TPTP Problem File: ITP034^1.p
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%------------------------------------------------------------------------------
% File : ITP034^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer BinaryTree problem prob_562__3255854_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_562__3255854_1 [Des21]
% Status : Theorem
% Rating : 0.30 v8.2.0, 0.15 v8.1.0, 0.18 v7.5.0
% Syntax : Number of formulae : 389 ( 192 unt; 36 typ; 0 def)
% Number of atoms : 955 ( 354 equ; 0 cnn)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 2500 ( 83 ~; 18 |; 75 &;2043 @)
% ( 0 <=>; 281 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 314 ( 314 >; 0 *; 0 +; 0 <<)
% Number of symbols : 34 ( 32 usr; 8 con; 0-4 aty)
% Number of variables : 944 ( 112 ^; 801 !; 31 ?; 944 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:31:47.772
%------------------------------------------------------------------------------
% 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 (32)
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_Obinsert_001tf__a,type,
binary1226383794sert_a: ( a > int ) > a > binary1439146945Tree_a > binary1439146945Tree_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_Oremove_001tf__a,type,
binary1804682569move_a: ( a > int ) > a > binary1439146945Tree_a > binary1439146945Tree_a ).
thf(sy_c_BinaryTree__Mirabelle__mlzyzwgbkd_Orm_001tf__a,type,
binary339557810e_rm_a: ( a > int ) > binary1439146945Tree_a > a ).
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_BinaryTree__Mirabelle__mlzyzwgbkd_Owrm_001tf__a,type,
binary1217730267_wrm_a: ( a > int ) > binary1439146945Tree_a > binary1439146945Tree_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_001_062_Itf__a_M_Eo_J,type,
ord_less_eq_a_o: ( a > $o ) > ( a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $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_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_h,type,
h: a > int ).
thf(sy_v_l____,type,
l: a ).
thf(sy_v_t1____,type,
t1: binary1439146945Tree_a ).
thf(sy_v_t2____,type,
t2: binary1439146945Tree_a ).
thf(sy_v_x____,type,
x: a ).
% Relevant facts (352)
thf(fact_0_hLess,axiom,
ord_less_int @ ( h @ l ) @ ( h @ ( binary339557810e_rm_a @ h @ t2 ) ) ).
% hLess
thf(fact_1_rm__res,axiom,
( ( binary339557810e_rm_a @ h @ ( binary717961607le_T_a @ t1 @ x @ t2 ) )
= ( binary339557810e_rm_a @ h @ t2 ) ) ).
% rm_res
thf(fact_2_t2nTip,axiom,
t2 != binary476621312_Tip_a ).
% t2nTip
thf(fact_3_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_4_s,axiom,
binary1721989714Tree_a @ h @ ( binary717961607le_T_a @ t1 @ x @ t2 ) ).
% s
thf(fact_5_wrm__res,axiom,
( ( binary1217730267_wrm_a @ h @ ( binary717961607le_T_a @ t1 @ x @ t2 ) )
= ( binary717961607le_T_a @ t1 @ x @ ( binary1217730267_wrm_a @ h @ t2 ) ) ) ).
% wrm_res
thf(fact_6_ldef,axiom,
member_a @ l @ ( binary945792244etOf_a @ ( binary1217730267_wrm_a @ h @ ( binary717961607le_T_a @ t1 @ x @ t2 ) ) ) ).
% ldef
thf(fact_7_s1,axiom,
binary1721989714Tree_a @ h @ t1 ).
% s1
thf(fact_8_s2,axiom,
binary1721989714Tree_a @ h @ t2 ).
% s2
thf(fact_9_binsert_Osimps_I2_J,axiom,
! [H: a > int,E: a,X: a,T1: binary1439146945Tree_a,T2: binary1439146945Tree_a] :
( ( ( ord_less_int @ ( H @ E ) @ ( H @ X ) )
=> ( ( binary1226383794sert_a @ H @ E @ ( binary717961607le_T_a @ T1 @ X @ T2 ) )
= ( binary717961607le_T_a @ ( binary1226383794sert_a @ H @ E @ T1 ) @ X @ T2 ) ) )
& ( ~ ( ord_less_int @ ( H @ E ) @ ( H @ X ) )
=> ( ( ( ord_less_int @ ( H @ X ) @ ( H @ E ) )
=> ( ( binary1226383794sert_a @ H @ E @ ( binary717961607le_T_a @ T1 @ X @ T2 ) )
= ( binary717961607le_T_a @ T1 @ X @ ( binary1226383794sert_a @ H @ E @ T2 ) ) ) )
& ( ~ ( ord_less_int @ ( H @ X ) @ ( H @ E ) )
=> ( ( binary1226383794sert_a @ H @ E @ ( binary717961607le_T_a @ T1 @ X @ T2 ) )
= ( binary717961607le_T_a @ T1 @ E @ T2 ) ) ) ) ) ) ).
% binsert.simps(2)
thf(fact_10_rm_Osimps,axiom,
! [T2: binary1439146945Tree_a,H: a > int,T1: binary1439146945Tree_a,X: a] :
( ( ( T2 = binary476621312_Tip_a )
=> ( ( binary339557810e_rm_a @ H @ ( binary717961607le_T_a @ T1 @ X @ T2 ) )
= X ) )
& ( ( T2 != binary476621312_Tip_a )
=> ( ( binary339557810e_rm_a @ H @ ( binary717961607le_T_a @ T1 @ X @ T2 ) )
= ( binary339557810e_rm_a @ H @ T2 ) ) ) ) ).
% rm.simps
thf(fact_11_minf_I7_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z )
=> ~ ( ord_less_int @ T @ X2 ) ) ).
% minf(7)
thf(fact_12_minf_I5_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z )
=> ( ord_less_int @ X2 @ T ) ) ).
% minf(5)
thf(fact_13_minf_I4_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z )
=> ( X2 != T ) ) ).
% minf(4)
thf(fact_14_minf_I3_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z )
=> ( X2 != T ) ) ).
% minf(3)
thf(fact_15_minf_I2_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z2: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z2 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z2: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z2 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P2 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(2)
thf(fact_16_sortedTree_Osimps_I1_J,axiom,
! [H: a > int] : ( binary1721989714Tree_a @ H @ binary476621312_Tip_a ) ).
% sortedTree.simps(1)
thf(fact_17_wrm__sort,axiom,
! [T: binary1439146945Tree_a,H: a > int] :
( ( ( T != binary476621312_Tip_a )
& ( binary1721989714Tree_a @ H @ T ) )
=> ( binary1721989714Tree_a @ H @ ( binary1217730267_wrm_a @ H @ T ) ) ) ).
% wrm_sort
thf(fact_18_binsert__sorted,axiom,
! [H: a > int,T: binary1439146945Tree_a,X: a] :
( ( binary1721989714Tree_a @ H @ T )
=> ( binary1721989714Tree_a @ H @ ( binary1226383794sert_a @ H @ X @ T ) ) ) ).
% binsert_sorted
thf(fact_19_rm__set,axiom,
! [T: binary1439146945Tree_a,H: a > int] :
( ( ( T != binary476621312_Tip_a )
& ( binary1721989714Tree_a @ H @ T ) )
=> ( member_a @ ( binary339557810e_rm_a @ H @ T ) @ ( binary945792244etOf_a @ T ) ) ) ).
% rm_set
thf(fact_20_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_21_wrm_Osimps,axiom,
! [T2: binary1439146945Tree_a,H: a > int,T1: binary1439146945Tree_a,X: a] :
( ( ( T2 = binary476621312_Tip_a )
=> ( ( binary1217730267_wrm_a @ H @ ( binary717961607le_T_a @ T1 @ X @ T2 ) )
= T1 ) )
& ( ( T2 != binary476621312_Tip_a )
=> ( ( binary1217730267_wrm_a @ H @ ( binary717961607le_T_a @ T1 @ X @ T2 ) )
= ( binary717961607le_T_a @ T1 @ X @ ( binary1217730267_wrm_a @ H @ T2 ) ) ) ) ) ).
% wrm.simps
thf(fact_22_sortedTree_Osimps_I2_J,axiom,
! [H: a > int,T1: binary1439146945Tree_a,X: a,T2: binary1439146945Tree_a] :
( ( binary1721989714Tree_a @ H @ ( binary717961607le_T_a @ T1 @ X @ T2 ) )
= ( ( binary1721989714Tree_a @ H @ T1 )
& ! [X4: a] :
( ( member_a @ X4 @ ( binary945792244etOf_a @ T1 ) )
=> ( ord_less_int @ ( H @ X4 ) @ ( H @ X ) ) )
& ! [X4: a] :
( ( member_a @ X4 @ ( binary945792244etOf_a @ T2 ) )
=> ( ord_less_int @ ( H @ X ) @ ( H @ X4 ) ) )
& ( binary1721989714Tree_a @ H @ T2 ) ) ) ).
% sortedTree.simps(2)
thf(fact_23_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_24_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_25_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_26_sortLemmaR,axiom,
! [H: a > int,T1: binary1439146945Tree_a,X: a,T2: binary1439146945Tree_a] :
( ( binary1721989714Tree_a @ H @ ( binary717961607le_T_a @ T1 @ X @ T2 ) )
=> ( binary1721989714Tree_a @ H @ T2 ) ) ).
% sortLemmaR
thf(fact_27_sortLemmaL,axiom,
! [H: a > int,T1: binary1439146945Tree_a,X: a,T2: binary1439146945Tree_a] :
( ( binary1721989714Tree_a @ H @ ( binary717961607le_T_a @ T1 @ X @ T2 ) )
=> ( binary1721989714Tree_a @ H @ T1 ) ) ).
% sortLemmaL
thf(fact_28_pinf_I1_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z2: int] :
! [X3: int] :
( ( ord_less_int @ Z2 @ X3 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z2: int] :
! [X3: int] :
( ( ord_less_int @ Z2 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: int] :
! [X2: int] :
( ( ord_less_int @ Z @ X2 )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P2 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_29_pinf_I2_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z2: int] :
! [X3: int] :
( ( ord_less_int @ Z2 @ X3 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z2: int] :
! [X3: int] :
( ( ord_less_int @ Z2 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: int] :
! [X2: int] :
( ( ord_less_int @ Z @ X2 )
=> ( ( ( P @ X2 )
| ( Q @ X2 ) )
= ( ( P2 @ X2 )
| ( Q2 @ X2 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_30_pinf_I3_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ Z @ X2 )
=> ( X2 != T ) ) ).
