TPTP Problem File: ITP196^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP196^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Sturm_Theorem problem prob_153__5878286_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : Sturm_Theorem/prob_153__5878286_1 [Des21]
% Status : Theorem
% Rating : 0.20 v8.2.0, 0.15 v8.1.0, 0.18 v7.5.0
% Syntax : Number of formulae : 529 ( 226 unt; 179 typ; 0 def)
% Number of atoms : 889 ( 625 equ; 0 cnn)
% Maximal formula atoms : 11 ( 2 avg)
% Number of connectives : 3412 ( 249 ~; 39 |; 117 &;2544 @)
% ( 0 <=>; 463 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 7 avg)
% Number of types : 28 ( 27 usr)
% Number of type conns : 702 ( 702 >; 0 *; 0 +; 0 <<)
% Number of symbols : 153 ( 152 usr; 11 con; 0-3 aty)
% Number of variables : 1259 ( 49 ^;1098 !; 112 ?;1259 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:40:52.335
%------------------------------------------------------------------------------
% Could-be-implicit typings (27)
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% Explicit typings (152)
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produc1867601537y_real: poly_real > poly_real > produc461822025y_real ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
collec1537941401y_real: ( list_poly_real > $o ) > set_list_poly_real ).
thf(sy_c_Set_OCollect_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
collect_poly_real: ( poly_real > $o ) > set_poly_real ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__List__Olist_It__Polynomial__Opoly_It__Real__Oreal_J_J_Mt__List__Olist_It__Polynomial__Opoly_It__Real__Oreal_J_J_J,type,
collec1459394516y_real: ( produc1426596841y_real > $o ) > set_Pr1405268809y_real ).
thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Polynomial__Opoly_It__Real__Oreal_J_Mt__Polynomial__Opoly_It__Real__Oreal_J_J,type,
collec329281332y_real: ( produc461822025y_real > $o ) > set_Pr483210409y_real ).
thf(sy_c_Sturm__Theorem__Mirabelle__ubahemswrm_Oquasi__sturm__seq,type,
sturm_891428828rm_seq: list_poly_real > $o ).
thf(sy_c_Sturm__Theorem__Mirabelle__ubahemswrm_Osturm__seq,type,
sturm_1664866327rm_seq: list_poly_real > poly_real > $o ).
thf(sy_c_Sturm__Theorem__Mirabelle__ubahemswrm_Osturm__seq__axioms,type,
sturm_1683633594axioms: list_poly_real > poly_real > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_I_062_It__Polynomial__Opoly_It__Real__Oreal_J_Mt__Polynomial__Opoly_It__Real__Oreal_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Polynomial__Opoly_It__Real__Oreal_J_J_Mt__List__Olist_It__Polynomial__Opoly_It__Real__Oreal_J_J_J_J,type,
accp_P948088014y_real: ( produc1456530199y_real > produc1456530199y_real > $o ) > produc1456530199y_real > $o ).
thf(sy_c_member_001t__List__Olist_It__Polynomial__Opoly_It__Real__Oreal_J_J,type,
member708095579y_real: list_poly_real > set_list_poly_real > $o ).
thf(sy_c_member_001t__Polynomial__Opoly_It__Real__Oreal_J,type,
member_poly_real2: poly_real > set_poly_real > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Polynomial__Opoly_It__Real__Oreal_J_J_Mt__List__Olist_It__Polynomial__Opoly_It__Real__Oreal_J_J_J,type,
member403334290y_real: produc1426596841y_real > set_Pr1405268809y_real > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Polynomial__Opoly_It__Real__Oreal_J_Mt__Polynomial__Opoly_It__Real__Oreal_J_J,type,
member1784523250y_real: produc461822025y_real > set_Pr483210409y_real > $o ).
% Relevant facts (349)
thf(fact_0_quasi__sturm__seq_Ops__not__Nil,axiom,
! [Ps: list_poly_real] :
( ( sturm_891428828rm_seq @ Ps )
=> ( Ps != nil_poly_real ) ) ).
% quasi_sturm_seq.ps_not_Nil
thf(fact_1_sturm__seq_Oquasi__sturm__seq,axiom,
! [Ps: list_poly_real,P: poly_real] :
( ( sturm_1664866327rm_seq @ Ps @ P )
=> ( sturm_891428828rm_seq @ Ps ) ) ).
% sturm_seq.quasi_sturm_seq
thf(fact_2_sturm__seq__def,axiom,
( sturm_1664866327rm_seq
= ( ^ [Ps2: list_poly_real,P2: poly_real] :
( ( sturm_891428828rm_seq @ Ps2 )
& ( sturm_1683633594axioms @ Ps2 @ P2 ) ) ) ) ).
% sturm_seq_def
thf(fact_3_sturm__seq_Ointro,axiom,
! [Ps: list_poly_real,P: poly_real] :
( ( sturm_891428828rm_seq @ Ps )
=> ( ( sturm_1683633594axioms @ Ps @ P )
=> ( sturm_1664866327rm_seq @ Ps @ P ) ) ) ).
% sturm_seq.intro
thf(fact_4_list__ex1__simps_I1_J,axiom,
! [P3: produc321349221y_real > $o] :
~ ( list_e1892536494y_real @ P3 @ nil_Pr1492526187y_real ) ).
% list_ex1_simps(1)
thf(fact_5_list__ex1__simps_I1_J,axiom,
! [P3: list_poly_real > $o] :
~ ( list_e367462459y_real @ P3 @ nil_list_poly_real ) ).
% list_ex1_simps(1)
thf(fact_6_list__ex1__simps_I1_J,axiom,
! [P3: produc461822025y_real > $o] :
~ ( list_e1507538962y_real @ P3 @ nil_Pr1207979855y_real ) ).
% list_ex1_simps(1)
thf(fact_7_list__ex1__simps_I1_J,axiom,
! [P3: poly_real > $o] :
~ ( list_ex1_poly_real @ P3 @ nil_poly_real ) ).
% list_ex1_simps(1)
thf(fact_8_bind__simps_I1_J,axiom,
! [F: poly_real > list_poly_real] :
( ( bind_p420216945y_real @ nil_poly_real @ F )
= nil_poly_real ) ).
% bind_simps(1)
thf(fact_9_bind__simps_I1_J,axiom,
! [F: poly_real > list_list_poly_real] :
( ( bind_p1226679937y_real @ nil_poly_real @ F )
= nil_list_poly_real ) ).
% bind_simps(1)
thf(fact_10_bind__simps_I1_J,axiom,
! [F: list_poly_real > list_poly_real] :
( ( bind_l146959617y_real @ nil_list_poly_real @ F )
= nil_poly_real ) ).
% bind_simps(1)
thf(fact_11_bind__simps_I1_J,axiom,
! [F: poly_real > list_P736648811y_real] :
( ( bind_p600434664y_real @ nil_poly_real @ F )
= nil_Pr1492526187y_real ) ).
% bind_simps(1)
thf(fact_12_bind__simps_I1_J,axiom,
! [F: produc321349221y_real > list_poly_real] :
( ( bind_P755425362y_real @ nil_Pr1492526187y_real @ F )
= nil_poly_real ) ).
% bind_simps(1)
thf(fact_13_bind__simps_I1_J,axiom,
! [F: list_poly_real > list_list_poly_real] :
( ( bind_l1810664209y_real @ nil_list_poly_real @ F )
= nil_list_poly_real ) ).
% bind_simps(1)
thf(fact_14_bind__simps_I1_J,axiom,
! [F: poly_real > list_P693436111y_real] :
( ( bind_p988182092y_real @ nil_poly_real @ F )
= nil_Pr1207979855y_real ) ).
% bind_simps(1)
thf(fact_15_bind__simps_I1_J,axiom,
! [F: produc321349221y_real > list_list_poly_real] :
( ( bind_P1861132770y_real @ nil_Pr1492526187y_real @ F )
= nil_list_poly_real ) ).
% bind_simps(1)
thf(fact_16_bind__simps_I1_J,axiom,
! [F: list_poly_real > list_P736648811y_real] :
( ( bind_l429098840y_real @ nil_list_poly_real @ F )
= nil_Pr1492526187y_real ) ).
% bind_simps(1)
thf(fact_17_bind__simps_I1_J,axiom,
! [F: produc461822025y_real > list_poly_real] :
( ( bind_P120560694y_real @ nil_Pr1207979855y_real @ F )
= nil_poly_real ) ).
% bind_simps(1)
thf(fact_18_member__rec_I2_J,axiom,
! [Y: produc321349221y_real] :
~ ( member1782690048y_real @ nil_Pr1492526187y_real @ Y ) ).
% member_rec(2)
thf(fact_19_member__rec_I2_J,axiom,
! [Y: list_poly_real] :
~ ( member1829603177y_real @ nil_list_poly_real @ Y ) ).
% member_rec(2)
thf(fact_20_member__rec_I2_J,axiom,
! [Y: produc461822025y_real] :
~ ( member202023780y_real @ nil_Pr1207979855y_real @ Y ) ).
% member_rec(2)
thf(fact_21_member__rec_I2_J,axiom,
! [Y: poly_real] :
~ ( member_poly_real @ nil_poly_real @ Y ) ).
% member_rec(2)
thf(fact_22_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_le904455032y_real @ N @ nil_Pr1492526187y_real )
= N ) ).
% gen_length_code(1)
thf(fact_23_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_le1160603377y_real @ N @ nil_list_poly_real )
= N ) ).
% gen_length_code(1)
thf(fact_24_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_le1524582876y_real @ N @ nil_Pr1207979855y_real )
= N ) ).
% gen_length_code(1)
thf(fact_25_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_poly_real @ N @ nil_poly_real )
= N ) ).
% gen_length_code(1)
thf(fact_26_maps__simps_I2_J,axiom,
! [F: poly_real > list_poly_real] :
( ( maps_p961674027y_real @ F @ nil_poly_real )
= nil_poly_real ) ).
% maps_simps(2)
thf(fact_27_maps__simps_I2_J,axiom,
! [F: poly_real > list_list_poly_real] :
( ( maps_p1417280571y_real @ F @ nil_poly_real )
= nil_list_poly_real ) ).
% maps_simps(2)
thf(fact_28_maps__simps_I2_J,axiom,
! [F: list_poly_real > list_poly_real] :
( ( maps_l337560251y_real @ F @ nil_list_poly_real )
= nil_poly_real ) ).
% maps_simps(2)
thf(fact_29_maps__simps_I2_J,axiom,
! [F: poly_real > list_P736648811y_real] :
( ( maps_p2108677294y_real @ F @ nil_poly_real )
= nil_Pr1492526187y_real ) ).
% maps_simps(2)
thf(fact_30_maps__simps_I2_J,axiom,
! [F: produc321349221y_real > list_poly_real] :
( ( maps_P116184344y_real @ F @ nil_Pr1492526187y_real )
= nil_poly_real ) ).
% maps_simps(2)
thf(fact_31_maps__simps_I2_J,axiom,
! [F: list_poly_real > list_list_poly_real] :
( ( maps_l502643659y_real @ F @ nil_list_poly_real )
= nil_list_poly_real ) ).
% maps_simps(2)
thf(fact_32_maps__simps_I2_J,axiom,
! [F: poly_real > list_P693436111y_real] :
( ( maps_p86983698y_real @ F @ nil_poly_real )
= nil_Pr1207979855y_real ) ).
% maps_simps(2)
thf(fact_33_maps__simps_I2_J,axiom,
! [F: produc321349221y_real > list_list_poly_real] :
( ( maps_P54557608y_real @ F @ nil_Pr1492526187y_real )
= nil_list_poly_real ) ).
% maps_simps(2)
thf(fact_34_maps__simps_I2_J,axiom,
! [F: list_poly_real > list_P736648811y_real] :
( ( maps_l770007326y_real @ F @ nil_list_poly_real )
= nil_Pr1492526187y_real ) ).
% maps_simps(2)
thf(fact_35_maps__simps_I2_J,axiom,
! [F: produc461822025y_real > list_poly_real] :
( ( maps_P1366845948y_real @ F @ nil_Pr1207979855y_real )
= nil_poly_real ) ).
% maps_simps(2)
thf(fact_36_map__filter__simps_I2_J,axiom,
! [F: poly_real > option_poly_real] :
( ( map_fi1183202740y_real @ F @ nil_poly_real )
= nil_poly_real ) ).
% map_filter_simps(2)
thf(fact_37_map__filter__simps_I2_J,axiom,
! [F: poly_real > option1330723604y_real] :
( ( map_fi920577604y_real @ F @ nil_poly_real )
= nil_list_poly_real ) ).
% map_filter_simps(2)
thf(fact_38_map__filter__simps_I2_J,axiom,
! [F: list_poly_real > option_poly_real] :
( ( map_fi1988340932y_real @ F @ nil_list_poly_real )
= nil_poly_real ) ).
% map_filter_simps(2)
thf(fact_39_map__filter__simps_I2_J,axiom,
! [F: poly_real > option988260395y_real] :
( ( map_fi740944613y_real @ F @ nil_poly_real )
= nil_Pr1492526187y_real ) ).
% map_filter_simps(2)
thf(fact_40_map__filter__simps_I2_J,axiom,
! [F: produc321349221y_real > option_poly_real] :
( ( map_fi895935311y_real @ F @ nil_Pr1492526187y_real )
= nil_poly_real ) ).
% map_filter_simps(2)
thf(fact_41_map__filter__simps_I2_J,axiom,
! [F: list_poly_real > option1330723604y_real] :
( ( map_fi357666132y_real @ F @ nil_list_poly_real )
= nil_list_poly_real ) ).
% map_filter_simps(2)
thf(fact_42_map__filter__simps_I2_J,axiom,
! [F: poly_real > option293660815y_real] :
( ( map_fi1117002953y_real @ F @ nil_poly_real )
= nil_Pr1207979855y_real ) ).
% map_filter_simps(2)
thf(fact_43_map__filter__simps_I2_J,axiom,
! [F: produc321349221y_real > option1330723604y_real] :
( ( map_fi1389003359y_real @ F @ nil_Pr1492526187y_real )
= nil_list_poly_real ) ).
% map_filter_simps(2)
thf(fact_44_map__filter__simps_I2_J,axiom,
! [F: list_poly_real > option988260395y_real] :
( ( map_fi2104453077y_real @ F @ nil_list_poly_real )
= nil_Pr1492526187y_real ) ).
% map_filter_simps(2)
thf(fact_45_map__filter__simps_I2_J,axiom,
! [F: produc461822025y_real > option_poly_real] :
( ( map_fi249381555y_real @ F @ nil_Pr1207979855y_real )
= nil_poly_real ) ).
% map_filter_simps(2)
thf(fact_46_null__rec_I2_J,axiom,
null_P89569907y_real @ nil_Pr1492526187y_real ).
% null_rec(2)
thf(fact_47_null__rec_I2_J,axiom,
null_list_poly_real @ nil_list_poly_real ).
% null_rec(2)
thf(fact_48_null__rec_I2_J,axiom,
null_P1782853975y_real @ nil_Pr1207979855y_real ).
% null_rec(2)
thf(fact_49_null__rec_I2_J,axiom,
null_poly_real @ nil_poly_real ).
% null_rec(2)
thf(fact_50_eq__Nil__null,axiom,
! [Xs: list_P736648811y_real] :
( ( Xs = nil_Pr1492526187y_real )
= ( null_P89569907y_real @ Xs ) ) ).
% eq_Nil_null
thf(fact_51_eq__Nil__null,axiom,
! [Xs: list_list_poly_real] :
( ( Xs = nil_list_poly_real )
= ( null_list_poly_real @ Xs ) ) ).
% eq_Nil_null
thf(fact_52_eq__Nil__null,axiom,
! [Xs: list_P693436111y_real] :
( ( Xs = nil_Pr1207979855y_real )
= ( null_P1782853975y_real @ Xs ) ) ).
% eq_Nil_null
thf(fact_53_eq__Nil__null,axiom,
! [Xs: list_poly_real] :
( ( Xs = nil_poly_real )
= ( null_poly_real @ Xs ) ) ).
% eq_Nil_null
thf(fact_54_sturm__seq_Oaxioms_I2_J,axiom,
! [Ps: list_poly_real,P: poly_real] :
( ( sturm_1664866327rm_seq @ Ps @ P )
=> ( sturm_1683633594axioms @ Ps @ P ) ) ).
% sturm_seq.axioms(2)
thf(fact_55_list__ex__simps_I2_J,axiom,
! [P3: produc321349221y_real > $o] :
~ ( list_e2093379497y_real @ P3 @ nil_Pr1492526187y_real ) ).
% list_ex_simps(2)
thf(fact_56_list__ex__simps_I2_J,axiom,
! [P3: list_poly_real > $o] :
~ ( list_e809719936y_real @ P3 @ nil_list_poly_real ) ).
% list_ex_simps(2)
thf(fact_57_list__ex__simps_I2_J,axiom,
! [P3: produc461822025y_real > $o] :
~ ( list_e1312221069y_real @ P3 @ nil_Pr1207979855y_real ) ).
% list_ex_simps(2)
thf(fact_58_list__ex__simps_I2_J,axiom,
! [P3: poly_real > $o] :
~ ( list_ex_poly_real @ P3 @ nil_poly_real ) ).
% list_ex_simps(2)
thf(fact_59_rotate1__is__Nil__conv,axiom,
! [Xs: list_P736648811y_real] :
( ( ( rotate623875384y_real @ Xs )
= nil_Pr1492526187y_real )
= ( Xs = nil_Pr1492526187y_real ) ) ).
% rotate1_is_Nil_conv
thf(fact_60_rotate1__is__Nil__conv,axiom,
! [Xs: list_list_poly_real] :
( ( ( rotate717473073y_real @ Xs )
= nil_list_poly_real )
= ( Xs = nil_list_poly_real ) ) ).
% rotate1_is_Nil_conv
thf(fact_61_rotate1__is__Nil__conv,axiom,
! [Xs: list_P693436111y_real] :
( ( ( rotate867081116y_real @ Xs )
= nil_Pr1207979855y_real )
= ( Xs = nil_Pr1207979855y_real ) ) ).
% rotate1_is_Nil_conv
thf(fact_62_rotate1__is__Nil__conv,axiom,
! [Xs: list_poly_real] :
( ( ( rotate1_poly_real @ Xs )
= nil_poly_real )
= ( Xs = nil_poly_real ) ) ).
% rotate1_is_Nil_conv
thf(fact_63_split__Nil__iff,axiom,
! [Xs: list_P736648811y_real,Ys: list_P736648811y_real] :
( ( ( splice1383748542y_real @ Xs @ Ys )
= nil_Pr1492526187y_real )
= ( ( Xs = nil_Pr1492526187y_real )
& ( Ys = nil_Pr1492526187y_real ) ) ) ).
% split_Nil_iff
thf(fact_64_split__Nil__iff,axiom,
! [Xs: list_list_poly_real,Ys: list_list_poly_real] :
( ( ( splice2080530731y_real @ Xs @ Ys )
= nil_list_poly_real )
= ( ( Xs = nil_list_poly_real )
& ( Ys = nil_list_poly_real ) ) ) ).
% split_Nil_iff
thf(fact_65_split__Nil__iff,axiom,
! [Xs: list_P693436111y_real,Ys: list_P693436111y_real] :
( ( ( splice1321784098y_real @ Xs @ Ys )
= nil_Pr1207979855y_real )
= ( ( Xs = nil_Pr1207979855y_real )
& ( Ys = nil_Pr1207979855y_real ) ) ) ).
% split_Nil_iff
thf(fact_66_split__Nil__iff,axiom,
! [Xs: list_poly_real,Ys: list_poly_real] :
( ( ( splice_poly_real @ Xs @ Ys )
= nil_poly_real )
= ( ( Xs = nil_poly_real )
& ( Ys = nil_poly_real ) ) ) ).
% split_Nil_iff
thf(fact_67_splice__Nil2,axiom,
! [Xs: list_P736648811y_real] :
( ( splice1383748542y_real @ Xs @ nil_Pr1492526187y_real )
= Xs ) ).
% splice_Nil2
thf(fact_68_splice__Nil2,axiom,
! [Xs: list_list_poly_real] :
( ( splice2080530731y_real @ Xs @ nil_list_poly_real )
= Xs ) ).
