TPTP Problem File: ITP033^1.p
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%------------------------------------------------------------------------------
% File : ITP033^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer BinaryTree problem prob_508__3255320_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_508__3255320_1 [Des21]
% Status : Theorem
% Rating : 0.30 v8.2.0, 0.23 v8.1.0, 0.27 v7.5.0
% Syntax : Number of formulae : 390 ( 96 unt; 36 typ; 0 def)
% Number of atoms : 1164 ( 241 equ; 0 cnn)
% Maximal formula atoms : 10 ( 3 avg)
% Number of connectives : 3147 ( 123 ~; 18 |; 53 &;2350 @)
% ( 0 <=>; 603 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 8 avg)
% Number of types : 6 ( 5 usr)
% Number of type conns : 570 ( 570 >; 0 *; 0 +; 0 <<)
% Number of symbols : 33 ( 31 usr; 7 con; 0-4 aty)
% Number of variables : 1135 ( 119 ^;1008 !; 8 ?;1135 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:31:27.980
%------------------------------------------------------------------------------
% Could-be-implicit typings (5)
thf(ty_n_t__Product____Type__Oprod_It__BinaryTree____Mirabelle____mlzyzwgbkd__OTree_Itf__a_J_Mtf__a_J,type,
produc376588196ee_a_a: $tType ).
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 (31)
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_BinaryTree__Mirabelle__mlzyzwgbkd_Owrmrm_001tf__a,type,
binary1774981152rmrm_a: ( a > int ) > binary1439146945Tree_a > produc376588196ee_a_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_001_Eo,type,
ord_less_eq_o: $o > $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_Orderings_Oordering__top_001_062_Itf__a_M_Eo_J,type,
ordering_top_a_o: ( ( a > $o ) > ( a > $o ) > $o ) > ( ( a > $o ) > ( a > $o ) > $o ) > ( a > $o ) > $o ).
thf(sy_c_Orderings_Oordering__top_001t__Set__Oset_Itf__a_J,type,
ordering_top_set_a: ( set_a > set_a > $o ) > ( set_a > set_a > $o ) > set_a > $o ).
thf(sy_c_Product__Type_OPair_001t__BinaryTree____Mirabelle____mlzyzwgbkd__OTree_Itf__a_J_001tf__a,type,
produc374321428ee_a_a: binary1439146945Tree_a > a > produc376588196ee_a_a ).
thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__BinaryTree____Mirabelle____mlzyzwgbkd__OTree_Itf__a_J_001tf__a_001t__BinaryTree____Mirabelle____mlzyzwgbkd__OTree_Itf__a_J,type,
produc1275654896Tree_a: ( binary1439146945Tree_a > a > binary1439146945Tree_a ) > produc376588196ee_a_a > binary1439146945Tree_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_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_t1____,type,
t1: binary1439146945Tree_a ).
thf(sy_v_t2____,type,
t2: binary1439146945Tree_a ).
thf(sy_v_x____,type,
x: a ).
% Relevant facts (353)
thf(fact_0_t2nTip,axiom,
t2 != binary476621312_Tip_a ).
% t2nTip
thf(fact_1_s,axiom,
binary1721989714Tree_a @ h @ ( binary717961607le_T_a @ t1 @ x @ t2 ) ).
% s
thf(fact_2_res,axiom,
( ( binary1217730267_wrm_a @ h @ ( binary717961607le_T_a @ t1 @ x @ t2 ) )
= ( binary717961607le_T_a @ t1 @ x @ ( binary1217730267_wrm_a @ h @ t2 ) ) ) ).
% res
thf(fact_3_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_4_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_5_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_6_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_7_h1,axiom,
( ( ( t1 != binary476621312_Tip_a )
& ( binary1721989714Tree_a @ h @ t1 ) )
=> ( binary1721989714Tree_a @ h @ ( binary1217730267_wrm_a @ h @ t1 ) ) ) ).
% h1
thf(fact_8_sortedTree_Osimps_I1_J,axiom,
! [H: a > int] : ( binary1721989714Tree_a @ H @ binary476621312_Tip_a ) ).
% sortedTree.simps(1)
thf(fact_9_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_10_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_11_h2,axiom,
( ( ( t2 != binary476621312_Tip_a )
& ( binary1721989714Tree_a @ h @ t2 ) )
=> ( binary1721989714Tree_a @ h @ ( binary1217730267_wrm_a @ h @ t2 ) ) ) ).
% h2
thf(fact_12_sorted__distinct,axiom,
! [H: a > int,A2: a,B2: a,T: binary1439146945Tree_a] : ( binary670562003pred_a @ H @ A2 @ B2 @ T ) ).
% sorted_distinct
thf(fact_13__092_060open_062Tip_A_092_060noteq_062_ATip_A_092_060and_062_AsortedTree_Ah_ATip_A_092_060longrightarrow_062_AsortedTree_Ah_A_Iwrm_Ah_ATip_J_092_060close_062,axiom,
( ( ( binary476621312_Tip_a != binary476621312_Tip_a )
& ( binary1721989714Tree_a @ h @ binary476621312_Tip_a ) )
=> ( binary1721989714Tree_a @ h @ ( binary1217730267_wrm_a @ h @ binary476621312_Tip_a ) ) ) ).
% \<open>Tip \<noteq> Tip \<and> sortedTree h Tip \<longrightarrow> sortedTree h (wrm h Tip)\<close>
thf(fact_14_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_15_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_16_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_17_Tree_Oinduct,axiom,
! [P: binary1439146945Tree_a > $o,Tree: binary1439146945Tree_a] :
( ( P @ binary476621312_Tip_a )
=> ( ! [X1: binary1439146945Tree_a,X2: a,X3: binary1439146945Tree_a] :
( ( P @ X1 )
=> ( ( P @ X3 )
=> ( P @ ( binary717961607le_T_a @ X1 @ X2 @ X3 ) ) ) )
=> ( P @ Tree ) ) ) ).
% Tree.induct
thf(fact_18_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_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_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_21_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_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_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_24_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_25_setOf_Osimps_I1_J,axiom,
( ( binary945792244etOf_a @ binary476621312_Tip_a )
= bot_bot_set_a ) ).
% setOf.simps(1)
thf(fact_26_empty__subsetI,axiom,
! [A3: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A3 ) ).
% empty_subsetI
thf(fact_27_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_28_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_29_subsetI,axiom,
! [A3: set_a,B3: set_a] :
( ! [X5: a] :
( ( member_a @ X5 @ A3 )
=> ( member_a @ X5 @ B3 ) )
=> ( ord_less_eq_set_a @ A3 @ B3 ) ) ).
% subsetI
thf(fact_30_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X4: a] :
~ ( P @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_31_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X4: a] :
~ ( P @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_32_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_33_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_34_bot__apply,axiom,
( bot_bot_a_o
= ( ^ [X4: a] : bot_bot_o ) ) ).
% bot_apply
thf(fact_35_order__refl,axiom,
! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).
% order_refl
thf(fact_36_order__refl,axiom,
! [X: a > $o] : ( ord_less_eq_a_o @ X @ X ) ).
% order_refl
thf(fact_37_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_38_le__funD,axiom,
! [F: a > $o,G: a > $o,X: a] :
( ( ord_less_eq_a_o @ F @ G )
=> ( ord_less_eq_o @ ( F @ X ) @ ( G @ X ) ) ) ).
% le_funD
thf(fact_39_le__funE,axiom,
! [F: a > $o,G: a > $o,X: a] :
( ( ord_less_eq_a_o @ F @ G )
=> ( ord_less_eq_o @ ( F @ X ) @ ( G @ X ) ) ) ).
% le_funE
thf(fact_40_le__funI,axiom,
! [F: a > $o,G: a > $o] :
( ! [X5: a] : ( ord_less_eq_o @ ( F @ X5 ) @ ( G @ X5 ) )
=> ( ord_less_eq_a_o @ F @ G ) ) ).
% le_funI
thf(fact_41_mem__Collect__eq,axiom,
! [A2: a,P: a > $o] :
( ( member_a @ A2 @ ( collect_a @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_42_Collect__mem__eq,axiom,
! [A3: set_a] :
( ( collect_a
@ ^ [X4: a] : ( member_a @ X4 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_43_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X5: a] :
( ( P @ X5 )
= ( Q @ X5 ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_44_le__fun__def,axiom,
( ord_less_eq_a_o
= ( ^ [F2: a > $o,G2: a > $o] :
! [X4: a] : ( ord_less_eq_o @ ( F2 @ X4 ) @ ( G2 @ X4 ) ) ) ) ).
