TPTP Problem File: ITP033^2.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : ITP033^2 : 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 : 0.00 v7.5.0
% Syntax : Number of formulae : 327 ( 90 unt; 47 typ; 0 def)
% Number of atoms : 862 ( 209 equ; 0 cnn)
% Maximal formula atoms : 13 ( 3 avg)
% Number of connectives : 3488 ( 101 ~; 19 |; 66 &;2845 @)
% ( 0 <=>; 457 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 9 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 315 ( 315 >; 0 *; 0 +; 0 <<)
% Number of symbols : 48 ( 45 usr; 5 con; 0-5 aty)
% Number of variables : 1166 ( 116 ^; 971 !; 34 ?;1166 :)
% ( 45 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:16:50.765
%------------------------------------------------------------------------------
% Could-be-implicit typings (5)
thf(ty_t_BinaryTree__Mirabelle__pchhvghoao_OTree,type,
binary1291135688e_Tree: $tType > $tType ).
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Int_Oint,type,
int: $tType ).
thf(ty_tf_a,type,
a: $tType ).
% Explicit typings (42)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Obot,type,
bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ono__bot,type,
no_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ono__top,type,
no_top:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder__bot,type,
order_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Owellorder,type,
wellorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Odense__order,type,
dense_order:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Odense__linorder,type,
dense_linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
condit1656338222tinuum:
!>[A: $tType] : $o ).
thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
condit1037483654norder:
!>[A: $tType] : $o ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type,
bNF_Ca1785829860lChain:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OTree_OT,type,
binary210054475elle_T:
!>[A: $tType] : ( ( binary1291135688e_Tree @ A ) > A > ( binary1291135688e_Tree @ A ) > ( binary1291135688e_Tree @ A ) ) ).
thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OTree_OTip,type,
binary1746293266le_Tip:
!>[A: $tType] : ( binary1291135688e_Tree @ A ) ).
thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OTree_Ocase__Tree,type,
binary536355927e_Tree:
!>[B: $tType,A: $tType] : ( B > ( ( binary1291135688e_Tree @ A ) > A > ( binary1291135688e_Tree @ A ) > B ) > ( binary1291135688e_Tree @ A ) > B ) ).
thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OTree_Orec__Tree,type,
binary1929596613c_Tree:
!>[C: $tType,A: $tType] : ( C > ( ( binary1291135688e_Tree @ A ) > A > ( binary1291135688e_Tree @ A ) > C > C > C ) > ( binary1291135688e_Tree @ A ) > C ) ).
thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Obinsert,type,
binary1830089824insert:
!>[A: $tType] : ( ( A > int ) > A > ( binary1291135688e_Tree @ A ) > ( binary1291135688e_Tree @ A ) ) ).
thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Omemb,type,
binary827270440e_memb:
!>[A: $tType] : ( ( A > int ) > A > ( binary1291135688e_Tree @ A ) > $o ) ).
thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Oremove,type,
binary997842527remove:
!>[A: $tType] : ( ( A > int ) > A > ( binary1291135688e_Tree @ A ) > ( binary1291135688e_Tree @ A ) ) ).
thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Orm,type,
binary576689334lle_rm:
!>[A: $tType] : ( ( A > int ) > ( binary1291135688e_Tree @ A ) > A ) ).
thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OsetOf,type,
binary1653327646_setOf:
!>[A: $tType] : ( ( binary1291135688e_Tree @ A ) > ( set @ A ) ) ).
thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_OsortedTree,type,
binary1610619414edTree:
!>[A: $tType] : ( ( A > int ) > ( binary1291135688e_Tree @ A ) > $o ) ).
thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Osorted__distinct__pred,type,
binary231205461t_pred:
!>[A: $tType] : ( ( A > int ) > A > A > ( binary1291135688e_Tree @ A ) > $o ) ).
thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Owrm,type,
binary213313527le_wrm:
!>[A: $tType] : ( ( A > int ) > ( binary1291135688e_Tree @ A ) > ( binary1291135688e_Tree @ A ) ) ).
thf(sy_c_BinaryTree__Mirabelle__pchhvghoao_Owrmrm,type,
binary1271298290_wrmrm:
!>[A: $tType] : ( ( A > int ) > ( binary1291135688e_Tree @ A ) > ( product_prod @ ( binary1291135688e_Tree @ A ) @ A ) ) ).
thf(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oordering__top,type,
ordering_top:
!>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > A > $o ) ).
thf(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
thf(sy_c_Product__Type_Oprod_Ocase__prod,type,
product_case_prod:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
thf(sy_c_Relation_Oinv__image,type,
inv_image:
!>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_Ois__empty,type,
is_empty:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_h,type,
h: a > int ).
thf(sy_v_t1____,type,
t1: binary1291135688e_Tree @ a ).
thf(sy_v_t2____,type,
t2: binary1291135688e_Tree @ a ).
thf(sy_v_x____,type,
x: a ).
% Relevant facts (256)
thf(fact_0_t2nTip,axiom,
( t2
!= ( binary1746293266le_Tip @ a ) ) ).
% t2nTip
thf(fact_1_s,axiom,
binary1610619414edTree @ a @ h @ ( binary210054475elle_T @ a @ t1 @ x @ t2 ) ).
% s
thf(fact_2_res,axiom,
( ( binary213313527le_wrm @ a @ h @ ( binary210054475elle_T @ a @ t1 @ x @ t2 ) )
= ( binary210054475elle_T @ a @ t1 @ x @ ( binary213313527le_wrm @ a @ h @ t2 ) ) ) ).
% res
thf(fact_3_sortLemmaL,axiom,
! [A: $tType,H: A > int,T1: binary1291135688e_Tree @ A,X: A,T2: binary1291135688e_Tree @ A] :
( ( binary1610619414edTree @ A @ H @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
=> ( binary1610619414edTree @ A @ H @ T1 ) ) ).
% sortLemmaL
thf(fact_4_sortLemmaR,axiom,
! [A: $tType,H: A > int,T1: binary1291135688e_Tree @ A,X: A,T2: binary1291135688e_Tree @ A] :
( ( binary1610619414edTree @ A @ H @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
=> ( binary1610619414edTree @ A @ H @ T2 ) ) ).
% sortLemmaR
thf(fact_5_Tree_Oinject,axiom,
! [A: $tType,X21: binary1291135688e_Tree @ A,X22: A,X23: binary1291135688e_Tree @ A,Y21: binary1291135688e_Tree @ A,Y22: A,Y23: binary1291135688e_Tree @ A] :
( ( ( binary210054475elle_T @ A @ X21 @ X22 @ X23 )
= ( binary210054475elle_T @ A @ Y21 @ Y22 @ Y23 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 )
& ( X23 = Y23 ) ) ) ).
% Tree.inject
thf(fact_6_binsert__sorted,axiom,
! [A: $tType,H: A > int,T: binary1291135688e_Tree @ A,X: A] :
( ( binary1610619414edTree @ A @ H @ T )
=> ( binary1610619414edTree @ A @ H @ ( binary1830089824insert @ A @ H @ X @ T ) ) ) ).
% binsert_sorted
thf(fact_7_h1,axiom,
( ( ( t1
!= ( binary1746293266le_Tip @ a ) )
& ( binary1610619414edTree @ a @ h @ t1 ) )
=> ( binary1610619414edTree @ a @ h @ ( binary213313527le_wrm @ a @ h @ t1 ) ) ) ).
% h1
thf(fact_8_sortedTree_Osimps_I1_J,axiom,
! [A: $tType,H: A > int] : ( binary1610619414edTree @ A @ H @ ( binary1746293266le_Tip @ A ) ) ).
% sortedTree.simps(1)
thf(fact_9_memb__spec,axiom,
! [A: $tType,H: A > int,T: binary1291135688e_Tree @ A,X: A] :
( ( binary1610619414edTree @ A @ H @ T )
=> ( ( binary827270440e_memb @ A @ H @ X @ T )
= ( member @ A @ X @ ( binary1653327646_setOf @ A @ T ) ) ) ) ).
% memb_spec
thf(fact_10_sorted__distinct__pred__def,axiom,
! [A: $tType] :
( ( binary231205461t_pred @ A )
= ( ^ [H2: A > int,A2: A,B2: A,T3: binary1291135688e_Tree @ A] :
( ( ( binary1610619414edTree @ A @ H2 @ T3 )
& ( member @ A @ A2 @ ( binary1653327646_setOf @ A @ T3 ) )
& ( member @ A @ B2 @ ( binary1653327646_setOf @ A @ T3 ) )
& ( ( H2 @ A2 )
= ( H2 @ B2 ) ) )
=> ( A2 = B2 ) ) ) ) ).
