TPTP Problem File: DAT159^1.p
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%------------------------------------------------------------------------------
% File : DAT159^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Hamming stream 232
% Version : [Bla16] axioms : Especial.
% English :
% Refs : [Loc10] Lochbihler (2010), Coinductive
% : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% Source : [Bla16]
% Names : hamming_stream__232.p [Bla16]
% Status : Theorem
% Rating : 0.00 v7.5.0, 0.67 v7.2.0, 0.50 v7.1.0
% Syntax : Number of formulae : 366 ( 104 unt; 43 typ; 0 def)
% Number of atoms : 971 ( 212 equ; 0 cnn)
% Maximal formula atoms : 13 ( 3 avg)
% Number of connectives : 3600 ( 103 ~; 22 |; 46 &;2908 @)
% ( 0 <=>; 521 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 8 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 125 ( 125 >; 0 *; 0 +; 0 <<)
% Number of symbols : 42 ( 39 usr; 3 con; 0-4 aty)
% Number of variables : 888 ( 26 ^; 778 !; 49 ?; 888 :)
% ( 35 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:42:29.978
%------------------------------------------------------------------------------
%----Could-be-implicit typings (5)
thf(ty_t_Code__Numeral_Onatural,type,
code_natural: $tType ).
thf(ty_t_Real_Oreal,type,
real: $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_t_Int_Oint,type,
int: $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
%----Explicit typings (38)
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_Fields_Ofield,type,
field:
!>[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_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_Rings_Osemiring__1,type,
semiring_1:
!>[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_Rings_Ozero__neq__one,type,
zero_neq_one:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Nat_Osemiring__char__0,type,
semiring_char_0:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Owellorder,type,
wellorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Fields_Odivision__ring,type,
division_ring:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Odense__order,type,
dense_order:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Fields_Olinordered__field,type,
linordered_field:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Odense__linorder,type,
dense_linorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Olinordered__semidom,type,
linordered_semidom:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Osemiring__1__no__zero__divisors,type,
semiri134348788visors:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Archimedean__Field_Oarchimedean__field,type,
archim1804426504_field:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni770627133id_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_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_Code__Numeral_Onatural_Ocase__natural,type,
code_case_natural:
!>[T: $tType] : ( T > ( code_natural > T ) > code_natural > T ) ).
thf(sy_c_Code__Numeral_Onatural_Osize__natural,type,
code_size_natural: code_natural > nat ).
thf(sy_c_Fields_Oinverse__class_Oinverse,type,
inverse_inverse:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_Hamming__Stream__Mirabelle__rwekfkwckg_Osmooth,type,
hammin1195055315smooth: nat > $o ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
semiring_1_of_nat:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : ( A > nat ) ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Power_Opower__class_Opower,type,
power_power:
!>[A: $tType] : ( A > nat > A ) ).
thf(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool,type,
zero_neq_one_of_bool:
!>[A: $tType] : ( $o > A ) ).
thf(sy_v_n,type,
n: nat ).
%----Relevant facts (254)
thf(fact_0_smooth__0,axiom,
~ ( hammin1195055315smooth @ ( zero_zero @ nat ) ) ).
% smooth_0
thf(fact_1_neq0__conv,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% neq0_conv
thf(fact_2_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
thf(fact_3_gr0I,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% gr0I
thf(fact_4_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% not_gr0
thf(fact_5_not__less0,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% not_less0
thf(fact_6_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% less_zeroE
thf(fact_7_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( N
!= ( zero_zero @ nat ) ) ) ).
% gr_implies_not0
thf(fact_8_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_9_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% less_nat_zero_code
thf(fact_10_infinite__descent0__measure,axiom,
! [A: $tType,V: A > nat,P: A > $o,X: A] :
( ! [X2: A] :
( ( ( V @ X2 )
= ( zero_zero @ nat ) )
=> ( P @ X2 ) )
=> ( ! [X2: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V @ X2 ) )
=> ( ~ ( P @ X2 )
=> ? [Y: A] :
( ( ord_less @ nat @ ( V @ Y ) @ ( V @ X2 ) )
& ~ ( P @ Y ) ) ) )
=> ( P @ X ) ) ) ).
% infinite_descent0_measure
thf(fact_11_less__numeral__extra_I3_J,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% less_numeral_extra(3)
thf(fact_12_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( ( zero_zero @ A )
= X )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_13_infinite__descent__measure,axiom,
! [A: $tType,P: A > $o,V: A > nat,X: A] :
( ! [X2: A] :
( ~ ( P @ X2 )
=> ? [Y: A] :
( ( ord_less @ nat @ ( V @ Y ) @ ( V @ X2 ) )
& ~ ( P @ Y ) ) )
=> ( P @ X ) ) ).
% infinite_descent_measure
thf(fact_14_measure__induct__rule,axiom,
! [A: $tType,F: A > nat,P: A > $o,A2: A] :
( ! [X2: A] :
( ! [Y: A] :
( ( ord_less @ nat @ ( F @ Y ) @ ( F @ X2 ) )
=> ( P @ Y ) )
=> ( P @ X2 ) )
=> ( P @ A2 ) ) ).
% measure_induct_rule
thf(fact_15_linorder__neqE__nat,axiom,
! [X: nat,Y2: nat] :
( ( X != Y2 )
=> ( ~ ( ord_less @ nat @ X @ Y2 )
=> ( ord_less @ nat @ Y2 @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_16_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_17_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_18_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_19_measure__induct,axiom,
! [A: $tType,F: A > nat,P: A > $o,A2: A] :
( ! [X2: A] :
( ! [Y: A] :
( ( ord_less @ nat @ ( F @ Y ) @ ( F @ X2 ) )
=> ( P @ Y ) )
=> ( P @ X2 ) )
=> ( P @ A2 ) ) ).
% measure_induct
thf(fact_20_less__not__refl3,axiom,
! [S: nat,T2: nat] :
( ( ord_less @ nat @ S @ T2 )
=> ( S != T2 ) ) ).
% less_not_refl3
thf(fact_21_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_22_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_not_refl
thf(fact_23_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less @ nat @ M @ N )
| ( ord_less @ nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_24_zero__less__iff__neq__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [N: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ N )
= ( N
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_iff_neq_zero
thf(fact_25_gr__implies__not__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [M: A,N: A] :
( ( ord_less @ A @ M @ N )
=> ( N
!= ( zero_zero @ A ) ) ) ) ).
% gr_implies_not_zero
thf(fact_26_not__less__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [N: A] :
~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% not_less_zero
thf(fact_27_gr__zeroI,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [N: A] :
( ( N
!= ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% gr_zeroI
thf(fact_28_of__nat__0__less__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% of_nat_0_less_iff
thf(fact_29_zero__natural_Orsp,axiom,
( ( zero_zero @ nat )
= ( zero_zero @ nat ) ) ).
% zero_natural.rsp
thf(fact_30_ex__gt__or__lt,axiom,
! [A: $tType] :
( ( condit1656338222tinuum @ A @ ( type2 @ A ) )
=> ! [A2: A] :
? [B: A] :
( ( ord_less @ A @ A2 @ B )
| ( ord_less @ A @ B @ A2 ) ) ) ).
