%------------------------------------------------------------------------------
% 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    : VYDRA_MDL/0011_Monitor/prob_00578_026655__17021920_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 2139 ( 588 unt; 880 typ;   0 def)
%            Number of atoms       : 3708 (1925 equ;   0 cnn)
%            Maximal formula atoms :   13 (   2 avg)
%            Number of connectives : 12540 ( 462   ~;  80   |; 451   &;10072   @)
%                                         (   0 <=>;1475  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   26 (   7 avg)
%            Number of types       :  221 ( 220 usr)
%            Number of type conns  : 1869 (1869   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  663 ( 660 usr;  48 con; 0-7 aty)
%            Number of variables   : 4244 ( 247   ^;3624   !; 373   ?;4244   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 14:54:34.613
%------------------------------------------------------------------------------
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thf(ty_n_t__Option__Ooption_Itf__t_J,type,
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thf(ty_n_t__Option__Ooption_Itf__a_J,type,
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thf(ty_n_t__IArray__Oiarray_Itf__t_J,type,
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thf(ty_n_t__IArray__Oiarray_I_Eo_J,type,
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thf(ty_n_t__List__Olist_Itf__t_J,type,
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thf(ty_n_t__List__Olist_Itf__h_J,type,
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thf(ty_n_t__List__Olist_Itf__a_J,type,
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thf(ty_n_t__Set__Oset_Itf__t_J,type,
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thf(ty_n_t__Set__Oset_Itf__h_J,type,
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thf(ty_n_t__Set__Oset_Itf__a_J,type,
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thf(ty_n_t__List__Olist_I_Eo_J,type,
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thf(ty_n_t__Set__Oset_I_Eo_J,type,
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thf(ty_n_t__Nat__Onat,type,
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thf(ty_n_tf__t,type,
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thf(ty_n_tf__h,type,
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thf(ty_n_tf__a,type,
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% Explicit typings (660)
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Option__Ooption_It__Nat__Onat_J,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_IArray_Oiarray_OIArray_001t__IArray__Oiarray_I_Eo_J,type,
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thf(sy_c_IArray_Oiarray_OIArray_001t__MDL__Oformula_Itf__a_Mtf__t_J,type,
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thf(sy_c_IArray_Oiarray_OIArray_001t__Monitor__Ovydra____aux_Itf__a_Mtf__t_Mtf__h_J,type,
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thf(sy_c_IArray_Oiarray_OIArray_001t__Product____Type__Oprod_Itf__t_Mt__Set__Oset_Itf__a_J_J,type,
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thf(sy_c_IArray_Oiarray_OIArray_001tf__t,type,
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thf(sy_c_IArray_Olength_001_Eo,type,
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thf(sy_c_IArray_Olength_001t__IArray__Oiarray_I_Eo_J,type,
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thf(sy_c_IArray_Olength_001t__MDL__Oformula_Itf__a_Mtf__t_J,type,
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thf(sy_c_IArray_Olength_001t__Monitor__Ovydra____aux_Itf__a_Mtf__t_Mtf__h_J,type,
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thf(sy_c_IArray_Olength_001t__Product____Type__Oprod_Itf__t_Mt__Set__Oset_Itf__a_J_J,type,
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thf(sy_c_IArray_Olength_001tf__t,type,
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thf(sy_c_IArray_Osub_001_Eo,type,
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thf(sy_c_IArray_Osub_001t__IArray__Oiarray_I_Eo_J,type,
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thf(sy_c_IArray_Osub_001t__MDL__Oformula_Itf__a_Mtf__t_J,type,
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thf(sy_c_IArray_Osub_001t__Monitor__Ovydra____aux_Itf__a_Mtf__t_Mtf__h_J,type,
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thf(sy_c_IArray_Osub_001t__Product____Type__Oprod_Itf__t_Mt__Set__Oset_Itf__a_J_J,type,
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thf(sy_c_IArray_Osub_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_IArray_Osub_001tf__t,type,
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thf(sy_c_Interval_Omem_001tf__t,type,
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thf(sy_c_List_Oappend_001_Eo,type,
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thf(sy_c_List_Oappend_001t__IArray__Oiarray_I_Eo_J,type,
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thf(sy_c_List_Oappend_001t__MDL__Oformula_Itf__a_Mtf__t_J,type,
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thf(sy_c_List_Oappend_001t__Monitor__Ovydra____aux_Itf__a_Mtf__t_Mtf__h_J,type,
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thf(sy_c_List_Oappend_001t__Nat__Onat,type,
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thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__IArray__Oiarray_I_Eo_J_Mt__IArray__Oiarray_I_Eo_J_J,type,
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thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__IArray__Oiarray_I_Eo_J_Mt__MDL__Oformula_Itf__a_Mtf__t_J_J,type,
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thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__IArray__Oiarray_I_Eo_J_Mtf__t_J,type,
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thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__MDL__Oformula_Itf__a_Mtf__t_J_Mt__IArray__Oiarray_I_Eo_J_J,type,
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thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__MDL__Oformula_Itf__a_Mtf__t_J_Mtf__t_J,type,
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thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Monitor__Ovydra____aux_Itf__a_Mtf__t_Mtf__h_J_Mtf__t_J,type,
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thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__t_Mt__Set__Oset_Itf__a_J_J_Mtf__t_J,type,
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thf(sy_c_List_Oappend_001t__Product____Type__Oprod_Itf__t_M_Eo_J,type,
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thf(sy_c_List_Oappend_001t__Product____Type__Oprod_Itf__t_Mt__IArray__Oiarray_I_Eo_J_J,type,
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thf(sy_c_List_Oappend_001t__Product____Type__Oprod_Itf__t_Mt__MDL__Oformula_Itf__a_Mtf__t_J_J,type,
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thf(sy_c_List_Oappend_001t__Product____Type__Oprod_Itf__t_Mt__Set__Oset_Itf__a_J_J,type,
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thf(sy_c_List_Oappend_001t__Product____Type__Oprod_Itf__t_Mtf__t_J,type,
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thf(sy_c_List_Oappend_001tf__a,type,
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thf(sy_c_List_Oappend_001tf__t,type,
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thf(sy_c_List_Odrop_001_Eo,type,
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thf(sy_c_List_Odrop_001t__IArray__Oiarray_I_Eo_J,type,
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thf(sy_c_List_Odrop_001t__MDL__Oformula_Itf__a_Mtf__t_J,type,
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thf(sy_c_List_Odrop_001t__Monitor__Ovydra____aux_Itf__a_Mtf__t_Mtf__h_J,type,
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thf(sy_c_List_Odrop_001t__Nat__Onat,type,
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thf(sy_c_List_Odrop_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Odrop_001t__Product____Type__Oprod_Itf__t_M_Eo_J,type,
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thf(sy_c_List_Odrop_001t__Product____Type__Oprod_Itf__t_Mt__Set__Oset_Itf__a_J_J,type,
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thf(sy_c_List_Odrop_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_List_Odrop_001tf__a,type,
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thf(sy_c_List_Odrop_001tf__t,type,
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thf(sy_c_List_Ofind_001t__IArray__Oiarray_I_Eo_J,type,
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thf(sy_c_List_Ofind_001t__MDL__Oformula_Itf__a_Mtf__t_J,type,
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thf(sy_c_List_Ofind_001t__Monitor__Ovydra____aux_Itf__a_Mtf__t_Mtf__h_J,type,
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thf(sy_c_List_Ofind_001t__Nat__Onat,type,
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thf(sy_c_List_Ofind_001t__Product____Type__Oprod_It__List__Olist_It__Monitor__Ovydra____aux_Itf__a_Mtf__t_Mtf__h_J_J_Mt__IArray__Oiarray_I_Eo_J_J,type,
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thf(sy_c_List_Ofind_001t__Product____Type__Oprod_Itf__t_Mt__Set__Oset_Itf__a_J_J,type,
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thf(sy_c_List_Ofind_001tf__a,type,
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thf(sy_c_List_Ofind_001tf__t,type,
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thf(sy_c_List_Olex_001t__IArray__Oiarray_I_Eo_J,type,
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thf(sy_c_List_Olex_001t__MDL__Oformula_Itf__a_Mtf__t_J,type,
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thf(sy_c_List_Olex_001t__Monitor__Ovydra____aux_Itf__a_Mtf__t_Mtf__h_J,type,
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thf(sy_c_List_Olex_001t__Nat__Onat,type,
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thf(sy_c_List_Olist_OCons_001_Eo,type,
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thf(sy_c_List_Olist_OCons_001t__IArray__Oiarray_I_Eo_J,type,
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thf(sy_c_List_Ozip_001tf__h_001tf__t,type,
    zip_h_t: list_h > list_t > list_P7520416276666729081od_h_t ).

thf(sy_c_List_Ozip_001tf__t_001_Eo,type,
    zip_t_o: list_t > list_o > list_P5575445741472356340od_t_o ).

thf(sy_c_List_Ozip_001tf__t_001t__IArray__Oiarray_I_Eo_J,type,
    zip_t_iarray_o: list_t > list_iarray_o > list_P7303666633494002394rray_o ).

thf(sy_c_List_Ozip_001tf__t_001t__MDL__Oformula_Itf__a_Mtf__t_J,type,
    zip_t_formula_a_t: list_t > list_formula_a_t > list_P4688085876415078003la_a_t ).

thf(sy_c_List_Ozip_001tf__t_001t__Nat__Onat,type,
    zip_t_nat: list_t > list_nat > list_P8463946597425410208_t_nat ).

thf(sy_c_List_Ozip_001tf__t_001t__Set__Oset_Itf__a_J,type,
    zip_t_set_a: list_t > list_set_a > list_P1095827921439705018_set_a ).

thf(sy_c_List_Ozip_001tf__t_001tf__a,type,
    zip_t_a: list_t > list_a > list_P7766353545381090906od_t_a ).

thf(sy_c_List_Ozip_001tf__t_001tf__t,type,
    zip_t_t: list_t > list_t > list_P9116000306261664621od_t_t ).

thf(sy_c_MDL_OMDL_Omatch_001tf__a_001tf__t,type,
    match_a_t: trace_a_t > regex_a_t > set_Pr1261947904930325089at_nat ).

thf(sy_c_MDL_OMDL_Osat_001tf__a_001tf__t,type,
    sat_a_t: trace_a_t > formula_a_t > nat > $o ).

thf(sy_c_MDL_Obounded__future__fmla_001tf__a_001tf__t,type,
    bounde209865433540700070la_a_t: formula_a_t > $o ).

thf(sy_c_MDL_Oeps_001tf__a_001tf__t,type,
    eps_a_t: regex_a_t > $o ).

thf(sy_c_MDL_Oformula_OAtom_001tf__a_001tf__t,type,
    atom_a_t: a > formula_a_t ).

thf(sy_c_MDL_Oformula_OBool_001tf__a_001tf__t,type,
    bool_a_t: $o > formula_a_t ).

thf(sy_c_MDL_Oformula_OMatchF_001tf__t_001tf__a,type,
    matchF_t_a: i_t > regex_a_t > formula_a_t ).

thf(sy_c_MDL_Oformula_OMatchP_001tf__t_001tf__a,type,
    matchP_t_a: i_t > regex_a_t > formula_a_t ).

thf(sy_c_MDL_Oformula_ONeg_001tf__a_001tf__t,type,
    neg_a_t: formula_a_t > formula_a_t ).

thf(sy_c_MDL_Oformula_ONext_001tf__t_001tf__a,type,
    next_t_a: i_t > formula_a_t > formula_a_t ).

thf(sy_c_MDL_Oformula_OPrev_001tf__t_001tf__a,type,
    prev_t_a: i_t > formula_a_t > formula_a_t ).

thf(sy_c_MDL_Orderive_001tf__a_001tf__t,type,
    rderive_a_t: regex_a_t > regex_a_t ).

thf(sy_c_MDL_Oregex_OStar_001tf__a_001tf__t,type,
    star_a_t: regex_a_t > regex_a_t ).

thf(sy_c_MDL_Owf__fmla_001tf__a_001tf__t,type,
    wf_fmla_a_t: formula_a_t > $o ).

thf(sy_c_MDL_Owf__regex_001tf__a_001tf__t,type,
    wf_regex_a_t: regex_a_t > $o ).

thf(sy_c_Monitor_OVYDRA__MDL_001tf__a_001tf__t_001tf__h,type,
    vYDRA_MDL_a_t_h: trace_a_t > h > ( h > option5775791656947114332_set_a ) > $o ).

thf(sy_c_Monitor_OVYDRA__MDL_Olast__before,type,
    vYDRA_last_before: ( nat > $o ) > nat > option_nat ).

thf(sy_c_Monitor_OVYDRA__MDL_Olast__before__rel,type,
    vYDRA_5175910486691664417re_rel: produc3074792404157404414_o_nat > produc3074792404157404414_o_nat > $o ).

thf(sy_c_Monitor_OVYDRA__MDL_Owf__vydra_001tf__a_001tf__t_001tf__h,type,
    vYDRA_wf_vydra_a_t_h: trace_a_t > h > ( h > option5775791656947114332_set_a ) > formula_a_t > nat > nat > vydra_aux_a_t_h > $o ).

thf(sy_c_Monitor_Oinit__vydra_001tf__h_001tf__t_001tf__a,type,
    init_vydra_h_t_a: h > ( h > option5775791656947114332_set_a ) > formula_a_t > produc5559660867765305033_a_t_h ).

thf(sy_c_Monitor_Omsize__fmla_001tf__a_001tf__t,type,
    msize_fmla_a_t: formula_a_t > nat ).

thf(sy_c_Monitor_Omsize__regex_001tf__a_001tf__t,type,
    msize_regex_a_t: regex_a_t > nat ).

thf(sy_c_Monitor_Oread__t_001_062_It__Nat__Onat_M_Eo_J_001t__Nat__Onat,type,
    read_t_nat_o_nat: option7421465107808516430_o_nat > option_nat ).

thf(sy_c_Monitor_Oread__t_001tf__h_001tf__t,type,
    read_t_h_t: option2110264179601050681od_h_t > option_t ).

thf(sy_c_Monitor_Orun_001tf__h_001tf__t_001tf__a,type,
    run_h_t_a: ( h > option5775791656947114332_set_a ) > nat > vydra_aux_a_t_h > option4580262698044125979od_t_o ).

thf(sy_c_Monitor_Orun__subs_001t__IArray__Oiarray_I_Eo_J_001tf__h_001tf__t_001t__Set__Oset_Itf__a_J,type,
    run_su853597514479236164_set_a: ( iarray_o > option5775791656947114332_set_a ) > list_iarray_o > option8761023800807099238_set_a ).

thf(sy_c_Monitor_Orun__subs_001t__IArray__Oiarray_I_Eo_J_001tf__t_001tf__h_001tf__t,type,
    run_su3655969966392431119_t_h_t: ( iarray_o > option6366625828353349007od_h_t ) > list_iarray_o > option2595117943684164237rray_t ).

thf(sy_c_Monitor_Orun__subs_001t__IArray__Oiarray_I_Eo_J_001tf__t_001tf__t_001_Eo,type,
    run_su98717407436750474_t_t_o: ( iarray_o > option5115755630998872842od_t_o ) > list_iarray_o > option6074195846726219284rray_o ).

thf(sy_c_Monitor_Orun__subs_001t__IArray__Oiarray_I_Eo_J_001tf__t_001tf__t_001t__Set__Oset_Itf__a_J,type,
    run_su6940939646075227728_set_a: ( iarray_o > option4136621186271108688_set_a ) > list_iarray_o > option325762455365084250_set_a ).

thf(sy_c_Monitor_Orun__subs_001t__MDL__Oformula_Itf__a_Mtf__t_J_001tf__t_001tf__h_001tf__t,type,
    run_su4105110139124697544_t_h_t: ( formula_a_t > option6366625828353349007od_h_t ) > list_formula_a_t > option2595117943684164237rray_t ).

thf(sy_c_Monitor_Orun__subs_001t__MDL__Oformula_Itf__a_Mtf__t_J_001tf__t_001tf__t_001_Eo,type,
    run_su7792916424282555075_t_t_o: ( formula_a_t > option5115755630998872842od_t_o ) > list_formula_a_t > option6074195846726219284rray_o ).

thf(sy_c_Monitor_Orun__subs_001t__Monitor__Ovydra____aux_Itf__a_Mtf__t_Mtf__h_J_001t__Monitor__Ovydra____aux_Itf__a_Mtf__t_Mtf__h_J_001tf__t_001_Eo,type,
    run_su3116026553151384105_h_t_o: ( vydra_aux_a_t_h > option4580262698044125979od_t_o ) > list_vydra_aux_a_t_h > option8553266494716013595rray_o ).

thf(sy_c_Monitor_Orun__subs_001tf__t_001tf__h_001tf__t_001t__Set__Oset_Itf__a_J,type,
    run_subs_t_h_t_set_a: ( t > option5775791656947114332_set_a ) > list_t > option8761023800807099238_set_a ).

thf(sy_c_Monitor_Orun__subs_001tf__t_001tf__t_001tf__h_001tf__t,type,
    run_subs_t_t_h_t: ( t > option6366625828353349007od_h_t ) > list_t > option2595117943684164237rray_t ).

thf(sy_c_Monitor_Orun__subs_001tf__t_001tf__t_001tf__t_001_Eo,type,
    run_subs_t_t_t_o: ( t > option5115755630998872842od_t_o ) > list_t > option6074195846726219284rray_o ).

thf(sy_c_Monitor_Orun__subs_001tf__t_001tf__t_001tf__t_001t__Set__Oset_Itf__a_J,type,
    run_subs_t_t_t_set_a: ( t > option4136621186271108688_set_a ) > list_t > option325762455365084250_set_a ).

thf(sy_c_Monitor_Orun__t_001tf__h_001tf__t_001tf__a,type,
    run_t_h_t_a: ( h > option5775791656947114332_set_a ) > option2110264179601050681od_h_t > option4241632866121027589_h_t_t ).

thf(sy_c_Monitor_Osub_001tf__h_001tf__t_001tf__a,type,
    sub_h_t_a: h > ( h > option5775791656947114332_set_a ) > nat > formula_a_t > vydra_aux_a_t_h ).

thf(sy_c_Monitor_Ot0_001tf__h_001tf__t_001tf__a,type,
    t0_h_t_a: h > ( h > option5775791656947114332_set_a ) > option2110264179601050681od_h_t ).

thf(sy_c_Monitor_Ovydra__aux_OVYDRA__Atom_001tf__a_001tf__h_001tf__t,type,
    vydra_3724560892389469594_a_h_t: a > h > vydra_aux_a_t_h ).

thf(sy_c_Monitor_Ovydra__aux_OVYDRA__Bool_001tf__h_001tf__a_001tf__t,type,
    vydra_2060541234293701615_h_a_t: $o > h > vydra_aux_a_t_h ).

thf(sy_c_Monitor_Ovydra__aux_OVYDRA__Neg_001tf__a_001tf__t_001tf__h,type,
    vydra_2331423191276895391_a_t_h: vydra_aux_a_t_h > vydra_aux_a_t_h ).

thf(sy_c_Monitor_Ovydra__aux_OVYDRA__None_001tf__a_001tf__t_001tf__h,type,
    vydra_4195442553028166619_a_t_h: vydra_aux_a_t_h ).

thf(sy_c_Monitor_Ovydra__aux_OVYDRA__Prev_001tf__t_001tf__a_001tf__h,type,
    vydra_3189297434896079910_t_a_h: i_t > vydra_aux_a_t_h > h > option8291060355354214836od_t_o > vydra_aux_a_t_h ).

thf(sy_c_NFA_Oiarray__of__list_001_Eo,type,
    iarray_of_list_o: list_o > iarray_o ).

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

thf(sy_c_Nat_Onat_Ocase__nat_001_Eo,type,
    case_nat_o: $o > ( nat > $o ) > nat > $o ).

thf(sy_c_Nat_Onat_Ocase__nat_001t__Nat__Onat,type,
    case_nat_nat: nat > ( nat > nat ) > nat > nat ).

thf(sy_c_Nat_Onat_Ocase__nat_001t__Option__Ooption_It__Product____Type__Oprod_Itf__t_M_Eo_J_J,type,
    case_n7548220105206534174od_t_o: option8291060355354214836od_t_o > ( nat > option8291060355354214836od_t_o ) > nat > option8291060355354214836od_t_o ).

thf(sy_c_Nat_Onat_Opred,type,
    pred: nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_I_062_It__Nat__Onat_M_Eo_J_J,type,
    size_size_list_nat_o: list_nat_o > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_I_Eo_J,type,
    size_size_list_o: list_o > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__IArray__Oiarray_I_Eo_J_J,type,
    size_s3243743627286111600rray_o: list_iarray_o > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__MDL__Oformula_Itf__a_Mtf__t_J_J,type,
    size_s8846756101701226951la_a_t: list_formula_a_t > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Monitor__Ovydra____aux_Itf__a_Mtf__t_Mtf__h_J_J,type,
    size_s8993897887676352572_a_t_h: list_vydra_aux_a_t_h > 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__List__Olist_It__Monitor__Ovydra____aux_Itf__a_Mtf__t_Mtf__h_J_J_Mt__IArray__Oiarray_I_Eo_J_J_J,type,
    size_s5675016999574124143rray_o: list_P1634102618077251035rray_o > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    size_s5460976970255530739at_nat: list_P6011104703257516679at_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_Itf__h_Mt__Product____Type__Oprod_Itf__t_Mt__Set__Oset_Itf__a_J_J_J_J,type,
    size_s2286189404704077320_set_a: list_P2035273040109328284_set_a > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_Itf__h_Mtf__t_J_J,type,
    size_s785782387481697509od_h_t: list_P7520416276666729081od_h_t > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_Itf__t_M_Eo_J_J,type,
    size_s4106575409306904672od_t_o: list_P5575445741472356340od_t_o > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_Itf__t_Mt__Set__Oset_Itf__a_J_J_J,type,
    size_s5362121731612881446_set_a: list_P1095827921439705018_set_a > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Set__Oset_Itf__a_J_J,type,
    size_size_list_set_a: list_set_a > 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__List__Olist_Itf__h_J,type,
    size_size_list_h: list_h > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__t_J,type,
    size_size_list_t: list_t > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__MDL__Oformula_Itf__a_Mtf__t_J,type,
    size_s4016968051272393527la_a_t: formula_a_t > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__MDL__Oregex_Itf__a_Mtf__t_J,type,
    size_size_regex_a_t: regex_a_t > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__Monitor__Ovydra____aux_Itf__a_Mtf__t_Mtf__h_J,type,
    size_s1533667335963039788_a_t_h: vydra_aux_a_t_h > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Nat__Onat_J,type,
    size_size_option_nat: option_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Product____Type__Oprod_It__List__Olist_It__Monitor__Ovydra____aux_Itf__a_Mtf__t_Mtf__h_J_J_Mt__IArray__Oiarray_I_Eo_J_J_J,type,
    size_s2704686997974730927rray_o: option8553266494716013595rray_o > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Product____Type__Oprod_It__Monitor__Ovydra____aux_Itf__a_Mtf__t_Mtf__h_J_Mt__Product____Type__Oprod_Itf__t_M_Eo_J_J_J,type,
    size_s8621177079540999087od_t_o: option4580262698044125979od_t_o > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Product____Type__Oprod_Itf__h_Mt__Product____Type__Oprod_Itf__t_Mt__Set__Oset_Itf__a_J_J_J_J,type,
    size_s5864537446109811656_set_a: option5775791656947114332_set_a > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Product____Type__Oprod_Itf__h_Mtf__t_J_J,type,
    size_s6536553198265154213od_h_t: option2110264179601050681od_h_t > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Product____Type__Oprod_Itf__t_M_Eo_J_J,type,
    size_s4900030450070283808od_t_o: option8291060355354214836od_t_o > nat ).

thf(sy_c_Option_Ooption_ONone_001t__MDL__Oformula_Itf__a_Mtf__t_J,type,
    none_formula_a_t: option_formula_a_t ).

thf(sy_c_Option_Ooption_ONone_001t__Nat__Onat,type,
    none_nat: option_nat ).

thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_I_062_It__Nat__Onat_M_Eo_J_Mt__Nat__Onat_J,type,
    none_P4770284390835442525_o_nat: option7421465107808516430_o_nat ).

thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__List__Olist_It__Monitor__Ovydra____aux_Itf__a_Mtf__t_Mtf__h_J_J_Mt__IArray__Oiarray_I_Eo_J_J,type,
    none_P945442321178488234rray_o: option8553266494716013595rray_o ).

thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Monitor__Ovydra____aux_Itf__a_Mtf__t_Mtf__h_J_Mt__Product____Type__Oprod_Itf__t_M_Eo_J_J,type,
    none_P3452577240289636010od_t_o: option4580262698044125979od_t_o ).

thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    none_P5556105721700978146at_nat: option4927543243414619207at_nat ).

thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Option__Ooption_It__Product____Type__Oprod_Itf__h_Mtf__t_J_J_Mtf__t_J,type,
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thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_Itf__h_Mt__Product____Type__Oprod_Itf__t_Mt__Set__Oset_Itf__a_J_J_J,type,
    none_P2868508613186038903_set_a: option5775791656947114332_set_a ).

thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_Itf__h_Mtf__t_J,type,
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thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_Itf__t_M_Eo_J,type,
    none_P7032077165169035727od_t_o: option8291060355354214836od_t_o ).

thf(sy_c_Option_Ooption_ONone_001tf__a,type,
    none_a: option_a ).

thf(sy_c_Option_Ooption_ONone_001tf__t,type,
    none_t: option_t ).

thf(sy_c_Option_Ooption_OSome_001t__IArray__Oiarray_I_Eo_J,type,
    some_iarray_o: iarray_o > option_iarray_o ).

thf(sy_c_Option_Ooption_OSome_001t__MDL__Oformula_Itf__a_Mtf__t_J,type,
    some_formula_a_t: formula_a_t > option_formula_a_t ).

thf(sy_c_Option_Ooption_OSome_001t__Monitor__Ovydra____aux_Itf__a_Mtf__t_Mtf__h_J,type,
    some_vydra_aux_a_t_h: vydra_aux_a_t_h > option4336620133012679656_a_t_h ).

thf(sy_c_Option_Ooption_OSome_001t__Nat__Onat,type,
    some_nat: nat > option_nat ).

thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_I_062_It__Nat__Onat_M_Eo_J_Mt__Nat__Onat_J,type,
    some_P2630223654056210137_o_nat: produc3074792404157404414_o_nat > option7421465107808516430_o_nat ).

thf(sy_c_Option_Ooption_OSome_001t__Product____Type__Oprod_It__List__Olist_It__Monitor__Ovydra____aux_Itf__a_Mtf__t_Mtf__h_J_J_Mt__IArray__Oiarray_I_Eo_J_J,type,
    some_P7279874174034738982rray_o: produc8228095596854190539rray_o > option8553266494716013595rray_o ).

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thf(sy_v__092_060phi_062,type,
    phi: formula_a_t ).

thf(sy_v__092_060sigma_062,type,
    sigma: trace_a_t ).

thf(sy_v_b,type,
    b: $o ).

thf(sy_v_ba____,type,
    ba: $o ).

thf(sy_v_bb____,type,
    bb: $o ).

thf(sy_v_i,type,
    i: nat ).

thf(sy_v_ia____,type,
    ia: nat ).

thf(sy_v_init__hd,type,
    init_hd: h ).

thf(sy_v_n,type,
    n: nat ).

thf(sy_v_na____,type,
    na: nat ).

thf(sy_v_run__hd,type,
    run_hd: h > option5775791656947114332_set_a ).

thf(sy_v_t,type,
    t2: t ).

thf(sy_v_ta____,type,
    ta: t ).

thf(sy_v_tb____,type,
    tb: t ).

thf(sy_v_v,type,
    v: vydra_aux_a_t_h ).

thf(sy_v_v_H,type,
    v2: vydra_aux_a_t_h ).

thf(sy_v_v_Ha____,type,
    v_a: vydra_aux_a_t_h ).

thf(sy_v_v_Hb____,type,
    v_b: vydra_aux_a_t_h ).

thf(sy_v_va____,type,
    va: vydra_aux_a_t_h ).

thf(sy_v_vb____,type,
    vb: vydra_aux_a_t_h ).

thf(sy_v_x____,type,
    x: formula_a_t ).

% Relevant facts (1256)
thf(fact_0_VYDRA__MDL__axioms,axiom,
    vYDRA_MDL_a_t_h @ sigma @ init_hd @ run_hd ).

% VYDRA_MDL_axioms
thf(fact_1_VYDRA__MDL_Owf__vydra_Ocong,axiom,
    vYDRA_wf_vydra_a_t_h = vYDRA_wf_vydra_a_t_h ).

% VYDRA_MDL.wf_vydra.cong
thf(fact_2_assms_I3_J,axiom,
    ( ( run_h_t_a @ run_hd @ n @ v )
    = ( some_P9127227083975130150od_t_o @ ( produc5956389175556509053od_t_o @ v2 @ ( product_Pair_t_o @ t2 @ b ) ) ) ) ).

% assms(3)
thf(fact_3_assms_I2_J,axiom,
    vYDRA_wf_vydra_a_t_h @ sigma @ init_hd @ run_hd @ phi @ i @ n @ v ).

% assms(2)
thf(fact_4_option_Oinject,axiom,
    ! [X2: produc8228095596854190539rray_o,Y2: produc8228095596854190539rray_o] :
      ( ( ( some_P7279874174034738982rray_o @ X2 )
        = ( some_P7279874174034738982rray_o @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% option.inject
thf(fact_5_option_Oinject,axiom,
    ! [X2: product_prod_t_o,Y2: product_prod_t_o] :
      ( ( ( some_P1805796590424124243od_t_o @ X2 )
        = ( some_P1805796590424124243od_t_o @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% option.inject
thf(fact_6_option_Oinject,axiom,
    ! [X2: produc8090089739038349590_set_a,Y2: produc8090089739038349590_set_a] :
      ( ( ( some_P2245991037696317435_set_a @ X2 )
        = ( some_P2245991037696317435_set_a @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% option.inject
thf(fact_7_option_Oinject,axiom,
    ! [X2: product_prod_h_t,Y2: product_prod_h_t] :
      ( ( ( some_P2526622376857543512od_h_t @ X2 )
        = ( some_P2526622376857543512od_h_t @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% option.inject
thf(fact_8_option_Oinject,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ( some_nat @ X2 )
        = ( some_nat @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% option.inject
thf(fact_9_nat_Oinject,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ( suc @ X2 )
        = ( suc @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% nat.inject
thf(fact_10_old_Onat_Oinject,axiom,
    ! [Nat: nat,Nat2: nat] :
      ( ( ( suc @ Nat )
        = ( suc @ Nat2 ) )
      = ( Nat = Nat2 ) ) ).

% old.nat.inject
thf(fact_11_prod_Oinject,axiom,
    ! [X1: nat > $o,X2: nat,Y1: nat > $o,Y2: nat] :
      ( ( ( produc7277522915581678840_o_nat @ X1 @ X2 )
        = ( produc7277522915581678840_o_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_12_prod_Oinject,axiom,
    ! [X1: t,X2: $o,Y1: t,Y2: $o] :
      ( ( ( product_Pair_t_o @ X1 @ X2 )
        = ( product_Pair_t_o @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_13_prod_Oinject,axiom,
    ! [X1: t,X2: set_a,Y1: t,Y2: set_a] :
      ( ( ( product_Pair_t_set_a @ X1 @ X2 )
        = ( product_Pair_t_set_a @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_14_prod_Oinject,axiom,
    ! [X1: h,X2: product_prod_t_set_a,Y1: h,Y2: product_prod_t_set_a] :
      ( ( ( produc3432067189782488838_set_a @ X1 @ X2 )
        = ( produc3432067189782488838_set_a @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_15_prod_Oinject,axiom,
    ! [X1: h,X2: t,Y1: h,Y2: t] :
      ( ( ( product_Pair_h_t @ X1 @ X2 )
        = ( product_Pair_h_t @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_16_old_Oprod_Oinject,axiom,
    ! [A: nat > $o,B: nat,A2: nat > $o,B2: nat] :
      ( ( ( produc7277522915581678840_o_nat @ A @ B )
        = ( produc7277522915581678840_o_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_17_old_Oprod_Oinject,axiom,
    ! [A: t,B: $o,A2: t,B2: $o] :
      ( ( ( product_Pair_t_o @ A @ B )
        = ( product_Pair_t_o @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_18_old_Oprod_Oinject,axiom,
    ! [A: t,B: set_a,A2: t,B2: set_a] :
      ( ( ( product_Pair_t_set_a @ A @ B )
        = ( product_Pair_t_set_a @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_19_old_Oprod_Oinject,axiom,
    ! [A: h,B: product_prod_t_set_a,A2: h,B2: product_prod_t_set_a] :
      ( ( ( produc3432067189782488838_set_a @ A @ B )
        = ( produc3432067189782488838_set_a @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_20_old_Oprod_Oinject,axiom,
    ! [A: h,B: t,A2: h,B2: t] :
      ( ( ( product_Pair_h_t @ A @ B )
        = ( product_Pair_h_t @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_21_reaches__on__inj,axiom,
    ! [Run: vydra_aux_a_t_h > option4580262698044125979od_t_o,S: vydra_aux_a_t_h,Vs: list_P5575445741472356340od_t_o,T: vydra_aux_a_t_h,Vs2: list_P5575445741472356340od_t_o,T2: vydra_aux_a_t_h] :
      ( ( reache8691153155344877136od_t_o @ Run @ S @ Vs @ T )
     => ( ( reache8691153155344877136od_t_o @ Run @ S @ Vs2 @ T2 )
       => ( ( ( size_s4106575409306904672od_t_o @ Vs )
            = ( size_s4106575409306904672od_t_o @ Vs2 ) )
         => ( ( Vs = Vs2 )
            & ( T = T2 ) ) ) ) ) ).

% reaches_on_inj
thf(fact_22_reaches__on__inj,axiom,
    ! [Run: h > option5775791656947114332_set_a,S: h,Vs: list_P1095827921439705018_set_a,T: h,Vs2: list_P1095827921439705018_set_a,T2: h] :
      ( ( reache3397181163414509427_set_a @ Run @ S @ Vs @ T )
     => ( ( reache3397181163414509427_set_a @ Run @ S @ Vs2 @ T2 )
       => ( ( ( size_s5362121731612881446_set_a @ Vs )
            = ( size_s5362121731612881446_set_a @ Vs2 ) )
         => ( ( Vs = Vs2 )
            & ( T = T2 ) ) ) ) ) ).

% reaches_on_inj
thf(fact_23_reaches__on__inj,axiom,
    ! [Run: list_vydra_aux_a_t_h > option8553266494716013595rray_o,S: list_vydra_aux_a_t_h,Vs: list_iarray_o,T: list_vydra_aux_a_t_h,Vs2: list_iarray_o,T2: list_vydra_aux_a_t_h] :
      ( ( reache2401979074735462992rray_o @ Run @ S @ Vs @ T )
     => ( ( reache2401979074735462992rray_o @ Run @ S @ Vs2 @ T2 )
       => ( ( ( size_s3243743627286111600rray_o @ Vs )
            = ( size_s3243743627286111600rray_o @ Vs2 ) )
         => ( ( Vs = Vs2 )
            & ( T = T2 ) ) ) ) ) ).

% reaches_on_inj
thf(fact_24_reaches__on__inj,axiom,
    ! [Run: option2110264179601050681od_h_t > option4241632866121027589_h_t_t,S: option2110264179601050681od_h_t,Vs: list_t,T: option2110264179601050681od_h_t,Vs2: list_t,T2: option2110264179601050681od_h_t] :
      ( ( reache645946503560514460_h_t_t @ Run @ S @ Vs @ T )
     => ( ( reache645946503560514460_h_t_t @ Run @ S @ Vs2 @ T2 )
       => ( ( ( size_size_list_t @ Vs )
            = ( size_size_list_t @ Vs2 ) )
         => ( ( Vs = Vs2 )
            & ( T = T2 ) ) ) ) ) ).

% reaches_on_inj
thf(fact_25_prod__cases3,axiom,
    ! [Y: produc8090089739038349590_set_a] :
      ~ ! [A3: h,B3: t,C: set_a] :
          ( Y
         != ( produc3432067189782488838_set_a @ A3 @ ( product_Pair_t_set_a @ B3 @ C ) ) ) ).

% prod_cases3
thf(fact_26_assms_I4_J,axiom,
    bounde209865433540700070la_a_t @ phi ).

% assms(4)
thf(fact_27_assms_I5_J,axiom,
    wf_fmla_a_t @ phi ).

% assms(5)
thf(fact_28_Pair__inject,axiom,
    ! [A: nat > $o,B: nat,A2: nat > $o,B2: nat] :
      ( ( ( produc7277522915581678840_o_nat @ A @ B )
        = ( produc7277522915581678840_o_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_29_Pair__inject,axiom,
    ! [A: t,B: $o,A2: t,B2: $o] :
      ( ( ( product_Pair_t_o @ A @ B )
        = ( product_Pair_t_o @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B = ~ B2 ) ) ) ).

% Pair_inject
thf(fact_30_Pair__inject,axiom,
    ! [A: t,B: set_a,A2: t,B2: set_a] :
      ( ( ( product_Pair_t_set_a @ A @ B )
        = ( product_Pair_t_set_a @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_31_Pair__inject,axiom,
    ! [A: h,B: product_prod_t_set_a,A2: h,B2: product_prod_t_set_a] :
      ( ( ( produc3432067189782488838_set_a @ A @ B )
        = ( produc3432067189782488838_set_a @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_32_Pair__inject,axiom,
    ! [A: h,B: t,A2: h,B2: t] :
      ( ( ( product_Pair_h_t @ A @ B )
        = ( product_Pair_h_t @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_33_prod__cases,axiom,
    ! [P: produc3074792404157404414_o_nat > $o,P2: produc3074792404157404414_o_nat] :
      ( ! [A3: nat > $o,B3: nat] : ( P @ ( produc7277522915581678840_o_nat @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_34_prod__cases,axiom,
    ! [P: product_prod_t_o > $o,P2: product_prod_t_o] :
      ( ! [A3: t,B3: $o] : ( P @ ( product_Pair_t_o @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_35_prod__cases,axiom,
    ! [P: product_prod_t_set_a > $o,P2: product_prod_t_set_a] :
      ( ! [A3: t,B3: set_a] : ( P @ ( product_Pair_t_set_a @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_36_prod__cases,axiom,
    ! [P: produc8090089739038349590_set_a > $o,P2: produc8090089739038349590_set_a] :
      ( ! [A3: h,B3: product_prod_t_set_a] : ( P @ ( produc3432067189782488838_set_a @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_37_prod__cases,axiom,
    ! [P: product_prod_h_t > $o,P2: product_prod_h_t] :
      ( ! [A3: h,B3: t] : ( P @ ( product_Pair_h_t @ A3 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_38_surj__pair,axiom,
    ! [P2: produc3074792404157404414_o_nat] :
    ? [X: nat > $o,Y3: nat] :
      ( P2
      = ( produc7277522915581678840_o_nat @ X @ Y3 ) ) ).

% surj_pair
thf(fact_39_surj__pair,axiom,
    ! [P2: product_prod_t_o] :
    ? [X: t,Y3: $o] :
      ( P2
      = ( product_Pair_t_o @ X @ Y3 ) ) ).

% surj_pair
thf(fact_40_surj__pair,axiom,
    ! [P2: product_prod_t_set_a] :
    ? [X: t,Y3: set_a] :
      ( P2
      = ( product_Pair_t_set_a @ X @ Y3 ) ) ).

% surj_pair
thf(fact_41_surj__pair,axiom,
    ! [P2: produc8090089739038349590_set_a] :
    ? [X: h,Y3: product_prod_t_set_a] :
      ( P2
      = ( produc3432067189782488838_set_a @ X @ Y3 ) ) ).

% surj_pair
thf(fact_42_surj__pair,axiom,
    ! [P2: product_prod_h_t] :
    ? [X: h,Y3: t] :
      ( P2
      = ( product_Pair_h_t @ X @ Y3 ) ) ).

% surj_pair
thf(fact_43_old_Oprod_Oexhaust,axiom,
    ! [Y: produc3074792404157404414_o_nat] :
      ~ ! [A3: nat > $o,B3: nat] :
          ( Y
         != ( produc7277522915581678840_o_nat @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_44_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_t_o] :
      ~ ! [A3: t,B3: $o] :
          ( Y
         != ( product_Pair_t_o @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_45_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_t_set_a] :
      ~ ! [A3: t,B3: set_a] :
          ( Y
         != ( product_Pair_t_set_a @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_46_old_Oprod_Oexhaust,axiom,
    ! [Y: produc8090089739038349590_set_a] :
      ~ ! [A3: h,B3: product_prod_t_set_a] :
          ( Y
         != ( produc3432067189782488838_set_a @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_47_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_h_t] :
      ~ ! [A3: h,B3: t] :
          ( Y
         != ( product_Pair_h_t @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_48_n__not__Suc__n,axiom,
    ! [N: nat] :
      ( N
     != ( suc @ N ) ) ).

% n_not_Suc_n
thf(fact_49_Suc__inject,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ( suc @ X3 )
        = ( suc @ Y ) )
     => ( X3 = Y ) ) ).

% Suc_inject
thf(fact_50_size__neq__size__imp__neq,axiom,
    ! [X3: list_P1095827921439705018_set_a,Y: list_P1095827921439705018_set_a] :
      ( ( ( size_s5362121731612881446_set_a @ X3 )
       != ( size_s5362121731612881446_set_a @ Y ) )
     => ( X3 != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_51_size__neq__size__imp__neq,axiom,
    ! [X3: list_vydra_aux_a_t_h,Y: list_vydra_aux_a_t_h] :
      ( ( ( size_s8993897887676352572_a_t_h @ X3 )
       != ( size_s8993897887676352572_a_t_h @ Y ) )
     => ( X3 != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_52_size__neq__size__imp__neq,axiom,
    ! [X3: list_iarray_o,Y: list_iarray_o] :
      ( ( ( size_s3243743627286111600rray_o @ X3 )
       != ( size_s3243743627286111600rray_o @ Y ) )
     => ( X3 != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_53_size__neq__size__imp__neq,axiom,
    ! [X3: list_formula_a_t,Y: list_formula_a_t] :
      ( ( ( size_s8846756101701226951la_a_t @ X3 )
       != ( size_s8846756101701226951la_a_t @ Y ) )
     => ( X3 != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_54_size__neq__size__imp__neq,axiom,
    ! [X3: list_t,Y: list_t] :
      ( ( ( size_size_list_t @ X3 )
       != ( size_size_list_t @ Y ) )
     => ( X3 != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_55_prod__induct3,axiom,
    ! [P: produc8090089739038349590_set_a > $o,X3: produc8090089739038349590_set_a] :
      ( ! [A3: h,B3: t,C: set_a] : ( P @ ( produc3432067189782488838_set_a @ A3 @ ( product_Pair_t_set_a @ B3 @ C ) ) )
     => ( P @ X3 ) ) ).

% prod_induct3
thf(fact_56_reaches__on__run__hd,axiom,
    ! [Es: list_P1095827921439705018_set_a,S: h,S2: h,T: t,X4: set_a] :
      ( ( reache3397181163414509427_set_a @ run_hd @ init_hd @ Es @ S )
     => ( ( ( run_hd @ S )
          = ( some_P2245991037696317435_set_a @ ( produc3432067189782488838_set_a @ S2 @ ( product_Pair_t_set_a @ T @ X4 ) ) ) )
       => ( ( T
            = ( tau_a_t @ sigma @ ( size_s5362121731612881446_set_a @ Es ) ) )
          & ( X4
            = ( gamma_a_t @ sigma @ ( size_s5362121731612881446_set_a @ Es ) ) ) ) ) ) ).

