TPTP Problem File: ITP136^1.p
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%------------------------------------------------------------------------------
% File : ITP136^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Paraconsistency problem prob_1104__3276536_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : Paraconsistency/prob_1104__3276536_1 [Des21]
% Status : Theorem
% Rating : 0.20 v8.2.0, 0.15 v8.1.0, 0.18 v7.5.0
% Syntax : Number of formulae : 555 ( 175 unt; 197 typ; 0 def)
% Number of atoms : 955 ( 534 equ; 0 cnn)
% Maximal formula atoms : 16 ( 2 avg)
% Number of connectives : 2679 ( 116 ~; 4 |; 29 &;2112 @)
% ( 0 <=>; 418 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 6 avg)
% Number of types : 37 ( 36 usr)
% Number of type conns : 797 ( 797 >; 0 *; 0 +; 0 <<)
% Number of symbols : 164 ( 161 usr; 16 con; 0-3 aty)
% Number of variables : 986 ( 25 ^; 934 !; 27 ?; 986 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:40:22.190
%------------------------------------------------------------------------------
% Could-be-implicit typings (36)
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Ofm_J_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Ofm_J_J_J,type,
set_Pr1475944804lle_fm: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Ofm_J_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Ofm_J_J,type,
produc1250255278lle_fm: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_Mt__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_J_J,type,
set_Pr2012908090lle_tv: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_Mt__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_J_J,type,
set_Pr698334450lle_tv: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_Mt__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_J,type,
produc1720769084lle_tv: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_M_062_It__List__Olist_It__String__Ochar_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_J_J,type,
set_Pr2061825550lle_tv: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Nat__Onat_J_J_J,type,
set_Pr1943077434tv_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_J_J,type,
set_Pr292927124lle_tv: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_M_062_It__List__Olist_It__String__Ochar_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_J,type,
produc1852595672lle_tv: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Ofm_M_062_It__List__Olist_It__String__Ochar_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_J_J,type,
set_Pr1723822845lle_tv: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Ofm_J_J,type,
set_Pr2120184979lle_fm: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Ofm_M_062_It__List__Olist_It__String__Ochar_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_J,type,
produc1443694919lle_tv: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Ofm_J,type,
produc696645085lle_fm: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_Pr1490359111at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_J,type,
set_Pr2014382200lle_tv: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Ofm_Mt__Paraconsistency____Mirabelle____qxuposogha__Ofm_J_J,type,
set_Pr1019238018lle_fm: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_Pr840833727at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_J,type,
set_Pr560526279lle_tv: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J,type,
produc1647334210lle_tv: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Ofm_Mt__Paraconsistency____Mirabelle____qxuposogha__Ofm_J,type,
produc450659916lle_fm: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
set_Pr1357842111at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
produc596913417at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J,type,
produc931990033lle_tv: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Nat__Onat_J_J,type,
set_Pr1725653840tv_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_J,type,
set_Pr13445366lle_tv: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
produc1281043295at_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Nat__Onat_J,type,
produc194167194tv_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J,type,
produc270694976lle_tv: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_Pr1986765409at_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Paraconsistency____Mirabelle____qxuposogha__Otv_J,type,
set_Pa1764573435lle_tv: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
product_prod_nat_nat: $tType ).
thf(ty_n_t__Paraconsistency____Mirabelle____qxuposogha__Otv,type,
paraco1605129243lle_tv: $tType ).
thf(ty_n_t__Paraconsistency____Mirabelle____qxuposogha__Ofm,type,
paraco1604210848lle_fm: $tType ).
thf(ty_n_t__List__Olist_It__String__Ochar_J,type,
list_char: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (161)
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Paraconsistency____Mirabelle____qxuposogha__Otv,type,
inj_on1955349912lle_tv: ( nat > paraco1605129243lle_tv ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Paraconsistency____Mirabelle____qxuposogha__Otv_001t__Nat__Onat,type,
inj_on1179712058tv_nat: ( paraco1605129243lle_tv > nat ) > set_Pa1764573435lle_tv > $o ).
thf(sy_c_Fun_Oinj__on_001t__Paraconsistency____Mirabelle____qxuposogha__Otv_001t__Paraconsistency____Mirabelle____qxuposogha__Otv,type,
inj_on772319074lle_tv: ( paraco1605129243lle_tv > paraco1605129243lle_tv ) > set_Pa1764573435lle_tv > $o ).
thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
inj_on26176689at_nat: ( product_prod_nat_nat > product_prod_nat_nat ) > set_Pr1986765409at_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J,type,
inj_on1745913142lle_tv: ( produc270694976lle_tv > produc270694976lle_tv ) > set_Pr13445366lle_tv > $o ).
thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Nat__Onat_J,type,
inj_on552895452tv_nat: ( produc194167194tv_nat > produc194167194tv_nat ) > set_Pr1725653840tv_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_001t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J,type,
inj_on2112487900lle_tv: ( produc1647334210lle_tv > produc1647334210lle_tv ) > set_Pr2014382200lle_tv > $o ).
thf(sy_c_Fun_Othe__inv__into_001t__Nat__Onat_001t__Nat__Onat,type,
the_inv_into_nat_nat: set_nat > ( nat > nat ) > nat > nat ).
thf(sy_c_Fun_Othe__inv__into_001t__Paraconsistency____Mirabelle____qxuposogha__Otv_001t__Paraconsistency____Mirabelle____qxuposogha__Otv,type,
the_in452116896lle_tv: set_Pa1764573435lle_tv > ( paraco1605129243lle_tv > paraco1605129243lle_tv ) > paraco1605129243lle_tv > paraco1605129243lle_tv ).
thf(sy_c_If_001t__Paraconsistency____Mirabelle____qxuposogha__Otv,type,
if_Par1134130401lle_tv: $o > paraco1605129243lle_tv > paraco1605129243lle_tv > paraco1605129243lle_tv ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Paraconsistency____Mirabelle____qxuposogha__Otv_J,type,
ord_le241064091lle_tv: set_Pa1764573435lle_tv > set_Pa1764573435lle_tv > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Nat__Onat_M_Eo_J,type,
top_top_nat_o: nat > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Paraconsistency____Mirabelle____qxuposogha__Otv_M_Eo_J,type,
top_to1299250770e_tv_o: paraco1605129243lle_tv > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
top_to632227436_nat_o: product_prod_nat_nat > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_M_Eo_J,type,
top_to1889334367e_tv_o: produc270694976lle_tv > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Nat__Onat_J_M_Eo_J,type,
top_to900819589_nat_o: produc194167194tv_nat > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_Eo,type,
top_top_o: $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
top_top_set_nat: set_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Paraconsistency____Mirabelle____qxuposogha__Otv_J,type,
top_to1057771083lle_tv: set_Pa1764573435lle_tv ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_M_062_It__List__Olist_It__String__Ochar_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_J_J,type,
top_to138697918lle_tv: set_Pr2061825550lle_tv ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Ofm_J_J,type,
top_to775056195lle_fm: set_Pr2120184979lle_fm ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
top_to1541413903at_nat: set_Pr1357842111at_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_J,type,
top_to449985143lle_tv: set_Pr560526279lle_tv ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
top_to1555900337at_nat: set_Pr1986765409at_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_J,type,
top_to1411004326lle_tv: set_Pr13445366lle_tv ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Nat__Onat_J_J,type,
top_to975729152tv_nat: set_Pr1725653840tv_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_J,type,
top_to1871154984lle_tv: set_Pr2014382200lle_tv ).
thf(sy_c_Paraconsistency__Mirabelle__qxuposogha_Ochange__int,type,
paraco761681009ge_int: ( nat > nat ) > ( list_char > paraco1605129243lle_tv ) > list_char > paraco1605129243lle_tv ).
thf(sy_c_Paraconsistency__Mirabelle__qxuposogha_Ochange__tv,type,
paraco1147068288nge_tv: ( nat > nat ) > paraco1605129243lle_tv > paraco1605129243lle_tv ).
thf(sy_c_Paraconsistency__Mirabelle__qxuposogha_Ochange__tv__rel,type,
paraco1632582009tv_rel: produc931990033lle_tv > produc931990033lle_tv > $o ).
thf(sy_c_Paraconsistency__Mirabelle__qxuposogha_Oeval,type,
paraco1325918602e_eval: ( list_char > paraco1605129243lle_tv ) > paraco1604210848lle_fm > paraco1605129243lle_tv ).
thf(sy_c_Paraconsistency__Mirabelle__qxuposogha_Ofm_OCon_H,type,
paraco69239110le_Con: paraco1604210848lle_fm > paraco1604210848lle_fm > paraco1604210848lle_fm ).
thf(sy_c_Paraconsistency__Mirabelle__qxuposogha_Ofm_OEql,type,
paraco1389086619le_Eql: paraco1604210848lle_fm > paraco1604210848lle_fm > paraco1604210848lle_fm ).
thf(sy_c_Paraconsistency__Mirabelle__qxuposogha_Ofm_OEql_H,type,
paraco1745535428le_Eql: paraco1604210848lle_fm > paraco1604210848lle_fm > paraco1604210848lle_fm ).
thf(sy_c_Paraconsistency__Mirabelle__qxuposogha_Ofm_ONeg_H,type,
paraco445776468le_Neg: paraco1604210848lle_fm > paraco1604210848lle_fm ).
thf(sy_c_Paraconsistency__Mirabelle__qxuposogha_Ofm_OPro,type,
paraco1480028776le_Pro: list_char > paraco1604210848lle_fm ).
thf(sy_c_Paraconsistency__Mirabelle__qxuposogha_Ofm_OTruth,type,
paraco367965286_Truth: paraco1604210848lle_fm ).
thf(sy_c_Paraconsistency__Mirabelle__qxuposogha_Otv_ODet,type,
paraco1344940915le_Det: $o > paraco1605129243lle_tv ).
thf(sy_c_Paraconsistency__Mirabelle__qxuposogha_Otv_OIndet,type,
paraco793048302_Indet: nat > paraco1605129243lle_tv ).
thf(sy_c_Paraconsistency__Mirabelle__qxuposogha_Otv_Ocase__tv_001t__Paraconsistency____Mirabelle____qxuposogha__Otv,type,
paraco2051596139lle_tv: ( $o > paraco1605129243lle_tv ) > ( nat > paraco1605129243lle_tv ) > paraco1605129243lle_tv > paraco1605129243lle_tv ).
thf(sy_c_Paraconsistency__Mirabelle__qxuposogha_Ovalid,type,
paraco327635598_valid: paraco1604210848lle_fm > $o ).
thf(sy_c_Paraconsistency__Mirabelle__qxuposogha_Ovalid__in,type,
paraco55203475lid_in: set_nat > paraco1604210848lle_fm > $o ).
thf(sy_c_Product__Type_OPair_001_062_It__List__Olist_It__String__Ochar_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_001_062_It__List__Olist_It__String__Ochar_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J,type,
produc1297923922lle_tv: ( list_char > paraco1605129243lle_tv ) > ( list_char > paraco1605129243lle_tv ) > produc1852595672lle_tv ).
thf(sy_c_Product__Type_OPair_001_062_It__List__Olist_It__String__Ochar_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_001t__Paraconsistency____Mirabelle____qxuposogha__Ofm,type,
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thf(sy_c_Product__Type_Obool_Ocase__bool_001t__Paraconsistency____Mirabelle____qxuposogha__Otv,type,
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produc1900967599lle_fm: ( ( list_char > paraco1605129243lle_tv ) > list_char > paraco1605129243lle_tv ) > ( paraco1604210848lle_fm > paraco1604210848lle_fm ) > produc696645085lle_fm > produc696645085lle_fm ).
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produc1566550641lle_tv: ( ( list_char > paraco1605129243lle_tv ) > nat > nat ) > ( paraco1604210848lle_fm > paraco1605129243lle_tv ) > produc696645085lle_fm > produc931990033lle_tv ).
thf(sy_c_Product__Type_Omap__prod_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__List__Olist_It__String__Ochar_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_001t__Paraconsistency____Mirabelle____qxuposogha__Otv_001t__Paraconsistency____Mirabelle____qxuposogha__Ofm,type,
produc951033137lle_fm: ( ( nat > nat ) > list_char > paraco1605129243lle_tv ) > ( paraco1605129243lle_tv > paraco1604210848lle_fm ) > produc931990033lle_tv > produc696645085lle_fm ).
thf(sy_c_Product__Type_Omap__prod_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Paraconsistency____Mirabelle____qxuposogha__Otv_001t__Paraconsistency____Mirabelle____qxuposogha__Otv,type,
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produc1250721397at_nat: ( nat > paraco1605129243lle_tv ) > ( nat > nat ) > product_prod_nat_nat > produc194167194tv_nat ).
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produc210389789lle_tv: ( nat > paraco1605129243lle_tv ) > ( nat > paraco1605129243lle_tv ) > product_prod_nat_nat > produc1647334210lle_tv ).
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produc1975500101lle_tv: ( nat > paraco1605129243lle_tv ) > ( paraco1605129243lle_tv > paraco1605129243lle_tv ) > produc270694976lle_tv > produc1647334210lle_tv ).
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produc72207551lle_tv: ( paraco1605129243lle_tv > nat ) > ( nat > paraco1605129243lle_tv ) > produc194167194tv_nat > produc270694976lle_tv ).
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produc1345917071lle_tv: ( paraco1605129243lle_tv > paraco1605129243lle_tv ) > ( paraco1605129243lle_tv > paraco1605129243lle_tv ) > produc1647334210lle_tv > produc1647334210lle_tv ).
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thf(sy_c_Set_Oimage_001t__Paraconsistency____Mirabelle____qxuposogha__Otv_001t__Paraconsistency____Mirabelle____qxuposogha__Otv,type,
image_335818702lle_tv: ( paraco1605129243lle_tv > paraco1605129243lle_tv ) > set_Pa1764573435lle_tv > set_Pa1764573435lle_tv ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Ofm_J_001t__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Ofm_J,type,
image_807538418lle_fm: ( produc696645085lle_fm > produc696645085lle_fm ) > set_Pr2120184979lle_fm > set_Pr2120184979lle_fm ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Ofm_J_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J,type,
image_1154804148lle_tv: ( produc696645085lle_fm > produc931990033lle_tv ) > set_Pr2120184979lle_fm > set_Pr560526279lle_tv ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Ofm_J_001t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Ofm_M_062_It__List__Olist_It__String__Ochar_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_J,type,
image_924970994lle_tv: ( produc696645085lle_fm > produc1443694919lle_tv ) > set_Pr2120184979lle_fm > set_Pr1723822845lle_tv ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_001t__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Ofm_J,type,
image_1585092542lle_fm: ( produc931990033lle_tv > produc696645085lle_fm ) > set_Pr560526279lle_tv > set_Pr2120184979lle_fm ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J,type,
image_1751508608lle_tv: ( produc931990033lle_tv > produc931990033lle_tv ) > set_Pr560526279lle_tv > set_Pr560526279lle_tv ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_001t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
image_1724396928at_nat: ( produc931990033lle_tv > produc596913417at_nat ) > set_Pr560526279lle_tv > set_Pr840833727at_nat ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
image_815163717at_nat: ( product_prod_nat_nat > product_prod_nat_nat ) > set_Pr1986765409at_nat > set_Pr1986765409at_nat ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J,type,
image_1674248956lle_tv: ( product_prod_nat_nat > produc270694976lle_tv ) > set_Pr1986765409at_nat > set_Pr13445366lle_tv ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Nat__Onat_J,type,
image_1597721174tv_nat: ( product_prod_nat_nat > produc194167194tv_nat ) > set_Pr1986765409at_nat > set_Pr1725653840tv_nat ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J,type,
image_1507992062lle_tv: ( product_prod_nat_nat > produc1647334210lle_tv ) > set_Pr1986765409at_nat > set_Pr2014382200lle_tv ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
image_1787705850at_nat: ( produc270694976lle_tv > product_prod_nat_nat ) > set_Pr13445366lle_tv > set_Pr1986765409at_nat ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J,type,
image_149830562lle_tv: ( produc270694976lle_tv > produc270694976lle_tv ) > set_Pr13445366lle_tv > set_Pr13445366lle_tv ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_001t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Nat__Onat_J,type,
image_53754658tv_nat: ( produc270694976lle_tv > produc194167194tv_nat ) > set_Pr13445366lle_tv > set_Pr1725653840tv_nat ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_001t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J,type,
image_988201162lle_tv: ( produc270694976lle_tv > produc1647334210lle_tv ) > set_Pr13445366lle_tv > set_Pr2014382200lle_tv ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Ofm_M_062_It__List__Olist_It__String__Ochar_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_J_001t__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Ofm_J,type,
image_326034184lle_fm: ( produc1443694919lle_tv > produc696645085lle_fm ) > set_Pr1723822845lle_tv > set_Pr2120184979lle_fm ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J,type,
image_769496456lle_tv: ( produc596913417at_nat > produc931990033lle_tv ) > set_Pr840833727at_nat > set_Pr560526279lle_tv ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
image_1805124128at_nat: ( produc194167194tv_nat > product_prod_nat_nat ) > set_Pr1725653840tv_nat > set_Pr1986765409at_nat ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J,type,
image_1200372424lle_tv: ( produc194167194tv_nat > produc270694976lle_tv ) > set_Pr1725653840tv_nat > set_Pr13445366lle_tv ).
thf(sy_c_Set_Oimage_001t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_001t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J,type,
image_1221801032lle_tv: ( produc1647334210lle_tv > produc1647334210lle_tv ) > set_Pr2014382200lle_tv > set_Pr2014382200lle_tv ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J,type,
accp_P2085826120lle_tv: ( produc931990033lle_tv > produc931990033lle_tv > $o ) > produc931990033lle_tv > $o ).
thf(sy_c_Wellfounded_Olex__prod_001_062_It__List__Olist_It__String__Ochar_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_001t__Paraconsistency____Mirabelle____qxuposogha__Ofm,type,
lex_pr2001015487lle_fm: set_Pr2061825550lle_tv > set_Pr1019238018lle_fm > set_Pr1475944804lle_fm ).
thf(sy_c_Wellfounded_Olex__prod_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Paraconsistency____Mirabelle____qxuposogha__Otv,type,
lex_pr1093414955lle_tv: set_Pr1357842111at_nat > set_Pr2014382200lle_tv > set_Pr698334450lle_tv ).
thf(sy_c_Wellfounded_Olex__prod_001t__Nat__Onat_001t__Nat__Onat,type,
lex_prod_nat_nat: set_Pr1986765409at_nat > set_Pr1986765409at_nat > set_Pr1490359111at_nat ).
thf(sy_c_Wellfounded_Olex__prod_001t__Nat__Onat_001t__Paraconsistency____Mirabelle____qxuposogha__Otv,type,
lex_pr413684186lle_tv: set_Pr1986765409at_nat > set_Pr2014382200lle_tv > set_Pr292927124lle_tv ).
thf(sy_c_Wellfounded_Olex__prod_001t__Paraconsistency____Mirabelle____qxuposogha__Otv_001t__Nat__Onat,type,
lex_pr1785529980tv_nat: set_Pr2014382200lle_tv > set_Pr1986765409at_nat > set_Pr1943077434tv_nat ).
thf(sy_c_Wellfounded_Olex__prod_001t__Paraconsistency____Mirabelle____qxuposogha__Otv_001t__Paraconsistency____Mirabelle____qxuposogha__Otv,type,
lex_pr567812260lle_tv: set_Pr2014382200lle_tv > set_Pr2014382200lle_tv > set_Pr2012908090lle_tv ).
thf(sy_c_Wellfounded_Owf_001t__Nat__Onat,type,
wf_nat: set_Pr1986765409at_nat > $o ).
thf(sy_c_Wellfounded_Owf_001t__Paraconsistency____Mirabelle____qxuposogha__Otv,type,
wf_Par1378404612lle_tv: set_Pr2014382200lle_tv > $o ).
thf(sy_c_Wfrec_Osame__fst_001_062_It__List__Olist_It__String__Ochar_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_001t__Paraconsistency____Mirabelle____qxuposogha__Ofm,type,
same_f1581085506lle_fm: ( ( list_char > paraco1605129243lle_tv ) > $o ) > ( ( list_char > paraco1605129243lle_tv ) > set_Pr1019238018lle_fm ) > set_Pr1475944804lle_fm ).
thf(sy_c_Wfrec_Osame__fst_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Paraconsistency____Mirabelle____qxuposogha__Otv,type,
same_f1138187694lle_tv: ( ( nat > nat ) > $o ) > ( ( nat > nat ) > set_Pr2014382200lle_tv ) > set_Pr698334450lle_tv ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Paraconsistency____Mirabelle____qxuposogha__Otv,type,
member266900804lle_tv: paraco1605129243lle_tv > set_Pa1764573435lle_tv > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_M_062_It__List__Olist_It__String__Ochar_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_J,type,
member146333935lle_tv: produc1852595672lle_tv > set_Pr2061825550lle_tv > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Ofm_J,type,
member1474794996lle_fm: produc696645085lle_fm > set_Pr2120184979lle_fm > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member112824072at_nat: produc1281043295at_nat > set_Pr1357842111at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J,type,
member1569366824lle_tv: produc931990033lle_tv > set_Pr560526279lle_tv > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
member701585322at_nat: product_prod_nat_nat > set_Pr1986765409at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J,type,
member813679063lle_tv: produc270694976lle_tv > set_Pr13445366lle_tv > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Ofm_M_062_It__List__Olist_It__String__Ochar_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_J,type,
member74361182lle_tv: produc1443694919lle_tv > set_Pr1723822845lle_tv > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Ofm_Mt__Paraconsistency____Mirabelle____qxuposogha__Ofm_J,type,
member334644067lle_fm: produc450659916lle_fm > set_Pr1019238018lle_fm > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member1234290208at_nat: produc596913417at_nat > set_Pr840833727at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Nat__Onat_J,type,
member737151281tv_nat: produc194167194tv_nat > set_Pr1725653840tv_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Paraconsistency____Mirabelle____qxuposogha__Otv_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J,type,
member1531318361lle_tv: produc1647334210lle_tv > set_Pr2014382200lle_tv > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Ofm_J_Mt__Product____Type__Oprod_I_062_It__List__Olist_It__String__Ochar_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Ofm_J_J,type,
member1592865093lle_fm: produc1250255278lle_fm > set_Pr1475944804lle_fm > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_Mt__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Paraconsistency____Mirabelle____qxuposogha__Otv_J_J,type,
member1339857875lle_tv: produc1720769084lle_tv > set_Pr698334450lle_tv > $o ).
thf(sy_v_f,type,
f: nat > nat ).
thf(sy_v_i,type,
i: list_char > paraco1605129243lle_tv ).
thf(sy_v_p1____,type,
p1: paraco1604210848lle_fm ).
thf(sy_v_p2____,type,
p2: paraco1604210848lle_fm ).