% splice_Nil2
thf(fact_69_splice__Nil2,axiom,
! [Xs: list_P693436111y_real] :
( ( splice1321784098y_real @ Xs @ nil_Pr1207979855y_real )
= Xs ) ).
% splice_Nil2
thf(fact_70_splice__Nil2,axiom,
! [Xs: list_poly_real] :
( ( splice_poly_real @ Xs @ nil_poly_real )
= Xs ) ).
% splice_Nil2
thf(fact_71_enumerate__simps_I1_J,axiom,
! [N: nat] :
( ( enumer1123002698y_real @ N @ nil_Pr1492526187y_real )
= nil_Pr570652676y_real ) ).
% enumerate_simps(1)
thf(fact_72_enumerate__simps_I1_J,axiom,
! [N: nat] :
( ( enumer1433313439y_real @ N @ nil_list_poly_real )
= nil_Pr1161137915y_real ) ).
% enumerate_simps(1)
thf(fact_73_enumerate__simps_I1_J,axiom,
! [N: nat] :
( ( enumer1015405230y_real @ N @ nil_Pr1207979855y_real )
= nil_Pr2056488424y_real ) ).
% enumerate_simps(1)
thf(fact_74_enumerate__simps_I1_J,axiom,
! [N: nat] :
( ( enumerate_poly_real @ N @ nil_poly_real )
= nil_Pr1492526187y_real ) ).
% enumerate_simps(1)
thf(fact_75_product_Osimps_I1_J,axiom,
! [Uu: list_poly_real] :
( ( produc1360353227y_real @ nil_nat @ Uu )
= nil_Pr1492526187y_real ) ).
% product.simps(1)
thf(fact_76_product_Osimps_I1_J,axiom,
! [Uu: list_poly_real] :
( ( produc1733309743y_real @ nil_poly_real @ Uu )
= nil_Pr1207979855y_real ) ).
% product.simps(1)
thf(fact_77_map__tailrec__rev_Osimps_I1_J,axiom,
! [F: poly_real > poly_real,Bs: list_poly_real] :
( ( map_ta1919289823y_real @ F @ nil_poly_real @ Bs )
= Bs ) ).
% map_tailrec_rev.simps(1)
thf(fact_78_ord_Olexordp__eq__simps_I1_J,axiom,
! [Less: produc321349221y_real > produc321349221y_real > $o,Ys: list_P736648811y_real] : ( lexord834705585y_real @ Less @ nil_Pr1492526187y_real @ Ys ) ).
% ord.lexordp_eq_simps(1)
thf(fact_79_ord_Olexordp__eq__simps_I1_J,axiom,
! [Less: list_poly_real > list_poly_real > $o,Ys: list_list_poly_real] : ( lexord1324236152y_real @ Less @ nil_list_poly_real @ Ys ) ).
% ord.lexordp_eq_simps(1)
thf(fact_80_ord_Olexordp__eq__simps_I1_J,axiom,
! [Less: produc461822025y_real > produc461822025y_real > $o,Ys: list_P693436111y_real] : ( lexord1053351061y_real @ Less @ nil_Pr1207979855y_real @ Ys ) ).
% ord.lexordp_eq_simps(1)
thf(fact_81_ord_Olexordp__eq__simps_I1_J,axiom,
! [Less: poly_real > poly_real > $o,Ys: list_poly_real] : ( lexordp_eq_poly_real @ Less @ nil_poly_real @ Ys ) ).
% ord.lexordp_eq_simps(1)
thf(fact_82_ord_Olexordp__eq__simps_I2_J,axiom,
! [Less: produc321349221y_real > produc321349221y_real > $o,Xs: list_P736648811y_real] :
( ( lexord834705585y_real @ Less @ Xs @ nil_Pr1492526187y_real )
= ( Xs = nil_Pr1492526187y_real ) ) ).
% ord.lexordp_eq_simps(2)
thf(fact_83_ord_Olexordp__eq__simps_I2_J,axiom,
! [Less: list_poly_real > list_poly_real > $o,Xs: list_list_poly_real] :
( ( lexord1324236152y_real @ Less @ Xs @ nil_list_poly_real )
= ( Xs = nil_list_poly_real ) ) ).
% ord.lexordp_eq_simps(2)
thf(fact_84_ord_Olexordp__eq__simps_I2_J,axiom,
! [Less: produc461822025y_real > produc461822025y_real > $o,Xs: list_P693436111y_real] :
( ( lexord1053351061y_real @ Less @ Xs @ nil_Pr1207979855y_real )
= ( Xs = nil_Pr1207979855y_real ) ) ).
% ord.lexordp_eq_simps(2)
thf(fact_85_ord_Olexordp__eq__simps_I2_J,axiom,
! [Less: poly_real > poly_real > $o,Xs: list_poly_real] :
( ( lexordp_eq_poly_real @ Less @ Xs @ nil_poly_real )
= ( Xs = nil_poly_real ) ) ).
% ord.lexordp_eq_simps(2)
thf(fact_86_listrelp_ONil,axiom,
! [R: poly_real > poly_real > $o] : ( listre1609905961y_real @ R @ nil_poly_real @ nil_poly_real ) ).
% listrelp.Nil
thf(fact_87_listrelp_ONil,axiom,
! [R: poly_real > list_poly_real > $o] : ( listre1793073465y_real @ R @ nil_poly_real @ nil_list_poly_real ) ).
% listrelp.Nil
thf(fact_88_listrelp_ONil,axiom,
! [R: list_poly_real > poly_real > $o] : ( listre713353145y_real @ R @ nil_list_poly_real @ nil_poly_real ) ).
% listrelp.Nil
thf(fact_89_listrelp_ONil,axiom,
! [R: poly_real > produc321349221y_real > $o] : ( listre889482032y_real @ R @ nil_poly_real @ nil_Pr1492526187y_real ) ).
% listrelp.Nil
thf(fact_90_listrelp_ONil,axiom,
! [R: produc321349221y_real > poly_real > $o] : ( listre1044472730y_real @ R @ nil_Pr1492526187y_real @ nil_poly_real ) ).
% listrelp.Nil
thf(fact_91_listrelp_ONil,axiom,
! [R: list_poly_real > list_poly_real > $o] : ( listre1763068361y_real @ R @ nil_list_poly_real @ nil_list_poly_real ) ).
% listrelp.Nil
thf(fact_92_listrelp_ONil,axiom,
! [R: poly_real > produc461822025y_real > $o] : ( listre471999892y_real @ R @ nil_poly_real @ nil_Pr1207979855y_real ) ).
% listrelp.Nil
thf(fact_93_listrelp_ONil,axiom,
! [R: produc321349221y_real > list_poly_real > $o] : ( listre321577258y_real @ R @ nil_Pr1492526187y_real @ nil_list_poly_real ) ).
% listrelp.Nil
thf(fact_94_listrelp_ONil,axiom,
! [R: list_poly_real > produc321349221y_real > $o] : ( listre1037026976y_real @ R @ nil_list_poly_real @ nil_Pr1492526187y_real ) ).
% listrelp.Nil
thf(fact_95_listrelp_ONil,axiom,
! [R: produc461822025y_real > poly_real > $o] : ( listre1751862142y_real @ R @ nil_Pr1207979855y_real @ nil_poly_real ) ).
% listrelp.Nil
thf(fact_96_ord_Olexordp__eq_Ocong,axiom,
lexordp_eq_poly_real = lexordp_eq_poly_real ).
% ord.lexordp_eq.cong
thf(fact_97_ord_Olexordp__eq__refl,axiom,
! [Less: poly_real > poly_real > $o,Xs: list_poly_real] : ( lexordp_eq_poly_real @ Less @ Xs @ Xs ) ).
% ord.lexordp_eq_refl
thf(fact_98_ord_Olexordp__eq_ONil,axiom,
! [Less: produc321349221y_real > produc321349221y_real > $o,Ys: list_P736648811y_real] : ( lexord834705585y_real @ Less @ nil_Pr1492526187y_real @ Ys ) ).
% ord.lexordp_eq.Nil
thf(fact_99_ord_Olexordp__eq_ONil,axiom,
! [Less: list_poly_real > list_poly_real > $o,Ys: list_list_poly_real] : ( lexord1324236152y_real @ Less @ nil_list_poly_real @ Ys ) ).
% ord.lexordp_eq.Nil
thf(fact_100_ord_Olexordp__eq_ONil,axiom,
! [Less: produc461822025y_real > produc461822025y_real > $o,Ys: list_P693436111y_real] : ( lexord1053351061y_real @ Less @ nil_Pr1207979855y_real @ Ys ) ).
% ord.lexordp_eq.Nil
thf(fact_101_ord_Olexordp__eq_ONil,axiom,
! [Less: poly_real > poly_real > $o,Ys: list_poly_real] : ( lexordp_eq_poly_real @ Less @ nil_poly_real @ Ys ) ).
% ord.lexordp_eq.Nil
thf(fact_102_splice_Osimps_I1_J,axiom,
! [Ys: list_P736648811y_real] :
( ( splice1383748542y_real @ nil_Pr1492526187y_real @ Ys )
= Ys ) ).
% splice.simps(1)
thf(fact_103_splice_Osimps_I1_J,axiom,
! [Ys: list_list_poly_real] :
( ( splice2080530731y_real @ nil_list_poly_real @ Ys )
= Ys ) ).
% splice.simps(1)
thf(fact_104_splice_Osimps_I1_J,axiom,
! [Ys: list_P693436111y_real] :
( ( splice1321784098y_real @ nil_Pr1207979855y_real @ Ys )
= Ys ) ).
% splice.simps(1)
thf(fact_105_splice_Osimps_I1_J,axiom,
! [Ys: list_poly_real] :
( ( splice_poly_real @ nil_poly_real @ Ys )
= Ys ) ).
% splice.simps(1)
thf(fact_106_rotate1_Osimps_I1_J,axiom,
( ( rotate623875384y_real @ nil_Pr1492526187y_real )
= nil_Pr1492526187y_real ) ).
% rotate1.simps(1)
thf(fact_107_rotate1_Osimps_I1_J,axiom,
( ( rotate717473073y_real @ nil_list_poly_real )
= nil_list_poly_real ) ).
% rotate1.simps(1)
thf(fact_108_rotate1_Osimps_I1_J,axiom,
( ( rotate867081116y_real @ nil_Pr1207979855y_real )
= nil_Pr1207979855y_real ) ).
% rotate1.simps(1)
thf(fact_109_rotate1_Osimps_I1_J,axiom,
( ( rotate1_poly_real @ nil_poly_real )
= nil_poly_real ) ).
% rotate1.simps(1)
thf(fact_110_ord_Olexordp__eq__simps_I3_J,axiom,
! [Less: list_poly_real > list_poly_real > $o,X: list_poly_real,Xs: list_list_poly_real] :
~ ( lexord1324236152y_real @ Less @ ( cons_list_poly_real @ X @ Xs ) @ nil_list_poly_real ) ).
% ord.lexordp_eq_simps(3)
thf(fact_111_ord_Olexordp__eq__simps_I3_J,axiom,
! [Less: produc321349221y_real > produc321349221y_real > $o,X: produc321349221y_real,Xs: list_P736648811y_real] :
~ ( lexord834705585y_real @ Less @ ( cons_P1027346459y_real @ X @ Xs ) @ nil_Pr1492526187y_real ) ).
% ord.lexordp_eq_simps(3)
thf(fact_112_ord_Olexordp__eq__simps_I3_J,axiom,
! [Less: produc461822025y_real > produc461822025y_real > $o,X: produc461822025y_real,Xs: list_P693436111y_real] :
~ ( lexord1053351061y_real @ Less @ ( cons_P1129399039y_real @ X @ Xs ) @ nil_Pr1207979855y_real ) ).
% ord.lexordp_eq_simps(3)
thf(fact_113_ord_Olexordp__eq__simps_I3_J,axiom,
! [Less: poly_real > poly_real > $o,X: poly_real,Xs: list_poly_real] :
~ ( lexordp_eq_poly_real @ Less @ ( cons_poly_real @ X @ Xs ) @ nil_poly_real ) ).
% ord.lexordp_eq_simps(3)
thf(fact_114_zip__Nil,axiom,
! [Ys: list_poly_real] :
( ( zip_nat_poly_real @ nil_nat @ Ys )
= nil_Pr1492526187y_real ) ).
% zip_Nil
thf(fact_115_zip__Nil,axiom,
! [Ys: list_poly_real] :
( ( zip_po657915297y_real @ nil_poly_real @ Ys )
= nil_Pr1207979855y_real ) ).
% zip_Nil
thf(fact_116_map__tailrec__rev_Oelims,axiom,
! [X: poly_real > poly_real,Xa: list_poly_real,Xb: list_poly_real,Y: list_poly_real] :
( ( ( map_ta1919289823y_real @ X @ Xa @ Xb )
= Y )
=> ( ( ( Xa = nil_poly_real )
=> ( Y != Xb ) )
=> ~ ! [A: poly_real,As: list_poly_real] :
( ( Xa
= ( cons_poly_real @ A @ As ) )
=> ( Y
!= ( map_ta1919289823y_real @ X @ As @ ( cons_poly_real @ ( X @ A ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_117_map__tailrec__rev_Oelims,axiom,
! [X: poly_real > list_poly_real,Xa: list_poly_real,Xb: list_list_poly_real,Y: list_list_poly_real] :
( ( ( map_ta1513955567y_real @ X @ Xa @ Xb )
= Y )
=> ( ( ( Xa = nil_poly_real )
=> ( Y != Xb ) )
=> ~ ! [A: poly_real,As: list_poly_real] :
( ( Xa
= ( cons_poly_real @ A @ As ) )
=> ( Y
!= ( map_ta1513955567y_real @ X @ As @ ( cons_list_poly_real @ ( X @ A ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_118_map__tailrec__rev_Oelims,axiom,
! [X: list_poly_real > poly_real,Xa: list_list_poly_real,Xb: list_poly_real,Y: list_poly_real] :
( ( ( map_ta434235247y_real @ X @ Xa @ Xb )
= Y )
=> ( ( ( Xa = nil_list_poly_real )
=> ( Y != Xb ) )
=> ~ ! [A: list_poly_real,As: list_list_poly_real] :
( ( Xa
= ( cons_list_poly_real @ A @ As ) )
=> ( Y
!= ( map_ta434235247y_real @ X @ As @ ( cons_poly_real @ ( X @ A ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_119_map__tailrec__rev_Oelims,axiom,
! [X: poly_real > produc321349221y_real,Xa: list_poly_real,Xb: list_P736648811y_real,Y: list_P736648811y_real] :
( ( ( map_ta632757498y_real @ X @ Xa @ Xb )
= Y )
=> ( ( ( Xa = nil_poly_real )
=> ( Y != Xb ) )
=> ~ ! [A: poly_real,As: list_poly_real] :
( ( Xa
= ( cons_poly_real @ A @ As ) )
=> ( Y
!= ( map_ta632757498y_real @ X @ As @ ( cons_P1027346459y_real @ ( X @ A ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_120_map__tailrec__rev_Oelims,axiom,
! [X: list_poly_real > list_poly_real,Xa: list_list_poly_real,Xb: list_list_poly_real,Y: list_list_poly_real] :
( ( ( map_ta316995199y_real @ X @ Xa @ Xb )
= Y )
=> ( ( ( Xa = nil_list_poly_real )
=> ( Y != Xb ) )
=> ~ ! [A: list_poly_real,As: list_list_poly_real] :
( ( Xa
= ( cons_list_poly_real @ A @ As ) )
=> ( Y
!= ( map_ta316995199y_real @ X @ As @ ( cons_list_poly_real @ ( X @ A ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_121_map__tailrec__rev_Oelims,axiom,
! [X: produc321349221y_real > poly_real,Xa: list_P736648811y_real,Xb: list_poly_real,Y: list_poly_real] :
( ( ( map_ta787748196y_real @ X @ Xa @ Xb )
= Y )
=> ( ( ( Xa = nil_Pr1492526187y_real )
=> ( Y != Xb ) )
=> ~ ! [A: produc321349221y_real,As: list_P736648811y_real] :
( ( Xa
= ( cons_P1027346459y_real @ A @ As ) )
=> ( Y
!= ( map_ta787748196y_real @ X @ As @ ( cons_poly_real @ ( X @ A ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_122_map__tailrec__rev_Oelims,axiom,
! [X: poly_real > produc461822025y_real,Xa: list_poly_real,Xb: list_P693436111y_real,Y: list_P693436111y_real] :
( ( ( map_ta920933982y_real @ X @ Xa @ Xb )
= Y )
=> ( ( ( Xa = nil_poly_real )
=> ( Y != Xb ) )
=> ~ ! [A: poly_real,As: list_poly_real] :
( ( Xa
= ( cons_poly_real @ A @ As ) )
=> ( Y
!= ( map_ta920933982y_real @ X @ As @ ( cons_P1129399039y_real @ ( X @ A ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_123_map__tailrec__rev_Oelims,axiom,
! [X: list_poly_real > produc321349221y_real,Xa: list_list_poly_real,Xb: list_P736648811y_real,Y: list_P736648811y_real] :
( ( ( map_ta1674185578y_real @ X @ Xa @ Xb )
= Y )
=> ( ( ( Xa = nil_list_poly_real )
=> ( Y != Xb ) )
=> ~ ! [A: list_poly_real,As: list_list_poly_real] :
( ( Xa
= ( cons_list_poly_real @ A @ As ) )
=> ( Y
!= ( map_ta1674185578y_real @ X @ As @ ( cons_P1027346459y_real @ ( X @ A ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_124_map__tailrec__rev_Oelims,axiom,
! [X: produc321349221y_real > list_poly_real,Xa: list_P736648811y_real,Xb: list_list_poly_real,Y: list_list_poly_real] :
( ( ( map_ta958735860y_real @ X @ Xa @ Xb )
= Y )
=> ( ( ( Xa = nil_Pr1492526187y_real )
=> ( Y != Xb ) )
=> ~ ! [A: produc321349221y_real,As: list_P736648811y_real] :
( ( Xa
= ( cons_P1027346459y_real @ A @ As ) )
=> ( Y
!= ( map_ta958735860y_real @ X @ As @ ( cons_list_poly_real @ ( X @ A ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_125_map__tailrec__rev_Oelims,axiom,
! [X: produc461822025y_real > poly_real,Xa: list_P693436111y_real,Xb: list_poly_real,Y: list_poly_real] :
( ( ( map_ta53312584y_real @ X @ Xa @ Xb )
= Y )
=> ( ( ( Xa = nil_Pr1207979855y_real )
=> ( Y != Xb ) )
=> ~ ! [A: produc461822025y_real,As: list_P693436111y_real] :
( ( Xa
= ( cons_P1129399039y_real @ A @ As ) )
=> ( Y
!= ( map_ta53312584y_real @ X @ As @ ( cons_poly_real @ ( X @ A ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_126_listrelp_Oinducts,axiom,
! [R: poly_real > poly_real > $o,X1: list_poly_real,X2: list_poly_real,P3: list_poly_real > list_poly_real > $o] :
( ( listre1609905961y_real @ R @ X1 @ X2 )
=> ( ( P3 @ nil_poly_real @ nil_poly_real )
=> ( ! [X3: poly_real,Y2: poly_real,Xs2: list_poly_real,Ys2: list_poly_real] :
( ( R @ X3 @ Y2 )
=> ( ( listre1609905961y_real @ R @ Xs2 @ Ys2 )
=> ( ( P3 @ Xs2 @ Ys2 )
=> ( P3 @ ( cons_poly_real @ X3 @ Xs2 ) @ ( cons_poly_real @ Y2 @ Ys2 ) ) ) ) )
=> ( P3 @ X1 @ X2 ) ) ) ) ).
% listrelp.inducts
thf(fact_127_listrelp_Oinducts,axiom,
! [R: poly_real > list_poly_real > $o,X1: list_poly_real,X2: list_list_poly_real,P3: list_poly_real > list_list_poly_real > $o] :
( ( listre1793073465y_real @ R @ X1 @ X2 )
=> ( ( P3 @ nil_poly_real @ nil_list_poly_real )
=> ( ! [X3: poly_real,Y2: list_poly_real,Xs2: list_poly_real,Ys2: list_list_poly_real] :
( ( R @ X3 @ Y2 )
=> ( ( listre1793073465y_real @ R @ Xs2 @ Ys2 )
=> ( ( P3 @ Xs2 @ Ys2 )
=> ( P3 @ ( cons_poly_real @ X3 @ Xs2 ) @ ( cons_list_poly_real @ Y2 @ Ys2 ) ) ) ) )
=> ( P3 @ X1 @ X2 ) ) ) ) ).
