact_1079_ex__subset__image__inj,axiom,
! [F2: int > option_list_o,S: set_int,P: set_option_list_o > $o] :
( ( ? [T4: set_option_list_o] :
( ( ord_le1162937763994921316list_o @ T4 @ ( image_3148138840367899121list_o @ F2 @ S ) )
& ( P @ T4 ) ) )
= ( ? [T4: set_int] :
( ( ord_less_eq_set_int @ T4 @ S )
& ( inj_on203032006962319749list_o @ F2 @ T4 )
& ( P @ ( image_3148138840367899121list_o @ F2 @ T4 ) ) ) ) ) ).
% ex_subset_image_inj
thf(fact_1080_all__subset__image__inj,axiom,
! [F2: nat > option_list_o,S: set_nat,P: set_option_list_o > $o] :
( ( ! [T4: set_option_list_o] :
( ( ord_le1162937763994921316list_o @ T4 @ ( image_4575287668734308173list_o @ F2 @ S ) )
=> ( P @ T4 ) ) )
= ( ! [T4: set_nat] :
( ( ( ord_less_eq_set_nat @ T4 @ S )
& ( inj_on1630180835328728801list_o @ F2 @ T4 ) )
=> ( P @ ( image_4575287668734308173list_o @ F2 @ T4 ) ) ) ) ) ).
% all_subset_image_inj
thf(fact_1081_all__subset__image__inj,axiom,
! [F2: int > option_list_o,S: set_int,P: set_option_list_o > $o] :
( ( ! [T4: set_option_list_o] :
( ( ord_le1162937763994921316list_o @ T4 @ ( image_3148138840367899121list_o @ F2 @ S ) )
=> ( P @ T4 ) ) )
= ( ! [T4: set_int] :
( ( ( ord_less_eq_set_int @ T4 @ S )
& ( inj_on203032006962319749list_o @ F2 @ T4 ) )
=> ( P @ ( image_3148138840367899121list_o @ F2 @ T4 ) ) ) ) ) ).
% all_subset_image_inj
thf(fact_1082_subset__emptyI,axiom,
! [A: set_nat] :
( ! [X2: nat] :
~ ( member_nat @ X2 @ A )
=> ( ord_less_eq_set_nat @ A @ bot_bot_set_nat ) ) ).
% subset_emptyI
thf(fact_1083_subset__emptyI,axiom,
! [A: set_int] :
( ! [X2: int] :
~ ( member_int @ X2 @ A )
=> ( ord_less_eq_set_int @ A @ bot_bot_set_int ) ) ).
% subset_emptyI
thf(fact_1084_finite__transitivity__chain,axiom,
! [A: set_nat,R2: nat > nat > $o] :
( ( finite_finite_nat @ A )
=> ( ! [X2: nat] :
~ ( R2 @ X2 @ X2 )
=> ( ! [X2: nat,Y2: nat,Z4: nat] :
( ( R2 @ X2 @ Y2 )
=> ( ( R2 @ Y2 @ Z4 )
=> ( R2 @ X2 @ Z4 ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ? [Y6: nat] :
( ( member_nat @ Y6 @ A )
& ( R2 @ X2 @ Y6 ) ) )
=> ( A = bot_bot_set_nat ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_1085_finite__transitivity__chain,axiom,
! [A: set_int,R2: int > int > $o] :
( ( finite_finite_int @ A )
=> ( ! [X2: int] :
~ ( R2 @ X2 @ X2 )
=> ( ! [X2: int,Y2: int,Z4: int] :
( ( R2 @ X2 @ Y2 )
=> ( ( R2 @ Y2 @ Z4 )
=> ( R2 @ X2 @ Z4 ) ) )
=> ( ! [X2: int] :
( ( member_int @ X2 @ A )
=> ? [Y6: int] :
( ( member_int @ Y6 @ A )
& ( R2 @ X2 @ Y6 ) ) )
=> ( A = bot_bot_set_int ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_1086_Sup__fin_Oin__idem,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ X @ A )
=> ( ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ A ) )
= ( lattic1093996805478795353in_nat @ A ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_1087_Sup__fin_Oin__idem,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ( member_int @ X @ A )
=> ( ( sup_sup_int @ X @ ( lattic1091506334969745077in_int @ A ) )
= ( lattic1091506334969745077in_int @ A ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_1088_Sup__fin_OboundedE,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ X )
=> ! [A9: nat] :
( ( member_nat @ A9 @ A )
=> ( ord_less_eq_nat @ A9 @ X ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_1089_Sup__fin_OboundedE,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ( ( ord_less_eq_int @ ( lattic1091506334969745077in_int @ A ) @ X )
=> ! [A9: int] :
( ( member_int @ A9 @ A )
=> ( ord_less_eq_int @ A9 @ X ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_1090_Sup__fin_OboundedI,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ A )
=> ( ord_less_eq_nat @ A4 @ X ) )
=> ( ord_less_eq_nat @ ( lattic1093996805478795353in_nat @ A ) @ X ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_1091_Sup__fin_OboundedI,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ( ! [A4: int] :
( ( member_int @ A4 @ A )
=> ( ord_less_eq_int @ A4 @ X ) )
=> ( ord_less_eq_int @ ( lattic1091506334969745077in_int @ A ) @ X ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_1092_Sup__fin_Oinfinite,axiom,
! [A: set_list_o] :
( ~ ( finite_finite_list_o @ A )
=> ( ( lattic42854817761806421list_o @ A )
= ( the_list_o @ none_list_o ) ) ) ).
% Sup_fin.infinite
thf(fact_1093_Sup__fin_Oinsert__not__elem,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ~ ( member_nat @ X @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ ( insert_nat @ X @ A ) )
= ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ A ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_1094_Sup__fin_Oinsert__not__elem,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ~ ( member_int @ X @ A )
=> ( ( A != bot_bot_set_int )
=> ( ( lattic1091506334969745077in_int @ ( insert_int @ X @ A ) )
= ( sup_sup_int @ X @ ( lattic1091506334969745077in_int @ A ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_1095_Sup__fin_Oclosed,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ! [X2: nat,Y2: nat] : ( member_nat @ ( sup_sup_nat @ X2 @ Y2 ) @ ( insert_nat @ X2 @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) )
=> ( member_nat @ ( lattic1093996805478795353in_nat @ A ) @ A ) ) ) ) ).
% Sup_fin.closed
thf(fact_1096_Sup__fin_Oclosed,axiom,
! [A: set_int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ( ! [X2: int,Y2: int] : ( member_int @ ( sup_sup_int @ X2 @ Y2 ) @ ( insert_int @ X2 @ ( insert_int @ Y2 @ bot_bot_set_int ) ) )
=> ( member_int @ ( lattic1091506334969745077in_int @ A ) @ A ) ) ) ) ).
% Sup_fin.closed
thf(fact_1097_Sup__fin_Oremove,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ X @ A )
=> ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ A )
= X ) )
& ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic1093996805478795353in_nat @ A )
= ( sup_sup_nat @ X @ ( lattic1093996805478795353in_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_1098_Sup__fin_Oremove,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ( member_int @ X @ A )
=> ( ( ( ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) )
= bot_bot_set_int )
=> ( ( lattic1091506334969745077in_int @ A )
= X ) )
& ( ( ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) )
!= bot_bot_set_int )
=> ( ( lattic1091506334969745077in_int @ A )
= ( sup_sup_int @ X @ ( lattic1091506334969745077in_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_1099_Inf__fin_Oremove,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ X @ A )
=> ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ A )
= X ) )
& ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ A )
= ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_1100_Inf__fin_Oremove,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ( member_int @ X @ A )
=> ( ( ( ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) )
= bot_bot_set_int )
=> ( ( lattic5235898064620869839in_int @ A )
= X ) )
& ( ( ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) )
!= bot_bot_set_int )
=> ( ( lattic5235898064620869839in_int @ A )
= ( inf_inf_int @ X @ ( lattic5235898064620869839in_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_1101_sup__Inf__absorb,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ( ( sup_sup_nat @ ( lattic5238388535129920115in_nat @ A ) @ A2 )
= A2 ) ) ) ).
% sup_Inf_absorb
thf(fact_1102_sup__Inf__absorb,axiom,
! [A: set_int,A2: int] :
( ( finite_finite_int @ A )
=> ( ( member_int @ A2 @ A )
=> ( ( sup_sup_int @ ( lattic5235898064620869839in_int @ A ) @ A2 )
= A2 ) ) ) ).
% sup_Inf_absorb
thf(fact_1103_Inf__fin_OboundedI,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ A )
=> ( ord_less_eq_nat @ X @ A4 ) )
=> ( ord_less_eq_nat @ X @ ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_1104_Inf__fin_OboundedI,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ( ! [A4: int] :
( ( member_int @ A4 @ A )
=> ( ord_less_eq_int @ X @ A4 ) )
=> ( ord_less_eq_int @ X @ ( lattic5235898064620869839in_int @ A ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_1105_Inf__fin_OboundedE,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X @ ( lattic5238388535129920115in_nat @ A ) )
=> ! [A9: nat] :
( ( member_nat @ A9 @ A )
=> ( ord_less_eq_nat @ X @ A9 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_1106_Inf__fin_OboundedE,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ( ( ord_less_eq_int @ X @ ( lattic5235898064620869839in_int @ A ) )
=> ! [A9: int] :
( ( member_int @ A9 @ A )
=> ( ord_less_eq_int @ X @ A9 ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_1107_Inf__fin_Oinfinite,axiom,
! [A: set_list_o] :
( ~ ( finite_finite_list_o @ A )
=> ( ( lattic5334489926099892411list_o @ A )
= ( the_list_o @ none_list_o ) ) ) ).
