TPTP Problem File: ITP032^1.p
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%------------------------------------------------------------------------------
% File : ITP032^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer BinaryTree problem prob_359__3253628_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_359__3253628_1 [Des21]
% Status : Theorem
% Rating : 0.50 v8.2.0, 0.46 v8.1.0, 0.45 v7.5.0
% Syntax : Number of formulae : 394 ( 179 unt; 43 typ; 0 def)
% Number of atoms : 957 ( 355 equ; 0 cnn)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 2711 ( 120 ~; 19 |; 56 &;2148 @)
% ( 0 <=>; 368 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 6 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 342 ( 342 >; 0 *; 0 +; 0 <<)
% Number of symbols : 41 ( 39 usr; 9 con; 0-4 aty)
% Number of variables : 987 ( 91 ^; 864 !; 32 ?; 987 :)
% SPC : TH0_CAX_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:30:56.100
%------------------------------------------------------------------------------
% Could-be-implicit typings (4)
thf(ty_n_t__BinaryTree____Mirabelle____mlzyzwgbkd__OTree_Itf__a_J,type,
binary1439146945Tree_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 (39)
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_001tf__a,type,
binary476621312_Tip_a: binary1439146945Tree_a ).
thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_OTree_Opred__Tree_001tf__a,type,
binary1452917696Tree_a: ( a > $o ) > binary1439146945Tree_a > $o ).
thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_OTree_Oset__Tree_001tf__a,type,
binary256242811Tree_a: binary1439146945Tree_a > set_a ).
thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_Obinsert_001tf__a,type,
binary1226383794sert_a: ( a > int ) > a > binary1439146945Tree_a > binary1439146945Tree_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_001tf__a,type,
binary945792244etOf_a: binary1439146945Tree_a > set_a ).
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_001tf__a,type,
binary670562003pred_a: ( a > int ) > a > a > binary1439146945Tree_a > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_062_Itf__a_M_Eo_J,type,
minus_minus_a_o: ( a > $o ) > ( a > $o ) > a > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001_Eo,type,
minus_minus_o: $o > $o > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
minus_minus_set_a: set_a > set_a > set_a ).
thf(sy_c_HOL_OThe_001tf__a,type,
the_a: ( a > $o ) > 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_001_Eo,type,
sup_sup_o: $o > $o > $o ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Int__Oint,type,
sup_sup_int: int > int > int ).
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_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_Itf__a_J,type,
bot_bot_set_a: set_a ).
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_Itf__a_J,type,
ord_less_set_a: set_a > set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Set_Ois__empty_001tf__a,type,
is_empty_a: set_a > $o ).
thf(sy_c_Set_Ois__singleton_001tf__a,type,
is_singleton_a: set_a > $o ).
thf(sy_c_Set_Oremove_001tf__a,type,
remove_a: a > set_a > set_a ).
thf(sy_c_Set_Othe__elem_001tf__a,type,
the_elem_a: set_a > a ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_e,type,
e: a ).
thf(sy_v_h,type,
h: a > int ).
thf(sy_v_t1____,type,
t1: binary1439146945Tree_a ).
thf(sy_v_t2____,type,
t2: binary1439146945Tree_a ).
thf(sy_v_w____,type,
w: a ).
thf(sy_v_x____,type,
x: a ).
% Relevant facts (350)
thf(fact_0_whSet,axiom,
member_a @ w @ ( binary945792244etOf_a @ t1 ) ).
% whSet
thf(fact_1_s,axiom,
binary1721989714Tree_a @ h @ ( binary717961607le_T_a @ t1 @ x @ t2 ) ).
% s
thf(fact_2_s2,axiom,
binary1721989714Tree_a @ h @ t2 ).
% s2
thf(fact_3_s1,axiom,
binary1721989714Tree_a @ h @ t1 ).
% s1
thf(fact_4_eNotLess,axiom,
~ ( ord_less_int @ ( h @ e ) @ ( h @ x ) ) ).
% eNotLess
thf(fact_5_xNotLess,axiom,
~ ( ord_less_int @ ( h @ x ) @ ( h @ e ) ) ).
% xNotLess
thf(fact_6_whEq,axiom,
member_a @ w @ ( binary504661350_eqs_a @ h @ e ) ).
% whEq
thf(fact_7_xeqe,axiom,
( ( h @ x )
= ( h @ e ) ) ).
% xeqe
thf(fact_8_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_9_calculation,axiom,
member_a @ x @ ( binary504661350_eqs_a @ h @ e ) ).
% calculation
thf(fact_10_minf_I7_J,axiom,
! [T: int] :
? [Z: int] :
! [X: int] :
( ( ord_less_int @ X @ Z )
=> ~ ( ord_less_int @ T @ X ) ) ).
% minf(7)
thf(fact_11_minf_I5_J,axiom,
! [T: int] :
? [Z: int] :
! [X: int] :
( ( ord_less_int @ X @ Z )
=> ( ord_less_int @ X @ T ) ) ).
% minf(5)
thf(fact_12_minf_I4_J,axiom,
! [T: int] :
? [Z: int] :
! [X: int] :
( ( ord_less_int @ X @ Z )
=> ( X != T ) ) ).
% minf(4)
thf(fact_13_minf_I3_J,axiom,
! [T: int] :
? [Z: int] :
! [X: int] :
( ( ord_less_int @ X @ Z )
=> ( X != T ) ) ).
% minf(3)
thf(fact_14_minf_I2_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z2: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z2: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: int] :
! [X: int] :
( ( ord_less_int @ X @ Z )
=> ( ( ( P @ X )
| ( Q @ X ) )
= ( ( P2 @ X )
| ( Q2 @ X ) ) ) ) ) ) ).
% minf(2)
thf(fact_15_minf_I1_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z2: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z2: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: int] :
! [X: int] :
( ( ord_less_int @ X @ Z )
=> ( ( ( P @ X )
& ( Q @ X ) )
= ( ( P2 @ X )
& ( Q2 @ X ) ) ) ) ) ) ).
% minf(1)
thf(fact_16_res,axiom,
( ( binary1226383794sert_a @ h @ e @ ( binary717961607le_T_a @ t1 @ x @ t2 ) )
= ( binary717961607le_T_a @ t1 @ e @ t2 ) ) ).
% res
thf(fact_17_sortLemmaR,axiom,
! [H: a > int,T1: binary1439146945Tree_a,X3: a,T2: binary1439146945Tree_a] :
( ( binary1721989714Tree_a @ H @ ( binary717961607le_T_a @ T1 @ X3 @ T2 ) )
=> ( binary1721989714Tree_a @ H @ T2 ) ) ).
% sortLemmaR
thf(fact_18_sortLemmaL,axiom,
! [H: a > int,T1: binary1439146945Tree_a,X3: a,T2: binary1439146945Tree_a] :
( ( binary1721989714Tree_a @ H @ ( binary717961607le_T_a @ T1 @ X3 @ T2 ) )
=> ( binary1721989714Tree_a @ H @ T1 ) ) ).
% sortLemmaL
thf(fact_19_sortedTree_Osimps_I2_J,axiom,
! [H: a > int,T1: binary1439146945Tree_a,X3: a,T2: binary1439146945Tree_a] :
( ( binary1721989714Tree_a @ H @ ( binary717961607le_T_a @ T1 @ X3 @ T2 ) )
= ( ( binary1721989714Tree_a @ H @ T1 )
& ! [X4: a] :
( ( member_a @ X4 @ ( binary945792244etOf_a @ T1 ) )
=> ( ord_less_int @ ( H @ X4 ) @ ( H @ X3 ) ) )
& ! [X4: a] :
( ( member_a @ X4 @ ( binary945792244etOf_a @ T2 ) )
=> ( ord_less_int @ ( H @ X3 ) @ ( H @ X4 ) ) )
& ( binary1721989714Tree_a @ H @ T2 ) ) ) ).
% sortedTree.simps(2)
thf(fact_20_pinf_I1_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z2: int] :
! [X2: int] :
( ( ord_less_int @ Z2 @ X2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z2: int] :
! [X2: int] :
( ( ord_less_int @ Z2 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: int] :
! [X: int] :
( ( ord_less_int @ Z @ X )
=> ( ( ( P @ X )
& ( Q @ X ) )
= ( ( P2 @ X )
& ( Q2 @ X ) ) ) ) ) ) ).
% pinf(1)
thf(fact_21_pinf_I2_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z2: int] :
! [X2: int] :
( ( ord_less_int @ Z2 @ X2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z2: int] :
! [X2: int] :
( ( ord_less_int @ Z2 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: int] :
! [X: int] :
( ( ord_less_int @ Z @ X )
=> ( ( ( P @ X )
| ( Q @ X ) )
= ( ( P2 @ X )
| ( Q2 @ X ) ) ) ) ) ) ).
% pinf(2)
thf(fact_22_pinf_I3_J,axiom,
! [T: int] :
? [Z: int] :
! [X: int] :
( ( ord_less_int @ Z @ X )
=> ( X != T ) ) ).
% pinf(3)
thf(fact_23_pinf_I4_J,axiom,
! [T: int] :
? [Z: int] :
! [X: int] :
( ( ord_less_int @ Z @ X )
=> ( X != T ) ) ).
% pinf(4)
thf(fact_24_pinf_I5_J,axiom,
! [T: int] :
? [Z: int] :
! [X: int] :
( ( ord_less_int @ Z @ X )
=> ~ ( ord_less_int @ X @ T ) ) ).
% pinf(5)
thf(fact_25_pinf_I7_J,axiom,
! [T: int] :
? [Z: int] :
! [X: int] :
( ( ord_less_int @ Z @ X )
=> ( ord_less_int @ T @ X ) ) ).
% pinf(7)
thf(fact_26_sorted__distinct__pred__def,axiom,
( binary670562003pred_a
= ( ^ [H2: a > int,A: a,B: a,T3: binary1439146945Tree_a] :
( ( ( binary1721989714Tree_a @ H2 @ T3 )
& ( member_a @ A @ ( binary945792244etOf_a @ T3 ) )
& ( member_a @ B @ ( binary945792244etOf_a @ T3 ) )
& ( ( H2 @ A )
= ( H2 @ B ) ) )
=> ( A = B ) ) ) ) ).
% sorted_distinct_pred_def
thf(fact_27_memb__spec,axiom,
! [H: a > int,T: binary1439146945Tree_a,X3: a] :
( ( binary1721989714Tree_a @ H @ T )
=> ( ( binary2053421120memb_a @ H @ X3 @ T )
= ( member_a @ X3 @ ( binary945792244etOf_a @ T ) ) ) ) ).
