TPTP Problem File: COM167^1.p
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%------------------------------------------------------------------------------
% File : COM167^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Computing Theory
% Problem : Binary decision diagram 250
% 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__250.p [Bla16]
% Status : Theorem
% Rating : 0.00 v7.2.0, 0.25 v7.1.0
% Syntax : Number of formulae : 353 ( 121 unt; 59 typ; 0 def)
% Number of atoms : 819 ( 290 equ; 0 cnn)
% Maximal formula atoms : 9 ( 2 avg)
% Number of connectives : 3553 ( 95 ~; 30 |; 63 &;3042 @)
% ( 0 <=>; 323 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 7 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 312 ( 312 >; 0 *; 0 +; 0 <<)
% Number of symbols : 60 ( 57 usr; 6 con; 0-6 aty)
% Number of variables : 1079 ( 81 ^; 900 !; 50 ?;1079 :)
% ( 48 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:45:46.829
%------------------------------------------------------------------------------
%----Could-be-implicit typings (4)
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_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
%----Explicit typings (55)
thf(sy_cl_Lattices_Obounded__lattice,type,
bounded_lattice:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
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_Lattices_Olattice,type,
lattice:
!>[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_Lattices_Osemilattice__sup,type,
semilattice_sup:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Obounded__lattice__bot,type,
bounded_lattice_bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Obounded__semilattice__sup__bot,type,
bounde1808546759up_bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
condit1656338222tinuum:
!>[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_Fun_Ooverride__on,type,
override_on:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( A > B ) > ( set @ A ) > A > B ) ).
thf(sy_c_Groups_Ocomm__monoid,type,
comm_monoid:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Groups_Omonoid,type,
monoid:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Lattices_Osemilattice__neutr,type,
semilattice_neutr:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Lattices_Osup__class_Osup,type,
sup_sup:
!>[A: $tType] : ( A > A > 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_Relation_Oinv__imagep,type,
inv_imagep:
!>[B: $tType,A: $tType] : ( ( B > B > $o ) > ( A > B ) > A > A > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_OPow,type,
pow:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
thf(sy_c_Set_Obind,type,
bind:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > ( set @ B ) ) > ( set @ B ) ) ).
thf(sy_c_Set_Oinsert,type,
insert:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Set_Ois__empty,type,
is_empty:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Set_Ois__singleton,type,
is_singleton:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Set_Opairwise,type,
pairwise:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Set_Othe__elem,type,
the_elem:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Simpl__Heap_ONull,type,
simpl_Null: simpl_ref ).
thf(sy_c_Zorn_Opred__on_Ochain,type,
pred_chain:
!>[A: $tType] : ( ( set @ A ) > ( A > A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Zorn_Opred__on_Omaxchain,type,
pred_maxchain:
!>[A: $tType] : ( ( set @ A ) > ( A > A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
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 ).
%----Relevant facts (256)
thf(fact_0_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_1_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_2_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_3_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_4_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_5_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_6_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] :
( ~ ( member @ 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_7_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] :
( ~ ( member @ 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_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 )
=> ~ ( member @ simpl_ref @ simpl_Null @ ( binDag1380252983set_of @ T ) ) ) ).
% Null_notin_Dag
thf(fact_9_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_10_Dag_Osimps_I2_J,axiom,
! [P: simpl_ref,L: simpl_ref > simpl_ref,R: simpl_ref > simpl_ref,Lt2: binDag_Mirabelle_dag,A2: simpl_ref,Rt2: binDag_Mirabelle_dag] :
( ( binDag_Mirabelle_Dag @ P @ L @ R @ ( binDag476092410e_Node @ Lt2 @ A2 @ Rt2 ) )
= ( ( P = A2 )
& ( P != simpl_Null )
& ( binDag_Mirabelle_Dag @ ( L @ P ) @ L @ R @ Lt2 )
& ( binDag_Mirabelle_Dag @ ( R @ P ) @ L @ R @ Rt2 ) ) ) ).
% Dag.simps(2)
thf(fact_11_fun__upd__def,axiom,
! [B: $tType,A: $tType] :
( ( fun_upd @ A @ B )
= ( ^ [F2: A > B,A3: A,B2: B,X2: A] : ( if @ B @ ( X2 = A3 ) @ B2 @ ( F2 @ X2 ) ) ) ) ).
% fun_upd_def
thf(fact_12_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_13_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_14_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_15_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_16_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_17_dag_Oinduct,axiom,
! [P2: binDag_Mirabelle_dag > $o,Dag: binDag_Mirabelle_dag] :
( ( P2 @ binDag_Mirabelle_Tip )
=> ( ! [X1: binDag_Mirabelle_dag,X24: simpl_ref,X3: binDag_Mirabelle_dag] :
( ( P2 @ X1 )
=> ( ( P2 @ X3 )
=> ( P2 @ ( binDag476092410e_Node @ X1 @ X24 @ X3 ) ) ) )
=> ( P2 @ Dag ) ) ) ).
% dag.induct
thf(fact_18_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_19_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_20_fun__upd__twist,axiom,
! [A: $tType,B: $tType,A2: A,C: A,M: A > B,B3: B,D: B] :
( ( A2 != C )
=> ( ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ M @ A2 @ B3 ) @ C @ D )
= ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ M @ C @ D ) @ A2 @ B3 ) ) ) ).
% fun_upd_twist
thf(fact_21_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_22_DAG_Osimps_I2_J,axiom,
! [L: binDag_Mirabelle_dag,A2: simpl_ref,R: binDag_Mirabelle_dag] :
( ( binDag_Mirabelle_DAG @ ( binDag476092410e_Node @ L @ A2 @ R ) )
= ( ~ ( member @ simpl_ref @ A2 @ ( binDag1380252983set_of @ L ) )
& ~ ( member @ simpl_ref @ A2 @ ( binDag1380252983set_of @ R ) )
& ( binDag_Mirabelle_DAG @ L )
& ( binDag_Mirabelle_DAG @ R ) ) ) ).
% DAG.simps(2)
thf(fact_23_in__set__of__decomp,axiom,
! [P: simpl_ref,T: binDag_Mirabelle_dag] :
( ( member @ 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_24_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_25_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_26_DAG_Osimps_I1_J,axiom,
binDag_Mirabelle_DAG @ binDag_Mirabelle_Tip ).
% DAG.simps(1)
thf(fact_27_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_28_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_29_set__of__Tip,axiom,
( ( binDag1380252983set_of @ binDag_Mirabelle_Tip )
= ( bot_bot @ ( set @ simpl_ref ) ) ) ).
% set_of_Tip
thf(fact_30_subdag_Osimps_I1_J,axiom,
! [T: binDag_Mirabelle_dag] :
~ ( binDag786255756subdag @ binDag_Mirabelle_Tip @ T ) ).
% subdag.simps(1)
thf(fact_31_subdag_Osimps_I2_J,axiom,
! [L: binDag_Mirabelle_dag,A2: simpl_ref,R: binDag_Mirabelle_dag,T: binDag_Mirabelle_dag] :
( ( binDag786255756subdag @ ( binDag476092410e_Node @ L @ A2 @ R ) @ T )
= ( ( T = L )
| ( T = R )
| ( binDag786255756subdag @ L @ T )
| ( binDag786255756subdag @ R @ T ) ) ) ).
% subdag.simps(2)
thf(fact_32_subdag__not__sym,axiom,
! [S: binDag_Mirabelle_dag,T: binDag_Mirabelle_dag] :
( ( binDag786255756subdag @ S @ T )
=> ~ ( binDag786255756subdag @ T @ S ) ) ).
% subdag_not_sym
thf(fact_33_subdag__trans,axiom,
! [T: binDag_Mirabelle_dag,S: binDag_Mirabelle_dag,R: binDag_Mirabelle_dag] :
( ( binDag786255756subdag @ T @ S )
=> ( ( binDag786255756subdag @ S @ R )
=> ( binDag786255756subdag @ T @ R ) ) ) ).
