TPTP Problem File: COM168^1.p
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%------------------------------------------------------------------------------
% File : COM168^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Computing Theory
% Problem : Binary decision diagram 276
% Version : [Bla16] axioms : Especial.
% English :
% Refs : [OS08] Ortner & Schirmer (2008), BDD Normalisation
% : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% Source : [Bla16]
% Names : bindag__276.p [Bla16]
% Status : Theorem
% Rating : 1.00 v7.5.0, 0.33 v7.2.0, 0.50 v7.1.0
% Syntax : Number of formulae : 347 ( 103 unt; 59 typ; 0 def)
% Number of atoms : 893 ( 227 equ; 0 cnn)
% Maximal formula atoms : 13 ( 3 avg)
% Number of connectives : 3414 ( 105 ~; 29 |; 54 &;2799 @)
% ( 0 <=>; 427 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 8 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 226 ( 226 >; 0 *; 0 +; 0 <<)
% Number of symbols : 59 ( 56 usr; 8 con; 0-6 aty)
% Number of variables : 1058 ( 60 ^; 915 !; 42 ?;1058 :)
% ( 41 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:45:57.605
%------------------------------------------------------------------------------
%----Could-be-implicit typings (6)
thf(ty_t_BinDag__Mirabelle__rybootvolr_Odag,type,
binDag_Mirabelle_dag: $tType ).
thf(ty_t_Simpl__Heap_Oref,type,
simpl_ref: $tType ).
thf(ty_t_List_Olist,type,
list: $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
%----Explicit typings (53)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Obot,type,
bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_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_Ono__bot,type,
no_bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Ono__top,type,
no_top:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oorder__bot,type,
order_bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Owellorder,type,
wellorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Odense__order,type,
dense_order:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Fields_Olinordered__field,type,
linordered_field:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Odense__linorder,type,
dense_linorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
condit1656338222tinuum:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
condit1037483654norder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_BinDag__Mirabelle__rybootvolr_ODAG,type,
binDag_Mirabelle_DAG: binDag_Mirabelle_dag > $o ).
thf(sy_c_BinDag__Mirabelle__rybootvolr_ODag,type,
binDag_Mirabelle_Dag: simpl_ref > ( simpl_ref > simpl_ref ) > ( simpl_ref > simpl_ref ) > binDag_Mirabelle_dag > $o ).
thf(sy_c_BinDag__Mirabelle__rybootvolr_Odag_ONode,type,
binDag476092410e_Node: binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > binDag_Mirabelle_dag ).
thf(sy_c_BinDag__Mirabelle__rybootvolr_Odag_OTip,type,
binDag_Mirabelle_Tip: binDag_Mirabelle_dag ).
thf(sy_c_BinDag__Mirabelle__rybootvolr_Odag_Ocase__dag,type,
binDag1297733282se_dag:
!>[A: $tType] : ( A > ( binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > A ) > binDag_Mirabelle_dag > A ) ).
thf(sy_c_BinDag__Mirabelle__rybootvolr_Odag_Orec__dag,type,
binDag1442713106ec_dag:
!>[A: $tType] : ( A > ( binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > A > A > A ) > binDag_Mirabelle_dag > A ) ).
thf(sy_c_BinDag__Mirabelle__rybootvolr_Oset__of,type,
binDag1380252983set_of: binDag_Mirabelle_dag > ( set @ simpl_ref ) ).
thf(sy_c_BinDag__Mirabelle__rybootvolr_Osubdag,type,
binDag786255756subdag: binDag_Mirabelle_dag > binDag_Mirabelle_dag > $o ).
thf(sy_c_Fun_Ofun__upd,type,
fun_upd:
!>[A: $tType,B: $tType] : ( ( A > B ) > A > B > A > B ) ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_List_Obind,type,
bind:
!>[A: $tType,B: $tType] : ( ( list @ A ) > ( A > ( list @ B ) ) > ( list @ B ) ) ).
thf(sy_c_List_Ocan__select,type,
can_select:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_List_Ocoset,type,
coset:
!>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
thf(sy_c_List_Oinsert,type,
insert:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist_OCons,type,
cons:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist_ONil,type,
nil:
!>[A: $tType] : ( list @ A ) ).
thf(sy_c_List_Olist_Oset,type,
set2:
!>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
thf(sy_c_List_Olist__ex1,type,
list_ex1:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Omember,type,
member:
!>[A: $tType] : ( ( list @ A ) > A > $o ) ).
thf(sy_c_List_OremoveAll,type,
removeAll:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Osublist,type,
sublist:
!>[A: $tType] : ( ( list @ A ) > ( set @ nat ) > ( list @ A ) ) ).
thf(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_Oremove,type,
remove:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Simpl__Heap_ONull,type,
simpl_Null: simpl_ref ).
thf(sy_c_Simpl__Heap_Onew,type,
simpl_new: ( set @ simpl_ref ) > simpl_ref ).
thf(sy_c_Zorn_Ochain__subset,type,
chain_subset:
!>[A: $tType] : ( ( set @ ( set @ A ) ) > $o ) ).
thf(sy_c_member,type,
member2:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_alloc,type,
alloc: list @ simpl_ref ).
thf(sy_v_l,type,
l: simpl_ref > simpl_ref ).
thf(sy_v_p,type,
p: simpl_ref ).
thf(sy_v_r,type,
r: simpl_ref > simpl_ref ).
thf(sy_v_t,type,
t: binDag_Mirabelle_dag ).
thf(sy_v_x,type,
x: simpl_ref ).
%----Relevant facts (255)
thf(fact_0_notin__Dag__update__l,axiom,
! [Q: simpl_ref,T: binDag_Mirabelle_dag,P: simpl_ref,L: simpl_ref > simpl_ref,Y: simpl_ref,R: simpl_ref > simpl_ref] :
( ~ ( member2 @ simpl_ref @ Q @ ( binDag1380252983set_of @ T ) )
=> ( ( binDag_Mirabelle_Dag @ P @ ( fun_upd @ simpl_ref @ simpl_ref @ L @ Q @ Y ) @ R @ T )
= ( binDag_Mirabelle_Dag @ P @ L @ R @ T ) ) ) ).
% notin_Dag_update_l
thf(fact_1_notin__Dag__update__r,axiom,
! [Q: simpl_ref,T: binDag_Mirabelle_dag,P: simpl_ref,L: simpl_ref > simpl_ref,R: simpl_ref > simpl_ref,Y: simpl_ref] :
( ~ ( member2 @ simpl_ref @ Q @ ( binDag1380252983set_of @ T ) )
=> ( ( binDag_Mirabelle_Dag @ P @ L @ ( fun_upd @ simpl_ref @ simpl_ref @ R @ Q @ Y ) @ T )
= ( binDag_Mirabelle_Dag @ P @ L @ R @ T ) ) ) ).
% notin_Dag_update_r
thf(fact_2_fun__upd__upd,axiom,
! [A: $tType,B: $tType,F: A > B,X: A,Y: B,Z: B] :
( ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ F @ X @ Y ) @ X @ Z )
= ( fun_upd @ A @ B @ F @ X @ Z ) ) ).
% fun_upd_upd
thf(fact_3_fun__upd__triv,axiom,
! [B: $tType,A: $tType,F: A > B,X: A] :
( ( fun_upd @ A @ B @ F @ X @ ( F @ X ) )
= F ) ).
% fun_upd_triv
thf(fact_4_fun__upd__apply,axiom,
! [A: $tType,B: $tType] :
( ( fun_upd @ B @ A )
= ( ^ [F2: B > A,X2: B,Y2: A,Z2: B] : ( if @ A @ ( Z2 = X2 ) @ Y2 @ ( F2 @ Z2 ) ) ) ) ).
% fun_upd_apply
thf(fact_5_subsetI,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ! [X3: A] :
( ( member2 @ A @ X3 @ A2 )
=> ( member2 @ A @ X3 @ B2 ) )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% subsetI
thf(fact_6_subset__antisym,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% subset_antisym
thf(fact_7_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_8_Null__notin__Dag,axiom,
! [P: simpl_ref,L: simpl_ref > simpl_ref,R: simpl_ref > simpl_ref,T: binDag_Mirabelle_dag] :
( ( binDag_Mirabelle_Dag @ P @ L @ R @ T )
=> ~ ( member2 @ simpl_ref @ simpl_Null @ ( binDag1380252983set_of @ T ) ) ) ).
% Null_notin_Dag
thf(fact_9_subset__code_I1_J,axiom,
! [A: $tType,Xs: list @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ B2 )
= ( ! [X2: A] :
( ( member2 @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( member2 @ A @ X2 @ B2 ) ) ) ) ).
% subset_code(1)
thf(fact_10_Dag__upd__same__l__lemma,axiom,
! [P: simpl_ref,L: simpl_ref > simpl_ref,R: simpl_ref > simpl_ref,T: binDag_Mirabelle_dag] :
( ( P != simpl_Null )
=> ~ ( binDag_Mirabelle_Dag @ P @ ( fun_upd @ simpl_ref @ simpl_ref @ L @ P @ P ) @ R @ T ) ) ).
