TPTP Problem File: ITP194^2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP194^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Sturm_Tarski problem prob_257__5870986_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : Sturm_Tarski/prob_257__5870986_1 [Des21]
% Status : Theorem
% Rating : 0.00 v7.5.0
% Syntax : Number of formulae : 371 ( 77 unt; 53 typ; 0 def)
% Number of atoms : 994 ( 224 equ; 0 cnn)
% Maximal formula atoms : 13 ( 3 avg)
% Number of connectives : 3143 ( 120 ~; 29 |; 65 &;2398 @)
% ( 0 <=>; 531 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 8 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 114 ( 114 >; 0 *; 0 +; 0 <<)
% Number of symbols : 54 ( 51 usr; 7 con; 0-7 aty)
% Number of variables : 978 ( 53 ^; 834 !; 47 ?; 978 :)
% ( 44 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:26:51.513
%------------------------------------------------------------------------------
% Could-be-implicit typings (4)
thf(ty_t_Polynomial_Opoly,type,
poly: $tType > $tType ).
thf(ty_t_Real_Oreal,type,
real: $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
% Explicit typings (49)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oidom,type,
idom:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Nat_Oring__char__0,type,
ring_char_0:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ono__bot,type,
no_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ono__top,type,
no_top:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__ring__1,type,
comm_ring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oidom__divide,type,
idom_divide:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oidom__abs__sgn,type,
idom_abs_sgn:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Owellorder,type,
wellorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Odense__order,type,
dense_order:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__semiring__0,type,
comm_semiring_0:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__semiring__1,type,
comm_semiring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Olinordered__field,type,
linordered_field:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Odense__linorder,type,
dense_linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oring__no__zero__divisors,type,
ring_n68954251visors:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
linord1659791738miring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni770627133id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__normed__vector,type,
real_V55928688vector:
!>[A: $tType] : $o ).
thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
condit1656338222tinuum:
!>[A: $tType] : $o ).
thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
condit1037483654norder:
!>[A: $tType] : $o ).
thf(sy_c_Groups_Oone__class_Oone,type,
one_one:
!>[A: $tType] : A ).
thf(sy_c_Groups_Osgn__class_Osgn,type,
sgn_sgn:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Omin,type,
ord_min:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Polynomial_Odivide__poly__main,type,
divide_poly_main:
!>[A: $tType] : ( A > ( poly @ A ) > ( poly @ A ) > ( poly @ A ) > nat > nat > ( poly @ A ) ) ).
thf(sy_c_Polynomial_Ois__zero,type,
is_zero:
!>[A: $tType] : ( ( poly @ A ) > $o ) ).
thf(sy_c_Polynomial_Oorder,type,
order2:
!>[A: $tType] : ( A > ( poly @ A ) > nat ) ).
thf(sy_c_Polynomial_Opoly,type,
poly2:
!>[A: $tType] : ( ( poly @ A ) > A > A ) ).
thf(sy_c_Polynomial_Opoly__cutoff,type,
poly_cutoff:
!>[A: $tType] : ( nat > ( poly @ A ) > ( poly @ A ) ) ).
thf(sy_c_Polynomial_Opoly__shift,type,
poly_shift:
!>[A: $tType] : ( nat > ( poly @ A ) > ( poly @ A ) ) ).
thf(sy_c_Polynomial_Oreflect__poly,type,
reflect_poly:
!>[A: $tType] : ( ( poly @ A ) > ( poly @ A ) ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Sturm__Tarski__Mirabelle__osyjafhhev_Osgn__neg__inf,type,
sturm_1439882693eg_inf:
!>[A: $tType] : ( ( poly @ A ) > A ) ).
thf(sy_c_Sturm__Tarski__Mirabelle__osyjafhhev_Osgn__pos__inf,type,
sturm_62398785os_inf:
!>[A: $tType] : ( ( poly @ A ) > A ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_lb1____,type,
lb1: real ).
thf(sy_v_lb2____,type,
lb2: real ).
thf(sy_v_lb____,type,
lb: real ).
thf(sy_v_p,type,
p: poly @ real ).
thf(sy_v_thesis,type,
thesis: $o ).
% Relevant facts (254)
thf(fact_0_lb1,axiom,
! [X: real] :
( ( ( poly2 @ real @ p @ X )
= ( zero_zero @ real ) )
=> ( ord_less @ real @ lb1 @ X ) ) ).
% lb1
thf(fact_1_lb__def,axiom,
( lb
= ( ord_min @ real @ lb1 @ lb2 ) ) ).
% lb_def
thf(fact_2__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062lb1_O_A_092_060forall_062x_O_Apoly_Ap_Ax_A_061_A0_A_092_060longrightarrow_062_Alb1_A_060_Ax_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Lb1: real] :
~ ! [X: real] :
( ( ( poly2 @ real @ p @ X )
= ( zero_zero @ real ) )
=> ( ord_less @ real @ Lb1 @ X ) ) ).
% \<open>\<And>thesis. (\<And>lb1. \<forall>x. poly p x = 0 \<longrightarrow> lb1 < x \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_3_assms,axiom,
( p
!= ( zero_zero @ ( poly @ real ) ) ) ).
% assms
thf(fact_4_poly__0,axiom,
! [A: $tType] :
( ( comm_semiring_0 @ A )
=> ! [X2: A] :
( ( poly2 @ A @ ( zero_zero @ ( poly @ A ) ) @ X2 )
= ( zero_zero @ A ) ) ) ).
% poly_0
thf(fact_5_not__gr__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N: A] :
( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
= ( N
= ( zero_zero @ A ) ) ) ) ).
% not_gr_zero
thf(fact_6_poly__IVT__neg,axiom,
! [A2: real,B: real,P: poly @ real] :
( ( ord_less @ real @ A2 @ B )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( poly2 @ real @ P @ A2 ) )
=> ( ( ord_less @ real @ ( poly2 @ real @ P @ B ) @ ( zero_zero @ real ) )
=> ? [X3: real] :
( ( ord_less @ real @ A2 @ X3 )
& ( ord_less @ real @ X3 @ B )
& ( ( poly2 @ real @ P @ X3 )
= ( zero_zero @ real ) ) ) ) ) ) ).
% poly_IVT_neg
thf(fact_7_poly__IVT__pos,axiom,
! [A2: real,B: real,P: poly @ real] :
( ( ord_less @ real @ A2 @ B )
=> ( ( ord_less @ real @ ( poly2 @ real @ P @ A2 ) @ ( zero_zero @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( poly2 @ real @ P @ B ) )
=> ? [X3: real] :
( ( ord_less @ real @ A2 @ X3 )
& ( ord_less @ real @ X3 @ B )
& ( ( poly2 @ real @ P @ X3 )
= ( zero_zero @ real ) ) ) ) ) ) ).
% poly_IVT_pos
thf(fact_8_poly__all__0__iff__0,axiom,
! [A: $tType] :
( ( ( ring_char_0 @ A )
& ( comm_ring_1 @ A )
& ( ring_n68954251visors @ A ) )
=> ! [P: poly @ A] :
( ( ! [X4: A] :
( ( poly2 @ A @ P @ X4 )
= ( zero_zero @ A ) ) )
= ( P
= ( zero_zero @ ( poly @ A ) ) ) ) ) ).
% poly_all_0_iff_0
thf(fact_9_less__numeral__extra_I3_J,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% less_numeral_extra(3)
thf(fact_10_field__lbound__gt__zero,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [D1: A,D2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ D1 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ D2 )
=> ? [E: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E )
& ( ord_less @ A @ E @ D1 )
& ( ord_less @ A @ E @ D2 ) ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_11_gr__zeroI,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N: A] :
( ( N
!= ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% gr_zeroI
thf(fact_12_not__less__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N: A] :
~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% not_less_zero
thf(fact_13_gr__implies__not__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [M: A,N: A] :
( ( ord_less @ A @ M @ N )
=> ( N
!= ( zero_zero @ A ) ) ) ) ).
% gr_implies_not_zero
thf(fact_14_zero__less__iff__neq__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ N )
= ( N
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_iff_neq_zero
thf(fact_15_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X2: A] :
( ( ( zero_zero @ A )
= X2 )
= ( X2
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_16_poly__eq__poly__eq__iff,axiom,
! [A: $tType] :
( ( ( ring_char_0 @ A )
& ( comm_ring_1 @ A )
& ( ring_n68954251visors @ A ) )
=> ! [P: poly @ A,Q: poly @ A] :
( ( ( poly2 @ A @ P )
= ( poly2 @ A @ Q ) )
= ( P = Q ) ) ) ).
% poly_eq_poly_eq_iff
thf(fact_17_min__less__iff__conj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Z: A,X2: A,Y: A] :
( ( ord_less @ A @ Z @ ( ord_min @ A @ X2 @ Y ) )
= ( ( ord_less @ A @ Z @ X2 )
& ( ord_less @ A @ Z @ Y ) ) ) ) ).
