TPTP Problem File: ITP030^1.p
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%------------------------------------------------------------------------------
% File : ITP030^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer BinaryTree problem prob_220__3252356_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 : BinaryTree/prob_220__3252356_1 [Des21]
% Status : ContradictoryAxioms
% Rating : 0.40 v8.2.0, 0.23 v8.1.0, 0.36 v7.5.0
% Syntax : Number of formulae : 431 ( 180 unt; 76 typ; 0 def)
% Number of atoms : 964 ( 427 equ; 0 cnn)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 2559 ( 158 ~; 17 |; 59 &;1941 @)
% ( 0 <=>; 384 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 6 avg)
% Number of types : 10 ( 9 usr)
% Number of type conns : 375 ( 375 >; 0 *; 0 +; 0 <<)
% Number of symbols : 70 ( 67 usr; 14 con; 0-4 aty)
% Number of variables : 963 ( 98 ^; 850 !; 15 ?; 963 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:30:19.933
%------------------------------------------------------------------------------
% Could-be-implicit typings (9)
thf(ty_n_t__BinaryTree____Mirabelle____mlzyzwgbkd__OTree_It__Option__Ooption_Itf__a_J_J,type,
binary1176601351tion_a: $tType ).
thf(ty_n_t__Set__Oset_It__Option__Ooption_It__Option__Ooption_Itf__a_J_J_J,type,
set_option_option_a: $tType ).
thf(ty_n_t__BinaryTree____Mirabelle____mlzyzwgbkd__OTree_Itf__a_J,type,
binary1439146945Tree_a: $tType ).
thf(ty_n_t__Option__Ooption_It__Option__Ooption_Itf__a_J_J,type,
option_option_a: $tType ).
thf(ty_n_t__Set__Oset_It__Option__Ooption_Itf__a_J_J,type,
set_option_a: $tType ).
thf(ty_n_t__Option__Ooption_Itf__a_J,type,
option_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (67)
thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_OTree_OT_001t__Option__Ooption_Itf__a_J,type,
binary1007279693tion_a: binary1176601351tion_a > option_a > binary1176601351tion_a > binary1176601351tion_a ).
thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_OTree_OT_001tf__a,type,
binary717961607le_T_a: binary1439146945Tree_a > a > binary1439146945Tree_a > binary1439146945Tree_a ).
thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_OTree_OTip_001t__Option__Ooption_Itf__a_J,type,
binary471442758tion_a: binary1176601351tion_a ).
thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_OTree_OTip_001tf__a,type,
binary476621312_Tip_a: binary1439146945Tree_a ).
thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_OTree_Oset__Tree_001tf__a,type,
binary256242811Tree_a: binary1439146945Tree_a > set_a ).
thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_Oeqs_001t__Option__Ooption_Itf__a_J,type,
binary301690796tion_a: ( option_a > int ) > option_a > set_option_a ).
thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_Oeqs_001tf__a,type,
binary504661350_eqs_a: ( a > int ) > a > set_a ).
thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_Omemb_001tf__a,type,
binary2053421120memb_a: ( a > int ) > a > binary1439146945Tree_a > $o ).
thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_OsetOf_001t__Option__Ooption_Itf__a_J,type,
binary1117200954tion_a: binary1176601351tion_a > set_option_a ).
thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_OsetOf_001tf__a,type,
binary945792244etOf_a: binary1439146945Tree_a > set_a ).
thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_OsortedTree_001t__Option__Ooption_Itf__a_J,type,
binary995005080tion_a: ( option_a > int ) > binary1176601351tion_a > $o ).
thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_OsortedTree_001tf__a,type,
binary1721989714Tree_a: ( a > int ) > binary1439146945Tree_a > $o ).
thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_Osorted__distinct__pred_001t__Option__Ooption_Itf__a_J,type,
binary1691793817tion_a: ( option_a > int ) > option_a > option_a > binary1176601351tion_a > $o ).
thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_Osorted__distinct__pred_001tf__a,type,
binary670562003pred_a: ( a > int ) > a > a > binary1439146945Tree_a > $o ).
thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_Otlookup_001t__Option__Ooption_Itf__a_J,type,
binary1040190117tion_a: ( option_a > int ) > int > binary1176601351tion_a > option_option_a ).
thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_Otlookup_001tf__a,type,
binary779405791okup_a: ( a > int ) > int > binary1439146945Tree_a > option_a ).
thf(sy_c_HOL_OThe_001t__Option__Ooption_Itf__a_J,type,
the_option_a: ( option_a > $o ) > option_a ).
thf(sy_c_HOL_OThe_001tf__a,type,
the_a: ( a > $o ) > a ).
thf(sy_c_HOL_Oundefined_001tf__a,type,
undefined_a: a ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_Itf__a_M_Eo_J,type,
sup_sup_a_o: ( a > $o ) > ( a > $o ) > a > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Option__Ooption_Itf__a_J_J,type,
sup_sup_set_option_a: set_option_a > set_option_a > set_option_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
sup_sup_set_a: set_a > set_a > set_a ).
thf(sy_c_Option_Ooption_ONone_001t__Option__Ooption_Itf__a_J,type,
none_option_a: option_option_a ).
thf(sy_c_Option_Ooption_ONone_001tf__a,type,
none_a: option_a ).
thf(sy_c_Option_Ooption_OSome_001t__Option__Ooption_Itf__a_J,type,
some_option_a: option_a > option_option_a ).
thf(sy_c_Option_Ooption_OSome_001tf__a,type,
some_a: a > option_a ).
thf(sy_c_Option_Ooption_Ocase__option_001_Eo_001tf__a,type,
case_option_o_a: $o > ( a > $o ) > option_a > $o ).
thf(sy_c_Option_Ooption_Ocase__option_001tf__a_001tf__a,type,
case_option_a_a: a > ( a > a ) > option_a > a ).
thf(sy_c_Option_Ooption_Oset__option_001t__Option__Ooption_Itf__a_J,type,
set_option_option_a2: option_option_a > set_option_a ).
thf(sy_c_Option_Ooption_Oset__option_001tf__a,type,
set_option_a2: option_a > set_a ).
thf(sy_c_Option_Ooption_Othe_001t__Option__Ooption_Itf__a_J,type,
the_option_a2: option_option_a > option_a ).
thf(sy_c_Option_Ooption_Othe_001tf__a,type,
the_a2: option_a > a ).
thf(sy_c_Option_Othese_001t__Option__Ooption_Itf__a_J,type,
these_option_a: set_option_option_a > set_option_a ).
thf(sy_c_Option_Othese_001tf__a,type,
these_a: set_option_a > set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Option__Ooption_Itf__a_J_M_Eo_J,type,
bot_bot_option_a_o: option_a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__a_M_Eo_J,type,
bot_bot_a_o: a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_Eo,type,
bot_bot_o: $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Option__Ooption_It__Option__Ooption_Itf__a_J_J_J,type,
bot_bo1607647958tion_a: set_option_option_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Option__Ooption_Itf__a_J_J,type,
bot_bot_set_option_a: set_option_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Option__Ooption_Itf__a_J_M_Eo_J,type,
ord_less_option_a_o: ( option_a > $o ) > ( option_a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001_062_Itf__a_M_Eo_J,type,
ord_less_a_o: ( a > $o ) > ( a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Option__Ooption_Itf__a_J_J,type,
ord_le1280890064tion_a: set_option_a > set_option_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__a_J,type,
ord_less_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__a_J,type,
top_top_set_a: set_a ).
thf(sy_c_Set_OCollect_001t__Option__Ooption_Itf__a_J,type,
collect_option_a: ( option_a > $o ) > set_option_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Oimage_001t__Option__Ooption_Itf__a_J_001t__Option__Ooption_Itf__a_J,type,
image_139503237tion_a: ( option_a > option_a ) > set_option_a > set_option_a ).
thf(sy_c_Set_Oimage_001t__Option__Ooption_Itf__a_J_001tf__a,type,
image_option_a_a: ( option_a > a ) > set_option_a > set_a ).
thf(sy_c_Set_Oimage_001tf__a_001t__Option__Ooption_Itf__a_J,type,
image_a_option_a: ( a > option_a ) > set_a > set_option_a ).
thf(sy_c_Set_Oimage_001tf__a_001tf__a,type,
image_a_a: ( a > a ) > set_a > set_a ).
thf(sy_c_Set_Oinsert_001t__Option__Ooption_It__Option__Ooption_Itf__a_J_J,type,
insert1662731738tion_a: option_option_a > set_option_option_a > set_option_option_a ).
thf(sy_c_Set_Oinsert_001t__Option__Ooption_Itf__a_J,type,
insert_option_a: option_a > set_option_a > set_option_a ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Set_Ois__empty_001t__Option__Ooption_Itf__a_J,type,
is_empty_option_a: set_option_a > $o ).
thf(sy_c_Set_Ois__empty_001tf__a,type,
is_empty_a: set_a > $o ).
thf(sy_c_Set_Ois__singleton_001t__Option__Ooption_Itf__a_J,type,
is_sin1624444472tion_a: set_option_a > $o ).
thf(sy_c_Set_Ois__singleton_001tf__a,type,
is_singleton_a: set_a > $o ).
thf(sy_c_Set_Othe__elem_001t__Option__Ooption_Itf__a_J,type,
the_elem_option_a: set_option_a > option_a ).
thf(sy_c_Set_Othe__elem_001tf__a,type,
the_elem_a: set_a > a ).
thf(sy_c_member_001t__Option__Ooption_It__Option__Ooption_Itf__a_J_J,type,
member553679923tion_a: option_option_a > set_option_option_a > $o ).
thf(sy_c_member_001t__Option__Ooption_Itf__a_J,type,
member_option_a: option_a > set_option_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_h,type,
h: a > int ).
thf(sy_v_t,type,
t: binary1439146945Tree_a ).
thf(sy_v_x,type,
x: a ).
% Relevant facts (354)
thf(fact_0_h,axiom,
member_a @ x @ ( binary945792244etOf_a @ t ) ).
% h
thf(fact_1_sorted__distinct,axiom,
! [H: a > int,A: a,B: a,T: binary1439146945Tree_a] : ( binary670562003pred_a @ H @ A @ B @ T ) ).
% sorted_distinct
thf(fact_2_s,axiom,
binary1721989714Tree_a @ h @ t ).
% s
thf(fact_3_res,axiom,
~ ( binary2053421120memb_a @ h @ x @ t ) ).
% res
thf(fact_4__092_060open_062h_Ax_A_092_060noteq_062_Ah_Ax_092_060close_062,axiom,
( ( h @ x )
!= ( h @ x ) ) ).
% \<open>h x \<noteq> h x\<close>
thf(fact_5_o1,axiom,
! [X: a] :
( ( member_a @ X @ ( binary945792244etOf_a @ t ) )
=> ( ( h @ X )
!= ( h @ x ) ) ) ).
% o1
thf(fact_6_sorted__distinct__pred__def,axiom,
( binary1691793817tion_a
= ( ^ [H2: option_a > int,A2: option_a,B2: option_a,T2: binary1176601351tion_a] :
( ( ( binary995005080tion_a @ H2 @ T2 )
& ( member_option_a @ A2 @ ( binary1117200954tion_a @ T2 ) )
& ( member_option_a @ B2 @ ( binary1117200954tion_a @ T2 ) )
& ( ( H2 @ A2 )
= ( H2 @ B2 ) ) )
=> ( A2 = B2 ) ) ) ) ).
% sorted_distinct_pred_def
thf(fact_7_sorted__distinct__pred__def,axiom,
( binary670562003pred_a
= ( ^ [H2: a > int,A2: a,B2: a,T2: binary1439146945Tree_a] :
( ( ( binary1721989714Tree_a @ H2 @ T2 )
& ( member_a @ A2 @ ( binary945792244etOf_a @ T2 ) )
& ( member_a @ B2 @ ( binary945792244etOf_a @ T2 ) )
& ( ( H2 @ A2 )
= ( H2 @ B2 ) ) )
=> ( A2 = B2 ) ) ) ) ).
% sorted_distinct_pred_def
thf(fact_8_tNone,axiom,
( ( binary779405791okup_a @ h @ ( h @ x ) @ t )
= none_a ) ).
% tNone
thf(fact_9_sortedTree_Osimps_I1_J,axiom,
! [H: a > int] : ( binary1721989714Tree_a @ H @ binary476621312_Tip_a ) ).