% pinf(3)
thf(fact_31_pinf_I4_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ Z @ X2 )
=> ( X2 != T ) ) ).
% pinf(4)
thf(fact_32_pinf_I5_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ Z @ X2 )
=> ~ ( ord_less_int @ X2 @ T ) ) ).
% pinf(5)
thf(fact_33_pinf_I7_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ Z @ X2 )
=> ( ord_less_int @ T @ X2 ) ) ).
% pinf(7)
thf(fact_34_minf_I1_J,axiom,
! [P: int > $o,P2: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z2: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z2 )
=> ( ( P @ X3 )
= ( P2 @ X3 ) ) )
=> ( ? [Z2: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z2 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z )
=> ( ( ( P @ X2 )
& ( Q @ X2 ) )
= ( ( P2 @ X2 )
& ( Q2 @ X2 ) ) ) ) ) ) ).
% minf(1)
thf(fact_35_h1,axiom,
( ( ( t1 != binary476621312_Tip_a )
& ( binary1721989714Tree_a @ h @ t1 ) )
=> ! [X2: a] :
( ( member_a @ X2 @ ( binary945792244etOf_a @ ( binary1217730267_wrm_a @ h @ t1 ) ) )
=> ( ord_less_int @ ( h @ X2 ) @ ( h @ ( binary339557810e_rm_a @ h @ t1 ) ) ) ) ) ).
% h1
thf(fact_36_h2,axiom,
( ( ( t2 != binary476621312_Tip_a )
& ( binary1721989714Tree_a @ h @ t2 ) )
=> ! [X2: a] :
( ( member_a @ X2 @ ( binary945792244etOf_a @ ( binary1217730267_wrm_a @ h @ t2 ) ) )
=> ( ord_less_int @ ( h @ X2 ) @ ( h @ ( binary339557810e_rm_a @ h @ t2 ) ) ) ) ) ).
% h2
thf(fact_37_memb__spec,axiom,
! [H: a > int,T: binary1439146945Tree_a,X: a] :
( ( binary1721989714Tree_a @ H @ T )
=> ( ( binary2053421120memb_a @ H @ X @ T )
= ( member_a @ X @ ( binary945792244etOf_a @ T ) ) ) ) ).
% memb_spec
thf(fact_38_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_39_l__scope,axiom,
member_a @ l @ ( sup_sup_set_a @ ( sup_sup_set_a @ ( insert_a @ x @ bot_bot_set_a ) @ ( binary945792244etOf_a @ t1 ) ) @ ( binary945792244etOf_a @ ( binary1217730267_wrm_a @ h @ t2 ) ) ) ).
% l_scope
thf(fact_40_wrm__set1,axiom,
! [T: binary1439146945Tree_a,H: a > int] :
( ( ( T != binary476621312_Tip_a )
& ( binary1721989714Tree_a @ H @ T ) )
=> ( ord_less_eq_set_a @ ( binary945792244etOf_a @ ( binary1217730267_wrm_a @ H @ T ) ) @ ( binary945792244etOf_a @ T ) ) ) ).
% wrm_set1
thf(fact_41_remove_Osimps_I1_J,axiom,
! [H: a > int,E: a] :
( ( binary1804682569move_a @ H @ E @ binary476621312_Tip_a )
= binary476621312_Tip_a ) ).
% remove.simps(1)
thf(fact_42_mem__Collect__eq,axiom,
! [A2: a,P: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_43_Collect__mem__eq,axiom,
! [A3: set_a] :
( ( collect_a
@ ^ [X4: a] : ( member_a @ X4 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_44_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X3: a] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_45_linorder__neqE__linordered__idom,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_46_complete__interval,axiom,
! [A2: int,B2: int,P: int > $o] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( P @ A2 )
=> ( ~ ( P @ B2 )
=> ? [C: int] :
( ( ord_less_eq_int @ A2 @ C )
& ( ord_less_eq_int @ C @ B2 )
& ! [X2: int] :
( ( ( ord_less_eq_int @ A2 @ X2 )
& ( ord_less_int @ X2 @ C ) )
=> ( P @ X2 ) )
& ! [D: int] :
( ! [X3: int] :
( ( ( ord_less_eq_int @ A2 @ X3 )
& ( ord_less_int @ X3 @ D ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_int @ D @ C ) ) ) ) ) ) ).
% complete_interval
thf(fact_47_setOf_Osimps_I2_J,axiom,
! [T1: binary1439146945Tree_a,X: a,T2: binary1439146945Tree_a] :
( ( binary945792244etOf_a @ ( binary717961607le_T_a @ T1 @ X @ T2 ) )
= ( sup_sup_set_a @ ( sup_sup_set_a @ ( binary945792244etOf_a @ T1 ) @ ( binary945792244etOf_a @ T2 ) ) @ ( insert_a @ X @ bot_bot_set_a ) ) ) ).
% setOf.simps(2)
thf(fact_48_minf_I8_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z )
=> ~ ( ord_less_eq_int @ T @ X2 ) ) ).
% minf(8)
thf(fact_49_minf_I6_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ X2 @ Z )
=> ( ord_less_eq_int @ X2 @ T ) ) ).
% minf(6)
thf(fact_50_pinf_I8_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ Z @ X2 )
=> ( ord_less_eq_int @ T @ X2 ) ) ).
% pinf(8)
thf(fact_51_pinf_I6_J,axiom,
! [T: int] :
? [Z: int] :
! [X2: int] :
( ( ord_less_int @ Z @ X2 )
=> ~ ( ord_less_eq_int @ X2 @ T ) ) ).
% pinf(6)
thf(fact_52_sorted__distinct,axiom,
! [H: a > int,A2: a,B2: a,T: binary1439146945Tree_a] : ( binary670562003pred_a @ H @ A2 @ B2 @ T ) ).
% sorted_distinct
thf(fact_53_setOf_Osimps_I1_J,axiom,
( ( binary945792244etOf_a @ binary476621312_Tip_a )
= bot_bot_set_a ) ).
% setOf.simps(1)
thf(fact_54_singleton__insert__inj__eq_H,axiom,
! [A2: a,A3: set_a,B2: a] :
( ( ( insert_a @ A2 @ A3 )
= ( insert_a @ B2 @ bot_bot_set_a ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_a @ A3 @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_55_singleton__insert__inj__eq,axiom,
! [B2: a,A2: a,A3: set_a] :
( ( ( insert_a @ B2 @ bot_bot_set_a )
= ( insert_a @ A2 @ A3 ) )
= ( ( A2 = B2 )
& ( ord_less_eq_set_a @ A3 @ ( insert_a @ B2 @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_56_singleton__conv2,axiom,
! [A2: a] :
( ( collect_a
@ ( ^ [Y2: a,Z3: a] : Y2 = Z3
@ A2 ) )
= ( insert_a @ A2 @ bot_bot_set_a ) ) ).
% singleton_conv2
thf(fact_57_singleton__conv,axiom,
! [A2: a] :
( ( collect_a
@ ^ [X4: a] : X4 = A2 )
= ( insert_a @ A2 @ bot_bot_set_a ) ) ).
% singleton_conv
thf(fact_58_Un__insert__right,axiom,
! [A3: set_a,A2: a,B3: set_a] :
( ( sup_sup_set_a @ A3 @ ( insert_a @ A2 @ B3 ) )
= ( insert_a @ A2 @ ( sup_sup_set_a @ A3 @ B3 ) ) ) ).
% Un_insert_right
thf(fact_59_Un__insert__left,axiom,
! [A2: a,B3: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( insert_a @ A2 @ B3 ) @ C2 )
= ( insert_a @ A2 @ ( sup_sup_set_a @ B3 @ C2 ) ) ) ).
% Un_insert_left
thf(fact_60_Un__subset__iff,axiom,
! [A3: set_a,B3: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A3 @ B3 ) @ C2 )
= ( ( ord_less_eq_set_a @ A3 @ C2 )
& ( ord_less_eq_set_a @ B3 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_61_Un__empty,axiom,
! [A3: set_a,B3: set_a] :
( ( ( sup_sup_set_a @ A3 @ B3 )
= bot_bot_set_a )
= ( ( A3 = bot_bot_set_a )
& ( B3 = bot_bot_set_a ) ) ) ).
% Un_empty
thf(fact_62_empty__iff,axiom,
! [C3: a] :
~ ( member_a @ C3 @ bot_bot_set_a ) ).
% empty_iff
thf(fact_63_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_64_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X4: a] :
~ ( P @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_65_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X4: a] :
~ ( P @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_66_subsetI,axiom,
! [A3: set_a,B3: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A3 )
=> ( member_a @ X3 @ B3 ) )
=> ( ord_less_eq_set_a @ A3 @ B3 ) ) ).
% subsetI
thf(fact_67_subset__antisym,axiom,
! [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% subset_antisym
thf(fact_68_psubsetI,axiom,
! [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( A3 != B3 )
=> ( ord_less_set_a @ A3 @ B3 ) ) ) ).
% psubsetI
thf(fact_69_insertCI,axiom,
! [A2: a,B3: set_a,B2: a] :
( ( ~ ( member_a @ A2 @ B3 )
=> ( A2 = B2 ) )
=> ( member_a @ A2 @ ( insert_a @ B2 @ B3 ) ) ) ).
% insertCI
thf(fact_70_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_71_insert__absorb2,axiom,
! [X: a,A3: set_a] :
( ( insert_a @ X @ ( insert_a @ X @ A3 ) )
= ( insert_a @ X @ A3 ) ) ).
% insert_absorb2
thf(fact_72_UnCI,axiom,
! [C3: a,B3: set_a,A3: set_a] :
( ( ~ ( member_a @ C3 @ B3 )
=> ( member_a @ C3 @ A3 ) )
=> ( member_a @ C3 @ ( sup_sup_set_a @ A3 @ B3 ) ) ) ).
% UnCI
thf(fact_73_Un__iff,axiom,
! [C3: a,A3: set_a,B3: set_a] :
( ( member_a @ C3 @ ( sup_sup_set_a @ A3 @ B3 ) )
= ( ( member_a @ C3 @ A3 )
| ( member_a @ C3 @ B3 ) ) ) ).
% Un_iff
thf(fact_74_subset__empty,axiom,
! [A3: set_a] :
( ( ord_less_eq_set_a @ A3 @ bot_bot_set_a )
= ( A3 = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_75_empty__subsetI,axiom,
! [A3: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A3 ) ).
