TPTP Problem File: DAT144^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : DAT144^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Coinductive stream 14
% Version : [Bla16] axioms : Especial.
% English :
% Refs : [Loc10] Lochbihler (2010), Coinductive
% : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% Source : [Bla16]
% Names : coinductive_stream__14.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 : 378 ( 141 unt; 59 typ; 0 def)
% Number of atoms : 828 ( 201 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 3312 ( 31 ~; 3 |; 42 &;2856 @)
% ( 0 <=>; 380 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 7 avg)
% Number of types : 7 ( 6 usr)
% Number of type conns : 402 ( 402 >; 0 *; 0 +; 0 <<)
% Number of symbols : 55 ( 53 usr; 2 con; 0-6 aty)
% Number of variables : 1016 ( 68 ^; 875 !; 17 ?;1016 :)
% ( 56 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 15:12:48.657
%------------------------------------------------------------------------------
%----Could-be-implicit typings (8)
thf(ty_t_Extended__Nat_Oenat,type,
extended_enat: $tType ).
thf(ty_t_Real_Oreal,type,
real: $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Num_Onum,type,
num: $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_t_Int_Oint,type,
int: $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
thf(ty_tf_a,type,
a: $tType ).
%----Explicit typings (51)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Oring__1,type,
ring_1:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Otop,type,
top:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[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__top,type,
order_top:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Oboolean__algebra,type,
boolean_algebra:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Olinordered__semidom,type,
linordered_semidom:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Olinordered__ab__group__add,type,
linord219039673up_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Archimedean__Field_Ofloor__ceiling,type,
archim1727834104eiling:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Topological__Spaces_Ouniform__space,type,
topolo47006728_space:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
comple187826305attice:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Archimedean__Field_Oarchimedean__field,type,
archim1804426504_field:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni770627133id_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_Archimedean__Field_Oceiling,type,
archimedean_ceiling:
!>[A: $tType] : ( A > int ) ).
thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor,type,
archim2085082626_floor:
!>[A: $tType] : ( A > int ) ).
thf(sy_c_Archimedean__Field_Oround,type,
archimedean_round:
!>[A: $tType] : ( A > int ) ).
thf(sy_c_BNF__Composition_ODEADID_Opred__DEADID,type,
bNF_pred_DEADID:
!>[A: $tType] : ( A > $o ) ).
thf(sy_c_BNF__Def_OGrp,type,
bNF_Grp:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > A > B > $o ) ).
thf(sy_c_BNF__Def_Oeq__onp,type,
bNF_eq_onp:
!>[A: $tType] : ( ( A > $o ) > A > A > $o ) ).
thf(sy_c_Extended__Nat_Oenat,type,
extended_enat2: nat > extended_enat ).
thf(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Ouminus__class_Ouminus,type,
uminus_uminus:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_HOL_Oundefined,type,
undefined:
!>[A: $tType] : A ).
thf(sy_c_Hilbert__Choice_OGreatestM,type,
hilbert_GreatestM:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( A > $o ) > A ) ).
thf(sy_c_Hilbert__Choice_OLeastM,type,
hilbert_LeastM:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( A > $o ) > A ) ).
thf(sy_c_Int_Onat,type,
nat2: int > nat ).
thf(sy_c_Lifting_ONEG,type,
neg:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( A > B > $o ) > $o ) ).
thf(sy_c_Lifting_OPOS,type,
pos:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( A > B > $o ) > $o ) ).
thf(sy_c_Lifting_OQuotient,type,
quotient:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > ( B > A ) > ( A > B > $o ) > $o ) ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
semiring_1_of_nat:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_Nat__Transfer_Otransfer__morphism,type,
nat_tr1645093318rphism:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( B > $o ) > $o ) ).
thf(sy_c_Nat__Transfer_Otsub,type,
nat_tsub: int > int > int ).
thf(sy_c_Num_Onumeral__class_Onumeral,type,
numeral_numeral:
!>[A: $tType] : ( num > A ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oorder__class_Oantimono,type,
order_antimono:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Orderings_Otop__class_Otop,type,
top_top:
!>[A: $tType] : A ).
thf(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Relation_ODomainp,type,
domainp:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > A > $o ) ).
thf(sy_c_Relation_Oinv__imagep,type,
inv_imagep:
!>[B: $tType,A: $tType] : ( ( B > B > $o ) > ( A > B ) > A > A > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Topological__Spaces_Omonoseq,type,
topological_monoseq:
!>[A: $tType] : ( ( nat > A ) > $o ) ).
thf(sy_c_Topological__Spaces_Ouniform__space__class_Ototally__bounded,type,
topolo406746546ounded:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_P,type,
p: a > $o ).
thf(sy_v_x,type,
x: a ).
%----Relevant facts (256)
thf(fact_0_eq__onp__eqD,axiom,
! [A: $tType,P: A > $o,Q: A > A > $o,X: A] :
( ( ( bNF_eq_onp @ A @ P )
= Q )
=> ( ( P @ X )
= ( Q @ X @ X ) ) ) ).
% eq_onp_eqD
thf(fact_1_eq__onp__mono0,axiom,
! [A: $tType,A2: set @ A,P: A > $o,Q: A > $o] :
( ! [X2: A] :
( ( member @ A @ X2 @ A2 )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ! [X3: A] :
( ( member @ A @ X3 @ A2 )
=> ! [Xa: A] :
( ( member @ A @ Xa @ A2 )
=> ( ( bNF_eq_onp @ A @ P @ X3 @ Xa )
=> ( bNF_eq_onp @ A @ Q @ X3 @ Xa ) ) ) ) ) ).
% eq_onp_mono0
thf(fact_2_eq__onp__to__eq,axiom,
! [A: $tType,P: A > $o,X: A,Y: A] :
( ( bNF_eq_onp @ A @ P @ X @ Y )
=> ( X = Y ) ) ).
% eq_onp_to_eq
thf(fact_3_eq__onp__same__args,axiom,
! [A: $tType,P: A > $o,X: A] :
( ( bNF_eq_onp @ A @ P @ X @ X )
= ( P @ X ) ) ).
% eq_onp_same_args
thf(fact_4_eq__onp__live__step,axiom,
! [A: $tType,X: $o,Y: $o,P: A > $o,A3: A] :
( ( X = Y )
=> ( ( ( bNF_eq_onp @ A @ P @ A3 @ A3 )
& X )
= ( ( P @ A3 )
& Y ) ) ) ).
% eq_onp_live_step
thf(fact_5_DEADID_Orel__eq__onp,axiom,
! [A: $tType] :
( ( ^ [Y2: A,Z: A] : Y2 = Z )
= ( bNF_eq_onp @ A @ ( bNF_pred_DEADID @ A ) ) ) ).
% DEADID.rel_eq_onp
thf(fact_6_eq__onp__le__eq,axiom,
! [A: $tType,P: A > $o] :
( ord_less_eq @ ( A > A > $o ) @ ( bNF_eq_onp @ A @ P )
@ ^ [Y2: A,Z: A] : Y2 = Z ) ).
% eq_onp_le_eq
thf(fact_7_type__definition__Quotient__not__empty,axiom,
! [B: $tType,A: $tType,P: A > $o,Abs: A > B,Rep: B > A,T: A > B > $o] :
( ( quotient @ A @ B @ ( bNF_eq_onp @ A @ P ) @ Abs @ Rep @ T )
=> ? [X1: A] : ( P @ X1 ) ) ).
% type_definition_Quotient_not_empty
thf(fact_8_eq__onp__top__eq__eq,axiom,
! [A: $tType] :
( ( bNF_eq_onp @ A @ ( top_top @ ( A > $o ) ) )
= ( ^ [Y2: A,Z: A] : Y2 = Z ) ) ).
% eq_onp_top_eq_eq
thf(fact_9_top__conj_I2_J,axiom,
! [A: $tType,P: $o,X: A] :
( ( P
& ( top_top @ ( A > $o ) @ X ) )
= P ) ).
% top_conj(2)
thf(fact_10_top__conj_I1_J,axiom,
! [A: $tType,X: A,P: $o] :
( ( ( top_top @ ( A > $o ) @ X )
& P )
= P ) ).
% top_conj(1)
thf(fact_11_ge__eq__refl,axiom,
! [A: $tType,R: A > A > $o,X: A] :
( ( ord_less_eq @ ( A > A > $o )
@ ^ [Y2: A,Z: A] : Y2 = Z
@ R )
=> ( R @ X @ X ) ) ).
% ge_eq_refl
thf(fact_12_refl__ge__eq,axiom,
! [A: $tType,R: A > A > $o] :
( ! [X2: A] : ( R @ X2 @ X2 )
=> ( ord_less_eq @ ( A > A > $o )
@ ^ [Y2: A,Z: A] : Y2 = Z
@ R ) ) ).
% refl_ge_eq
thf(fact_13_Quotient__rel,axiom,
! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T: A > B > $o,R2: A,S: A] :
( ( quotient @ A @ B @ R @ Abs @ Rep @ T )
=> ( ( ( R @ R2 @ R2 )
& ( R @ S @ S )
& ( ( Abs @ R2 )
= ( Abs @ S ) ) )
= ( R @ R2 @ S ) ) ) ).