% listrelp.inducts
thf(fact_128_listrelp_Oinducts,axiom,
! [R: list_poly_real > poly_real > $o,X1: list_list_poly_real,X2: list_poly_real,P3: list_list_poly_real > list_poly_real > $o] :
( ( listre713353145y_real @ R @ X1 @ X2 )
=> ( ( P3 @ nil_list_poly_real @ nil_poly_real )
=> ( ! [X3: list_poly_real,Y2: poly_real,Xs2: list_list_poly_real,Ys2: list_poly_real] :
( ( R @ X3 @ Y2 )
=> ( ( listre713353145y_real @ R @ Xs2 @ Ys2 )
=> ( ( P3 @ Xs2 @ Ys2 )
=> ( P3 @ ( cons_list_poly_real @ X3 @ Xs2 ) @ ( cons_poly_real @ Y2 @ Ys2 ) ) ) ) )
=> ( P3 @ X1 @ X2 ) ) ) ) ).
% listrelp.inducts
thf(fact_129_listrelp_Oinducts,axiom,
! [R: poly_real > produc321349221y_real > $o,X1: list_poly_real,X2: list_P736648811y_real,P3: list_poly_real > list_P736648811y_real > $o] :
( ( listre889482032y_real @ R @ X1 @ X2 )
=> ( ( P3 @ nil_poly_real @ nil_Pr1492526187y_real )
=> ( ! [X3: poly_real,Y2: produc321349221y_real,Xs2: list_poly_real,Ys2: list_P736648811y_real] :
( ( R @ X3 @ Y2 )
=> ( ( listre889482032y_real @ R @ Xs2 @ Ys2 )
=> ( ( P3 @ Xs2 @ Ys2 )
=> ( P3 @ ( cons_poly_real @ X3 @ Xs2 ) @ ( cons_P1027346459y_real @ Y2 @ Ys2 ) ) ) ) )
=> ( P3 @ X1 @ X2 ) ) ) ) ).
% listrelp.inducts
thf(fact_130_listrelp_Oinducts,axiom,
! [R: list_poly_real > list_poly_real > $o,X1: list_list_poly_real,X2: list_list_poly_real,P3: list_list_poly_real > list_list_poly_real > $o] :
( ( listre1763068361y_real @ R @ X1 @ X2 )
=> ( ( P3 @ nil_list_poly_real @ nil_list_poly_real )
=> ( ! [X3: list_poly_real,Y2: list_poly_real,Xs2: list_list_poly_real,Ys2: list_list_poly_real] :
( ( R @ X3 @ Y2 )
=> ( ( listre1763068361y_real @ R @ Xs2 @ Ys2 )
=> ( ( P3 @ Xs2 @ Ys2 )
=> ( P3 @ ( cons_list_poly_real @ X3 @ Xs2 ) @ ( cons_list_poly_real @ Y2 @ Ys2 ) ) ) ) )
=> ( P3 @ X1 @ X2 ) ) ) ) ).
% listrelp.inducts
thf(fact_131_listrelp_Oinducts,axiom,
! [R: produc321349221y_real > poly_real > $o,X1: list_P736648811y_real,X2: list_poly_real,P3: list_P736648811y_real > list_poly_real > $o] :
( ( listre1044472730y_real @ R @ X1 @ X2 )
=> ( ( P3 @ nil_Pr1492526187y_real @ nil_poly_real )
=> ( ! [X3: produc321349221y_real,Y2: poly_real,Xs2: list_P736648811y_real,Ys2: list_poly_real] :
( ( R @ X3 @ Y2 )
=> ( ( listre1044472730y_real @ R @ Xs2 @ Ys2 )
=> ( ( P3 @ Xs2 @ Ys2 )
=> ( P3 @ ( cons_P1027346459y_real @ X3 @ Xs2 ) @ ( cons_poly_real @ Y2 @ Ys2 ) ) ) ) )
=> ( P3 @ X1 @ X2 ) ) ) ) ).
% listrelp.inducts
thf(fact_132_listrelp_Oinducts,axiom,
! [R: poly_real > produc461822025y_real > $o,X1: list_poly_real,X2: list_P693436111y_real,P3: list_poly_real > list_P693436111y_real > $o] :
( ( listre471999892y_real @ R @ X1 @ X2 )
=> ( ( P3 @ nil_poly_real @ nil_Pr1207979855y_real )
=> ( ! [X3: poly_real,Y2: produc461822025y_real,Xs2: list_poly_real,Ys2: list_P693436111y_real] :
( ( R @ X3 @ Y2 )
=> ( ( listre471999892y_real @ R @ Xs2 @ Ys2 )
=> ( ( P3 @ Xs2 @ Ys2 )
=> ( P3 @ ( cons_poly_real @ X3 @ Xs2 ) @ ( cons_P1129399039y_real @ Y2 @ Ys2 ) ) ) ) )
=> ( P3 @ X1 @ X2 ) ) ) ) ).
% listrelp.inducts
thf(fact_133_listrelp_Oinducts,axiom,
! [R: list_poly_real > produc321349221y_real > $o,X1: list_list_poly_real,X2: list_P736648811y_real,P3: list_list_poly_real > list_P736648811y_real > $o] :
( ( listre1037026976y_real @ R @ X1 @ X2 )
=> ( ( P3 @ nil_list_poly_real @ nil_Pr1492526187y_real )
=> ( ! [X3: list_poly_real,Y2: produc321349221y_real,Xs2: list_list_poly_real,Ys2: list_P736648811y_real] :
( ( R @ X3 @ Y2 )
=> ( ( listre1037026976y_real @ R @ Xs2 @ Ys2 )
=> ( ( P3 @ Xs2 @ Ys2 )
=> ( P3 @ ( cons_list_poly_real @ X3 @ Xs2 ) @ ( cons_P1027346459y_real @ Y2 @ Ys2 ) ) ) ) )
=> ( P3 @ X1 @ X2 ) ) ) ) ).
% listrelp.inducts
thf(fact_134_listrelp_Oinducts,axiom,
! [R: produc321349221y_real > list_poly_real > $o,X1: list_P736648811y_real,X2: list_list_poly_real,P3: list_P736648811y_real > list_list_poly_real > $o] :
( ( listre321577258y_real @ R @ X1 @ X2 )
=> ( ( P3 @ nil_Pr1492526187y_real @ nil_list_poly_real )
=> ( ! [X3: produc321349221y_real,Y2: list_poly_real,Xs2: list_P736648811y_real,Ys2: list_list_poly_real] :
( ( R @ X3 @ Y2 )
=> ( ( listre321577258y_real @ R @ Xs2 @ Ys2 )
=> ( ( P3 @ Xs2 @ Ys2 )
=> ( P3 @ ( cons_P1027346459y_real @ X3 @ Xs2 ) @ ( cons_list_poly_real @ Y2 @ Ys2 ) ) ) ) )
=> ( P3 @ X1 @ X2 ) ) ) ) ).
% listrelp.inducts
thf(fact_135_listrelp_Oinducts,axiom,
! [R: produc461822025y_real > poly_real > $o,X1: list_P693436111y_real,X2: list_poly_real,P3: list_P693436111y_real > list_poly_real > $o] :
( ( listre1751862142y_real @ R @ X1 @ X2 )
=> ( ( P3 @ nil_Pr1207979855y_real @ nil_poly_real )
=> ( ! [X3: produc461822025y_real,Y2: poly_real,Xs2: list_P693436111y_real,Ys2: list_poly_real] :
( ( R @ X3 @ Y2 )
=> ( ( listre1751862142y_real @ R @ Xs2 @ Ys2 )
=> ( ( P3 @ Xs2 @ Ys2 )
=> ( P3 @ ( cons_P1129399039y_real @ X3 @ Xs2 ) @ ( cons_poly_real @ Y2 @ Ys2 ) ) ) ) )
=> ( P3 @ X1 @ X2 ) ) ) ) ).
% listrelp.inducts
thf(fact_136_listrelp_Osimps,axiom,
( listre1609905961y_real
= ( ^ [R2: poly_real > poly_real > $o,A1: list_poly_real,A2: list_poly_real] :
( ( ( A1 = nil_poly_real )
& ( A2 = nil_poly_real ) )
| ? [X4: poly_real,Y3: poly_real,Xs3: list_poly_real,Ys3: list_poly_real] :
( ( A1
= ( cons_poly_real @ X4 @ Xs3 ) )
& ( A2
= ( cons_poly_real @ Y3 @ Ys3 ) )
& ( R2 @ X4 @ Y3 )
& ( listre1609905961y_real @ R2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% listrelp.simps
thf(fact_137_listrelp_Osimps,axiom,
( listre1793073465y_real
= ( ^ [R2: poly_real > list_poly_real > $o,A1: list_poly_real,A2: list_list_poly_real] :
( ( ( A1 = nil_poly_real )
& ( A2 = nil_list_poly_real ) )
| ? [X4: poly_real,Y3: list_poly_real,Xs3: list_poly_real,Ys3: list_list_poly_real] :
( ( A1
= ( cons_poly_real @ X4 @ Xs3 ) )
& ( A2
= ( cons_list_poly_real @ Y3 @ Ys3 ) )
& ( R2 @ X4 @ Y3 )
& ( listre1793073465y_real @ R2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% listrelp.simps
thf(fact_138_listrelp_Osimps,axiom,
( listre713353145y_real
= ( ^ [R2: list_poly_real > poly_real > $o,A1: list_list_poly_real,A2: list_poly_real] :
( ( ( A1 = nil_list_poly_real )
& ( A2 = nil_poly_real ) )
| ? [X4: list_poly_real,Y3: poly_real,Xs3: list_list_poly_real,Ys3: list_poly_real] :
( ( A1
= ( cons_list_poly_real @ X4 @ Xs3 ) )
& ( A2
= ( cons_poly_real @ Y3 @ Ys3 ) )
& ( R2 @ X4 @ Y3 )
& ( listre713353145y_real @ R2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% listrelp.simps
thf(fact_139_listrelp_Osimps,axiom,
( listre889482032y_real
= ( ^ [R2: poly_real > produc321349221y_real > $o,A1: list_poly_real,A2: list_P736648811y_real] :
( ( ( A1 = nil_poly_real )
& ( A2 = nil_Pr1492526187y_real ) )
| ? [X4: poly_real,Y3: produc321349221y_real,Xs3: list_poly_real,Ys3: list_P736648811y_real] :
( ( A1
= ( cons_poly_real @ X4 @ Xs3 ) )
& ( A2
= ( cons_P1027346459y_real @ Y3 @ Ys3 ) )
& ( R2 @ X4 @ Y3 )
& ( listre889482032y_real @ R2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% listrelp.simps
thf(fact_140_listrelp_Osimps,axiom,
( listre1763068361y_real
= ( ^ [R2: list_poly_real > list_poly_real > $o,A1: list_list_poly_real,A2: list_list_poly_real] :
( ( ( A1 = nil_list_poly_real )
& ( A2 = nil_list_poly_real ) )
| ? [X4: list_poly_real,Y3: list_poly_real,Xs3: list_list_poly_real,Ys3: list_list_poly_real] :
( ( A1
= ( cons_list_poly_real @ X4 @ Xs3 ) )
& ( A2
= ( cons_list_poly_real @ Y3 @ Ys3 ) )
& ( R2 @ X4 @ Y3 )
& ( listre1763068361y_real @ R2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% listrelp.simps
thf(fact_141_listrelp_Osimps,axiom,
( listre1044472730y_real
= ( ^ [R2: produc321349221y_real > poly_real > $o,A1: list_P736648811y_real,A2: list_poly_real] :
( ( ( A1 = nil_Pr1492526187y_real )
& ( A2 = nil_poly_real ) )
| ? [X4: produc321349221y_real,Y3: poly_real,Xs3: list_P736648811y_real,Ys3: list_poly_real] :
( ( A1
= ( cons_P1027346459y_real @ X4 @ Xs3 ) )
& ( A2
= ( cons_poly_real @ Y3 @ Ys3 ) )
& ( R2 @ X4 @ Y3 )
& ( listre1044472730y_real @ R2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% listrelp.simps
thf(fact_142_listrelp_Osimps,axiom,
( listre471999892y_real
= ( ^ [R2: poly_real > produc461822025y_real > $o,A1: list_poly_real,A2: list_P693436111y_real] :
( ( ( A1 = nil_poly_real )
& ( A2 = nil_Pr1207979855y_real ) )
| ? [X4: poly_real,Y3: produc461822025y_real,Xs3: list_poly_real,Ys3: list_P693436111y_real] :
( ( A1
= ( cons_poly_real @ X4 @ Xs3 ) )
& ( A2
= ( cons_P1129399039y_real @ Y3 @ Ys3 ) )
& ( R2 @ X4 @ Y3 )
& ( listre471999892y_real @ R2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% listrelp.simps
thf(fact_143_listrelp_Osimps,axiom,
( listre1037026976y_real
= ( ^ [R2: list_poly_real > produc321349221y_real > $o,A1: list_list_poly_real,A2: list_P736648811y_real] :
( ( ( A1 = nil_list_poly_real )
& ( A2 = nil_Pr1492526187y_real ) )
| ? [X4: list_poly_real,Y3: produc321349221y_real,Xs3: list_list_poly_real,Ys3: list_P736648811y_real] :
( ( A1
= ( cons_list_poly_real @ X4 @ Xs3 ) )
& ( A2
= ( cons_P1027346459y_real @ Y3 @ Ys3 ) )
& ( R2 @ X4 @ Y3 )
& ( listre1037026976y_real @ R2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% listrelp.simps
thf(fact_144_listrelp_Osimps,axiom,
( listre321577258y_real
= ( ^ [R2: produc321349221y_real > list_poly_real > $o,A1: list_P736648811y_real,A2: list_list_poly_real] :
( ( ( A1 = nil_Pr1492526187y_real )
& ( A2 = nil_list_poly_real ) )
| ? [X4: produc321349221y_real,Y3: list_poly_real,Xs3: list_P736648811y_real,Ys3: list_list_poly_real] :
( ( A1
= ( cons_P1027346459y_real @ X4 @ Xs3 ) )
& ( A2
= ( cons_list_poly_real @ Y3 @ Ys3 ) )
& ( R2 @ X4 @ Y3 )
& ( listre321577258y_real @ R2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% listrelp.simps
thf(fact_145_listrelp_Osimps,axiom,
( listre1751862142y_real
= ( ^ [R2: produc461822025y_real > poly_real > $o,A1: list_P693436111y_real,A2: list_poly_real] :
( ( ( A1 = nil_Pr1207979855y_real )
& ( A2 = nil_poly_real ) )
| ? [X4: produc461822025y_real,Y3: poly_real,Xs3: list_P693436111y_real,Ys3: list_poly_real] :
( ( A1
= ( cons_P1129399039y_real @ X4 @ Xs3 ) )
& ( A2
= ( cons_poly_real @ Y3 @ Ys3 ) )
& ( R2 @ X4 @ Y3 )
& ( listre1751862142y_real @ R2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% listrelp.simps
thf(fact_146_listrelp_Ocases,axiom,
! [R: poly_real > poly_real > $o,A12: list_poly_real,A22: list_poly_real] :
( ( listre1609905961y_real @ R @ A12 @ A22 )
=> ( ( ( A12 = nil_poly_real )
=> ( A22 != nil_poly_real ) )
=> ~ ! [X3: poly_real,Y2: poly_real,Xs2: list_poly_real] :
( ( A12
= ( cons_poly_real @ X3 @ Xs2 ) )
=> ! [Ys2: list_poly_real] :
( ( A22
= ( cons_poly_real @ Y2 @ Ys2 ) )
=> ( ( R @ X3 @ Y2 )
=> ~ ( listre1609905961y_real @ R @ Xs2 @ Ys2 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_147_listrelp_Ocases,axiom,
! [R: poly_real > list_poly_real > $o,A12: list_poly_real,A22: list_list_poly_real] :
( ( listre1793073465y_real @ R @ A12 @ A22 )
=> ( ( ( A12 = nil_poly_real )
=> ( A22 != nil_list_poly_real ) )
=> ~ ! [X3: poly_real,Y2: list_poly_real,Xs2: list_poly_real] :
( ( A12
= ( cons_poly_real @ X3 @ Xs2 ) )
=> ! [Ys2: list_list_poly_real] :
( ( A22
= ( cons_list_poly_real @ Y2 @ Ys2 ) )
=> ( ( R @ X3 @ Y2 )
=> ~ ( listre1793073465y_real @ R @ Xs2 @ Ys2 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_148_listrelp_Ocases,axiom,
! [R: list_poly_real > poly_real > $o,A12: list_list_poly_real,A22: list_poly_real] :
( ( listre713353145y_real @ R @ A12 @ A22 )
=> ( ( ( A12 = nil_list_poly_real )
=> ( A22 != nil_poly_real ) )
=> ~ ! [X3: list_poly_real,Y2: poly_real,Xs2: list_list_poly_real] :
( ( A12
= ( cons_list_poly_real @ X3 @ Xs2 ) )
=> ! [Ys2: list_poly_real] :
( ( A22
= ( cons_poly_real @ Y2 @ Ys2 ) )
=> ( ( R @ X3 @ Y2 )
=> ~ ( listre713353145y_real @ R @ Xs2 @ Ys2 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_149_listrelp_Ocases,axiom,
! [R: poly_real > produc321349221y_real > $o,A12: list_poly_real,A22: list_P736648811y_real] :
( ( listre889482032y_real @ R @ A12 @ A22 )
=> ( ( ( A12 = nil_poly_real )
=> ( A22 != nil_Pr1492526187y_real ) )
=> ~ ! [X3: poly_real,Y2: produc321349221y_real,Xs2: list_poly_real] :
( ( A12
= ( cons_poly_real @ X3 @ Xs2 ) )
=> ! [Ys2: list_P736648811y_real] :
( ( A22
= ( cons_P1027346459y_real @ Y2 @ Ys2 ) )
=> ( ( R @ X3 @ Y2 )
=> ~ ( listre889482032y_real @ R @ Xs2 @ Ys2 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_150_listrelp_Ocases,axiom,
! [R: list_poly_real > list_poly_real > $o,A12: list_list_poly_real,A22: list_list_poly_real] :
( ( listre1763068361y_real @ R @ A12 @ A22 )
=> ( ( ( A12 = nil_list_poly_real )
=> ( A22 != nil_list_poly_real ) )
=> ~ ! [X3: list_poly_real,Y2: list_poly_real,Xs2: list_list_poly_real] :
( ( A12
= ( cons_list_poly_real @ X3 @ Xs2 ) )
=> ! [Ys2: list_list_poly_real] :
( ( A22
= ( cons_list_poly_real @ Y2 @ Ys2 ) )
=> ( ( R @ X3 @ Y2 )
=> ~ ( listre1763068361y_real @ R @ Xs2 @ Ys2 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_151_listrelp_Ocases,axiom,
! [R: produc321349221y_real > poly_real > $o,A12: list_P736648811y_real,A22: list_poly_real] :
( ( listre1044472730y_real @ R @ A12 @ A22 )
=> ( ( ( A12 = nil_Pr1492526187y_real )
=> ( A22 != nil_poly_real ) )
=> ~ ! [X3: produc321349221y_real,Y2: poly_real,Xs2: list_P736648811y_real] :
( ( A12
= ( cons_P1027346459y_real @ X3 @ Xs2 ) )
=> ! [Ys2: list_poly_real] :
( ( A22
= ( cons_poly_real @ Y2 @ Ys2 ) )
=> ( ( R @ X3 @ Y2 )
=> ~ ( listre1044472730y_real @ R @ Xs2 @ Ys2 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_152_listrelp_Ocases,axiom,
! [R: poly_real > produc461822025y_real > $o,A12: list_poly_real,A22: list_P693436111y_real] :
( ( listre471999892y_real @ R @ A12 @ A22 )
=> ( ( ( A12 = nil_poly_real )
=> ( A22 != nil_Pr1207979855y_real ) )
=> ~ ! [X3: poly_real,Y2: produc461822025y_real,Xs2: list_poly_real] :
( ( A12
= ( cons_poly_real @ X3 @ Xs2 ) )
=> ! [Ys2: list_P693436111y_real] :
( ( A22
= ( cons_P1129399039y_real @ Y2 @ Ys2 ) )
=> ( ( R @ X3 @ Y2 )
=> ~ ( listre471999892y_real @ R @ Xs2 @ Ys2 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_153_listrelp_Ocases,axiom,
! [R: list_poly_real > produc321349221y_real > $o,A12: list_list_poly_real,A22: list_P736648811y_real] :
( ( listre1037026976y_real @ R @ A12 @ A22 )
=> ( ( ( A12 = nil_list_poly_real )
=> ( A22 != nil_Pr1492526187y_real ) )
=> ~ ! [X3: list_poly_real,Y2: produc321349221y_real,Xs2: list_list_poly_real] :
( ( A12
= ( cons_list_poly_real @ X3 @ Xs2 ) )
=> ! [Ys2: list_P736648811y_real] :
( ( A22
= ( cons_P1027346459y_real @ Y2 @ Ys2 ) )
=> ( ( R @ X3 @ Y2 )
=> ~ ( listre1037026976y_real @ R @ Xs2 @ Ys2 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_154_listrelp_Ocases,axiom,
! [R: produc321349221y_real > list_poly_real > $o,A12: list_P736648811y_real,A22: list_list_poly_real] :
( ( listre321577258y_real @ R @ A12 @ A22 )
=> ( ( ( A12 = nil_Pr1492526187y_real )
=> ( A22 != nil_list_poly_real ) )
=> ~ ! [X3: produc321349221y_real,Y2: list_poly_real,Xs2: list_P736648811y_real] :
( ( A12
= ( cons_P1027346459y_real @ X3 @ Xs2 ) )
=> ! [Ys2: list_list_poly_real] :
( ( A22
= ( cons_list_poly_real @ Y2 @ Ys2 ) )
=> ( ( R @ X3 @ Y2 )
=> ~ ( listre321577258y_real @ R @ Xs2 @ Ys2 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_155_listrelp_Ocases,axiom,
! [R: produc461822025y_real > poly_real > $o,A12: list_P693436111y_real,A22: list_poly_real] :
( ( listre1751862142y_real @ R @ A12 @ A22 )
=> ( ( ( A12 = nil_Pr1207979855y_real )
=> ( A22 != nil_poly_real ) )
=> ~ ! [X3: produc461822025y_real,Y2: poly_real,Xs2: list_P693436111y_real] :
( ( A12
= ( cons_P1129399039y_real @ X3 @ Xs2 ) )
=> ! [Ys2: list_poly_real] :
( ( A22
= ( cons_poly_real @ Y2 @ Ys2 ) )
=> ( ( R @ X3 @ Y2 )
=> ~ ( listre1751862142y_real @ R @ Xs2 @ Ys2 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_156_splice_Oelims,axiom,
! [X: list_list_poly_real,Xa: list_list_poly_real,Y: list_list_poly_real] :
( ( ( splice2080530731y_real @ X @ Xa )
= Y )
=> ( ( ( X = nil_list_poly_real )
=> ( Y != Xa ) )
=> ~ ! [X3: list_poly_real,Xs2: list_list_poly_real] :
( ( X
= ( cons_list_poly_real @ X3 @ Xs2 ) )
=> ( Y
!= ( cons_list_poly_real @ X3 @ ( splice2080530731y_real @ Xa @ Xs2 ) ) ) ) ) ) ).
