TPTP Problem File: ITP044^1.p

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%------------------------------------------------------------------------------
% File     : ITP044^1 : TPTP v9.0.0. 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   : 0.38 v9.0.0, 0.40 v8.2.0, 0.31 v8.1.0, 0.36 v7.5.0
% Syntax   : Number of formulae    :  412 ( 240 unt;  65 typ;   0 def)
%            Number of atoms       :  790 ( 531 equ;   0 cnn)
%            Maximal formula atoms :   10 (   2 avg)
%            Number of connectives : 2132 ( 114   ~;  47   |;  46   &;1703   @)
%                                         (   0 <=>; 222  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   17 (   5 avg)
%            Number of types       :   10 (   9 usr)
%            Number of type conns  :  114 ( 114   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   57 (  56 usr;  17 con; 0-2 aty)
%            Number of variables   :  718 (  16   ^; 697   !;   5   ?; 718   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Sledgehammer 2021-02-23 15:42:44.488
%------------------------------------------------------------------------------
% Could-be-implicit typings (9)
thf(ty_n_t__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J,type,
    poly_poly_a: $tType ).

thf(ty_n_t__List__Olist_It__Polynomial__Opoly_Itf__a_J_J,type,
    list_poly_a: $tType ).

thf(ty_n_t__List__Olist_It__List__Olist_Itf__a_J_J,type,
    list_list_a: $tType ).

thf(ty_n_t__Polynomial__Opoly_It__Nat__Onat_J,type,
    poly_nat: $tType ).

thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
    list_nat: $tType ).

thf(ty_n_t__Polynomial__Opoly_Itf__a_J,type,
    poly_a: $tType ).

thf(ty_n_t__List__Olist_Itf__a_J,type,
    list_a: $tType ).

thf(ty_n_t__Nat__Onat,type,
    nat: $tType ).

thf(ty_n_tf__a,type,
    a: $tType ).

% Explicit typings (56)
thf(sy_c_Descartes__Sign__Rule__Mirabelle__gwrulepwnb_Opsums_001t__Nat__Onat,type,
    descar226543321ms_nat: list_nat > list_nat ).

thf(sy_c_Descartes__Sign__Rule__Mirabelle__gwrulepwnb_Opsums_001t__Polynomial__Opoly_Itf__a_J,type,
    descar282223555poly_a: list_poly_a > list_poly_a ).

thf(sy_c_Descartes__Sign__Rule__Mirabelle__gwrulepwnb_Opsums_001tf__a,type,
    descar1375166517sums_a: list_a > list_a ).

thf(sy_c_Descartes__Sign__Rule__Mirabelle__gwrulepwnb_Oreduce__root_001t__Polynomial__Opoly_Itf__a_J,type,
    descar434775507poly_a: poly_a > poly_poly_a > poly_poly_a ).

thf(sy_c_Descartes__Sign__Rule__Mirabelle__gwrulepwnb_Oreduce__root_001tf__a,type,
    descar466059845root_a: a > poly_a > poly_a ).

thf(sy_c_Descartes__Sign__Rule__Mirabelle__gwrulepwnb_Osign__changes_001t__Polynomial__Opoly_Itf__a_J,type,
    descar357075861poly_a: list_poly_a > nat ).

thf(sy_c_Descartes__Sign__Rule__Mirabelle__gwrulepwnb_Osign__changes_001tf__a,type,
    descar2095969287nges_a: list_a > nat ).

thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
    one_one_nat: nat ).

thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
    one_one_poly_nat: poly_nat ).

thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J,type,
    one_one_poly_poly_a: poly_poly_a ).

thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_Itf__a_J,type,
    one_one_poly_a: poly_a ).

thf(sy_c_Groups_Oone__class_Oone_001tf__a,type,
    one_one_a: a ).

thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
    times_times_nat: nat > nat > nat ).

thf(sy_c_Groups_Otimes__class_Otimes_001t__Polynomial__Opoly_Itf__a_J,type,
    times_times_poly_a: poly_a > poly_a > poly_a ).

thf(sy_c_Groups_Otimes__class_Otimes_001tf__a,type,
    times_times_a: a > a > a ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Polynomial__Opoly_Itf__a_J,type,
    uminus_uminus_poly_a: poly_a > poly_a ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001tf__a,type,
    uminus_uminus_a: a > a ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
    zero_zero_nat: nat ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Nat__Onat_J,type,
    zero_zero_poly_nat: poly_nat ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_It__Polynomial__Opoly_Itf__a_J_J,type,
    zero_z2096148049poly_a: poly_poly_a ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_Itf__a_J,type,
    zero_zero_poly_a: poly_a ).

thf(sy_c_Groups_Ozero__class_Ozero_001tf__a,type,
    zero_zero_a: a ).

thf(sy_c_List_Oappend_001t__Nat__Onat,type,
    append_nat: list_nat > list_nat > list_nat ).

thf(sy_c_List_Oappend_001t__Polynomial__Opoly_Itf__a_J,type,
    append_poly_a: list_poly_a > list_poly_a > list_poly_a ).

thf(sy_c_List_Oappend_001tf__a,type,
    append_a: list_a > list_a > list_a ).

thf(sy_c_List_Olist_OCons_001t__List__Olist_Itf__a_J,type,
    cons_list_a: list_a > list_list_a > list_list_a ).

thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
    cons_nat: nat > list_nat > list_nat ).

thf(sy_c_List_Olist_OCons_001t__Polynomial__Opoly_Itf__a_J,type,
    cons_poly_a: poly_a > list_poly_a > list_poly_a ).

thf(sy_c_List_Olist_OCons_001tf__a,type,
    cons_a: a > list_a > list_a ).

thf(sy_c_List_Olist_ONil_001t__List__Olist_Itf__a_J,type,
    nil_list_a: list_list_a ).

thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
    nil_nat: list_nat ).

thf(sy_c_List_Olist_ONil_001t__Polynomial__Opoly_Itf__a_J,type,
    nil_poly_a: list_poly_a ).

thf(sy_c_List_Olist_ONil_001tf__a,type,
    nil_a: list_a ).

thf(sy_c_List_Onull_001t__Polynomial__Opoly_Itf__a_J,type,
    null_poly_a: list_poly_a > $o ).

thf(sy_c_List_Onull_001tf__a,type,
    null_a: list_a > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
    ord_less_nat: nat > nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Polynomial__Opoly_Itf__a_J,type,
    ord_less_poly_a: poly_a > poly_a > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001tf__a,type,
    ord_less_a: a > a > $o ).

thf(sy_c_Polynomial_OPoly_001t__Nat__Onat,type,
    poly_nat2: list_nat > poly_nat ).

thf(sy_c_Polynomial_OPoly_001t__Polynomial__Opoly_Itf__a_J,type,
    poly_poly_a2: list_poly_a > poly_poly_a ).

thf(sy_c_Polynomial_OPoly_001tf__a,type,
    poly_a2: list_a > poly_a ).

thf(sy_c_Polynomial_Ocoeffs_001t__Nat__Onat,type,
    coeffs_nat: poly_nat > list_nat ).

thf(sy_c_Polynomial_Ocoeffs_001t__Polynomial__Opoly_Itf__a_J,type,
    coeffs_poly_a: poly_poly_a > list_poly_a ).

thf(sy_c_Polynomial_Ocoeffs_001tf__a,type,
    coeffs_a: poly_a > list_a ).

thf(sy_c_Polynomial_Ois__zero_001t__Polynomial__Opoly_Itf__a_J,type,
    is_zero_poly_a: poly_poly_a > $o ).

thf(sy_c_Polynomial_Ois__zero_001tf__a,type,
    is_zero_a: poly_a > $o ).

thf(sy_c_Polynomial_OpCons_001t__Nat__Onat,type,
    pCons_nat: nat > poly_nat > poly_nat ).

thf(sy_c_Polynomial_OpCons_001t__Polynomial__Opoly_Itf__a_J,type,
    pCons_poly_a: poly_a > poly_poly_a > poly_poly_a ).

thf(sy_c_Polynomial_OpCons_001tf__a,type,
    pCons_a: a > poly_a > poly_a ).

thf(sy_c_Polynomial_Osmult_001t__Nat__Onat,type,
    smult_nat: nat > poly_nat > poly_nat ).

thf(sy_c_Polynomial_Osmult_001t__Polynomial__Opoly_Itf__a_J,type,
    smult_poly_a: poly_a > poly_poly_a > poly_poly_a ).

thf(sy_c_Polynomial_Osmult_001tf__a,type,
    smult_a: 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 (346)
thf(fact_0_v__def,axiom,
    ( v
    = ( ^ [F: poly_a] : ( descar2095969287nges_a @ ( coeffs_a @ F ) ) ) ) ).

% v_def
thf(fact_1_nz,axiom,
    g != zero_zero_poly_a ).

% nz
thf(fact_2_coeffs__eq__iff,axiom,
    ( ( ^ [Y: poly_poly_a,Z: poly_poly_a] : ( Y = Z ) )
    = ( ^ [P: poly_poly_a,Q: poly_poly_a] :
          ( ( coeffs_poly_a @ P )
          = ( coeffs_poly_a @ Q ) ) ) ) ).

% coeffs_eq_iff
thf(fact_3_coeffs__eq__iff,axiom,
    ( ( ^ [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_4_ys,axiom,
    ( ys
    = ( descar1375166517sums_a @ xs ) ) ).

% ys
thf(fact_5_sign__changes__coeff__sign__changes,axiom,
    ! [Xs: list_poly_a,P2: poly_poly_a] :
      ( ( ( poly_poly_a2 @ Xs )
        = P2 )
     => ( ( descar357075861poly_a @ Xs )
        = ( descar357075861poly_a @ ( coeffs_poly_a @ P2 ) ) ) ) ).

% sign_changes_coeff_sign_changes
thf(fact_6_sign__changes__coeff__sign__changes,axiom,
    ! [Xs: list_a,P2: poly_a] :
      ( ( ( poly_a2 @ Xs )
        = P2 )
     => ( ( descar2095969287nges_a @ Xs )
        = ( descar2095969287nges_a @ ( coeffs_a @ P2 ) ) ) ) ).

% sign_changes_coeff_sign_changes
thf(fact_7_ys__def,axiom,
    ( ys
    = ( append_a @ ( coeffs_a @ g ) @ ( cons_a @ zero_zero_a @ nil_a ) ) ) ).

% ys_def
thf(fact_8_coeff__sign__changes__reduce__root,axiom,
    ! [A: poly_a,P2: poly_poly_a] :
      ( ( ord_less_poly_a @ zero_zero_poly_a @ A )
     => ( ( descar357075861poly_a @ ( coeffs_poly_a @ ( descar434775507poly_a @ A @ P2 ) ) )
        = ( descar357075861poly_a @ ( coeffs_poly_a @ P2 ) ) ) ) ).

% coeff_sign_changes_reduce_root
thf(fact_9_coeff__sign__changes__reduce__root,axiom,
    ! [A: a,P2: poly_a] :
      ( ( ord_less_a @ zero_zero_a @ A )
     => ( ( descar2095969287nges_a @ ( coeffs_a @ ( descar466059845root_a @ A @ P2 ) ) )
        = ( descar2095969287nges_a @ ( coeffs_a @ P2 ) ) ) ) ).