% memb_spec
thf(fact_28_binsert_Osimps_I2_J,axiom,
! [H: a > int,E: a,X3: a,T1: binary1439146945Tree_a,T2: binary1439146945Tree_a] :
( ( ( ord_less_int @ ( H @ E ) @ ( H @ X3 ) )
=> ( ( binary1226383794sert_a @ H @ E @ ( binary717961607le_T_a @ T1 @ X3 @ T2 ) )
= ( binary717961607le_T_a @ ( binary1226383794sert_a @ H @ E @ T1 ) @ X3 @ T2 ) ) )
& ( ~ ( ord_less_int @ ( H @ E ) @ ( H @ X3 ) )
=> ( ( ( ord_less_int @ ( H @ X3 ) @ ( H @ E ) )
=> ( ( binary1226383794sert_a @ H @ E @ ( binary717961607le_T_a @ T1 @ X3 @ T2 ) )
= ( binary717961607le_T_a @ T1 @ X3 @ ( binary1226383794sert_a @ H @ E @ T2 ) ) ) )
& ( ~ ( ord_less_int @ ( H @ X3 ) @ ( H @ E ) )
=> ( ( binary1226383794sert_a @ H @ E @ ( binary717961607le_T_a @ T1 @ X3 @ T2 ) )
= ( binary717961607le_T_a @ T1 @ E @ T2 ) ) ) ) ) ) ).
% binsert.simps(2)
thf(fact_29_linorder__neqE__linordered__idom,axiom,
! [X3: int,Y: int] :
( ( X3 != Y )
=> ( ~ ( ord_less_int @ X3 @ Y )
=> ( ord_less_int @ Y @ X3 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_30_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_set_a @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_31_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: a > $o,A2: a > $o] :
( ( ord_less_a_o @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_32_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_33_order_Ostrict__implies__not__eq,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_34_order_Ostrict__implies__not__eq,axiom,
! [A2: a > $o,B2: a > $o] :
( ( ord_less_a_o @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_35_order_Ostrict__implies__not__eq,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_36_sorted__distinct,axiom,
! [H: a > int,A2: a,B2: a,T: binary1439146945Tree_a] : ( binary670562003pred_a @ H @ A2 @ B2 @ T ) ).
% sorted_distinct
thf(fact_37_ord__eq__less__subst,axiom,
! [A2: set_a,F: int > set_a,B2: int,C: int] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_38_ord__eq__less__subst,axiom,
! [A2: a > $o,F: int > a > $o,B2: int,C: int] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_a_o @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_a_o @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_39_ord__eq__less__subst,axiom,
! [A2: int,F: set_a > int,B2: set_a,C: set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_set_a @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_40_ord__eq__less__subst,axiom,
! [A2: set_a,F: set_a > set_a,B2: set_a,C: set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_set_a @ X2 @ Y2 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_41_ord__eq__less__subst,axiom,
! [A2: a > $o,F: set_a > a > $o,B2: set_a,C: set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_set_a @ X2 @ Y2 )
=> ( ord_less_a_o @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_a_o @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_42_ord__eq__less__subst,axiom,
! [A2: int,F: ( a > $o ) > int,B2: a > $o,C: a > $o] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_a_o @ B2 @ C )
=> ( ! [X2: a > $o,Y2: a > $o] :
( ( ord_less_a_o @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_43_ord__eq__less__subst,axiom,
! [A2: set_a,F: ( a > $o ) > set_a,B2: a > $o,C: a > $o] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_a_o @ B2 @ C )
=> ( ! [X2: a > $o,Y2: a > $o] :
( ( ord_less_a_o @ X2 @ Y2 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_44_ord__eq__less__subst,axiom,
! [A2: a > $o,F: ( a > $o ) > a > $o,B2: a > $o,C: a > $o] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_a_o @ B2 @ C )
=> ( ! [X2: a > $o,Y2: a > $o] :
( ( ord_less_a_o @ X2 @ Y2 )
=> ( ord_less_a_o @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_a_o @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_45_ord__eq__less__subst,axiom,
! [A2: int,F: int > int,B2: int,C: int] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_46_ord__less__eq__subst,axiom,
! [A2: int,B2: int,F: int > set_a,C: set_a] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_47_ord__less__eq__subst,axiom,
! [A2: int,B2: int,F: int > a > $o,C: a > $o] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_a_o @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_a_o @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_48_ord__less__eq__subst,axiom,
! [A2: set_a,B2: set_a,F: set_a > int,C: int] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_set_a @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_49_ord__less__eq__subst,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_set_a @ X2 @ Y2 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_50_ord__less__eq__subst,axiom,
! [A2: set_a,B2: set_a,F: set_a > a > $o,C: a > $o] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_set_a @ X2 @ Y2 )
=> ( ord_less_a_o @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_a_o @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_51_ord__less__eq__subst,axiom,
! [A2: a > $o,B2: a > $o,F: ( a > $o ) > int,C: int] :
( ( ord_less_a_o @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: a > $o,Y2: a > $o] :
( ( ord_less_a_o @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_52_ord__less__eq__subst,axiom,
! [A2: a > $o,B2: a > $o,F: ( a > $o ) > set_a,C: set_a] :
( ( ord_less_a_o @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: a > $o,Y2: a > $o] :
( ( ord_less_a_o @ X2 @ Y2 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_53_ord__less__eq__subst,axiom,
! [A2: a > $o,B2: a > $o,F: ( a > $o ) > a > $o,C: a > $o] :
( ( ord_less_a_o @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: a > $o,Y2: a > $o] :
( ( ord_less_a_o @ X2 @ Y2 )
=> ( ord_less_a_o @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_a_o @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_54_ord__less__eq__subst,axiom,
! [A2: int,B2: int,F: int > int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_55_order__less__subst1,axiom,
! [A2: int,F: set_a > int,B2: set_a,C: set_a] :
( ( ord_less_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_set_a @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_56_order__less__subst1,axiom,
! [A2: int,F: ( a > $o ) > int,B2: a > $o,C: a > $o] :
( ( ord_less_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_a_o @ B2 @ C )
=> ( ! [X2: a > $o,Y2: a > $o] :
( ( ord_less_a_o @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_57_order__less__subst1,axiom,
! [A2: set_a,F: int > set_a,B2: int,C: int] :
( ( ord_less_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_58_order__less__subst1,axiom,
! [A2: set_a,F: set_a > set_a,B2: set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_set_a @ X2 @ Y2 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_59_order__less__subst1,axiom,
! [A2: set_a,F: ( a > $o ) > set_a,B2: a > $o,C: a > $o] :
( ( ord_less_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_a_o @ B2 @ C )
=> ( ! [X2: a > $o,Y2: a > $o] :
( ( ord_less_a_o @ X2 @ Y2 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_60_order__less__subst1,axiom,
! [A2: a > $o,F: int > a > $o,B2: int,C: int] :
( ( ord_less_a_o @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_a_o @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_a_o @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_61_order__less__subst1,axiom,
! [A2: a > $o,F: set_a > a > $o,B2: set_a,C: set_a] :
( ( ord_less_a_o @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_set_a @ X2 @ Y2 )
=> ( ord_less_a_o @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_a_o @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_62_order__less__subst1,axiom,
! [A2: a > $o,F: ( a > $o ) > a > $o,B2: a > $o,C: a > $o] :
( ( ord_less_a_o @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_a_o @ B2 @ C )
=> ( ! [X2: a > $o,Y2: a > $o] :
( ( ord_less_a_o @ X2 @ Y2 )
=> ( ord_less_a_o @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_a_o @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_63_order__less__subst1,axiom,
! [A2: int,F: int > int,B2: int,C: int] :
( ( ord_less_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_64_order__less__subst2,axiom,
! [A2: int,B2: int,F: int > set_a,C: set_a] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_set_a @ ( F @ B2 ) @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_65_order__less__subst2,axiom,
! [A2: int,B2: int,F: int > a > $o,C: a > $o] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_a_o @ ( F @ B2 ) @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_a_o @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_a_o @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_66_order__less__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > int,C: int] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_set_a @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_67_order__less__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ord_less_set_a @ ( F @ B2 ) @ C )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_set_a @ X2 @ Y2 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_68_order__less__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > a > $o,C: a > $o] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ord_less_a_o @ ( F @ B2 ) @ C )
=> ( ! [X2: set_a,Y2: set_a] :
( ( ord_less_set_a @ X2 @ Y2 )
=> ( ord_less_a_o @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_a_o @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_69_order__less__subst2,axiom,
! [A2: a > $o,B2: a > $o,F: ( a > $o ) > int,C: int] :
( ( ord_less_a_o @ A2 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X2: a > $o,Y2: a > $o] :
( ( ord_less_a_o @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_70_order__less__subst2,axiom,
! [A2: a > $o,B2: a > $o,F: ( a > $o ) > set_a,C: set_a] :
( ( ord_less_a_o @ A2 @ B2 )
=> ( ( ord_less_set_a @ ( F @ B2 ) @ C )
=> ( ! [X2: a > $o,Y2: a > $o] :
( ( ord_less_a_o @ X2 @ Y2 )
=> ( ord_less_set_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_71_order__less__subst2,axiom,
! [A2: a > $o,B2: a > $o,F: ( a > $o ) > a > $o,C: a > $o] :
( ( ord_less_a_o @ A2 @ B2 )
=> ( ( ord_less_a_o @ ( F @ B2 ) @ C )
=> ( ! [X2: a > $o,Y2: a > $o] :
( ( ord_less_a_o @ X2 @ Y2 )
=> ( ord_less_a_o @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_a_o @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_72_order__less__subst2,axiom,
! [A2: int,B2: int,F: int > int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X2: int,Y2: int] :
( ( ord_less_int @ X2 @ Y2 )
=> ( ord_less_int @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_73_lt__ex,axiom,
! [X3: int] :
? [Y2: int] : ( ord_less_int @ Y2 @ X3 ) ).
% lt_ex
thf(fact_74_gt__ex,axiom,
! [X3: int] :
? [X_1: int] : ( ord_less_int @ X3 @ X_1 ) ).
% gt_ex
thf(fact_75_mem__Collect__eq,axiom,
! [A2: a,P: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_76_Collect__mem__eq,axiom,
! [A3: set_a] :
( ( collect_a
@ ^ [X4: a] : ( member_a @ X4 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_77_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X2: a] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_78_neqE,axiom,
! [X3: int,Y: int] :
( ( X3 != Y )
=> ( ~ ( ord_less_int @ X3 @ Y )
=> ( ord_less_int @ Y @ X3 ) ) ) ).
