%------------------------------------------------------------------------------
% File     : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain   : Archive of Formal Proofs
% Problem  :
% Version  : Especial.
% English  :

% Refs     : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source   : [Des23]
% Names    : Multiset_Ordering_NPC/0003_Multiset_Ordering_NP_Hard/prob_00069_003111__14024990_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1104 ( 443 unt; 390 typ;   0 def)
%            Number of atoms       : 1770 (1679 equ;   0 cnn)
%            Maximal formula atoms :   11 (   2 avg)
%            Number of connectives : 6719 ( 491   ~;  29   |; 244   &;5272   @)
%                                         (   0 <=>; 683  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   29 (   6 avg)
%            Number of types       :  117 ( 116 usr)
%            Number of type conns  :  939 ( 939   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  277 ( 274 usr;  31 con; 0-3 aty)
%            Number of variables   : 2774 (  68   ^;2488   !; 218   ?;2774   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-18 16:30:26.131
%------------------------------------------------------------------------------
% Could-be-implicit typings (116)
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_M_Eo_J_J_J_J,type,
    produc9011176785203808030_nat_o: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_Mt__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_J_J_J,type,
    produc3053421305257255469on_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_J_J_J,type,
    produc3665718167116523309on_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_M_Eo_J_J_J_J,type,
    produc7775502106177927606_nat_o: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_J_J,type,
    produc7326124634926975417on_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_J_J,type,
    produc3060927313823239865on_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_J_J_J_J,type,
    produc5056679816642863195on_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_M_Eo_J_J_J,type,
    produc2753127401226480184_nat_o: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_Mt__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_J_J,type,
    produc8815734051188430663on_nat: $tType ).

thf(ty_n_t__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_J_J_J_J,type,
    list_l312409054758248841on_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_J_J_J,type,
    produc6588776972112385165on_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_J_J,type,
    produc6063590850804571341on_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_J_J_J,type,
    list_P6215481986061871491on_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_M_Eo_J_J_J,type,
    produc4594631521614525130_nat_o: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_J_J,type,
    produc8325617224762090749on_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_I_062_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_M_062_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_M_Eo_J_J_Mt__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_J,type,
    produc5764678590178156288on_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_J,type,
    produc8563703056922213325on_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_J,type,
    produc629934850357470163on_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_M_062_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_M_Eo_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J,type,
    produc2264144286178290682on_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_I_062_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_J_J,type,
    produc7012556527198387091on_nat: $tType ).

thf(ty_n_t__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_J_J_J,type,
    list_l2045081698307616223on_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_J_J,type,
    list_P5508476027986415439on_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_J,type,
    produc118098540238352959on_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_M_Eo_J_J_J_J,type,
    produc1666888912113777716_nat_o: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_I_062_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_J,type,
    produc1098090131618875051on_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_J_J,type,
    produc2892159590233455681on_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_J_J,type,
    produc2778868904637641793on_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_M_Eo_J_J,type,
    produc7496903103639383512_nat_o: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_M_Eo_J_J_J_J,type,
    produc8288325133855054004_nat_o: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_Mt__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_J,type,
    produc2417963485646110695on_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_M_Eo_J_J,type,
    produc4640512942233155940_nat_o: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_J,type,
    produc3977234058138274081on_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_J,type,
    produc4714912333100831009on_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_J,type,
    produc5827802662881200621on_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_J,type,
    list_P8734936519038225773on_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J,type,
    produc1714117091004681831on_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J,type,
    produc8413567841624424429on_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_I_062_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J,type,
    produc3717180461370874877on_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_M_Eo_J_J_J,type,
    produc3724276615328212598_nat_o: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_J,type,
    produc8096917655042337667on_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_J,type,
    produc41935680912008307on_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_M_Eo_J_J_J,type,
    produc8214513296305107590_nat_o: $tType ).

thf(ty_n_t__Product____Type__Oprod_I_062_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_M_062_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_M_Eo_J_J_Mt__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J,type,
    produc1732627278533666408on_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J,type,
    produc4387126377693847539on_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J,type,
    produc6827644173363976803on_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J,type,
    produc3922897021252936937on_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_I_062_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_M_062_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_M_Eo_J_J_Mt__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J,type,
    produc3191134443298988280on_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_M_Eo_J_J,type,
    produc1737863824307799982_nat_o: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J,type,
    produc6098023079705189563on_nat: $tType ).

thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_J_J,type,
    list_l5700991078898051731on_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_M_Eo_J_J,type,
    produc3152382218381312302_nat_o: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_M_Eo_J,type,
    produc6963029029708228856_nat_o: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J,type,
    produc630436911041420075on_nat: $tType ).

thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_J,type,
    list_l6719245731233385357on_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_M_Eo_J,type,
    produc1376987297294124798_nat_o: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J,type,
    produc5580075136493545115on_nat: $tType ).

thf(ty_n_t__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J,type,
    list_l1726306699111145479on_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J,type,
    produc952157413385281163on_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J,type,
    list_P9127870862658512897on_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J,type,
    produc1313295348606329723on_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_M_Eo_J_J_J_J,type,
    produc3630873615015234996_nat_o: $tType ).

thf(ty_n_t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_M_Eo_J_J_J_J,type,
    produc2716792995815215274st_a_o: $tType ).

thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_J,type,
    list_l7167270332776620429on_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J_J,type,
    produc7312811325926348135st_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__Nat__Onat_J_J_J_J,type,
    produc5923727742716231015st_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_M_Eo_J,type,
    produc7819550649547605728_nat_o: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_M_Eo_J_J_J,type,
    produc7413256671508406019_nat_o: $tType ).

thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J,type,
    list_l8365940747029150845on_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_M_Eo_J,type,
    produc103519031730445424_nat_o: $tType ).

thf(ty_n_t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J_J_J,type,
    produc8880992058355704947list_a: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_Itf__a_Mt__List__Olist_Itf__a_J_J_J_J,type,
    produc1874289995683674227list_a: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_M_Eo_J_J_J,type,
    produc8398638096451127297st_a_o: $tType ).

thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__List__Olist_It__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_J,type,
    produc1458239053189343596at_nat: $tType ).

thf(ty_n_t__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J,type,
    list_l8999537460041914605on_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
    produc5438398690206989622st_nat: $tType ).

thf(ty_n_t__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J,type,
    list_S5253404628753938141on_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_Itf__a_Mt__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_J_J,type,
    produc573162381751850459tation: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
    produc8196726482729697190st_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J_J,type,
    produc1553995403754578250list_a: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_M_Eo_J_J,type,
    produc430777597369845998_nat_o: $tType ).

thf(ty_n_t__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J,type,
    sum_su4907927656612468301on_nat: $tType ).

thf(ty_n_t__List__Olist_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_J_J,type,
    list_l3295615718716739902tation: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    produc8472197452120411308at_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_Itf__a_Mt__List__Olist_Itf__a_J_J_J,type,
    produc3854044251064639184list_a: $tType ).

thf(ty_n_t__List__Olist_It__List__Olist_It__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J_J,type,
    list_l5863831568797673340at_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_M_Eo_J_J,type,
    produc6745688853455182174st_a_o: $tType ).

thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    produc254973753779126261st_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_J,type,
    list_P7561118020193646638tation: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
    list_P5366131564312172150at_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    produc1828647624359046049st_nat: $tType ).

thf(ty_n_t__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    list_l3264859301627795341at_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_I_062_Itf__a_M_062_Itf__a_M_Eo_J_J_Mt__List__Olist_Itf__a_J_J,type,
    produc5032551385658279741list_a: $tType ).

thf(ty_n_t__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J,type,
    produc27318731269267614tation: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
    produc9164743771328383783list_a: $tType ).

thf(ty_n_t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
    produc8199716216217303280at_nat: $tType ).

thf(ty_n_t__List__Olist_It__List__Olist_It__Product____Type__Oprod_Itf__a_M_Eo_J_J_J,type,
    list_l8336264072933094989od_a_o: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__Nat__Onat_J_J,type,
    produc4575160907756185873st_nat: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    list_P6011104703257516679at_nat: $tType ).

thf(ty_n_t__List__Olist_It__Multiset____Ordering____NP____Hard__OAnnotation_J,type,
    list_M7834926499894101121tation: $tType ).

thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_It__Nat__Onat_J_J_J,type,
    list_list_list_nat: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
    produc8122078169606880685_nat_o: $tType ).

thf(ty_n_t__Product____Type__Oprod_Itf__a_Mt__List__Olist_Itf__a_J_J,type,
    produc8685980395799941037list_a: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__List__Olist_Itf__a_J_M_Eo_J,type,
    produc5207931094065476923st_a_o: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_Itf__a_M_Eo_J_J,type,
    list_P3196869577824546119od_a_o: $tType ).

thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    product_prod_nat_nat: $tType ).

thf(ty_n_t__List__Olist_I_062_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    list_nat_nat: $tType ).

thf(ty_n_t__Multiset____Ordering____NP____Hard__OAnnotation,type,
    multis8755981257395730801tation: $tType ).

thf(ty_n_t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
    list_list_nat: $tType ).

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

thf(ty_n_t__Product____Type__Oprod_Itf__a_M_Eo_J,type,
    product_prod_a_o: $tType ).

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

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

thf(ty_n_t__String__Ochar,type,
    char: $tType ).

thf(ty_n_t__Num__Onum,type,
    num: $tType ).

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

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

% Explicit typings (274)
thf(sy_c_Euclidean__Division_Odivmod__nat,type,
    euclidean_divmod_nat: nat > nat > product_prod_nat_nat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
    minus_minus_nat: nat > nat > nat ).

thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
    one_one_nat: nat ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
    plus_plus_nat: nat > nat > nat ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
    zero_zero_nat: nat ).

thf(sy_c_If_001t__List__Olist_It__Nat__Onat_J,type,
    if_list_nat: $o > list_nat > list_nat > list_nat ).

thf(sy_c_If_001t__Nat__Onat,type,
    if_nat: $o > nat > nat > nat ).

thf(sy_c_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    if_Pro6206227464963214023at_nat: $o > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ).

thf(sy_c_List_Oappend_001t__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J,type,
    append4931807731165890082on_nat: list_l6719245731233385357on_nat > list_l6719245731233385357on_nat > list_l6719245731233385357on_nat ).

thf(sy_c_List_Oappend_001t__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J,type,
    append7602222762569088146on_nat: list_l8365940747029150845on_nat > list_l8365940747029150845on_nat > list_l8365940747029150845on_nat ).

thf(sy_c_List_Oappend_001t__List__Olist_It__Nat__Onat_J,type,
    append_list_nat: list_list_nat > list_list_nat > list_list_nat ).

thf(sy_c_List_Oappend_001t__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J,type,
    append3839213209832495004on_nat: list_l1726306699111145479on_nat > list_l1726306699111145479on_nat > list_l1726306699111145479on_nat ).

thf(sy_c_List_Oappend_001t__List__Olist_It__Product____Type__Oprod_Itf__a_M_Eo_J_J,type,
    append2159120467191820514od_a_o: list_l8336264072933094989od_a_o > list_l8336264072933094989od_a_o > list_l8336264072933094989od_a_o ).

thf(sy_c_List_Oappend_001t__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J,type,
    append6174841615030841602on_nat: list_l8999537460041914605on_nat > list_l8999537460041914605on_nat > list_l8999537460041914605on_nat ).

thf(sy_c_List_Oappend_001t__List__Olist_Itf__a_J,type,
    append_list_a: list_list_a > list_list_a > list_list_a ).

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

thf(sy_c_List_Oappend_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
    append8880802010960057739at_nat: list_P5366131564312172150at_nat > list_P5366131564312172150at_nat > list_P5366131564312172150at_nat ).

thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_J,type,
    append858208364616284260on_nat: list_P5508476027986415439on_nat > list_P5508476027986415439on_nat > list_P5508476027986415439on_nat ).

thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_J_J,type,
    append3872231119213297432on_nat: list_P6215481986061871491on_nat > list_P6215481986061871491on_nat > list_P6215481986061871491on_nat ).

thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    append985823374593552924at_nat: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat ).

thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J,type,
    append2865821678206098582on_nat: list_P9127870862658512897on_nat > list_P9127870862658512897on_nat > list_P9127870862658512897on_nat ).

thf(sy_c_List_Oappend_001t__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J,type,
    append4134760898919571651tation: list_P7561118020193646638tation > list_P7561118020193646638tation > list_P7561118020193646638tation ).

thf(sy_c_List_Oappend_001t__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J,type,
    append3523812819975775986on_nat: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > list_S5253404628753938141on_nat ).

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

thf(sy_c_List_Oconcat_001t__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J,type,
    concat1602011024853954504on_nat: list_l5700991078898051731on_nat > list_l6719245731233385357on_nat ).

thf(sy_c_List_Oconcat_001t__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J,type,
    concat6293066666318574188on_nat: list_l7167270332776620429on_nat > list_l8365940747029150845on_nat ).

thf(sy_c_List_Oconcat_001t__List__Olist_It__Nat__Onat_J,type,
    concat_list_nat: list_list_list_nat > list_list_nat ).

thf(sy_c_List_Oconcat_001t__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J,type,
    concat4341727598008994882on_nat: list_l6719245731233385357on_nat > list_l1726306699111145479on_nat ).

thf(sy_c_List_Oconcat_001t__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J,type,
    concat4774721536428481500on_nat: list_l8365940747029150845on_nat > list_l8999537460041914605on_nat ).

thf(sy_c_List_Oconcat_001t__Nat__Onat,type,
    concat_nat: list_list_nat > list_nat ).

thf(sy_c_List_Oconcat_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
    concat821400627202602033at_nat: list_l5863831568797673340at_nat > list_P5366131564312172150at_nat ).

thf(sy_c_List_Oconcat_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_J,type,
    concat1127826427744615230on_nat: list_l2045081698307616223on_nat > list_P5508476027986415439on_nat ).

thf(sy_c_List_Oconcat_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_J_J,type,
    concat9214455607871246526on_nat: list_l312409054758248841on_nat > list_P6215481986061871491on_nat ).

thf(sy_c_List_Oconcat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    concat7691415812945658306at_nat: list_l3264859301627795341at_nat > list_P6011104703257516679at_nat ).

thf(sy_c_List_Oconcat_001t__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J,type,
    concat937493773106420796on_nat: list_l1726306699111145479on_nat > list_P9127870862658512897on_nat ).

thf(sy_c_List_Oconcat_001t__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J,type,
    concat1037798000090658717tation: list_l3295615718716739902tation > list_P7561118020193646638tation ).

thf(sy_c_List_Oconcat_001t__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J,type,
    concat6161509665823917772on_nat: list_l8999537460041914605on_nat > list_S5253404628753938141on_nat ).

thf(sy_c_List_Oconcat_001tf__a,type,
    concat_a: list_list_a > list_a ).

thf(sy_c_List_Odrop_001t__Nat__Onat,type,
    drop_nat: nat > list_nat > list_nat ).

thf(sy_c_List_Olast_001t__Nat__Onat,type,
    last_nat: list_nat > nat ).

thf(sy_c_List_Olist_OCons_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
    cons_nat_nat: ( nat > nat ) > list_nat_nat > list_nat_nat ).

thf(sy_c_List_Olist_OCons_001t__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J,type,
    cons_l4979741563022480061on_nat: list_l1726306699111145479on_nat > list_l6719245731233385357on_nat > list_l6719245731233385357on_nat ).

thf(sy_c_List_Olist_OCons_001t__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J,type,
    cons_l6725403660475259319on_nat: list_l8999537460041914605on_nat > list_l8365940747029150845on_nat > list_l8365940747029150845on_nat ).

thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Nat__Onat_J,type,
    cons_list_nat: list_nat > list_list_nat > list_list_nat ).

thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J,type,
    cons_l3028799188195208631on_nat: list_P9127870862658512897on_nat > list_l1726306699111145479on_nat > list_l1726306699111145479on_nat ).

thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Product____Type__Oprod_Itf__a_M_Eo_J_J,type,
    cons_l9179691953475787517od_a_o: list_P3196869577824546119od_a_o > list_l8336264072933094989od_a_o > list_l8336264072933094989od_a_o ).

thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J,type,
    cons_l5857885737047467175on_nat: list_S5253404628753938141on_nat > list_l8999537460041914605on_nat > list_l8999537460041914605on_nat ).

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

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

thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
    cons_P4219629788700907686at_nat: produc8199716216217303280at_nat > list_P5366131564312172150at_nat > list_P5366131564312172150at_nat ).

thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_J,type,
    cons_P4518942764240009225on_nat: produc118098540238352959on_nat > list_P5508476027986415439on_nat > list_P5508476027986415439on_nat ).

thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_J_J,type,
    cons_P8471348786669567923on_nat: produc8325617224762090749on_nat > list_P6215481986061871491on_nat > list_P6215481986061871491on_nat ).

thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    cons_P6512896166579812791at_nat: product_prod_nat_nat > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat ).

thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J,type,
    cons_P1582596238967110705on_nat: produc1313295348606329723on_nat > list_P9127870862658512897on_nat > list_P9127870862658512897on_nat ).

thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J,type,
    cons_P3184545829724658280tation: produc27318731269267614tation > list_P7561118020193646638tation > list_P7561118020193646638tation ).

thf(sy_c_List_Olist_OCons_001t__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J,type,
    cons_S6797538523646374167on_nat: sum_su4907927656612468301on_nat > list_S5253404628753938141on_nat > list_S5253404628753938141on_nat ).

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

thf(sy_c_List_Olist_ONil_001t__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J,type,
    nil_li985476221562969869on_nat: list_l6719245731233385357on_nat ).

thf(sy_c_List_Olist_ONil_001t__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J,type,
    nil_li3323240303193190759on_nat: list_l8365940747029150845on_nat ).

thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Nat__Onat_J,type,
    nil_list_nat: list_list_nat ).

thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J,type,
    nil_li2682182165152866823on_nat: list_l1726306699111145479on_nat ).

thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Product____Type__Oprod_Itf__a_M_Eo_J_J,type,
    nil_li1372968520647675725od_a_o: list_l8336264072933094989od_a_o ).

thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J,type,
    nil_li5563851662744376407on_nat: list_l8999537460041914605on_nat ).

thf(sy_c_List_Olist_ONil_001t__List__Olist_Itf__a_J,type,
    nil_list_a: list_list_a ).

thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
    nil_nat: list_nat ).

thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
    nil_Pr2223394031645031670at_nat: list_P5366131564312172150at_nat ).

thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_J,type,
    nil_Pr2927124501823088569on_nat: list_P5508476027986415439on_nat ).

thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_J_J,type,
    nil_Pr5986419849129872387on_nat: list_P6215481986061871491on_nat ).

thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    nil_Pr5478986624290739719at_nat: list_P6011104703257516679at_nat ).

thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J,type,
    nil_Pr8614093721167239297on_nat: list_P9127870862658512897on_nat ).

thf(sy_c_List_Olist_ONil_001t__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J,type,
    nil_Pr4033316107961887256tation: list_P7561118020193646638tation ).

thf(sy_c_List_Olist_ONil_001t__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J,type,
    nil_Su3078906344068544711on_nat: list_S5253404628753938141on_nat ).

thf(sy_c_List_Olist_ONil_001tf__a,type,
    nil_a: list_a ).

thf(sy_c_List_Olist_Ohd_001t__Nat__Onat,type,
    hd_nat: list_nat > nat ).

thf(sy_c_List_Olist_Omap_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__List__Olist_It__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
    map_na1011612288921435825at_nat: ( ( nat > nat ) > list_P5366131564312172150at_nat ) > list_nat_nat > list_l5863831568797673340at_nat ).

thf(sy_c_List_Olist_Omap_001t__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_001t__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J,type,
    map_li7104156278154209540on_nat: ( list_l1726306699111145479on_nat > list_l1726306699111145479on_nat ) > list_l6719245731233385357on_nat > list_l6719245731233385357on_nat ).

thf(sy_c_List_Olist_Omap_001t__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_001t__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J,type,
    map_li3413953685577837472on_nat: ( list_l8999537460041914605on_nat > list_l8999537460041914605on_nat ) > list_l8365940747029150845on_nat > list_l8365940747029150845on_nat ).

thf(sy_c_List_Olist_Omap_001t__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_001t__List__Olist_It__Nat__Onat_J,type,
    map_li2940836757168442131st_nat: ( list_l8999537460041914605on_nat > list_nat ) > list_l8365940747029150845on_nat > list_list_nat ).

thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
    map_li960784813134754710st_nat: ( list_nat > list_list_nat ) > list_list_nat > list_list_list_nat ).

thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J,type,
    map_li740583212113058067on_nat: ( list_nat > list_l8999537460041914605on_nat ) > list_list_nat > list_l8365940747029150845on_nat ).

thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
    map_li7225945977422193158st_nat: ( list_nat > list_nat ) > list_list_nat > list_list_nat ).

thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J,type,
    map_li1427182934947234307on_nat: ( list_nat > list_S5253404628753938141on_nat ) > list_list_nat > list_l8999537460041914605on_nat ).

thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_001t__List__Olist_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_J,type,
    map_li1709308397223692048on_nat: ( list_P9127870862658512897on_nat > list_l6719245731233385357on_nat ) > list_l1726306699111145479on_nat > list_l5700991078898051731on_nat ).

thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_001t__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J,type,
    map_li815642509948034692on_nat: ( list_P9127870862658512897on_nat > list_P9127870862658512897on_nat ) > list_l1726306699111145479on_nat > list_l1726306699111145479on_nat ).

thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Product____Type__Oprod_Itf__a_M_Eo_J_J_001t__List__Olist_It__Nat__Onat_J,type,
    map_li3237930340688975459st_nat: ( list_P3196869577824546119od_a_o > list_nat ) > list_l8336264072933094989od_a_o > list_list_nat ).

thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Product____Type__Oprod_Itf__a_M_Eo_J_J_001t__List__Olist_It__Product____Type__Oprod_Itf__a_M_Eo_J_J,type,
    map_li880244354692327940od_a_o: ( list_P3196869577824546119od_a_o > list_P3196869577824546119od_a_o ) > list_l8336264072933094989od_a_o > list_l8336264072933094989od_a_o ).

thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Product____Type__Oprod_Itf__a_M_Eo_J_J_001t__Nat__Onat,type,
    map_li381573783031109843_o_nat: ( list_P3196869577824546119od_a_o > nat ) > list_l8336264072933094989od_a_o > list_nat ).

thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_001t__List__Olist_It__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J,type,
    map_li2998272717593060896on_nat: ( list_S5253404628753938141on_nat > list_l8365940747029150845on_nat ) > list_l8999537460041914605on_nat > list_l7167270332776620429on_nat ).

thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_001t__List__Olist_It__Nat__Onat_J,type,
    map_li2358237010938824579st_nat: ( list_S5253404628753938141on_nat > list_nat ) > list_l8999537460041914605on_nat > list_list_nat ).

thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_001t__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_J_J,type,
    map_li6308922837919316594on_nat: ( list_S5253404628753938141on_nat > list_P5508476027986415439on_nat ) > list_l8999537460041914605on_nat > list_l2045081698307616223on_nat ).

thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_001t__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_J_J_J,type,
    map_li5621934489530828832on_nat: ( list_S5253404628753938141on_nat > list_P6215481986061871491on_nat ) > list_l8999537460041914605on_nat > list_l312409054758248841on_nat ).

thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_001t__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J,type,
    map_li4295621887133137792on_nat: ( list_S5253404628753938141on_nat > list_S5253404628753938141on_nat ) > list_l8999537460041914605on_nat > list_l8999537460041914605on_nat ).

thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
    map_li2142278717278591475at_nat: ( list_S5253404628753938141on_nat > nat ) > list_l8999537460041914605on_nat > list_nat ).

thf(sy_c_List_Olist_Omap_001t__List__Olist_Itf__a_J_001t__List__Olist_It__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J,type,
    map_li6744039989378561341on_nat: ( list_a > list_l8365940747029150845on_nat ) > list_list_a > list_l7167270332776620429on_nat ).

thf(sy_c_List_Olist_Omap_001t__List__Olist_Itf__a_J_001t__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J,type,
    map_li7277942993266323885on_nat: ( list_a > list_l8999537460041914605on_nat ) > list_list_a > list_l8365940747029150845on_nat ).

thf(sy_c_List_Olist_Omap_001t__List__Olist_Itf__a_J_001t__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J,type,
    map_li3793676233746214813on_nat: ( list_a > list_S5253404628753938141on_nat ) > list_list_a > list_l8999537460041914605on_nat ).

thf(sy_c_List_Olist_Omap_001t__Multiset____Ordering____NP____Hard__OAnnotation_001t__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J,type,
    map_Mu1807457972936786645tation: ( multis8755981257395730801tation > produc27318731269267614tation ) > list_M7834926499894101121tation > list_P7561118020193646638tation ).

thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__List__Olist_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_J,type,
    map_na5866636468935055223on_nat: ( nat > list_l6719245731233385357on_nat ) > list_nat > list_l5700991078898051731on_nat ).

thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
    map_na6205611841492582150st_nat: ( nat > list_list_nat ) > list_nat > list_list_list_nat ).

thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J,type,
    map_na8454508660465608945on_nat: ( nat > list_l1726306699111145479on_nat ) > list_nat > list_l6719245731233385357on_nat ).

thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
    map_nat_list_nat: ( nat > list_nat ) > list_nat > list_list_nat ).

thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    map_na4561905831291441265at_nat: ( nat > list_P6011104703257516679at_nat ) > list_nat > list_l3264859301627795341at_nat ).

thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J,type,
    map_na8715451351012060651on_nat: ( nat > list_P9127870862658512897on_nat ) > list_nat > list_l1726306699111145479on_nat ).

thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__List__Olist_It__Product____Type__Oprod_Itf__a_M_Eo_J_J,type,
    map_na764632570197019441od_a_o: ( nat > list_P3196869577824546119od_a_o ) > list_nat > list_l8336264072933094989od_a_o ).

thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J,type,
    map_na4864219606106146163on_nat: ( nat > list_S5253404628753938141on_nat ) > list_nat > list_l8999537460041914605on_nat ).

thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
    map_nat_nat: ( nat > nat ) > list_nat > list_nat ).

thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
    map_na6722963242033808602at_nat: ( nat > produc8199716216217303280at_nat ) > list_nat > list_P5366131564312172150at_nat ).

thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    map_na7298421622053143531at_nat: ( nat > product_prod_nat_nat ) > list_nat > list_P6011104703257516679at_nat ).

thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J,type,
    map_na2641141678045226597on_nat: ( nat > produc1313295348606329723on_nat ) > list_nat > list_P9127870862658512897on_nat ).

thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J,type,
    map_na5706685718985795555on_nat: ( nat > sum_su4907927656612468301on_nat ) > list_nat > list_S5253404628753938141on_nat ).

thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001tf__a,type,
    map_nat_a: ( nat > a ) > list_nat > list_a ).

thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_J,type,
    map_Pr5337824985771995362on_nat: ( produc1714117091004681831on_nat > produc118098540238352959on_nat ) > list_P8734936519038225773on_nat > list_P5508476027986415439on_nat ).

thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_J_J,type,
    map_Pr5195230461829504568on_nat: ( produc118098540238352959on_nat > produc8325617224762090749on_nat ) > list_P5508476027986415439on_nat > list_P6215481986061871491on_nat ).

thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_001t__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J,type,
    map_Pr9122214429247829392on_nat: ( produc1313295348606329723on_nat > list_l1726306699111145479on_nat ) > list_P9127870862658512897on_nat > list_l6719245731233385357on_nat ).

thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_001t__Nat__Onat,type,
    map_Pr1888482430906987655at_nat: ( produc1313295348606329723on_nat > nat ) > list_P9127870862658512897on_nat > list_nat ).

thf(sy_c_List_Olist_Omap_001t__Product____Type__Oprod_Itf__a_M_Eo_J_001t__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J,type,
    map_Pr6945619049609468478on_nat: ( product_prod_a_o > produc1313295348606329723on_nat ) > list_P3196869577824546119od_a_o > list_P9127870862658512897on_nat ).

thf(sy_c_List_Olist_Omap_001t__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_001t__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J,type,
    map_Su7921119544090670592on_nat: ( sum_su4907927656612468301on_nat > list_l8999537460041914605on_nat ) > list_S5253404628753938141on_nat > list_l8365940747029150845on_nat ).

thf(sy_c_List_Olist_Omap_001t__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_001t__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J,type,
    map_Su1406162919755814318on_nat: ( sum_su4907927656612468301on_nat > list_P9127870862658512897on_nat ) > list_S5253404628753938141on_nat > list_l1726306699111145479on_nat ).

thf(sy_c_List_Olist_Omap_001t__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_001t__Nat__Onat,type,
    map_Su4809418691747827683at_nat: ( sum_su4907927656612468301on_nat > nat ) > list_S5253404628753938141on_nat > list_nat ).

thf(sy_c_List_Olist_Omap_001t__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J,type,
    map_Su8438591925558430632on_nat: ( sum_su4907927656612468301on_nat > produc1313295348606329723on_nat ) > list_S5253404628753938141on_nat > list_P9127870862658512897on_nat ).

thf(sy_c_List_Olist_Omap_001t__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_001t__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J,type,
    map_Su241112079427911264on_nat: ( sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat ) > list_S5253404628753938141on_nat > list_S5253404628753938141on_nat ).

thf(sy_c_List_Olist_Omap_001t__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_001tf__a,type,
    map_Su5039174899830097323_nat_a: ( sum_su4907927656612468301on_nat > a ) > list_S5253404628753938141on_nat > list_a ).

thf(sy_c_List_Olist_Omap_001tf__a_001t__List__Olist_It__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J,type,
    map_a_5722733254325588791on_nat: ( a > list_l8365940747029150845on_nat ) > list_a > list_l7167270332776620429on_nat ).

thf(sy_c_List_Olist_Omap_001tf__a_001t__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J,type,
    map_a_8347779969077652429on_nat: ( a > list_l1726306699111145479on_nat ) > list_a > list_l6719245731233385357on_nat ).

thf(sy_c_List_Olist_Omap_001tf__a_001t__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J,type,
    map_a_3820070144034313383on_nat: ( a > list_l8999537460041914605on_nat ) > list_a > list_l8365940747029150845on_nat ).

thf(sy_c_List_Olist_Omap_001tf__a_001t__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J,type,
    map_a_7431629901395152583on_nat: ( a > list_P9127870862658512897on_nat ) > list_a > list_l1726306699111145479on_nat ).

thf(sy_c_List_Olist_Omap_001tf__a_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_J,type,
    map_a_2500903461722202088tation: ( a > list_P7561118020193646638tation ) > list_a > list_l3295615718716739902tation ).

thf(sy_c_List_Olist_Omap_001tf__a_001t__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J,type,
    map_a_3887198048842487703on_nat: ( a > list_S5253404628753938141on_nat ) > list_a > list_l8999537460041914605on_nat ).

thf(sy_c_List_Olist_Omap_001tf__a_001t__Nat__Onat,type,
    map_a_nat: ( a > nat ) > list_a > list_nat ).

thf(sy_c_List_Olist_Omap_001tf__a_001t__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J,type,
    map_a_2212652920793933121on_nat: ( a > produc1313295348606329723on_nat ) > list_a > list_P9127870862658512897on_nat ).

thf(sy_c_List_Olist_Omap_001tf__a_001t__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J,type,
    map_a_2711072540680412167on_nat: ( a > sum_su4907927656612468301on_nat ) > list_a > list_S5253404628753938141on_nat ).

thf(sy_c_List_Olist_Omap_001tf__a_001tf__a,type,
    map_a_a: ( a > a ) > list_a > list_a ).

thf(sy_c_List_Olist_Otl_001t__Nat__Onat,type,
    tl_nat: list_nat > list_nat ).

thf(sy_c_List_Onth_001t__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J,type,
    nth_li6929086678380665870on_nat: list_l6719245731233385357on_nat > nat > list_l1726306699111145479on_nat ).

thf(sy_c_List_Onth_001t__List__Olist_It__Nat__Onat_J,type,
    nth_list_nat: list_list_nat > nat > list_nat ).

thf(sy_c_List_Onth_001t__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J,type,
    nth_li1302981970230039432on_nat: list_l1726306699111145479on_nat > nat > list_P9127870862658512897on_nat ).

thf(sy_c_List_Onth_001t__List__Olist_It__Product____Type__Oprod_Itf__a_M_Eo_J_J,type,
    nth_li3248255784598043854od_a_o: list_l8336264072933094989od_a_o > nat > list_P3196869577824546119od_a_o ).

thf(sy_c_List_Onth_001t__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J,type,
    nth_li4091121329102240790on_nat: list_l8999537460041914605on_nat > nat > list_S5253404628753938141on_nat ).

thf(sy_c_List_Onth_001t__Nat__Onat,type,
    nth_nat: list_nat > nat > nat ).

thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J,type,
    nth_Pr3336893728888831618on_nat: list_P9127870862658512897on_nat > nat > produc1313295348606329723on_nat ).

thf(sy_c_List_Onth_001t__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J,type,
    nth_Su8471564416988646406on_nat: list_S5253404628753938141on_nat > nat > sum_su4907927656612468301on_nat ).

thf(sy_c_List_Oproduct_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
    product_nat_nat_nat: list_nat_nat > list_nat > list_P5366131564312172150at_nat ).

thf(sy_c_List_Oproduct_001t__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J,type,
    produc6587381211084932163on_nat: list_l8999537460041914605on_nat > list_P8734936519038225773on_nat > list_P5508476027986415439on_nat ).

thf(sy_c_List_Oproduct_001t__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_J,type,
    produc472613109841061731on_nat: list_l8999537460041914605on_nat > list_P5508476027986415439on_nat > list_P6215481986061871491on_nat ).

thf(sy_c_List_Oproduct_001t__Nat__Onat_001t__Nat__Onat,type,
    product_nat_nat: list_nat > list_nat > list_P6011104703257516679at_nat ).

thf(sy_c_List_Oproduct_001t__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_001t__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J,type,
    produc4987672674337023969on_nat: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > list_P9127870862658512897on_nat ).

thf(sy_c_List_Oproduct_001tf__a_001t__Multiset____Ordering____NP____Hard__OAnnotation,type,
    produc2053221683845580394tation: list_a > list_M7834926499894101121tation > list_P7561118020193646638tation ).

thf(sy_c_List_Oproduct__lists_001t__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J,type,
    produc1618157707243653360on_nat: list_l6719245731233385357on_nat > list_l6719245731233385357on_nat ).

thf(sy_c_List_Oproduct__lists_001t__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J,type,
    produc657118021824710830on_nat: list_l8365940747029150845on_nat > list_l8365940747029150845on_nat ).

thf(sy_c_List_Oproduct__lists_001t__Nat__Onat,type,
    product_lists_nat: list_list_nat > list_list_nat ).

thf(sy_c_List_Oproduct__lists_001t__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J,type,
    produc2357585119171201002on_nat: list_l1726306699111145479on_nat > list_l1726306699111145479on_nat ).

thf(sy_c_List_Oproduct__lists_001t__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J,type,
    produc1748266815408604318on_nat: list_l8999537460041914605on_nat > list_l8999537460041914605on_nat ).

thf(sy_c_List_Otake_001t__Nat__Onat,type,
    take_nat: nat > list_nat > list_nat ).

thf(sy_c_List_Oupt,type,
    upt: nat > nat > list_nat ).

thf(sy_c_Multiset__Ordering__NP__Hard_OAnnotation_ONegative,type,
    multis5977116398950053002gative: multis8755981257395730801tation ).

thf(sy_c_Multiset__Ordering__NP__Hard_OAnnotation_OPositive,type,
    multis8267843437418295310sitive: multis8755981257395730801tation ).

thf(sy_c_Multiset__Ordering__NP__Hard_OAnnotation_OUnsigned,type,
    multis5512618279175645674signed: multis8755981257395730801tation ).

thf(sy_c_Multiset__Ordering__NP__Hard_OAnnotation_Osize__Annotation,type,
    multis6502627171228953925tation: multis8755981257395730801tation > nat ).

thf(sy_c_Multiset__Ordering__NP__Hard_Oms__elem__of__lit_001tf__a,type,
    multis8230369766971345672_lit_a: product_prod_a_o > sum_su4907927656612468301on_nat ).

thf(sy_c_Multiset__Ordering__NP__Hard_Omultiset__problem__of__cnf_001tf__a,type,
    multis4221440410396274297_cnf_a: list_l8336264072933094989od_a_o > produc8325617224762090749on_nat ).

thf(sy_c_Multiset__Ordering__NP__Hard_Ovars__of__cnf_001tf__a,type,
    multis4741186478440321083_cnf_a: list_l8336264072933094989od_a_o > list_a ).

thf(sy_c_Nat_OSuc,type,
    suc: nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_J,type,
    size_s6092282996036616185on_nat: list_l6719245731233385357on_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J,type,
    size_s8175741288642522897on_nat: list_l8365940747029150845on_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
    size_s3023201423986296836st_nat: list_list_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J,type,
    size_s129639936549468787on_nat: list_l1726306699111145479on_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Product____Type__Oprod_Itf__a_M_Eo_J_J_J,type,
    size_s3379185846251495609od_a_o: list_l8336264072933094989od_a_o > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J,type,
    size_s650155532264187265on_nat: list_l8999537460041914605on_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
    size_size_list_nat: list_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J,type,
    size_s7490768528479217773on_nat: list_P9127870862658512897on_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J,type,
    size_s5686234912572693873on_nat: list_S5253404628753938141on_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__a_J,type,
    size_size_list_a: list_a > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__Multiset____Ordering____NP____Hard__OAnnotation,type,
    size_s7536720479165623301tation: multis8755981257395730801tation > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
    size_size_char: char > nat ).

thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
    numeral_numeral_nat: num > nat ).

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
    ord_less_eq_nat: nat > nat > $o ).

thf(sy_c_Product__Type_OPair_001_062_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_M_062_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_M_Eo_J_J_001t__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J,type,
    produc3961857940133396976on_nat: ( list_P9127870862658512897on_nat > list_P9127870862658512897on_nat > $o ) > list_l1726306699111145479on_nat > produc5764678590178156288on_nat ).

thf(sy_c_Product__Type_OPair_001_062_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_M_062_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_M_Eo_J_J_001t__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J,type,
    produc4619761231911769442on_nat: ( list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > $o ) > list_l8999537460041914605on_nat > produc1732627278533666408on_nat ).

thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J_001t__List__Olist_It__Nat__Onat_J,type,
    produc4727192421694094319st_nat: ( nat > nat > $o ) > list_nat > produc254973753779126261st_nat ).

thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__List__Olist_It__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J_J,type,
    produc17425388850148510at_nat: ( nat > nat ) > list_P5366131564312172150at_nat > produc1458239053189343596at_nat ).

thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
    produc72220940542539688at_nat: ( nat > nat ) > nat > produc8199716216217303280at_nat ).

thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_M_062_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_M_Eo_J_J_001t__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J,type,
    produc6237247764877134954on_nat: ( produc1313295348606329723on_nat > produc1313295348606329723on_nat > $o ) > list_P9127870862658512897on_nat > produc2264144286178290682on_nat ).

thf(sy_c_Product__Type_OPair_001_062_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_M_062_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_M_Eo_J_J_001t__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J,type,
    produc3262228428711020402on_nat: ( sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat > $o ) > list_S5253404628753938141on_nat > produc3191134443298988280on_nat ).

thf(sy_c_Product__Type_OPair_001_062_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_001t__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J,type,
    produc6419464700804365359on_nat: ( sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat ) > list_P9127870862658512897on_nat > produc3717180461370874877on_nat ).

thf(sy_c_Product__Type_OPair_001_062_Itf__a_M_062_Itf__a_M_Eo_J_J_001t__List__Olist_Itf__a_J,type,
    produc8111569692950616493list_a: ( a > a > $o ) > list_a > produc5032551385658279741list_a ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_001_Eo,type,
    produc715673817563144936_nat_o: list_l1726306699111145479on_nat > $o > produc6963029029708228856_nat_o ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_001t__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J,type,
    produc4739740463051851607on_nat: list_l1726306699111145479on_nat > list_l1726306699111145479on_nat > produc2417963485646110695on_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_001t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_M_Eo_J,type,
    produc2691072593788627016_nat_o: list_l1726306699111145479on_nat > produc6963029029708228856_nat_o > produc7496903103639383512_nat_o ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_001t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_Mt__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_J,type,
    produc162686122305543991on_nat: list_l1726306699111145479on_nat > produc2417963485646110695on_nat > produc8815734051188430663on_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_001t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_M_Eo_J_J,type,
    produc7274044877351478568_nat_o: list_l1726306699111145479on_nat > produc7496903103639383512_nat_o > produc2753127401226480184_nat_o ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_001t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_J_J,type,
    produc2763753542052437021on_nat: list_l1726306699111145479on_nat > produc6063590850804571341on_nat > produc3665718167116523309on_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_J,type,
    produc5719953898228879677on_nat: list_l1726306699111145479on_nat > produc5827802662881200621on_nat > produc6063590850804571341on_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_001_Eo,type,
    produc1964031788423343762_nat_o: list_l8999537460041914605on_nat > $o > produc7819550649547605728_nat_o ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_001t__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J,type,
    produc1341766448070154867on_nat: list_l8999537460041914605on_nat > list_l8999537460041914605on_nat > produc6098023079705189563on_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_001t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_M_Eo_J,type,
    produc5038552294322009446_nat_o: list_l8999537460041914605on_nat > produc7819550649547605728_nat_o > produc1737863824307799982_nat_o ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_001t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J,type,
    produc2560419634836715957on_nat: list_l8999537460041914605on_nat > produc6098023079705189563on_nat > produc8096917655042337667on_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_001t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_M_Eo_J_J,type,
    produc2951815235169433512_nat_o: list_l8999537460041914605on_nat > produc1737863824307799982_nat_o > produc3724276615328212598_nat_o ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_001t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_J,type,
    produc8625578076745794553on_nat: list_l8999537460041914605on_nat > produc41935680912008307on_nat > produc2778868904637641793on_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J,type,
    produc8536543913362324773on_nat: list_l8999537460041914605on_nat > produc630436911041420075on_nat > produc41935680912008307on_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Nat__Onat_J_001_Eo,type,
    produc1747790020158526943_nat_o: list_nat > $o > produc8122078169606880685_nat_o ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
    produc2694037385005941721st_nat: list_nat > list_nat > produc1828647624359046049st_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Nat__Onat_J_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
    produc5973304058656726566_nat_o: list_nat > produc8122078169606880685_nat_o > produc430777597369845998_nat_o ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Nat__Onat_J_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    produc4487115339913071592st_nat: list_nat > produc1828647624359046049st_nat > produc5438398690206989622st_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Nat__Onat_J_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_M_Eo_J_J,type,
    produc2959919240848031157_nat_o: list_nat > produc430777597369845998_nat_o > produc7413256671508406019_nat_o ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Nat__Onat_J_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
    produc6629933339985824415st_nat: list_nat > produc8196726482729697190st_nat > produc5923727742716231015st_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Nat__Onat_J_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__Nat__Onat_J_J,type,
    produc7518127839388293336st_nat: list_nat > produc4575160907756185873st_nat > produc8196726482729697190st_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_001_Eo,type,
    produc3679803167814765678_nat_o: list_P9127870862658512897on_nat > $o > produc1376987297294124798_nat_o ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_001t__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J,type,
    produc7310029628050611421on_nat: list_P9127870862658512897on_nat > list_l1726306699111145479on_nat > produc5827802662881200621on_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_001t__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J,type,
    produc7152764266822961111on_nat: list_P9127870862658512897on_nat > list_P9127870862658512897on_nat > produc1714117091004681831on_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_001t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_Mt__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_J_J,type,
    produc6960825939473561117on_nat: list_P9127870862658512897on_nat > produc8815734051188430663on_nat > produc3053421305257255469on_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_001t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_M_Eo_J_J_J,type,
    produc7597291304841350670_nat_o: list_P9127870862658512897on_nat > produc2753127401226480184_nat_o > produc9011176785203808030_nat_o ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_M_Eo_J,type,
    produc6066200345829616596_nat_o: list_P9127870862658512897on_nat > produc1376987297294124798_nat_o > produc4640512942233155940_nat_o ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J,type,
    produc1138821556964385341on_nat: list_P9127870862658512897on_nat > produc1714117091004681831on_nat > produc8563703056922213325on_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_M_Eo_J_J,type,
    produc2695279325889729082_nat_o: list_P9127870862658512897on_nat > produc4640512942233155940_nat_o > produc4594631521614525130_nat_o ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_J,type,
    produc7963087792460353193on_nat: list_P9127870862658512897on_nat > produc629934850357470163on_nat > produc3060927313823239865on_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J,type,
    produc2827541363413333699on_nat: list_P9127870862658512897on_nat > produc8413567841624424429on_nat > produc629934850357470163on_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_001_Eo,type,
    produc1279791631502317730_nat_o: list_S5253404628753938141on_nat > $o > produc103519031730445424_nat_o ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_001t__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J,type,
    produc9005406347315192931on_nat: list_S5253404628753938141on_nat > list_l8999537460041914605on_nat > produc630436911041420075on_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_001t__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_J_J,type,
    produc9043615093183382213on_nat: list_S5253404628753938141on_nat > list_P5508476027986415439on_nat > produc6588776972112385165on_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_001t__List__Olist_It__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_J_J_J,type,
    produc8252945450212337037on_nat: list_S5253404628753938141on_nat > list_P6215481986061871491on_nat > produc5056679816642863195on_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_001t__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J,type,
    produc2912653624306349395on_nat: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > produc5580075136493545115on_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_J,type,
    produc2491415968787288569on_nat: list_S5253404628753938141on_nat > produc8096917655042337667on_nat > produc2892159590233455681on_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_M_Eo_J_J_J,type,
    produc46917116878602732_nat_o: list_S5253404628753938141on_nat > produc3724276615328212598_nat_o > produc1666888912113777716_nat_o ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J,type,
    produc4176415733022414961on_nat: list_S5253404628753938141on_nat > produc1714117091004681831on_nat > produc118098540238352959on_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_M_Eo_J,type,
    produc4670265488131411942_nat_o: list_S5253404628753938141on_nat > produc103519031730445424_nat_o > produc3152382218381312302_nat_o ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J,type,
    produc7869692867729086373on_nat: list_S5253404628753938141on_nat > produc5580075136493545115on_nat > produc4387126377693847539on_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_J,type,
    produc1545487600272464053on_nat: list_S5253404628753938141on_nat > produc118098540238352959on_nat > produc8325617224762090749on_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_M_Eo_J_J,type,
    produc5518819433104342328_nat_o: list_S5253404628753938141on_nat > produc3152382218381312302_nat_o > produc8214513296305107590_nat_o ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J,type,
    produc8004745382299176153on_nat: list_S5253404628753938141on_nat > produc6827644173363976803on_nat > produc4714912333100831009on_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J,type,
    produc5847888561595751573on_nat: list_S5253404628753938141on_nat > produc952157413385281163on_nat > produc6827644173363976803on_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_Itf__a_J_001_Eo,type,
    produc4242187413542061611st_a_o: list_a > $o > produc5207931094065476923st_a_o ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
    produc6837034575241423639list_a: list_a > list_a > produc9164743771328383783list_a ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_Itf__a_J_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_M_Eo_J,type,
    produc5660821784244531022st_a_o: list_a > produc5207931094065476923st_a_o > produc6745688853455182174st_a_o ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_Itf__a_J_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
    produc1910438886824740410list_a: list_a > produc9164743771328383783list_a > produc1553995403754578250list_a ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_Itf__a_J_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_M_Eo_J_J,type,
    produc6327927755425424113st_a_o: list_a > produc6745688853455182174st_a_o > produc8398638096451127297st_a_o ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_Itf__a_J_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_Itf__a_Mt__List__Olist_Itf__a_J_J_J,type,
    produc1737125957862166243list_a: list_a > produc3854044251064639184list_a > produc1874289995683674227list_a ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_Itf__a_J_001t__Product____Type__Oprod_Itf__a_Mt__List__Olist_Itf__a_J_J,type,
    produc3204708664006668352list_a: list_a > produc8685980395799941037list_a > produc3854044251064639184list_a ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__List__Olist_It__Nat__Onat_J,type,
    produc8282810413953273033st_nat: nat > list_nat > produc4575160907756185873st_nat ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    produc6109913384486294878at_nat: nat > list_P6011104703257516679at_nat > produc8472197452120411308at_nat ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
    product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
    produc921417020989057183st_nat: nat > produc5438398690206989622st_nat > produc7312811325926348135st_nat ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_M_Eo_J_J_J,type,
    produc6858984136714788332_nat_o: nat > produc7413256671508406019_nat_o > produc3630873615015234996_nat_o ).

thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_001t__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J,type,
    produc6875097235463537885on_nat: produc1313295348606329723on_nat > list_P9127870862658512897on_nat > produc8413567841624424429on_nat ).

thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_J,type,
    produc2628373118283442345on_nat: produc1313295348606329723on_nat > produc8563703056922213325on_nat > produc7326124634926975417on_nat ).

thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_M_Eo_J_J_J,type,
    produc5930017123893236134_nat_o: produc1313295348606329723on_nat > produc4594631521614525130_nat_o > produc7775502106177927606_nat_o ).

thf(sy_c_Product__Type_OPair_001t__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_001t__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J,type,
    produc1755723437418862235on_nat: sum_su4907927656612468301on_nat > list_P9127870862658512897on_nat > produc3922897021252936937on_nat ).

thf(sy_c_Product__Type_OPair_001t__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_001t__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J,type,
    produc9137229129158211779on_nat: sum_su4907927656612468301on_nat > list_S5253404628753938141on_nat > produc952157413385281163on_nat ).

thf(sy_c_Product__Type_OPair_001t__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_001t__Product____Type__Oprod_I_062_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J,type,
    produc3053687648904799203on_nat: sum_su4907927656612468301on_nat > produc3717180461370874877on_nat > produc1098090131618875051on_nat ).

thf(sy_c_Product__Type_OPair_001t__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J,type,
    produc4095905013319207641on_nat: sum_su4907927656612468301on_nat > produc4387126377693847539on_nat > produc3977234058138274081on_nat ).

thf(sy_c_Product__Type_OPair_001t__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_M_Eo_J_J_J,type,
    produc1244398239520053868_nat_o: sum_su4907927656612468301on_nat > produc8214513296305107590_nat_o > produc8288325133855054004_nat_o ).

thf(sy_c_Product__Type_OPair_001t__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_I_062_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J_J,type,
    produc6662423293526568773on_nat: sum_su4907927656612468301on_nat > produc1098090131618875051on_nat > produc7012556527198387091on_nat ).

thf(sy_c_Product__Type_OPair_001t__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_001t__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J,type,
    produc6704844767664046899on_nat: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat > produc1313295348606329723on_nat ).

thf(sy_c_Product__Type_OPair_001tf__a_001_Eo,type,
    product_Pair_a_o: a > $o > product_prod_a_o ).

thf(sy_c_Product__Type_OPair_001tf__a_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_J,type,
    produc511914028911482581tation: a > list_P7561118020193646638tation > produc573162381751850459tation ).

thf(sy_c_Product__Type_OPair_001tf__a_001t__List__Olist_Itf__a_J,type,
    produc6670463072477821725list_a: a > list_a > produc8685980395799941037list_a ).

thf(sy_c_Product__Type_OPair_001tf__a_001t__Multiset____Ordering____NP____Hard__OAnnotation,type,
    produc7247797765490398872tation: a > multis8755981257395730801tation > produc27318731269267614tation ).

thf(sy_c_Product__Type_OPair_001tf__a_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J_J,type,
    produc8030850274662652387list_a: a > produc1553995403754578250list_a > produc8880992058355704947list_a ).

thf(sy_c_Product__Type_OPair_001tf__a_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Product____Type__Oprod_It__List__Olist_Itf__a_J_M_Eo_J_J_J,type,
    produc4030046113903262746st_a_o: a > produc8398638096451127297st_a_o > produc2716792995815215274st_a_o ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    produc2626176000494625587at_nat: ( nat > nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat ).

thf(sy_c_Seq_Oinf__concat__simple,type,
    inf_concat_simple: ( nat > nat ) > nat > product_prod_nat_nat ).

thf(sy_c_Seq_Oinf__concat__simple__rel,type,
    inf_co2821209343098213861le_rel: produc8199716216217303280at_nat > produc8199716216217303280at_nat > $o ).

thf(sy_c_String_Ochar_Osize__char,type,
    size_char: char > nat ).

thf(sy_c_Sum__Type_OInl_001t__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_001t__Nat__Onat,type,
    sum_In3878554002080803550on_nat: produc27318731269267614tation > sum_su4907927656612468301on_nat ).

thf(sy_c_Sum__Type_OInr_001t__Nat__Onat_001t__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J,type,
    sum_In255631617079920996tation: nat > sum_su4907927656612468301on_nat ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
    accp_P6412015083367660281at_nat: ( produc8199716216217303280at_nat > produc8199716216217303280at_nat > $o ) > produc8199716216217303280at_nat > $o ).

thf(sy_v_NSS,type,
    nss: list_P9127870862658512897on_nat ).

thf(sy_v_NS____,type,
    ns: list_P9127870862658512897on_nat ).

thf(sy_v_S,type,
    s: list_P9127870862658512897on_nat ).

thf(sy_v_cnf,type,
    cnf: list_l8336264072933094989od_a_o ).

thf(sy_v_cs____,type,
    cs: list_nat ).

thf(sy_v_left,type,
    left: list_S5253404628753938141on_nat ).

thf(sy_v_right,type,
    right: list_S5253404628753938141on_nat ).

thf(sy_v_xs____,type,
    xs: list_a ).