% Quotient_rel
thf(fact_14_Quotient__refl1,axiom,
! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T: A > B > $o,R2: A,S: A] :
( ( quotient @ A @ B @ R @ Abs @ Rep @ T )
=> ( ( R @ R2 @ S )
=> ( R @ R2 @ R2 ) ) ) ).
% Quotient_refl1
thf(fact_15_Quotient__refl2,axiom,
! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T: A > B > $o,R2: A,S: A] :
( ( quotient @ A @ B @ R @ Abs @ Rep @ T )
=> ( ( R @ R2 @ S )
=> ( R @ S @ S ) ) ) ).
% Quotient_refl2
thf(fact_16_Quotient__Rep__eq,axiom,
! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T: A > B > $o,X4: B,X: A] :
( ( quotient @ A @ B @ R @ Abs @ Rep @ T )
=> ( ( X4
= ( Abs @ X ) )
=> ( ( Rep @ X4 )
= ( Rep @ X4 ) ) ) ) ).
% Quotient_Rep_eq
thf(fact_17_Quotient__abs__rep,axiom,
! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T: A > B > $o,A3: B] :
( ( quotient @ A @ B @ R @ Abs @ Rep @ T )
=> ( ( Abs @ ( Rep @ A3 ) )
= A3 ) ) ).
% Quotient_abs_rep
thf(fact_18_Quotient__alt__def,axiom,
! [B: $tType,A: $tType] :
( ( quotient @ A @ B )
= ( ^ [R3: A > A > $o,Abs2: A > B,Rep2: B > A,T2: A > B > $o] :
( ! [A4: A,B2: B] :
( ( T2 @ A4 @ B2 )
=> ( ( Abs2 @ A4 )
= B2 ) )
& ! [B2: B] : ( T2 @ ( Rep2 @ B2 ) @ B2 )
& ! [X5: A,Y3: A] :
( ( R3 @ X5 @ Y3 )
= ( ( T2 @ X5 @ ( Abs2 @ X5 ) )
& ( T2 @ Y3 @ ( Abs2 @ Y3 ) )
& ( ( Abs2 @ X5 )
= ( Abs2 @ Y3 ) ) ) ) ) ) ) ).
% Quotient_alt_def
thf(fact_19_Quotient__rel__abs,axiom,
! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T: A > B > $o,R2: A,S: A] :
( ( quotient @ A @ B @ R @ Abs @ Rep @ T )
=> ( ( R @ R2 @ S )
=> ( ( Abs @ R2 )
= ( Abs @ S ) ) ) ) ).
% Quotient_rel_abs
thf(fact_20_Quotient__rel__rep,axiom,
! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T: A > B > $o,A3: B,B3: B] :
( ( quotient @ A @ B @ R @ Abs @ Rep @ T )
=> ( ( R @ ( Rep @ A3 ) @ ( Rep @ B3 ) )
= ( A3 = B3 ) ) ) ).
% Quotient_rel_rep
thf(fact_21_Quotient__rep__abs,axiom,
! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T: A > B > $o,R2: A] :
( ( quotient @ A @ B @ R @ Abs @ Rep @ T )
=> ( ( R @ R2 @ R2 )
=> ( R @ ( Rep @ ( Abs @ R2 ) ) @ R2 ) ) ) ).
% Quotient_rep_abs
thf(fact_22_Quotient__alt__def2,axiom,
! [B: $tType,A: $tType] :
( ( quotient @ A @ B )
= ( ^ [R3: A > A > $o,Abs2: A > B,Rep2: B > A,T2: A > B > $o] :
( ! [A4: A,B2: B] :
( ( T2 @ A4 @ B2 )
=> ( ( Abs2 @ A4 )
= B2 ) )
& ! [B2: B] : ( T2 @ ( Rep2 @ B2 ) @ B2 )
& ! [X5: A,Y3: A] :
( ( R3 @ X5 @ Y3 )
= ( ( T2 @ X5 @ ( Abs2 @ Y3 ) )
& ( T2 @ Y3 @ ( Abs2 @ X5 ) ) ) ) ) ) ) ).
% Quotient_alt_def2
thf(fact_23_Quotient__alt__def3,axiom,
! [B: $tType,A: $tType] :
( ( quotient @ A @ B )
= ( ^ [R3: A > A > $o,Abs2: A > B,Rep2: B > A,T2: A > B > $o] :
( ! [A4: A,B2: B] :
( ( T2 @ A4 @ B2 )
=> ( ( Abs2 @ A4 )
= B2 ) )
& ! [B2: B] : ( T2 @ ( Rep2 @ B2 ) @ B2 )
& ! [X5: A,Y3: A] :
( ( R3 @ X5 @ Y3 )
= ( ? [Z2: B] :
( ( T2 @ X5 @ Z2 )
& ( T2 @ Y3 @ Z2 ) ) ) ) ) ) ) ).
% Quotient_alt_def3
thf(fact_24_Quotient__rel__abs2,axiom,
! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T: A > B > $o,X: B,Y: A] :
( ( quotient @ A @ B @ R @ Abs @ Rep @ T )
=> ( ( R @ ( Rep @ X ) @ Y )
=> ( X
= ( Abs @ Y ) ) ) ) ).
% Quotient_rel_abs2
thf(fact_25_Quotient__rep__reflp,axiom,
! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T: A > B > $o,A3: B] :
( ( quotient @ A @ B @ R @ Abs @ Rep @ T )
=> ( R @ ( Rep @ A3 ) @ ( Rep @ A3 ) ) ) ).
% Quotient_rep_reflp
thf(fact_26_Quotient__abs__induct,axiom,
! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T: A > B > $o,P: B > $o,X: B] :
( ( quotient @ A @ B @ R @ Abs @ Rep @ T )
=> ( ! [Y4: A] :
( ( R @ Y4 @ Y4 )
=> ( P @ ( Abs @ Y4 ) ) )
=> ( P @ X ) ) ) ).
% Quotient_abs_induct
thf(fact_27_Quotient__rep__abs__eq,axiom,
! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T: A > B > $o,T3: A] :
( ( quotient @ A @ B @ R @ Abs @ Rep @ T )
=> ( ( R @ T3 @ T3 )
=> ( ( ord_less_eq @ ( A > A > $o ) @ R
@ ^ [Y2: A,Z: A] : Y2 = Z )
=> ( ( Rep @ ( Abs @ T3 ) )
= T3 ) ) ) ) ).
% Quotient_rep_abs_eq
thf(fact_28_Quotient__to__transfer,axiom,
! [A: $tType,B: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T: A > B > $o,C: A,C2: B] :
( ( quotient @ A @ B @ R @ Abs @ Rep @ T )
=> ( ( R @ C @ C )
=> ( ( C2
= ( Abs @ C ) )
=> ( T @ C @ C2 ) ) ) ) ).
% Quotient_to_transfer
thf(fact_29_DEADID_Opred__rel,axiom,
! [A: $tType] :
( ( bNF_pred_DEADID @ A )
= ( ^ [X5: A] : X5 = X5 ) ) ).
% DEADID.pred_rel
thf(fact_30_DEADID_Opred__True,axiom,
! [A: $tType] :
( ( bNF_pred_DEADID @ A )
= ( ^ [Uu: A] : $true ) ) ).
% DEADID.pred_True
thf(fact_31_predicate2D__conj,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,R: $o,X: A,Y: B] :
( ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
& R )
=> ( R
& ( ( P @ X @ Y )
=> ( Q @ X @ Y ) ) ) ) ).
% predicate2D_conj
thf(fact_32_Quotient__rep__abs__fold__unmap,axiom,
! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T: A > B > $o,X4: B,X: A,Rep3: B > A] :
( ( quotient @ A @ B @ R @ Abs @ Rep @ T )
=> ( ( X4
= ( Abs @ X ) )
=> ( ( R @ X @ X )
=> ( ( ( Rep @ X4 )
= ( Rep3 @ X4 ) )
=> ( R @ ( Rep3 @ X4 ) @ X ) ) ) ) ) ).
% Quotient_rep_abs_fold_unmap
thf(fact_33_DEADID_Opred__cong__simp,axiom,
! [A: $tType,X: A,Ya: A] :
( ( X = Ya )
=> ( ( bNF_pred_DEADID @ A @ X )
= ( bNF_pred_DEADID @ A @ Ya ) ) ) ).
% DEADID.pred_cong_simp
thf(fact_34_DEADID_Opred__mono__strong,axiom,
! [A: $tType,X: A] :
( ( bNF_pred_DEADID @ A @ X )
=> ( bNF_pred_DEADID @ A @ X ) ) ).
% DEADID.pred_mono_strong
thf(fact_35_top__apply,axiom,
! [C3: $tType,D: $tType] :
( ( top @ C3 @ ( type2 @ C3 ) )
=> ( ( top_top @ ( D > C3 ) )
= ( ^ [X5: D] : ( top_top @ C3 ) ) ) ) ).
% top_apply
thf(fact_36_top1I,axiom,
! [A: $tType,X: A] : ( top_top @ ( A > $o ) @ X ) ).
% top1I
thf(fact_37_predicate2I,axiom,
! [B: $tType,A: $tType,P: A > B > $o,Q: A > B > $o] :
( ! [X2: A,Y4: B] :
( ( P @ X2 @ Y4 )
=> ( Q @ X2 @ Y4 ) )
=> ( ord_less_eq @ ( A > B > $o ) @ P @ Q ) ) ).