% Relevant facts (354)
thf(fact_0_a_H_H_H,axiom,
( ( paraco1325918602e_eval @ i @ p2 )
!= ( paraco1344940915le_Det @ $true ) ) ).
% a'''
thf(fact_1_a_H_H,axiom,
( ( paraco1325918602e_eval @ i @ p1 )
!= ( paraco1344940915le_Det @ $true ) ) ).
% a''
thf(fact_2_a_H_H_H_H,axiom,
( ( paraco1325918602e_eval @ i @ p1 )
!= ( paraco1344940915le_Det @ $false ) ) ).
% a''''
thf(fact_3_b_H_H,axiom,
( ( paraco1325918602e_eval @ ( paraco761681009ge_int @ f @ i ) @ p1 )
!= ( paraco1344940915le_Det @ $true ) ) ).
% b''
thf(fact_4_a,axiom,
( ( paraco1325918602e_eval @ i @ p2 )
= ( paraco1344940915le_Det @ $false ) ) ).
% a
thf(fact_5_a_H,axiom,
( ( paraco1325918602e_eval @ i @ p1 )
!= ( paraco1325918602e_eval @ i @ p2 ) ) ).
% a'
thf(fact_6_b,axiom,
( ( paraco1325918602e_eval @ ( paraco761681009ge_int @ f @ i ) @ p2 )
= ( paraco1344940915le_Det @ $false ) ) ).
% b
thf(fact_7_b_H,axiom,
( ( paraco1325918602e_eval @ ( paraco761681009ge_int @ f @ i ) @ p1 )
!= ( paraco1325918602e_eval @ ( paraco761681009ge_int @ f @ i ) @ p2 ) ) ).
% b'
thf(fact_8_tv_Oinject_I1_J,axiom,
! [X1: $o,Y1: $o] :
( ( ( paraco1344940915le_Det @ X1 )
= ( paraco1344940915le_Det @ Y1 ) )
= ( X1 = Y1 ) ) ).
% tv.inject(1)
thf(fact_9_ih2,axiom,
( ( paraco1325918602e_eval @ ( paraco761681009ge_int @ f @ i ) @ p2 )
= ( paraco1147068288nge_tv @ f @ ( paraco1325918602e_eval @ i @ p2 ) ) ) ).
% ih2
thf(fact_10_ih1,axiom,
( ( paraco1325918602e_eval @ ( paraco761681009ge_int @ f @ i ) @ p1 )
= ( paraco1147068288nge_tv @ f @ ( paraco1325918602e_eval @ i @ p1 ) ) ) ).
% ih1
thf(fact_11__092_060open_062change__tv_Af_A_Ieval_Ai_Ap2_J_A_061_ADet_AFalse_092_060close_062,axiom,
( ( paraco1147068288nge_tv @ f @ ( paraco1325918602e_eval @ i @ p2 ) )
= ( paraco1344940915le_Det @ $false ) ) ).
% \<open>change_tv f (eval i p2) = Det False\<close>
thf(fact_12_valid__def,axiom,
( paraco327635598_valid
= ( ^ [P: paraco1604210848lle_fm] :
! [I: list_char > paraco1605129243lle_tv] :
( ( paraco1325918602e_eval @ I @ P )
= ( paraco1344940915le_Det @ $true ) ) ) ) ).
% valid_def
thf(fact_13_yes,axiom,
( ( paraco1325918602e_eval @ i @ ( paraco1745535428le_Eql @ p1 @ p2 ) )
= ( paraco1325918602e_eval @ i @ ( paraco445776468le_Neg @ p1 ) ) ) ).
% yes
thf(fact_14_eval_Osimps_I2_J,axiom,
! [I2: list_char > paraco1605129243lle_tv] :
( ( paraco1325918602e_eval @ I2 @ paraco367965286_Truth )
= ( paraco1344940915le_Det @ $true ) ) ).
% eval.simps(2)
thf(fact_15_eval_Osimps_I5_J,axiom,
! [I2: list_char > paraco1605129243lle_tv,P2: paraco1604210848lle_fm,Q: paraco1604210848lle_fm] :
( ( ( ( paraco1325918602e_eval @ I2 @ P2 )
= ( paraco1325918602e_eval @ I2 @ Q ) )
=> ( ( paraco1325918602e_eval @ I2 @ ( paraco1389086619le_Eql @ P2 @ Q ) )
= ( paraco1344940915le_Det @ $true ) ) )
& ( ( ( paraco1325918602e_eval @ I2 @ P2 )
!= ( paraco1325918602e_eval @ I2 @ Q ) )
=> ( ( paraco1325918602e_eval @ I2 @ ( paraco1389086619le_Eql @ P2 @ Q ) )
= ( paraco1344940915le_Det @ $false ) ) ) ) ).
% eval.simps(5)
thf(fact_16_eval__equality__simplify,axiom,
! [I2: list_char > paraco1605129243lle_tv,P2: paraco1604210848lle_fm,Q: paraco1604210848lle_fm] :
( ( paraco1325918602e_eval @ I2 @ ( paraco1389086619le_Eql @ P2 @ Q ) )
= ( paraco1344940915le_Det
@ ( ( paraco1325918602e_eval @ I2 @ P2 )
= ( paraco1325918602e_eval @ I2 @ Q ) ) ) ) ).
% eval_equality_simplify
thf(fact_17_change__int__def,axiom,
( paraco761681009ge_int
= ( ^ [F: nat > nat,I: list_char > paraco1605129243lle_tv,S: list_char] : ( paraco1147068288nge_tv @ F @ ( I @ S ) ) ) ) ).
% change_int_def
thf(fact_18_eval_Osimps_I4_J,axiom,
! [I2: list_char > paraco1605129243lle_tv,P2: paraco1604210848lle_fm,Q: paraco1604210848lle_fm] :
( ( ( ( paraco1325918602e_eval @ I2 @ P2 )
= ( paraco1325918602e_eval @ I2 @ Q ) )
=> ( ( paraco1325918602e_eval @ I2 @ ( paraco69239110le_Con @ P2 @ Q ) )
= ( paraco1325918602e_eval @ I2 @ P2 ) ) )
& ( ( ( paraco1325918602e_eval @ I2 @ P2 )
!= ( paraco1325918602e_eval @ I2 @ Q ) )
=> ( ( ( ( paraco1325918602e_eval @ I2 @ P2 )
= ( paraco1344940915le_Det @ $true ) )
=> ( ( paraco1325918602e_eval @ I2 @ ( paraco69239110le_Con @ P2 @ Q ) )
= ( paraco1325918602e_eval @ I2 @ Q ) ) )
& ( ( ( paraco1325918602e_eval @ I2 @ P2 )
!= ( paraco1344940915le_Det @ $true ) )
=> ( ( ( ( paraco1325918602e_eval @ I2 @ Q )
= ( paraco1344940915le_Det @ $true ) )
=> ( ( paraco1325918602e_eval @ I2 @ ( paraco69239110le_Con @ P2 @ Q ) )
= ( paraco1325918602e_eval @ I2 @ P2 ) ) )
& ( ( ( paraco1325918602e_eval @ I2 @ Q )
!= ( paraco1344940915le_Det @ $true ) )
=> ( ( paraco1325918602e_eval @ I2 @ ( paraco69239110le_Con @ P2 @ Q ) )
= ( paraco1344940915le_Det @ $false ) ) ) ) ) ) ) ) ).
% eval.simps(4)
thf(fact_19_tv_Osimps_I5_J,axiom,
! [F1: $o > paraco1605129243lle_tv,F2: nat > paraco1605129243lle_tv,X1: $o] :
( ( paraco2051596139lle_tv @ F1 @ F2 @ ( paraco1344940915le_Det @ X1 ) )
= ( F1 @ X1 ) ) ).
% tv.simps(5)
thf(fact_20_eval__negation,axiom,
! [I2: list_char > paraco1605129243lle_tv,P2: paraco1604210848lle_fm] :
( ( ( ( paraco1325918602e_eval @ I2 @ P2 )
= ( paraco1344940915le_Det @ $false ) )
=> ( ( paraco1325918602e_eval @ I2 @ ( paraco445776468le_Neg @ P2 ) )
= ( paraco1344940915le_Det @ $true ) ) )
& ( ( ( paraco1325918602e_eval @ I2 @ P2 )
!= ( paraco1344940915le_Det @ $false ) )
=> ( ( ( ( paraco1325918602e_eval @ I2 @ P2 )
= ( paraco1344940915le_Det @ $true ) )
=> ( ( paraco1325918602e_eval @ I2 @ ( paraco445776468le_Neg @ P2 ) )
= ( paraco1344940915le_Det @ $false ) ) )
& ( ( ( paraco1325918602e_eval @ I2 @ P2 )
!= ( paraco1344940915le_Det @ $true ) )
=> ( ( paraco1325918602e_eval @ I2 @ ( paraco445776468le_Neg @ P2 ) )
= ( paraco1325918602e_eval @ I2 @ P2 ) ) ) ) ) ) ).
% eval_negation
thf(fact_21_assms,axiom,
inj_on_nat_nat @ f @ top_top_set_nat ).
% assms
thf(fact_22_change__tv_Osimps_I1_J,axiom,
! [F3: nat > nat,B: $o] :
( ( paraco1147068288nge_tv @ F3 @ ( paraco1344940915le_Det @ B ) )
= ( paraco1344940915le_Det @ B ) ) ).
% change_tv.simps(1)
thf(fact_23_fm_Oinject_I2_J,axiom,
! [X3: paraco1604210848lle_fm,Y3: paraco1604210848lle_fm] :
( ( ( paraco445776468le_Neg @ X3 )
= ( paraco445776468le_Neg @ Y3 ) )
= ( X3 = Y3 ) ) ).
% fm.inject(2)
thf(fact_24_fm_Oinject_I3_J,axiom,
! [X41: paraco1604210848lle_fm,X42: paraco1604210848lle_fm,Y41: paraco1604210848lle_fm,Y42: paraco1604210848lle_fm] :
( ( ( paraco69239110le_Con @ X41 @ X42 )
= ( paraco69239110le_Con @ Y41 @ Y42 ) )
= ( ( X41 = Y41 )
& ( X42 = Y42 ) ) ) ).
% fm.inject(3)
thf(fact_25_fm_Oinject_I5_J,axiom,
! [X61: paraco1604210848lle_fm,X62: paraco1604210848lle_fm,Y61: paraco1604210848lle_fm,Y62: paraco1604210848lle_fm] :
( ( ( paraco1745535428le_Eql @ X61 @ X62 )
= ( paraco1745535428le_Eql @ Y61 @ Y62 ) )
= ( ( X61 = Y61 )
& ( X62 = Y62 ) ) ) ).
% fm.inject(5)
thf(fact_26_fm_Oinject_I4_J,axiom,
! [X51: paraco1604210848lle_fm,X52: paraco1604210848lle_fm,Y51: paraco1604210848lle_fm,Y52: paraco1604210848lle_fm] :
( ( ( paraco1389086619le_Eql @ X51 @ X52 )
= ( paraco1389086619le_Eql @ Y51 @ Y52 ) )
= ( ( X51 = Y51 )
& ( X52 = Y52 ) ) ) ).
% fm.inject(4)
thf(fact_27_fm_Odistinct_I29_J,axiom,
! [X51: paraco1604210848lle_fm,X52: paraco1604210848lle_fm,X61: paraco1604210848lle_fm,X62: paraco1604210848lle_fm] :
( ( paraco1389086619le_Eql @ X51 @ X52 )
!= ( paraco1745535428le_Eql @ X61 @ X62 ) ) ).
% fm.distinct(29)
thf(fact_28_fm_Odistinct_I27_J,axiom,
! [X41: paraco1604210848lle_fm,X42: paraco1604210848lle_fm,X61: paraco1604210848lle_fm,X62: paraco1604210848lle_fm] :
( ( paraco69239110le_Con @ X41 @ X42 )
!= ( paraco1745535428le_Eql @ X61 @ X62 ) ) ).
% fm.distinct(27)
thf(fact_29_fm_Odistinct_I25_J,axiom,
! [X41: paraco1604210848lle_fm,X42: paraco1604210848lle_fm,X51: paraco1604210848lle_fm,X52: paraco1604210848lle_fm] :
( ( paraco69239110le_Con @ X41 @ X42 )
!= ( paraco1389086619le_Eql @ X51 @ X52 ) ) ).
% fm.distinct(25)
thf(fact_30_fm_Odistinct_I23_J,axiom,
! [X3: paraco1604210848lle_fm,X61: paraco1604210848lle_fm,X62: paraco1604210848lle_fm] :
( ( paraco445776468le_Neg @ X3 )
!= ( paraco1745535428le_Eql @ X61 @ X62 ) ) ).
% fm.distinct(23)
thf(fact_31_fm_Odistinct_I21_J,axiom,
! [X3: paraco1604210848lle_fm,X51: paraco1604210848lle_fm,X52: paraco1604210848lle_fm] :
( ( paraco445776468le_Neg @ X3 )
!= ( paraco1389086619le_Eql @ X51 @ X52 ) ) ).
% fm.distinct(21)
thf(fact_32_fm_Odistinct_I19_J,axiom,
! [X3: paraco1604210848lle_fm,X41: paraco1604210848lle_fm,X42: paraco1604210848lle_fm] :
( ( paraco445776468le_Neg @ X3 )
!= ( paraco69239110le_Con @ X41 @ X42 ) ) ).
% fm.distinct(19)
thf(fact_33_fm_Odistinct_I17_J,axiom,
! [X61: paraco1604210848lle_fm,X62: paraco1604210848lle_fm] :
( paraco367965286_Truth
!= ( paraco1745535428le_Eql @ X61 @ X62 ) ) ).
% fm.distinct(17)
thf(fact_34_fm_Odistinct_I15_J,axiom,
! [X51: paraco1604210848lle_fm,X52: paraco1604210848lle_fm] :
( paraco367965286_Truth
!= ( paraco1389086619le_Eql @ X51 @ X52 ) ) ).
% fm.distinct(15)
thf(fact_35_fm_Odistinct_I13_J,axiom,
! [X41: paraco1604210848lle_fm,X42: paraco1604210848lle_fm] :
( paraco367965286_Truth
!= ( paraco69239110le_Con @ X41 @ X42 ) ) ).
% fm.distinct(13)
thf(fact_36_fm_Odistinct_I11_J,axiom,
! [X3: paraco1604210848lle_fm] :
( paraco367965286_Truth
!= ( paraco445776468le_Neg @ X3 ) ) ).
% fm.distinct(11)
thf(fact_37_conjunction,axiom,
! [P2: paraco1604210848lle_fm,Q: paraco1604210848lle_fm] :
( ( paraco327635598_valid @ ( paraco69239110le_Con @ P2 @ Q ) )
= ( ( paraco327635598_valid @ P2 )
& ( paraco327635598_valid @ Q ) ) ) ).
% conjunction
thf(fact_38_eval__equality,axiom,
! [I2: list_char > paraco1605129243lle_tv,P2: paraco1604210848lle_fm,Q: paraco1604210848lle_fm] :
( ( ( ( paraco1325918602e_eval @ I2 @ P2 )
= ( paraco1325918602e_eval @ I2 @ Q ) )
=> ( ( paraco1325918602e_eval @ I2 @ ( paraco1745535428le_Eql @ P2 @ Q ) )
= ( paraco1344940915le_Det @ $true ) ) )
& ( ( ( paraco1325918602e_eval @ I2 @ P2 )
!= ( paraco1325918602e_eval @ I2 @ Q ) )
=> ( ( ( ( paraco1325918602e_eval @ I2 @ P2 )
= ( paraco1344940915le_Det @ $true ) )
=> ( ( paraco1325918602e_eval @ I2 @ ( paraco1745535428le_Eql @ P2 @ Q ) )
= ( paraco1325918602e_eval @ I2 @ Q ) ) )
& ( ( ( paraco1325918602e_eval @ I2 @ P2 )
!= ( paraco1344940915le_Det @ $true ) )
=> ( ( ( ( paraco1325918602e_eval @ I2 @ Q )
= ( paraco1344940915le_Det @ $true ) )
=> ( ( paraco1325918602e_eval @ I2 @ ( paraco1745535428le_Eql @ P2 @ Q ) )
= ( paraco1325918602e_eval @ I2 @ P2 ) ) )
& ( ( ( paraco1325918602e_eval @ I2 @ Q )
!= ( paraco1344940915le_Det @ $true ) )
=> ( ( ( ( paraco1325918602e_eval @ I2 @ P2 )
= ( paraco1344940915le_Det @ $false ) )
=> ( ( paraco1325918602e_eval @ I2 @ ( paraco1745535428le_Eql @ P2 @ Q ) )
= ( paraco1325918602e_eval @ I2 @ ( paraco445776468le_Neg @ Q ) ) ) )
& ( ( ( paraco1325918602e_eval @ I2 @ P2 )
!= ( paraco1344940915le_Det @ $false ) )
=> ( ( ( ( paraco1325918602e_eval @ I2 @ Q )
= ( paraco1344940915le_Det @ $false ) )
=> ( ( paraco1325918602e_eval @ I2 @ ( paraco1745535428le_Eql @ P2 @ Q ) )
= ( paraco1325918602e_eval @ I2 @ ( paraco445776468le_Neg @ P2 ) ) ) )
& ( ( ( paraco1325918602e_eval @ I2 @ Q )
!= ( paraco1344940915le_Det @ $false ) )
=> ( ( paraco1325918602e_eval @ I2 @ ( paraco1745535428le_Eql @ P2 @ Q ) )
= ( paraco1344940915le_Det @ $false ) ) ) ) ) ) ) ) ) ) ) ) ).
% eval_equality
thf(fact_39_double__negation,axiom,
( paraco1325918602e_eval
= ( ^ [I: list_char > paraco1605129243lle_tv,P: paraco1604210848lle_fm] : ( paraco1325918602e_eval @ I @ ( paraco445776468le_Neg @ ( paraco445776468le_Neg @ P ) ) ) ) ) ).
% double_negation
thf(fact_40_UNIV__I,axiom,
! [X: produc696645085lle_fm] : ( member1474794996lle_fm @ X @ top_to775056195lle_fm ) ).
% UNIV_I
thf(fact_41_UNIV__I,axiom,
! [X: produc1852595672lle_tv] : ( member146333935lle_tv @ X @ top_to138697918lle_tv ) ).
% UNIV_I
thf(fact_42_UNIV__I,axiom,
! [X: produc931990033lle_tv] : ( member1569366824lle_tv @ X @ top_to449985143lle_tv ) ).
% UNIV_I
thf(fact_43_UNIV__I,axiom,
! [X: produc1281043295at_nat] : ( member112824072at_nat @ X @ top_to1541413903at_nat ) ).
% UNIV_I
thf(fact_44_UNIV__I,axiom,
! [X: produc194167194tv_nat] : ( member737151281tv_nat @ X @ top_to975729152tv_nat ) ).
% UNIV_I
thf(fact_45_UNIV__I,axiom,
! [X: produc270694976lle_tv] : ( member813679063lle_tv @ X @ top_to1411004326lle_tv ) ).
% UNIV_I
thf(fact_46_UNIV__I,axiom,
! [X: product_prod_nat_nat] : ( member701585322at_nat @ X @ top_to1555900337at_nat ) ).
% UNIV_I
thf(fact_47_UNIV__I,axiom,
! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% UNIV_I
thf(fact_48_UNIV__I,axiom,
! [X: paraco1605129243lle_tv] : ( member266900804lle_tv @ X @ top_to1057771083lle_tv ) ).
% UNIV_I
thf(fact_49_iso__tuple__UNIV__I,axiom,
! [X: produc696645085lle_fm] : ( member1474794996lle_fm @ X @ top_to775056195lle_fm ) ).