% coeff_sign_changes_reduce_root
thf(fact_10_is__zero__def,axiom,
    ( is_zero_poly_a
    = ( ^ [P: poly_poly_a] : ( null_poly_a @ ( coeffs_poly_a @ P ) ) ) ) ).

% is_zero_def
thf(fact_11_is__zero__def,axiom,
    ( is_zero_a
    = ( ^ [P: poly_a] : ( null_a @ ( coeffs_a @ P ) ) ) ) ).

% is_zero_def
thf(fact_12_Poly__coeffs,axiom,
    ! [P2: poly_poly_a] :
      ( ( poly_poly_a2 @ ( coeffs_poly_a @ P2 ) )
      = P2 ) ).

% Poly_coeffs
thf(fact_13_Poly__coeffs,axiom,
    ! [P2: poly_a] :
      ( ( poly_a2 @ ( coeffs_a @ P2 ) )
      = P2 ) ).

% Poly_coeffs
thf(fact_14_sign__changes__Nil,axiom,
    ( ( descar357075861poly_a @ nil_poly_a )
    = zero_zero_nat ) ).

% sign_changes_Nil
thf(fact_15_sign__changes__Nil,axiom,
    ( ( descar2095969287nges_a @ nil_a )
    = zero_zero_nat ) ).

% sign_changes_Nil
thf(fact_16_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_17_coeff__sign__changes__smult,axiom,
    ! [A: a,P2: poly_a] :
      ( ( ord_less_a @ zero_zero_a @ A )
     => ( ( descar2095969287nges_a @ ( coeffs_a @ ( smult_a @ A @ P2 ) ) )
        = ( descar2095969287nges_a @ ( coeffs_a @ P2 ) ) ) ) ).

% coeff_sign_changes_smult
thf(fact_18_coeff__sign__changes__smult,axiom,
    ! [A: poly_a,P2: poly_poly_a] :
      ( ( ord_less_poly_a @ zero_zero_poly_a @ A )
     => ( ( descar357075861poly_a @ ( coeffs_poly_a @ ( smult_poly_a @ A @ P2 ) ) )
        = ( descar357075861poly_a @ ( coeffs_poly_a @ P2 ) ) ) ) ).

% coeff_sign_changes_smult
thf(fact_19_sign__changes__0__Cons,axiom,
    ! [Xs: list_a] :
      ( ( descar2095969287nges_a @ ( cons_a @ zero_zero_a @ Xs ) )
      = ( descar2095969287nges_a @ Xs ) ) ).

% sign_changes_0_Cons
thf(fact_20_sign__changes__0__Cons,axiom,
    ! [Xs: list_poly_a] :
      ( ( descar357075861poly_a @ ( cons_poly_a @ zero_zero_poly_a @ Xs ) )
      = ( descar357075861poly_a @ Xs ) ) ).

% sign_changes_0_Cons
thf(fact_21_sign__changes__Cons__Cons__0,axiom,
    ! [X: a,Xs: list_a] :
      ( ( descar2095969287nges_a @ ( cons_a @ X @ ( cons_a @ zero_zero_a @ Xs ) ) )
      = ( descar2095969287nges_a @ ( cons_a @ X @ Xs ) ) ) ).

% sign_changes_Cons_Cons_0
thf(fact_22_sign__changes__Cons__Cons__0,axiom,
    ! [X: poly_a,Xs: list_poly_a] :
      ( ( descar357075861poly_a @ ( cons_poly_a @ X @ ( cons_poly_a @ zero_zero_poly_a @ Xs ) ) )
      = ( descar357075861poly_a @ ( cons_poly_a @ X @ Xs ) ) ) ).

% sign_changes_Cons_Cons_0
thf(fact_23_pCons__eq__iff,axiom,
    ! [A: a,P2: poly_a,B: a,Q2: poly_a] :
      ( ( ( pCons_a @ A @ P2 )
        = ( pCons_a @ B @ Q2 ) )
      = ( ( A = B )
        & ( P2 = Q2 ) ) ) ).

% pCons_eq_iff
thf(fact_24_minus__pCons,axiom,
    ! [A: a,P2: poly_a] :
      ( ( uminus_uminus_poly_a @ ( pCons_a @ A @ P2 ) )
      = ( pCons_a @ ( uminus_uminus_a @ A ) @ ( uminus_uminus_poly_a @ P2 ) ) ) ).

% minus_pCons
thf(fact_25_smult__smult,axiom,
    ! [A: poly_a,B: poly_a,P2: poly_poly_a] :
      ( ( smult_poly_a @ A @ ( smult_poly_a @ B @ P2 ) )
      = ( smult_poly_a @ ( times_times_poly_a @ A @ B ) @ P2 ) ) ).

% smult_smult
thf(fact_26_smult__smult,axiom,
    ! [A: nat,B: nat,P2: poly_nat] :
      ( ( smult_nat @ A @ ( smult_nat @ B @ P2 ) )
      = ( smult_nat @ ( times_times_nat @ A @ B ) @ P2 ) ) ).

% smult_smult
thf(fact_27_smult__1__left,axiom,
    ! [P2: poly_a] :
      ( ( smult_a @ one_one_a @ P2 )
      = P2 ) ).

% smult_1_left
thf(fact_28_smult__1__left,axiom,
    ! [P2: poly_nat] :
      ( ( smult_nat @ one_one_nat @ P2 )
      = P2 ) ).

% smult_1_left
thf(fact_29_smult__minus__left,axiom,
    ! [A: a,P2: poly_a] :
      ( ( smult_a @ ( uminus_uminus_a @ A ) @ P2 )
      = ( uminus_uminus_poly_a @ ( smult_a @ A @ P2 ) ) ) ).

% smult_minus_left
thf(fact_30_smult__0__right,axiom,
    ! [A: a] :
      ( ( smult_a @ A @ zero_zero_poly_a )
      = zero_zero_poly_a ) ).

% smult_0_right
thf(fact_31_mult__smult__left,axiom,
    ! [A: a,P2: poly_a,Q2: poly_a] :
      ( ( times_times_poly_a @ ( smult_a @ A @ P2 ) @ Q2 )
      = ( smult_a @ A @ ( times_times_poly_a @ P2 @ Q2 ) ) ) ).

% mult_smult_left
thf(fact_32_mult__smult__right,axiom,
    ! [P2: poly_a,A: a,Q2: poly_a] :
      ( ( times_times_poly_a @ P2 @ ( smult_a @ A @ Q2 ) )
      = ( smult_a @ A @ ( times_times_poly_a @ P2 @ Q2 ) ) ) ).

% mult_smult_right
thf(fact_33_pCons__0__0,axiom,
    ( ( pCons_poly_a @ zero_zero_poly_a @ zero_z2096148049poly_a )
    = zero_z2096148049poly_a ) ).

% pCons_0_0
thf(fact_34_pCons__0__0,axiom,
    ( ( pCons_a @ zero_zero_a @ zero_zero_poly_a )
    = zero_zero_poly_a ) ).

% pCons_0_0
thf(fact_35_pCons__0__0,axiom,
    ( ( pCons_nat @ zero_zero_nat @ zero_zero_poly_nat )
    = zero_zero_poly_nat ) ).

% pCons_0_0
thf(fact_36_pCons__eq__0__iff,axiom,
    ! [A: poly_a,P2: poly_poly_a] :
      ( ( ( pCons_poly_a @ A @ P2 )
        = zero_z2096148049poly_a )
      = ( ( A = zero_zero_poly_a )
        & ( P2 = zero_z2096148049poly_a ) ) ) ).

% pCons_eq_0_iff
thf(fact_37_pCons__eq__0__iff,axiom,
    ! [A: nat,P2: poly_nat] :
      ( ( ( pCons_nat @ A @ P2 )
        = zero_zero_poly_nat )
      = ( ( A = zero_zero_nat )
        & ( P2 = zero_zero_poly_nat ) ) ) ).

% pCons_eq_0_iff
thf(fact_38_pCons__eq__0__iff,axiom,
    ! [A: a,P2: poly_a] :
      ( ( ( pCons_a @ A @ P2 )
        = zero_zero_poly_a )
      = ( ( A = zero_zero_a )
        & ( P2 = zero_zero_poly_a ) ) ) ).

% pCons_eq_0_iff
thf(fact_39_one__poly__eq__simps_I2_J,axiom,
    ( ( pCons_nat @ one_one_nat @ zero_zero_poly_nat )
    = one_one_poly_nat ) ).

% one_poly_eq_simps(2)
thf(fact_40_one__poly__eq__simps_I2_J,axiom,
    ( ( pCons_a @ one_one_a @ zero_zero_poly_a )
    = one_one_poly_a ) ).

% one_poly_eq_simps(2)
thf(fact_41_one__poly__eq__simps_I1_J,axiom,
    ( one_one_poly_nat
    = ( pCons_nat @ one_one_nat @ zero_zero_poly_nat ) ) ).

% one_poly_eq_simps(1)
thf(fact_42_one__poly__eq__simps_I1_J,axiom,
    ( one_one_poly_a
    = ( pCons_a @ one_one_a @ zero_zero_poly_a ) ) ).

% one_poly_eq_simps(1)
thf(fact_43_smult__0__left,axiom,
    ! [P2: poly_poly_a] :
      ( ( smult_poly_a @ zero_zero_poly_a @ P2 )
      = zero_z2096148049poly_a ) ).

% smult_0_left
thf(fact_44_smult__0__left,axiom,
    ! [P2: poly_a] :
      ( ( smult_a @ zero_zero_a @ P2 )
      = zero_zero_poly_a ) ).

% smult_0_left
thf(fact_45_smult__0__left,axiom,
    ! [P2: poly_nat] :
      ( ( smult_nat @ zero_zero_nat @ P2 )
      = zero_zero_poly_nat ) ).

% smult_0_left
thf(fact_46_smult__eq__0__iff,axiom,
    ! [A: poly_a,P2: poly_poly_a] :
      ( ( ( smult_poly_a @ A @ P2 )
        = zero_z2096148049poly_a )
      = ( ( A = zero_zero_poly_a )
        | ( P2 = zero_z2096148049poly_a ) ) ) ).

% smult_eq_0_iff
thf(fact_47_smult__eq__0__iff,axiom,
    ! [A: nat,P2: poly_nat] :
      ( ( ( smult_nat @ A @ P2 )
        = zero_zero_poly_nat )
      = ( ( A = zero_zero_nat )
        | ( P2 = zero_zero_poly_nat ) ) ) ).

% smult_eq_0_iff
thf(fact_48_smult__eq__0__iff,axiom,
    ! [A: a,P2: poly_a] :
      ( ( ( smult_a @ A @ P2 )
        = zero_zero_poly_a )
      = ( ( A = zero_zero_a )
        | ( P2 = zero_zero_poly_a ) ) ) ).

% smult_eq_0_iff
thf(fact_49_smult__pCons,axiom,
    ! [A: a,B: a,P2: poly_a] :
      ( ( smult_a @ A @ ( pCons_a @ B @ P2 ) )
      = ( pCons_a @ ( times_times_a @ A @ B ) @ ( smult_a @ A @ P2 ) ) ) ).