% Dag_upd_same_l_lemma
thf(fact_11_Dag__upd__same__r__lemma,axiom,
! [P: simpl_ref,L: simpl_ref > simpl_ref,R: simpl_ref > simpl_ref,T: binDag_Mirabelle_dag] :
( ( P != simpl_Null )
=> ~ ( binDag_Mirabelle_Dag @ P @ L @ ( fun_upd @ simpl_ref @ simpl_ref @ R @ P @ P ) @ T ) ) ).
% Dag_upd_same_r_lemma
thf(fact_12_fun__upd__def,axiom,
! [B: $tType,A: $tType] :
( ( fun_upd @ A @ B )
= ( ^ [F2: A > B,A3: A,B3: B,X2: A] : ( if @ B @ ( X2 = A3 ) @ B3 @ ( F2 @ X2 ) ) ) ) ).
% fun_upd_def
thf(fact_13_fun__upd__eqD,axiom,
! [A: $tType,B: $tType,F: A > B,X: A,Y: B,G: A > B,Z: B] :
( ( ( fun_upd @ A @ B @ F @ X @ Y )
= ( fun_upd @ A @ B @ G @ X @ Z ) )
=> ( Y = Z ) ) ).
% fun_upd_eqD
thf(fact_14_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B4: A,A4: A] :
( ( ord_less_eq @ A @ B4 @ A4 )
=> ( ( ord_less_eq @ A @ A4 @ B4 )
=> ( A4 = B4 ) ) ) ) ).
% dual_order.antisym
thf(fact_15_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B4: A,A4: A,C: A] :
( ( ord_less_eq @ A @ B4 @ A4 )
=> ( ( ord_less_eq @ A @ C @ B4 )
=> ( ord_less_eq @ A @ C @ A4 ) ) ) ) ).
% dual_order.trans
thf(fact_16_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P2: A > A > $o,A4: A,B4: A] :
( ! [A5: A,B5: A] :
( ( ord_less_eq @ A @ A5 @ B5 )
=> ( P2 @ A5 @ B5 ) )
=> ( ! [A5: A,B5: A] :
( ( P2 @ B5 @ A5 )
=> ( P2 @ A5 @ B5 ) )
=> ( P2 @ A4 @ B4 ) ) ) ) ).
% linorder_wlog
thf(fact_17_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A] : ( ord_less_eq @ A @ A4 @ A4 ) ) ).
% dual_order.refl
thf(fact_18_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_19_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( ord_less_eq @ A @ B4 @ A4 )
=> ( A4 = B4 ) ) ) ) ).
% order_class.order.antisym
thf(fact_20_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A,C: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( B4 = C )
=> ( ord_less_eq @ A @ A4 @ C ) ) ) ) ).
% ord_le_eq_trans
thf(fact_21_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A,C: A] :
( ( A4 = B4 )
=> ( ( ord_less_eq @ A @ B4 @ C )
=> ( ord_less_eq @ A @ A4 @ C ) ) ) ) ).
% ord_eq_le_trans
thf(fact_22_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_23_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_24_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A,C: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( ord_less_eq @ A @ B4 @ C )
=> ( ord_less_eq @ A @ A4 @ C ) ) ) ) ).
% order.trans
thf(fact_25_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_26_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_27_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_28_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_29_eq__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ^ [Y3: A,Z3: A] : Y3 = Z3 )
= ( ^ [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
& ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ) ) ).
% eq_iff
thf(fact_30_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A4: A,B4: A,F: A > B,C: B] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( ( F @ B4 )
= C )
=> ( ! [X3: A,Y4: A] :
( ( ord_less_eq @ A @ X3 @ Y4 )
=> ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ B @ ( F @ A4 ) @ C ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_31_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A4: A,F: B > A,B4: B,C: B] :
( ( A4
= ( F @ B4 ) )
=> ( ( ord_less_eq @ B @ B4 @ C )
=> ( ! [X3: B,Y4: B] :
( ( ord_less_eq @ B @ X3 @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ A @ A4 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_32_order__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 @ ( type2 @ C2 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A4: A,B4: A,F: A > C2,C: C2] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( ord_less_eq @ C2 @ ( F @ B4 ) @ C )
=> ( ! [X3: A,Y4: A] :
( ( ord_less_eq @ A @ X3 @ Y4 )
=> ( ord_less_eq @ C2 @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ C2 @ ( F @ A4 ) @ C ) ) ) ) ) ).
% order_subst2
thf(fact_33_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A4: A,F: B > A,B4: B,C: B] :
( ( ord_less_eq @ A @ A4 @ ( F @ B4 ) )
=> ( ( ord_less_eq @ B @ B4 @ C )
=> ( ! [X3: B,Y4: B] :
( ( ord_less_eq @ B @ X3 @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ A @ A4 @ ( F @ C ) ) ) ) ) ) ).
% order_subst1
thf(fact_34_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B] :
! [X2: A] : ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% le_fun_def
thf(fact_35_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B] :
( ! [X3: A] : ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) )
=> ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% le_funI
thf(fact_36_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_37_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_38_Collect__mono__iff,axiom,
! [A: $tType,P2: A > $o,Q2: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q2 ) )
= ( ! [X2: A] :
( ( P2 @ X2 )
=> ( Q2 @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_39_contra__subsetD,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ~ ( member2 @ A @ C @ B2 )
=> ~ ( member2 @ A @ C @ A2 ) ) ) ).
% contra_subsetD
thf(fact_40_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y3: set @ A,Z3: set @ A] : Y3 = Z3 )
= ( ^ [A6: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
& ( ord_less_eq @ ( set @ A ) @ B6 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_41_subset__trans,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_42_Collect__mono,axiom,
! [A: $tType,P2: A > $o,Q2: A > $o] :
( ! [X3: A] :
( ( P2 @ X3 )
=> ( Q2 @ X3 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q2 ) ) ) ).
% Collect_mono
thf(fact_43_subset__refl,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% subset_refl
thf(fact_44_rev__subsetD,axiom,
! [A: $tType,C: A,A2: set @ A,B2: set @ A] :
( ( member2 @ A @ C @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( member2 @ A @ C @ B2 ) ) ) ).
% rev_subsetD
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A4: A,P2: A > $o] :
( ( member2 @ A @ A4 @ ( collect @ A @ P2 ) )
= ( P2 @ A4 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A2: set @ A] :
( ( collect @ A
@ ^ [X2: A] : ( member2 @ A @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P2: A > $o,Q2: A > $o] :
( ! [X3: A] :
( ( P2 @ X3 )
= ( Q2 @ X3 ) )
=> ( ( collect @ A @ P2 )
= ( collect @ A @ Q2 ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X3: A] :
( ( F @ X3 )
= ( G @ X3 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
! [T2: A] :
( ( member2 @ A @ T2 @ A6 )
=> ( member2 @ A @ T2 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_50_set__rev__mp,axiom,
! [A: $tType,X: A,A2: set @ A,B2: set @ A] :
( ( member2 @ A @ X @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( member2 @ A @ X @ B2 ) ) ) ).
% set_rev_mp
thf(fact_51_equalityD2,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ).
% equalityD2
thf(fact_52_equalityD1,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% equalityD1
thf(fact_53_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
! [X2: A] :
( ( member2 @ A @ X2 @ A6 )
=> ( member2 @ A @ X2 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_54_equalityE,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( A2 = B2 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ) ).
% equalityE
thf(fact_55_subsetCE,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( member2 @ A @ C @ A2 )
=> ( member2 @ A @ C @ B2 ) ) ) ).
% subsetCE
thf(fact_56_subsetD,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( member2 @ A @ C @ A2 )
=> ( member2 @ A @ C @ B2 ) ) ) ).
% subsetD
thf(fact_57_in__mono,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( member2 @ A @ X @ A2 )
=> ( member2 @ A @ X @ B2 ) ) ) ).
% in_mono
thf(fact_58_set__mp,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( member2 @ A @ X @ A2 )
=> ( member2 @ A @ X @ B2 ) ) ) ).
% set_mp
thf(fact_59_le__dag__set__of,axiom,
! [X: binDag_Mirabelle_dag,Y: binDag_Mirabelle_dag] :
( ( ord_less_eq @ binDag_Mirabelle_dag @ X @ Y )
=> ( ord_less_eq @ ( set @ simpl_ref ) @ ( binDag1380252983set_of @ X ) @ ( binDag1380252983set_of @ Y ) ) ) ).
% le_dag_set_of
thf(fact_60_fun__upd__idem__iff,axiom,
! [A: $tType,B: $tType,F: A > B,X: A,Y: B] :
( ( ( fun_upd @ A @ B @ F @ X @ Y )
= F )
= ( ( F @ X )
= Y ) ) ).