% splice.elims
thf(fact_157_splice_Oelims,axiom,
! [X: list_P736648811y_real,Xa: list_P736648811y_real,Y: list_P736648811y_real] :
( ( ( splice1383748542y_real @ X @ Xa )
= Y )
=> ( ( ( X = nil_Pr1492526187y_real )
=> ( Y != Xa ) )
=> ~ ! [X3: produc321349221y_real,Xs2: list_P736648811y_real] :
( ( X
= ( cons_P1027346459y_real @ X3 @ Xs2 ) )
=> ( Y
!= ( cons_P1027346459y_real @ X3 @ ( splice1383748542y_real @ Xa @ Xs2 ) ) ) ) ) ) ).
% splice.elims
thf(fact_158_splice_Oelims,axiom,
! [X: list_P693436111y_real,Xa: list_P693436111y_real,Y: list_P693436111y_real] :
( ( ( splice1321784098y_real @ X @ Xa )
= Y )
=> ( ( ( X = nil_Pr1207979855y_real )
=> ( Y != Xa ) )
=> ~ ! [X3: produc461822025y_real,Xs2: list_P693436111y_real] :
( ( X
= ( cons_P1129399039y_real @ X3 @ Xs2 ) )
=> ( Y
!= ( cons_P1129399039y_real @ X3 @ ( splice1321784098y_real @ Xa @ Xs2 ) ) ) ) ) ) ).
% splice.elims
thf(fact_159_splice_Oelims,axiom,
! [X: list_poly_real,Xa: list_poly_real,Y: list_poly_real] :
( ( ( splice_poly_real @ X @ Xa )
= Y )
=> ( ( ( X = nil_poly_real )
=> ( Y != Xa ) )
=> ~ ! [X3: poly_real,Xs2: list_poly_real] :
( ( X
= ( cons_poly_real @ X3 @ Xs2 ) )
=> ( Y
!= ( cons_poly_real @ X3 @ ( splice_poly_real @ Xa @ Xs2 ) ) ) ) ) ) ).
% splice.elims
thf(fact_160_ord_Olexordp__eq_Oinducts,axiom,
! [Less: list_poly_real > list_poly_real > $o,X1: list_list_poly_real,X2: list_list_poly_real,P3: list_list_poly_real > list_list_poly_real > $o] :
( ( lexord1324236152y_real @ Less @ X1 @ X2 )
=> ( ! [X_1: list_list_poly_real] : ( P3 @ nil_list_poly_real @ X_1 )
=> ( ! [X3: list_poly_real,Y2: list_poly_real,Xs2: list_list_poly_real,Ys2: list_list_poly_real] :
( ( Less @ X3 @ Y2 )
=> ( P3 @ ( cons_list_poly_real @ X3 @ Xs2 ) @ ( cons_list_poly_real @ Y2 @ Ys2 ) ) )
=> ( ! [X3: list_poly_real,Y2: list_poly_real,Xs2: list_list_poly_real,Ys2: list_list_poly_real] :
( ~ ( Less @ X3 @ Y2 )
=> ( ~ ( Less @ Y2 @ X3 )
=> ( ( lexord1324236152y_real @ Less @ Xs2 @ Ys2 )
=> ( ( P3 @ Xs2 @ Ys2 )
=> ( P3 @ ( cons_list_poly_real @ X3 @ Xs2 ) @ ( cons_list_poly_real @ Y2 @ Ys2 ) ) ) ) ) )
=> ( P3 @ X1 @ X2 ) ) ) ) ) ).
% ord.lexordp_eq.inducts
thf(fact_161_ord_Olexordp__eq_Oinducts,axiom,
! [Less: produc321349221y_real > produc321349221y_real > $o,X1: list_P736648811y_real,X2: list_P736648811y_real,P3: list_P736648811y_real > list_P736648811y_real > $o] :
( ( lexord834705585y_real @ Less @ X1 @ X2 )
=> ( ! [X_1: list_P736648811y_real] : ( P3 @ nil_Pr1492526187y_real @ X_1 )
=> ( ! [X3: produc321349221y_real,Y2: produc321349221y_real,Xs2: list_P736648811y_real,Ys2: list_P736648811y_real] :
( ( Less @ X3 @ Y2 )
=> ( P3 @ ( cons_P1027346459y_real @ X3 @ Xs2 ) @ ( cons_P1027346459y_real @ Y2 @ Ys2 ) ) )
=> ( ! [X3: produc321349221y_real,Y2: produc321349221y_real,Xs2: list_P736648811y_real,Ys2: list_P736648811y_real] :
( ~ ( Less @ X3 @ Y2 )
=> ( ~ ( Less @ Y2 @ X3 )
=> ( ( lexord834705585y_real @ Less @ Xs2 @ Ys2 )
=> ( ( P3 @ Xs2 @ Ys2 )
=> ( P3 @ ( cons_P1027346459y_real @ X3 @ Xs2 ) @ ( cons_P1027346459y_real @ Y2 @ Ys2 ) ) ) ) ) )
=> ( P3 @ X1 @ X2 ) ) ) ) ) ).
% ord.lexordp_eq.inducts
thf(fact_162_ord_Olexordp__eq_Oinducts,axiom,
! [Less: produc461822025y_real > produc461822025y_real > $o,X1: list_P693436111y_real,X2: list_P693436111y_real,P3: list_P693436111y_real > list_P693436111y_real > $o] :
( ( lexord1053351061y_real @ Less @ X1 @ X2 )
=> ( ! [X_1: list_P693436111y_real] : ( P3 @ nil_Pr1207979855y_real @ X_1 )
=> ( ! [X3: produc461822025y_real,Y2: produc461822025y_real,Xs2: list_P693436111y_real,Ys2: list_P693436111y_real] :
( ( Less @ X3 @ Y2 )
=> ( P3 @ ( cons_P1129399039y_real @ X3 @ Xs2 ) @ ( cons_P1129399039y_real @ Y2 @ Ys2 ) ) )
=> ( ! [X3: produc461822025y_real,Y2: produc461822025y_real,Xs2: list_P693436111y_real,Ys2: list_P693436111y_real] :
( ~ ( Less @ X3 @ Y2 )
=> ( ~ ( Less @ Y2 @ X3 )
=> ( ( lexord1053351061y_real @ Less @ Xs2 @ Ys2 )
=> ( ( P3 @ Xs2 @ Ys2 )
=> ( P3 @ ( cons_P1129399039y_real @ X3 @ Xs2 ) @ ( cons_P1129399039y_real @ Y2 @ Ys2 ) ) ) ) ) )
=> ( P3 @ X1 @ X2 ) ) ) ) ) ).
% ord.lexordp_eq.inducts
thf(fact_163_ord_Olexordp__eq_Oinducts,axiom,
! [Less: poly_real > poly_real > $o,X1: list_poly_real,X2: list_poly_real,P3: list_poly_real > list_poly_real > $o] :
( ( lexordp_eq_poly_real @ Less @ X1 @ X2 )
=> ( ! [X_1: list_poly_real] : ( P3 @ nil_poly_real @ X_1 )
=> ( ! [X3: poly_real,Y2: poly_real,Xs2: list_poly_real,Ys2: list_poly_real] :
( ( Less @ X3 @ Y2 )
=> ( P3 @ ( cons_poly_real @ X3 @ Xs2 ) @ ( cons_poly_real @ Y2 @ Ys2 ) ) )
=> ( ! [X3: poly_real,Y2: poly_real,Xs2: list_poly_real,Ys2: list_poly_real] :
( ~ ( Less @ X3 @ Y2 )
=> ( ~ ( Less @ Y2 @ X3 )
=> ( ( lexordp_eq_poly_real @ Less @ Xs2 @ Ys2 )
=> ( ( P3 @ Xs2 @ Ys2 )
=> ( P3 @ ( cons_poly_real @ X3 @ Xs2 ) @ ( cons_poly_real @ Y2 @ Ys2 ) ) ) ) ) )
=> ( P3 @ X1 @ X2 ) ) ) ) ) ).
% ord.lexordp_eq.inducts
thf(fact_164_ord_Olexordp__eq_Osimps,axiom,
( lexord1324236152y_real
= ( ^ [Less2: list_poly_real > list_poly_real > $o,A1: list_list_poly_real,A2: list_list_poly_real] :
( ? [Ys3: list_list_poly_real] :
( ( A1 = nil_list_poly_real )
& ( A2 = Ys3 ) )
| ? [X4: list_poly_real,Y3: list_poly_real,Xs3: list_list_poly_real,Ys3: list_list_poly_real] :
( ( A1
= ( cons_list_poly_real @ X4 @ Xs3 ) )
& ( A2
= ( cons_list_poly_real @ Y3 @ Ys3 ) )
& ( Less2 @ X4 @ Y3 ) )
| ? [X4: list_poly_real,Y3: list_poly_real,Xs3: list_list_poly_real,Ys3: list_list_poly_real] :
( ( A1
= ( cons_list_poly_real @ X4 @ Xs3 ) )
& ( A2
= ( cons_list_poly_real @ Y3 @ Ys3 ) )
& ~ ( Less2 @ X4 @ Y3 )
& ~ ( Less2 @ Y3 @ X4 )
& ( lexord1324236152y_real @ Less2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% ord.lexordp_eq.simps
thf(fact_165_ord_Olexordp__eq_Osimps,axiom,
( lexord834705585y_real
= ( ^ [Less2: produc321349221y_real > produc321349221y_real > $o,A1: list_P736648811y_real,A2: list_P736648811y_real] :
( ? [Ys3: list_P736648811y_real] :
( ( A1 = nil_Pr1492526187y_real )
& ( A2 = Ys3 ) )
| ? [X4: produc321349221y_real,Y3: produc321349221y_real,Xs3: list_P736648811y_real,Ys3: list_P736648811y_real] :
( ( A1
= ( cons_P1027346459y_real @ X4 @ Xs3 ) )
& ( A2
= ( cons_P1027346459y_real @ Y3 @ Ys3 ) )
& ( Less2 @ X4 @ Y3 ) )
| ? [X4: produc321349221y_real,Y3: produc321349221y_real,Xs3: list_P736648811y_real,Ys3: list_P736648811y_real] :
( ( A1
= ( cons_P1027346459y_real @ X4 @ Xs3 ) )
& ( A2
= ( cons_P1027346459y_real @ Y3 @ Ys3 ) )
& ~ ( Less2 @ X4 @ Y3 )
& ~ ( Less2 @ Y3 @ X4 )
& ( lexord834705585y_real @ Less2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% ord.lexordp_eq.simps
thf(fact_166_ord_Olexordp__eq_Osimps,axiom,
( lexord1053351061y_real
= ( ^ [Less2: produc461822025y_real > produc461822025y_real > $o,A1: list_P693436111y_real,A2: list_P693436111y_real] :
( ? [Ys3: list_P693436111y_real] :
( ( A1 = nil_Pr1207979855y_real )
& ( A2 = Ys3 ) )
| ? [X4: produc461822025y_real,Y3: produc461822025y_real,Xs3: list_P693436111y_real,Ys3: list_P693436111y_real] :
( ( A1
= ( cons_P1129399039y_real @ X4 @ Xs3 ) )
& ( A2
= ( cons_P1129399039y_real @ Y3 @ Ys3 ) )
& ( Less2 @ X4 @ Y3 ) )
| ? [X4: produc461822025y_real,Y3: produc461822025y_real,Xs3: list_P693436111y_real,Ys3: list_P693436111y_real] :
( ( A1
= ( cons_P1129399039y_real @ X4 @ Xs3 ) )
& ( A2
= ( cons_P1129399039y_real @ Y3 @ Ys3 ) )
& ~ ( Less2 @ X4 @ Y3 )
& ~ ( Less2 @ Y3 @ X4 )
& ( lexord1053351061y_real @ Less2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% ord.lexordp_eq.simps
thf(fact_167_ord_Olexordp__eq_Osimps,axiom,
( lexordp_eq_poly_real
= ( ^ [Less2: poly_real > poly_real > $o,A1: list_poly_real,A2: list_poly_real] :
( ? [Ys3: list_poly_real] :
( ( A1 = nil_poly_real )
& ( A2 = Ys3 ) )
| ? [X4: poly_real,Y3: poly_real,Xs3: list_poly_real,Ys3: list_poly_real] :
( ( A1
= ( cons_poly_real @ X4 @ Xs3 ) )
& ( A2
= ( cons_poly_real @ Y3 @ Ys3 ) )
& ( Less2 @ X4 @ Y3 ) )
| ? [X4: poly_real,Y3: poly_real,Xs3: list_poly_real,Ys3: list_poly_real] :
( ( A1
= ( cons_poly_real @ X4 @ Xs3 ) )
& ( A2
= ( cons_poly_real @ Y3 @ Ys3 ) )
& ~ ( Less2 @ X4 @ Y3 )
& ~ ( Less2 @ Y3 @ X4 )
& ( lexordp_eq_poly_real @ Less2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% ord.lexordp_eq.simps
thf(fact_168_ord_Olexordp__eq_Ocases,axiom,
! [Less: list_poly_real > list_poly_real > $o,A12: list_list_poly_real,A22: list_list_poly_real] :
( ( lexord1324236152y_real @ Less @ A12 @ A22 )
=> ( ( A12 != nil_list_poly_real )
=> ( ! [X3: list_poly_real] :
( ? [Xs2: list_list_poly_real] :
( A12
= ( cons_list_poly_real @ X3 @ Xs2 ) )
=> ! [Y2: list_poly_real] :
( ? [Ys2: list_list_poly_real] :
( A22
= ( cons_list_poly_real @ Y2 @ Ys2 ) )
=> ~ ( Less @ X3 @ Y2 ) ) )
=> ~ ! [X3: list_poly_real,Y2: list_poly_real,Xs2: list_list_poly_real] :
( ( A12
= ( cons_list_poly_real @ X3 @ Xs2 ) )
=> ! [Ys2: list_list_poly_real] :
( ( A22
= ( cons_list_poly_real @ Y2 @ Ys2 ) )
=> ( ~ ( Less @ X3 @ Y2 )
=> ( ~ ( Less @ Y2 @ X3 )
=> ~ ( lexord1324236152y_real @ Less @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ).
% ord.lexordp_eq.cases
thf(fact_169_ord_Olexordp__eq_Ocases,axiom,
! [Less: produc321349221y_real > produc321349221y_real > $o,A12: list_P736648811y_real,A22: list_P736648811y_real] :
( ( lexord834705585y_real @ Less @ A12 @ A22 )
=> ( ( A12 != nil_Pr1492526187y_real )
=> ( ! [X3: produc321349221y_real] :
( ? [Xs2: list_P736648811y_real] :
( A12
= ( cons_P1027346459y_real @ X3 @ Xs2 ) )
=> ! [Y2: produc321349221y_real] :
( ? [Ys2: list_P736648811y_real] :
( A22
= ( cons_P1027346459y_real @ Y2 @ Ys2 ) )
=> ~ ( Less @ X3 @ Y2 ) ) )
=> ~ ! [X3: produc321349221y_real,Y2: produc321349221y_real,Xs2: list_P736648811y_real] :
( ( A12
= ( cons_P1027346459y_real @ X3 @ Xs2 ) )
=> ! [Ys2: list_P736648811y_real] :
( ( A22
= ( cons_P1027346459y_real @ Y2 @ Ys2 ) )
=> ( ~ ( Less @ X3 @ Y2 )
=> ( ~ ( Less @ Y2 @ X3 )
=> ~ ( lexord834705585y_real @ Less @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ).
% ord.lexordp_eq.cases
thf(fact_170_ord_Olexordp__eq_Ocases,axiom,
! [Less: produc461822025y_real > produc461822025y_real > $o,A12: list_P693436111y_real,A22: list_P693436111y_real] :
( ( lexord1053351061y_real @ Less @ A12 @ A22 )
=> ( ( A12 != nil_Pr1207979855y_real )
=> ( ! [X3: produc461822025y_real] :
( ? [Xs2: list_P693436111y_real] :
( A12
= ( cons_P1129399039y_real @ X3 @ Xs2 ) )
=> ! [Y2: produc461822025y_real] :
( ? [Ys2: list_P693436111y_real] :
( A22
= ( cons_P1129399039y_real @ Y2 @ Ys2 ) )
=> ~ ( Less @ X3 @ Y2 ) ) )
=> ~ ! [X3: produc461822025y_real,Y2: produc461822025y_real,Xs2: list_P693436111y_real] :
( ( A12
= ( cons_P1129399039y_real @ X3 @ Xs2 ) )
=> ! [Ys2: list_P693436111y_real] :
( ( A22
= ( cons_P1129399039y_real @ Y2 @ Ys2 ) )
=> ( ~ ( Less @ X3 @ Y2 )
=> ( ~ ( Less @ Y2 @ X3 )
=> ~ ( lexord1053351061y_real @ Less @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ).
