TPTP Problem File: COM175^1.p
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%------------------------------------------------------------------------------
% File : COM175^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Computing Theory
% Problem : Koenig's lemma 115
% 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__115.p [Bla16]
% Status : Theorem
% Rating : 0.00 v8.1.0, 0.25 v7.5.0, 0.00 v7.3.0, 0.33 v7.2.0, 0.25 v7.1.0
% Syntax : Number of formulae : 326 ( 117 unt; 47 typ; 0 def)
% Number of atoms : 815 ( 241 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 4283 ( 86 ~; 39 |; 81 &;3671 @)
% ( 0 <=>; 406 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 9 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 283 ( 283 >; 0 *; 0 +; 0 <<)
% Number of symbols : 49 ( 46 usr; 8 con; 0-8 aty)
% Number of variables : 1143 ( 60 ^; 971 !; 74 ?;1143 :)
% ( 38 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:46:24.599
%------------------------------------------------------------------------------
%----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_Lattices_Olattice,type,
lattice:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Osemilattice__sup,type,
semilattice_sup:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
comple187826305attice:
!>[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_Oldistinct,type,
coindu351974385stinct:
!>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
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_Ollexord,type,
coinductive_llexord:
!>[A: $tType] : ( ( A > A > $o ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
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_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_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_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_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_n,type,
n: a ).
thf(sy_v_n_H____,type,
n2: a ).
thf(sy_v_ns,type,
ns: set @ a ).
thf(sy_v_x____,type,
x: a ).
thf(sy_v_xs_H____,type,
xs: coinductive_llist @ a ).
thf(sy_v_xs____,type,
xs2: coinductive_llist @ a ).
%----Relevant facts (256)
thf(fact_0__092_060open_062x_A_092_060in_062_Alset_Axs_092_060close_062,axiom,
member @ a @ x @ ( coinductive_lset @ a @ xs2 ) ).
% \<open>x \<in> lset xs\<close>
thf(fact_1__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062n_H_Axs_H_O_Axs_A_061_ALCons_An_H_Axs_H_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [N: a,Xs: coinductive_llist @ a] :
( xs2
!= ( coinductive_LCons @ a @ N @ Xs ) ) ).
% \<open>\<And>thesis. (\<And>n' xs'. xs = LCons n' xs' \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_2_xs,axiom,
( xs2
= ( coinductive_LCons @ a @ n2 @ xs ) ) ).
% xs
thf(fact_3__092_060open_062lset_Axs_A_092_060subseteq_062_Ans_092_060close_062,axiom,
ord_less_eq @ ( set @ a ) @ ( coinductive_lset @ a @ xs2 ) @ ns ).
% \<open>lset xs \<subseteq> ns\<close>
thf(fact_4_calculation,axiom,
graph @ n @ n2 ).
% calculation
thf(fact_5__092_060open_062x_A_092_060in_062_Areachable__via_Agraph_Ans_An_092_060close_062,axiom,
member @ a @ x @ ( koenig317145564le_via @ a @ graph @ ns @ n ) ).
% \<open>x \<in> reachable_via graph ns n\<close>
thf(fact_6__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062xs_O_A_092_060lbrakk_062LCons_An_Axs_A_092_060in_062_Apaths_Agraph_059_Ax_A_092_060in_062_Alset_Axs_059_Alset_Axs_A_092_060subseteq_062_Ans_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Xs2: coinductive_llist @ a] :
( ( member @ ( coinductive_llist @ a ) @ ( coinductive_LCons @ a @ n @ Xs2 ) @ ( koenig916195507_paths @ a @ graph ) )
=> ( ( member @ a @ x @ ( coinductive_lset @ a @ Xs2 ) )
=> ~ ( ord_less_eq @ ( set @ a ) @ ( coinductive_lset @ a @ Xs2 ) @ ns ) ) ) ).
% \<open>\<And>thesis. (\<And>xs. \<lbrakk>LCons n xs \<in> paths graph; x \<in> lset xs; lset xs \<subseteq> ns\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_7_path,axiom,
member @ ( coinductive_llist @ a ) @ ( coinductive_LCons @ a @ n @ xs2 ) @ ( koenig916195507_paths @ a @ graph ) ).
% path
thf(fact_8_paths_OLCons,axiom,
! [Node: $tType,Graph: Node > Node > $o,X: Node,Y: Node,Xs3: coinductive_llist @ Node] :
( ( Graph @ X @ Y )
=> ( ( member @ ( coinductive_llist @ Node ) @ ( coinductive_LCons @ Node @ Y @ Xs3 ) @ ( koenig916195507_paths @ Node @ Graph ) )
=> ( member @ ( coinductive_llist @ Node ) @ ( coinductive_LCons @ Node @ X @ ( coinductive_LCons @ Node @ Y @ Xs3 ) ) @ ( koenig916195507_paths @ Node @ Graph ) ) ) ) ).
% paths.LCons
thf(fact_9_paths__LConsD,axiom,
! [A: $tType,X: A,Xs3: coinductive_llist @ A,Graph: A > A > $o] :
( ( member @ ( coinductive_llist @ A ) @ ( coinductive_LCons @ A @ X @ Xs3 ) @ ( koenig916195507_paths @ A @ Graph ) )
=> ( member @ ( coinductive_llist @ A ) @ Xs3 @ ( koenig916195507_paths @ A @ Graph ) ) ) ).
% paths_LConsD
thf(fact_10_path__avoid__node,axiom,
! [A: $tType,N2: A,Xs3: coinductive_llist @ A,Graph: A > A > $o,X: A] :
( ( member @ ( coinductive_llist @ A ) @ ( coinductive_LCons @ A @ N2 @ Xs3 ) @ ( koenig916195507_paths @ A @ Graph ) )
=> ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs3 ) )
=> ( ( N2 != X )
=> ? [Xs: coinductive_llist @ A] :
( ( member @ ( coinductive_llist @ A ) @ ( coinductive_LCons @ A @ N2 @ Xs ) @ ( koenig916195507_paths @ A @ Graph ) )
& ( ord_less_eq @ ( set @ A ) @ ( coinductive_lset @ A @ Xs ) @ ( coinductive_lset @ A @ Xs3 ) )
& ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
& ~ ( member @ A @ N2 @ ( coinductive_lset @ A @ Xs ) ) ) ) ) ) ).
% path_avoid_node
thf(fact_11_reachable__via_Ocases,axiom,
! [Node: $tType,A2: Node,Graph: Node > Node > $o,Ns: set @ Node,N2: Node] :
( ( member @ Node @ A2 @ ( koenig317145564le_via @ Node @ Graph @ Ns @ N2 ) )
=> ~ ! [Xs2: coinductive_llist @ Node] :
( ( member @ ( coinductive_llist @ Node ) @ ( coinductive_LCons @ Node @ N2 @ 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_12_reachable__via_Osimps,axiom,
! [Node: $tType,A2: Node,Graph: Node > Node > $o,Ns: set @ Node,N2: Node] :
( ( member @ Node @ A2 @ ( koenig317145564le_via @ Node @ Graph @ Ns @ N2 ) )
= ( ? [Xs4: coinductive_llist @ Node,N3: Node] :
( ( A2 = N3 )
& ( member @ ( coinductive_llist @ Node ) @ ( coinductive_LCons @ Node @ N2 @ Xs4 ) @ ( koenig916195507_paths @ Node @ Graph ) )
& ( member @ Node @ N3 @ ( coinductive_lset @ Node @ Xs4 ) )
& ( ord_less_eq @ ( set @ Node ) @ ( coinductive_lset @ Node @ Xs4 ) @ Ns ) ) ) ) ).
% reachable_via.simps
thf(fact_13_reachable__via_Ointros,axiom,
! [Node: $tType,N2: Node,Xs3: coinductive_llist @ Node,Graph: Node > Node > $o,N4: Node,Ns: set @ Node] :
( ( member @ ( coinductive_llist @ Node ) @ ( coinductive_LCons @ Node @ N2 @ Xs3 ) @ ( koenig916195507_paths @ Node @ Graph ) )
=> ( ( member @ Node @ N4 @ ( coinductive_lset @ Node @ Xs3 ) )
=> ( ( ord_less_eq @ ( set @ Node ) @ ( coinductive_lset @ Node @ Xs3 ) @ Ns )
=> ( member @ Node @ N4 @ ( koenig317145564le_via @ Node @ Graph @ Ns @ N2 ) ) ) ) ) ).
% reachable_via.intros
thf(fact_14_reachable__via_Oinducts,axiom,
! [Node: $tType,X: Node,Graph: Node > Node > $o,Ns: set @ Node,N2: Node,P: Node > $o] :
( ( member @ Node @ X @ ( koenig317145564le_via @ Node @ Graph @ Ns @ N2 ) )
=> ( ! [Xs2: coinductive_llist @ Node,N: Node] :
( ( member @ ( coinductive_llist @ Node ) @ ( coinductive_LCons @ Node @ N2 @ Xs2 ) @ ( koenig916195507_paths @ Node @ Graph ) )
=> ( ( member @ Node @ N @ ( coinductive_lset @ Node @ Xs2 ) )
=> ( ( ord_less_eq @ ( set @ Node ) @ ( coinductive_lset @ Node @ Xs2 ) @ Ns )
=> ( P @ N ) ) ) )
=> ( P @ X ) ) ) ).
% reachable_via.inducts
thf(fact_15_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_16_subsetI,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( member @ A @ X2 @ B2 ) )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B2 ) ) ).
% subsetI
thf(fact_17_subset__antisym,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A3 )
=> ( A3 = B2 ) ) ) ).
% subset_antisym
thf(fact_18_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_19_lset__intros_I2_J,axiom,
! [A: $tType,X: A,Xs3: coinductive_llist @ A,X3: A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs3 ) )
=> ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ X3 @ Xs3 ) ) ) ) ).
% lset_intros(2)
thf(fact_20_lset__intros_I1_J,axiom,
! [A: $tType,X: A,Xs3: coinductive_llist @ A] : ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ X @ Xs3 ) ) ) ).
% lset_intros(1)
thf(fact_21_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_22_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_23_lset__cases,axiom,
! [A: $tType,X: A,Xs3: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs3 ) )
=> ( ! [Xs: coinductive_llist @ A] :
( Xs3
!= ( coinductive_LCons @ A @ X @ Xs ) )
=> ~ ! [X4: A,Xs: coinductive_llist @ A] :
( ( Xs3
= ( coinductive_LCons @ A @ X4 @ Xs ) )
=> ~ ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) ) ) ) ) ).