% fun_upd_idem_iff
thf(fact_61_fun__upd__twist,axiom,
! [A: $tType,B: $tType,A4: A,C: A,M: A > B,B4: B,D: B] :
( ( A4 != C )
=> ( ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ M @ A4 @ B4 ) @ C @ D )
= ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ M @ C @ D ) @ A4 @ B4 ) ) ) ).
% fun_upd_twist
thf(fact_62_fun__upd__other,axiom,
! [B: $tType,A: $tType,Z: A,X: A,F: A > B,Y: B] :
( ( Z != X )
=> ( ( fun_upd @ A @ B @ F @ X @ Y @ Z )
= ( F @ Z ) ) ) ).
% fun_upd_other
thf(fact_63_fun__upd__same,axiom,
! [B: $tType,A: $tType,F: B > A,X: B,Y: A] :
( ( fun_upd @ B @ A @ F @ X @ Y @ X )
= Y ) ).
% fun_upd_same
thf(fact_64_fun__upd__idem,axiom,
! [A: $tType,B: $tType,F: B > A,X: B,Y: A] :
( ( ( F @ X )
= Y )
=> ( ( fun_upd @ B @ A @ F @ X @ Y )
= F ) ) ).
% fun_upd_idem
thf(fact_65_Dag__upd__same__r,axiom,
! [P: simpl_ref,L: simpl_ref > simpl_ref,R: simpl_ref > simpl_ref,T: binDag_Mirabelle_dag] :
( ( binDag_Mirabelle_Dag @ P @ L @ ( fun_upd @ simpl_ref @ simpl_ref @ R @ P @ P ) @ T )
= ( ( P = simpl_Null )
& ( T = binDag_Mirabelle_Tip ) ) ) ).
% Dag_upd_same_r
thf(fact_66_Dag__upd__same__l,axiom,
! [P: simpl_ref,L: simpl_ref > simpl_ref,R: simpl_ref > simpl_ref,T: binDag_Mirabelle_dag] :
( ( binDag_Mirabelle_Dag @ P @ ( fun_upd @ simpl_ref @ simpl_ref @ L @ P @ P ) @ R @ T )
= ( ( P = simpl_Null )
& ( T = binDag_Mirabelle_Tip ) ) ) ).
% Dag_upd_same_l
thf(fact_67_Dag__Null,axiom,
! [L: simpl_ref > simpl_ref,R: simpl_ref > simpl_ref,T: binDag_Mirabelle_dag] :
( ( binDag_Mirabelle_Dag @ simpl_Null @ L @ R @ T )
= ( T = binDag_Mirabelle_Tip ) ) ).
% Dag_Null
thf(fact_68_Dag__Ref,axiom,
! [P: simpl_ref,L: simpl_ref > simpl_ref,R: simpl_ref > simpl_ref,T: binDag_Mirabelle_dag] :
( ( P != simpl_Null )
=> ( ( binDag_Mirabelle_Dag @ P @ L @ R @ T )
= ( ? [Lt: binDag_Mirabelle_dag,Rt: binDag_Mirabelle_dag] :
( ( T
= ( binDag476092410e_Node @ Lt @ P @ Rt ) )
& ( binDag_Mirabelle_Dag @ ( L @ P ) @ L @ R @ Lt )
& ( binDag_Mirabelle_Dag @ ( R @ P ) @ L @ R @ Rt ) ) ) ) ) ).
% Dag_Ref
thf(fact_69_Dag_Osimps_I1_J,axiom,
! [P: simpl_ref,L: simpl_ref > simpl_ref,R: simpl_ref > simpl_ref] :
( ( binDag_Mirabelle_Dag @ P @ L @ R @ binDag_Mirabelle_Tip )
= ( P = simpl_Null ) ) ).
% Dag.simps(1)
thf(fact_70_Dag_Osimps_I2_J,axiom,
! [P: simpl_ref,L: simpl_ref > simpl_ref,R: simpl_ref > simpl_ref,Lt2: binDag_Mirabelle_dag,A4: simpl_ref,Rt2: binDag_Mirabelle_dag] :
( ( binDag_Mirabelle_Dag @ P @ L @ R @ ( binDag476092410e_Node @ Lt2 @ A4 @ Rt2 ) )
= ( ( P = A4 )
& ( P != simpl_Null )
& ( binDag_Mirabelle_Dag @ ( L @ P ) @ L @ R @ Lt2 )
& ( binDag_Mirabelle_Dag @ ( R @ P ) @ L @ R @ Rt2 ) ) ) ).
% Dag.simps(2)
thf(fact_71_less__dag__set__of,axiom,
! [X: binDag_Mirabelle_dag,Y: binDag_Mirabelle_dag] :
( ( ord_less @ binDag_Mirabelle_dag @ X @ Y )
=> ( ord_less_eq @ ( set @ simpl_ref ) @ ( binDag1380252983set_of @ X ) @ ( binDag1380252983set_of @ Y ) ) ) ).
% less_dag_set_of
thf(fact_72_subset__code_I2_J,axiom,
! [B: $tType,A2: set @ B,Ys: list @ B] :
( ( ord_less_eq @ ( set @ B ) @ A2 @ ( coset @ B @ Ys ) )
= ( ! [X2: B] :
( ( member2 @ B @ X2 @ ( set2 @ B @ Ys ) )
=> ~ ( member2 @ B @ X2 @ A2 ) ) ) ) ).
% subset_code(2)
thf(fact_73_in__set__member,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
= ( member @ A @ Xs @ X ) ) ).
% in_set_member
thf(fact_74_set__sublist__subset,axiom,
! [A: $tType,Xs: list @ A,I: set @ nat] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( sublist @ A @ Xs @ I ) ) @ ( set2 @ A @ Xs ) ) ).
% set_sublist_subset
thf(fact_75_in__set__insert,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( insert @ A @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_76_dag_Oinject,axiom,
! [X21: binDag_Mirabelle_dag,X22: simpl_ref,X23: binDag_Mirabelle_dag,Y21: binDag_Mirabelle_dag,Y22: simpl_ref,Y23: binDag_Mirabelle_dag] :
( ( ( binDag476092410e_Node @ X21 @ X22 @ X23 )
= ( binDag476092410e_Node @ Y21 @ Y22 @ Y23 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 )
& ( X23 = Y23 ) ) ) ).
% dag.inject
thf(fact_77_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A4: A,F: B > A,B4: B,C: B] :
( ( A4
= ( F @ B4 ) )
=> ( ( ord_less @ B @ B4 @ C )
=> ( ! [X3: B,Y4: B] :
( ( ord_less @ B @ X3 @ Y4 )
=> ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ A @ A4 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_78_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A4: A,B4: A,F: A > B,C: B] :
( ( ord_less @ A @ A4 @ B4 )
=> ( ( ( F @ B4 )
= C )
=> ( ! [X3: A,Y4: A] :
( ( ord_less @ A @ X3 @ Y4 )
=> ( ord_less @ B @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ B @ ( F @ A4 ) @ C ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_79_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A4: A,F: B > A,B4: B,C: B] :
( ( ord_less @ A @ A4 @ ( F @ B4 ) )
=> ( ( ord_less @ B @ B4 @ C )
=> ( ! [X3: B,Y4: B] :
( ( ord_less @ B @ X3 @ Y4 )
=> ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ A @ A4 @ ( F @ C ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_80_order__less__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 @ ( type2 @ C2 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A4: A,B4: A,F: A > C2,C: C2] :
( ( ord_less @ A @ A4 @ B4 )
=> ( ( ord_less @ C2 @ ( F @ B4 ) @ C )
=> ( ! [X3: A,Y4: A] :
( ( ord_less @ A @ X3 @ Y4 )
=> ( ord_less @ C2 @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ C2 @ ( F @ A4 ) @ C ) ) ) ) ) ).
% order_less_subst2
thf(fact_81_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A @ ( type2 @ A ) )
=> ! [X: A] :
? [Y4: A] : ( ord_less @ A @ Y4 @ X ) ) ).
% lt_ex
thf(fact_82_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A @ ( type2 @ A ) )
=> ! [X: A] :
? [X1: A] : ( ord_less @ A @ X @ X1 ) ) ).
% gt_ex
thf(fact_83_neqE,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( X != Y )
=> ( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% neqE
thf(fact_84_neq__iff,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( X != Y )
= ( ( ord_less @ A @ X @ Y )
| ( ord_less @ A @ Y @ X ) ) ) ) ).
% neq_iff
thf(fact_85_order_Oasym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A] :
( ( ord_less @ A @ A4 @ B4 )
=> ~ ( ord_less @ A @ B4 @ A4 ) ) ) ).
% order.asym
thf(fact_86_dense,axiom,
! [A: $tType] :
( ( dense_order @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ? [Z4: A] :
( ( ord_less @ A @ X @ Z4 )
& ( ord_less @ A @ Z4 @ Y ) ) ) ) ).
% dense
thf(fact_87_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( X != Y ) ) ) ).
