TPTP Problem File: COM172^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : COM172^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Computing Theory
% Problem : Koenig's lemma 47
% 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 : koenigslemma__47.p [Bla16]
% Status : Theorem
% Rating : 0.00 v8.2.0, 0.33 v8.1.0, 0.25 v7.5.0, 0.67 v7.3.0, 1.00 v7.1.0
% Syntax : Number of formulae : 317 ( 135 unt; 47 typ; 0 def)
% Number of atoms : 747 ( 252 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 4075 ( 107 ~; 32 |; 78 &;3519 @)
% ( 0 <=>; 339 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 8 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 329 ( 329 >; 0 *; 0 +; 0 <<)
% Number of symbols : 49 ( 46 usr; 8 con; 0-8 aty)
% Number of variables : 1137 ( 92 ^; 952 !; 54 ?;1137 :)
% ( 39 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:45:38.000
%------------------------------------------------------------------------------
%----Could-be-implicit typings (4)
thf(ty_t_Coinductive__List_Ollist,type,
coinductive_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 (43)
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_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_c_Coinductive__List_Ofinite__lprefix,type,
coindu328551480prefix:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Ogen__lset,type,
coinductive_gen_lset:
!>[A: $tType] : ( ( set @ A ) > ( coinductive_llist @ A ) > ( set @ A ) ) ).
thf(sy_c_Coinductive__List_Olappend,type,
coinductive_lappend:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_OldropWhile,type,
coindu218763757pWhile:
!>[A: $tType] : ( ( A > $o ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_Olfilter,type,
coinductive_lfilter:
!>[A: $tType] : ( ( A > $o ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
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_Ollist_Ocase__llist,type,
coindu1381640503_llist:
!>[B: $tType,A: $tType] : ( B > ( A > ( coinductive_llist @ A ) > B ) > ( coinductive_llist @ A ) > B ) ).
thf(sy_c_Coinductive__List_Ollist_Ocorec__llist,type,
coindu1259883913_llist:
!>[C: $tType,A: $tType] : ( ( C > $o ) > ( C > A ) > ( C > $o ) > ( C > ( coinductive_llist @ A ) ) > ( C > C ) > C > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_Ollist_Olhd,type,
coinductive_lhd:
!>[A: $tType] : ( ( coinductive_llist @ A ) > A ) ).
thf(sy_c_Coinductive__List_Ollist_Olnull,type,
coinductive_lnull:
!>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Ollist_Olset,type,
coinductive_lset:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( set @ 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_Coinductive__List_OltakeWhile,type,
coindu501562517eWhile:
!>[A: $tType] : ( ( A > $o ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_Oord__class_Olsorted,type,
coindu63249387sorted:
!>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Ounfold__llist,type,
coindu1441602521_llist:
!>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > B ) > ( A > A ) > A > ( coinductive_llist @ B ) ) ).
thf(sy_c_HOL_Oundefined,type,
undefined:
!>[A: $tType] : A ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Koenigslemma__Mirabelle__aepjeeakgn_Opaths,type,
koenig916195507_paths:
!>[Node: $tType] : ( ( Node > Node > $o ) > ( set @ ( coinductive_llist @ Node ) ) ) ).
thf(sy_c_Koenigslemma__Mirabelle__aepjeeakgn_Opathsp,type,
koenig2031690877pathsp:
!>[Node: $tType] : ( ( Node > Node > $o ) > ( coinductive_llist @ Node ) > $o ) ).
thf(sy_c_Koenigslemma__Mirabelle__aepjeeakgn_Oreachable__via,type,
koenig317145564le_via:
!>[Node: $tType] : ( ( Node > Node > $o ) > ( set @ Node ) > Node > ( set @ Node ) ) ).
thf(sy_c_Koenigslemma__Mirabelle__aepjeeakgn_Oreachable__viap,type,
koenig1757754772e_viap:
!>[Node: $tType] : ( ( Node > Node > $o ) > ( set @ Node ) > Node > Node > $o ) ).
thf(sy_c_Lattices_Osup__class_Osup,type,
sup_sup:
!>[A: $tType] : ( A > A > 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_Relation_OPowp,type,
powp:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_graph,type,
graph: a > a > $o ).
thf(sy_v_x,type,
x: coinductive_llist @ a ).
thf(sy_v_x21,type,
x21: a ).
thf(sy_v_x22,type,
x22: coinductive_llist @ a ).
thf(sy_v_xa,type,
xa: a ).
thf(sy_v_xs,type,
xs: coinductive_llist @ a ).
thf(sy_v_ys,type,
ys: coinductive_llist @ a ).
%----Relevant facts (254)
thf(fact_0_assms,axiom,
member @ ( coinductive_llist @ a ) @ ( coinductive_lappend @ a @ xs @ ys ) @ ( koenig916195507_paths @ a @ graph ) ).
% assms
thf(fact_1_paths_OEmpty,axiom,
! [Node: $tType,Graph: Node > Node > $o] : ( member @ ( coinductive_llist @ Node ) @ ( coinductive_LNil @ Node ) @ ( koenig916195507_paths @ Node @ Graph ) ) ).
% paths.Empty
thf(fact_2_paths_OLCons,axiom,
! [Node: $tType,Graph: Node > Node > $o,X: Node,Y: Node,Xs: coinductive_llist @ Node] :
( ( Graph @ X @ Y )
=> ( ( member @ ( coinductive_llist @ Node ) @ ( coinductive_LCons @ Node @ Y @ Xs ) @ ( koenig916195507_paths @ Node @ Graph ) )
=> ( member @ ( coinductive_llist @ Node ) @ ( coinductive_LCons @ Node @ X @ ( coinductive_LCons @ Node @ Y @ Xs ) ) @ ( koenig916195507_paths @ Node @ Graph ) ) ) ) ).
% paths.LCons
thf(fact_3_paths_Ocases,axiom,
! [Node: $tType,A2: coinductive_llist @ Node,Graph: Node > Node > $o] :
( ( member @ ( coinductive_llist @ Node ) @ A2 @ ( koenig916195507_paths @ Node @ Graph ) )
=> ( ( A2
!= ( coinductive_LNil @ Node ) )
=> ( ! [X2: Node] :
( A2
!= ( coinductive_LCons @ Node @ X2 @ ( coinductive_LNil @ Node ) ) )
=> ~ ! [X2: Node,Y2: Node,Xs2: coinductive_llist @ Node] :
( ( A2
= ( coinductive_LCons @ Node @ X2 @ ( coinductive_LCons @ Node @ Y2 @ Xs2 ) ) )
=> ( ( Graph @ X2 @ Y2 )
=> ~ ( member @ ( coinductive_llist @ Node ) @ ( coinductive_LCons @ Node @ Y2 @ Xs2 ) @ ( koenig916195507_paths @ Node @ Graph ) ) ) ) ) ) ) ).
% paths.cases
thf(fact_4_paths_Osimps,axiom,
! [Node: $tType,A2: coinductive_llist @ Node,Graph: Node > Node > $o] :
( ( member @ ( coinductive_llist @ Node ) @ A2 @ ( koenig916195507_paths @ Node @ Graph ) )
= ( ( A2
= ( coinductive_LNil @ Node ) )
| ? [X3: Node] :
( A2
= ( coinductive_LCons @ Node @ X3 @ ( coinductive_LNil @ Node ) ) )
| ? [X3: Node,Y3: Node,Xs3: coinductive_llist @ Node] :
( ( A2
= ( coinductive_LCons @ Node @ X3 @ ( coinductive_LCons @ Node @ Y3 @ Xs3 ) ) )
& ( Graph @ X3 @ Y3 )
& ( member @ ( coinductive_llist @ Node ) @ ( coinductive_LCons @ Node @ Y3 @ Xs3 ) @ ( koenig916195507_paths @ Node @ Graph ) ) ) ) ) ).
% paths.simps
thf(fact_5_paths_OSingle,axiom,
! [Node: $tType,X: Node,Graph: Node > Node > $o] : ( member @ ( coinductive_llist @ Node ) @ ( coinductive_LCons @ Node @ X @ ( coinductive_LNil @ Node ) ) @ ( koenig916195507_paths @ Node @ Graph ) ) ).
% paths.Single
thf(fact_6_paths__LConsD,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A,Graph: A > A > $o] :
( ( member @ ( coinductive_llist @ A ) @ ( coinductive_LCons @ A @ X @ Xs ) @ ( koenig916195507_paths @ A @ Graph ) )
=> ( member @ ( coinductive_llist @ A ) @ Xs @ ( koenig916195507_paths @ A @ Graph ) ) ) ).
% paths_LConsD
thf(fact_7_paths_Ocoinduct,axiom,
! [Node: $tType,X4: ( coinductive_llist @ Node ) > $o,X: coinductive_llist @ Node,Graph: Node > Node > $o] :
( ( X4 @ X )
=> ( ! [X2: coinductive_llist @ Node] :
( ( X4 @ X2 )
=> ( ( X2
= ( coinductive_LNil @ Node ) )
| ? [Xa: Node] :
( X2
= ( coinductive_LCons @ Node @ Xa @ ( coinductive_LNil @ Node ) ) )
| ? [Xa: Node,Y4: Node,Xs4: coinductive_llist @ Node] :
( ( X2
= ( coinductive_LCons @ Node @ Xa @ ( coinductive_LCons @ Node @ Y4 @ Xs4 ) ) )
& ( Graph @ Xa @ Y4 )
& ( ( X4 @ ( coinductive_LCons @ Node @ Y4 @ Xs4 ) )
| ( member @ ( coinductive_llist @ Node ) @ ( coinductive_LCons @ Node @ Y4 @ Xs4 ) @ ( koenig916195507_paths @ Node @ Graph ) ) ) ) ) )
=> ( member @ ( coinductive_llist @ Node ) @ X @ ( koenig916195507_paths @ Node @ Graph ) ) ) ) ).
% paths.coinduct
thf(fact_8_lappend__code_I1_J,axiom,
! [A: $tType,Ys: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ ( coinductive_LNil @ A ) @ Ys )
= Ys ) ).
% lappend_code(1)
thf(fact_9_lappend__LNil2,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ Xs @ ( coinductive_LNil @ A ) )
= Xs ) ).
% lappend_LNil2
thf(fact_10_lappend__code_I2_J,axiom,
! [A: $tType,Xa2: A,X: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ ( coinductive_LCons @ A @ Xa2 @ X ) @ Ys )
= ( coinductive_LCons @ A @ Xa2 @ ( coinductive_lappend @ A @ X @ Ys ) ) ) ).
% lappend_code(2)
thf(fact_11_lappend__snocL1__conv__LCons2,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ Y @ ( coinductive_LNil @ A ) ) ) @ Ys )
= ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ Y @ Ys ) ) ) ).
% lappend_snocL1_conv_LCons2
thf(fact_12_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_13_lappend__LNil__LNil,axiom,
! [A: $tType] :
( ( coinductive_lappend @ A @ ( coinductive_LNil @ A ) @ ( coinductive_LNil @ A ) )
= ( coinductive_LNil @ A ) ) ).