% lset_cases
thf(fact_24_lset__induct,axiom,
! [A: $tType,X: A,Xs3: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs3 ) )
=> ( ! [Xs2: coinductive_llist @ A] : ( P @ ( coinductive_LCons @ A @ X @ Xs2 ) )
=> ( ! [X4: A,Xs2: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs2 ) )
=> ( ( X != X4 )
=> ( ( P @ Xs2 )
=> ( P @ ( coinductive_LCons @ A @ X4 @ Xs2 ) ) ) ) )
=> ( P @ Xs3 ) ) ) ) ).
% lset_induct
thf(fact_25_lset__induct_H,axiom,
! [A: $tType,X: A,Xs3: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs3 ) )
=> ( ! [Xs2: coinductive_llist @ A] : ( P @ ( coinductive_LCons @ A @ X @ Xs2 ) )
=> ( ! [X4: A,Xs2: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs2 ) )
=> ( ( P @ Xs2 )
=> ( P @ ( coinductive_LCons @ A @ X4 @ Xs2 ) ) ) )
=> ( P @ Xs3 ) ) ) ) ).
% lset_induct'
thf(fact_26_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A] :
( ( ord_less_eq @ A @ B3 @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ) ).
% dual_order.antisym
thf(fact_27_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B3 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ B3 )
=> ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% dual_order.trans
thf(fact_28_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > A > $o,A2: A,B3: A] :
( ! [A4: A,B4: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: A,B4: A] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A2 @ B3 ) ) ) ) ).
% linorder_wlog
thf(fact_29_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_30_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z )
=> ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% order_trans
thf(fact_31_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ) ).
% order_class.order.antisym
thf(fact_32_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_33_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A] :
( ( A2 = B3 )
=> ( ( ord_less_eq @ A @ B3 @ C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_34_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_35_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( ( ord_less_eq @ A @ X @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z ) )
=> ( ( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less_eq @ A @ X @ Z ) )
=> ( ( ( ord_less_eq @ A @ X @ Z )
=> ~ ( ord_less_eq @ A @ Z @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z )
=> ~ ( ord_less_eq @ A @ Z @ X ) )
=> ~ ( ( ord_less_eq @ A @ Z @ X )
=> ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_36_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% order.trans
thf(fact_37_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_38_wlog__linorder__le,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > A > $o,B3: A,A2: A] :
( ! [A4: A,B4: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ( ( P @ B3 @ A2 )
=> ( P @ A2 @ B3 ) )
=> ( P @ A2 @ B3 ) ) ) ) ).
% wlog_linorder_le
thf(fact_39_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_40_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_41_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_42_eq__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ^ [Y2: A,Z2: A] : Y2 = Z2 )
= ( ^ [X5: A,Y3: A] :
( ( ord_less_eq @ A @ X5 @ Y3 )
& ( ord_less_eq @ A @ Y3 @ X5 ) ) ) ) ) ).
% eq_iff
thf(fact_43_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B3: A,F: A > B,C2: B] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X2: A,Y4: A] :
( ( ord_less_eq @ A @ X2 @ Y4 )
=> ( ord_less_eq @ B @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ B @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_44_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B > A,B3: B,C2: B] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C2 )
=> ( ! [X2: B,Y4: B] :
( ( ord_less_eq @ B @ X2 @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% ord_eq_le_subst
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,A3: set @ A] :
( ( collect @ A
@ ^ [X5: A] : ( member @ A @ X5 @ A3 ) )
= A3 ) ).
% 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_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C @ ( type2 @ C ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B3: A,F: A > C,C2: C] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( ord_less_eq @ C @ ( F @ B3 ) @ C2 )
=> ( ! [X2: A,Y4: A] :
( ( ord_less_eq @ A @ X2 @ Y4 )
=> ( ord_less_eq @ C @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ C @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% order_subst2
thf(fact_50_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B > A,B3: B,C2: B] :
( ( ord_less_eq @ A @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C2 )
=> ( ! [X2: B,Y4: B] :
( ( ord_less_eq @ B @ X2 @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% order_subst1
thf(fact_51_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B] :
! [X5: A] : ( ord_less_eq @ B @ ( F2 @ X5 ) @ ( G2 @ X5 ) ) ) ) ) ).
% le_fun_def
thf(fact_52_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_53_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_54_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_55_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
= ( ! [X5: A] :
( ( P @ X5 )
=> ( Q @ X5 ) ) ) ) ).
% Collect_mono_iff
thf(fact_56_contra__subsetD,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ~ ( member @ A @ C2 @ B2 )
=> ~ ( member @ A @ C2 @ A3 ) ) ) ).
% contra_subsetD
thf(fact_57_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y2: set @ A,Z2: set @ A] : Y2 = Z2 )
= ( ^ [A5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
& ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_58_subset__trans,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% subset_trans
thf(fact_59_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_60_subset__refl,axiom,
! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ A3 ) ).
% subset_refl
thf(fact_61_rev__subsetD,axiom,
! [A: $tType,C2: A,A3: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( member @ A @ C2 @ B2 ) ) ) ).
% rev_subsetD
thf(fact_62_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A5: set @ A,B5: set @ A] :
! [T: A] :
( ( member @ A @ T @ A5 )
=> ( member @ A @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_63_set__rev__mp,axiom,
! [A: $tType,X: A,A3: set @ A,B2: set @ A] :
( ( member @ A @ X @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( member @ A @ X @ B2 ) ) ) ).
% set_rev_mp
thf(fact_64_equalityD2,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( A3 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ B2 @ A3 ) ) ).
% equalityD2
thf(fact_65_equalityD1,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( A3 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B2 ) ) ).
% equalityD1
thf(fact_66_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A5: set @ A,B5: set @ A] :
! [X5: A] :
( ( member @ A @ X5 @ A5 )
=> ( member @ A @ X5 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_67_equalityE,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( A3 = B2 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A3 ) ) ) ).
% equalityE
thf(fact_68_subsetCE,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ B2 ) ) ) ).
% subsetCE
thf(fact_69_subsetD,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ B2 ) ) ) ).
% subsetD
thf(fact_70_in__mono,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ( member @ A @ X @ A3 )
=> ( member @ A @ X @ B2 ) ) ) ).
% in_mono
thf(fact_71_set__mp,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ( member @ A @ X @ A3 )
=> ( member @ A @ X @ B2 ) ) ) ).
% set_mp
thf(fact_72_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,Z22: coinductive_llist @ A] : ( P @ Z1 @ ( coinductive_LCons @ A @ Z1 @ Z22 ) )
=> ( ! [Z1: A,Z22: coinductive_llist @ A,Xa: A] :
( ( member @ A @ Xa @ ( coinductive_lset @ A @ Z22 ) )
=> ( ( P @ Xa @ Z22 )
=> ( P @ Xa @ ( coinductive_LCons @ A @ Z1 @ Z22 ) ) ) )
=> ( P @ X @ A2 ) ) ) ) ).
% llist.set_induct
thf(fact_73_llist_Oset__cases,axiom,
! [A: $tType,E: A,A2: coinductive_llist @ A] :
( ( member @ A @ E @ ( coinductive_lset @ A @ A2 ) )
=> ( ! [Z22: coinductive_llist @ A] :
( A2
!= ( coinductive_LCons @ A @ E @ Z22 ) )
=> ~ ! [Z1: A,Z22: coinductive_llist @ A] :
( ( A2
= ( coinductive_LCons @ A @ Z1 @ Z22 ) )
=> ~ ( member @ A @ E @ ( coinductive_lset @ A @ Z22 ) ) ) ) ) ).
% llist.set_cases
thf(fact_74_reachable__viap__reachable__via__eq,axiom,
! [Node: $tType] :
( ( koenig1757754772e_viap @ Node )
= ( ^ [Graph2: Node > Node > $o,Ns2: set @ Node,N5: Node,X5: Node] : ( member @ Node @ X5 @ ( koenig317145564le_via @ Node @ Graph2 @ Ns2 @ N5 ) ) ) ) ).
% reachable_viap_reachable_via_eq
thf(fact_75_lset__lmember,axiom,
! [A: $tType,X: A,Xs3: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs3 ) )
= ( coinductive_lmember @ A @ X @ Xs3 ) ) ).
% lset_lmember
thf(fact_76_lsorted__LCons,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Xs3: coinductive_llist @ A] :
( ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ X @ Xs3 ) )
= ( ( coindu63249387sorted @ A @ Xs3 )
& ! [X5: A] :
( ( member @ A @ X5 @ ( coinductive_lset @ A @ Xs3 ) )
=> ( ord_less_eq @ A @ X @ X5 ) ) ) ) ) ).
% lsorted_LCons
thf(fact_77_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_78_lfp_Oleq__antisym,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ X )
=> ( X = Y ) ) ) ) ).
% lfp.leq_antisym
thf(fact_79_gfp_Oleq__antisym,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( X = Y ) ) ) ) ).
% gfp.leq_antisym
thf(fact_80_lfp_Oleq__trans,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z )
=> ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% lfp.leq_trans
thf(fact_81_gfp_Oleq__trans,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A,Z: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_less_eq @ A @ Z @ Y )
=> ( ord_less_eq @ A @ Z @ X ) ) ) ) ).
% gfp.leq_trans
thf(fact_82_lfp_Oleq__refl,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% lfp.leq_refl
thf(fact_83_lsorted__LCons__LCons,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Xs3: coinductive_llist @ A] :
( ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LCons @ A @ Y @ Xs3 ) ) )
= ( ( ord_less_eq @ A @ X @ Y )
& ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ Y @ Xs3 ) ) ) ) ) ).
% lsorted_LCons_LCons
thf(fact_84_LCons__LCons,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Xs3: coinductive_llist @ A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ Y @ Xs3 ) )
=> ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LCons @ A @ Y @ Xs3 ) ) ) ) ) ) ).
% LCons_LCons
thf(fact_85_reachable__viap_Ocases,axiom,
! [Node: $tType,Graph: Node > Node > $o,Ns: set @ Node,N2: Node,A2: Node] :
( ( koenig1757754772e_viap @ Node @ Graph @ Ns @ N2 @ A2 )
=> ~ ! [Xs2: coinductive_llist @ Node] :
( ( koenig2031690877pathsp @ Node @ Graph @ ( coinductive_LCons @ Node @ N2 @ Xs2 ) )
=> ( ( member @ Node @ A2 @ ( coinductive_lset @ Node @ Xs2 ) )
=> ~ ( ord_less_eq @ ( set @ Node ) @ ( coinductive_lset @ Node @ Xs2 ) @ Ns ) ) ) ) ).
