TPTP Problem File: ITP044^2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP044^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Descartes_Sign_Rule problem prob_761__5872108_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 : Descartes_Sign_Rule/prob_761__5872108_1 [Des21]
% Status : Theorem
% Rating : 0.00 v8.1.0, 0.25 v7.5.0
% Syntax : Number of formulae : 437 ( 94 unt; 72 typ; 0 def)
% Number of atoms : 901 ( 374 equ; 0 cnn)
% Maximal formula atoms : 11 ( 2 avg)
% Number of connectives : 3538 ( 96 ~; 27 |; 45 &;2942 @)
% ( 0 <=>; 428 =>; 0 <=; 0 <~>)
% Maximal formula depth : 20 ( 6 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 107 ( 107 >; 0 *; 0 +; 0 <<)
% Number of symbols : 71 ( 70 usr; 3 con; 0-3 aty)
% Number of variables : 942 ( 13 ^; 856 !; 9 ?; 942 :)
% ( 64 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:26:47.381
%------------------------------------------------------------------------------
% Could-be-implicit typings (4)
thf(ty_t_Polynomial_Opoly,type,
poly: $tType > $tType ).
thf(ty_t_List_Olist,type,
list: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_tf_a,type,
a: $tType ).
% Explicit typings (68)
thf(sy_cl_Rings_Ocomm__ring__1,type,
comm_ring_1:
!>[A: $tType] : $o ).
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oone,type,
one:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Osgn,type,
sgn:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oidom,type,
idom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oring,type,
ring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oplus,type,
plus:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oring__1,type,
ring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ouminus,type,
uminus:
!>[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_Num_Oneg__numeral,type,
neg_numeral:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__ring,type,
comm_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Omult__zero,type,
mult_zero:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Omonoid__add,type,
monoid_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Omonoid__mult,type,
monoid_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oidom__abs__sgn,type,
idom_abs_sgn:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ozero__neq__one,type,
zero_neq_one:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__group__add,type,
ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Owellorder,type,
wellorder:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ozero__less__one,type,
zero_less_one:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Osemigroup__mult,type,
semigroup_mult:
!>[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_Rings_Olinordered__ring,type,
linordered_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__add,type,
comm_monoid_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__mult,type,
comm_monoid_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__semigroup__mult,type,
ab_semigroup_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Oboolean__algebra,type,
boolean_algebra:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semidom,type,
linordered_semidom:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__ring__strict,type,
linord581940658strict:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oring__1__no__zero__divisors,type,
ring_11004092258visors:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Olinordered__ab__group__add,type,
linord219039673up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__no__zero__divisors,type,
semiri1193490041visors:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semiring__strict,type,
linord20386208strict:
!>[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_Rings_Olinordered__comm__semiring__strict,type,
linord893533164strict:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__no__zero__divisors__cancel,type,
semiri1923998003cancel:
!>[A: $tType] : $o ).
thf(sy_c_Descartes__Sign__Rule__Mirabelle__vuqjybseel_Opsums,type,
descar1668888542_psums:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Descartes__Sign__Rule__Mirabelle__vuqjybseel_Oreduce__root,type,
descar316357986e_root:
!>[A: $tType] : ( A > ( poly @ A ) > ( poly @ A ) ) ).
thf(sy_c_Descartes__Sign__Rule__Mirabelle__vuqjybseel_Osign__changes,type,
descar149487500hanges:
!>[A: $tType] : ( ( list @ A ) > nat ) ).
thf(sy_c_Groups_Oone__class_Oone,type,
one_one:
!>[A: $tType] : A ).
thf(sy_c_Groups_Otimes__class_Otimes,type,
times_times:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Ouminus__class_Ouminus,type,
uminus_uminus:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_List_Oappend,type,
append:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist_OCons,type,
cons:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist_ONil,type,
nil:
!>[A: $tType] : ( list @ A ) ).
thf(sy_c_List_On__lists,type,
n_lists:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Onull,type,
null:
!>[A: $tType] : ( ( list @ A ) > $o ) ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec,type,
neg_numeral_dbl_dec:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Polynomial_OPoly,type,
poly2:
!>[A: $tType] : ( ( list @ A ) > ( poly @ A ) ) ).
thf(sy_c_Polynomial_Ocoeffs,type,
coeffs:
!>[A: $tType] : ( ( poly @ A ) > ( list @ A ) ) ).
thf(sy_c_Polynomial_Ois__zero,type,
is_zero:
!>[A: $tType] : ( ( poly @ A ) > $o ) ).
thf(sy_c_Polynomial_OpCons,type,
pCons:
!>[A: $tType] : ( A > ( poly @ A ) > ( poly @ A ) ) ).
thf(sy_c_Polynomial_Opoly__cutoff,type,
poly_cutoff:
!>[A: $tType] : ( nat > ( poly @ A ) > ( poly @ A ) ) ).
thf(sy_c_Polynomial_Osmult,type,
smult:
!>[A: $tType] : ( A > ( poly @ A ) > ( poly @ A ) ) ).
thf(sy_v_g,type,
g: poly @ a ).
thf(sy_v_v,type,
v: ( poly @ a ) > nat ).
thf(sy_v_xs____,type,
xs: list @ a ).
thf(sy_v_ys____,type,
ys: list @ a ).
% Relevant facts (252)
thf(fact_0_v__def,axiom,
( v
= ( ^ [F: poly @ a] : ( descar149487500hanges @ a @ ( coeffs @ a @ F ) ) ) ) ).
% v_def
thf(fact_1_nz,axiom,
( g
!= ( zero_zero @ ( poly @ a ) ) ) ).
% nz
thf(fact_2_coeffs__eq__iff,axiom,
! [A: $tType] :
( ( zero @ A )
=> ( ( ^ [Y: poly @ A,Z: poly @ A] : Y = Z )
= ( ^ [P: poly @ A,Q: poly @ A] :
( ( coeffs @ A @ P )
= ( coeffs @ A @ Q ) ) ) ) ) ).
% coeffs_eq_iff
thf(fact_3_ys,axiom,
( ys
= ( descar1668888542_psums @ a @ xs ) ) ).
% ys
thf(fact_4_sign__changes__coeff__sign__changes,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Xs: list @ A,P2: poly @ A] :
( ( ( poly2 @ A @ Xs )
= P2 )
=> ( ( descar149487500hanges @ A @ Xs )
= ( descar149487500hanges @ A @ ( coeffs @ A @ P2 ) ) ) ) ) ).
% sign_changes_coeff_sign_changes
thf(fact_5_ys__def,axiom,
( ys
= ( append @ a @ ( coeffs @ a @ g ) @ ( cons @ a @ ( zero_zero @ a ) @ ( nil @ a ) ) ) ) ).
% ys_def
thf(fact_6_coeff__sign__changes__reduce__root,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,P2: poly @ A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( descar149487500hanges @ A @ ( coeffs @ A @ ( descar316357986e_root @ A @ A2 @ P2 ) ) )
= ( descar149487500hanges @ A @ ( coeffs @ A @ P2 ) ) ) ) ) ).
% coeff_sign_changes_reduce_root
thf(fact_7_is__zero__def,axiom,
! [A: $tType] :
( ( zero @ A )
=> ( ( is_zero @ A )
= ( ^ [P: poly @ A] : ( null @ A @ ( coeffs @ A @ P ) ) ) ) ) ).
% is_zero_def
thf(fact_8_Poly__coeffs,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [P2: poly @ A] :
( ( poly2 @ A @ ( coeffs @ A @ P2 ) )
= P2 ) ) ).
% Poly_coeffs
thf(fact_9_sign__changes__Nil,axiom,
! [A: $tType] :
( ( ( sgn @ A )
& ( zero @ A ) )
=> ( ( descar149487500hanges @ A @ ( nil @ A ) )
= ( zero_zero @ nat ) ) ) ).
% sign_changes_Nil
thf(fact_10_xs__def,axiom,
( xs
= ( coeffs @ a @ ( times_times @ ( poly @ a ) @ ( pCons @ a @ ( one_one @ a ) @ ( pCons @ a @ ( uminus_uminus @ a @ ( one_one @ a ) ) @ ( zero_zero @ ( poly @ a ) ) ) ) @ g ) ) ) ).
% xs_def
thf(fact_11_coeff__sign__changes__smult,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,P2: poly @ A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( descar149487500hanges @ A @ ( coeffs @ A @ ( smult @ A @ A2 @ P2 ) ) )
= ( descar149487500hanges @ A @ ( coeffs @ A @ P2 ) ) ) ) ) ).
% coeff_sign_changes_smult
thf(fact_12_sign__changes__0__Cons,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [Xs: list @ A] :
( ( descar149487500hanges @ A @ ( cons @ A @ ( zero_zero @ A ) @ Xs ) )
= ( descar149487500hanges @ A @ Xs ) ) ) ).
% sign_changes_0_Cons
thf(fact_13_sign__changes__Cons__Cons__0,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Xs: list @ A] :
( ( descar149487500hanges @ A @ ( cons @ A @ X @ ( cons @ A @ ( zero_zero @ A ) @ Xs ) ) )
= ( descar149487500hanges @ A @ ( cons @ A @ X @ Xs ) ) ) ) ).
% sign_changes_Cons_Cons_0
thf(fact_14_pCons__eq__iff,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [A2: A,P2: poly @ A,B: A,Q2: poly @ A] :
( ( ( pCons @ A @ A2 @ P2 )
= ( pCons @ A @ B @ Q2 ) )
= ( ( A2 = B )
& ( P2 = Q2 ) ) ) ) ).
% pCons_eq_iff
thf(fact_15_minus__pCons,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,P2: poly @ A] :
( ( uminus_uminus @ ( poly @ A ) @ ( pCons @ A @ A2 @ P2 ) )
= ( pCons @ A @ ( uminus_uminus @ A @ A2 ) @ ( uminus_uminus @ ( poly @ A ) @ P2 ) ) ) ) ).
% minus_pCons
thf(fact_16_smult__smult,axiom,
! [A: $tType] :
( ( comm_semiring_0 @ A )
=> ! [A2: A,B: A,P2: poly @ A] :
( ( smult @ A @ A2 @ ( smult @ A @ B @ P2 ) )
= ( smult @ A @ ( times_times @ A @ A2 @ B ) @ P2 ) ) ) ).
% smult_smult
thf(fact_17_smult__1__left,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [P2: poly @ A] :
( ( smult @ A @ ( one_one @ A ) @ P2 )
= P2 ) ) ).
% smult_1_left
thf(fact_18_smult__minus__left,axiom,
! [A: $tType] :
( ( comm_ring @ A )
=> ! [A2: A,P2: poly @ A] :
( ( smult @ A @ ( uminus_uminus @ A @ A2 ) @ P2 )
= ( uminus_uminus @ ( poly @ A ) @ ( smult @ A @ A2 @ P2 ) ) ) ) ).
