TPTP Problem File: ITP101^2.p
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%------------------------------------------------------------------------------
% File : ITP101^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer ListInf problem prob_63__5408460_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 : ListInf/prob_63__5408460_1 [Des21]
% Status : Theorem
% Rating : 0.00 v7.5.0
% Syntax : Number of formulae : 332 ( 61 unt; 45 typ; 0 def)
% Number of atoms : 955 ( 240 equ; 0 cnn)
% Maximal formula atoms : 13 ( 3 avg)
% Number of connectives : 3724 ( 86 ~; 25 |; 51 &;3024 @)
% ( 0 <=>; 538 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 9 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 136 ( 136 >; 0 *; 0 +; 0 <<)
% Number of symbols : 46 ( 43 usr; 5 con; 0-4 aty)
% Number of variables : 1092 ( 69 ^; 944 !; 43 ?;1092 :)
% ( 36 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:15:37.714
%------------------------------------------------------------------------------
% Could-be-implicit typings (4)
thf(ty_t_List_Olist,type,
list: $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_tf_a,type,
a: $tType ).
% Explicit typings (41)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Nat_Osize,type,
size:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oone,type,
one:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ominus,type,
minus:
!>[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_Groups_Ogroup__add,type,
group_add:
!>[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_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_Orderings_Odense__linorder,type,
dense_linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
cancel146912293up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
condit1037483654norder:
!>[A: $tType] : $o ).
thf(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Oone__class_Oone,type,
one_one:
!>[A: $tType] : A ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_List2_Olist__asc,type,
list_asc:
!>[A: $tType] : ( ( list @ A ) > $o ) ).
thf(sy_c_List2_Olist__desc,type,
list_desc:
!>[A: $tType] : ( ( list @ A ) > $o ) ).
thf(sy_c_List2_Olist__strict__asc,type,
list_strict_asc:
!>[A: $tType] : ( ( list @ A ) > $o ) ).
thf(sy_c_List2_Olist__strict__desc,type,
list_strict_desc:
!>[A: $tType] : ( ( list @ A ) > $o ) ).
thf(sy_c_ListInf__Mirabelle__nfmaokebij_Oi__append,type,
listIn521021761append:
!>[A: $tType] : ( ( list @ A ) > ( nat > A ) > nat > A ) ).
thf(sy_c_List_Oappend,type,
append:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Obutlast,type,
butlast:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist_Oset,type,
set2:
!>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
thf(sy_c_List_Olist__ex,type,
list_ex:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Olist__update,type,
list_update:
!>[A: $tType] : ( ( list @ A ) > nat > A > ( list @ A ) ) ).
thf(sy_c_List_Onth,type,
nth:
!>[A: $tType] : ( ( list @ A ) > nat > A ) ).
thf(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : ( A > nat ) ).
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_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_f,type,
f: nat > a ).
thf(sy_v_g,type,
g: nat > a ).
thf(sy_v_x,type,
x: nat ).
thf(sy_v_xs,type,
xs: list @ a ).
thf(sy_v_ys,type,
ys: list @ a ).
% Relevant facts (255)
thf(fact_0_nth__equalityI,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) )
=> ( ! [I: nat] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ Xs @ I )
= ( nth @ A @ Ys @ I ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_1_Skolem__list__nth,axiom,
! [A: $tType,K: nat,P: nat > A > $o] :
( ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ K )
=> ? [X: A] : ( P @ I2 @ X ) ) )
= ( ? [Xs2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= K )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ K )
=> ( P @ I2 @ ( nth @ A @ Xs2 @ I2 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_2_list__eq__iff__nth__eq,axiom,
! [A: $tType] :
( ( ^ [Y: list @ A,Z: list @ A] : Y = Z )
= ( ^ [Xs2: list @ A,Ys2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ A ) @ Ys2 ) )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( nth @ A @ Xs2 @ I2 )
= ( nth @ A @ Ys2 @ I2 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_3_minus__apply,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A2: A > B,B2: A > B,X2: A] : ( minus_minus @ B @ ( A2 @ X2 ) @ ( B2 @ X2 ) ) ) ) ) ).
% minus_apply
thf(fact_4_i__append__nth1,axiom,
! [A: $tType,N: nat,Xs: list @ A,F: nat > A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( listIn521021761append @ A @ Xs @ F @ N )
= ( nth @ A @ Xs @ N ) ) ) ).
% i_append_nth1
thf(fact_5_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ( ord_less @ nat @ M @ L )
=> ( ord_less @ nat @ ( minus_minus @ nat @ L @ N ) @ ( minus_minus @ nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_6_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less @ nat @ J @ K )
=> ( ord_less @ nat @ ( minus_minus @ nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_7_less__diff__imp__less,axiom,
! [I3: nat,J: nat,M: nat] :
( ( ord_less @ nat @ I3 @ ( minus_minus @ nat @ J @ M ) )
=> ( ord_less @ nat @ I3 @ J ) ) ).
% less_diff_imp_less
thf(fact_8_nat__diff__left__cancel__eq1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( minus_minus @ nat @ K @ M )
= ( minus_minus @ nat @ K @ N ) )
=> ( ( ord_less @ nat @ M @ K )
=> ( M = N ) ) ) ).
% nat_diff_left_cancel_eq1
thf(fact_9_nat__diff__left__cancel__eq2,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( minus_minus @ nat @ K @ M )
= ( minus_minus @ nat @ K @ N ) )
=> ( ( ord_less @ nat @ N @ K )
=> ( M = N ) ) ) ).
% nat_diff_left_cancel_eq2
thf(fact_10_nat__diff__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( minus_minus @ nat @ K @ M ) @ ( minus_minus @ nat @ K @ N ) )
=> ( ord_less @ nat @ N @ M ) ) ).
% nat_diff_left_cancel_less
thf(fact_11_nat__diff__right__cancel__eq1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( minus_minus @ nat @ M @ K )
= ( minus_minus @ nat @ N @ K ) )
=> ( ( ord_less @ nat @ K @ M )
=> ( M = N ) ) ) ).
% nat_diff_right_cancel_eq1
thf(fact_12_measure__induct__rule,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ B )
=> ! [F: A > B,P: A > $o,A3: A] :
( ! [X3: A] :
( ! [Y2: A] :
( ( ord_less @ B @ ( F @ Y2 ) @ ( F @ X3 ) )
=> ( P @ Y2 ) )
=> ( P @ X3 ) )
=> ( P @ A3 ) ) ) ).
% measure_induct_rule
thf(fact_13_measure__induct,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ B )
=> ! [F: A > B,P: A > $o,A3: A] :
( ! [X3: A] :
( ! [Y2: A] :
( ( ord_less @ B @ ( F @ Y2 ) @ ( F @ X3 ) )
=> ( P @ Y2 ) )
=> ( P @ X3 ) )
=> ( P @ A3 ) ) ) ).
% measure_induct
thf(fact_14_fun__diff__def,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A2: A > B,B2: A > B,X2: A] : ( minus_minus @ B @ ( A2 @ X2 ) @ ( B2 @ X2 ) ) ) ) ) ).
% fun_diff_def
thf(fact_15_infinite__descent__measure,axiom,
! [A: $tType,P: A > $o,V: A > nat,X4: A] :
( ! [X3: A] :
( ~ ( P @ X3 )
=> ? [Y2: A] :
( ( ord_less @ nat @ ( V @ Y2 ) @ ( V @ X3 ) )
& ~ ( P @ Y2 ) ) )
=> ( P @ X4 ) ) ).
% infinite_descent_measure
thf(fact_16_linorder__neqE__nat,axiom,
! [X4: nat,Y3: nat] :
( ( X4 != Y3 )
=> ( ~ ( ord_less @ nat @ X4 @ Y3 )
=> ( ord_less @ nat @ Y3 @ X4 ) ) ) ).