% min_less_iff_conj
thf(fact_18_min_Oidem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A] :
( ( ord_min @ A @ A2 @ A2 )
= A2 ) ) ).
% min.idem
thf(fact_19_min_Oleft__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B: A] :
( ( ord_min @ A @ A2 @ ( ord_min @ A @ A2 @ B ) )
= ( ord_min @ A @ A2 @ B ) ) ) ).
% min.left_idem
thf(fact_20_min_Oright__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B: A] :
( ( ord_min @ A @ ( ord_min @ A @ A2 @ B ) @ B )
= ( ord_min @ A @ A2 @ B ) ) ) ).
% min.right_idem
thf(fact_21_min_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B: A,C: A,A2: A] :
( ( ord_less @ A @ B @ C )
=> ( ord_less @ A @ ( ord_min @ A @ A2 @ B ) @ C ) ) ) ).
% min.strict_coboundedI2
thf(fact_22_min_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,C: A,B: A] :
( ( ord_less @ A @ A2 @ C )
=> ( ord_less @ A @ ( ord_min @ A @ A2 @ B ) @ C ) ) ) ).
% min.strict_coboundedI1
thf(fact_23_min_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less @ A )
= ( ^ [A3: A,B2: A] :
( ( A3
= ( ord_min @ A @ A3 @ B2 ) )
& ( A3 != B2 ) ) ) ) ) ).
% min.strict_order_iff
thf(fact_24_min_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B: A,C: A] :
( ( ord_less @ A @ A2 @ ( ord_min @ A @ B @ C ) )
=> ~ ( ( ord_less @ A @ A2 @ B )
=> ~ ( ord_less @ A @ A2 @ C ) ) ) ) ).
% min.strict_boundedE
thf(fact_25_min__less__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A,Z: A] :
( ( ord_less @ A @ ( ord_min @ A @ X2 @ Y ) @ Z )
= ( ( ord_less @ A @ X2 @ Z )
| ( ord_less @ A @ Y @ Z ) ) ) ) ).
% min_less_iff_disj
thf(fact_26_is__zero__null,axiom,
! [A: $tType] :
( ( zero @ A )
=> ( ( is_zero @ A )
= ( ^ [P2: poly @ A] :
( P2
= ( zero_zero @ ( poly @ A ) ) ) ) ) ) ).
% is_zero_null
thf(fact_27_min_Oleft__commute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B: A,A2: A,C: A] :
( ( ord_min @ A @ B @ ( ord_min @ A @ A2 @ C ) )
= ( ord_min @ A @ A2 @ ( ord_min @ A @ B @ C ) ) ) ) ).
% min.left_commute
thf(fact_28_min_Ocommute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_min @ A )
= ( ^ [A3: A,B2: A] : ( ord_min @ A @ B2 @ A3 ) ) ) ) ).
% min.commute
thf(fact_29_min_Oassoc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B: A,C: A] :
( ( ord_min @ A @ ( ord_min @ A @ A2 @ B ) @ C )
= ( ord_min @ A @ A2 @ ( ord_min @ A @ B @ C ) ) ) ) ).
% min.assoc
thf(fact_30_poly__cutoff__0,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [N: nat] :
( ( poly_cutoff @ A @ N @ ( zero_zero @ ( poly @ A ) ) )
= ( zero_zero @ ( poly @ A ) ) ) ) ).
% poly_cutoff_0
thf(fact_31_last__non__root__interval,axiom,
! [P: poly @ real,Ub: real] :
( ( P
!= ( zero_zero @ ( poly @ real ) ) )
=> ~ ! [Lb: real] :
( ( ord_less @ real @ Lb @ Ub )
=> ~ ! [Z2: real] :
( ( ( ord_less_eq @ real @ Lb @ Z2 )
& ( ord_less @ real @ Z2 @ Ub ) )
=> ( ( poly2 @ real @ P @ Z2 )
!= ( zero_zero @ real ) ) ) ) ) ).
% last_non_root_interval
thf(fact_32_next__non__root__interval,axiom,
! [P: poly @ real,Lb2: real] :
( ( P
!= ( zero_zero @ ( poly @ real ) ) )
=> ~ ! [Ub2: real] :
( ( ord_less @ real @ Lb2 @ Ub2 )
=> ~ ! [Z2: real] :
( ( ( ord_less @ real @ Lb2 @ Z2 )
& ( ord_less_eq @ real @ Z2 @ Ub2 ) )
=> ( ( poly2 @ real @ P @ Z2 )
!= ( zero_zero @ real ) ) ) ) ) ).
% next_non_root_interval
thf(fact_33_reflect__poly__at__0__eq__0__iff,axiom,
! [A: $tType] :
( ( comm_semiring_0 @ A )
=> ! [P: poly @ A] :
( ( ( poly2 @ A @ ( reflect_poly @ A @ P ) @ ( zero_zero @ A ) )
= ( zero_zero @ A ) )
= ( P
= ( zero_zero @ ( poly @ A ) ) ) ) ) ).
% reflect_poly_at_0_eq_0_iff
thf(fact_34_lb2,axiom,
! [X: real] :
( ( ord_less_eq @ real @ X @ lb2 )
=> ( ( sgn_sgn @ real @ ( poly2 @ real @ p @ X ) )
= ( sturm_1439882693eg_inf @ real @ p ) ) ) ).
% lb2
thf(fact_35_poly__shift__0,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [N: nat] :
( ( poly_shift @ A @ N @ ( zero_zero @ ( poly @ A ) ) )
= ( zero_zero @ ( poly @ A ) ) ) ) ).
% poly_shift_0
thf(fact_36_order__root,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [P: poly @ A,A2: A] :
( ( ( poly2 @ A @ P @ A2 )
= ( zero_zero @ A ) )
= ( ( P
= ( zero_zero @ ( poly @ A ) ) )
| ( ( order2 @ A @ A2 @ P )
!= ( zero_zero @ nat ) ) ) ) ) ).
% order_root
thf(fact_37_divide__poly__main__0,axiom,
! [A: $tType] :
( ( idom_divide @ A )
=> ! [R: poly @ A,D: poly @ A,Dr: nat,N: nat] :
( ( divide_poly_main @ A @ ( zero_zero @ A ) @ ( zero_zero @ ( poly @ A ) ) @ R @ D @ Dr @ N )
= ( zero_zero @ ( poly @ A ) ) ) ) ).
% divide_poly_main_0
thf(fact_38_not__eq__pos__or__neg__iff__1,axiom,
! [Lb2: real,Ub: real,P: poly @ real] :
( ( ! [Z3: real] :
( ( ( ord_less @ real @ Lb2 @ Z3 )
& ( ord_less_eq @ real @ Z3 @ Ub ) )
=> ( ( poly2 @ real @ P @ Z3 )
!= ( zero_zero @ real ) ) ) )
= ( ! [Z3: real] :
( ( ( ord_less @ real @ Lb2 @ Z3 )
& ( ord_less_eq @ real @ Z3 @ Ub ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( poly2 @ real @ P @ Z3 ) ) )
| ! [Z3: real] :
( ( ( ord_less @ real @ Lb2 @ Z3 )
& ( ord_less_eq @ real @ Z3 @ Ub ) )
=> ( ord_less @ real @ ( poly2 @ real @ P @ Z3 ) @ ( zero_zero @ real ) ) ) ) ) ).
% not_eq_pos_or_neg_iff_1
thf(fact_39__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062lb2_O_A_092_060forall_062x_092_060le_062lb2_O_Asgn_A_Ipoly_Ap_Ax_J_A_061_Asgn__neg__inf_Ap_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Lb22: real] :
~ ! [X: real] :
( ( ord_less_eq @ real @ X @ Lb22 )
=> ( ( sgn_sgn @ real @ ( poly2 @ real @ p @ X ) )
= ( sturm_1439882693eg_inf @ real @ p ) ) ) ).
% \<open>\<And>thesis. (\<And>lb2. \<forall>x\<le>lb2. sgn (poly p x) = sgn_neg_inf p \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_40_le__zero__eq,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N: A] :
( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
= ( N
= ( zero_zero @ A ) ) ) ) ).
% le_zero_eq
thf(fact_41_min_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B: A,C: A] :
( ( ord_less_eq @ A @ A2 @ ( ord_min @ A @ B @ C ) )
= ( ( ord_less_eq @ A @ A2 @ B )
& ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% min.bounded_iff
thf(fact_42_reflect__poly__0,axiom,
! [A: $tType] :
( ( zero @ A )
=> ( ( reflect_poly @ A @ ( zero_zero @ ( poly @ A ) ) )
= ( zero_zero @ ( poly @ A ) ) ) ) ).
% reflect_poly_0
thf(fact_43_poly__shift__id,axiom,
! [A: $tType] :
( ( zero @ A )
=> ( ( poly_shift @ A @ ( zero_zero @ nat ) )
= ( ^ [X4: poly @ A] : X4 ) ) ) ).
