TPTP Problem File: DAT114^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : DAT114^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Coinductive list 349
% Version : [Bla16] axioms : Especial.
% English :
% Refs : [Loc10] Lochbihler (2010), Coinductive
% : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% Source : [Bla16]
% Names : coinductive_list__349.p [Bla16]
% Status : Theorem
% Rating : 1.00 v7.5.0, 0.33 v7.2.0, 0.50 v7.1.0
% Syntax : Number of formulae : 308 ( 143 unt; 39 typ; 0 def)
% Number of atoms : 681 ( 292 equ; 0 cnn)
% Maximal formula atoms : 6 ( 2 avg)
% Number of connectives : 3761 ( 132 ~; 13 |; 49 &;3253 @)
% ( 0 <=>; 314 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 8 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 266 ( 266 >; 0 *; 0 +; 0 <<)
% Number of symbols : 41 ( 38 usr; 4 con; 0-8 aty)
% Number of variables : 1057 ( 41 ^; 960 !; 18 ?;1057 :)
% ( 38 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:43:42.886
%------------------------------------------------------------------------------
%----Could-be-implicit typings (4)
thf(ty_t_Coinductive__List__Mirabelle__kmikjhschf_Ollist,type,
coindu1593790203_llist: $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 (35)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Obot,type,
bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oorder__bot,type,
order_bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ogen__lset,type,
coindu1928975208n_lset:
!>[A: $tType] : ( ( set @ A ) > ( coindu1593790203_llist @ A ) > ( set @ A ) ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olappend,type,
coindu268472904append:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olfinite,type,
coindu1213758845finite:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_OLCons,type,
coindu1121789889_LCons:
!>[A: $tType] : ( A > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_OLNil,type,
coindu1598213697e_LNil:
!>[A: $tType] : ( coindu1593790203_llist @ A ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Ocase__llist,type,
coindu882539134_llist:
!>[B: $tType,A: $tType] : ( B > ( A > ( coindu1593790203_llist @ A ) > B ) > ( coindu1593790203_llist @ A ) > B ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Ocorec__llist,type,
coindu1244876290_llist:
!>[C: $tType,A: $tType] : ( ( C > $o ) > ( C > A ) > ( C > $o ) > ( C > ( coindu1593790203_llist @ A ) ) > ( C > C ) > C > ( coindu1593790203_llist @ A ) ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Olhd,type,
coindu1046438764le_lhd:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > A ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Ollist__all2,type,
coindu987187967t_all2:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ B ) > $o ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Olmap,type,
coindu1062782156e_lmap:
!>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ Aa ) ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Olnull,type,
coindu335574135_lnull:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Olset,type,
coindu1112613586e_lset:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > ( set @ A ) ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Oltl,type,
coindu1047225960le_ltl:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olmember,type,
coindu567634248member:
!>[A: $tType] : ( A > ( coindu1593790203_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ounfold__llist,type,
coindu1599971794_llist:
!>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > B ) > ( A > A ) > A > ( coindu1593790203_llist @ B ) ) ).
thf(sy_c_Finite__Set_Ofinite,type,
finite_finite2:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_Oinsert,type,
insert:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Set_Ois__empty,type,
is_empty:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Set_Ois__singleton,type,
is_singleton:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Set_Opairwise,type,
pairwise:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Set_Othe__elem,type,
the_elem:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_xs,type,
xs: coindu1593790203_llist @ a ).
thf(sy_v_ys,type,
ys: coindu1593790203_llist @ a ).
%----Relevant facts (255)
thf(fact_0__092_060open_062lfinite_A_Ilappend_Axs_Ays_J_092_060close_062,axiom,
coindu1213758845finite @ a @ ( coindu268472904append @ a @ xs @ ys ) ).
% \<open>lfinite (lappend xs ys)\<close>
thf(fact_1_lfinite__code_I1_J,axiom,
! [A: $tType] : ( coindu1213758845finite @ A @ ( coindu1598213697e_LNil @ A ) ) ).
% lfinite_code(1)
thf(fact_2_lfinite__ltl,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ ( coindu1047225960le_ltl @ A @ Xs ) )
= ( coindu1213758845finite @ A @ Xs ) ) ).
% lfinite_ltl
thf(fact_3_lfinite__code_I2_J,axiom,
! [B: $tType,X: B,Xs: coindu1593790203_llist @ B] :
( ( coindu1213758845finite @ B @ ( coindu1121789889_LCons @ B @ X @ Xs ) )
= ( coindu1213758845finite @ B @ Xs ) ) ).
% lfinite_code(2)
thf(fact_4_lfinite__LCons,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ ( coindu1121789889_LCons @ A @ X @ Xs ) )
= ( coindu1213758845finite @ A @ Xs ) ) ).
% lfinite_LCons
thf(fact_5_lfinite__LNil,axiom,
! [A: $tType] : ( coindu1213758845finite @ A @ ( coindu1598213697e_LNil @ A ) ) ).
% lfinite_LNil
thf(fact_6_lappend__assoc,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,Zs: coindu1593790203_llist @ A] :
( ( coindu268472904append @ A @ ( coindu268472904append @ A @ Xs @ Ys ) @ Zs )
= ( coindu268472904append @ A @ Xs @ ( coindu268472904append @ A @ Ys @ Zs ) ) ) ).
% lappend_assoc
thf(fact_7_lfinite__LConsI,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,X: A] :
( ( coindu1213758845finite @ A @ Xs )
=> ( coindu1213758845finite @ A @ ( coindu1121789889_LCons @ A @ X @ Xs ) ) ) ).
% lfinite_LConsI
thf(fact_8_lnull__imp__lfinite,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Xs )
=> ( coindu1213758845finite @ A @ Xs ) ) ).
% lnull_imp_lfinite
thf(fact_9_lfinite_Ocases,axiom,
! [A: $tType,A2: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ A2 )
=> ( ( A2
!= ( coindu1598213697e_LNil @ A ) )
=> ~ ! [Xs2: coindu1593790203_llist @ A] :
( ? [X2: A] :
( A2
= ( coindu1121789889_LCons @ A @ X2 @ Xs2 ) )
=> ~ ( coindu1213758845finite @ A @ Xs2 ) ) ) ) ).
% lfinite.cases
thf(fact_10_lfinite_Osimps,axiom,
! [A: $tType] :
( ( coindu1213758845finite @ A )
= ( ^ [A3: coindu1593790203_llist @ A] :
( ( A3
= ( coindu1598213697e_LNil @ A ) )
| ? [Xs3: coindu1593790203_llist @ A,X3: A] :
( ( A3
= ( coindu1121789889_LCons @ A @ X3 @ Xs3 ) )
& ( coindu1213758845finite @ A @ Xs3 ) ) ) ) ) ).
% lfinite.simps
thf(fact_11_lfinite_Oinducts,axiom,
! [A: $tType,X: coindu1593790203_llist @ A,P: ( coindu1593790203_llist @ A ) > $o] :
( ( coindu1213758845finite @ A @ X )
=> ( ( P @ ( coindu1598213697e_LNil @ A ) )
=> ( ! [Xs2: coindu1593790203_llist @ A,X2: A] :
( ( coindu1213758845finite @ A @ Xs2 )
=> ( ( P @ Xs2 )
=> ( P @ ( coindu1121789889_LCons @ A @ X2 @ Xs2 ) ) ) )
=> ( P @ X ) ) ) ) ).
% lfinite.inducts
thf(fact_12_lfinite__imp__finite__lset,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ Xs )
=> ( finite_finite2 @ A @ ( coindu1112613586e_lset @ A @ Xs ) ) ) ).
% lfinite_imp_finite_lset
thf(fact_13_llist_Oinject,axiom,
! [A: $tType,X21: A,X22: coindu1593790203_llist @ A,Y21: A,Y22: coindu1593790203_llist @ A] :
( ( ( coindu1121789889_LCons @ A @ X21 @ X22 )
= ( coindu1121789889_LCons @ A @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% llist.inject
thf(fact_14_lappend_Odisc__iff_I2_J,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( ~ ( coindu335574135_lnull @ A @ ( coindu268472904append @ A @ Xs @ Ys ) ) )
= ( ~ ( coindu335574135_lnull @ A @ Xs )
| ~ ( coindu335574135_lnull @ A @ Ys ) ) ) ).
% lappend.disc_iff(2)
thf(fact_15_lnull__lappend,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( ( coindu335574135_lnull @ A @ Xs )
& ( coindu335574135_lnull @ A @ Ys ) ) ) ).
% lnull_lappend
thf(fact_16_lappend__code_I2_J,axiom,
! [A: $tType,Xa: A,X: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu268472904append @ A @ ( coindu1121789889_LCons @ A @ Xa @ X ) @ Ys )
= ( coindu1121789889_LCons @ A @ Xa @ ( coindu268472904append @ A @ X @ Ys ) ) ) ).
% lappend_code(2)
thf(fact_17_lappend__code_I1_J,axiom,
! [A: $tType,Ys: coindu1593790203_llist @ A] :
( ( coindu268472904append @ A @ ( coindu1598213697e_LNil @ A ) @ Ys )
= Ys ) ).
% lappend_code(1)
thf(fact_18_lappend__LNil2,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu268472904append @ A @ Xs @ ( coindu1598213697e_LNil @ A ) )
= Xs ) ).
% lappend_LNil2
thf(fact_19_ltl__lappend,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu1047225960le_ltl @ A @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( coindu1047225960le_ltl @ A @ Ys ) ) )
& ( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu1047225960le_ltl @ A @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( coindu268472904append @ A @ ( coindu1047225960le_ltl @ A @ Xs ) @ Ys ) ) ) ) ).
% ltl_lappend
thf(fact_20_ltl__simps_I2_J,axiom,
! [A: $tType,X21: A,X22: coindu1593790203_llist @ A] :
( ( coindu1047225960le_ltl @ A @ ( coindu1121789889_LCons @ A @ X21 @ X22 ) )
= X22 ) ).
% ltl_simps(2)
thf(fact_21_ltl__simps_I1_J,axiom,
! [A: $tType] :
( ( coindu1047225960le_ltl @ A @ ( coindu1598213697e_LNil @ A ) )
= ( coindu1598213697e_LNil @ A ) ) ).
% ltl_simps(1)
thf(fact_22_llist_Odisc_I2_J,axiom,
! [A: $tType,X21: A,X22: coindu1593790203_llist @ A] :
~ ( coindu335574135_lnull @ A @ ( coindu1121789889_LCons @ A @ X21 @ X22 ) ) ).
% llist.disc(2)
thf(fact_23_llist_Odisc_I1_J,axiom,
! [A: $tType] : ( coindu335574135_lnull @ A @ ( coindu1598213697e_LNil @ A ) ) ).
% llist.disc(1)
thf(fact_24_lappend_Octr_I1_J,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu335574135_lnull @ A @ Ys )
=> ( ( coindu268472904append @ A @ Xs @ Ys )
= ( coindu1598213697e_LNil @ A ) ) ) ) ).
% lappend.ctr(1)
thf(fact_25_llist_OdiscI_I2_J,axiom,
! [A: $tType,Llist: coindu1593790203_llist @ A,X21: A,X22: coindu1593790203_llist @ A] :
( ( Llist
= ( coindu1121789889_LCons @ A @ X21 @ X22 ) )
=> ~ ( coindu335574135_lnull @ A @ Llist ) ) ).