% ord.lexordp_eq.cases
thf(fact_171_ord_Olexordp__eq_Ocases,axiom,
! [Less: poly_real > poly_real > $o,A12: list_poly_real,A22: list_poly_real] :
( ( lexordp_eq_poly_real @ Less @ A12 @ A22 )
=> ( ( A12 != nil_poly_real )
=> ( ! [X3: poly_real] :
( ? [Xs2: list_poly_real] :
( A12
= ( cons_poly_real @ X3 @ Xs2 ) )
=> ! [Y2: poly_real] :
( ? [Ys2: list_poly_real] :
( A22
= ( cons_poly_real @ Y2 @ Ys2 ) )
=> ~ ( Less @ X3 @ Y2 ) ) )
=> ~ ! [X3: poly_real,Y2: poly_real,Xs2: list_poly_real] :
( ( A12
= ( cons_poly_real @ X3 @ Xs2 ) )
=> ! [Ys2: list_poly_real] :
( ( A22
= ( cons_poly_real @ Y2 @ Ys2 ) )
=> ( ~ ( Less @ X3 @ Y2 )
=> ( ~ ( Less @ Y2 @ X3 )
=> ~ ( lexordp_eq_poly_real @ Less @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ).
% ord.lexordp_eq.cases
thf(fact_172_list_Oinject,axiom,
! [X21: list_poly_real,X22: list_list_poly_real,Y21: list_poly_real,Y22: list_list_poly_real] :
( ( ( cons_list_poly_real @ X21 @ X22 )
= ( cons_list_poly_real @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_173_list_Oinject,axiom,
! [X21: produc321349221y_real,X22: list_P736648811y_real,Y21: produc321349221y_real,Y22: list_P736648811y_real] :
( ( ( cons_P1027346459y_real @ X21 @ X22 )
= ( cons_P1027346459y_real @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_174_list_Oinject,axiom,
! [X21: produc461822025y_real,X22: list_P693436111y_real,Y21: produc461822025y_real,Y22: list_P693436111y_real] :
( ( ( cons_P1129399039y_real @ X21 @ X22 )
= ( cons_P1129399039y_real @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_175_list_Oinject,axiom,
! [X21: poly_real,X22: list_poly_real,Y21: poly_real,Y22: list_poly_real] :
( ( ( cons_poly_real @ X21 @ X22 )
= ( cons_poly_real @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_176_ord_Olexordp__eq__simps_I4_J,axiom,
! [Less: list_poly_real > list_poly_real > $o,X: list_poly_real,Xs: list_list_poly_real,Y: list_poly_real,Ys: list_list_poly_real] :
( ( lexord1324236152y_real @ Less @ ( cons_list_poly_real @ X @ Xs ) @ ( cons_list_poly_real @ Y @ Ys ) )
= ( ( Less @ X @ Y )
| ( ~ ( Less @ Y @ X )
& ( lexord1324236152y_real @ Less @ Xs @ Ys ) ) ) ) ).
% ord.lexordp_eq_simps(4)
thf(fact_177_ord_Olexordp__eq__simps_I4_J,axiom,
! [Less: produc321349221y_real > produc321349221y_real > $o,X: produc321349221y_real,Xs: list_P736648811y_real,Y: produc321349221y_real,Ys: list_P736648811y_real] :
( ( lexord834705585y_real @ Less @ ( cons_P1027346459y_real @ X @ Xs ) @ ( cons_P1027346459y_real @ Y @ Ys ) )
= ( ( Less @ X @ Y )
| ( ~ ( Less @ Y @ X )
& ( lexord834705585y_real @ Less @ Xs @ Ys ) ) ) ) ).
% ord.lexordp_eq_simps(4)
thf(fact_178_ord_Olexordp__eq__simps_I4_J,axiom,
! [Less: produc461822025y_real > produc461822025y_real > $o,X: produc461822025y_real,Xs: list_P693436111y_real,Y: produc461822025y_real,Ys: list_P693436111y_real] :
( ( lexord1053351061y_real @ Less @ ( cons_P1129399039y_real @ X @ Xs ) @ ( cons_P1129399039y_real @ Y @ Ys ) )
= ( ( Less @ X @ Y )
| ( ~ ( Less @ Y @ X )
& ( lexord1053351061y_real @ Less @ Xs @ Ys ) ) ) ) ).
% ord.lexordp_eq_simps(4)
thf(fact_179_ord_Olexordp__eq__simps_I4_J,axiom,
! [Less: poly_real > poly_real > $o,X: poly_real,Xs: list_poly_real,Y: poly_real,Ys: list_poly_real] :
( ( lexordp_eq_poly_real @ Less @ ( cons_poly_real @ X @ Xs ) @ ( cons_poly_real @ Y @ Ys ) )
= ( ( Less @ X @ Y )
| ( ~ ( Less @ Y @ X )
& ( lexordp_eq_poly_real @ Less @ Xs @ Ys ) ) ) ) ).
% ord.lexordp_eq_simps(4)
thf(fact_180_list__ex__simps_I1_J,axiom,
! [P3: list_poly_real > $o,X: list_poly_real,Xs: list_list_poly_real] :
( ( list_e809719936y_real @ P3 @ ( cons_list_poly_real @ X @ Xs ) )
= ( ( P3 @ X )
| ( list_e809719936y_real @ P3 @ Xs ) ) ) ).
% list_ex_simps(1)
thf(fact_181_list__ex__simps_I1_J,axiom,
! [P3: produc321349221y_real > $o,X: produc321349221y_real,Xs: list_P736648811y_real] :
( ( list_e2093379497y_real @ P3 @ ( cons_P1027346459y_real @ X @ Xs ) )
= ( ( P3 @ X )
| ( list_e2093379497y_real @ P3 @ Xs ) ) ) ).
% list_ex_simps(1)
thf(fact_182_list__ex__simps_I1_J,axiom,
! [P3: produc461822025y_real > $o,X: produc461822025y_real,Xs: list_P693436111y_real] :
( ( list_e1312221069y_real @ P3 @ ( cons_P1129399039y_real @ X @ Xs ) )
= ( ( P3 @ X )
| ( list_e1312221069y_real @ P3 @ Xs ) ) ) ).
% list_ex_simps(1)
thf(fact_183_list__ex__simps_I1_J,axiom,
! [P3: poly_real > $o,X: poly_real,Xs: list_poly_real] :
( ( list_ex_poly_real @ P3 @ ( cons_poly_real @ X @ Xs ) )
= ( ( P3 @ X )
| ( list_ex_poly_real @ P3 @ Xs ) ) ) ).
% list_ex_simps(1)
thf(fact_184_transpose_Ocases,axiom,
! [X: list_l1245274212y_real] :
( ( X != nil_li2088348750y_real )
=> ( ! [Xss: list_l1245274212y_real] :
( X
!= ( cons_l2022919326y_real @ nil_list_poly_real @ Xss ) )
=> ~ ! [X3: list_poly_real,Xs2: list_list_poly_real,Xss: list_l1245274212y_real] :
( X
!= ( cons_l2022919326y_real @ ( cons_list_poly_real @ X3 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_185_transpose_Ocases,axiom,
! [X: list_l6097393y_real] :
( ( X != nil_li579690865y_real )
=> ( ! [Xss: list_l6097393y_real] :
( X
!= ( cons_l730969377y_real @ nil_Pr1492526187y_real @ Xss ) )
=> ~ ! [X3: produc321349221y_real,Xs2: list_P736648811y_real,Xss: list_l6097393y_real] :
( X
!= ( cons_l730969377y_real @ ( cons_P1027346459y_real @ X3 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_186_transpose_Ocases,axiom,
! [X: list_l343759061y_real] :
( ( X != nil_li1299462869y_real )
=> ( ! [Xss: list_l343759061y_real] :
( X
!= ( cons_l677353093y_real @ nil_Pr1207979855y_real @ Xss ) )
=> ~ ! [X3: produc461822025y_real,Xs2: list_P693436111y_real,Xss: list_l343759061y_real] :
( X
!= ( cons_l677353093y_real @ ( cons_P1129399039y_real @ X3 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_187_transpose_Ocases,axiom,
! [X: list_list_poly_real] :
( ( X != nil_list_poly_real )
=> ( ! [Xss: list_list_poly_real] :
( X
!= ( cons_list_poly_real @ nil_poly_real @ Xss ) )
=> ~ ! [X3: poly_real,Xs2: list_poly_real,Xss: list_list_poly_real] :
( X
!= ( cons_list_poly_real @ ( cons_poly_real @ X3 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_188_not__Cons__self2,axiom,
! [X: list_poly_real,Xs: list_list_poly_real] :
( ( cons_list_poly_real @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_189_not__Cons__self2,axiom,
! [X: produc321349221y_real,Xs: list_P736648811y_real] :
( ( cons_P1027346459y_real @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_190_not__Cons__self2,axiom,
! [X: produc461822025y_real,Xs: list_P693436111y_real] :
( ( cons_P1129399039y_real @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_191_not__Cons__self2,axiom,
! [X: poly_real,Xs: list_poly_real] :
( ( cons_poly_real @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_192_mem__Collect__eq,axiom,
! [A3: produc1426596841y_real,P3: produc1426596841y_real > $o] :
( ( member403334290y_real @ A3 @ ( collec1459394516y_real @ P3 ) )
= ( P3 @ A3 ) ) ).
% mem_Collect_eq
thf(fact_193_mem__Collect__eq,axiom,
! [A3: produc461822025y_real,P3: produc461822025y_real > $o] :
( ( member1784523250y_real @ A3 @ ( collec329281332y_real @ P3 ) )
= ( P3 @ A3 ) ) ).
% mem_Collect_eq
thf(fact_194_mem__Collect__eq,axiom,
! [A3: poly_real,P3: poly_real > $o] :
( ( member_poly_real2 @ A3 @ ( collect_poly_real @ P3 ) )
= ( P3 @ A3 ) ) ).
% mem_Collect_eq
thf(fact_195_mem__Collect__eq,axiom,
! [A3: list_poly_real,P3: list_poly_real > $o] :
( ( member708095579y_real @ A3 @ ( collec1537941401y_real @ P3 ) )
= ( P3 @ A3 ) ) ).
% mem_Collect_eq
thf(fact_196_Collect__mem__eq,axiom,
! [A4: set_Pr1405268809y_real] :
( ( collec1459394516y_real
@ ^ [X4: produc1426596841y_real] : ( member403334290y_real @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_197_Collect__mem__eq,axiom,
! [A4: set_Pr483210409y_real] :
( ( collec329281332y_real
@ ^ [X4: produc461822025y_real] : ( member1784523250y_real @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_198_Collect__mem__eq,axiom,
! [A4: set_poly_real] :
( ( collect_poly_real
@ ^ [X4: poly_real] : ( member_poly_real2 @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_199_Collect__mem__eq,axiom,
! [A4: set_list_poly_real] :
( ( collec1537941401y_real
@ ^ [X4: list_poly_real] : ( member708095579y_real @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_200_strict__sorted_Oinduct,axiom,
! [P3: list_poly_real > $o,A0: list_poly_real] :
( ( P3 @ nil_poly_real )
=> ( ! [X3: poly_real,Ys2: list_poly_real] :
( ( P3 @ Ys2 )
=> ( P3 @ ( cons_poly_real @ X3 @ Ys2 ) ) )
=> ( P3 @ A0 ) ) ) ).
% strict_sorted.induct
thf(fact_201_strict__sorted_Ocases,axiom,
! [X: list_poly_real] :
( ( X != nil_poly_real )
=> ~ ! [X3: poly_real,Ys2: list_poly_real] :
( X
!= ( cons_poly_real @ X3 @ Ys2 ) ) ) ).
% strict_sorted.cases
thf(fact_202_map__tailrec__rev_Oinduct,axiom,
! [P3: ( list_poly_real > produc461822025y_real ) > list_list_poly_real > list_P693436111y_real > $o,A0: list_poly_real > produc461822025y_real,A12: list_list_poly_real,A22: list_P693436111y_real] :
( ! [F2: list_poly_real > produc461822025y_real,X_1: list_P693436111y_real] : ( P3 @ F2 @ nil_list_poly_real @ X_1 )
=> ( ! [F2: list_poly_real > produc461822025y_real,A: list_poly_real,As: list_list_poly_real,Bs2: list_P693436111y_real] :
( ( P3 @ F2 @ As @ ( cons_P1129399039y_real @ ( F2 @ A ) @ Bs2 ) )
=> ( P3 @ F2 @ ( cons_list_poly_real @ A @ As ) @ Bs2 ) )
=> ( P3 @ A0 @ A12 @ A22 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_203_map__tailrec__rev_Oinduct,axiom,
! [P3: ( produc321349221y_real > produc461822025y_real ) > list_P736648811y_real > list_P693436111y_real > $o,A0: produc321349221y_real > produc461822025y_real,A12: list_P736648811y_real,A22: list_P693436111y_real] :
( ! [F2: produc321349221y_real > produc461822025y_real,X_1: list_P693436111y_real] : ( P3 @ F2 @ nil_Pr1492526187y_real @ X_1 )
=> ( ! [F2: produc321349221y_real > produc461822025y_real,A: produc321349221y_real,As: list_P736648811y_real,Bs2: list_P693436111y_real] :
( ( P3 @ F2 @ As @ ( cons_P1129399039y_real @ ( F2 @ A ) @ Bs2 ) )
=> ( P3 @ F2 @ ( cons_P1027346459y_real @ A @ As ) @ Bs2 ) )
=> ( P3 @ A0 @ A12 @ A22 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_204_map__tailrec__rev_Oinduct,axiom,
! [P3: ( produc461822025y_real > produc461822025y_real ) > list_P693436111y_real > list_P693436111y_real > $o,A0: produc461822025y_real > produc461822025y_real,A12: list_P693436111y_real,A22: list_P693436111y_real] :
( ! [F2: produc461822025y_real > produc461822025y_real,X_1: list_P693436111y_real] : ( P3 @ F2 @ nil_Pr1207979855y_real @ X_1 )
=> ( ! [F2: produc461822025y_real > produc461822025y_real,A: produc461822025y_real,As: list_P693436111y_real,Bs2: list_P693436111y_real] :
( ( P3 @ F2 @ As @ ( cons_P1129399039y_real @ ( F2 @ A ) @ Bs2 ) )
=> ( P3 @ F2 @ ( cons_P1129399039y_real @ A @ As ) @ Bs2 ) )
=> ( P3 @ A0 @ A12 @ A22 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_205_map__tailrec__rev_Oinduct,axiom,
! [P3: ( poly_real > poly_real ) > list_poly_real > list_poly_real > $o,A0: poly_real > poly_real,A12: list_poly_real,A22: list_poly_real] :
( ! [F2: poly_real > poly_real,X_1: list_poly_real] : ( P3 @ F2 @ nil_poly_real @ X_1 )
=> ( ! [F2: poly_real > poly_real,A: poly_real,As: list_poly_real,Bs2: list_poly_real] :
( ( P3 @ F2 @ As @ ( cons_poly_real @ ( F2 @ A ) @ Bs2 ) )
=> ( P3 @ F2 @ ( cons_poly_real @ A @ As ) @ Bs2 ) )
=> ( P3 @ A0 @ A12 @ A22 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_206_list__nonempty__induct,axiom,
! [Xs: list_poly_real,P3: list_poly_real > $o] :
( ( Xs != nil_poly_real )
=> ( ! [X3: poly_real] : ( P3 @ ( cons_poly_real @ X3 @ nil_poly_real ) )
=> ( ! [X3: poly_real,Xs2: list_poly_real] :
( ( Xs2 != nil_poly_real )
=> ( ( P3 @ Xs2 )
=> ( P3 @ ( cons_poly_real @ X3 @ Xs2 ) ) ) )
=> ( P3 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_207_successively_Oinduct,axiom,
! [P3: ( poly_real > poly_real > $o ) > list_poly_real > $o,A0: poly_real > poly_real > $o,A12: list_poly_real] :
( ! [P4: poly_real > poly_real > $o] : ( P3 @ P4 @ nil_poly_real )
=> ( ! [P4: poly_real > poly_real > $o,X3: poly_real] : ( P3 @ P4 @ ( cons_poly_real @ X3 @ nil_poly_real ) )
=> ( ! [P4: poly_real > poly_real > $o,X3: poly_real,Y2: poly_real,Xs2: list_poly_real] :
( ( P3 @ P4 @ ( cons_poly_real @ Y2 @ Xs2 ) )
=> ( P3 @ P4 @ ( cons_poly_real @ X3 @ ( cons_poly_real @ Y2 @ Xs2 ) ) ) )
=> ( P3 @ A0 @ A12 ) ) ) ) ).
% successively.induct
thf(fact_208_remdups__adj_Oinduct,axiom,
! [P3: list_poly_real > $o,A0: list_poly_real] :
( ( P3 @ nil_poly_real )
=> ( ! [X3: poly_real] : ( P3 @ ( cons_poly_real @ X3 @ nil_poly_real ) )
=> ( ! [X3: poly_real,Y2: poly_real,Xs2: list_poly_real] :
( ( ( X3 = Y2 )
=> ( P3 @ ( cons_poly_real @ X3 @ Xs2 ) ) )
=> ( ( ( X3 != Y2 )
=> ( P3 @ ( cons_poly_real @ Y2 @ Xs2 ) ) )
=> ( P3 @ ( cons_poly_real @ X3 @ ( cons_poly_real @ Y2 @ Xs2 ) ) ) ) )
=> ( P3 @ A0 ) ) ) ) ).
% remdups_adj.induct
thf(fact_209_sorted__wrt_Oinduct,axiom,
! [P3: ( poly_real > poly_real > $o ) > list_poly_real > $o,A0: poly_real > poly_real > $o,A12: list_poly_real] :
( ! [P4: poly_real > poly_real > $o] : ( P3 @ P4 @ nil_poly_real )
=> ( ! [P4: poly_real > poly_real > $o,X3: poly_real,Ys2: list_poly_real] :
( ( P3 @ P4 @ Ys2 )
=> ( P3 @ P4 @ ( cons_poly_real @ X3 @ Ys2 ) ) )
=> ( P3 @ A0 @ A12 ) ) ) ).
% sorted_wrt.induct
thf(fact_210_remdups__adj_Ocases,axiom,
! [X: list_poly_real] :
( ( X != nil_poly_real )
=> ( ! [X3: poly_real] :
( X
!= ( cons_poly_real @ X3 @ nil_poly_real ) )
=> ~ ! [X3: poly_real,Y2: poly_real,Xs2: list_poly_real] :
( X
!= ( cons_poly_real @ X3 @ ( cons_poly_real @ Y2 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_211_shuffles_Oinduct,axiom,
! [P3: list_poly_real > list_poly_real > $o,A0: list_poly_real,A12: list_poly_real] :
( ! [X_1: list_poly_real] : ( P3 @ nil_poly_real @ X_1 )
=> ( ! [Xs2: list_poly_real] : ( P3 @ Xs2 @ nil_poly_real )
=> ( ! [X3: poly_real,Xs2: list_poly_real,Y2: poly_real,Ys2: list_poly_real] :
( ( P3 @ Xs2 @ ( cons_poly_real @ Y2 @ Ys2 ) )
=> ( ( P3 @ ( cons_poly_real @ X3 @ Xs2 ) @ Ys2 )
=> ( P3 @ ( cons_poly_real @ X3 @ Xs2 ) @ ( cons_poly_real @ Y2 @ Ys2 ) ) ) )
=> ( P3 @ A0 @ A12 ) ) ) ) ).
% shuffles.induct
thf(fact_212_min__list_Oinduct,axiom,
! [P3: list_poly_real > $o,A0: list_poly_real] :
( ! [X3: poly_real,Xs2: list_poly_real] :
( ! [X212: poly_real,X222: list_poly_real] :
( ( Xs2
= ( cons_poly_real @ X212 @ X222 ) )
=> ( P3 @ Xs2 ) )
=> ( P3 @ ( cons_poly_real @ X3 @ Xs2 ) ) )
=> ( ( P3 @ nil_poly_real )
=> ( P3 @ A0 ) ) ) ).
% min_list.induct
thf(fact_213_min__list_Ocases,axiom,
! [X: list_poly_real] :
( ! [X3: poly_real,Xs2: list_poly_real] :
( X
!= ( cons_poly_real @ X3 @ Xs2 ) )
=> ( X = nil_poly_real ) ) ).
