TPTP Problem File: ITP031^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : ITP031^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer BinaryTree problem prob_301__3253080_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_301__3253080_1 [Des21]
% Status : Theorem
% Rating : 0.30 v8.2.0, 0.23 v8.1.0, 0.27 v7.5.0
% Syntax : Number of formulae : 404 ( 207 unt; 45 typ; 0 def)
% Number of atoms : 982 ( 384 equ; 0 cnn)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 2678 ( 112 ~; 18 |; 67 &;2153 @)
% ( 0 <=>; 328 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 6 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 243 ( 243 >; 0 *; 0 +; 0 <<)
% Number of symbols : 44 ( 41 usr; 9 con; 0-4 aty)
% Number of variables : 972 ( 110 ^; 847 !; 15 ?; 972 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:30:37.987
%------------------------------------------------------------------------------
% 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 (41)
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_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_If_001tf__a,type,
if_a: $o > a > a > a ).
thf(sy_c_Lattices_Osup__class_Osup_001_062_Itf__a_M_Eo_J,type,
sup_sup_a_o: ( a > $o ) > ( a > $o ) > a > $o ).
thf(sy_c_Lattices_Osup__class_Osup_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_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_Opairwise_001tf__a,type,
pairwise_a: ( a > a > $o ) > 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_x____,type,
x: a ).
% Relevant facts (355)
thf(fact_0_eqsLessX,axiom,
! [X: a] :
( ( member_a @ X @ ( binary504661350_eqs_a @ h @ e ) )
=> ( ord_less_int @ ( h @ X ) @ ( h @ x ) ) ) ).
% eqsLessX
thf(fact_1_eLess,axiom,
ord_less_int @ ( h @ e ) @ ( h @ x ) ).
% eLess
thf(fact_2_res,axiom,
( ( binary1226383794sert_a @ h @ e @ ( binary717961607le_T_a @ t1 @ x @ t2 ) )
= ( binary717961607le_T_a @ ( binary1226383794sert_a @ h @ e @ t1 ) @ x @ t2 ) ) ).
% res
thf(fact_3_s1,axiom,
binary1721989714Tree_a @ h @ t1 ).
% s1
thf(fact_4_s2,axiom,
binary1721989714Tree_a @ h @ t2 ).
% s2
thf(fact_5_s,axiom,
binary1721989714Tree_a @ h @ ( binary717961607le_T_a @ t1 @ x @ t2 ) ).
% s
thf(fact_6_sorted__distinct,axiom,
! [H: a > int,A: a,B: a,T: binary1439146945Tree_a] : ( binary670562003pred_a @ H @ A @ B @ T ) ).
% sorted_distinct
thf(fact_7_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_8_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_9_eqs__def,axiom,
( binary504661350_eqs_a
= ( ^ [H2: a > int,X2: a] :
( collect_a
@ ^ [Y: a] :
( ( H2 @ Y )
= ( H2 @ X2 ) ) ) ) ) ).
% eqs_def
thf(fact_10_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_11_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_12__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_13_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_14_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_15_setOf_Osimps_I1_J,axiom,
( ( binary945792244etOf_a @ binary476621312_Tip_a )
= bot_bot_set_a ) ).
% setOf.simps(1)
thf(fact_16_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_17_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_18_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_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 )
& ! [X2: a] :
( ( member_a @ X2 @ ( binary945792244etOf_a @ T1 ) )
=> ( ord_less_int @ ( H @ X2 ) @ ( H @ X3 ) ) )
& ! [X2: a] :
( ( member_a @ X2 @ ( binary945792244etOf_a @ T2 ) )
=> ( ord_less_int @ ( H @ X3 ) @ ( H @ X2 ) ) )
& ( binary1721989714Tree_a @ H @ T2 ) ) ) ).
% sortedTree.simps(2)
thf(fact_20_sortedTree_Osimps_I1_J,axiom,
! [H: a > int] : ( binary1721989714Tree_a @ H @ binary476621312_Tip_a ) ).
% sortedTree.simps(1)
thf(fact_21_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_22_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_23_Tree_Oinduct,axiom,
! [P: binary1439146945Tree_a > $o,Tree: binary1439146945Tree_a] :
( ( P @ binary476621312_Tip_a )
=> ( ! [X1: binary1439146945Tree_a,X24: a,X32: binary1439146945Tree_a] :
( ( P @ X1 )
=> ( ( P @ X32 )
=> ( P @ ( binary717961607le_T_a @ X1 @ X24 @ X32 ) ) ) )
=> ( P @ Tree ) ) ) ).
% Tree.induct
thf(fact_24_Tree_Oexhaust,axiom,
! [Y2: binary1439146945Tree_a] :
( ( Y2 != binary476621312_Tip_a )
=> ~ ! [X212: binary1439146945Tree_a,X222: a,X232: binary1439146945Tree_a] :
( Y2
!= ( binary717961607le_T_a @ X212 @ X222 @ X232 ) ) ) ).
% Tree.exhaust
thf(fact_25_sorted__distinct__pred__def,axiom,
( binary670562003pred_a
= ( ^ [H2: a > int,A2: a,B2: a,T3: binary1439146945Tree_a] :
( ( ( binary1721989714Tree_a @ H2 @ T3 )
& ( member_a @ A2 @ ( binary945792244etOf_a @ T3 ) )
& ( member_a @ B2 @ ( binary945792244etOf_a @ T3 ) )
& ( ( H2 @ A2 )
= ( H2 @ B2 ) ) )
=> ( A2 = B2 ) ) ) ) ).
% sorted_distinct_pred_def
thf(fact_26_insert__Diff__single,axiom,
! [A: a,A3: set_a] :
( ( insert_a @ A @ ( minus_minus_set_a @ A3 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= ( insert_a @ A @ A3 ) ) ).
% insert_Diff_single
thf(fact_27_singleton__conv,axiom,
! [A: a] :
( ( collect_a
@ ^ [X2: a] : X2 = A )
= ( insert_a @ A @ bot_bot_set_a ) ) ).
% singleton_conv
thf(fact_28_singleton__conv2,axiom,
! [A: a] :
( ( collect_a
@ ( ^ [Y3: a,Z: a] : Y3 = Z
@ A ) )
= ( insert_a @ A @ bot_bot_set_a ) ) ).
% singleton_conv2
thf(fact_29_Un__Diff__cancel,axiom,
! [A3: set_a,B3: set_a] :
( ( sup_sup_set_a @ A3 @ ( minus_minus_set_a @ B3 @ A3 ) )
= ( sup_sup_set_a @ A3 @ B3 ) ) ).
% Un_Diff_cancel
thf(fact_30_Un__Diff__cancel2,axiom,
! [B3: set_a,A3: set_a] :
( ( sup_sup_set_a @ ( minus_minus_set_a @ B3 @ A3 ) @ A3 )
= ( sup_sup_set_a @ B3 @ A3 ) ) ).
% Un_Diff_cancel2
thf(fact_31_Diff__insert0,axiom,
! [X3: a,A3: set_a,B3: set_a] :
( ~ ( member_a @ X3 @ A3 )
=> ( ( minus_minus_set_a @ A3 @ ( insert_a @ X3 @ B3 ) )
= ( minus_minus_set_a @ A3 @ B3 ) ) ) ).
% Diff_insert0
thf(fact_32_insert__Diff1,axiom,
! [X3: a,B3: set_a,A3: set_a] :
( ( member_a @ X3 @ B3 )
=> ( ( minus_minus_set_a @ ( insert_a @ X3 @ A3 ) @ B3 )
= ( minus_minus_set_a @ A3 @ B3 ) ) ) ).
% insert_Diff1
thf(fact_33_Un__insert__left,axiom,
! [A: a,B3: set_a,C: set_a] :
( ( sup_sup_set_a @ ( insert_a @ A @ B3 ) @ C )
= ( insert_a @ A @ ( sup_sup_set_a @ B3 @ C ) ) ) ).
% Un_insert_left
thf(fact_34_Un__insert__right,axiom,
! [A3: set_a,A: a,B3: set_a] :
( ( sup_sup_set_a @ A3 @ ( insert_a @ A @ B3 ) )
= ( insert_a @ A @ ( sup_sup_set_a @ A3 @ B3 ) ) ) ).
% Un_insert_right
thf(fact_35_Diff__empty,axiom,
! [A3: set_a] :
( ( minus_minus_set_a @ A3 @ bot_bot_set_a )
= A3 ) ).
% Diff_empty
thf(fact_36_empty__Diff,axiom,
! [A3: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A3 )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_37_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X2: a] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_38_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X2: a] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_39_all__not__in__conv,axiom,
! [A3: set_a] :
( ( ! [X2: a] :
~ ( member_a @ X2 @ A3 ) )
= ( A3 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_40_empty__iff,axiom,
! [C2: a] :
~ ( member_a @ C2 @ bot_bot_set_a ) ).
% empty_iff
thf(fact_41_insert__absorb2,axiom,
! [X3: a,A3: set_a] :
( ( insert_a @ X3 @ ( insert_a @ X3 @ A3 ) )
= ( insert_a @ X3 @ A3 ) ) ).
% insert_absorb2
thf(fact_42_insert__iff,axiom,
! [A: a,B: a,A3: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A3 ) )
= ( ( A = B )
| ( member_a @ A @ A3 ) ) ) ).
% insert_iff
thf(fact_43_insertCI,axiom,
! [A: a,B3: set_a,B: a] :
( ( ~ ( member_a @ A @ B3 )
=> ( A = B ) )
=> ( member_a @ A @ ( insert_a @ B @ B3 ) ) ) ).
% insertCI
thf(fact_44_Un__iff,axiom,
! [C2: a,A3: set_a,B3: set_a] :
( ( member_a @ C2 @ ( sup_sup_set_a @ A3 @ B3 ) )
= ( ( member_a @ C2 @ A3 )
| ( member_a @ C2 @ B3 ) ) ) ).
% Un_iff
thf(fact_45_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A3: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X4: a] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_48_UnCI,axiom,
! [C2: a,B3: set_a,A3: set_a] :
( ( ~ ( member_a @ C2 @ B3 )
=> ( member_a @ C2 @ A3 ) )
=> ( member_a @ C2 @ ( sup_sup_set_a @ A3 @ B3 ) ) ) ).
% UnCI
thf(fact_49_Diff__idemp,axiom,
! [A3: set_a,B3: set_a] :
( ( minus_minus_set_a @ ( minus_minus_set_a @ A3 @ B3 ) @ B3 )
= ( minus_minus_set_a @ A3 @ B3 ) ) ).