% llist.discI(2)
thf(fact_26_llist_OdiscI_I1_J,axiom,
! [A: $tType,Llist: coindu1593790203_llist @ A] :
( ( Llist
= ( coindu1598213697e_LNil @ A ) )
=> ( coindu335574135_lnull @ A @ Llist ) ) ).
% llist.discI(1)
thf(fact_27_lset__intros_I2_J,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A,X4: A] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) )
=> ( member @ A @ X @ ( coindu1112613586e_lset @ A @ ( coindu1121789889_LCons @ A @ X4 @ Xs ) ) ) ) ).
% lset_intros(2)
thf(fact_28_lset__intros_I1_J,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A] : ( member @ A @ X @ ( coindu1112613586e_lset @ A @ ( coindu1121789889_LCons @ A @ X @ Xs ) ) ) ).
% lset_intros(1)
thf(fact_29_lappend_Odisc_I2_J,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( ~ ( coindu335574135_lnull @ A @ Xs )
| ~ ( coindu335574135_lnull @ A @ Ys ) )
=> ~ ( coindu335574135_lnull @ A @ ( coindu268472904append @ A @ Xs @ Ys ) ) ) ).
% lappend.disc(2)
thf(fact_30_lappend_Odisc_I1_J,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu335574135_lnull @ A @ Ys )
=> ( coindu335574135_lnull @ A @ ( coindu268472904append @ A @ Xs @ Ys ) ) ) ) ).
% lappend.disc(1)
thf(fact_31_llist_Oset__sel_I2_J,axiom,
! [A: $tType,A2: coindu1593790203_llist @ A,X: A] :
( ~ ( coindu335574135_lnull @ A @ A2 )
=> ( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ ( coindu1047225960le_ltl @ A @ A2 ) ) )
=> ( member @ A @ X @ ( coindu1112613586e_lset @ A @ A2 ) ) ) ) ).
% llist.set_sel(2)
thf(fact_32_llist_Ocollapse_I1_J,axiom,
! [A: $tType,Llist: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Llist )
=> ( Llist
= ( coindu1598213697e_LNil @ A ) ) ) ).
% llist.collapse(1)
thf(fact_33_llist_Odistinct_I1_J,axiom,
! [A: $tType,X21: A,X22: coindu1593790203_llist @ A] :
( ( coindu1598213697e_LNil @ A )
!= ( coindu1121789889_LCons @ A @ X21 @ X22 ) ) ).
% llist.distinct(1)
thf(fact_34_llist_Oset__intros_I2_J,axiom,
! [A: $tType,X: A,A22: coindu1593790203_llist @ A,A1: A] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ A22 ) )
=> ( member @ A @ X @ ( coindu1112613586e_lset @ A @ ( coindu1121789889_LCons @ A @ A1 @ A22 ) ) ) ) ).
% llist.set_intros(2)
thf(fact_35_llist_Oset__intros_I1_J,axiom,
! [A: $tType,A1: A,A22: coindu1593790203_llist @ A] : ( member @ A @ A1 @ ( coindu1112613586e_lset @ A @ ( coindu1121789889_LCons @ A @ A1 @ A22 ) ) ) ).
% llist.set_intros(1)
thf(fact_36_lnull__ltlI,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Xs )
=> ( coindu335574135_lnull @ A @ ( coindu1047225960le_ltl @ A @ Xs ) ) ) ).
% lnull_ltlI
thf(fact_37_lset__cases,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) )
=> ( ! [Xs4: coindu1593790203_llist @ A] :
( Xs
!= ( coindu1121789889_LCons @ A @ X @ Xs4 ) )
=> ~ ! [X5: A,Xs4: coindu1593790203_llist @ A] :
( ( Xs
= ( coindu1121789889_LCons @ A @ X5 @ Xs4 ) )
=> ~ ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs4 ) ) ) ) ) ).
% lset_cases
thf(fact_38_lappend__ltl,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu268472904append @ A @ ( coindu1047225960le_ltl @ A @ Xs ) @ Ys )
= ( coindu1047225960le_ltl @ A @ ( coindu268472904append @ A @ Xs @ Ys ) ) ) ) ).
% lappend_ltl
thf(fact_39_lset__induct,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A,P: ( coindu1593790203_llist @ A ) > $o] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) )
=> ( ! [Xs2: coindu1593790203_llist @ A] : ( P @ ( coindu1121789889_LCons @ A @ X @ Xs2 ) )
=> ( ! [X5: A,Xs2: coindu1593790203_llist @ A] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs2 ) )
=> ( ( X != X5 )
=> ( ( P @ Xs2 )
=> ( P @ ( coindu1121789889_LCons @ A @ X5 @ Xs2 ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% lset_induct
thf(fact_40_in__lset__ltlD,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ ( coindu1047225960le_ltl @ A @ Xs ) ) )
=> ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) ) ) ).
% in_lset_ltlD
thf(fact_41_lset__induct_H,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A,P: ( coindu1593790203_llist @ A ) > $o] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) )
=> ( ! [Xs2: coindu1593790203_llist @ A] : ( P @ ( coindu1121789889_LCons @ A @ X @ Xs2 ) )
=> ( ! [X5: A,Xs2: coindu1593790203_llist @ A] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs2 ) )
=> ( ( P @ Xs2 )
=> ( P @ ( coindu1121789889_LCons @ A @ X5 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% lset_induct'
thf(fact_42_llist_Oexhaust,axiom,
! [A: $tType,Y: coindu1593790203_llist @ A] :
( ( Y
!= ( coindu1598213697e_LNil @ A ) )
=> ~ ! [X212: A,X222: coindu1593790203_llist @ A] :
( Y
!= ( coindu1121789889_LCons @ A @ X212 @ X222 ) ) ) ).
% llist.exhaust
thf(fact_43_neq__LNil__conv,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( Xs
!= ( coindu1598213697e_LNil @ A ) )
= ( ? [X3: A,Xs5: coindu1593790203_llist @ A] :
( Xs
= ( coindu1121789889_LCons @ A @ X3 @ Xs5 ) ) ) ) ).
% neq_LNil_conv
thf(fact_44_lappend__lnull1,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu268472904append @ A @ Xs @ Ys )
= Ys ) ) ).
% lappend_lnull1
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( collect @ A
@ ^ [X3: A] : ( member @ A @ X3 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X2: A] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X2: A] :
( ( F @ X2 )
= ( G @ X2 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_lappend__lnull2,axiom,
! [A: $tType,Ys: coindu1593790203_llist @ A,Xs: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Ys )
=> ( ( coindu268472904append @ A @ Xs @ Ys )
= Xs ) ) ).
% lappend_lnull2
thf(fact_50_not__lnull__conv,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( ~ ( coindu335574135_lnull @ A @ Xs ) )
= ( ? [X3: A,Xs5: coindu1593790203_llist @ A] :
( Xs
= ( coindu1121789889_LCons @ A @ X3 @ Xs5 ) ) ) ) ).
% not_lnull_conv
thf(fact_51_lappend_Oexhaust,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( ( coindu335574135_lnull @ A @ Xs )
=> ~ ( coindu335574135_lnull @ A @ Ys ) )
=> ( ~ ( coindu335574135_lnull @ A @ Xs )
| ~ ( coindu335574135_lnull @ A @ Ys ) ) ) ).
% lappend.exhaust
thf(fact_52_lnull__def,axiom,
! [A: $tType] :
( ( coindu335574135_lnull @ A )
= ( ^ [Llist2: coindu1593790203_llist @ A] :
( Llist2
= ( coindu1598213697e_LNil @ A ) ) ) ) ).
% lnull_def
thf(fact_53_llist_Oset__cases,axiom,
! [A: $tType,E: A,A2: coindu1593790203_llist @ A] :
( ( member @ A @ E @ ( coindu1112613586e_lset @ A @ A2 ) )
=> ( ! [Z2: coindu1593790203_llist @ A] :
( A2
!= ( coindu1121789889_LCons @ A @ E @ Z2 ) )
=> ~ ! [Z1: A,Z2: coindu1593790203_llist @ A] :
( ( A2
= ( coindu1121789889_LCons @ A @ Z1 @ Z2 ) )
=> ~ ( member @ A @ E @ ( coindu1112613586e_lset @ A @ Z2 ) ) ) ) ) ).
% llist.set_cases
thf(fact_54_llist_Oset__induct,axiom,
! [A: $tType,X: A,A2: coindu1593790203_llist @ A,P: A > ( coindu1593790203_llist @ A ) > $o] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ A2 ) )
=> ( ! [Z1: A,Z2: coindu1593790203_llist @ A] : ( P @ Z1 @ ( coindu1121789889_LCons @ A @ Z1 @ Z2 ) )
=> ( ! [Z1: A,Z2: coindu1593790203_llist @ A,Xa2: A] :
( ( member @ A @ Xa2 @ ( coindu1112613586e_lset @ A @ Z2 ) )
=> ( ( P @ Xa2 @ Z2 )
=> ( P @ Xa2 @ ( coindu1121789889_LCons @ A @ Z1 @ Z2 ) ) ) )
=> ( P @ X @ A2 ) ) ) ) ).
% llist.set_induct
thf(fact_55_lappend__LNil__LNil,axiom,
! [A: $tType] :
( ( coindu268472904append @ A @ ( coindu1598213697e_LNil @ A ) @ ( coindu1598213697e_LNil @ A ) )
= ( coindu1598213697e_LNil @ A ) ) ).
% lappend_LNil_LNil
thf(fact_56_LNil__eq__lappend__iff,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( ( coindu1598213697e_LNil @ A )
= ( coindu268472904append @ A @ Xs @ Ys ) )
= ( ( Xs
= ( coindu1598213697e_LNil @ A ) )
& ( Ys
= ( coindu1598213697e_LNil @ A ) ) ) ) ).
% LNil_eq_lappend_iff
thf(fact_57_lappend__eq__LNil__iff,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( ( coindu268472904append @ A @ Xs @ Ys )
= ( coindu1598213697e_LNil @ A ) )
= ( ( Xs
= ( coindu1598213697e_LNil @ A ) )
& ( Ys
= ( coindu1598213697e_LNil @ A ) ) ) ) ).
% lappend_eq_LNil_iff
thf(fact_58_lappend__snocL1__conv__LCons2,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Y: A,Ys: coindu1593790203_llist @ A] :
( ( coindu268472904append @ A @ ( coindu268472904append @ A @ Xs @ ( coindu1121789889_LCons @ A @ Y @ ( coindu1598213697e_LNil @ A ) ) ) @ Ys )
= ( coindu268472904append @ A @ Xs @ ( coindu1121789889_LCons @ A @ Y @ Ys ) ) ) ).