% sortedTree.simps(1)
thf(fact_10_sortLemmaL,axiom,
! [H: a > int,T1: binary1439146945Tree_a,X2: a,T22: binary1439146945Tree_a] :
( ( binary1721989714Tree_a @ H @ ( binary717961607le_T_a @ T1 @ X2 @ T22 ) )
=> ( binary1721989714Tree_a @ H @ T1 ) ) ).
% sortLemmaL
thf(fact_11_sortLemmaR,axiom,
! [H: a > int,T1: binary1439146945Tree_a,X2: a,T22: binary1439146945Tree_a] :
( ( binary1721989714Tree_a @ H @ ( binary717961607le_T_a @ T1 @ X2 @ T22 ) )
=> ( binary1721989714Tree_a @ H @ T22 ) ) ).
% sortLemmaR
thf(fact_12_tlookup__none,axiom,
! [H: a > int,T: binary1439146945Tree_a,K: int] :
( ( ( binary1721989714Tree_a @ H @ T )
& ( ( binary779405791okup_a @ H @ K @ T )
= none_a ) )
=> ! [X: a] :
( ( member_a @ X @ ( binary945792244etOf_a @ T ) )
=> ( ( H @ X )
!= K ) ) ) ).
% tlookup_none
thf(fact_13_tlookup__some,axiom,
! [H: option_a > int,T: binary1176601351tion_a,K: int,X2: option_a] :
( ( ( binary995005080tion_a @ H @ T )
& ( ( binary1040190117tion_a @ H @ K @ T )
= ( some_option_a @ X2 ) ) )
=> ( ( member_option_a @ X2 @ ( binary1117200954tion_a @ T ) )
& ( ( H @ X2 )
= K ) ) ) ).
% tlookup_some
thf(fact_14_tlookup__some,axiom,
! [H: a > int,T: binary1439146945Tree_a,K: int,X2: a] :
( ( ( binary1721989714Tree_a @ H @ T )
& ( ( binary779405791okup_a @ H @ K @ T )
= ( some_a @ X2 ) ) )
=> ( ( member_a @ X2 @ ( binary945792244etOf_a @ T ) )
& ( ( H @ X2 )
= K ) ) ) ).
% tlookup_some
thf(fact_15_tlookup__finds,axiom,
! [H: option_a > int,T: binary1176601351tion_a,Y: option_a] :
( ( ( binary995005080tion_a @ H @ T )
& ( member_option_a @ Y @ ( binary1117200954tion_a @ T ) ) )
=> ( ( binary1040190117tion_a @ H @ ( H @ Y ) @ T )
= ( some_option_a @ Y ) ) ) ).
% tlookup_finds
thf(fact_16_tlookup__finds,axiom,
! [H: a > int,T: binary1439146945Tree_a,Y: a] :
( ( ( binary1721989714Tree_a @ H @ T )
& ( member_a @ Y @ ( binary945792244etOf_a @ T ) ) )
=> ( ( binary779405791okup_a @ H @ ( H @ Y ) @ T )
= ( some_a @ Y ) ) ) ).
% tlookup_finds
thf(fact_17_sortedTree_Osimps_I2_J,axiom,
! [H: a > int,T1: binary1439146945Tree_a,X2: a,T22: binary1439146945Tree_a] :
( ( binary1721989714Tree_a @ H @ ( binary717961607le_T_a @ T1 @ X2 @ T22 ) )
= ( ( binary1721989714Tree_a @ H @ T1 )
& ! [X3: a] :
( ( member_a @ X3 @ ( binary945792244etOf_a @ T1 ) )
=> ( ord_less_int @ ( H @ X3 ) @ ( H @ X2 ) ) )
& ! [X3: a] :
( ( member_a @ X3 @ ( binary945792244etOf_a @ T22 ) )
=> ( ord_less_int @ ( H @ X2 ) @ ( H @ X3 ) ) )
& ( binary1721989714Tree_a @ H @ T22 ) ) ) ).
% sortedTree.simps(2)
thf(fact_18_setOf_Osimps_I1_J,axiom,
( ( binary1117200954tion_a @ binary471442758tion_a )
= bot_bot_set_option_a ) ).
% setOf.simps(1)
thf(fact_19_setOf_Osimps_I1_J,axiom,
( ( binary945792244etOf_a @ binary476621312_Tip_a )
= bot_bot_set_a ) ).
% setOf.simps(1)
thf(fact_20_memb__def,axiom,
( binary2053421120memb_a
= ( ^ [H2: a > int,X3: a,T2: binary1439146945Tree_a] :
( case_option_o_a @ $false
@ ( ^ [Y2: a,Z: a] : Y2 = Z
@ X3 )
@ ( binary779405791okup_a @ H2 @ ( H2 @ X3 ) @ T2 ) ) ) ) ).
% memb_def
thf(fact_21_Tree_Oinject,axiom,
! [X21: binary1439146945Tree_a,X22: a,X23: binary1439146945Tree_a,Y21: binary1439146945Tree_a,Y22: a,Y23: binary1439146945Tree_a] :
( ( ( binary717961607le_T_a @ X21 @ X22 @ X23 )
= ( binary717961607le_T_a @ Y21 @ Y22 @ Y23 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 )
& ( X23 = Y23 ) ) ) ).
% Tree.inject
thf(fact_22_Tree_Oexhaust,axiom,
! [Y: binary1439146945Tree_a] :
( ( Y != binary476621312_Tip_a )
=> ~ ! [X212: binary1439146945Tree_a,X222: a,X232: binary1439146945Tree_a] :
( Y
!= ( binary717961607le_T_a @ X212 @ X222 @ X232 ) ) ) ).
% Tree.exhaust
thf(fact_23_Tree_Oinduct,axiom,
! [P: binary1439146945Tree_a > $o,Tree: binary1439146945Tree_a] :
( ( P @ binary476621312_Tip_a )
=> ( ! [X1: binary1439146945Tree_a,X24: a,X32: binary1439146945Tree_a] :
( ( P @ X1 )
=> ( ( P @ X32 )
=> ( P @ ( binary717961607le_T_a @ X1 @ X24 @ X32 ) ) ) )
=> ( P @ Tree ) ) ) ).
% Tree.induct
thf(fact_24_tlookup_Osimps_I1_J,axiom,
! [H: a > int,K: int] :
( ( binary779405791okup_a @ H @ K @ binary476621312_Tip_a )
= none_a ) ).
% tlookup.simps(1)
thf(fact_25_tlookup_Osimps_I2_J,axiom,
! [K: int,H: a > int,X2: a,T1: binary1439146945Tree_a,T22: binary1439146945Tree_a] :
( ( ( ord_less_int @ K @ ( H @ X2 ) )
=> ( ( binary779405791okup_a @ H @ K @ ( binary717961607le_T_a @ T1 @ X2 @ T22 ) )
= ( binary779405791okup_a @ H @ K @ T1 ) ) )
& ( ~ ( ord_less_int @ K @ ( H @ X2 ) )
=> ( ( ( ord_less_int @ ( H @ X2 ) @ K )
=> ( ( binary779405791okup_a @ H @ K @ ( binary717961607le_T_a @ T1 @ X2 @ T22 ) )
= ( binary779405791okup_a @ H @ K @ T22 ) ) )
& ( ~ ( ord_less_int @ ( H @ X2 ) @ K )
=> ( ( binary779405791okup_a @ H @ K @ ( binary717961607le_T_a @ T1 @ X2 @ T22 ) )
= ( some_a @ X2 ) ) ) ) ) ) ).
% tlookup.simps(2)
thf(fact_26_Tree_Odistinct_I1_J,axiom,
! [X21: binary1439146945Tree_a,X22: a,X23: binary1439146945Tree_a] :
( binary476621312_Tip_a
!= ( binary717961607le_T_a @ X21 @ X22 @ X23 ) ) ).
% Tree.distinct(1)
thf(fact_27_not__Some__eq,axiom,
! [X2: option_a] :
( ( ! [Y3: a] :
( X2
!= ( some_a @ Y3 ) ) )
= ( X2 = none_a ) ) ).
% not_Some_eq
thf(fact_28_not__None__eq,axiom,
! [X2: option_a] :
( ( X2 != none_a )
= ( ? [Y3: a] :
( X2
= ( some_a @ Y3 ) ) ) ) ).
% not_None_eq
thf(fact_29_case__optionE,axiom,
! [P: $o,Q: a > $o,X2: option_a] :
( ( case_option_o_a @ P @ Q @ X2 )
=> ( ( ( X2 = none_a )
=> ~ P )
=> ~ ! [Y4: a] :
( ( X2
= ( some_a @ Y4 ) )
=> ~ ( Q @ Y4 ) ) ) ) ).
% case_optionE
thf(fact_30_disjE__realizer2,axiom,
! [P: $o,Q: a > $o,X2: option_a,R: $o > $o,F: $o,G: a > $o] :
( ( case_option_o_a @ P @ Q @ X2 )
=> ( ( P
=> ( R @ F ) )
=> ( ! [Q2: a] :
( ( Q @ Q2 )
=> ( R @ ( G @ Q2 ) ) )
=> ( R @ ( case_option_o_a @ F @ G @ X2 ) ) ) ) ) ).
% disjE_realizer2
thf(fact_31_option_Odisc__eq__case_I1_J,axiom,
! [Option: option_a] :
( ( Option = none_a )
= ( case_option_o_a @ $true
@ ^ [Uu: a] : $false
@ Option ) ) ).
% option.disc_eq_case(1)
thf(fact_32_option_Odisc__eq__case_I2_J,axiom,
! [Option: option_a] :
( ( Option != none_a )
= ( case_option_o_a @ $false
@ ^ [Uu: a] : $true
@ Option ) ) ).
% option.disc_eq_case(2)
thf(fact_33_option_Oinject,axiom,
! [X25: a,Y24: a] :
( ( ( some_a @ X25 )
= ( some_a @ Y24 ) )
= ( X25 = Y24 ) ) ).
% option.inject
thf(fact_34_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_35_empty__Collect__eq,axiom,
! [P: option_a > $o] :
( ( bot_bot_set_option_a
= ( collect_option_a @ P ) )
= ( ! [X3: option_a] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_36_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X3: a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_37_Collect__empty__eq,axiom,
! [P: option_a > $o] :
( ( ( collect_option_a @ P )
= bot_bot_set_option_a )
= ( ! [X3: option_a] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_38_all__not__in__conv,axiom,
! [A3: set_option_a] :
( ( ! [X3: option_a] :
~ ( member_option_a @ X3 @ A3 ) )
= ( A3 = bot_bot_set_option_a ) ) ).
% all_not_in_conv
thf(fact_39_all__not__in__conv,axiom,
! [A3: set_a] :
( ( ! [X3: a] :
~ ( member_a @ X3 @ A3 ) )
= ( A3 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_40_empty__iff,axiom,
! [C: option_a] :
~ ( member_option_a @ C @ bot_bot_set_option_a ) ).
% empty_iff
thf(fact_41_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_42_emptyE,axiom,
! [A: option_a] :
~ ( member_option_a @ A @ bot_bot_set_option_a ) ).
% emptyE
thf(fact_43_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_44_equals0D,axiom,
! [A3: set_option_a,A: option_a] :
( ( A3 = bot_bot_set_option_a )
=> ~ ( member_option_a @ A @ A3 ) ) ).
% equals0D
thf(fact_45_equals0D,axiom,
! [A3: set_a,A: a] :
( ( A3 = bot_bot_set_a )
=> ~ ( member_a @ A @ A3 ) ) ).
% equals0D
thf(fact_46_equals0I,axiom,
! [A3: set_option_a] :
( ! [Y4: option_a] :
~ ( member_option_a @ Y4 @ A3 )
=> ( A3 = bot_bot_set_option_a ) ) ).
% equals0I
thf(fact_47_equals0I,axiom,
! [A3: set_a] :
( ! [Y4: a] :
~ ( member_a @ Y4 @ A3 )
=> ( A3 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_48_ex__in__conv,axiom,
! [A3: set_option_a] :
( ( ? [X3: option_a] : ( member_option_a @ X3 @ A3 ) )
= ( A3 != bot_bot_set_option_a ) ) ).
% ex_in_conv
thf(fact_49_ex__in__conv,axiom,
! [A3: set_a] :
( ( ? [X3: a] : ( member_a @ X3 @ A3 ) )
= ( A3 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_50_empty__def,axiom,
( bot_bot_set_a
= ( collect_a
@ ^ [X3: a] : $false ) ) ).
