TPTP Problem File: ITP099^2.p
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%------------------------------------------------------------------------------
% File : ITP099^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer ListInf problem prob_412__5414926_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_412__5414926_1 [Des21]
% Status : Theorem
% Rating : 0.00 v8.1.0, 0.25 v7.5.0
% Syntax : Number of formulae : 324 ( 53 unt; 42 typ; 0 def)
% Number of atoms : 946 ( 245 equ; 0 cnn)
% Maximal formula atoms : 13 ( 3 avg)
% Number of connectives : 3542 ( 80 ~; 25 |; 48 &;2865 @)
% ( 0 <=>; 524 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 8 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 138 ( 138 >; 0 *; 0 +; 0 <<)
% Number of symbols : 43 ( 40 usr; 4 con; 0-4 aty)
% Number of variables : 1085 ( 70 ^; 935 !; 44 ?;1085 :)
% ( 36 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:17:00.338
%------------------------------------------------------------------------------
% Could-be-implicit typings (3)
thf(ty_t_List_Olist,type,
list: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_tf_a,type,
a: $tType ).
% Explicit typings (39)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Nat_Osize,type,
size:
!>[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_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[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_Groups_Oordered__ab__semigroup__add__imp__le,type,
ordere236663937imp_le:
!>[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_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : ( A > A > 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_ListInf__Mirabelle__nfmaokebij_Oi__drop,type,
listIn1569887318i_drop:
!>[A: $tType] : ( nat > ( nat > A ) > nat > A ) ).
thf(sy_c_ListInf__Mirabelle__nfmaokebij_Oi__take,type,
listIn1033672622i_take:
!>[A: $tType] : ( nat > ( nat > A ) > ( list @ A ) ) ).
thf(sy_c_List_Oappend,type,
append:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Odrop,type,
drop:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olast,type,
last:
!>[A: $tType] : ( ( list @ A ) > 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_v_f,type,
f: nat > a ).
thf(sy_v_n,type,
n: nat ).
thf(sy_v_xs,type,
xs: list @ a ).
% Relevant facts (256)
thf(fact_0_same__i__append__eq,axiom,
! [A: $tType,Xs: list @ A,F: nat > A,G: nat > A] :
( ( ( listIn521021761append @ A @ Xs @ F )
= ( listIn521021761append @ A @ Xs @ G ) )
= ( F = G ) ) ).
% same_i_append_eq
thf(fact_1_i__append__eq__i__append__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,F: nat > A,G: nat > A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) )
=> ( ( ( listIn521021761append @ A @ Xs @ F )
= ( listIn521021761append @ A @ Ys @ G ) )
= ( ( Xs = Ys )
& ( F = G ) ) ) ) ).
% i_append_eq_i_append_conv
thf(fact_2_length__drop,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( drop @ A @ N @ Xs ) )
= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) ) ).
% length_drop
thf(fact_3_minus__apply,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A2: A > B,B2: A > B,X: A] : ( minus_minus @ B @ ( A2 @ X ) @ ( B2 @ X ) ) ) ) ) ).
% minus_apply
thf(fact_4_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_5_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_6_less__diff__imp__less,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less @ nat @ I @ ( minus_minus @ nat @ J @ M ) )
=> ( ord_less @ nat @ I @ J ) ) ).
% less_diff_imp_less
thf(fact_7_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_8_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_9_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_10_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_11_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_12_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_13_length__induct,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
( ! [Xs2: list @ A] :
( ! [Ys2: list @ A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ys2 ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_14_measure__induct__rule,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ B )
=> ! [F: A > B,P: A > $o,A3: A] :
( ! [X2: A] :
( ! [Y: A] :
( ( ord_less @ B @ ( F @ Y ) @ ( F @ X2 ) )
=> ( P @ Y ) )
=> ( P @ X2 ) )
=> ( P @ A3 ) ) ) ).
% measure_induct_rule
thf(fact_15_measure__induct,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ B )
=> ! [F: A > B,P: A > $o,A3: A] :
( ! [X2: A] :
( ! [Y: A] :
( ( ord_less @ B @ ( F @ Y ) @ ( F @ X2 ) )
=> ( P @ Y ) )
=> ( P @ X2 ) )
=> ( P @ A3 ) ) ) ).
% measure_induct
thf(fact_16_fun__diff__def,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A2: A > B,B2: A > B,X: A] : ( minus_minus @ B @ ( A2 @ X ) @ ( B2 @ X ) ) ) ) ) ).
% fun_diff_def
thf(fact_17_infinite__descent__measure,axiom,
! [A: $tType,P: A > $o,V: A > nat,X3: A] :
( ! [X2: A] :
( ~ ( P @ X2 )
=> ? [Y: A] :
( ( ord_less @ nat @ ( V @ Y ) @ ( V @ X2 ) )
& ~ ( P @ Y ) ) )
=> ( P @ X3 ) ) ).