% Relevant facts (706)
thf(fact_0_xs__def,axiom,
    ( xs
    = ( multis4741186478440321083_cnf_a @ cnf ) ) ).

% xs_def
thf(fact_1_Annotation_Odistinct_I5_J,axiom,
    multis8267843437418295310sitive != multis5977116398950053002gative ).

% Annotation.distinct(5)
thf(fact_2_concat__map__singleton,axiom,
    ! [F: nat > nat,Xs: list_nat] :
      ( ( concat_nat
        @ ( map_nat_list_nat
          @ ^ [X: nat] : ( cons_nat @ ( F @ X ) @ nil_nat )
          @ Xs ) )
      = ( map_nat_nat @ F @ Xs ) ) ).

% concat_map_singleton
thf(fact_3_concat__map__singleton,axiom,
    ! [F: a > sum_su4907927656612468301on_nat,Xs: list_a] :
      ( ( concat6161509665823917772on_nat
        @ ( map_a_3887198048842487703on_nat
          @ ^ [X: a] : ( cons_S6797538523646374167on_nat @ ( F @ X ) @ nil_Su3078906344068544711on_nat )
          @ Xs ) )
      = ( map_a_2711072540680412167on_nat @ F @ Xs ) ) ).

% concat_map_singleton
thf(fact_4_concat__map__singleton,axiom,
    ! [F: a > list_S5253404628753938141on_nat,Xs: list_a] :
      ( ( concat4774721536428481500on_nat
        @ ( map_a_3820070144034313383on_nat
          @ ^ [X: a] : ( cons_l5857885737047467175on_nat @ ( F @ X ) @ nil_li5563851662744376407on_nat )
          @ Xs ) )
      = ( map_a_3887198048842487703on_nat @ F @ Xs ) ) ).

% concat_map_singleton
thf(fact_5_concat__map__singleton,axiom,
    ! [F: nat > produc1313295348606329723on_nat,Xs: list_nat] :
      ( ( concat937493773106420796on_nat
        @ ( map_na8715451351012060651on_nat
          @ ^ [X: nat] : ( cons_P1582596238967110705on_nat @ ( F @ X ) @ nil_Pr8614093721167239297on_nat )
          @ Xs ) )
      = ( map_na2641141678045226597on_nat @ F @ Xs ) ) ).

% concat_map_singleton
thf(fact_6_concat__map__singleton,axiom,
    ! [F: a > produc1313295348606329723on_nat,Xs: list_a] :
      ( ( concat937493773106420796on_nat
        @ ( map_a_7431629901395152583on_nat
          @ ^ [X: a] : ( cons_P1582596238967110705on_nat @ ( F @ X ) @ nil_Pr8614093721167239297on_nat )
          @ Xs ) )
      = ( map_a_2212652920793933121on_nat @ F @ Xs ) ) ).

% concat_map_singleton
thf(fact_7_concat__map__singleton,axiom,
    ! [F: nat > list_P9127870862658512897on_nat,Xs: list_nat] :
      ( ( concat4341727598008994882on_nat
        @ ( map_na8454508660465608945on_nat
          @ ^ [X: nat] : ( cons_l3028799188195208631on_nat @ ( F @ X ) @ nil_li2682182165152866823on_nat )
          @ Xs ) )
      = ( map_na8715451351012060651on_nat @ F @ Xs ) ) ).

% concat_map_singleton
thf(fact_8_concat__map__singleton,axiom,
    ! [F: a > list_P9127870862658512897on_nat,Xs: list_a] :
      ( ( concat4341727598008994882on_nat
        @ ( map_a_8347779969077652429on_nat
          @ ^ [X: a] : ( cons_l3028799188195208631on_nat @ ( F @ X ) @ nil_li2682182165152866823on_nat )
          @ Xs ) )
      = ( map_a_7431629901395152583on_nat @ F @ Xs ) ) ).

% concat_map_singleton
thf(fact_9_concat__map__singleton,axiom,
    ! [F: nat > list_nat,Xs: list_nat] :
      ( ( concat_list_nat
        @ ( map_na6205611841492582150st_nat
          @ ^ [X: nat] : ( cons_list_nat @ ( F @ X ) @ nil_list_nat )
          @ Xs ) )
      = ( map_nat_list_nat @ F @ Xs ) ) ).

% concat_map_singleton
thf(fact_10_concat__map__singleton,axiom,
    ! [F: a > list_l8999537460041914605on_nat,Xs: list_a] :
      ( ( concat6293066666318574188on_nat
        @ ( map_a_5722733254325588791on_nat
          @ ^ [X: a] : ( cons_l6725403660475259319on_nat @ ( F @ X ) @ nil_li3323240303193190759on_nat )
          @ Xs ) )
      = ( map_a_3820070144034313383on_nat @ F @ Xs ) ) ).

% concat_map_singleton
thf(fact_11_concat__map__singleton,axiom,
    ! [F: nat > list_l1726306699111145479on_nat,Xs: list_nat] :
      ( ( concat1602011024853954504on_nat
        @ ( map_na5866636468935055223on_nat
          @ ^ [X: nat] : ( cons_l4979741563022480061on_nat @ ( F @ X ) @ nil_li985476221562969869on_nat )
          @ Xs ) )
      = ( map_na8454508660465608945on_nat @ F @ Xs ) ) ).

% concat_map_singleton
thf(fact_12_Annotation_Oexhaust,axiom,
    ! [Y: multis8755981257395730801tation] :
      ( ( Y != multis5512618279175645674signed )
     => ( ( Y != multis8267843437418295310sitive )
       => ( Y = multis5977116398950053002gative ) ) ) ).

% Annotation.exhaust
thf(fact_13_list_Omap__disc__iff,axiom,
    ! [F: nat > nat,A: list_nat] :
      ( ( ( map_nat_nat @ F @ A )
        = nil_nat )
      = ( A = nil_nat ) ) ).

% list.map_disc_iff
thf(fact_14_list_Omap__disc__iff,axiom,
    ! [F: nat > sum_su4907927656612468301on_nat,A: list_nat] :
      ( ( ( map_na5706685718985795555on_nat @ F @ A )
        = nil_Su3078906344068544711on_nat )
      = ( A = nil_nat ) ) ).

% list.map_disc_iff
thf(fact_15_list_Omap__disc__iff,axiom,
    ! [F: sum_su4907927656612468301on_nat > nat,A: list_S5253404628753938141on_nat] :
      ( ( ( map_Su4809418691747827683at_nat @ F @ A )
        = nil_nat )
      = ( A = nil_Su3078906344068544711on_nat ) ) ).

% list.map_disc_iff
thf(fact_16_list_Omap__disc__iff,axiom,
    ! [F: a > sum_su4907927656612468301on_nat,A: list_a] :
      ( ( ( map_a_2711072540680412167on_nat @ F @ A )
        = nil_Su3078906344068544711on_nat )
      = ( A = nil_a ) ) ).

% list.map_disc_iff
thf(fact_17_list_Omap__disc__iff,axiom,
    ! [F: a > list_S5253404628753938141on_nat,A: list_a] :
      ( ( ( map_a_3887198048842487703on_nat @ F @ A )
        = nil_li5563851662744376407on_nat )
      = ( A = nil_a ) ) ).

% list.map_disc_iff
thf(fact_18_list_Omap__disc__iff,axiom,
    ! [F: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat,A: list_S5253404628753938141on_nat] :
      ( ( ( map_Su241112079427911264on_nat @ F @ A )
        = nil_Su3078906344068544711on_nat )
      = ( A = nil_Su3078906344068544711on_nat ) ) ).

% list.map_disc_iff
thf(fact_19_list_Omap__disc__iff,axiom,
    ! [F: nat > produc1313295348606329723on_nat,A: list_nat] :
      ( ( ( map_na2641141678045226597on_nat @ F @ A )
        = nil_Pr8614093721167239297on_nat )
      = ( A = nil_nat ) ) ).

% list.map_disc_iff
thf(fact_20_list_Omap__disc__iff,axiom,
    ! [F: produc1313295348606329723on_nat > nat,A: list_P9127870862658512897on_nat] :
      ( ( ( map_Pr1888482430906987655at_nat @ F @ A )
        = nil_nat )
      = ( A = nil_Pr8614093721167239297on_nat ) ) ).

% list.map_disc_iff
thf(fact_21_list_Omap__disc__iff,axiom,
    ! [F: nat > list_P9127870862658512897on_nat,A: list_nat] :
      ( ( ( map_na8715451351012060651on_nat @ F @ A )
        = nil_li2682182165152866823on_nat )
      = ( A = nil_nat ) ) ).

% list.map_disc_iff
thf(fact_22_list_Omap__disc__iff,axiom,
    ! [F: a > list_P9127870862658512897on_nat,A: list_a] :
      ( ( ( map_a_7431629901395152583on_nat @ F @ A )
        = nil_li2682182165152866823on_nat )
      = ( A = nil_a ) ) ).

% list.map_disc_iff
thf(fact_23_Nil__is__map__conv,axiom,
    ! [F: nat > nat,Xs: list_nat] :
      ( ( nil_nat
        = ( map_nat_nat @ F @ Xs ) )
      = ( Xs = nil_nat ) ) ).

% Nil_is_map_conv
thf(fact_24_Nil__is__map__conv,axiom,
    ! [F: nat > sum_su4907927656612468301on_nat,Xs: list_nat] :
      ( ( nil_Su3078906344068544711on_nat
        = ( map_na5706685718985795555on_nat @ F @ Xs ) )
      = ( Xs = nil_nat ) ) ).

% Nil_is_map_conv
thf(fact_25_Nil__is__map__conv,axiom,
    ! [F: sum_su4907927656612468301on_nat > nat,Xs: list_S5253404628753938141on_nat] :
      ( ( nil_nat
        = ( map_Su4809418691747827683at_nat @ F @ Xs ) )
      = ( Xs = nil_Su3078906344068544711on_nat ) ) ).

% Nil_is_map_conv
thf(fact_26_Nil__is__map__conv,axiom,
    ! [F: a > sum_su4907927656612468301on_nat,Xs: list_a] :
      ( ( nil_Su3078906344068544711on_nat
        = ( map_a_2711072540680412167on_nat @ F @ Xs ) )
      = ( Xs = nil_a ) ) ).

% Nil_is_map_conv
thf(fact_27_Nil__is__map__conv,axiom,
    ! [F: a > list_S5253404628753938141on_nat,Xs: list_a] :
      ( ( nil_li5563851662744376407on_nat
        = ( map_a_3887198048842487703on_nat @ F @ Xs ) )
      = ( Xs = nil_a ) ) ).

% Nil_is_map_conv
thf(fact_28_Nil__is__map__conv,axiom,
    ! [F: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat] :
      ( ( nil_Su3078906344068544711on_nat
        = ( map_Su241112079427911264on_nat @ F @ Xs ) )
      = ( Xs = nil_Su3078906344068544711on_nat ) ) ).

% Nil_is_map_conv
thf(fact_29_Nil__is__map__conv,axiom,
    ! [F: nat > produc1313295348606329723on_nat,Xs: list_nat] :
      ( ( nil_Pr8614093721167239297on_nat
        = ( map_na2641141678045226597on_nat @ F @ Xs ) )
      = ( Xs = nil_nat ) ) ).

% Nil_is_map_conv
thf(fact_30_Nil__is__map__conv,axiom,
    ! [F: produc1313295348606329723on_nat > nat,Xs: list_P9127870862658512897on_nat] :
      ( ( nil_nat
        = ( map_Pr1888482430906987655at_nat @ F @ Xs ) )
      = ( Xs = nil_Pr8614093721167239297on_nat ) ) ).

% Nil_is_map_conv
thf(fact_31_Nil__is__map__conv,axiom,
    ! [F: nat > list_P9127870862658512897on_nat,Xs: list_nat] :
      ( ( nil_li2682182165152866823on_nat
        = ( map_na8715451351012060651on_nat @ F @ Xs ) )
      = ( Xs = nil_nat ) ) ).

% Nil_is_map_conv
thf(fact_32_Nil__is__map__conv,axiom,
    ! [F: a > list_P9127870862658512897on_nat,Xs: list_a] :
      ( ( nil_li2682182165152866823on_nat
        = ( map_a_7431629901395152583on_nat @ F @ Xs ) )
      = ( Xs = nil_a ) ) ).

% Nil_is_map_conv
thf(fact_33_map__is__Nil__conv,axiom,
    ! [F: nat > nat,Xs: list_nat] :
      ( ( ( map_nat_nat @ F @ Xs )
        = nil_nat )
      = ( Xs = nil_nat ) ) ).

% map_is_Nil_conv
thf(fact_34_map__is__Nil__conv,axiom,
    ! [F: nat > sum_su4907927656612468301on_nat,Xs: list_nat] :
      ( ( ( map_na5706685718985795555on_nat @ F @ Xs )
        = nil_Su3078906344068544711on_nat )
      = ( Xs = nil_nat ) ) ).

% map_is_Nil_conv
thf(fact_35_map__is__Nil__conv,axiom,
    ! [F: sum_su4907927656612468301on_nat > nat,Xs: list_S5253404628753938141on_nat] :
      ( ( ( map_Su4809418691747827683at_nat @ F @ Xs )
        = nil_nat )
      = ( Xs = nil_Su3078906344068544711on_nat ) ) ).

% map_is_Nil_conv
thf(fact_36_map__is__Nil__conv,axiom,
    ! [F: a > sum_su4907927656612468301on_nat,Xs: list_a] :
      ( ( ( map_a_2711072540680412167on_nat @ F @ Xs )
        = nil_Su3078906344068544711on_nat )
      = ( Xs = nil_a ) ) ).

% map_is_Nil_conv
thf(fact_37_map__is__Nil__conv,axiom,
    ! [F: a > list_S5253404628753938141on_nat,Xs: list_a] :
      ( ( ( map_a_3887198048842487703on_nat @ F @ Xs )
        = nil_li5563851662744376407on_nat )
      = ( Xs = nil_a ) ) ).

% map_is_Nil_conv
thf(fact_38_map__is__Nil__conv,axiom,
    ! [F: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat] :
      ( ( ( map_Su241112079427911264on_nat @ F @ Xs )
        = nil_Su3078906344068544711on_nat )
      = ( Xs = nil_Su3078906344068544711on_nat ) ) ).

% map_is_Nil_conv
thf(fact_39_map__is__Nil__conv,axiom,
    ! [F: nat > produc1313295348606329723on_nat,Xs: list_nat] :
      ( ( ( map_na2641141678045226597on_nat @ F @ Xs )
        = nil_Pr8614093721167239297on_nat )
      = ( Xs = nil_nat ) ) ).

% map_is_Nil_conv
thf(fact_40_map__is__Nil__conv,axiom,
    ! [F: produc1313295348606329723on_nat > nat,Xs: list_P9127870862658512897on_nat] :
      ( ( ( map_Pr1888482430906987655at_nat @ F @ Xs )
        = nil_nat )
      = ( Xs = nil_Pr8614093721167239297on_nat ) ) ).

% map_is_Nil_conv
thf(fact_41_map__is__Nil__conv,axiom,
    ! [F: nat > list_P9127870862658512897on_nat,Xs: list_nat] :
      ( ( ( map_na8715451351012060651on_nat @ F @ Xs )
        = nil_li2682182165152866823on_nat )
      = ( Xs = nil_nat ) ) ).

% map_is_Nil_conv
thf(fact_42_map__is__Nil__conv,axiom,
    ! [F: a > list_P9127870862658512897on_nat,Xs: list_a] :
      ( ( ( map_a_7431629901395152583on_nat @ F @ Xs )
        = nil_li2682182165152866823on_nat )
      = ( Xs = nil_a ) ) ).

% map_is_Nil_conv
thf(fact_43_map__ident,axiom,
    ( ( map_nat_nat
      @ ^ [X: nat] : X )
    = ( ^ [Xs2: list_nat] : Xs2 ) ) ).

% map_ident
thf(fact_44_Annotation_Odistinct_I3_J,axiom,
    multis5512618279175645674signed != multis5977116398950053002gative ).

% Annotation.distinct(3)
thf(fact_45_Annotation_Odistinct_I1_J,axiom,
    multis5512618279175645674signed != multis8267843437418295310sitive ).

% Annotation.distinct(1)
thf(fact_46_assms,axiom,
    ( ( multis4221440410396274297_cnf_a @ cnf )
    = ( produc1545487600272464053on_nat @ left @ ( produc4176415733022414961on_nat @ right @ ( produc7152764266822961111on_nat @ s @ nss ) ) ) ) ).

% assms
thf(fact_47_sum_Oinject_I1_J,axiom,
    ! [X1: produc27318731269267614tation,Y1: produc27318731269267614tation] :
      ( ( ( sum_In3878554002080803550on_nat @ X1 )
        = ( sum_In3878554002080803550on_nat @ Y1 ) )
      = ( X1 = Y1 ) ) ).

% sum.inject(1)
thf(fact_48_old_Osum_Oinject_I1_J,axiom,
    ! [A: produc27318731269267614tation,A2: produc27318731269267614tation] :
      ( ( ( sum_In3878554002080803550on_nat @ A )
        = ( sum_In3878554002080803550on_nat @ A2 ) )
      = ( A = A2 ) ) ).

% old.sum.inject(1)
thf(fact_49_list_Oinject,axiom,
    ! [X21: sum_su4907927656612468301on_nat,X22: list_S5253404628753938141on_nat,Y21: sum_su4907927656612468301on_nat,Y22: list_S5253404628753938141on_nat] :
      ( ( ( cons_S6797538523646374167on_nat @ X21 @ X22 )
        = ( cons_S6797538523646374167on_nat @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_50_list_Oinject,axiom,
    ! [X21: produc1313295348606329723on_nat,X22: list_P9127870862658512897on_nat,Y21: produc1313295348606329723on_nat,Y22: list_P9127870862658512897on_nat] :
      ( ( ( cons_P1582596238967110705on_nat @ X21 @ X22 )
        = ( cons_P1582596238967110705on_nat @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_51_list_Oinject,axiom,
    ! [X21: nat,X22: list_nat,Y21: nat,Y22: list_nat] :
      ( ( ( cons_nat @ X21 @ X22 )
        = ( cons_nat @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_52_list_Oinject,axiom,
    ! [X21: list_S5253404628753938141on_nat,X22: list_l8999537460041914605on_nat,Y21: list_S5253404628753938141on_nat,Y22: list_l8999537460041914605on_nat] :
      ( ( ( cons_l5857885737047467175on_nat @ X21 @ X22 )
        = ( cons_l5857885737047467175on_nat @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_53_list_Oinject,axiom,
    ! [X21: list_P9127870862658512897on_nat,X22: list_l1726306699111145479on_nat,Y21: list_P9127870862658512897on_nat,Y22: list_l1726306699111145479on_nat] :
      ( ( ( cons_l3028799188195208631on_nat @ X21 @ X22 )
        = ( cons_l3028799188195208631on_nat @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_54_res,axiom,
    ( ( produc1545487600272464053on_nat
      @ ( concat6161509665823917772on_nat
        @ ( map_a_3887198048842487703on_nat
          @ ^ [X: a] : ( cons_S6797538523646374167on_nat @ ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ X @ multis8267843437418295310sitive ) ) @ ( cons_S6797538523646374167on_nat @ ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ X @ multis5977116398950053002gative ) ) @ nil_Su3078906344068544711on_nat ) )
          @ xs ) )
      @ ( produc4176415733022414961on_nat
        @ ( append3523812819975775986on_nat
          @ ( map_a_2711072540680412167on_nat
            @ ^ [X: a] : ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ X @ multis5512618279175645674signed ) )
            @ xs )
          @ ( map_na5706685718985795555on_nat @ sum_In255631617079920996tation @ cs ) )
        @ ( produc7152764266822961111on_nat
          @ ( concat937493773106420796on_nat
            @ ( map_na8715451351012060651on_nat
              @ ^ [I: nat] :
                  ( map_Pr6945619049609468478on_nat
                  @ ^ [L: product_prod_a_o] : ( produc6704844767664046899on_nat @ ( multis8230369766971345672_lit_a @ L ) @ ( sum_In255631617079920996tation @ I ) )
                  @ ( nth_li3248255784598043854od_a_o @ cnf @ I ) )
              @ cs ) )
          @ ( append2865821678206098582on_nat
            @ ( concat937493773106420796on_nat
              @ ( map_a_7431629901395152583on_nat
                @ ^ [X: a] : ( cons_P1582596238967110705on_nat @ ( produc6704844767664046899on_nat @ ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ X @ multis8267843437418295310sitive ) ) @ ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ X @ multis5512618279175645674signed ) ) ) @ ( cons_P1582596238967110705on_nat @ ( produc6704844767664046899on_nat @ ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ X @ multis5977116398950053002gative ) ) @ ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ X @ multis5512618279175645674signed ) ) ) @ nil_Pr8614093721167239297on_nat ) )
                @ xs ) )
            @ ( concat937493773106420796on_nat
              @ ( map_na8715451351012060651on_nat
                @ ^ [I: nat] :
                    ( map_Pr6945619049609468478on_nat
                    @ ^ [L: product_prod_a_o] : ( produc6704844767664046899on_nat @ ( multis8230369766971345672_lit_a @ L ) @ ( sum_In255631617079920996tation @ I ) )
                    @ ( nth_li3248255784598043854od_a_o @ cnf @ I ) )
                @ cs ) ) ) ) ) )
    = ( produc1545487600272464053on_nat @ left @ ( produc4176415733022414961on_nat @ right @ ( produc7152764266822961111on_nat @ s @ nss ) ) ) ) ).

% res
thf(fact_55_not__Cons__self2,axiom,
    ! [X2: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat] :
      ( ( cons_S6797538523646374167on_nat @ X2 @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_56_not__Cons__self2,axiom,
    ! [X2: produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat] :
      ( ( cons_P1582596238967110705on_nat @ X2 @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_57_not__Cons__self2,axiom,
    ! [X2: nat,Xs: list_nat] :
      ( ( cons_nat @ X2 @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_58_not__Cons__self2,axiom,
    ! [X2: list_S5253404628753938141on_nat,Xs: list_l8999537460041914605on_nat] :
      ( ( cons_l5857885737047467175on_nat @ X2 @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_59_not__Cons__self2,axiom,
    ! [X2: list_P9127870862658512897on_nat,Xs: list_l1726306699111145479on_nat] :
      ( ( cons_l3028799188195208631on_nat @ X2 @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_60_shuffles_Ocases,axiom,
    ! [X2: produc9164743771328383783list_a] :
      ( ! [Ys: list_a] :
          ( X2
         != ( produc6837034575241423639list_a @ nil_a @ Ys ) )
     => ( ! [Xs3: list_a] :
            ( X2
           != ( produc6837034575241423639list_a @ Xs3 @ nil_a ) )
       => ~ ! [X3: a,Xs3: list_a,Y2: a,Ys: list_a] :
              ( X2
             != ( produc6837034575241423639list_a @ ( cons_a @ X3 @ Xs3 ) @ ( cons_a @ Y2 @ Ys ) ) ) ) ) ).

% shuffles.cases
thf(fact_61_shuffles_Ocases,axiom,
    ! [X2: produc5580075136493545115on_nat] :
      ( ! [Ys: list_S5253404628753938141on_nat] :
          ( X2
         != ( produc2912653624306349395on_nat @ nil_Su3078906344068544711on_nat @ Ys ) )
     => ( ! [Xs3: list_S5253404628753938141on_nat] :
            ( X2
           != ( produc2912653624306349395on_nat @ Xs3 @ nil_Su3078906344068544711on_nat ) )
       => ~ ! [X3: sum_su4907927656612468301on_nat,Xs3: list_S5253404628753938141on_nat,Y2: sum_su4907927656612468301on_nat,Ys: list_S5253404628753938141on_nat] :
              ( X2
             != ( produc2912653624306349395on_nat @ ( cons_S6797538523646374167on_nat @ X3 @ Xs3 ) @ ( cons_S6797538523646374167on_nat @ Y2 @ Ys ) ) ) ) ) ).

% shuffles.cases
thf(fact_62_shuffles_Ocases,axiom,
    ! [X2: produc1714117091004681831on_nat] :
      ( ! [Ys: list_P9127870862658512897on_nat] :
          ( X2
         != ( produc7152764266822961111on_nat @ nil_Pr8614093721167239297on_nat @ Ys ) )
     => ( ! [Xs3: list_P9127870862658512897on_nat] :
            ( X2
           != ( produc7152764266822961111on_nat @ Xs3 @ nil_Pr8614093721167239297on_nat ) )
       => ~ ! [X3: produc1313295348606329723on_nat,Xs3: list_P9127870862658512897on_nat,Y2: produc1313295348606329723on_nat,Ys: list_P9127870862658512897on_nat] :
              ( X2
             != ( produc7152764266822961111on_nat @ ( cons_P1582596238967110705on_nat @ X3 @ Xs3 ) @ ( cons_P1582596238967110705on_nat @ Y2 @ Ys ) ) ) ) ) ).

% shuffles.cases
thf(fact_63_shuffles_Ocases,axiom,
    ! [X2: produc1828647624359046049st_nat] :
      ( ! [Ys: list_nat] :
          ( X2
         != ( produc2694037385005941721st_nat @ nil_nat @ Ys ) )
     => ( ! [Xs3: list_nat] :
            ( X2
           != ( produc2694037385005941721st_nat @ Xs3 @ nil_nat ) )
       => ~ ! [X3: nat,Xs3: list_nat,Y2: nat,Ys: list_nat] :
              ( X2
             != ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_nat @ Y2 @ Ys ) ) ) ) ) ).

% shuffles.cases
thf(fact_64_shuffles_Ocases,axiom,
    ! [X2: produc6098023079705189563on_nat] :
      ( ! [Ys: list_l8999537460041914605on_nat] :
          ( X2
         != ( produc1341766448070154867on_nat @ nil_li5563851662744376407on_nat @ Ys ) )
     => ( ! [Xs3: list_l8999537460041914605on_nat] :
            ( X2
           != ( produc1341766448070154867on_nat @ Xs3 @ nil_li5563851662744376407on_nat ) )
       => ~ ! [X3: list_S5253404628753938141on_nat,Xs3: list_l8999537460041914605on_nat,Y2: list_S5253404628753938141on_nat,Ys: list_l8999537460041914605on_nat] :
              ( X2
             != ( produc1341766448070154867on_nat @ ( cons_l5857885737047467175on_nat @ X3 @ Xs3 ) @ ( cons_l5857885737047467175on_nat @ Y2 @ Ys ) ) ) ) ) ).

% shuffles.cases
thf(fact_65_shuffles_Ocases,axiom,
    ! [X2: produc2417963485646110695on_nat] :
      ( ! [Ys: list_l1726306699111145479on_nat] :
          ( X2
         != ( produc4739740463051851607on_nat @ nil_li2682182165152866823on_nat @ Ys ) )
     => ( ! [Xs3: list_l1726306699111145479on_nat] :
            ( X2
           != ( produc4739740463051851607on_nat @ Xs3 @ nil_li2682182165152866823on_nat ) )
       => ~ ! [X3: list_P9127870862658512897on_nat,Xs3: list_l1726306699111145479on_nat,Y2: list_P9127870862658512897on_nat,Ys: list_l1726306699111145479on_nat] :
              ( X2
             != ( produc4739740463051851607on_nat @ ( cons_l3028799188195208631on_nat @ X3 @ Xs3 ) @ ( cons_l3028799188195208631on_nat @ Y2 @ Ys ) ) ) ) ) ).

% shuffles.cases
thf(fact_66_Inl__inject,axiom,
    ! [X2: produc27318731269267614tation,Y: produc27318731269267614tation] :
      ( ( ( sum_In3878554002080803550on_nat @ X2 )
        = ( sum_In3878554002080803550on_nat @ Y ) )
     => ( X2 = Y ) ) ).

% Inl_inject
thf(fact_67_list_Omap__ident,axiom,
    ! [T: list_nat] :
      ( ( map_nat_nat
        @ ^ [X: nat] : X
        @ T )
      = T ) ).

% list.map_ident
thf(fact_68_list__nonempty__induct,axiom,
    ! [Xs: list_a,P: list_a > $o] :
      ( ( Xs != nil_a )
     => ( ! [X3: a] : ( P @ ( cons_a @ X3 @ nil_a ) )
       => ( ! [X3: a,Xs3: list_a] :
              ( ( Xs3 != nil_a )
             => ( ( P @ Xs3 )
               => ( P @ ( cons_a @ X3 @ Xs3 ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_69_list__nonempty__induct,axiom,
    ! [Xs: list_S5253404628753938141on_nat,P: list_S5253404628753938141on_nat > $o] :
      ( ( Xs != nil_Su3078906344068544711on_nat )
     => ( ! [X3: sum_su4907927656612468301on_nat] : ( P @ ( cons_S6797538523646374167on_nat @ X3 @ nil_Su3078906344068544711on_nat ) )
       => ( ! [X3: sum_su4907927656612468301on_nat,Xs3: list_S5253404628753938141on_nat] :
              ( ( Xs3 != nil_Su3078906344068544711on_nat )
             => ( ( P @ Xs3 )
               => ( P @ ( cons_S6797538523646374167on_nat @ X3 @ Xs3 ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_70_list__nonempty__induct,axiom,
    ! [Xs: list_P9127870862658512897on_nat,P: list_P9127870862658512897on_nat > $o] :
      ( ( Xs != nil_Pr8614093721167239297on_nat )
     => ( ! [X3: produc1313295348606329723on_nat] : ( P @ ( cons_P1582596238967110705on_nat @ X3 @ nil_Pr8614093721167239297on_nat ) )
       => ( ! [X3: produc1313295348606329723on_nat,Xs3: list_P9127870862658512897on_nat] :
              ( ( Xs3 != nil_Pr8614093721167239297on_nat )
             => ( ( P @ Xs3 )
               => ( P @ ( cons_P1582596238967110705on_nat @ X3 @ Xs3 ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_71_list__nonempty__induct,axiom,
    ! [Xs: list_nat,P: list_nat > $o] :
      ( ( Xs != nil_nat )
     => ( ! [X3: nat] : ( P @ ( cons_nat @ X3 @ nil_nat ) )
       => ( ! [X3: nat,Xs3: list_nat] :
              ( ( Xs3 != nil_nat )
             => ( ( P @ Xs3 )
               => ( P @ ( cons_nat @ X3 @ Xs3 ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_72_list__nonempty__induct,axiom,
    ! [Xs: list_l8999537460041914605on_nat,P: list_l8999537460041914605on_nat > $o] :
      ( ( Xs != nil_li5563851662744376407on_nat )
     => ( ! [X3: list_S5253404628753938141on_nat] : ( P @ ( cons_l5857885737047467175on_nat @ X3 @ nil_li5563851662744376407on_nat ) )
       => ( ! [X3: list_S5253404628753938141on_nat,Xs3: list_l8999537460041914605on_nat] :
              ( ( Xs3 != nil_li5563851662744376407on_nat )
             => ( ( P @ Xs3 )
               => ( P @ ( cons_l5857885737047467175on_nat @ X3 @ Xs3 ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_73_list__nonempty__induct,axiom,
    ! [Xs: list_l1726306699111145479on_nat,P: list_l1726306699111145479on_nat > $o] :
      ( ( Xs != nil_li2682182165152866823on_nat )
     => ( ! [X3: list_P9127870862658512897on_nat] : ( P @ ( cons_l3028799188195208631on_nat @ X3 @ nil_li2682182165152866823on_nat ) )
       => ( ! [X3: list_P9127870862658512897on_nat,Xs3: list_l1726306699111145479on_nat] :
              ( ( Xs3 != nil_li2682182165152866823on_nat )
             => ( ( P @ Xs3 )
               => ( P @ ( cons_l3028799188195208631on_nat @ X3 @ Xs3 ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_74_list__induct2_H,axiom,
    ! [P: list_a > list_a > $o,Xs: list_a,Ys2: list_a] :
      ( ( P @ nil_a @ nil_a )
     => ( ! [X3: a,Xs3: list_a] : ( P @ ( cons_a @ X3 @ Xs3 ) @ nil_a )
       => ( ! [Y2: a,Ys: list_a] : ( P @ nil_a @ ( cons_a @ Y2 @ Ys ) )
         => ( ! [X3: a,Xs3: list_a,Y2: a,Ys: list_a] :
                ( ( P @ Xs3 @ Ys )
               => ( P @ ( cons_a @ X3 @ Xs3 ) @ ( cons_a @ Y2 @ Ys ) ) )
           => ( P @ Xs @ Ys2 ) ) ) ) ) ).

% list_induct2'
thf(fact_75_list__induct2_H,axiom,
    ! [P: list_a > list_nat > $o,Xs: list_a,Ys2: list_nat] :
      ( ( P @ nil_a @ nil_nat )
     => ( ! [X3: a,Xs3: list_a] : ( P @ ( cons_a @ X3 @ Xs3 ) @ nil_nat )
       => ( ! [Y2: nat,Ys: list_nat] : ( P @ nil_a @ ( cons_nat @ Y2 @ Ys ) )
         => ( ! [X3: a,Xs3: list_a,Y2: nat,Ys: list_nat] :
                ( ( P @ Xs3 @ Ys )
               => ( P @ ( cons_a @ X3 @ Xs3 ) @ ( cons_nat @ Y2 @ Ys ) ) )
           => ( P @ Xs @ Ys2 ) ) ) ) ) ).

% list_induct2'
thf(fact_76_list__induct2_H,axiom,
    ! [P: list_nat > list_a > $o,Xs: list_nat,Ys2: list_a] :
      ( ( P @ nil_nat @ nil_a )
     => ( ! [X3: nat,Xs3: list_nat] : ( P @ ( cons_nat @ X3 @ Xs3 ) @ nil_a )
       => ( ! [Y2: a,Ys: list_a] : ( P @ nil_nat @ ( cons_a @ Y2 @ Ys ) )
         => ( ! [X3: nat,Xs3: list_nat,Y2: a,Ys: list_a] :
                ( ( P @ Xs3 @ Ys )
               => ( P @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_a @ Y2 @ Ys ) ) )
           => ( P @ Xs @ Ys2 ) ) ) ) ) ).

% list_induct2'
thf(fact_77_list__induct2_H,axiom,
    ! [P: list_nat > list_nat > $o,Xs: list_nat,Ys2: list_nat] :
      ( ( P @ nil_nat @ nil_nat )
     => ( ! [X3: nat,Xs3: list_nat] : ( P @ ( cons_nat @ X3 @ Xs3 ) @ nil_nat )
       => ( ! [Y2: nat,Ys: list_nat] : ( P @ nil_nat @ ( cons_nat @ Y2 @ Ys ) )
         => ( ! [X3: nat,Xs3: list_nat,Y2: nat,Ys: list_nat] :
                ( ( P @ Xs3 @ Ys )
               => ( P @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_nat @ Y2 @ Ys ) ) )
           => ( P @ Xs @ Ys2 ) ) ) ) ) ).

% list_induct2'
thf(fact_78_list__induct2_H,axiom,
    ! [P: list_a > list_S5253404628753938141on_nat > $o,Xs: list_a,Ys2: list_S5253404628753938141on_nat] :
      ( ( P @ nil_a @ nil_Su3078906344068544711on_nat )
     => ( ! [X3: a,Xs3: list_a] : ( P @ ( cons_a @ X3 @ Xs3 ) @ nil_Su3078906344068544711on_nat )
       => ( ! [Y2: sum_su4907927656612468301on_nat,Ys: list_S5253404628753938141on_nat] : ( P @ nil_a @ ( cons_S6797538523646374167on_nat @ Y2 @ Ys ) )
         => ( ! [X3: a,Xs3: list_a,Y2: sum_su4907927656612468301on_nat,Ys: list_S5253404628753938141on_nat] :
                ( ( P @ Xs3 @ Ys )
               => ( P @ ( cons_a @ X3 @ Xs3 ) @ ( cons_S6797538523646374167on_nat @ Y2 @ Ys ) ) )
           => ( P @ Xs @ Ys2 ) ) ) ) ) ).

% list_induct2'
thf(fact_79_list__induct2_H,axiom,
    ! [P: list_S5253404628753938141on_nat > list_a > $o,Xs: list_S5253404628753938141on_nat,Ys2: list_a] :
      ( ( P @ nil_Su3078906344068544711on_nat @ nil_a )
     => ( ! [X3: sum_su4907927656612468301on_nat,Xs3: list_S5253404628753938141on_nat] : ( P @ ( cons_S6797538523646374167on_nat @ X3 @ Xs3 ) @ nil_a )
       => ( ! [Y2: a,Ys: list_a] : ( P @ nil_Su3078906344068544711on_nat @ ( cons_a @ Y2 @ Ys ) )
         => ( ! [X3: sum_su4907927656612468301on_nat,Xs3: list_S5253404628753938141on_nat,Y2: a,Ys: list_a] :
                ( ( P @ Xs3 @ Ys )
               => ( P @ ( cons_S6797538523646374167on_nat @ X3 @ Xs3 ) @ ( cons_a @ Y2 @ Ys ) ) )
           => ( P @ Xs @ Ys2 ) ) ) ) ) ).

% list_induct2'
thf(fact_80_list__induct2_H,axiom,
    ! [P: list_S5253404628753938141on_nat > list_nat > $o,Xs: list_S5253404628753938141on_nat,Ys2: list_nat] :
      ( ( P @ nil_Su3078906344068544711on_nat @ nil_nat )
     => ( ! [X3: sum_su4907927656612468301on_nat,Xs3: list_S5253404628753938141on_nat] : ( P @ ( cons_S6797538523646374167on_nat @ X3 @ Xs3 ) @ nil_nat )
       => ( ! [Y2: nat,Ys: list_nat] : ( P @ nil_Su3078906344068544711on_nat @ ( cons_nat @ Y2 @ Ys ) )
         => ( ! [X3: sum_su4907927656612468301on_nat,Xs3: list_S5253404628753938141on_nat,Y2: nat,Ys: list_nat] :
                ( ( P @ Xs3 @ Ys )
               => ( P @ ( cons_S6797538523646374167on_nat @ X3 @ Xs3 ) @ ( cons_nat @ Y2 @ Ys ) ) )
           => ( P @ Xs @ Ys2 ) ) ) ) ) ).

% list_induct2'
thf(fact_81_list__induct2_H,axiom,
    ! [P: list_nat > list_S5253404628753938141on_nat > $o,Xs: list_nat,Ys2: list_S5253404628753938141on_nat] :
      ( ( P @ nil_nat @ nil_Su3078906344068544711on_nat )
     => ( ! [X3: nat,Xs3: list_nat] : ( P @ ( cons_nat @ X3 @ Xs3 ) @ nil_Su3078906344068544711on_nat )
       => ( ! [Y2: sum_su4907927656612468301on_nat,Ys: list_S5253404628753938141on_nat] : ( P @ nil_nat @ ( cons_S6797538523646374167on_nat @ Y2 @ Ys ) )
         => ( ! [X3: nat,Xs3: list_nat,Y2: sum_su4907927656612468301on_nat,Ys: list_S5253404628753938141on_nat] :
                ( ( P @ Xs3 @ Ys )
               => ( P @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_S6797538523646374167on_nat @ Y2 @ Ys ) ) )
           => ( P @ Xs @ Ys2 ) ) ) ) ) ).

% list_induct2'
thf(fact_82_list__induct2_H,axiom,
    ! [P: list_a > list_l8999537460041914605on_nat > $o,Xs: list_a,Ys2: list_l8999537460041914605on_nat] :
      ( ( P @ nil_a @ nil_li5563851662744376407on_nat )
     => ( ! [X3: a,Xs3: list_a] : ( P @ ( cons_a @ X3 @ Xs3 ) @ nil_li5563851662744376407on_nat )
       => ( ! [Y2: list_S5253404628753938141on_nat,Ys: list_l8999537460041914605on_nat] : ( P @ nil_a @ ( cons_l5857885737047467175on_nat @ Y2 @ Ys ) )
         => ( ! [X3: a,Xs3: list_a,Y2: list_S5253404628753938141on_nat,Ys: list_l8999537460041914605on_nat] :
                ( ( P @ Xs3 @ Ys )
               => ( P @ ( cons_a @ X3 @ Xs3 ) @ ( cons_l5857885737047467175on_nat @ Y2 @ Ys ) ) )
           => ( P @ Xs @ Ys2 ) ) ) ) ) ).

% list_induct2'
thf(fact_83_list__induct2_H,axiom,
    ! [P: list_nat > list_l8999537460041914605on_nat > $o,Xs: list_nat,Ys2: list_l8999537460041914605on_nat] :
      ( ( P @ nil_nat @ nil_li5563851662744376407on_nat )
     => ( ! [X3: nat,Xs3: list_nat] : ( P @ ( cons_nat @ X3 @ Xs3 ) @ nil_li5563851662744376407on_nat )
       => ( ! [Y2: list_S5253404628753938141on_nat,Ys: list_l8999537460041914605on_nat] : ( P @ nil_nat @ ( cons_l5857885737047467175on_nat @ Y2 @ Ys ) )
         => ( ! [X3: nat,Xs3: list_nat,Y2: list_S5253404628753938141on_nat,Ys: list_l8999537460041914605on_nat] :
                ( ( P @ Xs3 @ Ys )
               => ( P @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_l5857885737047467175on_nat @ Y2 @ Ys ) ) )
           => ( P @ Xs @ Ys2 ) ) ) ) ) ).

% list_induct2'
thf(fact_84_neq__Nil__conv,axiom,
    ! [Xs: list_a] :
      ( ( Xs != nil_a )
      = ( ? [Y3: a,Ys3: list_a] :
            ( Xs
            = ( cons_a @ Y3 @ Ys3 ) ) ) ) ).

% neq_Nil_conv
thf(fact_85_neq__Nil__conv,axiom,
    ! [Xs: list_S5253404628753938141on_nat] :
      ( ( Xs != nil_Su3078906344068544711on_nat )
      = ( ? [Y3: sum_su4907927656612468301on_nat,Ys3: list_S5253404628753938141on_nat] :
            ( Xs
            = ( cons_S6797538523646374167on_nat @ Y3 @ Ys3 ) ) ) ) ).

% neq_Nil_conv
thf(fact_86_neq__Nil__conv,axiom,
    ! [Xs: list_P9127870862658512897on_nat] :
      ( ( Xs != nil_Pr8614093721167239297on_nat )
      = ( ? [Y3: produc1313295348606329723on_nat,Ys3: list_P9127870862658512897on_nat] :
            ( Xs
            = ( cons_P1582596238967110705on_nat @ Y3 @ Ys3 ) ) ) ) ).

% neq_Nil_conv
thf(fact_87_neq__Nil__conv,axiom,
    ! [Xs: list_nat] :
      ( ( Xs != nil_nat )
      = ( ? [Y3: nat,Ys3: list_nat] :
            ( Xs
            = ( cons_nat @ Y3 @ Ys3 ) ) ) ) ).

% neq_Nil_conv
thf(fact_88_neq__Nil__conv,axiom,
    ! [Xs: list_l8999537460041914605on_nat] :
      ( ( Xs != nil_li5563851662744376407on_nat )
      = ( ? [Y3: list_S5253404628753938141on_nat,Ys3: list_l8999537460041914605on_nat] :
            ( Xs
            = ( cons_l5857885737047467175on_nat @ Y3 @ Ys3 ) ) ) ) ).

% neq_Nil_conv
thf(fact_89_neq__Nil__conv,axiom,
    ! [Xs: list_l1726306699111145479on_nat] :
      ( ( Xs != nil_li2682182165152866823on_nat )
      = ( ? [Y3: list_P9127870862658512897on_nat,Ys3: list_l1726306699111145479on_nat] :
            ( Xs
            = ( cons_l3028799188195208631on_nat @ Y3 @ Ys3 ) ) ) ) ).

% neq_Nil_conv
thf(fact_90_remdups__adj_Ocases,axiom,
    ! [X2: list_a] :
      ( ( X2 != nil_a )
     => ( ! [X3: a] :
            ( X2
           != ( cons_a @ X3 @ nil_a ) )
       => ~ ! [X3: a,Y2: a,Xs3: list_a] :
              ( X2
             != ( cons_a @ X3 @ ( cons_a @ Y2 @ Xs3 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_91_remdups__adj_Ocases,axiom,
    ! [X2: list_S5253404628753938141on_nat] :
      ( ( X2 != nil_Su3078906344068544711on_nat )
     => ( ! [X3: sum_su4907927656612468301on_nat] :
            ( X2
           != ( cons_S6797538523646374167on_nat @ X3 @ nil_Su3078906344068544711on_nat ) )
       => ~ ! [X3: sum_su4907927656612468301on_nat,Y2: sum_su4907927656612468301on_nat,Xs3: list_S5253404628753938141on_nat] :
              ( X2
             != ( cons_S6797538523646374167on_nat @ X3 @ ( cons_S6797538523646374167on_nat @ Y2 @ Xs3 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_92_remdups__adj_Ocases,axiom,
    ! [X2: list_P9127870862658512897on_nat] :
      ( ( X2 != nil_Pr8614093721167239297on_nat )
     => ( ! [X3: produc1313295348606329723on_nat] :
            ( X2
           != ( cons_P1582596238967110705on_nat @ X3 @ nil_Pr8614093721167239297on_nat ) )
       => ~ ! [X3: produc1313295348606329723on_nat,Y2: produc1313295348606329723on_nat,Xs3: list_P9127870862658512897on_nat] :
              ( X2
             != ( cons_P1582596238967110705on_nat @ X3 @ ( cons_P1582596238967110705on_nat @ Y2 @ Xs3 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_93_remdups__adj_Ocases,axiom,
    ! [X2: list_nat] :
      ( ( X2 != nil_nat )
     => ( ! [X3: nat] :
            ( X2
           != ( cons_nat @ X3 @ nil_nat ) )
       => ~ ! [X3: nat,Y2: nat,Xs3: list_nat] :
              ( X2
             != ( cons_nat @ X3 @ ( cons_nat @ Y2 @ Xs3 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_94_remdups__adj_Ocases,axiom,
    ! [X2: list_l8999537460041914605on_nat] :
      ( ( X2 != nil_li5563851662744376407on_nat )
     => ( ! [X3: list_S5253404628753938141on_nat] :
            ( X2
           != ( cons_l5857885737047467175on_nat @ X3 @ nil_li5563851662744376407on_nat ) )
       => ~ ! [X3: list_S5253404628753938141on_nat,Y2: list_S5253404628753938141on_nat,Xs3: list_l8999537460041914605on_nat] :
              ( X2
             != ( cons_l5857885737047467175on_nat @ X3 @ ( cons_l5857885737047467175on_nat @ Y2 @ Xs3 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_95_remdups__adj_Ocases,axiom,
    ! [X2: list_l1726306699111145479on_nat] :
      ( ( X2 != nil_li2682182165152866823on_nat )
     => ( ! [X3: list_P9127870862658512897on_nat] :
            ( X2
           != ( cons_l3028799188195208631on_nat @ X3 @ nil_li2682182165152866823on_nat ) )
       => ~ ! [X3: list_P9127870862658512897on_nat,Y2: list_P9127870862658512897on_nat,Xs3: list_l1726306699111145479on_nat] :
              ( X2
             != ( cons_l3028799188195208631on_nat @ X3 @ ( cons_l3028799188195208631on_nat @ Y2 @ Xs3 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_96_transpose_Ocases,axiom,
    ! [X2: list_list_a] :
      ( ( X2 != nil_list_a )
     => ( ! [Xss: list_list_a] :
            ( X2
           != ( cons_list_a @ nil_a @ Xss ) )
       => ~ ! [X3: a,Xs3: list_a,Xss: list_list_a] :
              ( X2
             != ( cons_list_a @ ( cons_a @ X3 @ Xs3 ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_97_transpose_Ocases,axiom,
    ! [X2: list_list_nat] :
      ( ( X2 != nil_list_nat )
     => ( ! [Xss: list_list_nat] :
            ( X2
           != ( cons_list_nat @ nil_nat @ Xss ) )
       => ~ ! [X3: nat,Xs3: list_nat,Xss: list_list_nat] :
              ( X2
             != ( cons_list_nat @ ( cons_nat @ X3 @ Xs3 ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_98_transpose_Ocases,axiom,
    ! [X2: list_l8365940747029150845on_nat] :
      ( ( X2 != nil_li3323240303193190759on_nat )
     => ( ! [Xss: list_l8365940747029150845on_nat] :
            ( X2
           != ( cons_l6725403660475259319on_nat @ nil_li5563851662744376407on_nat @ Xss ) )
       => ~ ! [X3: list_S5253404628753938141on_nat,Xs3: list_l8999537460041914605on_nat,Xss: list_l8365940747029150845on_nat] :
              ( X2
             != ( cons_l6725403660475259319on_nat @ ( cons_l5857885737047467175on_nat @ X3 @ Xs3 ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_99_transpose_Ocases,axiom,
    ! [X2: list_l6719245731233385357on_nat] :
      ( ( X2 != nil_li985476221562969869on_nat )
     => ( ! [Xss: list_l6719245731233385357on_nat] :
            ( X2
           != ( cons_l4979741563022480061on_nat @ nil_li2682182165152866823on_nat @ Xss ) )
       => ~ ! [X3: list_P9127870862658512897on_nat,Xs3: list_l1726306699111145479on_nat,Xss: list_l6719245731233385357on_nat] :
              ( X2
             != ( cons_l4979741563022480061on_nat @ ( cons_l3028799188195208631on_nat @ X3 @ Xs3 ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_100_transpose_Ocases,axiom,
    ! [X2: list_l8999537460041914605on_nat] :
      ( ( X2 != nil_li5563851662744376407on_nat )
     => ( ! [Xss: list_l8999537460041914605on_nat] :
            ( X2
           != ( cons_l5857885737047467175on_nat @ nil_Su3078906344068544711on_nat @ Xss ) )
       => ~ ! [X3: sum_su4907927656612468301on_nat,Xs3: list_S5253404628753938141on_nat,Xss: list_l8999537460041914605on_nat] :
              ( X2
             != ( cons_l5857885737047467175on_nat @ ( cons_S6797538523646374167on_nat @ X3 @ Xs3 ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_101_transpose_Ocases,axiom,
    ! [X2: list_l1726306699111145479on_nat] :
      ( ( X2 != nil_li2682182165152866823on_nat )
     => ( ! [Xss: list_l1726306699111145479on_nat] :
            ( X2
           != ( cons_l3028799188195208631on_nat @ nil_Pr8614093721167239297on_nat @ Xss ) )
       => ~ ! [X3: produc1313295348606329723on_nat,Xs3: list_P9127870862658512897on_nat,Xss: list_l1726306699111145479on_nat] :
              ( X2
             != ( cons_l3028799188195208631on_nat @ ( cons_P1582596238967110705on_nat @ X3 @ Xs3 ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_102_min__list_Ocases,axiom,
    ! [X2: list_nat] :
      ( ! [X3: nat,Xs3: list_nat] :
          ( X2
         != ( cons_nat @ X3 @ Xs3 ) )
     => ( X2 = nil_nat ) ) ).