% Inf_fin.infinite
thf(fact_1108_Inf__fin_Oinsert__not__elem,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ~ ( member_nat @ X @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( lattic5238388535129920115in_nat @ ( insert_nat @ X @ A ) )
= ( inf_inf_nat @ X @ ( lattic5238388535129920115in_nat @ A ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_1109_Inf__fin_Oinsert__not__elem,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ~ ( member_int @ X @ A )
=> ( ( A != bot_bot_set_int )
=> ( ( lattic5235898064620869839in_int @ ( insert_int @ X @ A ) )
= ( inf_inf_int @ X @ ( lattic5235898064620869839in_int @ A ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_1110_Inf__fin_Oclosed,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ! [X2: nat,Y2: nat] : ( member_nat @ ( inf_inf_nat @ X2 @ Y2 ) @ ( insert_nat @ X2 @ ( insert_nat @ Y2 @ bot_bot_set_nat ) ) )
=> ( member_nat @ ( lattic5238388535129920115in_nat @ A ) @ A ) ) ) ) ).
% Inf_fin.closed
thf(fact_1111_Inf__fin_Oclosed,axiom,
! [A: set_int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ( ! [X2: int,Y2: int] : ( member_int @ ( inf_inf_int @ X2 @ Y2 ) @ ( insert_int @ X2 @ ( insert_int @ Y2 @ bot_bot_set_int ) ) )
=> ( member_int @ ( lattic5235898064620869839in_int @ A ) @ A ) ) ) ) ).
% Inf_fin.closed
thf(fact_1112_linear__order__on__singleton,axiom,
! [X: option_list_o] : ( order_613716560800104601list_o @ ( insert_option_list_o @ X @ bot_bo3275880064859030064list_o ) @ ( insert7989621605229100023list_o @ ( produc5745850523778858007list_o @ X @ X ) @ bot_bo6949037146550090099list_o ) ) ).
% linear_order_on_singleton
thf(fact_1113_linear__order__on__singleton,axiom,
! [X: nat] : ( order_4473980167227706203on_nat @ ( insert_nat @ X @ bot_bot_set_nat ) @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ X @ X ) @ bot_bo2099793752762293965at_nat ) ) ).
% linear_order_on_singleton
thf(fact_1114_finite__dom__map__of,axiom,
! [L: list_P3508452081598505427list_o] : ( finite_finite_nat @ ( dom_nat_list_o @ ( map_of_nat_list_o @ L ) ) ) ).
% finite_dom_map_of
thf(fact_1115_finite__dom__map__of,axiom,
! [L: list_P5976111900331165815list_o] : ( finite_finite_int @ ( dom_int_list_o @ ( map_of_int_list_o @ L ) ) ) ).
% finite_dom_map_of
thf(fact_1116_Linear__order__Well__order__iff,axiom,
! [R: set_Pr7497825696620840711list_o] :
( ( order_613716560800104601list_o @ ( field_option_list_o @ R ) @ R )
=> ( ( order_2779118356246042754list_o @ ( field_option_list_o @ R ) @ R )
= ( ! [A5: set_option_list_o] :
( ( ord_le1162937763994921316list_o @ A5 @ ( field_option_list_o @ R ) )
=> ( ( A5 != bot_bo3275880064859030064list_o )
=> ? [X3: option_list_o] :
( ( member_option_list_o @ X3 @ A5 )
& ! [Y3: option_list_o] :
( ( member_option_list_o @ Y3 @ A5 )
=> ( member1589324699396745552list_o @ ( produc5745850523778858007list_o @ X3 @ Y3 ) @ R ) ) ) ) ) ) ) ) ).
% Linear_order_Well_order_iff
thf(fact_1117_Linear__order__Well__order__iff,axiom,
! [R: set_Pr1261947904930325089at_nat] :
( ( order_4473980167227706203on_nat @ ( field_nat @ R ) @ R )
=> ( ( order_2888998067076097458on_nat @ ( field_nat @ R ) @ R )
= ( ! [A5: set_nat] :
( ( ord_less_eq_set_nat @ A5 @ ( field_nat @ R ) )
=> ( ( A5 != bot_bot_set_nat )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A5 )
& ! [Y3: nat] :
( ( member_nat @ Y3 @ A5 )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R ) ) ) ) ) ) ) ) ).
% Linear_order_Well_order_iff
thf(fact_1118_the__inv__into__into,axiom,
! [F2: nat > option_list_o,A: set_nat,X: option_list_o,B4: set_nat] :
( ( inj_on1630180835328728801list_o @ F2 @ A )
=> ( ( member_option_list_o @ X @ ( image_4575287668734308173list_o @ F2 @ A ) )
=> ( ( ord_less_eq_set_nat @ A @ B4 )
=> ( member_nat @ ( the_in8802290381976579359list_o @ A @ F2 @ X ) @ B4 ) ) ) ) ).
% the_inv_into_into
thf(fact_1119_the__inv__into__into,axiom,
! [F2: int > option_list_o,A: set_int,X: option_list_o,B4: set_int] :
( ( inj_on203032006962319749list_o @ F2 @ A )
=> ( ( member_option_list_o @ X @ ( image_3148138840367899121list_o @ F2 @ A ) )
=> ( ( ord_less_eq_set_int @ A @ B4 )
=> ( member_int @ ( the_in7375141553610170307list_o @ A @ F2 @ X ) @ B4 ) ) ) ) ).
% the_inv_into_into
thf(fact_1120_the__inv__into__into,axiom,
! [F2: nat > nat,A: set_nat,X: nat,B4: set_nat] :
( ( inj_on_nat_nat @ F2 @ A )
=> ( ( member_nat @ X @ ( image_nat_nat @ F2 @ A ) )
=> ( ( ord_less_eq_set_nat @ A @ B4 )
=> ( member_nat @ ( the_inv_into_nat_nat @ A @ F2 @ X ) @ B4 ) ) ) ) ).
% the_inv_into_into
thf(fact_1121_the__inv__into__into,axiom,
! [F2: int > nat,A: set_int,X: nat,B4: set_int] :
( ( inj_on_int_nat @ F2 @ A )
=> ( ( member_nat @ X @ ( image_int_nat @ F2 @ A ) )
=> ( ( ord_less_eq_set_int @ A @ B4 )
=> ( member_int @ ( the_inv_into_int_nat @ A @ F2 @ X ) @ B4 ) ) ) ) ).
% the_inv_into_into
thf(fact_1122_the__inv__into__into,axiom,
! [F2: nat > int,A: set_nat,X: int,B4: set_nat] :
( ( inj_on_nat_int @ F2 @ A )
=> ( ( member_int @ X @ ( image_nat_int @ F2 @ A ) )
=> ( ( ord_less_eq_set_nat @ A @ B4 )
=> ( member_nat @ ( the_inv_into_nat_int @ A @ F2 @ X ) @ B4 ) ) ) ) ).
% the_inv_into_into
thf(fact_1123_the__inv__into__into,axiom,
! [F2: int > int,A: set_int,X: int,B4: set_int] :
( ( inj_on_int_int @ F2 @ A )
=> ( ( member_int @ X @ ( image_int_int @ F2 @ A ) )
=> ( ( ord_less_eq_set_int @ A @ B4 )
=> ( member_int @ ( the_inv_into_int_int @ A @ F2 @ X ) @ B4 ) ) ) ) ).
% the_inv_into_into
thf(fact_1124_the__inv__into__onto,axiom,
! [F2: nat > option_list_o,A: set_nat] :
( ( inj_on1630180835328728801list_o @ F2 @ A )
=> ( ( image_5401538521246155887_o_nat @ ( the_in8802290381976579359list_o @ A @ F2 ) @ ( image_4575287668734308173list_o @ F2 @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_1125_the__inv__into__onto,axiom,
! [F2: int > option_list_o,A: set_int] :
( ( inj_on203032006962319749list_o @ F2 @ A )
=> ( ( image_5399048050737105611_o_int @ ( the_in7375141553610170307list_o @ A @ F2 ) @ ( image_3148138840367899121list_o @ F2 @ A ) )
= A ) ) ).
% the_inv_into_onto
thf(fact_1126_the__inv__into__f__f,axiom,
! [F2: nat > option_list_o,A: set_nat,X: nat] :
( ( inj_on1630180835328728801list_o @ F2 @ A )
=> ( ( member_nat @ X @ A )
=> ( ( the_in8802290381976579359list_o @ A @ F2 @ ( F2 @ X ) )
= X ) ) ) ).
% the_inv_into_f_f
thf(fact_1127_the__inv__into__f__f,axiom,
! [F2: int > option_list_o,A: set_int,X: int] :
( ( inj_on203032006962319749list_o @ F2 @ A )
=> ( ( member_int @ X @ A )
=> ( ( the_in7375141553610170307list_o @ A @ F2 @ ( F2 @ X ) )
= X ) ) ) ).
% the_inv_into_f_f
thf(fact_1128_the__inv__into__f__eq,axiom,
! [F2: nat > option_list_o,A: set_nat,X: nat,Y: option_list_o] :
( ( inj_on1630180835328728801list_o @ F2 @ A )
=> ( ( ( F2 @ X )
= Y )
=> ( ( member_nat @ X @ A )
=> ( ( the_in8802290381976579359list_o @ A @ F2 @ Y )
= X ) ) ) ) ).