% infinite_descent_measure
thf(fact_18_linorder__neqE__nat,axiom,
! [X3: nat,Y2: nat] :
( ( X3 != Y2 )
=> ( ~ ( ord_less @ nat @ X3 @ Y2 )
=> ( ord_less @ nat @ Y2 @ X3 ) ) ) ).
% linorder_neqE_nat
thf(fact_19_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_20_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_21_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_22_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less @ nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_23_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_24_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_not_refl
thf(fact_25_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_26_size__neq__size__imp__neq,axiom,
! [A: $tType] :
( ( size @ A )
=> ! [X3: A,Y2: A] :
( ( ( size_size @ A @ X3 )
!= ( size_size @ A @ Y2 ) )
=> ( X3 != Y2 ) ) ) ).
% size_neq_size_imp_neq
thf(fact_27_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_28_Ex__list__of__length,axiom,
! [A: $tType,N: nat] :
? [Xs2: list @ A] :
( ( size_size @ ( list @ A ) @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_29_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_30_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus @ nat @ ( minus_minus @ nat @ I @ J ) @ K )
= ( minus_minus @ nat @ ( minus_minus @ nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_31_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_32_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_33_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_34_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_35_i__append__i__drop__eq1,axiom,
! [A: $tType,N: nat,Xs: list @ A,F: nat > A] :
( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( listIn1569887318i_drop @ A @ N @ ( listIn521021761append @ A @ Xs @ F ) )
= ( listIn521021761append @ A @ ( drop @ A @ N @ Xs ) @ F ) ) ) ).
% i_append_i_drop_eq1
thf(fact_36_i__append__i__drop__eq2,axiom,
! [A: $tType,Xs: list @ A,N: nat,F: nat > A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N )
=> ( ( listIn1569887318i_drop @ A @ N @ ( listIn521021761append @ A @ Xs @ F ) )
= ( listIn1569887318i_drop @ A @ ( minus_minus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) @ F ) ) ) ).
% i_append_i_drop_eq2
thf(fact_37_last__drop,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( last @ A @ ( drop @ A @ N @ Xs ) )
= ( last @ A @ Xs ) ) ) ).
% last_drop
thf(fact_38_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_39_i__append__nth,axiom,
! [A: $tType] :
( ( listIn521021761append @ A )
= ( ^ [Xs3: list @ A,F2: nat > A,N3: nat] : ( if @ A @ ( ord_less @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs3 ) ) @ ( nth @ A @ Xs3 @ N3 ) @ ( F2 @ ( minus_minus @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs3 ) ) ) ) ) ) ).
% i_append_nth
thf(fact_40_i__append__def,axiom,
! [A: $tType] :
( ( listIn521021761append @ A )
= ( ^ [Xs3: list @ A,F2: nat > A,N3: nat] : ( if @ A @ ( ord_less @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs3 ) ) @ ( nth @ A @ Xs3 @ N3 ) @ ( F2 @ ( minus_minus @ nat @ N3 @ ( size_size @ ( list @ A ) @ Xs3 ) ) ) ) ) ) ).
% i_append_def
thf(fact_41_drop__append,axiom,
! [A: $tType,N: nat,Xs: list @ A,Ys: list @ A] :
( ( drop @ A @ N @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( drop @ A @ N @ Xs ) @ ( drop @ A @ ( minus_minus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) @ Ys ) ) ) ).
% drop_append
thf(fact_42_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_43_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_44_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_45_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X2: A] :
( ( F @ X2 )
= ( G @ X2 ) )
=> ( F = G ) ) ).
% ext
thf(fact_46_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_47_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq @ nat @ I @ N )
=> ( ( minus_minus @ nat @ N @ ( minus_minus @ nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_48_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_49_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_50_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_51_le__refl,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% le_refl
thf(fact_52_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ nat @ J @ K )
=> ( ord_less_eq @ nat @ I @ K ) ) ) ).
% le_trans
thf(fact_53_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_54_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_55_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_56_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_57_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B3: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq @ nat @ Y3 @ B3 ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq @ nat @ Y @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_58_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_59_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_60_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_61_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_62_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_63_i__append__eq__i__append__conv2__aux,axiom,
! [A: $tType,Xs: list @ A,F: nat > A,Ys: list @ A,G: nat > A] :
( ( ( listIn521021761append @ A @ Xs @ F )
= ( listIn521021761append @ A @ Ys @ G ) )
=> ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys ) )
=> ? [Zs2: list @ A] :
( ( ( append @ A @ Xs @ Zs2 )
= Ys )
& ( F
= ( listIn521021761append @ A @ Zs2 @ G ) ) ) ) ) ).
% i_append_eq_i_append_conv2_aux
thf(fact_64_ge__less__neq__conv,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A4: A,N3: A] :
! [X: A] :
( ( ord_less @ A @ X @ A4 )
=> ( N3 != X ) ) ) ) ) ).