% min_list.cases
thf(fact_103_list_Oexhaust,axiom,
    ! [Y: list_a] :
      ( ( Y != nil_a )
     => ~ ! [X212: a,X222: list_a] :
            ( Y
           != ( cons_a @ X212 @ X222 ) ) ) ).

% list.exhaust
thf(fact_104_list_Oexhaust,axiom,
    ! [Y: list_S5253404628753938141on_nat] :
      ( ( Y != nil_Su3078906344068544711on_nat )
     => ~ ! [X212: sum_su4907927656612468301on_nat,X222: list_S5253404628753938141on_nat] :
            ( Y
           != ( cons_S6797538523646374167on_nat @ X212 @ X222 ) ) ) ).

% list.exhaust
thf(fact_105_list_Oexhaust,axiom,
    ! [Y: list_P9127870862658512897on_nat] :
      ( ( Y != nil_Pr8614093721167239297on_nat )
     => ~ ! [X212: produc1313295348606329723on_nat,X222: list_P9127870862658512897on_nat] :
            ( Y
           != ( cons_P1582596238967110705on_nat @ X212 @ X222 ) ) ) ).

% list.exhaust
thf(fact_106_list_Oexhaust,axiom,
    ! [Y: list_nat] :
      ( ( Y != nil_nat )
     => ~ ! [X212: nat,X222: list_nat] :
            ( Y
           != ( cons_nat @ X212 @ X222 ) ) ) ).

% list.exhaust
thf(fact_107_list_Oexhaust,axiom,
    ! [Y: list_l8999537460041914605on_nat] :
      ( ( Y != nil_li5563851662744376407on_nat )
     => ~ ! [X212: list_S5253404628753938141on_nat,X222: list_l8999537460041914605on_nat] :
            ( Y
           != ( cons_l5857885737047467175on_nat @ X212 @ X222 ) ) ) ).

% list.exhaust
thf(fact_108_list_Oexhaust,axiom,
    ! [Y: list_l1726306699111145479on_nat] :
      ( ( Y != nil_li2682182165152866823on_nat )
     => ~ ! [X212: list_P9127870862658512897on_nat,X222: list_l1726306699111145479on_nat] :
            ( Y
           != ( cons_l3028799188195208631on_nat @ X212 @ X222 ) ) ) ).

% list.exhaust
thf(fact_109_list_OdiscI,axiom,
    ! [List: list_a,X21: a,X22: list_a] :
      ( ( List
        = ( cons_a @ X21 @ X22 ) )
     => ( List != nil_a ) ) ).

% list.discI
thf(fact_110_list_OdiscI,axiom,
    ! [List: list_S5253404628753938141on_nat,X21: sum_su4907927656612468301on_nat,X22: list_S5253404628753938141on_nat] :
      ( ( List
        = ( cons_S6797538523646374167on_nat @ X21 @ X22 ) )
     => ( List != nil_Su3078906344068544711on_nat ) ) ).

% list.discI
thf(fact_111_list_OdiscI,axiom,
    ! [List: list_P9127870862658512897on_nat,X21: produc1313295348606329723on_nat,X22: list_P9127870862658512897on_nat] :
      ( ( List
        = ( cons_P1582596238967110705on_nat @ X21 @ X22 ) )
     => ( List != nil_Pr8614093721167239297on_nat ) ) ).

% list.discI
thf(fact_112_list_OdiscI,axiom,
    ! [List: list_nat,X21: nat,X22: list_nat] :
      ( ( List
        = ( cons_nat @ X21 @ X22 ) )
     => ( List != nil_nat ) ) ).

% list.discI
thf(fact_113_list_OdiscI,axiom,
    ! [List: list_l8999537460041914605on_nat,X21: list_S5253404628753938141on_nat,X22: list_l8999537460041914605on_nat] :
      ( ( List
        = ( cons_l5857885737047467175on_nat @ X21 @ X22 ) )
     => ( List != nil_li5563851662744376407on_nat ) ) ).

% list.discI
thf(fact_114_list_OdiscI,axiom,
    ! [List: list_l1726306699111145479on_nat,X21: list_P9127870862658512897on_nat,X22: list_l1726306699111145479on_nat] :
      ( ( List
        = ( cons_l3028799188195208631on_nat @ X21 @ X22 ) )
     => ( List != nil_li2682182165152866823on_nat ) ) ).

% list.discI
thf(fact_115_list_Odistinct_I1_J,axiom,
    ! [X21: a,X22: list_a] :
      ( nil_a
     != ( cons_a @ X21 @ X22 ) ) ).

% list.distinct(1)
thf(fact_116_list_Odistinct_I1_J,axiom,
    ! [X21: sum_su4907927656612468301on_nat,X22: list_S5253404628753938141on_nat] :
      ( nil_Su3078906344068544711on_nat
     != ( cons_S6797538523646374167on_nat @ X21 @ X22 ) ) ).

% list.distinct(1)
thf(fact_117_list_Odistinct_I1_J,axiom,
    ! [X21: produc1313295348606329723on_nat,X22: list_P9127870862658512897on_nat] :
      ( nil_Pr8614093721167239297on_nat
     != ( cons_P1582596238967110705on_nat @ X21 @ X22 ) ) ).

% list.distinct(1)
thf(fact_118_list_Odistinct_I1_J,axiom,
    ! [X21: nat,X22: list_nat] :
      ( nil_nat
     != ( cons_nat @ X21 @ X22 ) ) ).

% list.distinct(1)
thf(fact_119_list_Odistinct_I1_J,axiom,
    ! [X21: list_S5253404628753938141on_nat,X22: list_l8999537460041914605on_nat] :
      ( nil_li5563851662744376407on_nat
     != ( cons_l5857885737047467175on_nat @ X21 @ X22 ) ) ).

% list.distinct(1)
thf(fact_120_list_Odistinct_I1_J,axiom,
    ! [X21: list_P9127870862658512897on_nat,X22: list_l1726306699111145479on_nat] :
      ( nil_li2682182165152866823on_nat
     != ( cons_l3028799188195208631on_nat @ X21 @ X22 ) ) ).

% list.distinct(1)
thf(fact_121_map__eq__Cons__conv,axiom,
    ! [F: nat > nat,Xs: list_nat,Y: nat,Ys2: list_nat] :
      ( ( ( map_nat_nat @ F @ Xs )
        = ( cons_nat @ Y @ Ys2 ) )
      = ( ? [Z: nat,Zs: list_nat] :
            ( ( Xs
              = ( cons_nat @ Z @ Zs ) )
            & ( ( F @ Z )
              = Y )
            & ( ( map_nat_nat @ F @ Zs )
              = Ys2 ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_122_map__eq__Cons__conv,axiom,
    ! [F: nat > list_nat,Xs: list_nat,Y: list_nat,Ys2: list_list_nat] :
      ( ( ( map_nat_list_nat @ F @ Xs )
        = ( cons_list_nat @ Y @ Ys2 ) )
      = ( ? [Z: nat,Zs: list_nat] :
            ( ( Xs
              = ( cons_nat @ Z @ Zs ) )
            & ( ( F @ Z )
              = Y )
            & ( ( map_nat_list_nat @ F @ Zs )
              = Ys2 ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_123_map__eq__Cons__conv,axiom,
    ! [F: a > sum_su4907927656612468301on_nat,Xs: list_a,Y: sum_su4907927656612468301on_nat,Ys2: list_S5253404628753938141on_nat] :
      ( ( ( map_a_2711072540680412167on_nat @ F @ Xs )
        = ( cons_S6797538523646374167on_nat @ Y @ Ys2 ) )
      = ( ? [Z: a,Zs: list_a] :
            ( ( Xs
              = ( cons_a @ Z @ Zs ) )
            & ( ( F @ Z )
              = Y )
            & ( ( map_a_2711072540680412167on_nat @ F @ Zs )
              = Ys2 ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_124_map__eq__Cons__conv,axiom,
    ! [F: nat > sum_su4907927656612468301on_nat,Xs: list_nat,Y: sum_su4907927656612468301on_nat,Ys2: list_S5253404628753938141on_nat] :
      ( ( ( map_na5706685718985795555on_nat @ F @ Xs )
        = ( cons_S6797538523646374167on_nat @ Y @ Ys2 ) )
      = ( ? [Z: nat,Zs: list_nat] :
            ( ( Xs
              = ( cons_nat @ Z @ Zs ) )
            & ( ( F @ Z )
              = Y )
            & ( ( map_na5706685718985795555on_nat @ F @ Zs )
              = Ys2 ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_125_map__eq__Cons__conv,axiom,
    ! [F: sum_su4907927656612468301on_nat > nat,Xs: list_S5253404628753938141on_nat,Y: nat,Ys2: list_nat] :
      ( ( ( map_Su4809418691747827683at_nat @ F @ Xs )
        = ( cons_nat @ Y @ Ys2 ) )
      = ( ? [Z: sum_su4907927656612468301on_nat,Zs: list_S5253404628753938141on_nat] :
            ( ( Xs
              = ( cons_S6797538523646374167on_nat @ Z @ Zs ) )
            & ( ( F @ Z )
              = Y )
            & ( ( map_Su4809418691747827683at_nat @ F @ Zs )
              = Ys2 ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_126_map__eq__Cons__conv,axiom,
    ! [F: list_S5253404628753938141on_nat > nat,Xs: list_l8999537460041914605on_nat,Y: nat,Ys2: list_nat] :
      ( ( ( map_li2142278717278591475at_nat @ F @ Xs )
        = ( cons_nat @ Y @ Ys2 ) )
      = ( ? [Z: list_S5253404628753938141on_nat,Zs: list_l8999537460041914605on_nat] :
            ( ( Xs
              = ( cons_l5857885737047467175on_nat @ Z @ Zs ) )
            & ( ( F @ Z )
              = Y )
            & ( ( map_li2142278717278591475at_nat @ F @ Zs )
              = Ys2 ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_127_map__eq__Cons__conv,axiom,
    ! [F: a > list_S5253404628753938141on_nat,Xs: list_a,Y: list_S5253404628753938141on_nat,Ys2: list_l8999537460041914605on_nat] :
      ( ( ( map_a_3887198048842487703on_nat @ F @ Xs )
        = ( cons_l5857885737047467175on_nat @ Y @ Ys2 ) )
      = ( ? [Z: a,Zs: list_a] :
            ( ( Xs
              = ( cons_a @ Z @ Zs ) )
            & ( ( F @ Z )
              = Y )
            & ( ( map_a_3887198048842487703on_nat @ F @ Zs )
              = Ys2 ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_128_map__eq__Cons__conv,axiom,
    ! [F: nat > list_S5253404628753938141on_nat,Xs: list_nat,Y: list_S5253404628753938141on_nat,Ys2: list_l8999537460041914605on_nat] :
      ( ( ( map_na4864219606106146163on_nat @ F @ Xs )
        = ( cons_l5857885737047467175on_nat @ Y @ Ys2 ) )
      = ( ? [Z: nat,Zs: list_nat] :
            ( ( Xs
              = ( cons_nat @ Z @ Zs ) )
            & ( ( F @ Z )
              = Y )
            & ( ( map_na4864219606106146163on_nat @ F @ Zs )
              = Ys2 ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_129_map__eq__Cons__conv,axiom,
    ! [F: a > list_l8999537460041914605on_nat,Xs: list_a,Y: list_l8999537460041914605on_nat,Ys2: list_l8365940747029150845on_nat] :
      ( ( ( map_a_3820070144034313383on_nat @ F @ Xs )
        = ( cons_l6725403660475259319on_nat @ Y @ Ys2 ) )
      = ( ? [Z: a,Zs: list_a] :
            ( ( Xs
              = ( cons_a @ Z @ Zs ) )
            & ( ( F @ Z )
              = Y )
            & ( ( map_a_3820070144034313383on_nat @ F @ Zs )
              = Ys2 ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_130_map__eq__Cons__conv,axiom,
    ! [F: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat,Y: sum_su4907927656612468301on_nat,Ys2: list_S5253404628753938141on_nat] :
      ( ( ( map_Su241112079427911264on_nat @ F @ Xs )
        = ( cons_S6797538523646374167on_nat @ Y @ Ys2 ) )
      = ( ? [Z: sum_su4907927656612468301on_nat,Zs: list_S5253404628753938141on_nat] :
            ( ( Xs
              = ( cons_S6797538523646374167on_nat @ Z @ Zs ) )
            & ( ( F @ Z )
              = Y )
            & ( ( map_Su241112079427911264on_nat @ F @ Zs )
              = Ys2 ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_131_Cons__eq__map__conv,axiom,
    ! [X2: nat,Xs: list_nat,F: nat > nat,Ys2: list_nat] :
      ( ( ( cons_nat @ X2 @ Xs )
        = ( map_nat_nat @ F @ Ys2 ) )
      = ( ? [Z: nat,Zs: list_nat] :
            ( ( Ys2
              = ( cons_nat @ Z @ Zs ) )
            & ( X2
              = ( F @ Z ) )
            & ( Xs
              = ( map_nat_nat @ F @ Zs ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_132_Cons__eq__map__conv,axiom,
    ! [X2: list_nat,Xs: list_list_nat,F: nat > list_nat,Ys2: list_nat] :
      ( ( ( cons_list_nat @ X2 @ Xs )
        = ( map_nat_list_nat @ F @ Ys2 ) )
      = ( ? [Z: nat,Zs: list_nat] :
            ( ( Ys2
              = ( cons_nat @ Z @ Zs ) )
            & ( X2
              = ( F @ Z ) )
            & ( Xs
              = ( map_nat_list_nat @ F @ Zs ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_133_Cons__eq__map__conv,axiom,
    ! [X2: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat,F: a > sum_su4907927656612468301on_nat,Ys2: list_a] :
      ( ( ( cons_S6797538523646374167on_nat @ X2 @ Xs )
        = ( map_a_2711072540680412167on_nat @ F @ Ys2 ) )
      = ( ? [Z: a,Zs: list_a] :
            ( ( Ys2
              = ( cons_a @ Z @ Zs ) )
            & ( X2
              = ( F @ Z ) )
            & ( Xs
              = ( map_a_2711072540680412167on_nat @ F @ Zs ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_134_Cons__eq__map__conv,axiom,
    ! [X2: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat,F: nat > sum_su4907927656612468301on_nat,Ys2: list_nat] :
      ( ( ( cons_S6797538523646374167on_nat @ X2 @ Xs )
        = ( map_na5706685718985795555on_nat @ F @ Ys2 ) )
      = ( ? [Z: nat,Zs: list_nat] :
            ( ( Ys2
              = ( cons_nat @ Z @ Zs ) )
            & ( X2
              = ( F @ Z ) )
            & ( Xs
              = ( map_na5706685718985795555on_nat @ F @ Zs ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_135_Cons__eq__map__conv,axiom,
    ! [X2: nat,Xs: list_nat,F: sum_su4907927656612468301on_nat > nat,Ys2: list_S5253404628753938141on_nat] :
      ( ( ( cons_nat @ X2 @ Xs )
        = ( map_Su4809418691747827683at_nat @ F @ Ys2 ) )
      = ( ? [Z: sum_su4907927656612468301on_nat,Zs: list_S5253404628753938141on_nat] :
            ( ( Ys2
              = ( cons_S6797538523646374167on_nat @ Z @ Zs ) )
            & ( X2
              = ( F @ Z ) )
            & ( Xs
              = ( map_Su4809418691747827683at_nat @ F @ Zs ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_136_Cons__eq__map__conv,axiom,
    ! [X2: nat,Xs: list_nat,F: list_S5253404628753938141on_nat > nat,Ys2: list_l8999537460041914605on_nat] :
      ( ( ( cons_nat @ X2 @ Xs )
        = ( map_li2142278717278591475at_nat @ F @ Ys2 ) )
      = ( ? [Z: list_S5253404628753938141on_nat,Zs: list_l8999537460041914605on_nat] :
            ( ( Ys2
              = ( cons_l5857885737047467175on_nat @ Z @ Zs ) )
            & ( X2
              = ( F @ Z ) )
            & ( Xs
              = ( map_li2142278717278591475at_nat @ F @ Zs ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_137_Cons__eq__map__conv,axiom,
    ! [X2: list_S5253404628753938141on_nat,Xs: list_l8999537460041914605on_nat,F: a > list_S5253404628753938141on_nat,Ys2: list_a] :
      ( ( ( cons_l5857885737047467175on_nat @ X2 @ Xs )
        = ( map_a_3887198048842487703on_nat @ F @ Ys2 ) )
      = ( ? [Z: a,Zs: list_a] :
            ( ( Ys2
              = ( cons_a @ Z @ Zs ) )
            & ( X2
              = ( F @ Z ) )
            & ( Xs
              = ( map_a_3887198048842487703on_nat @ F @ Zs ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_138_Cons__eq__map__conv,axiom,
    ! [X2: list_S5253404628753938141on_nat,Xs: list_l8999537460041914605on_nat,F: nat > list_S5253404628753938141on_nat,Ys2: list_nat] :
      ( ( ( cons_l5857885737047467175on_nat @ X2 @ Xs )
        = ( map_na4864219606106146163on_nat @ F @ Ys2 ) )
      = ( ? [Z: nat,Zs: list_nat] :
            ( ( Ys2
              = ( cons_nat @ Z @ Zs ) )
            & ( X2
              = ( F @ Z ) )
            & ( Xs
              = ( map_na4864219606106146163on_nat @ F @ Zs ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_139_Cons__eq__map__conv,axiom,
    ! [X2: list_l8999537460041914605on_nat,Xs: list_l8365940747029150845on_nat,F: a > list_l8999537460041914605on_nat,Ys2: list_a] :
      ( ( ( cons_l6725403660475259319on_nat @ X2 @ Xs )
        = ( map_a_3820070144034313383on_nat @ F @ Ys2 ) )
      = ( ? [Z: a,Zs: list_a] :
            ( ( Ys2
              = ( cons_a @ Z @ Zs ) )
            & ( X2
              = ( F @ Z ) )
            & ( Xs
              = ( map_a_3820070144034313383on_nat @ F @ Zs ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_140_Cons__eq__map__conv,axiom,
    ! [X2: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat,F: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat,Ys2: list_S5253404628753938141on_nat] :
      ( ( ( cons_S6797538523646374167on_nat @ X2 @ Xs )
        = ( map_Su241112079427911264on_nat @ F @ Ys2 ) )
      = ( ? [Z: sum_su4907927656612468301on_nat,Zs: list_S5253404628753938141on_nat] :
            ( ( Ys2
              = ( cons_S6797538523646374167on_nat @ Z @ Zs ) )
            & ( X2
              = ( F @ Z ) )
            & ( Xs
              = ( map_Su241112079427911264on_nat @ F @ Zs ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_141_map__eq__Cons__D,axiom,
    ! [F: nat > nat,Xs: list_nat,Y: nat,Ys2: list_nat] :
      ( ( ( map_nat_nat @ F @ Xs )
        = ( cons_nat @ Y @ Ys2 ) )
     => ? [Z2: nat,Zs2: list_nat] :
          ( ( Xs
            = ( cons_nat @ Z2 @ Zs2 ) )
          & ( ( F @ Z2 )
            = Y )
          & ( ( map_nat_nat @ F @ Zs2 )
            = Ys2 ) ) ) ).

% map_eq_Cons_D
thf(fact_142_map__eq__Cons__D,axiom,
    ! [F: nat > list_nat,Xs: list_nat,Y: list_nat,Ys2: list_list_nat] :
      ( ( ( map_nat_list_nat @ F @ Xs )
        = ( cons_list_nat @ Y @ Ys2 ) )
     => ? [Z2: nat,Zs2: list_nat] :
          ( ( Xs
            = ( cons_nat @ Z2 @ Zs2 ) )
          & ( ( F @ Z2 )
            = Y )
          & ( ( map_nat_list_nat @ F @ Zs2 )
            = Ys2 ) ) ) ).

% map_eq_Cons_D
thf(fact_143_map__eq__Cons__D,axiom,
    ! [F: a > sum_su4907927656612468301on_nat,Xs: list_a,Y: sum_su4907927656612468301on_nat,Ys2: list_S5253404628753938141on_nat] :
      ( ( ( map_a_2711072540680412167on_nat @ F @ Xs )
        = ( cons_S6797538523646374167on_nat @ Y @ Ys2 ) )
     => ? [Z2: a,Zs2: list_a] :
          ( ( Xs
            = ( cons_a @ Z2 @ Zs2 ) )
          & ( ( F @ Z2 )
            = Y )
          & ( ( map_a_2711072540680412167on_nat @ F @ Zs2 )
            = Ys2 ) ) ) ).

% map_eq_Cons_D
thf(fact_144_map__eq__Cons__D,axiom,
    ! [F: nat > sum_su4907927656612468301on_nat,Xs: list_nat,Y: sum_su4907927656612468301on_nat,Ys2: list_S5253404628753938141on_nat] :
      ( ( ( map_na5706685718985795555on_nat @ F @ Xs )
        = ( cons_S6797538523646374167on_nat @ Y @ Ys2 ) )
     => ? [Z2: nat,Zs2: list_nat] :
          ( ( Xs
            = ( cons_nat @ Z2 @ Zs2 ) )
          & ( ( F @ Z2 )
            = Y )
          & ( ( map_na5706685718985795555on_nat @ F @ Zs2 )
            = Ys2 ) ) ) ).

% map_eq_Cons_D
thf(fact_145_map__eq__Cons__D,axiom,
    ! [F: sum_su4907927656612468301on_nat > nat,Xs: list_S5253404628753938141on_nat,Y: nat,Ys2: list_nat] :
      ( ( ( map_Su4809418691747827683at_nat @ F @ Xs )
        = ( cons_nat @ Y @ Ys2 ) )
     => ? [Z2: sum_su4907927656612468301on_nat,Zs2: list_S5253404628753938141on_nat] :
          ( ( Xs
            = ( cons_S6797538523646374167on_nat @ Z2 @ Zs2 ) )
          & ( ( F @ Z2 )
            = Y )
          & ( ( map_Su4809418691747827683at_nat @ F @ Zs2 )
            = Ys2 ) ) ) ).

% map_eq_Cons_D
thf(fact_146_map__eq__Cons__D,axiom,
    ! [F: list_S5253404628753938141on_nat > nat,Xs: list_l8999537460041914605on_nat,Y: nat,Ys2: list_nat] :
      ( ( ( map_li2142278717278591475at_nat @ F @ Xs )
        = ( cons_nat @ Y @ Ys2 ) )
     => ? [Z2: list_S5253404628753938141on_nat,Zs2: list_l8999537460041914605on_nat] :
          ( ( Xs
            = ( cons_l5857885737047467175on_nat @ Z2 @ Zs2 ) )
          & ( ( F @ Z2 )
            = Y )
          & ( ( map_li2142278717278591475at_nat @ F @ Zs2 )
            = Ys2 ) ) ) ).

% map_eq_Cons_D
thf(fact_147_map__eq__Cons__D,axiom,
    ! [F: a > list_S5253404628753938141on_nat,Xs: list_a,Y: list_S5253404628753938141on_nat,Ys2: list_l8999537460041914605on_nat] :
      ( ( ( map_a_3887198048842487703on_nat @ F @ Xs )
        = ( cons_l5857885737047467175on_nat @ Y @ Ys2 ) )
     => ? [Z2: a,Zs2: list_a] :
          ( ( Xs
            = ( cons_a @ Z2 @ Zs2 ) )
          & ( ( F @ Z2 )
            = Y )
          & ( ( map_a_3887198048842487703on_nat @ F @ Zs2 )
            = Ys2 ) ) ) ).

% map_eq_Cons_D
thf(fact_148_map__eq__Cons__D,axiom,
    ! [F: nat > list_S5253404628753938141on_nat,Xs: list_nat,Y: list_S5253404628753938141on_nat,Ys2: list_l8999537460041914605on_nat] :
      ( ( ( map_na4864219606106146163on_nat @ F @ Xs )
        = ( cons_l5857885737047467175on_nat @ Y @ Ys2 ) )
     => ? [Z2: nat,Zs2: list_nat] :
          ( ( Xs
            = ( cons_nat @ Z2 @ Zs2 ) )
          & ( ( F @ Z2 )
            = Y )
          & ( ( map_na4864219606106146163on_nat @ F @ Zs2 )
            = Ys2 ) ) ) ).

% map_eq_Cons_D
thf(fact_149_map__eq__Cons__D,axiom,
    ! [F: a > list_l8999537460041914605on_nat,Xs: list_a,Y: list_l8999537460041914605on_nat,Ys2: list_l8365940747029150845on_nat] :
      ( ( ( map_a_3820070144034313383on_nat @ F @ Xs )
        = ( cons_l6725403660475259319on_nat @ Y @ Ys2 ) )
     => ? [Z2: a,Zs2: list_a] :
          ( ( Xs
            = ( cons_a @ Z2 @ Zs2 ) )
          & ( ( F @ Z2 )
            = Y )
          & ( ( map_a_3820070144034313383on_nat @ F @ Zs2 )
            = Ys2 ) ) ) ).

% map_eq_Cons_D
thf(fact_150_map__eq__Cons__D,axiom,
    ! [F: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat,Y: sum_su4907927656612468301on_nat,Ys2: list_S5253404628753938141on_nat] :
      ( ( ( map_Su241112079427911264on_nat @ F @ Xs )
        = ( cons_S6797538523646374167on_nat @ Y @ Ys2 ) )
     => ? [Z2: sum_su4907927656612468301on_nat,Zs2: list_S5253404628753938141on_nat] :
          ( ( Xs
            = ( cons_S6797538523646374167on_nat @ Z2 @ Zs2 ) )
          & ( ( F @ Z2 )
            = Y )
          & ( ( map_Su241112079427911264on_nat @ F @ Zs2 )
            = Ys2 ) ) ) ).

% map_eq_Cons_D
thf(fact_151_Cons__eq__map__D,axiom,
    ! [X2: nat,Xs: list_nat,F: nat > nat,Ys2: list_nat] :
      ( ( ( cons_nat @ X2 @ Xs )
        = ( map_nat_nat @ F @ Ys2 ) )
     => ? [Z2: nat,Zs2: list_nat] :
          ( ( Ys2
            = ( cons_nat @ Z2 @ Zs2 ) )
          & ( X2
            = ( F @ Z2 ) )
          & ( Xs
            = ( map_nat_nat @ F @ Zs2 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_152_Cons__eq__map__D,axiom,
    ! [X2: list_nat,Xs: list_list_nat,F: nat > list_nat,Ys2: list_nat] :
      ( ( ( cons_list_nat @ X2 @ Xs )
        = ( map_nat_list_nat @ F @ Ys2 ) )
     => ? [Z2: nat,Zs2: list_nat] :
          ( ( Ys2
            = ( cons_nat @ Z2 @ Zs2 ) )
          & ( X2
            = ( F @ Z2 ) )
          & ( Xs
            = ( map_nat_list_nat @ F @ Zs2 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_153_Cons__eq__map__D,axiom,
    ! [X2: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat,F: a > sum_su4907927656612468301on_nat,Ys2: list_a] :
      ( ( ( cons_S6797538523646374167on_nat @ X2 @ Xs )
        = ( map_a_2711072540680412167on_nat @ F @ Ys2 ) )
     => ? [Z2: a,Zs2: list_a] :
          ( ( Ys2
            = ( cons_a @ Z2 @ Zs2 ) )
          & ( X2
            = ( F @ Z2 ) )
          & ( Xs
            = ( map_a_2711072540680412167on_nat @ F @ Zs2 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_154_Cons__eq__map__D,axiom,
    ! [X2: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat,F: nat > sum_su4907927656612468301on_nat,Ys2: list_nat] :
      ( ( ( cons_S6797538523646374167on_nat @ X2 @ Xs )
        = ( map_na5706685718985795555on_nat @ F @ Ys2 ) )
     => ? [Z2: nat,Zs2: list_nat] :
          ( ( Ys2
            = ( cons_nat @ Z2 @ Zs2 ) )
          & ( X2
            = ( F @ Z2 ) )
          & ( Xs
            = ( map_na5706685718985795555on_nat @ F @ Zs2 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_155_Cons__eq__map__D,axiom,
    ! [X2: nat,Xs: list_nat,F: sum_su4907927656612468301on_nat > nat,Ys2: list_S5253404628753938141on_nat] :
      ( ( ( cons_nat @ X2 @ Xs )
        = ( map_Su4809418691747827683at_nat @ F @ Ys2 ) )
     => ? [Z2: sum_su4907927656612468301on_nat,Zs2: list_S5253404628753938141on_nat] :
          ( ( Ys2
            = ( cons_S6797538523646374167on_nat @ Z2 @ Zs2 ) )
          & ( X2
            = ( F @ Z2 ) )
          & ( Xs
            = ( map_Su4809418691747827683at_nat @ F @ Zs2 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_156_Cons__eq__map__D,axiom,
    ! [X2: nat,Xs: list_nat,F: list_S5253404628753938141on_nat > nat,Ys2: list_l8999537460041914605on_nat] :
      ( ( ( cons_nat @ X2 @ Xs )
        = ( map_li2142278717278591475at_nat @ F @ Ys2 ) )
     => ? [Z2: list_S5253404628753938141on_nat,Zs2: list_l8999537460041914605on_nat] :
          ( ( Ys2
            = ( cons_l5857885737047467175on_nat @ Z2 @ Zs2 ) )
          & ( X2
            = ( F @ Z2 ) )
          & ( Xs
            = ( map_li2142278717278591475at_nat @ F @ Zs2 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_157_Cons__eq__map__D,axiom,
    ! [X2: list_S5253404628753938141on_nat,Xs: list_l8999537460041914605on_nat,F: a > list_S5253404628753938141on_nat,Ys2: list_a] :
      ( ( ( cons_l5857885737047467175on_nat @ X2 @ Xs )
        = ( map_a_3887198048842487703on_nat @ F @ Ys2 ) )
     => ? [Z2: a,Zs2: list_a] :
          ( ( Ys2
            = ( cons_a @ Z2 @ Zs2 ) )
          & ( X2
            = ( F @ Z2 ) )
          & ( Xs
            = ( map_a_3887198048842487703on_nat @ F @ Zs2 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_158_Cons__eq__map__D,axiom,
    ! [X2: list_S5253404628753938141on_nat,Xs: list_l8999537460041914605on_nat,F: nat > list_S5253404628753938141on_nat,Ys2: list_nat] :
      ( ( ( cons_l5857885737047467175on_nat @ X2 @ Xs )
        = ( map_na4864219606106146163on_nat @ F @ Ys2 ) )
     => ? [Z2: nat,Zs2: list_nat] :
          ( ( Ys2
            = ( cons_nat @ Z2 @ Zs2 ) )
          & ( X2
            = ( F @ Z2 ) )
          & ( Xs
            = ( map_na4864219606106146163on_nat @ F @ Zs2 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_159_Cons__eq__map__D,axiom,
    ! [X2: list_l8999537460041914605on_nat,Xs: list_l8365940747029150845on_nat,F: a > list_l8999537460041914605on_nat,Ys2: list_a] :
      ( ( ( cons_l6725403660475259319on_nat @ X2 @ Xs )
        = ( map_a_3820070144034313383on_nat @ F @ Ys2 ) )
     => ? [Z2: a,Zs2: list_a] :
          ( ( Ys2
            = ( cons_a @ Z2 @ Zs2 ) )
          & ( X2
            = ( F @ Z2 ) )
          & ( Xs
            = ( map_a_3820070144034313383on_nat @ F @ Zs2 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_160_Cons__eq__map__D,axiom,
    ! [X2: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat,F: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat,Ys2: list_S5253404628753938141on_nat] :
      ( ( ( cons_S6797538523646374167on_nat @ X2 @ Xs )
        = ( map_Su241112079427911264on_nat @ F @ Ys2 ) )
     => ? [Z2: sum_su4907927656612468301on_nat,Zs2: list_S5253404628753938141on_nat] :
          ( ( Ys2
            = ( cons_S6797538523646374167on_nat @ Z2 @ Zs2 ) )
          & ( X2
            = ( F @ Z2 ) )
          & ( Xs
            = ( map_Su241112079427911264on_nat @ F @ Zs2 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_161_list_Osimps_I9_J,axiom,
    ! [F: nat > nat,X21: nat,X22: list_nat] :
      ( ( map_nat_nat @ F @ ( cons_nat @ X21 @ X22 ) )
      = ( cons_nat @ ( F @ X21 ) @ ( map_nat_nat @ F @ X22 ) ) ) ).

% list.simps(9)
thf(fact_162_list_Osimps_I9_J,axiom,
    ! [F: nat > list_nat,X21: nat,X22: list_nat] :
      ( ( map_nat_list_nat @ F @ ( cons_nat @ X21 @ X22 ) )
      = ( cons_list_nat @ ( F @ X21 ) @ ( map_nat_list_nat @ F @ X22 ) ) ) ).

% list.simps(9)
thf(fact_163_list_Osimps_I9_J,axiom,
    ! [F: a > sum_su4907927656612468301on_nat,X21: a,X22: list_a] :
      ( ( map_a_2711072540680412167on_nat @ F @ ( cons_a @ X21 @ X22 ) )
      = ( cons_S6797538523646374167on_nat @ ( F @ X21 ) @ ( map_a_2711072540680412167on_nat @ F @ X22 ) ) ) ).

% list.simps(9)
thf(fact_164_list_Osimps_I9_J,axiom,
    ! [F: sum_su4907927656612468301on_nat > nat,X21: sum_su4907927656612468301on_nat,X22: list_S5253404628753938141on_nat] :
      ( ( map_Su4809418691747827683at_nat @ F @ ( cons_S6797538523646374167on_nat @ X21 @ X22 ) )
      = ( cons_nat @ ( F @ X21 ) @ ( map_Su4809418691747827683at_nat @ F @ X22 ) ) ) ).

% list.simps(9)
thf(fact_165_list_Osimps_I9_J,axiom,
    ! [F: nat > sum_su4907927656612468301on_nat,X21: nat,X22: list_nat] :
      ( ( map_na5706685718985795555on_nat @ F @ ( cons_nat @ X21 @ X22 ) )
      = ( cons_S6797538523646374167on_nat @ ( F @ X21 ) @ ( map_na5706685718985795555on_nat @ F @ X22 ) ) ) ).

% list.simps(9)
thf(fact_166_list_Osimps_I9_J,axiom,
    ! [F: a > list_S5253404628753938141on_nat,X21: a,X22: list_a] :
      ( ( map_a_3887198048842487703on_nat @ F @ ( cons_a @ X21 @ X22 ) )
      = ( cons_l5857885737047467175on_nat @ ( F @ X21 ) @ ( map_a_3887198048842487703on_nat @ F @ X22 ) ) ) ).

% list.simps(9)
thf(fact_167_list_Osimps_I9_J,axiom,
    ! [F: nat > list_S5253404628753938141on_nat,X21: nat,X22: list_nat] :
      ( ( map_na4864219606106146163on_nat @ F @ ( cons_nat @ X21 @ X22 ) )
      = ( cons_l5857885737047467175on_nat @ ( F @ X21 ) @ ( map_na4864219606106146163on_nat @ F @ X22 ) ) ) ).

% list.simps(9)
thf(fact_168_list_Osimps_I9_J,axiom,
    ! [F: list_S5253404628753938141on_nat > nat,X21: list_S5253404628753938141on_nat,X22: list_l8999537460041914605on_nat] :
      ( ( map_li2142278717278591475at_nat @ F @ ( cons_l5857885737047467175on_nat @ X21 @ X22 ) )
      = ( cons_nat @ ( F @ X21 ) @ ( map_li2142278717278591475at_nat @ F @ X22 ) ) ) ).

% list.simps(9)
thf(fact_169_list_Osimps_I9_J,axiom,
    ! [F: a > list_l8999537460041914605on_nat,X21: a,X22: list_a] :
      ( ( map_a_3820070144034313383on_nat @ F @ ( cons_a @ X21 @ X22 ) )
      = ( cons_l6725403660475259319on_nat @ ( F @ X21 ) @ ( map_a_3820070144034313383on_nat @ F @ X22 ) ) ) ).

% list.simps(9)
thf(fact_170_list_Osimps_I9_J,axiom,
    ! [F: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat,X21: sum_su4907927656612468301on_nat,X22: list_S5253404628753938141on_nat] :
      ( ( map_Su241112079427911264on_nat @ F @ ( cons_S6797538523646374167on_nat @ X21 @ X22 ) )
      = ( cons_S6797538523646374167on_nat @ ( F @ X21 ) @ ( map_Su241112079427911264on_nat @ F @ X22 ) ) ) ).

% list.simps(9)
thf(fact_171_list_Osimps_I8_J,axiom,
    ! [F: nat > a] :
      ( ( map_nat_a @ F @ nil_nat )
      = nil_a ) ).

% list.simps(8)
thf(fact_172_list_Osimps_I8_J,axiom,
    ! [F: a > nat] :
      ( ( map_a_nat @ F @ nil_a )
      = nil_nat ) ).

% list.simps(8)
thf(fact_173_list_Osimps_I8_J,axiom,
    ! [F: a > a] :
      ( ( map_a_a @ F @ nil_a )
      = nil_a ) ).

% list.simps(8)
thf(fact_174_list_Osimps_I8_J,axiom,
    ! [F: nat > nat] :
      ( ( map_nat_nat @ F @ nil_nat )
      = nil_nat ) ).

% list.simps(8)
thf(fact_175_list_Osimps_I8_J,axiom,
    ! [F: nat > list_nat] :
      ( ( map_nat_list_nat @ F @ nil_nat )
      = nil_list_nat ) ).

% list.simps(8)
thf(fact_176_list_Osimps_I8_J,axiom,
    ! [F: sum_su4907927656612468301on_nat > nat] :
      ( ( map_Su4809418691747827683at_nat @ F @ nil_Su3078906344068544711on_nat )
      = nil_nat ) ).

% list.simps(8)
thf(fact_177_list_Osimps_I8_J,axiom,
    ! [F: sum_su4907927656612468301on_nat > a] :
      ( ( map_Su5039174899830097323_nat_a @ F @ nil_Su3078906344068544711on_nat )
      = nil_a ) ).

% list.simps(8)
thf(fact_178_list_Osimps_I8_J,axiom,
    ! [F: nat > sum_su4907927656612468301on_nat] :
      ( ( map_na5706685718985795555on_nat @ F @ nil_nat )
      = nil_Su3078906344068544711on_nat ) ).

% list.simps(8)
thf(fact_179_list_Osimps_I8_J,axiom,
    ! [F: a > sum_su4907927656612468301on_nat] :
      ( ( map_a_2711072540680412167on_nat @ F @ nil_a )
      = nil_Su3078906344068544711on_nat ) ).

% list.simps(8)
thf(fact_180_list_Osimps_I8_J,axiom,
    ! [F: nat > list_S5253404628753938141on_nat] :
      ( ( map_na4864219606106146163on_nat @ F @ nil_nat )
      = nil_li5563851662744376407on_nat ) ).

% list.simps(8)
thf(fact_181_concat_Osimps_I1_J,axiom,
    ( ( concat_nat @ nil_list_nat )
    = nil_nat ) ).

% concat.simps(1)
thf(fact_182_concat_Osimps_I1_J,axiom,
    ( ( concat4774721536428481500on_nat @ nil_li3323240303193190759on_nat )
    = nil_li5563851662744376407on_nat ) ).

% concat.simps(1)
thf(fact_183_concat_Osimps_I1_J,axiom,
    ( ( concat4341727598008994882on_nat @ nil_li985476221562969869on_nat )
    = nil_li2682182165152866823on_nat ) ).

% concat.simps(1)
thf(fact_184_concat_Osimps_I1_J,axiom,
    ( ( concat_a @ nil_list_a )
    = nil_a ) ).

% concat.simps(1)
thf(fact_185_concat_Osimps_I1_J,axiom,
    ( ( concat6161509665823917772on_nat @ nil_li5563851662744376407on_nat )
    = nil_Su3078906344068544711on_nat ) ).

% concat.simps(1)
thf(fact_186_concat_Osimps_I1_J,axiom,
    ( ( concat937493773106420796on_nat @ nil_li2682182165152866823on_nat )
    = nil_Pr8614093721167239297on_nat ) ).

% concat.simps(1)
thf(fact_187_map__concat,axiom,
    ! [F: nat > nat,Xs: list_list_nat] :
      ( ( map_nat_nat @ F @ ( concat_nat @ Xs ) )
      = ( concat_nat @ ( map_li7225945977422193158st_nat @ ( map_nat_nat @ F ) @ Xs ) ) ) ).

% map_concat
thf(fact_188_map__concat,axiom,
    ! [F: nat > list_nat,Xs: list_list_nat] :
      ( ( map_nat_list_nat @ F @ ( concat_nat @ Xs ) )
      = ( concat_list_nat @ ( map_li960784813134754710st_nat @ ( map_nat_list_nat @ F ) @ Xs ) ) ) ).

% map_concat
thf(fact_189_map__concat,axiom,
    ! [F: sum_su4907927656612468301on_nat > nat,Xs: list_l8999537460041914605on_nat] :
      ( ( map_Su4809418691747827683at_nat @ F @ ( concat6161509665823917772on_nat @ Xs ) )
      = ( concat_nat @ ( map_li2358237010938824579st_nat @ ( map_Su4809418691747827683at_nat @ F ) @ Xs ) ) ) ).

% map_concat
thf(fact_190_map__concat,axiom,
    ! [F: nat > sum_su4907927656612468301on_nat,Xs: list_list_nat] :
      ( ( map_na5706685718985795555on_nat @ F @ ( concat_nat @ Xs ) )
      = ( concat6161509665823917772on_nat @ ( map_li1427182934947234307on_nat @ ( map_na5706685718985795555on_nat @ F ) @ Xs ) ) ) ).

% map_concat
thf(fact_191_map__concat,axiom,
    ! [F: a > sum_su4907927656612468301on_nat,Xs: list_list_a] :
      ( ( map_a_2711072540680412167on_nat @ F @ ( concat_a @ Xs ) )
      = ( concat6161509665823917772on_nat @ ( map_li3793676233746214813on_nat @ ( map_a_2711072540680412167on_nat @ F ) @ Xs ) ) ) ).

% map_concat
thf(fact_192_map__concat,axiom,
    ! [F: nat > list_S5253404628753938141on_nat,Xs: list_list_nat] :
      ( ( map_na4864219606106146163on_nat @ F @ ( concat_nat @ Xs ) )
      = ( concat4774721536428481500on_nat @ ( map_li740583212113058067on_nat @ ( map_na4864219606106146163on_nat @ F ) @ Xs ) ) ) ).

% map_concat
thf(fact_193_map__concat,axiom,
    ! [F: list_S5253404628753938141on_nat > nat,Xs: list_l8365940747029150845on_nat] :
      ( ( map_li2142278717278591475at_nat @ F @ ( concat4774721536428481500on_nat @ Xs ) )
      = ( concat_nat @ ( map_li2940836757168442131st_nat @ ( map_li2142278717278591475at_nat @ F ) @ Xs ) ) ) ).

% map_concat
thf(fact_194_map__concat,axiom,
    ! [F: a > list_S5253404628753938141on_nat,Xs: list_list_a] :
      ( ( map_a_3887198048842487703on_nat @ F @ ( concat_a @ Xs ) )
      = ( concat4774721536428481500on_nat @ ( map_li7277942993266323885on_nat @ ( map_a_3887198048842487703on_nat @ F ) @ Xs ) ) ) ).

% map_concat
thf(fact_195_map__concat,axiom,
    ! [F: a > list_l8999537460041914605on_nat,Xs: list_list_a] :
      ( ( map_a_3820070144034313383on_nat @ F @ ( concat_a @ Xs ) )
      = ( concat6293066666318574188on_nat @ ( map_li6744039989378561341on_nat @ ( map_a_3820070144034313383on_nat @ F ) @ Xs ) ) ) ).

% map_concat
thf(fact_196_map__concat,axiom,
    ! [F: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat,Xs: list_l8999537460041914605on_nat] :
      ( ( map_Su241112079427911264on_nat @ F @ ( concat6161509665823917772on_nat @ Xs ) )
      = ( concat6161509665823917772on_nat @ ( map_li4295621887133137792on_nat @ ( map_Su241112079427911264on_nat @ F ) @ Xs ) ) ) ).

% map_concat
thf(fact_197_NS__def,axiom,
    ( ns
    = ( concat937493773106420796on_nat
      @ ( map_a_7431629901395152583on_nat
        @ ^ [X: a] : ( cons_P1582596238967110705on_nat @ ( produc6704844767664046899on_nat @ ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ X @ multis8267843437418295310sitive ) ) @ ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ X @ multis5512618279175645674signed ) ) ) @ ( cons_P1582596238967110705on_nat @ ( produc6704844767664046899on_nat @ ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ X @ multis5977116398950053002gative ) ) @ ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ X @ multis5512618279175645674signed ) ) ) @ nil_Pr8614093721167239297on_nat ) )
        @ xs ) ) ) ).

% NS_def
thf(fact_198_NSS,axiom,
    ( nss
    = ( append2865821678206098582on_nat @ ns @ s ) ) ).

% NSS
thf(fact_199_append1__eq__conv,axiom,
    ! [Xs: list_a,X2: a,Ys2: list_a,Y: a] :
      ( ( ( append_a @ Xs @ ( cons_a @ X2 @ nil_a ) )
        = ( append_a @ Ys2 @ ( cons_a @ Y @ nil_a ) ) )
      = ( ( Xs = Ys2 )
        & ( X2 = Y ) ) ) ).

% append1_eq_conv
thf(fact_200_append1__eq__conv,axiom,
    ! [Xs: list_S5253404628753938141on_nat,X2: sum_su4907927656612468301on_nat,Ys2: list_S5253404628753938141on_nat,Y: sum_su4907927656612468301on_nat] :
      ( ( ( append3523812819975775986on_nat @ Xs @ ( cons_S6797538523646374167on_nat @ X2 @ nil_Su3078906344068544711on_nat ) )
        = ( append3523812819975775986on_nat @ Ys2 @ ( cons_S6797538523646374167on_nat @ Y @ nil_Su3078906344068544711on_nat ) ) )
      = ( ( Xs = Ys2 )
        & ( X2 = Y ) ) ) ).

% append1_eq_conv
thf(fact_201_append1__eq__conv,axiom,
    ! [Xs: list_P9127870862658512897on_nat,X2: produc1313295348606329723on_nat,Ys2: list_P9127870862658512897on_nat,Y: produc1313295348606329723on_nat] :
      ( ( ( append2865821678206098582on_nat @ Xs @ ( cons_P1582596238967110705on_nat @ X2 @ nil_Pr8614093721167239297on_nat ) )
        = ( append2865821678206098582on_nat @ Ys2 @ ( cons_P1582596238967110705on_nat @ Y @ nil_Pr8614093721167239297on_nat ) ) )
      = ( ( Xs = Ys2 )
        & ( X2 = Y ) ) ) ).

% append1_eq_conv
thf(fact_202_append1__eq__conv,axiom,
    ! [Xs: list_nat,X2: nat,Ys2: list_nat,Y: nat] :
      ( ( ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) )
        = ( append_nat @ Ys2 @ ( cons_nat @ Y @ nil_nat ) ) )
      = ( ( Xs = Ys2 )
        & ( X2 = Y ) ) ) ).

% append1_eq_conv
thf(fact_203_append1__eq__conv,axiom,
    ! [Xs: list_l8999537460041914605on_nat,X2: list_S5253404628753938141on_nat,Ys2: list_l8999537460041914605on_nat,Y: list_S5253404628753938141on_nat] :
      ( ( ( append6174841615030841602on_nat @ Xs @ ( cons_l5857885737047467175on_nat @ X2 @ nil_li5563851662744376407on_nat ) )
        = ( append6174841615030841602on_nat @ Ys2 @ ( cons_l5857885737047467175on_nat @ Y @ nil_li5563851662744376407on_nat ) ) )
      = ( ( Xs = Ys2 )
        & ( X2 = Y ) ) ) ).

% append1_eq_conv
thf(fact_204_append1__eq__conv,axiom,
    ! [Xs: list_l1726306699111145479on_nat,X2: list_P9127870862658512897on_nat,Ys2: list_l1726306699111145479on_nat,Y: list_P9127870862658512897on_nat] :
      ( ( ( append3839213209832495004on_nat @ Xs @ ( cons_l3028799188195208631on_nat @ X2 @ nil_li2682182165152866823on_nat ) )
        = ( append3839213209832495004on_nat @ Ys2 @ ( cons_l3028799188195208631on_nat @ Y @ nil_li2682182165152866823on_nat ) ) )
      = ( ( Xs = Ys2 )
        & ( X2 = Y ) ) ) ).

% append1_eq_conv
thf(fact_205_mult2__ex__dom_Ocases,axiom,
    ! [X2: produc8880992058355704947list_a] :
      ( ! [X3: a,Xs3: list_a,Ys4: list_a] :
          ( X2
         != ( produc8030850274662652387list_a @ X3 @ ( produc1910438886824740410list_a @ Xs3 @ ( produc6837034575241423639list_a @ nil_a @ Ys4 ) ) ) )
     => ~ ! [X3: a,Xs3: list_a,Y2: a,Ys: list_a,Ys4: list_a] :
            ( X2
           != ( produc8030850274662652387list_a @ X3 @ ( produc1910438886824740410list_a @ Xs3 @ ( produc6837034575241423639list_a @ ( cons_a @ Y2 @ Ys ) @ Ys4 ) ) ) ) ) ).

