TPTP Problem File: COM165^1.p
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%------------------------------------------------------------------------------
% File : COM165^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Computing Theory
% Problem : Binary decision diagram 172
% 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__172.p [Bla16]
% Status : Theorem
% Rating : 0.00 v7.2.0, 0.25 v7.1.0
% Syntax : Number of formulae : 350 ( 89 unt; 56 typ; 0 def)
% Number of atoms : 887 ( 197 equ; 0 cnn)
% Maximal formula atoms : 13 ( 3 avg)
% Number of connectives : 3292 ( 110 ~; 28 |; 48 &;2628 @)
% ( 0 <=>; 478 =>; 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 : 56 ( 53 usr; 9 con; 0-6 aty)
% Number of variables : 989 ( 55 ^; 858 !; 36 ?; 989 :)
% ( 40 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:45:25.154
%------------------------------------------------------------------------------
%----Could-be-implicit typings (5)
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_Nat_Onat,type,
nat: $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
%----Explicit typings (51)
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_Rings_Ozero__neq__one,type,
zero_neq_one:
!>[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_Rings_Ozero__less__one,type,
zero_less_one:
!>[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_Rings_Olinordered__nonzero__semiring,type,
linord1659791738miring:
!>[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_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_Groups_Oone__class_Oone,type,
one_one:
!>[A: $tType] : A ).
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_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_Oordering,type,
ordering:
!>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > $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_Rings_Ozero__neq__one__class_Oof__bool,type,
zero_neq_one_of_bool:
!>[A: $tType] : ( $o > 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_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_a____,type,
a: simpl_ref ).
thf(sy_v_l____,type,
l: binDag_Mirabelle_dag ).
thf(sy_v_r____,type,
r: binDag_Mirabelle_dag ).
thf(sy_v_thesis____,type,
thesis: $o ).
thf(sy_v_x,type,
x: binDag_Mirabelle_dag ).
thf(sy_v_y,type,
y: binDag_Mirabelle_dag ).
%----Relevant facts (256)
thf(fact_0_Node,axiom,
( y
= ( binDag476092410e_Node @ l @ a @ r ) ) ).
% Node
thf(fact_1_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_2_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_3_x__less__y,axiom,
ord_less @ binDag_Mirabelle_dag @ x @ y ).
% x_less_y
thf(fact_4_set__of__Tip,axiom,
( ( binDag1380252983set_of @ binDag_Mirabelle_Tip )
= ( bot_bot @ ( set @ simpl_ref ) ) ) ).
% set_of_Tip
thf(fact_5_le__dag__set__of,axiom,
! [X: binDag_Mirabelle_dag,Y: binDag_Mirabelle_dag] :
( ( ord_less_eq @ binDag_Mirabelle_dag @ X @ Y )
=> ( ord_less_eq @ ( set @ simpl_ref ) @ ( binDag1380252983set_of @ X ) @ ( binDag1380252983set_of @ Y ) ) ) ).
% le_dag_set_of
thf(fact_6_less__dag__set__of,axiom,
! [X: binDag_Mirabelle_dag,Y: binDag_Mirabelle_dag] :
( ( ord_less @ binDag_Mirabelle_dag @ X @ Y )
=> ( ord_less_eq @ ( set @ simpl_ref ) @ ( binDag1380252983set_of @ X ) @ ( binDag1380252983set_of @ Y ) ) ) ).
% less_dag_set_of
thf(fact_7_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_8_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_9_subdag__neq,axiom,
! [T: binDag_Mirabelle_dag,S: binDag_Mirabelle_dag] :
( ( binDag786255756subdag @ T @ S )
=> ( T != S ) ) ).
% subdag_neq
thf(fact_10_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_11_subdag__not__sym,axiom,
! [S: binDag_Mirabelle_dag,T: binDag_Mirabelle_dag] :
( ( binDag786255756subdag @ S @ T )
=> ~ ( binDag786255756subdag @ T @ S ) ) ).
% subdag_not_sym
thf(fact_12_DAG__y,axiom,
binDag_Mirabelle_DAG @ y ).
% DAG_y
thf(fact_13_DAG_Osimps_I1_J,axiom,
binDag_Mirabelle_DAG @ binDag_Mirabelle_Tip ).
% DAG.simps(1)
thf(fact_14_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_15_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_16_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_17_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_18_subdag_Osimps_I1_J,axiom,
! [T: binDag_Mirabelle_dag] :
~ ( binDag786255756subdag @ binDag_Mirabelle_Tip @ T ) ).
% subdag.simps(1)
thf(fact_19_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_20_dag_Oinduct,axiom,
! [P: binDag_Mirabelle_dag > $o,Dag: binDag_Mirabelle_dag] :
( ( P @ binDag_Mirabelle_Tip )
=> ( ! [X1: binDag_Mirabelle_dag,X2: simpl_ref,X3: binDag_Mirabelle_dag] :
( ( P @ X1 )
=> ( ( P @ X3 )
=> ( P @ ( binDag476092410e_Node @ X1 @ X2 @ X3 ) ) ) )
=> ( P @ Dag ) ) ) ).