% empty_subsetI
thf(fact_76_singletonI,axiom,
! [A2: a] : ( member_a @ A2 @ ( insert_a @ A2 @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_77_insert__subset,axiom,
! [X: a,A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X @ A3 ) @ B3 )
= ( ( member_a @ X @ B3 )
& ( ord_less_eq_set_a @ A3 @ B3 ) ) ) ).
% insert_subset
thf(fact_78_less__eq__set__def,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ord_less_eq_a_o
@ ^ [X4: a] : ( member_a @ X4 @ A4 )
@ ^ [X4: a] : ( member_a @ X4 @ B4 ) ) ) ) ).
% less_eq_set_def
thf(fact_79_emptyE,axiom,
! [A2: a] :
~ ( member_a @ A2 @ bot_bot_set_a ) ).
% emptyE
thf(fact_80_equals0D,axiom,
! [A3: set_a,A2: a] :
( ( A3 = bot_bot_set_a )
=> ~ ( member_a @ A2 @ A3 ) ) ).
% equals0D
thf(fact_81_equals0I,axiom,
! [A3: set_a] :
( ! [Y3: a] :
~ ( member_a @ Y3 @ A3 )
=> ( A3 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_82_ex__in__conv,axiom,
! [A3: set_a] :
( ( ? [X4: a] : ( member_a @ X4 @ A3 ) )
= ( A3 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_83_not__psubset__empty,axiom,
! [A3: set_a] :
~ ( ord_less_set_a @ A3 @ bot_bot_set_a ) ).
% not_psubset_empty
thf(fact_84_in__mono,axiom,
! [A3: set_a,B3: set_a,X: a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( member_a @ X @ A3 )
=> ( member_a @ X @ B3 ) ) ) ).
% in_mono
thf(fact_85_subsetD,axiom,
! [A3: set_a,B3: set_a,C3: a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( member_a @ C3 @ A3 )
=> ( member_a @ C3 @ B3 ) ) ) ).
% subsetD
thf(fact_86_equalityE,axiom,
! [A3: set_a,B3: set_a] :
( ( A3 = B3 )
=> ~ ( ( ord_less_eq_set_a @ A3 @ B3 )
=> ~ ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ).
% equalityE
thf(fact_87_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
! [X4: a] :
( ( member_a @ X4 @ A4 )
=> ( member_a @ X4 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_88_equalityD1,axiom,
! [A3: set_a,B3: set_a] :
( ( A3 = B3 )
=> ( ord_less_eq_set_a @ A3 @ B3 ) ) ).
% equalityD1
thf(fact_89_equalityD2,axiom,
! [A3: set_a,B3: set_a] :
( ( A3 = B3 )
=> ( ord_less_eq_set_a @ B3 @ A3 ) ) ).
% equalityD2
thf(fact_90_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
! [T3: a] :
( ( member_a @ T3 @ A4 )
=> ( member_a @ T3 @ B4 ) ) ) ) ).
% subset_iff
thf(fact_91_subset__refl,axiom,
! [A3: set_a] : ( ord_less_eq_set_a @ A3 @ A3 ) ).
% subset_refl
thf(fact_92_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X3: a] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_93_subset__trans,axiom,
! [A3: set_a,B3: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ C2 )
=> ( ord_less_eq_set_a @ A3 @ C2 ) ) ) ).
% subset_trans
thf(fact_94_set__eq__subset,axiom,
( ( ^ [Y2: set_a,Z3: set_a] : Y2 = Z3 )
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
& ( ord_less_eq_set_a @ B4 @ A4 ) ) ) ) ).
% set_eq_subset
thf(fact_95_Collect__mono__iff,axiom,
! [P: a > $o,Q: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) )
= ( ! [X4: a] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_96_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_set_a @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_97_subset__psubset__trans,axiom,
! [A3: set_a,B3: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( ord_less_set_a @ B3 @ C2 )
=> ( ord_less_set_a @ A3 @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_98_subset__not__subset__eq,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
& ~ ( ord_less_eq_set_a @ B4 @ A4 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_99_psubset__subset__trans,axiom,
! [A3: set_a,B3: set_a,C2: set_a] :
( ( ord_less_set_a @ A3 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ C2 )
=> ( ord_less_set_a @ A3 @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_100_psubset__imp__subset,axiom,
! [A3: set_a,B3: set_a] :
( ( ord_less_set_a @ A3 @ B3 )
=> ( ord_less_eq_set_a @ A3 @ B3 ) ) ).
% psubset_imp_subset
thf(fact_101_psubset__eq,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% psubset_eq
thf(fact_102_psubsetE,axiom,
! [A3: set_a,B3: set_a] :
( ( ord_less_set_a @ A3 @ B3 )
=> ~ ( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ord_less_eq_set_a @ B3 @ A3 ) ) ) ).
% psubsetE
thf(fact_103_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_104_insertI1,axiom,
! [A2: a,B3: set_a] : ( member_a @ A2 @ ( insert_a @ A2 @ B3 ) ) ).
% insertI1
thf(fact_105_insertI2,axiom,
! [A2: a,B3: set_a,B2: a] :
( ( member_a @ A2 @ B3 )
=> ( member_a @ A2 @ ( insert_a @ B2 @ B3 ) ) ) ).
% insertI2
thf(fact_106_Set_Oset__insert,axiom,
! [X: a,A3: set_a] :
( ( member_a @ X @ A3 )
=> ~ ! [B5: set_a] :
( ( A3
= ( insert_a @ X @ B5 ) )
=> ( member_a @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_107_insert__ident,axiom,
! [X: a,A3: set_a,B3: set_a] :
( ~ ( member_a @ X @ A3 )
=> ( ~ ( member_a @ X @ B3 )
=> ( ( ( insert_a @ X @ A3 )
= ( insert_a @ X @ B3 ) )
= ( A3 = B3 ) ) ) ) ).
% insert_ident
thf(fact_108_insert__absorb,axiom,
! [A2: a,A3: set_a] :
( ( member_a @ A2 @ A3 )
=> ( ( insert_a @ A2 @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_109_insert__eq__iff,axiom,
! [A2: a,A3: set_a,B2: a,B3: set_a] :
( ~ ( member_a @ A2 @ A3 )
=> ( ~ ( member_a @ B2 @ B3 )
=> ( ( ( insert_a @ A2 @ A3 )
= ( insert_a @ B2 @ B3 ) )
= ( ( ( A2 = B2 )
=> ( A3 = B3 ) )
& ( ( A2 != B2 )
=> ? [C4: set_a] :
( ( A3
= ( insert_a @ B2 @ C4 ) )
& ~ ( member_a @ B2 @ C4 )
& ( B3
= ( insert_a @ A2 @ C4 ) )
& ~ ( member_a @ A2 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_110_insert__commute,axiom,
! [X: a,Y: a,A3: set_a] :
( ( insert_a @ X @ ( insert_a @ Y @ A3 ) )
= ( insert_a @ Y @ ( insert_a @ X @ A3 ) ) ) ).
% insert_commute
thf(fact_111_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_112_UnE,axiom,
! [C3: a,A3: set_a,B3: set_a] :
( ( member_a @ C3 @ ( sup_sup_set_a @ A3 @ B3 ) )
=> ( ~ ( member_a @ C3 @ A3 )
=> ( member_a @ C3 @ B3 ) ) ) ).
% UnE
thf(fact_113_UnI1,axiom,
! [C3: a,A3: set_a,B3: set_a] :
( ( member_a @ C3 @ A3 )
=> ( member_a @ C3 @ ( sup_sup_set_a @ A3 @ B3 ) ) ) ).
% UnI1
thf(fact_114_UnI2,axiom,
! [C3: a,B3: set_a,A3: set_a] :
( ( member_a @ C3 @ B3 )
=> ( member_a @ C3 @ ( sup_sup_set_a @ A3 @ B3 ) ) ) ).
% UnI2
thf(fact_115_bex__Un,axiom,
! [A3: set_a,B3: set_a,P: a > $o] :
( ( ? [X4: a] :
( ( member_a @ X4 @ ( sup_sup_set_a @ A3 @ B3 ) )
& ( P @ X4 ) ) )
= ( ? [X4: a] :
( ( member_a @ X4 @ A3 )
& ( P @ X4 ) )
| ? [X4: a] :
( ( member_a @ X4 @ B3 )
& ( P @ X4 ) ) ) ) ).
% bex_Un
thf(fact_116_ball__Un,axiom,
! [A3: set_a,B3: set_a,P: a > $o] :
( ( ! [X4: a] :
( ( member_a @ X4 @ ( sup_sup_set_a @ A3 @ B3 ) )
=> ( P @ X4 ) ) )
= ( ! [X4: a] :
( ( member_a @ X4 @ A3 )
=> ( P @ X4 ) )
& ! [X4: a] :
( ( member_a @ X4 @ B3 )
=> ( P @ X4 ) ) ) ) ).
% ball_Un
thf(fact_117_Un__assoc,axiom,
! [A3: set_a,B3: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A3 @ B3 ) @ C2 )
= ( sup_sup_set_a @ A3 @ ( sup_sup_set_a @ B3 @ C2 ) ) ) ).
% Un_assoc
thf(fact_118_Un__absorb,axiom,
! [A3: set_a] :
( ( sup_sup_set_a @ A3 @ A3 )
= A3 ) ).
% Un_absorb
thf(fact_119_Un__commute,axiom,
( sup_sup_set_a
= ( ^ [A4: set_a,B4: set_a] : ( sup_sup_set_a @ B4 @ A4 ) ) ) ).
% Un_commute
thf(fact_120_Un__left__absorb,axiom,
! [A3: set_a,B3: set_a] :
( ( sup_sup_set_a @ A3 @ ( sup_sup_set_a @ A3 @ B3 ) )
= ( sup_sup_set_a @ A3 @ B3 ) ) ).
% Un_left_absorb
thf(fact_121_Un__left__commute,axiom,
! [A3: set_a,B3: set_a,C2: set_a] :
( ( sup_sup_set_a @ A3 @ ( sup_sup_set_a @ B3 @ C2 ) )
= ( sup_sup_set_a @ B3 @ ( sup_sup_set_a @ A3 @ C2 ) ) ) ).
% Un_left_commute
thf(fact_122_empty__def,axiom,
( bot_bot_set_a
= ( collect_a
@ ^ [X4: a] : $false ) ) ).