% mult2_ex_dom.cases
thf(fact_206_mult2__ex__dom_Ocases,axiom,
    ! [X2: produc3977234058138274081on_nat] :
      ( ! [X3: sum_su4907927656612468301on_nat,Xs3: list_S5253404628753938141on_nat,Ys4: list_S5253404628753938141on_nat] :
          ( X2
         != ( produc4095905013319207641on_nat @ X3 @ ( produc7869692867729086373on_nat @ Xs3 @ ( produc2912653624306349395on_nat @ nil_Su3078906344068544711on_nat @ Ys4 ) ) ) )
     => ~ ! [X3: sum_su4907927656612468301on_nat,Xs3: list_S5253404628753938141on_nat,Y2: sum_su4907927656612468301on_nat,Ys: list_S5253404628753938141on_nat,Ys4: list_S5253404628753938141on_nat] :
            ( X2
           != ( produc4095905013319207641on_nat @ X3 @ ( produc7869692867729086373on_nat @ Xs3 @ ( produc2912653624306349395on_nat @ ( cons_S6797538523646374167on_nat @ Y2 @ Ys ) @ Ys4 ) ) ) ) ) ).

% mult2_ex_dom.cases
thf(fact_207_mult2__ex__dom_Ocases,axiom,
    ! [X2: produc7326124634926975417on_nat] :
      ( ! [X3: produc1313295348606329723on_nat,Xs3: list_P9127870862658512897on_nat,Ys4: list_P9127870862658512897on_nat] :
          ( X2
         != ( produc2628373118283442345on_nat @ X3 @ ( produc1138821556964385341on_nat @ Xs3 @ ( produc7152764266822961111on_nat @ nil_Pr8614093721167239297on_nat @ Ys4 ) ) ) )
     => ~ ! [X3: produc1313295348606329723on_nat,Xs3: list_P9127870862658512897on_nat,Y2: produc1313295348606329723on_nat,Ys: list_P9127870862658512897on_nat,Ys4: list_P9127870862658512897on_nat] :
            ( X2
           != ( produc2628373118283442345on_nat @ X3 @ ( produc1138821556964385341on_nat @ Xs3 @ ( produc7152764266822961111on_nat @ ( cons_P1582596238967110705on_nat @ Y2 @ Ys ) @ Ys4 ) ) ) ) ) ).

% mult2_ex_dom.cases
thf(fact_208_mult2__ex__dom_Ocases,axiom,
    ! [X2: produc7312811325926348135st_nat] :
      ( ! [X3: nat,Xs3: list_nat,Ys4: list_nat] :
          ( X2
         != ( produc921417020989057183st_nat @ X3 @ ( produc4487115339913071592st_nat @ Xs3 @ ( produc2694037385005941721st_nat @ nil_nat @ Ys4 ) ) ) )
     => ~ ! [X3: nat,Xs3: list_nat,Y2: nat,Ys: list_nat,Ys4: list_nat] :
            ( X2
           != ( produc921417020989057183st_nat @ X3 @ ( produc4487115339913071592st_nat @ Xs3 @ ( produc2694037385005941721st_nat @ ( cons_nat @ Y2 @ Ys ) @ Ys4 ) ) ) ) ) ).

% mult2_ex_dom.cases
thf(fact_209_mult2__ex__dom_Ocases,axiom,
    ! [X2: produc2892159590233455681on_nat] :
      ( ! [X3: list_S5253404628753938141on_nat,Xs3: list_l8999537460041914605on_nat,Ys4: list_l8999537460041914605on_nat] :
          ( X2
         != ( produc2491415968787288569on_nat @ X3 @ ( produc2560419634836715957on_nat @ Xs3 @ ( produc1341766448070154867on_nat @ nil_li5563851662744376407on_nat @ Ys4 ) ) ) )
     => ~ ! [X3: list_S5253404628753938141on_nat,Xs3: list_l8999537460041914605on_nat,Y2: list_S5253404628753938141on_nat,Ys: list_l8999537460041914605on_nat,Ys4: list_l8999537460041914605on_nat] :
            ( X2
           != ( produc2491415968787288569on_nat @ X3 @ ( produc2560419634836715957on_nat @ Xs3 @ ( produc1341766448070154867on_nat @ ( cons_l5857885737047467175on_nat @ Y2 @ Ys ) @ Ys4 ) ) ) ) ) ).

% mult2_ex_dom.cases
thf(fact_210_mult2__ex__dom_Ocases,axiom,
    ! [X2: produc3053421305257255469on_nat] :
      ( ! [X3: list_P9127870862658512897on_nat,Xs3: list_l1726306699111145479on_nat,Ys4: list_l1726306699111145479on_nat] :
          ( X2
         != ( produc6960825939473561117on_nat @ X3 @ ( produc162686122305543991on_nat @ Xs3 @ ( produc4739740463051851607on_nat @ nil_li2682182165152866823on_nat @ Ys4 ) ) ) )
     => ~ ! [X3: list_P9127870862658512897on_nat,Xs3: list_l1726306699111145479on_nat,Y2: list_P9127870862658512897on_nat,Ys: list_l1726306699111145479on_nat,Ys4: list_l1726306699111145479on_nat] :
            ( X2
           != ( produc6960825939473561117on_nat @ X3 @ ( produc162686122305543991on_nat @ Xs3 @ ( produc4739740463051851607on_nat @ ( cons_l3028799188195208631on_nat @ Y2 @ Ys ) @ Ys4 ) ) ) ) ) ).

% mult2_ex_dom.cases
thf(fact_211_mul__ex__dom_Ocases,axiom,
    ! [X2: produc1874289995683674227list_a] :
      ( ! [Xs4: list_a,Y2: a,Ys: list_a] :
          ( X2
         != ( produc1737125957862166243list_a @ nil_a @ ( produc3204708664006668352list_a @ Xs4 @ ( produc6670463072477821725list_a @ Y2 @ Ys ) ) ) )
     => ~ ! [X3: a,Xs3: list_a,Xs4: list_a,Y2: a,Ys: list_a] :
            ( X2
           != ( produc1737125957862166243list_a @ ( cons_a @ X3 @ Xs3 ) @ ( produc3204708664006668352list_a @ Xs4 @ ( produc6670463072477821725list_a @ Y2 @ Ys ) ) ) ) ) ).

% mul_ex_dom.cases
thf(fact_212_mul__ex__dom_Ocases,axiom,
    ! [X2: produc4714912333100831009on_nat] :
      ( ! [Xs4: list_S5253404628753938141on_nat,Y2: sum_su4907927656612468301on_nat,Ys: list_S5253404628753938141on_nat] :
          ( X2
         != ( produc8004745382299176153on_nat @ nil_Su3078906344068544711on_nat @ ( produc5847888561595751573on_nat @ Xs4 @ ( produc9137229129158211779on_nat @ Y2 @ Ys ) ) ) )
     => ~ ! [X3: sum_su4907927656612468301on_nat,Xs3: list_S5253404628753938141on_nat,Xs4: list_S5253404628753938141on_nat,Y2: sum_su4907927656612468301on_nat,Ys: list_S5253404628753938141on_nat] :
            ( X2
           != ( produc8004745382299176153on_nat @ ( cons_S6797538523646374167on_nat @ X3 @ Xs3 ) @ ( produc5847888561595751573on_nat @ Xs4 @ ( produc9137229129158211779on_nat @ Y2 @ Ys ) ) ) ) ) ).

% mul_ex_dom.cases
thf(fact_213_mul__ex__dom_Ocases,axiom,
    ! [X2: produc3060927313823239865on_nat] :
      ( ! [Xs4: list_P9127870862658512897on_nat,Y2: produc1313295348606329723on_nat,Ys: list_P9127870862658512897on_nat] :
          ( X2
         != ( produc7963087792460353193on_nat @ nil_Pr8614093721167239297on_nat @ ( produc2827541363413333699on_nat @ Xs4 @ ( produc6875097235463537885on_nat @ Y2 @ Ys ) ) ) )
     => ~ ! [X3: produc1313295348606329723on_nat,Xs3: list_P9127870862658512897on_nat,Xs4: list_P9127870862658512897on_nat,Y2: produc1313295348606329723on_nat,Ys: list_P9127870862658512897on_nat] :
            ( X2
           != ( produc7963087792460353193on_nat @ ( cons_P1582596238967110705on_nat @ X3 @ Xs3 ) @ ( produc2827541363413333699on_nat @ Xs4 @ ( produc6875097235463537885on_nat @ Y2 @ Ys ) ) ) ) ) ).

% mul_ex_dom.cases
thf(fact_214_mul__ex__dom_Ocases,axiom,
    ! [X2: produc5923727742716231015st_nat] :
      ( ! [Xs4: list_nat,Y2: nat,Ys: list_nat] :
          ( X2
         != ( produc6629933339985824415st_nat @ nil_nat @ ( produc7518127839388293336st_nat @ Xs4 @ ( produc8282810413953273033st_nat @ Y2 @ Ys ) ) ) )
     => ~ ! [X3: nat,Xs3: list_nat,Xs4: list_nat,Y2: nat,Ys: list_nat] :
            ( X2
           != ( produc6629933339985824415st_nat @ ( cons_nat @ X3 @ Xs3 ) @ ( produc7518127839388293336st_nat @ Xs4 @ ( produc8282810413953273033st_nat @ Y2 @ Ys ) ) ) ) ) ).

% mul_ex_dom.cases
thf(fact_215_mul__ex__dom_Ocases,axiom,
    ! [X2: produc2778868904637641793on_nat] :
      ( ! [Xs4: list_l8999537460041914605on_nat,Y2: list_S5253404628753938141on_nat,Ys: list_l8999537460041914605on_nat] :
          ( X2
         != ( produc8625578076745794553on_nat @ nil_li5563851662744376407on_nat @ ( produc8536543913362324773on_nat @ Xs4 @ ( produc9005406347315192931on_nat @ Y2 @ Ys ) ) ) )
     => ~ ! [X3: list_S5253404628753938141on_nat,Xs3: list_l8999537460041914605on_nat,Xs4: list_l8999537460041914605on_nat,Y2: list_S5253404628753938141on_nat,Ys: list_l8999537460041914605on_nat] :
            ( X2
           != ( produc8625578076745794553on_nat @ ( cons_l5857885737047467175on_nat @ X3 @ Xs3 ) @ ( produc8536543913362324773on_nat @ Xs4 @ ( produc9005406347315192931on_nat @ Y2 @ Ys ) ) ) ) ) ).

% mul_ex_dom.cases
thf(fact_216_mul__ex__dom_Ocases,axiom,
    ! [X2: produc3665718167116523309on_nat] :
      ( ! [Xs4: list_l1726306699111145479on_nat,Y2: list_P9127870862658512897on_nat,Ys: list_l1726306699111145479on_nat] :
          ( X2
         != ( produc2763753542052437021on_nat @ nil_li2682182165152866823on_nat @ ( produc5719953898228879677on_nat @ Xs4 @ ( produc7310029628050611421on_nat @ Y2 @ Ys ) ) ) )
     => ~ ! [X3: list_P9127870862658512897on_nat,Xs3: list_l1726306699111145479on_nat,Xs4: list_l1726306699111145479on_nat,Y2: list_P9127870862658512897on_nat,Ys: list_l1726306699111145479on_nat] :
            ( X2
           != ( produc2763753542052437021on_nat @ ( cons_l3028799188195208631on_nat @ X3 @ Xs3 ) @ ( produc5719953898228879677on_nat @ Xs4 @ ( produc7310029628050611421on_nat @ Y2 @ Ys ) ) ) ) ) ).

% mul_ex_dom.cases
thf(fact_217_map__append,axiom,
    ! [F: nat > nat,Xs: list_nat,Ys2: list_nat] :
      ( ( map_nat_nat @ F @ ( append_nat @ Xs @ Ys2 ) )
      = ( append_nat @ ( map_nat_nat @ F @ Xs ) @ ( map_nat_nat @ F @ Ys2 ) ) ) ).

% map_append
thf(fact_218_map__append,axiom,
    ! [F: nat > list_nat,Xs: list_nat,Ys2: list_nat] :
      ( ( map_nat_list_nat @ F @ ( append_nat @ Xs @ Ys2 ) )
      = ( append_list_nat @ ( map_nat_list_nat @ F @ Xs ) @ ( map_nat_list_nat @ F @ Ys2 ) ) ) ).

% map_append
thf(fact_219_map__append,axiom,
    ! [F: sum_su4907927656612468301on_nat > nat,Xs: list_S5253404628753938141on_nat,Ys2: list_S5253404628753938141on_nat] :
      ( ( map_Su4809418691747827683at_nat @ F @ ( append3523812819975775986on_nat @ Xs @ Ys2 ) )
      = ( append_nat @ ( map_Su4809418691747827683at_nat @ F @ Xs ) @ ( map_Su4809418691747827683at_nat @ F @ Ys2 ) ) ) ).

% map_append
thf(fact_220_map__append,axiom,
    ! [F: nat > sum_su4907927656612468301on_nat,Xs: list_nat,Ys2: list_nat] :
      ( ( map_na5706685718985795555on_nat @ F @ ( append_nat @ Xs @ Ys2 ) )
      = ( append3523812819975775986on_nat @ ( map_na5706685718985795555on_nat @ F @ Xs ) @ ( map_na5706685718985795555on_nat @ F @ Ys2 ) ) ) ).

% map_append
thf(fact_221_map__append,axiom,
    ! [F: a > sum_su4907927656612468301on_nat,Xs: list_a,Ys2: list_a] :
      ( ( map_a_2711072540680412167on_nat @ F @ ( append_a @ Xs @ Ys2 ) )
      = ( append3523812819975775986on_nat @ ( map_a_2711072540680412167on_nat @ F @ Xs ) @ ( map_a_2711072540680412167on_nat @ F @ Ys2 ) ) ) ).

% map_append
thf(fact_222_map__append,axiom,
    ! [F: a > list_S5253404628753938141on_nat,Xs: list_a,Ys2: list_a] :
      ( ( map_a_3887198048842487703on_nat @ F @ ( append_a @ Xs @ Ys2 ) )
      = ( append6174841615030841602on_nat @ ( map_a_3887198048842487703on_nat @ F @ Xs ) @ ( map_a_3887198048842487703on_nat @ F @ Ys2 ) ) ) ).

% map_append
thf(fact_223_map__append,axiom,
    ! [F: a > list_l8999537460041914605on_nat,Xs: list_a,Ys2: list_a] :
      ( ( map_a_3820070144034313383on_nat @ F @ ( append_a @ Xs @ Ys2 ) )
      = ( append7602222762569088146on_nat @ ( map_a_3820070144034313383on_nat @ F @ Xs ) @ ( map_a_3820070144034313383on_nat @ F @ Ys2 ) ) ) ).

% map_append
thf(fact_224_map__append,axiom,
    ! [F: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat,Ys2: list_S5253404628753938141on_nat] :
      ( ( map_Su241112079427911264on_nat @ F @ ( append3523812819975775986on_nat @ Xs @ Ys2 ) )
      = ( append3523812819975775986on_nat @ ( map_Su241112079427911264on_nat @ F @ Xs ) @ ( map_Su241112079427911264on_nat @ F @ Ys2 ) ) ) ).

% map_append
thf(fact_225_map__append,axiom,
    ! [F: produc1313295348606329723on_nat > nat,Xs: list_P9127870862658512897on_nat,Ys2: list_P9127870862658512897on_nat] :
      ( ( map_Pr1888482430906987655at_nat @ F @ ( append2865821678206098582on_nat @ Xs @ Ys2 ) )
      = ( append_nat @ ( map_Pr1888482430906987655at_nat @ F @ Xs ) @ ( map_Pr1888482430906987655at_nat @ F @ Ys2 ) ) ) ).

% map_append
thf(fact_226_map__append,axiom,
    ! [F: nat > produc1313295348606329723on_nat,Xs: list_nat,Ys2: list_nat] :
      ( ( map_na2641141678045226597on_nat @ F @ ( append_nat @ Xs @ Ys2 ) )
      = ( append2865821678206098582on_nat @ ( map_na2641141678045226597on_nat @ F @ Xs ) @ ( map_na2641141678045226597on_nat @ F @ Ys2 ) ) ) ).

% map_append
thf(fact_227_append__is__Nil__conv,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys2: list_S5253404628753938141on_nat] :
      ( ( ( append3523812819975775986on_nat @ Xs @ Ys2 )
        = nil_Su3078906344068544711on_nat )
      = ( ( Xs = nil_Su3078906344068544711on_nat )
        & ( Ys2 = nil_Su3078906344068544711on_nat ) ) ) ).

% append_is_Nil_conv
thf(fact_228_append__is__Nil__conv,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys2: list_P9127870862658512897on_nat] :
      ( ( ( append2865821678206098582on_nat @ Xs @ Ys2 )
        = nil_Pr8614093721167239297on_nat )
      = ( ( Xs = nil_Pr8614093721167239297on_nat )
        & ( Ys2 = nil_Pr8614093721167239297on_nat ) ) ) ).

% append_is_Nil_conv
thf(fact_229_append__is__Nil__conv,axiom,
    ! [Xs: list_nat,Ys2: list_nat] :
      ( ( ( append_nat @ Xs @ Ys2 )
        = nil_nat )
      = ( ( Xs = nil_nat )
        & ( Ys2 = nil_nat ) ) ) ).

% append_is_Nil_conv
thf(fact_230_append__is__Nil__conv,axiom,
    ! [Xs: list_l8999537460041914605on_nat,Ys2: list_l8999537460041914605on_nat] :
      ( ( ( append6174841615030841602on_nat @ Xs @ Ys2 )
        = nil_li5563851662744376407on_nat )
      = ( ( Xs = nil_li5563851662744376407on_nat )
        & ( Ys2 = nil_li5563851662744376407on_nat ) ) ) ).

% append_is_Nil_conv
thf(fact_231_append__is__Nil__conv,axiom,
    ! [Xs: list_l1726306699111145479on_nat,Ys2: list_l1726306699111145479on_nat] :
      ( ( ( append3839213209832495004on_nat @ Xs @ Ys2 )
        = nil_li2682182165152866823on_nat )
      = ( ( Xs = nil_li2682182165152866823on_nat )
        & ( Ys2 = nil_li2682182165152866823on_nat ) ) ) ).

% append_is_Nil_conv
thf(fact_232_append__is__Nil__conv,axiom,
    ! [Xs: list_a,Ys2: list_a] :
      ( ( ( append_a @ Xs @ Ys2 )
        = nil_a )
      = ( ( Xs = nil_a )
        & ( Ys2 = nil_a ) ) ) ).

% append_is_Nil_conv
thf(fact_233_Nil__is__append__conv,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys2: list_S5253404628753938141on_nat] :
      ( ( nil_Su3078906344068544711on_nat
        = ( append3523812819975775986on_nat @ Xs @ Ys2 ) )
      = ( ( Xs = nil_Su3078906344068544711on_nat )
        & ( Ys2 = nil_Su3078906344068544711on_nat ) ) ) ).

% Nil_is_append_conv
thf(fact_234_Nil__is__append__conv,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys2: list_P9127870862658512897on_nat] :
      ( ( nil_Pr8614093721167239297on_nat
        = ( append2865821678206098582on_nat @ Xs @ Ys2 ) )
      = ( ( Xs = nil_Pr8614093721167239297on_nat )
        & ( Ys2 = nil_Pr8614093721167239297on_nat ) ) ) ).

% Nil_is_append_conv
thf(fact_235_Nil__is__append__conv,axiom,
    ! [Xs: list_nat,Ys2: list_nat] :
      ( ( nil_nat
        = ( append_nat @ Xs @ Ys2 ) )
      = ( ( Xs = nil_nat )
        & ( Ys2 = nil_nat ) ) ) ).

% Nil_is_append_conv
thf(fact_236_Nil__is__append__conv,axiom,
    ! [Xs: list_l8999537460041914605on_nat,Ys2: list_l8999537460041914605on_nat] :
      ( ( nil_li5563851662744376407on_nat
        = ( append6174841615030841602on_nat @ Xs @ Ys2 ) )
      = ( ( Xs = nil_li5563851662744376407on_nat )
        & ( Ys2 = nil_li5563851662744376407on_nat ) ) ) ).

% Nil_is_append_conv
thf(fact_237_Nil__is__append__conv,axiom,
    ! [Xs: list_l1726306699111145479on_nat,Ys2: list_l1726306699111145479on_nat] :
      ( ( nil_li2682182165152866823on_nat
        = ( append3839213209832495004on_nat @ Xs @ Ys2 ) )
      = ( ( Xs = nil_li2682182165152866823on_nat )
        & ( Ys2 = nil_li2682182165152866823on_nat ) ) ) ).

% Nil_is_append_conv
thf(fact_238_Nil__is__append__conv,axiom,
    ! [Xs: list_a,Ys2: list_a] :
      ( ( nil_a
        = ( append_a @ Xs @ Ys2 ) )
      = ( ( Xs = nil_a )
        & ( Ys2 = nil_a ) ) ) ).

% Nil_is_append_conv
thf(fact_239_self__append__conv2,axiom,
    ! [Y: list_S5253404628753938141on_nat,Xs: list_S5253404628753938141on_nat] :
      ( ( Y
        = ( append3523812819975775986on_nat @ Xs @ Y ) )
      = ( Xs = nil_Su3078906344068544711on_nat ) ) ).

% self_append_conv2
thf(fact_240_self__append__conv2,axiom,
    ! [Y: list_P9127870862658512897on_nat,Xs: list_P9127870862658512897on_nat] :
      ( ( Y
        = ( append2865821678206098582on_nat @ Xs @ Y ) )
      = ( Xs = nil_Pr8614093721167239297on_nat ) ) ).

% self_append_conv2
thf(fact_241_self__append__conv2,axiom,
    ! [Y: list_nat,Xs: list_nat] :
      ( ( Y
        = ( append_nat @ Xs @ Y ) )
      = ( Xs = nil_nat ) ) ).

% self_append_conv2
thf(fact_242_self__append__conv2,axiom,
    ! [Y: list_l8999537460041914605on_nat,Xs: list_l8999537460041914605on_nat] :
      ( ( Y
        = ( append6174841615030841602on_nat @ Xs @ Y ) )
      = ( Xs = nil_li5563851662744376407on_nat ) ) ).

% self_append_conv2
thf(fact_243_self__append__conv2,axiom,
    ! [Y: list_l1726306699111145479on_nat,Xs: list_l1726306699111145479on_nat] :
      ( ( Y
        = ( append3839213209832495004on_nat @ Xs @ Y ) )
      = ( Xs = nil_li2682182165152866823on_nat ) ) ).

% self_append_conv2
thf(fact_244_self__append__conv2,axiom,
    ! [Y: list_a,Xs: list_a] :
      ( ( Y
        = ( append_a @ Xs @ Y ) )
      = ( Xs = nil_a ) ) ).

% self_append_conv2
thf(fact_245_append__self__conv2,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys2: list_S5253404628753938141on_nat] :
      ( ( ( append3523812819975775986on_nat @ Xs @ Ys2 )
        = Ys2 )
      = ( Xs = nil_Su3078906344068544711on_nat ) ) ).

% append_self_conv2
thf(fact_246_append__self__conv2,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys2: list_P9127870862658512897on_nat] :
      ( ( ( append2865821678206098582on_nat @ Xs @ Ys2 )
        = Ys2 )
      = ( Xs = nil_Pr8614093721167239297on_nat ) ) ).

% append_self_conv2
thf(fact_247_append__self__conv2,axiom,
    ! [Xs: list_nat,Ys2: list_nat] :
      ( ( ( append_nat @ Xs @ Ys2 )
        = Ys2 )
      = ( Xs = nil_nat ) ) ).

% append_self_conv2
thf(fact_248_append__self__conv2,axiom,
    ! [Xs: list_l8999537460041914605on_nat,Ys2: list_l8999537460041914605on_nat] :
      ( ( ( append6174841615030841602on_nat @ Xs @ Ys2 )
        = Ys2 )
      = ( Xs = nil_li5563851662744376407on_nat ) ) ).

% append_self_conv2
thf(fact_249_append__self__conv2,axiom,
    ! [Xs: list_l1726306699111145479on_nat,Ys2: list_l1726306699111145479on_nat] :
      ( ( ( append3839213209832495004on_nat @ Xs @ Ys2 )
        = Ys2 )
      = ( Xs = nil_li2682182165152866823on_nat ) ) ).

% append_self_conv2
thf(fact_250_append__self__conv2,axiom,
    ! [Xs: list_a,Ys2: list_a] :
      ( ( ( append_a @ Xs @ Ys2 )
        = Ys2 )
      = ( Xs = nil_a ) ) ).

% append_self_conv2
thf(fact_251_self__append__conv,axiom,
    ! [Y: list_S5253404628753938141on_nat,Ys2: list_S5253404628753938141on_nat] :
      ( ( Y
        = ( append3523812819975775986on_nat @ Y @ Ys2 ) )
      = ( Ys2 = nil_Su3078906344068544711on_nat ) ) ).

% self_append_conv
thf(fact_252_self__append__conv,axiom,
    ! [Y: list_P9127870862658512897on_nat,Ys2: list_P9127870862658512897on_nat] :
      ( ( Y
        = ( append2865821678206098582on_nat @ Y @ Ys2 ) )
      = ( Ys2 = nil_Pr8614093721167239297on_nat ) ) ).

% self_append_conv
thf(fact_253_self__append__conv,axiom,
    ! [Y: list_nat,Ys2: list_nat] :
      ( ( Y
        = ( append_nat @ Y @ Ys2 ) )
      = ( Ys2 = nil_nat ) ) ).

% self_append_conv
thf(fact_254_self__append__conv,axiom,
    ! [Y: list_l8999537460041914605on_nat,Ys2: list_l8999537460041914605on_nat] :
      ( ( Y
        = ( append6174841615030841602on_nat @ Y @ Ys2 ) )
      = ( Ys2 = nil_li5563851662744376407on_nat ) ) ).

% self_append_conv
thf(fact_255_self__append__conv,axiom,
    ! [Y: list_l1726306699111145479on_nat,Ys2: list_l1726306699111145479on_nat] :
      ( ( Y
        = ( append3839213209832495004on_nat @ Y @ Ys2 ) )
      = ( Ys2 = nil_li2682182165152866823on_nat ) ) ).

% self_append_conv
thf(fact_256_self__append__conv,axiom,
    ! [Y: list_a,Ys2: list_a] :
      ( ( Y
        = ( append_a @ Y @ Ys2 ) )
      = ( Ys2 = nil_a ) ) ).

% self_append_conv
thf(fact_257_append__self__conv,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys2: list_S5253404628753938141on_nat] :
      ( ( ( append3523812819975775986on_nat @ Xs @ Ys2 )
        = Xs )
      = ( Ys2 = nil_Su3078906344068544711on_nat ) ) ).

% append_self_conv
thf(fact_258_append__self__conv,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys2: list_P9127870862658512897on_nat] :
      ( ( ( append2865821678206098582on_nat @ Xs @ Ys2 )
        = Xs )
      = ( Ys2 = nil_Pr8614093721167239297on_nat ) ) ).

% append_self_conv
thf(fact_259_append__self__conv,axiom,
    ! [Xs: list_nat,Ys2: list_nat] :
      ( ( ( append_nat @ Xs @ Ys2 )
        = Xs )
      = ( Ys2 = nil_nat ) ) ).

% append_self_conv
thf(fact_260_append__self__conv,axiom,
    ! [Xs: list_l8999537460041914605on_nat,Ys2: list_l8999537460041914605on_nat] :
      ( ( ( append6174841615030841602on_nat @ Xs @ Ys2 )
        = Xs )
      = ( Ys2 = nil_li5563851662744376407on_nat ) ) ).

% append_self_conv
thf(fact_261_append__self__conv,axiom,
    ! [Xs: list_l1726306699111145479on_nat,Ys2: list_l1726306699111145479on_nat] :
      ( ( ( append3839213209832495004on_nat @ Xs @ Ys2 )
        = Xs )
      = ( Ys2 = nil_li2682182165152866823on_nat ) ) ).

% append_self_conv
thf(fact_262_append__self__conv,axiom,
    ! [Xs: list_a,Ys2: list_a] :
      ( ( ( append_a @ Xs @ Ys2 )
        = Xs )
      = ( Ys2 = nil_a ) ) ).

% append_self_conv
thf(fact_263_same__append__eq,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys2: list_S5253404628753938141on_nat,Zs3: list_S5253404628753938141on_nat] :
      ( ( ( append3523812819975775986on_nat @ Xs @ Ys2 )
        = ( append3523812819975775986on_nat @ Xs @ Zs3 ) )
      = ( Ys2 = Zs3 ) ) ).

% same_append_eq
thf(fact_264_same__append__eq,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys2: list_P9127870862658512897on_nat,Zs3: list_P9127870862658512897on_nat] :
      ( ( ( append2865821678206098582on_nat @ Xs @ Ys2 )
        = ( append2865821678206098582on_nat @ Xs @ Zs3 ) )
      = ( Ys2 = Zs3 ) ) ).

% same_append_eq
thf(fact_265_same__append__eq,axiom,
    ! [Xs: list_nat,Ys2: list_nat,Zs3: list_nat] :
      ( ( ( append_nat @ Xs @ Ys2 )
        = ( append_nat @ Xs @ Zs3 ) )
      = ( Ys2 = Zs3 ) ) ).

% same_append_eq
thf(fact_266_append__same__eq,axiom,
    ! [Ys2: list_S5253404628753938141on_nat,Xs: list_S5253404628753938141on_nat,Zs3: list_S5253404628753938141on_nat] :
      ( ( ( append3523812819975775986on_nat @ Ys2 @ Xs )
        = ( append3523812819975775986on_nat @ Zs3 @ Xs ) )
      = ( Ys2 = Zs3 ) ) ).

% append_same_eq
thf(fact_267_append__same__eq,axiom,
    ! [Ys2: list_P9127870862658512897on_nat,Xs: list_P9127870862658512897on_nat,Zs3: list_P9127870862658512897on_nat] :
      ( ( ( append2865821678206098582on_nat @ Ys2 @ Xs )
        = ( append2865821678206098582on_nat @ Zs3 @ Xs ) )
      = ( Ys2 = Zs3 ) ) ).

% append_same_eq
thf(fact_268_append__same__eq,axiom,
    ! [Ys2: list_nat,Xs: list_nat,Zs3: list_nat] :
      ( ( ( append_nat @ Ys2 @ Xs )
        = ( append_nat @ Zs3 @ Xs ) )
      = ( Ys2 = Zs3 ) ) ).

% append_same_eq
thf(fact_269_append__assoc,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys2: list_S5253404628753938141on_nat,Zs3: list_S5253404628753938141on_nat] :
      ( ( append3523812819975775986on_nat @ ( append3523812819975775986on_nat @ Xs @ Ys2 ) @ Zs3 )
      = ( append3523812819975775986on_nat @ Xs @ ( append3523812819975775986on_nat @ Ys2 @ Zs3 ) ) ) ).

% append_assoc
thf(fact_270_append__assoc,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys2: list_P9127870862658512897on_nat,Zs3: list_P9127870862658512897on_nat] :
      ( ( append2865821678206098582on_nat @ ( append2865821678206098582on_nat @ Xs @ Ys2 ) @ Zs3 )
      = ( append2865821678206098582on_nat @ Xs @ ( append2865821678206098582on_nat @ Ys2 @ Zs3 ) ) ) ).

% append_assoc
thf(fact_271_append__assoc,axiom,
    ! [Xs: list_nat,Ys2: list_nat,Zs3: list_nat] :
      ( ( append_nat @ ( append_nat @ Xs @ Ys2 ) @ Zs3 )
      = ( append_nat @ Xs @ ( append_nat @ Ys2 @ Zs3 ) ) ) ).

% append_assoc
thf(fact_272_append_Oassoc,axiom,
    ! [A: list_S5253404628753938141on_nat,B: list_S5253404628753938141on_nat,C: list_S5253404628753938141on_nat] :
      ( ( append3523812819975775986on_nat @ ( append3523812819975775986on_nat @ A @ B ) @ C )
      = ( append3523812819975775986on_nat @ A @ ( append3523812819975775986on_nat @ B @ C ) ) ) ).

% append.assoc
thf(fact_273_append_Oassoc,axiom,
    ! [A: list_P9127870862658512897on_nat,B: list_P9127870862658512897on_nat,C: list_P9127870862658512897on_nat] :
      ( ( append2865821678206098582on_nat @ ( append2865821678206098582on_nat @ A @ B ) @ C )
      = ( append2865821678206098582on_nat @ A @ ( append2865821678206098582on_nat @ B @ C ) ) ) ).

% append.assoc
thf(fact_274_append_Oassoc,axiom,
    ! [A: list_nat,B: list_nat,C: list_nat] :
      ( ( append_nat @ ( append_nat @ A @ B ) @ C )
      = ( append_nat @ A @ ( append_nat @ B @ C ) ) ) ).

% append.assoc
thf(fact_275_old_Osum_Oinject_I2_J,axiom,
    ! [B: nat,B2: nat] :
      ( ( ( sum_In255631617079920996tation @ B )
        = ( sum_In255631617079920996tation @ B2 ) )
      = ( B = B2 ) ) ).

% old.sum.inject(2)
thf(fact_276_sum_Oinject_I2_J,axiom,
    ! [X23: nat,Y23: nat] :
      ( ( ( sum_In255631617079920996tation @ X23 )
        = ( sum_In255631617079920996tation @ Y23 ) )
      = ( X23 = Y23 ) ) ).

% sum.inject(2)
thf(fact_277_append_Oright__neutral,axiom,
    ! [A: list_S5253404628753938141on_nat] :
      ( ( append3523812819975775986on_nat @ A @ nil_Su3078906344068544711on_nat )
      = A ) ).

% append.right_neutral
thf(fact_278_append_Oright__neutral,axiom,
    ! [A: list_P9127870862658512897on_nat] :
      ( ( append2865821678206098582on_nat @ A @ nil_Pr8614093721167239297on_nat )
      = A ) ).

% append.right_neutral
thf(fact_279_append_Oright__neutral,axiom,
    ! [A: list_nat] :
      ( ( append_nat @ A @ nil_nat )
      = A ) ).

% append.right_neutral
thf(fact_280_append_Oright__neutral,axiom,
    ! [A: list_l8999537460041914605on_nat] :
      ( ( append6174841615030841602on_nat @ A @ nil_li5563851662744376407on_nat )
      = A ) ).

% append.right_neutral
thf(fact_281_append_Oright__neutral,axiom,
    ! [A: list_l1726306699111145479on_nat] :
      ( ( append3839213209832495004on_nat @ A @ nil_li2682182165152866823on_nat )
      = A ) ).

% append.right_neutral
thf(fact_282_append_Oright__neutral,axiom,
    ! [A: list_a] :
      ( ( append_a @ A @ nil_a )
      = A ) ).

% append.right_neutral
thf(fact_283_append__Nil2,axiom,
    ! [Xs: list_S5253404628753938141on_nat] :
      ( ( append3523812819975775986on_nat @ Xs @ nil_Su3078906344068544711on_nat )
      = Xs ) ).

% append_Nil2
thf(fact_284_append__Nil2,axiom,
    ! [Xs: list_P9127870862658512897on_nat] :
      ( ( append2865821678206098582on_nat @ Xs @ nil_Pr8614093721167239297on_nat )
      = Xs ) ).

% append_Nil2
thf(fact_285_append__Nil2,axiom,
    ! [Xs: list_nat] :
      ( ( append_nat @ Xs @ nil_nat )
      = Xs ) ).

% append_Nil2
thf(fact_286_append__Nil2,axiom,
    ! [Xs: list_l8999537460041914605on_nat] :
      ( ( append6174841615030841602on_nat @ Xs @ nil_li5563851662744376407on_nat )
      = Xs ) ).

% append_Nil2
thf(fact_287_append__Nil2,axiom,
    ! [Xs: list_l1726306699111145479on_nat] :
      ( ( append3839213209832495004on_nat @ Xs @ nil_li2682182165152866823on_nat )
      = Xs ) ).

% append_Nil2
thf(fact_288_append__Nil2,axiom,
    ! [Xs: list_a] :
      ( ( append_a @ Xs @ nil_a )
      = Xs ) ).

% append_Nil2
thf(fact_289_concat__append,axiom,
    ! [Xs: list_l8999537460041914605on_nat,Ys2: list_l8999537460041914605on_nat] :
      ( ( concat6161509665823917772on_nat @ ( append6174841615030841602on_nat @ Xs @ Ys2 ) )
      = ( append3523812819975775986on_nat @ ( concat6161509665823917772on_nat @ Xs ) @ ( concat6161509665823917772on_nat @ Ys2 ) ) ) ).

% concat_append
thf(fact_290_concat__append,axiom,
    ! [Xs: list_l1726306699111145479on_nat,Ys2: list_l1726306699111145479on_nat] :
      ( ( concat937493773106420796on_nat @ ( append3839213209832495004on_nat @ Xs @ Ys2 ) )
      = ( append2865821678206098582on_nat @ ( concat937493773106420796on_nat @ Xs ) @ ( concat937493773106420796on_nat @ Ys2 ) ) ) ).

% concat_append
thf(fact_291_concat__append,axiom,
    ! [Xs: list_list_nat,Ys2: list_list_nat] :
      ( ( concat_nat @ ( append_list_nat @ Xs @ Ys2 ) )
      = ( append_nat @ ( concat_nat @ Xs ) @ ( concat_nat @ Ys2 ) ) ) ).

% concat_append
thf(fact_292_concat__append,axiom,
    ! [Xs: list_l8365940747029150845on_nat,Ys2: list_l8365940747029150845on_nat] :
      ( ( concat4774721536428481500on_nat @ ( append7602222762569088146on_nat @ Xs @ Ys2 ) )
      = ( append6174841615030841602on_nat @ ( concat4774721536428481500on_nat @ Xs ) @ ( concat4774721536428481500on_nat @ Ys2 ) ) ) ).

% concat_append
thf(fact_293_concat__append,axiom,
    ! [Xs: list_l6719245731233385357on_nat,Ys2: list_l6719245731233385357on_nat] :
      ( ( concat4341727598008994882on_nat @ ( append4931807731165890082on_nat @ Xs @ Ys2 ) )
      = ( append3839213209832495004on_nat @ ( concat4341727598008994882on_nat @ Xs ) @ ( concat4341727598008994882on_nat @ Ys2 ) ) ) ).

% concat_append
thf(fact_294_S,axiom,
    ( s
    = ( concat937493773106420796on_nat
      @ ( map_na8715451351012060651on_nat
        @ ^ [I: nat] :
            ( map_Pr6945619049609468478on_nat
            @ ^ [L: product_prod_a_o] : ( produc6704844767664046899on_nat @ ( multis8230369766971345672_lit_a @ L ) @ ( sum_In255631617079920996tation @ I ) )
            @ ( nth_li3248255784598043854od_a_o @ cnf @ I ) )
        @ cs ) ) ) ).

% S
thf(fact_295_concat__eq__appendD,axiom,
    ! [Xss2: list_list_nat,Ys2: list_nat,Zs3: list_nat] :
      ( ( ( concat_nat @ Xss2 )
        = ( append_nat @ Ys2 @ Zs3 ) )
     => ( ( Xss2 != nil_list_nat )
       => ? [Xss1: list_list_nat,Xs3: list_nat,Xs4: list_nat,Xss22: list_list_nat] :
            ( ( Xss2
              = ( append_list_nat @ Xss1 @ ( cons_list_nat @ ( append_nat @ Xs3 @ Xs4 ) @ Xss22 ) ) )
            & ( Ys2
              = ( append_nat @ ( concat_nat @ Xss1 ) @ Xs3 ) )
            & ( Zs3
              = ( append_nat @ Xs4 @ ( concat_nat @ Xss22 ) ) ) ) ) ) ).

% concat_eq_appendD
thf(fact_296_concat__eq__appendD,axiom,
    ! [Xss2: list_l8365940747029150845on_nat,Ys2: list_l8999537460041914605on_nat,Zs3: list_l8999537460041914605on_nat] :
      ( ( ( concat4774721536428481500on_nat @ Xss2 )
        = ( append6174841615030841602on_nat @ Ys2 @ Zs3 ) )
     => ( ( Xss2 != nil_li3323240303193190759on_nat )
       => ? [Xss1: list_l8365940747029150845on_nat,Xs3: list_l8999537460041914605on_nat,Xs4: list_l8999537460041914605on_nat,Xss22: list_l8365940747029150845on_nat] :
            ( ( Xss2
              = ( append7602222762569088146on_nat @ Xss1 @ ( cons_l6725403660475259319on_nat @ ( append6174841615030841602on_nat @ Xs3 @ Xs4 ) @ Xss22 ) ) )
            & ( Ys2
              = ( append6174841615030841602on_nat @ ( concat4774721536428481500on_nat @ Xss1 ) @ Xs3 ) )
            & ( Zs3
              = ( append6174841615030841602on_nat @ Xs4 @ ( concat4774721536428481500on_nat @ Xss22 ) ) ) ) ) ) ).

% concat_eq_appendD
thf(fact_297_concat__eq__appendD,axiom,
    ! [Xss2: list_l6719245731233385357on_nat,Ys2: list_l1726306699111145479on_nat,Zs3: list_l1726306699111145479on_nat] :
      ( ( ( concat4341727598008994882on_nat @ Xss2 )
        = ( append3839213209832495004on_nat @ Ys2 @ Zs3 ) )
     => ( ( Xss2 != nil_li985476221562969869on_nat )
       => ? [Xss1: list_l6719245731233385357on_nat,Xs3: list_l1726306699111145479on_nat,Xs4: list_l1726306699111145479on_nat,Xss22: list_l6719245731233385357on_nat] :
            ( ( Xss2
              = ( append4931807731165890082on_nat @ Xss1 @ ( cons_l4979741563022480061on_nat @ ( append3839213209832495004on_nat @ Xs3 @ Xs4 ) @ Xss22 ) ) )
            & ( Ys2
              = ( append3839213209832495004on_nat @ ( concat4341727598008994882on_nat @ Xss1 ) @ Xs3 ) )
            & ( Zs3
              = ( append3839213209832495004on_nat @ Xs4 @ ( concat4341727598008994882on_nat @ Xss22 ) ) ) ) ) ) ).

% concat_eq_appendD
thf(fact_298_concat__eq__appendD,axiom,
    ! [Xss2: list_l8999537460041914605on_nat,Ys2: list_S5253404628753938141on_nat,Zs3: list_S5253404628753938141on_nat] :
      ( ( ( concat6161509665823917772on_nat @ Xss2 )
        = ( append3523812819975775986on_nat @ Ys2 @ Zs3 ) )
     => ( ( Xss2 != nil_li5563851662744376407on_nat )
       => ? [Xss1: list_l8999537460041914605on_nat,Xs3: list_S5253404628753938141on_nat,Xs4: list_S5253404628753938141on_nat,Xss22: list_l8999537460041914605on_nat] :
            ( ( Xss2
              = ( append6174841615030841602on_nat @ Xss1 @ ( cons_l5857885737047467175on_nat @ ( append3523812819975775986on_nat @ Xs3 @ Xs4 ) @ Xss22 ) ) )
            & ( Ys2
              = ( append3523812819975775986on_nat @ ( concat6161509665823917772on_nat @ Xss1 ) @ Xs3 ) )
            & ( Zs3
              = ( append3523812819975775986on_nat @ Xs4 @ ( concat6161509665823917772on_nat @ Xss22 ) ) ) ) ) ) ).

% concat_eq_appendD
thf(fact_299_concat__eq__appendD,axiom,
    ! [Xss2: list_l1726306699111145479on_nat,Ys2: list_P9127870862658512897on_nat,Zs3: list_P9127870862658512897on_nat] :
      ( ( ( concat937493773106420796on_nat @ Xss2 )
        = ( append2865821678206098582on_nat @ Ys2 @ Zs3 ) )
     => ( ( Xss2 != nil_li2682182165152866823on_nat )
       => ? [Xss1: list_l1726306699111145479on_nat,Xs3: list_P9127870862658512897on_nat,Xs4: list_P9127870862658512897on_nat,Xss22: list_l1726306699111145479on_nat] :
            ( ( Xss2
              = ( append3839213209832495004on_nat @ Xss1 @ ( cons_l3028799188195208631on_nat @ ( append2865821678206098582on_nat @ Xs3 @ Xs4 ) @ Xss22 ) ) )
            & ( Ys2
              = ( append2865821678206098582on_nat @ ( concat937493773106420796on_nat @ Xss1 ) @ Xs3 ) )
            & ( Zs3
              = ( append2865821678206098582on_nat @ Xs4 @ ( concat937493773106420796on_nat @ Xss22 ) ) ) ) ) ) ).

% concat_eq_appendD
thf(fact_300_Inr__inject,axiom,
    ! [X2: nat,Y: nat] :
      ( ( ( sum_In255631617079920996tation @ X2 )
        = ( sum_In255631617079920996tation @ Y ) )
     => ( X2 = Y ) ) ).

% Inr_inject
thf(fact_301_append__eq__append__conv2,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys2: list_S5253404628753938141on_nat,Zs3: list_S5253404628753938141on_nat,Ts: list_S5253404628753938141on_nat] :
      ( ( ( append3523812819975775986on_nat @ Xs @ Ys2 )
        = ( append3523812819975775986on_nat @ Zs3 @ Ts ) )
      = ( ? [Us: list_S5253404628753938141on_nat] :
            ( ( ( Xs
                = ( append3523812819975775986on_nat @ Zs3 @ Us ) )
              & ( ( append3523812819975775986on_nat @ Us @ Ys2 )
                = Ts ) )
            | ( ( ( append3523812819975775986on_nat @ Xs @ Us )
                = Zs3 )
              & ( Ys2
                = ( append3523812819975775986on_nat @ Us @ Ts ) ) ) ) ) ) ).

% append_eq_append_conv2
thf(fact_302_append__eq__append__conv2,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys2: list_P9127870862658512897on_nat,Zs3: list_P9127870862658512897on_nat,Ts: list_P9127870862658512897on_nat] :
      ( ( ( append2865821678206098582on_nat @ Xs @ Ys2 )
        = ( append2865821678206098582on_nat @ Zs3 @ Ts ) )
      = ( ? [Us: list_P9127870862658512897on_nat] :
            ( ( ( Xs
                = ( append2865821678206098582on_nat @ Zs3 @ Us ) )
              & ( ( append2865821678206098582on_nat @ Us @ Ys2 )
                = Ts ) )
            | ( ( ( append2865821678206098582on_nat @ Xs @ Us )
                = Zs3 )
              & ( Ys2
                = ( append2865821678206098582on_nat @ Us @ Ts ) ) ) ) ) ) ).

% append_eq_append_conv2
thf(fact_303_append__eq__append__conv2,axiom,
    ! [Xs: list_nat,Ys2: list_nat,Zs3: list_nat,Ts: list_nat] :
      ( ( ( append_nat @ Xs @ Ys2 )
        = ( append_nat @ Zs3 @ Ts ) )
      = ( ? [Us: list_nat] :
            ( ( ( Xs
                = ( append_nat @ Zs3 @ Us ) )
              & ( ( append_nat @ Us @ Ys2 )
                = Ts ) )
            | ( ( ( append_nat @ Xs @ Us )
                = Zs3 )
              & ( Ys2
                = ( append_nat @ Us @ Ts ) ) ) ) ) ) ).

% append_eq_append_conv2
thf(fact_304_append__eq__appendI,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Xs1: list_S5253404628753938141on_nat,Zs3: list_S5253404628753938141on_nat,Ys2: list_S5253404628753938141on_nat,Us2: list_S5253404628753938141on_nat] :
      ( ( ( append3523812819975775986on_nat @ Xs @ Xs1 )
        = Zs3 )
     => ( ( Ys2
          = ( append3523812819975775986on_nat @ Xs1 @ Us2 ) )
       => ( ( append3523812819975775986on_nat @ Xs @ Ys2 )
          = ( append3523812819975775986on_nat @ Zs3 @ Us2 ) ) ) ) ).

% append_eq_appendI
thf(fact_305_append__eq__appendI,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Xs1: list_P9127870862658512897on_nat,Zs3: list_P9127870862658512897on_nat,Ys2: list_P9127870862658512897on_nat,Us2: list_P9127870862658512897on_nat] :
      ( ( ( append2865821678206098582on_nat @ Xs @ Xs1 )
        = Zs3 )
     => ( ( Ys2
          = ( append2865821678206098582on_nat @ Xs1 @ Us2 ) )
       => ( ( append2865821678206098582on_nat @ Xs @ Ys2 )
          = ( append2865821678206098582on_nat @ Zs3 @ Us2 ) ) ) ) ).

% append_eq_appendI
thf(fact_306_append__eq__appendI,axiom,
    ! [Xs: list_nat,Xs1: list_nat,Zs3: list_nat,Ys2: list_nat,Us2: list_nat] :
      ( ( ( append_nat @ Xs @ Xs1 )
        = Zs3 )
     => ( ( Ys2
          = ( append_nat @ Xs1 @ Us2 ) )
       => ( ( append_nat @ Xs @ Ys2 )
          = ( append_nat @ Zs3 @ Us2 ) ) ) ) ).

% append_eq_appendI
thf(fact_307_concat__eq__append__conv,axiom,
    ! [Xss2: list_list_nat,Ys2: list_nat,Zs3: list_nat] :
      ( ( ( concat_nat @ Xss2 )
        = ( append_nat @ Ys2 @ Zs3 ) )
      = ( ( ( Xss2 = nil_list_nat )
         => ( ( Ys2 = nil_nat )
            & ( Zs3 = nil_nat ) ) )
        & ( ( Xss2 != nil_list_nat )
         => ? [Xss12: list_list_nat,Xs2: list_nat,Xs5: list_nat,Xss23: list_list_nat] :
              ( ( Xss2
                = ( append_list_nat @ Xss12 @ ( cons_list_nat @ ( append_nat @ Xs2 @ Xs5 ) @ Xss23 ) ) )
              & ( Ys2
                = ( append_nat @ ( concat_nat @ Xss12 ) @ Xs2 ) )
              & ( Zs3
                = ( append_nat @ Xs5 @ ( concat_nat @ Xss23 ) ) ) ) ) ) ) ).

% concat_eq_append_conv
thf(fact_308_concat__eq__append__conv,axiom,
    ! [Xss2: list_l8365940747029150845on_nat,Ys2: list_l8999537460041914605on_nat,Zs3: list_l8999537460041914605on_nat] :
      ( ( ( concat4774721536428481500on_nat @ Xss2 )
        = ( append6174841615030841602on_nat @ Ys2 @ Zs3 ) )
      = ( ( ( Xss2 = nil_li3323240303193190759on_nat )
         => ( ( Ys2 = nil_li5563851662744376407on_nat )
            & ( Zs3 = nil_li5563851662744376407on_nat ) ) )
        & ( ( Xss2 != nil_li3323240303193190759on_nat )
         => ? [Xss12: list_l8365940747029150845on_nat,Xs2: list_l8999537460041914605on_nat,Xs5: list_l8999537460041914605on_nat,Xss23: list_l8365940747029150845on_nat] :
              ( ( Xss2
                = ( append7602222762569088146on_nat @ Xss12 @ ( cons_l6725403660475259319on_nat @ ( append6174841615030841602on_nat @ Xs2 @ Xs5 ) @ Xss23 ) ) )
              & ( Ys2
                = ( append6174841615030841602on_nat @ ( concat4774721536428481500on_nat @ Xss12 ) @ Xs2 ) )
              & ( Zs3
                = ( append6174841615030841602on_nat @ Xs5 @ ( concat4774721536428481500on_nat @ Xss23 ) ) ) ) ) ) ) ).

