TPTP Problem File: DAT246^1.p
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%------------------------------------------------------------------------------
% File : DAT246^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Infinite streams (sequences/lists) 156
% Version : [Bla16] axioms : Especial.
% English :
% Refs : [BH+14] Blanchette et al. (2014), Truly Modular (Co)datatypes
% : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% Source : [Bla16]
% Names : stream__156.p [Bla16]
% Status : Theorem
% Rating : 0.00 v7.5.0, 0.67 v7.2.0, 1.00 v7.1.0
% Syntax : Number of formulae : 319 ( 68 unt; 44 typ; 0 def)
% Number of atoms : 875 ( 188 equ; 0 cnn)
% Maximal formula atoms : 13 ( 3 avg)
% Number of connectives : 3635 ( 86 ~; 21 |; 39 &;2963 @)
% ( 0 <=>; 526 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 9 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 163 ( 163 >; 0 *; 0 +; 0 <<)
% Number of symbols : 43 ( 42 usr; 3 con; 0-4 aty)
% Number of variables : 991 ( 41 ^; 873 !; 41 ?; 991 :)
% ( 36 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:41:00.835
%------------------------------------------------------------------------------
%----Could-be-implicit typings (5)
thf(ty_t_Stream__Mirabelle__hbrgyiwlrc_Ostream,type,
stream170649215stream: $tType > $tType ).
thf(ty_t_List_Olist,type,
list: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
thf(ty_tf_a,type,
a: $tType ).
%----Explicit typings (39)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ominus,type,
minus:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Ono__bot,type,
no_bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Ono__top,type,
no_top:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Owellorder,type,
wellorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Odense__order,type,
dense_order:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Odense__linorder,type,
dense_linorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
cancel146912293up_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni770627133id_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
condit1656338222tinuum:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
condit1037483654norder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_Hilbert__Choice_OGreatestM,type,
hilbert_GreatestM:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( A > $o ) > A ) ).
thf(sy_c_List_Ocount__list,type,
count_list:
!>[A: $tType] : ( ( list @ A ) > A > nat ) ).
thf(sy_c_List_Olinorder__class_Osorted,type,
linorder_sorted:
!>[A: $tType] : ( ( list @ A ) > $o ) ).
thf(sy_c_List_Olist_Otl,type,
tl:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist__ex,type,
list_ex:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Onth,type,
nth:
!>[A: $tType] : ( ( list @ A ) > nat > A ) ).
thf(sy_c_List_Oremdups__adj,type,
remdups_adj:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Orev,type,
rev:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_OtakeWhile,type,
takeWhile:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
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_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Stream__Mirabelle__hbrgyiwlrc_Oshift,type,
stream1035003186_shift:
!>[A: $tType] : ( ( list @ A ) > ( stream170649215stream @ A ) > ( stream170649215stream @ A ) ) ).
thf(sy_c_Stream__Mirabelle__hbrgyiwlrc_Osnth,type,
stream370371455e_snth:
!>[A: $tType] : ( ( stream170649215stream @ A ) > nat > A ) ).
thf(sy_v_n,type,
n: nat ).
thf(sy_v_s,type,
s: stream170649215stream @ a ).
thf(sy_v_xs,type,
xs: list @ a ).
%----Relevant facts (254)
thf(fact_0_shift__left__inj,axiom,
! [A: $tType,Xs: list @ A,S1: stream170649215stream @ A,S2: stream170649215stream @ A] :
( ( ( stream1035003186_shift @ A @ Xs @ S1 )
= ( stream1035003186_shift @ A @ Xs @ S2 ) )
= ( S1 = S2 ) ) ).
% shift_left_inj
thf(fact_1_shift__snth__less,axiom,
! [A: $tType,P: nat,Xs: list @ A,S: stream170649215stream @ A] :
( ( ord_less @ nat @ P @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( stream370371455e_snth @ A @ ( stream1035003186_shift @ A @ Xs @ S ) @ P )
= ( nth @ A @ Xs @ P ) ) ) ).
% shift_snth_less
thf(fact_2_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_3_Skolem__list__nth,axiom,
! [A: $tType,K: nat,P2: nat > A > $o] :
( ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ K )
=> ( ^ [P3: A > $o] :
? [X: A] : ( P3 @ X )
@ ( P2 @ I2 ) ) ) )
= ( ? [Xs2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= K )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ K )
=> ( P2 @ I2 @ ( nth @ A @ Xs2 @ I2 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_4_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_5_minus__apply,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B @ ( type2 @ B ) )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A2: A > B,B2: A > B,X2: A] : ( minus_minus @ B @ ( A2 @ X2 ) @ ( B2 @ X2 ) ) ) ) ) ).
% minus_apply
thf(fact_6_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_7_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_8_length__induct,axiom,
! [A: $tType,P2: ( 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 ) )
=> ( P2 @ Ys3 ) )
=> ( P2 @ Xs3 ) )
=> ( P2 @ Xs ) ) ).
% length_induct
thf(fact_9_diff__strict__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ 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_10_diff__eq__diff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ 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_11_diff__strict__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ 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_12_diff__strict__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ 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_13_shift__snth__ge,axiom,
! [A: $tType,Xs: list @ A,P: nat,S: stream170649215stream @ A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ P )
=> ( ( stream370371455e_snth @ A @ ( stream1035003186_shift @ A @ Xs @ S ) @ P )
= ( stream370371455e_snth @ A @ S @ ( minus_minus @ nat @ P @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ).
% shift_snth_ge
thf(fact_14_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_15_le__refl,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% le_refl
thf(fact_16_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_17_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_18_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_19_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_20_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ 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_21_diff__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A,C: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C ) @ ( minus_minus @ A @ B3 @ C ) ) ) ) ).