% the_inv_into_f_eq
thf(fact_1129_the__inv__into__f__eq,axiom,
! [F2: int > option_list_o,A: set_int,X: int,Y: option_list_o] :
( ( inj_on203032006962319749list_o @ F2 @ A )
=> ( ( ( F2 @ X )
= Y )
=> ( ( member_int @ X @ A )
=> ( ( the_in7375141553610170307list_o @ A @ F2 @ Y )
= X ) ) ) ) ).
% the_inv_into_f_eq
thf(fact_1130_the__inv__f__f,axiom,
! [F2: nat > option_list_o,X: nat] :
( ( inj_on1630180835328728801list_o @ F2 @ top_top_set_nat )
=> ( ( the_in8802290381976579359list_o @ top_top_set_nat @ F2 @ ( F2 @ X ) )
= X ) ) ).
% the_inv_f_f
thf(fact_1131_the__inv__f__f,axiom,
! [F2: int > option_list_o,X: int] :
( ( inj_on203032006962319749list_o @ F2 @ top_top_set_int )
=> ( ( the_in7375141553610170307list_o @ top_top_set_int @ F2 @ ( F2 @ X ) )
= X ) ) ).
% the_inv_f_f
thf(fact_1132_f__the__inv__into__f,axiom,
! [F2: nat > option_list_o,A: set_nat,Y: option_list_o] :
( ( inj_on1630180835328728801list_o @ F2 @ A )
=> ( ( member_option_list_o @ Y @ ( image_4575287668734308173list_o @ F2 @ A ) )
=> ( ( F2 @ ( the_in8802290381976579359list_o @ A @ F2 @ Y ) )
= Y ) ) ) ).
% f_the_inv_into_f
thf(fact_1133_f__the__inv__into__f,axiom,
! [F2: int > option_list_o,A: set_int,Y: option_list_o] :
( ( inj_on203032006962319749list_o @ F2 @ A )
=> ( ( member_option_list_o @ Y @ ( image_3148138840367899121list_o @ F2 @ A ) )
=> ( ( F2 @ ( the_in7375141553610170307list_o @ A @ F2 @ Y ) )
= Y ) ) ) ).
% f_the_inv_into_f
thf(fact_1134_inj__on__the__inv__into,axiom,
! [F2: option_list_o > nat,A: set_option_list_o] :
( ( inj_on2456431687840576515_o_nat @ F2 @ A )
=> ( inj_on1630180835328728801list_o @ ( the_in405169197633651265_o_nat @ A @ F2 ) @ ( image_5401538521246155887_o_nat @ F2 @ A ) ) ) ).
% inj_on_the_inv_into
thf(fact_1135_inj__on__the__inv__into,axiom,
! [F2: option_list_o > int,A: set_option_list_o] :
( ( inj_on2453941217331526239_o_int @ F2 @ A )
=> ( inj_on203032006962319749list_o @ ( the_in402678727124600989_o_int @ A @ F2 ) @ ( image_5399048050737105611_o_int @ F2 @ A ) ) ) ).
% inj_on_the_inv_into
thf(fact_1136_inj__on__the__inv__into,axiom,
! [F2: nat > option_list_o,A: set_nat] :
( ( inj_on1630180835328728801list_o @ F2 @ A )
=> ( inj_on2456431687840576515_o_nat @ ( the_in8802290381976579359list_o @ A @ F2 ) @ ( image_4575287668734308173list_o @ F2 @ A ) ) ) ).
% inj_on_the_inv_into
thf(fact_1137_inj__on__the__inv__into,axiom,
! [F2: int > option_list_o,A: set_int] :
( ( inj_on203032006962319749list_o @ F2 @ A )
=> ( inj_on2453941217331526239_o_int @ ( the_in7375141553610170307list_o @ A @ F2 ) @ ( image_3148138840367899121list_o @ F2 @ A ) ) ) ).
% inj_on_the_inv_into
thf(fact_1138_vimage__eq,axiom,
! [A2: nat,F2: nat > nat,B4: set_nat] :
( ( member_nat @ A2 @ ( vimage_nat_nat @ F2 @ B4 ) )
= ( member_nat @ ( F2 @ A2 ) @ B4 ) ) ).
% vimage_eq
thf(fact_1139_vimage__eq,axiom,
! [A2: nat,F2: nat > int,B4: set_int] :
( ( member_nat @ A2 @ ( vimage_nat_int @ F2 @ B4 ) )
= ( member_int @ ( F2 @ A2 ) @ B4 ) ) ).
% vimage_eq
thf(fact_1140_vimage__eq,axiom,
! [A2: int,F2: int > nat,B4: set_nat] :
( ( member_int @ A2 @ ( vimage_int_nat @ F2 @ B4 ) )
= ( member_nat @ ( F2 @ A2 ) @ B4 ) ) ).
% vimage_eq
thf(fact_1141_vimage__eq,axiom,
! [A2: int,F2: int > int,B4: set_int] :
( ( member_int @ A2 @ ( vimage_int_int @ F2 @ B4 ) )
= ( member_int @ ( F2 @ A2 ) @ B4 ) ) ).
% vimage_eq
thf(fact_1142_vimageI,axiom,
! [F2: nat > nat,A2: nat,B2: nat,B4: set_nat] :
( ( ( F2 @ A2 )
= B2 )
=> ( ( member_nat @ B2 @ B4 )
=> ( member_nat @ A2 @ ( vimage_nat_nat @ F2 @ B4 ) ) ) ) ).
% vimageI
thf(fact_1143_vimageI,axiom,
! [F2: int > nat,A2: int,B2: nat,B4: set_nat] :
( ( ( F2 @ A2 )
= B2 )
=> ( ( member_nat @ B2 @ B4 )
=> ( member_int @ A2 @ ( vimage_int_nat @ F2 @ B4 ) ) ) ) ).
% vimageI
thf(fact_1144_vimageI,axiom,
! [F2: nat > int,A2: nat,B2: int,B4: set_int] :
( ( ( F2 @ A2 )
= B2 )
=> ( ( member_int @ B2 @ B4 )
=> ( member_nat @ A2 @ ( vimage_nat_int @ F2 @ B4 ) ) ) ) ).
% vimageI
thf(fact_1145_vimageI,axiom,
! [F2: int > int,A2: int,B2: int,B4: set_int] :
( ( ( F2 @ A2 )
= B2 )
=> ( ( member_int @ B2 @ B4 )
=> ( member_int @ A2 @ ( vimage_int_int @ F2 @ B4 ) ) ) ) ).
% vimageI
thf(fact_1146_vimageI2,axiom,
! [F2: nat > nat,A2: nat,A: set_nat] :
( ( member_nat @ ( F2 @ A2 ) @ A )
=> ( member_nat @ A2 @ ( vimage_nat_nat @ F2 @ A ) ) ) ).
% vimageI2
thf(fact_1147_vimageI2,axiom,
! [F2: int > nat,A2: int,A: set_nat] :
( ( member_nat @ ( F2 @ A2 ) @ A )
=> ( member_int @ A2 @ ( vimage_int_nat @ F2 @ A ) ) ) ).
% vimageI2
thf(fact_1148_vimageI2,axiom,
! [F2: nat > int,A2: nat,A: set_int] :
( ( member_int @ ( F2 @ A2 ) @ A )
=> ( member_nat @ A2 @ ( vimage_nat_int @ F2 @ A ) ) ) ).
% vimageI2
thf(fact_1149_vimageI2,axiom,
! [F2: int > int,A2: int,A: set_int] :
( ( member_int @ ( F2 @ A2 ) @ A )
=> ( member_int @ A2 @ ( vimage_int_int @ F2 @ A ) ) ) ).
% vimageI2
thf(fact_1150_vimageE,axiom,
! [A2: nat,F2: nat > nat,B4: set_nat] :
( ( member_nat @ A2 @ ( vimage_nat_nat @ F2 @ B4 ) )
=> ( member_nat @ ( F2 @ A2 ) @ B4 ) ) ).
% vimageE
thf(fact_1151_vimageE,axiom,
! [A2: nat,F2: nat > int,B4: set_int] :
( ( member_nat @ A2 @ ( vimage_nat_int @ F2 @ B4 ) )
=> ( member_int @ ( F2 @ A2 ) @ B4 ) ) ).
% vimageE
thf(fact_1152_vimageE,axiom,
! [A2: int,F2: int > nat,B4: set_nat] :
( ( member_int @ A2 @ ( vimage_int_nat @ F2 @ B4 ) )
=> ( member_nat @ ( F2 @ A2 ) @ B4 ) ) ).
% vimageE
thf(fact_1153_vimageE,axiom,
! [A2: int,F2: int > int,B4: set_int] :
( ( member_int @ A2 @ ( vimage_int_int @ F2 @ B4 ) )
=> ( member_int @ ( F2 @ A2 ) @ B4 ) ) ).
% vimageE
thf(fact_1154_vimageD,axiom,
! [A2: nat,F2: nat > nat,A: set_nat] :
( ( member_nat @ A2 @ ( vimage_nat_nat @ F2 @ A ) )
=> ( member_nat @ ( F2 @ A2 ) @ A ) ) ).
% vimageD
thf(fact_1155_vimageD,axiom,
! [A2: nat,F2: nat > int,A: set_int] :
( ( member_nat @ A2 @ ( vimage_nat_int @ F2 @ A ) )
=> ( member_int @ ( F2 @ A2 ) @ A ) ) ).
% vimageD
thf(fact_1156_vimageD,axiom,
! [A2: int,F2: int > nat,A: set_nat] :
( ( member_int @ A2 @ ( vimage_int_nat @ F2 @ A ) )
=> ( member_nat @ ( F2 @ A2 ) @ A ) ) ).
