TPTP Problem File: COM169^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : COM169^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Computing Theory
% Problem : Binary decision diagram 318
% 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__318.p [Bla16]
% Status : Theorem
% Rating : 0.00 v8.1.0, 0.25 v7.5.0, 0.00 v7.2.0, 0.25 v7.1.0
% Syntax : Number of formulae : 349 ( 88 unt; 58 typ; 0 def)
% Number of atoms : 920 ( 216 equ; 0 cnn)
% Maximal formula atoms : 13 ( 3 avg)
% Number of connectives : 3415 ( 105 ~; 27 |; 53 &;2778 @)
% ( 0 <=>; 452 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 8 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 240 ( 240 >; 0 *; 0 +; 0 <<)
% Number of symbols : 58 ( 55 usr; 9 con; 0-6 aty)
% Number of variables : 1062 ( 60 ^; 921 !; 42 ?;1062 :)
% ( 39 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:46:08.355
%------------------------------------------------------------------------------
%----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 (52)
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_Rings_Olinordered__semidom,type,
linordered_semidom:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni770627133id_add:
!>[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_Odag_Osize__dag,type,
binDag1924123185ze_dag: binDag_Mirabelle_dag > nat ).
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_Osgn__class_Osgn,type,
sgn_sgn:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_List_Olist_Oset,type,
set2:
!>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
thf(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : ( A > nat ) ).
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_Orderings_Oorder__class_Ostrict__mono,type,
order_strict_mono:
!>[A: $tType,B: $tType] : ( ( A > B ) > $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_Ois__empty,type,
is_empty:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
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,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_l,type,
l: simpl_ref > simpl_ref ).
thf(sy_v_lt____,type,
lt: binDag_Mirabelle_dag ).
thf(sy_v_p,type,
p: simpl_ref ).
thf(sy_v_r,type,
r: simpl_ref > simpl_ref ).
thf(sy_v_rt____,type,
rt: binDag_Mirabelle_dag ).
thf(sy_v_t,type,
t: binDag_Mirabelle_dag ).
thf(sy_v_thesis____,type,
thesis: $o ).
%----Relevant facts (256)
thf(fact_0_assms,axiom,
binDag_Mirabelle_Dag @ ( l @ p ) @ l @ r @ t ).
% assms
thf(fact_1_calculation,axiom,
! [Lt: binDag_Mirabelle_dag,Rt: binDag_Mirabelle_dag] :
( t
!= ( binDag476092410e_Node @ Lt @ p @ Rt ) ) ).
% calculation
thf(fact_2_subdag,axiom,
binDag786255756subdag @ t @ ( binDag476092410e_Node @ lt @ p @ rt ) ).
% subdag
thf(fact_3_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_4_Dag__unique,axiom,
! [P: simpl_ref,L: simpl_ref > simpl_ref,R: simpl_ref > simpl_ref,T1: binDag_Mirabelle_dag,T2: binDag_Mirabelle_dag] :
( ( binDag_Mirabelle_Dag @ P @ L @ R @ T1 )
=> ( ( binDag_Mirabelle_Dag @ P @ L @ R @ T2 )
=> ( T1 = T2 ) ) ) ).
% Dag_unique
thf(fact_5_Dag__unique1,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 )
=> ? [X: binDag_Mirabelle_dag] :
( ( binDag_Mirabelle_Dag @ P @ L @ R @ X )
& ! [Y: binDag_Mirabelle_dag] :
( ( binDag_Mirabelle_Dag @ P @ L @ R @ Y )
=> ( Y = X ) ) ) ) ).
% Dag_unique1
thf(fact_6_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_7_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_8_Dag__subdag,axiom,
! [P: simpl_ref,L: simpl_ref > simpl_ref,R: simpl_ref > simpl_ref,T: binDag_Mirabelle_dag,S: binDag_Mirabelle_dag] :
( ( binDag_Mirabelle_Dag @ P @ L @ R @ T )
=> ( ( binDag786255756subdag @ T @ S )
=> ? [Q: simpl_ref] : ( binDag_Mirabelle_Dag @ Q @ L @ R @ S ) ) ) ).
% Dag_subdag
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 )
= ( ? [Lt3: binDag_Mirabelle_dag,Rt3: binDag_Mirabelle_dag] :
( ( T
= ( binDag476092410e_Node @ Lt3 @ P @ Rt3 ) )
& ( binDag_Mirabelle_Dag @ ( L @ P ) @ L @ R @ Lt3 )
& ( binDag_Mirabelle_Dag @ ( R @ P ) @ L @ R @ Rt3 ) ) ) ) ) ).
% 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_dag_Osimps_I7_J,axiom,
! [A: $tType,F1: A,F2: 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 @ F2 @ ( binDag476092410e_Node @ X21 @ X22 @ X23 ) )
= ( F2 @ X21 @ X22 @ X23 @ ( binDag1442713106ec_dag @ A @ F1 @ F2 @ X21 ) @ ( binDag1442713106ec_dag @ A @ F1 @ F2 @ X23 ) ) ) ).
% dag.simps(7)
thf(fact_12_dag_Osimps_I5_J,axiom,
! [A: $tType,F1: A,F2: binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > A,X21: binDag_Mirabelle_dag,X22: simpl_ref,X23: binDag_Mirabelle_dag] :
( ( binDag1297733282se_dag @ A @ F1 @ F2 @ ( binDag476092410e_Node @ X21 @ X22 @ X23 ) )
= ( F2 @ X21 @ X22 @ X23 ) ) ).
% dag.simps(5)
thf(fact_13_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_14_dag_Oinduct,axiom,
! [P2: binDag_Mirabelle_dag > $o,Dag: binDag_Mirabelle_dag] :
( ( P2 @ binDag_Mirabelle_Tip )
=> ( ! [X1: binDag_Mirabelle_dag,X2: simpl_ref,X3: binDag_Mirabelle_dag] :
( ( P2 @ X1 )
=> ( ( P2 @ X3 )
=> ( P2 @ ( binDag476092410e_Node @ X1 @ X2 @ X3 ) ) ) )
=> ( P2 @ Dag ) ) ) ).
% dag.induct
thf(fact_15_dag_Oexhaust,axiom,
! [Y2: binDag_Mirabelle_dag] :
( ( Y2 != binDag_Mirabelle_Tip )
=> ~ ! [X212: binDag_Mirabelle_dag,X222: simpl_ref,X232: binDag_Mirabelle_dag] :
( Y2
!= ( binDag476092410e_Node @ X212 @ X222 @ X232 ) ) ) ).
% dag.exhaust
thf(fact_16_subdag__neq,axiom,
! [T: binDag_Mirabelle_dag,S: binDag_Mirabelle_dag] :
( ( binDag786255756subdag @ T @ S )
=> ( T != S ) ) ).
% subdag_neq
thf(fact_17_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_18_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_19_dag_Osimps_I6_J,axiom,
! [A: $tType,F1: A,F2: binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > A > A > A] :
( ( binDag1442713106ec_dag @ A @ F1 @ F2 @ binDag_Mirabelle_Tip )
= F1 ) ).
% dag.simps(6)
thf(fact_20_dag_Osimps_I4_J,axiom,
! [A: $tType,F1: A,F2: binDag_Mirabelle_dag > simpl_ref > binDag_Mirabelle_dag > A] :
( ( binDag1297733282se_dag @ A @ F1 @ F2 @ binDag_Mirabelle_Tip )
= F1 ) ).
% dag.simps(4)
thf(fact_21_subdag_Osimps_I1_J,axiom,
! [T: binDag_Mirabelle_dag] :
~ ( binDag786255756subdag @ binDag_Mirabelle_Tip @ T ) ).
% subdag.simps(1)
thf(fact_22_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_23_subdag__not__sym,axiom,
! [S: binDag_Mirabelle_dag,T: binDag_Mirabelle_dag] :
( ( binDag786255756subdag @ S @ T )
=> ~ ( binDag786255756subdag @ T @ S ) ) ).
% subdag_not_sym
thf(fact_24_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_25_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_26_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_27_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_28_DAG_Osimps_I1_J,axiom,
binDag_Mirabelle_DAG @ binDag_Mirabelle_Tip ).
% DAG.simps(1)
thf(fact_29_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_30_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_31_less__dag__Tip,axiom,
! [X4: binDag_Mirabelle_dag] :
~ ( ord_less @ binDag_Mirabelle_dag @ X4 @ binDag_Mirabelle_Tip ) ).
% less_dag_Tip
thf(fact_32_less__dag__def,axiom,
( ( ord_less @ binDag_Mirabelle_dag )
= ( ^ [S2: binDag_Mirabelle_dag,T3: binDag_Mirabelle_dag] : ( binDag786255756subdag @ T3 @ S2 ) ) ) ).