% empty_def
thf(fact_51_empty__def,axiom,
( bot_bot_set_option_a
= ( collect_option_a
@ ^ [X3: option_a] : $false ) ) ).
% empty_def
thf(fact_52_option_Ocase__distrib,axiom,
! [H: $o > $o,F1: $o,F2: a > $o,Option: option_a] :
( ( H @ ( case_option_o_a @ F1 @ F2 @ Option ) )
= ( case_option_o_a @ ( H @ F1 )
@ ^ [X3: a] : ( H @ ( F2 @ X3 ) )
@ Option ) ) ).
% option.case_distrib
thf(fact_53_option_Odistinct_I1_J,axiom,
! [X25: a] :
( none_a
!= ( some_a @ X25 ) ) ).
% option.distinct(1)
thf(fact_54_option_OdiscI,axiom,
! [Option: option_a,X25: a] :
( ( Option
= ( some_a @ X25 ) )
=> ( Option != none_a ) ) ).
% option.discI
thf(fact_55_option_Oexhaust,axiom,
! [Y: option_a] :
( ( Y != none_a )
=> ~ ! [X24: a] :
( Y
!= ( some_a @ X24 ) ) ) ).
% option.exhaust
thf(fact_56_option_Oinducts,axiom,
! [P: option_a > $o,Option: option_a] :
( ( P @ none_a )
=> ( ! [X4: a] : ( P @ ( some_a @ X4 ) )
=> ( P @ Option ) ) ) ).
% option.inducts
thf(fact_57_mem__Collect__eq,axiom,
! [A: option_a,P: option_a > $o] :
( ( member_option_a @ A @ ( collect_option_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_58_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_59_Collect__mem__eq,axiom,
! [A3: set_option_a] :
( ( collect_option_a
@ ^ [X3: option_a] : ( member_option_a @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_60_Collect__mem__eq,axiom,
! [A3: set_a] :
( ( collect_a
@ ^ [X3: a] : ( member_a @ X3 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_61_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X4: a] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_62_Collect__cong,axiom,
! [P: option_a > $o,Q: option_a > $o] :
( ! [X4: option_a] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_option_a @ P )
= ( collect_option_a @ Q ) ) ) ).
% Collect_cong
thf(fact_63_split__option__ex,axiom,
( ( ^ [P2: option_a > $o] :
? [X5: option_a] : ( P2 @ X5 ) )
= ( ^ [P3: option_a > $o] :
( ( P3 @ none_a )
| ? [X3: a] : ( P3 @ ( some_a @ X3 ) ) ) ) ) ).
% split_option_ex
thf(fact_64_split__option__all,axiom,
( ( ^ [P2: option_a > $o] :
! [X5: option_a] : ( P2 @ X5 ) )
= ( ^ [P3: option_a > $o] :
( ( P3 @ none_a )
& ! [X3: a] : ( P3 @ ( some_a @ X3 ) ) ) ) ) ).
% split_option_all
thf(fact_65_combine__options__cases,axiom,
! [X2: option_a,P: option_a > option_a > $o,Y: option_a] :
( ( ( X2 = none_a )
=> ( P @ X2 @ Y ) )
=> ( ( ( Y = none_a )
=> ( P @ X2 @ Y ) )
=> ( ! [A4: a,B3: a] :
( ( X2
= ( some_a @ A4 ) )
=> ( ( Y
= ( some_a @ B3 ) )
=> ( P @ X2 @ Y ) ) )
=> ( P @ X2 @ Y ) ) ) ) ).
% combine_options_cases
thf(fact_66_option_Osimps_I4_J,axiom,
! [F1: $o,F2: a > $o] :
( ( case_option_o_a @ F1 @ F2 @ none_a )
= F1 ) ).
% option.simps(4)
thf(fact_67_option_Osimps_I5_J,axiom,
! [F1: $o,F2: a > $o,X25: a] :
( ( case_option_o_a @ F1 @ F2 @ ( some_a @ X25 ) )
= ( F2 @ X25 ) ) ).
% option.simps(5)
thf(fact_68_bot__apply,axiom,
( bot_bot_a_o
= ( ^ [X3: a] : bot_bot_o ) ) ).
% bot_apply
thf(fact_69_bot_Oextremum__strict,axiom,
! [A: set_option_a] :
~ ( ord_le1280890064tion_a @ A @ bot_bot_set_option_a ) ).
% bot.extremum_strict
thf(fact_70_bot_Oextremum__strict,axiom,
! [A: set_a] :
~ ( ord_less_set_a @ A @ bot_bot_set_a ) ).
% bot.extremum_strict
thf(fact_71_bot_Oextremum__strict,axiom,
! [A: a > $o] :
~ ( ord_less_a_o @ A @ bot_bot_a_o ) ).
% bot.extremum_strict
thf(fact_72_bot_Onot__eq__extremum,axiom,
! [A: set_option_a] :
( ( A != bot_bot_set_option_a )
= ( ord_le1280890064tion_a @ bot_bot_set_option_a @ A ) ) ).
% bot.not_eq_extremum
thf(fact_73_bot_Onot__eq__extremum,axiom,
! [A: set_a] :
( ( A != bot_bot_set_a )
= ( ord_less_set_a @ bot_bot_set_a @ A ) ) ).
% bot.not_eq_extremum
thf(fact_74_bot_Onot__eq__extremum,axiom,
! [A: a > $o] :
( ( A != bot_bot_a_o )
= ( ord_less_a_o @ bot_bot_a_o @ A ) ) ).
% bot.not_eq_extremum
thf(fact_75_Set_Ois__empty__def,axiom,
( is_empty_a
= ( ^ [A5: set_a] : A5 = bot_bot_set_a ) ) ).
% Set.is_empty_def
thf(fact_76_Set_Ois__empty__def,axiom,
( is_empty_option_a
= ( ^ [A5: set_option_a] : A5 = bot_bot_set_option_a ) ) ).
% Set.is_empty_def
thf(fact_77_eqs__def,axiom,
( binary504661350_eqs_a
= ( ^ [H2: a > int,X3: a] :
( collect_a
@ ^ [Y3: a] :
( ( H2 @ Y3 )
= ( H2 @ X3 ) ) ) ) ) ).
% eqs_def
thf(fact_78_eqs__def,axiom,
( binary301690796tion_a
= ( ^ [H2: option_a > int,X3: option_a] :
( collect_option_a
@ ^ [Y3: option_a] :
( ( H2 @ Y3 )
= ( H2 @ X3 ) ) ) ) ) ).
% eqs_def
thf(fact_79_option_Osplit__sel,axiom,
! [P: $o > $o,F1: $o,F2: a > $o,Option: option_a] :
( ( P @ ( case_option_o_a @ F1 @ F2 @ Option ) )
= ( ( ( Option = none_a )
=> ( P @ F1 ) )
& ( ( Option
= ( some_a @ ( the_a2 @ Option ) ) )
=> ( P @ ( F2 @ ( the_a2 @ Option ) ) ) ) ) ) ).
% option.split_sel
thf(fact_80_option_Osplit__sel__asm,axiom,
! [P: $o > $o,F1: $o,F2: a > $o,Option: option_a] :
( ( P @ ( case_option_o_a @ F1 @ F2 @ Option ) )
= ( ~ ( ( ( Option = none_a )
& ~ ( P @ F1 ) )
| ( ( Option
= ( some_a @ ( the_a2 @ Option ) ) )
& ~ ( P @ ( F2 @ ( the_a2 @ Option ) ) ) ) ) ) ) ).
% option.split_sel_asm
thf(fact_81_set__empty__eq,axiom,
! [Xo: option_option_a] :
( ( ( set_option_option_a2 @ Xo )
= bot_bot_set_option_a )
= ( Xo = none_option_a ) ) ).
% set_empty_eq
thf(fact_82_set__empty__eq,axiom,
! [Xo: option_a] :
( ( ( set_option_a2 @ Xo )
= bot_bot_set_a )
= ( Xo = none_a ) ) ).
% set_empty_eq
thf(fact_83_bot__fun__def,axiom,
( bot_bot_a_o
= ( ^ [X3: a] : bot_bot_o ) ) ).
% bot_fun_def
thf(fact_84_elem__set,axiom,
! [X2: option_a,Xo: option_option_a] :
( ( member_option_a @ X2 @ ( set_option_option_a2 @ Xo ) )
= ( Xo
= ( some_option_a @ X2 ) ) ) ).
% elem_set
thf(fact_85_elem__set,axiom,
! [X2: a,Xo: option_a] :
( ( member_a @ X2 @ ( set_option_a2 @ Xo ) )
= ( Xo
= ( some_a @ X2 ) ) ) ).
% elem_set
thf(fact_86_option_Ocollapse,axiom,
! [Option: option_a] :
( ( Option != none_a )
=> ( ( some_a @ ( the_a2 @ Option ) )
= Option ) ) ).
% option.collapse
thf(fact_87_option_Oset__sel,axiom,
! [A: option_option_a] :
( ( A != none_option_a )
=> ( member_option_a @ ( the_option_a2 @ A ) @ ( set_option_option_a2 @ A ) ) ) ).
% option.set_sel
thf(fact_88_option_Oset__sel,axiom,
! [A: option_a] :
( ( A != none_a )
=> ( member_a @ ( the_a2 @ A ) @ ( set_option_a2 @ A ) ) ) ).
% option.set_sel
thf(fact_89_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_90_bot__set__def,axiom,
( bot_bot_set_option_a
= ( collect_option_a @ bot_bot_option_a_o ) ) ).
% bot_set_def
thf(fact_91_not__psubset__empty,axiom,
! [A3: set_option_a] :
~ ( ord_le1280890064tion_a @ A3 @ bot_bot_set_option_a ) ).
% not_psubset_empty
thf(fact_92_not__psubset__empty,axiom,
! [A3: set_a] :
~ ( ord_less_set_a @ A3 @ bot_bot_set_a ) ).
% not_psubset_empty
thf(fact_93_option_Oexpand,axiom,
! [Option: option_a,Option2: option_a] :
( ( ( Option = none_a )
= ( Option2 = none_a ) )
=> ( ( ( Option != none_a )
=> ( ( Option2 != none_a )
=> ( ( the_a2 @ Option )
= ( the_a2 @ Option2 ) ) ) )
=> ( Option = Option2 ) ) ) ).
% option.expand
thf(fact_94_option_Osel,axiom,
! [X25: a] :
( ( the_a2 @ ( some_a @ X25 ) )
= X25 ) ).
% option.sel
thf(fact_95_option_Oset__intros,axiom,
! [X25: option_a] : ( member_option_a @ X25 @ ( set_option_option_a2 @ ( some_option_a @ X25 ) ) ) ).
% option.set_intros
thf(fact_96_option_Oset__intros,axiom,
! [X25: a] : ( member_a @ X25 @ ( set_option_a2 @ ( some_a @ X25 ) ) ) ).
% option.set_intros
thf(fact_97_option_Oset__cases,axiom,
! [E: option_a,A: option_option_a] :
( ( member_option_a @ E @ ( set_option_option_a2 @ A ) )
=> ( A
= ( some_option_a @ E ) ) ) ).
% option.set_cases
thf(fact_98_option_Oset__cases,axiom,
! [E: a,A: option_a] :
( ( member_a @ E @ ( set_option_a2 @ A ) )
=> ( A
= ( some_a @ E ) ) ) ).
% option.set_cases
thf(fact_99_ospec,axiom,
! [A3: option_a,P: a > $o,X2: a] :
( ! [X4: a] :
( ( member_a @ X4 @ ( set_option_a2 @ A3 ) )
=> ( P @ X4 ) )
=> ( ( A3
= ( some_a @ X2 ) )
=> ( P @ X2 ) ) ) ).
% ospec
thf(fact_100_option_Oexhaust__sel,axiom,
! [Option: option_a] :
( ( Option != none_a )
=> ( Option
= ( some_a @ ( the_a2 @ Option ) ) ) ) ).
% option.exhaust_sel
thf(fact_101_option_Osimps_I14_J,axiom,
( ( set_option_option_a2 @ none_option_a )
= bot_bot_set_option_a ) ).
% option.simps(14)
thf(fact_102_option_Osimps_I14_J,axiom,
( ( set_option_a2 @ none_a )
= bot_bot_set_a ) ).