% less_imp_neq
thf(fact_88_less__asym,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% less_asym
thf(fact_89_less__asym_H,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A] :
( ( ord_less @ A @ A4 @ B4 )
=> ~ ( ord_less @ A @ B4 @ A4 ) ) ) ).
% less_asym'
thf(fact_90_less__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ Z )
=> ( ord_less @ A @ X @ Z ) ) ) ) ).
% less_trans
thf(fact_91_less__linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
| ( X = Y )
| ( ord_less @ A @ Y @ X ) ) ) ).
% less_linear
thf(fact_92_less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A] :
~ ( ord_less @ A @ X @ X ) ) ).
% less_irrefl
thf(fact_93_ord__eq__less__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A,C: A] :
( ( A4 = B4 )
=> ( ( ord_less @ A @ B4 @ C )
=> ( ord_less @ A @ A4 @ C ) ) ) ) ).
% ord_eq_less_trans
thf(fact_94_ord__less__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A,C: A] :
( ( ord_less @ A @ A4 @ B4 )
=> ( ( B4 = C )
=> ( ord_less @ A @ A4 @ C ) ) ) ) ).
% ord_less_eq_trans
thf(fact_95_dual__order_Oasym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B4: A,A4: A] :
( ( ord_less @ A @ B4 @ A4 )
=> ~ ( ord_less @ A @ A4 @ B4 ) ) ) ).
% dual_order.asym
thf(fact_96_less__imp__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( X != Y ) ) ) ).
% less_imp_not_eq
thf(fact_97_less__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% less_not_sym
thf(fact_98_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A @ ( type2 @ A ) )
=> ! [P2: A > $o,A4: A] :
( ! [X3: A] :
( ! [Y5: A] :
( ( ord_less @ A @ Y5 @ X3 )
=> ( P2 @ Y5 ) )
=> ( P2 @ X3 ) )
=> ( P2 @ A4 ) ) ) ).
% less_induct
thf(fact_99_le__dag__def,axiom,
( ( ord_less_eq @ binDag_Mirabelle_dag )
= ( ^ [S: binDag_Mirabelle_dag,T2: binDag_Mirabelle_dag] :
( ( S = T2 )
| ( ord_less @ binDag_Mirabelle_dag @ S @ T2 ) ) ) ) ).
% le_dag_def
thf(fact_100_antisym__conv3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A] :
( ~ ( ord_less @ A @ Y @ X )
=> ( ( ~ ( ord_less @ A @ X @ Y ) )
= ( X = Y ) ) ) ) ).
% antisym_conv3
thf(fact_101_less__imp__not__eq2,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( Y != X ) ) ) ).
% less_imp_not_eq2
thf(fact_102_less__imp__triv,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,P2: $o] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ X )
=> P2 ) ) ) ).
% less_imp_triv
thf(fact_103_dag__less__le,axiom,
( ( ord_less @ binDag_Mirabelle_dag )
= ( ^ [X2: binDag_Mirabelle_dag,Y2: binDag_Mirabelle_dag] :
( ( ord_less_eq @ binDag_Mirabelle_dag @ X2 @ Y2 )
& ( X2 != Y2 ) ) ) ) ).
% dag_less_le
thf(fact_104_le__dag__refl,axiom,
! [X: binDag_Mirabelle_dag] : ( ord_less_eq @ binDag_Mirabelle_dag @ X @ X ) ).
% le_dag_refl
thf(fact_105_linorder__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ( X != Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% linorder_cases
thf(fact_106_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A] :
~ ( ord_less @ A @ A4 @ A4 ) ) ).
% dual_order.irrefl
thf(fact_107_le__dag__trans,axiom,
! [X: binDag_Mirabelle_dag,Y: binDag_Mirabelle_dag,Z: binDag_Mirabelle_dag] :
( ( ord_less_eq @ binDag_Mirabelle_dag @ X @ Y )
=> ( ( ord_less_eq @ binDag_Mirabelle_dag @ Y @ Z )
=> ( ord_less_eq @ binDag_Mirabelle_dag @ X @ Z ) ) ) ).
% le_dag_trans
thf(fact_108_order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A,C: A] :
( ( ord_less @ A @ A4 @ B4 )
=> ( ( ord_less @ A @ B4 @ C )
=> ( ord_less @ A @ A4 @ C ) ) ) ) ).
% order.strict_trans
thf(fact_109_less__dag__Node,axiom,
! [X: binDag_Mirabelle_dag,L: binDag_Mirabelle_dag,A4: simpl_ref,R: binDag_Mirabelle_dag] :
( ( ord_less @ binDag_Mirabelle_dag @ X @ ( binDag476092410e_Node @ L @ A4 @ R ) )
= ( ( ord_less_eq @ binDag_Mirabelle_dag @ X @ L )
| ( ord_less_eq @ binDag_Mirabelle_dag @ X @ R ) ) ) ).
% less_dag_Node
thf(fact_110_le__dag__antisym,axiom,
! [X: binDag_Mirabelle_dag,Y: binDag_Mirabelle_dag] :
( ( ord_less_eq @ binDag_Mirabelle_dag @ X @ Y )
=> ( ( ord_less_eq @ binDag_Mirabelle_dag @ Y @ X )
=> ( X = Y ) ) ) ).
% le_dag_antisym
thf(fact_111_less__imp__not__less,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% less_imp_not_less
thf(fact_112_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B4: A,A4: A,C: A] :
( ( ord_less @ A @ B4 @ A4 )
=> ( ( ord_less @ A @ C @ B4 )
=> ( ord_less @ A @ C @ A4 ) ) ) ) ).
% dual_order.strict_trans
thf(fact_113_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less @ A @ X @ Y ) )
= ( ( ord_less @ A @ Y @ X )
| ( X = Y ) ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_114_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A] :
( ( ord_less @ A @ A4 @ B4 )
=> ( A4 != B4 ) ) ) ).
% order.strict_implies_not_eq
thf(fact_115_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B4: A,A4: A] :
( ( ord_less @ A @ B4 @ A4 )
=> ( A4 != B4 ) ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_116_less__dag__Node_H,axiom,
! [X: binDag_Mirabelle_dag,L: binDag_Mirabelle_dag,A4: simpl_ref,R: binDag_Mirabelle_dag] :
( ( ord_less @ binDag_Mirabelle_dag @ X @ ( binDag476092410e_Node @ L @ A4 @ R ) )
= ( ( X = L )
| ( X = R )
| ( ord_less @ binDag_Mirabelle_dag @ X @ L )
| ( ord_less @ binDag_Mirabelle_dag @ X @ R ) ) ) ).
% less_dag_Node'
thf(fact_117_less__Node__dag,axiom,
! [L: binDag_Mirabelle_dag,A4: simpl_ref,R: binDag_Mirabelle_dag,X: binDag_Mirabelle_dag] :
( ( ord_less @ binDag_Mirabelle_dag @ ( binDag476092410e_Node @ L @ A4 @ R ) @ X )
=> ( ( ord_less @ binDag_Mirabelle_dag @ L @ X )
& ( ord_less @ binDag_Mirabelle_dag @ R @ X ) ) ) ).
% less_Node_dag
thf(fact_118_less__dag__Tip,axiom,
! [X: binDag_Mirabelle_dag] :
~ ( ord_less @ binDag_Mirabelle_dag @ X @ binDag_Mirabelle_Tip ) ).
% less_dag_Tip
thf(fact_119_dag_Oexhaust,axiom,
! [Y: binDag_Mirabelle_dag] :
( ( Y != binDag_Mirabelle_Tip )
=> ~ ! [X212: binDag_Mirabelle_dag,X222: simpl_ref,X232: binDag_Mirabelle_dag] :
( Y
!= ( binDag476092410e_Node @ X212 @ X222 @ X232 ) ) ) ).
% dag.exhaust
thf(fact_120_dag_Oinduct,axiom,
! [P2: binDag_Mirabelle_dag > $o,Dag: binDag_Mirabelle_dag] :
( ( P2 @ binDag_Mirabelle_Tip )
=> ( ! [X1: binDag_Mirabelle_dag,X24: simpl_ref,X32: binDag_Mirabelle_dag] :
( ( P2 @ X1 )
=> ( ( P2 @ X32 )
=> ( P2 @ ( binDag476092410e_Node @ X1 @ X24 @ X32 ) ) ) )
=> ( P2 @ Dag ) ) ) ).
% dag.induct
thf(fact_121_dag_Odistinct_I1_J,axiom,
! [X21: binDag_Mirabelle_dag,X22: simpl_ref,X23: binDag_Mirabelle_dag] :
( binDag_Mirabelle_Tip
!= ( binDag476092410e_Node @ X21 @ X22 @ X23 ) ) ).
% dag.distinct(1)
thf(fact_122_leD,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less @ A @ X @ Y ) ) ) ).
% leD
thf(fact_123_leI,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% leI
thf(fact_124_le__less,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [X2: A,Y2: A] :
( ( ord_less @ A @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ) ).
% le_less
thf(fact_125_less__le,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
& ( X2 != Y2 ) ) ) ) ) ).