% lappend_LNil_LNil
thf(fact_14_LNil__eq__lappend__iff,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ( coinductive_LNil @ A )
= ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( ( Xs
= ( coinductive_LNil @ A ) )
& ( Ys
= ( coinductive_LNil @ A ) ) ) ) ).
% LNil_eq_lappend_iff
thf(fact_15_lappend__eq__LNil__iff,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ( coinductive_lappend @ A @ Xs @ Ys )
= ( coinductive_LNil @ A ) )
= ( ( Xs
= ( coinductive_LNil @ A ) )
& ( Ys
= ( coinductive_LNil @ A ) ) ) ) ).
% lappend_eq_LNil_iff
thf(fact_16_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_17_llist_Oexhaust,axiom,
! [A: $tType,Y: coinductive_llist @ A] :
( ( Y
!= ( coinductive_LNil @ A ) )
=> ~ ! [X212: A,X222: coinductive_llist @ A] :
( Y
!= ( coinductive_LCons @ A @ X212 @ X222 ) ) ) ).
% llist.exhaust
thf(fact_18_neq__LNil__conv,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( Xs
!= ( coinductive_LNil @ A ) )
= ( ? [X3: A,Xs5: coinductive_llist @ A] :
( Xs
= ( coinductive_LCons @ A @ X3 @ Xs5 ) ) ) ) ).
% neq_LNil_conv
thf(fact_19_lappend__assoc,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) @ Zs )
= ( coinductive_lappend @ A @ Xs @ ( coinductive_lappend @ A @ Ys @ Zs ) ) ) ).
% lappend_assoc
thf(fact_20_Coinductive__List_Ofinite__lprefix__nitpick__simps_I3_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
( ( coindu328551480prefix @ A @ Xs @ ( coinductive_LCons @ A @ Y @ Ys ) )
= ( ( Xs
= ( coinductive_LNil @ A ) )
| ? [Xs5: coinductive_llist @ A] :
( ( Xs
= ( coinductive_LCons @ A @ Y @ Xs5 ) )
& ( coindu328551480prefix @ A @ Xs5 @ Ys ) ) ) ) ).
% Coinductive_List.finite_lprefix_nitpick_simps(3)
thf(fact_21_lstrict__prefix__code_I2_J,axiom,
! [B: $tType,Y: B,Ys: coinductive_llist @ B] : ( coindu1478340336prefix @ B @ ( coinductive_LNil @ B ) @ ( coinductive_LCons @ B @ Y @ Ys ) ) ).
% lstrict_prefix_code(2)
thf(fact_22_lstrict__prefix__code_I3_J,axiom,
! [B: $tType,X: B,Xs: coinductive_llist @ B] :
~ ( coindu1478340336prefix @ B @ ( coinductive_LCons @ B @ X @ Xs ) @ ( coinductive_LNil @ B ) ) ).
% lstrict_prefix_code(3)
thf(fact_23_llast__singleton,axiom,
! [A: $tType,X: A] :
( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LNil @ A ) ) )
= X ) ).
% llast_singleton
thf(fact_24_lmember__code_I1_J,axiom,
! [A: $tType,X: A] :
~ ( coinductive_lmember @ A @ X @ ( coinductive_LNil @ A ) ) ).
% lmember_code(1)
thf(fact_25_lmember__code_I2_J,axiom,
! [A: $tType,X: A,Y: A,Ys: coinductive_llist @ A] :
( ( coinductive_lmember @ A @ X @ ( coinductive_LCons @ A @ Y @ Ys ) )
= ( ( X = Y )
| ( coinductive_lmember @ A @ X @ Ys ) ) ) ).
% lmember_code(2)
thf(fact_26_pathsp_Ocoinduct,axiom,
! [Node: $tType,X4: ( coinductive_llist @ Node ) > $o,X: coinductive_llist @ Node,Graph: Node > Node > $o] :
( ( X4 @ X )
=> ( ! [X2: coinductive_llist @ Node] :
( ( X4 @ X2 )
=> ( ( X2
= ( coinductive_LNil @ Node ) )
| ? [Xa: Node] :
( X2
= ( coinductive_LCons @ Node @ Xa @ ( coinductive_LNil @ Node ) ) )
| ? [Xa: Node,Y4: Node,Xs4: coinductive_llist @ Node] :
( ( X2
= ( coinductive_LCons @ Node @ Xa @ ( coinductive_LCons @ Node @ Y4 @ Xs4 ) ) )
& ( Graph @ Xa @ Y4 )
& ( ( X4 @ ( coinductive_LCons @ Node @ Y4 @ Xs4 ) )
| ( koenig2031690877pathsp @ Node @ Graph @ ( coinductive_LCons @ Node @ Y4 @ Xs4 ) ) ) ) ) )
=> ( koenig2031690877pathsp @ Node @ Graph @ X ) ) ) ).
% pathsp.coinduct
thf(fact_27_pathsp_OSingle,axiom,
! [Node: $tType,Graph: Node > Node > $o,X: Node] : ( koenig2031690877pathsp @ Node @ Graph @ ( coinductive_LCons @ Node @ X @ ( coinductive_LNil @ Node ) ) ) ).
% pathsp.Single
thf(fact_28_pathsp_Osimps,axiom,
! [Node: $tType] :
( ( koenig2031690877pathsp @ Node )
= ( ^ [Graph2: Node > Node > $o,A3: coinductive_llist @ Node] :
( ( A3
= ( coinductive_LNil @ Node ) )
| ? [X3: Node] :
( A3
= ( coinductive_LCons @ Node @ X3 @ ( coinductive_LNil @ Node ) ) )
| ? [X3: Node,Y3: Node,Xs3: coinductive_llist @ Node] :
( ( A3
= ( coinductive_LCons @ Node @ X3 @ ( coinductive_LCons @ Node @ Y3 @ Xs3 ) ) )
& ( Graph2 @ X3 @ Y3 )
& ( koenig2031690877pathsp @ Node @ Graph2 @ ( coinductive_LCons @ Node @ Y3 @ Xs3 ) ) ) ) ) ) ).
% pathsp.simps
thf(fact_29_pathsp_Ocases,axiom,
! [Node: $tType,Graph: Node > Node > $o,A2: coinductive_llist @ Node] :
( ( koenig2031690877pathsp @ Node @ Graph @ A2 )
=> ( ( A2
!= ( coinductive_LNil @ Node ) )
=> ( ! [X2: Node] :
( A2
!= ( coinductive_LCons @ Node @ X2 @ ( coinductive_LNil @ Node ) ) )
=> ~ ! [X2: Node,Y2: Node,Xs2: coinductive_llist @ Node] :
( ( A2
= ( coinductive_LCons @ Node @ X2 @ ( coinductive_LCons @ Node @ Y2 @ Xs2 ) ) )
=> ( ( Graph @ X2 @ Y2 )
=> ~ ( koenig2031690877pathsp @ Node @ Graph @ ( coinductive_LCons @ Node @ Y2 @ Xs2 ) ) ) ) ) ) ) ).
% pathsp.cases
thf(fact_30_llast__LCons2,axiom,
! [A: $tType,X: A,Y: A,Xs: coinductive_llist @ A] :
( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LCons @ A @ Y @ Xs ) ) )
= ( coinductive_llast @ A @ ( coinductive_LCons @ A @ Y @ Xs ) ) ) ).
% llast_LCons2
thf(fact_31_lstrict__prefix__code_I4_J,axiom,
! [B: $tType,X: B,Xs: coinductive_llist @ B,Y: B,Ys: coinductive_llist @ B] :
( ( coindu1478340336prefix @ B @ ( coinductive_LCons @ B @ X @ Xs ) @ ( coinductive_LCons @ B @ Y @ Ys ) )
= ( ( X = Y )
& ( coindu1478340336prefix @ B @ Xs @ Ys ) ) ) ).
% lstrict_prefix_code(4)
thf(fact_32_lstrict__prefix__code_I1_J,axiom,
! [A: $tType] :
~ ( coindu1478340336prefix @ A @ ( coinductive_LNil @ A ) @ ( coinductive_LNil @ A ) ) ).
% lstrict_prefix_code(1)
thf(fact_33_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_34_pathsp_OLCons,axiom,
! [Node: $tType,Graph: Node > Node > $o,X: Node,Y: Node,Xs: coinductive_llist @ Node] :
( ( Graph @ X @ Y )
=> ( ( koenig2031690877pathsp @ Node @ Graph @ ( coinductive_LCons @ Node @ Y @ Xs ) )
=> ( koenig2031690877pathsp @ Node @ Graph @ ( coinductive_LCons @ Node @ X @ ( coinductive_LCons @ Node @ Y @ Xs ) ) ) ) ) ).
% pathsp.LCons
thf(fact_35_pathsp_OEmpty,axiom,
! [Node: $tType,Graph: Node > Node > $o] : ( koenig2031690877pathsp @ Node @ Graph @ ( coinductive_LNil @ Node ) ) ).
% pathsp.Empty
thf(fact_36_pathsp__paths__eq,axiom,
! [Node: $tType] :
( ( koenig2031690877pathsp @ Node )
= ( ^ [Graph2: Node > Node > $o,X3: coinductive_llist @ Node] : ( member @ ( coinductive_llist @ Node ) @ X3 @ ( koenig916195507_paths @ Node @ Graph2 ) ) ) ) ).
% pathsp_paths_eq
thf(fact_37_paths__def,axiom,
! [Node: $tType] :
( ( koenig916195507_paths @ Node )
= ( ^ [Graph2: Node > Node > $o] : ( collect @ ( coinductive_llist @ Node ) @ ( koenig2031690877pathsp @ Node @ Graph2 ) ) ) ) ).
% paths_def
thf(fact_38_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_39_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_40_llast__lappend__LCons,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_llast @ A @ ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ Y @ Ys ) ) )
= ( coinductive_llast @ A @ ( coinductive_LCons @ A @ Y @ Ys ) ) ) ) ).
% llast_lappend_LCons
thf(fact_41_llimit__induct,axiom,
! [A: $tType,P: ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A] :
( ( P @ ( coinductive_LNil @ A ) )
=> ( ! [X2: A,Xs2: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs2 )
=> ( ( P @ Xs2 )
=> ( P @ ( coinductive_LCons @ A @ X2 @ Xs2 ) ) ) )
=> ( ( ! [Ys2: coinductive_llist @ A] :
( ( coindu1478340336prefix @ A @ Ys2 @ Xs )
=> ( P @ Ys2 ) )
=> ( P @ Xs ) )
=> ( P @ Xs ) ) ) ) ).
% llimit_induct
thf(fact_42_gen__lset__code_I1_J,axiom,
! [A: $tType,A4: set @ A] :
( ( coinductive_gen_lset @ A @ A4 @ ( coinductive_LNil @ A ) )
= A4 ) ).
% gen_lset_code(1)
thf(fact_43_lsorted__code_I2_J,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X: A] : ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LNil @ A ) ) ) ) ).