% dag.induct
thf(fact_21_le__dag__def,axiom,
( ( ord_less_eq @ binDag_Mirabelle_dag )
= ( ^ [S2: binDag_Mirabelle_dag,T2: binDag_Mirabelle_dag] :
( ( S2 = T2 )
| ( ord_less @ binDag_Mirabelle_dag @ S2 @ T2 ) ) ) ) ).
% le_dag_def
thf(fact_22_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_23_dag__less__le,axiom,
( ( ord_less @ binDag_Mirabelle_dag )
= ( ^ [X4: binDag_Mirabelle_dag,Y2: binDag_Mirabelle_dag] :
( ( ord_less_eq @ binDag_Mirabelle_dag @ X4 @ Y2 )
& ( X4 != Y2 ) ) ) ) ).
% dag_less_le
thf(fact_24_le__dag__refl,axiom,
! [X: binDag_Mirabelle_dag] : ( ord_less_eq @ binDag_Mirabelle_dag @ X @ X ) ).
% le_dag_refl
thf(fact_25_le__dag__trans,axiom,
! [X: binDag_Mirabelle_dag,Y: binDag_Mirabelle_dag,Z: binDag_Mirabelle_dag] :
( ( ord_less_eq @ binDag_Mirabelle_dag @ X @ Y )
=> ( ( ord_less_eq @ binDag_Mirabelle_dag @ Y @ Z )
=> ( ord_less_eq @ binDag_Mirabelle_dag @ X @ Z ) ) ) ).
% le_dag_trans
thf(fact_26_less__dag__Tip,axiom,
! [X: binDag_Mirabelle_dag] :
~ ( ord_less @ binDag_Mirabelle_dag @ X @ binDag_Mirabelle_Tip ) ).
% less_dag_Tip
thf(fact_27_less__dag__def,axiom,
( ( ord_less @ binDag_Mirabelle_dag )
= ( ^ [S2: binDag_Mirabelle_dag,T2: binDag_Mirabelle_dag] : ( binDag786255756subdag @ T2 @ S2 ) ) ) ).
% less_dag_def
thf(fact_28_subdag__NodeD,axiom,
! [T: binDag_Mirabelle_dag,Lt: binDag_Mirabelle_dag,A2: simpl_ref,Rt: binDag_Mirabelle_dag] :
( ( binDag786255756subdag @ T @ ( binDag476092410e_Node @ Lt @ A2 @ Rt ) )
=> ( ( binDag786255756subdag @ T @ Lt )
& ( binDag786255756subdag @ T @ Rt ) ) ) ).
% subdag_NodeD
thf(fact_29_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_30_less__dag__Node,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 ) )
= ( ( ord_less_eq @ binDag_Mirabelle_dag @ X @ L )
| ( ord_less_eq @ binDag_Mirabelle_dag @ X @ R ) ) ) ).
% less_dag_Node
thf(fact_31_le__dag__antisym,axiom,
! [X: binDag_Mirabelle_dag,Y: binDag_Mirabelle_dag] :
( ( ord_less_eq @ binDag_Mirabelle_dag @ X @ Y )
=> ( ( ord_less_eq @ binDag_Mirabelle_dag @ Y @ X )
=> ( X = Y ) ) ) ).
% le_dag_antisym
thf(fact_32_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_33_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_34_empty__subsetI,axiom,
! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 ) ).
% empty_subsetI
thf(fact_35_subsetI,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ( member @ A @ X5 @ B2 ) )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B2 ) ) ).
% subsetI
thf(fact_36_subset__antisym,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ A3 )
=> ( A3 = B2 ) ) ) ).
% subset_antisym
thf(fact_37_empty__iff,axiom,
! [A: $tType,C: A] :
~ ( member @ A @ C @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_38_all__not__in__conv,axiom,
! [A: $tType,A3: set @ A] :
( ( ! [X4: A] :
~ ( member @ A @ X4 @ A3 ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_39_Collect__empty__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X4: A] :
~ ( P @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_40_empty__Collect__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P ) )
= ( ! [X4: A] :
~ ( P @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_41_bot__apply,axiom,
! [C2: $tType,D: $tType] :
( ( bot @ C2 @ ( type2 @ C2 ) )
=> ( ( bot_bot @ ( D > C2 ) )
= ( ^ [X4: D] : ( bot_bot @ C2 ) ) ) ) ).
% bot_apply
thf(fact_42_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_43_subset__emptyI,axiom,
! [A: $tType,A3: set @ A] :
( ! [X5: A] :
~ ( member @ A @ X5 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_emptyI
thf(fact_44_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( collect @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X5: A] :
( ( P @ X5 )
= ( Q @ X5 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X5: A] :
( ( F @ X5 )
= ( G @ X5 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A] :
( ( ord_less_eq @ A @ B3 @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ) ).
% dual_order.antisym
thf(fact_50_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A,C: A] :
( ( ord_less_eq @ A @ B3 @ A2 )
=> ( ( ord_less_eq @ A @ C @ B3 )
=> ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% dual_order.trans
thf(fact_51_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > A > $o,A2: A,B3: A] :
( ! [A4: A,B4: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: A,B4: A] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A2 @ B3 ) ) ) ) ).
% linorder_wlog
thf(fact_52_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_53_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z )
=> ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% order_trans
thf(fact_54_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ) ).