% smult_pCons
thf(fact_50_smult__pCons,axiom,
    ! [A: poly_a,B: poly_a,P2: poly_poly_a] :
      ( ( smult_poly_a @ A @ ( pCons_poly_a @ B @ P2 ) )
      = ( pCons_poly_a @ ( times_times_poly_a @ A @ B ) @ ( smult_poly_a @ A @ P2 ) ) ) ).

% smult_pCons
thf(fact_51_smult__pCons,axiom,
    ! [A: nat,B: nat,P2: poly_nat] :
      ( ( smult_nat @ A @ ( pCons_nat @ B @ P2 ) )
      = ( pCons_nat @ ( times_times_nat @ A @ B ) @ ( smult_nat @ A @ P2 ) ) ) ).

% smult_pCons
thf(fact_52_coeffs__eq__Nil,axiom,
    ! [P2: poly_poly_a] :
      ( ( ( coeffs_poly_a @ P2 )
        = nil_poly_a )
      = ( P2 = zero_z2096148049poly_a ) ) ).

% coeffs_eq_Nil
thf(fact_53_coeffs__eq__Nil,axiom,
    ! [P2: poly_a] :
      ( ( ( coeffs_a @ P2 )
        = nil_a )
      = ( P2 = zero_zero_poly_a ) ) ).

% coeffs_eq_Nil
thf(fact_54_coeffs__0__eq__Nil,axiom,
    ( ( coeffs_poly_a @ zero_z2096148049poly_a )
    = nil_poly_a ) ).

% coeffs_0_eq_Nil
thf(fact_55_coeffs__0__eq__Nil,axiom,
    ( ( coeffs_a @ zero_zero_poly_a )
    = nil_a ) ).

% coeffs_0_eq_Nil
thf(fact_56_psums__0__Cons,axiom,
    ! [Xs: list_poly_a] :
      ( ( descar282223555poly_a @ ( cons_poly_a @ zero_zero_poly_a @ Xs ) )
      = ( cons_poly_a @ zero_zero_poly_a @ ( descar282223555poly_a @ Xs ) ) ) ).

% psums_0_Cons
thf(fact_57_psums__0__Cons,axiom,
    ! [Xs: list_nat] :
      ( ( descar226543321ms_nat @ ( cons_nat @ zero_zero_nat @ Xs ) )
      = ( cons_nat @ zero_zero_nat @ ( descar226543321ms_nat @ Xs ) ) ) ).

% psums_0_Cons
thf(fact_58_psums__0__Cons,axiom,
    ! [Xs: list_a] :
      ( ( descar1375166517sums_a @ ( cons_a @ zero_zero_a @ Xs ) )
      = ( cons_a @ zero_zero_a @ ( descar1375166517sums_a @ Xs ) ) ) ).

% psums_0_Cons
thf(fact_59_coeffs__1__eq,axiom,
    ( ( coeffs_poly_a @ one_one_poly_poly_a )
    = ( cons_poly_a @ one_one_poly_a @ nil_poly_a ) ) ).

% coeffs_1_eq
thf(fact_60_coeffs__1__eq,axiom,
    ( ( coeffs_a @ one_one_poly_a )
    = ( cons_a @ one_one_a @ nil_a ) ) ).

% coeffs_1_eq
thf(fact_61_coeffs__1__eq,axiom,
    ( ( coeffs_nat @ one_one_poly_nat )
    = ( cons_nat @ one_one_nat @ nil_nat ) ) ).

% coeffs_1_eq
thf(fact_62_sign__changes__singleton,axiom,
    ! [X: a] :
      ( ( descar2095969287nges_a @ ( cons_a @ X @ nil_a ) )
      = zero_zero_nat ) ).

% sign_changes_singleton
thf(fact_63_sign__changes__singleton,axiom,
    ! [X: poly_a] :
      ( ( descar357075861poly_a @ ( cons_poly_a @ X @ nil_poly_a ) )
      = zero_zero_nat ) ).

% sign_changes_singleton
thf(fact_64__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,
    ( ( descar2095969287nges_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_65_Poly__snoc__zero,axiom,
    ! [As: list_poly_a] :
      ( ( poly_poly_a2 @ ( append_poly_a @ As @ ( cons_poly_a @ zero_zero_poly_a @ nil_poly_a ) ) )
      = ( poly_poly_a2 @ As ) ) ).

% Poly_snoc_zero
thf(fact_66_Poly__snoc__zero,axiom,
    ! [As: list_a] :
      ( ( poly_a2 @ ( append_a @ As @ ( cons_a @ zero_zero_a @ nil_a ) ) )
      = ( poly_a2 @ As ) ) ).

% Poly_snoc_zero
thf(fact_67_Poly__snoc__zero,axiom,
    ! [As: list_nat] :
      ( ( poly_nat2 @ ( append_nat @ As @ ( cons_nat @ zero_zero_nat @ nil_nat ) ) )
      = ( poly_nat2 @ As ) ) ).

% Poly_snoc_zero
thf(fact_68_Poly_Osimps_I2_J,axiom,
    ! [A: a,As: list_a] :
      ( ( poly_a2 @ ( cons_a @ A @ As ) )
      = ( pCons_a @ A @ ( poly_a2 @ As ) ) ) ).

% Poly.simps(2)
thf(fact_69_Poly_Osimps_I1_J,axiom,
    ( ( poly_a2 @ nil_a )
    = zero_zero_poly_a ) ).

% Poly.simps(1)
thf(fact_70_psums_Osimps_I2_J,axiom,
    ! [X: a] :
      ( ( descar1375166517sums_a @ ( cons_a @ X @ nil_a ) )
      = ( cons_a @ X @ nil_a ) ) ).

% psums.simps(2)
thf(fact_71_psums_Osimps_I1_J,axiom,
    ( ( descar1375166517sums_a @ nil_a )
    = nil_a ) ).

% psums.simps(1)
thf(fact_72_pCons__one,axiom,
    ( ( pCons_nat @ one_one_nat @ zero_zero_poly_nat )
    = one_one_poly_nat ) ).

% pCons_one
thf(fact_73_pCons__one,axiom,
    ( ( pCons_a @ one_one_a @ zero_zero_poly_a )
    = one_one_poly_a ) ).

% pCons_one
thf(fact_74_pCons__cases,axiom,
    ! [P2: poly_a] :
      ~ ! [A2: a,Q3: poly_a] :
          ( P2
         != ( pCons_a @ A2 @ Q3 ) ) ).

% pCons_cases
thf(fact_75_is__zero__null,axiom,
    ( is_zero_a
    = ( ^ [P: poly_a] : ( P = zero_zero_poly_a ) ) ) ).

% is_zero_null
thf(fact_76_pCons__induct,axiom,
    ! [P3: poly_poly_a > $o,P2: poly_poly_a] :
      ( ( P3 @ zero_z2096148049poly_a )
     => ( ! [A2: poly_a,P4: poly_poly_a] :
            ( ( ( A2 != zero_zero_poly_a )
              | ( P4 != zero_z2096148049poly_a ) )
           => ( ( P3 @ P4 )
             => ( P3 @ ( pCons_poly_a @ A2 @ P4 ) ) ) )
       => ( P3 @ P2 ) ) ) ).

% pCons_induct
thf(fact_77_pCons__induct,axiom,
    ! [P3: poly_nat > $o,P2: poly_nat] :
      ( ( P3 @ zero_zero_poly_nat )
     => ( ! [A2: nat,P4: poly_nat] :
            ( ( ( A2 != zero_zero_nat )
              | ( P4 != zero_zero_poly_nat ) )
           => ( ( P3 @ P4 )
             => ( P3 @ ( pCons_nat @ A2 @ P4 ) ) ) )
       => ( P3 @ P2 ) ) ) ).

% pCons_induct
thf(fact_78_pCons__induct,axiom,
    ! [P3: poly_a > $o,P2: poly_a] :
      ( ( P3 @ zero_zero_poly_a )
     => ( ! [A2: a,P4: poly_a] :
            ( ( ( A2 != zero_zero_a )
              | ( P4 != zero_zero_poly_a ) )
           => ( ( P3 @ P4 )
             => ( P3 @ ( pCons_a @ A2 @ P4 ) ) ) )
       => ( P3 @ P2 ) ) ) ).

% pCons_induct
thf(fact_79_pderiv_Ocases,axiom,
    ! [X: poly_a] :
      ~ ! [A2: a,P4: poly_a] :
          ( X
         != ( pCons_a @ A2 @ P4 ) ) ).

% pderiv.cases
thf(fact_80_poly__induct2,axiom,
    ! [P3: poly_a > poly_a > $o,P2: poly_a,Q2: poly_a] :
      ( ( P3 @ zero_zero_poly_a @ zero_zero_poly_a )
     => ( ! [A2: a,P4: poly_a,B2: a,Q3: poly_a] :
            ( ( P3 @ P4 @ Q3 )
           => ( P3 @ ( pCons_a @ A2 @ P4 ) @ ( pCons_a @ B2 @ Q3 ) ) )
       => ( P3 @ P2 @ Q2 ) ) ) ).

% poly_induct2
thf(fact_81_pderiv_Oinduct,axiom,
    ! [P3: poly_a > $o,A0: poly_a] :
      ( ! [A2: a,P4: poly_a] :
          ( ( ( P4 != zero_zero_poly_a )
           => ( P3 @ P4 ) )
         => ( P3 @ ( pCons_a @ A2 @ P4 ) ) )
     => ( P3 @ A0 ) ) ).

% pderiv.induct
thf(fact_82_mult__poly__0__left,axiom,
    ! [Q2: poly_a] :
      ( ( times_times_poly_a @ zero_zero_poly_a @ Q2 )
      = zero_zero_poly_a ) ).

% mult_poly_0_left
thf(fact_83_mult__poly__0__right,axiom,
    ! [P2: poly_a] :
      ( ( times_times_poly_a @ P2 @ zero_zero_poly_a )
      = zero_zero_poly_a ) ).

% mult_poly_0_right
thf(fact_84_plus__coeffs_Oinduct,axiom,
    ! [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_85_not__0__coeffs__not__Nil,axiom,
    ! [P2: poly_poly_a] :
      ( ( P2 != zero_z2096148049poly_a )
     => ( ( coeffs_poly_a @ P2 )
       != nil_poly_a ) ) ).

% not_0_coeffs_not_Nil
thf(fact_86_not__0__coeffs__not__Nil,axiom,
    ! [P2: poly_a] :
      ( ( P2 != zero_zero_poly_a )
     => ( ( coeffs_a @ P2 )
       != nil_a ) ) ).

% not_0_coeffs_not_Nil
thf(fact_87_minus__poly__rev__list_Oinduct,axiom,
    ! [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_88_synthetic__div__unique__lemma,axiom,
    ! [C: a,P2: poly_a,A: a] :
      ( ( ( smult_a @ C @ P2 )
        = ( pCons_a @ A @ P2 ) )
     => ( P2 = zero_zero_poly_a ) ) ).

% synthetic_div_unique_lemma
thf(fact_89_psums_Ocases,axiom,
    ! [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_90_reduce__root__pCons,axiom,
    ! [A: a,C: a,P2: poly_a] :
      ( ( descar466059845root_a @ A @ ( pCons_a @ C @ P2 ) )
      = ( pCons_a @ C @ ( smult_a @ A @ ( descar466059845root_a @ A @ P2 ) ) ) ) ).