% lappend_snocL1_conv_LCons2
thf(fact_59_lfinite__induct,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,P: ( coindu1593790203_llist @ A ) > $o] :
( ( coindu1213758845finite @ A @ Xs )
=> ( ! [Xs2: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Xs2 )
=> ( P @ Xs2 ) )
=> ( ! [Xs2: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ Xs2 )
=> ( ~ ( coindu335574135_lnull @ A @ Xs2 )
=> ( ( P @ ( coindu1047225960le_ltl @ A @ Xs2 ) )
=> ( P @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% lfinite_induct
thf(fact_60_finite__code,axiom,
! [A: $tType] :
( ( finite_finite @ A @ ( type2 @ A ) )
=> ( ( finite_finite2 @ A )
= ( ^ [A5: set @ A] : $true ) ) ) ).
% finite_code
thf(fact_61_lhd__LCons__ltl,axiom,
! [A: $tType,Llist: coindu1593790203_llist @ A] :
( ~ ( coindu335574135_lnull @ A @ Llist )
=> ( ( coindu1121789889_LCons @ A @ ( coindu1046438764le_lhd @ A @ Llist ) @ ( coindu1047225960le_ltl @ A @ Llist ) )
= Llist ) ) ).
% lhd_LCons_ltl
thf(fact_62_eq__LConsD,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Y: A,Ys: coindu1593790203_llist @ A] :
( ( Xs
= ( coindu1121789889_LCons @ A @ Y @ Ys ) )
=> ( ( Xs
!= ( coindu1598213697e_LNil @ A ) )
& ( ( coindu1046438764le_lhd @ A @ Xs )
= Y )
& ( ( coindu1047225960le_ltl @ A @ Xs )
= Ys ) ) ) ).
% eq_LConsD
thf(fact_63_llist_Oexhaust__sel,axiom,
! [A: $tType,Llist: coindu1593790203_llist @ A] :
( ( Llist
!= ( coindu1598213697e_LNil @ A ) )
=> ( Llist
= ( coindu1121789889_LCons @ A @ ( coindu1046438764le_lhd @ A @ Llist ) @ ( coindu1047225960le_ltl @ A @ Llist ) ) ) ) ).
% llist.exhaust_sel
thf(fact_64_llist__set__induct,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A,P: A > ( coindu1593790203_llist @ A ) > $o] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) )
=> ( ! [Xs2: coindu1593790203_llist @ A] :
( ~ ( coindu335574135_lnull @ A @ Xs2 )
=> ( P @ ( coindu1046438764le_lhd @ A @ Xs2 ) @ Xs2 ) )
=> ( ! [Xs2: coindu1593790203_llist @ A,Y2: A] :
( ~ ( coindu335574135_lnull @ A @ Xs2 )
=> ( ( member @ A @ Y2 @ ( coindu1112613586e_lset @ A @ ( coindu1047225960le_ltl @ A @ Xs2 ) ) )
=> ( ( P @ Y2 @ ( coindu1047225960le_ltl @ A @ Xs2 ) )
=> ( P @ Y2 @ Xs2 ) ) ) )
=> ( P @ X @ Xs ) ) ) ) ).
% llist_set_induct
thf(fact_65_lmember__code_I1_J,axiom,
! [A: $tType,X: A] :
~ ( coindu567634248member @ A @ X @ ( coindu1598213697e_LNil @ A ) ) ).
% lmember_code(1)
thf(fact_66_lset__lmember,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) )
= ( coindu567634248member @ A @ X @ Xs ) ) ).
% lset_lmember
thf(fact_67_lhd__lappend,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu1046438764le_lhd @ A @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( coindu1046438764le_lhd @ A @ Ys ) ) )
& ( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu1046438764le_lhd @ A @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( coindu1046438764le_lhd @ A @ Xs ) ) ) ) ).
% lhd_lappend
thf(fact_68_lmember__code_I2_J,axiom,
! [A: $tType,X: A,Y: A,Ys: coindu1593790203_llist @ A] :
( ( coindu567634248member @ A @ X @ ( coindu1121789889_LCons @ A @ Y @ Ys ) )
= ( ( X = Y )
| ( coindu567634248member @ A @ X @ Ys ) ) ) ).
% lmember_code(2)
thf(fact_69_finite__set__choice,axiom,
! [B: $tType,A: $tType,A4: set @ A,P: A > B > $o] :
( ( finite_finite2 @ A @ A4 )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ? [X1: B] : ( P @ X2 @ X1 ) )
=> ? [F2: A > B] :
! [X6: A] :
( ( member @ A @ X6 @ A4 )
=> ( P @ X6 @ ( F2 @ X6 ) ) ) ) ) ).
% finite_set_choice
thf(fact_70_finite,axiom,
! [A: $tType] :
( ( finite_finite @ A @ ( type2 @ A ) )
=> ! [A4: set @ A] : ( finite_finite2 @ A @ A4 ) ) ).
% finite
thf(fact_71_lhd__LCons,axiom,
! [A: $tType,X21: A,X22: coindu1593790203_llist @ A] :
( ( coindu1046438764le_lhd @ A @ ( coindu1121789889_LCons @ A @ X21 @ X22 ) )
= X21 ) ).
% lhd_LCons
thf(fact_72_llist_Oset__sel_I1_J,axiom,
! [A: $tType,A2: coindu1593790203_llist @ A] :
( ~ ( coindu335574135_lnull @ A @ A2 )
=> ( member @ A @ ( coindu1046438764le_lhd @ A @ A2 ) @ ( coindu1112613586e_lset @ A @ A2 ) ) ) ).
% llist.set_sel(1)
thf(fact_73_llist_Ocoinduct__strong,axiom,
! [A: $tType,R: ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o,Llist: coindu1593790203_llist @ A,Llist3: coindu1593790203_llist @ A] :
( ( R @ Llist @ Llist3 )
=> ( ! [Llist4: coindu1593790203_llist @ A,Llist5: coindu1593790203_llist @ A] :
( ( R @ Llist4 @ Llist5 )
=> ( ( ( coindu335574135_lnull @ A @ Llist4 )
= ( coindu335574135_lnull @ A @ Llist5 ) )
& ( ~ ( coindu335574135_lnull @ A @ Llist4 )
=> ( ~ ( coindu335574135_lnull @ A @ Llist5 )
=> ( ( ( coindu1046438764le_lhd @ A @ Llist4 )
= ( coindu1046438764le_lhd @ A @ Llist5 ) )
& ( ( R @ ( coindu1047225960le_ltl @ A @ Llist4 ) @ ( coindu1047225960le_ltl @ A @ Llist5 ) )
| ( ( coindu1047225960le_ltl @ A @ Llist4 )
= ( coindu1047225960le_ltl @ A @ Llist5 ) ) ) ) ) ) ) )
=> ( Llist = Llist3 ) ) ) ).
% llist.coinduct_strong
thf(fact_74_llist_Ocoinduct,axiom,
! [A: $tType,R: ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o,Llist: coindu1593790203_llist @ A,Llist3: coindu1593790203_llist @ A] :
( ( R @ Llist @ Llist3 )
=> ( ! [Llist4: coindu1593790203_llist @ A,Llist5: coindu1593790203_llist @ A] :
( ( R @ Llist4 @ Llist5 )
=> ( ( ( coindu335574135_lnull @ A @ Llist4 )
= ( coindu335574135_lnull @ A @ Llist5 ) )
& ( ~ ( coindu335574135_lnull @ A @ Llist4 )
=> ( ~ ( coindu335574135_lnull @ A @ Llist5 )
=> ( ( ( coindu1046438764le_lhd @ A @ Llist4 )
= ( coindu1046438764le_lhd @ A @ Llist5 ) )
& ( R @ ( coindu1047225960le_ltl @ A @ Llist4 ) @ ( coindu1047225960le_ltl @ A @ Llist5 ) ) ) ) ) ) )
=> ( Llist = Llist3 ) ) ) ).
% llist.coinduct
thf(fact_75_llist_Oexpand,axiom,
! [A: $tType,Llist: coindu1593790203_llist @ A,Llist3: coindu1593790203_llist @ A] :
( ( ( coindu335574135_lnull @ A @ Llist )
= ( coindu335574135_lnull @ A @ Llist3 ) )
=> ( ( ~ ( coindu335574135_lnull @ A @ Llist )
=> ( ~ ( coindu335574135_lnull @ A @ Llist3 )
=> ( ( ( coindu1046438764le_lhd @ A @ Llist )
= ( coindu1046438764le_lhd @ A @ Llist3 ) )
& ( ( coindu1047225960le_ltl @ A @ Llist )
= ( coindu1047225960le_ltl @ A @ Llist3 ) ) ) ) )
=> ( Llist = Llist3 ) ) ) ).
% llist.expand
thf(fact_76_llist_Osplit__sel__asm,axiom,
! [B: $tType,A: $tType,P: B > $o,F1: B,F22: A > ( coindu1593790203_llist @ A ) > B,Llist: coindu1593790203_llist @ A] :
( ( P @ ( coindu882539134_llist @ B @ A @ F1 @ F22 @ Llist ) )
= ( ~ ( ( ( Llist
= ( coindu1598213697e_LNil @ A ) )
& ~ ( P @ F1 ) )
| ( ( Llist
= ( coindu1121789889_LCons @ A @ ( coindu1046438764le_lhd @ A @ Llist ) @ ( coindu1047225960le_ltl @ A @ Llist ) ) )
& ~ ( P @ ( F22 @ ( coindu1046438764le_lhd @ A @ Llist ) @ ( coindu1047225960le_ltl @ A @ Llist ) ) ) ) ) ) ) ).
% llist.split_sel_asm
thf(fact_77_llist_Osplit__sel,axiom,
! [B: $tType,A: $tType,P: B > $o,F1: B,F22: A > ( coindu1593790203_llist @ A ) > B,Llist: coindu1593790203_llist @ A] :
( ( P @ ( coindu882539134_llist @ B @ A @ F1 @ F22 @ Llist ) )
= ( ( ( Llist
= ( coindu1598213697e_LNil @ A ) )
=> ( P @ F1 ) )
& ( ( Llist
= ( coindu1121789889_LCons @ A @ ( coindu1046438764le_lhd @ A @ Llist ) @ ( coindu1047225960le_ltl @ A @ Llist ) ) )
=> ( P @ ( F22 @ ( coindu1046438764le_lhd @ A @ Llist ) @ ( coindu1047225960le_ltl @ A @ Llist ) ) ) ) ) ) ).
% llist.split_sel
thf(fact_78_unfold__llist__id,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu1599971794_llist @ ( coindu1593790203_llist @ A ) @ A @ ( coindu335574135_lnull @ A ) @ ( coindu1046438764le_lhd @ A ) @ ( coindu1047225960le_ltl @ A ) @ Xs )
= Xs ) ).
% unfold_llist_id
thf(fact_79_llist_Ocase__eq__if,axiom,
! [A: $tType,B: $tType] :
( ( coindu882539134_llist @ B @ A )
= ( ^ [F12: B,F23: A > ( coindu1593790203_llist @ A ) > B,Llist2: coindu1593790203_llist @ A] : ( if @ B @ ( coindu335574135_lnull @ A @ Llist2 ) @ F12 @ ( F23 @ ( coindu1046438764le_lhd @ A @ Llist2 ) @ ( coindu1047225960le_ltl @ A @ Llist2 ) ) ) ) ) ).