% diff_right_mono
thf(fact_22_diff__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ 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_23_diff__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ 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_24_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I3: nat,J: nat] :
( ! [I: nat,J2: nat] :
( ( ord_less @ nat @ I @ J2 )
=> ( ord_less @ nat @ ( F @ I ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq @ nat @ I3 @ J )
=> ( ord_less_eq @ nat @ ( F @ I3 ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_25_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_26_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_27_le__eq__less__or__eq,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M2: nat,N2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
| ( M2 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_28_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_29_nat__less__le,axiom,
( ( ord_less @ nat )
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
& ( M2 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_30_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_31_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_32_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_33_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_34_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_35_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_36_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_37_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_38_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_39_diff__right__commute,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A @ ( type2 @ 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_40_diff__eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ 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_41_fun__diff__def,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B @ ( type2 @ 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_42_infinite__descent__measure,axiom,
! [A: $tType,P2: A > $o,V: A > nat,X3: A] :
( ! [X4: A] :
( ~ ( P2 @ X4 )
=> ? [Y2: A] :
( ( ord_less @ nat @ ( V @ Y2 ) @ ( V @ X4 ) )
& ~ ( P2 @ Y2 ) ) )
=> ( P2 @ X3 ) ) ).
% infinite_descent_measure
thf(fact_43_measure__induct__rule,axiom,
! [A: $tType,F: A > nat,P2: A > $o,A3: A] :
( ! [X4: A] :
( ! [Y2: A] :
( ( ord_less @ nat @ ( F @ Y2 ) @ ( F @ X4 ) )
=> ( P2 @ Y2 ) )
=> ( P2 @ X4 ) )
=> ( P2 @ A3 ) ) ).
% measure_induct_rule
thf(fact_44_linorder__neqE__nat,axiom,
! [X3: nat,Y3: nat] :
( ( X3 != Y3 )
=> ( ~ ( ord_less @ nat @ X3 @ Y3 )
=> ( ord_less @ nat @ Y3 @ X3 ) ) ) ).
% linorder_neqE_nat
thf(fact_45_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X4: A] :
( ( F @ X4 )
= ( G @ X4 ) )
=> ( F = G ) ) ).
% ext
thf(fact_46_infinite__descent,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P2 @ N3 )
=> ? [M3: nat] :
( ( ord_less @ nat @ M3 @ N3 )
& ~ ( P2 @ M3 ) ) )
=> ( P2 @ N ) ) ).
% infinite_descent
thf(fact_47_nat__less__induct,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less @ nat @ M3 @ N3 )
=> ( P2 @ M3 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ N ) ) ).
% nat_less_induct
thf(fact_48_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_49_measure__induct,axiom,
! [A: $tType,F: A > nat,P2: A > $o,A3: A] :
( ! [X4: A] :
( ! [Y2: A] :
( ( ord_less @ nat @ ( F @ Y2 ) @ ( F @ X4 ) )
=> ( P2 @ Y2 ) )
=> ( P2 @ X4 ) )
=> ( P2 @ A3 ) ) ).
% measure_induct
thf(fact_50_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less @ nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_51_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_52_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_not_refl
thf(fact_53_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_54_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_55_Ex__list__of__length,axiom,
! [A: $tType,N: nat] :
? [Xs3: list @ A] :
( ( size_size @ ( list @ A ) @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_56_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_57_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A] : ( ord_less_eq @ A @ X3 @ X3 ) ) ).
% order_refl
thf(fact_58_ex__has__greatest__nat,axiom,
! [A: $tType,P2: A > $o,K: A,M: A > nat,B3: nat] :
( ( P2 @ K )
=> ( ! [Y4: A] :
( ( P2 @ Y4 )
=> ( ord_less @ nat @ ( M @ Y4 ) @ B3 ) )
=> ? [X4: A] :
( ( P2 @ X4 )
& ! [Y2: A] :
( ( P2 @ Y2 )
=> ( ord_less_eq @ nat @ ( M @ Y2 ) @ ( M @ X4 ) ) ) ) ) ) ).
% ex_has_greatest_nat
thf(fact_59_complete__interval,axiom,
! [A: $tType] :
( ( condit1037483654norder @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A,P2: A > $o] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( P2 @ A3 )
=> ( ~ ( P2 @ B3 )
=> ? [C2: A] :
( ( ord_less_eq @ A @ A3 @ C2 )
& ( ord_less_eq @ A @ C2 @ B3 )
& ! [X5: A] :
( ( ( ord_less_eq @ A @ A3 @ X5 )
& ( ord_less @ A @ X5 @ C2 ) )
=> ( P2 @ X5 ) )
& ! [D2: A] :
( ! [X4: A] :
( ( ( ord_less_eq @ A @ A3 @ X4 )
& ( ord_less @ A @ X4 @ D2 ) )
=> ( P2 @ X4 ) )
=> ( ord_less_eq @ A @ D2 @ C2 ) ) ) ) ) ) ) ).
% complete_interval
thf(fact_60_order_Onot__eq__order__implies__strict,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ 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_61_dual__order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ( ord_less_eq @ A @ B3 @ A3 ) ) ) ).
% dual_order.strict_implies_order
thf(fact_62_dual__order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [B4: A,A4: A] :
( ( ord_less_eq @ A @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_63_dual__order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [B4: A,A4: A] :
( ( ord_less @ A @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_64_order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ord_less_eq @ A @ A3 @ B3 ) ) ) ).
% order.strict_implies_order
thf(fact_65_dense__le__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y3: A,Z2: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ! [W: A] :
( ( ord_less @ A @ X3 @ W )
=> ( ( ord_less @ A @ W @ Y3 )
=> ( ord_less_eq @ A @ W @ Z2 ) ) )
=> ( ord_less_eq @ A @ Y3 @ Z2 ) ) ) ) ).
% dense_le_bounded
thf(fact_66_dense__ge__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Z2: A,X3: A,Y3: A] :
( ( ord_less @ A @ Z2 @ X3 )
=> ( ! [W: A] :
( ( ord_less @ A @ Z2 @ W )
=> ( ( ord_less @ A @ W @ X3 )
=> ( ord_less_eq @ A @ Y3 @ W ) ) )
=> ( ord_less_eq @ A @ Y3 @ Z2 ) ) ) ) ).
% dense_ge_bounded
thf(fact_67_dual__order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ 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_68_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ) ).
% dual_order.antisym
thf(fact_69_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A3: A,C: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
=> ( ( ord_less_eq @ A @ C @ B3 )
=> ( ord_less_eq @ A @ C @ A3 ) ) ) ) ).