% sorted_distinct_pred_def
thf(fact_11_h2,axiom,
( ( ( t2
!= ( binary1746293266le_Tip @ a ) )
& ( binary1610619414edTree @ a @ h @ t2 ) )
=> ( binary1610619414edTree @ a @ h @ ( binary213313527le_wrm @ a @ h @ t2 ) ) ) ).
% h2
thf(fact_12_sorted__distinct,axiom,
! [A: $tType,H: A > int,A3: A,B3: A,T: binary1291135688e_Tree @ A] : ( binary231205461t_pred @ A @ H @ A3 @ B3 @ 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,
( ( ( ( binary1746293266le_Tip @ a )
!= ( binary1746293266le_Tip @ a ) )
& ( binary1610619414edTree @ a @ h @ ( binary1746293266le_Tip @ a ) ) )
=> ( binary1610619414edTree @ a @ h @ ( binary213313527le_wrm @ a @ h @ ( binary1746293266le_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,
! [A: $tType,X21: binary1291135688e_Tree @ A,X22: A,X23: binary1291135688e_Tree @ A] :
( ( binary1746293266le_Tip @ A )
!= ( binary210054475elle_T @ A @ X21 @ X22 @ X23 ) ) ).
% Tree.distinct(1)
thf(fact_15_binsert_Osimps_I1_J,axiom,
! [A: $tType,H: A > int,E: A] :
( ( binary1830089824insert @ A @ H @ E @ ( binary1746293266le_Tip @ A ) )
= ( binary210054475elle_T @ A @ ( binary1746293266le_Tip @ A ) @ E @ ( binary1746293266le_Tip @ A ) ) ) ).
% binsert.simps(1)
thf(fact_16_wrm_Osimps,axiom,
! [A: $tType,T2: binary1291135688e_Tree @ A,H: A > int,T1: binary1291135688e_Tree @ A,X: A] :
( ( ( T2
= ( binary1746293266le_Tip @ A ) )
=> ( ( binary213313527le_wrm @ A @ H @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
= T1 ) )
& ( ( T2
!= ( binary1746293266le_Tip @ A ) )
=> ( ( binary213313527le_wrm @ A @ H @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
= ( binary210054475elle_T @ A @ T1 @ X @ ( binary213313527le_wrm @ A @ H @ T2 ) ) ) ) ) ).
% wrm.simps
thf(fact_17_Tree_Oinduct,axiom,
! [A: $tType,P: ( binary1291135688e_Tree @ A ) > $o,Tree: binary1291135688e_Tree @ A] :
( ( P @ ( binary1746293266le_Tip @ A ) )
=> ( ! [X1: binary1291135688e_Tree @ A,X2: A,X3: binary1291135688e_Tree @ A] :
( ( P @ X1 )
=> ( ( P @ X3 )
=> ( P @ ( binary210054475elle_T @ A @ X1 @ X2 @ X3 ) ) ) )
=> ( P @ Tree ) ) ) ).
% Tree.induct
thf(fact_18_Tree_Oexhaust,axiom,
! [A: $tType,Y: binary1291135688e_Tree @ A] :
( ( Y
!= ( binary1746293266le_Tip @ A ) )
=> ~ ! [X212: binary1291135688e_Tree @ A,X222: A,X232: binary1291135688e_Tree @ A] :
( Y
!= ( binary210054475elle_T @ A @ X212 @ X222 @ X232 ) ) ) ).
% Tree.exhaust
thf(fact_19_rm__set,axiom,
! [A: $tType,T: binary1291135688e_Tree @ A,H: A > int] :
( ( ( T
!= ( binary1746293266le_Tip @ A ) )
& ( binary1610619414edTree @ A @ H @ T ) )
=> ( member @ A @ ( binary576689334lle_rm @ A @ H @ T ) @ ( binary1653327646_setOf @ A @ T ) ) ) ).
% rm_set
thf(fact_20_rm_Osimps,axiom,
! [A: $tType,T2: binary1291135688e_Tree @ A,H: A > int,T1: binary1291135688e_Tree @ A,X: A] :
( ( ( T2
= ( binary1746293266le_Tip @ A ) )
=> ( ( binary576689334lle_rm @ A @ H @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
= X ) )
& ( ( T2
!= ( binary1746293266le_Tip @ A ) )
=> ( ( binary576689334lle_rm @ A @ H @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
= ( binary576689334lle_rm @ A @ H @ T2 ) ) ) ) ).
% rm.simps
thf(fact_21_wrm__set1,axiom,
! [A: $tType,T: binary1291135688e_Tree @ A,H: A > int] :
( ( ( T
!= ( binary1746293266le_Tip @ A ) )
& ( binary1610619414edTree @ A @ H @ T ) )
=> ( ord_less_eq @ ( set @ A ) @ ( binary1653327646_setOf @ A @ ( binary213313527le_wrm @ A @ H @ T ) ) @ ( binary1653327646_setOf @ A @ T ) ) ) ).
% wrm_set1
thf(fact_22_sortedTree_Osimps_I2_J,axiom,
! [A: $tType,H: A > int,T1: binary1291135688e_Tree @ A,X: A,T2: binary1291135688e_Tree @ A] :
( ( binary1610619414edTree @ A @ H @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
= ( ( binary1610619414edTree @ A @ H @ T1 )
& ! [X4: A] :
( ( member @ A @ X4 @ ( binary1653327646_setOf @ A @ T1 ) )
=> ( ord_less @ int @ ( H @ X4 ) @ ( H @ X ) ) )
& ! [X4: A] :
( ( member @ A @ X4 @ ( binary1653327646_setOf @ A @ T2 ) )
=> ( ord_less @ int @ ( H @ X ) @ ( H @ X4 ) ) )
& ( binary1610619414edTree @ A @ H @ T2 ) ) ) ).
% sortedTree.simps(2)
thf(fact_23_remove_Osimps_I1_J,axiom,
! [A: $tType,H: A > int,E: A] :
( ( binary997842527remove @ A @ H @ E @ ( binary1746293266le_Tip @ A ) )
= ( binary1746293266le_Tip @ A ) ) ).
% remove.simps(1)
thf(fact_24_Tree_Osimps_I4_J,axiom,
! [A: $tType,B: $tType,F1: B,F2: ( binary1291135688e_Tree @ A ) > A > ( binary1291135688e_Tree @ A ) > B] :
( ( binary536355927e_Tree @ B @ A @ F1 @ F2 @ ( binary1746293266le_Tip @ A ) )
= F1 ) ).
% Tree.simps(4)
thf(fact_25_Tree_Osimps_I5_J,axiom,
! [B: $tType,A: $tType,F1: B,F2: ( binary1291135688e_Tree @ A ) > A > ( binary1291135688e_Tree @ A ) > B,X21: binary1291135688e_Tree @ A,X22: A,X23: binary1291135688e_Tree @ A] :
( ( binary536355927e_Tree @ B @ A @ F1 @ F2 @ ( binary210054475elle_T @ A @ X21 @ X22 @ X23 ) )
= ( F2 @ X21 @ X22 @ X23 ) ) ).
% Tree.simps(5)
thf(fact_26_binsert_Osimps_I2_J,axiom,
! [A: $tType,H: A > int,E: A,X: A,T1: binary1291135688e_Tree @ A,T2: binary1291135688e_Tree @ A] :
( ( ( ord_less @ int @ ( H @ E ) @ ( H @ X ) )
=> ( ( binary1830089824insert @ A @ H @ E @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
= ( binary210054475elle_T @ A @ ( binary1830089824insert @ A @ H @ E @ T1 ) @ X @ T2 ) ) )
& ( ~ ( ord_less @ int @ ( H @ E ) @ ( H @ X ) )
=> ( ( ( ord_less @ int @ ( H @ X ) @ ( H @ E ) )
=> ( ( binary1830089824insert @ A @ H @ E @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
= ( binary210054475elle_T @ A @ T1 @ X @ ( binary1830089824insert @ A @ H @ E @ T2 ) ) ) )
& ( ~ ( ord_less @ int @ ( H @ X ) @ ( H @ E ) )
=> ( ( binary1830089824insert @ A @ H @ E @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
= ( binary210054475elle_T @ A @ T1 @ E @ T2 ) ) ) ) ) ) ).