% empty_def
thf(fact_123_Collect__subset,axiom,
! [A3: set_a,P: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ A3 )
& ( P @ X4 ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_124_insert__compr,axiom,
( insert_a
= ( ^ [A: a,B4: set_a] :
( collect_a
@ ^ [X4: a] :
( ( X4 = A )
| ( member_a @ X4 @ B4 ) ) ) ) ) ).
% insert_compr
thf(fact_125_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_126_Un__def,axiom,
( sup_sup_set_a
= ( ^ [A4: set_a,B4: set_a] :
( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ A4 )
| ( member_a @ X4 @ B4 ) ) ) ) ) ).
% Un_def
thf(fact_127_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_128_singletonD,axiom,
! [B2: a,A2: a] :
( ( member_a @ B2 @ ( insert_a @ A2 @ bot_bot_set_a ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_129_singleton__iff,axiom,
! [B2: a,A2: a] :
( ( member_a @ B2 @ ( insert_a @ A2 @ bot_bot_set_a ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_130_doubleton__eq__iff,axiom,
! [A2: a,B2: a,C3: a,D2: a] :
( ( ( insert_a @ A2 @ ( insert_a @ B2 @ bot_bot_set_a ) )
= ( insert_a @ C3 @ ( insert_a @ D2 @ bot_bot_set_a ) ) )
= ( ( ( A2 = C3 )
& ( B2 = D2 ) )
| ( ( A2 = D2 )
& ( B2 = C3 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_131_insert__not__empty,axiom,
! [A2: a,A3: set_a] :
( ( insert_a @ A2 @ A3 )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_132_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_133_insert__mono,axiom,
! [C2: set_a,D3: set_a,A2: a] :
( ( ord_less_eq_set_a @ C2 @ D3 )
=> ( ord_less_eq_set_a @ ( insert_a @ A2 @ C2 ) @ ( insert_a @ A2 @ D3 ) ) ) ).
% insert_mono
thf(fact_134_subset__insert,axiom,
! [X: a,A3: set_a,B3: set_a] :
( ~ ( member_a @ X @ A3 )
=> ( ( ord_less_eq_set_a @ A3 @ ( insert_a @ X @ B3 ) )
= ( ord_less_eq_set_a @ A3 @ B3 ) ) ) ).
% subset_insert
thf(fact_135_subset__insertI,axiom,
! [B3: set_a,A2: a] : ( ord_less_eq_set_a @ B3 @ ( insert_a @ A2 @ B3 ) ) ).
% subset_insertI
thf(fact_136_subset__insertI2,axiom,
! [A3: set_a,B3: set_a,B2: a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ord_less_eq_set_a @ A3 @ ( insert_a @ B2 @ B3 ) ) ) ).
% subset_insertI2
thf(fact_137_Un__empty__left,axiom,
! [B3: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ B3 )
= B3 ) ).
% Un_empty_left
thf(fact_138_Un__empty__right,axiom,
! [A3: set_a] :
( ( sup_sup_set_a @ A3 @ bot_bot_set_a )
= A3 ) ).
% Un_empty_right
thf(fact_139_Un__mono,axiom,
! [A3: set_a,C2: set_a,B3: set_a,D3: set_a] :
( ( ord_less_eq_set_a @ A3 @ C2 )
=> ( ( ord_less_eq_set_a @ B3 @ D3 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A3 @ B3 ) @ ( sup_sup_set_a @ C2 @ D3 ) ) ) ) ).
% Un_mono
thf(fact_140_Un__least,axiom,
! [A3: set_a,C2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ C2 )
=> ( ( ord_less_eq_set_a @ B3 @ C2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A3 @ B3 ) @ C2 ) ) ) ).
% Un_least
thf(fact_141_Un__upper1,axiom,
! [A3: set_a,B3: set_a] : ( ord_less_eq_set_a @ A3 @ ( sup_sup_set_a @ A3 @ B3 ) ) ).
% Un_upper1
thf(fact_142_Un__upper2,axiom,
! [B3: set_a,A3: set_a] : ( ord_less_eq_set_a @ B3 @ ( sup_sup_set_a @ A3 @ B3 ) ) ).
% Un_upper2
thf(fact_143_Un__absorb1,axiom,
! [A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( sup_sup_set_a @ A3 @ B3 )
= B3 ) ) ).
% Un_absorb1
thf(fact_144_Un__absorb2,axiom,
! [B3: set_a,A3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A3 )
=> ( ( sup_sup_set_a @ A3 @ B3 )
= A3 ) ) ).
% Un_absorb2
thf(fact_145_subset__UnE,axiom,
! [C2: set_a,A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ C2 @ ( sup_sup_set_a @ A3 @ B3 ) )
=> ~ ! [A5: set_a] :
( ( ord_less_eq_set_a @ A5 @ A3 )
=> ! [B6: set_a] :
( ( ord_less_eq_set_a @ B6 @ B3 )
=> ( C2
!= ( sup_sup_set_a @ A5 @ B6 ) ) ) ) ) ).
% subset_UnE
thf(fact_146_subset__Un__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ( sup_sup_set_a @ A4 @ B4 )
= B4 ) ) ) ).
% subset_Un_eq
thf(fact_147_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_148_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_149_insert__def,axiom,
( insert_a
= ( ^ [A: a] :
( sup_sup_set_a
@ ( collect_a
@ ^ [X4: a] : X4 = A ) ) ) ) ).
% insert_def
thf(fact_150_subset__singletonD,axiom,
! [A3: set_a,X: a] :
( ( ord_less_eq_set_a @ A3 @ ( insert_a @ X @ bot_bot_set_a ) )
=> ( ( A3 = bot_bot_set_a )
| ( A3
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ).
% subset_singletonD
thf(fact_151_subset__singleton__iff,axiom,
! [X5: set_a,A2: a] :
( ( ord_less_eq_set_a @ X5 @ ( insert_a @ A2 @ bot_bot_set_a ) )
= ( ( X5 = bot_bot_set_a )
| ( X5
= ( insert_a @ A2 @ bot_bot_set_a ) ) ) ) ).
% subset_singleton_iff
thf(fact_152_insert__is__Un,axiom,
( insert_a
= ( ^ [A: a] : ( sup_sup_set_a @ ( insert_a @ A @ bot_bot_set_a ) ) ) ) ).
% insert_is_Un
thf(fact_153_Un__singleton__iff,axiom,
! [A3: set_a,B3: set_a,X: a] :
( ( ( sup_sup_set_a @ A3 @ B3 )
= ( insert_a @ X @ bot_bot_set_a ) )
= ( ( ( A3 = bot_bot_set_a )
& ( B3
= ( insert_a @ X @ bot_bot_set_a ) ) )
| ( ( A3
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B3 = bot_bot_set_a ) )
| ( ( A3
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B3
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_154_singleton__Un__iff,axiom,
! [X: a,A3: set_a,B3: set_a] :
( ( ( insert_a @ X @ bot_bot_set_a )
= ( sup_sup_set_a @ A3 @ B3 ) )
= ( ( ( A3 = bot_bot_set_a )
& ( B3
= ( insert_a @ X @ bot_bot_set_a ) ) )
| ( ( A3
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B3 = bot_bot_set_a ) )
| ( ( A3
= ( insert_a @ X @ bot_bot_set_a ) )
& ( B3
= ( insert_a @ X @ bot_bot_set_a ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_155_sup__bot__left,axiom,
! [X: a > $o] :
( ( sup_sup_a_o @ bot_bot_a_o @ X )
= X ) ).
% sup_bot_left
thf(fact_156_sup__bot__left,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ X )
= X ) ).
% sup_bot_left
thf(fact_157_sup__bot__right,axiom,
! [X: a > $o] :
( ( sup_sup_a_o @ X @ bot_bot_a_o )
= X ) ).
% sup_bot_right
thf(fact_158_sup__bot__right,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ bot_bot_set_a )
= X ) ).
% sup_bot_right
thf(fact_159_bot__eq__sup__iff,axiom,
! [X: a > $o,Y: a > $o] :
( ( bot_bot_a_o
= ( sup_sup_a_o @ X @ Y ) )
= ( ( X = bot_bot_a_o )
& ( Y = bot_bot_a_o ) ) ) ).
% bot_eq_sup_iff
thf(fact_160_bot__eq__sup__iff,axiom,
! [X: set_a,Y: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ X @ Y ) )
= ( ( X = bot_bot_set_a )
& ( Y = bot_bot_set_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_161_sup__eq__bot__iff,axiom,
! [X: a > $o,Y: a > $o] :
( ( ( sup_sup_a_o @ X @ Y )
= bot_bot_a_o )
= ( ( X = bot_bot_a_o )
& ( Y = bot_bot_a_o ) ) ) ).
% sup_eq_bot_iff
thf(fact_162_sup__eq__bot__iff,axiom,
! [X: set_a,Y: set_a] :
( ( ( sup_sup_set_a @ X @ Y )
= bot_bot_set_a )
= ( ( X = bot_bot_set_a )
& ( Y = bot_bot_set_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_163_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_164_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_165_sup__bot_Oleft__neutral,axiom,
! [A2: a > $o] :
( ( sup_sup_a_o @ bot_bot_a_o @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_166_sup__bot_Oleft__neutral,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_167_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_168_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_169_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_170_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_171_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_172_sup__left__idem,axiom,
! [X: a > $o,Y: a > $o] :
( ( sup_sup_a_o @ X @ ( sup_sup_a_o @ X @ Y ) )
= ( sup_sup_a_o @ X @ Y ) ) ).
% sup_left_idem
thf(fact_173_sup__left__idem,axiom,
! [X: int,Y: int] :
( ( sup_sup_int @ X @ ( sup_sup_int @ X @ Y ) )
= ( sup_sup_int @ X @ Y ) ) ).
% sup_left_idem
thf(fact_174_sup__left__idem,axiom,
! [X: set_a,Y: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ X @ Y ) )
= ( sup_sup_set_a @ X @ Y ) ) ).
% sup_left_idem
thf(fact_175_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_176_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_177_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_178_sup__idem,axiom,
! [X: a > $o] :
( ( sup_sup_a_o @ X @ X )
= X ) ).
% sup_idem
thf(fact_179_sup__idem,axiom,
! [X: int] :
( ( sup_sup_int @ X @ X )
= X ) ).