% neqE
thf(fact_79_neq__iff,axiom,
! [X3: int,Y: int] :
( ( X3 != Y )
= ( ( ord_less_int @ X3 @ Y )
| ( ord_less_int @ Y @ X3 ) ) ) ).
% neq_iff
thf(fact_80_order_Oasym,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ~ ( ord_less_set_a @ B2 @ A2 ) ) ).
% order.asym
thf(fact_81_order_Oasym,axiom,
! [A2: a > $o,B2: a > $o] :
( ( ord_less_a_o @ A2 @ B2 )
=> ~ ( ord_less_a_o @ B2 @ A2 ) ) ).
% order.asym
thf(fact_82_order_Oasym,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ~ ( ord_less_int @ B2 @ A2 ) ) ).
% order.asym
thf(fact_83_less__imp__neq,axiom,
! [X3: set_a,Y: set_a] :
( ( ord_less_set_a @ X3 @ Y )
=> ( X3 != Y ) ) ).
% less_imp_neq
thf(fact_84_less__imp__neq,axiom,
! [X3: a > $o,Y: a > $o] :
( ( ord_less_a_o @ X3 @ Y )
=> ( X3 != Y ) ) ).
% less_imp_neq
thf(fact_85_less__imp__neq,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ( X3 != Y ) ) ).
% less_imp_neq
thf(fact_86_less__asym,axiom,
! [X3: set_a,Y: set_a] :
( ( ord_less_set_a @ X3 @ Y )
=> ~ ( ord_less_set_a @ Y @ X3 ) ) ).
% less_asym
thf(fact_87_less__asym,axiom,
! [X3: a > $o,Y: a > $o] :
( ( ord_less_a_o @ X3 @ Y )
=> ~ ( ord_less_a_o @ Y @ X3 ) ) ).
% less_asym
thf(fact_88_less__asym,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ~ ( ord_less_int @ Y @ X3 ) ) ).
% less_asym
thf(fact_89_less__asym_H,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ~ ( ord_less_set_a @ B2 @ A2 ) ) ).
% less_asym'
thf(fact_90_less__asym_H,axiom,
! [A2: a > $o,B2: a > $o] :
( ( ord_less_a_o @ A2 @ B2 )
=> ~ ( ord_less_a_o @ B2 @ A2 ) ) ).
% less_asym'
thf(fact_91_less__asym_H,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ~ ( ord_less_int @ B2 @ A2 ) ) ).
% less_asym'
thf(fact_92_less__trans,axiom,
! [X3: set_a,Y: set_a,Z3: set_a] :
( ( ord_less_set_a @ X3 @ Y )
=> ( ( ord_less_set_a @ Y @ Z3 )
=> ( ord_less_set_a @ X3 @ Z3 ) ) ) ).
% less_trans
thf(fact_93_less__trans,axiom,
! [X3: a > $o,Y: a > $o,Z3: a > $o] :
( ( ord_less_a_o @ X3 @ Y )
=> ( ( ord_less_a_o @ Y @ Z3 )
=> ( ord_less_a_o @ X3 @ Z3 ) ) ) ).
% less_trans
thf(fact_94_less__trans,axiom,
! [X3: int,Y: int,Z3: int] :
( ( ord_less_int @ X3 @ Y )
=> ( ( ord_less_int @ Y @ Z3 )
=> ( ord_less_int @ X3 @ Z3 ) ) ) ).
% less_trans
thf(fact_95_less__linear,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
| ( X3 = Y )
| ( ord_less_int @ Y @ X3 ) ) ).
% less_linear
thf(fact_96_less__irrefl,axiom,
! [X3: set_a] :
~ ( ord_less_set_a @ X3 @ X3 ) ).
% less_irrefl
thf(fact_97_less__irrefl,axiom,
! [X3: a > $o] :
~ ( ord_less_a_o @ X3 @ X3 ) ).
% less_irrefl
thf(fact_98_less__irrefl,axiom,
! [X3: int] :
~ ( ord_less_int @ X3 @ X3 ) ).
% less_irrefl
thf(fact_99_ord__eq__less__trans,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( A2 = B2 )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ord_less_set_a @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_100_ord__eq__less__trans,axiom,
! [A2: a > $o,B2: a > $o,C: a > $o] :
( ( A2 = B2 )
=> ( ( ord_less_a_o @ B2 @ C )
=> ( ord_less_a_o @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_101_ord__eq__less__trans,axiom,
! [A2: int,B2: int,C: int] :
( ( A2 = B2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_102_ord__less__eq__trans,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_set_a @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_103_ord__less__eq__trans,axiom,
! [A2: a > $o,B2: a > $o,C: a > $o] :
( ( ord_less_a_o @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_a_o @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_104_ord__less__eq__trans,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_105_dual__order_Oasym,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_set_a @ B2 @ A2 )
=> ~ ( ord_less_set_a @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_106_dual__order_Oasym,axiom,
! [B2: a > $o,A2: a > $o] :
( ( ord_less_a_o @ B2 @ A2 )
=> ~ ( ord_less_a_o @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_107_dual__order_Oasym,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ B2 @ A2 )
=> ~ ( ord_less_int @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_108_less__imp__not__eq,axiom,
! [X3: set_a,Y: set_a] :
( ( ord_less_set_a @ X3 @ Y )
=> ( X3 != Y ) ) ).
% less_imp_not_eq
thf(fact_109_less__imp__not__eq,axiom,
! [X3: a > $o,Y: a > $o] :
( ( ord_less_a_o @ X3 @ Y )
=> ( X3 != Y ) ) ).
% less_imp_not_eq
thf(fact_110_less__imp__not__eq,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ( X3 != Y ) ) ).
% less_imp_not_eq
thf(fact_111_less__not__sym,axiom,
! [X3: set_a,Y: set_a] :
( ( ord_less_set_a @ X3 @ Y )
=> ~ ( ord_less_set_a @ Y @ X3 ) ) ).
% less_not_sym
thf(fact_112_less__not__sym,axiom,
! [X3: a > $o,Y: a > $o] :
( ( ord_less_a_o @ X3 @ Y )
=> ~ ( ord_less_a_o @ Y @ X3 ) ) ).
% less_not_sym
thf(fact_113_less__not__sym,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ~ ( ord_less_int @ Y @ X3 ) ) ).
% less_not_sym
thf(fact_114_antisym__conv3,axiom,
! [Y: int,X3: int] :
( ~ ( ord_less_int @ Y @ X3 )
=> ( ( ~ ( ord_less_int @ X3 @ Y ) )
= ( X3 = Y ) ) ) ).
% antisym_conv3
thf(fact_115_less__imp__not__eq2,axiom,
! [X3: set_a,Y: set_a] :
( ( ord_less_set_a @ X3 @ Y )
=> ( Y != X3 ) ) ).
% less_imp_not_eq2
thf(fact_116_less__imp__not__eq2,axiom,
! [X3: a > $o,Y: a > $o] :
( ( ord_less_a_o @ X3 @ Y )
=> ( Y != X3 ) ) ).
% less_imp_not_eq2
thf(fact_117_less__imp__not__eq2,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ( Y != X3 ) ) ).
% less_imp_not_eq2
thf(fact_118_less__imp__triv,axiom,
! [X3: set_a,Y: set_a,P: $o] :
( ( ord_less_set_a @ X3 @ Y )
=> ( ( ord_less_set_a @ Y @ X3 )
=> P ) ) ).
% less_imp_triv
thf(fact_119_less__imp__triv,axiom,
! [X3: a > $o,Y: a > $o,P: $o] :
( ( ord_less_a_o @ X3 @ Y )
=> ( ( ord_less_a_o @ Y @ X3 )
=> P ) ) ).
% less_imp_triv
thf(fact_120_less__imp__triv,axiom,
! [X3: int,Y: int,P: $o] :
( ( ord_less_int @ X3 @ Y )
=> ( ( ord_less_int @ Y @ X3 )
=> P ) ) ).
% less_imp_triv
thf(fact_121_linorder__cases,axiom,
! [X3: int,Y: int] :
( ~ ( ord_less_int @ X3 @ Y )
=> ( ( X3 != Y )
=> ( ord_less_int @ Y @ X3 ) ) ) ).
% linorder_cases
thf(fact_122_dual__order_Oirrefl,axiom,
! [A2: set_a] :
~ ( ord_less_set_a @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_123_dual__order_Oirrefl,axiom,
! [A2: a > $o] :
~ ( ord_less_a_o @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_124_dual__order_Oirrefl,axiom,
! [A2: int] :
~ ( ord_less_int @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_125_order_Ostrict__trans,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ord_less_set_a @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_126_order_Ostrict__trans,axiom,
! [A2: a > $o,B2: a > $o,C: a > $o] :
( ( ord_less_a_o @ A2 @ B2 )
=> ( ( ord_less_a_o @ B2 @ C )
=> ( ord_less_a_o @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_127_order_Ostrict__trans,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_128_less__imp__not__less,axiom,
! [X3: set_a,Y: set_a] :
( ( ord_less_set_a @ X3 @ Y )
=> ~ ( ord_less_set_a @ Y @ X3 ) ) ).
% less_imp_not_less
thf(fact_129_less__imp__not__less,axiom,
! [X3: a > $o,Y: a > $o] :
( ( ord_less_a_o @ X3 @ Y )
=> ~ ( ord_less_a_o @ Y @ X3 ) ) ).
% less_imp_not_less
thf(fact_130_less__imp__not__less,axiom,
! [X3: int,Y: int] :
( ( ord_less_int @ X3 @ Y )
=> ~ ( ord_less_int @ Y @ X3 ) ) ).
% less_imp_not_less
thf(fact_131_linorder__less__wlog,axiom,
! [P: int > int > $o,A2: int,B2: 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 @ A2 @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_132_dual__order_Ostrict__trans,axiom,
! [B2: set_a,A2: set_a,C: set_a] :
( ( ord_less_set_a @ B2 @ A2 )
=> ( ( ord_less_set_a @ C @ B2 )
=> ( ord_less_set_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_133_dual__order_Ostrict__trans,axiom,
! [B2: a > $o,A2: a > $o,C: a > $o] :
( ( ord_less_a_o @ B2 @ A2 )
=> ( ( ord_less_a_o @ C @ B2 )
=> ( ord_less_a_o @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_134_dual__order_Ostrict__trans,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( ( ord_less_int @ C @ B2 )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_135_not__less__iff__gr__or__eq,axiom,
! [X3: int,Y: int] :
( ( ~ ( ord_less_int @ X3 @ Y ) )
= ( ( ord_less_int @ Y @ X3 )
| ( X3 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_136_binsert_Osimps_I1_J,axiom,
! [H: a > int,E: a] :
( ( binary1226383794sert_a @ H @ E @ binary476621312_Tip_a )
= ( binary717961607le_T_a @ binary476621312_Tip_a @ E @ binary476621312_Tip_a ) ) ).