% reachable_viap.cases
thf(fact_86_reachable__viap_Osimps,axiom,
! [Node: $tType] :
( ( koenig1757754772e_viap @ Node )
= ( ^ [Graph2: Node > Node > $o,Ns2: set @ Node,N5: Node,A6: Node] :
? [Xs4: coinductive_llist @ Node,N3: Node] :
( ( A6 = N3 )
& ( koenig2031690877pathsp @ Node @ Graph2 @ ( coinductive_LCons @ Node @ N5 @ Xs4 ) )
& ( member @ Node @ N3 @ ( coinductive_lset @ Node @ Xs4 ) )
& ( ord_less_eq @ ( set @ Node ) @ ( coinductive_lset @ Node @ Xs4 ) @ Ns2 ) ) ) ) ).
% reachable_viap.simps
thf(fact_87_reachable__viap_Ointros,axiom,
! [Node: $tType,Graph: Node > Node > $o,N2: Node,Xs3: coinductive_llist @ Node,N4: Node,Ns: set @ Node] :
( ( koenig2031690877pathsp @ Node @ Graph @ ( coinductive_LCons @ Node @ N2 @ Xs3 ) )
=> ( ( member @ Node @ N4 @ ( coinductive_lset @ Node @ Xs3 ) )
=> ( ( ord_less_eq @ ( set @ Node ) @ ( coinductive_lset @ Node @ Xs3 ) @ Ns )
=> ( koenig1757754772e_viap @ Node @ Graph @ Ns @ N2 @ N4 ) ) ) ) ).
% reachable_viap.intros
thf(fact_88_reachable__viap_Oinducts,axiom,
! [Node: $tType,Graph: Node > Node > $o,Ns: set @ Node,N2: Node,X: Node,P: Node > $o] :
( ( koenig1757754772e_viap @ Node @ Graph @ Ns @ N2 @ X )
=> ( ! [Xs2: coinductive_llist @ Node,N: Node] :
( ( koenig2031690877pathsp @ Node @ Graph @ ( coinductive_LCons @ Node @ N2 @ Xs2 ) )
=> ( ( member @ Node @ N @ ( coinductive_lset @ Node @ Xs2 ) )
=> ( ( ord_less_eq @ ( set @ Node ) @ ( coinductive_lset @ Node @ Xs2 ) @ Ns )
=> ( P @ N ) ) ) )
=> ( P @ X ) ) ) ).
% reachable_viap.inducts
thf(fact_89_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,Y4: A,Xs2: coinductive_llist @ A] :
( ( A2
= ( coinductive_LCons @ A @ X2 @ ( coinductive_LCons @ A @ Y4 @ Xs2 ) ) )
=> ( ( ord_less_eq @ A @ X2 @ Y4 )
=> ~ ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ Y4 @ Xs2 ) ) ) ) ) ) ) ) ).
% lsorted.cases
thf(fact_90_lsorted_Osimps,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ( ( coindu63249387sorted @ A )
= ( ^ [A6: coinductive_llist @ A] :
( ( A6
= ( coinductive_LNil @ A ) )
| ? [X5: A] :
( A6
= ( coinductive_LCons @ A @ X5 @ ( coinductive_LNil @ A ) ) )
| ? [X5: A,Y3: A,Xs4: coinductive_llist @ A] :
( ( A6
= ( coinductive_LCons @ A @ X5 @ ( coinductive_LCons @ A @ Y3 @ Xs4 ) ) )
& ( ord_less_eq @ A @ X5 @ Y3 )
& ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ Y3 @ Xs4 ) ) ) ) ) ) ) ).
% lsorted.simps
thf(fact_91_lsorted__code_I1_J,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ( coindu63249387sorted @ A @ ( coinductive_LNil @ A ) ) ) ).
% lsorted_code(1)
thf(fact_92_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_93_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,Y4: Node,Xs2: coinductive_llist @ Node] :
( ( A2
= ( coinductive_LCons @ Node @ X2 @ ( coinductive_LCons @ Node @ Y4 @ Xs2 ) ) )
=> ( ( Graph @ X2 @ Y4 )
=> ~ ( koenig2031690877pathsp @ Node @ Graph @ ( coinductive_LCons @ Node @ Y4 @ Xs2 ) ) ) ) ) ) ) ).
% pathsp.cases
thf(fact_94_pathsp_Osimps,axiom,
! [Node: $tType] :
( ( koenig2031690877pathsp @ Node )
= ( ^ [Graph2: Node > Node > $o,A6: coinductive_llist @ Node] :
( ( A6
= ( coinductive_LNil @ Node ) )
| ? [X5: Node] :
( A6
= ( coinductive_LCons @ Node @ X5 @ ( coinductive_LNil @ Node ) ) )
| ? [X5: Node,Y3: Node,Xs4: coinductive_llist @ Node] :
( ( A6
= ( coinductive_LCons @ Node @ X5 @ ( coinductive_LCons @ Node @ Y3 @ Xs4 ) ) )
& ( Graph2 @ X5 @ Y3 )
& ( koenig2031690877pathsp @ Node @ Graph2 @ ( coinductive_LCons @ Node @ Y3 @ Xs4 ) ) ) ) ) ) ).
% pathsp.simps
thf(fact_95_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_96_pathsp_Ocoinduct,axiom,
! [Node: $tType,X6: ( coinductive_llist @ Node ) > $o,X: coinductive_llist @ Node,Graph: Node > Node > $o] :
( ( X6 @ X )
=> ( ! [X2: coinductive_llist @ Node] :
( ( X6 @ X2 )
=> ( ( X2
= ( coinductive_LNil @ Node ) )
| ? [Xa2: Node] :
( X2
= ( coinductive_LCons @ Node @ Xa2 @ ( coinductive_LNil @ Node ) ) )
| ? [Xa2: Node,Y5: Node,Xs5: coinductive_llist @ Node] :
( ( X2
= ( coinductive_LCons @ Node @ Xa2 @ ( coinductive_LCons @ Node @ Y5 @ Xs5 ) ) )
& ( Graph @ Xa2 @ Y5 )
& ( ( X6 @ ( coinductive_LCons @ Node @ Y5 @ Xs5 ) )
| ( koenig2031690877pathsp @ Node @ Graph @ ( coinductive_LCons @ Node @ Y5 @ Xs5 ) ) ) ) ) )
=> ( koenig2031690877pathsp @ Node @ Graph @ X ) ) ) ).
% pathsp.coinduct
thf(fact_97_pathsp_OEmpty,axiom,
! [Node: $tType,Graph: Node > Node > $o] : ( koenig2031690877pathsp @ Node @ Graph @ ( coinductive_LNil @ Node ) ) ).
% pathsp.Empty
thf(fact_98_neq__LNil__conv,axiom,
! [A: $tType,Xs3: coinductive_llist @ A] :
( ( Xs3
!= ( coinductive_LNil @ A ) )
= ( ? [X5: A,Xs6: coinductive_llist @ A] :
( Xs3
= ( coinductive_LCons @ A @ X5 @ Xs6 ) ) ) ) ).
% neq_LNil_conv
thf(fact_99_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_100_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_101_LNil,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ( coindu63249387sorted @ A @ ( coinductive_LNil @ A ) ) ) ).
% LNil
thf(fact_102_paths_OEmpty,axiom,
! [Node: $tType,Graph: Node > Node > $o] : ( member @ ( coinductive_llist @ Node ) @ ( coinductive_LNil @ Node ) @ ( koenig916195507_paths @ Node @ Graph ) ) ).
% paths.Empty
thf(fact_103_pathsp_OLCons,axiom,
! [Node: $tType,Graph: Node > Node > $o,X: Node,Y: Node,Xs3: coinductive_llist @ Node] :
( ( Graph @ X @ Y )
=> ( ( koenig2031690877pathsp @ Node @ Graph @ ( coinductive_LCons @ Node @ Y @ Xs3 ) )
=> ( koenig2031690877pathsp @ Node @ Graph @ ( coinductive_LCons @ Node @ X @ ( coinductive_LCons @ Node @ Y @ Xs3 ) ) ) ) ) ).
% pathsp.LCons
thf(fact_104_lmember__code_I1_J,axiom,
! [A: $tType,X: A] :
~ ( coinductive_lmember @ A @ X @ ( coinductive_LNil @ A ) ) ).
% lmember_code(1)
thf(fact_105_pathsp__paths__eq,axiom,
! [Node: $tType] :
( ( koenig2031690877pathsp @ Node )
= ( ^ [Graph2: Node > Node > $o,X5: coinductive_llist @ Node] : ( member @ ( coinductive_llist @ Node ) @ X5 @ ( koenig916195507_paths @ Node @ Graph2 ) ) ) ) ).
% pathsp_paths_eq
thf(fact_106_Singleton,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X: A] : ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LNil @ A ) ) ) ) ).
% Singleton
thf(fact_107_paths_Ocoinduct,axiom,
! [Node: $tType,X6: ( coinductive_llist @ Node ) > $o,X: coinductive_llist @ Node,Graph: Node > Node > $o] :
( ( X6 @ X )
=> ( ! [X2: coinductive_llist @ Node] :
( ( X6 @ X2 )
=> ( ( X2
= ( coinductive_LNil @ Node ) )
| ? [Xa2: Node] :
( X2
= ( coinductive_LCons @ Node @ Xa2 @ ( coinductive_LNil @ Node ) ) )
| ? [Xa2: Node,Y5: Node,Xs5: coinductive_llist @ Node] :
( ( X2
= ( coinductive_LCons @ Node @ Xa2 @ ( coinductive_LCons @ Node @ Y5 @ Xs5 ) ) )
& ( Graph @ Xa2 @ Y5 )
& ( ( X6 @ ( coinductive_LCons @ Node @ Y5 @ Xs5 ) )
| ( member @ ( coinductive_llist @ Node ) @ ( coinductive_LCons @ Node @ Y5 @ Xs5 ) @ ( koenig916195507_paths @ Node @ Graph ) ) ) ) ) )
=> ( member @ ( coinductive_llist @ Node ) @ X @ ( koenig916195507_paths @ Node @ Graph ) ) ) ) ).
% paths.coinduct
thf(fact_108_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_109_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 ) )
| ? [X5: Node] :
( A2
= ( coinductive_LCons @ Node @ X5 @ ( coinductive_LNil @ Node ) ) )
| ? [X5: Node,Y3: Node,Xs4: coinductive_llist @ Node] :
( ( A2
= ( coinductive_LCons @ Node @ X5 @ ( coinductive_LCons @ Node @ Y3 @ Xs4 ) ) )
& ( Graph @ X5 @ Y3 )
& ( member @ ( coinductive_llist @ Node ) @ ( coinductive_LCons @ Node @ Y3 @ Xs4 ) @ ( koenig916195507_paths @ Node @ Graph ) ) ) ) ) ).