% sup_idem
thf(fact_180_sup__idem,axiom,
! [X: set_a] :
( ( sup_sup_set_a @ X @ X )
= X ) ).
% sup_idem
thf(fact_181_sup_Oidem,axiom,
! [A2: a > $o] :
( ( sup_sup_a_o @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_182_sup_Oidem,axiom,
! [A2: int] :
( ( sup_sup_int @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_183_sup_Oidem,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_184_sup__apply,axiom,
( sup_sup_a_o
= ( ^ [F: a > $o,G: a > $o,X4: a] : ( sup_sup_o @ ( F @ X4 ) @ ( G @ X4 ) ) ) ) ).
% sup_apply
thf(fact_185_sup_Obounded__iff,axiom,
! [B2: int,C3: int,A2: int] :
( ( ord_less_eq_int @ ( sup_sup_int @ B2 @ C3 ) @ A2 )
= ( ( ord_less_eq_int @ B2 @ A2 )
& ( ord_less_eq_int @ C3 @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_186_sup_Obounded__iff,axiom,
! [B2: a > $o,C3: a > $o,A2: a > $o] :
( ( ord_less_eq_a_o @ ( sup_sup_a_o @ B2 @ C3 ) @ A2 )
= ( ( ord_less_eq_a_o @ B2 @ A2 )
& ( ord_less_eq_a_o @ C3 @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_187_sup_Obounded__iff,axiom,
! [B2: set_a,C3: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C3 ) @ A2 )
= ( ( ord_less_eq_set_a @ B2 @ A2 )
& ( ord_less_eq_set_a @ C3 @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_188_le__sup__iff,axiom,
! [X: int,Y: int,Z4: int] :
( ( ord_less_eq_int @ ( sup_sup_int @ X @ Y ) @ Z4 )
= ( ( ord_less_eq_int @ X @ Z4 )
& ( ord_less_eq_int @ Y @ Z4 ) ) ) ).
% le_sup_iff
thf(fact_189_le__sup__iff,axiom,
! [X: a > $o,Y: a > $o,Z4: a > $o] :
( ( ord_less_eq_a_o @ ( sup_sup_a_o @ X @ Y ) @ Z4 )
= ( ( ord_less_eq_a_o @ X @ Z4 )
& ( ord_less_eq_a_o @ Y @ Z4 ) ) ) ).
% le_sup_iff
thf(fact_190_le__sup__iff,axiom,
! [X: set_a,Y: set_a,Z4: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z4 )
= ( ( ord_less_eq_set_a @ X @ Z4 )
& ( ord_less_eq_set_a @ Y @ Z4 ) ) ) ).
% le_sup_iff
thf(fact_191_sup__bot_Oright__neutral,axiom,
! [A2: a > $o] :
( ( sup_sup_a_o @ A2 @ bot_bot_a_o )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_192_sup__bot_Oright__neutral,axiom,
! [A2: set_a] :
( ( sup_sup_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_193_less__set__def,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B4: set_a] :
( ord_less_a_o
@ ^ [X4: a] : ( member_a @ X4 @ A4 )
@ ^ [X4: a] : ( member_a @ X4 @ B4 ) ) ) ) ).
% less_set_def
thf(fact_194_psubset__trans,axiom,
! [A3: set_a,B3: set_a,C2: set_a] :
( ( ord_less_set_a @ A3 @ B3 )
=> ( ( ord_less_set_a @ B3 @ C2 )
=> ( ord_less_set_a @ A3 @ C2 ) ) ) ).
% psubset_trans
thf(fact_195_psubsetD,axiom,
! [A3: set_a,B3: set_a,C3: a] :
( ( ord_less_set_a @ A3 @ B3 )
=> ( ( member_a @ C3 @ A3 )
=> ( member_a @ C3 @ B3 ) ) ) ).
% psubsetD
thf(fact_196_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_197_sup__set__def,axiom,
( sup_sup_set_a
= ( ^ [A4: set_a,B4: set_a] :
( collect_a
@ ( sup_sup_a_o
@ ^ [X4: a] : ( member_a @ X4 @ A4 )
@ ^ [X4: a] : ( member_a @ X4 @ B4 ) ) ) ) ) ).
% sup_set_def
thf(fact_198_sup__left__commute,axiom,
! [X: a > $o,Y: a > $o,Z4: a > $o] :
( ( sup_sup_a_o @ X @ ( sup_sup_a_o @ Y @ Z4 ) )
= ( sup_sup_a_o @ Y @ ( sup_sup_a_o @ X @ Z4 ) ) ) ).
% sup_left_commute
thf(fact_199_sup__left__commute,axiom,
! [X: int,Y: int,Z4: int] :
( ( sup_sup_int @ X @ ( sup_sup_int @ Y @ Z4 ) )
= ( sup_sup_int @ Y @ ( sup_sup_int @ X @ Z4 ) ) ) ).
% sup_left_commute
thf(fact_200_sup__left__commute,axiom,
! [X: set_a,Y: set_a,Z4: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z4 ) )
= ( sup_sup_set_a @ Y @ ( sup_sup_set_a @ X @ Z4 ) ) ) ).
% sup_left_commute
thf(fact_201_sup_Oleft__commute,axiom,
! [B2: a > $o,A2: a > $o,C3: a > $o] :
( ( sup_sup_a_o @ B2 @ ( sup_sup_a_o @ A2 @ C3 ) )
= ( sup_sup_a_o @ A2 @ ( sup_sup_a_o @ B2 @ C3 ) ) ) ).
% sup.left_commute
thf(fact_202_sup_Oleft__commute,axiom,
! [B2: int,A2: int,C3: int] :
( ( sup_sup_int @ B2 @ ( sup_sup_int @ A2 @ C3 ) )
= ( sup_sup_int @ A2 @ ( sup_sup_int @ B2 @ C3 ) ) ) ).
% sup.left_commute
thf(fact_203_sup_Oleft__commute,axiom,
! [B2: set_a,A2: set_a,C3: set_a] :
( ( sup_sup_set_a @ B2 @ ( sup_sup_set_a @ A2 @ C3 ) )
= ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B2 @ C3 ) ) ) ).
% sup.left_commute
thf(fact_204_sup__commute,axiom,
( sup_sup_a_o
= ( ^ [X4: a > $o,Y4: a > $o] : ( sup_sup_a_o @ Y4 @ X4 ) ) ) ).
% sup_commute
thf(fact_205_sup__commute,axiom,
( sup_sup_int
= ( ^ [X4: int,Y4: int] : ( sup_sup_int @ Y4 @ X4 ) ) ) ).
% sup_commute
thf(fact_206_sup__commute,axiom,
( sup_sup_set_a
= ( ^ [X4: set_a,Y4: set_a] : ( sup_sup_set_a @ Y4 @ X4 ) ) ) ).
% sup_commute
thf(fact_207_sup_Ocommute,axiom,
( sup_sup_a_o
= ( ^ [A: a > $o,B: a > $o] : ( sup_sup_a_o @ B @ A ) ) ) ).
% sup.commute
thf(fact_208_sup_Ocommute,axiom,
( sup_sup_int
= ( ^ [A: int,B: int] : ( sup_sup_int @ B @ A ) ) ) ).
% sup.commute
thf(fact_209_sup_Ocommute,axiom,
( sup_sup_set_a
= ( ^ [A: set_a,B: set_a] : ( sup_sup_set_a @ B @ A ) ) ) ).
% sup.commute
thf(fact_210_sup__assoc,axiom,
! [X: a > $o,Y: a > $o,Z4: a > $o] :
( ( sup_sup_a_o @ ( sup_sup_a_o @ X @ Y ) @ Z4 )
= ( sup_sup_a_o @ X @ ( sup_sup_a_o @ Y @ Z4 ) ) ) ).
% sup_assoc
thf(fact_211_sup__assoc,axiom,
! [X: int,Y: int,Z4: int] :
( ( sup_sup_int @ ( sup_sup_int @ X @ Y ) @ Z4 )
= ( sup_sup_int @ X @ ( sup_sup_int @ Y @ Z4 ) ) ) ).
% sup_assoc
thf(fact_212_sup__assoc,axiom,
! [X: set_a,Y: set_a,Z4: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z4 )
= ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z4 ) ) ) ).
% sup_assoc
thf(fact_213_sup_Oassoc,axiom,
! [A2: a > $o,B2: a > $o,C3: a > $o] :
( ( sup_sup_a_o @ ( sup_sup_a_o @ A2 @ B2 ) @ C3 )
= ( sup_sup_a_o @ A2 @ ( sup_sup_a_o @ B2 @ C3 ) ) ) ).
% sup.assoc
thf(fact_214_sup_Oassoc,axiom,
! [A2: int,B2: int,C3: int] :
( ( sup_sup_int @ ( sup_sup_int @ A2 @ B2 ) @ C3 )
= ( sup_sup_int @ A2 @ ( sup_sup_int @ B2 @ C3 ) ) ) ).
% sup.assoc
thf(fact_215_sup_Oassoc,axiom,
! [A2: set_a,B2: set_a,C3: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ C3 )
= ( sup_sup_set_a @ A2 @ ( sup_sup_set_a @ B2 @ C3 ) ) ) ).
% sup.assoc
thf(fact_216_boolean__algebra__cancel_Osup2,axiom,
! [B3: a > $o,K: a > $o,B2: a > $o,A2: a > $o] :
( ( B3
= ( sup_sup_a_o @ K @ B2 ) )
=> ( ( sup_sup_a_o @ A2 @ B3 )
= ( sup_sup_a_o @ K @ ( sup_sup_a_o @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_217_boolean__algebra__cancel_Osup2,axiom,
! [B3: int,K: int,B2: int,A2: int] :
( ( B3
= ( sup_sup_int @ K @ B2 ) )
=> ( ( sup_sup_int @ A2 @ B3 )
= ( sup_sup_int @ K @ ( sup_sup_int @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_218_boolean__algebra__cancel_Osup2,axiom,
! [B3: set_a,K: set_a,B2: set_a,A2: set_a] :
( ( B3
= ( sup_sup_set_a @ K @ B2 ) )
=> ( ( sup_sup_set_a @ A2 @ B3 )
= ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_219_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_220_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_221_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_222_sup__fun__def,axiom,
( sup_sup_a_o
= ( ^ [F: a > $o,G: a > $o,X4: a] : ( sup_sup_o @ ( F @ X4 ) @ ( G @ X4 ) ) ) ) ).