% subdag_trans
thf(fact_34_subdag__neq,axiom,
! [T: binDag_Mirabelle_dag,S: binDag_Mirabelle_dag] :
( ( binDag786255756subdag @ T @ S )
=> ( T != S ) ) ).
% subdag_neq
thf(fact_35_subdag__NodeD,axiom,
! [T: binDag_Mirabelle_dag,Lt2: binDag_Mirabelle_dag,A2: simpl_ref,Rt2: binDag_Mirabelle_dag] :
( ( binDag786255756subdag @ T @ ( binDag476092410e_Node @ Lt2 @ A2 @ Rt2 ) )
=> ( ( binDag786255756subdag @ T @ Lt2 )
& ( binDag786255756subdag @ T @ Rt2 ) ) ) ).
% subdag_NodeD
thf(fact_36_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_37_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_38_all__not__in__conv,axiom,
! [A: $tType,A4: set @ A] :
( ( ! [X2: A] :
~ ( member @ A @ X2 @ A4 ) )
= ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_39_empty__iff,axiom,
! [A: $tType,C: A] :
~ ( member @ A @ C @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_40_bot__apply,axiom,
! [C2: $tType,D2: $tType] :
( ( bot @ C2 @ ( type2 @ C2 ) )
=> ( ( bot_bot @ ( D2 > C2 ) )
= ( ^ [X2: D2] : ( bot_bot @ C2 ) ) ) ) ).
% bot_apply
thf(fact_41_override__on__emptyset,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ( override_on @ A @ B @ F @ G @ ( bot_bot @ ( set @ A ) ) )
= F ) ).
% override_on_emptyset
thf(fact_42_ex__in__conv,axiom,
! [A: $tType,A4: set @ A] :
( ( ? [X2: A] : ( member @ A @ X2 @ A4 ) )
= ( A4
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_43_equals0I,axiom,
! [A: $tType,A4: set @ A] :
( ! [Y3: A] :
~ ( member @ A @ Y3 @ A4 )
=> ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_44_equals0D,axiom,
! [A: $tType,A4: set @ A,A2: A] :
( ( A4
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A2 @ A4 ) ) ).
% equals0D
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P2: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( collect @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P2: A > $o,Q2: A > $o] :
( ! [X4: A] :
( ( P2 @ X4 )
= ( Q2 @ X4 ) )
=> ( ( collect @ A @ P2 )
= ( collect @ A @ Q2 ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X4: A] :
( ( F @ X4 )
= ( G @ X4 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_override__on__apply__in,axiom,
! [B: $tType,A: $tType,A2: A,A4: set @ A,F: A > B,G: A > B] :
( ( member @ A @ A2 @ A4 )
=> ( ( override_on @ A @ B @ F @ G @ A4 @ A2 )
= ( G @ A2 ) ) ) ).
% override_on_apply_in
thf(fact_50_override__on__apply__notin,axiom,
! [B: $tType,A: $tType,A2: A,A4: set @ A,F: A > B,G: A > B] :
( ~ ( member @ A @ A2 @ A4 )
=> ( ( override_on @ A @ B @ F @ G @ A4 @ A2 )
= ( F @ A2 ) ) ) ).
% override_on_apply_notin
thf(fact_51_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_52_override__on__def,axiom,
! [B: $tType,A: $tType] :
( ( override_on @ A @ B )
= ( ^ [F2: A > B,G2: A > B,A5: set @ A,A3: A] : ( if @ B @ ( member @ A @ A3 @ A5 ) @ ( G2 @ A3 ) @ ( F2 @ A3 ) ) ) ) ).
% override_on_def
thf(fact_53_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_54_emptyE,axiom,
! [A: $tType,A2: A] :
~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_55_Set_Ois__empty__def,axiom,
! [A: $tType] :
( ( is_empty @ A )
= ( ^ [A5: set @ A] :
( A5
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Set.is_empty_def
thf(fact_56_Collect__empty__eq__bot,axiom,
! [A: $tType,P2: A > $o] :
( ( ( collect @ A @ P2 )
= ( bot_bot @ ( set @ A ) ) )
= ( P2
= ( bot_bot @ ( A > $o ) ) ) ) ).
% Collect_empty_eq_bot
thf(fact_57_bot__empty__eq,axiom,
! [A: $tType] :
( ( bot_bot @ ( A > $o ) )
= ( ^ [X2: A] : ( member @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% bot_empty_eq
thf(fact_58_is__singletonI_H,axiom,
! [A: $tType,A4: set @ A] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X4: A,Y3: A] :
( ( member @ A @ X4 @ A4 )
=> ( ( member @ A @ Y3 @ A4 )
=> ( X4 = Y3 ) ) )
=> ( is_singleton @ A @ A4 ) ) ) ).
% is_singletonI'
thf(fact_59_empty__bind,axiom,
! [B: $tType,A: $tType,F: B > ( set @ A )] :
( ( bind @ B @ A @ ( bot_bot @ ( set @ B ) ) @ F )
= ( bot_bot @ ( set @ A ) ) ) ).
% empty_bind
thf(fact_60_set__of__Node,axiom,
! [Lt2: binDag_Mirabelle_dag,A2: simpl_ref,Rt2: binDag_Mirabelle_dag] :
( ( binDag1380252983set_of @ ( binDag476092410e_Node @ Lt2 @ A2 @ Rt2 ) )
= ( sup_sup @ ( set @ simpl_ref ) @ ( sup_sup @ ( set @ simpl_ref ) @ ( insert @ simpl_ref @ A2 @ ( bot_bot @ ( set @ simpl_ref ) ) ) @ ( binDag1380252983set_of @ Lt2 ) ) @ ( binDag1380252983set_of @ Rt2 ) ) ) ).
% set_of_Node
thf(fact_61_pairwise__empty,axiom,
! [A: $tType,P2: A > A > $o] : ( pairwise @ A @ P2 @ ( bot_bot @ ( set @ A ) ) ) ).
% pairwise_empty
thf(fact_62_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_63_insertCI,axiom,
! [A: $tType,A2: A,B4: set @ A,B3: A] :
( ( ~ ( member @ A @ A2 @ B4 )
=> ( A2 = B3 ) )
=> ( member @ A @ A2 @ ( insert @ A @ B3 @ B4 ) ) ) ).
% insertCI
thf(fact_64_insert__iff,axiom,
! [A: $tType,A2: A,B3: A,A4: set @ A] :
( ( member @ A @ A2 @ ( insert @ A @ B3 @ A4 ) )
= ( ( A2 = B3 )
| ( member @ A @ A2 @ A4 ) ) ) ).
% insert_iff
thf(fact_65_insert__absorb2,axiom,
! [A: $tType,X: A,A4: set @ A] :
( ( insert @ A @ X @ ( insert @ A @ X @ A4 ) )
= ( insert @ A @ X @ A4 ) ) ).
% insert_absorb2
thf(fact_66_UnCI,axiom,
! [A: $tType,C: A,B4: set @ A,A4: set @ A] :
( ( ~ ( member @ A @ C @ B4 )
=> ( member @ A @ C @ A4 ) )
=> ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) ) ) ).
% UnCI
thf(fact_67_Un__iff,axiom,
! [A: $tType,C: A,A4: set @ A,B4: set @ A] :
( ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) )
= ( ( member @ A @ C @ A4 )
| ( member @ A @ C @ B4 ) ) ) ).