% iso_tuple_UNIV_I
thf(fact_50_iso__tuple__UNIV__I,axiom,
! [X: produc1852595672lle_tv] : ( member146333935lle_tv @ X @ top_to138697918lle_tv ) ).
% iso_tuple_UNIV_I
thf(fact_51_iso__tuple__UNIV__I,axiom,
! [X: produc931990033lle_tv] : ( member1569366824lle_tv @ X @ top_to449985143lle_tv ) ).
% iso_tuple_UNIV_I
thf(fact_52_iso__tuple__UNIV__I,axiom,
! [X: produc1281043295at_nat] : ( member112824072at_nat @ X @ top_to1541413903at_nat ) ).
% iso_tuple_UNIV_I
thf(fact_53_iso__tuple__UNIV__I,axiom,
! [X: produc194167194tv_nat] : ( member737151281tv_nat @ X @ top_to975729152tv_nat ) ).
% iso_tuple_UNIV_I
thf(fact_54_iso__tuple__UNIV__I,axiom,
! [X: produc270694976lle_tv] : ( member813679063lle_tv @ X @ top_to1411004326lle_tv ) ).
% iso_tuple_UNIV_I
thf(fact_55_iso__tuple__UNIV__I,axiom,
! [X: product_prod_nat_nat] : ( member701585322at_nat @ X @ top_to1555900337at_nat ) ).
% iso_tuple_UNIV_I
thf(fact_56_iso__tuple__UNIV__I,axiom,
! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% iso_tuple_UNIV_I
thf(fact_57_iso__tuple__UNIV__I,axiom,
! [X: paraco1605129243lle_tv] : ( member266900804lle_tv @ X @ top_to1057771083lle_tv ) ).
% iso_tuple_UNIV_I
thf(fact_58_top__apply,axiom,
( top_to1299250770e_tv_o
= ( ^ [X2: paraco1605129243lle_tv] : top_top_o ) ) ).
% top_apply
thf(fact_59_top__apply,axiom,
( top_top_nat_o
= ( ^ [X2: nat] : top_top_o ) ) ).
% top_apply
thf(fact_60_injD,axiom,
! [F3: produc1647334210lle_tv > produc1647334210lle_tv,X: produc1647334210lle_tv,Y: produc1647334210lle_tv] :
( ( inj_on2112487900lle_tv @ F3 @ top_to1871154984lle_tv )
=> ( ( ( F3 @ X )
= ( F3 @ Y ) )
=> ( X = Y ) ) ) ).
% injD
thf(fact_61_injD,axiom,
! [F3: nat > paraco1605129243lle_tv,X: nat,Y: nat] :
( ( inj_on1955349912lle_tv @ F3 @ top_top_set_nat )
=> ( ( ( F3 @ X )
= ( F3 @ Y ) )
=> ( X = Y ) ) ) ).
% injD
thf(fact_62_injD,axiom,
! [F3: produc194167194tv_nat > produc194167194tv_nat,X: produc194167194tv_nat,Y: produc194167194tv_nat] :
( ( inj_on552895452tv_nat @ F3 @ top_to975729152tv_nat )
=> ( ( ( F3 @ X )
= ( F3 @ Y ) )
=> ( X = Y ) ) ) ).
% injD
thf(fact_63_injD,axiom,
! [F3: produc270694976lle_tv > produc270694976lle_tv,X: produc270694976lle_tv,Y: produc270694976lle_tv] :
( ( inj_on1745913142lle_tv @ F3 @ top_to1411004326lle_tv )
=> ( ( ( F3 @ X )
= ( F3 @ Y ) )
=> ( X = Y ) ) ) ).
% injD
thf(fact_64_injD,axiom,
! [F3: product_prod_nat_nat > product_prod_nat_nat,X: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( inj_on26176689at_nat @ F3 @ top_to1555900337at_nat )
=> ( ( ( F3 @ X )
= ( F3 @ Y ) )
=> ( X = Y ) ) ) ).
% injD
thf(fact_65_injD,axiom,
! [F3: nat > nat,X: nat,Y: nat] :
( ( inj_on_nat_nat @ F3 @ top_top_set_nat )
=> ( ( ( F3 @ X )
= ( F3 @ Y ) )
=> ( X = Y ) ) ) ).
% injD
thf(fact_66_injD,axiom,
! [F3: paraco1605129243lle_tv > paraco1605129243lle_tv,X: paraco1605129243lle_tv,Y: paraco1605129243lle_tv] :
( ( inj_on772319074lle_tv @ F3 @ top_to1057771083lle_tv )
=> ( ( ( F3 @ X )
= ( F3 @ Y ) )
=> ( X = Y ) ) ) ).
% injD
thf(fact_67_mem__Collect__eq,axiom,
! [A: produc696645085lle_fm,P3: produc696645085lle_fm > $o] :
( ( member1474794996lle_fm @ A @ ( collec835549234lle_fm @ P3 ) )
= ( P3 @ A ) ) ).
% mem_Collect_eq
thf(fact_68_mem__Collect__eq,axiom,
! [A: produc1852595672lle_tv,P3: produc1852595672lle_tv > $o] :
( ( member146333935lle_tv @ A @ ( collec1102043437lle_tv @ P3 ) )
= ( P3 @ A ) ) ).
% mem_Collect_eq
thf(fact_69_mem__Collect__eq,axiom,
! [A: produc931990033lle_tv,P3: produc931990033lle_tv > $o] :
( ( member1569366824lle_tv @ A @ ( collec1624074086lle_tv @ P3 ) )
= ( P3 @ A ) ) ).
% mem_Collect_eq
thf(fact_70_mem__Collect__eq,axiom,
! [A: produc1281043295at_nat,P3: produc1281043295at_nat > $o] :
( ( member112824072at_nat @ A @ ( collec791369802at_nat @ P3 ) )
= ( P3 @ A ) ) ).
% mem_Collect_eq
thf(fact_71_mem__Collect__eq,axiom,
! [A: paraco1605129243lle_tv,P3: paraco1605129243lle_tv > $o] :
( ( member266900804lle_tv @ A @ ( collec1111687558lle_tv @ P3 ) )
= ( P3 @ A ) ) ).
% mem_Collect_eq
thf(fact_72_mem__Collect__eq,axiom,
! [A: nat,P3: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P3 ) )
= ( P3 @ A ) ) ).
% mem_Collect_eq
thf(fact_73_Collect__mem__eq,axiom,
! [A2: set_Pr2120184979lle_fm] :
( ( collec835549234lle_fm
@ ^ [X2: produc696645085lle_fm] : ( member1474794996lle_fm @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_74_Collect__mem__eq,axiom,
! [A2: set_Pr2061825550lle_tv] :
( ( collec1102043437lle_tv
@ ^ [X2: produc1852595672lle_tv] : ( member146333935lle_tv @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_75_Collect__mem__eq,axiom,
! [A2: set_Pr560526279lle_tv] :
( ( collec1624074086lle_tv
@ ^ [X2: produc931990033lle_tv] : ( member1569366824lle_tv @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_76_Collect__mem__eq,axiom,
! [A2: set_Pr1357842111at_nat] :
( ( collec791369802at_nat
@ ^ [X2: produc1281043295at_nat] : ( member112824072at_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_77_Collect__mem__eq,axiom,
! [A2: set_Pa1764573435lle_tv] :
( ( collec1111687558lle_tv
@ ^ [X2: paraco1605129243lle_tv] : ( member266900804lle_tv @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_78_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_79_Collect__cong,axiom,
! [P3: paraco1605129243lle_tv > $o,Q2: paraco1605129243lle_tv > $o] :
( ! [X4: paraco1605129243lle_tv] :
( ( P3 @ X4 )
= ( Q2 @ X4 ) )
=> ( ( collec1111687558lle_tv @ P3 )
= ( collec1111687558lle_tv @ Q2 ) ) ) ).
% Collect_cong
thf(fact_80_Collect__cong,axiom,
! [P3: nat > $o,Q2: nat > $o] :
( ! [X4: nat] :
( ( P3 @ X4 )
= ( Q2 @ X4 ) )
=> ( ( collect_nat @ P3 )
= ( collect_nat @ Q2 ) ) ) ).
% Collect_cong
thf(fact_81_injI,axiom,
! [F3: produc1647334210lle_tv > produc1647334210lle_tv] :
( ! [X4: produc1647334210lle_tv,Y2: produc1647334210lle_tv] :
( ( ( F3 @ X4 )
= ( F3 @ Y2 ) )
=> ( X4 = Y2 ) )
=> ( inj_on2112487900lle_tv @ F3 @ top_to1871154984lle_tv ) ) ).
% injI
thf(fact_82_injI,axiom,
! [F3: nat > paraco1605129243lle_tv] :
( ! [X4: nat,Y2: nat] :
( ( ( F3 @ X4 )
= ( F3 @ Y2 ) )
=> ( X4 = Y2 ) )
=> ( inj_on1955349912lle_tv @ F3 @ top_top_set_nat ) ) ).
% injI
thf(fact_83_injI,axiom,
! [F3: produc194167194tv_nat > produc194167194tv_nat] :
( ! [X4: produc194167194tv_nat,Y2: produc194167194tv_nat] :
( ( ( F3 @ X4 )
= ( F3 @ Y2 ) )
=> ( X4 = Y2 ) )
=> ( inj_on552895452tv_nat @ F3 @ top_to975729152tv_nat ) ) ).
% injI
thf(fact_84_injI,axiom,
! [F3: produc270694976lle_tv > produc270694976lle_tv] :
( ! [X4: produc270694976lle_tv,Y2: produc270694976lle_tv] :
( ( ( F3 @ X4 )
= ( F3 @ Y2 ) )
=> ( X4 = Y2 ) )
=> ( inj_on1745913142lle_tv @ F3 @ top_to1411004326lle_tv ) ) ).
% injI
thf(fact_85_injI,axiom,
! [F3: product_prod_nat_nat > product_prod_nat_nat] :
( ! [X4: product_prod_nat_nat,Y2: product_prod_nat_nat] :
( ( ( F3 @ X4 )
= ( F3 @ Y2 ) )
=> ( X4 = Y2 ) )
=> ( inj_on26176689at_nat @ F3 @ top_to1555900337at_nat ) ) ).
% injI
thf(fact_86_injI,axiom,
! [F3: nat > nat] :
( ! [X4: nat,Y2: nat] :
( ( ( F3 @ X4 )
= ( F3 @ Y2 ) )
=> ( X4 = Y2 ) )
=> ( inj_on_nat_nat @ F3 @ top_top_set_nat ) ) ).
% injI
thf(fact_87_injI,axiom,
! [F3: paraco1605129243lle_tv > paraco1605129243lle_tv] :
( ! [X4: paraco1605129243lle_tv,Y2: paraco1605129243lle_tv] :
( ( ( F3 @ X4 )
= ( F3 @ Y2 ) )
=> ( X4 = Y2 ) )
=> ( inj_on772319074lle_tv @ F3 @ top_to1057771083lle_tv ) ) ).
% injI
thf(fact_88_inj__eq,axiom,
! [F3: produc1647334210lle_tv > produc1647334210lle_tv,X: produc1647334210lle_tv,Y: produc1647334210lle_tv] :
( ( inj_on2112487900lle_tv @ F3 @ top_to1871154984lle_tv )
=> ( ( ( F3 @ X )
= ( F3 @ Y ) )
= ( X = Y ) ) ) ).
% inj_eq
thf(fact_89_inj__eq,axiom,
! [F3: nat > paraco1605129243lle_tv,X: nat,Y: nat] :
( ( inj_on1955349912lle_tv @ F3 @ top_top_set_nat )
=> ( ( ( F3 @ X )
= ( F3 @ Y ) )
= ( X = Y ) ) ) ).
% inj_eq
thf(fact_90_inj__eq,axiom,
! [F3: produc194167194tv_nat > produc194167194tv_nat,X: produc194167194tv_nat,Y: produc194167194tv_nat] :
( ( inj_on552895452tv_nat @ F3 @ top_to975729152tv_nat )
=> ( ( ( F3 @ X )
= ( F3 @ Y ) )
= ( X = Y ) ) ) ).
% inj_eq
thf(fact_91_inj__eq,axiom,
! [F3: produc270694976lle_tv > produc270694976lle_tv,X: produc270694976lle_tv,Y: produc270694976lle_tv] :
( ( inj_on1745913142lle_tv @ F3 @ top_to1411004326lle_tv )
=> ( ( ( F3 @ X )
= ( F3 @ Y ) )
= ( X = Y ) ) ) ).
% inj_eq
thf(fact_92_inj__eq,axiom,
! [F3: product_prod_nat_nat > product_prod_nat_nat,X: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( inj_on26176689at_nat @ F3 @ top_to1555900337at_nat )
=> ( ( ( F3 @ X )
= ( F3 @ Y ) )
= ( X = Y ) ) ) ).
% inj_eq
thf(fact_93_inj__eq,axiom,
! [F3: nat > nat,X: nat,Y: nat] :
( ( inj_on_nat_nat @ F3 @ top_top_set_nat )
=> ( ( ( F3 @ X )
= ( F3 @ Y ) )
= ( X = Y ) ) ) ).
% inj_eq
thf(fact_94_inj__eq,axiom,
! [F3: paraco1605129243lle_tv > paraco1605129243lle_tv,X: paraco1605129243lle_tv,Y: paraco1605129243lle_tv] :
( ( inj_on772319074lle_tv @ F3 @ top_to1057771083lle_tv )
=> ( ( ( F3 @ X )
= ( F3 @ Y ) )
= ( X = Y ) ) ) ).
% inj_eq
thf(fact_95_inj__def,axiom,
! [F3: produc1647334210lle_tv > produc1647334210lle_tv] :
( ( inj_on2112487900lle_tv @ F3 @ top_to1871154984lle_tv )
= ( ! [X2: produc1647334210lle_tv,Y4: produc1647334210lle_tv] :
( ( ( F3 @ X2 )
= ( F3 @ Y4 ) )
=> ( X2 = Y4 ) ) ) ) ).
% inj_def
thf(fact_96_inj__def,axiom,
! [F3: nat > paraco1605129243lle_tv] :
( ( inj_on1955349912lle_tv @ F3 @ top_top_set_nat )
= ( ! [X2: nat,Y4: nat] :
( ( ( F3 @ X2 )
= ( F3 @ Y4 ) )
=> ( X2 = Y4 ) ) ) ) ).
% inj_def
thf(fact_97_inj__def,axiom,
! [F3: produc194167194tv_nat > produc194167194tv_nat] :
( ( inj_on552895452tv_nat @ F3 @ top_to975729152tv_nat )
= ( ! [X2: produc194167194tv_nat,Y4: produc194167194tv_nat] :
( ( ( F3 @ X2 )
= ( F3 @ Y4 ) )
=> ( X2 = Y4 ) ) ) ) ).
% inj_def
thf(fact_98_inj__def,axiom,
! [F3: produc270694976lle_tv > produc270694976lle_tv] :
( ( inj_on1745913142lle_tv @ F3 @ top_to1411004326lle_tv )
= ( ! [X2: produc270694976lle_tv,Y4: produc270694976lle_tv] :
( ( ( F3 @ X2 )
= ( F3 @ Y4 ) )
=> ( X2 = Y4 ) ) ) ) ).
% inj_def
thf(fact_99_inj__def,axiom,
! [F3: product_prod_nat_nat > product_prod_nat_nat] :
( ( inj_on26176689at_nat @ F3 @ top_to1555900337at_nat )
= ( ! [X2: product_prod_nat_nat,Y4: product_prod_nat_nat] :
( ( ( F3 @ X2 )
= ( F3 @ Y4 ) )
=> ( X2 = Y4 ) ) ) ) ).
% inj_def
thf(fact_100_inj__def,axiom,
! [F3: nat > nat] :
( ( inj_on_nat_nat @ F3 @ top_top_set_nat )
= ( ! [X2: nat,Y4: nat] :
( ( ( F3 @ X2 )
= ( F3 @ Y4 ) )
=> ( X2 = Y4 ) ) ) ) ).
% inj_def
thf(fact_101_inj__def,axiom,
! [F3: paraco1605129243lle_tv > paraco1605129243lle_tv] :
( ( inj_on772319074lle_tv @ F3 @ top_to1057771083lle_tv )
= ( ! [X2: paraco1605129243lle_tv,Y4: paraco1605129243lle_tv] :
( ( ( F3 @ X2 )
= ( F3 @ Y4 ) )
=> ( X2 = Y4 ) ) ) ) ).
% inj_def
thf(fact_102_fm_Oinduct,axiom,
! [P3: paraco1604210848lle_fm > $o,Fm: paraco1604210848lle_fm] :
( ! [X4: list_char] : ( P3 @ ( paraco1480028776le_Pro @ X4 ) )
=> ( ( P3 @ paraco367965286_Truth )
=> ( ! [X4: paraco1604210848lle_fm] :
( ( P3 @ X4 )
=> ( P3 @ ( paraco445776468le_Neg @ X4 ) ) )
=> ( ! [X1a: paraco1604210848lle_fm,X22: paraco1604210848lle_fm] :
( ( P3 @ X1a )
=> ( ( P3 @ X22 )
=> ( P3 @ ( paraco69239110le_Con @ X1a @ X22 ) ) ) )
=> ( ! [X1a: paraco1604210848lle_fm,X22: paraco1604210848lle_fm] :
( ( P3 @ X1a )
=> ( ( P3 @ X22 )
=> ( P3 @ ( paraco1389086619le_Eql @ X1a @ X22 ) ) ) )
=> ( ! [X1a: paraco1604210848lle_fm,X22: paraco1604210848lle_fm] :
( ( P3 @ X1a )
=> ( ( P3 @ X22 )
=> ( P3 @ ( paraco1745535428le_Eql @ X1a @ X22 ) ) ) )
=> ( P3 @ Fm ) ) ) ) ) ) ) ).
% fm.induct
thf(fact_103_fm_Oexhaust,axiom,
! [Y: paraco1604210848lle_fm] :
( ! [X12: list_char] :
( Y
!= ( paraco1480028776le_Pro @ X12 ) )
=> ( ( Y != paraco367965286_Truth )
=> ( ! [X32: paraco1604210848lle_fm] :
( Y
!= ( paraco445776468le_Neg @ X32 ) )
=> ( ! [X412: paraco1604210848lle_fm,X422: paraco1604210848lle_fm] :
( Y
!= ( paraco69239110le_Con @ X412 @ X422 ) )
=> ( ! [X512: paraco1604210848lle_fm,X522: paraco1604210848lle_fm] :
( Y
!= ( paraco1389086619le_Eql @ X512 @ X522 ) )
=> ~ ! [X612: paraco1604210848lle_fm,X622: paraco1604210848lle_fm] :
( Y
!= ( paraco1745535428le_Eql @ X612 @ X622 ) ) ) ) ) ) ) ).
% fm.exhaust
thf(fact_104_props_Ocases,axiom,
! [X: paraco1604210848lle_fm] :
( ( X != paraco367965286_Truth )
=> ( ! [S2: list_char] :
( X
!= ( paraco1480028776le_Pro @ S2 ) )
=> ( ! [P4: paraco1604210848lle_fm] :
( X
!= ( paraco445776468le_Neg @ P4 ) )
=> ( ! [P4: paraco1604210848lle_fm,Q3: paraco1604210848lle_fm] :
( X
!= ( paraco69239110le_Con @ P4 @ Q3 ) )
=> ( ! [P4: paraco1604210848lle_fm,Q3: paraco1604210848lle_fm] :
( X
!= ( paraco1389086619le_Eql @ P4 @ Q3 ) )
=> ~ ! [P4: paraco1604210848lle_fm,Q3: paraco1604210848lle_fm] :
( X
!= ( paraco1745535428le_Eql @ P4 @ Q3 ) ) ) ) ) ) ) ).
% props.cases
thf(fact_105_fm_Oinject_I1_J,axiom,
! [X1: list_char,Y1: list_char] :
( ( ( paraco1480028776le_Pro @ X1 )
= ( paraco1480028776le_Pro @ Y1 ) )
= ( X1 = Y1 ) ) ).
% fm.inject(1)
thf(fact_106_top__set__def,axiom,
( top_to975729152tv_nat
= ( collec848462447tv_nat @ top_to900819589_nat_o ) ) ).
% top_set_def
thf(fact_107_top__set__def,axiom,
( top_to1411004326lle_tv
= ( collec924990229lle_tv @ top_to1889334367e_tv_o ) ) ).
% top_set_def
thf(fact_108_top__set__def,axiom,
( top_to1555900337at_nat
= ( collec7649004at_nat @ top_to632227436_nat_o ) ) ).
% top_set_def
thf(fact_109_top__set__def,axiom,
( top_top_set_nat
= ( collect_nat @ top_top_nat_o ) ) ).
% top_set_def
thf(fact_110_top__set__def,axiom,
( top_to1057771083lle_tv
= ( collec1111687558lle_tv @ top_to1299250770e_tv_o ) ) ).
% top_set_def
thf(fact_111_eval_Osimps_I1_J,axiom,
! [I2: list_char > paraco1605129243lle_tv,S3: list_char] :
( ( paraco1325918602e_eval @ I2 @ ( paraco1480028776le_Pro @ S3 ) )
= ( I2 @ S3 ) ) ).
