TPTP Problem File: DAT175^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : DAT175^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Lazy lists II 183
% Version : [Bla16] axioms : Especial.
% English :
% Refs : [Fri04] Friedrich (2004), Lazy Lists II
% : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% Source : [Bla16]
% Names : llist2__183.p [Bla16]
% Status : Theorem
% Rating : 0.67 v8.1.0, 0.75 v7.5.0, 1.00 v7.1.0
% Syntax : Number of formulae : 326 ( 131 unt; 53 typ; 0 def)
% Number of atoms : 688 ( 197 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 4840 ( 88 ~; 14 |; 59 &;4340 @)
% ( 0 <=>; 339 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 9 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 305 ( 305 >; 0 *; 0 +; 0 <<)
% Number of symbols : 54 ( 52 usr; 4 con; 0-9 aty)
% Number of variables : 1369 ( 124 ^;1129 !; 48 ?;1369 :)
% ( 68 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:44:38.061
%------------------------------------------------------------------------------
%----Could-be-implicit typings (5)
thf(ty_t_Coinductive__List_Ollist,type,
coinductive_llist: $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_itself,type,
itself: $tType > $tType ).
thf(ty_tf_a,type,
a: $tType ).
%----Explicit typings (48)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Otop,type,
top:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oorder__top,type,
order_top:
!>[A: $tType] : ( ( itself @ A ) > $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_BNF__Greatest__Fixpoint_Oimage2,type,
bNF_Greatest_image2:
!>[C: $tType,A: $tType,B: $tType] : ( ( set @ C ) > ( C > A ) > ( C > B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_BNF__Greatest__Fixpoint_Oimage2p,type,
bNF_Greatest_image2p:
!>[C: $tType,A: $tType,D: $tType,B: $tType] : ( ( C > A ) > ( D > B ) > ( C > D > $o ) > A > B > $o ) ).
thf(sy_c_BNF__Greatest__Fixpoint_OrelImage,type,
bNF_Gr1317331620lImage:
!>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( B > A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_BNF__Greatest__Fixpoint_OrelInvImage,type,
bNF_Gr2107612801vImage:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Coinductive__List_Ofinite__lprefix,type,
coindu328551480prefix:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Olfinite,type,
coinductive_lfinite:
!>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Ollast,type,
coinductive_llast:
!>[A: $tType] : ( ( coinductive_llist @ A ) > A ) ).
thf(sy_c_Coinductive__List_Ollist_OLCons,type,
coinductive_LCons:
!>[A: $tType] : ( A > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_Ollist_OLNil,type,
coinductive_LNil:
!>[A: $tType] : ( coinductive_llist @ A ) ).
thf(sy_c_Coinductive__List_Olmember,type,
coinductive_lmember:
!>[A: $tType] : ( A > ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Olstrict__prefix,type,
coindu1478340336prefix:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Fun__Def_Oreduction__pair,type,
fun_reduction_pair:
!>[A: $tType] : ( ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) > $o ) ).
thf(sy_c_Fun__Def_Orp__inv__image,type,
fun_rp_inv_image:
!>[A: $tType,B: $tType] : ( ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) > ( B > A ) > ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Oalllsts,type,
lList2435255213lllsts:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( coinductive_llist @ A ) ) ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Oalllstsp,type,
lList21511617539llstsp:
!>[A: $tType] : ( ( A > $o ) > ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Ofinlsts,type,
lList2236698231inlsts:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( coinductive_llist @ A ) ) ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Ofinlsts__pred,type,
lList22005681144s_pred:
!>[A: $tType] : ( set @ ( product_prod @ ( coinductive_llist @ A ) @ ( coinductive_llist @ A ) ) ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Ofinlstsp,type,
lList2860480441nlstsp:
!>[A: $tType] : ( ( A > $o ) > ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_LList2__Mirabelle__hamjzmohle_Oposlsts,type,
lList21148268032oslsts:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( coinductive_llist @ A ) ) ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Otop__class_Otop,type,
top_top:
!>[A: $tType] : A ).
thf(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
thf(sy_c_Product__Type_Ointernal__case__prod,type,
produc2004651681e_prod:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
product_rec_prod:
!>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
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_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Relation_ODomain,type,
domain:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ A ) ) ).
thf(sy_c_Relation_ORange,type,
range:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ B ) ) ).
thf(sy_c_Relation_ORangep,type,
rangep:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > B > $o ) ).
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_Relation_Orelcomp,type,
relcomp:
!>[A: $tType,B: $tType,C: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ B @ C ) ) > ( set @ ( product_prod @ A @ C ) ) ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Wellfounded_Olex__prod,type,
lex_prod:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
thf(sy_c_Wellfounded_Owf,type,
wf:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Wfrec_Osame__fst,type,
same_fst:
!>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > ( set @ ( product_prod @ B @ B ) ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_P____,type,
p: ( coinductive_llist @ a ) > $o ).
thf(sy_v_a____,type,
a2: a ).
thf(sy_v_l____,type,
l: coinductive_llist @ a ).
thf(sy_v_x____,type,
x: coinductive_llist @ a ).
%----Relevant facts (256)
thf(fact_0_xfin,axiom,
member @ ( coinductive_llist @ a ) @ x @ ( lList2236698231inlsts @ a @ ( top_top @ ( set @ a ) ) ) ).
% xfin
thf(fact_1_LCons__fin_Ohyps_I1_J,axiom,
member @ ( coinductive_llist @ a ) @ l @ ( lList2236698231inlsts @ a @ ( top_top @ ( set @ a ) ) ) ).
% LCons_fin.hyps(1)
thf(fact_2_LCons__fin_Ohyps_I3_J,axiom,
member @ a @ a2 @ ( top_top @ ( set @ a ) ) ).
% LCons_fin.hyps(3)
thf(fact_3_LCons__fin_Ohyps_I2_J,axiom,
p @ l ).
% LCons_fin.hyps(2)
thf(fact_4_llist_Oinject,axiom,
! [A: $tType,X21: A,X22: coinductive_llist @ A,Y21: A,Y22: coinductive_llist @ A] :
( ( ( coinductive_LCons @ A @ X21 @ X22 )
= ( coinductive_LCons @ A @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% llist.inject
thf(fact_5_finlsts_OLCons__fin,axiom,
! [A: $tType,L: coinductive_llist @ A,A2: set @ A,A3: A] :
( ( member @ ( coinductive_llist @ A ) @ L @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( member @ A @ A3 @ A2 )
=> ( member @ ( coinductive_llist @ A ) @ ( coinductive_LCons @ A @ A3 @ L ) @ ( lList2236698231inlsts @ A @ A2 ) ) ) ) ).
% finlsts.LCons_fin
thf(fact_6_H,axiom,
! [X: coinductive_llist @ a] :
( ! [Y: coinductive_llist @ a] :
( ( member @ ( product_prod @ ( coinductive_llist @ a ) @ ( coinductive_llist @ a ) ) @ ( product_Pair @ ( coinductive_llist @ a ) @ ( coinductive_llist @ a ) @ Y @ X )
@ ( collect @ ( product_prod @ ( coinductive_llist @ a ) @ ( coinductive_llist @ a ) )
@ ( product_case_prod @ ( coinductive_llist @ a ) @ ( coinductive_llist @ a ) @ $o
@ ^ [R: coinductive_llist @ a,S: coinductive_llist @ a] :
( ( member @ ( coinductive_llist @ a ) @ R @ ( lList2236698231inlsts @ a @ ( top_top @ ( set @ a ) ) ) )
& ? [A4: a] :
( ( coinductive_LCons @ a @ A4 @ R )
= S ) ) ) ) )
=> ( p @ Y ) )
=> ( p @ X ) ) ).
% H
thf(fact_7_LConsE,axiom,
! [A: $tType,X2: A,Xs: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ ( coinductive_LCons @ A @ X2 @ Xs ) @ ( lList2435255213lllsts @ A @ A2 ) )
= ( ( member @ A @ X2 @ A2 )
& ( member @ ( coinductive_llist @ A ) @ Xs @ ( lList2435255213lllsts @ A @ A2 ) ) ) ) ).
% LConsE
thf(fact_8_alllstsp_OLCons__all,axiom,
! [A: $tType,A2: A > $o,L: coinductive_llist @ A,A3: A] :
( ( lList21511617539llstsp @ A @ A2 @ L )
=> ( ( A2 @ A3 )
=> ( lList21511617539llstsp @ A @ A2 @ ( coinductive_LCons @ A @ A3 @ L ) ) ) ) ).
% alllstsp.LCons_all
thf(fact_9_finlstsp_OLCons__fin,axiom,
! [A: $tType,A2: A > $o,L: coinductive_llist @ A,A3: A] :
( ( lList2860480441nlstsp @ A @ A2 @ L )
=> ( ( A2 @ A3 )
=> ( lList2860480441nlstsp @ A @ A2 @ ( coinductive_LCons @ A @ A3 @ L ) ) ) ) ).
% finlstsp.LCons_fin
thf(fact_10_alllsts_OLCons__all,axiom,
! [A: $tType,L: coinductive_llist @ A,A2: set @ A,A3: A] :
( ( member @ ( coinductive_llist @ A ) @ L @ ( lList2435255213lllsts @ A @ A2 ) )
=> ( ( member @ A @ A3 @ A2 )
=> ( member @ ( coinductive_llist @ A ) @ ( coinductive_LCons @ A @ A3 @ L ) @ ( lList2435255213lllsts @ A @ A2 ) ) ) ) ).
% alllsts.LCons_all
thf(fact_11_finlsts__predI,axiom,
! [A: $tType,R2: coinductive_llist @ A,A2: set @ A,A3: A] :
( ( member @ ( coinductive_llist @ A ) @ R2 @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( member @ ( product_prod @ ( coinductive_llist @ A ) @ ( coinductive_llist @ A ) ) @ ( product_Pair @ ( coinductive_llist @ A ) @ ( coinductive_llist @ A ) @ R2 @ ( coinductive_LCons @ A @ A3 @ R2 ) ) @ ( lList22005681144s_pred @ A ) ) ) ).
% finlsts_predI
thf(fact_12_lmember__code_I2_J,axiom,
! [A: $tType,X2: A,Y2: A,Ys: coinductive_llist @ A] :
( ( coinductive_lmember @ A @ X2 @ ( coinductive_LCons @ A @ Y2 @ Ys ) )
= ( ( X2 = Y2 )
| ( coinductive_lmember @ A @ X2 @ Ys ) ) ) ).
% lmember_code(2)
thf(fact_13_llistE,axiom,
! [A: $tType,Y2: coinductive_llist @ A] :
( ( Y2
!= ( coinductive_LNil @ A ) )
=> ~ ! [X212: A,X222: coinductive_llist @ A] :
( Y2
!= ( coinductive_LCons @ A @ X212 @ X222 ) ) ) ).
% llistE
thf(fact_14_lstrict__prefix__code_I4_J,axiom,
! [B: $tType,X2: B,Xs: coinductive_llist @ B,Y2: B,Ys: coinductive_llist @ B] :
( ( coindu1478340336prefix @ B @ ( coinductive_LCons @ B @ X2 @ Xs ) @ ( coinductive_LCons @ B @ Y2 @ Ys ) )
= ( ( X2 = Y2 )
& ( coindu1478340336prefix @ B @ Xs @ Ys ) ) ) ).