% lsorted_code(2)
thf(fact_44_lfinite__rev__induct,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
( ( coinductive_lfinite @ A @ Xs )
=> ( ( P @ ( coinductive_LNil @ A ) )
=> ( ! [X2: A,Xs2: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs2 )
=> ( ( P @ Xs2 )
=> ( P @ ( coinductive_lappend @ A @ Xs2 @ ( coinductive_LCons @ A @ X2 @ ( coinductive_LNil @ A ) ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% lfinite_rev_induct
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_llist_Ocorec__code,axiom,
! [A: $tType,C: $tType] :
( ( coindu1259883913_llist @ C @ A )
= ( ^ [P2: C > $o,G21: C > A,Q22: C > $o,G221: C > ( coinductive_llist @ A ),G222: C > C,A3: C] : ( if @ ( coinductive_llist @ A ) @ ( P2 @ A3 ) @ ( coinductive_LNil @ A ) @ ( coinductive_LCons @ A @ ( G21 @ A3 ) @ ( if @ ( coinductive_llist @ A ) @ ( Q22 @ A3 ) @ ( G221 @ A3 ) @ ( coindu1259883913_llist @ C @ A @ P2 @ G21 @ Q22 @ G221 @ G222 @ ( G222 @ A3 ) ) ) ) ) ) ) ).
% llist.corec_code
thf(fact_50_unfold__llist_Ocode,axiom,
! [B: $tType,A: $tType] :
( ( coindu1441602521_llist @ A @ B )
= ( ^ [P2: A > $o,G21: A > B,G22: A > A,A3: A] : ( if @ ( coinductive_llist @ B ) @ ( P2 @ A3 ) @ ( coinductive_LNil @ B ) @ ( coinductive_LCons @ B @ ( G21 @ A3 ) @ ( coindu1441602521_llist @ A @ B @ P2 @ G21 @ G22 @ ( G22 @ A3 ) ) ) ) ) ) ).
% unfold_llist.code
thf(fact_51_lappend_Ocode,axiom,
! [A: $tType] :
( ( coinductive_lappend @ A )
= ( ^ [Xs3: coinductive_llist @ A,Ys3: coinductive_llist @ A] :
( coindu1381640503_llist @ ( coinductive_llist @ A ) @ A @ Ys3
@ ^ [X3: A,Xs5: coinductive_llist @ A] : ( coinductive_LCons @ A @ X3 @ ( coinductive_lappend @ A @ Xs5 @ Ys3 ) )
@ Xs3 ) ) ) ).
% lappend.code
thf(fact_52_ltakeWhile__LCons,axiom,
! [A: $tType,P: A > $o,X: A,Xs: coinductive_llist @ A] :
( ( ( P @ X )
=> ( ( coindu501562517eWhile @ A @ P @ ( coinductive_LCons @ A @ X @ Xs ) )
= ( coinductive_LCons @ A @ X @ ( coindu501562517eWhile @ A @ P @ Xs ) ) ) )
& ( ~ ( P @ X )
=> ( ( coindu501562517eWhile @ A @ P @ ( coinductive_LCons @ A @ X @ Xs ) )
= ( coinductive_LNil @ A ) ) ) ) ).
% ltakeWhile_LCons
thf(fact_53_ltakeWhile__K__True,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coindu501562517eWhile @ A
@ ^ [Uu: A] : $true
@ Xs )
= Xs ) ).
% ltakeWhile_K_True
thf(fact_54_lfinite__LCons,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
= ( coinductive_lfinite @ A @ Xs ) ) ).
% lfinite_LCons
thf(fact_55_lfinite__code_I2_J,axiom,
! [B: $tType,X: B,Xs: coinductive_llist @ B] :
( ( coinductive_lfinite @ B @ ( coinductive_LCons @ B @ X @ Xs ) )
= ( coinductive_lfinite @ B @ Xs ) ) ).
% lfinite_code(2)
thf(fact_56_lfinite__code_I1_J,axiom,
! [A: $tType] : ( coinductive_lfinite @ A @ ( coinductive_LNil @ A ) ) ).
% lfinite_code(1)
thf(fact_57_lfinite__lappend,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( ( coinductive_lfinite @ A @ Xs )
& ( coinductive_lfinite @ A @ Ys ) ) ) ).
% lfinite_lappend
thf(fact_58_ltakeWhile__LNil,axiom,
! [A: $tType,P: A > $o] :
( ( coindu501562517eWhile @ A @ P @ ( coinductive_LNil @ A ) )
= ( coinductive_LNil @ A ) ) ).
% ltakeWhile_LNil
thf(fact_59_lsorted__code_I1_J,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ( coindu63249387sorted @ A @ ( coinductive_LNil @ A ) ) ) ).
% lsorted_code(1)
thf(fact_60_unfold__llist__eq__LCons,axiom,
! [A: $tType,B: $tType,IS_LNIL: B > $o,LHD: B > A,LTL: B > B,B2: B,X: A,Xs: coinductive_llist @ A] :
( ( ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ B2 )
= ( coinductive_LCons @ A @ X @ Xs ) )
= ( ~ ( IS_LNIL @ B2 )
& ( X
= ( LHD @ B2 ) )
& ( Xs
= ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ ( LTL @ B2 ) ) ) ) ) ).
% unfold_llist_eq_LCons
thf(fact_61_ltakeWhile__K__False,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coindu501562517eWhile @ A
@ ^ [Uu: A] : $false
@ Xs )
= ( coinductive_LNil @ A ) ) ).
% ltakeWhile_K_False
thf(fact_62_corec__llist__never__stop,axiom,
! [A: $tType,B: $tType,IS_LNIL: B > $o,LHD: B > A,MORE: B > ( coinductive_llist @ A ),LTL: B > B,X: B] :
( ( coindu1259883913_llist @ B @ A @ IS_LNIL @ LHD
@ ^ [Uu: B] : $false
@ MORE
@ LTL
@ X )
= ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ X ) ) ).
% corec_llist_never_stop
thf(fact_63_llist_Ocase__distrib,axiom,
! [B: $tType,C: $tType,A: $tType,H: B > C,F1: B,F2: A > ( coinductive_llist @ A ) > B,Llist: coinductive_llist @ A] :
( ( H @ ( coindu1381640503_llist @ B @ A @ F1 @ F2 @ Llist ) )
= ( coindu1381640503_llist @ C @ A @ ( H @ F1 )
@ ^ [X1: A,X23: coinductive_llist @ A] : ( H @ ( F2 @ X1 @ X23 ) )
@ Llist ) ) ).
% llist.case_distrib
thf(fact_64_lfinite__LConsI,axiom,
! [A: $tType,Xs: coinductive_llist @ A,X: A] :
( ( coinductive_lfinite @ A @ Xs )
=> ( coinductive_lfinite @ A @ ( coinductive_LCons @ A @ X @ Xs ) ) ) ).
% lfinite_LConsI
thf(fact_65_lfinite__LNil,axiom,
! [A: $tType] : ( coinductive_lfinite @ A @ ( coinductive_LNil @ A ) ) ).
% lfinite_LNil
thf(fact_66_lappend__inf,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ~ ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_lappend @ A @ Xs @ Ys )
= Xs ) ) ).
% lappend_inf
thf(fact_67_LNil,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ( coindu63249387sorted @ A @ ( coinductive_LNil @ A ) ) ) ).
% LNil
thf(fact_68_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_69_unfold__llist_Octr_I2_J,axiom,
! [B: $tType,A: $tType,P3: A > $o,A2: A,G212: A > B,G223: A > A] :
( ~ ( P3 @ A2 )
=> ( ( coindu1441602521_llist @ A @ B @ P3 @ G212 @ G223 @ A2 )
= ( coinductive_LCons @ B @ ( G212 @ A2 ) @ ( coindu1441602521_llist @ A @ B @ P3 @ G212 @ G223 @ ( G223 @ A2 ) ) ) ) ) ).
% unfold_llist.ctr(2)
thf(fact_70_unfold__llist_Octr_I1_J,axiom,
! [A: $tType,B: $tType,P3: A > $o,A2: A,G212: A > B,G223: A > A] :
( ( P3 @ A2 )
=> ( ( coindu1441602521_llist @ A @ B @ P3 @ G212 @ G223 @ A2 )
= ( coinductive_LNil @ B ) ) ) ).
% unfold_llist.ctr(1)
thf(fact_71_llist_Ocorec_I2_J,axiom,
! [A: $tType,C: $tType,P3: C > $o,A2: C,G212: C > A,Q222: C > $o,G2212: C > ( coinductive_llist @ A ),G2222: C > C] :
( ~ ( P3 @ A2 )
=> ( ( coindu1259883913_llist @ C @ A @ P3 @ G212 @ Q222 @ G2212 @ G2222 @ A2 )
= ( coinductive_LCons @ A @ ( G212 @ A2 ) @ ( if @ ( coinductive_llist @ A ) @ ( Q222 @ A2 ) @ ( G2212 @ A2 ) @ ( coindu1259883913_llist @ C @ A @ P3 @ G212 @ Q222 @ G2212 @ G2222 @ ( G2222 @ A2 ) ) ) ) ) ) ).
% llist.corec(2)
thf(fact_72_llist_Ocorec_I1_J,axiom,
! [C: $tType,A: $tType,P3: C > $o,A2: C,G212: C > A,Q222: C > $o,G2212: C > ( coinductive_llist @ A ),G2222: C > C] :
( ( P3 @ A2 )
=> ( ( coindu1259883913_llist @ C @ A @ P3 @ G212 @ Q222 @ G2212 @ G2222 @ A2 )
= ( coinductive_LNil @ A ) ) ) ).
% llist.corec(1)
thf(fact_73_llist_Osimps_I5_J,axiom,
! [B: $tType,A: $tType,F1: B,F2: A > ( coinductive_llist @ A ) > B,X21: A,X22: coinductive_llist @ A] :
( ( coindu1381640503_llist @ B @ A @ F1 @ F2 @ ( coinductive_LCons @ A @ X21 @ X22 ) )
= ( F2 @ X21 @ X22 ) ) ).
% llist.simps(5)
thf(fact_74_llist_Osimps_I4_J,axiom,
! [A: $tType,B: $tType,F1: B,F2: A > ( coinductive_llist @ A ) > B] :
( ( coindu1381640503_llist @ B @ A @ F1 @ F2 @ ( coinductive_LNil @ A ) )
= F1 ) ).
% llist.simps(4)
thf(fact_75_lfinite_Oinducts,axiom,
! [A: $tType,X: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
( ( coinductive_lfinite @ A @ X )
=> ( ( P @ ( coinductive_LNil @ A ) )
=> ( ! [Xs2: coinductive_llist @ A,X2: A] :
( ( coinductive_lfinite @ A @ Xs2 )
=> ( ( P @ Xs2 )
=> ( P @ ( coinductive_LCons @ A @ X2 @ Xs2 ) ) ) )
=> ( P @ X ) ) ) ) ).
% lfinite.inducts
thf(fact_76_lfinite_Osimps,axiom,
! [A: $tType] :
( ( coinductive_lfinite @ A )
= ( ^ [A3: coinductive_llist @ A] :
( ( A3
= ( coinductive_LNil @ A ) )
| ? [Xs3: coinductive_llist @ A,X3: A] :
( ( A3
= ( coinductive_LCons @ A @ X3 @ Xs3 ) )
& ( coinductive_lfinite @ A @ Xs3 ) ) ) ) ) ).