% binsert.simps(1)
thf(fact_137_verit__comp__simplify1_I1_J,axiom,
! [A2: set_a] :
~ ( ord_less_set_a @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_138_verit__comp__simplify1_I1_J,axiom,
! [A2: a > $o] :
~ ( ord_less_a_o @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_139_verit__comp__simplify1_I1_J,axiom,
! [A2: int] :
~ ( ord_less_int @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_140_c2,axiom,
( ( binary945792244etOf_a @ ( binary1226383794sert_a @ h @ e @ t2 ) )
= ( sup_sup_set_a @ ( minus_minus_set_a @ ( binary945792244etOf_a @ t2 ) @ ( binary504661350_eqs_a @ h @ e ) ) @ ( insert_a @ e @ bot_bot_set_a ) ) ) ).
% c2
thf(fact_141_c1,axiom,
( ( binary945792244etOf_a @ ( binary1226383794sert_a @ h @ e @ t1 ) )
= ( sup_sup_set_a @ ( minus_minus_set_a @ ( binary945792244etOf_a @ t1 ) @ ( binary504661350_eqs_a @ h @ e ) ) @ ( insert_a @ e @ bot_bot_set_a ) ) ) ).
% c1
thf(fact_142_Tree_Opred__inject_I2_J,axiom,
! [P: a > $o,A2: binary1439146945Tree_a,Aa: a,Ab: binary1439146945Tree_a] :
( ( binary1452917696Tree_a @ P @ ( binary717961607le_T_a @ A2 @ Aa @ Ab ) )
= ( ( binary1452917696Tree_a @ P @ A2 )
& ( P @ Aa )
& ( binary1452917696Tree_a @ P @ Ab ) ) ) ).
% Tree.pred_inject(2)
thf(fact_143_bot__apply,axiom,
( bot_bot_a_o
= ( ^ [X4: a] : bot_bot_o ) ) ).
% bot_apply
thf(fact_144_bot__fun__def,axiom,
( bot_bot_a_o
= ( ^ [X4: a] : bot_bot_o ) ) ).
% bot_fun_def
thf(fact_145_Tree_Opred__inject_I1_J,axiom,
! [P: a > $o] : ( binary1452917696Tree_a @ P @ binary476621312_Tip_a ) ).
% Tree.pred_inject(1)
thf(fact_146_setOf_Osimps_I2_J,axiom,
! [T1: binary1439146945Tree_a,X3: a,T2: binary1439146945Tree_a] :
( ( binary945792244etOf_a @ ( binary717961607le_T_a @ T1 @ X3 @ T2 ) )
= ( sup_sup_set_a @ ( sup_sup_set_a @ ( binary945792244etOf_a @ T1 ) @ ( binary945792244etOf_a @ T2 ) ) @ ( insert_a @ X3 @ bot_bot_set_a ) ) ) ).
% setOf.simps(2)
thf(fact_147_setOf_Osimps_I1_J,axiom,
( ( binary945792244etOf_a @ binary476621312_Tip_a )
= bot_bot_set_a ) ).
% setOf.simps(1)
thf(fact_148_bot_Onot__eq__extremum,axiom,
! [A2: a > $o] :
( ( A2 != bot_bot_a_o )
= ( ord_less_a_o @ bot_bot_a_o @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_149_bot_Onot__eq__extremum,axiom,
! [A2: set_a] :
( ( A2 != bot_bot_set_a )
= ( ord_less_set_a @ bot_bot_set_a @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_150_bot_Oextremum__strict,axiom,
! [A2: a > $o] :
~ ( ord_less_a_o @ A2 @ bot_bot_a_o ) ).
% bot.extremum_strict
thf(fact_151_bot_Oextremum__strict,axiom,
! [A2: set_a] :
~ ( ord_less_set_a @ A2 @ bot_bot_set_a ) ).
% bot.extremum_strict
thf(fact_152_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_153_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_154_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_155_sortedTree_Osimps_I1_J,axiom,
! [H: a > int] : ( binary1721989714Tree_a @ H @ binary476621312_Tip_a ) ).
% sortedTree.simps(1)
thf(fact_156_insert__Diff__single,axiom,
! [A2: a,A3: set_a] :
( ( insert_a @ A2 @ ( minus_minus_set_a @ A3 @ ( insert_a @ A2 @ bot_bot_set_a ) ) )
= ( insert_a @ A2 @ A3 ) ) ).
% insert_Diff_single
thf(fact_157_Un__Diff__cancel,axiom,
! [A3: set_a,B4: set_a] :
( ( sup_sup_set_a @ A3 @ ( minus_minus_set_a @ B4 @ A3 ) )
= ( sup_sup_set_a @ A3 @ B4 ) ) ).
% Un_Diff_cancel
thf(fact_158_Un__Diff__cancel2,axiom,
! [B4: set_a,A3: set_a] :
( ( sup_sup_set_a @ ( minus_minus_set_a @ B4 @ A3 ) @ A3 )
= ( sup_sup_set_a @ B4 @ A3 ) ) ).
% Un_Diff_cancel2
thf(fact_159_Diff__insert0,axiom,
! [X3: a,A3: set_a,B4: set_a] :
( ~ ( member_a @ X3 @ A3 )
=> ( ( minus_minus_set_a @ A3 @ ( insert_a @ X3 @ B4 ) )
= ( minus_minus_set_a @ A3 @ B4 ) ) ) ).
% Diff_insert0
thf(fact_160_insert__Diff1,axiom,
! [X3: a,B4: set_a,A3: set_a] :
( ( member_a @ X3 @ B4 )
=> ( ( minus_minus_set_a @ ( insert_a @ X3 @ A3 ) @ B4 )
= ( minus_minus_set_a @ A3 @ B4 ) ) ) ).
% insert_Diff1
thf(fact_161_Un__insert__left,axiom,
! [A2: a,B4: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( insert_a @ A2 @ B4 ) @ C2 )
= ( insert_a @ A2 @ ( sup_sup_set_a @ B4 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_162_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X4: a] :
~ ( P @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_163_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X4: a] :
~ ( P @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_164_all__not__in__conv,axiom,
! [A3: set_a] :
( ( ! [X4: a] :
~ ( member_a @ X4 @ A3 ) )
= ( A3 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_165_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_166_insert__absorb2,axiom,
! [X3: a,A3: set_a] :
( ( insert_a @ X3 @ ( insert_a @ X3 @ A3 ) )
= ( insert_a @ X3 @ A3 ) ) ).
% insert_absorb2
thf(fact_167_insert__iff,axiom,
! [A2: a,B2: a,A3: set_a] :
( ( member_a @ A2 @ ( insert_a @ B2 @ A3 ) )
= ( ( A2 = B2 )
| ( member_a @ A2 @ A3 ) ) ) ).
% insert_iff
thf(fact_168_insertCI,axiom,
! [A2: a,B4: set_a,B2: a] :
( ( ~ ( member_a @ A2 @ B4 )
=> ( A2 = B2 ) )
=> ( member_a @ A2 @ ( insert_a @ B2 @ B4 ) ) ) ).
% insertCI
thf(fact_169_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_170_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_171_Diff__idemp,axiom,
! [A3: set_a,B4: set_a] :
( ( minus_minus_set_a @ ( minus_minus_set_a @ A3 @ B4 ) @ B4 )
= ( minus_minus_set_a @ A3 @ B4 ) ) ).
% Diff_idemp
thf(fact_172_Diff__iff,axiom,
! [C: a,A3: set_a,B4: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A3 @ B4 ) )
= ( ( member_a @ C @ A3 )
& ~ ( member_a @ C @ B4 ) ) ) ).
% Diff_iff
thf(fact_173_DiffI,axiom,
! [C: a,A3: set_a,B4: set_a] :
( ( member_a @ C @ A3 )
=> ( ~ ( member_a @ C @ B4 )
=> ( member_a @ C @ ( minus_minus_set_a @ A3 @ B4 ) ) ) ) ).
% DiffI
thf(fact_174_singletonI,axiom,
! [A2: a] : ( member_a @ A2 @ ( insert_a @ A2 @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_175_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_176_Diff__cancel,axiom,
! [A3: set_a] :
( ( minus_minus_set_a @ A3 @ A3 )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_177_empty__Diff,axiom,
! [A3: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A3 )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_178_Diff__empty,axiom,
! [A3: set_a] :
( ( minus_minus_set_a @ A3 @ bot_bot_set_a )
= A3 ) ).
% Diff_empty
thf(fact_179_Un__insert__right,axiom,
! [A3: set_a,A2: a,B4: set_a] :
( ( sup_sup_set_a @ A3 @ ( insert_a @ A2 @ B4 ) )
= ( insert_a @ A2 @ ( sup_sup_set_a @ A3 @ B4 ) ) ) ).
% Un_insert_right
thf(fact_180_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_181_not__psubset__empty,axiom,
! [A3: set_a] :
~ ( ord_less_set_a @ A3 @ bot_bot_set_a ) ).
% not_psubset_empty
thf(fact_182_ex__in__conv,axiom,
! [A3: set_a] :
( ( ? [X4: a] : ( member_a @ X4 @ A3 ) )
= ( A3 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_183_equals0I,axiom,
! [A3: set_a] :
( ! [Y2: a] :
~ ( member_a @ Y2 @ A3 )
=> ( A3 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_184_equals0D,axiom,
! [A3: set_a,A2: a] :
( ( A3 = bot_bot_set_a )
=> ~ ( member_a @ A2 @ A3 ) ) ).
% equals0D
thf(fact_185_emptyE,axiom,
! [A2: a] :
~ ( member_a @ A2 @ bot_bot_set_a ) ).
% emptyE
thf(fact_186_mk__disjoint__insert,axiom,
! [A2: a,A3: set_a] :
( ( member_a @ A2 @ A3 )
=> ? [B5: set_a] :
( ( A3
= ( insert_a @ A2 @ B5 ) )
& ~ ( member_a @ A2 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_187_insert__commute,axiom,
! [X3: a,Y: a,A3: set_a] :
( ( insert_a @ X3 @ ( insert_a @ Y @ A3 ) )
= ( insert_a @ Y @ ( insert_a @ X3 @ A3 ) ) ) ).