% llist.case_eq_if
thf(fact_80_llist_Orel__coinduct,axiom,
! [A: $tType,B: $tType,P: ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ B ) > $o,X: coindu1593790203_llist @ A,Y: coindu1593790203_llist @ B,R: A > B > $o] :
( ( P @ X @ Y )
=> ( ! [Llist4: coindu1593790203_llist @ A,Llist5: coindu1593790203_llist @ B] :
( ( P @ Llist4 @ Llist5 )
=> ( ( ( coindu335574135_lnull @ A @ Llist4 )
= ( coindu335574135_lnull @ B @ Llist5 ) )
& ( ~ ( coindu335574135_lnull @ A @ Llist4 )
=> ( ~ ( coindu335574135_lnull @ B @ Llist5 )
=> ( ( R @ ( coindu1046438764le_lhd @ A @ Llist4 ) @ ( coindu1046438764le_lhd @ B @ Llist5 ) )
& ( P @ ( coindu1047225960le_ltl @ A @ Llist4 ) @ ( coindu1047225960le_ltl @ B @ Llist5 ) ) ) ) ) ) )
=> ( coindu987187967t_all2 @ A @ B @ R @ X @ Y ) ) ) ).
% llist.rel_coinduct
thf(fact_81_llist_Orel__sel,axiom,
! [B: $tType,A: $tType] :
( ( coindu987187967t_all2 @ A @ B )
= ( ^ [R2: A > B > $o,A3: coindu1593790203_llist @ A,B2: coindu1593790203_llist @ B] :
( ( ( coindu335574135_lnull @ A @ A3 )
= ( coindu335574135_lnull @ B @ B2 ) )
& ( ~ ( coindu335574135_lnull @ A @ A3 )
=> ( ~ ( coindu335574135_lnull @ B @ B2 )
=> ( ( R2 @ ( coindu1046438764le_lhd @ A @ A3 ) @ ( coindu1046438764le_lhd @ B @ B2 ) )
& ( coindu987187967t_all2 @ A @ B @ R2 @ ( coindu1047225960le_ltl @ A @ A3 ) @ ( coindu1047225960le_ltl @ B @ B2 ) ) ) ) ) ) ) ) ).
% llist.rel_sel
thf(fact_82_gen__lset__code_I1_J,axiom,
! [A: $tType,A4: set @ A] :
( ( coindu1928975208n_lset @ A @ A4 @ ( coindu1598213697e_LNil @ A ) )
= A4 ) ).
% gen_lset_code(1)
thf(fact_83_llist_Orel__inject_I2_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X21: A,X22: coindu1593790203_llist @ A,Y21: B,Y22: coindu1593790203_llist @ B] :
( ( coindu987187967t_all2 @ A @ B @ R @ ( coindu1121789889_LCons @ A @ X21 @ X22 ) @ ( coindu1121789889_LCons @ B @ Y21 @ Y22 ) )
= ( ( R @ X21 @ Y21 )
& ( coindu987187967t_all2 @ A @ B @ R @ X22 @ Y22 ) ) ) ).
% llist.rel_inject(2)
thf(fact_84_unfold__llist_Odisc__iff_I2_J,axiom,
! [B: $tType,A: $tType,P2: A > $o,G21: A > B,G22: A > A,A2: A] :
( ( ~ ( coindu335574135_lnull @ B @ ( coindu1599971794_llist @ A @ B @ P2 @ G21 @ G22 @ A2 ) ) )
= ( ~ ( P2 @ A2 ) ) ) ).
% unfold_llist.disc_iff(2)
thf(fact_85_unfold__llist_Odisc__iff_I1_J,axiom,
! [B: $tType,A: $tType,P2: A > $o,G21: A > B,G22: A > A,A2: A] :
( ( coindu335574135_lnull @ B @ ( coindu1599971794_llist @ A @ B @ P2 @ G21 @ G22 @ A2 ) )
= ( P2 @ A2 ) ) ).
% unfold_llist.disc_iff(1)
thf(fact_86_unfold__llist__eq__LCons,axiom,
! [A: $tType,B: $tType,IS_LNIL: B > $o,LHD: B > A,LTL: B > B,B3: B,X: A,Xs: coindu1593790203_llist @ A] :
( ( ( coindu1599971794_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ B3 )
= ( coindu1121789889_LCons @ A @ X @ Xs ) )
= ( ~ ( IS_LNIL @ B3 )
& ( X
= ( LHD @ B3 ) )
& ( Xs
= ( coindu1599971794_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ ( LTL @ B3 ) ) ) ) ) ).
% unfold_llist_eq_LCons
thf(fact_87_llist_Orel__eq,axiom,
! [A: $tType] :
( ( coindu987187967t_all2 @ A @ A
@ ^ [Y3: A,Z: A] : Y3 = Z )
= ( ^ [Y3: coindu1593790203_llist @ A,Z: coindu1593790203_llist @ A] : Y3 = Z ) ) ).
% llist.rel_eq
thf(fact_88_llist_Orel__refl,axiom,
! [B: $tType,Ra: B > B > $o,X: coindu1593790203_llist @ B] :
( ! [X2: B] : ( Ra @ X2 @ X2 )
=> ( coindu987187967t_all2 @ B @ B @ Ra @ X @ X ) ) ).
% llist.rel_refl
thf(fact_89_llist_Orel__intros_I2_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X21: A,Y21: B,X22: coindu1593790203_llist @ A,Y22: coindu1593790203_llist @ B] :
( ( R @ X21 @ Y21 )
=> ( ( coindu987187967t_all2 @ A @ B @ R @ X22 @ Y22 )
=> ( coindu987187967t_all2 @ A @ B @ R @ ( coindu1121789889_LCons @ A @ X21 @ X22 ) @ ( coindu1121789889_LCons @ B @ Y21 @ Y22 ) ) ) ) ).
% llist.rel_intros(2)
thf(fact_90_llist_Orel__cong,axiom,
! [A: $tType,B: $tType,X: coindu1593790203_llist @ A,Ya: coindu1593790203_llist @ A,Y: coindu1593790203_llist @ B,Xa: coindu1593790203_llist @ B,R: A > B > $o,Ra: A > B > $o] :
( ( X = Ya )
=> ( ( Y = Xa )
=> ( ! [Z3: A,Yb: B] :
( ( member @ A @ Z3 @ ( coindu1112613586e_lset @ A @ Ya ) )
=> ( ( member @ B @ Yb @ ( coindu1112613586e_lset @ B @ Xa ) )
=> ( ( R @ Z3 @ Yb )
= ( Ra @ Z3 @ Yb ) ) ) )
=> ( ( coindu987187967t_all2 @ A @ B @ R @ X @ Y )
= ( coindu987187967t_all2 @ A @ B @ Ra @ Ya @ Xa ) ) ) ) ) ).
% llist.rel_cong
thf(fact_91_llist_Orel__mono__strong,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X: coindu1593790203_llist @ A,Y: coindu1593790203_llist @ B,Ra: A > B > $o] :
( ( coindu987187967t_all2 @ A @ B @ R @ X @ Y )
=> ( ! [Z3: A,Yb: B] :
( ( member @ A @ Z3 @ ( coindu1112613586e_lset @ A @ X ) )
=> ( ( member @ B @ Yb @ ( coindu1112613586e_lset @ B @ Y ) )
=> ( ( R @ Z3 @ Yb )
=> ( Ra @ Z3 @ Yb ) ) ) )
=> ( coindu987187967t_all2 @ A @ B @ Ra @ X @ Y ) ) ) ).
% llist.rel_mono_strong
thf(fact_92_llist_Orel__refl__strong,axiom,
! [A: $tType,X: coindu1593790203_llist @ A,Ra: A > A > $o] :
( ! [Z3: A] :
( ( member @ A @ Z3 @ ( coindu1112613586e_lset @ A @ X ) )
=> ( Ra @ Z3 @ Z3 ) )
=> ( coindu987187967t_all2 @ A @ A @ Ra @ X @ X ) ) ).
% llist.rel_refl_strong
thf(fact_93_llist_Octr__transfer_I1_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o] : ( coindu987187967t_all2 @ A @ B @ R @ ( coindu1598213697e_LNil @ A ) @ ( coindu1598213697e_LNil @ B ) ) ).
% llist.ctr_transfer(1)
thf(fact_94_unfold__llist_Odisc_I2_J,axiom,
! [B: $tType,A: $tType,P2: A > $o,A2: A,G21: A > B,G22: A > A] :
( ~ ( P2 @ A2 )
=> ~ ( coindu335574135_lnull @ B @ ( coindu1599971794_llist @ A @ B @ P2 @ G21 @ G22 @ A2 ) ) ) ).
% unfold_llist.disc(2)
thf(fact_95_unfold__llist_Odisc_I1_J,axiom,
! [B: $tType,A: $tType,P2: A > $o,A2: A,G21: A > B,G22: A > A] :
( ( P2 @ A2 )
=> ( coindu335574135_lnull @ B @ ( coindu1599971794_llist @ A @ B @ P2 @ G21 @ G22 @ A2 ) ) ) ).
% unfold_llist.disc(1)
thf(fact_96_unfold__llist_Octr_I2_J,axiom,
! [B: $tType,A: $tType,P2: A > $o,A2: A,G21: A > B,G22: A > A] :
( ~ ( P2 @ A2 )
=> ( ( coindu1599971794_llist @ A @ B @ P2 @ G21 @ G22 @ A2 )
= ( coindu1121789889_LCons @ B @ ( G21 @ A2 ) @ ( coindu1599971794_llist @ A @ B @ P2 @ G21 @ G22 @ ( G22 @ A2 ) ) ) ) ) ).
% unfold_llist.ctr(2)
thf(fact_97_unfold__llist_Octr_I1_J,axiom,
! [A: $tType,B: $tType,P2: A > $o,A2: A,G21: A > B,G22: A > A] :
( ( P2 @ A2 )
=> ( ( coindu1599971794_llist @ A @ B @ P2 @ G21 @ G22 @ A2 )
= ( coindu1598213697e_LNil @ B ) ) ) ).
% unfold_llist.ctr(1)
thf(fact_98_unfold__llist_Osimps_I4_J,axiom,
! [B: $tType,A: $tType,P2: A > $o,A2: A,G21: A > B,G22: A > A] :
( ~ ( P2 @ A2 )
=> ( ( coindu1047225960le_ltl @ B @ ( coindu1599971794_llist @ A @ B @ P2 @ G21 @ G22 @ A2 ) )
= ( coindu1599971794_llist @ A @ B @ P2 @ G21 @ G22 @ ( G22 @ A2 ) ) ) ) ).
% unfold_llist.simps(4)
thf(fact_99_unfold__llist_Osimps_I3_J,axiom,
! [B: $tType,A: $tType,P2: A > $o,A2: A,G21: A > B,G22: A > A] :
( ~ ( P2 @ A2 )
=> ( ( coindu1046438764le_lhd @ B @ ( coindu1599971794_llist @ A @ B @ P2 @ G21 @ G22 @ A2 ) )
= ( G21 @ A2 ) ) ) ).
% unfold_llist.simps(3)
thf(fact_100_llist_Osimps_I5_J,axiom,
! [B: $tType,A: $tType,F1: B,F22: A > ( coindu1593790203_llist @ A ) > B,X21: A,X22: coindu1593790203_llist @ A] :
( ( coindu882539134_llist @ B @ A @ F1 @ F22 @ ( coindu1121789889_LCons @ A @ X21 @ X22 ) )
= ( F22 @ X21 @ X22 ) ) ).