% min_list.cases
thf(fact_214_induct__list012,axiom,
! [P3: list_poly_real > $o,Xs: list_poly_real] :
( ( P3 @ nil_poly_real )
=> ( ! [X3: poly_real] : ( P3 @ ( cons_poly_real @ X3 @ nil_poly_real ) )
=> ( ! [X3: poly_real,Y2: poly_real,Zs: list_poly_real] :
( ( P3 @ Zs )
=> ( ( P3 @ ( cons_poly_real @ Y2 @ Zs ) )
=> ( P3 @ ( cons_poly_real @ X3 @ ( cons_poly_real @ Y2 @ Zs ) ) ) ) )
=> ( P3 @ Xs ) ) ) ) ).
% induct_list012
thf(fact_215_splice_Oinduct,axiom,
! [P3: list_poly_real > list_poly_real > $o,A0: list_poly_real,A12: list_poly_real] :
( ! [X_1: list_poly_real] : ( P3 @ nil_poly_real @ X_1 )
=> ( ! [X3: poly_real,Xs2: list_poly_real,Ys2: list_poly_real] :
( ( P3 @ Ys2 @ Xs2 )
=> ( P3 @ ( cons_poly_real @ X3 @ Xs2 ) @ Ys2 ) )
=> ( P3 @ A0 @ A12 ) ) ) ).
% splice.induct
thf(fact_216_list__induct2_H,axiom,
! [P3: list_poly_real > list_poly_real > $o,Xs: list_poly_real,Ys: list_poly_real] :
( ( P3 @ nil_poly_real @ nil_poly_real )
=> ( ! [X3: poly_real,Xs2: list_poly_real] : ( P3 @ ( cons_poly_real @ X3 @ Xs2 ) @ nil_poly_real )
=> ( ! [Y2: poly_real,Ys2: list_poly_real] : ( P3 @ nil_poly_real @ ( cons_poly_real @ Y2 @ Ys2 ) )
=> ( ! [X3: poly_real,Xs2: list_poly_real,Y2: poly_real,Ys2: list_poly_real] :
( ( P3 @ Xs2 @ Ys2 )
=> ( P3 @ ( cons_poly_real @ X3 @ Xs2 ) @ ( cons_poly_real @ Y2 @ Ys2 ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_217_neq__Nil__conv,axiom,
! [Xs: list_poly_real] :
( ( Xs != nil_poly_real )
= ( ? [Y3: poly_real,Ys3: list_poly_real] :
( Xs
= ( cons_poly_real @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_218_list_Oinducts,axiom,
! [P3: list_poly_real > $o,List: list_poly_real] :
( ( P3 @ nil_poly_real )
=> ( ! [X12: poly_real,X23: list_poly_real] :
( ( P3 @ X23 )
=> ( P3 @ ( cons_poly_real @ X12 @ X23 ) ) )
=> ( P3 @ List ) ) ) ).
% list.inducts
thf(fact_219_list_Oexhaust,axiom,
! [Y: list_poly_real] :
( ( Y != nil_poly_real )
=> ~ ! [X213: poly_real,X223: list_poly_real] :
( Y
!= ( cons_poly_real @ X213 @ X223 ) ) ) ).
% list.exhaust
thf(fact_220_list_OdiscI,axiom,
! [List: list_poly_real,X21: poly_real,X22: list_poly_real] :
( ( List
= ( cons_poly_real @ X21 @ X22 ) )
=> ( List != nil_poly_real ) ) ).
% list.discI
thf(fact_221_list_Odistinct_I1_J,axiom,
! [X21: poly_real,X22: list_poly_real] :
( nil_poly_real
!= ( cons_poly_real @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_222_ord_Olexordp__eq_OCons__eq,axiom,
! [Less: poly_real > poly_real > $o,X: poly_real,Y: poly_real,Xs: list_poly_real,Ys: list_poly_real] :
( ~ ( Less @ X @ Y )
=> ( ~ ( Less @ Y @ X )
=> ( ( lexordp_eq_poly_real @ Less @ Xs @ Ys )
=> ( lexordp_eq_poly_real @ Less @ ( cons_poly_real @ X @ Xs ) @ ( cons_poly_real @ Y @ Ys ) ) ) ) ) ).
% ord.lexordp_eq.Cons_eq
thf(fact_223_ord_Olexordp__eq_OCons,axiom,
! [Less: poly_real > poly_real > $o,X: poly_real,Y: poly_real,Xs: list_poly_real,Ys: list_poly_real] :
( ( Less @ X @ Y )
=> ( lexordp_eq_poly_real @ Less @ ( cons_poly_real @ X @ Xs ) @ ( cons_poly_real @ Y @ Ys ) ) ) ).
% ord.lexordp_eq.Cons
thf(fact_224_splice_Osimps_I2_J,axiom,
! [X: poly_real,Xs: list_poly_real,Ys: list_poly_real] :
( ( splice_poly_real @ ( cons_poly_real @ X @ Xs ) @ Ys )
= ( cons_poly_real @ X @ ( splice_poly_real @ Ys @ Xs ) ) ) ).
% splice.simps(2)
thf(fact_225_listrelp_OCons,axiom,
! [R: poly_real > poly_real > $o,X: poly_real,Y: poly_real,Xs: list_poly_real,Ys: list_poly_real] :
( ( R @ X @ Y )
=> ( ( listre1609905961y_real @ R @ Xs @ Ys )
=> ( listre1609905961y_real @ R @ ( cons_poly_real @ X @ Xs ) @ ( cons_poly_real @ Y @ Ys ) ) ) ) ).
% listrelp.Cons
thf(fact_226_sturm__seq_Ops__first__two,axiom,
! [Ps: list_poly_real,P: poly_real] :
( ( sturm_1664866327rm_seq @ Ps @ P )
=> ~ ! [Q: poly_real,Ps3: list_poly_real] :
( Ps
!= ( cons_poly_real @ P @ ( cons_poly_real @ Q @ Ps3 ) ) ) ) ).
% sturm_seq.ps_first_two
thf(fact_227_map__tailrec__rev_Osimps_I2_J,axiom,
! [F: poly_real > poly_real,A3: poly_real,As2: list_poly_real,Bs: list_poly_real] :
( ( map_ta1919289823y_real @ F @ ( cons_poly_real @ A3 @ As2 ) @ Bs )
= ( map_ta1919289823y_real @ F @ As2 @ ( cons_poly_real @ ( F @ A3 ) @ Bs ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_228_null__rec_I1_J,axiom,
! [X: poly_real,Xs: list_poly_real] :
~ ( null_poly_real @ ( cons_poly_real @ X @ Xs ) ) ).
% null_rec(1)
thf(fact_229_member__rec_I1_J,axiom,
! [X: poly_real,Xs: list_poly_real,Y: poly_real] :
( ( member_poly_real @ ( cons_poly_real @ X @ Xs ) @ Y )
= ( ( X = Y )
| ( member_poly_real @ Xs @ Y ) ) ) ).
% member_rec(1)
thf(fact_230_zip__eq__Nil__iff,axiom,
! [Xs: list_poly_real,Ys: list_poly_real] :
( ( ( zip_po657915297y_real @ Xs @ Ys )
= nil_Pr1207979855y_real )
= ( ( Xs = nil_poly_real )
| ( Ys = nil_poly_real ) ) ) ).
% zip_eq_Nil_iff
thf(fact_231_minus__poly__rev__list_Oinduct,axiom,
! [P3: list_poly_real > list_poly_real > $o,A0: list_poly_real,A12: list_poly_real] :
( ! [X3: poly_real,Xs2: list_poly_real,Y2: poly_real,Ys2: list_poly_real] :
( ( P3 @ Xs2 @ Ys2 )
=> ( P3 @ ( cons_poly_real @ X3 @ Xs2 ) @ ( cons_poly_real @ Y2 @ Ys2 ) ) )
=> ( ! [Xs2: list_poly_real] : ( P3 @ Xs2 @ nil_poly_real )
=> ( ! [Y2: poly_real,Ys2: list_poly_real] : ( P3 @ nil_poly_real @ ( cons_poly_real @ Y2 @ Ys2 ) )
=> ( P3 @ A0 @ A12 ) ) ) ) ).
% minus_poly_rev_list.induct
thf(fact_232_plus__coeffs_Oinduct,axiom,
! [P3: list_poly_real > list_poly_real > $o,A0: list_poly_real,A12: list_poly_real] :
( ! [Xs2: list_poly_real] : ( P3 @ Xs2 @ nil_poly_real )
=> ( ! [V: poly_real,Va: list_poly_real] : ( P3 @ nil_poly_real @ ( cons_poly_real @ V @ Va ) )
=> ( ! [X3: poly_real,Xs2: list_poly_real,Y2: poly_real,Ys2: list_poly_real] :
( ( P3 @ Xs2 @ Ys2 )
=> ( P3 @ ( cons_poly_real @ X3 @ Xs2 ) @ ( cons_poly_real @ Y2 @ Ys2 ) ) )
=> ( P3 @ A0 @ A12 ) ) ) ) ).
% plus_coeffs.induct
thf(fact_233_insert__Nil,axiom,
! [X: poly_real] :
( ( insert_poly_real @ X @ nil_poly_real )
= ( cons_poly_real @ X @ nil_poly_real ) ) ).
% insert_Nil
thf(fact_234_lexordp__eq__simps_I3_J,axiom,
! [X: poly_real,Xs: list_poly_real] :
~ ( ord_le1233681790y_real @ ( cons_poly_real @ X @ Xs ) @ nil_poly_real ) ).
% lexordp_eq_simps(3)
thf(fact_235_rotate1_Osimps_I2_J,axiom,
! [X: poly_real,Xs: list_poly_real] :
( ( rotate1_poly_real @ ( cons_poly_real @ X @ Xs ) )
= ( append_poly_real @ Xs @ ( cons_poly_real @ X @ nil_poly_real ) ) ) ).
% rotate1.simps(2)
thf(fact_236_bind__simps_I2_J,axiom,
! [X: poly_real,Xs: list_poly_real,F: poly_real > list_poly_real] :
( ( bind_p420216945y_real @ ( cons_poly_real @ X @ Xs ) @ F )
= ( append_poly_real @ ( F @ X ) @ ( bind_p420216945y_real @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_237_gen__length__code_I2_J,axiom,
! [N: nat,X: poly_real,Xs: list_poly_real] :
( ( gen_length_poly_real @ N @ ( cons_poly_real @ X @ Xs ) )
= ( gen_length_poly_real @ ( suc @ N ) @ Xs ) ) ).
% gen_length_code(2)
thf(fact_238_append__Nil2,axiom,
! [Xs: list_poly_real] :
( ( append_poly_real @ Xs @ nil_poly_real )
= Xs ) ).
% append_Nil2
thf(fact_239_append__self__conv,axiom,
! [Xs: list_poly_real,Ys: list_poly_real] :
( ( ( append_poly_real @ Xs @ Ys )
= Xs )
= ( Ys = nil_poly_real ) ) ).
% append_self_conv
thf(fact_240_self__append__conv,axiom,
! [Xs: list_poly_real,Ys: list_poly_real] :
( ( Xs
= ( append_poly_real @ Xs @ Ys ) )
= ( Ys = nil_poly_real ) ) ).
% self_append_conv
thf(fact_241_append__self__conv2,axiom,
! [Xs: list_poly_real,Ys: list_poly_real] :
( ( ( append_poly_real @ Xs @ Ys )
= Ys )
= ( Xs = nil_poly_real ) ) ).
% append_self_conv2
thf(fact_242_self__append__conv2,axiom,
! [Ys: list_poly_real,Xs: list_poly_real] :
( ( Ys
= ( append_poly_real @ Xs @ Ys ) )
= ( Xs = nil_poly_real ) ) ).
% self_append_conv2
thf(fact_243_Nil__is__append__conv,axiom,
! [Xs: list_poly_real,Ys: list_poly_real] :
( ( nil_poly_real
= ( append_poly_real @ Xs @ Ys ) )
= ( ( Xs = nil_poly_real )
& ( Ys = nil_poly_real ) ) ) ).
% Nil_is_append_conv
thf(fact_244_append__is__Nil__conv,axiom,
! [Xs: list_poly_real,Ys: list_poly_real] :
( ( ( append_poly_real @ Xs @ Ys )
= nil_poly_real )
= ( ( Xs = nil_poly_real )
& ( Ys = nil_poly_real ) ) ) ).
% append_is_Nil_conv
thf(fact_245_append_Oright__neutral,axiom,
! [A3: list_poly_real] :
( ( append_poly_real @ A3 @ nil_poly_real )
= A3 ) ).
% append.right_neutral
thf(fact_246_lexordp__eq__simps_I2_J,axiom,
! [Xs: list_poly_real] :
( ( ord_le1233681790y_real @ Xs @ nil_poly_real )
= ( Xs = nil_poly_real ) ) ).
% lexordp_eq_simps(2)
thf(fact_247_lexordp__eq__simps_I1_J,axiom,
! [Ys: list_poly_real] : ( ord_le1233681790y_real @ nil_poly_real @ Ys ) ).
% lexordp_eq_simps(1)
thf(fact_248_append1__eq__conv,axiom,
! [Xs: list_poly_real,X: poly_real,Ys: list_poly_real,Y: poly_real] :
( ( ( append_poly_real @ Xs @ ( cons_poly_real @ X @ nil_poly_real ) )
= ( append_poly_real @ Ys @ ( cons_poly_real @ Y @ nil_poly_real ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_249_Cons__eq__appendI,axiom,
! [X: poly_real,Xs1: list_poly_real,Ys: list_poly_real,Xs: list_poly_real,Zs2: list_poly_real] :
( ( ( cons_poly_real @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_poly_real @ Xs1 @ Zs2 ) )
=> ( ( cons_poly_real @ X @ Xs )
= ( append_poly_real @ Ys @ Zs2 ) ) ) ) ).
% Cons_eq_appendI
thf(fact_250_append__Cons,axiom,
! [X: poly_real,Xs: list_poly_real,Ys: list_poly_real] :
( ( append_poly_real @ ( cons_poly_real @ X @ Xs ) @ Ys )
= ( cons_poly_real @ X @ ( append_poly_real @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_251_eq__Nil__appendI,axiom,
! [Xs: list_poly_real,Ys: list_poly_real] :
( ( Xs = Ys )
=> ( Xs
= ( append_poly_real @ nil_poly_real @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_252_append__Nil,axiom,
! [Ys: list_poly_real] :
( ( append_poly_real @ nil_poly_real @ Ys )
= Ys ) ).
% append_Nil
thf(fact_253_append_Oleft__neutral,axiom,
! [A3: list_poly_real] :
( ( append_poly_real @ nil_poly_real @ A3 )
= A3 ) ).
% append.left_neutral
thf(fact_254_lexordp__eq_ONil,axiom,
! [Ys: list_poly_real] : ( ord_le1233681790y_real @ nil_poly_real @ Ys ) ).
% lexordp_eq.Nil
thf(fact_255_rev__induct,axiom,
! [P3: list_poly_real > $o,Xs: list_poly_real] :
( ( P3 @ nil_poly_real )
=> ( ! [X3: poly_real,Xs2: list_poly_real] :
( ( P3 @ Xs2 )
=> ( P3 @ ( append_poly_real @ Xs2 @ ( cons_poly_real @ X3 @ nil_poly_real ) ) ) )
=> ( P3 @ Xs ) ) ) ).
% rev_induct
thf(fact_256_rev__exhaust,axiom,
! [Xs: list_poly_real] :
( ( Xs != nil_poly_real )
=> ~ ! [Ys2: list_poly_real,Y2: poly_real] :
( Xs
!= ( append_poly_real @ Ys2 @ ( cons_poly_real @ Y2 @ nil_poly_real ) ) ) ) ).
% rev_exhaust
thf(fact_257_Cons__eq__append__conv,axiom,
! [X: poly_real,Xs: list_poly_real,Ys: list_poly_real,Zs2: list_poly_real] :
( ( ( cons_poly_real @ X @ Xs )
= ( append_poly_real @ Ys @ Zs2 ) )
= ( ( ( Ys = nil_poly_real )
& ( ( cons_poly_real @ X @ Xs )
= Zs2 ) )
| ? [Ys4: list_poly_real] :
( ( ( cons_poly_real @ X @ Ys4 )
= Ys )
& ( Xs
= ( append_poly_real @ Ys4 @ Zs2 ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_258_append__eq__Cons__conv,axiom,
! [Ys: list_poly_real,Zs2: list_poly_real,X: poly_real,Xs: list_poly_real] :
( ( ( append_poly_real @ Ys @ Zs2 )
= ( cons_poly_real @ X @ Xs ) )
= ( ( ( Ys = nil_poly_real )
& ( Zs2
= ( cons_poly_real @ X @ Xs ) ) )
| ? [Ys4: list_poly_real] :
( ( Ys
= ( cons_poly_real @ X @ Ys4 ) )
& ( ( append_poly_real @ Ys4 @ Zs2 )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_259_rev__nonempty__induct,axiom,
! [Xs: list_poly_real,P3: list_poly_real > $o] :
( ( Xs != nil_poly_real )
=> ( ! [X3: poly_real] : ( P3 @ ( cons_poly_real @ X3 @ nil_poly_real ) )
=> ( ! [X3: poly_real,Xs2: list_poly_real] :
( ( Xs2 != nil_poly_real )
=> ( ( P3 @ Xs2 )
=> ( P3 @ ( append_poly_real @ Xs2 @ ( cons_poly_real @ X3 @ nil_poly_real ) ) ) ) )
=> ( P3 @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_260_maps__simps_I1_J,axiom,
! [F: poly_real > list_poly_real,X: poly_real,Xs: list_poly_real] :
( ( maps_p961674027y_real @ F @ ( cons_poly_real @ X @ Xs ) )
= ( append_poly_real @ ( F @ X ) @ ( maps_p961674027y_real @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_261_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_262_nat_Oinject,axiom,
! [X2: nat,Y23: nat] :
( ( ( suc @ X2 )
= ( suc @ Y23 ) )
= ( X2 = Y23 ) ) ).
% nat.inject
thf(fact_263_product__lists_Osimps_I1_J,axiom,
( ( produc523415045y_real @ nil_list_poly_real )
= ( cons_list_poly_real @ nil_poly_real @ nil_list_poly_real ) ) ).
% product_lists.simps(1)
thf(fact_264_subseqs_Osimps_I1_J,axiom,
( ( subseqs_poly_real @ nil_poly_real )
= ( cons_list_poly_real @ nil_poly_real @ nil_list_poly_real ) ) ).
% subseqs.simps(1)
thf(fact_265_concat__eq__append__conv,axiom,
! [Xss2: list_list_poly_real,Ys: list_poly_real,Zs2: list_poly_real] :
( ( ( concat_poly_real @ Xss2 )
= ( append_poly_real @ Ys @ Zs2 ) )
= ( ( ( Xss2 = nil_list_poly_real )
=> ( ( Ys = nil_poly_real )
& ( Zs2 = nil_poly_real ) ) )
& ( ( Xss2 != nil_list_poly_real )
=> ? [Xss1: list_list_poly_real,Xs3: list_poly_real,Xs4: list_poly_real,Xss22: list_list_poly_real] :
( ( Xss2
= ( append527544425y_real @ Xss1 @ ( cons_list_poly_real @ ( append_poly_real @ Xs3 @ Xs4 ) @ Xss22 ) ) )
& ( Ys
= ( append_poly_real @ ( concat_poly_real @ Xss1 ) @ Xs3 ) )
& ( Zs2
= ( append_poly_real @ Xs4 @ ( concat_poly_real @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_266_concat_Osimps_I1_J,axiom,
( ( concat_poly_real @ nil_list_poly_real )
= nil_poly_real ) ).
% concat.simps(1)
thf(fact_267_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_268_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_269_butlast__snoc,axiom,
! [Xs: list_poly_real,X: poly_real] :
( ( butlast_poly_real @ ( append_poly_real @ Xs @ ( cons_poly_real @ X @ nil_poly_real ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_270_enumerate__simps_I2_J,axiom,
! [N: nat,X: poly_real,Xs: list_poly_real] :
( ( enumerate_poly_real @ N @ ( cons_poly_real @ X @ Xs ) )
= ( cons_P1027346459y_real @ ( produc404352541y_real @ N @ X ) @ ( enumerate_poly_real @ ( suc @ N ) @ Xs ) ) ) ).