% order_class.order.antisym
thf(fact_55_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% ord_le_eq_trans
thf(fact_56_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C: A] :
( ( A2 = B3 )
=> ( ( ord_less_eq @ A @ B3 @ C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% ord_eq_le_trans
thf(fact_57_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_less_eq @ A @ X @ Y )
= ( X = Y ) ) ) ) ).
% antisym_conv
thf(fact_58_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( ( ord_less_eq @ A @ X @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z ) )
=> ( ( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less_eq @ A @ X @ Z ) )
=> ( ( ( ord_less_eq @ A @ X @ Z )
=> ~ ( ord_less_eq @ A @ Z @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z )
=> ~ ( ord_less_eq @ A @ Z @ X ) )
=> ~ ( ( ord_less_eq @ A @ Z @ X )
=> ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_59_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% order.trans
thf(fact_60_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ~ ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% le_cases
thf(fact_61_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( X = Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% eq_refl
thf(fact_62_linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
| ( ord_less_eq @ A @ Y @ X ) ) ) ).
% linear
thf(fact_63_antisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ X )
=> ( X = Y ) ) ) ) ).
% antisym
thf(fact_64_eq__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ^ [Y3: A,Z2: A] : Y3 = Z2 )
= ( ^ [X4: A,Y2: A] :
( ( ord_less_eq @ A @ X4 @ Y2 )
& ( ord_less_eq @ A @ Y2 @ X4 ) ) ) ) ) ).
% eq_iff
thf(fact_65_ord__le__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_eq @ A @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X5: A,Y4: A] :
( ( ord_less_eq @ A @ X5 @ Y4 )
=> ( ord_less_eq @ B @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_66_ord__eq__le__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_eq @ B @ B3 @ C )
=> ( ! [X5: B,Y4: B] :
( ( ord_less_eq @ B @ X5 @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_67_order__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_eq @ A @ A2 @ B3 )
=> ( ( ord_less_eq @ C2 @ ( F @ B3 ) @ C )
=> ( ! [X5: A,Y4: A] :
( ( ord_less_eq @ A @ X5 @ Y4 )
=> ( ord_less_eq @ C2 @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_subst2
thf(fact_68_order__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_eq @ A @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C )
=> ( ! [X5: B,Y4: B] :
( ( ord_less_eq @ B @ X5 @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_subst1
thf(fact_69_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F3: A > B,G2: A > B] :
! [X4: A] : ( ord_less_eq @ B @ ( F3 @ X4 ) @ ( G2 @ X4 ) ) ) ) ) ).
% le_fun_def
thf(fact_70_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B] :
( ! [X5: A] : ( ord_less_eq @ B @ ( F @ X5 ) @ ( G @ X5 ) )
=> ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% le_funI
thf(fact_71_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% le_funE
thf(fact_72_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% le_funD
thf(fact_73_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_74_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_75_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_76_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_77_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_78_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_79_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ A2 ) ) ).
% dual_order.irrefl
thf(fact_80_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_81_less__imp__triv,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,P: $o] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ X )
=> P ) ) ) ).
% less_imp_triv
thf(fact_82_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_83_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_84_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,A2: A] :
( ! [X5: A] :
( ! [Y5: A] :
( ( ord_less @ A @ Y5 @ X5 )
=> ( P @ Y5 ) )
=> ( P @ X5 ) )
=> ( P @ A2 ) ) ) ).
% less_induct
thf(fact_85_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_86_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_87_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_88_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_89_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_90_less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A] :
~ ( ord_less @ A @ X @ X ) ) ).
% less_irrefl
thf(fact_91_less__linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
| ( X = Y )
| ( ord_less @ A @ Y @ X ) ) ) ).
% less_linear
thf(fact_92_less__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_93_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_94_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_95_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_96_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_97_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_98_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_99_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_100_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A @ ( type2 @ A ) )
=> ! [X: A] :
? [X1: A] : ( ord_less @ A @ X @ X1 ) ) ).
% gt_ex
thf(fact_101_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A @ ( type2 @ A ) )
=> ! [X: A] :
? [Y4: A] : ( ord_less @ A @ Y4 @ X ) ) ).
% lt_ex
thf(fact_102_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 )
=> ( ! [X5: A,Y4: A] :
( ( ord_less @ A @ X5 @ Y4 )
=> ( ord_less @ C2 @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_less_subst2
thf(fact_103_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 )
=> ( ! [X5: B,Y4: B] :
( ( ord_less @ B @ X5 @ Y4 )
=> ( ord_less @ A @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_104_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 )
=> ( ! [X5: A,Y4: A] :
( ( ord_less @ A @ X5 @ Y4 )
=> ( ord_less @ B @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_105_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 )
=> ( ! [X5: B,Y4: B] :
( ( ord_less @ B @ X5 @ Y4 )
=> ( ord_less @ A @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_106_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B @ ( type2 @ B ) )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X4: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_107_ex__in__conv,axiom,
! [A: $tType,A3: set @ A] :
( ( ? [X4: A] : ( member @ A @ X4 @ A3 ) )
= ( A3
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_108_equals0I,axiom,
! [A: $tType,A3: set @ A] :
( ! [Y4: A] :
~ ( member @ A @ Y4 @ A3 )
=> ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_109_equals0D,axiom,
! [A: $tType,A3: set @ A,A2: A] :
( ( A3
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A2 @ A3 ) ) ).