% ex_gt_or_lt
thf(fact_31_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( X != Y2 )
=> ( ~ ( ord_less @ A @ X @ Y2 )
=> ( ord_less @ A @ Y2 @ X ) ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_32_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( A2 != B2 ) ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_33_linordered__field__no__ub,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [X3: A] :
? [X1: A] : ( ord_less @ A @ X3 @ X1 ) ) ).
% linordered_field_no_ub
thf(fact_34_linordered__field__no__lb,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [X3: A] :
? [Y3: A] : ( ord_less @ A @ Y3 @ X3 ) ) ).
% linordered_field_no_lb
thf(fact_35_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( A2 != B2 ) ) ) ).
% order.strict_implies_not_eq
thf(fact_36_dependent__wellorder__choice,axiom,
! [B3: $tType,A: $tType] :
( ( wellorder @ A @ ( type2 @ A ) )
=> ! [P: ( A > B3 ) > A > B3 > $o] :
( ! [R: B3,F2: A > B3,G: A > B3,X2: A] :
( ! [Y: A] :
( ( ord_less @ A @ Y @ X2 )
=> ( ( F2 @ Y )
= ( G @ Y ) ) )
=> ( ( P @ F2 @ X2 @ R )
= ( P @ G @ X2 @ R ) ) )
=> ( ! [X2: A,F2: A > B3] :
( ! [Y: A] :
( ( ord_less @ A @ Y @ X2 )
=> ( P @ F2 @ Y @ ( F2 @ Y ) ) )
=> ? [X12: B3] : ( P @ F2 @ X2 @ X12 ) )
=> ? [F2: A > B3] :
! [X3: A] : ( P @ F2 @ X3 @ ( F2 @ X3 ) ) ) ) ) ).
% dependent_wellorder_choice
thf(fact_37_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ~ ( ord_less @ A @ X @ Y2 ) )
= ( ( ord_less @ A @ Y2 @ X )
| ( X = Y2 ) ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_38_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_less @ A @ C @ B2 )
=> ( ord_less @ A @ C @ A2 ) ) ) ) ).
% dual_order.strict_trans
thf(fact_39_of__nat__eq__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A @ ( type2 @ A ) )
=> ! [M: nat,N: nat] :
( ( ( semiring_1_of_nat @ A @ M )
= ( semiring_1_of_nat @ A @ N ) )
= ( M = N ) ) ) ).
% of_nat_eq_iff
thf(fact_40_of__nat__0,axiom,
! [A: $tType] :
( ( semiring_1 @ A @ ( type2 @ A ) )
=> ( ( semiring_1_of_nat @ A @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) ) ) ).
% of_nat_0
thf(fact_41_of__nat__0__eq__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A @ ( type2 @ A ) )
=> ! [N: nat] :
( ( ( zero_zero @ A )
= ( semiring_1_of_nat @ A @ N ) )
= ( ( zero_zero @ nat )
= N ) ) ) ).
% of_nat_0_eq_iff
thf(fact_42_of__nat__eq__0__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A @ ( type2 @ A ) )
=> ! [M: nat] :
( ( ( semiring_1_of_nat @ A @ M )
= ( zero_zero @ A ) )
= ( M
= ( zero_zero @ nat ) ) ) ) ).
% of_nat_eq_0_iff
thf(fact_43_of__nat__less__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [M: nat,N: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
= ( ord_less @ nat @ M @ N ) ) ) ).
% of_nat_less_iff
thf(fact_44_of__nat__less__0__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [M: nat] :
~ ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( zero_zero @ A ) ) ) ).
% of_nat_less_0_iff
thf(fact_45_less__imp__of__nat__less,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% less_imp_of_nat_less
thf(fact_46_of__nat__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [M: nat,N: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
=> ( ord_less @ nat @ M @ N ) ) ) ).
% of_nat_less_imp_less
thf(fact_47_ord__eq__less__subst,axiom,
! [A: $tType,B3: $tType] :
( ( ( ord @ B3 @ ( type2 @ B3 ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B3 > A,B2: B3,C: B3] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less @ B3 @ B2 @ C )
=> ( ! [X2: B3,Y3: B3] :
( ( ord_less @ B3 @ X2 @ Y3 )
=> ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_48_ord__less__eq__subst,axiom,
! [A: $tType,B3: $tType] :
( ( ( ord @ B3 @ ( type2 @ B3 ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B2: A,F: A > B3,C: B3] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: A,Y3: A] :
( ( ord_less @ A @ X2 @ Y3 )
=> ( ord_less @ B3 @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ B3 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_49_order__less__subst1,axiom,
! [A: $tType,B3: $tType] :
( ( ( order @ B3 @ ( type2 @ B3 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B3 > A,B2: B3,C: B3] :
( ( ord_less @ A @ A2 @ ( F @ B2 ) )
=> ( ( ord_less @ B3 @ B2 @ C )
=> ( ! [X2: B3,Y3: B3] :
( ( ord_less @ B3 @ X2 @ Y3 )
=> ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_50_order__less__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 @ ( type2 @ C2 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B2: A,F: A > C2,C: C2] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ C2 @ ( F @ B2 ) @ C )
=> ( ! [X2: A,Y3: A] :
( ( ord_less @ A @ X2 @ Y3 )
=> ( ord_less @ C2 @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_less_subst2
thf(fact_51_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A @ ( type2 @ A ) )
=> ! [X: A] :
? [Y3: A] : ( ord_less @ A @ Y3 @ X ) ) ).
% lt_ex
thf(fact_52_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A @ ( type2 @ A ) )
=> ! [X: A] :
? [X1: A] : ( ord_less @ A @ X @ X1 ) ) ).
% gt_ex
thf(fact_53_neqE,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( X != Y2 )
=> ( ~ ( ord_less @ A @ X @ Y2 )
=> ( ord_less @ A @ Y2 @ X ) ) ) ) ).
% neqE
thf(fact_54_neq__iff,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( X != Y2 )
= ( ( ord_less @ A @ X @ Y2 )
| ( ord_less @ A @ Y2 @ X ) ) ) ) ).
% neq_iff
thf(fact_55_order_Oasym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% order.asym
thf(fact_56_dense,axiom,
! [A: $tType] :
( ( dense_order @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ? [Z: A] :
( ( ord_less @ A @ X @ Z )
& ( ord_less @ A @ Z @ Y2 ) ) ) ) ).
% dense
thf(fact_57_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ( X != Y2 ) ) ) ).
% less_imp_neq
thf(fact_58_less__asym,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ~ ( ord_less @ A @ Y2 @ X ) ) ) ).
% less_asym
thf(fact_59_less__asym_H,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% less_asym'
thf(fact_60_less__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A,Z2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ( ( ord_less @ A @ Y2 @ Z2 )
=> ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% less_trans
thf(fact_61_less__linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
| ( X = Y2 )
| ( ord_less @ A @ Y2 @ X ) ) ) ).