% smult_minus_left
thf(fact_19_smult__0__right,axiom,
! [A: $tType] :
( ( comm_semiring_0 @ A )
=> ! [A2: A] :
( ( smult @ A @ A2 @ ( zero_zero @ ( poly @ A ) ) )
= ( zero_zero @ ( poly @ A ) ) ) ) ).
% smult_0_right
thf(fact_20_mult__smult__left,axiom,
! [A: $tType] :
( ( comm_semiring_0 @ A )
=> ! [A2: A,P2: poly @ A,Q2: poly @ A] :
( ( times_times @ ( poly @ A ) @ ( smult @ A @ A2 @ P2 ) @ Q2 )
= ( smult @ A @ A2 @ ( times_times @ ( poly @ A ) @ P2 @ Q2 ) ) ) ) ).
% mult_smult_left
thf(fact_21_mult__smult__right,axiom,
! [A: $tType] :
( ( comm_semiring_0 @ A )
=> ! [P2: poly @ A,A2: A,Q2: poly @ A] :
( ( times_times @ ( poly @ A ) @ P2 @ ( smult @ A @ A2 @ Q2 ) )
= ( smult @ A @ A2 @ ( times_times @ ( poly @ A ) @ P2 @ Q2 ) ) ) ) ).
% mult_smult_right
thf(fact_22_pCons__0__0,axiom,
! [A: $tType] :
( ( zero @ A )
=> ( ( pCons @ A @ ( zero_zero @ A ) @ ( zero_zero @ ( poly @ A ) ) )
= ( zero_zero @ ( poly @ A ) ) ) ) ).
% pCons_0_0
thf(fact_23_pCons__eq__0__iff,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [A2: A,P2: poly @ A] :
( ( ( pCons @ A @ A2 @ P2 )
= ( zero_zero @ ( poly @ A ) ) )
= ( ( A2
= ( zero_zero @ A ) )
& ( P2
= ( zero_zero @ ( poly @ A ) ) ) ) ) ) ).
% pCons_eq_0_iff
thf(fact_24_one__poly__eq__simps_I2_J,axiom,
! [B2: $tType] :
( ( comm_semiring_1 @ B2 )
=> ( ( pCons @ B2 @ ( one_one @ B2 ) @ ( zero_zero @ ( poly @ B2 ) ) )
= ( one_one @ ( poly @ B2 ) ) ) ) ).
% one_poly_eq_simps(2)
thf(fact_25_one__poly__eq__simps_I1_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( one_one @ ( poly @ A ) )
= ( pCons @ A @ ( one_one @ A ) @ ( zero_zero @ ( poly @ A ) ) ) ) ) ).
% one_poly_eq_simps(1)
thf(fact_26_smult__0__left,axiom,
! [A: $tType] :
( ( comm_semiring_0 @ A )
=> ! [P2: poly @ A] :
( ( smult @ A @ ( zero_zero @ A ) @ P2 )
= ( zero_zero @ ( poly @ A ) ) ) ) ).
% smult_0_left
thf(fact_27_smult__eq__0__iff,axiom,
! [A: $tType] :
( ( ( comm_semiring_0 @ A )
& ( semiri1193490041visors @ A ) )
=> ! [A2: A,P2: poly @ A] :
( ( ( smult @ A @ A2 @ P2 )
= ( zero_zero @ ( poly @ A ) ) )
= ( ( A2
= ( zero_zero @ A ) )
| ( P2
= ( zero_zero @ ( poly @ A ) ) ) ) ) ) ).
% smult_eq_0_iff
thf(fact_28_smult__pCons,axiom,
! [A: $tType] :
( ( comm_semiring_0 @ A )
=> ! [A2: A,B: A,P2: poly @ A] :
( ( smult @ A @ A2 @ ( pCons @ A @ B @ P2 ) )
= ( pCons @ A @ ( times_times @ A @ A2 @ B ) @ ( smult @ A @ A2 @ P2 ) ) ) ) ).
% smult_pCons
thf(fact_29_coeffs__eq__Nil,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [P2: poly @ A] :
( ( ( coeffs @ A @ P2 )
= ( nil @ A ) )
= ( P2
= ( zero_zero @ ( poly @ A ) ) ) ) ) ).
% coeffs_eq_Nil
thf(fact_30_coeffs__0__eq__Nil,axiom,
! [A: $tType] :
( ( zero @ A )
=> ( ( coeffs @ A @ ( zero_zero @ ( poly @ A ) ) )
= ( nil @ A ) ) ) ).
% coeffs_0_eq_Nil
thf(fact_31_psums__0__Cons,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [Xs: list @ A] :
( ( descar1668888542_psums @ A @ ( cons @ A @ ( zero_zero @ A ) @ Xs ) )
= ( cons @ A @ ( zero_zero @ A ) @ ( descar1668888542_psums @ A @ Xs ) ) ) ) ).
% psums_0_Cons
thf(fact_32_coeffs__1__eq,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( coeffs @ A @ ( one_one @ ( poly @ A ) ) )
= ( cons @ A @ ( one_one @ A ) @ ( nil @ A ) ) ) ) ).
% coeffs_1_eq
thf(fact_33_sign__changes__singleton,axiom,
! [A: $tType] :
( ( ( sgn @ A )
& ( zero @ A ) )
=> ! [X: A] :
( ( descar149487500hanges @ A @ ( cons @ A @ X @ ( nil @ A ) ) )
= ( zero_zero @ nat ) ) ) ).
% sign_changes_singleton
thf(fact_34__092_060open_062sign__changes_Axs_A_061_Av_A_I_091_0581_058_058_Ha_M_A_N_A_I1_058_058_Ha_J_058_093_A_K_Ag_J_092_060close_062,axiom,
( ( descar149487500hanges @ a @ xs )
= ( v @ ( times_times @ ( poly @ a ) @ ( pCons @ a @ ( one_one @ a ) @ ( pCons @ a @ ( uminus_uminus @ a @ ( one_one @ a ) ) @ ( zero_zero @ ( poly @ a ) ) ) ) @ g ) ) ) ).
% \<open>sign_changes xs = v ([:1::'a, - (1::'a):] * g)\<close>
thf(fact_35_Poly__snoc__zero,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [As: list @ A] :
( ( poly2 @ A @ ( append @ A @ As @ ( cons @ A @ ( zero_zero @ A ) @ ( nil @ A ) ) ) )
= ( poly2 @ A @ As ) ) ) ).
% Poly_snoc_zero
thf(fact_36_Poly_Osimps_I2_J,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [A2: A,As: list @ A] :
( ( poly2 @ A @ ( cons @ A @ A2 @ As ) )
= ( pCons @ A @ A2 @ ( poly2 @ A @ As ) ) ) ) ).
% Poly.simps(2)
thf(fact_37_Poly_Osimps_I1_J,axiom,
! [A: $tType] :
( ( zero @ A )
=> ( ( poly2 @ A @ ( nil @ A ) )
= ( zero_zero @ ( poly @ A ) ) ) ) ).
% Poly.simps(1)
thf(fact_38_psums_Osimps_I2_J,axiom,
! [A: $tType] :
( ( plus @ A )
=> ! [X: A] :
( ( descar1668888542_psums @ A @ ( cons @ A @ X @ ( nil @ A ) ) )
= ( cons @ A @ X @ ( nil @ A ) ) ) ) ).
% psums.simps(2)
thf(fact_39_psums_Osimps_I1_J,axiom,
! [A: $tType] :
( ( plus @ A )
=> ( ( descar1668888542_psums @ A @ ( nil @ A ) )
= ( nil @ A ) ) ) ).
% psums.simps(1)
thf(fact_40_pCons__one,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( pCons @ A @ ( one_one @ A ) @ ( zero_zero @ ( poly @ A ) ) )
= ( one_one @ ( poly @ A ) ) ) ) ).
% pCons_one
thf(fact_41_pCons__cases,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [P2: poly @ A] :
~ ! [A3: A,Q3: poly @ A] :
( P2
!= ( pCons @ A @ A3 @ Q3 ) ) ) ).
% pCons_cases
thf(fact_42_is__zero__null,axiom,
! [A: $tType] :
( ( zero @ A )
=> ( ( is_zero @ A )
= ( ^ [P: poly @ A] :
( P
= ( zero_zero @ ( poly @ A ) ) ) ) ) ) ).
% is_zero_null
thf(fact_43_pCons__induct,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [P3: ( poly @ A ) > $o,P2: poly @ A] :
( ( P3 @ ( zero_zero @ ( poly @ A ) ) )
=> ( ! [A3: A,P4: poly @ A] :
( ( ( A3
!= ( zero_zero @ A ) )
| ( P4
!= ( zero_zero @ ( poly @ A ) ) ) )
=> ( ( P3 @ P4 )
=> ( P3 @ ( pCons @ A @ A3 @ P4 ) ) ) )
=> ( P3 @ P2 ) ) ) ) ).
% pCons_induct
thf(fact_44_pderiv_Ocases,axiom,
! [A: $tType] :
( ( ( comm_semiring_1 @ A )
& ( semiri1193490041visors @ A ) )
=> ! [X: poly @ A] :
~ ! [A3: A,P4: poly @ A] :
( X
!= ( pCons @ A @ A3 @ P4 ) ) ) ).
% pderiv.cases
thf(fact_45_ext,axiom,
! [B2: $tType,A: $tType,F2: A > B2,G: A > B2] :
( ! [X2: A] :
( ( F2 @ X2 )
= ( G @ X2 ) )
=> ( F2 = G ) ) ).
% ext
thf(fact_46_poly__induct2,axiom,
! [A: $tType,B2: $tType] :
( ( ( zero @ B2 )
& ( zero @ A ) )
=> ! [P3: ( poly @ A ) > ( poly @ B2 ) > $o,P2: poly @ A,Q2: poly @ B2] :
( ( P3 @ ( zero_zero @ ( poly @ A ) ) @ ( zero_zero @ ( poly @ B2 ) ) )
=> ( ! [A3: A,P4: poly @ A,B3: B2,Q3: poly @ B2] :
( ( P3 @ P4 @ Q3 )
=> ( P3 @ ( pCons @ A @ A3 @ P4 ) @ ( pCons @ B2 @ B3 @ Q3 ) ) )
=> ( P3 @ P2 @ Q2 ) ) ) ) ).
% poly_induct2
thf(fact_47_pderiv_Oinduct,axiom,
! [A: $tType] :
( ( ( comm_semiring_1 @ A )
& ( semiri1193490041visors @ A ) )
=> ! [P3: ( poly @ A ) > $o,A0: poly @ A] :
( ! [A3: A,P4: poly @ A] :
( ( ( P4
!= ( zero_zero @ ( poly @ A ) ) )
=> ( P3 @ P4 ) )
=> ( P3 @ ( pCons @ A @ A3 @ P4 ) ) )
=> ( P3 @ A0 ) ) ) ).
% pderiv.induct
thf(fact_48_mult__poly__0__left,axiom,
! [A: $tType] :
( ( comm_semiring_0 @ A )
=> ! [Q2: poly @ A] :
( ( times_times @ ( poly @ A ) @ ( zero_zero @ ( poly @ A ) ) @ Q2 )
= ( zero_zero @ ( poly @ A ) ) ) ) ).