% binsert.simps(2)
thf(fact_27_setOf_Osimps_I1_J,axiom,
! [A: $tType] :
( ( binary1653327646_setOf @ A @ ( binary1746293266le_Tip @ A ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% setOf.simps(1)
thf(fact_28_Tree_Osimps_I6_J,axiom,
! [A: $tType,C: $tType,F1: C,F2: ( binary1291135688e_Tree @ A ) > A > ( binary1291135688e_Tree @ A ) > C > C > C] :
( ( binary1929596613c_Tree @ C @ A @ F1 @ F2 @ ( binary1746293266le_Tip @ A ) )
= F1 ) ).
% Tree.simps(6)
thf(fact_29_Tree_Osimps_I7_J,axiom,
! [C: $tType,A: $tType,F1: C,F2: ( binary1291135688e_Tree @ A ) > A > ( binary1291135688e_Tree @ A ) > C > C > C,X21: binary1291135688e_Tree @ A,X22: A,X23: binary1291135688e_Tree @ A] :
( ( binary1929596613c_Tree @ C @ A @ F1 @ F2 @ ( binary210054475elle_T @ A @ X21 @ X22 @ X23 ) )
= ( F2 @ X21 @ X22 @ X23 @ ( binary1929596613c_Tree @ C @ A @ F1 @ F2 @ X21 ) @ ( binary1929596613c_Tree @ C @ A @ F1 @ F2 @ X23 ) ) ) ).
% Tree.simps(7)
thf(fact_30_Tree_Ocase__distrib,axiom,
! [C: $tType,B: $tType,A: $tType,H: B > C,F1: B,F2: ( binary1291135688e_Tree @ A ) > A > ( binary1291135688e_Tree @ A ) > B,Tree: binary1291135688e_Tree @ A] :
( ( H @ ( binary536355927e_Tree @ B @ A @ F1 @ F2 @ Tree ) )
= ( binary536355927e_Tree @ C @ A @ ( H @ F1 )
@ ^ [X12: binary1291135688e_Tree @ A,X24: A,X32: binary1291135688e_Tree @ A] : ( H @ ( F2 @ X12 @ X24 @ X32 ) )
@ Tree ) ) ).
% Tree.case_distrib
thf(fact_31_empty__subsetI,axiom,
! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A4 ) ).
% empty_subsetI
thf(fact_32_subset__empty,axiom,
! [A: $tType,A4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
= ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_empty
thf(fact_33_subset__antisym,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ A4 )
=> ( A4 = B4 ) ) ) ).
% subset_antisym
thf(fact_34_subsetI,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ! [X5: A] :
( ( member @ A @ X5 @ A4 )
=> ( member @ A @ X5 @ B4 ) )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ).
% subsetI
thf(fact_35_empty__Collect__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P ) )
= ( ! [X4: A] :
~ ( P @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_36_Collect__empty__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X4: A] :
~ ( P @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_37_all__not__in__conv,axiom,
! [A: $tType,A4: set @ A] :
( ( ! [X4: A] :
~ ( member @ A @ X4 @ A4 ) )
= ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_38_empty__iff,axiom,
! [A: $tType,C2: A] :
~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_39_bot__apply,axiom,
! [C: $tType,D: $tType] :
( ( bot @ C )
=> ( ( bot_bot @ ( D > C ) )
= ( ^ [X4: D] : ( bot_bot @ C ) ) ) ) ).
% bot_apply
thf(fact_40_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_41_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_42_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% le_funD
thf(fact_43_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% le_funE
thf(fact_44_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B] :
( ! [X5: A] : ( ord_less_eq @ B @ ( F @ X5 ) @ ( G @ X5 ) )
=> ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% le_funI
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A3: A,P: A > $o] :
( ( member @ A @ A3 @ ( collect @ A @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( collect @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X5: A] :
( ( P @ X5 )
= ( Q @ X5 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X5: A] :
( ( F @ X5 )
= ( G @ X5 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F3: A > B,G2: A > B] :
! [X4: A] : ( ord_less_eq @ B @ ( F3 @ X4 ) @ ( G2 @ X4 ) ) ) ) ) ).
% le_fun_def
thf(fact_50_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F: B > A,B3: B,C2: B] :
( ( ord_less_eq @ A @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C2 )
=> ( ! [X5: B,Y2: B] :
( ( ord_less_eq @ B @ X5 @ Y2 )
=> ( ord_less_eq @ A @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% order_subst1
thf(fact_51_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A3: A,B3: A,F: A > C,C2: C] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ C @ ( F @ B3 ) @ C2 )
=> ( ! [X5: A,Y2: A] :
( ( ord_less_eq @ A @ X5 @ Y2 )
=> ( ord_less_eq @ C @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ C @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% order_subst2
thf(fact_52_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,F: B > A,B3: B,C2: B] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C2 )
=> ( ! [X5: B,Y2: B] :
( ( ord_less_eq @ B @ X5 @ Y2 )
=> ( ord_less_eq @ A @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_53_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,B3: A,F: A > B,C2: B] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X5: A,Y2: A] :
( ( ord_less_eq @ A @ X5 @ Y2 )
=> ( ord_less_eq @ B @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ B @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_54_eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y3: A,Z: A] : Y3 = Z )
= ( ^ [X4: A,Y4: A] :
( ( ord_less_eq @ A @ X4 @ Y4 )
& ( ord_less_eq @ A @ Y4 @ X4 ) ) ) ) ) ).
% eq_iff
thf(fact_55_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ X )
=> ( X = Y ) ) ) ) ).
% antisym
thf(fact_56_linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
| ( ord_less_eq @ A @ Y @ X ) ) ) ).
% linear
thf(fact_57_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( X = Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% eq_refl
thf(fact_58_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% le_cases
thf(fact_59_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ C2 )
=> ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% order.trans
thf(fact_60_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( ( ord_less_eq @ A @ X @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less_eq @ A @ X @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ X @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z2 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ X ) )
=> ~ ( ( ord_less_eq @ A @ Z2 @ X )
=> ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_61_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_less_eq @ A @ X @ Y )
= ( X = Y ) ) ) ) ).
% antisym_conv
thf(fact_62_order__class_Oorder_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y3: A,Z: A] : Y3 = Z )
= ( ^ [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
& ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_63_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( A3 = B3 )
=> ( ( ord_less_eq @ A @ B3 @ C2 )
=> ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_64_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_65_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ) ).
% order_class.order.antisym
thf(fact_66_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z2 )
=> ( ord_less_eq @ A @ X @ Z2 ) ) ) ) ).
% order_trans
thf(fact_67_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% dual_order.refl
thf(fact_68_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A3: A,B3: A] :
( ! [A5: A,B5: A] :
( ( ord_less_eq @ A @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: A,B5: A] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A3 @ B3 ) ) ) ) ).
% linorder_wlog
thf(fact_69_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A3: A,C2: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ C2 @ B3 )
=> ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% dual_order.trans
thf(fact_70_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y3: A,Z: A] : Y3 = Z )
= ( ^ [A2: A,B2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
& ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_71_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ) ).
% dual_order.antisym
thf(fact_72_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,F: B > A,B3: B,C2: B] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less @ B @ B3 @ C2 )
=> ( ! [X5: B,Y2: B] :
( ( ord_less @ B @ X5 @ Y2 )
=> ( ord_less @ A @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_73_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,B3: A,F: A > B,C2: B] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X5: A,Y2: A] :
( ( ord_less @ A @ X5 @ Y2 )
=> ( ord_less @ B @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ B @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_74_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F: B > A,B3: B,C2: B] :
( ( ord_less @ A @ A3 @ ( F @ B3 ) )
=> ( ( ord_less @ B @ B3 @ C2 )
=> ( ! [X5: B,Y2: B] :
( ( ord_less @ B @ X5 @ Y2 )
=> ( ord_less @ A @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_75_order__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A3: A,B3: A,F: A > C,C2: C] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less @ C @ ( F @ B3 ) @ C2 )
=> ( ! [X5: A,Y2: A] :
( ( ord_less @ A @ X5 @ Y2 )
=> ( ord_less @ C @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ C @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% order_less_subst2
thf(fact_76_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [X: A] :
? [Y2: A] : ( ord_less @ A @ Y2 @ X ) ) ).