% less_linear
thf(fact_62_less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A] :
~ ( ord_less @ A @ X @ X ) ) ).
% less_irrefl
thf(fact_63_ord__eq__less__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( A2 = B2 )
=> ( ( ord_less @ A @ B2 @ C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% ord_eq_less_trans
thf(fact_64_ord__less__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% ord_less_eq_trans
thf(fact_65_dual__order_Oasym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ~ ( ord_less @ A @ A2 @ B2 ) ) ) ).
% dual_order.asym
thf(fact_66_less__imp__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ( X != Y2 ) ) ) ).
% less_imp_not_eq
thf(fact_67_less__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ~ ( ord_less @ A @ Y2 @ X ) ) ) ).
% less_not_sym
thf(fact_68_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,A2: A] :
( ! [X2: A] :
( ! [Y: A] :
( ( ord_less @ A @ Y @ X2 )
=> ( P @ Y ) )
=> ( P @ X2 ) )
=> ( P @ A2 ) ) ) ).
% less_induct
thf(fact_69_antisym__conv3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y2: A,X: A] :
( ~ ( ord_less @ A @ Y2 @ X )
=> ( ( ~ ( ord_less @ A @ X @ Y2 ) )
= ( X = Y2 ) ) ) ) ).
% antisym_conv3
thf(fact_70_less__imp__not__eq2,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ( Y2 != X ) ) ) ).
% less_imp_not_eq2
thf(fact_71_less__imp__triv,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A,P: $o] :
( ( ord_less @ A @ X @ Y2 )
=> ( ( ord_less @ A @ Y2 @ X )
=> P ) ) ) ).
% less_imp_triv
thf(fact_72_linorder__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ~ ( ord_less @ A @ X @ Y2 )
=> ( ( X != Y2 )
=> ( ord_less @ A @ Y2 @ X ) ) ) ) ).
% linorder_cases
thf(fact_73_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ A2 ) ) ).
% dual_order.irrefl
thf(fact_74_order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ B2 @ C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% order.strict_trans
thf(fact_75_less__imp__not__less,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ~ ( ord_less @ A @ Y2 @ X ) ) ) ).
% less_imp_not_less
thf(fact_76_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ? [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
& ( K
= ( semiring_1_of_nat @ int @ N2 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_77_reals__Archimedean2,axiom,
! [A: $tType] :
( ( archim1804426504_field @ A @ ( type2 @ A ) )
=> ! [X: A] :
? [N2: nat] : ( ord_less @ A @ X @ ( semiring_1_of_nat @ A @ N2 ) ) ) ).
% reals_Archimedean2
thf(fact_78_natural_Osize_I1_J,axiom,
( ( code_size_natural @ ( zero_zero @ code_natural ) )
= ( zero_zero @ nat ) ) ).
% natural.size(1)
thf(fact_79_of__nat__zero__less__power__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [X: nat,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ X ) @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_80_of__nat__le__0__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [M: nat] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M ) @ ( zero_zero @ A ) )
= ( M
= ( zero_zero @ nat ) ) ) ) ).
% of_nat_le_0_iff
thf(fact_81_natural_Osize_I3_J,axiom,
( ( size_size @ code_natural @ ( zero_zero @ code_natural ) )
= ( zero_zero @ nat ) ) ).
% natural.size(3)
thf(fact_82_of__bool__eq_I1_J,axiom,
! [A: $tType] :
( ( zero_neq_one @ A @ ( type2 @ A ) )
=> ( ( zero_neq_one_of_bool @ A @ $false )
= ( zero_zero @ A ) ) ) ).
% of_bool_eq(1)
thf(fact_83_ex__inverse__of__nat__less,axiom,
! [A: $tType] :
( ( archim1804426504_field @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ? [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
& ( ord_less @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ N2 ) ) @ X ) ) ) ) ).
% ex_inverse_of_nat_less
thf(fact_84_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_85_le0,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% le0
thf(fact_86_inverse__inverse__eq,axiom,
! [A: $tType] :
( ( division_ring @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( inverse_inverse @ A @ ( inverse_inverse @ A @ A2 ) )
= A2 ) ) ).
% inverse_inverse_eq
thf(fact_87_inverse__eq__iff__eq,axiom,
! [A: $tType] :
( ( division_ring @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ( inverse_inverse @ A @ A2 )
= ( inverse_inverse @ A @ B2 ) )
= ( A2 = B2 ) ) ) ).
% inverse_eq_iff_eq
thf(fact_88_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_89_of__nat__le__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [M: nat,N: nat] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
= ( ord_less_eq @ nat @ M @ N ) ) ) ).
% of_nat_le_iff
thf(fact_90_inverse__nonzero__iff__nonzero,axiom,
! [A: $tType] :
( ( division_ring @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ( inverse_inverse @ A @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% inverse_nonzero_iff_nonzero
thf(fact_91_inverse__zero,axiom,
! [A: $tType] :
( ( division_ring @ A @ ( type2 @ A ) )
=> ( ( inverse_inverse @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% inverse_zero
thf(fact_92_nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( power_power @ nat @ X @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X )
| ( N
= ( zero_zero @ nat ) ) ) ) ).
% nat_zero_less_power_iff
thf(fact_93_of__nat__power,axiom,
! [A: $tType] :
( ( semiring_1 @ A @ ( type2 @ A ) )
=> ! [M: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( power_power @ nat @ M @ N ) )
= ( power_power @ A @ ( semiring_1_of_nat @ A @ M ) @ N ) ) ) ).
% of_nat_power
thf(fact_94_inverse__nonnegative__iff__nonnegative,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% inverse_nonnegative_iff_nonnegative
thf(fact_95_inverse__nonpositive__iff__nonpositive,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% inverse_nonpositive_iff_nonpositive
thf(fact_96_inverse__positive__iff__positive,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% inverse_positive_iff_positive
thf(fact_97_inverse__negative__iff__negative,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% inverse_negative_iff_negative
thf(fact_98_inverse__less__iff__less__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
= ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).
% inverse_less_iff_less_neg
thf(fact_99_inverse__less__iff__less,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
= ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).
% inverse_less_iff_less
thf(fact_100_inverse__le__iff__le__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).
% inverse_le_iff_le_neg
thf(fact_101_inverse__le__iff__le,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).
% inverse_le_iff_le
thf(fact_102_power__eq__0__iff,axiom,
! [A: $tType] :
( ( semiri134348788visors @ A @ ( type2 @ A ) )
=> ! [A2: A,N: nat] :
( ( ( power_power @ A @ A2 @ N )
= ( zero_zero @ A ) )
= ( ( A2
= ( zero_zero @ A ) )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% power_eq_0_iff
thf(fact_103_nonneg__eq__int,axiom,
! [Z2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
=> ~ ! [M3: nat] :
( Z2
!= ( semiring_1_of_nat @ int @ M3 ) ) ) ).
% nonneg_eq_int
thf(fact_104_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ~ ! [N2: nat] :
( K
!= ( semiring_1_of_nat @ int @ N2 ) ) ) ).