% llist.simps(5)
thf(fact_101_llist_Osimps_I4_J,axiom,
! [A: $tType,B: $tType,F1: B,F22: A > ( coindu1593790203_llist @ A ) > B] :
( ( coindu882539134_llist @ B @ A @ F1 @ F22 @ ( coindu1598213697e_LNil @ A ) )
= F1 ) ).
% llist.simps(4)
thf(fact_102_llist_Orel__distinct_I2_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o,Y21: A,Y22: coindu1593790203_llist @ A] :
~ ( coindu987187967t_all2 @ A @ B @ R @ ( coindu1121789889_LCons @ A @ Y21 @ Y22 ) @ ( coindu1598213697e_LNil @ B ) ) ).
% llist.rel_distinct(2)
thf(fact_103_llist_Orel__distinct_I1_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o,Y21: B,Y22: coindu1593790203_llist @ B] :
~ ( coindu987187967t_all2 @ A @ B @ R @ ( coindu1598213697e_LNil @ A ) @ ( coindu1121789889_LCons @ B @ Y21 @ Y22 ) ) ).
% llist.rel_distinct(1)
thf(fact_104_llist_Orel__cases,axiom,
! [A: $tType,B: $tType,R: A > B > $o,A2: coindu1593790203_llist @ A,B3: coindu1593790203_llist @ B] :
( ( coindu987187967t_all2 @ A @ B @ R @ A2 @ B3 )
=> ( ( ( A2
= ( coindu1598213697e_LNil @ A ) )
=> ( B3
!= ( coindu1598213697e_LNil @ B ) ) )
=> ~ ! [X12: A,X23: coindu1593790203_llist @ A] :
( ( A2
= ( coindu1121789889_LCons @ A @ X12 @ X23 ) )
=> ! [Y1: B,Y23: coindu1593790203_llist @ B] :
( ( B3
= ( coindu1121789889_LCons @ B @ Y1 @ Y23 ) )
=> ( ( R @ X12 @ Y1 )
=> ~ ( coindu987187967t_all2 @ A @ B @ R @ X23 @ Y23 ) ) ) ) ) ) ).
% llist.rel_cases
thf(fact_105_unfold__llist_Ocode,axiom,
! [B: $tType,A: $tType] :
( ( coindu1599971794_llist @ A @ B )
= ( ^ [P3: A > $o,G212: A > B,G222: A > A,A3: A] : ( if @ ( coindu1593790203_llist @ B ) @ ( P3 @ A3 ) @ ( coindu1598213697e_LNil @ B ) @ ( coindu1121789889_LCons @ B @ ( G212 @ A3 ) @ ( coindu1599971794_llist @ A @ B @ P3 @ G212 @ G222 @ ( G222 @ A3 ) ) ) ) ) ) ).
% unfold_llist.code
thf(fact_106_ltl__unfold__llist,axiom,
! [A: $tType,B: $tType,IS_LNIL: B > $o,A2: B,LHD: B > A,LTL: B > B] :
( ( ( IS_LNIL @ A2 )
=> ( ( coindu1047225960le_ltl @ A @ ( coindu1599971794_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ A2 ) )
= ( coindu1598213697e_LNil @ A ) ) )
& ( ~ ( IS_LNIL @ A2 )
=> ( ( coindu1047225960le_ltl @ A @ ( coindu1599971794_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ A2 ) )
= ( coindu1599971794_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ ( LTL @ A2 ) ) ) ) ) ).
% ltl_unfold_llist
thf(fact_107_llist_Ocorec__code,axiom,
! [A: $tType,C: $tType] :
( ( coindu1244876290_llist @ C @ A )
= ( ^ [P3: C > $o,G212: C > A,Q22: C > $o,G221: C > ( coindu1593790203_llist @ A ),G2222: C > C,A3: C] : ( if @ ( coindu1593790203_llist @ A ) @ ( P3 @ A3 ) @ ( coindu1598213697e_LNil @ A ) @ ( coindu1121789889_LCons @ A @ ( G212 @ A3 ) @ ( if @ ( coindu1593790203_llist @ A ) @ ( Q22 @ A3 ) @ ( G221 @ A3 ) @ ( coindu1244876290_llist @ C @ A @ P3 @ G212 @ Q22 @ G221 @ G2222 @ ( G2222 @ A3 ) ) ) ) ) ) ) ).
% llist.corec_code
thf(fact_108_lset__eq__empty,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( ( coindu1112613586e_lset @ A @ Xs )
= ( bot_bot @ ( set @ A ) ) )
= ( coindu335574135_lnull @ A @ Xs ) ) ).
% lset_eq_empty
thf(fact_109_llist_Omap__sel_I1_J,axiom,
! [B: $tType,A: $tType,A2: coindu1593790203_llist @ A,F: A > B] :
( ~ ( coindu335574135_lnull @ A @ A2 )
=> ( ( coindu1046438764le_lhd @ B @ ( coindu1062782156e_lmap @ A @ B @ F @ A2 ) )
= ( F @ ( coindu1046438764le_lhd @ A @ A2 ) ) ) ) ).
% llist.map_sel(1)
thf(fact_110_lset__code,axiom,
! [A: $tType] :
( ( coindu1112613586e_lset @ A )
= ( coindu1928975208n_lset @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% lset_code
thf(fact_111_llist_Osimps_I19_J,axiom,
! [A: $tType,X21: A,X22: coindu1593790203_llist @ A] :
( ( coindu1112613586e_lset @ A @ ( coindu1121789889_LCons @ A @ X21 @ X22 ) )
= ( insert @ A @ X21 @ ( coindu1112613586e_lset @ A @ X22 ) ) ) ).
% llist.simps(19)
thf(fact_112_lset__LCons,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A] :
( ( coindu1112613586e_lset @ A @ ( coindu1121789889_LCons @ A @ X @ Xs ) )
= ( insert @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) ) ) ).
% lset_LCons
thf(fact_113_finite__insert,axiom,
! [A: $tType,A2: A,A4: set @ A] :
( ( finite_finite2 @ A @ ( insert @ A @ A2 @ A4 ) )
= ( finite_finite2 @ A @ A4 ) ) ).
% finite_insert
thf(fact_114_llist_Omap__disc__iff,axiom,
! [B: $tType,A: $tType,F: A > B,A2: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ B @ ( coindu1062782156e_lmap @ A @ B @ F @ A2 ) )
= ( coindu335574135_lnull @ A @ A2 ) ) ).
% llist.map_disc_iff
thf(fact_115_ltl__lmap,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: coindu1593790203_llist @ B] :
( ( coindu1047225960le_ltl @ A @ ( coindu1062782156e_lmap @ B @ A @ F @ Xs ) )
= ( coindu1062782156e_lmap @ B @ A @ F @ ( coindu1047225960le_ltl @ B @ Xs ) ) ) ).
% ltl_lmap
thf(fact_116_llist_Ocorec__disc__iff_I1_J,axiom,
! [A: $tType,C: $tType,P2: C > $o,G21: C > A,Q222: C > $o,G2212: C > ( coindu1593790203_llist @ A ),G2223: C > C,A2: C] :
( ( coindu335574135_lnull @ A @ ( coindu1244876290_llist @ C @ A @ P2 @ G21 @ Q222 @ G2212 @ G2223 @ A2 ) )
= ( P2 @ A2 ) ) ).
% llist.corec_disc_iff(1)
thf(fact_117_llist_Ocorec__disc__iff_I2_J,axiom,
! [A: $tType,C: $tType,P2: C > $o,G21: C > A,Q222: C > $o,G2212: C > ( coindu1593790203_llist @ A ),G2223: C > C,A2: C] :
( ( ~ ( coindu335574135_lnull @ A @ ( coindu1244876290_llist @ C @ A @ P2 @ G21 @ Q222 @ G2212 @ G2223 @ A2 ) ) )
= ( ~ ( P2 @ A2 ) ) ) ).
% llist.corec_disc_iff(2)
thf(fact_118_infinite__finite__induct,axiom,
! [A: $tType,P: ( set @ A ) > $o,A4: set @ A] :
( ! [A6: set @ A] :
( ~ ( finite_finite2 @ A @ A6 )
=> ( P @ A6 ) )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [X2: A,F3: set @ A] :
( ( finite_finite2 @ A @ F3 )
=> ( ~ ( member @ A @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert @ A @ X2 @ F3 ) ) ) ) )
=> ( P @ A4 ) ) ) ) ).
% infinite_finite_induct
thf(fact_119_finite__ne__induct,axiom,
! [A: $tType,F4: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F4 )
=> ( ( F4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X2: A] : ( P @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ! [X2: A,F3: set @ A] :
( ( finite_finite2 @ A @ F3 )
=> ( ( F3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ~ ( member @ A @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert @ A @ X2 @ F3 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_120_finite_Oinducts,axiom,
! [A: $tType,X: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ X )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A6: set @ A,A7: A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( P @ A6 )
=> ( P @ ( insert @ A @ A7 @ A6 ) ) ) )
=> ( P @ X ) ) ) ) ).
% finite.inducts
thf(fact_121_finite__induct,axiom,
! [A: $tType,F4: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F4 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [X2: A,F3: set @ A] :
( ( finite_finite2 @ A @ F3 )
=> ( ~ ( member @ A @ X2 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert @ A @ X2 @ F3 ) ) ) ) )
=> ( P @ F4 ) ) ) ) ).
% finite_induct
thf(fact_122_finite_Osimps,axiom,
! [A: $tType] :
( ( finite_finite2 @ A )
= ( ^ [A3: set @ A] :
( ( A3
= ( bot_bot @ ( set @ A ) ) )
| ? [A5: set @ A,B2: A] :
( ( A3
= ( insert @ A @ B2 @ A5 ) )
& ( finite_finite2 @ A @ A5 ) ) ) ) ) ).
% finite.simps
thf(fact_123_finite_Ocases,axiom,
! [A: $tType,A2: set @ A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [A6: set @ A] :
( ? [A7: A] :
( A2
= ( insert @ A @ A7 @ A6 ) )
=> ~ ( finite_finite2 @ A @ A6 ) ) ) ) ).
% finite.cases
thf(fact_124_finite_OinsertI,axiom,
! [A: $tType,A4: set @ A,A2: A] :
( ( finite_finite2 @ A @ A4 )
=> ( finite_finite2 @ A @ ( insert @ A @ A2 @ A4 ) ) ) ).
% finite.insertI
thf(fact_125_infinite__imp__nonempty,axiom,
! [A: $tType,S: set @ A] :
( ~ ( finite_finite2 @ A @ S )
=> ( S
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% infinite_imp_nonempty
thf(fact_126_finite_OemptyI,axiom,
! [A: $tType] : ( finite_finite2 @ A @ ( bot_bot @ ( set @ A ) ) ) ).