% lt_ex
thf(fact_77_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [X: A] :
? [X_1: A] : ( ord_less @ A @ X @ X_1 ) ) ).
% gt_ex
thf(fact_78_neqE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
=> ( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% neqE
thf(fact_79_neq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
= ( ( ord_less @ A @ X @ Y )
| ( ord_less @ A @ Y @ X ) ) ) ) ).
% neq_iff
thf(fact_80_order_Oasym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ~ ( ord_less @ A @ B3 @ A3 ) ) ) ).
% order.asym
thf(fact_81_dense,axiom,
! [A: $tType] :
( ( dense_order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ? [Z3: A] :
( ( ord_less @ A @ X @ Z3 )
& ( ord_less @ A @ Z3 @ Y ) ) ) ) ).
% dense
thf(fact_82_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( X != Y ) ) ) ).
% less_imp_neq
thf(fact_83_less__asym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% less_asym
thf(fact_84_less__asym_H,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ~ ( ord_less @ A @ B3 @ A3 ) ) ) ).
% less_asym'
thf(fact_85_less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ Z2 )
=> ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% less_trans
thf(fact_86_less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
| ( X = Y )
| ( ord_less @ A @ Y @ X ) ) ) ).
% less_linear
thf(fact_87_less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] :
~ ( ord_less @ A @ X @ X ) ) ).
% less_irrefl
thf(fact_88_ord__eq__less__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( A3 = B3 )
=> ( ( ord_less @ A @ B3 @ C2 )
=> ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% ord_eq_less_trans
thf(fact_89_ord__less__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% ord_less_eq_trans
thf(fact_90_dual__order_Oasym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ~ ( ord_less @ A @ A3 @ B3 ) ) ) ).
% dual_order.asym
thf(fact_91_less__imp__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( X != Y ) ) ) ).
% less_imp_not_eq
thf(fact_92_less__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% less_not_sym
thf(fact_93_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,A3: A] :
( ! [X5: A] :
( ! [Y5: A] :
( ( ord_less @ A @ Y5 @ X5 )
=> ( P @ Y5 ) )
=> ( P @ X5 ) )
=> ( P @ A3 ) ) ) ).
% less_induct
thf(fact_94_antisym__conv3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X: A] :
( ~ ( ord_less @ A @ Y @ X )
=> ( ( ~ ( ord_less @ A @ X @ Y ) )
= ( X = Y ) ) ) ) ).
% antisym_conv3
thf(fact_95_less__imp__not__eq2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( Y != X ) ) ) ).
% less_imp_not_eq2
thf(fact_96_less__imp__triv,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,P: $o] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ X )
=> P ) ) ) ).
% less_imp_triv
thf(fact_97_linorder__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ( X != Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% linorder_cases
thf(fact_98_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ A3 @ A3 ) ) ).
% dual_order.irrefl
thf(fact_99_order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ B3 @ C2 )
=> ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% order.strict_trans
thf(fact_100_less__imp__not__less,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% less_imp_not_less
thf(fact_101_exists__least__iff,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ( ( ^ [P2: A > $o] :
? [X6: A] : ( P2 @ X6 ) )
= ( ^ [P3: A > $o] :
? [N: A] :
( ( P3 @ N )
& ! [M: A] :
( ( ord_less @ A @ M @ N )
=> ~ ( P3 @ M ) ) ) ) ) ) ).
% exists_least_iff
thf(fact_102_linorder__less__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A3: A,B3: A] :
( ! [A5: A,B5: A] :
( ( ord_less @ A @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: A] : ( P @ A5 @ A5 )
=> ( ! [A5: A,B5: A] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A3 @ B3 ) ) ) ) ) ).
% linorder_less_wlog
thf(fact_103_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A3: A,C2: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ( ( ord_less @ A @ C2 @ B3 )
=> ( ord_less @ A @ C2 @ A3 ) ) ) ) ).
% dual_order.strict_trans
thf(fact_104_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less @ A @ X @ Y ) )
= ( ( ord_less @ A @ Y @ X )
| ( X = Y ) ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_105_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( A3 != B3 ) ) ) ).
% order.strict_implies_not_eq
thf(fact_106_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ( A3 != B3 ) ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_107_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X4: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_108_emptyE,axiom,
! [A: $tType,A3: A] :
~ ( member @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_109_equals0D,axiom,
! [A: $tType,A4: set @ A,A3: A] :
( ( A4
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A3 @ A4 ) ) ).
% equals0D
thf(fact_110_equals0I,axiom,
! [A: $tType,A4: set @ A] :
( ! [Y2: A] :
~ ( member @ A @ Y2 @ A4 )
=> ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_111_ex__in__conv,axiom,
! [A: $tType,A4: set @ A] :
( ( ? [X4: A] : ( member @ A @ X4 @ A4 ) )
= ( A4
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_112_in__mono,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ( member @ A @ X @ A4 )
=> ( member @ A @ X @ B4 ) ) ) ).
% in_mono
thf(fact_113_subsetD,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ( member @ A @ C2 @ A4 )
=> ( member @ A @ C2 @ B4 ) ) ) ).
% subsetD
thf(fact_114_equalityE,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( A4 = B4 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ) ).
% equalityE
thf(fact_115_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
! [X4: A] :
( ( member @ A @ X4 @ A6 )
=> ( member @ A @ X4 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_116_equalityD1,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( A4 = B4 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ).
% equalityD1
thf(fact_117_equalityD2,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( A4 = B4 )
=> ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ).
% equalityD2
thf(fact_118_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
! [T3: A] :
( ( member @ A @ T3 @ A6 )
=> ( member @ A @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_119_subset__refl,axiom,
! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ A4 @ A4 ) ).
% subset_refl
thf(fact_120_Collect__mono,axiom,
! [A: $tType,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_121_subset__trans,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ C3 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ C3 ) ) ) ).
% subset_trans
thf(fact_122_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y3: set @ A,Z: set @ A] : Y3 = Z )
= ( ^ [A6: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
& ( ord_less_eq @ ( set @ A ) @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_123_Collect__mono__iff,axiom,
! [A: $tType,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_124_empty__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A
@ ^ [X4: A] : $false ) ) ).
% empty_def
thf(fact_125_Collect__subset,axiom,
! [A: $tType,A4: set @ A,P: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A4 )
& ( P @ X4 ) ) )
@ A4 ) ).
% Collect_subset
thf(fact_126_leD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less @ A @ X @ Y ) ) ) ).
% leD
thf(fact_127_leI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% leI
thf(fact_128_le__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X4: A,Y4: A] :
( ( ord_less @ A @ X4 @ Y4 )
| ( X4 = Y4 ) ) ) ) ) ).
% le_less
thf(fact_129_less__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [X4: A,Y4: A] :
( ( ord_less_eq @ A @ X4 @ Y4 )
& ( X4 != Y4 ) ) ) ) ) ).
% less_le
thf(fact_130_order__le__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F: B > A,B3: B,C2: B] :
( ( ord_less_eq @ A @ A3 @ ( F @ B3 ) )
=> ( ( ord_less @ B @ B3 @ C2 )
=> ( ! [X5: B,Y2: B] :
( ( ord_less @ B @ X5 @ Y2 )
=> ( ord_less @ A @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_131_order__le__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A3: A,B3: A,F: A > C,C2: C] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less @ C @ ( F @ B3 ) @ C2 )
=> ( ! [X5: A,Y2: A] :
( ( ord_less_eq @ A @ X5 @ Y2 )
=> ( ord_less_eq @ C @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ C @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% order_le_less_subst2
thf(fact_132_order__less__le__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F: B > A,B3: B,C2: B] :
( ( ord_less @ A @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C2 )
=> ( ! [X5: B,Y2: B] :
( ( ord_less_eq @ B @ X5 @ Y2 )
=> ( ord_less_eq @ A @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_133_order__less__le__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A3: A,B3: A,F: A > C,C2: C] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ C @ ( F @ B3 ) @ C2 )
=> ( ! [X5: A,Y2: A] :
( ( ord_less @ A @ X5 @ Y2 )
=> ( ord_less @ C @ ( F @ X5 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ C @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% order_less_le_subst2
thf(fact_134_not__le,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less_eq @ A @ X @ Y ) )
= ( ord_less @ A @ Y @ X ) ) ) ).
% not_le
thf(fact_135_not__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less @ A @ X @ Y ) )
= ( ord_less_eq @ A @ Y @ X ) ) ) ).