% insert_commute
thf(fact_188_insert__eq__iff,axiom,
! [A2: a,A3: set_a,B2: a,B4: set_a] :
( ~ ( member_a @ A2 @ A3 )
=> ( ~ ( member_a @ B2 @ B4 )
=> ( ( ( insert_a @ A2 @ A3 )
= ( insert_a @ B2 @ B4 ) )
= ( ( ( A2 = B2 )
=> ( A3 = B4 ) )
& ( ( A2 != B2 )
=> ? [C3: set_a] :
( ( A3
= ( insert_a @ B2 @ C3 ) )
& ~ ( member_a @ B2 @ C3 )
& ( B4
= ( insert_a @ A2 @ C3 ) )
& ~ ( member_a @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_189_insert__absorb,axiom,
! [A2: a,A3: set_a] :
( ( member_a @ A2 @ A3 )
=> ( ( insert_a @ A2 @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_190_insert__ident,axiom,
! [X3: a,A3: set_a,B4: set_a] :
( ~ ( member_a @ X3 @ A3 )
=> ( ~ ( member_a @ X3 @ B4 )
=> ( ( ( insert_a @ X3 @ A3 )
= ( insert_a @ X3 @ B4 ) )
= ( A3 = B4 ) ) ) ) ).
% insert_ident
thf(fact_191_Set_Oset__insert,axiom,
! [X3: a,A3: set_a] :
( ( member_a @ X3 @ A3 )
=> ~ ! [B5: set_a] :
( ( A3
= ( insert_a @ X3 @ B5 ) )
=> ( member_a @ X3 @ B5 ) ) ) ).
% Set.set_insert
thf(fact_192_insertI2,axiom,
! [A2: a,B4: set_a,B2: a] :
( ( member_a @ A2 @ B4 )
=> ( member_a @ A2 @ ( insert_a @ B2 @ B4 ) ) ) ).
% insertI2
thf(fact_193_insertI1,axiom,
! [A2: a,B4: set_a] : ( member_a @ A2 @ ( insert_a @ A2 @ B4 ) ) ).
% insertI1
thf(fact_194_insertE,axiom,
! [A2: a,B2: a,A3: set_a] :
( ( member_a @ A2 @ ( insert_a @ B2 @ A3 ) )
=> ( ( A2 != B2 )
=> ( member_a @ A2 @ A3 ) ) ) ).
% insertE
thf(fact_195_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_196_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_197_Un__commute,axiom,
( sup_sup_set_a
= ( ^ [A5: set_a,B6: set_a] : ( sup_sup_set_a @ B6 @ A5 ) ) ) ).
% Un_commute
thf(fact_198_Un__absorb,axiom,
! [A3: set_a] :
( ( sup_sup_set_a @ A3 @ A3 )
= A3 ) ).
% Un_absorb
thf(fact_199_Un__assoc,axiom,
! [A3: set_a,B4: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A3 @ B4 ) @ C2 )
= ( sup_sup_set_a @ A3 @ ( sup_sup_set_a @ B4 @ C2 ) ) ) ).
% Un_assoc
thf(fact_200_ball__Un,axiom,
! [A3: set_a,B4: set_a,P: a > $o] :
( ( ! [X4: a] :
( ( member_a @ X4 @ ( sup_sup_set_a @ A3 @ B4 ) )
=> ( P @ X4 ) ) )
= ( ! [X4: a] :
( ( member_a @ X4 @ A3 )
=> ( P @ X4 ) )
& ! [X4: a] :
( ( member_a @ X4 @ B4 )
=> ( P @ X4 ) ) ) ) ).
% ball_Un
thf(fact_201_bex__Un,axiom,
! [A3: set_a,B4: set_a,P: a > $o] :
( ( ? [X4: a] :
( ( member_a @ X4 @ ( sup_sup_set_a @ A3 @ B4 ) )
& ( P @ X4 ) ) )
= ( ? [X4: a] :
( ( member_a @ X4 @ A3 )
& ( P @ X4 ) )
| ? [X4: a] :
( ( member_a @ X4 @ B4 )
& ( P @ X4 ) ) ) ) ).
% bex_Un
thf(fact_202_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_203_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_204_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_205_psubset__imp__ex__mem,axiom,
! [A3: set_a,B4: set_a] :
( ( ord_less_set_a @ A3 @ B4 )
=> ? [B3: a] : ( member_a @ B3 @ ( minus_minus_set_a @ B4 @ A3 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_206_DiffD2,axiom,
! [C: a,A3: set_a,B4: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A3 @ B4 ) )
=> ~ ( member_a @ C @ B4 ) ) ).
% DiffD2
thf(fact_207_DiffD1,axiom,
! [C: a,A3: set_a,B4: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A3 @ B4 ) )
=> ( member_a @ C @ A3 ) ) ).
% DiffD1
thf(fact_208_DiffE,axiom,
! [C: a,A3: set_a,B4: set_a] :
( ( member_a @ C @ ( minus_minus_set_a @ A3 @ B4 ) )
=> ~ ( ( member_a @ C @ A3 )
=> ( member_a @ C @ B4 ) ) ) ).
% DiffE
thf(fact_209_singleton__inject,axiom,
! [A2: a,B2: a] :
( ( ( insert_a @ A2 @ bot_bot_set_a )
= ( insert_a @ B2 @ bot_bot_set_a ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_210_insert__not__empty,axiom,
! [A2: a,A3: set_a] :
( ( insert_a @ A2 @ A3 )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_211_doubleton__eq__iff,axiom,
! [A2: a,B2: a,C: a,D: a] :
( ( ( insert_a @ A2 @ ( insert_a @ B2 @ bot_bot_set_a ) )
= ( insert_a @ C @ ( insert_a @ D @ bot_bot_set_a ) ) )
= ( ( ( A2 = C )
& ( B2 = D ) )
| ( ( A2 = D )
& ( B2 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_212_singleton__iff,axiom,
! [B2: a,A2: a] :
( ( member_a @ B2 @ ( insert_a @ A2 @ bot_bot_set_a ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_213_singletonD,axiom,
! [B2: a,A2: a] :
( ( member_a @ B2 @ ( insert_a @ A2 @ bot_bot_set_a ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_214_Un__empty__right,axiom,
! [A3: set_a] :
( ( sup_sup_set_a @ A3 @ bot_bot_set_a )
= A3 ) ).
% Un_empty_right
thf(fact_215_Un__empty__left,axiom,
! [B4: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ B4 )
= B4 ) ).
% Un_empty_left
thf(fact_216_insert__Diff__if,axiom,
! [X3: a,B4: set_a,A3: set_a] :
( ( ( member_a @ X3 @ B4 )
=> ( ( minus_minus_set_a @ ( insert_a @ X3 @ A3 ) @ B4 )
= ( minus_minus_set_a @ A3 @ B4 ) ) )
& ( ~ ( member_a @ X3 @ B4 )
=> ( ( minus_minus_set_a @ ( insert_a @ X3 @ A3 ) @ B4 )
= ( insert_a @ X3 @ ( minus_minus_set_a @ A3 @ B4 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_217_Un__Diff,axiom,
! [A3: set_a,B4: set_a,C2: set_a] :
( ( minus_minus_set_a @ ( sup_sup_set_a @ A3 @ B4 ) @ C2 )
= ( sup_sup_set_a @ ( minus_minus_set_a @ A3 @ C2 ) @ ( minus_minus_set_a @ B4 @ C2 ) ) ) ).
% Un_Diff
thf(fact_218_singleton__Un__iff,axiom,
! [X3: a,A3: set_a,B4: set_a] :
( ( ( insert_a @ X3 @ bot_bot_set_a )
= ( sup_sup_set_a @ A3 @ B4 ) )
= ( ( ( A3 = bot_bot_set_a )
& ( B4
= ( insert_a @ X3 @ bot_bot_set_a ) ) )
| ( ( A3
= ( insert_a @ X3 @ bot_bot_set_a ) )
& ( B4 = bot_bot_set_a ) )
| ( ( A3
= ( insert_a @ X3 @ bot_bot_set_a ) )
& ( B4
= ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_219_Un__singleton__iff,axiom,
! [A3: set_a,B4: set_a,X3: a] :
( ( ( sup_sup_set_a @ A3 @ B4 )
= ( insert_a @ X3 @ bot_bot_set_a ) )
= ( ( ( A3 = bot_bot_set_a )
& ( B4
= ( insert_a @ X3 @ bot_bot_set_a ) ) )
| ( ( A3
= ( insert_a @ X3 @ bot_bot_set_a ) )
& ( B4 = bot_bot_set_a ) )
| ( ( A3
= ( insert_a @ X3 @ bot_bot_set_a ) )
& ( B4
= ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_220_insert__is__Un,axiom,
( insert_a
= ( ^ [A: a] : ( sup_sup_set_a @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% insert_is_Un
thf(fact_221_Diff__insert__absorb,axiom,
! [X3: a,A3: set_a] :
( ~ ( member_a @ X3 @ A3 )
=> ( ( minus_minus_set_a @ ( insert_a @ X3 @ A3 ) @ ( insert_a @ X3 @ bot_bot_set_a ) )
= A3 ) ) ).
% Diff_insert_absorb
thf(fact_222_Diff__insert2,axiom,
! [A3: set_a,A2: a,B4: set_a] :
( ( minus_minus_set_a @ A3 @ ( insert_a @ A2 @ B4 ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A3 @ ( insert_a @ A2 @ bot_bot_set_a ) ) @ B4 ) ) ).
% Diff_insert2
thf(fact_223_insert__Diff,axiom,
! [A2: a,A3: set_a] :
( ( member_a @ A2 @ A3 )
=> ( ( insert_a @ A2 @ ( minus_minus_set_a @ A3 @ ( insert_a @ A2 @ bot_bot_set_a ) ) )
= A3 ) ) ).
% insert_Diff
thf(fact_224_Diff__insert,axiom,
! [A3: set_a,A2: a,B4: set_a] :
( ( minus_minus_set_a @ A3 @ ( insert_a @ A2 @ B4 ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A3 @ B4 ) @ ( insert_a @ A2 @ bot_bot_set_a ) ) ) ).