% sup_fun_def
thf(fact_223_inf__sup__aci_I5_J,axiom,
( sup_sup_a_o
= ( ^ [X4: a > $o,Y4: a > $o] : ( sup_sup_a_o @ Y4 @ X4 ) ) ) ).
% inf_sup_aci(5)
thf(fact_224_inf__sup__aci_I5_J,axiom,
( sup_sup_int
= ( ^ [X4: int,Y4: int] : ( sup_sup_int @ Y4 @ X4 ) ) ) ).
% inf_sup_aci(5)
thf(fact_225_inf__sup__aci_I5_J,axiom,
( sup_sup_set_a
= ( ^ [X4: set_a,Y4: set_a] : ( sup_sup_set_a @ Y4 @ X4 ) ) ) ).
% inf_sup_aci(5)
thf(fact_226_inf__sup__aci_I6_J,axiom,
! [X: a > $o,Y: a > $o,Z4: a > $o] :
( ( sup_sup_a_o @ ( sup_sup_a_o @ X @ Y ) @ Z4 )
= ( sup_sup_a_o @ X @ ( sup_sup_a_o @ Y @ Z4 ) ) ) ).
% inf_sup_aci(6)
thf(fact_227_inf__sup__aci_I6_J,axiom,
! [X: int,Y: int,Z4: int] :
( ( sup_sup_int @ ( sup_sup_int @ X @ Y ) @ Z4 )
= ( sup_sup_int @ X @ ( sup_sup_int @ Y @ Z4 ) ) ) ).
% inf_sup_aci(6)
thf(fact_228_inf__sup__aci_I6_J,axiom,
! [X: set_a,Y: set_a,Z4: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X @ Y ) @ Z4 )
= ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z4 ) ) ) ).
% inf_sup_aci(6)
thf(fact_229_inf__sup__aci_I7_J,axiom,
! [X: a > $o,Y: a > $o,Z4: a > $o] :
( ( sup_sup_a_o @ X @ ( sup_sup_a_o @ Y @ Z4 ) )
= ( sup_sup_a_o @ Y @ ( sup_sup_a_o @ X @ Z4 ) ) ) ).
% inf_sup_aci(7)
thf(fact_230_inf__sup__aci_I7_J,axiom,
! [X: int,Y: int,Z4: int] :
( ( sup_sup_int @ X @ ( sup_sup_int @ Y @ Z4 ) )
= ( sup_sup_int @ Y @ ( sup_sup_int @ X @ Z4 ) ) ) ).
% inf_sup_aci(7)
thf(fact_231_inf__sup__aci_I7_J,axiom,
! [X: set_a,Y: set_a,Z4: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ Y @ Z4 ) )
= ( sup_sup_set_a @ Y @ ( sup_sup_set_a @ X @ Z4 ) ) ) ).
% inf_sup_aci(7)
thf(fact_232_inf__sup__aci_I8_J,axiom,
! [X: a > $o,Y: a > $o] :
( ( sup_sup_a_o @ X @ ( sup_sup_a_o @ X @ Y ) )
= ( sup_sup_a_o @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_233_inf__sup__aci_I8_J,axiom,
! [X: int,Y: int] :
( ( sup_sup_int @ X @ ( sup_sup_int @ X @ Y ) )
= ( sup_sup_int @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_234_inf__sup__aci_I8_J,axiom,
! [X: set_a,Y: set_a] :
( ( sup_sup_set_a @ X @ ( sup_sup_set_a @ X @ Y ) )
= ( sup_sup_set_a @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_235_sup_OcoboundedI2,axiom,
! [C3: int,B2: int,A2: int] :
( ( ord_less_eq_int @ C3 @ B2 )
=> ( ord_less_eq_int @ C3 @ ( sup_sup_int @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_236_sup_OcoboundedI2,axiom,
! [C3: a > $o,B2: a > $o,A2: a > $o] :
( ( ord_less_eq_a_o @ C3 @ B2 )
=> ( ord_less_eq_a_o @ C3 @ ( sup_sup_a_o @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_237_sup_OcoboundedI2,axiom,
! [C3: set_a,B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ C3 @ B2 )
=> ( ord_less_eq_set_a @ C3 @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_238_sup_OcoboundedI1,axiom,
! [C3: int,A2: int,B2: int] :
( ( ord_less_eq_int @ C3 @ A2 )
=> ( ord_less_eq_int @ C3 @ ( sup_sup_int @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_239_sup_OcoboundedI1,axiom,
! [C3: a > $o,A2: a > $o,B2: a > $o] :
( ( ord_less_eq_a_o @ C3 @ A2 )
=> ( ord_less_eq_a_o @ C3 @ ( sup_sup_a_o @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_240_sup_OcoboundedI1,axiom,
! [C3: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C3 @ A2 )
=> ( ord_less_eq_set_a @ C3 @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_241_sup_Oabsorb__iff2,axiom,
( ord_less_eq_int
= ( ^ [A: int,B: int] :
( ( sup_sup_int @ A @ B )
= B ) ) ) ).
% sup.absorb_iff2
thf(fact_242_sup_Oabsorb__iff2,axiom,
( ord_less_eq_a_o
= ( ^ [A: a > $o,B: a > $o] :
( ( sup_sup_a_o @ A @ B )
= B ) ) ) ).
% sup.absorb_iff2
thf(fact_243_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_a
= ( ^ [A: set_a,B: set_a] :
( ( sup_sup_set_a @ A @ B )
= B ) ) ) ).
% sup.absorb_iff2
thf(fact_244_sup_Oabsorb__iff1,axiom,
( ord_less_eq_int
= ( ^ [B: int,A: int] :
( ( sup_sup_int @ A @ B )
= A ) ) ) ).
% sup.absorb_iff1
thf(fact_245_sup_Oabsorb__iff1,axiom,
( ord_less_eq_a_o
= ( ^ [B: a > $o,A: a > $o] :
( ( sup_sup_a_o @ A @ B )
= A ) ) ) ).
% sup.absorb_iff1
thf(fact_246_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_a
= ( ^ [B: set_a,A: set_a] :
( ( sup_sup_set_a @ A @ B )
= A ) ) ) ).
% sup.absorb_iff1
thf(fact_247_sup_Ocobounded2,axiom,
! [B2: int,A2: int] : ( ord_less_eq_int @ B2 @ ( sup_sup_int @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_248_sup_Ocobounded2,axiom,
! [B2: a > $o,A2: a > $o] : ( ord_less_eq_a_o @ B2 @ ( sup_sup_a_o @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_249_sup_Ocobounded2,axiom,
! [B2: set_a,A2: set_a] : ( ord_less_eq_set_a @ B2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_250_sup_Ocobounded1,axiom,
! [A2: int,B2: int] : ( ord_less_eq_int @ A2 @ ( sup_sup_int @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_251_sup_Ocobounded1,axiom,
! [A2: a > $o,B2: a > $o] : ( ord_less_eq_a_o @ A2 @ ( sup_sup_a_o @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_252_sup_Ocobounded1,axiom,
! [A2: set_a,B2: set_a] : ( ord_less_eq_set_a @ A2 @ ( sup_sup_set_a @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_253_sup_Oorder__iff,axiom,
( ord_less_eq_int
= ( ^ [B: int,A: int] :
( A
= ( sup_sup_int @ A @ B ) ) ) ) ).
% sup.order_iff
thf(fact_254_sup_Oorder__iff,axiom,
( ord_less_eq_a_o
= ( ^ [B: a > $o,A: a > $o] :
( A
= ( sup_sup_a_o @ A @ B ) ) ) ) ).
% sup.order_iff
thf(fact_255_sup_Oorder__iff,axiom,
( ord_less_eq_set_a
= ( ^ [B: set_a,A: set_a] :
( A
= ( sup_sup_set_a @ A @ B ) ) ) ) ).
% sup.order_iff
thf(fact_256_sup_OboundedI,axiom,
! [B2: int,A2: int,C3: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ( ord_less_eq_int @ C3 @ A2 )
=> ( ord_less_eq_int @ ( sup_sup_int @ B2 @ C3 ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_257_sup_OboundedI,axiom,
! [B2: a > $o,A2: a > $o,C3: a > $o] :
( ( ord_less_eq_a_o @ B2 @ A2 )
=> ( ( ord_less_eq_a_o @ C3 @ A2 )
=> ( ord_less_eq_a_o @ ( sup_sup_a_o @ B2 @ C3 ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_258_sup_OboundedI,axiom,
! [B2: set_a,A2: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( ord_less_eq_set_a @ C3 @ A2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C3 ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_259_sup_OboundedE,axiom,
! [B2: int,C3: int,A2: int] :
( ( ord_less_eq_int @ ( sup_sup_int @ B2 @ C3 ) @ A2 )
=> ~ ( ( ord_less_eq_int @ B2 @ A2 )
=> ~ ( ord_less_eq_int @ C3 @ A2 ) ) ) ).
% sup.boundedE
thf(fact_260_sup_OboundedE,axiom,
! [B2: a > $o,C3: a > $o,A2: a > $o] :
( ( ord_less_eq_a_o @ ( sup_sup_a_o @ B2 @ C3 ) @ A2 )
=> ~ ( ( ord_less_eq_a_o @ B2 @ A2 )
=> ~ ( ord_less_eq_a_o @ C3 @ A2 ) ) ) ).
% sup.boundedE
thf(fact_261_sup_OboundedE,axiom,
! [B2: set_a,C3: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B2 @ C3 ) @ A2 )
=> ~ ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ~ ( ord_less_eq_set_a @ C3 @ A2 ) ) ) ).