% reaches_on_run_hd
thf(fact_57_VYDRA__MDL_Oreaches__on__run__hd,axiom,
    ! [Sigma: trace_a_t,Init_hd: h,Run_hd: h > option5775791656947114332_set_a,Es: list_P1095827921439705018_set_a,S: h,S2: h,T: t,X4: set_a] :
      ( ( vYDRA_MDL_a_t_h @ Sigma @ Init_hd @ Run_hd )
     => ( ( reache3397181163414509427_set_a @ Run_hd @ Init_hd @ Es @ S )
       => ( ( ( Run_hd @ S )
            = ( some_P2245991037696317435_set_a @ ( produc3432067189782488838_set_a @ S2 @ ( product_Pair_t_set_a @ T @ X4 ) ) ) )
         => ( ( T
              = ( tau_a_t @ Sigma @ ( size_s5362121731612881446_set_a @ Es ) ) )
            & ( X4
              = ( gamma_a_t @ Sigma @ ( size_s5362121731612881446_set_a @ Es ) ) ) ) ) ) ) ).

% VYDRA_MDL.reaches_on_run_hd
thf(fact_58_assms_I1_J,axiom,
    ord_less_eq_nat @ ( msize_fmla_a_t @ phi ) @ n ).

% assms(1)
thf(fact_59_run__hd__sound,axiom,
    ! [N: nat,S: h,S2: h,T: t,X4: set_a] :
      ( ( reache8480716032150177652_set_a @ run_hd @ init_hd @ N @ S )
     => ( ( ( run_hd @ S )
          = ( some_P2245991037696317435_set_a @ ( produc3432067189782488838_set_a @ S2 @ ( product_Pair_t_set_a @ T @ X4 ) ) ) )
       => ( ( product_Pair_t_set_a @ T @ X4 )
          = ( product_Pair_t_set_a @ ( tau_a_t @ sigma @ N ) @ ( gamma_a_t @ sigma @ N ) ) ) ) ) ).

% run_hd_sound
thf(fact_60_Neg_Oprems_I1_J,axiom,
    vYDRA_wf_vydra_a_t_h @ sigma @ init_hd @ run_hd @ ( neg_a_t @ x ) @ ia @ ( suc @ na ) @ va ).

% Neg.prems(1)
thf(fact_61_sat_Osimps_I3_J,axiom,
    ! [Phi: formula_a_t,I: nat] :
      ( ( sat_a_t @ sigma @ ( neg_a_t @ Phi ) @ I )
      = ( ~ ( sat_a_t @ sigma @ Phi @ I ) ) ) ).

% sat.simps(3)
thf(fact_62_sat_Osimps_I1_J,axiom,
    ! [B: $o,I: nat] :
      ( ( sat_a_t @ sigma @ ( bool_a_t @ B ) @ I )
      = B ) ).

% sat.simps(1)
thf(fact_63_ru__t__event,axiom,
    ! [T: option2110264179601050681od_h_t,Ts: list_t,T2: option2110264179601050681od_h_t,T3: option2110264179601050681od_h_t,X3: t] :
      ( ( reache645946503560514460_h_t_t @ ( run_t_h_t_a @ run_hd ) @ T @ Ts @ T2 )
     => ( ( T
          = ( t0_h_t_a @ init_hd @ run_hd ) )
       => ( ( ( run_t_h_t_a @ run_hd @ T2 )
            = ( some_P8406129549712030756_h_t_t @ ( produc770200333395398319_h_t_t @ T3 @ X3 ) ) )
         => ? [Rho: list_P1095827921439705018_set_a,E: h,Tt: t] :
              ( ( T2
                = ( some_P2526622376857543512od_h_t @ ( product_Pair_h_t @ E @ Tt ) ) )
              & ( reache3397181163414509427_set_a @ run_hd @ init_hd @ Rho @ E )
              & ( ( size_s5362121731612881446_set_a @ Rho )
                = ( suc @ ( size_size_list_t @ Ts ) ) )
              & ( X3
                = ( tau_a_t @ sigma @ ( size_size_list_t @ Ts ) ) ) ) ) ) ) ).

% ru_t_event
thf(fact_64_wf__vydra_Ointros_I3_J,axiom,
    ! [Es: list_P1095827921439705018_set_a,Sub: h,I: nat,B: $o,N: nat] :
      ( ( reache3397181163414509427_set_a @ run_hd @ init_hd @ Es @ Sub )
     => ( ( ( size_s5362121731612881446_set_a @ Es )
          = I )
       => ( vYDRA_wf_vydra_a_t_h @ sigma @ init_hd @ run_hd @ ( bool_a_t @ B ) @ I @ N @ ( vydra_2060541234293701615_h_a_t @ B @ Sub ) ) ) ) ).

% wf_vydra.intros(3)
thf(fact_65_wf__vydra_Ointros_I4_J,axiom,
    ! [Es: list_P1095827921439705018_set_a,Sub: h,I: nat,A: a,N: nat] :
      ( ( reache3397181163414509427_set_a @ run_hd @ init_hd @ Es @ Sub )
     => ( ( ( size_s5362121731612881446_set_a @ Es )
          = I )
       => ( vYDRA_wf_vydra_a_t_h @ sigma @ init_hd @ run_hd @ ( atom_a_t @ A ) @ I @ N @ ( vydra_3724560892389469594_a_h_t @ A @ Sub ) ) ) ) ).

% wf_vydra.intros(4)
thf(fact_66_mem__Collect__eq,axiom,
    ! [A: a,P: a > $o] :
      ( ( member_a @ A @ ( collect_a @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_67_mem__Collect__eq,axiom,
    ! [A: formula_a_t,P: formula_a_t > $o] :
      ( ( member_formula_a_t @ A @ ( collect_formula_a_t @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_68_mem__Collect__eq,axiom,
    ! [A: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ A @ ( collec3392354462482085612at_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_69_mem__Collect__eq,axiom,
    ! [A: nat,P: nat > $o] :
      ( ( member_nat @ A @ ( collect_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_70_Collect__mem__eq,axiom,
    ! [A4: set_a] :
      ( ( collect_a
        @ ^ [X5: a] : ( member_a @ X5 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_71_Collect__mem__eq,axiom,
    ! [A4: set_formula_a_t] :
      ( ( collect_formula_a_t
        @ ^ [X5: formula_a_t] : ( member_formula_a_t @ X5 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_72_Collect__mem__eq,axiom,
    ! [A4: set_Pr1261947904930325089at_nat] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X5: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X5 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_73_Collect__mem__eq,axiom,
    ! [A4: set_nat] :
      ( ( collect_nat
        @ ^ [X5: nat] : ( member_nat @ X5 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_74_Neg_OIH,axiom,
    ! [I: nat,V: vydra_aux_a_t_h,V2: vydra_aux_a_t_h,T: t,B: $o] :
      ( ( vYDRA_wf_vydra_a_t_h @ sigma @ init_hd @ run_hd @ x @ I @ na @ V )
     => ( ( ( run_h_t_a @ run_hd @ na @ V )
          = ( some_P9127227083975130150od_t_o @ ( produc5956389175556509053od_t_o @ V2 @ ( product_Pair_t_o @ T @ B ) ) ) )
       => ( ( bounde209865433540700070la_a_t @ x )
         => ( ( wf_fmla_a_t @ x )
           => ( ( vYDRA_wf_vydra_a_t_h @ sigma @ init_hd @ run_hd @ x @ ( suc @ I ) @ na @ V2 )
              & ? [Es2: list_P1095827921439705018_set_a,E: h] :
                  ( ( reache3397181163414509427_set_a @ run_hd @ init_hd @ Es2 @ E )
                  & ( ( size_s5362121731612881446_set_a @ Es2 )
                    = ( suc @ I ) ) )
              & ( T
                = ( tau_a_t @ sigma @ I ) )
              & ( B
                = ( sat_a_t @ sigma @ x @ I ) ) ) ) ) ) ) ).

% Neg.IH
thf(fact_75_Neg_Oprems_I4_J,axiom,
    wf_fmla_a_t @ ( neg_a_t @ x ) ).

% Neg.prems(4)
thf(fact_76_Neg_Oprems_I3_J,axiom,
    bounde209865433540700070la_a_t @ ( neg_a_t @ x ) ).

% Neg.prems(3)
thf(fact_77_Neg_Ohyps,axiom,
    ord_less_eq_nat @ ( msize_fmla_a_t @ x ) @ na ).

% Neg.hyps
thf(fact_78_vydra__aux_Oinject_I1_J,axiom,
    ! [X21: $o,X22: h,Y21: $o,Y22: h] :
      ( ( ( vydra_2060541234293701615_h_a_t @ X21 @ X22 )
        = ( vydra_2060541234293701615_h_a_t @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% vydra_aux.inject(1)
thf(fact_79_vydra__aux_Oinject_I2_J,axiom,
    ! [X31: a,X32: h,Y31: a,Y32: h] :
      ( ( ( vydra_3724560892389469594_a_h_t @ X31 @ X32 )
        = ( vydra_3724560892389469594_a_h_t @ Y31 @ Y32 ) )
      = ( ( X31 = Y31 )
        & ( X32 = Y32 ) ) ) ).

% vydra_aux.inject(2)
thf(fact_80_sat_Osimps_I2_J,axiom,
    ! [A: a,I: nat] :
      ( ( sat_a_t @ sigma @ ( atom_a_t @ A ) @ I )
      = ( member_a @ A @ ( gamma_a_t @ sigma @ I ) ) ) ).

% sat.simps(2)
thf(fact_81_Suc__le__mono,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
      = ( ord_less_eq_nat @ N @ M ) ) ).

% Suc_le_mono
thf(fact_82_ru__t__tau,axiom,
    ! [Ts: list_t,T2: option2110264179601050681od_h_t,T3: option2110264179601050681od_h_t,X3: t] :
      ( ( reache645946503560514460_h_t_t @ ( run_t_h_t_a @ run_hd ) @ ( t0_h_t_a @ init_hd @ run_hd ) @ Ts @ T2 )
     => ( ( ( run_t_h_t_a @ run_hd @ T2 )
          = ( some_P8406129549712030756_h_t_t @ ( produc770200333395398319_h_t_t @ T3 @ X3 ) ) )
       => ( X3
          = ( tau_a_t @ sigma @ ( size_size_list_t @ Ts ) ) ) ) ) ).

% ru_t_tau
thf(fact_83_ru__t__Some__tau,axiom,
    ! [Ts: list_t,E2: h,T: t] :
      ( ( reache645946503560514460_h_t_t @ ( run_t_h_t_a @ run_hd ) @ ( t0_h_t_a @ init_hd @ run_hd ) @ Ts @ ( some_P2526622376857543512od_h_t @ ( product_Pair_h_t @ E2 @ T ) ) )
     => ( T
        = ( tau_a_t @ sigma @ ( size_size_list_t @ Ts ) ) ) ) ).

% ru_t_Some_tau
thf(fact_84_ru__t__Some,axiom,
    ! [Fs: list_P1095827921439705018_set_a,F: h,I: nat,Es: list_t,E2: option2110264179601050681od_h_t] :
      ( ( reache3397181163414509427_set_a @ run_hd @ init_hd @ Fs @ F )
     => ( ( ( size_s5362121731612881446_set_a @ Fs )
          = ( suc @ I ) )
       => ( ( reache645946503560514460_h_t_t @ ( run_t_h_t_a @ run_hd ) @ ( t0_h_t_a @ init_hd @ run_hd ) @ Es @ E2 )
         => ( ( ord_less_eq_nat @ ( size_size_list_t @ Es ) @ I )
           => ? [E3: option2110264179601050681od_h_t,Et: t] :
                ( ( run_t_h_t_a @ run_hd @ E2 )
                = ( some_P8406129549712030756_h_t_t @ ( produc770200333395398319_h_t_t @ E3 @ Et ) ) ) ) ) ) ) ).

% ru_t_Some
thf(fact_85_Neg_Oprems_I2_J,axiom,
    ( ( run_h_t_a @ run_hd @ ( suc @ na ) @ va )
    = ( some_P9127227083975130150od_t_o @ ( produc5956389175556509053od_t_o @ v_a @ ( product_Pair_t_o @ ta @ ba ) ) ) ) ).

% Neg.prems(2)
thf(fact_86_le__refl,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).

% le_refl
thf(fact_87_le__trans,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ J @ K )
       => ( ord_less_eq_nat @ I @ K ) ) ) ).

% le_trans
thf(fact_88_eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( M = N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% eq_imp_le
thf(fact_89_le__antisym,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_eq_nat @ N @ M )
       => ( M = N ) ) ) ).

% le_antisym
thf(fact_90_nat__le__linear,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
      | ( ord_less_eq_nat @ N @ M ) ) ).

% nat_le_linear
thf(fact_91_Nat_Oex__has__greatest__nat,axiom,
    ! [P: nat > $o,K: nat,B: nat] :
      ( ( P @ K )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B ) )
       => ? [X: nat] :
            ( ( P @ X )
            & ! [Y4: nat] :
                ( ( P @ Y4 )
               => ( ord_less_eq_nat @ Y4 @ X ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_92_vydra__aux_Odistinct_I21_J,axiom,
    ! [X21: $o,X22: h,X31: a,X32: h] :
      ( ( vydra_2060541234293701615_h_a_t @ X21 @ X22 )
     != ( vydra_3724560892389469594_a_h_t @ X31 @ X32 ) ) ).

% vydra_aux.distinct(21)
thf(fact_93_msize__fmla_Osimps_I3_J,axiom,
    ! [Phi2: formula_a_t] :
      ( ( msize_fmla_a_t @ ( neg_a_t @ Phi2 ) )
      = ( suc @ ( msize_fmla_a_t @ Phi2 ) ) ) ).

% msize_fmla.simps(3)
thf(fact_94_lift__Suc__antimono__le,axiom,
    ! [F: nat > nat,N: nat,N2: nat] :
      ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
     => ( ( ord_less_eq_nat @ N @ N2 )
       => ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ N ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_95_lift__Suc__mono__le,axiom,
    ! [F: nat > nat,N: nat,N2: nat] :
      ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_eq_nat @ N @ N2 )
       => ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_96_Suc__leD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% Suc_leD
thf(fact_97_le__SucE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
     => ( ~ ( ord_less_eq_nat @ M @ N )
       => ( M
          = ( suc @ N ) ) ) ) ).

% le_SucE
thf(fact_98_le__SucI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).

% le_SucI
thf(fact_99_Suc__le__D,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ M2 )
     => ? [M3: nat] :
          ( M2
          = ( suc @ M3 ) ) ) ).

% Suc_le_D
thf(fact_100_le__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N ) )
      = ( ( ord_less_eq_nat @ M @ N )
        | ( M
          = ( suc @ N ) ) ) ) ).

% le_Suc_eq
thf(fact_101_Suc__n__not__le__n,axiom,
    ! [N: nat] :
      ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).

% Suc_n_not_le_n
thf(fact_102_not__less__eq__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ~ ( ord_less_eq_nat @ M @ N ) )
      = ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).

% not_less_eq_eq
thf(fact_103_full__nat__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N3: nat] :
          ( ! [M4: nat] :
              ( ( ord_less_eq_nat @ ( suc @ M4 ) @ N3 )
             => ( P @ M4 ) )
         => ( P @ N3 ) )
     => ( P @ N ) ) ).

% full_nat_induct
thf(fact_104_nat__induct__at__least,axiom,
    ! [M: nat,N: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( P @ M )
       => ( ! [N3: nat] :
              ( ( ord_less_eq_nat @ M @ N3 )
             => ( ( P @ N3 )
               => ( P @ ( suc @ N3 ) ) ) )
         => ( P @ N ) ) ) ) ).

% nat_induct_at_least
thf(fact_105_transitive__stepwise__le,axiom,
    ! [M: nat,N: nat,R: nat > nat > $o] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ! [X: nat] : ( R @ X @ X )
       => ( ! [X: nat,Y3: nat,Z: nat] :
              ( ( R @ X @ Y3 )
             => ( ( R @ Y3 @ Z )
               => ( R @ X @ Z ) ) )
         => ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
           => ( R @ M @ N ) ) ) ) ) ).

% transitive_stepwise_le
thf(fact_106_VYDRA__MDL_Oru__t__tau,axiom,
    ! [Sigma: trace_a_t,Init_hd: h,Run_hd: h > option5775791656947114332_set_a,Ts: list_t,T2: option2110264179601050681od_h_t,T3: option2110264179601050681od_h_t,X3: t] :
      ( ( vYDRA_MDL_a_t_h @ Sigma @ Init_hd @ Run_hd )
     => ( ( reache645946503560514460_h_t_t @ ( run_t_h_t_a @ Run_hd ) @ ( t0_h_t_a @ Init_hd @ Run_hd ) @ Ts @ T2 )
       => ( ( ( run_t_h_t_a @ Run_hd @ T2 )
            = ( some_P8406129549712030756_h_t_t @ ( produc770200333395398319_h_t_t @ T3 @ X3 ) ) )
         => ( X3
            = ( tau_a_t @ Sigma @ ( size_size_list_t @ Ts ) ) ) ) ) ) ).

% VYDRA_MDL.ru_t_tau
thf(fact_107_VYDRA__MDL__def,axiom,
    ( vYDRA_MDL_a_t_h
    = ( ^ [Sigma2: trace_a_t,Init_hd2: h,Run_hd2: h > option5775791656947114332_set_a] :
        ! [N4: nat,S3: h,S4: h,T4: t,X6: set_a] :
          ( ( reache8480716032150177652_set_a @ Run_hd2 @ Init_hd2 @ N4 @ S3 )
         => ( ( ( Run_hd2 @ S3 )
              = ( some_P2245991037696317435_set_a @ ( produc3432067189782488838_set_a @ S4 @ ( product_Pair_t_set_a @ T4 @ X6 ) ) ) )
           => ( ( product_Pair_t_set_a @ T4 @ X6 )
              = ( product_Pair_t_set_a @ ( tau_a_t @ Sigma2 @ N4 ) @ ( gamma_a_t @ Sigma2 @ N4 ) ) ) ) ) ) ) ).

% VYDRA_MDL_def
thf(fact_108_VYDRA__MDL_Orun__hd__sound,axiom,
    ! [Sigma: trace_a_t,Init_hd: h,Run_hd: h > option5775791656947114332_set_a,N: nat,S: h,S2: h,T: t,X4: set_a] :
      ( ( vYDRA_MDL_a_t_h @ Sigma @ Init_hd @ Run_hd )
     => ( ( reache8480716032150177652_set_a @ Run_hd @ Init_hd @ N @ S )
       => ( ( ( Run_hd @ S )
            = ( some_P2245991037696317435_set_a @ ( produc3432067189782488838_set_a @ S2 @ ( product_Pair_t_set_a @ T @ X4 ) ) ) )
         => ( ( product_Pair_t_set_a @ T @ X4 )
            = ( product_Pair_t_set_a @ ( tau_a_t @ Sigma @ N ) @ ( gamma_a_t @ Sigma @ N ) ) ) ) ) ) ).

% VYDRA_MDL.run_hd_sound
thf(fact_109_VYDRA__MDL_Ointro,axiom,
    ! [Run_hd: h > option5775791656947114332_set_a,Init_hd: h,Sigma: trace_a_t] :
      ( ! [N3: nat,S5: h,S6: h,T5: t,X7: set_a] :
          ( ( reache8480716032150177652_set_a @ Run_hd @ Init_hd @ N3 @ S5 )
         => ( ( ( Run_hd @ S5 )
              = ( some_P2245991037696317435_set_a @ ( produc3432067189782488838_set_a @ S6 @ ( product_Pair_t_set_a @ T5 @ X7 ) ) ) )
           => ( ( product_Pair_t_set_a @ T5 @ X7 )
              = ( product_Pair_t_set_a @ ( tau_a_t @ Sigma @ N3 ) @ ( gamma_a_t @ Sigma @ N3 ) ) ) ) )
     => ( vYDRA_MDL_a_t_h @ Sigma @ Init_hd @ Run_hd ) ) ).

% VYDRA_MDL.intro
thf(fact_110_VYDRA__MDL_Oru__t__Some,axiom,
    ! [Sigma: trace_a_t,Init_hd: h,Run_hd: h > option5775791656947114332_set_a,Fs: list_P1095827921439705018_set_a,F: h,I: nat,Es: list_t,E2: option2110264179601050681od_h_t] :
      ( ( vYDRA_MDL_a_t_h @ Sigma @ Init_hd @ Run_hd )
     => ( ( reache3397181163414509427_set_a @ Run_hd @ Init_hd @ Fs @ F )
       => ( ( ( size_s5362121731612881446_set_a @ Fs )
            = ( suc @ I ) )
         => ( ( reache645946503560514460_h_t_t @ ( run_t_h_t_a @ Run_hd ) @ ( t0_h_t_a @ Init_hd @ Run_hd ) @ Es @ E2 )
           => ( ( ord_less_eq_nat @ ( size_size_list_t @ Es ) @ I )
             => ? [E3: option2110264179601050681od_h_t,Et: t] :
                  ( ( run_t_h_t_a @ Run_hd @ E2 )
                  = ( some_P8406129549712030756_h_t_t @ ( produc770200333395398319_h_t_t @ E3 @ Et ) ) ) ) ) ) ) ) ).

% VYDRA_MDL.ru_t_Some
thf(fact_111_VYDRA__MDL_Owf__vydra_Ointros_I4_J,axiom,
    ! [Sigma: trace_a_t,Init_hd: h,Run_hd: h > option5775791656947114332_set_a,Es: list_P1095827921439705018_set_a,Sub: h,I: nat,A: a,N: nat] :
      ( ( vYDRA_MDL_a_t_h @ Sigma @ Init_hd @ Run_hd )
     => ( ( reache3397181163414509427_set_a @ Run_hd @ Init_hd @ Es @ Sub )
       => ( ( ( size_s5362121731612881446_set_a @ Es )
            = I )
         => ( vYDRA_wf_vydra_a_t_h @ Sigma @ Init_hd @ Run_hd @ ( atom_a_t @ A ) @ I @ N @ ( vydra_3724560892389469594_a_h_t @ A @ Sub ) ) ) ) ) ).

% VYDRA_MDL.wf_vydra.intros(4)
thf(fact_112_VYDRA__MDL_Owf__vydra_Ointros_I3_J,axiom,
    ! [Sigma: trace_a_t,Init_hd: h,Run_hd: h > option5775791656947114332_set_a,Es: list_P1095827921439705018_set_a,Sub: h,I: nat,B: $o,N: nat] :
      ( ( vYDRA_MDL_a_t_h @ Sigma @ Init_hd @ Run_hd )
     => ( ( reache3397181163414509427_set_a @ Run_hd @ Init_hd @ Es @ Sub )
       => ( ( ( size_s5362121731612881446_set_a @ Es )
            = I )
         => ( vYDRA_wf_vydra_a_t_h @ Sigma @ Init_hd @ Run_hd @ ( bool_a_t @ B ) @ I @ N @ ( vydra_2060541234293701615_h_a_t @ B @ Sub ) ) ) ) ) ).

% VYDRA_MDL.wf_vydra.intros(3)
thf(fact_113_VYDRA__MDL_Oru__t__Some__tau,axiom,
    ! [Sigma: trace_a_t,Init_hd: h,Run_hd: h > option5775791656947114332_set_a,Ts: list_t,E2: h,T: t] :
      ( ( vYDRA_MDL_a_t_h @ Sigma @ Init_hd @ Run_hd )
     => ( ( reache645946503560514460_h_t_t @ ( run_t_h_t_a @ Run_hd ) @ ( t0_h_t_a @ Init_hd @ Run_hd ) @ Ts @ ( some_P2526622376857543512od_h_t @ ( product_Pair_h_t @ E2 @ T ) ) )
       => ( T
          = ( tau_a_t @ Sigma @ ( size_size_list_t @ Ts ) ) ) ) ) ).

% VYDRA_MDL.ru_t_Some_tau
thf(fact_114_reaches__invar,axiom,
    ! [F: ( nat > $o ) > option7421465107808516430_o_nat,X3: nat > $o,N: nat,Y: nat > $o,P: ( nat > $o ) > $o] :
      ( ( reaches_nat_o_nat @ F @ X3 @ N @ Y )
     => ( ( P @ X3 )
       => ( ! [Z: nat > $o,Z2: nat > $o,V3: nat] :
              ( ( P @ Z )
             => ( ( ( F @ Z )
                  = ( some_P2630223654056210137_o_nat @ ( produc7277522915581678840_o_nat @ Z2 @ V3 ) ) )
               => ( P @ Z2 ) ) )
         => ( P @ Y ) ) ) ) ).

% reaches_invar
thf(fact_115_reaches__invar,axiom,
    ! [F: t > option4621139365931258234_set_a,X3: t,N: nat,Y: t,P: t > $o] :
      ( ( reaches_t_set_a @ F @ X3 @ N @ Y )
     => ( ( P @ X3 )
       => ( ! [Z: t,Z2: t,V3: set_a] :
              ( ( P @ Z )
             => ( ( ( F @ Z )
                  = ( some_P4667681669429350681_set_a @ ( product_Pair_t_set_a @ Z2 @ V3 ) ) )
               => ( P @ Z2 ) ) )
         => ( P @ Y ) ) ) ) ).

% reaches_invar
thf(fact_116_reaches__invar,axiom,
    ! [F: list_vydra_aux_a_t_h > option8553266494716013595rray_o,X3: list_vydra_aux_a_t_h,N: nat,Y: list_vydra_aux_a_t_h,P: list_vydra_aux_a_t_h > $o] :
      ( ( reache2947816147551113615rray_o @ F @ X3 @ N @ Y )
     => ( ( P @ X3 )
       => ( ! [Z: list_vydra_aux_a_t_h,Z2: list_vydra_aux_a_t_h,V3: iarray_o] :
              ( ( P @ Z )
             => ( ( ( F @ Z )
                  = ( some_P7279874174034738982rray_o @ ( produc8472968102330193021rray_o @ Z2 @ V3 ) ) )
               => ( P @ Z2 ) ) )
         => ( P @ Y ) ) ) ) ).

% reaches_invar
thf(fact_117_reaches__invar,axiom,
    ! [F: t > option8291060355354214836od_t_o,X3: t,N: nat,Y: t,P: t > $o] :
      ( ( reaches_t_o @ F @ X3 @ N @ Y )
     => ( ( P @ X3 )
       => ( ! [Z: t,Z2: t,V3: $o] :
              ( ( P @ Z )
             => ( ( ( F @ Z )
                  = ( some_P1805796590424124243od_t_o @ ( product_Pair_t_o @ Z2 @ V3 ) ) )
               => ( P @ Z2 ) ) )
         => ( P @ Y ) ) ) ) ).

% reaches_invar
thf(fact_118_reaches__invar,axiom,
    ! [F: h > option5775791656947114332_set_a,X3: h,N: nat,Y: h,P: h > $o] :
      ( ( reache8480716032150177652_set_a @ F @ X3 @ N @ Y )
     => ( ( P @ X3 )
       => ( ! [Z: h,Z2: h,V3: product_prod_t_set_a] :
              ( ( P @ Z )
             => ( ( ( F @ Z )
                  = ( some_P2245991037696317435_set_a @ ( produc3432067189782488838_set_a @ Z2 @ V3 ) ) )
               => ( P @ Z2 ) ) )
         => ( P @ Y ) ) ) ) ).

% reaches_invar
thf(fact_119_reaches__invar,axiom,
    ! [F: h > option2110264179601050681od_h_t,X3: h,N: nat,Y: h,P: h > $o] :
      ( ( reaches_h_t @ F @ X3 @ N @ Y )
     => ( ( P @ X3 )
       => ( ! [Z: h,Z2: h,V3: t] :
              ( ( P @ Z )
             => ( ( ( F @ Z )
                  = ( some_P2526622376857543512od_h_t @ ( product_Pair_h_t @ Z2 @ V3 ) ) )
               => ( P @ Z2 ) ) )
         => ( P @ Y ) ) ) ) ).

% reaches_invar
thf(fact_120_reaches__on,axiom,
    ! [Run: vydra_aux_a_t_h > option4580262698044125979od_t_o,S: vydra_aux_a_t_h,N: nat,S2: vydra_aux_a_t_h] :
      ( ( reache549850244866497679od_t_o @ Run @ S @ N @ S2 )
     => ? [Vs3: list_P5575445741472356340od_t_o] :
          ( ( reache8691153155344877136od_t_o @ Run @ S @ Vs3 @ S2 )
          & ( ( size_s4106575409306904672od_t_o @ Vs3 )
            = N ) ) ) ).

% reaches_on
thf(fact_121_reaches__on,axiom,
    ! [Run: h > option5775791656947114332_set_a,S: h,N: nat,S2: h] :
      ( ( reache8480716032150177652_set_a @ Run @ S @ N @ S2 )
     => ? [Vs3: list_P1095827921439705018_set_a] :
          ( ( reache3397181163414509427_set_a @ Run @ S @ Vs3 @ S2 )
          & ( ( size_s5362121731612881446_set_a @ Vs3 )
            = N ) ) ) ).

% reaches_on
thf(fact_122_reaches__on,axiom,
    ! [Run: list_vydra_aux_a_t_h > option8553266494716013595rray_o,S: list_vydra_aux_a_t_h,N: nat,S2: list_vydra_aux_a_t_h] :
      ( ( reache2947816147551113615rray_o @ Run @ S @ N @ S2 )
     => ? [Vs3: list_iarray_o] :
          ( ( reache2401979074735462992rray_o @ Run @ S @ Vs3 @ S2 )
          & ( ( size_s3243743627286111600rray_o @ Vs3 )
            = N ) ) ) ).

% reaches_on
thf(fact_123_reaches__on,axiom,
    ! [Run: option2110264179601050681od_h_t > option4241632866121027589_h_t_t,S: option2110264179601050681od_h_t,N: nat,S2: option2110264179601050681od_h_t] :
      ( ( reache5944868495896439325_h_t_t @ Run @ S @ N @ S2 )
     => ? [Vs3: list_t] :
          ( ( reache645946503560514460_h_t_t @ Run @ S @ Vs3 @ S2 )
          & ( ( size_size_list_t @ Vs3 )
            = N ) ) ) ).

% reaches_on
thf(fact_124_reaches__on__n,axiom,
    ! [Run: vydra_aux_a_t_h > option4580262698044125979od_t_o,S: vydra_aux_a_t_h,Vs: list_P5575445741472356340od_t_o,S2: vydra_aux_a_t_h] :
      ( ( reache8691153155344877136od_t_o @ Run @ S @ Vs @ S2 )
     => ( reache549850244866497679od_t_o @ Run @ S @ ( size_s4106575409306904672od_t_o @ Vs ) @ S2 ) ) ).

% reaches_on_n
thf(fact_125_reaches__on__n,axiom,
    ! [Run: h > option5775791656947114332_set_a,S: h,Vs: list_P1095827921439705018_set_a,S2: h] :
      ( ( reache3397181163414509427_set_a @ Run @ S @ Vs @ S2 )
     => ( reache8480716032150177652_set_a @ Run @ S @ ( size_s5362121731612881446_set_a @ Vs ) @ S2 ) ) ).

% reaches_on_n
thf(fact_126_reaches__on__n,axiom,
    ! [Run: list_vydra_aux_a_t_h > option8553266494716013595rray_o,S: list_vydra_aux_a_t_h,Vs: list_iarray_o,S2: list_vydra_aux_a_t_h] :
      ( ( reache2401979074735462992rray_o @ Run @ S @ Vs @ S2 )
     => ( reache2947816147551113615rray_o @ Run @ S @ ( size_s3243743627286111600rray_o @ Vs ) @ S2 ) ) ).

% reaches_on_n
thf(fact_127_reaches__on__n,axiom,
    ! [Run: option2110264179601050681od_h_t > option4241632866121027589_h_t_t,S: option2110264179601050681od_h_t,Vs: list_t,S2: option2110264179601050681od_h_t] :
      ( ( reache645946503560514460_h_t_t @ Run @ S @ Vs @ S2 )
     => ( reache5944868495896439325_h_t_t @ Run @ S @ ( size_size_list_t @ Vs ) @ S2 ) ) ).

% reaches_on_n
thf(fact_128_VYDRA__MDL_Oru__t__event,axiom,
    ! [Sigma: trace_a_t,Init_hd: h,Run_hd: h > option5775791656947114332_set_a,T: option2110264179601050681od_h_t,Ts: list_t,T2: option2110264179601050681od_h_t,T3: option2110264179601050681od_h_t,X3: t] :
      ( ( vYDRA_MDL_a_t_h @ Sigma @ Init_hd @ Run_hd )
     => ( ( reache645946503560514460_h_t_t @ ( run_t_h_t_a @ Run_hd ) @ T @ Ts @ T2 )
       => ( ( T
            = ( t0_h_t_a @ Init_hd @ Run_hd ) )
         => ( ( ( run_t_h_t_a @ Run_hd @ T2 )
              = ( some_P8406129549712030756_h_t_t @ ( produc770200333395398319_h_t_t @ T3 @ X3 ) ) )
           => ? [Rho: list_P1095827921439705018_set_a,E: h,Tt: t] :
                ( ( T2
                  = ( some_P2526622376857543512od_h_t @ ( product_Pair_h_t @ E @ Tt ) ) )
                & ( reache3397181163414509427_set_a @ Run_hd @ Init_hd @ Rho @ E )
                & ( ( size_s5362121731612881446_set_a @ Rho )
                  = ( suc @ ( size_size_list_t @ Ts ) ) )
                & ( X3
                  = ( tau_a_t @ Sigma @ ( size_size_list_t @ Ts ) ) ) ) ) ) ) ) ).

% VYDRA_MDL.ru_t_event
thf(fact_129_reaches_Ointros_I2_J,axiom,
    ! [Run: ( nat > $o ) > option7421465107808516430_o_nat,S: nat > $o,S2: nat > $o,V: nat,N: nat,S7: nat > $o] :
      ( ( ( Run @ S )
        = ( some_P2630223654056210137_o_nat @ ( produc7277522915581678840_o_nat @ S2 @ V ) ) )
     => ( ( reaches_nat_o_nat @ Run @ S2 @ N @ S7 )
       => ( reaches_nat_o_nat @ Run @ S @ ( suc @ N ) @ S7 ) ) ) ).

% reaches.intros(2)
thf(fact_130_reaches_Ointros_I2_J,axiom,
    ! [Run: t > option4621139365931258234_set_a,S: t,S2: t,V: set_a,N: nat,S7: t] :
      ( ( ( Run @ S )
        = ( some_P4667681669429350681_set_a @ ( product_Pair_t_set_a @ S2 @ V ) ) )
     => ( ( reaches_t_set_a @ Run @ S2 @ N @ S7 )
       => ( reaches_t_set_a @ Run @ S @ ( suc @ N ) @ S7 ) ) ) ).

% reaches.intros(2)
thf(fact_131_reaches_Ointros_I2_J,axiom,
    ! [Run: list_vydra_aux_a_t_h > option8553266494716013595rray_o,S: list_vydra_aux_a_t_h,S2: list_vydra_aux_a_t_h,V: iarray_o,N: nat,S7: list_vydra_aux_a_t_h] :
      ( ( ( Run @ S )
        = ( some_P7279874174034738982rray_o @ ( produc8472968102330193021rray_o @ S2 @ V ) ) )
     => ( ( reache2947816147551113615rray_o @ Run @ S2 @ N @ S7 )
       => ( reache2947816147551113615rray_o @ Run @ S @ ( suc @ N ) @ S7 ) ) ) ).

% reaches.intros(2)
thf(fact_132_reaches_Ointros_I2_J,axiom,
    ! [Run: t > option8291060355354214836od_t_o,S: t,S2: t,V: $o,N: nat,S7: t] :
      ( ( ( Run @ S )
        = ( some_P1805796590424124243od_t_o @ ( product_Pair_t_o @ S2 @ V ) ) )
     => ( ( reaches_t_o @ Run @ S2 @ N @ S7 )
       => ( reaches_t_o @ Run @ S @ ( suc @ N ) @ S7 ) ) ) ).

% reaches.intros(2)
thf(fact_133_reaches_Ointros_I2_J,axiom,
    ! [Run: h > option5775791656947114332_set_a,S: h,S2: h,V: product_prod_t_set_a,N: nat,S7: h] :
      ( ( ( Run @ S )
        = ( some_P2245991037696317435_set_a @ ( produc3432067189782488838_set_a @ S2 @ V ) ) )
     => ( ( reache8480716032150177652_set_a @ Run @ S2 @ N @ S7 )
       => ( reache8480716032150177652_set_a @ Run @ S @ ( suc @ N ) @ S7 ) ) ) ).

% reaches.intros(2)
thf(fact_134_reaches_Ointros_I2_J,axiom,
    ! [Run: h > option2110264179601050681od_h_t,S: h,S2: h,V: t,N: nat,S7: h] :
      ( ( ( Run @ S )
        = ( some_P2526622376857543512od_h_t @ ( product_Pair_h_t @ S2 @ V ) ) )
     => ( ( reaches_h_t @ Run @ S2 @ N @ S7 )
       => ( reaches_h_t @ Run @ S @ ( suc @ N ) @ S7 ) ) ) ).

% reaches.intros(2)
thf(fact_135_reaches__Suc__split__last,axiom,
    ! [Run: ( nat > $o ) > option7421465107808516430_o_nat,S: nat > $o,N: nat,S2: nat > $o] :
      ( ( reaches_nat_o_nat @ Run @ S @ ( suc @ N ) @ S2 )
     => ? [S8: nat > $o,X: nat] :
          ( ( reaches_nat_o_nat @ Run @ S @ N @ S8 )
          & ( ( Run @ S8 )
            = ( some_P2630223654056210137_o_nat @ ( produc7277522915581678840_o_nat @ S2 @ X ) ) ) ) ) ).

% reaches_Suc_split_last
thf(fact_136_reaches__Suc__split__last,axiom,
    ! [Run: t > option4621139365931258234_set_a,S: t,N: nat,S2: t] :
      ( ( reaches_t_set_a @ Run @ S @ ( suc @ N ) @ S2 )
     => ? [S8: t,X: set_a] :
          ( ( reaches_t_set_a @ Run @ S @ N @ S8 )
          & ( ( Run @ S8 )
            = ( some_P4667681669429350681_set_a @ ( product_Pair_t_set_a @ S2 @ X ) ) ) ) ) ).

% reaches_Suc_split_last
thf(fact_137_reaches__Suc__split__last,axiom,
    ! [Run: list_vydra_aux_a_t_h > option8553266494716013595rray_o,S: list_vydra_aux_a_t_h,N: nat,S2: list_vydra_aux_a_t_h] :
      ( ( reache2947816147551113615rray_o @ Run @ S @ ( suc @ N ) @ S2 )
     => ? [S8: list_vydra_aux_a_t_h,X: iarray_o] :
          ( ( reache2947816147551113615rray_o @ Run @ S @ N @ S8 )
          & ( ( Run @ S8 )
            = ( some_P7279874174034738982rray_o @ ( produc8472968102330193021rray_o @ S2 @ X ) ) ) ) ) ).

% reaches_Suc_split_last
thf(fact_138_reaches__Suc__split__last,axiom,
    ! [Run: t > option8291060355354214836od_t_o,S: t,N: nat,S2: t] :
      ( ( reaches_t_o @ Run @ S @ ( suc @ N ) @ S2 )
     => ? [S8: t,X: $o] :
          ( ( reaches_t_o @ Run @ S @ N @ S8 )
          & ( ( Run @ S8 )
            = ( some_P1805796590424124243od_t_o @ ( product_Pair_t_o @ S2 @ X ) ) ) ) ) ).

% reaches_Suc_split_last
thf(fact_139_reaches__Suc__split__last,axiom,
    ! [Run: h > option5775791656947114332_set_a,S: h,N: nat,S2: h] :
      ( ( reache8480716032150177652_set_a @ Run @ S @ ( suc @ N ) @ S2 )
     => ? [S8: h,X: product_prod_t_set_a] :
          ( ( reache8480716032150177652_set_a @ Run @ S @ N @ S8 )
          & ( ( Run @ S8 )
            = ( some_P2245991037696317435_set_a @ ( produc3432067189782488838_set_a @ S2 @ X ) ) ) ) ) ).

% reaches_Suc_split_last
thf(fact_140_reaches__Suc__split__last,axiom,
    ! [Run: h > option2110264179601050681od_h_t,S: h,N: nat,S2: h] :
      ( ( reaches_h_t @ Run @ S @ ( suc @ N ) @ S2 )
     => ? [S8: h,X: t] :
          ( ( reaches_h_t @ Run @ S @ N @ S8 )
          & ( ( Run @ S8 )
            = ( some_P2526622376857543512od_h_t @ ( product_Pair_h_t @ S2 @ X ) ) ) ) ) ).

% reaches_Suc_split_last
thf(fact_141_wf__vydra_Ointros_I5_J,axiom,
    ! [Phi2: formula_a_t,I: nat,N: nat,V: vydra_aux_a_t_h] :
      ( ( vYDRA_wf_vydra_a_t_h @ sigma @ init_hd @ run_hd @ Phi2 @ I @ N @ V )
     => ( vYDRA_wf_vydra_a_t_h @ sigma @ init_hd @ run_hd @ ( neg_a_t @ Phi2 ) @ I @ ( suc @ N ) @ ( vydra_2331423191276895391_a_t_h @ V ) ) ) ).

% wf_vydra.intros(5)
thf(fact_142_MDL_Osat_Osimps_I2_J,axiom,
    ! [Sigma: trace_a_t,A: a,I: nat] :
      ( ( sat_a_t @ Sigma @ ( atom_a_t @ A ) @ I )
      = ( member_a @ A @ ( gamma_a_t @ Sigma @ I ) ) ) ).

% MDL.sat.simps(2)
thf(fact_143__092_060tau_062__mono,axiom,
    ! [I: nat,J: nat,S: trace_a_t] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_t @ ( tau_a_t @ S @ I ) @ ( tau_a_t @ S @ J ) ) ) ).

% \<tau>_mono
thf(fact_144_formula_Oinject_I2_J,axiom,
    ! [X2: a,Y2: a] :
      ( ( ( atom_a_t @ X2 )
        = ( atom_a_t @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% formula.inject(2)
thf(fact_145_formula_Oinject_I1_J,axiom,
    ! [X1: $o,Y1: $o] :
      ( ( ( bool_a_t @ X1 )
        = ( bool_a_t @ Y1 ) )
      = ( X1 = Y1 ) ) ).

% formula.inject(1)
thf(fact_146_formula_Oinject_I3_J,axiom,
    ! [X33: formula_a_t,Y33: formula_a_t] :
      ( ( ( neg_a_t @ X33 )
        = ( neg_a_t @ Y33 ) )
      = ( X33 = Y33 ) ) ).

% formula.inject(3)
thf(fact_147_dual__order_Orefl,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% dual_order.refl
thf(fact_148_order__refl,axiom,
    ! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).