% le_fun_def
thf(fact_45_order__subst1,axiom,
! [A2: set_a,F: set_a > set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X5: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_46_order__subst1,axiom,
! [A2: set_a,F: ( a > $o ) > set_a,B2: a > $o,C: a > $o] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_a_o @ B2 @ C )
=> ( ! [X5: a > $o,Y2: a > $o] :
( ( ord_less_eq_a_o @ X5 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_47_order__subst1,axiom,
! [A2: a > $o,F: set_a > a > $o,B2: set_a,C: set_a] :
( ( ord_less_eq_a_o @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X5: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y2 )
=> ( ord_less_eq_a_o @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a_o @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_48_order__subst1,axiom,
! [A2: a > $o,F: ( a > $o ) > a > $o,B2: a > $o,C: a > $o] :
( ( ord_less_eq_a_o @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_a_o @ B2 @ C )
=> ( ! [X5: a > $o,Y2: a > $o] :
( ( ord_less_eq_a_o @ X5 @ Y2 )
=> ( ord_less_eq_a_o @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a_o @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_49_order__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C )
=> ( ! [X5: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_50_order__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > a > $o,C: a > $o] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_a_o @ ( F @ B2 ) @ C )
=> ( ! [X5: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y2 )
=> ( ord_less_eq_a_o @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a_o @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_51_order__subst2,axiom,
! [A2: a > $o,B2: a > $o,F: ( a > $o ) > set_a,C: set_a] :
( ( ord_less_eq_a_o @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C )
=> ( ! [X5: a > $o,Y2: a > $o] :
( ( ord_less_eq_a_o @ X5 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_52_order__subst2,axiom,
! [A2: a > $o,B2: a > $o,F: ( a > $o ) > a > $o,C: a > $o] :
( ( ord_less_eq_a_o @ A2 @ B2 )
=> ( ( ord_less_eq_a_o @ ( F @ B2 ) @ C )
=> ( ! [X5: a > $o,Y2: a > $o] :
( ( ord_less_eq_a_o @ X5 @ Y2 )
=> ( ord_less_eq_a_o @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a_o @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_53_ord__eq__le__subst,axiom,
! [A2: set_a,F: set_a > set_a,B2: set_a,C: set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X5: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_54_ord__eq__le__subst,axiom,
! [A2: a > $o,F: set_a > a > $o,B2: set_a,C: set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X5: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y2 )
=> ( ord_less_eq_a_o @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a_o @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_55_ord__eq__le__subst,axiom,
! [A2: set_a,F: ( a > $o ) > set_a,B2: a > $o,C: a > $o] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_a_o @ B2 @ C )
=> ( ! [X5: a > $o,Y2: a > $o] :
( ( ord_less_eq_a_o @ X5 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_56_ord__eq__le__subst,axiom,
! [A2: a > $o,F: ( a > $o ) > a > $o,B2: a > $o,C: a > $o] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_a_o @ B2 @ C )
=> ( ! [X5: a > $o,Y2: a > $o] :
( ( ord_less_eq_a_o @ X5 @ Y2 )
=> ( ord_less_eq_a_o @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a_o @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_57_ord__le__eq__subst,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X5: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_58_ord__le__eq__subst,axiom,
! [A2: set_a,B2: set_a,F: set_a > a > $o,C: a > $o] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X5: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y2 )
=> ( ord_less_eq_a_o @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a_o @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_59_ord__le__eq__subst,axiom,
! [A2: a > $o,B2: a > $o,F: ( a > $o ) > set_a,C: set_a] :
( ( ord_less_eq_a_o @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X5: a > $o,Y2: a > $o] :
( ( ord_less_eq_a_o @ X5 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_60_ord__le__eq__subst,axiom,
! [A2: a > $o,B2: a > $o,F: ( a > $o ) > a > $o,C: a > $o] :
( ( ord_less_eq_a_o @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X5: a > $o,Y2: a > $o] :
( ( ord_less_eq_a_o @ X5 @ Y2 )
=> ( ord_less_eq_a_o @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_a_o @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_61_eq__iff,axiom,
( ( ^ [Y3: set_a,Z: set_a] : Y3 = Z )
= ( ^ [X4: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y4 )
& ( ord_less_eq_set_a @ Y4 @ X4 ) ) ) ) ).
% eq_iff
thf(fact_62_eq__iff,axiom,
( ( ^ [Y3: a > $o,Z: a > $o] : Y3 = Z )
= ( ^ [X4: a > $o,Y4: a > $o] :
( ( ord_less_eq_a_o @ X4 @ Y4 )
& ( ord_less_eq_a_o @ Y4 @ X4 ) ) ) ) ).
% eq_iff
thf(fact_63_antisym,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ X )
=> ( X = Y ) ) ) ).
% antisym
thf(fact_64_antisym,axiom,
! [X: a > $o,Y: a > $o] :
( ( ord_less_eq_a_o @ X @ Y )
=> ( ( ord_less_eq_a_o @ Y @ X )
=> ( X = Y ) ) ) ).
% antisym
thf(fact_65_eq__refl,axiom,
! [X: set_a,Y: set_a] :
( ( X = Y )
=> ( ord_less_eq_set_a @ X @ Y ) ) ).
% eq_refl
thf(fact_66_eq__refl,axiom,
! [X: a > $o,Y: a > $o] :
( ( X = Y )
=> ( ord_less_eq_a_o @ X @ Y ) ) ).
% eq_refl
thf(fact_67_order_Otrans,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% order.trans
thf(fact_68_order_Otrans,axiom,
! [A2: a > $o,B2: a > $o,C: a > $o] :
( ( ord_less_eq_a_o @ A2 @ B2 )
=> ( ( ord_less_eq_a_o @ B2 @ C )
=> ( ord_less_eq_a_o @ A2 @ C ) ) ) ).
% order.trans
thf(fact_69_antisym__conv,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ( ( ord_less_eq_set_a @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv
thf(fact_70_antisym__conv,axiom,
! [Y: a > $o,X: a > $o] :
( ( ord_less_eq_a_o @ Y @ X )
=> ( ( ord_less_eq_a_o @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv
thf(fact_71_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y3: set_a,Z: set_a] : Y3 = Z )
= ( ^ [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
& ( ord_less_eq_set_a @ B @ A ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_72_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y3: a > $o,Z: a > $o] : Y3 = Z )
= ( ^ [A: a > $o,B: a > $o] :
( ( ord_less_eq_a_o @ A @ B )
& ( ord_less_eq_a_o @ B @ A ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_73_ord__eq__le__trans,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( A2 = B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_74_ord__eq__le__trans,axiom,
! [A2: a > $o,B2: a > $o,C: a > $o] :
( ( A2 = B2 )
=> ( ( ord_less_eq_a_o @ B2 @ C )
=> ( ord_less_eq_a_o @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_75_ord__le__eq__trans,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_set_a @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_76_ord__le__eq__trans,axiom,
! [A2: a > $o,B2: a > $o,C: a > $o] :
( ( ord_less_eq_a_o @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_a_o @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_77_order__class_Oorder_Oantisym,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% order_class.order.antisym
thf(fact_78_order__class_Oorder_Oantisym,axiom,
! [A2: a > $o,B2: a > $o] :
( ( ord_less_eq_a_o @ A2 @ B2 )
=> ( ( ord_less_eq_a_o @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% order_class.order.antisym
thf(fact_79_order__trans,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z2 )
=> ( ord_less_eq_set_a @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_80_order__trans,axiom,
! [X: a > $o,Y: a > $o,Z2: a > $o] :
( ( ord_less_eq_a_o @ X @ Y )
=> ( ( ord_less_eq_a_o @ Y @ Z2 )
=> ( ord_less_eq_a_o @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_81_dual__order_Orefl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_82_dual__order_Orefl,axiom,
! [A2: a > $o] : ( ord_less_eq_a_o @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_83_dual__order_Otrans,axiom,
! [B2: set_a,A2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( ord_less_eq_set_a @ C @ B2 )
=> ( ord_less_eq_set_a @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_84_dual__order_Otrans,axiom,
! [B2: a > $o,A2: a > $o,C: a > $o] :
( ( ord_less_eq_a_o @ B2 @ A2 )
=> ( ( ord_less_eq_a_o @ C @ B2 )
=> ( ord_less_eq_a_o @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_85_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: set_a,Z: set_a] : Y3 = Z )
= ( ^ [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ B @ A )
& ( ord_less_eq_set_a @ A @ B ) ) ) ) ).
% dual_order.eq_iff
thf(fact_86_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: a > $o,Z: a > $o] : Y3 = Z )
= ( ^ [A: a > $o,B: a > $o] :
( ( ord_less_eq_a_o @ B @ A )
& ( ord_less_eq_a_o @ A @ B ) ) ) ) ).