% less_le
thf(fact_126_order__le__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A4: A,F: B > A,B4: B,C: B] :
( ( ord_less_eq @ A @ A4 @ ( F @ B4 ) )
=> ( ( ord_less @ B @ B4 @ C )
=> ( ! [X3: B,Y4: B] :
( ( ord_less @ B @ X3 @ Y4 )
=> ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ A @ A4 @ ( F @ C ) ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_127_order__le__less__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 @ ( type2 @ C2 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A4: A,B4: A,F: A > C2,C: C2] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( ord_less @ C2 @ ( F @ B4 ) @ C )
=> ( ! [X3: A,Y4: A] :
( ( ord_less_eq @ A @ X3 @ Y4 )
=> ( ord_less_eq @ C2 @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ C2 @ ( F @ A4 ) @ C ) ) ) ) ) ).
% order_le_less_subst2
thf(fact_128_order__less__le__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A4: A,F: B > A,B4: B,C: B] :
( ( ord_less @ A @ A4 @ ( F @ B4 ) )
=> ( ( ord_less_eq @ B @ B4 @ C )
=> ( ! [X3: B,Y4: B] :
( ( ord_less_eq @ B @ X3 @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ A @ A4 @ ( F @ C ) ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_129_order__less__le__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 @ ( type2 @ C2 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A4: A,B4: A,F: A > C2,C: C2] :
( ( ord_less @ A @ A4 @ B4 )
=> ( ( ord_less_eq @ C2 @ ( F @ B4 ) @ C )
=> ( ! [X3: A,Y4: A] :
( ( ord_less @ A @ X3 @ Y4 )
=> ( ord_less @ C2 @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ C2 @ ( F @ A4 ) @ C ) ) ) ) ) ).
% order_less_le_subst2
thf(fact_130_not__le,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less_eq @ A @ X @ Y ) )
= ( ord_less @ A @ Y @ X ) ) ) ).
% not_le
thf(fact_131_not__less,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less @ A @ X @ Y ) )
= ( ord_less_eq @ A @ Y @ X ) ) ) ).
% not_less
thf(fact_132_le__neq__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( A4 != B4 )
=> ( ord_less @ A @ A4 @ B4 ) ) ) ) ).
% le_neq_trans
thf(fact_133_less__imp__le,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% less_imp_le
thf(fact_134_antisym__conv1,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ X @ Y )
= ( X = Y ) ) ) ) ).
% antisym_conv1
thf(fact_135_antisym__conv2,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ~ ( ord_less @ A @ X @ Y ) )
= ( X = Y ) ) ) ) ).
% antisym_conv2
thf(fact_136_le__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ Z )
=> ( ord_less @ A @ X @ Z ) ) ) ) ).
% le_less_trans
thf(fact_137_less__le__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z )
=> ( ord_less @ A @ X @ Z ) ) ) ) ).
% less_le_trans
thf(fact_138_dense__ge,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Z: A,Y: A] :
( ! [X3: A] :
( ( ord_less @ A @ Z @ X3 )
=> ( ord_less_eq @ A @ Y @ X3 ) )
=> ( ord_less_eq @ A @ Y @ Z ) ) ) ).
% dense_ge
thf(fact_139_dense__le,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Y: A,Z: A] :
( ! [X3: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ord_less_eq @ A @ X3 @ Z ) )
=> ( ord_less_eq @ A @ Y @ Z ) ) ) ).
% dense_le
thf(fact_140_le__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
| ( ord_less @ A @ Y @ X ) ) ) ).
% le_less_linear
thf(fact_141_le__imp__less__or__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ X @ Y )
| ( X = Y ) ) ) ) ).
% le_imp_less_or_eq
thf(fact_142_less__le__not__le,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
& ~ ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ) ) ).
% less_le_not_le
thf(fact_143_not__le__imp__less,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A] :
( ~ ( ord_less_eq @ A @ Y @ X )
=> ( ord_less @ A @ X @ Y ) ) ) ).
% not_le_imp_less
thf(fact_144_order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A,C: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ( ord_less @ A @ B4 @ C )
=> ( ord_less @ A @ A4 @ C ) ) ) ) ).
% order.strict_trans1
thf(fact_145_order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A,C: A] :
( ( ord_less @ A @ A4 @ B4 )
=> ( ( ord_less_eq @ A @ B4 @ C )
=> ( ord_less @ A @ A4 @ C ) ) ) ) ).
% order.strict_trans2
thf(fact_146_order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ) ).
% order.order_iff_strict
thf(fact_147_order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ) ).
% order.strict_iff_order
thf(fact_148_dual__order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B4: A,A4: A,C: A] :
( ( ord_less_eq @ A @ B4 @ A4 )
=> ( ( ord_less @ A @ C @ B4 )
=> ( ord_less @ A @ C @ A4 ) ) ) ) ).
% dual_order.strict_trans1
thf(fact_149_dual__order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B4: A,A4: A,C: A] :
( ( ord_less @ A @ B4 @ A4 )
=> ( ( ord_less_eq @ A @ C @ B4 )
=> ( ord_less @ A @ C @ A4 ) ) ) ) ).
% dual_order.strict_trans2
thf(fact_150_dense__ge__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Z: A,X: A,Y: A] :
( ( ord_less @ A @ Z @ X )
=> ( ! [W: A] :
( ( ord_less @ A @ Z @ W )
=> ( ( ord_less @ A @ W @ X )
=> ( ord_less_eq @ A @ Y @ W ) ) )
=> ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% dense_ge_bounded
thf(fact_151_dense__le__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less @ A @ X @ Y )
=> ( ! [W: A] :
( ( ord_less @ A @ X @ W )
=> ( ( ord_less @ A @ W @ Y )
=> ( ord_less_eq @ A @ W @ Z ) ) )
=> ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% dense_le_bounded
thf(fact_152_order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A] :
( ( ord_less @ A @ A4 @ B4 )
=> ( ord_less_eq @ A @ A4 @ B4 ) ) ) ).
% order.strict_implies_order
thf(fact_153_dual__order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [B3: A,A3: A] :
( ( ord_less @ A @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_154_dual__order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [B3: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_155_dual__order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B4: A,A4: A] :
( ( ord_less @ A @ B4 @ A4 )
=> ( ord_less_eq @ A @ B4 @ A4 ) ) ) ).
% dual_order.strict_implies_order
thf(fact_156_order_Onot__eq__order__implies__strict,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A] :
( ( A4 != B4 )
=> ( ( ord_less_eq @ A @ A4 @ B4 )
=> ( ord_less @ A @ A4 @ B4 ) ) ) ) ).
% order.not_eq_order_implies_strict
thf(fact_157_in__set__sublistD,axiom,
! [A: $tType,X: A,Xs: list @ A,I: set @ nat] :
( ( member2 @ A @ X @ ( set2 @ A @ ( sublist @ A @ Xs @ I ) ) )
=> ( member2 @ A @ X @ ( set2 @ A @ Xs ) ) ) ).
% in_set_sublistD
thf(fact_158_notin__set__sublistI,axiom,
! [A: $tType,X: A,Xs: list @ A,I: set @ nat] :
( ~ ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
=> ~ ( member2 @ A @ X @ ( set2 @ A @ ( sublist @ A @ Xs @ I ) ) ) ) ).
% notin_set_sublistI
thf(fact_159_minf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ~ ( ord_less_eq @ A @ T @ X4 ) ) ) ).
% minf(8)
thf(fact_160_minf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ( ord_less_eq @ A @ X4 @ T ) ) ) ).
% minf(6)
thf(fact_161_pinf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ( ord_less_eq @ A @ T @ X4 ) ) ) ).
% pinf(8)
thf(fact_162_pinf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ~ ( ord_less_eq @ A @ X4 @ T ) ) ) ).
% pinf(6)
thf(fact_163_complete__interval,axiom,
! [A: $tType] :
( ( condit1037483654norder @ A @ ( type2 @ A ) )
=> ! [A4: A,B4: A,P2: A > $o] :
( ( ord_less @ A @ A4 @ B4 )
=> ( ( P2 @ A4 )
=> ( ~ ( P2 @ B4 )
=> ? [C4: A] :
( ( ord_less_eq @ A @ A4 @ C4 )
& ( ord_less_eq @ A @ C4 @ B4 )
& ! [X4: A] :
( ( ( ord_less_eq @ A @ A4 @ X4 )
& ( ord_less @ A @ X4 @ C4 ) )
=> ( P2 @ X4 ) )
& ! [D2: A] :
( ! [X3: A] :
( ( ( ord_less_eq @ A @ A4 @ X3 )
& ( ord_less @ A @ X3 @ D2 ) )
=> ( P2 @ X3 ) )
=> ( ord_less_eq @ A @ D2 @ C4 ) ) ) ) ) ) ) ).
