TPTP Problem File: ITP199^1.p
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%------------------------------------------------------------------------------
% File : ITP199^1 : TPTP v9.0.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer USubst problem prob_1246__6349986_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 : USubst/prob_1246__6349986_1 [Des21]
% Status : Theorem
% Rating : 0.25 v9.0.0, 0.20 v8.2.0, 0.15 v8.1.0, 0.18 v7.5.0
% Syntax : Number of formulae : 462 ( 256 unt; 105 typ; 0 def)
% Number of atoms : 834 ( 531 equ; 0 cnn)
% Maximal formula atoms : 9 ( 2 avg)
% Number of connectives : 2475 ( 116 ~; 23 |; 68 &;2092 @)
% ( 0 <=>; 176 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 5 avg)
% Number of types : 19 ( 18 usr)
% Number of type conns : 373 ( 373 >; 0 *; 0 +; 0 <<)
% Number of symbols : 90 ( 87 usr; 22 con; 0-7 aty)
% Number of variables : 849 ( 54 ^; 774 !; 21 ?; 849 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:37:46.391
%------------------------------------------------------------------------------
% Could-be-implicit typings (18)
thf(ty_n_t__Product____Type__Oprod_I_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Otrm_J_J_Mt__Product____Type__Oprod_I_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Otrm_J_J_Mt__Product____Type__Oprod_I_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ofml_J_J_M_062_It__String__Ochar_Mt__Option__Ooption_It__Syntax__Ogame_J_J_J_J_J,type,
produc1418842292n_game: $tType ).
thf(ty_n_t__Set__Oset_It__Option__Ooption_I_062_It__Syntax__Ovariable_Mt__Real__Oreal_J_J_J,type,
set_op12188086e_real: $tType ).
thf(ty_n_t__Option__Ooption_I_062_It__Syntax__Ovariable_Mt__Real__Oreal_J_J,type,
option_variable_real: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__Syntax__Ovariable_Mt__Real__Oreal_J_J,type,
set_variable_real: $tType ).
thf(ty_n_t__Set__Oset_It__Option__Ooption_It__Syntax__Ovariable_J_J,type,
set_option_variable: $tType ).
thf(ty_n_t__Set__Oset_It__Option__Ooption_It__Syntax__Otrm_J_J,type,
set_option_trm: $tType ).
thf(ty_n_t__Option__Ooption_It__Syntax__Ovariable_J,type,
option_variable: $tType ).
thf(ty_n_t__Option__Ooption_It__Syntax__Otrm_J,type,
option_trm: $tType ).
thf(ty_n_t__Denotational____Semantics__Ointerp,type,
denotational_interp: $tType ).
thf(ty_n_t__Set__Oset_It__Syntax__Ovariable_J,type,
set_variable: $tType ).
thf(ty_n_t__Set__Oset_It__Syntax__Otrm_J,type,
set_trm: $tType ).
thf(ty_n_t__Set__Oset_It__Real__Oreal_J,type,
set_real: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Syntax__Ovariable,type,
variable: $tType ).
thf(ty_n_t__String__Ochar,type,
char: $tType ).
thf(ty_n_t__Syntax__Otrm,type,
trm: $tType ).
thf(ty_n_t__Real__Oreal,type,
real: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (87)
thf(sy_c_Coincidence_Ostatediff,type,
statediff: ( variable > real ) > ( variable > real ) > set_variable ).
thf(sy_c_Coincidence_Ostateinterpol,type,
stateinterpol: ( variable > real ) > ( variable > real ) > set_variable > variable > real ).
thf(sy_c_Denotational__Semantics_OUvariation,type,
denota1419872369iation: ( variable > real ) > ( variable > real ) > set_variable > $o ).
thf(sy_c_Denotational__Semantics_OVagree,type,
denotational_Vagree: ( variable > real ) > ( variable > real ) > set_variable > $o ).
thf(sy_c_Denotational__Semantics_Orepv,type,
denotational_repv: ( variable > real ) > variable > real > variable > real ).
thf(sy_c_Denotational__Semantics_Osolves__ODE,type,
denota1778088425es_ODE: denotational_interp > ( real > variable > real ) > char > trm > $o ).
thf(sy_c_Denotational__Semantics_Oterm__sem,type,
denota1863255036rm_sem: denotational_interp > trm > ( variable > real ) > real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
uminus_uminus_real: real > real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_I_062_It__Syntax__Ovariable_Mt__Real__Oreal_J_J,type,
uminus430703407e_real: set_variable_real > set_variable_real ).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Syntax__Ovariable_J,type,
uminus1851247844riable: set_variable > set_variable ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal,type,
zero_zero_real: real ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Set__Oset_It__Nat__Onat_J,type,
zero_zero_set_nat: set_nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Set__Oset_It__Real__Oreal_J,type,
zero_zero_set_real: set_real ).
thf(sy_c_If_001t__Real__Oreal,type,
if_real: $o > real > real > real ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_I_062_It__Syntax__Ovariable_Mt__Real__Oreal_J_J,type,
inf_in1556002680e_real: set_variable_real > set_variable_real > set_variable_real ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Syntax__Ovariable_J,type,
inf_inf_set_variable: set_variable > set_variable > set_variable ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_I_062_It__Syntax__Ovariable_Mt__Real__Oreal_J_J,type,
sup_su1685293586e_real: set_variable_real > set_variable_real > set_variable_real ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Syntax__Ovariable_J,type,
sup_sup_set_variable: set_variable > set_variable > set_variable ).
thf(sy_c_Nat_Osize__class_Osize_001t__Syntax__Ovariable,type,
size_size_variable: variable > nat ).
thf(sy_c_Option_Ooption_ONone_001_062_It__Syntax__Ovariable_Mt__Real__Oreal_J,type,
none_variable_real: option_variable_real ).
thf(sy_c_Option_Ooption_ONone_001t__Syntax__Otrm,type,
none_trm: option_trm ).
thf(sy_c_Option_Ooption_ONone_001t__Syntax__Ovariable,type,
none_variable: option_variable ).
thf(sy_c_Option_Ooption_OSome_001_062_It__Syntax__Ovariable_Mt__Real__Oreal_J,type,
some_variable_real: ( variable > real ) > option_variable_real ).
thf(sy_c_Option_Ooption_OSome_001t__Syntax__Otrm,type,
some_trm: trm > option_trm ).
thf(sy_c_Option_Ooption_OSome_001t__Syntax__Ovariable,type,
some_variable: variable > option_variable ).
thf(sy_c_Option_Ooption_Oset__option_001_062_It__Syntax__Ovariable_Mt__Real__Oreal_J,type,
set_op697949218e_real: option_variable_real > set_variable_real ).
thf(sy_c_Option_Ooption_Oset__option_001t__Syntax__Otrm,type,
set_option_trm2: option_trm > set_trm ).
thf(sy_c_Option_Ooption_Oset__option_001t__Syntax__Ovariable,type,
set_option_variable2: option_variable > set_variable ).
thf(sy_c_Option_Ooption_Othe_001_062_It__Syntax__Ovariable_Mt__Real__Oreal_J,type,
the_variable_real: option_variable_real > variable > real ).
thf(sy_c_Option_Ooption_Othe_001t__Syntax__Otrm,type,
the_trm: option_trm > trm ).
thf(sy_c_Option_Ooption_Othe_001t__Syntax__Ovariable,type,
the_variable: option_variable > variable ).
thf(sy_c_Option_Othese_001_062_It__Syntax__Ovariable_Mt__Real__Oreal_J,type,
these_variable_real: set_op12188086e_real > set_variable_real ).
thf(sy_c_Option_Othese_001t__Syntax__Otrm,type,
these_trm: set_option_trm > set_trm ).
thf(sy_c_Option_Othese_001t__Syntax__Ovariable,type,
these_variable: set_option_variable > set_variable ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_I_062_It__Syntax__Ovariable_Mt__Real__Oreal_J_M_Eo_J,type,
bot_bo1661475211real_o: ( variable > real ) > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Syntax__Ovariable_M_Eo_J,type,
bot_bot_variable_o: variable > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__Syntax__Ovariable_Mt__Real__Oreal_J_J,type,
bot_bo721182586e_real: set_variable_real ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Option__Ooption_I_062_It__Syntax__Ovariable_Mt__Real__Oreal_J_J_J,type,
bot_bo1411475018e_real: set_op12188086e_real ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Option__Ooption_It__Syntax__Otrm_J_J,type,
bot_bo946428664on_trm: set_option_trm ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Option__Ooption_It__Syntax__Ovariable_J_J,type,
bot_bo266290559riable: set_option_variable ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Real__Oreal_J,type,
bot_bot_set_real: set_real ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Syntax__Otrm_J,type,
bot_bot_set_trm: set_trm ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Syntax__Ovariable_J,type,
bot_bot_set_variable: set_variable ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__Syntax__Ovariable_Mt__Real__Oreal_J_J,type,
ord_le1113654598e_real: set_variable_real > set_variable_real > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Syntax__Ovariable_J,type,
ord_le282106107riable: set_variable > set_variable > $o ).
thf(sy_c_Set_OCollect_001_062_It__Syntax__Ovariable_Mt__Real__Oreal_J,type,
collec633296133e_real: ( ( variable > real ) > $o ) > set_variable_real ).
thf(sy_c_Set_OCollect_001t__Syntax__Ovariable,type,
collect_variable: ( variable > $o ) > set_variable ).
thf(sy_c_Set_Oinsert_001_062_It__Syntax__Ovariable_Mt__Real__Oreal_J,type,
insert_variable_real: ( variable > real ) > set_variable_real > set_variable_real ).
thf(sy_c_Set_Oinsert_001t__Nat__Onat,type,
insert_nat: nat > set_nat > set_nat ).
thf(sy_c_Set_Oinsert_001t__Option__Ooption_I_062_It__Syntax__Ovariable_Mt__Real__Oreal_J_J,type,
insert526581936e_real: option_variable_real > set_op12188086e_real > set_op12188086e_real ).
thf(sy_c_Set_Oinsert_001t__Option__Ooption_It__Syntax__Otrm_J,type,
insert_option_trm: option_trm > set_option_trm > set_option_trm ).
thf(sy_c_Set_Oinsert_001t__Option__Ooption_It__Syntax__Ovariable_J,type,
insert1340772453riable: option_variable > set_option_variable > set_option_variable ).
thf(sy_c_Set_Oinsert_001t__Real__Oreal,type,
insert_real: real > set_real > set_real ).
thf(sy_c_Set_Oinsert_001t__Syntax__Otrm,type,
insert_trm: trm > set_trm > set_trm ).
thf(sy_c_Set_Oinsert_001t__Syntax__Ovariable,type,
insert_variable: variable > set_variable > set_variable ).
thf(sy_c_Set_Ois__empty_001_062_It__Syntax__Ovariable_Mt__Real__Oreal_J,type,
is_emp227886046e_real: set_variable_real > $o ).
thf(sy_c_Set_Ois__empty_001t__Syntax__Ovariable,type,
is_empty_variable: set_variable > $o ).
thf(sy_c_Set_Ois__singleton_001_062_It__Syntax__Ovariable_Mt__Real__Oreal_J,type,
is_sin524757308e_real: set_variable_real > $o ).
thf(sy_c_Set_Ois__singleton_001t__Syntax__Ovariable,type,
is_sin155454833riable: set_variable > $o ).
thf(sy_c_Set_Othe__elem_001_062_It__Syntax__Ovariable_Mt__Real__Oreal_J,type,
the_el1059226619e_real: set_variable_real > variable > real ).
thf(sy_c_Set_Othe__elem_001t__Syntax__Ovariable,type,
the_elem_variable: set_variable > variable ).
thf(sy_c_Static__Semantics_OFVT,type,
static_FVT: trm > set_variable ).
thf(sy_c_Static__Semantics_Oselectlike,type,
static_selectlike: set_variable_real > ( variable > real ) > set_variable > set_variable_real ).
thf(sy_c_Syntax_Otrm_OTimes,type,
times: trm > trm > trm ).
thf(sy_c_Syntax_Ovariable_ODVar,type,
dVar: char > variable ).
thf(sy_c_Syntax_Ovariable_ORVar,type,
rVar: char > variable ).
thf(sy_c_Syntax_Ovariable_Osize__variable,type,
size_variable: variable > nat ).
thf(sy_c_USubst__Mirabelle__vidvnmlwwz_OTimeso,type,
uSubst918876924Timeso: option_trm > option_trm > option_trm ).
thf(sy_c_USubst__Mirabelle__vidvnmlwwz_Oadjoint,type,
uSubst1599435252djoint: produc1418842292n_game > denotational_interp > ( variable > real ) > denotational_interp ).
thf(sy_c_USubst__Mirabelle__vidvnmlwwz_Ousubstappt,type,
uSubst516392818stappt: produc1418842292n_game > set_variable > trm > option_trm ).
thf(sy_c_member_001_062_It__Syntax__Ovariable_Mt__Real__Oreal_J,type,
member_variable_real: ( variable > real ) > set_variable_real > $o ).
thf(sy_c_member_001t__Option__Ooption_I_062_It__Syntax__Ovariable_Mt__Real__Oreal_J_J,type,
member523846807e_real: option_variable_real > set_op12188086e_real > $o ).
thf(sy_c_member_001t__Option__Ooption_It__Syntax__Otrm_J,type,
member_option_trm: option_trm > set_option_trm > $o ).
thf(sy_c_member_001t__Option__Ooption_It__Syntax__Ovariable_J,type,
member814448204riable: option_variable > set_option_variable > $o ).
thf(sy_c_member_001t__Syntax__Otrm,type,
member_trm: trm > set_trm > $o ).
thf(sy_c_member_001t__Syntax__Ovariable,type,
member_variable: variable > set_variable > $o ).
thf(sy_v_F,type,
f: real > variable > real ).
thf(sy_v_I,type,
i: denotational_interp ).
thf(sy_v_U,type,
u: set_variable ).
thf(sy_v__092_060omega_062,type,
omega: variable > real ).
thf(sy_v__092_060sigma_062,type,
sigma: produc1418842292n_game ).
thf(sy_v__092_060theta_062,type,
theta: trm ).
thf(sy_v_rr____,type,
rr: trm > denotational_interp > trm > denotational_interp > char > ( real > variable > real ) > variable > real ).
thf(sy_v_x,type,
x: char ).
% Relevant facts (353)
thf(fact_0_uv,axiom,
denota1419872369iation @ ( f @ zero_zero_real ) @ omega @ ( sup_sup_set_variable @ u @ ( insert_variable @ ( rVar @ x ) @ ( insert_variable @ ( dVar @ x ) @ bot_bot_set_variable ) ) ) ).
% uv
thf(fact_1_Uvariation__empty,axiom,
! [Nu: variable > real,Nu2: variable > real] :
( ( denota1419872369iation @ Nu @ Nu2 @ bot_bot_set_variable )
= ( Nu = Nu2 ) ) ).
% Uvariation_empty
thf(fact_2_Un__insert__left,axiom,
! [A: variable,B: set_variable,C: set_variable] :
( ( sup_sup_set_variable @ ( insert_variable @ A @ B ) @ C )
= ( insert_variable @ A @ ( sup_sup_set_variable @ B @ C ) ) ) ).
% Un_insert_left
thf(fact_3_Un__insert__left,axiom,
! [A: variable > real,B: set_variable_real,C: set_variable_real] :
( ( sup_su1685293586e_real @ ( insert_variable_real @ A @ B ) @ C )
= ( insert_variable_real @ A @ ( sup_su1685293586e_real @ B @ C ) ) ) ).
% Un_insert_left
thf(fact_4_Un__insert__right,axiom,
! [A2: set_variable,A: variable,B: set_variable] :
( ( sup_sup_set_variable @ A2 @ ( insert_variable @ A @ B ) )
= ( insert_variable @ A @ ( sup_sup_set_variable @ A2 @ B ) ) ) ).