% less_dag_def
thf(fact_33_less__dag__Node_H,axiom,
! [X4: binDag_Mirabelle_dag,L: binDag_Mirabelle_dag,A2: simpl_ref,R: binDag_Mirabelle_dag] :
( ( ord_less @ binDag_Mirabelle_dag @ X4 @ ( binDag476092410e_Node @ L @ A2 @ R ) )
= ( ( X4 = L )
| ( X4 = R )
| ( ord_less @ binDag_Mirabelle_dag @ X4 @ L )
| ( ord_less @ binDag_Mirabelle_dag @ X4 @ R ) ) ) ).
% less_dag_Node'
thf(fact_34_Dag__update__lI,axiom,
! [P: simpl_ref,L: simpl_ref > simpl_ref,R: simpl_ref > simpl_ref,T: binDag_Mirabelle_dag,Q2: simpl_ref,Y2: simpl_ref] :
( ( binDag_Mirabelle_Dag @ P @ L @ R @ T )
=> ( ~ ( member @ simpl_ref @ Q2 @ ( binDag1380252983set_of @ T ) )
=> ( binDag_Mirabelle_Dag @ P @ ( fun_upd @ simpl_ref @ simpl_ref @ L @ Q2 @ Y2 ) @ R @ T ) ) ) ).
% Dag_update_lI
thf(fact_35_Dag__update__rI,axiom,
! [P: simpl_ref,L: simpl_ref > simpl_ref,R: simpl_ref > simpl_ref,T: binDag_Mirabelle_dag,Q2: simpl_ref,Y2: simpl_ref] :
( ( binDag_Mirabelle_Dag @ P @ L @ R @ T )
=> ( ~ ( member @ simpl_ref @ Q2 @ ( binDag1380252983set_of @ T ) )
=> ( binDag_Mirabelle_Dag @ P @ L @ ( fun_upd @ simpl_ref @ simpl_ref @ R @ Q2 @ Y2 ) @ T ) ) ) ).
% Dag_update_rI
thf(fact_36_notin__Dag__update__l,axiom,
! [Q2: simpl_ref,T: binDag_Mirabelle_dag,P: simpl_ref,L: simpl_ref > simpl_ref,Y2: simpl_ref,R: simpl_ref > simpl_ref] :
( ~ ( member @ simpl_ref @ Q2 @ ( binDag1380252983set_of @ T ) )
=> ( ( binDag_Mirabelle_Dag @ P @ ( fun_upd @ simpl_ref @ simpl_ref @ L @ Q2 @ Y2 ) @ R @ T )
= ( binDag_Mirabelle_Dag @ P @ L @ R @ T ) ) ) ).
% notin_Dag_update_l
thf(fact_37_notin__Dag__update__r,axiom,
! [Q2: simpl_ref,T: binDag_Mirabelle_dag,P: simpl_ref,L: simpl_ref > simpl_ref,R: simpl_ref > simpl_ref,Y2: simpl_ref] :
( ~ ( member @ simpl_ref @ Q2 @ ( binDag1380252983set_of @ T ) )
=> ( ( binDag_Mirabelle_Dag @ P @ L @ ( fun_upd @ simpl_ref @ simpl_ref @ R @ Q2 @ Y2 ) @ T )
= ( binDag_Mirabelle_Dag @ P @ L @ R @ T ) ) ) ).
% notin_Dag_update_r
thf(fact_38_DAG__less,axiom,
! [Y2: binDag_Mirabelle_dag,X4: binDag_Mirabelle_dag] :
( ( binDag_Mirabelle_DAG @ Y2 )
=> ( ( ord_less @ binDag_Mirabelle_dag @ X4 @ Y2 )
=> ( binDag_Mirabelle_DAG @ X4 ) ) ) ).
% DAG_less
thf(fact_39_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_40_less__Node__dag,axiom,
! [L: binDag_Mirabelle_dag,A2: simpl_ref,R: binDag_Mirabelle_dag,X4: binDag_Mirabelle_dag] :
( ( ord_less @ binDag_Mirabelle_dag @ ( binDag476092410e_Node @ L @ A2 @ R ) @ X4 )
=> ( ( ord_less @ binDag_Mirabelle_dag @ L @ X4 )
& ( ord_less @ binDag_Mirabelle_dag @ R @ X4 ) ) ) ).
% less_Node_dag
thf(fact_41_fun__upd__apply,axiom,
! [A: $tType,B: $tType] :
( ( fun_upd @ B @ A )
= ( ^ [F: B > A,X5: B,Y3: A,Z: B] : ( if @ A @ ( Z = X5 ) @ Y3 @ ( F @ Z ) ) ) ) ).
% fun_upd_apply
thf(fact_42_fun__upd__triv,axiom,
! [B: $tType,A: $tType,F3: A > B,X4: A] :
( ( fun_upd @ A @ B @ F3 @ X4 @ ( F3 @ X4 ) )
= F3 ) ).
% fun_upd_triv
thf(fact_43_fun__upd__upd,axiom,
! [A: $tType,B: $tType,F3: A > B,X4: A,Y2: B,Z2: B] :
( ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ F3 @ X4 @ Y2 ) @ X4 @ Z2 )
= ( fun_upd @ A @ B @ F3 @ X4 @ Z2 ) ) ).
% fun_upd_upd
thf(fact_44_less__DAG__set__of,axiom,
! [X4: binDag_Mirabelle_dag,Y2: binDag_Mirabelle_dag] :
( ( ord_less @ binDag_Mirabelle_dag @ X4 @ Y2 )
=> ( ( binDag_Mirabelle_DAG @ Y2 )
=> ( ord_less @ ( set @ simpl_ref ) @ ( binDag1380252983set_of @ X4 ) @ ( binDag1380252983set_of @ Y2 ) ) ) ) ).
% less_DAG_set_of
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,A3: set @ A] :
( ( collect @ A
@ ^ [X5: A] : ( member @ A @ X5 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P2: A > $o,Q3: A > $o] :
( ! [X: A] :
( ( P2 @ X )
= ( Q3 @ X ) )
=> ( ( collect @ A @ P2 )
= ( collect @ A @ Q3 ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F3: A > B,G: A > B] :
( ! [X: A] :
( ( F3 @ X )
= ( G @ X ) )
=> ( F3 = G ) ) ).
% ext
thf(fact_49_fun__upd__idem__iff,axiom,
! [A: $tType,B: $tType,F3: A > B,X4: A,Y2: B] :
( ( ( fun_upd @ A @ B @ F3 @ X4 @ Y2 )
= F3 )
= ( ( F3 @ X4 )
= Y2 ) ) ).
% fun_upd_idem_iff
thf(fact_50_fun__upd__twist,axiom,
! [A: $tType,B: $tType,A2: A,C: A,M: A > B,B2: B,D: B] :
( ( A2 != C )
=> ( ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ M @ A2 @ B2 ) @ C @ D )
= ( fun_upd @ A @ B @ ( fun_upd @ A @ B @ M @ C @ D ) @ A2 @ B2 ) ) ) ).
% fun_upd_twist
thf(fact_51_fun__upd__other,axiom,
! [B: $tType,A: $tType,Z2: A,X4: A,F3: A > B,Y2: B] :
( ( Z2 != X4 )
=> ( ( fun_upd @ A @ B @ F3 @ X4 @ Y2 @ Z2 )
= ( F3 @ Z2 ) ) ) ).
% fun_upd_other
thf(fact_52_fun__upd__same,axiom,
! [B: $tType,A: $tType,F3: B > A,X4: B,Y2: A] :
( ( fun_upd @ B @ A @ F3 @ X4 @ Y2 @ X4 )
= Y2 ) ).
% fun_upd_same
thf(fact_53_fun__upd__idem,axiom,
! [A: $tType,B: $tType,F3: B > A,X4: B,Y2: A] :
( ( ( F3 @ X4 )
= Y2 )
=> ( ( fun_upd @ B @ A @ F3 @ X4 @ Y2 )
= F3 ) ) ).
% fun_upd_idem
thf(fact_54_fun__upd__def,axiom,
! [B: $tType,A: $tType] :
( ( fun_upd @ A @ B )
= ( ^ [F: A > B,A4: A,B3: B,X5: A] : ( if @ B @ ( X5 = A4 ) @ B3 @ ( F @ X5 ) ) ) ) ).
% fun_upd_def
thf(fact_55_fun__upd__eqD,axiom,
! [A: $tType,B: $tType,F3: A > B,X4: A,Y2: B,G: A > B,Z2: B] :
( ( ( fun_upd @ A @ B @ F3 @ X4 @ Y2 )
= ( fun_upd @ A @ B @ G @ X4 @ Z2 ) )
=> ( Y2 = Z2 ) ) ).