% mult_poly_0_left
thf(fact_49_mult__poly__0__right,axiom,
! [A: $tType] :
( ( comm_semiring_0 @ A )
=> ! [P2: poly @ A] :
( ( times_times @ ( poly @ A ) @ P2 @ ( zero_zero @ ( poly @ A ) ) )
= ( zero_zero @ ( poly @ A ) ) ) ) ).
% mult_poly_0_right
thf(fact_50_plus__coeffs_Oinduct,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [P3: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
( ! [Xs2: list @ A] : ( P3 @ Xs2 @ ( nil @ A ) )
=> ( ! [V: A,Va: list @ A] : ( P3 @ ( nil @ A ) @ ( cons @ A @ V @ Va ) )
=> ( ! [X2: A,Xs2: list @ A,Y2: A,Ys: list @ A] :
( ( P3 @ Xs2 @ Ys )
=> ( P3 @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ A @ Y2 @ Ys ) ) )
=> ( P3 @ A0 @ A1 ) ) ) ) ) ).
% plus_coeffs.induct
thf(fact_51_not__0__coeffs__not__Nil,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [P2: poly @ A] :
( ( P2
!= ( zero_zero @ ( poly @ A ) ) )
=> ( ( coeffs @ A @ P2 )
!= ( nil @ A ) ) ) ) ).
% not_0_coeffs_not_Nil
thf(fact_52_minus__poly__rev__list_Oinduct,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [P3: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
( ! [X2: A,Xs2: list @ A,Y2: A,Ys: list @ A] :
( ( P3 @ Xs2 @ Ys )
=> ( P3 @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ A @ Y2 @ Ys ) ) )
=> ( ! [Xs2: list @ A] : ( P3 @ Xs2 @ ( nil @ A ) )
=> ( ! [Y2: A,Ys: list @ A] : ( P3 @ ( nil @ A ) @ ( cons @ A @ Y2 @ Ys ) )
=> ( P3 @ A0 @ A1 ) ) ) ) ) ).
% minus_poly_rev_list.induct
thf(fact_53_synthetic__div__unique__lemma,axiom,
! [A: $tType] :
( ( comm_semiring_0 @ A )
=> ! [C: A,P2: poly @ A,A2: A] :
( ( ( smult @ A @ C @ P2 )
= ( pCons @ A @ A2 @ P2 ) )
=> ( P2
= ( zero_zero @ ( poly @ A ) ) ) ) ) ).
% synthetic_div_unique_lemma
thf(fact_54_psums_Ocases,axiom,
! [A: $tType] :
( ( plus @ A )
=> ! [X: list @ A] :
( ( X
!= ( nil @ A ) )
=> ( ! [X2: A] :
( X
!= ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ~ ! [X2: A,Y2: A,Xs2: list @ A] :
( X
!= ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Xs2 ) ) ) ) ) ) ).
% psums.cases
thf(fact_55_reduce__root__pCons,axiom,
! [A: $tType] :
( ( comm_semiring_0 @ A )
=> ! [A2: A,C: A,P2: poly @ A] :
( ( descar316357986e_root @ A @ A2 @ ( pCons @ A @ C @ P2 ) )
= ( pCons @ A @ C @ ( smult @ A @ A2 @ ( descar316357986e_root @ A @ A2 @ P2 ) ) ) ) ) ).
% reduce_root_pCons
thf(fact_56_reduce__root__nonzero,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [A2: A,P2: poly @ A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( P2
!= ( zero_zero @ ( poly @ A ) ) )
=> ( ( descar316357986e_root @ A @ A2 @ P2 )
!= ( zero_zero @ ( poly @ A ) ) ) ) ) ) ).
% reduce_root_nonzero
thf(fact_57_sign__changes__Cons__Cons__same,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y3: A,Xs: list @ A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ X @ Y3 ) )
=> ( ( descar149487500hanges @ A @ ( cons @ A @ X @ ( cons @ A @ Y3 @ Xs ) ) )
= ( descar149487500hanges @ A @ ( cons @ A @ Y3 @ Xs ) ) ) ) ) ).
% sign_changes_Cons_Cons_same
thf(fact_58_append1__eq__conv,axiom,
! [A: $tType,Xs: list @ A,X: A,Ys2: list @ A,Y3: A] :
( ( ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) )
= ( append @ A @ Ys2 @ ( cons @ A @ Y3 @ ( nil @ A ) ) ) )
= ( ( Xs = Ys2 )
& ( X = Y3 ) ) ) ).
% append1_eq_conv
thf(fact_59_mult__minus1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z2: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ Z2 )
= ( uminus_uminus @ A @ Z2 ) ) ) ).
% mult_minus1
thf(fact_60_mult__minus1__right,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z2: A] :
( ( times_times @ A @ Z2 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ Z2 ) ) ) ).
% mult_minus1_right
thf(fact_61_less__neg__neg,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A2: A] :
( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% less_neg_neg
thf(fact_62_neg__less__pos,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% neg_less_pos
thf(fact_63_neg__0__less__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% neg_0_less_iff_less
thf(fact_64_neg__less__0__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% neg_less_0_iff_less
thf(fact_65_mult__cancel__left1,axiom,
! [A: $tType] :
( ( ring_11004092258visors @ A )
=> ! [C: A,B: A] :
( ( C
= ( times_times @ A @ C @ B ) )
= ( ( C
= ( zero_zero @ A ) )
| ( B
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_left1
thf(fact_66_mult__cancel__left2,axiom,
! [A: $tType] :
( ( ring_11004092258visors @ A )
=> ! [C: A,A2: A] :
( ( ( times_times @ A @ C @ A2 )
= C )
= ( ( C
= ( zero_zero @ A ) )
| ( A2
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_left2
thf(fact_67_mult__cancel__right1,axiom,
! [A: $tType] :
( ( ring_11004092258visors @ A )
=> ! [C: A,B: A] :
( ( C
= ( times_times @ A @ B @ C ) )
= ( ( C
= ( zero_zero @ A ) )
| ( B
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_right1
thf(fact_68_mult__cancel__right2,axiom,
! [A: $tType] :
( ( ring_11004092258visors @ A )
=> ! [A2: A,C: A] :
( ( ( times_times @ A @ A2 @ C )
= C )
= ( ( C
= ( zero_zero @ A ) )
| ( A2
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_right2
thf(fact_69_neg__equal__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B: A] :
( ( ( uminus_uminus @ A @ A2 )
= ( uminus_uminus @ A @ B ) )
= ( A2 = B ) ) ) ).
% neg_equal_iff_equal
thf(fact_70_add_Oinverse__inverse,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A2 ) )
= A2 ) ) ).
% add.inverse_inverse
thf(fact_71_list_Oinject,axiom,
! [A: $tType,X21: A,X22: list @ A,Y21: A,Y22: list @ A] :
( ( ( cons @ A @ X21 @ X22 )
= ( cons @ A @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_72_same__append__eq,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A,Zs: list @ A] :
( ( ( append @ A @ Xs @ Ys2 )
= ( append @ A @ Xs @ Zs ) )
= ( Ys2 = Zs ) ) ).
% same_append_eq
thf(fact_73_append__same__eq,axiom,
! [A: $tType,Ys2: list @ A,Xs: list @ A,Zs: list @ A] :
( ( ( append @ A @ Ys2 @ Xs )
= ( append @ A @ Zs @ Xs ) )
= ( Ys2 = Zs ) ) ).
% append_same_eq
thf(fact_74_append__assoc,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A,Zs: list @ A] :
( ( append @ A @ ( append @ A @ Xs @ Ys2 ) @ Zs )
= ( append @ A @ Xs @ ( append @ A @ Ys2 @ Zs ) ) ) ).
% append_assoc
thf(fact_75_append_Oassoc,axiom,
! [A: $tType,A2: list @ A,B: list @ A,C: list @ A] :
( ( append @ A @ ( append @ A @ A2 @ B ) @ C )
= ( append @ A @ A2 @ ( append @ A @ B @ C ) ) ) ).
% append.assoc
thf(fact_76_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_77_mult__cancel__right,axiom,
! [A: $tType] :
( ( semiri1923998003cancel @ A )
=> ! [A2: A,C: A,B: A] :
( ( ( times_times @ A @ A2 @ C )
= ( times_times @ A @ B @ C ) )
= ( ( C
= ( zero_zero @ A ) )
| ( A2 = B ) ) ) ) ).
% mult_cancel_right
thf(fact_78_mult__cancel__left,axiom,
! [A: $tType] :
( ( semiri1923998003cancel @ A )
=> ! [C: A,A2: A,B: A] :
( ( ( times_times @ A @ C @ A2 )
= ( times_times @ A @ C @ B ) )
= ( ( C
= ( zero_zero @ A ) )
| ( A2 = B ) ) ) ) ).
% mult_cancel_left
thf(fact_79_mult__eq__0__iff,axiom,
! [A: $tType] :
( ( semiri1193490041visors @ A )
=> ! [A2: A,B: A] :
( ( ( times_times @ A @ A2 @ B )
= ( zero_zero @ A ) )
= ( ( A2
= ( zero_zero @ A ) )
| ( B
= ( zero_zero @ A ) ) ) ) ) ).
% mult_eq_0_iff
thf(fact_80_mult__zero__right,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A2: A] :
( ( times_times @ A @ A2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% mult_zero_right
thf(fact_81_mult__zero__left,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A2: A] :
( ( times_times @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% mult_zero_left
thf(fact_82_neg__equal__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A2: A] :
( ( ( uminus_uminus @ A @ A2 )
= A2 )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% neg_equal_zero
thf(fact_83_equal__neg__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A )
=> ! [A2: A] :
( ( A2
= ( uminus_uminus @ A @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% equal_neg_zero
thf(fact_84_neg__equal__0__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( ( uminus_uminus @ A @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% neg_equal_0_iff_equal
thf(fact_85_neg__0__equal__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( ( zero_zero @ A )
= ( uminus_uminus @ A @ A2 ) )
= ( ( zero_zero @ A )
= A2 ) ) ) ).
% neg_0_equal_iff_equal
thf(fact_86_add_Oinverse__neutral,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( uminus_uminus @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% add.inverse_neutral
thf(fact_87_mult_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A] :
( ( times_times @ A @ A2 @ ( one_one @ A ) )
= A2 ) ) ).
% mult.right_neutral
thf(fact_88_mult_Oleft__neutral,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A] :
( ( times_times @ A @ ( one_one @ A ) @ A2 )
= A2 ) ) ).
% mult.left_neutral
thf(fact_89_neg__less__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B: A,A2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ B ) @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less @ A @ A2 @ B ) ) ) ).
% neg_less_iff_less
thf(fact_90_mult__minus__right,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,B: A] :
( ( times_times @ A @ A2 @ ( uminus_uminus @ A @ B ) )
= ( uminus_uminus @ A @ ( times_times @ A @ A2 @ B ) ) ) ) ).