% dual_order.eq_iff
thf(fact_87_dual__order_Oantisym,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_88_dual__order_Oantisym,axiom,
! [B2: a > $o,A2: a > $o] :
( ( ord_less_eq_a_o @ B2 @ A2 )
=> ( ( ord_less_eq_a_o @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_89_ord__eq__less__subst,axiom,
! [A2: int,F: int > int,B2: int,C: int] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X5: int,Y2: int] :
( ( ord_less_int @ X5 @ Y2 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_90_ord__eq__less__subst,axiom,
! [A2: set_a,F: int > set_a,B2: int,C: int] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X5: int,Y2: int] :
( ( ord_less_int @ X5 @ Y2 )
=> ( ord_less_set_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_91_ord__eq__less__subst,axiom,
! [A2: a > $o,F: int > a > $o,B2: int,C: int] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X5: int,Y2: int] :
( ( ord_less_int @ X5 @ Y2 )
=> ( ord_less_a_o @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_a_o @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_92_ord__eq__less__subst,axiom,
! [A2: int,F: set_a > int,B2: set_a,C: set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ! [X5: set_a,Y2: set_a] :
( ( ord_less_set_a @ X5 @ Y2 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_93_ord__eq__less__subst,axiom,
! [A2: set_a,F: set_a > set_a,B2: set_a,C: set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ! [X5: set_a,Y2: set_a] :
( ( ord_less_set_a @ X5 @ Y2 )
=> ( ord_less_set_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_94_ord__eq__less__subst,axiom,
! [A2: a > $o,F: set_a > a > $o,B2: set_a,C: set_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ! [X5: set_a,Y2: set_a] :
( ( ord_less_set_a @ X5 @ Y2 )
=> ( ord_less_a_o @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_a_o @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_95_ord__eq__less__subst,axiom,
! [A2: int,F: ( a > $o ) > int,B2: a > $o,C: a > $o] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_a_o @ B2 @ C )
=> ( ! [X5: a > $o,Y2: a > $o] :
( ( ord_less_a_o @ X5 @ Y2 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_96_ord__eq__less__subst,axiom,
! [A2: set_a,F: ( a > $o ) > set_a,B2: a > $o,C: a > $o] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_a_o @ B2 @ C )
=> ( ! [X5: a > $o,Y2: a > $o] :
( ( ord_less_a_o @ X5 @ Y2 )
=> ( ord_less_set_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_97_ord__eq__less__subst,axiom,
! [A2: a > $o,F: ( a > $o ) > a > $o,B2: a > $o,C: a > $o] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_a_o @ B2 @ C )
=> ( ! [X5: a > $o,Y2: a > $o] :
( ( ord_less_a_o @ X5 @ Y2 )
=> ( ord_less_a_o @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_a_o @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_98_ord__less__eq__subst,axiom,
! [A2: int,B2: int,F: int > int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X5: int,Y2: int] :
( ( ord_less_int @ X5 @ Y2 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_99_ord__less__eq__subst,axiom,
! [A2: int,B2: int,F: int > set_a,C: set_a] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X5: int,Y2: int] :
( ( ord_less_int @ X5 @ Y2 )
=> ( ord_less_set_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_100_ord__less__eq__subst,axiom,
! [A2: int,B2: int,F: int > a > $o,C: a > $o] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X5: int,Y2: int] :
( ( ord_less_int @ X5 @ Y2 )
=> ( ord_less_a_o @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_a_o @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_101_ord__less__eq__subst,axiom,
! [A2: set_a,B2: set_a,F: set_a > int,C: int] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X5: set_a,Y2: set_a] :
( ( ord_less_set_a @ X5 @ Y2 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_102_ord__less__eq__subst,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X5: set_a,Y2: set_a] :
( ( ord_less_set_a @ X5 @ Y2 )
=> ( ord_less_set_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_103_ord__less__eq__subst,axiom,
! [A2: set_a,B2: set_a,F: set_a > a > $o,C: a > $o] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X5: set_a,Y2: set_a] :
( ( ord_less_set_a @ X5 @ Y2 )
=> ( ord_less_a_o @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_a_o @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_104_ord__less__eq__subst,axiom,
! [A2: a > $o,B2: a > $o,F: ( a > $o ) > int,C: int] :
( ( ord_less_a_o @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X5: a > $o,Y2: a > $o] :
( ( ord_less_a_o @ X5 @ Y2 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_105_ord__less__eq__subst,axiom,
! [A2: a > $o,B2: a > $o,F: ( a > $o ) > set_a,C: set_a] :
( ( ord_less_a_o @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X5: a > $o,Y2: a > $o] :
( ( ord_less_a_o @ X5 @ Y2 )
=> ( ord_less_set_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_106_ord__less__eq__subst,axiom,
! [A2: a > $o,B2: a > $o,F: ( a > $o ) > a > $o,C: a > $o] :
( ( ord_less_a_o @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X5: a > $o,Y2: a > $o] :
( ( ord_less_a_o @ X5 @ Y2 )
=> ( ord_less_a_o @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_a_o @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_107_order__less__subst1,axiom,
! [A2: int,F: int > int,B2: int,C: int] :
( ( ord_less_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X5: int,Y2: int] :
( ( ord_less_int @ X5 @ Y2 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_108_order__less__subst1,axiom,
! [A2: int,F: set_a > int,B2: set_a,C: set_a] :
( ( ord_less_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ! [X5: set_a,Y2: set_a] :
( ( ord_less_set_a @ X5 @ Y2 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_109_order__less__subst1,axiom,
! [A2: int,F: ( a > $o ) > int,B2: a > $o,C: a > $o] :
( ( ord_less_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_a_o @ B2 @ C )
=> ( ! [X5: a > $o,Y2: a > $o] :
( ( ord_less_a_o @ X5 @ Y2 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_110_order__less__subst1,axiom,
! [A2: set_a,F: int > set_a,B2: int,C: int] :
( ( ord_less_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X5: int,Y2: int] :
( ( ord_less_int @ X5 @ Y2 )
=> ( ord_less_set_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_111_order__less__subst1,axiom,
! [A2: set_a,F: set_a > set_a,B2: set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ! [X5: set_a,Y2: set_a] :
( ( ord_less_set_a @ X5 @ Y2 )
=> ( ord_less_set_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_112_order__less__subst1,axiom,
! [A2: set_a,F: ( a > $o ) > set_a,B2: a > $o,C: a > $o] :
( ( ord_less_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_a_o @ B2 @ C )
=> ( ! [X5: a > $o,Y2: a > $o] :
( ( ord_less_a_o @ X5 @ Y2 )
=> ( ord_less_set_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_113_order__less__subst1,axiom,
! [A2: a > $o,F: int > a > $o,B2: int,C: int] :
( ( ord_less_a_o @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X5: int,Y2: int] :
( ( ord_less_int @ X5 @ Y2 )
=> ( ord_less_a_o @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_a_o @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_114_order__less__subst1,axiom,
! [A2: a > $o,F: set_a > a > $o,B2: set_a,C: set_a] :
( ( ord_less_a_o @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ! [X5: set_a,Y2: set_a] :
( ( ord_less_set_a @ X5 @ Y2 )
=> ( ord_less_a_o @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_a_o @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_115_order__less__subst1,axiom,
! [A2: a > $o,F: ( a > $o ) > a > $o,B2: a > $o,C: a > $o] :
( ( ord_less_a_o @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_a_o @ B2 @ C )
=> ( ! [X5: a > $o,Y2: a > $o] :
( ( ord_less_a_o @ X5 @ Y2 )
=> ( ord_less_a_o @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_a_o @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_116_order__less__subst2,axiom,
! [A2: int,B2: int,F: int > int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X5: int,Y2: int] :
( ( ord_less_int @ X5 @ Y2 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_117_order__less__subst2,axiom,
! [A2: int,B2: int,F: int > set_a,C: set_a] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_set_a @ ( F @ B2 ) @ C )
=> ( ! [X5: int,Y2: int] :
( ( ord_less_int @ X5 @ Y2 )
=> ( ord_less_set_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_118_order__less__subst2,axiom,
! [A2: int,B2: int,F: int > a > $o,C: a > $o] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_a_o @ ( F @ B2 ) @ C )
=> ( ! [X5: int,Y2: int] :
( ( ord_less_int @ X5 @ Y2 )
=> ( ord_less_a_o @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_a_o @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_119_order__less__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > int,C: int] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X5: set_a,Y2: set_a] :
( ( ord_less_set_a @ X5 @ Y2 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_120_order__less__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ord_less_set_a @ ( F @ B2 ) @ C )
=> ( ! [X5: set_a,Y2: set_a] :
( ( ord_less_set_a @ X5 @ Y2 )
=> ( ord_less_set_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_121_order__less__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > a > $o,C: a > $o] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ord_less_a_o @ ( F @ B2 ) @ C )
=> ( ! [X5: set_a,Y2: set_a] :
( ( ord_less_set_a @ X5 @ Y2 )
=> ( ord_less_a_o @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_a_o @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_122_order__less__subst2,axiom,
! [A2: a > $o,B2: a > $o,F: ( a > $o ) > int,C: int] :
( ( ord_less_a_o @ A2 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X5: a > $o,Y2: a > $o] :
( ( ord_less_a_o @ X5 @ Y2 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_123_order__less__subst2,axiom,
! [A2: a > $o,B2: a > $o,F: ( a > $o ) > set_a,C: set_a] :
( ( ord_less_a_o @ A2 @ B2 )
=> ( ( ord_less_set_a @ ( F @ B2 ) @ C )
=> ( ! [X5: a > $o,Y2: a > $o] :
( ( ord_less_a_o @ X5 @ Y2 )
=> ( ord_less_set_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_124_order__less__subst2,axiom,
! [A2: a > $o,B2: a > $o,F: ( a > $o ) > a > $o,C: a > $o] :
( ( ord_less_a_o @ A2 @ B2 )
=> ( ( ord_less_a_o @ ( F @ B2 ) @ C )
=> ( ! [X5: a > $o,Y2: a > $o] :
( ( ord_less_a_o @ X5 @ Y2 )
=> ( ord_less_a_o @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_a_o @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_125_lt__ex,axiom,
! [X: int] :
? [Y2: int] : ( ord_less_int @ Y2 @ X ) ).