% fun_upd_eqD
thf(fact_56_psubset__trans,axiom,
! [A: $tType,A3: set @ A,B4: set @ A,C2: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ B4 )
=> ( ( ord_less @ ( set @ A ) @ B4 @ C2 )
=> ( ord_less @ ( set @ A ) @ A3 @ C2 ) ) ) ).
% psubset_trans
thf(fact_57_psubsetD,axiom,
! [A: $tType,A3: set @ A,B4: set @ A,C: A] :
( ( ord_less @ ( set @ A ) @ A3 @ B4 )
=> ( ( member @ A @ C @ A3 )
=> ( member @ A @ C @ B4 ) ) ) ).
% psubsetD
thf(fact_58_minf_I11_J,axiom,
! [C3: $tType,D2: $tType] :
( ( ord @ C3 @ ( type2 @ C3 ) )
=> ! [F4: D2] :
? [Z3: C3] :
! [X6: C3] :
( ( ord_less @ C3 @ X6 @ Z3 )
=> ( F4 = F4 ) ) ) ).
% minf(11)
thf(fact_59_minf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z3: A] :
! [X6: A] :
( ( ord_less @ A @ X6 @ Z3 )
=> ~ ( ord_less @ A @ T @ X6 ) ) ) ).
% minf(7)
thf(fact_60_minf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z3: A] :
! [X6: A] :
( ( ord_less @ A @ X6 @ Z3 )
=> ( ord_less @ A @ X6 @ T ) ) ) ).
% minf(5)
thf(fact_61_minf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z3: A] :
! [X6: A] :
( ( ord_less @ A @ X6 @ Z3 )
=> ( X6 != T ) ) ) ).
% minf(4)
thf(fact_62_minf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z3: A] :
! [X6: A] :
( ( ord_less @ A @ X6 @ Z3 )
=> ( X6 != T ) ) ) ).
% minf(3)
thf(fact_63_minf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P2: A > $o,P3: A > $o,Q3: A > $o,Q4: A > $o] :
( ? [Z4: A] :
! [X: A] :
( ( ord_less @ A @ X @ Z4 )
=> ( ( P2 @ X )
= ( P3 @ X ) ) )
=> ( ? [Z4: A] :
! [X: A] :
( ( ord_less @ A @ X @ Z4 )
=> ( ( Q3 @ X )
= ( Q4 @ X ) ) )
=> ? [Z3: A] :
! [X6: A] :
( ( ord_less @ A @ X6 @ Z3 )
=> ( ( ( P2 @ X6 )
| ( Q3 @ X6 ) )
= ( ( P3 @ X6 )
| ( Q4 @ X6 ) ) ) ) ) ) ) ).
% minf(2)
thf(fact_64_pinf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P2: A > $o,P3: A > $o,Q3: A > $o,Q4: A > $o] :
( ? [Z4: A] :
! [X: A] :
( ( ord_less @ A @ Z4 @ X )
=> ( ( P2 @ X )
= ( P3 @ X ) ) )
=> ( ? [Z4: A] :
! [X: A] :
( ( ord_less @ A @ Z4 @ X )
=> ( ( Q3 @ X )
= ( Q4 @ X ) ) )
=> ? [Z3: A] :
! [X6: A] :
( ( ord_less @ A @ Z3 @ X6 )
=> ( ( ( P2 @ X6 )
& ( Q3 @ X6 ) )
= ( ( P3 @ X6 )
& ( Q4 @ X6 ) ) ) ) ) ) ) ).
% pinf(1)
thf(fact_65_pinf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P2: A > $o,P3: A > $o,Q3: A > $o,Q4: A > $o] :
( ? [Z4: A] :
! [X: A] :
( ( ord_less @ A @ Z4 @ X )
=> ( ( P2 @ X )
= ( P3 @ X ) ) )
=> ( ? [Z4: A] :
! [X: A] :
( ( ord_less @ A @ Z4 @ X )
=> ( ( Q3 @ X )
= ( Q4 @ X ) ) )
=> ? [Z3: A] :
! [X6: A] :
( ( ord_less @ A @ Z3 @ X6 )
=> ( ( ( P2 @ X6 )
| ( Q3 @ X6 ) )
= ( ( P3 @ X6 )
| ( Q4 @ X6 ) ) ) ) ) ) ) ).
% pinf(2)
thf(fact_66_pinf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z3: A] :
! [X6: A] :
( ( ord_less @ A @ Z3 @ X6 )
=> ( X6 != T ) ) ) ).
% pinf(3)
thf(fact_67_pinf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z3: A] :
! [X6: A] :
( ( ord_less @ A @ Z3 @ X6 )
=> ( X6 != T ) ) ) ).
% pinf(4)
thf(fact_68_pinf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z3: A] :
! [X6: A] :
( ( ord_less @ A @ Z3 @ X6 )
=> ~ ( ord_less @ A @ X6 @ T ) ) ) ).
% pinf(5)
thf(fact_69_pinf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z3: A] :
! [X6: A] :
( ( ord_less @ A @ Z3 @ X6 )
=> ( ord_less @ A @ T @ X6 ) ) ) ).
% pinf(7)
thf(fact_70_pinf_I11_J,axiom,
! [C3: $tType,D2: $tType] :
( ( ord @ C3 @ ( type2 @ C3 ) )
=> ! [F4: D2] :
? [Z3: C3] :
! [X6: C3] :
( ( ord_less @ C3 @ Z3 @ X6 )
=> ( F4 = F4 ) ) ) ).
% pinf(11)
thf(fact_71_minf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P2: A > $o,P3: A > $o,Q3: A > $o,Q4: A > $o] :
( ? [Z4: A] :
! [X: A] :
( ( ord_less @ A @ X @ Z4 )
=> ( ( P2 @ X )
= ( P3 @ X ) ) )
=> ( ? [Z4: A] :
! [X: A] :
( ( ord_less @ A @ X @ Z4 )
=> ( ( Q3 @ X )
= ( Q4 @ X ) ) )
=> ? [Z3: A] :
! [X6: A] :
( ( ord_less @ A @ X6 @ Z3 )
=> ( ( ( P2 @ X6 )
& ( Q3 @ X6 ) )
= ( ( P3 @ X6 )
& ( Q4 @ X6 ) ) ) ) ) ) ) ).
% minf(1)
thf(fact_72_ex__gt__or__lt,axiom,
! [A: $tType] :
( ( condit1656338222tinuum @ A @ ( type2 @ A ) )
=> ! [A2: A] :
? [B5: A] :
( ( ord_less @ A @ A2 @ B5 )
| ( ord_less @ A @ B5 @ A2 ) ) ) ).
% ex_gt_or_lt
thf(fact_73_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type2 @ A ) )
=> ! [X4: A,Y2: A] :
( ( X4 != Y2 )
=> ( ~ ( ord_less @ A @ X4 @ Y2 )
=> ( ord_less @ A @ Y2 @ X4 ) ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_74_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( A2 != B2 ) ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_75_linordered__field__no__ub,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [X6: A] :
? [X1: A] : ( ord_less @ A @ X6 @ X1 ) ) ).
% linordered_field_no_ub
thf(fact_76_linordered__field__no__lb,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [X6: A] :
? [Y4: A] : ( ord_less @ A @ Y4 @ X6 ) ) ).
% linordered_field_no_lb
thf(fact_77_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( A2 != B2 ) ) ) ).
% order.strict_implies_not_eq
thf(fact_78_dependent__wellorder__choice,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ A @ ( type2 @ A ) )
=> ! [P2: ( A > B ) > A > B > $o] :
( ! [R3: B,F5: A > B,G2: A > B,X: A] :
( ! [Y: A] :
( ( ord_less @ A @ Y @ X )
=> ( ( F5 @ Y )
= ( G2 @ Y ) ) )
=> ( ( P2 @ F5 @ X @ R3 )
= ( P2 @ G2 @ X @ R3 ) ) )
=> ( ! [X: A,F5: A > B] :
( ! [Y: A] :
( ( ord_less @ A @ Y @ X )
=> ( P2 @ F5 @ Y @ ( F5 @ Y ) ) )
=> ? [X12: B] : ( P2 @ F5 @ X @ X12 ) )
=> ? [F5: A > B] :
! [X6: A] : ( P2 @ F5 @ X6 @ ( F5 @ X6 ) ) ) ) ) ).