% Un_iff
thf(fact_68_singletonI,axiom,
! [A: $tType,A2: A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singletonI
thf(fact_69_Un__empty,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( ( sup_sup @ ( set @ A ) @ A4 @ B4 )
= ( bot_bot @ ( set @ A ) ) )
= ( ( A4
= ( bot_bot @ ( set @ A ) ) )
& ( B4
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Un_empty
thf(fact_70_Un__insert__left,axiom,
! [A: $tType,A2: A,B4: set @ A,C3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( insert @ A @ A2 @ B4 ) @ C3 )
= ( insert @ A @ A2 @ ( sup_sup @ ( set @ A ) @ B4 @ C3 ) ) ) ).
% Un_insert_left
thf(fact_71_Un__insert__right,axiom,
! [A: $tType,A4: set @ A,A2: A,B4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( insert @ A @ A2 @ B4 ) )
= ( insert @ A @ A2 @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) ) ) ).
% Un_insert_right
thf(fact_72_is__singletonI,axiom,
! [A: $tType,X: A] : ( is_singleton @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% is_singletonI
thf(fact_73_pairwise__singleton,axiom,
! [A: $tType,P2: A > A > $o,A4: A] : ( pairwise @ A @ P2 @ ( insert @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% pairwise_singleton
thf(fact_74_singleton__Un__iff,axiom,
! [A: $tType,X: A,A4: set @ A,B4: set @ A] :
( ( ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) )
= ( sup_sup @ ( set @ A ) @ A4 @ B4 ) )
= ( ( ( A4
= ( bot_bot @ ( set @ A ) ) )
& ( B4
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
| ( ( A4
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B4
= ( bot_bot @ ( set @ A ) ) ) )
| ( ( A4
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B4
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_75_Un__singleton__iff,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,X: A] :
( ( ( sup_sup @ ( set @ A ) @ A4 @ B4 )
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( ( A4
= ( bot_bot @ ( set @ A ) ) )
& ( B4
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
| ( ( A4
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B4
= ( bot_bot @ ( set @ A ) ) ) )
| ( ( A4
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B4
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_76_insert__is__Un,axiom,
! [A: $tType] :
( ( insert @ A )
= ( ^ [A3: A] : ( sup_sup @ ( set @ A ) @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% insert_is_Un
thf(fact_77_UnE,axiom,
! [A: $tType,C: A,A4: set @ A,B4: set @ A] :
( ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) )
=> ( ~ ( member @ A @ C @ A4 )
=> ( member @ A @ C @ B4 ) ) ) ).
% UnE
thf(fact_78_UnI1,axiom,
! [A: $tType,C: A,A4: set @ A,B4: set @ A] :
( ( member @ A @ C @ A4 )
=> ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) ) ) ).
% UnI1
thf(fact_79_UnI2,axiom,
! [A: $tType,C: A,B4: set @ A,A4: set @ A] :
( ( member @ A @ C @ B4 )
=> ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) ) ) ).
% UnI2
thf(fact_80_bex__Un,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,P2: A > $o] :
( ( ? [X2: A] :
( ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) )
& ( P2 @ X2 ) ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ A4 )
& ( P2 @ X2 ) )
| ? [X2: A] :
( ( member @ A @ X2 @ B4 )
& ( P2 @ X2 ) ) ) ) ).
% bex_Un
thf(fact_81_ball__Un,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,P2: A > $o] :
( ( ! [X2: A] :
( ( member @ A @ X2 @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) )
=> ( P2 @ X2 ) ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( P2 @ X2 ) )
& ! [X2: A] :
( ( member @ A @ X2 @ B4 )
=> ( P2 @ X2 ) ) ) ) ).
% ball_Un
thf(fact_82_insertE,axiom,
! [A: $tType,A2: A,B3: A,A4: set @ A] :
( ( member @ A @ A2 @ ( insert @ A @ B3 @ A4 ) )
=> ( ( A2 != B3 )
=> ( member @ A @ A2 @ A4 ) ) ) ).
% insertE
thf(fact_83_Un__assoc,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,C3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) @ C3 )
= ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B4 @ C3 ) ) ) ).
% Un_assoc
thf(fact_84_insertI1,axiom,
! [A: $tType,A2: A,B4: set @ A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ B4 ) ) ).
% insertI1
thf(fact_85_insertI2,axiom,
! [A: $tType,A2: A,B4: set @ A,B3: A] :
( ( member @ A @ A2 @ B4 )
=> ( member @ A @ A2 @ ( insert @ A @ B3 @ B4 ) ) ) ).
% insertI2
thf(fact_86_Un__absorb,axiom,
! [A: $tType,A4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ A4 )
= A4 ) ).
% Un_absorb
thf(fact_87_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_88_Set_Oset__insert,axiom,
! [A: $tType,X: A,A4: set @ A] :
( ( member @ A @ X @ A4 )
=> ~ ! [B6: set @ A] :
( ( A4
= ( insert @ A @ X @ B6 ) )
=> ( member @ A @ X @ B6 ) ) ) ).
% Set.set_insert
thf(fact_89_insert__ident,axiom,
! [A: $tType,X: A,A4: set @ A,B4: set @ A] :
( ~ ( member @ A @ X @ A4 )
=> ( ~ ( member @ A @ X @ B4 )
=> ( ( ( insert @ A @ X @ A4 )
= ( insert @ A @ X @ B4 ) )
= ( A4 = B4 ) ) ) ) ).
% insert_ident
thf(fact_90_pairwise__def,axiom,
! [A: $tType] :
( ( pairwise @ A )
= ( ^ [R3: A > A > $o,S2: set @ A] :
! [X2: A] :
( ( member @ A @ X2 @ S2 )
=> ! [Y2: A] :
( ( member @ A @ Y2 @ S2 )
=> ( ( X2 != Y2 )
=> ( R3 @ X2 @ Y2 ) ) ) ) ) ) ).
% pairwise_def
thf(fact_91_insert__absorb,axiom,
! [A: $tType,A2: A,A4: set @ A] :
( ( member @ A @ A2 @ A4 )
=> ( ( insert @ A @ A2 @ A4 )
= A4 ) ) ).
% insert_absorb
thf(fact_92_insert__eq__iff,axiom,
! [A: $tType,A2: A,A4: set @ A,B3: A,B4: set @ A] :
( ~ ( member @ A @ A2 @ A4 )
=> ( ~ ( member @ A @ B3 @ B4 )
=> ( ( ( insert @ A @ A2 @ A4 )
= ( insert @ A @ B3 @ B4 ) )
= ( ( ( A2 = B3 )
=> ( A4 = B4 ) )
& ( ( A2 != B3 )
=> ? [C4: set @ A] :
( ( A4
= ( insert @ A @ B3 @ C4 ) )
& ~ ( member @ A @ B3 @ C4 )
& ( B4
= ( insert @ A @ A2 @ C4 ) )
& ~ ( member @ A @ A2 @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_93_Un__left__absorb,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) )
= ( sup_sup @ ( set @ A ) @ A4 @ B4 ) ) ).
% Un_left_absorb
thf(fact_94_insert__commute,axiom,
! [A: $tType,X: A,Y: A,A4: set @ A] :
( ( insert @ A @ X @ ( insert @ A @ Y @ A4 ) )
= ( insert @ A @ Y @ ( insert @ A @ X @ A4 ) ) ) ).
% insert_commute
thf(fact_95_Un__left__commute,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,C3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B4 @ C3 ) )
= ( sup_sup @ ( set @ A ) @ B4 @ ( sup_sup @ ( set @ A ) @ A4 @ C3 ) ) ) ).
% Un_left_commute
thf(fact_96_pairwise__insert,axiom,
! [A: $tType,R: A > A > $o,X: A,S: set @ A] :
( ( pairwise @ A @ R @ ( insert @ A @ X @ S ) )
= ( ! [Y2: A] :
( ( ( member @ A @ Y2 @ S )
& ( Y2 != X ) )
=> ( ( R @ X @ Y2 )
& ( R @ Y2 @ X ) ) )
& ( pairwise @ A @ R @ S ) ) ) ).