% lt_ex
thf(fact_126_gt__ex,axiom,
! [X: int] :
? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% gt_ex
thf(fact_127_neqE,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% neqE
thf(fact_128_neq__iff,axiom,
! [X: int,Y: int] :
( ( X != Y )
= ( ( ord_less_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ) ).
% neq_iff
thf(fact_129_order_Oasym,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ~ ( ord_less_int @ B2 @ A2 ) ) ).
% order.asym
thf(fact_130_order_Oasym,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ~ ( ord_less_set_a @ B2 @ A2 ) ) ).
% order.asym
thf(fact_131_order_Oasym,axiom,
! [A2: a > $o,B2: a > $o] :
( ( ord_less_a_o @ A2 @ B2 )
=> ~ ( ord_less_a_o @ B2 @ A2 ) ) ).
% order.asym
thf(fact_132_less__imp__neq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_133_less__imp__neq,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_134_less__imp__neq,axiom,
! [X: a > $o,Y: a > $o] :
( ( ord_less_a_o @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_135_less__asym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% less_asym
thf(fact_136_less__asym,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ~ ( ord_less_set_a @ Y @ X ) ) ).
% less_asym
thf(fact_137_less__asym,axiom,
! [X: a > $o,Y: a > $o] :
( ( ord_less_a_o @ X @ Y )
=> ~ ( ord_less_a_o @ Y @ X ) ) ).
% less_asym
thf(fact_138_less__asym_H,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ~ ( ord_less_int @ B2 @ A2 ) ) ).
% less_asym'
thf(fact_139_less__asym_H,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ~ ( ord_less_set_a @ B2 @ A2 ) ) ).
% less_asym'
thf(fact_140_less__asym_H,axiom,
! [A2: a > $o,B2: a > $o] :
( ( ord_less_a_o @ A2 @ B2 )
=> ~ ( ord_less_a_o @ B2 @ A2 ) ) ).
% less_asym'
thf(fact_141_less__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% less_trans
thf(fact_142_less__trans,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ( ( ord_less_set_a @ Y @ Z2 )
=> ( ord_less_set_a @ X @ Z2 ) ) ) ).
% less_trans
thf(fact_143_less__trans,axiom,
! [X: a > $o,Y: a > $o,Z2: a > $o] :
( ( ord_less_a_o @ X @ Y )
=> ( ( ord_less_a_o @ Y @ Z2 )
=> ( ord_less_a_o @ X @ Z2 ) ) ) ).
% less_trans
thf(fact_144_less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
| ( X = Y )
| ( ord_less_int @ Y @ X ) ) ).
% less_linear
thf(fact_145_less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% less_irrefl
thf(fact_146_less__irrefl,axiom,
! [X: set_a] :
~ ( ord_less_set_a @ X @ X ) ).
% less_irrefl
thf(fact_147_less__irrefl,axiom,
! [X: a > $o] :
~ ( ord_less_a_o @ X @ X ) ).
% less_irrefl
thf(fact_148_ord__eq__less__trans,axiom,
! [A2: int,B2: int,C: int] :
( ( A2 = B2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_149_ord__eq__less__trans,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( A2 = B2 )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ord_less_set_a @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_150_ord__eq__less__trans,axiom,
! [A2: a > $o,B2: a > $o,C: a > $o] :
( ( A2 = B2 )
=> ( ( ord_less_a_o @ B2 @ C )
=> ( ord_less_a_o @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_151_ord__less__eq__trans,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_152_ord__less__eq__trans,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_set_a @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_153_ord__less__eq__trans,axiom,
! [A2: a > $o,B2: a > $o,C: a > $o] :
( ( ord_less_a_o @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_a_o @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_154_dual__order_Oasym,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ B2 @ A2 )
=> ~ ( ord_less_int @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_155_dual__order_Oasym,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_set_a @ B2 @ A2 )
=> ~ ( ord_less_set_a @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_156_dual__order_Oasym,axiom,
! [B2: a > $o,A2: a > $o] :
( ( ord_less_a_o @ B2 @ A2 )
=> ~ ( ord_less_a_o @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_157_less__imp__not__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_not_eq
thf(fact_158_less__imp__not__eq,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_not_eq
thf(fact_159_less__imp__not__eq,axiom,
! [X: a > $o,Y: a > $o] :
( ( ord_less_a_o @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_not_eq
thf(fact_160_less__not__sym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% less_not_sym
thf(fact_161_less__not__sym,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ~ ( ord_less_set_a @ Y @ X ) ) ).
% less_not_sym
thf(fact_162_less__not__sym,axiom,
! [X: a > $o,Y: a > $o] :
( ( ord_less_a_o @ X @ Y )
=> ~ ( ord_less_a_o @ Y @ X ) ) ).
% less_not_sym
thf(fact_163_antisym__conv3,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_int @ Y @ X )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_164_less__imp__not__eq2,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( Y != X ) ) ).
% less_imp_not_eq2
thf(fact_165_less__imp__not__eq2,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ( Y != X ) ) ).
% less_imp_not_eq2
thf(fact_166_less__imp__not__eq2,axiom,
! [X: a > $o,Y: a > $o] :
( ( ord_less_a_o @ X @ Y )
=> ( Y != X ) ) ).
% less_imp_not_eq2
thf(fact_167_less__imp__triv,axiom,
! [X: int,Y: int,P: $o] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ X )
=> P ) ) ).
% less_imp_triv
thf(fact_168_less__imp__triv,axiom,
! [X: set_a,Y: set_a,P: $o] :
( ( ord_less_set_a @ X @ Y )
=> ( ( ord_less_set_a @ Y @ X )
=> P ) ) ).
% less_imp_triv
thf(fact_169_less__imp__triv,axiom,
! [X: a > $o,Y: a > $o,P: $o] :
( ( ord_less_a_o @ X @ Y )
=> ( ( ord_less_a_o @ Y @ X )
=> P ) ) ).
% less_imp_triv
thf(fact_170_linorder__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_171_dual__order_Oirrefl,axiom,
! [A2: int] :
~ ( ord_less_int @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_172_dual__order_Oirrefl,axiom,
! [A2: set_a] :
~ ( ord_less_set_a @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_173_dual__order_Oirrefl,axiom,
! [A2: a > $o] :
~ ( ord_less_a_o @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_174_order_Ostrict__trans,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_175_order_Ostrict__trans,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ord_less_set_a @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_176_order_Ostrict__trans,axiom,
! [A2: a > $o,B2: a > $o,C: a > $o] :
( ( ord_less_a_o @ A2 @ B2 )
=> ( ( ord_less_a_o @ B2 @ C )
=> ( ord_less_a_o @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_177_less__imp__not__less,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% less_imp_not_less
thf(fact_178_less__imp__not__less,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ~ ( ord_less_set_a @ Y @ X ) ) ).
% less_imp_not_less
thf(fact_179_less__imp__not__less,axiom,
! [X: a > $o,Y: a > $o] :
( ( ord_less_a_o @ X @ Y )
=> ~ ( ord_less_a_o @ Y @ X ) ) ).