% eval.simps(1)
thf(fact_112_fm_Odistinct_I3_J,axiom,
! [X1: list_char,X3: paraco1604210848lle_fm] :
( ( paraco1480028776le_Pro @ X1 )
!= ( paraco445776468le_Neg @ X3 ) ) ).
% fm.distinct(3)
thf(fact_113_fm_Odistinct_I5_J,axiom,
! [X1: list_char,X41: paraco1604210848lle_fm,X42: paraco1604210848lle_fm] :
( ( paraco1480028776le_Pro @ X1 )
!= ( paraco69239110le_Con @ X41 @ X42 ) ) ).
% fm.distinct(5)
thf(fact_114_fm_Odistinct_I9_J,axiom,
! [X1: list_char,X61: paraco1604210848lle_fm,X62: paraco1604210848lle_fm] :
( ( paraco1480028776le_Pro @ X1 )
!= ( paraco1745535428le_Eql @ X61 @ X62 ) ) ).
% fm.distinct(9)
thf(fact_115_fm_Odistinct_I7_J,axiom,
! [X1: list_char,X51: paraco1604210848lle_fm,X52: paraco1604210848lle_fm] :
( ( paraco1480028776le_Pro @ X1 )
!= ( paraco1389086619le_Eql @ X51 @ X52 ) ) ).
% fm.distinct(7)
thf(fact_116_fm_Odistinct_I1_J,axiom,
! [X1: list_char] :
( ( paraco1480028776le_Pro @ X1 )
!= paraco367965286_Truth ) ).
% fm.distinct(1)
thf(fact_117_change__tv__injection,axiom,
! [F3: nat > nat] :
( ( inj_on_nat_nat @ F3 @ top_top_set_nat )
=> ( inj_on772319074lle_tv @ ( paraco1147068288nge_tv @ F3 ) @ top_to1057771083lle_tv ) ) ).
% change_tv_injection
thf(fact_118_UNIV__witness,axiom,
? [X4: produc696645085lle_fm] : ( member1474794996lle_fm @ X4 @ top_to775056195lle_fm ) ).
% UNIV_witness
thf(fact_119_UNIV__witness,axiom,
? [X4: produc1852595672lle_tv] : ( member146333935lle_tv @ X4 @ top_to138697918lle_tv ) ).
% UNIV_witness
thf(fact_120_UNIV__witness,axiom,
? [X4: produc931990033lle_tv] : ( member1569366824lle_tv @ X4 @ top_to449985143lle_tv ) ).
% UNIV_witness
thf(fact_121_UNIV__witness,axiom,
? [X4: produc1281043295at_nat] : ( member112824072at_nat @ X4 @ top_to1541413903at_nat ) ).
% UNIV_witness
thf(fact_122_UNIV__witness,axiom,
? [X4: produc194167194tv_nat] : ( member737151281tv_nat @ X4 @ top_to975729152tv_nat ) ).
% UNIV_witness
thf(fact_123_UNIV__witness,axiom,
? [X4: produc270694976lle_tv] : ( member813679063lle_tv @ X4 @ top_to1411004326lle_tv ) ).
% UNIV_witness
thf(fact_124_UNIV__witness,axiom,
? [X4: product_prod_nat_nat] : ( member701585322at_nat @ X4 @ top_to1555900337at_nat ) ).
% UNIV_witness
thf(fact_125_UNIV__witness,axiom,
? [X4: nat] : ( member_nat @ X4 @ top_top_set_nat ) ).
% UNIV_witness
thf(fact_126_UNIV__witness,axiom,
? [X4: paraco1605129243lle_tv] : ( member266900804lle_tv @ X4 @ top_to1057771083lle_tv ) ).
% UNIV_witness
thf(fact_127_UNIV__eq__I,axiom,
! [A2: set_Pr2120184979lle_fm] :
( ! [X4: produc696645085lle_fm] : ( member1474794996lle_fm @ X4 @ A2 )
=> ( top_to775056195lle_fm = A2 ) ) ).
% UNIV_eq_I
thf(fact_128_UNIV__eq__I,axiom,
! [A2: set_Pr2061825550lle_tv] :
( ! [X4: produc1852595672lle_tv] : ( member146333935lle_tv @ X4 @ A2 )
=> ( top_to138697918lle_tv = A2 ) ) ).
% UNIV_eq_I
thf(fact_129_UNIV__eq__I,axiom,
! [A2: set_Pr560526279lle_tv] :
( ! [X4: produc931990033lle_tv] : ( member1569366824lle_tv @ X4 @ A2 )
=> ( top_to449985143lle_tv = A2 ) ) ).
% UNIV_eq_I
thf(fact_130_UNIV__eq__I,axiom,
! [A2: set_Pr1357842111at_nat] :
( ! [X4: produc1281043295at_nat] : ( member112824072at_nat @ X4 @ A2 )
=> ( top_to1541413903at_nat = A2 ) ) ).
% UNIV_eq_I
thf(fact_131_UNIV__eq__I,axiom,
! [A2: set_Pr1725653840tv_nat] :
( ! [X4: produc194167194tv_nat] : ( member737151281tv_nat @ X4 @ A2 )
=> ( top_to975729152tv_nat = A2 ) ) ).
% UNIV_eq_I
thf(fact_132_UNIV__eq__I,axiom,
! [A2: set_Pr13445366lle_tv] :
( ! [X4: produc270694976lle_tv] : ( member813679063lle_tv @ X4 @ A2 )
=> ( top_to1411004326lle_tv = A2 ) ) ).
% UNIV_eq_I
thf(fact_133_UNIV__eq__I,axiom,
! [A2: set_Pr1986765409at_nat] :
( ! [X4: product_prod_nat_nat] : ( member701585322at_nat @ X4 @ A2 )
=> ( top_to1555900337at_nat = A2 ) ) ).
% UNIV_eq_I
thf(fact_134_UNIV__eq__I,axiom,
! [A2: set_nat] :
( ! [X4: nat] : ( member_nat @ X4 @ A2 )
=> ( top_top_set_nat = A2 ) ) ).
% UNIV_eq_I
thf(fact_135_UNIV__eq__I,axiom,
! [A2: set_Pa1764573435lle_tv] :
( ! [X4: paraco1605129243lle_tv] : ( member266900804lle_tv @ X4 @ A2 )
=> ( top_to1057771083lle_tv = A2 ) ) ).
% UNIV_eq_I
thf(fact_136_inj__on__inverseI,axiom,
! [A2: set_Pr2014382200lle_tv,G: produc1647334210lle_tv > produc1647334210lle_tv,F3: produc1647334210lle_tv > produc1647334210lle_tv] :
( ! [X4: produc1647334210lle_tv] :
( ( member1531318361lle_tv @ X4 @ A2 )
=> ( ( G @ ( F3 @ X4 ) )
= X4 ) )
=> ( inj_on2112487900lle_tv @ F3 @ A2 ) ) ).
% inj_on_inverseI
thf(fact_137_inj__on__inverseI,axiom,
! [A2: set_Pr1725653840tv_nat,G: produc194167194tv_nat > produc194167194tv_nat,F3: produc194167194tv_nat > produc194167194tv_nat] :
( ! [X4: produc194167194tv_nat] :
( ( member737151281tv_nat @ X4 @ A2 )
=> ( ( G @ ( F3 @ X4 ) )
= X4 ) )
=> ( inj_on552895452tv_nat @ F3 @ A2 ) ) ).
% inj_on_inverseI
thf(fact_138_inj__on__inverseI,axiom,
! [A2: set_Pr13445366lle_tv,G: produc270694976lle_tv > produc270694976lle_tv,F3: produc270694976lle_tv > produc270694976lle_tv] :
( ! [X4: produc270694976lle_tv] :
( ( member813679063lle_tv @ X4 @ A2 )
=> ( ( G @ ( F3 @ X4 ) )
= X4 ) )
=> ( inj_on1745913142lle_tv @ F3 @ A2 ) ) ).
% inj_on_inverseI
thf(fact_139_inj__on__inverseI,axiom,
! [A2: set_Pr1986765409at_nat,G: product_prod_nat_nat > product_prod_nat_nat,F3: product_prod_nat_nat > product_prod_nat_nat] :
( ! [X4: product_prod_nat_nat] :
( ( member701585322at_nat @ X4 @ A2 )
=> ( ( G @ ( F3 @ X4 ) )
= X4 ) )
=> ( inj_on26176689at_nat @ F3 @ A2 ) ) ).
% inj_on_inverseI
thf(fact_140_inj__on__inverseI,axiom,
! [A2: set_nat,G: paraco1605129243lle_tv > nat,F3: nat > paraco1605129243lle_tv] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( G @ ( F3 @ X4 ) )
= X4 ) )
=> ( inj_on1955349912lle_tv @ F3 @ A2 ) ) ).
% inj_on_inverseI
thf(fact_141_inj__on__inverseI,axiom,
! [A2: set_nat,G: nat > nat,F3: nat > nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( G @ ( F3 @ X4 ) )
= X4 ) )
=> ( inj_on_nat_nat @ F3 @ A2 ) ) ).
% inj_on_inverseI
thf(fact_142_inj__on__inverseI,axiom,
! [A2: set_Pa1764573435lle_tv,G: paraco1605129243lle_tv > paraco1605129243lle_tv,F3: paraco1605129243lle_tv > paraco1605129243lle_tv] :
( ! [X4: paraco1605129243lle_tv] :
( ( member266900804lle_tv @ X4 @ A2 )
=> ( ( G @ ( F3 @ X4 ) )
= X4 ) )
=> ( inj_on772319074lle_tv @ F3 @ A2 ) ) ).
% inj_on_inverseI
thf(fact_143_inj__on__contraD,axiom,
! [F3: produc1647334210lle_tv > produc1647334210lle_tv,A2: set_Pr2014382200lle_tv,X: produc1647334210lle_tv,Y: produc1647334210lle_tv] :
( ( inj_on2112487900lle_tv @ F3 @ A2 )
=> ( ( X != Y )
=> ( ( member1531318361lle_tv @ X @ A2 )
=> ( ( member1531318361lle_tv @ Y @ A2 )
=> ( ( F3 @ X )
!= ( F3 @ Y ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_144_inj__on__contraD,axiom,
! [F3: produc194167194tv_nat > produc194167194tv_nat,A2: set_Pr1725653840tv_nat,X: produc194167194tv_nat,Y: produc194167194tv_nat] :
( ( inj_on552895452tv_nat @ F3 @ A2 )
=> ( ( X != Y )
=> ( ( member737151281tv_nat @ X @ A2 )
=> ( ( member737151281tv_nat @ Y @ A2 )
=> ( ( F3 @ X )
!= ( F3 @ Y ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_145_inj__on__contraD,axiom,
! [F3: produc270694976lle_tv > produc270694976lle_tv,A2: set_Pr13445366lle_tv,X: produc270694976lle_tv,Y: produc270694976lle_tv] :
( ( inj_on1745913142lle_tv @ F3 @ A2 )
=> ( ( X != Y )
=> ( ( member813679063lle_tv @ X @ A2 )
=> ( ( member813679063lle_tv @ Y @ A2 )
=> ( ( F3 @ X )
!= ( F3 @ Y ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_146_inj__on__contraD,axiom,
! [F3: product_prod_nat_nat > product_prod_nat_nat,A2: set_Pr1986765409at_nat,X: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( inj_on26176689at_nat @ F3 @ A2 )
=> ( ( X != Y )
=> ( ( member701585322at_nat @ X @ A2 )
=> ( ( member701585322at_nat @ Y @ A2 )
=> ( ( F3 @ X )
!= ( F3 @ Y ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_147_inj__on__contraD,axiom,
! [F3: nat > paraco1605129243lle_tv,A2: set_nat,X: nat,Y: nat] :
( ( inj_on1955349912lle_tv @ F3 @ A2 )
=> ( ( X != Y )
=> ( ( member_nat @ X @ A2 )
=> ( ( member_nat @ Y @ A2 )
=> ( ( F3 @ X )
!= ( F3 @ Y ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_148_inj__on__contraD,axiom,
! [F3: nat > nat,A2: set_nat,X: nat,Y: nat] :
( ( inj_on_nat_nat @ F3 @ A2 )
=> ( ( X != Y )
=> ( ( member_nat @ X @ A2 )
=> ( ( member_nat @ Y @ A2 )
=> ( ( F3 @ X )
!= ( F3 @ Y ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_149_inj__on__contraD,axiom,
! [F3: paraco1605129243lle_tv > paraco1605129243lle_tv,A2: set_Pa1764573435lle_tv,X: paraco1605129243lle_tv,Y: paraco1605129243lle_tv] :
( ( inj_on772319074lle_tv @ F3 @ A2 )
=> ( ( X != Y )
=> ( ( member266900804lle_tv @ X @ A2 )
=> ( ( member266900804lle_tv @ Y @ A2 )
=> ( ( F3 @ X )
!= ( F3 @ Y ) ) ) ) ) ) ).
% inj_on_contraD
thf(fact_150_inj__on__eq__iff,axiom,
! [F3: produc1647334210lle_tv > produc1647334210lle_tv,A2: set_Pr2014382200lle_tv,X: produc1647334210lle_tv,Y: produc1647334210lle_tv] :
( ( inj_on2112487900lle_tv @ F3 @ A2 )
=> ( ( member1531318361lle_tv @ X @ A2 )
=> ( ( member1531318361lle_tv @ Y @ A2 )
=> ( ( ( F3 @ X )
= ( F3 @ Y ) )
= ( X = Y ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_151_inj__on__eq__iff,axiom,
! [F3: produc194167194tv_nat > produc194167194tv_nat,A2: set_Pr1725653840tv_nat,X: produc194167194tv_nat,Y: produc194167194tv_nat] :
( ( inj_on552895452tv_nat @ F3 @ A2 )
=> ( ( member737151281tv_nat @ X @ A2 )
=> ( ( member737151281tv_nat @ Y @ A2 )
=> ( ( ( F3 @ X )
= ( F3 @ Y ) )
= ( X = Y ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_152_inj__on__eq__iff,axiom,
! [F3: produc270694976lle_tv > produc270694976lle_tv,A2: set_Pr13445366lle_tv,X: produc270694976lle_tv,Y: produc270694976lle_tv] :
( ( inj_on1745913142lle_tv @ F3 @ A2 )
=> ( ( member813679063lle_tv @ X @ A2 )
=> ( ( member813679063lle_tv @ Y @ A2 )
=> ( ( ( F3 @ X )
= ( F3 @ Y ) )
= ( X = Y ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_153_inj__on__eq__iff,axiom,
! [F3: product_prod_nat_nat > product_prod_nat_nat,A2: set_Pr1986765409at_nat,X: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( inj_on26176689at_nat @ F3 @ A2 )
=> ( ( member701585322at_nat @ X @ A2 )
=> ( ( member701585322at_nat @ Y @ A2 )
=> ( ( ( F3 @ X )
= ( F3 @ Y ) )
= ( X = Y ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_154_inj__on__eq__iff,axiom,
! [F3: nat > paraco1605129243lle_tv,A2: set_nat,X: nat,Y: nat] :
( ( inj_on1955349912lle_tv @ F3 @ A2 )
=> ( ( member_nat @ X @ A2 )
=> ( ( member_nat @ Y @ A2 )
=> ( ( ( F3 @ X )
= ( F3 @ Y ) )
= ( X = Y ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_155_inj__on__eq__iff,axiom,
! [F3: nat > nat,A2: set_nat,X: nat,Y: nat] :
( ( inj_on_nat_nat @ F3 @ A2 )
=> ( ( member_nat @ X @ A2 )
=> ( ( member_nat @ Y @ A2 )
=> ( ( ( F3 @ X )
= ( F3 @ Y ) )
= ( X = Y ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_156_inj__on__eq__iff,axiom,
! [F3: paraco1605129243lle_tv > paraco1605129243lle_tv,A2: set_Pa1764573435lle_tv,X: paraco1605129243lle_tv,Y: paraco1605129243lle_tv] :
( ( inj_on772319074lle_tv @ F3 @ A2 )
=> ( ( member266900804lle_tv @ X @ A2 )
=> ( ( member266900804lle_tv @ Y @ A2 )
=> ( ( ( F3 @ X )
= ( F3 @ Y ) )
= ( X = Y ) ) ) ) ) ).
% inj_on_eq_iff
thf(fact_157_inj__on__cong,axiom,
! [A2: set_Pr2014382200lle_tv,F3: produc1647334210lle_tv > produc1647334210lle_tv,G: produc1647334210lle_tv > produc1647334210lle_tv] :
( ! [A3: produc1647334210lle_tv] :
( ( member1531318361lle_tv @ A3 @ A2 )
=> ( ( F3 @ A3 )
= ( G @ A3 ) ) )
=> ( ( inj_on2112487900lle_tv @ F3 @ A2 )
= ( inj_on2112487900lle_tv @ G @ A2 ) ) ) ).
% inj_on_cong
thf(fact_158_inj__on__cong,axiom,
! [A2: set_Pr1725653840tv_nat,F3: produc194167194tv_nat > produc194167194tv_nat,G: produc194167194tv_nat > produc194167194tv_nat] :
( ! [A3: produc194167194tv_nat] :
( ( member737151281tv_nat @ A3 @ A2 )
=> ( ( F3 @ A3 )
= ( G @ A3 ) ) )
=> ( ( inj_on552895452tv_nat @ F3 @ A2 )
= ( inj_on552895452tv_nat @ G @ A2 ) ) ) ).
% inj_on_cong
thf(fact_159_inj__on__cong,axiom,
! [A2: set_Pr13445366lle_tv,F3: produc270694976lle_tv > produc270694976lle_tv,G: produc270694976lle_tv > produc270694976lle_tv] :
( ! [A3: produc270694976lle_tv] :
( ( member813679063lle_tv @ A3 @ A2 )
=> ( ( F3 @ A3 )
= ( G @ A3 ) ) )
=> ( ( inj_on1745913142lle_tv @ F3 @ A2 )
= ( inj_on1745913142lle_tv @ G @ A2 ) ) ) ).
% inj_on_cong
thf(fact_160_inj__on__cong,axiom,
! [A2: set_Pr1986765409at_nat,F3: product_prod_nat_nat > product_prod_nat_nat,G: product_prod_nat_nat > product_prod_nat_nat] :
( ! [A3: product_prod_nat_nat] :
( ( member701585322at_nat @ A3 @ A2 )
=> ( ( F3 @ A3 )
= ( G @ A3 ) ) )
=> ( ( inj_on26176689at_nat @ F3 @ A2 )
= ( inj_on26176689at_nat @ G @ A2 ) ) ) ).
% inj_on_cong
thf(fact_161_inj__on__cong,axiom,
! [A2: set_nat,F3: nat > paraco1605129243lle_tv,G: nat > paraco1605129243lle_tv] :
( ! [A3: nat] :
( ( member_nat @ A3 @ A2 )
=> ( ( F3 @ A3 )
= ( G @ A3 ) ) )
=> ( ( inj_on1955349912lle_tv @ F3 @ A2 )
= ( inj_on1955349912lle_tv @ G @ A2 ) ) ) ).
% inj_on_cong
thf(fact_162_inj__on__cong,axiom,
! [A2: set_nat,F3: nat > nat,G: nat > nat] :
( ! [A3: nat] :
( ( member_nat @ A3 @ A2 )
=> ( ( F3 @ A3 )
= ( G @ A3 ) ) )
=> ( ( inj_on_nat_nat @ F3 @ A2 )
= ( inj_on_nat_nat @ G @ A2 ) ) ) ).
% inj_on_cong
thf(fact_163_inj__on__cong,axiom,
! [A2: set_Pa1764573435lle_tv,F3: paraco1605129243lle_tv > paraco1605129243lle_tv,G: paraco1605129243lle_tv > paraco1605129243lle_tv] :
( ! [A3: paraco1605129243lle_tv] :
( ( member266900804lle_tv @ A3 @ A2 )
=> ( ( F3 @ A3 )
= ( G @ A3 ) ) )
=> ( ( inj_on772319074lle_tv @ F3 @ A2 )
= ( inj_on772319074lle_tv @ G @ A2 ) ) ) ).
% inj_on_cong
thf(fact_164_inj__on__def,axiom,
( inj_on_nat_nat
= ( ^ [F: nat > nat,A4: set_nat] :
! [X2: nat] :
( ( member_nat @ X2 @ A4 )
=> ! [Y4: nat] :
( ( member_nat @ Y4 @ A4 )
=> ( ( ( F @ X2 )
= ( F @ Y4 ) )
=> ( X2 = Y4 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_165_inj__on__def,axiom,
( inj_on772319074lle_tv
= ( ^ [F: paraco1605129243lle_tv > paraco1605129243lle_tv,A4: set_Pa1764573435lle_tv] :
! [X2: paraco1605129243lle_tv] :
( ( member266900804lle_tv @ X2 @ A4 )
=> ! [Y4: paraco1605129243lle_tv] :
( ( member266900804lle_tv @ Y4 @ A4 )
=> ( ( ( F @ X2 )
= ( F @ Y4 ) )
=> ( X2 = Y4 ) ) ) ) ) ) ).
% inj_on_def
thf(fact_166_inj__onI,axiom,
! [A2: set_nat,F3: nat > nat] :
( ! [X4: nat,Y2: nat] :
( ( member_nat @ X4 @ A2 )
=> ( ( member_nat @ Y2 @ A2 )
=> ( ( ( F3 @ X4 )
= ( F3 @ Y2 ) )
=> ( X4 = Y2 ) ) ) )
=> ( inj_on_nat_nat @ F3 @ A2 ) ) ).