% equals0D
thf(fact_110_emptyE,axiom,
! [A: $tType,A2: A] :
~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_111_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
= ( ! [X4: A] :
( ( P @ X4 )
=> ( Q @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_112_contra__subsetD,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,C: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ~ ( member @ A @ C @ B2 )
=> ~ ( member @ A @ C @ A3 ) ) ) ).
% contra_subsetD
thf(fact_113_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y3: set @ A,Z2: set @ A] : Y3 = Z2 )
= ( ^ [A5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
& ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_114_subset__trans,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% subset_trans
thf(fact_115_Collect__mono,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X5: A] :
( ( P @ X5 )
=> ( Q @ X5 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_116_subset__refl,axiom,
! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ A3 ) ).
% subset_refl
thf(fact_117_rev__subsetD,axiom,
! [A: $tType,C: A,A3: set @ A,B2: set @ A] :
( ( member @ A @ C @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( member @ A @ C @ B2 ) ) ) ).
% rev_subsetD
thf(fact_118_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A5: set @ A,B5: set @ A] :
! [T2: A] :
( ( member @ A @ T2 @ A5 )
=> ( member @ A @ T2 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_119_set__rev__mp,axiom,
! [A: $tType,X: A,A3: set @ A,B2: set @ A] :
( ( member @ A @ X @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( member @ A @ X @ B2 ) ) ) ).
% set_rev_mp
thf(fact_120_equalityD2,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( A3 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ B2 @ A3 ) ) ).
% equalityD2
thf(fact_121_equalityD1,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( A3 = B2 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B2 ) ) ).
% equalityD1
thf(fact_122_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A5: set @ A,B5: set @ A] :
! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( member @ A @ X4 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_123_equalityE,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( A3 = B2 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B2 @ A3 ) ) ) ).
% equalityE
thf(fact_124_subsetCE,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,C: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ( member @ A @ C @ A3 )
=> ( member @ A @ C @ B2 ) ) ) ).
% subsetCE
thf(fact_125_subsetD,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,C: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ( member @ A @ C @ A3 )
=> ( member @ A @ C @ B2 ) ) ) ).
% subsetD
thf(fact_126_in__mono,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ( member @ A @ X @ A3 )
=> ( member @ A @ X @ B2 ) ) ) ).
% in_mono
thf(fact_127_set__mp,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ( member @ A @ X @ A3 )
=> ( member @ A @ X @ B2 ) ) ) ).
% set_mp
thf(fact_128_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_129_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_130_dual__order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [B6: A,A6: A] :
( ( ord_less_eq @ A @ B6 @ A6 )
& ( A6 != B6 ) ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_131_dual__order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [B6: A,A6: A] :
( ( ord_less @ A @ B6 @ A6 )
| ( A6 = B6 ) ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_132_order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( ord_less @ A @ A2 @ B3 )
=> ( ord_less_eq @ A @ A2 @ B3 ) ) ) ).
% order.strict_implies_order
thf(fact_133_dense__le__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less @ A @ X @ Y )
=> ( ! [W: A] :
( ( ord_less @ A @ X @ W )
=> ( ( ord_less @ A @ W @ Y )
=> ( ord_less_eq @ A @ W @ Z ) ) )
=> ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% dense_le_bounded
thf(fact_134_dense__ge__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Z: A,X: A,Y: A] :
( ( ord_less @ A @ Z @ X )
=> ( ! [W: A] :
( ( ord_less @ A @ Z @ W )
=> ( ( ord_less @ A @ W @ X )
=> ( ord_less_eq @ A @ Y @ W ) ) )
=> ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% dense_ge_bounded
thf(fact_135_dual__order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A,C: A] :
( ( ord_less @ A @ B3 @ A2 )
=> ( ( ord_less_eq @ A @ C @ B3 )
=> ( ord_less @ A @ C @ A2 ) ) ) ) ).
% dual_order.strict_trans2
thf(fact_136_dual__order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A,C: A] :
( ( ord_less_eq @ A @ B3 @ A2 )
=> ( ( ord_less @ A @ C @ B3 )
=> ( ord_less @ A @ C @ A2 ) ) ) ) ).
% dual_order.strict_trans1
thf(fact_137_order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [A6: A,B6: A] :
( ( ord_less_eq @ A @ A6 @ B6 )
& ( A6 != B6 ) ) ) ) ) ).
% order.strict_iff_order
thf(fact_138_order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [A6: A,B6: A] :
( ( ord_less @ A @ A6 @ B6 )
| ( A6 = B6 ) ) ) ) ) ).
% order.order_iff_strict
thf(fact_139_order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C: A] :
( ( ord_less @ A @ A2 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% order.strict_trans2
thf(fact_140_order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( ord_less @ A @ B3 @ C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% order.strict_trans1
thf(fact_141_not__le__imp__less,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A] :
( ~ ( ord_less_eq @ A @ Y @ X )
=> ( ord_less @ A @ X @ Y ) ) ) ).
% not_le_imp_less
thf(fact_142_less__le__not__le,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [X4: A,Y2: A] :
( ( ord_less_eq @ A @ X4 @ Y2 )
& ~ ( ord_less_eq @ A @ Y2 @ X4 ) ) ) ) ) ).
