TPTP Problem File: ITP073^2.p
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%------------------------------------------------------------------------------
% File : ITP073^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer HF problem prob_473__5331678_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : HF/prob_473__5331678_1 [Des21]
% Status : Theorem
% Rating : 0.67 v8.1.0, 0.50 v7.5.0
% Syntax : Number of formulae : 324 ( 67 unt; 43 typ; 0 def)
% Number of atoms : 977 ( 194 equ; 0 cnn)
% Maximal formula atoms : 13 ( 3 avg)
% Number of connectives : 3417 ( 95 ~; 21 |; 81 &;2637 @)
% ( 0 <=>; 583 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 9 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 259 ( 259 >; 0 *; 0 +; 0 <<)
% Number of symbols : 44 ( 41 usr; 4 con; 0-4 aty)
% Number of variables : 1111 ( 51 ^; 961 !; 66 ?;1111 :)
% ( 33 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:22:11.962
%------------------------------------------------------------------------------
% Could-be-implicit typings (3)
thf(ty_t_HF__Mirabelle__fsbjehakzm_Ohf,type,
hF_Mirabelle_hf: $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
% Explicit typings (40)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ono__bot,type,
no_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ono__top,type,
no_top:
!>[A: $tType] : $o ).
thf(sy_cl_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Owellorder,type,
wellorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Odense__order,type,
dense_order:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Olinordered__field,type,
linordered_field:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Odense__linorder,type,
dense_linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni770627133id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
condit1656338222tinuum:
!>[A: $tType] : $o ).
thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
condit1037483654norder:
!>[A: $tType] : $o ).
thf(sy_c_Finite__Set_OFpow,type,
finite_Fpow:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
thf(sy_c_Finite__Set_Ocard,type,
finite_card:
!>[B: $tType] : ( ( set @ B ) > nat ) ).
thf(sy_c_Finite__Set_Ofinite,type,
finite_finite2:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Fun_Oinj__on,type,
inj_on:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_HF__Mirabelle__fsbjehakzm_OHCollect,type,
hF_Mir1687042746ollect: ( hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(sy_c_HF__Mirabelle__fsbjehakzm_OHF,type,
hF_Mirabelle_HF: ( set @ hF_Mirabelle_hf ) > hF_Mirabelle_hf ).
thf(sy_c_HF__Mirabelle__fsbjehakzm_OHUnion,type,
hF_Mirabelle_HUnion: hF_Mirabelle_hf > hF_Mirabelle_hf ).
thf(sy_c_HF__Mirabelle__fsbjehakzm_OPrimReplace,type,
hF_Mir569462966eplace: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
thf(sy_c_HF__Mirabelle__fsbjehakzm_ORepFun,type,
hF_Mirabelle_RepFun: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf ) > hF_Mirabelle_hf ).
thf(sy_c_HF__Mirabelle__fsbjehakzm_OReplace,type,
hF_Mirabelle_Replace: hF_Mirabelle_hf > ( hF_Mirabelle_hf > hF_Mirabelle_hf > $o ) > hF_Mirabelle_hf ).
thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohfset,type,
hF_Mirabelle_hfset: hF_Mirabelle_hf > ( set @ hF_Mirabelle_hf ) ).
thf(sy_c_HF__Mirabelle__fsbjehakzm_Ohmem,type,
hF_Mirabelle_hmem: hF_Mirabelle_hf > hF_Mirabelle_hf > $o ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oorder__class_OGreatest,type,
order_Greatest:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_Orderings_Oorder__class_Oantimono,type,
order_antimono:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Orderings_Oordering,type,
ordering:
!>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_Oimage,type,
image:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan,type,
set_or578182835ssThan:
!>[A: $tType] : ( A > A > ( set @ A ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_A,type,
a: hF_Mirabelle_hf ).
thf(sy_v_B,type,
b: hF_Mirabelle_hf ).
% Relevant facts (255)
thf(fact_0_hf__equalityI,axiom,
! [A2: hF_Mirabelle_hf,B2: hF_Mirabelle_hf] :
( ! [X: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X @ A2 )
= ( hF_Mirabelle_hmem @ X @ B2 ) )
=> ( A2 = B2 ) ) ).
% hf_equalityI
thf(fact_1_hf__ext,axiom,
( ( ^ [Y: hF_Mirabelle_hf,Z: hF_Mirabelle_hf] : Y = Z )
= ( ^ [A3: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] :
! [X2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X2 @ A3 )
= ( hF_Mirabelle_hmem @ X2 @ B3 ) ) ) ) ).
% hf_ext
thf(fact_2_replacement,axiom,
! [X3: hF_Mirabelle_hf,R: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
( ! [U: hF_Mirabelle_hf,V: hF_Mirabelle_hf,V2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ U @ X3 )
=> ( ( R @ U @ V )
=> ( ( R @ U @ V2 )
=> ( V2 = V ) ) ) )
=> ? [Z2: hF_Mirabelle_hf] :
! [V3: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ V3 @ Z2 )
= ( ? [U2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ U2 @ X3 )
& ( R @ U2 @ V3 ) ) ) ) ) ).
% replacement
thf(fact_3_binary__union,axiom,
! [X3: hF_Mirabelle_hf,Y2: hF_Mirabelle_hf] :
? [Z2: hF_Mirabelle_hf] :
! [U3: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ U3 @ Z2 )
= ( ( hF_Mirabelle_hmem @ U3 @ X3 )
| ( hF_Mirabelle_hmem @ U3 @ Y2 ) ) ) ).
% binary_union
thf(fact_4_union__of__set,axiom,
! [X3: hF_Mirabelle_hf] :
? [Z2: hF_Mirabelle_hf] :
! [U3: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ U3 @ Z2 )
= ( ? [Y3: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ Y3 @ X3 )
& ( hF_Mirabelle_hmem @ U3 @ Y3 ) ) ) ) ).
% union_of_set
thf(fact_5_comprehension,axiom,
! [X3: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o] :
? [Z2: hF_Mirabelle_hf] :
! [U3: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ U3 @ Z2 )
= ( ( hF_Mirabelle_hmem @ U3 @ X3 )
& ( P @ U3 ) ) ) ).
% comprehension
thf(fact_6_less__eq__hf__def,axiom,
( ( ord_less_eq @ hF_Mirabelle_hf )
= ( ^ [A4: hF_Mirabelle_hf,B4: hF_Mirabelle_hf] :
! [X2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X2 @ A4 )
=> ( hF_Mirabelle_hmem @ X2 @ B4 ) ) ) ) ).
% less_eq_hf_def
thf(fact_7_replacement__fun,axiom,
! [X3: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
? [Z2: hF_Mirabelle_hf] :
! [V3: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ V3 @ Z2 )
= ( ? [U2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ U2 @ X3 )
& ( V3
= ( F @ U2 ) ) ) ) ) ).
% replacement_fun
thf(fact_8_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A] : ( ord_less_eq @ A @ X3 @ X3 ) ) ).
% order_refl
thf(fact_9_PrimReplace__iff,axiom,
! [A5: hF_Mirabelle_hf,R: hF_Mirabelle_hf > hF_Mirabelle_hf > $o,V4: hF_Mirabelle_hf] :
( ! [U: hF_Mirabelle_hf,V: hF_Mirabelle_hf,V2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ U @ A5 )
=> ( ( R @ U @ V )
=> ( ( R @ U @ V2 )
=> ( V2 = V ) ) ) )
=> ( ( hF_Mirabelle_hmem @ V4 @ ( hF_Mir569462966eplace @ A5 @ R ) )
= ( ? [U2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ U2 @ A5 )
& ( R @ U2 @ V4 ) ) ) ) ) ).
% PrimReplace_iff
thf(fact_10_HCollect__iff,axiom,
! [X3: hF_Mirabelle_hf,P: hF_Mirabelle_hf > $o,A5: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X3 @ ( hF_Mir1687042746ollect @ P @ A5 ) )
= ( ( P @ X3 )
& ( hF_Mirabelle_hmem @ X3 @ A5 ) ) ) ).
% HCollect_iff
thf(fact_11_HUnion__iff,axiom,
! [X3: hF_Mirabelle_hf,A5: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X3 @ ( hF_Mirabelle_HUnion @ A5 ) )
= ( ? [Y3: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ Y3 @ A5 )
& ( hF_Mirabelle_hmem @ X3 @ Y3 ) ) ) ) ).
% HUnion_iff
thf(fact_12_Replace__iff,axiom,
! [V4: hF_Mirabelle_hf,A5: hF_Mirabelle_hf,R: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
( ( hF_Mirabelle_hmem @ V4 @ ( hF_Mirabelle_Replace @ A5 @ R ) )
= ( ? [U2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ U2 @ A5 )
& ( R @ U2 @ V4 )
& ! [Y3: hF_Mirabelle_hf] :
( ( R @ U2 @ Y3 )
=> ( Y3 = V4 ) ) ) ) ) ).