% mult_minus_right
thf(fact_91_minus__mult__minus,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,B: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ A2 ) @ ( uminus_uminus @ A @ B ) )
= ( times_times @ A @ A2 @ B ) ) ) ).
% minus_mult_minus
thf(fact_92_mult__minus__left,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,B: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ A2 ) @ B )
= ( uminus_uminus @ A @ ( times_times @ A @ A2 @ B ) ) ) ) ).
% mult_minus_left
thf(fact_93_append_Oright__neutral,axiom,
! [A: $tType,A2: list @ A] :
( ( append @ A @ A2 @ ( nil @ A ) )
= A2 ) ).
% append.right_neutral
thf(fact_94_append__is__Nil__conv,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( ( append @ A @ Xs @ Ys2 )
= ( nil @ A ) )
= ( ( Xs
= ( nil @ A ) )
& ( Ys2
= ( nil @ A ) ) ) ) ).
% append_is_Nil_conv
thf(fact_95_Nil__is__append__conv,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( ( nil @ A )
= ( append @ A @ Xs @ Ys2 ) )
= ( ( Xs
= ( nil @ A ) )
& ( Ys2
= ( nil @ A ) ) ) ) ).
% Nil_is_append_conv
thf(fact_96_self__append__conv2,axiom,
! [A: $tType,Ys2: list @ A,Xs: list @ A] :
( ( Ys2
= ( append @ A @ Xs @ Ys2 ) )
= ( Xs
= ( nil @ A ) ) ) ).
% self_append_conv2
thf(fact_97_append__self__conv2,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( ( append @ A @ Xs @ Ys2 )
= Ys2 )
= ( Xs
= ( nil @ A ) ) ) ).
% append_self_conv2
thf(fact_98_self__append__conv,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( Xs
= ( append @ A @ Xs @ Ys2 ) )
= ( Ys2
= ( nil @ A ) ) ) ).
% self_append_conv
thf(fact_99_append__self__conv,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( ( append @ A @ Xs @ Ys2 )
= Xs )
= ( Ys2
= ( nil @ A ) ) ) ).
% append_self_conv
thf(fact_100_append__Nil2,axiom,
! [A: $tType,Xs: list @ A] :
( ( append @ A @ Xs @ ( nil @ A ) )
= Xs ) ).
% append_Nil2
thf(fact_101_smult__minus__right,axiom,
! [A: $tType] :
( ( comm_ring @ A )
=> ! [A2: A,P2: poly @ A] :
( ( smult @ A @ A2 @ ( uminus_uminus @ ( poly @ A ) @ P2 ) )
= ( uminus_uminus @ ( poly @ A ) @ ( smult @ A @ A2 @ P2 ) ) ) ) ).
% smult_minus_right
thf(fact_102_smult__one,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [C: A] :
( ( smult @ A @ C @ ( one_one @ ( poly @ A ) ) )
= ( pCons @ A @ C @ ( zero_zero @ ( poly @ A ) ) ) ) ) ).
% smult_one
thf(fact_103_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X: A] :
( ( ( zero_zero @ A )
= X )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_104_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y3: A] :
( ( X != Y3 )
=> ( ~ ( ord_less @ A @ X @ Y3 )
=> ( ord_less @ A @ Y3 @ X ) ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_105_mult_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [B: A,A2: A,C: A] :
( ( times_times @ A @ B @ ( times_times @ A @ A2 @ C ) )
= ( times_times @ A @ A2 @ ( times_times @ A @ B @ C ) ) ) ) ).
% mult.left_commute
thf(fact_106_mult_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ( ( times_times @ A )
= ( ^ [A4: A,B4: A] : ( times_times @ A @ B4 @ A4 ) ) ) ) ).
% mult.commute
thf(fact_107_mult_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_mult @ A )
=> ! [A2: A,B: A,C: A] :
( ( times_times @ A @ ( times_times @ A @ A2 @ B ) @ C )
= ( times_times @ A @ A2 @ ( times_times @ A @ B @ C ) ) ) ) ).
% mult.assoc
thf(fact_108_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [A2: A,B: A,C: A] :
( ( times_times @ A @ ( times_times @ A @ A2 @ B ) @ C )
= ( times_times @ A @ A2 @ ( times_times @ A @ B @ C ) ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_109_one__reorient,axiom,
! [A: $tType] :
( ( one @ A )
=> ! [X: A] :
( ( ( one_one @ A )
= X )
= ( X
= ( one_one @ A ) ) ) ) ).
% one_reorient
thf(fact_110_minus__equation__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B: A] :
( ( ( uminus_uminus @ A @ A2 )
= B )
= ( ( uminus_uminus @ A @ B )
= A2 ) ) ) ).
% minus_equation_iff
thf(fact_111_equation__minus__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B: A] :
( ( A2
= ( uminus_uminus @ A @ B ) )
= ( B
= ( uminus_uminus @ A @ A2 ) ) ) ) ).
% equation_minus_iff
thf(fact_112_not__Cons__self2,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( cons @ A @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_113_append__eq__append__conv2,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A,Zs: list @ A,Ts: list @ A] :
( ( ( append @ A @ Xs @ Ys2 )
= ( append @ A @ Zs @ Ts ) )
= ( ? [Us: list @ A] :
( ( ( Xs
= ( append @ A @ Zs @ Us ) )
& ( ( append @ A @ Us @ Ys2 )
= Ts ) )
| ( ( ( append @ A @ Xs @ Us )
= Zs )
& ( Ys2
= ( append @ A @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_114_append__eq__appendI,axiom,
! [A: $tType,Xs: list @ A,Xs1: list @ A,Zs: list @ A,Ys2: list @ A,Us2: list @ A] :
( ( ( append @ A @ Xs @ Xs1 )
= Zs )
=> ( ( Ys2
= ( append @ A @ Xs1 @ Us2 ) )
=> ( ( append @ A @ Xs @ Ys2 )
= ( append @ A @ Zs @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_115_sign__changes__two,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y3: A] :
( ( ( ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
& ( ord_less @ A @ Y3 @ ( zero_zero @ A ) ) )
| ( ( ord_less @ A @ X @ ( zero_zero @ A ) )
& ( ord_less @ A @ ( zero_zero @ A ) @ Y3 ) ) )
=> ( ( descar149487500hanges @ A @ ( cons @ A @ X @ ( cons @ A @ Y3 @ ( nil @ A ) ) ) )
= ( one_one @ nat ) ) )
& ( ~ ( ( ( ord_less @ A @ ( zero_zero @ A ) @ X )
& ( ord_less @ A @ Y3 @ ( zero_zero @ A ) ) )
| ( ( ord_less @ A @ X @ ( zero_zero @ A ) )
& ( ord_less @ A @ ( zero_zero @ A ) @ Y3 ) ) )
=> ( ( descar149487500hanges @ A @ ( cons @ A @ X @ ( cons @ A @ Y3 @ ( nil @ A ) ) ) )
= ( zero_zero @ nat ) ) ) ) ) ).
% sign_changes_two
thf(fact_116_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_117_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_118_not__less__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N: A] :
~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% not_less_zero
thf(fact_119_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_120_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_121_mult__right__cancel,axiom,
! [A: $tType] :
( ( semiri1923998003cancel @ A )
=> ! [C: A,A2: A,B: A] :
( ( C
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ A2 @ C )
= ( times_times @ A @ B @ C ) )
= ( A2 = B ) ) ) ) ).
% mult_right_cancel
thf(fact_122_mult__left__cancel,axiom,
! [A: $tType] :
( ( semiri1923998003cancel @ A )
=> ! [C: A,A2: A,B: A] :
( ( C
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ C @ A2 )
= ( times_times @ A @ C @ B ) )
= ( A2 = B ) ) ) ) ).
% mult_left_cancel
thf(fact_123_no__zero__divisors,axiom,
! [A: $tType] :
( ( semiri1193490041visors @ A )
=> ! [A2: A,B: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( B
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A2 @ B )
!= ( zero_zero @ A ) ) ) ) ) ).
% no_zero_divisors
thf(fact_124_divisors__zero,axiom,
! [A: $tType] :
( ( semiri1193490041visors @ A )
=> ! [A2: A,B: A] :
( ( ( times_times @ A @ A2 @ B )
= ( zero_zero @ A ) )
=> ( ( A2
= ( zero_zero @ A ) )
| ( B
= ( zero_zero @ A ) ) ) ) ) ).
% divisors_zero
thf(fact_125_mult__not__zero,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A2: A,B: A] :
( ( ( times_times @ A @ A2 @ B )
!= ( zero_zero @ A ) )
=> ( ( A2
!= ( zero_zero @ A ) )
& ( B
!= ( zero_zero @ A ) ) ) ) ) ).
% mult_not_zero
thf(fact_126_zero__neq__one,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ( ( zero_zero @ A )
!= ( one_one @ A ) ) ) ).
% zero_neq_one
thf(fact_127_less__numeral__extra_I4_J,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% less_numeral_extra(4)
thf(fact_128_mult_Ocomm__neutral,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A] :
( ( times_times @ A @ A2 @ ( one_one @ A ) )
= A2 ) ) ).
% mult.comm_neutral
thf(fact_129_comm__monoid__mult__class_Omult__1,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A] :
( ( times_times @ A @ ( one_one @ A ) @ A2 )
= A2 ) ) ).
% comm_monoid_mult_class.mult_1
thf(fact_130_minus__less__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ B )
= ( ord_less @ A @ ( uminus_uminus @ A @ B ) @ A2 ) ) ) ).
% minus_less_iff
thf(fact_131_less__minus__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ B ) )
= ( ord_less @ A @ B @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% less_minus_iff
thf(fact_132_minus__mult__commute,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,B: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ A2 ) @ B )
= ( times_times @ A @ A2 @ ( uminus_uminus @ A @ B ) ) ) ) ).
% minus_mult_commute
thf(fact_133_square__eq__iff,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [A2: A,B: A] :
( ( ( times_times @ A @ A2 @ A2 )
= ( times_times @ A @ B @ B ) )
= ( ( A2 = B )
| ( A2
= ( uminus_uminus @ A @ B ) ) ) ) ) ).
% square_eq_iff
thf(fact_134_one__neq__neg__one,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ( ( one_one @ A )
!= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% one_neq_neg_one
thf(fact_135_strict__sorted_Oinduct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P3: ( list @ A ) > $o,A0: list @ A] :
( ( P3 @ ( nil @ A ) )
=> ( ! [X2: A,Ys: list @ A] :
( ( P3 @ Ys )
=> ( P3 @ ( cons @ A @ X2 @ Ys ) ) )
=> ( P3 @ A0 ) ) ) ) ).
% strict_sorted.induct
thf(fact_136_strict__sorted_Ocases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: list @ A] :
( ( X
!= ( nil @ A ) )
=> ~ ! [X2: A,Ys: list @ A] :
( X
!= ( cons @ A @ X2 @ Ys ) ) ) ) ).