% option.simps(14)
thf(fact_103_option_Ocase__eq__if,axiom,
( case_option_o_a
= ( ^ [F12: $o,F22: a > $o,Option3: option_a] :
( ( ( Option3 = none_a )
=> F12 )
& ( ( Option3 != none_a )
=> ( F22 @ ( the_a2 @ Option3 ) ) ) ) ) ) ).
% option.case_eq_if
thf(fact_104_dual__order_Ostrict__implies__not__eq,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_105_dual__order_Ostrict__implies__not__eq,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_set_a @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_106_dual__order_Ostrict__implies__not__eq,axiom,
! [B: a > $o,A: a > $o] :
( ( ord_less_a_o @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_107_order_Ostrict__implies__not__eq,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_108_order_Ostrict__implies__not__eq,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_109_order_Ostrict__implies__not__eq,axiom,
! [A: a > $o,B: a > $o] :
( ( ord_less_a_o @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_110_not__less__iff__gr__or__eq,axiom,
! [X2: int,Y: int] :
( ( ~ ( ord_less_int @ X2 @ Y ) )
= ( ( ord_less_int @ Y @ X2 )
| ( X2 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_111_dual__order_Ostrict__trans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_112_dual__order_Ostrict__trans,axiom,
! [B: set_a,A: set_a,C: set_a] :
( ( ord_less_set_a @ B @ A )
=> ( ( ord_less_set_a @ C @ B )
=> ( ord_less_set_a @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_113_dual__order_Ostrict__trans,axiom,
! [B: a > $o,A: a > $o,C: a > $o] :
( ( ord_less_a_o @ B @ A )
=> ( ( ord_less_a_o @ C @ B )
=> ( ord_less_a_o @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_114_linorder__less__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A4: int,B3: int] :
( ( ord_less_int @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: int] : ( P @ A4 @ A4 )
=> ( ! [A4: int,B3: int] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_115_less__imp__not__less,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ~ ( ord_less_int @ Y @ X2 ) ) ).
% less_imp_not_less
thf(fact_116_less__imp__not__less,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_less_set_a @ X2 @ Y )
=> ~ ( ord_less_set_a @ Y @ X2 ) ) ).
% less_imp_not_less
thf(fact_117_less__imp__not__less,axiom,
! [X2: a > $o,Y: a > $o] :
( ( ord_less_a_o @ X2 @ Y )
=> ~ ( ord_less_a_o @ Y @ X2 ) ) ).
% less_imp_not_less
thf(fact_118_order_Ostrict__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_119_order_Ostrict__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_set_a @ B @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_120_order_Ostrict__trans,axiom,
! [A: a > $o,B: a > $o,C: a > $o] :
( ( ord_less_a_o @ A @ B )
=> ( ( ord_less_a_o @ B @ C )
=> ( ord_less_a_o @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_121_dual__order_Oirrefl,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% dual_order.irrefl
thf(fact_122_dual__order_Oirrefl,axiom,
! [A: set_a] :
~ ( ord_less_set_a @ A @ A ) ).
% dual_order.irrefl
thf(fact_123_dual__order_Oirrefl,axiom,
! [A: a > $o] :
~ ( ord_less_a_o @ A @ A ) ).
% dual_order.irrefl
thf(fact_124_linorder__cases,axiom,
! [X2: int,Y: int] :
( ~ ( ord_less_int @ X2 @ Y )
=> ( ( X2 != Y )
=> ( ord_less_int @ Y @ X2 ) ) ) ).
% linorder_cases
thf(fact_125_less__imp__triv,axiom,
! [X2: int,Y: int,P: $o] :
( ( ord_less_int @ X2 @ Y )
=> ( ( ord_less_int @ Y @ X2 )
=> P ) ) ).
% less_imp_triv
thf(fact_126_less__imp__triv,axiom,
! [X2: set_a,Y: set_a,P: $o] :
( ( ord_less_set_a @ X2 @ Y )
=> ( ( ord_less_set_a @ Y @ X2 )
=> P ) ) ).
% less_imp_triv
thf(fact_127_less__imp__triv,axiom,
! [X2: a > $o,Y: a > $o,P: $o] :
( ( ord_less_a_o @ X2 @ Y )
=> ( ( ord_less_a_o @ Y @ X2 )
=> P ) ) ).
% less_imp_triv
thf(fact_128_less__imp__not__eq2,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( Y != X2 ) ) ).
% less_imp_not_eq2
thf(fact_129_less__imp__not__eq2,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_less_set_a @ X2 @ Y )
=> ( Y != X2 ) ) ).
% less_imp_not_eq2
thf(fact_130_less__imp__not__eq2,axiom,
! [X2: a > $o,Y: a > $o] :
( ( ord_less_a_o @ X2 @ Y )
=> ( Y != X2 ) ) ).
% less_imp_not_eq2
thf(fact_131_antisym__conv3,axiom,
! [Y: int,X2: int] :
( ~ ( ord_less_int @ Y @ X2 )
=> ( ( ~ ( ord_less_int @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv3
thf(fact_132_less__not__sym,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ~ ( ord_less_int @ Y @ X2 ) ) ).
% less_not_sym
thf(fact_133_less__not__sym,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_less_set_a @ X2 @ Y )
=> ~ ( ord_less_set_a @ Y @ X2 ) ) ).
% less_not_sym
thf(fact_134_less__not__sym,axiom,
! [X2: a > $o,Y: a > $o] :
( ( ord_less_a_o @ X2 @ Y )
=> ~ ( ord_less_a_o @ Y @ X2 ) ) ).
% less_not_sym
thf(fact_135_less__imp__not__eq,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_not_eq
thf(fact_136_less__imp__not__eq,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_less_set_a @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_not_eq
thf(fact_137_less__imp__not__eq,axiom,
! [X2: a > $o,Y: a > $o] :
( ( ord_less_a_o @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_not_eq
thf(fact_138_dual__order_Oasym,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ A @ B ) ) ).
% dual_order.asym
thf(fact_139_dual__order_Oasym,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_set_a @ B @ A )
=> ~ ( ord_less_set_a @ A @ B ) ) ).
% dual_order.asym
thf(fact_140_dual__order_Oasym,axiom,
! [B: a > $o,A: a > $o] :
( ( ord_less_a_o @ B @ A )
=> ~ ( ord_less_a_o @ A @ B ) ) ).
% dual_order.asym
thf(fact_141_ord__less__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_142_ord__less__eq__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( B = C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_143_ord__less__eq__trans,axiom,
! [A: a > $o,B: a > $o,C: a > $o] :
( ( ord_less_a_o @ A @ B )
=> ( ( B = C )
=> ( ord_less_a_o @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_144_ord__eq__less__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_145_ord__eq__less__trans,axiom,
! [A: set_a,B: set_a,C: set_a] :
( ( A = B )
=> ( ( ord_less_set_a @ B @ C )
=> ( ord_less_set_a @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_146_ord__eq__less__trans,axiom,
! [A: a > $o,B: a > $o,C: a > $o] :
( ( A = B )
=> ( ( ord_less_a_o @ B @ C )
=> ( ord_less_a_o @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_147_less__irrefl,axiom,
! [X2: int] :
~ ( ord_less_int @ X2 @ X2 ) ).
% less_irrefl
thf(fact_148_less__irrefl,axiom,
! [X2: set_a] :
~ ( ord_less_set_a @ X2 @ X2 ) ).
% less_irrefl
thf(fact_149_less__irrefl,axiom,
! [X2: a > $o] :
~ ( ord_less_a_o @ X2 @ X2 ) ).
% less_irrefl
thf(fact_150_less__linear,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
| ( X2 = Y )
| ( ord_less_int @ Y @ X2 ) ) ).
% less_linear
thf(fact_151_less__trans,axiom,
! [X2: int,Y: int,Z2: int] :
( ( ord_less_int @ X2 @ Y )
=> ( ( ord_less_int @ Y @ Z2 )
=> ( ord_less_int @ X2 @ Z2 ) ) ) ).
% less_trans
thf(fact_152_less__trans,axiom,
! [X2: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_set_a @ X2 @ Y )
=> ( ( ord_less_set_a @ Y @ Z2 )
=> ( ord_less_set_a @ X2 @ Z2 ) ) ) ).
% less_trans
thf(fact_153_less__trans,axiom,
! [X2: a > $o,Y: a > $o,Z2: a > $o] :
( ( ord_less_a_o @ X2 @ Y )
=> ( ( ord_less_a_o @ Y @ Z2 )
=> ( ord_less_a_o @ X2 @ Z2 ) ) ) ).
% less_trans
thf(fact_154_less__asym_H,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% less_asym'
thf(fact_155_less__asym_H,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ~ ( ord_less_set_a @ B @ A ) ) ).
% less_asym'
thf(fact_156_less__asym_H,axiom,
! [A: a > $o,B: a > $o] :
( ( ord_less_a_o @ A @ B )
=> ~ ( ord_less_a_o @ B @ A ) ) ).
% less_asym'
thf(fact_157_less__asym,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ~ ( ord_less_int @ Y @ X2 ) ) ).
% less_asym
thf(fact_158_less__asym,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_less_set_a @ X2 @ Y )
=> ~ ( ord_less_set_a @ Y @ X2 ) ) ).
% less_asym
thf(fact_159_less__asym,axiom,
! [X2: a > $o,Y: a > $o] :
( ( ord_less_a_o @ X2 @ Y )
=> ~ ( ord_less_a_o @ Y @ X2 ) ) ).
% less_asym
thf(fact_160_less__imp__neq,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_neq
thf(fact_161_less__imp__neq,axiom,
! [X2: set_a,Y: set_a] :
( ( ord_less_set_a @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_neq
thf(fact_162_less__imp__neq,axiom,
! [X2: a > $o,Y: a > $o] :
( ( ord_less_a_o @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_neq
thf(fact_163_order_Oasym,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order.asym
thf(fact_164_order_Oasym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ~ ( ord_less_set_a @ B @ A ) ) ).
% order.asym
thf(fact_165_order_Oasym,axiom,
! [A: a > $o,B: a > $o] :
( ( ord_less_a_o @ A @ B )
=> ~ ( ord_less_a_o @ B @ A ) ) ).
% order.asym
thf(fact_166_neq__iff,axiom,
! [X2: int,Y: int] :
( ( X2 != Y )
= ( ( ord_less_int @ X2 @ Y )
| ( ord_less_int @ Y @ X2 ) ) ) ).
% neq_iff
thf(fact_167_neqE,axiom,
! [X2: int,Y: int] :
( ( X2 != Y )
=> ( ~ ( ord_less_int @ X2 @ Y )
=> ( ord_less_int @ Y @ X2 ) ) ) ).
% neqE
thf(fact_168_gt__ex,axiom,
! [X2: int] :
? [X_1: int] : ( ord_less_int @ X2 @ X_1 ) ).
% gt_ex
thf(fact_169_lt__ex,axiom,
! [X2: int] :
? [Y4: int] : ( ord_less_int @ Y4 @ X2 ) ).