% lstrict_prefix_code(4)
thf(fact_15_llast__LCons2,axiom,
! [A: $tType,X2: A,Y2: A,Xs: coinductive_llist @ A] :
( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X2 @ ( coinductive_LCons @ A @ Y2 @ Xs ) ) )
= ( coinductive_llast @ A @ ( coinductive_LCons @ A @ Y2 @ Xs ) ) ) ).
% llast_LCons2
thf(fact_16_alllsts__UNIV,axiom,
! [A: $tType,S2: coinductive_llist @ A] : ( member @ ( coinductive_llist @ A ) @ S2 @ ( lList2435255213lllsts @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% alllsts_UNIV
thf(fact_17_lstrict__prefix__code_I1_J,axiom,
! [A: $tType] :
~ ( coindu1478340336prefix @ A @ ( coinductive_LNil @ A ) @ ( coinductive_LNil @ A ) ) ).
% lstrict_prefix_code(1)
thf(fact_18_llast__singleton,axiom,
! [A: $tType,X2: A] :
( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X2 @ ( coinductive_LNil @ A ) ) )
= X2 ) ).
% llast_singleton
thf(fact_19_lstrict__prefix__code_I2_J,axiom,
! [B: $tType,Y2: B,Ys: coinductive_llist @ B] : ( coindu1478340336prefix @ B @ ( coinductive_LNil @ B ) @ ( coinductive_LCons @ B @ Y2 @ Ys ) ) ).
% lstrict_prefix_code(2)
thf(fact_20_lstrict__prefix__code_I3_J,axiom,
! [B: $tType,X2: B,Xs: coinductive_llist @ B] :
~ ( coindu1478340336prefix @ B @ ( coinductive_LCons @ B @ X2 @ Xs ) @ ( coinductive_LNil @ B ) ) ).
% lstrict_prefix_code(3)
thf(fact_21_lmember__code_I1_J,axiom,
! [A: $tType,X2: A] :
~ ( coinductive_lmember @ A @ X2 @ ( coinductive_LNil @ A ) ) ).
% lmember_code(1)
thf(fact_22_llist__less__induct,axiom,
! [A: $tType,P: ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A] :
( ! [Xs2: coinductive_llist @ A] :
( ! [Ys2: coinductive_llist @ A] :
( ( coindu1478340336prefix @ A @ Ys2 @ Xs2 )
=> ( P @ Ys2 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% llist_less_induct
thf(fact_23_alllsts__def,axiom,
! [A: $tType] :
( ( lList2435255213lllsts @ A )
= ( ^ [A5: set @ A] :
( collect @ ( coinductive_llist @ A )
@ ( lList21511617539llstsp @ A
@ ^ [X3: A] : ( member @ A @ X3 @ A5 ) ) ) ) ) ).
% alllsts_def
thf(fact_24_finlsts__def,axiom,
! [A: $tType] :
( ( lList2236698231inlsts @ A )
= ( ^ [A5: set @ A] :
( collect @ ( coinductive_llist @ A )
@ ( lList2860480441nlstsp @ A
@ ^ [X3: A] : ( member @ A @ X3 @ A5 ) ) ) ) ) ).
% finlsts_def
thf(fact_25_finite__lemma,axiom,
! [A: $tType,X2: coinductive_llist @ A,A2: set @ A,B2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ X2 @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( member @ ( coinductive_llist @ A ) @ X2 @ ( lList2435255213lllsts @ A @ B2 ) )
=> ( member @ ( coinductive_llist @ A ) @ X2 @ ( lList2236698231inlsts @ A @ B2 ) ) ) ) ).
% finite_lemma
thf(fact_26_finsubsetall,axiom,
! [A: $tType,X2: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ X2 @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ X2 @ ( lList2435255213lllsts @ A @ A2 ) ) ) ).
% finsubsetall
thf(fact_27_alllsts_OLNil__all,axiom,
! [A: $tType,A2: set @ A] : ( member @ ( coinductive_llist @ A ) @ ( coinductive_LNil @ A ) @ ( lList2435255213lllsts @ A @ A2 ) ) ).
% alllsts.LNil_all
thf(fact_28_finlsts_OLNil__fin,axiom,
! [A: $tType,A2: set @ A] : ( member @ ( coinductive_llist @ A ) @ ( coinductive_LNil @ A ) @ ( lList2236698231inlsts @ A @ A2 ) ) ).
% finlsts.LNil_fin
thf(fact_29_finlsts__pred__def,axiom,
! [A: $tType] :
( ( lList22005681144s_pred @ A )
= ( collect @ ( product_prod @ ( coinductive_llist @ A ) @ ( coinductive_llist @ A ) )
@ ( product_case_prod @ ( coinductive_llist @ A ) @ ( coinductive_llist @ A ) @ $o
@ ^ [R: coinductive_llist @ A,S: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ R @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) )
& ? [A4: A] :
( ( coinductive_LCons @ A @ A4 @ R )
= S ) ) ) ) ) ).
% finlsts_pred_def
thf(fact_30_alllstsp_OLNil__all,axiom,
! [A: $tType,A2: A > $o] : ( lList21511617539llstsp @ A @ A2 @ ( coinductive_LNil @ A ) ) ).
% alllstsp.LNil_all
thf(fact_31_finlstsp_OLNil__fin,axiom,
! [A: $tType,A2: A > $o] : ( lList2860480441nlstsp @ A @ A2 @ ( coinductive_LNil @ A ) ) ).
% finlstsp.LNil_fin
thf(fact_32_alllstsp__alllsts__eq,axiom,
! [A: $tType,A2: set @ A] :
( ( lList21511617539llstsp @ A
@ ^ [X3: A] : ( member @ A @ X3 @ A2 ) )
= ( ^ [X3: coinductive_llist @ A] : ( member @ ( coinductive_llist @ A ) @ X3 @ ( lList2435255213lllsts @ A @ A2 ) ) ) ) ).
% alllstsp_alllsts_eq
thf(fact_33_finlstsp__finlsts__eq,axiom,
! [A: $tType,A2: set @ A] :
( ( lList2860480441nlstsp @ A
@ ^ [X3: A] : ( member @ A @ X3 @ A2 ) )
= ( ^ [X3: coinductive_llist @ A] : ( member @ ( coinductive_llist @ A ) @ X3 @ ( lList2236698231inlsts @ A @ A2 ) ) ) ) ).
% finlstsp_finlsts_eq
thf(fact_34_alllsts_Ocoinduct,axiom,
! [A: $tType,X4: ( coinductive_llist @ A ) > $o,X2: coinductive_llist @ A,A2: set @ A] :
( ( X4 @ X2 )
=> ( ! [X5: coinductive_llist @ A] :
( ( X4 @ X5 )
=> ( ( X5
= ( coinductive_LNil @ A ) )
| ? [L2: coinductive_llist @ A,A6: A] :
( ( X5
= ( coinductive_LCons @ A @ A6 @ L2 ) )
& ( ( X4 @ L2 )
| ( member @ ( coinductive_llist @ A ) @ L2 @ ( lList2435255213lllsts @ A @ A2 ) ) )
& ( member @ A @ A6 @ A2 ) ) ) )
=> ( member @ ( coinductive_llist @ A ) @ X2 @ ( lList2435255213lllsts @ A @ A2 ) ) ) ) ).
% alllsts.coinduct
thf(fact_35_alllsts_Osimps,axiom,
! [A: $tType,A3: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ A3 @ ( lList2435255213lllsts @ A @ A2 ) )
= ( ( A3
= ( coinductive_LNil @ A ) )
| ? [L3: coinductive_llist @ A,A4: A] :
( ( A3
= ( coinductive_LCons @ A @ A4 @ L3 ) )
& ( member @ ( coinductive_llist @ A ) @ L3 @ ( lList2435255213lllsts @ A @ A2 ) )
& ( member @ A @ A4 @ A2 ) ) ) ) ).
% alllsts.simps
thf(fact_36_alllsts_Ocases,axiom,
! [A: $tType,A3: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ A3 @ ( lList2435255213lllsts @ A @ A2 ) )
=> ( ( A3
!= ( coinductive_LNil @ A ) )
=> ~ ! [L4: coinductive_llist @ A,A7: A] :
( ( A3
= ( coinductive_LCons @ A @ A7 @ L4 ) )
=> ( ( member @ ( coinductive_llist @ A ) @ L4 @ ( lList2435255213lllsts @ A @ A2 ) )
=> ~ ( member @ A @ A7 @ A2 ) ) ) ) ) ).
% alllsts.cases
thf(fact_37_neq__LNil__conv,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( Xs
!= ( coinductive_LNil @ A ) )
= ( ? [X3: A,Xs3: coinductive_llist @ A] :
( Xs
= ( coinductive_LCons @ A @ X3 @ Xs3 ) ) ) ) ).
% neq_LNil_conv
thf(fact_38_llist_Odistinct_I1_J,axiom,
! [A: $tType,X21: A,X22: coinductive_llist @ A] :
( ( coinductive_LNil @ A )
!= ( coinductive_LCons @ A @ X21 @ X22 ) ) ).
% llist.distinct(1)
thf(fact_39_alllstsp_Ocoinduct,axiom,
! [A: $tType,X4: ( coinductive_llist @ A ) > $o,X2: coinductive_llist @ A,A2: A > $o] :
( ( X4 @ X2 )
=> ( ! [X5: coinductive_llist @ A] :
( ( X4 @ X5 )
=> ( ( X5
= ( coinductive_LNil @ A ) )
| ? [L2: coinductive_llist @ A,A6: A] :
( ( X5
= ( coinductive_LCons @ A @ A6 @ L2 ) )
& ( ( X4 @ L2 )
| ( lList21511617539llstsp @ A @ A2 @ L2 ) )
& ( A2 @ A6 ) ) ) )
=> ( lList21511617539llstsp @ A @ A2 @ X2 ) ) ) ).
% alllstsp.coinduct
thf(fact_40_finlstsp_Oinducts,axiom,
! [A: $tType,A2: A > $o,X2: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
( ( lList2860480441nlstsp @ A @ A2 @ X2 )
=> ( ( P @ ( coinductive_LNil @ A ) )
=> ( ! [L4: coinductive_llist @ A,A7: A] :
( ( lList2860480441nlstsp @ A @ A2 @ L4 )
=> ( ( P @ L4 )
=> ( ( A2 @ A7 )
=> ( P @ ( coinductive_LCons @ A @ A7 @ L4 ) ) ) ) )
=> ( P @ X2 ) ) ) ) ).
% finlstsp.inducts
thf(fact_41_finlstsp_Osimps,axiom,
! [A: $tType] :
( ( lList2860480441nlstsp @ A )
= ( ^ [A5: A > $o,A4: coinductive_llist @ A] :
( ( A4
= ( coinductive_LNil @ A ) )
| ? [L3: coinductive_llist @ A,B3: A] :
( ( A4
= ( coinductive_LCons @ A @ B3 @ L3 ) )
& ( lList2860480441nlstsp @ A @ A5 @ L3 )
& ( A5 @ B3 ) ) ) ) ) ).