% pairwise_insert
thf(fact_97_mk__disjoint__insert,axiom,
! [A: $tType,A2: A,A4: set @ A] :
( ( member @ A @ A2 @ A4 )
=> ? [B6: set @ A] :
( ( A4
= ( insert @ A @ A2 @ B6 ) )
& ~ ( member @ A @ A2 @ B6 ) ) ) ).
% mk_disjoint_insert
thf(fact_98_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B > A,B3: B,C: B] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less @ B @ B3 @ C )
=> ( ! [X4: B,Y3: B] :
( ( ord_less @ B @ X4 @ Y3 )
=> ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_99_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B3: A,F: A > B,C: B] :
( ( ord_less @ A @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X4: A,Y3: A] :
( ( ord_less @ A @ X4 @ Y3 )
=> ( ord_less @ B @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_100_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B > A,B3: B,C: B] :
( ( ord_less @ A @ A2 @ ( F @ B3 ) )
=> ( ( ord_less @ B @ B3 @ C )
=> ( ! [X4: B,Y3: B] :
( ( ord_less @ B @ X4 @ Y3 )
=> ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_101_order__less__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 @ ( type2 @ C2 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B3: A,F: A > C2,C: C2] :
( ( ord_less @ A @ A2 @ B3 )
=> ( ( ord_less @ C2 @ ( F @ B3 ) @ C )
=> ( ! [X4: A,Y3: A] :
( ( ord_less @ A @ X4 @ Y3 )
=> ( ord_less @ C2 @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_less_subst2
thf(fact_102_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A @ ( type2 @ A ) )
=> ! [X: A] :
? [Y3: A] : ( ord_less @ A @ Y3 @ X ) ) ).
% lt_ex
thf(fact_103_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A @ ( type2 @ A ) )
=> ! [X: A] :
? [X1: A] : ( ord_less @ A @ X @ X1 ) ) ).
% gt_ex
thf(fact_104_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_105_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_106_order_Oasym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( ord_less @ A @ A2 @ B3 )
=> ~ ( ord_less @ A @ B3 @ A2 ) ) ) ).
% order.asym
thf(fact_107_dense,axiom,
! [A: $tType] :
( ( dense_order @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ? [Z3: A] :
( ( ord_less @ A @ X @ Z3 )
& ( ord_less @ A @ Z3 @ Y ) ) ) ) ).
% dense
thf(fact_108_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_109_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_110_less__asym_H,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( ord_less @ A @ A2 @ B3 )
=> ~ ( ord_less @ A @ B3 @ A2 ) ) ) ).
% less_asym'
thf(fact_111_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_112_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_113_less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A] :
~ ( ord_less @ A @ X @ X ) ) ).
% less_irrefl
thf(fact_114_ord__eq__less__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C: A] :
( ( A2 = B3 )
=> ( ( ord_less @ A @ B3 @ C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% ord_eq_less_trans
thf(fact_115_ord__less__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C: A] :
( ( ord_less @ A @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% ord_less_eq_trans
thf(fact_116_dual__order_Oasym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A] :
( ( ord_less @ A @ B3 @ A2 )
=> ~ ( ord_less @ A @ A2 @ B3 ) ) ) ).
% dual_order.asym
thf(fact_117_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_118_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_119_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A @ ( type2 @ A ) )
=> ! [P2: A > $o,A2: A] :
( ! [X4: A] :
( ! [Y4: A] :
( ( ord_less @ A @ Y4 @ X4 )
=> ( P2 @ Y4 ) )
=> ( P2 @ X4 ) )
=> ( P2 @ A2 ) ) ) ).
% less_induct
thf(fact_120_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_121_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_122_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_123_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_124_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ A2 ) ) ).
% dual_order.irrefl
thf(fact_125_order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C: A] :
( ( ord_less @ A @ A2 @ B3 )
=> ( ( ord_less @ A @ B3 @ C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% order.strict_trans
thf(fact_126_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_127_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A,C: A] :
( ( ord_less @ A @ B3 @ A2 )
=> ( ( ord_less @ A @ C @ B3 )
=> ( ord_less @ A @ C @ A2 ) ) ) ) ).
% dual_order.strict_trans
thf(fact_128_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_129_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( ord_less @ A @ A2 @ B3 )
=> ( A2 != B3 ) ) ) ).
% order.strict_implies_not_eq
thf(fact_130_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A] :
( ( ord_less @ A @ B3 @ A2 )
=> ( A2 != B3 ) ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_131_is__singletonE,axiom,
! [A: $tType,A4: set @ A] :
( ( is_singleton @ A @ A4 )
=> ~ ! [X4: A] :
( A4
!= ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% is_singletonE
thf(fact_132_is__singleton__def,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A5: set @ A] :
? [X2: A] :
( A5
= ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_def
thf(fact_133_Un__empty__left,axiom,
! [A: $tType,B4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B4 )
= B4 ) ).
% Un_empty_left
thf(fact_134_Un__empty__right,axiom,
! [A: $tType,A4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
= A4 ) ).
% Un_empty_right
thf(fact_135_singletonD,axiom,
! [A: $tType,B3: A,A2: A] :
( ( member @ A @ B3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( B3 = A2 ) ) ).
% singletonD
thf(fact_136_singleton__iff,axiom,
! [A: $tType,B3: A,A2: A] :
( ( member @ A @ B3 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( B3 = A2 ) ) ).
% singleton_iff
thf(fact_137_doubleton__eq__iff,axiom,
! [A: $tType,A2: A,B3: A,C: A,D: A] :
( ( ( insert @ A @ A2 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ C @ ( insert @ A @ D @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ( ( A2 = C )
& ( B3 = D ) )
| ( ( A2 = D )
& ( B3 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_138_insert__not__empty,axiom,
! [A: $tType,A2: A,A4: set @ A] :
( ( insert @ A @ A2 @ A4 )
!= ( bot_bot @ ( set @ A ) ) ) ).
% insert_not_empty
thf(fact_139_singleton__inject,axiom,
! [A: $tType,A2: A,B3: A] :
( ( ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( A2 = B3 ) ) ).
% singleton_inject
thf(fact_140_bot_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ ( bot_bot @ A ) ) ) ).
% bot.extremum_strict
thf(fact_141_bot_Onot__eq__extremum,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( A2
!= ( bot_bot @ A ) )
= ( ord_less @ A @ ( bot_bot @ A ) @ A2 ) ) ) ).
% bot.not_eq_extremum
thf(fact_142_less__dag__Node_H,axiom,
! [X: binDag_Mirabelle_dag,L: binDag_Mirabelle_dag,A2: simpl_ref,R: binDag_Mirabelle_dag] :
( ( ord_less @ binDag_Mirabelle_dag @ X @ ( binDag476092410e_Node @ L @ A2 @ R ) )
= ( ( X = L )
| ( X = R )
| ( ord_less @ binDag_Mirabelle_dag @ X @ L )
| ( ord_less @ binDag_Mirabelle_dag @ X @ R ) ) ) ).
% less_dag_Node'
thf(fact_143_less__Node__dag,axiom,
! [L: binDag_Mirabelle_dag,A2: simpl_ref,R: binDag_Mirabelle_dag,X: binDag_Mirabelle_dag] :
( ( ord_less @ binDag_Mirabelle_dag @ ( binDag476092410e_Node @ L @ A2 @ R ) @ X )
=> ( ( ord_less @ binDag_Mirabelle_dag @ L @ X )
& ( ord_less @ binDag_Mirabelle_dag @ R @ X ) ) ) ).
% less_Node_dag
thf(fact_144_less__dag__def,axiom,
( ( ord_less @ binDag_Mirabelle_dag )
= ( ^ [S3: binDag_Mirabelle_dag,T2: binDag_Mirabelle_dag] : ( binDag786255756subdag @ T2 @ S3 ) ) ) ).