% Replace_iff
thf(fact_13_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B,X3: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) ) ) ) ).
% le_funD
thf(fact_14_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B,X3: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) ) ) ) ).
% le_funE
thf(fact_15_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B] :
( ! [X: A] : ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) )
=> ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% le_funI
thf(fact_16_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B] :
! [X2: A] : ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% le_fun_def
thf(fact_17_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F: B > A,B2: B,C: B] :
( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C )
=> ( ! [X: B,Y4: B] :
( ( ord_less_eq @ B @ X @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_subst1
thf(fact_18_order__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 )
& ( order @ A ) )
=> ! [A2: A,B2: A,F: A > C2,C: C2] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ C2 @ ( F @ B2 ) @ C )
=> ( ! [X: A,Y4: A] :
( ( ord_less_eq @ A @ X @ Y4 )
=> ( ord_less_eq @ C2 @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_subst2
thf(fact_19_verit__la__disequality,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( A2 = B2 )
| ~ ( ord_less_eq @ A @ A2 @ B2 )
| ~ ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% verit_la_disequality
thf(fact_20_Replace__cong,axiom,
! [A5: hF_Mirabelle_hf,B5: hF_Mirabelle_hf,P: hF_Mirabelle_hf > hF_Mirabelle_hf > $o,Q: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
( ( A5 = B5 )
=> ( ! [X: hF_Mirabelle_hf,Y4: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X @ B5 )
=> ( ( P @ X @ Y4 )
= ( Q @ X @ Y4 ) ) )
=> ( ( hF_Mirabelle_Replace @ A5 @ P )
= ( hF_Mirabelle_Replace @ B5 @ Q ) ) ) ) ).
% Replace_cong
thf(fact_21_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ) ).
% dual_order.antisym
thf(fact_22_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y: A,Z: A] : Y = Z )
= ( ^ [A3: A,B3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
& ( ord_less_eq @ A @ A3 @ B3 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_23_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A,C: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C @ B2 )
=> ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% dual_order.trans
thf(fact_24_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A2: A,B2: A] :
( ! [A6: A,B6: A] :
( ( ord_less_eq @ A @ A6 @ B6 )
=> ( P @ A6 @ B6 ) )
=> ( ! [A6: A,B6: A] :
( ( P @ B6 @ A6 )
=> ( P @ A6 @ B6 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_wlog
thf(fact_25_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_26_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y2: A,Z3: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ Z3 )
=> ( ord_less_eq @ A @ X3 @ Z3 ) ) ) ) ).
% order_trans
thf(fact_27_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ) ).
% order_class.order.antisym
thf(fact_28_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% ord_le_eq_trans
thf(fact_29_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C: A] :
( ( A2 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% ord_eq_le_trans
thf(fact_30_order__class_Oorder_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y: A,Z: A] : Y = Z )
= ( ^ [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
& ( ord_less_eq @ A @ B3 @ A3 ) ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_31_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y2: A,X3: A] :
( ( ord_less_eq @ A @ Y2 @ X3 )
=> ( ( ord_less_eq @ A @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ) ).
% antisym_conv
thf(fact_32_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y2: A,Z3: A] :
( ( ( ord_less_eq @ A @ X3 @ Y2 )
=> ~ ( ord_less_eq @ A @ Y2 @ Z3 ) )
=> ( ( ( ord_less_eq @ A @ Y2 @ X3 )
=> ~ ( ord_less_eq @ A @ X3 @ Z3 ) )
=> ( ( ( ord_less_eq @ A @ X3 @ Z3 )
=> ~ ( ord_less_eq @ A @ Z3 @ Y2 ) )
=> ( ( ( ord_less_eq @ A @ Z3 @ Y2 )
=> ~ ( ord_less_eq @ A @ Y2 @ X3 ) )
=> ( ( ( ord_less_eq @ A @ Y2 @ Z3 )
=> ~ ( ord_less_eq @ A @ Z3 @ X3 ) )
=> ~ ( ( ord_less_eq @ A @ Z3 @ X3 )
=> ~ ( ord_less_eq @ A @ X3 @ Y2 ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_33_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% order.trans
thf(fact_34_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y2: A] :
( ~ ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ).
% le_cases
thf(fact_35_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y2: A] :
( ( X3 = Y2 )
=> ( ord_less_eq @ A @ X3 @ Y2 ) ) ) ).
% eq_refl
thf(fact_36_linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
| ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ).
% linear
thf(fact_37_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ X3 )
=> ( X3 = Y2 ) ) ) ) ).
% antisym
thf(fact_38_eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y: A,Z: A] : Y = Z )
= ( ^ [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
& ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ) ) ).
% eq_iff
thf(fact_39_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,B2: A,F: A > B,C: B] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: A,Y4: A] :
( ( ord_less_eq @ A @ X @ Y4 )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_40_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,F: B > A,B2: B,C: B] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C )
=> ( ! [X: B,Y4: B] :
( ( ord_less_eq @ B @ X @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_41_Greatest__equality,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X3: A] :
( ( P @ X3 )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ A @ Y4 @ X3 ) )
=> ( ( order_Greatest @ A @ P )
= X3 ) ) ) ) ).
% Greatest_equality
thf(fact_42_GreatestI2__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X3: A,Q: A > $o] :
( ( P @ X3 )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ A @ Y4 @ X3 ) )
=> ( ! [X: A] :
( ( P @ X )
=> ( ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ A @ Y5 @ X ) )
=> ( Q @ X ) ) )
=> ( Q @ ( order_Greatest @ A @ P ) ) ) ) ) ) ).
% GreatestI2_order
thf(fact_43_le__rel__bool__arg__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_less_eq @ ( $o > A ) )
= ( ^ [X4: $o > A,Y6: $o > A] :
( ( ord_less_eq @ A @ ( X4 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq @ A @ ( X4 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_44_hmem__def,axiom,
( hF_Mirabelle_hmem
= ( ^ [A3: hF_Mirabelle_hf,B3: hF_Mirabelle_hf] : ( member @ hF_Mirabelle_hf @ A3 @ ( hF_Mirabelle_hfset @ B3 ) ) ) ) ).
% hmem_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,A5: set @ A] :
( ( collect @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X: A] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X: A] :
( ( F @ X )
= ( G @ X ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_RepFun__iff,axiom,
! [V4: hF_Mirabelle_hf,A5: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ V4 @ ( hF_Mirabelle_RepFun @ A5 @ F ) )
= ( ? [U2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ U2 @ A5 )
& ( V4
= ( F @ U2 ) ) ) ) ) ).
% RepFun_iff
thf(fact_50_less__hf__def,axiom,
( ( ord_less @ hF_Mirabelle_hf )
= ( ^ [A4: hF_Mirabelle_hf,B4: hF_Mirabelle_hf] :
( ( ord_less_eq @ hF_Mirabelle_hf @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% less_hf_def
thf(fact_51_antimono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ( ( order_antimono @ A @ B )
= ( ^ [F2: A > B] :
! [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
=> ( ord_less_eq @ B @ ( F2 @ Y3 ) @ ( F2 @ X2 ) ) ) ) ) ) ).
% antimono_def
thf(fact_52_antimonoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F: A > B] :
( ! [X: A,Y4: A] :
( ( ord_less_eq @ A @ X @ Y4 )
=> ( ord_less_eq @ B @ ( F @ Y4 ) @ ( F @ X ) ) )
=> ( order_antimono @ A @ B @ F ) ) ) ).
% antimonoI
thf(fact_53_antimonoE,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F: A > B,X3: A,Y2: A] :
( ( order_antimono @ A @ B @ F )
=> ( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ord_less_eq @ B @ ( F @ Y2 ) @ ( F @ X3 ) ) ) ) ) ).
% antimonoE
thf(fact_54_antimonoD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F: A > B,X3: A,Y2: A] :
( ( order_antimono @ A @ B @ F )
=> ( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ord_less_eq @ B @ ( F @ Y2 ) @ ( F @ X3 ) ) ) ) ) ).
% antimonoD
thf(fact_55_verit__comp__simplify1_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ A2 ) ) ).
% verit_comp_simplify1(1)
thf(fact_56_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,F: B > A,B2: B,C: B] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C )
=> ( ! [X: B,Y4: B] :
( ( ord_less @ B @ X @ Y4 )
=> ( ord_less @ A @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_57_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,B2: A,F: A > B,C: B] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: A,Y4: A] :
( ( ord_less @ A @ X @ Y4 )
=> ( ord_less @ B @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_58_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F: B > A,B2: B,C: B] :
( ( ord_less @ A @ A2 @ ( F @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C )
=> ( ! [X: B,Y4: B] :
( ( ord_less @ B @ X @ Y4 )
=> ( ord_less @ A @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_59_order__less__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 )
& ( order @ A ) )
=> ! [A2: A,B2: A,F: A > C2,C: C2] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ C2 @ ( F @ B2 ) @ C )
=> ( ! [X: A,Y4: A] :
( ( ord_less @ A @ X @ Y4 )
=> ( ord_less @ C2 @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_less_subst2
thf(fact_60_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [X3: A] :
? [Y4: A] : ( ord_less @ A @ Y4 @ X3 ) ) ).