% finlstsp.simps
thf(fact_42_finlstsp_Ocases,axiom,
! [A: $tType,A2: A > $o,A3: coinductive_llist @ A] :
( ( lList2860480441nlstsp @ A @ A2 @ A3 )
=> ( ( A3
!= ( coinductive_LNil @ A ) )
=> ~ ! [L4: coinductive_llist @ A,A7: A] :
( ( A3
= ( coinductive_LCons @ A @ A7 @ L4 ) )
=> ( ( lList2860480441nlstsp @ A @ A2 @ L4 )
=> ~ ( A2 @ A7 ) ) ) ) ) ).
% finlstsp.cases
thf(fact_43_alllstsp_Osimps,axiom,
! [A: $tType] :
( ( lList21511617539llstsp @ A )
= ( ^ [A5: A > $o,A4: coinductive_llist @ A] :
( ( A4
= ( coinductive_LNil @ A ) )
| ? [L3: coinductive_llist @ A,B3: A] :
( ( A4
= ( coinductive_LCons @ A @ B3 @ L3 ) )
& ( lList21511617539llstsp @ A @ A5 @ L3 )
& ( A5 @ B3 ) ) ) ) ) ).
% alllstsp.simps
thf(fact_44_alllstsp_Ocases,axiom,
! [A: $tType,A2: A > $o,A3: coinductive_llist @ A] :
( ( lList21511617539llstsp @ A @ A2 @ A3 )
=> ( ( A3
!= ( coinductive_LNil @ A ) )
=> ~ ! [L4: coinductive_llist @ A,A7: A] :
( ( A3
= ( coinductive_LCons @ A @ A7 @ L4 ) )
=> ( ( lList21511617539llstsp @ A @ A2 @ L4 )
=> ~ ( A2 @ A7 ) ) ) ) ) ).
% alllstsp.cases
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,A2: set @ A] :
( ( collect @ A
@ ^ [X3: A] : ( member @ A @ X3 @ A2 ) )
= A2 ) ).
% 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_fin__finite,axiom,
! [A: $tType,R2: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ R2 @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ R2 @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% fin_finite
thf(fact_50_finT__simp,axiom,
! [A: $tType,R2: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ R2 @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( member @ ( coinductive_llist @ A ) @ R2 @ ( lList2236698231inlsts @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% finT_simp
thf(fact_51_finlsts_Oinducts,axiom,
! [A: $tType,X2: coinductive_llist @ A,A2: set @ A,P: ( coinductive_llist @ A ) > $o] :
( ( member @ ( coinductive_llist @ A ) @ X2 @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( P @ ( coinductive_LNil @ A ) )
=> ( ! [L4: coinductive_llist @ A,A7: A] :
( ( member @ ( coinductive_llist @ A ) @ L4 @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( P @ L4 )
=> ( ( member @ A @ A7 @ A2 )
=> ( P @ ( coinductive_LCons @ A @ A7 @ L4 ) ) ) ) )
=> ( P @ X2 ) ) ) ) ).
% finlsts.inducts
thf(fact_52_finlsts__induct,axiom,
! [A: $tType,X2: coinductive_llist @ A,A2: set @ A,P: ( coinductive_llist @ A ) > $o] :
( ( member @ ( coinductive_llist @ A ) @ X2 @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ! [L4: coinductive_llist @ A] :
( ( L4
= ( coinductive_LNil @ A ) )
=> ( P @ L4 ) )
=> ( ! [A7: A,L4: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ L4 @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( P @ L4 )
=> ( ( member @ A @ A7 @ A2 )
=> ( P @ ( coinductive_LCons @ A @ A7 @ L4 ) ) ) ) )
=> ( P @ X2 ) ) ) ) ).
% finlsts_induct
thf(fact_53_finlsts_Osimps,axiom,
! [A: $tType,A3: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ A3 @ ( lList2236698231inlsts @ A @ A2 ) )
= ( ( A3
= ( coinductive_LNil @ A ) )
| ? [L3: coinductive_llist @ A,A4: A] :
( ( A3
= ( coinductive_LCons @ A @ A4 @ L3 ) )
& ( member @ ( coinductive_llist @ A ) @ L3 @ ( lList2236698231inlsts @ A @ A2 ) )
& ( member @ A @ A4 @ A2 ) ) ) ) ).
% finlsts.simps
thf(fact_54_finlsts_Ocases,axiom,
! [A: $tType,A3: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ A3 @ ( lList2236698231inlsts @ A @ A2 ) )
=> ( ( A3
!= ( coinductive_LNil @ A ) )
=> ~ ! [L4: coinductive_llist @ A,A7: A] :
( ( A3
= ( coinductive_LCons @ A @ A7 @ L4 ) )
=> ( ( member @ ( coinductive_llist @ A ) @ L4 @ ( lList2236698231inlsts @ A @ A2 ) )
=> ~ ( member @ A @ A7 @ A2 ) ) ) ) ) ).
% finlsts.cases
thf(fact_55_case__prodI,axiom,
! [A: $tType,B: $tType,F: A > B > $o,A3: A,B4: B] :
( ( F @ A3 @ B4 )
=> ( product_case_prod @ A @ B @ $o @ F @ ( product_Pair @ A @ B @ A3 @ B4 ) ) ) ).
% case_prodI
thf(fact_56_case__prodI2,axiom,
! [B: $tType,A: $tType,P2: product_prod @ A @ B,C2: A > B > $o] :
( ! [A7: A,B5: B] :
( ( P2
= ( product_Pair @ A @ B @ A7 @ B5 ) )
=> ( C2 @ A7 @ B5 ) )
=> ( product_case_prod @ A @ B @ $o @ C2 @ P2 ) ) ).
% case_prodI2
thf(fact_57_case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,F: B > C > A,A3: B,B4: C] :
( ( product_case_prod @ B @ C @ A @ F @ ( product_Pair @ B @ C @ A3 @ B4 ) )
= ( F @ A3 @ B4 ) ) ).
% case_prod_conv
thf(fact_58_split__part,axiom,
! [B: $tType,A: $tType,P: $o,Q: A > B > $o] :
( ( product_case_prod @ A @ B @ $o
@ ^ [A4: A,B3: B] :
( P
& ( Q @ A4 @ B3 ) ) )
= ( ^ [Ab: product_prod @ A @ B] :
( P
& ( product_case_prod @ A @ B @ $o @ Q @ Ab ) ) ) ) ).
% split_part
thf(fact_59_UNIV__I,axiom,
! [A: $tType,X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_I
thf(fact_60_iso__tuple__UNIV__I,axiom,
! [A: $tType,X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% iso_tuple_UNIV_I
thf(fact_61_poslsts__iff,axiom,
! [A: $tType,S2: coinductive_llist @ A,A2: set @ A] :
( ( member @ ( coinductive_llist @ A ) @ S2 @ ( lList21148268032oslsts @ A @ A2 ) )
= ( ( member @ ( coinductive_llist @ A ) @ S2 @ ( lList2435255213lllsts @ A @ A2 ) )
& ( S2
!= ( coinductive_LNil @ A ) ) ) ) ).
% poslsts_iff
thf(fact_62_top__apply,axiom,
! [C: $tType,D: $tType] :
( ( top @ C @ ( type2 @ C ) )
=> ( ( top_top @ ( D > C ) )
= ( ^ [X3: D] : ( top_top @ C ) ) ) ) ).
% top_apply
thf(fact_63_prod_Oinject,axiom,
! [A: $tType,B: $tType,X1: A,X23: B,Y1: A,Y23: B] :
( ( ( product_Pair @ A @ B @ X1 @ X23 )
= ( product_Pair @ A @ B @ Y1 @ Y23 ) )
= ( ( X1 = Y1 )
& ( X23 = Y23 ) ) ) ).
% prod.inject
thf(fact_64_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A3: A,B4: B,A8: A,B6: B] :
( ( ( product_Pair @ A @ B @ A3 @ B4 )
= ( product_Pair @ A @ B @ A8 @ B6 ) )
= ( ( A3 = A8 )
& ( B4 = B6 ) ) ) ).
% old.prod.inject
thf(fact_65_mem__case__prodI2,axiom,
! [C: $tType,B: $tType,A: $tType,P2: product_prod @ A @ B,Z: C,C2: A > B > ( set @ C )] :
( ! [A7: A,B5: B] :
( ( P2
= ( product_Pair @ A @ B @ A7 @ B5 ) )
=> ( member @ C @ Z @ ( C2 @ A7 @ B5 ) ) )
=> ( member @ C @ Z @ ( product_case_prod @ A @ B @ ( set @ C ) @ C2 @ P2 ) ) ) ).
% mem_case_prodI2
thf(fact_66_mem__case__prodI,axiom,
! [A: $tType,B: $tType,C: $tType,Z: A,C2: B > C > ( set @ A ),A3: B,B4: C] :
( ( member @ A @ Z @ ( C2 @ A3 @ B4 ) )
=> ( member @ A @ Z @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ ( product_Pair @ B @ C @ A3 @ B4 ) ) ) ) ).
% mem_case_prodI
thf(fact_67_case__prodI2_H,axiom,
! [A: $tType,B: $tType,C: $tType,P2: product_prod @ A @ B,C2: A > B > C > $o,X2: C] :
( ! [A7: A,B5: B] :
( ( ( product_Pair @ A @ B @ A7 @ B5 )
= P2 )
=> ( C2 @ A7 @ B5 @ X2 ) )
=> ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P2 @ X2 ) ) ).
% case_prodI2'
thf(fact_68_poslsts__UNIV,axiom,
! [A: $tType,S2: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ S2 @ ( lList21148268032oslsts @ A @ ( top_top @ ( set @ A ) ) ) )
= ( S2
!= ( coinductive_LNil @ A ) ) ) ).
% poslsts_UNIV
thf(fact_69_mem__case__prodE,axiom,
! [B: $tType,A: $tType,C: $tType,Z: A,C2: B > C > ( set @ A ),P2: product_prod @ B @ C] :
( ( member @ A @ Z @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ P2 ) )
=> ~ ! [X5: B,Y: C] :
( ( P2
= ( product_Pair @ B @ C @ X5 @ Y ) )
=> ~ ( member @ A @ Z @ ( C2 @ X5 @ Y ) ) ) ) ).
% mem_case_prodE
thf(fact_70_top__set__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% top_set_def
thf(fact_71_case__prodE_H,axiom,
! [B: $tType,A: $tType,C: $tType,C2: A > B > C > $o,P2: product_prod @ A @ B,Z: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P2 @ Z )
=> ~ ! [X5: A,Y: B] :
( ( P2
= ( product_Pair @ A @ B @ X5 @ Y ) )
=> ~ ( C2 @ X5 @ Y @ Z ) ) ) ).
% case_prodE'
thf(fact_72_case__prodD_H,axiom,
! [B: $tType,A: $tType,C: $tType,R3: A > B > C > $o,A3: A,B4: B,C2: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ R3 @ ( product_Pair @ A @ B @ A3 @ B4 ) @ C2 )
=> ( R3 @ A3 @ B4 @ C2 ) ) ).