% strict_sorted.cases
thf(fact_137_map__tailrec__rev_Oinduct,axiom,
! [A: $tType,B2: $tType,P3: ( A > B2 ) > ( list @ A ) > ( list @ B2 ) > $o,A0: A > B2,A1: list @ A,A22: list @ B2] :
( ! [F3: A > B2,X_1: list @ B2] : ( P3 @ F3 @ ( nil @ A ) @ X_1 )
=> ( ! [F3: A > B2,A3: A,As2: list @ A,Bs: list @ B2] :
( ( P3 @ F3 @ As2 @ ( cons @ B2 @ ( F3 @ A3 ) @ Bs ) )
=> ( P3 @ F3 @ ( cons @ A @ A3 @ As2 ) @ Bs ) )
=> ( P3 @ A0 @ A1 @ A22 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_138_list__nonempty__induct,axiom,
! [A: $tType,Xs: list @ A,P3: ( list @ A ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X2: A] : ( P3 @ ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ( ! [X2: A,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( P3 @ Xs2 )
=> ( P3 @ ( cons @ A @ X2 @ Xs2 ) ) ) )
=> ( P3 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_139_successively_Oinduct,axiom,
! [A: $tType,P3: ( A > A > $o ) > ( list @ A ) > $o,A0: A > A > $o,A1: list @ A] :
( ! [P5: A > A > $o] : ( P3 @ P5 @ ( nil @ A ) )
=> ( ! [P5: A > A > $o,X2: A] : ( P3 @ P5 @ ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ( ! [P5: A > A > $o,X2: A,Y2: A,Xs2: list @ A] :
( ( P3 @ P5 @ ( cons @ A @ Y2 @ Xs2 ) )
=> ( P3 @ P5 @ ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Xs2 ) ) ) )
=> ( P3 @ A0 @ A1 ) ) ) ) ).
% successively.induct
thf(fact_140_arg__min__list_Oinduct,axiom,
! [B2: $tType,A: $tType] :
( ( linorder @ B2 )
=> ! [P3: ( A > B2 ) > ( list @ A ) > $o,A0: A > B2,A1: list @ A] :
( ! [F3: A > B2,X2: A] : ( P3 @ F3 @ ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ( ! [F3: A > B2,X2: A,Y2: A,Zs2: list @ A] :
( ( P3 @ F3 @ ( cons @ A @ Y2 @ Zs2 ) )
=> ( P3 @ F3 @ ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Zs2 ) ) ) )
=> ( ! [A3: A > B2] : ( P3 @ A3 @ ( nil @ A ) )
=> ( P3 @ A0 @ A1 ) ) ) ) ) ).
% arg_min_list.induct
thf(fact_141_remdups__adj_Oinduct,axiom,
! [A: $tType,P3: ( list @ A ) > $o,A0: list @ A] :
( ( P3 @ ( nil @ A ) )
=> ( ! [X2: A] : ( P3 @ ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ( ! [X2: A,Y2: A,Xs2: list @ A] :
( ( ( X2 = Y2 )
=> ( P3 @ ( cons @ A @ X2 @ Xs2 ) ) )
=> ( ( ( X2 != Y2 )
=> ( P3 @ ( cons @ A @ Y2 @ Xs2 ) ) )
=> ( P3 @ ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Xs2 ) ) ) ) )
=> ( P3 @ A0 ) ) ) ) ).
% remdups_adj.induct
thf(fact_142_sorted__wrt_Oinduct,axiom,
! [A: $tType,P3: ( A > A > $o ) > ( list @ A ) > $o,A0: A > A > $o,A1: list @ A] :
( ! [P5: A > A > $o] : ( P3 @ P5 @ ( nil @ A ) )
=> ( ! [P5: A > A > $o,X2: A,Ys: list @ A] :
( ( P3 @ P5 @ Ys )
=> ( P3 @ P5 @ ( cons @ A @ X2 @ Ys ) ) )
=> ( P3 @ A0 @ A1 ) ) ) ).
% sorted_wrt.induct
thf(fact_143_remdups__adj_Ocases,axiom,
! [A: $tType,X: list @ A] :
( ( X
!= ( nil @ A ) )
=> ( ! [X2: A] :
( X
!= ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ~ ! [X2: A,Y2: A,Xs2: list @ A] :
( X
!= ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_144_transpose_Ocases,axiom,
! [A: $tType,X: list @ ( list @ A )] :
( ( X
!= ( nil @ ( list @ A ) ) )
=> ( ! [Xss: list @ ( list @ A )] :
( X
!= ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
=> ~ ! [X2: A,Xs2: list @ A,Xss: list @ ( list @ A )] :
( X
!= ( cons @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_145_shuffles_Oinduct,axiom,
! [A: $tType,P3: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
( ! [X_1: list @ A] : ( P3 @ ( nil @ A ) @ X_1 )
=> ( ! [Xs2: list @ A] : ( P3 @ Xs2 @ ( nil @ A ) )
=> ( ! [X2: A,Xs2: list @ A,Y2: A,Ys: list @ A] :
( ( P3 @ Xs2 @ ( cons @ A @ Y2 @ Ys ) )
=> ( ( P3 @ ( cons @ A @ X2 @ Xs2 ) @ Ys )
=> ( P3 @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ A @ Y2 @ Ys ) ) ) )
=> ( P3 @ A0 @ A1 ) ) ) ) ).
% shuffles.induct
thf(fact_146_min__list_Oinduct,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [P3: ( list @ A ) > $o,A0: list @ A] :
( ! [X2: A,Xs2: list @ A] :
( ! [X212: A,X222: list @ A] :
( ( Xs2
= ( cons @ A @ X212 @ X222 ) )
=> ( P3 @ Xs2 ) )
=> ( P3 @ ( cons @ A @ X2 @ Xs2 ) ) )
=> ( ( P3 @ ( nil @ A ) )
=> ( P3 @ A0 ) ) ) ) ).
% min_list.induct
thf(fact_147_min__list_Ocases,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: list @ A] :
( ! [X2: A,Xs2: list @ A] :
( X
!= ( cons @ A @ X2 @ Xs2 ) )
=> ( X
= ( nil @ A ) ) ) ) ).
% min_list.cases
thf(fact_148_induct__list012,axiom,
! [A: $tType,P3: ( list @ A ) > $o,Xs: list @ A] :
( ( P3 @ ( nil @ A ) )
=> ( ! [X2: A] : ( P3 @ ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ( ! [X2: A,Y2: A,Zs2: list @ A] :
( ( P3 @ Zs2 )
=> ( ( P3 @ ( cons @ A @ Y2 @ Zs2 ) )
=> ( P3 @ ( cons @ A @ X2 @ ( cons @ A @ Y2 @ Zs2 ) ) ) ) )
=> ( P3 @ Xs ) ) ) ) ).
% induct_list012
thf(fact_149_splice_Oinduct,axiom,
! [A: $tType,P3: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
( ! [X_1: list @ A] : ( P3 @ ( nil @ A ) @ X_1 )
=> ( ! [X2: A,Xs2: list @ A,Ys: list @ A] :
( ( P3 @ Ys @ Xs2 )
=> ( P3 @ ( cons @ A @ X2 @ Xs2 ) @ Ys ) )
=> ( P3 @ A0 @ A1 ) ) ) ).
% splice.induct
thf(fact_150_list__induct2_H,axiom,
! [A: $tType,B2: $tType,P3: ( list @ A ) > ( list @ B2 ) > $o,Xs: list @ A,Ys2: list @ B2] :
( ( P3 @ ( nil @ A ) @ ( nil @ B2 ) )
=> ( ! [X2: A,Xs2: list @ A] : ( P3 @ ( cons @ A @ X2 @ Xs2 ) @ ( nil @ B2 ) )
=> ( ! [Y2: B2,Ys: list @ B2] : ( P3 @ ( nil @ A ) @ ( cons @ B2 @ Y2 @ Ys ) )
=> ( ! [X2: A,Xs2: list @ A,Y2: B2,Ys: list @ B2] :
( ( P3 @ Xs2 @ Ys )
=> ( P3 @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ B2 @ Y2 @ Ys ) ) )
=> ( P3 @ Xs @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_151_neq__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
= ( ? [Y4: A,Ys3: list @ A] :
( Xs
= ( cons @ A @ Y4 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_152_list_Oinducts,axiom,
! [A: $tType,P3: ( list @ A ) > $o,List: list @ A] :
( ( P3 @ ( nil @ A ) )
=> ( ! [X1: A,X23: list @ A] :
( ( P3 @ X23 )
=> ( P3 @ ( cons @ A @ X1 @ X23 ) ) )
=> ( P3 @ List ) ) ) ).
% list.inducts
thf(fact_153_list_Oexhaust,axiom,
! [A: $tType,Y3: list @ A] :
( ( Y3
!= ( nil @ A ) )
=> ~ ! [X213: A,X223: list @ A] :
( Y3
!= ( cons @ A @ X213 @ X223 ) ) ) ).
% list.exhaust
thf(fact_154_list_OdiscI,axiom,
! [A: $tType,List: list @ A,X21: A,X22: list @ A] :
( ( List
= ( cons @ A @ X21 @ X22 ) )
=> ( List
!= ( nil @ A ) ) ) ).
% list.discI
thf(fact_155_list_Odistinct_I1_J,axiom,
! [A: $tType,X21: A,X22: list @ A] :
( ( nil @ A )
!= ( cons @ A @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_156_append__Cons,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys2: list @ A] :
( ( append @ A @ ( cons @ A @ X @ Xs ) @ Ys2 )
= ( cons @ A @ X @ ( append @ A @ Xs @ Ys2 ) ) ) ).
% append_Cons
thf(fact_157_Cons__eq__appendI,axiom,
! [A: $tType,X: A,Xs1: list @ A,Ys2: list @ A,Xs: list @ A,Zs: list @ A] :
( ( ( cons @ A @ X @ Xs1 )
= Ys2 )
=> ( ( Xs
= ( append @ A @ Xs1 @ Zs ) )
=> ( ( cons @ A @ X @ Xs )
= ( append @ A @ Ys2 @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_158_append_Oleft__neutral,axiom,
! [A: $tType,A2: list @ A] :
( ( append @ A @ ( nil @ A ) @ A2 )
= A2 ) ).
% append.left_neutral
thf(fact_159_append__Nil,axiom,
! [A: $tType,Ys2: list @ A] :
( ( append @ A @ ( nil @ A ) @ Ys2 )
= Ys2 ) ).
% append_Nil
thf(fact_160_eq__Nil__appendI,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( Xs = Ys2 )
=> ( Xs
= ( append @ A @ ( nil @ A ) @ Ys2 ) ) ) ).
% eq_Nil_appendI
thf(fact_161_null__rec_I1_J,axiom,
! [A: $tType,X: A,Xs: list @ A] :
~ ( null @ A @ ( cons @ A @ X @ Xs ) ) ).