% poly_shift_id
thf(fact_44_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P3: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P3 ) )
= ( P3 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( collect @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_46_Collect__cong,axiom,
! [A: $tType,P3: A > $o,Q2: A > $o] :
( ! [X3: A] :
( ( P3 @ X3 )
= ( Q2 @ X3 ) )
=> ( ( collect @ A @ P3 )
= ( collect @ A @ Q2 ) ) ) ).
% Collect_cong
thf(fact_47_ext,axiom,
! [B3: $tType,A: $tType,F: A > B3,G: A > B3] :
( ! [X3: A] :
( ( F @ X3 )
= ( G @ X3 ) )
=> ( F = G ) ) ).
% ext
thf(fact_48_that,axiom,
! [Lb2: real] :
( ! [X3: real] :
( ( ( poly2 @ real @ p @ X3 )
= ( zero_zero @ real ) )
=> ( ord_less @ real @ Lb2 @ X3 ) )
=> ( ! [X3: real] :
( ( ord_less_eq @ real @ X3 @ Lb2 )
=> ( ( sgn_sgn @ real @ ( poly2 @ real @ p @ X3 ) )
= ( sturm_1439882693eg_inf @ real @ p ) ) )
=> thesis ) ) ).
% that
thf(fact_49_complete__real,axiom,
! [S: set @ real] :
( ? [X: real] : ( member @ real @ X @ S )
=> ( ? [Z2: real] :
! [X3: real] :
( ( member @ real @ X3 @ S )
=> ( ord_less_eq @ real @ X3 @ Z2 ) )
=> ? [Y2: real] :
( ! [X: real] :
( ( member @ real @ X @ S )
=> ( ord_less_eq @ real @ X @ Y2 ) )
& ! [Z2: real] :
( ! [X3: real] :
( ( member @ real @ X3 @ S )
=> ( ord_less_eq @ real @ X3 @ Z2 ) )
=> ( ord_less_eq @ real @ Y2 @ Z2 ) ) ) ) ) ).
% complete_real
thf(fact_50_divide__poly__main_Osimps_I2_J,axiom,
! [A: $tType] :
( ( idom_divide @ A )
=> ! [Lc: A,Q: poly @ A,R: poly @ A,D: poly @ A,Dr: nat] :
( ( divide_poly_main @ A @ Lc @ Q @ R @ D @ Dr @ ( zero_zero @ nat ) )
= Q ) ) ).
% divide_poly_main.simps(2)
thf(fact_51_zero__le,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [X2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 ) ) ).
% zero_le
thf(fact_52_le__numeral__extra_I3_J,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% le_numeral_extra(3)
thf(fact_53_less__eq__real__def,axiom,
( ( ord_less_eq @ real )
= ( ^ [X4: real,Y3: real] :
( ( ord_less @ real @ X4 @ Y3 )
| ( X4 = Y3 ) ) ) ) ).
% less_eq_real_def
thf(fact_54_min_Omono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,C: A,B: A,D: A] :
( ( ord_less_eq @ A @ A2 @ C )
=> ( ( ord_less_eq @ A @ B @ D )
=> ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B ) @ ( ord_min @ A @ C @ D ) ) ) ) ) ).
% min.mono
thf(fact_55_min_OorderE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( A2
= ( ord_min @ A @ A2 @ B ) ) ) ) ).
% min.orderE
thf(fact_56_min_OorderI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B: A] :
( ( A2
= ( ord_min @ A @ A2 @ B ) )
=> ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% min.orderI
thf(fact_57_min_Oabsorb1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( ord_min @ A @ A2 @ B )
= A2 ) ) ) ).
% min.absorb1
thf(fact_58_min_Oabsorb2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B: A,A2: A] :
( ( ord_less_eq @ A @ B @ A2 )
=> ( ( ord_min @ A @ A2 @ B )
= B ) ) ) ).
% min.absorb2
thf(fact_59_min_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B: A,C: A] :
( ( ord_less_eq @ A @ A2 @ ( ord_min @ A @ B @ C ) )
=> ~ ( ( ord_less_eq @ A @ A2 @ B )
=> ~ ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% min.boundedE
thf(fact_60_min_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( ord_less_eq @ A @ A2 @ C )
=> ( ord_less_eq @ A @ A2 @ ( ord_min @ A @ B @ C ) ) ) ) ) ).
% min.boundedI
thf(fact_61_min_Oorder__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A3: A,B2: A] :
( A3
= ( ord_min @ A @ A3 @ B2 ) ) ) ) ) ).
% min.order_iff
thf(fact_62_min_Ocobounded1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B: A] : ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B ) @ A2 ) ) ).
% min.cobounded1
thf(fact_63_min_Ocobounded2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B: A] : ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B ) @ B ) ) ).
% min.cobounded2
thf(fact_64_min_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A3: A,B2: A] :
( ( ord_min @ A @ A3 @ B2 )
= A3 ) ) ) ) ).
% min.absorb_iff1
thf(fact_65_min_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B2: A,A3: A] :
( ( ord_min @ A @ A3 @ B2 )
= B2 ) ) ) ) ).
% min.absorb_iff2
thf(fact_66_min_OcoboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,C: A,B: A] :
( ( ord_less_eq @ A @ A2 @ C )
=> ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B ) @ C ) ) ) ).
% min.coboundedI1
thf(fact_67_min_OcoboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B: A,C: A,A2: A] :
( ( ord_less_eq @ A @ B @ C )
=> ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B ) @ C ) ) ) ).
% min.coboundedI2
thf(fact_68_min__le__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ ( ord_min @ A @ X2 @ Y ) @ Z )
= ( ( ord_less_eq @ A @ X2 @ Z )
| ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% min_le_iff_disj
thf(fact_69_order__0I,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [P: poly @ A,A2: A] :
( ( ( poly2 @ A @ P @ A2 )
!= ( zero_zero @ A ) )
=> ( ( order2 @ A @ A2 @ P )
= ( zero_zero @ nat ) ) ) ) ).
% order_0I
thf(fact_70_not__eq__pos__or__neg__iff__2,axiom,
! [Lb2: real,Ub: real,P: poly @ real] :
( ( ! [Z3: real] :
( ( ( ord_less_eq @ real @ Lb2 @ Z3 )
& ( ord_less @ real @ Z3 @ Ub ) )
=> ( ( poly2 @ real @ P @ Z3 )
!= ( zero_zero @ real ) ) ) )
= ( ! [Z3: real] :
( ( ( ord_less_eq @ real @ Lb2 @ Z3 )
& ( ord_less @ real @ Z3 @ Ub ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( poly2 @ real @ P @ Z3 ) ) )
| ! [Z3: real] :
( ( ( ord_less_eq @ real @ Lb2 @ Z3 )
& ( ord_less @ real @ Z3 @ Ub ) )
=> ( ord_less @ real @ ( poly2 @ real @ P @ Z3 ) @ ( zero_zero @ real ) ) ) ) ) ).
% not_eq_pos_or_neg_iff_2
thf(fact_71_zero__le__sgn__iff,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sgn_sgn @ real @ X2 ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 ) ) ).
% zero_le_sgn_iff
thf(fact_72_sgn__le__0__iff,axiom,
! [X2: real] :
( ( ord_less_eq @ real @ ( sgn_sgn @ real @ X2 ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X2 @ ( zero_zero @ real ) ) ) ).
% sgn_le_0_iff
thf(fact_73_sgn__greater,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( sgn_sgn @ A @ A2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% sgn_greater
thf(fact_74_sgn__less,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( sgn_sgn @ A @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% sgn_less
thf(fact_75_root__ub,axiom,
! [P: poly @ real] :
( ( P
!= ( zero_zero @ ( poly @ real ) ) )
=> ~ ! [Ub2: real] :
( ! [X: real] :
( ( ( poly2 @ real @ P @ X )
= ( zero_zero @ real ) )
=> ( ord_less @ real @ X @ Ub2 ) )
=> ~ ! [X: real] :
( ( ord_less_eq @ real @ Ub2 @ X )
=> ( ( sgn_sgn @ real @ ( poly2 @ real @ P @ X ) )
= ( sturm_62398785os_inf @ real @ P ) ) ) ) ) ).
% root_ub
thf(fact_76_sgn__0,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( sgn_sgn @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% sgn_0
thf(fact_77_sgn__zero,axiom,
! [A: $tType] :
( ( real_V55928688vector @ A )
=> ( ( sgn_sgn @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% sgn_zero
thf(fact_78_sgn__sgn,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A2: A] :
( ( sgn_sgn @ A @ ( sgn_sgn @ A @ A2 ) )
= ( sgn_sgn @ A @ A2 ) ) ) ).