% inj_onI
thf(fact_167_inj__onI,axiom,
! [A2: set_Pa1764573435lle_tv,F3: paraco1605129243lle_tv > paraco1605129243lle_tv] :
( ! [X4: paraco1605129243lle_tv,Y2: paraco1605129243lle_tv] :
( ( member266900804lle_tv @ X4 @ A2 )
=> ( ( member266900804lle_tv @ Y2 @ A2 )
=> ( ( ( F3 @ X4 )
= ( F3 @ Y2 ) )
=> ( X4 = Y2 ) ) ) )
=> ( inj_on772319074lle_tv @ F3 @ A2 ) ) ).
% inj_onI
thf(fact_168_inj__onD,axiom,
! [F3: nat > nat,A2: set_nat,X: nat,Y: nat] :
( ( inj_on_nat_nat @ F3 @ A2 )
=> ( ( ( F3 @ X )
= ( F3 @ Y ) )
=> ( ( member_nat @ X @ A2 )
=> ( ( member_nat @ Y @ A2 )
=> ( X = Y ) ) ) ) ) ).
% inj_onD
thf(fact_169_inj__onD,axiom,
! [F3: paraco1605129243lle_tv > paraco1605129243lle_tv,A2: set_Pa1764573435lle_tv,X: paraco1605129243lle_tv,Y: paraco1605129243lle_tv] :
( ( inj_on772319074lle_tv @ F3 @ A2 )
=> ( ( ( F3 @ X )
= ( F3 @ Y ) )
=> ( ( member266900804lle_tv @ X @ A2 )
=> ( ( member266900804lle_tv @ Y @ A2 )
=> ( X = Y ) ) ) ) ) ).
% inj_onD
thf(fact_170_props_Oinduct,axiom,
! [P3: paraco1604210848lle_fm > $o,A0: paraco1604210848lle_fm] :
( ( P3 @ paraco367965286_Truth )
=> ( ! [S2: list_char] : ( P3 @ ( paraco1480028776le_Pro @ S2 ) )
=> ( ! [P4: paraco1604210848lle_fm] :
( ( P3 @ P4 )
=> ( P3 @ ( paraco445776468le_Neg @ P4 ) ) )
=> ( ! [P4: paraco1604210848lle_fm,Q3: paraco1604210848lle_fm] :
( ( P3 @ P4 )
=> ( ( P3 @ Q3 )
=> ( P3 @ ( paraco69239110le_Con @ P4 @ Q3 ) ) ) )
=> ( ! [P4: paraco1604210848lle_fm,Q3: paraco1604210848lle_fm] :
( ( P3 @ P4 )
=> ( ( P3 @ Q3 )
=> ( P3 @ ( paraco1389086619le_Eql @ P4 @ Q3 ) ) ) )
=> ( ! [P4: paraco1604210848lle_fm,Q3: paraco1604210848lle_fm] :
( ( P3 @ P4 )
=> ( ( P3 @ Q3 )
=> ( P3 @ ( paraco1745535428le_Eql @ P4 @ Q3 ) ) ) )
=> ( P3 @ A0 ) ) ) ) ) ) ) ).
% props.induct
thf(fact_171_eval_Ocases,axiom,
! [X: produc696645085lle_fm] :
( ! [I3: list_char > paraco1605129243lle_tv,S2: list_char] :
( X
!= ( produc1710735575lle_fm @ I3 @ ( paraco1480028776le_Pro @ S2 ) ) )
=> ( ! [I3: list_char > paraco1605129243lle_tv] :
( X
!= ( produc1710735575lle_fm @ I3 @ paraco367965286_Truth ) )
=> ( ! [I3: list_char > paraco1605129243lle_tv,P4: paraco1604210848lle_fm] :
( X
!= ( produc1710735575lle_fm @ I3 @ ( paraco445776468le_Neg @ P4 ) ) )
=> ( ! [I3: list_char > paraco1605129243lle_tv,P4: paraco1604210848lle_fm,Q3: paraco1604210848lle_fm] :
( X
!= ( produc1710735575lle_fm @ I3 @ ( paraco69239110le_Con @ P4 @ Q3 ) ) )
=> ( ! [I3: list_char > paraco1605129243lle_tv,P4: paraco1604210848lle_fm,Q3: paraco1604210848lle_fm] :
( X
!= ( produc1710735575lle_fm @ I3 @ ( paraco1389086619le_Eql @ P4 @ Q3 ) ) )
=> ~ ! [I3: list_char > paraco1605129243lle_tv,P4: paraco1604210848lle_fm,Q3: paraco1604210848lle_fm] :
( X
!= ( produc1710735575lle_fm @ I3 @ ( paraco1745535428le_Eql @ P4 @ Q3 ) ) ) ) ) ) ) ) ).
% eval.cases
thf(fact_172_top__empty__eq,axiom,
( top_top_nat_o
= ( ^ [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ) ) ).
% top_empty_eq
thf(fact_173_top__empty__eq,axiom,
( top_to1299250770e_tv_o
= ( ^ [X2: paraco1605129243lle_tv] : ( member266900804lle_tv @ X2 @ top_to1057771083lle_tv ) ) ) ).
% top_empty_eq
thf(fact_174_change__tv_Oelims,axiom,
! [X: nat > nat,Xa: paraco1605129243lle_tv,Y: paraco1605129243lle_tv] :
( ( ( paraco1147068288nge_tv @ X @ Xa )
= Y )
=> ( ! [B2: $o] :
( ( Xa
= ( paraco1344940915le_Det @ B2 ) )
=> ( Y
!= ( paraco1344940915le_Det @ B2 ) ) )
=> ~ ! [N: nat] :
( ( Xa
= ( paraco793048302_Indet @ N ) )
=> ( Y
!= ( paraco793048302_Indet @ ( X @ N ) ) ) ) ) ) ).
% change_tv.elims
thf(fact_175_valid__valid__in,axiom,
! [P2: paraco1604210848lle_fm,U: set_nat] :
( ( paraco327635598_valid @ P2 )
=> ( paraco55203475lid_in @ U @ P2 ) ) ).
% valid_valid_in
thf(fact_176_transfer,axiom,
! [U: set_nat,P2: paraco1604210848lle_fm] :
( ~ ( paraco55203475lid_in @ U @ P2 )
=> ~ ( paraco327635598_valid @ P2 ) ) ).
% transfer
thf(fact_177_the__inv__f__f,axiom,
! [F3: nat > nat,X: nat] :
( ( inj_on_nat_nat @ F3 @ top_top_set_nat )
=> ( ( the_inv_into_nat_nat @ top_top_set_nat @ F3 @ ( F3 @ X ) )
= X ) ) ).
% the_inv_f_f
thf(fact_178_the__inv__f__f,axiom,
! [F3: paraco1605129243lle_tv > paraco1605129243lle_tv,X: paraco1605129243lle_tv] :
( ( inj_on772319074lle_tv @ F3 @ top_to1057771083lle_tv )
=> ( ( the_in452116896lle_tv @ top_to1057771083lle_tv @ F3 @ ( F3 @ X ) )
= X ) ) ).
% the_inv_f_f
thf(fact_179_conjunction__in,axiom,
! [U: set_nat,P2: paraco1604210848lle_fm,Q: paraco1604210848lle_fm] :
( ( paraco55203475lid_in @ U @ ( paraco69239110le_Con @ P2 @ Q ) )
= ( ( paraco55203475lid_in @ U @ P2 )
& ( paraco55203475lid_in @ U @ Q ) ) ) ).
% conjunction_in
thf(fact_180_tv_Oinject_I2_J,axiom,
! [X23: nat,Y22: nat] :
( ( ( paraco793048302_Indet @ X23 )
= ( paraco793048302_Indet @ Y22 ) )
= ( X23 = Y22 ) ) ).
% tv.inject(2)
thf(fact_181_string__tv_Oinduct,axiom,
! [P3: paraco1605129243lle_tv > $o,A0: paraco1605129243lle_tv] :
( ( P3 @ ( paraco1344940915le_Det @ $true ) )
=> ( ( P3 @ ( paraco1344940915le_Det @ $false ) )
=> ( ! [N: nat] : ( P3 @ ( paraco793048302_Indet @ N ) )
=> ( P3 @ A0 ) ) ) ) ).
% string_tv.induct
thf(fact_182_change__tv_Oinduct,axiom,
! [P3: ( nat > nat ) > paraco1605129243lle_tv > $o,A0: nat > nat,A1: paraco1605129243lle_tv] :
( ! [F4: nat > nat,B2: $o] : ( P3 @ F4 @ ( paraco1344940915le_Det @ B2 ) )
=> ( ! [F4: nat > nat,N: nat] : ( P3 @ F4 @ ( paraco793048302_Indet @ N ) )
=> ( P3 @ A0 @ A1 ) ) ) ).
% change_tv.induct
thf(fact_183_string__tv_Ocases,axiom,
! [X: paraco1605129243lle_tv] :
( ( X
!= ( paraco1344940915le_Det @ $true ) )
=> ( ( X
!= ( paraco1344940915le_Det @ $false ) )
=> ~ ! [N: nat] :
( X
!= ( paraco793048302_Indet @ N ) ) ) ) ).
% string_tv.cases
thf(fact_184_tv_Oexhaust,axiom,
! [Y: paraco1605129243lle_tv] :
( ! [X12: $o] :
( Y
!= ( paraco1344940915le_Det @ X12 ) )
=> ~ ! [X22: nat] :
( Y
!= ( paraco793048302_Indet @ X22 ) ) ) ).
% tv.exhaust
thf(fact_185_tv_Oinduct,axiom,
! [P3: paraco1605129243lle_tv > $o,Tv: paraco1605129243lle_tv] :
( ! [X4: $o] : ( P3 @ ( paraco1344940915le_Det @ X4 ) )
=> ( ! [X4: nat] : ( P3 @ ( paraco793048302_Indet @ X4 ) )
=> ( P3 @ Tv ) ) ) ).
% tv.induct
thf(fact_186_tv_Odistinct_I1_J,axiom,
! [X1: $o,X23: nat] :
( ( paraco1344940915le_Det @ X1 )
!= ( paraco793048302_Indet @ X23 ) ) ).
% tv.distinct(1)
thf(fact_187_the__inv__into__f__f,axiom,
! [F3: nat > nat,A2: set_nat,X: nat] :
( ( inj_on_nat_nat @ F3 @ A2 )
=> ( ( member_nat @ X @ A2 )
=> ( ( the_inv_into_nat_nat @ A2 @ F3 @ ( F3 @ X ) )
= X ) ) ) ).
% the_inv_into_f_f
thf(fact_188_the__inv__into__f__f,axiom,
! [F3: paraco1605129243lle_tv > paraco1605129243lle_tv,A2: set_Pa1764573435lle_tv,X: paraco1605129243lle_tv] :
( ( inj_on772319074lle_tv @ F3 @ A2 )
=> ( ( member266900804lle_tv @ X @ A2 )
=> ( ( the_in452116896lle_tv @ A2 @ F3 @ ( F3 @ X ) )
= X ) ) ) ).
% the_inv_into_f_f
thf(fact_189_the__inv__into__f__eq,axiom,
! [F3: nat > nat,A2: set_nat,X: nat,Y: nat] :
( ( inj_on_nat_nat @ F3 @ A2 )
=> ( ( ( F3 @ X )
= Y )
=> ( ( member_nat @ X @ A2 )
=> ( ( the_inv_into_nat_nat @ A2 @ F3 @ Y )
= X ) ) ) ) ).
% the_inv_into_f_eq
thf(fact_190_the__inv__into__f__eq,axiom,
! [F3: paraco1605129243lle_tv > paraco1605129243lle_tv,A2: set_Pa1764573435lle_tv,X: paraco1605129243lle_tv,Y: paraco1605129243lle_tv] :
( ( inj_on772319074lle_tv @ F3 @ A2 )
=> ( ( ( F3 @ X )
= Y )
=> ( ( member266900804lle_tv @ X @ A2 )
=> ( ( the_in452116896lle_tv @ A2 @ F3 @ Y )
= X ) ) ) ) ).
% the_inv_into_f_eq
thf(fact_191_tv_Osimps_I6_J,axiom,
! [F1: $o > paraco1605129243lle_tv,F2: nat > paraco1605129243lle_tv,X23: nat] :
( ( paraco2051596139lle_tv @ F1 @ F2 @ ( paraco793048302_Indet @ X23 ) )
= ( F2 @ X23 ) ) ).
% tv.simps(6)
thf(fact_192_change__tv_Osimps_I2_J,axiom,
! [F3: nat > nat,N2: nat] :
( ( paraco1147068288nge_tv @ F3 @ ( paraco793048302_Indet @ N2 ) )
= ( paraco793048302_Indet @ ( F3 @ N2 ) ) ) ).
% change_tv.simps(2)
thf(fact_193_prod_Oinject,axiom,
! [X1: list_char > paraco1605129243lle_tv,X23: paraco1604210848lle_fm,Y1: list_char > paraco1605129243lle_tv,Y22: paraco1604210848lle_fm] :
( ( ( produc1710735575lle_fm @ X1 @ X23 )
= ( produc1710735575lle_fm @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X23 = Y22 ) ) ) ).
% prod.inject
thf(fact_194_prod_Oinject,axiom,
! [X1: nat > nat,X23: paraco1605129243lle_tv,Y1: nat > nat,Y22: paraco1605129243lle_tv] :
( ( ( produc1680557635lle_tv @ X1 @ X23 )
= ( produc1680557635lle_tv @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X23 = Y22 ) ) ) ).
% prod.inject
thf(fact_195_old_Oprod_Oinject,axiom,
! [A: list_char > paraco1605129243lle_tv,B: paraco1604210848lle_fm,A5: list_char > paraco1605129243lle_tv,B3: paraco1604210848lle_fm] :
( ( ( produc1710735575lle_fm @ A @ B )
= ( produc1710735575lle_fm @ A5 @ B3 ) )
= ( ( A = A5 )
& ( B = B3 ) ) ) ).
% old.prod.inject
thf(fact_196_old_Oprod_Oinject,axiom,
! [A: nat > nat,B: paraco1605129243lle_tv,A5: nat > nat,B3: paraco1605129243lle_tv] :
( ( ( produc1680557635lle_tv @ A @ B )
= ( produc1680557635lle_tv @ A5 @ B3 ) )
= ( ( A = A5 )
& ( B = B3 ) ) ) ).
% old.prod.inject
thf(fact_197_eval_Osimps_I3_J,axiom,
! [I2: list_char > paraco1605129243lle_tv,P2: paraco1604210848lle_fm] :
( ( paraco1325918602e_eval @ I2 @ ( paraco445776468le_Neg @ P2 ) )
= ( paraco2051596139lle_tv @ ( produc234482697lle_tv @ ( paraco1344940915le_Det @ $false ) @ ( paraco1344940915le_Det @ $true ) ) @ paraco793048302_Indet @ ( paraco1325918602e_eval @ I2 @ P2 ) ) ) ).
% eval.simps(3)
thf(fact_198_surj__pair,axiom,
! [P2: produc696645085lle_fm] :
? [X4: list_char > paraco1605129243lle_tv,Y2: paraco1604210848lle_fm] :
( P2
= ( produc1710735575lle_fm @ X4 @ Y2 ) ) ).
% surj_pair
thf(fact_199_surj__pair,axiom,
! [P2: produc931990033lle_tv] :
? [X4: nat > nat,Y2: paraco1605129243lle_tv] :
( P2
= ( produc1680557635lle_tv @ X4 @ Y2 ) ) ).
% surj_pair
thf(fact_200_bool_Osplit__sel__asm,axiom,
! [P3: paraco1605129243lle_tv > $o,F1: paraco1605129243lle_tv,F2: paraco1605129243lle_tv,Bool: $o] :
( ( P3 @ ( produc234482697lle_tv @ F1 @ F2 @ Bool ) )
= ( ~ ( ( Bool
& ~ ( P3 @ F1 ) )
| ( ~ Bool
& ~ ( P3 @ F2 ) ) ) ) ) ).
% bool.split_sel_asm
thf(fact_201_bool_Ocase__distrib,axiom,
! [H: paraco1605129243lle_tv > paraco1605129243lle_tv,F1: paraco1605129243lle_tv,F2: paraco1605129243lle_tv,Bool: $o] :
( ( H @ ( produc234482697lle_tv @ F1 @ F2 @ Bool ) )
= ( produc234482697lle_tv @ ( H @ F1 ) @ ( H @ F2 ) @ Bool ) ) ).
% bool.case_distrib
thf(fact_202_bool_Ocase__eq__if,axiom,
( produc234482697lle_tv
= ( ^ [F12: paraco1605129243lle_tv,F22: paraco1605129243lle_tv,Bool2: $o] : ( if_Par1134130401lle_tv @ Bool2 @ F12 @ F22 ) ) ) ).
% bool.case_eq_if
thf(fact_203_bool_Osplit__sel,axiom,
! [P3: paraco1605129243lle_tv > $o,F1: paraco1605129243lle_tv,F2: paraco1605129243lle_tv,Bool: $o] :
( ( P3 @ ( produc234482697lle_tv @ F1 @ F2 @ Bool ) )
= ( ( Bool
=> ( P3 @ F1 ) )
& ( ~ Bool
=> ( P3 @ F2 ) ) ) ) ).
% bool.split_sel
thf(fact_204_old_Obool_Osimps_I3_J,axiom,
! [F1: paraco1605129243lle_tv,F2: paraco1605129243lle_tv] :
( ( produc234482697lle_tv @ F1 @ F2 @ $true )
= F1 ) ).
% old.bool.simps(3)
thf(fact_205_old_Obool_Osimps_I4_J,axiom,
! [F1: paraco1605129243lle_tv,F2: paraco1605129243lle_tv] :
( ( produc234482697lle_tv @ F1 @ F2 @ $false )
= F2 ) ).
% old.bool.simps(4)
thf(fact_206_change__tv_Ocases,axiom,
! [X: produc931990033lle_tv] :
( ! [F4: nat > nat,B2: $o] :
( X
!= ( produc1680557635lle_tv @ F4 @ ( paraco1344940915le_Det @ B2 ) ) )
=> ~ ! [F4: nat > nat,N: nat] :
( X
!= ( produc1680557635lle_tv @ F4 @ ( paraco793048302_Indet @ N ) ) ) ) ).
% change_tv.cases
thf(fact_207_old_Oprod_Oinducts,axiom,
! [P3: produc696645085lle_fm > $o,Prod: produc696645085lle_fm] :
( ! [A3: list_char > paraco1605129243lle_tv,B2: paraco1604210848lle_fm] : ( P3 @ ( produc1710735575lle_fm @ A3 @ B2 ) )
=> ( P3 @ Prod ) ) ).
% old.prod.inducts
thf(fact_208_old_Oprod_Oinducts,axiom,
! [P3: produc931990033lle_tv > $o,Prod: produc931990033lle_tv] :
( ! [A3: nat > nat,B2: paraco1605129243lle_tv] : ( P3 @ ( produc1680557635lle_tv @ A3 @ B2 ) )
=> ( P3 @ Prod ) ) ).
% old.prod.inducts
thf(fact_209_old_Oprod_Oexhaust,axiom,
! [Y: produc696645085lle_fm] :
~ ! [A3: list_char > paraco1605129243lle_tv,B2: paraco1604210848lle_fm] :
( Y
!= ( produc1710735575lle_fm @ A3 @ B2 ) ) ).
% old.prod.exhaust
thf(fact_210_old_Oprod_Oexhaust,axiom,
! [Y: produc931990033lle_tv] :
~ ! [A3: nat > nat,B2: paraco1605129243lle_tv] :
( Y
!= ( produc1680557635lle_tv @ A3 @ B2 ) ) ).
% old.prod.exhaust
thf(fact_211_Pair__inject,axiom,
! [A: list_char > paraco1605129243lle_tv,B: paraco1604210848lle_fm,A5: list_char > paraco1605129243lle_tv,B3: paraco1604210848lle_fm] :
( ( ( produc1710735575lle_fm @ A @ B )
= ( produc1710735575lle_fm @ A5 @ B3 ) )
=> ~ ( ( A = A5 )
=> ( B != B3 ) ) ) ).
% Pair_inject
thf(fact_212_Pair__inject,axiom,
! [A: nat > nat,B: paraco1605129243lle_tv,A5: nat > nat,B3: paraco1605129243lle_tv] :
( ( ( produc1680557635lle_tv @ A @ B )
= ( produc1680557635lle_tv @ A5 @ B3 ) )
=> ~ ( ( A = A5 )
=> ( B != B3 ) ) ) ).
% Pair_inject
thf(fact_213_prod__cases,axiom,
! [P3: produc696645085lle_fm > $o,P2: produc696645085lle_fm] :
( ! [A3: list_char > paraco1605129243lle_tv,B2: paraco1604210848lle_fm] : ( P3 @ ( produc1710735575lle_fm @ A3 @ B2 ) )
=> ( P3 @ P2 ) ) ).