% finite.emptyI
thf(fact_127_lmap__eq__LCons__conv,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: coindu1593790203_llist @ B,Y: A,Ys: coindu1593790203_llist @ A] :
( ( ( coindu1062782156e_lmap @ B @ A @ F @ Xs )
= ( coindu1121789889_LCons @ A @ Y @ Ys ) )
= ( ? [X3: B,Xs5: coindu1593790203_llist @ B] :
( ( Xs
= ( coindu1121789889_LCons @ B @ X3 @ Xs5 ) )
& ( Y
= ( F @ X3 ) )
& ( Ys
= ( coindu1062782156e_lmap @ B @ A @ F @ Xs5 ) ) ) ) ) ).
% lmap_eq_LCons_conv
thf(fact_128_llist_Osimps_I13_J,axiom,
! [B: $tType,A: $tType,F: A > B,X21: A,X22: coindu1593790203_llist @ A] :
( ( coindu1062782156e_lmap @ A @ B @ F @ ( coindu1121789889_LCons @ A @ X21 @ X22 ) )
= ( coindu1121789889_LCons @ B @ ( F @ X21 ) @ ( coindu1062782156e_lmap @ A @ B @ F @ X22 ) ) ) ).
% llist.simps(13)
thf(fact_129_llist_Oinj__map__strong,axiom,
! [B: $tType,A: $tType,X: coindu1593790203_llist @ A,Xa: coindu1593790203_llist @ A,F: A > B,Fa: A > B] :
( ! [Z3: A,Za: A] :
( ( member @ A @ Z3 @ ( coindu1112613586e_lset @ A @ X ) )
=> ( ( member @ A @ Za @ ( coindu1112613586e_lset @ A @ Xa ) )
=> ( ( ( F @ Z3 )
= ( Fa @ Za ) )
=> ( Z3 = Za ) ) ) )
=> ( ( ( coindu1062782156e_lmap @ A @ B @ F @ X )
= ( coindu1062782156e_lmap @ A @ B @ Fa @ Xa ) )
=> ( X = Xa ) ) ) ).
% llist.inj_map_strong
thf(fact_130_llist_Omap__cong0,axiom,
! [B: $tType,A: $tType,X: coindu1593790203_llist @ A,F: A > B,G: A > B] :
( ! [Z3: A] :
( ( member @ A @ Z3 @ ( coindu1112613586e_lset @ A @ X ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( coindu1062782156e_lmap @ A @ B @ F @ X )
= ( coindu1062782156e_lmap @ A @ B @ G @ X ) ) ) ).
% llist.map_cong0
thf(fact_131_llist_Omap__cong,axiom,
! [B: $tType,A: $tType,X: coindu1593790203_llist @ A,Ya: coindu1593790203_llist @ A,F: A > B,G: A > B] :
( ( X = Ya )
=> ( ! [Z3: A] :
( ( member @ A @ Z3 @ ( coindu1112613586e_lset @ A @ Ya ) )
=> ( ( F @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( coindu1062782156e_lmap @ A @ B @ F @ X )
= ( coindu1062782156e_lmap @ A @ B @ G @ Ya ) ) ) ) ).
% llist.map_cong
thf(fact_132_lmap__eq__LNil,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: coindu1593790203_llist @ B] :
( ( ( coindu1062782156e_lmap @ B @ A @ F @ Xs )
= ( coindu1598213697e_LNil @ A ) )
= ( Xs
= ( coindu1598213697e_LNil @ B ) ) ) ).
% lmap_eq_LNil
thf(fact_133_LNil__eq__lmap,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: coindu1593790203_llist @ B] :
( ( ( coindu1598213697e_LNil @ A )
= ( coindu1062782156e_lmap @ B @ A @ F @ Xs ) )
= ( Xs
= ( coindu1598213697e_LNil @ B ) ) ) ).
% LNil_eq_lmap
thf(fact_134_llist_Osimps_I12_J,axiom,
! [A: $tType,B: $tType,F: A > B] :
( ( coindu1062782156e_lmap @ A @ B @ F @ ( coindu1598213697e_LNil @ A ) )
= ( coindu1598213697e_LNil @ B ) ) ).
% llist.simps(12)
thf(fact_135_lmap__lappend__distrib,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: coindu1593790203_llist @ B,Ys: coindu1593790203_llist @ B] :
( ( coindu1062782156e_lmap @ B @ A @ F @ ( coindu268472904append @ B @ Xs @ Ys ) )
= ( coindu268472904append @ A @ ( coindu1062782156e_lmap @ B @ A @ F @ Xs ) @ ( coindu1062782156e_lmap @ B @ A @ F @ Ys ) ) ) ).
% lmap_lappend_distrib
thf(fact_136_llist_Ocorec__disc_I1_J,axiom,
! [A: $tType,C: $tType,P2: C > $o,A2: C,G21: C > A,Q222: C > $o,G2212: C > ( coindu1593790203_llist @ A ),G2223: C > C] :
( ( P2 @ A2 )
=> ( coindu335574135_lnull @ A @ ( coindu1244876290_llist @ C @ A @ P2 @ G21 @ Q222 @ G2212 @ G2223 @ A2 ) ) ) ).
% llist.corec_disc(1)
thf(fact_137_llist_Ocorec__disc_I2_J,axiom,
! [A: $tType,C: $tType,P2: C > $o,A2: C,G21: C > A,Q222: C > $o,G2212: C > ( coindu1593790203_llist @ A ),G2223: C > C] :
( ~ ( P2 @ A2 )
=> ~ ( coindu335574135_lnull @ A @ ( coindu1244876290_llist @ C @ A @ P2 @ G21 @ Q222 @ G2212 @ G2223 @ A2 ) ) ) ).
% llist.corec_disc(2)
thf(fact_138_llist_Ocorec_I2_J,axiom,
! [A: $tType,C: $tType,P2: C > $o,A2: C,G21: C > A,Q222: C > $o,G2212: C > ( coindu1593790203_llist @ A ),G2223: C > C] :
( ~ ( P2 @ A2 )
=> ( ( coindu1244876290_llist @ C @ A @ P2 @ G21 @ Q222 @ G2212 @ G2223 @ A2 )
= ( coindu1121789889_LCons @ A @ ( G21 @ A2 ) @ ( if @ ( coindu1593790203_llist @ A ) @ ( Q222 @ A2 ) @ ( G2212 @ A2 ) @ ( coindu1244876290_llist @ C @ A @ P2 @ G21 @ Q222 @ G2212 @ G2223 @ ( G2223 @ A2 ) ) ) ) ) ) ).
% llist.corec(2)
thf(fact_139_llist_Ocorec_I1_J,axiom,
! [C: $tType,A: $tType,P2: C > $o,A2: C,G21: C > A,Q222: C > $o,G2212: C > ( coindu1593790203_llist @ A ),G2223: C > C] :
( ( P2 @ A2 )
=> ( ( coindu1244876290_llist @ C @ A @ P2 @ G21 @ Q222 @ G2212 @ G2223 @ A2 )
= ( coindu1598213697e_LNil @ A ) ) ) ).
% llist.corec(1)
thf(fact_140_llist_Ocorec__sel_I2_J,axiom,
! [A: $tType,C: $tType,P2: C > $o,A2: C,Q222: C > $o,G21: C > A,G2212: C > ( coindu1593790203_llist @ A ),G2223: C > C] :
( ~ ( P2 @ A2 )
=> ( ( ( Q222 @ A2 )
=> ( ( coindu1047225960le_ltl @ A @ ( coindu1244876290_llist @ C @ A @ P2 @ G21 @ Q222 @ G2212 @ G2223 @ A2 ) )
= ( G2212 @ A2 ) ) )
& ( ~ ( Q222 @ A2 )
=> ( ( coindu1047225960le_ltl @ A @ ( coindu1244876290_llist @ C @ A @ P2 @ G21 @ Q222 @ G2212 @ G2223 @ A2 ) )
= ( coindu1244876290_llist @ C @ A @ P2 @ G21 @ Q222 @ G2212 @ G2223 @ ( G2223 @ A2 ) ) ) ) ) ) ).
% llist.corec_sel(2)
thf(fact_141_llist_Ocorec__sel_I1_J,axiom,
! [A: $tType,C: $tType,P2: C > $o,A2: C,G21: C > A,Q222: C > $o,G2212: C > ( coindu1593790203_llist @ A ),G2223: C > C] :
( ~ ( P2 @ A2 )
=> ( ( coindu1046438764le_lhd @ A @ ( coindu1244876290_llist @ C @ A @ P2 @ G21 @ Q222 @ G2212 @ G2223 @ A2 ) )
= ( G21 @ A2 ) ) ) ).
% llist.corec_sel(1)
thf(fact_142_llist_Omap__sel_I2_J,axiom,
! [B: $tType,A: $tType,A2: coindu1593790203_llist @ A,F: A > B] :
( ~ ( coindu335574135_lnull @ A @ A2 )
=> ( ( coindu1047225960le_ltl @ B @ ( coindu1062782156e_lmap @ A @ B @ F @ A2 ) )
= ( coindu1062782156e_lmap @ A @ B @ F @ ( coindu1047225960le_ltl @ A @ A2 ) ) ) ) ).
% llist.map_sel(2)
thf(fact_143_lset__lnull,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu1112613586e_lset @ A @ Xs )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% lset_lnull
thf(fact_144_lset__LNil,axiom,
! [A: $tType] :
( ( coindu1112613586e_lset @ A @ ( coindu1598213697e_LNil @ A ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% lset_LNil
thf(fact_145_gen__lset__code_I2_J,axiom,
! [A: $tType,A4: set @ A,X: A,Xs: coindu1593790203_llist @ A] :
( ( coindu1928975208n_lset @ A @ A4 @ ( coindu1121789889_LCons @ A @ X @ Xs ) )
= ( coindu1928975208n_lset @ A @ ( insert @ A @ X @ A4 ) @ Xs ) ) ).
% gen_lset_code(2)
thf(fact_146_singletonI,axiom,
! [A: $tType,A2: A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singletonI
thf(fact_147_insert__absorb2,axiom,
! [A: $tType,X: A,A4: set @ A] :
( ( insert @ A @ X @ ( insert @ A @ X @ A4 ) )
= ( insert @ A @ X @ A4 ) ) ).
% insert_absorb2
thf(fact_148_insert__iff,axiom,
! [A: $tType,A2: A,B3: A,A4: set @ A] :
( ( member @ A @ A2 @ ( insert @ A @ B3 @ A4 ) )
= ( ( A2 = B3 )
| ( member @ A @ A2 @ A4 ) ) ) ).
% insert_iff
thf(fact_149_insertCI,axiom,
! [A: $tType,A2: A,B4: set @ A,B3: A] :
( ( ~ ( member @ A @ A2 @ B4 )
=> ( A2 = B3 ) )
=> ( member @ A @ A2 @ ( insert @ A @ B3 @ B4 ) ) ) ).
% insertCI
thf(fact_150_empty__Collect__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P ) )
= ( ! [X3: A] :
~ ( P @ X3 ) ) ) ).
% empty_Collect_eq
thf(fact_151_empty__iff,axiom,
! [A: $tType,C2: A] :
~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_152_all__not__in__conv,axiom,
! [A: $tType,A4: set @ A] :
( ( ! [X3: A] :
~ ( member @ A @ X3 @ A4 ) )
= ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_153_Collect__empty__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X3: A] :
~ ( P @ X3 ) ) ) ).