% vimageD
thf(fact_1157_vimageD,axiom,
! [A2: int,F2: int > int,A: set_int] :
( ( member_int @ A2 @ ( vimage_int_int @ F2 @ A ) )
=> ( member_int @ ( F2 @ A2 ) @ A ) ) ).
% vimageD
thf(fact_1158_inj__vimage__image__eq,axiom,
! [F2: nat > option_list_o,A: set_nat] :
( ( inj_on1630180835328728801list_o @ F2 @ top_top_set_nat )
=> ( ( vimage323003453082605847list_o @ F2 @ ( image_4575287668734308173list_o @ F2 @ A ) )
= A ) ) ).
% inj_vimage_image_eq
thf(fact_1159_inj__vimage__image__eq,axiom,
! [F2: int > option_list_o,A: set_int] :
( ( inj_on203032006962319749list_o @ F2 @ top_top_set_int )
=> ( ( vimage8119226661570972603list_o @ F2 @ ( image_3148138840367899121list_o @ F2 @ A ) )
= A ) ) ).
% inj_vimage_image_eq
thf(fact_1160_finite__vimageI,axiom,
! [F3: set_option_list_o,H: nat > option_list_o] :
( ( finite7007496012504252301list_o @ F3 )
=> ( ( inj_on1630180835328728801list_o @ H @ top_top_set_nat )
=> ( finite_finite_nat @ ( vimage323003453082605847list_o @ H @ F3 ) ) ) ) ).
% finite_vimageI
thf(fact_1161_finite__vimageI,axiom,
! [F3: set_option_list_o,H: int > option_list_o] :
( ( finite7007496012504252301list_o @ F3 )
=> ( ( inj_on203032006962319749list_o @ H @ top_top_set_int )
=> ( finite_finite_int @ ( vimage8119226661570972603list_o @ H @ F3 ) ) ) ) ).
% finite_vimageI
thf(fact_1162_finite__vimage__IntI,axiom,
! [F3: set_option_list_o,H: nat > option_list_o,A: set_nat] :
( ( finite7007496012504252301list_o @ F3 )
=> ( ( inj_on1630180835328728801list_o @ H @ A )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ ( vimage323003453082605847list_o @ H @ F3 ) @ A ) ) ) ) ).
% finite_vimage_IntI
thf(fact_1163_finite__vimage__IntI,axiom,
! [F3: set_option_list_o,H: int > option_list_o,A: set_int] :
( ( finite7007496012504252301list_o @ F3 )
=> ( ( inj_on203032006962319749list_o @ H @ A )
=> ( finite_finite_int @ ( inf_inf_set_int @ ( vimage8119226661570972603list_o @ H @ F3 ) @ A ) ) ) ) ).
% finite_vimage_IntI
thf(fact_1164_vimage__subsetI,axiom,
! [F2: nat > option_list_o,B4: set_option_list_o,A: set_nat] :
( ( inj_on1630180835328728801list_o @ F2 @ top_top_set_nat )
=> ( ( ord_le1162937763994921316list_o @ B4 @ ( image_4575287668734308173list_o @ F2 @ A ) )
=> ( ord_less_eq_set_nat @ ( vimage323003453082605847list_o @ F2 @ B4 ) @ A ) ) ) ).
% vimage_subsetI
thf(fact_1165_vimage__subsetI,axiom,
! [F2: int > option_list_o,B4: set_option_list_o,A: set_int] :
( ( inj_on203032006962319749list_o @ F2 @ top_top_set_int )
=> ( ( ord_le1162937763994921316list_o @ B4 @ ( image_3148138840367899121list_o @ F2 @ A ) )
=> ( ord_less_eq_set_int @ ( vimage8119226661570972603list_o @ F2 @ B4 ) @ A ) ) ) ).
% vimage_subsetI
thf(fact_1166_card__vimage__inj,axiom,
! [F2: nat > option_list_o,A: set_option_list_o] :
( ( inj_on1630180835328728801list_o @ F2 @ top_top_set_nat )
=> ( ( ord_le1162937763994921316list_o @ A @ ( image_4575287668734308173list_o @ F2 @ top_top_set_nat ) )
=> ( ( finite_card_nat @ ( vimage323003453082605847list_o @ F2 @ A ) )
= ( finite3362998479529755404list_o @ A ) ) ) ) ).
% card_vimage_inj
thf(fact_1167_card__vimage__inj,axiom,
! [F2: int > option_list_o,A: set_option_list_o] :
( ( inj_on203032006962319749list_o @ F2 @ top_top_set_int )
=> ( ( ord_le1162937763994921316list_o @ A @ ( image_3148138840367899121list_o @ F2 @ top_top_set_int ) )
=> ( ( finite_card_int @ ( vimage8119226661570972603list_o @ F2 @ A ) )
= ( finite3362998479529755404list_o @ A ) ) ) ) ).
% card_vimage_inj
thf(fact_1168_trans__singleton,axiom,
! [A2: option_list_o] : ( trans_4700657668258775172list_o @ top_to633166595683317524list_o @ ( insert7989621605229100023list_o @ ( produc5745850523778858007list_o @ A2 @ A2 ) @ bot_bo6949037146550090099list_o ) ) ).
% trans_singleton
thf(fact_1169_trans__singleton,axiom,
! [A2: nat] : ( trans_on_nat @ top_top_set_nat @ ( insert8211810215607154385at_nat @ ( product_Pair_nat_nat @ A2 @ A2 ) @ bot_bo2099793752762293965at_nat ) ) ).
% trans_singleton
thf(fact_1170_card__image,axiom,
! [F2: nat > option_list_o,A: set_nat] :
( ( inj_on1630180835328728801list_o @ F2 @ A )
=> ( ( finite3362998479529755404list_o @ ( image_4575287668734308173list_o @ F2 @ A ) )
= ( finite_card_nat @ A ) ) ) ).
% card_image
thf(fact_1171_card__image,axiom,
! [F2: int > option_list_o,A: set_int] :
( ( inj_on203032006962319749list_o @ F2 @ A )
=> ( ( finite3362998479529755404list_o @ ( image_3148138840367899121list_o @ F2 @ A ) )
= ( finite_card_int @ A ) ) ) ).
% card_image
thf(fact_1172_inj__on__iff__eq__card,axiom,
! [A: set_nat,F2: nat > option_list_o] :
( ( finite_finite_nat @ A )
=> ( ( inj_on1630180835328728801list_o @ F2 @ A )
= ( ( finite3362998479529755404list_o @ ( image_4575287668734308173list_o @ F2 @ A ) )
= ( finite_card_nat @ A ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_1173_inj__on__iff__eq__card,axiom,
! [A: set_int,F2: int > option_list_o] :
( ( finite_finite_int @ A )
=> ( ( inj_on203032006962319749list_o @ F2 @ A )
= ( ( finite3362998479529755404list_o @ ( image_3148138840367899121list_o @ F2 @ A ) )
= ( finite_card_int @ A ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_1174_eq__card__imp__inj__on,axiom,
! [A: set_nat,F2: nat > option_list_o] :
( ( finite_finite_nat @ A )
=> ( ( ( finite3362998479529755404list_o @ ( image_4575287668734308173list_o @ F2 @ A ) )
= ( finite_card_nat @ A ) )
=> ( inj_on1630180835328728801list_o @ F2 @ A ) ) ) ).
% eq_card_imp_inj_on
thf(fact_1175_eq__card__imp__inj__on,axiom,
! [A: set_int,F2: int > option_list_o] :
( ( finite_finite_int @ A )
=> ( ( ( finite3362998479529755404list_o @ ( image_3148138840367899121list_o @ F2 @ A ) )
= ( finite_card_int @ A ) )
=> ( inj_on203032006962319749list_o @ F2 @ A ) ) ) ).
% eq_card_imp_inj_on
thf(fact_1176_card__le__inj,axiom,
! [A: set_nat,B4: set_option_list_o] :
( ( finite_finite_nat @ A )
=> ( ( finite7007496012504252301list_o @ B4 )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite3362998479529755404list_o @ B4 ) )
=> ? [F5: nat > option_list_o] :
( ( ord_le1162937763994921316list_o @ ( image_4575287668734308173list_o @ F5 @ A ) @ B4 )
& ( inj_on1630180835328728801list_o @ F5 @ A ) ) ) ) ) ).
% card_le_inj
thf(fact_1177_card__le__inj,axiom,
! [A: set_int,B4: set_option_list_o] :
( ( finite_finite_int @ A )
=> ( ( finite7007496012504252301list_o @ B4 )
=> ( ( ord_less_eq_nat @ ( finite_card_int @ A ) @ ( finite3362998479529755404list_o @ B4 ) )
=> ? [F5: int > option_list_o] :
( ( ord_le1162937763994921316list_o @ ( image_3148138840367899121list_o @ F5 @ A ) @ B4 )
& ( inj_on203032006962319749list_o @ F5 @ A ) ) ) ) ) ).