% ge_less_neq_conv
thf(fact_65_less__ge__neq__conv,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less @ A )
= ( ^ [N3: A,A4: A] :
! [X: A] :
( ( ord_less_eq @ A @ A4 @ X )
=> ( N3 != X ) ) ) ) ) ).
% less_ge_neq_conv
thf(fact_66_greater__le__neq__conv,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less @ A )
= ( ^ [A4: A,N3: A] :
! [X: A] :
( ( ord_less_eq @ A @ X @ A4 )
=> ( N3 != X ) ) ) ) ) ).
% greater_le_neq_conv
thf(fact_67_le__greater__neq__conv,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [N3: A,A4: A] :
! [X: A] :
( ( ord_less @ A @ A4 @ X )
=> ( N3 != X ) ) ) ) ) ).
% le_greater_neq_conv
thf(fact_68_nat__less__le,axiom,
( ( ord_less @ nat )
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_eq @ nat @ M3 @ N3 )
& ( M3 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_69_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_70_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_71_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_72_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_73_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less @ nat @ I2 @ J2 )
=> ( ord_less @ nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_74_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_75_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_76_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_77_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_78_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_79_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_80_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_81_le__diff__swap,axiom,
! [I: nat,K: nat,J: nat] :
( ( ord_less_eq @ nat @ I @ K )
=> ( ( ord_less_eq @ nat @ J @ K )
=> ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ K @ J ) @ I )
= ( ord_less_eq @ nat @ ( minus_minus @ nat @ K @ I ) @ J ) ) ) ) ).
% le_diff_swap
thf(fact_82_le__diff__imp__le,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq @ nat @ I @ ( minus_minus @ nat @ J @ M ) )
=> ( ord_less_eq @ nat @ I @ J ) ) ).
% le_diff_imp_le
thf(fact_83_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_84_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_85_le__diff__le__imp__le,axiom,
! [I: nat,J: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ I @ ( minus_minus @ nat @ J @ M ) )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( ord_less_eq @ nat @ I @ ( minus_minus @ nat @ J @ N ) ) ) ) ).
% le_diff_le_imp_le
thf(fact_86_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_87_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_88_i__append__eq__i__appendI,axiom,
! [A: $tType,Xs: list @ A,Xs4: list @ A,Ys: list @ A,F: nat > A,G: nat > A] :
( ( ( append @ A @ Xs @ Xs4 )
= Ys )
=> ( ( F
= ( listIn521021761append @ A @ Xs4 @ G ) )
=> ( ( listIn521021761append @ A @ Xs @ F )
= ( listIn521021761append @ A @ Ys @ G ) ) ) ) ).
% i_append_eq_i_appendI
thf(fact_89_i__append__eq__i__append__conv2,axiom,
! [A: $tType,Xs: list @ A,F: nat > A,Ys: list @ A,G: nat > A] :
( ( ( listIn521021761append @ A @ Xs @ F )
= ( listIn521021761append @ A @ Ys @ G ) )
= ( ? [Zs3: list @ A] :
( ( ( Xs
= ( append @ A @ Ys @ Zs3 ) )
& ( ( listIn521021761append @ A @ Zs3 @ F )
= G ) )
| ( ( ( append @ A @ Xs @ Zs3 )
= Ys )
& ( F
= ( listIn521021761append @ A @ Zs3 @ G ) ) ) ) ) ) ).
% i_append_eq_i_append_conv2
thf(fact_90_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_91_nth__equalityI,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ Xs @ I2 )
= ( nth @ A @ Ys @ I2 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_92_Skolem__list__nth,axiom,
! [A: $tType,K: nat,P: nat > A > $o] :
( ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ K )
=> ? [X4: A] : ( P @ I3 @ X4 ) ) )
= ( ? [Xs3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= K )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ K )
=> ( P @ I3 @ ( nth @ A @ Xs3 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_93_list__eq__iff__nth__eq,axiom,
! [A: $tType] :
( ( ^ [Y4: list @ A,Z: list @ A] : Y4 = Z )
= ( ^ [Xs3: list @ A,Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ A ) @ Ys3 ) )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ( ( nth @ A @ Xs3 @ I3 )
= ( nth @ A @ Ys3 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_94_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_95_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_96_less__diff__le__imp__less,axiom,
! [I: nat,J: nat,M: nat,N: nat] :
( ( ord_less @ nat @ I @ ( minus_minus @ nat @ J @ M ) )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( ord_less @ nat @ I @ ( minus_minus @ nat @ J @ N ) ) ) ) ).
% less_diff_le_imp_less
thf(fact_97_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_98_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_99_i__drop__nth__sub,axiom,
! [A: $tType,N: nat,X3: nat,S: nat > A] :
( ( ord_less_eq @ nat @ N @ X3 )
=> ( ( listIn1569887318i_drop @ A @ N @ S @ ( minus_minus @ nat @ X3 @ N ) )
= ( S @ X3 ) ) ) ).