% nonneg_int_cases
thf(fact_105_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ? [N2: nat] :
( K
= ( semiring_1_of_nat @ int @ N2 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_106_zero__integer_Orsp,axiom,
( ( zero_zero @ int )
= ( zero_zero @ int ) ) ).
% zero_integer.rsp
thf(fact_107_inverse__eq__imp__eq,axiom,
! [A: $tType] :
( ( division_ring @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ( inverse_inverse @ A @ A2 )
= ( inverse_inverse @ A @ B2 ) )
=> ( A2 = B2 ) ) ) ).
% inverse_eq_imp_eq
thf(fact_108_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ) ).
% dual_order.antisym
thf(fact_109_of__bool__eq__iff,axiom,
! [A: $tType] :
( ( zero_neq_one @ A @ ( type2 @ A ) )
=> ! [P2: $o,Q: $o] :
( ( ( zero_neq_one_of_bool @ A @ P2 )
= ( zero_neq_one_of_bool @ A @ Q ) )
= ( P2 = Q ) ) ) ).
% of_bool_eq_iff
thf(fact_110_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C @ B2 )
=> ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% dual_order.trans
thf(fact_111_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > A > $o,A2: A,B2: A] :
( ! [A3: A,B: A] :
( ( ord_less_eq @ A @ A3 @ B )
=> ( P @ A3 @ B ) )
=> ( ! [A3: A,B: A] :
( ( P @ B @ A3 )
=> ( P @ A3 @ B ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_wlog
thf(fact_112_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_113_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A,Z2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ Z2 )
=> ( ord_less_eq @ A @ X @ Z2 ) ) ) ) ).
% order_trans
thf(fact_114_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ) ).
% order_class.order.antisym
thf(fact_115_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% ord_le_eq_trans
thf(fact_116_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( A2 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% ord_eq_le_trans
thf(fact_117_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [Y2: A,X: A] :
( ( ord_less_eq @ A @ Y2 @ X )
=> ( ( ord_less_eq @ A @ X @ Y2 )
= ( X = Y2 ) ) ) ) ).
% antisym_conv
thf(fact_118_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A,Z2: A] :
( ( ( ord_less_eq @ A @ X @ Y2 )
=> ~ ( ord_less_eq @ A @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ Y2 @ X )
=> ~ ( ord_less_eq @ A @ X @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ X @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq @ A @ Z2 @ Y2 )
=> ~ ( ord_less_eq @ A @ Y2 @ X ) )
=> ( ( ( ord_less_eq @ A @ Y2 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ X ) )
=> ~ ( ( ord_less_eq @ A @ Z2 @ X )
=> ~ ( ord_less_eq @ A @ X @ Y2 ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_119_real__arch__simple,axiom,
! [A: $tType] :
( ( archim1804426504_field @ A @ ( type2 @ A ) )
=> ! [X: A] :
? [N2: nat] : ( ord_less_eq @ A @ X @ ( semiring_1_of_nat @ A @ N2 ) ) ) ).
% real_arch_simple
thf(fact_120_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% order.trans
thf(fact_121_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ~ ( ord_less_eq @ A @ X @ Y2 )
=> ( ord_less_eq @ A @ Y2 @ X ) ) ) ).
% le_cases
thf(fact_122_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( X = Y2 )
=> ( ord_less_eq @ A @ X @ Y2 ) ) ) ).
% eq_refl
thf(fact_123_linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
| ( ord_less_eq @ A @ Y2 @ X ) ) ) ).
% linear
thf(fact_124_antisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ X )
=> ( X = Y2 ) ) ) ) ).
% antisym
thf(fact_125_eq__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ^ [Y4: A,Z3: A] : Y4 = Z3 )
= ( ^ [X4: A,Y5: A] :
( ( ord_less_eq @ A @ X4 @ Y5 )
& ( ord_less_eq @ A @ Y5 @ X4 ) ) ) ) ) ).
% eq_iff
thf(fact_126_ord__le__eq__subst,axiom,
! [A: $tType,B3: $tType] :
( ( ( ord @ B3 @ ( type2 @ B3 ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B2: A,F: A > B3,C: B3] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
=> ( ord_less_eq @ B3 @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ B3 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_127_ord__eq__le__subst,axiom,
! [A: $tType,B3: $tType] :
( ( ( ord @ B3 @ ( type2 @ B3 ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B3 > A,B2: B3,C: B3] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq @ B3 @ B2 @ C )
=> ( ! [X2: B3,Y3: B3] :
( ( ord_less_eq @ B3 @ X2 @ Y3 )
=> ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_128_order__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 @ ( type2 @ C2 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B2: A,F: A > C2,C: C2] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ C2 @ ( F @ B2 ) @ C )
=> ( ! [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
=> ( ord_less_eq @ C2 @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_subst2
thf(fact_129_order__subst1,axiom,
! [A: $tType,B3: $tType] :
( ( ( order @ B3 @ ( type2 @ B3 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B3 > A,B2: B3,C: B3] :
( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq @ B3 @ B2 @ C )
=> ( ! [X2: B3,Y3: B3] :
( ( ord_less_eq @ B3 @ X2 @ Y3 )
=> ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_subst1
thf(fact_130_le__fun__def,axiom,
! [B3: $tType,A: $tType] :
( ( ord @ B3 @ ( type2 @ B3 ) )
=> ( ( ord_less_eq @ ( A > B3 ) )
= ( ^ [F3: A > B3,G2: A > B3] :
! [X4: A] : ( ord_less_eq @ B3 @ ( F3 @ X4 ) @ ( G2 @ X4 ) ) ) ) ) ).
% le_fun_def
thf(fact_131_le__funI,axiom,
! [B3: $tType,A: $tType] :
( ( ord @ B3 @ ( type2 @ B3 ) )
=> ! [F: A > B3,G3: A > B3] :
( ! [X2: A] : ( ord_less_eq @ B3 @ ( F @ X2 ) @ ( G3 @ X2 ) )
=> ( ord_less_eq @ ( A > B3 ) @ F @ G3 ) ) ) ).
% le_funI
thf(fact_132_le__funE,axiom,
! [B3: $tType,A: $tType] :
( ( ord @ B3 @ ( type2 @ B3 ) )
=> ! [F: A > B3,G3: A > B3,X: A] :
( ( ord_less_eq @ ( A > B3 ) @ F @ G3 )
=> ( ord_less_eq @ B3 @ ( F @ X ) @ ( G3 @ X ) ) ) ) ).
% le_funE
thf(fact_133_le__funD,axiom,
! [B3: $tType,A: $tType] :
( ( ord @ B3 @ ( type2 @ B3 ) )
=> ! [F: A > B3,G3: A > B3,X: A] :
( ( ord_less_eq @ ( A > B3 ) @ F @ G3 )
=> ( ord_less_eq @ B3 @ ( F @ X ) @ ( G3 @ X ) ) ) ) ).
% le_funD
thf(fact_134_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiring_1_of_nat @ int @ M )
= ( semiring_1_of_nat @ int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_135_zle__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N ) )
= ( ord_less_eq @ nat @ M @ N ) ) ).