% card_le_inj
thf(fact_1178_card__inj__on__le,axiom,
! [F2: nat > option_list_o,A: set_nat,B4: set_option_list_o] :
( ( inj_on1630180835328728801list_o @ F2 @ A )
=> ( ( ord_le1162937763994921316list_o @ ( image_4575287668734308173list_o @ F2 @ A ) @ B4 )
=> ( ( finite7007496012504252301list_o @ B4 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite3362998479529755404list_o @ B4 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_1179_card__inj__on__le,axiom,
! [F2: int > option_list_o,A: set_int,B4: set_option_list_o] :
( ( inj_on203032006962319749list_o @ F2 @ A )
=> ( ( ord_le1162937763994921316list_o @ ( image_3148138840367899121list_o @ F2 @ A ) @ B4 )
=> ( ( finite7007496012504252301list_o @ B4 )
=> ( ord_less_eq_nat @ ( finite_card_int @ A ) @ ( finite3362998479529755404list_o @ B4 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_1180_inj__on__iff__card__le,axiom,
! [A: set_nat,B4: set_option_list_o] :
( ( finite_finite_nat @ A )
=> ( ( finite7007496012504252301list_o @ B4 )
=> ( ( ? [F: nat > option_list_o] :
( ( inj_on1630180835328728801list_o @ F @ A )
& ( ord_le1162937763994921316list_o @ ( image_4575287668734308173list_o @ F @ A ) @ B4 ) ) )
= ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite3362998479529755404list_o @ B4 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_1181_inj__on__iff__card__le,axiom,
! [A: set_int,B4: set_option_list_o] :
( ( finite_finite_int @ A )
=> ( ( finite7007496012504252301list_o @ B4 )
=> ( ( ? [F: int > option_list_o] :
( ( inj_on203032006962319749list_o @ F @ A )
& ( ord_le1162937763994921316list_o @ ( image_3148138840367899121list_o @ F @ A ) @ B4 ) ) )
= ( ord_less_eq_nat @ ( finite_card_int @ A ) @ ( finite3362998479529755404list_o @ B4 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_1182_surjective__iff__injective__gen,axiom,
! [S: set_nat,T3: set_option_list_o,F2: nat > option_list_o] :
( ( finite_finite_nat @ S )
=> ( ( finite7007496012504252301list_o @ T3 )
=> ( ( ( finite_card_nat @ S )
= ( finite3362998479529755404list_o @ T3 ) )
=> ( ( ord_le1162937763994921316list_o @ ( image_4575287668734308173list_o @ F2 @ S ) @ T3 )
=> ( ( ! [X3: option_list_o] :
( ( member_option_list_o @ X3 @ T3 )
=> ? [Y3: nat] :
( ( member_nat @ Y3 @ S )
& ( ( F2 @ Y3 )
= X3 ) ) ) )
= ( inj_on1630180835328728801list_o @ F2 @ S ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_1183_surjective__iff__injective__gen,axiom,
! [S: set_int,T3: set_option_list_o,F2: int > option_list_o] :
( ( finite_finite_int @ S )
=> ( ( finite7007496012504252301list_o @ T3 )
=> ( ( ( finite_card_int @ S )
= ( finite3362998479529755404list_o @ T3 ) )
=> ( ( ord_le1162937763994921316list_o @ ( image_3148138840367899121list_o @ F2 @ S ) @ T3 )
=> ( ( ! [X3: option_list_o] :
( ( member_option_list_o @ X3 @ T3 )
=> ? [Y3: int] :
( ( member_int @ Y3 @ S )
& ( ( F2 @ Y3 )
= X3 ) ) ) )
= ( inj_on203032006962319749list_o @ F2 @ S ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_1184_card__bij__eq,axiom,
! [F2: option_list_o > nat,A: set_option_list_o,B4: set_nat,G: nat > option_list_o] :
( ( inj_on2456431687840576515_o_nat @ F2 @ A )
=> ( ( ord_less_eq_set_nat @ ( image_5401538521246155887_o_nat @ F2 @ A ) @ B4 )
=> ( ( inj_on1630180835328728801list_o @ G @ B4 )
=> ( ( ord_le1162937763994921316list_o @ ( image_4575287668734308173list_o @ G @ B4 ) @ A )
=> ( ( finite7007496012504252301list_o @ A )
=> ( ( finite_finite_nat @ B4 )
=> ( ( finite3362998479529755404list_o @ A )
= ( finite_card_nat @ B4 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_1185_card__bij__eq,axiom,
! [F2: option_list_o > int,A: set_option_list_o,B4: set_int,G: int > option_list_o] :
( ( inj_on2453941217331526239_o_int @ F2 @ A )
=> ( ( ord_less_eq_set_int @ ( image_5399048050737105611_o_int @ F2 @ A ) @ B4 )
=> ( ( inj_on203032006962319749list_o @ G @ B4 )
=> ( ( ord_le1162937763994921316list_o @ ( image_3148138840367899121list_o @ G @ B4 ) @ A )
=> ( ( finite7007496012504252301list_o @ A )
=> ( ( finite_finite_int @ B4 )
=> ( ( finite3362998479529755404list_o @ A )
= ( finite_card_int @ B4 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_1186_card__bij__eq,axiom,
! [F2: nat > option_list_o,A: set_nat,B4: set_option_list_o,G: option_list_o > nat] :
( ( inj_on1630180835328728801list_o @ F2 @ A )
=> ( ( ord_le1162937763994921316list_o @ ( image_4575287668734308173list_o @ F2 @ A ) @ B4 )
=> ( ( inj_on2456431687840576515_o_nat @ G @ B4 )
=> ( ( ord_less_eq_set_nat @ ( image_5401538521246155887_o_nat @ G @ B4 ) @ A )
=> ( ( finite_finite_nat @ A )
=> ( ( finite7007496012504252301list_o @ B4 )
=> ( ( finite_card_nat @ A )
= ( finite3362998479529755404list_o @ B4 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_1187_card__bij__eq,axiom,
! [F2: int > option_list_o,A: set_int,B4: set_option_list_o,G: option_list_o > int] :
( ( inj_on203032006962319749list_o @ F2 @ A )
=> ( ( ord_le1162937763994921316list_o @ ( image_3148138840367899121list_o @ F2 @ A ) @ B4 )
=> ( ( inj_on2453941217331526239_o_int @ G @ B4 )
=> ( ( ord_less_eq_set_int @ ( image_5399048050737105611_o_int @ G @ B4 ) @ A )
=> ( ( finite_finite_int @ A )
=> ( ( finite7007496012504252301list_o @ B4 )
=> ( ( finite_card_int @ A )
= ( finite3362998479529755404list_o @ B4 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_1188_card__vimage__inj__on__le,axiom,
! [F2: nat > option_list_o,D2: set_nat,A: set_option_list_o] :
( ( inj_on1630180835328728801list_o @ F2 @ D2 )
=> ( ( finite7007496012504252301list_o @ A )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( inf_inf_set_nat @ ( vimage323003453082605847list_o @ F2 @ A ) @ D2 ) ) @ ( finite3362998479529755404list_o @ A ) ) ) ) ).
% card_vimage_inj_on_le
thf(fact_1189_card__vimage__inj__on__le,axiom,
! [F2: int > option_list_o,D2: set_int,A: set_option_list_o] :
( ( inj_on203032006962319749list_o @ F2 @ D2 )
=> ( ( finite7007496012504252301list_o @ A )
=> ( ord_less_eq_nat @ ( finite_card_int @ ( inf_inf_set_int @ ( vimage8119226661570972603list_o @ F2 @ A ) @ D2 ) ) @ ( finite3362998479529755404list_o @ A ) ) ) ) ).
% card_vimage_inj_on_le
thf(fact_1190_card__Diff1__less__iff,axiom,
! [A: set_nat,X: nat] :
( ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A ) )
= ( ( finite_finite_nat @ A )
& ( member_nat @ X @ A ) ) ) ).
% card_Diff1_less_iff
thf(fact_1191_card__Diff1__less__iff,axiom,
! [A: set_int,X: int] :
( ( ord_less_nat @ ( finite_card_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) @ ( finite_card_int @ A ) )
= ( ( finite_finite_int @ A )
& ( member_int @ X @ A ) ) ) ).
% card_Diff1_less_iff
thf(fact_1192_card__Diff2__less,axiom,
! [A: set_nat,X: nat,Y: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ X @ A )
=> ( ( member_nat @ Y @ A )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ ( insert_nat @ Y @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A ) ) ) ) ) ).
% card_Diff2_less
thf(fact_1193_card__Diff2__less,axiom,
! [A: set_int,X: int,Y: int] :
( ( finite_finite_int @ A )
=> ( ( member_int @ X @ A )
=> ( ( member_int @ Y @ A )
=> ( ord_less_nat @ ( finite_card_int @ ( minus_minus_set_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) @ ( insert_int @ Y @ bot_bot_set_int ) ) ) @ ( finite_card_int @ A ) ) ) ) ) ).
% card_Diff2_less
thf(fact_1194_linorder__injI,axiom,
! [F2: nat > option_list_o] :
( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ( F2 @ X2 )
!= ( F2 @ Y2 ) ) )
=> ( inj_on1630180835328728801list_o @ F2 @ top_top_set_nat ) ) ).
% linorder_injI
thf(fact_1195_linorder__injI,axiom,
! [F2: int > option_list_o] :
( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ( F2 @ X2 )
!= ( F2 @ Y2 ) ) )
=> ( inj_on203032006962319749list_o @ F2 @ top_top_set_int ) ) ).
% linorder_injI
thf(fact_1196_linorder__inj__onI_H,axiom,
! [A: set_nat,F2: nat > option_list_o] :
( ! [I: nat,J: nat] :
( ( member_nat @ I @ A )
=> ( ( member_nat @ J @ A )
=> ( ( ord_less_nat @ I @ J )
=> ( ( F2 @ I )
!= ( F2 @ J ) ) ) ) )
=> ( inj_on1630180835328728801list_o @ F2 @ A ) ) ).