% dependent_wellorder_choice
thf(fact_79_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F3: B > A,B2: B,C: B] :
( ( A2
= ( F3 @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C )
=> ( ! [X: B,Y4: B] :
( ( ord_less @ B @ X @ Y4 )
=> ( ord_less @ A @ ( F3 @ X ) @ ( F3 @ Y4 ) ) )
=> ( ord_less @ A @ A2 @ ( F3 @ C ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_80_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B2: A,F3: A > B,C: B] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ( F3 @ B2 )
= C )
=> ( ! [X: A,Y4: A] :
( ( ord_less @ A @ X @ Y4 )
=> ( ord_less @ B @ ( F3 @ X ) @ ( F3 @ Y4 ) ) )
=> ( ord_less @ B @ ( F3 @ A2 ) @ C ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_81_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F3: B > A,B2: B,C: B] :
( ( ord_less @ A @ A2 @ ( F3 @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C )
=> ( ! [X: B,Y4: B] :
( ( ord_less @ B @ X @ Y4 )
=> ( ord_less @ A @ ( F3 @ X ) @ ( F3 @ Y4 ) ) )
=> ( ord_less @ A @ A2 @ ( F3 @ C ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_82_order__less__subst2,axiom,
! [A: $tType,C3: $tType] :
( ( ( order @ C3 @ ( type2 @ C3 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B2: A,F3: A > C3,C: C3] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ C3 @ ( F3 @ B2 ) @ C )
=> ( ! [X: A,Y4: A] :
( ( ord_less @ A @ X @ Y4 )
=> ( ord_less @ C3 @ ( F3 @ X ) @ ( F3 @ Y4 ) ) )
=> ( ord_less @ C3 @ ( F3 @ A2 ) @ C ) ) ) ) ) ).
% order_less_subst2
thf(fact_83_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A @ ( type2 @ A ) )
=> ! [X4: A] :
? [Y4: A] : ( ord_less @ A @ Y4 @ X4 ) ) ).
% lt_ex
thf(fact_84_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A @ ( type2 @ A ) )
=> ! [X4: A] :
? [X1: A] : ( ord_less @ A @ X4 @ X1 ) ) ).
% gt_ex
thf(fact_85_neqE,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X4: A,Y2: A] :
( ( X4 != Y2 )
=> ( ~ ( ord_less @ A @ X4 @ Y2 )
=> ( ord_less @ A @ Y2 @ X4 ) ) ) ) ).
% neqE
thf(fact_86_neq__iff,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X4: A,Y2: A] :
( ( X4 != Y2 )
= ( ( ord_less @ A @ X4 @ Y2 )
| ( ord_less @ A @ Y2 @ X4 ) ) ) ) ).
% neq_iff
thf(fact_87_order_Oasym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% order.asym
thf(fact_88_dense,axiom,
! [A: $tType] :
( ( dense_order @ A @ ( type2 @ A ) )
=> ! [X4: A,Y2: A] :
( ( ord_less @ A @ X4 @ Y2 )
=> ? [Z3: A] :
( ( ord_less @ A @ X4 @ Z3 )
& ( ord_less @ A @ Z3 @ Y2 ) ) ) ) ).
% dense
thf(fact_89_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X4: A,Y2: A] :
( ( ord_less @ A @ X4 @ Y2 )
=> ( X4 != Y2 ) ) ) ).
% less_imp_neq
thf(fact_90_less__asym,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X4: A,Y2: A] :
( ( ord_less @ A @ X4 @ Y2 )
=> ~ ( ord_less @ A @ Y2 @ X4 ) ) ) ).
% less_asym
thf(fact_91_less__asym_H,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% less_asym'
thf(fact_92_less__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X4: A,Y2: A,Z2: A] :
( ( ord_less @ A @ X4 @ Y2 )
=> ( ( ord_less @ A @ Y2 @ Z2 )
=> ( ord_less @ A @ X4 @ Z2 ) ) ) ) ).
% less_trans
thf(fact_93_less__linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X4: A,Y2: A] :
( ( ord_less @ A @ X4 @ Y2 )
| ( X4 = Y2 )
| ( ord_less @ A @ Y2 @ X4 ) ) ) ).
% less_linear
thf(fact_94_less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X4: A] :
~ ( ord_less @ A @ X4 @ X4 ) ) ).
% less_irrefl
thf(fact_95_ord__eq__less__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( A2 = B2 )
=> ( ( ord_less @ A @ B2 @ C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% ord_eq_less_trans
thf(fact_96_ord__less__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% ord_less_eq_trans
thf(fact_97_dual__order_Oasym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ~ ( ord_less @ A @ A2 @ B2 ) ) ) ).
% dual_order.asym
thf(fact_98_less__imp__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X4: A,Y2: A] :
( ( ord_less @ A @ X4 @ Y2 )
=> ( X4 != Y2 ) ) ) ).
% less_imp_not_eq
thf(fact_99_less__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X4: A,Y2: A] :
( ( ord_less @ A @ X4 @ Y2 )
=> ~ ( ord_less @ A @ Y2 @ X4 ) ) ) ).
% less_not_sym
thf(fact_100_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A @ ( type2 @ A ) )
=> ! [P2: A > $o,A2: A] :
( ! [X: A] :
( ! [Y: A] :
( ( ord_less @ A @ Y @ X )
=> ( P2 @ Y ) )
=> ( P2 @ X ) )
=> ( P2 @ A2 ) ) ) ).
% less_induct
thf(fact_101_antisym__conv3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y2: A,X4: A] :
( ~ ( ord_less @ A @ Y2 @ X4 )
=> ( ( ~ ( ord_less @ A @ X4 @ Y2 ) )
= ( X4 = Y2 ) ) ) ) ).
% antisym_conv3
thf(fact_102_less__imp__not__eq2,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X4: A,Y2: A] :
( ( ord_less @ A @ X4 @ Y2 )
=> ( Y2 != X4 ) ) ) ).
% less_imp_not_eq2
thf(fact_103_less__imp__triv,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X4: A,Y2: A,P2: $o] :
( ( ord_less @ A @ X4 @ Y2 )
=> ( ( ord_less @ A @ Y2 @ X4 )
=> P2 ) ) ) ).
% less_imp_triv
thf(fact_104_linorder__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X4: A,Y2: A] :
( ~ ( ord_less @ A @ X4 @ Y2 )
=> ( ( X4 != Y2 )
=> ( ord_less @ A @ Y2 @ X4 ) ) ) ) ).
% linorder_cases
thf(fact_105_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ A2 ) ) ).
% dual_order.irrefl
thf(fact_106_order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ B2 @ C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% order.strict_trans
thf(fact_107_less__imp__not__less,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X4: A,Y2: A] :
( ( ord_less @ A @ X4 @ Y2 )
=> ~ ( ord_less @ A @ Y2 @ X4 ) ) ) ).
% less_imp_not_less
thf(fact_108_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_less @ A @ C @ B2 )
=> ( ord_less @ A @ C @ A2 ) ) ) ) ).
% dual_order.strict_trans
thf(fact_109_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X4: A,Y2: A] :
( ( ~ ( ord_less @ A @ X4 @ Y2 ) )
= ( ( ord_less @ A @ Y2 @ X4 )
| ( X4 = Y2 ) ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_110_set__of__Tip,axiom,
( ( binDag1380252983set_of @ binDag_Mirabelle_Tip )
= ( bot_bot @ ( set @ simpl_ref ) ) ) ).
% set_of_Tip
thf(fact_111_less__dag__set__of,axiom,
! [X4: binDag_Mirabelle_dag,Y2: binDag_Mirabelle_dag] :
( ( ord_less @ binDag_Mirabelle_dag @ X4 @ Y2 )
=> ( ord_less_eq @ ( set @ simpl_ref ) @ ( binDag1380252983set_of @ X4 ) @ ( binDag1380252983set_of @ Y2 ) ) ) ).
% less_dag_set_of
thf(fact_112_subdag__size,axiom,
! [T: binDag_Mirabelle_dag,S: binDag_Mirabelle_dag] :
( ( binDag786255756subdag @ T @ S )
=> ( ord_less @ nat @ ( size_size @ binDag_Mirabelle_dag @ S ) @ ( size_size @ binDag_Mirabelle_dag @ T ) ) ) ).
% subdag_size
thf(fact_113_less__dag__Node,axiom,
! [X4: binDag_Mirabelle_dag,L: binDag_Mirabelle_dag,A2: simpl_ref,R: binDag_Mirabelle_dag] :
( ( ord_less @ binDag_Mirabelle_dag @ X4 @ ( binDag476092410e_Node @ L @ A2 @ R ) )
= ( ( ord_less_eq @ binDag_Mirabelle_dag @ X4 @ L )
| ( ord_less_eq @ binDag_Mirabelle_dag @ X4 @ R ) ) ) ).