% not_less
thf(fact_136_le__neq__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( A3 != B3 )
=> ( ord_less @ A @ A3 @ B3 ) ) ) ) ).
% le_neq_trans
thf(fact_137_antisym__conv1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ X @ Y )
= ( X = Y ) ) ) ) ).
% antisym_conv1
thf(fact_138_antisym__conv2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ~ ( ord_less @ A @ X @ Y ) )
= ( X = Y ) ) ) ) ).
% antisym_conv2
thf(fact_139_less__imp__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% less_imp_le
thf(fact_140_le__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ Z2 )
=> ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% le_less_trans
thf(fact_141_less__le__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z2 )
=> ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% less_le_trans
thf(fact_142_dense__ge,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z2: A,Y: A] :
( ! [X5: A] :
( ( ord_less @ A @ Z2 @ X5 )
=> ( ord_less_eq @ A @ Y @ X5 ) )
=> ( ord_less_eq @ A @ Y @ Z2 ) ) ) ).
% dense_ge
thf(fact_143_dense__le,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Y: A,Z2: A] :
( ! [X5: A] :
( ( ord_less @ A @ X5 @ Y )
=> ( ord_less_eq @ A @ X5 @ Z2 ) )
=> ( ord_less_eq @ A @ Y @ Z2 ) ) ) ).
% dense_le
thf(fact_144_le__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
| ( ord_less @ A @ Y @ X ) ) ) ).
% le_less_linear
thf(fact_145_le__imp__less__or__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ X @ Y )
| ( X = Y ) ) ) ) ).
% le_imp_less_or_eq
thf(fact_146_less__le__not__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [X4: A,Y4: A] :
( ( ord_less_eq @ A @ X4 @ Y4 )
& ~ ( ord_less_eq @ A @ Y4 @ X4 ) ) ) ) ) ).
% less_le_not_le
thf(fact_147_not__le__imp__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X: A] :
( ~ ( ord_less_eq @ A @ Y @ X )
=> ( ord_less @ A @ X @ Y ) ) ) ).
% not_le_imp_less
thf(fact_148_order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ B3 @ C2 )
=> ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% order.strict_trans1
thf(fact_149_order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A,C2: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ C2 )
=> ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% order.strict_trans2
thf(fact_150_order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ) ).
% order.order_iff_strict
thf(fact_151_order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ) ).
% order.strict_iff_order
thf(fact_152_dual__order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A3: A,C2: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less @ A @ C2 @ B3 )
=> ( ord_less @ A @ C2 @ A3 ) ) ) ) ).
% dual_order.strict_trans1
thf(fact_153_dual__order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A3: A,C2: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ C2 @ B3 )
=> ( ord_less @ A @ C2 @ A3 ) ) ) ) ).
% dual_order.strict_trans2
thf(fact_154_dense__ge__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z2: A,X: A,Y: A] :
( ( ord_less @ A @ Z2 @ X )
=> ( ! [W: A] :
( ( ord_less @ A @ Z2 @ W )
=> ( ( ord_less @ A @ W @ X )
=> ( ord_less_eq @ A @ Y @ W ) ) )
=> ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% dense_ge_bounded
thf(fact_155_dense__le__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less @ A @ X @ Y )
=> ( ! [W: A] :
( ( ord_less @ A @ X @ W )
=> ( ( ord_less @ A @ W @ Y )
=> ( ord_less_eq @ A @ W @ Z2 ) ) )
=> ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% dense_le_bounded
thf(fact_156_order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ord_less_eq @ A @ A3 @ B3 ) ) ) ).
% order.strict_implies_order
thf(fact_157_dual__order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_158_dual__order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_159_dual__order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ( ord_less_eq @ A @ B3 @ A3 ) ) ) ).
% dual_order.strict_implies_order
thf(fact_160_order_Onot__eq__order__implies__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A] :
( ( A3 != B3 )
=> ( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ord_less @ A @ A3 @ B3 ) ) ) ) ).
% order.not_eq_order_implies_strict
thf(fact_161_bot_Oextremum,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A3 ) ) ).
% bot.extremum
thf(fact_162_bot_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
= ( A3
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_unique
thf(fact_163_bot_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
=> ( A3
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_uniqueI
thf(fact_164_bot_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ A3 @ ( bot_bot @ A ) ) ) ).
% bot.extremum_strict
thf(fact_165_bot_Onot__eq__extremum,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] :
( ( A3
!= ( bot_bot @ A ) )
= ( ord_less @ A @ ( bot_bot @ A ) @ A3 ) ) ) ).
% bot.not_eq_extremum
thf(fact_166_subset__emptyI,axiom,
! [A: $tType,A4: set @ A] :
( ! [X5: A] :
~ ( member @ A @ X5 @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_emptyI
thf(fact_167_minf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z3: A] :
! [X7: A] :
( ( ord_less @ A @ X7 @ Z3 )
=> ~ ( ord_less_eq @ A @ T @ X7 ) ) ) ).
% minf(8)
thf(fact_168_minf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z3: A] :
! [X7: A] :
( ( ord_less @ A @ X7 @ Z3 )
=> ( ord_less_eq @ A @ X7 @ T ) ) ) ).
% minf(6)
thf(fact_169_pinf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z3: A] :
! [X7: A] :
( ( ord_less @ A @ Z3 @ X7 )
=> ( ord_less_eq @ A @ T @ X7 ) ) ) ).
% pinf(8)
thf(fact_170_pinf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z3: A] :
! [X7: A] :
( ( ord_less @ A @ Z3 @ X7 )
=> ~ ( ord_less_eq @ A @ X7 @ T ) ) ) ).
% pinf(6)
thf(fact_171_verit__comp__simplify1_I3_J,axiom,
! [B: $tType] :
( ( linorder @ B )
=> ! [B7: B,A7: B] :
( ( ~ ( ord_less_eq @ B @ B7 @ A7 ) )
= ( ord_less @ B @ A7 @ B7 ) ) ) ).
% verit_comp_simplify1(3)
thf(fact_172_complete__interval,axiom,
! [A: $tType] :
( ( condit1037483654norder @ A )
=> ! [A3: A,B3: A,P: A > $o] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( P @ A3 )
=> ( ~ ( P @ B3 )
=> ? [C4: A] :
( ( ord_less_eq @ A @ A3 @ C4 )
& ( ord_less_eq @ A @ C4 @ B3 )
& ! [X7: A] :
( ( ( ord_less_eq @ A @ A3 @ X7 )
& ( ord_less @ A @ X7 @ C4 ) )
=> ( P @ X7 ) )
& ! [D2: A] :
( ! [X5: A] :
( ( ( ord_less_eq @ A @ A3 @ X5 )
& ( ord_less @ A @ X5 @ D2 ) )
=> ( P @ X5 ) )
=> ( ord_less_eq @ A @ D2 @ C4 ) ) ) ) ) ) ) ).
% complete_interval
thf(fact_173_bot_Oordering__top__axioms,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ( ordering_top @ A
@ ^ [X4: A,Y4: A] : ( ord_less_eq @ A @ Y4 @ X4 )
@ ^ [X4: A,Y4: A] : ( ord_less @ A @ Y4 @ X4 )
@ ( bot_bot @ A ) ) ) ).
% bot.ordering_top_axioms
thf(fact_174_psubsetI,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ( A4 != B4 )
=> ( ord_less @ ( set @ A ) @ A4 @ B4 ) ) ) ).
% psubsetI
thf(fact_175_less__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less @ ( A > B ) )
= ( ^ [F3: A > B,G2: A > B] :
( ( ord_less_eq @ ( A > B ) @ F3 @ G2 )
& ~ ( ord_less_eq @ ( A > B ) @ G2 @ F3 ) ) ) ) ) ).
% less_fun_def
thf(fact_176_ordering__top_Oextremum,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A3: A] :
( ( ordering_top @ A @ Less_eq @ Less @ Top )
=> ( Less_eq @ A3 @ Top ) ) ).
% ordering_top.extremum
thf(fact_177_ordering__top_Oextremum__strict,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A3: A] :
( ( ordering_top @ A @ Less_eq @ Less @ Top )
=> ~ ( Less @ Top @ A3 ) ) ).
% ordering_top.extremum_strict
thf(fact_178_ordering__top_Oextremum__unique,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A3: A] :
( ( ordering_top @ A @ Less_eq @ Less @ Top )
=> ( ( Less_eq @ Top @ A3 )
= ( A3 = Top ) ) ) ).