% case_prodD'
thf(fact_73_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A7: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A7 @ B5 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_74_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y2: product_prod @ A @ B] :
~ ! [A7: A,B5: B] :
( Y2
!= ( product_Pair @ A @ B @ A7 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_75_prod__induct7,axiom,
! [G2: $tType,F2: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) )] :
( ! [A7: A,B5: B,C3: C,D2: D,E2: E,F3: F2,G3: G2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) ) @ A7 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F2 @ G2 ) @ E2 @ ( product_Pair @ F2 @ G2 @ F3 @ G3 ) ) ) ) ) ) )
=> ( P @ X2 ) ) ).
% prod_induct7
thf(fact_76_prod__induct6,axiom,
! [F2: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) )] :
( ! [A7: A,B5: B,C3: C,D2: D,E2: E,F3: F2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) ) @ A7 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F2 ) @ D2 @ ( product_Pair @ E @ F2 @ E2 @ F3 ) ) ) ) ) )
=> ( P @ X2 ) ) ).
% prod_induct6
thf(fact_77_prod__induct5,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
( ! [A7: A,B5: B,C3: C,D2: D,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A7 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
=> ( P @ X2 ) ) ).
% prod_induct5
thf(fact_78_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
( ! [A7: A,B5: B,C3: C,D2: D] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A7 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) )
=> ( P @ X2 ) ) ).
% prod_induct4
thf(fact_79_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X2: product_prod @ A @ ( product_prod @ B @ C )] :
( ! [A7: A,B5: B,C3: C] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A7 @ ( product_Pair @ B @ C @ B5 @ C3 ) ) )
=> ( P @ X2 ) ) ).
% prod_induct3
thf(fact_80_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F2: $tType,G2: $tType,Y2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) )] :
~ ! [A7: A,B5: B,C3: C,D2: D,E2: E,F3: F2,G3: G2] :
( Y2
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) ) @ A7 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F2 @ G2 ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F2 @ G2 ) @ E2 @ ( product_Pair @ F2 @ G2 @ F3 @ G3 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_81_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F2: $tType,Y2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) )] :
~ ! [A7: A,B5: B,C3: C,D2: D,E2: E,F3: F2] :
( Y2
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) ) @ A7 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F2 ) ) @ C3 @ ( product_Pair @ D @ ( product_prod @ E @ F2 ) @ D2 @ ( product_Pair @ E @ F2 @ E2 @ F3 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_82_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
~ ! [A7: A,B5: B,C3: C,D2: D,E2: E] :
( Y2
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A7 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B5 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C3 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_83_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y2: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
~ ! [A7: A,B5: B,C3: C,D2: D] :
( Y2
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A7 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B5 @ ( product_Pair @ C @ D @ C3 @ D2 ) ) ) ) ).
% prod_cases4
thf(fact_84_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y2: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A7: A,B5: B,C3: C] :
( Y2
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A7 @ ( product_Pair @ B @ C @ B5 @ C3 ) ) ) ).
% prod_cases3
thf(fact_85_Pair__inject,axiom,
! [A: $tType,B: $tType,A3: A,B4: B,A8: A,B6: B] :
( ( ( product_Pair @ A @ B @ A3 @ B4 )
= ( product_Pair @ A @ B @ A8 @ B6 ) )
=> ~ ( ( A3 = A8 )
=> ( B4 != B6 ) ) ) ).
% Pair_inject
thf(fact_86_prod__cases,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
( ! [A7: A,B5: B] : ( P @ ( product_Pair @ A @ B @ A7 @ B5 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_87_surj__pair,axiom,
! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
? [X5: A,Y: B] :
( P2
= ( product_Pair @ A @ B @ X5 @ Y ) ) ).
% surj_pair
thf(fact_88_UNIV__witness,axiom,
! [A: $tType] :
? [X5: A] : ( member @ A @ X5 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_witness
thf(fact_89_UNIV__eq__I,axiom,
! [A: $tType,A2: set @ A] :
( ! [X5: A] : ( member @ A @ X5 @ A2 )
=> ( ( top_top @ ( set @ A ) )
= A2 ) ) ).
% UNIV_eq_I
thf(fact_90_UNIV__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A
@ ^ [X3: A] : $true ) ) ).
% UNIV_def
thf(fact_91_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_92_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_93_old_Oprod_Ocase,axiom,
! [A: $tType,C: $tType,B: $tType,F: A > B > C,X1: A,X23: B] :
( ( product_case_prod @ A @ B @ C @ F @ ( product_Pair @ A @ B @ X1 @ X23 ) )
= ( F @ X1 @ X23 ) ) ).
% old.prod.case
thf(fact_94_cond__case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F: A > B > C,G: ( product_prod @ A @ B ) > C] :
( ! [X5: A,Y: B] :
( ( F @ X5 @ Y )
= ( G @ ( product_Pair @ A @ B @ X5 @ Y ) ) )
=> ( ( product_case_prod @ A @ B @ C @ F )
= G ) ) ).
% cond_case_prod_eta
thf(fact_95_case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F: ( product_prod @ A @ B ) > C] :
( ( product_case_prod @ A @ B @ C
@ ^ [X3: A,Y3: B] : ( F @ ( product_Pair @ A @ B @ X3 @ Y3 ) ) )
= F ) ).
% case_prod_eta
thf(fact_96_case__prodE2,axiom,
! [B: $tType,A: $tType,C: $tType,Q: A > $o,P: B > C > A,Z: product_prod @ B @ C] :
( ( Q @ ( product_case_prod @ B @ C @ A @ P @ Z ) )
=> ~ ! [X5: B,Y: C] :
( ( Z
= ( product_Pair @ B @ C @ X5 @ Y ) )
=> ~ ( Q @ ( P @ X5 @ Y ) ) ) ) ).
% case_prodE2
thf(fact_97_case__prodE,axiom,
! [A: $tType,B: $tType,C2: A > B > $o,P2: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ $o @ C2 @ P2 )
=> ~ ! [X5: A,Y: B] :
( ( P2
= ( product_Pair @ A @ B @ X5 @ Y ) )
=> ~ ( C2 @ X5 @ Y ) ) ) ).
% case_prodE
thf(fact_98_case__prodD,axiom,
! [A: $tType,B: $tType,F: A > B > $o,A3: A,B4: B] :
( ( product_case_prod @ A @ B @ $o @ F @ ( product_Pair @ A @ B @ A3 @ B4 ) )
=> ( F @ A3 @ B4 ) ) ).
% case_prodD
thf(fact_99_old_Oprod_Orec,axiom,
! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A3: A,B4: B] :
( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A3 @ B4 ) )
= ( F1 @ A3 @ B4 ) ) ).
% old.prod.rec
thf(fact_100_split__cong,axiom,
! [C: $tType,B: $tType,A: $tType,Q2: product_prod @ A @ B,F: A > B > C,G: A > B > C,P2: product_prod @ A @ B] :
( ! [X5: A,Y: B] :
( ( ( product_Pair @ A @ B @ X5 @ Y )
= Q2 )
=> ( ( F @ X5 @ Y )
= ( G @ X5 @ Y ) ) )
=> ( ( P2 = Q2 )
=> ( ( product_case_prod @ A @ B @ C @ F @ P2 )
= ( product_case_prod @ A @ B @ C @ G @ Q2 ) ) ) ) ).
% split_cong
thf(fact_101_Coinductive__List_Ofinite__lprefix__nitpick__simps_I3_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Y2: A,Ys: coinductive_llist @ A] :
( ( coindu328551480prefix @ A @ Xs @ ( coinductive_LCons @ A @ Y2 @ Ys ) )
= ( ( Xs
= ( coinductive_LNil @ A ) )
| ? [Xs3: coinductive_llist @ A] :
( ( Xs
= ( coinductive_LCons @ A @ Y2 @ Xs3 ) )
& ( coindu328551480prefix @ A @ Xs3 @ Ys ) ) ) ) ).
% Coinductive_List.finite_lprefix_nitpick_simps(3)
thf(fact_102_case__prod__app,axiom,
! [A: $tType,D: $tType,C: $tType,B: $tType] :
( ( product_case_prod @ B @ C @ ( D > A ) )
= ( ^ [F4: B > C > D > A,X3: product_prod @ B @ C,Y3: D] :
( product_case_prod @ B @ C @ A
@ ^ [L3: B,R: C] : ( F4 @ L3 @ R @ Y3 )
@ X3 ) ) ) ).
% case_prod_app
thf(fact_103_internal__case__prod__def,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( produc2004651681e_prod @ A @ B @ C )
= ( product_case_prod @ A @ B @ C ) ) ).
% internal_case_prod_def
thf(fact_104_top__empty__eq2,axiom,
! [B: $tType,A: $tType] :
( ( top_top @ ( A > B > $o ) )
= ( ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% top_empty_eq2
thf(fact_105_case__prod__Pair__iden,axiom,
! [B: $tType,A: $tType,P2: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) @ P2 )
= P2 ) ).
% case_prod_Pair_iden
thf(fact_106_top1I,axiom,
! [A: $tType,X2: A] : ( top_top @ ( A > $o ) @ X2 ) ).
% top1I
thf(fact_107_top2I,axiom,
! [A: $tType,B: $tType,X2: A,Y2: B] : ( top_top @ ( A > B > $o ) @ X2 @ Y2 ) ).
% top2I
thf(fact_108_top__conj_I2_J,axiom,
! [A: $tType,P: $o,X2: A] :
( ( P
& ( top_top @ ( A > $o ) @ X2 ) )
= P ) ).
% top_conj(2)
thf(fact_109_top__conj_I1_J,axiom,
! [A: $tType,X2: A,P: $o] :
( ( ( top_top @ ( A > $o ) @ X2 )
& P )
= P ) ).
% top_conj(1)
thf(fact_110_Coinductive__List_Ofinite__lprefix__nitpick__simps_I2_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A] : ( coindu328551480prefix @ A @ ( coinductive_LNil @ A ) @ Xs ) ).
% Coinductive_List.finite_lprefix_nitpick_simps(2)
thf(fact_111_Coinductive__List_Ofinite__lprefix__nitpick__simps_I1_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coindu328551480prefix @ A @ Xs @ ( coinductive_LNil @ A ) )
= ( Xs
= ( coinductive_LNil @ A ) ) ) ).
% Coinductive_List.finite_lprefix_nitpick_simps(1)
thf(fact_112_top__empty__eq,axiom,
! [A: $tType] :
( ( top_top @ ( A > $o ) )
= ( ^ [X3: A] : ( member @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% top_empty_eq
thf(fact_113_internal__case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,C2: B > C > A,A3: B,B4: C] :
( ( produc2004651681e_prod @ B @ C @ A @ C2 @ ( product_Pair @ B @ C @ A3 @ B4 ) )
= ( C2 @ A3 @ B4 ) ) ).
% internal_case_prod_conv
thf(fact_114_pred__equals__eq2,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
( ( ( ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R3 ) )
= ( ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ S3 ) ) )
= ( R3 = S3 ) ) ).
% pred_equals_eq2
thf(fact_115_inv__image__def,axiom,
! [A: $tType,B: $tType] :
( ( inv_image @ B @ A )
= ( ^ [R: set @ ( product_prod @ B @ B ),F4: A > B] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X3: A,Y3: A] : ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F4 @ X3 ) @ ( F4 @ Y3 ) ) @ R ) ) ) ) ) ).