% linorder_neqE_nat
thf(fact_17_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_18_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_19_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_20_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less @ nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_21_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_22_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_not_refl
thf(fact_23_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less @ nat @ M @ N )
| ( ord_less @ nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_24_size__neq__size__imp__neq,axiom,
! [A: $tType] :
( ( size @ A )
=> ! [X4: A,Y3: A] :
( ( ( size_size @ A @ X4 )
!= ( size_size @ A @ Y3 ) )
=> ( X4 != Y3 ) ) ) ).
% size_neq_size_imp_neq
thf(fact_25_neq__if__length__neq,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
!= ( size_size @ ( list @ A ) @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_26_Ex__list__of__length,axiom,
! [A: $tType,N: nat] :
? [Xs3: list @ A] :
( ( size_size @ ( list @ A ) @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_27_eq__imp__diff__eq,axiom,
! [M: nat,N: nat,K: nat] :
( ( M = N )
=> ( ( minus_minus @ nat @ M @ K )
= ( minus_minus @ nat @ N @ K ) ) ) ).
% eq_imp_diff_eq
thf(fact_28_diff__commute,axiom,
! [I3: nat,J: nat,K: nat] :
( ( minus_minus @ nat @ ( minus_minus @ nat @ I3 @ J ) @ K )
= ( minus_minus @ nat @ ( minus_minus @ nat @ I3 @ K ) @ J ) ) ).
% diff_commute
thf(fact_29_i__append__nth,axiom,
! [A: $tType] :
( ( listIn521021761append @ A )
= ( ^ [Xs2: list @ A,F2: nat > A,N3: nat] : ( if @ A @ ( ord_less @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs2 ) ) @ ( nth @ A @ Xs2 @ N3 ) @ ( F2 @ ( minus_minus @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) ) ) ).
% i_append_nth
thf(fact_30_i__append__def,axiom,
! [A: $tType] :
( ( listIn521021761append @ A )
= ( ^ [Xs2: list @ A,F2: nat > A,N3: nat] : ( if @ A @ ( ord_less @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs2 ) ) @ ( nth @ A @ Xs2 @ N3 ) @ ( F2 @ ( minus_minus @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) ) ) ).
% i_append_def
thf(fact_31_length__induct,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
( ! [Xs3: list @ A] :
( ! [Ys3: list @ A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ys3 ) @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ( P @ Ys3 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_32_nat__diff__right__cancel__less,axiom,
! [N: nat,K: nat,M: nat] :
( ( ord_less @ nat @ ( minus_minus @ nat @ N @ K ) @ ( minus_minus @ nat @ M @ K ) )
=> ( ord_less @ nat @ N @ M ) ) ).
% nat_diff_right_cancel_less
thf(fact_33_nat__diff__right__cancel__eq2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( minus_minus @ nat @ M @ K )
= ( minus_minus @ nat @ N @ K ) )
=> ( ( ord_less @ nat @ K @ N )
=> ( M = N ) ) ) ).
% nat_diff_right_cancel_eq2
thf(fact_34_diff__strict__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B3: A,C: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ord_less @ A @ ( minus_minus @ A @ A3 @ C ) @ ( minus_minus @ A @ B3 @ C ) ) ) ) ).
% diff_strict_right_mono
thf(fact_35_diff__strict__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B3: A,A3: A,C: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ( ord_less @ A @ ( minus_minus @ A @ C @ A3 ) @ ( minus_minus @ A @ C @ B3 ) ) ) ) ).
% diff_strict_left_mono
thf(fact_36_diff__eq__diff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B3: A,C: A,D: A] :
( ( ( minus_minus @ A @ A3 @ B3 )
= ( minus_minus @ A @ C @ D ) )
=> ( ( ord_less @ A @ A3 @ B3 )
= ( ord_less @ A @ C @ D ) ) ) ) ).
% diff_eq_diff_less
thf(fact_37_diff__strict__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B3: A,D: A,C: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ D @ C )
=> ( ord_less @ A @ ( minus_minus @ A @ A3 @ C ) @ ( minus_minus @ A @ B3 @ D ) ) ) ) ) ).
% diff_strict_mono
thf(fact_38_list__strict__desc__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( list_strict_desc @ A )
= ( ^ [Xs2: list @ A] :
! [J2: nat] :
( ( ord_less @ nat @ J2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ! [I2: nat] :
( ( ord_less @ nat @ I2 @ J2 )
=> ( ord_less @ A @ ( nth @ A @ Xs2 @ J2 ) @ ( nth @ A @ Xs2 @ I2 ) ) ) ) ) ) ) ).
% list_strict_desc_trans
thf(fact_39_list__strict__asc__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( list_strict_asc @ A )
= ( ^ [Xs2: list @ A] :
! [J2: nat] :
( ( ord_less @ nat @ J2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ! [I2: nat] :
( ( ord_less @ nat @ I2 @ J2 )
=> ( ord_less @ A @ ( nth @ A @ Xs2 @ I2 ) @ ( nth @ A @ Xs2 @ J2 ) ) ) ) ) ) ) ).
% list_strict_asc_trans
thf(fact_40_list__ex__length,axiom,
! [A: $tType] :
( ( list_ex @ A )
= ( ^ [P2: A > $o,Xs2: list @ A] :
? [N3: nat] :
( ( ord_less @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
& ( P2 @ ( nth @ A @ Xs2 @ N3 ) ) ) ) ) ).
% list_ex_length
thf(fact_41_i__append__nth2,axiom,
! [A: $tType,Xs: list @ A,N: nat,F: nat > A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N )
=> ( ( listIn521021761append @ A @ Xs @ F @ N )
= ( F @ ( minus_minus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ).
% i_append_nth2
thf(fact_42_nth__append,axiom,
! [A: $tType,N: nat,Xs: list @ A,Ys: list @ A] :
( ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( append @ A @ Xs @ Ys ) @ N )
= ( nth @ A @ Xs @ N ) ) )
& ( ~ ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( append @ A @ Xs @ Ys ) @ N )
= ( nth @ A @ Ys @ ( minus_minus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_43_nth__list__update__eq,axiom,
! [A: $tType,I3: nat,Xs: list @ A,X4: A] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( list_update @ A @ Xs @ I3 @ X4 ) @ I3 )
= X4 ) ) ).
% nth_list_update_eq
thf(fact_44_nth__butlast,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ ( butlast @ A @ Xs ) ) )
=> ( ( nth @ A @ ( butlast @ A @ Xs ) @ N )
= ( nth @ A @ Xs @ N ) ) ) ).
% nth_butlast
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,A4: set @ A] :
( ( collect @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X3: A] :
( ( F @ X3 )
= ( G @ X3 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_append_Oassoc,axiom,
! [A: $tType,A3: list @ A,B3: list @ A,C: list @ A] :
( ( append @ A @ ( append @ A @ A3 @ B3 ) @ C )
= ( append @ A @ A3 @ ( append @ A @ B3 @ C ) ) ) ).
% append.assoc
thf(fact_50_append__assoc,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( append @ A @ ( append @ A @ Xs @ Ys ) @ Zs )
= ( append @ A @ Xs @ ( append @ A @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_51_append__same__eq,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A,Zs: list @ A] :
( ( ( append @ A @ Ys @ Xs )
= ( append @ A @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_52_same__append__eq,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_53_list__update__overwrite,axiom,
! [A: $tType,Xs: list @ A,I3: nat,X4: A,Y3: A] :
( ( list_update @ A @ ( list_update @ A @ Xs @ I3 @ X4 ) @ I3 @ Y3 )
= ( list_update @ A @ Xs @ I3 @ Y3 ) ) ).
% list_update_overwrite
thf(fact_54_diff__diff__cancel,axiom,
! [I3: nat,N: nat] :
( ( ord_less_eq @ nat @ I3 @ N )
=> ( ( minus_minus @ nat @ N @ ( minus_minus @ nat @ N @ I3 ) )
= I3 ) ) ).