% Diff_idemp
thf(fact_50_Diff__iff,axiom,
! [C2: a,A3: set_a,B3: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A3 @ B3 ) )
= ( ( member_a @ C2 @ A3 )
& ~ ( member_a @ C2 @ B3 ) ) ) ).
% Diff_iff
thf(fact_51_DiffI,axiom,
! [C2: a,A3: set_a,B3: set_a] :
( ( member_a @ C2 @ A3 )
=> ( ~ ( member_a @ C2 @ B3 )
=> ( member_a @ C2 @ ( minus_minus_set_a @ A3 @ B3 ) ) ) ) ).
% DiffI
thf(fact_52_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_53_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_54_Diff__cancel,axiom,
! [A3: set_a] :
( ( minus_minus_set_a @ A3 @ A3 )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_55_ex__in__conv,axiom,
! [A3: set_a] :
( ( ? [X2: a] : ( member_a @ X2 @ A3 ) )
= ( A3 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_56_equals0I,axiom,
! [A3: set_a] :
( ! [Y4: a] :
~ ( member_a @ Y4 @ A3 )
=> ( A3 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_57_equals0D,axiom,
! [A3: set_a,A: a] :
( ( A3 = bot_bot_set_a )
=> ~ ( member_a @ A @ A3 ) ) ).
% equals0D
thf(fact_58_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_59_mk__disjoint__insert,axiom,
! [A: a,A3: set_a] :
( ( member_a @ A @ A3 )
=> ? [B4: set_a] :
( ( A3
= ( insert_a @ A @ B4 ) )
& ~ ( member_a @ A @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_60_insert__commute,axiom,
! [X3: a,Y2: a,A3: set_a] :
( ( insert_a @ X3 @ ( insert_a @ Y2 @ A3 ) )
= ( insert_a @ Y2 @ ( insert_a @ X3 @ A3 ) ) ) ).
% insert_commute
thf(fact_61_insert__eq__iff,axiom,
! [A: a,A3: set_a,B: a,B3: set_a] :
( ~ ( member_a @ A @ A3 )
=> ( ~ ( member_a @ B @ B3 )
=> ( ( ( insert_a @ A @ A3 )
= ( insert_a @ B @ B3 ) )
= ( ( ( A = B )
=> ( A3 = B3 ) )
& ( ( A != B )
=> ? [C3: set_a] :
( ( A3
= ( insert_a @ B @ C3 ) )
& ~ ( member_a @ B @ C3 )
& ( B3
= ( insert_a @ A @ C3 ) )
& ~ ( member_a @ A @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_62_insert__absorb,axiom,
! [A: a,A3: set_a] :
( ( member_a @ A @ A3 )
=> ( ( insert_a @ A @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_63_insert__ident,axiom,
! [X3: a,A3: set_a,B3: set_a] :
( ~ ( member_a @ X3 @ A3 )
=> ( ~ ( member_a @ X3 @ B3 )
=> ( ( ( insert_a @ X3 @ A3 )
= ( insert_a @ X3 @ B3 ) )
= ( A3 = B3 ) ) ) ) ).
% insert_ident
thf(fact_64_Set_Oset__insert,axiom,
! [X3: a,A3: set_a] :
( ( member_a @ X3 @ A3 )
=> ~ ! [B4: set_a] :
( ( A3
= ( insert_a @ X3 @ B4 ) )
=> ( member_a @ X3 @ B4 ) ) ) ).
% Set.set_insert
thf(fact_65_insertI2,axiom,
! [A: a,B3: set_a,B: a] :
( ( member_a @ A @ B3 )
=> ( member_a @ A @ ( insert_a @ B @ B3 ) ) ) ).
% insertI2
thf(fact_66_insertI1,axiom,
! [A: a,B3: set_a] : ( member_a @ A @ ( insert_a @ A @ B3 ) ) ).
% insertI1
thf(fact_67_insertE,axiom,
! [A: a,B: a,A3: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A3 ) )
=> ( ( A != B )
=> ( member_a @ A @ A3 ) ) ) ).
% insertE
thf(fact_68_Un__left__commute,axiom,
! [A3: set_a,B3: set_a,C: set_a] :
( ( sup_sup_set_a @ A3 @ ( sup_sup_set_a @ B3 @ C ) )
= ( sup_sup_set_a @ B3 @ ( sup_sup_set_a @ A3 @ C ) ) ) ).
% Un_left_commute
thf(fact_69_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_70_Un__commute,axiom,
( sup_sup_set_a
= ( ^ [A4: set_a,B5: set_a] : ( sup_sup_set_a @ B5 @ A4 ) ) ) ).
% Un_commute
thf(fact_71_Un__absorb,axiom,
! [A3: set_a] :
( ( sup_sup_set_a @ A3 @ A3 )
= A3 ) ).
% Un_absorb
thf(fact_72_Un__assoc,axiom,
! [A3: set_a,B3: set_a,C: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A3 @ B3 ) @ C )
= ( sup_sup_set_a @ A3 @ ( sup_sup_set_a @ B3 @ C ) ) ) ).
% Un_assoc
thf(fact_73_ball__Un,axiom,
! [A3: set_a,B3: set_a,P: a > $o] :
( ( ! [X2: a] :
( ( member_a @ X2 @ ( sup_sup_set_a @ A3 @ B3 ) )
=> ( P @ X2 ) ) )
= ( ! [X2: a] :
( ( member_a @ X2 @ A3 )
=> ( P @ X2 ) )
& ! [X2: a] :
( ( member_a @ X2 @ B3 )
=> ( P @ X2 ) ) ) ) ).
% ball_Un
thf(fact_74_bex__Un,axiom,
! [A3: set_a,B3: set_a,P: a > $o] :
( ( ? [X2: a] :
( ( member_a @ X2 @ ( sup_sup_set_a @ A3 @ B3 ) )
& ( P @ X2 ) ) )
= ( ? [X2: a] :
( ( member_a @ X2 @ A3 )
& ( P @ X2 ) )
| ? [X2: a] :
( ( member_a @ X2 @ B3 )
& ( P @ X2 ) ) ) ) ).
% bex_Un
thf(fact_75_UnI2,axiom,
! [C2: a,B3: set_a,A3: set_a] :
( ( member_a @ C2 @ B3 )
=> ( member_a @ C2 @ ( sup_sup_set_a @ A3 @ B3 ) ) ) ).
% UnI2
thf(fact_76_UnI1,axiom,
! [C2: a,A3: set_a,B3: set_a] :
( ( member_a @ C2 @ A3 )
=> ( member_a @ C2 @ ( sup_sup_set_a @ A3 @ B3 ) ) ) ).
% UnI1
thf(fact_77_UnE,axiom,
! [C2: a,A3: set_a,B3: set_a] :
( ( member_a @ C2 @ ( sup_sup_set_a @ A3 @ B3 ) )
=> ( ~ ( member_a @ C2 @ A3 )
=> ( member_a @ C2 @ B3 ) ) ) ).
% UnE
thf(fact_78_DiffD2,axiom,
! [C2: a,A3: set_a,B3: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A3 @ B3 ) )
=> ~ ( member_a @ C2 @ B3 ) ) ).
% DiffD2
thf(fact_79_DiffD1,axiom,
! [C2: a,A3: set_a,B3: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A3 @ B3 ) )
=> ( member_a @ C2 @ A3 ) ) ).
% DiffD1
thf(fact_80_DiffE,axiom,
! [C2: a,A3: set_a,B3: set_a] :
( ( member_a @ C2 @ ( minus_minus_set_a @ A3 @ B3 ) )
=> ~ ( ( member_a @ C2 @ A3 )
=> ( member_a @ C2 @ B3 ) ) ) ).
% DiffE
thf(fact_81_empty__def,axiom,
( bot_bot_set_a
= ( collect_a
@ ^ [X2: a] : $false ) ) ).
% empty_def
thf(fact_82_insert__Collect,axiom,
! [A: a,P: a > $o] :
( ( insert_a @ A @ ( collect_a @ P ) )
= ( collect_a
@ ^ [U: a] :
( ( U != A )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_83_insert__compr,axiom,
( insert_a
= ( ^ [A2: a,B5: set_a] :
( collect_a
@ ^ [X2: a] :
( ( X2 = A2 )
| ( member_a @ X2 @ B5 ) ) ) ) ) ).
% insert_compr
thf(fact_84_Collect__disj__eq,axiom,
! [P: a > $o,Q: a > $o] :
( ( collect_a
@ ^ [X2: a] :
( ( P @ X2 )
| ( Q @ X2 ) ) )
= ( sup_sup_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_85_Un__def,axiom,
( sup_sup_set_a
= ( ^ [A4: set_a,B5: set_a] :
( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A4 )
| ( member_a @ X2 @ B5 ) ) ) ) ) ).
% Un_def
thf(fact_86_set__diff__eq,axiom,
( minus_minus_set_a
= ( ^ [A4: set_a,B5: set_a] :
( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A4 )
& ~ ( member_a @ X2 @ B5 ) ) ) ) ) ).
% set_diff_eq
thf(fact_87_singleton__inject,axiom,
! [A: a,B: a] :
( ( ( insert_a @ A @ bot_bot_set_a )
= ( insert_a @ B @ bot_bot_set_a ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_88_insert__not__empty,axiom,
! [A: a,A3: set_a] :
( ( insert_a @ A @ A3 )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_89_doubleton__eq__iff,axiom,
! [A: a,B: a,C2: a,D: a] :
( ( ( insert_a @ A @ ( insert_a @ B @ bot_bot_set_a ) )
= ( insert_a @ C2 @ ( insert_a @ D @ bot_bot_set_a ) ) )
= ( ( ( A = C2 )
& ( B = D ) )
| ( ( A = D )
& ( B = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_90_singleton__iff,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_91_singletonD,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_92_Un__empty__right,axiom,
! [A3: set_a] :
( ( sup_sup_set_a @ A3 @ bot_bot_set_a )
= A3 ) ).
% Un_empty_right
thf(fact_93_Un__empty__left,axiom,
! [B3: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ B3 )
= B3 ) ).
% Un_empty_left
thf(fact_94_insert__Diff__if,axiom,
! [X3: a,B3: set_a,A3: set_a] :
( ( ( member_a @ X3 @ B3 )
=> ( ( minus_minus_set_a @ ( insert_a @ X3 @ A3 ) @ B3 )
= ( minus_minus_set_a @ A3 @ B3 ) ) )
& ( ~ ( member_a @ X3 @ B3 )
=> ( ( minus_minus_set_a @ ( insert_a @ X3 @ A3 ) @ B3 )
= ( insert_a @ X3 @ ( minus_minus_set_a @ A3 @ B3 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_95_Un__Diff,axiom,
! [A3: set_a,B3: set_a,C: set_a] :
( ( minus_minus_set_a @ ( sup_sup_set_a @ A3 @ B3 ) @ C )
= ( sup_sup_set_a @ ( minus_minus_set_a @ A3 @ C ) @ ( minus_minus_set_a @ B3 @ C ) ) ) ).