% less_dag_Node
thf(fact_114_strict__monoD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A @ ( type2 @ A ) )
& ( order @ B @ ( type2 @ B ) ) )
=> ! [F3: A > B,X4: A,Y2: A] :
( ( order_strict_mono @ A @ B @ F3 )
=> ( ( ord_less @ A @ X4 @ Y2 )
=> ( ord_less @ B @ ( F3 @ X4 ) @ ( F3 @ Y2 ) ) ) ) ) ).
% strict_monoD
thf(fact_115_strict__monoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A @ ( type2 @ A ) )
& ( order @ B @ ( type2 @ B ) ) )
=> ! [F3: A > B] :
( ! [X: A,Y4: A] :
( ( ord_less @ A @ X @ Y4 )
=> ( ord_less @ B @ ( F3 @ X ) @ ( F3 @ Y4 ) ) )
=> ( order_strict_mono @ A @ B @ F3 ) ) ) ).
% strict_monoI
thf(fact_116_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X4: A] : ( ord_less_eq @ A @ X4 @ X4 ) ) ).
% order_refl
thf(fact_117_bot__apply,axiom,
! [C3: $tType,D2: $tType] :
( ( bot @ C3 @ ( type2 @ C3 ) )
=> ( ( bot_bot @ ( D2 > C3 ) )
= ( ^ [X5: D2] : ( bot_bot @ C3 ) ) ) ) ).
% bot_apply
thf(fact_118_empty__iff,axiom,
! [A: $tType,C: A] :
~ ( member @ A @ C @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_119_all__not__in__conv,axiom,
! [A: $tType,A3: set @ A] :
( ( ! [X5: A] :
~ ( member @ A @ X5 @ A3 ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_120_Collect__empty__eq,axiom,
! [A: $tType,P2: A > $o] :
( ( ( collect @ A @ P2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X5: A] :
~ ( P2 @ X5 ) ) ) ).
% Collect_empty_eq
thf(fact_121_empty__Collect__eq,axiom,
! [A: $tType,P2: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P2 ) )
= ( ! [X5: A] :
~ ( P2 @ X5 ) ) ) ).
% empty_Collect_eq
thf(fact_122_subsetI,axiom,
! [A: $tType,A3: set @ A,B4: set @ A] :
( ! [X: A] :
( ( member @ A @ X @ A3 )
=> ( member @ A @ X @ B4 ) )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B4 ) ) ).
% subsetI
thf(fact_123_subset__antisym,axiom,
! [A: $tType,A3: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ A3 )
=> ( A3 = B4 ) ) ) ).
% subset_antisym
thf(fact_124_subset__empty,axiom,
! [A: $tType,A3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_empty
thf(fact_125_empty__subsetI,axiom,
! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 ) ).
% empty_subsetI
thf(fact_126_psubsetI,axiom,
! [A: $tType,A3: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
=> ( ( A3 != B4 )
=> ( ord_less @ ( set @ A ) @ A3 @ B4 ) ) ) ).
% psubsetI
thf(fact_127_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F3: A > B,G: A > B,X4: A] :
( ( ord_less_eq @ ( A > B ) @ F3 @ G )
=> ( ord_less_eq @ B @ ( F3 @ X4 ) @ ( G @ X4 ) ) ) ) ).
% le_funD
thf(fact_128_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F3: A > B,G: A > B,X4: A] :
( ( ord_less_eq @ ( A > B ) @ F3 @ G )
=> ( ord_less_eq @ B @ ( F3 @ X4 ) @ ( G @ X4 ) ) ) ) ).
% le_funE
thf(fact_129_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F3: A > B,G: A > B] :
( ! [X: A] : ( ord_less_eq @ B @ ( F3 @ X ) @ ( G @ X ) )
=> ( ord_less_eq @ ( A > B ) @ F3 @ G ) ) ) ).
% le_funI
thf(fact_130_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F: A > B,G3: A > B] :
! [X5: A] : ( ord_less_eq @ B @ ( F @ X5 ) @ ( G3 @ X5 ) ) ) ) ) ).
% le_fun_def
thf(fact_131_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B @ ( type2 @ B ) )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X5: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_132_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F3: B > A,B2: B,C: B] :
( ( ord_less_eq @ A @ A2 @ ( F3 @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C )
=> ( ! [X: B,Y4: B] :
( ( ord_less_eq @ B @ X @ Y4 )
=> ( ord_less_eq @ A @ ( F3 @ X ) @ ( F3 @ Y4 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F3 @ C ) ) ) ) ) ) ).
% order_subst1
thf(fact_133_order__subst2,axiom,
! [A: $tType,C3: $tType] :
( ( ( order @ C3 @ ( type2 @ C3 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B2: A,F3: A > C3,C: C3] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ C3 @ ( F3 @ B2 ) @ C )
=> ( ! [X: A,Y4: A] :
( ( ord_less_eq @ A @ X @ Y4 )
=> ( ord_less_eq @ C3 @ ( F3 @ X ) @ ( F3 @ Y4 ) ) )
=> ( ord_less_eq @ C3 @ ( F3 @ A2 ) @ C ) ) ) ) ) ).
% order_subst2
thf(fact_134_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F3: B > A,B2: B,C: B] :
( ( A2
= ( F3 @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C )
=> ( ! [X: B,Y4: B] :
( ( ord_less_eq @ B @ X @ Y4 )
=> ( ord_less_eq @ A @ ( F3 @ X ) @ ( F3 @ Y4 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F3 @ C ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_135_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B2: A,F3: A > B,C: B] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ( F3 @ B2 )
= C )
=> ( ! [X: A,Y4: A] :
( ( ord_less_eq @ A @ X @ Y4 )
=> ( ord_less_eq @ B @ ( F3 @ X ) @ ( F3 @ Y4 ) ) )
=> ( ord_less_eq @ B @ ( F3 @ A2 ) @ C ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_136_eq__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ^ [Y5: A,Z5: A] : Y5 = Z5 )
= ( ^ [X5: A,Y3: A] :
( ( ord_less_eq @ A @ X5 @ Y3 )
& ( ord_less_eq @ A @ Y3 @ X5 ) ) ) ) ) ).
% eq_iff
thf(fact_137_antisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X4: A,Y2: A] :
( ( ord_less_eq @ A @ X4 @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ X4 )
=> ( X4 = Y2 ) ) ) ) ).
% antisym
thf(fact_138_linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X4: A,Y2: A] :
( ( ord_less_eq @ A @ X4 @ Y2 )
| ( ord_less_eq @ A @ Y2 @ X4 ) ) ) ).
% linear
thf(fact_139_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X4: A,Y2: A] :
( ( X4 = Y2 )
=> ( ord_less_eq @ A @ X4 @ Y2 ) ) ) ).
% eq_refl
thf(fact_140_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X4: A,Y2: A] :
( ~ ( ord_less_eq @ A @ X4 @ Y2 )
=> ( ord_less_eq @ A @ Y2 @ X4 ) ) ) ).
% le_cases
thf(fact_141_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% order.trans
thf(fact_142_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X4: A,Y2: A,Z2: A] :
( ( ( ord_less_eq @ A @ X4 @ Y2 )
=> ~ ( ord_less_eq @ A @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ Y2 @ X4 )
=> ~ ( ord_less_eq @ A @ X4 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ X4 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq @ A @ Z2 @ Y2 )
=> ~ ( ord_less_eq @ A @ Y2 @ X4 ) )
=> ( ( ( ord_less_eq @ A @ Y2 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ X4 ) )
=> ~ ( ( ord_less_eq @ A @ Z2 @ X4 )
=> ~ ( ord_less_eq @ A @ X4 @ Y2 ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_143_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [Y2: A,X4: A] :
( ( ord_less_eq @ A @ Y2 @ X4 )
=> ( ( ord_less_eq @ A @ X4 @ Y2 )
= ( X4 = Y2 ) ) ) ) ).
% antisym_conv
thf(fact_144_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( A2 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% ord_eq_le_trans
thf(fact_145_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% ord_le_eq_trans
thf(fact_146_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ) ).
% order_class.order.antisym
thf(fact_147_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X4: A,Y2: A,Z2: A] :
( ( ord_less_eq @ A @ X4 @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ Z2 )
=> ( ord_less_eq @ A @ X4 @ Z2 ) ) ) ) ).
% order_trans
thf(fact_148_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_149_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P2: A > A > $o,A2: A,B2: 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 @ A2 @ B2 ) ) ) ) ).
% linorder_wlog
thf(fact_150_bot_Oextremum,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A2: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A2 ) ) ).