% concat_eq_append_conv
thf(fact_309_concat__eq__append__conv,axiom,
    ! [Xss2: list_l6719245731233385357on_nat,Ys2: list_l1726306699111145479on_nat,Zs3: list_l1726306699111145479on_nat] :
      ( ( ( concat4341727598008994882on_nat @ Xss2 )
        = ( append3839213209832495004on_nat @ Ys2 @ Zs3 ) )
      = ( ( ( Xss2 = nil_li985476221562969869on_nat )
         => ( ( Ys2 = nil_li2682182165152866823on_nat )
            & ( Zs3 = nil_li2682182165152866823on_nat ) ) )
        & ( ( Xss2 != nil_li985476221562969869on_nat )
         => ? [Xss12: list_l6719245731233385357on_nat,Xs2: list_l1726306699111145479on_nat,Xs5: list_l1726306699111145479on_nat,Xss23: list_l6719245731233385357on_nat] :
              ( ( Xss2
                = ( append4931807731165890082on_nat @ Xss12 @ ( cons_l4979741563022480061on_nat @ ( append3839213209832495004on_nat @ Xs2 @ Xs5 ) @ Xss23 ) ) )
              & ( Ys2
                = ( append3839213209832495004on_nat @ ( concat4341727598008994882on_nat @ Xss12 ) @ Xs2 ) )
              & ( Zs3
                = ( append3839213209832495004on_nat @ Xs5 @ ( concat4341727598008994882on_nat @ Xss23 ) ) ) ) ) ) ) ).

% concat_eq_append_conv
thf(fact_310_concat__eq__append__conv,axiom,
    ! [Xss2: list_list_a,Ys2: list_a,Zs3: list_a] :
      ( ( ( concat_a @ Xss2 )
        = ( append_a @ Ys2 @ Zs3 ) )
      = ( ( ( Xss2 = nil_list_a )
         => ( ( Ys2 = nil_a )
            & ( Zs3 = nil_a ) ) )
        & ( ( Xss2 != nil_list_a )
         => ? [Xss12: list_list_a,Xs2: list_a,Xs5: list_a,Xss23: list_list_a] :
              ( ( Xss2
                = ( append_list_a @ Xss12 @ ( cons_list_a @ ( append_a @ Xs2 @ Xs5 ) @ Xss23 ) ) )
              & ( Ys2
                = ( append_a @ ( concat_a @ Xss12 ) @ Xs2 ) )
              & ( Zs3
                = ( append_a @ Xs5 @ ( concat_a @ Xss23 ) ) ) ) ) ) ) ).

% concat_eq_append_conv
thf(fact_311_concat__eq__append__conv,axiom,
    ! [Xss2: list_l8999537460041914605on_nat,Ys2: list_S5253404628753938141on_nat,Zs3: list_S5253404628753938141on_nat] :
      ( ( ( concat6161509665823917772on_nat @ Xss2 )
        = ( append3523812819975775986on_nat @ Ys2 @ Zs3 ) )
      = ( ( ( Xss2 = nil_li5563851662744376407on_nat )
         => ( ( Ys2 = nil_Su3078906344068544711on_nat )
            & ( Zs3 = nil_Su3078906344068544711on_nat ) ) )
        & ( ( Xss2 != nil_li5563851662744376407on_nat )
         => ? [Xss12: list_l8999537460041914605on_nat,Xs2: list_S5253404628753938141on_nat,Xs5: list_S5253404628753938141on_nat,Xss23: list_l8999537460041914605on_nat] :
              ( ( Xss2
                = ( append6174841615030841602on_nat @ Xss12 @ ( cons_l5857885737047467175on_nat @ ( append3523812819975775986on_nat @ Xs2 @ Xs5 ) @ Xss23 ) ) )
              & ( Ys2
                = ( append3523812819975775986on_nat @ ( concat6161509665823917772on_nat @ Xss12 ) @ Xs2 ) )
              & ( Zs3
                = ( append3523812819975775986on_nat @ Xs5 @ ( concat6161509665823917772on_nat @ Xss23 ) ) ) ) ) ) ) ).

% concat_eq_append_conv
thf(fact_312_concat__eq__append__conv,axiom,
    ! [Xss2: list_l1726306699111145479on_nat,Ys2: list_P9127870862658512897on_nat,Zs3: list_P9127870862658512897on_nat] :
      ( ( ( concat937493773106420796on_nat @ Xss2 )
        = ( append2865821678206098582on_nat @ Ys2 @ Zs3 ) )
      = ( ( ( Xss2 = nil_li2682182165152866823on_nat )
         => ( ( Ys2 = nil_Pr8614093721167239297on_nat )
            & ( Zs3 = nil_Pr8614093721167239297on_nat ) ) )
        & ( ( Xss2 != nil_li2682182165152866823on_nat )
         => ? [Xss12: list_l1726306699111145479on_nat,Xs2: list_P9127870862658512897on_nat,Xs5: list_P9127870862658512897on_nat,Xss23: list_l1726306699111145479on_nat] :
              ( ( Xss2
                = ( append3839213209832495004on_nat @ Xss12 @ ( cons_l3028799188195208631on_nat @ ( append2865821678206098582on_nat @ Xs2 @ Xs5 ) @ Xss23 ) ) )
              & ( Ys2
                = ( append2865821678206098582on_nat @ ( concat937493773106420796on_nat @ Xss12 ) @ Xs2 ) )
              & ( Zs3
                = ( append2865821678206098582on_nat @ Xs5 @ ( concat937493773106420796on_nat @ Xss23 ) ) ) ) ) ) ) ).

% concat_eq_append_conv
thf(fact_313_sorted__wrt_Ocases,axiom,
    ! [X2: produc5032551385658279741list_a] :
      ( ! [P2: a > a > $o] :
          ( X2
         != ( produc8111569692950616493list_a @ P2 @ nil_a ) )
     => ~ ! [P2: a > a > $o,X3: a,Ys: list_a] :
            ( X2
           != ( produc8111569692950616493list_a @ P2 @ ( cons_a @ X3 @ Ys ) ) ) ) ).

% sorted_wrt.cases
thf(fact_314_sorted__wrt_Ocases,axiom,
    ! [X2: produc3191134443298988280on_nat] :
      ( ! [P2: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat > $o] :
          ( X2
         != ( produc3262228428711020402on_nat @ P2 @ nil_Su3078906344068544711on_nat ) )
     => ~ ! [P2: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat > $o,X3: sum_su4907927656612468301on_nat,Ys: list_S5253404628753938141on_nat] :
            ( X2
           != ( produc3262228428711020402on_nat @ P2 @ ( cons_S6797538523646374167on_nat @ X3 @ Ys ) ) ) ) ).

% sorted_wrt.cases
thf(fact_315_sorted__wrt_Ocases,axiom,
    ! [X2: produc2264144286178290682on_nat] :
      ( ! [P2: produc1313295348606329723on_nat > produc1313295348606329723on_nat > $o] :
          ( X2
         != ( produc6237247764877134954on_nat @ P2 @ nil_Pr8614093721167239297on_nat ) )
     => ~ ! [P2: produc1313295348606329723on_nat > produc1313295348606329723on_nat > $o,X3: produc1313295348606329723on_nat,Ys: list_P9127870862658512897on_nat] :
            ( X2
           != ( produc6237247764877134954on_nat @ P2 @ ( cons_P1582596238967110705on_nat @ X3 @ Ys ) ) ) ) ).

% sorted_wrt.cases
thf(fact_316_sorted__wrt_Ocases,axiom,
    ! [X2: produc254973753779126261st_nat] :
      ( ! [P2: nat > nat > $o] :
          ( X2
         != ( produc4727192421694094319st_nat @ P2 @ nil_nat ) )
     => ~ ! [P2: nat > nat > $o,X3: nat,Ys: list_nat] :
            ( X2
           != ( produc4727192421694094319st_nat @ P2 @ ( cons_nat @ X3 @ Ys ) ) ) ) ).

% sorted_wrt.cases
thf(fact_317_sorted__wrt_Ocases,axiom,
    ! [X2: produc1732627278533666408on_nat] :
      ( ! [P2: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > $o] :
          ( X2
         != ( produc4619761231911769442on_nat @ P2 @ nil_li5563851662744376407on_nat ) )
     => ~ ! [P2: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > $o,X3: list_S5253404628753938141on_nat,Ys: list_l8999537460041914605on_nat] :
            ( X2
           != ( produc4619761231911769442on_nat @ P2 @ ( cons_l5857885737047467175on_nat @ X3 @ Ys ) ) ) ) ).

% sorted_wrt.cases
thf(fact_318_sorted__wrt_Ocases,axiom,
    ! [X2: produc5764678590178156288on_nat] :
      ( ! [P2: list_P9127870862658512897on_nat > list_P9127870862658512897on_nat > $o] :
          ( X2
         != ( produc3961857940133396976on_nat @ P2 @ nil_li2682182165152866823on_nat ) )
     => ~ ! [P2: list_P9127870862658512897on_nat > list_P9127870862658512897on_nat > $o,X3: list_P9127870862658512897on_nat,Ys: list_l1726306699111145479on_nat] :
            ( X2
           != ( produc3961857940133396976on_nat @ P2 @ ( cons_l3028799188195208631on_nat @ X3 @ Ys ) ) ) ) ).

% sorted_wrt.cases
thf(fact_319_successively_Ocases,axiom,
    ! [X2: produc5032551385658279741list_a] :
      ( ! [P2: a > a > $o] :
          ( X2
         != ( produc8111569692950616493list_a @ P2 @ nil_a ) )
     => ( ! [P2: a > a > $o,X3: a] :
            ( X2
           != ( produc8111569692950616493list_a @ P2 @ ( cons_a @ X3 @ nil_a ) ) )
       => ~ ! [P2: a > a > $o,X3: a,Y2: a,Xs3: list_a] :
              ( X2
             != ( produc8111569692950616493list_a @ P2 @ ( cons_a @ X3 @ ( cons_a @ Y2 @ Xs3 ) ) ) ) ) ) ).

% successively.cases
thf(fact_320_successively_Ocases,axiom,
    ! [X2: produc3191134443298988280on_nat] :
      ( ! [P2: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat > $o] :
          ( X2
         != ( produc3262228428711020402on_nat @ P2 @ nil_Su3078906344068544711on_nat ) )
     => ( ! [P2: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat > $o,X3: sum_su4907927656612468301on_nat] :
            ( X2
           != ( produc3262228428711020402on_nat @ P2 @ ( cons_S6797538523646374167on_nat @ X3 @ nil_Su3078906344068544711on_nat ) ) )
       => ~ ! [P2: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat > $o,X3: sum_su4907927656612468301on_nat,Y2: sum_su4907927656612468301on_nat,Xs3: list_S5253404628753938141on_nat] :
              ( X2
             != ( produc3262228428711020402on_nat @ P2 @ ( cons_S6797538523646374167on_nat @ X3 @ ( cons_S6797538523646374167on_nat @ Y2 @ Xs3 ) ) ) ) ) ) ).

% successively.cases
thf(fact_321_successively_Ocases,axiom,
    ! [X2: produc2264144286178290682on_nat] :
      ( ! [P2: produc1313295348606329723on_nat > produc1313295348606329723on_nat > $o] :
          ( X2
         != ( produc6237247764877134954on_nat @ P2 @ nil_Pr8614093721167239297on_nat ) )
     => ( ! [P2: produc1313295348606329723on_nat > produc1313295348606329723on_nat > $o,X3: produc1313295348606329723on_nat] :
            ( X2
           != ( produc6237247764877134954on_nat @ P2 @ ( cons_P1582596238967110705on_nat @ X3 @ nil_Pr8614093721167239297on_nat ) ) )
       => ~ ! [P2: produc1313295348606329723on_nat > produc1313295348606329723on_nat > $o,X3: produc1313295348606329723on_nat,Y2: produc1313295348606329723on_nat,Xs3: list_P9127870862658512897on_nat] :
              ( X2
             != ( produc6237247764877134954on_nat @ P2 @ ( cons_P1582596238967110705on_nat @ X3 @ ( cons_P1582596238967110705on_nat @ Y2 @ Xs3 ) ) ) ) ) ) ).

% successively.cases
thf(fact_322_successively_Ocases,axiom,
    ! [X2: produc254973753779126261st_nat] :
      ( ! [P2: nat > nat > $o] :
          ( X2
         != ( produc4727192421694094319st_nat @ P2 @ nil_nat ) )
     => ( ! [P2: nat > nat > $o,X3: nat] :
            ( X2
           != ( produc4727192421694094319st_nat @ P2 @ ( cons_nat @ X3 @ nil_nat ) ) )
       => ~ ! [P2: nat > nat > $o,X3: nat,Y2: nat,Xs3: list_nat] :
              ( X2
             != ( produc4727192421694094319st_nat @ P2 @ ( cons_nat @ X3 @ ( cons_nat @ Y2 @ Xs3 ) ) ) ) ) ) ).

% successively.cases
thf(fact_323_successively_Ocases,axiom,
    ! [X2: produc1732627278533666408on_nat] :
      ( ! [P2: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > $o] :
          ( X2
         != ( produc4619761231911769442on_nat @ P2 @ nil_li5563851662744376407on_nat ) )
     => ( ! [P2: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > $o,X3: list_S5253404628753938141on_nat] :
            ( X2
           != ( produc4619761231911769442on_nat @ P2 @ ( cons_l5857885737047467175on_nat @ X3 @ nil_li5563851662744376407on_nat ) ) )
       => ~ ! [P2: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > $o,X3: list_S5253404628753938141on_nat,Y2: list_S5253404628753938141on_nat,Xs3: list_l8999537460041914605on_nat] :
              ( X2
             != ( produc4619761231911769442on_nat @ P2 @ ( cons_l5857885737047467175on_nat @ X3 @ ( cons_l5857885737047467175on_nat @ Y2 @ Xs3 ) ) ) ) ) ) ).

% successively.cases
thf(fact_324_successively_Ocases,axiom,
    ! [X2: produc5764678590178156288on_nat] :
      ( ! [P2: list_P9127870862658512897on_nat > list_P9127870862658512897on_nat > $o] :
          ( X2
         != ( produc3961857940133396976on_nat @ P2 @ nil_li2682182165152866823on_nat ) )
     => ( ! [P2: list_P9127870862658512897on_nat > list_P9127870862658512897on_nat > $o,X3: list_P9127870862658512897on_nat] :
            ( X2
           != ( produc3961857940133396976on_nat @ P2 @ ( cons_l3028799188195208631on_nat @ X3 @ nil_li2682182165152866823on_nat ) ) )
       => ~ ! [P2: list_P9127870862658512897on_nat > list_P9127870862658512897on_nat > $o,X3: list_P9127870862658512897on_nat,Y2: list_P9127870862658512897on_nat,Xs3: list_l1726306699111145479on_nat] :
              ( X2
             != ( produc3961857940133396976on_nat @ P2 @ ( cons_l3028799188195208631on_nat @ X3 @ ( cons_l3028799188195208631on_nat @ Y2 @ Xs3 ) ) ) ) ) ) ).

% successively.cases
thf(fact_325_append__Cons,axiom,
    ! [X2: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat,Ys2: list_S5253404628753938141on_nat] :
      ( ( append3523812819975775986on_nat @ ( cons_S6797538523646374167on_nat @ X2 @ Xs ) @ Ys2 )
      = ( cons_S6797538523646374167on_nat @ X2 @ ( append3523812819975775986on_nat @ Xs @ Ys2 ) ) ) ).

% append_Cons
thf(fact_326_append__Cons,axiom,
    ! [X2: produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat,Ys2: list_P9127870862658512897on_nat] :
      ( ( append2865821678206098582on_nat @ ( cons_P1582596238967110705on_nat @ X2 @ Xs ) @ Ys2 )
      = ( cons_P1582596238967110705on_nat @ X2 @ ( append2865821678206098582on_nat @ Xs @ Ys2 ) ) ) ).

% append_Cons
thf(fact_327_append__Cons,axiom,
    ! [X2: nat,Xs: list_nat,Ys2: list_nat] :
      ( ( append_nat @ ( cons_nat @ X2 @ Xs ) @ Ys2 )
      = ( cons_nat @ X2 @ ( append_nat @ Xs @ Ys2 ) ) ) ).

% append_Cons
thf(fact_328_append__Cons,axiom,
    ! [X2: list_S5253404628753938141on_nat,Xs: list_l8999537460041914605on_nat,Ys2: list_l8999537460041914605on_nat] :
      ( ( append6174841615030841602on_nat @ ( cons_l5857885737047467175on_nat @ X2 @ Xs ) @ Ys2 )
      = ( cons_l5857885737047467175on_nat @ X2 @ ( append6174841615030841602on_nat @ Xs @ Ys2 ) ) ) ).

% append_Cons
thf(fact_329_append__Cons,axiom,
    ! [X2: list_P9127870862658512897on_nat,Xs: list_l1726306699111145479on_nat,Ys2: list_l1726306699111145479on_nat] :
      ( ( append3839213209832495004on_nat @ ( cons_l3028799188195208631on_nat @ X2 @ Xs ) @ Ys2 )
      = ( cons_l3028799188195208631on_nat @ X2 @ ( append3839213209832495004on_nat @ Xs @ Ys2 ) ) ) ).

% append_Cons
thf(fact_330_Cons__eq__appendI,axiom,
    ! [X2: sum_su4907927656612468301on_nat,Xs1: list_S5253404628753938141on_nat,Ys2: list_S5253404628753938141on_nat,Xs: list_S5253404628753938141on_nat,Zs3: list_S5253404628753938141on_nat] :
      ( ( ( cons_S6797538523646374167on_nat @ X2 @ Xs1 )
        = Ys2 )
     => ( ( Xs
          = ( append3523812819975775986on_nat @ Xs1 @ Zs3 ) )
       => ( ( cons_S6797538523646374167on_nat @ X2 @ Xs )
          = ( append3523812819975775986on_nat @ Ys2 @ Zs3 ) ) ) ) ).

% Cons_eq_appendI
thf(fact_331_Cons__eq__appendI,axiom,
    ! [X2: produc1313295348606329723on_nat,Xs1: list_P9127870862658512897on_nat,Ys2: list_P9127870862658512897on_nat,Xs: list_P9127870862658512897on_nat,Zs3: list_P9127870862658512897on_nat] :
      ( ( ( cons_P1582596238967110705on_nat @ X2 @ Xs1 )
        = Ys2 )
     => ( ( Xs
          = ( append2865821678206098582on_nat @ Xs1 @ Zs3 ) )
       => ( ( cons_P1582596238967110705on_nat @ X2 @ Xs )
          = ( append2865821678206098582on_nat @ Ys2 @ Zs3 ) ) ) ) ).

% Cons_eq_appendI
thf(fact_332_Cons__eq__appendI,axiom,
    ! [X2: nat,Xs1: list_nat,Ys2: list_nat,Xs: list_nat,Zs3: list_nat] :
      ( ( ( cons_nat @ X2 @ Xs1 )
        = Ys2 )
     => ( ( Xs
          = ( append_nat @ Xs1 @ Zs3 ) )
       => ( ( cons_nat @ X2 @ Xs )
          = ( append_nat @ Ys2 @ Zs3 ) ) ) ) ).

% Cons_eq_appendI
thf(fact_333_Cons__eq__appendI,axiom,
    ! [X2: list_S5253404628753938141on_nat,Xs1: list_l8999537460041914605on_nat,Ys2: list_l8999537460041914605on_nat,Xs: list_l8999537460041914605on_nat,Zs3: list_l8999537460041914605on_nat] :
      ( ( ( cons_l5857885737047467175on_nat @ X2 @ Xs1 )
        = Ys2 )
     => ( ( Xs
          = ( append6174841615030841602on_nat @ Xs1 @ Zs3 ) )
       => ( ( cons_l5857885737047467175on_nat @ X2 @ Xs )
          = ( append6174841615030841602on_nat @ Ys2 @ Zs3 ) ) ) ) ).

% Cons_eq_appendI
thf(fact_334_Cons__eq__appendI,axiom,
    ! [X2: list_P9127870862658512897on_nat,Xs1: list_l1726306699111145479on_nat,Ys2: list_l1726306699111145479on_nat,Xs: list_l1726306699111145479on_nat,Zs3: list_l1726306699111145479on_nat] :
      ( ( ( cons_l3028799188195208631on_nat @ X2 @ Xs1 )
        = Ys2 )
     => ( ( Xs
          = ( append3839213209832495004on_nat @ Xs1 @ Zs3 ) )
       => ( ( cons_l3028799188195208631on_nat @ X2 @ Xs )
          = ( append3839213209832495004on_nat @ Ys2 @ Zs3 ) ) ) ) ).

% Cons_eq_appendI
thf(fact_335_append__Nil,axiom,
    ! [Ys2: list_S5253404628753938141on_nat] :
      ( ( append3523812819975775986on_nat @ nil_Su3078906344068544711on_nat @ Ys2 )
      = Ys2 ) ).

% append_Nil
thf(fact_336_append__Nil,axiom,
    ! [Ys2: list_P9127870862658512897on_nat] :
      ( ( append2865821678206098582on_nat @ nil_Pr8614093721167239297on_nat @ Ys2 )
      = Ys2 ) ).

% append_Nil
thf(fact_337_append__Nil,axiom,
    ! [Ys2: list_nat] :
      ( ( append_nat @ nil_nat @ Ys2 )
      = Ys2 ) ).

% append_Nil
thf(fact_338_append__Nil,axiom,
    ! [Ys2: list_l8999537460041914605on_nat] :
      ( ( append6174841615030841602on_nat @ nil_li5563851662744376407on_nat @ Ys2 )
      = Ys2 ) ).

% append_Nil
thf(fact_339_append__Nil,axiom,
    ! [Ys2: list_l1726306699111145479on_nat] :
      ( ( append3839213209832495004on_nat @ nil_li2682182165152866823on_nat @ Ys2 )
      = Ys2 ) ).

% append_Nil
thf(fact_340_append__Nil,axiom,
    ! [Ys2: list_a] :
      ( ( append_a @ nil_a @ Ys2 )
      = Ys2 ) ).

% append_Nil
thf(fact_341_append_Oleft__neutral,axiom,
    ! [A: list_S5253404628753938141on_nat] :
      ( ( append3523812819975775986on_nat @ nil_Su3078906344068544711on_nat @ A )
      = A ) ).

% append.left_neutral
thf(fact_342_append_Oleft__neutral,axiom,
    ! [A: list_P9127870862658512897on_nat] :
      ( ( append2865821678206098582on_nat @ nil_Pr8614093721167239297on_nat @ A )
      = A ) ).

% append.left_neutral
thf(fact_343_append_Oleft__neutral,axiom,
    ! [A: list_nat] :
      ( ( append_nat @ nil_nat @ A )
      = A ) ).

% append.left_neutral
thf(fact_344_append_Oleft__neutral,axiom,
    ! [A: list_l8999537460041914605on_nat] :
      ( ( append6174841615030841602on_nat @ nil_li5563851662744376407on_nat @ A )
      = A ) ).

% append.left_neutral
thf(fact_345_append_Oleft__neutral,axiom,
    ! [A: list_l1726306699111145479on_nat] :
      ( ( append3839213209832495004on_nat @ nil_li2682182165152866823on_nat @ A )
      = A ) ).

% append.left_neutral
thf(fact_346_append_Oleft__neutral,axiom,
    ! [A: list_a] :
      ( ( append_a @ nil_a @ A )
      = A ) ).

% append.left_neutral
thf(fact_347_eq__Nil__appendI,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys2: list_S5253404628753938141on_nat] :
      ( ( Xs = Ys2 )
     => ( Xs
        = ( append3523812819975775986on_nat @ nil_Su3078906344068544711on_nat @ Ys2 ) ) ) ).

% eq_Nil_appendI
thf(fact_348_eq__Nil__appendI,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys2: list_P9127870862658512897on_nat] :
      ( ( Xs = Ys2 )
     => ( Xs
        = ( append2865821678206098582on_nat @ nil_Pr8614093721167239297on_nat @ Ys2 ) ) ) ).

% eq_Nil_appendI
thf(fact_349_eq__Nil__appendI,axiom,
    ! [Xs: list_nat,Ys2: list_nat] :
      ( ( Xs = Ys2 )
     => ( Xs
        = ( append_nat @ nil_nat @ Ys2 ) ) ) ).

% eq_Nil_appendI
thf(fact_350_eq__Nil__appendI,axiom,
    ! [Xs: list_l8999537460041914605on_nat,Ys2: list_l8999537460041914605on_nat] :
      ( ( Xs = Ys2 )
     => ( Xs
        = ( append6174841615030841602on_nat @ nil_li5563851662744376407on_nat @ Ys2 ) ) ) ).

% eq_Nil_appendI
thf(fact_351_eq__Nil__appendI,axiom,
    ! [Xs: list_l1726306699111145479on_nat,Ys2: list_l1726306699111145479on_nat] :
      ( ( Xs = Ys2 )
     => ( Xs
        = ( append3839213209832495004on_nat @ nil_li2682182165152866823on_nat @ Ys2 ) ) ) ).

% eq_Nil_appendI
thf(fact_352_eq__Nil__appendI,axiom,
    ! [Xs: list_a,Ys2: list_a] :
      ( ( Xs = Ys2 )
     => ( Xs
        = ( append_a @ nil_a @ Ys2 ) ) ) ).

% eq_Nil_appendI
thf(fact_353_sum_Odistinct_I1_J,axiom,
    ! [X1: produc27318731269267614tation,X23: nat] :
      ( ( sum_In3878554002080803550on_nat @ X1 )
     != ( sum_In255631617079920996tation @ X23 ) ) ).

% sum.distinct(1)
thf(fact_354_old_Osum_Odistinct_I2_J,axiom,
    ! [B2: nat,A: produc27318731269267614tation] :
      ( ( sum_In255631617079920996tation @ B2 )
     != ( sum_In3878554002080803550on_nat @ A ) ) ).

% old.sum.distinct(2)
thf(fact_355_old_Osum_Odistinct_I1_J,axiom,
    ! [A: produc27318731269267614tation,B2: nat] :
      ( ( sum_In3878554002080803550on_nat @ A )
     != ( sum_In255631617079920996tation @ B2 ) ) ).

% old.sum.distinct(1)
thf(fact_356_old_Osum_Oexhaust,axiom,
    ! [Y: sum_su4907927656612468301on_nat] :
      ( ! [A3: produc27318731269267614tation] :
          ( Y
         != ( sum_In3878554002080803550on_nat @ A3 ) )
     => ~ ! [B3: nat] :
            ( Y
           != ( sum_In255631617079920996tation @ B3 ) ) ) ).

% old.sum.exhaust
thf(fact_357_sumE,axiom,
    ! [S: sum_su4907927656612468301on_nat] :
      ( ! [X3: produc27318731269267614tation] :
          ( S
         != ( sum_In3878554002080803550on_nat @ X3 ) )
     => ~ ! [Y2: nat] :
            ( S
           != ( sum_In255631617079920996tation @ Y2 ) ) ) ).

% sumE
thf(fact_358_Inr__not__Inl,axiom,
    ! [B: nat,A: produc27318731269267614tation] :
      ( ( sum_In255631617079920996tation @ B )
     != ( sum_In3878554002080803550on_nat @ A ) ) ).

% Inr_not_Inl
thf(fact_359_split__sum__ex,axiom,
    ( ( ^ [P3: sum_su4907927656612468301on_nat > $o] :
        ? [X4: sum_su4907927656612468301on_nat] : ( P3 @ X4 ) )
    = ( ^ [P4: sum_su4907927656612468301on_nat > $o] :
          ( ? [X: produc27318731269267614tation] : ( P4 @ ( sum_In3878554002080803550on_nat @ X ) )
          | ? [X: nat] : ( P4 @ ( sum_In255631617079920996tation @ X ) ) ) ) ) ).

% split_sum_ex
thf(fact_360_split__sum__all,axiom,
    ( ( ^ [P3: sum_su4907927656612468301on_nat > $o] :
        ! [X4: sum_su4907927656612468301on_nat] : ( P3 @ X4 ) )
    = ( ^ [P4: sum_su4907927656612468301on_nat > $o] :
          ( ! [X: produc27318731269267614tation] : ( P4 @ ( sum_In3878554002080803550on_nat @ X ) )
          & ! [X: nat] : ( P4 @ ( sum_In255631617079920996tation @ X ) ) ) ) ) ).

% split_sum_all
thf(fact_361_append__eq__map__conv,axiom,
    ! [Ys2: list_nat,Zs3: list_nat,F: nat > nat,Xs: list_nat] :
      ( ( ( append_nat @ Ys2 @ Zs3 )
        = ( map_nat_nat @ F @ Xs ) )
      = ( ? [Us: list_nat,Vs: list_nat] :
            ( ( Xs
              = ( append_nat @ Us @ Vs ) )
            & ( Ys2
              = ( map_nat_nat @ F @ Us ) )
            & ( Zs3
              = ( map_nat_nat @ F @ Vs ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_362_append__eq__map__conv,axiom,
    ! [Ys2: list_list_nat,Zs3: list_list_nat,F: nat > list_nat,Xs: list_nat] :
      ( ( ( append_list_nat @ Ys2 @ Zs3 )
        = ( map_nat_list_nat @ F @ Xs ) )
      = ( ? [Us: list_nat,Vs: list_nat] :
            ( ( Xs
              = ( append_nat @ Us @ Vs ) )
            & ( Ys2
              = ( map_nat_list_nat @ F @ Us ) )
            & ( Zs3
              = ( map_nat_list_nat @ F @ Vs ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_363_append__eq__map__conv,axiom,
    ! [Ys2: list_S5253404628753938141on_nat,Zs3: list_S5253404628753938141on_nat,F: nat > sum_su4907927656612468301on_nat,Xs: list_nat] :
      ( ( ( append3523812819975775986on_nat @ Ys2 @ Zs3 )
        = ( map_na5706685718985795555on_nat @ F @ Xs ) )
      = ( ? [Us: list_nat,Vs: list_nat] :
            ( ( Xs
              = ( append_nat @ Us @ Vs ) )
            & ( Ys2
              = ( map_na5706685718985795555on_nat @ F @ Us ) )
            & ( Zs3
              = ( map_na5706685718985795555on_nat @ F @ Vs ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_364_append__eq__map__conv,axiom,
    ! [Ys2: list_nat,Zs3: list_nat,F: sum_su4907927656612468301on_nat > nat,Xs: list_S5253404628753938141on_nat] :
      ( ( ( append_nat @ Ys2 @ Zs3 )
        = ( map_Su4809418691747827683at_nat @ F @ Xs ) )
      = ( ? [Us: list_S5253404628753938141on_nat,Vs: list_S5253404628753938141on_nat] :
            ( ( Xs
              = ( append3523812819975775986on_nat @ Us @ Vs ) )
            & ( Ys2
              = ( map_Su4809418691747827683at_nat @ F @ Us ) )
            & ( Zs3
              = ( map_Su4809418691747827683at_nat @ F @ Vs ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_365_append__eq__map__conv,axiom,
    ! [Ys2: list_S5253404628753938141on_nat,Zs3: list_S5253404628753938141on_nat,F: a > sum_su4907927656612468301on_nat,Xs: list_a] :
      ( ( ( append3523812819975775986on_nat @ Ys2 @ Zs3 )
        = ( map_a_2711072540680412167on_nat @ F @ Xs ) )
      = ( ? [Us: list_a,Vs: list_a] :
            ( ( Xs
              = ( append_a @ Us @ Vs ) )
            & ( Ys2
              = ( map_a_2711072540680412167on_nat @ F @ Us ) )
            & ( Zs3
              = ( map_a_2711072540680412167on_nat @ F @ Vs ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_366_append__eq__map__conv,axiom,
    ! [Ys2: list_l8999537460041914605on_nat,Zs3: list_l8999537460041914605on_nat,F: a > list_S5253404628753938141on_nat,Xs: list_a] :
      ( ( ( append6174841615030841602on_nat @ Ys2 @ Zs3 )
        = ( map_a_3887198048842487703on_nat @ F @ Xs ) )
      = ( ? [Us: list_a,Vs: list_a] :
            ( ( Xs
              = ( append_a @ Us @ Vs ) )
            & ( Ys2
              = ( map_a_3887198048842487703on_nat @ F @ Us ) )
            & ( Zs3
              = ( map_a_3887198048842487703on_nat @ F @ Vs ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_367_append__eq__map__conv,axiom,
    ! [Ys2: list_l8365940747029150845on_nat,Zs3: list_l8365940747029150845on_nat,F: a > list_l8999537460041914605on_nat,Xs: list_a] :
      ( ( ( append7602222762569088146on_nat @ Ys2 @ Zs3 )
        = ( map_a_3820070144034313383on_nat @ F @ Xs ) )
      = ( ? [Us: list_a,Vs: list_a] :
            ( ( Xs
              = ( append_a @ Us @ Vs ) )
            & ( Ys2
              = ( map_a_3820070144034313383on_nat @ F @ Us ) )
            & ( Zs3
              = ( map_a_3820070144034313383on_nat @ F @ Vs ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_368_append__eq__map__conv,axiom,
    ! [Ys2: list_S5253404628753938141on_nat,Zs3: list_S5253404628753938141on_nat,F: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat] :
      ( ( ( append3523812819975775986on_nat @ Ys2 @ Zs3 )
        = ( map_Su241112079427911264on_nat @ F @ Xs ) )
      = ( ? [Us: list_S5253404628753938141on_nat,Vs: list_S5253404628753938141on_nat] :
            ( ( Xs
              = ( append3523812819975775986on_nat @ Us @ Vs ) )
            & ( Ys2
              = ( map_Su241112079427911264on_nat @ F @ Us ) )
            & ( Zs3
              = ( map_Su241112079427911264on_nat @ F @ Vs ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_369_append__eq__map__conv,axiom,
    ! [Ys2: list_P9127870862658512897on_nat,Zs3: list_P9127870862658512897on_nat,F: nat > produc1313295348606329723on_nat,Xs: list_nat] :
      ( ( ( append2865821678206098582on_nat @ Ys2 @ Zs3 )
        = ( map_na2641141678045226597on_nat @ F @ Xs ) )
      = ( ? [Us: list_nat,Vs: list_nat] :
            ( ( Xs
              = ( append_nat @ Us @ Vs ) )
            & ( Ys2
              = ( map_na2641141678045226597on_nat @ F @ Us ) )
            & ( Zs3
              = ( map_na2641141678045226597on_nat @ F @ Vs ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_370_append__eq__map__conv,axiom,
    ! [Ys2: list_nat,Zs3: list_nat,F: produc1313295348606329723on_nat > nat,Xs: list_P9127870862658512897on_nat] :
      ( ( ( append_nat @ Ys2 @ Zs3 )
        = ( map_Pr1888482430906987655at_nat @ F @ Xs ) )
      = ( ? [Us: list_P9127870862658512897on_nat,Vs: list_P9127870862658512897on_nat] :
            ( ( Xs
              = ( append2865821678206098582on_nat @ Us @ Vs ) )
            & ( Ys2
              = ( map_Pr1888482430906987655at_nat @ F @ Us ) )
            & ( Zs3
              = ( map_Pr1888482430906987655at_nat @ F @ Vs ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_371_map__eq__append__conv,axiom,
    ! [F: nat > nat,Xs: list_nat,Ys2: list_nat,Zs3: list_nat] :
      ( ( ( map_nat_nat @ F @ Xs )
        = ( append_nat @ Ys2 @ Zs3 ) )
      = ( ? [Us: list_nat,Vs: list_nat] :
            ( ( Xs
              = ( append_nat @ Us @ Vs ) )
            & ( Ys2
              = ( map_nat_nat @ F @ Us ) )
            & ( Zs3
              = ( map_nat_nat @ F @ Vs ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_372_map__eq__append__conv,axiom,
    ! [F: nat > list_nat,Xs: list_nat,Ys2: list_list_nat,Zs3: list_list_nat] :
      ( ( ( map_nat_list_nat @ F @ Xs )
        = ( append_list_nat @ Ys2 @ Zs3 ) )
      = ( ? [Us: list_nat,Vs: list_nat] :
            ( ( Xs
              = ( append_nat @ Us @ Vs ) )
            & ( Ys2
              = ( map_nat_list_nat @ F @ Us ) )
            & ( Zs3
              = ( map_nat_list_nat @ F @ Vs ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_373_map__eq__append__conv,axiom,
    ! [F: nat > sum_su4907927656612468301on_nat,Xs: list_nat,Ys2: list_S5253404628753938141on_nat,Zs3: list_S5253404628753938141on_nat] :
      ( ( ( map_na5706685718985795555on_nat @ F @ Xs )
        = ( append3523812819975775986on_nat @ Ys2 @ Zs3 ) )
      = ( ? [Us: list_nat,Vs: list_nat] :
            ( ( Xs
              = ( append_nat @ Us @ Vs ) )
            & ( Ys2
              = ( map_na5706685718985795555on_nat @ F @ Us ) )
            & ( Zs3
              = ( map_na5706685718985795555on_nat @ F @ Vs ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_374_map__eq__append__conv,axiom,
    ! [F: sum_su4907927656612468301on_nat > nat,Xs: list_S5253404628753938141on_nat,Ys2: list_nat,Zs3: list_nat] :
      ( ( ( map_Su4809418691747827683at_nat @ F @ Xs )
        = ( append_nat @ Ys2 @ Zs3 ) )
      = ( ? [Us: list_S5253404628753938141on_nat,Vs: list_S5253404628753938141on_nat] :
            ( ( Xs
              = ( append3523812819975775986on_nat @ Us @ Vs ) )
            & ( Ys2
              = ( map_Su4809418691747827683at_nat @ F @ Us ) )
            & ( Zs3
              = ( map_Su4809418691747827683at_nat @ F @ Vs ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_375_map__eq__append__conv,axiom,
    ! [F: a > sum_su4907927656612468301on_nat,Xs: list_a,Ys2: list_S5253404628753938141on_nat,Zs3: list_S5253404628753938141on_nat] :
      ( ( ( map_a_2711072540680412167on_nat @ F @ Xs )
        = ( append3523812819975775986on_nat @ Ys2 @ Zs3 ) )
      = ( ? [Us: list_a,Vs: list_a] :
            ( ( Xs
              = ( append_a @ Us @ Vs ) )
            & ( Ys2
              = ( map_a_2711072540680412167on_nat @ F @ Us ) )
            & ( Zs3
              = ( map_a_2711072540680412167on_nat @ F @ Vs ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_376_map__eq__append__conv,axiom,
    ! [F: a > list_S5253404628753938141on_nat,Xs: list_a,Ys2: list_l8999537460041914605on_nat,Zs3: list_l8999537460041914605on_nat] :
      ( ( ( map_a_3887198048842487703on_nat @ F @ Xs )
        = ( append6174841615030841602on_nat @ Ys2 @ Zs3 ) )
      = ( ? [Us: list_a,Vs: list_a] :
            ( ( Xs
              = ( append_a @ Us @ Vs ) )
            & ( Ys2
              = ( map_a_3887198048842487703on_nat @ F @ Us ) )
            & ( Zs3
              = ( map_a_3887198048842487703on_nat @ F @ Vs ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_377_map__eq__append__conv,axiom,
    ! [F: a > list_l8999537460041914605on_nat,Xs: list_a,Ys2: list_l8365940747029150845on_nat,Zs3: list_l8365940747029150845on_nat] :
      ( ( ( map_a_3820070144034313383on_nat @ F @ Xs )
        = ( append7602222762569088146on_nat @ Ys2 @ Zs3 ) )
      = ( ? [Us: list_a,Vs: list_a] :
            ( ( Xs
              = ( append_a @ Us @ Vs ) )
            & ( Ys2
              = ( map_a_3820070144034313383on_nat @ F @ Us ) )
            & ( Zs3
              = ( map_a_3820070144034313383on_nat @ F @ Vs ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_378_map__eq__append__conv,axiom,
    ! [F: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat,Ys2: list_S5253404628753938141on_nat,Zs3: list_S5253404628753938141on_nat] :
      ( ( ( map_Su241112079427911264on_nat @ F @ Xs )
        = ( append3523812819975775986on_nat @ Ys2 @ Zs3 ) )
      = ( ? [Us: list_S5253404628753938141on_nat,Vs: list_S5253404628753938141on_nat] :
            ( ( Xs
              = ( append3523812819975775986on_nat @ Us @ Vs ) )
            & ( Ys2
              = ( map_Su241112079427911264on_nat @ F @ Us ) )
            & ( Zs3
              = ( map_Su241112079427911264on_nat @ F @ Vs ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_379_map__eq__append__conv,axiom,
    ! [F: nat > produc1313295348606329723on_nat,Xs: list_nat,Ys2: list_P9127870862658512897on_nat,Zs3: list_P9127870862658512897on_nat] :
      ( ( ( map_na2641141678045226597on_nat @ F @ Xs )
        = ( append2865821678206098582on_nat @ Ys2 @ Zs3 ) )
      = ( ? [Us: list_nat,Vs: list_nat] :
            ( ( Xs
              = ( append_nat @ Us @ Vs ) )
            & ( Ys2
              = ( map_na2641141678045226597on_nat @ F @ Us ) )
            & ( Zs3
              = ( map_na2641141678045226597on_nat @ F @ Vs ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_380_map__eq__append__conv,axiom,
    ! [F: produc1313295348606329723on_nat > nat,Xs: list_P9127870862658512897on_nat,Ys2: list_nat,Zs3: list_nat] :
      ( ( ( map_Pr1888482430906987655at_nat @ F @ Xs )
        = ( append_nat @ Ys2 @ Zs3 ) )
      = ( ? [Us: list_P9127870862658512897on_nat,Vs: list_P9127870862658512897on_nat] :
            ( ( Xs
              = ( append2865821678206098582on_nat @ Us @ Vs ) )
            & ( Ys2
              = ( map_Pr1888482430906987655at_nat @ F @ Us ) )
            & ( Zs3
              = ( map_Pr1888482430906987655at_nat @ F @ Vs ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_381_concat_Osimps_I2_J,axiom,
    ! [X2: list_nat,Xs: list_list_nat] :
      ( ( concat_nat @ ( cons_list_nat @ X2 @ Xs ) )
      = ( append_nat @ X2 @ ( concat_nat @ Xs ) ) ) ).

% concat.simps(2)
thf(fact_382_concat_Osimps_I2_J,axiom,
    ! [X2: list_l8999537460041914605on_nat,Xs: list_l8365940747029150845on_nat] :
      ( ( concat4774721536428481500on_nat @ ( cons_l6725403660475259319on_nat @ X2 @ Xs ) )
      = ( append6174841615030841602on_nat @ X2 @ ( concat4774721536428481500on_nat @ Xs ) ) ) ).

% concat.simps(2)
thf(fact_383_concat_Osimps_I2_J,axiom,
    ! [X2: list_l1726306699111145479on_nat,Xs: list_l6719245731233385357on_nat] :
      ( ( concat4341727598008994882on_nat @ ( cons_l4979741563022480061on_nat @ X2 @ Xs ) )
      = ( append3839213209832495004on_nat @ X2 @ ( concat4341727598008994882on_nat @ Xs ) ) ) ).

% concat.simps(2)
thf(fact_384_concat_Osimps_I2_J,axiom,
    ! [X2: list_S5253404628753938141on_nat,Xs: list_l8999537460041914605on_nat] :
      ( ( concat6161509665823917772on_nat @ ( cons_l5857885737047467175on_nat @ X2 @ Xs ) )
      = ( append3523812819975775986on_nat @ X2 @ ( concat6161509665823917772on_nat @ Xs ) ) ) ).

% concat.simps(2)
thf(fact_385_concat_Osimps_I2_J,axiom,
    ! [X2: list_P9127870862658512897on_nat,Xs: list_l1726306699111145479on_nat] :
      ( ( concat937493773106420796on_nat @ ( cons_l3028799188195208631on_nat @ X2 @ Xs ) )
      = ( append2865821678206098582on_nat @ X2 @ ( concat937493773106420796on_nat @ Xs ) ) ) ).

% concat.simps(2)
thf(fact_386_rev__induct,axiom,
    ! [P: list_a > $o,Xs: list_a] :
      ( ( P @ nil_a )
     => ( ! [X3: a,Xs3: list_a] :
            ( ( P @ Xs3 )
           => ( P @ ( append_a @ Xs3 @ ( cons_a @ X3 @ nil_a ) ) ) )
       => ( P @ Xs ) ) ) ).

% rev_induct
thf(fact_387_rev__induct,axiom,
    ! [P: list_S5253404628753938141on_nat > $o,Xs: list_S5253404628753938141on_nat] :
      ( ( P @ nil_Su3078906344068544711on_nat )
     => ( ! [X3: sum_su4907927656612468301on_nat,Xs3: list_S5253404628753938141on_nat] :
            ( ( P @ Xs3 )
           => ( P @ ( append3523812819975775986on_nat @ Xs3 @ ( cons_S6797538523646374167on_nat @ X3 @ nil_Su3078906344068544711on_nat ) ) ) )
       => ( P @ Xs ) ) ) ).

% rev_induct
thf(fact_388_rev__induct,axiom,
    ! [P: list_P9127870862658512897on_nat > $o,Xs: list_P9127870862658512897on_nat] :
      ( ( P @ nil_Pr8614093721167239297on_nat )
     => ( ! [X3: produc1313295348606329723on_nat,Xs3: list_P9127870862658512897on_nat] :
            ( ( P @ Xs3 )
           => ( P @ ( append2865821678206098582on_nat @ Xs3 @ ( cons_P1582596238967110705on_nat @ X3 @ nil_Pr8614093721167239297on_nat ) ) ) )
       => ( P @ Xs ) ) ) ).

% rev_induct
thf(fact_389_rev__induct,axiom,
    ! [P: list_nat > $o,Xs: list_nat] :
      ( ( P @ nil_nat )
     => ( ! [X3: nat,Xs3: list_nat] :
            ( ( P @ Xs3 )
           => ( P @ ( append_nat @ Xs3 @ ( cons_nat @ X3 @ nil_nat ) ) ) )
       => ( P @ Xs ) ) ) ).

% rev_induct
thf(fact_390_rev__induct,axiom,
    ! [P: list_l8999537460041914605on_nat > $o,Xs: list_l8999537460041914605on_nat] :
      ( ( P @ nil_li5563851662744376407on_nat )
     => ( ! [X3: list_S5253404628753938141on_nat,Xs3: list_l8999537460041914605on_nat] :
            ( ( P @ Xs3 )
           => ( P @ ( append6174841615030841602on_nat @ Xs3 @ ( cons_l5857885737047467175on_nat @ X3 @ nil_li5563851662744376407on_nat ) ) ) )
       => ( P @ Xs ) ) ) ).

% rev_induct
thf(fact_391_rev__induct,axiom,
    ! [P: list_l1726306699111145479on_nat > $o,Xs: list_l1726306699111145479on_nat] :
      ( ( P @ nil_li2682182165152866823on_nat )
     => ( ! [X3: list_P9127870862658512897on_nat,Xs3: list_l1726306699111145479on_nat] :
            ( ( P @ Xs3 )
           => ( P @ ( append3839213209832495004on_nat @ Xs3 @ ( cons_l3028799188195208631on_nat @ X3 @ nil_li2682182165152866823on_nat ) ) ) )
       => ( P @ Xs ) ) ) ).

% rev_induct
thf(fact_392_rev__exhaust,axiom,
    ! [Xs: list_a] :
      ( ( Xs != nil_a )
     => ~ ! [Ys: list_a,Y2: a] :
            ( Xs
           != ( append_a @ Ys @ ( cons_a @ Y2 @ nil_a ) ) ) ) ).

% rev_exhaust
thf(fact_393_rev__exhaust,axiom,
    ! [Xs: list_S5253404628753938141on_nat] :
      ( ( Xs != nil_Su3078906344068544711on_nat )
     => ~ ! [Ys: list_S5253404628753938141on_nat,Y2: sum_su4907927656612468301on_nat] :
            ( Xs
           != ( append3523812819975775986on_nat @ Ys @ ( cons_S6797538523646374167on_nat @ Y2 @ nil_Su3078906344068544711on_nat ) ) ) ) ).

% rev_exhaust
thf(fact_394_rev__exhaust,axiom,
    ! [Xs: list_P9127870862658512897on_nat] :
      ( ( Xs != nil_Pr8614093721167239297on_nat )
     => ~ ! [Ys: list_P9127870862658512897on_nat,Y2: produc1313295348606329723on_nat] :
            ( Xs
           != ( append2865821678206098582on_nat @ Ys @ ( cons_P1582596238967110705on_nat @ Y2 @ nil_Pr8614093721167239297on_nat ) ) ) ) ).

% rev_exhaust
thf(fact_395_rev__exhaust,axiom,
    ! [Xs: list_nat] :
      ( ( Xs != nil_nat )
     => ~ ! [Ys: list_nat,Y2: nat] :
            ( Xs
           != ( append_nat @ Ys @ ( cons_nat @ Y2 @ nil_nat ) ) ) ) ).

% rev_exhaust
thf(fact_396_rev__exhaust,axiom,
    ! [Xs: list_l8999537460041914605on_nat] :
      ( ( Xs != nil_li5563851662744376407on_nat )
     => ~ ! [Ys: list_l8999537460041914605on_nat,Y2: list_S5253404628753938141on_nat] :
            ( Xs
           != ( append6174841615030841602on_nat @ Ys @ ( cons_l5857885737047467175on_nat @ Y2 @ nil_li5563851662744376407on_nat ) ) ) ) ).

% rev_exhaust
thf(fact_397_rev__exhaust,axiom,
    ! [Xs: list_l1726306699111145479on_nat] :
      ( ( Xs != nil_li2682182165152866823on_nat )
     => ~ ! [Ys: list_l1726306699111145479on_nat,Y2: list_P9127870862658512897on_nat] :
            ( Xs
           != ( append3839213209832495004on_nat @ Ys @ ( cons_l3028799188195208631on_nat @ Y2 @ nil_li2682182165152866823on_nat ) ) ) ) ).

% rev_exhaust
thf(fact_398_Cons__eq__append__conv,axiom,
    ! [X2: a,Xs: list_a,Ys2: list_a,Zs3: list_a] :
      ( ( ( cons_a @ X2 @ Xs )
        = ( append_a @ Ys2 @ Zs3 ) )
      = ( ( ( Ys2 = nil_a )
          & ( ( cons_a @ X2 @ Xs )
            = Zs3 ) )
        | ? [Ys5: list_a] :
            ( ( ( cons_a @ X2 @ Ys5 )
              = Ys2 )
            & ( Xs
              = ( append_a @ Ys5 @ Zs3 ) ) ) ) ) ).