% lt_ex
thf(fact_61_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [X3: A] :
? [X_1: A] : ( ord_less @ A @ X3 @ X_1 ) ) ).
% gt_ex
thf(fact_62_neqE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y2: A] :
( ( X3 != Y2 )
=> ( ~ ( ord_less @ A @ X3 @ Y2 )
=> ( ord_less @ A @ Y2 @ X3 ) ) ) ) ).
% neqE
thf(fact_63_neq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y2: A] :
( ( X3 != Y2 )
= ( ( ord_less @ A @ X3 @ Y2 )
| ( ord_less @ A @ Y2 @ X3 ) ) ) ) ).
% neq_iff
thf(fact_64_order_Oasym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% order.asym
thf(fact_65_dense,axiom,
! [A: $tType] :
( ( dense_order @ A )
=> ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ? [Z2: A] :
( ( ord_less @ A @ X3 @ Z2 )
& ( ord_less @ A @ Z2 @ Y2 ) ) ) ) ).
% dense
thf(fact_66_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( X3 != Y2 ) ) ) ).
% less_imp_neq
thf(fact_67_less__asym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ~ ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% less_asym
thf(fact_68_less__asym_H,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% less_asym'
thf(fact_69_less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y2: A,Z3: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( ( ord_less @ A @ Y2 @ Z3 )
=> ( ord_less @ A @ X3 @ Z3 ) ) ) ) ).
% less_trans
thf(fact_70_less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
| ( X3 = Y2 )
| ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% less_linear
thf(fact_71_less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A] :
~ ( ord_less @ A @ X3 @ X3 ) ) ).
% less_irrefl
thf(fact_72_ord__eq__less__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C: A] :
( ( A2 = B2 )
=> ( ( ord_less @ A @ B2 @ C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% ord_eq_less_trans
thf(fact_73_ord__less__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% ord_less_eq_trans
thf(fact_74_dual__order_Oasym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ~ ( ord_less @ A @ A2 @ B2 ) ) ) ).
% dual_order.asym
thf(fact_75_less__imp__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( X3 != Y2 ) ) ) ).
% less_imp_not_eq
thf(fact_76_less__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ~ ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% less_not_sym
thf(fact_77_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,A2: A] :
( ! [X: A] :
( ! [Y5: A] :
( ( ord_less @ A @ Y5 @ X )
=> ( P @ Y5 ) )
=> ( P @ X ) )
=> ( P @ A2 ) ) ) ).
% less_induct
thf(fact_78_antisym__conv3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y2: A,X3: A] :
( ~ ( ord_less @ A @ Y2 @ X3 )
=> ( ( ~ ( ord_less @ A @ X3 @ Y2 ) )
= ( X3 = Y2 ) ) ) ) ).
% antisym_conv3
thf(fact_79_less__imp__not__eq2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( Y2 != X3 ) ) ) ).
% less_imp_not_eq2
thf(fact_80_less__imp__triv,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y2: A,P: $o] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( ( ord_less @ A @ Y2 @ X3 )
=> P ) ) ) ).
% less_imp_triv
thf(fact_81_linorder__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y2: A] :
( ~ ( ord_less @ A @ X3 @ Y2 )
=> ( ( X3 != Y2 )
=> ( ord_less @ A @ Y2 @ X3 ) ) ) ) ).
% linorder_cases
thf(fact_82_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ A2 ) ) ).
% dual_order.irrefl
thf(fact_83_order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ B2 @ C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% order.strict_trans
thf(fact_84_less__imp__not__less,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ~ ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% less_imp_not_less
thf(fact_85_exists__least__iff,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ( ( ^ [P2: A > $o] :
? [X5: A] : ( P2 @ X5 ) )
= ( ^ [P3: A > $o] :
? [N: A] :
( ( P3 @ N )
& ! [M: A] :
( ( ord_less @ A @ M @ N )
=> ~ ( P3 @ M ) ) ) ) ) ) ).
% exists_least_iff
thf(fact_86_linorder__less__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A2: A,B2: A] :
( ! [A6: A,B6: A] :
( ( ord_less @ A @ A6 @ B6 )
=> ( P @ A6 @ B6 ) )
=> ( ! [A6: A] : ( P @ A6 @ A6 )
=> ( ! [A6: A,B6: A] :
( ( P @ B6 @ A6 )
=> ( P @ A6 @ B6 ) )
=> ( P @ A2 @ B2 ) ) ) ) ) ).
% linorder_less_wlog
thf(fact_87_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A,C: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_less @ A @ C @ B2 )
=> ( ord_less @ A @ C @ A2 ) ) ) ) ).
% dual_order.strict_trans
thf(fact_88_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y2: A] :
( ( ~ ( ord_less @ A @ X3 @ Y2 ) )
= ( ( ord_less @ A @ Y2 @ X3 )
| ( X3 = Y2 ) ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_89_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( A2 != B2 ) ) ) ).
% order.strict_implies_not_eq
thf(fact_90_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( A2 != B2 ) ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_91_verit__comp__simplify1_I3_J,axiom,
! [B: $tType] :
( ( linorder @ B )
=> ! [B7: B,A7: B] :
( ( ~ ( ord_less_eq @ B @ B7 @ A7 ) )
= ( ord_less @ B @ A7 @ B7 ) ) ) ).
% verit_comp_simplify1(3)
thf(fact_92_leD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y2: A,X3: A] :
( ( ord_less_eq @ A @ Y2 @ X3 )
=> ~ ( ord_less @ A @ X3 @ Y2 ) ) ) ).
% leD
thf(fact_93_leI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y2: A] :
( ~ ( ord_less @ A @ X3 @ Y2 )
=> ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ).
% leI
thf(fact_94_le__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X2: A,Y3: A] :
( ( ord_less @ A @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ) ).
% le_less
thf(fact_95_less__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
& ( X2 != Y3 ) ) ) ) ) ).
% less_le
thf(fact_96_order__le__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F: B > A,B2: B,C: B] :
( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C )
=> ( ! [X: B,Y4: B] :
( ( ord_less @ B @ X @ Y4 )
=> ( ord_less @ A @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_97_order__le__less__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 )
& ( order @ A ) )
=> ! [A2: A,B2: A,F: A > C2,C: C2] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ C2 @ ( F @ B2 ) @ C )
=> ( ! [X: A,Y4: A] :
( ( ord_less_eq @ A @ X @ Y4 )
=> ( ord_less_eq @ C2 @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_le_less_subst2
thf(fact_98_order__less__le__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F: B > A,B2: B,C: B] :
( ( ord_less @ A @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C )
=> ( ! [X: B,Y4: B] :
( ( ord_less_eq @ B @ X @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_99_order__less__le__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 )
& ( order @ A ) )
=> ! [A2: A,B2: A,F: A > C2,C: C2] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ C2 @ ( F @ B2 ) @ C )
=> ( ! [X: A,Y4: A] :
( ( ord_less @ A @ X @ Y4 )
=> ( ord_less @ C2 @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_less_le_subst2
thf(fact_100_not__le,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y2: A] :
( ( ~ ( ord_less_eq @ A @ X3 @ Y2 ) )
= ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% not_le
thf(fact_101_not__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y2: A] :
( ( ~ ( ord_less @ A @ X3 @ Y2 ) )
= ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ).
% not_less
thf(fact_102_le__neq__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% le_neq_trans
thf(fact_103_antisym__conv1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y2: A] :
( ~ ( ord_less @ A @ X3 @ Y2 )
=> ( ( ord_less_eq @ A @ X3 @ Y2 )
= ( X3 = Y2 ) ) ) ) ).
% antisym_conv1
thf(fact_104_antisym__conv2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ( ~ ( ord_less @ A @ X3 @ Y2 ) )
= ( X3 = Y2 ) ) ) ) ).
% antisym_conv2
thf(fact_105_less__imp__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( ord_less_eq @ A @ X3 @ Y2 ) ) ) ).
% less_imp_le
thf(fact_106_le__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y2: A,Z3: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ( ord_less @ A @ Y2 @ Z3 )
=> ( ord_less @ A @ X3 @ Z3 ) ) ) ) ).
% le_less_trans
thf(fact_107_less__le__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y2: A,Z3: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ Z3 )
=> ( ord_less @ A @ X3 @ Z3 ) ) ) ) ).
% less_le_trans
thf(fact_108_dense__ge,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z3: A,Y2: A] :
( ! [X: A] :
( ( ord_less @ A @ Z3 @ X )
=> ( ord_less_eq @ A @ Y2 @ X ) )
=> ( ord_less_eq @ A @ Y2 @ Z3 ) ) ) ).