% Un_Diff
thf(fact_96_Collect__conv__if2,axiom,
! [P: a > $o,A: a] :
( ( ( P @ A )
=> ( ( collect_a
@ ^ [X2: a] :
( ( A = X2 )
& ( P @ X2 ) ) )
= ( insert_a @ A @ bot_bot_set_a ) ) )
& ( ~ ( P @ A )
=> ( ( collect_a
@ ^ [X2: a] :
( ( A = X2 )
& ( P @ X2 ) ) )
= bot_bot_set_a ) ) ) ).
% Collect_conv_if2
thf(fact_97_Collect__conv__if,axiom,
! [P: a > $o,A: a] :
( ( ( P @ A )
=> ( ( collect_a
@ ^ [X2: a] :
( ( X2 = A )
& ( P @ X2 ) ) )
= ( insert_a @ A @ bot_bot_set_a ) ) )
& ( ~ ( P @ A )
=> ( ( collect_a
@ ^ [X2: a] :
( ( X2 = A )
& ( P @ X2 ) ) )
= bot_bot_set_a ) ) ) ).
% Collect_conv_if
thf(fact_98_insert__def,axiom,
( insert_a
= ( ^ [A2: a] :
( sup_sup_set_a
@ ( collect_a
@ ^ [X2: a] : X2 = A2 ) ) ) ) ).
% insert_def
thf(fact_99_singleton__Un__iff,axiom,
! [X3: a,A3: set_a,B3: set_a] :
( ( ( insert_a @ X3 @ bot_bot_set_a )
= ( sup_sup_set_a @ A3 @ B3 ) )
= ( ( ( A3 = bot_bot_set_a )
& ( B3
= ( insert_a @ X3 @ bot_bot_set_a ) ) )
| ( ( A3
= ( insert_a @ X3 @ bot_bot_set_a ) )
& ( B3 = bot_bot_set_a ) )
| ( ( A3
= ( insert_a @ X3 @ bot_bot_set_a ) )
& ( B3
= ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_100_Un__singleton__iff,axiom,
! [A3: set_a,B3: set_a,X3: a] :
( ( ( sup_sup_set_a @ A3 @ B3 )
= ( insert_a @ X3 @ bot_bot_set_a ) )
= ( ( ( A3 = bot_bot_set_a )
& ( B3
= ( insert_a @ X3 @ bot_bot_set_a ) ) )
| ( ( A3
= ( insert_a @ X3 @ bot_bot_set_a ) )
& ( B3 = bot_bot_set_a ) )
| ( ( A3
= ( insert_a @ X3 @ bot_bot_set_a ) )
& ( B3
= ( insert_a @ X3 @ bot_bot_set_a ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_101_insert__is__Un,axiom,
( insert_a
= ( ^ [A2: a] : ( sup_sup_set_a @ ( insert_a @ A2 @ bot_bot_set_a ) ) ) ) ).
% insert_is_Un
thf(fact_102_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_103_Diff__insert2,axiom,
! [A3: set_a,A: a,B3: set_a] :
( ( minus_minus_set_a @ A3 @ ( insert_a @ A @ B3 ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A3 @ ( insert_a @ A @ bot_bot_set_a ) ) @ B3 ) ) ).
% Diff_insert2
thf(fact_104_insert__Diff,axiom,
! [A: a,A3: set_a] :
( ( member_a @ A @ A3 )
=> ( ( insert_a @ A @ ( minus_minus_set_a @ A3 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= A3 ) ) ).
% insert_Diff
thf(fact_105_Diff__insert,axiom,
! [A3: set_a,A: a,B3: set_a] :
( ( minus_minus_set_a @ A3 @ ( insert_a @ A @ B3 ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A3 @ B3 ) @ ( insert_a @ A @ bot_bot_set_a ) ) ) ).
% Diff_insert
thf(fact_106_sup__bot_Oright__neutral,axiom,
! [A: a > $o] :
( ( sup_sup_a_o @ A @ bot_bot_a_o )
= A ) ).
% sup_bot.right_neutral
thf(fact_107_sup__bot_Oright__neutral,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ bot_bot_set_a )
= A ) ).
% sup_bot.right_neutral
thf(fact_108_sup__bot_Oneutr__eq__iff,axiom,
! [A: a > $o,B: a > $o] :
( ( bot_bot_a_o
= ( sup_sup_a_o @ A @ B ) )
= ( ( A = bot_bot_a_o )
& ( B = bot_bot_a_o ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_109_sup__bot_Oneutr__eq__iff,axiom,
! [A: set_a,B: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ A @ B ) )
= ( ( A = bot_bot_set_a )
& ( B = bot_bot_set_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_110_sup__bot_Oleft__neutral,axiom,
! [A: a > $o] :
( ( sup_sup_a_o @ bot_bot_a_o @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_111_sup__bot_Oleft__neutral,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ A )
= A ) ).
% sup_bot.left_neutral
thf(fact_112_sup__bot_Oeq__neutr__iff,axiom,
! [A: a > $o,B: a > $o] :
( ( ( sup_sup_a_o @ A @ B )
= bot_bot_a_o )
= ( ( A = bot_bot_a_o )
& ( B = bot_bot_a_o ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_113_sup__bot_Oeq__neutr__iff,axiom,
! [A: set_a,B: set_a] :
( ( ( sup_sup_set_a @ A @ B )
= bot_bot_set_a )
= ( ( A = bot_bot_set_a )
& ( B = bot_bot_set_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_114_sup__eq__bot__iff,axiom,
! [X3: a > $o,Y2: a > $o] :
( ( ( sup_sup_a_o @ X3 @ Y2 )
= bot_bot_a_o )
= ( ( X3 = bot_bot_a_o )
& ( Y2 = bot_bot_a_o ) ) ) ).
% sup_eq_bot_iff
thf(fact_115_sup__eq__bot__iff,axiom,
! [X3: set_a,Y2: set_a] :
( ( ( sup_sup_set_a @ X3 @ Y2 )
= bot_bot_set_a )
= ( ( X3 = bot_bot_set_a )
& ( Y2 = bot_bot_set_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_116_bot__eq__sup__iff,axiom,
! [X3: a > $o,Y2: a > $o] :
( ( bot_bot_a_o
= ( sup_sup_a_o @ X3 @ Y2 ) )
= ( ( X3 = bot_bot_a_o )
& ( Y2 = bot_bot_a_o ) ) ) ).
% bot_eq_sup_iff
thf(fact_117_bot__eq__sup__iff,axiom,
! [X3: set_a,Y2: set_a] :
( ( bot_bot_set_a
= ( sup_sup_set_a @ X3 @ Y2 ) )
= ( ( X3 = bot_bot_set_a )
& ( Y2 = bot_bot_set_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_118_sup__bot__right,axiom,
! [X3: a > $o] :
( ( sup_sup_a_o @ X3 @ bot_bot_a_o )
= X3 ) ).
% sup_bot_right
thf(fact_119_sup__bot__right,axiom,
! [X3: set_a] :
( ( sup_sup_set_a @ X3 @ bot_bot_set_a )
= X3 ) ).
% sup_bot_right
thf(fact_120_sup__bot__left,axiom,
! [X3: a > $o] :
( ( sup_sup_a_o @ bot_bot_a_o @ X3 )
= X3 ) ).
% sup_bot_left
thf(fact_121_sup__bot__left,axiom,
! [X3: set_a] :
( ( sup_sup_set_a @ bot_bot_set_a @ X3 )
= X3 ) ).
% sup_bot_left
thf(fact_122_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_123_minus__apply,axiom,
( minus_minus_a_o
= ( ^ [A4: a > $o,B5: a > $o,X2: a] : ( minus_minus_o @ ( A4 @ X2 ) @ ( B5 @ X2 ) ) ) ) ).
% minus_apply
thf(fact_124_sup__apply,axiom,
( sup_sup_a_o
= ( ^ [F: a > $o,G: a > $o,X2: a] : ( sup_sup_o @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% sup_apply
thf(fact_125_sup_Oidem,axiom,
! [A: a > $o] :
( ( sup_sup_a_o @ A @ A )
= A ) ).
% sup.idem
thf(fact_126_sup_Oidem,axiom,
! [A: int] :
( ( sup_sup_int @ A @ A )
= A ) ).
% sup.idem
thf(fact_127_sup_Oidem,axiom,
! [A: set_a] :
( ( sup_sup_set_a @ A @ A )
= A ) ).
% sup.idem
thf(fact_128_sup__idem,axiom,
! [X3: a > $o] :
( ( sup_sup_a_o @ X3 @ X3 )
= X3 ) ).
% sup_idem
thf(fact_129_sup__idem,axiom,
! [X3: int] :
( ( sup_sup_int @ X3 @ X3 )
= X3 ) ).
% sup_idem
thf(fact_130_sup__idem,axiom,
! [X3: set_a] :
( ( sup_sup_set_a @ X3 @ X3 )
= X3 ) ).
% sup_idem
thf(fact_131_sup_Oleft__idem,axiom,
! [A: a > $o,B: a > $o] :
( ( sup_sup_a_o @ A @ ( sup_sup_a_o @ A @ B ) )
= ( sup_sup_a_o @ A @ B ) ) ).
% sup.left_idem
thf(fact_132_sup_Oleft__idem,axiom,
! [A: int,B: int] :
( ( sup_sup_int @ A @ ( sup_sup_int @ A @ B ) )
= ( sup_sup_int @ A @ B ) ) ).
% sup.left_idem
thf(fact_133_sup_Oleft__idem,axiom,
! [A: set_a,B: set_a] :
( ( sup_sup_set_a @ A @ ( sup_sup_set_a @ A @ B ) )
= ( sup_sup_set_a @ A @ B ) ) ).
% sup.left_idem
thf(fact_134_sup__left__idem,axiom,
! [X3: a > $o,Y2: a > $o] :
( ( sup_sup_a_o @ X3 @ ( sup_sup_a_o @ X3 @ Y2 ) )
= ( sup_sup_a_o @ X3 @ Y2 ) ) ).
% sup_left_idem
thf(fact_135_sup__left__idem,axiom,
! [X3: int,Y2: int] :
( ( sup_sup_int @ X3 @ ( sup_sup_int @ X3 @ Y2 ) )
= ( sup_sup_int @ X3 @ Y2 ) ) ).