% reduce_root_pCons
thf(fact_91_reduce__root__pCons,axiom,
    ! [A: poly_a,C: poly_a,P2: poly_poly_a] :
      ( ( descar434775507poly_a @ A @ ( pCons_poly_a @ C @ P2 ) )
      = ( pCons_poly_a @ C @ ( smult_poly_a @ A @ ( descar434775507poly_a @ A @ P2 ) ) ) ) ).

% reduce_root_pCons
thf(fact_92_reduce__root__nonzero,axiom,
    ! [A: a,P2: poly_a] :
      ( ( A != zero_zero_a )
     => ( ( P2 != zero_zero_poly_a )
       => ( ( descar466059845root_a @ A @ P2 )
         != zero_zero_poly_a ) ) ) ).

% reduce_root_nonzero
thf(fact_93_reduce__root__nonzero,axiom,
    ! [A: poly_a,P2: poly_poly_a] :
      ( ( A != zero_zero_poly_a )
     => ( ( P2 != zero_z2096148049poly_a )
       => ( ( descar434775507poly_a @ A @ P2 )
         != zero_z2096148049poly_a ) ) ) ).

% reduce_root_nonzero
thf(fact_94_sign__changes__Cons__Cons__same,axiom,
    ! [X: a,Y3: a,Xs: list_a] :
      ( ( ord_less_a @ zero_zero_a @ ( times_times_a @ X @ Y3 ) )
     => ( ( descar2095969287nges_a @ ( cons_a @ X @ ( cons_a @ Y3 @ Xs ) ) )
        = ( descar2095969287nges_a @ ( cons_a @ Y3 @ Xs ) ) ) ) ).

% sign_changes_Cons_Cons_same
thf(fact_95_sign__changes__Cons__Cons__same,axiom,
    ! [X: poly_a,Y3: poly_a,Xs: list_poly_a] :
      ( ( ord_less_poly_a @ zero_zero_poly_a @ ( times_times_poly_a @ X @ Y3 ) )
     => ( ( descar357075861poly_a @ ( cons_poly_a @ X @ ( cons_poly_a @ Y3 @ Xs ) ) )
        = ( descar357075861poly_a @ ( cons_poly_a @ Y3 @ Xs ) ) ) ) ).

% sign_changes_Cons_Cons_same
thf(fact_96_append1__eq__conv,axiom,
    ! [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_97_mult__minus1,axiom,
    ! [Z2: poly_a] :
      ( ( times_times_poly_a @ ( uminus_uminus_poly_a @ one_one_poly_a ) @ Z2 )
      = ( uminus_uminus_poly_a @ Z2 ) ) ).

% mult_minus1
thf(fact_98_mult__minus1,axiom,
    ! [Z2: a] :
      ( ( times_times_a @ ( uminus_uminus_a @ one_one_a ) @ Z2 )
      = ( uminus_uminus_a @ Z2 ) ) ).

% mult_minus1
thf(fact_99_mult__minus1__right,axiom,
    ! [Z2: poly_a] :
      ( ( times_times_poly_a @ Z2 @ ( uminus_uminus_poly_a @ one_one_poly_a ) )
      = ( uminus_uminus_poly_a @ Z2 ) ) ).

% mult_minus1_right
thf(fact_100_mult__minus1__right,axiom,
    ! [Z2: a] :
      ( ( times_times_a @ Z2 @ ( uminus_uminus_a @ one_one_a ) )
      = ( uminus_uminus_a @ Z2 ) ) ).

% mult_minus1_right
thf(fact_101_less__neg__neg,axiom,
    ! [A: poly_a] :
      ( ( ord_less_poly_a @ A @ ( uminus_uminus_poly_a @ A ) )
      = ( ord_less_poly_a @ A @ zero_zero_poly_a ) ) ).

% less_neg_neg
thf(fact_102_less__neg__neg,axiom,
    ! [A: a] :
      ( ( ord_less_a @ A @ ( uminus_uminus_a @ A ) )
      = ( ord_less_a @ A @ zero_zero_a ) ) ).

% less_neg_neg
thf(fact_103_neg__less__pos,axiom,
    ! [A: poly_a] :
      ( ( ord_less_poly_a @ ( uminus_uminus_poly_a @ A ) @ A )
      = ( ord_less_poly_a @ zero_zero_poly_a @ A ) ) ).

% neg_less_pos
thf(fact_104_neg__less__pos,axiom,
    ! [A: a] :
      ( ( ord_less_a @ ( uminus_uminus_a @ A ) @ A )
      = ( ord_less_a @ zero_zero_a @ A ) ) ).

% neg_less_pos
thf(fact_105_neg__0__less__iff__less,axiom,
    ! [A: poly_a] :
      ( ( ord_less_poly_a @ zero_zero_poly_a @ ( uminus_uminus_poly_a @ A ) )
      = ( ord_less_poly_a @ A @ zero_zero_poly_a ) ) ).

% neg_0_less_iff_less
thf(fact_106_neg__0__less__iff__less,axiom,
    ! [A: a] :
      ( ( ord_less_a @ zero_zero_a @ ( uminus_uminus_a @ A ) )
      = ( ord_less_a @ A @ zero_zero_a ) ) ).

% neg_0_less_iff_less
thf(fact_107_neg__less__0__iff__less,axiom,
    ! [A: poly_a] :
      ( ( ord_less_poly_a @ ( uminus_uminus_poly_a @ A ) @ zero_zero_poly_a )
      = ( ord_less_poly_a @ zero_zero_poly_a @ A ) ) ).

% neg_less_0_iff_less
thf(fact_108_neg__less__0__iff__less,axiom,
    ! [A: a] :
      ( ( ord_less_a @ ( uminus_uminus_a @ A ) @ zero_zero_a )
      = ( ord_less_a @ zero_zero_a @ A ) ) ).

% neg_less_0_iff_less
thf(fact_109_mult__cancel__left1,axiom,
    ! [C: a,B: a] :
      ( ( C
        = ( times_times_a @ C @ B ) )
      = ( ( C = zero_zero_a )
        | ( B = one_one_a ) ) ) ).

% mult_cancel_left1
thf(fact_110_mult__cancel__left1,axiom,
    ! [C: poly_a,B: poly_a] :
      ( ( C
        = ( times_times_poly_a @ C @ B ) )
      = ( ( C = zero_zero_poly_a )
        | ( B = one_one_poly_a ) ) ) ).

% mult_cancel_left1
thf(fact_111_mult__cancel__left2,axiom,
    ! [C: a,A: a] :
      ( ( ( times_times_a @ C @ A )
        = C )
      = ( ( C = zero_zero_a )
        | ( A = one_one_a ) ) ) ).

% mult_cancel_left2
thf(fact_112_mult__cancel__left2,axiom,
    ! [C: poly_a,A: poly_a] :
      ( ( ( times_times_poly_a @ C @ A )
        = C )
      = ( ( C = zero_zero_poly_a )
        | ( A = one_one_poly_a ) ) ) ).

% mult_cancel_left2
thf(fact_113_mult__cancel__right1,axiom,
    ! [C: a,B: a] :
      ( ( C
        = ( times_times_a @ B @ C ) )
      = ( ( C = zero_zero_a )
        | ( B = one_one_a ) ) ) ).

% mult_cancel_right1
thf(fact_114_mult__cancel__right1,axiom,
    ! [C: poly_a,B: poly_a] :
      ( ( C
        = ( times_times_poly_a @ B @ C ) )
      = ( ( C = zero_zero_poly_a )
        | ( B = one_one_poly_a ) ) ) ).

% mult_cancel_right1
thf(fact_115_mult__cancel__right2,axiom,
    ! [A: a,C: a] :
      ( ( ( times_times_a @ A @ C )
        = C )
      = ( ( C = zero_zero_a )
        | ( A = one_one_a ) ) ) ).

% mult_cancel_right2
thf(fact_116_mult__cancel__right2,axiom,
    ! [A: poly_a,C: poly_a] :
      ( ( ( times_times_poly_a @ A @ C )
        = C )
      = ( ( C = zero_zero_poly_a )
        | ( A = one_one_poly_a ) ) ) ).

% mult_cancel_right2
thf(fact_117_neg__equal__iff__equal,axiom,
    ! [A: a,B: a] :
      ( ( ( uminus_uminus_a @ A )
        = ( uminus_uminus_a @ B ) )
      = ( A = B ) ) ).

% neg_equal_iff_equal
thf(fact_118_add_Oinverse__inverse,axiom,
    ! [A: a] :
      ( ( uminus_uminus_a @ ( uminus_uminus_a @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_119_list_Oinject,axiom,
    ! [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_120_same__append__eq,axiom,
    ! [Xs: list_a,Ys2: list_a,Zs: list_a] :
      ( ( ( append_a @ Xs @ Ys2 )
        = ( append_a @ Xs @ Zs ) )
      = ( Ys2 = Zs ) ) ).

% same_append_eq
thf(fact_121_append__same__eq,axiom,
    ! [Ys2: list_a,Xs: list_a,Zs: list_a] :
      ( ( ( append_a @ Ys2 @ Xs )
        = ( append_a @ Zs @ Xs ) )
      = ( Ys2 = Zs ) ) ).

% append_same_eq
thf(fact_122_append__assoc,axiom,
    ! [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_123_append_Oassoc,axiom,
    ! [A: list_a,B: list_a,C: list_a] :
      ( ( append_a @ ( append_a @ A @ B ) @ C )
      = ( append_a @ A @ ( append_a @ B @ C ) ) ) ).

% append.assoc
thf(fact_124_not__gr__zero,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr_zero
thf(fact_125_mult__cancel__right,axiom,
    ! [A: a,C: a,B: a] :
      ( ( ( times_times_a @ A @ C )
        = ( times_times_a @ B @ C ) )
      = ( ( C = zero_zero_a )
        | ( A = B ) ) ) ).

% mult_cancel_right
thf(fact_126_mult__cancel__right,axiom,
    ! [A: poly_a,C: poly_a,B: poly_a] :
      ( ( ( times_times_poly_a @ A @ C )
        = ( times_times_poly_a @ B @ C ) )
      = ( ( C = zero_zero_poly_a )
        | ( A = B ) ) ) ).

% mult_cancel_right
thf(fact_127_mult__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ( times_times_nat @ A @ C )
        = ( times_times_nat @ B @ C ) )
      = ( ( C = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_right
thf(fact_128_mult__cancel__left,axiom,
    ! [C: a,A: a,B: a] :
      ( ( ( times_times_a @ C @ A )
        = ( times_times_a @ C @ B ) )
      = ( ( C = zero_zero_a )
        | ( A = B ) ) ) ).

% mult_cancel_left
thf(fact_129_mult__cancel__left,axiom,
    ! [C: poly_a,A: poly_a,B: poly_a] :
      ( ( ( times_times_poly_a @ C @ A )
        = ( times_times_poly_a @ C @ B ) )
      = ( ( C = zero_zero_poly_a )
        | ( A = B ) ) ) ).

% mult_cancel_left
thf(fact_130_mult__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ( times_times_nat @ C @ A )
        = ( times_times_nat @ C @ B ) )
      = ( ( C = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_left
thf(fact_131_mult__eq__0__iff,axiom,
    ! [A: a,B: a] :
      ( ( ( times_times_a @ A @ B )
        = zero_zero_a )
      = ( ( A = zero_zero_a )
        | ( B = zero_zero_a ) ) ) ).