% dense_ge
thf(fact_109_dense__le,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Y2: A,Z3: A] :
( ! [X: A] :
( ( ord_less @ A @ X @ Y2 )
=> ( ord_less_eq @ A @ X @ Z3 ) )
=> ( ord_less_eq @ A @ Y2 @ Z3 ) ) ) ).
% dense_le
thf(fact_110_le__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
| ( ord_less @ A @ Y2 @ X3 ) ) ) ).
% le_less_linear
thf(fact_111_le__imp__less__or__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ( ord_less @ A @ X3 @ Y2 )
| ( X3 = Y2 ) ) ) ) ).
% le_imp_less_or_eq
thf(fact_112_less__le__not__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
& ~ ( ord_less_eq @ A @ Y3 @ X2 ) ) ) ) ) ).
% less_le_not_le
thf(fact_113_not__le__imp__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y2: A,X3: A] :
( ~ ( ord_less_eq @ A @ Y2 @ X3 )
=> ( ord_less @ A @ X3 @ Y2 ) ) ) ).
% not_le_imp_less
thf(fact_114_order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ B2 @ C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% order.strict_trans1
thf(fact_115_order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% order.strict_trans2
thf(fact_116_order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ) ).
% order.order_iff_strict
thf(fact_117_order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ) ).
% order.strict_iff_order
thf(fact_118_dual__order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A,C: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less @ A @ C @ B2 )
=> ( ord_less @ A @ C @ A2 ) ) ) ) ).
% dual_order.strict_trans1
thf(fact_119_dual__order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A,C: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C @ B2 )
=> ( ord_less @ A @ C @ A2 ) ) ) ) ).
% dual_order.strict_trans2
thf(fact_120_dense__ge__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z3: A,X3: A,Y2: A] :
( ( ord_less @ A @ Z3 @ X3 )
=> ( ! [W: A] :
( ( ord_less @ A @ Z3 @ W )
=> ( ( ord_less @ A @ W @ X3 )
=> ( ord_less_eq @ A @ Y2 @ W ) ) )
=> ( ord_less_eq @ A @ Y2 @ Z3 ) ) ) ) ).
% dense_ge_bounded
thf(fact_121_dense__le__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [X3: A,Y2: A,Z3: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( ! [W: A] :
( ( ord_less @ A @ X3 @ W )
=> ( ( ord_less @ A @ W @ Y2 )
=> ( ord_less_eq @ A @ W @ Z3 ) ) )
=> ( ord_less_eq @ A @ Y2 @ Z3 ) ) ) ) ).
% dense_le_bounded
thf(fact_122_order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% order.strict_implies_order
thf(fact_123_dual__order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B3: A,A3: A] :
( ( ord_less @ A @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_124_dual__order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [B3: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_125_dual__order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% dual_order.strict_implies_order
thf(fact_126_order_Onot__eq__order__implies__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( A2 != B2 )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% order.not_eq_order_implies_strict
thf(fact_127_RepFun__cong,axiom,
! [A5: hF_Mirabelle_hf,B5: hF_Mirabelle_hf,F: hF_Mirabelle_hf > hF_Mirabelle_hf,G: hF_Mirabelle_hf > hF_Mirabelle_hf] :
( ( A5 = B5 )
=> ( ! [X: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X @ B5 )
=> ( ( F @ X )
= ( G @ X ) ) )
=> ( ( hF_Mirabelle_RepFun @ A5 @ F )
= ( hF_Mirabelle_RepFun @ B5 @ G ) ) ) ) ).
% RepFun_cong
thf(fact_128_minf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z2: A] :
! [X6: A] :
( ( ord_less @ A @ X6 @ Z2 )
=> ~ ( ord_less_eq @ A @ T @ X6 ) ) ) ).
% minf(8)
thf(fact_129_minf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z2: A] :
! [X6: A] :
( ( ord_less @ A @ X6 @ Z2 )
=> ( ord_less_eq @ A @ X6 @ T ) ) ) ).
% minf(6)
thf(fact_130_pinf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z2: A] :
! [X6: A] :
( ( ord_less @ A @ Z2 @ X6 )
=> ( ord_less_eq @ A @ T @ X6 ) ) ) ).
% pinf(8)
thf(fact_131_pinf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z2: A] :
! [X6: A] :
( ( ord_less @ A @ Z2 @ X6 )
=> ~ ( ord_less_eq @ A @ X6 @ T ) ) ) ).
% pinf(6)
thf(fact_132_complete__interval,axiom,
! [A: $tType] :
( ( condit1037483654norder @ A )
=> ! [A2: A,B2: A,P: A > $o] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( P @ A2 )
=> ( ~ ( P @ B2 )
=> ? [C3: A] :
( ( ord_less_eq @ A @ A2 @ C3 )
& ( ord_less_eq @ A @ C3 @ B2 )
& ! [X6: A] :
( ( ( ord_less_eq @ A @ A2 @ X6 )
& ( ord_less @ A @ X6 @ C3 ) )
=> ( P @ X6 ) )
& ! [D: A] :
( ! [X: A] :
( ( ( ord_less_eq @ A @ A2 @ X )
& ( ord_less @ A @ X @ D ) )
=> ( P @ X ) )
=> ( ord_less_eq @ A @ D @ C3 ) ) ) ) ) ) ) ).
% complete_interval
thf(fact_133_HF__hfset,axiom,
! [A2: hF_Mirabelle_hf] :
( ( hF_Mirabelle_HF @ ( hF_Mirabelle_hfset @ A2 ) )
= A2 ) ).
% HF_hfset
thf(fact_134_ex__gt__or__lt,axiom,
! [A: $tType] :
( ( condit1656338222tinuum @ A )
=> ! [A2: A] :
? [B6: A] :
( ( ord_less @ A @ A2 @ B6 )
| ( ord_less @ A @ B6 @ A2 ) ) ) ).
% ex_gt_or_lt
thf(fact_135_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X3: A,Y2: A] :
( ( X3 != Y2 )
=> ( ~ ( ord_less @ A @ X3 @ Y2 )
=> ( ord_less @ A @ Y2 @ X3 ) ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_136_less__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B] :
( ( ord_less_eq @ ( A > B ) @ F2 @ G2 )
& ~ ( ord_less_eq @ ( A > B ) @ G2 @ F2 ) ) ) ) ) ).
% less_fun_def
thf(fact_137_minf_I11_J,axiom,
! [C2: $tType,D2: $tType] :
( ( ord @ C2 )
=> ! [F3: D2] :
? [Z2: C2] :
! [X6: C2] :
( ( ord_less @ C2 @ X6 @ Z2 )
=> ( F3 = F3 ) ) ) ).
% minf(11)
thf(fact_138_minf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z2: A] :
! [X6: A] :
( ( ord_less @ A @ X6 @ Z2 )
=> ~ ( ord_less @ A @ T @ X6 ) ) ) ).
% minf(7)
thf(fact_139_minf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z2: A] :
! [X6: A] :
( ( ord_less @ A @ X6 @ Z2 )
=> ( ord_less @ A @ X6 @ T ) ) ) ).
% minf(5)
thf(fact_140_minf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z2: A] :
! [X6: A] :
( ( ord_less @ A @ X6 @ Z2 )
=> ( X6 != T ) ) ) ).
% minf(4)
thf(fact_141_minf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z2: A] :
! [X6: A] :
( ( ord_less @ A @ X6 @ Z2 )
=> ( X6 != T ) ) ) ).
% minf(3)
thf(fact_142_minf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X: A] :
( ( ord_less @ A @ X @ Z4 )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: A] :
! [X: A] :
( ( ord_less @ A @ X @ Z4 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: A] :
! [X6: A] :
( ( ord_less @ A @ X6 @ Z2 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ) ).
% minf(2)
thf(fact_143_minf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X: A] :
( ( ord_less @ A @ X @ Z4 )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: A] :
! [X: A] :
( ( ord_less @ A @ X @ Z4 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: A] :
! [X6: A] :
( ( ord_less @ A @ X6 @ Z2 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ) ).
% minf(1)
thf(fact_144_pinf_I11_J,axiom,
! [C2: $tType,D2: $tType] :
( ( ord @ C2 )
=> ! [F3: D2] :
? [Z2: C2] :
! [X6: C2] :
( ( ord_less @ C2 @ Z2 @ X6 )
=> ( F3 = F3 ) ) ) ).
% pinf(11)
thf(fact_145_pinf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z2: A] :
! [X6: A] :
( ( ord_less @ A @ Z2 @ X6 )
=> ( ord_less @ A @ T @ X6 ) ) ) ).