% inv_image_def
thf(fact_116_Range__Collect__case__prod,axiom,
! [B: $tType,A: $tType,P: B > A > $o] :
( ( range @ B @ A @ ( collect @ ( product_prod @ B @ A ) @ ( product_case_prod @ B @ A @ $o @ P ) ) )
= ( collect @ A
@ ^ [Y3: A] :
? [X3: B] : ( P @ X3 @ Y3 ) ) ) ).
% Range_Collect_case_prod
thf(fact_117_same__fst__def,axiom,
! [B: $tType,A: $tType] :
( ( same_fst @ A @ B )
= ( ^ [P3: A > $o,R4: A > ( set @ ( product_prod @ B @ B ) )] :
( collect @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) )
@ ( product_case_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ $o
@ ( product_case_prod @ A @ B @ ( ( product_prod @ A @ B ) > $o )
@ ^ [X6: A,Y4: B] :
( product_case_prod @ A @ B @ $o
@ ^ [X3: A,Y3: B] :
( ( X6 = X3 )
& ( P3 @ X3 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y4 @ Y3 ) @ ( R4 @ X3 ) ) ) ) ) ) ) ) ) ).
% same_fst_def
thf(fact_118_Domain__Collect__case__prod,axiom,
! [B: $tType,A: $tType,P: A > B > $o] :
( ( domain @ A @ B @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P ) ) )
= ( collect @ A
@ ^ [X3: A] :
( ^ [P4: B > $o] :
? [X7: B] : ( P4 @ X7 )
@ ( P @ X3 ) ) ) ) ).
% Domain_Collect_case_prod
thf(fact_119_image2__def,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( bNF_Greatest_image2 @ C @ A @ B )
= ( ^ [A5: set @ C,F4: C > A,G4: C > B] :
( collect @ ( product_prod @ A @ B )
@ ^ [Uu: product_prod @ A @ B] :
? [A4: C] :
( ( Uu
= ( product_Pair @ A @ B @ ( F4 @ A4 ) @ ( G4 @ A4 ) ) )
& ( member @ C @ A4 @ A5 ) ) ) ) ) ).
% image2_def
thf(fact_120_in__inv__image,axiom,
! [A: $tType,B: $tType,X2: A,Y2: A,R2: set @ ( product_prod @ B @ B ),F: A > B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y2 ) @ ( inv_image @ B @ A @ R2 @ F ) )
= ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F @ X2 ) @ ( F @ Y2 ) ) @ R2 ) ) ).
% in_inv_image
thf(fact_121_same__fstI,axiom,
! [B: $tType,A: $tType,P: A > $o,X2: A,Y5: B,Y2: B,R3: A > ( set @ ( product_prod @ B @ B ) )] :
( ( P @ X2 )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y5 @ Y2 ) @ ( R3 @ X2 ) )
=> ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X2 @ Y5 ) @ ( product_Pair @ A @ B @ X2 @ Y2 ) ) @ ( same_fst @ A @ B @ P @ R3 ) ) ) ) ).
% same_fstI
thf(fact_122_DomainE,axiom,
! [B: $tType,A: $tType,A3: A,R2: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A3 @ ( domain @ A @ B @ R2 ) )
=> ~ ! [B5: B] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B5 ) @ R2 ) ) ).
% DomainE
thf(fact_123_Domain__iff,axiom,
! [A: $tType,B: $tType,A3: A,R2: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A3 @ ( domain @ A @ B @ R2 ) )
= ( ? [Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ Y3 ) @ R2 ) ) ) ).
% Domain_iff
thf(fact_124_Domain_Ocases,axiom,
! [B: $tType,A: $tType,A3: A,R2: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A3 @ ( domain @ A @ B @ R2 ) )
=> ~ ! [B5: B] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B5 ) @ R2 ) ) ).
% Domain.cases
thf(fact_125_Domain_Osimps,axiom,
! [B: $tType,A: $tType,A3: A,R2: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A3 @ ( domain @ A @ B @ R2 ) )
= ( ? [A4: A,B3: B] :
( ( A3 = A4 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ R2 ) ) ) ) ).
% Domain.simps
thf(fact_126_Domain_ODomainI,axiom,
! [B: $tType,A: $tType,A3: A,B4: B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B4 ) @ R2 )
=> ( member @ A @ A3 @ ( domain @ A @ B @ R2 ) ) ) ).
% Domain.DomainI
thf(fact_127_Domain_Oinducts,axiom,
! [B: $tType,A: $tType,X2: A,R2: set @ ( product_prod @ A @ B ),P: A > $o] :
( ( member @ A @ X2 @ ( domain @ A @ B @ R2 ) )
=> ( ! [A7: A,B5: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A7 @ B5 ) @ R2 )
=> ( P @ A7 ) )
=> ( P @ X2 ) ) ) ).
% Domain.inducts
thf(fact_128_RangeE,axiom,
! [A: $tType,B: $tType,B4: A,R2: set @ ( product_prod @ B @ A )] :
( ( member @ A @ B4 @ ( range @ B @ A @ R2 ) )
=> ~ ! [A7: B] :
~ ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A7 @ B4 ) @ R2 ) ) ).
% RangeE
thf(fact_129_Range__iff,axiom,
! [A: $tType,B: $tType,A3: A,R2: set @ ( product_prod @ B @ A )] :
( ( member @ A @ A3 @ ( range @ B @ A @ R2 ) )
= ( ? [Y3: B] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ Y3 @ A3 ) @ R2 ) ) ) ).
% Range_iff
thf(fact_130_Range_Ocases,axiom,
! [B: $tType,A: $tType,A3: B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ B @ A3 @ ( range @ A @ B @ R2 ) )
=> ~ ! [A7: A] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A7 @ A3 ) @ R2 ) ) ).
% Range.cases
thf(fact_131_Range_Osimps,axiom,
! [B: $tType,A: $tType,A3: B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ B @ A3 @ ( range @ A @ B @ R2 ) )
= ( ? [A4: A,B3: B] :
( ( A3 = B3 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ R2 ) ) ) ) ).
% Range.simps
thf(fact_132_Range_Ointros,axiom,
! [B: $tType,A: $tType,A3: A,B4: B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B4 ) @ R2 )
=> ( member @ B @ B4 @ ( range @ A @ B @ R2 ) ) ) ).
% Range.intros
thf(fact_133_Range_Oinducts,axiom,
! [A: $tType,B: $tType,X2: B,R2: set @ ( product_prod @ A @ B ),P: B > $o] :
( ( member @ B @ X2 @ ( range @ A @ B @ R2 ) )
=> ( ! [A7: A,B5: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A7 @ B5 ) @ R2 )
=> ( P @ B5 ) )
=> ( P @ X2 ) ) ) ).
% Range.inducts
thf(fact_134_image2__eqI,axiom,
! [A: $tType,C: $tType,B: $tType,B4: A,F: B > A,X2: B,C2: C,G: B > C,A2: set @ B] :
( ( B4
= ( F @ X2 ) )
=> ( ( C2
= ( G @ X2 ) )
=> ( ( member @ B @ X2 @ A2 )
=> ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ B4 @ C2 ) @ ( bNF_Greatest_image2 @ B @ A @ C @ A2 @ F @ G ) ) ) ) ) ).
% image2_eqI
thf(fact_135_Domain__unfold,axiom,
! [B: $tType,A: $tType] :
( ( domain @ A @ B )
= ( ^ [R: set @ ( product_prod @ A @ B )] :
( collect @ A
@ ^ [X3: A] :
? [Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R ) ) ) ) ).
% Domain_unfold
thf(fact_136_rp__inv__image__def,axiom,
! [B: $tType,A: $tType] :
( ( fun_rp_inv_image @ A @ B )
= ( product_case_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( ( B > A ) > ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) )
@ ^ [R4: set @ ( product_prod @ A @ A ),S4: set @ ( product_prod @ A @ A ),F4: B > A] : ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( inv_image @ A @ B @ R4 @ F4 ) @ ( inv_image @ A @ B @ S4 @ F4 ) ) ) ) ).
% rp_inv_image_def
thf(fact_137_lex__prod__def,axiom,
! [B: $tType,A: $tType] :
( ( lex_prod @ A @ B )
= ( ^ [Ra: set @ ( product_prod @ A @ A ),Rb: set @ ( product_prod @ B @ B )] :
( collect @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) )
@ ( product_case_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ $o
@ ( product_case_prod @ A @ B @ ( ( product_prod @ A @ B ) > $o )
@ ^ [A4: A,B3: B] :
( product_case_prod @ A @ B @ $o
@ ^ [A9: A,B7: B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ A9 ) @ Ra )
| ( ( A4 = A9 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B3 @ B7 ) @ Rb ) ) ) ) ) ) ) ) ) ).
% lex_prod_def
thf(fact_138_relImage__def,axiom,
! [A: $tType,B: $tType] :
( ( bNF_Gr1317331620lImage @ B @ A )
= ( ^ [R4: set @ ( product_prod @ B @ B ),F4: B > A] :
( collect @ ( product_prod @ A @ A )
@ ^ [Uu: product_prod @ A @ A] :
? [A1: B,A22: B] :
( ( Uu
= ( product_Pair @ A @ A @ ( F4 @ A1 ) @ ( F4 @ A22 ) ) )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A1 @ A22 ) @ R4 ) ) ) ) ) ).
% relImage_def
thf(fact_139_Rangep__Range__eq,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ A @ B )] :
( ( rangep @ A @ B
@ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R2 ) )
= ( ^ [X3: B] : ( member @ B @ X3 @ ( range @ A @ B @ R2 ) ) ) ) ).
% Rangep_Range_eq
thf(fact_140_in__lex__prod,axiom,
! [A: $tType,B: $tType,A3: A,B4: B,A8: A,B6: B,R2: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B4 ) @ ( product_Pair @ A @ B @ A8 @ B6 ) ) @ ( lex_prod @ A @ B @ R2 @ S2 ) )
= ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A8 ) @ R2 )
| ( ( A3 = A8 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B4 @ B6 ) @ S2 ) ) ) ) ).
% in_lex_prod
thf(fact_141_Rangep_Oinducts,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,X2: B,P: B > $o] :
( ( rangep @ A @ B @ R2 @ X2 )
=> ( ! [A7: A,B5: B] :
( ( R2 @ A7 @ B5 )
=> ( P @ B5 ) )
=> ( P @ X2 ) ) ) ).
% Rangep.inducts
thf(fact_142_Rangep_Ointros,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,A3: A,B4: B] :
( ( R2 @ A3 @ B4 )
=> ( rangep @ A @ B @ R2 @ B4 ) ) ).
% Rangep.intros
thf(fact_143_Rangep_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( rangep @ A @ B )
= ( ^ [R: A > B > $o,A4: B] :
? [B3: A,C4: B] :
( ( A4 = C4 )
& ( R @ B3 @ C4 ) ) ) ) ).
% Rangep.simps
thf(fact_144_Rangep_Ocases,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,A3: B] :
( ( rangep @ A @ B @ R2 @ A3 )
=> ~ ! [A7: A] :
~ ( R2 @ A7 @ A3 ) ) ).
% Rangep.cases
thf(fact_145_RangepE,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,B4: B] :
( ( rangep @ A @ B @ R2 @ B4 )
=> ~ ! [A7: A] :
~ ( R2 @ A7 @ B4 ) ) ).