% diff_diff_cancel
thf(fact_55_append__eq__append__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Us: list @ A,Vs: list @ A] :
( ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) )
| ( ( size_size @ ( list @ A ) @ Us )
= ( size_size @ ( list @ A ) @ Vs ) ) )
=> ( ( ( append @ A @ Xs @ Us )
= ( append @ A @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_56_length__list__update,axiom,
! [A: $tType,Xs: list @ A,I3: nat,X4: A] :
( ( size_size @ ( list @ A ) @ ( list_update @ A @ Xs @ I3 @ X4 ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_list_update
thf(fact_57_list__update__id,axiom,
! [A: $tType,Xs: list @ A,I3: nat] :
( ( list_update @ A @ Xs @ I3 @ ( nth @ A @ Xs @ I3 ) )
= Xs ) ).
% list_update_id
thf(fact_58_nth__list__update__neq,axiom,
! [A: $tType,I3: nat,J: nat,Xs: list @ A,X4: A] :
( ( I3 != J )
=> ( ( nth @ A @ ( list_update @ A @ Xs @ I3 @ X4 ) @ J )
= ( nth @ A @ Xs @ J ) ) ) ).
% nth_list_update_neq
thf(fact_59_list__ex__append,axiom,
! [A: $tType,P: A > $o,Xs: list @ A,Ys: list @ A] :
( ( list_ex @ A @ P @ ( append @ A @ Xs @ Ys ) )
= ( ( list_ex @ A @ P @ Xs )
| ( list_ex @ A @ P @ Ys ) ) ) ).
% list_ex_append
thf(fact_60_i__append__assoc,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,F: nat > A] :
( ( listIn521021761append @ A @ Xs @ ( listIn521021761append @ A @ Ys @ F ) )
= ( listIn521021761append @ A @ ( append @ A @ Xs @ Ys ) @ F ) ) ).
% i_append_assoc
thf(fact_61_list__update__beyond,axiom,
! [A: $tType,Xs: list @ A,I3: nat,X4: A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ I3 )
=> ( ( list_update @ A @ Xs @ I3 @ X4 )
= Xs ) ) ).
% list_update_beyond
thf(fact_62_le__refl,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% le_refl
thf(fact_63_le__trans,axiom,
! [I3: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I3 @ J )
=> ( ( ord_less_eq @ nat @ J @ K )
=> ( ord_less_eq @ nat @ I3 @ K ) ) ) ).
% le_trans
thf(fact_64_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_65_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_66_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
| ( ord_less_eq @ nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_67_list__update__swap,axiom,
! [A: $tType,I3: nat,I4: nat,Xs: list @ A,X4: A,X5: A] :
( ( I3 != I4 )
=> ( ( list_update @ A @ ( list_update @ A @ Xs @ I3 @ X4 ) @ I4 @ X5 )
= ( list_update @ A @ ( list_update @ A @ Xs @ I4 @ X5 ) @ I3 @ X4 ) ) ) ).
% list_update_swap
thf(fact_68_append__eq__appendI,axiom,
! [A: $tType,Xs: list @ A,Xs1: list @ A,Zs: list @ A,Ys: list @ A,Us: list @ A] :
( ( ( append @ A @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append @ A @ Xs1 @ Us ) )
=> ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_69_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B3: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq @ nat @ Y4 @ B3 ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq @ nat @ Y2 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_70_append__eq__append__conv2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A,Ts: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Zs @ Ts ) )
= ( ? [Us2: list @ A] :
( ( ( Xs
= ( append @ A @ Zs @ Us2 ) )
& ( ( append @ A @ Us2 @ Ys )
= Ts ) )
| ( ( ( append @ A @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append @ A @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_71_diff__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B3: A,D: A,C: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ D @ C )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C ) @ ( minus_minus @ A @ B3 @ D ) ) ) ) ) ).
% diff_mono
thf(fact_72_diff__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B3: A,A3: A,C: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ C @ A3 ) @ ( minus_minus @ A @ C @ B3 ) ) ) ) ).
% diff_left_mono
thf(fact_73_diff__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ 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
thf(fact_74_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B3: A,C: A,D: A] :
( ( ( minus_minus @ A @ A3 @ B3 )
= ( minus_minus @ A @ C @ D ) )
=> ( ( ord_less_eq @ A @ A3 @ B3 )
= ( ord_less_eq @ A @ C @ D ) ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_75_list__strict__asc__trans__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [Xs: list @ A] :
( ( list_strict_asc @ A @ Xs )
=> ! [J3: nat] :
( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ! [I5: nat] :
( ( ord_less_eq @ nat @ I5 @ J3 )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ I5 ) @ ( nth @ A @ Xs @ J3 ) ) ) ) ) ) ).
% list_strict_asc_trans_le
thf(fact_76_list__update__append1,axiom,
! [A: $tType,I3: nat,Xs: list @ A,Ys: list @ A,X4: A] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( list_update @ A @ ( append @ A @ Xs @ Ys ) @ I3 @ X4 )
= ( append @ A @ ( list_update @ A @ Xs @ I3 @ X4 ) @ Ys ) ) ) ).
% list_update_append1
thf(fact_77_nth__append2,axiom,
! [A: $tType,Xs: list @ A,N: nat,Ys: list @ A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N )
=> ( ( nth @ A @ ( append @ A @ Xs @ Ys ) @ N )
= ( nth @ A @ Ys @ ( minus_minus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ).
% nth_append2
thf(fact_78_ge__less__neq__conv,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,N3: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ A5 )
=> ( N3 != X2 ) ) ) ) ) ).
% ge_less_neq_conv
thf(fact_79_less__ge__neq__conv,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less @ A )
= ( ^ [N3: A,A5: A] :
! [X2: A] :
( ( ord_less_eq @ A @ A5 @ X2 )
=> ( N3 != X2 ) ) ) ) ) ).
% less_ge_neq_conv
thf(fact_80_greater__le__neq__conv,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less @ A )
= ( ^ [A5: A,N3: A] :
! [X2: A] :
( ( ord_less_eq @ A @ X2 @ A5 )
=> ( N3 != X2 ) ) ) ) ) ).
% greater_le_neq_conv
thf(fact_81_le__greater__neq__conv,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [N3: A,A5: A] :
! [X2: A] :
( ( ord_less @ A @ A5 @ X2 )
=> ( N3 != X2 ) ) ) ) ) ).
% le_greater_neq_conv
thf(fact_82_nat__less__le,axiom,
( ( ord_less @ nat )
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_eq @ nat @ M3 @ N3 )
& ( M3 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_83_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_84_le__eq__less__or__eq,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M3: nat,N3: nat] :
( ( ord_less @ nat @ M3 @ N3 )
| ( M3 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_85_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less @ nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_86_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( M != N )
=> ( ord_less @ nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_87_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I3: nat,J: nat] :
( ! [I: nat,J4: nat] :
( ( ord_less @ nat @ I @ J4 )
=> ( ord_less @ nat @ ( F @ I ) @ ( F @ J4 ) ) )
=> ( ( ord_less_eq @ nat @ I3 @ J )
=> ( ord_less_eq @ nat @ ( F @ I3 ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_88_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( ( ( minus_minus @ nat @ M @ K )
= ( minus_minus @ nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_89_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
= ( ord_less_eq @ nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_90_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( ( minus_minus @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
= ( minus_minus @ nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_91_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ L ) @ ( minus_minus @ nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_92_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_93_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_94_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ L @ N ) @ ( minus_minus @ nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_95_le__diff__swap,axiom,
! [I3: nat,K: nat,J: nat] :
( ( ord_less_eq @ nat @ I3 @ K )
=> ( ( ord_less_eq @ nat @ J @ K )
=> ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ K @ J ) @ I3 )
= ( ord_less_eq @ nat @ ( minus_minus @ nat @ K @ I3 ) @ J ) ) ) ) ).
% le_diff_swap
thf(fact_96_le__diff__imp__le,axiom,
! [I3: nat,J: nat,M: nat] :
( ( ord_less_eq @ nat @ I3 @ ( minus_minus @ nat @ J @ M ) )
=> ( ord_less_eq @ nat @ I3 @ J ) ) ).