% predicate2I
thf(fact_38_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_39_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_40_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_41_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B] :
( ! [X2: A] : ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) )
=> ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% le_funI
thf(fact_42_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B] :
! [X5: A] : ( ord_less_eq @ B @ ( F2 @ X5 ) @ ( G2 @ X5 ) ) ) ) ) ).
% le_fun_def
thf(fact_43_top__greatest,axiom,
! [A: $tType] :
( ( order_top @ A @ ( type2 @ A ) )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ ( top_top @ A ) ) ) ).
% top_greatest
thf(fact_44_top_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_top @ A @ ( type2 @ A ) )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
= ( A3
= ( top_top @ A ) ) ) ) ).
% top.extremum_unique
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A3: A,P: A > $o] :
( ( member @ A @ A3 @ ( collect @ A @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A2: set @ A] :
( ( collect @ A
@ ^ [X5: A] : ( member @ A @ X5 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X2: A] :
( ( P @ X2 )
= ( Q @ X2 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X2: A] :
( ( F @ X2 )
= ( G @ X2 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_top_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_top @ A @ ( type2 @ A ) )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A3 )
=> ( A3
= ( top_top @ A ) ) ) ) ).
% top.extremum_uniqueI
thf(fact_50_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ) ).
% dual_order.antisym
thf(fact_51_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A3: A,C: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ C @ B3 )
=> ( ord_less_eq @ A @ C @ A3 ) ) ) ) ).
% dual_order.trans
thf(fact_52_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > A > $o,A3: A,B3: A] :
( ! [A5: A,B4: A] :
( ( ord_less_eq @ A @ A5 @ B4 )
=> ( P @ A5 @ B4 ) )
=> ( ! [A5: A,B4: A] :
( ( P @ B4 @ A5 )
=> ( P @ A5 @ B4 ) )
=> ( P @ A3 @ B3 ) ) ) ) ).
% linorder_wlog
thf(fact_53_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% dual_order.refl
thf(fact_54_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z3: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z3 )
=> ( ord_less_eq @ A @ X @ Z3 ) ) ) ) ).
% order_trans
thf(fact_55_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ) ).
% order_class.order.antisym
thf(fact_56_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A,C: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% ord_le_eq_trans
thf(fact_57_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A,C: A] :
( ( A3 = B3 )
=> ( ( ord_less_eq @ A @ B3 @ C )
=> ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% ord_eq_le_trans
thf(fact_58_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_59_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z3: A] :
( ( ( ord_less_eq @ A @ X @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z3 ) )
=> ( ( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less_eq @ A @ X @ Z3 ) )
=> ( ( ( ord_less_eq @ A @ X @ Z3 )
=> ~ ( ord_less_eq @ A @ Z3 @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z3 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z3 )
=> ~ ( ord_less_eq @ A @ Z3 @ X ) )
=> ~ ( ( ord_less_eq @ A @ Z3 @ X )
=> ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_60_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A,C: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ C )
=> ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% order.trans
thf(fact_61_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_62_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_63_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_64_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_65_eq__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ^ [Y2: A,Z: A] : Y2 = Z )
= ( ^ [X5: A,Y3: A] :
( ( ord_less_eq @ A @ X5 @ Y3 )
& ( ord_less_eq @ A @ Y3 @ X5 ) ) ) ) ) ).
% eq_iff
thf(fact_66_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A3: A,B3: A,F: A > B,C: B] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: A,Y4: A] :
( ( ord_less_eq @ A @ X2 @ Y4 )
=> ( ord_less_eq @ B @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ B @ ( F @ A3 ) @ C ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_67_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A3: A,F: B > A,B3: B,C: B] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C )
=> ( ! [X2: B,Y4: B] :
( ( ord_less_eq @ B @ X2 @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_68_order__subst2,axiom,
! [A: $tType,C3: $tType] :
( ( ( order @ C3 @ ( type2 @ C3 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A3: A,B3: A,F: A > C3,C: C3] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ C3 @ ( F @ B3 ) @ C )
=> ( ! [X2: A,Y4: A] :
( ( ord_less_eq @ A @ X2 @ Y4 )
=> ( ord_less_eq @ C3 @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ C3 @ ( F @ A3 ) @ C ) ) ) ) ) ).
% order_subst2
thf(fact_69_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A3: A,F: B > A,B3: B,C: B] :
( ( ord_less_eq @ A @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C )
=> ( ! [X2: B,Y4: B] :
( ( ord_less_eq @ B @ X2 @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% order_subst1
thf(fact_70_rev__predicate2D,axiom,
! [A: $tType,B: $tType,P: A > B > $o,X: A,Y: B,Q: A > B > $o] :
( ( P @ X @ Y )
=> ( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
=> ( Q @ X @ Y ) ) ) ).
% rev_predicate2D
thf(fact_71_predicate2D,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Q: A > B > $o,X: A,Y: B] :
( ( ord_less_eq @ ( A > B > $o ) @ P @ Q )
=> ( ( P @ X @ Y )
=> ( Q @ X @ Y ) ) ) ).
% predicate2D
thf(fact_72_top__empty__eq,axiom,
! [A: $tType] :
( ( top_top @ ( A > $o ) )
= ( ^ [X5: A] : ( member @ A @ X5 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% top_empty_eq
thf(fact_73_top__set__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A @ ( top_top @ ( A > $o ) ) ) ) ).
% top_set_def
thf(fact_74_lfp_Oleq__antisym,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ X )
=> ( X = Y ) ) ) ) ).
% lfp.leq_antisym
thf(fact_75_gfp_Oleq__antisym,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( X = Y ) ) ) ) ).
% gfp.leq_antisym
thf(fact_76_lfp_Oleq__trans,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z3: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z3 )
=> ( ord_less_eq @ A @ X @ Z3 ) ) ) ) ).
% lfp.leq_trans
thf(fact_77_gfp_Oleq__trans,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A,Z3: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_less_eq @ A @ Z3 @ Y )
=> ( ord_less_eq @ A @ Z3 @ X ) ) ) ) ).
% gfp.leq_trans
thf(fact_78_lfp_Oleq__refl,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% lfp.leq_refl
thf(fact_79_wlog__linorder__le,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > A > $o,B3: A,A3: A] :
( ! [A5: A,B4: A] :
( ( ord_less_eq @ A @ A5 @ B4 )
=> ( P @ A5 @ B4 ) )
=> ( ( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A3 @ B3 ) ) ) ) ).
% wlog_linorder_le
thf(fact_80_UNIV__I,axiom,
! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_I
thf(fact_81_UNIV__eq__I,axiom,
! [A: $tType,A2: set @ A] :
( ! [X2: A] : ( member @ A @ X2 @ A2 )
=> ( ( top_top @ ( set @ A ) )
= A2 ) ) ).
% UNIV_eq_I
thf(fact_82_subset__UNIV,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ ( top_top @ ( set @ A ) ) ) ).
% subset_UNIV
thf(fact_83_UNIV__witness,axiom,
! [A: $tType] :
? [X2: A] : ( member @ A @ X2 @ ( top_top @ ( set @ A ) ) ) ).
% UNIV_witness
thf(fact_84_iso__tuple__UNIV__I,axiom,
! [A: $tType,X: A] : ( member @ A @ X @ ( top_top @ ( set @ A ) ) ) ).
% iso_tuple_UNIV_I
thf(fact_85_in__inv__imagep,axiom,
! [B: $tType,A: $tType] :
( ( inv_imagep @ A @ B )
= ( ^ [R4: A > A > $o,F2: B > A,X5: B,Y3: B] : ( R4 @ ( F2 @ X5 ) @ ( F2 @ Y3 ) ) ) ) ).
% in_inv_imagep
thf(fact_86_eq__onp__to__Domainp,axiom,
! [B: $tType,A: $tType,P: A > $o,Abs: A > B,Rep: B > A,T: A > B > $o] :
( ( quotient @ A @ B @ ( bNF_eq_onp @ A @ P ) @ Abs @ Rep @ T )
=> ( ( domainp @ A @ B @ T )
= P ) ) ).
% eq_onp_to_Domainp
thf(fact_87_type__definition__Quotient__not__empty__witness,axiom,
! [A: $tType,B: $tType,P: A > $o,Abs: A > B,Rep: B > A,T: A > B > $o] :
( ( quotient @ A @ B @ ( bNF_eq_onp @ A @ P ) @ Abs @ Rep @ T )
=> ( P @ ( Rep @ ( undefined @ B ) ) ) ) ).
% type_definition_Quotient_not_empty_witness
thf(fact_88_NEG__def,axiom,
! [B: $tType,A: $tType] :
( ( neg @ A @ B )
= ( ^ [A6: A > B > $o,B5: A > B > $o] : ( ord_less_eq @ ( A > B > $o ) @ B5 @ A6 ) ) ) ).
% NEG_def
thf(fact_89_antimonoD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A @ ( type2 @ A ) )
& ( order @ B @ ( type2 @ B ) ) )
=> ! [F: A > B,X: A,Y: A] :
( ( order_antimono @ A @ B @ F )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ B @ ( F @ Y ) @ ( F @ X ) ) ) ) ) ).