% order_refl
thf(fact_149_ru__t__tau__in,axiom,
    ! [Ts: list_t,T: option2110264179601050681od_h_t,J: nat] :
      ( ( reache645946503560514460_h_t_t @ ( run_t_h_t_a @ run_hd ) @ ( t0_h_t_a @ init_hd @ run_hd ) @ Ts @ T )
     => ( ( ord_less_nat @ J @ ( size_size_list_t @ Ts ) )
       => ( ( nth_t @ Ts @ J )
          = ( tau_a_t @ sigma @ J ) ) ) ) ).

% ru_t_tau_in
thf(fact_150_bounded__future__fmla_Osimps_I2_J,axiom,
    ! [A: a] : ( bounde209865433540700070la_a_t @ ( atom_a_t @ A ) ) ).

% bounded_future_fmla.simps(2)
thf(fact_151_vydra__aux_Oinject_I3_J,axiom,
    ! [X42: vydra_aux_a_t_h,Y42: vydra_aux_a_t_h] :
      ( ( ( vydra_2331423191276895391_a_t_h @ X42 )
        = ( vydra_2331423191276895391_a_t_h @ Y42 ) )
      = ( X42 = Y42 ) ) ).

% vydra_aux.inject(3)
thf(fact_152_Suc__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
      = ( ord_less_nat @ M @ N ) ) ).

% Suc_less_eq
thf(fact_153_Suc__mono,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).

% Suc_mono
thf(fact_154_lessI,axiom,
    ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).

% lessI
thf(fact_155_less___092_060tau_062D,axiom,
    ! [Sigma: trace_a_t,I: nat,J: nat] :
      ( ( ord_less_t @ ( tau_a_t @ Sigma @ I ) @ ( tau_a_t @ Sigma @ J ) )
     => ( ord_less_nat @ I @ J ) ) ).

% less_\<tau>D
thf(fact_156_gt__ex,axiom,
    ! [X3: nat] :
    ? [X_1: nat] : ( ord_less_nat @ X3 @ X_1 ) ).

% gt_ex
thf(fact_157_less__imp__neq,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_nat @ X3 @ Y )
     => ( X3 != Y ) ) ).

% less_imp_neq
thf(fact_158_order_Oasym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order.asym
thf(fact_159_ord__eq__less__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( A = B )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ord_less_nat @ A @ C2 ) ) ) ).

% ord_eq_less_trans
thf(fact_160_ord__less__eq__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( B = C2 )
       => ( ord_less_nat @ A @ C2 ) ) ) ).

% ord_less_eq_trans
thf(fact_161_less__induct,axiom,
    ! [P: nat > $o,A: nat] :
      ( ! [X: nat] :
          ( ! [Y4: nat] :
              ( ( ord_less_nat @ Y4 @ X )
             => ( P @ Y4 ) )
         => ( P @ X ) )
     => ( P @ A ) ) ).

% less_induct
thf(fact_162_antisym__conv3,axiom,
    ! [Y: nat,X3: nat] :
      ( ~ ( ord_less_nat @ Y @ X3 )
     => ( ( ~ ( ord_less_nat @ X3 @ Y ) )
        = ( X3 = Y ) ) ) ).

% antisym_conv3
thf(fact_163_linorder__cases,axiom,
    ! [X3: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X3 @ Y )
     => ( ( X3 != Y )
       => ( ord_less_nat @ Y @ X3 ) ) ) ).

% linorder_cases
thf(fact_164_dual__order_Oasym,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ~ ( ord_less_nat @ A @ B ) ) ).

% dual_order.asym
thf(fact_165_dual__order_Oirrefl,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% dual_order.irrefl
thf(fact_166_exists__least__iff,axiom,
    ( ( ^ [P3: nat > $o] :
        ? [X8: nat] : ( P3 @ X8 ) )
    = ( ^ [P4: nat > $o] :
        ? [N4: nat] :
          ( ( P4 @ N4 )
          & ! [M5: nat] :
              ( ( ord_less_nat @ M5 @ N4 )
             => ~ ( P4 @ M5 ) ) ) ) ) ).

% exists_least_iff
thf(fact_167_linorder__less__wlog,axiom,
    ! [P: nat > nat > $o,A: nat,B: nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( ord_less_nat @ A3 @ B3 )
         => ( P @ A3 @ B3 ) )
     => ( ! [A3: nat] : ( P @ A3 @ A3 )
       => ( ! [A3: nat,B3: nat] :
              ( ( P @ B3 @ A3 )
             => ( P @ A3 @ B3 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_168_order_Ostrict__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ord_less_nat @ A @ C2 ) ) ) ).

% order.strict_trans
thf(fact_169_not__less__iff__gr__or__eq,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X3 @ Y ) )
      = ( ( ord_less_nat @ Y @ X3 )
        | ( X3 = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_170_dual__order_Ostrict__trans,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_less_nat @ C2 @ B )
       => ( ord_less_nat @ C2 @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_171_order_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_172_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_173_linorder__neqE,axiom,
    ! [X3: nat,Y: nat] :
      ( ( X3 != Y )
     => ( ~ ( ord_less_nat @ X3 @ Y )
       => ( ord_less_nat @ Y @ X3 ) ) ) ).

% linorder_neqE
thf(fact_174_order__less__asym,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_nat @ X3 @ Y )
     => ~ ( ord_less_nat @ Y @ X3 ) ) ).

% order_less_asym
thf(fact_175_linorder__neq__iff,axiom,
    ! [X3: nat,Y: nat] :
      ( ( X3 != Y )
      = ( ( ord_less_nat @ X3 @ Y )
        | ( ord_less_nat @ Y @ X3 ) ) ) ).

% linorder_neq_iff
thf(fact_176_order__less__asym_H,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order_less_asym'
thf(fact_177_order__less__trans,axiom,
    ! [X3: nat,Y: nat,Z3: nat] :
      ( ( ord_less_nat @ X3 @ Y )
     => ( ( ord_less_nat @ Y @ Z3 )
       => ( ord_less_nat @ X3 @ Z3 ) ) ) ).

% order_less_trans
thf(fact_178_ord__eq__less__subst,axiom,
    ! [A: nat,F: nat > nat,B: nat,C2: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ! [X: nat,Y3: nat] :
              ( ( ord_less_nat @ X @ Y3 )
             => ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_179_ord__less__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X: nat,Y3: nat] :
              ( ( ord_less_nat @ X @ Y3 )
             => ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_180_order__less__irrefl,axiom,
    ! [X3: nat] :
      ~ ( ord_less_nat @ X3 @ X3 ) ).

% order_less_irrefl
thf(fact_181_order__less__subst1,axiom,
    ! [A: nat,F: nat > nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ! [X: nat,Y3: nat] :
              ( ( ord_less_nat @ X @ Y3 )
             => ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_182_order__less__subst2,axiom,
    ! [A: nat,B: nat,F: nat > nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C2 )
       => ( ! [X: nat,Y3: nat] :
              ( ( ord_less_nat @ X @ Y3 )
             => ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_183_order__less__not__sym,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_nat @ X3 @ Y )
     => ~ ( ord_less_nat @ Y @ X3 ) ) ).

% order_less_not_sym
thf(fact_184_order__less__imp__triv,axiom,
    ! [X3: nat,Y: nat,P: $o] :
      ( ( ord_less_nat @ X3 @ Y )
     => ( ( ord_less_nat @ Y @ X3 )
       => P ) ) ).

% order_less_imp_triv
thf(fact_185_linorder__less__linear,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_nat @ X3 @ Y )
      | ( X3 = Y )
      | ( ord_less_nat @ Y @ X3 ) ) ).

% linorder_less_linear
thf(fact_186_order__less__imp__not__eq,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_nat @ X3 @ Y )
     => ( X3 != Y ) ) ).

% order_less_imp_not_eq
thf(fact_187_order__less__imp__not__eq2,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_nat @ X3 @ Y )
     => ( Y != X3 ) ) ).

% order_less_imp_not_eq2
thf(fact_188_order__less__imp__not__less,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_nat @ X3 @ Y )
     => ~ ( ord_less_nat @ Y @ X3 ) ) ).

% order_less_imp_not_less
thf(fact_189_linorder__neqE__nat,axiom,
    ! [X3: nat,Y: nat] :
      ( ( X3 != Y )
     => ( ~ ( ord_less_nat @ X3 @ Y )
       => ( ord_less_nat @ Y @ X3 ) ) ) ).

% linorder_neqE_nat
thf(fact_190_infinite__descent,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N3: nat] :
          ( ~ ( P @ N3 )
         => ? [M4: nat] :
              ( ( ord_less_nat @ M4 @ N3 )
              & ~ ( P @ M4 ) ) )
     => ( P @ N ) ) ).

% infinite_descent
thf(fact_191_nat__less__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N3: nat] :
          ( ! [M4: nat] :
              ( ( ord_less_nat @ M4 @ N3 )
             => ( P @ M4 ) )
         => ( P @ N3 ) )
     => ( P @ N ) ) ).

% nat_less_induct
thf(fact_192_less__irrefl__nat,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_irrefl_nat
thf(fact_193_less__not__refl3,axiom,
    ! [S: nat,T: nat] :
      ( ( ord_less_nat @ S @ T )
     => ( S != T ) ) ).

% less_not_refl3
thf(fact_194_less__not__refl2,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ N @ M )
     => ( M != N ) ) ).

% less_not_refl2
thf(fact_195_less__not__refl,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_not_refl
thf(fact_196_nat__neq__iff,axiom,
    ! [M: nat,N: nat] :
      ( ( M != N )
      = ( ( ord_less_nat @ M @ N )
        | ( ord_less_nat @ N @ M ) ) ) ).

% nat_neq_iff
thf(fact_197_lift__Suc__mono__less,axiom,
    ! [F: nat > nat,N: nat,N2: nat] :
      ( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_nat @ N @ N2 )
       => ( ord_less_nat @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_198_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > nat,N: nat,M: nat] :
      ( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
        = ( ord_less_nat @ N @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_199_leD,axiom,
    ! [Y: nat,X3: nat] :
      ( ( ord_less_eq_nat @ Y @ X3 )
     => ~ ( ord_less_nat @ X3 @ Y ) ) ).

% leD
thf(fact_200_leI,axiom,
    ! [X3: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X3 @ Y )
     => ( ord_less_eq_nat @ Y @ X3 ) ) ).

% leI
thf(fact_201_nless__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_nat @ A @ B ) )
      = ( ~ ( ord_less_eq_nat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_202_antisym__conv1,axiom,
    ! [X3: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X3 @ Y )
     => ( ( ord_less_eq_nat @ X3 @ Y )
        = ( X3 = Y ) ) ) ).

% antisym_conv1
thf(fact_203_antisym__conv2,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y )
     => ( ( ~ ( ord_less_nat @ X3 @ Y ) )
        = ( X3 = Y ) ) ) ).

% antisym_conv2
thf(fact_204_less__le__not__le,axiom,
    ( ord_less_nat
    = ( ^ [X5: nat,Y5: nat] :
          ( ( ord_less_eq_nat @ X5 @ Y5 )
          & ~ ( ord_less_eq_nat @ Y5 @ X5 ) ) ) ) ).

% less_le_not_le
thf(fact_205_not__le__imp__less,axiom,
    ! [Y: nat,X3: nat] :
      ( ~ ( ord_less_eq_nat @ Y @ X3 )
     => ( ord_less_nat @ X3 @ Y ) ) ).

% not_le_imp_less
thf(fact_206_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [A5: nat,B4: nat] :
          ( ( ord_less_nat @ A5 @ B4 )
          | ( A5 = B4 ) ) ) ) ).

% order.order_iff_strict
thf(fact_207_order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [A5: nat,B4: nat] :
          ( ( ord_less_eq_nat @ A5 @ B4 )
          & ( A5 != B4 ) ) ) ) ).

% order.strict_iff_order
thf(fact_208_order_Ostrict__trans1,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ord_less_nat @ A @ C2 ) ) ) ).

% order.strict_trans1
thf(fact_209_order_Ostrict__trans2,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ord_less_nat @ A @ C2 ) ) ) ).

% order.strict_trans2
thf(fact_210_order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [A5: nat,B4: nat] :
          ( ( ord_less_eq_nat @ A5 @ B4 )
          & ~ ( ord_less_eq_nat @ B4 @ A5 ) ) ) ) ).

% order.strict_iff_not
thf(fact_211_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [B4: nat,A5: nat] :
          ( ( ord_less_nat @ B4 @ A5 )
          | ( A5 = B4 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_212_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [B4: nat,A5: nat] :
          ( ( ord_less_eq_nat @ B4 @ A5 )
          & ( A5 != B4 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_213_dual__order_Ostrict__trans1,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_nat @ C2 @ B )
       => ( ord_less_nat @ C2 @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_214_dual__order_Ostrict__trans2,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C2 @ B )
       => ( ord_less_nat @ C2 @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_215_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [B4: nat,A5: nat] :
          ( ( ord_less_eq_nat @ B4 @ A5 )
          & ~ ( ord_less_eq_nat @ A5 @ B4 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_216_order_Ostrict__implies__order,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_217_dual__order_Ostrict__implies__order,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ord_less_eq_nat @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_218_order__le__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [X5: nat,Y5: nat] :
          ( ( ord_less_nat @ X5 @ Y5 )
          | ( X5 = Y5 ) ) ) ) ).

% order_le_less
thf(fact_219_order__less__le,axiom,
    ( ord_less_nat
    = ( ^ [X5: nat,Y5: nat] :
          ( ( ord_less_eq_nat @ X5 @ Y5 )
          & ( X5 != Y5 ) ) ) ) ).

% order_less_le
thf(fact_220_linorder__not__le,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ~ ( ord_less_eq_nat @ X3 @ Y ) )
      = ( ord_less_nat @ Y @ X3 ) ) ).

% linorder_not_le
thf(fact_221_linorder__not__less,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X3 @ Y ) )
      = ( ord_less_eq_nat @ Y @ X3 ) ) ).

% linorder_not_less
thf(fact_222_order__less__imp__le,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_nat @ X3 @ Y )
     => ( ord_less_eq_nat @ X3 @ Y ) ) ).

% order_less_imp_le
thf(fact_223_order__le__neq__trans,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( A != B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_224_order__neq__le__trans,axiom,
    ! [A: nat,B: nat] :
      ( ( A != B )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_225_order__le__less__trans,axiom,
    ! [X3: nat,Y: nat,Z3: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y )
     => ( ( ord_less_nat @ Y @ Z3 )
       => ( ord_less_nat @ X3 @ Z3 ) ) ) ).

% order_le_less_trans
thf(fact_226_order__less__le__trans,axiom,
    ! [X3: nat,Y: nat,Z3: nat] :
      ( ( ord_less_nat @ X3 @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z3 )
       => ( ord_less_nat @ X3 @ Z3 ) ) ) ).

% order_less_le_trans
thf(fact_227_order__le__less__subst1,axiom,
    ! [A: nat,F: nat > nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ! [X: nat,Y3: nat] :
              ( ( ord_less_nat @ X @ Y3 )
             => ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_228_order__le__less__subst2,axiom,
    ! [A: nat,B: nat,F: nat > nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C2 )
       => ( ! [X: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_229_order__less__le__subst1,axiom,
    ! [A: nat,F: nat > nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ! [X: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_230_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: nat > nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
       => ( ! [X: nat,Y3: nat] :
              ( ( ord_less_nat @ X @ Y3 )
             => ( ord_less_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_231_linorder__le__less__linear,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y )
      | ( ord_less_nat @ Y @ X3 ) ) ).

% linorder_le_less_linear
thf(fact_232_order__le__imp__less__or__eq,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y )
     => ( ( ord_less_nat @ X3 @ Y )
        | ( X3 = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_233_not__less__less__Suc__eq,axiom,
    ! [N: nat,M: nat] :
      ( ~ ( ord_less_nat @ N @ M )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
        = ( N = M ) ) ) ).

% not_less_less_Suc_eq
thf(fact_234_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_nat @ I @ J )
     => ( ! [I2: nat] :
            ( ( J
              = ( suc @ I2 ) )
           => ( P @ I2 ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ J )
             => ( ( P @ ( suc @ I2 ) )
               => ( P @ I2 ) ) )
         => ( P @ I ) ) ) ) ).

% strict_inc_induct
thf(fact_235_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P: nat > nat > $o] :
      ( ( ord_less_nat @ I @ J )
     => ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
       => ( ! [I2: nat,J2: nat,K2: nat] :
              ( ( ord_less_nat @ I2 @ J2 )
             => ( ( ord_less_nat @ J2 @ K2 )
               => ( ( P @ I2 @ J2 )
                 => ( ( P @ J2 @ K2 )
                   => ( P @ I2 @ K2 ) ) ) ) )
         => ( P @ I @ J ) ) ) ) ).

% less_Suc_induct
thf(fact_236_less__trans__Suc,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ J @ K )
       => ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).

% less_trans_Suc
thf(fact_237_Suc__less__SucD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_less_SucD
thf(fact_238_less__antisym,axiom,
    ! [N: nat,M: nat] :
      ( ~ ( ord_less_nat @ N @ M )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
       => ( M = N ) ) ) ).

% less_antisym
thf(fact_239_Suc__less__eq2,axiom,
    ! [N: nat,M: nat] :
      ( ( ord_less_nat @ ( suc @ N ) @ M )
      = ( ? [M6: nat] :
            ( ( M
              = ( suc @ M6 ) )
            & ( ord_less_nat @ N @ M6 ) ) ) ) ).

% Suc_less_eq2
thf(fact_240_All__less__Suc,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N ) )
           => ( P @ I3 ) ) )
      = ( ( P @ N )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ N )
           => ( P @ I3 ) ) ) ) ).

% All_less_Suc
thf(fact_241_not__less__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ~ ( ord_less_nat @ M @ N ) )
      = ( ord_less_nat @ N @ ( suc @ M ) ) ) ).

% not_less_eq
thf(fact_242_less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
      = ( ( ord_less_nat @ M @ N )
        | ( M = N ) ) ) ).

% less_Suc_eq
thf(fact_243_Ex__less__Suc,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N ) )
            & ( P @ I3 ) ) )
      = ( ( P @ N )
        | ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ N )
            & ( P @ I3 ) ) ) ) ).

% Ex_less_Suc
thf(fact_244_less__SucI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).

% less_SucI
thf(fact_245_less__SucE,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
     => ( ~ ( ord_less_nat @ M @ N )
       => ( M = N ) ) ) ).

% less_SucE
thf(fact_246_Suc__lessI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ( ( suc @ M )
         != N )
       => ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).

% Suc_lessI
thf(fact_247_Suc__lessE,axiom,
    ! [I: nat,K: nat] :
      ( ( ord_less_nat @ ( suc @ I ) @ K )
     => ~ ! [J2: nat] :
            ( ( ord_less_nat @ I @ J2 )
           => ( K
             != ( suc @ J2 ) ) ) ) ).

% Suc_lessE
thf(fact_248_Suc__lessD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ N )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_lessD
thf(fact_249_Nat_OlessE,axiom,
    ! [I: nat,K: nat] :
      ( ( ord_less_nat @ I @ K )
     => ( ( K
         != ( suc @ I ) )
       => ~ ! [J2: nat] :
              ( ( ord_less_nat @ I @ J2 )
             => ( K
               != ( suc @ J2 ) ) ) ) ) ).

% Nat.lessE
thf(fact_250_less__mono__imp__le__mono,axiom,
    ! [F: nat > nat,I: nat,J: nat] :
      ( ! [I2: nat,J2: nat] :
          ( ( ord_less_nat @ I2 @ J2 )
         => ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).

% less_mono_imp_le_mono
thf(fact_251_le__neq__implies__less,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( M != N )
       => ( ord_less_nat @ M @ N ) ) ) ).

% le_neq_implies_less
thf(fact_252_less__or__eq__imp__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ( ord_less_nat @ M @ N )
        | ( M = N ) )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% less_or_eq_imp_le
thf(fact_253_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M5: nat,N4: nat] :
          ( ( ord_less_nat @ M5 @ N4 )
          | ( M5 = N4 ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_254_less__imp__le__nat,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_eq_nat @ M @ N ) ) ).

% less_imp_le_nat
thf(fact_255_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M5: nat,N4: nat] :
          ( ( ord_less_eq_nat @ M5 @ N4 )
          & ( M5 != N4 ) ) ) ) ).

% nat_less_le
thf(fact_256_vydra__aux_Odistinct_I39_J,axiom,
    ! [X31: a,X32: h,X42: vydra_aux_a_t_h] :
      ( ( vydra_3724560892389469594_a_h_t @ X31 @ X32 )
     != ( vydra_2331423191276895391_a_t_h @ X42 ) ) ).

% vydra_aux.distinct(39)
thf(fact_257_vydra__aux_Odistinct_I23_J,axiom,
    ! [X21: $o,X22: h,X42: vydra_aux_a_t_h] :
      ( ( vydra_2060541234293701615_h_a_t @ X21 @ X22 )
     != ( vydra_2331423191276895391_a_t_h @ X42 ) ) ).

% vydra_aux.distinct(23)
thf(fact_258_le___092_060tau_062__less,axiom,
    ! [Sigma: trace_a_t,I: nat,J: nat] :
      ( ( ord_less_eq_t @ ( tau_a_t @ Sigma @ I ) @ ( tau_a_t @ Sigma @ J ) )
     => ( ( ord_less_nat @ J @ I )
       => ( ( tau_a_t @ Sigma @ I )
          = ( tau_a_t @ Sigma @ J ) ) ) ) ).

% le_\<tau>_less
thf(fact_259_Suc__leI,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).

% Suc_leI
thf(fact_260_Suc__le__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
      = ( ord_less_nat @ M @ N ) ) ).

% Suc_le_eq
thf(fact_261_dec__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P @ I )
       => ( ! [N3: nat] :
              ( ( ord_less_eq_nat @ I @ N3 )
             => ( ( ord_less_nat @ N3 @ J )
               => ( ( P @ N3 )
                 => ( P @ ( suc @ N3 ) ) ) ) )
         => ( P @ J ) ) ) ) ).

% dec_induct
thf(fact_262_inc__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P @ J )
       => ( ! [N3: nat] :
              ( ( ord_less_eq_nat @ I @ N3 )
             => ( ( ord_less_nat @ N3 @ J )
               => ( ( P @ ( suc @ N3 ) )
                 => ( P @ N3 ) ) ) )
         => ( P @ I ) ) ) ) ).

% inc_induct
thf(fact_263_Suc__le__lessD,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
     => ( ord_less_nat @ M @ N ) ) ).

% Suc_le_lessD
thf(fact_264_le__less__Suc__eq,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ( ord_less_nat @ N @ ( suc @ M ) )
        = ( N = M ) ) ) ).

% le_less_Suc_eq
thf(fact_265_less__Suc__eq__le,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
      = ( ord_less_eq_nat @ M @ N ) ) ).

% less_Suc_eq_le
thf(fact_266_less__eq__Suc__le,axiom,
    ( ord_less_nat
    = ( ^ [N4: nat] : ( ord_less_eq_nat @ ( suc @ N4 ) ) ) ) ).

% less_eq_Suc_le
thf(fact_267_le__imp__less__Suc,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_less_nat @ M @ ( suc @ N ) ) ) ).

% le_imp_less_Suc
thf(fact_268_nle__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_eq_nat @ A @ B ) )
      = ( ( ord_less_eq_nat @ B @ A )
        & ( B != A ) ) ) ).

% nle_le
thf(fact_269_le__cases3,axiom,
    ! [X3: nat,Y: nat,Z3: nat] :
      ( ( ( ord_less_eq_nat @ X3 @ Y )
       => ~ ( ord_less_eq_nat @ Y @ Z3 ) )
     => ( ( ( ord_less_eq_nat @ Y @ X3 )
         => ~ ( ord_less_eq_nat @ X3 @ Z3 ) )
       => ( ( ( ord_less_eq_nat @ X3 @ Z3 )
           => ~ ( ord_less_eq_nat @ Z3 @ Y ) )
         => ( ( ( ord_less_eq_nat @ Z3 @ Y )
             => ~ ( ord_less_eq_nat @ Y @ X3 ) )
           => ( ( ( ord_less_eq_nat @ Y @ Z3 )
               => ~ ( ord_less_eq_nat @ Z3 @ X3 ) )
             => ~ ( ( ord_less_eq_nat @ Z3 @ X3 )
                 => ~ ( ord_less_eq_nat @ X3 @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_270_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
    = ( ^ [X5: nat,Y5: nat] :
          ( ( ord_less_eq_nat @ X5 @ Y5 )
          & ( ord_less_eq_nat @ Y5 @ X5 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_271_ord__eq__le__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( A = B )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ord_less_eq_nat @ A @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_272_ord__le__eq__trans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( B = C2 )
       => ( ord_less_eq_nat @ A @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_273_order__antisym,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y )
     => ( ( ord_less_eq_nat @ Y @ X3 )
       => ( X3 = Y ) ) ) ).

% order_antisym
thf(fact_274_order_Otrans,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ord_less_eq_nat @ A @ C2 ) ) ) ).

% order.trans
thf(fact_275_order__trans,axiom,
    ! [X3: nat,Y: nat,Z3: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z3 )
       => ( ord_less_eq_nat @ X3 @ Z3 ) ) ) ).

% order_trans
thf(fact_276_linorder__wlog,axiom,
    ! [P: nat > nat > $o,A: nat,B: nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A3 @ B3 )
         => ( P @ A3 @ B3 ) )
     => ( ! [A3: nat,B3: nat] :
            ( ( P @ B3 @ A3 )
           => ( P @ A3 @ B3 ) )
       => ( P @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_277_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
    = ( ^ [A5: nat,B4: nat] :
          ( ( ord_less_eq_nat @ B4 @ A5 )
          & ( ord_less_eq_nat @ A5 @ B4 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_278_dual__order_Oantisym,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_279_dual__order_Otrans,axiom,
    ! [B: nat,A: nat,C2: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C2 @ B )
       => ( ord_less_eq_nat @ C2 @ A ) ) ) ).

% dual_order.trans
thf(fact_280_antisym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_281_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: nat,Z4: nat] : ( Y6 = Z4 ) )
    = ( ^ [A5: nat,B4: nat] :
          ( ( ord_less_eq_nat @ A5 @ B4 )
          & ( ord_less_eq_nat @ B4 @ A5 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_282_order__subst1,axiom,
    ! [A: nat,F: nat > nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ! [X: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_283_order__subst2,axiom,
    ! [A: nat,B: nat,F: nat > nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
       => ( ! [X: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_284_order__eq__refl,axiom,
    ! [X3: nat,Y: nat] :
      ( ( X3 = Y )
     => ( ord_less_eq_nat @ X3 @ Y ) ) ).

% order_eq_refl
thf(fact_285_linorder__linear,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y )
      | ( ord_less_eq_nat @ Y @ X3 ) ) ).

% linorder_linear
thf(fact_286_ord__eq__le__subst,axiom,
    ! [A: nat,F: nat > nat,B: nat,C2: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ! [X: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_287_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F @ B )
          = C2 )
       => ( ! [X: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_288_linorder__le__cases,axiom,
    ! [X3: nat,Y: nat] :
      ( ~ ( ord_less_eq_nat @ X3 @ Y )
     => ( ord_less_eq_nat @ Y @ X3 ) ) ).

% linorder_le_cases
thf(fact_289_order__antisym__conv,axiom,
    ! [Y: nat,X3: nat] :
      ( ( ord_less_eq_nat @ Y @ X3 )
     => ( ( ord_less_eq_nat @ X3 @ Y )
        = ( X3 = Y ) ) ) ).

% order_antisym_conv
thf(fact_290_VYDRA__MDL_Oru__t__tau__in,axiom,
    ! [Sigma: trace_a_t,Init_hd: h,Run_hd: h > option5775791656947114332_set_a,Ts: list_t,T: option2110264179601050681od_h_t,J: nat] :
      ( ( vYDRA_MDL_a_t_h @ Sigma @ Init_hd @ Run_hd )
     => ( ( reache645946503560514460_h_t_t @ ( run_t_h_t_a @ Run_hd ) @ ( t0_h_t_a @ Init_hd @ Run_hd ) @ Ts @ T )
       => ( ( ord_less_nat @ J @ ( size_size_list_t @ Ts ) )
         => ( ( nth_t @ Ts @ J )
            = ( tau_a_t @ Sigma @ J ) ) ) ) ) ).

% VYDRA_MDL.ru_t_tau_in
thf(fact_291_MDL_Osat_Ocong,axiom,
    sat_a_t = sat_a_t ).

% MDL.sat.cong
thf(fact_292_VYDRA__MDL_Owf__vydra_Ointros_I5_J,axiom,
    ! [Sigma: trace_a_t,Init_hd: h,Run_hd: h > option5775791656947114332_set_a,Phi2: formula_a_t,I: nat,N: nat,V: vydra_aux_a_t_h] :
      ( ( vYDRA_MDL_a_t_h @ Sigma @ Init_hd @ Run_hd )
     => ( ( vYDRA_wf_vydra_a_t_h @ Sigma @ Init_hd @ Run_hd @ Phi2 @ I @ N @ V )
       => ( vYDRA_wf_vydra_a_t_h @ Sigma @ Init_hd @ Run_hd @ ( neg_a_t @ Phi2 ) @ I @ ( suc @ N ) @ ( vydra_2331423191276895391_a_t_h @ V ) ) ) ) ).

% VYDRA_MDL.wf_vydra.intros(5)
thf(fact_293_MDL_Osat_Osimps_I3_J,axiom,
    ! [Sigma: trace_a_t,Phi: formula_a_t,I: nat] :
      ( ( sat_a_t @ Sigma @ ( neg_a_t @ Phi ) @ I )
      = ( ~ ( sat_a_t @ Sigma @ Phi @ I ) ) ) ).

% MDL.sat.simps(3)
thf(fact_294_MDL_Osat_Osimps_I1_J,axiom,
    ! [Sigma: trace_a_t,B: $o,I: nat] :
      ( ( sat_a_t @ Sigma @ ( bool_a_t @ B ) @ I )
      = B ) ).

% MDL.sat.simps(1)
thf(fact_295_formula_Odistinct_I3_J,axiom,
    ! [X1: $o,X33: formula_a_t] :
      ( ( bool_a_t @ X1 )
     != ( neg_a_t @ X33 ) ) ).

% formula.distinct(3)
thf(fact_296_formula_Odistinct_I19_J,axiom,
    ! [X2: a,X33: formula_a_t] :
      ( ( atom_a_t @ X2 )
     != ( neg_a_t @ X33 ) ) ).

% formula.distinct(19)
thf(fact_297_formula_Odistinct_I1_J,axiom,
    ! [X1: $o,X2: a] :
      ( ( bool_a_t @ X1 )
     != ( atom_a_t @ X2 ) ) ).

% formula.distinct(1)
thf(fact_298_wf__fmla_Osimps_I3_J,axiom,
    ! [Phi2: formula_a_t] :
      ( ( wf_fmla_a_t @ ( neg_a_t @ Phi2 ) )
      = ( wf_fmla_a_t @ Phi2 ) ) ).

% wf_fmla.simps(3)
thf(fact_299_bounded__future__fmla_Osimps_I3_J,axiom,
    ! [Phi2: formula_a_t] :
      ( ( bounde209865433540700070la_a_t @ ( neg_a_t @ Phi2 ) )
      = ( bounde209865433540700070la_a_t @ Phi2 ) ) ).

% bounded_future_fmla.simps(3)
thf(fact_300_wf__fmla_Osimps_I1_J,axiom,
    ! [B: $o] : ( wf_fmla_a_t @ ( bool_a_t @ B ) ) ).

% wf_fmla.simps(1)
thf(fact_301_wf__fmla_Osimps_I2_J,axiom,
    ! [A: a] : ( wf_fmla_a_t @ ( atom_a_t @ A ) ) ).

% wf_fmla.simps(2)
thf(fact_302_bounded__future__fmla_Osimps_I1_J,axiom,
    ! [B: $o] : ( bounde209865433540700070la_a_t @ ( bool_a_t @ B ) ) ).

% bounded_future_fmla.simps(1)
thf(fact_303_less__prod__simp,axiom,
    ! [X1: t,Y1: $o,X2: t,Y2: $o] :
      ( ( ord_le3945363144663698842od_t_o @ ( product_Pair_t_o @ X1 @ Y1 ) @ ( product_Pair_t_o @ X2 @ Y2 ) )
      = ( ( ord_less_t @ X1 @ X2 )
        | ( ( ord_less_eq_t @ X1 @ X2 )
          & ( ord_less_o @ Y1 @ Y2 ) ) ) ) ).

% less_prod_simp
thf(fact_304_less__prod__simp,axiom,
    ! [X1: t,Y1: set_a,X2: t,Y2: set_a] :
      ( ( ord_le7650071679307129824_set_a @ ( product_Pair_t_set_a @ X1 @ Y1 ) @ ( product_Pair_t_set_a @ X2 @ Y2 ) )
      = ( ( ord_less_t @ X1 @ X2 )
        | ( ( ord_less_eq_t @ X1 @ X2 )
          & ( ord_less_set_a @ Y1 @ Y2 ) ) ) ) ).

% less_prod_simp
thf(fact_305_less__prod__simp,axiom,
    ! [X1: nat > $o,Y1: nat,X2: nat > $o,Y2: nat] :
      ( ( ord_le1440161272000278354_o_nat @ ( produc7277522915581678840_o_nat @ X1 @ Y1 ) @ ( produc7277522915581678840_o_nat @ X2 @ Y2 ) )
      = ( ( ord_less_nat_o @ X1 @ X2 )
        | ( ( ord_less_eq_nat_o @ X1 @ X2 )
          & ( ord_less_nat @ Y1 @ Y2 ) ) ) ) ).

% less_prod_simp
thf(fact_306_less__prod__simp,axiom,
    ! [X1: nat,Y1: nat,X2: nat,Y2: nat] :
      ( ( ord_le1203424502768444845at_nat @ ( product_Pair_nat_nat @ X1 @ Y1 ) @ ( product_Pair_nat_nat @ X2 @ Y2 ) )
      = ( ( ord_less_nat @ X1 @ X2 )
        | ( ( ord_less_eq_nat @ X1 @ X2 )
          & ( ord_less_nat @ Y1 @ Y2 ) ) ) ) ).

% less_prod_simp
thf(fact_307_less__eq__prod__simp,axiom,
    ! [X1: t,Y1: $o,X2: t,Y2: $o] :
      ( ( ord_le1999770283289415822od_t_o @ ( product_Pair_t_o @ X1 @ Y1 ) @ ( product_Pair_t_o @ X2 @ Y2 ) )
      = ( ( ord_less_t @ X1 @ X2 )
        | ( ( ord_less_eq_t @ X1 @ X2 )
          & ( ord_less_eq_o @ Y1 @ Y2 ) ) ) ) ).

% less_eq_prod_simp
thf(fact_308_less__eq__prod__simp,axiom,
    ! [X1: t,Y1: set_a,X2: t,Y2: set_a] :
      ( ( ord_le1576776917030837460_set_a @ ( product_Pair_t_set_a @ X1 @ Y1 ) @ ( product_Pair_t_set_a @ X2 @ Y2 ) )
      = ( ( ord_less_t @ X1 @ X2 )
        | ( ( ord_less_eq_t @ X1 @ X2 )
          & ( ord_less_eq_set_a @ Y1 @ Y2 ) ) ) ) ).

% less_eq_prod_simp
thf(fact_309_less__eq__prod__simp,axiom,
    ! [X1: nat > $o,Y1: nat,X2: nat > $o,Y2: nat] :
      ( ( ord_le1595970101268698462_o_nat @ ( produc7277522915581678840_o_nat @ X1 @ Y1 ) @ ( produc7277522915581678840_o_nat @ X2 @ Y2 ) )
      = ( ( ord_less_nat_o @ X1 @ X2 )
        | ( ( ord_less_eq_nat_o @ X1 @ X2 )
          & ( ord_less_eq_nat @ Y1 @ Y2 ) ) ) ) ).

% less_eq_prod_simp
thf(fact_310_less__eq__prod__simp,axiom,
    ! [X1: nat,Y1: nat,X2: nat,Y2: nat] :
      ( ( ord_le8460144461188290721at_nat @ ( product_Pair_nat_nat @ X1 @ Y1 ) @ ( product_Pair_nat_nat @ X2 @ Y2 ) )
      = ( ( ord_less_nat @ X1 @ X2 )
        | ( ( ord_less_eq_nat @ X1 @ X2 )
          & ( ord_less_eq_nat @ Y1 @ Y2 ) ) ) ) ).

% less_eq_prod_simp
thf(fact_311_run__hd__tau__in,axiom,
    ! [Vs: list_P1095827921439705018_set_a,E4: h,E5: h,T2: t,X9: set_a,J: nat] :
      ( ( reache3397181163414509427_set_a @ run_hd @ init_hd @ Vs @ E4 )
     => ( ( ( run_hd @ E4 )
          = ( some_P2245991037696317435_set_a @ ( produc3432067189782488838_set_a @ E5 @ ( product_Pair_t_set_a @ T2 @ X9 ) ) ) )
       => ( ( ord_less_nat @ J @ ( size_s5362121731612881446_set_a @ Vs ) )
         => ( ( nth_t @ ( map_Pr119846518281778721et_a_t @ product_fst_t_set_a @ Vs ) @ J )
            = ( tau_a_t @ sigma @ J ) ) ) ) ) ).

% run_hd_tau_in
thf(fact_312_nth__equalityI,axiom,
    ! [Xs: list_P1095827921439705018_set_a,Ys: list_P1095827921439705018_set_a] :
      ( ( ( size_s5362121731612881446_set_a @ Xs )
        = ( size_s5362121731612881446_set_a @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s5362121731612881446_set_a @ Xs ) )
           => ( ( nth_Pr633927046780761275_set_a @ Xs @ I2 )
              = ( nth_Pr633927046780761275_set_a @ Ys @ I2 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_313_nth__equalityI,axiom,
    ! [Xs: list_vydra_aux_a_t_h,Ys: list_vydra_aux_a_t_h] :
      ( ( ( size_s8993897887676352572_a_t_h @ Xs )
        = ( size_s8993897887676352572_a_t_h @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s8993897887676352572_a_t_h @ Xs ) )
           => ( ( nth_vydra_aux_a_t_h @ Xs @ I2 )
              = ( nth_vydra_aux_a_t_h @ Ys @ I2 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_314_nth__equalityI,axiom,
    ! [Xs: list_iarray_o,Ys: list_iarray_o] :
      ( ( ( size_s3243743627286111600rray_o @ Xs )
        = ( size_s3243743627286111600rray_o @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s3243743627286111600rray_o @ Xs ) )
           => ( ( nth_iarray_o @ Xs @ I2 )
              = ( nth_iarray_o @ Ys @ I2 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_315_nth__equalityI,axiom,
    ! [Xs: list_formula_a_t,Ys: list_formula_a_t] :
      ( ( ( size_s8846756101701226951la_a_t @ Xs )
        = ( size_s8846756101701226951la_a_t @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s8846756101701226951la_a_t @ Xs ) )
           => ( ( nth_formula_a_t @ Xs @ I2 )
              = ( nth_formula_a_t @ Ys @ I2 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_316_nth__equalityI,axiom,
    ! [Xs: list_t,Ys: list_t] :
      ( ( ( size_size_list_t @ Xs )
        = ( size_size_list_t @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_t @ Xs ) )
           => ( ( nth_t @ Xs @ I2 )
              = ( nth_t @ Ys @ I2 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_317_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > product_prod_t_set_a > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K )
           => ? [X6: product_prod_t_set_a] : ( P @ I3 @ X6 ) ) )
      = ( ? [Xs2: list_P1095827921439705018_set_a] :
            ( ( ( size_s5362121731612881446_set_a @ Xs2 )
              = K )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K )
               => ( P @ I3 @ ( nth_Pr633927046780761275_set_a @ Xs2 @ I3 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_318_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > vydra_aux_a_t_h > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K )
           => ? [X6: vydra_aux_a_t_h] : ( P @ I3 @ X6 ) ) )
      = ( ? [Xs2: list_vydra_aux_a_t_h] :
            ( ( ( size_s8993897887676352572_a_t_h @ Xs2 )
              = K )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K )
               => ( P @ I3 @ ( nth_vydra_aux_a_t_h @ Xs2 @ I3 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_319_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > iarray_o > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K )
           => ? [X6: iarray_o] : ( P @ I3 @ X6 ) ) )
      = ( ? [Xs2: list_iarray_o] :
            ( ( ( size_s3243743627286111600rray_o @ Xs2 )
              = K )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K )
               => ( P @ I3 @ ( nth_iarray_o @ Xs2 @ I3 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_320_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > formula_a_t > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K )
           => ? [X6: formula_a_t] : ( P @ I3 @ X6 ) ) )
      = ( ? [Xs2: list_formula_a_t] :
            ( ( ( size_s8846756101701226951la_a_t @ Xs2 )
              = K )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K )
               => ( P @ I3 @ ( nth_formula_a_t @ Xs2 @ I3 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_321_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > t > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K )
           => ? [X6: t] : ( P @ I3 @ X6 ) ) )
      = ( ? [Xs2: list_t] :
            ( ( ( size_size_list_t @ Xs2 )
              = K )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K )
               => ( P @ I3 @ ( nth_t @ Xs2 @ I3 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_322_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y6: list_P1095827921439705018_set_a,Z4: list_P1095827921439705018_set_a] : ( Y6 = Z4 ) )
    = ( ^ [Xs2: list_P1095827921439705018_set_a,Ys2: list_P1095827921439705018_set_a] :
          ( ( ( size_s5362121731612881446_set_a @ Xs2 )
            = ( size_s5362121731612881446_set_a @ Ys2 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_s5362121731612881446_set_a @ Xs2 ) )
             => ( ( nth_Pr633927046780761275_set_a @ Xs2 @ I3 )
                = ( nth_Pr633927046780761275_set_a @ Ys2 @ I3 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_323_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y6: list_vydra_aux_a_t_h,Z4: list_vydra_aux_a_t_h] : ( Y6 = Z4 ) )
    = ( ^ [Xs2: list_vydra_aux_a_t_h,Ys2: list_vydra_aux_a_t_h] :
          ( ( ( size_s8993897887676352572_a_t_h @ Xs2 )
            = ( size_s8993897887676352572_a_t_h @ Ys2 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_s8993897887676352572_a_t_h @ Xs2 ) )
             => ( ( nth_vydra_aux_a_t_h @ Xs2 @ I3 )
                = ( nth_vydra_aux_a_t_h @ Ys2 @ I3 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_324_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y6: list_iarray_o,Z4: list_iarray_o] : ( Y6 = Z4 ) )
    = ( ^ [Xs2: list_iarray_o,Ys2: list_iarray_o] :
          ( ( ( size_s3243743627286111600rray_o @ Xs2 )
            = ( size_s3243743627286111600rray_o @ Ys2 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_s3243743627286111600rray_o @ Xs2 ) )
             => ( ( nth_iarray_o @ Xs2 @ I3 )
                = ( nth_iarray_o @ Ys2 @ I3 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_325_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y6: list_formula_a_t,Z4: list_formula_a_t] : ( Y6 = Z4 ) )
    = ( ^ [Xs2: list_formula_a_t,Ys2: list_formula_a_t] :
          ( ( ( size_s8846756101701226951la_a_t @ Xs2 )
            = ( size_s8846756101701226951la_a_t @ Ys2 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_s8846756101701226951la_a_t @ Xs2 ) )
             => ( ( nth_formula_a_t @ Xs2 @ I3 )
                = ( nth_formula_a_t @ Ys2 @ I3 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_326_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y6: list_t,Z4: list_t] : ( Y6 = Z4 ) )
    = ( ^ [Xs2: list_t,Ys2: list_t] :
          ( ( ( size_size_list_t @ Xs2 )
            = ( size_size_list_t @ Ys2 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_size_list_t @ Xs2 ) )
             => ( ( nth_t @ Xs2 @ I3 )
                = ( nth_t @ Ys2 @ I3 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_327_chain__le,axiom,
    ! [Xs: list_t,I: nat,J: nat] :
      ( ( chain_le_t @ Xs )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( ( ord_less_nat @ J @ ( size_size_list_t @ Xs ) )
         => ( ord_less_eq_t @ ( nth_t @ Xs @ I ) @ ( nth_t @ Xs @ J ) ) ) ) ) ).