% less_dag_def
thf(fact_145_less__dag__Tip,axiom,
! [X: binDag_Mirabelle_dag] :
~ ( ord_less @ binDag_Mirabelle_dag @ X @ binDag_Mirabelle_Tip ) ).
% less_dag_Tip
thf(fact_146_bot__eq__sup__iff,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ( bot_bot @ A )
= ( sup_sup @ A @ X @ Y ) )
= ( ( X
= ( bot_bot @ A ) )
& ( Y
= ( bot_bot @ A ) ) ) ) ) ).
% bot_eq_sup_iff
thf(fact_147_sup__eq__bot__iff,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ( sup_sup @ A @ X @ Y )
= ( bot_bot @ A ) )
= ( ( X
= ( bot_bot @ A ) )
& ( Y
= ( bot_bot @ A ) ) ) ) ) ).
% sup_eq_bot_iff
thf(fact_148_sup__bot_Oleft__neutral,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( sup_sup @ A @ ( bot_bot @ A ) @ A2 )
= A2 ) ) ).
% sup_bot.left_neutral
thf(fact_149_sup__bot_Oright__neutral,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( sup_sup @ A @ A2 @ ( bot_bot @ A ) )
= A2 ) ) ).
% sup_bot.right_neutral
thf(fact_150_is__singleton__the__elem,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A5: set @ A] :
( A5
= ( insert @ A @ ( the_elem @ A @ A5 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_the_elem
thf(fact_151_sup_Oright__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B3 ) @ B3 )
= ( sup_sup @ A @ A2 @ B3 ) ) ) ).
% sup.right_idem
thf(fact_152_sup__left__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
= ( sup_sup @ A @ X @ Y ) ) ) ).
% sup_left_idem
thf(fact_153_sup__apply,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B @ ( type2 @ B ) )
=> ( ( sup_sup @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B,X2: A] : ( sup_sup @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% sup_apply
thf(fact_154_sup_Oidem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( sup_sup @ A @ A2 @ A2 )
= A2 ) ) ).
% sup.idem
thf(fact_155_sup__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( sup_sup @ A @ X @ X )
= X ) ) ).
% sup_idem
thf(fact_156_sup_Oleft__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( sup_sup @ A @ A2 @ ( sup_sup @ A @ A2 @ B3 ) )
= ( sup_sup @ A @ A2 @ B3 ) ) ) ).
% sup.left_idem
thf(fact_157_the__elem__eq,axiom,
! [A: $tType,X: A] :
( ( the_elem @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ).
% the_elem_eq
thf(fact_158_not__psubset__empty,axiom,
! [A: $tType,A4: set @ A] :
~ ( ord_less @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) ) ).
% not_psubset_empty
thf(fact_159_inf__sup__aci_I8_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
= ( sup_sup @ A @ X @ Y ) ) ) ).
% inf_sup_aci(8)
thf(fact_160_inf__sup__aci_I7_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) )
= ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% inf_sup_aci(7)
thf(fact_161_inf__sup__aci_I6_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z )
= ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) ) ) ) ).
% inf_sup_aci(6)
thf(fact_162_inf__sup__aci_I5_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ( ( sup_sup @ A )
= ( ^ [X2: A,Y2: A] : ( sup_sup @ A @ Y2 @ X2 ) ) ) ) ).
% inf_sup_aci(5)
thf(fact_163_sup__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B @ ( type2 @ B ) )
=> ( ( sup_sup @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B,X2: A] : ( sup_sup @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% sup_fun_def
thf(fact_164_sup_Oassoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B3 ) @ C )
= ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B3 @ C ) ) ) ) ).
% sup.assoc
thf(fact_165_sup__assoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z )
= ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) ) ) ) ).
% sup_assoc
thf(fact_166_sup_Ocommute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( sup_sup @ A )
= ( ^ [A3: A,B2: A] : ( sup_sup @ A @ B2 @ A3 ) ) ) ) ).
% sup.commute
thf(fact_167_sup__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( sup_sup @ A )
= ( ^ [X2: A,Y2: A] : ( sup_sup @ A @ Y2 @ X2 ) ) ) ) ).
% sup_commute
thf(fact_168_sup_Oleft__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A,C: A] :
( ( sup_sup @ A @ B3 @ ( sup_sup @ A @ A2 @ C ) )
= ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B3 @ C ) ) ) ) ).
% sup.left_commute
thf(fact_169_sup__left__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) )
= ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% sup_left_commute
thf(fact_170_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_171_less__supI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,A2: A,B3: A] :
( ( ord_less @ A @ X @ A2 )
=> ( ord_less @ A @ X @ ( sup_sup @ A @ A2 @ B3 ) ) ) ) ).
% less_supI1
thf(fact_172_less__supI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,B3: A,A2: A] :
( ( ord_less @ A @ X @ B3 )
=> ( ord_less @ A @ X @ ( sup_sup @ A @ A2 @ B3 ) ) ) ) ).
% less_supI2
thf(fact_173_sup_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B3: A,C: A,A2: A] :
( ( ord_less @ A @ ( sup_sup @ A @ B3 @ C ) @ A2 )
=> ~ ( ( ord_less @ A @ B3 @ A2 )
=> ~ ( ord_less @ A @ C @ A2 ) ) ) ) ).
% sup.strict_boundedE
thf(fact_174_sup_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [B2: A,A3: A] :
( ( A3
= ( sup_sup @ A @ A3 @ B2 ) )
& ( A3 != B2 ) ) ) ) ) ).
% sup.strict_order_iff
thf(fact_175_sup_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [C: A,A2: A,B3: A] :
( ( ord_less @ A @ C @ A2 )
=> ( ord_less @ A @ C @ ( sup_sup @ A @ A2 @ B3 ) ) ) ) ).
% sup.strict_coboundedI1
thf(fact_176_sup_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [C: A,B3: A,A2: A] :
( ( ord_less @ A @ C @ B3 )
=> ( ord_less @ A @ C @ ( sup_sup @ A @ A2 @ B3 ) ) ) ) ).
% sup.strict_coboundedI2
thf(fact_177_sup__bot__left,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( sup_sup @ A @ ( bot_bot @ A ) @ X )
= X ) ) ).
% sup_bot_left
thf(fact_178_sup__bot__right,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( sup_sup @ A @ X @ ( bot_bot @ A ) )
= X ) ) ).
% sup_bot_right
thf(fact_179_sup__bot_Osemilattice__neutr__axioms,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A @ ( type2 @ A ) )
=> ( semilattice_neutr @ A @ ( sup_sup @ A ) @ ( bot_bot @ A ) ) ) ).
% sup_bot.semilattice_neutr_axioms
thf(fact_180_sup__bot_Omonoid__axioms,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A @ ( type2 @ A ) )
=> ( monoid @ A @ ( sup_sup @ A ) @ ( bot_bot @ A ) ) ) ).
% sup_bot.monoid_axioms
thf(fact_181_minf_I11_J,axiom,
! [C2: $tType,D2: $tType] :
( ( ord @ C2 @ ( type2 @ C2 ) )
=> ! [F3: D2] :
? [Z3: C2] :
! [X5: C2] :
( ( ord_less @ C2 @ X5 @ Z3 )
=> ( F3 = F3 ) ) ) ).
% minf(11)
thf(fact_182_psubsetD,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,C: A] :
( ( ord_less @ ( set @ A ) @ A4 @ B4 )
=> ( ( member @ A @ C @ A4 )
=> ( member @ A @ C @ B4 ) ) ) ).
% psubsetD
thf(fact_183_psubset__trans,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,C3: set @ A] :
( ( ord_less @ ( set @ A ) @ A4 @ B4 )
=> ( ( ord_less @ ( set @ A ) @ B4 @ C3 )
=> ( ord_less @ ( set @ A ) @ A4 @ C3 ) ) ) ).