% Diff_insert
thf(fact_225_sup__bot_Oright__neutral,axiom,
! [A2: a > $o] :
( ( sup_sup_a_o @ A2 @ bot_bot_a_o )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_226_sup__bot_Oright__neutral,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_227_sup__bot_Oneutr__eq__iff,axiom,
! [A2: a > $o,B2: a > $o] :
( ( bot_bot_a_o
= ( sup_sup_a_o @ A2 @ B2 ) )
= ( ( A2 = bot_bot_a_o )
& ( B2 = bot_bot_a_o ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_228_sup__bot_Oneutr__eq__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ A2 @ B2 ) )
= ( ( A2 = bot_bot_set_a )
& ( B2 = bot_bot_set_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_229_sup__bot_Oleft__neutral,axiom,
! [A2: a > $o] :
( ( sup_sup_a_o @ bot_bot_a_o @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_230_sup__bot_Oleft__neutral,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_231_sup__bot_Oeq__neutr__iff,axiom,
! [A2: a > $o,B2: a > $o] :
( ( ( sup_sup_a_o @ A2 @ B2 )
= bot_bot_a_o )
= ( ( A2 = bot_bot_a_o )
& ( B2 = bot_bot_a_o ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_232_sup__bot_Oeq__neutr__iff,axiom,
! [A2: set_a,B2: set_a] :
( ( ( sup_sup_set_a @ A2 @ B2 )
= bot_bot_set_a )
= ( ( A2 = bot_bot_set_a )
& ( B2 = bot_bot_set_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_233_sup__eq__bot__iff,axiom,
! [X3: a > $o,Y: a > $o] :
( ( ( sup_sup_a_o @ X3 @ Y )
= bot_bot_a_o )
= ( ( X3 = bot_bot_a_o )
& ( Y = bot_bot_a_o ) ) ) ).
% sup_eq_bot_iff
thf(fact_234_sup__eq__bot__iff,axiom,
! [X3: set_a,Y: set_a] :
( ( ( sup_sup_set_a @ X3 @ Y )
= bot_bot_set_a )
= ( ( X3 = bot_bot_set_a )
& ( Y = bot_bot_set_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_235_minus__apply,axiom,
( minus_minus_a_o
= ( ^ [A5: a > $o,B6: a > $o,X4: a] : ( minus_minus_o @ ( A5 @ X4 ) @ ( B6 @ X4 ) ) ) ) ).
% minus_apply
thf(fact_236_sup__apply,axiom,
( sup_sup_a_o
= ( ^ [F2: a > $o,G: a > $o,X4: a] : ( sup_sup_o @ ( F2 @ X4 ) @ ( G @ X4 ) ) ) ) ).
% sup_apply
thf(fact_237_sup_Oidem,axiom,
! [A2: int] :
( ( sup_sup_int @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_238_sup_Oidem,axiom,
! [A2: a > $o] :
( ( sup_sup_a_o @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_239_sup_Oidem,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_240_sup__idem,axiom,
! [X3: int] :
( ( sup_sup_int @ X3 @ X3 )
= X3 ) ).
% sup_idem
thf(fact_241_sup__idem,axiom,
! [X3: a > $o] :
( ( sup_sup_a_o @ X3 @ X3 )
= X3 ) ).
% sup_idem
thf(fact_242_sup__idem,axiom,
! [X3: set_a] :
( ( sup_sup_set_a @ X3 @ X3 )
= X3 ) ).
% sup_idem
thf(fact_243_sup_Oleft__idem,axiom,
! [A2: int,B2: int] :
( ( sup_sup_int @ A2 @ ( sup_sup_int @ A2 @ B2 ) )
= ( sup_sup_int @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_244_sup_Oleft__idem,axiom,
! [A2: a > $o,B2: a > $o] :
( ( sup_sup_a_o @ A2 @ ( sup_sup_a_o @ A2 @ B2 ) )
= ( sup_sup_a_o @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_245_sup_Oleft__idem,axiom,
! [A2: set_a,B2: set_a] :
( ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B2 ) )
= ( sup_sup_set_a @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_246_sup__left__idem,axiom,
! [X3: int,Y: int] :
( ( sup_sup_int @ X3 @ ( sup_sup_int @ X3 @ Y ) )
= ( sup_sup_int @ X3 @ Y ) ) ).
% sup_left_idem
thf(fact_247_sup__left__idem,axiom,
! [X3: a > $o,Y: a > $o] :
( ( sup_sup_a_o @ X3 @ ( sup_sup_a_o @ X3 @ Y ) )
= ( sup_sup_a_o @ X3 @ Y ) ) ).
% sup_left_idem
thf(fact_248_sup__left__idem,axiom,
! [X3: set_a,Y: set_a] :
( ( sup_sup_set_a @ X3 @ ( sup_sup_set_a @ X3 @ Y ) )
= ( sup_sup_set_a @ X3 @ Y ) ) ).
% sup_left_idem
thf(fact_249_sup_Oright__idem,axiom,
! [A2: int,B2: int] :
( ( sup_sup_int @ ( sup_sup_int @ A2 @ B2 ) @ B2 )
= ( sup_sup_int @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_250_sup_Oright__idem,axiom,
! [A2: a > $o,B2: a > $o] :
( ( sup_sup_a_o @ ( sup_sup_a_o @ A2 @ B2 ) @ B2 )
= ( sup_sup_a_o @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_251_sup_Oright__idem,axiom,
! [A2: set_a,B2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ B2 )
= ( sup_sup_set_a @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_252_sup__bot__left,axiom,
! [X3: a > $o] :
( ( sup_sup_a_o @ bot_bot_a_o @ X3 )
= X3 ) ).
% sup_bot_left
thf(fact_253_sup__bot__left,axiom,
! [X3: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ X3 )
= X3 ) ).
% sup_bot_left
thf(fact_254_sup__bot__right,axiom,
! [X3: a > $o] :
( ( sup_sup_a_o @ X3 @ bot_bot_a_o )
= X3 ) ).
% sup_bot_right
thf(fact_255_sup__bot__right,axiom,
! [X3: set_a] :
( ( sup_sup_set_a @ X3 @ bot_bot_set_a )
= X3 ) ).
% sup_bot_right
thf(fact_256_bot__eq__sup__iff,axiom,
! [X3: a > $o,Y: a > $o] :
( ( bot_bot_a_o
= ( sup_sup_a_o @ X3 @ Y ) )
= ( ( X3 = bot_bot_a_o )
& ( Y = bot_bot_a_o ) ) ) ).
% bot_eq_sup_iff
thf(fact_257_bot__eq__sup__iff,axiom,
! [X3: set_a,Y: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ X3 @ Y ) )
= ( ( X3 = bot_bot_set_a )
& ( Y = bot_bot_set_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_258_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_259_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_260_fun__diff__def,axiom,
( minus_minus_a_o
= ( ^ [A5: a > $o,B6: a > $o,X4: a] : ( minus_minus_o @ ( A5 @ X4 ) @ ( B6 @ X4 ) ) ) ) ).
% fun_diff_def
thf(fact_261_inf__sup__aci_I8_J,axiom,
! [X3: int,Y: int] :
( ( sup_sup_int @ X3 @ ( sup_sup_int @ X3 @ Y ) )
= ( sup_sup_int @ X3 @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_262_inf__sup__aci_I8_J,axiom,
! [X3: a > $o,Y: a > $o] :
( ( sup_sup_a_o @ X3 @ ( sup_sup_a_o @ X3 @ Y ) )
= ( sup_sup_a_o @ X3 @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_263_inf__sup__aci_I8_J,axiom,
! [X3: set_a,Y: set_a] :
( ( sup_sup_set_a @ X3 @ ( sup_sup_set_a @ X3 @ Y ) )
= ( sup_sup_set_a @ X3 @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_264_inf__sup__aci_I7_J,axiom,
! [X3: int,Y: int,Z3: int] :
( ( sup_sup_int @ X3 @ ( sup_sup_int @ Y @ Z3 ) )
= ( sup_sup_int @ Y @ ( sup_sup_int @ X3 @ Z3 ) ) ) ).
% inf_sup_aci(7)
thf(fact_265_inf__sup__aci_I7_J,axiom,
! [X3: a > $o,Y: a > $o,Z3: a > $o] :
( ( sup_sup_a_o @ X3 @ ( sup_sup_a_o @ Y @ Z3 ) )
= ( sup_sup_a_o @ Y @ ( sup_sup_a_o @ X3 @ Z3 ) ) ) ).
% inf_sup_aci(7)
thf(fact_266_inf__sup__aci_I7_J,axiom,
! [X3: set_a,Y: set_a,Z3: set_a] :
( ( sup_sup_set_a @ X3 @ ( sup_sup_set_a @ Y @ Z3 ) )
= ( sup_sup_set_a @ Y @ ( sup_sup_set_a @ X3 @ Z3 ) ) ) ).
% inf_sup_aci(7)
thf(fact_267_inf__sup__aci_I6_J,axiom,
! [X3: int,Y: int,Z3: int] :
( ( sup_sup_int @ ( sup_sup_int @ X3 @ Y ) @ Z3 )
= ( sup_sup_int @ X3 @ ( sup_sup_int @ Y @ Z3 ) ) ) ).
% inf_sup_aci(6)
thf(fact_268_inf__sup__aci_I6_J,axiom,
! [X3: a > $o,Y: a > $o,Z3: a > $o] :
( ( sup_sup_a_o @ ( sup_sup_a_o @ X3 @ Y ) @ Z3 )
= ( sup_sup_a_o @ X3 @ ( sup_sup_a_o @ Y @ Z3 ) ) ) ).
% inf_sup_aci(6)
thf(fact_269_inf__sup__aci_I6_J,axiom,
! [X3: set_a,Y: set_a,Z3: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X3 @ Y ) @ Z3 )
= ( sup_sup_set_a @ X3 @ ( sup_sup_set_a @ Y @ Z3 ) ) ) ).
% inf_sup_aci(6)
thf(fact_270_inf__sup__aci_I5_J,axiom,
( sup_sup_int
= ( ^ [X4: int,Y3: int] : ( sup_sup_int @ Y3 @ X4 ) ) ) ).
% inf_sup_aci(5)
thf(fact_271_inf__sup__aci_I5_J,axiom,
( sup_sup_a_o
= ( ^ [X4: a > $o,Y3: a > $o] : ( sup_sup_a_o @ Y3 @ X4 ) ) ) ).
% inf_sup_aci(5)
thf(fact_272_inf__sup__aci_I5_J,axiom,
( sup_sup_set_a
= ( ^ [X4: set_a,Y3: set_a] : ( sup_sup_set_a @ Y3 @ X4 ) ) ) ).
% inf_sup_aci(5)
thf(fact_273_sup__fun__def,axiom,
( sup_sup_a_o
= ( ^ [F2: a > $o,G: a > $o,X4: a] : ( sup_sup_o @ ( F2 @ X4 ) @ ( G @ X4 ) ) ) ) ).