% less_le_not_le
thf(fact_143_le__imp__less__or__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ X @ Y )
| ( X = Y ) ) ) ) ).
% le_imp_less_or_eq
thf(fact_144_le__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
| ( ord_less @ A @ Y @ X ) ) ) ).
% le_less_linear
thf(fact_145_dense__le,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Y: A,Z: A] :
( ! [X5: A] :
( ( ord_less @ A @ X5 @ Y )
=> ( ord_less_eq @ A @ X5 @ Z ) )
=> ( ord_less_eq @ A @ Y @ Z ) ) ) ).
% dense_le
thf(fact_146_dense__ge,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Z: A,Y: A] :
( ! [X5: A] :
( ( ord_less @ A @ Z @ X5 )
=> ( ord_less_eq @ A @ Y @ X5 ) )
=> ( ord_less_eq @ A @ Y @ Z ) ) ) ).
% dense_ge
thf(fact_147_less__le__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z )
=> ( ord_less @ A @ X @ Z ) ) ) ) ).
% less_le_trans
thf(fact_148_le__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ Z )
=> ( ord_less @ A @ X @ Z ) ) ) ) ).
% le_less_trans
thf(fact_149_antisym__conv2,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ~ ( ord_less @ A @ X @ Y ) )
= ( X = Y ) ) ) ) ).
% antisym_conv2
thf(fact_150_antisym__conv1,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ X @ Y )
= ( X = Y ) ) ) ) ).
% antisym_conv1
thf(fact_151_less__imp__le,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% less_imp_le
thf(fact_152_le__neq__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( A2 != B3 )
=> ( ord_less @ A @ A2 @ B3 ) ) ) ) ).
% le_neq_trans
thf(fact_153_not__less,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less @ A @ X @ Y ) )
= ( ord_less_eq @ A @ Y @ X ) ) ) ).
% not_less
thf(fact_154_not__le,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less_eq @ A @ X @ Y ) )
= ( ord_less @ A @ Y @ X ) ) ) ).
% not_le
thf(fact_155_order__less__le__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_eq @ C2 @ ( F @ B3 ) @ C )
=> ( ! [X5: A,Y4: A] :
( ( ord_less @ A @ X5 @ Y4 )
=> ( ord_less @ C2 @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_less_le_subst2
thf(fact_156_order__less__le__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_eq @ B @ B3 @ C )
=> ( ! [X5: B,Y4: B] :
( ( ord_less_eq @ B @ X5 @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_157_order__le__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_eq @ A @ A2 @ B3 )
=> ( ( ord_less @ C2 @ ( F @ B3 ) @ C )
=> ( ! [X5: A,Y4: A] :
( ( ord_less_eq @ A @ X5 @ Y4 )
=> ( ord_less_eq @ C2 @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_le_less_subst2
thf(fact_158_order__le__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_eq @ A @ A2 @ ( F @ B3 ) )
=> ( ( ord_less @ B @ B3 @ C )
=> ( ! [X5: B,Y4: B] :
( ( ord_less @ B @ X5 @ Y4 )
=> ( ord_less @ A @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_159_less__le,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [X4: A,Y2: A] :
( ( ord_less_eq @ A @ X4 @ Y2 )
& ( X4 != Y2 ) ) ) ) ) ).
% less_le
thf(fact_160_le__less,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [X4: A,Y2: A] :
( ( ord_less @ A @ X4 @ Y2 )
| ( X4 = Y2 ) ) ) ) ) ).
% le_less
thf(fact_161_leI,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% leI
thf(fact_162_leD,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less @ A @ X @ Y ) ) ) ).
% leD
thf(fact_163_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_164_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_165_bot_Oextremum,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A2: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A2 ) ) ).
% bot.extremum
thf(fact_166_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_167_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_168_complete__interval,axiom,
! [A: $tType] :
( ( condit1037483654norder @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,P: A > $o] :
( ( ord_less @ A @ A2 @ B3 )
=> ( ( P @ A2 )
=> ( ~ ( P @ B3 )
=> ? [C4: A] :
( ( ord_less_eq @ A @ A2 @ C4 )
& ( ord_less_eq @ A @ C4 @ B3 )
& ! [X6: A] :
( ( ( ord_less_eq @ A @ A2 @ X6 )
& ( ord_less @ A @ X6 @ C4 ) )
=> ( P @ X6 ) )
& ! [D2: A] :
( ! [X5: A] :
( ( ( ord_less_eq @ A @ A2 @ X5 )
& ( ord_less @ A @ X5 @ D2 ) )
=> ( P @ X5 ) )
=> ( ord_less_eq @ A @ D2 @ C4 ) ) ) ) ) ) ) ).
% complete_interval
thf(fact_169_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_170_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_171_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_172_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_173_Set_Ois__empty__def,axiom,
! [A: $tType] :
( ( is_empty @ A )
= ( ^ [A5: set @ A] :
( A5
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Set.is_empty_def
thf(fact_174_Collect__empty__eq__bot,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( P
= ( bot_bot @ ( A > $o ) ) ) ) ).
% Collect_empty_eq_bot
thf(fact_175_bot__empty__eq,axiom,
! [A: $tType] :
( ( bot_bot @ ( A > $o ) )
= ( ^ [X4: A] : ( member @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% bot_empty_eq
thf(fact_176_psubsetI,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ( A3 != B2 )
=> ( ord_less @ ( set @ A ) @ A3 @ B2 ) ) ) ).