% sup_left_idem
thf(fact_136_sup__left__idem,axiom,
! [X3: set_a,Y2: set_a] :
( ( sup_sup_set_a @ X3 @ ( sup_sup_set_a @ X3 @ Y2 ) )
= ( sup_sup_set_a @ X3 @ Y2 ) ) ).
% sup_left_idem
thf(fact_137_sup_Oright__idem,axiom,
! [A: a > $o,B: a > $o] :
( ( sup_sup_a_o @ ( sup_sup_a_o @ A @ B ) @ B )
= ( sup_sup_a_o @ A @ B ) ) ).
% sup.right_idem
thf(fact_138_sup_Oright__idem,axiom,
! [A: int,B: int] :
( ( sup_sup_int @ ( sup_sup_int @ A @ B ) @ B )
= ( sup_sup_int @ A @ B ) ) ).
% sup.right_idem
thf(fact_139_sup_Oright__idem,axiom,
! [A: set_a,B: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A @ B ) @ B )
= ( sup_sup_set_a @ A @ B ) ) ).
% sup.right_idem
thf(fact_140_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_141_sup__set__def,axiom,
( sup_sup_set_a
= ( ^ [A4: set_a,B5: set_a] :
( collect_a
@ ( sup_sup_a_o
@ ^ [X2: a] : ( member_a @ X2 @ A4 )
@ ^ [X2: a] : ( member_a @ X2 @ B5 ) ) ) ) ) ).
% sup_set_def
thf(fact_142_fun__diff__def,axiom,
( minus_minus_a_o
= ( ^ [A4: a > $o,B5: a > $o,X2: a] : ( minus_minus_o @ ( A4 @ X2 ) @ ( B5 @ X2 ) ) ) ) ).
% fun_diff_def
thf(fact_143_inf__sup__aci_I8_J,axiom,
! [X3: a > $o,Y2: a > $o] :
( ( sup_sup_a_o @ X3 @ ( sup_sup_a_o @ X3 @ Y2 ) )
= ( sup_sup_a_o @ X3 @ Y2 ) ) ).
% inf_sup_aci(8)
thf(fact_144_inf__sup__aci_I8_J,axiom,
! [X3: int,Y2: int] :
( ( sup_sup_int @ X3 @ ( sup_sup_int @ X3 @ Y2 ) )
= ( sup_sup_int @ X3 @ Y2 ) ) ).
% inf_sup_aci(8)
thf(fact_145_inf__sup__aci_I8_J,axiom,
! [X3: set_a,Y2: set_a] :
( ( sup_sup_set_a @ X3 @ ( sup_sup_set_a @ X3 @ Y2 ) )
= ( sup_sup_set_a @ X3 @ Y2 ) ) ).
% inf_sup_aci(8)
thf(fact_146_inf__sup__aci_I7_J,axiom,
! [X3: a > $o,Y2: a > $o,Z2: a > $o] :
( ( sup_sup_a_o @ X3 @ ( sup_sup_a_o @ Y2 @ Z2 ) )
= ( sup_sup_a_o @ Y2 @ ( sup_sup_a_o @ X3 @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_147_inf__sup__aci_I7_J,axiom,
! [X3: int,Y2: int,Z2: int] :
( ( sup_sup_int @ X3 @ ( sup_sup_int @ Y2 @ Z2 ) )
= ( sup_sup_int @ Y2 @ ( sup_sup_int @ X3 @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_148_inf__sup__aci_I7_J,axiom,
! [X3: set_a,Y2: set_a,Z2: set_a] :
( ( sup_sup_set_a @ X3 @ ( sup_sup_set_a @ Y2 @ Z2 ) )
= ( sup_sup_set_a @ Y2 @ ( sup_sup_set_a @ X3 @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_149_inf__sup__aci_I6_J,axiom,
! [X3: a > $o,Y2: a > $o,Z2: a > $o] :
( ( sup_sup_a_o @ ( sup_sup_a_o @ X3 @ Y2 ) @ Z2 )
= ( sup_sup_a_o @ X3 @ ( sup_sup_a_o @ Y2 @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_150_inf__sup__aci_I6_J,axiom,
! [X3: int,Y2: int,Z2: int] :
( ( sup_sup_int @ ( sup_sup_int @ X3 @ Y2 ) @ Z2 )
= ( sup_sup_int @ X3 @ ( sup_sup_int @ Y2 @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_151_inf__sup__aci_I6_J,axiom,
! [X3: set_a,Y2: set_a,Z2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X3 @ Y2 ) @ Z2 )
= ( sup_sup_set_a @ X3 @ ( sup_sup_set_a @ Y2 @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_152_inf__sup__aci_I5_J,axiom,
( sup_sup_a_o
= ( ^ [X2: a > $o,Y: a > $o] : ( sup_sup_a_o @ Y @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_153_inf__sup__aci_I5_J,axiom,
( sup_sup_int
= ( ^ [X2: int,Y: int] : ( sup_sup_int @ Y @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_154_inf__sup__aci_I5_J,axiom,
( sup_sup_set_a
= ( ^ [X2: set_a,Y: set_a] : ( sup_sup_set_a @ Y @ X2 ) ) ) ).
% inf_sup_aci(5)
thf(fact_155_sup__fun__def,axiom,
( sup_sup_a_o
= ( ^ [F: a > $o,G: a > $o,X2: a] : ( sup_sup_o @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% sup_fun_def
thf(fact_156_boolean__algebra__cancel_Osup1,axiom,
! [A3: a > $o,K: a > $o,A: a > $o,B: a > $o] :
( ( A3
= ( sup_sup_a_o @ K @ A ) )
=> ( ( sup_sup_a_o @ A3 @ B )
= ( sup_sup_a_o @ K @ ( sup_sup_a_o @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_157_boolean__algebra__cancel_Osup1,axiom,
! [A3: int,K: int,A: int,B: int] :
( ( A3
= ( sup_sup_int @ K @ A ) )
=> ( ( sup_sup_int @ A3 @ B )
= ( sup_sup_int @ K @ ( sup_sup_int @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_158_boolean__algebra__cancel_Osup1,axiom,
! [A3: set_a,K: set_a,A: set_a,B: set_a] :
( ( A3
= ( sup_sup_set_a @ K @ A ) )
=> ( ( sup_sup_set_a @ A3 @ B )
= ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_159_boolean__algebra__cancel_Osup2,axiom,
! [B3: a > $o,K: a > $o,B: a > $o,A: a > $o] :
( ( B3
= ( sup_sup_a_o @ K @ B ) )
=> ( ( sup_sup_a_o @ A @ B3 )
= ( sup_sup_a_o @ K @ ( sup_sup_a_o @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_160_boolean__algebra__cancel_Osup2,axiom,
! [B3: int,K: int,B: int,A: int] :
( ( B3
= ( sup_sup_int @ K @ B ) )
=> ( ( sup_sup_int @ A @ B3 )
= ( sup_sup_int @ K @ ( sup_sup_int @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_161_boolean__algebra__cancel_Osup2,axiom,
! [B3: set_a,K: set_a,B: set_a,A: set_a] :
( ( B3
= ( sup_sup_set_a @ K @ B ) )
=> ( ( sup_sup_set_a @ A @ B3 )
= ( sup_sup_set_a @ K @ ( sup_sup_set_a @ A @ B ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_162_sup_Oassoc,axiom,
! [A: a > $o,B: a > $o,C2: a > $o] :
( ( sup_sup_a_o @ ( sup_sup_a_o @ A @ B ) @ C2 )
= ( sup_sup_a_o @ A @ ( sup_sup_a_o @ B @ C2 ) ) ) ).
% sup.assoc
thf(fact_163_sup_Oassoc,axiom,
! [A: int,B: int,C2: int] :
( ( sup_sup_int @ ( sup_sup_int @ A @ B ) @ C2 )
= ( sup_sup_int @ A @ ( sup_sup_int @ B @ C2 ) ) ) ).
% sup.assoc
thf(fact_164_sup_Oassoc,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ A @ B ) @ C2 )
= ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B @ C2 ) ) ) ).
% sup.assoc
thf(fact_165_sup__assoc,axiom,
! [X3: a > $o,Y2: a > $o,Z2: a > $o] :
( ( sup_sup_a_o @ ( sup_sup_a_o @ X3 @ Y2 ) @ Z2 )
= ( sup_sup_a_o @ X3 @ ( sup_sup_a_o @ Y2 @ Z2 ) ) ) ).
% sup_assoc
thf(fact_166_sup__assoc,axiom,
! [X3: int,Y2: int,Z2: int] :
( ( sup_sup_int @ ( sup_sup_int @ X3 @ Y2 ) @ Z2 )
= ( sup_sup_int @ X3 @ ( sup_sup_int @ Y2 @ Z2 ) ) ) ).
% sup_assoc
thf(fact_167_sup__assoc,axiom,
! [X3: set_a,Y2: set_a,Z2: set_a] :
( ( sup_sup_set_a @ ( sup_sup_set_a @ X3 @ Y2 ) @ Z2 )
= ( sup_sup_set_a @ X3 @ ( sup_sup_set_a @ Y2 @ Z2 ) ) ) ).
% sup_assoc
thf(fact_168_sup_Ocommute,axiom,
( sup_sup_a_o
= ( ^ [A2: a > $o,B2: a > $o] : ( sup_sup_a_o @ B2 @ A2 ) ) ) ).
% sup.commute
thf(fact_169_sup_Ocommute,axiom,
( sup_sup_int
= ( ^ [A2: int,B2: int] : ( sup_sup_int @ B2 @ A2 ) ) ) ).
% sup.commute
thf(fact_170_sup_Ocommute,axiom,
( sup_sup_set_a
= ( ^ [A2: set_a,B2: set_a] : ( sup_sup_set_a @ B2 @ A2 ) ) ) ).
% sup.commute
thf(fact_171_sup__commute,axiom,
( sup_sup_a_o
= ( ^ [X2: a > $o,Y: a > $o] : ( sup_sup_a_o @ Y @ X2 ) ) ) ).
% sup_commute
thf(fact_172_sup__commute,axiom,
( sup_sup_int
= ( ^ [X2: int,Y: int] : ( sup_sup_int @ Y @ X2 ) ) ) ).
% sup_commute
thf(fact_173_sup__commute,axiom,
( sup_sup_set_a
= ( ^ [X2: set_a,Y: set_a] : ( sup_sup_set_a @ Y @ X2 ) ) ) ).