% null_rec(1)
thf(fact_162_eq__Nil__null,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
= ( nil @ A ) )
= ( null @ A @ Xs ) ) ).
% eq_Nil_null
thf(fact_163_null__rec_I2_J,axiom,
! [B2: $tType] : ( null @ B2 @ ( nil @ B2 ) ) ).
% null_rec(2)
thf(fact_164_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: $tType] :
( ( linord893533164strict @ A )
=> ! [A2: A,B: A,C: A] :
( ( ord_less @ A @ A2 @ B )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C )
=> ( ord_less @ A @ ( times_times @ A @ C @ A2 ) @ ( times_times @ A @ C @ B ) ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_165_mult__less__cancel__right__disj,axiom,
! [A: $tType] :
( ( linord581940658strict @ A )
=> ! [A2: A,C: A,B: A] :
( ( ord_less @ A @ ( times_times @ A @ A2 @ C ) @ ( times_times @ A @ B @ C ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C )
& ( ord_less @ A @ A2 @ B ) )
| ( ( ord_less @ A @ C @ ( zero_zero @ A ) )
& ( ord_less @ A @ B @ A2 ) ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_166_mult__strict__right__mono,axiom,
! [A: $tType] :
( ( linord20386208strict @ A )
=> ! [A2: A,B: A,C: A] :
( ( ord_less @ A @ A2 @ B )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C ) @ ( times_times @ A @ B @ C ) ) ) ) ) ).
% mult_strict_right_mono
thf(fact_167_mult__strict__right__mono__neg,axiom,
! [A: $tType] :
( ( linord581940658strict @ A )
=> ! [B: A,A2: A,C: A] :
( ( ord_less @ A @ B @ A2 )
=> ( ( ord_less @ A @ C @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C ) @ ( times_times @ A @ B @ C ) ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_168_mult__less__cancel__left__disj,axiom,
! [A: $tType] :
( ( linord581940658strict @ A )
=> ! [C: A,A2: A,B: A] :
( ( ord_less @ A @ ( times_times @ A @ C @ A2 ) @ ( times_times @ A @ C @ B ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C )
& ( ord_less @ A @ A2 @ B ) )
| ( ( ord_less @ A @ C @ ( zero_zero @ A ) )
& ( ord_less @ A @ B @ A2 ) ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_169_mult__strict__left__mono,axiom,
! [A: $tType] :
( ( linord20386208strict @ A )
=> ! [A2: A,B: A,C: A] :
( ( ord_less @ A @ A2 @ B )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C )
=> ( ord_less @ A @ ( times_times @ A @ C @ A2 ) @ ( times_times @ A @ C @ B ) ) ) ) ) ).
% mult_strict_left_mono
thf(fact_170_mult__strict__left__mono__neg,axiom,
! [A: $tType] :
( ( linord581940658strict @ A )
=> ! [B: A,A2: A,C: A] :
( ( ord_less @ A @ B @ A2 )
=> ( ( ord_less @ A @ C @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ C @ A2 ) @ ( times_times @ A @ C @ B ) ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_171_mult__less__cancel__left__pos,axiom,
! [A: $tType] :
( ( linord581940658strict @ A )
=> ! [C: A,A2: A,B: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C )
=> ( ( ord_less @ A @ ( times_times @ A @ C @ A2 ) @ ( times_times @ A @ C @ B ) )
= ( ord_less @ A @ A2 @ B ) ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_172_mult__less__cancel__left__neg,axiom,
! [A: $tType] :
( ( linord581940658strict @ A )
=> ! [C: A,A2: A,B: A] :
( ( ord_less @ A @ C @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( times_times @ A @ C @ A2 ) @ ( times_times @ A @ C @ B ) )
= ( ord_less @ A @ B @ A2 ) ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_173_zero__less__mult__pos2,axiom,
! [A: $tType] :
( ( linord20386208strict @ A )
=> ! [B: A,A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ B @ A2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ B ) ) ) ) ).
% zero_less_mult_pos2
thf(fact_174_zero__less__mult__pos,axiom,
! [A: $tType] :
( ( linord20386208strict @ A )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ B ) ) ) ) ).
% zero_less_mult_pos
thf(fact_175_zero__less__mult__iff,axiom,
! [A: $tType] :
( ( linord581940658strict @ A )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less @ A @ ( zero_zero @ A ) @ B ) )
| ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_less_mult_iff
thf(fact_176_mult__pos__neg2,axiom,
! [A: $tType] :
( ( linord20386208strict @ A )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ B @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ B @ A2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_pos_neg2
thf(fact_177_mult__pos__pos,axiom,
! [A: $tType] :
( ( linord20386208strict @ A )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B ) ) ) ) ) ).
% mult_pos_pos
thf(fact_178_mult__pos__neg,axiom,
! [A: $tType] :
( ( linord20386208strict @ A )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ B @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ B ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_pos_neg
thf(fact_179_mult__neg__pos,axiom,
! [A: $tType] :
( ( linord20386208strict @ A )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ B ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_neg_pos
thf(fact_180_mult__less__0__iff,axiom,
! [A: $tType] :
( ( linord581940658strict @ A )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ ( times_times @ A @ A2 @ B ) @ ( zero_zero @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less @ A @ B @ ( zero_zero @ A ) ) )
| ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less @ A @ ( zero_zero @ A ) @ B ) ) ) ) ) ).
% mult_less_0_iff
thf(fact_181_not__square__less__zero,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ ( times_times @ A @ A2 @ A2 ) @ ( zero_zero @ A ) ) ) ).
% not_square_less_zero
thf(fact_182_mult__neg__neg,axiom,
! [A: $tType] :
( ( linord581940658strict @ A )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B ) ) ) ) ) ).
% mult_neg_neg
thf(fact_183_not__one__less__zero,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% not_one_less_zero
thf(fact_184_zero__less__one,axiom,
! [A: $tType] :
( ( zero_less_one @ A )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% zero_less_one
thf(fact_185_less__numeral__extra_I1_J,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% less_numeral_extra(1)
thf(fact_186_less__1__mult,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [M: A,N: A] :
( ( ord_less @ A @ ( one_one @ A ) @ M )
=> ( ( ord_less @ A @ ( one_one @ A ) @ N )
=> ( ord_less @ A @ ( one_one @ A ) @ ( times_times @ A @ M @ N ) ) ) ) ) ).
% less_1_mult
thf(fact_187_zero__neq__neg__one,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ( ( zero_zero @ A )
!= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% zero_neq_neg_one
thf(fact_188_less__minus__one__simps_I2_J,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( one_one @ A ) ) ) ).
% less_minus_one_simps(2)
thf(fact_189_less__minus__one__simps_I4_J,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% less_minus_one_simps(4)
thf(fact_190_square__eq__1__iff,axiom,
! [A: $tType] :
( ( ring_11004092258visors @ A )
=> ! [X: A] :
( ( ( times_times @ A @ X @ X )
= ( one_one @ A ) )
= ( ( X
= ( one_one @ A ) )
| ( X
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% square_eq_1_iff
thf(fact_191_rev__nonempty__induct,axiom,
! [A: $tType,Xs: list @ A,P3: ( list @ A ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X2: A] : ( P3 @ ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ( ! [X2: A,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( P3 @ Xs2 )
=> ( P3 @ ( append @ A @ Xs2 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) ) )
=> ( P3 @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_192_append__eq__Cons__conv,axiom,
! [A: $tType,Ys2: list @ A,Zs: list @ A,X: A,Xs: list @ A] :
( ( ( append @ A @ Ys2 @ Zs )
= ( cons @ A @ X @ Xs ) )
= ( ( ( Ys2
= ( nil @ A ) )
& ( Zs
= ( cons @ A @ X @ Xs ) ) )
| ? [Ys4: list @ A] :
( ( Ys2
= ( cons @ A @ X @ Ys4 ) )
& ( ( append @ A @ Ys4 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_193_Cons__eq__append__conv,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys2: list @ A,Zs: list @ A] :
( ( ( cons @ A @ X @ Xs )
= ( append @ A @ Ys2 @ Zs ) )
= ( ( ( Ys2
= ( nil @ A ) )
& ( ( cons @ A @ X @ Xs )
= Zs ) )
| ? [Ys4: list @ A] :
( ( ( cons @ A @ X @ Ys4 )
= Ys2 )
& ( Xs
= ( append @ A @ Ys4 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_194_rev__exhaust,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ~ ! [Ys: list @ A,Y2: A] :
( Xs
!= ( append @ A @ Ys @ ( cons @ A @ Y2 @ ( nil @ A ) ) ) ) ) ).
% rev_exhaust
thf(fact_195_rev__induct,axiom,
! [A: $tType,P3: ( list @ A ) > $o,Xs: list @ A] :
( ( P3 @ ( nil @ A ) )
=> ( ! [X2: A,Xs2: list @ A] :
( ( P3 @ Xs2 )
=> ( P3 @ ( append @ A @ Xs2 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) )
=> ( P3 @ Xs ) ) ) ).
% rev_induct
thf(fact_196_less__minus__one__simps_I1_J,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ).
% less_minus_one_simps(1)
thf(fact_197_less__minus__one__simps_I3_J,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% less_minus_one_simps(3)
thf(fact_198_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
thf(fact_199_mult__less__iff1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: A,X: A,Y3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z2 )
=> ( ( ord_less @ A @ ( times_times @ A @ X @ Z2 ) @ ( times_times @ A @ Y3 @ Z2 ) )
= ( ord_less @ A @ X @ Y3 ) ) ) ) ).
% mult_less_iff1
thf(fact_200_dbl__dec__simps_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_dec @ A @ ( zero_zero @ A ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% dbl_dec_simps(2)
thf(fact_201_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
& ( ord_less @ nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_202_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( times_times @ nat @ M @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_203_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N @ K ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
& ( ord_less @ nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_204_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times @ nat @ M @ K )
= ( times_times @ nat @ N @ K ) )
= ( ( M = N )
| ( K
= ( zero_zero @ nat ) ) ) ) ).
% mult_cancel2
thf(fact_205_neq0__conv,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% neq0_conv
thf(fact_206_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% less_nat_zero_code
thf(fact_207_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_208_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times @ nat @ M @ N )
= ( zero_zero @ nat ) )
= ( ( M
= ( zero_zero @ nat ) )
| ( N
= ( zero_zero @ nat ) ) ) ) ).
% mult_is_0
thf(fact_209_mult__0__right,axiom,
! [M: nat] :
( ( times_times @ nat @ M @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% mult_0_right
thf(fact_210_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times @ nat @ K @ M )
= ( times_times @ nat @ K @ N ) )
= ( ( M = N )
| ( K
= ( zero_zero @ nat ) ) ) ) ).