% sgn_sgn
thf(fact_79_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X2: A,Y: A] :
( ( X2 != Y )
=> ( ~ ( ord_less @ A @ X2 @ Y )
=> ( ord_less @ A @ Y @ X2 ) ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_80_sgn__eq__0__iff,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A2: A] :
( ( ( sgn_sgn @ A @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% sgn_eq_0_iff
thf(fact_81_sgn__0__0,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( ( sgn_sgn @ A @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% sgn_0_0
thf(fact_82_sgn__zero__iff,axiom,
! [A: $tType] :
( ( real_V55928688vector @ A )
=> ! [X2: A] :
( ( ( sgn_sgn @ A @ X2 )
= ( zero_zero @ A ) )
= ( X2
= ( zero_zero @ A ) ) ) ) ).
% sgn_zero_iff
thf(fact_83_le0,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% le0
thf(fact_84_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A2 ) ).
% bot_nat_0.extremum
thf(fact_85_neq0__conv,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% neq0_conv
thf(fact_86_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% less_nat_zero_code
thf(fact_87_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_88_min__0R,axiom,
! [N: nat] :
( ( ord_min @ nat @ N @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% min_0R
thf(fact_89_min__0L,axiom,
! [N: nat] :
( ( ord_min @ nat @ ( zero_zero @ nat ) @ N )
= ( zero_zero @ nat ) ) ).
% min_0L
thf(fact_90_le__refl,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% le_refl
thf(fact_91_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_92_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_93_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_94_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_95_Nat_Oex__has__greatest__nat,axiom,
! [P3: nat > $o,K: nat,B: nat] :
( ( P3 @ K )
=> ( ! [Y2: nat] :
( ( P3 @ Y2 )
=> ( ord_less_eq @ nat @ Y2 @ B ) )
=> ? [X3: nat] :
( ( P3 @ X3 )
& ! [Y4: nat] :
( ( P3 @ Y4 )
=> ( ord_less_eq @ nat @ Y4 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_96_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_97_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_98_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_99_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_100_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_101_nat__less__le,axiom,
( ( ord_less @ nat )
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
& ( M2 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_102_infinite__descent__measure,axiom,
! [A: $tType,P3: A > $o,V: A > nat,X2: A] :
( ! [X3: A] :
( ~ ( P3 @ X3 )
=> ? [Y4: A] :
( ( ord_less @ nat @ ( V @ Y4 ) @ ( V @ X3 ) )
& ~ ( P3 @ Y4 ) ) )
=> ( P3 @ X2 ) ) ).
% infinite_descent_measure
thf(fact_103_linorder__neqE__nat,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
=> ( ~ ( ord_less @ nat @ X2 @ Y )
=> ( ord_less @ nat @ Y @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_104_infinite__descent,axiom,
! [P3: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P3 @ N3 )
=> ? [M3: nat] :
( ( ord_less @ nat @ M3 @ N3 )
& ~ ( P3 @ M3 ) ) )
=> ( P3 @ N ) ) ).
% infinite_descent
thf(fact_105_nat__less__induct,axiom,
! [P3: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less @ nat @ M3 @ N3 )
=> ( P3 @ M3 ) )
=> ( P3 @ N3 ) )
=> ( P3 @ N ) ) ).
% nat_less_induct
thf(fact_106_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_107_less__not__refl3,axiom,
! [S2: nat,T: nat] :
( ( ord_less @ nat @ S2 @ T )
=> ( S2 != T ) ) ).
% less_not_refl3
thf(fact_108_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_109_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_not_refl
thf(fact_110_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_111_measure__induct__rule,axiom,
! [B3: $tType,A: $tType] :
( ( wellorder @ B3 )
=> ! [F: A > B3,P3: A > $o,A2: A] :
( ! [X3: A] :
( ! [Y4: A] :
( ( ord_less @ B3 @ ( F @ Y4 ) @ ( F @ X3 ) )
=> ( P3 @ Y4 ) )
=> ( P3 @ X3 ) )
=> ( P3 @ A2 ) ) ) ).
% measure_induct_rule
thf(fact_112_measure__induct,axiom,
! [B3: $tType,A: $tType] :
( ( wellorder @ B3 )
=> ! [F: A > B3,P3: A > $o,A2: A] :
( ! [X3: A] :
( ! [Y4: A] :
( ( ord_less @ B3 @ ( F @ Y4 ) @ ( F @ X3 ) )
=> ( P3 @ Y4 ) )
=> ( P3 @ X3 ) )
=> ( P3 @ A2 ) ) ) ).
% measure_induct
thf(fact_113_infinite__descent0__measure,axiom,
! [A: $tType,V: A > nat,P3: A > $o,X2: A] :
( ! [X3: A] :
( ( ( V @ X3 )
= ( zero_zero @ nat ) )
=> ( P3 @ X3 ) )
=> ( ! [X3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V @ X3 ) )
=> ( ~ ( P3 @ X3 )
=> ? [Y4: A] :
( ( ord_less @ nat @ ( V @ Y4 ) @ ( V @ X3 ) )
& ~ ( P3 @ Y4 ) ) ) )
=> ( P3 @ X2 ) ) ) ).
% infinite_descent0_measure
thf(fact_114_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less @ nat @ A2 @ ( zero_zero @ nat ) ) ).
% bot_nat_0.extremum_strict
thf(fact_115_infinite__descent0,axiom,
! [P3: nat > $o,N: nat] :
( ( P3 @ ( zero_zero @ nat ) )
=> ( ! [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ~ ( P3 @ N3 )
=> ? [M3: nat] :
( ( ord_less @ nat @ M3 @ N3 )
& ~ ( P3 @ M3 ) ) ) )
=> ( P3 @ N ) ) ) ).
% infinite_descent0
thf(fact_116_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( N
!= ( zero_zero @ nat ) ) ) ).
% gr_implies_not0
thf(fact_117_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% less_zeroE
thf(fact_118_not__less0,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% not_less0
thf(fact_119_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% not_gr0
thf(fact_120_gr0I,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% gr0I
thf(fact_121_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% less_eq_nat.simps(1)
thf(fact_122_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% le_0_eq
thf(fact_123_ex__least__nat__le,axiom,
! [P3: nat > $o,N: nat] :
( ( P3 @ N )
=> ( ~ ( P3 @ ( zero_zero @ nat ) )
=> ? [K2: nat] :
( ( ord_less_eq @ nat @ K2 @ N )
& ! [I3: nat] :
( ( ord_less @ nat @ I3 @ K2 )
=> ~ ( P3 @ I3 ) )
& ( P3 @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_124_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
= ( A2
= ( zero_zero @ nat ) ) ) ).
% bot_nat_0.extremum_unique
thf(fact_125_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
=> ( A2
= ( zero_zero @ nat ) ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_126_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% order_refl
thf(fact_127_min__absorb1,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ( ord_min @ A @ X2 @ Y )
= X2 ) ) ) ).
% min_absorb1
thf(fact_128_min__absorb2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X2: A] :
( ( ord_less_eq @ A @ Y @ X2 )
=> ( ( ord_min @ A @ X2 @ Y )
= Y ) ) ) ).
% min_absorb2
thf(fact_129_min__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_min @ A )
= ( ^ [A3: A,B2: A] : ( if @ A @ ( ord_less_eq @ A @ A3 @ B2 ) @ A3 @ B2 ) ) ) ) ).
% min_def
thf(fact_130_Lattices__Big_Oex__has__greatest__nat,axiom,
! [A: $tType,P3: A > $o,K: A,F: A > nat,B: nat] :
( ( P3 @ K )
=> ( ! [Y2: A] :
( ( P3 @ Y2 )
=> ( ord_less @ nat @ ( F @ Y2 ) @ B ) )
=> ? [X3: A] :
( ( P3 @ X3 )
& ! [Y4: A] :
( ( P3 @ Y4 )
=> ( ord_less_eq @ nat @ ( F @ Y4 ) @ ( F @ X3 ) ) ) ) ) ) ).
% Lattices_Big.ex_has_greatest_nat
thf(fact_131_nat__descend__induct,axiom,
! [N: nat,P3: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less @ nat @ N @ K2 )
=> ( P3 @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq @ nat @ K2 @ N )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ K2 @ I3 )
=> ( P3 @ I3 ) )
=> ( P3 @ K2 ) ) )
=> ( P3 @ M ) ) ) ).
% nat_descend_induct
thf(fact_132_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B: A,A2: A] :
( ( ord_less_eq @ A @ B @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ B )
=> ( A2 = B ) ) ) ) ).
% dual_order.antisym
thf(fact_133_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y5: A,Z4: A] : Y5 = Z4 )
= ( ^ [A3: A,B2: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
& ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_134_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B: A,A2: A,C: A] :
( ( ord_less_eq @ A @ B @ A2 )
=> ( ( ord_less_eq @ A @ C @ B )
=> ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% dual_order.trans
thf(fact_135_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P3: A > A > $o,A2: A,B: A] :
( ! [A5: A,B4: A] :
( ( ord_less_eq @ A @ A5 @ B4 )
=> ( P3 @ A5 @ B4 ) )
=> ( ! [A5: A,B4: A] :
( ( P3 @ B4 @ A5 )
=> ( P3 @ A5 @ B4 ) )
=> ( P3 @ A2 @ B ) ) ) ) ).