% sup.boundedE
thf(fact_262_sup__absorb2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( sup_sup_int @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_263_sup__absorb2,axiom,
! [X: a > $o,Y: a > $o] :
( ( ord_less_eq_a_o @ X @ Y )
=> ( ( sup_sup_a_o @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_264_sup__absorb2,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( sup_sup_set_a @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_265_sup__absorb1,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( sup_sup_int @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_266_sup__absorb1,axiom,
! [Y: a > $o,X: a > $o] :
( ( ord_less_eq_a_o @ Y @ X )
=> ( ( sup_sup_a_o @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_267_sup__absorb1,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( sup_sup_set_a @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_268_sup_Oabsorb2,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( sup_sup_int @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_269_sup_Oabsorb2,axiom,
! [A2: a > $o,B2: a > $o] :
( ( ord_less_eq_a_o @ A2 @ B2 )
=> ( ( sup_sup_a_o @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_270_sup_Oabsorb2,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( sup_sup_set_a @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_271_sup_Oabsorb1,axiom,
! [B2: int,A2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ( sup_sup_int @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_272_sup_Oabsorb1,axiom,
! [B2: a > $o,A2: a > $o] :
( ( ord_less_eq_a_o @ B2 @ A2 )
=> ( ( sup_sup_a_o @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_273_sup_Oabsorb1,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( sup_sup_set_a @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_274_sup__unique,axiom,
! [F2: int > int > int,X: int,Y: int] :
( ! [X3: int,Y3: int] : ( ord_less_eq_int @ X3 @ ( F2 @ X3 @ Y3 ) )
=> ( ! [X3: int,Y3: int] : ( ord_less_eq_int @ Y3 @ ( F2 @ X3 @ Y3 ) )
=> ( ! [X3: int,Y3: int,Z: int] :
( ( ord_less_eq_int @ Y3 @ X3 )
=> ( ( ord_less_eq_int @ Z @ X3 )
=> ( ord_less_eq_int @ ( F2 @ Y3 @ Z ) @ X3 ) ) )
=> ( ( sup_sup_int @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_275_sup__unique,axiom,
! [F2: ( a > $o ) > ( a > $o ) > a > $o,X: a > $o,Y: a > $o] :
( ! [X3: a > $o,Y3: a > $o] : ( ord_less_eq_a_o @ X3 @ ( F2 @ X3 @ Y3 ) )
=> ( ! [X3: a > $o,Y3: a > $o] : ( ord_less_eq_a_o @ Y3 @ ( F2 @ X3 @ Y3 ) )
=> ( ! [X3: a > $o,Y3: a > $o,Z: a > $o] :
( ( ord_less_eq_a_o @ Y3 @ X3 )
=> ( ( ord_less_eq_a_o @ Z @ X3 )
=> ( ord_less_eq_a_o @ ( F2 @ Y3 @ Z ) @ X3 ) ) )
=> ( ( sup_sup_a_o @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_276_sup__unique,axiom,
! [F2: set_a > set_a > set_a,X: set_a,Y: set_a] :
( ! [X3: set_a,Y3: set_a] : ( ord_less_eq_set_a @ X3 @ ( F2 @ X3 @ Y3 ) )
=> ( ! [X3: set_a,Y3: set_a] : ( ord_less_eq_set_a @ Y3 @ ( F2 @ X3 @ Y3 ) )
=> ( ! [X3: set_a,Y3: set_a,Z: set_a] :
( ( ord_less_eq_set_a @ Y3 @ X3 )
=> ( ( ord_less_eq_set_a @ Z @ X3 )
=> ( ord_less_eq_set_a @ ( F2 @ Y3 @ Z ) @ X3 ) ) )
=> ( ( sup_sup_set_a @ X @ Y )
= ( F2 @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_277_sup_OorderI,axiom,
! [A2: int,B2: int] :
( ( A2
= ( sup_sup_int @ A2 @ B2 ) )
=> ( ord_less_eq_int @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_278_sup_OorderI,axiom,
! [A2: a > $o,B2: a > $o] :
( ( A2
= ( sup_sup_a_o @ A2 @ B2 ) )
=> ( ord_less_eq_a_o @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_279_sup_OorderI,axiom,
! [A2: set_a,B2: set_a] :
( ( A2
= ( sup_sup_set_a @ A2 @ B2 ) )
=> ( ord_less_eq_set_a @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_280_sup_OorderE,axiom,
! [B2: int,A2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( A2
= ( sup_sup_int @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_281_sup_OorderE,axiom,
! [B2: a > $o,A2: a > $o] :
( ( ord_less_eq_a_o @ B2 @ A2 )
=> ( A2
= ( sup_sup_a_o @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_282_sup_OorderE,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( A2
= ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_283_le__iff__sup,axiom,
( ord_less_eq_int
= ( ^ [X4: int,Y4: int] :
( ( sup_sup_int @ X4 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_284_le__iff__sup,axiom,
( ord_less_eq_a_o
= ( ^ [X4: a > $o,Y4: a > $o] :
( ( sup_sup_a_o @ X4 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_285_le__iff__sup,axiom,
( ord_less_eq_set_a
= ( ^ [X4: set_a,Y4: set_a] :
( ( sup_sup_set_a @ X4 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_286_sup__least,axiom,
! [Y: int,X: int,Z4: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( ord_less_eq_int @ Z4 @ X )
=> ( ord_less_eq_int @ ( sup_sup_int @ Y @ Z4 ) @ X ) ) ) ).
% sup_least
thf(fact_287_sup__least,axiom,
! [Y: a > $o,X: a > $o,Z4: a > $o] :
( ( ord_less_eq_a_o @ Y @ X )
=> ( ( ord_less_eq_a_o @ Z4 @ X )
=> ( ord_less_eq_a_o @ ( sup_sup_a_o @ Y @ Z4 ) @ X ) ) ) ).
% sup_least
thf(fact_288_sup__least,axiom,
! [Y: set_a,X: set_a,Z4: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( ord_less_eq_set_a @ Z4 @ X )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ Y @ Z4 ) @ X ) ) ) ).
% sup_least
thf(fact_289_sup__mono,axiom,
! [A2: int,C3: int,B2: int,D2: int] :
( ( ord_less_eq_int @ A2 @ C3 )
=> ( ( ord_less_eq_int @ B2 @ D2 )
=> ( ord_less_eq_int @ ( sup_sup_int @ A2 @ B2 ) @ ( sup_sup_int @ C3 @ D2 ) ) ) ) ).
% sup_mono
thf(fact_290_sup__mono,axiom,
! [A2: a > $o,C3: a > $o,B2: a > $o,D2: a > $o] :
( ( ord_less_eq_a_o @ A2 @ C3 )
=> ( ( ord_less_eq_a_o @ B2 @ D2 )
=> ( ord_less_eq_a_o @ ( sup_sup_a_o @ A2 @ B2 ) @ ( sup_sup_a_o @ C3 @ D2 ) ) ) ) ).
% sup_mono
thf(fact_291_sup__mono,axiom,
! [A2: set_a,C3: set_a,B2: set_a,D2: set_a] :
( ( ord_less_eq_set_a @ A2 @ C3 )
=> ( ( ord_less_eq_set_a @ B2 @ D2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ ( sup_sup_set_a @ C3 @ D2 ) ) ) ) ).
% sup_mono
thf(fact_292_sup_Omono,axiom,
! [C3: int,A2: int,D2: int,B2: int] :
( ( ord_less_eq_int @ C3 @ A2 )
=> ( ( ord_less_eq_int @ D2 @ B2 )
=> ( ord_less_eq_int @ ( sup_sup_int @ C3 @ D2 ) @ ( sup_sup_int @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_293_sup_Omono,axiom,
! [C3: a > $o,A2: a > $o,D2: a > $o,B2: a > $o] :
( ( ord_less_eq_a_o @ C3 @ A2 )
=> ( ( ord_less_eq_a_o @ D2 @ B2 )
=> ( ord_less_eq_a_o @ ( sup_sup_a_o @ C3 @ D2 ) @ ( sup_sup_a_o @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_294_sup_Omono,axiom,
! [C3: set_a,A2: set_a,D2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ C3 @ A2 )
=> ( ( ord_less_eq_set_a @ D2 @ B2 )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ C3 @ D2 ) @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_295_le__supI2,axiom,
! [X: int,B2: int,A2: int] :
( ( ord_less_eq_int @ X @ B2 )
=> ( ord_less_eq_int @ X @ ( sup_sup_int @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_296_le__supI2,axiom,
! [X: a > $o,B2: a > $o,A2: a > $o] :
( ( ord_less_eq_a_o @ X @ B2 )
=> ( ord_less_eq_a_o @ X @ ( sup_sup_a_o @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_297_le__supI2,axiom,
! [X: set_a,B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ X @ B2 )
=> ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_298_le__supI1,axiom,
! [X: int,A2: int,B2: int] :
( ( ord_less_eq_int @ X @ A2 )
=> ( ord_less_eq_int @ X @ ( sup_sup_int @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_299_le__supI1,axiom,
! [X: a > $o,A2: a > $o,B2: a > $o] :
( ( ord_less_eq_a_o @ X @ A2 )
=> ( ord_less_eq_a_o @ X @ ( sup_sup_a_o @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_300_le__supI1,axiom,
! [X: set_a,A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ X @ A2 )
=> ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_301_sup__ge2,axiom,
! [Y: int,X: int] : ( ord_less_eq_int @ Y @ ( sup_sup_int @ X @ Y ) ) ).
% sup_ge2
thf(fact_302_sup__ge2,axiom,
! [Y: a > $o,X: a > $o] : ( ord_less_eq_a_o @ Y @ ( sup_sup_a_o @ X @ Y ) ) ).
% sup_ge2
thf(fact_303_sup__ge2,axiom,
! [Y: set_a,X: set_a] : ( ord_less_eq_set_a @ Y @ ( sup_sup_set_a @ X @ Y ) ) ).
% sup_ge2
thf(fact_304_sup__ge1,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ X @ ( sup_sup_int @ X @ Y ) ) ).
% sup_ge1
thf(fact_305_sup__ge1,axiom,
! [X: a > $o,Y: a > $o] : ( ord_less_eq_a_o @ X @ ( sup_sup_a_o @ X @ Y ) ) ).
% sup_ge1
thf(fact_306_sup__ge1,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ X @ Y ) ) ).
% sup_ge1
thf(fact_307_le__supI,axiom,
! [A2: int,X: int,B2: int] :
( ( ord_less_eq_int @ A2 @ X )
=> ( ( ord_less_eq_int @ B2 @ X )
=> ( ord_less_eq_int @ ( sup_sup_int @ A2 @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_308_le__supI,axiom,
! [A2: a > $o,X: a > $o,B2: a > $o] :
( ( ord_less_eq_a_o @ A2 @ X )
=> ( ( ord_less_eq_a_o @ B2 @ X )
=> ( ord_less_eq_a_o @ ( sup_sup_a_o @ A2 @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_309_le__supI,axiom,
! [A2: set_a,X: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ X )
=> ( ( ord_less_eq_set_a @ B2 @ X )
=> ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_310_le__supE,axiom,
! [A2: int,B2: int,X: int] :
( ( ord_less_eq_int @ ( sup_sup_int @ A2 @ B2 ) @ X )
=> ~ ( ( ord_less_eq_int @ A2 @ X )
=> ~ ( ord_less_eq_int @ B2 @ X ) ) ) ).