% chain_le
thf(fact_328_chain__le,axiom,
    ! [Xs: list_nat,I: nat,J: nat] :
      ( ( chain_le_nat @ Xs )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
         => ( ord_less_eq_nat @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Xs @ J ) ) ) ) ) ).

% chain_le
thf(fact_329_sorted__list__atD,axiom,
    ! [Xs: list_P1095827921439705018_set_a,I: nat,J: nat] :
      ( ( sorted5003863949113869663_set_a @ Xs )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( ( ord_less_nat @ J @ ( size_s5362121731612881446_set_a @ Xs ) )
         => ( ord_le1576776917030837460_set_a @ ( nth_Pr633927046780761275_set_a @ Xs @ I ) @ ( nth_Pr633927046780761275_set_a @ Xs @ J ) ) ) ) ) ).

% sorted_list_atD
thf(fact_330_sorted__list__atD,axiom,
    ! [Xs: list_t,I: nat,J: nat] :
      ( ( sorted_list_t @ Xs )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( ( ord_less_nat @ J @ ( size_size_list_t @ Xs ) )
         => ( ord_less_eq_t @ ( nth_t @ Xs @ I ) @ ( nth_t @ Xs @ J ) ) ) ) ) ).

% sorted_list_atD
thf(fact_331_sorted__list__atD,axiom,
    ! [Xs: list_nat,I: nat,J: nat] :
      ( ( sorted_list_nat @ Xs )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
         => ( ord_less_eq_nat @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Xs @ J ) ) ) ) ) ).

% sorted_list_atD
thf(fact_332_run__subs__vD,axiom,
    ! [Run: t > option5115755630998872842od_t_o,Vs: list_t,Vs2: list_t,Bs: iarray_o,J: nat] :
      ( ( ( run_subs_t_t_t_o @ Run @ Vs )
        = ( some_P7501796307291990963rray_o @ ( produc8281831009199921854rray_o @ Vs2 @ Bs ) ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_t @ Vs ) )
       => ? [Vj: t,Tj: t,Bj: $o] :
            ( ( ( Run @ ( nth_t @ Vs @ J ) )
              = ( some_P5782336797119413673od_t_o @ ( produc7539267677312392372od_t_o @ Vj @ ( product_Pair_t_o @ Tj @ Bj ) ) ) )
            & ( ( nth_t @ Vs2 @ J )
              = Vj )
            & ( ( sub_o @ Bs @ J )
              = Bj ) ) ) ) ).

% run_subs_vD
thf(fact_333_run__subs__vD,axiom,
    ! [Run: t > option6366625828353349007od_h_t,Vs: list_t,Vs2: list_t,Bs: iarray_t,J: nat] :
      ( ( ( run_subs_t_t_h_t @ Run @ Vs )
        = ( some_P7066414042217547692rray_t @ ( produc4777086648503855159rray_t @ Vs2 @ Bs ) ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_t @ Vs ) )
       => ? [Vj: t,Tj: h,Bj: t] :
            ( ( ( Run @ ( nth_t @ Vs @ J ) )
              = ( some_P3412453634688163246od_h_t @ ( produc5663778790490485817od_h_t @ Vj @ ( product_Pair_h_t @ Tj @ Bj ) ) ) )
            & ( ( nth_t @ Vs2 @ J )
              = Vj )
            & ( ( sub_t @ Bs @ J )
              = Bj ) ) ) ) ).

% run_subs_vD
thf(fact_334_run__subs__vD,axiom,
    ! [Run: iarray_o > option5115755630998872842od_t_o,Vs: list_iarray_o,Vs2: list_t,Bs: iarray_o,J: nat] :
      ( ( ( run_su98717407436750474_t_t_o @ Run @ Vs )
        = ( some_P7501796307291990963rray_o @ ( produc8281831009199921854rray_o @ Vs2 @ Bs ) ) )
     => ( ( ord_less_nat @ J @ ( size_s3243743627286111600rray_o @ Vs ) )
       => ? [Vj: t,Tj: t,Bj: $o] :
            ( ( ( Run @ ( nth_iarray_o @ Vs @ J ) )
              = ( some_P5782336797119413673od_t_o @ ( produc7539267677312392372od_t_o @ Vj @ ( product_Pair_t_o @ Tj @ Bj ) ) ) )
            & ( ( nth_t @ Vs2 @ J )
              = Vj )
            & ( ( sub_o @ Bs @ J )
              = Bj ) ) ) ) ).

% run_subs_vD
thf(fact_335_run__subs__vD,axiom,
    ! [Run: iarray_o > option6366625828353349007od_h_t,Vs: list_iarray_o,Vs2: list_t,Bs: iarray_t,J: nat] :
      ( ( ( run_su3655969966392431119_t_h_t @ Run @ Vs )
        = ( some_P7066414042217547692rray_t @ ( produc4777086648503855159rray_t @ Vs2 @ Bs ) ) )
     => ( ( ord_less_nat @ J @ ( size_s3243743627286111600rray_o @ Vs ) )
       => ? [Vj: t,Tj: h,Bj: t] :
            ( ( ( Run @ ( nth_iarray_o @ Vs @ J ) )
              = ( some_P3412453634688163246od_h_t @ ( produc5663778790490485817od_h_t @ Vj @ ( product_Pair_h_t @ Tj @ Bj ) ) ) )
            & ( ( nth_t @ Vs2 @ J )
              = Vj )
            & ( ( sub_t @ Bs @ J )
              = Bj ) ) ) ) ).

% run_subs_vD
thf(fact_336_run__subs__vD,axiom,
    ! [Run: t > option4136621186271108688_set_a,Vs: list_t,Vs2: list_t,Bs: iarray_set_a,J: nat] :
      ( ( ( run_subs_t_t_t_set_a @ Run @ Vs )
        = ( some_P331467869274976249_set_a @ ( produc4688466989509365508_set_a @ Vs2 @ Bs ) ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_t @ Vs ) )
       => ? [Vj: t,Tj: t,Bj: set_a] :
            ( ( ( Run @ ( nth_t @ Vs @ J ) )
              = ( some_P7916121082815958767_set_a @ ( produc9102197234902130170_set_a @ Vj @ ( product_Pair_t_set_a @ Tj @ Bj ) ) ) )
            & ( ( nth_t @ Vs2 @ J )
              = Vj )
            & ( ( sub_set_a @ Bs @ J )
              = Bj ) ) ) ) ).

% run_subs_vD
thf(fact_337_run__subs__vD,axiom,
    ! [Run: t > option5775791656947114332_set_a,Vs: list_t,Vs2: list_h,Bs: iarray_set_a,J: nat] :
      ( ( ( run_subs_t_h_t_set_a @ Run @ Vs )
        = ( some_P7355459033391471877_set_a @ ( produc2489086116771085328_set_a @ Vs2 @ Bs ) ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_t @ Vs ) )
       => ? [Vj: h,Tj: t,Bj: set_a] :
            ( ( ( Run @ ( nth_t @ Vs @ J ) )
              = ( some_P2245991037696317435_set_a @ ( produc3432067189782488838_set_a @ Vj @ ( product_Pair_t_set_a @ Tj @ Bj ) ) ) )
            & ( ( nth_h @ Vs2 @ J )
              = Vj )
            & ( ( sub_set_a @ Bs @ J )
              = Bj ) ) ) ) ).

% run_subs_vD
thf(fact_338_run__subs__vD,axiom,
    ! [Run: iarray_o > option4136621186271108688_set_a,Vs: list_iarray_o,Vs2: list_t,Bs: iarray_set_a,J: nat] :
      ( ( ( run_su6940939646075227728_set_a @ Run @ Vs )
        = ( some_P331467869274976249_set_a @ ( produc4688466989509365508_set_a @ Vs2 @ Bs ) ) )
     => ( ( ord_less_nat @ J @ ( size_s3243743627286111600rray_o @ Vs ) )
       => ? [Vj: t,Tj: t,Bj: set_a] :
            ( ( ( Run @ ( nth_iarray_o @ Vs @ J ) )
              = ( some_P7916121082815958767_set_a @ ( produc9102197234902130170_set_a @ Vj @ ( product_Pair_t_set_a @ Tj @ Bj ) ) ) )
            & ( ( nth_t @ Vs2 @ J )
              = Vj )
            & ( ( sub_set_a @ Bs @ J )
              = Bj ) ) ) ) ).

% run_subs_vD
thf(fact_339_run__subs__vD,axiom,
    ! [Run: iarray_o > option5775791656947114332_set_a,Vs: list_iarray_o,Vs2: list_h,Bs: iarray_set_a,J: nat] :
      ( ( ( run_su853597514479236164_set_a @ Run @ Vs )
        = ( some_P7355459033391471877_set_a @ ( produc2489086116771085328_set_a @ Vs2 @ Bs ) ) )
     => ( ( ord_less_nat @ J @ ( size_s3243743627286111600rray_o @ Vs ) )
       => ? [Vj: h,Tj: t,Bj: set_a] :
            ( ( ( Run @ ( nth_iarray_o @ Vs @ J ) )
              = ( some_P2245991037696317435_set_a @ ( produc3432067189782488838_set_a @ Vj @ ( product_Pair_t_set_a @ Tj @ Bj ) ) ) )
            & ( ( nth_h @ Vs2 @ J )
              = Vj )
            & ( ( sub_set_a @ Bs @ J )
              = Bj ) ) ) ) ).

% run_subs_vD
thf(fact_340_run__subs__vD,axiom,
    ! [Run: formula_a_t > option5115755630998872842od_t_o,Vs: list_formula_a_t,Vs2: list_t,Bs: iarray_o,J: nat] :
      ( ( ( run_su7792916424282555075_t_t_o @ Run @ Vs )
        = ( some_P7501796307291990963rray_o @ ( produc8281831009199921854rray_o @ Vs2 @ Bs ) ) )
     => ( ( ord_less_nat @ J @ ( size_s8846756101701226951la_a_t @ Vs ) )
       => ? [Vj: t,Tj: t,Bj: $o] :
            ( ( ( Run @ ( nth_formula_a_t @ Vs @ J ) )
              = ( some_P5782336797119413673od_t_o @ ( produc7539267677312392372od_t_o @ Vj @ ( product_Pair_t_o @ Tj @ Bj ) ) ) )
            & ( ( nth_t @ Vs2 @ J )
              = Vj )
            & ( ( sub_o @ Bs @ J )
              = Bj ) ) ) ) ).

% run_subs_vD
thf(fact_341_run__subs__vD,axiom,
    ! [Run: formula_a_t > option6366625828353349007od_h_t,Vs: list_formula_a_t,Vs2: list_t,Bs: iarray_t,J: nat] :
      ( ( ( run_su4105110139124697544_t_h_t @ Run @ Vs )
        = ( some_P7066414042217547692rray_t @ ( produc4777086648503855159rray_t @ Vs2 @ Bs ) ) )
     => ( ( ord_less_nat @ J @ ( size_s8846756101701226951la_a_t @ Vs ) )
       => ? [Vj: t,Tj: h,Bj: t] :
            ( ( ( Run @ ( nth_formula_a_t @ Vs @ J ) )
              = ( some_P3412453634688163246od_h_t @ ( produc5663778790490485817od_h_t @ Vj @ ( product_Pair_h_t @ Tj @ Bj ) ) ) )
            & ( ( nth_t @ Vs2 @ J )
              = Vj )
            & ( ( sub_t @ Bs @ J )
              = Bj ) ) ) ) ).

% run_subs_vD
thf(fact_342_length__induct,axiom,
    ! [P: list_P1095827921439705018_set_a > $o,Xs: list_P1095827921439705018_set_a] :
      ( ! [Xs3: list_P1095827921439705018_set_a] :
          ( ! [Ys3: list_P1095827921439705018_set_a] :
              ( ( ord_less_nat @ ( size_s5362121731612881446_set_a @ Ys3 ) @ ( size_s5362121731612881446_set_a @ Xs3 ) )
             => ( P @ Ys3 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_343_length__induct,axiom,
    ! [P: list_vydra_aux_a_t_h > $o,Xs: list_vydra_aux_a_t_h] :
      ( ! [Xs3: list_vydra_aux_a_t_h] :
          ( ! [Ys3: list_vydra_aux_a_t_h] :
              ( ( ord_less_nat @ ( size_s8993897887676352572_a_t_h @ Ys3 ) @ ( size_s8993897887676352572_a_t_h @ Xs3 ) )
             => ( P @ Ys3 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_344_length__induct,axiom,
    ! [P: list_iarray_o > $o,Xs: list_iarray_o] :
      ( ! [Xs3: list_iarray_o] :
          ( ! [Ys3: list_iarray_o] :
              ( ( ord_less_nat @ ( size_s3243743627286111600rray_o @ Ys3 ) @ ( size_s3243743627286111600rray_o @ Xs3 ) )
             => ( P @ Ys3 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_345_length__induct,axiom,
    ! [P: list_formula_a_t > $o,Xs: list_formula_a_t] :
      ( ! [Xs3: list_formula_a_t] :
          ( ! [Ys3: list_formula_a_t] :
              ( ( ord_less_nat @ ( size_s8846756101701226951la_a_t @ Ys3 ) @ ( size_s8846756101701226951la_a_t @ Xs3 ) )
             => ( P @ Ys3 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_346_length__induct,axiom,
    ! [P: list_t > $o,Xs: list_t] :
      ( ! [Xs3: list_t] :
          ( ! [Ys3: list_t] :
              ( ( ord_less_nat @ ( size_size_list_t @ Ys3 ) @ ( size_size_list_t @ Xs3 ) )
             => ( P @ Ys3 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_347_length__map,axiom,
    ! [F: t > t,Xs: list_t] :
      ( ( size_size_list_t @ ( map_t_t @ F @ Xs ) )
      = ( size_size_list_t @ Xs ) ) ).

% length_map
thf(fact_348_length__map,axiom,
    ! [F: t > iarray_o,Xs: list_t] :
      ( ( size_s3243743627286111600rray_o @ ( map_t_iarray_o @ F @ Xs ) )
      = ( size_size_list_t @ Xs ) ) ).

% length_map
thf(fact_349_length__map,axiom,
    ! [F: iarray_o > t,Xs: list_iarray_o] :
      ( ( size_size_list_t @ ( map_iarray_o_t @ F @ Xs ) )
      = ( size_s3243743627286111600rray_o @ Xs ) ) ).

% length_map
thf(fact_350_length__map,axiom,
    ! [F: formula_a_t > $o,Xs: list_formula_a_t] :
      ( ( size_size_list_o @ ( map_formula_a_t_o @ F @ Xs ) )
      = ( size_s8846756101701226951la_a_t @ Xs ) ) ).

% length_map
thf(fact_351_length__map,axiom,
    ! [F: iarray_o > iarray_o,Xs: list_iarray_o] :
      ( ( size_s3243743627286111600rray_o @ ( map_ia4191468537823275204rray_o @ F @ Xs ) )
      = ( size_s3243743627286111600rray_o @ Xs ) ) ).

% length_map
thf(fact_352_length__map,axiom,
    ! [F: t > formula_a_t,Xs: list_t] :
      ( ( size_s8846756101701226951la_a_t @ ( map_t_formula_a_t @ F @ Xs ) )
      = ( size_size_list_t @ Xs ) ) ).

% length_map
thf(fact_353_length__map,axiom,
    ! [F: formula_a_t > t,Xs: list_formula_a_t] :
      ( ( size_size_list_t @ ( map_formula_a_t_t @ F @ Xs ) )
      = ( size_s8846756101701226951la_a_t @ Xs ) ) ).

% length_map
thf(fact_354_length__map,axiom,
    ! [F: t > product_prod_t_set_a,Xs: list_t] :
      ( ( size_s5362121731612881446_set_a @ ( map_t_7579227405386970535_set_a @ F @ Xs ) )
      = ( size_size_list_t @ Xs ) ) ).

% length_map
thf(fact_355_length__map,axiom,
    ! [F: t > vydra_aux_a_t_h,Xs: list_t] :
      ( ( size_s8993897887676352572_a_t_h @ ( map_t_1712355122357082213_a_t_h @ F @ Xs ) )
      = ( size_size_list_t @ Xs ) ) ).

% length_map
thf(fact_356_length__map,axiom,
    ! [F: formula_a_t > iarray_o,Xs: list_formula_a_t] :
      ( ( size_s3243743627286111600rray_o @ ( map_fo7929784389399151445rray_o @ F @ Xs ) )
      = ( size_s8846756101701226951la_a_t @ Xs ) ) ).

% length_map
thf(fact_357_nth__map,axiom,
    ! [N: nat,Xs: list_t,F: t > t] :
      ( ( ord_less_nat @ N @ ( size_size_list_t @ Xs ) )
     => ( ( nth_t @ ( map_t_t @ F @ Xs ) @ N )
        = ( F @ ( nth_t @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_358_nth__map,axiom,
    ! [N: nat,Xs: list_iarray_o,F: iarray_o > t] :
      ( ( ord_less_nat @ N @ ( size_s3243743627286111600rray_o @ Xs ) )
     => ( ( nth_t @ ( map_iarray_o_t @ F @ Xs ) @ N )
        = ( F @ ( nth_iarray_o @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_359_nth__map,axiom,
    ! [N: nat,Xs: list_formula_a_t,F: formula_a_t > t] :
      ( ( ord_less_nat @ N @ ( size_s8846756101701226951la_a_t @ Xs ) )
     => ( ( nth_t @ ( map_formula_a_t_t @ F @ Xs ) @ N )
        = ( F @ ( nth_formula_a_t @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_360_nth__map,axiom,
    ! [N: nat,Xs: list_formula_a_t,F: formula_a_t > $o] :
      ( ( ord_less_nat @ N @ ( size_s8846756101701226951la_a_t @ Xs ) )
     => ( ( nth_o @ ( map_formula_a_t_o @ F @ Xs ) @ N )
        = ( F @ ( nth_formula_a_t @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_361_nth__map,axiom,
    ! [N: nat,Xs: list_t,F: t > formula_a_t] :
      ( ( ord_less_nat @ N @ ( size_size_list_t @ Xs ) )
     => ( ( nth_formula_a_t @ ( map_t_formula_a_t @ F @ Xs ) @ N )
        = ( F @ ( nth_t @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_362_nth__map,axiom,
    ! [N: nat,Xs: list_P1095827921439705018_set_a,F: product_prod_t_set_a > t] :
      ( ( ord_less_nat @ N @ ( size_s5362121731612881446_set_a @ Xs ) )
     => ( ( nth_t @ ( map_Pr119846518281778721et_a_t @ F @ Xs ) @ N )
        = ( F @ ( nth_Pr633927046780761275_set_a @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_363_nth__map,axiom,
    ! [N: nat,Xs: list_vydra_aux_a_t_h,F: vydra_aux_a_t_h > t] :
      ( ( ord_less_nat @ N @ ( size_s8993897887676352572_a_t_h @ Xs ) )
     => ( ( nth_t @ ( map_vy4786242547320179251_t_h_t @ F @ Xs ) @ N )
        = ( F @ ( nth_vydra_aux_a_t_h @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_364_nth__map,axiom,
    ! [N: nat,Xs: list_iarray_o,F: iarray_o > formula_a_t] :
      ( ( ord_less_nat @ N @ ( size_s3243743627286111600rray_o @ Xs ) )
     => ( ( nth_formula_a_t @ ( map_ia8297430680906214877la_a_t @ F @ Xs ) @ N )
        = ( F @ ( nth_iarray_o @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_365_nth__map,axiom,
    ! [N: nat,Xs: list_t,F: t > vydra_aux_a_t_h] :
      ( ( ord_less_nat @ N @ ( size_size_list_t @ Xs ) )
     => ( ( nth_vydra_aux_a_t_h @ ( map_t_1712355122357082213_a_t_h @ F @ Xs ) @ N )
        = ( F @ ( nth_t @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_366_nth__map,axiom,
    ! [N: nat,Xs: list_iarray_o,F: iarray_o > vydra_aux_a_t_h] :
      ( ( ord_less_nat @ N @ ( size_s3243743627286111600rray_o @ Xs ) )
     => ( ( nth_vydra_aux_a_t_h @ ( map_ia7517763552769034962_a_t_h @ F @ Xs ) @ N )
        = ( F @ ( nth_iarray_o @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_367_map__eq__imp__length__eq,axiom,
    ! [F: formula_a_t > $o,Xs: list_formula_a_t,G: t > $o,Ys: list_t] :
      ( ( ( map_formula_a_t_o @ F @ Xs )
        = ( map_t_o @ G @ Ys ) )
     => ( ( size_s8846756101701226951la_a_t @ Xs )
        = ( size_size_list_t @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_368_map__eq__imp__length__eq,axiom,
    ! [F: t > $o,Xs: list_t,G: formula_a_t > $o,Ys: list_formula_a_t] :
      ( ( ( map_t_o @ F @ Xs )
        = ( map_formula_a_t_o @ G @ Ys ) )
     => ( ( size_size_list_t @ Xs )
        = ( size_s8846756101701226951la_a_t @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_369_map__eq__imp__length__eq,axiom,
    ! [F: product_prod_t_set_a > t,Xs: list_P1095827921439705018_set_a,G: t > t,Ys: list_t] :
      ( ( ( map_Pr119846518281778721et_a_t @ F @ Xs )
        = ( map_t_t @ G @ Ys ) )
     => ( ( size_s5362121731612881446_set_a @ Xs )
        = ( size_size_list_t @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_370_map__eq__imp__length__eq,axiom,
    ! [F: iarray_o > $o,Xs: list_iarray_o,G: formula_a_t > $o,Ys: list_formula_a_t] :
      ( ( ( map_iarray_o_o @ F @ Xs )
        = ( map_formula_a_t_o @ G @ Ys ) )
     => ( ( size_s3243743627286111600rray_o @ Xs )
        = ( size_s8846756101701226951la_a_t @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_371_map__eq__imp__length__eq,axiom,
    ! [F: formula_a_t > $o,Xs: list_formula_a_t,G: iarray_o > $o,Ys: list_iarray_o] :
      ( ( ( map_formula_a_t_o @ F @ Xs )
        = ( map_iarray_o_o @ G @ Ys ) )
     => ( ( size_s8846756101701226951la_a_t @ Xs )
        = ( size_s3243743627286111600rray_o @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_372_map__eq__imp__length__eq,axiom,
    ! [F: t > t,Xs: list_t,G: product_prod_t_set_a > t,Ys: list_P1095827921439705018_set_a] :
      ( ( ( map_t_t @ F @ Xs )
        = ( map_Pr119846518281778721et_a_t @ G @ Ys ) )
     => ( ( size_size_list_t @ Xs )
        = ( size_s5362121731612881446_set_a @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_373_map__eq__imp__length__eq,axiom,
    ! [F: product_prod_t_set_a > t,Xs: list_P1095827921439705018_set_a,G: iarray_o > t,Ys: list_iarray_o] :
      ( ( ( map_Pr119846518281778721et_a_t @ F @ Xs )
        = ( map_iarray_o_t @ G @ Ys ) )
     => ( ( size_s5362121731612881446_set_a @ Xs )
        = ( size_s3243743627286111600rray_o @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_374_map__eq__imp__length__eq,axiom,
    ! [F: iarray_o > t,Xs: list_iarray_o,G: product_prod_t_set_a > t,Ys: list_P1095827921439705018_set_a] :
      ( ( ( map_iarray_o_t @ F @ Xs )
        = ( map_Pr119846518281778721et_a_t @ G @ Ys ) )
     => ( ( size_s3243743627286111600rray_o @ Xs )
        = ( size_s5362121731612881446_set_a @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_375_map__eq__imp__length__eq,axiom,
    ! [F: formula_a_t > $o,Xs: list_formula_a_t,G: formula_a_t > $o,Ys: list_formula_a_t] :
      ( ( ( map_formula_a_t_o @ F @ Xs )
        = ( map_formula_a_t_o @ G @ Ys ) )
     => ( ( size_s8846756101701226951la_a_t @ Xs )
        = ( size_s8846756101701226951la_a_t @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_376_map__eq__imp__length__eq,axiom,
    ! [F: product_prod_t_set_a > $o,Xs: list_P1095827921439705018_set_a,G: formula_a_t > $o,Ys: list_formula_a_t] :
      ( ( ( map_Pr3138614915695031976et_a_o @ F @ Xs )
        = ( map_formula_a_t_o @ G @ Ys ) )
     => ( ( size_s5362121731612881446_set_a @ Xs )
        = ( size_s8846756101701226951la_a_t @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_377_fst__eqD,axiom,
    ! [X3: nat > $o,Y: nat,A: nat > $o] :
      ( ( ( produc7643701372303132684_o_nat @ ( produc7277522915581678840_o_nat @ X3 @ Y ) )
        = A )
     => ( X3 = A ) ) ).

% fst_eqD
thf(fact_378_fst__eqD,axiom,
    ! [X3: t,Y: $o,A: t] :
      ( ( ( product_fst_t_o @ ( product_Pair_t_o @ X3 @ Y ) )
        = A )
     => ( X3 = A ) ) ).

% fst_eqD
thf(fact_379_fst__eqD,axiom,
    ! [X3: h,Y: product_prod_t_set_a,A: h] :
      ( ( ( produc6892702156181598450_set_a @ ( produc3432067189782488838_set_a @ X3 @ Y ) )
        = A )
     => ( X3 = A ) ) ).

% fst_eqD
thf(fact_380_fst__eqD,axiom,
    ! [X3: h,Y: t,A: h] :
      ( ( ( product_fst_h_t @ ( product_Pair_h_t @ X3 @ Y ) )
        = A )
     => ( X3 = A ) ) ).

% fst_eqD
thf(fact_381_fst__eqD,axiom,
    ! [X3: t,Y: set_a,A: t] :
      ( ( ( product_fst_t_set_a @ ( product_Pair_t_set_a @ X3 @ Y ) )
        = A )
     => ( X3 = A ) ) ).

% fst_eqD
thf(fact_382_fst__conv,axiom,
    ! [X1: nat > $o,X2: nat] :
      ( ( produc7643701372303132684_o_nat @ ( produc7277522915581678840_o_nat @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_383_fst__conv,axiom,
    ! [X1: t,X2: $o] :
      ( ( product_fst_t_o @ ( product_Pair_t_o @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_384_fst__conv,axiom,
    ! [X1: h,X2: product_prod_t_set_a] :
      ( ( produc6892702156181598450_set_a @ ( produc3432067189782488838_set_a @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_385_fst__conv,axiom,
    ! [X1: h,X2: t] :
      ( ( product_fst_h_t @ ( product_Pair_h_t @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_386_fst__conv,axiom,
    ! [X1: t,X2: set_a] :
      ( ( product_fst_t_set_a @ ( product_Pair_t_set_a @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_387_map__equality__iff,axiom,
    ! [F: formula_a_t > $o,Xs: list_formula_a_t,G: t > $o,Ys: list_t] :
      ( ( ( map_formula_a_t_o @ F @ Xs )
        = ( map_t_o @ G @ Ys ) )
      = ( ( ( size_s8846756101701226951la_a_t @ Xs )
          = ( size_size_list_t @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_t @ Ys ) )
           => ( ( F @ ( nth_formula_a_t @ Xs @ I3 ) )
              = ( G @ ( nth_t @ Ys @ I3 ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_388_map__equality__iff,axiom,
    ! [F: t > $o,Xs: list_t,G: formula_a_t > $o,Ys: list_formula_a_t] :
      ( ( ( map_t_o @ F @ Xs )
        = ( map_formula_a_t_o @ G @ Ys ) )
      = ( ( ( size_size_list_t @ Xs )
          = ( size_s8846756101701226951la_a_t @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s8846756101701226951la_a_t @ Ys ) )
           => ( ( F @ ( nth_t @ Xs @ I3 ) )
              = ( G @ ( nth_formula_a_t @ Ys @ I3 ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_389_map__equality__iff,axiom,
    ! [F: product_prod_t_set_a > t,Xs: list_P1095827921439705018_set_a,G: t > t,Ys: list_t] :
      ( ( ( map_Pr119846518281778721et_a_t @ F @ Xs )
        = ( map_t_t @ G @ Ys ) )
      = ( ( ( size_s5362121731612881446_set_a @ Xs )
          = ( size_size_list_t @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_t @ Ys ) )
           => ( ( F @ ( nth_Pr633927046780761275_set_a @ Xs @ I3 ) )
              = ( G @ ( nth_t @ Ys @ I3 ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_390_map__equality__iff,axiom,
    ! [F: iarray_o > $o,Xs: list_iarray_o,G: formula_a_t > $o,Ys: list_formula_a_t] :
      ( ( ( map_iarray_o_o @ F @ Xs )
        = ( map_formula_a_t_o @ G @ Ys ) )
      = ( ( ( size_s3243743627286111600rray_o @ Xs )
          = ( size_s8846756101701226951la_a_t @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s8846756101701226951la_a_t @ Ys ) )
           => ( ( F @ ( nth_iarray_o @ Xs @ I3 ) )
              = ( G @ ( nth_formula_a_t @ Ys @ I3 ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_391_map__equality__iff,axiom,
    ! [F: formula_a_t > $o,Xs: list_formula_a_t,G: iarray_o > $o,Ys: list_iarray_o] :
      ( ( ( map_formula_a_t_o @ F @ Xs )
        = ( map_iarray_o_o @ G @ Ys ) )
      = ( ( ( size_s8846756101701226951la_a_t @ Xs )
          = ( size_s3243743627286111600rray_o @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s3243743627286111600rray_o @ Ys ) )
           => ( ( F @ ( nth_formula_a_t @ Xs @ I3 ) )
              = ( G @ ( nth_iarray_o @ Ys @ I3 ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_392_map__equality__iff,axiom,
    ! [F: t > t,Xs: list_t,G: product_prod_t_set_a > t,Ys: list_P1095827921439705018_set_a] :
      ( ( ( map_t_t @ F @ Xs )
        = ( map_Pr119846518281778721et_a_t @ G @ Ys ) )
      = ( ( ( size_size_list_t @ Xs )
          = ( size_s5362121731612881446_set_a @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s5362121731612881446_set_a @ Ys ) )
           => ( ( F @ ( nth_t @ Xs @ I3 ) )
              = ( G @ ( nth_Pr633927046780761275_set_a @ Ys @ I3 ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_393_map__equality__iff,axiom,
    ! [F: product_prod_t_set_a > t,Xs: list_P1095827921439705018_set_a,G: iarray_o > t,Ys: list_iarray_o] :
      ( ( ( map_Pr119846518281778721et_a_t @ F @ Xs )
        = ( map_iarray_o_t @ G @ Ys ) )
      = ( ( ( size_s5362121731612881446_set_a @ Xs )
          = ( size_s3243743627286111600rray_o @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s3243743627286111600rray_o @ Ys ) )
           => ( ( F @ ( nth_Pr633927046780761275_set_a @ Xs @ I3 ) )
              = ( G @ ( nth_iarray_o @ Ys @ I3 ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_394_map__equality__iff,axiom,
    ! [F: iarray_o > t,Xs: list_iarray_o,G: product_prod_t_set_a > t,Ys: list_P1095827921439705018_set_a] :
      ( ( ( map_iarray_o_t @ F @ Xs )
        = ( map_Pr119846518281778721et_a_t @ G @ Ys ) )
      = ( ( ( size_s3243743627286111600rray_o @ Xs )
          = ( size_s5362121731612881446_set_a @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s5362121731612881446_set_a @ Ys ) )
           => ( ( F @ ( nth_iarray_o @ Xs @ I3 ) )
              = ( G @ ( nth_Pr633927046780761275_set_a @ Ys @ I3 ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_395_map__equality__iff,axiom,
    ! [F: formula_a_t > $o,Xs: list_formula_a_t,G: formula_a_t > $o,Ys: list_formula_a_t] :
      ( ( ( map_formula_a_t_o @ F @ Xs )
        = ( map_formula_a_t_o @ G @ Ys ) )
      = ( ( ( size_s8846756101701226951la_a_t @ Xs )
          = ( size_s8846756101701226951la_a_t @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s8846756101701226951la_a_t @ Ys ) )
           => ( ( F @ ( nth_formula_a_t @ Xs @ I3 ) )
              = ( G @ ( nth_formula_a_t @ Ys @ I3 ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_396_map__equality__iff,axiom,
    ! [F: product_prod_t_set_a > $o,Xs: list_P1095827921439705018_set_a,G: formula_a_t > $o,Ys: list_formula_a_t] :
      ( ( ( map_Pr3138614915695031976et_a_o @ F @ Xs )
        = ( map_formula_a_t_o @ G @ Ys ) )
      = ( ( ( size_s5362121731612881446_set_a @ Xs )
          = ( size_s8846756101701226951la_a_t @ Ys ) )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s8846756101701226951la_a_t @ Ys ) )
           => ( ( F @ ( nth_Pr633927046780761275_set_a @ Xs @ I3 ) )
              = ( G @ ( nth_formula_a_t @ Ys @ I3 ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_397_reaches__on__run__hd__t,axiom,
    ! [Run_hd: h > option5775791656947114332_set_a,Init_hd: h,Vs: list_P1095827921439705018_set_a,E2: h] :
      ( ( reache3397181163414509427_set_a @ Run_hd @ Init_hd @ Vs @ E2 )
     => ? [X_1: option2110264179601050681od_h_t] : ( reache645946503560514460_h_t_t @ ( run_t_h_t_a @ Run_hd ) @ ( t0_h_t_a @ Init_hd @ Run_hd ) @ ( map_Pr119846518281778721et_a_t @ product_fst_t_set_a @ Vs ) @ X_1 ) ) ).

% reaches_on_run_hd_t
thf(fact_398_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_P1095827921439705018_set_a] :
      ( ( size_s5362121731612881446_set_a @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_399_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_vydra_aux_a_t_h] :
      ( ( size_s8993897887676352572_a_t_h @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_400_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_iarray_o] :
      ( ( size_s3243743627286111600rray_o @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_401_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_formula_a_t] :
      ( ( size_s8846756101701226951la_a_t @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_402_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_t] :
      ( ( size_size_list_t @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_403_neq__if__length__neq,axiom,
    ! [Xs: list_P1095827921439705018_set_a,Ys: list_P1095827921439705018_set_a] :
      ( ( ( size_s5362121731612881446_set_a @ Xs )
       != ( size_s5362121731612881446_set_a @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_404_neq__if__length__neq,axiom,
    ! [Xs: list_vydra_aux_a_t_h,Ys: list_vydra_aux_a_t_h] :
      ( ( ( size_s8993897887676352572_a_t_h @ Xs )
       != ( size_s8993897887676352572_a_t_h @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_405_neq__if__length__neq,axiom,
    ! [Xs: list_iarray_o,Ys: list_iarray_o] :
      ( ( ( size_s3243743627286111600rray_o @ Xs )
       != ( size_s3243743627286111600rray_o @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_406_neq__if__length__neq,axiom,
    ! [Xs: list_formula_a_t,Ys: list_formula_a_t] :
      ( ( ( size_s8846756101701226951la_a_t @ Xs )
       != ( size_s8846756101701226951la_a_t @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_407_neq__if__length__neq,axiom,
    ! [Xs: list_t,Ys: list_t] :
      ( ( ( size_size_list_t @ Xs )
       != ( size_size_list_t @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_408_reach__event__t,axiom,
    ! [Run_hd: h > option5775791656947114332_set_a,E2: h,Vs: list_P1095827921439705018_set_a,E4: h,E6: h,T: t,X4: set_a,E5: h,T2: t,X9: set_a] :
      ( ( reache3397181163414509427_set_a @ Run_hd @ E2 @ Vs @ E4 )
     => ( ( ( Run_hd @ E2 )
          = ( some_P2245991037696317435_set_a @ ( produc3432067189782488838_set_a @ E6 @ ( product_Pair_t_set_a @ T @ X4 ) ) ) )
       => ( ( ( Run_hd @ E4 )
            = ( some_P2245991037696317435_set_a @ ( produc3432067189782488838_set_a @ E5 @ ( product_Pair_t_set_a @ T2 @ X9 ) ) ) )
         => ( reache645946503560514460_h_t_t @ ( run_t_h_t_a @ Run_hd ) @ ( some_P2526622376857543512od_h_t @ ( product_Pair_h_t @ E6 @ T ) ) @ ( map_Pr119846518281778721et_a_t @ product_fst_t_set_a @ Vs ) @ ( some_P2526622376857543512od_h_t @ ( product_Pair_h_t @ E5 @ T2 ) ) ) ) ) ) ).

% reach_event_t
thf(fact_409_VYDRA__MDL_Orun__hd__tau__in,axiom,
    ! [Sigma: trace_a_t,Init_hd: h,Run_hd: h > option5775791656947114332_set_a,Vs: list_P1095827921439705018_set_a,E4: h,E5: h,T2: t,X9: set_a,J: nat] :
      ( ( vYDRA_MDL_a_t_h @ Sigma @ Init_hd @ Run_hd )
     => ( ( reache3397181163414509427_set_a @ Run_hd @ Init_hd @ Vs @ E4 )
       => ( ( ( Run_hd @ E4 )
            = ( some_P2245991037696317435_set_a @ ( produc3432067189782488838_set_a @ E5 @ ( product_Pair_t_set_a @ T2 @ X9 ) ) ) )
         => ( ( ord_less_nat @ J @ ( size_s5362121731612881446_set_a @ Vs ) )
           => ( ( nth_t @ ( map_Pr119846518281778721et_a_t @ product_fst_t_set_a @ Vs ) @ J )
              = ( tau_a_t @ Sigma @ J ) ) ) ) ) ) ).

% VYDRA_MDL.run_hd_tau_in
thf(fact_410_reach__event__t0__t,axiom,
    ! [Run_hd: h > option5775791656947114332_set_a,Init_hd: h,Vs: list_P1095827921439705018_set_a,E4: h,E5: h,T2: t,X9: set_a] :
      ( ( reache3397181163414509427_set_a @ Run_hd @ Init_hd @ Vs @ E4 )
     => ( ( ( Run_hd @ E4 )
          = ( some_P2245991037696317435_set_a @ ( produc3432067189782488838_set_a @ E5 @ ( product_Pair_t_set_a @ T2 @ X9 ) ) ) )
       => ( reache645946503560514460_h_t_t @ ( run_t_h_t_a @ Run_hd ) @ ( t0_h_t_a @ Init_hd @ Run_hd ) @ ( map_Pr119846518281778721et_a_t @ product_fst_t_set_a @ Vs ) @ ( some_P2526622376857543512od_h_t @ ( product_Pair_h_t @ E5 @ T2 ) ) ) ) ) ).

% reach_event_t0_t
thf(fact_411_run__subs__lD,axiom,
    ! [Run: vydra_aux_a_t_h > option4580262698044125979od_t_o,Vs: list_vydra_aux_a_t_h,Vs2: list_vydra_aux_a_t_h,Bs: iarray_o] :
      ( ( ( run_su3116026553151384105_h_t_o @ Run @ Vs )
        = ( some_P7279874174034738982rray_o @ ( produc8472968102330193021rray_o @ Vs2 @ Bs ) ) )
     => ( ( ( size_s8993897887676352572_a_t_h @ Vs2 )
          = ( size_s8993897887676352572_a_t_h @ Vs ) )
        & ( ( length_o @ Bs )
          = ( size_s8993897887676352572_a_t_h @ Vs ) ) ) ) ).

% run_subs_lD
thf(fact_412_nat__descend__induct,axiom,
    ! [N: nat,P: nat > $o,M: nat] :
      ( ! [K2: nat] :
          ( ( ord_less_nat @ N @ K2 )
         => ( P @ K2 ) )
     => ( ! [K2: nat] :
            ( ( ord_less_eq_nat @ K2 @ N )
           => ( ! [I4: nat] :
                  ( ( ord_less_nat @ K2 @ I4 )
                 => ( P @ I4 ) )
             => ( P @ K2 ) ) )
       => ( P @ M ) ) ) ).

% nat_descend_induct
thf(fact_413_eq__fst__iff,axiom,
    ! [A: nat > $o,P2: produc3074792404157404414_o_nat] :
      ( ( A
        = ( produc7643701372303132684_o_nat @ P2 ) )
      = ( ? [B4: nat] :
            ( P2
            = ( produc7277522915581678840_o_nat @ A @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_414_eq__fst__iff,axiom,
    ! [A: t,P2: product_prod_t_o] :
      ( ( A
        = ( product_fst_t_o @ P2 ) )
      = ( ? [B4: $o] :
            ( P2
            = ( product_Pair_t_o @ A @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_415_eq__fst__iff,axiom,
    ! [A: h,P2: produc8090089739038349590_set_a] :
      ( ( A
        = ( produc6892702156181598450_set_a @ P2 ) )
      = ( ? [B4: product_prod_t_set_a] :
            ( P2
            = ( produc3432067189782488838_set_a @ A @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_416_eq__fst__iff,axiom,
    ! [A: h,P2: product_prod_h_t] :
      ( ( A
        = ( product_fst_h_t @ P2 ) )
      = ( ? [B4: t] :
            ( P2
            = ( product_Pair_h_t @ A @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_417_eq__fst__iff,axiom,
    ! [A: t,P2: product_prod_t_set_a] :
      ( ( A
        = ( product_fst_t_set_a @ P2 ) )
      = ( ? [B4: set_a] :
            ( P2
            = ( product_Pair_t_set_a @ A @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_418_fstI,axiom,
    ! [X3: produc3074792404157404414_o_nat,Y: nat > $o,Z3: nat] :
      ( ( X3
        = ( produc7277522915581678840_o_nat @ Y @ Z3 ) )
     => ( ( produc7643701372303132684_o_nat @ X3 )
        = Y ) ) ).

% fstI
thf(fact_419_fstI,axiom,
    ! [X3: product_prod_t_o,Y: t,Z3: $o] :
      ( ( X3
        = ( product_Pair_t_o @ Y @ Z3 ) )
     => ( ( product_fst_t_o @ X3 )
        = Y ) ) ).

% fstI
thf(fact_420_fstI,axiom,
    ! [X3: produc8090089739038349590_set_a,Y: h,Z3: product_prod_t_set_a] :
      ( ( X3
        = ( produc3432067189782488838_set_a @ Y @ Z3 ) )
     => ( ( produc6892702156181598450_set_a @ X3 )
        = Y ) ) ).

% fstI
thf(fact_421_fstI,axiom,
    ! [X3: product_prod_h_t,Y: h,Z3: t] :
      ( ( X3
        = ( product_Pair_h_t @ Y @ Z3 ) )
     => ( ( product_fst_h_t @ X3 )
        = Y ) ) ).

% fstI
thf(fact_422_fstI,axiom,
    ! [X3: product_prod_t_set_a,Y: t,Z3: set_a] :
      ( ( X3
        = ( product_Pair_t_set_a @ Y @ Z3 ) )
     => ( ( product_fst_t_set_a @ X3 )
        = Y ) ) ).