% antimonoD
thf(fact_90_subsetI,axiom,
! [A: $tType,A2: set @ A,B6: set @ A] :
( ! [X2: A] :
( ( member @ A @ X2 @ A2 )
=> ( member @ A @ X2 @ B6 ) )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B6 ) ) ).
% subsetI
thf(fact_91_subset__antisym,axiom,
! [A: $tType,A2: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ A2 )
=> ( A2 = B6 ) ) ) ).
% subset_antisym
thf(fact_92_set__mp,axiom,
! [A: $tType,A2: set @ A,B6: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B6 )
=> ( ( member @ A @ X @ A2 )
=> ( member @ A @ X @ B6 ) ) ) ).
% set_mp
thf(fact_93_in__mono,axiom,
! [A: $tType,A2: set @ A,B6: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B6 )
=> ( ( member @ A @ X @ A2 )
=> ( member @ A @ X @ B6 ) ) ) ).
% in_mono
thf(fact_94_subsetD,axiom,
! [A: $tType,A2: set @ A,B6: set @ A,C: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B6 )
=> ( ( member @ A @ C @ A2 )
=> ( member @ A @ C @ B6 ) ) ) ).
% subsetD
thf(fact_95_subsetCE,axiom,
! [A: $tType,A2: set @ A,B6: set @ A,C: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B6 )
=> ( ( member @ A @ C @ A2 )
=> ( member @ A @ C @ B6 ) ) ) ).
% subsetCE
thf(fact_96_equalityE,axiom,
! [A: $tType,A2: set @ A,B6: set @ A] :
( ( A2 = B6 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A2 @ B6 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B6 @ A2 ) ) ) ).
% equalityE
thf(fact_97_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B5: set @ A] :
! [X5: A] :
( ( member @ A @ X5 @ A6 )
=> ( member @ A @ X5 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_98_equalityD1,axiom,
! [A: $tType,A2: set @ A,B6: set @ A] :
( ( A2 = B6 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ B6 ) ) ).
% equalityD1
thf(fact_99_equalityD2,axiom,
! [A: $tType,A2: set @ A,B6: set @ A] :
( ( A2 = B6 )
=> ( ord_less_eq @ ( set @ A ) @ B6 @ A2 ) ) ).
% equalityD2
thf(fact_100_set__rev__mp,axiom,
! [A: $tType,X: A,A2: set @ A,B6: set @ A] :
( ( member @ A @ X @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ B6 )
=> ( member @ A @ X @ B6 ) ) ) ).
% set_rev_mp
thf(fact_101_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A6: set @ A,B5: set @ A] :
! [T4: A] :
( ( member @ A @ T4 @ A6 )
=> ( member @ A @ T4 @ B5 ) ) ) ) ).
% subset_iff
thf(fact_102_rev__subsetD,axiom,
! [A: $tType,C: A,A2: set @ A,B6: set @ A] :
( ( member @ A @ C @ A2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A2 @ B6 )
=> ( member @ A @ C @ B6 ) ) ) ).
% rev_subsetD
thf(fact_103_subset__refl,axiom,
! [A: $tType,A2: set @ A] : ( ord_less_eq @ ( set @ A ) @ A2 @ A2 ) ).
% subset_refl
thf(fact_104_Collect__mono,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X2: A] :
( ( P @ X2 )
=> ( Q @ X2 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_105_subset__trans,axiom,
! [A: $tType,A2: set @ A,B6: set @ A,C4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ C4 )
=> ( ord_less_eq @ ( set @ A ) @ A2 @ C4 ) ) ) ).
% subset_trans
thf(fact_106_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y2: set @ A,Z: set @ A] : Y2 = Z )
= ( ^ [A6: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A6 @ B5 )
& ( ord_less_eq @ ( set @ A ) @ B5 @ A6 ) ) ) ) ).
% set_eq_subset
thf(fact_107_contra__subsetD,axiom,
! [A: $tType,A2: set @ A,B6: set @ A,C: A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B6 )
=> ( ~ ( member @ A @ C @ B6 )
=> ~ ( member @ A @ C @ A2 ) ) ) ).
% contra_subsetD
thf(fact_108_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
= ( ! [X5: A] :
( ( P @ X5 )
=> ( Q @ X5 ) ) ) ) ).
% Collect_mono_iff
thf(fact_109_DomainpE,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,A3: A] :
( ( domainp @ A @ B @ R2 @ A3 )
=> ~ ! [B4: B] :
~ ( R2 @ A3 @ B4 ) ) ).
% DomainpE
thf(fact_110_Domainp_Ocases,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,A3: A] :
( ( domainp @ A @ B @ R2 @ A3 )
=> ~ ! [B4: B] :
~ ( R2 @ A3 @ B4 ) ) ).
% Domainp.cases
thf(fact_111_Domainp_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( domainp @ A @ B )
= ( ^ [R4: A > B > $o,A4: A] :
? [B2: A,C5: B] :
( ( A4 = B2 )
& ( R4 @ B2 @ C5 ) ) ) ) ).
% Domainp.simps
thf(fact_112_Domainp_ODomainI,axiom,
! [B: $tType,A: $tType,R2: A > B > $o,A3: A,B3: B] :
( ( R2 @ A3 @ B3 )
=> ( domainp @ A @ B @ R2 @ A3 ) ) ).
% Domainp.DomainI
thf(fact_113_Domainp_Oinducts,axiom,
! [B: $tType,A: $tType,R2: A > B > $o,X: A,P: A > $o] :
( ( domainp @ A @ B @ R2 @ X )
=> ( ! [A5: A,B4: B] :
( ( R2 @ A5 @ B4 )
=> ( P @ A5 ) )
=> ( P @ X ) ) ) ).
% Domainp.inducts
thf(fact_114_NEG__trans,axiom,
! [B: $tType,A: $tType,A2: A > B > $o,B6: A > B > $o,C4: A > B > $o] :
( ( neg @ A @ B @ A2 @ B6 )
=> ( ( neg @ A @ B @ B6 @ C4 )
=> ( neg @ A @ B @ A2 @ C4 ) ) ) ).
% NEG_trans
thf(fact_115_Quotient__to__Domainp,axiom,
! [B: $tType,A: $tType,R: A > A > $o,Abs: A > B,Rep: B > A,T: A > B > $o] :
( ( quotient @ A @ B @ R @ Abs @ Rep @ T )
=> ( ( domainp @ A @ B @ T )
= ( ^ [X5: A] : ( R @ X5 @ X5 ) ) ) ) ).
% Quotient_to_Domainp
thf(fact_116_antimono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A @ ( type2 @ A ) )
& ( order @ B @ ( type2 @ B ) ) )
=> ( ( order_antimono @ A @ B )
= ( ^ [F2: A > B] :
! [X5: A,Y3: A] :
( ( ord_less_eq @ A @ X5 @ Y3 )
=> ( ord_less_eq @ B @ ( F2 @ Y3 ) @ ( F2 @ X5 ) ) ) ) ) ) ).
% antimono_def
thf(fact_117_antimonoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A @ ( type2 @ A ) )
& ( order @ B @ ( type2 @ B ) ) )
=> ! [F: A > B] :
( ! [X2: A,Y4: A] :
( ( ord_less_eq @ A @ X2 @ Y4 )
=> ( ord_less_eq @ B @ ( F @ Y4 ) @ ( F @ X2 ) ) )
=> ( order_antimono @ A @ B @ F ) ) ) ).
% antimonoI
thf(fact_118_antimonoE,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A @ ( type2 @ A ) )
& ( order @ B @ ( type2 @ B ) ) )
=> ! [F: A > B,X: A,Y: A] :
( ( order_antimono @ A @ B @ F )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ B @ ( F @ Y ) @ ( F @ X ) ) ) ) ) ).
% antimonoE
thf(fact_119_Domain__eq__top,axiom,
! [A: $tType] :
( ( domainp @ A @ A
@ ^ [Y2: A,Z: A] : Y2 = Z )
= ( top_top @ ( A > $o ) ) ) ).
% Domain_eq_top
thf(fact_120_Domainp__refl,axiom,
! [B: $tType,A: $tType] :
( ( domainp @ A @ B )
= ( domainp @ A @ B ) ) ).
% Domainp_refl
thf(fact_121_Domainp__iff,axiom,
! [B: $tType,A: $tType] :
( ( domainp @ A @ B )
= ( ^ [T2: A > B > $o,X5: A] :
( ^ [P2: B > $o] :
? [X6: B] : ( P2 @ X6 )
@ ( T2 @ X5 ) ) ) ) ).
% Domainp_iff
thf(fact_122_decseqD,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [F: nat > A,I: nat,J: nat] :
( ( order_antimono @ nat @ A @ F )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ A @ ( F @ J ) @ ( F @ I ) ) ) ) ) ).
% decseqD
thf(fact_123_decseq__def,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( order_antimono @ nat @ A )
= ( ^ [X7: nat > A] :
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ A @ ( X7 @ N ) @ ( X7 @ M ) ) ) ) ) ) ).
% decseq_def
thf(fact_124_NEG__POS,axiom,
! [B: $tType,A: $tType] :
( ( neg @ A @ B )
= ( ^ [A6: A > B > $o,B5: A > B > $o] : ( pos @ A @ B @ B5 @ A6 ) ) ) ).