% Un_insert_right
thf(fact_5_Un__insert__right,axiom,
! [A2: set_variable_real,A: variable > real,B: set_variable_real] :
( ( sup_su1685293586e_real @ A2 @ ( insert_variable_real @ A @ B ) )
= ( insert_variable_real @ A @ ( sup_su1685293586e_real @ A2 @ B ) ) ) ).
% Un_insert_right
thf(fact_6_Un__empty,axiom,
! [A2: set_variable_real,B: set_variable_real] :
( ( ( sup_su1685293586e_real @ A2 @ B )
= bot_bo721182586e_real )
= ( ( A2 = bot_bo721182586e_real )
& ( B = bot_bo721182586e_real ) ) ) ).
% Un_empty
thf(fact_7_Un__empty,axiom,
! [A2: set_variable,B: set_variable] :
( ( ( sup_sup_set_variable @ A2 @ B )
= bot_bot_set_variable )
= ( ( A2 = bot_bot_set_variable )
& ( B = bot_bot_set_variable ) ) ) ).
% Un_empty
thf(fact_8_sup__bot__left,axiom,
! [X: set_variable_real] :
( ( sup_su1685293586e_real @ bot_bo721182586e_real @ X )
= X ) ).
% sup_bot_left
thf(fact_9_sup__bot__left,axiom,
! [X: set_variable] :
( ( sup_sup_set_variable @ bot_bot_set_variable @ X )
= X ) ).
% sup_bot_left
thf(fact_10_sup__bot__right,axiom,
! [X: set_variable_real] :
( ( sup_su1685293586e_real @ X @ bot_bo721182586e_real )
= X ) ).
% sup_bot_right
thf(fact_11_sup__bot__right,axiom,
! [X: set_variable] :
( ( sup_sup_set_variable @ X @ bot_bot_set_variable )
= X ) ).
% sup_bot_right
thf(fact_12_bot__eq__sup__iff,axiom,
! [X: set_variable_real,Y: set_variable_real] :
( ( bot_bo721182586e_real
= ( sup_su1685293586e_real @ X @ Y ) )
= ( ( X = bot_bo721182586e_real )
& ( Y = bot_bo721182586e_real ) ) ) ).
% bot_eq_sup_iff
thf(fact_13_bot__eq__sup__iff,axiom,
! [X: set_variable,Y: set_variable] :
( ( bot_bot_set_variable
= ( sup_sup_set_variable @ X @ Y ) )
= ( ( X = bot_bot_set_variable )
& ( Y = bot_bot_set_variable ) ) ) ).
% bot_eq_sup_iff
thf(fact_14_sup__eq__bot__iff,axiom,
! [X: set_variable_real,Y: set_variable_real] :
( ( ( sup_su1685293586e_real @ X @ Y )
= bot_bo721182586e_real )
= ( ( X = bot_bo721182586e_real )
& ( Y = bot_bo721182586e_real ) ) ) ).
% sup_eq_bot_iff
thf(fact_15_sup__eq__bot__iff,axiom,
! [X: set_variable,Y: set_variable] :
( ( ( sup_sup_set_variable @ X @ Y )
= bot_bot_set_variable )
= ( ( X = bot_bot_set_variable )
& ( Y = bot_bot_set_variable ) ) ) ).
% sup_eq_bot_iff
thf(fact_16_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_variable_real,B2: set_variable_real] :
( ( ( sup_su1685293586e_real @ A @ B2 )
= bot_bo721182586e_real )
= ( ( A = bot_bo721182586e_real )
& ( B2 = bot_bo721182586e_real ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_17_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_variable,B2: set_variable] :
( ( ( sup_sup_set_variable @ A @ B2 )
= bot_bot_set_variable )
= ( ( A = bot_bot_set_variable )
& ( B2 = bot_bot_set_variable ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_18_sup__bot_Oleft__neutral,axiom,
! [A: set_variable] :
( ( sup_sup_set_variable @ bot_bot_set_variable @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_19_sup__bot_Oleft__neutral,axiom,
! [A: set_variable_real] :
( ( sup_su1685293586e_real @ bot_bo721182586e_real @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_20_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_variable,B2: set_variable] :
( ( bot_bot_set_variable
= ( sup_sup_set_variable @ A @ B2 ) )
= ( ( A = bot_bot_set_variable )
& ( B2 = bot_bot_set_variable ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_21_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_variable_real,B2: set_variable_real] :
( ( bot_bo721182586e_real
= ( sup_su1685293586e_real @ A @ B2 ) )
= ( ( A = bot_bo721182586e_real )
& ( B2 = bot_bo721182586e_real ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_22_sup__bot_Oright__neutral,axiom,
! [A: set_variable] :
( ( sup_sup_set_variable @ A @ bot_bot_set_variable )
= A ) ).
% sup_bot.right_neutral
thf(fact_23_sup__bot_Oright__neutral,axiom,
! [A: set_variable_real] :
( ( sup_su1685293586e_real @ A @ bot_bo721182586e_real )
= A ) ).
% sup_bot.right_neutral
thf(fact_24_empty__Collect__eq,axiom,
! [P: variable > $o] :
( ( bot_bot_set_variable
= ( collect_variable @ P ) )
= ( ! [X2: variable] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_25_empty__Collect__eq,axiom,
! [P: ( variable > real ) > $o] :
( ( bot_bo721182586e_real
= ( collec633296133e_real @ P ) )
= ( ! [X2: variable > real] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_26_Collect__empty__eq,axiom,
! [P: variable > $o] :
( ( ( collect_variable @ P )
= bot_bot_set_variable )
= ( ! [X2: variable] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_27_Collect__empty__eq,axiom,
! [P: ( variable > real ) > $o] :
( ( ( collec633296133e_real @ P )
= bot_bo721182586e_real )
= ( ! [X2: variable > real] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_28_all__not__in__conv,axiom,
! [A2: set_variable] :
( ( ! [X2: variable] :
~ ( member_variable @ X2 @ A2 ) )
= ( A2 = bot_bot_set_variable ) ) ).
% all_not_in_conv
thf(fact_29_all__not__in__conv,axiom,
! [A2: set_variable_real] :
( ( ! [X2: variable > real] :
~ ( member_variable_real @ X2 @ A2 ) )
= ( A2 = bot_bo721182586e_real ) ) ).
% all_not_in_conv
thf(fact_30_empty__iff,axiom,
! [C2: variable] :
~ ( member_variable @ C2 @ bot_bot_set_variable ) ).
% empty_iff
thf(fact_31_empty__iff,axiom,
! [C2: variable > real] :
~ ( member_variable_real @ C2 @ bot_bo721182586e_real ) ).
% empty_iff
thf(fact_32_insert__absorb2,axiom,
! [X: variable,A2: set_variable] :
( ( insert_variable @ X @ ( insert_variable @ X @ A2 ) )
= ( insert_variable @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_33_insert__absorb2,axiom,
! [X: variable > real,A2: set_variable_real] :
( ( insert_variable_real @ X @ ( insert_variable_real @ X @ A2 ) )
= ( insert_variable_real @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_34_insert__iff,axiom,
! [A: variable,B2: variable,A2: set_variable] :
( ( member_variable @ A @ ( insert_variable @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member_variable @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_35_insert__iff,axiom,
! [A: variable > real,B2: variable > real,A2: set_variable_real] :
( ( member_variable_real @ A @ ( insert_variable_real @ B2 @ A2 ) )
= ( ( A = B2 )
| ( member_variable_real @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_36_insertCI,axiom,
! [A: variable,B: set_variable,B2: variable] :
( ( ~ ( member_variable @ A @ B )
=> ( A = B2 ) )
=> ( member_variable @ A @ ( insert_variable @ B2 @ B ) ) ) ).
% insertCI
thf(fact_37_insertCI,axiom,
! [A: variable > real,B: set_variable_real,B2: variable > real] :
( ( ~ ( member_variable_real @ A @ B )
=> ( A = B2 ) )
=> ( member_variable_real @ A @ ( insert_variable_real @ B2 @ B ) ) ) ).
% insertCI
thf(fact_38_sup_Oright__idem,axiom,
! [A: set_variable,B2: set_variable] :
( ( sup_sup_set_variable @ ( sup_sup_set_variable @ A @ B2 ) @ B2 )
= ( sup_sup_set_variable @ A @ B2 ) ) ).
% sup.right_idem
thf(fact_39_sup_Oright__idem,axiom,
! [A: set_variable_real,B2: set_variable_real] :
( ( sup_su1685293586e_real @ ( sup_su1685293586e_real @ A @ B2 ) @ B2 )
= ( sup_su1685293586e_real @ A @ B2 ) ) ).
% sup.right_idem
thf(fact_40_sup__left__idem,axiom,
! [X: set_variable,Y: set_variable] :
( ( sup_sup_set_variable @ X @ ( sup_sup_set_variable @ X @ Y ) )
= ( sup_sup_set_variable @ X @ Y ) ) ).
% sup_left_idem
thf(fact_41_sup__left__idem,axiom,
! [X: set_variable_real,Y: set_variable_real] :
( ( sup_su1685293586e_real @ X @ ( sup_su1685293586e_real @ X @ Y ) )
= ( sup_su1685293586e_real @ X @ Y ) ) ).
% sup_left_idem
thf(fact_42_sup_Oleft__idem,axiom,
! [A: set_variable,B2: set_variable] :
( ( sup_sup_set_variable @ A @ ( sup_sup_set_variable @ A @ B2 ) )
= ( sup_sup_set_variable @ A @ B2 ) ) ).
% sup.left_idem
thf(fact_43_sup_Oleft__idem,axiom,
! [A: set_variable_real,B2: set_variable_real] :
( ( sup_su1685293586e_real @ A @ ( sup_su1685293586e_real @ A @ B2 ) )
= ( sup_su1685293586e_real @ A @ B2 ) ) ).
% sup.left_idem
thf(fact_44_sup__idem,axiom,
! [X: set_variable] :
( ( sup_sup_set_variable @ X @ X )
= X ) ).
% sup_idem
thf(fact_45_sup__idem,axiom,
! [X: set_variable_real] :
( ( sup_su1685293586e_real @ X @ X )
= X ) ).
% sup_idem
thf(fact_46_sup_Oidem,axiom,
! [A: set_variable] :
( ( sup_sup_set_variable @ A @ A )
= A ) ).
% sup.idem
thf(fact_47_sup_Oidem,axiom,
! [A: set_variable_real] :
( ( sup_su1685293586e_real @ A @ A )
= A ) ).
% sup.idem
thf(fact_48_UnCI,axiom,
! [C2: variable,B: set_variable,A2: set_variable] :
( ( ~ ( member_variable @ C2 @ B )
=> ( member_variable @ C2 @ A2 ) )
=> ( member_variable @ C2 @ ( sup_sup_set_variable @ A2 @ B ) ) ) ).
% UnCI
thf(fact_49_UnCI,axiom,
! [C2: variable > real,B: set_variable_real,A2: set_variable_real] :
( ( ~ ( member_variable_real @ C2 @ B )
=> ( member_variable_real @ C2 @ A2 ) )
=> ( member_variable_real @ C2 @ ( sup_su1685293586e_real @ A2 @ B ) ) ) ).
% UnCI
thf(fact_50_singletonI,axiom,
! [A: variable] : ( member_variable @ A @ ( insert_variable @ A @ bot_bot_set_variable ) ) ).
% singletonI
thf(fact_51_singletonI,axiom,
! [A: variable > real] : ( member_variable_real @ A @ ( insert_variable_real @ A @ bot_bo721182586e_real ) ) ).
% singletonI
thf(fact_52_bot__set__def,axiom,
( bot_bot_set_variable
= ( collect_variable @ bot_bot_variable_o ) ) ).
% bot_set_def
thf(fact_53_bot__set__def,axiom,
( bot_bo721182586e_real
= ( collec633296133e_real @ bot_bo1661475211real_o ) ) ).
% bot_set_def
thf(fact_54_ex__in__conv,axiom,
! [A2: set_variable] :
( ( ? [X2: variable] : ( member_variable @ X2 @ A2 ) )
= ( A2 != bot_bot_set_variable ) ) ).
% ex_in_conv
thf(fact_55_ex__in__conv,axiom,
! [A2: set_variable_real] :
( ( ? [X2: variable > real] : ( member_variable_real @ X2 @ A2 ) )
= ( A2 != bot_bo721182586e_real ) ) ).
% ex_in_conv
thf(fact_56_equals0I,axiom,
! [A2: set_variable] :
( ! [Y2: variable] :
~ ( member_variable @ Y2 @ A2 )
=> ( A2 = bot_bot_set_variable ) ) ).
% equals0I
thf(fact_57_equals0I,axiom,
! [A2: set_variable_real] :
( ! [Y2: variable > real] :
~ ( member_variable_real @ Y2 @ A2 )
=> ( A2 = bot_bo721182586e_real ) ) ).
% equals0I
thf(fact_58_equals0D,axiom,
! [A2: set_variable,A: variable] :
( ( A2 = bot_bot_set_variable )
=> ~ ( member_variable @ A @ A2 ) ) ).
% equals0D
thf(fact_59_equals0D,axiom,
! [A2: set_variable_real,A: variable > real] :
( ( A2 = bot_bo721182586e_real )
=> ~ ( member_variable_real @ A @ A2 ) ) ).
% equals0D
thf(fact_60_emptyE,axiom,
! [A: variable] :
~ ( member_variable @ A @ bot_bot_set_variable ) ).
% emptyE
thf(fact_61_emptyE,axiom,
! [A: variable > real] :
~ ( member_variable_real @ A @ bot_bo721182586e_real ) ).
% emptyE
thf(fact_62_mk__disjoint__insert,axiom,
! [A: variable,A2: set_variable] :
( ( member_variable @ A @ A2 )
=> ? [B3: set_variable] :
( ( A2
= ( insert_variable @ A @ B3 ) )
& ~ ( member_variable @ A @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_63_mk__disjoint__insert,axiom,
! [A: variable > real,A2: set_variable_real] :
( ( member_variable_real @ A @ A2 )
=> ? [B3: set_variable_real] :
( ( A2
= ( insert_variable_real @ A @ B3 ) )
& ~ ( member_variable_real @ A @ B3 ) ) ) ).
% mk_disjoint_insert
thf(fact_64_insert__commute,axiom,
! [X: variable,Y: variable,A2: set_variable] :
( ( insert_variable @ X @ ( insert_variable @ Y @ A2 ) )
= ( insert_variable @ Y @ ( insert_variable @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_65_insert__commute,axiom,
! [X: variable > real,Y: variable > real,A2: set_variable_real] :
( ( insert_variable_real @ X @ ( insert_variable_real @ Y @ A2 ) )
= ( insert_variable_real @ Y @ ( insert_variable_real @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_66_insert__eq__iff,axiom,
! [A: variable,A2: set_variable,B2: variable,B: set_variable] :
( ~ ( member_variable @ A @ A2 )
=> ( ~ ( member_variable @ B2 @ B )
=> ( ( ( insert_variable @ A @ A2 )
= ( insert_variable @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C3: set_variable] :
( ( A2
= ( insert_variable @ B2 @ C3 ) )
& ~ ( member_variable @ B2 @ C3 )
& ( B
= ( insert_variable @ A @ C3 ) )
& ~ ( member_variable @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_67_insert__eq__iff,axiom,
! [A: variable > real,A2: set_variable_real,B2: variable > real,B: set_variable_real] :
( ~ ( member_variable_real @ A @ A2 )
=> ( ~ ( member_variable_real @ B2 @ B )
=> ( ( ( insert_variable_real @ A @ A2 )
= ( insert_variable_real @ B2 @ B ) )
= ( ( ( A = B2 )
=> ( A2 = B ) )
& ( ( A != B2 )
=> ? [C3: set_variable_real] :
( ( A2
= ( insert_variable_real @ B2 @ C3 ) )
& ~ ( member_variable_real @ B2 @ C3 )
& ( B
= ( insert_variable_real @ A @ C3 ) )
& ~ ( member_variable_real @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_68_insert__absorb,axiom,
! [A: variable,A2: set_variable] :
( ( member_variable @ A @ A2 )
=> ( ( insert_variable @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_69_insert__absorb,axiom,
! [A: variable > real,A2: set_variable_real] :
( ( member_variable_real @ A @ A2 )
=> ( ( insert_variable_real @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_70_insert__ident,axiom,
! [X: variable,A2: set_variable,B: set_variable] :
( ~ ( member_variable @ X @ A2 )
=> ( ~ ( member_variable @ X @ B )
=> ( ( ( insert_variable @ X @ A2 )
= ( insert_variable @ X @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_71_insert__ident,axiom,
! [X: variable > real,A2: set_variable_real,B: set_variable_real] :
( ~ ( member_variable_real @ X @ A2 )
=> ( ~ ( member_variable_real @ X @ B )
=> ( ( ( insert_variable_real @ X @ A2 )
= ( insert_variable_real @ X @ B ) )
= ( A2 = B ) ) ) ) ).