% paths.simps
thf(fact_110_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,Y4: Node,Xs2: coinductive_llist @ Node] :
( ( A2
= ( coinductive_LCons @ Node @ X2 @ ( coinductive_LCons @ Node @ Y4 @ Xs2 ) ) )
=> ( ( Graph @ X2 @ Y4 )
=> ~ ( member @ ( coinductive_llist @ Node ) @ ( coinductive_LCons @ Node @ Y4 @ Xs2 ) @ ( koenig916195507_paths @ Node @ Graph ) ) ) ) ) ) ) ).
% paths.cases
thf(fact_111_lsorted_Ocoinduct,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X6: ( coinductive_llist @ A ) > $o,X: coinductive_llist @ A] :
( ( X6 @ X )
=> ( ! [X2: coinductive_llist @ A] :
( ( X6 @ X2 )
=> ( ( X2
= ( coinductive_LNil @ A ) )
| ? [Xa2: A] :
( X2
= ( coinductive_LCons @ A @ Xa2 @ ( coinductive_LNil @ A ) ) )
| ? [Xa2: A,Y5: A,Xs5: coinductive_llist @ A] :
( ( X2
= ( coinductive_LCons @ A @ Xa2 @ ( coinductive_LCons @ A @ Y5 @ Xs5 ) ) )
& ( ord_less_eq @ A @ Xa2 @ Y5 )
& ( ( X6 @ ( coinductive_LCons @ A @ Y5 @ Xs5 ) )
| ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ Y5 @ Xs5 ) ) ) ) ) )
=> ( coindu63249387sorted @ A @ X ) ) ) ) ).
% lsorted.coinduct
thf(fact_112_Coinductive__List_Ofinite__lprefix__nitpick__simps_I3_J,axiom,
! [A: $tType,Xs3: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
( ( coindu328551480prefix @ A @ Xs3 @ ( coinductive_LCons @ A @ Y @ Ys ) )
= ( ( Xs3
= ( coinductive_LNil @ A ) )
| ? [Xs6: coinductive_llist @ A] :
( ( Xs3
= ( coinductive_LCons @ A @ Y @ Xs6 ) )
& ( coindu328551480prefix @ A @ Xs6 @ Ys ) ) ) ) ).
% Coinductive_List.finite_lprefix_nitpick_simps(3)
thf(fact_113_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_114_lstrict__prefix__code_I3_J,axiom,
! [B: $tType,X: B,Xs3: coinductive_llist @ B] :
~ ( coindu1478340336prefix @ B @ ( coinductive_LCons @ B @ X @ Xs3 ) @ ( coinductive_LNil @ B ) ) ).
% lstrict_prefix_code(3)
thf(fact_115_llast__singleton,axiom,
! [A: $tType,X: A] :
( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LNil @ A ) ) )
= X ) ).
% llast_singleton
thf(fact_116_ldistinct_Ocoinduct,axiom,
! [A: $tType,X6: ( coinductive_llist @ A ) > $o,X: coinductive_llist @ A] :
( ( X6 @ X )
=> ( ! [X2: coinductive_llist @ A] :
( ( X6 @ X2 )
=> ( ( X2
= ( coinductive_LNil @ A ) )
| ? [Xa2: A,Xs5: coinductive_llist @ A] :
( ( X2
= ( coinductive_LCons @ A @ Xa2 @ Xs5 ) )
& ~ ( member @ A @ Xa2 @ ( coinductive_lset @ A @ Xs5 ) )
& ( ( X6 @ Xs5 )
| ( coindu351974385stinct @ A @ Xs5 ) ) ) ) )
=> ( coindu351974385stinct @ A @ X ) ) ) ).
% ldistinct.coinduct
thf(fact_117_ldistinct_Osimps,axiom,
! [A: $tType] :
( ( coindu351974385stinct @ A )
= ( ^ [A6: coinductive_llist @ A] :
( ( A6
= ( coinductive_LNil @ A ) )
| ? [X5: A,Xs4: coinductive_llist @ A] :
( ( A6
= ( coinductive_LCons @ A @ X5 @ Xs4 ) )
& ~ ( member @ A @ X5 @ ( coinductive_lset @ A @ Xs4 ) )
& ( coindu351974385stinct @ A @ Xs4 ) ) ) ) ) ).
% ldistinct.simps
thf(fact_118_ldistinct__LNil__code,axiom,
! [A: $tType] : ( coindu351974385stinct @ A @ ( coinductive_LNil @ A ) ) ).
% ldistinct_LNil_code
thf(fact_119_llast__LCons2,axiom,
! [A: $tType,X: A,Y: A,Xs3: coinductive_llist @ A] :
( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LCons @ A @ Y @ Xs3 ) ) )
= ( coinductive_llast @ A @ ( coinductive_LCons @ A @ Y @ Xs3 ) ) ) ).
% llast_LCons2
thf(fact_120_lstrict__prefix__code_I4_J,axiom,
! [B: $tType,X: B,Xs3: coinductive_llist @ B,Y: B,Ys: coinductive_llist @ B] :
( ( coindu1478340336prefix @ B @ ( coinductive_LCons @ B @ X @ Xs3 ) @ ( coinductive_LCons @ B @ Y @ Ys ) )
= ( ( X = Y )
& ( coindu1478340336prefix @ B @ Xs3 @ Ys ) ) ) ).
% lstrict_prefix_code(4)
thf(fact_121_lstrict__prefix__code_I1_J,axiom,
! [A: $tType] :
~ ( coindu1478340336prefix @ A @ ( coinductive_LNil @ A ) @ ( coinductive_LNil @ A ) ) ).
% lstrict_prefix_code(1)
thf(fact_122_ldistinct__LCons,axiom,
! [A: $tType,X: A,Xs3: coinductive_llist @ A] :
( ( coindu351974385stinct @ A @ ( coinductive_LCons @ A @ X @ Xs3 ) )
= ( ~ ( member @ A @ X @ ( coinductive_lset @ A @ Xs3 ) )
& ( coindu351974385stinct @ A @ Xs3 ) ) ) ).
% ldistinct_LCons
thf(fact_123_llist__less__induct,axiom,
! [A: $tType,P: ( coinductive_llist @ A ) > $o,Xs3: coinductive_llist @ A] :
( ! [Xs2: coinductive_llist @ A] :
( ! [Ys2: coinductive_llist @ A] :
( ( coindu1478340336prefix @ A @ Ys2 @ Xs2 )
=> ( P @ Ys2 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs3 ) ) ).
% llist_less_induct
thf(fact_124_ldistinct_OLNil,axiom,
! [A: $tType] : ( coindu351974385stinct @ A @ ( coinductive_LNil @ A ) ) ).
% ldistinct.LNil
thf(fact_125_ldistinct_OLCons,axiom,
! [A: $tType,X: A,Xs3: coinductive_llist @ A] :
( ~ ( member @ A @ X @ ( coinductive_lset @ A @ Xs3 ) )
=> ( ( coindu351974385stinct @ A @ Xs3 )
=> ( coindu351974385stinct @ A @ ( coinductive_LCons @ A @ X @ Xs3 ) ) ) ) ).
% ldistinct.LCons
thf(fact_126_lsorted__ldistinct__lset__unique,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Xs3: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coindu63249387sorted @ A @ Xs3 )
=> ( ( coindu351974385stinct @ A @ Xs3 )
=> ( ( coindu63249387sorted @ A @ Ys )
=> ( ( coindu351974385stinct @ A @ Ys )
=> ( ( ( coinductive_lset @ A @ Xs3 )
= ( coinductive_lset @ A @ Ys ) )
=> ( Xs3 = Ys ) ) ) ) ) ) ) ).
% lsorted_ldistinct_lset_unique
thf(fact_127_Coinductive__List_Ofinite__lprefix__nitpick__simps_I1_J,axiom,
! [A: $tType,Xs3: coinductive_llist @ A] :
( ( coindu328551480prefix @ A @ Xs3 @ ( coinductive_LNil @ A ) )
= ( Xs3
= ( coinductive_LNil @ A ) ) ) ).
% Coinductive_List.finite_lprefix_nitpick_simps(1)
thf(fact_128_Coinductive__List_Ofinite__lprefix__nitpick__simps_I2_J,axiom,
! [A: $tType,Xs3: coinductive_llist @ A] : ( coindu328551480prefix @ A @ ( coinductive_LNil @ A ) @ Xs3 ) ).
% Coinductive_List.finite_lprefix_nitpick_simps(2)
thf(fact_129_ldistinct_Ocases,axiom,
! [A: $tType,A2: coinductive_llist @ A] :
( ( coindu351974385stinct @ A @ A2 )
=> ( ( A2
!= ( coinductive_LNil @ A ) )
=> ~ ! [X2: A,Xs2: coinductive_llist @ A] :
( ( A2
= ( coinductive_LCons @ A @ X2 @ Xs2 ) )
=> ( ~ ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs2 ) )
=> ~ ( coindu351974385stinct @ A @ Xs2 ) ) ) ) ) ).
% ldistinct.cases
thf(fact_130_llimit__induct,axiom,
! [A: $tType,P: ( coinductive_llist @ A ) > $o,Xs3: 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 @ Xs3 )
=> ( P @ Ys2 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs3 ) ) ) ) ).
% llimit_induct
thf(fact_131_gen__lset__code_I1_J,axiom,
! [A: $tType,A3: set @ A] :
( ( coinductive_gen_lset @ A @ A3 @ ( coinductive_LNil @ A ) )
= A3 ) ).
% gen_lset_code(1)
thf(fact_132_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,A6: C] : ( if @ ( coinductive_llist @ A ) @ ( P2 @ A6 ) @ ( coinductive_LNil @ A ) @ ( coinductive_LCons @ A @ ( G21 @ A6 ) @ ( if @ ( coinductive_llist @ A ) @ ( Q22 @ A6 ) @ ( G221 @ A6 ) @ ( coindu1259883913_llist @ C @ A @ P2 @ G21 @ Q22 @ G221 @ G222 @ ( G222 @ A6 ) ) ) ) ) ) ) ).
% llist.corec_code
thf(fact_133_unfold__llist_Ocode,axiom,
! [B: $tType,A: $tType] :
( ( coindu1441602521_llist @ A @ B )
= ( ^ [P2: A > $o,G21: A > B,G22: A > A,A6: A] : ( if @ ( coinductive_llist @ B ) @ ( P2 @ A6 ) @ ( coinductive_LNil @ B ) @ ( coinductive_LCons @ B @ ( G21 @ A6 ) @ ( coindu1441602521_llist @ A @ B @ P2 @ G21 @ G22 @ ( G22 @ A6 ) ) ) ) ) ) ).