% dual_order.trans
thf(fact_70_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P2: A > A > $o,A3: A,B3: A] :
( ! [A5: A,B5: A] :
( ( ord_less_eq @ A @ A5 @ B5 )
=> ( P2 @ A5 @ B5 ) )
=> ( ! [A5: A,B5: A] :
( ( P2 @ B5 @ A5 )
=> ( P2 @ A5 @ B5 ) )
=> ( P2 @ A3 @ B3 ) ) ) ) ).
% linorder_wlog
thf(fact_71_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% dual_order.refl
thf(fact_72_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y3: A,Z2: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ( ord_less_eq @ A @ Y3 @ Z2 )
=> ( ord_less_eq @ A @ X3 @ Z2 ) ) ) ) ).
% order_trans
thf(fact_73_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ) ).
% order_class.order.antisym
thf(fact_74_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A,C: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% ord_le_eq_trans
thf(fact_75_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A,C: A] :
( ( A3 = B3 )
=> ( ( ord_less_eq @ A @ B3 @ C )
=> ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% ord_eq_le_trans
thf(fact_76_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [Y3: A,X3: A] :
( ( ord_less_eq @ A @ Y3 @ X3 )
=> ( ( ord_less_eq @ A @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ) ).
% antisym_conv
thf(fact_77_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y3: A,Z2: A] :
( ( ( ord_less_eq @ A @ X3 @ Y3 )
=> ~ ( ord_less_eq @ A @ Y3 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ Y3 @ X3 )
=> ~ ( ord_less_eq @ A @ X3 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ X3 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ Y3 ) )
=> ( ( ( ord_less_eq @ A @ Z2 @ Y3 )
=> ~ ( ord_less_eq @ A @ Y3 @ X3 ) )
=> ( ( ( ord_less_eq @ A @ Y3 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ X3 ) )
=> ~ ( ( ord_less_eq @ A @ Z2 @ X3 )
=> ~ ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_78_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A,C: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ C )
=> ( ord_less_eq @ A @ A3 @ C ) ) ) ) ).
% order.trans
thf(fact_79_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y3: A] :
( ~ ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% le_cases
thf(fact_80_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y3: A] :
( ( X3 = Y3 )
=> ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ).
% eq_refl
thf(fact_81_linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
| ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% linear
thf(fact_82_antisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ( ord_less_eq @ A @ Y3 @ X3 )
=> ( X3 = Y3 ) ) ) ) ).
% antisym
thf(fact_83_eq__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ 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_84_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A3: A,B3: A,F: A > B,C: B] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X4: A,Y4: A] :
( ( ord_less_eq @ A @ X4 @ Y4 )
=> ( ord_less_eq @ B @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ B @ ( F @ A3 ) @ C ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_85_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A3: A,F: B > A,B3: B,C: B] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C )
=> ( ! [X4: B,Y4: B] :
( ( ord_less_eq @ B @ X4 @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_86_order__subst2,axiom,
! [A: $tType,C3: $tType] :
( ( ( order @ C3 @ ( type2 @ C3 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A3: A,B3: A,F: A > C3,C: C3] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ C3 @ ( F @ B3 ) @ C )
=> ( ! [X4: A,Y4: A] :
( ( ord_less_eq @ A @ X4 @ Y4 )
=> ( ord_less_eq @ C3 @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ C3 @ ( F @ A3 ) @ C ) ) ) ) ) ).
% order_subst2
thf(fact_87_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A3: A,F: B > A,B3: B,C: B] :
( ( ord_less_eq @ A @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C )
=> ( ! [X4: B,Y4: B] :
( ( ord_less_eq @ B @ X4 @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% order_subst1
thf(fact_88_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ 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_89_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B] :
( ! [X4: A] : ( ord_less_eq @ B @ ( F @ X4 ) @ ( G @ X4 ) )
=> ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% le_funI
thf(fact_90_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ 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_91_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ 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_92_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A3: A,F: B > A,B3: B,C: B] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less @ B @ B3 @ C )
=> ( ! [X4: B,Y4: B] :
( ( ord_less @ B @ X4 @ Y4 )
=> ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_93_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A3: A,B3: A,F: A > B,C: B] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X4: A,Y4: A] :
( ( ord_less @ A @ X4 @ Y4 )
=> ( ord_less @ B @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ B @ ( F @ A3 ) @ C ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_94_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A3: A,F: B > A,B3: B,C: B] :
( ( ord_less @ A @ A3 @ ( F @ B3 ) )
=> ( ( ord_less @ B @ B3 @ C )
=> ( ! [X4: B,Y4: B] :
( ( ord_less @ B @ X4 @ Y4 )
=> ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_95_order__less__subst2,axiom,
! [A: $tType,C3: $tType] :
( ( ( order @ C3 @ ( type2 @ C3 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A3: A,B3: A,F: A > C3,C: C3] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less @ C3 @ ( F @ B3 ) @ C )
=> ( ! [X4: A,Y4: A] :
( ( ord_less @ A @ X4 @ Y4 )
=> ( ord_less @ C3 @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ C3 @ ( F @ A3 ) @ C ) ) ) ) ) ).
% order_less_subst2
thf(fact_96_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A @ ( type2 @ A ) )
=> ! [X3: A] :
? [Y4: A] : ( ord_less @ A @ Y4 @ X3 ) ) ).
% lt_ex
thf(fact_97_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A @ ( type2 @ A ) )
=> ! [X3: A] :
? [X1: A] : ( ord_less @ A @ X3 @ X1 ) ) ).
% gt_ex
thf(fact_98_neqE,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y3: A] :
( ( X3 != Y3 )
=> ( ~ ( ord_less @ A @ X3 @ Y3 )
=> ( ord_less @ A @ Y3 @ X3 ) ) ) ) ).
% neqE
thf(fact_99_neq__iff,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y3: A] :
( ( X3 != Y3 )
= ( ( ord_less @ A @ X3 @ Y3 )
| ( ord_less @ A @ Y3 @ X3 ) ) ) ) ).
% neq_iff
thf(fact_100_order_Oasym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ~ ( ord_less @ A @ B3 @ A3 ) ) ) ).