% insert_ident
thf(fact_72_Set_Oset__insert,axiom,
! [X: variable,A2: set_variable] :
( ( member_variable @ X @ A2 )
=> ~ ! [B3: set_variable] :
( ( A2
= ( insert_variable @ X @ B3 ) )
=> ( member_variable @ X @ B3 ) ) ) ).
% Set.set_insert
thf(fact_73_Set_Oset__insert,axiom,
! [X: variable > real,A2: set_variable_real] :
( ( member_variable_real @ X @ A2 )
=> ~ ! [B3: set_variable_real] :
( ( A2
= ( insert_variable_real @ X @ B3 ) )
=> ( member_variable_real @ X @ B3 ) ) ) ).
% Set.set_insert
thf(fact_74_insertI2,axiom,
! [A: variable,B: set_variable,B2: variable] :
( ( member_variable @ A @ B )
=> ( member_variable @ A @ ( insert_variable @ B2 @ B ) ) ) ).
% insertI2
thf(fact_75_insertI2,axiom,
! [A: variable > real,B: set_variable_real,B2: variable > real] :
( ( member_variable_real @ A @ B )
=> ( member_variable_real @ A @ ( insert_variable_real @ B2 @ B ) ) ) ).
% insertI2
thf(fact_76_insertI1,axiom,
! [A: variable,B: set_variable] : ( member_variable @ A @ ( insert_variable @ A @ B ) ) ).
% insertI1
thf(fact_77_insertI1,axiom,
! [A: variable > real,B: set_variable_real] : ( member_variable_real @ A @ ( insert_variable_real @ A @ B ) ) ).
% insertI1
thf(fact_78_insertE,axiom,
! [A: variable,B2: variable,A2: set_variable] :
( ( member_variable @ A @ ( insert_variable @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( member_variable @ A @ A2 ) ) ) ).
% insertE
thf(fact_79_insertE,axiom,
! [A: variable > real,B2: variable > real,A2: set_variable_real] :
( ( member_variable_real @ A @ ( insert_variable_real @ B2 @ A2 ) )
=> ( ( A != B2 )
=> ( member_variable_real @ A @ A2 ) ) ) ).
% insertE
thf(fact_80_sup__left__commute,axiom,
! [X: set_variable,Y: set_variable,Z: set_variable] :
( ( sup_sup_set_variable @ X @ ( sup_sup_set_variable @ Y @ Z ) )
= ( sup_sup_set_variable @ Y @ ( sup_sup_set_variable @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_81_sup__left__commute,axiom,
! [X: set_variable_real,Y: set_variable_real,Z: set_variable_real] :
( ( sup_su1685293586e_real @ X @ ( sup_su1685293586e_real @ Y @ Z ) )
= ( sup_su1685293586e_real @ Y @ ( sup_su1685293586e_real @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_82_sup_Oleft__commute,axiom,
! [B2: set_variable,A: set_variable,C2: set_variable] :
( ( sup_sup_set_variable @ B2 @ ( sup_sup_set_variable @ A @ C2 ) )
= ( sup_sup_set_variable @ A @ ( sup_sup_set_variable @ B2 @ C2 ) ) ) ).
% sup.left_commute
thf(fact_83_sup_Oleft__commute,axiom,
! [B2: set_variable_real,A: set_variable_real,C2: set_variable_real] :
( ( sup_su1685293586e_real @ B2 @ ( sup_su1685293586e_real @ A @ C2 ) )
= ( sup_su1685293586e_real @ A @ ( sup_su1685293586e_real @ B2 @ C2 ) ) ) ).
% sup.left_commute
thf(fact_84_sup__commute,axiom,
( sup_sup_set_variable
= ( ^ [X2: set_variable,Y3: set_variable] : ( sup_sup_set_variable @ Y3 @ X2 ) ) ) ).
% sup_commute
thf(fact_85_sup__commute,axiom,
( sup_su1685293586e_real
= ( ^ [X2: set_variable_real,Y3: set_variable_real] : ( sup_su1685293586e_real @ Y3 @ X2 ) ) ) ).
% sup_commute
thf(fact_86_mem__Collect__eq,axiom,
! [A: variable,P: variable > $o] :
( ( member_variable @ A @ ( collect_variable @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_87_mem__Collect__eq,axiom,
! [A: variable > real,P: ( variable > real ) > $o] :
( ( member_variable_real @ A @ ( collec633296133e_real @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_88_Collect__mem__eq,axiom,
! [A2: set_variable] :
( ( collect_variable
@ ^ [X2: variable] : ( member_variable @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_89_Collect__mem__eq,axiom,
! [A2: set_variable_real] :
( ( collec633296133e_real
@ ^ [X2: variable > real] : ( member_variable_real @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_90_sup_Ocommute,axiom,
( sup_sup_set_variable
= ( ^ [A3: set_variable,B4: set_variable] : ( sup_sup_set_variable @ B4 @ A3 ) ) ) ).
% sup.commute
thf(fact_91_sup_Ocommute,axiom,
( sup_su1685293586e_real
= ( ^ [A3: set_variable_real,B4: set_variable_real] : ( sup_su1685293586e_real @ B4 @ A3 ) ) ) ).
% sup.commute
thf(fact_92_sup__assoc,axiom,
! [X: set_variable,Y: set_variable,Z: set_variable] :
( ( sup_sup_set_variable @ ( sup_sup_set_variable @ X @ Y ) @ Z )
= ( sup_sup_set_variable @ X @ ( sup_sup_set_variable @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_93_sup__assoc,axiom,
! [X: set_variable_real,Y: set_variable_real,Z: set_variable_real] :
( ( sup_su1685293586e_real @ ( sup_su1685293586e_real @ X @ Y ) @ Z )
= ( sup_su1685293586e_real @ X @ ( sup_su1685293586e_real @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_94_sup_Oassoc,axiom,
! [A: set_variable,B2: set_variable,C2: set_variable] :
( ( sup_sup_set_variable @ ( sup_sup_set_variable @ A @ B2 ) @ C2 )
= ( sup_sup_set_variable @ A @ ( sup_sup_set_variable @ B2 @ C2 ) ) ) ).
% sup.assoc
thf(fact_95_sup_Oassoc,axiom,
! [A: set_variable_real,B2: set_variable_real,C2: set_variable_real] :
( ( sup_su1685293586e_real @ ( sup_su1685293586e_real @ A @ B2 ) @ C2 )
= ( sup_su1685293586e_real @ A @ ( sup_su1685293586e_real @ B2 @ C2 ) ) ) ).
% sup.assoc
thf(fact_96_boolean__algebra__cancel_Osup2,axiom,
! [B: set_variable,K: set_variable,B2: set_variable,A: set_variable] :
( ( B
= ( sup_sup_set_variable @ K @ B2 ) )
=> ( ( sup_sup_set_variable @ A @ B )
= ( sup_sup_set_variable @ K @ ( sup_sup_set_variable @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_97_boolean__algebra__cancel_Osup2,axiom,
! [B: set_variable_real,K: set_variable_real,B2: set_variable_real,A: set_variable_real] :
( ( B
= ( sup_su1685293586e_real @ K @ B2 ) )
=> ( ( sup_su1685293586e_real @ A @ B )
= ( sup_su1685293586e_real @ K @ ( sup_su1685293586e_real @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_98_boolean__algebra__cancel_Osup1,axiom,
! [A2: set_variable,K: set_variable,A: set_variable,B2: set_variable] :
( ( A2
= ( sup_sup_set_variable @ K @ A ) )
=> ( ( sup_sup_set_variable @ A2 @ B2 )
= ( sup_sup_set_variable @ K @ ( sup_sup_set_variable @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_99_boolean__algebra__cancel_Osup1,axiom,
! [A2: set_variable_real,K: set_variable_real,A: set_variable_real,B2: set_variable_real] :
( ( A2
= ( sup_su1685293586e_real @ K @ A ) )
=> ( ( sup_su1685293586e_real @ A2 @ B2 )
= ( sup_su1685293586e_real @ K @ ( sup_su1685293586e_real @ A @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_100_inf__sup__aci_I5_J,axiom,
( sup_sup_set_variable
= ( ^ [X2: set_variable,Y3: set_variable] : ( sup_sup_set_variable @ Y3 @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_101_inf__sup__aci_I5_J,axiom,
( sup_su1685293586e_real
= ( ^ [X2: set_variable_real,Y3: set_variable_real] : ( sup_su1685293586e_real @ Y3 @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_102_inf__sup__aci_I6_J,axiom,
! [X: set_variable,Y: set_variable,Z: set_variable] :
( ( sup_sup_set_variable @ ( sup_sup_set_variable @ X @ Y ) @ Z )
= ( sup_sup_set_variable @ X @ ( sup_sup_set_variable @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_103_inf__sup__aci_I6_J,axiom,
! [X: set_variable_real,Y: set_variable_real,Z: set_variable_real] :
( ( sup_su1685293586e_real @ ( sup_su1685293586e_real @ X @ Y ) @ Z )
= ( sup_su1685293586e_real @ X @ ( sup_su1685293586e_real @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_104_inf__sup__aci_I7_J,axiom,
! [X: set_variable,Y: set_variable,Z: set_variable] :
( ( sup_sup_set_variable @ X @ ( sup_sup_set_variable @ Y @ Z ) )
= ( sup_sup_set_variable @ Y @ ( sup_sup_set_variable @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_105_inf__sup__aci_I7_J,axiom,
! [X: set_variable_real,Y: set_variable_real,Z: set_variable_real] :
( ( sup_su1685293586e_real @ X @ ( sup_su1685293586e_real @ Y @ Z ) )
= ( sup_su1685293586e_real @ Y @ ( sup_su1685293586e_real @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_106_inf__sup__aci_I8_J,axiom,
! [X: set_variable,Y: set_variable] :
( ( sup_sup_set_variable @ X @ ( sup_sup_set_variable @ X @ Y ) )
= ( sup_sup_set_variable @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_107_inf__sup__aci_I8_J,axiom,
! [X: set_variable_real,Y: set_variable_real] :
( ( sup_su1685293586e_real @ X @ ( sup_su1685293586e_real @ X @ Y ) )
= ( sup_su1685293586e_real @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_108_Un__left__commute,axiom,
! [A2: set_variable,B: set_variable,C: set_variable] :
( ( sup_sup_set_variable @ A2 @ ( sup_sup_set_variable @ B @ C ) )
= ( sup_sup_set_variable @ B @ ( sup_sup_set_variable @ A2 @ C ) ) ) ).
% Un_left_commute
thf(fact_109_Un__left__commute,axiom,
! [A2: set_variable_real,B: set_variable_real,C: set_variable_real] :
( ( sup_su1685293586e_real @ A2 @ ( sup_su1685293586e_real @ B @ C ) )
= ( sup_su1685293586e_real @ B @ ( sup_su1685293586e_real @ A2 @ C ) ) ) ).
% Un_left_commute
thf(fact_110_Un__left__absorb,axiom,
! [A2: set_variable,B: set_variable] :
( ( sup_sup_set_variable @ A2 @ ( sup_sup_set_variable @ A2 @ B ) )
= ( sup_sup_set_variable @ A2 @ B ) ) ).
% Un_left_absorb
thf(fact_111_Un__left__absorb,axiom,
! [A2: set_variable_real,B: set_variable_real] :
( ( sup_su1685293586e_real @ A2 @ ( sup_su1685293586e_real @ A2 @ B ) )
= ( sup_su1685293586e_real @ A2 @ B ) ) ).
% Un_left_absorb
thf(fact_112_Un__commute,axiom,
( sup_sup_set_variable
= ( ^ [A4: set_variable,B5: set_variable] : ( sup_sup_set_variable @ B5 @ A4 ) ) ) ).
% Un_commute
thf(fact_113_Un__commute,axiom,
( sup_su1685293586e_real
= ( ^ [A4: set_variable_real,B5: set_variable_real] : ( sup_su1685293586e_real @ B5 @ A4 ) ) ) ).
% Un_commute
thf(fact_114_Un__absorb,axiom,
! [A2: set_variable] :
( ( sup_sup_set_variable @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_115_Un__absorb,axiom,
! [A2: set_variable_real] :
( ( sup_su1685293586e_real @ A2 @ A2 )
= A2 ) ).
% Un_absorb
thf(fact_116_Un__assoc,axiom,
! [A2: set_variable,B: set_variable,C: set_variable] :
( ( sup_sup_set_variable @ ( sup_sup_set_variable @ A2 @ B ) @ C )
= ( sup_sup_set_variable @ A2 @ ( sup_sup_set_variable @ B @ C ) ) ) ).
% Un_assoc
thf(fact_117_Un__assoc,axiom,
! [A2: set_variable_real,B: set_variable_real,C: set_variable_real] :
( ( sup_su1685293586e_real @ ( sup_su1685293586e_real @ A2 @ B ) @ C )
= ( sup_su1685293586e_real @ A2 @ ( sup_su1685293586e_real @ B @ C ) ) ) ).
% Un_assoc
thf(fact_118_ball__Un,axiom,
! [A2: set_variable,B: set_variable,P: variable > $o] :
( ( ! [X2: variable] :
( ( member_variable @ X2 @ ( sup_sup_set_variable @ A2 @ B ) )
=> ( P @ X2 ) ) )
= ( ! [X2: variable] :
( ( member_variable @ X2 @ A2 )
=> ( P @ X2 ) )
& ! [X2: variable] :
( ( member_variable @ X2 @ B )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_119_ball__Un,axiom,
! [A2: set_variable_real,B: set_variable_real,P: ( variable > real ) > $o] :
( ( ! [X2: variable > real] :
( ( member_variable_real @ X2 @ ( sup_su1685293586e_real @ A2 @ B ) )
=> ( P @ X2 ) ) )
= ( ! [X2: variable > real] :
( ( member_variable_real @ X2 @ A2 )
=> ( P @ X2 ) )
& ! [X2: variable > real] :
( ( member_variable_real @ X2 @ B )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_120_bex__Un,axiom,
! [A2: set_variable,B: set_variable,P: variable > $o] :
( ( ? [X2: variable] :
( ( member_variable @ X2 @ ( sup_sup_set_variable @ A2 @ B ) )
& ( P @ X2 ) ) )
= ( ? [X2: variable] :
( ( member_variable @ X2 @ A2 )
& ( P @ X2 ) )
| ? [X2: variable] :
( ( member_variable @ X2 @ B )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_121_bex__Un,axiom,
! [A2: set_variable_real,B: set_variable_real,P: ( variable > real ) > $o] :
( ( ? [X2: variable > real] :
( ( member_variable_real @ X2 @ ( sup_su1685293586e_real @ A2 @ B ) )
& ( P @ X2 ) ) )
= ( ? [X2: variable > real] :
( ( member_variable_real @ X2 @ A2 )
& ( P @ X2 ) )
| ? [X2: variable > real] :
( ( member_variable_real @ X2 @ B )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_122_UnI2,axiom,
! [C2: variable,B: set_variable,A2: set_variable] :
( ( member_variable @ C2 @ B )
=> ( member_variable @ C2 @ ( sup_sup_set_variable @ A2 @ B ) ) ) ).