% complete_interval
thf(fact_164_DAG_Osimps_I2_J,axiom,
! [L: binDag_Mirabelle_dag,A4: simpl_ref,R: binDag_Mirabelle_dag] :
( ( binDag_Mirabelle_DAG @ ( binDag476092410e_Node @ L @ A4 @ R ) )
= ( ~ ( member2 @ simpl_ref @ A4 @ ( binDag1380252983set_of @ L ) )
& ~ ( member2 @ simpl_ref @ A4 @ ( binDag1380252983set_of @ R ) )
& ( binDag_Mirabelle_DAG @ L )
& ( binDag_Mirabelle_DAG @ R ) ) ) ).
% DAG.simps(2)
thf(fact_165_psubsetI,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less @ ( set @ A ) @ A2 @ B2 ) ) ) ).
% psubsetI
thf(fact_166_subset__iff__psubset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
( ( ord_less @ ( set @ A ) @ A6 @ B6 )
| ( A6 = B6 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_167_subset__psubset__trans,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ( ord_less @ ( set @ A ) @ B2 @ C3 )
=> ( ord_less @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% subset_psubset_trans
thf(fact_168_subset__not__subset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
& ~ ( ord_less_eq @ ( set @ A ) @ B6 @ A6 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_169_psubset__subset__trans,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
( ( ord_less @ ( set @ A ) @ A2 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
=> ( ord_less @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% psubset_subset_trans
thf(fact_170_psubset__imp__subset,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less @ ( set @ A ) @ A2 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B2 ) ) ).
% psubset_imp_subset
thf(fact_171_psubset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B6 )
& ( A6 != B6 ) ) ) ) ).
% psubset_eq
thf(fact_172_psubsetE,axiom,
! [A: $tType,A2: set @ A,B2: set @ A] :
( ( ord_less @ ( set @ A ) @ A2 @ B2 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ B2 @ A2 ) ) ) ).
% psubsetE
thf(fact_173_less__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ( ( ord_less @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B] :
( ( ord_less_eq @ ( A > B ) @ F2 @ G2 )
& ~ ( ord_less_eq @ ( A > B ) @ G2 @ F2 ) ) ) ) ) ).
% less_fun_def
thf(fact_174_less__DAG__set__of,axiom,
! [X: binDag_Mirabelle_dag,Y: binDag_Mirabelle_dag] :
( ( ord_less @ binDag_Mirabelle_dag @ X @ Y )
=> ( ( binDag_Mirabelle_DAG @ Y )
=> ( ord_less @ ( set @ simpl_ref ) @ ( binDag1380252983set_of @ X ) @ ( binDag1380252983set_of @ Y ) ) ) ) ).
% less_DAG_set_of
thf(fact_175_DAG_Osimps_I1_J,axiom,
binDag_Mirabelle_DAG @ binDag_Mirabelle_Tip ).
% DAG.simps(1)
thf(fact_176_DAG__less,axiom,
! [Y: binDag_Mirabelle_dag,X: binDag_Mirabelle_dag] :
( ( binDag_Mirabelle_DAG @ Y )
=> ( ( ord_less @ binDag_Mirabelle_dag @ X @ Y )
=> ( binDag_Mirabelle_DAG @ X ) ) ) ).
% DAG_less
thf(fact_177_minf_I11_J,axiom,
! [C2: $tType,D3: $tType] :
( ( ord @ C2 @ ( type2 @ C2 ) )
=> ! [F3: D3] :
? [Z4: C2] :
! [X4: C2] :
( ( ord_less @ C2 @ X4 @ Z4 )
=> ( F3 = F3 ) ) ) ).
% minf(11)
thf(fact_178_minf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ~ ( ord_less @ A @ T @ X4 ) ) ) ).
% minf(7)
thf(fact_179_minf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ( ord_less @ A @ X4 @ T ) ) ) ).
% minf(5)
thf(fact_180_minf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ( X4 != T ) ) ) ).
% minf(4)
thf(fact_181_minf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ( X4 != T ) ) ) ).
% minf(3)
thf(fact_182_minf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P2: A > $o,P3: A > $o,Q2: A > $o,Q3: A > $o] :
( ? [Z5: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z5 )
=> ( ( P2 @ X3 )
= ( P3 @ X3 ) ) )
=> ( ? [Z5: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z5 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ( ( ( P2 @ X4 )
| ( Q2 @ X4 ) )
= ( ( P3 @ X4 )
| ( Q3 @ X4 ) ) ) ) ) ) ) ).
% minf(2)
thf(fact_183_minf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P2: A > $o,P3: A > $o,Q2: A > $o,Q3: A > $o] :
( ? [Z5: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z5 )
=> ( ( P2 @ X3 )
= ( P3 @ X3 ) ) )
=> ( ? [Z5: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z5 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ( ( ( P2 @ X4 )
& ( Q2 @ X4 ) )
= ( ( P3 @ X4 )
& ( Q3 @ X4 ) ) ) ) ) ) ) ).
% minf(1)
thf(fact_184_pinf_I11_J,axiom,
! [C2: $tType,D3: $tType] :
( ( ord @ C2 @ ( type2 @ C2 ) )
=> ! [F3: D3] :
? [Z4: C2] :
! [X4: C2] :
( ( ord_less @ C2 @ Z4 @ X4 )
=> ( F3 = F3 ) ) ) ).
% pinf(11)
thf(fact_185_pinf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ( ord_less @ A @ T @ X4 ) ) ) ).
% pinf(7)
thf(fact_186_pinf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ~ ( ord_less @ A @ X4 @ T ) ) ) ).
% pinf(5)
thf(fact_187_pinf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ( X4 != T ) ) ) ).
% pinf(4)
thf(fact_188_pinf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ( X4 != T ) ) ) ).
% pinf(3)
thf(fact_189_pinf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P2: A > $o,P3: A > $o,Q2: A > $o,Q3: A > $o] :
( ? [Z5: A] :
! [X3: A] :
( ( ord_less @ A @ Z5 @ X3 )
=> ( ( P2 @ X3 )
= ( P3 @ X3 ) ) )
=> ( ? [Z5: A] :
! [X3: A] :
( ( ord_less @ A @ Z5 @ X3 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ( ( ( P2 @ X4 )
| ( Q2 @ X4 ) )
= ( ( P3 @ X4 )
| ( Q3 @ X4 ) ) ) ) ) ) ) ).
% pinf(2)
thf(fact_190_pinf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P2: A > $o,P3: A > $o,Q2: A > $o,Q3: A > $o] :
( ? [Z5: A] :
! [X3: A] :
( ( ord_less @ A @ Z5 @ X3 )
=> ( ( P2 @ X3 )
= ( P3 @ X3 ) ) )
=> ( ? [Z5: A] :
! [X3: A] :
( ( ord_less @ A @ Z5 @ X3 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ( ( ( P2 @ X4 )
& ( Q2 @ X4 ) )
= ( ( P3 @ X4 )
& ( Q3 @ X4 ) ) ) ) ) ) ) ).
% pinf(1)
thf(fact_191_ex__gt__or__lt,axiom,
! [A: $tType] :
( ( condit1656338222tinuum @ A @ ( type2 @ A ) )
=> ! [A4: A] :
? [B5: A] :
( ( ord_less @ A @ A4 @ B5 )
| ( ord_less @ A @ B5 @ A4 ) ) ) ).
% ex_gt_or_lt
thf(fact_192_remove__code_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( remove @ A @ X @ ( coset @ A @ Xs ) )
= ( coset @ A @ ( insert @ A @ X @ Xs ) ) ) ).
% remove_code(2)
thf(fact_193_in__set__of__decomp,axiom,
! [P: simpl_ref,T: binDag_Mirabelle_dag] :
( ( member2 @ simpl_ref @ P @ ( binDag1380252983set_of @ T ) )
= ( ? [L2: binDag_Mirabelle_dag,R2: binDag_Mirabelle_dag] :
( ( T
= ( binDag476092410e_Node @ L2 @ P @ R2 ) )
| ( binDag786255756subdag @ T @ ( binDag476092410e_Node @ L2 @ P @ R2 ) ) ) ) ) ).
% in_set_of_decomp
thf(fact_194_subset__code_I3_J,axiom,
! [C2: $tType] :
~ ( ord_less_eq @ ( set @ C2 ) @ ( coset @ C2 @ ( nil @ C2 ) ) @ ( set2 @ C2 @ ( nil @ C2 ) ) ) ).
% subset_code(3)
thf(fact_195_member__remove,axiom,
! [A: $tType,X: A,Y: A,A2: set @ A] :
( ( member2 @ A @ X @ ( remove @ A @ Y @ A2 ) )
= ( ( member2 @ A @ X @ A2 )
& ( X != Y ) ) ) ).
% member_remove
thf(fact_196_sublist__nil,axiom,
! [A: $tType,A2: set @ nat] :
( ( sublist @ A @ ( nil @ A ) @ A2 )
= ( nil @ A ) ) ).