% psubset_trans
thf(fact_184_pinf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P2: A > $o,P3: A > $o,Q2: A > $o,Q3: A > $o] :
( ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ( ( P2 @ X4 )
= ( P3 @ X4 ) ) )
=> ( ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ( ( Q2 @ X4 )
= ( Q3 @ X4 ) ) )
=> ? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ( ( ( P2 @ X5 )
& ( Q2 @ X5 ) )
= ( ( P3 @ X5 )
& ( Q3 @ X5 ) ) ) ) ) ) ) ).
% pinf(1)
thf(fact_185_pinf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P2: A > $o,P3: A > $o,Q2: A > $o,Q3: A > $o] :
( ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ( ( P2 @ X4 )
= ( P3 @ X4 ) ) )
=> ( ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ( ( Q2 @ X4 )
= ( Q3 @ X4 ) ) )
=> ? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ( ( ( P2 @ X5 )
| ( Q2 @ X5 ) )
= ( ( P3 @ X5 )
| ( Q3 @ X5 ) ) ) ) ) ) ) ).
% pinf(2)
thf(fact_186_pinf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ( X5 != T ) ) ) ).
% pinf(3)
thf(fact_187_pinf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ( X5 != T ) ) ) ).
% pinf(4)
thf(fact_188_pinf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ~ ( ord_less @ A @ X5 @ T ) ) ) ).
% pinf(5)
thf(fact_189_pinf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ( ord_less @ A @ T @ X5 ) ) ) ).
% pinf(7)
thf(fact_190_pinf_I11_J,axiom,
! [C2: $tType,D2: $tType] :
( ( ord @ C2 @ ( type2 @ C2 ) )
=> ! [F3: D2] :
? [Z3: C2] :
! [X5: C2] :
( ( ord_less @ C2 @ Z3 @ X5 )
=> ( F3 = F3 ) ) ) ).
% pinf(11)
thf(fact_191_minf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P2: A > $o,P3: A > $o,Q2: A > $o,Q3: A > $o] :
( ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ( ( P2 @ X4 )
= ( P3 @ X4 ) ) )
=> ( ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ( ( Q2 @ X4 )
= ( Q3 @ X4 ) ) )
=> ? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ( ( ( P2 @ X5 )
& ( Q2 @ X5 ) )
= ( ( P3 @ X5 )
& ( Q3 @ X5 ) ) ) ) ) ) ) ).
% minf(1)
thf(fact_192_minf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P2: A > $o,P3: A > $o,Q2: A > $o,Q3: A > $o] :
( ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ( ( P2 @ X4 )
= ( P3 @ X4 ) ) )
=> ( ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ( ( Q2 @ X4 )
= ( Q3 @ X4 ) ) )
=> ? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ( ( ( P2 @ X5 )
| ( Q2 @ X5 ) )
= ( ( P3 @ X5 )
| ( Q3 @ X5 ) ) ) ) ) ) ) ).
% minf(2)
thf(fact_193_minf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ( X5 != T ) ) ) ).
% minf(3)
thf(fact_194_minf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ( X5 != T ) ) ) ).
% minf(4)
thf(fact_195_minf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ( ord_less @ A @ X5 @ T ) ) ) ).
% minf(5)
thf(fact_196_minf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ~ ( ord_less @ A @ T @ X5 ) ) ) ).
% minf(7)
thf(fact_197_ex__gt__or__lt,axiom,
! [A: $tType] :
( ( condit1656338222tinuum @ A @ ( type2 @ A ) )
=> ! [A2: A] :
? [B7: A] :
( ( ord_less @ A @ A2 @ B7 )
| ( ord_less @ A @ B7 @ A2 ) ) ) ).
% ex_gt_or_lt
thf(fact_198_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_199_linordered__field__no__ub,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [X5: A] :
? [X1: A] : ( ord_less @ A @ X5 @ X1 ) ) ).
% linordered_field_no_ub
thf(fact_200_linordered__field__no__lb,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [X5: A] :
? [Y3: A] : ( ord_less @ A @ Y3 @ X5 ) ) ).
% linordered_field_no_lb
thf(fact_201_dependent__wellorder__choice,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ A @ ( type2 @ A ) )
=> ! [P2: ( A > B ) > A > B > $o] :
( ! [R4: B,F4: A > B,G3: A > B,X4: A] :
( ! [Y4: A] :
( ( ord_less @ A @ Y4 @ X4 )
=> ( ( F4 @ Y4 )
= ( G3 @ Y4 ) ) )
=> ( ( P2 @ F4 @ X4 @ R4 )
= ( P2 @ G3 @ X4 @ R4 ) ) )
=> ( ! [X4: A,F4: A > B] :
( ! [Y4: A] :
( ( ord_less @ A @ Y4 @ X4 )
=> ( P2 @ F4 @ Y4 @ ( F4 @ Y4 ) ) )
=> ? [X12: B] : ( P2 @ F4 @ X4 @ X12 ) )
=> ? [F4: A > B] :
! [X5: A] : ( P2 @ F4 @ X5 @ ( F4 @ X5 ) ) ) ) ) ).
% dependent_wellorder_choice
thf(fact_202_in__inv__imagep,axiom,
! [B: $tType,A: $tType] :
( ( inv_imagep @ A @ B )
= ( ^ [R2: A > A > $o,F2: B > A,X2: B,Y2: B] : ( R2 @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) ) ) ).
% in_inv_imagep
thf(fact_203_pred__on_Ochain__extend,axiom,
! [A: $tType,A4: set @ A,P2: A > A > $o,C3: set @ A,Z: A] :
( ( pred_chain @ A @ A4 @ P2 @ C3 )
=> ( ( member @ A @ Z @ A4 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ C3 )
=> ( sup_sup @ ( A > A > $o ) @ P2
@ ^ [Y5: A,Z5: A] : Y5 = Z5
@ X4
@ Z ) )
=> ( pred_chain @ A @ A4 @ P2 @ ( sup_sup @ ( set @ A ) @ ( insert @ A @ Z @ ( bot_bot @ ( set @ A ) ) ) @ C3 ) ) ) ) ) ).
% pred_on.chain_extend
thf(fact_204_sup__bot_Ocomm__monoid__axioms,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A @ ( type2 @ A ) )
=> ( comm_monoid @ A @ ( sup_sup @ A ) @ ( bot_bot @ A ) ) ) ).
% sup_bot.comm_monoid_axioms
thf(fact_205_sup2CI,axiom,
! [A: $tType,B: $tType,B4: A > B > $o,X: A,Y: B,A4: A > B > $o] :
( ( ~ ( B4 @ X @ Y )
=> ( A4 @ X @ Y ) )
=> ( sup_sup @ ( A > B > $o ) @ A4 @ B4 @ X @ Y ) ) ).
% sup2CI
thf(fact_206_pred__on_Ochain__empty,axiom,
! [A: $tType,A4: set @ A,P2: A > A > $o] : ( pred_chain @ A @ A4 @ P2 @ ( bot_bot @ ( set @ A ) ) ) ).
% pred_on.chain_empty
thf(fact_207_subset_Ochain__total,axiom,
! [A: $tType,A4: set @ ( set @ A ),C3: set @ ( set @ A ),X: set @ A,Y: set @ A] :
( ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C3 )
=> ( ( member @ ( set @ A ) @ X @ C3 )
=> ( ( member @ ( set @ A ) @ Y @ C3 )
=> ( ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
@ ^ [Y5: set @ A,Z5: set @ A] : Y5 = Z5
@ X
@ Y )
| ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
@ ^ [Y5: set @ A,Z5: set @ A] : Y5 = Z5
@ Y
@ X ) ) ) ) ) ).