% UnI2
thf(fact_123_UnI2,axiom,
! [C2: variable > real,B: set_variable_real,A2: set_variable_real] :
( ( member_variable_real @ C2 @ B )
=> ( member_variable_real @ C2 @ ( sup_su1685293586e_real @ A2 @ B ) ) ) ).
% UnI2
thf(fact_124_UnI1,axiom,
! [C2: variable,A2: set_variable,B: set_variable] :
( ( member_variable @ C2 @ A2 )
=> ( member_variable @ C2 @ ( sup_sup_set_variable @ A2 @ B ) ) ) ).
% UnI1
thf(fact_125_UnI1,axiom,
! [C2: variable > real,A2: set_variable_real,B: set_variable_real] :
( ( member_variable_real @ C2 @ A2 )
=> ( member_variable_real @ C2 @ ( sup_su1685293586e_real @ A2 @ B ) ) ) ).
% UnI1
thf(fact_126_UnE,axiom,
! [C2: variable,A2: set_variable,B: set_variable] :
( ( member_variable @ C2 @ ( sup_sup_set_variable @ A2 @ B ) )
=> ( ~ ( member_variable @ C2 @ A2 )
=> ( member_variable @ C2 @ B ) ) ) ).
% UnE
thf(fact_127_UnE,axiom,
! [C2: variable > real,A2: set_variable_real,B: set_variable_real] :
( ( member_variable_real @ C2 @ ( sup_su1685293586e_real @ A2 @ B ) )
=> ( ~ ( member_variable_real @ C2 @ A2 )
=> ( member_variable_real @ C2 @ B ) ) ) ).
% UnE
thf(fact_128_Uvariation__sym__rel,axiom,
! [Omega: variable > real,Nu: variable > real,U: set_variable] :
( ( denota1419872369iation @ Omega @ Nu @ U )
=> ( denota1419872369iation @ Nu @ Omega @ U ) ) ).
% Uvariation_sym_rel
thf(fact_129_Uvariation__refl,axiom,
! [Nu: variable > real,V: set_variable] : ( denota1419872369iation @ Nu @ Nu @ V ) ).
% Uvariation_refl
thf(fact_130_Uvariation__sym,axiom,
( denota1419872369iation
= ( ^ [Omega2: variable > real,Nu3: variable > real] : ( denota1419872369iation @ Nu3 @ Omega2 ) ) ) ).
% Uvariation_sym
thf(fact_131_Uvariation__def,axiom,
( denota1419872369iation
= ( ^ [Nu3: variable > real,Nu4: variable > real,U2: set_variable] :
! [I: variable] :
( ~ ( member_variable @ I @ U2 )
=> ( ( Nu3 @ I )
= ( Nu4 @ I ) ) ) ) ) ).
% Uvariation_def
thf(fact_132_singleton__inject,axiom,
! [A: variable,B2: variable] :
( ( ( insert_variable @ A @ bot_bot_set_variable )
= ( insert_variable @ B2 @ bot_bot_set_variable ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_133_singleton__inject,axiom,
! [A: variable > real,B2: variable > real] :
( ( ( insert_variable_real @ A @ bot_bo721182586e_real )
= ( insert_variable_real @ B2 @ bot_bo721182586e_real ) )
=> ( A = B2 ) ) ).
% singleton_inject
thf(fact_134_insert__not__empty,axiom,
! [A: variable,A2: set_variable] :
( ( insert_variable @ A @ A2 )
!= bot_bot_set_variable ) ).
% insert_not_empty
thf(fact_135_insert__not__empty,axiom,
! [A: variable > real,A2: set_variable_real] :
( ( insert_variable_real @ A @ A2 )
!= bot_bo721182586e_real ) ).
% insert_not_empty
thf(fact_136_doubleton__eq__iff,axiom,
! [A: variable,B2: variable,C2: variable,D: variable] :
( ( ( insert_variable @ A @ ( insert_variable @ B2 @ bot_bot_set_variable ) )
= ( insert_variable @ C2 @ ( insert_variable @ D @ bot_bot_set_variable ) ) )
= ( ( ( A = C2 )
& ( B2 = D ) )
| ( ( A = D )
& ( B2 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_137_doubleton__eq__iff,axiom,
! [A: variable > real,B2: variable > real,C2: variable > real,D: variable > real] :
( ( ( insert_variable_real @ A @ ( insert_variable_real @ B2 @ bot_bo721182586e_real ) )
= ( insert_variable_real @ C2 @ ( insert_variable_real @ D @ bot_bo721182586e_real ) ) )
= ( ( ( A = C2 )
& ( B2 = D ) )
| ( ( A = D )
& ( B2 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_138_singleton__iff,axiom,
! [B2: variable,A: variable] :
( ( member_variable @ B2 @ ( insert_variable @ A @ bot_bot_set_variable ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_139_singleton__iff,axiom,
! [B2: variable > real,A: variable > real] :
( ( member_variable_real @ B2 @ ( insert_variable_real @ A @ bot_bo721182586e_real ) )
= ( B2 = A ) ) ).
% singleton_iff
thf(fact_140_singletonD,axiom,
! [B2: variable,A: variable] :
( ( member_variable @ B2 @ ( insert_variable @ A @ bot_bot_set_variable ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_141_singletonD,axiom,
! [B2: variable > real,A: variable > real] :
( ( member_variable_real @ B2 @ ( insert_variable_real @ A @ bot_bo721182586e_real ) )
=> ( B2 = A ) ) ).
% singletonD
thf(fact_142_Un__empty__right,axiom,
! [A2: set_variable] :
( ( sup_sup_set_variable @ A2 @ bot_bot_set_variable )
= A2 ) ).
% Un_empty_right
thf(fact_143_Un__empty__right,axiom,
! [A2: set_variable_real] :
( ( sup_su1685293586e_real @ A2 @ bot_bo721182586e_real )
= A2 ) ).
% Un_empty_right
thf(fact_144_Un__empty__left,axiom,
! [B: set_variable] :
( ( sup_sup_set_variable @ bot_bot_set_variable @ B )
= B ) ).
% Un_empty_left
thf(fact_145_Un__empty__left,axiom,
! [B: set_variable_real] :
( ( sup_su1685293586e_real @ bot_bo721182586e_real @ B )
= B ) ).
% Un_empty_left
thf(fact_146_Uvariation__trans,axiom,
! [Omega: variable > real,Nu: variable > real,U: set_variable,Mu: variable > real,V: set_variable] :
( ( denota1419872369iation @ Omega @ Nu @ U )
=> ( ( denota1419872369iation @ Nu @ Mu @ V )
=> ( denota1419872369iation @ Omega @ Mu @ ( sup_sup_set_variable @ U @ V ) ) ) ) ).
% Uvariation_trans
thf(fact_147_singleton__Un__iff,axiom,
! [X: variable,A2: set_variable,B: set_variable] :
( ( ( insert_variable @ X @ bot_bot_set_variable )
= ( sup_sup_set_variable @ A2 @ B ) )
= ( ( ( A2 = bot_bot_set_variable )
& ( B
= ( insert_variable @ X @ bot_bot_set_variable ) ) )
| ( ( A2
= ( insert_variable @ X @ bot_bot_set_variable ) )
& ( B = bot_bot_set_variable ) )
| ( ( A2
= ( insert_variable @ X @ bot_bot_set_variable ) )
& ( B
= ( insert_variable @ X @ bot_bot_set_variable ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_148_singleton__Un__iff,axiom,
! [X: variable > real,A2: set_variable_real,B: set_variable_real] :
( ( ( insert_variable_real @ X @ bot_bo721182586e_real )
= ( sup_su1685293586e_real @ A2 @ B ) )
= ( ( ( A2 = bot_bo721182586e_real )
& ( B
= ( insert_variable_real @ X @ bot_bo721182586e_real ) ) )
| ( ( A2
= ( insert_variable_real @ X @ bot_bo721182586e_real ) )
& ( B = bot_bo721182586e_real ) )
| ( ( A2
= ( insert_variable_real @ X @ bot_bo721182586e_real ) )
& ( B
= ( insert_variable_real @ X @ bot_bo721182586e_real ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_149_Un__singleton__iff,axiom,
! [A2: set_variable,B: set_variable,X: variable] :
( ( ( sup_sup_set_variable @ A2 @ B )
= ( insert_variable @ X @ bot_bot_set_variable ) )
= ( ( ( A2 = bot_bot_set_variable )
& ( B
= ( insert_variable @ X @ bot_bot_set_variable ) ) )
| ( ( A2
= ( insert_variable @ X @ bot_bot_set_variable ) )
& ( B = bot_bot_set_variable ) )
| ( ( A2
= ( insert_variable @ X @ bot_bot_set_variable ) )
& ( B
= ( insert_variable @ X @ bot_bot_set_variable ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_150_Un__singleton__iff,axiom,
! [A2: set_variable_real,B: set_variable_real,X: variable > real] :
( ( ( sup_su1685293586e_real @ A2 @ B )
= ( insert_variable_real @ X @ bot_bo721182586e_real ) )
= ( ( ( A2 = bot_bo721182586e_real )
& ( B
= ( insert_variable_real @ X @ bot_bo721182586e_real ) ) )
| ( ( A2
= ( insert_variable_real @ X @ bot_bo721182586e_real ) )
& ( B = bot_bo721182586e_real ) )
| ( ( A2
= ( insert_variable_real @ X @ bot_bo721182586e_real ) )
& ( B
= ( insert_variable_real @ X @ bot_bo721182586e_real ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_151_insert__is__Un,axiom,
( insert_variable
= ( ^ [A3: variable] : ( sup_sup_set_variable @ ( insert_variable @ A3 @ bot_bot_set_variable ) ) ) ) ).
% insert_is_Un
thf(fact_152_insert__is__Un,axiom,
( insert_variable_real
= ( ^ [A3: variable > real] : ( sup_su1685293586e_real @ ( insert_variable_real @ A3 @ bot_bo721182586e_real ) ) ) ) ).
% insert_is_Un
thf(fact_153__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062rr_O_A_092_060forall_062x0_Ax1_Ax2_Ax3_Ax4_Ax5_O_A_I_092_060exists_062v6_O_AUvariation_Av6_A_Ix5_A0_J_A_123RVar_Ax4_M_ADVar_Ax4_125_A_092_060and_062_Aterm__sem_Ax3_Ax2_Av6_A_092_060noteq_062_Aterm__sem_Ax1_Ax0_Av6_J_A_061_A_IUvariation_A_Irr_Ax0_Ax1_Ax2_Ax3_Ax4_Ax5_J_A_Ix5_A0_J_A_123RVar_Ax4_M_ADVar_Ax4_125_A_092_060and_062_Aterm__sem_Ax3_Ax2_A_Irr_Ax0_Ax1_Ax2_Ax3_Ax4_Ax5_J_A_092_060noteq_062_Aterm__sem_Ax1_Ax0_A_Irr_Ax0_Ax1_Ax2_Ax3_Ax4_Ax5_J_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Rr: trm > denotational_interp > trm > denotational_interp > char > ( real > variable > real ) > variable > real] :
~ ! [X0: trm,X1: denotational_interp,X22: trm,X3: denotational_interp,X4: char,X5: real > variable > real] :
( ( ? [V6: variable > real] :
( ( denota1419872369iation @ V6 @ ( X5 @ zero_zero_real ) @ ( insert_variable @ ( rVar @ X4 ) @ ( insert_variable @ ( dVar @ X4 ) @ bot_bot_set_variable ) ) )
& ( ( denota1863255036rm_sem @ X3 @ X22 @ V6 )
!= ( denota1863255036rm_sem @ X1 @ X0 @ V6 ) ) ) )
= ( ( denota1419872369iation @ ( Rr @ X0 @ X1 @ X22 @ X3 @ X4 @ X5 ) @ ( X5 @ zero_zero_real ) @ ( insert_variable @ ( rVar @ X4 ) @ ( insert_variable @ ( dVar @ X4 ) @ bot_bot_set_variable ) ) )
& ( ( denota1863255036rm_sem @ X3 @ X22 @ ( Rr @ X0 @ X1 @ X22 @ X3 @ X4 @ X5 ) )
!= ( denota1863255036rm_sem @ X1 @ X0 @ ( Rr @ X0 @ X1 @ X22 @ X3 @ X4 @ X5 ) ) ) ) ) ).
% \<open>\<And>thesis. (\<And>rr. \<forall>x0 x1 x2 x3 x4 x5. (\<exists>v6. Uvariation v6 (x5 0) {RVar x4, DVar x4} \<and> term_sem x3 x2 v6 \<noteq> term_sem x1 x0 v6) = (Uvariation (rr x0 x1 x2 x3 x4 x5) (x5 0) {RVar x4, DVar x4} \<and> term_sem x3 x2 (rr x0 x1 x2 x3 x4 x5) \<noteq> term_sem x1 x0 (rr x0 x1 x2 x3 x4 x5)) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_154_solves__Vagree__trans,axiom,
! [F: real > variable > real,Omega: variable > real,U: set_variable,I2: denotational_interp,X: char,Theta: trm,Zeta: real] :
( ( denota1419872369iation @ ( F @ zero_zero_real ) @ Omega @ U )
=> ( ( denota1778088425es_ODE @ I2 @ F @ X @ Theta )
=> ( denota1419872369iation @ ( F @ Zeta ) @ Omega @ ( sup_sup_set_variable @ U @ ( insert_variable @ ( rVar @ X ) @ ( insert_variable @ ( dVar @ X ) @ bot_bot_set_variable ) ) ) ) ) ) ).
% solves_Vagree_trans
thf(fact_155__092_060open_062_092_060forall_062x0_Ax1_Ax2_Ax3_Ax4_Ax5_O_A_I_092_060exists_062v6_O_AUvariation_Av6_A_Ix5_A0_J_A_123RVar_Ax4_M_ADVar_Ax4_125_A_092_060and_062_Aterm__sem_Ax3_Ax2_Av6_A_092_060noteq_062_Aterm__sem_Ax1_Ax0_Av6_J_A_061_A_IUvariation_A_Irr_Ax0_Ax1_Ax2_Ax3_Ax4_Ax5_J_A_Ix5_A0_J_A_123RVar_Ax4_M_ADVar_Ax4_125_A_092_060and_062_Aterm__sem_Ax3_Ax2_A_Irr_Ax0_Ax1_Ax2_Ax3_Ax4_Ax5_J_A_092_060noteq_062_Aterm__sem_Ax1_Ax0_A_Irr_Ax0_Ax1_Ax2_Ax3_Ax4_Ax5_J_J_092_060close_062,axiom,
! [X0: trm,X1: denotational_interp,X22: trm,X3: denotational_interp,X4: char,X5: real > variable > real] :
( ( ? [V6: variable > real] :
( ( denota1419872369iation @ V6 @ ( X5 @ zero_zero_real ) @ ( insert_variable @ ( rVar @ X4 ) @ ( insert_variable @ ( dVar @ X4 ) @ bot_bot_set_variable ) ) )
& ( ( denota1863255036rm_sem @ X3 @ X22 @ V6 )
!= ( denota1863255036rm_sem @ X1 @ X0 @ V6 ) ) ) )
= ( ( denota1419872369iation @ ( rr @ X0 @ X1 @ X22 @ X3 @ X4 @ X5 ) @ ( X5 @ zero_zero_real ) @ ( insert_variable @ ( rVar @ X4 ) @ ( insert_variable @ ( dVar @ X4 ) @ bot_bot_set_variable ) ) )
& ( ( denota1863255036rm_sem @ X3 @ X22 @ ( rr @ X0 @ X1 @ X22 @ X3 @ X4 @ X5 ) )
!= ( denota1863255036rm_sem @ X1 @ X0 @ ( rr @ X0 @ X1 @ X22 @ X3 @ X4 @ X5 ) ) ) ) ) ).