% order.asym
thf(fact_101_dense,axiom,
! [A: $tType] :
( ( dense_order @ A @ ( type2 @ A ) )
=> ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ? [Z3: A] :
( ( ord_less @ A @ X3 @ Z3 )
& ( ord_less @ A @ Z3 @ Y3 ) ) ) ) ).
% dense
thf(fact_102_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( X3 != Y3 ) ) ) ).
% less_imp_neq
thf(fact_103_less__asym,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ~ ( ord_less @ A @ Y3 @ X3 ) ) ) ).
% less_asym
thf(fact_104_less__asym_H,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ~ ( ord_less @ A @ B3 @ A3 ) ) ) ).
% less_asym'
thf(fact_105_less__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y3: A,Z2: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ( ord_less @ A @ Y3 @ Z2 )
=> ( ord_less @ A @ X3 @ Z2 ) ) ) ) ).
% less_trans
thf(fact_106_less__linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
| ( X3 = Y3 )
| ( ord_less @ A @ Y3 @ X3 ) ) ) ).
% less_linear
thf(fact_107_less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A] :
~ ( ord_less @ A @ X3 @ X3 ) ) ).
% less_irrefl
thf(fact_108_ord__eq__less__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ 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_109_ord__less__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ 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_110_dual__order_Oasym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ~ ( ord_less @ A @ A3 @ B3 ) ) ) ).
% dual_order.asym
thf(fact_111_less__imp__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( X3 != Y3 ) ) ) ).
% less_imp_not_eq
thf(fact_112_less__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ~ ( ord_less @ A @ Y3 @ X3 ) ) ) ).
% less_not_sym
thf(fact_113_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A @ ( type2 @ A ) )
=> ! [P2: A > $o,A3: A] :
( ! [X4: A] :
( ! [Y2: A] :
( ( ord_less @ A @ Y2 @ X4 )
=> ( P2 @ Y2 ) )
=> ( P2 @ X4 ) )
=> ( P2 @ A3 ) ) ) ).
% less_induct
thf(fact_114_antisym__conv3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y3: A,X3: A] :
( ~ ( ord_less @ A @ Y3 @ X3 )
=> ( ( ~ ( ord_less @ A @ X3 @ Y3 ) )
= ( X3 = Y3 ) ) ) ) ).
% antisym_conv3
thf(fact_115_less__imp__not__eq2,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( Y3 != X3 ) ) ) ).
% less_imp_not_eq2
thf(fact_116_less__imp__triv,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y3: A,P2: $o] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ( ord_less @ A @ Y3 @ X3 )
=> P2 ) ) ) ).
% less_imp_triv
thf(fact_117_linorder__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y3: A] :
( ~ ( ord_less @ A @ X3 @ Y3 )
=> ( ( X3 != Y3 )
=> ( ord_less @ A @ Y3 @ X3 ) ) ) ) ).
% linorder_cases
thf(fact_118_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A3: A] :
~ ( ord_less @ A @ A3 @ A3 ) ) ).
% dual_order.irrefl
thf(fact_119_order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ 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_120_less__imp__not__less,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ~ ( ord_less @ A @ Y3 @ X3 ) ) ) ).
% less_imp_not_less
thf(fact_121_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ 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_122_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y3: A] :
( ( ~ ( ord_less @ A @ X3 @ Y3 ) )
= ( ( ord_less @ A @ Y3 @ X3 )
| ( X3 = Y3 ) ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_123_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
=> ( A3 != B3 ) ) ) ).
% order.strict_implies_not_eq
thf(fact_124_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A3: A] :
( ( ord_less @ A @ B3 @ A3 )
=> ( A3 != B3 ) ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_125_ex__gt__or__lt,axiom,
! [A: $tType] :
( ( condit1656338222tinuum @ A @ ( type2 @ A ) )
=> ! [A3: A] :
? [B5: A] :
( ( ord_less @ A @ A3 @ B5 )
| ( ord_less @ A @ B5 @ A3 ) ) ) ).
% ex_gt_or_lt
thf(fact_126_ex__has__least__nat,axiom,
! [A: $tType,P2: A > $o,K: A,M: A > nat] :
( ( P2 @ K )
=> ? [X4: A] :
( ( P2 @ X4 )
& ! [Y2: A] :
( ( P2 @ Y2 )
=> ( ord_less_eq @ nat @ ( M @ X4 ) @ ( M @ Y2 ) ) ) ) ) ).
% ex_has_least_nat
thf(fact_127_leD,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y3: A,X3: A] :
( ( ord_less_eq @ A @ Y3 @ X3 )
=> ~ ( ord_less @ A @ X3 @ Y3 ) ) ) ).
% leD
thf(fact_128_leI,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y3: A] :
( ~ ( ord_less @ A @ X3 @ Y3 )
=> ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% leI
thf(fact_129_le__less,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [X2: A,Y5: A] :
( ( ord_less @ A @ X2 @ Y5 )
| ( X2 = Y5 ) ) ) ) ) ).
% le_less
thf(fact_130_less__le,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [X2: A,Y5: A] :
( ( ord_less_eq @ A @ X2 @ Y5 )
& ( X2 != Y5 ) ) ) ) ) ).
% less_le
thf(fact_131_order__le__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A3: A,F: B > A,B3: B,C: B] :
( ( ord_less_eq @ A @ A3 @ ( F @ B3 ) )
=> ( ( ord_less @ B @ B3 @ C )
=> ( ! [X4: B,Y4: B] :
( ( ord_less @ B @ X4 @ Y4 )
=> ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_132_order__le__less__subst2,axiom,
! [A: $tType,C3: $tType] :
( ( ( order @ C3 @ ( type2 @ C3 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A3: A,B3: A,F: A > C3,C: C3] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( ord_less @ C3 @ ( F @ B3 ) @ C )
=> ( ! [X4: A,Y4: A] :
( ( ord_less_eq @ A @ X4 @ Y4 )
=> ( ord_less_eq @ C3 @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ C3 @ ( F @ A3 ) @ C ) ) ) ) ) ).