% zle_int
thf(fact_136_power__not__zero,axiom,
! [A: $tType] :
( ( semiri134348788visors @ A @ ( type2 @ A ) )
=> ! [A2: A,N: nat] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( power_power @ A @ A2 @ N )
!= ( zero_zero @ A ) ) ) ) ).
% power_not_zero
thf(fact_137_power__less__imp__less__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [A2: A,N: nat,B2: A] :
( ( ord_less @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% power_less_imp_less_base
thf(fact_138_power__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A @ ( type2 @ A ) )
=> ! [A2: A,N: nat] :
( ( power_power @ A @ ( inverse_inverse @ A @ A2 ) @ N )
= ( inverse_inverse @ A @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% power_inverse
thf(fact_139_zero__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [A2: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% zero_le_power
thf(fact_140_power__mono,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,N: nat] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ) ).
% power_mono
thf(fact_141_less__eq__int__code_I1_J,axiom,
ord_less_eq @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ).
% less_eq_int_code(1)
thf(fact_142_less__int__code_I1_J,axiom,
~ ( ord_less @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ) ).
% less_int_code(1)
thf(fact_143_le__imp__inverse__le__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% le_imp_inverse_le_neg
thf(fact_144_inverse__le__imp__le__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ).
% inverse_le_imp_le_neg
thf(fact_145_le__imp__inverse__le,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less_eq @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% le_imp_inverse_le
thf(fact_146_inverse__le__imp__le,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ).
% inverse_le_imp_le
thf(fact_147_nonzero__imp__inverse__nonzero,axiom,
! [A: $tType] :
( ( division_ring @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ A2 )
!= ( zero_zero @ A ) ) ) ) ).
% nonzero_imp_inverse_nonzero
thf(fact_148_nonzero__inverse__inverse__eq,axiom,
! [A: $tType] :
( ( division_ring @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ ( inverse_inverse @ A @ A2 ) )
= A2 ) ) ) ).
% nonzero_inverse_inverse_eq
thf(fact_149_nonzero__inverse__eq__imp__eq,axiom,
! [A: $tType] :
( ( division_ring @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ( inverse_inverse @ A @ A2 )
= ( inverse_inverse @ A @ B2 ) )
=> ( ( A2
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( A2 = B2 ) ) ) ) ) ).
% nonzero_inverse_eq_imp_eq
thf(fact_150_inverse__zero__imp__zero,axiom,
! [A: $tType] :
( ( division_ring @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ( inverse_inverse @ A @ A2 )
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ).
% inverse_zero_imp_zero
thf(fact_151_field__class_Ofield__inverse__zero,axiom,
! [A: $tType] :
( ( field @ A @ ( type2 @ A ) )
=> ( ( inverse_inverse @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% field_class.field_inverse_zero
thf(fact_152_power__eq__imp__eq__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [A2: A,N: nat,B2: A] :
( ( ( power_power @ A @ A2 @ N )
= ( power_power @ A @ B2 @ N ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( A2 = B2 ) ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_153_power__eq__iff__eq__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [N: nat,A2: A,B2: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ( power_power @ A @ A2 @ N )
= ( power_power @ A @ B2 @ N ) )
= ( A2 = B2 ) ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_154_zero__le,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% zero_le
thf(fact_155_le__numeral__extra_I3_J,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% le_numeral_extra(3)
thf(fact_156_complete__interval,axiom,
! [A: $tType] :
( ( condit1037483654norder @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,P: A > $o] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( P @ A2 )
=> ( ~ ( P @ B2 )
=> ? [C3: A] :
( ( ord_less_eq @ A @ A2 @ C3 )
& ( ord_less_eq @ A @ C3 @ B2 )
& ! [X3: A] :
( ( ( ord_less_eq @ A @ A2 @ X3 )
& ( ord_less @ A @ X3 @ C3 ) )
=> ( P @ X3 ) )
& ! [D: A] :
( ! [X2: A] :
( ( ( ord_less_eq @ A @ A2 @ X2 )
& ( ord_less @ A @ X2 @ D ) )
=> ( P @ X2 ) )
=> ( ord_less_eq @ A @ D @ C3 ) ) ) ) ) ) ) ).
% complete_interval
thf(fact_157_order_Onot__eq__order__implies__strict,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( A2 != B2 )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% order.not_eq_order_implies_strict
thf(fact_158_dual__order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% dual_order.strict_implies_order
thf(fact_159_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_160_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_161_order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% order.strict_implies_order
thf(fact_162_dense__le__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A,Z2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ( ! [W: A] :
( ( ord_less @ A @ X @ W )
=> ( ( ord_less @ A @ W @ Y2 )
=> ( ord_less_eq @ A @ W @ Z2 ) ) )
=> ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ) ).
% dense_le_bounded
thf(fact_163_dense__ge__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Z2: A,X: A,Y2: A] :
( ( ord_less @ A @ Z2 @ X )
=> ( ! [W: A] :
( ( ord_less @ A @ Z2 @ W )
=> ( ( ord_less @ A @ W @ X )
=> ( ord_less_eq @ A @ Y2 @ W ) ) )
=> ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ) ).
% dense_ge_bounded
thf(fact_164_dual__order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C @ B2 )
=> ( ord_less @ A @ C @ A2 ) ) ) ) ).
% dual_order.strict_trans2
thf(fact_165_dual__order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less @ A @ C @ B2 )
=> ( ord_less @ A @ C @ A2 ) ) ) ) ).
% dual_order.strict_trans1
thf(fact_166_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_167_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_168_order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% order.strict_trans2
thf(fact_169_order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ B2 @ C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% order.strict_trans1
thf(fact_170_not__le__imp__less,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y2: A,X: A] :
( ~ ( ord_less_eq @ A @ Y2 @ X )
=> ( ord_less @ A @ X @ Y2 ) ) ) ).
% not_le_imp_less
thf(fact_171_less__le__not__le,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [X4: A,Y5: A] :
( ( ord_less_eq @ A @ X4 @ Y5 )
& ~ ( ord_less_eq @ A @ Y5 @ X4 ) ) ) ) ) ).
% less_le_not_le
thf(fact_172_le__imp__less__or__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
=> ( ( ord_less @ A @ X @ Y2 )
| ( X = Y2 ) ) ) ) ).
% le_imp_less_or_eq
thf(fact_173_le__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
| ( ord_less @ A @ Y2 @ X ) ) ) ).
% le_less_linear
thf(fact_174_dense__le,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ 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_175_dense__ge,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ 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_176_less__le__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A,Z2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ Z2 )
=> ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% less_le_trans
thf(fact_177_le__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A,Z2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
=> ( ( ord_less @ A @ Y2 @ Z2 )
=> ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% le_less_trans
thf(fact_178_antisym__conv2,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
=> ( ( ~ ( ord_less @ A @ X @ Y2 ) )
= ( X = Y2 ) ) ) ) ).
% antisym_conv2
thf(fact_179_antisym__conv1,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ~ ( ord_less @ A @ X @ Y2 )
=> ( ( ord_less_eq @ A @ X @ Y2 )
= ( X = Y2 ) ) ) ) ).