% psubsetI
thf(fact_177_less__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ( ( ord_less @ ( A > B ) )
= ( ^ [F3: A > B,G2: A > B] :
( ( ord_less_eq @ ( A > B ) @ F3 @ G2 )
& ~ ( ord_less_eq @ ( A > B ) @ G2 @ F3 ) ) ) ) ) ).
% less_fun_def
thf(fact_178_not__psubset__empty,axiom,
! [A: $tType,A3: set @ A] :
~ ( ord_less @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% not_psubset_empty
thf(fact_179_psubsetE,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ B2 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ B2 @ A3 ) ) ) ).
% psubsetE
thf(fact_180_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_181_psubset__imp__subset,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ B2 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B2 ) ) ).
% psubset_imp_subset
thf(fact_182_psubset__subset__trans,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,C3: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ B2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B2 @ C3 )
=> ( ord_less @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% psubset_subset_trans
thf(fact_183_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_184_subset__psubset__trans,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,C3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B2 )
=> ( ( ord_less @ ( set @ A ) @ B2 @ C3 )
=> ( ord_less @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% subset_psubset_trans
thf(fact_185_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_186_minf_I11_J,axiom,
! [C2: $tType,D: $tType] :
( ( ord @ C2 @ ( type2 @ C2 ) )
=> ! [F4: D] :
? [Z3: C2] :
! [X6: C2] :
( ( ord_less @ C2 @ X6 @ Z3 )
=> ( F4 = F4 ) ) ) ).
% minf(11)
thf(fact_187_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_188_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_189_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_190_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_191_minf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z4 )
=> ( ( P @ X5 )
= ( P2 @ X5 ) ) )
=> ( ? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z4 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z3: A] :
! [X6: A] :
( ( ord_less @ A @ X6 @ Z3 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P2 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ) ).
% minf(2)
thf(fact_192_minf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z4 )
=> ( ( P @ X5 )
= ( P2 @ X5 ) ) )
=> ( ? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z4 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z3: A] :
! [X6: A] :
( ( ord_less @ A @ X6 @ Z3 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P2 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ) ).
% minf(1)
thf(fact_193_pinf_I11_J,axiom,
! [C2: $tType,D: $tType] :
( ( ord @ C2 @ ( type2 @ C2 ) )
=> ! [F4: D] :
? [Z3: C2] :
! [X6: C2] :
( ( ord_less @ C2 @ Z3 @ X6 )
=> ( F4 = F4 ) ) ) ).
% pinf(11)
thf(fact_194_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_195_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_196_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_197_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_198_pinf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ Z4 @ X5 )
=> ( ( P @ X5 )
= ( P2 @ X5 ) ) )
=> ( ? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ Z4 @ X5 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z3: A] :
! [X6: A] :
( ( ord_less @ A @ Z3 @ X6 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P2 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ) ).
% pinf(2)
thf(fact_199_pinf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ Z4 @ X5 )
=> ( ( P @ X5 )
= ( P2 @ X5 ) ) )
=> ( ? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ Z4 @ X5 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z3: A] :
! [X6: A] :
( ( ord_less @ A @ Z3 @ X6 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P2 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ) ).
% pinf(1)
thf(fact_200_ex__gt__or__lt,axiom,
! [A: $tType] :
( ( condit1656338222tinuum @ A @ ( type2 @ A ) )
=> ! [A2: A] :
? [B4: A] :
( ( ord_less @ A @ A2 @ B4 )
| ( ord_less @ A @ B4 @ A2 ) ) ) ).
% ex_gt_or_lt
thf(fact_201_dependent__wellorder__choice,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ A @ ( type2 @ A ) )
=> ! [P: ( A > B ) > A > B > $o] :
( ! [R2: B,F5: A > B,G3: A > B,X5: A] :
( ! [Y5: A] :
( ( ord_less @ A @ Y5 @ X5 )
=> ( ( F5 @ Y5 )
= ( G3 @ Y5 ) ) )
=> ( ( P @ F5 @ X5 @ R2 )
= ( P @ G3 @ X5 @ R2 ) ) )
=> ( ! [X5: A,F5: A > B] :
( ! [Y5: A] :
( ( ord_less @ A @ Y5 @ X5 )
=> ( P @ F5 @ Y5 @ ( F5 @ Y5 ) ) )
=> ? [X12: B] : ( P @ F5 @ X5 @ X12 ) )
=> ? [F5: A > B] :
! [X6: A] : ( P @ F5 @ X6 @ ( F5 @ X6 ) ) ) ) ) ).
% dependent_wellorder_choice
thf(fact_202_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_203_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_204_psubsetD,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,C: A] :
( ( ord_less @ ( set @ A ) @ A3 @ B2 )
=> ( ( member @ A @ C @ A3 )
=> ( member @ A @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_205_psubset__trans,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,C3: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ B2 )
=> ( ( ord_less @ ( set @ A ) @ B2 @ C3 )
=> ( ord_less @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% psubset_trans
thf(fact_206_chain__subset__def,axiom,
! [A: $tType] :
( ( chain_subset @ A )
= ( ^ [C5: set @ ( set @ A )] :
! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ C5 )
=> ! [Y2: set @ A] :
( ( member @ ( set @ A ) @ Y2 @ C5 )
=> ( ( ord_less_eq @ ( set @ A ) @ X4 @ Y2 )
| ( ord_less_eq @ ( set @ A ) @ Y2 @ X4 ) ) ) ) ) ) ).