% le_diff_imp_le
thf(fact_97_le__imp__diff__le,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_eq @ nat @ J @ K )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ J @ N ) @ K ) ) ).
% le_imp_diff_le
thf(fact_98_eq__diff__left__iff,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ K )
=> ( ( ord_less_eq @ nat @ N @ K )
=> ( ( ( minus_minus @ nat @ K @ M )
= ( minus_minus @ nat @ K @ N ) )
= ( M = N ) ) ) ) ).
% eq_diff_left_iff
thf(fact_99_le__diff__le__imp__le,axiom,
! [I3: nat,J: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ I3 @ ( minus_minus @ nat @ J @ M ) )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( ord_less_eq @ nat @ I3 @ ( minus_minus @ nat @ J @ N ) ) ) ) ).
% le_diff_le_imp_le
thf(fact_100_nat__diff__left__cancel__le2,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( minus_minus @ nat @ K @ M ) @ ( minus_minus @ nat @ K @ N ) )
=> ( ( ord_less_eq @ nat @ N @ K )
=> ( ord_less_eq @ nat @ N @ M ) ) ) ).
% nat_diff_left_cancel_le2
thf(fact_101_nat__diff__right__cancel__le2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( ord_less_eq @ nat @ M @ N ) ) ) ).
% nat_diff_right_cancel_le2
thf(fact_102_list__update__append,axiom,
! [A: $tType,N: nat,Xs: list @ A,Ys: list @ A,X4: A] :
( ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( list_update @ A @ ( append @ A @ Xs @ Ys ) @ N @ X4 )
= ( append @ A @ ( list_update @ A @ Xs @ N @ X4 ) @ Ys ) ) )
& ( ~ ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( list_update @ A @ ( append @ A @ Xs @ Ys ) @ N @ X4 )
= ( append @ A @ Xs @ ( list_update @ A @ Ys @ ( minus_minus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) @ X4 ) ) ) ) ) ).
% list_update_append
thf(fact_103_nth__append1,axiom,
! [A: $tType,N: nat,Xs: list @ A,Ys: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( append @ A @ Xs @ Ys ) @ N )
= ( nth @ A @ Xs @ N ) ) ) ).
% nth_append1
thf(fact_104_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( ( ord_less @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
= ( ord_less @ nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_105_diff__less__mono,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less @ nat @ A3 @ B3 )
=> ( ( ord_less_eq @ nat @ C @ A3 )
=> ( ord_less @ nat @ ( minus_minus @ nat @ A3 @ C ) @ ( minus_minus @ nat @ B3 @ C ) ) ) ) ).
% diff_less_mono
thf(fact_106_less__diff__le__imp__less,axiom,
! [I3: nat,J: nat,M: nat,N: nat] :
( ( ord_less @ nat @ I3 @ ( minus_minus @ nat @ J @ M ) )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( ord_less @ nat @ I3 @ ( minus_minus @ nat @ J @ N ) ) ) ) ).
% less_diff_le_imp_less
thf(fact_107_nat__diff__left__cancel__le1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( minus_minus @ nat @ K @ M ) @ ( minus_minus @ nat @ K @ N ) )
=> ( ( ord_less @ nat @ M @ K )
=> ( ord_less_eq @ nat @ N @ M ) ) ) ).
% nat_diff_left_cancel_le1
thf(fact_108_nat__diff__right__cancel__le1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
=> ( ( ord_less @ nat @ K @ M )
=> ( ord_less_eq @ nat @ M @ N ) ) ) ).
% nat_diff_right_cancel_le1
thf(fact_109_nth__list__update,axiom,
! [A: $tType,I3: nat,Xs: list @ A,J: nat,X4: A] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( I3 = J )
=> ( ( nth @ A @ ( list_update @ A @ Xs @ I3 @ X4 ) @ J )
= X4 ) )
& ( ( I3 != J )
=> ( ( nth @ A @ ( list_update @ A @ Xs @ I3 @ X4 ) @ J )
= ( nth @ A @ Xs @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_110_list__update__same__conv,axiom,
! [A: $tType,I3: nat,Xs: list @ A,X4: A] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( list_update @ A @ Xs @ I3 @ X4 )
= Xs )
= ( ( nth @ A @ Xs @ I3 )
= X4 ) ) ) ).
% list_update_same_conv
thf(fact_111_diff__eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B3: A,C: A,D: A] :
( ( ( minus_minus @ A @ A3 @ B3 )
= ( minus_minus @ A @ C @ D ) )
=> ( ( A3 = B3 )
= ( C = D ) ) ) ) ).
% diff_eq_diff_eq
thf(fact_112_diff__right__commute,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A )
=> ! [A3: A,C: A,B3: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A3 @ C ) @ B3 )
= ( minus_minus @ A @ ( minus_minus @ A @ A3 @ B3 ) @ C ) ) ) ).
% diff_right_commute
thf(fact_113_list__desc__trans__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( list_desc @ A )
= ( ^ [Xs2: list @ A] :
! [J2: nat] :
( ( ord_less @ nat @ J2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ! [I2: nat] :
( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs2 @ J2 ) @ ( nth @ A @ Xs2 @ I2 ) ) ) ) ) ) ) ).
% list_desc_trans_le
thf(fact_114_list__asc__trans__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( list_asc @ A )
= ( ^ [Xs2: list @ A] :
! [J2: nat] :
( ( ord_less @ nat @ J2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ! [I2: nat] :
( ( ord_less_eq @ nat @ I2 @ J2 )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs2 @ I2 ) @ ( nth @ A @ Xs2 @ J2 ) ) ) ) ) ) ) ).
% list_asc_trans_le
thf(fact_115_list__desc__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( list_desc @ A )
= ( ^ [Xs2: list @ A] :
! [J2: nat] :
( ( ord_less @ nat @ J2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ! [I2: nat] :
( ( ord_less @ nat @ I2 @ J2 )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs2 @ J2 ) @ ( nth @ A @ Xs2 @ I2 ) ) ) ) ) ) ) ).
% list_desc_trans
thf(fact_116_list__asc__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( list_asc @ A )
= ( ^ [Xs2: list @ A] :
! [J2: nat] :
( ( ord_less @ nat @ J2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ! [I2: nat] :
( ( ord_less @ nat @ I2 @ J2 )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs2 @ I2 ) @ ( nth @ A @ Xs2 @ J2 ) ) ) ) ) ) ) ).
% list_asc_trans
thf(fact_117_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X4: A] : ( ord_less_eq @ A @ X4 @ X4 ) ) ).
% order_refl
thf(fact_118_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less @ nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq @ nat @ K2 @ N )
=> ( ! [I5: nat] :
( ( ord_less @ nat @ K2 @ I5 )
=> ( P @ I5 ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_119_Lattices__Big_Oex__has__greatest__nat,axiom,
! [A: $tType,P: A > $o,K: A,F: A > nat,B3: nat] :
( ( P @ K )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less @ nat @ ( F @ Y4 ) @ B3 ) )
=> ? [X3: A] :
( ( P @ X3 )
& ! [Y2: A] :
( ( P @ Y2 )
=> ( ord_less_eq @ nat @ ( F @ Y2 ) @ ( F @ X3 ) ) ) ) ) ) ).
% Lattices_Big.ex_has_greatest_nat
thf(fact_120_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_121_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B,X4: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X4 ) @ ( G @ X4 ) ) ) ) ).
% le_funD
thf(fact_122_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B,X4: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X4 ) @ ( G @ X4 ) ) ) ) ).
% le_funE
thf(fact_123_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B] :
( ! [X3: A] : ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) )
=> ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% le_funI
thf(fact_124_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_125_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F: B > A,B3: B,C: B] :
( ( ord_less_eq @ A @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C )
=> ( ! [X3: B,Y4: B] :
( ( ord_less_eq @ B @ X3 @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% order_subst1
thf(fact_126_order__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 )
& ( order @ A ) )
=> ! [A3: A,B3: A,F: A > C2,C: C2] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ C2 @ ( F @ B3 ) @ C )
=> ( ! [X3: A,Y4: A] :
( ( ord_less_eq @ A @ X3 @ Y4 )
=> ( ord_less_eq @ C2 @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ C2 @ ( F @ A3 ) @ C ) ) ) ) ) ).