% linorder_wlog
thf(fact_136_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_137_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z )
=> ( ord_less_eq @ A @ X2 @ Z ) ) ) ) ).
% order_trans
thf(fact_138_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( ord_less_eq @ A @ B @ A2 )
=> ( A2 = B ) ) ) ) ).
% order_class.order.antisym
thf(fact_139_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% ord_le_eq_trans
thf(fact_140_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B: A,C: A] :
( ( A2 = B )
=> ( ( ord_less_eq @ A @ B @ C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% ord_eq_le_trans
thf(fact_141_order__class_Oorder_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y5: A,Z4: A] : Y5 = Z4 )
= ( ^ [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
& ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_142_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X2: A] :
( ( ord_less_eq @ A @ Y @ X2 )
=> ( ( ord_less_eq @ A @ X2 @ Y )
= ( X2 = Y ) ) ) ) ).
% antisym_conv
thf(fact_143_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A,Z: A] :
( ( ( ord_less_eq @ A @ X2 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z ) )
=> ( ( ( ord_less_eq @ A @ Y @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ Z ) )
=> ( ( ( ord_less_eq @ A @ X2 @ Z )
=> ~ ( ord_less_eq @ A @ Z @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X2 ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z )
=> ~ ( ord_less_eq @ A @ Z @ X2 ) )
=> ~ ( ( ord_less_eq @ A @ Z @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_144_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( ord_less_eq @ A @ B @ C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% order.trans
thf(fact_145_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ~ ( ord_less_eq @ A @ X2 @ Y )
=> ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% le_cases
thf(fact_146_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A] :
( ( X2 = Y )
=> ( ord_less_eq @ A @ X2 @ Y ) ) ) ).
% eq_refl
thf(fact_147_linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
| ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% linear
thf(fact_148_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ( ord_less_eq @ A @ Y @ X2 )
=> ( X2 = Y ) ) ) ) ).
% antisym
thf(fact_149_eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y5: A,Z4: A] : Y5 = Z4 )
= ( ^ [X4: A,Y3: A] :
( ( ord_less_eq @ A @ X4 @ Y3 )
& ( ord_less_eq @ A @ Y3 @ X4 ) ) ) ) ) ).
% eq_iff
thf(fact_150_ord__le__eq__subst,axiom,
! [A: $tType,B3: $tType] :
( ( ( ord @ B3 )
& ( ord @ A ) )
=> ! [A2: A,B: A,F: A > B3,C: B3] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ord_less_eq @ B3 @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ B3 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_151_ord__eq__le__subst,axiom,
! [A: $tType,B3: $tType] :
( ( ( ord @ B3 )
& ( ord @ A ) )
=> ! [A2: A,F: B3 > A,B: B3,C: B3] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq @ B3 @ B @ C )
=> ( ! [X3: B3,Y2: B3] :
( ( ord_less_eq @ B3 @ X3 @ Y2 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_152_order__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 )
& ( order @ A ) )
=> ! [A2: A,B: A,F: A > C2,C: C2] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( ord_less_eq @ C2 @ ( F @ B ) @ C )
=> ( ! [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ord_less_eq @ C2 @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_subst2
thf(fact_153_order__subst1,axiom,
! [A: $tType,B3: $tType] :
( ( ( order @ B3 )
& ( order @ A ) )
=> ! [A2: A,F: B3 > A,B: B3,C: B3] :
( ( ord_less_eq @ A @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq @ B3 @ B @ C )
=> ( ! [X3: B3,Y2: B3] :
( ( ord_less_eq @ B3 @ X3 @ Y2 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_subst1
thf(fact_154_le__fun__def,axiom,
! [B3: $tType,A: $tType] :
( ( ord @ B3 )
=> ( ( ord_less_eq @ ( A > B3 ) )
= ( ^ [F2: A > B3,G2: A > B3] :
! [X4: A] : ( ord_less_eq @ B3 @ ( F2 @ X4 ) @ ( G2 @ X4 ) ) ) ) ) ).
% le_fun_def
thf(fact_155_le__funI,axiom,
! [B3: $tType,A: $tType] :
( ( ord @ B3 )
=> ! [F: A > B3,G: A > B3] :
( ! [X3: A] : ( ord_less_eq @ B3 @ ( F @ X3 ) @ ( G @ X3 ) )
=> ( ord_less_eq @ ( A > B3 ) @ F @ G ) ) ) ).
% le_funI
thf(fact_156_le__funE,axiom,
! [B3: $tType,A: $tType] :
( ( ord @ B3 )
=> ! [F: A > B3,G: A > B3,X2: A] :
( ( ord_less_eq @ ( A > B3 ) @ F @ G )
=> ( ord_less_eq @ B3 @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% le_funE
thf(fact_157_le__funD,axiom,
! [B3: $tType,A: $tType] :
( ( ord @ B3 )
=> ! [F: A > B3,G: A > B3,X2: A] :
( ( ord_less_eq @ ( A > B3 ) @ F @ G )
=> ( ord_less_eq @ B3 @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% le_funD
thf(fact_158_ord__eq__less__subst,axiom,
! [A: $tType,B3: $tType] :
( ( ( ord @ B3 )
& ( ord @ A ) )
=> ! [A2: A,F: B3 > A,B: B3,C: B3] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less @ B3 @ B @ C )
=> ( ! [X3: B3,Y2: B3] :
( ( ord_less @ B3 @ X3 @ Y2 )
=> ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_159_ord__less__eq__subst,axiom,
! [A: $tType,B3: $tType] :
( ( ( ord @ B3 )
& ( ord @ A ) )
=> ! [A2: A,B: A,F: A > B3,C: B3] :
( ( ord_less @ A @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( ord_less @ B3 @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ B3 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_160_order__less__subst1,axiom,
! [A: $tType,B3: $tType] :
( ( ( order @ B3 )
& ( order @ A ) )
=> ! [A2: A,F: B3 > A,B: B3,C: B3] :
( ( ord_less @ A @ A2 @ ( F @ B ) )
=> ( ( ord_less @ B3 @ B @ C )
=> ( ! [X3: B3,Y2: B3] :
( ( ord_less @ B3 @ X3 @ Y2 )
=> ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_161_order__less__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 )
& ( order @ A ) )
=> ! [A2: A,B: A,F: A > C2,C: C2] :
( ( ord_less @ A @ A2 @ B )
=> ( ( ord_less @ C2 @ ( F @ B ) @ C )
=> ( ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( ord_less @ C2 @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_less_subst2
thf(fact_162_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [X2: A] :
? [Y2: A] : ( ord_less @ A @ Y2 @ X2 ) ) ).
% lt_ex
thf(fact_163_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [X2: A] :
? [X_1: A] : ( ord_less @ A @ X2 @ X_1 ) ) ).
% gt_ex
thf(fact_164_neqE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ( X2 != Y )
=> ( ~ ( ord_less @ A @ X2 @ Y )
=> ( ord_less @ A @ Y @ X2 ) ) ) ) ).
% neqE
thf(fact_165_neq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ( X2 != Y )
= ( ( ord_less @ A @ X2 @ Y )
| ( ord_less @ A @ Y @ X2 ) ) ) ) ).
% neq_iff
thf(fact_166_order_Oasym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ A2 @ B )
=> ~ ( ord_less @ A @ B @ A2 ) ) ) ).
% order.asym
thf(fact_167_dense,axiom,
! [A: $tType] :
( ( dense_order @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ Y )
=> ? [Z5: A] :
( ( ord_less @ A @ X2 @ Z5 )
& ( ord_less @ A @ Z5 @ Y ) ) ) ) ).
% dense
thf(fact_168_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ Y )
=> ( X2 != Y ) ) ) ).
% less_imp_neq
thf(fact_169_less__asym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ Y )
=> ~ ( ord_less @ A @ Y @ X2 ) ) ) ).
% less_asym
thf(fact_170_less__asym_H,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ A2 @ B )
=> ~ ( ord_less @ A @ B @ A2 ) ) ) ).
% less_asym'
thf(fact_171_less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A,Z: A] :
( ( ord_less @ A @ X2 @ Y )
=> ( ( ord_less @ A @ Y @ Z )
=> ( ord_less @ A @ X2 @ Z ) ) ) ) ).
% less_trans
thf(fact_172_less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ Y )
| ( X2 = Y )
| ( ord_less @ A @ Y @ X2 ) ) ) ).
% less_linear
thf(fact_173_less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A] :
~ ( ord_less @ A @ X2 @ X2 ) ) ).
% less_irrefl
thf(fact_174_ord__eq__less__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B: A,C: A] :
( ( A2 = B )
=> ( ( ord_less @ A @ B @ C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% ord_eq_less_trans
thf(fact_175_ord__less__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B: A,C: A] :
( ( ord_less @ A @ A2 @ B )
=> ( ( B = C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% ord_less_eq_trans
thf(fact_176_dual__order_Oasym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B: A,A2: A] :
( ( ord_less @ A @ B @ A2 )
=> ~ ( ord_less @ A @ A2 @ B ) ) ) ).