% sup_commute
thf(fact_174_sup_Oleft__commute,axiom,
! [B: a > $o,A: a > $o,C2: a > $o] :
( ( sup_sup_a_o @ B @ ( sup_sup_a_o @ A @ C2 ) )
= ( sup_sup_a_o @ A @ ( sup_sup_a_o @ B @ C2 ) ) ) ).
% sup.left_commute
thf(fact_175_sup_Oleft__commute,axiom,
! [B: int,A: int,C2: int] :
( ( sup_sup_int @ B @ ( sup_sup_int @ A @ C2 ) )
= ( sup_sup_int @ A @ ( sup_sup_int @ B @ C2 ) ) ) ).
% sup.left_commute
thf(fact_176_sup_Oleft__commute,axiom,
! [B: set_a,A: set_a,C2: set_a] :
( ( sup_sup_set_a @ B @ ( sup_sup_set_a @ A @ C2 ) )
= ( sup_sup_set_a @ A @ ( sup_sup_set_a @ B @ C2 ) ) ) ).
% sup.left_commute
thf(fact_177_sup__left__commute,axiom,
! [X3: a > $o,Y2: a > $o,Z2: a > $o] :
( ( sup_sup_a_o @ X3 @ ( sup_sup_a_o @ Y2 @ Z2 ) )
= ( sup_sup_a_o @ Y2 @ ( sup_sup_a_o @ X3 @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_178_sup__left__commute,axiom,
! [X3: int,Y2: int,Z2: int] :
( ( sup_sup_int @ X3 @ ( sup_sup_int @ Y2 @ Z2 ) )
= ( sup_sup_int @ Y2 @ ( sup_sup_int @ X3 @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_179_sup__left__commute,axiom,
! [X3: set_a,Y2: set_a,Z2: set_a] :
( ( sup_sup_set_a @ X3 @ ( sup_sup_set_a @ Y2 @ Z2 ) )
= ( sup_sup_set_a @ Y2 @ ( sup_sup_set_a @ X3 @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_180_less__supI1,axiom,
! [X3: a > $o,A: a > $o,B: a > $o] :
( ( ord_less_a_o @ X3 @ A )
=> ( ord_less_a_o @ X3 @ ( sup_sup_a_o @ A @ B ) ) ) ).
% less_supI1
thf(fact_181_less__supI1,axiom,
! [X3: set_a,A: set_a,B: set_a] :
( ( ord_less_set_a @ X3 @ A )
=> ( ord_less_set_a @ X3 @ ( sup_sup_set_a @ A @ B ) ) ) ).
% less_supI1
thf(fact_182_less__supI1,axiom,
! [X3: int,A: int,B: int] :
( ( ord_less_int @ X3 @ A )
=> ( ord_less_int @ X3 @ ( sup_sup_int @ A @ B ) ) ) ).
% less_supI1
thf(fact_183_less__supI2,axiom,
! [X3: a > $o,B: a > $o,A: a > $o] :
( ( ord_less_a_o @ X3 @ B )
=> ( ord_less_a_o @ X3 @ ( sup_sup_a_o @ A @ B ) ) ) ).
% less_supI2
thf(fact_184_less__supI2,axiom,
! [X3: set_a,B: set_a,A: set_a] :
( ( ord_less_set_a @ X3 @ B )
=> ( ord_less_set_a @ X3 @ ( sup_sup_set_a @ A @ B ) ) ) ).
% less_supI2
thf(fact_185_less__supI2,axiom,
! [X3: int,B: int,A: int] :
( ( ord_less_int @ X3 @ B )
=> ( ord_less_int @ X3 @ ( sup_sup_int @ A @ B ) ) ) ).
% less_supI2
thf(fact_186_sup_Ostrict__boundedE,axiom,
! [B: a > $o,C2: a > $o,A: a > $o] :
( ( ord_less_a_o @ ( sup_sup_a_o @ B @ C2 ) @ A )
=> ~ ( ( ord_less_a_o @ B @ A )
=> ~ ( ord_less_a_o @ C2 @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_187_sup_Ostrict__boundedE,axiom,
! [B: set_a,C2: set_a,A: set_a] :
( ( ord_less_set_a @ ( sup_sup_set_a @ B @ C2 ) @ A )
=> ~ ( ( ord_less_set_a @ B @ A )
=> ~ ( ord_less_set_a @ C2 @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_188_sup_Ostrict__boundedE,axiom,
! [B: int,C2: int,A: int] :
( ( ord_less_int @ ( sup_sup_int @ B @ C2 ) @ A )
=> ~ ( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ C2 @ A ) ) ) ).
% sup.strict_boundedE
thf(fact_189_sup_Ostrict__order__iff,axiom,
( ord_less_a_o
= ( ^ [B2: a > $o,A2: a > $o] :
( ( A2
= ( sup_sup_a_o @ A2 @ B2 ) )
& ( A2 != B2 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_190_sup_Ostrict__order__iff,axiom,
( ord_less_set_a
= ( ^ [B2: set_a,A2: set_a] :
( ( A2
= ( sup_sup_set_a @ A2 @ B2 ) )
& ( A2 != B2 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_191_sup_Ostrict__order__iff,axiom,
( ord_less_int
= ( ^ [B2: int,A2: int] :
( ( A2
= ( sup_sup_int @ A2 @ B2 ) )
& ( A2 != B2 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_192_sup_Ostrict__coboundedI1,axiom,
! [C2: a > $o,A: a > $o,B: a > $o] :
( ( ord_less_a_o @ C2 @ A )
=> ( ord_less_a_o @ C2 @ ( sup_sup_a_o @ A @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_193_sup_Ostrict__coboundedI1,axiom,
! [C2: set_a,A: set_a,B: set_a] :
( ( ord_less_set_a @ C2 @ A )
=> ( ord_less_set_a @ C2 @ ( sup_sup_set_a @ A @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_194_sup_Ostrict__coboundedI1,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_int @ C2 @ A )
=> ( ord_less_int @ C2 @ ( sup_sup_int @ A @ B ) ) ) ).
% sup.strict_coboundedI1
thf(fact_195_sup_Ostrict__coboundedI2,axiom,
! [C2: a > $o,B: a > $o,A: a > $o] :
( ( ord_less_a_o @ C2 @ B )
=> ( ord_less_a_o @ C2 @ ( sup_sup_a_o @ A @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_196_sup_Ostrict__coboundedI2,axiom,
! [C2: set_a,B: set_a,A: set_a] :
( ( ord_less_set_a @ C2 @ B )
=> ( ord_less_set_a @ C2 @ ( sup_sup_set_a @ A @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_197_sup_Ostrict__coboundedI2,axiom,
! [C2: int,B: int,A: int] :
( ( ord_less_int @ C2 @ B )
=> ( ord_less_int @ C2 @ ( sup_sup_int @ A @ B ) ) ) ).
% sup.strict_coboundedI2
thf(fact_198_the__elem__eq,axiom,
! [X3: a] :
( ( the_elem_a @ ( insert_a @ X3 @ bot_bot_set_a ) )
= X3 ) ).
% the_elem_eq
thf(fact_199_bot__apply,axiom,
( bot_bot_a_o
= ( ^ [X2: a] : bot_bot_o ) ) ).
% bot_apply
thf(fact_200_is__singletonI,axiom,
! [X3: a] : ( is_singleton_a @ ( insert_a @ X3 @ bot_bot_set_a ) ) ).
% is_singletonI
thf(fact_201_diff__strict__mono,axiom,
! [A: int,B: int,D: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ D @ C2 )
=> ( ord_less_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_202_sup1CI,axiom,
! [B3: a > $o,X3: a,A3: a > $o] :
( ( ~ ( B3 @ X3 )
=> ( A3 @ X3 ) )
=> ( sup_sup_a_o @ A3 @ B3 @ X3 ) ) ).
% sup1CI
thf(fact_203_sup1E,axiom,
! [A3: a > $o,B3: a > $o,X3: a] :
( ( sup_sup_a_o @ A3 @ B3 @ X3 )
=> ( ~ ( A3 @ X3 )
=> ( B3 @ X3 ) ) ) ).
% sup1E
thf(fact_204_sup1I1,axiom,
! [A3: a > $o,X3: a,B3: a > $o] :
( ( A3 @ X3 )
=> ( sup_sup_a_o @ A3 @ B3 @ X3 ) ) ).
% sup1I1
thf(fact_205_sup1I2,axiom,
! [B3: a > $o,X3: a,A3: a > $o] :
( ( B3 @ X3 )
=> ( sup_sup_a_o @ A3 @ B3 @ X3 ) ) ).
% sup1I2
thf(fact_206_not__psubset__empty,axiom,
! [A3: set_a] :
~ ( ord_less_set_a @ A3 @ bot_bot_set_a ) ).
% not_psubset_empty
thf(fact_207_psubset__imp__ex__mem,axiom,
! [A3: set_a,B3: set_a] :
( ( ord_less_set_a @ A3 @ B3 )
=> ? [B6: a] : ( member_a @ B6 @ ( minus_minus_set_a @ B3 @ A3 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_208_minus__set__def,axiom,
( minus_minus_set_a
= ( ^ [A4: set_a,B5: set_a] :
( collect_a
@ ( minus_minus_a_o
@ ^ [X2: a] : ( member_a @ X2 @ A4 )
@ ^ [X2: a] : ( member_a @ X2 @ B5 ) ) ) ) ) ).
% minus_set_def
thf(fact_209_is__singleton__the__elem,axiom,
( is_singleton_a
= ( ^ [A4: set_a] :
( A4
= ( insert_a @ ( the_elem_a @ A4 ) @ bot_bot_set_a ) ) ) ) ).
% is_singleton_the_elem
thf(fact_210_is__singletonI_H,axiom,
! [A3: set_a] :
( ( A3 != bot_bot_set_a )
=> ( ! [X4: a,Y4: a] :
( ( member_a @ X4 @ A3 )
=> ( ( member_a @ Y4 @ A3 )
=> ( X4 = Y4 ) ) )
=> ( is_singleton_a @ A3 ) ) ) ).
% is_singletonI'
thf(fact_211_dual__order_Ostrict__implies__not__eq,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_set_a @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_212_dual__order_Ostrict__implies__not__eq,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_213_order_Ostrict__implies__not__eq,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_214_order_Ostrict__implies__not__eq,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_215_not__less__iff__gr__or__eq,axiom,
! [X3: int,Y2: int] :
( ( ~ ( ord_less_int @ X3 @ Y2 ) )
= ( ( ord_less_int @ Y2 @ X3 )
| ( X3 = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_216_dual__order_Ostrict__trans,axiom,
! [B: set_a,A: set_a,C2: set_a] :
( ( ord_less_set_a @ B @ A )
=> ( ( ord_less_set_a @ C2 @ B )
=> ( ord_less_set_a @ C2 @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_217_dual__order_Ostrict__trans,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C2 @ B )
=> ( ord_less_int @ C2 @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_218_linorder__less__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A5: int,B6: int] :
( ( ord_less_int @ A5 @ B6 )
=> ( P @ A5 @ B6 ) )
=> ( ! [A5: int] : ( P @ A5 @ A5 )
=> ( ! [A5: int,B6: int] :
( ( P @ B6 @ A5 )
=> ( P @ A5 @ B6 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_219_less__imp__not__less,axiom,
! [X3: set_a,Y2: set_a] :
( ( ord_less_set_a @ X3 @ Y2 )
=> ~ ( ord_less_set_a @ Y2 @ X3 ) ) ).