% sup_fun_def
thf(fact_274_boolean__algebra__cancel_Osup1,axiom,
! [A3: int,K: int,A2: int,B2: int] :
( ( A3
= ( sup_sup_int @ K @ A2 ) )
=> ( ( sup_sup_int @ A3 @ B2 )
= ( sup_sup_int @ K @ ( sup_sup_int @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_275_boolean__algebra__cancel_Osup1,axiom,
! [A3: a > $o,K: a > $o,A2: a > $o,B2: a > $o] :
( ( A3
= ( sup_sup_a_o @ K @ A2 ) )
=> ( ( sup_sup_a_o @ A3 @ B2 )
= ( sup_sup_a_o @ K @ ( sup_sup_a_o @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_276_boolean__algebra__cancel_Osup1,axiom,
! [A3: set_a,K: set_a,A2: set_a,B2: set_a] :
( ( A3
= ( sup_sup_set_a @ K @ A2 ) )
=> ( ( sup_sup_set_a @ A3 @ B2 )
= ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_277_boolean__algebra__cancel_Osup2,axiom,
! [B4: int,K: int,B2: int,A2: int] :
( ( B4
= ( sup_sup_int @ K @ B2 ) )
=> ( ( sup_sup_int @ A2 @ B4 )
= ( sup_sup_int @ K @ ( sup_sup_int @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_278_boolean__algebra__cancel_Osup2,axiom,
! [B4: a > $o,K: a > $o,B2: a > $o,A2: a > $o] :
( ( B4
= ( sup_sup_a_o @ K @ B2 ) )
=> ( ( sup_sup_a_o @ A2 @ B4 )
= ( sup_sup_a_o @ K @ ( sup_sup_a_o @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_279_boolean__algebra__cancel_Osup2,axiom,
! [B4: set_a,K: set_a,B2: set_a,A2: set_a] :
( ( B4
= ( sup_sup_set_a @ K @ B2 ) )
=> ( ( sup_sup_set_a @ A2 @ B4 )
= ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_280_sup_Oassoc,axiom,
! [A2: int,B2: int,C: int] :
( ( sup_sup_int @ ( sup_sup_int @ A2 @ B2 ) @ C )
= ( sup_sup_int @ A2 @ ( sup_sup_int @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_281_sup_Oassoc,axiom,
! [A2: a > $o,B2: a > $o,C: a > $o] :
( ( sup_sup_a_o @ ( sup_sup_a_o @ A2 @ B2 ) @ C )
= ( sup_sup_a_o @ A2 @ ( sup_sup_a_o @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_282_sup_Oassoc,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ C )
= ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_283_sup__assoc,axiom,
! [X3: int,Y: int,Z3: int] :
( ( sup_sup_int @ ( sup_sup_int @ X3 @ Y ) @ Z3 )
= ( sup_sup_int @ X3 @ ( sup_sup_int @ Y @ Z3 ) ) ) ).
% sup_assoc
thf(fact_284_sup__assoc,axiom,
! [X3: a > $o,Y: a > $o,Z3: a > $o] :
( ( sup_sup_a_o @ ( sup_sup_a_o @ X3 @ Y ) @ Z3 )
= ( sup_sup_a_o @ X3 @ ( sup_sup_a_o @ Y @ Z3 ) ) ) ).
% sup_assoc
thf(fact_285_sup__assoc,axiom,
! [X3: set_a,Y: set_a,Z3: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X3 @ Y ) @ Z3 )
= ( sup_sup_set_a @ X3 @ ( sup_sup_set_a @ Y @ Z3 ) ) ) ).
% sup_assoc
thf(fact_286_sup_Ocommute,axiom,
( sup_sup_int
= ( ^ [A: int,B: int] : ( sup_sup_int @ B @ A ) ) ) ).
% sup.commute
thf(fact_287_sup_Ocommute,axiom,
( sup_sup_a_o
= ( ^ [A: a > $o,B: a > $o] : ( sup_sup_a_o @ B @ A ) ) ) ).
% sup.commute
thf(fact_288_sup_Ocommute,axiom,
( sup_sup_set_a
= ( ^ [A: set_a,B: set_a] : ( sup_sup_set_a @ B @ A ) ) ) ).
% sup.commute
thf(fact_289_sup__commute,axiom,
( sup_sup_int
= ( ^ [X4: int,Y3: int] : ( sup_sup_int @ Y3 @ X4 ) ) ) ).
% sup_commute
thf(fact_290_sup__commute,axiom,
( sup_sup_a_o
= ( ^ [X4: a > $o,Y3: a > $o] : ( sup_sup_a_o @ Y3 @ X4 ) ) ) ).
% sup_commute
thf(fact_291_sup__commute,axiom,
( sup_sup_set_a
= ( ^ [X4: set_a,Y3: set_a] : ( sup_sup_set_a @ Y3 @ X4 ) ) ) ).
% sup_commute
thf(fact_292_sup_Oleft__commute,axiom,
! [B2: set_a,A2: set_a,C: set_a] :
( ( sup_sup_set_a @ B2 @ ( sup_sup_set_a @ A2 @ C ) )
= ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_293_sup__left__commute,axiom,
! [X3: set_a,Y: set_a,Z3: set_a] :
( ( sup_sup_set_a @ X3 @ ( sup_sup_set_a @ Y @ Z3 ) )
= ( sup_sup_set_a @ Y @ ( sup_sup_set_a @ X3 @ Z3 ) ) ) ).
% sup_left_commute
thf(fact_294_less__supI1,axiom,
! [X3: set_a,A2: set_a,B2: set_a] :
( ( ord_less_set_a @ X3 @ A2 )
=> ( ord_less_set_a @ X3 @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% less_supI1
thf(fact_295_less__supI1,axiom,
! [X3: int,A2: int,B2: int] :
( ( ord_less_int @ X3 @ A2 )
=> ( ord_less_int @ X3 @ ( sup_sup_int @ A2 @ B2 ) ) ) ).
% less_supI1
thf(fact_296_less__supI2,axiom,
! [X3: set_a,B2: set_a,A2: set_a] :
( ( ord_less_set_a @ X3 @ B2 )
=> ( ord_less_set_a @ X3 @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% less_supI2
thf(fact_297_less__supI2,axiom,
! [X3: int,B2: int,A2: int] :
( ( ord_less_int @ X3 @ B2 )
=> ( ord_less_int @ X3 @ ( sup_sup_int @ A2 @ B2 ) ) ) ).
% less_supI2
thf(fact_298_sup_Ostrict__boundedE,axiom,
! [B2: set_a,C: set_a,A2: set_a] :
( ( ord_less_set_a @ ( sup_sup_set_a @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_set_a @ B2 @ A2 )
=> ~ ( ord_less_set_a @ C @ A2 ) ) ) ).
% sup.strict_boundedE
thf(fact_299_sup_Ostrict__boundedE,axiom,
! [B2: int,C: int,A2: int] :
( ( ord_less_int @ ( sup_sup_int @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_int @ B2 @ A2 )
=> ~ ( ord_less_int @ C @ A2 ) ) ) ).
% sup.strict_boundedE
thf(fact_300_sup_Ostrict__order__iff,axiom,
( ord_less_set_a
= ( ^ [B: set_a,A: set_a] :
( ( A
= ( sup_sup_set_a @ A @ B ) )
& ( A != B ) ) ) ) ).
% sup.strict_order_iff
thf(fact_301_sup_Ostrict__order__iff,axiom,
( ord_less_int
= ( ^ [B: int,A: int] :
( ( A
= ( sup_sup_int @ A @ B ) )
& ( A != B ) ) ) ) ).
% sup.strict_order_iff
thf(fact_302_sup_Ostrict__coboundedI1,axiom,
! [C: set_a,A2: set_a,B2: set_a] :
( ( ord_less_set_a @ C @ A2 )
=> ( ord_less_set_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_303_sup_Ostrict__coboundedI1,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_int @ C @ A2 )
=> ( ord_less_int @ C @ ( sup_sup_int @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_304_sup_Ostrict__coboundedI2,axiom,
! [C: set_a,B2: set_a,A2: set_a] :
( ( ord_less_set_a @ C @ B2 )
=> ( ord_less_set_a @ C @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_305_sup_Ostrict__coboundedI2,axiom,
! [C: int,B2: int,A2: int] :
( ( ord_less_int @ C @ B2 )
=> ( ord_less_int @ C @ ( sup_sup_int @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_306_h2,axiom,
( ( binary1721989714Tree_a @ h @ t2 )
=> ( ( binary945792244etOf_a @ ( binary1226383794sert_a @ h @ e @ t2 ) )
= ( sup_sup_set_a @ ( minus_minus_set_a @ ( binary945792244etOf_a @ t2 ) @ ( binary504661350_eqs_a @ h @ e ) ) @ ( insert_a @ e @ bot_bot_set_a ) ) ) ) ).
% h2
thf(fact_307_h1,axiom,
( ( binary1721989714Tree_a @ h @ t1 )
=> ( ( binary945792244etOf_a @ ( binary1226383794sert_a @ h @ e @ t1 ) )
= ( sup_sup_set_a @ ( minus_minus_set_a @ ( binary945792244etOf_a @ t1 ) @ ( binary504661350_eqs_a @ h @ e ) ) @ ( insert_a @ e @ bot_bot_set_a ) ) ) ) ).
% h1
thf(fact_308__092_060open_062sortedTree_Ah_ATip_A_092_060longrightarrow_062_AsetOf_A_Ibinsert_Ah_Ae_ATip_J_A_061_AsetOf_ATip_A_N_Aeqs_Ah_Ae_A_092_060union_062_A_123e_125_092_060close_062,axiom,
( ( binary1721989714Tree_a @ h @ binary476621312_Tip_a )
=> ( ( binary945792244etOf_a @ ( binary1226383794sert_a @ h @ e @ binary476621312_Tip_a ) )
= ( sup_sup_set_a @ ( minus_minus_set_a @ ( binary945792244etOf_a @ binary476621312_Tip_a ) @ ( binary504661350_eqs_a @ h @ e ) ) @ ( insert_a @ e @ bot_bot_set_a ) ) ) ) ).
% \<open>sortedTree h Tip \<longrightarrow> setOf (binsert h e Tip) = setOf Tip - eqs h e \<union> {e}\<close>
thf(fact_309_singleton__conv,axiom,
! [A2: a] :
( ( collect_a
@ ^ [X4: a] : X4 = A2 )
= ( insert_a @ A2 @ bot_bot_set_a ) ) ).
% singleton_conv
thf(fact_310_singleton__conv2,axiom,
! [A2: a] :
( ( collect_a
@ ( ^ [Y4: a,Z4: a] : Y4 = Z4
@ A2 ) )
= ( insert_a @ A2 @ bot_bot_set_a ) ) ).