% i_drop_nth_sub
thf(fact_100_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_101_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_102_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_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_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A] : ( ord_less_eq @ A @ X3 @ X3 ) ) ).
% order_refl
thf(fact_105_list__desc__trans__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( list_desc @ A )
= ( ^ [Xs3: list @ A] :
! [J3: nat] :
( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ! [I3: nat] :
( ( ord_less_eq @ nat @ I3 @ J3 )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs3 @ J3 ) @ ( nth @ A @ Xs3 @ I3 ) ) ) ) ) ) ) ).
% list_desc_trans_le
thf(fact_106_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 )
=> ( ! [I4: nat] :
( ( ord_less @ nat @ K2 @ I4 )
=> ( P @ I4 ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_107_Lattices__Big_Oex__has__greatest__nat,axiom,
! [A: $tType,P: A > $o,K: A,F: A > nat,B3: nat] :
( ( P @ K )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less @ nat @ ( F @ Y3 ) @ B3 ) )
=> ? [X2: A] :
( ( P @ X2 )
& ! [Y: A] :
( ( P @ Y )
=> ( ord_less_eq @ nat @ ( F @ Y ) @ ( F @ X2 ) ) ) ) ) ) ).
% Lattices_Big.ex_has_greatest_nat
thf(fact_108_list__asc__trans__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( list_asc @ A )
= ( ^ [Xs3: list @ A] :
! [J3: nat] :
( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ! [I3: nat] :
( ( ord_less_eq @ nat @ I3 @ J3 )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs3 @ I3 ) @ ( nth @ A @ Xs3 @ J3 ) ) ) ) ) ) ) ).
% list_asc_trans_le
thf(fact_109_list__strict__asc__trans__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [Xs: list @ A] :
( ( list_strict_asc @ A @ Xs )
=> ! [J4: nat] :
( ( ord_less @ nat @ J4 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ! [I4: nat] :
( ( ord_less_eq @ nat @ I4 @ J4 )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ I4 ) @ ( nth @ A @ Xs @ J4 ) ) ) ) ) ) ).
% list_strict_asc_trans_le
thf(fact_110_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_111_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_112_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_113_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G: A > B] :
( ! [X2: A] : ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) )
=> ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% le_funI
thf(fact_114_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B] :
! [X: A] : ( ord_less_eq @ B @ ( F2 @ X ) @ ( G2 @ X ) ) ) ) ) ).
% le_fun_def
thf(fact_115_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 )
=> ( ! [X2: B,Y3: B] :
( ( ord_less_eq @ B @ X2 @ Y3 )
=> ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% order_subst1
thf(fact_116_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 )
=> ( ! [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
=> ( ord_less_eq @ C2 @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ C2 @ ( F @ A3 ) @ C ) ) ) ) ) ).
% order_subst2
thf(fact_117_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 )
=> ( ! [X2: B,Y3: B] :
( ( ord_less_eq @ B @ X2 @ Y3 )
=> ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_118_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 )
=> ( ! [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
=> ( ord_less_eq @ B @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ B @ ( F @ A3 ) @ C ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_119_eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z: A] : Y4 = Z )
= ( ^ [X: A,Y5: A] :
( ( ord_less_eq @ A @ X @ Y5 )
& ( ord_less_eq @ A @ Y5 @ X ) ) ) ) ) ).
% eq_iff
thf(fact_120_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_121_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_122_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y2: A] :
( ( X3 = Y2 )
=> ( ord_less_eq @ A @ X3 @ Y2 ) ) ) ).
% eq_refl
thf(fact_123_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_124_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_125_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y2: A,Z2: A] :
( ( ( ord_less_eq @ A @ X3 @ Y2 )
=> ~ ( ord_less_eq @ A @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ Y2 @ X3 )
=> ~ ( ord_less_eq @ A @ X3 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ X3 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq @ A @ Z2 @ Y2 )
=> ~ ( ord_less_eq @ A @ Y2 @ X3 ) )
=> ( ( ( ord_less_eq @ A @ Y2 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ X3 ) )
=> ~ ( ( ord_less_eq @ A @ Z2 @ X3 )
=> ~ ( ord_less_eq @ A @ X3 @ Y2 ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_126_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_127_order__class_Oorder_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z: A] : Y4 = Z )
= ( ^ [A4: A,B4: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
& ( ord_less_eq @ A @ B4 @ A4 ) ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_128_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_129_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_130_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_131_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y2: A,Z2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ Z2 )
=> ( ord_less_eq @ A @ X3 @ Z2 ) ) ) ) ).
% order_trans
thf(fact_132_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% dual_order.refl
thf(fact_133_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A3: A,B3: A] :
( ! [A5: A,B5: A] :
( ( ord_less_eq @ A @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: A,B5: A] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A3 @ B3 ) ) ) ) ).