% order_subst2
thf(fact_127_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,F: B > A,B3: B,C: B] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C )
=> ( ! [X3: B,Y4: B] :
( ( ord_less_eq @ B @ X3 @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_128_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,B3: A,F: A > B,C: B] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X3: A,Y4: A] :
( ( ord_less_eq @ A @ X3 @ Y4 )
=> ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ B @ ( F @ A3 ) @ C ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_129_eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y: A,Z: A] : Y = Z )
= ( ^ [X2: A,Y5: A] :
( ( ord_less_eq @ A @ X2 @ Y5 )
& ( ord_less_eq @ A @ Y5 @ X2 ) ) ) ) ) ).
% eq_iff
thf(fact_130_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X4: A,Y3: A] :
( ( ord_less_eq @ A @ X4 @ Y3 )
=> ( ( ord_less_eq @ A @ Y3 @ X4 )
=> ( X4 = Y3 ) ) ) ) ).
% antisym
thf(fact_131_linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X4: A,Y3: A] :
( ( ord_less_eq @ A @ X4 @ Y3 )
| ( ord_less_eq @ A @ Y3 @ X4 ) ) ) ).
% linear
thf(fact_132_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X4: A,Y3: A] :
( ( X4 = Y3 )
=> ( ord_less_eq @ A @ X4 @ Y3 ) ) ) ).
% eq_refl
thf(fact_133_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X4: A,Y3: A] :
( ~ ( ord_less_eq @ A @ X4 @ Y3 )
=> ( ord_less_eq @ A @ Y3 @ X4 ) ) ) ).
% le_cases
thf(fact_134_order_Otrans,axiom,
! [A: $tType] :
( ( order @ 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_135_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X4: A,Y3: A,Z3: A] :
( ( ( ord_less_eq @ A @ X4 @ Y3 )
=> ~ ( ord_less_eq @ A @ Y3 @ Z3 ) )
=> ( ( ( ord_less_eq @ A @ Y3 @ X4 )
=> ~ ( ord_less_eq @ A @ X4 @ Z3 ) )
=> ( ( ( ord_less_eq @ A @ X4 @ Z3 )
=> ~ ( ord_less_eq @ A @ Z3 @ Y3 ) )
=> ( ( ( ord_less_eq @ A @ Z3 @ Y3 )
=> ~ ( ord_less_eq @ A @ Y3 @ X4 ) )
=> ( ( ( ord_less_eq @ A @ Y3 @ Z3 )
=> ~ ( ord_less_eq @ A @ Z3 @ X4 ) )
=> ~ ( ( ord_less_eq @ A @ Z3 @ X4 )
=> ~ ( ord_less_eq @ A @ X4 @ Y3 ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_136_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y3: A,X4: A] :
( ( ord_less_eq @ A @ Y3 @ X4 )
=> ( ( ord_less_eq @ A @ X4 @ Y3 )
= ( X4 = Y3 ) ) ) ) ).
% antisym_conv
thf(fact_137_order__class_Oorder_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y: A,Z: A] : Y = Z )
= ( ^ [A5: A,B4: A] :
( ( ord_less_eq @ A @ A5 @ B4 )
& ( ord_less_eq @ A @ B4 @ A5 ) ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_138_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ 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_139_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ 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_140_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ 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_141_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X4: A,Y3: A,Z3: A] :
( ( ord_less_eq @ A @ X4 @ Y3 )
=> ( ( ord_less_eq @ A @ Y3 @ Z3 )
=> ( ord_less_eq @ A @ X4 @ Z3 ) ) ) ) ).
% order_trans
thf(fact_142_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% dual_order.refl
thf(fact_143_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A3: A,B3: A] :
( ! [A6: A,B5: A] :
( ( ord_less_eq @ A @ A6 @ B5 )
=> ( P @ A6 @ B5 ) )
=> ( ! [A6: A,B5: A] :
( ( P @ B5 @ A6 )
=> ( P @ A6 @ B5 ) )
=> ( P @ A3 @ B3 ) ) ) ) ).
% linorder_wlog
thf(fact_144_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ 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_145_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y: A,Z: A] : Y = Z )
= ( ^ [A5: A,B4: A] :
( ( ord_less_eq @ A @ B4 @ A5 )
& ( ord_less_eq @ A @ A5 @ B4 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_146_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ) ).
% dual_order.antisym
thf(fact_147_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ( A3 != B3 ) ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_148_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( A3 != B3 ) ) ) ).
% order.strict_implies_not_eq
thf(fact_149_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X4: A,Y3: A] :
( ( ~ ( ord_less @ A @ X4 @ Y3 ) )
= ( ( ord_less @ A @ Y3 @ X4 )
| ( X4 = Y3 ) ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_150_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A3: A,C: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ( ( ord_less @ A @ C @ B3 )
=> ( ord_less @ A @ C @ A3 ) ) ) ) ).
% dual_order.strict_trans
thf(fact_151_linorder__less__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A3: A,B3: A] :
( ! [A6: A,B5: A] :
( ( ord_less @ A @ A6 @ B5 )
=> ( P @ A6 @ B5 ) )
=> ( ! [A6: A] : ( P @ A6 @ A6 )
=> ( ! [A6: A,B5: A] :
( ( P @ B5 @ A6 )
=> ( P @ A6 @ B5 ) )
=> ( P @ A3 @ B3 ) ) ) ) ) ).
% linorder_less_wlog
thf(fact_152_exists__least__iff,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ( ( ^ [P3: A > $o] :
? [X: A] : ( P3 @ X ) )
= ( ^ [P2: A > $o] :
? [N3: A] :
( ( P2 @ N3 )
& ! [M3: A] :
( ( ord_less @ A @ M3 @ N3 )
=> ~ ( P2 @ M3 ) ) ) ) ) ) ).
% exists_least_iff
thf(fact_153_less__imp__not__less,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X4: A,Y3: A] :
( ( ord_less @ A @ X4 @ Y3 )
=> ~ ( ord_less @ A @ Y3 @ X4 ) ) ) ).
% less_imp_not_less
thf(fact_154_order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A,C: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ B3 @ C )
=> ( ord_less @ A @ A3 @ C ) ) ) ) ).
% order.strict_trans
thf(fact_155_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ A3 @ A3 ) ) ).
% dual_order.irrefl
thf(fact_156_linorder__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X4: A,Y3: A] :
( ~ ( ord_less @ A @ X4 @ Y3 )
=> ( ( X4 != Y3 )
=> ( ord_less @ A @ Y3 @ X4 ) ) ) ) ).
% linorder_cases
thf(fact_157_less__imp__triv,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X4: A,Y3: A,P: $o] :
( ( ord_less @ A @ X4 @ Y3 )
=> ( ( ord_less @ A @ Y3 @ X4 )
=> P ) ) ) ).
% less_imp_triv
thf(fact_158_less__imp__not__eq2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X4: A,Y3: A] :
( ( ord_less @ A @ X4 @ Y3 )
=> ( Y3 != X4 ) ) ) ).
% less_imp_not_eq2
thf(fact_159_antisym__conv3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y3: A,X4: A] :
( ~ ( ord_less @ A @ Y3 @ X4 )
=> ( ( ~ ( ord_less @ A @ X4 @ Y3 ) )
= ( X4 = Y3 ) ) ) ) ).
% antisym_conv3
thf(fact_160_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,A3: A] :
( ! [X3: A] :
( ! [Y2: A] :
( ( ord_less @ A @ Y2 @ X3 )
=> ( P @ Y2 ) )
=> ( P @ X3 ) )
=> ( P @ A3 ) ) ) ).
% less_induct
thf(fact_161_less__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X4: A,Y3: A] :
( ( ord_less @ A @ X4 @ Y3 )
=> ~ ( ord_less @ A @ Y3 @ X4 ) ) ) ).