% Collect_empty_eq
thf(fact_154_emptyE,axiom,
! [A: $tType,A2: A] :
~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_155_equals0D,axiom,
! [A: $tType,A4: set @ A,A2: A] :
( ( A4
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A2 @ A4 ) ) ).
% equals0D
thf(fact_156_equals0I,axiom,
! [A: $tType,A4: set @ A] :
( ! [Y2: A] :
~ ( member @ A @ Y2 @ A4 )
=> ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_157_ex__in__conv,axiom,
! [A: $tType,A4: set @ A] :
( ( ? [X3: A] : ( member @ A @ X3 @ A4 ) )
= ( A4
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_158_insertE,axiom,
! [A: $tType,A2: A,B3: A,A4: set @ A] :
( ( member @ A @ A2 @ ( insert @ A @ B3 @ A4 ) )
=> ( ( A2 != B3 )
=> ( member @ A @ A2 @ A4 ) ) ) ).
% insertE
thf(fact_159_insertI1,axiom,
! [A: $tType,A2: A,B4: set @ A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ B4 ) ) ).
% insertI1
thf(fact_160_insertI2,axiom,
! [A: $tType,A2: A,B4: set @ A,B3: A] :
( ( member @ A @ A2 @ B4 )
=> ( member @ A @ A2 @ ( insert @ A @ B3 @ B4 ) ) ) ).
% insertI2
thf(fact_161_Set_Oset__insert,axiom,
! [A: $tType,X: A,A4: set @ A] :
( ( member @ A @ X @ A4 )
=> ~ ! [B5: set @ A] :
( ( A4
= ( insert @ A @ X @ B5 ) )
=> ( member @ A @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_162_insert__ident,axiom,
! [A: $tType,X: A,A4: set @ A,B4: set @ A] :
( ~ ( member @ A @ X @ A4 )
=> ( ~ ( member @ A @ X @ B4 )
=> ( ( ( insert @ A @ X @ A4 )
= ( insert @ A @ X @ B4 ) )
= ( A4 = B4 ) ) ) ) ).
% insert_ident
thf(fact_163_insert__absorb,axiom,
! [A: $tType,A2: A,A4: set @ A] :
( ( member @ A @ A2 @ A4 )
=> ( ( insert @ A @ A2 @ A4 )
= A4 ) ) ).
% insert_absorb
thf(fact_164_insert__eq__iff,axiom,
! [A: $tType,A2: A,A4: set @ A,B3: A,B4: set @ A] :
( ~ ( member @ A @ A2 @ A4 )
=> ( ~ ( member @ A @ B3 @ B4 )
=> ( ( ( insert @ A @ A2 @ A4 )
= ( insert @ A @ B3 @ B4 ) )
= ( ( ( A2 = B3 )
=> ( A4 = B4 ) )
& ( ( A2 != B3 )
=> ? [C3: set @ A] :
( ( A4
= ( insert @ A @ B3 @ C3 ) )
& ~ ( member @ A @ B3 @ C3 )
& ( B4
= ( insert @ A @ A2 @ C3 ) )
& ~ ( member @ A @ A2 @ C3 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_165_insert__commute,axiom,
! [A: $tType,X: A,Y: A,A4: set @ A] :
( ( insert @ A @ X @ ( insert @ A @ Y @ A4 ) )
= ( insert @ A @ Y @ ( insert @ A @ X @ A4 ) ) ) ).
% insert_commute
thf(fact_166_mk__disjoint__insert,axiom,
! [A: $tType,A2: A,A4: set @ A] :
( ( member @ A @ A2 @ A4 )
=> ? [B5: set @ A] :
( ( A4
= ( insert @ A @ A2 @ B5 ) )
& ~ ( member @ A @ A2 @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_167_singletonD,axiom,
! [A: $tType,B3: A,A2: A] :
( ( member @ A @ B3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( B3 = A2 ) ) ).
% singletonD
thf(fact_168_singleton__iff,axiom,
! [A: $tType,B3: A,A2: A] :
( ( member @ A @ B3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( B3 = A2 ) ) ).
% singleton_iff
thf(fact_169_doubleton__eq__iff,axiom,
! [A: $tType,A2: A,B3: A,C2: A,D: A] :
( ( ( insert @ A @ A2 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ C2 @ ( insert @ A @ D @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ( ( A2 = C2 )
& ( B3 = D ) )
| ( ( A2 = D )
& ( B3 = C2 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_170_insert__not__empty,axiom,
! [A: $tType,A2: A,A4: set @ A] :
( ( insert @ A @ A2 @ A4 )
!= ( bot_bot @ ( set @ A ) ) ) ).
% insert_not_empty
thf(fact_171_singleton__inject,axiom,
! [A: $tType,A2: A,B3: A] :
( ( ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( A2 = B3 ) ) ).
% singleton_inject
thf(fact_172_the__elem__eq,axiom,
! [A: $tType,X: A] :
( ( the_elem @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ).
% the_elem_eq
thf(fact_173_bot__apply,axiom,
! [C: $tType,D2: $tType] :
( ( bot @ C @ ( type2 @ C ) )
=> ( ( bot_bot @ ( D2 > C ) )
= ( ^ [X3: D2] : ( bot_bot @ C ) ) ) ) ).
% bot_apply
thf(fact_174_is__singletonI,axiom,
! [A: $tType,X: A] : ( is_singleton @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% is_singletonI
thf(fact_175_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_176_is__singleton__the__elem,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A5: set @ A] :
( A5
= ( insert @ A @ ( the_elem @ A @ A5 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_the_elem
thf(fact_177_is__singletonI_H,axiom,
! [A: $tType,A4: set @ A] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X2: A,Y2: A] :
( ( member @ A @ X2 @ A4 )
=> ( ( member @ A @ Y2 @ A4 )
=> ( X2 = Y2 ) ) )
=> ( is_singleton @ A @ A4 ) ) ) ).
% is_singletonI'
thf(fact_178_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B @ ( type2 @ B ) )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X3: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_179_is__singleton__def,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A5: set @ A] :
? [X3: A] :
( A5
= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_def
thf(fact_180_is__singletonE,axiom,
! [A: $tType,A4: set @ A] :
( ( is_singleton @ A @ A4 )
=> ~ ! [X2: A] :
( A4
!= ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% is_singletonE
thf(fact_181_Set_Ois__empty__def,axiom,
! [A: $tType] :
( ( is_empty @ A )
= ( ^ [A5: set @ A] :
( A5
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Set.is_empty_def
thf(fact_182_pairwise__singleton,axiom,
! [A: $tType,P: A > A > $o,A4: A] : ( pairwise @ A @ P @ ( insert @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% pairwise_singleton
thf(fact_183_bot__empty__eq,axiom,
! [A: $tType] :
( ( bot_bot @ ( A > $o ) )
= ( ^ [X3: A] : ( member @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% bot_empty_eq
thf(fact_184_pairwise__def,axiom,
! [A: $tType] :
( ( pairwise @ A )
= ( ^ [R2: A > A > $o,S2: set @ A] :
! [X3: A] :
( ( member @ A @ X3 @ S2 )
=> ! [Y4: A] :
( ( member @ A @ Y4 @ S2 )
=> ( ( X3 != Y4 )
=> ( R2 @ X3 @ Y4 ) ) ) ) ) ) ).
% pairwise_def
thf(fact_185_pairwise__empty,axiom,
! [A: $tType,P: A > A > $o] : ( pairwise @ A @ P @ ( bot_bot @ ( set @ A ) ) ) ).
% pairwise_empty
thf(fact_186_pairwise__insert,axiom,
! [A: $tType,R3: A > A > $o,X: A,S3: set @ A] :
( ( pairwise @ A @ R3 @ ( insert @ A @ X @ S3 ) )
= ( ! [Y4: A] :
( ( ( member @ A @ Y4 @ S3 )
& ( Y4 != X ) )
=> ( ( R3 @ X @ Y4 )
& ( R3 @ Y4 @ X ) ) )
& ( pairwise @ A @ R3 @ S3 ) ) ) ).
% pairwise_insert
thf(fact_187_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_188_finite__subset__induct,axiom,
! [A: $tType,F4: set @ A,A4: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F4 )
=> ( ( ord_less_eq @ ( set @ A ) @ F4 @ A4 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A7: A,F3: set @ A] :
( ( finite_finite2 @ A @ F3 )
=> ( ( member @ A @ A7 @ A4 )
=> ( ~ ( member @ A @ A7 @ F3 )
=> ( ( P @ F3 )
=> ( P @ ( insert @ A @ A7 @ F3 ) ) ) ) ) )
=> ( P @ F4 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_189_infinite__remove,axiom,
! [A: $tType,S: set @ A,A2: A] :
( ~ ( finite_finite2 @ A @ S )
=> ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ S @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% infinite_remove
thf(fact_190_subsetI,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( member @ A @ X2 @ B4 ) )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ).
% subsetI
thf(fact_191_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_192_DiffI,axiom,
! [A: $tType,C2: A,A4: set @ A,B4: set @ A] :
( ( member @ A @ C2 @ A4 )
=> ( ~ ( member @ A @ C2 @ B4 )
=> ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A4 @ B4 ) ) ) ) ).
% DiffI
thf(fact_193_Diff__iff,axiom,
! [A: $tType,C2: A,A4: set @ A,B4: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A4 @ B4 ) )
= ( ( member @ A @ C2 @ A4 )
& ~ ( member @ A @ C2 @ B4 ) ) ) ).
% Diff_iff
thf(fact_194_Diff__idemp,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B4 ) @ B4 )
= ( minus_minus @ ( set @ A ) @ A4 @ B4 ) ) ).
% Diff_idemp
thf(fact_195_empty__subsetI,axiom,
! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A4 ) ).
% empty_subsetI
thf(fact_196_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_197_insert__subset,axiom,
! [A: $tType,X: A,A4: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A4 ) @ B4 )
= ( ( member @ A @ X @ B4 )
& ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ) ).
% insert_subset
thf(fact_198_Diff__cancel,axiom,
! [A: $tType,A4: set @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ A4 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_cancel
thf(fact_199_empty__Diff,axiom,
! [A: $tType,A4: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A4 )
= ( bot_bot @ ( set @ A ) ) ) ).
% empty_Diff
thf(fact_200_Diff__empty,axiom,
! [A: $tType,A4: set @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
= A4 ) ).
% Diff_empty
thf(fact_201_finite__Diff,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( finite_finite2 @ A @ A4 )
=> ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A4 @ B4 ) ) ) ).
% finite_Diff
thf(fact_202_finite__Diff2,axiom,
! [A: $tType,B4: set @ A,A4: set @ A] :
( ( finite_finite2 @ A @ B4 )
=> ( ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A4 @ B4 ) )
= ( finite_finite2 @ A @ A4 ) ) ) ).
% finite_Diff2
thf(fact_203_insert__Diff1,axiom,
! [A: $tType,X: A,B4: set @ A,A4: set @ A] :
( ( member @ A @ X @ B4 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A4 ) @ B4 )
= ( minus_minus @ ( set @ A ) @ A4 @ B4 ) ) ) ).