% sublist_nil
thf(fact_197_psubsetD,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C: A] :
( ( ord_less @ ( set @ A ) @ A2 @ B2 )
=> ( ( member2 @ A @ C @ A2 )
=> ( member2 @ A @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_198_psubset__trans,axiom,
! [A: $tType,A2: set @ A,B2: set @ A,C3: set @ A] :
( ( ord_less @ ( set @ A ) @ A2 @ B2 )
=> ( ( ord_less @ ( set @ A ) @ B2 @ C3 )
=> ( ord_less @ ( set @ A ) @ A2 @ C3 ) ) ) ).
% psubset_trans
thf(fact_199_subdag__not__sym,axiom,
! [S2: binDag_Mirabelle_dag,T: binDag_Mirabelle_dag] :
( ( binDag786255756subdag @ S2 @ T )
=> ~ ( binDag786255756subdag @ T @ S2 ) ) ).
% subdag_not_sym
thf(fact_200_subdag__trans,axiom,
! [T: binDag_Mirabelle_dag,S2: binDag_Mirabelle_dag,R: binDag_Mirabelle_dag] :
( ( binDag786255756subdag @ T @ S2 )
=> ( ( binDag786255756subdag @ S2 @ R )
=> ( binDag786255756subdag @ T @ R ) ) ) ).
% subdag_trans
thf(fact_201_subdag__neq,axiom,
! [T: binDag_Mirabelle_dag,S2: binDag_Mirabelle_dag] :
( ( binDag786255756subdag @ T @ S2 )
=> ( T != S2 ) ) ).
% subdag_neq
thf(fact_202_member__rec_I2_J,axiom,
! [A: $tType,Y: A] :
~ ( member @ A @ ( nil @ A ) @ Y ) ).
% member_rec(2)
thf(fact_203_subdag__NodeD,axiom,
! [T: binDag_Mirabelle_dag,Lt2: binDag_Mirabelle_dag,A4: simpl_ref,Rt2: binDag_Mirabelle_dag] :
( ( binDag786255756subdag @ T @ ( binDag476092410e_Node @ Lt2 @ A4 @ Rt2 ) )
=> ( ( binDag786255756subdag @ T @ Lt2 )
& ( binDag786255756subdag @ T @ Rt2 ) ) ) ).
% subdag_NodeD
thf(fact_204_subdag_Osimps_I2_J,axiom,
! [L: binDag_Mirabelle_dag,A4: simpl_ref,R: binDag_Mirabelle_dag,T: binDag_Mirabelle_dag] :
( ( binDag786255756subdag @ ( binDag476092410e_Node @ L @ A4 @ R ) @ T )
= ( ( T = L )
| ( T = R )
| ( binDag786255756subdag @ L @ T )
| ( binDag786255756subdag @ R @ T ) ) ) ).
% subdag.simps(2)
thf(fact_205_subdag_Osimps_I1_J,axiom,
! [T: binDag_Mirabelle_dag] :
~ ( binDag786255756subdag @ binDag_Mirabelle_Tip @ T ) ).
% subdag.simps(1)
thf(fact_206_less__dag__def,axiom,
( ( ord_less @ binDag_Mirabelle_dag )
= ( ^ [S: binDag_Mirabelle_dag,T2: binDag_Mirabelle_dag] : ( binDag786255756subdag @ T2 @ S ) ) ) ).
% less_dag_def
thf(fact_207_list__ex1__simps_I1_J,axiom,
! [A: $tType,P2: A > $o] :
~ ( list_ex1 @ A @ P2 @ ( nil @ A ) ) ).
% list_ex1_simps(1)
thf(fact_208_bind__simps_I1_J,axiom,
! [B: $tType,A: $tType,F: B > ( list @ A )] :
( ( bind @ B @ A @ ( nil @ B ) @ F )
= ( nil @ A ) ) ).
% bind_simps(1)
thf(fact_209_list__bind__cong,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ A,F: A > ( list @ B ),G: A > ( list @ B )] :
( ( Xs = Ys )
=> ( ! [X3: A] :
( ( member2 @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( bind @ A @ B @ Xs @ F )
= ( bind @ A @ B @ Ys @ G ) ) ) ) ).
% list_bind_cong
thf(fact_210_list__ex1__iff,axiom,
! [A: $tType] :
( ( list_ex1 @ A )
= ( ^ [P4: A > $o,Xs2: list @ A] :
? [X2: A] :
( ( member2 @ A @ X2 @ ( set2 @ A @ Xs2 ) )
& ( P4 @ X2 )
& ! [Y2: A] :
( ( ( member2 @ A @ Y2 @ ( set2 @ A @ Xs2 ) )
& ( P4 @ Y2 ) )
=> ( Y2 = X2 ) ) ) ) ) ).
% list_ex1_iff
thf(fact_211_can__select__set__list__ex1,axiom,
! [A: $tType,P2: A > $o,A2: list @ A] :
( ( can_select @ A @ P2 @ ( set2 @ A @ A2 ) )
= ( list_ex1 @ A @ P2 @ A2 ) ) ).
% can_select_set_list_ex1
thf(fact_212_remove__code_I1_J,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( remove @ A @ X @ ( set2 @ A @ Xs ) )
= ( set2 @ A @ ( removeAll @ A @ X @ Xs ) ) ) ).
% remove_code(1)
thf(fact_213_removeAll__id,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ~ ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( removeAll @ A @ X @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_214_can__select__def,axiom,
! [A: $tType] :
( ( can_select @ A )
= ( ^ [P4: A > $o,A6: set @ A] :
? [X2: A] :
( ( member2 @ A @ X2 @ A6 )
& ( P4 @ X2 )
& ! [Y2: A] :
( ( ( member2 @ A @ Y2 @ A6 )
& ( P4 @ Y2 ) )
=> ( Y2 = X2 ) ) ) ) ) ).
% can_select_def
thf(fact_215_removeAll_Osimps_I1_J,axiom,
! [A: $tType,X: A] :
( ( removeAll @ A @ X @ ( nil @ A ) )
= ( nil @ A ) ) ).
% removeAll.simps(1)
thf(fact_216_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( X != Y )
=> ( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_217_linordered__field__no__ub,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [X4: A] :
? [X1: A] : ( ord_less @ A @ X4 @ X1 ) ) ).
% linordered_field_no_ub
thf(fact_218_linordered__field__no__lb,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [X4: A] :
? [Y4: A] : ( ord_less @ A @ Y4 @ X4 ) ) ).
% linordered_field_no_lb
thf(fact_219_dependent__wellorder__choice,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ A @ ( type2 @ A ) )
=> ! [P2: ( A > B ) > A > B > $o] :
( ! [R3: B,F4: A > B,G3: A > B,X3: A] :
( ! [Y5: A] :
( ( ord_less @ A @ Y5 @ X3 )
=> ( ( F4 @ Y5 )
= ( G3 @ Y5 ) ) )
=> ( ( P2 @ F4 @ X3 @ R3 )
= ( P2 @ G3 @ X3 @ R3 ) ) )
=> ( ! [X3: A,F4: A > B] :
( ! [Y5: A] :
( ( ord_less @ A @ Y5 @ X3 )
=> ( P2 @ F4 @ Y5 @ ( F4 @ Y5 ) ) )
=> ? [X12: B] : ( P2 @ F4 @ X3 @ X12 ) )
=> ? [F4: A > B] :
! [X4: A] : ( P2 @ F4 @ X4 @ ( F4 @ X4 ) ) ) ) ) ).
% dependent_wellorder_choice
thf(fact_220_dag_Osimps_I6_J,axiom,
! [A: $tType,F1: A,F22: binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > A > A > A] :
( ( binDag1442713106ec_dag @ A @ F1 @ F22 @ binDag_Mirabelle_Tip )
= F1 ) ).
% dag.simps(6)
thf(fact_221_dag_Osimps_I7_J,axiom,
! [A: $tType,F1: A,F22: binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > A > A > A,X21: binDag_Mirabelle_dag,X22: simpl_ref,X23: binDag_Mirabelle_dag] :
( ( binDag1442713106ec_dag @ A @ F1 @ F22 @ ( binDag476092410e_Node @ X21 @ X22 @ X23 ) )
= ( F22 @ X21 @ X22 @ X23 @ ( binDag1442713106ec_dag @ A @ F1 @ F22 @ X21 ) @ ( binDag1442713106ec_dag @ A @ F1 @ F22 @ X23 ) ) ) ).
% dag.simps(7)
thf(fact_222_chain__subset__def,axiom,
! [A: $tType] :
( ( chain_subset @ A )
= ( ^ [C5: set @ ( set @ A )] :
! [X2: set @ A] :
( ( member2 @ ( set @ A ) @ X2 @ C5 )
=> ! [Y2: set @ A] :
( ( member2 @ ( set @ A ) @ Y2 @ C5 )
=> ( ( ord_less_eq @ ( set @ A ) @ X2 @ Y2 )
| ( ord_less_eq @ ( set @ A ) @ Y2 @ X2 ) ) ) ) ) ) ).
% chain_subset_def
thf(fact_223_dag_Osimps_I4_J,axiom,
! [A: $tType,F1: A,F22: binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > A] :
( ( binDag1297733282se_dag @ A @ F1 @ F22 @ binDag_Mirabelle_Tip )
= F1 ) ).