% NEG_POS
thf(fact_125_POS__trans,axiom,
! [B: $tType,A: $tType,A2: A > B > $o,B6: A > B > $o,C4: A > B > $o] :
( ( pos @ A @ B @ A2 @ B6 )
=> ( ( pos @ A @ B @ B6 @ C4 )
=> ( pos @ A @ B @ A2 @ C4 ) ) ) ).
% POS_trans
thf(fact_126_POS__apply,axiom,
! [A: $tType,B: $tType,R: A > B > $o,R5: A > B > $o,F: A,G: B] :
( ( pos @ A @ B @ R @ R5 )
=> ( ( R @ F @ G )
=> ( R5 @ F @ G ) ) ) ).
% POS_apply
thf(fact_127_POS__def,axiom,
! [B: $tType,A: $tType] :
( ( pos @ A @ B )
= ( ord_less_eq @ ( A > B > $o ) ) ) ).
% POS_def
thf(fact_128_POS__NEG,axiom,
! [B: $tType,A: $tType] :
( ( pos @ A @ B )
= ( ^ [A6: A > B > $o,B5: A > B > $o] : ( neg @ A @ B @ B5 @ A6 ) ) ) ).
% POS_NEG
thf(fact_129_ex__has__least__nat,axiom,
! [A: $tType,P: A > $o,K: A,M2: A > nat] :
( ( P @ K )
=> ? [X2: A] :
( ( P @ X2 )
& ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ nat @ ( M2 @ X2 ) @ ( M2 @ Y5 ) ) ) ) ) ).
% ex_has_least_nat
thf(fact_130_nat__le__linear,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
| ( ord_less_eq @ nat @ N2 @ M2 ) ) ).
% nat_le_linear
thf(fact_131_le__antisym,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ( ord_less_eq @ nat @ N2 @ M2 )
=> ( M2 = N2 ) ) ) ).
% le_antisym
thf(fact_132_eq__imp__le,axiom,
! [M2: nat,N2: nat] :
( ( M2 = N2 )
=> ( ord_less_eq @ nat @ M2 @ N2 ) ) ).
% eq_imp_le
thf(fact_133_le__refl,axiom,
! [N2: nat] : ( ord_less_eq @ nat @ N2 @ N2 ) ).
% le_refl
thf(fact_134_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ nat @ J @ K )
=> ( ord_less_eq @ nat @ I @ K ) ) ) ).
% le_trans
thf(fact_135_LeastMI2,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [P: A > $o,X: A,M2: A > B,Q: A > $o] :
( ( P @ X )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ B @ ( M2 @ X ) @ ( M2 @ Y4 ) ) )
=> ( ! [X2: A] :
( ( P @ X2 )
=> ( ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ B @ ( M2 @ X2 ) @ ( M2 @ Y5 ) ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( hilbert_LeastM @ A @ B @ M2 @ P ) ) ) ) ) ) ).
% LeastMI2
thf(fact_136_monoseq__def,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( topological_monoseq @ A )
= ( ^ [X7: nat > A] :
( ! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ A @ ( X7 @ M ) @ ( X7 @ N ) ) )
| ! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ A @ ( X7 @ N ) @ ( X7 @ M ) ) ) ) ) ) ) ).
% monoseq_def
thf(fact_137_monoI2,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X8: nat > A] :
( ! [M3: nat,N3: nat] :
( ( ord_less_eq @ nat @ M3 @ N3 )
=> ( ord_less_eq @ A @ ( X8 @ N3 ) @ ( X8 @ M3 ) ) )
=> ( topological_monoseq @ A @ X8 ) ) ) ).
% monoI2
thf(fact_138_monoI1,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X8: nat > A] :
( ! [M3: nat,N3: nat] :
( ( ord_less_eq @ nat @ M3 @ N3 )
=> ( ord_less_eq @ A @ ( X8 @ M3 ) @ ( X8 @ N3 ) ) )
=> ( topological_monoseq @ A @ X8 ) ) ) ).
% monoI1
thf(fact_139_decseq__imp__monoseq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X8: nat > A] :
( ( order_antimono @ nat @ A @ X8 )
=> ( topological_monoseq @ A @ X8 ) ) ) ).
% decseq_imp_monoseq
thf(fact_140_LeastM__nat__le,axiom,
! [A: $tType,P: A > $o,X: A,M2: A > nat] :
( ( P @ X )
=> ( ord_less_eq @ nat @ ( M2 @ ( hilbert_LeastM @ A @ nat @ M2 @ P ) ) @ ( M2 @ X ) ) ) ).
% LeastM_nat_le
thf(fact_141_LeastM__nat__lemma,axiom,
! [A: $tType,P: A > $o,K: A,M2: A > nat] :
( ( P @ K )
=> ( ( P @ ( hilbert_LeastM @ A @ nat @ M2 @ P ) )
& ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ nat @ ( M2 @ ( hilbert_LeastM @ A @ nat @ M2 @ P ) ) @ ( M2 @ Y5 ) ) ) ) ) ).
% LeastM_nat_lemma
thf(fact_142_GreatestMI2,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [P: A > $o,X: A,M2: A > B,Q: A > $o] :
( ( P @ X )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ B @ ( M2 @ Y4 ) @ ( M2 @ X ) ) )
=> ( ! [X2: A] :
( ( P @ X2 )
=> ( ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ B @ ( M2 @ Y5 ) @ ( M2 @ X2 ) ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( hilbert_GreatestM @ A @ B @ M2 @ P ) ) ) ) ) ) ).
% GreatestMI2
thf(fact_143_totally__bounded__subset,axiom,
! [A: $tType] :
( ( topolo47006728_space @ A @ ( type2 @ A ) )
=> ! [S2: set @ A,T: set @ A] :
( ( topolo406746546ounded @ A @ S2 )
=> ( ( ord_less_eq @ ( set @ A ) @ T @ S2 )
=> ( topolo406746546ounded @ A @ T ) ) ) ) ).
% totally_bounded_subset
thf(fact_144_Grp__mono,axiom,
! [B: $tType,A: $tType,A2: set @ A,B6: set @ A,F: A > B] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B6 )
=> ( ord_less_eq @ ( A > B > $o ) @ ( bNF_Grp @ A @ B @ A2 @ F ) @ ( bNF_Grp @ A @ B @ B6 @ F ) ) ) ).
% Grp_mono
thf(fact_145_of__nat__le__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [M2: nat,N2: nat] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N2 ) )
= ( ord_less_eq @ nat @ M2 @ N2 ) ) ) ).
% of_nat_le_iff
thf(fact_146_GrpE,axiom,
! [B: $tType,A: $tType,A2: set @ A,F: A > B,X: A,Y: B] :
( ( bNF_Grp @ A @ B @ A2 @ F @ X @ Y )
=> ~ ( ( ( F @ X )
= Y )
=> ~ ( member @ A @ X @ A2 ) ) ) ).
% GrpE
thf(fact_147_GrpI,axiom,
! [B: $tType,A: $tType,F: B > A,X: B,Y: A,A2: set @ B] :
( ( ( F @ X )
= Y )
=> ( ( member @ B @ X @ A2 )
=> ( bNF_Grp @ B @ A @ A2 @ F @ X @ Y ) ) ) ).
% GrpI
thf(fact_148_real__arch__simple,axiom,
! [A: $tType] :
( ( archim1804426504_field @ A @ ( type2 @ A ) )
=> ! [X: A] :
? [N3: nat] : ( ord_less_eq @ A @ X @ ( semiring_1_of_nat @ A @ N3 ) ) ) ).
% real_arch_simple
thf(fact_149_zle__int,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N2 ) )
= ( ord_less_eq @ nat @ M2 @ N2 ) ) ).
% zle_int
thf(fact_150_transfer__int__nat__relations_I3_J,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ X ) @ ( semiring_1_of_nat @ int @ Y ) )
= ( ord_less_eq @ nat @ X @ Y ) ) ).
% transfer_int_nat_relations(3)
thf(fact_151_round__mono,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ int @ ( archimedean_round @ A @ X ) @ ( archimedean_round @ A @ Y ) ) ) ) ).
% round_mono
thf(fact_152_nat__le__iff,axiom,
! [X: int,N2: nat] :
( ( ord_less_eq @ nat @ ( nat2 @ X ) @ N2 )
= ( ord_less_eq @ int @ X @ ( semiring_1_of_nat @ int @ N2 ) ) ) ).
% nat_le_iff
thf(fact_153_negative__zle,axiom,
! [N2: nat,M2: nat] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N2 ) ) @ ( semiring_1_of_nat @ int @ M2 ) ) ).
% negative_zle
thf(fact_154_nat__mono,axiom,
! [X: int,Y: int] :
( ( ord_less_eq @ int @ X @ Y )
=> ( ord_less_eq @ nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ).
% nat_mono
thf(fact_155_compl__le__compl__iff,axiom,
! [A: $tType] :
( ( boolean_algebra @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y ) )
= ( ord_less_eq @ A @ Y @ X ) ) ) ).
% compl_le_compl_iff
thf(fact_156_neg__le__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less_eq @ A @ A3 @ B3 ) ) ) ).