% less_imp_not_less
thf(fact_180_linorder__less__wlog,axiom,
! [P: int > int > $o,A2: int,B2: int] :
( ! [A4: int,B4: int] :
( ( ord_less_int @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: int] : ( P @ A4 @ A4 )
=> ( ! [A4: int,B4: int] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_181_dual__order_Ostrict__trans,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( ( ord_less_int @ C @ B2 )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_182_dual__order_Ostrict__trans,axiom,
! [B2: set_a,A2: set_a,C: set_a] :
( ( ord_less_set_a @ B2 @ A2 )
=> ( ( ord_less_set_a @ C @ B2 )
=> ( ord_less_set_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_183_dual__order_Ostrict__trans,axiom,
! [B2: a > $o,A2: a > $o,C: a > $o] :
( ( ord_less_a_o @ B2 @ A2 )
=> ( ( ord_less_a_o @ C @ B2 )
=> ( ord_less_a_o @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_184_not__less__iff__gr__or__eq,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ( ord_less_int @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_185_order_Ostrict__implies__not__eq,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_186_order_Ostrict__implies__not__eq,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_187_order_Ostrict__implies__not__eq,axiom,
! [A2: a > $o,B2: a > $o] :
( ( ord_less_a_o @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_188_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_189_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_set_a @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_190_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: a > $o,A2: a > $o] :
( ( ord_less_a_o @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_191_bot__fun__def,axiom,
( bot_bot_a_o
= ( ^ [X4: a] : bot_bot_o ) ) ).
% bot_fun_def
thf(fact_192_emptyE,axiom,
! [A2: a] :
~ ( member_a @ A2 @ bot_bot_set_a ) ).
% emptyE
thf(fact_193_equals0D,axiom,
! [A3: set_a,A2: a] :
( ( A3 = bot_bot_set_a )
=> ~ ( member_a @ A2 @ A3 ) ) ).
% equals0D
thf(fact_194_equals0I,axiom,
! [A3: set_a] :
( ! [Y2: a] :
~ ( member_a @ Y2 @ A3 )
=> ( A3 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_195_ex__in__conv,axiom,
! [A3: set_a] :
( ( ? [X4: a] : ( member_a @ X4 @ A3 ) )
= ( A3 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_196_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_197_subsetD,axiom,
! [A3: set_a,B3: set_a,C: a] :
( ( ord_less_eq_set_a @ A3 @ B3 )
=> ( ( member_a @ C @ A3 )
=> ( member_a @ C @ B3 ) ) ) ).
% subsetD
thf(fact_198_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_199_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
! [X4: a] :
( ( member_a @ X4 @ A5 )
=> ( member_a @ X4 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_200_equalityD1,axiom,
! [A3: set_a,B3: set_a] :
( ( A3 = B3 )
=> ( ord_less_eq_set_a @ A3 @ B3 ) ) ).
% equalityD1
thf(fact_201_equalityD2,axiom,
! [A3: set_a,B3: set_a] :
( ( A3 = B3 )
=> ( ord_less_eq_set_a @ B3 @ A3 ) ) ).
% equalityD2
thf(fact_202_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
! [T3: a] :
( ( member_a @ T3 @ A5 )
=> ( member_a @ T3 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_203_subset__refl,axiom,
! [A3: set_a] : ( ord_less_eq_set_a @ A3 @ A3 ) ).
% subset_refl
thf(fact_204_Collect__mono,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X5: a] :
( ( P @ X5 )
=> ( Q @ X5 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q ) ) ) ).
% Collect_mono
thf(fact_205_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_206_set__eq__subset,axiom,
( ( ^ [Y3: set_a,Z: set_a] : Y3 = Z )
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
& ( ord_less_eq_set_a @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_207_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_208_empty__def,axiom,
( bot_bot_set_a
= ( collect_a
@ ^ [X4: a] : $false ) ) ).
% empty_def
thf(fact_209_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_210_leD,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_int @ X @ Y ) ) ).
% leD
thf(fact_211_leD,axiom,
! [Y: set_a,X: set_a] :
( ( ord_less_eq_set_a @ Y @ X )
=> ~ ( ord_less_set_a @ X @ Y ) ) ).
% leD
thf(fact_212_leD,axiom,
! [Y: a > $o,X: a > $o] :
( ( ord_less_eq_a_o @ Y @ X )
=> ~ ( ord_less_a_o @ X @ Y ) ) ).
% leD
thf(fact_213_leI,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% leI
thf(fact_214_le__less,axiom,
( ord_less_eq_int
= ( ^ [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% le_less
thf(fact_215_le__less,axiom,
( ord_less_eq_set_a
= ( ^ [X4: set_a,Y4: set_a] :
( ( ord_less_set_a @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% le_less
thf(fact_216_le__less,axiom,
( ord_less_eq_a_o
= ( ^ [X4: a > $o,Y4: a > $o] :
( ( ord_less_a_o @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ).
% le_less
thf(fact_217_less__le,axiom,
( ord_less_int
= ( ^ [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% less_le
thf(fact_218_less__le,axiom,
( ord_less_set_a
= ( ^ [X4: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% less_le
thf(fact_219_less__le,axiom,
( ord_less_a_o
= ( ^ [X4: a > $o,Y4: a > $o] :
( ( ord_less_eq_a_o @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ).
% less_le
thf(fact_220_order__le__less__subst1,axiom,
! [A2: int,F: int > int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X5: int,Y2: int] :
( ( ord_less_int @ X5 @ Y2 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_221_order__le__less__subst1,axiom,
! [A2: int,F: set_a > int,B2: set_a,C: set_a] :
( ( ord_less_eq_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ! [X5: set_a,Y2: set_a] :
( ( ord_less_set_a @ X5 @ Y2 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_222_order__le__less__subst1,axiom,
! [A2: int,F: ( a > $o ) > int,B2: a > $o,C: a > $o] :
( ( ord_less_eq_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_a_o @ B2 @ C )
=> ( ! [X5: a > $o,Y2: a > $o] :
( ( ord_less_a_o @ X5 @ Y2 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_223_order__le__less__subst1,axiom,
! [A2: set_a,F: int > set_a,B2: int,C: int] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X5: int,Y2: int] :
( ( ord_less_int @ X5 @ Y2 )
=> ( ord_less_set_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_224_order__le__less__subst1,axiom,
! [A2: set_a,F: set_a > set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ! [X5: set_a,Y2: set_a] :
( ( ord_less_set_a @ X5 @ Y2 )
=> ( ord_less_set_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_225_order__le__less__subst1,axiom,
! [A2: set_a,F: ( a > $o ) > set_a,B2: a > $o,C: a > $o] :
( ( ord_less_eq_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_a_o @ B2 @ C )
=> ( ! [X5: a > $o,Y2: a > $o] :
( ( ord_less_a_o @ X5 @ Y2 )
=> ( ord_less_set_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_226_order__le__less__subst1,axiom,
! [A2: a > $o,F: int > a > $o,B2: int,C: int] :
( ( ord_less_eq_a_o @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X5: int,Y2: int] :
( ( ord_less_int @ X5 @ Y2 )
=> ( ord_less_a_o @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_a_o @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_227_order__le__less__subst1,axiom,
! [A2: a > $o,F: set_a > a > $o,B2: set_a,C: set_a] :
( ( ord_less_eq_a_o @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ! [X5: set_a,Y2: set_a] :
( ( ord_less_set_a @ X5 @ Y2 )
=> ( ord_less_a_o @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_a_o @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_228_order__le__less__subst1,axiom,
! [A2: a > $o,F: ( a > $o ) > a > $o,B2: a > $o,C: a > $o] :
( ( ord_less_eq_a_o @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_a_o @ B2 @ C )
=> ( ! [X5: a > $o,Y2: a > $o] :
( ( ord_less_a_o @ X5 @ Y2 )
=> ( ord_less_a_o @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_a_o @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_229_order__le__less__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > int,C: int] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X5: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_230_order__le__less__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_set_a @ ( F @ B2 ) @ C )
=> ( ! [X5: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_231_order__le__less__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > a > $o,C: a > $o] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_a_o @ ( F @ B2 ) @ C )
=> ( ! [X5: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y2 )
=> ( ord_less_eq_a_o @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_a_o @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_232_order__le__less__subst2,axiom,