% lfinite.simps
thf(fact_77_lfinite_Ocases,axiom,
! [A: $tType,A2: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ A2 )
=> ( ( A2
!= ( coinductive_LNil @ A ) )
=> ~ ! [Xs2: coinductive_llist @ A] :
( ? [X2: A] :
( A2
= ( coinductive_LCons @ A @ X2 @ Xs2 ) )
=> ~ ( coinductive_lfinite @ A @ Xs2 ) ) ) ) ).
% lfinite.cases
thf(fact_78_Singleton,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X: A] : ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LNil @ A ) ) ) ) ).
% Singleton
thf(fact_79_lfilter_Osimps,axiom,
! [A: $tType] :
( ( coinductive_lfilter @ A )
= ( ^ [P4: A > $o] :
( coindu1381640503_llist @ ( coinductive_llist @ A ) @ A @ ( coinductive_LNil @ A )
@ ^ [X3: A,Xs5: coinductive_llist @ A] : ( if @ ( coinductive_llist @ A ) @ ( P4 @ X3 ) @ ( coinductive_LCons @ A @ X3 @ ( coinductive_lfilter @ A @ P4 @ Xs5 ) ) @ ( coinductive_lfilter @ A @ P4 @ Xs5 ) ) ) ) ) ).
% lfilter.simps
thf(fact_80_lsorted_Ocoinduct,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X4: ( coinductive_llist @ A ) > $o,X: coinductive_llist @ A] :
( ( X4 @ X )
=> ( ! [X2: coinductive_llist @ A] :
( ( X4 @ X2 )
=> ( ( X2
= ( coinductive_LNil @ A ) )
| ? [Xa: A] :
( X2
= ( coinductive_LCons @ A @ Xa @ ( coinductive_LNil @ A ) ) )
| ? [Xa: A,Y4: A,Xs4: coinductive_llist @ A] :
( ( X2
= ( coinductive_LCons @ A @ Xa @ ( coinductive_LCons @ A @ Y4 @ Xs4 ) ) )
& ( ord_less_eq @ A @ Xa @ Y4 )
& ( ( X4 @ ( coinductive_LCons @ A @ Y4 @ Xs4 ) )
| ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ Y4 @ Xs4 ) ) ) ) ) )
=> ( coindu63249387sorted @ A @ X ) ) ) ) ).
% lsorted.coinduct
thf(fact_81_lsorted_Osimps,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ( ( coindu63249387sorted @ A )
= ( ^ [A3: coinductive_llist @ A] :
( ( A3
= ( coinductive_LNil @ A ) )
| ? [X3: A] :
( A3
= ( coinductive_LCons @ A @ X3 @ ( coinductive_LNil @ A ) ) )
| ? [X3: A,Y3: A,Xs3: coinductive_llist @ A] :
( ( A3
= ( coinductive_LCons @ A @ X3 @ ( coinductive_LCons @ A @ Y3 @ Xs3 ) ) )
& ( ord_less_eq @ A @ X3 @ Y3 )
& ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ Y3 @ Xs3 ) ) ) ) ) ) ) ).
% lsorted.simps
thf(fact_82_lsorted_Ocases,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: coinductive_llist @ A] :
( ( coindu63249387sorted @ A @ A2 )
=> ( ( A2
!= ( coinductive_LNil @ A ) )
=> ( ! [X2: A] :
( A2
!= ( coinductive_LCons @ A @ X2 @ ( coinductive_LNil @ A ) ) )
=> ~ ! [X2: A,Y2: A,Xs2: coinductive_llist @ A] :
( ( A2
= ( coinductive_LCons @ A @ X2 @ ( coinductive_LCons @ A @ Y2 @ Xs2 ) ) )
=> ( ( ord_less_eq @ A @ X2 @ Y2 )
=> ~ ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ Y2 @ Xs2 ) ) ) ) ) ) ) ) ).
% lsorted.cases
thf(fact_83_lstrict__prefix__lappend__conv,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coindu1478340336prefix @ A @ Xs @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( ( coinductive_lfinite @ A @ Xs )
& ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% lstrict_prefix_lappend_conv
thf(fact_84_split__llist__first,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
=> ? [Ys4: coinductive_llist @ A,Zs2: coinductive_llist @ A] :
( ( Xs
= ( coinductive_lappend @ A @ Ys4 @ ( coinductive_LCons @ A @ X @ Zs2 ) ) )
& ( coinductive_lfinite @ A @ Ys4 )
& ~ ( member @ A @ X @ ( coinductive_lset @ A @ Ys4 ) ) ) ) ).
% split_llist_first
thf(fact_85_split__llist,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
=> ? [Ys4: coinductive_llist @ A,Zs2: coinductive_llist @ A] :
( ( Xs
= ( coinductive_lappend @ A @ Ys4 @ ( coinductive_LCons @ A @ X @ Zs2 ) ) )
& ( coinductive_lfinite @ A @ Ys4 ) ) ) ).
% split_llist
thf(fact_86_lappend__ltakeWhile__ldropWhile,axiom,
! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) @ ( coindu218763757pWhile @ A @ P @ Xs ) )
= Xs ) ).
% lappend_ltakeWhile_ldropWhile
thf(fact_87_lfilter__idem,axiom,
! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
( ( coinductive_lfilter @ A @ P @ ( coinductive_lfilter @ A @ P @ Xs ) )
= ( coinductive_lfilter @ A @ P @ Xs ) ) ).
% lfilter_idem
thf(fact_88_lfilter__K__True,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lfilter @ A
@ ^ [Uu: A] : $true
@ Xs )
= Xs ) ).
% lfilter_K_True
thf(fact_89_lappend_Odisc__iff_I2_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ~ ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) )
= ( ~ ( coinductive_lnull @ A @ Xs )
| ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% lappend.disc_iff(2)
thf(fact_90_lnull__lappend,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( ( coinductive_lnull @ A @ Xs )
& ( coinductive_lnull @ A @ Ys ) ) ) ).
% lnull_lappend
thf(fact_91_lfilter__LCons,axiom,
! [A: $tType,P: A > $o,X: A,Xs: coinductive_llist @ A] :
( ( ( P @ X )
=> ( ( coinductive_lfilter @ A @ P @ ( coinductive_LCons @ A @ X @ Xs ) )
= ( coinductive_LCons @ A @ X @ ( coinductive_lfilter @ A @ P @ Xs ) ) ) )
& ( ~ ( P @ X )
=> ( ( coinductive_lfilter @ A @ P @ ( coinductive_LCons @ A @ X @ Xs ) )
= ( coinductive_lfilter @ A @ P @ Xs ) ) ) ) ).
% lfilter_LCons
thf(fact_92_lfilter__LNil,axiom,
! [A: $tType,P: A > $o] :
( ( coinductive_lfilter @ A @ P @ ( coinductive_LNil @ A ) )
= ( coinductive_LNil @ A ) ) ).
% lfilter_LNil
thf(fact_93_ldropWhile__LCons,axiom,
! [A: $tType,P: A > $o,X: A,Xs: coinductive_llist @ A] :
( ( ( P @ X )
=> ( ( coindu218763757pWhile @ A @ P @ ( coinductive_LCons @ A @ X @ Xs ) )
= ( coindu218763757pWhile @ A @ P @ Xs ) ) )
& ( ~ ( P @ X )
=> ( ( coindu218763757pWhile @ A @ P @ ( coinductive_LCons @ A @ X @ Xs ) )
= ( coinductive_LCons @ A @ X @ Xs ) ) ) ) ).
% ldropWhile_LCons
thf(fact_94_ldropWhile__LNil,axiom,
! [A: $tType,P: A > $o] :
( ( coindu218763757pWhile @ A @ P @ ( coinductive_LNil @ A ) )
= ( coinductive_LNil @ A ) ) ).
% ldropWhile_LNil
thf(fact_95_unfold__llist_Odisc__iff_I2_J,axiom,
! [B: $tType,A: $tType,P3: A > $o,G212: A > B,G223: A > A,A2: A] :
( ( ~ ( coinductive_lnull @ B @ ( coindu1441602521_llist @ A @ B @ P3 @ G212 @ G223 @ A2 ) ) )
= ( ~ ( P3 @ A2 ) ) ) ).
% unfold_llist.disc_iff(2)
thf(fact_96_unfold__llist_Odisc__iff_I1_J,axiom,
! [B: $tType,A: $tType,P3: A > $o,G212: A > B,G223: A > A,A2: A] :
( ( coinductive_lnull @ B @ ( coindu1441602521_llist @ A @ B @ P3 @ G212 @ G223 @ A2 ) )
= ( P3 @ A2 ) ) ).
% unfold_llist.disc_iff(1)
thf(fact_97_llist_Ocorec__disc__iff_I2_J,axiom,
! [A: $tType,C: $tType,P3: C > $o,G212: C > A,Q222: C > $o,G2212: C > ( coinductive_llist @ A ),G2222: C > C,A2: C] :
( ( ~ ( coinductive_lnull @ A @ ( coindu1259883913_llist @ C @ A @ P3 @ G212 @ Q222 @ G2212 @ G2222 @ A2 ) ) )
= ( ~ ( P3 @ A2 ) ) ) ).
% llist.corec_disc_iff(2)
thf(fact_98_llist_Ocorec__disc__iff_I1_J,axiom,
! [A: $tType,C: $tType,P3: C > $o,G212: C > A,Q222: C > $o,G2212: C > ( coinductive_llist @ A ),G2222: C > C,A2: C] :
( ( coinductive_lnull @ A @ ( coindu1259883913_llist @ C @ A @ P3 @ G212 @ Q222 @ G2212 @ G2222 @ A2 ) )
= ( P3 @ A2 ) ) ).
% llist.corec_disc_iff(1)
thf(fact_99_lfitler__K__False,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lfilter @ A
@ ^ [Uu: A] : $false
@ Xs )
= ( coinductive_LNil @ A ) ) ).
% lfitler_K_False
thf(fact_100_lset__lfilter,axiom,
! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
( ( coinductive_lset @ A @ ( coinductive_lfilter @ A @ P @ Xs ) )
= ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
& ( P @ X3 ) ) ) ) ).
% lset_lfilter
thf(fact_101_ldropWhile__K__True,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coindu218763757pWhile @ A
@ ^ [Uu: A] : $true
@ Xs )
= ( coinductive_LNil @ A ) ) ).
% ldropWhile_K_True
thf(fact_102_lsorted__LCons__LCons,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Xs: coinductive_llist @ A] :
( ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LCons @ A @ Y @ Xs ) ) )
= ( ( ord_less_eq @ A @ X @ Y )
& ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ Y @ Xs ) ) ) ) ) ).
% lsorted_LCons_LCons
thf(fact_103_diverge__lfilter__LNil,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
( ! [X2: A] :
( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
=> ~ ( P @ X2 ) )
=> ( ( coinductive_lfilter @ A @ P @ Xs )
= ( coinductive_LNil @ A ) ) ) ).
% diverge_lfilter_LNil
thf(fact_104_lnull__lfilter,axiom,
! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ ( coinductive_lfilter @ A @ P @ Xs ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
=> ~ ( P @ X3 ) ) ) ) ).
% lnull_lfilter
thf(fact_105_lnull__ldropWhile,axiom,
! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ ( coindu218763757pWhile @ A @ P @ Xs ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
=> ( P @ X3 ) ) ) ) ).