% \<open>\<forall>x0 x1 x2 x3 x4 x5. (\<exists>v6. Uvariation v6 (x5 0) {RVar x4, DVar x4} \<and> term_sem x3 x2 v6 \<noteq> term_sem x1 x0 v6) = (Uvariation (rr x0 x1 x2 x3 x4 x5) (x5 0) {RVar x4, DVar x4} \<and> term_sem x3 x2 (rr x0 x1 x2 x3 x4 x5) \<noteq> term_sem x1 x0 (rr x0 x1 x2 x3 x4 x5))\<close>
thf(fact_156_variable_Oinject_I1_J,axiom,
! [X12: char,Y1: char] :
( ( ( rVar @ X12 )
= ( rVar @ Y1 ) )
= ( X12 = Y1 ) ) ).
% variable.inject(1)
thf(fact_157_set__zero,axiom,
( zero_zero_set_real
= ( insert_real @ zero_zero_real @ bot_bot_set_real ) ) ).
% set_zero
thf(fact_158_set__zero,axiom,
( zero_zero_set_nat
= ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).
% set_zero
thf(fact_159_variable_Oinject_I2_J,axiom,
! [X23: char,Y22: char] :
( ( ( dVar @ X23 )
= ( dVar @ Y22 ) )
= ( X23 = Y22 ) ) ).
% variable.inject(2)
thf(fact_160_union__or,axiom,
! [C2: variable,A2: set_variable,B: set_variable] :
( ( member_variable @ C2 @ ( sup_sup_set_variable @ A2 @ B ) )
= ( ( member_variable @ C2 @ A2 )
| ( member_variable @ C2 @ B ) ) ) ).
% union_or
thf(fact_161_union__or,axiom,
! [C2: variable > real,A2: set_variable_real,B: set_variable_real] :
( ( member_variable_real @ C2 @ ( sup_su1685293586e_real @ A2 @ B ) )
= ( ( member_variable_real @ C2 @ A2 )
| ( member_variable_real @ C2 @ B ) ) ) ).
% union_or
thf(fact_162_ball__insert,axiom,
! [A: variable,B: set_variable,P: variable > $o] :
( ( ! [X2: variable] :
( ( member_variable @ X2 @ ( insert_variable @ A @ B ) )
=> ( P @ X2 ) ) )
= ( ( P @ A )
& ! [X2: variable] :
( ( member_variable @ X2 @ B )
=> ( P @ X2 ) ) ) ) ).
% ball_insert
thf(fact_163_ball__insert,axiom,
! [A: variable > real,B: set_variable_real,P: ( variable > real ) > $o] :
( ( ! [X2: variable > real] :
( ( member_variable_real @ X2 @ ( insert_variable_real @ A @ B ) )
=> ( P @ X2 ) ) )
= ( ( P @ A )
& ! [X2: variable > real] :
( ( member_variable_real @ X2 @ B )
=> ( P @ X2 ) ) ) ) ).
% ball_insert
thf(fact_164_the__elem__eq,axiom,
! [X: variable] :
( ( the_elem_variable @ ( insert_variable @ X @ bot_bot_set_variable ) )
= X ) ).
% the_elem_eq
thf(fact_165_the__elem__eq,axiom,
! [X: variable > real] :
( ( the_el1059226619e_real @ ( insert_variable_real @ X @ bot_bo721182586e_real ) )
= X ) ).
% the_elem_eq
thf(fact_166_f2,axiom,
! [F2: real > variable > real,C4: char,I3: denotational_interp,T: trm,Ia: denotational_interp,Ta: trm] :
( ( ( denota1419872369iation @ ( rr @ Ta @ Ia @ T @ I3 @ C4 @ F2 ) @ ( F2 @ zero_zero_real ) @ ( insert_variable @ ( rVar @ C4 ) @ ( insert_variable @ ( dVar @ C4 ) @ bot_bot_set_variable ) ) )
& ( ( denota1863255036rm_sem @ I3 @ T @ ( rr @ Ta @ Ia @ T @ I3 @ C4 @ F2 ) )
!= ( denota1863255036rm_sem @ Ia @ Ta @ ( rr @ Ta @ Ia @ T @ I3 @ C4 @ F2 ) ) ) )
| ( ( denota1778088425es_ODE @ I3 @ F2 @ C4 @ T )
= ( denota1778088425es_ODE @ Ia @ F2 @ C4 @ Ta ) ) ) ).
% f2
thf(fact_167_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_168_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_169_not__union__or,axiom,
! [X: variable,A2: set_variable,B: set_variable] :
( ( ~ ( member_variable @ X @ ( sup_sup_set_variable @ A2 @ B ) ) )
= ( ~ ( member_variable @ X @ A2 )
& ~ ( member_variable @ X @ B ) ) ) ).
% not_union_or
thf(fact_170_not__union__or,axiom,
! [X: variable > real,A2: set_variable_real,B: set_variable_real] :
( ( ~ ( member_variable_real @ X @ ( sup_su1685293586e_real @ A2 @ B ) ) )
= ( ~ ( member_variable_real @ X @ A2 )
& ~ ( member_variable_real @ X @ B ) ) ) ).
% not_union_or
thf(fact_171_same__ODE__same__sol,axiom,
! [F: real > variable > real,X: char,I2: denotational_interp,Theta: trm,J: denotational_interp,Eta: trm] :
( ! [Nu5: variable > real] :
( ( denota1419872369iation @ Nu5 @ ( F @ zero_zero_real ) @ ( insert_variable @ ( rVar @ X ) @ ( insert_variable @ ( dVar @ X ) @ bot_bot_set_variable ) ) )
=> ( ( denota1863255036rm_sem @ I2 @ Theta @ Nu5 )
= ( denota1863255036rm_sem @ J @ Eta @ Nu5 ) ) )
=> ( ( denota1778088425es_ODE @ I2 @ F @ X @ Theta )
= ( denota1778088425es_ODE @ J @ F @ X @ Eta ) ) ) ).
% same_ODE_same_sol
thf(fact_172_variable_Odistinct_I1_J,axiom,
! [X12: char,X23: char] :
( ( rVar @ X12 )
!= ( dVar @ X23 ) ) ).
% variable.distinct(1)
thf(fact_173_variable_Oinduct,axiom,
! [P: variable > $o,Variable: variable] :
( ! [X6: char] : ( P @ ( rVar @ X6 ) )
=> ( ! [X6: char] : ( P @ ( dVar @ X6 ) )
=> ( P @ Variable ) ) ) ).
% variable.induct
thf(fact_174_variable_Oexhaust,axiom,
! [Y: variable] :
( ! [X13: char] :
( Y
!= ( rVar @ X13 ) )
=> ~ ! [X24: char] :
( Y
!= ( dVar @ X24 ) ) ) ).
% variable.exhaust
thf(fact_175_bot__empty__eq,axiom,
( bot_bot_variable_o
= ( ^ [X2: variable] : ( member_variable @ X2 @ bot_bot_set_variable ) ) ) ).
% bot_empty_eq
thf(fact_176_bot__empty__eq,axiom,
( bot_bo1661475211real_o
= ( ^ [X2: variable > real] : ( member_variable_real @ X2 @ bot_bo721182586e_real ) ) ) ).
% bot_empty_eq
thf(fact_177_Collect__empty__eq__bot,axiom,
! [P: variable > $o] :
( ( ( collect_variable @ P )
= bot_bot_set_variable )
= ( P = bot_bot_variable_o ) ) ).
% Collect_empty_eq_bot
thf(fact_178_Collect__empty__eq__bot,axiom,
! [P: ( variable > real ) > $o] :
( ( ( collec633296133e_real @ P )
= bot_bo721182586e_real )
= ( P = bot_bo1661475211real_o ) ) ).
% Collect_empty_eq_bot
thf(fact_179_is__singleton__the__elem,axiom,
( is_sin155454833riable
= ( ^ [A4: set_variable] :
( A4
= ( insert_variable @ ( the_elem_variable @ A4 ) @ bot_bot_set_variable ) ) ) ) ).
% is_singleton_the_elem
thf(fact_180_is__singleton__the__elem,axiom,
( is_sin524757308e_real
= ( ^ [A4: set_variable_real] :
( A4
= ( insert_variable_real @ ( the_el1059226619e_real @ A4 ) @ bot_bo721182586e_real ) ) ) ) ).
% is_singleton_the_elem
thf(fact_181_vaflow2,axiom,
! [F: real > variable > real,Theta: trm,Zeta: real] :
( ( denota1778088425es_ODE @ ( uSubst1599435252djoint @ sigma @ i @ omega ) @ F @ x @ Theta )
=> ( denota1419872369iation @ ( F @ Zeta ) @ ( F @ zero_zero_real ) @ ( insert_variable @ ( rVar @ x ) @ ( insert_variable @ ( dVar @ x ) @ bot_bot_set_variable ) ) ) ) ).
% vaflow2
thf(fact_182_is__singletonI,axiom,
! [X: variable] : ( is_sin155454833riable @ ( insert_variable @ X @ bot_bot_set_variable ) ) ).
% is_singletonI
thf(fact_183_is__singletonI,axiom,
! [X: variable > real] : ( is_sin524757308e_real @ ( insert_variable_real @ X @ bot_bo721182586e_real ) ) ).
% is_singletonI
thf(fact_184_a1,axiom,
denota1778088425es_ODE @ ( uSubst1599435252djoint @ sigma @ i @ omega ) @ f @ x @ theta ).
% a1
thf(fact_185_is__singletonI_H,axiom,
! [A2: set_variable] :
( ( A2 != bot_bot_set_variable )
=> ( ! [X6: variable,Y2: variable] :
( ( member_variable @ X6 @ A2 )
=> ( ( member_variable @ Y2 @ A2 )
=> ( X6 = Y2 ) ) )
=> ( is_sin155454833riable @ A2 ) ) ) ).
% is_singletonI'
thf(fact_186_is__singletonI_H,axiom,
! [A2: set_variable_real] :
( ( A2 != bot_bo721182586e_real )
=> ( ! [X6: variable > real,Y2: variable > real] :
( ( member_variable_real @ X6 @ A2 )
=> ( ( member_variable_real @ Y2 @ A2 )
=> ( X6 = Y2 ) ) )
=> ( is_sin524757308e_real @ A2 ) ) ) ).
% is_singletonI'
thf(fact_187_is__singleton__def,axiom,
( is_sin155454833riable
= ( ^ [A4: set_variable] :
? [X2: variable] :
( A4
= ( insert_variable @ X2 @ bot_bot_set_variable ) ) ) ) ).
% is_singleton_def
thf(fact_188_is__singleton__def,axiom,
( is_sin524757308e_real
= ( ^ [A4: set_variable_real] :
? [X2: variable > real] :
( A4
= ( insert_variable_real @ X2 @ bot_bo721182586e_real ) ) ) ) ).
% is_singleton_def
thf(fact_189_is__singletonE,axiom,
! [A2: set_variable] :
( ( is_sin155454833riable @ A2 )
=> ~ ! [X6: variable] :
( A2
!= ( insert_variable @ X6 @ bot_bot_set_variable ) ) ) ).
% is_singletonE
thf(fact_190_is__singletonE,axiom,
! [A2: set_variable_real] :
( ( is_sin524757308e_real @ A2 )
=> ~ ! [X6: variable > real] :
( A2
!= ( insert_variable_real @ X6 @ bot_bo721182586e_real ) ) ) ).
% is_singletonE
thf(fact_191_vaflow1,axiom,
! [F: real > variable > real,Theta: trm,Zeta: real] :
( ( denota1778088425es_ODE @ i @ F @ x @ ( the_trm @ ( uSubst516392818stappt @ sigma @ ( sup_sup_set_variable @ u @ ( insert_variable @ ( rVar @ x ) @ ( insert_variable @ ( dVar @ x ) @ bot_bot_set_variable ) ) ) @ Theta ) ) )
=> ( denota1419872369iation @ ( F @ Zeta ) @ ( F @ zero_zero_real ) @ ( insert_variable @ ( rVar @ x ) @ ( insert_variable @ ( dVar @ x ) @ bot_bot_set_variable ) ) ) ) ).
% vaflow1
thf(fact_192_l2r,axiom,
( ( denota1778088425es_ODE @ i @ f @ x @ ( the_trm @ ( uSubst516392818stappt @ sigma @ ( sup_sup_set_variable @ u @ ( insert_variable @ ( rVar @ x ) @ ( insert_variable @ ( dVar @ x ) @ bot_bot_set_variable ) ) ) @ theta ) ) )
=> ( denota1778088425es_ODE @ ( uSubst1599435252djoint @ sigma @ i @ omega ) @ f @ x @ theta ) ) ).
% l2r
thf(fact_193_IH,axiom,
! [Nu: variable > real] :
( ( denota1419872369iation @ Nu @ ( f @ zero_zero_real ) @ ( sup_sup_set_variable @ u @ ( insert_variable @ ( rVar @ x ) @ ( insert_variable @ ( dVar @ x ) @ bot_bot_set_variable ) ) ) )
=> ( ( denota1863255036rm_sem @ i @ ( the_trm @ ( uSubst516392818stappt @ sigma @ ( sup_sup_set_variable @ u @ ( insert_variable @ ( rVar @ x ) @ ( insert_variable @ ( dVar @ x ) @ bot_bot_set_variable ) ) ) @ theta ) ) @ Nu )
= ( denota1863255036rm_sem @ ( uSubst1599435252djoint @ sigma @ i @ ( f @ zero_zero_real ) ) @ theta @ Nu ) ) ) ).
% IH
thf(fact_194_subdef,axiom,
( ( uSubst516392818stappt @ sigma @ ( sup_sup_set_variable @ u @ ( insert_variable @ ( rVar @ x ) @ ( insert_variable @ ( dVar @ x ) @ bot_bot_set_variable ) ) ) @ theta )
!= none_trm ) ).
% subdef
thf(fact_195_variable_Osize__gen_I1_J,axiom,
! [X12: char] :
( ( size_variable @ ( rVar @ X12 ) )
= zero_zero_nat ) ).
% variable.size_gen(1)
thf(fact_196_usubst__term,axiom,
! [Nu: variable > real,Omega: variable > real,U: set_variable,Sigma: produc1418842292n_game,Theta: trm,I2: denotational_interp] :
( ( denota1419872369iation @ Nu @ Omega @ U )
=> ( ( ( uSubst516392818stappt @ Sigma @ U @ Theta )
!= none_trm )
=> ( ( denota1863255036rm_sem @ I2 @ ( the_trm @ ( uSubst516392818stappt @ Sigma @ U @ Theta ) ) @ Nu )
= ( denota1863255036rm_sem @ ( uSubst1599435252djoint @ Sigma @ I2 @ Omega ) @ Theta @ Nu ) ) ) ) ).
% usubst_term
thf(fact_197_usubstappt__det,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,Theta: trm,V: set_variable] :
( ( ( uSubst516392818stappt @ Sigma @ U @ Theta )
!= none_trm )
=> ( ( ( uSubst516392818stappt @ Sigma @ V @ Theta )
!= none_trm )
=> ( ( uSubst516392818stappt @ Sigma @ U @ Theta )
= ( uSubst516392818stappt @ Sigma @ V @ Theta ) ) ) ) ).
% usubstappt_det
thf(fact_198_undeft__None,axiom,
none_trm = none_trm ).
% undeft_None
thf(fact_199_usubst__ode,axiom,
! [Sigma: produc1418842292n_game,X: char,Theta: trm,I2: denotational_interp,F: real > variable > real] :
( ( ( uSubst516392818stappt @ Sigma @ ( insert_variable @ ( rVar @ X ) @ ( insert_variable @ ( dVar @ X ) @ bot_bot_set_variable ) ) @ Theta )
!= none_trm )
=> ( ( denota1778088425es_ODE @ I2 @ F @ X @ ( the_trm @ ( uSubst516392818stappt @ Sigma @ ( insert_variable @ ( rVar @ X ) @ ( insert_variable @ ( dVar @ X ) @ bot_bot_set_variable ) ) @ Theta ) ) )
= ( denota1778088425es_ODE @ ( uSubst1599435252djoint @ Sigma @ I2 @ ( F @ zero_zero_real ) ) @ F @ X @ Theta ) ) ) ).