% dag.simps(4)
thf(fact_224_dag_Osimps_I5_J,axiom,
! [A: $tType,F1: A,F22: binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > A,X21: binDag_Mirabelle_dag,X22: simpl_ref,X23: binDag_Mirabelle_dag] :
( ( binDag1297733282se_dag @ A @ F1 @ F22 @ ( binDag476092410e_Node @ X21 @ X22 @ X23 ) )
= ( F22 @ X21 @ X22 @ X23 ) ) ).
% dag.simps(5)
thf(fact_225_sublist__empty,axiom,
! [A: $tType,Xs: list @ A] :
( ( sublist @ A @ Xs @ ( bot_bot @ ( set @ nat ) ) )
= ( nil @ A ) ) ).
% sublist_empty
thf(fact_226_not__in__set__insert,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ~ ( member2 @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( insert @ A @ X @ Xs )
= ( cons @ A @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_227_bot__apply,axiom,
! [C2: $tType,D3: $tType] :
( ( bot @ C2 @ ( type2 @ C2 ) )
=> ( ( bot_bot @ ( D3 > C2 ) )
= ( ^ [X2: D3] : ( bot_bot @ C2 ) ) ) ) ).
% bot_apply
thf(fact_228_empty__iff,axiom,
! [A: $tType,C: A] :
~ ( member2 @ A @ C @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_229_all__not__in__conv,axiom,
! [A: $tType,A2: set @ A] :
( ( ! [X2: A] :
~ ( member2 @ A @ X2 @ A2 ) )
= ( A2
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_230_Collect__empty__eq,axiom,
! [A: $tType,P2: A > $o] :
( ( ( collect @ A @ P2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X2: A] :
~ ( P2 @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_231_empty__Collect__eq,axiom,
! [A: $tType,P2: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P2 ) )
= ( ! [X2: A] :
~ ( P2 @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_232_list_Oinject,axiom,
! [A: $tType,X21: A,X22: list @ A,Y21: A,Y22: list @ A] :
( ( ( cons @ A @ X21 @ X22 )
= ( cons @ A @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_233_empty__subsetI,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A2 ) ).
% empty_subsetI
thf(fact_234_subset__empty,axiom,
! [A: $tType,A2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ ( bot_bot @ ( set @ A ) ) )
= ( A2
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_empty
thf(fact_235_set__empty2,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set2 @ A @ Xs ) )
= ( Xs
= ( nil @ A ) ) ) ).
% set_empty2
thf(fact_236_set__empty,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( set2 @ A @ Xs )
= ( bot_bot @ ( set @ A ) ) )
= ( Xs
= ( nil @ A ) ) ) ).
% set_empty
thf(fact_237_insert__Nil,axiom,
! [A: $tType,X: A] :
( ( insert @ A @ X @ ( nil @ A ) )
= ( cons @ A @ X @ ( nil @ A ) ) ) ).
% insert_Nil
thf(fact_238_List_Oinsert__def,axiom,
! [A: $tType] :
( ( insert @ A )
= ( ^ [X2: A,Xs2: list @ A] : ( if @ ( list @ A ) @ ( member2 @ A @ X2 @ ( set2 @ A @ Xs2 ) ) @ Xs2 @ ( cons @ A @ X2 @ Xs2 ) ) ) ) ).
% List.insert_def
thf(fact_239_bot_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A4: A] :
( ( ord_less_eq @ A @ A4 @ ( bot_bot @ A ) )
=> ( A4
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_uniqueI
thf(fact_240_bot_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A4: A] :
( ( ord_less_eq @ A @ A4 @ ( bot_bot @ A ) )
= ( A4
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_unique
thf(fact_241_bot_Oextremum,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A4: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A4 ) ) ).
% bot.extremum
thf(fact_242_list_Oset__cases,axiom,
! [A: $tType,E: A,A4: list @ A] :
( ( member2 @ A @ E @ ( set2 @ A @ A4 ) )
=> ( ! [Z22: list @ A] :
( A4
!= ( cons @ A @ E @ Z22 ) )
=> ~ ! [Z1: A,Z22: list @ A] :
( ( A4
= ( cons @ A @ Z1 @ Z22 ) )
=> ~ ( member2 @ A @ E @ ( set2 @ A @ Z22 ) ) ) ) ) ).
% list.set_cases
thf(fact_243_set__ConsD,axiom,
! [A: $tType,Y: A,X: A,Xs: list @ A] :
( ( member2 @ A @ Y @ ( set2 @ A @ ( cons @ A @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member2 @ A @ Y @ ( set2 @ A @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_244_list_Oset__intros_I1_J,axiom,
! [A: $tType,A1: A,A22: list @ A] : ( member2 @ A @ A1 @ ( set2 @ A @ ( cons @ A @ A1 @ A22 ) ) ) ).
% list.set_intros(1)
thf(fact_245_list_Oset__intros_I2_J,axiom,
! [A: $tType,X: A,A22: list @ A,A1: A] :
( ( member2 @ A @ X @ ( set2 @ A @ A22 ) )
=> ( member2 @ A @ X @ ( set2 @ A @ ( cons @ A @ A1 @ A22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_246_member__rec_I1_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: A] :
( ( member @ A @ ( cons @ A @ X @ Xs ) @ Y )
= ( ( X = Y )
| ( member @ A @ Xs @ Y ) ) ) ).
% member_rec(1)
thf(fact_247_emptyE,axiom,
! [A: $tType,A4: A] :
~ ( member2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_248_equals0D,axiom,
! [A: $tType,A2: set @ A,A4: A] :
( ( A2
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member2 @ A @ A4 @ A2 ) ) ).
% equals0D
thf(fact_249_equals0I,axiom,
! [A: $tType,A2: set @ A] :
( ! [Y4: A] :
~ ( member2 @ A @ Y4 @ A2 )
=> ( A2
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_250_ex__in__conv,axiom,
! [A: $tType,A2: set @ A] :
( ( ? [X2: A] : ( member2 @ A @ X2 @ A2 ) )
= ( A2
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_251_not__Cons__self2,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( cons @ A @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_252_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B @ ( type2 @ B ) )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X2: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_253_set__subset__Cons,axiom,
! [A: $tType,Xs: list @ A,X: A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ ( cons @ A @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_254_empty__set,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( set2 @ A @ ( nil @ A ) ) ) ).
% empty_set
%----Type constructors (28)
thf(tcon_fun___Orderings_Oorder__bot,axiom,
! [A7: $tType,A8: $tType] :
( ( order_bot @ A8 @ ( type2 @ A8 ) )
=> ( order_bot @ ( 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___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_fun___Orderings_Obot,axiom,
! [A7: $tType,A8: $tType] :
( ( bot @ A8 @ ( type2 @ A8 ) )
=> ( bot @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
condit1037483654norder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
wellorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oorder__bot_1,axiom,
order_bot @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Opreorder_2,axiom,
preorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
linorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Ono__top,axiom,
no_top @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oorder_3,axiom,
order @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oord_4,axiom,
ord @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Obot_5,axiom,
bot @ nat @ ( type2 @ nat ) ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_6,axiom,
! [A7: $tType] : ( order_bot @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_7,axiom,
! [A7: $tType] : ( preorder @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_8,axiom,
! [A7: $tType] : ( order @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_9,axiom,
! [A7: $tType] : ( ord @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_10,axiom,
! [A7: $tType] : ( bot @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_11,axiom,
order_bot @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_12,axiom,
preorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder_13,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_14,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_15,axiom,
ord @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Obot_16,axiom,
bot @ $o @ ( type2 @ $o ) ).
thf(tcon_BinDag__Mirabelle__rybootvolr_Odag___Orderings_Opreorder_17,axiom,
preorder @ binDag_Mirabelle_dag @ ( type2 @ binDag_Mirabelle_dag ) ).
thf(tcon_BinDag__Mirabelle__rybootvolr_Odag___Orderings_Oorder_18,axiom,
order @ binDag_Mirabelle_dag @ ( type2 @ binDag_Mirabelle_dag ) ).
thf(tcon_BinDag__Mirabelle__rybootvolr_Odag___Orderings_Oord_19,axiom,
ord @ binDag_Mirabelle_dag @ ( type2 @ binDag_Mirabelle_dag ) ).
%----Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $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 (2)
thf(conj_0,hypothesis,
ord_less_eq @ ( set @ simpl_ref ) @ ( binDag1380252983set_of @ t ) @ ( set2 @ simpl_ref @ alloc ) ).
thf(conj_1,conjecture,
( ( binDag_Mirabelle_Dag @ p @ ( fun_upd @ simpl_ref @ simpl_ref @ l @ ( simpl_new @ ( set2 @ simpl_ref @ alloc ) ) @ x ) @ r @ t )
= ( binDag_Mirabelle_Dag @ p @ l @ r @ t ) ) ).
%------------------------------------------------------------------------------