% linorder_wlog
thf(fact_134_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_135_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z: A] : Y4 = Z )
= ( ^ [A4: A,B4: A] :
( ( ord_less_eq @ A @ B4 @ A4 )
& ( ord_less_eq @ A @ A4 @ B4 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_136_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_137_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_138_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_139_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_140_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_141_linorder__less__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A3: A,B3: A] :
( ! [A5: A,B5: A] :
( ( ord_less @ A @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: A] : ( P @ A5 @ A5 )
=> ( ! [A5: A,B5: A] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A3 @ B3 ) ) ) ) ) ).
% linorder_less_wlog
thf(fact_142_exists__least__iff,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ( ( ^ [P2: A > $o] :
? [X4: A] : ( P2 @ X4 ) )
= ( ^ [P3: A > $o] :
? [N3: A] :
( ( P3 @ N3 )
& ! [M3: A] :
( ( ord_less @ A @ M3 @ N3 )
=> ~ ( P3 @ M3 ) ) ) ) ) ) ).
% exists_least_iff
thf(fact_143_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_144_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_145_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ A3 @ A3 ) ) ).
% dual_order.irrefl
thf(fact_146_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_147_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_148_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_149_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_150_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,A3: A] :
( ! [X2: A] :
( ! [Y: A] :
( ( ord_less @ A @ Y @ X2 )
=> ( P @ Y ) )
=> ( P @ X2 ) )
=> ( P @ A3 ) ) ) ).
% less_induct
thf(fact_151_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_152_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_153_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_154_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_155_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_156_less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A] :
~ ( ord_less @ A @ X3 @ X3 ) ) ).
% less_irrefl
thf(fact_157_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_158_less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y2: A,Z2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( ( ord_less @ A @ Y2 @ Z2 )
=> ( ord_less @ A @ X3 @ Z2 ) ) ) ) ).
% less_trans
thf(fact_159_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_160_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_161_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( X3 != Y2 ) ) ) ).
% less_imp_neq
thf(fact_162_dense,axiom,
! [A: $tType] :
( ( dense_order @ A )
=> ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ? [Z3: A] :
( ( ord_less @ A @ X3 @ Z3 )
& ( ord_less @ A @ Z3 @ Y2 ) ) ) ) ).
% dense
thf(fact_163_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_164_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_165_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_166_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [X3: A] :
? [X_1: A] : ( ord_less @ A @ X3 @ X_1 ) ) ).
% gt_ex
thf(fact_167_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [X3: A] :
? [Y3: A] : ( ord_less @ A @ Y3 @ X3 ) ) ).
% lt_ex
thf(fact_168_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 )
=> ( ! [X2: A,Y3: A] :
( ( ord_less @ A @ X2 @ Y3 )
=> ( ord_less @ C2 @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ C2 @ ( F @ A3 ) @ C ) ) ) ) ) ).
% order_less_subst2
thf(fact_169_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 )
=> ( ! [X2: B,Y3: B] :
( ( ord_less @ B @ X2 @ Y3 )
=> ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_170_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 )
=> ( ! [X2: A,Y3: A] :
( ( ord_less @ A @ X2 @ Y3 )
=> ( ord_less @ B @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ B @ ( F @ A3 ) @ C ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_171_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 )
=> ( ! [X2: B,Y3: B] :
( ( ord_less @ B @ X2 @ Y3 )
=> ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_172_ex__has__least__nat,axiom,
! [A: $tType,P: A > $o,K: A,M: A > nat] :
( ( P @ K )
=> ? [X2: A] :
( ( P @ X2 )
& ! [Y: A] :
( ( P @ Y )
=> ( ord_less_eq @ nat @ ( M @ X2 ) @ ( M @ Y ) ) ) ) ) ).
% ex_has_least_nat
thf(fact_173_list__drop__eq__conv,axiom,
! [A: $tType] :
( ( ^ [Y4: list @ A,Z: list @ A] : Y4 = Z )
= ( ^ [Xs3: list @ A,Ys3: list @ A] :
! [N3: nat] :
( ( drop @ A @ N3 @ Xs3 )
= ( drop @ A @ N3 @ Ys3 ) ) ) ) ).
% list_drop_eq_conv
thf(fact_174_list__asc__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( list_asc @ A )
= ( ^ [Xs3: list @ A] :
! [J3: nat] :
( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ! [I3: nat] :
( ( ord_less @ nat @ I3 @ J3 )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs3 @ I3 ) @ ( nth @ A @ Xs3 @ J3 ) ) ) ) ) ) ) ).
% list_asc_trans
thf(fact_175_list__strict__asc__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( list_strict_asc @ A )
= ( ^ [Xs3: list @ A] :
! [J3: nat] :
( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ! [I3: nat] :
( ( ord_less @ nat @ I3 @ J3 )
=> ( ord_less @ A @ ( nth @ A @ Xs3 @ I3 ) @ ( nth @ A @ Xs3 @ J3 ) ) ) ) ) ) ) ).