% order_le_less_subst2
thf(fact_133_order__less__le__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A3: A,F: B > A,B3: B,C: B] :
( ( ord_less @ A @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C )
=> ( ! [X4: B,Y4: B] :
( ( ord_less_eq @ B @ X4 @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ A @ A3 @ ( F @ C ) ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_134_order__less__le__subst2,axiom,
! [A: $tType,C3: $tType] :
( ( ( order @ C3 @ ( type2 @ C3 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A3: A,B3: A,F: A > C3,C: C3] :
( ( ord_less @ A @ A3 @ B3 )
=> ( ( ord_less_eq @ C3 @ ( F @ B3 ) @ C )
=> ( ! [X4: A,Y4: A] :
( ( ord_less @ A @ X4 @ Y4 )
=> ( ord_less @ C3 @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ C3 @ ( F @ A3 ) @ C ) ) ) ) ) ).
% order_less_le_subst2
thf(fact_135_not__le,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y3: A] :
( ( ~ ( ord_less_eq @ A @ X3 @ Y3 ) )
= ( ord_less @ A @ Y3 @ X3 ) ) ) ).
% not_le
thf(fact_136_not__less,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y3: A] :
( ( ~ ( ord_less @ A @ X3 @ Y3 ) )
= ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ).
% not_less
thf(fact_137_le__neq__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( ( A3 != B3 )
=> ( ord_less @ A @ A3 @ B3 ) ) ) ) ).
% le_neq_trans
thf(fact_138_less__imp__le,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ord_less_eq @ A @ X3 @ Y3 ) ) ) ).
% less_imp_le
thf(fact_139_antisym__conv1,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y3: A] :
( ~ ( ord_less @ A @ X3 @ Y3 )
=> ( ( ord_less_eq @ A @ X3 @ Y3 )
= ( X3 = Y3 ) ) ) ) ).
% antisym_conv1
thf(fact_140_antisym__conv2,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ( ~ ( ord_less @ A @ X3 @ Y3 ) )
= ( X3 = Y3 ) ) ) ) ).
% antisym_conv2
thf(fact_141_le__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y3: A,Z2: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ( ord_less @ A @ Y3 @ Z2 )
=> ( ord_less @ A @ X3 @ Z2 ) ) ) ) ).
% le_less_trans
thf(fact_142_less__le__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y3: A,Z2: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ( ord_less_eq @ A @ Y3 @ Z2 )
=> ( ord_less @ A @ X3 @ Z2 ) ) ) ) ).
% less_le_trans
thf(fact_143_dense__ge,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Z2: A,Y3: A] :
( ! [X4: A] :
( ( ord_less @ A @ Z2 @ X4 )
=> ( ord_less_eq @ A @ Y3 @ X4 ) )
=> ( ord_less_eq @ A @ Y3 @ Z2 ) ) ) ).
% dense_ge
thf(fact_144_dense__le,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Y3: A,Z2: A] :
( ! [X4: A] :
( ( ord_less @ A @ X4 @ Y3 )
=> ( ord_less_eq @ A @ X4 @ Z2 ) )
=> ( ord_less_eq @ A @ Y3 @ Z2 ) ) ) ).
% dense_le
thf(fact_145_le__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
| ( ord_less @ A @ Y3 @ X3 ) ) ) ).
% le_less_linear
thf(fact_146_le__imp__less__or__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ( ord_less @ A @ X3 @ Y3 )
| ( X3 = Y3 ) ) ) ) ).
% le_imp_less_or_eq
thf(fact_147_less__le__not__le,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ 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_148_not__le__imp__less,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y3: A,X3: A] :
( ~ ( ord_less_eq @ A @ Y3 @ X3 )
=> ( ord_less @ A @ X3 @ Y3 ) ) ) ).
% not_le_imp_less
thf(fact_149_order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ 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_150_order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ 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_151_order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [A4: A,B4: A] :
( ( ord_less @ A @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ) ).
% order.order_iff_strict
thf(fact_152_order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [A4: A,B4: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ) ).
% order.strict_iff_order
thf(fact_153_dual__order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ 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_154_minf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ~ ( ord_less_eq @ A @ T @ X5 ) ) ) ).
% minf(8)
thf(fact_155_minf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ( ord_less_eq @ A @ X5 @ T ) ) ) ).
% minf(6)
thf(fact_156_pinf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ( ord_less_eq @ A @ T @ X5 ) ) ) ).
% pinf(8)
thf(fact_157_pinf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ~ ( ord_less_eq @ A @ X5 @ T ) ) ) ).
% pinf(6)
thf(fact_158_sorted__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Xs: list @ A,I3: nat,J: nat] :
( ( linorder_sorted @ A @ Xs )
=> ( ( ord_less_eq @ nat @ I3 @ J )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ I3 ) @ ( nth @ A @ Xs @ J ) ) ) ) ) ) ).
% sorted_nth_mono
thf(fact_159_sorted__nth__monoI,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Xs: list @ A] :
( ! [I: nat,J2: nat] :
( ( ord_less_eq @ nat @ I @ J2 )
=> ( ( ord_less @ nat @ J2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ I ) @ ( nth @ A @ Xs @ J2 ) ) ) )
=> ( linorder_sorted @ A @ Xs ) ) ) ).
% sorted_nth_monoI
thf(fact_160_sorted__equals__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ( ( linorder_sorted @ A )
= ( ^ [Xs2: list @ A] :
! [J3: nat] :
( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ! [I2: nat] :
( ( ord_less_eq @ nat @ I2 @ J3 )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs2 @ I2 ) @ ( nth @ A @ Xs2 @ J3 ) ) ) ) ) ) ) ).
% sorted_equals_nth_mono
thf(fact_161_less__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ 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_162_pinf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P2: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ( ( P2 @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ( ( ( P2 @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ) ).
% pinf(1)
thf(fact_163_pinf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P2: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ( ( P2 @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ( ( ( P2 @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ) ).
% pinf(2)
thf(fact_164_pinf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ( X5 != T ) ) ) ).
% pinf(3)
thf(fact_165_pinf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ( X5 != T ) ) ) ).
% pinf(4)
thf(fact_166_pinf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ~ ( ord_less @ A @ X5 @ T ) ) ) ).
% pinf(5)
thf(fact_167_pinf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ( ord_less @ A @ T @ X5 ) ) ) ).