% Cons_eq_append_conv
thf(fact_399_Cons__eq__append__conv,axiom,
    ! [X2: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat,Ys2: list_S5253404628753938141on_nat,Zs3: list_S5253404628753938141on_nat] :
      ( ( ( cons_S6797538523646374167on_nat @ X2 @ Xs )
        = ( append3523812819975775986on_nat @ Ys2 @ Zs3 ) )
      = ( ( ( Ys2 = nil_Su3078906344068544711on_nat )
          & ( ( cons_S6797538523646374167on_nat @ X2 @ Xs )
            = Zs3 ) )
        | ? [Ys5: list_S5253404628753938141on_nat] :
            ( ( ( cons_S6797538523646374167on_nat @ X2 @ Ys5 )
              = Ys2 )
            & ( Xs
              = ( append3523812819975775986on_nat @ Ys5 @ Zs3 ) ) ) ) ) ).

% Cons_eq_append_conv
thf(fact_400_Cons__eq__append__conv,axiom,
    ! [X2: produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat,Ys2: list_P9127870862658512897on_nat,Zs3: list_P9127870862658512897on_nat] :
      ( ( ( cons_P1582596238967110705on_nat @ X2 @ Xs )
        = ( append2865821678206098582on_nat @ Ys2 @ Zs3 ) )
      = ( ( ( Ys2 = nil_Pr8614093721167239297on_nat )
          & ( ( cons_P1582596238967110705on_nat @ X2 @ Xs )
            = Zs3 ) )
        | ? [Ys5: list_P9127870862658512897on_nat] :
            ( ( ( cons_P1582596238967110705on_nat @ X2 @ Ys5 )
              = Ys2 )
            & ( Xs
              = ( append2865821678206098582on_nat @ Ys5 @ Zs3 ) ) ) ) ) ).

% Cons_eq_append_conv
thf(fact_401_Cons__eq__append__conv,axiom,
    ! [X2: nat,Xs: list_nat,Ys2: list_nat,Zs3: list_nat] :
      ( ( ( cons_nat @ X2 @ Xs )
        = ( append_nat @ Ys2 @ Zs3 ) )
      = ( ( ( Ys2 = nil_nat )
          & ( ( cons_nat @ X2 @ Xs )
            = Zs3 ) )
        | ? [Ys5: list_nat] :
            ( ( ( cons_nat @ X2 @ Ys5 )
              = Ys2 )
            & ( Xs
              = ( append_nat @ Ys5 @ Zs3 ) ) ) ) ) ).

% Cons_eq_append_conv
thf(fact_402_Cons__eq__append__conv,axiom,
    ! [X2: list_S5253404628753938141on_nat,Xs: list_l8999537460041914605on_nat,Ys2: list_l8999537460041914605on_nat,Zs3: list_l8999537460041914605on_nat] :
      ( ( ( cons_l5857885737047467175on_nat @ X2 @ Xs )
        = ( append6174841615030841602on_nat @ Ys2 @ Zs3 ) )
      = ( ( ( Ys2 = nil_li5563851662744376407on_nat )
          & ( ( cons_l5857885737047467175on_nat @ X2 @ Xs )
            = Zs3 ) )
        | ? [Ys5: list_l8999537460041914605on_nat] :
            ( ( ( cons_l5857885737047467175on_nat @ X2 @ Ys5 )
              = Ys2 )
            & ( Xs
              = ( append6174841615030841602on_nat @ Ys5 @ Zs3 ) ) ) ) ) ).

% Cons_eq_append_conv
thf(fact_403_Cons__eq__append__conv,axiom,
    ! [X2: list_P9127870862658512897on_nat,Xs: list_l1726306699111145479on_nat,Ys2: list_l1726306699111145479on_nat,Zs3: list_l1726306699111145479on_nat] :
      ( ( ( cons_l3028799188195208631on_nat @ X2 @ Xs )
        = ( append3839213209832495004on_nat @ Ys2 @ Zs3 ) )
      = ( ( ( Ys2 = nil_li2682182165152866823on_nat )
          & ( ( cons_l3028799188195208631on_nat @ X2 @ Xs )
            = Zs3 ) )
        | ? [Ys5: list_l1726306699111145479on_nat] :
            ( ( ( cons_l3028799188195208631on_nat @ X2 @ Ys5 )
              = Ys2 )
            & ( Xs
              = ( append3839213209832495004on_nat @ Ys5 @ Zs3 ) ) ) ) ) ).

% Cons_eq_append_conv
thf(fact_404_append__eq__Cons__conv,axiom,
    ! [Ys2: list_a,Zs3: list_a,X2: a,Xs: list_a] :
      ( ( ( append_a @ Ys2 @ Zs3 )
        = ( cons_a @ X2 @ Xs ) )
      = ( ( ( Ys2 = nil_a )
          & ( Zs3
            = ( cons_a @ X2 @ Xs ) ) )
        | ? [Ys5: list_a] :
            ( ( Ys2
              = ( cons_a @ X2 @ Ys5 ) )
            & ( ( append_a @ Ys5 @ Zs3 )
              = Xs ) ) ) ) ).

% append_eq_Cons_conv
thf(fact_405_append__eq__Cons__conv,axiom,
    ! [Ys2: list_S5253404628753938141on_nat,Zs3: list_S5253404628753938141on_nat,X2: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat] :
      ( ( ( append3523812819975775986on_nat @ Ys2 @ Zs3 )
        = ( cons_S6797538523646374167on_nat @ X2 @ Xs ) )
      = ( ( ( Ys2 = nil_Su3078906344068544711on_nat )
          & ( Zs3
            = ( cons_S6797538523646374167on_nat @ X2 @ Xs ) ) )
        | ? [Ys5: list_S5253404628753938141on_nat] :
            ( ( Ys2
              = ( cons_S6797538523646374167on_nat @ X2 @ Ys5 ) )
            & ( ( append3523812819975775986on_nat @ Ys5 @ Zs3 )
              = Xs ) ) ) ) ).

% append_eq_Cons_conv
thf(fact_406_append__eq__Cons__conv,axiom,
    ! [Ys2: list_P9127870862658512897on_nat,Zs3: list_P9127870862658512897on_nat,X2: produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat] :
      ( ( ( append2865821678206098582on_nat @ Ys2 @ Zs3 )
        = ( cons_P1582596238967110705on_nat @ X2 @ Xs ) )
      = ( ( ( Ys2 = nil_Pr8614093721167239297on_nat )
          & ( Zs3
            = ( cons_P1582596238967110705on_nat @ X2 @ Xs ) ) )
        | ? [Ys5: list_P9127870862658512897on_nat] :
            ( ( Ys2
              = ( cons_P1582596238967110705on_nat @ X2 @ Ys5 ) )
            & ( ( append2865821678206098582on_nat @ Ys5 @ Zs3 )
              = Xs ) ) ) ) ).

% append_eq_Cons_conv
thf(fact_407_append__eq__Cons__conv,axiom,
    ! [Ys2: list_nat,Zs3: list_nat,X2: nat,Xs: list_nat] :
      ( ( ( append_nat @ Ys2 @ Zs3 )
        = ( cons_nat @ X2 @ Xs ) )
      = ( ( ( Ys2 = nil_nat )
          & ( Zs3
            = ( cons_nat @ X2 @ Xs ) ) )
        | ? [Ys5: list_nat] :
            ( ( Ys2
              = ( cons_nat @ X2 @ Ys5 ) )
            & ( ( append_nat @ Ys5 @ Zs3 )
              = Xs ) ) ) ) ).

% append_eq_Cons_conv
thf(fact_408_append__eq__Cons__conv,axiom,
    ! [Ys2: list_l8999537460041914605on_nat,Zs3: list_l8999537460041914605on_nat,X2: list_S5253404628753938141on_nat,Xs: list_l8999537460041914605on_nat] :
      ( ( ( append6174841615030841602on_nat @ Ys2 @ Zs3 )
        = ( cons_l5857885737047467175on_nat @ X2 @ Xs ) )
      = ( ( ( Ys2 = nil_li5563851662744376407on_nat )
          & ( Zs3
            = ( cons_l5857885737047467175on_nat @ X2 @ Xs ) ) )
        | ? [Ys5: list_l8999537460041914605on_nat] :
            ( ( Ys2
              = ( cons_l5857885737047467175on_nat @ X2 @ Ys5 ) )
            & ( ( append6174841615030841602on_nat @ Ys5 @ Zs3 )
              = Xs ) ) ) ) ).

% append_eq_Cons_conv
thf(fact_409_append__eq__Cons__conv,axiom,
    ! [Ys2: list_l1726306699111145479on_nat,Zs3: list_l1726306699111145479on_nat,X2: list_P9127870862658512897on_nat,Xs: list_l1726306699111145479on_nat] :
      ( ( ( append3839213209832495004on_nat @ Ys2 @ Zs3 )
        = ( cons_l3028799188195208631on_nat @ X2 @ Xs ) )
      = ( ( ( Ys2 = nil_li2682182165152866823on_nat )
          & ( Zs3
            = ( cons_l3028799188195208631on_nat @ X2 @ Xs ) ) )
        | ? [Ys5: list_l1726306699111145479on_nat] :
            ( ( Ys2
              = ( cons_l3028799188195208631on_nat @ X2 @ Ys5 ) )
            & ( ( append3839213209832495004on_nat @ Ys5 @ Zs3 )
              = Xs ) ) ) ) ).

% append_eq_Cons_conv
thf(fact_410_rev__nonempty__induct,axiom,
    ! [Xs: list_a,P: list_a > $o] :
      ( ( Xs != nil_a )
     => ( ! [X3: a] : ( P @ ( cons_a @ X3 @ nil_a ) )
       => ( ! [X3: a,Xs3: list_a] :
              ( ( Xs3 != nil_a )
             => ( ( P @ Xs3 )
               => ( P @ ( append_a @ Xs3 @ ( cons_a @ X3 @ nil_a ) ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% rev_nonempty_induct
thf(fact_411_rev__nonempty__induct,axiom,
    ! [Xs: list_S5253404628753938141on_nat,P: list_S5253404628753938141on_nat > $o] :
      ( ( Xs != nil_Su3078906344068544711on_nat )
     => ( ! [X3: sum_su4907927656612468301on_nat] : ( P @ ( cons_S6797538523646374167on_nat @ X3 @ nil_Su3078906344068544711on_nat ) )
       => ( ! [X3: sum_su4907927656612468301on_nat,Xs3: list_S5253404628753938141on_nat] :
              ( ( Xs3 != nil_Su3078906344068544711on_nat )
             => ( ( P @ Xs3 )
               => ( P @ ( append3523812819975775986on_nat @ Xs3 @ ( cons_S6797538523646374167on_nat @ X3 @ nil_Su3078906344068544711on_nat ) ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% rev_nonempty_induct
thf(fact_412_rev__nonempty__induct,axiom,
    ! [Xs: list_P9127870862658512897on_nat,P: list_P9127870862658512897on_nat > $o] :
      ( ( Xs != nil_Pr8614093721167239297on_nat )
     => ( ! [X3: produc1313295348606329723on_nat] : ( P @ ( cons_P1582596238967110705on_nat @ X3 @ nil_Pr8614093721167239297on_nat ) )
       => ( ! [X3: produc1313295348606329723on_nat,Xs3: list_P9127870862658512897on_nat] :
              ( ( Xs3 != nil_Pr8614093721167239297on_nat )
             => ( ( P @ Xs3 )
               => ( P @ ( append2865821678206098582on_nat @ Xs3 @ ( cons_P1582596238967110705on_nat @ X3 @ nil_Pr8614093721167239297on_nat ) ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% rev_nonempty_induct
thf(fact_413_rev__nonempty__induct,axiom,
    ! [Xs: list_nat,P: list_nat > $o] :
      ( ( Xs != nil_nat )
     => ( ! [X3: nat] : ( P @ ( cons_nat @ X3 @ nil_nat ) )
       => ( ! [X3: nat,Xs3: list_nat] :
              ( ( Xs3 != nil_nat )
             => ( ( P @ Xs3 )
               => ( P @ ( append_nat @ Xs3 @ ( cons_nat @ X3 @ nil_nat ) ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% rev_nonempty_induct
thf(fact_414_rev__nonempty__induct,axiom,
    ! [Xs: list_l8999537460041914605on_nat,P: list_l8999537460041914605on_nat > $o] :
      ( ( Xs != nil_li5563851662744376407on_nat )
     => ( ! [X3: list_S5253404628753938141on_nat] : ( P @ ( cons_l5857885737047467175on_nat @ X3 @ nil_li5563851662744376407on_nat ) )
       => ( ! [X3: list_S5253404628753938141on_nat,Xs3: list_l8999537460041914605on_nat] :
              ( ( Xs3 != nil_li5563851662744376407on_nat )
             => ( ( P @ Xs3 )
               => ( P @ ( append6174841615030841602on_nat @ Xs3 @ ( cons_l5857885737047467175on_nat @ X3 @ nil_li5563851662744376407on_nat ) ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% rev_nonempty_induct
thf(fact_415_rev__nonempty__induct,axiom,
    ! [Xs: list_l1726306699111145479on_nat,P: list_l1726306699111145479on_nat > $o] :
      ( ( Xs != nil_li2682182165152866823on_nat )
     => ( ! [X3: list_P9127870862658512897on_nat] : ( P @ ( cons_l3028799188195208631on_nat @ X3 @ nil_li2682182165152866823on_nat ) )
       => ( ! [X3: list_P9127870862658512897on_nat,Xs3: list_l1726306699111145479on_nat] :
              ( ( Xs3 != nil_li2682182165152866823on_nat )
             => ( ( P @ Xs3 )
               => ( P @ ( append3839213209832495004on_nat @ Xs3 @ ( cons_l3028799188195208631on_nat @ X3 @ nil_li2682182165152866823on_nat ) ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% rev_nonempty_induct
thf(fact_416_mul__ext__impl_Ocases,axiom,
    ! [X2: produc9164743771328383783list_a] :
      ( ( X2
       != ( produc6837034575241423639list_a @ nil_a @ nil_a ) )
     => ( ! [V: a,Va: list_a] :
            ( X2
           != ( produc6837034575241423639list_a @ nil_a @ ( cons_a @ V @ Va ) ) )
       => ( ! [V: a,Va: list_a] :
              ( X2
             != ( produc6837034575241423639list_a @ ( cons_a @ V @ Va ) @ nil_a ) )
         => ~ ! [V: a,Va: list_a,Y2: a,Ys: list_a] :
                ( X2
               != ( produc6837034575241423639list_a @ ( cons_a @ V @ Va ) @ ( cons_a @ Y2 @ Ys ) ) ) ) ) ) ).

% mul_ext_impl.cases
thf(fact_417_mul__ext__impl_Ocases,axiom,
    ! [X2: produc5580075136493545115on_nat] :
      ( ( X2
       != ( produc2912653624306349395on_nat @ nil_Su3078906344068544711on_nat @ nil_Su3078906344068544711on_nat ) )
     => ( ! [V: sum_su4907927656612468301on_nat,Va: list_S5253404628753938141on_nat] :
            ( X2
           != ( produc2912653624306349395on_nat @ nil_Su3078906344068544711on_nat @ ( cons_S6797538523646374167on_nat @ V @ Va ) ) )
       => ( ! [V: sum_su4907927656612468301on_nat,Va: list_S5253404628753938141on_nat] :
              ( X2
             != ( produc2912653624306349395on_nat @ ( cons_S6797538523646374167on_nat @ V @ Va ) @ nil_Su3078906344068544711on_nat ) )
         => ~ ! [V: sum_su4907927656612468301on_nat,Va: list_S5253404628753938141on_nat,Y2: sum_su4907927656612468301on_nat,Ys: list_S5253404628753938141on_nat] :
                ( X2
               != ( produc2912653624306349395on_nat @ ( cons_S6797538523646374167on_nat @ V @ Va ) @ ( cons_S6797538523646374167on_nat @ Y2 @ Ys ) ) ) ) ) ) ).

% mul_ext_impl.cases
thf(fact_418_mul__ext__impl_Ocases,axiom,
    ! [X2: produc1714117091004681831on_nat] :
      ( ( X2
       != ( produc7152764266822961111on_nat @ nil_Pr8614093721167239297on_nat @ nil_Pr8614093721167239297on_nat ) )
     => ( ! [V: produc1313295348606329723on_nat,Va: list_P9127870862658512897on_nat] :
            ( X2
           != ( produc7152764266822961111on_nat @ nil_Pr8614093721167239297on_nat @ ( cons_P1582596238967110705on_nat @ V @ Va ) ) )
       => ( ! [V: produc1313295348606329723on_nat,Va: list_P9127870862658512897on_nat] :
              ( X2
             != ( produc7152764266822961111on_nat @ ( cons_P1582596238967110705on_nat @ V @ Va ) @ nil_Pr8614093721167239297on_nat ) )
         => ~ ! [V: produc1313295348606329723on_nat,Va: list_P9127870862658512897on_nat,Y2: produc1313295348606329723on_nat,Ys: list_P9127870862658512897on_nat] :
                ( X2
               != ( produc7152764266822961111on_nat @ ( cons_P1582596238967110705on_nat @ V @ Va ) @ ( cons_P1582596238967110705on_nat @ Y2 @ Ys ) ) ) ) ) ) ).

% mul_ext_impl.cases
thf(fact_419_mul__ext__impl_Ocases,axiom,
    ! [X2: produc1828647624359046049st_nat] :
      ( ( X2
       != ( produc2694037385005941721st_nat @ nil_nat @ nil_nat ) )
     => ( ! [V: nat,Va: list_nat] :
            ( X2
           != ( produc2694037385005941721st_nat @ nil_nat @ ( cons_nat @ V @ Va ) ) )
       => ( ! [V: nat,Va: list_nat] :
              ( X2
             != ( produc2694037385005941721st_nat @ ( cons_nat @ V @ Va ) @ nil_nat ) )
         => ~ ! [V: nat,Va: list_nat,Y2: nat,Ys: list_nat] :
                ( X2
               != ( produc2694037385005941721st_nat @ ( cons_nat @ V @ Va ) @ ( cons_nat @ Y2 @ Ys ) ) ) ) ) ) ).

% mul_ext_impl.cases
thf(fact_420_mul__ext__impl_Ocases,axiom,
    ! [X2: produc6098023079705189563on_nat] :
      ( ( X2
       != ( produc1341766448070154867on_nat @ nil_li5563851662744376407on_nat @ nil_li5563851662744376407on_nat ) )
     => ( ! [V: list_S5253404628753938141on_nat,Va: list_l8999537460041914605on_nat] :
            ( X2
           != ( produc1341766448070154867on_nat @ nil_li5563851662744376407on_nat @ ( cons_l5857885737047467175on_nat @ V @ Va ) ) )
       => ( ! [V: list_S5253404628753938141on_nat,Va: list_l8999537460041914605on_nat] :
              ( X2
             != ( produc1341766448070154867on_nat @ ( cons_l5857885737047467175on_nat @ V @ Va ) @ nil_li5563851662744376407on_nat ) )
         => ~ ! [V: list_S5253404628753938141on_nat,Va: list_l8999537460041914605on_nat,Y2: list_S5253404628753938141on_nat,Ys: list_l8999537460041914605on_nat] :
                ( X2
               != ( produc1341766448070154867on_nat @ ( cons_l5857885737047467175on_nat @ V @ Va ) @ ( cons_l5857885737047467175on_nat @ Y2 @ Ys ) ) ) ) ) ) ).

% mul_ext_impl.cases
thf(fact_421_mul__ext__impl_Ocases,axiom,
    ! [X2: produc2417963485646110695on_nat] :
      ( ( X2
       != ( produc4739740463051851607on_nat @ nil_li2682182165152866823on_nat @ nil_li2682182165152866823on_nat ) )
     => ( ! [V: list_P9127870862658512897on_nat,Va: list_l1726306699111145479on_nat] :
            ( X2
           != ( produc4739740463051851607on_nat @ nil_li2682182165152866823on_nat @ ( cons_l3028799188195208631on_nat @ V @ Va ) ) )
       => ( ! [V: list_P9127870862658512897on_nat,Va: list_l1726306699111145479on_nat] :
              ( X2
             != ( produc4739740463051851607on_nat @ ( cons_l3028799188195208631on_nat @ V @ Va ) @ nil_li2682182165152866823on_nat ) )
         => ~ ! [V: list_P9127870862658512897on_nat,Va: list_l1726306699111145479on_nat,Y2: list_P9127870862658512897on_nat,Ys: list_l1726306699111145479on_nat] :
                ( X2
               != ( produc4739740463051851607on_nat @ ( cons_l3028799188195208631on_nat @ V @ Va ) @ ( cons_l3028799188195208631on_nat @ Y2 @ Ys ) ) ) ) ) ) ).

% mul_ext_impl.cases
thf(fact_422_mult2__impl_Ocases,axiom,
    ! [X2: produc9164743771328383783list_a] :
      ( ( X2
       != ( produc6837034575241423639list_a @ nil_a @ nil_a ) )
     => ( ! [V: a,Va: list_a] :
            ( X2
           != ( produc6837034575241423639list_a @ ( cons_a @ V @ Va ) @ nil_a ) )
       => ( ! [V: a,Va: list_a] :
              ( X2
             != ( produc6837034575241423639list_a @ nil_a @ ( cons_a @ V @ Va ) ) )
         => ~ ! [X3: a,Xs3: list_a,V: a,Va: list_a] :
                ( X2
               != ( produc6837034575241423639list_a @ ( cons_a @ X3 @ Xs3 ) @ ( cons_a @ V @ Va ) ) ) ) ) ) ).

% mult2_impl.cases
thf(fact_423_mult2__impl_Ocases,axiom,
    ! [X2: produc5580075136493545115on_nat] :
      ( ( X2
       != ( produc2912653624306349395on_nat @ nil_Su3078906344068544711on_nat @ nil_Su3078906344068544711on_nat ) )
     => ( ! [V: sum_su4907927656612468301on_nat,Va: list_S5253404628753938141on_nat] :
            ( X2
           != ( produc2912653624306349395on_nat @ ( cons_S6797538523646374167on_nat @ V @ Va ) @ nil_Su3078906344068544711on_nat ) )
       => ( ! [V: sum_su4907927656612468301on_nat,Va: list_S5253404628753938141on_nat] :
              ( X2
             != ( produc2912653624306349395on_nat @ nil_Su3078906344068544711on_nat @ ( cons_S6797538523646374167on_nat @ V @ Va ) ) )
         => ~ ! [X3: sum_su4907927656612468301on_nat,Xs3: list_S5253404628753938141on_nat,V: sum_su4907927656612468301on_nat,Va: list_S5253404628753938141on_nat] :
                ( X2
               != ( produc2912653624306349395on_nat @ ( cons_S6797538523646374167on_nat @ X3 @ Xs3 ) @ ( cons_S6797538523646374167on_nat @ V @ Va ) ) ) ) ) ) ).

% mult2_impl.cases
thf(fact_424_mult2__impl_Ocases,axiom,
    ! [X2: produc1714117091004681831on_nat] :
      ( ( X2
       != ( produc7152764266822961111on_nat @ nil_Pr8614093721167239297on_nat @ nil_Pr8614093721167239297on_nat ) )
     => ( ! [V: produc1313295348606329723on_nat,Va: list_P9127870862658512897on_nat] :
            ( X2
           != ( produc7152764266822961111on_nat @ ( cons_P1582596238967110705on_nat @ V @ Va ) @ nil_Pr8614093721167239297on_nat ) )
       => ( ! [V: produc1313295348606329723on_nat,Va: list_P9127870862658512897on_nat] :
              ( X2
             != ( produc7152764266822961111on_nat @ nil_Pr8614093721167239297on_nat @ ( cons_P1582596238967110705on_nat @ V @ Va ) ) )
         => ~ ! [X3: produc1313295348606329723on_nat,Xs3: list_P9127870862658512897on_nat,V: produc1313295348606329723on_nat,Va: list_P9127870862658512897on_nat] :
                ( X2
               != ( produc7152764266822961111on_nat @ ( cons_P1582596238967110705on_nat @ X3 @ Xs3 ) @ ( cons_P1582596238967110705on_nat @ V @ Va ) ) ) ) ) ) ).

% mult2_impl.cases
thf(fact_425_mult2__impl_Ocases,axiom,
    ! [X2: produc1828647624359046049st_nat] :
      ( ( X2
       != ( produc2694037385005941721st_nat @ nil_nat @ nil_nat ) )
     => ( ! [V: nat,Va: list_nat] :
            ( X2
           != ( produc2694037385005941721st_nat @ ( cons_nat @ V @ Va ) @ nil_nat ) )
       => ( ! [V: nat,Va: list_nat] :
              ( X2
             != ( produc2694037385005941721st_nat @ nil_nat @ ( cons_nat @ V @ Va ) ) )
         => ~ ! [X3: nat,Xs3: list_nat,V: nat,Va: list_nat] :
                ( X2
               != ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_nat @ V @ Va ) ) ) ) ) ) ).

% mult2_impl.cases
thf(fact_426_mult2__impl_Ocases,axiom,
    ! [X2: produc6098023079705189563on_nat] :
      ( ( X2
       != ( produc1341766448070154867on_nat @ nil_li5563851662744376407on_nat @ nil_li5563851662744376407on_nat ) )
     => ( ! [V: list_S5253404628753938141on_nat,Va: list_l8999537460041914605on_nat] :
            ( X2
           != ( produc1341766448070154867on_nat @ ( cons_l5857885737047467175on_nat @ V @ Va ) @ nil_li5563851662744376407on_nat ) )
       => ( ! [V: list_S5253404628753938141on_nat,Va: list_l8999537460041914605on_nat] :
              ( X2
             != ( produc1341766448070154867on_nat @ nil_li5563851662744376407on_nat @ ( cons_l5857885737047467175on_nat @ V @ Va ) ) )
         => ~ ! [X3: list_S5253404628753938141on_nat,Xs3: list_l8999537460041914605on_nat,V: list_S5253404628753938141on_nat,Va: list_l8999537460041914605on_nat] :
                ( X2
               != ( produc1341766448070154867on_nat @ ( cons_l5857885737047467175on_nat @ X3 @ Xs3 ) @ ( cons_l5857885737047467175on_nat @ V @ Va ) ) ) ) ) ) ).

% mult2_impl.cases
thf(fact_427_mult2__impl_Ocases,axiom,
    ! [X2: produc2417963485646110695on_nat] :
      ( ( X2
       != ( produc4739740463051851607on_nat @ nil_li2682182165152866823on_nat @ nil_li2682182165152866823on_nat ) )
     => ( ! [V: list_P9127870862658512897on_nat,Va: list_l1726306699111145479on_nat] :
            ( X2
           != ( produc4739740463051851607on_nat @ ( cons_l3028799188195208631on_nat @ V @ Va ) @ nil_li2682182165152866823on_nat ) )
       => ( ! [V: list_P9127870862658512897on_nat,Va: list_l1726306699111145479on_nat] :
              ( X2
             != ( produc4739740463051851607on_nat @ nil_li2682182165152866823on_nat @ ( cons_l3028799188195208631on_nat @ V @ Va ) ) )
         => ~ ! [X3: list_P9127870862658512897on_nat,Xs3: list_l1726306699111145479on_nat,V: list_P9127870862658512897on_nat,Va: list_l1726306699111145479on_nat] :
                ( X2
               != ( produc4739740463051851607on_nat @ ( cons_l3028799188195208631on_nat @ X3 @ Xs3 ) @ ( cons_l3028799188195208631on_nat @ V @ Va ) ) ) ) ) ) ).

% mult2_impl.cases
thf(fact_428_Inl__Inr__False,axiom,
    ! [X2: produc27318731269267614tation,Y: nat] :
      ( ( sum_In3878554002080803550on_nat @ X2 )
     != ( sum_In255631617079920996tation @ Y ) ) ).

% Inl_Inr_False
thf(fact_429_Inr__Inl__False,axiom,
    ! [X2: nat,Y: produc27318731269267614tation] :
      ( ( sum_In255631617079920996tation @ X2 )
     != ( sum_In3878554002080803550on_nat @ Y ) ) ).

% Inr_Inl_False
thf(fact_430_delete__aux_Ocases,axiom,
    ! [X2: produc3922897021252936937on_nat] :
      ( ! [K: sum_su4907927656612468301on_nat] :
          ( X2
         != ( produc1755723437418862235on_nat @ K @ nil_Pr8614093721167239297on_nat ) )
     => ~ ! [K: sum_su4907927656612468301on_nat,K2: sum_su4907927656612468301on_nat,V: sum_su4907927656612468301on_nat,Xs3: list_P9127870862658512897on_nat] :
            ( X2
           != ( produc1755723437418862235on_nat @ K @ ( cons_P1582596238967110705on_nat @ ( produc6704844767664046899on_nat @ K2 @ V ) @ Xs3 ) ) ) ) ).

% delete_aux.cases
thf(fact_431_delete__aux_Ocases,axiom,
    ! [X2: produc6588776972112385165on_nat] :
      ( ! [K: list_S5253404628753938141on_nat] :
          ( X2
         != ( produc9043615093183382213on_nat @ K @ nil_Pr2927124501823088569on_nat ) )
     => ~ ! [K: list_S5253404628753938141on_nat,K2: list_S5253404628753938141on_nat,V: produc1714117091004681831on_nat,Xs3: list_P5508476027986415439on_nat] :
            ( X2
           != ( produc9043615093183382213on_nat @ K @ ( cons_P4518942764240009225on_nat @ ( produc4176415733022414961on_nat @ K2 @ V ) @ Xs3 ) ) ) ) ).

% delete_aux.cases
thf(fact_432_delete__aux_Ocases,axiom,
    ! [X2: produc5056679816642863195on_nat] :
      ( ! [K: list_S5253404628753938141on_nat] :
          ( X2
         != ( produc8252945450212337037on_nat @ K @ nil_Pr5986419849129872387on_nat ) )
     => ~ ! [K: list_S5253404628753938141on_nat,K2: list_S5253404628753938141on_nat,V: produc118098540238352959on_nat,Xs3: list_P6215481986061871491on_nat] :
            ( X2
           != ( produc8252945450212337037on_nat @ K @ ( cons_P8471348786669567923on_nat @ ( produc1545487600272464053on_nat @ K2 @ V ) @ Xs3 ) ) ) ) ).

% delete_aux.cases
thf(fact_433_delete__aux_Ocases,axiom,
    ! [X2: produc8472197452120411308at_nat] :
      ( ! [K: nat] :
          ( X2
         != ( produc6109913384486294878at_nat @ K @ nil_Pr5478986624290739719at_nat ) )
     => ~ ! [K: nat,K2: nat,V: nat,Xs3: list_P6011104703257516679at_nat] :
            ( X2
           != ( produc6109913384486294878at_nat @ K @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V ) @ Xs3 ) ) ) ) ).

% delete_aux.cases
thf(fact_434_delete__aux_Ocases,axiom,
    ! [X2: produc1458239053189343596at_nat] :
      ( ! [K: nat > nat] :
          ( X2
         != ( produc17425388850148510at_nat @ K @ nil_Pr2223394031645031670at_nat ) )
     => ~ ! [K: nat > nat,K2: nat > nat,V: nat,Xs3: list_P5366131564312172150at_nat] :
            ( X2
           != ( produc17425388850148510at_nat @ K @ ( cons_P4219629788700907686at_nat @ ( produc72220940542539688at_nat @ K2 @ V ) @ Xs3 ) ) ) ) ).

% delete_aux.cases
thf(fact_435_delete__aux_Ocases,axiom,
    ! [X2: produc573162381751850459tation] :
      ( ! [K: a] :
          ( X2
         != ( produc511914028911482581tation @ K @ nil_Pr4033316107961887256tation ) )
     => ~ ! [K: a,K2: a,V: multis8755981257395730801tation,Xs3: list_P7561118020193646638tation] :
            ( X2
           != ( produc511914028911482581tation @ K @ ( cons_P3184545829724658280tation @ ( produc7247797765490398872tation @ K2 @ V ) @ Xs3 ) ) ) ) ).

% delete_aux.cases
thf(fact_436_prod__cases4,axiom,
    ! [Y: produc8325617224762090749on_nat] :
      ~ ! [A3: list_S5253404628753938141on_nat,B3: list_S5253404628753938141on_nat,C2: list_P9127870862658512897on_nat,D: list_P9127870862658512897on_nat] :
          ( Y
         != ( produc1545487600272464053on_nat @ A3 @ ( produc4176415733022414961on_nat @ B3 @ ( produc7152764266822961111on_nat @ C2 @ D ) ) ) ) ).

% prod_cases4
thf(fact_437_prod__induct4,axiom,
    ! [P: produc8325617224762090749on_nat > $o,X2: produc8325617224762090749on_nat] :
      ( ! [A3: list_S5253404628753938141on_nat,B3: list_S5253404628753938141on_nat,C2: list_P9127870862658512897on_nat,D: list_P9127870862658512897on_nat] : ( P @ ( produc1545487600272464053on_nat @ A3 @ ( produc4176415733022414961on_nat @ B3 @ ( produc7152764266822961111on_nat @ C2 @ D ) ) ) )
     => ( P @ X2 ) ) ).

% prod_induct4
thf(fact_438_old_Oprod_Oinject,axiom,
    ! [A: list_S5253404628753938141on_nat,B: produc1714117091004681831on_nat,A2: list_S5253404628753938141on_nat,B2: produc1714117091004681831on_nat] :
      ( ( ( produc4176415733022414961on_nat @ A @ B )
        = ( produc4176415733022414961on_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_439_old_Oprod_Oinject,axiom,
    ! [A: list_S5253404628753938141on_nat,B: produc118098540238352959on_nat,A2: list_S5253404628753938141on_nat,B2: produc118098540238352959on_nat] :
      ( ( ( produc1545487600272464053on_nat @ A @ B )
        = ( produc1545487600272464053on_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_440_old_Oprod_Oinject,axiom,
    ! [A: nat,B: nat,A2: nat,B2: nat] :
      ( ( ( product_Pair_nat_nat @ A @ B )
        = ( product_Pair_nat_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_441_old_Oprod_Oinject,axiom,
    ! [A: nat > nat,B: nat,A2: nat > nat,B2: nat] :
      ( ( ( produc72220940542539688at_nat @ A @ B )
        = ( produc72220940542539688at_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_442_old_Oprod_Oinject,axiom,
    ! [A: a,B: multis8755981257395730801tation,A2: a,B2: multis8755981257395730801tation] :
      ( ( ( produc7247797765490398872tation @ A @ B )
        = ( produc7247797765490398872tation @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_443_prod_Oinject,axiom,
    ! [X1: list_S5253404628753938141on_nat,X23: produc1714117091004681831on_nat,Y1: list_S5253404628753938141on_nat,Y23: produc1714117091004681831on_nat] :
      ( ( ( produc4176415733022414961on_nat @ X1 @ X23 )
        = ( produc4176415733022414961on_nat @ Y1 @ Y23 ) )
      = ( ( X1 = Y1 )
        & ( X23 = Y23 ) ) ) ).

% prod.inject
thf(fact_444_prod_Oinject,axiom,
    ! [X1: list_S5253404628753938141on_nat,X23: produc118098540238352959on_nat,Y1: list_S5253404628753938141on_nat,Y23: produc118098540238352959on_nat] :
      ( ( ( produc1545487600272464053on_nat @ X1 @ X23 )
        = ( produc1545487600272464053on_nat @ Y1 @ Y23 ) )
      = ( ( X1 = Y1 )
        & ( X23 = Y23 ) ) ) ).

% prod.inject
thf(fact_445_prod_Oinject,axiom,
    ! [X1: nat,X23: nat,Y1: nat,Y23: nat] :
      ( ( ( product_Pair_nat_nat @ X1 @ X23 )
        = ( product_Pair_nat_nat @ Y1 @ Y23 ) )
      = ( ( X1 = Y1 )
        & ( X23 = Y23 ) ) ) ).

% prod.inject
thf(fact_446_prod_Oinject,axiom,
    ! [X1: nat > nat,X23: nat,Y1: nat > nat,Y23: nat] :
      ( ( ( produc72220940542539688at_nat @ X1 @ X23 )
        = ( produc72220940542539688at_nat @ Y1 @ Y23 ) )
      = ( ( X1 = Y1 )
        & ( X23 = Y23 ) ) ) ).

% prod.inject
thf(fact_447_prod_Oinject,axiom,
    ! [X1: a,X23: multis8755981257395730801tation,Y1: a,Y23: multis8755981257395730801tation] :
      ( ( ( produc7247797765490398872tation @ X1 @ X23 )
        = ( produc7247797765490398872tation @ Y1 @ Y23 ) )
      = ( ( X1 = Y1 )
        & ( X23 = Y23 ) ) ) ).

% prod.inject
thf(fact_448_mult2__ex__dom0_Ocases,axiom,
    ! [X2: produc2716792995815215274st_a_o] :
      ( ! [X3: a,Xs3: list_a,Ys4: list_a,B3: $o] :
          ( X2
         != ( produc4030046113903262746st_a_o @ X3 @ ( produc6327927755425424113st_a_o @ Xs3 @ ( produc5660821784244531022st_a_o @ nil_a @ ( produc4242187413542061611st_a_o @ Ys4 @ B3 ) ) ) ) )
     => ~ ! [X3: a,Xs3: list_a,Y2: a,Ys: list_a,Ys4: list_a,B3: $o] :
            ( X2
           != ( produc4030046113903262746st_a_o @ X3 @ ( produc6327927755425424113st_a_o @ Xs3 @ ( produc5660821784244531022st_a_o @ ( cons_a @ Y2 @ Ys ) @ ( produc4242187413542061611st_a_o @ Ys4 @ B3 ) ) ) ) ) ) ).

% mult2_ex_dom0.cases
thf(fact_449_mult2__ex__dom0_Ocases,axiom,
    ! [X2: produc8288325133855054004_nat_o] :
      ( ! [X3: sum_su4907927656612468301on_nat,Xs3: list_S5253404628753938141on_nat,Ys4: list_S5253404628753938141on_nat,B3: $o] :
          ( X2
         != ( produc1244398239520053868_nat_o @ X3 @ ( produc5518819433104342328_nat_o @ Xs3 @ ( produc4670265488131411942_nat_o @ nil_Su3078906344068544711on_nat @ ( produc1279791631502317730_nat_o @ Ys4 @ B3 ) ) ) ) )
     => ~ ! [X3: sum_su4907927656612468301on_nat,Xs3: list_S5253404628753938141on_nat,Y2: sum_su4907927656612468301on_nat,Ys: list_S5253404628753938141on_nat,Ys4: list_S5253404628753938141on_nat,B3: $o] :
            ( X2
           != ( produc1244398239520053868_nat_o @ X3 @ ( produc5518819433104342328_nat_o @ Xs3 @ ( produc4670265488131411942_nat_o @ ( cons_S6797538523646374167on_nat @ Y2 @ Ys ) @ ( produc1279791631502317730_nat_o @ Ys4 @ B3 ) ) ) ) ) ) ).

% mult2_ex_dom0.cases
thf(fact_450_mult2__ex__dom0_Ocases,axiom,
    ! [X2: produc7775502106177927606_nat_o] :
      ( ! [X3: produc1313295348606329723on_nat,Xs3: list_P9127870862658512897on_nat,Ys4: list_P9127870862658512897on_nat,B3: $o] :
          ( X2
         != ( produc5930017123893236134_nat_o @ X3 @ ( produc2695279325889729082_nat_o @ Xs3 @ ( produc6066200345829616596_nat_o @ nil_Pr8614093721167239297on_nat @ ( produc3679803167814765678_nat_o @ Ys4 @ B3 ) ) ) ) )
     => ~ ! [X3: produc1313295348606329723on_nat,Xs3: list_P9127870862658512897on_nat,Y2: produc1313295348606329723on_nat,Ys: list_P9127870862658512897on_nat,Ys4: list_P9127870862658512897on_nat,B3: $o] :
            ( X2
           != ( produc5930017123893236134_nat_o @ X3 @ ( produc2695279325889729082_nat_o @ Xs3 @ ( produc6066200345829616596_nat_o @ ( cons_P1582596238967110705on_nat @ Y2 @ Ys ) @ ( produc3679803167814765678_nat_o @ Ys4 @ B3 ) ) ) ) ) ) ).

% mult2_ex_dom0.cases
thf(fact_451_mult2__ex__dom0_Ocases,axiom,
    ! [X2: produc3630873615015234996_nat_o] :
      ( ! [X3: nat,Xs3: list_nat,Ys4: list_nat,B3: $o] :
          ( X2
         != ( produc6858984136714788332_nat_o @ X3 @ ( produc2959919240848031157_nat_o @ Xs3 @ ( produc5973304058656726566_nat_o @ nil_nat @ ( produc1747790020158526943_nat_o @ Ys4 @ B3 ) ) ) ) )
     => ~ ! [X3: nat,Xs3: list_nat,Y2: nat,Ys: list_nat,Ys4: list_nat,B3: $o] :
            ( X2
           != ( produc6858984136714788332_nat_o @ X3 @ ( produc2959919240848031157_nat_o @ Xs3 @ ( produc5973304058656726566_nat_o @ ( cons_nat @ Y2 @ Ys ) @ ( produc1747790020158526943_nat_o @ Ys4 @ B3 ) ) ) ) ) ) ).

% mult2_ex_dom0.cases
thf(fact_452_mult2__ex__dom0_Ocases,axiom,
    ! [X2: produc1666888912113777716_nat_o] :
      ( ! [X3: list_S5253404628753938141on_nat,Xs3: list_l8999537460041914605on_nat,Ys4: list_l8999537460041914605on_nat,B3: $o] :
          ( X2
         != ( produc46917116878602732_nat_o @ X3 @ ( produc2951815235169433512_nat_o @ Xs3 @ ( produc5038552294322009446_nat_o @ nil_li5563851662744376407on_nat @ ( produc1964031788423343762_nat_o @ Ys4 @ B3 ) ) ) ) )
     => ~ ! [X3: list_S5253404628753938141on_nat,Xs3: list_l8999537460041914605on_nat,Y2: list_S5253404628753938141on_nat,Ys: list_l8999537460041914605on_nat,Ys4: list_l8999537460041914605on_nat,B3: $o] :
            ( X2
           != ( produc46917116878602732_nat_o @ X3 @ ( produc2951815235169433512_nat_o @ Xs3 @ ( produc5038552294322009446_nat_o @ ( cons_l5857885737047467175on_nat @ Y2 @ Ys ) @ ( produc1964031788423343762_nat_o @ Ys4 @ B3 ) ) ) ) ) ) ).

% mult2_ex_dom0.cases
thf(fact_453_mult2__ex__dom0_Ocases,axiom,
    ! [X2: produc9011176785203808030_nat_o] :
      ( ! [X3: list_P9127870862658512897on_nat,Xs3: list_l1726306699111145479on_nat,Ys4: list_l1726306699111145479on_nat,B3: $o] :
          ( X2
         != ( produc7597291304841350670_nat_o @ X3 @ ( produc7274044877351478568_nat_o @ Xs3 @ ( produc2691072593788627016_nat_o @ nil_li2682182165152866823on_nat @ ( produc715673817563144936_nat_o @ Ys4 @ B3 ) ) ) ) )
     => ~ ! [X3: list_P9127870862658512897on_nat,Xs3: list_l1726306699111145479on_nat,Y2: list_P9127870862658512897on_nat,Ys: list_l1726306699111145479on_nat,Ys4: list_l1726306699111145479on_nat,B3: $o] :
            ( X2
           != ( produc7597291304841350670_nat_o @ X3 @ ( produc7274044877351478568_nat_o @ Xs3 @ ( produc2691072593788627016_nat_o @ ( cons_l3028799188195208631on_nat @ Y2 @ Ys ) @ ( produc715673817563144936_nat_o @ Ys4 @ B3 ) ) ) ) ) ) ).

% mult2_ex_dom0.cases
thf(fact_454_map__default_Ocases,axiom,
    ! [X2: produc7012556527198387091on_nat] :
      ( ! [K: sum_su4907927656612468301on_nat,V: sum_su4907927656612468301on_nat,F2: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat] :
          ( X2
         != ( produc6662423293526568773on_nat @ K @ ( produc3053687648904799203on_nat @ V @ ( produc6419464700804365359on_nat @ F2 @ nil_Pr8614093721167239297on_nat ) ) ) )
     => ~ ! [K: sum_su4907927656612468301on_nat,V: sum_su4907927656612468301on_nat,F2: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat,P5: produc1313295348606329723on_nat,Ps: list_P9127870862658512897on_nat] :
            ( X2
           != ( produc6662423293526568773on_nat @ K @ ( produc3053687648904799203on_nat @ V @ ( produc6419464700804365359on_nat @ F2 @ ( cons_P1582596238967110705on_nat @ P5 @ Ps ) ) ) ) ) ) ).

% map_default.cases
thf(fact_455_mult2__impl0_Ocases,axiom,
    ! [X2: produc6745688853455182174st_a_o] :
      ( ! [B3: $o] :
          ( X2
         != ( produc5660821784244531022st_a_o @ nil_a @ ( produc4242187413542061611st_a_o @ nil_a @ B3 ) ) )
     => ( ! [V: a,Va: list_a,B3: $o] :
            ( X2
           != ( produc5660821784244531022st_a_o @ ( cons_a @ V @ Va ) @ ( produc4242187413542061611st_a_o @ nil_a @ B3 ) ) )
       => ( ! [V: a,Va: list_a,B3: $o] :
              ( X2
             != ( produc5660821784244531022st_a_o @ nil_a @ ( produc4242187413542061611st_a_o @ ( cons_a @ V @ Va ) @ B3 ) ) )
         => ~ ! [X3: a,Xs3: list_a,V: a,Va: list_a,B3: $o] :
                ( X2
               != ( produc5660821784244531022st_a_o @ ( cons_a @ X3 @ Xs3 ) @ ( produc4242187413542061611st_a_o @ ( cons_a @ V @ Va ) @ B3 ) ) ) ) ) ) ).

% mult2_impl0.cases
thf(fact_456_mult2__impl0_Ocases,axiom,
    ! [X2: produc3152382218381312302_nat_o] :
      ( ! [B3: $o] :
          ( X2
         != ( produc4670265488131411942_nat_o @ nil_Su3078906344068544711on_nat @ ( produc1279791631502317730_nat_o @ nil_Su3078906344068544711on_nat @ B3 ) ) )
     => ( ! [V: sum_su4907927656612468301on_nat,Va: list_S5253404628753938141on_nat,B3: $o] :
            ( X2
           != ( produc4670265488131411942_nat_o @ ( cons_S6797538523646374167on_nat @ V @ Va ) @ ( produc1279791631502317730_nat_o @ nil_Su3078906344068544711on_nat @ B3 ) ) )
       => ( ! [V: sum_su4907927656612468301on_nat,Va: list_S5253404628753938141on_nat,B3: $o] :
              ( X2
             != ( produc4670265488131411942_nat_o @ nil_Su3078906344068544711on_nat @ ( produc1279791631502317730_nat_o @ ( cons_S6797538523646374167on_nat @ V @ Va ) @ B3 ) ) )
         => ~ ! [X3: sum_su4907927656612468301on_nat,Xs3: list_S5253404628753938141on_nat,V: sum_su4907927656612468301on_nat,Va: list_S5253404628753938141on_nat,B3: $o] :
                ( X2
               != ( produc4670265488131411942_nat_o @ ( cons_S6797538523646374167on_nat @ X3 @ Xs3 ) @ ( produc1279791631502317730_nat_o @ ( cons_S6797538523646374167on_nat @ V @ Va ) @ B3 ) ) ) ) ) ) ).

% mult2_impl0.cases
thf(fact_457_mult2__impl0_Ocases,axiom,
    ! [X2: produc4640512942233155940_nat_o] :
      ( ! [B3: $o] :
          ( X2
         != ( produc6066200345829616596_nat_o @ nil_Pr8614093721167239297on_nat @ ( produc3679803167814765678_nat_o @ nil_Pr8614093721167239297on_nat @ B3 ) ) )
     => ( ! [V: produc1313295348606329723on_nat,Va: list_P9127870862658512897on_nat,B3: $o] :
            ( X2
           != ( produc6066200345829616596_nat_o @ ( cons_P1582596238967110705on_nat @ V @ Va ) @ ( produc3679803167814765678_nat_o @ nil_Pr8614093721167239297on_nat @ B3 ) ) )
       => ( ! [V: produc1313295348606329723on_nat,Va: list_P9127870862658512897on_nat,B3: $o] :
              ( X2
             != ( produc6066200345829616596_nat_o @ nil_Pr8614093721167239297on_nat @ ( produc3679803167814765678_nat_o @ ( cons_P1582596238967110705on_nat @ V @ Va ) @ B3 ) ) )
         => ~ ! [X3: produc1313295348606329723on_nat,Xs3: list_P9127870862658512897on_nat,V: produc1313295348606329723on_nat,Va: list_P9127870862658512897on_nat,B3: $o] :
                ( X2
               != ( produc6066200345829616596_nat_o @ ( cons_P1582596238967110705on_nat @ X3 @ Xs3 ) @ ( produc3679803167814765678_nat_o @ ( cons_P1582596238967110705on_nat @ V @ Va ) @ B3 ) ) ) ) ) ) ).

% mult2_impl0.cases
thf(fact_458_mult2__impl0_Ocases,axiom,
    ! [X2: produc430777597369845998_nat_o] :
      ( ! [B3: $o] :
          ( X2
         != ( produc5973304058656726566_nat_o @ nil_nat @ ( produc1747790020158526943_nat_o @ nil_nat @ B3 ) ) )
     => ( ! [V: nat,Va: list_nat,B3: $o] :
            ( X2
           != ( produc5973304058656726566_nat_o @ ( cons_nat @ V @ Va ) @ ( produc1747790020158526943_nat_o @ nil_nat @ B3 ) ) )
       => ( ! [V: nat,Va: list_nat,B3: $o] :
              ( X2
             != ( produc5973304058656726566_nat_o @ nil_nat @ ( produc1747790020158526943_nat_o @ ( cons_nat @ V @ Va ) @ B3 ) ) )
         => ~ ! [X3: nat,Xs3: list_nat,V: nat,Va: list_nat,B3: $o] :
                ( X2
               != ( produc5973304058656726566_nat_o @ ( cons_nat @ X3 @ Xs3 ) @ ( produc1747790020158526943_nat_o @ ( cons_nat @ V @ Va ) @ B3 ) ) ) ) ) ) ).

% mult2_impl0.cases
thf(fact_459_mult2__impl0_Ocases,axiom,
    ! [X2: produc1737863824307799982_nat_o] :
      ( ! [B3: $o] :
          ( X2
         != ( produc5038552294322009446_nat_o @ nil_li5563851662744376407on_nat @ ( produc1964031788423343762_nat_o @ nil_li5563851662744376407on_nat @ B3 ) ) )
     => ( ! [V: list_S5253404628753938141on_nat,Va: list_l8999537460041914605on_nat,B3: $o] :
            ( X2
           != ( produc5038552294322009446_nat_o @ ( cons_l5857885737047467175on_nat @ V @ Va ) @ ( produc1964031788423343762_nat_o @ nil_li5563851662744376407on_nat @ B3 ) ) )
       => ( ! [V: list_S5253404628753938141on_nat,Va: list_l8999537460041914605on_nat,B3: $o] :
              ( X2
             != ( produc5038552294322009446_nat_o @ nil_li5563851662744376407on_nat @ ( produc1964031788423343762_nat_o @ ( cons_l5857885737047467175on_nat @ V @ Va ) @ B3 ) ) )
         => ~ ! [X3: list_S5253404628753938141on_nat,Xs3: list_l8999537460041914605on_nat,V: list_S5253404628753938141on_nat,Va: list_l8999537460041914605on_nat,B3: $o] :
                ( X2
               != ( produc5038552294322009446_nat_o @ ( cons_l5857885737047467175on_nat @ X3 @ Xs3 ) @ ( produc1964031788423343762_nat_o @ ( cons_l5857885737047467175on_nat @ V @ Va ) @ B3 ) ) ) ) ) ) ).