% fstI
thf(fact_423_complete__interval,axiom,
    ! [A: nat,B: nat,P: nat > $o] :
      ( ( ord_less_nat @ A @ B )
     => ( ( P @ A )
       => ( ~ ( P @ B )
         => ? [C: nat] :
              ( ( ord_less_eq_nat @ A @ C )
              & ( ord_less_eq_nat @ C @ B )
              & ! [X10: nat] :
                  ( ( ( ord_less_eq_nat @ A @ X10 )
                    & ( ord_less_nat @ X10 @ C ) )
                 => ( P @ X10 ) )
              & ! [D: nat] :
                  ( ! [X: nat] :
                      ( ( ( ord_less_eq_nat @ A @ X )
                        & ( ord_less_nat @ X @ D ) )
                     => ( P @ X ) )
                 => ( ord_less_eq_nat @ D @ C ) ) ) ) ) ) ).

% complete_interval
thf(fact_424_IArray__nth__equalityI,axiom,
    ! [Xs: iarray1745998091029370010_set_a,Ys: list_P1095827921439705018_set_a] :
      ( ( ( length4184113030062493425_set_a @ Xs )
        = ( size_s5362121731612881446_set_a @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( length4184113030062493425_set_a @ Xs ) )
           => ( ( sub_Pr3630787706679140143_set_a @ Xs @ I2 )
              = ( nth_Pr633927046780761275_set_a @ Ys @ I2 ) ) )
       => ( Xs
          = ( iArray8998427052846395271_set_a @ Ys ) ) ) ) ).

% IArray_nth_equalityI
thf(fact_425_IArray__nth__equalityI,axiom,
    ! [Xs: iarray4851489445666185416_a_t_h,Ys: list_vydra_aux_a_t_h] :
      ( ( ( length242649669638338715_a_t_h @ Xs )
        = ( size_s8993897887676352572_a_t_h @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( length242649669638338715_a_t_h @ Xs ) )
           => ( ( sub_vydra_aux_a_t_h @ Xs @ I2 )
              = ( nth_vydra_aux_a_t_h @ Ys @ I2 ) ) )
       => ( Xs
          = ( iArray3337576414570620549_a_t_h @ Ys ) ) ) ) ).

% IArray_nth_equalityI
thf(fact_426_IArray__nth__equalityI,axiom,
    ! [Xs: iarray_iarray_o,Ys: list_iarray_o] :
      ( ( ( length_iarray_o @ Xs )
        = ( size_s3243743627286111600rray_o @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( length_iarray_o @ Xs ) )
           => ( ( sub_iarray_o @ Xs @ I2 )
              = ( nth_iarray_o @ Ys @ I2 ) ) )
       => ( Xs
          = ( iArray_iarray_o @ Ys ) ) ) ) ).

% IArray_nth_equalityI
thf(fact_427_IArray__nth__equalityI,axiom,
    ! [Xs: iarray_formula_a_t,Ys: list_formula_a_t] :
      ( ( ( length_formula_a_t @ Xs )
        = ( size_s8846756101701226951la_a_t @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( length_formula_a_t @ Xs ) )
           => ( ( sub_formula_a_t @ Xs @ I2 )
              = ( nth_formula_a_t @ Ys @ I2 ) ) )
       => ( Xs
          = ( iArray_formula_a_t @ Ys ) ) ) ) ).

% IArray_nth_equalityI
thf(fact_428_IArray__nth__equalityI,axiom,
    ! [Xs: iarray_t,Ys: list_t] :
      ( ( ( length_t @ Xs )
        = ( size_size_list_t @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( length_t @ Xs ) )
           => ( ( sub_t @ Xs @ I2 )
              = ( nth_t @ Ys @ I2 ) ) )
       => ( Xs
          = ( iArray_t @ Ys ) ) ) ) ).

% IArray_nth_equalityI
thf(fact_429_VYDRA__MDL_OIArray__nth__equalityI,axiom,
    ! [Sigma: trace_a_t,Init_hd: h,Run_hd: h > option5775791656947114332_set_a,Xs: iarray1745998091029370010_set_a,Ys: list_P1095827921439705018_set_a] :
      ( ( vYDRA_MDL_a_t_h @ Sigma @ Init_hd @ Run_hd )
     => ( ( ( length4184113030062493425_set_a @ Xs )
          = ( size_s5362121731612881446_set_a @ Ys ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( length4184113030062493425_set_a @ Xs ) )
             => ( ( sub_Pr3630787706679140143_set_a @ Xs @ I2 )
                = ( nth_Pr633927046780761275_set_a @ Ys @ I2 ) ) )
         => ( Xs
            = ( iArray8998427052846395271_set_a @ Ys ) ) ) ) ) ).

% VYDRA_MDL.IArray_nth_equalityI
thf(fact_430_VYDRA__MDL_OIArray__nth__equalityI,axiom,
    ! [Sigma: trace_a_t,Init_hd: h,Run_hd: h > option5775791656947114332_set_a,Xs: iarray4851489445666185416_a_t_h,Ys: list_vydra_aux_a_t_h] :
      ( ( vYDRA_MDL_a_t_h @ Sigma @ Init_hd @ Run_hd )
     => ( ( ( length242649669638338715_a_t_h @ Xs )
          = ( size_s8993897887676352572_a_t_h @ Ys ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( length242649669638338715_a_t_h @ Xs ) )
             => ( ( sub_vydra_aux_a_t_h @ Xs @ I2 )
                = ( nth_vydra_aux_a_t_h @ Ys @ I2 ) ) )
         => ( Xs
            = ( iArray3337576414570620549_a_t_h @ Ys ) ) ) ) ) ).

% VYDRA_MDL.IArray_nth_equalityI
thf(fact_431_VYDRA__MDL_OIArray__nth__equalityI,axiom,
    ! [Sigma: trace_a_t,Init_hd: h,Run_hd: h > option5775791656947114332_set_a,Xs: iarray_iarray_o,Ys: list_iarray_o] :
      ( ( vYDRA_MDL_a_t_h @ Sigma @ Init_hd @ Run_hd )
     => ( ( ( length_iarray_o @ Xs )
          = ( size_s3243743627286111600rray_o @ Ys ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( length_iarray_o @ Xs ) )
             => ( ( sub_iarray_o @ Xs @ I2 )
                = ( nth_iarray_o @ Ys @ I2 ) ) )
         => ( Xs
            = ( iArray_iarray_o @ Ys ) ) ) ) ) ).

% VYDRA_MDL.IArray_nth_equalityI
thf(fact_432_VYDRA__MDL_OIArray__nth__equalityI,axiom,
    ! [Sigma: trace_a_t,Init_hd: h,Run_hd: h > option5775791656947114332_set_a,Xs: iarray_formula_a_t,Ys: list_formula_a_t] :
      ( ( vYDRA_MDL_a_t_h @ Sigma @ Init_hd @ Run_hd )
     => ( ( ( length_formula_a_t @ Xs )
          = ( size_s8846756101701226951la_a_t @ Ys ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( length_formula_a_t @ Xs ) )
             => ( ( sub_formula_a_t @ Xs @ I2 )
                = ( nth_formula_a_t @ Ys @ I2 ) ) )
         => ( Xs
            = ( iArray_formula_a_t @ Ys ) ) ) ) ) ).

% VYDRA_MDL.IArray_nth_equalityI
thf(fact_433_VYDRA__MDL_OIArray__nth__equalityI,axiom,
    ! [Sigma: trace_a_t,Init_hd: h,Run_hd: h > option5775791656947114332_set_a,Xs: iarray_t,Ys: list_t] :
      ( ( vYDRA_MDL_a_t_h @ Sigma @ Init_hd @ Run_hd )
     => ( ( ( length_t @ Xs )
          = ( size_size_list_t @ Ys ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( length_t @ Xs ) )
             => ( ( sub_t @ Xs @ I2 )
                = ( nth_t @ Ys @ I2 ) ) )
         => ( Xs
            = ( iArray_t @ Ys ) ) ) ) ) ).

% VYDRA_MDL.IArray_nth_equalityI
thf(fact_434_minf_I8_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X10: nat] :
      ( ( ord_less_nat @ X10 @ Z )
     => ~ ( ord_less_eq_nat @ T @ X10 ) ) ).

% minf(8)
thf(fact_435_minf_I6_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X10: nat] :
      ( ( ord_less_nat @ X10 @ Z )
     => ( ord_less_eq_nat @ X10 @ T ) ) ).

% minf(6)
thf(fact_436_pinf_I8_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X10: nat] :
      ( ( ord_less_nat @ Z @ X10 )
     => ( ord_less_eq_nat @ T @ X10 ) ) ).

% pinf(8)
thf(fact_437_pinf_I1_J,axiom,
    ! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
      ( ? [Z5: nat] :
        ! [X: nat] :
          ( ( ord_less_nat @ Z5 @ X )
         => ( ( P @ X )
            = ( P5 @ X ) ) )
     => ( ? [Z5: nat] :
          ! [X: nat] :
            ( ( ord_less_nat @ Z5 @ X )
           => ( ( Q @ X )
              = ( Q2 @ X ) ) )
       => ? [Z: nat] :
          ! [X10: nat] :
            ( ( ord_less_nat @ Z @ X10 )
           => ( ( ( P @ X10 )
                & ( Q @ X10 ) )
              = ( ( P5 @ X10 )
                & ( Q2 @ X10 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_438_pinf_I2_J,axiom,
    ! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
      ( ? [Z5: nat] :
        ! [X: nat] :
          ( ( ord_less_nat @ Z5 @ X )
         => ( ( P @ X )
            = ( P5 @ X ) ) )
     => ( ? [Z5: nat] :
          ! [X: nat] :
            ( ( ord_less_nat @ Z5 @ X )
           => ( ( Q @ X )
              = ( Q2 @ X ) ) )
       => ? [Z: nat] :
          ! [X10: nat] :
            ( ( ord_less_nat @ Z @ X10 )
           => ( ( ( P @ X10 )
                | ( Q @ X10 ) )
              = ( ( P5 @ X10 )
                | ( Q2 @ X10 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_439_pinf_I3_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X10: nat] :
      ( ( ord_less_nat @ Z @ X10 )
     => ( X10 != T ) ) ).

% pinf(3)
thf(fact_440_pinf_I4_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X10: nat] :
      ( ( ord_less_nat @ Z @ X10 )
     => ( X10 != T ) ) ).

% pinf(4)
thf(fact_441_pinf_I5_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X10: nat] :
      ( ( ord_less_nat @ Z @ X10 )
     => ~ ( ord_less_nat @ X10 @ T ) ) ).

% pinf(5)
thf(fact_442_pinf_I7_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X10: nat] :
      ( ( ord_less_nat @ Z @ X10 )
     => ( ord_less_nat @ T @ X10 ) ) ).

% pinf(7)
thf(fact_443_minf_I1_J,axiom,
    ! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
      ( ? [Z5: nat] :
        ! [X: nat] :
          ( ( ord_less_nat @ X @ Z5 )
         => ( ( P @ X )
            = ( P5 @ X ) ) )
     => ( ? [Z5: nat] :
          ! [X: nat] :
            ( ( ord_less_nat @ X @ Z5 )
           => ( ( Q @ X )
              = ( Q2 @ X ) ) )
       => ? [Z: nat] :
          ! [X10: nat] :
            ( ( ord_less_nat @ X10 @ Z )
           => ( ( ( P @ X10 )
                & ( Q @ X10 ) )
              = ( ( P5 @ X10 )
                & ( Q2 @ X10 ) ) ) ) ) ) ).

% minf(1)
thf(fact_444_minf_I2_J,axiom,
    ! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
      ( ? [Z5: nat] :
        ! [X: nat] :
          ( ( ord_less_nat @ X @ Z5 )
         => ( ( P @ X )
            = ( P5 @ X ) ) )
     => ( ? [Z5: nat] :
          ! [X: nat] :
            ( ( ord_less_nat @ X @ Z5 )
           => ( ( Q @ X )
              = ( Q2 @ X ) ) )
       => ? [Z: nat] :
          ! [X10: nat] :
            ( ( ord_less_nat @ X10 @ Z )
           => ( ( ( P @ X10 )
                | ( Q @ X10 ) )
              = ( ( P5 @ X10 )
                | ( Q2 @ X10 ) ) ) ) ) ) ).

% minf(2)
thf(fact_445_minf_I3_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X10: nat] :
      ( ( ord_less_nat @ X10 @ Z )
     => ( X10 != T ) ) ).

% minf(3)
thf(fact_446_minf_I4_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X10: nat] :
      ( ( ord_less_nat @ X10 @ Z )
     => ( X10 != T ) ) ).

% minf(4)
thf(fact_447_minf_I5_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X10: nat] :
      ( ( ord_less_nat @ X10 @ Z )
     => ( ord_less_nat @ X10 @ T ) ) ).

% minf(5)
thf(fact_448_minf_I7_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X10: nat] :
      ( ( ord_less_nat @ X10 @ Z )
     => ~ ( ord_less_nat @ T @ X10 ) ) ).

% minf(7)
thf(fact_449_pinf_I6_J,axiom,
    ! [T: nat] :
    ? [Z: nat] :
    ! [X10: nat] :
      ( ( ord_less_nat @ Z @ X10 )
     => ~ ( ord_less_eq_nat @ X10 @ T ) ) ).

% pinf(6)
thf(fact_450_verit__comp__simplify1_I3_J,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ~ ( ord_less_eq_nat @ B2 @ A2 ) )
      = ( ord_less_nat @ A2 @ B2 ) ) ).

% verit_comp_simplify1(3)
thf(fact_451_find__Some__iff,axiom,
    ! [P: produc8228095596854190539rray_o > $o,Xs: list_P1634102618077251035rray_o,X3: produc8228095596854190539rray_o] :
      ( ( ( find_P1694764551063991983rray_o @ P @ Xs )
        = ( some_P7279874174034738982rray_o @ X3 ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s5675016999574124143rray_o @ Xs ) )
            & ( P @ ( nth_Pr8962777788532761604rray_o @ Xs @ I3 ) )
            & ( X3
              = ( nth_Pr8962777788532761604rray_o @ Xs @ I3 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I3 )
               => ~ ( P @ ( nth_Pr8962777788532761604rray_o @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff
thf(fact_452_find__Some__iff,axiom,
    ! [P: product_prod_t_o > $o,Xs: list_P5575445741472356340od_t_o,X3: product_prod_t_o] :
      ( ( ( find_P3690432262111017226od_t_o @ P @ Xs )
        = ( some_P1805796590424124243od_t_o @ X3 ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s4106575409306904672od_t_o @ Xs ) )
            & ( P @ ( nth_Product_prod_t_o @ Xs @ I3 ) )
            & ( X3
              = ( nth_Product_prod_t_o @ Xs @ I3 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I3 )
               => ~ ( P @ ( nth_Product_prod_t_o @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff
thf(fact_453_find__Some__iff,axiom,
    ! [P: produc8090089739038349590_set_a > $o,Xs: list_P2035273040109328284_set_a,X3: produc8090089739038349590_set_a] :
      ( ( ( find_P6060379126387363634_set_a @ P @ Xs )
        = ( some_P2245991037696317435_set_a @ X3 ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s2286189404704077320_set_a @ Xs ) )
            & ( P @ ( nth_Pr2651153946192415133_set_a @ Xs @ I3 ) )
            & ( X3
              = ( nth_Pr2651153946192415133_set_a @ Xs @ I3 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I3 )
               => ~ ( P @ ( nth_Pr2651153946192415133_set_a @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff
thf(fact_454_find__Some__iff,axiom,
    ! [P: product_prod_h_t > $o,Xs: list_P7520416276666729081od_h_t,X3: product_prod_h_t] :
      ( ( ( find_P1183509874644452751od_h_t @ P @ Xs )
        = ( some_P2526622376857543512od_h_t @ X3 ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s785782387481697509od_h_t @ Xs ) )
            & ( P @ ( nth_Product_prod_h_t @ Xs @ I3 ) )
            & ( X3
              = ( nth_Product_prod_h_t @ Xs @ I3 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I3 )
               => ~ ( P @ ( nth_Product_prod_h_t @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff
thf(fact_455_find__Some__iff,axiom,
    ! [P: nat > $o,Xs: list_nat,X3: nat] :
      ( ( ( find_nat @ P @ Xs )
        = ( some_nat @ X3 ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
            & ( P @ ( nth_nat @ Xs @ I3 ) )
            & ( X3
              = ( nth_nat @ Xs @ I3 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I3 )
               => ~ ( P @ ( nth_nat @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff
thf(fact_456_find__Some__iff,axiom,
    ! [P: product_prod_t_set_a > $o,Xs: list_P1095827921439705018_set_a,X3: product_prod_t_set_a] :
      ( ( ( find_P3657267623437522768_set_a @ P @ Xs )
        = ( some_P4667681669429350681_set_a @ X3 ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s5362121731612881446_set_a @ Xs ) )
            & ( P @ ( nth_Pr633927046780761275_set_a @ Xs @ I3 ) )
            & ( X3
              = ( nth_Pr633927046780761275_set_a @ Xs @ I3 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I3 )
               => ~ ( P @ ( nth_Pr633927046780761275_set_a @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff
thf(fact_457_find__Some__iff,axiom,
    ! [P: vydra_aux_a_t_h > $o,Xs: list_vydra_aux_a_t_h,X3: vydra_aux_a_t_h] :
      ( ( ( find_vydra_aux_a_t_h @ P @ Xs )
        = ( some_vydra_aux_a_t_h @ X3 ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s8993897887676352572_a_t_h @ Xs ) )
            & ( P @ ( nth_vydra_aux_a_t_h @ Xs @ I3 ) )
            & ( X3
              = ( nth_vydra_aux_a_t_h @ Xs @ I3 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I3 )
               => ~ ( P @ ( nth_vydra_aux_a_t_h @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff
thf(fact_458_find__Some__iff,axiom,
    ! [P: iarray_o > $o,Xs: list_iarray_o,X3: iarray_o] :
      ( ( ( find_iarray_o @ P @ Xs )
        = ( some_iarray_o @ X3 ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s3243743627286111600rray_o @ Xs ) )
            & ( P @ ( nth_iarray_o @ Xs @ I3 ) )
            & ( X3
              = ( nth_iarray_o @ Xs @ I3 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I3 )
               => ~ ( P @ ( nth_iarray_o @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff
thf(fact_459_find__Some__iff,axiom,
    ! [P: formula_a_t > $o,Xs: list_formula_a_t,X3: formula_a_t] :
      ( ( ( find_formula_a_t @ P @ Xs )
        = ( some_formula_a_t @ X3 ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s8846756101701226951la_a_t @ Xs ) )
            & ( P @ ( nth_formula_a_t @ Xs @ I3 ) )
            & ( X3
              = ( nth_formula_a_t @ Xs @ I3 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I3 )
               => ~ ( P @ ( nth_formula_a_t @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff
thf(fact_460_find__Some__iff,axiom,
    ! [P: t > $o,Xs: list_t,X3: t] :
      ( ( ( find_t @ P @ Xs )
        = ( some_t @ X3 ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_t @ Xs ) )
            & ( P @ ( nth_t @ Xs @ I3 ) )
            & ( X3
              = ( nth_t @ Xs @ I3 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I3 )
               => ~ ( P @ ( nth_t @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff
thf(fact_461_find__Some__iff2,axiom,
    ! [X3: produc8228095596854190539rray_o,P: produc8228095596854190539rray_o > $o,Xs: list_P1634102618077251035rray_o] :
      ( ( ( some_P7279874174034738982rray_o @ X3 )
        = ( find_P1694764551063991983rray_o @ P @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s5675016999574124143rray_o @ Xs ) )
            & ( P @ ( nth_Pr8962777788532761604rray_o @ Xs @ I3 ) )
            & ( X3
              = ( nth_Pr8962777788532761604rray_o @ Xs @ I3 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I3 )
               => ~ ( P @ ( nth_Pr8962777788532761604rray_o @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff2
thf(fact_462_find__Some__iff2,axiom,
    ! [X3: product_prod_t_o,P: product_prod_t_o > $o,Xs: list_P5575445741472356340od_t_o] :
      ( ( ( some_P1805796590424124243od_t_o @ X3 )
        = ( find_P3690432262111017226od_t_o @ P @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s4106575409306904672od_t_o @ Xs ) )
            & ( P @ ( nth_Product_prod_t_o @ Xs @ I3 ) )
            & ( X3
              = ( nth_Product_prod_t_o @ Xs @ I3 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I3 )
               => ~ ( P @ ( nth_Product_prod_t_o @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff2
thf(fact_463_find__Some__iff2,axiom,
    ! [X3: produc8090089739038349590_set_a,P: produc8090089739038349590_set_a > $o,Xs: list_P2035273040109328284_set_a] :
      ( ( ( some_P2245991037696317435_set_a @ X3 )
        = ( find_P6060379126387363634_set_a @ P @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s2286189404704077320_set_a @ Xs ) )
            & ( P @ ( nth_Pr2651153946192415133_set_a @ Xs @ I3 ) )
            & ( X3
              = ( nth_Pr2651153946192415133_set_a @ Xs @ I3 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I3 )
               => ~ ( P @ ( nth_Pr2651153946192415133_set_a @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff2
thf(fact_464_find__Some__iff2,axiom,
    ! [X3: product_prod_h_t,P: product_prod_h_t > $o,Xs: list_P7520416276666729081od_h_t] :
      ( ( ( some_P2526622376857543512od_h_t @ X3 )
        = ( find_P1183509874644452751od_h_t @ P @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s785782387481697509od_h_t @ Xs ) )
            & ( P @ ( nth_Product_prod_h_t @ Xs @ I3 ) )
            & ( X3
              = ( nth_Product_prod_h_t @ Xs @ I3 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I3 )
               => ~ ( P @ ( nth_Product_prod_h_t @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff2
thf(fact_465_find__Some__iff2,axiom,
    ! [X3: nat,P: nat > $o,Xs: list_nat] :
      ( ( ( some_nat @ X3 )
        = ( find_nat @ P @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
            & ( P @ ( nth_nat @ Xs @ I3 ) )
            & ( X3
              = ( nth_nat @ Xs @ I3 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I3 )
               => ~ ( P @ ( nth_nat @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff2
thf(fact_466_find__Some__iff2,axiom,
    ! [X3: product_prod_t_set_a,P: product_prod_t_set_a > $o,Xs: list_P1095827921439705018_set_a] :
      ( ( ( some_P4667681669429350681_set_a @ X3 )
        = ( find_P3657267623437522768_set_a @ P @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s5362121731612881446_set_a @ Xs ) )
            & ( P @ ( nth_Pr633927046780761275_set_a @ Xs @ I3 ) )
            & ( X3
              = ( nth_Pr633927046780761275_set_a @ Xs @ I3 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I3 )
               => ~ ( P @ ( nth_Pr633927046780761275_set_a @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff2
thf(fact_467_find__Some__iff2,axiom,
    ! [X3: vydra_aux_a_t_h,P: vydra_aux_a_t_h > $o,Xs: list_vydra_aux_a_t_h] :
      ( ( ( some_vydra_aux_a_t_h @ X3 )
        = ( find_vydra_aux_a_t_h @ P @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s8993897887676352572_a_t_h @ Xs ) )
            & ( P @ ( nth_vydra_aux_a_t_h @ Xs @ I3 ) )
            & ( X3
              = ( nth_vydra_aux_a_t_h @ Xs @ I3 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I3 )
               => ~ ( P @ ( nth_vydra_aux_a_t_h @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff2
thf(fact_468_find__Some__iff2,axiom,
    ! [X3: iarray_o,P: iarray_o > $o,Xs: list_iarray_o] :
      ( ( ( some_iarray_o @ X3 )
        = ( find_iarray_o @ P @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s3243743627286111600rray_o @ Xs ) )
            & ( P @ ( nth_iarray_o @ Xs @ I3 ) )
            & ( X3
              = ( nth_iarray_o @ Xs @ I3 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I3 )
               => ~ ( P @ ( nth_iarray_o @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff2
thf(fact_469_find__Some__iff2,axiom,
    ! [X3: formula_a_t,P: formula_a_t > $o,Xs: list_formula_a_t] :
      ( ( ( some_formula_a_t @ X3 )
        = ( find_formula_a_t @ P @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s8846756101701226951la_a_t @ Xs ) )
            & ( P @ ( nth_formula_a_t @ Xs @ I3 ) )
            & ( X3
              = ( nth_formula_a_t @ Xs @ I3 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I3 )
               => ~ ( P @ ( nth_formula_a_t @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff2
thf(fact_470_find__Some__iff2,axiom,
    ! [X3: t,P: t > $o,Xs: list_t] :
      ( ( ( some_t @ X3 )
        = ( find_t @ P @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_t @ Xs ) )
            & ( P @ ( nth_t @ Xs @ I3 ) )
            & ( X3
              = ( nth_t @ Xs @ I3 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I3 )
               => ~ ( P @ ( nth_t @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff2
thf(fact_471_wf__vydra__sub,axiom,
    ! [Phi: formula_a_t,N: nat] :
      ( ( ord_less_eq_nat @ ( msize_fmla_a_t @ Phi ) @ N )
     => ( vYDRA_wf_vydra_a_t_h @ sigma @ init_hd @ run_hd @ Phi @ zero_zero_nat @ N @ ( sub_h_t_a @ init_hd @ run_hd @ N @ Phi ) ) ) ).

% wf_vydra_sub
thf(fact_472_last__before_Ocases,axiom,
    ! [X3: produc3074792404157404414_o_nat] :
      ( ! [P6: nat > $o] :
          ( X3
         != ( produc7277522915581678840_o_nat @ P6 @ zero_zero_nat ) )
     => ~ ! [P6: nat > $o,N3: nat] :
            ( X3
           != ( produc7277522915581678840_o_nat @ P6 @ ( suc @ N3 ) ) ) ) ).

% last_before.cases
thf(fact_473_le0,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% le0
thf(fact_474_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).

% bot_nat_0.extremum
thf(fact_475_less__nat__zero__code,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_nat_zero_code
thf(fact_476_neq0__conv,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% neq0_conv
thf(fact_477_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_478_less__Suc0,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( N = zero_zero_nat ) ) ).

% less_Suc0
thf(fact_479_zero__less__Suc,axiom,
    ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).

% zero_less_Suc
thf(fact_480_prod_Ocollapse,axiom,
    ! [Prod: produc3074792404157404414_o_nat] :
      ( ( produc7277522915581678840_o_nat @ ( produc7643701372303132684_o_nat @ Prod ) @ ( produc3470649411511297610_o_nat @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_481_prod_Ocollapse,axiom,
    ! [Prod: produc8090089739038349590_set_a] :
      ( ( produc3432067189782488838_set_a @ ( produc6892702156181598450_set_a @ Prod ) @ ( produc5260167487839082036_set_a @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_482_prod_Ocollapse,axiom,
    ! [Prod: product_prod_h_t] :
      ( ( product_Pair_h_t @ ( product_fst_h_t @ Prod ) @ ( product_snd_h_t @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_483_prod_Ocollapse,axiom,
    ! [Prod: product_prod_t_set_a] :
      ( ( product_Pair_t_set_a @ ( product_fst_t_set_a @ Prod ) @ ( product_snd_t_set_a @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_484_prod_Ocollapse,axiom,
    ! [Prod: product_prod_t_o] :
      ( ( product_Pair_t_o @ ( product_fst_t_o @ Prod ) @ ( product_snd_t_o @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_485_take__all__iff,axiom,
    ! [N: nat,Xs: list_P1095827921439705018_set_a] :
      ( ( ( take_P2822414627813279522_set_a @ N @ Xs )
        = Xs )
      = ( ord_less_eq_nat @ ( size_s5362121731612881446_set_a @ Xs ) @ N ) ) ).

% take_all_iff
thf(fact_486_take__all__iff,axiom,
    ! [N: nat,Xs: list_vydra_aux_a_t_h] :
      ( ( ( take_vydra_aux_a_t_h @ N @ Xs )
        = Xs )
      = ( ord_less_eq_nat @ ( size_s8993897887676352572_a_t_h @ Xs ) @ N ) ) ).

% take_all_iff
thf(fact_487_take__all__iff,axiom,
    ! [N: nat,Xs: list_iarray_o] :
      ( ( ( take_iarray_o @ N @ Xs )
        = Xs )
      = ( ord_less_eq_nat @ ( size_s3243743627286111600rray_o @ Xs ) @ N ) ) ).

% take_all_iff
thf(fact_488_take__all__iff,axiom,
    ! [N: nat,Xs: list_formula_a_t] :
      ( ( ( take_formula_a_t @ N @ Xs )
        = Xs )
      = ( ord_less_eq_nat @ ( size_s8846756101701226951la_a_t @ Xs ) @ N ) ) ).

% take_all_iff
thf(fact_489_take__all__iff,axiom,
    ! [N: nat,Xs: list_t] :
      ( ( ( take_t @ N @ Xs )
        = Xs )
      = ( ord_less_eq_nat @ ( size_size_list_t @ Xs ) @ N ) ) ).

% take_all_iff
thf(fact_490_take__all,axiom,
    ! [Xs: list_P1095827921439705018_set_a,N: nat] :
      ( ( ord_less_eq_nat @ ( size_s5362121731612881446_set_a @ Xs ) @ N )
     => ( ( take_P2822414627813279522_set_a @ N @ Xs )
        = Xs ) ) ).

% take_all
thf(fact_491_take__all,axiom,
    ! [Xs: list_vydra_aux_a_t_h,N: nat] :
      ( ( ord_less_eq_nat @ ( size_s8993897887676352572_a_t_h @ Xs ) @ N )
     => ( ( take_vydra_aux_a_t_h @ N @ Xs )
        = Xs ) ) ).

% take_all
thf(fact_492_take__all,axiom,
    ! [Xs: list_iarray_o,N: nat] :
      ( ( ord_less_eq_nat @ ( size_s3243743627286111600rray_o @ Xs ) @ N )
     => ( ( take_iarray_o @ N @ Xs )
        = Xs ) ) ).

% take_all
thf(fact_493_take__all,axiom,
    ! [Xs: list_formula_a_t,N: nat] :
      ( ( ord_less_eq_nat @ ( size_s8846756101701226951la_a_t @ Xs ) @ N )
     => ( ( take_formula_a_t @ N @ Xs )
        = Xs ) ) ).

% take_all
thf(fact_494_take__all,axiom,
    ! [Xs: list_t,N: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_t @ Xs ) @ N )
     => ( ( take_t @ N @ Xs )
        = Xs ) ) ).

% take_all
thf(fact_495_nth__take,axiom,
    ! [I: nat,N: nat,Xs: list_t] :
      ( ( ord_less_nat @ I @ N )
     => ( ( nth_t @ ( take_t @ N @ Xs ) @ I )
        = ( nth_t @ Xs @ I ) ) ) ).

% nth_take
thf(fact_496_nth__take,axiom,
    ! [I: nat,N: nat,Xs: list_formula_a_t] :
      ( ( ord_less_nat @ I @ N )
     => ( ( nth_formula_a_t @ ( take_formula_a_t @ N @ Xs ) @ I )
        = ( nth_formula_a_t @ Xs @ I ) ) ) ).

% nth_take
thf(fact_497_nth__take,axiom,
    ! [I: nat,N: nat,Xs: list_vydra_aux_a_t_h] :
      ( ( ord_less_nat @ I @ N )
     => ( ( nth_vydra_aux_a_t_h @ ( take_vydra_aux_a_t_h @ N @ Xs ) @ I )
        = ( nth_vydra_aux_a_t_h @ Xs @ I ) ) ) ).

% nth_take
thf(fact_498_eq__snd__iff,axiom,
    ! [B: nat,P2: produc3074792404157404414_o_nat] :
      ( ( B
        = ( produc3470649411511297610_o_nat @ P2 ) )
      = ( ? [A5: nat > $o] :
            ( P2
            = ( produc7277522915581678840_o_nat @ A5 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_499_eq__snd__iff,axiom,
    ! [B: set_a,P2: product_prod_t_set_a] :
      ( ( B
        = ( product_snd_t_set_a @ P2 ) )
      = ( ? [A5: t] :
            ( P2
            = ( product_Pair_t_set_a @ A5 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_500_eq__snd__iff,axiom,
    ! [B: product_prod_t_set_a,P2: produc8090089739038349590_set_a] :
      ( ( B
        = ( produc5260167487839082036_set_a @ P2 ) )
      = ( ? [A5: h] :
            ( P2
            = ( produc3432067189782488838_set_a @ A5 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_501_eq__snd__iff,axiom,
    ! [B: t,P2: product_prod_h_t] :
      ( ( B
        = ( product_snd_h_t @ P2 ) )
      = ( ? [A5: h] :
            ( P2
            = ( product_Pair_h_t @ A5 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_502_eq__snd__iff,axiom,
    ! [B: $o,P2: product_prod_t_o] :
      ( ( B
        = ( product_snd_t_o @ P2 ) )
      = ( ? [A5: t] :
            ( P2
            = ( product_Pair_t_o @ A5 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_503_sndI,axiom,
    ! [X3: produc3074792404157404414_o_nat,Y: nat > $o,Z3: nat] :
      ( ( X3
        = ( produc7277522915581678840_o_nat @ Y @ Z3 ) )
     => ( ( produc3470649411511297610_o_nat @ X3 )
        = Z3 ) ) ).

% sndI
thf(fact_504_sndI,axiom,
    ! [X3: product_prod_t_set_a,Y: t,Z3: set_a] :
      ( ( X3
        = ( product_Pair_t_set_a @ Y @ Z3 ) )
     => ( ( product_snd_t_set_a @ X3 )
        = Z3 ) ) ).

% sndI
thf(fact_505_sndI,axiom,
    ! [X3: produc8090089739038349590_set_a,Y: h,Z3: product_prod_t_set_a] :
      ( ( X3
        = ( produc3432067189782488838_set_a @ Y @ Z3 ) )
     => ( ( produc5260167487839082036_set_a @ X3 )
        = Z3 ) ) ).

% sndI
thf(fact_506_sndI,axiom,
    ! [X3: product_prod_h_t,Y: h,Z3: t] :
      ( ( X3
        = ( product_Pair_h_t @ Y @ Z3 ) )
     => ( ( product_snd_h_t @ X3 )
        = Z3 ) ) ).

% sndI
thf(fact_507_sndI,axiom,
    ! [X3: product_prod_t_o,Y: t,Z3: $o] :
      ( ( X3
        = ( product_Pair_t_o @ Y @ Z3 ) )
     => ( ( product_snd_t_o @ X3 )
        = Z3 ) ) ).

% sndI
thf(fact_508_snd__conv,axiom,
    ! [X1: nat > $o,X2: nat] :
      ( ( produc3470649411511297610_o_nat @ ( produc7277522915581678840_o_nat @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_509_snd__conv,axiom,
    ! [X1: t,X2: set_a] :
      ( ( product_snd_t_set_a @ ( product_Pair_t_set_a @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_510_snd__conv,axiom,
    ! [X1: h,X2: product_prod_t_set_a] :
      ( ( produc5260167487839082036_set_a @ ( produc3432067189782488838_set_a @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_511_snd__conv,axiom,
    ! [X1: h,X2: t] :
      ( ( product_snd_h_t @ ( product_Pair_h_t @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_512_snd__conv,axiom,
    ! [X1: t,X2: $o] :
      ( ( product_snd_t_o @ ( product_Pair_t_o @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_513_snd__eqD,axiom,
    ! [X3: nat > $o,Y: nat,A: nat] :
      ( ( ( produc3470649411511297610_o_nat @ ( produc7277522915581678840_o_nat @ X3 @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_514_snd__eqD,axiom,
    ! [X3: t,Y: set_a,A: set_a] :
      ( ( ( product_snd_t_set_a @ ( product_Pair_t_set_a @ X3 @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_515_snd__eqD,axiom,
    ! [X3: h,Y: product_prod_t_set_a,A: product_prod_t_set_a] :
      ( ( ( produc5260167487839082036_set_a @ ( produc3432067189782488838_set_a @ X3 @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_516_snd__eqD,axiom,
    ! [X3: h,Y: t,A: t] :
      ( ( ( product_snd_h_t @ ( product_Pair_h_t @ X3 @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_517_snd__eqD,axiom,
    ! [X3: t,Y: $o,A: $o] :
      ( ( ( product_snd_t_o @ ( product_Pair_t_o @ X3 @ Y ) )
        = A )
     => ( Y = A ) ) ).

% snd_eqD
thf(fact_518_take__map,axiom,
    ! [N: nat,F: product_prod_t_set_a > t,Xs: list_P1095827921439705018_set_a] :
      ( ( take_t @ N @ ( map_Pr119846518281778721et_a_t @ F @ Xs ) )
      = ( map_Pr119846518281778721et_a_t @ F @ ( take_P2822414627813279522_set_a @ N @ Xs ) ) ) ).

% take_map
thf(fact_519_take__map,axiom,
    ! [N: nat,F: formula_a_t > $o,Xs: list_formula_a_t] :
      ( ( take_o @ N @ ( map_formula_a_t_o @ F @ Xs ) )
      = ( map_formula_a_t_o @ F @ ( take_formula_a_t @ N @ Xs ) ) ) ).

% take_map
thf(fact_520_take__map,axiom,
    ! [N: nat,F: formula_a_t > vydra_aux_a_t_h,Xs: list_formula_a_t] :
      ( ( take_vydra_aux_a_t_h @ N @ ( map_fo5746072578778194433_a_t_h @ F @ Xs ) )
      = ( map_fo5746072578778194433_a_t_h @ F @ ( take_formula_a_t @ N @ Xs ) ) ) ).

% take_map
thf(fact_521_prod__eq__iff,axiom,
    ( ( ^ [Y6: product_prod_t_set_a,Z4: product_prod_t_set_a] : ( Y6 = Z4 ) )
    = ( ^ [S3: product_prod_t_set_a,T4: product_prod_t_set_a] :
          ( ( ( product_fst_t_set_a @ S3 )
            = ( product_fst_t_set_a @ T4 ) )
          & ( ( product_snd_t_set_a @ S3 )
            = ( product_snd_t_set_a @ T4 ) ) ) ) ) ).

% prod_eq_iff
thf(fact_522_prod__eq__iff,axiom,
    ( ( ^ [Y6: product_prod_t_o,Z4: product_prod_t_o] : ( Y6 = Z4 ) )
    = ( ^ [S3: product_prod_t_o,T4: product_prod_t_o] :
          ( ( ( product_fst_t_o @ S3 )
            = ( product_fst_t_o @ T4 ) )
          & ( ( product_snd_t_o @ S3 )
            = ( product_snd_t_o @ T4 ) ) ) ) ) ).

% prod_eq_iff
thf(fact_523_prod__eqI,axiom,
    ! [P2: product_prod_t_set_a,Q3: product_prod_t_set_a] :
      ( ( ( product_fst_t_set_a @ P2 )
        = ( product_fst_t_set_a @ Q3 ) )
     => ( ( ( product_snd_t_set_a @ P2 )
          = ( product_snd_t_set_a @ Q3 ) )
       => ( P2 = Q3 ) ) ) ).

% prod_eqI
thf(fact_524_prod__eqI,axiom,
    ! [P2: product_prod_t_o,Q3: product_prod_t_o] :
      ( ( ( product_fst_t_o @ P2 )
        = ( product_fst_t_o @ Q3 ) )
     => ( ( ( product_snd_t_o @ P2 )
          = ( product_snd_t_o @ Q3 ) )
       => ( P2 = Q3 ) ) ) ).

% prod_eqI
thf(fact_525_prod_Oexpand,axiom,
    ! [Prod: product_prod_t_set_a,Prod2: product_prod_t_set_a] :
      ( ( ( ( product_fst_t_set_a @ Prod )
          = ( product_fst_t_set_a @ Prod2 ) )
        & ( ( product_snd_t_set_a @ Prod )
          = ( product_snd_t_set_a @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_526_prod_Oexpand,axiom,
    ! [Prod: product_prod_t_o,Prod2: product_prod_t_o] :
      ( ( ( ( product_fst_t_o @ Prod )
          = ( product_fst_t_o @ Prod2 ) )
        & ( ( product_snd_t_o @ Prod )
          = ( product_snd_t_o @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_527_not0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ? [M3: nat] :
          ( N
          = ( suc @ M3 ) ) ) ).

% not0_implies_Suc
thf(fact_528_Zero__not__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_not_Suc
thf(fact_529_Zero__neq__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_neq_Suc
thf(fact_530_Suc__neq__Zero,axiom,
    ! [M: nat] :
      ( ( suc @ M )
     != zero_zero_nat ) ).

% Suc_neq_Zero
thf(fact_531_zero__induct,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( P @ K )
     => ( ! [N3: nat] :
            ( ( P @ ( suc @ N3 ) )
           => ( P @ N3 ) )
       => ( P @ zero_zero_nat ) ) ) ).

% zero_induct
thf(fact_532_diff__induct,axiom,
    ! [P: nat > nat > $o,M: nat,N: nat] :
      ( ! [X: nat] : ( P @ X @ zero_zero_nat )
     => ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
       => ( ! [X: nat,Y3: nat] :
              ( ( P @ X @ Y3 )
             => ( P @ ( suc @ X ) @ ( suc @ Y3 ) ) )
         => ( P @ M @ N ) ) ) ) ).

% diff_induct
thf(fact_533_nat__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N3: nat] :
            ( ( P @ N3 )
           => ( P @ ( suc @ N3 ) ) )
       => ( P @ N ) ) ) ).

% nat_induct
thf(fact_534_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat3: nat] :
            ( Y
           != ( suc @ Nat3 ) ) ) ).

% old.nat.exhaust
thf(fact_535_nat_OdiscI,axiom,
    ! [Nat: nat,X2: nat] :
      ( ( Nat
        = ( suc @ X2 ) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
thf(fact_536_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] :
      ( zero_zero_nat
     != ( suc @ Nat2 ) ) ).

% old.nat.distinct(1)
thf(fact_537_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] :
      ( ( suc @ Nat2 )
     != zero_zero_nat ) ).

% old.nat.distinct(2)
thf(fact_538_nat_Odistinct_I1_J,axiom,
    ! [X2: nat] :
      ( zero_zero_nat
     != ( suc @ X2 ) ) ).

% nat.distinct(1)
thf(fact_539_le__0__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_0_eq
thf(fact_540_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_541_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_542_less__eq__nat_Osimps_I1_J,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% less_eq_nat.simps(1)
thf(fact_543_infinite__descent0,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N3: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N3 )
           => ( ~ ( P @ N3 )
             => ? [M4: nat] :
                  ( ( ord_less_nat @ M4 @ N3 )
                  & ~ ( P @ M4 ) ) ) )
       => ( P @ N ) ) ) ).

% infinite_descent0
thf(fact_544_gr__implies__not0,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not0
thf(fact_545_less__zeroE,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_zeroE
thf(fact_546_not__less0,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less0
thf(fact_547_not__gr0,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr0
thf(fact_548_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr0I
thf(fact_549_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ zero_zero_nat ) ).

% bot_nat_0.extremum_strict
thf(fact_550_reaches_Ointros_I1_J,axiom,
    ! [Run: h > option5775791656947114332_set_a,S: h] : ( reache8480716032150177652_set_a @ Run @ S @ zero_zero_nat @ S ) ).