% bot.extremum
thf(fact_151_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C @ B2 )
=> ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% dual_order.trans
thf(fact_152_strict__mono__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A @ ( type2 @ A ) )
& ( order @ B @ ( type2 @ B ) ) )
=> ! [F3: A > B,X4: A,Y2: A] :
( ( order_strict_mono @ A @ B @ F3 )
=> ( ( ( F3 @ X4 )
= ( F3 @ Y2 ) )
= ( X4 = Y2 ) ) ) ) ).
% strict_mono_eq
thf(fact_153_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ) ).
% dual_order.antisym
thf(fact_154_strict__mono__less__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A @ ( type2 @ A ) )
& ( order @ B @ ( type2 @ B ) ) )
=> ! [F3: A > B,X4: A,Y2: A] :
( ( order_strict_mono @ A @ B @ F3 )
=> ( ( ord_less_eq @ B @ ( F3 @ X4 ) @ ( F3 @ Y2 ) )
= ( ord_less_eq @ A @ X4 @ Y2 ) ) ) ) ).
% strict_mono_less_eq
thf(fact_155_bot_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
= ( A2
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_unique
thf(fact_156_bot_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
=> ( A2
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_uniqueI
thf(fact_157_le__dag__set__of,axiom,
! [X4: binDag_Mirabelle_dag,Y2: binDag_Mirabelle_dag] :
( ( ord_less_eq @ binDag_Mirabelle_dag @ X4 @ Y2 )
=> ( ord_less_eq @ ( set @ simpl_ref ) @ ( binDag1380252983set_of @ X4 ) @ ( binDag1380252983set_of @ Y2 ) ) ) ).
% le_dag_set_of
thf(fact_158_le__dag__antisym,axiom,
! [X4: binDag_Mirabelle_dag,Y2: binDag_Mirabelle_dag] :
( ( ord_less_eq @ binDag_Mirabelle_dag @ X4 @ Y2 )
=> ( ( ord_less_eq @ binDag_Mirabelle_dag @ Y2 @ X4 )
=> ( X4 = Y2 ) ) ) ).
% le_dag_antisym
thf(fact_159_le__dag__trans,axiom,
! [X4: binDag_Mirabelle_dag,Y2: binDag_Mirabelle_dag,Z2: binDag_Mirabelle_dag] :
( ( ord_less_eq @ binDag_Mirabelle_dag @ X4 @ Y2 )
=> ( ( ord_less_eq @ binDag_Mirabelle_dag @ Y2 @ Z2 )
=> ( ord_less_eq @ binDag_Mirabelle_dag @ X4 @ Z2 ) ) ) ).
% le_dag_trans
thf(fact_160_le__dag__refl,axiom,
! [X4: binDag_Mirabelle_dag] : ( ord_less_eq @ binDag_Mirabelle_dag @ X4 @ X4 ) ).
% le_dag_refl
thf(fact_161_emptyE,axiom,
! [A: $tType,A2: A] :
~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_162_set__mp,axiom,
! [A: $tType,A3: set @ A,B4: set @ A,X4: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
=> ( ( member @ A @ X4 @ A3 )
=> ( member @ A @ X4 @ B4 ) ) ) ).
% set_mp
thf(fact_163_in__mono,axiom,
! [A: $tType,A3: set @ A,B4: set @ A,X4: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
=> ( ( member @ A @ X4 @ A3 )
=> ( member @ A @ X4 @ B4 ) ) ) ).
% in_mono
thf(fact_164_subsetD,axiom,
! [A: $tType,A3: set @ A,B4: set @ A,C: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
=> ( ( member @ A @ C @ A3 )
=> ( member @ A @ C @ B4 ) ) ) ).
% subsetD
thf(fact_165_equals0D,axiom,
! [A: $tType,A3: set @ A,A2: A] :
( ( A3
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A2 @ A3 ) ) ).
% equals0D
thf(fact_166_equals0I,axiom,
! [A: $tType,A3: set @ A] :
( ! [Y4: A] :
~ ( member @ A @ Y4 @ A3 )
=> ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_167_subsetCE,axiom,
! [A: $tType,A3: set @ A,B4: set @ A,C: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
=> ( ( member @ A @ C @ A3 )
=> ( member @ A @ C @ B4 ) ) ) ).
% subsetCE
thf(fact_168_equalityE,axiom,
! [A: $tType,A3: set @ A,B4: set @ A] :
( ( A3 = B4 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B4 @ A3 ) ) ) ).
% equalityE
thf(fact_169_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
! [X5: A] :
( ( member @ A @ X5 @ A6 )
=> ( member @ A @ X5 @ B6 ) ) ) ) ).
% subset_eq
thf(fact_170_equalityD1,axiom,
! [A: $tType,A3: set @ A,B4: set @ A] :
( ( A3 = B4 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B4 ) ) ).
% equalityD1
thf(fact_171_equalityD2,axiom,
! [A: $tType,A3: set @ A,B4: set @ A] :
( ( A3 = B4 )
=> ( ord_less_eq @ ( set @ A ) @ B4 @ A3 ) ) ).
% equalityD2
thf(fact_172_ex__in__conv,axiom,
! [A: $tType,A3: set @ A] :
( ( ? [X5: A] : ( member @ A @ X5 @ A3 ) )
= ( A3
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_173_set__rev__mp,axiom,
! [A: $tType,X4: A,A3: set @ A,B4: set @ A] :
( ( member @ A @ X4 @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
=> ( member @ A @ X4 @ B4 ) ) ) ).
% set_rev_mp
thf(fact_174_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B6: set @ A] :
! [T3: A] :
( ( member @ A @ T3 @ A6 )
=> ( member @ A @ T3 @ B6 ) ) ) ) ).
% subset_iff
thf(fact_175_rev__subsetD,axiom,
! [A: $tType,C: A,A3: set @ A,B4: set @ A] :
( ( member @ A @ C @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
=> ( member @ A @ C @ B4 ) ) ) ).
% rev_subsetD
thf(fact_176_subset__refl,axiom,
! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ A3 ) ).
% subset_refl
thf(fact_177_Collect__mono,axiom,
! [A: $tType,P2: A > $o,Q3: A > $o] :
( ! [X: A] :
( ( P2 @ X )
=> ( Q3 @ X ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q3 ) ) ) ).
% Collect_mono
thf(fact_178_subset__trans,axiom,
! [A: $tType,A3: set @ A,B4: set @ A,C2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ C2 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ C2 ) ) ) ).
% subset_trans
thf(fact_179_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y5: set @ A,Z5: set @ A] : Y5 = Z5 )
= ( ^ [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_180_contra__subsetD,axiom,
! [A: $tType,A3: set @ A,B4: set @ A,C: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
=> ( ~ ( member @ A @ C @ B4 )
=> ~ ( member @ A @ C @ A3 ) ) ) ).
% contra_subsetD
thf(fact_181_Collect__mono__iff,axiom,
! [A: $tType,P2: A > $o,Q3: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( collect @ A @ Q3 ) )
= ( ! [X5: A] :
( ( P2 @ X5 )
=> ( Q3 @ X5 ) ) ) ) ).
% Collect_mono_iff
thf(fact_182_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_183_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_184_pinf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z3: A] :
! [X6: A] :
( ( ord_less @ A @ Z3 @ X6 )
=> ~ ( ord_less_eq @ A @ X6 @ T ) ) ) ).
% pinf(6)
thf(fact_185_pinf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z3: A] :
! [X6: A] :
( ( ord_less @ A @ Z3 @ X6 )
=> ( ord_less_eq @ A @ T @ X6 ) ) ) ).
% pinf(8)
thf(fact_186_minf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z3: A] :
! [X6: A] :
( ( ord_less @ A @ X6 @ Z3 )
=> ( ord_less_eq @ A @ X6 @ T ) ) ) ).
% minf(6)
thf(fact_187_minf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z3: A] :
! [X6: A] :
( ( ord_less @ A @ X6 @ Z3 )
=> ~ ( ord_less_eq @ A @ T @ X6 ) ) ) ).
% minf(8)
thf(fact_188_leD,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y2: A,X4: A] :
( ( ord_less_eq @ A @ Y2 @ X4 )
=> ~ ( ord_less @ A @ X4 @ Y2 ) ) ) ).
% leD
thf(fact_189_leI,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X4: A,Y2: A] :
( ~ ( ord_less @ A @ X4 @ Y2 )
=> ( ord_less_eq @ A @ Y2 @ X4 ) ) ) ).
% leI
thf(fact_190_le__less,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [X5: A,Y3: A] :
( ( ord_less @ A @ X5 @ Y3 )
| ( X5 = Y3 ) ) ) ) ) ).
% le_less
thf(fact_191_less__le,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [X5: A,Y3: A] :
( ( ord_less_eq @ A @ X5 @ Y3 )
& ( X5 != Y3 ) ) ) ) ) ).