% ordering_top.extremum_unique
thf(fact_179_ordering__top_Onot__eq__extremum,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A3: A] :
( ( ordering_top @ A @ Less_eq @ Less @ Top )
=> ( ( A3 != Top )
= ( Less @ A3 @ Top ) ) ) ).
% ordering_top.not_eq_extremum
thf(fact_180_ordering__top_Oextremum__uniqueI,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,Top: A,A3: A] :
( ( ordering_top @ A @ Less_eq @ Less @ Top )
=> ( ( Less_eq @ Top @ A3 )
=> ( A3 = Top ) ) ) ).
% ordering_top.extremum_uniqueI
thf(fact_181_not__psubset__empty,axiom,
! [A: $tType,A4: set @ A] :
~ ( ord_less @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) ) ).
% not_psubset_empty
thf(fact_182_subset__iff__psubset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
( ( ord_less @ ( set @ A ) @ A6 @ B6 )
| ( A6 = B6 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_183_subset__psubset__trans,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ( ord_less @ ( set @ A ) @ B4 @ C3 )
=> ( ord_less @ ( set @ A ) @ A4 @ C3 ) ) ) ).
% subset_psubset_trans
thf(fact_184_subset__not__subset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
& ~ ( ord_less_eq @ ( set @ A ) @ B6 @ A6 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_185_psubset__subset__trans,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,C3: set @ A] :
( ( ord_less @ ( set @ A ) @ A4 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ C3 )
=> ( ord_less @ ( set @ A ) @ A4 @ C3 ) ) ) ).
% psubset_subset_trans
thf(fact_186_psubset__imp__subset,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( ord_less @ ( set @ A ) @ A4 @ B4 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ).
% psubset_imp_subset
thf(fact_187_psubset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
& ( A6 != B6 ) ) ) ) ).
% psubset_eq
thf(fact_188_psubsetE,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( ord_less @ ( set @ A ) @ A4 @ B4 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ) ).
% psubsetE
thf(fact_189_less__eq__set__def,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
( ord_less_eq @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ A6 )
@ ^ [X4: A] : ( member @ A @ X4 @ B6 ) ) ) ) ).
% less_eq_set_def
thf(fact_190_verit__la__disequality,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] :
( ( A3 = B3 )
| ~ ( ord_less_eq @ A @ A3 @ B3 )
| ~ ( ord_less_eq @ A @ B3 @ A3 ) ) ) ).
% verit_la_disequality
thf(fact_191_ex__gt__or__lt,axiom,
! [A: $tType] :
( ( condit1656338222tinuum @ A )
=> ! [A3: A] :
? [B5: A] :
( ( ord_less @ A @ A3 @ B5 )
| ( ord_less @ A @ B5 @ A3 ) ) ) ).
% ex_gt_or_lt
thf(fact_192_verit__comp__simplify1_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ A3 @ A3 ) ) ).
% verit_comp_simplify1(1)
thf(fact_193_pinf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ Z4 @ X5 )
=> ( ( P @ X5 )
= ( P4 @ X5 ) ) )
=> ( ? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ Z4 @ X5 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z3: A] :
! [X7: A] :
( ( ord_less @ A @ Z3 @ X7 )
=> ( ( ( P @ X7 )
& ( Q @ X7 ) )
= ( ( P4 @ X7 )
& ( Q2 @ X7 ) ) ) ) ) ) ) ).
% pinf(1)
thf(fact_194_pinf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ Z4 @ X5 )
=> ( ( P @ X5 )
= ( P4 @ X5 ) ) )
=> ( ? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ Z4 @ X5 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z3: A] :
! [X7: A] :
( ( ord_less @ A @ Z3 @ X7 )
=> ( ( ( P @ X7 )
| ( Q @ X7 ) )
= ( ( P4 @ X7 )
| ( Q2 @ X7 ) ) ) ) ) ) ) ).
% pinf(2)
thf(fact_195_pinf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z3: A] :
! [X7: A] :
( ( ord_less @ A @ Z3 @ X7 )
=> ( X7 != T ) ) ) ).
% pinf(3)
thf(fact_196_pinf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z3: A] :
! [X7: A] :
( ( ord_less @ A @ Z3 @ X7 )
=> ( X7 != T ) ) ) ).
% pinf(4)
thf(fact_197_pinf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z3: A] :
! [X7: A] :
( ( ord_less @ A @ Z3 @ X7 )
=> ~ ( ord_less @ A @ X7 @ T ) ) ) ).
% pinf(5)
thf(fact_198_pinf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z3: A] :
! [X7: A] :
( ( ord_less @ A @ Z3 @ X7 )
=> ( ord_less @ A @ T @ X7 ) ) ) ).
% pinf(7)
thf(fact_199_pinf_I11_J,axiom,
! [C: $tType,D: $tType] :
( ( ord @ C )
=> ! [F4: D] :
? [Z3: C] :
! [X7: C] :
( ( ord_less @ C @ Z3 @ X7 )
=> ( F4 = F4 ) ) ) ).
% pinf(11)
thf(fact_200_minf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z4 )
=> ( ( P @ X5 )
= ( P4 @ X5 ) ) )
=> ( ? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z4 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z3: A] :
! [X7: A] :
( ( ord_less @ A @ X7 @ Z3 )
=> ( ( ( P @ X7 )
& ( Q @ X7 ) )
= ( ( P4 @ X7 )
& ( Q2 @ X7 ) ) ) ) ) ) ) ).
% minf(1)
thf(fact_201_minf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z4 )
=> ( ( P @ X5 )
= ( P4 @ X5 ) ) )
=> ( ? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z4 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z3: A] :
! [X7: A] :
( ( ord_less @ A @ X7 @ Z3 )
=> ( ( ( P @ X7 )
| ( Q @ X7 ) )
= ( ( P4 @ X7 )
| ( Q2 @ X7 ) ) ) ) ) ) ) ).
% minf(2)
thf(fact_202_minf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z3: A] :
! [X7: A] :
( ( ord_less @ A @ X7 @ Z3 )
=> ( X7 != T ) ) ) ).
% minf(3)
thf(fact_203_minf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z3: A] :
! [X7: A] :
( ( ord_less @ A @ X7 @ Z3 )
=> ( X7 != T ) ) ) ).
% minf(4)
thf(fact_204_minf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z3: A] :
! [X7: A] :
( ( ord_less @ A @ X7 @ Z3 )
=> ( ord_less @ A @ X7 @ T ) ) ) ).
% minf(5)
thf(fact_205_minf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z3: A] :
! [X7: A] :
( ( ord_less @ A @ X7 @ Z3 )
=> ~ ( ord_less @ A @ T @ X7 ) ) ) ).
% minf(7)
thf(fact_206_minf_I11_J,axiom,
! [C: $tType,D: $tType] :
( ( ord @ C )
=> ! [F4: D] :
? [Z3: C] :
! [X7: C] :
( ( ord_less @ C @ X7 @ Z3 )
=> ( F4 = F4 ) ) ) ).
% minf(11)
thf(fact_207_prop__restrict,axiom,
! [A: $tType,X: A,Z5: set @ A,X8: set @ A,P: A > $o] :
( ( member @ A @ X @ Z5 )
=> ( ( ord_less_eq @ ( set @ A ) @ Z5
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ X8 )
& ( P @ X4 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_208_Collect__restrict,axiom,
! [A: $tType,X8: set @ A,P: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ X8 )
& ( P @ X4 ) ) )
@ X8 ) ).
% Collect_restrict
thf(fact_209_bot__empty__eq,axiom,
! [A: $tType] :
( ( bot_bot @ ( A > $o ) )
= ( ^ [X4: A] : ( member @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% bot_empty_eq
thf(fact_210_Collect__empty__eq__bot,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( P
= ( bot_bot @ ( A > $o ) ) ) ) ).
% Collect_empty_eq_bot
thf(fact_211_Set_Ois__empty__def,axiom,
! [A: $tType] :
( ( is_empty @ A )
= ( ^ [A6: set @ A] :
( A6
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Set.is_empty_def
thf(fact_212_predicate1I,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X5: A] :
( ( P @ X5 )
=> ( Q @ X5 ) )
=> ( ord_less_eq @ ( A > $o ) @ P @ Q ) ) ).