% mult_eq_0_iff
thf(fact_132_mult__eq__0__iff,axiom,
    ! [A: poly_a,B: poly_a] :
      ( ( ( times_times_poly_a @ A @ B )
        = zero_zero_poly_a )
      = ( ( A = zero_zero_poly_a )
        | ( B = zero_zero_poly_a ) ) ) ).

% mult_eq_0_iff
thf(fact_133_mult__eq__0__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( ( times_times_nat @ A @ B )
        = zero_zero_nat )
      = ( ( A = zero_zero_nat )
        | ( B = zero_zero_nat ) ) ) ).

% mult_eq_0_iff
thf(fact_134_mult__zero__right,axiom,
    ! [A: a] :
      ( ( times_times_a @ A @ zero_zero_a )
      = zero_zero_a ) ).

% mult_zero_right
thf(fact_135_mult__zero__right,axiom,
    ! [A: poly_a] :
      ( ( times_times_poly_a @ A @ zero_zero_poly_a )
      = zero_zero_poly_a ) ).

% mult_zero_right
thf(fact_136_mult__zero__right,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_zero_right
thf(fact_137_mult__zero__left,axiom,
    ! [A: a] :
      ( ( times_times_a @ zero_zero_a @ A )
      = zero_zero_a ) ).

% mult_zero_left
thf(fact_138_mult__zero__left,axiom,
    ! [A: poly_a] :
      ( ( times_times_poly_a @ zero_zero_poly_a @ A )
      = zero_zero_poly_a ) ).

% mult_zero_left
thf(fact_139_mult__zero__left,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% mult_zero_left
thf(fact_140_neg__equal__zero,axiom,
    ! [A: poly_a] :
      ( ( ( uminus_uminus_poly_a @ A )
        = A )
      = ( A = zero_zero_poly_a ) ) ).

% neg_equal_zero
thf(fact_141_neg__equal__zero,axiom,
    ! [A: a] :
      ( ( ( uminus_uminus_a @ A )
        = A )
      = ( A = zero_zero_a ) ) ).

% neg_equal_zero
thf(fact_142_equal__neg__zero,axiom,
    ! [A: poly_a] :
      ( ( A
        = ( uminus_uminus_poly_a @ A ) )
      = ( A = zero_zero_poly_a ) ) ).

% equal_neg_zero
thf(fact_143_equal__neg__zero,axiom,
    ! [A: a] :
      ( ( A
        = ( uminus_uminus_a @ A ) )
      = ( A = zero_zero_a ) ) ).

% equal_neg_zero
thf(fact_144_neg__equal__0__iff__equal,axiom,
    ! [A: poly_a] :
      ( ( ( uminus_uminus_poly_a @ A )
        = zero_zero_poly_a )
      = ( A = zero_zero_poly_a ) ) ).

% neg_equal_0_iff_equal
thf(fact_145_neg__equal__0__iff__equal,axiom,
    ! [A: a] :
      ( ( ( uminus_uminus_a @ A )
        = zero_zero_a )
      = ( A = zero_zero_a ) ) ).

% neg_equal_0_iff_equal
thf(fact_146_neg__0__equal__iff__equal,axiom,
    ! [A: poly_a] :
      ( ( zero_zero_poly_a
        = ( uminus_uminus_poly_a @ A ) )
      = ( zero_zero_poly_a = A ) ) ).

% neg_0_equal_iff_equal
thf(fact_147_neg__0__equal__iff__equal,axiom,
    ! [A: a] :
      ( ( zero_zero_a
        = ( uminus_uminus_a @ A ) )
      = ( zero_zero_a = A ) ) ).

% neg_0_equal_iff_equal
thf(fact_148_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_poly_a @ zero_zero_poly_a )
    = zero_zero_poly_a ) ).

% add.inverse_neutral
thf(fact_149_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_a @ zero_zero_a )
    = zero_zero_a ) ).

% add.inverse_neutral
thf(fact_150_mult_Oright__neutral,axiom,
    ! [A: a] :
      ( ( times_times_a @ A @ one_one_a )
      = A ) ).

% mult.right_neutral
thf(fact_151_mult_Oright__neutral,axiom,
    ! [A: poly_a] :
      ( ( times_times_poly_a @ A @ one_one_poly_a )
      = A ) ).

% mult.right_neutral
thf(fact_152_mult_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ one_one_nat )
      = A ) ).

% mult.right_neutral
thf(fact_153_mult_Oleft__neutral,axiom,
    ! [A: a] :
      ( ( times_times_a @ one_one_a @ A )
      = A ) ).

% mult.left_neutral
thf(fact_154_mult_Oleft__neutral,axiom,
    ! [A: poly_a] :
      ( ( times_times_poly_a @ one_one_poly_a @ A )
      = A ) ).

% mult.left_neutral
thf(fact_155_mult_Oleft__neutral,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ one_one_nat @ A )
      = A ) ).

% mult.left_neutral
thf(fact_156_neg__less__iff__less,axiom,
    ! [B: poly_a,A: poly_a] :
      ( ( ord_less_poly_a @ ( uminus_uminus_poly_a @ B ) @ ( uminus_uminus_poly_a @ A ) )
      = ( ord_less_poly_a @ A @ B ) ) ).

% neg_less_iff_less
thf(fact_157_neg__less__iff__less,axiom,
    ! [B: a,A: a] :
      ( ( ord_less_a @ ( uminus_uminus_a @ B ) @ ( uminus_uminus_a @ A ) )
      = ( ord_less_a @ A @ B ) ) ).

% neg_less_iff_less
thf(fact_158_mult__minus__right,axiom,
    ! [A: poly_a,B: poly_a] :
      ( ( times_times_poly_a @ A @ ( uminus_uminus_poly_a @ B ) )
      = ( uminus_uminus_poly_a @ ( times_times_poly_a @ A @ B ) ) ) ).

% mult_minus_right
thf(fact_159_mult__minus__right,axiom,
    ! [A: a,B: a] :
      ( ( times_times_a @ A @ ( uminus_uminus_a @ B ) )
      = ( uminus_uminus_a @ ( times_times_a @ A @ B ) ) ) ).

% mult_minus_right
thf(fact_160_minus__mult__minus,axiom,
    ! [A: poly_a,B: poly_a] :
      ( ( times_times_poly_a @ ( uminus_uminus_poly_a @ A ) @ ( uminus_uminus_poly_a @ B ) )
      = ( times_times_poly_a @ A @ B ) ) ).

% minus_mult_minus
thf(fact_161_minus__mult__minus,axiom,
    ! [A: a,B: a] :
      ( ( times_times_a @ ( uminus_uminus_a @ A ) @ ( uminus_uminus_a @ B ) )
      = ( times_times_a @ A @ B ) ) ).

% minus_mult_minus
thf(fact_162_mult__minus__left,axiom,
    ! [A: poly_a,B: poly_a] :
      ( ( times_times_poly_a @ ( uminus_uminus_poly_a @ A ) @ B )
      = ( uminus_uminus_poly_a @ ( times_times_poly_a @ A @ B ) ) ) ).

% mult_minus_left
thf(fact_163_mult__minus__left,axiom,
    ! [A: a,B: a] :
      ( ( times_times_a @ ( uminus_uminus_a @ A ) @ B )
      = ( uminus_uminus_a @ ( times_times_a @ A @ B ) ) ) ).

% mult_minus_left
thf(fact_164_append_Oright__neutral,axiom,
    ! [A: list_a] :
      ( ( append_a @ A @ nil_a )
      = A ) ).

% append.right_neutral
thf(fact_165_append__is__Nil__conv,axiom,
    ! [Xs: list_a,Ys2: list_a] :
      ( ( ( append_a @ Xs @ Ys2 )
        = nil_a )
      = ( ( Xs = nil_a )
        & ( Ys2 = nil_a ) ) ) ).

% append_is_Nil_conv
thf(fact_166_Nil__is__append__conv,axiom,
    ! [Xs: list_a,Ys2: list_a] :
      ( ( nil_a
        = ( append_a @ Xs @ Ys2 ) )
      = ( ( Xs = nil_a )
        & ( Ys2 = nil_a ) ) ) ).

% Nil_is_append_conv
thf(fact_167_self__append__conv2,axiom,
    ! [Ys2: list_a,Xs: list_a] :
      ( ( Ys2
        = ( append_a @ Xs @ Ys2 ) )
      = ( Xs = nil_a ) ) ).

% self_append_conv2
thf(fact_168_append__self__conv2,axiom,
    ! [Xs: list_a,Ys2: list_a] :
      ( ( ( append_a @ Xs @ Ys2 )
        = Ys2 )
      = ( Xs = nil_a ) ) ).

% append_self_conv2
thf(fact_169_self__append__conv,axiom,
    ! [Xs: list_a,Ys2: list_a] :
      ( ( Xs
        = ( append_a @ Xs @ Ys2 ) )
      = ( Ys2 = nil_a ) ) ).

% self_append_conv
thf(fact_170_append__self__conv,axiom,
    ! [Xs: list_a,Ys2: list_a] :
      ( ( ( append_a @ Xs @ Ys2 )
        = Xs )
      = ( Ys2 = nil_a ) ) ).

% append_self_conv
thf(fact_171_append__Nil2,axiom,
    ! [Xs: list_a] :
      ( ( append_a @ Xs @ nil_a )
      = Xs ) ).

% append_Nil2
thf(fact_172_smult__one,axiom,
    ! [C: a] :
      ( ( smult_a @ C @ one_one_poly_a )
      = ( pCons_a @ C @ zero_zero_poly_a ) ) ).

% smult_one
thf(fact_173_zero__reorient,axiom,
    ! [X: poly_a] :
      ( ( zero_zero_poly_a = X )
      = ( X = zero_zero_poly_a ) ) ).

% zero_reorient
thf(fact_174_zero__reorient,axiom,
    ! [X: a] :
      ( ( zero_zero_a = X )
      = ( X = zero_zero_a ) ) ).

% zero_reorient
thf(fact_175_zero__reorient,axiom,
    ! [X: nat] :
      ( ( zero_zero_nat = X )
      = ( X = zero_zero_nat ) ) ).

% zero_reorient
thf(fact_176_linorder__neqE__linordered__idom,axiom,
    ! [X: a,Y3: a] :
      ( ( X != Y3 )
     => ( ~ ( ord_less_a @ X @ Y3 )
       => ( ord_less_a @ Y3 @ X ) ) ) ).

% linorder_neqE_linordered_idom
thf(fact_177_linorder__neqE__linordered__idom,axiom,
    ! [X: poly_a,Y3: poly_a] :
      ( ( X != Y3 )
     => ( ~ ( ord_less_poly_a @ X @ Y3 )
       => ( ord_less_poly_a @ Y3 @ X ) ) ) ).

% linorder_neqE_linordered_idom
thf(fact_178_mult_Oleft__commute,axiom,
    ! [B: poly_a,A: poly_a,C: poly_a] :
      ( ( times_times_poly_a @ B @ ( times_times_poly_a @ A @ C ) )
      = ( times_times_poly_a @ A @ ( times_times_poly_a @ B @ C ) ) ) ).