% antisym_conv1
thf(fact_180_less__imp__le,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ( ord_less_eq @ A @ X @ Y2 ) ) ) ).
% less_imp_le
thf(fact_181_le__neq__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% le_neq_trans
thf(fact_182_not__less,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ~ ( ord_less @ A @ X @ Y2 ) )
= ( ord_less_eq @ A @ Y2 @ X ) ) ) ).
% not_less
thf(fact_183_not__le,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ( ~ ( ord_less_eq @ A @ X @ Y2 ) )
= ( ord_less @ A @ Y2 @ X ) ) ) ).
% not_le
thf(fact_184_order__less__le__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 @ ( type2 @ C2 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B2: A,F: A > C2,C: C2] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ C2 @ ( F @ B2 ) @ C )
=> ( ! [X2: A,Y3: A] :
( ( ord_less @ A @ X2 @ Y3 )
=> ( ord_less @ C2 @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_less_le_subst2
thf(fact_185_order__less__le__subst1,axiom,
! [A: $tType,B3: $tType] :
( ( ( order @ B3 @ ( type2 @ B3 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B3 > A,B2: B3,C: B3] :
( ( ord_less @ A @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq @ B3 @ B2 @ C )
=> ( ! [X2: B3,Y3: B3] :
( ( ord_less_eq @ B3 @ X2 @ Y3 )
=> ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_186_order__le__less__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 @ ( type2 @ C2 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B2: A,F: A > C2,C: C2] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ C2 @ ( F @ B2 ) @ C )
=> ( ! [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
=> ( ord_less_eq @ C2 @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_le_less_subst2
thf(fact_187_order__le__less__subst1,axiom,
! [A: $tType,B3: $tType] :
( ( ( order @ B3 @ ( type2 @ B3 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B3 > A,B2: B3,C: B3] :
( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
=> ( ( ord_less @ B3 @ B2 @ C )
=> ( ! [X2: B3,Y3: B3] :
( ( ord_less @ B3 @ X2 @ Y3 )
=> ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_188_less__le,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [X4: A,Y5: A] :
( ( ord_less_eq @ A @ X4 @ Y5 )
& ( X4 != Y5 ) ) ) ) ) ).
% less_le
thf(fact_189_le__less,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [X4: A,Y5: A] :
( ( ord_less @ A @ X4 @ Y5 )
| ( X4 = Y5 ) ) ) ) ) ).
% le_less
thf(fact_190_leI,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y2: A] :
( ~ ( ord_less @ A @ X @ Y2 )
=> ( ord_less_eq @ A @ Y2 @ X ) ) ) ).
% leI
thf(fact_191_leD,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y2: A,X: A] :
( ( ord_less_eq @ A @ Y2 @ X )
=> ~ ( ord_less @ A @ X @ Y2 ) ) ) ).
% leD
thf(fact_192_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% le_0_eq
thf(fact_193_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% less_eq_nat.simps(1)
thf(fact_194_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_195_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_196_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_197_le__eq__less__or__eq,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M4: nat,N3: nat] :
( ( ord_less @ nat @ M4 @ N3 )
| ( M4 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_198_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_199_nat__less__le,axiom,
( ( ord_less @ nat )
= ( ^ [M4: nat,N3: nat] :
( ( ord_less_eq @ nat @ M4 @ N3 )
& ( M4 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_200_zero__less__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [A2: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% zero_less_power
thf(fact_201_nat__power__less__imp__less,axiom,
! [I: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ I )
=> ( ( ord_less @ nat @ ( power_power @ nat @ I @ M ) @ ( power_power @ nat @ I @ N ) )
=> ( ord_less @ nat @ M @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_202_power__strict__mono,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,N: nat] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ) ) ).
% power_strict_mono
thf(fact_203_positive__imp__inverse__positive,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ).
% positive_imp_inverse_positive
thf(fact_204_negative__imp__inverse__negative,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) ) ) ) ).
% negative_imp_inverse_negative
thf(fact_205_inverse__positive__imp__positive,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) )
=> ( ( A2
!= ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ).
% inverse_positive_imp_positive
thf(fact_206_inverse__negative__imp__negative,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) )
=> ( ( A2
!= ( zero_zero @ A ) )
=> ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ).
% inverse_negative_imp_negative
thf(fact_207_less__imp__inverse__less__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% less_imp_inverse_less_neg
thf(fact_208_inverse__less__imp__less__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ A2 ) ) ) ) ).
% inverse_less_imp_less_neg
thf(fact_209_less__imp__inverse__less,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% less_imp_inverse_less
thf(fact_210_inverse__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less @ A @ B2 @ A2 ) ) ) ) ).
% inverse_less_imp_less
thf(fact_211_zero__power,axiom,
! [A: $tType] :
( ( semiring_1 @ A @ ( type2 @ A ) )
=> ! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( power_power @ A @ ( zero_zero @ A ) @ N )
= ( zero_zero @ A ) ) ) ) ).
% zero_power
thf(fact_212_of__nat__0__le__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [N: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N ) ) ) ).
% of_nat_0_le_iff
thf(fact_213_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ~ ( P @ ( zero_zero @ nat ) )
=> ( ( P @ N )
=> ? [K2: nat] :
( ( ord_less_eq @ nat @ K2 @ N )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_214_transfer__int__nat__relations_I2_J,axiom,
! [X: nat,Y2: nat] :
( ( ord_less @ int @ ( semiring_1_of_nat @ int @ X ) @ ( semiring_1_of_nat @ int @ Y2 ) )
= ( ord_less @ nat @ X @ Y2 ) ) ).
% transfer_int_nat_relations(2)
thf(fact_215_transfer__int__nat__numerals_I1_J,axiom,
( ( zero_zero @ int )
= ( semiring_1_of_nat @ int @ ( zero_zero @ nat ) ) ) ).
% transfer_int_nat_numerals(1)
thf(fact_216_natural_Osimps_I4_J,axiom,
! [T: $tType,F1: T,F22: code_natural > T] :
( ( code_case_natural @ T @ F1 @ F22 @ ( zero_zero @ code_natural ) )
= F1 ) ).
% natural.simps(4)
thf(fact_217_transfer__int__nat__quantifiers_I2_J,axiom,
! [P: int > $o] :
( ( ? [X4: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X4 )
& ( P @ X4 ) ) )
= ( ? [X4: nat] : ( P @ ( semiring_1_of_nat @ int @ X4 ) ) ) ) ).
% transfer_int_nat_quantifiers(2)
thf(fact_218_transfer__int__nat__quantifiers_I1_J,axiom,
! [P: int > $o] :
( ( ! [X4: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X4 )
=> ( P @ X4 ) ) )
= ( ! [X4: nat] : ( P @ ( semiring_1_of_nat @ int @ X4 ) ) ) ) ).
% transfer_int_nat_quantifiers(1)
thf(fact_219_Nat__Transfer_Otransfer__nat__int__function__closures_I9_J,axiom,
! [Z2: nat] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( semiring_1_of_nat @ int @ Z2 ) ) ).