% pinf(7)
thf(fact_146_pinf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z2: A] :
! [X6: A] :
( ( ord_less @ A @ Z2 @ X6 )
=> ~ ( ord_less @ A @ X6 @ T ) ) ) ).
% pinf(5)
thf(fact_147_pinf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z2: A] :
! [X6: A] :
( ( ord_less @ A @ Z2 @ X6 )
=> ( X6 != T ) ) ) ).
% pinf(4)
thf(fact_148_pinf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z2: A] :
! [X6: A] :
( ( ord_less @ A @ Z2 @ X6 )
=> ( X6 != T ) ) ) ).
% pinf(3)
thf(fact_149_pinf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X: A] :
( ( ord_less @ A @ Z4 @ X )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: A] :
! [X: A] :
( ( ord_less @ A @ Z4 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: A] :
! [X6: A] :
( ( ord_less @ A @ Z2 @ X6 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ) ).
% pinf(2)
thf(fact_150_pinf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X: A] :
( ( ord_less @ A @ Z4 @ X )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: A] :
! [X: A] :
( ( ord_less @ A @ Z4 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: A] :
! [X6: A] :
( ( ord_less @ A @ Z2 @ X6 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ) ).
% pinf(1)
thf(fact_151_hfset__HF,axiom,
! [A5: set @ hF_Mirabelle_hf] :
( ( finite_finite2 @ hF_Mirabelle_hf @ A5 )
=> ( ( hF_Mirabelle_hfset @ ( hF_Mirabelle_HF @ A5 ) )
= A5 ) ) ).
% hfset_HF
thf(fact_152_hmem__HF__iff,axiom,
! [X3: hF_Mirabelle_hf,A5: set @ hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X3 @ ( hF_Mirabelle_HF @ A5 ) )
= ( ( member @ hF_Mirabelle_hf @ X3 @ A5 )
& ( finite_finite2 @ hF_Mirabelle_hf @ A5 ) ) ) ).
% hmem_HF_iff
thf(fact_153_measure__induct,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ B )
=> ! [F: A > B,P: A > $o,A2: A] :
( ! [X: A] :
( ! [Y5: A] :
( ( ord_less @ B @ ( F @ Y5 ) @ ( F @ X ) )
=> ( P @ Y5 ) )
=> ( P @ X ) )
=> ( P @ A2 ) ) ) ).
% measure_induct
thf(fact_154_measure__induct__rule,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ B )
=> ! [F: A > B,P: A > $o,A2: A] :
( ! [X: A] :
( ! [Y5: A] :
( ( ord_less @ B @ ( F @ Y5 ) @ ( F @ X ) )
=> ( P @ Y5 ) )
=> ( P @ X ) )
=> ( P @ A2 ) ) ) ).
% measure_induct_rule
thf(fact_155_finite__hfset,axiom,
! [A2: hF_Mirabelle_hf] : ( finite_finite2 @ hF_Mirabelle_hf @ ( hF_Mirabelle_hfset @ A2 ) ) ).
% finite_hfset
thf(fact_156_finite__code,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( finite_finite2 @ A )
= ( ^ [A4: set @ A] : $true ) ) ) ).
% finite_code
thf(fact_157_finite__has__minimal2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A5: set @ A,A2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ A2 @ A5 )
=> ? [X: A] :
( ( member @ A @ X @ A5 )
& ( ord_less_eq @ A @ X @ A2 )
& ! [Xa: A] :
( ( member @ A @ Xa @ A5 )
=> ( ( ord_less_eq @ A @ Xa @ X )
=> ( X = Xa ) ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_158_finite__has__maximal2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A5: set @ A,A2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ A2 @ A5 )
=> ? [X: A] :
( ( member @ A @ X @ A5 )
& ( ord_less_eq @ A @ A2 @ X )
& ! [Xa: A] :
( ( member @ A @ Xa @ A5 )
=> ( ( ord_less_eq @ A @ X @ Xa )
=> ( X = Xa ) ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_159_finite__subset,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ( finite_finite2 @ A @ B5 )
=> ( finite_finite2 @ A @ A5 ) ) ) ).
% finite_subset
thf(fact_160_infinite__super,axiom,
! [A: $tType,S: set @ A,T2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ S @ T2 )
=> ( ~ ( finite_finite2 @ A @ S )
=> ~ ( finite_finite2 @ A @ T2 ) ) ) ).
% infinite_super
thf(fact_161_rev__finite__subset,axiom,
! [A: $tType,B5: set @ A,A5: set @ A] :
( ( finite_finite2 @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( finite_finite2 @ A @ A5 ) ) ) ).
% rev_finite_subset
thf(fact_162_finite__set__choice,axiom,
! [B: $tType,A: $tType,A5: set @ A,P: A > B > $o] :
( ( finite_finite2 @ A @ A5 )
=> ( ! [X: A] :
( ( member @ A @ X @ A5 )
=> ? [X_12: B] : ( P @ X @ X_12 ) )
=> ? [F4: A > B] :
! [X6: A] :
( ( member @ A @ X6 @ A5 )
=> ( P @ X6 @ ( F4 @ X6 ) ) ) ) ) ).
% finite_set_choice
thf(fact_163_finite,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [A5: set @ A] : ( finite_finite2 @ A @ A5 ) ) ).
% finite
thf(fact_164_finite__psubset__induct,axiom,
! [A: $tType,A5: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A5 )
=> ( ! [A8: set @ A] :
( ( finite_finite2 @ A @ A8 )
=> ( ! [B8: set @ A] :
( ( ord_less @ ( set @ A ) @ B8 @ A8 )
=> ( P @ B8 ) )
=> ( P @ A8 ) ) )
=> ( P @ A5 ) ) ) ).
% finite_psubset_induct
thf(fact_165_dependent__wellorder__choice,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ A )
=> ! [P: ( A > B ) > A > B > $o] :
( ! [R2: B,F4: A > B,G3: A > B,X: A] :
( ! [Y5: A] :
( ( ord_less @ A @ Y5 @ X )
=> ( ( F4 @ Y5 )
= ( G3 @ Y5 ) ) )
=> ( ( P @ F4 @ X @ R2 )
= ( P @ G3 @ X @ R2 ) ) )
=> ( ! [X: A,F4: A > B] :
( ! [Y5: A] :
( ( ord_less @ A @ Y5 @ X )
=> ( P @ F4 @ Y5 @ ( F4 @ Y5 ) ) )
=> ? [X_12: B] : ( P @ F4 @ X @ X_12 ) )
=> ? [F4: A > B] :
! [X6: A] : ( P @ F4 @ X6 @ ( F4 @ X6 ) ) ) ) ) ).
% dependent_wellorder_choice
thf(fact_166_linordered__field__no__lb,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X6: A] :
? [Y4: A] : ( ord_less @ A @ Y4 @ X6 ) ) ).
% linordered_field_no_lb
thf(fact_167_linordered__field__no__ub,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X6: A] :
? [X_1: A] : ( ord_less @ A @ X6 @ X_1 ) ) ).
% linordered_field_no_ub
thf(fact_168_card__psubset,axiom,
! [A: $tType,B5: set @ A,A5: set @ A] :
( ( finite_finite2 @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ( ord_less @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ A @ B5 ) )
=> ( ord_less @ ( set @ A ) @ A5 @ B5 ) ) ) ) ).
% card_psubset
thf(fact_169_inj__on__HF,axiom,
inj_on @ ( set @ hF_Mirabelle_hf ) @ hF_Mirabelle_hf @ hF_Mirabelle_HF @ ( collect @ ( set @ hF_Mirabelle_hf ) @ ( finite_finite2 @ hF_Mirabelle_hf ) ) ).
% inj_on_HF
thf(fact_170_order_Oordering__axioms,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ordering @ A @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% order.ordering_axioms
thf(fact_171_ordering_Onot__eq__order__implies__strict,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( A2 != B2 )
=> ( ( Less_eq @ A2 @ B2 )
=> ( Less @ A2 @ B2 ) ) ) ) ).
% ordering.not_eq_order_implies_strict
thf(fact_172_ordering_Ostrict__implies__not__eq,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( Less @ A2 @ B2 )
=> ( A2 != B2 ) ) ) ).
% ordering.strict_implies_not_eq
thf(fact_173_ordering_Ostrict__implies__order,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( Less @ A2 @ B2 )
=> ( Less_eq @ A2 @ B2 ) ) ) ).
% ordering.strict_implies_order
thf(fact_174_ordering_Ostrict__iff__order,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( Less @ A2 @ B2 )
= ( ( Less_eq @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% ordering.strict_iff_order
thf(fact_175_ordering_Oorder__iff__strict,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( Less_eq @ A2 @ B2 )
= ( ( Less @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% ordering.order_iff_strict
thf(fact_176_ordering_Ostrict__trans2,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A,C: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( Less @ A2 @ B2 )
=> ( ( Less_eq @ B2 @ C )
=> ( Less @ A2 @ C ) ) ) ) ).