% prod_cases
thf(fact_214_prod__cases,axiom,
! [P3: produc931990033lle_tv > $o,P2: produc931990033lle_tv] :
( ! [A3: nat > nat,B2: paraco1605129243lle_tv] : ( P3 @ ( produc1680557635lle_tv @ A3 @ B2 ) )
=> ( P3 @ P2 ) ) ).
% prod_cases
thf(fact_215_ssubst__Pair__rhs,axiom,
! [R: list_char > paraco1605129243lle_tv,S3: paraco1604210848lle_fm,R2: set_Pr2120184979lle_fm,S4: paraco1604210848lle_fm] :
( ( member1474794996lle_fm @ ( produc1710735575lle_fm @ R @ S3 ) @ R2 )
=> ( ( S4 = S3 )
=> ( member1474794996lle_fm @ ( produc1710735575lle_fm @ R @ S4 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_216_ssubst__Pair__rhs,axiom,
! [R: nat > nat,S3: paraco1605129243lle_tv,R2: set_Pr560526279lle_tv,S4: paraco1605129243lle_tv] :
( ( member1569366824lle_tv @ ( produc1680557635lle_tv @ R @ S3 ) @ R2 )
=> ( ( S4 = S3 )
=> ( member1569366824lle_tv @ ( produc1680557635lle_tv @ R @ S4 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_217_apfst__conv,axiom,
! [F3: ( list_char > paraco1605129243lle_tv ) > list_char > paraco1605129243lle_tv,X: list_char > paraco1605129243lle_tv,Y: paraco1604210848lle_fm] :
( ( produc1942165868lle_fm @ F3 @ ( produc1710735575lle_fm @ X @ Y ) )
= ( produc1710735575lle_fm @ ( F3 @ X ) @ Y ) ) ).
% apfst_conv
thf(fact_218_apfst__conv,axiom,
! [F3: ( nat > nat ) > nat > nat,X: nat > nat,Y: paraco1605129243lle_tv] :
( ( produc939316703lle_tv @ F3 @ ( produc1680557635lle_tv @ X @ Y ) )
= ( produc1680557635lle_tv @ ( F3 @ X ) @ Y ) ) ).
% apfst_conv
thf(fact_219_apsnd__conv,axiom,
! [F3: paraco1604210848lle_fm > paraco1604210848lle_fm,X: list_char > paraco1605129243lle_tv,Y: paraco1604210848lle_fm] :
( ( produc1807466521lle_tv @ F3 @ ( produc1710735575lle_fm @ X @ Y ) )
= ( produc1710735575lle_fm @ X @ ( F3 @ Y ) ) ) ).
% apsnd_conv
thf(fact_220_apsnd__conv,axiom,
! [F3: paraco1605129243lle_tv > paraco1605129243lle_tv,X: nat > nat,Y: paraco1605129243lle_tv] :
( ( produc109093078at_nat @ F3 @ ( produc1680557635lle_tv @ X @ Y ) )
= ( produc1680557635lle_tv @ X @ ( F3 @ Y ) ) ) ).
% apsnd_conv
thf(fact_221_change__tv_Opelims,axiom,
! [X: nat > nat,Xa: paraco1605129243lle_tv,Y: paraco1605129243lle_tv] :
( ( ( paraco1147068288nge_tv @ X @ Xa )
= Y )
=> ( ( accp_P2085826120lle_tv @ paraco1632582009tv_rel @ ( produc1680557635lle_tv @ X @ Xa ) )
=> ( ! [B2: $o] :
( ( Xa
= ( paraco1344940915le_Det @ B2 ) )
=> ( ( Y
= ( paraco1344940915le_Det @ B2 ) )
=> ~ ( accp_P2085826120lle_tv @ paraco1632582009tv_rel @ ( produc1680557635lle_tv @ X @ ( paraco1344940915le_Det @ B2 ) ) ) ) )
=> ~ ! [N: nat] :
( ( Xa
= ( paraco793048302_Indet @ N ) )
=> ( ( Y
= ( paraco793048302_Indet @ ( X @ N ) ) )
=> ~ ( accp_P2085826120lle_tv @ paraco1632582009tv_rel @ ( produc1680557635lle_tv @ X @ ( paraco793048302_Indet @ N ) ) ) ) ) ) ) ) ).
% change_tv.pelims
thf(fact_222_prod_Oinj__map,axiom,
! [F1: nat > nat,F2: nat > nat] :
( ( inj_on_nat_nat @ F1 @ top_top_set_nat )
=> ( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
=> ( inj_on26176689at_nat @ ( produc505105722at_nat @ F1 @ F2 ) @ top_to1555900337at_nat ) ) ) ).
% prod.inj_map
thf(fact_223_prod_Oinj__map,axiom,
! [F1: nat > nat,F2: paraco1605129243lle_tv > paraco1605129243lle_tv] :
( ( inj_on_nat_nat @ F1 @ top_top_set_nat )
=> ( ( inj_on772319074lle_tv @ F2 @ top_to1057771083lle_tv )
=> ( inj_on1745913142lle_tv @ ( produc2101297531lle_tv @ F1 @ F2 ) @ top_to1411004326lle_tv ) ) ) ).
% prod.inj_map
thf(fact_224_prod_Oinj__map,axiom,
! [F1: paraco1605129243lle_tv > paraco1605129243lle_tv,F2: nat > nat] :
( ( inj_on772319074lle_tv @ F1 @ top_to1057771083lle_tv )
=> ( ( inj_on_nat_nat @ F2 @ top_top_set_nat )
=> ( inj_on552895452tv_nat @ ( produc42536895at_nat @ F1 @ F2 ) @ top_to975729152tv_nat ) ) ) ).
% prod.inj_map
thf(fact_225_prod_Oinj__map,axiom,
! [F1: paraco1605129243lle_tv > paraco1605129243lle_tv,F2: paraco1605129243lle_tv > paraco1605129243lle_tv] :
( ( inj_on772319074lle_tv @ F1 @ top_to1057771083lle_tv )
=> ( ( inj_on772319074lle_tv @ F2 @ top_to1057771083lle_tv )
=> ( inj_on2112487900lle_tv @ ( produc1345917071lle_tv @ F1 @ F2 ) @ top_to1871154984lle_tv ) ) ) ).
% prod.inj_map
thf(fact_226_curryI,axiom,
! [F3: produc696645085lle_fm > $o,A: list_char > paraco1605129243lle_tv,B: paraco1604210848lle_fm] :
( ( F3 @ ( produc1710735575lle_fm @ A @ B ) )
=> ( produc275353523e_fm_o @ F3 @ A @ B ) ) ).
% curryI
thf(fact_227_curryI,axiom,
! [F3: produc931990033lle_tv > $o,A: nat > nat,B: paraco1605129243lle_tv] :
( ( F3 @ ( produc1680557635lle_tv @ A @ B ) )
=> ( produc752649287e_tv_o @ F3 @ A @ B ) ) ).
% curryI
thf(fact_228_map__prod__simp,axiom,
! [F3: ( list_char > paraco1605129243lle_tv ) > list_char > paraco1605129243lle_tv,G: paraco1604210848lle_fm > paraco1604210848lle_fm,A: list_char > paraco1605129243lle_tv,B: paraco1604210848lle_fm] :
( ( produc1900967599lle_fm @ F3 @ G @ ( produc1710735575lle_fm @ A @ B ) )
= ( produc1710735575lle_fm @ ( F3 @ A ) @ ( G @ B ) ) ) ).
% map_prod_simp
thf(fact_229_map__prod__simp,axiom,
! [F3: ( list_char > paraco1605129243lle_tv ) > nat > nat,G: paraco1604210848lle_fm > paraco1605129243lle_tv,A: list_char > paraco1605129243lle_tv,B: paraco1604210848lle_fm] :
( ( produc1566550641lle_tv @ F3 @ G @ ( produc1710735575lle_fm @ A @ B ) )
= ( produc1680557635lle_tv @ ( F3 @ A ) @ ( G @ B ) ) ) ).
% map_prod_simp
thf(fact_230_map__prod__simp,axiom,
! [F3: ( nat > nat ) > list_char > paraco1605129243lle_tv,G: paraco1605129243lle_tv > paraco1604210848lle_fm,A: nat > nat,B: paraco1605129243lle_tv] :
( ( produc951033137lle_fm @ F3 @ G @ ( produc1680557635lle_tv @ A @ B ) )
= ( produc1710735575lle_fm @ ( F3 @ A ) @ ( G @ B ) ) ) ).
% map_prod_simp
thf(fact_231_map__prod__simp,axiom,
! [F3: ( nat > nat ) > nat > nat,G: paraco1605129243lle_tv > paraco1605129243lle_tv,A: nat > nat,B: paraco1605129243lle_tv] :
( ( produc532909085lle_tv @ F3 @ G @ ( produc1680557635lle_tv @ A @ B ) )
= ( produc1680557635lle_tv @ ( F3 @ A ) @ ( G @ B ) ) ) ).
% map_prod_simp
thf(fact_232_curryD,axiom,
! [F3: produc696645085lle_fm > $o,A: list_char > paraco1605129243lle_tv,B: paraco1604210848lle_fm] :
( ( produc275353523e_fm_o @ F3 @ A @ B )
=> ( F3 @ ( produc1710735575lle_fm @ A @ B ) ) ) ).
% curryD
thf(fact_233_curryD,axiom,
! [F3: produc931990033lle_tv > $o,A: nat > nat,B: paraco1605129243lle_tv] :
( ( produc752649287e_tv_o @ F3 @ A @ B )
=> ( F3 @ ( produc1680557635lle_tv @ A @ B ) ) ) ).
% curryD
thf(fact_234_curryE,axiom,
! [F3: produc696645085lle_fm > $o,A: list_char > paraco1605129243lle_tv,B: paraco1604210848lle_fm] :
( ( produc275353523e_fm_o @ F3 @ A @ B )
=> ( F3 @ ( produc1710735575lle_fm @ A @ B ) ) ) ).
% curryE
thf(fact_235_curryE,axiom,
! [F3: produc931990033lle_tv > $o,A: nat > nat,B: paraco1605129243lle_tv] :
( ( produc752649287e_tv_o @ F3 @ A @ B )
=> ( F3 @ ( produc1680557635lle_tv @ A @ B ) ) ) ).
% curryE
thf(fact_236_the__inv__into__onto,axiom,
! [F3: nat > nat,A2: set_nat] :
( ( inj_on_nat_nat @ F3 @ A2 )
=> ( ( image_nat_nat @ ( the_inv_into_nat_nat @ A2 @ F3 ) @ ( image_nat_nat @ F3 @ A2 ) )
= A2 ) ) ).
% the_inv_into_onto
thf(fact_237_the__inv__into__onto,axiom,
! [F3: paraco1605129243lle_tv > paraco1605129243lle_tv,A2: set_Pa1764573435lle_tv] :
( ( inj_on772319074lle_tv @ F3 @ A2 )
=> ( ( image_335818702lle_tv @ ( the_in452116896lle_tv @ A2 @ F3 ) @ ( image_335818702lle_tv @ F3 @ A2 ) )
= A2 ) ) ).
% the_inv_into_onto
thf(fact_238_map__prod__imageI,axiom,
! [A: list_char > paraco1605129243lle_tv,B: paraco1604210848lle_fm,R2: set_Pr2120184979lle_fm,F3: ( list_char > paraco1605129243lle_tv ) > list_char > paraco1605129243lle_tv,G: paraco1604210848lle_fm > paraco1604210848lle_fm] :
( ( member1474794996lle_fm @ ( produc1710735575lle_fm @ A @ B ) @ R2 )
=> ( member1474794996lle_fm @ ( produc1710735575lle_fm @ ( F3 @ A ) @ ( G @ B ) ) @ ( image_807538418lle_fm @ ( produc1900967599lle_fm @ F3 @ G ) @ R2 ) ) ) ).
% map_prod_imageI
thf(fact_239_map__prod__imageI,axiom,
! [A: list_char > paraco1605129243lle_tv,B: paraco1604210848lle_fm,R2: set_Pr2120184979lle_fm,F3: ( list_char > paraco1605129243lle_tv ) > nat > nat,G: paraco1604210848lle_fm > paraco1605129243lle_tv] :
( ( member1474794996lle_fm @ ( produc1710735575lle_fm @ A @ B ) @ R2 )
=> ( member1569366824lle_tv @ ( produc1680557635lle_tv @ ( F3 @ A ) @ ( G @ B ) ) @ ( image_1154804148lle_tv @ ( produc1566550641lle_tv @ F3 @ G ) @ R2 ) ) ) ).
% map_prod_imageI
thf(fact_240_map__prod__imageI,axiom,
! [A: nat > nat,B: paraco1605129243lle_tv,R2: set_Pr560526279lle_tv,F3: ( nat > nat ) > list_char > paraco1605129243lle_tv,G: paraco1605129243lle_tv > paraco1604210848lle_fm] :
( ( member1569366824lle_tv @ ( produc1680557635lle_tv @ A @ B ) @ R2 )
=> ( member1474794996lle_fm @ ( produc1710735575lle_fm @ ( F3 @ A ) @ ( G @ B ) ) @ ( image_1585092542lle_fm @ ( produc951033137lle_fm @ F3 @ G ) @ R2 ) ) ) ).
% map_prod_imageI
thf(fact_241_map__prod__imageI,axiom,
! [A: nat > nat,B: paraco1605129243lle_tv,R2: set_Pr560526279lle_tv,F3: ( nat > nat ) > nat > nat,G: paraco1605129243lle_tv > paraco1605129243lle_tv] :
( ( member1569366824lle_tv @ ( produc1680557635lle_tv @ A @ B ) @ R2 )
=> ( member1569366824lle_tv @ ( produc1680557635lle_tv @ ( F3 @ A ) @ ( G @ B ) ) @ ( image_1751508608lle_tv @ ( produc532909085lle_tv @ F3 @ G ) @ R2 ) ) ) ).
% map_prod_imageI
thf(fact_242_swap__simp,axiom,
! [X: paraco1604210848lle_fm,Y: list_char > paraco1605129243lle_tv] :
( ( produc1232814113lle_tv @ ( produc1943344577lle_tv @ X @ Y ) )
= ( produc1710735575lle_fm @ Y @ X ) ) ).
% swap_simp
thf(fact_243_swap__simp,axiom,
! [X: paraco1605129243lle_tv,Y: nat > nat] :
( ( produc178635363at_nat @ ( produc1633679363at_nat @ X @ Y ) )
= ( produc1680557635lle_tv @ Y @ X ) ) ).
% swap_simp
thf(fact_244_swap__simp,axiom,
! [X: list_char > paraco1605129243lle_tv,Y: paraco1604210848lle_fm] :
( ( produc1000205111lle_fm @ ( produc1710735575lle_fm @ X @ Y ) )
= ( produc1943344577lle_tv @ Y @ X ) ) ).
% swap_simp
thf(fact_245_swap__simp,axiom,
! [X: nat > nat,Y: paraco1605129243lle_tv] :
( ( produc225513635lle_tv @ ( produc1680557635lle_tv @ X @ Y ) )
= ( produc1633679363at_nat @ Y @ X ) ) ).
% swap_simp
thf(fact_246_pair__in__swap__image,axiom,
! [Y: paraco1604210848lle_fm,X: list_char > paraco1605129243lle_tv,A2: set_Pr2120184979lle_fm] :
( ( member74361182lle_tv @ ( produc1943344577lle_tv @ Y @ X ) @ ( image_924970994lle_tv @ produc1000205111lle_fm @ A2 ) )
= ( member1474794996lle_fm @ ( produc1710735575lle_fm @ X @ Y ) @ A2 ) ) ).
% pair_in_swap_image
thf(fact_247_pair__in__swap__image,axiom,
! [Y: paraco1605129243lle_tv,X: nat > nat,A2: set_Pr560526279lle_tv] :
( ( member1234290208at_nat @ ( produc1633679363at_nat @ Y @ X ) @ ( image_1724396928at_nat @ produc225513635lle_tv @ A2 ) )
= ( member1569366824lle_tv @ ( produc1680557635lle_tv @ X @ Y ) @ A2 ) ) ).
% pair_in_swap_image
thf(fact_248_pair__in__swap__image,axiom,
! [Y: list_char > paraco1605129243lle_tv,X: paraco1604210848lle_fm,A2: set_Pr1723822845lle_tv] :
( ( member1474794996lle_fm @ ( produc1710735575lle_fm @ Y @ X ) @ ( image_326034184lle_fm @ produc1232814113lle_tv @ A2 ) )
= ( member74361182lle_tv @ ( produc1943344577lle_tv @ X @ Y ) @ A2 ) ) ).
% pair_in_swap_image
thf(fact_249_pair__in__swap__image,axiom,
! [Y: nat > nat,X: paraco1605129243lle_tv,A2: set_Pr840833727at_nat] :
( ( member1569366824lle_tv @ ( produc1680557635lle_tv @ Y @ X ) @ ( image_769496456lle_tv @ produc178635363at_nat @ A2 ) )
= ( member1234290208at_nat @ ( produc1633679363at_nat @ X @ Y ) @ A2 ) ) ).
% pair_in_swap_image
thf(fact_250_map__prod__surj,axiom,
! [F3: nat > nat,G: nat > nat] :
( ( ( image_nat_nat @ F3 @ top_top_set_nat )
= top_top_set_nat )
=> ( ( ( image_nat_nat @ G @ top_top_set_nat )
= top_top_set_nat )
=> ( ( image_815163717at_nat @ ( produc505105722at_nat @ F3 @ G ) @ top_to1555900337at_nat )
= top_to1555900337at_nat ) ) ) ).
% map_prod_surj
thf(fact_251_map__prod__surj,axiom,
! [F3: nat > nat,G: nat > paraco1605129243lle_tv] :
( ( ( image_nat_nat @ F3 @ top_top_set_nat )
= top_top_set_nat )
=> ( ( ( image_1151963140lle_tv @ G @ top_top_set_nat )
= top_to1057771083lle_tv )
=> ( ( image_1674248956lle_tv @ ( produc313957681lle_tv @ F3 @ G ) @ top_to1555900337at_nat )
= top_to1411004326lle_tv ) ) ) ).
% map_prod_surj
thf(fact_252_map__prod__surj,axiom,
! [F3: nat > nat,G: paraco1605129243lle_tv > nat] :
( ( ( image_nat_nat @ F3 @ top_top_set_nat )
= top_top_set_nat )
=> ( ( ( image_376325286tv_nat @ G @ top_to1057771083lle_tv )
= top_top_set_nat )
=> ( ( image_1787705850at_nat @ ( produc1685803475tv_nat @ F3 @ G ) @ top_to1411004326lle_tv )
= top_to1555900337at_nat ) ) ) ).
% map_prod_surj
thf(fact_253_map__prod__surj,axiom,
! [F3: nat > nat,G: paraco1605129243lle_tv > paraco1605129243lle_tv] :
( ( ( image_nat_nat @ F3 @ top_top_set_nat )
= top_top_set_nat )
=> ( ( ( image_335818702lle_tv @ G @ top_to1057771083lle_tv )
= top_to1057771083lle_tv )
=> ( ( image_149830562lle_tv @ ( produc2101297531lle_tv @ F3 @ G ) @ top_to1411004326lle_tv )
= top_to1411004326lle_tv ) ) ) ).
% map_prod_surj
thf(fact_254_map__prod__surj,axiom,
! [F3: nat > paraco1605129243lle_tv,G: nat > nat] :
( ( ( image_1151963140lle_tv @ F3 @ top_top_set_nat )
= top_to1057771083lle_tv )
=> ( ( ( image_nat_nat @ G @ top_top_set_nat )
= top_top_set_nat )
=> ( ( image_1597721174tv_nat @ ( produc1250721397at_nat @ F3 @ G ) @ top_to1555900337at_nat )
= top_to975729152tv_nat ) ) ) ).
% map_prod_surj
thf(fact_255_map__prod__surj,axiom,
! [F3: nat > paraco1605129243lle_tv,G: nat > paraco1605129243lle_tv] :
( ( ( image_1151963140lle_tv @ F3 @ top_top_set_nat )
= top_to1057771083lle_tv )
=> ( ( ( image_1151963140lle_tv @ G @ top_top_set_nat )
= top_to1057771083lle_tv )
=> ( ( image_1507992062lle_tv @ ( produc210389789lle_tv @ F3 @ G ) @ top_to1555900337at_nat )
= top_to1871154984lle_tv ) ) ) ).
% map_prod_surj
thf(fact_256_map__prod__surj,axiom,
! [F3: nat > paraco1605129243lle_tv,G: paraco1605129243lle_tv > nat] :
( ( ( image_1151963140lle_tv @ F3 @ top_top_set_nat )
= top_to1057771083lle_tv )
=> ( ( ( image_376325286tv_nat @ G @ top_to1057771083lle_tv )
= top_top_set_nat )
=> ( ( image_53754658tv_nat @ ( produc189459613tv_nat @ F3 @ G ) @ top_to1411004326lle_tv )
= top_to975729152tv_nat ) ) ) ).
% map_prod_surj
thf(fact_257_map__prod__surj,axiom,
! [F3: nat > paraco1605129243lle_tv,G: paraco1605129243lle_tv > paraco1605129243lle_tv] :
( ( ( image_1151963140lle_tv @ F3 @ top_top_set_nat )
= top_to1057771083lle_tv )
=> ( ( ( image_335818702lle_tv @ G @ top_to1057771083lle_tv )
= top_to1057771083lle_tv )
=> ( ( image_988201162lle_tv @ ( produc1975500101lle_tv @ F3 @ G ) @ top_to1411004326lle_tv )
= top_to1871154984lle_tv ) ) ) ).