% less_le
thf(fact_192_order__le__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F3: B > A,B2: B,C: B] :
( ( ord_less_eq @ A @ A2 @ ( F3 @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C )
=> ( ! [X: B,Y4: B] :
( ( ord_less @ B @ X @ Y4 )
=> ( ord_less @ A @ ( F3 @ X ) @ ( F3 @ Y4 ) ) )
=> ( ord_less @ A @ A2 @ ( F3 @ C ) ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_193_order__le__less__subst2,axiom,
! [A: $tType,C3: $tType] :
( ( ( order @ C3 @ ( type2 @ C3 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B2: A,F3: A > C3,C: C3] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ C3 @ ( F3 @ B2 ) @ C )
=> ( ! [X: A,Y4: A] :
( ( ord_less_eq @ A @ X @ Y4 )
=> ( ord_less_eq @ C3 @ ( F3 @ X ) @ ( F3 @ Y4 ) ) )
=> ( ord_less @ C3 @ ( F3 @ A2 ) @ C ) ) ) ) ) ).
% order_le_less_subst2
thf(fact_194_order__less__le__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F3: B > A,B2: B,C: B] :
( ( ord_less @ A @ A2 @ ( F3 @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C )
=> ( ! [X: B,Y4: B] :
( ( ord_less_eq @ B @ X @ Y4 )
=> ( ord_less_eq @ A @ ( F3 @ X ) @ ( F3 @ Y4 ) ) )
=> ( ord_less @ A @ A2 @ ( F3 @ C ) ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_195_order__less__le__subst2,axiom,
! [A: $tType,C3: $tType] :
( ( ( order @ C3 @ ( type2 @ C3 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B2: A,F3: A > C3,C: C3] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ C3 @ ( F3 @ B2 ) @ C )
=> ( ! [X: A,Y4: A] :
( ( ord_less @ A @ X @ Y4 )
=> ( ord_less @ C3 @ ( F3 @ X ) @ ( F3 @ Y4 ) ) )
=> ( ord_less @ C3 @ ( F3 @ A2 ) @ C ) ) ) ) ) ).
% order_less_le_subst2
thf(fact_196_not__le,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X4: A,Y2: A] :
( ( ~ ( ord_less_eq @ A @ X4 @ Y2 ) )
= ( ord_less @ A @ Y2 @ X4 ) ) ) ).
% not_le
thf(fact_197_not__less,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X4: A,Y2: A] :
( ( ~ ( ord_less @ A @ X4 @ Y2 ) )
= ( ord_less_eq @ A @ Y2 @ X4 ) ) ) ).
% not_less
thf(fact_198_le__neq__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% le_neq_trans
thf(fact_199_less__imp__le,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X4: A,Y2: A] :
( ( ord_less @ A @ X4 @ Y2 )
=> ( ord_less_eq @ A @ X4 @ Y2 ) ) ) ).
% less_imp_le
thf(fact_200_antisym__conv1,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X4: A,Y2: A] :
( ~ ( ord_less @ A @ X4 @ Y2 )
=> ( ( ord_less_eq @ A @ X4 @ Y2 )
= ( X4 = Y2 ) ) ) ) ).
% antisym_conv1
thf(fact_201_antisym__conv2,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X4: A,Y2: A] :
( ( ord_less_eq @ A @ X4 @ Y2 )
=> ( ( ~ ( ord_less @ A @ X4 @ Y2 ) )
= ( X4 = Y2 ) ) ) ) ).
% antisym_conv2
thf(fact_202_le__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X4: A,Y2: A,Z2: A] :
( ( ord_less_eq @ A @ X4 @ Y2 )
=> ( ( ord_less @ A @ Y2 @ Z2 )
=> ( ord_less @ A @ X4 @ Z2 ) ) ) ) ).
% le_less_trans
thf(fact_203_less__le__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X4: A,Y2: A,Z2: A] :
( ( ord_less @ A @ X4 @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ Z2 )
=> ( ord_less @ A @ X4 @ Z2 ) ) ) ) ).
% less_le_trans
thf(fact_204_dense__ge,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Z2: A,Y2: A] :
( ! [X: A] :
( ( ord_less @ A @ Z2 @ X )
=> ( ord_less_eq @ A @ Y2 @ X ) )
=> ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ).
% dense_ge
thf(fact_205_dense__le,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Y2: A,Z2: A] :
( ! [X: A] :
( ( ord_less @ A @ X @ Y2 )
=> ( ord_less_eq @ A @ X @ Z2 ) )
=> ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ).
% dense_le
thf(fact_206_le__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X4: A,Y2: A] :
( ( ord_less_eq @ A @ X4 @ Y2 )
| ( ord_less @ A @ Y2 @ X4 ) ) ) ).
% le_less_linear
thf(fact_207_le__imp__less__or__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X4: A,Y2: A] :
( ( ord_less_eq @ A @ X4 @ Y2 )
=> ( ( ord_less @ A @ X4 @ Y2 )
| ( X4 = Y2 ) ) ) ) ).
% le_imp_less_or_eq
thf(fact_208_less__le__not__le,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [X5: A,Y3: A] :
( ( ord_less_eq @ A @ X5 @ Y3 )
& ~ ( ord_less_eq @ A @ Y3 @ X5 ) ) ) ) ) ).
% less_le_not_le
thf(fact_209_not__le__imp__less,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y2: A,X4: A] :
( ~ ( ord_less_eq @ A @ Y2 @ X4 )
=> ( ord_less @ A @ X4 @ Y2 ) ) ) ).
% not_le_imp_less
thf(fact_210_order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ B2 @ C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% order.strict_trans1
thf(fact_211_order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% order.strict_trans2
thf(fact_212_order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ) ).
% order.order_iff_strict
thf(fact_213_order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ) ).
% order.strict_iff_order
thf(fact_214_dual__order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less @ A @ C @ B2 )
=> ( ord_less @ A @ C @ A2 ) ) ) ) ).
% dual_order.strict_trans1
thf(fact_215_dual__order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C @ B2 )
=> ( ord_less @ A @ C @ A2 ) ) ) ) ).
% dual_order.strict_trans2
thf(fact_216_dense__ge__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Z2: A,X4: A,Y2: A] :
( ( ord_less @ A @ Z2 @ X4 )
=> ( ! [W: A] :
( ( ord_less @ A @ Z2 @ W )
=> ( ( ord_less @ A @ W @ X4 )
=> ( ord_less_eq @ A @ Y2 @ W ) ) )
=> ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ) ).
% dense_ge_bounded
thf(fact_217_dense__le__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [X4: A,Y2: A,Z2: A] :
( ( ord_less @ A @ X4 @ Y2 )
=> ( ! [W: A] :
( ( ord_less @ A @ X4 @ W )
=> ( ( ord_less @ A @ W @ Y2 )
=> ( ord_less_eq @ A @ W @ Z2 ) ) )
=> ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ) ).
% dense_le_bounded
thf(fact_218_order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% order.strict_implies_order
thf(fact_219_dual__order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [B3: A,A4: A] :
( ( ord_less @ A @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_220_dual__order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [B3: A,A4: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_221_dual__order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% dual_order.strict_implies_order
thf(fact_222_order_Onot__eq__order__implies__strict,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( A2 != B2 )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% order.not_eq_order_implies_strict
thf(fact_223_complete__interval,axiom,
! [A: $tType] :
( ( condit1037483654norder @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,P2: A > $o] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( P2 @ A2 )
=> ( ~ ( P2 @ B2 )
=> ? [C4: A] :
( ( ord_less_eq @ A @ A2 @ C4 )
& ( ord_less_eq @ A @ C4 @ B2 )
& ! [X6: A] :
( ( ( ord_less_eq @ A @ A2 @ X6 )
& ( ord_less @ A @ X6 @ C4 ) )
=> ( P2 @ X6 ) )
& ! [D3: A] :
( ! [X: A] :
( ( ( ord_less_eq @ A @ A2 @ X )
& ( ord_less @ A @ X @ D3 ) )
=> ( P2 @ X ) )
=> ( ord_less_eq @ A @ D3 @ C4 ) ) ) ) ) ) ) ).
% complete_interval
thf(fact_224_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_225_subset__psubset__trans,axiom,
! [A: $tType,A3: set @ A,B4: set @ A,C2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
=> ( ( ord_less @ ( set @ A ) @ B4 @ C2 )
=> ( ord_less @ ( set @ A ) @ A3 @ C2 ) ) ) ).
% subset_psubset_trans
thf(fact_226_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_227_psubset__subset__trans,axiom,
! [A: $tType,A3: set @ A,B4: set @ A,C2: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ C2 )
=> ( ord_less @ ( set @ A ) @ A3 @ C2 ) ) ) ).