% lnull_ldropWhile
thf(fact_106_lfilter__lappend__lfinite,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P: A > $o,Ys: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_lfilter @ A @ P @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( coinductive_lappend @ A @ ( coinductive_lfilter @ A @ P @ Xs ) @ ( coinductive_lfilter @ A @ P @ Ys ) ) ) ) ).
% lfilter_lappend_lfinite
thf(fact_107_lfinite__ldropWhile,axiom,
! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ ( coindu218763757pWhile @ A @ P @ Xs ) )
= ( ? [X3: A] :
( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
& ~ ( P @ X3 ) )
=> ( coinductive_lfinite @ A @ Xs ) ) ) ).
% lfinite_ldropWhile
thf(fact_108_llist_Odisc__eq__case_I1_J,axiom,
! [A: $tType] :
( ( coinductive_lnull @ A )
= ( coindu1381640503_llist @ $o @ A @ $true
@ ^ [Uu: A,Uv: coinductive_llist @ A] : $false ) ) ).
% llist.disc_eq_case(1)
thf(fact_109_llist_Odisc__eq__case_I2_J,axiom,
! [A: $tType,Llist: coinductive_llist @ A] :
( ( ~ ( coinductive_lnull @ A @ Llist ) )
= ( coindu1381640503_llist @ $o @ A @ $false
@ ^ [Uu: A,Uv: coinductive_llist @ A] : $true
@ Llist ) ) ).
% llist.disc_eq_case(2)
thf(fact_110_lfilter__cong,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,P: A > $o,Q: A > $o] :
( ( Xs = Ys )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ ( coinductive_lset @ A @ Ys ) )
=> ( ( P @ X2 )
= ( Q @ X2 ) ) )
=> ( ( coinductive_lfilter @ A @ P @ Xs )
= ( coinductive_lfilter @ A @ Q @ Ys ) ) ) ) ).
% lfilter_cong
thf(fact_111_lzip_Oexhaust,axiom,
! [A: $tType,B: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
( ~ ( ( coinductive_lnull @ A @ Xs )
| ( coinductive_lnull @ B @ Ys ) )
=> ~ ( ~ ( coinductive_lnull @ A @ Xs )
=> ( coinductive_lnull @ B @ Ys ) ) ) ).
% lzip.exhaust
thf(fact_112_lappend_Oexhaust,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ( coinductive_lnull @ A @ Xs )
=> ~ ( coinductive_lnull @ A @ Ys ) )
=> ( ~ ( coinductive_lnull @ A @ Xs )
| ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% lappend.exhaust
thf(fact_113_ldropWhile__cong,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,P: A > $o,Q: A > $o] :
( ( Xs = Ys )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ ( coinductive_lset @ A @ Ys ) )
=> ( ( P @ X2 )
= ( Q @ X2 ) ) )
=> ( ( coindu218763757pWhile @ A @ P @ Xs )
= ( coindu218763757pWhile @ A @ Q @ Ys ) ) ) ) ).
% ldropWhile_cong
thf(fact_114_lfilter__id__conv,axiom,
! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
( ( ( coinductive_lfilter @ A @ P @ Xs )
= Xs )
= ( ! [X3: A] :
( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
=> ( P @ X3 ) ) ) ) ).
% lfilter_id_conv
thf(fact_115_wlog__linorder__le,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > A > $o,B2: A,A2: A] :
( ! [A5: A,B3: A] :
( ( ord_less_eq @ A @ A5 @ B3 )
=> ( P @ A5 @ B3 ) )
=> ( ( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% wlog_linorder_le
thf(fact_116_lfilter__empty__conv,axiom,
! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
( ( ( coinductive_lfilter @ A @ P @ Xs )
= ( coinductive_LNil @ A ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
=> ~ ( P @ X3 ) ) ) ) ).
% lfilter_empty_conv
thf(fact_117_in__lset__ldropWhileD,axiom,
! [A: $tType,X: A,P: A > $o,Xs: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ ( coindu218763757pWhile @ A @ P @ Xs ) ) )
=> ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) ) ) ).
% in_lset_ldropWhileD
thf(fact_118_lset__ldropWhile__subset,axiom,
! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] : ( ord_less_eq @ ( set @ A ) @ ( coinductive_lset @ A @ ( coindu218763757pWhile @ A @ P @ Xs ) ) @ ( coinductive_lset @ A @ Xs ) ) ).
% lset_ldropWhile_subset
thf(fact_119_lset__lappend1,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] : ( ord_less_eq @ ( set @ A ) @ ( coinductive_lset @ A @ Xs ) @ ( coinductive_lset @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) ) ).
% lset_lappend1
thf(fact_120_lfilter__lfilter,axiom,
! [A: $tType,P: A > $o,Q: A > $o,Xs: coinductive_llist @ A] :
( ( coinductive_lfilter @ A @ P @ ( coinductive_lfilter @ A @ Q @ Xs ) )
= ( coinductive_lfilter @ A
@ ^ [X3: A] :
( ( P @ X3 )
& ( Q @ X3 ) )
@ Xs ) ) ).
% lfilter_lfilter
thf(fact_121_ldropWhile__eq__LNil__iff,axiom,
! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
( ( ( coindu218763757pWhile @ A @ P @ Xs )
= ( coinductive_LNil @ A ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
=> ( P @ X3 ) ) ) ) ).
% ldropWhile_eq_LNil_iff
thf(fact_122_ltakeWhile__all__conv,axiom,
! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
( ( ( coindu501562517eWhile @ A @ P @ Xs )
= Xs )
= ( ord_less_eq @ ( set @ A ) @ ( coinductive_lset @ A @ Xs ) @ ( collect @ A @ P ) ) ) ).
% ltakeWhile_all_conv
thf(fact_123_lsorted__LCons,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Xs: coinductive_llist @ A] :
( ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
= ( ( coindu63249387sorted @ A @ Xs )
& ! [X3: A] :
( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
=> ( ord_less_eq @ A @ X @ X3 ) ) ) ) ) ).
% lsorted_LCons
thf(fact_124_lfilter__LCons__seek,axiom,
! [A: $tType,P3: A > $o,X: A,L: coinductive_llist @ A] :
( ~ ( P3 @ X )
=> ( ( coinductive_lfilter @ A @ P3 @ ( coinductive_LCons @ A @ X @ L ) )
= ( coinductive_lfilter @ A @ P3 @ L ) ) ) ).
% lfilter_LCons_seek
thf(fact_125_lfilter__LCons__found,axiom,
! [A: $tType,P: A > $o,X: A,Xs: coinductive_llist @ A] :
( ( P @ X )
=> ( ( coinductive_lfilter @ A @ P @ ( coinductive_LCons @ A @ X @ Xs ) )
= ( coinductive_LCons @ A @ X @ ( coinductive_lfilter @ A @ P @ Xs ) ) ) ) ).
% lfilter_LCons_found
thf(fact_126_lset__intros_I2_J,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A,X5: A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
=> ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ X5 @ Xs ) ) ) ) ).
% lset_intros(2)
thf(fact_127_lset__intros_I1_J,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A] : ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ X @ Xs ) ) ) ).
% lset_intros(1)
thf(fact_128_llist_Oset__intros_I2_J,axiom,
! [A: $tType,X: A,A22: coinductive_llist @ A,A1: A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ A22 ) )
=> ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ A1 @ A22 ) ) ) ) ).
% llist.set_intros(2)
thf(fact_129_llist_Oset__intros_I1_J,axiom,
! [A: $tType,A1: A,A22: coinductive_llist @ A] : ( member @ A @ A1 @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ A1 @ A22 ) ) ) ).
% llist.set_intros(1)
thf(fact_130_lset__cases,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
=> ( ! [Xs6: coinductive_llist @ A] :
( Xs
!= ( coinductive_LCons @ A @ X @ Xs6 ) )
=> ~ ! [X6: A,Xs6: coinductive_llist @ A] :
( ( Xs
= ( coinductive_LCons @ A @ X6 @ Xs6 ) )
=> ~ ( member @ A @ X @ ( coinductive_lset @ A @ Xs6 ) ) ) ) ) ).
% lset_cases
thf(fact_131_lset__induct,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
=> ( ! [Xs2: coinductive_llist @ A] : ( P @ ( coinductive_LCons @ A @ X @ Xs2 ) )
=> ( ! [X6: A,Xs2: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs2 ) )
=> ( ( X != X6 )
=> ( ( P @ Xs2 )
=> ( P @ ( coinductive_LCons @ A @ X6 @ Xs2 ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% lset_induct
thf(fact_132_lset__induct_H,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
=> ( ! [Xs2: coinductive_llist @ A] : ( P @ ( coinductive_LCons @ A @ X @ Xs2 ) )
=> ( ! [X6: A,Xs2: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs2 ) )
=> ( ( P @ Xs2 )
=> ( P @ ( coinductive_LCons @ A @ X6 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% lset_induct'
thf(fact_133_llist_Oset__cases,axiom,
! [A: $tType,E: A,A2: coinductive_llist @ A] :
( ( member @ A @ E @ ( coinductive_lset @ A @ A2 ) )
=> ( ! [Z2: coinductive_llist @ A] :
( A2
!= ( coinductive_LCons @ A @ E @ Z2 ) )
=> ~ ! [Z1: A,Z2: coinductive_llist @ A] :
( ( A2
= ( coinductive_LCons @ A @ Z1 @ Z2 ) )
=> ~ ( member @ A @ E @ ( coinductive_lset @ A @ Z2 ) ) ) ) ) ).
% llist.set_cases
thf(fact_134_llist_Oset__induct,axiom,
! [A: $tType,X: A,A2: coinductive_llist @ A,P: A > ( coinductive_llist @ A ) > $o] :
( ( member @ A @ X @ ( coinductive_lset @ A @ A2 ) )
=> ( ! [Z1: A,Z2: coinductive_llist @ A] : ( P @ Z1 @ ( coinductive_LCons @ A @ Z1 @ Z2 ) )
=> ( ! [Z1: A,Z2: coinductive_llist @ A,Xa3: A] :
( ( member @ A @ Xa3 @ ( coinductive_lset @ A @ Z2 ) )
=> ( ( P @ Xa3 @ Z2 )
=> ( P @ Xa3 @ ( coinductive_LCons @ A @ Z1 @ Z2 ) ) ) )
=> ( P @ X @ A2 ) ) ) ) ).
% llist.set_induct
thf(fact_135_llist_Odisc_I2_J,axiom,
! [A: $tType,X21: A,X22: coinductive_llist @ A] :
~ ( coinductive_lnull @ A @ ( coinductive_LCons @ A @ X21 @ X22 ) ) ).
% llist.disc(2)
thf(fact_136_llist_OdiscI_I2_J,axiom,
! [A: $tType,Llist: coinductive_llist @ A,X21: A,X22: coinductive_llist @ A] :
( ( Llist
= ( coinductive_LCons @ A @ X21 @ X22 ) )
=> ~ ( coinductive_lnull @ A @ Llist ) ) ).
% llist.discI(2)
thf(fact_137_not__lnull__conv,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( ~ ( coinductive_lnull @ A @ Xs ) )
= ( ? [X3: A,Xs5: coinductive_llist @ A] :
( Xs
= ( coinductive_LCons @ A @ X3 @ Xs5 ) ) ) ) ).