% pinf(7)
thf(fact_168_pinf_I11_J,axiom,
! [C3: $tType,D3: $tType] :
( ( ord @ C3 @ ( type2 @ C3 ) )
=> ! [F3: D3] :
? [Z3: C3] :
! [X5: C3] :
( ( ord_less @ C3 @ Z3 @ X5 )
=> ( F3 = F3 ) ) ) ).
% pinf(11)
thf(fact_169_minf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P2: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ( ( P2 @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ( ( ( P2 @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ) ).
% minf(1)
thf(fact_170_minf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P2: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ( ( P2 @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ( ( ( P2 @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ) ).
% minf(2)
thf(fact_171_minf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ( X5 != T ) ) ) ).
% minf(3)
thf(fact_172_minf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ( X5 != T ) ) ) ).
% minf(4)
thf(fact_173_minf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ( ord_less @ A @ X5 @ T ) ) ) ).
% minf(5)
thf(fact_174_minf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ~ ( ord_less @ A @ T @ X5 ) ) ) ).
% minf(7)
thf(fact_175_minf_I11_J,axiom,
! [C3: $tType,D3: $tType] :
( ( ord @ C3 @ ( type2 @ C3 ) )
=> ! [F3: D3] :
? [Z3: C3] :
! [X5: C3] :
( ( ord_less @ C3 @ X5 @ Z3 )
=> ( F3 = F3 ) ) ) ).
% minf(11)
thf(fact_176_sorted__rev__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Xs: list @ A,I3: nat,J: nat] :
( ( linorder_sorted @ A @ ( rev @ A @ Xs ) )
=> ( ( ord_less_eq @ nat @ I3 @ J )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ J ) @ ( nth @ A @ Xs @ I3 ) ) ) ) ) ) ).
% sorted_rev_nth_mono
thf(fact_177_list__ex__length,axiom,
! [A: $tType] :
( ( list_ex @ A )
= ( ^ [P5: A > $o,Xs2: list @ A] :
? [N2: nat] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
& ( P5 @ ( nth @ A @ Xs2 @ N2 ) ) ) ) ) ).
% list_ex_length
thf(fact_178_length__takeWhile__less__P__nth,axiom,
! [A: $tType,J: nat,P2: A > $o,Xs: list @ A] :
( ! [I: nat] :
( ( ord_less @ nat @ I @ J )
=> ( P2 @ ( nth @ A @ Xs @ I ) ) )
=> ( ( ord_less_eq @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ nat @ J @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P2 @ Xs ) ) ) ) ) ).
% length_takeWhile_less_P_nth
thf(fact_179_rev__rev__ident,axiom,
! [A: $tType,Xs: list @ A] :
( ( rev @ A @ ( rev @ A @ Xs ) )
= Xs ) ).
% rev_rev_ident
thf(fact_180_rev__is__rev__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( rev @ A @ Xs )
= ( rev @ A @ Ys ) )
= ( Xs = Ys ) ) ).
% rev_is_rev_conv
thf(fact_181_takeWhile__idem,axiom,
! [A: $tType,P2: A > $o,Xs: list @ A] :
( ( takeWhile @ A @ P2 @ ( takeWhile @ A @ P2 @ Xs ) )
= ( takeWhile @ A @ P2 @ Xs ) ) ).
% takeWhile_idem
thf(fact_182_length__rev,axiom,
! [A: $tType,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( rev @ A @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_rev
thf(fact_183_list__ex__rev,axiom,
! [A: $tType,P2: A > $o,Xs: list @ A] :
( ( list_ex @ A @ P2 @ ( rev @ A @ Xs ) )
= ( list_ex @ A @ P2 @ Xs ) ) ).
% list_ex_rev
thf(fact_184_sorted__takeWhile,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Xs: list @ A,P2: A > $o] :
( ( linorder_sorted @ A @ Xs )
=> ( linorder_sorted @ A @ ( takeWhile @ A @ P2 @ Xs ) ) ) ) ).
% sorted_takeWhile
thf(fact_185_rev__swap,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( rev @ A @ Xs )
= Ys )
= ( Xs
= ( rev @ A @ Ys ) ) ) ).
% rev_swap
thf(fact_186_length__takeWhile__le,axiom,
! [A: $tType,P2: A > $o,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P2 @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_takeWhile_le
thf(fact_187_takeWhile__nth,axiom,
! [A: $tType,J: nat,P2: A > $o,Xs: list @ A] :
( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P2 @ Xs ) ) )
=> ( ( nth @ A @ ( takeWhile @ A @ P2 @ Xs ) @ J )
= ( nth @ A @ Xs @ J ) ) ) ).
% takeWhile_nth
thf(fact_188_nth__length__takeWhile,axiom,
! [A: $tType,P2: A > $o,Xs: list @ A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P2 @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) )
=> ~ ( P2 @ ( nth @ A @ Xs @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P2 @ Xs ) ) ) ) ) ).
% nth_length_takeWhile
thf(fact_189_rev__nth,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( rev @ A @ Xs ) @ N )
= ( nth @ A @ Xs @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( suc @ N ) ) ) ) ) ).
% rev_nth
thf(fact_190_count__le__length,axiom,
! [A: $tType,Xs: list @ A,X3: A] : ( ord_less_eq @ nat @ ( count_list @ A @ Xs @ X3 ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% count_le_length
thf(fact_191_GreatestM__nat__lemma,axiom,
! [A: $tType,P2: A > $o,K: A,M: A > nat,B3: nat] :
( ( P2 @ K )
=> ( ! [Y4: A] :
( ( P2 @ Y4 )
=> ( ord_less @ nat @ ( M @ Y4 ) @ B3 ) )
=> ( ( P2 @ ( hilbert_GreatestM @ A @ nat @ M @ P2 ) )
& ! [Y2: A] :
( ( P2 @ Y2 )
=> ( ord_less_eq @ nat @ ( M @ Y2 ) @ ( M @ ( hilbert_GreatestM @ A @ nat @ M @ P2 ) ) ) ) ) ) ) ).