% lt_ex
thf(fact_170_order__less__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_171_order__less__subst2,axiom,
! [A: int,B: int,F: int > set_a,C: set_a] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_set_a @ ( F @ B ) @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_set_a @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_172_order__less__subst2,axiom,
! [A: int,B: int,F: int > a > $o,C: a > $o] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_a_o @ ( F @ B ) @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_a_o @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a_o @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_173_order__less__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > int,C: int] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X4: set_a,Y4: set_a] :
( ( ord_less_set_a @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_174_order__less__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_set_a @ ( F @ B ) @ C )
=> ( ! [X4: set_a,Y4: set_a] :
( ( ord_less_set_a @ X4 @ Y4 )
=> ( ord_less_set_a @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_175_order__less__subst2,axiom,
! [A: set_a,B: set_a,F: set_a > a > $o,C: a > $o] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_a_o @ ( F @ B ) @ C )
=> ( ! [X4: set_a,Y4: set_a] :
( ( ord_less_set_a @ X4 @ Y4 )
=> ( ord_less_a_o @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a_o @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_176_order__less__subst2,axiom,
! [A: a > $o,B: a > $o,F: ( a > $o ) > int,C: int] :
( ( ord_less_a_o @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X4: a > $o,Y4: a > $o] :
( ( ord_less_a_o @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_177_order__less__subst2,axiom,
! [A: a > $o,B: a > $o,F: ( a > $o ) > set_a,C: set_a] :
( ( ord_less_a_o @ A @ B )
=> ( ( ord_less_set_a @ ( F @ B ) @ C )
=> ( ! [X4: a > $o,Y4: a > $o] :
( ( ord_less_a_o @ X4 @ Y4 )
=> ( ord_less_set_a @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_178_order__less__subst2,axiom,
! [A: a > $o,B: a > $o,F: ( a > $o ) > a > $o,C: a > $o] :
( ( ord_less_a_o @ A @ B )
=> ( ( ord_less_a_o @ ( F @ B ) @ C )
=> ( ! [X4: a > $o,Y4: a > $o] :
( ( ord_less_a_o @ X4 @ Y4 )
=> ( ord_less_a_o @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a_o @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_179_order__less__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_180_order__less__subst1,axiom,
! [A: int,F: set_a > int,B: set_a,C: set_a] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_set_a @ B @ C )
=> ( ! [X4: set_a,Y4: set_a] :
( ( ord_less_set_a @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_181_order__less__subst1,axiom,
! [A: int,F: ( a > $o ) > int,B: a > $o,C: a > $o] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_a_o @ B @ C )
=> ( ! [X4: a > $o,Y4: a > $o] :
( ( ord_less_a_o @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_182_order__less__subst1,axiom,
! [A: set_a,F: int > set_a,B: int,C: int] :
( ( ord_less_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_set_a @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_183_order__less__subst1,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( ord_less_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_set_a @ B @ C )
=> ( ! [X4: set_a,Y4: set_a] :
( ( ord_less_set_a @ X4 @ Y4 )
=> ( ord_less_set_a @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_184_order__less__subst1,axiom,
! [A: set_a,F: ( a > $o ) > set_a,B: a > $o,C: a > $o] :
( ( ord_less_set_a @ A @ ( F @ B ) )
=> ( ( ord_less_a_o @ B @ C )
=> ( ! [X4: a > $o,Y4: a > $o] :
( ( ord_less_a_o @ X4 @ Y4 )
=> ( ord_less_set_a @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_185_order__less__subst1,axiom,
! [A: a > $o,F: int > a > $o,B: int,C: int] :
( ( ord_less_a_o @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_a_o @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a_o @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_186_order__less__subst1,axiom,
! [A: a > $o,F: set_a > a > $o,B: set_a,C: set_a] :
( ( ord_less_a_o @ A @ ( F @ B ) )
=> ( ( ord_less_set_a @ B @ C )
=> ( ! [X4: set_a,Y4: set_a] :
( ( ord_less_set_a @ X4 @ Y4 )
=> ( ord_less_a_o @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a_o @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_187_order__less__subst1,axiom,
! [A: a > $o,F: ( a > $o ) > a > $o,B: a > $o,C: a > $o] :
( ( ord_less_a_o @ A @ ( F @ B ) )
=> ( ( ord_less_a_o @ B @ C )
=> ( ! [X4: a > $o,Y4: a > $o] :
( ( ord_less_a_o @ X4 @ Y4 )
=> ( ord_less_a_o @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a_o @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_188_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_189_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > set_a,C: set_a] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_set_a @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_190_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > a > $o,C: a > $o] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_a_o @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a_o @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_191_ord__less__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > int,C: int] :
( ( ord_less_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: set_a,Y4: set_a] :
( ( ord_less_set_a @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_192_ord__less__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: set_a,Y4: set_a] :
( ( ord_less_set_a @ X4 @ Y4 )
=> ( ord_less_set_a @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_193_ord__less__eq__subst,axiom,
! [A: set_a,B: set_a,F: set_a > a > $o,C: a > $o] :
( ( ord_less_set_a @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: set_a,Y4: set_a] :
( ( ord_less_set_a @ X4 @ Y4 )
=> ( ord_less_a_o @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a_o @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_194_ord__less__eq__subst,axiom,
! [A: a > $o,B: a > $o,F: ( a > $o ) > int,C: int] :
( ( ord_less_a_o @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: a > $o,Y4: a > $o] :
( ( ord_less_a_o @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_195_ord__less__eq__subst,axiom,
! [A: a > $o,B: a > $o,F: ( a > $o ) > set_a,C: set_a] :
( ( ord_less_a_o @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: a > $o,Y4: a > $o] :
( ( ord_less_a_o @ X4 @ Y4 )
=> ( ord_less_set_a @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_196_ord__less__eq__subst,axiom,
! [A: a > $o,B: a > $o,F: ( a > $o ) > a > $o,C: a > $o] :
( ( ord_less_a_o @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: a > $o,Y4: a > $o] :
( ( ord_less_a_o @ X4 @ Y4 )
=> ( ord_less_a_o @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a_o @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_197_ord__eq__less__subst,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_198_ord__eq__less__subst,axiom,
! [A: set_a,F: int > set_a,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_set_a @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_199_ord__eq__less__subst,axiom,
! [A: a > $o,F: int > a > $o,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_a_o @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a_o @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_200_ord__eq__less__subst,axiom,
! [A: int,F: set_a > int,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_set_a @ B @ C )
=> ( ! [X4: set_a,Y4: set_a] :
( ( ord_less_set_a @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_201_ord__eq__less__subst,axiom,
! [A: set_a,F: set_a > set_a,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_set_a @ B @ C )
=> ( ! [X4: set_a,Y4: set_a] :
( ( ord_less_set_a @ X4 @ Y4 )
=> ( ord_less_set_a @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_202_ord__eq__less__subst,axiom,
! [A: a > $o,F: set_a > a > $o,B: set_a,C: set_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_set_a @ B @ C )
=> ( ! [X4: set_a,Y4: set_a] :
( ( ord_less_set_a @ X4 @ Y4 )
=> ( ord_less_a_o @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a_o @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_203_ord__eq__less__subst,axiom,
! [A: int,F: ( a > $o ) > int,B: a > $o,C: a > $o] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_a_o @ B @ C )
=> ( ! [X4: a > $o,Y4: a > $o] :
( ( ord_less_a_o @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_204_ord__eq__less__subst,axiom,
! [A: set_a,F: ( a > $o ) > set_a,B: a > $o,C: a > $o] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_a_o @ B @ C )
=> ( ! [X4: a > $o,Y4: a > $o] :
( ( ord_less_a_o @ X4 @ Y4 )
=> ( ord_less_set_a @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_set_a @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_205_ord__eq__less__subst,axiom,
! [A: a > $o,F: ( a > $o ) > a > $o,B: a > $o,C: a > $o] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_a_o @ B @ C )
=> ( ! [X4: a > $o,Y4: a > $o] :
( ( ord_less_a_o @ X4 @ Y4 )
=> ( ord_less_a_o @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_a_o @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_206_option_Osimps_I15_J,axiom,
! [X25: option_a] :
( ( set_option_option_a2 @ ( some_option_a @ X25 ) )
= ( insert_option_a @ X25 @ bot_bot_set_option_a ) ) ).
% option.simps(15)
thf(fact_207_option_Osimps_I15_J,axiom,
! [X25: a] :
( ( set_option_a2 @ ( some_a @ X25 ) )
= ( insert_a @ X25 @ bot_bot_set_a ) ) ).
% option.simps(15)
thf(fact_208_option_Othe__def,axiom,
( the_a2
= ( case_option_a_a @ undefined_a
@ ^ [X26: a] : X26 ) ) ).
% option.the_def
thf(fact_209_bot__empty__eq,axiom,
( bot_bot_option_a_o
= ( ^ [X3: option_a] : ( member_option_a @ X3 @ bot_bot_set_option_a ) ) ) ).
% bot_empty_eq
thf(fact_210_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X3: a] : ( member_a @ X3 @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_211_Collect__empty__eq__bot,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( P = bot_bot_a_o ) ) ).
% Collect_empty_eq_bot
thf(fact_212_Collect__empty__eq__bot,axiom,
! [P: option_a > $o] :
( ( ( collect_option_a @ P )
= bot_bot_set_option_a )
= ( P = bot_bot_option_a_o ) ) ).
% Collect_empty_eq_bot
thf(fact_213_linorder__neqE__linordered__idom,axiom,
! [X2: int,Y: int] :
( ( X2 != Y )
=> ( ~ ( ord_less_int @ X2 @ Y )
=> ( ord_less_int @ Y @ X2 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_214_insert__absorb2,axiom,
! [X2: a,A3: set_a] :
( ( insert_a @ X2 @ ( insert_a @ X2 @ A3 ) )
= ( insert_a @ X2 @ A3 ) ) ).
% insert_absorb2
thf(fact_215_insert__absorb2,axiom,
! [X2: option_a,A3: set_option_a] :
( ( insert_option_a @ X2 @ ( insert_option_a @ X2 @ A3 ) )
= ( insert_option_a @ X2 @ A3 ) ) ).
% insert_absorb2
thf(fact_216_insert__iff,axiom,
! [A: option_a,B: option_a,A3: set_option_a] :
( ( member_option_a @ A @ ( insert_option_a @ B @ A3 ) )
= ( ( A = B )
| ( member_option_a @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_217_insert__iff,axiom,
! [A: a,B: a,A3: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A3 ) )
= ( ( A = B )
| ( member_a @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_218_insertCI,axiom,
! [A: option_a,B4: set_option_a,B: option_a] :
( ( ~ ( member_option_a @ A @ B4 )
=> ( A = B ) )
=> ( member_option_a @ A @ ( insert_option_a @ B @ B4 ) ) ) ).
% insertCI
thf(fact_219_insertCI,axiom,
! [A: a,B4: set_a,B: a] :
( ( ~ ( member_a @ A @ B4 )
=> ( A = B ) )
=> ( member_a @ A @ ( insert_a @ B @ B4 ) ) ) ).
% insertCI
thf(fact_220_singletonI,axiom,
! [A: option_a] : ( member_option_a @ A @ ( insert_option_a @ A @ bot_bot_set_option_a ) ) ).
% singletonI
thf(fact_221_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_222_singleton__conv,axiom,
! [A: a] :
( ( collect_a
@ ^ [X3: a] : X3 = A )
= ( insert_a @ A @ bot_bot_set_a ) ) ).
% singleton_conv
thf(fact_223_singleton__conv,axiom,
! [A: option_a] :
( ( collect_option_a
@ ^ [X3: option_a] : X3 = A )
= ( insert_option_a @ A @ bot_bot_set_option_a ) ) ).
% singleton_conv
thf(fact_224_singleton__conv2,axiom,
! [A: a] :
( ( collect_a
@ ( ^ [Y2: a,Z: a] : Y2 = Z
@ A ) )
= ( insert_a @ A @ bot_bot_set_a ) ) ).
% singleton_conv2
thf(fact_225_singleton__conv2,axiom,
! [A: option_a] :
( ( collect_option_a
@ ( ^ [Y2: option_a,Z: option_a] : Y2 = Z
@ A ) )
= ( insert_option_a @ A @ bot_bot_set_option_a ) ) ).
% singleton_conv2
thf(fact_226_psubset__trans,axiom,
! [A3: set_a,B4: set_a,C2: set_a] :
( ( ord_less_set_a @ A3 @ B4 )
=> ( ( ord_less_set_a @ B4 @ C2 )
=> ( ord_less_set_a @ A3 @ C2 ) ) ) ).
% psubset_trans
thf(fact_227_less__set__def,axiom,
( ord_le1280890064tion_a
= ( ^ [A5: set_option_a,B5: set_option_a] :
( ord_less_option_a_o
@ ^ [X3: option_a] : ( member_option_a @ X3 @ A5 )
@ ^ [X3: option_a] : ( member_option_a @ X3 @ B5 ) ) ) ) ).
% less_set_def
thf(fact_228_less__set__def,axiom,
( ord_less_set_a
= ( ^ [A5: set_a,B5: set_a] :
( ord_less_a_o
@ ^ [X3: a] : ( member_a @ X3 @ A5 )
@ ^ [X3: a] : ( member_a @ X3 @ B5 ) ) ) ) ).
% less_set_def
thf(fact_229_psubsetD,axiom,
! [A3: set_option_a,B4: set_option_a,C: option_a] :
( ( ord_le1280890064tion_a @ A3 @ B4 )
=> ( ( member_option_a @ C @ A3 )
=> ( member_option_a @ C @ B4 ) ) ) ).
% psubsetD
thf(fact_230_psubsetD,axiom,
! [A3: set_a,B4: set_a,C: a] :
( ( ord_less_set_a @ A3 @ B4 )
=> ( ( member_a @ C @ A3 )
=> ( member_a @ C @ B4 ) ) ) ).