% unfold_llist.code
thf(fact_134_lfinite__code_I2_J,axiom,
! [B: $tType,X: B,Xs3: coinductive_llist @ B] :
( ( coinductive_lfinite @ B @ ( coinductive_LCons @ B @ X @ Xs3 ) )
= ( coinductive_lfinite @ B @ Xs3 ) ) ).
% lfinite_code(2)
thf(fact_135_lfinite__LCons,axiom,
! [A: $tType,X: A,Xs3: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ ( coinductive_LCons @ A @ X @ Xs3 ) )
= ( coinductive_lfinite @ A @ Xs3 ) ) ).
% lfinite_LCons
thf(fact_136_lfinite__code_I1_J,axiom,
! [A: $tType] : ( coinductive_lfinite @ A @ ( coinductive_LNil @ A ) ) ).
% lfinite_code(1)
thf(fact_137_unfold__llist__eq__LCons,axiom,
! [A: $tType,B: $tType,IS_LNIL: B > $o,LHD: B > A,LTL: B > B,B3: B,X: A,Xs3: coinductive_llist @ A] :
( ( ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ B3 )
= ( coinductive_LCons @ A @ X @ Xs3 ) )
= ( ~ ( IS_LNIL @ B3 )
& ( X
= ( LHD @ B3 ) )
& ( Xs3
= ( coindu1441602521_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ ( LTL @ B3 ) ) ) ) ) ).
% unfold_llist_eq_LCons
thf(fact_138_lfinite__LConsI,axiom,
! [A: $tType,Xs3: coinductive_llist @ A,X: A] :
( ( coinductive_lfinite @ A @ Xs3 )
=> ( coinductive_lfinite @ A @ ( coinductive_LCons @ A @ X @ Xs3 ) ) ) ).
% lfinite_LConsI
thf(fact_139_lfinite__LNil,axiom,
! [A: $tType] : ( coinductive_lfinite @ A @ ( coinductive_LNil @ A ) ) ).
% lfinite_LNil
thf(fact_140_lstrict__prefix__lfinite1,axiom,
! [A: $tType,Xs3: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coindu1478340336prefix @ A @ Xs3 @ Ys )
=> ( coinductive_lfinite @ A @ Xs3 ) ) ).
% lstrict_prefix_lfinite1
thf(fact_141_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_142_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_143_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_144_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_145_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_146_lfinite_Osimps,axiom,
! [A: $tType] :
( ( coinductive_lfinite @ A )
= ( ^ [A6: coinductive_llist @ A] :
( ( A6
= ( coinductive_LNil @ A ) )
| ? [Xs4: coinductive_llist @ A,X5: A] :
( ( A6
= ( coinductive_LCons @ A @ X5 @ Xs4 ) )
& ( coinductive_lfinite @ A @ Xs4 ) ) ) ) ) ).
% lfinite.simps
thf(fact_147_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_148_llast__lappend__LCons,axiom,
! [A: $tType,Xs3: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs3 )
=> ( ( coinductive_llast @ A @ ( coinductive_lappend @ A @ Xs3 @ ( coinductive_LCons @ A @ Y @ Ys ) ) )
= ( coinductive_llast @ A @ ( coinductive_LCons @ A @ Y @ Ys ) ) ) ) ).
% llast_lappend_LCons
thf(fact_149_split__llist,axiom,
! [A: $tType,X: A,Xs3: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs3 ) )
=> ? [Ys3: coinductive_llist @ A,Zs: coinductive_llist @ A] :
( ( Xs3
= ( coinductive_lappend @ A @ Ys3 @ ( coinductive_LCons @ A @ X @ Zs ) ) )
& ( coinductive_lfinite @ A @ Ys3 ) ) ) ).
% split_llist
thf(fact_150_split__llist__first,axiom,
! [A: $tType,X: A,Xs3: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs3 ) )
=> ? [Ys3: coinductive_llist @ A,Zs: coinductive_llist @ A] :
( ( Xs3
= ( coinductive_lappend @ A @ Ys3 @ ( coinductive_LCons @ A @ X @ Zs ) ) )
& ( coinductive_lfinite @ A @ Ys3 )
& ~ ( member @ A @ X @ ( coinductive_lset @ A @ Ys3 ) ) ) ) ).
% split_llist_first
thf(fact_151_lfinite__rev__induct,axiom,
! [A: $tType,Xs3: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
( ( coinductive_lfinite @ A @ Xs3 )
=> ( ( 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 @ Xs3 ) ) ) ) ).
% lfinite_rev_induct
thf(fact_152_lappend__code_I2_J,axiom,
! [A: $tType,Xa3: A,X: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ ( coinductive_LCons @ A @ Xa3 @ X ) @ Ys )
= ( coinductive_LCons @ A @ Xa3 @ ( coinductive_lappend @ A @ X @ Ys ) ) ) ).
% lappend_code(2)
thf(fact_153_lappend__code_I1_J,axiom,
! [A: $tType,Ys: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ ( coinductive_LNil @ A ) @ Ys )
= Ys ) ).
% lappend_code(1)
thf(fact_154_lappend__LNil2,axiom,
! [A: $tType,Xs3: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ Xs3 @ ( coinductive_LNil @ A ) )
= Xs3 ) ).
% lappend_LNil2
thf(fact_155_lfinite__lappend,axiom,
! [A: $tType,Xs3: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ ( coinductive_lappend @ A @ Xs3 @ Ys ) )
= ( ( coinductive_lfinite @ A @ Xs3 )
& ( coinductive_lfinite @ A @ Ys ) ) ) ).
% lfinite_lappend
thf(fact_156_lappend__inf,axiom,
! [A: $tType,Xs3: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ~ ( coinductive_lfinite @ A @ Xs3 )
=> ( ( coinductive_lappend @ A @ Xs3 @ Ys )
= Xs3 ) ) ).
% lappend_inf
thf(fact_157_lappend__assoc,axiom,
! [A: $tType,Xs3: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs2: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ ( coinductive_lappend @ A @ Xs3 @ Ys ) @ Zs2 )
= ( coinductive_lappend @ A @ Xs3 @ ( coinductive_lappend @ A @ Ys @ Zs2 ) ) ) ).
% lappend_assoc
thf(fact_158_paths__lappendD1,axiom,
! [A: $tType,Xs3: coinductive_llist @ A,Ys: coinductive_llist @ A,Graph: A > A > $o] :
( ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ Xs3 @ Ys ) @ ( koenig916195507_paths @ A @ Graph ) )
=> ( member @ ( coinductive_llist @ A ) @ Xs3 @ ( koenig916195507_paths @ A @ Graph ) ) ) ).
% paths_lappendD1
thf(fact_159_lappend__LNil__LNil,axiom,
! [A: $tType] :
( ( coinductive_lappend @ A @ ( coinductive_LNil @ A ) @ ( coinductive_LNil @ A ) )
= ( coinductive_LNil @ A ) ) ).
% lappend_LNil_LNil
thf(fact_160_LNil__eq__lappend__iff,axiom,
! [A: $tType,Xs3: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ( coinductive_LNil @ A )
= ( coinductive_lappend @ A @ Xs3 @ Ys ) )
= ( ( Xs3
= ( coinductive_LNil @ A ) )
& ( Ys
= ( coinductive_LNil @ A ) ) ) ) ).
% LNil_eq_lappend_iff
thf(fact_161_lappend__eq__LNil__iff,axiom,
! [A: $tType,Xs3: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ( coinductive_lappend @ A @ Xs3 @ Ys )
= ( coinductive_LNil @ A ) )
= ( ( Xs3
= ( coinductive_LNil @ A ) )
& ( Ys
= ( coinductive_LNil @ A ) ) ) ) ).
% lappend_eq_LNil_iff
thf(fact_162_lset__lappend1,axiom,
! [A: $tType,Xs3: coinductive_llist @ A,Ys: coinductive_llist @ A] : ( ord_less_eq @ ( set @ A ) @ ( coinductive_lset @ A @ Xs3 ) @ ( coinductive_lset @ A @ ( coinductive_lappend @ A @ Xs3 @ Ys ) ) ) ).
% lset_lappend1
thf(fact_163_lappend__snocL1__conv__LCons2,axiom,
! [A: $tType,Xs3: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ ( coinductive_lappend @ A @ Xs3 @ ( coinductive_LCons @ A @ Y @ ( coinductive_LNil @ A ) ) ) @ Ys )
= ( coinductive_lappend @ A @ Xs3 @ ( coinductive_LCons @ A @ Y @ Ys ) ) ) ).
% lappend_snocL1_conv_LCons2
thf(fact_164_in__lset__lappend__iff,axiom,
! [A: $tType,X: A,Xs3: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_lappend @ A @ Xs3 @ Ys ) ) )
= ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs3 ) )
| ( ( coinductive_lfinite @ A @ Xs3 )
& ( member @ A @ X @ ( coinductive_lset @ A @ Ys ) ) ) ) ) ).
% in_lset_lappend_iff
thf(fact_165_paths__lappendD2,axiom,
! [A: $tType,Xs3: coinductive_llist @ A,Ys: coinductive_llist @ A,Graph: A > A > $o] :
( ( coinductive_lfinite @ A @ Xs3 )
=> ( ( member @ ( coinductive_llist @ A ) @ ( coinductive_lappend @ A @ Xs3 @ Ys ) @ ( koenig916195507_paths @ A @ Graph ) )
=> ( member @ ( coinductive_llist @ A ) @ Ys @ ( koenig916195507_paths @ A @ Graph ) ) ) ) ).
% paths_lappendD2
thf(fact_166_lfilter__eq__LConsD,axiom,
! [A: $tType,P: A > $o,Ys: coinductive_llist @ A,X: A,Xs3: coinductive_llist @ A] :
( ( ( coinductive_lfilter @ A @ P @ Ys )
= ( coinductive_LCons @ A @ X @ Xs3 ) )
=> ? [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 )
& ( Xs3
= ( coinductive_lfilter @ A @ P @ Vs ) ) ) ) ).
% lfilter_eq_LConsD
thf(fact_167_lset__lappend__lfinite,axiom,
! [A: $tType,Xs3: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs3 )
=> ( ( coinductive_lset @ A @ ( coinductive_lappend @ A @ Xs3 @ Ys ) )
= ( sup_sup @ ( set @ A ) @ ( coinductive_lset @ A @ Xs3 ) @ ( coinductive_lset @ A @ Ys ) ) ) ) ).