% neg_le_iff_le
thf(fact_157_le__imp__neg__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% le_imp_neg_le
thf(fact_158_Compl__anti__mono,axiom,
! [A: $tType,A2: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A2 @ B6 )
=> ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B6 ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) ) ) ).
% Compl_anti_mono
thf(fact_159_Compl__subset__Compl__iff,axiom,
! [A: $tType,A2: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) @ ( uminus_uminus @ ( set @ A ) @ B6 ) )
= ( ord_less_eq @ ( set @ A ) @ B6 @ A2 ) ) ).
% Compl_subset_Compl_iff
thf(fact_160_compl__mono,axiom,
! [A: $tType] :
( ( boolean_algebra @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% compl_mono
thf(fact_161_compl__le__swap1,axiom,
! [A: $tType] :
( ( boolean_algebra @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ ( uminus_uminus @ A @ X ) )
=> ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% compl_le_swap1
thf(fact_162_compl__le__swap2,axiom,
! [A: $tType] :
( ( boolean_algebra @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ X )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ Y ) ) ) ).
% compl_le_swap2
thf(fact_163_le__minus__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ B3 ) )
= ( ord_less_eq @ A @ B3 @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% le_minus_iff
thf(fact_164_minus__le__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B3 )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B3 ) @ A3 ) ) ) ).
% minus_le_iff
thf(fact_165_le__nat__iff,axiom,
! [K: int,N2: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less_eq @ nat @ N2 @ ( nat2 @ K ) )
= ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ N2 ) @ K ) ) ) ).
% le_nat_iff
thf(fact_166_int__nat__eq,axiom,
! [Z3: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ( ( semiring_1_of_nat @ int @ ( nat2 @ Z3 ) )
= Z3 ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ( ( semiring_1_of_nat @ int @ ( nat2 @ Z3 ) )
= ( zero_zero @ int ) ) ) ) ).
% int_nat_eq
thf(fact_167_int__zle__neg,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ N2 ) @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ M2 ) ) )
= ( ( N2
= ( zero_zero @ nat ) )
& ( M2
= ( zero_zero @ nat ) ) ) ) ).
% int_zle_neg
thf(fact_168_ComplI,axiom,
! [A: $tType,C: A,A2: set @ A] :
( ~ ( member @ A @ C @ A2 )
=> ( member @ A @ C @ ( uminus_uminus @ ( set @ A ) @ A2 ) ) ) ).
% ComplI
thf(fact_169_Compl__iff,axiom,
! [A: $tType,C: A,A2: set @ A] :
( ( member @ A @ C @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
= ( ~ ( member @ A @ C @ A2 ) ) ) ).
% Compl_iff
thf(fact_170_Compl__eq__Compl__iff,axiom,
! [A: $tType,A2: set @ A,B6: set @ A] :
( ( ( uminus_uminus @ ( set @ A ) @ A2 )
= ( uminus_uminus @ ( set @ A ) @ B6 ) )
= ( A2 = B6 ) ) ).
% Compl_eq_Compl_iff
thf(fact_171_le__zero__eq,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [N2: A] :
( ( ord_less_eq @ A @ N2 @ ( zero_zero @ A ) )
= ( N2
= ( zero_zero @ A ) ) ) ) ).
% le_zero_eq
thf(fact_172_le0,axiom,
! [N2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N2 ) ).
% le0
thf(fact_173_neg__less__eq__nonneg,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A @ ( type2 @ A ) )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ A3 )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% neg_less_eq_nonneg
thf(fact_174_less__eq__neg__nonpos,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A @ ( type2 @ A ) )
=> ! [A3: A] :
( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% less_eq_neg_nonpos
thf(fact_175_neg__le__0__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% neg_le_0_iff_le
thf(fact_176_neg__0__le__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% neg_0_le_iff_le
thf(fact_177_of__nat__le__0__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [M2: nat] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( zero_zero @ A ) )
= ( M2
= ( zero_zero @ nat ) ) ) ) ).
% of_nat_le_0_iff
thf(fact_178_nat__0__iff,axiom,
! [I: int] :
( ( ( nat2 @ I )
= ( zero_zero @ nat ) )
= ( ord_less_eq @ int @ I @ ( zero_zero @ int ) ) ) ).
% nat_0_iff
thf(fact_179_nat__le__0,axiom,
! [Z3: int] :
( ( ord_less_eq @ int @ Z3 @ ( zero_zero @ int ) )
=> ( ( nat2 @ Z3 )
= ( zero_zero @ nat ) ) ) ).
% nat_le_0
thf(fact_180_ComplD,axiom,
! [A: $tType,C: A,A2: set @ A] :
( ( member @ A @ C @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
=> ~ ( member @ A @ C @ A2 ) ) ).
% ComplD
thf(fact_181_double__complement,axiom,
! [A: $tType,A2: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A2 ) )
= A2 ) ).
% double_complement
thf(fact_182_zero__le,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% zero_le
thf(fact_183_le__0__eq,axiom,
! [N2: nat] :
( ( ord_less_eq @ nat @ N2 @ ( zero_zero @ nat ) )
= ( N2
= ( zero_zero @ nat ) ) ) ).
% le_0_eq
thf(fact_184_less__eq__nat_Osimps_I1_J,axiom,
! [N2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N2 ) ).
% less_eq_nat.simps(1)
thf(fact_185_less__eq__int__code_I1_J,axiom,
ord_less_eq @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ).
% less_eq_int_code(1)
thf(fact_186_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J,axiom,
ord_less_eq @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ).
% Nat_Transfer.transfer_nat_int_function_closures(5)
thf(fact_187_nat__eq__iff,axiom,
! [W: int,M2: nat] :
( ( ( nat2 @ W )
= M2 )
= ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
=> ( W
= ( semiring_1_of_nat @ int @ M2 ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
=> ( M2
= ( zero_zero @ nat ) ) ) ) ) ).
% nat_eq_iff
thf(fact_188_nat__eq__iff2,axiom,
! [M2: nat,W: int] :
( ( M2
= ( nat2 @ W ) )
= ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
=> ( W
= ( semiring_1_of_nat @ int @ M2 ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
=> ( M2
= ( zero_zero @ nat ) ) ) ) ) ).
% nat_eq_iff2
thf(fact_189_of__nat__0__le__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N2 ) ) ) ).
% of_nat_0_le_iff
thf(fact_190_transfer__nat__int__relations_I1_J,axiom,
! [X: int,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ( ( nat2 @ X )
= ( nat2 @ Y ) )
= ( X = Y ) ) ) ) ).
% transfer_nat_int_relations(1)
thf(fact_191_eq__nat__nat__iff,axiom,
! [Z3: int,Z4: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z4 )
=> ( ( ( nat2 @ Z3 )
= ( nat2 @ Z4 ) )
= ( Z3 = Z4 ) ) ) ) ).
% eq_nat_nat_iff
thf(fact_192_ex__nat,axiom,
( ( ^ [P2: nat > $o] :
? [X6: nat] : ( P2 @ X6 ) )
= ( ^ [P3: nat > $o] :
? [X5: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X5 )
& ( P3 @ ( nat2 @ X5 ) ) ) ) ) ).
% ex_nat
thf(fact_193_all__nat,axiom,
( ( ^ [P2: nat > $o] :
! [X6: nat] : ( P2 @ X6 ) )
= ( ^ [P3: nat > $o] :
! [X5: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X5 )
=> ( P3 @ ( nat2 @ X5 ) ) ) ) ) ).
% all_nat
thf(fact_194_transfer__int__nat__quantifiers_I2_J,axiom,
! [P: int > $o] :
( ( ? [X5: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X5 )
& ( P @ X5 ) ) )
= ( ? [X5: nat] : ( P @ ( semiring_1_of_nat @ int @ X5 ) ) ) ) ).
% transfer_int_nat_quantifiers(2)
thf(fact_195_transfer__int__nat__quantifiers_I1_J,axiom,
! [P: int > $o] :
( ( ! [X5: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X5 )
=> ( P @ X5 ) ) )
= ( ! [X5: nat] : ( P @ ( semiring_1_of_nat @ int @ X5 ) ) ) ) ).
% transfer_int_nat_quantifiers(1)
thf(fact_196_Nat__Transfer_Otransfer__nat__int__function__closures_I9_J,axiom,
! [Z3: nat] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( semiring_1_of_nat @ int @ Z3 ) ) ).
% Nat_Transfer.transfer_nat_int_function_closures(9)
thf(fact_197_nonneg__eq__int,axiom,
! [Z3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ~ ! [M3: nat] :
( Z3
!= ( semiring_1_of_nat @ int @ M3 ) ) ) ).
% nonneg_eq_int
thf(fact_198_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ~ ! [N3: nat] :
( K
!= ( semiring_1_of_nat @ int @ N3 ) ) ) ).
% nonneg_int_cases
thf(fact_199_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ? [N3: nat] :
( K
= ( semiring_1_of_nat @ int @ N3 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_200_transfer__nat__int__relations_I3_J,axiom,
! [X: int,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ( ord_less_eq @ nat @ ( nat2 @ X ) @ ( nat2 @ Y ) )
= ( ord_less_eq @ int @ X @ Y ) ) ) ) ).