% GreatestM_nat_lemma
thf(fact_192_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_193_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_194_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less @ nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_195_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_196_lessI,axiom,
! [N: nat] : ( ord_less @ nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_197_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq @ nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_198_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus @ nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus @ nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_199_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus @ nat @ ( minus_minus @ nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus @ nat @ ( minus_minus @ nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_200_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less @ nat @ N @ M )
=> ( ( ord_less @ nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_201_strict__inc__induct,axiom,
! [I3: nat,J: nat,P2: nat > $o] :
( ( ord_less @ nat @ I3 @ J )
=> ( ! [I: nat] :
( ( J
= ( suc @ I ) )
=> ( P2 @ I ) )
=> ( ! [I: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( P2 @ ( suc @ I ) )
=> ( P2 @ I ) ) )
=> ( P2 @ I3 ) ) ) ) ).
% strict_inc_induct
thf(fact_202_less__Suc__induct,axiom,
! [I3: nat,J: nat,P2: nat > nat > $o] :
( ( ord_less @ nat @ I3 @ J )
=> ( ! [I: nat] : ( P2 @ I @ ( suc @ I ) )
=> ( ! [I: nat,J2: nat,K2: nat] :
( ( ord_less @ nat @ I @ J2 )
=> ( ( ord_less @ nat @ J2 @ K2 )
=> ( ( P2 @ I @ J2 )
=> ( ( P2 @ J2 @ K2 )
=> ( P2 @ I @ K2 ) ) ) ) )
=> ( P2 @ I3 @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_203_less__trans__Suc,axiom,
! [I3: nat,J: nat,K: nat] :
( ( ord_less @ nat @ I3 @ J )
=> ( ( ord_less @ nat @ J @ K )
=> ( ord_less @ nat @ ( suc @ I3 ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_204_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less @ nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_205_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less @ nat @ N @ M )
=> ( ( ord_less @ nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_206_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( suc @ N ) @ M )
= ( ? [M4: nat] :
( ( M
= ( suc @ M4 ) )
& ( ord_less @ nat @ N @ M4 ) ) ) ) ).
% Suc_less_eq2
thf(fact_207_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less @ nat @ M @ N ) )
= ( ord_less @ nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_208_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ ( suc @ N ) )
= ( ( ord_less @ nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_209_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less @ nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_210_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less @ nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_211_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less @ nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_212_Suc__lessE,axiom,
! [I3: nat,K: nat] :
( ( ord_less @ nat @ ( suc @ I3 ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less @ nat @ I3 @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_213_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ M ) @ N )
=> ( ord_less @ nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_214_lessE,axiom,
! [I3: nat,K: nat] :
( ( ord_less @ nat @ I3 @ K )
=> ( ( K
!= ( suc @ I3 ) )
=> ~ ! [J2: nat] :
( ( ord_less @ nat @ I3 @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% lessE
thf(fact_215_lift__Suc__mono__less,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [F: nat > A,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less @ A @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less @ nat @ N @ N4 )
=> ( ord_less @ A @ ( F @ N ) @ ( F @ N4 ) ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_216_lift__Suc__mono__less__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [F: nat > A,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less @ A @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less @ A @ ( F @ N ) @ ( F @ M ) )
= ( ord_less @ nat @ N @ M ) ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_217_lift__Suc__antimono__le,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [F: nat > A,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq @ nat @ N @ N4 )
=> ( ord_less_eq @ A @ ( F @ N4 ) @ ( F @ N ) ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_218_lift__Suc__mono__le,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [F: nat > A,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq @ nat @ N @ N4 )
=> ( ord_less_eq @ A @ ( F @ N ) @ ( F @ N4 ) ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_219_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less @ nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_220_less__eq__Suc__le,axiom,
( ( ord_less @ nat )
= ( ^ [N2: nat] : ( ord_less_eq @ nat @ ( suc @ N2 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_221_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ ( suc @ N ) )
= ( ord_less_eq @ nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_222_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less @ nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_223_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
=> ( ord_less @ nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_224_inc__induct,axiom,
! [I3: nat,J: nat,P2: nat > $o] :
( ( ord_less_eq @ nat @ I3 @ J )
=> ( ( P2 @ J )
=> ( ! [N3: nat] :
( ( ord_less_eq @ nat @ I3 @ N3 )
=> ( ( ord_less @ nat @ N3 @ J )
=> ( ( P2 @ ( suc @ N3 ) )
=> ( P2 @ N3 ) ) ) )
=> ( P2 @ I3 ) ) ) ) ).
% inc_induct
thf(fact_225_dec__induct,axiom,
! [I3: nat,J: nat,P2: nat > $o] :
( ( ord_less_eq @ nat @ I3 @ J )
=> ( ( P2 @ I3 )
=> ( ! [N3: nat] :
( ( ord_less_eq @ nat @ I3 @ N3 )
=> ( ( ord_less @ nat @ N3 @ J )
=> ( ( P2 @ N3 )
=> ( P2 @ ( suc @ N3 ) ) ) ) )
=> ( P2 @ J ) ) ) ) ).
% dec_induct
thf(fact_226_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
= ( ord_less @ nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_227_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less_eq @ nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_228_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_229_Suc__inject,axiom,
! [X3: nat,Y3: nat] :
( ( ( suc @ X3 )
= ( suc @ Y3 ) )
=> ( X3 = Y3 ) ) ).
% Suc_inject
thf(fact_230_zero__induct__lemma,axiom,
! [P2: nat > $o,K: nat,I3: nat] :
( ( P2 @ K )
=> ( ! [N3: nat] :
( ( P2 @ ( suc @ N3 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ ( minus_minus @ nat @ K @ I3 ) ) ) ) ).
% zero_induct_lemma
thf(fact_231_full__nat__induct,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_eq @ nat @ ( suc @ M3 ) @ N3 )
=> ( P2 @ M3 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ N ) ) ).
% full_nat_induct
thf(fact_232_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq @ nat @ M @ N ) )
= ( ord_less_eq @ nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_233_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq @ nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_234_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq @ nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_235_Suc__le__D,axiom,
! [N: nat,M5: nat] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ M5 )
=> ? [M6: nat] :
( M5
= ( suc @ M6 ) ) ) ).