% RangepE
thf(fact_146_Range__def,axiom,
! [B: $tType,A: $tType] :
( ( range @ A @ B )
= ( ^ [R: set @ ( product_prod @ A @ B )] :
( collect @ B
@ ( rangep @ A @ B
@ ^ [X3: A,Y3: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y3 ) @ R ) ) ) ) ) ).
% Range_def
thf(fact_147_rp__inv__image__rp,axiom,
! [A: $tType,B: $tType,P: product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ),F: B > A] :
( ( fun_reduction_pair @ A @ P )
=> ( fun_reduction_pair @ B @ ( fun_rp_inv_image @ A @ B @ P @ F ) ) ) ).
% rp_inv_image_rp
thf(fact_148_relInvImage__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Gr2107612801vImage @ A @ B )
= ( ^ [A5: set @ A,R4: set @ ( product_prod @ B @ B ),F4: A > B] :
( collect @ ( product_prod @ A @ A )
@ ^ [Uu: product_prod @ A @ A] :
? [A1: A,A22: A] :
( ( Uu
= ( product_Pair @ A @ A @ A1 @ A22 ) )
& ( member @ A @ A1 @ A5 )
& ( member @ A @ A22 @ A5 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F4 @ A1 ) @ ( F4 @ A22 ) ) @ R4 ) ) ) ) ) ).
% relInvImage_def
thf(fact_149_image2p__def,axiom,
! [D: $tType,B: $tType,A: $tType,C: $tType] :
( ( bNF_Greatest_image2p @ C @ A @ D @ B )
= ( ^ [F4: C > A,G4: D > B,R4: C > D > $o,X3: A,Y3: B] :
? [X6: C,Y4: D] :
( ( R4 @ X6 @ Y4 )
& ( ( F4 @ X6 )
= X3 )
& ( ( G4 @ Y4 )
= Y3 ) ) ) ) ).
% image2p_def
thf(fact_150_relcomp__unfold,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( relcomp @ A @ C @ B )
= ( ^ [R: set @ ( product_prod @ A @ C ),S: set @ ( product_prod @ C @ B )] :
( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X3: A,Z2: B] :
? [Y3: C] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X3 @ Y3 ) @ R )
& ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ Y3 @ Z2 ) @ S ) ) ) ) ) ) ).
% relcomp_unfold
thf(fact_151_relcompE,axiom,
! [A: $tType,B: $tType,C: $tType,Xz: product_prod @ A @ B,R2: set @ ( product_prod @ A @ C ),S2: set @ ( product_prod @ C @ B )] :
( ( member @ ( product_prod @ A @ B ) @ Xz @ ( relcomp @ A @ C @ B @ R2 @ S2 ) )
=> ~ ! [X5: A,Y: C,Z3: B] :
( ( Xz
= ( product_Pair @ A @ B @ X5 @ Z3 ) )
=> ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X5 @ Y ) @ R2 )
=> ~ ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ Y @ Z3 ) @ S2 ) ) ) ) ).
% relcompE
thf(fact_152_relcompEpair,axiom,
! [A: $tType,B: $tType,C: $tType,A3: A,C2: B,R2: set @ ( product_prod @ A @ C ),S2: set @ ( product_prod @ C @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ C2 ) @ ( relcomp @ A @ C @ B @ R2 @ S2 ) )
=> ~ ! [B5: C] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A3 @ B5 ) @ R2 )
=> ~ ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ B5 @ C2 ) @ S2 ) ) ) ).
% relcompEpair
thf(fact_153_relcomp_Ocases,axiom,
! [A: $tType,C: $tType,B: $tType,A12: A,A23: C,R2: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ B @ C )] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A12 @ A23 ) @ ( relcomp @ A @ B @ C @ R2 @ S2 ) )
=> ~ ! [B5: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A12 @ B5 ) @ R2 )
=> ~ ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B5 @ A23 ) @ S2 ) ) ) ).
% relcomp.cases
thf(fact_154_relcomp_Osimps,axiom,
! [B: $tType,C: $tType,A: $tType,A12: A,A23: C,R2: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ B @ C )] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A12 @ A23 ) @ ( relcomp @ A @ B @ C @ R2 @ S2 ) )
= ( ? [A4: A,B3: B,C4: C] :
( ( A12 = A4 )
& ( A23 = C4 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B3 ) @ R2 )
& ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B3 @ C4 ) @ S2 ) ) ) ) ).
% relcomp.simps
thf(fact_155_relcomp_Oinducts,axiom,
! [B: $tType,A: $tType,C: $tType,X1: A,X23: C,R2: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ B @ C ),P: A > C > $o] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X1 @ X23 ) @ ( relcomp @ A @ B @ C @ R2 @ S2 ) )
=> ( ! [A7: A,B5: B,C3: C] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A7 @ B5 ) @ R2 )
=> ( ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B5 @ C3 ) @ S2 )
=> ( P @ A7 @ C3 ) ) )
=> ( P @ X1 @ X23 ) ) ) ).
% relcomp.inducts
thf(fact_156_relcomp_OrelcompI,axiom,
! [A: $tType,C: $tType,B: $tType,A3: A,B4: B,R2: set @ ( product_prod @ A @ B ),C2: C,S2: set @ ( product_prod @ B @ C )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A3 @ B4 ) @ R2 )
=> ( ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B4 @ C2 ) @ S2 )
=> ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A3 @ C2 ) @ ( relcomp @ A @ B @ C @ R2 @ S2 ) ) ) ) ).
% relcomp.relcompI
thf(fact_157_O__assoc,axiom,
! [A: $tType,D: $tType,B: $tType,C: $tType,R3: set @ ( product_prod @ A @ D ),S3: set @ ( product_prod @ D @ C ),T2: set @ ( product_prod @ C @ B )] :
( ( relcomp @ A @ C @ B @ ( relcomp @ A @ D @ C @ R3 @ S3 ) @ T2 )
= ( relcomp @ A @ D @ B @ R3 @ ( relcomp @ D @ C @ B @ S3 @ T2 ) ) ) ).
% O_assoc
thf(fact_158_image2pE,axiom,
! [D: $tType,B: $tType,A: $tType,C: $tType,F: A > B,G: C > D,R3: A > C > $o,Fx: B,Gy: D] :
( ( bNF_Greatest_image2p @ A @ B @ C @ D @ F @ G @ R3 @ Fx @ Gy )
=> ~ ! [X5: A] :
( ( Fx
= ( F @ X5 ) )
=> ! [Y: C] :
( ( Gy
= ( G @ Y ) )
=> ~ ( R3 @ X5 @ Y ) ) ) ) ).
% image2pE
thf(fact_159_image2pI,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,R3: A > B > $o,X2: A,Y2: B,F: A > C,G: B > D] :
( ( R3 @ X2 @ Y2 )
=> ( bNF_Greatest_image2p @ A @ C @ B @ D @ F @ G @ R3 @ ( F @ X2 ) @ ( G @ Y2 ) ) ) ).
% image2pI
thf(fact_160_relInvImage__UNIV__relImage,axiom,
! [B: $tType,A: $tType,R3: set @ ( product_prod @ A @ A ),F: A > B] : ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ ( bNF_Gr2107612801vImage @ A @ B @ ( top_top @ ( set @ A ) ) @ ( bNF_Gr1317331620lImage @ A @ B @ R3 @ F ) @ F ) ) ).
% relInvImage_UNIV_relImage
thf(fact_161_ssubst__Pair__rhs,axiom,
! [B: $tType,A: $tType,R2: A,S2: B,R3: set @ ( product_prod @ A @ B ),S5: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S2 ) @ R3 )
=> ( ( S5 = S2 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S5 ) @ R3 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_162_llimit__induct,axiom,
! [A: $tType,P: ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A] :
( ( P @ ( coinductive_LNil @ A ) )
=> ( ! [X5: A,Xs2: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs2 )
=> ( ( P @ Xs2 )
=> ( P @ ( coinductive_LCons @ A @ X5 @ Xs2 ) ) ) )
=> ( ( ! [Ys2: coinductive_llist @ A] :
( ( coindu1478340336prefix @ A @ Ys2 @ Xs )
=> ( P @ Ys2 ) )
=> ( P @ Xs ) )
=> ( P @ Xs ) ) ) ) ).
% llimit_induct
thf(fact_163_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% order_refl
thf(fact_164_subset__antisym,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_165_subsetI,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ! [X5: A] :
( ( member @ A @ X5 @ A2 )
=> ( member @ A @ X5 @ B2 ) )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% subsetI
thf(fact_166_lfinite__code_I2_J,axiom,
! [B: $tType,X2: B,Xs: coinductive_llist @ B] :
( ( coinductive_lfinite @ B @ ( coinductive_LCons @ B @ X2 @ Xs ) )
= ( coinductive_lfinite @ B @ Xs ) ) ).
% lfinite_code(2)
thf(fact_167_lfinite__LCons,axiom,
! [A: $tType,X2: A,Xs: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ ( coinductive_LCons @ A @ X2 @ Xs ) )
= ( coinductive_lfinite @ A @ Xs ) ) ).
% lfinite_LCons
thf(fact_168_lfinite__code_I1_J,axiom,
! [A: $tType] : ( coinductive_lfinite @ A @ ( coinductive_LNil @ A ) ) ).
% lfinite_code(1)
thf(fact_169_relcomp__mono,axiom,
! [A: $tType,C: $tType,B: $tType,R5: set @ ( product_prod @ A @ B ),R2: set @ ( product_prod @ A @ B ),S5: set @ ( product_prod @ B @ C ),S2: set @ ( product_prod @ B @ C )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R5 @ R2 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ B @ C ) ) @ S5 @ S2 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ C ) ) @ ( relcomp @ A @ B @ C @ R5 @ S5 ) @ ( relcomp @ A @ B @ C @ R2 @ S2 ) ) ) ) ).
% relcomp_mono
thf(fact_170_relInvImage__mono,axiom,
! [A: $tType,B: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ A @ A ),A2: set @ B,F: B > A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R1 @ R22 )
=> ( ord_less_eq @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Gr2107612801vImage @ B @ A @ A2 @ R1 @ F ) @ ( bNF_Gr2107612801vImage @ B @ A @ A2 @ R22 @ F ) ) ) ).
% relInvImage_mono
thf(fact_171_relImage__mono,axiom,
! [B: $tType,A: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ A @ A ),F: A > B] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R1 @ R22 )
=> ( ord_less_eq @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_Gr1317331620lImage @ A @ B @ R1 @ F ) @ ( bNF_Gr1317331620lImage @ A @ B @ R22 @ F ) ) ) ).
% relImage_mono
thf(fact_172_Range__mono,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S2 )
=> ( ord_less_eq @ ( set @ B ) @ ( range @ A @ B @ R2 ) @ ( range @ A @ B @ S2 ) ) ) ).
% Range_mono
thf(fact_173_Domain__mono,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S2 )
=> ( ord_less_eq @ ( set @ A ) @ ( domain @ A @ B @ R2 ) @ ( domain @ A @ B @ S2 ) ) ) ).