% map_prod_surj
thf(fact_258_map__prod__surj,axiom,
! [F3: paraco1605129243lle_tv > nat,G: nat > nat] :
( ( ( image_376325286tv_nat @ F3 @ top_to1057771083lle_tv )
= top_top_set_nat )
=> ( ( ( image_nat_nat @ G @ top_top_set_nat )
= top_top_set_nat )
=> ( ( image_1805124128at_nat @ ( produc386540311at_nat @ F3 @ G ) @ top_to975729152tv_nat )
= top_to1555900337at_nat ) ) ) ).
% map_prod_surj
thf(fact_259_map__prod__surj,axiom,
! [F3: paraco1605129243lle_tv > nat,G: nat > paraco1605129243lle_tv] :
( ( ( image_376325286tv_nat @ F3 @ top_to1057771083lle_tv )
= top_top_set_nat )
=> ( ( ( image_1151963140lle_tv @ G @ top_top_set_nat )
= top_to1057771083lle_tv )
=> ( ( image_1200372424lle_tv @ ( produc72207551lle_tv @ F3 @ G ) @ top_to975729152tv_nat )
= top_to1411004326lle_tv ) ) ) ).
% map_prod_surj
thf(fact_260_inj__on__image__iff,axiom,
! [A2: set_nat,G: nat > nat,F3: nat > nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A2 )
=> ( ( ( G @ ( F3 @ X4 ) )
= ( G @ ( F3 @ Xa2 ) ) )
= ( ( G @ X4 )
= ( G @ Xa2 ) ) ) ) )
=> ( ( inj_on_nat_nat @ F3 @ A2 )
=> ( ( inj_on_nat_nat @ G @ ( image_nat_nat @ F3 @ A2 ) )
= ( inj_on_nat_nat @ G @ A2 ) ) ) ) ).
% inj_on_image_iff
thf(fact_261_inj__on__image__iff,axiom,
! [A2: set_Pa1764573435lle_tv,G: paraco1605129243lle_tv > paraco1605129243lle_tv,F3: paraco1605129243lle_tv > paraco1605129243lle_tv] :
( ! [X4: paraco1605129243lle_tv] :
( ( member266900804lle_tv @ X4 @ A2 )
=> ! [Xa2: paraco1605129243lle_tv] :
( ( member266900804lle_tv @ Xa2 @ A2 )
=> ( ( ( G @ ( F3 @ X4 ) )
= ( G @ ( F3 @ Xa2 ) ) )
= ( ( G @ X4 )
= ( G @ Xa2 ) ) ) ) )
=> ( ( inj_on772319074lle_tv @ F3 @ A2 )
=> ( ( inj_on772319074lle_tv @ G @ ( image_335818702lle_tv @ F3 @ A2 ) )
= ( inj_on772319074lle_tv @ G @ A2 ) ) ) ) ).
% inj_on_image_iff
thf(fact_262_surjD,axiom,
! [F3: nat > nat,Y: nat] :
( ( ( image_nat_nat @ F3 @ top_top_set_nat )
= top_top_set_nat )
=> ? [X4: nat] :
( Y
= ( F3 @ X4 ) ) ) ).
% surjD
thf(fact_263_surjD,axiom,
! [F3: nat > paraco1605129243lle_tv,Y: paraco1605129243lle_tv] :
( ( ( image_1151963140lle_tv @ F3 @ top_top_set_nat )
= top_to1057771083lle_tv )
=> ? [X4: nat] :
( Y
= ( F3 @ X4 ) ) ) ).
% surjD
thf(fact_264_surjD,axiom,
! [F3: paraco1605129243lle_tv > nat,Y: nat] :
( ( ( image_376325286tv_nat @ F3 @ top_to1057771083lle_tv )
= top_top_set_nat )
=> ? [X4: paraco1605129243lle_tv] :
( Y
= ( F3 @ X4 ) ) ) ).
% surjD
thf(fact_265_surjD,axiom,
! [F3: paraco1605129243lle_tv > paraco1605129243lle_tv,Y: paraco1605129243lle_tv] :
( ( ( image_335818702lle_tv @ F3 @ top_to1057771083lle_tv )
= top_to1057771083lle_tv )
=> ? [X4: paraco1605129243lle_tv] :
( Y
= ( F3 @ X4 ) ) ) ).
% surjD
thf(fact_266_surjE,axiom,
! [F3: nat > nat,Y: nat] :
( ( ( image_nat_nat @ F3 @ top_top_set_nat )
= top_top_set_nat )
=> ~ ! [X4: nat] :
( Y
!= ( F3 @ X4 ) ) ) ).
% surjE
thf(fact_267_surjE,axiom,
! [F3: nat > paraco1605129243lle_tv,Y: paraco1605129243lle_tv] :
( ( ( image_1151963140lle_tv @ F3 @ top_top_set_nat )
= top_to1057771083lle_tv )
=> ~ ! [X4: nat] :
( Y
!= ( F3 @ X4 ) ) ) ).
% surjE
thf(fact_268_surjE,axiom,
! [F3: paraco1605129243lle_tv > nat,Y: nat] :
( ( ( image_376325286tv_nat @ F3 @ top_to1057771083lle_tv )
= top_top_set_nat )
=> ~ ! [X4: paraco1605129243lle_tv] :
( Y
!= ( F3 @ X4 ) ) ) ).
% surjE
thf(fact_269_surjE,axiom,
! [F3: paraco1605129243lle_tv > paraco1605129243lle_tv,Y: paraco1605129243lle_tv] :
( ( ( image_335818702lle_tv @ F3 @ top_to1057771083lle_tv )
= top_to1057771083lle_tv )
=> ~ ! [X4: paraco1605129243lle_tv] :
( Y
!= ( F3 @ X4 ) ) ) ).
% surjE
thf(fact_270_surjI,axiom,
! [G: nat > nat,F3: nat > nat] :
( ! [X4: nat] :
( ( G @ ( F3 @ X4 ) )
= X4 )
=> ( ( image_nat_nat @ G @ top_top_set_nat )
= top_top_set_nat ) ) ).
% surjI
thf(fact_271_surjI,axiom,
! [G: nat > paraco1605129243lle_tv,F3: paraco1605129243lle_tv > nat] :
( ! [X4: paraco1605129243lle_tv] :
( ( G @ ( F3 @ X4 ) )
= X4 )
=> ( ( image_1151963140lle_tv @ G @ top_top_set_nat )
= top_to1057771083lle_tv ) ) ).
% surjI
thf(fact_272_surjI,axiom,
! [G: paraco1605129243lle_tv > nat,F3: nat > paraco1605129243lle_tv] :
( ! [X4: nat] :
( ( G @ ( F3 @ X4 ) )
= X4 )
=> ( ( image_376325286tv_nat @ G @ top_to1057771083lle_tv )
= top_top_set_nat ) ) ).
% surjI
thf(fact_273_surjI,axiom,
! [G: paraco1605129243lle_tv > paraco1605129243lle_tv,F3: paraco1605129243lle_tv > paraco1605129243lle_tv] :
( ! [X4: paraco1605129243lle_tv] :
( ( G @ ( F3 @ X4 ) )
= X4 )
=> ( ( image_335818702lle_tv @ G @ top_to1057771083lle_tv )
= top_to1057771083lle_tv ) ) ).
% surjI
thf(fact_274_surj__def,axiom,
! [F3: nat > nat] :
( ( ( image_nat_nat @ F3 @ top_top_set_nat )
= top_top_set_nat )
= ( ! [Y4: nat] :
? [X2: nat] :
( Y4
= ( F3 @ X2 ) ) ) ) ).
% surj_def
thf(fact_275_surj__def,axiom,
! [F3: nat > paraco1605129243lle_tv] :
( ( ( image_1151963140lle_tv @ F3 @ top_top_set_nat )
= top_to1057771083lle_tv )
= ( ! [Y4: paraco1605129243lle_tv] :
? [X2: nat] :
( Y4
= ( F3 @ X2 ) ) ) ) ).
% surj_def
thf(fact_276_surj__def,axiom,
! [F3: paraco1605129243lle_tv > nat] :
( ( ( image_376325286tv_nat @ F3 @ top_to1057771083lle_tv )
= top_top_set_nat )
= ( ! [Y4: nat] :
? [X2: paraco1605129243lle_tv] :
( Y4
= ( F3 @ X2 ) ) ) ) ).
% surj_def
thf(fact_277_surj__def,axiom,
! [F3: paraco1605129243lle_tv > paraco1605129243lle_tv] :
( ( ( image_335818702lle_tv @ F3 @ top_to1057771083lle_tv )
= top_to1057771083lle_tv )
= ( ! [Y4: paraco1605129243lle_tv] :
? [X2: paraco1605129243lle_tv] :
( Y4
= ( F3 @ X2 ) ) ) ) ).
% surj_def
thf(fact_278_range__ex1__eq,axiom,
! [F3: nat > nat,B: nat] :
( ( inj_on_nat_nat @ F3 @ top_top_set_nat )
=> ( ( member_nat @ B @ ( image_nat_nat @ F3 @ top_top_set_nat ) )
= ( ? [X2: nat] :
( ( B
= ( F3 @ X2 ) )
& ! [Y4: nat] :
( ( B
= ( F3 @ Y4 ) )
=> ( Y4 = X2 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_279_range__ex1__eq,axiom,
! [F3: paraco1605129243lle_tv > paraco1605129243lle_tv,B: paraco1605129243lle_tv] :
( ( inj_on772319074lle_tv @ F3 @ top_to1057771083lle_tv )
=> ( ( member266900804lle_tv @ B @ ( image_335818702lle_tv @ F3 @ top_to1057771083lle_tv ) )
= ( ? [X2: paraco1605129243lle_tv] :
( ( B
= ( F3 @ X2 ) )
& ! [Y4: paraco1605129243lle_tv] :
( ( B
= ( F3 @ Y4 ) )
=> ( Y4 = X2 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_280_inj__image__eq__iff,axiom,
! [F3: nat > nat,A2: set_nat,B4: set_nat] :
( ( inj_on_nat_nat @ F3 @ top_top_set_nat )
=> ( ( ( image_nat_nat @ F3 @ A2 )
= ( image_nat_nat @ F3 @ B4 ) )
= ( A2 = B4 ) ) ) ).
% inj_image_eq_iff
thf(fact_281_inj__image__eq__iff,axiom,
! [F3: paraco1605129243lle_tv > paraco1605129243lle_tv,A2: set_Pa1764573435lle_tv,B4: set_Pa1764573435lle_tv] :
( ( inj_on772319074lle_tv @ F3 @ top_to1057771083lle_tv )
=> ( ( ( image_335818702lle_tv @ F3 @ A2 )
= ( image_335818702lle_tv @ F3 @ B4 ) )
= ( A2 = B4 ) ) ) ).
% inj_image_eq_iff
thf(fact_282_inj__image__mem__iff,axiom,
! [F3: nat > nat,A: nat,A2: set_nat] :
( ( inj_on_nat_nat @ F3 @ top_top_set_nat )
=> ( ( member_nat @ ( F3 @ A ) @ ( image_nat_nat @ F3 @ A2 ) )
= ( member_nat @ A @ A2 ) ) ) ).
% inj_image_mem_iff
thf(fact_283_inj__image__mem__iff,axiom,
! [F3: paraco1605129243lle_tv > paraco1605129243lle_tv,A: paraco1605129243lle_tv,A2: set_Pa1764573435lle_tv] :
( ( inj_on772319074lle_tv @ F3 @ top_to1057771083lle_tv )
=> ( ( member266900804lle_tv @ ( F3 @ A ) @ ( image_335818702lle_tv @ F3 @ A2 ) )
= ( member266900804lle_tv @ A @ A2 ) ) ) ).
% inj_image_mem_iff
thf(fact_284_prod__fun__imageE,axiom,
! [C: produc696645085lle_fm,F3: ( list_char > paraco1605129243lle_tv ) > list_char > paraco1605129243lle_tv,G: paraco1604210848lle_fm > paraco1604210848lle_fm,R2: set_Pr2120184979lle_fm] :
( ( member1474794996lle_fm @ C @ ( image_807538418lle_fm @ ( produc1900967599lle_fm @ F3 @ G ) @ R2 ) )
=> ~ ! [X4: list_char > paraco1605129243lle_tv,Y2: paraco1604210848lle_fm] :
( ( C
= ( produc1710735575lle_fm @ ( F3 @ X4 ) @ ( G @ Y2 ) ) )
=> ~ ( member1474794996lle_fm @ ( produc1710735575lle_fm @ X4 @ Y2 ) @ R2 ) ) ) ).
% prod_fun_imageE
thf(fact_285_prod__fun__imageE,axiom,
! [C: produc696645085lle_fm,F3: ( nat > nat ) > list_char > paraco1605129243lle_tv,G: paraco1605129243lle_tv > paraco1604210848lle_fm,R2: set_Pr560526279lle_tv] :
( ( member1474794996lle_fm @ C @ ( image_1585092542lle_fm @ ( produc951033137lle_fm @ F3 @ G ) @ R2 ) )
=> ~ ! [X4: nat > nat,Y2: paraco1605129243lle_tv] :
( ( C
= ( produc1710735575lle_fm @ ( F3 @ X4 ) @ ( G @ Y2 ) ) )
=> ~ ( member1569366824lle_tv @ ( produc1680557635lle_tv @ X4 @ Y2 ) @ R2 ) ) ) ).
% prod_fun_imageE
thf(fact_286_prod__fun__imageE,axiom,
! [C: produc931990033lle_tv,F3: ( list_char > paraco1605129243lle_tv ) > nat > nat,G: paraco1604210848lle_fm > paraco1605129243lle_tv,R2: set_Pr2120184979lle_fm] :
( ( member1569366824lle_tv @ C @ ( image_1154804148lle_tv @ ( produc1566550641lle_tv @ F3 @ G ) @ R2 ) )
=> ~ ! [X4: list_char > paraco1605129243lle_tv,Y2: paraco1604210848lle_fm] :
( ( C
= ( produc1680557635lle_tv @ ( F3 @ X4 ) @ ( G @ Y2 ) ) )
=> ~ ( member1474794996lle_fm @ ( produc1710735575lle_fm @ X4 @ Y2 ) @ R2 ) ) ) ).
% prod_fun_imageE
thf(fact_287_prod__fun__imageE,axiom,
! [C: produc931990033lle_tv,F3: ( nat > nat ) > nat > nat,G: paraco1605129243lle_tv > paraco1605129243lle_tv,R2: set_Pr560526279lle_tv] :
( ( member1569366824lle_tv @ C @ ( image_1751508608lle_tv @ ( produc532909085lle_tv @ F3 @ G ) @ R2 ) )
=> ~ ! [X4: nat > nat,Y2: paraco1605129243lle_tv] :
( ( C
= ( produc1680557635lle_tv @ ( F3 @ X4 ) @ ( G @ Y2 ) ) )
=> ~ ( member1569366824lle_tv @ ( produc1680557635lle_tv @ X4 @ Y2 ) @ R2 ) ) ) ).
% prod_fun_imageE
thf(fact_288_f__the__inv__into__f,axiom,
! [F3: nat > nat,A2: set_nat,Y: nat] :
( ( inj_on_nat_nat @ F3 @ A2 )
=> ( ( member_nat @ Y @ ( image_nat_nat @ F3 @ A2 ) )
=> ( ( F3 @ ( the_inv_into_nat_nat @ A2 @ F3 @ Y ) )
= Y ) ) ) ).
% f_the_inv_into_f
thf(fact_289_f__the__inv__into__f,axiom,
! [F3: paraco1605129243lle_tv > paraco1605129243lle_tv,A2: set_Pa1764573435lle_tv,Y: paraco1605129243lle_tv] :
( ( inj_on772319074lle_tv @ F3 @ A2 )
=> ( ( member266900804lle_tv @ Y @ ( image_335818702lle_tv @ F3 @ A2 ) )
=> ( ( F3 @ ( the_in452116896lle_tv @ A2 @ F3 @ Y ) )
= Y ) ) ) ).
% f_the_inv_into_f
thf(fact_290_inj__on__the__inv__into,axiom,
! [F3: nat > nat,A2: set_nat] :
( ( inj_on_nat_nat @ F3 @ A2 )
=> ( inj_on_nat_nat @ ( the_inv_into_nat_nat @ A2 @ F3 ) @ ( image_nat_nat @ F3 @ A2 ) ) ) ).
% inj_on_the_inv_into
thf(fact_291_inj__on__the__inv__into,axiom,
! [F3: paraco1605129243lle_tv > paraco1605129243lle_tv,A2: set_Pa1764573435lle_tv] :
( ( inj_on772319074lle_tv @ F3 @ A2 )
=> ( inj_on772319074lle_tv @ ( the_in452116896lle_tv @ A2 @ F3 ) @ ( image_335818702lle_tv @ F3 @ A2 ) ) ) ).
% inj_on_the_inv_into
thf(fact_292_total__inv__image,axiom,
! [F3: nat > nat,R: set_Pr1986765409at_nat] :
( ( inj_on_nat_nat @ F3 @ top_top_set_nat )
=> ( ( total_on_nat @ top_top_set_nat @ R )
=> ( total_on_nat @ top_top_set_nat @ ( inv_image_nat_nat @ R @ F3 ) ) ) ) ).
% total_inv_image
thf(fact_293_total__inv__image,axiom,
! [F3: nat > paraco1605129243lle_tv,R: set_Pr2014382200lle_tv] :
( ( inj_on1955349912lle_tv @ F3 @ top_top_set_nat )
=> ( ( total_1781621663lle_tv @ top_to1057771083lle_tv @ R )
=> ( total_on_nat @ top_top_set_nat @ ( inv_im516950809tv_nat @ R @ F3 ) ) ) ) ).
% total_inv_image
thf(fact_294_total__inv__image,axiom,
! [F3: paraco1605129243lle_tv > nat,R: set_Pr1986765409at_nat] :
( ( inj_on1179712058tv_nat @ F3 @ top_to1057771083lle_tv )
=> ( ( total_on_nat @ top_top_set_nat @ R )
=> ( total_1781621663lle_tv @ top_to1057771083lle_tv @ ( inv_im1292588663lle_tv @ R @ F3 ) ) ) ) ).
% total_inv_image
thf(fact_295_total__inv__image,axiom,
! [F3: paraco1605129243lle_tv > paraco1605129243lle_tv,R: set_Pr2014382200lle_tv] :
( ( inj_on772319074lle_tv @ F3 @ top_to1057771083lle_tv )
=> ( ( total_1781621663lle_tv @ top_to1057771083lle_tv @ R )
=> ( total_1781621663lle_tv @ top_to1057771083lle_tv @ ( inv_im719552961lle_tv @ R @ F3 ) ) ) ) ).
% total_inv_image
thf(fact_296_prod_Ocollapse,axiom,
! [Prod: produc696645085lle_fm] :
( ( produc1710735575lle_fm @ ( produc29804011lle_fm @ Prod ) @ ( produc148705065lle_fm @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_297_prod_Ocollapse,axiom,
! [Prod: produc931990033lle_tv] :
( ( produc1680557635lle_tv @ ( produc1636169559lle_tv @ Prod ) @ ( produc547496597lle_tv @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_298_prod_Oswap__def,axiom,
( produc1232814113lle_tv
= ( ^ [P: produc1443694919lle_tv] : ( produc1710735575lle_fm @ ( produc381314067lle_tv @ P ) @ ( produc262413013lle_tv @ P ) ) ) ) ).
% prod.swap_def
thf(fact_299_prod_Oswap__def,axiom,
( produc178635363at_nat
= ( ^ [P: produc596913417at_nat] : ( produc1680557635lle_tv @ ( produc500618325at_nat @ P ) @ ( produc1589291287at_nat @ P ) ) ) ) ).
% prod.swap_def
thf(fact_300_fst__eqD,axiom,
! [X: list_char > paraco1605129243lle_tv,Y: paraco1604210848lle_fm,A: list_char > paraco1605129243lle_tv] :
( ( ( produc29804011lle_fm @ ( produc1710735575lle_fm @ X @ Y ) )
= A )
=> ( X = A ) ) ).
% fst_eqD
thf(fact_301_fst__eqD,axiom,
! [X: nat > nat,Y: paraco1605129243lle_tv,A: nat > nat] :
( ( ( produc1636169559lle_tv @ ( produc1680557635lle_tv @ X @ Y ) )
= A )
=> ( X = A ) ) ).
% fst_eqD
thf(fact_302_snd__eqD,axiom,
! [X: list_char > paraco1605129243lle_tv,Y: paraco1604210848lle_fm,A: paraco1604210848lle_fm] :
( ( ( produc148705065lle_fm @ ( produc1710735575lle_fm @ X @ Y ) )
= A )
=> ( Y = A ) ) ).
% snd_eqD
thf(fact_303_snd__eqD,axiom,
! [X: nat > nat,Y: paraco1605129243lle_tv,A: paraco1605129243lle_tv] :
( ( ( produc547496597lle_tv @ ( produc1680557635lle_tv @ X @ Y ) )
= A )
=> ( Y = A ) ) ).