% less_not_sym
thf(fact_162_less__imp__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X4: A,Y3: A] :
( ( ord_less @ A @ X4 @ Y3 )
=> ( X4 != Y3 ) ) ) ).
% less_imp_not_eq
thf(fact_163_dual__order_Oasym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ~ ( ord_less @ A @ A3 @ B3 ) ) ) ).
% dual_order.asym
thf(fact_164_ord__less__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B3: A,C: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( B3 = C )
=> ( ord_less @ A @ A3 @ C ) ) ) ) ).
% ord_less_eq_trans
thf(fact_165_ord__eq__less__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B3: A,C: A] :
( ( A3 = B3 )
=> ( ( ord_less @ A @ B3 @ C )
=> ( ord_less @ A @ A3 @ C ) ) ) ) ).
% ord_eq_less_trans
thf(fact_166_less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X4: A] :
~ ( ord_less @ A @ X4 @ X4 ) ) ).
% less_irrefl
thf(fact_167_less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X4: A,Y3: A] :
( ( ord_less @ A @ X4 @ Y3 )
| ( X4 = Y3 )
| ( ord_less @ A @ Y3 @ X4 ) ) ) ).
% less_linear
thf(fact_168_less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X4: A,Y3: A,Z3: A] :
( ( ord_less @ A @ X4 @ Y3 )
=> ( ( ord_less @ A @ Y3 @ Z3 )
=> ( ord_less @ A @ X4 @ Z3 ) ) ) ) ).
% less_trans
thf(fact_169_less__asym_H,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ~ ( ord_less @ A @ B3 @ A3 ) ) ) ).
% less_asym'
thf(fact_170_less__asym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X4: A,Y3: A] :
( ( ord_less @ A @ X4 @ Y3 )
=> ~ ( ord_less @ A @ Y3 @ X4 ) ) ) ).
% less_asym
thf(fact_171_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X4: A,Y3: A] :
( ( ord_less @ A @ X4 @ Y3 )
=> ( X4 != Y3 ) ) ) ).
% less_imp_neq
thf(fact_172_dense,axiom,
! [A: $tType] :
( ( dense_order @ A )
=> ! [X4: A,Y3: A] :
( ( ord_less @ A @ X4 @ Y3 )
=> ? [Z2: A] :
( ( ord_less @ A @ X4 @ Z2 )
& ( ord_less @ A @ Z2 @ Y3 ) ) ) ) ).
% dense
thf(fact_173_order_Oasym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ~ ( ord_less @ A @ B3 @ A3 ) ) ) ).
% order.asym
thf(fact_174_neq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X4: A,Y3: A] :
( ( X4 != Y3 )
= ( ( ord_less @ A @ X4 @ Y3 )
| ( ord_less @ A @ Y3 @ X4 ) ) ) ) ).
% neq_iff
thf(fact_175_neqE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X4: A,Y3: A] :
( ( X4 != Y3 )
=> ( ~ ( ord_less @ A @ X4 @ Y3 )
=> ( ord_less @ A @ Y3 @ X4 ) ) ) ) ).
% neqE
thf(fact_176_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [X4: A] :
? [X_1: A] : ( ord_less @ A @ X4 @ X_1 ) ) ).
% gt_ex
thf(fact_177_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [X4: A] :
? [Y4: A] : ( ord_less @ A @ Y4 @ X4 ) ) ).
% lt_ex
thf(fact_178_order__less__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 )
& ( order @ A ) )
=> ! [A3: A,B3: A,F: A > C2,C: C2] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less @ C2 @ ( F @ B3 ) @ C )
=> ( ! [X3: A,Y4: A] :
( ( ord_less @ A @ X3 @ Y4 )
=> ( ord_less @ C2 @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ C2 @ ( F @ A3 ) @ C ) ) ) ) ) ).
% order_less_subst2
thf(fact_179_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F: B > A,B3: B,C: B] :
( ( ord_less @ A @ A3 @ ( F @ B3 ) )
=> ( ( ord_less @ B @ B3 @ C )
=> ( ! [X3: B,Y4: B] :
( ( ord_less @ B @ X3 @ Y4 )
=> ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_180_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,B3: A,F: A > B,C: B] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X3: A,Y4: A] :
( ( ord_less @ A @ X3 @ Y4 )
=> ( ord_less @ B @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ B @ ( F @ A3 ) @ C ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_181_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,F: B > A,B3: B,C: B] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less @ B @ B3 @ C )
=> ( ! [X3: B,Y4: B] :
( ( ord_less @ B @ X3 @ Y4 )
=> ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_182_pinf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X3: A] :
( ( ord_less @ A @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: A] :
! [X3: A] :
( ( ord_less @ A @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: A] :
! [X6: A] :
( ( ord_less @ A @ Z2 @ X6 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ) ).
% pinf(1)
thf(fact_183_pinf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X3: A] :
( ( ord_less @ A @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: A] :
! [X3: A] :
( ( ord_less @ A @ Z4 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: A] :
! [X6: A] :
( ( ord_less @ A @ Z2 @ X6 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ) ).
% pinf(2)
thf(fact_184_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_185_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_186_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_187_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_188_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_189_minf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: A] :
! [X6: A] :
( ( ord_less @ A @ X6 @ Z2 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P4 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ) ).
% minf(1)
thf(fact_190_minf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P4 @ X3 ) ) )
=> ( ? [Z4: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z4 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z2: A] :
! [X6: A] :
( ( ord_less @ A @ X6 @ Z2 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P4 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ) ).
% minf(2)
thf(fact_191_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_192_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_193_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_194_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_195_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_196_ex__has__least__nat,axiom,
! [A: $tType,P: A > $o,K: A,M: A > nat] :
( ( P @ K )
=> ? [X3: A] :
( ( P @ X3 )
& ! [Y2: A] :
( ( P @ Y2 )
=> ( ord_less_eq @ nat @ ( M @ X3 ) @ ( M @ Y2 ) ) ) ) ) ).
% ex_has_least_nat
thf(fact_197_list__strict__asc__imp__list__asc,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [Xs: list @ A] :
( ( list_strict_asc @ A @ Xs )
=> ( list_asc @ A @ Xs ) ) ) ).
% list_strict_asc_imp_list_asc
thf(fact_198_list__strict__desc__imp__list__desc,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [Xs: list @ A] :
( ( list_strict_desc @ A @ Xs )
=> ( list_desc @ A @ Xs ) ) ) ).
% list_strict_desc_imp_list_desc
thf(fact_199_order_Onot__eq__order__implies__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A] :
( ( A3 != B3 )
=> ( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ord_less @ A @ A3 @ B3 ) ) ) ) ).
% order.not_eq_order_implies_strict
thf(fact_200_dual__order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ( ord_less_eq @ A @ B3 @ A3 ) ) ) ).
% dual_order.strict_implies_order
thf(fact_201_dual__order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [B4: A,A5: A] :
( ( ord_less_eq @ A @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_202_dual__order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B4: A,A5: A] :
( ( ord_less @ A @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_203_order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ord_less_eq @ A @ A3 @ B3 ) ) ) ).
% order.strict_implies_order
thf(fact_204_dense__le__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [X4: A,Y3: A,Z3: A] :
( ( ord_less @ A @ X4 @ Y3 )
=> ( ! [W: A] :
( ( ord_less @ A @ X4 @ W )
=> ( ( ord_less @ A @ W @ Y3 )
=> ( ord_less_eq @ A @ W @ Z3 ) ) )
=> ( ord_less_eq @ A @ Y3 @ Z3 ) ) ) ) ).
% dense_le_bounded
thf(fact_205_dense__ge__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z3: A,X4: A,Y3: A] :
( ( ord_less @ A @ Z3 @ X4 )
=> ( ! [W: A] :
( ( ord_less @ A @ Z3 @ W )
=> ( ( ord_less @ A @ W @ X4 )
=> ( ord_less_eq @ A @ Y3 @ W ) ) )
=> ( ord_less_eq @ A @ Y3 @ Z3 ) ) ) ) ).