% subset.chain_total
thf(fact_208_pred__on_Ochain__total,axiom,
! [A: $tType,A4: set @ A,P2: A > A > $o,C3: set @ A,X: A,Y: A] :
( ( pred_chain @ A @ A4 @ P2 @ C3 )
=> ( ( member @ A @ X @ C3 )
=> ( ( member @ A @ Y @ C3 )
=> ( ( sup_sup @ ( A > A > $o ) @ P2
@ ^ [Y5: A,Z5: A] : Y5 = Z5
@ X
@ Y )
| ( sup_sup @ ( A > A > $o ) @ P2
@ ^ [Y5: A,Z5: A] : Y5 = Z5
@ Y
@ X ) ) ) ) ) ).
% pred_on.chain_total
thf(fact_209_subset_Ochain__empty,axiom,
! [A: $tType,A4: set @ ( set @ A )] : ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ).
% subset.chain_empty
thf(fact_210_subset_Ochain__extend,axiom,
! [A: $tType,A4: set @ ( set @ A ),C3: set @ ( set @ A ),Z: set @ A] :
( ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C3 )
=> ( ( member @ ( set @ A ) @ Z @ A4 )
=> ( ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ C3 )
=> ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
@ ^ [Y5: set @ A,Z5: set @ A] : Y5 = Z5
@ X4
@ Z ) )
=> ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ ( sup_sup @ ( set @ ( set @ A ) ) @ ( insert @ ( set @ A ) @ Z @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) @ C3 ) ) ) ) ) ).
% subset.chain_extend
thf(fact_211_sup2I2,axiom,
! [A: $tType,B: $tType,B4: A > B > $o,X: A,Y: B,A4: A > B > $o] :
( ( B4 @ X @ Y )
=> ( sup_sup @ ( A > B > $o ) @ A4 @ B4 @ X @ Y ) ) ).
% sup2I2
thf(fact_212_sup2I1,axiom,
! [A: $tType,B: $tType,A4: A > B > $o,X: A,Y: B,B4: A > B > $o] :
( ( A4 @ X @ Y )
=> ( sup_sup @ ( A > B > $o ) @ A4 @ B4 @ X @ Y ) ) ).
% sup2I1
thf(fact_213_sup2E,axiom,
! [A: $tType,B: $tType,A4: A > B > $o,B4: A > B > $o,X: A,Y: B] :
( ( sup_sup @ ( A > B > $o ) @ A4 @ B4 @ X @ Y )
=> ( ~ ( A4 @ X @ Y )
=> ( B4 @ X @ Y ) ) ) ).
% sup2E
thf(fact_214_semilattice__neutr_Oaxioms_I2_J,axiom,
! [A: $tType,F: A > A > A,Z: A] :
( ( semilattice_neutr @ A @ F @ Z )
=> ( comm_monoid @ A @ F @ Z ) ) ).
% semilattice_neutr.axioms(2)
thf(fact_215_reflclp__idemp,axiom,
! [A: $tType,P2: A > A > $o] :
( ( sup_sup @ ( A > A > $o )
@ ( sup_sup @ ( A > A > $o ) @ P2
@ ^ [Y5: A,Z5: A] : Y5 = Z5 )
@ ^ [Y5: A,Z5: A] : Y5 = Z5 )
= ( sup_sup @ ( A > A > $o ) @ P2
@ ^ [Y5: A,Z5: A] : Y5 = Z5 ) ) ).
% reflclp_idemp
thf(fact_216_pred__on_Omaxchain__def,axiom,
! [A: $tType] :
( ( pred_maxchain @ A )
= ( ^ [A5: set @ A,P4: A > A > $o,C4: set @ A] :
( ( pred_chain @ A @ A5 @ P4 @ C4 )
& ~ ? [S2: set @ A] :
( ( pred_chain @ A @ A5 @ P4 @ S2 )
& ( ord_less @ ( set @ A ) @ C4 @ S2 ) ) ) ) ) ).
% pred_on.maxchain_def
thf(fact_217_subset_Omaxchain__def,axiom,
! [A: $tType,A4: set @ ( set @ A ),C3: set @ ( set @ A )] :
( ( pred_maxchain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C3 )
= ( ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C3 )
& ~ ? [S2: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ S2 )
& ( ord_less @ ( set @ ( set @ A ) ) @ C3 @ S2 ) ) ) ) ).
% subset.maxchain_def
thf(fact_218_subset_Omaxchain__imp__chain,axiom,
! [A: $tType,A4: set @ ( set @ A ),C3: set @ ( set @ A )] :
( ( pred_maxchain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C3 )
=> ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C3 ) ) ).
% subset.maxchain_imp_chain
thf(fact_219_subset_OHausdorff,axiom,
! [A: $tType,A4: set @ ( set @ A )] :
? [X1: set @ ( set @ A )] : ( pred_maxchain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ X1 ) ).
% subset.Hausdorff
thf(fact_220_Pow__singleton__iff,axiom,
! [A: $tType,X6: set @ A,Y6: set @ A] :
( ( ( pow @ A @ X6 )
= ( insert @ ( set @ A ) @ Y6 @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) )
= ( ( X6
= ( bot_bot @ ( set @ A ) ) )
& ( Y6
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Pow_singleton_iff
thf(fact_221_Pow__empty,axiom,
! [A: $tType] :
( ( pow @ A @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ).
% Pow_empty
thf(fact_222_Pow__bottom,axiom,
! [A: $tType,B4: set @ A] : ( member @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( pow @ A @ B4 ) ) ).
% Pow_bottom
thf(fact_223_Pow__not__empty,axiom,
! [A: $tType,A4: set @ A] :
( ( pow @ A @ A4 )
!= ( bot_bot @ ( set @ ( set @ A ) ) ) ) ).
% Pow_not_empty
thf(fact_224_Pow__top,axiom,
! [A: $tType,A4: set @ A] : ( member @ ( set @ A ) @ A4 @ ( pow @ A @ A4 ) ) ).
% Pow_top
thf(fact_225_subset_Ochain__def,axiom,
! [A: $tType,A4: set @ ( set @ A ),C3: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C3 )
= ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ C3 @ A4 )
& ! [X2: set @ A] :
( ( member @ ( set @ A ) @ X2 @ C3 )
=> ! [Y2: set @ A] :
( ( member @ ( set @ A ) @ Y2 @ C3 )
=> ( ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
@ ^ [Y5: set @ A,Z5: set @ A] : Y5 = Z5
@ X2
@ Y2 )
| ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
@ ^ [Y5: set @ A,Z5: set @ A] : Y5 = Z5
@ Y2
@ X2 ) ) ) ) ) ) ).
% subset.chain_def
thf(fact_226_subset_OchainI,axiom,
! [A: $tType,C3: set @ ( set @ A ),A4: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ C3 @ A4 )
=> ( ! [X4: set @ A,Y3: set @ A] :
( ( member @ ( set @ A ) @ X4 @ C3 )
=> ( ( member @ ( set @ A ) @ Y3 @ C3 )
=> ( ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
@ ^ [Y5: set @ A,Z5: set @ A] : Y5 = Z5
@ X4
@ Y3 )
| ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
@ ^ [Y5: set @ A,Z5: set @ A] : Y5 = Z5
@ Y3
@ X4 ) ) ) )
=> ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C3 ) ) ) ).
% subset.chainI
thf(fact_227_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_228_subsetI,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ! [X4: A] :
( ( member @ A @ X4 @ A4 )
=> ( member @ A @ X4 @ B4 ) )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ).
% subsetI
thf(fact_229_subset__antisym,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ A4 )
=> ( A4 = B4 ) ) ) ).
% subset_antisym
thf(fact_230_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_231_sup_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B3: A,C: A,A2: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B3 @ C ) @ A2 )
= ( ( ord_less_eq @ A @ B3 @ A2 )
& ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% sup.bounded_iff
thf(fact_232_empty__subsetI,axiom,
! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A4 ) ).