% reaches.intros(1)
thf(fact_551_prod_Oexhaust__sel,axiom,
    ! [Prod: produc3074792404157404414_o_nat] :
      ( Prod
      = ( produc7277522915581678840_o_nat @ ( produc7643701372303132684_o_nat @ Prod ) @ ( produc3470649411511297610_o_nat @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_552_prod_Oexhaust__sel,axiom,
    ! [Prod: produc8090089739038349590_set_a] :
      ( Prod
      = ( produc3432067189782488838_set_a @ ( produc6892702156181598450_set_a @ Prod ) @ ( produc5260167487839082036_set_a @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_553_prod_Oexhaust__sel,axiom,
    ! [Prod: product_prod_h_t] :
      ( Prod
      = ( product_Pair_h_t @ ( product_fst_h_t @ Prod ) @ ( product_snd_h_t @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_554_prod_Oexhaust__sel,axiom,
    ! [Prod: product_prod_t_set_a] :
      ( Prod
      = ( product_Pair_t_set_a @ ( product_fst_t_set_a @ Prod ) @ ( product_snd_t_set_a @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_555_prod_Oexhaust__sel,axiom,
    ! [Prod: product_prod_t_o] :
      ( Prod
      = ( product_Pair_t_o @ ( product_fst_t_o @ Prod ) @ ( product_snd_t_o @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_556_surjective__pairing,axiom,
    ! [T: produc3074792404157404414_o_nat] :
      ( T
      = ( produc7277522915581678840_o_nat @ ( produc7643701372303132684_o_nat @ T ) @ ( produc3470649411511297610_o_nat @ T ) ) ) ).

% surjective_pairing
thf(fact_557_surjective__pairing,axiom,
    ! [T: produc8090089739038349590_set_a] :
      ( T
      = ( produc3432067189782488838_set_a @ ( produc6892702156181598450_set_a @ T ) @ ( produc5260167487839082036_set_a @ T ) ) ) ).

% surjective_pairing
thf(fact_558_surjective__pairing,axiom,
    ! [T: product_prod_h_t] :
      ( T
      = ( product_Pair_h_t @ ( product_fst_h_t @ T ) @ ( product_snd_h_t @ T ) ) ) ).

% surjective_pairing
thf(fact_559_surjective__pairing,axiom,
    ! [T: product_prod_t_set_a] :
      ( T
      = ( product_Pair_t_set_a @ ( product_fst_t_set_a @ T ) @ ( product_snd_t_set_a @ T ) ) ) ).

% surjective_pairing
thf(fact_560_surjective__pairing,axiom,
    ! [T: product_prod_t_o] :
      ( T
      = ( product_Pair_t_o @ ( product_fst_t_o @ T ) @ ( product_snd_t_o @ T ) ) ) ).

% surjective_pairing
thf(fact_561_pair__list__eqI,axiom,
    ! [Xs: list_P1095827921439705018_set_a,Ys: list_P1095827921439705018_set_a] :
      ( ( ( map_Pr119846518281778721et_a_t @ product_fst_t_set_a @ Xs )
        = ( map_Pr119846518281778721et_a_t @ product_fst_t_set_a @ Ys ) )
     => ( ( ( map_Pr7710007540447715182_set_a @ product_snd_t_set_a @ Xs )
          = ( map_Pr7710007540447715182_set_a @ product_snd_t_set_a @ Ys ) )
       => ( Xs = Ys ) ) ) ).

% pair_list_eqI
thf(fact_562_pair__list__eqI,axiom,
    ! [Xs: list_P5575445741472356340od_t_o,Ys: list_P5575445741472356340od_t_o] :
      ( ( ( map_Pr4056436085500758887_t_o_t @ product_fst_t_o @ Xs )
        = ( map_Pr4056436085500758887_t_o_t @ product_fst_t_o @ Ys ) )
     => ( ( ( map_Pr913679237327294190_t_o_o @ product_snd_t_o @ Xs )
          = ( map_Pr913679237327294190_t_o_o @ product_snd_t_o @ Ys ) )
       => ( Xs = Ys ) ) ) ).

% pair_list_eqI
thf(fact_563_VYDRA__MDL_Owf__vydra__sub,axiom,
    ! [Sigma: trace_a_t,Init_hd: h,Run_hd: h > option5775791656947114332_set_a,Phi: formula_a_t,N: nat] :
      ( ( vYDRA_MDL_a_t_h @ Sigma @ Init_hd @ Run_hd )
     => ( ( ord_less_eq_nat @ ( msize_fmla_a_t @ Phi ) @ N )
       => ( vYDRA_wf_vydra_a_t_h @ Sigma @ Init_hd @ Run_hd @ Phi @ zero_zero_nat @ N @ ( sub_h_t_a @ Init_hd @ Run_hd @ N @ Phi ) ) ) ) ).

% VYDRA_MDL.wf_vydra_sub
thf(fact_564_sub_Osimps_I1_J,axiom,
    ! [Init_hd: h,Run_hd: h > option5775791656947114332_set_a,N: nat,B: $o] :
      ( ( sub_h_t_a @ Init_hd @ Run_hd @ N @ ( bool_a_t @ B ) )
      = ( vydra_2060541234293701615_h_a_t @ B @ Init_hd ) ) ).

% sub.simps(1)
thf(fact_565_sub_Osimps_I2_J,axiom,
    ! [Init_hd: h,Run_hd: h > option5775791656947114332_set_a,N: nat,A: a] :
      ( ( sub_h_t_a @ Init_hd @ Run_hd @ N @ ( atom_a_t @ A ) )
      = ( vydra_3724560892389469594_a_h_t @ A @ Init_hd ) ) ).

% sub.simps(2)
thf(fact_566_Ex__less__Suc2,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N ) )
            & ( P @ I3 ) ) )
      = ( ( P @ zero_zero_nat )
        | ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ N )
            & ( P @ ( suc @ I3 ) ) ) ) ) ).

% Ex_less_Suc2
thf(fact_567_gr0__conv__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
      = ( ? [M5: nat] :
            ( N
            = ( suc @ M5 ) ) ) ) ).

% gr0_conv_Suc
thf(fact_568_All__less__Suc2,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N ) )
           => ( P @ I3 ) ) )
      = ( ( P @ zero_zero_nat )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ N )
           => ( P @ ( suc @ I3 ) ) ) ) ) ).

% All_less_Suc2
thf(fact_569_gr0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ? [M3: nat] :
          ( N
          = ( suc @ M3 ) ) ) ).

% gr0_implies_Suc
thf(fact_570_less__Suc__eq__0__disj,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N ) )
      = ( ( M = zero_zero_nat )
        | ? [J3: nat] :
            ( ( M
              = ( suc @ J3 ) )
            & ( ord_less_nat @ J3 @ N ) ) ) ) ).

% less_Suc_eq_0_disj
thf(fact_571_option_Osize_I4_J,axiom,
    ! [X2: produc8228095596854190539rray_o] :
      ( ( size_s2704686997974730927rray_o @ ( some_P7279874174034738982rray_o @ X2 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_572_option_Osize_I4_J,axiom,
    ! [X2: product_prod_t_o] :
      ( ( size_s4900030450070283808od_t_o @ ( some_P1805796590424124243od_t_o @ X2 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_573_option_Osize_I4_J,axiom,
    ! [X2: produc8090089739038349590_set_a] :
      ( ( size_s5864537446109811656_set_a @ ( some_P2245991037696317435_set_a @ X2 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_574_option_Osize_I4_J,axiom,
    ! [X2: product_prod_h_t] :
      ( ( size_s6536553198265154213od_h_t @ ( some_P2526622376857543512od_h_t @ X2 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_575_option_Osize_I4_J,axiom,
    ! [X2: nat] :
      ( ( size_size_option_nat @ ( some_nat @ X2 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_576_ex__least__nat__le,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ N )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K2: nat] :
            ( ( ord_less_eq_nat @ K2 @ N )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K2 )
               => ~ ( P @ I4 ) )
            & ( P @ K2 ) ) ) ) ).

% ex_least_nat_le
thf(fact_577_formula_Osize_I11_J,axiom,
    ! [X11: $o] :
      ( ( size_s4016968051272393527la_a_t @ ( bool_a_t @ X11 ) )
      = ( suc @ zero_zero_nat ) ) ).

% formula.size(11)
thf(fact_578_formula_Osize_I12_J,axiom,
    ! [X12: a] :
      ( ( size_s4016968051272393527la_a_t @ ( atom_a_t @ X12 ) )
      = ( suc @ zero_zero_nat ) ) ).

% formula.size(12)
thf(fact_579_vydra__aux_Osize_I13_J,axiom,
    ! [X21: $o,X22: h] :
      ( ( size_s1533667335963039788_a_t_h @ ( vydra_2060541234293701615_h_a_t @ X21 @ X22 ) )
      = ( suc @ zero_zero_nat ) ) ).

% vydra_aux.size(13)
thf(fact_580_vydra__aux_Osize_I14_J,axiom,
    ! [X31: a,X32: h] :
      ( ( size_s1533667335963039788_a_t_h @ ( vydra_3724560892389469594_a_h_t @ X31 @ X32 ) )
      = ( suc @ zero_zero_nat ) ) ).

% vydra_aux.size(14)
thf(fact_581_VYDRA__MDL_Olast__before_Ocases,axiom,
    ! [Sigma: trace_a_t,Init_hd: h,Run_hd: h > option5775791656947114332_set_a,X3: produc3074792404157404414_o_nat] :
      ( ( vYDRA_MDL_a_t_h @ Sigma @ Init_hd @ Run_hd )
     => ( ! [P6: nat > $o] :
            ( X3
           != ( produc7277522915581678840_o_nat @ P6 @ zero_zero_nat ) )
       => ~ ! [P6: nat > $o,N3: nat] :
              ( X3
             != ( produc7277522915581678840_o_nat @ P6 @ ( suc @ N3 ) ) ) ) ) ).

% VYDRA_MDL.last_before.cases
thf(fact_582_msize__fmla_Osimps_I1_J,axiom,
    ! [B: $o] :
      ( ( msize_fmla_a_t @ ( bool_a_t @ B ) )
      = zero_zero_nat ) ).

% msize_fmla.simps(1)
thf(fact_583_msize__fmla_Osimps_I2_J,axiom,
    ! [A: a] :
      ( ( msize_fmla_a_t @ ( atom_a_t @ A ) )
      = zero_zero_nat ) ).

% msize_fmla.simps(2)
thf(fact_584_sub_Osimps_I3_J,axiom,
    ! [Init_hd: h,Run_hd: h > option5775791656947114332_set_a,N: nat,Phi2: formula_a_t] :
      ( ( sub_h_t_a @ Init_hd @ Run_hd @ ( suc @ N ) @ ( neg_a_t @ Phi2 ) )
      = ( vydra_2331423191276895391_a_t_h @ ( sub_h_t_a @ Init_hd @ Run_hd @ N @ Phi2 ) ) ) ).

% sub.simps(3)
thf(fact_585_verit__comp__simplify1_I2_J,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_586_verit__la__disequality,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
      | ~ ( ord_less_eq_nat @ A @ B )
      | ~ ( ord_less_eq_nat @ B @ A ) ) ).

% verit_la_disequality
thf(fact_587_verit__comp__simplify1_I1_J,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_588_ex__least__nat__less,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ N )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K2: nat] :
            ( ( ord_less_nat @ K2 @ N )
            & ! [I4: nat] :
                ( ( ord_less_eq_nat @ I4 @ K2 )
               => ~ ( P @ I4 ) )
            & ( P @ ( suc @ K2 ) ) ) ) ) ).

% ex_least_nat_less
thf(fact_589_less__prod__def_H,axiom,
    ( ord_le7650071679307129824_set_a
    = ( ^ [X5: product_prod_t_set_a,Y5: product_prod_t_set_a] :
          ( ( ord_less_t @ ( product_fst_t_set_a @ X5 ) @ ( product_fst_t_set_a @ Y5 ) )
          | ( ( ( product_fst_t_set_a @ X5 )
              = ( product_fst_t_set_a @ Y5 ) )
            & ( ord_less_set_a @ ( product_snd_t_set_a @ X5 ) @ ( product_snd_t_set_a @ Y5 ) ) ) ) ) ) ).

% less_prod_def'
thf(fact_590_less__prod__def_H,axiom,
    ( ord_le3945363144663698842od_t_o
    = ( ^ [X5: product_prod_t_o,Y5: product_prod_t_o] :
          ( ( ord_less_t @ ( product_fst_t_o @ X5 ) @ ( product_fst_t_o @ Y5 ) )
          | ( ( ( product_fst_t_o @ X5 )
              = ( product_fst_t_o @ Y5 ) )
            & ( ord_less_o @ ( product_snd_t_o @ X5 ) @ ( product_snd_t_o @ Y5 ) ) ) ) ) ) ).

% less_prod_def'
thf(fact_591_less__prod__def_H,axiom,
    ( ord_le1203424502768444845at_nat
    = ( ^ [X5: product_prod_nat_nat,Y5: product_prod_nat_nat] :
          ( ( ord_less_nat @ ( product_fst_nat_nat @ X5 ) @ ( product_fst_nat_nat @ Y5 ) )
          | ( ( ( product_fst_nat_nat @ X5 )
              = ( product_fst_nat_nat @ Y5 ) )
            & ( ord_less_nat @ ( product_snd_nat_nat @ X5 ) @ ( product_snd_nat_nat @ Y5 ) ) ) ) ) ) ).

% less_prod_def'
thf(fact_592_nth__take__lemma,axiom,
    ! [K: nat,Xs: list_P1095827921439705018_set_a,Ys: list_P1095827921439705018_set_a] :
      ( ( ord_less_eq_nat @ K @ ( size_s5362121731612881446_set_a @ Xs ) )
     => ( ( ord_less_eq_nat @ K @ ( size_s5362121731612881446_set_a @ Ys ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ K )
             => ( ( nth_Pr633927046780761275_set_a @ Xs @ I2 )
                = ( nth_Pr633927046780761275_set_a @ Ys @ I2 ) ) )
         => ( ( take_P2822414627813279522_set_a @ K @ Xs )
            = ( take_P2822414627813279522_set_a @ K @ Ys ) ) ) ) ) ).

% nth_take_lemma
thf(fact_593_nth__take__lemma,axiom,
    ! [K: nat,Xs: list_vydra_aux_a_t_h,Ys: list_vydra_aux_a_t_h] :
      ( ( ord_less_eq_nat @ K @ ( size_s8993897887676352572_a_t_h @ Xs ) )
     => ( ( ord_less_eq_nat @ K @ ( size_s8993897887676352572_a_t_h @ Ys ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ K )
             => ( ( nth_vydra_aux_a_t_h @ Xs @ I2 )
                = ( nth_vydra_aux_a_t_h @ Ys @ I2 ) ) )
         => ( ( take_vydra_aux_a_t_h @ K @ Xs )
            = ( take_vydra_aux_a_t_h @ K @ Ys ) ) ) ) ) ).

% nth_take_lemma
thf(fact_594_nth__take__lemma,axiom,
    ! [K: nat,Xs: list_iarray_o,Ys: list_iarray_o] :
      ( ( ord_less_eq_nat @ K @ ( size_s3243743627286111600rray_o @ Xs ) )
     => ( ( ord_less_eq_nat @ K @ ( size_s3243743627286111600rray_o @ Ys ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ K )
             => ( ( nth_iarray_o @ Xs @ I2 )
                = ( nth_iarray_o @ Ys @ I2 ) ) )
         => ( ( take_iarray_o @ K @ Xs )
            = ( take_iarray_o @ K @ Ys ) ) ) ) ) ).

% nth_take_lemma
thf(fact_595_nth__take__lemma,axiom,
    ! [K: nat,Xs: list_formula_a_t,Ys: list_formula_a_t] :
      ( ( ord_less_eq_nat @ K @ ( size_s8846756101701226951la_a_t @ Xs ) )
     => ( ( ord_less_eq_nat @ K @ ( size_s8846756101701226951la_a_t @ Ys ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ K )
             => ( ( nth_formula_a_t @ Xs @ I2 )
                = ( nth_formula_a_t @ Ys @ I2 ) ) )
         => ( ( take_formula_a_t @ K @ Xs )
            = ( take_formula_a_t @ K @ Ys ) ) ) ) ) ).

% nth_take_lemma
thf(fact_596_nth__take__lemma,axiom,
    ! [K: nat,Xs: list_t,Ys: list_t] :
      ( ( ord_less_eq_nat @ K @ ( size_size_list_t @ Xs ) )
     => ( ( ord_less_eq_nat @ K @ ( size_size_list_t @ Ys ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ K )
             => ( ( nth_t @ Xs @ I2 )
                = ( nth_t @ Ys @ I2 ) ) )
         => ( ( take_t @ K @ Xs )
            = ( take_t @ K @ Ys ) ) ) ) ) ).

% nth_take_lemma
thf(fact_597_prod__less__def,axiom,
    ( ord_le7650071679307129824_set_a
    = ( ^ [X5: product_prod_t_set_a,Y5: product_prod_t_set_a] :
          ( ( ord_less_t @ ( product_fst_t_set_a @ X5 ) @ ( product_fst_t_set_a @ Y5 ) )
          | ( ( ord_less_eq_t @ ( product_fst_t_set_a @ X5 ) @ ( product_fst_t_set_a @ Y5 ) )
            & ( ord_less_set_a @ ( product_snd_t_set_a @ X5 ) @ ( product_snd_t_set_a @ Y5 ) ) ) ) ) ) ).

% prod_less_def
thf(fact_598_prod__less__def,axiom,
    ( ord_le3945363144663698842od_t_o
    = ( ^ [X5: product_prod_t_o,Y5: product_prod_t_o] :
          ( ( ord_less_t @ ( product_fst_t_o @ X5 ) @ ( product_fst_t_o @ Y5 ) )
          | ( ( ord_less_eq_t @ ( product_fst_t_o @ X5 ) @ ( product_fst_t_o @ Y5 ) )
            & ( ord_less_o @ ( product_snd_t_o @ X5 ) @ ( product_snd_t_o @ Y5 ) ) ) ) ) ) ).

% prod_less_def
thf(fact_599_prod__less__def,axiom,
    ( ord_le1203424502768444845at_nat
    = ( ^ [X5: product_prod_nat_nat,Y5: product_prod_nat_nat] :
          ( ( ord_less_nat @ ( product_fst_nat_nat @ X5 ) @ ( product_fst_nat_nat @ Y5 ) )
          | ( ( ord_less_eq_nat @ ( product_fst_nat_nat @ X5 ) @ ( product_fst_nat_nat @ Y5 ) )
            & ( ord_less_nat @ ( product_snd_nat_nat @ X5 ) @ ( product_snd_nat_nat @ Y5 ) ) ) ) ) ) ).

% prod_less_def
thf(fact_600_reaches_Osimps,axiom,
    ( reaches_nat_o_nat
    = ( ^ [Run2: ( nat > $o ) > option7421465107808516430_o_nat,A1: nat > $o,A22: nat,A32: nat > $o] :
          ( ? [S3: nat > $o] :
              ( ( A1 = S3 )
              & ( A22 = zero_zero_nat )
              & ( A32 = S3 ) )
          | ? [S3: nat > $o,S4: nat > $o,V4: nat,N4: nat,S9: nat > $o] :
              ( ( A1 = S3 )
              & ( A22
                = ( suc @ N4 ) )
              & ( A32 = S9 )
              & ( ( Run2 @ S3 )
                = ( some_P2630223654056210137_o_nat @ ( produc7277522915581678840_o_nat @ S4 @ V4 ) ) )
              & ( reaches_nat_o_nat @ Run2 @ S4 @ N4 @ S9 ) ) ) ) ) ).

% reaches.simps
thf(fact_601_reaches_Osimps,axiom,
    ( reaches_t_set_a
    = ( ^ [Run2: t > option4621139365931258234_set_a,A1: t,A22: nat,A32: t] :
          ( ? [S3: t] :
              ( ( A1 = S3 )
              & ( A22 = zero_zero_nat )
              & ( A32 = S3 ) )
          | ? [S3: t,S4: t,V4: set_a,N4: nat,S9: t] :
              ( ( A1 = S3 )
              & ( A22
                = ( suc @ N4 ) )
              & ( A32 = S9 )
              & ( ( Run2 @ S3 )
                = ( some_P4667681669429350681_set_a @ ( product_Pair_t_set_a @ S4 @ V4 ) ) )
              & ( reaches_t_set_a @ Run2 @ S4 @ N4 @ S9 ) ) ) ) ) ).

% reaches.simps
thf(fact_602_reaches_Osimps,axiom,
    ( reache2947816147551113615rray_o
    = ( ^ [Run2: list_vydra_aux_a_t_h > option8553266494716013595rray_o,A1: list_vydra_aux_a_t_h,A22: nat,A32: list_vydra_aux_a_t_h] :
          ( ? [S3: list_vydra_aux_a_t_h] :
              ( ( A1 = S3 )
              & ( A22 = zero_zero_nat )
              & ( A32 = S3 ) )
          | ? [S3: list_vydra_aux_a_t_h,S4: list_vydra_aux_a_t_h,V4: iarray_o,N4: nat,S9: list_vydra_aux_a_t_h] :
              ( ( A1 = S3 )
              & ( A22
                = ( suc @ N4 ) )
              & ( A32 = S9 )
              & ( ( Run2 @ S3 )
                = ( some_P7279874174034738982rray_o @ ( produc8472968102330193021rray_o @ S4 @ V4 ) ) )
              & ( reache2947816147551113615rray_o @ Run2 @ S4 @ N4 @ S9 ) ) ) ) ) ).

% reaches.simps
thf(fact_603_reaches_Osimps,axiom,
    ( reaches_t_o
    = ( ^ [Run2: t > option8291060355354214836od_t_o,A1: t,A22: nat,A32: t] :
          ( ? [S3: t] :
              ( ( A1 = S3 )
              & ( A22 = zero_zero_nat )
              & ( A32 = S3 ) )
          | ? [S3: t,S4: t,V4: $o,N4: nat,S9: t] :
              ( ( A1 = S3 )
              & ( A22
                = ( suc @ N4 ) )
              & ( A32 = S9 )
              & ( ( Run2 @ S3 )
                = ( some_P1805796590424124243od_t_o @ ( product_Pair_t_o @ S4 @ V4 ) ) )
              & ( reaches_t_o @ Run2 @ S4 @ N4 @ S9 ) ) ) ) ) ).

% reaches.simps
thf(fact_604_reaches_Osimps,axiom,
    ( reache8480716032150177652_set_a
    = ( ^ [Run2: h > option5775791656947114332_set_a,A1: h,A22: nat,A32: h] :
          ( ? [S3: h] :
              ( ( A1 = S3 )
              & ( A22 = zero_zero_nat )
              & ( A32 = S3 ) )
          | ? [S3: h,S4: h,V4: product_prod_t_set_a,N4: nat,S9: h] :
              ( ( A1 = S3 )
              & ( A22
                = ( suc @ N4 ) )
              & ( A32 = S9 )
              & ( ( Run2 @ S3 )
                = ( some_P2245991037696317435_set_a @ ( produc3432067189782488838_set_a @ S4 @ V4 ) ) )
              & ( reache8480716032150177652_set_a @ Run2 @ S4 @ N4 @ S9 ) ) ) ) ) ).

% reaches.simps
thf(fact_605_reaches_Osimps,axiom,
    ( reaches_h_t
    = ( ^ [Run2: h > option2110264179601050681od_h_t,A1: h,A22: nat,A32: h] :
          ( ? [S3: h] :
              ( ( A1 = S3 )
              & ( A22 = zero_zero_nat )
              & ( A32 = S3 ) )
          | ? [S3: h,S4: h,V4: t,N4: nat,S9: h] :
              ( ( A1 = S3 )
              & ( A22
                = ( suc @ N4 ) )
              & ( A32 = S9 )
              & ( ( Run2 @ S3 )
                = ( some_P2526622376857543512od_h_t @ ( product_Pair_h_t @ S4 @ V4 ) ) )
              & ( reaches_h_t @ Run2 @ S4 @ N4 @ S9 ) ) ) ) ) ).

% reaches.simps
thf(fact_606_reaches_Ocases,axiom,
    ! [Run: ( nat > $o ) > option7421465107808516430_o_nat,A12: nat > $o,A23: nat,A33: nat > $o] :
      ( ( reaches_nat_o_nat @ Run @ A12 @ A23 @ A33 )
     => ( ( ( A23 = zero_zero_nat )
         => ( A33 != A12 ) )
       => ~ ! [S6: nat > $o,V3: nat,N3: nat] :
              ( ( A23
                = ( suc @ N3 ) )
             => ( ( ( Run @ A12 )
                  = ( some_P2630223654056210137_o_nat @ ( produc7277522915581678840_o_nat @ S6 @ V3 ) ) )
               => ~ ( reaches_nat_o_nat @ Run @ S6 @ N3 @ A33 ) ) ) ) ) ).

% reaches.cases
thf(fact_607_reaches_Ocases,axiom,
    ! [Run: t > option4621139365931258234_set_a,A12: t,A23: nat,A33: t] :
      ( ( reaches_t_set_a @ Run @ A12 @ A23 @ A33 )
     => ( ( ( A23 = zero_zero_nat )
         => ( A33 != A12 ) )
       => ~ ! [S6: t,V3: set_a,N3: nat] :
              ( ( A23
                = ( suc @ N3 ) )
             => ( ( ( Run @ A12 )
                  = ( some_P4667681669429350681_set_a @ ( product_Pair_t_set_a @ S6 @ V3 ) ) )
               => ~ ( reaches_t_set_a @ Run @ S6 @ N3 @ A33 ) ) ) ) ) ).

% reaches.cases
thf(fact_608_reaches_Ocases,axiom,
    ! [Run: list_vydra_aux_a_t_h > option8553266494716013595rray_o,A12: list_vydra_aux_a_t_h,A23: nat,A33: list_vydra_aux_a_t_h] :
      ( ( reache2947816147551113615rray_o @ Run @ A12 @ A23 @ A33 )
     => ( ( ( A23 = zero_zero_nat )
         => ( A33 != A12 ) )
       => ~ ! [S6: list_vydra_aux_a_t_h,V3: iarray_o,N3: nat] :
              ( ( A23
                = ( suc @ N3 ) )
             => ( ( ( Run @ A12 )
                  = ( some_P7279874174034738982rray_o @ ( produc8472968102330193021rray_o @ S6 @ V3 ) ) )
               => ~ ( reache2947816147551113615rray_o @ Run @ S6 @ N3 @ A33 ) ) ) ) ) ).

% reaches.cases
thf(fact_609_reaches_Ocases,axiom,
    ! [Run: t > option8291060355354214836od_t_o,A12: t,A23: nat,A33: t] :
      ( ( reaches_t_o @ Run @ A12 @ A23 @ A33 )
     => ( ( ( A23 = zero_zero_nat )
         => ( A33 != A12 ) )
       => ~ ! [S6: t,V3: $o,N3: nat] :
              ( ( A23
                = ( suc @ N3 ) )
             => ( ( ( Run @ A12 )
                  = ( some_P1805796590424124243od_t_o @ ( product_Pair_t_o @ S6 @ V3 ) ) )
               => ~ ( reaches_t_o @ Run @ S6 @ N3 @ A33 ) ) ) ) ) ).

% reaches.cases
thf(fact_610_reaches_Ocases,axiom,
    ! [Run: h > option5775791656947114332_set_a,A12: h,A23: nat,A33: h] :
      ( ( reache8480716032150177652_set_a @ Run @ A12 @ A23 @ A33 )
     => ( ( ( A23 = zero_zero_nat )
         => ( A33 != A12 ) )
       => ~ ! [S6: h,V3: product_prod_t_set_a,N3: nat] :
              ( ( A23
                = ( suc @ N3 ) )
             => ( ( ( Run @ A12 )
                  = ( some_P2245991037696317435_set_a @ ( produc3432067189782488838_set_a @ S6 @ V3 ) ) )
               => ~ ( reache8480716032150177652_set_a @ Run @ S6 @ N3 @ A33 ) ) ) ) ) ).

% reaches.cases
thf(fact_611_reaches_Ocases,axiom,
    ! [Run: h > option2110264179601050681od_h_t,A12: h,A23: nat,A33: h] :
      ( ( reaches_h_t @ Run @ A12 @ A23 @ A33 )
     => ( ( ( A23 = zero_zero_nat )
         => ( A33 != A12 ) )
       => ~ ! [S6: h,V3: t,N3: nat] :
              ( ( A23
                = ( suc @ N3 ) )
             => ( ( ( Run @ A12 )
                  = ( some_P2526622376857543512od_h_t @ ( product_Pair_h_t @ S6 @ V3 ) ) )
               => ~ ( reaches_h_t @ Run @ S6 @ N3 @ A33 ) ) ) ) ) ).

% reaches.cases
thf(fact_612_less__eq__prod__def,axiom,
    ( ord_le1576776917030837460_set_a
    = ( ^ [X5: product_prod_t_set_a,Y5: product_prod_t_set_a] :
          ( ( ord_less_t @ ( product_fst_t_set_a @ X5 ) @ ( product_fst_t_set_a @ Y5 ) )
          | ( ( ord_less_eq_t @ ( product_fst_t_set_a @ X5 ) @ ( product_fst_t_set_a @ Y5 ) )
            & ( ord_less_eq_set_a @ ( product_snd_t_set_a @ X5 ) @ ( product_snd_t_set_a @ Y5 ) ) ) ) ) ) ).

% less_eq_prod_def
thf(fact_613_less__eq__prod__def,axiom,
    ( ord_le1999770283289415822od_t_o
    = ( ^ [X5: product_prod_t_o,Y5: product_prod_t_o] :
          ( ( ord_less_t @ ( product_fst_t_o @ X5 ) @ ( product_fst_t_o @ Y5 ) )
          | ( ( ord_less_eq_t @ ( product_fst_t_o @ X5 ) @ ( product_fst_t_o @ Y5 ) )
            & ( ord_less_eq_o @ ( product_snd_t_o @ X5 ) @ ( product_snd_t_o @ Y5 ) ) ) ) ) ) ).

% less_eq_prod_def
thf(fact_614_less__eq__prod__def,axiom,
    ( ord_le8460144461188290721at_nat
    = ( ^ [X5: product_prod_nat_nat,Y5: product_prod_nat_nat] :
          ( ( ord_less_nat @ ( product_fst_nat_nat @ X5 ) @ ( product_fst_nat_nat @ Y5 ) )
          | ( ( ord_less_eq_nat @ ( product_fst_nat_nat @ X5 ) @ ( product_fst_nat_nat @ Y5 ) )
            & ( ord_less_eq_nat @ ( product_snd_nat_nat @ X5 ) @ ( product_snd_nat_nat @ Y5 ) ) ) ) ) ) ).

% less_eq_prod_def
thf(fact_615_init__vydra__def,axiom,
    ( init_vydra_h_t_a
    = ( ^ [Init_hd2: h,Run_hd2: h > option5775791656947114332_set_a,Phi3: formula_a_t] : ( produc184169161053073665_a_t_h @ ( msize_fmla_a_t @ Phi3 ) @ ( sub_h_t_a @ Init_hd2 @ Run_hd2 @ ( msize_fmla_a_t @ Phi3 ) @ Phi3 ) ) ) ) ).

% init_vydra_def
thf(fact_616_not__gr__zero,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr_zero
thf(fact_617_le__zero__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_zero_eq
thf(fact_618_drop__map,axiom,
    ! [N: nat,F: product_prod_t_set_a > t,Xs: list_P1095827921439705018_set_a] :
      ( ( drop_t @ N @ ( map_Pr119846518281778721et_a_t @ F @ Xs ) )
      = ( map_Pr119846518281778721et_a_t @ F @ ( drop_P283403782220237946_set_a @ N @ Xs ) ) ) ).

% drop_map
thf(fact_619_drop__map,axiom,
    ! [N: nat,F: formula_a_t > $o,Xs: list_formula_a_t] :
      ( ( drop_o @ N @ ( map_formula_a_t_o @ F @ Xs ) )
      = ( map_formula_a_t_o @ F @ ( drop_formula_a_t @ N @ Xs ) ) ) ).

% drop_map
thf(fact_620_drop__map,axiom,
    ! [N: nat,F: formula_a_t > vydra_aux_a_t_h,Xs: list_formula_a_t] :
      ( ( drop_vydra_aux_a_t_h @ N @ ( map_fo5746072578778194433_a_t_h @ F @ Xs ) )
      = ( map_fo5746072578778194433_a_t_h @ F @ ( drop_formula_a_t @ N @ Xs ) ) ) ).

% drop_map
thf(fact_621_reaches__on__split_H,axiom,
    ! [Run: vydra_aux_a_t_h > option4580262698044125979od_t_o,S: vydra_aux_a_t_h,Vs: list_P5575445741472356340od_t_o,S2: vydra_aux_a_t_h,I: nat] :
      ( ( reache8691153155344877136od_t_o @ Run @ S @ Vs @ S2 )
     => ( ( ord_less_eq_nat @ I @ ( size_s4106575409306904672od_t_o @ Vs ) )
       => ? [S8: vydra_aux_a_t_h] :
            ( ( reache8691153155344877136od_t_o @ Run @ S @ ( take_P7695396673888565724od_t_o @ I @ Vs ) @ S8 )
            & ( reache8691153155344877136od_t_o @ Run @ S8 @ ( drop_P3066774390924266292od_t_o @ I @ Vs ) @ S2 ) ) ) ) ).

% reaches_on_split'
thf(fact_622_reaches__on__split_H,axiom,
    ! [Run: h > option5775791656947114332_set_a,S: h,Vs: list_P1095827921439705018_set_a,S2: h,I: nat] :
      ( ( reache3397181163414509427_set_a @ Run @ S @ Vs @ S2 )
     => ( ( ord_less_eq_nat @ I @ ( size_s5362121731612881446_set_a @ Vs ) )
       => ? [S8: h] :
            ( ( reache3397181163414509427_set_a @ Run @ S @ ( take_P2822414627813279522_set_a @ I @ Vs ) @ S8 )
            & ( reache3397181163414509427_set_a @ Run @ S8 @ ( drop_P283403782220237946_set_a @ I @ Vs ) @ S2 ) ) ) ) ).

% reaches_on_split'
thf(fact_623_reaches__on__split_H,axiom,
    ! [Run: list_vydra_aux_a_t_h > option8553266494716013595rray_o,S: list_vydra_aux_a_t_h,Vs: list_iarray_o,S2: list_vydra_aux_a_t_h,I: nat] :
      ( ( reache2401979074735462992rray_o @ Run @ S @ Vs @ S2 )
     => ( ( ord_less_eq_nat @ I @ ( size_s3243743627286111600rray_o @ Vs ) )
       => ? [S8: list_vydra_aux_a_t_h] :
            ( ( reache2401979074735462992rray_o @ Run @ S @ ( take_iarray_o @ I @ Vs ) @ S8 )
            & ( reache2401979074735462992rray_o @ Run @ S8 @ ( drop_iarray_o @ I @ Vs ) @ S2 ) ) ) ) ).

% reaches_on_split'
thf(fact_624_reaches__on__split_H,axiom,
    ! [Run: option2110264179601050681od_h_t > option4241632866121027589_h_t_t,S: option2110264179601050681od_h_t,Vs: list_t,S2: option2110264179601050681od_h_t,I: nat] :
      ( ( reache645946503560514460_h_t_t @ Run @ S @ Vs @ S2 )
     => ( ( ord_less_eq_nat @ I @ ( size_size_list_t @ Vs ) )
       => ? [S8: option2110264179601050681od_h_t] :
            ( ( reache645946503560514460_h_t_t @ Run @ S @ ( take_t @ I @ Vs ) @ S8 )
            & ( reache645946503560514460_h_t_t @ Run @ S8 @ ( drop_t @ I @ Vs ) @ S2 ) ) ) ) ).

% reaches_on_split'
thf(fact_625_zero__le,axiom,
    ! [X3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X3 ) ).

% zero_le
thf(fact_626_zero__less__iff__neq__zero,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
      = ( N != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
thf(fact_627_gr__implies__not__zero,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_nat @ M @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not_zero
thf(fact_628_not__less__zero,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less_zero
thf(fact_629_gr__zeroI,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr_zeroI
thf(fact_630_reaches__on__split,axiom,
    ! [Run: ( nat > $o ) > option7421465107808516430_o_nat,S: nat > $o,Vs: list_nat,S2: nat > $o,I: nat] :
      ( ( reaches_on_nat_o_nat @ Run @ S @ Vs @ S2 )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Vs ) )
       => ? [S8: nat > $o,S10: nat > $o] :
            ( ( reaches_on_nat_o_nat @ Run @ S @ ( take_nat @ I @ Vs ) @ S8 )
            & ( ( Run @ S8 )
              = ( some_P2630223654056210137_o_nat @ ( produc7277522915581678840_o_nat @ S10 @ ( nth_nat @ Vs @ I ) ) ) )
            & ( reaches_on_nat_o_nat @ Run @ S10 @ ( drop_nat @ ( suc @ I ) @ Vs ) @ S2 ) ) ) ) ).

% reaches_on_split
thf(fact_631_reaches__on__split,axiom,
    ! [Run: t > option4621139365931258234_set_a,S: t,Vs: list_set_a,S2: t,I: nat] :
      ( ( reaches_on_t_set_a @ Run @ S @ Vs @ S2 )
     => ( ( ord_less_nat @ I @ ( size_size_list_set_a @ Vs ) )
       => ? [S8: t,S10: t] :
            ( ( reaches_on_t_set_a @ Run @ S @ ( take_set_a @ I @ Vs ) @ S8 )
            & ( ( Run @ S8 )
              = ( some_P4667681669429350681_set_a @ ( product_Pair_t_set_a @ S10 @ ( nth_set_a @ Vs @ I ) ) ) )
            & ( reaches_on_t_set_a @ Run @ S10 @ ( drop_set_a @ ( suc @ I ) @ Vs ) @ S2 ) ) ) ) ).

% reaches_on_split
thf(fact_632_reaches__on__split,axiom,
    ! [Run: vydra_aux_a_t_h > option4580262698044125979od_t_o,S: vydra_aux_a_t_h,Vs: list_P5575445741472356340od_t_o,S2: vydra_aux_a_t_h,I: nat] :
      ( ( reache8691153155344877136od_t_o @ Run @ S @ Vs @ S2 )
     => ( ( ord_less_nat @ I @ ( size_s4106575409306904672od_t_o @ Vs ) )
       => ? [S8: vydra_aux_a_t_h,S10: vydra_aux_a_t_h] :
            ( ( reache8691153155344877136od_t_o @ Run @ S @ ( take_P7695396673888565724od_t_o @ I @ Vs ) @ S8 )
            & ( ( Run @ S8 )
              = ( some_P9127227083975130150od_t_o @ ( produc5956389175556509053od_t_o @ S10 @ ( nth_Product_prod_t_o @ Vs @ I ) ) ) )
            & ( reache8691153155344877136od_t_o @ Run @ S10 @ ( drop_P3066774390924266292od_t_o @ ( suc @ I ) @ Vs ) @ S2 ) ) ) ) ).

% reaches_on_split
thf(fact_633_reaches__on__split,axiom,
    ! [Run: t > option8291060355354214836od_t_o,S: t,Vs: list_o,S2: t,I: nat] :
      ( ( reaches_on_t_o @ Run @ S @ Vs @ S2 )
     => ( ( ord_less_nat @ I @ ( size_size_list_o @ Vs ) )
       => ? [S8: t,S10: t] :
            ( ( reaches_on_t_o @ Run @ S @ ( take_o @ I @ Vs ) @ S8 )
            & ( ( Run @ S8 )
              = ( some_P1805796590424124243od_t_o @ ( product_Pair_t_o @ S10 @ ( nth_o @ Vs @ I ) ) ) )
            & ( reaches_on_t_o @ Run @ S10 @ ( drop_o @ ( suc @ I ) @ Vs ) @ S2 ) ) ) ) ).

% reaches_on_split
thf(fact_634_reaches__on__split,axiom,
    ! [Run: h > option5775791656947114332_set_a,S: h,Vs: list_P1095827921439705018_set_a,S2: h,I: nat] :
      ( ( reache3397181163414509427_set_a @ Run @ S @ Vs @ S2 )
     => ( ( ord_less_nat @ I @ ( size_s5362121731612881446_set_a @ Vs ) )
       => ? [S8: h,S10: h] :
            ( ( reache3397181163414509427_set_a @ Run @ S @ ( take_P2822414627813279522_set_a @ I @ Vs ) @ S8 )
            & ( ( Run @ S8 )
              = ( some_P2245991037696317435_set_a @ ( produc3432067189782488838_set_a @ S10 @ ( nth_Pr633927046780761275_set_a @ Vs @ I ) ) ) )
            & ( reache3397181163414509427_set_a @ Run @ S10 @ ( drop_P283403782220237946_set_a @ ( suc @ I ) @ Vs ) @ S2 ) ) ) ) ).

% reaches_on_split
thf(fact_635_reaches__on__split,axiom,
    ! [Run: list_vydra_aux_a_t_h > option8553266494716013595rray_o,S: list_vydra_aux_a_t_h,Vs: list_iarray_o,S2: list_vydra_aux_a_t_h,I: nat] :
      ( ( reache2401979074735462992rray_o @ Run @ S @ Vs @ S2 )
     => ( ( ord_less_nat @ I @ ( size_s3243743627286111600rray_o @ Vs ) )
       => ? [S8: list_vydra_aux_a_t_h,S10: list_vydra_aux_a_t_h] :
            ( ( reache2401979074735462992rray_o @ Run @ S @ ( take_iarray_o @ I @ Vs ) @ S8 )
            & ( ( Run @ S8 )
              = ( some_P7279874174034738982rray_o @ ( produc8472968102330193021rray_o @ S10 @ ( nth_iarray_o @ Vs @ I ) ) ) )
            & ( reache2401979074735462992rray_o @ Run @ S10 @ ( drop_iarray_o @ ( suc @ I ) @ Vs ) @ S2 ) ) ) ) ).

% reaches_on_split
thf(fact_636_reaches__on__split,axiom,
    ! [Run: option2110264179601050681od_h_t > option4241632866121027589_h_t_t,S: option2110264179601050681od_h_t,Vs: list_t,S2: option2110264179601050681od_h_t,I: nat] :
      ( ( reache645946503560514460_h_t_t @ Run @ S @ Vs @ S2 )
     => ( ( ord_less_nat @ I @ ( size_size_list_t @ Vs ) )
       => ? [S8: option2110264179601050681od_h_t,S10: option2110264179601050681od_h_t] :
            ( ( reache645946503560514460_h_t_t @ Run @ S @ ( take_t @ I @ Vs ) @ S8 )
            & ( ( Run @ S8 )
              = ( some_P8406129549712030756_h_t_t @ ( produc770200333395398319_h_t_t @ S10 @ ( nth_t @ Vs @ I ) ) ) )
            & ( reache645946503560514460_h_t_t @ Run @ S10 @ ( drop_t @ ( suc @ I ) @ Vs ) @ S2 ) ) ) ) ).

% reaches_on_split
thf(fact_637_reaches__on__split,axiom,
    ! [Run: h > option2110264179601050681od_h_t,S: h,Vs: list_t,S2: h,I: nat] :
      ( ( reaches_on_h_t @ Run @ S @ Vs @ S2 )
     => ( ( ord_less_nat @ I @ ( size_size_list_t @ Vs ) )
       => ? [S8: h,S10: h] :
            ( ( reaches_on_h_t @ Run @ S @ ( take_t @ I @ Vs ) @ S8 )
            & ( ( Run @ S8 )
              = ( some_P2526622376857543512od_h_t @ ( product_Pair_h_t @ S10 @ ( nth_t @ Vs @ I ) ) ) )
            & ( reaches_on_h_t @ Run @ S10 @ ( drop_t @ ( suc @ I ) @ Vs ) @ S2 ) ) ) ) ).