% Nat_Transfer.transfer_nat_int_function_closures(9)
thf(fact_220_less__fun__def,axiom,
! [B3: $tType,A: $tType] :
( ( ord @ B3 @ ( type2 @ B3 ) )
=> ( ( ord_less @ ( A > B3 ) )
= ( ^ [F3: A > B3,G2: A > B3] :
( ( ord_less_eq @ ( A > B3 ) @ F3 @ G2 )
& ~ ( ord_less_eq @ ( A > B3 ) @ G2 @ F3 ) ) ) ) ) ).
% less_fun_def
thf(fact_221_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_222_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_223_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_224_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_225_le__refl,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% le_refl
thf(fact_226_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J,axiom,
ord_less_eq @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ).
% Nat_Transfer.transfer_nat_int_function_closures(5)
thf(fact_227_Nat__Transfer_Otransfer__nat__int__function__closures_I4_J,axiom,
! [X: int,N: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( power_power @ int @ X @ N ) ) ) ).
% Nat_Transfer.transfer_nat_int_function_closures(4)
thf(fact_228_Nat__Transfer_Otransfer__int__nat__functions_I4_J,axiom,
! [X: nat,N: nat] :
( ( power_power @ int @ ( semiring_1_of_nat @ int @ X ) @ N )
= ( semiring_1_of_nat @ int @ ( power_power @ nat @ X @ N ) ) ) ).
% Nat_Transfer.transfer_int_nat_functions(4)
thf(fact_229_transfer__int__nat__relations_I3_J,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ X ) @ ( semiring_1_of_nat @ int @ Y2 ) )
= ( ord_less_eq @ nat @ X @ Y2 ) ) ).
% transfer_int_nat_relations(3)
thf(fact_230_transfer__int__nat__relations_I1_J,axiom,
! [X: nat,Y2: nat] :
( ( ( semiring_1_of_nat @ int @ X )
= ( semiring_1_of_nat @ int @ Y2 ) )
= ( X = Y2 ) ) ).
% transfer_int_nat_relations(1)
thf(fact_231_int__if__cong,axiom,
! [P: $o,X: nat,Y2: nat] :
( ( P
=> ( ( semiring_1_of_nat @ int @ X )
= ( semiring_1_of_nat @ int @ ( if @ nat @ P @ X @ Y2 ) ) ) )
& ( ~ P
=> ( ( semiring_1_of_nat @ int @ Y2 )
= ( semiring_1_of_nat @ int @ ( if @ nat @ P @ X @ Y2 ) ) ) ) ) ).
% int_if_cong
thf(fact_232_imp__le__cong,axiom,
! [X: int,X5: int,P: $o,P3: $o] :
( ( X = X5 )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X5 )
=> ( P = P3 ) )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> P )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X5 )
=> P3 ) ) ) ) ).
% imp_le_cong
thf(fact_233_conj__le__cong,axiom,
! [X: int,X5: int,P: $o,P3: $o] :
( ( X = X5 )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X5 )
=> ( P = P3 ) )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
& P )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X5 )
& P3 ) ) ) ) ).
% conj_le_cong
thf(fact_234_pinf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P3: A > $o,Q2: A > $o,Q3: A > $o] :
( ? [Z4: A] :
! [X2: A] :
( ( ord_less @ A @ Z4 @ X2 )
=> ( ( P @ X2 )
= ( P3 @ X2 ) ) )
=> ( ? [Z4: A] :
! [X2: A] :
( ( ord_less @ A @ Z4 @ X2 )
=> ( ( Q2 @ X2 )
= ( Q3 @ X2 ) ) )
=> ? [Z: A] :
! [X3: A] :
( ( ord_less @ A @ Z @ X3 )
=> ( ( ( P @ X3 )
& ( Q2 @ X3 ) )
= ( ( P3 @ X3 )
& ( Q3 @ X3 ) ) ) ) ) ) ) ).
% pinf(1)
thf(fact_235_pinf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P3: A > $o,Q2: A > $o,Q3: A > $o] :
( ? [Z4: A] :
! [X2: A] :
( ( ord_less @ A @ Z4 @ X2 )
=> ( ( P @ X2 )
= ( P3 @ X2 ) ) )
=> ( ? [Z4: A] :
! [X2: A] :
( ( ord_less @ A @ Z4 @ X2 )
=> ( ( Q2 @ X2 )
= ( Q3 @ X2 ) ) )
=> ? [Z: A] :
! [X3: A] :
( ( ord_less @ A @ Z @ X3 )
=> ( ( ( P @ X3 )
| ( Q2 @ X3 ) )
= ( ( P3 @ X3 )
| ( Q3 @ X3 ) ) ) ) ) ) ) ).
% pinf(2)
thf(fact_236_pinf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z: A] :
! [X3: A] :
( ( ord_less @ A @ Z @ X3 )
=> ( X3 != T2 ) ) ) ).
% pinf(3)
thf(fact_237_pinf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z: A] :
! [X3: A] :
( ( ord_less @ A @ Z @ X3 )
=> ( X3 != T2 ) ) ) ).
% pinf(4)
thf(fact_238_pinf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z: A] :
! [X3: A] :
( ( ord_less @ A @ Z @ X3 )
=> ~ ( ord_less @ A @ X3 @ T2 ) ) ) ).
% pinf(5)
thf(fact_239_pinf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z: A] :
! [X3: A] :
( ( ord_less @ A @ Z @ X3 )
=> ( ord_less @ A @ T2 @ X3 ) ) ) ).
% pinf(7)
thf(fact_240_pinf_I11_J,axiom,
! [C2: $tType,D2: $tType] :
( ( ord @ C2 @ ( type2 @ C2 ) )
=> ! [F4: D2] :
? [Z: C2] :
! [X3: C2] :
( ( ord_less @ C2 @ Z @ X3 )
=> ( F4 = F4 ) ) ) ).
% pinf(11)
thf(fact_241_minf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P3: A > $o,Q2: A > $o,Q3: A > $o] :
( ? [Z4: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z4 )
=> ( ( P @ X2 )
= ( P3 @ X2 ) ) )
=> ( ? [Z4: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z4 )
=> ( ( Q2 @ X2 )
= ( Q3 @ X2 ) ) )
=> ? [Z: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z )
=> ( ( ( P @ X3 )
& ( Q2 @ X3 ) )
= ( ( P3 @ X3 )
& ( Q3 @ X3 ) ) ) ) ) ) ) ).
% minf(1)
thf(fact_242_minf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P3: A > $o,Q2: A > $o,Q3: A > $o] :
( ? [Z4: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z4 )
=> ( ( P @ X2 )
= ( P3 @ X2 ) ) )
=> ( ? [Z4: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z4 )
=> ( ( Q2 @ X2 )
= ( Q3 @ X2 ) ) )
=> ? [Z: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z )
=> ( ( ( P @ X3 )
| ( Q2 @ X3 ) )
= ( ( P3 @ X3 )
| ( Q3 @ X3 ) ) ) ) ) ) ) ).