% usubst_ode
thf(fact_200_variable_Osize__gen_I2_J,axiom,
! [X23: char] :
( ( size_variable @ ( dVar @ X23 ) )
= zero_zero_nat ) ).
% variable.size_gen(2)
thf(fact_201_option_Oexpand,axiom,
! [Option: option_trm,Option2: option_trm] :
( ( ( Option = none_trm )
= ( Option2 = none_trm ) )
=> ( ( ( Option != none_trm )
=> ( ( Option2 != none_trm )
=> ( ( the_trm @ Option )
= ( the_trm @ Option2 ) ) ) )
=> ( Option = Option2 ) ) ) ).
% option.expand
thf(fact_202_variable_Osize_I3_J,axiom,
! [X12: char] :
( ( size_size_variable @ ( rVar @ X12 ) )
= zero_zero_nat ) ).
% variable.size(3)
thf(fact_203_variable_Osize_I4_J,axiom,
! [X23: char] :
( ( size_size_variable @ ( dVar @ X23 ) )
= zero_zero_nat ) ).
% variable.size(4)
thf(fact_204_Set_Ois__empty__def,axiom,
( is_empty_variable
= ( ^ [A4: set_variable] : ( A4 = bot_bot_set_variable ) ) ) ).
% Set.is_empty_def
thf(fact_205_Set_Ois__empty__def,axiom,
( is_emp227886046e_real
= ( ^ [A4: set_variable_real] : ( A4 = bot_bo721182586e_real ) ) ) ).
% Set.is_empty_def
thf(fact_206_Uvariation__repv,axiom,
! [Omega: variable > real,X: variable,D: real] : ( denota1419872369iation @ Omega @ ( denotational_repv @ Omega @ X @ D ) @ ( insert_variable @ X @ bot_bot_set_variable ) ) ).
% Uvariation_repv
thf(fact_207_these__empty__eq,axiom,
! [B: set_option_trm] :
( ( ( these_trm @ B )
= bot_bot_set_trm )
= ( ( B = bot_bo946428664on_trm )
| ( B
= ( insert_option_trm @ none_trm @ bot_bo946428664on_trm ) ) ) ) ).
% these_empty_eq
thf(fact_208_these__empty__eq,axiom,
! [B: set_option_variable] :
( ( ( these_variable @ B )
= bot_bot_set_variable )
= ( ( B = bot_bo266290559riable )
| ( B
= ( insert1340772453riable @ none_variable @ bot_bo266290559riable ) ) ) ) ).
% these_empty_eq
thf(fact_209_these__empty__eq,axiom,
! [B: set_op12188086e_real] :
( ( ( these_variable_real @ B )
= bot_bo721182586e_real )
= ( ( B = bot_bo1411475018e_real )
| ( B
= ( insert526581936e_real @ none_variable_real @ bot_bo1411475018e_real ) ) ) ) ).
% these_empty_eq
thf(fact_210_these__not__empty__eq,axiom,
! [B: set_option_trm] :
( ( ( these_trm @ B )
!= bot_bot_set_trm )
= ( ( B != bot_bo946428664on_trm )
& ( B
!= ( insert_option_trm @ none_trm @ bot_bo946428664on_trm ) ) ) ) ).
% these_not_empty_eq
thf(fact_211_these__not__empty__eq,axiom,
! [B: set_option_variable] :
( ( ( these_variable @ B )
!= bot_bot_set_variable )
= ( ( B != bot_bo266290559riable )
& ( B
!= ( insert1340772453riable @ none_variable @ bot_bo266290559riable ) ) ) ) ).
% these_not_empty_eq
thf(fact_212_these__not__empty__eq,axiom,
! [B: set_op12188086e_real] :
( ( ( these_variable_real @ B )
!= bot_bo721182586e_real )
= ( ( B != bot_bo1411475018e_real )
& ( B
!= ( insert526581936e_real @ none_variable_real @ bot_bo1411475018e_real ) ) ) ) ).
% these_not_empty_eq
thf(fact_213_repv__self,axiom,
! [Omega: variable > real,X: variable] :
( ( denotational_repv @ Omega @ X @ ( Omega @ X ) )
= Omega ) ).
% repv_self
thf(fact_214_repv__access,axiom,
( denotational_repv
= ( ^ [Omega2: variable > real,X2: variable,R: real,Y3: variable] : ( if_real @ ( X2 = Y3 ) @ R @ ( Omega2 @ Y3 ) ) ) ) ).
% repv_access
thf(fact_215_these__empty,axiom,
( ( these_variable @ bot_bo266290559riable )
= bot_bot_set_variable ) ).
% these_empty
thf(fact_216_these__empty,axiom,
( ( these_variable_real @ bot_bo1411475018e_real )
= bot_bo721182586e_real ) ).
% these_empty
thf(fact_217_these__insert__None,axiom,
! [A2: set_option_trm] :
( ( these_trm @ ( insert_option_trm @ none_trm @ A2 ) )
= ( these_trm @ A2 ) ) ).
% these_insert_None
thf(fact_218_repv__def__correct,axiom,
( denotational_repv
= ( ^ [Omega2: variable > real,X2: variable,R: real,Y3: variable] : ( if_real @ ( X2 = Y3 ) @ R @ ( Omega2 @ Y3 ) ) ) ) ).
% repv_def_correct
thf(fact_219_these__insert__Some,axiom,
! [X: variable,A2: set_option_variable] :
( ( these_variable @ ( insert1340772453riable @ ( some_variable @ X ) @ A2 ) )
= ( insert_variable @ X @ ( these_variable @ A2 ) ) ) ).
% these_insert_Some
thf(fact_220_these__insert__Some,axiom,
! [X: variable > real,A2: set_op12188086e_real] :
( ( these_variable_real @ ( insert526581936e_real @ ( some_variable_real @ X ) @ A2 ) )
= ( insert_variable_real @ X @ ( these_variable_real @ A2 ) ) ) ).
% these_insert_Some
thf(fact_221_these__insert__Some,axiom,
! [X: trm,A2: set_option_trm] :
( ( these_trm @ ( insert_option_trm @ ( some_trm @ X ) @ A2 ) )
= ( insert_trm @ X @ ( these_trm @ A2 ) ) ) ).
% these_insert_Some
thf(fact_222_repv__selectlike__other__converse,axiom,
! [X: variable,Y: variable,Omega: variable > real,D: real,X7: set_variable_real] :
( ( X != Y )
=> ( ( member_variable_real @ ( denotational_repv @ Omega @ X @ D ) @ X7 )
= ( member_variable_real @ ( denotational_repv @ Omega @ X @ D ) @ ( static_selectlike @ X7 @ Omega @ ( insert_variable @ Y @ bot_bot_set_variable ) ) ) ) ) ).
% repv_selectlike_other_converse
thf(fact_223_repv__selectlike__other,axiom,
! [X: variable,Y: variable,Omega: variable > real,D: real,X7: set_variable_real] :
( ( X != Y )
=> ( ( member_variable_real @ ( denotational_repv @ Omega @ X @ D ) @ ( static_selectlike @ X7 @ Omega @ ( insert_variable @ Y @ bot_bot_set_variable ) ) )
= ( member_variable_real @ ( denotational_repv @ Omega @ X @ D ) @ X7 ) ) ) ).
% repv_selectlike_other
thf(fact_224_repv__selectlike__self,axiom,
! [Omega: variable > real,X: variable,D: real,X7: set_variable_real] :
( ( member_variable_real @ ( denotational_repv @ Omega @ X @ D ) @ ( static_selectlike @ X7 @ Omega @ ( insert_variable @ X @ bot_bot_set_variable ) ) )
= ( ( D
= ( Omega @ X ) )
& ( member_variable_real @ Omega @ X7 ) ) ) ).
% repv_selectlike_self
thf(fact_225_option_Oinject,axiom,
! [X23: trm,Y22: trm] :
( ( ( some_trm @ X23 )
= ( some_trm @ Y22 ) )
= ( X23 = Y22 ) ) ).
% option.inject
thf(fact_226_selectlike__self,axiom,
! [Nu: variable > real,X7: set_variable_real,V: set_variable] :
( ( member_variable_real @ Nu @ ( static_selectlike @ X7 @ Nu @ V ) )
= ( member_variable_real @ Nu @ X7 ) ) ).
% selectlike_self
thf(fact_227_not__None__eq,axiom,
! [X: option_trm] :
( ( X != none_trm )
= ( ? [Y3: trm] :
( X
= ( some_trm @ Y3 ) ) ) ) ).
% not_None_eq
thf(fact_228_not__Some__eq,axiom,
! [X: option_trm] :
( ( ! [Y3: trm] :
( X
!= ( some_trm @ Y3 ) ) )
= ( X = none_trm ) ) ).
% not_Some_eq
thf(fact_229_selectlike__empty,axiom,
! [X7: set_variable_real,Nu: variable > real] :
( ( static_selectlike @ X7 @ Nu @ bot_bot_set_variable )
= X7 ) ).
% selectlike_empty
thf(fact_230_selectlike__compose,axiom,
! [X7: set_variable_real,Nu: variable > real,V: set_variable,W: set_variable] :
( ( static_selectlike @ ( static_selectlike @ X7 @ Nu @ V ) @ Nu @ W )
= ( static_selectlike @ X7 @ Nu @ ( sup_sup_set_variable @ V @ W ) ) ) ).
% selectlike_compose
thf(fact_231_option_Ocollapse,axiom,
! [Option: option_trm] :
( ( Option != none_trm )
=> ( ( some_trm @ ( the_trm @ Option ) )
= Option ) ) ).
% option.collapse
thf(fact_232_in__these__eq,axiom,
! [X: variable,A2: set_option_variable] :
( ( member_variable @ X @ ( these_variable @ A2 ) )
= ( member814448204riable @ ( some_variable @ X ) @ A2 ) ) ).
% in_these_eq
thf(fact_233_in__these__eq,axiom,
! [X: variable > real,A2: set_op12188086e_real] :
( ( member_variable_real @ X @ ( these_variable_real @ A2 ) )
= ( member523846807e_real @ ( some_variable_real @ X ) @ A2 ) ) ).
% in_these_eq
thf(fact_234_in__these__eq,axiom,
! [X: trm,A2: set_option_trm] :
( ( member_trm @ X @ ( these_trm @ A2 ) )
= ( member_option_trm @ ( some_trm @ X ) @ A2 ) ) ).
% in_these_eq
thf(fact_235_selectlike__union,axiom,
! [X7: set_variable_real,Y4: set_variable_real,Nu: variable > real,V: set_variable] :
( ( static_selectlike @ ( sup_su1685293586e_real @ X7 @ Y4 ) @ Nu @ V )
= ( sup_su1685293586e_real @ ( static_selectlike @ X7 @ Nu @ V ) @ ( static_selectlike @ Y4 @ Nu @ V ) ) ) ).
% selectlike_union
thf(fact_236_option_Odistinct_I1_J,axiom,
! [X23: trm] :
( none_trm
!= ( some_trm @ X23 ) ) ).
% option.distinct(1)
thf(fact_237_option_Osel,axiom,
! [X23: trm] :
( ( the_trm @ ( some_trm @ X23 ) )
= X23 ) ).
% option.sel
thf(fact_238_option_OdiscI,axiom,
! [Option: option_trm,X23: trm] :
( ( Option
= ( some_trm @ X23 ) )
=> ( Option != none_trm ) ) ).
% option.discI
thf(fact_239_option_Oexhaust,axiom,
! [Y: option_trm] :
( ( Y != none_trm )
=> ~ ! [X24: trm] :
( Y
!= ( some_trm @ X24 ) ) ) ).
% option.exhaust
thf(fact_240_option_Oinducts,axiom,
! [P: option_trm > $o,Option: option_trm] :
( ( P @ none_trm )
=> ( ! [X6: trm] : ( P @ ( some_trm @ X6 ) )
=> ( P @ Option ) ) ) ).
% option.inducts
thf(fact_241_split__option__ex,axiom,
( ( ^ [P2: option_trm > $o] :
? [X8: option_trm] : ( P2 @ X8 ) )
= ( ^ [P3: option_trm > $o] :
( ( P3 @ none_trm )
| ? [X2: trm] : ( P3 @ ( some_trm @ X2 ) ) ) ) ) ).
% split_option_ex
thf(fact_242_split__option__all,axiom,
( ( ^ [P2: option_trm > $o] :
! [X8: option_trm] : ( P2 @ X8 ) )
= ( ^ [P3: option_trm > $o] :
( ( P3 @ none_trm )
& ! [X2: trm] : ( P3 @ ( some_trm @ X2 ) ) ) ) ) ).
% split_option_all
thf(fact_243_combine__options__cases,axiom,
! [X: option_trm,P: option_trm > option_trm > $o,Y: option_trm] :
( ( ( X = none_trm )
=> ( P @ X @ Y ) )
=> ( ( ( Y = none_trm )
=> ( P @ X @ Y ) )
=> ( ! [A5: trm,B6: trm] :
( ( X
= ( some_trm @ A5 ) )
=> ( ( Y
= ( some_trm @ B6 ) )
=> ( P @ X @ Y ) ) )
=> ( P @ X @ Y ) ) ) ) ).
% combine_options_cases
thf(fact_244_ODEo_Oinduct,axiom,
! [P: char > option_trm > $o,A0: char,A1: option_trm] :
( ! [X6: char,Theta2: trm] : ( P @ X6 @ ( some_trm @ Theta2 ) )
=> ( ! [X6: char] : ( P @ X6 @ none_trm )
=> ( P @ A0 @ A1 ) ) ) ).
% ODEo.induct
thf(fact_245_undeft__equiv,axiom,
! [Theta: option_trm] :
( ( Theta != none_trm )
= ( ? [T2: trm] :
( Theta
= ( some_trm @ T2 ) ) ) ) ).
% undeft_equiv
thf(fact_246_Timeso_Oinduct,axiom,
! [P: option_trm > option_trm > $o,A0: option_trm,A1: option_trm] :
( ! [Theta2: trm,Eta2: trm] : ( P @ ( some_trm @ Theta2 ) @ ( some_trm @ Eta2 ) )
=> ( ! [X_1: option_trm] : ( P @ none_trm @ X_1 )
=> ( ! [V2: trm] : ( P @ ( some_trm @ V2 ) @ none_trm )
=> ( P @ A0 @ A1 ) ) ) ) ).
% Timeso.induct
thf(fact_247_Assigno_Oinduct,axiom,
! [P: variable > option_trm > $o,A0: variable,A1: option_trm] :
( ! [X6: variable,Theta2: trm] : ( P @ X6 @ ( some_trm @ Theta2 ) )
=> ( ! [X6: variable] : ( P @ X6 @ none_trm )
=> ( P @ A0 @ A1 ) ) ) ).
% Assigno.induct
thf(fact_248_Differentialo_Ocases,axiom,
! [X: option_trm] :
( ! [Theta2: trm] :
( X
!= ( some_trm @ Theta2 ) )
=> ( X = none_trm ) ) ).
% Differentialo.cases
thf(fact_249_Differentialo_Oinduct,axiom,
! [P: option_trm > $o,A0: option_trm] :
( ! [Theta2: trm] : ( P @ ( some_trm @ Theta2 ) )
=> ( ( P @ none_trm )
=> ( P @ A0 ) ) ) ).
% Differentialo.induct
thf(fact_250_option_Oexhaust__sel,axiom,
! [Option: option_trm] :
( ( Option != none_trm )
=> ( Option
= ( some_trm @ ( the_trm @ Option ) ) ) ) ).