% not_lnull_conv
thf(fact_138_llist_Odisc_I1_J,axiom,
! [A: $tType] : ( coinductive_lnull @ A @ ( coinductive_LNil @ A ) ) ).
% llist.disc(1)
thf(fact_139_llist_OdiscI_I1_J,axiom,
! [A: $tType,Llist: coinductive_llist @ A] :
( ( Llist
= ( coinductive_LNil @ A ) )
=> ( coinductive_lnull @ A @ Llist ) ) ).
% llist.discI(1)
thf(fact_140_llist_Ocollapse_I1_J,axiom,
! [A: $tType,Llist: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Llist )
=> ( Llist
= ( coinductive_LNil @ A ) ) ) ).
% llist.collapse(1)
thf(fact_141_lnull__def,axiom,
! [A: $tType] :
( ( coinductive_lnull @ A )
= ( ^ [Llist2: coinductive_llist @ A] :
( Llist2
= ( coinductive_LNil @ A ) ) ) ) ).
% lnull_def
thf(fact_142_lfinite__lfilterI,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
( ( coinductive_lfinite @ A @ Xs )
=> ( coinductive_lfinite @ A @ ( coinductive_lfilter @ A @ P @ Xs ) ) ) ).
% lfinite_lfilterI
thf(fact_143_lappend_Odisc_I2_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ~ ( coinductive_lnull @ A @ Xs )
| ~ ( coinductive_lnull @ A @ Ys ) )
=> ~ ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) ) ).
% lappend.disc(2)
thf(fact_144_lappend_Odisc_I1_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_lnull @ A @ Ys )
=> ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) ) ) ).
% lappend.disc(1)
thf(fact_145_lappend__lnull1,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_lappend @ A @ Xs @ Ys )
= Ys ) ) ).
% lappend_lnull1
thf(fact_146_lappend__lnull2,axiom,
! [A: $tType,Ys: coinductive_llist @ A,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Ys )
=> ( ( coinductive_lappend @ A @ Xs @ Ys )
= Xs ) ) ).
% lappend_lnull2
thf(fact_147_lnull__imp__lfinite,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( coinductive_lfinite @ A @ Xs ) ) ).
% lnull_imp_lfinite
thf(fact_148_ltakeWhile__all,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
( ! [X2: A] :
( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
=> ( P @ X2 ) )
=> ( ( coindu501562517eWhile @ A @ P @ Xs )
= Xs ) ) ).
% ltakeWhile_all
thf(fact_149_ltakeWhile__cong,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,P: A > $o,Q: A > $o] :
( ( Xs = Ys )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ ( coinductive_lset @ A @ Ys ) )
=> ( ( P @ X2 )
= ( Q @ X2 ) ) )
=> ( ( coindu501562517eWhile @ A @ P @ Xs )
= ( coindu501562517eWhile @ A @ Q @ Ys ) ) ) ) ).
% ltakeWhile_cong
thf(fact_150_lset__ltakeWhileD,axiom,
! [A: $tType,X: A,P: A > $o,Xs: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) ) )
=> ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
& ( P @ X ) ) ) ).
% lset_ltakeWhileD
thf(fact_151_lsorted__lfilterI,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [Xs: coinductive_llist @ A,P: A > $o] :
( ( coindu63249387sorted @ A @ Xs )
=> ( coindu63249387sorted @ A @ ( coinductive_lfilter @ A @ P @ Xs ) ) ) ) ).
% lsorted_lfilterI
thf(fact_152_ltakeWhile__lappend2,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P: A > $o,Ys: coinductive_llist @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( coinductive_lset @ A @ Xs ) @ ( collect @ A @ P ) )
=> ( ( coindu501562517eWhile @ A @ P @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( coinductive_lappend @ A @ Xs @ ( coindu501562517eWhile @ A @ P @ Ys ) ) ) ) ).
% ltakeWhile_lappend2
thf(fact_153_unfold__llist_Odisc_I2_J,axiom,
! [B: $tType,A: $tType,P3: A > $o,A2: A,G212: A > B,G223: A > A] :
( ~ ( P3 @ A2 )
=> ~ ( coinductive_lnull @ B @ ( coindu1441602521_llist @ A @ B @ P3 @ G212 @ G223 @ A2 ) ) ) ).
% unfold_llist.disc(2)
thf(fact_154_unfold__llist_Odisc_I1_J,axiom,
! [B: $tType,A: $tType,P3: A > $o,A2: A,G212: A > B,G223: A > A] :
( ( P3 @ A2 )
=> ( coinductive_lnull @ B @ ( coindu1441602521_llist @ A @ B @ P3 @ G212 @ G223 @ A2 ) ) ) ).
% unfold_llist.disc(1)
thf(fact_155_llist_Ocorec__disc_I2_J,axiom,
! [A: $tType,C: $tType,P3: C > $o,A2: C,G212: C > A,Q222: C > $o,G2212: C > ( coinductive_llist @ A ),G2222: C > C] :
( ~ ( P3 @ A2 )
=> ~ ( coinductive_lnull @ A @ ( coindu1259883913_llist @ C @ A @ P3 @ G212 @ Q222 @ G2212 @ G2222 @ A2 ) ) ) ).
% llist.corec_disc(2)
thf(fact_156_llist_Ocorec__disc_I1_J,axiom,
! [A: $tType,C: $tType,P3: C > $o,A2: C,G212: C > A,Q222: C > $o,G2212: C > ( coinductive_llist @ A ),G2222: C > C] :
( ( P3 @ A2 )
=> ( coinductive_lnull @ A @ ( coindu1259883913_llist @ C @ A @ P3 @ G212 @ Q222 @ G2212 @ G2222 @ A2 ) ) ) ).
% llist.corec_disc(1)
thf(fact_157_lfilter__eq__LConsD,axiom,
! [A: $tType,P: A > $o,Ys: coinductive_llist @ A,X: A,Xs: coinductive_llist @ A] :
( ( ( coinductive_lfilter @ A @ P @ Ys )
= ( coinductive_LCons @ A @ X @ Xs ) )
=> ? [Us: coinductive_llist @ A,Vs: coinductive_llist @ A] :
( ( Ys
= ( coinductive_lappend @ A @ Us @ ( coinductive_LCons @ A @ X @ Vs ) ) )
& ( coinductive_lfinite @ A @ Us )
& ! [X7: A] :
( ( member @ A @ X7 @ ( coinductive_lset @ A @ Us ) )
=> ~ ( P @ X7 ) )
& ( P @ X )
& ( Xs
= ( coinductive_lfilter @ A @ P @ Vs ) ) ) ) ).
% lfilter_eq_LConsD
thf(fact_158_lsorted__lfilter__same,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [C2: A,Xs: coinductive_llist @ A] :
( coindu63249387sorted @ A
@ ( coinductive_lfilter @ A
@ ^ [X3: A] : X3 = C2
@ Xs ) ) ) ).
% lsorted_lfilter_same
thf(fact_159_lset__lmember,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
= ( coinductive_lmember @ A @ X @ Xs ) ) ).
% lset_lmember
thf(fact_160_lfilter__eq__lappend__lfiniteD,axiom,
! [A: $tType,P: A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
( ( ( coinductive_lfilter @ A @ P @ Xs )
= ( coinductive_lappend @ A @ Ys @ Zs ) )
=> ( ( coinductive_lfinite @ A @ Ys )
=> ? [Us: coinductive_llist @ A,Vs: coinductive_llist @ A] :
( ( Xs
= ( coinductive_lappend @ A @ Us @ Vs ) )
& ( coinductive_lfinite @ A @ Us )
& ( Ys
= ( coinductive_lfilter @ A @ P @ Us ) )
& ( Zs
= ( coinductive_lfilter @ A @ P @ Vs ) ) ) ) ) ).
% lfilter_eq_lappend_lfiniteD
thf(fact_161_in__lset__lappend__iff,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) )
= ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
| ( ( coinductive_lfinite @ A @ Xs )
& ( member @ A @ X @ ( coinductive_lset @ A @ Ys ) ) ) ) ) ).
% in_lset_lappend_iff
thf(fact_162_lappend_Octr_I1_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_lnull @ A @ Ys )
=> ( ( coinductive_lappend @ A @ Xs @ Ys )
= ( coinductive_LNil @ A ) ) ) ) ).
% lappend.ctr(1)
thf(fact_163_ltakeWhile__lappend1,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A,P: A > $o,Ys: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
=> ( ~ ( P @ X )
=> ( ( coindu501562517eWhile @ A @ P @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( coindu501562517eWhile @ A @ P @ Xs ) ) ) ) ).
% ltakeWhile_lappend1
thf(fact_164_lfinite__ltakeWhile,axiom,
! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) )
= ( ( coinductive_lfinite @ A @ Xs )
| ? [X3: A] :
( ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs ) )
& ~ ( P @ X3 ) ) ) ) ).
% lfinite_ltakeWhile
thf(fact_165_llast__LCons,axiom,
! [A: $tType,Xs: coinductive_llist @ A,X: A] :
( ( ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
= X ) )
& ( ~ ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
= ( coinductive_llast @ A @ Xs ) ) ) ) ).
% llast_LCons
thf(fact_166_LCons__LCons,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Xs: coinductive_llist @ A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ Y @ Xs ) )
=> ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LCons @ A @ Y @ Xs ) ) ) ) ) ) ).
% LCons_LCons
thf(fact_167_ldropWhile_Osimps,axiom,
! [A: $tType] :
( ( coindu218763757pWhile @ A )
= ( ^ [P4: A > $o,Xs3: coinductive_llist @ A] :
( coindu1381640503_llist @ ( coinductive_llist @ A ) @ A @ ( coinductive_LNil @ A )
@ ^ [X3: A,Xs5: coinductive_llist @ A] : ( if @ ( coinductive_llist @ A ) @ ( P4 @ X3 ) @ ( coindu218763757pWhile @ A @ P4 @ Xs5 ) @ Xs3 )
@ Xs3 ) ) ) ).
% ldropWhile.simps
thf(fact_168_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_169_reachable__viap_Ocases,axiom,
! [Node: $tType,Graph: Node > Node > $o,Ns: set @ Node,N: Node,A2: Node] :
( ( koenig1757754772e_viap @ Node @ Graph @ Ns @ N @ A2 )
=> ~ ! [Xs2: coinductive_llist @ Node] :
( ( koenig2031690877pathsp @ Node @ Graph @ ( coinductive_LCons @ Node @ N @ Xs2 ) )
=> ( ( member @ Node @ A2 @ ( coinductive_lset @ Node @ Xs2 ) )
=> ~ ( ord_less_eq @ ( set @ Node ) @ ( coinductive_lset @ Node @ Xs2 ) @ Ns ) ) ) ) ).