% mult2_impl0.cases
thf(fact_460_mult2__impl0_Ocases,axiom,
    ! [X2: produc7496903103639383512_nat_o] :
      ( ! [B3: $o] :
          ( X2
         != ( produc2691072593788627016_nat_o @ nil_li2682182165152866823on_nat @ ( produc715673817563144936_nat_o @ nil_li2682182165152866823on_nat @ B3 ) ) )
     => ( ! [V: list_P9127870862658512897on_nat,Va: list_l1726306699111145479on_nat,B3: $o] :
            ( X2
           != ( produc2691072593788627016_nat_o @ ( cons_l3028799188195208631on_nat @ V @ Va ) @ ( produc715673817563144936_nat_o @ nil_li2682182165152866823on_nat @ B3 ) ) )
       => ( ! [V: list_P9127870862658512897on_nat,Va: list_l1726306699111145479on_nat,B3: $o] :
              ( X2
             != ( produc2691072593788627016_nat_o @ nil_li2682182165152866823on_nat @ ( produc715673817563144936_nat_o @ ( cons_l3028799188195208631on_nat @ V @ Va ) @ B3 ) ) )
         => ~ ! [X3: list_P9127870862658512897on_nat,Xs3: list_l1726306699111145479on_nat,V: list_P9127870862658512897on_nat,Va: list_l1726306699111145479on_nat,B3: $o] :
                ( X2
               != ( produc2691072593788627016_nat_o @ ( cons_l3028799188195208631on_nat @ X3 @ Xs3 ) @ ( produc715673817563144936_nat_o @ ( cons_l3028799188195208631on_nat @ V @ Va ) @ B3 ) ) ) ) ) ) ).

% mult2_impl0.cases
thf(fact_461_map__entry_Ocases,axiom,
    ! [X2: produc1098090131618875051on_nat] :
      ( ! [K: sum_su4907927656612468301on_nat,F2: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat] :
          ( X2
         != ( produc3053687648904799203on_nat @ K @ ( produc6419464700804365359on_nat @ F2 @ nil_Pr8614093721167239297on_nat ) ) )
     => ~ ! [K: sum_su4907927656612468301on_nat,F2: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat,P5: produc1313295348606329723on_nat,Ps: list_P9127870862658512897on_nat] :
            ( X2
           != ( produc3053687648904799203on_nat @ K @ ( produc6419464700804365359on_nat @ F2 @ ( cons_P1582596238967110705on_nat @ P5 @ Ps ) ) ) ) ) ).

% map_entry.cases
thf(fact_462_Pair__inject,axiom,
    ! [A: list_S5253404628753938141on_nat,B: produc1714117091004681831on_nat,A2: list_S5253404628753938141on_nat,B2: produc1714117091004681831on_nat] :
      ( ( ( produc4176415733022414961on_nat @ A @ B )
        = ( produc4176415733022414961on_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_463_Pair__inject,axiom,
    ! [A: list_S5253404628753938141on_nat,B: produc118098540238352959on_nat,A2: list_S5253404628753938141on_nat,B2: produc118098540238352959on_nat] :
      ( ( ( produc1545487600272464053on_nat @ A @ B )
        = ( produc1545487600272464053on_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_464_Pair__inject,axiom,
    ! [A: nat,B: nat,A2: nat,B2: nat] :
      ( ( ( product_Pair_nat_nat @ A @ B )
        = ( product_Pair_nat_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_465_Pair__inject,axiom,
    ! [A: nat > nat,B: nat,A2: nat > nat,B2: nat] :
      ( ( ( produc72220940542539688at_nat @ A @ B )
        = ( produc72220940542539688at_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_466_Pair__inject,axiom,
    ! [A: a,B: multis8755981257395730801tation,A2: a,B2: multis8755981257395730801tation] :
      ( ( ( produc7247797765490398872tation @ A @ B )
        = ( produc7247797765490398872tation @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_467_prod__cases,axiom,
    ! [P: produc118098540238352959on_nat > $o,P6: produc118098540238352959on_nat] :
      ( ! [A3: list_S5253404628753938141on_nat,B3: produc1714117091004681831on_nat] : ( P @ ( produc4176415733022414961on_nat @ A3 @ B3 ) )
     => ( P @ P6 ) ) ).

% prod_cases
thf(fact_468_prod__cases,axiom,
    ! [P: produc8325617224762090749on_nat > $o,P6: produc8325617224762090749on_nat] :
      ( ! [A3: list_S5253404628753938141on_nat,B3: produc118098540238352959on_nat] : ( P @ ( produc1545487600272464053on_nat @ A3 @ B3 ) )
     => ( P @ P6 ) ) ).

% prod_cases
thf(fact_469_prod__cases,axiom,
    ! [P: product_prod_nat_nat > $o,P6: product_prod_nat_nat] :
      ( ! [A3: nat,B3: nat] : ( P @ ( product_Pair_nat_nat @ A3 @ B3 ) )
     => ( P @ P6 ) ) ).

% prod_cases
thf(fact_470_prod__cases,axiom,
    ! [P: produc8199716216217303280at_nat > $o,P6: produc8199716216217303280at_nat] :
      ( ! [A3: nat > nat,B3: nat] : ( P @ ( produc72220940542539688at_nat @ A3 @ B3 ) )
     => ( P @ P6 ) ) ).

% prod_cases
thf(fact_471_prod__cases,axiom,
    ! [P: produc27318731269267614tation > $o,P6: produc27318731269267614tation] :
      ( ! [A3: a,B3: multis8755981257395730801tation] : ( P @ ( produc7247797765490398872tation @ A3 @ B3 ) )
     => ( P @ P6 ) ) ).

% prod_cases
thf(fact_472_surj__pair,axiom,
    ! [P6: produc118098540238352959on_nat] :
    ? [X3: list_S5253404628753938141on_nat,Y2: produc1714117091004681831on_nat] :
      ( P6
      = ( produc4176415733022414961on_nat @ X3 @ Y2 ) ) ).

% surj_pair
thf(fact_473_surj__pair,axiom,
    ! [P6: produc8325617224762090749on_nat] :
    ? [X3: list_S5253404628753938141on_nat,Y2: produc118098540238352959on_nat] :
      ( P6
      = ( produc1545487600272464053on_nat @ X3 @ Y2 ) ) ).

% surj_pair
thf(fact_474_surj__pair,axiom,
    ! [P6: product_prod_nat_nat] :
    ? [X3: nat,Y2: nat] :
      ( P6
      = ( product_Pair_nat_nat @ X3 @ Y2 ) ) ).

% surj_pair
thf(fact_475_surj__pair,axiom,
    ! [P6: produc8199716216217303280at_nat] :
    ? [X3: nat > nat,Y2: nat] :
      ( P6
      = ( produc72220940542539688at_nat @ X3 @ Y2 ) ) ).

% surj_pair
thf(fact_476_surj__pair,axiom,
    ! [P6: produc27318731269267614tation] :
    ? [X3: a,Y2: multis8755981257395730801tation] :
      ( P6
      = ( produc7247797765490398872tation @ X3 @ Y2 ) ) ).

% surj_pair
thf(fact_477_old_Oprod_Oexhaust,axiom,
    ! [Y: produc118098540238352959on_nat] :
      ~ ! [A3: list_S5253404628753938141on_nat,B3: produc1714117091004681831on_nat] :
          ( Y
         != ( produc4176415733022414961on_nat @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_478_old_Oprod_Oexhaust,axiom,
    ! [Y: produc8325617224762090749on_nat] :
      ~ ! [A3: list_S5253404628753938141on_nat,B3: produc118098540238352959on_nat] :
          ( Y
         != ( produc1545487600272464053on_nat @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_479_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_nat_nat] :
      ~ ! [A3: nat,B3: nat] :
          ( Y
         != ( product_Pair_nat_nat @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_480_old_Oprod_Oexhaust,axiom,
    ! [Y: produc8199716216217303280at_nat] :
      ~ ! [A3: nat > nat,B3: nat] :
          ( Y
         != ( produc72220940542539688at_nat @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_481_old_Oprod_Oexhaust,axiom,
    ! [Y: produc27318731269267614tation] :
      ~ ! [A3: a,B3: multis8755981257395730801tation] :
          ( Y
         != ( produc7247797765490398872tation @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_482_ms__elem__of__lit_Osimps_I1_J,axiom,
    ! [X2: a] :
      ( ( multis8230369766971345672_lit_a @ ( product_Pair_a_o @ X2 @ $true ) )
      = ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ X2 @ multis8267843437418295310sitive ) ) ) ).

% ms_elem_of_lit.simps(1)
thf(fact_483_ms__elem__of__lit_Osimps_I2_J,axiom,
    ! [X2: a] :
      ( ( multis8230369766971345672_lit_a @ ( product_Pair_a_o @ X2 @ $false ) )
      = ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ X2 @ multis5977116398950053002gative ) ) ) ).

% ms_elem_of_lit.simps(2)
thf(fact_484_ms__elem__of__lit_Oelims,axiom,
    ! [X2: product_prod_a_o,Y: sum_su4907927656612468301on_nat] :
      ( ( ( multis8230369766971345672_lit_a @ X2 )
        = Y )
     => ( ! [X3: a] :
            ( ( X2
              = ( product_Pair_a_o @ X3 @ $true ) )
           => ( Y
             != ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ X3 @ multis8267843437418295310sitive ) ) ) )
       => ~ ! [X3: a] :
              ( ( X2
                = ( product_Pair_a_o @ X3 @ $false ) )
             => ( Y
               != ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ X3 @ multis5977116398950053002gative ) ) ) ) ) ) ).

% ms_elem_of_lit.elims
thf(fact_485_prod__induct3,axiom,
    ! [P: produc118098540238352959on_nat > $o,X2: produc118098540238352959on_nat] :
      ( ! [A3: list_S5253404628753938141on_nat,B3: list_P9127870862658512897on_nat,C2: list_P9127870862658512897on_nat] : ( P @ ( produc4176415733022414961on_nat @ A3 @ ( produc7152764266822961111on_nat @ B3 @ C2 ) ) )
     => ( P @ X2 ) ) ).

% prod_induct3
thf(fact_486_prod__induct3,axiom,
    ! [P: produc8325617224762090749on_nat > $o,X2: produc8325617224762090749on_nat] :
      ( ! [A3: list_S5253404628753938141on_nat,B3: list_S5253404628753938141on_nat,C2: produc1714117091004681831on_nat] : ( P @ ( produc1545487600272464053on_nat @ A3 @ ( produc4176415733022414961on_nat @ B3 @ C2 ) ) )
     => ( P @ X2 ) ) ).

% prod_induct3
thf(fact_487_prod__cases3,axiom,
    ! [Y: produc118098540238352959on_nat] :
      ~ ! [A3: list_S5253404628753938141on_nat,B3: list_P9127870862658512897on_nat,C2: list_P9127870862658512897on_nat] :
          ( Y
         != ( produc4176415733022414961on_nat @ A3 @ ( produc7152764266822961111on_nat @ B3 @ C2 ) ) ) ).

% prod_cases3
thf(fact_488_prod__cases3,axiom,
    ! [Y: produc8325617224762090749on_nat] :
      ~ ! [A3: list_S5253404628753938141on_nat,B3: list_S5253404628753938141on_nat,C2: produc1714117091004681831on_nat] :
          ( Y
         != ( produc1545487600272464053on_nat @ A3 @ ( produc4176415733022414961on_nat @ B3 @ C2 ) ) ) ).

% prod_cases3
thf(fact_489_clearjunk_Ocases,axiom,
    ! [X2: list_P9127870862658512897on_nat] :
      ( ( X2 != nil_Pr8614093721167239297on_nat )
     => ~ ! [P5: produc1313295348606329723on_nat,Ps: list_P9127870862658512897on_nat] :
            ( X2
           != ( cons_P1582596238967110705on_nat @ P5 @ Ps ) ) ) ).

% clearjunk.cases
thf(fact_490_subset__eq__mset__impl_Ocases,axiom,
    ! [X2: produc9164743771328383783list_a] :
      ( ! [Ys: list_a] :
          ( X2
         != ( produc6837034575241423639list_a @ nil_a @ Ys ) )
     => ~ ! [X3: a,Xs3: list_a,Ys: list_a] :
            ( X2
           != ( produc6837034575241423639list_a @ ( cons_a @ X3 @ Xs3 ) @ Ys ) ) ) ).

% subset_eq_mset_impl.cases
thf(fact_491_subset__eq__mset__impl_Ocases,axiom,
    ! [X2: produc5580075136493545115on_nat] :
      ( ! [Ys: list_S5253404628753938141on_nat] :
          ( X2
         != ( produc2912653624306349395on_nat @ nil_Su3078906344068544711on_nat @ Ys ) )
     => ~ ! [X3: sum_su4907927656612468301on_nat,Xs3: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat] :
            ( X2
           != ( produc2912653624306349395on_nat @ ( cons_S6797538523646374167on_nat @ X3 @ Xs3 ) @ Ys ) ) ) ).

% subset_eq_mset_impl.cases
thf(fact_492_subset__eq__mset__impl_Ocases,axiom,
    ! [X2: produc1714117091004681831on_nat] :
      ( ! [Ys: list_P9127870862658512897on_nat] :
          ( X2
         != ( produc7152764266822961111on_nat @ nil_Pr8614093721167239297on_nat @ Ys ) )
     => ~ ! [X3: produc1313295348606329723on_nat,Xs3: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat] :
            ( X2
           != ( produc7152764266822961111on_nat @ ( cons_P1582596238967110705on_nat @ X3 @ Xs3 ) @ Ys ) ) ) ).

% subset_eq_mset_impl.cases
thf(fact_493_subset__eq__mset__impl_Ocases,axiom,
    ! [X2: produc1828647624359046049st_nat] :
      ( ! [Ys: list_nat] :
          ( X2
         != ( produc2694037385005941721st_nat @ nil_nat @ Ys ) )
     => ~ ! [X3: nat,Xs3: list_nat,Ys: list_nat] :
            ( X2
           != ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs3 ) @ Ys ) ) ) ).

% subset_eq_mset_impl.cases
thf(fact_494_subset__eq__mset__impl_Ocases,axiom,
    ! [X2: produc6098023079705189563on_nat] :
      ( ! [Ys: list_l8999537460041914605on_nat] :
          ( X2
         != ( produc1341766448070154867on_nat @ nil_li5563851662744376407on_nat @ Ys ) )
     => ~ ! [X3: list_S5253404628753938141on_nat,Xs3: list_l8999537460041914605on_nat,Ys: list_l8999537460041914605on_nat] :
            ( X2
           != ( produc1341766448070154867on_nat @ ( cons_l5857885737047467175on_nat @ X3 @ Xs3 ) @ Ys ) ) ) ).

% subset_eq_mset_impl.cases
thf(fact_495_subset__eq__mset__impl_Ocases,axiom,
    ! [X2: produc2417963485646110695on_nat] :
      ( ! [Ys: list_l1726306699111145479on_nat] :
          ( X2
         != ( produc4739740463051851607on_nat @ nil_li2682182165152866823on_nat @ Ys ) )
     => ~ ! [X3: list_P9127870862658512897on_nat,Xs3: list_l1726306699111145479on_nat,Ys: list_l1726306699111145479on_nat] :
            ( X2
           != ( produc4739740463051851607on_nat @ ( cons_l3028799188195208631on_nat @ X3 @ Xs3 ) @ Ys ) ) ) ).

% subset_eq_mset_impl.cases
thf(fact_496_product__concat__map,axiom,
    ( produc4987672674337023969on_nat
    = ( ^ [Xs2: list_S5253404628753938141on_nat,Ys3: list_S5253404628753938141on_nat] :
          ( concat937493773106420796on_nat
          @ ( map_Su1406162919755814318on_nat
            @ ^ [X: sum_su4907927656612468301on_nat] : ( map_Su8438591925558430632on_nat @ ( produc6704844767664046899on_nat @ X ) @ Ys3 )
            @ Xs2 ) ) ) ) ).

% product_concat_map
thf(fact_497_product__concat__map,axiom,
    ( produc6587381211084932163on_nat
    = ( ^ [Xs2: list_l8999537460041914605on_nat,Ys3: list_P8734936519038225773on_nat] :
          ( concat1127826427744615230on_nat
          @ ( map_li6308922837919316594on_nat
            @ ^ [X: list_S5253404628753938141on_nat] : ( map_Pr5337824985771995362on_nat @ ( produc4176415733022414961on_nat @ X ) @ Ys3 )
            @ Xs2 ) ) ) ) ).

% product_concat_map
thf(fact_498_product__concat__map,axiom,
    ( produc472613109841061731on_nat
    = ( ^ [Xs2: list_l8999537460041914605on_nat,Ys3: list_P5508476027986415439on_nat] :
          ( concat9214455607871246526on_nat
          @ ( map_li5621934489530828832on_nat
            @ ^ [X: list_S5253404628753938141on_nat] : ( map_Pr5195230461829504568on_nat @ ( produc1545487600272464053on_nat @ X ) @ Ys3 )
            @ Xs2 ) ) ) ) ).

% product_concat_map
thf(fact_499_product__concat__map,axiom,
    ( product_nat_nat
    = ( ^ [Xs2: list_nat,Ys3: list_nat] :
          ( concat7691415812945658306at_nat
          @ ( map_na4561905831291441265at_nat
            @ ^ [X: nat] : ( map_na7298421622053143531at_nat @ ( product_Pair_nat_nat @ X ) @ Ys3 )
            @ Xs2 ) ) ) ) ).

% product_concat_map
thf(fact_500_product__concat__map,axiom,
    ( product_nat_nat_nat
    = ( ^ [Xs2: list_nat_nat,Ys3: list_nat] :
          ( concat821400627202602033at_nat
          @ ( map_na1011612288921435825at_nat
            @ ^ [X: nat > nat] : ( map_na6722963242033808602at_nat @ ( produc72220940542539688at_nat @ X ) @ Ys3 )
            @ Xs2 ) ) ) ) ).

% product_concat_map
thf(fact_501_product__concat__map,axiom,
    ( produc2053221683845580394tation
    = ( ^ [Xs2: list_a,Ys3: list_M7834926499894101121tation] :
          ( concat1037798000090658717tation
          @ ( map_a_2500903461722202088tation
            @ ^ [X: a] : ( map_Mu1807457972936786645tation @ ( produc7247797765490398872tation @ X ) @ Ys3 )
            @ Xs2 ) ) ) ) ).

% product_concat_map
thf(fact_502_product_Osimps_I2_J,axiom,
    ! [X2: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat,Ys2: list_S5253404628753938141on_nat] :
      ( ( produc4987672674337023969on_nat @ ( cons_S6797538523646374167on_nat @ X2 @ Xs ) @ Ys2 )
      = ( append2865821678206098582on_nat @ ( map_Su8438591925558430632on_nat @ ( produc6704844767664046899on_nat @ X2 ) @ Ys2 ) @ ( produc4987672674337023969on_nat @ Xs @ Ys2 ) ) ) ).

% product.simps(2)
thf(fact_503_product_Osimps_I2_J,axiom,
    ! [X2: list_S5253404628753938141on_nat,Xs: list_l8999537460041914605on_nat,Ys2: list_P8734936519038225773on_nat] :
      ( ( produc6587381211084932163on_nat @ ( cons_l5857885737047467175on_nat @ X2 @ Xs ) @ Ys2 )
      = ( append858208364616284260on_nat @ ( map_Pr5337824985771995362on_nat @ ( produc4176415733022414961on_nat @ X2 ) @ Ys2 ) @ ( produc6587381211084932163on_nat @ Xs @ Ys2 ) ) ) ).

% product.simps(2)
thf(fact_504_product_Osimps_I2_J,axiom,
    ! [X2: list_S5253404628753938141on_nat,Xs: list_l8999537460041914605on_nat,Ys2: list_P5508476027986415439on_nat] :
      ( ( produc472613109841061731on_nat @ ( cons_l5857885737047467175on_nat @ X2 @ Xs ) @ Ys2 )
      = ( append3872231119213297432on_nat @ ( map_Pr5195230461829504568on_nat @ ( produc1545487600272464053on_nat @ X2 ) @ Ys2 ) @ ( produc472613109841061731on_nat @ Xs @ Ys2 ) ) ) ).

% product.simps(2)
thf(fact_505_product_Osimps_I2_J,axiom,
    ! [X2: nat,Xs: list_nat,Ys2: list_nat] :
      ( ( product_nat_nat @ ( cons_nat @ X2 @ Xs ) @ Ys2 )
      = ( append985823374593552924at_nat @ ( map_na7298421622053143531at_nat @ ( product_Pair_nat_nat @ X2 ) @ Ys2 ) @ ( product_nat_nat @ Xs @ Ys2 ) ) ) ).

% product.simps(2)
thf(fact_506_product_Osimps_I2_J,axiom,
    ! [X2: nat > nat,Xs: list_nat_nat,Ys2: list_nat] :
      ( ( product_nat_nat_nat @ ( cons_nat_nat @ X2 @ Xs ) @ Ys2 )
      = ( append8880802010960057739at_nat @ ( map_na6722963242033808602at_nat @ ( produc72220940542539688at_nat @ X2 ) @ Ys2 ) @ ( product_nat_nat_nat @ Xs @ Ys2 ) ) ) ).

% product.simps(2)
thf(fact_507_product_Osimps_I2_J,axiom,
    ! [X2: a,Xs: list_a,Ys2: list_M7834926499894101121tation] :
      ( ( produc2053221683845580394tation @ ( cons_a @ X2 @ Xs ) @ Ys2 )
      = ( append4134760898919571651tation @ ( map_Mu1807457972936786645tation @ ( produc7247797765490398872tation @ X2 ) @ Ys2 ) @ ( produc2053221683845580394tation @ Xs @ Ys2 ) ) ) ).

% product.simps(2)
thf(fact_508_obj__sumE,axiom,
    ! [S: sum_su4907927656612468301on_nat] :
      ( ! [X3: produc27318731269267614tation] :
          ( S
         != ( sum_In3878554002080803550on_nat @ X3 ) )
     => ~ ! [X3: nat] :
            ( S
           != ( sum_In255631617079920996tation @ X3 ) ) ) ).

% obj_sumE
thf(fact_509_product__lists_Osimps_I2_J,axiom,
    ! [Xs: list_nat,Xss2: list_list_nat] :
      ( ( product_lists_nat @ ( cons_list_nat @ Xs @ Xss2 ) )
      = ( concat_list_nat
        @ ( map_na6205611841492582150st_nat
          @ ^ [X: nat] : ( map_li7225945977422193158st_nat @ ( cons_nat @ X ) @ ( product_lists_nat @ Xss2 ) )
          @ Xs ) ) ) ).

% product_lists.simps(2)
thf(fact_510_product__lists_Osimps_I2_J,axiom,
    ! [Xs: list_l8999537460041914605on_nat,Xss2: list_l8365940747029150845on_nat] :
      ( ( produc657118021824710830on_nat @ ( cons_l6725403660475259319on_nat @ Xs @ Xss2 ) )
      = ( concat6293066666318574188on_nat
        @ ( map_li2998272717593060896on_nat
          @ ^ [X: list_S5253404628753938141on_nat] : ( map_li3413953685577837472on_nat @ ( cons_l5857885737047467175on_nat @ X ) @ ( produc657118021824710830on_nat @ Xss2 ) )
          @ Xs ) ) ) ).

% product_lists.simps(2)
thf(fact_511_product__lists_Osimps_I2_J,axiom,
    ! [Xs: list_l1726306699111145479on_nat,Xss2: list_l6719245731233385357on_nat] :
      ( ( produc1618157707243653360on_nat @ ( cons_l4979741563022480061on_nat @ Xs @ Xss2 ) )
      = ( concat1602011024853954504on_nat
        @ ( map_li1709308397223692048on_nat
          @ ^ [X: list_P9127870862658512897on_nat] : ( map_li7104156278154209540on_nat @ ( cons_l3028799188195208631on_nat @ X ) @ ( produc1618157707243653360on_nat @ Xss2 ) )
          @ Xs ) ) ) ).

% product_lists.simps(2)
thf(fact_512_product__lists_Osimps_I2_J,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Xss2: list_l8999537460041914605on_nat] :
      ( ( produc1748266815408604318on_nat @ ( cons_l5857885737047467175on_nat @ Xs @ Xss2 ) )
      = ( concat4774721536428481500on_nat
        @ ( map_Su7921119544090670592on_nat
          @ ^ [X: sum_su4907927656612468301on_nat] : ( map_li4295621887133137792on_nat @ ( cons_S6797538523646374167on_nat @ X ) @ ( produc1748266815408604318on_nat @ Xss2 ) )
          @ Xs ) ) ) ).

% product_lists.simps(2)
thf(fact_513_product__lists_Osimps_I2_J,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Xss2: list_l1726306699111145479on_nat] :
      ( ( produc2357585119171201002on_nat @ ( cons_l3028799188195208631on_nat @ Xs @ Xss2 ) )
      = ( concat4341727598008994882on_nat
        @ ( map_Pr9122214429247829392on_nat
          @ ^ [X: produc1313295348606329723on_nat] : ( map_li815642509948034692on_nat @ ( cons_P1582596238967110705on_nat @ X ) @ ( produc2357585119171201002on_nat @ Xss2 ) )
          @ Xs ) ) ) ).

% product_lists.simps(2)
thf(fact_514_cs__def,axiom,
    ( cs
    = ( upt @ zero_zero_nat @ ( size_s3379185846251495609od_a_o @ cnf ) ) ) ).

% cs_def
thf(fact_515_length__map,axiom,
    ! [F: nat > nat,Xs: list_nat] :
      ( ( size_size_list_nat @ ( map_nat_nat @ F @ Xs ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_map
thf(fact_516_length__map,axiom,
    ! [F: nat > list_nat,Xs: list_nat] :
      ( ( size_s3023201423986296836st_nat @ ( map_nat_list_nat @ F @ Xs ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_map
thf(fact_517_length__map,axiom,
    ! [F: nat > list_P3196869577824546119od_a_o,Xs: list_nat] :
      ( ( size_s3379185846251495609od_a_o @ ( map_na764632570197019441od_a_o @ F @ Xs ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_map
thf(fact_518_length__map,axiom,
    ! [F: list_P3196869577824546119od_a_o > nat,Xs: list_l8336264072933094989od_a_o] :
      ( ( size_size_list_nat @ ( map_li381573783031109843_o_nat @ F @ Xs ) )
      = ( size_s3379185846251495609od_a_o @ Xs ) ) ).

% length_map
thf(fact_519_length__map,axiom,
    ! [F: a > sum_su4907927656612468301on_nat,Xs: list_a] :
      ( ( size_s5686234912572693873on_nat @ ( map_a_2711072540680412167on_nat @ F @ Xs ) )
      = ( size_size_list_a @ Xs ) ) ).

% length_map
thf(fact_520_length__map,axiom,
    ! [F: a > list_S5253404628753938141on_nat,Xs: list_a] :
      ( ( size_s650155532264187265on_nat @ ( map_a_3887198048842487703on_nat @ F @ Xs ) )
      = ( size_size_list_a @ Xs ) ) ).

% length_map
thf(fact_521_length__map,axiom,
    ! [F: a > list_l8999537460041914605on_nat,Xs: list_a] :
      ( ( size_s8175741288642522897on_nat @ ( map_a_3820070144034313383on_nat @ F @ Xs ) )
      = ( size_size_list_a @ Xs ) ) ).

% length_map
thf(fact_522_length__map,axiom,
    ! [F: list_P3196869577824546119od_a_o > list_P3196869577824546119od_a_o,Xs: list_l8336264072933094989od_a_o] :
      ( ( size_s3379185846251495609od_a_o @ ( map_li880244354692327940od_a_o @ F @ Xs ) )
      = ( size_s3379185846251495609od_a_o @ Xs ) ) ).

% length_map
thf(fact_523_length__map,axiom,
    ! [F: a > produc1313295348606329723on_nat,Xs: list_a] :
      ( ( size_s7490768528479217773on_nat @ ( map_a_2212652920793933121on_nat @ F @ Xs ) )
      = ( size_size_list_a @ Xs ) ) ).

% length_map
thf(fact_524_length__map,axiom,
    ! [F: a > list_P9127870862658512897on_nat,Xs: list_a] :
      ( ( size_s129639936549468787on_nat @ ( map_a_7431629901395152583on_nat @ F @ Xs ) )
      = ( size_size_list_a @ Xs ) ) ).

% length_map
thf(fact_525_append__eq__append__conv,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys2: list_S5253404628753938141on_nat,Us2: list_S5253404628753938141on_nat,Vs2: list_S5253404628753938141on_nat] :
      ( ( ( ( size_s5686234912572693873on_nat @ Xs )
          = ( size_s5686234912572693873on_nat @ Ys2 ) )
        | ( ( size_s5686234912572693873on_nat @ Us2 )
          = ( size_s5686234912572693873on_nat @ Vs2 ) ) )
     => ( ( ( append3523812819975775986on_nat @ Xs @ Us2 )
          = ( append3523812819975775986on_nat @ Ys2 @ Vs2 ) )
        = ( ( Xs = Ys2 )
          & ( Us2 = Vs2 ) ) ) ) ).

% append_eq_append_conv
thf(fact_526_append__eq__append__conv,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys2: list_P9127870862658512897on_nat,Us2: list_P9127870862658512897on_nat,Vs2: list_P9127870862658512897on_nat] :
      ( ( ( ( size_s7490768528479217773on_nat @ Xs )
          = ( size_s7490768528479217773on_nat @ Ys2 ) )
        | ( ( size_s7490768528479217773on_nat @ Us2 )
          = ( size_s7490768528479217773on_nat @ Vs2 ) ) )
     => ( ( ( append2865821678206098582on_nat @ Xs @ Us2 )
          = ( append2865821678206098582on_nat @ Ys2 @ Vs2 ) )
        = ( ( Xs = Ys2 )
          & ( Us2 = Vs2 ) ) ) ) ).

% append_eq_append_conv
thf(fact_527_append__eq__append__conv,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys2: list_l8336264072933094989od_a_o,Us2: list_l8336264072933094989od_a_o,Vs2: list_l8336264072933094989od_a_o] :
      ( ( ( ( size_s3379185846251495609od_a_o @ Xs )
          = ( size_s3379185846251495609od_a_o @ Ys2 ) )
        | ( ( size_s3379185846251495609od_a_o @ Us2 )
          = ( size_s3379185846251495609od_a_o @ Vs2 ) ) )
     => ( ( ( append2159120467191820514od_a_o @ Xs @ Us2 )
          = ( append2159120467191820514od_a_o @ Ys2 @ Vs2 ) )
        = ( ( Xs = Ys2 )
          & ( Us2 = Vs2 ) ) ) ) ).

% append_eq_append_conv
thf(fact_528_append__eq__append__conv,axiom,
    ! [Xs: list_nat,Ys2: list_nat,Us2: list_nat,Vs2: list_nat] :
      ( ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_nat @ Ys2 ) )
        | ( ( size_size_list_nat @ Us2 )
          = ( size_size_list_nat @ Vs2 ) ) )
     => ( ( ( append_nat @ Xs @ Us2 )
          = ( append_nat @ Ys2 @ Vs2 ) )
        = ( ( Xs = Ys2 )
          & ( Us2 = Vs2 ) ) ) ) ).

% append_eq_append_conv
thf(fact_529_length__0__conv,axiom,
    ! [Xs: list_S5253404628753938141on_nat] :
      ( ( ( size_s5686234912572693873on_nat @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_Su3078906344068544711on_nat ) ) ).

% length_0_conv
thf(fact_530_length__0__conv,axiom,
    ! [Xs: list_P9127870862658512897on_nat] :
      ( ( ( size_s7490768528479217773on_nat @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_Pr8614093721167239297on_nat ) ) ).

% length_0_conv
thf(fact_531_length__0__conv,axiom,
    ! [Xs: list_l8999537460041914605on_nat] :
      ( ( ( size_s650155532264187265on_nat @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_li5563851662744376407on_nat ) ) ).

% length_0_conv
thf(fact_532_length__0__conv,axiom,
    ! [Xs: list_l1726306699111145479on_nat] :
      ( ( ( size_s129639936549468787on_nat @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_li2682182165152866823on_nat ) ) ).

% length_0_conv
thf(fact_533_length__0__conv,axiom,
    ! [Xs: list_a] :
      ( ( ( size_size_list_a @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_a ) ) ).

% length_0_conv
thf(fact_534_length__0__conv,axiom,
    ! [Xs: list_l8336264072933094989od_a_o] :
      ( ( ( size_s3379185846251495609od_a_o @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_li1372968520647675725od_a_o ) ) ).

% length_0_conv
thf(fact_535_length__0__conv,axiom,
    ! [Xs: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_nat ) ) ).

% length_0_conv
thf(fact_536_nth__Cons__0,axiom,
    ! [X2: list_P3196869577824546119od_a_o,Xs: list_l8336264072933094989od_a_o] :
      ( ( nth_li3248255784598043854od_a_o @ ( cons_l9179691953475787517od_a_o @ X2 @ Xs ) @ zero_zero_nat )
      = X2 ) ).

% nth_Cons_0
thf(fact_537_nth__Cons__0,axiom,
    ! [X2: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat] :
      ( ( nth_Su8471564416988646406on_nat @ ( cons_S6797538523646374167on_nat @ X2 @ Xs ) @ zero_zero_nat )
      = X2 ) ).

% nth_Cons_0
thf(fact_538_nth__Cons__0,axiom,
    ! [X2: produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat] :
      ( ( nth_Pr3336893728888831618on_nat @ ( cons_P1582596238967110705on_nat @ X2 @ Xs ) @ zero_zero_nat )
      = X2 ) ).

% nth_Cons_0
thf(fact_539_nth__Cons__0,axiom,
    ! [X2: nat,Xs: list_nat] :
      ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ zero_zero_nat )
      = X2 ) ).

% nth_Cons_0
thf(fact_540_nth__Cons__0,axiom,
    ! [X2: list_S5253404628753938141on_nat,Xs: list_l8999537460041914605on_nat] :
      ( ( nth_li4091121329102240790on_nat @ ( cons_l5857885737047467175on_nat @ X2 @ Xs ) @ zero_zero_nat )
      = X2 ) ).

% nth_Cons_0
thf(fact_541_nth__Cons__0,axiom,
    ! [X2: list_P9127870862658512897on_nat,Xs: list_l1726306699111145479on_nat] :
      ( ( nth_li1302981970230039432on_nat @ ( cons_l3028799188195208631on_nat @ X2 @ Xs ) @ zero_zero_nat )
      = X2 ) ).

% nth_Cons_0
thf(fact_542_nth__append__length,axiom,
    ! [Xs: list_S5253404628753938141on_nat,X2: sum_su4907927656612468301on_nat,Ys2: list_S5253404628753938141on_nat] :
      ( ( nth_Su8471564416988646406on_nat @ ( append3523812819975775986on_nat @ Xs @ ( cons_S6797538523646374167on_nat @ X2 @ Ys2 ) ) @ ( size_s5686234912572693873on_nat @ Xs ) )
      = X2 ) ).

% nth_append_length
thf(fact_543_nth__append__length,axiom,
    ! [Xs: list_P9127870862658512897on_nat,X2: produc1313295348606329723on_nat,Ys2: list_P9127870862658512897on_nat] :
      ( ( nth_Pr3336893728888831618on_nat @ ( append2865821678206098582on_nat @ Xs @ ( cons_P1582596238967110705on_nat @ X2 @ Ys2 ) ) @ ( size_s7490768528479217773on_nat @ Xs ) )
      = X2 ) ).

% nth_append_length
thf(fact_544_nth__append__length,axiom,
    ! [Xs: list_l8999537460041914605on_nat,X2: list_S5253404628753938141on_nat,Ys2: list_l8999537460041914605on_nat] :
      ( ( nth_li4091121329102240790on_nat @ ( append6174841615030841602on_nat @ Xs @ ( cons_l5857885737047467175on_nat @ X2 @ Ys2 ) ) @ ( size_s650155532264187265on_nat @ Xs ) )
      = X2 ) ).

% nth_append_length
thf(fact_545_nth__append__length,axiom,
    ! [Xs: list_l1726306699111145479on_nat,X2: list_P9127870862658512897on_nat,Ys2: list_l1726306699111145479on_nat] :
      ( ( nth_li1302981970230039432on_nat @ ( append3839213209832495004on_nat @ Xs @ ( cons_l3028799188195208631on_nat @ X2 @ Ys2 ) ) @ ( size_s129639936549468787on_nat @ Xs ) )
      = X2 ) ).

% nth_append_length
thf(fact_546_nth__append__length,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,X2: list_P3196869577824546119od_a_o,Ys2: list_l8336264072933094989od_a_o] :
      ( ( nth_li3248255784598043854od_a_o @ ( append2159120467191820514od_a_o @ Xs @ ( cons_l9179691953475787517od_a_o @ X2 @ Ys2 ) ) @ ( size_s3379185846251495609od_a_o @ Xs ) )
      = X2 ) ).

% nth_append_length
thf(fact_547_nth__append__length,axiom,
    ! [Xs: list_nat,X2: nat,Ys2: list_nat] :
      ( ( nth_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ Ys2 ) ) @ ( size_size_list_nat @ Xs ) )
      = X2 ) ).

% nth_append_length
thf(fact_548_upt__0,axiom,
    ! [I2: nat] :
      ( ( upt @ I2 @ zero_zero_nat )
      = nil_nat ) ).

% upt_0
thf(fact_549_list_Osize_I3_J,axiom,
    ( ( size_s5686234912572693873on_nat @ nil_Su3078906344068544711on_nat )
    = zero_zero_nat ) ).

% list.size(3)
thf(fact_550_list_Osize_I3_J,axiom,
    ( ( size_s7490768528479217773on_nat @ nil_Pr8614093721167239297on_nat )
    = zero_zero_nat ) ).

% list.size(3)
thf(fact_551_list_Osize_I3_J,axiom,
    ( ( size_s650155532264187265on_nat @ nil_li5563851662744376407on_nat )
    = zero_zero_nat ) ).

% list.size(3)
thf(fact_552_list_Osize_I3_J,axiom,
    ( ( size_s129639936549468787on_nat @ nil_li2682182165152866823on_nat )
    = zero_zero_nat ) ).

% list.size(3)
thf(fact_553_list_Osize_I3_J,axiom,
    ( ( size_size_list_a @ nil_a )
    = zero_zero_nat ) ).

% list.size(3)
thf(fact_554_list_Osize_I3_J,axiom,
    ( ( size_s3379185846251495609od_a_o @ nil_li1372968520647675725od_a_o )
    = zero_zero_nat ) ).

% list.size(3)
thf(fact_555_list_Osize_I3_J,axiom,
    ( ( size_size_list_nat @ nil_nat )
    = zero_zero_nat ) ).

% list.size(3)
thf(fact_556_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_l8336264072933094989od_a_o] :
      ( ( size_s3379185846251495609od_a_o @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_557_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_nat] :
      ( ( size_size_list_nat @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_558_neq__if__length__neq,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys2: list_l8336264072933094989od_a_o] :
      ( ( ( size_s3379185846251495609od_a_o @ Xs )
       != ( size_s3379185846251495609od_a_o @ Ys2 ) )
     => ( Xs != Ys2 ) ) ).

% neq_if_length_neq
thf(fact_559_neq__if__length__neq,axiom,
    ! [Xs: list_nat,Ys2: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
       != ( size_size_list_nat @ Ys2 ) )
     => ( Xs != Ys2 ) ) ).

% neq_if_length_neq
thf(fact_560_map__nth,axiom,
    ! [Xs: list_l1726306699111145479on_nat] :
      ( ( map_na8715451351012060651on_nat @ ( nth_li1302981970230039432on_nat @ Xs ) @ ( upt @ zero_zero_nat @ ( size_s129639936549468787on_nat @ Xs ) ) )
      = Xs ) ).

% map_nth
thf(fact_561_map__nth,axiom,
    ! [Xs: list_l6719245731233385357on_nat] :
      ( ( map_na8454508660465608945on_nat @ ( nth_li6929086678380665870on_nat @ Xs ) @ ( upt @ zero_zero_nat @ ( size_s6092282996036616185on_nat @ Xs ) ) )
      = Xs ) ).

% map_nth
thf(fact_562_map__nth,axiom,
    ! [Xs: list_list_nat] :
      ( ( map_nat_list_nat @ ( nth_list_nat @ Xs ) @ ( upt @ zero_zero_nat @ ( size_s3023201423986296836st_nat @ Xs ) ) )
      = Xs ) ).

% map_nth
thf(fact_563_map__nth,axiom,
    ! [Xs: list_l8336264072933094989od_a_o] :
      ( ( map_na764632570197019441od_a_o @ ( nth_li3248255784598043854od_a_o @ Xs ) @ ( upt @ zero_zero_nat @ ( size_s3379185846251495609od_a_o @ Xs ) ) )
      = Xs ) ).

% map_nth
thf(fact_564_map__nth,axiom,
    ! [Xs: list_nat] :
      ( ( map_nat_nat @ ( nth_nat @ Xs ) @ ( upt @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) )
      = Xs ) ).

% map_nth
thf(fact_565_map__eq__imp__length__eq,axiom,
    ! [F: nat > nat,Xs: list_nat,G: nat > nat,Ys2: list_nat] :
      ( ( ( map_nat_nat @ F @ Xs )
        = ( map_nat_nat @ G @ Ys2 ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys2 ) ) ) ).

% map_eq_imp_length_eq
thf(fact_566_map__eq__imp__length__eq,axiom,
    ! [F: nat > list_nat,Xs: list_nat,G: nat > list_nat,Ys2: list_nat] :
      ( ( ( map_nat_list_nat @ F @ Xs )
        = ( map_nat_list_nat @ G @ Ys2 ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys2 ) ) ) ).

% map_eq_imp_length_eq
thf(fact_567_map__eq__imp__length__eq,axiom,
    ! [F: list_P3196869577824546119od_a_o > nat,Xs: list_l8336264072933094989od_a_o,G: nat > nat,Ys2: list_nat] :
      ( ( ( map_li381573783031109843_o_nat @ F @ Xs )
        = ( map_nat_nat @ G @ Ys2 ) )
     => ( ( size_s3379185846251495609od_a_o @ Xs )
        = ( size_size_list_nat @ Ys2 ) ) ) ).

% map_eq_imp_length_eq
thf(fact_568_map__eq__imp__length__eq,axiom,
    ! [F: nat > nat,Xs: list_nat,G: list_P3196869577824546119od_a_o > nat,Ys2: list_l8336264072933094989od_a_o] :
      ( ( ( map_nat_nat @ F @ Xs )
        = ( map_li381573783031109843_o_nat @ G @ Ys2 ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_s3379185846251495609od_a_o @ Ys2 ) ) ) ).

% map_eq_imp_length_eq
thf(fact_569_map__eq__imp__length__eq,axiom,
    ! [F: a > sum_su4907927656612468301on_nat,Xs: list_a,G: a > sum_su4907927656612468301on_nat,Ys2: list_a] :
      ( ( ( map_a_2711072540680412167on_nat @ F @ Xs )
        = ( map_a_2711072540680412167on_nat @ G @ Ys2 ) )
     => ( ( size_size_list_a @ Xs )
        = ( size_size_list_a @ Ys2 ) ) ) ).

% map_eq_imp_length_eq
thf(fact_570_map__eq__imp__length__eq,axiom,
    ! [F: a > sum_su4907927656612468301on_nat,Xs: list_a,G: nat > sum_su4907927656612468301on_nat,Ys2: list_nat] :
      ( ( ( map_a_2711072540680412167on_nat @ F @ Xs )
        = ( map_na5706685718985795555on_nat @ G @ Ys2 ) )
     => ( ( size_size_list_a @ Xs )
        = ( size_size_list_nat @ Ys2 ) ) ) ).

% map_eq_imp_length_eq
thf(fact_571_map__eq__imp__length__eq,axiom,
    ! [F: list_P3196869577824546119od_a_o > list_nat,Xs: list_l8336264072933094989od_a_o,G: nat > list_nat,Ys2: list_nat] :
      ( ( ( map_li3237930340688975459st_nat @ F @ Xs )
        = ( map_nat_list_nat @ G @ Ys2 ) )
     => ( ( size_s3379185846251495609od_a_o @ Xs )
        = ( size_size_list_nat @ Ys2 ) ) ) ).

% map_eq_imp_length_eq
thf(fact_572_map__eq__imp__length__eq,axiom,
    ! [F: nat > sum_su4907927656612468301on_nat,Xs: list_nat,G: a > sum_su4907927656612468301on_nat,Ys2: list_a] :
      ( ( ( map_na5706685718985795555on_nat @ F @ Xs )
        = ( map_a_2711072540680412167on_nat @ G @ Ys2 ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_size_list_a @ Ys2 ) ) ) ).

% map_eq_imp_length_eq
thf(fact_573_map__eq__imp__length__eq,axiom,
    ! [F: nat > list_nat,Xs: list_nat,G: list_P3196869577824546119od_a_o > list_nat,Ys2: list_l8336264072933094989od_a_o] :
      ( ( ( map_nat_list_nat @ F @ Xs )
        = ( map_li3237930340688975459st_nat @ G @ Ys2 ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_s3379185846251495609od_a_o @ Ys2 ) ) ) ).

% map_eq_imp_length_eq
thf(fact_574_map__eq__imp__length__eq,axiom,
    ! [F: a > list_S5253404628753938141on_nat,Xs: list_a,G: a > list_S5253404628753938141on_nat,Ys2: list_a] :
      ( ( ( map_a_3887198048842487703on_nat @ F @ Xs )
        = ( map_a_3887198048842487703on_nat @ G @ Ys2 ) )
     => ( ( size_size_list_a @ Xs )
        = ( size_size_list_a @ Ys2 ) ) ) ).

% map_eq_imp_length_eq
thf(fact_575_list__induct2,axiom,
    ! [Xs: list_a,Ys2: list_a,P: list_a > list_a > $o] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_a @ Ys2 ) )
     => ( ( P @ nil_a @ nil_a )
       => ( ! [X3: a,Xs3: list_a,Y2: a,Ys: list_a] :
              ( ( ( size_size_list_a @ Xs3 )
                = ( size_size_list_a @ Ys ) )
             => ( ( P @ Xs3 @ Ys )
               => ( P @ ( cons_a @ X3 @ Xs3 ) @ ( cons_a @ Y2 @ Ys ) ) ) )
         => ( P @ Xs @ Ys2 ) ) ) ) ).

% list_induct2
thf(fact_576_list__induct2,axiom,
    ! [Xs: list_a,Ys2: list_nat,P: list_a > list_nat > $o] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_nat @ Ys2 ) )
     => ( ( P @ nil_a @ nil_nat )
       => ( ! [X3: a,Xs3: list_a,Y2: nat,Ys: list_nat] :
              ( ( ( size_size_list_a @ Xs3 )
                = ( size_size_list_nat @ Ys ) )
             => ( ( P @ Xs3 @ Ys )
               => ( P @ ( cons_a @ X3 @ Xs3 ) @ ( cons_nat @ Y2 @ Ys ) ) ) )
         => ( P @ Xs @ Ys2 ) ) ) ) ).

% list_induct2
thf(fact_577_list__induct2,axiom,
    ! [Xs: list_nat,Ys2: list_a,P: list_nat > list_a > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_a @ Ys2 ) )
     => ( ( P @ nil_nat @ nil_a )
       => ( ! [X3: nat,Xs3: list_nat,Y2: a,Ys: list_a] :
              ( ( ( size_size_list_nat @ Xs3 )
                = ( size_size_list_a @ Ys ) )
             => ( ( P @ Xs3 @ Ys )
               => ( P @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_a @ Y2 @ Ys ) ) ) )
         => ( P @ Xs @ Ys2 ) ) ) ) ).

% list_induct2
thf(fact_578_list__induct2,axiom,
    ! [Xs: list_nat,Ys2: list_nat,P: list_nat > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys2 ) )
     => ( ( P @ nil_nat @ nil_nat )
       => ( ! [X3: nat,Xs3: list_nat,Y2: nat,Ys: list_nat] :
              ( ( ( size_size_list_nat @ Xs3 )
                = ( size_size_list_nat @ Ys ) )
             => ( ( P @ Xs3 @ Ys )
               => ( P @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_nat @ Y2 @ Ys ) ) ) )
         => ( P @ Xs @ Ys2 ) ) ) ) ).

% list_induct2
thf(fact_579_list__induct2,axiom,
    ! [Xs: list_a,Ys2: list_l8336264072933094989od_a_o,P: list_a > list_l8336264072933094989od_a_o > $o] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_s3379185846251495609od_a_o @ Ys2 ) )
     => ( ( P @ nil_a @ nil_li1372968520647675725od_a_o )
       => ( ! [X3: a,Xs3: list_a,Y2: list_P3196869577824546119od_a_o,Ys: list_l8336264072933094989od_a_o] :
              ( ( ( size_size_list_a @ Xs3 )
                = ( size_s3379185846251495609od_a_o @ Ys ) )
             => ( ( P @ Xs3 @ Ys )
               => ( P @ ( cons_a @ X3 @ Xs3 ) @ ( cons_l9179691953475787517od_a_o @ Y2 @ Ys ) ) ) )
         => ( P @ Xs @ Ys2 ) ) ) ) ).

% list_induct2
thf(fact_580_list__induct2,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys2: list_a,P: list_l8336264072933094989od_a_o > list_a > $o] :
      ( ( ( size_s3379185846251495609od_a_o @ Xs )
        = ( size_size_list_a @ Ys2 ) )
     => ( ( P @ nil_li1372968520647675725od_a_o @ nil_a )
       => ( ! [X3: list_P3196869577824546119od_a_o,Xs3: list_l8336264072933094989od_a_o,Y2: a,Ys: list_a] :
              ( ( ( size_s3379185846251495609od_a_o @ Xs3 )
                = ( size_size_list_a @ Ys ) )
             => ( ( P @ Xs3 @ Ys )
               => ( P @ ( cons_l9179691953475787517od_a_o @ X3 @ Xs3 ) @ ( cons_a @ Y2 @ Ys ) ) ) )
         => ( P @ Xs @ Ys2 ) ) ) ) ).

% list_induct2
thf(fact_581_list__induct2,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys2: list_nat,P: list_l8336264072933094989od_a_o > list_nat > $o] :
      ( ( ( size_s3379185846251495609od_a_o @ Xs )
        = ( size_size_list_nat @ Ys2 ) )
     => ( ( P @ nil_li1372968520647675725od_a_o @ nil_nat )
       => ( ! [X3: list_P3196869577824546119od_a_o,Xs3: list_l8336264072933094989od_a_o,Y2: nat,Ys: list_nat] :
              ( ( ( size_s3379185846251495609od_a_o @ Xs3 )
                = ( size_size_list_nat @ Ys ) )
             => ( ( P @ Xs3 @ Ys )
               => ( P @ ( cons_l9179691953475787517od_a_o @ X3 @ Xs3 ) @ ( cons_nat @ Y2 @ Ys ) ) ) )
         => ( P @ Xs @ Ys2 ) ) ) ) ).