% dual_order.asym
thf(fact_177_less__imp__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ Y )
=> ( X2 != Y ) ) ) ).
% less_imp_not_eq
thf(fact_178_less__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ Y )
=> ~ ( ord_less @ A @ Y @ X2 ) ) ) ).
% less_not_sym
thf(fact_179_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P3: A > $o,A2: A] :
( ! [X3: A] :
( ! [Y4: A] :
( ( ord_less @ A @ Y4 @ X3 )
=> ( P3 @ Y4 ) )
=> ( P3 @ X3 ) )
=> ( P3 @ A2 ) ) ) ).
% less_induct
thf(fact_180_antisym__conv3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X2: A] :
( ~ ( ord_less @ A @ Y @ X2 )
=> ( ( ~ ( ord_less @ A @ X2 @ Y ) )
= ( X2 = Y ) ) ) ) ).
% antisym_conv3
thf(fact_181_less__imp__not__eq2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ Y )
=> ( Y != X2 ) ) ) ).
% less_imp_not_eq2
thf(fact_182_less__imp__triv,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A,P3: $o] :
( ( ord_less @ A @ X2 @ Y )
=> ( ( ord_less @ A @ Y @ X2 )
=> P3 ) ) ) ).
% less_imp_triv
thf(fact_183_linorder__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ~ ( ord_less @ A @ X2 @ Y )
=> ( ( X2 != Y )
=> ( ord_less @ A @ Y @ X2 ) ) ) ) ).
% linorder_cases
thf(fact_184_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ A2 ) ) ).
% dual_order.irrefl
thf(fact_185_order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B: A,C: A] :
( ( ord_less @ A @ A2 @ B )
=> ( ( ord_less @ A @ B @ C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% order.strict_trans
thf(fact_186_less__imp__not__less,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ Y )
=> ~ ( ord_less @ A @ Y @ X2 ) ) ) ).
% less_imp_not_less
thf(fact_187_exists__least__iff,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ( ( ^ [P4: A > $o] :
? [X5: A] : ( P4 @ X5 ) )
= ( ^ [P5: A > $o] :
? [N2: A] :
( ( P5 @ N2 )
& ! [M2: A] :
( ( ord_less @ A @ M2 @ N2 )
=> ~ ( P5 @ M2 ) ) ) ) ) ) ).
% exists_least_iff
thf(fact_188_linorder__less__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P3: A > A > $o,A2: A,B: A] :
( ! [A5: A,B4: A] :
( ( ord_less @ A @ A5 @ B4 )
=> ( P3 @ A5 @ B4 ) )
=> ( ! [A5: A] : ( P3 @ A5 @ A5 )
=> ( ! [A5: A,B4: A] :
( ( P3 @ B4 @ A5 )
=> ( P3 @ A5 @ B4 ) )
=> ( P3 @ A2 @ B ) ) ) ) ) ).
% linorder_less_wlog
thf(fact_189_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B: A,A2: A,C: A] :
( ( ord_less @ A @ B @ A2 )
=> ( ( ord_less @ A @ C @ B )
=> ( ord_less @ A @ C @ A2 ) ) ) ) ).
% dual_order.strict_trans
thf(fact_190_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ( ~ ( ord_less @ A @ X2 @ Y ) )
= ( ( ord_less @ A @ Y @ X2 )
| ( X2 = Y ) ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_191_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ A2 @ B )
=> ( A2 != B ) ) ) ).
% order.strict_implies_not_eq
thf(fact_192_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B: A,A2: A] :
( ( ord_less @ A @ B @ A2 )
=> ( A2 != B ) ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_193_ex__has__least__nat,axiom,
! [A: $tType,P3: A > $o,K: A,M: A > nat] :
( ( P3 @ K )
=> ? [X3: A] :
( ( P3 @ X3 )
& ! [Y4: A] :
( ( P3 @ Y4 )
=> ( ord_less_eq @ nat @ ( M @ X3 ) @ ( M @ Y4 ) ) ) ) ) ).
% ex_has_least_nat
thf(fact_194_leD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X2: A] :
( ( ord_less_eq @ A @ Y @ X2 )
=> ~ ( ord_less @ A @ X2 @ Y ) ) ) ).
% leD
thf(fact_195_leI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ~ ( ord_less @ A @ X2 @ Y )
=> ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% leI
thf(fact_196_le__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X4: A,Y3: A] :
( ( ord_less @ A @ X4 @ Y3 )
| ( X4 = Y3 ) ) ) ) ) ).
% le_less
thf(fact_197_less__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [X4: A,Y3: A] :
( ( ord_less_eq @ A @ X4 @ Y3 )
& ( X4 != Y3 ) ) ) ) ) ).
% less_le
thf(fact_198_order__le__less__subst1,axiom,
! [A: $tType,B3: $tType] :
( ( ( order @ B3 )
& ( order @ A ) )
=> ! [A2: A,F: B3 > A,B: B3,C: B3] :
( ( ord_less_eq @ A @ A2 @ ( F @ B ) )
=> ( ( ord_less @ B3 @ B @ C )
=> ( ! [X3: B3,Y2: B3] :
( ( ord_less @ B3 @ X3 @ Y2 )
=> ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_199_order__le__less__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 )
& ( order @ A ) )
=> ! [A2: A,B: A,F: A > C2,C: C2] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( ord_less @ C2 @ ( F @ B ) @ C )
=> ( ! [X3: A,Y2: A] :
( ( ord_less_eq @ A @ X3 @ Y2 )
=> ( ord_less_eq @ C2 @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_le_less_subst2
thf(fact_200_order__less__le__subst1,axiom,
! [A: $tType,B3: $tType] :
( ( ( order @ B3 )
& ( order @ A ) )
=> ! [A2: A,F: B3 > A,B: B3,C: B3] :
( ( ord_less @ A @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq @ B3 @ B @ C )
=> ( ! [X3: B3,Y2: B3] :
( ( ord_less_eq @ B3 @ X3 @ Y2 )
=> ( ord_less_eq @ A @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_201_order__less__le__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 )
& ( order @ A ) )
=> ! [A2: A,B: A,F: A > C2,C: C2] :
( ( ord_less @ A @ A2 @ B )
=> ( ( ord_less_eq @ C2 @ ( F @ B ) @ C )
=> ( ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( ord_less @ C2 @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_less_le_subst2
thf(fact_202_not__le,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ( ~ ( ord_less_eq @ A @ X2 @ Y ) )
= ( ord_less @ A @ Y @ X2 ) ) ) ).
% not_le
thf(fact_203_not__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ( ~ ( ord_less @ A @ X2 @ Y ) )
= ( ord_less_eq @ A @ Y @ X2 ) ) ) ).
% not_less
thf(fact_204_le__neq__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less @ A @ A2 @ B ) ) ) ) ).
% le_neq_trans
thf(fact_205_antisym__conv1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y: A] :
( ~ ( ord_less @ A @ X2 @ Y )
=> ( ( ord_less_eq @ A @ X2 @ Y )
= ( X2 = Y ) ) ) ) ).
% antisym_conv1
thf(fact_206_antisym__conv2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ( ~ ( ord_less @ A @ X2 @ Y ) )
= ( X2 = Y ) ) ) ) ).
% antisym_conv2
thf(fact_207_less__imp__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A] :
( ( ord_less @ A @ X2 @ Y )
=> ( ord_less_eq @ A @ X2 @ Y ) ) ) ).
% less_imp_le
thf(fact_208_le__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ( ord_less @ A @ Y @ Z )
=> ( ord_less @ A @ X2 @ Z ) ) ) ) ).
% le_less_trans
thf(fact_209_less__le__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y: A,Z: A] :
( ( ord_less @ A @ X2 @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z )
=> ( ord_less @ A @ X2 @ Z ) ) ) ) ).
% less_le_trans
thf(fact_210_dense__ge,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z: A,Y: A] :
( ! [X3: A] :
( ( ord_less @ A @ Z @ X3 )
=> ( ord_less_eq @ A @ Y @ X3 ) )
=> ( ord_less_eq @ A @ Y @ Z ) ) ) ).
% dense_ge
thf(fact_211_dense__le,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Y: A,Z: A] :
( ! [X3: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ord_less_eq @ A @ X3 @ Z ) )
=> ( ord_less_eq @ A @ Y @ Z ) ) ) ).
% dense_le
thf(fact_212_le__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
| ( ord_less @ A @ Y @ X2 ) ) ) ).
% le_less_linear
thf(fact_213_le__imp__less__or__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ Y )
=> ( ( ord_less @ A @ X2 @ Y )
| ( X2 = Y ) ) ) ) ).
% le_imp_less_or_eq
thf(fact_214_less__le__not__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [X4: A,Y3: A] :
( ( ord_less_eq @ A @ X4 @ Y3 )
& ~ ( ord_less_eq @ A @ Y3 @ X4 ) ) ) ) ) ).