% predicate1I
thf(fact_213_less__set__def,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
( ord_less @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ A6 )
@ ^ [X4: A] : ( member @ A @ X4 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_214_rev__predicate1D,axiom,
! [A: $tType,P: A > $o,X: A,Q: A > $o] :
( ( P @ X )
=> ( ( ord_less_eq @ ( A > $o ) @ P @ Q )
=> ( Q @ X ) ) ) ).
% rev_predicate1D
thf(fact_215_predicate1D,axiom,
! [A: $tType,P: A > $o,Q: A > $o,X: A] :
( ( ord_less_eq @ ( A > $o ) @ P @ Q )
=> ( ( P @ X )
=> ( Q @ X ) ) ) ).
% predicate1D
thf(fact_216_psubset__trans,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,C3: set @ A] :
( ( ord_less @ ( set @ A ) @ A4 @ B4 )
=> ( ( ord_less @ ( set @ A ) @ B4 @ C3 )
=> ( ord_less @ ( set @ A ) @ A4 @ C3 ) ) ) ).
% psubset_trans
thf(fact_217_psubsetD,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,C2: A] :
( ( ord_less @ ( set @ A ) @ A4 @ B4 )
=> ( ( member @ A @ C2 @ A4 )
=> ( member @ A @ C2 @ B4 ) ) ) ).
% psubsetD
thf(fact_218_pred__subset__eq,axiom,
! [A: $tType,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_219_subset__Collect__iff,axiom,
! [A: $tType,B4: set @ A,A4: set @ A,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B4 @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A4 )
& ( P @ X4 ) ) ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ B4 )
=> ( P @ X4 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_220_subset__CollectI,axiom,
! [A: $tType,B4: set @ A,A4: set @ A,Q: A > $o,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B4 @ A4 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ B4 )
=> ( ( Q @ X5 )
=> ( P @ X5 ) ) )
=> ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ B4 )
& ( Q @ X4 ) ) )
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A4 )
& ( P @ X4 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_221_conj__subset__def,axiom,
! [A: $tType,A4: set @ A,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ A4
@ ( collect @ A
@ ^ [X4: A] :
( ( P @ X4 )
& ( Q @ X4 ) ) ) )
= ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( collect @ A @ P ) )
& ( ord_less_eq @ ( set @ A ) @ A4 @ ( collect @ A @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_222_remove_Osimps_I2_J,axiom,
! [A: $tType,H: A > int,E: A,X: A,T1: binary1291135688e_Tree @ A,T2: binary1291135688e_Tree @ A] :
( ( ( ord_less @ int @ ( H @ E ) @ ( H @ X ) )
=> ( ( binary997842527remove @ A @ H @ E @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
= ( binary210054475elle_T @ A @ ( binary997842527remove @ A @ H @ E @ T1 ) @ X @ T2 ) ) )
& ( ~ ( ord_less @ int @ ( H @ E ) @ ( H @ X ) )
=> ( ( ( ord_less @ int @ ( H @ X ) @ ( H @ E ) )
=> ( ( binary997842527remove @ A @ H @ E @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
= ( binary210054475elle_T @ A @ T1 @ X @ ( binary997842527remove @ A @ H @ E @ T2 ) ) ) )
& ( ~ ( ord_less @ int @ ( H @ X ) @ ( H @ E ) )
=> ( ( ( T1
= ( binary1746293266le_Tip @ A ) )
=> ( ( binary997842527remove @ A @ H @ E @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
= T2 ) )
& ( ( T1
!= ( binary1746293266le_Tip @ A ) )
=> ( ( binary997842527remove @ A @ H @ E @ ( binary210054475elle_T @ A @ T1 @ X @ T2 ) )
= ( product_case_prod @ ( binary1291135688e_Tree @ A ) @ A @ ( binary1291135688e_Tree @ A )
@ ^ [T1p: binary1291135688e_Tree @ A,R2: A] : ( binary210054475elle_T @ A @ T1p @ R2 @ T2 )
@ ( binary1271298290_wrmrm @ A @ H @ T1 ) ) ) ) ) ) ) ) ) ).
% remove.simps(2)
thf(fact_223_wrmrm__decomp,axiom,
! [A: $tType,T: binary1291135688e_Tree @ A,H: A > int] :
( ( T
!= ( binary1746293266le_Tip @ A ) )
=> ( ( binary1271298290_wrmrm @ A @ H @ T )
= ( product_Pair @ ( binary1291135688e_Tree @ A ) @ A @ ( binary213313527le_wrm @ A @ H @ T ) @ ( binary576689334lle_rm @ A @ H @ T ) ) ) ) ).
% wrmrm_decomp
thf(fact_224_case__prod__app,axiom,
! [A: $tType,D: $tType,C: $tType,B: $tType] :
( ( product_case_prod @ B @ C @ ( D > A ) )
= ( ^ [F3: B > C > D > A,X4: product_prod @ B @ C,Y4: D] :
( product_case_prod @ B @ C @ A
@ ^ [L: B,R2: C] : ( F3 @ L @ R2 @ Y4 )
@ X4 ) ) ) ).
% case_prod_app
thf(fact_225_case__prod__Pair__iden,axiom,
! [B: $tType,A: $tType,P5: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) @ P5 )
= P5 ) ).
% case_prod_Pair_iden
thf(fact_226_bot__empty__eq2,axiom,
! [B: $tType,A: $tType] :
( ( bot_bot @ ( A > B > $o ) )
= ( ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% bot_empty_eq2
thf(fact_227_subrelI,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
( ! [X5: A,Y2: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y2 ) @ R3 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y2 ) @ S2 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R3 @ S2 ) ) ).
% subrelI
thf(fact_228_pred__subset__eq2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( A > B > $o )
@ ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ R )
@ ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ S ) )
= ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S ) ) ).
% pred_subset_eq2
thf(fact_229_pred__equals__eq2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ( ( ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ R ) )
= ( ^ [X4: A,Y4: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y4 ) @ S ) ) )
= ( R = S ) ) ).
% pred_equals_eq2
thf(fact_230_ssubst__Pair__rhs,axiom,
! [B: $tType,A: $tType,R3: A,S2: B,R: set @ ( product_prod @ A @ B ),S3: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R3 @ S2 ) @ R )
=> ( ( S3 = S2 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R3 @ S3 ) @ R ) ) ) ).
% ssubst_Pair_rhs
thf(fact_231_case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,F: B > C > A,A3: B,B3: C] :
( ( product_case_prod @ B @ C @ A @ F @ ( product_Pair @ B @ C @ A3 @ B3 ) )
= ( F @ A3 @ B3 ) ) ).
% case_prod_conv
thf(fact_232_mem__case__prodI2,axiom,
! [C: $tType,B: $tType,A: $tType,P5: product_prod @ A @ B,Z2: C,C2: A > B > ( set @ C )] :
( ! [A5: A,B5: B] :
( ( P5
= ( product_Pair @ A @ B @ A5 @ B5 ) )
=> ( member @ C @ Z2 @ ( C2 @ A5 @ B5 ) ) )
=> ( member @ C @ Z2 @ ( product_case_prod @ A @ B @ ( set @ C ) @ C2 @ P5 ) ) ) ).
% mem_case_prodI2
thf(fact_233_predicate2I,axiom,
! [B: $tType,A: $tType,P: A > B > $o,Q: A > B > $o] :
( ! [X5: A,Y2: B] :
( ( P @ X5 @ Y2 )
=> ( Q @ X5 @ Y2 ) )
=> ( ord_less_eq @ ( A > B > $o ) @ P @ Q ) ) ).
% predicate2I
thf(fact_234_case__prodI,axiom,
! [A: $tType,B: $tType,F: A > B > $o,A3: A,B3: B] :
( ( F @ A3 @ B3 )
=> ( product_case_prod @ A @ B @ $o @ F @ ( product_Pair @ A @ B @ A3 @ B3 ) ) ) ).
% case_prodI
thf(fact_235_case__prodI2,axiom,
! [B: $tType,A: $tType,P5: product_prod @ A @ B,C2: A > B > $o] :
( ! [A5: A,B5: B] :
( ( P5
= ( product_Pair @ A @ B @ A5 @ B5 ) )
=> ( C2 @ A5 @ B5 ) )
=> ( product_case_prod @ A @ B @ $o @ C2 @ P5 ) ) ).
% case_prodI2
thf(fact_236_case__prodI2_H,axiom,
! [A: $tType,B: $tType,C: $tType,P5: product_prod @ A @ B,C2: A > B > C > $o,X: C] :
( ! [A5: A,B5: B] :
( ( ( product_Pair @ A @ B @ A5 @ B5 )
= P5 )
=> ( C2 @ A5 @ B5 @ X ) )
=> ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P5 @ X ) ) ).