% less_imp_not_less
thf(fact_220_less__imp__not__less,axiom,
! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ~ ( ord_less_int @ Y2 @ X3 ) ) ).
% less_imp_not_less
thf(fact_221_order_Ostrict__trans,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_set_a @ B @ C2 )
=> ( ord_less_set_a @ A @ C2 ) ) ) ).
% order.strict_trans
thf(fact_222_order_Ostrict__trans,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ B @ C2 )
=> ( ord_less_int @ A @ C2 ) ) ) ).
% order.strict_trans
thf(fact_223_dual__order_Oirrefl,axiom,
! [A: set_a] :
~ ( ord_less_set_a @ A @ A ) ).
% dual_order.irrefl
thf(fact_224_dual__order_Oirrefl,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% dual_order.irrefl
thf(fact_225_linorder__cases,axiom,
! [X3: int,Y2: int] :
( ~ ( ord_less_int @ X3 @ Y2 )
=> ( ( X3 != Y2 )
=> ( ord_less_int @ Y2 @ X3 ) ) ) ).
% linorder_cases
thf(fact_226_less__imp__triv,axiom,
! [X3: set_a,Y2: set_a,P: $o] :
( ( ord_less_set_a @ X3 @ Y2 )
=> ( ( ord_less_set_a @ Y2 @ X3 )
=> P ) ) ).
% less_imp_triv
thf(fact_227_less__imp__triv,axiom,
! [X3: int,Y2: int,P: $o] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ( ord_less_int @ Y2 @ X3 )
=> P ) ) ).
% less_imp_triv
thf(fact_228_less__imp__not__eq2,axiom,
! [X3: set_a,Y2: set_a] :
( ( ord_less_set_a @ X3 @ Y2 )
=> ( Y2 != X3 ) ) ).
% less_imp_not_eq2
thf(fact_229_less__imp__not__eq2,axiom,
! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( Y2 != X3 ) ) ).
% less_imp_not_eq2
thf(fact_230_antisym__conv3,axiom,
! [Y2: int,X3: int] :
( ~ ( ord_less_int @ Y2 @ X3 )
=> ( ( ~ ( ord_less_int @ X3 @ Y2 ) )
= ( X3 = Y2 ) ) ) ).
% antisym_conv3
thf(fact_231_less__not__sym,axiom,
! [X3: set_a,Y2: set_a] :
( ( ord_less_set_a @ X3 @ Y2 )
=> ~ ( ord_less_set_a @ Y2 @ X3 ) ) ).
% less_not_sym
thf(fact_232_less__not__sym,axiom,
! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ~ ( ord_less_int @ Y2 @ X3 ) ) ).
% less_not_sym
thf(fact_233_less__imp__not__eq,axiom,
! [X3: set_a,Y2: set_a] :
( ( ord_less_set_a @ X3 @ Y2 )
=> ( X3 != Y2 ) ) ).
% less_imp_not_eq
thf(fact_234_less__imp__not__eq,axiom,
! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( X3 != Y2 ) ) ).
% less_imp_not_eq
thf(fact_235_dual__order_Oasym,axiom,
! [B: set_a,A: set_a] :
( ( ord_less_set_a @ B @ A )
=> ~ ( ord_less_set_a @ A @ B ) ) ).
% dual_order.asym
thf(fact_236_dual__order_Oasym,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ A @ B ) ) ).
% dual_order.asym
thf(fact_237_ord__less__eq__trans,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_set_a @ A @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_238_ord__less__eq__trans,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_int @ A @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_239_ord__eq__less__trans,axiom,
! [A: set_a,B: set_a,C2: set_a] :
( ( A = B )
=> ( ( ord_less_set_a @ B @ C2 )
=> ( ord_less_set_a @ A @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_240_ord__eq__less__trans,axiom,
! [A: int,B: int,C2: int] :
( ( A = B )
=> ( ( ord_less_int @ B @ C2 )
=> ( ord_less_int @ A @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_241_less__irrefl,axiom,
! [X3: set_a] :
~ ( ord_less_set_a @ X3 @ X3 ) ).
% less_irrefl
thf(fact_242_less__irrefl,axiom,
! [X3: int] :
~ ( ord_less_int @ X3 @ X3 ) ).
% less_irrefl
thf(fact_243_less__linear,axiom,
! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
| ( X3 = Y2 )
| ( ord_less_int @ Y2 @ X3 ) ) ).
% less_linear
thf(fact_244_less__trans,axiom,
! [X3: set_a,Y2: set_a,Z2: set_a] :
( ( ord_less_set_a @ X3 @ Y2 )
=> ( ( ord_less_set_a @ Y2 @ Z2 )
=> ( ord_less_set_a @ X3 @ Z2 ) ) ) ).
% less_trans
thf(fact_245_less__trans,axiom,
! [X3: int,Y2: int,Z2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ( ord_less_int @ Y2 @ Z2 )
=> ( ord_less_int @ X3 @ Z2 ) ) ) ).
% less_trans
thf(fact_246_less__asym_H,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ~ ( ord_less_set_a @ B @ A ) ) ).
% less_asym'
thf(fact_247_less__asym_H,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% less_asym'
thf(fact_248_less__asym,axiom,
! [X3: set_a,Y2: set_a] :
( ( ord_less_set_a @ X3 @ Y2 )
=> ~ ( ord_less_set_a @ Y2 @ X3 ) ) ).
% less_asym
thf(fact_249_less__asym,axiom,
! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ~ ( ord_less_int @ Y2 @ X3 ) ) ).
% less_asym
thf(fact_250_less__imp__neq,axiom,
! [X3: set_a,Y2: set_a] :
( ( ord_less_set_a @ X3 @ Y2 )
=> ( X3 != Y2 ) ) ).
% less_imp_neq
thf(fact_251_less__imp__neq,axiom,
! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( X3 != Y2 ) ) ).
% less_imp_neq
thf(fact_252_order_Oasym,axiom,
! [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
=> ~ ( ord_less_set_a @ B @ A ) ) ).
% order.asym
thf(fact_253_order_Oasym,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order.asym
thf(fact_254_neq__iff,axiom,
! [X3: int,Y2: int] :
( ( X3 != Y2 )
= ( ( ord_less_int @ X3 @ Y2 )
| ( ord_less_int @ Y2 @ X3 ) ) ) ).
% neq_iff
thf(fact_255_neqE,axiom,
! [X3: int,Y2: int] :
( ( X3 != Y2 )
=> ( ~ ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_int @ Y2 @ X3 ) ) ) ).
% neqE
thf(fact_256_gt__ex,axiom,
! [X3: int] :
? [X_1: int] : ( ord_less_int @ X3 @ X_1 ) ).
% gt_ex
thf(fact_257_lt__ex,axiom,
! [X3: int] :
? [Y4: int] : ( ord_less_int @ Y4 @ X3 ) ).
% lt_ex
thf(fact_258_order__less__subst2,axiom,
! [A: int,B: int,F2: int > set_a,C2: set_a] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_set_a @ ( F2 @ B ) @ C2 )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_set_a @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_set_a @ ( F2 @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_259_order__less__subst2,axiom,
! [A: set_a,B: set_a,F2: set_a > int,C2: int] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_int @ ( F2 @ B ) @ C2 )
=> ( ! [X4: set_a,Y4: set_a] :
( ( ord_less_set_a @ X4 @ Y4 )
=> ( ord_less_int @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_int @ ( F2 @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_260_order__less__subst2,axiom,
! [A: set_a,B: set_a,F2: set_a > set_a,C2: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ord_less_set_a @ ( F2 @ B ) @ C2 )
=> ( ! [X4: set_a,Y4: set_a] :
( ( ord_less_set_a @ X4 @ Y4 )
=> ( ord_less_set_a @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_set_a @ ( F2 @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_261_order__less__subst2,axiom,
! [A: int,B: int,F2: int > int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ ( F2 @ B ) @ C2 )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_int @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_int @ ( F2 @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_262_order__less__subst1,axiom,
! [A: int,F2: set_a > int,B: set_a,C2: set_a] :
( ( ord_less_int @ A @ ( F2 @ B ) )
=> ( ( ord_less_set_a @ B @ C2 )
=> ( ! [X4: set_a,Y4: set_a] :
( ( ord_less_set_a @ X4 @ Y4 )
=> ( ord_less_int @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F2 @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_263_order__less__subst1,axiom,
! [A: set_a,F2: int > set_a,B: int,C2: int] :
( ( ord_less_set_a @ A @ ( F2 @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_set_a @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_set_a @ A @ ( F2 @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_264_order__less__subst1,axiom,
! [A: set_a,F2: set_a > set_a,B: set_a,C2: set_a] :
( ( ord_less_set_a @ A @ ( F2 @ B ) )
=> ( ( ord_less_set_a @ B @ C2 )
=> ( ! [X4: set_a,Y4: set_a] :
( ( ord_less_set_a @ X4 @ Y4 )
=> ( ord_less_set_a @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_set_a @ A @ ( F2 @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_265_order__less__subst1,axiom,
! [A: int,F2: int > int,B: int,C2: int] :
( ( ord_less_int @ A @ ( F2 @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_int @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F2 @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_266_ord__less__eq__subst,axiom,
! [A: int,B: int,F2: int > set_a,C2: set_a] :
( ( ord_less_int @ A @ B )
=> ( ( ( F2 @ B )
= C2 )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_set_a @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_set_a @ ( F2 @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_267_ord__less__eq__subst,axiom,
! [A: set_a,B: set_a,F2: set_a > int,C2: int] :
( ( ord_less_set_a @ A @ B )
=> ( ( ( F2 @ B )
= C2 )
=> ( ! [X4: set_a,Y4: set_a] :
( ( ord_less_set_a @ X4 @ Y4 )
=> ( ord_less_int @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_int @ ( F2 @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_268_ord__less__eq__subst,axiom,
! [A: set_a,B: set_a,F2: set_a > set_a,C2: set_a] :
( ( ord_less_set_a @ A @ B )
=> ( ( ( F2 @ B )
= C2 )
=> ( ! [X4: set_a,Y4: set_a] :
( ( ord_less_set_a @ X4 @ Y4 )
=> ( ord_less_set_a @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_set_a @ ( F2 @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_269_ord__less__eq__subst,axiom,
! [A: int,B: int,F2: int > int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ( F2 @ B )
= C2 )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_int @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_int @ ( F2 @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_270_ord__eq__less__subst,axiom,
! [A: set_a,F2: int > set_a,B: int,C2: int] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_set_a @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_set_a @ A @ ( F2 @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_271_ord__eq__less__subst,axiom,
! [A: int,F2: set_a > int,B: set_a,C2: set_a] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_set_a @ B @ C2 )
=> ( ! [X4: set_a,Y4: set_a] :
( ( ord_less_set_a @ X4 @ Y4 )
=> ( ord_less_int @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F2 @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_272_ord__eq__less__subst,axiom,
! [A: set_a,F2: set_a > set_a,B: set_a,C2: set_a] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_set_a @ B @ C2 )
=> ( ! [X4: set_a,Y4: set_a] :
( ( ord_less_set_a @ X4 @ Y4 )
=> ( ord_less_set_a @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_set_a @ A @ ( F2 @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_273_ord__eq__less__subst,axiom,
! [A: int,F2: int > int,B: int,C2: int] :
( ( A
= ( F2 @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_int @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F2 @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_274_bot__fun__def,axiom,
( bot_bot_a_o
= ( ^ [X2: a] : bot_bot_o ) ) ).