% psubsetD
thf(fact_231_mk__disjoint__insert,axiom,
! [A: option_a,A3: set_option_a] :
( ( member_option_a @ A @ A3 )
=> ? [B6: set_option_a] :
( ( A3
= ( insert_option_a @ A @ B6 ) )
& ~ ( member_option_a @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_232_mk__disjoint__insert,axiom,
! [A: a,A3: set_a] :
( ( member_a @ A @ A3 )
=> ? [B6: set_a] :
( ( A3
= ( insert_a @ A @ B6 ) )
& ~ ( member_a @ A @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_233_insert__commute,axiom,
! [X2: a,Y: a,A3: set_a] :
( ( insert_a @ X2 @ ( insert_a @ Y @ A3 ) )
= ( insert_a @ Y @ ( insert_a @ X2 @ A3 ) ) ) ).
% insert_commute
thf(fact_234_insert__commute,axiom,
! [X2: option_a,Y: option_a,A3: set_option_a] :
( ( insert_option_a @ X2 @ ( insert_option_a @ Y @ A3 ) )
= ( insert_option_a @ Y @ ( insert_option_a @ X2 @ A3 ) ) ) ).
% insert_commute
thf(fact_235_insert__eq__iff,axiom,
! [A: option_a,A3: set_option_a,B: option_a,B4: set_option_a] :
( ~ ( member_option_a @ A @ A3 )
=> ( ~ ( member_option_a @ B @ B4 )
=> ( ( ( insert_option_a @ A @ A3 )
= ( insert_option_a @ B @ B4 ) )
= ( ( ( A = B )
=> ( A3 = B4 ) )
& ( ( A != B )
=> ? [C3: set_option_a] :
( ( A3
= ( insert_option_a @ B @ C3 ) )
& ~ ( member_option_a @ B @ C3 )
& ( B4
= ( insert_option_a @ A @ C3 ) )
& ~ ( member_option_a @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_236_insert__eq__iff,axiom,
! [A: a,A3: set_a,B: a,B4: set_a] :
( ~ ( member_a @ A @ A3 )
=> ( ~ ( member_a @ B @ B4 )
=> ( ( ( insert_a @ A @ A3 )
= ( insert_a @ B @ B4 ) )
= ( ( ( A = B )
=> ( A3 = B4 ) )
& ( ( A != B )
=> ? [C3: set_a] :
( ( A3
= ( insert_a @ B @ C3 ) )
& ~ ( member_a @ B @ C3 )
& ( B4
= ( insert_a @ A @ C3 ) )
& ~ ( member_a @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_237_insert__absorb,axiom,
! [A: option_a,A3: set_option_a] :
( ( member_option_a @ A @ A3 )
=> ( ( insert_option_a @ A @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_238_insert__absorb,axiom,
! [A: a,A3: set_a] :
( ( member_a @ A @ A3 )
=> ( ( insert_a @ A @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_239_insert__ident,axiom,
! [X2: option_a,A3: set_option_a,B4: set_option_a] :
( ~ ( member_option_a @ X2 @ A3 )
=> ( ~ ( member_option_a @ X2 @ B4 )
=> ( ( ( insert_option_a @ X2 @ A3 )
= ( insert_option_a @ X2 @ B4 ) )
= ( A3 = B4 ) ) ) ) ).
% insert_ident
thf(fact_240_insert__ident,axiom,
! [X2: a,A3: set_a,B4: set_a] :
( ~ ( member_a @ X2 @ A3 )
=> ( ~ ( member_a @ X2 @ B4 )
=> ( ( ( insert_a @ X2 @ A3 )
= ( insert_a @ X2 @ B4 ) )
= ( A3 = B4 ) ) ) ) ).
% insert_ident
thf(fact_241_Set_Oset__insert,axiom,
! [X2: option_a,A3: set_option_a] :
( ( member_option_a @ X2 @ A3 )
=> ~ ! [B6: set_option_a] :
( ( A3
= ( insert_option_a @ X2 @ B6 ) )
=> ( member_option_a @ X2 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_242_Set_Oset__insert,axiom,
! [X2: a,A3: set_a] :
( ( member_a @ X2 @ A3 )
=> ~ ! [B6: set_a] :
( ( A3
= ( insert_a @ X2 @ B6 ) )
=> ( member_a @ X2 @ B6 ) ) ) ).
% Set.set_insert
thf(fact_243_insertI2,axiom,
! [A: option_a,B4: set_option_a,B: option_a] :
( ( member_option_a @ A @ B4 )
=> ( member_option_a @ A @ ( insert_option_a @ B @ B4 ) ) ) ).
% insertI2
thf(fact_244_insertI2,axiom,
! [A: a,B4: set_a,B: a] :
( ( member_a @ A @ B4 )
=> ( member_a @ A @ ( insert_a @ B @ B4 ) ) ) ).
% insertI2
thf(fact_245_insertI1,axiom,
! [A: option_a,B4: set_option_a] : ( member_option_a @ A @ ( insert_option_a @ A @ B4 ) ) ).
% insertI1
thf(fact_246_insertI1,axiom,
! [A: a,B4: set_a] : ( member_a @ A @ ( insert_a @ A @ B4 ) ) ).
% insertI1
thf(fact_247_insertE,axiom,
! [A: option_a,B: option_a,A3: set_option_a] :
( ( member_option_a @ A @ ( insert_option_a @ B @ A3 ) )
=> ( ( A != B )
=> ( member_option_a @ A @ A3 ) ) ) ).
% insertE
thf(fact_248_insertE,axiom,
! [A: a,B: a,A3: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A3 ) )
=> ( ( A != B )
=> ( member_a @ A @ A3 ) ) ) ).
% insertE
thf(fact_249_insert__compr,axiom,
( insert_option_a
= ( ^ [A2: option_a,B5: set_option_a] :
( collect_option_a
@ ^ [X3: option_a] :
( ( X3 = A2 )
| ( member_option_a @ X3 @ B5 ) ) ) ) ) ).
% insert_compr
thf(fact_250_insert__compr,axiom,
( insert_a
= ( ^ [A2: a,B5: set_a] :
( collect_a
@ ^ [X3: a] :
( ( X3 = A2 )
| ( member_a @ X3 @ B5 ) ) ) ) ) ).
% insert_compr
thf(fact_251_insert__Collect,axiom,
! [A: a,P: a > $o] :
( ( insert_a @ A @ ( collect_a @ P ) )
= ( collect_a
@ ^ [U: a] :
( ( U != A )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_252_insert__Collect,axiom,
! [A: option_a,P: option_a > $o] :
( ( insert_option_a @ A @ ( collect_option_a @ P ) )
= ( collect_option_a
@ ^ [U: option_a] :
( ( U != A )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_253_singletonD,axiom,
! [B: option_a,A: option_a] :
( ( member_option_a @ B @ ( insert_option_a @ A @ bot_bot_set_option_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_254_singletonD,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_255_singleton__iff,axiom,
! [B: option_a,A: option_a] :
( ( member_option_a @ B @ ( insert_option_a @ A @ bot_bot_set_option_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_256_singleton__iff,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_257_doubleton__eq__iff,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ( insert_a @ A @ ( insert_a @ B @ bot_bot_set_a ) )
= ( insert_a @ C @ ( insert_a @ D @ bot_bot_set_a ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_258_doubleton__eq__iff,axiom,
! [A: option_a,B: option_a,C: option_a,D: option_a] :
( ( ( insert_option_a @ A @ ( insert_option_a @ B @ bot_bot_set_option_a ) )
= ( insert_option_a @ C @ ( insert_option_a @ D @ bot_bot_set_option_a ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_259_insert__not__empty,axiom,
! [A: a,A3: set_a] :
( ( insert_a @ A @ A3 )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_260_insert__not__empty,axiom,
! [A: option_a,A3: set_option_a] :
( ( insert_option_a @ A @ A3 )
!= bot_bot_set_option_a ) ).
% insert_not_empty
thf(fact_261_singleton__inject,axiom,
! [A: a,B: a] :
( ( ( insert_a @ A @ bot_bot_set_a )
= ( insert_a @ B @ bot_bot_set_a ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_262_singleton__inject,axiom,
! [A: option_a,B: option_a] :
( ( ( insert_option_a @ A @ bot_bot_set_option_a )
= ( insert_option_a @ B @ bot_bot_set_option_a ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_263_Collect__conv__if,axiom,
! [P: a > $o,A: a] :
( ( ( P @ A )
=> ( ( collect_a
@ ^ [X3: a] :
( ( X3 = A )
& ( P @ X3 ) ) )
= ( insert_a @ A @ bot_bot_set_a ) ) )
& ( ~ ( P @ A )
=> ( ( collect_a
@ ^ [X3: a] :
( ( X3 = A )
& ( P @ X3 ) ) )
= bot_bot_set_a ) ) ) ).
% Collect_conv_if
thf(fact_264_Collect__conv__if,axiom,
! [P: option_a > $o,A: option_a] :
( ( ( P @ A )
=> ( ( collect_option_a
@ ^ [X3: option_a] :
( ( X3 = A )
& ( P @ X3 ) ) )
= ( insert_option_a @ A @ bot_bot_set_option_a ) ) )
& ( ~ ( P @ A )
=> ( ( collect_option_a
@ ^ [X3: option_a] :
( ( X3 = A )
& ( P @ X3 ) ) )
= bot_bot_set_option_a ) ) ) ).
% Collect_conv_if
thf(fact_265_Collect__conv__if2,axiom,
! [P: a > $o,A: a] :
( ( ( P @ A )
=> ( ( collect_a
@ ^ [X3: a] :
( ( A = X3 )
& ( P @ X3 ) ) )
= ( insert_a @ A @ bot_bot_set_a ) ) )
& ( ~ ( P @ A )
=> ( ( collect_a
@ ^ [X3: a] :
( ( A = X3 )
& ( P @ X3 ) ) )
= bot_bot_set_a ) ) ) ).
% Collect_conv_if2
thf(fact_266_Collect__conv__if2,axiom,
! [P: option_a > $o,A: option_a] :
( ( ( P @ A )
=> ( ( collect_option_a
@ ^ [X3: option_a] :
( ( A = X3 )
& ( P @ X3 ) ) )
= ( insert_option_a @ A @ bot_bot_set_option_a ) ) )
& ( ~ ( P @ A )
=> ( ( collect_option_a
@ ^ [X3: option_a] :
( ( A = X3 )
& ( P @ X3 ) ) )
= bot_bot_set_option_a ) ) ) ).
% Collect_conv_if2
thf(fact_267_the__elem__eq,axiom,
! [X2: a] :
( ( the_elem_a @ ( insert_a @ X2 @ bot_bot_set_a ) )
= X2 ) ).
% the_elem_eq
thf(fact_268_the__elem__eq,axiom,
! [X2: option_a] :
( ( the_elem_option_a @ ( insert_option_a @ X2 @ bot_bot_set_option_a ) )
= X2 ) ).
% the_elem_eq
thf(fact_269_is__singletonI,axiom,
! [X2: a] : ( is_singleton_a @ ( insert_a @ X2 @ bot_bot_set_a ) ) ).
% is_singletonI
thf(fact_270_is__singletonI,axiom,
! [X2: option_a] : ( is_sin1624444472tion_a @ ( insert_option_a @ X2 @ bot_bot_set_option_a ) ) ).
% is_singletonI
thf(fact_271_these__insert__Some,axiom,
! [X2: option_a,A3: set_option_option_a] :
( ( these_option_a @ ( insert1662731738tion_a @ ( some_option_a @ X2 ) @ A3 ) )
= ( insert_option_a @ X2 @ ( these_option_a @ A3 ) ) ) ).
% these_insert_Some
thf(fact_272_these__insert__Some,axiom,
! [X2: a,A3: set_option_a] :
( ( these_a @ ( insert_option_a @ ( some_a @ X2 ) @ A3 ) )
= ( insert_a @ X2 @ ( these_a @ A3 ) ) ) ).
% these_insert_Some
thf(fact_273_is__singletonE,axiom,
! [A3: set_a] :
( ( is_singleton_a @ A3 )
=> ~ ! [X4: a] :
( A3
!= ( insert_a @ X4 @ bot_bot_set_a ) ) ) ).