% option.exhaust_sel
thf(fact_251_option_Osimps_I15_J,axiom,
! [X23: trm] :
( ( set_option_trm2 @ ( some_trm @ X23 ) )
= ( insert_trm @ X23 @ bot_bot_set_trm ) ) ).
% option.simps(15)
thf(fact_252_option_Osimps_I15_J,axiom,
! [X23: variable] :
( ( set_option_variable2 @ ( some_variable @ X23 ) )
= ( insert_variable @ X23 @ bot_bot_set_variable ) ) ).
% option.simps(15)
thf(fact_253_option_Osimps_I15_J,axiom,
! [X23: variable > real] :
( ( set_op697949218e_real @ ( some_variable_real @ X23 ) )
= ( insert_variable_real @ X23 @ bot_bo721182586e_real ) ) ).
% option.simps(15)
thf(fact_254_Vagree__repv__self,axiom,
! [Omega: variable > real,X: variable,D: real] :
( ( denotational_Vagree @ Omega @ ( denotational_repv @ Omega @ X @ D ) @ ( insert_variable @ X @ bot_bot_set_variable ) )
= ( D
= ( Omega @ X ) ) ) ).
% Vagree_repv_self
thf(fact_255_elem__set,axiom,
! [X: variable,Xo: option_variable] :
( ( member_variable @ X @ ( set_option_variable2 @ Xo ) )
= ( Xo
= ( some_variable @ X ) ) ) ).
% elem_set
thf(fact_256_elem__set,axiom,
! [X: variable > real,Xo: option_variable_real] :
( ( member_variable_real @ X @ ( set_op697949218e_real @ Xo ) )
= ( Xo
= ( some_variable_real @ X ) ) ) ).
% elem_set
thf(fact_257_elem__set,axiom,
! [X: trm,Xo: option_trm] :
( ( member_trm @ X @ ( set_option_trm2 @ Xo ) )
= ( Xo
= ( some_trm @ X ) ) ) ).
% elem_set
thf(fact_258_Vagree__and,axiom,
! [Nu: variable > real,Nu2: variable > real,V: set_variable,W: set_variable] :
( ( ( denotational_Vagree @ Nu @ Nu2 @ V )
& ( denotational_Vagree @ Nu @ Nu2 @ W ) )
= ( denotational_Vagree @ Nu @ Nu2 @ ( sup_sup_set_variable @ V @ W ) ) ) ).
% Vagree_and
thf(fact_259_set__empty__eq,axiom,
! [Xo: option_trm] :
( ( ( set_option_trm2 @ Xo )
= bot_bot_set_trm )
= ( Xo = none_trm ) ) ).
% set_empty_eq
thf(fact_260_set__empty__eq,axiom,
! [Xo: option_variable] :
( ( ( set_option_variable2 @ Xo )
= bot_bot_set_variable )
= ( Xo = none_variable ) ) ).
% set_empty_eq
thf(fact_261_set__empty__eq,axiom,
! [Xo: option_variable_real] :
( ( ( set_op697949218e_real @ Xo )
= bot_bo721182586e_real )
= ( Xo = none_variable_real ) ) ).
% set_empty_eq
thf(fact_262_similar__selectlike__mem,axiom,
! [Nu: variable > real,Omega: variable > real,V: set_variable,X7: set_variable_real] :
( ( denotational_Vagree @ Nu @ Omega @ V )
=> ( ( member_variable_real @ Nu @ ( static_selectlike @ X7 @ Omega @ V ) )
= ( member_variable_real @ Nu @ X7 ) ) ) ).
% similar_selectlike_mem
thf(fact_263_selectlike__Vagree,axiom,
! [Nu: variable > real,Omega: variable > real,V: set_variable,X7: set_variable_real] :
( ( denotational_Vagree @ Nu @ Omega @ V )
=> ( ( static_selectlike @ X7 @ Nu @ V )
= ( static_selectlike @ X7 @ Omega @ V ) ) ) ).
% selectlike_Vagree
thf(fact_264_Aterm__Some,axiom,
some_trm = some_trm ).
% Aterm_Some
thf(fact_265_option_Oset__intros,axiom,
! [X23: variable] : ( member_variable @ X23 @ ( set_option_variable2 @ ( some_variable @ X23 ) ) ) ).
% option.set_intros
thf(fact_266_option_Oset__intros,axiom,
! [X23: variable > real] : ( member_variable_real @ X23 @ ( set_op697949218e_real @ ( some_variable_real @ X23 ) ) ) ).
% option.set_intros
thf(fact_267_option_Oset__intros,axiom,
! [X23: trm] : ( member_trm @ X23 @ ( set_option_trm2 @ ( some_trm @ X23 ) ) ) ).
% option.set_intros
thf(fact_268_option_Oset__cases,axiom,
! [E: variable,A: option_variable] :
( ( member_variable @ E @ ( set_option_variable2 @ A ) )
=> ( A
= ( some_variable @ E ) ) ) ).
% option.set_cases
thf(fact_269_option_Oset__cases,axiom,
! [E: variable > real,A: option_variable_real] :
( ( member_variable_real @ E @ ( set_op697949218e_real @ A ) )
=> ( A
= ( some_variable_real @ E ) ) ) ).
% option.set_cases
thf(fact_270_option_Oset__cases,axiom,
! [E: trm,A: option_trm] :
( ( member_trm @ E @ ( set_option_trm2 @ A ) )
=> ( A
= ( some_trm @ E ) ) ) ).
% option.set_cases
thf(fact_271_ospec,axiom,
! [A2: option_trm,P: trm > $o,X: trm] :
( ! [X6: trm] :
( ( member_trm @ X6 @ ( set_option_trm2 @ A2 ) )
=> ( P @ X6 ) )
=> ( ( A2
= ( some_trm @ X ) )
=> ( P @ X ) ) ) ).
% ospec
thf(fact_272_Vagree__union,axiom,
! [Nu: variable > real,Nu2: variable > real,V: set_variable,W: set_variable] :
( ( denotational_Vagree @ Nu @ Nu2 @ V )
=> ( ( denotational_Vagree @ Nu @ Nu2 @ W )
=> ( denotational_Vagree @ Nu @ Nu2 @ ( sup_sup_set_variable @ V @ W ) ) ) ) ).
% Vagree_union
thf(fact_273_Vagree__def,axiom,
( denotational_Vagree
= ( ^ [Nu3: variable > real,Nu4: variable > real,V3: set_variable] :
! [I: variable] :
( ( member_variable @ I @ V3 )
=> ( ( Nu3 @ I )
= ( Nu4 @ I ) ) ) ) ) ).
% Vagree_def
thf(fact_274_Vagree__sym,axiom,
( denotational_Vagree
= ( ^ [Nu3: variable > real,Nu4: variable > real] : ( denotational_Vagree @ Nu4 @ Nu3 ) ) ) ).
% Vagree_sym
thf(fact_275_Vagree__refl,axiom,
! [Nu: variable > real,V: set_variable] : ( denotational_Vagree @ Nu @ Nu @ V ) ).
% Vagree_refl
thf(fact_276_Vagree__sym__rel,axiom,
! [Nu: variable > real,Nu2: variable > real,V: set_variable] :
( ( denotational_Vagree @ Nu @ Nu2 @ V )
=> ( denotational_Vagree @ Nu2 @ Nu @ V ) ) ).
% Vagree_sym_rel
thf(fact_277_Vagree__empty,axiom,
! [Nu: variable > real,Nu2: variable > real] : ( denotational_Vagree @ Nu @ Nu2 @ bot_bot_set_variable ) ).
% Vagree_empty
thf(fact_278_option_Osimps_I14_J,axiom,
( ( set_option_trm2 @ none_trm )
= bot_bot_set_trm ) ).
% option.simps(14)
thf(fact_279_option_Osimps_I14_J,axiom,
( ( set_option_variable2 @ none_variable )
= bot_bot_set_variable ) ).
% option.simps(14)
thf(fact_280_option_Osimps_I14_J,axiom,
( ( set_op697949218e_real @ none_variable_real )
= bot_bo721182586e_real ) ).
% option.simps(14)
thf(fact_281_option_Oset__sel,axiom,
! [A: option_variable] :
( ( A != none_variable )
=> ( member_variable @ ( the_variable @ A ) @ ( set_option_variable2 @ A ) ) ) ).
% option.set_sel
thf(fact_282_option_Oset__sel,axiom,
! [A: option_variable_real] :
( ( A != none_variable_real )
=> ( member_variable_real @ ( the_variable_real @ A ) @ ( set_op697949218e_real @ A ) ) ) ).
% option.set_sel
thf(fact_283_option_Oset__sel,axiom,
! [A: option_trm] :
( ( A != none_trm )
=> ( member_trm @ ( the_trm @ A ) @ ( set_option_trm2 @ A ) ) ) ).
% option.set_sel
thf(fact_284_solves__Vagree,axiom,
! [I2: denotational_interp,F: real > variable > real,X: char,Theta: trm,Zeta: real] :
( ( denota1778088425es_ODE @ I2 @ F @ X @ Theta )
=> ( denotational_Vagree @ ( F @ Zeta ) @ ( F @ zero_zero_real ) @ ( uminus1851247844riable @ ( insert_variable @ ( rVar @ X ) @ ( insert_variable @ ( dVar @ X ) @ bot_bot_set_variable ) ) ) ) ) ).
% solves_Vagree
thf(fact_285_Vagree__repv,axiom,
! [Omega: variable > real,X: variable,D: real] : ( denotational_Vagree @ Omega @ ( denotational_repv @ Omega @ X @ D ) @ ( uminus1851247844riable @ ( insert_variable @ X @ bot_bot_set_variable ) ) ) ).
% Vagree_repv
thf(fact_286_Timeso_Osimps_I3_J,axiom,
! [V4: trm] :
( ( uSubst918876924Timeso @ ( some_trm @ V4 ) @ none_trm )
= none_trm ) ).
% Timeso.simps(3)
thf(fact_287_compl__eq__compl__iff,axiom,
! [X: set_variable,Y: set_variable] :
( ( ( uminus1851247844riable @ X )
= ( uminus1851247844riable @ Y ) )
= ( X = Y ) ) ).
% compl_eq_compl_iff
thf(fact_288_compl__eq__compl__iff,axiom,
! [X: set_variable_real,Y: set_variable_real] :
( ( ( uminus430703407e_real @ X )
= ( uminus430703407e_real @ Y ) )
= ( X = Y ) ) ).
% compl_eq_compl_iff
thf(fact_289_double__compl,axiom,
! [X: set_variable] :
( ( uminus1851247844riable @ ( uminus1851247844riable @ X ) )
= X ) ).
% double_compl
thf(fact_290_double__compl,axiom,
! [X: set_variable_real] :
( ( uminus430703407e_real @ ( uminus430703407e_real @ X ) )
= X ) ).
% double_compl
thf(fact_291_Compl__eq__Compl__iff,axiom,
! [A2: set_variable,B: set_variable] :
( ( ( uminus1851247844riable @ A2 )
= ( uminus1851247844riable @ B ) )
= ( A2 = B ) ) ).
% Compl_eq_Compl_iff
thf(fact_292_Compl__eq__Compl__iff,axiom,
! [A2: set_variable_real,B: set_variable_real] :
( ( ( uminus430703407e_real @ A2 )
= ( uminus430703407e_real @ B ) )
= ( A2 = B ) ) ).
% Compl_eq_Compl_iff
thf(fact_293_Compl__iff,axiom,
! [C2: variable,A2: set_variable] :
( ( member_variable @ C2 @ ( uminus1851247844riable @ A2 ) )
= ( ~ ( member_variable @ C2 @ A2 ) ) ) ).
% Compl_iff
thf(fact_294_Compl__iff,axiom,
! [C2: variable > real,A2: set_variable_real] :
( ( member_variable_real @ C2 @ ( uminus430703407e_real @ A2 ) )
= ( ~ ( member_variable_real @ C2 @ A2 ) ) ) ).
% Compl_iff
thf(fact_295_ComplI,axiom,
! [C2: variable,A2: set_variable] :
( ~ ( member_variable @ C2 @ A2 )
=> ( member_variable @ C2 @ ( uminus1851247844riable @ A2 ) ) ) ).
% ComplI
thf(fact_296_ComplI,axiom,
! [C2: variable > real,A2: set_variable_real] :
( ~ ( member_variable_real @ C2 @ A2 )
=> ( member_variable_real @ C2 @ ( uminus430703407e_real @ A2 ) ) ) ).
% ComplI
thf(fact_297_add_Oinverse__neutral,axiom,
( ( uminus_uminus_real @ zero_zero_real )
= zero_zero_real ) ).
% add.inverse_neutral
thf(fact_298_neg__0__equal__iff__equal,axiom,
! [A: real] :
( ( zero_zero_real
= ( uminus_uminus_real @ A ) )
= ( zero_zero_real = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_299_neg__equal__0__iff__equal,axiom,
! [A: real] :
( ( ( uminus_uminus_real @ A )
= zero_zero_real )
= ( A = zero_zero_real ) ) ).
% neg_equal_0_iff_equal
thf(fact_300_equal__neg__zero,axiom,
! [A: real] :
( ( A
= ( uminus_uminus_real @ A ) )
= ( A = zero_zero_real ) ) ).
% equal_neg_zero
thf(fact_301_neg__equal__zero,axiom,
! [A: real] :
( ( ( uminus_uminus_real @ A )
= A )
= ( A = zero_zero_real ) ) ).
% neg_equal_zero
thf(fact_302_Uvariation__Vagree,axiom,
! [Nu: variable > real,Nu2: variable > real,V: set_variable] :
( ( denota1419872369iation @ Nu @ Nu2 @ ( uminus1851247844riable @ V ) )
= ( denotational_Vagree @ Nu @ Nu2 @ V ) ) ).
% Uvariation_Vagree
thf(fact_303_double__complement,axiom,
! [A2: set_variable] :
( ( uminus1851247844riable @ ( uminus1851247844riable @ A2 ) )
= A2 ) ).
% double_complement
thf(fact_304_double__complement,axiom,
! [A2: set_variable_real] :
( ( uminus430703407e_real @ ( uminus430703407e_real @ A2 ) )
= A2 ) ).
% double_complement
thf(fact_305_ComplD,axiom,
! [C2: variable,A2: set_variable] :
( ( member_variable @ C2 @ ( uminus1851247844riable @ A2 ) )
=> ~ ( member_variable @ C2 @ A2 ) ) ).
% ComplD
thf(fact_306_ComplD,axiom,
! [C2: variable > real,A2: set_variable_real] :
( ( member_variable_real @ C2 @ ( uminus430703407e_real @ A2 ) )
=> ~ ( member_variable_real @ C2 @ A2 ) ) ).
% ComplD
thf(fact_307_Timeso_Osimps_I2_J,axiom,
! [Eta: option_trm] :
( ( uSubst918876924Timeso @ none_trm @ Eta )
= none_trm ) ).
% Timeso.simps(2)
thf(fact_308_Timeso__undef,axiom,
! [Theta: option_trm,Eta: option_trm] :
( ( ( uSubst918876924Timeso @ Theta @ Eta )
= none_trm )
= ( ( Theta = none_trm )
| ( Eta = none_trm ) ) ) ).
% Timeso_undef
thf(fact_309_selectlike__equal__cocond__corule,axiom,
! [Nu: variable > real,V: set_variable,X7: set_variable_real,Y4: set_variable_real] :
( ! [Mu2: variable > real] :
( ( denota1419872369iation @ Mu2 @ Nu @ V )
=> ( ( member_variable_real @ Mu2 @ X7 )
= ( member_variable_real @ Mu2 @ Y4 ) ) )
=> ( ( static_selectlike @ X7 @ Nu @ ( uminus1851247844riable @ V ) )
= ( static_selectlike @ Y4 @ Nu @ ( uminus1851247844riable @ V ) ) ) ) ).