! [A2: a > $o,B2: a > $o,F: ( a > $o ) > int,C: int] :
( ( ord_less_eq_a_o @ A2 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X5: a > $o,Y2: a > $o] :
( ( ord_less_eq_a_o @ X5 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_233_order__le__less__subst2,axiom,
! [A2: a > $o,B2: a > $o,F: ( a > $o ) > set_a,C: set_a] :
( ( ord_less_eq_a_o @ A2 @ B2 )
=> ( ( ord_less_set_a @ ( F @ B2 ) @ C )
=> ( ! [X5: a > $o,Y2: a > $o] :
( ( ord_less_eq_a_o @ X5 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_234_order__le__less__subst2,axiom,
! [A2: a > $o,B2: a > $o,F: ( a > $o ) > a > $o,C: a > $o] :
( ( ord_less_eq_a_o @ A2 @ B2 )
=> ( ( ord_less_a_o @ ( F @ B2 ) @ C )
=> ( ! [X5: a > $o,Y2: a > $o] :
( ( ord_less_eq_a_o @ X5 @ Y2 )
=> ( ord_less_eq_a_o @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_a_o @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_235_order__less__le__subst1,axiom,
! [A2: int,F: set_a > int,B2: set_a,C: set_a] :
( ( ord_less_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X5: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_236_order__less__le__subst1,axiom,
! [A2: set_a,F: set_a > set_a,B2: set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X5: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_237_order__less__le__subst1,axiom,
! [A2: a > $o,F: set_a > a > $o,B2: set_a,C: set_a] :
( ( ord_less_a_o @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ! [X5: set_a,Y2: set_a] :
( ( ord_less_eq_set_a @ X5 @ Y2 )
=> ( ord_less_eq_a_o @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_a_o @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_238_order__less__le__subst1,axiom,
! [A2: int,F: ( a > $o ) > int,B2: a > $o,C: a > $o] :
( ( ord_less_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_a_o @ B2 @ C )
=> ( ! [X5: a > $o,Y2: a > $o] :
( ( ord_less_eq_a_o @ X5 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_239_order__less__le__subst1,axiom,
! [A2: set_a,F: ( a > $o ) > set_a,B2: a > $o,C: a > $o] :
( ( ord_less_set_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_a_o @ B2 @ C )
=> ( ! [X5: a > $o,Y2: a > $o] :
( ( ord_less_eq_a_o @ X5 @ Y2 )
=> ( ord_less_eq_set_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_240_order__less__le__subst1,axiom,
! [A2: a > $o,F: ( a > $o ) > a > $o,B2: a > $o,C: a > $o] :
( ( ord_less_a_o @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_a_o @ B2 @ C )
=> ( ! [X5: a > $o,Y2: a > $o] :
( ( ord_less_eq_a_o @ X5 @ Y2 )
=> ( ord_less_eq_a_o @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_a_o @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_241_order__less__le__subst2,axiom,
! [A2: int,B2: int,F: int > int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
=> ( ! [X5: int,Y2: int] :
( ( ord_less_int @ X5 @ Y2 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_242_order__less__le__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > int,C: int] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
=> ( ! [X5: set_a,Y2: set_a] :
( ( ord_less_set_a @ X5 @ Y2 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_243_order__less__le__subst2,axiom,
! [A2: a > $o,B2: a > $o,F: ( a > $o ) > int,C: int] :
( ( ord_less_a_o @ A2 @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
=> ( ! [X5: a > $o,Y2: a > $o] :
( ( ord_less_a_o @ X5 @ Y2 )
=> ( ord_less_int @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_244_order__less__le__subst2,axiom,
! [A2: int,B2: int,F: int > set_a,C: set_a] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C )
=> ( ! [X5: int,Y2: int] :
( ( ord_less_int @ X5 @ Y2 )
=> ( ord_less_set_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_245_order__less__le__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C )
=> ( ! [X5: set_a,Y2: set_a] :
( ( ord_less_set_a @ X5 @ Y2 )
=> ( ord_less_set_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_246_order__less__le__subst2,axiom,
! [A2: a > $o,B2: a > $o,F: ( a > $o ) > set_a,C: set_a] :
( ( ord_less_a_o @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ ( F @ B2 ) @ C )
=> ( ! [X5: a > $o,Y2: a > $o] :
( ( ord_less_a_o @ X5 @ Y2 )
=> ( ord_less_set_a @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_set_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_247_order__less__le__subst2,axiom,
! [A2: int,B2: int,F: int > a > $o,C: a > $o] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_eq_a_o @ ( F @ B2 ) @ C )
=> ( ! [X5: int,Y2: int] :
( ( ord_less_int @ X5 @ Y2 )
=> ( ord_less_a_o @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_a_o @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_248_order__less__le__subst2,axiom,
! [A2: set_a,B2: set_a,F: set_a > a > $o,C: a > $o] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_a_o @ ( F @ B2 ) @ C )
=> ( ! [X5: set_a,Y2: set_a] :
( ( ord_less_set_a @ X5 @ Y2 )
=> ( ord_less_a_o @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_a_o @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_249_order__less__le__subst2,axiom,
! [A2: a > $o,B2: a > $o,F: ( a > $o ) > a > $o,C: a > $o] :
( ( ord_less_a_o @ A2 @ B2 )
=> ( ( ord_less_eq_a_o @ ( F @ B2 ) @ C )
=> ( ! [X5: a > $o,Y2: a > $o] :
( ( ord_less_a_o @ X5 @ Y2 )
=> ( ord_less_a_o @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_a_o @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_250_not__le,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X @ Y ) )
= ( ord_less_int @ Y @ X ) ) ).
% not_le
thf(fact_251_not__less,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ord_less_eq_int @ Y @ X ) ) ).
% not_less
thf(fact_252_le__neq__trans,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_int @ A2 @ B2 ) ) ) ).
% le_neq_trans
thf(fact_253_le__neq__trans,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_set_a @ A2 @ B2 ) ) ) ).
% le_neq_trans
thf(fact_254_le__neq__trans,axiom,
! [A2: a > $o,B2: a > $o] :
( ( ord_less_eq_a_o @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_a_o @ A2 @ B2 ) ) ) ).
% le_neq_trans
thf(fact_255_antisym__conv1,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_256_antisym__conv1,axiom,
! [X: set_a,Y: set_a] :
( ~ ( ord_less_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_257_antisym__conv1,axiom,
! [X: a > $o,Y: a > $o] :
( ~ ( ord_less_a_o @ X @ Y )
=> ( ( ord_less_eq_a_o @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_258_antisym__conv2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_259_antisym__conv2,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ~ ( ord_less_set_a @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_260_antisym__conv2,axiom,
! [X: a > $o,Y: a > $o] :
( ( ord_less_eq_a_o @ X @ Y )
=> ( ( ~ ( ord_less_a_o @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_261_less__imp__le,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% less_imp_le
thf(fact_262_less__imp__le,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ( ord_less_eq_set_a @ X @ Y ) ) ).
% less_imp_le
thf(fact_263_less__imp__le,axiom,
! [X: a > $o,Y: a > $o] :
( ( ord_less_a_o @ X @ Y )
=> ( ord_less_eq_a_o @ X @ Y ) ) ).
% less_imp_le
thf(fact_264_le__less__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% le_less_trans
thf(fact_265_le__less__trans,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_set_a @ Y @ Z2 )
=> ( ord_less_set_a @ X @ Z2 ) ) ) ).
% le_less_trans
thf(fact_266_le__less__trans,axiom,
! [X: a > $o,Y: a > $o,Z2: a > $o] :
( ( ord_less_eq_a_o @ X @ Y )
=> ( ( ord_less_a_o @ Y @ Z2 )
=> ( ord_less_a_o @ X @ Z2 ) ) ) ).
% le_less_trans
thf(fact_267_less__le__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% less_le_trans
thf(fact_268_less__le__trans,axiom,
! [X: set_a,Y: set_a,Z2: set_a] :
( ( ord_less_set_a @ X @ Y )
=> ( ( ord_less_eq_set_a @ Y @ Z2 )
=> ( ord_less_set_a @ X @ Z2 ) ) ) ).
% less_le_trans
thf(fact_269_less__le__trans,axiom,
! [X: a > $o,Y: a > $o,Z2: a > $o] :
( ( ord_less_a_o @ X @ Y )
=> ( ( ord_less_eq_a_o @ Y @ Z2 )
=> ( ord_less_a_o @ X @ Z2 ) ) ) ).
% less_le_trans
thf(fact_270_le__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ).
% le_less_linear
thf(fact_271_le__imp__less__or__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ X @ Y )
| ( X = Y ) ) ) ).
% le_imp_less_or_eq
thf(fact_272_le__imp__less__or__eq,axiom,
! [X: set_a,Y: set_a] :
( ( ord_less_eq_set_a @ X @ Y )
=> ( ( ord_less_set_a @ X @ Y )
| ( X = Y ) ) ) ).
% le_imp_less_or_eq
thf(fact_273_le__imp__less__or__eq,axiom,
! [X: a > $o,Y: a > $o] :
( ( ord_less_eq_a_o @ X @ Y )
=> ( ( ord_less_a_o @ X @ Y )
| ( X = Y ) ) ) ).