% list_strict_asc_trans
thf(fact_176_list__desc__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( list_desc @ A )
= ( ^ [Xs3: list @ A] :
! [J3: nat] :
( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ! [I3: nat] :
( ( ord_less @ nat @ I3 @ J3 )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs3 @ J3 ) @ ( nth @ A @ Xs3 @ I3 ) ) ) ) ) ) ) ).
% list_desc_trans
thf(fact_177_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_178_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_179_dual__order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [B4: A,A4: A] :
( ( ord_less_eq @ A @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_180_dual__order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B4: A,A4: A] :
( ( ord_less @ A @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_181_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_182_dense__le__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [X3: A,Y2: A,Z2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( ! [W: A] :
( ( ord_less @ A @ X3 @ W )
=> ( ( ord_less @ A @ W @ Y2 )
=> ( ord_less_eq @ A @ W @ Z2 ) ) )
=> ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ) ).
% dense_le_bounded
thf(fact_183_dense__ge__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z2: A,X3: A,Y2: A] :
( ( ord_less @ A @ Z2 @ X3 )
=> ( ! [W: A] :
( ( ord_less @ A @ Z2 @ W )
=> ( ( ord_less @ A @ W @ X3 )
=> ( ord_less_eq @ A @ Y2 @ W ) ) )
=> ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ) ).
% dense_ge_bounded
thf(fact_184_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_185_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_186_order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [A4: A,B4: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ) ).
% order.strict_iff_order
thf(fact_187_order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A4: A,B4: A] :
( ( ord_less @ A @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ) ).
% order.order_iff_strict
thf(fact_188_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_189_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_190_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_191_less__le__not__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [X: A,Y5: A] :
( ( ord_less_eq @ A @ X @ Y5 )
& ~ ( ord_less_eq @ A @ Y5 @ X ) ) ) ) ) ).
% less_le_not_le
thf(fact_192_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_193_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_194_dense__le,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Y2: A,Z2: A] :
( ! [X2: A] :
( ( ord_less @ A @ X2 @ Y2 )
=> ( ord_less_eq @ A @ X2 @ Z2 ) )
=> ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ).
% dense_le
thf(fact_195_dense__ge,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z2: A,Y2: A] :
( ! [X2: A] :
( ( ord_less @ A @ Z2 @ X2 )
=> ( ord_less_eq @ A @ Y2 @ X2 ) )
=> ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ).
% dense_ge
thf(fact_196_less__le__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y2: A,Z2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ Z2 )
=> ( ord_less @ A @ X3 @ Z2 ) ) ) ) ).
% less_le_trans
thf(fact_197_le__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y2: A,Z2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ( ord_less @ A @ Y2 @ Z2 )
=> ( ord_less @ A @ X3 @ Z2 ) ) ) ) ).
% le_less_trans
thf(fact_198_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_199_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_200_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_201_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_202_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_203_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_204_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 )
=> ( ! [X2: A,Y3: A] :
( ( ord_less @ A @ X2 @ Y3 )
=> ( ord_less @ C2 @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ C2 @ ( F @ A3 ) @ C ) ) ) ) ) ).
% order_less_le_subst2
thf(fact_205_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 )
=> ( ! [X2: B,Y3: B] :
( ( ord_less_eq @ B @ X2 @ Y3 )
=> ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_206_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 )
=> ( ! [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
=> ( ord_less_eq @ C2 @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ C2 @ ( F @ A3 ) @ C ) ) ) ) ) ).
% order_le_less_subst2
thf(fact_207_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 )
=> ( ! [X2: B,Y3: B] :
( ( ord_less @ B @ X2 @ Y3 )
=> ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_208_less__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [X: A,Y5: A] :
( ( ord_less_eq @ A @ X @ Y5 )
& ( X != Y5 ) ) ) ) ) ).
% less_le
thf(fact_209_le__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X: A,Y5: A] :
( ( ord_less @ A @ X @ Y5 )
| ( X = Y5 ) ) ) ) ) ).
% le_less
thf(fact_210_leI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y2: A] :
( ~ ( ord_less @ A @ X3 @ Y2 )
=> ( ord_less_eq @ A @ Y2 @ X3 ) ) ) ).
% leI
thf(fact_211_leD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y2: A,X3: A] :
( ( ord_less_eq @ A @ Y2 @ X3 )
=> ~ ( ord_less @ A @ X3 @ Y2 ) ) ) ).
% leD
thf(fact_212_list__strict__desc__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( list_strict_desc @ A )
= ( ^ [Xs3: list @ A] :
! [J3: nat] :
( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ! [I3: nat] :
( ( ord_less @ nat @ I3 @ J3 )
=> ( ord_less @ A @ ( nth @ A @ Xs3 @ J3 ) @ ( nth @ A @ Xs3 @ I3 ) ) ) ) ) ) ) ).