% linorder_inj_onI'
thf(fact_1197_linorder__inj__onI_H,axiom,
! [A: set_int,F2: int > option_list_o] :
( ! [I: int,J: int] :
( ( member_int @ I @ A )
=> ( ( member_int @ J @ A )
=> ( ( ord_less_int @ I @ J )
=> ( ( F2 @ I )
!= ( F2 @ J ) ) ) ) )
=> ( inj_on203032006962319749list_o @ F2 @ A ) ) ).
% linorder_inj_onI'
thf(fact_1198_linorder__inj__onI,axiom,
! [A: set_nat,F2: nat > option_list_o] :
( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ( member_nat @ X2 @ A )
=> ( ( member_nat @ Y2 @ A )
=> ( ( F2 @ X2 )
!= ( F2 @ Y2 ) ) ) ) )
=> ( ! [X2: nat,Y2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( member_nat @ Y2 @ A )
=> ( ( ord_less_eq_nat @ X2 @ Y2 )
| ( ord_less_eq_nat @ Y2 @ X2 ) ) ) )
=> ( inj_on1630180835328728801list_o @ F2 @ A ) ) ) ).
% linorder_inj_onI
thf(fact_1199_linorder__inj__onI,axiom,
! [A: set_int,F2: int > option_list_o] :
( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ( member_int @ X2 @ A )
=> ( ( member_int @ Y2 @ A )
=> ( ( F2 @ X2 )
!= ( F2 @ Y2 ) ) ) ) )
=> ( ! [X2: int,Y2: int] :
( ( member_int @ X2 @ A )
=> ( ( member_int @ Y2 @ A )
=> ( ( ord_less_eq_int @ X2 @ Y2 )
| ( ord_less_eq_int @ Y2 @ X2 ) ) ) )
=> ( inj_on203032006962319749list_o @ F2 @ A ) ) ) ).
% linorder_inj_onI
thf(fact_1200_infinite__growing,axiom,
! [X7: set_nat] :
( ( X7 != bot_bot_set_nat )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ X7 )
=> ? [Xa3: nat] :
( ( member_nat @ Xa3 @ X7 )
& ( ord_less_nat @ X2 @ Xa3 ) ) )
=> ~ ( finite_finite_nat @ X7 ) ) ) ).
% infinite_growing
thf(fact_1201_infinite__growing,axiom,
! [X7: set_int] :
( ( X7 != bot_bot_set_int )
=> ( ! [X2: int] :
( ( member_int @ X2 @ X7 )
=> ? [Xa3: int] :
( ( member_int @ Xa3 @ X7 )
& ( ord_less_int @ X2 @ Xa3 ) ) )
=> ~ ( finite_finite_int @ X7 ) ) ) ).
% infinite_growing
thf(fact_1202_pigeonhole,axiom,
! [F2: nat > option_list_o,A: set_nat] :
( ( ord_less_nat @ ( finite3362998479529755404list_o @ ( image_4575287668734308173list_o @ F2 @ A ) ) @ ( finite_card_nat @ A ) )
=> ~ ( inj_on1630180835328728801list_o @ F2 @ A ) ) ).
% pigeonhole
thf(fact_1203_pigeonhole,axiom,
! [F2: int > option_list_o,A: set_int] :
( ( ord_less_nat @ ( finite3362998479529755404list_o @ ( image_3148138840367899121list_o @ F2 @ A ) ) @ ( finite_card_int @ A ) )
=> ~ ( inj_on203032006962319749list_o @ F2 @ A ) ) ).
% pigeonhole
thf(fact_1204_card__Diff1__less,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ X @ A )
=> ( ord_less_nat @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) @ ( finite_card_nat @ A ) ) ) ) ).
% card_Diff1_less
thf(fact_1205_card__Diff1__less,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ( member_int @ X @ A )
=> ( ord_less_nat @ ( finite_card_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) @ ( finite_card_int @ A ) ) ) ) ).
% card_Diff1_less
thf(fact_1206_card_Oremove,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ X @ A )
=> ( ( finite_card_nat @ A )
= ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ).
% card.remove
thf(fact_1207_card_Oremove,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ( member_int @ X @ A )
=> ( ( finite_card_int @ A )
= ( suc @ ( finite_card_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ).
% card.remove
thf(fact_1208_prefix__order_Olift__Suc__mono__le,axiom,
! [F2: nat > list_o,N2: nat,N3: nat] :
( ! [N4: nat] : ( prefix_o @ ( F2 @ N4 ) @ ( F2 @ ( suc @ N4 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N3 )
=> ( prefix_o @ ( F2 @ N2 ) @ ( F2 @ N3 ) ) ) ) ).
% prefix_order.lift_Suc_mono_le
thf(fact_1209_prefix__order_Olift__Suc__antimono__le,axiom,
! [F2: nat > list_o,N2: nat,N3: nat] :
( ! [N4: nat] : ( prefix_o @ ( F2 @ ( suc @ N4 ) ) @ ( F2 @ N4 ) )
=> ( ( ord_less_eq_nat @ N2 @ N3 )
=> ( prefix_o @ ( F2 @ N3 ) @ ( F2 @ N2 ) ) ) ) ).
% prefix_order.lift_Suc_antimono_le
thf(fact_1210_psubset__imp__ex__mem,axiom,
! [A: set_nat,B4: set_nat] :
( ( ord_less_set_nat @ A @ B4 )
=> ? [B: nat] : ( member_nat @ B @ ( minus_minus_set_nat @ B4 @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1211_psubset__imp__ex__mem,axiom,
! [A: set_int,B4: set_int] :
( ( ord_less_set_int @ A @ B4 )
=> ? [B: int] : ( member_int @ B @ ( minus_minus_set_int @ B4 @ A ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1212_image__strict__mono,axiom,
! [F2: nat > option_list_o,B4: set_nat,A: set_nat] :
( ( inj_on1630180835328728801list_o @ F2 @ B4 )
=> ( ( ord_less_set_nat @ A @ B4 )
=> ( ord_le4476516537835661936list_o @ ( image_4575287668734308173list_o @ F2 @ A ) @ ( image_4575287668734308173list_o @ F2 @ B4 ) ) ) ) ).
% image_strict_mono
thf(fact_1213_image__strict__mono,axiom,
! [F2: int > option_list_o,B4: set_int,A: set_int] :
( ( inj_on203032006962319749list_o @ F2 @ B4 )
=> ( ( ord_less_set_int @ A @ B4 )
=> ( ord_le4476516537835661936list_o @ ( image_3148138840367899121list_o @ F2 @ A ) @ ( image_3148138840367899121list_o @ F2 @ B4 ) ) ) ) ).
% image_strict_mono
thf(fact_1214_psubset__insert__iff,axiom,
! [A: set_nat,X: nat,B4: set_nat] :
( ( ord_less_set_nat @ A @ ( insert_nat @ X @ B4 ) )
= ( ( ( member_nat @ X @ B4 )
=> ( ord_less_set_nat @ A @ B4 ) )
& ( ~ ( member_nat @ X @ B4 )
=> ( ( ( member_nat @ X @ A )
=> ( ord_less_set_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) @ B4 ) )
& ( ~ ( member_nat @ X @ A )
=> ( ord_less_eq_set_nat @ A @ B4 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1215_psubset__insert__iff,axiom,
! [A: set_int,X: int,B4: set_int] :
( ( ord_less_set_int @ A @ ( insert_int @ X @ B4 ) )
= ( ( ( member_int @ X @ B4 )
=> ( ord_less_set_int @ A @ B4 ) )
& ( ~ ( member_int @ X @ B4 )
=> ( ( ( member_int @ X @ A )
=> ( ord_less_set_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) @ B4 ) )
& ( ~ ( member_int @ X @ A )
=> ( ord_less_eq_set_int @ A @ B4 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_1216_card__Suc__Diff1,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ X @ A )
=> ( ( suc @ ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) )
= ( finite_card_nat @ A ) ) ) ) ).
% card_Suc_Diff1
thf(fact_1217_card__Suc__Diff1,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ( member_int @ X @ A )
=> ( ( suc @ ( finite_card_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) )
= ( finite_card_int @ A ) ) ) ) ).
% card_Suc_Diff1
thf(fact_1218_N_092_060_094sub_062e__def,axiom,
( prefix_Free_Code_N_e
= ( ^ [X3: nat] : ( prefix1649127329469935890e_Ng_e @ ( plus_plus_nat @ X3 @ one_one_nat ) ) ) ) ).
% N\<^sub>e_def
thf(fact_1219_disjnt__insert1,axiom,
! [A2: nat,X7: set_nat,Y7: set_nat] :
( ( disjnt_nat @ ( insert_nat @ A2 @ X7 ) @ Y7 )
= ( ~ ( member_nat @ A2 @ Y7 )
& ( disjnt_nat @ X7 @ Y7 ) ) ) ).
% disjnt_insert1
thf(fact_1220_disjnt__insert1,axiom,
! [A2: int,X7: set_int,Y7: set_int] :
( ( disjnt_int @ ( insert_int @ A2 @ X7 ) @ Y7 )
= ( ~ ( member_int @ A2 @ Y7 )
& ( disjnt_int @ X7 @ Y7 ) ) ) ).
% disjnt_insert1
thf(fact_1221_disjnt__insert2,axiom,
! [Y7: set_nat,A2: nat,X7: set_nat] :
( ( disjnt_nat @ Y7 @ ( insert_nat @ A2 @ X7 ) )
= ( ~ ( member_nat @ A2 @ Y7 )
& ( disjnt_nat @ Y7 @ X7 ) ) ) ).