% case_prodI2'
thf(fact_237_mem__case__prodI,axiom,
! [A: $tType,B: $tType,C: $tType,Z2: A,C2: B > C > ( set @ A ),A3: B,B3: C] :
( ( member @ A @ Z2 @ ( C2 @ A3 @ B3 ) )
=> ( member @ A @ Z2 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ ( product_Pair @ B @ C @ A3 @ B3 ) ) ) ) ).
% mem_case_prodI
thf(fact_238_Collect__case__prod__mono,axiom,
! [B: $tType,A: $tType,A4: A > B > $o,B4: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ A4 @ B4 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A4 ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ B4 ) ) ) ) ).
% Collect_case_prod_mono
thf(fact_239_predicate2D__conj,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,R: $o,X: A,Y: B] :
( ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
& R )
=> ( R
& ( ( P @ X @ Y )
=> ( Q @ X @ Y ) ) ) ) ).
% predicate2D_conj
thf(fact_240_eq__subset,axiom,
! [A: $tType,P: A > A > $o] :
( ord_less_eq @ ( A > A > $o )
@ ^ [Y3: A,Z: A] : Y3 = Z
@ ^ [A2: A,B2: A] :
( ( P @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% eq_subset
thf(fact_241_predicate2D,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,X: A,Y: B] :
( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
=> ( ( P @ X @ Y )
=> ( Q @ X @ Y ) ) ) ).
% predicate2D
thf(fact_242_rev__predicate2D,axiom,
! [A: $tType,B: $tType,P: A > B > $o,X: A,Y: B,Q: A > B > $o] :
( ( P @ X @ Y )
=> ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
=> ( Q @ X @ Y ) ) ) ).
% rev_predicate2D
thf(fact_243_case__prodD,axiom,
! [A: $tType,B: $tType,F: A > B > $o,A3: A,B3: B] :
( ( product_case_prod @ A @ B @ $o @ F @ ( product_Pair @ A @ B @ A3 @ B3 ) )
=> ( F @ A3 @ B3 ) ) ).
% case_prodD
thf(fact_244_case__prodE,axiom,
! [A: $tType,B: $tType,C2: A > B > $o,P5: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ $o @ C2 @ P5 )
=> ~ ! [X5: A,Y2: B] :
( ( P5
= ( product_Pair @ A @ B @ X5 @ Y2 ) )
=> ~ ( C2 @ X5 @ Y2 ) ) ) ).
% case_prodE
thf(fact_245_case__prodD_H,axiom,
! [B: $tType,A: $tType,C: $tType,R: A > B > C > $o,A3: A,B3: B,C2: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ R @ ( product_Pair @ A @ B @ A3 @ B3 ) @ C2 )
=> ( R @ A3 @ B3 @ C2 ) ) ).
% case_prodD'
thf(fact_246_case__prodE_H,axiom,
! [B: $tType,A: $tType,C: $tType,C2: A > B > C > $o,P5: product_prod @ A @ B,Z2: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P5 @ Z2 )
=> ~ ! [X5: A,Y2: B] :
( ( P5
= ( product_Pair @ A @ B @ X5 @ Y2 ) )
=> ~ ( C2 @ X5 @ Y2 @ Z2 ) ) ) ).
% case_prodE'
thf(fact_247_prod_Ocase__distrib,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,H: C > D,F: A > B > C,Prod: product_prod @ A @ B] :
( ( H @ ( product_case_prod @ A @ B @ C @ F @ Prod ) )
= ( product_case_prod @ A @ B @ D
@ ^ [X12: A,X24: B] : ( H @ ( F @ X12 @ X24 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_248_case__prodE2,axiom,
! [B: $tType,A: $tType,C: $tType,Q: A > $o,P: B > C > A,Z2: product_prod @ B @ C] :
( ( Q @ ( product_case_prod @ B @ C @ A @ P @ Z2 ) )
=> ~ ! [X5: B,Y2: C] :
( ( Z2
= ( product_Pair @ B @ C @ X5 @ Y2 ) )
=> ~ ( Q @ ( P @ X5 @ Y2 ) ) ) ) ).
% case_prodE2
thf(fact_249_case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F: ( product_prod @ A @ B ) > C] :
( ( product_case_prod @ A @ B @ C
@ ^ [X4: A,Y4: B] : ( F @ ( product_Pair @ A @ B @ X4 @ Y4 ) ) )
= F ) ).
% case_prod_eta
thf(fact_250_cond__case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > B > C,G: ( product_prod @ A @ B ) > C] :
( ! [X5: A,Y2: B] :
( ( F @ X5 @ Y2 )
= ( G @ ( product_Pair @ A @ B @ X5 @ Y2 ) ) )
=> ( ( product_case_prod @ A @ B @ C @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_251_less__by__empty,axiom,
! [A: $tType,A4: set @ ( product_prod @ A @ A ),B4: set @ ( product_prod @ A @ A )] :
( ( A4
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ A4 @ B4 ) ) ).
% less_by_empty
thf(fact_252_relChain__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( bNF_Ca1785829860lChain @ A @ B )
= ( ^ [R2: set @ ( product_prod @ A @ A ),As: A > B] :
! [I: A,J: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J ) @ R2 )
=> ( ord_less_eq @ B @ ( As @ I ) @ ( As @ J ) ) ) ) ) ) ).
% relChain_def
thf(fact_253_split__part,axiom,
! [B: $tType,A: $tType,P: $o,Q: A > B > $o] :
( ( product_case_prod @ A @ B @ $o
@ ^ [A2: A,B2: B] :
( P
& ( Q @ A2 @ B2 ) ) )
= ( ^ [Ab: product_prod @ A @ B] :
( P
& ( product_case_prod @ A @ B @ $o @ Q @ Ab ) ) ) ) ).
% split_part
thf(fact_254_prod_Odisc__eq__case,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
( product_case_prod @ A @ B @ $o
@ ^ [Uu: A,Uv: B] : $true
@ Prod ) ).
% prod.disc_eq_case
thf(fact_255_inv__image__def,axiom,
! [A: $tType,B: $tType] :
( ( inv_image @ B @ A )
= ( ^ [R2: set @ ( product_prod @ B @ B ),F3: A > B] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X4: A,Y4: A] : ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F3 @ X4 ) @ ( F3 @ Y4 ) ) @ R2 ) ) ) ) ) ).
% inv_image_def
% Type constructors (23)
thf(tcon_fun___Orderings_Oorder__bot,axiom,
! [A8: $tType,A9: $tType] :
( ( order_bot @ A9 )
=> ( order_bot @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A8: $tType,A9: $tType] :
( ( preorder @ A9 )
=> ( preorder @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A8: $tType,A9: $tType] :
( ( order @ A9 )
=> ( order @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A8: $tType,A9: $tType] :
( ( ord @ A9 )
=> ( ord @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Obot,axiom,
! [A8: $tType,A9: $tType] :
( ( bot @ A9 )
=> ( bot @ ( A8 > A9 ) ) ) ).
thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
condit1037483654norder @ int ).
thf(tcon_Int_Oint___Orderings_Opreorder_1,axiom,
preorder @ int ).
thf(tcon_Int_Oint___Orderings_Olinorder,axiom,
linorder @ int ).
thf(tcon_Int_Oint___Orderings_Ono__top,axiom,
no_top @ int ).
thf(tcon_Int_Oint___Orderings_Ono__bot,axiom,
no_bot @ int ).
thf(tcon_Int_Oint___Orderings_Oorder_2,axiom,
order @ int ).
thf(tcon_Int_Oint___Orderings_Oord_3,axiom,
ord @ int ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_4,axiom,
! [A8: $tType] : ( order_bot @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_5,axiom,
! [A8: $tType] : ( preorder @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_6,axiom,
! [A8: $tType] : ( order @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_7,axiom,
! [A8: $tType] : ( ord @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_8,axiom,
! [A8: $tType] : ( bot @ ( set @ A8 ) ) ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_9,axiom,
order_bot @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_10,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder_11,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_12,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_13,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Orderings_Obot_14,axiom,
bot @ $o ).
% Conjectures (1)
thf(conj_0,conjecture,
binary1610619414edTree @ a @ h @ t1 ).
%------------------------------------------------------------------------------