% ordering.strict_trans2
thf(fact_177_ordering_Ostrict__trans1,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A,C: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( Less_eq @ A2 @ B2 )
=> ( ( Less @ B2 @ C )
=> ( Less @ A2 @ C ) ) ) ) ).
% ordering.strict_trans1
thf(fact_178_ordering_Ostrict__trans,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A,C: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( Less @ A2 @ B2 )
=> ( ( Less @ B2 @ C )
=> ( Less @ A2 @ C ) ) ) ) ).
% ordering.strict_trans
thf(fact_179_ordering__strictI,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o] :
( ! [A6: A,B6: A] :
( ( Less_eq @ A6 @ B6 )
= ( ( Less @ A6 @ B6 )
| ( A6 = B6 ) ) )
=> ( ! [A6: A,B6: A] :
( ( Less @ A6 @ B6 )
=> ~ ( Less @ B6 @ A6 ) )
=> ( ! [A6: A] :
~ ( Less @ A6 @ A6 )
=> ( ! [A6: A,B6: A,C3: A] :
( ( Less @ A6 @ B6 )
=> ( ( Less @ B6 @ C3 )
=> ( Less @ A6 @ C3 ) ) )
=> ( ordering @ A @ Less_eq @ Less ) ) ) ) ) ).
% ordering_strictI
thf(fact_180_ordering_Oantisym,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( Less_eq @ A2 @ B2 )
=> ( ( Less_eq @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ) ).
% ordering.antisym
thf(fact_181_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_182_ordering_Oeq__iff,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( A2 = B2 )
= ( ( Less_eq @ A2 @ B2 )
& ( Less_eq @ B2 @ A2 ) ) ) ) ).
% ordering.eq_iff
thf(fact_183_ordering_Otrans,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A,C: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( Less_eq @ A2 @ B2 )
=> ( ( Less_eq @ B2 @ C )
=> ( Less_eq @ A2 @ C ) ) ) ) ).
% ordering.trans
thf(fact_184_ordering_Ointro,axiom,
! [A: $tType,Less: A > A > $o,Less_eq: A > A > $o] :
( ! [A6: A,B6: A] :
( ( Less @ A6 @ B6 )
= ( ( Less_eq @ A6 @ B6 )
& ( A6 != B6 ) ) )
=> ( ! [A6: A] : ( Less_eq @ A6 @ A6 )
=> ( ! [A6: A,B6: A] :
( ( Less_eq @ A6 @ B6 )
=> ( ( Less_eq @ B6 @ A6 )
=> ( A6 = B6 ) ) )
=> ( ! [A6: A,B6: A,C3: A] :
( ( Less_eq @ A6 @ B6 )
=> ( ( Less_eq @ B6 @ C3 )
=> ( Less_eq @ A6 @ C3 ) ) )
=> ( ordering @ A @ Less_eq @ Less ) ) ) ) ) ).
% ordering.intro
thf(fact_185_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_186_ordering_Oasym,axiom,
! [A: $tType,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A] :
( ( ordering @ A @ Less_eq @ Less )
=> ( ( Less @ A2 @ B2 )
=> ~ ( Less @ B2 @ A2 ) ) ) ).
% ordering.asym
thf(fact_187_ordering__def,axiom,
! [A: $tType] :
( ( ordering @ A )
= ( ^ [Less_eq2: A > A > $o,Less2: A > A > $o] :
( ! [A3: A,B3: A] :
( ( Less2 @ A3 @ B3 )
= ( ( Less_eq2 @ A3 @ B3 )
& ( A3 != B3 ) ) )
& ! [A3: A] : ( Less_eq2 @ A3 @ A3 )
& ! [A3: A,B3: A] :
( ( Less_eq2 @ A3 @ B3 )
=> ( ( Less_eq2 @ B3 @ A3 )
=> ( A3 = B3 ) ) )
& ! [A3: A,B3: A,C4: A] :
( ( Less_eq2 @ A3 @ B3 )
=> ( ( Less_eq2 @ B3 @ C4 )
=> ( Less_eq2 @ A3 @ C4 ) ) ) ) ) ) ).
% ordering_def
thf(fact_188_card__mono,axiom,
! [A: $tType,B5: set @ A,A5: set @ A] :
( ( finite_finite2 @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ A @ B5 ) ) ) ) ).
% card_mono
thf(fact_189_card__seteq,axiom,
! [A: $tType,B5: set @ A,A5: set @ A] :
( ( finite_finite2 @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ B5 ) @ ( finite_card @ A @ A5 ) )
=> ( A5 = B5 ) ) ) ) ).
% card_seteq
thf(fact_190_card__subset__eq,axiom,
! [A: $tType,B5: set @ A,A5: set @ A] :
( ( finite_finite2 @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ( ( finite_card @ A @ A5 )
= ( finite_card @ A @ B5 ) )
=> ( A5 = B5 ) ) ) ) ).
% card_subset_eq
thf(fact_191_infinite__arbitrarily__large,axiom,
! [A: $tType,A5: set @ A,N2: nat] :
( ~ ( finite_finite2 @ A @ A5 )
=> ? [B9: set @ A] :
( ( finite_finite2 @ A @ B9 )
& ( ( finite_card @ A @ B9 )
= N2 )
& ( ord_less_eq @ ( set @ A ) @ B9 @ A5 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_192_finite__if__finite__subsets__card__bdd,axiom,
! [A: $tType,F3: set @ A,C5: nat] :
( ! [G4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ G4 @ F3 )
=> ( ( finite_finite2 @ A @ G4 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ G4 ) @ C5 ) ) )
=> ( ( finite_finite2 @ A @ F3 )
& ( ord_less_eq @ nat @ ( finite_card @ A @ F3 ) @ C5 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_193_psubset__card__mono,axiom,
! [A: $tType,B5: set @ A,A5: set @ A] :
( ( finite_finite2 @ A @ B5 )
=> ( ( ord_less @ ( set @ A ) @ A5 @ B5 )
=> ( ord_less @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ A @ B5 ) ) ) ) ).
% psubset_card_mono
thf(fact_194_obtain__subset__with__card__n,axiom,
! [A: $tType,N2: nat,S: set @ A] :
( ( ord_less_eq @ nat @ N2 @ ( finite_card @ A @ S ) )
=> ~ ! [T3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ T3 @ S )
=> ( ( ( finite_card @ A @ T3 )
= N2 )
=> ~ ( finite_finite2 @ A @ T3 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_195_linorder__inj__onI,axiom,
! [B: $tType,A: $tType] :
( ( order @ A )
=> ! [A5: set @ A,F: A > B] :
( ! [X: A,Y4: A] :
( ( ord_less @ A @ X @ Y4 )
=> ( ( member @ A @ X @ A5 )
=> ( ( member @ A @ Y4 @ A5 )
=> ( ( F @ X )
!= ( F @ Y4 ) ) ) ) )
=> ( ! [X: A,Y4: A] :
( ( member @ A @ X @ A5 )
=> ( ( member @ A @ Y4 @ A5 )
=> ( ( ord_less_eq @ A @ X @ Y4 )
| ( ord_less_eq @ A @ Y4 @ X ) ) ) )
=> ( inj_on @ A @ B @ F @ A5 ) ) ) ) ).
% linorder_inj_onI
thf(fact_196_card__le__if__inj__on__rel,axiom,
! [B: $tType,A: $tType,B5: set @ A,A5: set @ B,R3: B > A > $o] :
( ( finite_finite2 @ A @ B5 )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ A5 )
=> ? [B10: A] :
( ( member @ A @ B10 @ B5 )
& ( R3 @ A6 @ B10 ) ) )
=> ( ! [A1: B,A22: B,B6: A] :
( ( member @ B @ A1 @ A5 )
=> ( ( member @ B @ A22 @ A5 )
=> ( ( member @ A @ B6 @ B5 )
=> ( ( R3 @ A1 @ B6 )
=> ( ( R3 @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ A5 ) @ ( finite_card @ A @ B5 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_197_finite__nat__set__iff__bounded__le,axiom,
( ( finite_finite2 @ nat )
= ( ^ [N3: set @ nat] :
? [M: nat] :
! [X2: nat] :
( ( member @ nat @ X2 @ N3 )
=> ( ord_less_eq @ nat @ X2 @ M ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_198_bounded__nat__set__is__finite,axiom,
! [N4: set @ nat,N2: nat] :
( ! [X: nat] :
( ( member @ nat @ X @ N4 )
=> ( ord_less @ nat @ X @ N2 ) )
=> ( finite_finite2 @ nat @ N4 ) ) ).