% transfer_nat_int_relations(3)
thf(fact_201_int__eq__iff,axiom,
! [M2: nat,Z3: int] :
( ( ( semiring_1_of_nat @ int @ M2 )
= Z3 )
= ( ( M2
= ( nat2 @ Z3 ) )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 ) ) ) ).
% int_eq_iff
thf(fact_202_nat__0__le,axiom,
! [Z3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ( ( semiring_1_of_nat @ int @ ( nat2 @ Z3 ) )
= Z3 ) ) ).
% nat_0_le
thf(fact_203_negative__zle__0,axiom,
! [N2: nat] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N2 ) ) @ ( zero_zero @ int ) ) ).
% negative_zle_0
thf(fact_204_transfer__morphism__nat__int,axiom,
nat_tr1645093318rphism @ int @ nat @ nat2 @ ( ord_less_eq @ int @ ( zero_zero @ int ) ) ).
% transfer_morphism_nat_int
thf(fact_205_imp__le__cong,axiom,
! [X: int,X4: int,P: $o,P4: $o] :
( ( X = X4 )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X4 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> P )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X4 )
=> P4 ) ) ) ) ).
% imp_le_cong
thf(fact_206_conj__le__cong,axiom,
! [X: int,X4: int,P: $o,P4: $o] :
( ( X = X4 )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X4 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
& P )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X4 )
& P4 ) ) ) ) ).
% conj_le_cong
thf(fact_207_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_208_int__eq__iff__numeral,axiom,
! [M2: nat,V: num] :
( ( ( semiring_1_of_nat @ int @ M2 )
= ( numeral_numeral @ int @ V ) )
= ( ( M2
= ( nat2 @ ( numeral_numeral @ int @ V ) ) )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( numeral_numeral @ int @ V ) ) ) ) ).
% int_eq_iff_numeral
thf(fact_209_numeral__le__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [M2: num,N2: num] :
( ( ord_less_eq @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ N2 ) )
= ( ord_less_eq @ num @ M2 @ N2 ) ) ) ).
% numeral_le_iff
thf(fact_210_neg__numeral__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type2 @ A ) )
=> ! [M2: num,N2: num] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) )
= ( ord_less_eq @ num @ N2 @ M2 ) ) ) ).
% neg_numeral_le_iff
thf(fact_211_zero__le__numeral,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [N2: num] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ N2 ) ) ) ).
% zero_le_numeral
thf(fact_212_not__numeral__le__zero,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [N2: num] :
~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ N2 ) @ ( zero_zero @ A ) ) ) ).
% not_numeral_le_zero
thf(fact_213_not__zero__le__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type2 @ A ) )
=> ! [N2: num] :
~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) ) ) ).
% not_zero_le_neg_numeral
thf(fact_214_neg__numeral__le__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type2 @ A ) )
=> ! [N2: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) @ ( zero_zero @ A ) ) ) ).
% neg_numeral_le_zero
thf(fact_215_not__numeral__le__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type2 @ A ) )
=> ! [M2: num,N2: num] :
~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ M2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N2 ) ) ) ) ).
% not_numeral_le_neg_numeral
thf(fact_216_neg__numeral__le__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type2 @ A ) )
=> ! [M2: num,N2: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( numeral_numeral @ A @ N2 ) ) ) ).
% neg_numeral_le_numeral
thf(fact_217_ceiling__le__neg__numeral,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A @ ( type2 @ A ) )
=> ! [X: A,V: num] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) )
= ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) ) ) ) ).
% ceiling_le_neg_numeral
thf(fact_218_neg__numeral__le__floor,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A @ ( type2 @ A ) )
=> ! [V: num,X: A] :
( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) @ ( archim2085082626_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ X ) ) ) ).
% neg_numeral_le_floor
thf(fact_219_zero__le__floor,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( archim2085082626_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ) ).
% zero_le_floor
thf(fact_220_ceiling__le__zero,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( zero_zero @ int ) )
= ( ord_less_eq @ A @ X @ ( zero_zero @ A ) ) ) ) ).
% ceiling_le_zero
thf(fact_221_numeral__le__floor,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A @ ( type2 @ A ) )
=> ! [V: num,X: A] :
( ( ord_less_eq @ int @ ( numeral_numeral @ int @ V ) @ ( archim2085082626_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( numeral_numeral @ A @ V ) @ X ) ) ) ).
% numeral_le_floor
thf(fact_222_ceiling__le__numeral,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A @ ( type2 @ A ) )
=> ! [X: A,V: num] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( numeral_numeral @ int @ V ) )
= ( ord_less_eq @ A @ X @ ( numeral_numeral @ A @ V ) ) ) ) ).
% ceiling_le_numeral
thf(fact_223_floor__mono,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ int @ ( archim2085082626_floor @ A @ X ) @ ( archim2085082626_floor @ A @ Y ) ) ) ) ).
% floor_mono
thf(fact_224_floor__le__ceiling,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A @ ( type2 @ A ) )
=> ! [X: A] : ( ord_less_eq @ int @ ( archim2085082626_floor @ A @ X ) @ ( archimedean_ceiling @ A @ X ) ) ) ).
% floor_le_ceiling
thf(fact_225_ceiling__ge__round,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A @ ( type2 @ A ) )
=> ! [X: A] : ( ord_less_eq @ int @ ( archimedean_round @ A @ X ) @ ( archimedean_ceiling @ A @ X ) ) ) ).
% ceiling_ge_round
thf(fact_226_floor__le__round,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A @ ( type2 @ A ) )
=> ! [X: A] : ( ord_less_eq @ int @ ( archim2085082626_floor @ A @ X ) @ ( archimedean_round @ A @ X ) ) ) ).
% floor_le_round
thf(fact_227_ceiling__mono,axiom,
! [A: $tType] :
( ( archim1727834104eiling @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ Y ) @ ( archimedean_ceiling @ A @ X ) ) ) ) ).
% ceiling_mono
thf(fact_228_nat__ceiling__le__eq,axiom,
! [X: real,A3: nat] :
( ( ord_less_eq @ nat @ ( nat2 @ ( archimedean_ceiling @ real @ X ) ) @ A3 )
= ( ord_less_eq @ real @ X @ ( semiring_1_of_nat @ real @ A3 ) ) ) ).
% nat_ceiling_le_eq
thf(fact_229_numeral__le__real__of__nat__iff,axiom,
! [N2: num,M2: nat] :
( ( ord_less_eq @ real @ ( numeral_numeral @ real @ N2 ) @ ( semiring_1_of_nat @ real @ M2 ) )
= ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ N2 ) @ M2 ) ) ).
% numeral_le_real_of_nat_iff
thf(fact_230_complete__real,axiom,
! [S2: set @ real] :
( ? [X3: real] : ( member @ real @ X3 @ S2 )
=> ( ? [Z5: real] :
! [X2: real] :
( ( member @ real @ X2 @ S2 )
=> ( ord_less_eq @ real @ X2 @ Z5 ) )
=> ? [Y4: real] :
( ! [X3: real] :
( ( member @ real @ X3 @ S2 )
=> ( ord_less_eq @ real @ X3 @ Y4 ) )
& ! [Z5: real] :
( ! [X2: real] :
( ( member @ real @ X2 @ S2 )
=> ( ord_less_eq @ real @ X2 @ Z5 ) )
=> ( ord_less_eq @ real @ Y4 @ Z5 ) ) ) ) ) ).
% complete_real
thf(fact_231_real__nat__ceiling__ge,axiom,
! [X: real] : ( ord_less_eq @ real @ X @ ( semiring_1_of_nat @ real @ ( nat2 @ ( archimedean_ceiling @ real @ X ) ) ) ) ).
% real_nat_ceiling_ge
thf(fact_232_nat__floor__neg,axiom,
! [X: real] :
( ( ord_less_eq @ real @ X @ ( zero_zero @ real ) )
=> ( ( nat2 @ ( archim2085082626_floor @ real @ X ) )
= ( zero_zero @ nat ) ) ) ).
% nat_floor_neg
thf(fact_233_le__nat__floor,axiom,
! [X: nat,A3: real] :
( ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ X ) @ A3 )
=> ( ord_less_eq @ nat @ X @ ( nat2 @ ( archim2085082626_floor @ real @ A3 ) ) ) ) ).
% le_nat_floor
thf(fact_234_enat__ord__number_I1_J,axiom,
! [M2: num,N2: num] :
( ( ord_less_eq @ extended_enat @ ( numeral_numeral @ extended_enat @ M2 ) @ ( numeral_numeral @ extended_enat @ N2 ) )
= ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ M2 ) @ ( numeral_numeral @ nat @ N2 ) ) ) ).
% enat_ord_number(1)
thf(fact_235_Nat__Transfer_Otransfer__nat__int__function__closures_I3_J,axiom,
! [X: int,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( nat_tsub @ X @ Y ) ) ) ) ).
% Nat_Transfer.transfer_nat_int_function_closures(3)
thf(fact_236_ile0__eq,axiom,
! [N2: extended_enat] :
( ( ord_less_eq @ extended_enat @ N2 @ ( zero_zero @ extended_enat ) )
= ( N2
= ( zero_zero @ extended_enat ) ) ) ).
% ile0_eq
thf(fact_237_i0__lb,axiom,
! [N2: extended_enat] : ( ord_less_eq @ extended_enat @ ( zero_zero @ extended_enat ) @ N2 ) ).