% insert_Diff1
thf(fact_204_Diff__insert0,axiom,
! [A: $tType,X: A,A4: set @ A,B4: set @ A] :
( ~ ( member @ A @ X @ A4 )
=> ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ B4 ) )
= ( minus_minus @ ( set @ A ) @ A4 @ B4 ) ) ) ).
% Diff_insert0
thf(fact_205_singleton__insert__inj__eq_H,axiom,
! [A: $tType,A2: A,A4: set @ A,B3: A] :
( ( ( insert @ A @ A2 @ A4 )
= ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A2 = B3 )
& ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_206_singleton__insert__inj__eq,axiom,
! [A: $tType,B3: A,A2: A,A4: set @ A] :
( ( ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ A2 @ A4 ) )
= ( ( A2 = B3 )
& ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_207_Diff__eq__empty__iff,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( ( minus_minus @ ( set @ A ) @ A4 @ B4 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ).
% Diff_eq_empty_iff
thf(fact_208_insert__Diff__single,axiom,
! [A: $tType,A2: A,A4: set @ A] :
( ( insert @ A @ A2 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( insert @ A @ A2 @ A4 ) ) ).
% insert_Diff_single
thf(fact_209_finite__Diff__insert,axiom,
! [A: $tType,A4: set @ A,A2: A,B4: set @ A] :
( ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ A2 @ B4 ) ) )
= ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A4 @ B4 ) ) ) ).
% finite_Diff_insert
thf(fact_210_Diff__single__insert,axiom,
! [A: $tType,A4: set @ A,X: A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B4 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ X @ B4 ) ) ) ).
% Diff_single_insert
thf(fact_211_subset__insert__iff,axiom,
! [A: $tType,A4: set @ A,X: A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ X @ B4 ) )
= ( ( ( member @ A @ X @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B4 ) )
& ( ~ ( member @ A @ X @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ) ) ).
% subset_insert_iff
thf(fact_212_remove__induct,axiom,
! [A: $tType,P: ( set @ A ) > $o,B4: set @ A] :
( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ( ~ ( finite_finite2 @ A @ B4 )
=> ( P @ B4 ) )
=> ( ! [A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ B4 )
=> ( ! [X6: A] :
( ( member @ A @ X6 @ A6 )
=> ( P @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B4 ) ) ) ) ).
% remove_induct
thf(fact_213_finite__remove__induct,axiom,
! [A: $tType,B4: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ B4 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A6: set @ A] :
( ( finite_finite2 @ A @ A6 )
=> ( ( A6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A6 @ B4 )
=> ( ! [X6: A] :
( ( member @ A @ X6 @ A6 )
=> ( P @ ( minus_minus @ ( set @ A ) @ A6 @ ( insert @ A @ X6 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( P @ A6 ) ) ) ) )
=> ( P @ B4 ) ) ) ) ).
% finite_remove_induct
thf(fact_214_bot_Oextremum,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A2: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A2 ) ) ).
% bot.extremum
thf(fact_215_bot_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
= ( A2
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_unique
thf(fact_216_bot_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
=> ( A2
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_uniqueI
thf(fact_217_pairwise__subset,axiom,
! [A: $tType,P: A > A > $o,S: set @ A,T: set @ A] :
( ( pairwise @ A @ P @ S )
=> ( ( ord_less_eq @ ( set @ A ) @ T @ S )
=> ( pairwise @ A @ P @ T ) ) ) ).
% pairwise_subset
thf(fact_218_Diff__insert,axiom,
! [A: $tType,A4: set @ A,A2: A,B4: set @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ A2 @ B4 ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B4 ) @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Diff_insert
thf(fact_219_insert__Diff,axiom,
! [A: $tType,A2: A,A4: set @ A] :
( ( member @ A @ A2 @ A4 )
=> ( ( insert @ A @ A2 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= A4 ) ) ).
% insert_Diff
thf(fact_220_Diff__insert2,axiom,
! [A: $tType,A4: set @ A,A2: A,B4: set @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ A2 @ B4 ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) @ B4 ) ) ).
% Diff_insert2
thf(fact_221_Diff__insert__absorb,axiom,
! [A: $tType,X: A,A4: set @ A] :
( ~ ( member @ A @ X @ A4 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A4 ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= A4 ) ) ).
% Diff_insert_absorb
thf(fact_222_subset__singletonD,axiom,
! [A: $tType,A4: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( A4
= ( bot_bot @ ( set @ A ) ) )
| ( A4
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singletonD
thf(fact_223_subset__singleton__iff,axiom,
! [A: $tType,X7: set @ A,A2: A] :
( ( ord_less_eq @ ( set @ A ) @ X7 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( X7
= ( bot_bot @ ( set @ A ) ) )
| ( X7
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singleton_iff
thf(fact_224_subset__Diff__insert,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,X: A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ ( minus_minus @ ( set @ A ) @ B4 @ ( insert @ A @ X @ C4 ) ) )
= ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( minus_minus @ ( set @ A ) @ B4 @ C4 ) )
& ~ ( member @ A @ X @ A4 ) ) ) ).
% subset_Diff_insert
thf(fact_225_subset__insertI2,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,B3: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ B3 @ B4 ) ) ) ).
% subset_insertI2
thf(fact_226_subset__insertI,axiom,
! [A: $tType,B4: set @ A,A2: A] : ( ord_less_eq @ ( set @ A ) @ B4 @ ( insert @ A @ A2 @ B4 ) ) ).
% subset_insertI
thf(fact_227_insert__Diff__if,axiom,
! [A: $tType,X: A,B4: set @ A,A4: set @ A] :
( ( ( member @ A @ X @ B4 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A4 ) @ B4 )
= ( minus_minus @ ( set @ A ) @ A4 @ B4 ) ) )
& ( ~ ( member @ A @ X @ B4 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A4 ) @ B4 )
= ( insert @ A @ X @ ( minus_minus @ ( set @ A ) @ A4 @ B4 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_228_subset__insert,axiom,
! [A: $tType,X: A,A4: set @ A,B4: set @ A] :
( ~ ( member @ A @ X @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ X @ B4 ) )
= ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ) ).
% subset_insert
thf(fact_229_Set_Oinsert__mono,axiom,
! [A: $tType,C4: set @ A,D3: set @ A,A2: A] :
( ( ord_less_eq @ ( set @ A ) @ C4 @ D3 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A2 @ C4 ) @ ( insert @ A @ A2 @ D3 ) ) ) ).
% Set.insert_mono
thf(fact_230_DiffE,axiom,
! [A: $tType,C2: A,A4: set @ A,B4: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A4 @ B4 ) )
=> ~ ( ( member @ A @ C2 @ A4 )
=> ( member @ A @ C2 @ B4 ) ) ) ).
% DiffE
thf(fact_231_DiffD1,axiom,
! [A: $tType,C2: A,A4: set @ A,B4: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A4 @ B4 ) )
=> ( member @ A @ C2 @ A4 ) ) ).
% DiffD1
thf(fact_232_DiffD2,axiom,
! [A: $tType,C2: A,A4: set @ A,B4: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A4 @ B4 ) )
=> ~ ( member @ A @ C2 @ B4 ) ) ).
% DiffD2
thf(fact_233_set__mp,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 ) ) ) ).
% set_mp
thf(fact_234_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_235_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_236_subsetCE,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 ) ) ) ).
% subsetCE
thf(fact_237_Diff__mono,axiom,
! [A: $tType,A4: set @ A,C4: set @ A,D3: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ C4 )
=> ( ( ord_less_eq @ ( set @ A ) @ D3 @ B4 )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B4 ) @ ( minus_minus @ ( set @ A ) @ C4 @ D3 ) ) ) ) ).
% Diff_mono
thf(fact_238_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_239_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A5: set @ A,B6: set @ A] :
! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( member @ A @ X3 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_240_equalityD1,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( A4 = B4 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ).
% equalityD1
thf(fact_241_equalityD2,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( A4 = B4 )
=> ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ).
% equalityD2
thf(fact_242_set__rev__mp,axiom,
! [A: $tType,X: A,A4: set @ A,B4: set @ A] :
( ( member @ A @ X @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( member @ A @ X @ B4 ) ) ) ).
% set_rev_mp
thf(fact_243_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A5: set @ A,B6: set @ A] :
! [T2: A] :
( ( member @ A @ T2 @ A5 )
=> ( member @ A @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_244_Diff__subset,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B4 ) @ A4 ) ).
% Diff_subset
thf(fact_245_double__diff,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ C4 )
=> ( ( minus_minus @ ( set @ A ) @ B4 @ ( minus_minus @ ( set @ A ) @ C4 @ A4 ) )
= A4 ) ) ) ).
% double_diff
thf(fact_246_rev__subsetD,axiom,
! [A: $tType,C2: A,A4: set @ A,B4: set @ A] :
( ( member @ A @ C2 @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( member @ A @ C2 @ B4 ) ) ) ).
% rev_subsetD
thf(fact_247_subset__refl,axiom,
! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ A4 @ A4 ) ).
% subset_refl
thf(fact_248_Collect__mono,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X2: A] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_249_subset__trans,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ C4 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ C4 ) ) ) ).
% subset_trans
thf(fact_250_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y3: set @ A,Z: set @ A] : Y3 = Z )
= ( ^ [A5: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
& ( ord_less_eq @ ( set @ A ) @ B6 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_251_contra__subsetD,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ~ ( member @ A @ C2 @ B4 )
=> ~ ( member @ A @ C2 @ A4 ) ) ) ).
% contra_subsetD
thf(fact_252_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_253_wlog__linorder__le,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > A > $o,B3: A,A2: A] :
( ! [A7: A,B7: A] :
( ( ord_less_eq @ A @ A7 @ B7 )
=> ( P @ A7 @ B7 ) )
=> ( ( ( P @ B3 @ A2 )
=> ( P @ A2 @ B3 ) )
=> ( P @ A2 @ B3 ) ) ) ) ).
% wlog_linorder_le
thf(fact_254_finite__subset,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ( finite_finite2 @ A @ B4 )
=> ( finite_finite2 @ A @ A4 ) ) ) ).
% finite_subset
%----Type constructors (10)
thf(tcon_fun___Orderings_Oorder__bot,axiom,
! [A8: $tType,A9: $tType] :
( ( order_bot @ A9 @ ( type2 @ A9 ) )
=> ( order_bot @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Finite__Set_Ofinite,axiom,
! [A8: $tType,A9: $tType] :
( ( ( finite_finite @ A8 @ ( type2 @ A8 ) )
& ( finite_finite @ A9 @ ( type2 @ A9 ) ) )
=> ( finite_finite @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Orderings_Obot,axiom,
! [A8: $tType,A9: $tType] :
( ( bot @ A9 @ ( type2 @ A9 ) )
=> ( bot @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_1,axiom,
! [A8: $tType] : ( order_bot @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_2,axiom,
! [A8: $tType] :
( ( finite_finite @ A8 @ ( type2 @ A8 ) )
=> ( finite_finite @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_3,axiom,
! [A8: $tType] : ( bot @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_4,axiom,
order_bot @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_5,axiom,
finite_finite @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Obot_6,axiom,
bot @ $o @ ( type2 @ $o ) ).
%----Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $true @ X @ Y )
= X ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
( ( coindu1213758845finite @ a @ xs )
& ( coindu1213758845finite @ a @ ys ) ) ).
%------------------------------------------------------------------------------