% chain_subset_def
thf(fact_207_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_208_in__inv__imagep,axiom,
! [B: $tType,A: $tType] :
( ( inv_imagep @ A @ B )
= ( ^ [R3: A > A > $o,F3: B > A,X4: B,Y2: B] : ( R3 @ ( F3 @ X4 ) @ ( F3 @ Y2 ) ) ) ) ).
% in_inv_imagep
thf(fact_209_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_210_dag_Osize_I3_J,axiom,
( ( size_size @ binDag_Mirabelle_dag @ binDag_Mirabelle_Tip )
= ( zero_zero @ nat ) ) ).
% dag.size(3)
thf(fact_211_order_Oordering__axioms,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ordering @ A @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% order.ordering_axioms
thf(fact_212_ordering_Onot__eq__order__implies__strict,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B3: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( A2 != B3 )
=> ( ( Less_eq @ A2 @ B3 )
=> ( Less @ A2 @ B3 ) ) ) ) ).
% ordering.not_eq_order_implies_strict
thf(fact_213_ordering_Ostrict__implies__not__eq,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B3: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( Less @ A2 @ B3 )
=> ( A2 != B3 ) ) ) ).
% ordering.strict_implies_not_eq
thf(fact_214_ordering_Ostrict__implies__order,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B3: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( Less @ A2 @ B3 )
=> ( Less_eq @ A2 @ B3 ) ) ) ).
% ordering.strict_implies_order
thf(fact_215_ordering_Ostrict__iff__order,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B3: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( Less @ A2 @ B3 )
= ( ( Less_eq @ A2 @ B3 )
& ( A2 != B3 ) ) ) ) ).
% ordering.strict_iff_order
thf(fact_216_ordering_Oorder__iff__strict,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B3: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( Less_eq @ A2 @ B3 )
= ( ( Less @ A2 @ B3 )
| ( A2 = B3 ) ) ) ) ).
% ordering.order_iff_strict
thf(fact_217_ordering_Ostrict__trans2,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B3: A,C: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( Less @ A2 @ B3 )
=> ( ( Less_eq @ B3 @ C )
=> ( Less @ A2 @ C ) ) ) ) ).
% ordering.strict_trans2
thf(fact_218_ordering_Ostrict__trans1,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B3: A,C: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( Less_eq @ A2 @ B3 )
=> ( ( Less @ B3 @ C )
=> ( Less @ A2 @ C ) ) ) ) ).
% ordering.strict_trans1
thf(fact_219_ordering_Ostrict__trans,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B3: A,C: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( Less @ A2 @ B3 )
=> ( ( Less @ B3 @ C )
=> ( Less @ A2 @ C ) ) ) ) ).
% ordering.strict_trans
thf(fact_220_ordering_Oantisym,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B3: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( Less_eq @ A2 @ B3 )
=> ( ( Less_eq @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ) ).
% ordering.antisym
thf(fact_221_ordering_Oirrefl,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ~ ( Less @ A2 @ A2 ) ) ).
% ordering.irrefl
thf(fact_222_ordering_Otrans,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B3: A,C: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( Less_eq @ A2 @ B3 )
=> ( ( Less_eq @ B3 @ C )
=> ( Less_eq @ A2 @ C ) ) ) ) ).
% ordering.trans
thf(fact_223_ordering_Ointro,axiom,
! [A: $tType,Less: A > A > $o,Less_eq: A > A > $o] :
( ! [A4: A,B4: A] :
( ( Less @ A4 @ B4 )
= ( ( Less_eq @ A4 @ B4 )
& ( A4 != B4 ) ) )
=> ( ! [A4: A] : ( Less_eq @ A4 @ A4 )
=> ( ! [A4: A,B4: A] :
( ( Less_eq @ A4 @ B4 )
=> ( ( Less_eq @ B4 @ A4 )
=> ( A4 = B4 ) ) )
=> ( ! [A4: A,B4: A,C4: A] :
( ( Less_eq @ A4 @ B4 )
=> ( ( Less_eq @ B4 @ C4 )
=> ( Less_eq @ A4 @ C4 ) ) )
=> ( ordering @ A @ Less_eq @ Less ) ) ) ) ) ).
% ordering.intro
thf(fact_224_ordering_Orefl,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( Less_eq @ A2 @ A2 ) ) ).
% ordering.refl
thf(fact_225_ordering_Oasym,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B3: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( Less @ A2 @ B3 )
=> ~ ( Less @ B3 @ A2 ) ) ) ).
% ordering.asym
thf(fact_226_ordering__def,axiom,
! [A: $tType] :
( ( ordering @ A )
= ( ^ [Less_eq2: A > A > $o,Less2: A > A > $o] :
( ! [A6: A,B6: A] :
( ( Less2 @ A6 @ B6 )
= ( ( Less_eq2 @ A6 @ B6 )
& ( A6 != B6 ) ) )
& ! [A6: A] : ( Less_eq2 @ A6 @ A6 )
& ! [A6: A,B6: A] :
( ( Less_eq2 @ A6 @ B6 )
=> ( ( Less_eq2 @ B6 @ A6 )
=> ( A6 = B6 ) ) )
& ! [A6: A,B6: A,C6: A] :
( ( Less_eq2 @ A6 @ B6 )
=> ( ( Less_eq2 @ B6 @ C6 )
=> ( Less_eq2 @ A6 @ C6 ) ) ) ) ) ) ).