% mult.left_commute
thf(fact_179_mult_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
      = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).

% mult.left_commute
thf(fact_180_mult_Ocommute,axiom,
    ( times_times_poly_a
    = ( ^ [A3: poly_a,B3: poly_a] : ( times_times_poly_a @ B3 @ A3 ) ) ) ).

% mult.commute
thf(fact_181_mult_Ocommute,axiom,
    ( times_times_nat
    = ( ^ [A3: nat,B3: nat] : ( times_times_nat @ B3 @ A3 ) ) ) ).

% mult.commute
thf(fact_182_mult_Oassoc,axiom,
    ! [A: poly_a,B: poly_a,C: poly_a] :
      ( ( times_times_poly_a @ ( times_times_poly_a @ A @ B ) @ C )
      = ( times_times_poly_a @ A @ ( times_times_poly_a @ B @ C ) ) ) ).

% mult.assoc
thf(fact_183_mult_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
      = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).

% mult.assoc
thf(fact_184_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: poly_a,B: poly_a,C: poly_a] :
      ( ( times_times_poly_a @ ( times_times_poly_a @ A @ B ) @ C )
      = ( times_times_poly_a @ A @ ( times_times_poly_a @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_185_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
      = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_186_one__reorient,axiom,
    ! [X: a] :
      ( ( one_one_a = X )
      = ( X = one_one_a ) ) ).

% one_reorient
thf(fact_187_one__reorient,axiom,
    ! [X: nat] :
      ( ( one_one_nat = X )
      = ( X = one_one_nat ) ) ).

% one_reorient
thf(fact_188_minus__equation__iff,axiom,
    ! [A: a,B: a] :
      ( ( ( uminus_uminus_a @ A )
        = B )
      = ( ( uminus_uminus_a @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_189_equation__minus__iff,axiom,
    ! [A: a,B: a] :
      ( ( A
        = ( uminus_uminus_a @ B ) )
      = ( B
        = ( uminus_uminus_a @ A ) ) ) ).

% equation_minus_iff
thf(fact_190_not__Cons__self2,axiom,
    ! [X: a,Xs: list_a] :
      ( ( cons_a @ X @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_191_append__eq__append__conv2,axiom,
    ! [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_192_append__eq__appendI,axiom,
    ! [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_193_sign__changes__two,axiom,
    ! [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 ) ) )
       => ( ( descar2095969287nges_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 ) ) )
       => ( ( descar2095969287nges_a @ ( cons_a @ X @ ( cons_a @ Y3 @ nil_a ) ) )
          = zero_zero_nat ) ) ) ).

% sign_changes_two
thf(fact_194_sign__changes__two,axiom,
    ! [X: poly_a,Y3: poly_a] :
      ( ( ( ( ( ord_less_poly_a @ zero_zero_poly_a @ X )
            & ( ord_less_poly_a @ Y3 @ zero_zero_poly_a ) )
          | ( ( ord_less_poly_a @ X @ zero_zero_poly_a )
            & ( ord_less_poly_a @ zero_zero_poly_a @ Y3 ) ) )
       => ( ( descar357075861poly_a @ ( cons_poly_a @ X @ ( cons_poly_a @ Y3 @ nil_poly_a ) ) )
          = one_one_nat ) )
      & ( ~ ( ( ( ord_less_poly_a @ zero_zero_poly_a @ X )
              & ( ord_less_poly_a @ Y3 @ zero_zero_poly_a ) )
            | ( ( ord_less_poly_a @ X @ zero_zero_poly_a )
              & ( ord_less_poly_a @ zero_zero_poly_a @ Y3 ) ) )
       => ( ( descar357075861poly_a @ ( cons_poly_a @ X @ ( cons_poly_a @ Y3 @ nil_poly_a ) ) )
          = zero_zero_nat ) ) ) ).

% sign_changes_two
thf(fact_195_zero__less__iff__neq__zero,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
      = ( N != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
thf(fact_196_gr__implies__not__zero,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not_zero
thf(fact_197_not__less__zero,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less_zero
thf(fact_198_gr__zeroI,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr_zeroI
thf(fact_199_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).

% less_numeral_extra(3)
thf(fact_200_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_a @ zero_zero_a @ zero_zero_a ) ).

% less_numeral_extra(3)
thf(fact_201_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_poly_a @ zero_zero_poly_a @ zero_zero_poly_a ) ).

% less_numeral_extra(3)
thf(fact_202_mult__right__cancel,axiom,
    ! [C: a,A: a,B: a] :
      ( ( C != zero_zero_a )
     => ( ( ( times_times_a @ A @ C )
          = ( times_times_a @ B @ C ) )
        = ( A = B ) ) ) ).

% mult_right_cancel
thf(fact_203_mult__right__cancel,axiom,
    ! [C: poly_a,A: poly_a,B: poly_a] :
      ( ( C != zero_zero_poly_a )
     => ( ( ( times_times_poly_a @ A @ C )
          = ( times_times_poly_a @ B @ C ) )
        = ( A = B ) ) ) ).

% mult_right_cancel
thf(fact_204_mult__right__cancel,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( ( times_times_nat @ A @ C )
          = ( times_times_nat @ B @ C ) )
        = ( A = B ) ) ) ).

% mult_right_cancel
thf(fact_205_mult__left__cancel,axiom,
    ! [C: a,A: a,B: a] :
      ( ( C != zero_zero_a )
     => ( ( ( times_times_a @ C @ A )
          = ( times_times_a @ C @ B ) )
        = ( A = B ) ) ) ).

% mult_left_cancel
thf(fact_206_mult__left__cancel,axiom,
    ! [C: poly_a,A: poly_a,B: poly_a] :
      ( ( C != zero_zero_poly_a )
     => ( ( ( times_times_poly_a @ C @ A )
          = ( times_times_poly_a @ C @ B ) )
        = ( A = B ) ) ) ).

% mult_left_cancel
thf(fact_207_mult__left__cancel,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( ( times_times_nat @ C @ A )
          = ( times_times_nat @ C @ B ) )
        = ( A = B ) ) ) ).

% mult_left_cancel
thf(fact_208_no__zero__divisors,axiom,
    ! [A: a,B: a] :
      ( ( A != zero_zero_a )
     => ( ( B != zero_zero_a )
       => ( ( times_times_a @ A @ B )
         != zero_zero_a ) ) ) ).

% no_zero_divisors
thf(fact_209_no__zero__divisors,axiom,
    ! [A: poly_a,B: poly_a] :
      ( ( A != zero_zero_poly_a )
     => ( ( B != zero_zero_poly_a )
       => ( ( times_times_poly_a @ A @ B )
         != zero_zero_poly_a ) ) ) ).

% no_zero_divisors
thf(fact_210_no__zero__divisors,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( B != zero_zero_nat )
       => ( ( times_times_nat @ A @ B )
         != zero_zero_nat ) ) ) ).

% no_zero_divisors
thf(fact_211_divisors__zero,axiom,
    ! [A: a,B: a] :
      ( ( ( times_times_a @ A @ B )
        = zero_zero_a )
     => ( ( A = zero_zero_a )
        | ( B = zero_zero_a ) ) ) ).

% divisors_zero
thf(fact_212_divisors__zero,axiom,
    ! [A: poly_a,B: poly_a] :
      ( ( ( times_times_poly_a @ A @ B )
        = zero_zero_poly_a )
     => ( ( A = zero_zero_poly_a )
        | ( B = zero_zero_poly_a ) ) ) ).

% divisors_zero
thf(fact_213_divisors__zero,axiom,
    ! [A: nat,B: nat] :
      ( ( ( times_times_nat @ A @ B )
        = zero_zero_nat )
     => ( ( A = zero_zero_nat )
        | ( B = zero_zero_nat ) ) ) ).

% divisors_zero
thf(fact_214_mult__not__zero,axiom,
    ! [A: a,B: a] :
      ( ( ( times_times_a @ A @ B )
       != zero_zero_a )
     => ( ( A != zero_zero_a )
        & ( B != zero_zero_a ) ) ) ).

% mult_not_zero
thf(fact_215_mult__not__zero,axiom,
    ! [A: poly_a,B: poly_a] :
      ( ( ( times_times_poly_a @ A @ B )
       != zero_zero_poly_a )
     => ( ( A != zero_zero_poly_a )
        & ( B != zero_zero_poly_a ) ) ) ).

% mult_not_zero
thf(fact_216_mult__not__zero,axiom,
    ! [A: nat,B: nat] :
      ( ( ( times_times_nat @ A @ B )
       != zero_zero_nat )
     => ( ( A != zero_zero_nat )
        & ( B != zero_zero_nat ) ) ) ).

% mult_not_zero
thf(fact_217_zero__neq__one,axiom,
    zero_zero_poly_a != one_one_poly_a ).

% zero_neq_one
thf(fact_218_zero__neq__one,axiom,
    zero_zero_a != one_one_a ).

% zero_neq_one
thf(fact_219_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
thf(fact_220_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).

% less_numeral_extra(4)
thf(fact_221_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_a @ one_one_a @ one_one_a ) ).

% less_numeral_extra(4)
thf(fact_222_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_poly_a @ one_one_poly_a @ one_one_poly_a ) ).

% less_numeral_extra(4)
thf(fact_223_mult_Ocomm__neutral,axiom,
    ! [A: a] :
      ( ( times_times_a @ A @ one_one_a )
      = A ) ).

% mult.comm_neutral
thf(fact_224_mult_Ocomm__neutral,axiom,
    ! [A: poly_a] :
      ( ( times_times_poly_a @ A @ one_one_poly_a )
      = A ) ).

% mult.comm_neutral
thf(fact_225_mult_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ one_one_nat )
      = A ) ).

% mult.comm_neutral
thf(fact_226_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: a] :
      ( ( times_times_a @ one_one_a @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_227_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: poly_a] :
      ( ( times_times_poly_a @ one_one_poly_a @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_228_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ one_one_nat @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_229_minus__less__iff,axiom,
    ! [A: poly_a,B: poly_a] :
      ( ( ord_less_poly_a @ ( uminus_uminus_poly_a @ A ) @ B )
      = ( ord_less_poly_a @ ( uminus_uminus_poly_a @ B ) @ A ) ) ).

% minus_less_iff
thf(fact_230_minus__less__iff,axiom,
    ! [A: a,B: a] :
      ( ( ord_less_a @ ( uminus_uminus_a @ A ) @ B )
      = ( ord_less_a @ ( uminus_uminus_a @ B ) @ A ) ) ).

% minus_less_iff
thf(fact_231_less__minus__iff,axiom,
    ! [A: poly_a,B: poly_a] :
      ( ( ord_less_poly_a @ A @ ( uminus_uminus_poly_a @ B ) )
      = ( ord_less_poly_a @ B @ ( uminus_uminus_poly_a @ A ) ) ) ).

% less_minus_iff
thf(fact_232_less__minus__iff,axiom,
    ! [A: a,B: a] :
      ( ( ord_less_a @ A @ ( uminus_uminus_a @ B ) )
      = ( ord_less_a @ B @ ( uminus_uminus_a @ A ) ) ) ).

% less_minus_iff
thf(fact_233_minus__mult__commute,axiom,
    ! [A: poly_a,B: poly_a] :
      ( ( times_times_poly_a @ ( uminus_uminus_poly_a @ A ) @ B )
      = ( times_times_poly_a @ A @ ( uminus_uminus_poly_a @ B ) ) ) ).