% enumerate_simps(2)
thf(fact_271_last__snoc,axiom,
! [Xs: list_poly_real,X: poly_real] :
( ( last_poly_real @ ( append_poly_real @ Xs @ ( cons_poly_real @ X @ nil_poly_real ) ) )
= X ) ).
% last_snoc
thf(fact_272_plus__coeffs_Osimps_I2_J,axiom,
! [V2: poly_real,Va2: list_poly_real] :
( ( plus_c288199817y_real @ nil_poly_real @ ( cons_poly_real @ V2 @ Va2 ) )
= ( cons_poly_real @ V2 @ Va2 ) ) ).
% plus_coeffs.simps(2)
thf(fact_273_last__appendL,axiom,
! [Ys: list_poly_real,Xs: list_poly_real] :
( ( Ys = nil_poly_real )
=> ( ( last_poly_real @ ( append_poly_real @ Xs @ Ys ) )
= ( last_poly_real @ Xs ) ) ) ).
% last_appendL
thf(fact_274_last__appendR,axiom,
! [Ys: list_poly_real,Xs: list_poly_real] :
( ( Ys != nil_poly_real )
=> ( ( last_poly_real @ ( append_poly_real @ Xs @ Ys ) )
= ( last_poly_real @ Ys ) ) ) ).
% last_appendR
thf(fact_275_zip__Cons__Cons,axiom,
! [X: poly_real,Xs: list_poly_real,Y: poly_real,Ys: list_poly_real] :
( ( zip_po657915297y_real @ ( cons_poly_real @ X @ Xs ) @ ( cons_poly_real @ Y @ Ys ) )
= ( cons_P1129399039y_real @ ( produc1867601537y_real @ X @ Y ) @ ( zip_po657915297y_real @ Xs @ Ys ) ) ) ).
% zip_Cons_Cons
thf(fact_276_append__butlast__last__id,axiom,
! [Xs: list_poly_real] :
( ( Xs != nil_poly_real )
=> ( ( append_poly_real @ ( butlast_poly_real @ Xs ) @ ( cons_poly_real @ ( last_poly_real @ Xs ) @ nil_poly_real ) )
= Xs ) ) ).
% append_butlast_last_id
thf(fact_277_butlast_Osimps_I1_J,axiom,
( ( butlast_poly_real @ nil_poly_real )
= nil_poly_real ) ).
% butlast.simps(1)
thf(fact_278_snoc__eq__iff__butlast,axiom,
! [Xs: list_poly_real,X: poly_real,Ys: list_poly_real] :
( ( ( append_poly_real @ Xs @ ( cons_poly_real @ X @ nil_poly_real ) )
= Ys )
= ( ( Ys != nil_poly_real )
& ( ( butlast_poly_real @ Ys )
= Xs )
& ( ( last_poly_real @ Ys )
= X ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_279_last_Osimps,axiom,
! [Xs: list_poly_real,X: poly_real] :
( ( ( Xs = nil_poly_real )
=> ( ( last_poly_real @ ( cons_poly_real @ X @ Xs ) )
= X ) )
& ( ( Xs != nil_poly_real )
=> ( ( last_poly_real @ ( cons_poly_real @ X @ Xs ) )
= ( last_poly_real @ Xs ) ) ) ) ).
% last.simps
thf(fact_280_last__ConsL,axiom,
! [Xs: list_poly_real,X: poly_real] :
( ( Xs = nil_poly_real )
=> ( ( last_poly_real @ ( cons_poly_real @ X @ Xs ) )
= X ) ) ).
% last_ConsL
thf(fact_281_last__ConsR,axiom,
! [Xs: list_poly_real,X: poly_real] :
( ( Xs != nil_poly_real )
=> ( ( last_poly_real @ ( cons_poly_real @ X @ Xs ) )
= ( last_poly_real @ Xs ) ) ) ).
% last_ConsR
thf(fact_282_zip__eq__ConsE,axiom,
! [Xs: list_poly_real,Ys: list_poly_real,Xy: produc461822025y_real,Xys: list_P693436111y_real] :
( ( ( zip_po657915297y_real @ Xs @ Ys )
= ( cons_P1129399039y_real @ Xy @ Xys ) )
=> ~ ! [X3: poly_real,Xs5: list_poly_real] :
( ( Xs
= ( cons_poly_real @ X3 @ Xs5 ) )
=> ! [Y2: poly_real,Ys5: list_poly_real] :
( ( Ys
= ( cons_poly_real @ Y2 @ Ys5 ) )
=> ( ( Xy
= ( produc1867601537y_real @ X3 @ Y2 ) )
=> ( Xys
!= ( zip_po657915297y_real @ Xs5 @ Ys5 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_283_plus__coeffs_Osimps_I1_J,axiom,
! [Xs: list_poly_real] :
( ( plus_c288199817y_real @ Xs @ nil_poly_real )
= Xs ) ).
% plus_coeffs.simps(1)
thf(fact_284_last__append,axiom,
! [Ys: list_poly_real,Xs: list_poly_real] :
( ( ( Ys = nil_poly_real )
=> ( ( last_poly_real @ ( append_poly_real @ Xs @ Ys ) )
= ( last_poly_real @ Xs ) ) )
& ( ( Ys != nil_poly_real )
=> ( ( last_poly_real @ ( append_poly_real @ Xs @ Ys ) )
= ( last_poly_real @ Ys ) ) ) ) ).
% last_append
thf(fact_285_longest__common__suffix,axiom,
! [Xs: list_poly_real,Ys: list_poly_real] :
? [Ss: list_poly_real,Xs5: list_poly_real,Ys5: list_poly_real] :
( ( Xs
= ( append_poly_real @ Xs5 @ Ss ) )
& ( Ys
= ( append_poly_real @ Ys5 @ Ss ) )
& ( ( Xs5 = nil_poly_real )
| ( Ys5 = nil_poly_real )
| ( ( last_poly_real @ Xs5 )
!= ( last_poly_real @ Ys5 ) ) ) ) ).
% longest_common_suffix
thf(fact_286_butlast_Osimps_I2_J,axiom,
! [Xs: list_poly_real,X: poly_real] :
( ( ( Xs = nil_poly_real )
=> ( ( butlast_poly_real @ ( cons_poly_real @ X @ Xs ) )
= nil_poly_real ) )
& ( ( Xs != nil_poly_real )
=> ( ( butlast_poly_real @ ( cons_poly_real @ X @ Xs ) )
= ( cons_poly_real @ X @ ( butlast_poly_real @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_287_butlast__append,axiom,
! [Ys: list_poly_real,Xs: list_poly_real] :
( ( ( Ys = nil_poly_real )
=> ( ( butlast_poly_real @ ( append_poly_real @ Xs @ Ys ) )
= ( butlast_poly_real @ Xs ) ) )
& ( ( Ys != nil_poly_real )
=> ( ( butlast_poly_real @ ( append_poly_real @ Xs @ Ys ) )
= ( append_poly_real @ Xs @ ( butlast_poly_real @ Ys ) ) ) ) ) ).
% butlast_append
thf(fact_288_snoc__listrel1__snoc__iff,axiom,
! [Xs: list_poly_real,X: poly_real,Ys: list_poly_real,Y: poly_real,R: set_Pr483210409y_real] :
( ( member403334290y_real @ ( produc1000427809y_real @ ( append_poly_real @ Xs @ ( cons_poly_real @ X @ nil_poly_real ) ) @ ( append_poly_real @ Ys @ ( cons_poly_real @ Y @ nil_poly_real ) ) ) @ ( listrel1_poly_real @ R ) )
= ( ( ( member403334290y_real @ ( produc1000427809y_real @ Xs @ Ys ) @ ( listrel1_poly_real @ R ) )
& ( X = Y ) )
| ( ( Xs = Ys )
& ( member1784523250y_real @ ( produc1867601537y_real @ X @ Y ) @ R ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_289_last__zip,axiom,
! [Xs: list_poly_real,Ys: list_poly_real] :
( ( Xs != nil_poly_real )
=> ( ( Ys != nil_poly_real )
=> ( ( ( size_s259235672y_real @ Xs )
= ( size_s259235672y_real @ Ys ) )
=> ( ( last_P21901160y_real @ ( zip_po657915297y_real @ Xs @ Ys ) )
= ( produc1867601537y_real @ ( last_poly_real @ Xs ) @ ( last_poly_real @ Ys ) ) ) ) ) ) ).
% last_zip
thf(fact_290_concat__conv__foldr,axiom,
( concat_poly_real
= ( ^ [Xss3: list_list_poly_real] : ( foldr_2117681201y_real @ append_poly_real @ Xss3 @ nil_poly_real ) ) ) ).
% concat_conv_foldr
thf(fact_291_rotate__is__Nil__conv,axiom,
! [N: nat,Xs: list_poly_real] :
( ( ( rotate_poly_real @ N @ Xs )
= nil_poly_real )
= ( Xs = nil_poly_real ) ) ).
% rotate_is_Nil_conv
thf(fact_292_Cons__listrel1__Cons,axiom,
! [X: poly_real,Xs: list_poly_real,Y: poly_real,Ys: list_poly_real,R: set_Pr483210409y_real] :
( ( member403334290y_real @ ( produc1000427809y_real @ ( cons_poly_real @ X @ Xs ) @ ( cons_poly_real @ Y @ Ys ) ) @ ( listrel1_poly_real @ R ) )
= ( ( ( member1784523250y_real @ ( produc1867601537y_real @ X @ Y ) @ R )
& ( Xs = Ys ) )
| ( ( X = Y )
& ( member403334290y_real @ ( produc1000427809y_real @ Xs @ Ys ) @ ( listrel1_poly_real @ R ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_293_not__listrel1__Nil,axiom,
! [Xs: list_poly_real,R: set_Pr483210409y_real] :
~ ( member403334290y_real @ ( produc1000427809y_real @ Xs @ nil_poly_real ) @ ( listrel1_poly_real @ R ) ) ).
% not_listrel1_Nil
thf(fact_294_not__Nil__listrel1,axiom,
! [Xs: list_poly_real,R: set_Pr483210409y_real] :
~ ( member403334290y_real @ ( produc1000427809y_real @ nil_poly_real @ Xs ) @ ( listrel1_poly_real @ R ) ) ).
% not_Nil_listrel1
thf(fact_295_listrel1I2,axiom,
! [Xs: list_poly_real,Ys: list_poly_real,R: set_Pr483210409y_real,X: poly_real] :
( ( member403334290y_real @ ( produc1000427809y_real @ Xs @ Ys ) @ ( listrel1_poly_real @ R ) )
=> ( member403334290y_real @ ( produc1000427809y_real @ ( cons_poly_real @ X @ Xs ) @ ( cons_poly_real @ X @ Ys ) ) @ ( listrel1_poly_real @ R ) ) ) ).
% listrel1I2
thf(fact_296_successively_Ocases,axiom,
! [X: produc1060019729y_real] :
( ! [P4: poly_real > poly_real > $o] :
( X
!= ( produc1345584779y_real @ P4 @ nil_poly_real ) )
=> ( ! [P4: poly_real > poly_real > $o,X3: poly_real] :
( X
!= ( produc1345584779y_real @ P4 @ ( cons_poly_real @ X3 @ nil_poly_real ) ) )
=> ~ ! [P4: poly_real > poly_real > $o,X3: poly_real,Y2: poly_real,Xs2: list_poly_real] :
( X
!= ( produc1345584779y_real @ P4 @ ( cons_poly_real @ X3 @ ( cons_poly_real @ Y2 @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_297_sorted__wrt_Ocases,axiom,
! [X: produc1060019729y_real] :
( ! [P4: poly_real > poly_real > $o] :
( X
!= ( produc1345584779y_real @ P4 @ nil_poly_real ) )
=> ~ ! [P4: poly_real > poly_real > $o,X3: poly_real,Ys2: list_poly_real] :
( X
!= ( produc1345584779y_real @ P4 @ ( cons_poly_real @ X3 @ Ys2 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_298_Suc__length__conv,axiom,
! [N: nat,Xs: list_poly_real] :
( ( ( suc @ N )
= ( size_s259235672y_real @ Xs ) )
= ( ? [Y3: poly_real,Ys3: list_poly_real] :
( ( Xs
= ( cons_poly_real @ Y3 @ Ys3 ) )
& ( ( size_s259235672y_real @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_299_length__Suc__conv,axiom,
! [Xs: list_poly_real,N: nat] :
( ( ( size_s259235672y_real @ Xs )
= ( suc @ N ) )
= ( ? [Y3: poly_real,Ys3: list_poly_real] :
( ( Xs
= ( cons_poly_real @ Y3 @ Ys3 ) )
& ( ( size_s259235672y_real @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_300_list__induct4,axiom,
! [Xs: list_poly_real,Ys: list_poly_real,Zs2: list_poly_real,Ws: list_poly_real,P3: list_poly_real > list_poly_real > list_poly_real > list_poly_real > $o] :
( ( ( size_s259235672y_real @ Xs )
= ( size_s259235672y_real @ Ys ) )
=> ( ( ( size_s259235672y_real @ Ys )
= ( size_s259235672y_real @ Zs2 ) )
=> ( ( ( size_s259235672y_real @ Zs2 )
= ( size_s259235672y_real @ Ws ) )
=> ( ( P3 @ nil_poly_real @ nil_poly_real @ nil_poly_real @ nil_poly_real )
=> ( ! [X3: poly_real,Xs2: list_poly_real,Y2: poly_real,Ys2: list_poly_real,Z: poly_real,Zs: list_poly_real,W: poly_real,Ws2: list_poly_real] :
( ( ( size_s259235672y_real @ Xs2 )
= ( size_s259235672y_real @ Ys2 ) )
=> ( ( ( size_s259235672y_real @ Ys2 )
= ( size_s259235672y_real @ Zs ) )
=> ( ( ( size_s259235672y_real @ Zs )
= ( size_s259235672y_real @ Ws2 ) )
=> ( ( P3 @ Xs2 @ Ys2 @ Zs @ Ws2 )
=> ( P3 @ ( cons_poly_real @ X3 @ Xs2 ) @ ( cons_poly_real @ Y2 @ Ys2 ) @ ( cons_poly_real @ Z @ Zs ) @ ( cons_poly_real @ W @ Ws2 ) ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs2 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_301_list__induct3,axiom,
! [Xs: list_poly_real,Ys: list_poly_real,Zs2: list_poly_real,P3: list_poly_real > list_poly_real > list_poly_real > $o] :
( ( ( size_s259235672y_real @ Xs )
= ( size_s259235672y_real @ Ys ) )
=> ( ( ( size_s259235672y_real @ Ys )
= ( size_s259235672y_real @ Zs2 ) )
=> ( ( P3 @ nil_poly_real @ nil_poly_real @ nil_poly_real )
=> ( ! [X3: poly_real,Xs2: list_poly_real,Y2: poly_real,Ys2: list_poly_real,Z: poly_real,Zs: list_poly_real] :
( ( ( size_s259235672y_real @ Xs2 )
= ( size_s259235672y_real @ Ys2 ) )
=> ( ( ( size_s259235672y_real @ Ys2 )
= ( size_s259235672y_real @ Zs ) )
=> ( ( P3 @ Xs2 @ Ys2 @ Zs )
=> ( P3 @ ( cons_poly_real @ X3 @ Xs2 ) @ ( cons_poly_real @ Y2 @ Ys2 ) @ ( cons_poly_real @ Z @ Zs ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs2 ) ) ) ) ) ).
% list_induct3
thf(fact_302_list__induct2,axiom,
! [Xs: list_poly_real,Ys: list_poly_real,P3: list_poly_real > list_poly_real > $o] :
( ( ( size_s259235672y_real @ Xs )
= ( size_s259235672y_real @ Ys ) )
=> ( ( P3 @ nil_poly_real @ nil_poly_real )
=> ( ! [X3: poly_real,Xs2: list_poly_real,Y2: poly_real,Ys2: list_poly_real] :
( ( ( size_s259235672y_real @ Xs2 )
= ( size_s259235672y_real @ Ys2 ) )
=> ( ( P3 @ Xs2 @ Ys2 )
=> ( P3 @ ( cons_poly_real @ X3 @ Xs2 ) @ ( cons_poly_real @ Y2 @ Ys2 ) ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_303_pderiv__coeffs__code_Ocases,axiom,
! [X: produc796638169y_real] :
( ! [F2: poly_real,X3: poly_real,Xs2: list_poly_real] :
( X
!= ( produc490255505y_real @ F2 @ ( cons_poly_real @ X3 @ Xs2 ) ) )
=> ~ ! [F2: poly_real] :
( X
!= ( produc490255505y_real @ F2 @ nil_poly_real ) ) ) ).
% pderiv_coeffs_code.cases
thf(fact_304_shuffles_Ocases,axiom,
! [X: produc1426596841y_real] :
( ! [Ys2: list_poly_real] :
( X
!= ( produc1000427809y_real @ nil_poly_real @ Ys2 ) )
=> ( ! [Xs2: list_poly_real] :
( X
!= ( produc1000427809y_real @ Xs2 @ nil_poly_real ) )
=> ~ ! [X3: poly_real,Xs2: list_poly_real,Y2: poly_real,Ys2: list_poly_real] :
( X
!= ( produc1000427809y_real @ ( cons_poly_real @ X3 @ Xs2 ) @ ( cons_poly_real @ Y2 @ Ys2 ) ) ) ) ) ).
% shuffles.cases
thf(fact_305_plus__coeffs_Ocases,axiom,
! [X: produc1426596841y_real] :
( ! [Xs2: list_poly_real] :
( X
!= ( produc1000427809y_real @ Xs2 @ nil_poly_real ) )
=> ( ! [V: poly_real,Va: list_poly_real] :
( X
!= ( produc1000427809y_real @ nil_poly_real @ ( cons_poly_real @ V @ Va ) ) )
=> ~ ! [X3: poly_real,Xs2: list_poly_real,Y2: poly_real,Ys2: list_poly_real] :
( X
!= ( produc1000427809y_real @ ( cons_poly_real @ X3 @ Xs2 ) @ ( cons_poly_real @ Y2 @ Ys2 ) ) ) ) ) ).
% plus_coeffs.cases
thf(fact_306_minus__poly__rev__list_Ocases,axiom,
! [X: produc1426596841y_real] :
( ! [X3: poly_real,Xs2: list_poly_real,Y2: poly_real,Ys2: list_poly_real] :
( X
!= ( produc1000427809y_real @ ( cons_poly_real @ X3 @ Xs2 ) @ ( cons_poly_real @ Y2 @ Ys2 ) ) )
=> ( ! [Xs2: list_poly_real] :
( X
!= ( produc1000427809y_real @ Xs2 @ nil_poly_real ) )
=> ~ ! [Y2: poly_real,Ys2: list_poly_real] :
( X
!= ( produc1000427809y_real @ nil_poly_real @ ( cons_poly_real @ Y2 @ Ys2 ) ) ) ) ) ).
% minus_poly_rev_list.cases
thf(fact_307_Cons__listrel1E2,axiom,
! [Xs: list_poly_real,Y: poly_real,Ys: list_poly_real,R: set_Pr483210409y_real] :
( ( member403334290y_real @ ( produc1000427809y_real @ Xs @ ( cons_poly_real @ Y @ Ys ) ) @ ( listrel1_poly_real @ R ) )
=> ( ! [X3: poly_real] :
( ( Xs
= ( cons_poly_real @ X3 @ Ys ) )
=> ~ ( member1784523250y_real @ ( produc1867601537y_real @ X3 @ Y ) @ R ) )
=> ~ ! [Zs: list_poly_real] :
( ( Xs
= ( cons_poly_real @ Y @ Zs ) )
=> ~ ( member403334290y_real @ ( produc1000427809y_real @ Zs @ Ys ) @ ( listrel1_poly_real @ R ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_308_Cons__listrel1E1,axiom,
! [X: poly_real,Xs: list_poly_real,Ys: list_poly_real,R: set_Pr483210409y_real] :
( ( member403334290y_real @ ( produc1000427809y_real @ ( cons_poly_real @ X @ Xs ) @ Ys ) @ ( listrel1_poly_real @ R ) )
=> ( ! [Y2: poly_real] :
( ( Ys
= ( cons_poly_real @ Y2 @ Xs ) )
=> ~ ( member1784523250y_real @ ( produc1867601537y_real @ X @ Y2 ) @ R ) )
=> ~ ! [Zs: list_poly_real] :
( ( Ys
= ( cons_poly_real @ X @ Zs ) )
=> ~ ( member403334290y_real @ ( produc1000427809y_real @ Xs @ Zs ) @ ( listrel1_poly_real @ R ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_309_listrel1I1,axiom,
! [X: poly_real,Y: poly_real,R: set_Pr483210409y_real,Xs: list_poly_real] :
( ( member1784523250y_real @ ( produc1867601537y_real @ X @ Y ) @ R )
=> ( member403334290y_real @ ( produc1000427809y_real @ ( cons_poly_real @ X @ Xs ) @ ( cons_poly_real @ Y @ Xs ) ) @ ( listrel1_poly_real @ R ) ) ) ).