% Domain_mono
thf(fact_174_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B4: A,A3: A] :
( ( ord_less_eq @ A @ B4 @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ B4 )
=> ( A3 = B4 ) ) ) ) ).
% dual_order.antisym
thf(fact_175_poslsts__mono,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ord_less_eq @ ( set @ ( coinductive_llist @ A ) ) @ ( lList21148268032oslsts @ A @ A2 ) @ ( lList21148268032oslsts @ A @ B2 ) ) ) ).
% poslsts_mono
thf(fact_176_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B4: A,A3: A,C2: A] :
( ( ord_less_eq @ A @ B4 @ A3 )
=> ( ( ord_less_eq @ A @ C2 @ B4 )
=> ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% dual_order.trans
thf(fact_177_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > A > $o,A3: A,B4: A] :
( ! [A7: A,B5: A] :
( ( ord_less_eq @ A @ A7 @ B5 )
=> ( P @ A7 @ B5 ) )
=> ( ! [A7: A,B5: A] :
( ( P @ B5 @ A7 )
=> ( P @ A7 @ B5 ) )
=> ( P @ A3 @ B4 ) ) ) ) ).
% linorder_wlog
thf(fact_178_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% dual_order.refl
thf(fact_179_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X2: A,Y2: A,Z: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ Z )
=> ( ord_less_eq @ A @ X2 @ Z ) ) ) ) ).
% order_trans
thf(fact_180_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A3: A,B4: A] :
( ( ord_less_eq @ A @ A3 @ B4 )
=> ( ( ord_less_eq @ A @ B4 @ A3 )
=> ( A3 = B4 ) ) ) ) ).
% order_class.order.antisym
thf(fact_181_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A3: A,B4: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B4 )
=> ( ( B4 = C2 )
=> ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_182_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A3: A,B4: A,C2: A] :
( ( A3 = B4 )
=> ( ( ord_less_eq @ A @ B4 @ C2 )
=> ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_183_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [Y2: A,X2: A] :
( ( ord_less_eq @ A @ Y2 @ X2 )
=> ( ( ord_less_eq @ A @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ) ).
% antisym_conv
thf(fact_184_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X2: A,Y2: A,Z: A] :
( ( ( ord_less_eq @ A @ X2 @ Y2 )
=> ~ ( ord_less_eq @ A @ Y2 @ Z ) )
=> ( ( ( ord_less_eq @ A @ Y2 @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ Z ) )
=> ( ( ( ord_less_eq @ A @ X2 @ Z )
=> ~ ( ord_less_eq @ A @ Z @ Y2 ) )
=> ( ( ( ord_less_eq @ A @ Z @ Y2 )
=> ~ ( ord_less_eq @ A @ Y2 @ X2 ) )
=> ( ( ( ord_less_eq @ A @ Y2 @ Z )
=> ~ ( ord_less_eq @ A @ Z @ X2 ) )
=> ~ ( ( ord_less_eq @ A @ Z @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_185_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A3: A,B4: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B4 )
=> ( ( ord_less_eq @ A @ B4 @ C2 )
=> ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% order.trans
thf(fact_186_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X2: A,Y2: A] :
( ~ ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ).
% le_cases
thf(fact_187_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X2: A,Y2: A] :
( ( X2 = Y2 )
=> ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ).
% eq_refl
thf(fact_188_linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
| ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ).
% linear
thf(fact_189_antisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ) ).
% antisym
thf(fact_190_eq__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ^ [Y6: A,Z4: A] : Y6 = Z4 )
= ( ^ [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
& ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ) ) ).
% eq_iff
thf(fact_191_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A3: A,B4: A,F: A > B,C2: B] :
( ( ord_less_eq @ A @ A3 @ B4 )
=> ( ( ( F @ B4 )
= C2 )
=> ( ! [X5: A,Y: A] :
( ( ord_less_eq @ A @ X5 @ Y )
=> ( ord_less_eq @ B @ ( F @ X5 ) @ ( F @ Y ) ) )
=> ( ord_less_eq @ B @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_192_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A3: A,F: B > A,B4: B,C2: B] :
( ( A3
= ( F @ B4 ) )
=> ( ( ord_less_eq @ B @ B4 @ C2 )
=> ( ! [X5: B,Y: B] :
( ( ord_less_eq @ B @ X5 @ Y )
=> ( ord_less_eq @ A @ ( F @ X5 ) @ ( F @ Y ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_193_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C @ ( type2 @ C ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A3: A,B4: A,F: A > C,C2: C] :
( ( ord_less_eq @ A @ A3 @ B4 )
=> ( ( ord_less_eq @ C @ ( F @ B4 ) @ C2 )
=> ( ! [X5: A,Y: A] :
( ( ord_less_eq @ A @ X5 @ Y )
=> ( ord_less_eq @ C @ ( F @ X5 ) @ ( F @ Y ) ) )
=> ( ord_less_eq @ C @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% order_subst2
thf(fact_194_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A3: A,F: B > A,B4: B,C2: B] :
( ( ord_less_eq @ A @ A3 @ ( F @ B4 ) )
=> ( ( ord_less_eq @ B @ B4 @ C2 )
=> ( ! [X5: B,Y: B] :
( ( ord_less_eq @ B @ X5 @ Y )
=> ( ord_less_eq @ A @ ( F @ X5 ) @ ( F @ Y ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% order_subst1
thf(fact_195_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
= ( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) ) ) ) ).
% Collect_mono_iff
thf(fact_196_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F4: A > B,G4: A > B] :
! [X3: A] : ( ord_less_eq @ B @ ( F4 @ X3 ) @ ( G4 @ X3 ) ) ) ) ) ).
% le_fun_def
thf(fact_197_contra__subsetD,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ~ ( member @ A @ C2 @ B2 )
=> ~ ( member @ A @ C2 @ A2 ) ) ) ).
% contra_subsetD
thf(fact_198_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y6: set @ A,Z4: set @ A] : Y6 = Z4 )
= ( ^ [A5: set @ A,B8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B8 )
& ( ord_less_eq @ ( set @ A ) @ B8 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_199_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ 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_200_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B,X2: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% le_funE
thf(fact_201_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B,X2: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% le_funD
thf(fact_202_subset__trans,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C5 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ C5 ) ) ) ).
% subset_trans
thf(fact_203_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_204_subset__refl,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% subset_refl
thf(fact_205_rev__subsetD,axiom,
! [A: $tType,C2: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( member @ A @ C2 @ B2 ) ) ) ).
% rev_subsetD
thf(fact_206_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A5: set @ A,B8: set @ A] :
! [T3: A] :
( ( member @ A @ T3 @ A5 )
=> ( member @ A @ T3 @ B8 ) ) ) ) ).
% subset_iff
thf(fact_207_set__rev__mp,axiom,
! [A: $tType,X2: A,A2: set @ A,B2: set @ A] :
( ( member @ A @ X2 @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( member @ A @ X2 @ B2 ) ) ) ).
% set_rev_mp
thf(fact_208_equalityD2,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_209_equalityD1,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_210_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A5: set @ A,B8: set @ A] :
! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( member @ A @ X3 @ B8 ) ) ) ) ).
% subset_eq
thf(fact_211_equalityE,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_212_subsetCE,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( member @ A @ C2 @ A2 )
=> ( member @ A @ C2 @ B2 ) ) ) ).
% subsetCE
thf(fact_213_subsetD,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( member @ A @ C2 @ A2 )
=> ( member @ A @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_214_in__mono,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,X2: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( member @ A @ X2 @ A2 )
=> ( member @ A @ X2 @ B2 ) ) ) ).
% in_mono
thf(fact_215_set__mp,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,X2: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( member @ A @ X2 @ A2 )
=> ( member @ A @ X2 @ B2 ) ) ) ).
% set_mp
thf(fact_216_finlsts__mono,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ord_less_eq @ ( set @ ( coinductive_llist @ A ) ) @ ( lList2236698231inlsts @ A @ A2 ) @ ( lList2236698231inlsts @ A @ B2 ) ) ) ).
% finlsts_mono
thf(fact_217_alllsts__mono,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ord_less_eq @ ( set @ ( coinductive_llist @ A ) ) @ ( lList2435255213lllsts @ A @ A2 ) @ ( lList2435255213lllsts @ A @ B2 ) ) ) ).
% alllsts_mono
thf(fact_218_wlog__linorder__le,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > A > $o,B4: A,A3: A] :
( ! [A7: A,B5: A] :
( ( ord_less_eq @ A @ A7 @ B5 )
=> ( P @ A7 @ B5 ) )
=> ( ( ( P @ B4 @ A3 )
=> ( P @ A3 @ B4 ) )
=> ( P @ A3 @ B4 ) ) ) ) ).
% wlog_linorder_le
thf(fact_219_subset__UNIV,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ ( top_top @ ( set @ A ) ) ) ).
% subset_UNIV
thf(fact_220_top_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_top @ A @ ( type2 @ A ) )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
=> ( A3
= ( top_top @ A ) ) ) ) ).
% top.extremum_uniqueI
thf(fact_221_top_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_top @ A @ ( type2 @ A ) )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
= ( A3
= ( top_top @ A ) ) ) ) ).
% top.extremum_unique
thf(fact_222_top__greatest,axiom,
! [A: $tType] :
( ( order_top @ A @ ( type2 @ A ) )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ ( top_top @ A ) ) ) ).
% top_greatest
thf(fact_223_lfinite__LNil,axiom,
! [A: $tType] : ( coinductive_lfinite @ A @ ( coinductive_LNil @ A ) ) ).
% lfinite_LNil
thf(fact_224_lfinite__LConsI,axiom,
! [A: $tType,Xs: coinductive_llist @ A,X2: A] :
( ( coinductive_lfinite @ A @ Xs )
=> ( coinductive_lfinite @ A @ ( coinductive_LCons @ A @ X2 @ Xs ) ) ) ).
% lfinite_LConsI
thf(fact_225_lstrict__prefix__lfinite1,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coindu1478340336prefix @ A @ Xs @ Ys )
=> ( coinductive_lfinite @ A @ Xs ) ) ).
% lstrict_prefix_lfinite1
thf(fact_226_subrelI,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
( ! [X5: A,Y: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y ) @ R2 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y ) @ S2 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S2 ) ) ).
% subrelI
thf(fact_227_prop__restrict,axiom,
! [A: $tType,X2: A,Z5: set @ A,X4: set @ A,P: A > $o] :
( ( member @ A @ X2 @ Z5 )
=> ( ( ord_less_eq @ ( set @ A ) @ Z5
@ ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ X4 )
& ( P @ X3 ) ) ) )
=> ( P @ X2 ) ) ) ).
% prop_restrict
thf(fact_228_Collect__restrict,axiom,
! [A: $tType,X4: set @ A,P: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ X4 )
& ( P @ X3 ) ) )
@ X4 ) ).