% minus_mult_commute
thf(fact_234_minus__mult__commute,axiom,
    ! [A: a,B: a] :
      ( ( times_times_a @ ( uminus_uminus_a @ A ) @ B )
      = ( times_times_a @ A @ ( uminus_uminus_a @ B ) ) ) ).

% minus_mult_commute
thf(fact_235_square__eq__iff,axiom,
    ! [A: poly_a,B: poly_a] :
      ( ( ( times_times_poly_a @ A @ A )
        = ( times_times_poly_a @ B @ B ) )
      = ( ( A = B )
        | ( A
          = ( uminus_uminus_poly_a @ B ) ) ) ) ).

% square_eq_iff
thf(fact_236_square__eq__iff,axiom,
    ! [A: a,B: a] :
      ( ( ( times_times_a @ A @ A )
        = ( times_times_a @ B @ B ) )
      = ( ( A = B )
        | ( A
          = ( uminus_uminus_a @ B ) ) ) ) ).

% square_eq_iff
thf(fact_237_one__neq__neg__one,axiom,
    ( one_one_a
   != ( uminus_uminus_a @ one_one_a ) ) ).

% one_neq_neg_one
thf(fact_238_strict__sorted_Oinduct,axiom,
    ! [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_239_strict__sorted_Ocases,axiom,
    ! [X: list_a] :
      ( ( X != nil_a )
     => ~ ! [X2: a,Ys: list_a] :
            ( X
           != ( cons_a @ X2 @ Ys ) ) ) ).

% strict_sorted.cases
thf(fact_240_map__tailrec__rev_Oinduct,axiom,
    ! [P3: ( a > a ) > list_a > list_a > $o,A0: a > a,A1: list_a,A22: list_a] :
      ( ! [F2: a > a,X_1: list_a] : ( P3 @ F2 @ nil_a @ X_1 )
     => ( ! [F2: a > a,A2: a,As2: list_a,Bs: list_a] :
            ( ( P3 @ F2 @ As2 @ ( cons_a @ ( F2 @ A2 ) @ Bs ) )
           => ( P3 @ F2 @ ( cons_a @ A2 @ As2 ) @ Bs ) )
       => ( P3 @ A0 @ A1 @ A22 ) ) ) ).

% map_tailrec_rev.induct
thf(fact_241_list__nonempty__induct,axiom,
    ! [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_242_successively_Oinduct,axiom,
    ! [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_243_remdups__adj_Oinduct,axiom,
    ! [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_244_sorted__wrt_Oinduct,axiom,
    ! [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_245_remdups__adj_Ocases,axiom,
    ! [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_246_transpose_Ocases,axiom,
    ! [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_247_shuffles_Oinduct,axiom,
    ! [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_248_min__list_Oinduct,axiom,
    ! [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_249_min__list_Ocases,axiom,
    ! [X: list_a] :
      ( ! [X2: a,Xs2: list_a] :
          ( X
         != ( cons_a @ X2 @ Xs2 ) )
     => ( X = nil_a ) ) ).

% min_list.cases
thf(fact_250_induct__list012,axiom,
    ! [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_251_splice_Oinduct,axiom,
    ! [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_252_list__induct2_H,axiom,
    ! [P3: list_a > list_a > $o,Xs: list_a,Ys2: list_a] :
      ( ( P3 @ nil_a @ nil_a )
     => ( ! [X2: a,Xs2: list_a] : ( P3 @ ( cons_a @ X2 @ Xs2 ) @ nil_a )
       => ( ! [Y2: a,Ys: list_a] : ( P3 @ nil_a @ ( cons_a @ Y2 @ Ys ) )
         => ( ! [X2: a,Xs2: list_a,Y2: a,Ys: list_a] :
                ( ( P3 @ Xs2 @ Ys )
               => ( P3 @ ( cons_a @ X2 @ Xs2 ) @ ( cons_a @ Y2 @ Ys ) ) )
           => ( P3 @ Xs @ Ys2 ) ) ) ) ) ).

% list_induct2'
thf(fact_253_neq__Nil__conv,axiom,
    ! [Xs: list_a] :
      ( ( Xs != nil_a )
      = ( ? [Y4: a,Ys3: list_a] :
            ( Xs
            = ( cons_a @ Y4 @ Ys3 ) ) ) ) ).

% neq_Nil_conv
thf(fact_254_list_Oinducts,axiom,
    ! [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_255_list_Oexhaust,axiom,
    ! [Y3: list_a] :
      ( ( Y3 != nil_a )
     => ~ ! [X213: a,X223: list_a] :
            ( Y3
           != ( cons_a @ X213 @ X223 ) ) ) ).

% list.exhaust
thf(fact_256_list_OdiscI,axiom,
    ! [List: list_a,X21: a,X22: list_a] :
      ( ( List
        = ( cons_a @ X21 @ X22 ) )
     => ( List != nil_a ) ) ).

% list.discI
thf(fact_257_list_Odistinct_I1_J,axiom,
    ! [X21: a,X22: list_a] :
      ( nil_a
     != ( cons_a @ X21 @ X22 ) ) ).

% list.distinct(1)
thf(fact_258_append__Cons,axiom,
    ! [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_259_Cons__eq__appendI,axiom,
    ! [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_260_append_Oleft__neutral,axiom,
    ! [A: list_a] :
      ( ( append_a @ nil_a @ A )
      = A ) ).

% append.left_neutral
thf(fact_261_append__Nil,axiom,
    ! [Ys2: list_a] :
      ( ( append_a @ nil_a @ Ys2 )
      = Ys2 ) ).

% append_Nil
thf(fact_262_eq__Nil__appendI,axiom,
    ! [Xs: list_a,Ys2: list_a] :
      ( ( Xs = Ys2 )
     => ( Xs
        = ( append_a @ nil_a @ Ys2 ) ) ) ).

% eq_Nil_appendI
thf(fact_263_null__rec_I1_J,axiom,
    ! [X: a,Xs: list_a] :
      ~ ( null_a @ ( cons_a @ X @ Xs ) ) ).

% null_rec(1)
thf(fact_264_eq__Nil__null,axiom,
    ! [Xs: list_a] :
      ( ( Xs = nil_a )
      = ( null_a @ Xs ) ) ).

% eq_Nil_null
thf(fact_265_null__rec_I2_J,axiom,
    null_a @ nil_a ).

% null_rec(2)
thf(fact_266_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_267_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: a,B: a,C: a] :
      ( ( ord_less_a @ A @ B )
     => ( ( ord_less_a @ zero_zero_a @ C )
       => ( ord_less_a @ ( times_times_a @ C @ A ) @ ( times_times_a @ C @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_268_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: poly_a,B: poly_a,C: poly_a] :
      ( ( ord_less_poly_a @ A @ B )
     => ( ( ord_less_poly_a @ zero_zero_poly_a @ C )
       => ( ord_less_poly_a @ ( times_times_poly_a @ C @ A ) @ ( times_times_poly_a @ C @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_269_mult__less__cancel__right__disj,axiom,
    ! [A: a,C: a,B: a] :
      ( ( ord_less_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ B @ C ) )
      = ( ( ( ord_less_a @ zero_zero_a @ C )
          & ( ord_less_a @ A @ B ) )
        | ( ( ord_less_a @ C @ zero_zero_a )
          & ( ord_less_a @ B @ A ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_270_mult__less__cancel__right__disj,axiom,
    ! [A: poly_a,C: poly_a,B: poly_a] :
      ( ( ord_less_poly_a @ ( times_times_poly_a @ A @ C ) @ ( times_times_poly_a @ B @ C ) )
      = ( ( ( ord_less_poly_a @ zero_zero_poly_a @ C )
          & ( ord_less_poly_a @ A @ B ) )
        | ( ( ord_less_poly_a @ C @ zero_zero_poly_a )
          & ( ord_less_poly_a @ B @ A ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_271_mult__strict__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_272_mult__strict__right__mono,axiom,
    ! [A: a,B: a,C: a] :
      ( ( ord_less_a @ A @ B )
     => ( ( ord_less_a @ zero_zero_a @ C )
       => ( ord_less_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ B @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_273_mult__strict__right__mono,axiom,
    ! [A: poly_a,B: poly_a,C: poly_a] :
      ( ( ord_less_poly_a @ A @ B )
     => ( ( ord_less_poly_a @ zero_zero_poly_a @ C )
       => ( ord_less_poly_a @ ( times_times_poly_a @ A @ C ) @ ( times_times_poly_a @ B @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_274_mult__strict__right__mono__neg,axiom,
    ! [B: a,A: a,C: a] :
      ( ( ord_less_a @ B @ A )
     => ( ( ord_less_a @ C @ zero_zero_a )
       => ( ord_less_a @ ( times_times_a @ A @ C ) @ ( times_times_a @ B @ C ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_275_mult__strict__right__mono__neg,axiom,
    ! [B: poly_a,A: poly_a,C: poly_a] :
      ( ( ord_less_poly_a @ B @ A )
     => ( ( ord_less_poly_a @ C @ zero_zero_poly_a )
       => ( ord_less_poly_a @ ( times_times_poly_a @ A @ C ) @ ( times_times_poly_a @ B @ C ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_276_mult__less__cancel__left__disj,axiom,
    ! [C: a,A: a,B: a] :
      ( ( ord_less_a @ ( times_times_a @ C @ A ) @ ( times_times_a @ C @ B ) )
      = ( ( ( ord_less_a @ zero_zero_a @ C )
          & ( ord_less_a @ A @ B ) )
        | ( ( ord_less_a @ C @ zero_zero_a )
          & ( ord_less_a @ B @ A ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_277_mult__less__cancel__left__disj,axiom,
    ! [C: poly_a,A: poly_a,B: poly_a] :
      ( ( ord_less_poly_a @ ( times_times_poly_a @ C @ A ) @ ( times_times_poly_a @ C @ B ) )
      = ( ( ( ord_less_poly_a @ zero_zero_poly_a @ C )
          & ( ord_less_poly_a @ A @ B ) )
        | ( ( ord_less_poly_a @ C @ zero_zero_poly_a )
          & ( ord_less_poly_a @ B @ A ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_278_mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_279_mult__strict__left__mono,axiom,
    ! [A: a,B: a,C: a] :
      ( ( ord_less_a @ A @ B )
     => ( ( ord_less_a @ zero_zero_a @ C )
       => ( ord_less_a @ ( times_times_a @ C @ A ) @ ( times_times_a @ C @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_280_mult__strict__left__mono,axiom,
    ! [A: poly_a,B: poly_a,C: poly_a] :
      ( ( ord_less_poly_a @ A @ B )
     => ( ( ord_less_poly_a @ zero_zero_poly_a @ C )
       => ( ord_less_poly_a @ ( times_times_poly_a @ C @ A ) @ ( times_times_poly_a @ C @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_281_mult__strict__left__mono__neg,axiom,
    ! [B: a,A: a,C: a] :
      ( ( ord_less_a @ B @ A )
     => ( ( ord_less_a @ C @ zero_zero_a )
       => ( ord_less_a @ ( times_times_a @ C @ A ) @ ( times_times_a @ C @ B ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_282_mult__strict__left__mono__neg,axiom,
    ! [B: poly_a,A: poly_a,C: poly_a] :
      ( ( ord_less_poly_a @ B @ A )
     => ( ( ord_less_poly_a @ C @ zero_zero_poly_a )
       => ( ord_less_poly_a @ ( times_times_poly_a @ C @ A ) @ ( times_times_poly_a @ C @ B ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_283_mult__less__cancel__left__pos,axiom,
    ! [C: a,A: a,B: a] :
      ( ( ord_less_a @ zero_zero_a @ C )
     => ( ( ord_less_a @ ( times_times_a @ C @ A ) @ ( times_times_a @ C @ B ) )
        = ( ord_less_a @ A @ B ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_284_mult__less__cancel__left__pos,axiom,
    ! [C: poly_a,A: poly_a,B: poly_a] :
      ( ( ord_less_poly_a @ zero_zero_poly_a @ C )
     => ( ( ord_less_poly_a @ ( times_times_poly_a @ C @ A ) @ ( times_times_poly_a @ C @ B ) )
        = ( ord_less_poly_a @ A @ B ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_285_mult__less__cancel__left__neg,axiom,
    ! [C: a,A: a,B: a] :
      ( ( ord_less_a @ C @ zero_zero_a )
     => ( ( ord_less_a @ ( times_times_a @ C @ A ) @ ( times_times_a @ C @ B ) )
        = ( ord_less_a @ B @ A ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_286_mult__less__cancel__left__neg,axiom,
    ! [C: poly_a,A: poly_a,B: poly_a] :
      ( ( ord_less_poly_a @ C @ zero_zero_poly_a )
     => ( ( ord_less_poly_a @ ( times_times_poly_a @ C @ A ) @ ( times_times_poly_a @ C @ B ) )
        = ( ord_less_poly_a @ B @ A ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_287_zero__less__mult__pos2,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
     => ( ( ord_less_nat @ zero_zero_nat @ A )
       => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).