% bounded_nat_set_is_finite
thf(fact_199_finite__nat__set__iff__bounded,axiom,
( ( finite_finite2 @ nat )
= ( ^ [N3: set @ nat] :
? [M: nat] :
! [X2: nat] :
( ( member @ nat @ X2 @ N3 )
=> ( ord_less @ nat @ X2 @ M ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_200_card__le__inj,axiom,
! [B: $tType,A: $tType,A5: set @ A,B5: set @ B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ B @ B5 ) )
=> ? [F4: A > B] :
( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F4 @ A5 ) @ B5 )
& ( inj_on @ A @ B @ F4 @ A5 ) ) ) ) ) ).
% card_le_inj
thf(fact_201_card__inj__on__le,axiom,
! [A: $tType,B: $tType,F: A > B,A5: set @ A,B5: set @ B] :
( ( inj_on @ A @ B @ F @ A5 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A5 ) @ B5 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ B @ B5 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_202_finite__imageI,axiom,
! [B: $tType,A: $tType,F3: set @ A,H: A > B] :
( ( finite_finite2 @ A @ F3 )
=> ( finite_finite2 @ B @ ( image @ A @ B @ H @ F3 ) ) ) ).
% finite_imageI
thf(fact_203_all__subset__image,axiom,
! [A: $tType,B: $tType,F: B > A,A5: set @ B,P: ( set @ A ) > $o] :
( ( ! [B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B4 @ ( image @ B @ A @ F @ A5 ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ B4 @ A5 )
=> ( P @ ( image @ B @ A @ F @ B4 ) ) ) ) ) ).
% all_subset_image
thf(fact_204_all__finite__subset__image,axiom,
! [A: $tType,B: $tType,F: B > A,A5: set @ B,P: ( set @ A ) > $o] :
( ( ! [B4: set @ A] :
( ( ( finite_finite2 @ A @ B4 )
& ( ord_less_eq @ ( set @ A ) @ B4 @ ( image @ B @ A @ F @ A5 ) ) )
=> ( P @ B4 ) ) )
= ( ! [B4: set @ B] :
( ( ( finite_finite2 @ B @ B4 )
& ( ord_less_eq @ ( set @ B ) @ B4 @ A5 ) )
=> ( P @ ( image @ B @ A @ F @ B4 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_205_ex__finite__subset__image,axiom,
! [A: $tType,B: $tType,F: B > A,A5: set @ B,P: ( set @ A ) > $o] :
( ( ? [B4: set @ A] :
( ( finite_finite2 @ A @ B4 )
& ( ord_less_eq @ ( set @ A ) @ B4 @ ( image @ B @ A @ F @ A5 ) )
& ( P @ B4 ) ) )
= ( ? [B4: set @ B] :
( ( finite_finite2 @ B @ B4 )
& ( ord_less_eq @ ( set @ B ) @ B4 @ A5 )
& ( P @ ( image @ B @ A @ F @ B4 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_206_finite__subset__image,axiom,
! [A: $tType,B: $tType,B5: set @ A,F: B > A,A5: set @ B] :
( ( finite_finite2 @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ ( image @ B @ A @ F @ A5 ) )
=> ? [C6: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ C6 @ A5 )
& ( finite_finite2 @ B @ C6 )
& ( B5
= ( image @ B @ A @ F @ C6 ) ) ) ) ) ).
% finite_subset_image
thf(fact_207_finite__surj,axiom,
! [A: $tType,B: $tType,A5: set @ A,B5: set @ B,F: A > B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ord_less_eq @ ( set @ B ) @ B5 @ ( image @ A @ B @ F @ A5 ) )
=> ( finite_finite2 @ B @ B5 ) ) ) ).
% finite_surj
thf(fact_208_finite__image__iff,axiom,
! [B: $tType,A: $tType,F: A > B,A5: set @ A] :
( ( inj_on @ A @ B @ F @ A5 )
=> ( ( finite_finite2 @ B @ ( image @ A @ B @ F @ A5 ) )
= ( finite_finite2 @ A @ A5 ) ) ) ).
% finite_image_iff
thf(fact_209_finite__imageD,axiom,
! [A: $tType,B: $tType,F: B > A,A5: set @ B] :
( ( finite_finite2 @ A @ ( image @ B @ A @ F @ A5 ) )
=> ( ( inj_on @ B @ A @ F @ A5 )
=> ( finite_finite2 @ B @ A5 ) ) ) ).
% finite_imageD
thf(fact_210_card__image,axiom,
! [B: $tType,A: $tType,F: A > B,A5: set @ A] :
( ( inj_on @ A @ B @ F @ A5 )
=> ( ( finite_card @ B @ ( image @ A @ B @ F @ A5 ) )
= ( finite_card @ A @ A5 ) ) ) ).
% card_image
thf(fact_211_finite__surj__inj,axiom,
! [A: $tType,A5: set @ A,F: A > A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( image @ A @ A @ F @ A5 ) )
=> ( inj_on @ A @ A @ F @ A5 ) ) ) ).
% finite_surj_inj
thf(fact_212_inj__on__finite,axiom,
! [B: $tType,A: $tType,F: A > B,A5: set @ A,B5: set @ B] :
( ( inj_on @ A @ B @ F @ A5 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A5 ) @ B5 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( finite_finite2 @ A @ A5 ) ) ) ) ).
% inj_on_finite
thf(fact_213_endo__inj__surj,axiom,
! [A: $tType,A5: set @ A,F: A > A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ F @ A5 ) @ A5 )
=> ( ( inj_on @ A @ A @ F @ A5 )
=> ( ( image @ A @ A @ F @ A5 )
= A5 ) ) ) ) ).
% endo_inj_surj
thf(fact_214_card__image__le,axiom,
! [B: $tType,A: $tType,A5: set @ A,F: A > B] :
( ( finite_finite2 @ A @ A5 )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ ( image @ A @ B @ F @ A5 ) ) @ ( finite_card @ A @ A5 ) ) ) ).
% card_image_le
thf(fact_215_eq__card__imp__inj__on,axiom,
! [B: $tType,A: $tType,A5: set @ A,F: A > B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ( finite_card @ B @ ( image @ A @ B @ F @ A5 ) )
= ( finite_card @ A @ A5 ) )
=> ( inj_on @ A @ B @ F @ A5 ) ) ) ).
% eq_card_imp_inj_on
thf(fact_216_inj__on__iff__eq__card,axiom,
! [B: $tType,A: $tType,A5: set @ A,F: A > B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( inj_on @ A @ B @ F @ A5 )
= ( ( finite_card @ B @ ( image @ A @ B @ F @ A5 ) )
= ( finite_card @ A @ A5 ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_217_pigeonhole,axiom,
! [A: $tType,B: $tType,F: B > A,A5: set @ B] :
( ( ord_less @ nat @ ( finite_card @ A @ ( image @ B @ A @ F @ A5 ) ) @ ( finite_card @ B @ A5 ) )
=> ~ ( inj_on @ B @ A @ F @ A5 ) ) ).
% pigeonhole
thf(fact_218_surj__card__le,axiom,
! [B: $tType,A: $tType,A5: set @ A,B5: set @ B,F: A > B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ord_less_eq @ ( set @ B ) @ B5 @ ( image @ A @ B @ F @ A5 ) )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ B5 ) @ ( finite_card @ A @ A5 ) ) ) ) ).
% surj_card_le
thf(fact_219_surjective__iff__injective__gen,axiom,
! [B: $tType,A: $tType,S: set @ A,T2: set @ B,F: A > B] :
( ( finite_finite2 @ A @ S )
=> ( ( finite_finite2 @ B @ T2 )
=> ( ( ( finite_card @ A @ S )
= ( finite_card @ B @ T2 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ S ) @ T2 )
=> ( ( ! [X2: B] :
( ( member @ B @ X2 @ T2 )
=> ? [Y3: A] :
( ( member @ A @ Y3 @ S )
& ( ( F @ Y3 )
= X2 ) ) ) )
= ( inj_on @ A @ B @ F @ S ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_220_card__bij__eq,axiom,
! [A: $tType,B: $tType,F: A > B,A5: set @ A,B5: set @ B,G: B > A] :
( ( inj_on @ A @ B @ F @ A5 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F @ A5 ) @ B5 )
=> ( ( inj_on @ B @ A @ G @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ G @ B5 ) @ A5 )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ( finite_card @ A @ A5 )
= ( finite_card @ B @ B5 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_221_inj__on__iff__card__le,axiom,
! [A: $tType,B: $tType,A5: set @ A,B5: set @ B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ( ? [F2: A > B] :
( ( inj_on @ A @ B @ F2 @ A5 )
& ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ A5 ) @ B5 ) ) )
= ( ord_less_eq @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ B @ B5 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_222_inj__on__image__Fpow,axiom,
! [B: $tType,A: $tType,F: A > B,A5: set @ A] :
( ( inj_on @ A @ B @ F @ A5 )
=> ( inj_on @ ( set @ A ) @ ( set @ B ) @ ( image @ A @ B @ F ) @ ( finite_Fpow @ A @ A5 ) ) ) ).