% less_le_not_le
thf(fact_215_not__le__imp__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X2: A] :
( ~ ( ord_less_eq @ A @ Y @ X2 )
=> ( ord_less @ A @ X2 @ Y ) ) ) ).
% not_le_imp_less
thf(fact_216_order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( ord_less @ A @ B @ C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% order.strict_trans1
thf(fact_217_order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B: A,C: A] :
( ( ord_less @ A @ A2 @ B )
=> ( ( ord_less_eq @ A @ B @ C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% order.strict_trans2
thf(fact_218_order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ) ).
% order.order_iff_strict
thf(fact_219_order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ) ).
% order.strict_iff_order
thf(fact_220_dual__order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B: A,A2: A,C: A] :
( ( ord_less_eq @ A @ B @ A2 )
=> ( ( ord_less @ A @ C @ B )
=> ( ord_less @ A @ C @ A2 ) ) ) ) ).
% dual_order.strict_trans1
thf(fact_221_dual__order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B: A,A2: A,C: A] :
( ( ord_less @ A @ B @ A2 )
=> ( ( ord_less_eq @ A @ C @ B )
=> ( ord_less @ A @ C @ A2 ) ) ) ) ).
% dual_order.strict_trans2
thf(fact_222_dense__ge__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z: A,X2: A,Y: A] :
( ( ord_less @ A @ Z @ X2 )
=> ( ! [W: A] :
( ( ord_less @ A @ Z @ W )
=> ( ( ord_less @ A @ W @ X2 )
=> ( ord_less_eq @ A @ Y @ W ) ) )
=> ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% dense_ge_bounded
thf(fact_223_dense__le__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [X2: A,Y: A,Z: A] :
( ( ord_less @ A @ X2 @ Y )
=> ( ! [W: A] :
( ( ord_less @ A @ X2 @ W )
=> ( ( ord_less @ A @ W @ Y )
=> ( ord_less_eq @ A @ W @ Z ) ) )
=> ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% dense_le_bounded
thf(fact_224_order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ A2 @ B )
=> ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% order.strict_implies_order
thf(fact_225_dual__order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B2: A,A3: A] :
( ( ord_less @ A @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_226_dual__order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_227_dual__order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B: A,A2: A] :
( ( ord_less @ A @ B @ A2 )
=> ( ord_less_eq @ A @ B @ A2 ) ) ) ).
% dual_order.strict_implies_order
thf(fact_228_order_Onot__eq__order__implies__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B: A] :
( ( A2 != B )
=> ( ( ord_less_eq @ A @ A2 @ B )
=> ( ord_less @ A @ A2 @ B ) ) ) ) ).
% order.not_eq_order_implies_strict
thf(fact_229_minf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z5: A] :
! [X: A] :
( ( ord_less @ A @ X @ Z5 )
=> ~ ( ord_less_eq @ A @ T @ X ) ) ) ).
% minf(8)
thf(fact_230_minf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z5: A] :
! [X: A] :
( ( ord_less @ A @ X @ Z5 )
=> ( ord_less_eq @ A @ X @ T ) ) ) ).
% minf(6)
thf(fact_231_pinf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z5: A] :
! [X: A] :
( ( ord_less @ A @ Z5 @ X )
=> ( ord_less_eq @ A @ T @ X ) ) ) ).
% pinf(8)
thf(fact_232_pinf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z5: A] :
! [X: A] :
( ( ord_less @ A @ Z5 @ X )
=> ~ ( ord_less_eq @ A @ X @ T ) ) ) ).
% pinf(6)
thf(fact_233_less__fun__def,axiom,
! [B3: $tType,A: $tType] :
( ( ord @ B3 )
=> ( ( ord_less @ ( A > B3 ) )
= ( ^ [F2: A > B3,G2: A > B3] :
( ( ord_less_eq @ ( A > B3 ) @ F2 @ G2 )
& ~ ( ord_less_eq @ ( A > B3 ) @ G2 @ F2 ) ) ) ) ) ).
% less_fun_def
thf(fact_234_pinf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P3: A > $o,P6: A > $o,Q2: A > $o,Q3: A > $o] :
( ? [Z2: A] :
! [X3: A] :
( ( ord_less @ A @ Z2 @ X3 )
=> ( ( P3 @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z2: A] :
! [X3: A] :
( ( ord_less @ A @ Z2 @ X3 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z5: A] :
! [X: A] :
( ( ord_less @ A @ Z5 @ X )
=> ( ( ( P3 @ X )
& ( Q2 @ X ) )
= ( ( P6 @ X )
& ( Q3 @ X ) ) ) ) ) ) ) ).
% pinf(1)
thf(fact_235_pinf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P3: A > $o,P6: A > $o,Q2: A > $o,Q3: A > $o] :
( ? [Z2: A] :
! [X3: A] :
( ( ord_less @ A @ Z2 @ X3 )
=> ( ( P3 @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z2: A] :
! [X3: A] :
( ( ord_less @ A @ Z2 @ X3 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z5: A] :
! [X: A] :
( ( ord_less @ A @ Z5 @ X )
=> ( ( ( P3 @ X )
| ( Q2 @ X ) )
= ( ( P6 @ X )
| ( Q3 @ X ) ) ) ) ) ) ) ).
% pinf(2)
thf(fact_236_pinf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z5: A] :
! [X: A] :
( ( ord_less @ A @ Z5 @ X )
=> ( X != T ) ) ) ).
% pinf(3)
thf(fact_237_pinf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z5: A] :
! [X: A] :
( ( ord_less @ A @ Z5 @ X )
=> ( X != T ) ) ) ).
% pinf(4)
thf(fact_238_pinf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z5: A] :
! [X: A] :
( ( ord_less @ A @ Z5 @ X )
=> ~ ( ord_less @ A @ X @ T ) ) ) ).
% pinf(5)
thf(fact_239_pinf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z5: A] :
! [X: A] :
( ( ord_less @ A @ Z5 @ X )
=> ( ord_less @ A @ T @ X ) ) ) ).
% pinf(7)
thf(fact_240_pinf_I11_J,axiom,
! [C2: $tType,D3: $tType] :
( ( ord @ C2 )
=> ! [F3: D3] :
? [Z5: C2] :
! [X: C2] :
( ( ord_less @ C2 @ Z5 @ X )
=> ( F3 = F3 ) ) ) ).
% pinf(11)
thf(fact_241_minf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P3: A > $o,P6: A > $o,Q2: A > $o,Q3: A > $o] :
( ? [Z2: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z2 )
=> ( ( P3 @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z2: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z2 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z5: A] :
! [X: A] :
( ( ord_less @ A @ X @ Z5 )
=> ( ( ( P3 @ X )
& ( Q2 @ X ) )
= ( ( P6 @ X )
& ( Q3 @ X ) ) ) ) ) ) ) ).
% minf(1)
thf(fact_242_minf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P3: A > $o,P6: A > $o,Q2: A > $o,Q3: A > $o] :
( ? [Z2: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z2 )
=> ( ( P3 @ X3 )
= ( P6 @ X3 ) ) )
=> ( ? [Z2: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z2 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z5: A] :
! [X: A] :
( ( ord_less @ A @ X @ Z5 )
=> ( ( ( P3 @ X )
| ( Q2 @ X ) )
= ( ( P6 @ X )
| ( Q3 @ X ) ) ) ) ) ) ) ).
% minf(2)
thf(fact_243_minf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z5: A] :
! [X: A] :
( ( ord_less @ A @ X @ Z5 )
=> ( X != T ) ) ) ).
% minf(3)
thf(fact_244_minf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z5: A] :
! [X: A] :
( ( ord_less @ A @ X @ Z5 )
=> ( X != T ) ) ) ).
% minf(4)
thf(fact_245_minf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z5: A] :
! [X: A] :
( ( ord_less @ A @ X @ Z5 )
=> ( ord_less @ A @ X @ T ) ) ) ).
% minf(5)
thf(fact_246_minf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z5: A] :
! [X: A] :
( ( ord_less @ A @ X @ Z5 )
=> ~ ( ord_less @ A @ T @ X ) ) ) ).
% minf(7)
thf(fact_247_minf_I11_J,axiom,
! [C2: $tType,D3: $tType] :
( ( ord @ C2 )
=> ! [F3: D3] :
? [Z5: C2] :
! [X: C2] :
( ( ord_less @ C2 @ X @ Z5 )
=> ( F3 = F3 ) ) ) ).
% minf(11)
thf(fact_248_complete__interval,axiom,
! [A: $tType] :
( ( condit1037483654norder @ A )
=> ! [A2: A,B: A,P3: A > $o] :
( ( ord_less @ A @ A2 @ B )
=> ( ( P3 @ A2 )
=> ( ~ ( P3 @ B )
=> ? [C3: A] :
( ( ord_less_eq @ A @ A2 @ C3 )
& ( ord_less_eq @ A @ C3 @ B )
& ! [X: A] :
( ( ( ord_less_eq @ A @ A2 @ X )
& ( ord_less @ A @ X @ C3 ) )
=> ( P3 @ X ) )
& ! [D4: A] :
( ! [X3: A] :
( ( ( ord_less_eq @ A @ A2 @ X3 )
& ( ord_less @ A @ X3 @ D4 ) )
=> ( P3 @ X3 ) )
=> ( ord_less_eq @ A @ D4 @ C3 ) ) ) ) ) ) ) ).