% bot_fun_def
thf(fact_275_diff__right__commute,axiom,
! [A: int,C2: int,B: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ C2 ) @ B )
= ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C2 ) ) ).
% diff_right_commute
thf(fact_276_diff__eq__diff__eq,axiom,
! [A: int,B: int,C2: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C2 @ D ) )
=> ( ( A = B )
= ( C2 = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_277_is__singleton__def,axiom,
( is_singleton_a
= ( ^ [A4: set_a] :
? [X2: a] :
( A4
= ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ).
% is_singleton_def
thf(fact_278_is__singletonE,axiom,
! [A3: set_a] :
( ( is_singleton_a @ A3 )
=> ~ ! [X4: a] :
( A3
!= ( insert_a @ X4 @ bot_bot_set_a ) ) ) ).
% is_singletonE
thf(fact_279_bot_Onot__eq__extremum,axiom,
! [A: a > $o] :
( ( A != bot_bot_a_o )
= ( ord_less_a_o @ bot_bot_a_o @ A ) ) ).
% bot.not_eq_extremum
thf(fact_280_bot_Onot__eq__extremum,axiom,
! [A: set_a] :
( ( A != bot_bot_set_a )
= ( ord_less_set_a @ bot_bot_set_a @ A ) ) ).
% bot.not_eq_extremum
thf(fact_281_bot_Oextremum__strict,axiom,
! [A: a > $o] :
~ ( ord_less_a_o @ A @ bot_bot_a_o ) ).
% bot.extremum_strict
thf(fact_282_bot_Oextremum__strict,axiom,
! [A: set_a] :
~ ( ord_less_set_a @ A @ bot_bot_set_a ) ).
% bot.extremum_strict
thf(fact_283_diff__strict__right__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B @ C2 ) ) ) ).
% diff_strict_right_mono
thf(fact_284_diff__strict__left__mono,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_int @ ( minus_minus_int @ C2 @ A ) @ ( minus_minus_int @ C2 @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_285_diff__eq__diff__less,axiom,
! [A: int,B: int,C2: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C2 @ D ) )
=> ( ( ord_less_int @ A @ B )
= ( ord_less_int @ C2 @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_286_sup__Un__eq,axiom,
! [R: set_a,S: set_a] :
( ( sup_sup_a_o
@ ^ [X2: a] : ( member_a @ X2 @ R )
@ ^ [X2: a] : ( member_a @ X2 @ S ) )
= ( ^ [X2: a] : ( member_a @ X2 @ ( sup_sup_set_a @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_287_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X2: a] : ( member_a @ X2 @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_288_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_289_psubset__trans,axiom,
! [A3: set_a,B3: set_a,C: set_a] :
( ( ord_less_set_a @ A3 @ B3 )
=> ( ( ord_less_set_a @ B3 @ C )
=> ( ord_less_set_a @ A3 @ C ) ) ) ).
% psubset_trans
thf(fact_290_less__set__def,axiom,
( ord_less_set_a
= ( ^ [A4: set_a,B5: set_a] :
( ord_less_a_o
@ ^ [X2: a] : ( member_a @ X2 @ A4 )
@ ^ [X2: a] : ( member_a @ X2 @ B5 ) ) ) ) ).
% less_set_def
thf(fact_291_psubsetD,axiom,
! [A3: set_a,B3: set_a,C2: a] :
( ( ord_less_set_a @ A3 @ B3 )
=> ( ( member_a @ C2 @ A3 )
=> ( member_a @ C2 @ B3 ) ) ) ).
% psubsetD
thf(fact_292_remove__def,axiom,
( remove_a
= ( ^ [X2: a,A4: set_a] : ( minus_minus_set_a @ A4 @ ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ).
% remove_def
thf(fact_293_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_294_member__remove,axiom,
! [X3: a,Y2: a,A3: set_a] :
( ( member_a @ X3 @ ( remove_a @ Y2 @ A3 ) )
= ( ( member_a @ X3 @ A3 )
& ( X3 != Y2 ) ) ) ).
% member_remove
thf(fact_295_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_296_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_297_Tree_Oset__intros_I1_J,axiom,
! [Y2: a,X21: binary1439146945Tree_a,X22: a,X23: binary1439146945Tree_a] :
( ( member_a @ Y2 @ ( binary256242811Tree_a @ X21 ) )
=> ( member_a @ Y2 @ ( binary256242811Tree_a @ ( binary717961607le_T_a @ X21 @ X22 @ X23 ) ) ) ) ).
% Tree.set_intros(1)
thf(fact_298_Tree_Oset__cases,axiom,
! [E: a,A: binary1439146945Tree_a] :
( ( member_a @ E @ ( binary256242811Tree_a @ A ) )
=> ( ! [Z1: binary1439146945Tree_a] :
( ? [Z22: a,Z3: binary1439146945Tree_a] :
( A
= ( binary717961607le_T_a @ Z1 @ Z22 @ Z3 ) )
=> ~ ( member_a @ E @ ( binary256242811Tree_a @ Z1 ) ) )
=> ( ! [Z1: binary1439146945Tree_a,Z3: binary1439146945Tree_a] :
( A
!= ( binary717961607le_T_a @ Z1 @ E @ Z3 ) )
=> ~ ! [Z1: binary1439146945Tree_a,Z22: a,Z3: binary1439146945Tree_a] :
( ( A
= ( binary717961607le_T_a @ Z1 @ Z22 @ Z3 ) )
=> ~ ( member_a @ E @ ( binary256242811Tree_a @ Z3 ) ) ) ) ) ) ).
% Tree.set_cases
thf(fact_299_Tree_Osimps_I14_J,axiom,
( ( binary256242811Tree_a @ binary476621312_Tip_a )
= bot_bot_set_a ) ).
% Tree.simps(14)
thf(fact_300_Set_Ois__empty__def,axiom,
( is_empty_a
= ( ^ [A4: set_a] : A4 = bot_bot_set_a ) ) ).
% Set.is_empty_def
thf(fact_301_the__elem__def,axiom,
( the_elem_a
= ( ^ [X5: set_a] :
( the_a
@ ^ [X2: a] :
( X5
= ( insert_a @ X2 @ bot_bot_set_a ) ) ) ) ) ).
% the_elem_def
thf(fact_302_the__sym__eq__trivial,axiom,
! [X3: a] :
( ( the_a
@ ( ^ [Y3: a,Z: a] : Y3 = Z
@ X3 ) )
= X3 ) ).
% the_sym_eq_trivial
thf(fact_303_the__eq__trivial,axiom,
! [A: a] :
( ( the_a
@ ^ [X2: a] : X2 = A )
= A ) ).
% the_eq_trivial
thf(fact_304_the__equality,axiom,
! [P: a > $o,A: a] :
( ( P @ A )
=> ( ! [X4: a] :
( ( P @ X4 )
=> ( X4 = A ) )
=> ( ( the_a @ P )
= A ) ) ) ).
% the_equality
thf(fact_305_theI,axiom,
! [P: a > $o,A: a] :
( ( P @ A )
=> ( ! [X4: a] :
( ( P @ X4 )
=> ( X4 = A ) )
=> ( P @ ( the_a @ P ) ) ) ) ).
% theI
thf(fact_306_theI_H,axiom,
! [P: a > $o] :
( ? [X: a] :
( ( P @ X )
& ! [Y4: a] :
( ( P @ Y4 )
=> ( Y4 = X ) ) )
=> ( P @ ( the_a @ P ) ) ) ).
% theI'
thf(fact_307_theI2,axiom,
! [P: a > $o,A: a,Q: a > $o] :
( ( P @ A )
=> ( ! [X4: a] :
( ( P @ X4 )
=> ( X4 = A ) )
=> ( ! [X4: a] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( Q @ ( the_a @ P ) ) ) ) ) ).
% theI2
thf(fact_308_If__def,axiom,
( if_a
= ( ^ [P2: $o,X2: a,Y: a] :
( the_a
@ ^ [Z4: a] :
( ( P2
=> ( Z4 = X2 ) )
& ( ~ P2
=> ( Z4 = Y ) ) ) ) ) ) ).
% If_def
thf(fact_309_the1I2,axiom,
! [P: a > $o,Q: a > $o] :
( ? [X: a] :
( ( P @ X )
& ! [Y4: a] :
( ( P @ Y4 )
=> ( Y4 = X ) ) )
=> ( ! [X4: a] :
( ( P @ X4 )
=> ( Q @ X4 ) )
=> ( Q @ ( the_a @ P ) ) ) ) ).
% the1I2
thf(fact_310_the1__equality,axiom,
! [P: a > $o,A: a] :
( ? [X: a] :
( ( P @ X )
& ! [Y4: a] :
( ( P @ Y4 )
=> ( Y4 = X ) ) )
=> ( ( P @ A )
=> ( ( the_a @ P )
= A ) ) ) ).