% empty_subsetI
thf(fact_233_subset__empty,axiom,
! [A: $tType,A4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
= ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_empty
thf(fact_234_insert__subset,axiom,
! [A: $tType,X: A,A4: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A4 ) @ B4 )
= ( ( member @ A @ X @ B4 )
& ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ) ).
% insert_subset
thf(fact_235_Un__subset__iff,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) @ C3 )
= ( ( ord_less_eq @ ( set @ A ) @ A4 @ C3 )
& ( ord_less_eq @ ( set @ A ) @ B4 @ C3 ) ) ) ).
% Un_subset_iff
thf(fact_236_psubsetI,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ( A4 != B4 )
=> ( ord_less @ ( set @ A ) @ A4 @ B4 ) ) ) ).
% psubsetI
thf(fact_237_Pow__iff,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( member @ ( set @ A ) @ A4 @ ( pow @ A @ B4 ) )
= ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ).
% Pow_iff
thf(fact_238_PowI,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( member @ ( set @ A ) @ A4 @ ( pow @ A @ B4 ) ) ) ).
% PowI
thf(fact_239_singleton__insert__inj__eq,axiom,
! [A: $tType,B3: A,A2: A,A4: set @ A] :
( ( ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ A2 @ A4 ) )
= ( ( A2 = B3 )
& ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_240_singleton__insert__inj__eq_H,axiom,
! [A: $tType,A2: A,A4: set @ A,B3: A] :
( ( ( insert @ A @ A2 @ A4 )
= ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A2 = B3 )
& ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_241_subset__Zorn,axiom,
! [A: $tType,A4: set @ ( set @ A )] :
( ! [C5: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A4 @ ( ord_less @ ( set @ A ) ) @ C5 )
=> ? [X5: set @ A] :
( ( member @ ( set @ A ) @ X5 @ A4 )
& ! [Xa: set @ A] :
( ( member @ ( set @ A ) @ Xa @ C5 )
=> ( ord_less_eq @ ( set @ A ) @ Xa @ X5 ) ) ) )
=> ? [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ A4 )
& ! [Xa2: set @ A] :
( ( member @ ( set @ A ) @ Xa2 @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ X4 @ Xa2 )
=> ( Xa2 = X4 ) ) ) ) ) ).
% subset_Zorn
thf(fact_242_psubsetE,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( ord_less @ ( set @ A ) @ A4 @ B4 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ) ).
% psubsetE
thf(fact_243_psubset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ).
% psubset_eq
thf(fact_244_psubset__imp__subset,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( ord_less @ ( set @ A ) @ A4 @ B4 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ).
% psubset_imp_subset
thf(fact_245_psubset__subset__trans,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,C3: set @ A] :
( ( ord_less @ ( set @ A ) @ A4 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ C3 )
=> ( ord_less @ ( set @ A ) @ A4 @ C3 ) ) ) ).
% psubset_subset_trans
thf(fact_246_subset__not__subset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
& ~ ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_247_subset__psubset__trans,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ( ord_less @ ( set @ A ) @ B4 @ C3 )
=> ( ord_less @ ( set @ A ) @ A4 @ C3 ) ) ) ).
% subset_psubset_trans
thf(fact_248_subset__iff__psubset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A5: set @ A,B5: set @ A] :
( ( ord_less @ ( set @ A ) @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_249_pinf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ~ ( ord_less_eq @ A @ X5 @ T ) ) ) ).
% pinf(6)
thf(fact_250_pinf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ( ord_less_eq @ A @ T @ X5 ) ) ) ).
% pinf(8)
thf(fact_251_minf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ( ord_less_eq @ A @ X5 @ T ) ) ) ).
% minf(6)
thf(fact_252_minf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ~ ( ord_less_eq @ A @ T @ X5 ) ) ) ).
% minf(8)
thf(fact_253_order_Onot__eq__order__implies__strict,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( A2 != B3 )
=> ( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ord_less @ A @ A2 @ B3 ) ) ) ) ).
% order.not_eq_order_implies_strict
thf(fact_254_dual__order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A] :
( ( ord_less @ A @ B3 @ A2 )
=> ( ord_less_eq @ A @ B3 @ A2 ) ) ) ).
% dual_order.strict_implies_order
thf(fact_255_dual__order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ) ).
% dual_order.strict_iff_order
%----Type constructors (34)
thf(tcon_HOL_Obool___Lattices_Obounded__lattice,axiom,
bounded_lattice @ $o @ ( type2 @ $o ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice_1,axiom,
! [A6: $tType] : ( bounded_lattice @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice_2,axiom,
! [A6: $tType,A7: $tType] :
( ( bounded_lattice @ A7 @ ( type2 @ A7 ) )
=> ( bounded_lattice @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
! [A6: $tType,A7: $tType] :
( ( bounded_lattice @ A7 @ ( type2 @ A7 ) )
=> ( bounde1808546759up_bot @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice__bot,axiom,
! [A6: $tType,A7: $tType] :
( ( bounded_lattice @ A7 @ ( type2 @ A7 ) )
=> ( bounded_lattice_bot @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A6: $tType,A7: $tType] :
( ( semilattice_sup @ A7 @ ( type2 @ A7 ) )
=> ( semilattice_sup @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder__bot,axiom,
! [A6: $tType,A7: $tType] :
( ( order_bot @ A7 @ ( type2 @ A7 ) )
=> ( order_bot @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A6: $tType,A7: $tType] :
( ( preorder @ A7 @ ( type2 @ A7 ) )
=> ( preorder @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Lattices_Olattice,axiom,
! [A6: $tType,A7: $tType] :
( ( lattice @ A7 @ ( type2 @ A7 ) )
=> ( lattice @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A6: $tType,A7: $tType] :
( ( order @ A7 @ ( type2 @ A7 ) )
=> ( order @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A6: $tType,A7: $tType] :
( ( ord @ A7 @ ( type2 @ A7 ) )
=> ( ord @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Orderings_Obot,axiom,
! [A6: $tType,A7: $tType] :
( ( bot @ A7 @ ( type2 @ A7 ) )
=> ( bot @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_3,axiom,
! [A6: $tType] : ( bounde1808546759up_bot @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice__bot_4,axiom,
! [A6: $tType] : ( bounded_lattice_bot @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__sup_5,axiom,
! [A6: $tType] : ( semilattice_sup @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_6,axiom,
! [A6: $tType] : ( order_bot @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_7,axiom,
! [A6: $tType] : ( preorder @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Olattice_8,axiom,
! [A6: $tType] : ( lattice @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_9,axiom,
! [A6: $tType] : ( order @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_10,axiom,
! [A6: $tType] : ( ord @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_11,axiom,
! [A6: $tType] : ( bot @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_12,axiom,
bounde1808546759up_bot @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_13,axiom,
bounded_lattice_bot @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_14,axiom,
semilattice_sup @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_15,axiom,
order_bot @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_16,axiom,
preorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Lattices_Olattice_17,axiom,
lattice @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_18,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_19,axiom,
ord @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Obot_20,axiom,
bot @ $o @ ( type2 @ $o ) ).
thf(tcon_BinDag__Mirabelle__rybootvolr_Odag___Orderings_Opreorder_21,axiom,
preorder @ binDag_Mirabelle_dag @ ( type2 @ binDag_Mirabelle_dag ) ).
thf(tcon_BinDag__Mirabelle__rybootvolr_Odag___Orderings_Oorder_22,axiom,
order @ binDag_Mirabelle_dag @ ( type2 @ binDag_Mirabelle_dag ) ).
thf(tcon_BinDag__Mirabelle__rybootvolr_Odag___Orderings_Oord_23,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 (1)
thf(conj_0,conjecture,
( ( binDag_Mirabelle_Dag @ p @ ( fun_upd @ simpl_ref @ simpl_ref @ l @ p @ p ) @ r @ t )
= ( ( p = simpl_Null )
& ( t = binDag_Mirabelle_Tip ) ) ) ).
%------------------------------------------------------------------------------