% listrel1I1
thf(fact_310_same__length__different,axiom,
! [Xs: list_poly_real,Ys: list_poly_real] :
( ( Xs != Ys )
=> ( ( ( size_s259235672y_real @ Xs )
= ( size_s259235672y_real @ Ys ) )
=> ? [Pre: list_poly_real,X3: poly_real,Xs5: list_poly_real,Y2: poly_real,Ys5: list_poly_real] :
( ( X3 != Y2 )
& ( Xs
= ( append_poly_real @ Pre @ ( append_poly_real @ ( cons_poly_real @ X3 @ nil_poly_real ) @ Xs5 ) ) )
& ( Ys
= ( append_poly_real @ Pre @ ( append_poly_real @ ( cons_poly_real @ Y2 @ nil_poly_real ) @ Ys5 ) ) ) ) ) ) ).
% same_length_different
thf(fact_311_listrel1E,axiom,
! [Xs: list_poly_real,Ys: list_poly_real,R: set_Pr483210409y_real] :
( ( member403334290y_real @ ( produc1000427809y_real @ Xs @ Ys ) @ ( listrel1_poly_real @ R ) )
=> ~ ! [X3: poly_real,Y2: poly_real] :
( ( member1784523250y_real @ ( produc1867601537y_real @ X3 @ Y2 ) @ R )
=> ! [Us: list_poly_real,Vs: list_poly_real] :
( ( Xs
= ( append_poly_real @ Us @ ( cons_poly_real @ X3 @ Vs ) ) )
=> ( Ys
!= ( append_poly_real @ Us @ ( cons_poly_real @ Y2 @ Vs ) ) ) ) ) ) ).
% listrel1E
thf(fact_312_listrel1I,axiom,
! [X: poly_real,Y: poly_real,R: set_Pr483210409y_real,Xs: list_poly_real,Us2: list_poly_real,Vs2: list_poly_real,Ys: list_poly_real] :
( ( member1784523250y_real @ ( produc1867601537y_real @ X @ Y ) @ R )
=> ( ( Xs
= ( append_poly_real @ Us2 @ ( cons_poly_real @ X @ Vs2 ) ) )
=> ( ( Ys
= ( append_poly_real @ Us2 @ ( cons_poly_real @ Y @ Vs2 ) ) )
=> ( member403334290y_real @ ( produc1000427809y_real @ Xs @ Ys ) @ ( listrel1_poly_real @ R ) ) ) ) ) ).
% listrel1I
thf(fact_313_length__append__singleton,axiom,
! [Xs: list_poly_real,X: poly_real] :
( ( size_s259235672y_real @ ( append_poly_real @ Xs @ ( cons_poly_real @ X @ nil_poly_real ) ) )
= ( suc @ ( size_s259235672y_real @ Xs ) ) ) ).
% length_append_singleton
thf(fact_314_subset__eq__mset__impl_Ocases,axiom,
! [X: produc1426596841y_real] :
( ! [Ys2: list_poly_real] :
( X
!= ( produc1000427809y_real @ nil_poly_real @ Ys2 ) )
=> ~ ! [X3: poly_real,Xs2: list_poly_real,Ys2: list_poly_real] :
( X
!= ( produc1000427809y_real @ ( cons_poly_real @ X3 @ Xs2 ) @ Ys2 ) ) ) ).
% subset_eq_mset_impl.cases
thf(fact_315_length__Cons,axiom,
! [X: poly_real,Xs: list_poly_real] :
( ( size_s259235672y_real @ ( cons_poly_real @ X @ Xs ) )
= ( suc @ ( size_s259235672y_real @ Xs ) ) ) ).
% length_Cons
thf(fact_316_Cons__in__lex,axiom,
! [X: poly_real,Xs: list_poly_real,Y: poly_real,Ys: list_poly_real,R: set_Pr483210409y_real] :
( ( member403334290y_real @ ( produc1000427809y_real @ ( cons_poly_real @ X @ Xs ) @ ( cons_poly_real @ Y @ Ys ) ) @ ( lex_poly_real @ R ) )
= ( ( ( member1784523250y_real @ ( produc1867601537y_real @ X @ Y ) @ R )
& ( ( size_s259235672y_real @ Xs )
= ( size_s259235672y_real @ Ys ) ) )
| ( ( X = Y )
& ( member403334290y_real @ ( produc1000427809y_real @ Xs @ Ys ) @ ( lex_poly_real @ R ) ) ) ) ) ).
% Cons_in_lex
thf(fact_317_Nil2__notin__lex,axiom,
! [Xs: list_poly_real,R: set_Pr483210409y_real] :
~ ( member403334290y_real @ ( produc1000427809y_real @ Xs @ nil_poly_real ) @ ( lex_poly_real @ R ) ) ).
% Nil2_notin_lex
thf(fact_318_Nil__notin__lex,axiom,
! [Ys: list_poly_real,R: set_Pr483210409y_real] :
~ ( member403334290y_real @ ( produc1000427809y_real @ nil_poly_real @ Ys ) @ ( lex_poly_real @ R ) ) ).
% Nil_notin_lex
thf(fact_319_SuccD,axiom,
! [K: poly_real,Kl: set_list_poly_real,Kl2: list_poly_real] :
( ( member_poly_real2 @ K @ ( bNF_Gr1644259033y_real @ Kl @ Kl2 ) )
=> ( member708095579y_real @ ( append_poly_real @ Kl2 @ ( cons_poly_real @ K @ nil_poly_real ) ) @ Kl ) ) ).
% SuccD
thf(fact_320_SuccI,axiom,
! [Kl2: list_poly_real,K: poly_real,Kl: set_list_poly_real] :
( ( member708095579y_real @ ( append_poly_real @ Kl2 @ ( cons_poly_real @ K @ nil_poly_real ) ) @ Kl )
=> ( member_poly_real2 @ K @ ( bNF_Gr1644259033y_real @ Kl @ Kl2 ) ) ) ).
% SuccI
thf(fact_321_empty__Shift,axiom,
! [Kl: set_list_poly_real,K: poly_real] :
( ( member708095579y_real @ nil_poly_real @ Kl )
=> ( ( member_poly_real2 @ K @ ( bNF_Gr1644259033y_real @ Kl @ nil_poly_real ) )
=> ( member708095579y_real @ nil_poly_real @ ( bNF_Gr851617109y_real @ Kl @ K ) ) ) ) ).
% empty_Shift
thf(fact_322_Succ__Shift,axiom,
! [Kl: set_list_poly_real,K: poly_real,Kl2: list_poly_real] :
( ( bNF_Gr1644259033y_real @ ( bNF_Gr851617109y_real @ Kl @ K ) @ Kl2 )
= ( bNF_Gr1644259033y_real @ Kl @ ( cons_poly_real @ K @ Kl2 ) ) ) ).
% Succ_Shift
thf(fact_323_ShiftD,axiom,
! [Kl2: list_poly_real,Kl: set_list_poly_real,K: poly_real] :
( ( member708095579y_real @ Kl2 @ ( bNF_Gr851617109y_real @ Kl @ K ) )
=> ( member708095579y_real @ ( cons_poly_real @ K @ Kl2 ) @ Kl ) ) ).
% ShiftD
thf(fact_324_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M )
= ( times_times_nat @ ( suc @ K ) @ N ) )
= ( M = N ) ) ).
% Suc_mult_cancel1
thf(fact_325_lexord__append__left__rightI,axiom,
! [A3: poly_real,B: poly_real,R: set_Pr483210409y_real,U: list_poly_real,X: list_poly_real,Y: list_poly_real] :
( ( member1784523250y_real @ ( produc1867601537y_real @ A3 @ B ) @ R )
=> ( member403334290y_real @ ( produc1000427809y_real @ ( append_poly_real @ U @ ( cons_poly_real @ A3 @ X ) ) @ ( append_poly_real @ U @ ( cons_poly_real @ B @ Y ) ) ) @ ( lexord_poly_real @ R ) ) ) ).
% lexord_append_left_rightI
thf(fact_326_listrel_Oinducts,axiom,
! [X1: list_poly_real,X2: list_poly_real,R: set_Pr483210409y_real,P3: list_poly_real > list_poly_real > $o] :
( ( member403334290y_real @ ( produc1000427809y_real @ X1 @ X2 ) @ ( listre572247515y_real @ R ) )
=> ( ( P3 @ nil_poly_real @ nil_poly_real )
=> ( ! [X3: poly_real,Y2: poly_real,Xs2: list_poly_real,Ys2: list_poly_real] :
( ( member1784523250y_real @ ( produc1867601537y_real @ X3 @ Y2 ) @ R )
=> ( ( member403334290y_real @ ( produc1000427809y_real @ Xs2 @ Ys2 ) @ ( listre572247515y_real @ R ) )
=> ( ( P3 @ Xs2 @ Ys2 )
=> ( P3 @ ( cons_poly_real @ X3 @ Xs2 ) @ ( cons_poly_real @ Y2 @ Ys2 ) ) ) ) )
=> ( P3 @ X1 @ X2 ) ) ) ) ).
% listrel.inducts
thf(fact_327_lexord__cons__cons,axiom,
! [A3: poly_real,X: list_poly_real,B: poly_real,Y: list_poly_real,R: set_Pr483210409y_real] :
( ( member403334290y_real @ ( produc1000427809y_real @ ( cons_poly_real @ A3 @ X ) @ ( cons_poly_real @ B @ Y ) ) @ ( lexord_poly_real @ R ) )
= ( ( member1784523250y_real @ ( produc1867601537y_real @ A3 @ B ) @ R )
| ( ( A3 = B )
& ( member403334290y_real @ ( produc1000427809y_real @ X @ Y ) @ ( lexord_poly_real @ R ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_328_lexord__Nil__left,axiom,
! [Y: list_poly_real,R: set_Pr483210409y_real] :
( ( member403334290y_real @ ( produc1000427809y_real @ nil_poly_real @ Y ) @ ( lexord_poly_real @ R ) )
= ( ? [A5: poly_real,X4: list_poly_real] :
( Y
= ( cons_poly_real @ A5 @ X4 ) ) ) ) ).
% lexord_Nil_left
thf(fact_329_lexord__Nil__right,axiom,
! [X: list_poly_real,R: set_Pr483210409y_real] :
~ ( member403334290y_real @ ( produc1000427809y_real @ X @ nil_poly_real ) @ ( lexord_poly_real @ R ) ) ).
% lexord_Nil_right
thf(fact_330_listrel__Nil2,axiom,
! [Xs: list_poly_real,R: set_Pr483210409y_real] :
( ( member403334290y_real @ ( produc1000427809y_real @ Xs @ nil_poly_real ) @ ( listre572247515y_real @ R ) )
=> ( Xs = nil_poly_real ) ) ).
% listrel_Nil2
thf(fact_331_listrel__Nil1,axiom,
! [Xs: list_poly_real,R: set_Pr483210409y_real] :
( ( member403334290y_real @ ( produc1000427809y_real @ nil_poly_real @ Xs ) @ ( listre572247515y_real @ R ) )
=> ( Xs = nil_poly_real ) ) ).
% listrel_Nil1
thf(fact_332_listrel_ONil,axiom,
! [R: set_Pr483210409y_real] : ( member403334290y_real @ ( produc1000427809y_real @ nil_poly_real @ nil_poly_real ) @ ( listre572247515y_real @ R ) ) ).
% listrel.Nil
thf(fact_333_listrel__Cons2,axiom,
! [Xs: list_poly_real,Y: poly_real,Ys: list_poly_real,R: set_Pr483210409y_real] :
( ( member403334290y_real @ ( produc1000427809y_real @ Xs @ ( cons_poly_real @ Y @ Ys ) ) @ ( listre572247515y_real @ R ) )
=> ~ ! [X3: poly_real,Xs2: list_poly_real] :
( ( Xs
= ( cons_poly_real @ X3 @ Xs2 ) )
=> ( ( member1784523250y_real @ ( produc1867601537y_real @ X3 @ Y ) @ R )
=> ~ ( member403334290y_real @ ( produc1000427809y_real @ Xs2 @ Ys ) @ ( listre572247515y_real @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_334_listrel__Cons1,axiom,
! [Y: poly_real,Ys: list_poly_real,Xs: list_poly_real,R: set_Pr483210409y_real] :
( ( member403334290y_real @ ( produc1000427809y_real @ ( cons_poly_real @ Y @ Ys ) @ Xs ) @ ( listre572247515y_real @ R ) )
=> ~ ! [Y2: poly_real,Ys2: list_poly_real] :
( ( Xs
= ( cons_poly_real @ Y2 @ Ys2 ) )
=> ( ( member1784523250y_real @ ( produc1867601537y_real @ Y @ Y2 ) @ R )
=> ~ ( member403334290y_real @ ( produc1000427809y_real @ Ys @ Ys2 ) @ ( listre572247515y_real @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_335_listrel_OCons,axiom,
! [X: poly_real,Y: poly_real,R: set_Pr483210409y_real,Xs: list_poly_real,Ys: list_poly_real] :
( ( member1784523250y_real @ ( produc1867601537y_real @ X @ Y ) @ R )
=> ( ( member403334290y_real @ ( produc1000427809y_real @ Xs @ Ys ) @ ( listre572247515y_real @ R ) )
=> ( member403334290y_real @ ( produc1000427809y_real @ ( cons_poly_real @ X @ Xs ) @ ( cons_poly_real @ Y @ Ys ) ) @ ( listre572247515y_real @ R ) ) ) ) ).
% listrel.Cons
thf(fact_336_lexord__append__rightI,axiom,
! [Y: list_poly_real,X: list_poly_real,R: set_Pr483210409y_real] :
( ? [B2: poly_real,Z2: list_poly_real] :
( Y
= ( cons_poly_real @ B2 @ Z2 ) )
=> ( member403334290y_real @ ( produc1000427809y_real @ X @ ( append_poly_real @ X @ Y ) ) @ ( lexord_poly_real @ R ) ) ) ).
% lexord_append_rightI
thf(fact_337_listrel_Ocases,axiom,
! [A12: list_poly_real,A22: list_poly_real,R: set_Pr483210409y_real] :
( ( member403334290y_real @ ( produc1000427809y_real @ A12 @ A22 ) @ ( listre572247515y_real @ R ) )
=> ( ( ( A12 = nil_poly_real )
=> ( A22 != nil_poly_real ) )
=> ~ ! [X3: poly_real,Y2: poly_real,Xs2: list_poly_real] :
( ( A12
= ( cons_poly_real @ X3 @ Xs2 ) )
=> ! [Ys2: list_poly_real] :
( ( A22
= ( cons_poly_real @ Y2 @ Ys2 ) )
=> ( ( member1784523250y_real @ ( produc1867601537y_real @ X3 @ Y2 ) @ R )
=> ~ ( member403334290y_real @ ( produc1000427809y_real @ Xs2 @ Ys2 ) @ ( listre572247515y_real @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_338_listrel_Osimps,axiom,
! [A12: list_poly_real,A22: list_poly_real,R: set_Pr483210409y_real] :
( ( member403334290y_real @ ( produc1000427809y_real @ A12 @ A22 ) @ ( listre572247515y_real @ R ) )
= ( ( ( A12 = nil_poly_real )
& ( A22 = nil_poly_real ) )
| ? [X4: poly_real,Y3: poly_real,Xs3: list_poly_real,Ys3: list_poly_real] :
( ( A12
= ( cons_poly_real @ X4 @ Xs3 ) )
& ( A22
= ( cons_poly_real @ Y3 @ Ys3 ) )
& ( member1784523250y_real @ ( produc1867601537y_real @ X4 @ Y3 ) @ R )
& ( member403334290y_real @ ( produc1000427809y_real @ Xs3 @ Ys3 ) @ ( listre572247515y_real @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_339_map__tailrec__rev_Opelims,axiom,
! [X: poly_real > poly_real,Xa: list_poly_real,Xb: list_poly_real,Y: list_poly_real] :
( ( ( map_ta1919289823y_real @ X @ Xa @ Xb )
= Y )
=> ( ( accp_P948088014y_real @ map_ta954657966y_real @ ( produc2087464649y_real @ X @ ( produc1000427809y_real @ Xa @ Xb ) ) )
=> ( ( ( Xa = nil_poly_real )
=> ( ( Y = Xb )
=> ~ ( accp_P948088014y_real @ map_ta954657966y_real @ ( produc2087464649y_real @ X @ ( produc1000427809y_real @ nil_poly_real @ Xb ) ) ) ) )
=> ~ ! [A: poly_real,As: list_poly_real] :
( ( Xa
= ( cons_poly_real @ A @ As ) )
=> ( ( Y
= ( map_ta1919289823y_real @ X @ As @ ( cons_poly_real @ ( X @ A ) @ Xb ) ) )
=> ~ ( accp_P948088014y_real @ map_ta954657966y_real @ ( produc2087464649y_real @ X @ ( produc1000427809y_real @ ( cons_poly_real @ A @ As ) @ Xb ) ) ) ) ) ) ) ) ).
% map_tailrec_rev.pelims
thf(fact_340_Nil__lenlex__iff1,axiom,
! [Ns: list_poly_real,R: set_Pr483210409y_real] :
( ( member403334290y_real @ ( produc1000427809y_real @ nil_poly_real @ Ns ) @ ( lenlex_poly_real @ R ) )
= ( Ns != nil_poly_real ) ) ).
% Nil_lenlex_iff1
thf(fact_341_Nil__lenlex__iff2,axiom,
! [Ns: list_poly_real,R: set_Pr483210409y_real] :
~ ( member403334290y_real @ ( produc1000427809y_real @ Ns @ nil_poly_real ) @ ( lenlex_poly_real @ R ) ) ).
% Nil_lenlex_iff2
thf(fact_342_Cons__lenlex__iff,axiom,
! [M: poly_real,Ms: list_poly_real,N: poly_real,Ns: list_poly_real,R: set_Pr483210409y_real] :
( ( member403334290y_real @ ( produc1000427809y_real @ ( cons_poly_real @ M @ Ms ) @ ( cons_poly_real @ N @ Ns ) ) @ ( lenlex_poly_real @ R ) )
= ( ( ord_less_nat @ ( size_s259235672y_real @ Ms ) @ ( size_s259235672y_real @ Ns ) )
| ( ( ( size_s259235672y_real @ Ms )
= ( size_s259235672y_real @ Ns ) )
& ( member1784523250y_real @ ( produc1867601537y_real @ M @ N ) @ R ) )
| ( ( M = N )
& ( member403334290y_real @ ( produc1000427809y_real @ Ms @ Ns ) @ ( lenlex_poly_real @ R ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_343_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_344_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_345_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_346_list_Omap__disc__iff,axiom,
! [F: poly_real > poly_real,A3: list_poly_real] :
( ( ( map_po2132607150y_real @ F @ A3 )
= nil_poly_real )
= ( A3 = nil_poly_real ) ) ).
% list.map_disc_iff
thf(fact_347_map__is__Nil__conv,axiom,
! [F: poly_real > poly_real,Xs: list_poly_real] :
( ( ( map_po2132607150y_real @ F @ Xs )
= nil_poly_real )
= ( Xs = nil_poly_real ) ) ).
% map_is_Nil_conv
thf(fact_348_Nil__is__map__conv,axiom,
! [F: poly_real > poly_real,Xs: list_poly_real] :
( ( nil_poly_real
= ( map_po2132607150y_real @ F @ Xs ) )
= ( Xs = nil_poly_real ) ) ).
% Nil_is_map_conv
% Conjectures (1)
thf(conj_0,conjecture,
~ ( sturm_891428828rm_seq @ nil_poly_real ) ).
%------------------------------------------------------------------------------