% reachable_viap.cases
thf(fact_170_reachable__viap_Osimps,axiom,
! [Node: $tType] :
( ( koenig1757754772e_viap @ Node )
= ( ^ [Graph2: Node > Node > $o,Ns2: set @ Node,N2: Node,A3: Node] :
? [Xs3: coinductive_llist @ Node,N3: Node] :
( ( A3 = N3 )
& ( koenig2031690877pathsp @ Node @ Graph2 @ ( coinductive_LCons @ Node @ N2 @ Xs3 ) )
& ( member @ Node @ N3 @ ( coinductive_lset @ Node @ Xs3 ) )
& ( ord_less_eq @ ( set @ Node ) @ ( coinductive_lset @ Node @ Xs3 ) @ Ns2 ) ) ) ) ).
% reachable_viap.simps
thf(fact_171_reachable__viap_Ointros,axiom,
! [Node: $tType,Graph: Node > Node > $o,N: Node,Xs: coinductive_llist @ Node,N4: Node,Ns: set @ Node] :
( ( koenig2031690877pathsp @ Node @ Graph @ ( coinductive_LCons @ Node @ N @ Xs ) )
=> ( ( member @ Node @ N4 @ ( coinductive_lset @ Node @ Xs ) )
=> ( ( ord_less_eq @ ( set @ Node ) @ ( coinductive_lset @ Node @ Xs ) @ Ns )
=> ( koenig1757754772e_viap @ Node @ Graph @ Ns @ N @ N4 ) ) ) ) ).
% reachable_viap.intros
thf(fact_172_reachable__viap_Oinducts,axiom,
! [Node: $tType,Graph: Node > Node > $o,Ns: set @ Node,N: Node,X: Node,P: Node > $o] :
( ( koenig1757754772e_viap @ Node @ Graph @ Ns @ N @ X )
=> ( ! [Xs2: coinductive_llist @ Node,N5: Node] :
( ( koenig2031690877pathsp @ Node @ Graph @ ( coinductive_LCons @ Node @ N @ Xs2 ) )
=> ( ( member @ Node @ N5 @ ( coinductive_lset @ Node @ Xs2 ) )
=> ( ( ord_less_eq @ ( set @ Node ) @ ( coinductive_lset @ Node @ Xs2 ) @ Ns )
=> ( P @ N5 ) ) ) )
=> ( P @ X ) ) ) ).
% reachable_viap.inducts
thf(fact_173_reachable__via_Ocases,axiom,
! [Node: $tType,A2: Node,Graph: Node > Node > $o,Ns: set @ Node,N: Node] :
( ( member @ Node @ A2 @ ( koenig317145564le_via @ Node @ Graph @ Ns @ N ) )
=> ~ ! [Xs2: coinductive_llist @ Node] :
( ( member @ ( coinductive_llist @ Node ) @ ( coinductive_LCons @ Node @ N @ Xs2 ) @ ( koenig916195507_paths @ Node @ Graph ) )
=> ( ( member @ Node @ A2 @ ( coinductive_lset @ Node @ Xs2 ) )
=> ~ ( ord_less_eq @ ( set @ Node ) @ ( coinductive_lset @ Node @ Xs2 ) @ Ns ) ) ) ) ).
% reachable_via.cases
thf(fact_174_reachable__viap__reachable__via__eq,axiom,
! [Node: $tType] :
( ( koenig1757754772e_viap @ Node )
= ( ^ [Graph2: Node > Node > $o,Ns2: set @ Node,N2: Node,X3: Node] : ( member @ Node @ X3 @ ( koenig317145564le_via @ Node @ Graph2 @ Ns2 @ N2 ) ) ) ) ).
% reachable_viap_reachable_via_eq
thf(fact_175_reachable__via__def,axiom,
! [Node: $tType] :
( ( koenig317145564le_via @ Node )
= ( ^ [Graph2: Node > Node > $o,Ns2: set @ Node,N2: Node] : ( collect @ Node @ ( koenig1757754772e_viap @ Node @ Graph2 @ Ns2 @ N2 ) ) ) ) ).
% reachable_via_def
thf(fact_176_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% le_funD
thf(fact_177_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% le_funE
thf(fact_178_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B] :
( ! [X2: A] : ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) )
=> ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% le_funI
thf(fact_179_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F3: A > B,G2: A > B] :
! [X3: A] : ( ord_less_eq @ B @ ( F3 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% le_fun_def
thf(fact_180_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B > A,B2: B,C2: B] :
( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C2 )
=> ( ! [X2: B,Y2: B] :
( ( ord_less_eq @ B @ X2 @ Y2 )
=> ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% order_subst1
thf(fact_181_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C @ ( type2 @ C ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B2: A,F: A > C,C2: C] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ C @ ( F @ B2 ) @ C2 )
=> ( ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ord_less_eq @ C @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ C @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% order_subst2
thf(fact_182_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B > A,B2: B,C2: B] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C2 )
=> ( ! [X2: B,Y2: B] :
( ( ord_less_eq @ B @ X2 @ Y2 )
=> ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_183_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B2: A,F: A > B,C2: B] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ord_less_eq @ B @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ B @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_184_eq__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ^ [Y5: A,Z: A] : Y5 = Z )
= ( ^ [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
& ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ) ) ).
% eq_iff
thf(fact_185_antisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ X )
=> ( X = Y ) ) ) ) ).
% antisym
thf(fact_186_linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
| ( ord_less_eq @ A @ Y @ X ) ) ) ).
% linear
thf(fact_187_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( X = Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% eq_refl
thf(fact_188_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ~ ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% le_cases
thf(fact_189_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% order.trans
thf(fact_190_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z3: A] :
( ( ( ord_less_eq @ A @ X @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z3 ) )
=> ( ( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less_eq @ A @ X @ Z3 ) )
=> ( ( ( ord_less_eq @ A @ X @ Z3 )
=> ~ ( ord_less_eq @ A @ Z3 @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z3 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z3 )
=> ~ ( ord_less_eq @ A @ Z3 @ X ) )
=> ~ ( ( ord_less_eq @ A @ Z3 @ X )
=> ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_191_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_less_eq @ A @ X @ Y )
= ( X = Y ) ) ) ) ).
% antisym_conv
thf(fact_192_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C2: A] :
( ( A2 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_193_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_194_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ) ).
% order_class.order.antisym
thf(fact_195_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z3: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z3 )
=> ( ord_less_eq @ A @ X @ Z3 ) ) ) ) ).
% order_trans
thf(fact_196_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_197_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > A > $o,A2: A,B2: A] :
( ! [A5: A,B3: A] :
( ( ord_less_eq @ A @ A5 @ B3 )
=> ( P @ A5 @ B3 ) )
=> ( ! [A5: A,B3: A] :
( ( P @ B3 @ A5 )
=> ( P @ A5 @ B3 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_wlog
thf(fact_198_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% dual_order.trans
thf(fact_199_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ) ).
% dual_order.antisym
thf(fact_200_reachable__via_Oinducts,axiom,
! [Node: $tType,X: Node,Graph: Node > Node > $o,Ns: set @ Node,N: Node,P: Node > $o] :
( ( member @ Node @ X @ ( koenig317145564le_via @ Node @ Graph @ Ns @ N ) )
=> ( ! [Xs2: coinductive_llist @ Node,N5: Node] :
( ( member @ ( coinductive_llist @ Node ) @ ( coinductive_LCons @ Node @ N @ Xs2 ) @ ( koenig916195507_paths @ Node @ Graph ) )
=> ( ( member @ Node @ N5 @ ( coinductive_lset @ Node @ Xs2 ) )
=> ( ( ord_less_eq @ ( set @ Node ) @ ( coinductive_lset @ Node @ Xs2 ) @ Ns )
=> ( P @ N5 ) ) ) )
=> ( P @ X ) ) ) ).
% reachable_via.inducts
thf(fact_201_reachable__via_Ointros,axiom,
! [Node: $tType,N: Node,Xs: coinductive_llist @ Node,Graph: Node > Node > $o,N4: Node,Ns: set @ Node] :
( ( member @ ( coinductive_llist @ Node ) @ ( coinductive_LCons @ Node @ N @ Xs ) @ ( koenig916195507_paths @ Node @ Graph ) )
=> ( ( member @ Node @ N4 @ ( coinductive_lset @ Node @ Xs ) )
=> ( ( ord_less_eq @ ( set @ Node ) @ ( coinductive_lset @ Node @ Xs ) @ Ns )
=> ( member @ Node @ N4 @ ( koenig317145564le_via @ Node @ Graph @ Ns @ N ) ) ) ) ) ).
% reachable_via.intros
thf(fact_202_reachable__via_Osimps,axiom,
! [Node: $tType,A2: Node,Graph: Node > Node > $o,Ns: set @ Node,N: Node] :
( ( member @ Node @ A2 @ ( koenig317145564le_via @ Node @ Graph @ Ns @ N ) )
= ( ? [Xs3: coinductive_llist @ Node,N3: Node] :
( ( A2 = N3 )
& ( member @ ( coinductive_llist @ Node ) @ ( coinductive_LCons @ Node @ N @ Xs3 ) @ ( koenig916195507_paths @ Node @ Graph ) )
& ( member @ Node @ N3 @ ( coinductive_lset @ Node @ Xs3 ) )
& ( ord_less_eq @ ( set @ Node ) @ ( coinductive_lset @ Node @ Xs3 ) @ Ns ) ) ) ) ).
% reachable_via.simps
thf(fact_203_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_204_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_205_ldropWhile__lappend,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P: A > $o,Ys: coinductive_llist @ A] :
( ( ? [X7: A] :
( ( member @ A @ X7 @ ( coinductive_lset @ A @ Xs ) )
& ~ ( P @ X7 ) )
=> ( ( coindu218763757pWhile @ A @ P @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( coinductive_lappend @ A @ ( coindu218763757pWhile @ A @ P @ Xs ) @ Ys ) ) )
& ( ~ ? [X2: A] :
( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
& ~ ( P @ X2 ) )
=> ( ( ( coinductive_lfinite @ A @ Xs )
=> ( ( coindu218763757pWhile @ A @ P @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( coindu218763757pWhile @ A @ P @ Ys ) ) )
& ( ~ ( coinductive_lfinite @ A @ Xs )
=> ( ( coindu218763757pWhile @ A @ P @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( coinductive_LNil @ A ) ) ) ) ) ) ).
% ldropWhile_lappend
thf(fact_206_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_207_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_208_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_209_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_210_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_211_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B5: set @ A] :
! [X3: A] :
( ( member @ A @ X3 @ A6 )
=> ( member @ A @ X3 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_212_equalityD1,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( A4 = B4 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ).
% equalityD1
thf(fact_213_equalityD2,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( A4 = B4 )
=> ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ).
% equalityD2
thf(fact_214_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_215_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B5: set @ A] :
! [T: A] :
( ( member @ A @ T @ A6 )
=> ( member @ A @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_216_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_217_subset__refl,axiom,
! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ A4 @ A4 ) ).
% subset_refl
thf(fact_218_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_219_subset__trans,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ C3 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ C3 ) ) ) ).