% lset_lappend_lfinite
thf(fact_168_llexord__lappend__leftD,axiom,
! [A: $tType,R: A > A > $o,Xs3: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs2: coinductive_llist @ A] :
( ( coinductive_llexord @ A @ R @ ( coinductive_lappend @ A @ Xs3 @ Ys ) @ ( coinductive_lappend @ A @ Xs3 @ Zs2 ) )
=> ( ( coinductive_lfinite @ A @ Xs3 )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs3 ) )
=> ~ ( R @ X2 @ X2 ) )
=> ( coinductive_llexord @ A @ R @ Ys @ Zs2 ) ) ) ) ).
% llexord_lappend_leftD
thf(fact_169_UnCI,axiom,
! [A: $tType,C2: A,B2: set @ A,A3: set @ A] :
( ( ~ ( member @ A @ C2 @ B2 )
=> ( member @ A @ C2 @ A3 ) )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) ) ) ).
% UnCI
thf(fact_170_Un__iff,axiom,
! [A: $tType,C2: A,A3: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) )
= ( ( member @ A @ C2 @ A3 )
| ( member @ A @ C2 @ B2 ) ) ) ).
% Un_iff
thf(fact_171_lfilter__idem,axiom,
! [A: $tType,P: A > $o,Xs3: coinductive_llist @ A] :
( ( coinductive_lfilter @ A @ P @ ( coinductive_lfilter @ A @ P @ Xs3 ) )
= ( coinductive_lfilter @ A @ P @ Xs3 ) ) ).
% lfilter_idem
thf(fact_172_llexord__refl,axiom,
! [A: $tType,R: A > A > $o,Xs3: coinductive_llist @ A] : ( coinductive_llexord @ A @ R @ Xs3 @ Xs3 ) ).
% llexord_refl
thf(fact_173_Un__subset__iff,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) @ C3 )
= ( ( ord_less_eq @ ( set @ A ) @ A3 @ C3 )
& ( ord_less_eq @ ( set @ A ) @ B2 @ C3 ) ) ) ).
% Un_subset_iff
thf(fact_174_lfilter__LCons,axiom,
! [A: $tType,P: A > $o,X: A,Xs3: coinductive_llist @ A] :
( ( ( P @ X )
=> ( ( coinductive_lfilter @ A @ P @ ( coinductive_LCons @ A @ X @ Xs3 ) )
= ( coinductive_LCons @ A @ X @ ( coinductive_lfilter @ A @ P @ Xs3 ) ) ) )
& ( ~ ( P @ X )
=> ( ( coinductive_lfilter @ A @ P @ ( coinductive_LCons @ A @ X @ Xs3 ) )
= ( coinductive_lfilter @ A @ P @ Xs3 ) ) ) ) ).
% lfilter_LCons
thf(fact_175_lfilter__LNil,axiom,
! [A: $tType,P: A > $o] :
( ( coinductive_lfilter @ A @ P @ ( coinductive_LNil @ A ) )
= ( coinductive_LNil @ A ) ) ).
% lfilter_LNil
thf(fact_176_llexord__LCons__LCons,axiom,
! [A: $tType,R: A > A > $o,X: A,Xs3: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
( ( coinductive_llexord @ A @ R @ ( coinductive_LCons @ A @ X @ Xs3 ) @ ( coinductive_LCons @ A @ Y @ Ys ) )
= ( ( ( X = Y )
& ( coinductive_llexord @ A @ R @ Xs3 @ Ys ) )
| ( R @ X @ Y ) ) ) ).
% llexord_LCons_LCons
thf(fact_177_llexord__code_I1_J,axiom,
! [A: $tType,R: A > A > $o,Ys: coinductive_llist @ A] : ( coinductive_llexord @ A @ R @ ( coinductive_LNil @ A ) @ Ys ) ).
% llexord_code(1)
thf(fact_178_diverge__lfilter__LNil,axiom,
! [A: $tType,Xs3: coinductive_llist @ A,P: A > $o] :
( ! [X2: A] :
( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs3 ) )
=> ~ ( P @ X2 ) )
=> ( ( coinductive_lfilter @ A @ P @ Xs3 )
= ( coinductive_LNil @ A ) ) ) ).
% diverge_lfilter_LNil
thf(fact_179_lfilter__lappend__lfinite,axiom,
! [A: $tType,Xs3: coinductive_llist @ A,P: A > $o,Ys: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs3 )
=> ( ( coinductive_lfilter @ A @ P @ ( coinductive_lappend @ A @ Xs3 @ Ys ) )
= ( coinductive_lappend @ A @ ( coinductive_lfilter @ A @ P @ Xs3 ) @ ( coinductive_lfilter @ A @ P @ Ys ) ) ) ) ).
% lfilter_lappend_lfinite
thf(fact_180_llexord__lappend__leftI,axiom,
! [A: $tType,R: A > A > $o,Ys: coinductive_llist @ A,Zs2: coinductive_llist @ A,Xs3: coinductive_llist @ A] :
( ( coinductive_llexord @ A @ R @ Ys @ Zs2 )
=> ( coinductive_llexord @ A @ R @ ( coinductive_lappend @ A @ Xs3 @ Ys ) @ ( coinductive_lappend @ A @ Xs3 @ Zs2 ) ) ) ).
% llexord_lappend_leftI
thf(fact_181_llexord__append__right,axiom,
! [A: $tType,R: A > A > $o,Xs3: coinductive_llist @ A,Ys: coinductive_llist @ A] : ( coinductive_llexord @ A @ R @ Xs3 @ ( coinductive_lappend @ A @ Xs3 @ Ys ) ) ).
% llexord_append_right
thf(fact_182_ldistinct__lfilterI,axiom,
! [A: $tType,Xs3: coinductive_llist @ A,P: A > $o] :
( ( coindu351974385stinct @ A @ Xs3 )
=> ( coindu351974385stinct @ A @ ( coinductive_lfilter @ A @ P @ Xs3 ) ) ) ).
% ldistinct_lfilterI
thf(fact_183_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_184_lfilter__LCons__found,axiom,
! [A: $tType,P: A > $o,X: A,Xs3: coinductive_llist @ A] :
( ( P @ X )
=> ( ( coinductive_lfilter @ A @ P @ ( coinductive_LCons @ A @ X @ Xs3 ) )
= ( coinductive_LCons @ A @ X @ ( coinductive_lfilter @ A @ P @ Xs3 ) ) ) ) ).
% lfilter_LCons_found
thf(fact_185_lfilter__cong,axiom,
! [A: $tType,Xs3: coinductive_llist @ A,Ys: coinductive_llist @ A,P: A > $o,Q: A > $o] :
( ( Xs3 = Ys )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ ( coinductive_lset @ A @ Ys ) )
=> ( ( P @ X2 )
= ( Q @ X2 ) ) )
=> ( ( coinductive_lfilter @ A @ P @ Xs3 )
= ( coinductive_lfilter @ A @ Q @ Ys ) ) ) ) ).
% lfilter_cong
thf(fact_186_lfilter__id__conv,axiom,
! [A: $tType,P: A > $o,Xs3: coinductive_llist @ A] :
( ( ( coinductive_lfilter @ A @ P @ Xs3 )
= Xs3 )
= ( ! [X5: A] :
( ( member @ A @ X5 @ ( coinductive_lset @ A @ Xs3 ) )
=> ( P @ X5 ) ) ) ) ).
% lfilter_id_conv
thf(fact_187_Un__mono,axiom,
! [A: $tType,A3: set @ A,C3: set @ A,B2: set @ A,D: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ C3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ D )
=> ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) @ ( sup_sup @ ( set @ A ) @ C3 @ D ) ) ) ) ).
% Un_mono
thf(fact_188_Un__least,axiom,
! [A: $tType,A3: set @ A,C3: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ C3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
=> ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) @ C3 ) ) ) ).
% Un_least
thf(fact_189_Un__upper1,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) ) ).
% Un_upper1
thf(fact_190_Un__upper2,axiom,
! [A: $tType,B2: set @ A,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ B2 @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) ) ).
% Un_upper2
thf(fact_191_Un__absorb1,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ( sup_sup @ ( set @ A ) @ A3 @ B2 )
= B2 ) ) ).
% Un_absorb1
thf(fact_192_Un__absorb2,axiom,
! [A: $tType,B2: set @ A,A3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B2 @ A3 )
=> ( ( sup_sup @ ( set @ A ) @ A3 @ B2 )
= A3 ) ) ).
% Un_absorb2
thf(fact_193_subset__Un__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A5: set @ A,B5: set @ A] :
( ( sup_sup @ ( set @ A ) @ A5 @ B5 )
= B5 ) ) ) ).
% subset_Un_eq
thf(fact_194_lsorted__lfilterI,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [Xs3: coinductive_llist @ A,P: A > $o] :
( ( coindu63249387sorted @ A @ Xs3 )
=> ( coindu63249387sorted @ A @ ( coinductive_lfilter @ A @ P @ Xs3 ) ) ) ) ).
% lsorted_lfilterI
thf(fact_195_llexord__code_I3_J,axiom,
! [A: $tType,R: A > A > $o,X: A,Xs3: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
( ( coinductive_llexord @ A @ R @ ( coinductive_LCons @ A @ X @ Xs3 ) @ ( coinductive_LCons @ A @ Y @ Ys ) )
= ( ( R @ X @ Y )
| ( ( X = Y )
& ( coinductive_llexord @ A @ R @ Xs3 @ Ys ) ) ) ) ).
% llexord_code(3)
thf(fact_196_llexord__LCons__left,axiom,
! [A: $tType,R: A > A > $o,X: A,Xs3: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_llexord @ A @ R @ ( coinductive_LCons @ A @ X @ Xs3 ) @ Ys )
= ( ? [Y3: A,Ys4: coinductive_llist @ A] :
( ( Ys
= ( coinductive_LCons @ A @ Y3 @ Ys4 ) )
& ( ( ( X = Y3 )
& ( coinductive_llexord @ A @ R @ Xs3 @ Ys4 ) )
| ( R @ X @ Y3 ) ) ) ) ) ).
% llexord_LCons_left
thf(fact_197_llexord__LCons__eq,axiom,
! [A: $tType,R: A > A > $o,Xs3: coinductive_llist @ A,Ys: coinductive_llist @ A,X: A] :
( ( coinductive_llexord @ A @ R @ Xs3 @ Ys )
=> ( coinductive_llexord @ A @ R @ ( coinductive_LCons @ A @ X @ Xs3 ) @ ( coinductive_LCons @ A @ X @ Ys ) ) ) ).
% llexord_LCons_eq
thf(fact_198_llexord__LCons__less,axiom,
! [A: $tType,R: A > A > $o,X: A,Y: A,Xs3: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( R @ X @ Y )
=> ( coinductive_llexord @ A @ R @ ( coinductive_LCons @ A @ X @ Xs3 ) @ ( coinductive_LCons @ A @ Y @ Ys ) ) ) ).