% ordering_def
thf(fact_227_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_228_le__zero__eq,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [N: A] :
( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
= ( N
= ( zero_zero @ A ) ) ) ) ).
% le_zero_eq
thf(fact_229_zero__le,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% zero_le
thf(fact_230_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_231_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_232_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_233_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_234_dag_Osize__gen_I1_J,axiom,
( ( binDag1924123185ze_dag @ binDag_Mirabelle_Tip )
= ( zero_zero @ nat ) ) ).
% dag.size_gen(1)
thf(fact_235_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_236_le__numeral__extra_I3_J,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% le_numeral_extra(3)
thf(fact_237_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_238_sgn__less,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less @ A @ ( sgn_sgn @ A @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% sgn_less
thf(fact_239_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
thf(fact_240_sgn__0__0,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ( sgn_sgn @ A @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% sgn_0_0
thf(fact_241_sgn__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( sgn_sgn @ A @ A2 )
= ( one_one @ A ) ) ) ) ).
% sgn_pos
thf(fact_242_of__bool__eq_I1_J,axiom,
! [A: $tType] :
( ( zero_neq_one @ A @ ( type2 @ A ) )
=> ( ( zero_neq_one_of_bool @ A @ $false )
= ( zero_zero @ A ) ) ) ).
% of_bool_eq(1)
thf(fact_243_of__bool__eq_I2_J,axiom,
! [A: $tType] :
( ( zero_neq_one @ A @ ( type2 @ A ) )
=> ( ( zero_neq_one_of_bool @ A @ $true )
= ( one_one @ A ) ) ) ).
% of_bool_eq(2)
thf(fact_244_not__one__less__zero,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A @ ( type2 @ A ) )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% not_one_less_zero
thf(fact_245_zero__less__one,axiom,
! [A: $tType] :
( ( zero_less_one @ A @ ( type2 @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% zero_less_one
thf(fact_246_zero__le__one,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A @ ( type2 @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% zero_le_one
thf(fact_247_not__one__le__zero,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A @ ( type2 @ A ) )
=> ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% not_one_le_zero
thf(fact_248_split__of__bool__asm,axiom,
! [A: $tType] :
( ( zero_neq_one @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P3: $o] :
( ( P @ ( zero_neq_one_of_bool @ A @ P3 ) )
= ( ~ ( ( P3
& ~ ( P @ ( one_one @ A ) ) )
| ( ~ P3
& ~ ( P @ ( zero_zero @ A ) ) ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_249_split__of__bool,axiom,
! [A: $tType] :
( ( zero_neq_one @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P3: $o] :
( ( P @ ( zero_neq_one_of_bool @ A @ P3 ) )
= ( ( P3
=> ( P @ ( one_one @ A ) ) )
& ( ~ P3
=> ( P @ ( zero_zero @ A ) ) ) ) ) ) ).
% split_of_bool
thf(fact_250_zero__neq__one,axiom,
! [A: $tType] :
( ( zero_neq_one @ A @ ( type2 @ A ) )
=> ( ( zero_zero @ A )
!= ( one_one @ A ) ) ) ).
% zero_neq_one
thf(fact_251_of__bool__def,axiom,
! [A: $tType] :
( ( zero_neq_one @ A @ ( type2 @ A ) )
=> ( ( zero_neq_one_of_bool @ A )
= ( ^ [P4: $o] : ( if @ A @ P4 @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ) ) ).
% of_bool_def
thf(fact_252_of__bool__eq__iff,axiom,
! [A: $tType] :
( ( zero_neq_one @ A @ ( type2 @ A ) )
=> ! [P3: $o,Q3: $o] :
( ( ( zero_neq_one_of_bool @ A @ P3 )
= ( zero_neq_one_of_bool @ A @ Q3 ) )
= ( P3 = Q3 ) ) ) ).
% of_bool_eq_iff
thf(fact_253_le__numeral__extra_I4_J,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% le_numeral_extra(4)
thf(fact_254_less__numeral__extra_I4_J,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% less_numeral_extra(4)
thf(fact_255_le__numeral__extra_I2_J,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% le_numeral_extra(2)
%----Type constructors (33)
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__nonzero__semiring,axiom,
linord1659791738miring @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Olinordered__semidom,axiom,
linordered_semidom @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Ozero__less__one,axiom,
zero_less_one @ 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___Rings_Ozero__neq__one,axiom,
zero_neq_one @ 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,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $true @ X @ Y )
= X ) ).
%----Conjectures (2)
thf(conj_0,hypothesis,
( ~ ( member @ simpl_ref @ a @ ( binDag1380252983set_of @ l ) )
=> ( ~ ( member @ simpl_ref @ a @ ( binDag1380252983set_of @ r ) )
=> thesis ) ) ).
thf(conj_1,conjecture,
thesis ).
%------------------------------------------------------------------------------