% reaches_on_split
thf(fact_638_ts__at__def,axiom,
    ( ts_at_t_set_a
    = ( ^ [Rho2: list_P1095827921439705018_set_a,I3: nat] : ( product_fst_t_set_a @ ( nth_Pr633927046780761275_set_a @ Rho2 @ I3 ) ) ) ) ).

% ts_at_def
thf(fact_639_bs__at__def,axiom,
    ( bs_at_t_o
    = ( ^ [Rho2: list_P5575445741472356340od_t_o,I3: nat] : ( product_snd_t_o @ ( nth_Product_prod_t_o @ Rho2 @ I3 ) ) ) ) ).

% bs_at_def
thf(fact_640_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: ( nat > $o ) > nat > $o,X3: nat > $o,Y: nat,A: produc3074792404157404414_o_nat] :
      ( ( P @ X3 @ Y )
     => ( ( A
          = ( produc7277522915581678840_o_nat @ X3 @ Y ) )
       => ( P @ ( produc7643701372303132684_o_nat @ A ) @ ( produc3470649411511297610_o_nat @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_641_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: h > product_prod_t_set_a > $o,X3: h,Y: product_prod_t_set_a,A: produc8090089739038349590_set_a] :
      ( ( P @ X3 @ Y )
     => ( ( A
          = ( produc3432067189782488838_set_a @ X3 @ Y ) )
       => ( P @ ( produc6892702156181598450_set_a @ A ) @ ( produc5260167487839082036_set_a @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_642_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: h > t > $o,X3: h,Y: t,A: product_prod_h_t] :
      ( ( P @ X3 @ Y )
     => ( ( A
          = ( product_Pair_h_t @ X3 @ Y ) )
       => ( P @ ( product_fst_h_t @ A ) @ ( product_snd_h_t @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_643_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: t > set_a > $o,X3: t,Y: set_a,A: product_prod_t_set_a] :
      ( ( P @ X3 @ Y )
     => ( ( A
          = ( product_Pair_t_set_a @ X3 @ Y ) )
       => ( P @ ( product_fst_t_set_a @ A ) @ ( product_snd_t_set_a @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_644_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P: t > $o > $o,X3: t,Y: $o,A: product_prod_t_o] :
      ( ( P @ X3 @ Y )
     => ( ( A
          = ( product_Pair_t_o @ X3 @ Y ) )
       => ( P @ ( product_fst_t_o @ A ) @ ( product_snd_t_o @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_645_conjI__realizer,axiom,
    ! [P: ( nat > $o ) > $o,P2: nat > $o,Q: nat > $o,Q3: nat] :
      ( ( P @ P2 )
     => ( ( Q @ Q3 )
       => ( ( P @ ( produc7643701372303132684_o_nat @ ( produc7277522915581678840_o_nat @ P2 @ Q3 ) ) )
          & ( Q @ ( produc3470649411511297610_o_nat @ ( produc7277522915581678840_o_nat @ P2 @ Q3 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_646_conjI__realizer,axiom,
    ! [P: h > $o,P2: h,Q: product_prod_t_set_a > $o,Q3: product_prod_t_set_a] :
      ( ( P @ P2 )
     => ( ( Q @ Q3 )
       => ( ( P @ ( produc6892702156181598450_set_a @ ( produc3432067189782488838_set_a @ P2 @ Q3 ) ) )
          & ( Q @ ( produc5260167487839082036_set_a @ ( produc3432067189782488838_set_a @ P2 @ Q3 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_647_conjI__realizer,axiom,
    ! [P: h > $o,P2: h,Q: t > $o,Q3: t] :
      ( ( P @ P2 )
     => ( ( Q @ Q3 )
       => ( ( P @ ( product_fst_h_t @ ( product_Pair_h_t @ P2 @ Q3 ) ) )
          & ( Q @ ( product_snd_h_t @ ( product_Pair_h_t @ P2 @ Q3 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_648_conjI__realizer,axiom,
    ! [P: t > $o,P2: t,Q: set_a > $o,Q3: set_a] :
      ( ( P @ P2 )
     => ( ( Q @ Q3 )
       => ( ( P @ ( product_fst_t_set_a @ ( product_Pair_t_set_a @ P2 @ Q3 ) ) )
          & ( Q @ ( product_snd_t_set_a @ ( product_Pair_t_set_a @ P2 @ Q3 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_649_conjI__realizer,axiom,
    ! [P: t > $o,P2: t,Q: $o > $o,Q3: $o] :
      ( ( P @ P2 )
     => ( ( Q @ Q3 )
       => ( ( P @ ( product_fst_t_o @ ( product_Pair_t_o @ P2 @ Q3 ) ) )
          & ( Q @ ( product_snd_t_o @ ( product_Pair_t_o @ P2 @ Q3 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_650_exI__realizer,axiom,
    ! [P: nat > ( nat > $o ) > $o,Y: nat,X3: nat > $o] :
      ( ( P @ Y @ X3 )
     => ( P @ ( produc3470649411511297610_o_nat @ ( produc7277522915581678840_o_nat @ X3 @ Y ) ) @ ( produc7643701372303132684_o_nat @ ( produc7277522915581678840_o_nat @ X3 @ Y ) ) ) ) ).

% exI_realizer
thf(fact_651_exI__realizer,axiom,
    ! [P: product_prod_t_set_a > h > $o,Y: product_prod_t_set_a,X3: h] :
      ( ( P @ Y @ X3 )
     => ( P @ ( produc5260167487839082036_set_a @ ( produc3432067189782488838_set_a @ X3 @ Y ) ) @ ( produc6892702156181598450_set_a @ ( produc3432067189782488838_set_a @ X3 @ Y ) ) ) ) ).

% exI_realizer
thf(fact_652_exI__realizer,axiom,
    ! [P: t > h > $o,Y: t,X3: h] :
      ( ( P @ Y @ X3 )
     => ( P @ ( product_snd_h_t @ ( product_Pair_h_t @ X3 @ Y ) ) @ ( product_fst_h_t @ ( product_Pair_h_t @ X3 @ Y ) ) ) ) ).

% exI_realizer
thf(fact_653_exI__realizer,axiom,
    ! [P: set_a > t > $o,Y: set_a,X3: t] :
      ( ( P @ Y @ X3 )
     => ( P @ ( product_snd_t_set_a @ ( product_Pair_t_set_a @ X3 @ Y ) ) @ ( product_fst_t_set_a @ ( product_Pair_t_set_a @ X3 @ Y ) ) ) ) ).

% exI_realizer
thf(fact_654_exI__realizer,axiom,
    ! [P: $o > t > $o,Y: $o,X3: t] :
      ( ( P @ Y @ X3 )
     => ( P @ ( product_snd_t_o @ ( product_Pair_t_o @ X3 @ Y ) ) @ ( product_fst_t_o @ ( product_Pair_t_o @ X3 @ Y ) ) ) ) ).

% exI_realizer
thf(fact_655_exE__realizer_H,axiom,
    ! [P: set_a > t > $o,P2: product_prod_t_set_a] :
      ( ( P @ ( product_snd_t_set_a @ P2 ) @ ( product_fst_t_set_a @ P2 ) )
     => ~ ! [X: t,Y3: set_a] :
            ~ ( P @ Y3 @ X ) ) ).

% exE_realizer'
thf(fact_656_exE__realizer_H,axiom,
    ! [P: $o > t > $o,P2: product_prod_t_o] :
      ( ( P @ ( product_snd_t_o @ P2 ) @ ( product_fst_t_o @ P2 ) )
     => ~ ! [X: t,Y3: $o] :
            ~ ( P @ Y3 @ X ) ) ).

% exE_realizer'
thf(fact_657_lex__take__index,axiom,
    ! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( lex_nat @ R2 ) )
     => ~ ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
           => ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
             => ( ( ( take_nat @ I2 @ Xs )
                  = ( take_nat @ I2 @ Ys ) )
               => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Ys @ I2 ) ) @ R2 ) ) ) ) ) ).

% lex_take_index
thf(fact_658_lex__take__index,axiom,
    ! [Xs: list_P1095827921439705018_set_a,Ys: list_P1095827921439705018_set_a,R2: set_Pr2751536223740281991_set_a] :
      ( ( member7105239521983867088_set_a @ ( produc3154345580163335831_set_a @ Xs @ Ys ) @ ( lex_Pr3546421067709956510_set_a @ R2 ) )
     => ~ ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s5362121731612881446_set_a @ Xs ) )
           => ( ( ord_less_nat @ I2 @ ( size_s5362121731612881446_set_a @ Ys ) )
             => ( ( ( take_P2822414627813279522_set_a @ I2 @ Xs )
                  = ( take_P2822414627813279522_set_a @ I2 @ Ys ) )
               => ~ ( member6953777265722981072_set_a @ ( produc4993190531567155607_set_a @ ( nth_Pr633927046780761275_set_a @ Xs @ I2 ) @ ( nth_Pr633927046780761275_set_a @ Ys @ I2 ) ) @ R2 ) ) ) ) ) ).

% lex_take_index
thf(fact_659_lex__take__index,axiom,
    ! [Xs: list_vydra_aux_a_t_h,Ys: list_vydra_aux_a_t_h,R2: set_Pr5596025548626726129_a_t_h] :
      ( ( member2805742798364816858_a_t_h @ ( produc7080071857504742505_a_t_h @ Xs @ Ys ) @ ( lex_vydra_aux_a_t_h @ R2 ) )
     => ~ ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s8993897887676352572_a_t_h @ Xs ) )
           => ( ( ord_less_nat @ I2 @ ( size_s8993897887676352572_a_t_h @ Ys ) )
             => ( ( ( take_vydra_aux_a_t_h @ I2 @ Xs )
                  = ( take_vydra_aux_a_t_h @ I2 @ Ys ) )
               => ~ ( member3443537796814201402_a_t_h @ ( produc6881572889943414985_a_t_h @ ( nth_vydra_aux_a_t_h @ Xs @ I2 ) @ ( nth_vydra_aux_a_t_h @ Ys @ I2 ) ) @ R2 ) ) ) ) ) ).

% lex_take_index
thf(fact_660_lex__take__index,axiom,
    ! [Xs: list_iarray_o,Ys: list_iarray_o,R2: set_Pr3159307878933344263rray_o] :
      ( ( member8606389602990657872rray_o @ ( produc2869105288327869463rray_o @ Xs @ Ys ) @ ( lex_iarray_o @ R2 ) )
     => ~ ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s3243743627286111600rray_o @ Xs ) )
           => ( ( ord_less_nat @ I2 @ ( size_s3243743627286111600rray_o @ Ys ) )
             => ( ( ( take_iarray_o @ I2 @ Xs )
                  = ( take_iarray_o @ I2 @ Ys ) )
               => ~ ( member3227708950830999120rray_o @ ( produc513795717669825047rray_o @ ( nth_iarray_o @ Xs @ I2 ) @ ( nth_iarray_o @ Ys @ I2 ) ) @ R2 ) ) ) ) ) ).

% lex_take_index
thf(fact_661_lex__take__index,axiom,
    ! [Xs: list_formula_a_t,Ys: list_formula_a_t,R2: set_Pr3703585533673849735la_a_t] :
      ( ( member5181786688746205936la_a_t @ ( produc7251354486509092607la_a_t @ Xs @ Ys ) @ ( lex_formula_a_t @ R2 ) )
     => ~ ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s8846756101701226951la_a_t @ Xs ) )
           => ( ( ord_less_nat @ I2 @ ( size_s8846756101701226951la_a_t @ Ys ) )
             => ( ( ( take_formula_a_t @ I2 @ Xs )
                  = ( take_formula_a_t @ I2 @ Ys ) )
               => ~ ( member9065953759106212816la_a_t @ ( produc8281305675666354143la_a_t @ ( nth_formula_a_t @ Xs @ I2 ) @ ( nth_formula_a_t @ Ys @ I2 ) ) @ R2 ) ) ) ) ) ).

% lex_take_index
thf(fact_662_lex__take__index,axiom,
    ! [Xs: list_t,Ys: list_t,R2: set_Product_prod_t_t] :
      ( ( member1598007745118852880list_t @ ( produc8116999438917745495list_t @ Xs @ Ys ) @ ( lex_t @ R2 ) )
     => ~ ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_t @ Xs ) )
           => ( ( ord_less_nat @ I2 @ ( size_size_list_t @ Ys ) )
             => ( ( ( take_t @ I2 @ Xs )
                  = ( take_t @ I2 @ Ys ) )
               => ~ ( member3805107722934592656od_t_t @ ( product_Pair_t_t @ ( nth_t @ Xs @ I2 ) @ ( nth_t @ Ys @ I2 ) ) @ R2 ) ) ) ) ) ).

% lex_take_index
thf(fact_663_exists__least__lemma,axiom,
    ! [P: nat > $o] :
      ( ~ ( P @ zero_zero_nat )
     => ( ? [X_12: nat] : ( P @ X_12 )
       => ? [N3: nat] :
            ( ~ ( P @ N3 )
            & ( P @ ( suc @ N3 ) ) ) ) ) ).

% exists_least_lemma
thf(fact_664_list__decode_Ocases,axiom,
    ! [X3: nat] :
      ( ( X3 != zero_zero_nat )
     => ~ ! [N3: nat] :
            ( X3
           != ( suc @ N3 ) ) ) ).

% list_decode.cases
thf(fact_665_zero__option__def,axiom,
    ( zero_zero_option_nat
    = ( some_nat @ zero_zero_nat ) ) ).

% zero_option_def
thf(fact_666_zero__prod__def,axiom,
    ( zero_z3979849011205770936at_nat
    = ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) ).

% zero_prod_def
thf(fact_667_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).

% less_numeral_extra(3)
thf(fact_668_timestamp__total,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
      | ( ord_less_eq_nat @ B @ A ) ) ).

% timestamp_total
thf(fact_669_le__numeral__extra_I3_J,axiom,
    ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).

% le_numeral_extra(3)
thf(fact_670_nth__zip,axiom,
    ! [I: nat,Xs: list_h,Ys: list_t] :
      ( ( ord_less_nat @ I @ ( size_size_list_h @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_t @ Ys ) )
       => ( ( nth_Product_prod_h_t @ ( zip_h_t @ Xs @ Ys ) @ I )
          = ( product_Pair_h_t @ ( nth_h @ Xs @ I ) @ ( nth_t @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_671_nth__zip,axiom,
    ! [I: nat,Xs: list_t,Ys: list_o] :
      ( ( ord_less_nat @ I @ ( size_size_list_t @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_o @ Ys ) )
       => ( ( nth_Product_prod_t_o @ ( zip_t_o @ Xs @ Ys ) @ I )
          = ( product_Pair_t_o @ ( nth_t @ Xs @ I ) @ ( nth_o @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_672_nth__zip,axiom,
    ! [I: nat,Xs: list_t,Ys: list_t] :
      ( ( ord_less_nat @ I @ ( size_size_list_t @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_t @ Ys ) )
       => ( ( nth_Product_prod_t_t @ ( zip_t_t @ Xs @ Ys ) @ I )
          = ( product_Pair_t_t @ ( nth_t @ Xs @ I ) @ ( nth_t @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_673_nth__zip,axiom,
    ! [I: nat,Xs: list_iarray_o,Ys: list_t] :
      ( ( ord_less_nat @ I @ ( size_s3243743627286111600rray_o @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_t @ Ys ) )
       => ( ( nth_Pr6673551354596039873ay_o_t @ ( zip_iarray_o_t @ Xs @ Ys ) @ I )
          = ( produc1379057907826328746ay_o_t @ ( nth_iarray_o @ Xs @ I ) @ ( nth_t @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_674_nth__zip,axiom,
    ! [I: nat,Xs: list_t,Ys: list_set_a] :
      ( ( ord_less_nat @ I @ ( size_size_list_t @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_set_a @ Ys ) )
       => ( ( nth_Pr633927046780761275_set_a @ ( zip_t_set_a @ Xs @ Ys ) @ I )
          = ( product_Pair_t_set_a @ ( nth_t @ Xs @ I ) @ ( nth_set_a @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_675_nth__zip,axiom,
    ! [I: nat,Xs: list_t,Ys: list_iarray_o] :
      ( ( ord_less_nat @ I @ ( size_size_list_t @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_s3243743627286111600rray_o @ Ys ) )
       => ( ( nth_Pr3658878690291239899rray_o @ ( zip_t_iarray_o @ Xs @ Ys ) @ I )
          = ( produc4914121904368913988rray_o @ ( nth_t @ Xs @ I ) @ ( nth_iarray_o @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_676_nth__zip,axiom,
    ! [I: nat,Xs: list_nat_o,Ys: list_nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat_o @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
       => ( ( nth_Pr8224016110953687223_o_nat @ ( zip_nat_o_nat @ Xs @ Ys ) @ I )
          = ( produc7277522915581678840_o_nat @ ( nth_nat_o @ Xs @ I ) @ ( nth_nat @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_677_nth__zip,axiom,
    ! [I: nat,Xs: list_iarray_o,Ys: list_iarray_o] :
      ( ( ord_less_nat @ I @ ( size_s3243743627286111600rray_o @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_s3243743627286111600rray_o @ Ys ) )
       => ( ( nth_Pr4184477330912886702rray_o @ ( zip_ia7658775312918225207rray_o @ Xs @ Ys ) @ I )
          = ( produc513795717669825047rray_o @ ( nth_iarray_o @ Xs @ I ) @ ( nth_iarray_o @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_678_nth__zip,axiom,
    ! [I: nat,Xs: list_formula_a_t,Ys: list_t] :
      ( ( ord_less_nat @ I @ ( size_s8846756101701226951la_a_t @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_t @ Ys ) )
       => ( ( nth_Pr8255060927036994268_a_t_t @ ( zip_formula_a_t_t @ Xs @ Ys ) @ I )
          = ( produc8102335687358248149_a_t_t @ ( nth_formula_a_t @ Xs @ I ) @ ( nth_t @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_679_nth__zip,axiom,
    ! [I: nat,Xs: list_t,Ys: list_formula_a_t] :
      ( ( ord_less_nat @ I @ ( size_size_list_t @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_s8846756101701226951la_a_t @ Ys ) )
       => ( ( nth_Pr3292926552878659228la_a_t @ ( zip_t_formula_a_t @ Xs @ Ys ) @ I )
          = ( produc910217554375469149la_a_t @ ( nth_t @ Xs @ I ) @ ( nth_formula_a_t @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_680_pos__sound,axiom,
    ! [A: product_prod_t_set_a,Xs: list_P1095827921439705018_set_a,I: nat] :
      ( ( ( pos_Pr9217298455217426541_set_a @ A @ Xs )
        = ( some_nat @ I ) )
     => ( ( ord_less_nat @ I @ ( size_s5362121731612881446_set_a @ Xs ) )
        & ( ( nth_Pr633927046780761275_set_a @ Xs @ I )
          = A ) ) ) ).

% pos_sound
thf(fact_681_pos__sound,axiom,
    ! [A: vydra_aux_a_t_h,Xs: list_vydra_aux_a_t_h,I: nat] :
      ( ( ( pos_vydra_aux_a_t_h @ A @ Xs )
        = ( some_nat @ I ) )
     => ( ( ord_less_nat @ I @ ( size_s8993897887676352572_a_t_h @ Xs ) )
        & ( ( nth_vydra_aux_a_t_h @ Xs @ I )
          = A ) ) ) ).

% pos_sound
thf(fact_682_pos__sound,axiom,
    ! [A: iarray_o,Xs: list_iarray_o,I: nat] :
      ( ( ( pos_iarray_o @ A @ Xs )
        = ( some_nat @ I ) )
     => ( ( ord_less_nat @ I @ ( size_s3243743627286111600rray_o @ Xs ) )
        & ( ( nth_iarray_o @ Xs @ I )
          = A ) ) ) ).

% pos_sound
thf(fact_683_pos__sound,axiom,
    ! [A: formula_a_t,Xs: list_formula_a_t,I: nat] :
      ( ( ( pos_formula_a_t @ A @ Xs )
        = ( some_nat @ I ) )
     => ( ( ord_less_nat @ I @ ( size_s8846756101701226951la_a_t @ Xs ) )
        & ( ( nth_formula_a_t @ Xs @ I )
          = A ) ) ) ).

% pos_sound
thf(fact_684_pos__sound,axiom,
    ! [A: t,Xs: list_t,I: nat] :
      ( ( ( pos_t @ A @ Xs )
        = ( some_nat @ I ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_t @ Xs ) )
        & ( ( nth_t @ Xs @ I )
          = A ) ) ) ).

% pos_sound
thf(fact_685_listrel__iff__nth,axiom,
    ! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel_nat_nat @ R2 ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [N4: nat] :
            ( ( ord_less_nat @ N4 @ ( size_size_list_nat @ Xs ) )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ N4 ) @ ( nth_nat @ Ys @ N4 ) ) @ R2 ) ) ) ) ).

% listrel_iff_nth
thf(fact_686_listrel__iff__nth,axiom,
    ! [Xs: list_h,Ys: list_t,R2: set_Product_prod_h_t] :
      ( ( member304168301969670172list_t @ ( produc6823159995768562787list_t @ Xs @ Ys ) @ ( listrel_h_t @ R2 ) )
      = ( ( ( size_size_list_h @ Xs )
          = ( size_size_list_t @ Ys ) )
        & ! [N4: nat] :
            ( ( ord_less_nat @ N4 @ ( size_size_list_h @ Xs ) )
           => ( member361086559187601820od_h_t @ ( product_Pair_h_t @ ( nth_h @ Xs @ N4 ) @ ( nth_t @ Ys @ N4 ) ) @ R2 ) ) ) ) ).

% listrel_iff_nth
thf(fact_687_listrel__iff__nth,axiom,
    ! [Xs: list_t,Ys: list_o,R2: set_Product_prod_t_o] :
      ( ( member6485259358765100311list_o @ ( produc1599052279520165342list_o @ Xs @ Ys ) @ ( listrel_t_o @ R2 ) )
      = ( ( ( size_size_list_t @ Xs )
          = ( size_size_list_o @ Ys ) )
        & ! [N4: nat] :
            ( ( ord_less_nat @ N4 @ ( size_size_list_t @ Xs ) )
           => ( member3252940038344094615od_t_o @ ( product_Pair_t_o @ ( nth_t @ Xs @ N4 ) @ ( nth_o @ Ys @ N4 ) ) @ R2 ) ) ) ) ).

% listrel_iff_nth
thf(fact_688_listrel__iff__nth,axiom,
    ! [Xs: list_t,Ys: list_t,R2: set_Product_prod_t_t] :
      ( ( member1598007745118852880list_t @ ( produc8116999438917745495list_t @ Xs @ Ys ) @ ( listrel_t_t @ R2 ) )
      = ( ( ( size_size_list_t @ Xs )
          = ( size_size_list_t @ Ys ) )
        & ! [N4: nat] :
            ( ( ord_less_nat @ N4 @ ( size_size_list_t @ Xs ) )
           => ( member3805107722934592656od_t_t @ ( product_Pair_t_t @ ( nth_t @ Xs @ N4 ) @ ( nth_t @ Ys @ N4 ) ) @ R2 ) ) ) ) ).

% listrel_iff_nth
thf(fact_689_listrel__iff__nth,axiom,
    ! [Xs: list_iarray_o,Ys: list_t,R2: set_Pr6227552764373916058ay_o_t] :
      ( ( member5803030360786749411list_t @ ( produc2633676126830968362list_t @ Xs @ Ys ) @ ( listrel_iarray_o_t @ R2 ) )
      = ( ( ( size_s3243743627286111600rray_o @ Xs )
          = ( size_size_list_t @ Ys ) )
        & ! [N4: nat] :
            ( ( ord_less_nat @ N4 @ ( size_s3243743627286111600rray_o @ Xs ) )
           => ( member7622367568820579683ay_o_t @ ( produc1379057907826328746ay_o_t @ ( nth_iarray_o @ Xs @ N4 ) @ ( nth_t @ Ys @ N4 ) ) @ R2 ) ) ) ) ).

% listrel_iff_nth
thf(fact_690_listrel__iff__nth,axiom,
    ! [Xs: list_t,Ys: list_set_a,R2: set_Pr6994796776334525076_set_a] :
      ( ( member4811356929818619741_set_a @ ( produc7167514779196891684_set_a @ Xs @ Ys ) @ ( listrel_t_set_a @ R2 ) )
      = ( ( ( size_size_list_t @ Xs )
          = ( size_size_list_set_a @ Ys ) )
        & ! [N4: nat] :
            ( ( ord_less_nat @ N4 @ ( size_size_list_t @ Xs ) )
           => ( member2646690471337469789_set_a @ ( product_Pair_t_set_a @ ( nth_t @ Xs @ N4 ) @ ( nth_set_a @ Ys @ N4 ) ) @ R2 ) ) ) ) ).

% listrel_iff_nth
thf(fact_691_listrel__iff__nth,axiom,
    ! [Xs: list_t,Ys: list_iarray_o,R2: set_Pr9127296274279734196rray_o] :
      ( ( member8325202025224052605rray_o @ ( produc4102859334763393092rray_o @ Xs @ Ys ) @ ( listrel_t_iarray_o @ R2 ) )
      = ( ( ( size_size_list_t @ Xs )
          = ( size_s3243743627286111600rray_o @ Ys ) )
        & ! [N4: nat] :
            ( ( ord_less_nat @ N4 @ ( size_size_list_t @ Xs ) )
           => ( member4607694904515779709rray_o @ ( produc4914121904368913988rray_o @ ( nth_t @ Xs @ N4 ) @ ( nth_iarray_o @ Ys @ N4 ) ) @ R2 ) ) ) ) ).

% listrel_iff_nth
thf(fact_692_listrel__iff__nth,axiom,
    ! [Xs: list_nat_o,Ys: list_nat,R2: set_Pr4354077987950702516_o_nat] :
      ( ( member69777164926368939st_nat @ ( produc1237022150899190286st_nat @ Xs @ Ys ) @ ( listrel_nat_o_nat @ R2 ) )
      = ( ( ( size_size_list_nat_o @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [N4: nat] :
            ( ( ord_less_nat @ N4 @ ( size_size_list_nat_o @ Xs ) )
           => ( member2897238310647649173_o_nat @ ( produc7277522915581678840_o_nat @ ( nth_nat_o @ Xs @ N4 ) @ ( nth_nat @ Ys @ N4 ) ) @ R2 ) ) ) ) ).

% listrel_iff_nth
thf(fact_693_listrel__iff__nth,axiom,
    ! [Xs: list_iarray_o,Ys: list_iarray_o,R2: set_Pr3159307878933344263rray_o] :
      ( ( member8606389602990657872rray_o @ ( produc2869105288327869463rray_o @ Xs @ Ys ) @ ( listre8053432375812573809rray_o @ R2 ) )
      = ( ( ( size_s3243743627286111600rray_o @ Xs )
          = ( size_s3243743627286111600rray_o @ Ys ) )
        & ! [N4: nat] :
            ( ( ord_less_nat @ N4 @ ( size_s3243743627286111600rray_o @ Xs ) )
           => ( member3227708950830999120rray_o @ ( produc513795717669825047rray_o @ ( nth_iarray_o @ Xs @ N4 ) @ ( nth_iarray_o @ Ys @ N4 ) ) @ R2 ) ) ) ) ).

% listrel_iff_nth
thf(fact_694_listrel__iff__nth,axiom,
    ! [Xs: list_formula_a_t,Ys: list_t,R2: set_Pr7150229820077291353_a_t_t] :
      ( ( member1539210528405594928list_t @ ( produc3206563442672280907list_t @ Xs @ Ys ) @ ( listre1607657222427758587_a_t_t @ R2 ) )
      = ( ( ( size_s8846756101701226951la_a_t @ Xs )
          = ( size_size_list_t @ Ys ) )
        & ! [N4: nat] :
            ( ( ord_less_nat @ N4 @ ( size_s8846756101701226951la_a_t @ Xs ) )
           => ( member2928283126730956218_a_t_t @ ( produc8102335687358248149_a_t_t @ ( nth_formula_a_t @ Xs @ N4 ) @ ( nth_t @ Ys @ N4 ) ) @ R2 ) ) ) ) ).

% listrel_iff_nth
thf(fact_695_map__fst__zip,axiom,
    ! [Xs: list_t,Ys: list_t] :
      ( ( ( size_size_list_t @ Xs )
        = ( size_size_list_t @ Ys ) )
     => ( ( map_Pr5914473941742019502_t_t_t @ product_fst_t_t @ ( zip_t_t @ Xs @ Ys ) )
        = Xs ) ) ).

% map_fst_zip
thf(fact_696_map__fst__zip,axiom,
    ! [Xs: list_iarray_o,Ys: list_t] :
      ( ( ( size_s3243743627286111600rray_o @ Xs )
        = ( size_size_list_t @ Ys ) )
     => ( ( map_Pr1987961106661228552rray_o @ produc8772693979502063254ay_o_t @ ( zip_iarray_o_t @ Xs @ Ys ) )
        = Xs ) ) ).

% map_fst_zip
thf(fact_697_map__fst__zip,axiom,
    ! [Xs: list_t,Ys: list_iarray_o] :
      ( ( ( size_size_list_t @ Xs )
        = ( size_s3243743627286111600rray_o @ Ys ) )
     => ( ( map_Pr4602466608603904769ay_o_t @ produc3084385939189872688rray_o @ ( zip_t_iarray_o @ Xs @ Ys ) )
        = Xs ) ) ).

% map_fst_zip
thf(fact_698_map__fst__zip,axiom,
    ! [Xs: list_t,Ys: list_set_a] :
      ( ( ( size_size_list_t @ Xs )
        = ( size_size_list_set_a @ Ys ) )
     => ( ( map_Pr119846518281778721et_a_t @ product_fst_t_set_a @ ( zip_t_set_a @ Xs @ Ys ) )
        = Xs ) ) ).

% map_fst_zip
thf(fact_699_map__fst__zip,axiom,
    ! [Xs: list_iarray_o,Ys: list_iarray_o] :
      ( ( ( size_s3243743627286111600rray_o @ Xs )
        = ( size_s3243743627286111600rray_o @ Ys ) )
     => ( ( map_Pr4316704043788064603rray_o @ produc1785037579375327235rray_o @ ( zip_ia7658775312918225207rray_o @ Xs @ Ys ) )
        = Xs ) ) ).

% map_fst_zip
thf(fact_700_map__fst__zip,axiom,
    ! [Xs: list_formula_a_t,Ys: list_t] :
      ( ( ( size_s8846756101701226951la_a_t @ Xs )
        = ( size_size_list_t @ Ys ) )
     => ( ( map_Pr4841008491744877708la_a_t @ produc8468514144079701993_a_t_t @ ( zip_formula_a_t_t @ Xs @ Ys ) )
        = Xs ) ) ).

% map_fst_zip
thf(fact_701_map__fst__zip,axiom,
    ! [Xs: list_t,Ys: list_formula_a_t] :
      ( ( ( size_size_list_t @ Xs )
        = ( size_s8846756101701226951la_a_t @ Ys ) )
     => ( ( map_Pr4224834645403059368_a_t_t @ produc1276396011096922993la_a_t @ ( zip_t_formula_a_t @ Xs @ Ys ) )
        = Xs ) ) ).

% map_fst_zip
thf(fact_702_map__fst__zip,axiom,
    ! [Xs: list_P1095827921439705018_set_a,Ys: list_t] :
      ( ( ( size_s5362121731612881446_set_a @ Xs )
        = ( size_size_list_t @ Ys ) )
     => ( ( map_Pr5476362019341647476_set_a @ produc5103451314196047968et_a_t @ ( zip_Pr6246384304659936660et_a_t @ Xs @ Ys ) )
        = Xs ) ) ).

% map_fst_zip
thf(fact_703_map__fst__zip,axiom,
    ! [Xs: list_vydra_aux_a_t_h,Ys: list_t] :
      ( ( ( size_s8993897887676352572_a_t_h @ Xs )
        = ( size_size_list_t @ Ys ) )
     => ( ( map_Pr480469552278880460_a_t_h @ produc1169508987236047412_t_h_t @ ( zip_vy170765141419497472_t_h_t @ Xs @ Ys ) )
        = Xs ) ) ).

% map_fst_zip
thf(fact_704_map__fst__zip,axiom,
    ! [Xs: list_iarray_o,Ys: list_formula_a_t] :
      ( ( ( size_s3243743627286111600rray_o @ Xs )
        = ( size_s8846756101701226951la_a_t @ Ys ) )
     => ( ( map_Pr7503474447545825512rray_o @ produc5171939130915396830la_a_t @ ( zip_ia3729555385294884522la_a_t @ Xs @ Ys ) )
        = Xs ) ) ).

% map_fst_zip
thf(fact_705_map__snd__zip,axiom,
    ! [Xs: list_t,Ys: list_t] :
      ( ( ( size_size_list_t @ Xs )
        = ( size_size_list_t @ Ys ) )
     => ( ( map_Pr5914473941742019502_t_t_t @ product_snd_t_t @ ( zip_t_t @ Xs @ Ys ) )
        = Ys ) ) ).

% map_snd_zip
thf(fact_706_map__snd__zip,axiom,
    ! [Xs: list_t,Ys: list_o] :
      ( ( ( size_size_list_t @ Xs )
        = ( size_size_list_o @ Ys ) )
     => ( ( map_Pr913679237327294190_t_o_o @ product_snd_t_o @ ( zip_t_o @ Xs @ Ys ) )
        = Ys ) ) ).

% map_snd_zip
thf(fact_707_map__snd__zip,axiom,
    ! [Xs: list_iarray_o,Ys: list_t] :
      ( ( ( size_s3243743627286111600rray_o @ Xs )
        = ( size_size_list_t @ Ys ) )
     => ( ( map_Pr6542469619230011291_o_t_t @ produc6442735128552203224ay_o_t @ ( zip_iarray_o_t @ Xs @ Ys ) )
        = Ys ) ) ).

% map_snd_zip
thf(fact_708_map__snd__zip,axiom,
    ! [Xs: list_t,Ys: list_iarray_o] :
      ( ( ( size_size_list_t @ Xs )
        = ( size_s3243743627286111600rray_o @ Ys ) )
     => ( ( map_Pr2443951592686308718rray_o @ produc754427088240012658rray_o @ ( zip_t_iarray_o @ Xs @ Ys ) )
        = Ys ) ) ).

% map_snd_zip
thf(fact_709_map__snd__zip,axiom,
    ! [Xs: list_iarray_o,Ys: list_iarray_o] :
      ( ( ( size_s3243743627286111600rray_o @ Xs )
        = ( size_s3243743627286111600rray_o @ Ys ) )
     => ( ( map_Pr4316704043788064603rray_o @ produc1607185793151689541rray_o @ ( zip_ia7658775312918225207rray_o @ Xs @ Ys ) )
        = Ys ) ) ).

% map_snd_zip
thf(fact_710_map__snd__zip,axiom,
    ! [Xs: list_formula_a_t,Ys: list_t] :
      ( ( ( size_s8846756101701226951la_a_t @ Xs )
        = ( size_size_list_t @ Ys ) )
     => ( ( map_Pr8067792060164626536_t_t_t @ produc4295462183287866919_a_t_t @ ( zip_formula_a_t_t @ Xs @ Ys ) )
        = Ys ) ) ).

% map_snd_zip
thf(fact_711_map__snd__zip,axiom,
    ! [Xs: list_t,Ys: list_formula_a_t] :
      ( ( ( size_size_list_t @ Xs )
        = ( size_s8846756101701226951la_a_t @ Ys ) )
     => ( ( map_Pr4729283454584889420la_a_t @ produc6326716087159863727la_a_t @ ( zip_t_formula_a_t @ Xs @ Ys ) )
        = Ys ) ) ).

% map_snd_zip
thf(fact_712_map__snd__zip,axiom,
    ! [Xs: list_P1095827921439705018_set_a,Ys: list_t] :
      ( ( ( size_s5362121731612881446_set_a @ Xs )
        = ( size_size_list_t @ Ys ) )
     => ( ( map_Pr4846382328090868817_a_t_t @ produc3470916645853531554et_a_t @ ( zip_Pr6246384304659936660et_a_t @ Xs @ Ys ) )
        = Ys ) ) ).

% map_snd_zip
thf(fact_713_map__snd__zip,axiom,
    ! [Xs: list_vydra_aux_a_t_h,Ys: list_t] :
      ( ( ( size_s8993897887676352572_a_t_h @ Xs )
        = ( size_size_list_t @ Ys ) )
     => ( ( map_Pr6132075605078445789_h_t_t @ produc1722491146242769010_t_h_t @ ( zip_vy170765141419497472_t_h_t @ Xs @ Ys ) )
        = Ys ) ) ).

% map_snd_zip
thf(fact_714_map__snd__zip,axiom,
    ! [Xs: list_iarray_o,Ys: list_formula_a_t] :
      ( ( ( size_s3243743627286111600rray_o @ Xs )
        = ( size_s8846756101701226951la_a_t @ Ys ) )
     => ( ( map_Pr4281774288685423801la_a_t @ produc3853001391928378652la_a_t @ ( zip_ia3729555385294884522la_a_t @ Xs @ Ys ) )
        = Ys ) ) ).

% map_snd_zip
thf(fact_715_listrel__eq__len,axiom,
    ! [Xs: list_t,Ys: list_t,R2: set_Product_prod_t_t] :
      ( ( member1598007745118852880list_t @ ( produc8116999438917745495list_t @ Xs @ Ys ) @ ( listrel_t_t @ R2 ) )
     => ( ( size_size_list_t @ Xs )
        = ( size_size_list_t @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_716_listrel__eq__len,axiom,
    ! [Xs: list_iarray_o,Ys: list_t,R2: set_Pr6227552764373916058ay_o_t] :
      ( ( member5803030360786749411list_t @ ( produc2633676126830968362list_t @ Xs @ Ys ) @ ( listrel_iarray_o_t @ R2 ) )
     => ( ( size_s3243743627286111600rray_o @ Xs )
        = ( size_size_list_t @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_717_listrel__eq__len,axiom,
    ! [Xs: list_t,Ys: list_iarray_o,R2: set_Pr9127296274279734196rray_o] :
      ( ( member8325202025224052605rray_o @ ( produc4102859334763393092rray_o @ Xs @ Ys ) @ ( listrel_t_iarray_o @ R2 ) )
     => ( ( size_size_list_t @ Xs )
        = ( size_s3243743627286111600rray_o @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_718_listrel__eq__len,axiom,
    ! [Xs: list_iarray_o,Ys: list_iarray_o,R2: set_Pr3159307878933344263rray_o] :
      ( ( member8606389602990657872rray_o @ ( produc2869105288327869463rray_o @ Xs @ Ys ) @ ( listre8053432375812573809rray_o @ R2 ) )
     => ( ( size_s3243743627286111600rray_o @ Xs )
        = ( size_s3243743627286111600rray_o @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_719_listrel__eq__len,axiom,
    ! [Xs: list_formula_a_t,Ys: list_t,R2: set_Pr7150229820077291353_a_t_t] :
      ( ( member1539210528405594928list_t @ ( produc3206563442672280907list_t @ Xs @ Ys ) @ ( listre1607657222427758587_a_t_t @ R2 ) )
     => ( ( size_s8846756101701226951la_a_t @ Xs )
        = ( size_size_list_t @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_720_listrel__eq__len,axiom,
    ! [Xs: list_t,Ys: list_formula_a_t,R2: set_Pr2339794881257429785la_a_t] :
      ( ( member4477015913101731600la_a_t @ ( produc1741912214820895219la_a_t @ Xs @ Ys ) @ ( listre3638911126299755395la_a_t @ R2 ) )
     => ( ( size_size_list_t @ Xs )
        = ( size_s8846756101701226951la_a_t @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_721_listrel__eq__len,axiom,
    ! [Xs: list_P1095827921439705018_set_a,Ys: list_t,R2: set_Pr9005033546302094948et_a_t] :
      ( ( member6641401218325034925list_t @ ( produc3836631461499960820list_t @ Xs @ Ys ) @ ( listre6033618074479554766et_a_t @ R2 ) )
     => ( ( size_s5362121731612881446_set_a @ Xs )
        = ( size_size_list_t @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_722_listrel__eq__len,axiom,
    ! [Xs: list_vydra_aux_a_t_h,Ys: list_t,R2: set_Pr5515790935469808676_t_h_t] :
      ( ( member8293557410520320635list_t @ ( produc1516534556286100630list_t @ Xs @ Ys ) @ ( listre3777388779728501318_t_h_t @ R2 ) )
     => ( ( size_s8993897887676352572_a_t_h @ Xs )
        = ( size_size_list_t @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_723_listrel__eq__len,axiom,
    ! [Xs: list_iarray_o,Ys: list_formula_a_t,R2: set_Pr1869755387858020358la_a_t] :
      ( ( member1806728011820988029la_a_t @ ( produc1118928471642896224la_a_t @ Xs @ Ys ) @ ( listre5068381310365788912la_a_t @ R2 ) )
     => ( ( size_s3243743627286111600rray_o @ Xs )
        = ( size_s8846756101701226951la_a_t @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_724_listrel__eq__len,axiom,
    ! [Xs: list_formula_a_t,Ys: list_iarray_o,R2: set_Pr425776304706548422rray_o] :
      ( ( member5596048621676874653rray_o @ ( produc8257717237541426744rray_o @ Xs @ Ys ) @ ( listre4700735018858725480rray_o @ R2 ) )
     => ( ( size_s8846756101701226951la_a_t @ Xs )
        = ( size_s3243743627286111600rray_o @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_725_zip__map__fst__snd,axiom,
    ! [Zs: list_P1095827921439705018_set_a] :
      ( ( zip_t_set_a @ ( map_Pr119846518281778721et_a_t @ product_fst_t_set_a @ Zs ) @ ( map_Pr7710007540447715182_set_a @ product_snd_t_set_a @ Zs ) )
      = Zs ) ).

% zip_map_fst_snd
thf(fact_726_zip__map__fst__snd,axiom,
    ! [Zs: list_P5575445741472356340od_t_o] :
      ( ( zip_t_o @ ( map_Pr4056436085500758887_t_o_t @ product_fst_t_o @ Zs ) @ ( map_Pr913679237327294190_t_o_o @ product_snd_t_o @ Zs ) )
      = Zs ) ).

% zip_map_fst_snd
thf(fact_727_zip__eq__conv,axiom,
    ! [Xs: list_t,Ys: list_t,Zs: list_P9116000306261664621od_t_t] :
      ( ( ( size_size_list_t @ Xs )
        = ( size_size_list_t @ Ys ) )
     => ( ( ( zip_t_t @ Xs @ Ys )
          = Zs )
        = ( ( ( map_Pr5914473941742019502_t_t_t @ product_fst_t_t @ Zs )
            = Xs )
          & ( ( map_Pr5914473941742019502_t_t_t @ product_snd_t_t @ Zs )
            = Ys ) ) ) ) ).

% zip_eq_conv
thf(fact_728_zip__eq__conv,axiom,
    ! [Xs: list_t,Ys: list_o,Zs: list_P5575445741472356340od_t_o] :
      ( ( ( size_size_list_t @ Xs )
        = ( size_size_list_o @ Ys ) )
     => ( ( ( zip_t_o @ Xs @ Ys )
          = Zs )
        = ( ( ( map_Pr4056436085500758887_t_o_t @ product_fst_t_o @ Zs )
            = Xs )
          & ( ( map_Pr913679237327294190_t_o_o @ product_snd_t_o @ Zs )
            = Ys ) ) ) ) ).