% i0_lb
thf(fact_238_numeral__le__enat__iff,axiom,
! [M2: num,N2: nat] :
( ( ord_less_eq @ extended_enat @ ( numeral_numeral @ extended_enat @ M2 ) @ ( extended_enat2 @ N2 ) )
= ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ M2 ) @ N2 ) ) ).
% numeral_le_enat_iff
thf(fact_239_Nat__Transfer_Otransfer__nat__int__functions_I3_J,axiom,
! [X: int,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ( minus_minus @ nat @ ( nat2 @ X ) @ ( nat2 @ Y ) )
= ( nat2 @ ( nat_tsub @ X @ Y ) ) ) ) ) ).
% Nat_Transfer.transfer_nat_int_functions(3)
thf(fact_240_diff__diff__cancel,axiom,
! [I: nat,N2: nat] :
( ( ord_less_eq @ nat @ I @ N2 )
=> ( ( minus_minus @ nat @ N2 @ ( minus_minus @ nat @ N2 @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_241_diff__ge__0__iff__ge,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A3 @ B3 ) )
= ( ord_less_eq @ A @ B3 @ A3 ) ) ) ).
% diff_ge_0_iff_ge
thf(fact_242_diff__is__0__eq,axiom,
! [M2: nat,N2: nat] :
( ( ( minus_minus @ nat @ M2 @ N2 )
= ( zero_zero @ nat ) )
= ( ord_less_eq @ nat @ M2 @ N2 ) ) ).
% diff_is_0_eq
thf(fact_243_diff__is__0__eq_H,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ( minus_minus @ nat @ M2 @ N2 )
= ( zero_zero @ nat ) ) ) ).
% diff_is_0_eq'
thf(fact_244_enat__ord__simps_I1_J,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq @ extended_enat @ ( extended_enat2 @ M2 ) @ ( extended_enat2 @ N2 ) )
= ( ord_less_eq @ nat @ M2 @ N2 ) ) ).
% enat_ord_simps(1)
thf(fact_245_enat__ile,axiom,
! [N2: extended_enat,M2: nat] :
( ( ord_less_eq @ extended_enat @ N2 @ ( extended_enat2 @ M2 ) )
=> ? [K2: nat] :
( N2
= ( extended_enat2 @ K2 ) ) ) ).
% enat_ile
thf(fact_246_eq__diff__iff,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ M2 )
=> ( ( ord_less_eq @ nat @ K @ N2 )
=> ( ( ( minus_minus @ nat @ M2 @ K )
= ( minus_minus @ nat @ N2 @ K ) )
= ( M2 = N2 ) ) ) ) ).
% eq_diff_iff
thf(fact_247_le__diff__iff,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ M2 )
=> ( ( ord_less_eq @ nat @ K @ N2 )
=> ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ M2 @ K ) @ ( minus_minus @ nat @ N2 @ K ) )
= ( ord_less_eq @ nat @ M2 @ N2 ) ) ) ) ).
% le_diff_iff
thf(fact_248_Nat_Odiff__diff__eq,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ K @ M2 )
=> ( ( ord_less_eq @ nat @ K @ N2 )
=> ( ( minus_minus @ nat @ ( minus_minus @ nat @ M2 @ K ) @ ( minus_minus @ nat @ N2 @ K ) )
= ( minus_minus @ nat @ M2 @ N2 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_249_diff__le__mono,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ M2 @ L ) @ ( minus_minus @ nat @ N2 @ L ) ) ) ).
% diff_le_mono
thf(fact_250_diff__le__self,axiom,
! [M2: nat,N2: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ M2 @ N2 ) @ M2 ) ).
% diff_le_self
thf(fact_251_le__diff__iff_H,axiom,
! [A3: nat,C: nat,B3: nat] :
( ( ord_less_eq @ nat @ A3 @ C )
=> ( ( ord_less_eq @ nat @ B3 @ C )
=> ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ C @ A3 ) @ ( minus_minus @ nat @ C @ B3 ) )
= ( ord_less_eq @ nat @ B3 @ A3 ) ) ) ) ).
% le_diff_iff'
thf(fact_252_diff__le__mono2,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ L @ N2 ) @ ( minus_minus @ nat @ L @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_253_of__nat__diff,axiom,
! [A: $tType] :
( ( ring_1 @ A @ ( type2 @ A ) )
=> ! [N2: nat,M2: nat] :
( ( ord_less_eq @ nat @ N2 @ M2 )
=> ( ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ M2 @ N2 ) )
= ( minus_minus @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N2 ) ) ) ) ) ).
% of_nat_diff
thf(fact_254_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A,C: A,D2: A] :
( ( ( minus_minus @ A @ A3 @ B3 )
= ( minus_minus @ A @ C @ D2 ) )
=> ( ( ord_less_eq @ A @ A3 @ B3 )
= ( ord_less_eq @ A @ C @ D2 ) ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_255_diff__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A,C: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C ) @ ( minus_minus @ A @ B3 @ C ) ) ) ) ).
% diff_right_mono
%----Type constructors (61)
thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
! [A7: $tType,A8: $tType] :
( ( comple187826305attice @ A8 @ ( type2 @ A8 ) )
=> ( comple187826305attice @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_fun___Lattices_Oboolean__algebra,axiom,
! [A7: $tType,A8: $tType] :
( ( boolean_algebra @ A8 @ ( type2 @ A8 ) )
=> ( boolean_algebra @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder__top,axiom,
! [A7: $tType,A8: $tType] :
( ( order_top @ A8 @ ( type2 @ A8 ) )
=> ( order_top @ ( 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_Otop,axiom,
! [A7: $tType,A8: $tType] :
( ( top @ A8 @ ( type2 @ A8 ) )
=> ( top @ ( 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_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
linord219039673up_add @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Groups_Oordered__ab__group__add,axiom,
ordered_ab_group_add @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
linordered_semidom @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
linordered_idom @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Orderings_Opreorder_1,axiom,
preorder @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Orderings_Olinorder,axiom,
linorder @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Orderings_Oorder_2,axiom,
order @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Orderings_Oord_3,axiom,
ord @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Rings_Oring__1,axiom,
ring_1 @ int @ ( type2 @ int ) ).
thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
canoni770627133id_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Olinordered__semidom_4,axiom,
linordered_semidom @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Opreorder_5,axiom,
preorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Olinorder_6,axiom,
linorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oorder_7,axiom,
order @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oord_8,axiom,
ord @ nat @ ( type2 @ nat ) ).
thf(tcon_Num_Onum___Orderings_Opreorder_9,axiom,
preorder @ num @ ( type2 @ num ) ).
thf(tcon_Num_Onum___Orderings_Olinorder_10,axiom,
linorder @ num @ ( type2 @ num ) ).
thf(tcon_Num_Onum___Orderings_Oorder_11,axiom,
order @ num @ ( type2 @ num ) ).
thf(tcon_Num_Onum___Orderings_Oord_12,axiom,
ord @ num @ ( type2 @ num ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_13,axiom,
! [A7: $tType] : ( comple187826305attice @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Oboolean__algebra_14,axiom,
! [A7: $tType] : ( boolean_algebra @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__top_15,axiom,
! [A7: $tType] : ( order_top @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_16,axiom,
! [A7: $tType] : ( preorder @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_17,axiom,
! [A7: $tType] : ( order @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Otop_18,axiom,
! [A7: $tType] : ( top @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_19,axiom,
! [A7: $tType] : ( ord @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_20,axiom,
comple187826305attice @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_21,axiom,
boolean_algebra @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder__top_22,axiom,
order_top @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_23,axiom,
preorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder_24,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_25,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Otop_26,axiom,
top @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_27,axiom,
ord @ $o @ ( type2 @ $o ) ).
thf(tcon_Real_Oreal___Archimedean__Field_Oarchimedean__field,axiom,
archim1804426504_field @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Topological__Spaces_Ouniform__space,axiom,
topolo47006728_space @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Archimedean__Field_Ofloor__ceiling,axiom,
archim1727834104eiling @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Groups_Olinordered__ab__group__add_28,axiom,
linord219039673up_add @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add_29,axiom,
ordered_ab_group_add @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Rings_Olinordered__semidom_30,axiom,
linordered_semidom @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Rings_Olinordered__idom_31,axiom,
linordered_idom @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Orderings_Opreorder_32,axiom,
preorder @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Orderings_Olinorder_33,axiom,
linorder @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Orderings_Oorder_34,axiom,
order @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Orderings_Oord_35,axiom,
ord @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Rings_Oring__1_36,axiom,
ring_1 @ real @ ( type2 @ real ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_37,axiom,
canoni770627133id_add @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__lattice_38,axiom,
comple187826305attice @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oorder__top_39,axiom,
order_top @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Orderings_Opreorder_40,axiom,
preorder @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Orderings_Olinorder_41,axiom,
linorder @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oorder_42,axiom,
order @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Orderings_Otop_43,axiom,
top @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oord_44,axiom,
ord @ extended_enat @ ( type2 @ extended_enat ) ).
%----Conjectures (2)
thf(conj_0,hypothesis,
p @ x ).
thf(conj_1,conjecture,
bNF_eq_onp @ a @ p @ x @ x ).
%------------------------------------------------------------------------------