% selectlike_equal_cocond_corule
thf(fact_310_selectlike__equal__cocond__rule,axiom,
! [Nu: variable > real,V: set_variable,X7: set_variable_real,Y4: set_variable_real] :
( ! [Mu2: variable > real] :
( ( denota1419872369iation @ Mu2 @ Nu @ ( uminus1851247844riable @ V ) )
=> ( ( member_variable_real @ Mu2 @ X7 )
= ( member_variable_real @ Mu2 @ Y4 ) ) )
=> ( ( static_selectlike @ X7 @ Nu @ V )
= ( static_selectlike @ Y4 @ Nu @ V ) ) ) ).
% selectlike_equal_cocond_rule
thf(fact_311_selectlike__equal__cocond,axiom,
! [X7: set_variable_real,Nu: variable > real,V: set_variable,Y4: set_variable_real] :
( ( ( static_selectlike @ X7 @ Nu @ ( uminus1851247844riable @ V ) )
= ( static_selectlike @ Y4 @ Nu @ ( uminus1851247844riable @ V ) ) )
= ( ! [Mu3: variable > real] :
( ( denota1419872369iation @ Mu3 @ Nu @ V )
=> ( ( member_variable_real @ Mu3 @ X7 )
= ( member_variable_real @ Mu3 @ Y4 ) ) ) ) ) ).
% selectlike_equal_cocond
thf(fact_312_selectlike__equal__cond,axiom,
! [X7: set_variable_real,Nu: variable > real,V: set_variable,Y4: set_variable_real] :
( ( ( static_selectlike @ X7 @ Nu @ V )
= ( static_selectlike @ Y4 @ Nu @ V ) )
= ( ! [Mu3: variable > real] :
( ( denota1419872369iation @ Mu3 @ Nu @ ( uminus1851247844riable @ V ) )
=> ( ( member_variable_real @ Mu3 @ X7 )
= ( member_variable_real @ Mu3 @ Y4 ) ) ) ) ) ).
% selectlike_equal_cond
thf(fact_313_stateinterpol__insert,axiom,
! [V4: variable > real,W2: variable > real,S: set_variable,Z: variable] : ( denotational_Vagree @ ( stateinterpol @ V4 @ W2 @ S ) @ ( stateinterpol @ V4 @ W2 @ ( insert_variable @ Z @ S ) ) @ ( uminus1851247844riable @ ( insert_variable @ Z @ bot_bot_set_variable ) ) ) ).
% stateinterpol_insert
thf(fact_314_Timeso_Oelims,axiom,
! [X: option_trm,Xa: option_trm,Y: option_trm] :
( ( ( uSubst918876924Timeso @ X @ Xa )
= Y )
=> ( ! [Theta2: trm] :
( ( X
= ( some_trm @ Theta2 ) )
=> ! [Eta2: trm] :
( ( Xa
= ( some_trm @ Eta2 ) )
=> ( Y
!= ( some_trm @ ( times @ Theta2 @ Eta2 ) ) ) ) )
=> ( ( ( X = none_trm )
=> ( Y != none_trm ) )
=> ~ ( ? [V2: trm] :
( X
= ( some_trm @ V2 ) )
=> ( ( Xa = none_trm )
=> ( Y != none_trm ) ) ) ) ) ) ).
% Timeso.elims
thf(fact_315_trm_Oinject_I6_J,axiom,
! [X61: trm,X62: trm,Y61: trm,Y62: trm] :
( ( ( times @ X61 @ X62 )
= ( times @ Y61 @ Y62 ) )
= ( ( X61 = Y61 )
& ( X62 = Y62 ) ) ) ).
% trm.inject(6)
thf(fact_316_stateinterpol__left,axiom,
! [X: variable,S: set_variable,Nu: variable > real,Omega: variable > real] :
( ( member_variable @ X @ S )
=> ( ( stateinterpol @ Nu @ Omega @ S @ X )
= ( Nu @ X ) ) ) ).
% stateinterpol_left
thf(fact_317_stateinterpol__right,axiom,
! [X: variable,S: set_variable,Nu: variable > real,Omega: variable > real] :
( ~ ( member_variable @ X @ S )
=> ( ( stateinterpol @ Nu @ Omega @ S @ X )
= ( Omega @ X ) ) ) ).
% stateinterpol_right
thf(fact_318_selectlike__co__selectlike,axiom,
! [X7: set_variable_real,Nu: variable > real,V: set_variable] :
( ( static_selectlike @ ( uminus430703407e_real @ ( static_selectlike @ X7 @ Nu @ V ) ) @ Nu @ V )
= ( static_selectlike @ ( uminus430703407e_real @ X7 ) @ Nu @ V ) ) ).
% selectlike_co_selectlike
thf(fact_319_stateinterpol__empty,axiom,
! [Nu: variable > real,Omega: variable > real] :
( ( stateinterpol @ Nu @ Omega @ bot_bot_set_variable )
= Omega ) ).
% stateinterpol_empty
thf(fact_320_Vagree__stateinterpol,axiom,
! [Nu: variable > real,Omega: variable > real,S: set_variable] : ( denotational_Vagree @ ( stateinterpol @ Nu @ Omega @ S ) @ Nu @ S ) ).
% Vagree_stateinterpol
thf(fact_321_stateinterpol__def,axiom,
( stateinterpol
= ( ^ [Nu3: variable > real,Omega2: variable > real,S2: set_variable,X2: variable] : ( if_real @ ( member_variable @ X2 @ S2 ) @ ( Nu3 @ X2 ) @ ( Omega2 @ X2 ) ) ) ) ).
% stateinterpol_def
thf(fact_322_usubstappt__times__conv,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,Theta: trm,Eta: trm] :
( ( ( uSubst516392818stappt @ Sigma @ U @ ( times @ Theta @ Eta ) )
!= none_trm )
=> ( ( ( uSubst516392818stappt @ Sigma @ U @ Theta )
!= none_trm )
& ( ( uSubst516392818stappt @ Sigma @ U @ Eta )
!= none_trm ) ) ) ).
% usubstappt_times_conv
thf(fact_323_usubstappt_Osimps_I6_J,axiom,
! [Sigma: produc1418842292n_game,U: set_variable,Theta: trm,Eta: trm] :
( ( uSubst516392818stappt @ Sigma @ U @ ( times @ Theta @ Eta ) )
= ( uSubst918876924Timeso @ ( uSubst516392818stappt @ Sigma @ U @ Theta ) @ ( uSubst516392818stappt @ Sigma @ U @ Eta ) ) ) ).
% usubstappt.simps(6)
thf(fact_324_Timeso_Osimps_I1_J,axiom,
! [Theta: trm,Eta: trm] :
( ( uSubst918876924Timeso @ ( some_trm @ Theta ) @ ( some_trm @ Eta ) )
= ( some_trm @ ( times @ Theta @ Eta ) ) ) ).
% Timeso.simps(1)
thf(fact_325_stateinterpol__diff,axiom,
! [Nu: variable > real,Omega: variable > real] :
( ( stateinterpol @ Nu @ Omega @ ( statediff @ Nu @ Omega ) )
= Nu ) ).
% stateinterpol_diff
thf(fact_326_stateinterpol__FVT,axiom,
! [Z: variable,T3: trm,I2: denotational_interp,Omega: variable > real,Omega3: variable > real,S: set_variable] :
( ~ ( member_variable @ Z @ ( static_FVT @ T3 ) )
=> ( ( denota1863255036rm_sem @ I2 @ T3 @ ( stateinterpol @ Omega @ Omega3 @ S ) )
= ( denota1863255036rm_sem @ I2 @ T3 @ ( stateinterpol @ Omega @ Omega3 @ ( insert_variable @ Z @ S ) ) ) ) ) ).
% stateinterpol_FVT
thf(fact_327_nostatediff,axiom,
! [X: variable,Nu: variable > real,Omega: variable > real] :
( ~ ( member_variable @ X @ ( statediff @ Nu @ Omega ) )
=> ( ( Nu @ X )
= ( Omega @ X ) ) ) ).
% nostatediff
thf(fact_328_coincidence__term,axiom,
! [Omega: variable > real,Omega3: variable > real,Theta: trm,I2: denotational_interp] :
( ( denotational_Vagree @ Omega @ Omega3 @ ( static_FVT @ Theta ) )
=> ( ( denota1863255036rm_sem @ I2 @ Theta @ Omega )
= ( denota1863255036rm_sem @ I2 @ Theta @ Omega3 ) ) ) ).
% coincidence_term
thf(fact_329_coincidence__term__cor,axiom,
! [Omega: variable > real,Omega3: variable > real,U: set_variable,Theta: trm,I2: denotational_interp] :
( ( denota1419872369iation @ Omega @ Omega3 @ U )
=> ( ( ( inf_inf_set_variable @ ( static_FVT @ Theta ) @ U )
= bot_bot_set_variable )
=> ( ( denota1863255036rm_sem @ I2 @ Theta @ Omega )
= ( denota1863255036rm_sem @ I2 @ Theta @ Omega3 ) ) ) ) ).
% coincidence_term_cor
thf(fact_330_Vagree__statediff,axiom,
! [Omega: variable > real,Omega3: variable > real,S: set_variable] :
( ( denotational_Vagree @ Omega @ Omega3 @ S )
=> ( ord_le282106107riable @ ( statediff @ Omega @ Omega3 ) @ ( uminus1851247844riable @ S ) ) ) ).
% Vagree_statediff
thf(fact_331_order__refl,axiom,
! [X: set_variable] : ( ord_le282106107riable @ X @ X ) ).
% order_refl
thf(fact_332_subsetI,axiom,
! [A2: set_variable_real,B: set_variable_real] :
( ! [X6: variable > real] :
( ( member_variable_real @ X6 @ A2 )
=> ( member_variable_real @ X6 @ B ) )
=> ( ord_le1113654598e_real @ A2 @ B ) ) ).
% subsetI
thf(fact_333_subsetI,axiom,
! [A2: set_variable,B: set_variable] :
( ! [X6: variable] :
( ( member_variable @ X6 @ A2 )
=> ( member_variable @ X6 @ B ) )
=> ( ord_le282106107riable @ A2 @ B ) ) ).
% subsetI
thf(fact_334_subset__antisym,axiom,
! [A2: set_variable,B: set_variable] :
( ( ord_le282106107riable @ A2 @ B )
=> ( ( ord_le282106107riable @ B @ A2 )
=> ( A2 = B ) ) ) ).
% subset_antisym
thf(fact_335_inf_Oidem,axiom,
! [A: set_variable] :
( ( inf_inf_set_variable @ A @ A )
= A ) ).
% inf.idem
thf(fact_336_inf__idem,axiom,
! [X: set_variable] :
( ( inf_inf_set_variable @ X @ X )
= X ) ).
% inf_idem
thf(fact_337_inf_Oleft__idem,axiom,
! [A: set_variable,B2: set_variable] :
( ( inf_inf_set_variable @ A @ ( inf_inf_set_variable @ A @ B2 ) )
= ( inf_inf_set_variable @ A @ B2 ) ) ).
% inf.left_idem
thf(fact_338_inf__left__idem,axiom,
! [X: set_variable,Y: set_variable] :
( ( inf_inf_set_variable @ X @ ( inf_inf_set_variable @ X @ Y ) )
= ( inf_inf_set_variable @ X @ Y ) ) ).
% inf_left_idem
thf(fact_339_inf_Oright__idem,axiom,
! [A: set_variable,B2: set_variable] :
( ( inf_inf_set_variable @ ( inf_inf_set_variable @ A @ B2 ) @ B2 )
= ( inf_inf_set_variable @ A @ B2 ) ) ).
% inf.right_idem
thf(fact_340_inf__right__idem,axiom,
! [X: set_variable,Y: set_variable] :
( ( inf_inf_set_variable @ ( inf_inf_set_variable @ X @ Y ) @ Y )
= ( inf_inf_set_variable @ X @ Y ) ) ).
% inf_right_idem
thf(fact_341_IntI,axiom,
! [C2: variable > real,A2: set_variable_real,B: set_variable_real] :
( ( member_variable_real @ C2 @ A2 )
=> ( ( member_variable_real @ C2 @ B )
=> ( member_variable_real @ C2 @ ( inf_in1556002680e_real @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_342_IntI,axiom,
! [C2: variable,A2: set_variable,B: set_variable] :
( ( member_variable @ C2 @ A2 )
=> ( ( member_variable @ C2 @ B )
=> ( member_variable @ C2 @ ( inf_inf_set_variable @ A2 @ B ) ) ) ) ).
% IntI
thf(fact_343_Int__iff,axiom,
! [C2: variable > real,A2: set_variable_real,B: set_variable_real] :
( ( member_variable_real @ C2 @ ( inf_in1556002680e_real @ A2 @ B ) )
= ( ( member_variable_real @ C2 @ A2 )
& ( member_variable_real @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_344_Int__iff,axiom,
! [C2: variable,A2: set_variable,B: set_variable] :
( ( member_variable @ C2 @ ( inf_inf_set_variable @ A2 @ B ) )
= ( ( member_variable @ C2 @ A2 )
& ( member_variable @ C2 @ B ) ) ) ).
% Int_iff
thf(fact_345_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_346_compl__le__compl__iff,axiom,
! [X: set_variable_real,Y: set_variable_real] :
( ( ord_le1113654598e_real @ ( uminus430703407e_real @ X ) @ ( uminus430703407e_real @ Y ) )
= ( ord_le1113654598e_real @ Y @ X ) ) ).
% compl_le_compl_iff
thf(fact_347_compl__le__compl__iff,axiom,
! [X: set_variable,Y: set_variable] :
( ( ord_le282106107riable @ ( uminus1851247844riable @ X ) @ ( uminus1851247844riable @ Y ) )
= ( ord_le282106107riable @ Y @ X ) ) ).
% compl_le_compl_iff
thf(fact_348_le__inf__iff,axiom,
! [X: set_variable,Y: set_variable,Z: set_variable] :
( ( ord_le282106107riable @ X @ ( inf_inf_set_variable @ Y @ Z ) )
= ( ( ord_le282106107riable @ X @ Y )
& ( ord_le282106107riable @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_349_inf_Obounded__iff,axiom,
! [A: set_variable,B2: set_variable,C2: set_variable] :
( ( ord_le282106107riable @ A @ ( inf_inf_set_variable @ B2 @ C2 ) )
= ( ( ord_le282106107riable @ A @ B2 )
& ( ord_le282106107riable @ A @ C2 ) ) ) ).
% inf.bounded_iff
thf(fact_350_le__sup__iff,axiom,
! [X: set_variable_real,Y: set_variable_real,Z: set_variable_real] :
( ( ord_le1113654598e_real @ ( sup_su1685293586e_real @ X @ Y ) @ Z )
= ( ( ord_le1113654598e_real @ X @ Z )
& ( ord_le1113654598e_real @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_351_le__sup__iff,axiom,
! [X: set_variable,Y: set_variable,Z: set_variable] :
( ( ord_le282106107riable @ ( sup_sup_set_variable @ X @ Y ) @ Z )
= ( ( ord_le282106107riable @ X @ Z )
& ( ord_le282106107riable @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_352_sup_Obounded__iff,axiom,
! [B2: set_variable,C2: set_variable,A: set_variable] :
( ( ord_le282106107riable @ ( sup_sup_set_variable @ B2 @ C2 ) @ A )
= ( ( ord_le282106107riable @ B2 @ A )
& ( ord_le282106107riable @ C2 @ A ) ) ) ).
% sup.bounded_iff
% Helper facts (3)
thf(help_If_3_1_If_001t__Real__Oreal_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y: real] :
( ( if_real @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
denota1419872369iation @ ( f @ zero_zero_real ) @ omega @ ( sup_sup_set_variable @ ( insert_variable @ ( rVar @ x ) @ ( insert_variable @ ( dVar @ x ) @ bot_bot_set_variable ) ) @ u ) ).
%------------------------------------------------------------------------------