% list_induct2
thf(fact_582_list__induct2,axiom,
    ! [Xs: list_nat,Ys2: list_l8336264072933094989od_a_o,P: list_nat > list_l8336264072933094989od_a_o > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s3379185846251495609od_a_o @ Ys2 ) )
     => ( ( P @ nil_nat @ nil_li1372968520647675725od_a_o )
       => ( ! [X3: nat,Xs3: list_nat,Y2: list_P3196869577824546119od_a_o,Ys: list_l8336264072933094989od_a_o] :
              ( ( ( size_size_list_nat @ Xs3 )
                = ( size_s3379185846251495609od_a_o @ Ys ) )
             => ( ( P @ Xs3 @ Ys )
               => ( P @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_l9179691953475787517od_a_o @ Y2 @ Ys ) ) ) )
         => ( P @ Xs @ Ys2 ) ) ) ) ).

% list_induct2
thf(fact_583_list__induct2,axiom,
    ! [Xs: list_a,Ys2: list_S5253404628753938141on_nat,P: list_a > list_S5253404628753938141on_nat > $o] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_s5686234912572693873on_nat @ Ys2 ) )
     => ( ( P @ nil_a @ nil_Su3078906344068544711on_nat )
       => ( ! [X3: a,Xs3: list_a,Y2: sum_su4907927656612468301on_nat,Ys: list_S5253404628753938141on_nat] :
              ( ( ( size_size_list_a @ Xs3 )
                = ( size_s5686234912572693873on_nat @ Ys ) )
             => ( ( P @ Xs3 @ Ys )
               => ( P @ ( cons_a @ X3 @ Xs3 ) @ ( cons_S6797538523646374167on_nat @ Y2 @ Ys ) ) ) )
         => ( P @ Xs @ Ys2 ) ) ) ) ).

% list_induct2
thf(fact_584_list__induct2,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys2: list_a,P: list_S5253404628753938141on_nat > list_a > $o] :
      ( ( ( size_s5686234912572693873on_nat @ Xs )
        = ( size_size_list_a @ Ys2 ) )
     => ( ( P @ nil_Su3078906344068544711on_nat @ nil_a )
       => ( ! [X3: sum_su4907927656612468301on_nat,Xs3: list_S5253404628753938141on_nat,Y2: a,Ys: list_a] :
              ( ( ( size_s5686234912572693873on_nat @ Xs3 )
                = ( size_size_list_a @ Ys ) )
             => ( ( P @ Xs3 @ Ys )
               => ( P @ ( cons_S6797538523646374167on_nat @ X3 @ Xs3 ) @ ( cons_a @ Y2 @ Ys ) ) ) )
         => ( P @ Xs @ Ys2 ) ) ) ) ).

% list_induct2
thf(fact_585_list__induct3,axiom,
    ! [Xs: list_a,Ys2: list_a,Zs3: list_a,P: list_a > list_a > list_a > $o] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_a @ Ys2 ) )
     => ( ( ( size_size_list_a @ Ys2 )
          = ( size_size_list_a @ Zs3 ) )
       => ( ( P @ nil_a @ nil_a @ nil_a )
         => ( ! [X3: a,Xs3: list_a,Y2: a,Ys: list_a,Z2: a,Zs2: list_a] :
                ( ( ( size_size_list_a @ Xs3 )
                  = ( size_size_list_a @ Ys ) )
               => ( ( ( size_size_list_a @ Ys )
                    = ( size_size_list_a @ Zs2 ) )
                 => ( ( P @ Xs3 @ Ys @ Zs2 )
                   => ( P @ ( cons_a @ X3 @ Xs3 ) @ ( cons_a @ Y2 @ Ys ) @ ( cons_a @ Z2 @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys2 @ Zs3 ) ) ) ) ) ).

% list_induct3
thf(fact_586_list__induct3,axiom,
    ! [Xs: list_a,Ys2: list_a,Zs3: list_nat,P: list_a > list_a > list_nat > $o] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_a @ Ys2 ) )
     => ( ( ( size_size_list_a @ Ys2 )
          = ( size_size_list_nat @ Zs3 ) )
       => ( ( P @ nil_a @ nil_a @ nil_nat )
         => ( ! [X3: a,Xs3: list_a,Y2: a,Ys: list_a,Z2: nat,Zs2: list_nat] :
                ( ( ( size_size_list_a @ Xs3 )
                  = ( size_size_list_a @ Ys ) )
               => ( ( ( size_size_list_a @ Ys )
                    = ( size_size_list_nat @ Zs2 ) )
                 => ( ( P @ Xs3 @ Ys @ Zs2 )
                   => ( P @ ( cons_a @ X3 @ Xs3 ) @ ( cons_a @ Y2 @ Ys ) @ ( cons_nat @ Z2 @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys2 @ Zs3 ) ) ) ) ) ).

% list_induct3
thf(fact_587_list__induct3,axiom,
    ! [Xs: list_a,Ys2: list_nat,Zs3: list_a,P: list_a > list_nat > list_a > $o] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_nat @ Ys2 ) )
     => ( ( ( size_size_list_nat @ Ys2 )
          = ( size_size_list_a @ Zs3 ) )
       => ( ( P @ nil_a @ nil_nat @ nil_a )
         => ( ! [X3: a,Xs3: list_a,Y2: nat,Ys: list_nat,Z2: a,Zs2: list_a] :
                ( ( ( size_size_list_a @ Xs3 )
                  = ( size_size_list_nat @ Ys ) )
               => ( ( ( size_size_list_nat @ Ys )
                    = ( size_size_list_a @ Zs2 ) )
                 => ( ( P @ Xs3 @ Ys @ Zs2 )
                   => ( P @ ( cons_a @ X3 @ Xs3 ) @ ( cons_nat @ Y2 @ Ys ) @ ( cons_a @ Z2 @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys2 @ Zs3 ) ) ) ) ) ).

% list_induct3
thf(fact_588_list__induct3,axiom,
    ! [Xs: list_a,Ys2: list_nat,Zs3: list_nat,P: list_a > list_nat > list_nat > $o] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_nat @ Ys2 ) )
     => ( ( ( size_size_list_nat @ Ys2 )
          = ( size_size_list_nat @ Zs3 ) )
       => ( ( P @ nil_a @ nil_nat @ nil_nat )
         => ( ! [X3: a,Xs3: list_a,Y2: nat,Ys: list_nat,Z2: nat,Zs2: list_nat] :
                ( ( ( size_size_list_a @ Xs3 )
                  = ( size_size_list_nat @ Ys ) )
               => ( ( ( size_size_list_nat @ Ys )
                    = ( size_size_list_nat @ Zs2 ) )
                 => ( ( P @ Xs3 @ Ys @ Zs2 )
                   => ( P @ ( cons_a @ X3 @ Xs3 ) @ ( cons_nat @ Y2 @ Ys ) @ ( cons_nat @ Z2 @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys2 @ Zs3 ) ) ) ) ) ).

% list_induct3
thf(fact_589_list__induct3,axiom,
    ! [Xs: list_nat,Ys2: list_a,Zs3: list_a,P: list_nat > list_a > list_a > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_a @ Ys2 ) )
     => ( ( ( size_size_list_a @ Ys2 )
          = ( size_size_list_a @ Zs3 ) )
       => ( ( P @ nil_nat @ nil_a @ nil_a )
         => ( ! [X3: nat,Xs3: list_nat,Y2: a,Ys: list_a,Z2: a,Zs2: list_a] :
                ( ( ( size_size_list_nat @ Xs3 )
                  = ( size_size_list_a @ Ys ) )
               => ( ( ( size_size_list_a @ Ys )
                    = ( size_size_list_a @ Zs2 ) )
                 => ( ( P @ Xs3 @ Ys @ Zs2 )
                   => ( P @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_a @ Y2 @ Ys ) @ ( cons_a @ Z2 @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys2 @ Zs3 ) ) ) ) ) ).

% list_induct3
thf(fact_590_list__induct3,axiom,
    ! [Xs: list_nat,Ys2: list_a,Zs3: list_nat,P: list_nat > list_a > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_a @ Ys2 ) )
     => ( ( ( size_size_list_a @ Ys2 )
          = ( size_size_list_nat @ Zs3 ) )
       => ( ( P @ nil_nat @ nil_a @ nil_nat )
         => ( ! [X3: nat,Xs3: list_nat,Y2: a,Ys: list_a,Z2: nat,Zs2: list_nat] :
                ( ( ( size_size_list_nat @ Xs3 )
                  = ( size_size_list_a @ Ys ) )
               => ( ( ( size_size_list_a @ Ys )
                    = ( size_size_list_nat @ Zs2 ) )
                 => ( ( P @ Xs3 @ Ys @ Zs2 )
                   => ( P @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_a @ Y2 @ Ys ) @ ( cons_nat @ Z2 @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys2 @ Zs3 ) ) ) ) ) ).

% list_induct3
thf(fact_591_list__induct3,axiom,
    ! [Xs: list_nat,Ys2: list_nat,Zs3: list_a,P: list_nat > list_nat > list_a > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys2 ) )
     => ( ( ( size_size_list_nat @ Ys2 )
          = ( size_size_list_a @ Zs3 ) )
       => ( ( P @ nil_nat @ nil_nat @ nil_a )
         => ( ! [X3: nat,Xs3: list_nat,Y2: nat,Ys: list_nat,Z2: a,Zs2: list_a] :
                ( ( ( size_size_list_nat @ Xs3 )
                  = ( size_size_list_nat @ Ys ) )
               => ( ( ( size_size_list_nat @ Ys )
                    = ( size_size_list_a @ Zs2 ) )
                 => ( ( P @ Xs3 @ Ys @ Zs2 )
                   => ( P @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_nat @ Y2 @ Ys ) @ ( cons_a @ Z2 @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys2 @ Zs3 ) ) ) ) ) ).

% list_induct3
thf(fact_592_list__induct3,axiom,
    ! [Xs: list_nat,Ys2: list_nat,Zs3: list_nat,P: list_nat > list_nat > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys2 ) )
     => ( ( ( size_size_list_nat @ Ys2 )
          = ( size_size_list_nat @ Zs3 ) )
       => ( ( P @ nil_nat @ nil_nat @ nil_nat )
         => ( ! [X3: nat,Xs3: list_nat,Y2: nat,Ys: list_nat,Z2: nat,Zs2: list_nat] :
                ( ( ( size_size_list_nat @ Xs3 )
                  = ( size_size_list_nat @ Ys ) )
               => ( ( ( size_size_list_nat @ Ys )
                    = ( size_size_list_nat @ Zs2 ) )
                 => ( ( P @ Xs3 @ Ys @ Zs2 )
                   => ( P @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_nat @ Y2 @ Ys ) @ ( cons_nat @ Z2 @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys2 @ Zs3 ) ) ) ) ) ).

% list_induct3
thf(fact_593_list__induct3,axiom,
    ! [Xs: list_a,Ys2: list_a,Zs3: list_l8336264072933094989od_a_o,P: list_a > list_a > list_l8336264072933094989od_a_o > $o] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_a @ Ys2 ) )
     => ( ( ( size_size_list_a @ Ys2 )
          = ( size_s3379185846251495609od_a_o @ Zs3 ) )
       => ( ( P @ nil_a @ nil_a @ nil_li1372968520647675725od_a_o )
         => ( ! [X3: a,Xs3: list_a,Y2: a,Ys: list_a,Z2: list_P3196869577824546119od_a_o,Zs2: list_l8336264072933094989od_a_o] :
                ( ( ( size_size_list_a @ Xs3 )
                  = ( size_size_list_a @ Ys ) )
               => ( ( ( size_size_list_a @ Ys )
                    = ( size_s3379185846251495609od_a_o @ Zs2 ) )
                 => ( ( P @ Xs3 @ Ys @ Zs2 )
                   => ( P @ ( cons_a @ X3 @ Xs3 ) @ ( cons_a @ Y2 @ Ys ) @ ( cons_l9179691953475787517od_a_o @ Z2 @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys2 @ Zs3 ) ) ) ) ) ).

% list_induct3
thf(fact_594_list__induct3,axiom,
    ! [Xs: list_a,Ys2: list_l8336264072933094989od_a_o,Zs3: list_a,P: list_a > list_l8336264072933094989od_a_o > list_a > $o] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_s3379185846251495609od_a_o @ Ys2 ) )
     => ( ( ( size_s3379185846251495609od_a_o @ Ys2 )
          = ( size_size_list_a @ Zs3 ) )
       => ( ( P @ nil_a @ nil_li1372968520647675725od_a_o @ nil_a )
         => ( ! [X3: a,Xs3: list_a,Y2: list_P3196869577824546119od_a_o,Ys: list_l8336264072933094989od_a_o,Z2: a,Zs2: list_a] :
                ( ( ( size_size_list_a @ Xs3 )
                  = ( size_s3379185846251495609od_a_o @ Ys ) )
               => ( ( ( size_s3379185846251495609od_a_o @ Ys )
                    = ( size_size_list_a @ Zs2 ) )
                 => ( ( P @ Xs3 @ Ys @ Zs2 )
                   => ( P @ ( cons_a @ X3 @ Xs3 ) @ ( cons_l9179691953475787517od_a_o @ Y2 @ Ys ) @ ( cons_a @ Z2 @ Zs2 ) ) ) ) )
           => ( P @ Xs @ Ys2 @ Zs3 ) ) ) ) ) ).

% list_induct3
thf(fact_595_list__induct4,axiom,
    ! [Xs: list_nat,Ys2: list_a,Zs3: list_a,Ws: list_nat,P: list_nat > list_a > list_a > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_a @ Ys2 ) )
     => ( ( ( size_size_list_a @ Ys2 )
          = ( size_size_list_a @ Zs3 ) )
       => ( ( ( size_size_list_a @ Zs3 )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P @ nil_nat @ nil_a @ nil_a @ nil_nat )
           => ( ! [X3: nat,Xs3: list_nat,Y2: a,Ys: list_a,Z2: a,Zs2: list_a,W: nat,Ws2: list_nat] :
                  ( ( ( size_size_list_nat @ Xs3 )
                    = ( size_size_list_a @ Ys ) )
                 => ( ( ( size_size_list_a @ Ys )
                      = ( size_size_list_a @ Zs2 ) )
                   => ( ( ( size_size_list_a @ Zs2 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P @ Xs3 @ Ys @ Zs2 @ Ws2 )
                       => ( P @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_a @ Y2 @ Ys ) @ ( cons_a @ Z2 @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys2 @ Zs3 @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_596_list__induct4,axiom,
    ! [Xs: list_nat,Ys2: list_a,Zs3: list_nat,Ws: list_a,P: list_nat > list_a > list_nat > list_a > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_a @ Ys2 ) )
     => ( ( ( size_size_list_a @ Ys2 )
          = ( size_size_list_nat @ Zs3 ) )
       => ( ( ( size_size_list_nat @ Zs3 )
            = ( size_size_list_a @ Ws ) )
         => ( ( P @ nil_nat @ nil_a @ nil_nat @ nil_a )
           => ( ! [X3: nat,Xs3: list_nat,Y2: a,Ys: list_a,Z2: nat,Zs2: list_nat,W: a,Ws2: list_a] :
                  ( ( ( size_size_list_nat @ Xs3 )
                    = ( size_size_list_a @ Ys ) )
                 => ( ( ( size_size_list_a @ Ys )
                      = ( size_size_list_nat @ Zs2 ) )
                   => ( ( ( size_size_list_nat @ Zs2 )
                        = ( size_size_list_a @ Ws2 ) )
                     => ( ( P @ Xs3 @ Ys @ Zs2 @ Ws2 )
                       => ( P @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_a @ Y2 @ Ys ) @ ( cons_nat @ Z2 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys2 @ Zs3 @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_597_list__induct4,axiom,
    ! [Xs: list_nat,Ys2: list_a,Zs3: list_nat,Ws: list_nat,P: list_nat > list_a > list_nat > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_a @ Ys2 ) )
     => ( ( ( size_size_list_a @ Ys2 )
          = ( size_size_list_nat @ Zs3 ) )
       => ( ( ( size_size_list_nat @ Zs3 )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P @ nil_nat @ nil_a @ nil_nat @ nil_nat )
           => ( ! [X3: nat,Xs3: list_nat,Y2: a,Ys: list_a,Z2: nat,Zs2: list_nat,W: nat,Ws2: list_nat] :
                  ( ( ( size_size_list_nat @ Xs3 )
                    = ( size_size_list_a @ Ys ) )
                 => ( ( ( size_size_list_a @ Ys )
                      = ( size_size_list_nat @ Zs2 ) )
                   => ( ( ( size_size_list_nat @ Zs2 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P @ Xs3 @ Ys @ Zs2 @ Ws2 )
                       => ( P @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_a @ Y2 @ Ys ) @ ( cons_nat @ Z2 @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys2 @ Zs3 @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_598_list__induct4,axiom,
    ! [Xs: list_nat,Ys2: list_nat,Zs3: list_a,Ws: list_a,P: list_nat > list_nat > list_a > list_a > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys2 ) )
     => ( ( ( size_size_list_nat @ Ys2 )
          = ( size_size_list_a @ Zs3 ) )
       => ( ( ( size_size_list_a @ Zs3 )
            = ( size_size_list_a @ Ws ) )
         => ( ( P @ nil_nat @ nil_nat @ nil_a @ nil_a )
           => ( ! [X3: nat,Xs3: list_nat,Y2: nat,Ys: list_nat,Z2: a,Zs2: list_a,W: a,Ws2: list_a] :
                  ( ( ( size_size_list_nat @ Xs3 )
                    = ( size_size_list_nat @ Ys ) )
                 => ( ( ( size_size_list_nat @ Ys )
                      = ( size_size_list_a @ Zs2 ) )
                   => ( ( ( size_size_list_a @ Zs2 )
                        = ( size_size_list_a @ Ws2 ) )
                     => ( ( P @ Xs3 @ Ys @ Zs2 @ Ws2 )
                       => ( P @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_nat @ Y2 @ Ys ) @ ( cons_a @ Z2 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys2 @ Zs3 @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_599_list__induct4,axiom,
    ! [Xs: list_nat,Ys2: list_nat,Zs3: list_a,Ws: list_nat,P: list_nat > list_nat > list_a > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys2 ) )
     => ( ( ( size_size_list_nat @ Ys2 )
          = ( size_size_list_a @ Zs3 ) )
       => ( ( ( size_size_list_a @ Zs3 )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P @ nil_nat @ nil_nat @ nil_a @ nil_nat )
           => ( ! [X3: nat,Xs3: list_nat,Y2: nat,Ys: list_nat,Z2: a,Zs2: list_a,W: nat,Ws2: list_nat] :
                  ( ( ( size_size_list_nat @ Xs3 )
                    = ( size_size_list_nat @ Ys ) )
                 => ( ( ( size_size_list_nat @ Ys )
                      = ( size_size_list_a @ Zs2 ) )
                   => ( ( ( size_size_list_a @ Zs2 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P @ Xs3 @ Ys @ Zs2 @ Ws2 )
                       => ( P @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_nat @ Y2 @ Ys ) @ ( cons_a @ Z2 @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys2 @ Zs3 @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_600_list__induct4,axiom,
    ! [Xs: list_nat,Ys2: list_nat,Zs3: list_nat,Ws: list_a,P: list_nat > list_nat > list_nat > list_a > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys2 ) )
     => ( ( ( size_size_list_nat @ Ys2 )
          = ( size_size_list_nat @ Zs3 ) )
       => ( ( ( size_size_list_nat @ Zs3 )
            = ( size_size_list_a @ Ws ) )
         => ( ( P @ nil_nat @ nil_nat @ nil_nat @ nil_a )
           => ( ! [X3: nat,Xs3: list_nat,Y2: nat,Ys: list_nat,Z2: nat,Zs2: list_nat,W: a,Ws2: list_a] :
                  ( ( ( size_size_list_nat @ Xs3 )
                    = ( size_size_list_nat @ Ys ) )
                 => ( ( ( size_size_list_nat @ Ys )
                      = ( size_size_list_nat @ Zs2 ) )
                   => ( ( ( size_size_list_nat @ Zs2 )
                        = ( size_size_list_a @ Ws2 ) )
                     => ( ( P @ Xs3 @ Ys @ Zs2 @ Ws2 )
                       => ( P @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_nat @ Y2 @ Ys ) @ ( cons_nat @ Z2 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys2 @ Zs3 @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_601_list__induct4,axiom,
    ! [Xs: list_nat,Ys2: list_nat,Zs3: list_nat,Ws: list_nat,P: list_nat > list_nat > list_nat > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys2 ) )
     => ( ( ( size_size_list_nat @ Ys2 )
          = ( size_size_list_nat @ Zs3 ) )
       => ( ( ( size_size_list_nat @ Zs3 )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P @ nil_nat @ nil_nat @ nil_nat @ nil_nat )
           => ( ! [X3: nat,Xs3: list_nat,Y2: nat,Ys: list_nat,Z2: nat,Zs2: list_nat,W: nat,Ws2: list_nat] :
                  ( ( ( size_size_list_nat @ Xs3 )
                    = ( size_size_list_nat @ Ys ) )
                 => ( ( ( size_size_list_nat @ Ys )
                      = ( size_size_list_nat @ Zs2 ) )
                   => ( ( ( size_size_list_nat @ Zs2 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P @ Xs3 @ Ys @ Zs2 @ Ws2 )
                       => ( P @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_nat @ Y2 @ Ys ) @ ( cons_nat @ Z2 @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys2 @ Zs3 @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_602_list__induct4,axiom,
    ! [Xs: list_nat,Ys2: list_a,Zs3: list_a,Ws: list_l8336264072933094989od_a_o,P: list_nat > list_a > list_a > list_l8336264072933094989od_a_o > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_a @ Ys2 ) )
     => ( ( ( size_size_list_a @ Ys2 )
          = ( size_size_list_a @ Zs3 ) )
       => ( ( ( size_size_list_a @ Zs3 )
            = ( size_s3379185846251495609od_a_o @ Ws ) )
         => ( ( P @ nil_nat @ nil_a @ nil_a @ nil_li1372968520647675725od_a_o )
           => ( ! [X3: nat,Xs3: list_nat,Y2: a,Ys: list_a,Z2: a,Zs2: list_a,W: list_P3196869577824546119od_a_o,Ws2: list_l8336264072933094989od_a_o] :
                  ( ( ( size_size_list_nat @ Xs3 )
                    = ( size_size_list_a @ Ys ) )
                 => ( ( ( size_size_list_a @ Ys )
                      = ( size_size_list_a @ Zs2 ) )
                   => ( ( ( size_size_list_a @ Zs2 )
                        = ( size_s3379185846251495609od_a_o @ Ws2 ) )
                     => ( ( P @ Xs3 @ Ys @ Zs2 @ Ws2 )
                       => ( P @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_a @ Y2 @ Ys ) @ ( cons_a @ Z2 @ Zs2 ) @ ( cons_l9179691953475787517od_a_o @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys2 @ Zs3 @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_603_list__induct4,axiom,
    ! [Xs: list_nat,Ys2: list_a,Zs3: list_l8336264072933094989od_a_o,Ws: list_a,P: list_nat > list_a > list_l8336264072933094989od_a_o > list_a > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_a @ Ys2 ) )
     => ( ( ( size_size_list_a @ Ys2 )
          = ( size_s3379185846251495609od_a_o @ Zs3 ) )
       => ( ( ( size_s3379185846251495609od_a_o @ Zs3 )
            = ( size_size_list_a @ Ws ) )
         => ( ( P @ nil_nat @ nil_a @ nil_li1372968520647675725od_a_o @ nil_a )
           => ( ! [X3: nat,Xs3: list_nat,Y2: a,Ys: list_a,Z2: list_P3196869577824546119od_a_o,Zs2: list_l8336264072933094989od_a_o,W: a,Ws2: list_a] :
                  ( ( ( size_size_list_nat @ Xs3 )
                    = ( size_size_list_a @ Ys ) )
                 => ( ( ( size_size_list_a @ Ys )
                      = ( size_s3379185846251495609od_a_o @ Zs2 ) )
                   => ( ( ( size_s3379185846251495609od_a_o @ Zs2 )
                        = ( size_size_list_a @ Ws2 ) )
                     => ( ( P @ Xs3 @ Ys @ Zs2 @ Ws2 )
                       => ( P @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_a @ Y2 @ Ys ) @ ( cons_l9179691953475787517od_a_o @ Z2 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys2 @ Zs3 @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_604_list__induct4,axiom,
    ! [Xs: list_nat,Ys2: list_a,Zs3: list_l8336264072933094989od_a_o,Ws: list_nat,P: list_nat > list_a > list_l8336264072933094989od_a_o > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_a @ Ys2 ) )
     => ( ( ( size_size_list_a @ Ys2 )
          = ( size_s3379185846251495609od_a_o @ Zs3 ) )
       => ( ( ( size_s3379185846251495609od_a_o @ Zs3 )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P @ nil_nat @ nil_a @ nil_li1372968520647675725od_a_o @ nil_nat )
           => ( ! [X3: nat,Xs3: list_nat,Y2: a,Ys: list_a,Z2: list_P3196869577824546119od_a_o,Zs2: list_l8336264072933094989od_a_o,W: nat,Ws2: list_nat] :
                  ( ( ( size_size_list_nat @ Xs3 )
                    = ( size_size_list_a @ Ys ) )
                 => ( ( ( size_size_list_a @ Ys )
                      = ( size_s3379185846251495609od_a_o @ Zs2 ) )
                   => ( ( ( size_s3379185846251495609od_a_o @ Zs2 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P @ Xs3 @ Ys @ Zs2 @ Ws2 )
                       => ( P @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_a @ Y2 @ Ys ) @ ( cons_l9179691953475787517od_a_o @ Z2 @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys2 @ Zs3 @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_605_Annotation_Osize__gen_I3_J,axiom,
    ( ( multis6502627171228953925tation @ multis5977116398950053002gative )
    = zero_zero_nat ) ).

% Annotation.size_gen(3)
thf(fact_606_Annotation_Osize__gen_I2_J,axiom,
    ( ( multis6502627171228953925tation @ multis8267843437418295310sitive )
    = zero_zero_nat ) ).

% Annotation.size_gen(2)
thf(fact_607_Annotation_Osize__gen_I1_J,axiom,
    ( ( multis6502627171228953925tation @ multis5512618279175645674signed )
    = zero_zero_nat ) ).

% Annotation.size_gen(1)
thf(fact_608_less__nat__zero__code,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_nat_zero_code
thf(fact_609_neq0__conv,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% neq0_conv
thf(fact_610_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ A ) ) ).

% bot_nat_0.not_eq_extremum
thf(fact_611_less__Suc0,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( N = zero_zero_nat ) ) ).

% less_Suc0
thf(fact_612_zero__less__Suc,axiom,
    ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).

% zero_less_Suc
thf(fact_613_nat_Odistinct_I1_J,axiom,
    ! [X23: nat] :
      ( zero_zero_nat
     != ( suc @ X23 ) ) ).

% nat.distinct(1)
thf(fact_614_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat: nat] :
      ( ( suc @ Nat )
     != zero_zero_nat ) ).

% old.nat.distinct(2)
thf(fact_615_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat: nat] :
      ( zero_zero_nat
     != ( suc @ Nat ) ) ).

% old.nat.distinct(1)
thf(fact_616_nat_OdiscI,axiom,
    ! [Nat2: nat,X23: nat] :
      ( ( Nat2
        = ( suc @ X23 ) )
     => ( Nat2 != zero_zero_nat ) ) ).

% nat.discI
thf(fact_617_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat3: nat] :
            ( Y
           != ( suc @ Nat3 ) ) ) ).

% old.nat.exhaust
thf(fact_618_nat__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N2: nat] :
            ( ( P @ N2 )
           => ( P @ ( suc @ N2 ) ) )
       => ( P @ N ) ) ) ).

% nat_induct
thf(fact_619_diff__induct,axiom,
    ! [P: nat > nat > $o,M: nat,N: nat] :
      ( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
     => ( ! [Y2: nat] : ( P @ zero_zero_nat @ ( suc @ Y2 ) )
       => ( ! [X3: nat,Y2: nat] :
              ( ( P @ X3 @ Y2 )
             => ( P @ ( suc @ X3 ) @ ( suc @ Y2 ) ) )
         => ( P @ M @ N ) ) ) ) ).

% diff_induct
thf(fact_620_zero__induct,axiom,
    ! [P: nat > $o,K3: nat] :
      ( ( P @ K3 )
     => ( ! [N2: nat] :
            ( ( P @ ( suc @ N2 ) )
           => ( P @ N2 ) )
       => ( P @ zero_zero_nat ) ) ) ).

% zero_induct
thf(fact_621_Suc__neq__Zero,axiom,
    ! [M: nat] :
      ( ( suc @ M )
     != zero_zero_nat ) ).

% Suc_neq_Zero
thf(fact_622_Zero__neq__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_neq_Suc
thf(fact_623_Zero__not__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_not_Suc
thf(fact_624_not0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ? [M2: nat] :
          ( N
          = ( suc @ M2 ) ) ) ).

% not0_implies_Suc
thf(fact_625_map__Suc__upt,axiom,
    ! [M: nat,N: nat] :
      ( ( map_nat_nat @ suc @ ( upt @ M @ N ) )
      = ( upt @ ( suc @ M ) @ ( suc @ N ) ) ) ).

% map_Suc_upt
thf(fact_626_upt__conv__Cons__Cons,axiom,
    ! [M: nat,N: nat,Ns: list_nat,Q: nat] :
      ( ( ( cons_nat @ M @ ( cons_nat @ N @ Ns ) )
        = ( upt @ M @ Q ) )
      = ( ( cons_nat @ N @ Ns )
        = ( upt @ ( suc @ M ) @ Q ) ) ) ).

% upt_conv_Cons_Cons
thf(fact_627_Ex__less__Suc2,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( suc @ N ) )
            & ( P @ I ) ) )
      = ( ( P @ zero_zero_nat )
        | ? [I: nat] :
            ( ( ord_less_nat @ I @ N )
            & ( P @ ( suc @ I ) ) ) ) ) ).

% Ex_less_Suc2
thf(fact_628_gr0__conv__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
      = ( ? [M3: nat] :
            ( N
            = ( suc @ M3 ) ) ) ) ).

% gr0_conv_Suc
thf(fact_629_All__less__Suc2,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( suc @ N ) )
           => ( P @ I ) ) )
      = ( ( P @ zero_zero_nat )
        & ! [I: nat] :
            ( ( ord_less_nat @ I @ N )
           => ( P @ ( suc @ I ) ) ) ) ) ).

% All_less_Suc2
thf(fact_630_gr0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ? [M2: nat] :
          ( N
          = ( suc @ M2 ) ) ) ).

% gr0_implies_Suc
thf(fact_631_less__Suc__eq__0__disj,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
      = ( ( M = zero_zero_nat )
        | ? [J: nat] :
            ( ( M
              = ( suc @ J ) )
            & ( ord_less_nat @ J @ N ) ) ) ) ).

% less_Suc_eq_0_disj
thf(fact_632_upt__conv__Cons,axiom,
    ! [I2: nat,J2: nat] :
      ( ( ord_less_nat @ I2 @ J2 )
     => ( ( upt @ I2 @ J2 )
        = ( cons_nat @ I2 @ ( upt @ ( suc @ I2 ) @ J2 ) ) ) ) ).

% upt_conv_Cons
thf(fact_633_upt__rec,axiom,
    ( upt
    = ( ^ [I: nat,J: nat] : ( if_list_nat @ ( ord_less_nat @ I @ J ) @ ( cons_nat @ I @ ( upt @ ( suc @ I ) @ J ) ) @ nil_nat ) ) ) ).

% upt_rec
thf(fact_634_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ zero_zero_nat ) ).

% bot_nat_0.extremum_strict
thf(fact_635_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr0I
thf(fact_636_not__gr0,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr0
thf(fact_637_not__less0,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less0
thf(fact_638_less__zeroE,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_zeroE
thf(fact_639_gr__implies__not0,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not0
thf(fact_640_infinite__descent0,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N2: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( ~ ( P @ N2 )
             => ? [M4: nat] :
                  ( ( ord_less_nat @ M4 @ N2 )
                  & ~ ( P @ M4 ) ) ) )
       => ( P @ N ) ) ) ).

% infinite_descent0
thf(fact_641_inf__concat__simple_Ocases,axiom,
    ! [X2: produc8199716216217303280at_nat] :
      ( ! [F2: nat > nat] :
          ( X2
         != ( produc72220940542539688at_nat @ F2 @ zero_zero_nat ) )
     => ~ ! [F2: nat > nat,N2: nat] :
            ( X2
           != ( produc72220940542539688at_nat @ F2 @ ( suc @ N2 ) ) ) ) ).

% inf_concat_simple.cases
thf(fact_642_tl__upt,axiom,
    ! [M: nat,N: nat] :
      ( ( tl_nat @ ( upt @ M @ N ) )
      = ( upt @ ( suc @ M ) @ N ) ) ).

% tl_upt
thf(fact_643_hd__upt,axiom,
    ! [I2: nat,J2: nat] :
      ( ( ord_less_nat @ I2 @ J2 )
     => ( ( hd_nat @ ( upt @ I2 @ J2 ) )
        = I2 ) ) ).

% hd_upt
thf(fact_644_Annotation_Osize_I6_J,axiom,
    ( ( size_s7536720479165623301tation @ multis5977116398950053002gative )
    = zero_zero_nat ) ).

% Annotation.size(6)
thf(fact_645_Annotation_Osize_I5_J,axiom,
    ( ( size_s7536720479165623301tation @ multis8267843437418295310sitive )
    = zero_zero_nat ) ).

% Annotation.size(5)
thf(fact_646_Annotation_Osize_I4_J,axiom,
    ( ( size_s7536720479165623301tation @ multis5512618279175645674signed )
    = zero_zero_nat ) ).

% Annotation.size(4)
thf(fact_647_size__char__eq__0,axiom,
    ( size_size_char
    = ( ^ [C3: char] : zero_zero_nat ) ) ).

% size_char_eq_0
thf(fact_648_upt__rec__numeral,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
       => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
          = ( cons_nat @ ( numeral_numeral_nat @ M ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) ) ) ) )
      & ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
       => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
          = nil_nat ) ) ) ).

% upt_rec_numeral
thf(fact_649_size_H__char__eq__0,axiom,
    ( size_char
    = ( ^ [C3: char] : zero_zero_nat ) ) ).

% size'_char_eq_0
thf(fact_650_map__decr__upt,axiom,
    ! [M: nat,N: nat] :
      ( ( map_nat_nat
        @ ^ [N3: nat] : ( minus_minus_nat @ N3 @ ( suc @ zero_zero_nat ) )
        @ ( upt @ ( suc @ M ) @ ( suc @ N ) ) )
      = ( upt @ M @ N ) ) ).

% map_decr_upt
thf(fact_651_diff__0__eq__0,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% diff_0_eq_0
thf(fact_652_diff__self__eq__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ M )
      = zero_zero_nat ) ).

% diff_self_eq_0
thf(fact_653_length__upt,axiom,
    ! [I2: nat,J2: nat] :
      ( ( size_size_list_nat @ ( upt @ I2 @ J2 ) )
      = ( minus_minus_nat @ J2 @ I2 ) ) ).

% length_upt
thf(fact_654_zero__less__diff,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
      = ( ord_less_nat @ M @ N ) ) ).

% zero_less_diff
thf(fact_655_Suc__pred,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
        = N ) ) ).

% Suc_pred
thf(fact_656_minus__nat_Odiff__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ zero_zero_nat )
      = M ) ).

% minus_nat.diff_0
thf(fact_657_diffs0__imp__equal,axiom,
    ! [M: nat,N: nat] :
      ( ( ( minus_minus_nat @ M @ N )
        = zero_zero_nat )
     => ( ( ( minus_minus_nat @ N @ M )
          = zero_zero_nat )
       => ( M = N ) ) ) ).

% diffs0_imp_equal
thf(fact_658_diff__less,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).

% diff_less
thf(fact_659_diff__Suc__less,axiom,
    ! [N: nat,I2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I2 ) ) @ N ) ) ).

% diff_Suc_less
thf(fact_660_Euclidean__Division_Odivmod__nat__if,axiom,
    ( euclidean_divmod_nat
    = ( ^ [M3: nat,N3: nat] :
          ( if_Pro6206227464963214023at_nat
          @ ( ( N3 = zero_zero_nat )
            | ( ord_less_nat @ M3 @ N3 ) )
          @ ( product_Pair_nat_nat @ zero_zero_nat @ M3 )
          @ ( produc2626176000494625587at_nat
            @ ^ [Q2: nat] : ( product_Pair_nat_nat @ ( suc @ Q2 ) )
            @ ( euclidean_divmod_nat @ ( minus_minus_nat @ M3 @ N3 ) @ N3 ) ) ) ) ) ).

% Euclidean_Division.divmod_nat_if
thf(fact_661_less__one,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ one_one_nat )
      = ( N = zero_zero_nat ) ) ).

% less_one
thf(fact_662_Suc__diff__1,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
        = N ) ) ).

% Suc_diff_1
thf(fact_663_One__nat__def,axiom,
    ( one_one_nat
    = ( suc @ zero_zero_nat ) ) ).

% One_nat_def
thf(fact_664_nat__induct__non__zero,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( P @ one_one_nat )
       => ( ! [N2: nat] :
              ( ( ord_less_nat @ zero_zero_nat @ N2 )
             => ( ( P @ N2 )
               => ( P @ ( suc @ N2 ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_induct_non_zero
thf(fact_665_Suc__diff__eq__diff__pred,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( minus_minus_nat @ ( suc @ M ) @ N )
        = ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% Suc_diff_eq_diff_pred
thf(fact_666_Suc__pred_H,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( N
        = ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% Suc_pred'
thf(fact_667_inf__concat__simple_Osimps_I2_J,axiom,
    ! [F: nat > nat,N: nat] :
      ( ( inf_concat_simple @ F @ ( suc @ N ) )
      = ( produc2626176000494625587at_nat
        @ ^ [I: nat,J: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_nat @ ( suc @ J ) @ ( F @ I ) ) @ ( product_Pair_nat_nat @ I @ ( suc @ J ) ) @ ( product_Pair_nat_nat @ ( suc @ I ) @ zero_zero_nat ) )
        @ ( inf_concat_simple @ F @ N ) ) ) ).

% inf_concat_simple.simps(2)
thf(fact_668_inf__concat__simple_Oelims,axiom,
    ! [X2: nat > nat,Xa: nat,Y: product_prod_nat_nat] :
      ( ( ( inf_concat_simple @ X2 @ Xa )
        = Y )
     => ( ( ( Xa = zero_zero_nat )
         => ( Y
           != ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) )
       => ~ ! [N2: nat] :
              ( ( Xa
                = ( suc @ N2 ) )
             => ( Y
               != ( produc2626176000494625587at_nat
                  @ ^ [I: nat,J: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_nat @ ( suc @ J ) @ ( X2 @ I ) ) @ ( product_Pair_nat_nat @ I @ ( suc @ J ) ) @ ( product_Pair_nat_nat @ ( suc @ I ) @ zero_zero_nat ) )
                  @ ( inf_concat_simple @ X2 @ N2 ) ) ) ) ) ) ).

% inf_concat_simple.elims
thf(fact_669_inf__concat__simple_Osimps_I1_J,axiom,
    ! [F: nat > nat] :
      ( ( inf_concat_simple @ F @ zero_zero_nat )
      = ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) ).

% inf_concat_simple.simps(1)
thf(fact_670_inf__concat__simple__surj__zero,axiom,
    ! [F: nat > nat,I2: nat] :
    ? [K: nat] :
      ( ( inf_concat_simple @ F @ K )
      = ( product_Pair_nat_nat @ I2 @ zero_zero_nat ) ) ).

% inf_concat_simple_surj_zero
thf(fact_671_inf__concat__simple_Opelims,axiom,
    ! [X2: nat > nat,Xa: nat,Y: product_prod_nat_nat] :
      ( ( ( inf_concat_simple @ X2 @ Xa )
        = Y )
     => ( ( accp_P6412015083367660281at_nat @ inf_co2821209343098213861le_rel @ ( produc72220940542539688at_nat @ X2 @ Xa ) )
       => ( ( ( Xa = zero_zero_nat )
           => ( ( Y
                = ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) )
             => ~ ( accp_P6412015083367660281at_nat @ inf_co2821209343098213861le_rel @ ( produc72220940542539688at_nat @ X2 @ zero_zero_nat ) ) ) )
         => ~ ! [N2: nat] :
                ( ( Xa
                  = ( suc @ N2 ) )
               => ( ( Y
                    = ( produc2626176000494625587at_nat
                      @ ^ [I: nat,J: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_nat @ ( suc @ J ) @ ( X2 @ I ) ) @ ( product_Pair_nat_nat @ I @ ( suc @ J ) ) @ ( product_Pair_nat_nat @ ( suc @ I ) @ zero_zero_nat ) )
                      @ ( inf_concat_simple @ X2 @ N2 ) ) )
                 => ~ ( accp_P6412015083367660281at_nat @ inf_co2821209343098213861le_rel @ ( produc72220940542539688at_nat @ X2 @ ( suc @ N2 ) ) ) ) ) ) ) ) ).

% inf_concat_simple.pelims
thf(fact_672_add__is__0,axiom,
    ! [M: nat,N: nat] :
      ( ( ( plus_plus_nat @ M @ N )
        = zero_zero_nat )
      = ( ( M = zero_zero_nat )
        & ( N = zero_zero_nat ) ) ) ).

% add_is_0
thf(fact_673_Nat_Oadd__0__right,axiom,
    ! [M: nat] :
      ( ( plus_plus_nat @ M @ zero_zero_nat )
      = M ) ).

% Nat.add_0_right
thf(fact_674_drop__upt,axiom,
    ! [M: nat,I2: nat,J2: nat] :
      ( ( drop_nat @ M @ ( upt @ I2 @ J2 ) )
      = ( upt @ ( plus_plus_nat @ I2 @ M ) @ J2 ) ) ).

% drop_upt
thf(fact_675_add__gr__0,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M )
        | ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% add_gr_0
thf(fact_676_nth__upt,axiom,
    ! [I2: nat,K3: nat,J2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I2 @ K3 ) @ J2 )
     => ( ( nth_nat @ ( upt @ I2 @ J2 ) @ K3 )
        = ( plus_plus_nat @ I2 @ K3 ) ) ) ).

% nth_upt
thf(fact_677_add__eq__self__zero,axiom,
    ! [M: nat,N: nat] :
      ( ( ( plus_plus_nat @ M @ N )
        = M )
     => ( N = zero_zero_nat ) ) ).

% add_eq_self_zero
thf(fact_678_plus__nat_Oadd__0,axiom,
    ! [N: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ N )
      = N ) ).

% plus_nat.add_0
thf(fact_679_add__is__1,axiom,
    ! [M: nat,N: nat] :
      ( ( ( plus_plus_nat @ M @ N )
        = ( suc @ zero_zero_nat ) )
      = ( ( ( M
            = ( suc @ zero_zero_nat ) )
          & ( N = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% add_is_1
thf(fact_680_one__is__add,axiom,
    ! [M: nat,N: nat] :
      ( ( ( suc @ zero_zero_nat )
        = ( plus_plus_nat @ M @ N ) )
      = ( ( ( M
            = ( suc @ zero_zero_nat ) )
          & ( N = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% one_is_add
thf(fact_681_less__imp__add__positive,axiom,
    ! [I2: nat,J2: nat] :
      ( ( ord_less_nat @ I2 @ J2 )
     => ? [K: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ K )
          & ( ( plus_plus_nat @ I2 @ K )
            = J2 ) ) ) ).

% less_imp_add_positive
thf(fact_682_diff__add__0,axiom,
    ! [N: nat,M: nat] :
      ( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
      = zero_zero_nat ) ).

% diff_add_0
thf(fact_683_nat__diff__split,axiom,
    ! [P: nat > $o,A: nat,B: nat] :
      ( ( P @ ( minus_minus_nat @ A @ B ) )
      = ( ( ( ord_less_nat @ A @ B )
         => ( P @ zero_zero_nat ) )
        & ! [D2: nat] :
            ( ( A
              = ( plus_plus_nat @ B @ D2 ) )
           => ( P @ D2 ) ) ) ) ).

% nat_diff_split
thf(fact_684_nat__diff__split__asm,axiom,
    ! [P: nat > $o,A: nat,B: nat] :
      ( ( P @ ( minus_minus_nat @ A @ B ) )
      = ( ~ ( ( ( ord_less_nat @ A @ B )
              & ~ ( P @ zero_zero_nat ) )
            | ? [D2: nat] :
                ( ( A
                  = ( plus_plus_nat @ B @ D2 ) )
                & ~ ( P @ D2 ) ) ) ) ) ).

% nat_diff_split_asm
thf(fact_685_add__eq__if,axiom,
    ( plus_plus_nat
    = ( ^ [M3: nat,N3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N3 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N3 ) ) ) ) ) ).

% add_eq_if
thf(fact_686_map__add__upt,axiom,
    ! [N: nat,M: nat] :
      ( ( map_nat_nat
        @ ^ [I: nat] : ( plus_plus_nat @ I @ N )
        @ ( upt @ zero_zero_nat @ M ) )
      = ( upt @ N @ ( plus_plus_nat @ M @ N ) ) ) ).

% map_add_upt
thf(fact_687_upt__eq__Cons__conv,axiom,
    ! [I2: nat,J2: nat,X2: nat,Xs: list_nat] :
      ( ( ( upt @ I2 @ J2 )
        = ( cons_nat @ X2 @ Xs ) )
      = ( ( ord_less_nat @ I2 @ J2 )
        & ( I2 = X2 )
        & ( ( upt @ ( plus_plus_nat @ I2 @ one_one_nat ) @ J2 )
          = Xs ) ) ) ).

% upt_eq_Cons_conv
thf(fact_688_Euclid__induct,axiom,
    ! [P: nat > nat > $o,A: nat,B: nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( P @ A3 @ B3 )
          = ( P @ B3 @ A3 ) )
     => ( ! [A3: nat] : ( P @ A3 @ zero_zero_nat )
       => ( ! [A3: nat,B3: nat] :
              ( ( P @ A3 @ B3 )
             => ( P @ A3 @ ( plus_plus_nat @ A3 @ B3 ) ) )
         => ( P @ A @ B ) ) ) ) ).

% Euclid_induct
thf(fact_689_last__upt,axiom,
    ! [I2: nat,J2: nat] :
      ( ( ord_less_nat @ I2 @ J2 )
     => ( ( last_nat @ ( upt @ I2 @ J2 ) )
        = ( minus_minus_nat @ J2 @ one_one_nat ) ) ) ).

% last_upt
thf(fact_690_le0,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% le0
thf(fact_691_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).

% bot_nat_0.extremum
thf(fact_692_diff__is__0__eq_H,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( minus_minus_nat @ M @ N )
        = zero_zero_nat ) ) ).

% diff_is_0_eq'
thf(fact_693_diff__is__0__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ( minus_minus_nat @ M @ N )
        = zero_zero_nat )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% diff_is_0_eq
thf(fact_694_take__upt,axiom,
    ! [I2: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ M ) @ N )
     => ( ( take_nat @ M @ ( upt @ I2 @ N ) )
        = ( upt @ I2 @ ( plus_plus_nat @ I2 @ M ) ) ) ) ).

% take_upt
thf(fact_695_upt__conv__Nil,axiom,
    ! [J2: nat,I2: nat] :
      ( ( ord_less_eq_nat @ J2 @ I2 )
     => ( ( upt @ I2 @ J2 )
        = nil_nat ) ) ).

% upt_conv_Nil
thf(fact_696_upt__eq__Nil__conv,axiom,
    ! [I2: nat,J2: nat] :
      ( ( ( upt @ I2 @ J2 )
        = nil_nat )
      = ( ( J2 = zero_zero_nat )
        | ( ord_less_eq_nat @ J2 @ I2 ) ) ) ).

% upt_eq_Nil_conv
thf(fact_697_ex__least__nat__le,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ N )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K: nat] :
            ( ( ord_less_eq_nat @ K @ N )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K )
               => ~ ( P @ I3 ) )
            & ( P @ K ) ) ) ) ).

% ex_least_nat_le
thf(fact_698_upt__add__eq__append,axiom,
    ! [I2: nat,J2: nat,K3: nat] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ( upt @ I2 @ ( plus_plus_nat @ J2 @ K3 ) )
        = ( append_nat @ ( upt @ I2 @ J2 ) @ ( upt @ J2 @ ( plus_plus_nat @ J2 @ K3 ) ) ) ) ) ).

% upt_add_eq_append
thf(fact_699_le__0__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_0_eq
thf(fact_700_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_uniqueI
thf(fact_701_bot__nat__0_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
      = ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_unique
thf(fact_702_less__eq__nat_Osimps_I1_J,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% less_eq_nat.simps(1)
thf(fact_703_ex__least__nat__less,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ N )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K: nat] :
            ( ( ord_less_nat @ K @ N )
            & ! [I3: nat] :
                ( ( ord_less_eq_nat @ I3 @ K )
               => ~ ( P @ I3 ) )
            & ( P @ ( suc @ K ) ) ) ) ) ).

% ex_least_nat_less
thf(fact_704_upt__Suc,axiom,
    ! [I2: nat,J2: nat] :
      ( ( ( ord_less_eq_nat @ I2 @ J2 )
       => ( ( upt @ I2 @ ( suc @ J2 ) )
          = ( append_nat @ ( upt @ I2 @ J2 ) @ ( cons_nat @ J2 @ nil_nat ) ) ) )
      & ( ~ ( ord_less_eq_nat @ I2 @ J2 )
       => ( ( upt @ I2 @ ( suc @ J2 ) )
          = nil_nat ) ) ) ).

% upt_Suc
thf(fact_705_upt__Suc__append,axiom,
    ! [I2: nat,J2: nat] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ( upt @ I2 @ ( suc @ J2 ) )
        = ( append_nat @ ( upt @ I2 @ J2 ) @ ( cons_nat @ J2 @ nil_nat ) ) ) ) ).

% upt_Suc_append

% Helper facts (7)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X2: nat,Y: nat] :
      ( ( if_nat @ $false @ X2 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X2: nat,Y: nat] :
      ( ( if_nat @ $true @ X2 @ Y )
      = X2 ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X2: list_nat,Y: list_nat] :
      ( ( if_list_nat @ $false @ X2 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X2: list_nat,Y: list_nat] :
      ( ( if_list_nat @ $true @ X2 @ Y )
      = X2 ) ).

thf(help_If_3_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X2: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $false @ X2 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X2: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $true @ X2 @ Y )
      = X2 ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ( left
    = ( concat6161509665823917772on_nat
      @ ( map_a_3887198048842487703on_nat
        @ ^ [X: a] : ( cons_S6797538523646374167on_nat @ ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ X @ multis8267843437418295310sitive ) ) @ ( cons_S6797538523646374167on_nat @ ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ X @ multis5977116398950053002gative ) ) @ nil_Su3078906344068544711on_nat ) )
        @ xs ) ) ) ).

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