% le_imp_less_or_eq
thf(fact_274_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
& ~ ( ord_less_eq_int @ Y4 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_275_less__le__not__le,axiom,
( ord_less_set_a
= ( ^ [X4: set_a,Y4: set_a] :
( ( ord_less_eq_set_a @ X4 @ Y4 )
& ~ ( ord_less_eq_set_a @ Y4 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_276_less__le__not__le,axiom,
( ord_less_a_o
= ( ^ [X4: a > $o,Y4: a > $o] :
( ( ord_less_eq_a_o @ X4 @ Y4 )
& ~ ( ord_less_eq_a_o @ Y4 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_277_not__le__imp__less,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_eq_int @ Y @ X )
=> ( ord_less_int @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_278_order_Ostrict__trans1,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_279_order_Ostrict__trans1,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ( ord_less_set_a @ B2 @ C )
=> ( ord_less_set_a @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_280_order_Ostrict__trans1,axiom,
! [A2: a > $o,B2: a > $o,C: a > $o] :
( ( ord_less_eq_a_o @ A2 @ B2 )
=> ( ( ord_less_a_o @ B2 @ C )
=> ( ord_less_a_o @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_281_order_Ostrict__trans2,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_282_order_Ostrict__trans2,axiom,
! [A2: set_a,B2: set_a,C: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ( ord_less_eq_set_a @ B2 @ C )
=> ( ord_less_set_a @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_283_order_Ostrict__trans2,axiom,
! [A2: a > $o,B2: a > $o,C: a > $o] :
( ( ord_less_a_o @ A2 @ B2 )
=> ( ( ord_less_eq_a_o @ B2 @ C )
=> ( ord_less_a_o @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_284_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A: int,B: int] :
( ( ord_less_int @ A @ B )
| ( A = B ) ) ) ) ).
% order.order_iff_strict
thf(fact_285_order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [A: set_a,B: set_a] :
( ( ord_less_set_a @ A @ B )
| ( A = B ) ) ) ) ).
% order.order_iff_strict
thf(fact_286_order_Oorder__iff__strict,axiom,
( ord_less_eq_a_o
= ( ^ [A: a > $o,B: a > $o] :
( ( ord_less_a_o @ A @ B )
| ( A = B ) ) ) ) ).
% order.order_iff_strict
thf(fact_287_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
& ( A != B ) ) ) ) ).
% order.strict_iff_order
thf(fact_288_order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [A: set_a,B: set_a] :
( ( ord_less_eq_set_a @ A @ B )
& ( A != B ) ) ) ) ).
% order.strict_iff_order
thf(fact_289_order_Ostrict__iff__order,axiom,
( ord_less_a_o
= ( ^ [A: a > $o,B: a > $o] :
( ( ord_less_eq_a_o @ A @ B )
& ( A != B ) ) ) ) ).
% order.strict_iff_order
thf(fact_290_dual__order_Ostrict__trans1,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ( ord_less_int @ C @ B2 )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_291_dual__order_Ostrict__trans1,axiom,
! [B2: set_a,A2: set_a,C: set_a] :
( ( ord_less_eq_set_a @ B2 @ A2 )
=> ( ( ord_less_set_a @ C @ B2 )
=> ( ord_less_set_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_292_dual__order_Ostrict__trans1,axiom,
! [B2: a > $o,A2: a > $o,C: a > $o] :
( ( ord_less_eq_a_o @ B2 @ A2 )
=> ( ( ord_less_a_o @ C @ B2 )
=> ( ord_less_a_o @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_293_dual__order_Ostrict__trans2,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( ( ord_less_eq_int @ C @ B2 )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_294_dual__order_Ostrict__trans2,axiom,
! [B2: set_a,A2: set_a,C: set_a] :
( ( ord_less_set_a @ B2 @ A2 )
=> ( ( ord_less_eq_set_a @ C @ B2 )
=> ( ord_less_set_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_295_dual__order_Ostrict__trans2,axiom,
! [B2: a > $o,A2: a > $o,C: a > $o] :
( ( ord_less_a_o @ B2 @ A2 )
=> ( ( ord_less_eq_a_o @ C @ B2 )
=> ( ord_less_a_o @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_296_order_Ostrict__implies__order,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ord_less_eq_int @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_297_order_Ostrict__implies__order,axiom,
! [A2: set_a,B2: set_a] :
( ( ord_less_set_a @ A2 @ B2 )
=> ( ord_less_eq_set_a @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_298_order_Ostrict__implies__order,axiom,
! [A2: a > $o,B2: a > $o] :
( ( ord_less_a_o @ A2 @ B2 )
=> ( ord_less_eq_a_o @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_299_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B: int,A: int] :
( ( ord_less_int @ B @ A )
| ( A = B ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_300_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_set_a
= ( ^ [B: set_a,A: set_a] :
( ( ord_less_set_a @ B @ A )
| ( A = B ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_301_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_a_o
= ( ^ [B: a > $o,A: a > $o] :
( ( ord_less_a_o @ B @ A )
| ( A = B ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_302_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
& ( A != B ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_303_dual__order_Ostrict__iff__order,axiom,
( ord_less_set_a
= ( ^ [B: set_a,A: set_a] :
( ( ord_less_eq_set_a @ B @ A )
& ( A != B ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_304_dual__order_Ostrict__iff__order,axiom,
( ord_less_a_o
= ( ^ [B: a > $o,A: a > $o] :
( ( ord_less_eq_a_o @ B @ A )
& ( A != B ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_305_dual__order_Ostrict__implies__order,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( ord_less_eq_int @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_306_dual__order_Ostrict__implies__order,axiom,
! [B2: set_a,A2: set_a] :
( ( ord_less_set_a @ B2 @ A2 )
=> ( ord_less_eq_set_a @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_307_dual__order_Ostrict__implies__order,axiom,
! [B2: a > $o,A2: a > $o] :
( ( ord_less_a_o @ B2 @ A2 )
=> ( ord_less_eq_a_o @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_308_order_Onot__eq__order__implies__strict,axiom,
! [A2: int,B2: int] :
( ( A2 != B2 )
=> ( ( ord_less_eq_int @ A2 @ B2 )
=> ( ord_less_int @ A2 @ B2 ) ) ) ).
% order.not_eq_order_implies_strict
thf(fact_309_order_Onot__eq__order__implies__strict,axiom,
! [A2: set_a,B2: set_a] :
( ( A2 != B2 )
=> ( ( ord_less_eq_set_a @ A2 @ B2 )
=> ( ord_less_set_a @ A2 @ B2 ) ) ) ).
% order.not_eq_order_implies_strict
thf(fact_310_order_Onot__eq__order__implies__strict,axiom,
! [A2: a > $o,B2: a > $o] :
( ( A2 != B2 )
=> ( ( ord_less_eq_a_o @ A2 @ B2 )
=> ( ord_less_a_o @ A2 @ B2 ) ) ) ).
% order.not_eq_order_implies_strict
thf(fact_311_bot_Oextremum,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% bot.extremum
thf(fact_312_bot_Oextremum,axiom,
! [A2: a > $o] : ( ord_less_eq_a_o @ bot_bot_a_o @ A2 ) ).
% bot.extremum
thf(fact_313_bot_Oextremum__unique,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% bot.extremum_unique
thf(fact_314_bot_Oextremum__unique,axiom,
! [A2: a > $o] :
( ( ord_less_eq_a_o @ A2 @ bot_bot_a_o )
= ( A2 = bot_bot_a_o ) ) ).
% bot.extremum_unique
thf(fact_315_bot_Oextremum__uniqueI,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
=> ( A2 = bot_bot_set_a ) ) ).
% bot.extremum_uniqueI
thf(fact_316_bot_Oextremum__uniqueI,axiom,
! [A2: a > $o] :
( ( ord_less_eq_a_o @ A2 @ bot_bot_a_o )
=> ( A2 = bot_bot_a_o ) ) ).
% bot.extremum_uniqueI
thf(fact_317_bot_Oextremum__strict,axiom,
! [A2: set_a] :
~ ( ord_less_set_a @ A2 @ bot_bot_set_a ) ).
% bot.extremum_strict
thf(fact_318_bot_Oextremum__strict,axiom,
! [A2: a > $o] :
~ ( ord_less_a_o @ A2 @ bot_bot_a_o ) ).
% bot.extremum_strict
thf(fact_319_bot_Onot__eq__extremum,axiom,
! [A2: set_a] :
( ( A2 != bot_bot_set_a )
= ( ord_less_set_a @ bot_bot_set_a @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_320_bot_Onot__eq__extremum,axiom,
! [A2: a > $o] :
( ( A2 != bot_bot_a_o )
= ( ord_less_a_o @ bot_bot_a_o @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_321_subset__emptyI,axiom,
! [A3: set_a] :
( ! [X5: a] :
~ ( member_a @ X5 @ A3 )
=> ( ord_less_eq_set_a @ A3 @ bot_bot_set_a ) ) ).