% the1_equality
thf(fact_311_psubset__insert__iff,axiom,
! [A3: set_a,X3: a,B3: set_a] :
( ( ord_less_set_a @ A3 @ ( insert_a @ X3 @ B3 ) )
= ( ( ( member_a @ X3 @ B3 )
=> ( ord_less_set_a @ A3 @ B3 ) )
& ( ~ ( member_a @ X3 @ B3 )
=> ( ( ( member_a @ X3 @ A3 )
=> ( ord_less_set_a @ ( minus_minus_set_a @ A3 @ ( insert_a @ X3 @ bot_bot_set_a ) ) @ B3 ) )
& ( ~ ( member_a @ X3 @ A3 )
=> ( ord_less_eq_set_a @ A3 @ B3 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_312_pairwise__alt,axiom,
( pairwise_a
= ( ^ [R2: a > a > $o,S2: set_a] :
! [X2: a] :
( ( member_a @ X2 @ S2 )
=> ! [Y: a] :
( ( member_a @ Y @ ( minus_minus_set_a @ S2 @ ( insert_a @ X2 @ bot_bot_set_a ) ) )
=> ( R2 @ X2 @ Y ) ) ) ) ) ).
% pairwise_alt
thf(fact_313_order__refl,axiom,
! [X3: set_a] : ( ord_less_eq_set_a @ X3 @ X3 ) ).
% order_refl
thf(fact_314_order__refl,axiom,
! [X3: int] : ( ord_less_eq_int @ X3 @ X3 ) ).
% order_refl
thf(fact_315_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_316_subsetI,axiom,
! [A3: set_a,B3: set_a] :
( ! [X4: a] :
( ( member_a @ X4 @ A3 )
=> ( member_a @ X4 @ B3 ) )
=> ( ord_less_eq_set_a @ A3 @ B3 ) ) ).
% subsetI
thf(fact_317_sup_Obounded__iff,axiom,
! [B: a > $o,C2: a > $o,A: a > $o] :
( ( ord_less_eq_a_o @ ( sup_sup_a_o @ B @ C2 ) @ A )
= ( ( ord_less_eq_a_o @ B @ A )
& ( ord_less_eq_a_o @ C2 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_318_sup_Obounded__iff,axiom,
! [B: int,C2: int,A: int] :
( ( ord_less_eq_int @ ( sup_sup_int @ B @ C2 ) @ A )
= ( ( ord_less_eq_int @ B @ A )
& ( ord_less_eq_int @ C2 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_319_sup_Obounded__iff,axiom,
! [B: set_a,C2: set_a,A: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ B @ C2 ) @ A )
= ( ( ord_less_eq_set_a @ B @ A )
& ( ord_less_eq_set_a @ C2 @ A ) ) ) ).
% sup.bounded_iff
thf(fact_320_le__sup__iff,axiom,
! [X3: a > $o,Y2: a > $o,Z2: a > $o] :
( ( ord_less_eq_a_o @ ( sup_sup_a_o @ X3 @ Y2 ) @ Z2 )
= ( ( ord_less_eq_a_o @ X3 @ Z2 )
& ( ord_less_eq_a_o @ Y2 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_321_le__sup__iff,axiom,
! [X3: int,Y2: int,Z2: int] :
( ( ord_less_eq_int @ ( sup_sup_int @ X3 @ Y2 ) @ Z2 )
= ( ( ord_less_eq_int @ X3 @ Z2 )
& ( ord_less_eq_int @ Y2 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_322_le__sup__iff,axiom,
! [X3: set_a,Y2: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ X3 @ Y2 ) @ Z2 )
= ( ( ord_less_eq_set_a @ X3 @ Z2 )
& ( ord_less_eq_set_a @ Y2 @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_323_empty__subsetI,axiom,
! [A3: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A3 ) ).
% empty_subsetI
thf(fact_324_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_325_insert__subset,axiom,
! [X3: a,A3: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X3 @ A3 ) @ B3 )
= ( ( member_a @ X3 @ B3 )
& ( ord_less_eq_set_a @ A3 @ B3 ) ) ) ).
% insert_subset
thf(fact_326_Un__subset__iff,axiom,
! [A3: set_a,B3: set_a,C: set_a] :
( ( ord_less_eq_set_a @ ( sup_sup_set_a @ A3 @ B3 ) @ C )
= ( ( ord_less_eq_set_a @ A3 @ C )
& ( ord_less_eq_set_a @ B3 @ C ) ) ) ).
% Un_subset_iff
thf(fact_327_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_328_singleton__insert__inj__eq_H,axiom,
! [A: a,A3: set_a,B: a] :
( ( ( insert_a @ A @ A3 )
= ( insert_a @ B @ bot_bot_set_a ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A3 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_329_singleton__insert__inj__eq,axiom,
! [B: a,A: a,A3: set_a] :
( ( ( insert_a @ B @ bot_bot_set_a )
= ( insert_a @ A @ A3 ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A3 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_330_Diff__eq__empty__iff,axiom,
! [A3: set_a,B3: set_a] :
( ( ( minus_minus_set_a @ A3 @ B3 )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A3 @ B3 ) ) ).
% Diff_eq_empty_iff
thf(fact_331_leD,axiom,
! [Y2: set_a,X3: set_a] :
( ( ord_less_eq_set_a @ Y2 @ X3 )
=> ~ ( ord_less_set_a @ X3 @ Y2 ) ) ).
% leD
thf(fact_332_leD,axiom,
! [Y2: int,X3: int] :
( ( ord_less_eq_int @ Y2 @ X3 )
=> ~ ( ord_less_int @ X3 @ Y2 ) ) ).
% leD
thf(fact_333_leI,axiom,
! [X3: int,Y2: int] :
( ~ ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_eq_int @ Y2 @ X3 ) ) ).
% leI
thf(fact_334_le__less,axiom,
( ord_less_eq_set_a
= ( ^ [X2: set_a,Y: set_a] :
( ( ord_less_set_a @ X2 @ Y )
| ( X2 = Y ) ) ) ) ).
% le_less
thf(fact_335_le__less,axiom,
( ord_less_eq_int
= ( ^ [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
| ( X2 = Y ) ) ) ) ).
% le_less
thf(fact_336_less__le,axiom,
( ord_less_set_a
= ( ^ [X2: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X2 @ Y )
& ( X2 != Y ) ) ) ) ).
% less_le
thf(fact_337_less__le,axiom,
( ord_less_int
= ( ^ [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
& ( X2 != Y ) ) ) ) ).
% less_le
thf(fact_338_order__le__less__subst1,axiom,
! [A: set_a,F2: int > set_a,B: int,C2: int] :
( ( ord_less_eq_set_a @ A @ ( F2 @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_set_a @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_set_a @ A @ ( F2 @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_339_order__le__less__subst1,axiom,
! [A: set_a,F2: set_a > set_a,B: set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ ( F2 @ B ) )
=> ( ( ord_less_set_a @ B @ C2 )
=> ( ! [X4: set_a,Y4: set_a] :
( ( ord_less_set_a @ X4 @ Y4 )
=> ( ord_less_set_a @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_set_a @ A @ ( F2 @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_340_order__le__less__subst1,axiom,
! [A: int,F2: set_a > int,B: set_a,C2: set_a] :
( ( ord_less_eq_int @ A @ ( F2 @ B ) )
=> ( ( ord_less_set_a @ B @ C2 )
=> ( ! [X4: set_a,Y4: set_a] :
( ( ord_less_set_a @ X4 @ Y4 )
=> ( ord_less_int @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F2 @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_341_order__le__less__subst1,axiom,
! [A: int,F2: int > int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ ( F2 @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_int @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F2 @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_342_order__le__less__subst2,axiom,
! [A: set_a,B: set_a,F2: set_a > set_a,C2: set_a] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_set_a @ ( F2 @ B ) @ C2 )
=> ( ! [X4: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y4 )
=> ( ord_less_eq_set_a @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_set_a @ ( F2 @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_343_order__le__less__subst2,axiom,
! [A: set_a,B: set_a,F2: set_a > int,C2: int] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ( ord_less_int @ ( F2 @ B ) @ C2 )
=> ( ! [X4: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_int @ ( F2 @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_344_order__le__less__subst2,axiom,
! [A: int,B: int,F2: int > set_a,C2: set_a] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_set_a @ ( F2 @ B ) @ C2 )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_set_a @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_set_a @ ( F2 @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_345_order__le__less__subst2,axiom,
! [A: int,B: int,F2: int > int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ ( F2 @ B ) @ C2 )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_int @ ( F2 @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_346_order__less__le__subst1,axiom,
! [A: int,F2: set_a > int,B: set_a,C2: set_a] :
( ( ord_less_int @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_set_a @ B @ C2 )
=> ( ! [X4: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F2 @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_347_order__less__le__subst1,axiom,
! [A: set_a,F2: int > set_a,B: int,C2: int] :
( ( ord_less_set_a @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_set_a @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_set_a @ A @ ( F2 @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_348_order__less__le__subst1,axiom,
! [A: int,F2: int > int,B: int,C2: int] :
( ( ord_less_int @ A @ ( F2 @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F2 @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_349_order__less__le__subst2,axiom,
! [A: int,B: int,F2: int > int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F2 @ B ) @ C2 )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_int @ ( F2 @ X4 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_int @ ( F2 @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_350_not__le,axiom,
! [X3: int,Y2: int] :
( ( ~ ( ord_less_eq_int @ X3 @ Y2 ) )
= ( ord_less_int @ Y2 @ X3 ) ) ).
% not_le
thf(fact_351_not__less,axiom,
! [X3: int,Y2: int] :
( ( ~ ( ord_less_int @ X3 @ Y2 ) )
= ( ord_less_eq_int @ Y2 @ X3 ) ) ).
% not_less
thf(fact_352_le__neq__trans,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( A != B )
=> ( ord_less_int @ A @ B ) ) ) ).
% le_neq_trans
thf(fact_353_antisym__conv1,axiom,
! [X3: int,Y2: int] :
( ~ ( ord_less_int @ X3 @ Y2 )
=> ( ( ord_less_eq_int @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ).
% antisym_conv1
thf(fact_354_antisym__conv2,axiom,
! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ( ~ ( ord_less_int @ X3 @ Y2 ) )
= ( X3 = Y2 ) ) ) ).
% antisym_conv2
% Helper facts (3)
thf(help_If_3_1_If_001tf__a_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001tf__a_T,axiom,
! [X3: a,Y2: a] :
( ( if_a @ $false @ X3 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001tf__a_T,axiom,
! [X3: a,Y2: a] :
( ( if_a @ $true @ X3 @ Y2 )
= X3 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
! [X4: a] :
( ( member_a @ X4 @ ( binary504661350_eqs_a @ h @ e ) )
=> ( ( h @ X4 )
!= ( h @ x ) ) ) ).
%------------------------------------------------------------------------------