% llexord_LCons_less
thf(fact_199_llexord__LNil,axiom,
! [A: $tType,R: A > A > $o,Ys: coinductive_llist @ A] : ( coinductive_llexord @ A @ R @ ( coinductive_LNil @ A ) @ Ys ) ).
% llexord_LNil
thf(fact_200_UnE,axiom,
! [A: $tType,C2: A,A3: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) )
=> ( ~ ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ B2 ) ) ) ).
% UnE
thf(fact_201_UnI1,axiom,
! [A: $tType,C2: A,A3: set @ A,B2: set @ A] :
( ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) ) ) ).
% UnI1
thf(fact_202_UnI2,axiom,
! [A: $tType,C2: A,B2: set @ A,A3: set @ A] :
( ( member @ A @ C2 @ B2 )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) ) ) ).
% UnI2
thf(fact_203_bex__Un,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,P: A > $o] :
( ( ? [X5: A] :
( ( member @ A @ X5 @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) )
& ( P @ X5 ) ) )
= ( ? [X5: A] :
( ( member @ A @ X5 @ A3 )
& ( P @ X5 ) )
| ? [X5: A] :
( ( member @ A @ X5 @ B2 )
& ( P @ X5 ) ) ) ) ).
% bex_Un
thf(fact_204_ball__Un,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,P: A > $o] :
( ( ! [X5: A] :
( ( member @ A @ X5 @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) )
=> ( P @ X5 ) ) )
= ( ! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ( P @ X5 ) )
& ! [X5: A] :
( ( member @ A @ X5 @ B2 )
=> ( P @ X5 ) ) ) ) ).
% ball_Un
thf(fact_205_Un__assoc,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,C3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) @ C3 )
= ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B2 @ C3 ) ) ) ).
% Un_assoc
thf(fact_206_Un__absorb,axiom,
! [A: $tType,A3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ A3 )
= A3 ) ).
% Un_absorb
thf(fact_207_Un__commute,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A5: set @ A,B5: set @ A] : ( sup_sup @ ( set @ A ) @ B5 @ A5 ) ) ) ).
% Un_commute
thf(fact_208_Un__left__absorb,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) )
= ( sup_sup @ ( set @ A ) @ A3 @ B2 ) ) ).
% Un_left_absorb
thf(fact_209_Un__left__commute,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,C3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B2 @ C3 ) )
= ( sup_sup @ ( set @ A ) @ B2 @ ( sup_sup @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% Un_left_commute
thf(fact_210_llexord__trans,axiom,
! [A: $tType,R: A > A > $o,Xs3: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs2: coinductive_llist @ A] :
( ( coinductive_llexord @ A @ R @ Xs3 @ Ys )
=> ( ( coinductive_llexord @ A @ R @ Ys @ Zs2 )
=> ( ! [A4: A,B4: A,C4: A] :
( ( R @ A4 @ B4 )
=> ( ( R @ B4 @ C4 )
=> ( R @ A4 @ C4 ) ) )
=> ( coinductive_llexord @ A @ R @ Xs3 @ Zs2 ) ) ) ) ).
% llexord_trans
thf(fact_211_llexord__linear,axiom,
! [A: $tType,R: A > A > $o,Xs3: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ! [X2: A,Y4: A] :
( ( R @ X2 @ Y4 )
| ( X2 = Y4 )
| ( R @ Y4 @ X2 ) )
=> ( ( coinductive_llexord @ A @ R @ Xs3 @ Ys )
| ( coinductive_llexord @ A @ R @ Ys @ Xs3 ) ) ) ).
% llexord_linear
thf(fact_212_llexord__antisym,axiom,
! [A: $tType,R: A > A > $o,Xs3: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_llexord @ A @ R @ Xs3 @ Ys )
=> ( ( coinductive_llexord @ A @ R @ Ys @ Xs3 )
=> ( ! [A4: A,B4: A] :
( ( R @ A4 @ B4 )
=> ~ ( R @ B4 @ A4 ) )
=> ( Xs3 = Ys ) ) ) ) ).
% llexord_antisym
thf(fact_213_lfinite__lfilterI,axiom,
! [A: $tType,Xs3: coinductive_llist @ A,P: A > $o] :
( ( coinductive_lfinite @ A @ Xs3 )
=> ( coinductive_lfinite @ A @ ( coinductive_lfilter @ A @ P @ Xs3 ) ) ) ).
% lfinite_lfilterI
thf(fact_214_lfilter__empty__conv,axiom,
! [A: $tType,P: A > $o,Xs3: coinductive_llist @ A] :
( ( ( coinductive_lfilter @ A @ P @ Xs3 )
= ( coinductive_LNil @ A ) )
= ( ! [X5: A] :
( ( member @ A @ X5 @ ( coinductive_lset @ A @ Xs3 ) )
=> ~ ( P @ X5 ) ) ) ) ).
% lfilter_empty_conv
thf(fact_215_lfilter__eq__lappend__lfiniteD,axiom,
! [A: $tType,P: A > $o,Xs3: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs2: coinductive_llist @ A] :
( ( ( coinductive_lfilter @ A @ P @ Xs3 )
= ( coinductive_lappend @ A @ Ys @ Zs2 ) )
=> ( ( coinductive_lfinite @ A @ Ys )
=> ? [Us: coinductive_llist @ A,Vs: coinductive_llist @ A] :
( ( Xs3
= ( coinductive_lappend @ A @ Us @ Vs ) )
& ( coinductive_lfinite @ A @ Us )
& ( Ys
= ( coinductive_lfilter @ A @ P @ Us ) )
& ( Zs2
= ( coinductive_lfilter @ A @ P @ Vs ) ) ) ) ) ).
% lfilter_eq_lappend_lfiniteD
thf(fact_216_llexord_Ocoinduct,axiom,
! [A: $tType,X6: ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o,X: coinductive_llist @ A,Xa3: coinductive_llist @ A,R: A > A > $o] :
( ( X6 @ X @ Xa3 )
=> ( ! [X2: coinductive_llist @ A,Xa: coinductive_llist @ A] :
( ( X6 @ X2 @ Xa )
=> ( ? [Xs5: coinductive_llist @ A,Ys2: coinductive_llist @ A,Xb: A] :
( ( X2
= ( coinductive_LCons @ A @ Xb @ Xs5 ) )
& ( Xa
= ( coinductive_LCons @ A @ Xb @ Ys2 ) )
& ( ( X6 @ Xs5 @ Ys2 )
| ( coinductive_llexord @ A @ R @ Xs5 @ Ys2 ) ) )
| ? [Xb: A,Y5: A,Xs5: coinductive_llist @ A,Ys2: coinductive_llist @ A] :
( ( X2
= ( coinductive_LCons @ A @ Xb @ Xs5 ) )
& ( Xa
= ( coinductive_LCons @ A @ Y5 @ Ys2 ) )
& ( R @ Xb @ Y5 ) )
| ? [Ys2: coinductive_llist @ A] :
( ( X2
= ( coinductive_LNil @ A ) )
& ( Xa = Ys2 ) ) ) )
=> ( coinductive_llexord @ A @ R @ X @ Xa3 ) ) ) ).
% llexord.coinduct
thf(fact_217_llexord_Osimps,axiom,
! [A: $tType] :
( ( coinductive_llexord @ A )
= ( ^ [R2: A > A > $o,A12: coinductive_llist @ A,A23: coinductive_llist @ A] :
( ? [Xs4: coinductive_llist @ A,Ys5: coinductive_llist @ A,X5: A] :
( ( A12
= ( coinductive_LCons @ A @ X5 @ Xs4 ) )
& ( A23
= ( coinductive_LCons @ A @ X5 @ Ys5 ) )
& ( coinductive_llexord @ A @ R2 @ Xs4 @ Ys5 ) )
| ? [X5: A,Y3: A,Xs4: coinductive_llist @ A,Ys5: coinductive_llist @ A] :
( ( A12
= ( coinductive_LCons @ A @ X5 @ Xs4 ) )
& ( A23
= ( coinductive_LCons @ A @ Y3 @ Ys5 ) )
& ( R2 @ X5 @ Y3 ) )
| ? [Ys5: coinductive_llist @ A] :
( ( A12
= ( coinductive_LNil @ A ) )
& ( A23 = Ys5 ) ) ) ) ) ).
% llexord.simps
thf(fact_218_llexord_Ocases,axiom,
! [A: $tType,R: A > A > $o,A1: coinductive_llist @ A,A22: coinductive_llist @ A] :
( ( coinductive_llexord @ A @ R @ A1 @ A22 )
=> ( ! [Xs2: coinductive_llist @ A,Ys3: coinductive_llist @ A,X2: A] :
( ( A1
= ( coinductive_LCons @ A @ X2 @ Xs2 ) )
=> ( ( A22
= ( coinductive_LCons @ A @ X2 @ Ys3 ) )
=> ~ ( coinductive_llexord @ A @ R @ Xs2 @ Ys3 ) ) )
=> ( ! [X2: A] :
( ? [Xs2: coinductive_llist @ A] :
( A1
= ( coinductive_LCons @ A @ X2 @ Xs2 ) )
=> ! [Y4: A] :
( ? [Ys3: coinductive_llist @ A] :
( A22
= ( coinductive_LCons @ A @ Y4 @ Ys3 ) )
=> ~ ( R @ X2 @ Y4 ) ) )
=> ~ ( ( A1
= ( coinductive_LNil @ A ) )
=> ! [Ys3: coinductive_llist @ A] : A22 != Ys3 ) ) ) ) ).
% llexord.cases
thf(fact_219_llexord__code_I2_J,axiom,
! [A: $tType,R: A > A > $o,X: A,Xs3: coinductive_llist @ A] :
~ ( coinductive_llexord @ A @ R @ ( coinductive_LCons @ A @ X @ Xs3 ) @ ( coinductive_LNil @ A ) ) ).
% llexord_code(2)
thf(fact_220_gen__lset__def,axiom,
! [A: $tType] :
( ( coinductive_gen_lset @ A )
= ( ^ [A5: set @ A,Xs4: coinductive_llist @ A] : ( sup_sup @ ( set @ A ) @ A5 @ ( coinductive_lset @ A @ Xs4 ) ) ) ) ).
% gen_lset_def
thf(fact_221_lset__lappend,axiom,
! [A: $tType,Xs3: coinductive_llist @ A,Ys: coinductive_llist @ A] : ( ord_less_eq @ ( set @ A ) @ ( coinductive_lset @ A @ ( coinductive_lappend @ A @ Xs3 @ Ys ) ) @ ( sup_sup @ ( set @ A ) @ ( coinductive_lset @ A @ Xs3 ) @ ( coinductive_lset @ A @ Ys ) ) ) ).