% mult_cancel1
thf(fact_211_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ( one_one @ nat )
= ( times_times @ nat @ M @ N ) )
= ( ( M
= ( one_one @ nat ) )
& ( N
= ( one_one @ nat ) ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_212_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times @ nat @ M @ N )
= ( one_one @ nat ) )
= ( ( M
= ( one_one @ nat ) )
& ( N
= ( one_one @ nat ) ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_213_dbl__dec__simps_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_dec @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% dbl_dec_simps(3)
thf(fact_214_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_215_less__one,axiom,
! [N: nat] :
( ( ord_less @ nat @ N @ ( one_one @ nat ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% less_one
thf(fact_216_nat__mult__1,axiom,
! [N: nat] :
( ( times_times @ nat @ ( one_one @ nat ) @ N )
= N ) ).
% nat_mult_1
thf(fact_217_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times @ nat @ N @ ( one_one @ nat ) )
= N ) ).
% nat_mult_1_right
thf(fact_218_infinite__descent__measure,axiom,
! [A: $tType,P3: A > $o,V2: A > nat,X: A] :
( ! [X2: A] :
( ~ ( P3 @ X2 )
=> ? [Y5: A] :
( ( ord_less @ nat @ ( V2 @ Y5 ) @ ( V2 @ X2 ) )
& ~ ( P3 @ Y5 ) ) )
=> ( P3 @ X ) ) ).
% infinite_descent_measure
thf(fact_219_linorder__neqE__nat,axiom,
! [X: nat,Y3: nat] :
( ( X != Y3 )
=> ( ~ ( ord_less @ nat @ X @ Y3 )
=> ( ord_less @ nat @ Y3 @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_220_infinite__descent,axiom,
! [P3: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P3 @ N2 )
=> ? [M2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
& ~ ( P3 @ M2 ) ) )
=> ( P3 @ N ) ) ).
% infinite_descent
thf(fact_221_nat__less__induct,axiom,
! [P3: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
=> ( P3 @ M2 ) )
=> ( P3 @ N2 ) )
=> ( P3 @ N ) ) ).
% nat_less_induct
thf(fact_222_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_223_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less @ nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_224_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_225_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_not_refl
thf(fact_226_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_227_measure__induct,axiom,
! [B2: $tType,A: $tType] :
( ( wellorder @ B2 )
=> ! [F2: A > B2,P3: A > $o,A2: A] :
( ! [X2: A] :
( ! [Y5: A] :
( ( ord_less @ B2 @ ( F2 @ Y5 ) @ ( F2 @ X2 ) )
=> ( P3 @ Y5 ) )
=> ( P3 @ X2 ) )
=> ( P3 @ A2 ) ) ) ).
% measure_induct
thf(fact_228_measure__induct__rule,axiom,
! [B2: $tType,A: $tType] :
( ( wellorder @ B2 )
=> ! [F2: A > B2,P3: A > $o,A2: A] :
( ! [X2: A] :
( ! [Y5: A] :
( ( ord_less @ B2 @ ( F2 @ Y5 ) @ ( F2 @ X2 ) )
=> ( P3 @ Y5 ) )
=> ( P3 @ X2 ) )
=> ( P3 @ A2 ) ) ) ).
% measure_induct_rule
thf(fact_229_gr0I,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% gr0I
thf(fact_230_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% not_gr0
thf(fact_231_not__less0,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% not_less0
thf(fact_232_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% less_zeroE
thf(fact_233_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( N
!= ( zero_zero @ nat ) ) ) ).
% gr_implies_not0
thf(fact_234_infinite__descent0,axiom,
! [P3: nat > $o,N: nat] :
( ( P3 @ ( zero_zero @ nat ) )
=> ( ! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ~ ( P3 @ N2 )
=> ? [M2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
& ~ ( P3 @ M2 ) ) ) )
=> ( P3 @ N ) ) ) ).
% infinite_descent0
thf(fact_235_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less @ nat @ A2 @ ( zero_zero @ nat ) ) ).
% bot_nat_0.extremum_strict
thf(fact_236_infinite__descent0__measure,axiom,
! [A: $tType,V2: A > nat,P3: A > $o,X: A] :
( ! [X2: A] :
( ( ( V2 @ X2 )
= ( zero_zero @ nat ) )
=> ( P3 @ X2 ) )
=> ( ! [X2: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V2 @ X2 ) )
=> ( ~ ( P3 @ X2 )
=> ? [Y5: A] :
( ( ord_less @ nat @ ( V2 @ Y5 ) @ ( V2 @ X2 ) )
& ~ ( P3 @ Y5 ) ) ) )
=> ( P3 @ X ) ) ) ).
% infinite_descent0_measure
thf(fact_237_mult__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ord_less @ nat @ ( times_times @ nat @ I @ K ) @ ( times_times @ nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_238_mult__less__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ord_less @ nat @ ( times_times @ nat @ K @ I ) @ ( times_times @ nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_239_mult__0,axiom,
! [N: nat] :
( ( times_times @ nat @ ( zero_zero @ nat ) @ N )
= ( zero_zero @ nat ) ) ).
% mult_0
thf(fact_240_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( ( times_times @ nat @ K @ M )
= ( times_times @ nat @ K @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_241_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( ord_less @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) )
= ( ord_less @ nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_242_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times @ nat @ K @ M )
= ( times_times @ nat @ K @ N ) )
= ( ( K
= ( zero_zero @ nat ) )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_243_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times @ nat @ M @ N ) )
=> ( ( N
= ( one_one @ nat ) )
| ( M
= ( zero_zero @ nat ) ) ) ) ).
% mult_eq_self_implies_10
thf(fact_244_n__lists__Nil,axiom,
! [A: $tType,N: nat] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( n_lists @ A @ N @ ( nil @ A ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( n_lists @ A @ N @ ( nil @ A ) )
= ( nil @ ( list @ A ) ) ) ) ) ).
% n_lists_Nil
thf(fact_245_n__lists_Osimps_I1_J,axiom,
! [A: $tType,Xs: list @ A] :
( ( n_lists @ A @ ( zero_zero @ nat ) @ Xs )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% n_lists.simps(1)
thf(fact_246_compl__eq__compl__iff,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [X: A,Y3: A] :
( ( ( uminus_uminus @ A @ X )
= ( uminus_uminus @ A @ Y3 ) )
= ( X = Y3 ) ) ) ).
% compl_eq_compl_iff
thf(fact_247_double__compl,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [X: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ X ) )
= X ) ) ).
% double_compl
thf(fact_248_uminus__apply,axiom,
! [B2: $tType,A: $tType] :
( ( uminus @ B2 )
=> ( ( uminus_uminus @ ( A > B2 ) )
= ( ^ [A5: A > B2,X3: A] : ( uminus_uminus @ B2 @ ( A5 @ X3 ) ) ) ) ) ).
% uminus_apply
thf(fact_249_fun__Compl__def,axiom,
! [B2: $tType,A: $tType] :
( ( uminus @ B2 )
=> ( ( uminus_uminus @ ( A > B2 ) )
= ( ^ [A5: A > B2,X3: A] : ( uminus_uminus @ B2 @ ( A5 @ X3 ) ) ) ) ) ).
% fun_Compl_def
thf(fact_250_compl__less__swap1,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [Y3: A,X: A] :
( ( ord_less @ A @ Y3 @ ( uminus_uminus @ A @ X ) )
=> ( ord_less @ A @ X @ ( uminus_uminus @ A @ Y3 ) ) ) ) ).
% compl_less_swap1
thf(fact_251_compl__less__swap2,axiom,
! [A: $tType] :
( ( boolean_algebra @ A )
=> ! [Y3: A,X: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ Y3 ) @ X )
=> ( ord_less @ A @ ( uminus_uminus @ A @ X ) @ Y3 ) ) ) ).
% compl_less_swap2
% Subclasses (40)
thf(subcl_Rings_Olinordered__idom___Rings_Ocomm__ring__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( comm_ring_1 @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___HOL_Otype,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( type @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Groups_Oone,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( one @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Groups_Osgn,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( sgn @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Rings_Oidom,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( idom @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Rings_Oring,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ring @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Groups_Oplus,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( plus @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Groups_Ozero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( zero @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Rings_Oring__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ring_1 @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Groups_Ouminus,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( uminus @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Orderings_Oord,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ord @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Nat_Oring__char__0,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ring_char_0 @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Num_Oneg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( neg_numeral @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Rings_Ocomm__ring,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( comm_ring @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Rings_Omult__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( mult_zero @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Groups_Ogroup__add,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( group_add @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Groups_Omonoid__add,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( monoid_add @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Groups_Omonoid__mult,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( monoid_mult @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Orderings_Olinorder,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( linorder @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Rings_Oidom__abs__sgn,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( idom_abs_sgn @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Rings_Ozero__neq__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( zero_neq_one @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Groups_Oab__group__add,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ab_group_add @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Rings_Ozero__less__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( zero_less_one @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Groups_Osemigroup__mult,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( semigroup_mult @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Rings_Ocomm__semiring__0,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( comm_semiring_0 @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Rings_Ocomm__semiring__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( comm_semiring_1 @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Rings_Olinordered__ring,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( linordered_ring @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Groups_Ocomm__monoid__add,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( comm_monoid_add @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Groups_Ocomm__monoid__mult,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( comm_monoid_mult @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Groups_Oab__semigroup__mult,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ab_semigroup_mult @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Rings_Olinordered__semidom,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( linordered_semidom @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Groups_Oordered__ab__group__add,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ordered_ab_group_add @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Rings_Olinordered__ring__strict,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( linord581940658strict @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Rings_Oring__1__no__zero__divisors,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ring_11004092258visors @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Groups_Olinordered__ab__group__add,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( linord219039673up_add @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Rings_Osemiring__no__zero__divisors,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( semiri1193490041visors @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Rings_Olinordered__semiring__strict,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( linord20386208strict @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Rings_Olinordered__nonzero__semiring,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( linord1659791738miring @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Rings_Olinordered__comm__semiring__strict,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( linord893533164strict @ A ) ) ).
thf(subcl_Rings_Olinordered__idom___Rings_Osemiring__no__zero__divisors__cancel,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( semiri1923998003cancel @ A ) ) ).