% minf(2)
thf(fact_243_minf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z )
=> ( X3 != T2 ) ) ) ).
% minf(3)
thf(fact_244_minf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z )
=> ( X3 != T2 ) ) ) ).
% minf(4)
thf(fact_245_minf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z )
=> ( ord_less @ A @ X3 @ T2 ) ) ) ).
% minf(5)
thf(fact_246_minf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z )
=> ~ ( ord_less @ A @ T2 @ X3 ) ) ) ).
% minf(7)
thf(fact_247_minf_I11_J,axiom,
! [C2: $tType,D2: $tType] :
( ( ord @ C2 @ ( type2 @ C2 ) )
=> ! [F4: D2] :
? [Z: C2] :
! [X3: C2] :
( ( ord_less @ C2 @ X3 @ Z )
=> ( F4 = F4 ) ) ) ).
% minf(11)
thf(fact_248_minf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z )
=> ~ ( ord_less_eq @ A @ T2 @ X3 ) ) ) ).
% minf(8)
thf(fact_249_minf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z )
=> ( ord_less_eq @ A @ X3 @ T2 ) ) ) ).
% minf(6)
thf(fact_250_pinf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z: A] :
! [X3: A] :
( ( ord_less @ A @ Z @ X3 )
=> ( ord_less_eq @ A @ T2 @ X3 ) ) ) ).
% pinf(8)
thf(fact_251_pinf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z: A] :
! [X3: A] :
( ( ord_less @ A @ Z @ X3 )
=> ~ ( ord_less_eq @ A @ X3 @ T2 ) ) ) ).
% pinf(6)
thf(fact_252_ex__has__greatest__nat,axiom,
! [A: $tType,P: A > $o,K: A,M: A > nat,B2: nat] :
( ( P @ K )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less @ nat @ ( M @ Y3 ) @ B2 ) )
=> ? [X2: A] :
( ( P @ X2 )
& ! [Y: A] :
( ( P @ Y )
=> ( ord_less_eq @ nat @ ( M @ Y ) @ ( M @ X2 ) ) ) ) ) ) ).
% ex_has_greatest_nat
thf(fact_253_realpow__pos__nth__unique,axiom,
! [N: nat,A2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ? [X2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X2 )
& ( ( power_power @ real @ X2 @ N )
= A2 )
& ! [Y: real] :
( ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
& ( ( power_power @ real @ Y @ N )
= A2 ) )
=> ( Y = X2 ) ) ) ) ) ).
% realpow_pos_nth_unique
%----Type constructors (64)
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A5: $tType,A6: $tType] :
( ( preorder @ A6 @ ( type2 @ A6 ) )
=> ( preorder @ ( A5 > A6 ) @ ( type2 @ ( A5 > A6 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A5: $tType,A6: $tType] :
( ( order @ A6 @ ( type2 @ A6 ) )
=> ( order @ ( A5 > A6 ) @ ( type2 @ ( A5 > A6 ) ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A5: $tType,A6: $tType] :
( ( ord @ A6 @ ( type2 @ A6 ) )
=> ( ord @ ( A5 > A6 ) @ ( type2 @ ( A5 > A6 ) ) ) ) ).
thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
condit1037483654norder @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Rings_Osemiring__1__no__zero__divisors,axiom,
semiri134348788visors @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
linordered_semidom @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
linordered_idom @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
semiring_char_0 @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Rings_Ozero__neq__one,axiom,
zero_neq_one @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Orderings_Opreorder_1,axiom,
preorder @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Orderings_Olinorder,axiom,
linorder @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Rings_Osemiring__1,axiom,
semiring_1 @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Orderings_Ono__top,axiom,
no_top @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Orderings_Ono__bot,axiom,
no_bot @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Orderings_Oorder_2,axiom,
order @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Orderings_Oord_3,axiom,
ord @ int @ ( type2 @ int ) ).
thf(tcon_Int_Oint___Groups_Ozero,axiom,
zero @ int @ ( type2 @ int ) ).
thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_4,axiom,
condit1037483654norder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
canoni770627133id_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_5,axiom,
semiri134348788visors @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Olinordered__semidom_6,axiom,
linordered_semidom @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
wellorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Nat_Osemiring__char__0_7,axiom,
semiring_char_0 @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Ozero__neq__one_8,axiom,
zero_neq_one @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Opreorder_9,axiom,
preorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Olinorder_10,axiom,
linorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Osemiring__1_11,axiom,
semiring_1 @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Ono__top_12,axiom,
no_top @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oorder_13,axiom,
order @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oord_14,axiom,
ord @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ozero_15,axiom,
zero @ nat @ ( type2 @ nat ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_16,axiom,
preorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder_17,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_18,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_19,axiom,
ord @ $o @ ( type2 @ $o ) ).
thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_20,axiom,
condit1037483654norder @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Olinear__continuum,axiom,
condit1656338222tinuum @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Archimedean__Field_Oarchimedean__field,axiom,
archim1804426504_field @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Rings_Osemiring__1__no__zero__divisors_21,axiom,
semiri134348788visors @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Rings_Olinordered__semidom_22,axiom,
linordered_semidom @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Orderings_Odense__linorder,axiom,
dense_linorder @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Fields_Olinordered__field,axiom,
linordered_field @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Rings_Olinordered__idom_23,axiom,
linordered_idom @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Orderings_Odense__order,axiom,
dense_order @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Fields_Odivision__ring,axiom,
division_ring @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Nat_Osemiring__char__0_24,axiom,
semiring_char_0 @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Rings_Ozero__neq__one_25,axiom,
zero_neq_one @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Orderings_Opreorder_26,axiom,
preorder @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Orderings_Olinorder_27,axiom,
linorder @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Rings_Osemiring__1_28,axiom,
semiring_1 @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Orderings_Ono__top_29,axiom,
no_top @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Orderings_Ono__bot_30,axiom,
no_bot @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Orderings_Oorder_31,axiom,
order @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Orderings_Oord_32,axiom,
ord @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Fields_Ofield,axiom,
field @ real @ ( type2 @ real ) ).
thf(tcon_Real_Oreal___Groups_Ozero_33,axiom,
zero @ real @ ( type2 @ real ) ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__1__no__zero__divisors_34,axiom,
semiri134348788visors @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Rings_Ozero__neq__one_35,axiom,
zero_neq_one @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Orderings_Opreorder_36,axiom,
preorder @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Orderings_Olinorder_37,axiom,
linorder @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__1_38,axiom,
semiring_1 @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Orderings_Oorder_39,axiom,
order @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Orderings_Oord_40,axiom,
ord @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Ozero_41,axiom,
zero @ code_natural @ ( type2 @ code_natural ) ).
%----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,X: A,Y2: A] :
( ( if @ A @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X: A,Y2: A] :
( ( if @ A @ $true @ X @ Y2 )
= X ) ).
%----Conjectures (2)
thf(conj_0,hypothesis,
hammin1195055315smooth @ n ).
thf(conj_1,conjecture,
ord_less @ nat @ ( zero_zero @ nat ) @ n ).
%------------------------------------------------------------------------------