TPTP Problem File: ITP044^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% 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 ) ) ) ).
%------------------------------------------------------------------------------