% zero_less_mult_pos2
thf(fact_288_zero__less__mult__pos2,axiom,
    ! [B: a,A: a] :
      ( ( ord_less_a @ zero_zero_a @ ( times_times_a @ B @ A ) )
     => ( ( ord_less_a @ zero_zero_a @ A )
       => ( ord_less_a @ zero_zero_a @ B ) ) ) ).

% zero_less_mult_pos2
thf(fact_289_zero__less__mult__pos2,axiom,
    ! [B: poly_a,A: poly_a] :
      ( ( ord_less_poly_a @ zero_zero_poly_a @ ( times_times_poly_a @ B @ A ) )
     => ( ( ord_less_poly_a @ zero_zero_poly_a @ A )
       => ( ord_less_poly_a @ zero_zero_poly_a @ B ) ) ) ).

% zero_less_mult_pos2
thf(fact_290_zero__less__mult__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
     => ( ( ord_less_nat @ zero_zero_nat @ A )
       => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).

% zero_less_mult_pos
thf(fact_291_zero__less__mult__pos,axiom,
    ! [A: a,B: a] :
      ( ( ord_less_a @ zero_zero_a @ ( times_times_a @ A @ B ) )
     => ( ( ord_less_a @ zero_zero_a @ A )
       => ( ord_less_a @ zero_zero_a @ B ) ) ) ).

% zero_less_mult_pos
thf(fact_292_zero__less__mult__pos,axiom,
    ! [A: poly_a,B: poly_a] :
      ( ( ord_less_poly_a @ zero_zero_poly_a @ ( times_times_poly_a @ A @ B ) )
     => ( ( ord_less_poly_a @ zero_zero_poly_a @ A )
       => ( ord_less_poly_a @ zero_zero_poly_a @ B ) ) ) ).

% zero_less_mult_pos
thf(fact_293_zero__less__mult__iff,axiom,
    ! [A: a,B: a] :
      ( ( ord_less_a @ zero_zero_a @ ( times_times_a @ A @ B ) )
      = ( ( ( ord_less_a @ zero_zero_a @ A )
          & ( ord_less_a @ zero_zero_a @ B ) )
        | ( ( ord_less_a @ A @ zero_zero_a )
          & ( ord_less_a @ B @ zero_zero_a ) ) ) ) ).

% zero_less_mult_iff
thf(fact_294_zero__less__mult__iff,axiom,
    ! [A: poly_a,B: poly_a] :
      ( ( ord_less_poly_a @ zero_zero_poly_a @ ( times_times_poly_a @ A @ B ) )
      = ( ( ( ord_less_poly_a @ zero_zero_poly_a @ A )
          & ( ord_less_poly_a @ zero_zero_poly_a @ B ) )
        | ( ( ord_less_poly_a @ A @ zero_zero_poly_a )
          & ( ord_less_poly_a @ B @ zero_zero_poly_a ) ) ) ) ).

% zero_less_mult_iff
thf(fact_295_mult__pos__neg2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).

% mult_pos_neg2
thf(fact_296_mult__pos__neg2,axiom,
    ! [A: a,B: a] :
      ( ( ord_less_a @ zero_zero_a @ A )
     => ( ( ord_less_a @ B @ zero_zero_a )
       => ( ord_less_a @ ( times_times_a @ B @ A ) @ zero_zero_a ) ) ) ).

% mult_pos_neg2
thf(fact_297_mult__pos__neg2,axiom,
    ! [A: poly_a,B: poly_a] :
      ( ( ord_less_poly_a @ zero_zero_poly_a @ A )
     => ( ( ord_less_poly_a @ B @ zero_zero_poly_a )
       => ( ord_less_poly_a @ ( times_times_poly_a @ B @ A ) @ zero_zero_poly_a ) ) ) ).

% mult_pos_neg2
thf(fact_298_mult__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).

% mult_pos_pos
thf(fact_299_mult__pos__pos,axiom,
    ! [A: a,B: a] :
      ( ( ord_less_a @ zero_zero_a @ A )
     => ( ( ord_less_a @ zero_zero_a @ B )
       => ( ord_less_a @ zero_zero_a @ ( times_times_a @ A @ B ) ) ) ) ).

% mult_pos_pos
thf(fact_300_mult__pos__pos,axiom,
    ! [A: poly_a,B: poly_a] :
      ( ( ord_less_poly_a @ zero_zero_poly_a @ A )
     => ( ( ord_less_poly_a @ zero_zero_poly_a @ B )
       => ( ord_less_poly_a @ zero_zero_poly_a @ ( times_times_poly_a @ A @ B ) ) ) ) ).

% mult_pos_pos
thf(fact_301_mult__pos__neg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% mult_pos_neg
thf(fact_302_mult__pos__neg,axiom,
    ! [A: a,B: a] :
      ( ( ord_less_a @ zero_zero_a @ A )
     => ( ( ord_less_a @ B @ zero_zero_a )
       => ( ord_less_a @ ( times_times_a @ A @ B ) @ zero_zero_a ) ) ) ).

% mult_pos_neg
thf(fact_303_mult__pos__neg,axiom,
    ! [A: poly_a,B: poly_a] :
      ( ( ord_less_poly_a @ zero_zero_poly_a @ A )
     => ( ( ord_less_poly_a @ B @ zero_zero_poly_a )
       => ( ord_less_poly_a @ ( times_times_poly_a @ A @ B ) @ zero_zero_poly_a ) ) ) ).

% mult_pos_neg
thf(fact_304_mult__neg__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ zero_zero_nat )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% mult_neg_pos
thf(fact_305_mult__neg__pos,axiom,
    ! [A: a,B: a] :
      ( ( ord_less_a @ A @ zero_zero_a )
     => ( ( ord_less_a @ zero_zero_a @ B )
       => ( ord_less_a @ ( times_times_a @ A @ B ) @ zero_zero_a ) ) ) ).

% mult_neg_pos
thf(fact_306_mult__neg__pos,axiom,
    ! [A: poly_a,B: poly_a] :
      ( ( ord_less_poly_a @ A @ zero_zero_poly_a )
     => ( ( ord_less_poly_a @ zero_zero_poly_a @ B )
       => ( ord_less_poly_a @ ( times_times_poly_a @ A @ B ) @ zero_zero_poly_a ) ) ) ).

% mult_neg_pos
thf(fact_307_mult__less__0__iff,axiom,
    ! [A: a,B: a] :
      ( ( ord_less_a @ ( times_times_a @ A @ B ) @ zero_zero_a )
      = ( ( ( ord_less_a @ zero_zero_a @ A )
          & ( ord_less_a @ B @ zero_zero_a ) )
        | ( ( ord_less_a @ A @ zero_zero_a )
          & ( ord_less_a @ zero_zero_a @ B ) ) ) ) ).

% mult_less_0_iff
thf(fact_308_mult__less__0__iff,axiom,
    ! [A: poly_a,B: poly_a] :
      ( ( ord_less_poly_a @ ( times_times_poly_a @ A @ B ) @ zero_zero_poly_a )
      = ( ( ( ord_less_poly_a @ zero_zero_poly_a @ A )
          & ( ord_less_poly_a @ B @ zero_zero_poly_a ) )
        | ( ( ord_less_poly_a @ A @ zero_zero_poly_a )
          & ( ord_less_poly_a @ zero_zero_poly_a @ B ) ) ) ) ).

% mult_less_0_iff
thf(fact_309_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_310_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_311_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_312_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_313_neq0__conv,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% neq0_conv
thf(fact_314_less__nat__zero__code,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_nat_zero_code
thf(fact_315_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ A ) ) ).

% bot_nat_0.not_eq_extremum
thf(fact_316_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_317_mult__0__right,axiom,
    ! [M: nat] :
      ( ( times_times_nat @ M @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_0_right
thf(fact_318_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_319_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_320_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_321_less__one,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ one_one_nat )
      = ( N = zero_zero_nat ) ) ).

% less_one
thf(fact_322_nat__mult__1,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ one_one_nat @ N )
      = N ) ).

% nat_mult_1
thf(fact_323_nat__mult__1__right,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ N @ one_one_nat )
      = N ) ).

% nat_mult_1_right
thf(fact_324_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_325_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_326_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_327_less__irrefl__nat,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_irrefl_nat
thf(fact_328_less__not__refl3,axiom,
    ! [S: nat,T: nat] :
      ( ( ord_less_nat @ S @ T )
     => ( S != T ) ) ).

% less_not_refl3
thf(fact_329_less__not__refl2,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ N @ M )
     => ( M != N ) ) ).

% less_not_refl2
thf(fact_330_less__not__refl,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_not_refl
thf(fact_331_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_332_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr0I
thf(fact_333_not__gr0,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr0
thf(fact_334_not__less0,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less0
thf(fact_335_less__zeroE,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_zeroE
thf(fact_336_gr__implies__not0,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not0
thf(fact_337_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_338_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ zero_zero_nat ) ).

% bot_nat_0.extremum_strict
thf(fact_339_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_340_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_341_mult__0,axiom,
    ! [N: nat] :
      ( ( times_times_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% mult_0
thf(fact_342_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_343_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_344_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_345_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

% Conjectures (1)
thf(conj_0,conjecture,
    ( ( descar2095969287nges_a @ ys )
    = ( descar2095969287nges_a @ ( coeffs_a @ g ) ) ) ).

%------------------------------------------------------------------------------