% dense_ge_bounded
thf(fact_206_dual__order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A3: A,C: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ C @ B3 )
=> ( ord_less @ A @ C @ A3 ) ) ) ) ).
% dual_order.strict_trans2
thf(fact_207_dual__order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A3: A,C: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less @ A @ C @ B3 )
=> ( ord_less @ A @ C @ A3 ) ) ) ) ).
% dual_order.strict_trans1
thf(fact_208_order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [A5: A,B4: A] :
( ( ord_less_eq @ A @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ) ).
% order.strict_iff_order
thf(fact_209_order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B4: A] :
( ( ord_less @ A @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ) ).
% order.order_iff_strict
thf(fact_210_order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A,C: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ C )
=> ( ord_less @ A @ A3 @ C ) ) ) ) ).
% order.strict_trans2
thf(fact_211_order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A,C: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less @ A @ B3 @ C )
=> ( ord_less @ A @ A3 @ C ) ) ) ) ).
% order.strict_trans1
thf(fact_212_not__le__imp__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y3: A,X4: A] :
( ~ ( ord_less_eq @ A @ Y3 @ X4 )
=> ( ord_less @ A @ X4 @ Y3 ) ) ) ).
% not_le_imp_less
thf(fact_213_less__le__not__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [X2: A,Y5: A] :
( ( ord_less_eq @ A @ X2 @ Y5 )
& ~ ( ord_less_eq @ A @ Y5 @ X2 ) ) ) ) ) ).
% less_le_not_le
thf(fact_214_le__imp__less__or__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X4: A,Y3: A] :
( ( ord_less_eq @ A @ X4 @ Y3 )
=> ( ( ord_less @ A @ X4 @ Y3 )
| ( X4 = Y3 ) ) ) ) ).
% le_imp_less_or_eq
thf(fact_215_le__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X4: A,Y3: A] :
( ( ord_less_eq @ A @ X4 @ Y3 )
| ( ord_less @ A @ Y3 @ X4 ) ) ) ).
% le_less_linear
thf(fact_216_dense__le,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Y3: A,Z3: A] :
( ! [X3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ord_less_eq @ A @ X3 @ Z3 ) )
=> ( ord_less_eq @ A @ Y3 @ Z3 ) ) ) ).
% dense_le
thf(fact_217_dense__ge,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z3: A,Y3: A] :
( ! [X3: A] :
( ( ord_less @ A @ Z3 @ X3 )
=> ( ord_less_eq @ A @ Y3 @ X3 ) )
=> ( ord_less_eq @ A @ Y3 @ Z3 ) ) ) ).
% dense_ge
thf(fact_218_less__le__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X4: A,Y3: A,Z3: A] :
( ( ord_less @ A @ X4 @ Y3 )
=> ( ( ord_less_eq @ A @ Y3 @ Z3 )
=> ( ord_less @ A @ X4 @ Z3 ) ) ) ) ).
% less_le_trans
thf(fact_219_le__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X4: A,Y3: A,Z3: A] :
( ( ord_less_eq @ A @ X4 @ Y3 )
=> ( ( ord_less @ A @ Y3 @ Z3 )
=> ( ord_less @ A @ X4 @ Z3 ) ) ) ) ).
% le_less_trans
thf(fact_220_less__imp__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X4: A,Y3: A] :
( ( ord_less @ A @ X4 @ Y3 )
=> ( ord_less_eq @ A @ X4 @ Y3 ) ) ) ).
% less_imp_le
thf(fact_221_antisym__conv2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X4: A,Y3: A] :
( ( ord_less_eq @ A @ X4 @ Y3 )
=> ( ( ~ ( ord_less @ A @ X4 @ Y3 ) )
= ( X4 = Y3 ) ) ) ) ).
% antisym_conv2
thf(fact_222_antisym__conv1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X4: A,Y3: A] :
( ~ ( ord_less @ A @ X4 @ Y3 )
=> ( ( ord_less_eq @ A @ X4 @ Y3 )
= ( X4 = Y3 ) ) ) ) ).
% antisym_conv1
thf(fact_223_le__neq__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( A3 != B3 )
=> ( ord_less @ A @ A3 @ B3 ) ) ) ) ).
% le_neq_trans
thf(fact_224_not__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X4: A,Y3: A] :
( ( ~ ( ord_less @ A @ X4 @ Y3 ) )
= ( ord_less_eq @ A @ Y3 @ X4 ) ) ) ).
% not_less
thf(fact_225_not__le,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X4: A,Y3: A] :
( ( ~ ( ord_less_eq @ A @ X4 @ Y3 ) )
= ( ord_less @ A @ Y3 @ X4 ) ) ) ).
% not_le
thf(fact_226_order__less__le__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 )
& ( order @ A ) )
=> ! [A3: A,B3: A,F: A > C2,C: C2] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ C2 @ ( F @ B3 ) @ C )
=> ( ! [X3: A,Y4: A] :
( ( ord_less @ A @ X3 @ Y4 )
=> ( ord_less @ C2 @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ C2 @ ( F @ A3 ) @ C ) ) ) ) ) ).
% order_less_le_subst2
thf(fact_227_order__less__le__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F: B > A,B3: B,C: B] :
( ( ord_less @ A @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C )
=> ( ! [X3: B,Y4: B] :
( ( ord_less_eq @ B @ X3 @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_228_order__le__less__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 )
& ( order @ A ) )
=> ! [A3: A,B3: A,F: A > C2,C: C2] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less @ C2 @ ( F @ B3 ) @ C )
=> ( ! [X3: A,Y4: A] :
( ( ord_less_eq @ A @ X3 @ Y4 )
=> ( ord_less_eq @ C2 @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ C2 @ ( F @ A3 ) @ C ) ) ) ) ) ).
% order_le_less_subst2
thf(fact_229_order__le__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F: B > A,B3: B,C: B] :
( ( ord_less_eq @ A @ A3 @ ( F @ B3 ) )
=> ( ( ord_less @ B @ B3 @ C )
=> ( ! [X3: B,Y4: B] :
( ( ord_less @ B @ X3 @ Y4 )
=> ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_230_less__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [X2: A,Y5: A] :
( ( ord_less_eq @ A @ X2 @ Y5 )
& ( X2 != Y5 ) ) ) ) ) ).
% less_le
thf(fact_231_le__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X2: A,Y5: A] :
( ( ord_less @ A @ X2 @ Y5 )
| ( X2 = Y5 ) ) ) ) ) ).
% le_less
thf(fact_232_leI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X4: A,Y3: A] :
( ~ ( ord_less @ A @ X4 @ Y3 )
=> ( ord_less_eq @ A @ Y3 @ X4 ) ) ) ).
% leI
thf(fact_233_leD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y3: A,X4: A] :
( ( ord_less_eq @ A @ Y3 @ X4 )
=> ~ ( ord_less @ A @ X4 @ Y3 ) ) ) ).
% leD
thf(fact_234_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_235_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_236_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_237_complete__interval,axiom,
! [A: $tType] :
( ( condit1037483654norder @ A )
=> ! [A3: A,B3: A,P: A > $o] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( P @ A3 )
=> ( ~ ( P @ B3 )
=> ? [C3: A] :
( ( ord_less_eq @ A @ A3 @ C3 )
& ( ord_less_eq @ A @ C3 @ B3 )
& ! [X6: A] :
( ( ( ord_less_eq @ A @ A3 @ X6 )
& ( ord_less @ A @ X6 @ C3 ) )
=> ( P @ X6 ) )
& ! [D3: A] :
( ! [X3: A] :
( ( ( ord_less_eq @ A @ A3 @ X3 )
& ( ord_less @ A @ X3 @ D3 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq @ A @ D3 @ C3 ) ) ) ) ) ) ) ).
% complete_interval
thf(fact_238_verit__comp__simplify1_I3_J,axiom,
! [B: $tType] :
( ( linorder @ B )
=> ! [B6: B,A7: B] :
( ( ~ ( ord_less_eq @ B @ B6 @ A7 ) )
= ( ord_less @ B @ A7 @ B6 ) ) ) ).