% inj_on_image_Fpow
thf(fact_223_image__Fpow__mono,axiom,
! [B: $tType,A: $tType,F: B > A,A5: set @ B,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F @ A5 ) @ B5 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( image @ ( set @ B ) @ ( set @ A ) @ ( image @ B @ A @ F ) @ ( finite_Fpow @ B @ A5 ) ) @ ( finite_Fpow @ A @ B5 ) ) ) ).
% image_Fpow_mono
thf(fact_224_Fpow__mono,axiom,
! [A: $tType,A5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( finite_Fpow @ A @ A5 ) @ ( finite_Fpow @ A @ B5 ) ) ) ).
% Fpow_mono
thf(fact_225_card__ge__0__finite,axiom,
! [A: $tType,A5: set @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ A5 ) )
=> ( finite_finite2 @ A @ A5 ) ) ).
% card_ge_0_finite
thf(fact_226_greaterThanLessThan__subseteq__greaterThanLessThan,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or578182835ssThan @ A @ A2 @ B2 ) @ ( set_or578182835ssThan @ A @ C @ D3 ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C @ A2 )
& ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_greaterThanLessThan
thf(fact_227_finite__greaterThanLessThan,axiom,
! [L: nat,U4: nat] : ( finite_finite2 @ nat @ ( set_or578182835ssThan @ nat @ L @ U4 ) ) ).
% finite_greaterThanLessThan
thf(fact_228_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_229_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less @ nat @ N2 @ ( zero_zero @ nat ) ) ).
% less_nat_zero_code
thf(fact_230_neq0__conv,axiom,
! [N2: nat] :
( ( N2
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ).
% neq0_conv
thf(fact_231_le0,axiom,
! [N2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N2 ) ).
% le0
thf(fact_232_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A2 ) ).
% bot_nat_0.extremum
thf(fact_233_greaterThanLessThan__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,L: A,U4: A] :
( ( member @ A @ I @ ( set_or578182835ssThan @ A @ L @ U4 ) )
= ( ( ord_less @ A @ L @ I )
& ( ord_less @ A @ I @ U4 ) ) ) ) ).
% greaterThanLessThan_iff
thf(fact_234_card_Oinfinite,axiom,
! [A: $tType,A5: set @ A] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ( finite_card @ A @ A5 )
= ( zero_zero @ nat ) ) ) ).
% card.infinite
thf(fact_235_infinite__Ioo__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ~ ( finite_finite2 @ A @ ( set_or578182835ssThan @ A @ A2 @ B2 ) ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% infinite_Ioo_iff
thf(fact_236_less__eq__nat_Osimps_I1_J,axiom,
! [N2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N2 ) ).
% less_eq_nat.simps(1)
thf(fact_237_le__0__eq,axiom,
! [N2: nat] :
( ( ord_less_eq @ nat @ N2 @ ( zero_zero @ nat ) )
= ( N2
= ( zero_zero @ nat ) ) ) ).
% le_0_eq
thf(fact_238_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
= ( A2
= ( zero_zero @ nat ) ) ) ).
% bot_nat_0.extremum_unique
thf(fact_239_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
=> ( A2
= ( zero_zero @ nat ) ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_240_ex__least__nat__le,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ? [K: nat] :
( ( ord_less_eq @ nat @ K @ N2 )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ K )
=> ~ ( P @ I2 ) )
& ( P @ K ) ) ) ) ).
% ex_least_nat_le
thf(fact_241_infinite__descent0__measure,axiom,
! [A: $tType,V5: A > nat,P: A > $o,X3: A] :
( ! [X: A] :
( ( ( V5 @ X )
= ( zero_zero @ nat ) )
=> ( P @ X ) )
=> ( ! [X: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V5 @ X ) )
=> ( ~ ( P @ X )
=> ? [Y5: A] :
( ( ord_less @ nat @ ( V5 @ Y5 ) @ ( V5 @ X ) )
& ~ ( P @ Y5 ) ) ) )
=> ( P @ X3 ) ) ) ).
% infinite_descent0_measure
thf(fact_242_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less @ nat @ A2 @ ( zero_zero @ nat ) ) ).
% bot_nat_0.extremum_strict
thf(fact_243_infinite__descent0,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N5: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N5 )
=> ( ~ ( P @ N5 )
=> ? [M2: nat] :
( ( ord_less @ nat @ M2 @ N5 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N2 ) ) ) ).
% infinite_descent0
thf(fact_244_gr__implies__not0,axiom,
! [M3: nat,N2: nat] :
( ( ord_less @ nat @ M3 @ N2 )
=> ( N2
!= ( zero_zero @ nat ) ) ) ).
% gr_implies_not0
thf(fact_245_less__zeroE,axiom,
! [N2: nat] :
~ ( ord_less @ nat @ N2 @ ( zero_zero @ nat ) ) ).
% less_zeroE
thf(fact_246_not__less0,axiom,
! [N2: nat] :
~ ( ord_less @ nat @ N2 @ ( zero_zero @ nat ) ) ).
% not_less0
thf(fact_247_not__gr0,axiom,
! [N2: nat] :
( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) )
= ( N2
= ( zero_zero @ nat ) ) ) ).
% not_gr0
thf(fact_248_gr0I,axiom,
! [N2: nat] :
( ( N2
!= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ).
% gr0I
thf(fact_249_infinite__Ioo,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ( finite_finite2 @ A @ ( set_or578182835ssThan @ A @ A2 @ B2 ) ) ) ) ).
% infinite_Ioo
thf(fact_250_not__gr__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N2: A] :
( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N2 ) )
= ( N2
= ( zero_zero @ A ) ) ) ) ).
% not_gr_zero
thf(fact_251_le__zero__eq,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N2: A] :
( ( ord_less_eq @ A @ N2 @ ( zero_zero @ A ) )
= ( N2
= ( zero_zero @ A ) ) ) ) ).
% le_zero_eq
thf(fact_252_RepFun__0,axiom,
! [F: hF_Mirabelle_hf > hF_Mirabelle_hf] :
( ( hF_Mirabelle_RepFun @ ( zero_zero @ hF_Mirabelle_hf ) @ F )
= ( zero_zero @ hF_Mirabelle_hf ) ) ).
% RepFun_0
thf(fact_253_HUnion__hempty,axiom,
( ( hF_Mirabelle_HUnion @ ( zero_zero @ hF_Mirabelle_hf ) )
= ( zero_zero @ hF_Mirabelle_hf ) ) ).
% HUnion_hempty
thf(fact_254_Replace__0,axiom,
! [R: hF_Mirabelle_hf > hF_Mirabelle_hf > $o] :
( ( hF_Mirabelle_Replace @ ( zero_zero @ hF_Mirabelle_hf ) @ R )
= ( zero_zero @ hF_Mirabelle_hf ) ) ).
% Replace_0
% Type constructors (24)
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A9: $tType,A10: $tType] :
( ( preorder @ A10 )
=> ( preorder @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Finite__Set_Ofinite,axiom,
! [A9: $tType,A10: $tType] :
( ( ( finite_finite @ A9 )
& ( finite_finite @ A10 ) )
=> ( finite_finite @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A9: $tType,A10: $tType] :
( ( order @ A10 )
=> ( order @ ( A9 > A10 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A9: $tType,A10: $tType] :
( ( ord @ A10 )
=> ( ord @ ( A9 > A10 ) ) ) ).
thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
condit1037483654norder @ nat ).
thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
canoni770627133id_add @ nat ).
thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
wellorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
preorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
linorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Ono__top,axiom,
no_top @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
order @ nat ).
thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
ord @ nat ).
thf(tcon_Set_Oset___Orderings_Opreorder_4,axiom,
! [A9: $tType] : ( preorder @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_5,axiom,
! [A9: $tType] :
( ( finite_finite @ A9 )
=> ( finite_finite @ ( set @ A9 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_6,axiom,
! [A9: $tType] : ( order @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_7,axiom,
! [A9: $tType] : ( ord @ ( set @ A9 ) ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_8,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder_9,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_10,axiom,
finite_finite @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_11,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_12,axiom,
ord @ $o ).
thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Orderings_Opreorder_13,axiom,
preorder @ hF_Mirabelle_hf ).
thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Orderings_Oorder_14,axiom,
order @ hF_Mirabelle_hf ).
thf(tcon_HF__Mirabelle__fsbjehakzm_Ohf___Orderings_Oord_15,axiom,
ord @ hF_Mirabelle_hf ).
% Conjectures (2)
thf(conj_0,hypothesis,
! [X6: hF_Mirabelle_hf] :
( ( hF_Mirabelle_hmem @ X6 @ a )
=> ( hF_Mirabelle_hmem @ X6 @ b ) ) ).
thf(conj_1,conjecture,
ord_less_eq @ hF_Mirabelle_hf @ a @ b ).
%------------------------------------------------------------------------------