% psubset_subset_trans
thf(fact_228_psubset__imp__subset,axiom,
! [A: $tType,A3: set @ A,B4: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ B4 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B4 ) ) ).
% psubset_imp_subset
thf(fact_229_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_230_psubsetE,axiom,
! [A: $tType,A3: set @ A,B4: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ B4 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B4 )
=> ( ord_less_eq @ ( set @ A ) @ B4 @ A3 ) ) ) ).
% psubsetE
thf(fact_231_not__psubset__empty,axiom,
! [A: $tType,A3: set @ A] :
~ ( ord_less @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% not_psubset_empty
thf(fact_232_dag__less__le,axiom,
( ( ord_less @ binDag_Mirabelle_dag )
= ( ^ [X5: binDag_Mirabelle_dag,Y3: binDag_Mirabelle_dag] :
( ( ord_less_eq @ binDag_Mirabelle_dag @ X5 @ Y3 )
& ( X5 != Y3 ) ) ) ) ).
% dag_less_le
thf(fact_233_le__dag__def,axiom,
( ( ord_less_eq @ binDag_Mirabelle_dag )
= ( ^ [S2: binDag_Mirabelle_dag,T3: binDag_Mirabelle_dag] :
( ( S2 = T3 )
| ( ord_less @ binDag_Mirabelle_dag @ S2 @ T3 ) ) ) ) ).
% le_dag_def
thf(fact_234_strict__mono__less,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A @ ( type2 @ A ) )
& ( order @ B @ ( type2 @ B ) ) )
=> ! [F3: A > B,X4: A,Y2: A] :
( ( order_strict_mono @ A @ B @ F3 )
=> ( ( ord_less @ B @ ( F3 @ X4 ) @ ( F3 @ Y2 ) )
= ( ord_less @ A @ X4 @ Y2 ) ) ) ) ).
% strict_mono_less
thf(fact_235_strict__mono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A @ ( type2 @ A ) )
& ( order @ B @ ( type2 @ B ) ) )
=> ( ( order_strict_mono @ A @ B )
= ( ^ [F: A > B] :
! [X5: A,Y3: A] :
( ( ord_less @ A @ X5 @ Y3 )
=> ( ord_less @ B @ ( F @ X5 ) @ ( F @ Y3 ) ) ) ) ) ) ).
% strict_mono_def
thf(fact_236_override__on__emptyset,axiom,
! [B: $tType,A: $tType,F3: A > B,G: A > B] :
( ( override_on @ A @ B @ F3 @ G @ ( bot_bot @ ( set @ A ) ) )
= F3 ) ).
% override_on_emptyset
thf(fact_237_Set_Ois__empty__def,axiom,
! [A: $tType] :
( ( is_empty @ A )
= ( ^ [A6: set @ A] :
( A6
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Set.is_empty_def
thf(fact_238_Dag__update__r__new,axiom,
! [T: binDag_Mirabelle_dag,Alloc: list @ simpl_ref,P: simpl_ref,L: simpl_ref > simpl_ref,R: simpl_ref > simpl_ref,X4: simpl_ref] :
( ( ord_less_eq @ ( set @ simpl_ref ) @ ( binDag1380252983set_of @ T ) @ ( set2 @ simpl_ref @ Alloc ) )
=> ( ( binDag_Mirabelle_Dag @ P @ L @ ( fun_upd @ simpl_ref @ simpl_ref @ R @ ( simpl_new @ ( set2 @ simpl_ref @ Alloc ) ) @ X4 ) @ T )
= ( binDag_Mirabelle_Dag @ P @ L @ R @ T ) ) ) ).
% Dag_update_r_new
thf(fact_239_Dag__update__l__new,axiom,
! [T: binDag_Mirabelle_dag,Alloc: list @ simpl_ref,P: simpl_ref,L: simpl_ref > simpl_ref,X4: simpl_ref,R: simpl_ref > simpl_ref] :
( ( ord_less_eq @ ( set @ simpl_ref ) @ ( binDag1380252983set_of @ T ) @ ( set2 @ simpl_ref @ Alloc ) )
=> ( ( binDag_Mirabelle_Dag @ P @ ( fun_upd @ simpl_ref @ simpl_ref @ L @ ( simpl_new @ ( set2 @ simpl_ref @ Alloc ) ) @ X4 ) @ R @ T )
= ( binDag_Mirabelle_Dag @ P @ L @ R @ T ) ) ) ).
% Dag_update_l_new
thf(fact_240_override__on__apply__in,axiom,
! [B: $tType,A: $tType,A2: A,A3: set @ A,F3: A > B,G: A > B] :
( ( member @ A @ A2 @ A3 )
=> ( ( override_on @ A @ B @ F3 @ G @ A3 @ A2 )
= ( G @ A2 ) ) ) ).
% override_on_apply_in
thf(fact_241_override__on__apply__notin,axiom,
! [B: $tType,A: $tType,A2: A,A3: set @ A,F3: A > B,G: A > B] :
( ~ ( member @ A @ A2 @ A3 )
=> ( ( override_on @ A @ B @ F3 @ G @ A3 @ A2 )
= ( F3 @ A2 ) ) ) ).
% override_on_apply_notin
thf(fact_242_less__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ( ( ord_less @ ( A > B ) )
= ( ^ [F: A > B,G3: A > B] :
( ( ord_less_eq @ ( A > B ) @ F @ G3 )
& ~ ( ord_less_eq @ ( A > B ) @ G3 @ F ) ) ) ) ) ).
% less_fun_def
thf(fact_243_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_244_override__on__def,axiom,
! [B: $tType,A: $tType] :
( ( override_on @ A @ B )
= ( ^ [F: A > B,G3: A > B,A6: set @ A,A4: A] : ( if @ B @ ( member @ A @ A4 @ A6 ) @ ( G3 @ A4 ) @ ( F @ A4 ) ) ) ) ).
% override_on_def
thf(fact_245_chain__subset__def,axiom,
! [A: $tType] :
( ( chain_subset @ A )
= ( ^ [C5: set @ ( set @ A )] :
! [X5: set @ A] :
( ( member @ ( set @ A ) @ X5 @ C5 )
=> ! [Y3: set @ A] :
( ( member @ ( set @ A ) @ Y3 @ C5 )
=> ( ( ord_less_eq @ ( set @ A ) @ X5 @ Y3 )
| ( ord_less_eq @ ( set @ A ) @ Y3 @ X5 ) ) ) ) ) ) ).
% chain_subset_def
thf(fact_246_dag_Osize_I3_J,axiom,
( ( size_size @ binDag_Mirabelle_dag @ binDag_Mirabelle_Tip )
= ( zero_zero @ nat ) ) ).
% dag.size(3)
thf(fact_247_not__gr__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [N: A] :
( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
= ( N
= ( zero_zero @ A ) ) ) ) ).
% not_gr_zero
thf(fact_248_dag_Osize__gen_I1_J,axiom,
( ( binDag1924123185ze_dag @ binDag_Mirabelle_Tip )
= ( zero_zero @ nat ) ) ).
% dag.size_gen(1)
thf(fact_249_gr__zeroI,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [N: A] :
( ( N
!= ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% gr_zeroI
thf(fact_250_not__less__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [N: A] :
~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% not_less_zero
thf(fact_251_gr__implies__not__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [M: A,N: A] :
( ( ord_less @ A @ M @ N )
=> ( N
!= ( zero_zero @ A ) ) ) ) ).
% gr_implies_not_zero
thf(fact_252_zero__less__iff__neq__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [N: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ N )
= ( N
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_iff_neq_zero
thf(fact_253_less__numeral__extra_I3_J,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% less_numeral_extra(3)
thf(fact_254_sgn__greater,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( sgn_sgn @ A @ A2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% sgn_greater
thf(fact_255_sgn__sgn,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( sgn_sgn @ A @ ( sgn_sgn @ A @ A2 ) )
= ( sgn_sgn @ A @ A2 ) ) ) ).
% sgn_sgn
%----Type constructors (30)
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___Groups_Ocanonically__ordered__monoid__add,axiom,
canoni770627133id_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Olinordered__semidom,axiom,
linordered_semidom @ 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,X4: A,Y2: A] :
( ( if @ A @ $false @ X4 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X4: A,Y2: A] :
( ( if @ A @ $true @ X4 @ Y2 )
= X4 ) ).
%----Conjectures (2)
thf(conj_0,hypothesis,
! [Q5: simpl_ref] :
( ( binDag_Mirabelle_Dag @ Q5 @ l @ r @ ( binDag476092410e_Node @ lt @ p @ rt ) )
=> thesis ) ).
thf(conj_1,conjecture,
thesis ).
%------------------------------------------------------------------------------