% complete_interval
thf(fact_249_verit__comp__simplify1_I3_J,axiom,
! [B3: $tType] :
( ( linorder @ B3 )
=> ! [B5: B3,A6: B3] :
( ( ~ ( ord_less_eq @ B3 @ B5 @ A6 ) )
= ( ord_less @ B3 @ A6 @ B5 ) ) ) ).
% verit_comp_simplify1(3)
thf(fact_250_verit__la__disequality,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B: A] :
( ( A2 = B )
| ~ ( ord_less_eq @ A @ A2 @ B )
| ~ ( ord_less_eq @ A @ B @ A2 ) ) ) ).
% verit_la_disequality
thf(fact_251_verit__comp__simplify1_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ A2 ) ) ).
% verit_comp_simplify1(1)
thf(fact_252_ex__gt__or__lt,axiom,
! [A: $tType] :
( ( condit1656338222tinuum @ A )
=> ! [A2: A] :
? [B4: A] :
( ( ord_less @ A @ A2 @ B4 )
| ( ord_less @ A @ B4 @ A2 ) ) ) ).
% ex_gt_or_lt
thf(fact_253_poly__cutoff__1,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [N: nat] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( poly_cutoff @ A @ N @ ( one_one @ ( poly @ A ) ) )
= ( zero_zero @ ( poly @ A ) ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( poly_cutoff @ A @ N @ ( one_one @ ( poly @ A ) ) )
= ( one_one @ ( poly @ A ) ) ) ) ) ) ).
% poly_cutoff_1
% Type constructors (60)
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A7: $tType,A8: $tType] :
( ( preorder @ A8 )
=> ( preorder @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A7: $tType,A8: $tType] :
( ( order @ A8 )
=> ( order @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A7: $tType,A8: $tType] :
( ( ord @ A8 )
=> ( ord @ ( A7 > A8 ) ) ) ).
thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
condit1037483654norder @ nat ).
thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
canoni770627133id_add @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring,axiom,
linord1659791738miring @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1 @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__0,axiom,
comm_semiring_0 @ nat ).
thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
wellorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
preorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
linorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Ono__top,axiom,
no_top @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
order @ nat ).
thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
ord @ nat ).
thf(tcon_Nat_Onat___Groups_Ozero,axiom,
zero @ nat ).
thf(tcon_Set_Oset___Orderings_Opreorder_4,axiom,
! [A7: $tType] : ( preorder @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_5,axiom,
! [A7: $tType] : ( order @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_6,axiom,
! [A7: $tType] : ( ord @ ( set @ A7 ) ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_7,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder_8,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_9,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_10,axiom,
ord @ $o ).
thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_11,axiom,
condit1037483654norder @ real ).
thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Olinear__continuum,axiom,
condit1656338222tinuum @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__vector,axiom,
real_V55928688vector @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_12,axiom,
linord1659791738miring @ real ).
thf(tcon_Real_Oreal___Rings_Oring__no__zero__divisors,axiom,
ring_n68954251visors @ real ).
thf(tcon_Real_Oreal___Orderings_Odense__linorder,axiom,
dense_linorder @ real ).
thf(tcon_Real_Oreal___Fields_Olinordered__field,axiom,
linordered_field @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__idom,axiom,
linordered_idom @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring__1_13,axiom,
comm_semiring_1 @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring__0_14,axiom,
comm_semiring_0 @ real ).
thf(tcon_Real_Oreal___Orderings_Odense__order,axiom,
dense_order @ real ).
thf(tcon_Real_Oreal___Rings_Oidom__abs__sgn,axiom,
idom_abs_sgn @ real ).
thf(tcon_Real_Oreal___Orderings_Opreorder_15,axiom,
preorder @ real ).
thf(tcon_Real_Oreal___Orderings_Olinorder_16,axiom,
linorder @ real ).
thf(tcon_Real_Oreal___Rings_Oidom__divide,axiom,
idom_divide @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__ring__1,axiom,
comm_ring_1 @ real ).
thf(tcon_Real_Oreal___Orderings_Ono__top_17,axiom,
no_top @ real ).
thf(tcon_Real_Oreal___Orderings_Ono__bot,axiom,
no_bot @ real ).
thf(tcon_Real_Oreal___Orderings_Oorder_18,axiom,
order @ real ).
thf(tcon_Real_Oreal___Nat_Oring__char__0,axiom,
ring_char_0 @ real ).
thf(tcon_Real_Oreal___Orderings_Oord_19,axiom,
ord @ real ).
thf(tcon_Real_Oreal___Groups_Ozero_20,axiom,
zero @ real ).
thf(tcon_Real_Oreal___Rings_Oidom,axiom,
idom @ real ).
thf(tcon_Polynomial_Opoly___Rings_Olinordered__nonzero__semiring_21,axiom,
! [A7: $tType] :
( ( linordered_idom @ A7 )
=> ( linord1659791738miring @ ( poly @ A7 ) ) ) ).
thf(tcon_Polynomial_Opoly___Rings_Oring__no__zero__divisors_22,axiom,
! [A7: $tType] :
( ( idom @ A7 )
=> ( ring_n68954251visors @ ( poly @ A7 ) ) ) ).
thf(tcon_Polynomial_Opoly___Rings_Olinordered__idom_23,axiom,
! [A7: $tType] :
( ( linordered_idom @ A7 )
=> ( linordered_idom @ ( poly @ A7 ) ) ) ).
thf(tcon_Polynomial_Opoly___Rings_Ocomm__semiring__1_24,axiom,
! [A7: $tType] :
( ( comm_semiring_1 @ A7 )
=> ( comm_semiring_1 @ ( poly @ A7 ) ) ) ).
thf(tcon_Polynomial_Opoly___Rings_Ocomm__semiring__0_25,axiom,
! [A7: $tType] :
( ( comm_semiring_0 @ A7 )
=> ( comm_semiring_0 @ ( poly @ A7 ) ) ) ).
thf(tcon_Polynomial_Opoly___Rings_Oidom__abs__sgn_26,axiom,
! [A7: $tType] :
( ( linordered_idom @ A7 )
=> ( idom_abs_sgn @ ( poly @ A7 ) ) ) ).
thf(tcon_Polynomial_Opoly___Orderings_Opreorder_27,axiom,
! [A7: $tType] :
( ( linordered_idom @ A7 )
=> ( preorder @ ( poly @ A7 ) ) ) ).
thf(tcon_Polynomial_Opoly___Orderings_Olinorder_28,axiom,
! [A7: $tType] :
( ( linordered_idom @ A7 )
=> ( linorder @ ( poly @ A7 ) ) ) ).
thf(tcon_Polynomial_Opoly___Rings_Oidom__divide_29,axiom,
! [A7: $tType] :
( ( idom_divide @ A7 )
=> ( idom_divide @ ( poly @ A7 ) ) ) ).
thf(tcon_Polynomial_Opoly___Rings_Ocomm__ring__1_30,axiom,
! [A7: $tType] :
( ( comm_ring_1 @ A7 )
=> ( comm_ring_1 @ ( poly @ A7 ) ) ) ).
thf(tcon_Polynomial_Opoly___Orderings_Oorder_31,axiom,
! [A7: $tType] :
( ( linordered_idom @ A7 )
=> ( order @ ( poly @ A7 ) ) ) ).
thf(tcon_Polynomial_Opoly___Nat_Oring__char__0_32,axiom,
! [A7: $tType] :
( ( ( ring_char_0 @ A7 )
& ( comm_ring_1 @ A7 ) )
=> ( ring_char_0 @ ( poly @ A7 ) ) ) ).
thf(tcon_Polynomial_Opoly___Orderings_Oord_33,axiom,
! [A7: $tType] :
( ( linordered_idom @ A7 )
=> ( ord @ ( poly @ A7 ) ) ) ).
thf(tcon_Polynomial_Opoly___Groups_Ozero_34,axiom,
! [A7: $tType] :
( ( zero @ A7 )
=> ( zero @ ( poly @ A7 ) ) ) ).
thf(tcon_Polynomial_Opoly___Rings_Oidom_35,axiom,
! [A7: $tType] :
( ( idom @ A7 )
=> ( idom @ ( poly @ A7 ) ) ) ).
% Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P3: $o] :
( ( P3 = $true )
| ( P3 = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X2: A,Y: A] :
( ( if @ A @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X2: A,Y: A] :
( ( if @ A @ $true @ X2 @ Y )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
! [X3: real] :
( ( ( poly2 @ real @ p @ X3 )
!= ( zero_zero @ real ) )
| ( ord_less @ real @ lb @ X3 ) ) ).
%------------------------------------------------------------------------------