% disjnt_insert2
thf(fact_1222_disjnt__insert2,axiom,
! [Y7: set_int,A2: int,X7: set_int] :
( ( disjnt_int @ Y7 @ ( insert_int @ A2 @ X7 ) )
= ( ~ ( member_int @ A2 @ Y7 )
& ( disjnt_int @ Y7 @ X7 ) ) ) ).
% disjnt_insert2
thf(fact_1223_psubsetD,axiom,
! [A: set_nat,B4: set_nat,C: nat] :
( ( ord_less_set_nat @ A @ B4 )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B4 ) ) ) ).
% psubsetD
thf(fact_1224_psubsetD,axiom,
! [A: set_int,B4: set_int,C: int] :
( ( ord_less_set_int @ A @ B4 )
=> ( ( member_int @ C @ A )
=> ( member_int @ C @ B4 ) ) ) ).
% psubsetD
thf(fact_1225_disjnt__iff,axiom,
( disjnt_nat
= ( ^ [A5: set_nat,B5: set_nat] :
! [X3: nat] :
~ ( ( member_nat @ X3 @ A5 )
& ( member_nat @ X3 @ B5 ) ) ) ) ).
% disjnt_iff
thf(fact_1226_disjnt__iff,axiom,
( disjnt_int
= ( ^ [A5: set_int,B5: set_int] :
! [X3: int] :
~ ( ( member_int @ X3 @ A5 )
& ( member_int @ X3 @ B5 ) ) ) ) ).
% disjnt_iff
thf(fact_1227_disjnt__insert,axiom,
! [X: nat,N5: set_nat,M5: set_nat] :
( ~ ( member_nat @ X @ N5 )
=> ( ( disjnt_nat @ M5 @ N5 )
=> ( disjnt_nat @ ( insert_nat @ X @ M5 ) @ N5 ) ) ) ).
% disjnt_insert
thf(fact_1228_disjnt__insert,axiom,
! [X: int,N5: set_int,M5: set_int] :
( ~ ( member_int @ X @ N5 )
=> ( ( disjnt_int @ M5 @ N5 )
=> ( disjnt_int @ ( insert_int @ X @ M5 ) @ N5 ) ) ) ).
% disjnt_insert
thf(fact_1229_card__Diff__singleton,axiom,
! [X: nat,A: set_nat] :
( ( member_nat @ X @ A )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ A ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_1230_card__Diff__singleton,axiom,
! [X: int,A: set_int] :
( ( member_int @ X @ A )
=> ( ( finite_card_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) )
= ( minus_minus_nat @ ( finite_card_int @ A ) @ one_one_nat ) ) ) ).
% card_Diff_singleton
thf(fact_1231_card__Diff__singleton__if,axiom,
! [X: nat,A: set_nat] :
( ( ( member_nat @ X @ A )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
= ( minus_minus_nat @ ( finite_card_nat @ A ) @ one_one_nat ) ) )
& ( ~ ( member_nat @ X @ A )
=> ( ( finite_card_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) )
= ( finite_card_nat @ A ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_1232_card__Diff__singleton__if,axiom,
! [X: int,A: set_int] :
( ( ( member_int @ X @ A )
=> ( ( finite_card_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) )
= ( minus_minus_nat @ ( finite_card_int @ A ) @ one_one_nat ) ) )
& ( ~ ( member_int @ X @ A )
=> ( ( finite_card_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) )
= ( finite_card_int @ A ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_1233_card__Suc__eq,axiom,
! [A: set_nat,K: nat] :
( ( ( finite_card_nat @ A )
= ( suc @ K ) )
= ( ? [B3: nat,B5: set_nat] :
( ( A
= ( insert_nat @ B3 @ B5 ) )
& ~ ( member_nat @ B3 @ B5 )
& ( ( finite_card_nat @ B5 )
= K )
& ( ( K = zero_zero_nat )
=> ( B5 = bot_bot_set_nat ) ) ) ) ) ).
% card_Suc_eq
thf(fact_1234_card__Suc__eq,axiom,
! [A: set_int,K: nat] :
( ( ( finite_card_int @ A )
= ( suc @ K ) )
= ( ? [B3: int,B5: set_int] :
( ( A
= ( insert_int @ B3 @ B5 ) )
& ~ ( member_int @ B3 @ B5 )
& ( ( finite_card_int @ B5 )
= K )
& ( ( K = zero_zero_nat )
=> ( B5 = bot_bot_set_int ) ) ) ) ) ).
% card_Suc_eq
thf(fact_1235_card__eq__SucD,axiom,
! [A: set_nat,K: nat] :
( ( ( finite_card_nat @ A )
= ( suc @ K ) )
=> ? [B: nat,B6: set_nat] :
( ( A
= ( insert_nat @ B @ B6 ) )
& ~ ( member_nat @ B @ B6 )
& ( ( finite_card_nat @ B6 )
= K )
& ( ( K = zero_zero_nat )
=> ( B6 = bot_bot_set_nat ) ) ) ) ).
% card_eq_SucD
thf(fact_1236_card__eq__SucD,axiom,
! [A: set_int,K: nat] :
( ( ( finite_card_int @ A )
= ( suc @ K ) )
=> ? [B: int,B6: set_int] :
( ( A
= ( insert_int @ B @ B6 ) )
& ~ ( member_int @ B @ B6 )
& ( ( finite_card_int @ B6 )
= K )
& ( ( K = zero_zero_nat )
=> ( B6 = bot_bot_set_int ) ) ) ) ).
% card_eq_SucD
thf(fact_1237_image__add__0,axiom,
! [S: set_nat] :
( ( image_nat_nat @ ( plus_plus_nat @ zero_zero_nat ) @ S )
= S ) ).
% image_add_0
thf(fact_1238_encode__bounded__nat_Ocases,axiom,
! [X: product_prod_nat_nat] :
( ! [L3: nat,N4: nat] :
( X
!= ( product_Pair_nat_nat @ ( suc @ L3 ) @ N4 ) )
=> ~ ! [Uu2: nat] :
( X
!= ( product_Pair_nat_nat @ zero_zero_nat @ Uu2 ) ) ) ).
% encode_bounded_nat.cases
thf(fact_1239_encode__unary__nat_Ocases,axiom,
! [X: nat] :
( ! [L3: nat] :
( X
!= ( suc @ L3 ) )
=> ( X = zero_zero_nat ) ) ).
% encode_unary_nat.cases
thf(fact_1240_option_Osize__gen_I2_J,axiom,
! [X: list_o > nat,X22: list_o] :
( ( size_option_list_o @ X @ ( some_list_o @ X22 ) )
= ( plus_plus_nat @ ( X @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% option.size_gen(2)
thf(fact_1241_option_Osize__gen_I1_J,axiom,
! [X: list_o > nat] :
( ( size_option_list_o @ X @ none_list_o )
= ( suc @ zero_zero_nat ) ) ).
% option.size_gen(1)
thf(fact_1242_option_Osize_I3_J,axiom,
( ( size_s9086679804311813456list_o @ none_list_o )
= ( suc @ zero_zero_nat ) ) ).
% option.size(3)
thf(fact_1243_option_Osize_I4_J,axiom,
! [X22: list_o] :
( ( size_s9086679804311813456list_o @ ( some_list_o @ X22 ) )
= ( suc @ zero_zero_nat ) ) ).
% option.size(4)
thf(fact_1244_pairwise__insert,axiom,
! [R: nat > nat > $o,X: nat,S2: set_nat] :
( ( pairwise_nat @ R @ ( insert_nat @ X @ S2 ) )
= ( ! [Y3: nat] :
( ( ( member_nat @ Y3 @ S2 )
& ( Y3 != X ) )
=> ( ( R @ X @ Y3 )
& ( R @ Y3 @ X ) ) )
& ( pairwise_nat @ R @ S2 ) ) ) ).
% pairwise_insert
thf(fact_1245_pairwise__insert,axiom,
! [R: int > int > $o,X: int,S2: set_int] :
( ( pairwise_int @ R @ ( insert_int @ X @ S2 ) )
= ( ! [Y3: int] :
( ( ( member_int @ Y3 @ S2 )
& ( Y3 != X ) )
=> ( ( R @ X @ Y3 )
& ( R @ Y3 @ X ) ) )
& ( pairwise_int @ R @ S2 ) ) ) ).
% pairwise_insert
thf(fact_1246_pairwiseI,axiom,
! [S: set_nat,R2: nat > nat > $o] :
( ! [X2: nat,Y2: nat] :
( ( member_nat @ X2 @ S )
=> ( ( member_nat @ Y2 @ S )
=> ( ( X2 != Y2 )
=> ( R2 @ X2 @ Y2 ) ) ) )
=> ( pairwise_nat @ R2 @ S ) ) ).
% pairwiseI
thf(fact_1247_pairwiseI,axiom,
! [S: set_int,R2: int > int > $o] :
( ! [X2: int,Y2: int] :
( ( member_int @ X2 @ S )
=> ( ( member_int @ Y2 @ S )
=> ( ( X2 != Y2 )
=> ( R2 @ X2 @ Y2 ) ) ) )
=> ( pairwise_int @ R2 @ S ) ) ).
% pairwiseI
thf(fact_1248_pairwiseD,axiom,
! [R2: nat > nat > $o,S: set_nat,X: nat,Y: nat] :
( ( pairwise_nat @ R2 @ S )
=> ( ( member_nat @ X @ S )
=> ( ( member_nat @ Y @ S )
=> ( ( X != Y )
=> ( R2 @ X @ Y ) ) ) ) ) ).