% is_singletonE
thf(fact_274_is__singletonE,axiom,
! [A3: set_option_a] :
( ( is_sin1624444472tion_a @ A3 )
=> ~ ! [X4: option_a] :
( A3
!= ( insert_option_a @ X4 @ bot_bot_set_option_a ) ) ) ).
% is_singletonE
thf(fact_275_these__empty,axiom,
( ( these_option_a @ bot_bo1607647958tion_a )
= bot_bot_set_option_a ) ).
% these_empty
thf(fact_276_these__empty,axiom,
( ( these_a @ bot_bot_set_option_a )
= bot_bot_set_a ) ).
% these_empty
thf(fact_277_these__insert__None,axiom,
! [A3: set_option_a] :
( ( these_a @ ( insert_option_a @ none_a @ A3 ) )
= ( these_a @ A3 ) ) ).
% these_insert_None
thf(fact_278_in__these__eq,axiom,
! [X2: option_a,A3: set_option_option_a] :
( ( member_option_a @ X2 @ ( these_option_a @ A3 ) )
= ( member553679923tion_a @ ( some_option_a @ X2 ) @ A3 ) ) ).
% in_these_eq
thf(fact_279_in__these__eq,axiom,
! [X2: a,A3: set_option_a] :
( ( member_a @ X2 @ ( these_a @ A3 ) )
= ( member_option_a @ ( some_a @ X2 ) @ A3 ) ) ).
% in_these_eq
thf(fact_280_is__singleton__the__elem,axiom,
( is_singleton_a
= ( ^ [A5: set_a] :
( A5
= ( insert_a @ ( the_elem_a @ A5 ) @ bot_bot_set_a ) ) ) ) ).
% is_singleton_the_elem
thf(fact_281_is__singleton__the__elem,axiom,
( is_sin1624444472tion_a
= ( ^ [A5: set_option_a] :
( A5
= ( insert_option_a @ ( the_elem_option_a @ A5 ) @ bot_bot_set_option_a ) ) ) ) ).
% is_singleton_the_elem
thf(fact_282_is__singletonI_H,axiom,
! [A3: set_option_a] :
( ( A3 != bot_bot_set_option_a )
=> ( ! [X4: option_a,Y4: option_a] :
( ( member_option_a @ X4 @ A3 )
=> ( ( member_option_a @ Y4 @ A3 )
=> ( X4 = Y4 ) ) )
=> ( is_sin1624444472tion_a @ A3 ) ) ) ).
% is_singletonI'
thf(fact_283_is__singletonI_H,axiom,
! [A3: set_a] :
( ( A3 != bot_bot_set_a )
=> ( ! [X4: a,Y4: a] :
( ( member_a @ X4 @ A3 )
=> ( ( member_a @ Y4 @ A3 )
=> ( X4 = Y4 ) ) )
=> ( is_singleton_a @ A3 ) ) ) ).
% is_singletonI'
thf(fact_284_these__not__empty__eq,axiom,
! [B4: set_option_option_a] :
( ( ( these_option_a @ B4 )
!= bot_bot_set_option_a )
= ( ( B4 != bot_bo1607647958tion_a )
& ( B4
!= ( insert1662731738tion_a @ none_option_a @ bot_bo1607647958tion_a ) ) ) ) ).
% these_not_empty_eq
thf(fact_285_these__not__empty__eq,axiom,
! [B4: set_option_a] :
( ( ( these_a @ B4 )
!= bot_bot_set_a )
= ( ( B4 != bot_bot_set_option_a )
& ( B4
!= ( insert_option_a @ none_a @ bot_bot_set_option_a ) ) ) ) ).
% these_not_empty_eq
thf(fact_286_these__empty__eq,axiom,
! [B4: set_option_option_a] :
( ( ( these_option_a @ B4 )
= bot_bot_set_option_a )
= ( ( B4 = bot_bo1607647958tion_a )
| ( B4
= ( insert1662731738tion_a @ none_option_a @ bot_bo1607647958tion_a ) ) ) ) ).
% these_empty_eq
thf(fact_287_these__empty__eq,axiom,
! [B4: set_option_a] :
( ( ( these_a @ B4 )
= bot_bot_set_a )
= ( ( B4 = bot_bot_set_option_a )
| ( B4
= ( insert_option_a @ none_a @ bot_bot_set_option_a ) ) ) ) ).
% these_empty_eq
thf(fact_288_is__singleton__def,axiom,
( is_singleton_a
= ( ^ [A5: set_a] :
? [X3: a] :
( A5
= ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ).
% is_singleton_def
thf(fact_289_is__singleton__def,axiom,
( is_sin1624444472tion_a
= ( ^ [A5: set_option_a] :
? [X3: option_a] :
( A5
= ( insert_option_a @ X3 @ bot_bot_set_option_a ) ) ) ) ).
% is_singleton_def
thf(fact_290_the__elem__def,axiom,
( the_elem_a
= ( ^ [X6: set_a] :
( the_a
@ ^ [X3: a] :
( X6
= ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ) ).
% the_elem_def
thf(fact_291_the__elem__def,axiom,
( the_elem_option_a
= ( ^ [X6: set_option_a] :
( the_option_a
@ ^ [X3: option_a] :
( X6
= ( insert_option_a @ X3 @ bot_bot_set_option_a ) ) ) ) ) ).
% the_elem_def
thf(fact_292_setOf_Osimps_I2_J,axiom,
! [T1: binary1176601351tion_a,X2: option_a,T22: binary1176601351tion_a] :
( ( binary1117200954tion_a @ ( binary1007279693tion_a @ T1 @ X2 @ T22 ) )
= ( sup_sup_set_option_a @ ( sup_sup_set_option_a @ ( binary1117200954tion_a @ T1 ) @ ( binary1117200954tion_a @ T22 ) ) @ ( insert_option_a @ X2 @ bot_bot_set_option_a ) ) ) ).
% setOf.simps(2)
thf(fact_293_setOf_Osimps_I2_J,axiom,
! [T1: binary1439146945Tree_a,X2: a,T22: binary1439146945Tree_a] :
( ( binary945792244etOf_a @ ( binary717961607le_T_a @ T1 @ X2 @ T22 ) )
= ( sup_sup_set_a @ ( sup_sup_set_a @ ( binary945792244etOf_a @ T1 ) @ ( binary945792244etOf_a @ T22 ) ) @ ( insert_a @ X2 @ bot_bot_set_a ) ) ) ).
% setOf.simps(2)
thf(fact_294_Option_Othese__def,axiom,
( these_a
= ( ^ [A5: set_option_a] :
( image_option_a_a @ the_a2
@ ( collect_option_a
@ ^ [X3: option_a] :
( ( member_option_a @ X3 @ A5 )
& ( X3 != none_a ) ) ) ) ) ) ).
% Option.these_def
thf(fact_295_image__eqI,axiom,
! [B: option_a,F: a > option_a,X2: a,A3: set_a] :
( ( B
= ( F @ X2 ) )
=> ( ( member_a @ X2 @ A3 )
=> ( member_option_a @ B @ ( image_a_option_a @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_296_image__eqI,axiom,
! [B: a,F: option_a > a,X2: option_a,A3: set_option_a] :
( ( B
= ( F @ X2 ) )
=> ( ( member_option_a @ X2 @ A3 )
=> ( member_a @ B @ ( image_option_a_a @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_297_image__eqI,axiom,
! [B: option_a,F: option_a > option_a,X2: option_a,A3: set_option_a] :
( ( B
= ( F @ X2 ) )
=> ( ( member_option_a @ X2 @ A3 )
=> ( member_option_a @ B @ ( image_139503237tion_a @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_298_image__eqI,axiom,
! [B: a,F: a > a,X2: a,A3: set_a] :
( ( B
= ( F @ X2 ) )
=> ( ( member_a @ X2 @ A3 )
=> ( member_a @ B @ ( image_a_a @ F @ A3 ) ) ) ) ).
% image_eqI
thf(fact_299_Un__iff,axiom,
! [C: option_a,A3: set_option_a,B4: set_option_a] :
( ( member_option_a @ C @ ( sup_sup_set_option_a @ A3 @ B4 ) )
= ( ( member_option_a @ C @ A3 )
| ( member_option_a @ C @ B4 ) ) ) ).
% Un_iff
thf(fact_300_Un__iff,axiom,
! [C: a,A3: set_a,B4: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A3 @ B4 ) )
= ( ( member_a @ C @ A3 )
| ( member_a @ C @ B4 ) ) ) ).
% Un_iff
thf(fact_301_UnCI,axiom,
! [C: option_a,B4: set_option_a,A3: set_option_a] :
( ( ~ ( member_option_a @ C @ B4 )
=> ( member_option_a @ C @ A3 ) )
=> ( member_option_a @ C @ ( sup_sup_set_option_a @ A3 @ B4 ) ) ) ).
% UnCI
thf(fact_302_UnCI,axiom,
! [C: a,B4: set_a,A3: set_a] :
( ( ~ ( member_a @ C @ B4 )
=> ( member_a @ C @ A3 ) )
=> ( member_a @ C @ ( sup_sup_set_a @ A3 @ B4 ) ) ) ).
% UnCI
thf(fact_303_image__ident,axiom,
! [Y5: set_a] :
( ( image_a_a
@ ^ [X3: a] : X3
@ Y5 )
= Y5 ) ).
% image_ident
thf(fact_304_image__is__empty,axiom,
! [F: a > a,A3: set_a] :
( ( ( image_a_a @ F @ A3 )
= bot_bot_set_a )
= ( A3 = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_305_image__is__empty,axiom,
! [F: option_a > a,A3: set_option_a] :
( ( ( image_option_a_a @ F @ A3 )
= bot_bot_set_a )
= ( A3 = bot_bot_set_option_a ) ) ).
% image_is_empty
thf(fact_306_image__is__empty,axiom,
! [F: a > option_a,A3: set_a] :
( ( ( image_a_option_a @ F @ A3 )
= bot_bot_set_option_a )
= ( A3 = bot_bot_set_a ) ) ).
% image_is_empty
thf(fact_307_image__is__empty,axiom,
! [F: option_a > option_a,A3: set_option_a] :
( ( ( image_139503237tion_a @ F @ A3 )
= bot_bot_set_option_a )
= ( A3 = bot_bot_set_option_a ) ) ).
% image_is_empty
thf(fact_308_empty__is__image,axiom,
! [F: a > a,A3: set_a] :
( ( bot_bot_set_a
= ( image_a_a @ F @ A3 ) )
= ( A3 = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_309_empty__is__image,axiom,
! [F: option_a > a,A3: set_option_a] :
( ( bot_bot_set_a
= ( image_option_a_a @ F @ A3 ) )
= ( A3 = bot_bot_set_option_a ) ) ).
% empty_is_image
thf(fact_310_empty__is__image,axiom,
! [F: a > option_a,A3: set_a] :
( ( bot_bot_set_option_a
= ( image_a_option_a @ F @ A3 ) )
= ( A3 = bot_bot_set_a ) ) ).
% empty_is_image
thf(fact_311_empty__is__image,axiom,
! [F: option_a > option_a,A3: set_option_a] :
( ( bot_bot_set_option_a
= ( image_139503237tion_a @ F @ A3 ) )
= ( A3 = bot_bot_set_option_a ) ) ).
% empty_is_image
thf(fact_312_image__empty,axiom,
! [F: a > a] :
( ( image_a_a @ F @ bot_bot_set_a )
= bot_bot_set_a ) ).
% image_empty
thf(fact_313_image__empty,axiom,
! [F: a > option_a] :
( ( image_a_option_a @ F @ bot_bot_set_a )
= bot_bot_set_option_a ) ).
% image_empty
thf(fact_314_image__empty,axiom,
! [F: option_a > a] :
( ( image_option_a_a @ F @ bot_bot_set_option_a )
= bot_bot_set_a ) ).
% image_empty
thf(fact_315_image__empty,axiom,
! [F: option_a > option_a] :
( ( image_139503237tion_a @ F @ bot_bot_set_option_a )
= bot_bot_set_option_a ) ).
% image_empty
thf(fact_316_image__insert,axiom,
! [F: a > a,A: a,B4: set_a] :
( ( image_a_a @ F @ ( insert_a @ A @ B4 ) )
= ( insert_a @ ( F @ A ) @ ( image_a_a @ F @ B4 ) ) ) ).