% lset_lappend
thf(fact_222_lset__lappend__conv,axiom,
! [A: $tType,Xs3: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ( coinductive_lfinite @ A @ Xs3 )
=> ( ( coinductive_lset @ A @ ( coinductive_lappend @ A @ Xs3 @ Ys ) )
= ( sup_sup @ ( set @ A ) @ ( coinductive_lset @ A @ Xs3 ) @ ( coinductive_lset @ A @ Ys ) ) ) )
& ( ~ ( coinductive_lfinite @ A @ Xs3 )
=> ( ( coinductive_lset @ A @ ( coinductive_lappend @ A @ Xs3 @ Ys ) )
= ( coinductive_lset @ A @ Xs3 ) ) ) ) ).
% lset_lappend_conv
thf(fact_223_llexord__lappend__left,axiom,
! [A: $tType,Xs3: coinductive_llist @ A,R: A > A > $o,Ys: coinductive_llist @ A,Zs2: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs3 )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs3 ) )
=> ~ ( R @ X2 @ X2 ) )
=> ( ( coinductive_llexord @ A @ R @ ( coinductive_lappend @ A @ Xs3 @ Ys ) @ ( coinductive_lappend @ A @ Xs3 @ Zs2 ) )
= ( coinductive_llexord @ A @ R @ Ys @ Zs2 ) ) ) ) ).
% llexord_lappend_left
thf(fact_224_le__sup__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ X @ Y ) @ Z )
= ( ( ord_less_eq @ A @ X @ Z )
& ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% le_sup_iff
thf(fact_225_sup_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B3: A,C2: A,A2: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B3 @ C2 ) @ A2 )
= ( ( ord_less_eq @ A @ B3 @ A2 )
& ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% sup.bounded_iff
thf(fact_226_sup_OcoboundedI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [C2: A,B3: A,A2: A] :
( ( ord_less_eq @ A @ C2 @ B3 )
=> ( ord_less_eq @ A @ C2 @ ( sup_sup @ A @ A2 @ B3 ) ) ) ) ).
% sup.coboundedI2
thf(fact_227_inf__sup__ord_I4_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X @ Y ) ) ) ).
% inf_sup_ord(4)
thf(fact_228_inf__sup__ord_I3_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ X @ ( sup_sup @ A @ X @ Y ) ) ) ).
% inf_sup_ord(3)
thf(fact_229_le__supE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,X: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B3 ) @ X )
=> ~ ( ( ord_less_eq @ A @ A2 @ X )
=> ~ ( ord_less_eq @ A @ B3 @ X ) ) ) ) ).
% le_supE
thf(fact_230_le__supI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A,X: A,B3: A] :
( ( ord_less_eq @ A @ A2 @ X )
=> ( ( ord_less_eq @ A @ B3 @ X )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B3 ) @ X ) ) ) ) ).
% le_supI
thf(fact_231_sup__ge1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ X @ ( sup_sup @ A @ X @ Y ) ) ) ).
% sup_ge1
thf(fact_232_sup__ge2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X @ Y ) ) ) ).
% sup_ge2
thf(fact_233_le__supI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,A2: A,B3: A] :
( ( ord_less_eq @ A @ X @ A2 )
=> ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A2 @ B3 ) ) ) ) ).
% le_supI1
thf(fact_234_le__supI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,B3: A,A2: A] :
( ( ord_less_eq @ A @ X @ B3 )
=> ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A2 @ B3 ) ) ) ) ).
% le_supI2
thf(fact_235_sup_Omono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [C2: A,A2: A,D2: A,B3: A] :
( ( ord_less_eq @ A @ C2 @ A2 )
=> ( ( ord_less_eq @ A @ D2 @ B3 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ C2 @ D2 ) @ ( sup_sup @ A @ A2 @ B3 ) ) ) ) ) ).
% sup.mono
thf(fact_236_sup__mono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A,C2: A,B3: A,D2: A] :
( ( ord_less_eq @ A @ A2 @ C2 )
=> ( ( ord_less_eq @ A @ B3 @ D2 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B3 ) @ ( sup_sup @ A @ C2 @ D2 ) ) ) ) ) ).
% sup_mono
thf(fact_237_sup__least,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A,Z: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_less_eq @ A @ Z @ X )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ Y @ Z ) @ X ) ) ) ) ).
% sup_least
thf(fact_238_le__iff__sup,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [X5: A,Y3: A] :
( ( sup_sup @ A @ X5 @ Y3 )
= Y3 ) ) ) ) ).
% le_iff_sup
thf(fact_239_sup_OorderE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A] :
( ( ord_less_eq @ A @ B3 @ A2 )
=> ( A2
= ( sup_sup @ A @ A2 @ B3 ) ) ) ) ).
% sup.orderE
thf(fact_240_sup_OorderI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( A2
= ( sup_sup @ A @ A2 @ B3 ) )
=> ( ord_less_eq @ A @ B3 @ A2 ) ) ) ).
% sup.orderI
thf(fact_241_sup__unique,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [F: A > A > A,X: A,Y: A] :
( ! [X2: A,Y4: A] : ( ord_less_eq @ A @ X2 @ ( F @ X2 @ Y4 ) )
=> ( ! [X2: A,Y4: A] : ( ord_less_eq @ A @ Y4 @ ( F @ X2 @ Y4 ) )
=> ( ! [X2: A,Y4: A,Z3: A] :
( ( ord_less_eq @ A @ Y4 @ X2 )
=> ( ( ord_less_eq @ A @ Z3 @ X2 )
=> ( ord_less_eq @ A @ ( F @ Y4 @ Z3 ) @ X2 ) ) )
=> ( ( sup_sup @ A @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ) ).
% sup_unique
thf(fact_242_sup_Oabsorb1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A] :
( ( ord_less_eq @ A @ B3 @ A2 )
=> ( ( sup_sup @ A @ A2 @ B3 )
= A2 ) ) ) ).
% sup.absorb1
thf(fact_243_sup_Oabsorb2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( sup_sup @ A @ A2 @ B3 )
= B3 ) ) ) ).
% sup.absorb2
thf(fact_244_sup__absorb1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( sup_sup @ A @ X @ Y )
= X ) ) ) ).
% sup_absorb1
thf(fact_245_sup__absorb2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( sup_sup @ A @ X @ Y )
= Y ) ) ) ).
% sup_absorb2
thf(fact_246_sup_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B3: A,C2: A,A2: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B3 @ C2 ) @ A2 )
=> ~ ( ( ord_less_eq @ A @ B3 @ A2 )
=> ~ ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% sup.boundedE
thf(fact_247_sup_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B3 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ B3 @ C2 ) @ A2 ) ) ) ) ).
% sup.boundedI
thf(fact_248_sup_Oorder__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [B6: A,A6: A] :
( A6
= ( sup_sup @ A @ A6 @ B6 ) ) ) ) ) ).
% sup.order_iff
thf(fact_249_sup_Ocobounded1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] : ( ord_less_eq @ A @ A2 @ ( sup_sup @ A @ A2 @ B3 ) ) ) ).
% sup.cobounded1
thf(fact_250_sup_Ocobounded2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A] : ( ord_less_eq @ A @ B3 @ ( sup_sup @ A @ A2 @ B3 ) ) ) ).
% sup.cobounded2
thf(fact_251_sup_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [B6: A,A6: A] :
( ( sup_sup @ A @ A6 @ B6 )
= A6 ) ) ) ) ).
% sup.absorb_iff1
thf(fact_252_sup_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [A6: A,B6: A] :
( ( sup_sup @ A @ A6 @ B6 )
= B6 ) ) ) ) ).
% sup.absorb_iff2
thf(fact_253_sup_OcoboundedI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [C2: A,A2: A,B3: A] :
( ( ord_less_eq @ A @ C2 @ A2 )
=> ( ord_less_eq @ A @ C2 @ ( sup_sup @ A @ A2 @ B3 ) ) ) ) ).
% sup.coboundedI1
thf(fact_254_llexord__conv,axiom,
! [A: $tType] :
( ( coinductive_llexord @ A )
= ( ^ [R2: A > A > $o,Xs4: coinductive_llist @ A,Ys5: coinductive_llist @ A] :
( ( Xs4 = Ys5 )
| ? [Zs3: coinductive_llist @ A,Xs6: coinductive_llist @ A,Y3: A,Ys4: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Zs3 )
& ( Xs4
= ( coinductive_lappend @ A @ Zs3 @ Xs6 ) )
& ( Ys5
= ( coinductive_lappend @ A @ Zs3 @ ( coinductive_LCons @ A @ Y3 @ Ys4 ) ) )
& ( ( Xs6
= ( coinductive_LNil @ A ) )
| ( R2 @ ( coinductive_lhd @ A @ Xs6 ) @ Y3 ) ) ) ) ) ) ).
% llexord_conv
thf(fact_255_lstrict__prefix__lappend__conv,axiom,
! [A: $tType,Xs3: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coindu1478340336prefix @ A @ Xs3 @ ( coinductive_lappend @ A @ Xs3 @ Ys ) )
= ( ( coinductive_lfinite @ A @ Xs3 )
& ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% lstrict_prefix_lappend_conv
%----Type constructors (19)
thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
! [A7: $tType,A8: $tType] :
( ( comple187826305attice @ A8 @ ( type2 @ A8 ) )
=> ( comple187826305attice @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A7: $tType,A8: $tType] :
( ( semilattice_sup @ A8 @ ( type2 @ A8 ) )
=> ( semilattice_sup @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A7: $tType,A8: $tType] :
( ( preorder @ A8 @ ( type2 @ A8 ) )
=> ( preorder @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_fun___Lattices_Olattice,axiom,
! [A7: $tType,A8: $tType] :
( ( lattice @ A8 @ ( type2 @ A8 ) )
=> ( lattice @ ( 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___Complete__Lattices_Ocomplete__lattice_1,axiom,
! [A7: $tType] : ( comple187826305attice @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__sup_2,axiom,
! [A7: $tType] : ( semilattice_sup @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_3,axiom,
! [A7: $tType] : ( preorder @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Olattice_4,axiom,
! [A7: $tType] : ( lattice @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_5,axiom,
! [A7: $tType] : ( order @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_6,axiom,
! [A7: $tType] : ( ord @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_7,axiom,
comple187826305attice @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_8,axiom,
semilattice_sup @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_9,axiom,
preorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Lattices_Olattice_10,axiom,
lattice @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_11,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_12,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 (1)
thf(conj_0,conjecture,
member @ a @ n2 @ ns ).
%------------------------------------------------------------------------------