% singleton_conv2
thf(fact_311_Un__def,axiom,
( sup_sup_set_a
= ( ^ [A5: set_a,B6: set_a] :
( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ A5 )
| ( member_a @ X4 @ B6 ) ) ) ) ) ).
% Un_def
thf(fact_312_empty__def,axiom,
( bot_bot_set_a
= ( collect_a
@ ^ [X4: a] : $false ) ) ).
% empty_def
thf(fact_313_insert__def,axiom,
( insert_a
= ( ^ [A: a] :
( sup_sup_set_a
@ ( collect_a
@ ^ [X4: a] : X4 = A ) ) ) ) ).
% insert_def
thf(fact_314_sup__set__def,axiom,
( sup_sup_set_a
= ( ^ [A5: set_a,B6: set_a] :
( collect_a
@ ( sup_sup_a_o
@ ^ [X4: a] : ( member_a @ X4 @ A5 )
@ ^ [X4: a] : ( member_a @ X4 @ B6 ) ) ) ) ) ).
% sup_set_def
thf(fact_315_Collect__conv__if,axiom,
! [P: a > $o,A2: a] :
( ( ( P @ A2 )
=> ( ( collect_a
@ ^ [X4: a] :
( ( X4 = A2 )
& ( P @ X4 ) ) )
= ( insert_a @ A2 @ bot_bot_set_a ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect_a
@ ^ [X4: a] :
( ( X4 = A2 )
& ( P @ X4 ) ) )
= bot_bot_set_a ) ) ) ).
% Collect_conv_if
thf(fact_316_Collect__disj__eq,axiom,
! [P: a > $o,Q: a > $o] :
( ( collect_a
@ ^ [X4: a] :
( ( P @ X4 )
| ( Q @ X4 ) ) )
= ( sup_sup_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_317_Collect__conv__if2,axiom,
! [P: a > $o,A2: a] :
( ( ( P @ A2 )
=> ( ( collect_a
@ ^ [X4: a] :
( ( A2 = X4 )
& ( P @ X4 ) ) )
= ( insert_a @ A2 @ bot_bot_set_a ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect_a
@ ^ [X4: a] :
( ( A2 = X4 )
& ( P @ X4 ) ) )
= bot_bot_set_a ) ) ) ).
% Collect_conv_if2
thf(fact_318_less__set__def,axiom,
( ord_less_set_a
= ( ^ [A5: set_a,B6: set_a] :
( ord_less_a_o
@ ^ [X4: a] : ( member_a @ X4 @ A5 )
@ ^ [X4: a] : ( member_a @ X4 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_319_eqs__def,axiom,
( binary504661350_eqs_a
= ( ^ [H2: a > int,X4: a] :
( collect_a
@ ^ [Y3: a] :
( ( H2 @ Y3 )
= ( H2 @ X4 ) ) ) ) ) ).
% eqs_def
thf(fact_320_insert__Collect,axiom,
! [A2: a,P: a > $o] :
( ( insert_a @ A2 @ ( collect_a @ P ) )
= ( collect_a
@ ^ [U: a] :
( ( U != A2 )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_321_minus__set__def,axiom,
( minus_minus_set_a
= ( ^ [A5: set_a,B6: set_a] :
( collect_a
@ ( minus_minus_a_o
@ ^ [X4: a] : ( member_a @ X4 @ A5 )
@ ^ [X4: a] : ( member_a @ X4 @ B6 ) ) ) ) ) ).
% minus_set_def
thf(fact_322_insert__compr,axiom,
( insert_a
= ( ^ [A: a,B6: set_a] :
( collect_a
@ ^ [X4: a] :
( ( X4 = A )
| ( member_a @ X4 @ B6 ) ) ) ) ) ).
% insert_compr
thf(fact_323_set__diff__eq,axiom,
( minus_minus_set_a
= ( ^ [A5: set_a,B6: set_a] :
( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ A5 )
& ~ ( member_a @ X4 @ B6 ) ) ) ) ) ).
% set_diff_eq
thf(fact_324_the__elem__eq,axiom,
! [X3: a] :
( ( the_elem_a @ ( insert_a @ X3 @ bot_bot_set_a ) )
= X3 ) ).
% the_elem_eq
thf(fact_325_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X4: a] : ( member_a @ X4 @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_326_sup__Un__eq,axiom,
! [R: set_a,S: set_a] :
( ( sup_sup_a_o
@ ^ [X4: a] : ( member_a @ X4 @ R )
@ ^ [X4: a] : ( member_a @ X4 @ S ) )
= ( ^ [X4: a] : ( member_a @ X4 @ ( sup_sup_set_a @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_327_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_328_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_329_is__singletonI,axiom,
! [X3: a] : ( is_singleton_a @ ( insert_a @ X3 @ bot_bot_set_a ) ) ).
% is_singletonI
thf(fact_330_is__singletonI_H,axiom,
! [A3: set_a] :
( ( A3 != bot_bot_set_a )
=> ( ! [X2: a,Y2: a] :
( ( member_a @ X2 @ A3 )
=> ( ( member_a @ Y2 @ A3 )
=> ( X2 = Y2 ) ) )
=> ( is_singleton_a @ A3 ) ) ) ).
% is_singletonI'
thf(fact_331_is__singleton__def,axiom,
( is_singleton_a
= ( ^ [A5: set_a] :
? [X4: a] :
( A5
= ( insert_a @ X4 @ bot_bot_set_a ) ) ) ) ).
% is_singleton_def
thf(fact_332_is__singletonE,axiom,
! [A3: set_a] :
( ( is_singleton_a @ A3 )
=> ~ ! [X2: a] :
( A3
!= ( insert_a @ X2 @ bot_bot_set_a ) ) ) ).
% is_singletonE
thf(fact_333_diff__strict__right__mono,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ord_less_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B2 @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_334_diff__strict__left__mono,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( ord_less_int @ ( minus_minus_int @ C @ A2 ) @ ( minus_minus_int @ C @ B2 ) ) ) ).
% diff_strict_left_mono
thf(fact_335_diff__strict__mono,axiom,
! [A2: int,B2: int,D: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ D @ C )
=> ( ord_less_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B2 @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_336_diff__eq__diff__less,axiom,
! [A2: int,B2: int,C: int,D: int] :
( ( ( minus_minus_int @ A2 @ B2 )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_int @ A2 @ B2 )
= ( ord_less_int @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_337_remove__def,axiom,
( remove_a
= ( ^ [X4: a,A5: set_a] : ( minus_minus_set_a @ A5 @ ( insert_a @ X4 @ bot_bot_set_a ) ) ) ) ).
% remove_def
thf(fact_338_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_339_member__remove,axiom,
! [X3: a,Y: a,A3: set_a] :
( ( member_a @ X3 @ ( remove_a @ Y @ A3 ) )
= ( ( member_a @ X3 @ A3 )
& ( X3 != Y ) ) ) ).
% member_remove
thf(fact_340_Tree_Osimps_I14_J,axiom,
( ( binary256242811Tree_a @ binary476621312_Tip_a )
= bot_bot_set_a ) ).
% Tree.simps(14)
thf(fact_341_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_342_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_343_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_344_Tree_Oset__cases,axiom,
! [E: a,A2: binary1439146945Tree_a] :
( ( member_a @ E @ ( binary256242811Tree_a @ A2 ) )
=> ( ! [Z1: binary1439146945Tree_a] :
( ? [Z22: a,Z32: binary1439146945Tree_a] :
( A2
= ( binary717961607le_T_a @ Z1 @ Z22 @ Z32 ) )
=> ~ ( member_a @ E @ ( binary256242811Tree_a @ Z1 ) ) )
=> ( ! [Z1: binary1439146945Tree_a,Z32: binary1439146945Tree_a] :
( A2
!= ( binary717961607le_T_a @ Z1 @ E @ Z32 ) )
=> ~ ! [Z1: binary1439146945Tree_a,Z22: a,Z32: binary1439146945Tree_a] :
( ( A2
= ( binary717961607le_T_a @ Z1 @ Z22 @ Z32 ) )
=> ~ ( member_a @ E @ ( binary256242811Tree_a @ Z32 ) ) ) ) ) ) ).
% Tree.set_cases
thf(fact_345_Tree_Opred__cong,axiom,
! [X3: binary1439146945Tree_a,Ya: binary1439146945Tree_a,P: a > $o,Pa: a > $o] :
( ( X3 = Ya )
=> ( ! [Z: a] :
( ( member_a @ Z @ ( binary256242811Tree_a @ Ya ) )
=> ( ( P @ Z )
= ( Pa @ Z ) ) )
=> ( ( binary1452917696Tree_a @ P @ X3 )
= ( binary1452917696Tree_a @ Pa @ Ya ) ) ) ) ).
% Tree.pred_cong
thf(fact_346_Tree_Opred__mono__strong,axiom,
! [P: a > $o,X3: binary1439146945Tree_a,Pa: a > $o] :
( ( binary1452917696Tree_a @ P @ X3 )
=> ( ! [Z: a] :
( ( member_a @ Z @ ( binary256242811Tree_a @ X3 ) )
=> ( ( P @ Z )
=> ( Pa @ Z ) ) )
=> ( binary1452917696Tree_a @ Pa @ X3 ) ) ) ).
% Tree.pred_mono_strong
thf(fact_347_Set_Ois__empty__def,axiom,
( is_empty_a
= ( ^ [A5: set_a] : A5 = bot_bot_set_a ) ) ).
% Set.is_empty_def
thf(fact_348_the__elem__def,axiom,
( the_elem_a
= ( ^ [X5: set_a] :
( the_a
@ ^ [X4: a] :
( X5
= ( insert_a @ X4 @ bot_bot_set_a ) ) ) ) ) ).
% the_elem_def
thf(fact_349_psubset__insert__iff,axiom,
! [A3: set_a,X3: a,B4: set_a] :
( ( ord_less_set_a @ A3 @ ( insert_a @ X3 @ B4 ) )
= ( ( ( member_a @ X3 @ B4 )
=> ( ord_less_set_a @ A3 @ B4 ) )
& ( ~ ( member_a @ X3 @ B4 )
=> ( ( ( member_a @ X3 @ A3 )
=> ( ord_less_set_a @ ( minus_minus_set_a @ A3 @ ( insert_a @ X3 @ bot_bot_set_a ) ) @ B4 ) )
& ( ~ ( member_a @ X3 @ A3 )
=> ( ord_less_eq_set_a @ A3 @ B4 ) ) ) ) ) ) ).
% psubset_insert_iff
% Conjectures (1)
thf(conj_0,conjecture,
ord_less_int @ ( h @ w ) @ ( h @ x ) ).
%------------------------------------------------------------------------------