% subset_trans
thf(fact_220_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y5: set @ A,Z: set @ A] : Y5 = Z )
= ( ^ [A6: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
& ( ord_less_eq @ ( set @ A ) @ B5 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_221_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_222_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_223_ltakeWhile__lappend,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P: A > $o,Ys: coinductive_llist @ A] :
( ( ? [X7: A] :
( ( member @ A @ X7 @ ( coinductive_lset @ A @ Xs ) )
& ~ ( P @ X7 ) )
=> ( ( coindu501562517eWhile @ A @ P @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( coindu501562517eWhile @ A @ P @ Xs ) ) )
& ( ~ ? [X2: A] :
( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs ) )
& ~ ( P @ X2 ) )
=> ( ( coindu501562517eWhile @ A @ P @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( coinductive_lappend @ A @ Xs @ ( coindu501562517eWhile @ A @ P @ Ys ) ) ) ) ) ).
% ltakeWhile_lappend
thf(fact_224_less__eq__set__def,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B5: set @ A] :
( ord_less_eq @ ( A > $o )
@ ^ [X3: A] : ( member @ A @ X3 @ A6 )
@ ^ [X3: A] : ( member @ A @ X3 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_225_pred__subset__eq,axiom,
! [A: $tType,R: set @ A,S: set @ A] :
( ( ord_less_eq @ ( A > $o )
@ ^ [X3: A] : ( member @ A @ X3 @ R )
@ ^ [X3: A] : ( member @ A @ X3 @ S ) )
= ( ord_less_eq @ ( set @ A ) @ R @ S ) ) ).
% pred_subset_eq
thf(fact_226_conj__subset__def,axiom,
! [A: $tType,A4: set @ A,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ A4
@ ( collect @ A
@ ^ [X3: A] :
( ( P @ X3 )
& ( Q @ X3 ) ) ) )
= ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( collect @ A @ P ) )
& ( ord_less_eq @ ( set @ A ) @ A4 @ ( collect @ A @ Q ) ) ) ) ).
% conj_subset_def
thf(fact_227_predicate1I,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X2: A] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq @ ( A > $o ) @ P @ Q ) ) ).
% predicate1I
thf(fact_228_rev__predicate1D,axiom,
! [A: $tType,P: A > $o,X: A,Q: A > $o] :
( ( P @ X )
=> ( ( ord_less_eq @ ( A > $o ) @ P @ Q )
=> ( Q @ X ) ) ) ).
% rev_predicate1D
thf(fact_229_predicate1D,axiom,
! [A: $tType,P: A > $o,Q: A > $o,X: A] :
( ( ord_less_eq @ ( A > $o ) @ P @ Q )
=> ( ( P @ X )
=> ( Q @ X ) ) ) ).
% predicate1D
thf(fact_230_subset__Collect__iff,axiom,
! [A: $tType,B4: set @ A,A4: set @ A,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B4 @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4
@ ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ A4 )
& ( P @ X3 ) ) ) )
= ( ! [X3: A] :
( ( member @ A @ X3 @ B4 )
=> ( P @ X3 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_231_subset__CollectI,axiom,
! [A: $tType,B4: set @ A,A4: set @ A,Q: A > $o,P: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B4 @ A4 )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ B4 )
=> ( ( Q @ X2 )
=> ( P @ X2 ) ) )
=> ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ B4 )
& ( Q @ X3 ) ) )
@ ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ A4 )
& ( P @ X3 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_232_prop__restrict,axiom,
! [A: $tType,X: A,Z4: set @ A,X4: set @ A,P: A > $o] :
( ( member @ A @ X @ Z4 )
=> ( ( ord_less_eq @ ( set @ A ) @ Z4
@ ( collect @ A
@ ^ [X3: A] :
( ( member @ A @ X3 @ X4 )
& ( P @ X3 ) ) ) )
=> ( P @ X ) ) ) ).
% prop_restrict
thf(fact_233_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_234_llast__lappend,axiom,
! [A: $tType,Ys: coinductive_llist @ A,Xs: coinductive_llist @ A] :
( ( ( coinductive_lnull @ A @ Ys )
=> ( ( coinductive_llast @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( coinductive_llast @ A @ Xs ) ) )
& ( ~ ( coinductive_lnull @ A @ Ys )
=> ( ( ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_llast @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( coinductive_llast @ A @ Ys ) ) )
& ( ~ ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_llast @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( undefined @ A ) ) ) ) ) ) ).
% llast_lappend
thf(fact_235_Powp__mono,axiom,
! [A: $tType,A4: A > $o,B4: A > $o] :
( ( ord_less_eq @ ( A > $o ) @ A4 @ B4 )
=> ( ord_less_eq @ ( ( set @ A ) > $o ) @ ( powp @ A @ A4 ) @ ( powp @ A @ B4 ) ) ) ).
% Powp_mono
thf(fact_236_llast__linfinite,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ~ ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_llast @ A @ Xs )
= ( undefined @ A ) ) ) ).
% llast_linfinite
thf(fact_237_llast__LNil,axiom,
! [A: $tType] :
( ( coinductive_llast @ A @ ( coinductive_LNil @ A ) )
= ( undefined @ A ) ) ).
% llast_LNil
thf(fact_238_lappend_Octr_I2_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ~ ( coinductive_lnull @ A @ Xs )
| ~ ( coinductive_lnull @ A @ Ys ) )
=> ( ( coinductive_lappend @ A @ Xs @ Ys )
= ( coinductive_LCons @ A
@ ( coindu1381640503_llist @ A @ A @ ( coinductive_lhd @ A @ Ys )
@ ^ [X3: A,Xs5: coinductive_llist @ A] : X3
@ Xs )
@ ( coindu1381640503_llist @ ( coinductive_llist @ A ) @ A
@ ( coindu1381640503_llist @ ( coinductive_llist @ A ) @ A @ ( undefined @ ( coinductive_llist @ A ) )
@ ^ [Uu2: A,Uv2: coinductive_llist @ A] : Uv2
@ Ys )
@ ^ [X3: A,Xs5: coinductive_llist @ A] : ( coinductive_lappend @ A @ Xs5 @ Ys )
@ Xs ) ) ) ) ).
% lappend.ctr(2)
thf(fact_239_lset__lappend__lfinite,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_lset @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( sup_sup @ ( set @ A ) @ ( coinductive_lset @ A @ Xs ) @ ( coinductive_lset @ A @ Ys ) ) ) ) ).
% lset_lappend_lfinite
thf(fact_240_Un__subset__iff,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) @ C3 )
= ( ( ord_less_eq @ ( set @ A ) @ A4 @ C3 )
& ( ord_less_eq @ ( set @ A ) @ B4 @ C3 ) ) ) ).
% Un_subset_iff
thf(fact_241_lhd__lappend,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_lhd @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( coinductive_lhd @ A @ Ys ) ) )
& ( ~ ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_lhd @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% lhd_lappend
thf(fact_242_lnull__ltakeWhile,axiom,
! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) )
= ( ~ ( coinductive_lnull @ A @ Xs )
=> ~ ( P @ ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% lnull_ltakeWhile
thf(fact_243_ltakeWhile_Odisc__iff_I1_J,axiom,
! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) )
= ( ( coinductive_lnull @ A @ Xs )
| ~ ( P @ ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% ltakeWhile.disc_iff(1)
thf(fact_244_ltakeWhile_Odisc__iff_I2_J,axiom,
! [A: $tType,P: A > $o,Xs: coinductive_llist @ A] :
( ( ~ ( coinductive_lnull @ A @ ( coindu501562517eWhile @ A @ P @ Xs ) ) )
= ( ~ ( coinductive_lnull @ A @ Xs )
& ( P @ ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% ltakeWhile.disc_iff(2)
thf(fact_245_subset__Un__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B5: set @ A] :
( ( sup_sup @ ( set @ A ) @ A6 @ B5 )
= B5 ) ) ) ).
% subset_Un_eq
thf(fact_246_Un__absorb2,axiom,
! [A: $tType,B4: set @ A,A4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B4 @ A4 )
=> ( ( sup_sup @ ( set @ A ) @ A4 @ B4 )
= A4 ) ) ).
% Un_absorb2
thf(fact_247_Un__absorb1,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ( sup_sup @ ( set @ A ) @ A4 @ B4 )
= B4 ) ) ).
% Un_absorb1
thf(fact_248_Un__upper2,axiom,
! [A: $tType,B4: set @ A,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ B4 @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) ) ).
% Un_upper2
thf(fact_249_Un__upper1,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] : ( ord_less_eq @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) ) ).
% Un_upper1
thf(fact_250_Un__least,axiom,
! [A: $tType,A4: set @ A,C3: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ C3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ C3 )
=> ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) @ C3 ) ) ) ).
% Un_least
thf(fact_251_Un__mono,axiom,
! [A: $tType,A4: set @ A,C3: set @ A,B4: set @ A,D: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ C3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ D )
=> ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) @ ( sup_sup @ ( set @ A ) @ C3 @ D ) ) ) ) ).
% Un_mono
thf(fact_252_lhd__def,axiom,
! [A: $tType] :
( ( coinductive_lhd @ A )
= ( coindu1381640503_llist @ A @ A @ ( undefined @ A )
@ ^ [X213: A,X223: coinductive_llist @ A] : X213 ) ) ).
% lhd_def
thf(fact_253_ltakeWhile_Octr_I1_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
( ( ( coinductive_lnull @ A @ Xs )
| ~ ( P @ ( coinductive_lhd @ A @ Xs ) ) )
=> ( ( coindu501562517eWhile @ A @ P @ Xs )
= ( coinductive_LNil @ A ) ) ) ).
% ltakeWhile.ctr(1)
%----Type constructors (10)
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A7: $tType,A8: $tType] :
( ( preorder @ A8 @ ( type2 @ A8 ) )
=> ( preorder @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A7: $tType,A8: $tType] :
( ( order @ A8 @ ( type2 @ A8 ) )
=> ( order @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A7: $tType,A8: $tType] :
( ( ord @ A8 @ ( type2 @ A8 ) )
=> ( ord @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_1,axiom,
! [A7: $tType] : ( preorder @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_2,axiom,
! [A7: $tType] : ( order @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_3,axiom,
! [A7: $tType] : ( ord @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_4,axiom,
preorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_5,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_6,axiom,
ord @ $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 (3)
thf(conj_0,hypothesis,
( ( coinductive_lappend @ a @ x @ ys )
= ( coinductive_LCons @ a @ xa @ ( coinductive_LNil @ a ) ) ) ).
thf(conj_1,hypothesis,
( x
= ( coinductive_LCons @ a @ x21 @ x22 ) ) ).
thf(conj_2,conjecture,
( ( x
= ( coinductive_LNil @ a ) )
| ? [X7: a] :
( x
= ( coinductive_LCons @ a @ X7 @ ( coinductive_LNil @ a ) ) )
| ? [X7: a,Y4: a,Xs4: coinductive_llist @ a] :
( ( x
= ( coinductive_LCons @ a @ X7 @ ( coinductive_LCons @ a @ Y4 @ Xs4 ) ) )
& ( graph @ X7 @ Y4 )
& ( ( member @ ( coinductive_llist @ a ) @ ( coinductive_lappend @ a @ ( coinductive_LCons @ a @ Y4 @ Xs4 ) @ ys ) @ ( koenig916195507_paths @ a @ graph ) )
| ( member @ ( coinductive_llist @ a ) @ ( coinductive_LCons @ a @ Y4 @ Xs4 ) @ ( koenig916195507_paths @ a @ graph ) ) ) ) ) ).
%------------------------------------------------------------------------------