% Collect_restrict
thf(fact_229_subset__Collect__iff,axiom,
! [A: $tType,B2: set @ A,A2: set @ A,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2
@ ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ A2 )
& ( P @ X3 ) ) ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ B2 )
=> ( P @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_230_subset__CollectI,axiom,
! [A: $tType,B2: set @ A,A2: set @ A,Q: A > $o,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ B2 )
=> ( ( Q @ X5 )
=> ( P @ X5 ) ) )
=> ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ B2 )
& ( Q @ X3 ) ) )
@ ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ A2 )
& ( P @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_231_lfinite_Ocases,axiom,
! [A: $tType,A3: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ A3 )
=> ( ( A3
!= ( coinductive_LNil @ A ) )
=> ~ ! [Xs2: coinductive_llist @ A] :
( ? [X5: A] :
( A3
= ( coinductive_LCons @ A @ X5 @ Xs2 ) )
=> ~ ( coinductive_lfinite @ A @ Xs2 ) ) ) ) ).
% lfinite.cases
thf(fact_232_lfinite_Osimps,axiom,
! [A: $tType] :
( ( coinductive_lfinite @ A )
= ( ^ [A4: coinductive_llist @ A] :
( ( A4
= ( coinductive_LNil @ A ) )
| ? [Xs4: coinductive_llist @ A,X3: A] :
( ( A4
= ( coinductive_LCons @ A @ X3 @ Xs4 ) )
& ( coinductive_lfinite @ A @ Xs4 ) ) ) ) ) ).
% lfinite.simps
thf(fact_233_lfinite_Oinducts,axiom,
! [A: $tType,X2: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
( ( coinductive_lfinite @ A @ X2 )
=> ( ( P @ ( coinductive_LNil @ A ) )
=> ( ! [Xs2: coinductive_llist @ A,X5: A] :
( ( coinductive_lfinite @ A @ Xs2 )
=> ( ( P @ Xs2 )
=> ( P @ ( coinductive_LCons @ A @ X5 @ Xs2 ) ) ) )
=> ( P @ X2 ) ) ) ) ).
% lfinite.inducts
thf(fact_234_reduction__pairI,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ R3 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R3 @ S3 ) @ R3 )
=> ( fun_reduction_pair @ A @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R3 @ S3 ) ) ) ) ).
% reduction_pairI
thf(fact_235_conj__subset__def,axiom,
! [A: $tType,A2: set @ A,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ A2
@ ( collect @ A
@ ^ [X3: A] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) )
= ( ( ord_less_eq @ ( set @ A ) @ A2 @ ( collect @ A @ P ) )
& ( ord_less_eq @ ( set @ A ) @ A2 @ ( collect @ A @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_236_relChain__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ( ( bNF_Ca1785829860lChain @ A @ B )
= ( ^ [R: set @ ( product_prod @ A @ A ),As: A > B] :
! [I: A,J: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I @ J ) @ R )
=> ( ord_less_eq @ B @ ( As @ I ) @ ( As @ J ) ) ) ) ) ) ).
% relChain_def
thf(fact_237_wf__inv__image,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ B @ B ),F: A > B] :
( ( wf @ B @ R2 )
=> ( wf @ A @ ( inv_image @ B @ A @ R2 @ F ) ) ) ).
% wf_inv_image
thf(fact_238_wf__lex__prod,axiom,
! [A: $tType,B: $tType,Ra2: set @ ( product_prod @ A @ A ),Rb2: set @ ( product_prod @ B @ B )] :
( ( wf @ A @ Ra2 )
=> ( ( wf @ B @ Rb2 )
=> ( wf @ ( product_prod @ A @ B ) @ ( lex_prod @ A @ B @ Ra2 @ Rb2 ) ) ) ) ).
% wf_lex_prod
thf(fact_239_finlstsp__mono,axiom,
! [A: $tType,A2: A > $o,B2: A > $o] :
( ( ord_less_eq @ ( A > $o ) @ A2 @ B2 )
=> ( ord_less_eq @ ( ( coinductive_llist @ A ) > $o ) @ ( lList2860480441nlstsp @ A @ A2 ) @ ( lList2860480441nlstsp @ A @ B2 ) ) ) ).
% finlstsp_mono
thf(fact_240_alllstsp__mono,axiom,
! [A: $tType,A2: A > $o,B2: A > $o] :
( ( ord_less_eq @ ( A > $o ) @ A2 @ B2 )
=> ( ord_less_eq @ ( ( coinductive_llist @ A ) > $o ) @ ( lList21511617539llstsp @ A @ A2 ) @ ( lList21511617539llstsp @ A @ B2 ) ) ) ).
% alllstsp_mono
thf(fact_241_pred__subset__eq,axiom,
! [A: $tType,R3: set @ A,S3: set @ A] :
( ( ord_less_eq @ ( A > $o )
@ ^ [X3: A] : ( member @ A @ X3 @ R3 )
@ ^ [X3: A] : ( member @ A @ X3 @ S3 ) )
= ( ord_less_eq @ ( set @ A ) @ R3 @ S3 ) ) ).
% pred_subset_eq
thf(fact_242_wf__subset,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),P2: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ R2 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ P2 @ R2 )
=> ( wf @ A @ P2 ) ) ) ).
% wf_subset
thf(fact_243_less__eq__set__def,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A5: set @ A,B8: set @ A] :
( ord_less_eq @ ( A > $o )
@ ^ [X3: A] : ( member @ A @ X3 @ A5 )
@ ^ [X3: A] : ( member @ A @ X3 @ B8 ) ) ) ) ).
% less_eq_set_def
thf(fact_244_wf__def,axiom,
! [A: $tType] :
( ( wf @ A )
= ( ^ [R: set @ ( product_prod @ A @ A )] :
! [P3: A > $o] :
( ! [X3: A] :
( ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R )
=> ( P3 @ Y3 ) )
=> ( P3 @ X3 ) )
=> ( ^ [P4: A > $o] :
! [X7: A] : ( P4 @ X7 )
@ P3 ) ) ) ) ).
% wf_def
thf(fact_245_wfE__min,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),X2: A,Q: set @ A] :
( ( wf @ A @ R3 )
=> ( ( member @ A @ X2 @ Q )
=> ~ ! [Z3: A] :
( ( member @ A @ Z3 @ Q )
=> ~ ! [Y7: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y7 @ Z3 ) @ R3 )
=> ~ ( member @ A @ Y7 @ Q ) ) ) ) ) ).
% wfE_min
thf(fact_246_wfI__min,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ! [X5: A,Q3: set @ A] :
( ( member @ A @ X5 @ Q3 )
=> ? [Xa: A] :
( ( member @ A @ Xa @ Q3 )
& ! [Y: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Xa ) @ R3 )
=> ~ ( member @ A @ Y @ Q3 ) ) ) )
=> ( wf @ A @ R3 ) ) ).
% wfI_min
thf(fact_247_wfUNIVI,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ! [P5: A > $o,X5: A] :
( ! [Xa: A] :
( ! [Y: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Xa ) @ R2 )
=> ( P5 @ Y ) )
=> ( P5 @ Xa ) )
=> ( P5 @ X5 ) )
=> ( wf @ A @ R2 ) ) ).
% wfUNIVI
thf(fact_248_wf__asym,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A,X2: A] :
( ( wf @ A @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ X2 ) @ R2 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ A3 ) @ R2 ) ) ) ).
% wf_asym
thf(fact_249_wf__induct,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),P: A > $o,A3: A] :
( ( wf @ A @ R2 )
=> ( ! [X5: A] :
( ! [Y7: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y7 @ X5 ) @ R2 )
=> ( P @ Y7 ) )
=> ( P @ X5 ) )
=> ( P @ A3 ) ) ) ).
% wf_induct
thf(fact_250_wf__irrefl,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A] :
( ( wf @ A @ R2 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ R2 ) ) ).
% wf_irrefl
thf(fact_251_wf__not__sym,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A,X2: A] :
( ( wf @ A @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ X2 ) @ R2 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ A3 ) @ R2 ) ) ) ).
% wf_not_sym
thf(fact_252_wf__not__refl,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A3: A] :
( ( wf @ A @ R2 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ A3 ) @ R2 ) ) ).
% wf_not_refl
thf(fact_253_wf__eq__minimal,axiom,
! [A: $tType] :
( ( wf @ A )
= ( ^ [R: set @ ( product_prod @ A @ A )] :
! [Q4: set @ A] :
( ? [X3: A] : ( member @ A @ X3 @ Q4 )
=> ? [X3: A] :
( ( member @ A @ X3 @ Q4 )
& ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X3 ) @ R )
=> ~ ( member @ A @ Y3 @ Q4 ) ) ) ) ) ) ).
% wf_eq_minimal
thf(fact_254_wf__induct__rule,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),P: A > $o,A3: A] :
( ( wf @ A @ R2 )
=> ( ! [X5: A] :
( ! [Y7: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y7 @ X5 ) @ R2 )
=> ( P @ Y7 ) )
=> ( P @ X5 ) )
=> ( P @ A3 ) ) ) ).
% wf_induct_rule
thf(fact_255_wf__same__fst,axiom,
! [B: $tType,A: $tType,P: A > $o,R3: A > ( set @ ( product_prod @ B @ B ) )] :
( ! [X5: A] :
( ( P @ X5 )
=> ( wf @ B @ ( R3 @ X5 ) ) )
=> ( wf @ ( product_prod @ A @ B ) @ ( same_fst @ A @ B @ P @ R3 ) ) ) ).
% wf_same_fst
%----Type constructors (16)
thf(tcon_fun___Orderings_Oorder__top,axiom,
! [A10: $tType,A11: $tType] :
( ( order_top @ A11 @ ( type2 @ A11 ) )
=> ( order_top @ ( A10 > A11 ) @ ( type2 @ ( A10 > A11 ) ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A10: $tType,A11: $tType] :
( ( preorder @ A11 @ ( type2 @ A11 ) )
=> ( preorder @ ( A10 > A11 ) @ ( type2 @ ( A10 > A11 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A10: $tType,A11: $tType] :
( ( order @ A11 @ ( type2 @ A11 ) )
=> ( order @ ( A10 > A11 ) @ ( type2 @ ( A10 > A11 ) ) ) ) ).
thf(tcon_fun___Orderings_Otop,axiom,
! [A10: $tType,A11: $tType] :
( ( top @ A11 @ ( type2 @ A11 ) )
=> ( top @ ( A10 > A11 ) @ ( type2 @ ( A10 > A11 ) ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A10: $tType,A11: $tType] :
( ( ord @ A11 @ ( type2 @ A11 ) )
=> ( ord @ ( A10 > A11 ) @ ( type2 @ ( A10 > A11 ) ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__top_1,axiom,
! [A10: $tType] : ( order_top @ ( set @ A10 ) @ ( type2 @ ( set @ A10 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_2,axiom,
! [A10: $tType] : ( preorder @ ( set @ A10 ) @ ( type2 @ ( set @ A10 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_3,axiom,
! [A10: $tType] : ( order @ ( set @ A10 ) @ ( type2 @ ( set @ A10 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Otop_4,axiom,
! [A10: $tType] : ( top @ ( set @ A10 ) @ ( type2 @ ( set @ A10 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_5,axiom,
! [A10: $tType] : ( ord @ ( set @ A10 ) @ ( type2 @ ( set @ A10 ) ) ) ).
thf(tcon_HOL_Obool___Orderings_Oorder__top_6,axiom,
order_top @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_7,axiom,
preorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_8,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Otop_9,axiom,
top @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_10,axiom,
ord @ $o @ ( type2 @ $o ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
p @ ( coinductive_LCons @ a @ a2 @ l ) ).
%------------------------------------------------------------------------------