% subset_emptyI
thf(fact_322_minf_I8_J,axiom,
! [T: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z3 )
=> ~ ( ord_less_eq_int @ T @ X6 ) ) ).
% minf(8)
thf(fact_323_minf_I6_J,axiom,
! [T: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z3 )
=> ( ord_less_eq_int @ X6 @ T ) ) ).
% minf(6)
thf(fact_324_pinf_I8_J,axiom,
! [T: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ Z3 @ X6 )
=> ( ord_less_eq_int @ T @ X6 ) ) ).
% pinf(8)
thf(fact_325_pinf_I6_J,axiom,
! [T: int] :
? [Z3: int] :
! [X6: int] :
( ( ord_less_int @ Z3 @ X6 )
=> ~ ( ord_less_eq_int @ X6 @ T ) ) ).
% pinf(6)
thf(fact_326_verit__comp__simplify1_I3_J,axiom,
! [B6: int,A6: int] :
( ( ~ ( ord_less_eq_int @ B6 @ A6 ) )
= ( ord_less_int @ A6 @ B6 ) ) ).
% verit_comp_simplify1(3)
thf(fact_327_complete__interval,axiom,
! [A2: int,B2: int,P: int > $o] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( P @ A2 )
=> ( ~ ( P @ B2 )
=> ? [C3: int] :
( ( ord_less_eq_int @ A2 @ C3 )
& ( ord_less_eq_int @ C3 @ B2 )
& ! [X6: int] :
( ( ( ord_less_eq_int @ A2 @ X6 )
& ( ord_less_int @ X6 @ C3 ) )
=> ( P @ X6 ) )
& ! [D: int] :
( ! [X5: int] :
( ( ( ord_less_eq_int @ A2 @ X5 )
& ( ord_less_int @ X5 @ D ) )
=> ( P @ X5 ) )
=> ( ord_less_eq_int @ D @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_328_bot_Oordering__top__axioms,axiom,
( ordering_top_set_a
@ ^ [X4: set_a,Y4: set_a] : ( ord_less_eq_set_a @ Y4 @ X4 )
@ ^ [X4: set_a,Y4: set_a] : ( ord_less_set_a @ Y4 @ X4 )
@ bot_bot_set_a ) ).
% bot.ordering_top_axioms
thf(fact_329_bot_Oordering__top__axioms,axiom,
( ordering_top_a_o
@ ^ [X4: a > $o,Y4: a > $o] : ( ord_less_eq_a_o @ Y4 @ X4 )
@ ^ [X4: a > $o,Y4: a > $o] : ( ord_less_a_o @ Y4 @ X4 )
@ bot_bot_a_o ) ).
% bot.ordering_top_axioms
thf(fact_330_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_331_less__fun__def,axiom,
( ord_less_a_o
= ( ^ [F2: a > $o,G2: a > $o] :
( ( ord_less_eq_a_o @ F2 @ G2 )
& ~ ( ord_less_eq_a_o @ G2 @ F2 ) ) ) ) ).
% less_fun_def
thf(fact_332_not__psubset__empty,axiom,
! [A3: set_a] :
~ ( ord_less_set_a @ A3 @ bot_bot_set_a ) ).
% not_psubset_empty
thf(fact_333_subset__iff__psubset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_set_a @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_334_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_335_subset__not__subset__eq,axiom,
( ord_less_set_a
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
& ~ ( ord_less_eq_set_a @ B5 @ A5 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_336_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_337_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_338_psubset__eq,axiom,
( ord_less_set_a
= ( ^ [A5: set_a,B5: set_a] :
( ( ord_less_eq_set_a @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% psubset_eq
thf(fact_339_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_340_less__eq__set__def,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B5: set_a] :
( ord_less_eq_a_o
@ ^ [X4: a] : ( member_a @ X4 @ A5 )
@ ^ [X4: a] : ( member_a @ X4 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_341_verit__comp__simplify1_I1_J,axiom,
! [A2: set_a] :
~ ( ord_less_set_a @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_342_verit__comp__simplify1_I1_J,axiom,
! [A2: a > $o] :
~ ( ord_less_a_o @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_343_prop__restrict,axiom,
! [X: a,Z4: set_a,X7: set_a,P: a > $o] :
( ( member_a @ X @ Z4 )
=> ( ( ord_less_eq_set_a @ Z4
@ ( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ X7 )
& ( P @ X4 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_344_Collect__restrict,axiom,
! [X7: set_a,P: a > $o] :
( ord_less_eq_set_a
@ ( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ X7 )
& ( P @ X4 ) ) )
@ X7 ) ).
% Collect_restrict
thf(fact_345_bot__empty__eq,axiom,
( bot_bot_a_o
= ( ^ [X4: a] : ( member_a @ X4 @ bot_bot_set_a ) ) ) ).
% bot_empty_eq
thf(fact_346_less__set__def,axiom,
( ord_less_set_a
= ( ^ [A5: set_a,B5: set_a] :
( ord_less_a_o
@ ^ [X4: a] : ( member_a @ X4 @ A5 )
@ ^ [X4: a] : ( member_a @ X4 @ B5 ) ) ) ) ).
% less_set_def
thf(fact_347_psubsetD,axiom,
! [A3: set_a,B3: set_a,C: a] :
( ( ord_less_set_a @ A3 @ B3 )
=> ( ( member_a @ C @ A3 )
=> ( member_a @ C @ B3 ) ) ) ).
% psubsetD
thf(fact_348_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_349_subset__Collect__iff,axiom,
! [B3: set_a,A3: set_a,P: a > $o] :
( ( ord_less_eq_set_a @ B3 @ A3 )
=> ( ( ord_less_eq_set_a @ B3
@ ( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ A3 )
& ( P @ X4 ) ) ) )
= ( ! [X4: a] :
( ( member_a @ X4 @ B3 )
=> ( P @ X4 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_350_subset__CollectI,axiom,
! [B3: set_a,A3: set_a,Q: a > $o,P: a > $o] :
( ( ord_less_eq_set_a @ B3 @ A3 )
=> ( ! [X5: a] :
( ( member_a @ X5 @ B3 )
=> ( ( Q @ X5 )
=> ( P @ X5 ) ) )
=> ( ord_less_eq_set_a
@ ( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ B3 )
& ( Q @ X4 ) ) )
@ ( collect_a
@ ^ [X4: a] :
( ( member_a @ X4 @ A3 )
& ( P @ X4 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_351_remove_Osimps_I2_J,axiom,
! [H: a > int,E: a,X: a,T1: binary1439146945Tree_a,T2: binary1439146945Tree_a] :
( ( ( ord_less_int @ ( H @ E ) @ ( H @ X ) )
=> ( ( binary1804682569move_a @ H @ E @ ( binary717961607le_T_a @ T1 @ X @ T2 ) )
= ( binary717961607le_T_a @ ( binary1804682569move_a @ H @ E @ T1 ) @ X @ T2 ) ) )
& ( ~ ( ord_less_int @ ( H @ E ) @ ( H @ X ) )
=> ( ( ( ord_less_int @ ( H @ X ) @ ( H @ E ) )
=> ( ( binary1804682569move_a @ H @ E @ ( binary717961607le_T_a @ T1 @ X @ T2 ) )
= ( binary717961607le_T_a @ T1 @ X @ ( binary1804682569move_a @ H @ E @ T2 ) ) ) )
& ( ~ ( ord_less_int @ ( H @ X ) @ ( H @ E ) )
=> ( ( ( T1 = binary476621312_Tip_a )
=> ( ( binary1804682569move_a @ H @ E @ ( binary717961607le_T_a @ T1 @ X @ T2 ) )
= T2 ) )
& ( ( T1 != binary476621312_Tip_a )
=> ( ( binary1804682569move_a @ H @ E @ ( binary717961607le_T_a @ T1 @ X @ T2 ) )
= ( produc1275654896Tree_a
@ ^ [T1p: binary1439146945Tree_a,R2: a] : ( binary717961607le_T_a @ T1p @ R2 @ T2 )
@ ( binary1774981152rmrm_a @ H @ T1 ) ) ) ) ) ) ) ) ) ).
% remove.simps(2)
thf(fact_352_wrmrm__decomp,axiom,
! [T: binary1439146945Tree_a,H: a > int] :
( ( T != binary476621312_Tip_a )
=> ( ( binary1774981152rmrm_a @ H @ T )
= ( produc374321428ee_a_a @ ( binary1217730267_wrm_a @ H @ T ) @ ( binary339557810e_rm_a @ H @ T ) ) ) ) ).
% wrmrm_decomp
% Conjectures (1)
thf(conj_0,conjecture,
binary1721989714Tree_a @ h @ t1 ).
%------------------------------------------------------------------------------