% list_strict_desc_trans
thf(fact_213_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 )
& ! [X5: A] :
( ( ( ord_less_eq @ A @ A3 @ X5 )
& ( ord_less @ A @ X5 @ C3 ) )
=> ( P @ X5 ) )
& ! [D2: A] :
( ! [X2: A] :
( ( ( ord_less_eq @ A @ A3 @ X2 )
& ( ord_less @ A @ X2 @ D2 ) )
=> ( P @ X2 ) )
=> ( ord_less_eq @ A @ D2 @ C3 ) ) ) ) ) ) ) ).
% complete_interval
thf(fact_214_verit__comp__simplify1_I3_J,axiom,
! [B: $tType] :
( ( linorder @ B )
=> ! [B6: B,A6: B] :
( ( ~ ( ord_less_eq @ B @ B6 @ A6 ) )
= ( ord_less @ B @ A6 @ B6 ) ) ) ).
% verit_comp_simplify1(3)
thf(fact_215_pinf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ~ ( ord_less_eq @ A @ X5 @ T ) ) ) ).
% pinf(6)
thf(fact_216_pinf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ( ord_less_eq @ A @ T @ X5 ) ) ) ).
% pinf(8)
thf(fact_217_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_218_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
thf(fact_219_minf_I11_J,axiom,
! [C2: $tType,D3: $tType] :
( ( ord @ C2 )
=> ! [F3: D3] :
? [Z3: C2] :
! [X5: C2] :
( ( ord_less @ C2 @ X5 @ Z3 )
=> ( F3 = F3 ) ) ) ).
% minf(11)
thf(fact_220_minf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ~ ( ord_less @ A @ T @ X5 ) ) ) ).
% minf(7)
thf(fact_221_minf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ( ord_less @ A @ X5 @ T ) ) ) ).
% minf(5)
thf(fact_222_minf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ( X5 != T ) ) ) ).
% minf(4)
thf(fact_223_minf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ( X5 != T ) ) ) ).
% minf(3)
thf(fact_224_minf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z4 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z4: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z4 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ) ).
% minf(2)
thf(fact_225_minf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z4 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z4: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z4 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ) ).
% minf(1)
thf(fact_226_pinf_I11_J,axiom,
! [C2: $tType,D3: $tType] :
( ( ord @ C2 )
=> ! [F3: D3] :
? [Z3: C2] :
! [X5: C2] :
( ( ord_less @ C2 @ Z3 @ X5 )
=> ( F3 = F3 ) ) ) ).
% pinf(11)
thf(fact_227_pinf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ( ord_less @ A @ T @ X5 ) ) ) ).
% pinf(7)
thf(fact_228_pinf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ~ ( ord_less @ A @ X5 @ T ) ) ) ).
% pinf(5)
thf(fact_229_pinf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ( X5 != T ) ) ) ).
% pinf(4)
thf(fact_230_pinf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ( X5 != T ) ) ) ).
% pinf(3)
thf(fact_231_pinf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X2: A] :
( ( ord_less @ A @ Z4 @ X2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z4: A] :
! [X2: A] :
( ( ord_less @ A @ Z4 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ) ).
% pinf(2)
thf(fact_232_pinf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X2: A] :
( ( ord_less @ A @ Z4 @ X2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z4: A] :
! [X2: A] :
( ( ord_less @ A @ Z4 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ) ).
% pinf(1)
thf(fact_233_verit__comp__simplify1_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ A3 @ A3 ) ) ).
% verit_comp_simplify1(1)
thf(fact_234_ex__gt__or__lt,axiom,
! [A: $tType] :
( ( condit1656338222tinuum @ A )
=> ! [A3: A] :
? [B5: A] :
( ( ord_less @ A @ A3 @ B5 )
| ( ord_less @ A @ B5 @ A3 ) ) ) ).
% ex_gt_or_lt
thf(fact_235_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_236_minf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ~ ( ord_less_eq @ A @ T @ X5 ) ) ) ).
% minf(8)
thf(fact_237_minf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ( ord_less_eq @ A @ X5 @ T ) ) ) ).
% minf(6)
thf(fact_238_i__append__i__take__eq2,axiom,
! [A: $tType,Xs: list @ A,N: nat,F: nat > A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N )
=> ( ( listIn1033672622i_take @ A @ N @ ( listIn521021761append @ A @ Xs @ F ) )
= ( append @ A @ Xs @ ( listIn1033672622i_take @ A @ ( minus_minus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) @ F ) ) ) ) ).
% i_append_i_take_eq2
thf(fact_239_nth__drop,axiom,
! [A: $tType,N: nat,Xs: list @ A,I: nat] :
( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( drop @ A @ N @ Xs ) @ I )
= ( nth @ A @ Xs @ ( plus_plus @ nat @ N @ I ) ) ) ) ).
% nth_drop
thf(fact_240_add__right__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B3: A,A3: A,C: A] :
( ( ( plus_plus @ A @ B3 @ A3 )
= ( plus_plus @ A @ C @ A3 ) )
= ( B3 = C ) ) ) ).
% add_right_cancel
thf(fact_241_add__left__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A3: A,B3: A,C: A] :
( ( ( plus_plus @ A @ A3 @ B3 )
= ( plus_plus @ A @ A3 @ C ) )
= ( B3 = C ) ) ) ).