% le_supE
thf(fact_311_le__supE,axiom,
! [A2: a > $o,B2: a > $o,X: a > $o] :
( ( ord_less_eq_a_o @ ( sup_sup_a_o @ A2 @ B2 ) @ X )
=> ~ ( ( ord_less_eq_a_o @ A2 @ X )
=> ~ ( ord_less_eq_a_o @ B2 @ X ) ) ) ).
% le_supE
thf(fact_312_le__supE,axiom,
! [A2: set_a,B2: set_a,X: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A2 @ B2 ) @ X )
=> ~ ( ( ord_less_eq_set_a @ A2 @ X )
=> ~ ( ord_less_eq_set_a @ B2 @ X ) ) ) ).
% le_supE
thf(fact_313_inf__sup__ord_I3_J,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ X @ ( sup_sup_int @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_314_inf__sup__ord_I3_J,axiom,
! [X: a > $o,Y: a > $o] : ( ord_less_eq_a_o @ X @ ( sup_sup_a_o @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_315_inf__sup__ord_I3_J,axiom,
! [X: set_a,Y: set_a] : ( ord_less_eq_set_a @ X @ ( sup_sup_set_a @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_316_inf__sup__ord_I4_J,axiom,
! [Y: int,X: int] : ( ord_less_eq_int @ Y @ ( sup_sup_int @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_317_inf__sup__ord_I4_J,axiom,
! [Y: a > $o,X: a > $o] : ( ord_less_eq_a_o @ Y @ ( sup_sup_a_o @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_318_inf__sup__ord_I4_J,axiom,
! [Y: set_a,X: set_a] : ( ord_less_eq_set_a @ Y @ ( sup_sup_set_a @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_319_less__supI1,axiom,
! [X: a > $o,A2: a > $o,B2: a > $o] :
( ( ord_less_a_o @ X @ A2 )
=> ( ord_less_a_o @ X @ ( sup_sup_a_o @ A2 @ B2 ) ) ) ).
% less_supI1
thf(fact_320_less__supI1,axiom,
! [X: set_a,A2: set_a,B2: set_a] :
( ( ord_less_set_a @ X @ A2 )
=> ( ord_less_set_a @ X @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% less_supI1
thf(fact_321_less__supI1,axiom,
! [X: int,A2: int,B2: int] :
( ( ord_less_int @ X @ A2 )
=> ( ord_less_int @ X @ ( sup_sup_int @ A2 @ B2 ) ) ) ).
% less_supI1
thf(fact_322_less__supI2,axiom,
! [X: a > $o,B2: a > $o,A2: a > $o] :
( ( ord_less_a_o @ X @ B2 )
=> ( ord_less_a_o @ X @ ( sup_sup_a_o @ A2 @ B2 ) ) ) ).
% less_supI2
thf(fact_323_less__supI2,axiom,
! [X: set_a,B2: set_a,A2: set_a] :
( ( ord_less_set_a @ X @ B2 )
=> ( ord_less_set_a @ X @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% less_supI2
thf(fact_324_less__supI2,axiom,
! [X: int,B2: int,A2: int] :
( ( ord_less_int @ X @ B2 )
=> ( ord_less_int @ X @ ( sup_sup_int @ A2 @ B2 ) ) ) ).
% less_supI2
thf(fact_325_sup_Ostrict__boundedE,axiom,
! [B2: a > $o,C3: a > $o,A2: a > $o] :
( ( ord_less_a_o @ ( sup_sup_a_o @ B2 @ C3 ) @ A2 )
=> ~ ( ( ord_less_a_o @ B2 @ A2 )
=> ~ ( ord_less_a_o @ C3 @ A2 ) ) ) ).
% sup.strict_boundedE
thf(fact_326_sup_Ostrict__boundedE,axiom,
! [B2: set_a,C3: set_a,A2: set_a] :
( ( ord_less_set_a @ ( sup_sup_set_a @ B2 @ C3 ) @ A2 )
=> ~ ( ( ord_less_set_a @ B2 @ A2 )
=> ~ ( ord_less_set_a @ C3 @ A2 ) ) ) ).
% sup.strict_boundedE
thf(fact_327_sup_Ostrict__boundedE,axiom,
! [B2: int,C3: int,A2: int] :
( ( ord_less_int @ ( sup_sup_int @ B2 @ C3 ) @ A2 )
=> ~ ( ( ord_less_int @ B2 @ A2 )
=> ~ ( ord_less_int @ C3 @ A2 ) ) ) ).
% sup.strict_boundedE
thf(fact_328_sup_Ostrict__order__iff,axiom,
( ord_less_a_o
= ( ^ [B: a > $o,A: a > $o] :
( ( A
= ( sup_sup_a_o @ A @ B ) )
& ( A != B ) ) ) ) ).
% sup.strict_order_iff
thf(fact_329_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_330_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_331_sup_Ostrict__coboundedI1,axiom,
! [C3: a > $o,A2: a > $o,B2: a > $o] :
( ( ord_less_a_o @ C3 @ A2 )
=> ( ord_less_a_o @ C3 @ ( sup_sup_a_o @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_332_sup_Ostrict__coboundedI1,axiom,
! [C3: set_a,A2: set_a,B2: set_a] :
( ( ord_less_set_a @ C3 @ A2 )
=> ( ord_less_set_a @ C3 @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_333_sup_Ostrict__coboundedI1,axiom,
! [C3: int,A2: int,B2: int] :
( ( ord_less_int @ C3 @ A2 )
=> ( ord_less_int @ C3 @ ( sup_sup_int @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_334_sup_Ostrict__coboundedI2,axiom,
! [C3: a > $o,B2: a > $o,A2: a > $o] :
( ( ord_less_a_o @ C3 @ B2 )
=> ( ord_less_a_o @ C3 @ ( sup_sup_a_o @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_335_sup_Ostrict__coboundedI2,axiom,
! [C3: set_a,B2: set_a,A2: set_a] :
( ( ord_less_set_a @ C3 @ B2 )
=> ( ord_less_set_a @ C3 @ ( sup_sup_set_a @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_336_sup_Ostrict__coboundedI2,axiom,
! [C3: int,B2: int,A2: int] :
( ( ord_less_int @ C3 @ B2 )
=> ( ord_less_int @ C3 @ ( sup_sup_int @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_337_predicate1I,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X3: a] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq_a_o @ P @ Q ) ) ).
% predicate1I
thf(fact_338_pred__subset__eq,axiom,
! [R: set_a,S: set_a] :
( ( ord_less_eq_a_o
@ ^ [X4: a] : ( member_a @ X4 @ R )
@ ^ [X4: a] : ( member_a @ X4 @ S ) )
= ( ord_less_eq_set_a @ R @ S ) ) ).
% pred_subset_eq
thf(fact_339_the__elem__eq,axiom,
! [X: a] :
( ( the_elem_a @ ( insert_a @ X @ bot_bot_set_a ) )
= X ) ).
% the_elem_eq
thf(fact_340_order__refl,axiom,
! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).
% order_refl
thf(fact_341_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_342_order__refl,axiom,
! [X: a > $o] : ( ord_less_eq_a_o @ X @ X ) ).
% order_refl
thf(fact_343_sup1CI,axiom,
! [B3: a > $o,X: a,A3: a > $o] :
( ( ~ ( B3 @ X )
=> ( A3 @ X ) )
=> ( sup_sup_a_o @ A3 @ B3 @ X ) ) ).
% sup1CI
thf(fact_344_bot__apply,axiom,
( bot_bot_a_o
= ( ^ [X4: a] : bot_bot_o ) ) ).
% bot_apply
thf(fact_345_sup1I2,axiom,
! [B3: a > $o,X: a,A3: a > $o] :
( ( B3 @ X )
=> ( sup_sup_a_o @ A3 @ B3 @ X ) ) ).
% sup1I2
thf(fact_346_sup1I1,axiom,
! [A3: a > $o,X: a,B3: a > $o] :
( ( A3 @ X )
=> ( sup_sup_a_o @ A3 @ B3 @ X ) ) ).
% sup1I1
thf(fact_347_sup1E,axiom,
! [A3: a > $o,B3: a > $o,X: a] :
( ( sup_sup_a_o @ A3 @ B3 @ X )
=> ( ~ ( A3 @ X )
=> ( B3 @ X ) ) ) ).
% sup1E
thf(fact_348_order__subst1,axiom,
! [A2: int,F2: ( a > $o ) > int,B2: a > $o,C3: a > $o] :
( ( ord_less_eq_int @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_less_eq_a_o @ B2 @ C3 )
=> ( ! [X3: a > $o,Y3: a > $o] :
( ( ord_less_eq_a_o @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F2 @ C3 ) ) ) ) ) ).
% order_subst1
thf(fact_349_order__subst1,axiom,
! [A2: a > $o,F2: set_a > a > $o,B2: set_a,C3: set_a] :
( ( ord_less_eq_a_o @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C3 )
=> ( ! [X3: set_a,Y3: set_a] :
( ( ord_less_eq_set_a @ X3 @ Y3 )
=> ( ord_less_eq_a_o @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_a_o @ A2 @ ( F2 @ C3 ) ) ) ) ) ).
% order_subst1
thf(fact_350_order__subst1,axiom,
! [A2: a > $o,F2: int > a > $o,B2: int,C3: int] :
( ( ord_less_eq_a_o @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C3 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_a_o @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_a_o @ A2 @ ( F2 @ C3 ) ) ) ) ) ).
% order_subst1
thf(fact_351_order__subst1,axiom,
! [A2: a > $o,F2: ( a > $o ) > a > $o,B2: a > $o,C3: a > $o] :
( ( ord_less_eq_a_o @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_less_eq_a_o @ B2 @ C3 )
=> ( ! [X3: a > $o,Y3: a > $o] :
( ( ord_less_eq_a_o @ X3 @ Y3 )
=> ( ord_less_eq_a_o @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq_a_o @ A2 @ ( F2 @ C3 ) ) ) ) ) ).
% order_subst1
% Conjectures (1)
thf(conj_0,conjecture,
ord_less_int @ ( h @ l ) @ ( h @ ( binary339557810e_rm_a @ h @ ( binary717961607le_T_a @ t1 @ x @ t2 ) ) ) ).
%------------------------------------------------------------------------------