% Suc_le_D
thf(fact_236_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_237_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq @ nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_238_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% Suc_leD
thf(fact_239_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less @ nat @ ( minus_minus @ nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_240_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ N @ M )
=> ( ( suc @ ( minus_minus @ nat @ M @ ( suc @ N ) ) )
= ( minus_minus @ nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_241_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( minus_minus @ nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus @ nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_242_GreatestM__natI,axiom,
! [A: $tType,P2: A > $o,K: A,M: A > nat,B3: nat] :
( ( P2 @ K )
=> ( ! [Y4: A] :
( ( P2 @ Y4 )
=> ( ord_less @ nat @ ( M @ Y4 ) @ B3 ) )
=> ( P2 @ ( hilbert_GreatestM @ A @ nat @ M @ P2 ) ) ) ) ).
% GreatestM_natI
thf(fact_243_GreatestMI2,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [P2: A > $o,X3: A,M: A > B,Q: A > $o] :
( ( P2 @ X3 )
=> ( ! [Y4: A] :
( ( P2 @ Y4 )
=> ( ord_less_eq @ B @ ( M @ Y4 ) @ ( M @ X3 ) ) )
=> ( ! [X4: A] :
( ( P2 @ X4 )
=> ( ! [Y2: A] :
( ( P2 @ Y2 )
=> ( ord_less_eq @ B @ ( M @ Y2 ) @ ( M @ X4 ) ) )
=> ( Q @ X4 ) ) )
=> ( Q @ ( hilbert_GreatestM @ A @ B @ M @ P2 ) ) ) ) ) ) ).
% GreatestMI2
thf(fact_244_GreatestM__nat__le,axiom,
! [A: $tType,P2: A > $o,X3: A,M: A > nat,B3: nat] :
( ( P2 @ X3 )
=> ( ! [Y4: A] :
( ( P2 @ Y4 )
=> ( ord_less @ nat @ ( M @ Y4 ) @ B3 ) )
=> ( ord_less_eq @ nat @ ( M @ X3 ) @ ( M @ ( hilbert_GreatestM @ A @ nat @ M @ P2 ) ) ) ) ) ).
% GreatestM_nat_le
thf(fact_245_prod__decode__aux_Oinduct,axiom,
! [P2: nat > nat > $o,A0: nat,A1: nat] :
( ! [K2: nat,M6: nat] :
( ( ~ ( ord_less_eq @ nat @ M6 @ K2 )
=> ( P2 @ ( suc @ K2 ) @ ( minus_minus @ nat @ M6 @ ( suc @ K2 ) ) ) )
=> ( P2 @ K2 @ M6 ) )
=> ( P2 @ A0 @ A1 ) ) ).
% prod_decode_aux.induct
thf(fact_246_remdups__adj__adjacent,axiom,
! [A: $tType,I3: nat,Xs: list @ A] :
( ( ord_less @ nat @ ( suc @ I3 ) @ ( size_size @ ( list @ A ) @ ( remdups_adj @ A @ Xs ) ) )
=> ( ( nth @ A @ ( remdups_adj @ A @ Xs ) @ I3 )
!= ( nth @ A @ ( remdups_adj @ A @ Xs ) @ ( suc @ I3 ) ) ) ) ).
% remdups_adj_adjacent
thf(fact_247_remdups__adj__rev,axiom,
! [A: $tType,Xs: list @ A] :
( ( remdups_adj @ A @ ( rev @ A @ Xs ) )
= ( rev @ A @ ( remdups_adj @ A @ Xs ) ) ) ).
% remdups_adj_rev
thf(fact_248_sorted__remdups__adj,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Xs: list @ A] :
( ( linorder_sorted @ A @ Xs )
=> ( linorder_sorted @ A @ ( remdups_adj @ A @ Xs ) ) ) ) ).
% sorted_remdups_adj
thf(fact_249_remdups__adj__length,axiom,
! [A: $tType,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( remdups_adj @ A @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% remdups_adj_length
thf(fact_250_nth__tl,axiom,
! [A: $tType,N: nat,X3: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ ( tl @ A @ X3 ) ) )
=> ( ( nth @ A @ ( tl @ A @ X3 ) @ N )
= ( nth @ A @ X3 @ ( suc @ N ) ) ) ) ).
% nth_tl
thf(fact_251_Suc__pred,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( suc @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) )
= N ) ) ).
% Suc_pred
thf(fact_252_le__zero__eq,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [N: A] :
( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
= ( N
= ( zero_zero @ A ) ) ) ) ).
% le_zero_eq
thf(fact_253_not__gr__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [N: A] :
( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
= ( N
= ( zero_zero @ A ) ) ) ) ).
% not_gr_zero
%----Type constructors (20)
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A6: $tType,A7: $tType] :
( ( preorder @ A7 @ ( type2 @ A7 ) )
=> ( preorder @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A6: $tType,A7: $tType] :
( ( order @ A7 @ ( type2 @ A7 ) )
=> ( order @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A6: $tType,A7: $tType] :
( ( ord @ A7 @ ( type2 @ A7 ) )
=> ( ord @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Groups_Ominus,axiom,
! [A6: $tType,A7: $tType] :
( ( minus @ A7 @ ( type2 @ A7 ) )
=> ( minus @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
condit1037483654norder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
canoni770627133id_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
wellorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
preorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
linorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Ono__top,axiom,
no_top @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
order @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
ord @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ominus_4,axiom,
minus @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ozero,axiom,
zero @ nat @ ( type2 @ nat ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_5,axiom,
preorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder_6,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_7,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_8,axiom,
ord @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Groups_Ominus_9,axiom,
minus @ $o @ ( type2 @ $o ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
( ( ( ord_less @ nat @ n @ ( size_size @ ( list @ a ) @ xs ) )
=> ( ( stream370371455e_snth @ a @ ( stream1035003186_shift @ a @ xs @ s ) @ n )
= ( nth @ a @ xs @ n ) ) )
& ( ~ ( ord_less @ nat @ n @ ( size_size @ ( list @ a ) @ xs ) )
=> ( ( stream370371455e_snth @ a @ ( stream1035003186_shift @ a @ xs @ s ) @ n )
= ( stream370371455e_snth @ a @ s @ ( minus_minus @ nat @ n @ ( size_size @ ( list @ a ) @ xs ) ) ) ) ) ) ).
%------------------------------------------------------------------------------