% add_left_cancel
thf(fact_242_add__le__cancel__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [A3: A,C: A,B3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C ) @ ( plus_plus @ A @ B3 @ C ) )
= ( ord_less_eq @ A @ A3 @ B3 ) ) ) ).
% add_le_cancel_right
thf(fact_243_add__le__cancel__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [C: A,A3: A,B3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A3 ) @ ( plus_plus @ A @ C @ B3 ) )
= ( ord_less_eq @ A @ A3 @ B3 ) ) ) ).
% add_le_cancel_left
thf(fact_244_add__less__cancel__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [C: A,A3: A,B3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C @ A3 ) @ ( plus_plus @ A @ C @ B3 ) )
= ( ord_less @ A @ A3 @ B3 ) ) ) ).
% add_less_cancel_left
thf(fact_245_add__less__cancel__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [A3: A,C: A,B3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A3 @ C ) @ ( plus_plus @ A @ B3 @ C ) )
= ( ord_less @ A @ A3 @ B3 ) ) ) ).
% add_less_cancel_right
thf(fact_246_add__diff__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B3: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B3 ) @ B3 )
= A3 ) ) ).
% add_diff_cancel
thf(fact_247_diff__add__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B3: A] :
( ( plus_plus @ A @ ( minus_minus @ A @ A3 @ B3 ) @ B3 )
= A3 ) ) ).
% diff_add_cancel
thf(fact_248_add__diff__cancel__left,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A )
=> ! [C: A,A3: A,B3: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ C @ A3 ) @ ( plus_plus @ A @ C @ B3 ) )
= ( minus_minus @ A @ A3 @ B3 ) ) ) ).
% add_diff_cancel_left
thf(fact_249_add__diff__cancel__left_H,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A )
=> ! [A3: A,B3: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B3 ) @ A3 )
= B3 ) ) ).
% add_diff_cancel_left'
thf(fact_250_add__diff__cancel__right,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A )
=> ! [A3: A,C: A,B3: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ C ) @ ( plus_plus @ A @ B3 @ C ) )
= ( minus_minus @ A @ A3 @ B3 ) ) ) ).
% add_diff_cancel_right
thf(fact_251_add__diff__cancel__right_H,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A )
=> ! [A3: A,B3: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B3 ) @ B3 )
= A3 ) ) ).
% add_diff_cancel_right'
thf(fact_252_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
= ( ord_less @ nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_253_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
= ( ord_less_eq @ nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_254_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus @ nat @ ( minus_minus @ nat @ I @ J ) @ K )
= ( minus_minus @ nat @ I @ ( plus_plus @ nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_255_drop__drop,axiom,
! [A: $tType,N: nat,M: nat,Xs: list @ A] :
( ( drop @ A @ N @ ( drop @ A @ M @ Xs ) )
= ( drop @ A @ ( plus_plus @ nat @ N @ M ) @ Xs ) ) ).
% drop_drop
% Type constructors (22)
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A7: $tType,A8: $tType] :
( ( preorder @ A8 )
=> ( preorder @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A7: $tType,A8: $tType] :
( ( order @ A8 )
=> ( order @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A7: $tType,A8: $tType] :
( ( ord @ A8 )
=> ( ord @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Groups_Ominus,axiom,
! [A7: $tType,A8: $tType] :
( ( minus @ A8 )
=> ( minus @ ( A7 > A8 ) ) ) ).
thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
condit1037483654norder @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere236663937imp_le @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_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___Nat_Osize,axiom,
size @ nat ).
thf(tcon_HOL_Obool___Orderings_Opreorder_5,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder_6,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_7,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_8,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Groups_Ominus_9,axiom,
minus @ $o ).
thf(tcon_List_Olist___Nat_Osize_10,axiom,
! [A7: $tType] : ( size @ ( list @ A7 ) ) ).
% 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,X3: A,Y2: A] :
( ( if @ A @ $false @ X3 @ Y2 )
= Y2 ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X3: A,Y2: A] :
( ( if @ A @ $true @ X3 @ Y2 )
= X3 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( ( ord_less @ nat @ n @ ( size_size @ ( list @ a ) @ xs ) )
=> ( ( listIn1569887318i_drop @ a @ n @ ( listIn521021761append @ a @ xs @ f ) )
= ( listIn521021761append @ a @ ( drop @ a @ n @ xs ) @ f ) ) )
& ( ~ ( ord_less @ nat @ n @ ( size_size @ ( list @ a ) @ xs ) )
=> ( ( listIn1569887318i_drop @ a @ n @ ( listIn521021761append @ a @ xs @ f ) )
= ( listIn1569887318i_drop @ a @ ( minus_minus @ nat @ n @ ( size_size @ ( list @ a ) @ xs ) ) @ f ) ) ) ) ).
%------------------------------------------------------------------------------