% snd_eqD
thf(fact_304_fst__conv,axiom,
! [X1: list_char > paraco1605129243lle_tv,X23: paraco1604210848lle_fm] :
( ( produc29804011lle_fm @ ( produc1710735575lle_fm @ X1 @ X23 ) )
= X1 ) ).
% fst_conv
thf(fact_305_fst__conv,axiom,
! [X1: nat > nat,X23: paraco1605129243lle_tv] :
( ( produc1636169559lle_tv @ ( produc1680557635lle_tv @ X1 @ X23 ) )
= X1 ) ).
% fst_conv
thf(fact_306_snd__conv,axiom,
! [X1: list_char > paraco1605129243lle_tv,X23: paraco1604210848lle_fm] :
( ( produc148705065lle_fm @ ( produc1710735575lle_fm @ X1 @ X23 ) )
= X23 ) ).
% snd_conv
thf(fact_307_snd__conv,axiom,
! [X1: nat > nat,X23: paraco1605129243lle_tv] :
( ( produc547496597lle_tv @ ( produc1680557635lle_tv @ X1 @ X23 ) )
= X23 ) ).
% snd_conv
thf(fact_308_prod_Oexhaust__sel,axiom,
! [Prod: produc696645085lle_fm] :
( Prod
= ( produc1710735575lle_fm @ ( produc29804011lle_fm @ Prod ) @ ( produc148705065lle_fm @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_309_prod_Oexhaust__sel,axiom,
! [Prod: produc931990033lle_tv] :
( Prod
= ( produc1680557635lle_tv @ ( produc1636169559lle_tv @ Prod ) @ ( produc547496597lle_tv @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_310_surjective__pairing,axiom,
! [T: produc696645085lle_fm] :
( T
= ( produc1710735575lle_fm @ ( produc29804011lle_fm @ T ) @ ( produc148705065lle_fm @ T ) ) ) ).
% surjective_pairing
thf(fact_311_surjective__pairing,axiom,
! [T: produc931990033lle_tv] :
( T
= ( produc1680557635lle_tv @ ( produc1636169559lle_tv @ T ) @ ( produc547496597lle_tv @ T ) ) ) ).
% surjective_pairing
thf(fact_312_exI__realizer,axiom,
! [P3: paraco1604210848lle_fm > ( list_char > paraco1605129243lle_tv ) > $o,Y: paraco1604210848lle_fm,X: list_char > paraco1605129243lle_tv] :
( ( P3 @ Y @ X )
=> ( P3 @ ( produc148705065lle_fm @ ( produc1710735575lle_fm @ X @ Y ) ) @ ( produc29804011lle_fm @ ( produc1710735575lle_fm @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_313_exI__realizer,axiom,
! [P3: paraco1605129243lle_tv > ( nat > nat ) > $o,Y: paraco1605129243lle_tv,X: nat > nat] :
( ( P3 @ Y @ X )
=> ( P3 @ ( produc547496597lle_tv @ ( produc1680557635lle_tv @ X @ Y ) ) @ ( produc1636169559lle_tv @ ( produc1680557635lle_tv @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_314_conjI__realizer,axiom,
! [P3: ( list_char > paraco1605129243lle_tv ) > $o,P2: list_char > paraco1605129243lle_tv,Q2: paraco1604210848lle_fm > $o,Q: paraco1604210848lle_fm] :
( ( P3 @ P2 )
=> ( ( Q2 @ Q )
=> ( ( P3 @ ( produc29804011lle_fm @ ( produc1710735575lle_fm @ P2 @ Q ) ) )
& ( Q2 @ ( produc148705065lle_fm @ ( produc1710735575lle_fm @ P2 @ Q ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_315_conjI__realizer,axiom,
! [P3: ( nat > nat ) > $o,P2: nat > nat,Q2: paraco1605129243lle_tv > $o,Q: paraco1605129243lle_tv] :
( ( P3 @ P2 )
=> ( ( Q2 @ Q )
=> ( ( P3 @ ( produc1636169559lle_tv @ ( produc1680557635lle_tv @ P2 @ Q ) ) )
& ( Q2 @ ( produc547496597lle_tv @ ( produc1680557635lle_tv @ P2 @ Q ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_316_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P3: ( list_char > paraco1605129243lle_tv ) > paraco1604210848lle_fm > $o,X: list_char > paraco1605129243lle_tv,Y: paraco1604210848lle_fm,A: produc696645085lle_fm] :
( ( P3 @ X @ Y )
=> ( ( A
= ( produc1710735575lle_fm @ X @ Y ) )
=> ( P3 @ ( produc29804011lle_fm @ A ) @ ( produc148705065lle_fm @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_317_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P3: ( nat > nat ) > paraco1605129243lle_tv > $o,X: nat > nat,Y: paraco1605129243lle_tv,A: produc931990033lle_tv] :
( ( P3 @ X @ Y )
=> ( ( A
= ( produc1680557635lle_tv @ X @ Y ) )
=> ( P3 @ ( produc1636169559lle_tv @ A ) @ ( produc547496597lle_tv @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_318_eq__snd__iff,axiom,
! [B: paraco1604210848lle_fm,P2: produc696645085lle_fm] :
( ( B
= ( produc148705065lle_fm @ P2 ) )
= ( ? [A6: list_char > paraco1605129243lle_tv] :
( P2
= ( produc1710735575lle_fm @ A6 @ B ) ) ) ) ).
% eq_snd_iff
thf(fact_319_eq__snd__iff,axiom,
! [B: paraco1605129243lle_tv,P2: produc931990033lle_tv] :
( ( B
= ( produc547496597lle_tv @ P2 ) )
= ( ? [A6: nat > nat] :
( P2
= ( produc1680557635lle_tv @ A6 @ B ) ) ) ) ).
% eq_snd_iff
thf(fact_320_sndI,axiom,
! [X: produc696645085lle_fm,Y: list_char > paraco1605129243lle_tv,Z: paraco1604210848lle_fm] :
( ( X
= ( produc1710735575lle_fm @ Y @ Z ) )
=> ( ( produc148705065lle_fm @ X )
= Z ) ) ).
% sndI
thf(fact_321_sndI,axiom,
! [X: produc931990033lle_tv,Y: nat > nat,Z: paraco1605129243lle_tv] :
( ( X
= ( produc1680557635lle_tv @ Y @ Z ) )
=> ( ( produc547496597lle_tv @ X )
= Z ) ) ).
% sndI
thf(fact_322_fstI,axiom,
! [X: produc696645085lle_fm,Y: list_char > paraco1605129243lle_tv,Z: paraco1604210848lle_fm] :
( ( X
= ( produc1710735575lle_fm @ Y @ Z ) )
=> ( ( produc29804011lle_fm @ X )
= Y ) ) ).
% fstI
thf(fact_323_fstI,axiom,
! [X: produc931990033lle_tv,Y: nat > nat,Z: paraco1605129243lle_tv] :
( ( X
= ( produc1680557635lle_tv @ Y @ Z ) )
=> ( ( produc1636169559lle_tv @ X )
= Y ) ) ).
% fstI
thf(fact_324_eq__fst__iff,axiom,
! [A: list_char > paraco1605129243lle_tv,P2: produc696645085lle_fm] :
( ( A
= ( produc29804011lle_fm @ P2 ) )
= ( ? [B5: paraco1604210848lle_fm] :
( P2
= ( produc1710735575lle_fm @ A @ B5 ) ) ) ) ).
% eq_fst_iff
thf(fact_325_eq__fst__iff,axiom,
! [A: nat > nat,P2: produc931990033lle_tv] :
( ( A
= ( produc1636169559lle_tv @ P2 ) )
= ( ? [B5: paraco1605129243lle_tv] :
( P2
= ( produc1680557635lle_tv @ A @ B5 ) ) ) ) ).
% eq_fst_iff
thf(fact_326_total__lex__prod,axiom,
! [R: set_Pr1986765409at_nat,S3: set_Pr1986765409at_nat] :
( ( total_on_nat @ top_top_set_nat @ R )
=> ( ( total_on_nat @ top_top_set_nat @ S3 )
=> ( total_1775715973at_nat @ top_to1555900337at_nat @ ( lex_prod_nat_nat @ R @ S3 ) ) ) ) ).
% total_lex_prod
thf(fact_327_total__lex__prod,axiom,
! [R: set_Pr1986765409at_nat,S3: set_Pr2014382200lle_tv] :
( ( total_on_nat @ top_top_set_nat @ R )
=> ( ( total_1781621663lle_tv @ top_to1057771083lle_tv @ S3 )
=> ( total_517371324lle_tv @ top_to1411004326lle_tv @ ( lex_pr413684186lle_tv @ R @ S3 ) ) ) ) ).
% total_lex_prod
thf(fact_328_total__lex__prod,axiom,
! [R: set_Pr2014382200lle_tv,S3: set_Pr1986765409at_nat] :
( ( total_1781621663lle_tv @ top_to1057771083lle_tv @ R )
=> ( ( total_on_nat @ top_top_set_nat @ S3 )
=> ( total_440843542tv_nat @ top_to975729152tv_nat @ ( lex_pr1785529980tv_nat @ R @ S3 ) ) ) ) ).
% total_lex_prod
thf(fact_329_total__lex__prod,axiom,
! [R: set_Pr2014382200lle_tv,S3: set_Pr2014382200lle_tv] :
( ( total_1781621663lle_tv @ top_to1057771083lle_tv @ R )
=> ( ( total_1781621663lle_tv @ top_to1057771083lle_tv @ S3 )
=> ( total_92239038lle_tv @ top_to1871154984lle_tv @ ( lex_pr567812260lle_tv @ R @ S3 ) ) ) ) ).
% total_lex_prod
thf(fact_330_in__lex__prod,axiom,
! [A: list_char > paraco1605129243lle_tv,B: paraco1604210848lle_fm,A5: list_char > paraco1605129243lle_tv,B3: paraco1604210848lle_fm,R: set_Pr2061825550lle_tv,S3: set_Pr1019238018lle_fm] :
( ( member1592865093lle_fm @ ( produc407632608lle_fm @ ( produc1710735575lle_fm @ A @ B ) @ ( produc1710735575lle_fm @ A5 @ B3 ) ) @ ( lex_pr2001015487lle_fm @ R @ S3 ) )
= ( ( member146333935lle_tv @ ( produc1297923922lle_tv @ A @ A5 ) @ R )
| ( ( A = A5 )
& ( member334644067lle_fm @ ( produc1126104134lle_fm @ B @ B3 ) @ S3 ) ) ) ) ).
% in_lex_prod
thf(fact_331_in__lex__prod,axiom,
! [A: nat > nat,B: paraco1605129243lle_tv,A5: nat > nat,B3: paraco1605129243lle_tv,R: set_Pr1357842111at_nat,S3: set_Pr2014382200lle_tv] :
( ( member1339857875lle_tv @ ( produc2057188974lle_tv @ ( produc1680557635lle_tv @ A @ B ) @ ( produc1680557635lle_tv @ A5 @ B3 ) ) @ ( lex_pr1093414955lle_tv @ R @ S3 ) )
= ( ( member112824072at_nat @ ( produc780837143at_nat @ A @ A5 ) @ R )
| ( ( A = A5 )
& ( member1531318361lle_tv @ ( produc2058540732lle_tv @ B @ B3 ) @ S3 ) ) ) ) ).
% in_lex_prod
thf(fact_332_wf__map__prod__image,axiom,
! [R: set_Pr1986765409at_nat,F3: nat > nat] :
( ( wf_nat @ R )
=> ( ( inj_on_nat_nat @ F3 @ top_top_set_nat )
=> ( wf_nat @ ( image_815163717at_nat @ ( produc505105722at_nat @ F3 @ F3 ) @ R ) ) ) ) ).
% wf_map_prod_image
thf(fact_333_wf__map__prod__image,axiom,
! [R: set_Pr2014382200lle_tv,F3: paraco1605129243lle_tv > paraco1605129243lle_tv] :
( ( wf_Par1378404612lle_tv @ R )
=> ( ( inj_on772319074lle_tv @ F3 @ top_to1057771083lle_tv )
=> ( wf_Par1378404612lle_tv @ ( image_1221801032lle_tv @ ( produc1345917071lle_tv @ F3 @ F3 ) @ R ) ) ) ) ).
% wf_map_prod_image
thf(fact_334_same__fstI,axiom,
! [P3: ( list_char > paraco1605129243lle_tv ) > $o,X: list_char > paraco1605129243lle_tv,Y5: paraco1604210848lle_fm,Y: paraco1604210848lle_fm,R2: ( list_char > paraco1605129243lle_tv ) > set_Pr1019238018lle_fm] :
( ( P3 @ X )
=> ( ( member334644067lle_fm @ ( produc1126104134lle_fm @ Y5 @ Y ) @ ( R2 @ X ) )
=> ( member1592865093lle_fm @ ( produc407632608lle_fm @ ( produc1710735575lle_fm @ X @ Y5 ) @ ( produc1710735575lle_fm @ X @ Y ) ) @ ( same_f1581085506lle_fm @ P3 @ R2 ) ) ) ) ).
% same_fstI
thf(fact_335_same__fstI,axiom,
! [P3: ( nat > nat ) > $o,X: nat > nat,Y5: paraco1605129243lle_tv,Y: paraco1605129243lle_tv,R2: ( nat > nat ) > set_Pr2014382200lle_tv] :
( ( P3 @ X )
=> ( ( member1531318361lle_tv @ ( produc2058540732lle_tv @ Y5 @ Y ) @ ( R2 @ X ) )
=> ( member1339857875lle_tv @ ( produc2057188974lle_tv @ ( produc1680557635lle_tv @ X @ Y5 ) @ ( produc1680557635lle_tv @ X @ Y ) ) @ ( same_f1138187694lle_tv @ P3 @ R2 ) ) ) ) ).
% same_fstI
thf(fact_336_the__inv__into__into,axiom,
! [F3: nat > nat,A2: set_nat,X: nat,B4: set_nat] :
( ( inj_on_nat_nat @ F3 @ A2 )
=> ( ( member_nat @ X @ ( image_nat_nat @ F3 @ A2 ) )
=> ( ( ord_less_eq_set_nat @ A2 @ B4 )
=> ( member_nat @ ( the_inv_into_nat_nat @ A2 @ F3 @ X ) @ B4 ) ) ) ) ).
% the_inv_into_into
thf(fact_337_the__inv__into__into,axiom,
! [F3: paraco1605129243lle_tv > paraco1605129243lle_tv,A2: set_Pa1764573435lle_tv,X: paraco1605129243lle_tv,B4: set_Pa1764573435lle_tv] :
( ( inj_on772319074lle_tv @ F3 @ A2 )
=> ( ( member266900804lle_tv @ X @ ( image_335818702lle_tv @ F3 @ A2 ) )
=> ( ( ord_le241064091lle_tv @ A2 @ B4 )
=> ( member266900804lle_tv @ ( the_in452116896lle_tv @ A2 @ F3 @ X ) @ B4 ) ) ) ) ).
% the_inv_into_into
thf(fact_338_inj__on__image__eq__iff,axiom,
! [F3: nat > nat,C2: set_nat,A2: set_nat,B4: set_nat] :
( ( inj_on_nat_nat @ F3 @ C2 )
=> ( ( ord_less_eq_set_nat @ A2 @ C2 )
=> ( ( ord_less_eq_set_nat @ B4 @ C2 )
=> ( ( ( image_nat_nat @ F3 @ A2 )
= ( image_nat_nat @ F3 @ B4 ) )
= ( A2 = B4 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_339_inj__on__image__eq__iff,axiom,
! [F3: paraco1605129243lle_tv > paraco1605129243lle_tv,C2: set_Pa1764573435lle_tv,A2: set_Pa1764573435lle_tv,B4: set_Pa1764573435lle_tv] :
( ( inj_on772319074lle_tv @ F3 @ C2 )
=> ( ( ord_le241064091lle_tv @ A2 @ C2 )
=> ( ( ord_le241064091lle_tv @ B4 @ C2 )
=> ( ( ( image_335818702lle_tv @ F3 @ A2 )
= ( image_335818702lle_tv @ F3 @ B4 ) )
= ( A2 = B4 ) ) ) ) ) ).
% inj_on_image_eq_iff
thf(fact_340_inj__on__image__mem__iff,axiom,
! [F3: nat > nat,B4: set_nat,A: nat,A2: set_nat] :
( ( inj_on_nat_nat @ F3 @ B4 )
=> ( ( member_nat @ A @ B4 )
=> ( ( ord_less_eq_set_nat @ A2 @ B4 )
=> ( ( member_nat @ ( F3 @ A ) @ ( image_nat_nat @ F3 @ A2 ) )
= ( member_nat @ A @ A2 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_341_inj__on__image__mem__iff,axiom,
! [F3: paraco1605129243lle_tv > paraco1605129243lle_tv,B4: set_Pa1764573435lle_tv,A: paraco1605129243lle_tv,A2: set_Pa1764573435lle_tv] :
( ( inj_on772319074lle_tv @ F3 @ B4 )
=> ( ( member266900804lle_tv @ A @ B4 )
=> ( ( ord_le241064091lle_tv @ A2 @ B4 )
=> ( ( member266900804lle_tv @ ( F3 @ A ) @ ( image_335818702lle_tv @ F3 @ A2 ) )
= ( member266900804lle_tv @ A @ A2 ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_342_top_Oextremum__uniqueI,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A )
=> ( A = top_top_set_nat ) ) ).
% top.extremum_uniqueI
thf(fact_343_top_Oextremum__uniqueI,axiom,
! [A: set_Pa1764573435lle_tv] :
( ( ord_le241064091lle_tv @ top_to1057771083lle_tv @ A )
=> ( A = top_to1057771083lle_tv ) ) ).
% top.extremum_uniqueI
thf(fact_344_top_Oextremum__unique,axiom,
! [A: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A )
= ( A = top_top_set_nat ) ) ).
% top.extremum_unique
thf(fact_345_top_Oextremum__unique,axiom,
! [A: set_Pa1764573435lle_tv] :
( ( ord_le241064091lle_tv @ top_to1057771083lle_tv @ A )
= ( A = top_to1057771083lle_tv ) ) ).
% top.extremum_unique
thf(fact_346_top__greatest,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ top_top_set_nat ) ).
% top_greatest
thf(fact_347_top__greatest,axiom,
! [A: set_Pa1764573435lle_tv] : ( ord_le241064091lle_tv @ A @ top_to1057771083lle_tv ) ).
% top_greatest
thf(fact_348_subset__UNIV,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ top_top_set_nat ) ).
% subset_UNIV
thf(fact_349_subset__UNIV,axiom,
! [A2: set_Pa1764573435lle_tv] : ( ord_le241064091lle_tv @ A2 @ top_to1057771083lle_tv ) ).
% subset_UNIV
thf(fact_350_inj__on__subset,axiom,
! [F3: nat > nat,A2: set_nat,B4: set_nat] :
( ( inj_on_nat_nat @ F3 @ A2 )
=> ( ( ord_less_eq_set_nat @ B4 @ A2 )
=> ( inj_on_nat_nat @ F3 @ B4 ) ) ) ).
% inj_on_subset
thf(fact_351_inj__on__subset,axiom,
! [F3: paraco1605129243lle_tv > paraco1605129243lle_tv,A2: set_Pa1764573435lle_tv,B4: set_Pa1764573435lle_tv] :
( ( inj_on772319074lle_tv @ F3 @ A2 )
=> ( ( ord_le241064091lle_tv @ B4 @ A2 )
=> ( inj_on772319074lle_tv @ F3 @ B4 ) ) ) ).
% inj_on_subset
thf(fact_352_subset__inj__on,axiom,
! [F3: nat > nat,B4: set_nat,A2: set_nat] :
( ( inj_on_nat_nat @ F3 @ B4 )
=> ( ( ord_less_eq_set_nat @ A2 @ B4 )
=> ( inj_on_nat_nat @ F3 @ A2 ) ) ) ).
% subset_inj_on
thf(fact_353_subset__inj__on,axiom,
! [F3: paraco1605129243lle_tv > paraco1605129243lle_tv,B4: set_Pa1764573435lle_tv,A2: set_Pa1764573435lle_tv] :
( ( inj_on772319074lle_tv @ F3 @ B4 )
=> ( ( ord_le241064091lle_tv @ A2 @ B4 )
=> ( inj_on772319074lle_tv @ F3 @ A2 ) ) ) ).
% subset_inj_on
% Helper facts (3)
thf(help_If_3_1_If_001t__Paraconsistency____Mirabelle____qxuposogha__Otv_T,axiom,
! [P3: $o] :
( ( P3 = $true )
| ( P3 = $false ) ) ).
thf(help_If_2_1_If_001t__Paraconsistency____Mirabelle____qxuposogha__Otv_T,axiom,
! [X: paraco1605129243lle_tv,Y: paraco1605129243lle_tv] :
( ( if_Par1134130401lle_tv @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Paraconsistency____Mirabelle____qxuposogha__Otv_T,axiom,
! [X: paraco1605129243lle_tv,Y: paraco1605129243lle_tv] :
( ( if_Par1134130401lle_tv @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( paraco1325918602e_eval @ ( paraco761681009ge_int @ f @ i ) @ p2 )
!= ( paraco1344940915le_Det @ $true ) ) ).
%------------------------------------------------------------------------------