% pairwiseD
thf(fact_1249_pairwiseD,axiom,
! [R2: int > int > $o,S: set_int,X: int,Y: int] :
( ( pairwise_int @ R2 @ S )
=> ( ( member_int @ X @ S )
=> ( ( member_int @ Y @ S )
=> ( ( X != Y )
=> ( R2 @ X @ Y ) ) ) ) ) ).
% pairwiseD
thf(fact_1250_Max__in,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( member_nat @ ( lattic8265883725875713057ax_nat @ A ) @ A ) ) ) ).
% Max_in
thf(fact_1251_Max__in,axiom,
! [A: set_int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ( member_int @ ( lattic8263393255366662781ax_int @ A ) @ A ) ) ) ).
% Max_in
thf(fact_1252_Max_OboundedI,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ A )
=> ( ord_less_eq_nat @ A4 @ X ) )
=> ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A ) @ X ) ) ) ) ).
% Max.boundedI
thf(fact_1253_Max_OboundedI,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ( ! [A4: int] :
( ( member_int @ A4 @ A )
=> ( ord_less_eq_int @ A4 @ X ) )
=> ( ord_less_eq_int @ ( lattic8263393255366662781ax_int @ A ) @ X ) ) ) ) ).
% Max.boundedI
thf(fact_1254_Max_OboundedE,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ ( lattic8265883725875713057ax_nat @ A ) @ X )
=> ! [A9: nat] :
( ( member_nat @ A9 @ A )
=> ( ord_less_eq_nat @ A9 @ X ) ) ) ) ) ).
% Max.boundedE
thf(fact_1255_Max_OboundedE,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ( ( ord_less_eq_int @ ( lattic8263393255366662781ax_int @ A ) @ X )
=> ! [A9: int] :
( ( member_int @ A9 @ A )
=> ( ord_less_eq_int @ A9 @ X ) ) ) ) ) ).
% Max.boundedE
thf(fact_1256_eq__Max__iff,axiom,
! [A: set_nat,M: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( M
= ( lattic8265883725875713057ax_nat @ A ) )
= ( ( member_nat @ M @ A )
& ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_nat @ X3 @ M ) ) ) ) ) ) ).
% eq_Max_iff
thf(fact_1257_eq__Max__iff,axiom,
! [A: set_int,M: int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ( ( M
= ( lattic8263393255366662781ax_int @ A ) )
= ( ( member_int @ M @ A )
& ! [X3: int] :
( ( member_int @ X3 @ A )
=> ( ord_less_eq_int @ X3 @ M ) ) ) ) ) ) ).
% eq_Max_iff
thf(fact_1258_Max__eq__iff,axiom,
! [A: set_nat,M: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ( lattic8265883725875713057ax_nat @ A )
= M )
= ( ( member_nat @ M @ A )
& ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_nat @ X3 @ M ) ) ) ) ) ) ).
% Max_eq_iff
thf(fact_1259_Max__eq__iff,axiom,
! [A: set_int,M: int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ( ( ( lattic8263393255366662781ax_int @ A )
= M )
= ( ( member_int @ M @ A )
& ! [X3: int] :
( ( member_int @ X3 @ A )
=> ( ord_less_eq_int @ X3 @ M ) ) ) ) ) ) ).
% Max_eq_iff
thf(fact_1260_Max_Oinfinite,axiom,
! [A: set_list_o] :
( ~ ( finite_finite_list_o @ A )
=> ( ( lattic8010587763684703885list_o @ A )
= ( the_list_o @ none_list_o ) ) ) ).
% Max.infinite
thf(fact_1261_Min__eq__iff,axiom,
! [A: set_nat,M: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ( lattic8721135487736765967in_nat @ A )
= M )
= ( ( member_nat @ M @ A )
& ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_nat @ M @ X3 ) ) ) ) ) ) ).
% Min_eq_iff
thf(fact_1262_Min__eq__iff,axiom,
! [A: set_int,M: int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ( ( ( lattic8718645017227715691in_int @ A )
= M )
= ( ( member_int @ M @ A )
& ! [X3: int] :
( ( member_int @ X3 @ A )
=> ( ord_less_eq_int @ M @ X3 ) ) ) ) ) ) ).
% Min_eq_iff
thf(fact_1263_eq__Min__iff,axiom,
! [A: set_nat,M: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( M
= ( lattic8721135487736765967in_nat @ A ) )
= ( ( member_nat @ M @ A )
& ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( ord_less_eq_nat @ M @ X3 ) ) ) ) ) ) ).
% eq_Min_iff
thf(fact_1264_eq__Min__iff,axiom,
! [A: set_int,M: int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ( ( M
= ( lattic8718645017227715691in_int @ A ) )
= ( ( member_int @ M @ A )
& ! [X3: int] :
( ( member_int @ X3 @ A )
=> ( ord_less_eq_int @ M @ X3 ) ) ) ) ) ) ).
% eq_Min_iff
thf(fact_1265_Min_OboundedE,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ( ord_less_eq_nat @ X @ ( lattic8721135487736765967in_nat @ A ) )
=> ! [A9: nat] :
( ( member_nat @ A9 @ A )
=> ( ord_less_eq_nat @ X @ A9 ) ) ) ) ) ).
% Min.boundedE
thf(fact_1266_Min_OboundedE,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ( ( ord_less_eq_int @ X @ ( lattic8718645017227715691in_int @ A ) )
=> ! [A9: int] :
( ( member_int @ A9 @ A )
=> ( ord_less_eq_int @ X @ A9 ) ) ) ) ) ).
% Min.boundedE
thf(fact_1267_Min_OboundedI,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( ! [A4: nat] :
( ( member_nat @ A4 @ A )
=> ( ord_less_eq_nat @ X @ A4 ) )
=> ( ord_less_eq_nat @ X @ ( lattic8721135487736765967in_nat @ A ) ) ) ) ) ).
% Min.boundedI
thf(fact_1268_Min_OboundedI,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ( ! [A4: int] :
( ( member_int @ A4 @ A )
=> ( ord_less_eq_int @ X @ A4 ) )
=> ( ord_less_eq_int @ X @ ( lattic8718645017227715691in_int @ A ) ) ) ) ) ).
% Min.boundedI
thf(fact_1269_Min__in,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( A != bot_bot_set_nat )
=> ( member_nat @ ( lattic8721135487736765967in_nat @ A ) @ A ) ) ) ).
% Min_in
thf(fact_1270_Min__in,axiom,
! [A: set_int] :
( ( finite_finite_int @ A )
=> ( ( A != bot_bot_set_int )
=> ( member_int @ ( lattic8718645017227715691in_int @ A ) @ A ) ) ) ).
% Min_in
thf(fact_1271_Min_Oinfinite,axiom,
! [A: set_list_o] :
( ~ ( finite_finite_list_o @ A )
=> ( ( lattic7994203477452000287list_o @ A )
= ( the_list_o @ none_list_o ) ) ) ).
% Min.infinite
thf(fact_1272_mono__add,axiom,
! [A2: nat] : ( monotone_on_nat_nat @ top_top_set_nat @ ord_less_eq_nat @ ord_less_eq_nat @ ( plus_plus_nat @ A2 ) ) ).
% mono_add
thf(fact_1273_Min_Oremove,axiom,
! [A: set_nat,X: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ X @ A )
=> ( ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
= bot_bot_set_nat )
=> ( ( lattic8721135487736765967in_nat @ A )
= X ) )
& ( ( ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) )
!= bot_bot_set_nat )
=> ( ( lattic8721135487736765967in_nat @ A )
= ( ord_min_nat @ X @ ( lattic8721135487736765967in_nat @ ( minus_minus_set_nat @ A @ ( insert_nat @ X @ bot_bot_set_nat ) ) ) ) ) ) ) ) ) ).
% Min.remove
thf(fact_1274_Min_Oremove,axiom,
! [A: set_int,X: int] :
( ( finite_finite_int @ A )
=> ( ( member_int @ X @ A )
=> ( ( ( ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) )
= bot_bot_set_int )
=> ( ( lattic8718645017227715691in_int @ A )
= X ) )
& ( ( ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) )
!= bot_bot_set_int )
=> ( ( lattic8718645017227715691in_int @ A )
= ( ord_min_int @ X @ ( lattic8718645017227715691in_int @ ( minus_minus_set_int @ A @ ( insert_int @ X @ bot_bot_set_int ) ) ) ) ) ) ) ) ) ).
% Min.remove
% Helper facts (7)
thf(help_If_2_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
! [X: option_nat,Y: option_nat] :
( ( if_option_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Option__Ooption_It__Nat__Onat_J_T,axiom,
! [X: option_nat,Y: option_nat] :
( ( if_option_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Option__Ooption_It__List__Olist_I_Eo_J_J_T,axiom,
! [X: option_list_o,Y: option_list_o] :
( ( if_option_list_o @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Option__Ooption_It__List__Olist_I_Eo_J_J_T,axiom,
! [X: option_list_o,Y: option_list_o] :
( ( if_option_list_o @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__Option__Ooption_It__Option__Ooption_It__List__Olist_I_Eo_J_J_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Option__Ooption_It__Option__Ooption_It__List__Olist_I_Eo_J_J_J_T,axiom,
! [X: option_option_list_o,Y: option_option_list_o] :
( ( if_opt4852585628301394032list_o @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Option__Ooption_It__Option__Ooption_It__List__Olist_I_Eo_J_J_J_T,axiom,
! [X: option_option_list_o,Y: option_option_list_o] :
( ( if_opt4852585628301394032list_o @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
prefix3555694663680348106ng_int @ prefix_Free_Code_I_e ).
%------------------------------------------------------------------------------