% Type constructors (71)
thf(tcon_Polynomial_Opoly___Rings_Ocomm__ring__1,axiom,
! [A6: $tType] :
( ( comm_ring_1 @ A6 )
=> ( comm_ring_1 @ ( poly @ A6 ) ) ) ).
thf(tcon_fun___Lattices_Oboolean__algebra,axiom,
! [A6: $tType,A7: $tType] :
( ( boolean_algebra @ A7 )
=> ( boolean_algebra @ ( A6 > A7 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A6: $tType,A7: $tType] :
( ( ord @ A7 )
=> ( ord @ ( A6 > A7 ) ) ) ).
thf(tcon_fun___Groups_Ouminus,axiom,
! [A6: $tType,A7: $tType] :
( ( uminus @ A7 )
=> ( uminus @ ( A6 > A7 ) ) ) ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel,axiom,
semiri1923998003cancel @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict,axiom,
linord893533164strict @ 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_Olinordered__semiring__strict,axiom,
linord20386208strict @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors,axiom,
semiri1193490041visors @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semidom,axiom,
linordered_semidom @ nat ).
thf(tcon_Nat_Onat___Groups_Oab__semigroup__mult,axiom,
ab_semigroup_mult @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__mult,axiom,
comm_monoid_mult @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add @ 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___Groups_Osemigroup__mult,axiom,
semigroup_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Ozero__less__one,axiom,
zero_less_one @ nat ).
thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
wellorder @ nat ).
thf(tcon_Nat_Onat___Rings_Ozero__neq__one,axiom,
zero_neq_one @ nat ).
thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
linorder @ nat ).
thf(tcon_Nat_Onat___Groups_Omonoid__mult,axiom,
monoid_mult @ nat ).
thf(tcon_Nat_Onat___Groups_Omonoid__add,axiom,
monoid_add @ nat ).
thf(tcon_Nat_Onat___Rings_Omult__zero,axiom,
mult_zero @ nat ).
thf(tcon_Nat_Onat___Orderings_Oord_1,axiom,
ord @ nat ).
thf(tcon_Nat_Onat___Groups_Ozero,axiom,
zero @ nat ).
thf(tcon_Nat_Onat___Groups_Oplus,axiom,
plus @ nat ).
thf(tcon_Nat_Onat___Groups_Oone,axiom,
one @ nat ).
thf(tcon_HOL_Obool___Lattices_Oboolean__algebra_2,axiom,
boolean_algebra @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder_3,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_4,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Groups_Ouminus_5,axiom,
uminus @ $o ).
thf(tcon_Polynomial_Opoly___Rings_Osemiring__no__zero__divisors__cancel_6,axiom,
! [A6: $tType] :
( ( idom @ A6 )
=> ( semiri1923998003cancel @ ( poly @ A6 ) ) ) ).
thf(tcon_Polynomial_Opoly___Rings_Olinordered__comm__semiring__strict_7,axiom,
! [A6: $tType] :
( ( linordered_idom @ A6 )
=> ( linord893533164strict @ ( poly @ A6 ) ) ) ).
thf(tcon_Polynomial_Opoly___Rings_Olinordered__nonzero__semiring_8,axiom,
! [A6: $tType] :
( ( linordered_idom @ A6 )
=> ( linord1659791738miring @ ( poly @ A6 ) ) ) ).
thf(tcon_Polynomial_Opoly___Rings_Olinordered__semiring__strict_9,axiom,
! [A6: $tType] :
( ( linordered_idom @ A6 )
=> ( linord20386208strict @ ( poly @ A6 ) ) ) ).
thf(tcon_Polynomial_Opoly___Rings_Osemiring__no__zero__divisors_10,axiom,
! [A6: $tType] :
( ( ( comm_semiring_0 @ A6 )
& ( semiri1193490041visors @ A6 ) )
=> ( semiri1193490041visors @ ( poly @ A6 ) ) ) ).
thf(tcon_Polynomial_Opoly___Groups_Olinordered__ab__group__add,axiom,
! [A6: $tType] :
( ( linordered_idom @ A6 )
=> ( linord219039673up_add @ ( poly @ A6 ) ) ) ).
thf(tcon_Polynomial_Opoly___Rings_Oring__1__no__zero__divisors,axiom,
! [A6: $tType] :
( ( idom @ A6 )
=> ( ring_11004092258visors @ ( poly @ A6 ) ) ) ).
thf(tcon_Polynomial_Opoly___Rings_Olinordered__ring__strict,axiom,
! [A6: $tType] :
( ( linordered_idom @ A6 )
=> ( linord581940658strict @ ( poly @ A6 ) ) ) ).
thf(tcon_Polynomial_Opoly___Groups_Oordered__ab__group__add,axiom,
! [A6: $tType] :
( ( linordered_idom @ A6 )
=> ( ordered_ab_group_add @ ( poly @ A6 ) ) ) ).
thf(tcon_Polynomial_Opoly___Rings_Olinordered__semidom_11,axiom,
! [A6: $tType] :
( ( linordered_idom @ A6 )
=> ( linordered_semidom @ ( poly @ A6 ) ) ) ).
thf(tcon_Polynomial_Opoly___Groups_Oab__semigroup__mult_12,axiom,
! [A6: $tType] :
( ( comm_semiring_0 @ A6 )
=> ( ab_semigroup_mult @ ( poly @ A6 ) ) ) ).
thf(tcon_Polynomial_Opoly___Groups_Ocomm__monoid__mult_13,axiom,
! [A6: $tType] :
( ( comm_semiring_1 @ A6 )
=> ( comm_monoid_mult @ ( poly @ A6 ) ) ) ).
thf(tcon_Polynomial_Opoly___Groups_Ocomm__monoid__add_14,axiom,
! [A6: $tType] :
( ( comm_monoid_add @ A6 )
=> ( comm_monoid_add @ ( poly @ A6 ) ) ) ).
thf(tcon_Polynomial_Opoly___Rings_Olinordered__ring,axiom,
! [A6: $tType] :
( ( linordered_idom @ A6 )
=> ( linordered_ring @ ( poly @ A6 ) ) ) ).
thf(tcon_Polynomial_Opoly___Rings_Olinordered__idom,axiom,
! [A6: $tType] :
( ( linordered_idom @ A6 )
=> ( linordered_idom @ ( poly @ A6 ) ) ) ).
thf(tcon_Polynomial_Opoly___Rings_Ocomm__semiring__1_15,axiom,
! [A6: $tType] :
( ( comm_semiring_1 @ A6 )
=> ( comm_semiring_1 @ ( poly @ A6 ) ) ) ).
thf(tcon_Polynomial_Opoly___Rings_Ocomm__semiring__0_16,axiom,
! [A6: $tType] :
( ( comm_semiring_0 @ A6 )
=> ( comm_semiring_0 @ ( poly @ A6 ) ) ) ).
thf(tcon_Polynomial_Opoly___Groups_Osemigroup__mult_17,axiom,
! [A6: $tType] :
( ( comm_semiring_0 @ A6 )
=> ( semigroup_mult @ ( poly @ A6 ) ) ) ).
thf(tcon_Polynomial_Opoly___Rings_Ozero__less__one_18,axiom,
! [A6: $tType] :
( ( linordered_idom @ A6 )
=> ( zero_less_one @ ( poly @ A6 ) ) ) ).
thf(tcon_Polynomial_Opoly___Groups_Oab__group__add,axiom,
! [A6: $tType] :
( ( ab_group_add @ A6 )
=> ( ab_group_add @ ( poly @ A6 ) ) ) ).
thf(tcon_Polynomial_Opoly___Rings_Ozero__neq__one_19,axiom,
! [A6: $tType] :
( ( comm_semiring_1 @ A6 )
=> ( zero_neq_one @ ( poly @ A6 ) ) ) ).
thf(tcon_Polynomial_Opoly___Rings_Oidom__abs__sgn,axiom,
! [A6: $tType] :
( ( linordered_idom @ A6 )
=> ( idom_abs_sgn @ ( poly @ A6 ) ) ) ).
thf(tcon_Polynomial_Opoly___Orderings_Olinorder_20,axiom,
! [A6: $tType] :
( ( linordered_idom @ A6 )
=> ( linorder @ ( poly @ A6 ) ) ) ).
thf(tcon_Polynomial_Opoly___Groups_Omonoid__mult_21,axiom,
! [A6: $tType] :
( ( comm_semiring_1 @ A6 )
=> ( monoid_mult @ ( poly @ A6 ) ) ) ).
thf(tcon_Polynomial_Opoly___Groups_Omonoid__add_22,axiom,
! [A6: $tType] :
( ( comm_monoid_add @ A6 )
=> ( monoid_add @ ( poly @ A6 ) ) ) ).
thf(tcon_Polynomial_Opoly___Groups_Ogroup__add,axiom,
! [A6: $tType] :
( ( ab_group_add @ A6 )
=> ( group_add @ ( poly @ A6 ) ) ) ).
thf(tcon_Polynomial_Opoly___Rings_Omult__zero_23,axiom,
! [A6: $tType] :
( ( comm_semiring_0 @ A6 )
=> ( mult_zero @ ( poly @ A6 ) ) ) ).
thf(tcon_Polynomial_Opoly___Rings_Ocomm__ring,axiom,
! [A6: $tType] :
( ( comm_ring @ A6 )
=> ( comm_ring @ ( poly @ A6 ) ) ) ).
thf(tcon_Polynomial_Opoly___Num_Oneg__numeral,axiom,
! [A6: $tType] :
( ( comm_ring_1 @ A6 )
=> ( neg_numeral @ ( poly @ A6 ) ) ) ).
thf(tcon_Polynomial_Opoly___Nat_Oring__char__0,axiom,
! [A6: $tType] :
( ( ( ring_char_0 @ A6 )
& ( comm_ring_1 @ A6 ) )
=> ( ring_char_0 @ ( poly @ A6 ) ) ) ).
thf(tcon_Polynomial_Opoly___Orderings_Oord_24,axiom,
! [A6: $tType] :
( ( linordered_idom @ A6 )
=> ( ord @ ( poly @ A6 ) ) ) ).
thf(tcon_Polynomial_Opoly___Groups_Ouminus_25,axiom,
! [A6: $tType] :
( ( ab_group_add @ A6 )
=> ( uminus @ ( poly @ A6 ) ) ) ).
thf(tcon_Polynomial_Opoly___Rings_Oring__1,axiom,
! [A6: $tType] :
( ( comm_ring_1 @ A6 )
=> ( ring_1 @ ( poly @ A6 ) ) ) ).
thf(tcon_Polynomial_Opoly___Groups_Ozero_26,axiom,
! [A6: $tType] :
( ( zero @ A6 )
=> ( zero @ ( poly @ A6 ) ) ) ).
thf(tcon_Polynomial_Opoly___Groups_Oplus_27,axiom,
! [A6: $tType] :
( ( comm_monoid_add @ A6 )
=> ( plus @ ( poly @ A6 ) ) ) ).
thf(tcon_Polynomial_Opoly___Rings_Oring,axiom,
! [A6: $tType] :
( ( comm_ring @ A6 )
=> ( ring @ ( poly @ A6 ) ) ) ).
thf(tcon_Polynomial_Opoly___Rings_Oidom,axiom,
! [A6: $tType] :
( ( idom @ A6 )
=> ( idom @ ( poly @ A6 ) ) ) ).
thf(tcon_Polynomial_Opoly___Groups_Osgn,axiom,
! [A6: $tType] :
( ( linordered_idom @ A6 )
=> ( sgn @ ( poly @ A6 ) ) ) ).
thf(tcon_Polynomial_Opoly___Groups_Oone_28,axiom,
! [A6: $tType] :
( ( comm_semiring_1 @ A6 )
=> ( one @ ( poly @ A6 ) ) ) ).
% Free types (1)
thf(tfree_0,hypothesis,
linordered_idom @ a ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( descar149487500hanges @ a @ ys )
= ( descar149487500hanges @ a @ ( coeffs @ a @ g ) ) ) ).
%------------------------------------------------------------------------------