% image_insert
thf(fact_317_image__insert,axiom,
! [F: a > option_a,A: a,B4: set_a] :
( ( image_a_option_a @ F @ ( insert_a @ A @ B4 ) )
= ( insert_option_a @ ( F @ A ) @ ( image_a_option_a @ F @ B4 ) ) ) ).
% image_insert
thf(fact_318_image__insert,axiom,
! [F: option_a > a,A: option_a,B4: set_option_a] :
( ( image_option_a_a @ F @ ( insert_option_a @ A @ B4 ) )
= ( insert_a @ ( F @ A ) @ ( image_option_a_a @ F @ B4 ) ) ) ).
% image_insert
thf(fact_319_image__insert,axiom,
! [F: option_a > option_a,A: option_a,B4: set_option_a] :
( ( image_139503237tion_a @ F @ ( insert_option_a @ A @ B4 ) )
= ( insert_option_a @ ( F @ A ) @ ( image_139503237tion_a @ F @ B4 ) ) ) ).
% image_insert
thf(fact_320_insert__image,axiom,
! [X2: a,A3: set_a,F: a > a] :
( ( member_a @ X2 @ A3 )
=> ( ( insert_a @ ( F @ X2 ) @ ( image_a_a @ F @ A3 ) )
= ( image_a_a @ F @ A3 ) ) ) ).
% insert_image
thf(fact_321_insert__image,axiom,
! [X2: a,A3: set_a,F: a > option_a] :
( ( member_a @ X2 @ A3 )
=> ( ( insert_option_a @ ( F @ X2 ) @ ( image_a_option_a @ F @ A3 ) )
= ( image_a_option_a @ F @ A3 ) ) ) ).
% insert_image
thf(fact_322_insert__image,axiom,
! [X2: option_a,A3: set_option_a,F: option_a > a] :
( ( member_option_a @ X2 @ A3 )
=> ( ( insert_a @ ( F @ X2 ) @ ( image_option_a_a @ F @ A3 ) )
= ( image_option_a_a @ F @ A3 ) ) ) ).
% insert_image
thf(fact_323_insert__image,axiom,
! [X2: option_a,A3: set_option_a,F: option_a > option_a] :
( ( member_option_a @ X2 @ A3 )
=> ( ( insert_option_a @ ( F @ X2 ) @ ( image_139503237tion_a @ F @ A3 ) )
= ( image_139503237tion_a @ F @ A3 ) ) ) ).
% insert_image
thf(fact_324_Un__empty,axiom,
! [A3: set_a,B4: set_a] :
( ( ( sup_sup_set_a @ A3 @ B4 )
= bot_bot_set_a )
= ( ( A3 = bot_bot_set_a )
& ( B4 = bot_bot_set_a ) ) ) ).
% Un_empty
thf(fact_325_Un__empty,axiom,
! [A3: set_option_a,B4: set_option_a] :
( ( ( sup_sup_set_option_a @ A3 @ B4 )
= bot_bot_set_option_a )
= ( ( A3 = bot_bot_set_option_a )
& ( B4 = bot_bot_set_option_a ) ) ) ).
% Un_empty
thf(fact_326_Un__insert__left,axiom,
! [A: option_a,B4: set_option_a,C2: set_option_a] :
( ( sup_sup_set_option_a @ ( insert_option_a @ A @ B4 ) @ C2 )
= ( insert_option_a @ A @ ( sup_sup_set_option_a @ B4 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_327_Un__insert__left,axiom,
! [A: a,B4: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( insert_a @ A @ B4 ) @ C2 )
= ( insert_a @ A @ ( sup_sup_set_a @ B4 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_328_Un__insert__right,axiom,
! [A3: set_option_a,A: option_a,B4: set_option_a] :
( ( sup_sup_set_option_a @ A3 @ ( insert_option_a @ A @ B4 ) )
= ( insert_option_a @ A @ ( sup_sup_set_option_a @ A3 @ B4 ) ) ) ).
% Un_insert_right
thf(fact_329_Un__insert__right,axiom,
! [A3: set_a,A: a,B4: set_a] :
( ( sup_sup_set_a @ A3 @ ( insert_a @ A @ B4 ) )
= ( insert_a @ A @ ( sup_sup_set_a @ A3 @ B4 ) ) ) ).
% Un_insert_right
thf(fact_330_insert__def,axiom,
( insert_option_a
= ( ^ [A2: option_a] :
( sup_sup_set_option_a
@ ( collect_option_a
@ ^ [X3: option_a] : X3 = A2 ) ) ) ) ).
% insert_def
thf(fact_331_insert__def,axiom,
( insert_a
= ( ^ [A2: a] :
( sup_sup_set_a
@ ( collect_a
@ ^ [X3: a] : X3 = A2 ) ) ) ) ).
% insert_def
thf(fact_332_Un__left__commute,axiom,
! [A3: set_a,B4: set_a,C2: set_a] :
( ( sup_sup_set_a @ A3 @ ( sup_sup_set_a @ B4 @ C2 ) )
= ( sup_sup_set_a @ B4 @ ( sup_sup_set_a @ A3 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_333_Un__left__absorb,axiom,
! [A3: set_a,B4: set_a] :
( ( sup_sup_set_a @ A3 @ ( sup_sup_set_a @ A3 @ B4 ) )
= ( sup_sup_set_a @ A3 @ B4 ) ) ).
% Un_left_absorb
thf(fact_334_rev__image__eqI,axiom,
! [X2: option_a,A3: set_option_a,B: option_a,F: option_a > option_a] :
( ( member_option_a @ X2 @ A3 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_option_a @ B @ ( image_139503237tion_a @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_335_rev__image__eqI,axiom,
! [X2: a,A3: set_a,B: a,F: a > a] :
( ( member_a @ X2 @ A3 )
=> ( ( B
= ( F @ X2 ) )
=> ( member_a @ B @ ( image_a_a @ F @ A3 ) ) ) ) ).
% rev_image_eqI
thf(fact_336_imageI,axiom,
! [X2: a,A3: set_a,F: a > a] :
( ( member_a @ X2 @ A3 )
=> ( member_a @ ( F @ X2 ) @ ( image_a_a @ F @ A3 ) ) ) ).
% imageI
thf(fact_337_UnI2,axiom,
! [C: a,B4: set_a,A3: set_a] :
( ( member_a @ C @ B4 )
=> ( member_a @ C @ ( sup_sup_set_a @ A3 @ B4 ) ) ) ).
% UnI2
thf(fact_338_UnI1,axiom,
! [C: a,A3: set_a,B4: set_a] :
( ( member_a @ C @ A3 )
=> ( member_a @ C @ ( sup_sup_set_a @ A3 @ B4 ) ) ) ).
% UnI1
thf(fact_339_UnE,axiom,
! [C: a,A3: set_a,B4: set_a] :
( ( member_a @ C @ ( sup_sup_set_a @ A3 @ B4 ) )
=> ( ~ ( member_a @ C @ A3 )
=> ( member_a @ C @ B4 ) ) ) ).
% UnE
thf(fact_340_Un__def,axiom,
( sup_sup_set_a
= ( ^ [A5: set_a,B5: set_a] :
( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A5 )
| ( member_a @ X3 @ B5 ) ) ) ) ) ).
% Un_def
thf(fact_341_imageE,axiom,
! [B: a,F: a > a,A3: set_a] :
( ( member_a @ B @ ( image_a_a @ F @ A3 ) )
=> ~ ! [X4: a] :
( ( B
= ( F @ X4 ) )
=> ~ ( member_a @ X4 @ A3 ) ) ) ).
% imageE
thf(fact_342_Compr__image__eq,axiom,
! [F: a > a,A3: set_a,P: a > $o] :
( ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ ( image_a_a @ F @ A3 ) )
& ( P @ X3 ) ) )
= ( image_a_a @ F
@ ( collect_a
@ ^ [X3: a] :
( ( member_a @ X3 @ A3 )
& ( P @ ( F @ X3 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_343_sup__set__def,axiom,
( sup_sup_set_a
= ( ^ [A5: set_a,B5: set_a] :
( collect_a
@ ( sup_sup_a_o
@ ^ [X3: a] : ( member_a @ X3 @ A5 )
@ ^ [X3: a] : ( member_a @ X3 @ B5 ) ) ) ) ) ).
% sup_set_def
thf(fact_344_sup__Un__eq,axiom,
! [R: set_a,S: set_a] :
( ( sup_sup_a_o
@ ^ [X3: a] : ( member_a @ X3 @ R )
@ ^ [X3: a] : ( member_a @ X3 @ S ) )
= ( ^ [X3: a] : ( member_a @ X3 @ ( sup_sup_set_a @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_345_None__notin__image__Some,axiom,
! [A3: set_a] :
~ ( member_option_a @ none_a @ ( image_a_option_a @ some_a @ A3 ) ) ).
% None_notin_image_Some
thf(fact_346_Some__image__these__eq,axiom,
! [A3: set_option_a] :
( ( image_a_option_a @ some_a @ ( these_a @ A3 ) )
= ( collect_option_a
@ ^ [X3: option_a] :
( ( member_option_a @ X3 @ A3 )
& ( X3 != none_a ) ) ) ) ).
% Some_image_these_eq
thf(fact_347_Tree_Osimps_I15_J,axiom,
! [X21: binary1439146945Tree_a,X22: a,X23: binary1439146945Tree_a] :
( ( binary256242811Tree_a @ ( binary717961607le_T_a @ X21 @ X22 @ X23 ) )
= ( sup_sup_set_a @ ( sup_sup_set_a @ ( binary256242811Tree_a @ X21 ) @ ( insert_a @ X22 @ bot_bot_set_a ) ) @ ( binary256242811Tree_a @ X23 ) ) ) ).
% Tree.simps(15)
thf(fact_348_Tree_Oset__intros_I3_J,axiom,
! [Ya: a,X23: binary1439146945Tree_a,X21: binary1439146945Tree_a,X22: a] :
( ( member_a @ Ya @ ( binary256242811Tree_a @ X23 ) )
=> ( member_a @ Ya @ ( binary256242811Tree_a @ ( binary717961607le_T_a @ X21 @ X22 @ X23 ) ) ) ) ).
% Tree.set_intros(3)
thf(fact_349_Tree_Oset__intros_I2_J,axiom,
! [X22: a,X21: binary1439146945Tree_a,X23: binary1439146945Tree_a] : ( member_a @ X22 @ ( binary256242811Tree_a @ ( binary717961607le_T_a @ X21 @ X22 @ X23 ) ) ) ).
% Tree.set_intros(2)
thf(fact_350_Tree_Oset__intros_I1_J,axiom,
! [Y: a,X21: binary1439146945Tree_a,X22: a,X23: binary1439146945Tree_a] :
( ( member_a @ Y @ ( binary256242811Tree_a @ X21 ) )
=> ( member_a @ Y @ ( binary256242811Tree_a @ ( binary717961607le_T_a @ X21 @ X22 @ X23 ) ) ) ) ).
% Tree.set_intros(1)
thf(fact_351_Tree_Oset__cases,axiom,
! [E: a,A: binary1439146945Tree_a] :
( ( member_a @ E @ ( binary256242811Tree_a @ A ) )
=> ( ! [Z1: binary1439146945Tree_a] :
( ? [Z22: a,Z3: binary1439146945Tree_a] :
( A
= ( binary717961607le_T_a @ Z1 @ Z22 @ Z3 ) )
=> ~ ( member_a @ E @ ( binary256242811Tree_a @ Z1 ) ) )
=> ( ! [Z1: binary1439146945Tree_a,Z3: binary1439146945Tree_a] :
( A
!= ( binary717961607le_T_a @ Z1 @ E @ Z3 ) )
=> ~ ! [Z1: binary1439146945Tree_a,Z22: a,Z3: binary1439146945Tree_a] :
( ( A
= ( binary717961607le_T_a @ Z1 @ Z22 @ Z3 ) )
=> ~ ( member_a @ E @ ( binary256242811Tree_a @ Z3 ) ) ) ) ) ) ).
% Tree.set_cases
thf(fact_352_Tree_Osimps_I14_J,axiom,
( ( binary256242811Tree_a @ binary476621312_Tip_a )
= bot_bot_set_a ) ).
% Tree.simps(14)
thf(fact_353_UNIV__I,axiom,
! [X2: a] : ( member_a @ X2 @ top_top_set_a ) ).
% UNIV_I
% Conjectures (1)
thf(conj_0,conjecture,
$false ).
%------------------------------------------------------------------------------