% verit_comp_simplify1(3)
thf(fact_239_set__swap,axiom,
! [A: $tType,I3: nat,Xs: list @ A,J: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( set2 @ A @ ( list_update @ A @ ( list_update @ A @ Xs @ I3 @ ( nth @ A @ Xs @ J ) ) @ J @ ( nth @ A @ Xs @ I3 ) ) )
= ( set2 @ A @ Xs ) ) ) ) ).
% set_swap
thf(fact_240_butlast__list__update,axiom,
! [A: $tType,K: nat,Xs: list @ A,X4: A] :
( ( ( K
= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) )
=> ( ( butlast @ A @ ( list_update @ A @ Xs @ K @ X4 ) )
= ( butlast @ A @ Xs ) ) )
& ( ( K
!= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) )
=> ( ( butlast @ A @ ( list_update @ A @ Xs @ K @ X4 ) )
= ( list_update @ A @ ( butlast @ A @ Xs ) @ K @ X4 ) ) ) ) ).
% butlast_list_update
thf(fact_241_length__butlast,axiom,
! [A: $tType,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( butlast @ A @ Xs ) )
= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) ) ).
% length_butlast
thf(fact_242_subset__code_I1_J,axiom,
! [A: $tType,Xs: list @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ B7 )
= ( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( member @ A @ X2 @ B7 ) ) ) ) ).
% subset_code(1)
thf(fact_243_set__update__subsetI,axiom,
! [A: $tType,Xs: list @ A,A4: set @ A,X4: A,I3: nat] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A4 )
=> ( ( member @ A @ X4 @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( list_update @ A @ Xs @ I3 @ X4 ) ) @ A4 ) ) ) ).
% set_update_subsetI
thf(fact_244_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_245_in__set__butlastD,axiom,
! [A: $tType,X4: A,Xs: list @ A] :
( ( member @ A @ X4 @ ( set2 @ A @ ( butlast @ A @ Xs ) ) )
=> ( member @ A @ X4 @ ( set2 @ A @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_246_one__reorient,axiom,
! [A: $tType] :
( ( one @ A )
=> ! [X4: A] :
( ( ( one_one @ A )
= X4 )
= ( X4
= ( one_one @ A ) ) ) ) ).
% one_reorient
thf(fact_247_list__ex__cong,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,F: A > $o,G: A > $o] :
( ( Xs = Ys )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Ys ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( list_ex @ A @ F @ Xs )
= ( list_ex @ A @ G @ Ys ) ) ) ) ).
% list_ex_cong
thf(fact_248_in__set__butlast__appendI,axiom,
! [A: $tType,X4: A,Xs: list @ A,Ys: list @ A] :
( ( ( member @ A @ X4 @ ( set2 @ A @ ( butlast @ A @ Xs ) ) )
| ( member @ A @ X4 @ ( set2 @ A @ ( butlast @ A @ Ys ) ) ) )
=> ( member @ A @ X4 @ ( set2 @ A @ ( butlast @ A @ ( append @ A @ Xs @ Ys ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_249_nth__mem,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( member @ A @ ( nth @ A @ Xs @ N ) @ ( set2 @ A @ Xs ) ) ) ).
% nth_mem
thf(fact_250_list__ball__nth,axiom,
! [A: $tType,N: nat,Xs: list @ A,P: A > $o] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( P @ X3 ) )
=> ( P @ ( nth @ A @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_251_in__set__conv__nth,axiom,
! [A: $tType,X4: A,Xs: list @ A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
= ( ? [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( ( nth @ A @ Xs @ I2 )
= X4 ) ) ) ) ).
% in_set_conv_nth
thf(fact_252_all__nth__imp__all__set,axiom,
! [A: $tType,Xs: list @ A,P: A > $o,X4: A] :
( ! [I: nat] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ I ) ) )
=> ( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( P @ X4 ) ) ) ).
% all_nth_imp_all_set
thf(fact_253_all__set__conv__all__nth,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( P @ X2 ) ) )
= ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ I2 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_254_verit__la__disequality,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B3: A] :
( ( A3 = B3 )
| ~ ( ord_less_eq @ A @ A3 @ B3 )
| ~ ( ord_less_eq @ A @ B3 @ A3 ) ) ) ).
% verit_la_disequality
% Type constructors (25)
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A8: $tType,A9: $tType] :
( ( preorder @ A9 )
=> ( preorder @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A8: $tType,A9: $tType] :
( ( order @ A9 )
=> ( order @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A8: $tType,A9: $tType] :
( ( ord @ A9 )
=> ( ord @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Groups_Ominus,axiom,
! [A8: $tType,A9: $tType] :
( ( minus @ A9 )
=> ( minus @ ( A8 > A9 ) ) ) ).
thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
condit1037483654norder @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_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_Nat_Onat___Groups_Ominus_4,axiom,
minus @ nat ).
thf(tcon_Nat_Onat___Groups_Oone,axiom,
one @ nat ).
thf(tcon_Nat_Onat___Nat_Osize,axiom,
size @ nat ).
thf(tcon_Set_Oset___Orderings_Opreorder_5,axiom,
! [A8: $tType] : ( preorder @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_6,axiom,
! [A8: $tType] : ( order @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_7,axiom,
! [A8: $tType] : ( ord @ ( set @ A8 ) ) ).
thf(tcon_Set_Oset___Groups_Ominus_8,axiom,
! [A8: $tType] : ( minus @ ( set @ A8 ) ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_9,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder_10,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_11,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_12,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Groups_Ominus_13,axiom,
minus @ $o ).
thf(tcon_List_Olist___Nat_Osize_14,axiom,
! [A8: $tType] : ( size @ ( list @ A8 ) ) ).
% Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X4: A,Y3: A] :
( ( if @ A @ $false @ X4 @ Y3 )
= Y3 ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X4: A,Y3: A] :
( ( if @ A @ $true @ X4 @ Y3 )
= X4 ) ).
% Conjectures (4)
thf(conj_0,hypothesis,
( ( size_size @ ( list @ a ) @ xs )
= ( size_size @ ( list @ a ) @ ys ) ) ).
thf(conj_1,hypothesis,
! [X6: nat] :
( ( ( ord_less @ nat @ X6 @ ( size_size @ ( list @ a ) @ ys ) )
=> ( ( ( ord_less @ nat @ X6 @ ( size_size @ ( list @ a ) @ ys ) )
=> ( ( nth @ a @ xs @ X6 )
= ( nth @ a @ ys @ X6 ) ) )
& ( ~ ( ord_less @ nat @ X6 @ ( size_size @ ( list @ a ) @ ys ) )
=> ( ( f @ ( minus_minus @ nat @ X6 @ ( size_size @ ( list @ a ) @ xs ) ) )
= ( nth @ a @ ys @ X6 ) ) ) ) )
& ( ~ ( ord_less @ nat @ X6 @ ( size_size @ ( list @ a ) @ ys ) )
=> ( ( ( ord_less @ nat @ X6 @ ( size_size @ ( list @ a ) @ ys ) )
=> ( ( nth @ a @ xs @ X6 )
= ( g @ ( minus_minus @ nat @ X6 @ ( size_size @ ( list @ a ) @ ys ) ) ) ) )
& ( ~ ( ord_less @ nat @ X6 @ ( size_size @ ( list @ a ) @ ys ) )
=> ( ( f @ ( minus_minus @ nat @ X6 @ ( size_size @ ( list @ a ) @ xs ) ) )
= ( g @ ( minus_minus @ nat @ X6 @ ( size_size @ ( list @ a ) @ ys ) ) ) ) ) ) ) ) ).
thf(conj_2,hypothesis,
ord_less @ nat @ x @ ( size_size @ ( list @ a ) @ ys ) ).
thf(conj_3,conjecture,
( ( nth @ a @ xs @ x )
= ( nth @ a @ ys @ x ) ) ).
%------------------------------------------------------------------------------