% zip_eq_conv
thf(fact_729_zip__eq__conv,axiom,
    ! [Xs: list_iarray_o,Ys: list_t,Zs: list_P4403923123588184256ay_o_t] :
      ( ( ( size_s3243743627286111600rray_o @ Xs )
        = ( size_size_list_t @ Ys ) )
     => ( ( ( zip_iarray_o_t @ Xs @ Ys )
          = Zs )
        = ( ( ( map_Pr1987961106661228552rray_o @ produc8772693979502063254ay_o_t @ Zs )
            = Xs )
          & ( ( map_Pr6542469619230011291_o_t_t @ produc6442735128552203224ay_o_t @ Zs )
            = Ys ) ) ) ) ).

% zip_eq_conv
thf(fact_730_zip__eq__conv,axiom,
    ! [Xs: list_t,Ys: list_iarray_o,Zs: list_P7303666633494002394rray_o] :
      ( ( ( size_size_list_t @ Xs )
        = ( size_s3243743627286111600rray_o @ Ys ) )
     => ( ( ( zip_t_iarray_o @ Xs @ Ys )
          = Zs )
        = ( ( ( map_Pr4602466608603904769ay_o_t @ produc3084385939189872688rray_o @ Zs )
            = Xs )
          & ( ( map_Pr2443951592686308718rray_o @ produc754427088240012658rray_o @ Zs )
            = Ys ) ) ) ) ).

% zip_eq_conv
thf(fact_731_zip__eq__conv,axiom,
    ! [Xs: list_t,Ys: list_set_a,Zs: list_P1095827921439705018_set_a] :
      ( ( ( size_size_list_t @ Xs )
        = ( size_size_list_set_a @ Ys ) )
     => ( ( ( zip_t_set_a @ Xs @ Ys )
          = Zs )
        = ( ( ( map_Pr119846518281778721et_a_t @ product_fst_t_set_a @ Zs )
            = Xs )
          & ( ( map_Pr7710007540447715182_set_a @ product_snd_t_set_a @ Zs )
            = Ys ) ) ) ) ).

% zip_eq_conv
thf(fact_732_zip__eq__conv,axiom,
    ! [Xs: list_iarray_o,Ys: list_iarray_o,Zs: list_P6105256026958827821rray_o] :
      ( ( ( size_s3243743627286111600rray_o @ Xs )
        = ( size_s3243743627286111600rray_o @ Ys ) )
     => ( ( ( zip_ia7658775312918225207rray_o @ Xs @ Ys )
          = Zs )
        = ( ( ( map_Pr4316704043788064603rray_o @ produc1785037579375327235rray_o @ Zs )
            = Xs )
          & ( ( map_Pr4316704043788064603rray_o @ produc1607185793151689541rray_o @ Zs )
            = Ys ) ) ) ) ).

% zip_eq_conv
thf(fact_733_zip__eq__conv,axiom,
    ! [Xs: list_formula_a_t,Ys: list_t,Zs: list_P275148778380163763_a_t_t] :
      ( ( ( size_s8846756101701226951la_a_t @ Xs )
        = ( size_size_list_t @ Ys ) )
     => ( ( ( zip_formula_a_t_t @ Xs @ Ys )
          = Zs )
        = ( ( ( map_Pr4841008491744877708la_a_t @ produc8468514144079701993_a_t_t @ Zs )
            = Xs )
          & ( ( map_Pr8067792060164626536_t_t_t @ produc4295462183287866919_a_t_t @ Zs )
            = Ys ) ) ) ) ).

% zip_eq_conv
thf(fact_734_zip__eq__conv,axiom,
    ! [Xs: list_t,Ys: list_formula_a_t,Zs: list_P4688085876415078003la_a_t] :
      ( ( ( size_size_list_t @ Xs )
        = ( size_s8846756101701226951la_a_t @ Ys ) )
     => ( ( ( zip_t_formula_a_t @ Xs @ Ys )
          = Zs )
        = ( ( ( map_Pr4224834645403059368_a_t_t @ produc1276396011096922993la_a_t @ Zs )
            = Xs )
          & ( ( map_Pr4729283454584889420la_a_t @ produc6326716087159863727la_a_t @ Zs )
            = Ys ) ) ) ) ).

% zip_eq_conv
thf(fact_735_zip__eq__conv,axiom,
    ! [Xs: list_P1095827921439705018_set_a,Ys: list_t,Zs: list_P6689957890503519626et_a_t] :
      ( ( ( size_s5362121731612881446_set_a @ Xs )
        = ( size_size_list_t @ Ys ) )
     => ( ( ( zip_Pr6246384304659936660et_a_t @ Xs @ Ys )
          = Zs )
        = ( ( ( map_Pr5476362019341647476_set_a @ produc5103451314196047968et_a_t @ Zs )
            = Xs )
          & ( ( map_Pr4846382328090868817_a_t_t @ produc3470916645853531554et_a_t @ Zs )
            = Ys ) ) ) ) ).

% zip_eq_conv
thf(fact_736_zip__eq__conv,axiom,
    ! [Xs: list_vydra_aux_a_t_h,Ys: list_t,Zs: list_P4939240235957729150_t_h_t] :
      ( ( ( size_s8993897887676352572_a_t_h @ Xs )
        = ( size_size_list_t @ Ys ) )
     => ( ( ( zip_vy170765141419497472_t_h_t @ Xs @ Ys )
          = Zs )
        = ( ( ( map_Pr480469552278880460_a_t_h @ produc1169508987236047412_t_h_t @ Zs )
            = Xs )
          & ( ( map_Pr6132075605078445789_h_t_t @ produc1722491146242769010_t_h_t @ Zs )
            = Ys ) ) ) ) ).

% zip_eq_conv
thf(fact_737_last__before_Osimps_I2_J,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( ( P @ N )
       => ( ( vYDRA_last_before @ P @ ( suc @ N ) )
          = ( some_nat @ N ) ) )
      & ( ~ ( P @ N )
       => ( ( vYDRA_last_before @ P @ ( suc @ N ) )
          = ( vYDRA_last_before @ P @ N ) ) ) ) ).

% last_before.simps(2)
thf(fact_738_in__set__zip,axiom,
    ! [P2: product_prod_nat_nat,Xs: list_nat,Ys: list_nat] :
      ( ( member8440522571783428010at_nat @ P2 @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) )
      = ( ? [N4: nat] :
            ( ( ( nth_nat @ Xs @ N4 )
              = ( product_fst_nat_nat @ P2 ) )
            & ( ( nth_nat @ Ys @ N4 )
              = ( product_snd_nat_nat @ P2 ) )
            & ( ord_less_nat @ N4 @ ( size_size_list_nat @ Xs ) )
            & ( ord_less_nat @ N4 @ ( size_size_list_nat @ Ys ) ) ) ) ) ).

% in_set_zip
thf(fact_739_in__set__zip,axiom,
    ! [P2: product_prod_t_t,Xs: list_t,Ys: list_t] :
      ( ( member3805107722934592656od_t_t @ P2 @ ( set_Product_prod_t_t2 @ ( zip_t_t @ Xs @ Ys ) ) )
      = ( ? [N4: nat] :
            ( ( ( nth_t @ Xs @ N4 )
              = ( product_fst_t_t @ P2 ) )
            & ( ( nth_t @ Ys @ N4 )
              = ( product_snd_t_t @ P2 ) )
            & ( ord_less_nat @ N4 @ ( size_size_list_t @ Xs ) )
            & ( ord_less_nat @ N4 @ ( size_size_list_t @ Ys ) ) ) ) ) ).

% in_set_zip
thf(fact_740_in__set__zip,axiom,
    ! [P2: product_prod_t_o,Xs: list_t,Ys: list_o] :
      ( ( member3252940038344094615od_t_o @ P2 @ ( set_Product_prod_t_o2 @ ( zip_t_o @ Xs @ Ys ) ) )
      = ( ? [N4: nat] :
            ( ( ( nth_t @ Xs @ N4 )
              = ( product_fst_t_o @ P2 ) )
            & ( ( nth_o @ Ys @ N4 )
              = ( product_snd_t_o @ P2 ) )
            & ( ord_less_nat @ N4 @ ( size_size_list_t @ Xs ) )
            & ( ord_less_nat @ N4 @ ( size_size_list_o @ Ys ) ) ) ) ) ).

% in_set_zip
thf(fact_741_in__set__zip,axiom,
    ! [P2: produc6664940298603869242ay_o_t,Xs: list_iarray_o,Ys: list_t] :
      ( ( member7622367568820579683ay_o_t @ P2 @ ( set_Pr383680581274750927ay_o_t @ ( zip_iarray_o_t @ Xs @ Ys ) ) )
      = ( ? [N4: nat] :
            ( ( ( nth_iarray_o @ Xs @ N4 )
              = ( produc8772693979502063254ay_o_t @ P2 ) )
            & ( ( nth_t @ Ys @ N4 )
              = ( produc6442735128552203224ay_o_t @ P2 ) )
            & ( ord_less_nat @ N4 @ ( size_s3243743627286111600rray_o @ Xs ) )
            & ( ord_less_nat @ N4 @ ( size_size_list_t @ Ys ) ) ) ) ) ).

% in_set_zip
thf(fact_742_in__set__zip,axiom,
    ! [P2: produc3650267634299069268rray_o,Xs: list_t,Ys: list_iarray_o] :
      ( ( member4607694904515779709rray_o @ P2 @ ( set_Pr6592379953824726761rray_o @ ( zip_t_iarray_o @ Xs @ Ys ) ) )
      = ( ? [N4: nat] :
            ( ( ( nth_t @ Xs @ N4 )
              = ( produc3084385939189872688rray_o @ P2 ) )
            & ( ( nth_iarray_o @ Ys @ N4 )
              = ( produc754427088240012658rray_o @ P2 ) )
            & ( ord_less_nat @ N4 @ ( size_size_list_t @ Xs ) )
            & ( ord_less_nat @ N4 @ ( size_s3243743627286111600rray_o @ Ys ) ) ) ) ) ).

% in_set_zip
thf(fact_743_in__set__zip,axiom,
    ! [P2: product_prod_t_set_a,Xs: list_t,Ys: list_set_a] :
      ( ( member2646690471337469789_set_a @ P2 @ ( set_Pr15456023412717001_set_a @ ( zip_t_set_a @ Xs @ Ys ) ) )
      = ( ? [N4: nat] :
            ( ( ( nth_t @ Xs @ N4 )
              = ( product_fst_t_set_a @ P2 ) )
            & ( ( nth_set_a @ Ys @ N4 )
              = ( product_snd_t_set_a @ P2 ) )
            & ( ord_less_nat @ N4 @ ( size_size_list_t @ Xs ) )
            & ( ord_less_nat @ N4 @ ( size_size_list_set_a @ Ys ) ) ) ) ) ).

% in_set_zip
thf(fact_744_in__set__zip,axiom,
    ! [P2: produc3866506239541402151rray_o,Xs: list_iarray_o,Ys: list_iarray_o] :
      ( ( member3227708950830999120rray_o @ P2 @ ( set_Pr1876133337737620668rray_o @ ( zip_ia7658775312918225207rray_o @ Xs @ Ys ) ) )
      = ( ? [N4: nat] :
            ( ( ( nth_iarray_o @ Xs @ N4 )
              = ( produc1785037579375327235rray_o @ P2 ) )
            & ( ( nth_iarray_o @ Ys @ N4 )
              = ( produc1607185793151689541rray_o @ P2 ) )
            & ( ord_less_nat @ N4 @ ( size_s3243743627286111600rray_o @ Xs ) )
            & ( ord_less_nat @ N4 @ ( size_s3243743627286111600rray_o @ Ys ) ) ) ) ) ).

% in_set_zip
thf(fact_745_in__set__zip,axiom,
    ! [P2: produc3105837220240711459_a_t_t,Xs: list_formula_a_t,Ys: list_t] :
      ( ( member2928283126730956218_a_t_t @ P2 @ ( set_Pr6786568251656468558_a_t_t @ ( zip_formula_a_t_t @ Xs @ Ys ) ) )
      = ( ? [N4: nat] :
            ( ( ( nth_formula_a_t @ Xs @ N4 )
              = ( produc8468514144079701993_a_t_t @ P2 ) )
            & ( ( nth_t @ Ys @ N4 )
              = ( produc4295462183287866919_a_t_t @ P2 ) )
            & ( ord_less_nat @ N4 @ ( size_s8846756101701226951la_a_t @ Xs ) )
            & ( ord_less_nat @ N4 @ ( size_size_list_t @ Ys ) ) ) ) ) ).

% in_set_zip
thf(fact_746_in__set__zip,axiom,
    ! [P2: produc7367074882937152227la_a_t,Xs: list_t,Ys: list_formula_a_t] :
      ( ( member7189520789427396986la_a_t @ P2 @ ( set_Pr1824433877498133518la_a_t @ ( zip_t_formula_a_t @ Xs @ Ys ) ) )
      = ( ? [N4: nat] :
            ( ( ( nth_t @ Xs @ N4 )
              = ( produc1276396011096922993la_a_t @ P2 ) )
            & ( ( nth_formula_a_t @ Ys @ N4 )
              = ( produc6326716087159863727la_a_t @ P2 ) )
            & ( ord_less_nat @ N4 @ ( size_size_list_t @ Xs ) )
            & ( ord_less_nat @ N4 @ ( size_s8846756101701226951la_a_t @ Ys ) ) ) ) ) ).

% in_set_zip
thf(fact_747_in__set__zip,axiom,
    ! [P2: produc267757839698675204et_a_t,Xs: list_P1095827921439705018_set_a,Ys: list_t] :
      ( ( member8171993688707725101et_a_t @ P2 @ ( set_Pr8523454786824026521et_a_t @ ( zip_Pr6246384304659936660et_a_t @ Xs @ Ys ) ) )
      = ( ? [N4: nat] :
            ( ( ( nth_Pr633927046780761275_set_a @ Xs @ N4 )
              = ( produc5103451314196047968et_a_t @ P2 ) )
            & ( ( nth_t @ Ys @ N4 )
              = ( produc3470916645853531554et_a_t @ P2 ) )
            & ( ord_less_nat @ N4 @ ( size_s5362121731612881446_set_a @ Xs ) )
            & ( ord_less_nat @ N4 @ ( size_size_list_t @ Ys ) ) ) ) ) ).

% in_set_zip
thf(fact_748_listrelp__listrel__eq,axiom,
    ! [R2: set_Pr1261947904930325089at_nat] :
      ( ( listrelp_nat_nat
        @ ^ [X5: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y5 ) @ R2 ) )
      = ( ^ [X5: list_nat,Y5: list_nat] : ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X5 @ Y5 ) @ ( listrel_nat_nat @ R2 ) ) ) ) ).

% listrelp_listrel_eq
thf(fact_749_listrelp__listrel__eq,axiom,
    ! [R2: set_Pr4354077987950702516_o_nat] :
      ( ( listrelp_nat_o_nat
        @ ^ [X5: nat > $o,Y5: nat] : ( member2897238310647649173_o_nat @ ( produc7277522915581678840_o_nat @ X5 @ Y5 ) @ R2 ) )
      = ( ^ [X5: list_nat_o,Y5: list_nat] : ( member69777164926368939st_nat @ ( produc1237022150899190286st_nat @ X5 @ Y5 ) @ ( listrel_nat_o_nat @ R2 ) ) ) ) ).

% listrelp_listrel_eq
thf(fact_750_listrelp__listrel__eq,axiom,
    ! [R2: set_Product_prod_t_o] :
      ( ( listrelp_t_o
        @ ^ [X5: t,Y5: $o] : ( member3252940038344094615od_t_o @ ( product_Pair_t_o @ X5 @ Y5 ) @ R2 ) )
      = ( ^ [X5: list_t,Y5: list_o] : ( member6485259358765100311list_o @ ( produc1599052279520165342list_o @ X5 @ Y5 ) @ ( listrel_t_o @ R2 ) ) ) ) ).

% listrelp_listrel_eq
thf(fact_751_listrelp__listrel__eq,axiom,
    ! [R2: set_Pr6994796776334525076_set_a] :
      ( ( listrelp_t_set_a
        @ ^ [X5: t,Y5: set_a] : ( member2646690471337469789_set_a @ ( product_Pair_t_set_a @ X5 @ Y5 ) @ R2 ) )
      = ( ^ [X5: list_t,Y5: list_set_a] : ( member4811356929818619741_set_a @ ( produc7167514779196891684_set_a @ X5 @ Y5 ) @ ( listrel_t_set_a @ R2 ) ) ) ) ).

% listrelp_listrel_eq
thf(fact_752_listrelp__listrel__eq,axiom,
    ! [R2: set_Pr4350348695907903606_set_a] :
      ( ( listre1962977712775025582_set_a
        @ ^ [X5: h,Y5: product_prod_t_set_a] : ( member6770953551192623679_set_a @ ( produc3432067189782488838_set_a @ X5 @ Y5 ) @ R2 ) )
      = ( ^ [X5: list_h,Y5: list_P1095827921439705018_set_a] : ( member7622607899657334335_set_a @ ( produc3834712951359203846_set_a @ X5 @ Y5 ) @ ( listre7822868916465105248_set_a @ R2 ) ) ) ) ).

% listrelp_listrel_eq
thf(fact_753_listrelp__listrel__eq,axiom,
    ! [R2: set_Product_prod_h_t] :
      ( ( listrelp_h_t
        @ ^ [X5: h,Y5: t] : ( member361086559187601820od_h_t @ ( product_Pair_h_t @ X5 @ Y5 ) @ R2 ) )
      = ( ^ [X5: list_h,Y5: list_t] : ( member304168301969670172list_t @ ( produc6823159995768562787list_t @ X5 @ Y5 ) @ ( listrel_h_t @ R2 ) ) ) ) ).

% listrelp_listrel_eq
thf(fact_754_mmap__map__fst,axiom,
    ! [F: t > set_a > set_a,M: list_P1095827921439705018_set_a] :
      ( ( map_Pr119846518281778721et_a_t @ product_fst_t_set_a @ ( mmap_m4827790735783412863_set_a @ F @ M ) )
      = ( map_Pr119846518281778721et_a_t @ product_fst_t_set_a @ M ) ) ).

% mmap_map_fst
thf(fact_755_map__eq__conv,axiom,
    ! [F: product_prod_t_set_a > t,Xs: list_P1095827921439705018_set_a,G: product_prod_t_set_a > t] :
      ( ( ( map_Pr119846518281778721et_a_t @ F @ Xs )
        = ( map_Pr119846518281778721et_a_t @ G @ Xs ) )
      = ( ! [X5: product_prod_t_set_a] :
            ( ( member2646690471337469789_set_a @ X5 @ ( set_Pr15456023412717001_set_a @ Xs ) )
           => ( ( F @ X5 )
              = ( G @ X5 ) ) ) ) ) ).

% map_eq_conv
thf(fact_756_map__eq__conv,axiom,
    ! [F: formula_a_t > $o,Xs: list_formula_a_t,G: formula_a_t > $o] :
      ( ( ( map_formula_a_t_o @ F @ Xs )
        = ( map_formula_a_t_o @ G @ Xs ) )
      = ( ! [X5: formula_a_t] :
            ( ( member_formula_a_t @ X5 @ ( set_formula_a_t2 @ Xs ) )
           => ( ( F @ X5 )
              = ( G @ X5 ) ) ) ) ) ).

% map_eq_conv
thf(fact_757_map__eq__conv,axiom,
    ! [F: formula_a_t > vydra_aux_a_t_h,Xs: list_formula_a_t,G: formula_a_t > vydra_aux_a_t_h] :
      ( ( ( map_fo5746072578778194433_a_t_h @ F @ Xs )
        = ( map_fo5746072578778194433_a_t_h @ G @ Xs ) )
      = ( ! [X5: formula_a_t] :
            ( ( member_formula_a_t @ X5 @ ( set_formula_a_t2 @ Xs ) )
           => ( ( F @ X5 )
              = ( G @ X5 ) ) ) ) ) ).

% map_eq_conv
thf(fact_758_zip__same,axiom,
    ! [A: a,B: a,Xs: list_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( set_Product_prod_a_a2 @ ( zip_a_a @ Xs @ Xs ) ) )
      = ( ( member_a @ A @ ( set_a2 @ Xs ) )
        & ( A = B ) ) ) ).

% zip_same
thf(fact_759_zip__same,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( set_Pr5518436109238095868at_nat @ ( zip_Pr4664179122662387191at_nat @ Xs @ Xs ) ) )
      = ( ( member8440522571783428010at_nat @ A @ ( set_Pr5648618587558075414at_nat @ Xs ) )
        & ( A = B ) ) ) ).

% zip_same
thf(fact_760_zip__same,axiom,
    ! [A: nat,B: nat,Xs: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Xs ) ) )
      = ( ( member_nat @ A @ ( set_nat2 @ Xs ) )
        & ( A = B ) ) ) ).

% zip_same
thf(fact_761_zip__same,axiom,
    ! [A: formula_a_t,B: formula_a_t,Xs: list_formula_a_t] :
      ( ( member9065953759106212816la_a_t @ ( produc8281305675666354143la_a_t @ A @ B ) @ ( set_Pr507822682862612540la_a_t @ ( zip_fo8638544115243153151la_a_t @ Xs @ Xs ) ) )
      = ( ( member_formula_a_t @ A @ ( set_formula_a_t2 @ Xs ) )
        & ( A = B ) ) ) ).

% zip_same
thf(fact_762_in__set__zipE,axiom,
    ! [X3: a,Y: a,Xs: list_a,Ys: list_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ ( set_Product_prod_a_a2 @ ( zip_a_a @ Xs @ Ys ) ) )
     => ~ ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
         => ~ ( member_a @ Y @ ( set_a2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_763_in__set__zipE,axiom,
    ! [X3: a,Y: nat,Xs: list_a,Ys: list_nat] :
      ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X3 @ Y ) @ ( set_Pr924983374503034536_a_nat @ ( zip_a_nat @ Xs @ Ys ) ) )
     => ~ ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
         => ~ ( member_nat @ Y @ ( set_nat2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_764_in__set__zipE,axiom,
    ! [X3: nat,Y: a,Xs: list_nat,Ys: list_a] :
      ( ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ X3 @ Y ) @ ( set_Pr4163146838226711502_nat_a @ ( zip_nat_a @ Xs @ Ys ) ) )
     => ~ ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
         => ~ ( member_a @ Y @ ( set_a2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_765_in__set__zipE,axiom,
    ! [X3: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) )
     => ~ ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
         => ~ ( member_nat @ Y @ ( set_nat2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_766_in__set__zipE,axiom,
    ! [X3: t,Y: $o,Xs: list_t,Ys: list_o] :
      ( ( member3252940038344094615od_t_o @ ( product_Pair_t_o @ X3 @ Y ) @ ( set_Product_prod_t_o2 @ ( zip_t_o @ Xs @ Ys ) ) )
     => ~ ( ( member_t @ X3 @ ( set_t2 @ Xs ) )
         => ~ ( member_o @ Y @ ( set_o2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_767_in__set__zipE,axiom,
    ! [X3: h,Y: t,Xs: list_h,Ys: list_t] :
      ( ( member361086559187601820od_h_t @ ( product_Pair_h_t @ X3 @ Y ) @ ( set_Product_prod_h_t2 @ ( zip_h_t @ Xs @ Ys ) ) )
     => ~ ( ( member_h @ X3 @ ( set_h2 @ Xs ) )
         => ~ ( member_t @ Y @ ( set_t2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_768_in__set__zipE,axiom,
    ! [X3: t,Y: set_a,Xs: list_t,Ys: list_set_a] :
      ( ( member2646690471337469789_set_a @ ( product_Pair_t_set_a @ X3 @ Y ) @ ( set_Pr15456023412717001_set_a @ ( zip_t_set_a @ Xs @ Ys ) ) )
     => ~ ( ( member_t @ X3 @ ( set_t2 @ Xs ) )
         => ~ ( member_set_a @ Y @ ( set_set_a2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_769_in__set__zipE,axiom,
    ! [X3: a,Y: product_prod_nat_nat,Xs: list_a,Ys: list_P6011104703257516679at_nat] :
      ( ( member4275837660285810771at_nat @ ( produc164408500721353882at_nat @ X3 @ Y ) @ ( set_Pr777002202233794751at_nat @ ( zip_a_4627834845020498874at_nat @ Xs @ Ys ) ) )
     => ~ ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
         => ~ ( member8440522571783428010at_nat @ Y @ ( set_Pr5648618587558075414at_nat @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_770_in__set__zipE,axiom,
    ! [X3: product_prod_nat_nat,Y: a,Xs: list_P6011104703257516679at_nat,Ys: list_a] :
      ( ( member8515758769482883853_nat_a @ ( produc712563270363020244_nat_a @ X3 @ Y ) @ ( set_Pr5016923311430867833_nat_a @ ( zip_Pr5175989614662165236_nat_a @ Xs @ Ys ) ) )
     => ~ ( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
         => ~ ( member_a @ Y @ ( set_a2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_771_in__set__zipE,axiom,
    ! [X3: product_prod_nat_nat,Y: nat,Xs: list_P6011104703257516679at_nat,Ys: list_nat] :
      ( ( member3348759134392003351at_nat @ ( produc6350711070570205562at_nat @ X3 @ Y ) @ ( set_Pr7836445846575771563at_nat @ ( zip_Pr6869450617852699226at_nat @ Xs @ Ys ) ) )
     => ~ ( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
         => ~ ( member_nat @ Y @ ( set_nat2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_772_set__zip__leftD,axiom,
    ! [X3: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) )
     => ( member_nat @ X3 @ ( set_nat2 @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_773_set__zip__leftD,axiom,
    ! [X3: nat > $o,Y: nat,Xs: list_nat_o,Ys: list_nat] :
      ( ( member2897238310647649173_o_nat @ ( produc7277522915581678840_o_nat @ X3 @ Y ) @ ( set_Pr6755523435573161513_o_nat @ ( zip_nat_o_nat @ Xs @ Ys ) ) )
     => ( member_nat_o @ X3 @ ( set_nat_o2 @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_774_set__zip__leftD,axiom,
    ! [X3: t,Y: $o,Xs: list_t,Ys: list_o] :
      ( ( member3252940038344094615od_t_o @ ( product_Pair_t_o @ X3 @ Y ) @ ( set_Product_prod_t_o2 @ ( zip_t_o @ Xs @ Ys ) ) )
     => ( member_t @ X3 @ ( set_t2 @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_775_set__zip__leftD,axiom,
    ! [X3: t,Y: set_a,Xs: list_t,Ys: list_set_a] :
      ( ( member2646690471337469789_set_a @ ( product_Pair_t_set_a @ X3 @ Y ) @ ( set_Pr15456023412717001_set_a @ ( zip_t_set_a @ Xs @ Ys ) ) )
     => ( member_t @ X3 @ ( set_t2 @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_776_set__zip__leftD,axiom,
    ! [X3: h,Y: product_prod_t_set_a,Xs: list_h,Ys: list_P1095827921439705018_set_a] :
      ( ( member6770953551192623679_set_a @ ( produc3432067189782488838_set_a @ X3 @ Y ) @ ( set_Pr7122414649308925099_set_a @ ( zip_h_8035635146645487142_set_a @ Xs @ Ys ) ) )
     => ( member_h @ X3 @ ( set_h2 @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_777_set__zip__leftD,axiom,
    ! [X3: h,Y: t,Xs: list_h,Ys: list_t] :
      ( ( member361086559187601820od_h_t @ ( product_Pair_h_t @ X3 @ Y ) @ ( set_Product_prod_h_t2 @ ( zip_h_t @ Xs @ Ys ) ) )
     => ( member_h @ X3 @ ( set_h2 @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_778_set__zip__rightD,axiom,
    ! [X3: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) )
     => ( member_nat @ Y @ ( set_nat2 @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_779_set__zip__rightD,axiom,
    ! [X3: nat > $o,Y: nat,Xs: list_nat_o,Ys: list_nat] :
      ( ( member2897238310647649173_o_nat @ ( produc7277522915581678840_o_nat @ X3 @ Y ) @ ( set_Pr6755523435573161513_o_nat @ ( zip_nat_o_nat @ Xs @ Ys ) ) )
     => ( member_nat @ Y @ ( set_nat2 @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_780_set__zip__rightD,axiom,
    ! [X3: t,Y: $o,Xs: list_t,Ys: list_o] :
      ( ( member3252940038344094615od_t_o @ ( product_Pair_t_o @ X3 @ Y ) @ ( set_Product_prod_t_o2 @ ( zip_t_o @ Xs @ Ys ) ) )
     => ( member_o @ Y @ ( set_o2 @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_781_set__zip__rightD,axiom,
    ! [X3: t,Y: set_a,Xs: list_t,Ys: list_set_a] :
      ( ( member2646690471337469789_set_a @ ( product_Pair_t_set_a @ X3 @ Y ) @ ( set_Pr15456023412717001_set_a @ ( zip_t_set_a @ Xs @ Ys ) ) )
     => ( member_set_a @ Y @ ( set_set_a2 @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_782_set__zip__rightD,axiom,
    ! [X3: h,Y: product_prod_t_set_a,Xs: list_h,Ys: list_P1095827921439705018_set_a] :
      ( ( member6770953551192623679_set_a @ ( produc3432067189782488838_set_a @ X3 @ Y ) @ ( set_Pr7122414649308925099_set_a @ ( zip_h_8035635146645487142_set_a @ Xs @ Ys ) ) )
     => ( member2646690471337469789_set_a @ Y @ ( set_Pr15456023412717001_set_a @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_783_set__zip__rightD,axiom,
    ! [X3: h,Y: t,Xs: list_h,Ys: list_t] :
      ( ( member361086559187601820od_h_t @ ( product_Pair_h_t @ X3 @ Y ) @ ( set_Product_prod_h_t2 @ ( zip_h_t @ Xs @ Ys ) ) )
     => ( member_t @ Y @ ( set_t2 @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_784_set__drop__subset,axiom,
    ! [N: nat,Xs: list_formula_a_t] : ( ord_le7457455060544393785la_a_t @ ( set_formula_a_t2 @ ( drop_formula_a_t @ N @ Xs ) ) @ ( set_formula_a_t2 @ Xs ) ) ).

% set_drop_subset
thf(fact_785_in__set__dropD,axiom,
    ! [X3: a,N: nat,Xs: list_a] :
      ( ( member_a @ X3 @ ( set_a2 @ ( drop_a @ N @ Xs ) ) )
     => ( member_a @ X3 @ ( set_a2 @ Xs ) ) ) ).

% in_set_dropD
thf(fact_786_in__set__dropD,axiom,
    ! [X3: product_prod_nat_nat,N: nat,Xs: list_P6011104703257516679at_nat] :
      ( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ ( drop_P8868858903918902087at_nat @ N @ Xs ) ) )
     => ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ) ).

% in_set_dropD
thf(fact_787_in__set__dropD,axiom,
    ! [X3: nat,N: nat,Xs: list_nat] :
      ( ( member_nat @ X3 @ ( set_nat2 @ ( drop_nat @ N @ Xs ) ) )
     => ( member_nat @ X3 @ ( set_nat2 @ Xs ) ) ) ).

% in_set_dropD
thf(fact_788_in__set__dropD,axiom,
    ! [X3: formula_a_t,N: nat,Xs: list_formula_a_t] :
      ( ( member_formula_a_t @ X3 @ ( set_formula_a_t2 @ ( drop_formula_a_t @ N @ Xs ) ) )
     => ( member_formula_a_t @ X3 @ ( set_formula_a_t2 @ Xs ) ) ) ).

% in_set_dropD
thf(fact_789_subset__code_I1_J,axiom,
    ! [Xs: list_a,B5: set_a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ B5 )
      = ( ! [X5: a] :
            ( ( member_a @ X5 @ ( set_a2 @ Xs ) )
           => ( member_a @ X5 @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_790_subset__code_I1_J,axiom,
    ! [Xs: list_P6011104703257516679at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) @ B5 )
      = ( ! [X5: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X5 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
           => ( member8440522571783428010at_nat @ X5 @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_791_subset__code_I1_J,axiom,
    ! [Xs: list_nat,B5: set_nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B5 )
      = ( ! [X5: nat] :
            ( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
           => ( member_nat @ X5 @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_792_subset__code_I1_J,axiom,
    ! [Xs: list_formula_a_t,B5: set_formula_a_t] :
      ( ( ord_le7457455060544393785la_a_t @ ( set_formula_a_t2 @ Xs ) @ B5 )
      = ( ! [X5: formula_a_t] :
            ( ( member_formula_a_t @ X5 @ ( set_formula_a_t2 @ Xs ) )
           => ( member_formula_a_t @ X5 @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_793_set__take__subset,axiom,
    ! [N: nat,Xs: list_formula_a_t] : ( ord_le7457455060544393785la_a_t @ ( set_formula_a_t2 @ ( take_formula_a_t @ N @ Xs ) ) @ ( set_formula_a_t2 @ Xs ) ) ).

% set_take_subset
thf(fact_794_in__set__takeD,axiom,
    ! [X3: a,N: nat,Xs: list_a] :
      ( ( member_a @ X3 @ ( set_a2 @ ( take_a @ N @ Xs ) ) )
     => ( member_a @ X3 @ ( set_a2 @ Xs ) ) ) ).

% in_set_takeD
thf(fact_795_in__set__takeD,axiom,
    ! [X3: product_prod_nat_nat,N: nat,Xs: list_P6011104703257516679at_nat] :
      ( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ ( take_P2173866234530122223at_nat @ N @ Xs ) ) )
     => ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ) ).

% in_set_takeD
thf(fact_796_in__set__takeD,axiom,
    ! [X3: nat,N: nat,Xs: list_nat] :
      ( ( member_nat @ X3 @ ( set_nat2 @ ( take_nat @ N @ Xs ) ) )
     => ( member_nat @ X3 @ ( set_nat2 @ Xs ) ) ) ).

% in_set_takeD
thf(fact_797_in__set__takeD,axiom,
    ! [X3: formula_a_t,N: nat,Xs: list_formula_a_t] :
      ( ( member_formula_a_t @ X3 @ ( set_formula_a_t2 @ ( take_formula_a_t @ N @ Xs ) ) )
     => ( member_formula_a_t @ X3 @ ( set_formula_a_t2 @ Xs ) ) ) ).

% in_set_takeD
thf(fact_798_list_Omap__cong,axiom,
    ! [X3: list_P1095827921439705018_set_a,Ya: list_P1095827921439705018_set_a,F: product_prod_t_set_a > t,G: product_prod_t_set_a > t] :
      ( ( X3 = Ya )
     => ( ! [Z: product_prod_t_set_a] :
            ( ( member2646690471337469789_set_a @ Z @ ( set_Pr15456023412717001_set_a @ Ya ) )
           => ( ( F @ Z )
              = ( G @ Z ) ) )
       => ( ( map_Pr119846518281778721et_a_t @ F @ X3 )
          = ( map_Pr119846518281778721et_a_t @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_799_list_Omap__cong,axiom,
    ! [X3: list_formula_a_t,Ya: list_formula_a_t,F: formula_a_t > $o,G: formula_a_t > $o] :
      ( ( X3 = Ya )
     => ( ! [Z: formula_a_t] :
            ( ( member_formula_a_t @ Z @ ( set_formula_a_t2 @ Ya ) )
           => ( ( F @ Z )
              = ( G @ Z ) ) )
       => ( ( map_formula_a_t_o @ F @ X3 )
          = ( map_formula_a_t_o @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_800_list_Omap__cong,axiom,
    ! [X3: list_formula_a_t,Ya: list_formula_a_t,F: formula_a_t > vydra_aux_a_t_h,G: formula_a_t > vydra_aux_a_t_h] :
      ( ( X3 = Ya )
     => ( ! [Z: formula_a_t] :
            ( ( member_formula_a_t @ Z @ ( set_formula_a_t2 @ Ya ) )
           => ( ( F @ Z )
              = ( G @ Z ) ) )
       => ( ( map_fo5746072578778194433_a_t_h @ F @ X3 )
          = ( map_fo5746072578778194433_a_t_h @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_801_list_Omap__cong0,axiom,
    ! [X3: list_P1095827921439705018_set_a,F: product_prod_t_set_a > t,G: product_prod_t_set_a > t] :
      ( ! [Z: product_prod_t_set_a] :
          ( ( member2646690471337469789_set_a @ Z @ ( set_Pr15456023412717001_set_a @ X3 ) )
         => ( ( F @ Z )
            = ( G @ Z ) ) )
     => ( ( map_Pr119846518281778721et_a_t @ F @ X3 )
        = ( map_Pr119846518281778721et_a_t @ G @ X3 ) ) ) ).

% list.map_cong0
thf(fact_802_list_Omap__cong0,axiom,
    ! [X3: list_formula_a_t,F: formula_a_t > $o,G: formula_a_t > $o] :
      ( ! [Z: formula_a_t] :
          ( ( member_formula_a_t @ Z @ ( set_formula_a_t2 @ X3 ) )
         => ( ( F @ Z )
            = ( G @ Z ) ) )
     => ( ( map_formula_a_t_o @ F @ X3 )
        = ( map_formula_a_t_o @ G @ X3 ) ) ) ).

% list.map_cong0
thf(fact_803_list_Omap__cong0,axiom,
    ! [X3: list_formula_a_t,F: formula_a_t > vydra_aux_a_t_h,G: formula_a_t > vydra_aux_a_t_h] :
      ( ! [Z: formula_a_t] :
          ( ( member_formula_a_t @ Z @ ( set_formula_a_t2 @ X3 ) )
         => ( ( F @ Z )
            = ( G @ Z ) ) )
     => ( ( map_fo5746072578778194433_a_t_h @ F @ X3 )
        = ( map_fo5746072578778194433_a_t_h @ G @ X3 ) ) ) ).

% list.map_cong0
thf(fact_804_list_Oinj__map__strong,axiom,
    ! [X3: list_P1095827921439705018_set_a,Xa: list_P1095827921439705018_set_a,F: product_prod_t_set_a > t,Fa: product_prod_t_set_a > t] :
      ( ! [Z: product_prod_t_set_a,Za: product_prod_t_set_a] :
          ( ( member2646690471337469789_set_a @ Z @ ( set_Pr15456023412717001_set_a @ X3 ) )
         => ( ( member2646690471337469789_set_a @ Za @ ( set_Pr15456023412717001_set_a @ Xa ) )
           => ( ( ( F @ Z )
                = ( Fa @ Za ) )
             => ( Z = Za ) ) ) )
     => ( ( ( map_Pr119846518281778721et_a_t @ F @ X3 )
          = ( map_Pr119846518281778721et_a_t @ Fa @ Xa ) )
       => ( X3 = Xa ) ) ) ).

% list.inj_map_strong
thf(fact_805_list_Oinj__map__strong,axiom,
    ! [X3: list_formula_a_t,Xa: list_formula_a_t,F: formula_a_t > $o,Fa: formula_a_t > $o] :
      ( ! [Z: formula_a_t,Za: formula_a_t] :
          ( ( member_formula_a_t @ Z @ ( set_formula_a_t2 @ X3 ) )
         => ( ( member_formula_a_t @ Za @ ( set_formula_a_t2 @ Xa ) )
           => ( ( ( F @ Z )
                = ( Fa @ Za ) )
             => ( Z = Za ) ) ) )
     => ( ( ( map_formula_a_t_o @ F @ X3 )
          = ( map_formula_a_t_o @ Fa @ Xa ) )
       => ( X3 = Xa ) ) ) ).

% list.inj_map_strong
thf(fact_806_list_Oinj__map__strong,axiom,
    ! [X3: list_formula_a_t,Xa: list_formula_a_t,F: formula_a_t > vydra_aux_a_t_h,Fa: formula_a_t > vydra_aux_a_t_h] :
      ( ! [Z: formula_a_t,Za: formula_a_t] :
          ( ( member_formula_a_t @ Z @ ( set_formula_a_t2 @ X3 ) )
         => ( ( member_formula_a_t @ Za @ ( set_formula_a_t2 @ Xa ) )
           => ( ( ( F @ Z )
                = ( Fa @ Za ) )
             => ( Z = Za ) ) ) )
     => ( ( ( map_fo5746072578778194433_a_t_h @ F @ X3 )
          = ( map_fo5746072578778194433_a_t_h @ Fa @ Xa ) )
       => ( X3 = Xa ) ) ) ).

% list.inj_map_strong
thf(fact_807_list_Omap__ident__strong,axiom,
    ! [T: list_a,F: a > a] :
      ( ! [Z: a] :
          ( ( member_a @ Z @ ( set_a2 @ T ) )
         => ( ( F @ Z )
            = Z ) )
     => ( ( map_a_a @ F @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_808_list_Omap__ident__strong,axiom,
    ! [T: list_P6011104703257516679at_nat,F: product_prod_nat_nat > product_prod_nat_nat] :
      ( ! [Z: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ Z @ ( set_Pr5648618587558075414at_nat @ T ) )
         => ( ( F @ Z )
            = Z ) )
     => ( ( map_Pr8058819605623181956at_nat @ F @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_809_list_Omap__ident__strong,axiom,
    ! [T: list_nat,F: nat > nat] :
      ( ! [Z: nat] :
          ( ( member_nat @ Z @ ( set_nat2 @ T ) )
         => ( ( F @ Z )
            = Z ) )
     => ( ( map_nat_nat @ F @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_810_list_Omap__ident__strong,axiom,
    ! [T: list_formula_a_t,F: formula_a_t > formula_a_t] :
      ( ! [Z: formula_a_t] :
          ( ( member_formula_a_t @ Z @ ( set_formula_a_t2 @ T ) )
         => ( ( F @ Z )
            = Z ) )
     => ( ( map_fo8328018262729589004la_a_t @ F @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_811_map__ext,axiom,
    ! [Xs: list_P1095827921439705018_set_a,F: product_prod_t_set_a > t,G: product_prod_t_set_a > t] :
      ( ! [X: product_prod_t_set_a] :
          ( ( member2646690471337469789_set_a @ X @ ( set_Pr15456023412717001_set_a @ Xs ) )
         => ( ( F @ X )
            = ( G @ X ) ) )
     => ( ( map_Pr119846518281778721et_a_t @ F @ Xs )
        = ( map_Pr119846518281778721et_a_t @ G @ Xs ) ) ) ).

% map_ext
thf(fact_812_map__ext,axiom,
    ! [Xs: list_formula_a_t,F: formula_a_t > $o,G: formula_a_t > $o] :
      ( ! [X: formula_a_t] :
          ( ( member_formula_a_t @ X @ ( set_formula_a_t2 @ Xs ) )
         => ( ( F @ X )
            = ( G @ X ) ) )
     => ( ( map_formula_a_t_o @ F @ Xs )
        = ( map_formula_a_t_o @ G @ Xs ) ) ) ).

% map_ext
thf(fact_813_map__ext,axiom,
    ! [Xs: list_formula_a_t,F: formula_a_t > vydra_aux_a_t_h,G: formula_a_t > vydra_aux_a_t_h] :
      ( ! [X: formula_a_t] :
          ( ( member_formula_a_t @ X @ ( set_formula_a_t2 @ Xs ) )
         => ( ( F @ X )
            = ( G @ X ) ) )
     => ( ( map_fo5746072578778194433_a_t_h @ F @ Xs )
        = ( map_fo5746072578778194433_a_t_h @ G @ Xs ) ) ) ).

% map_ext
thf(fact_814_map__idI,axiom,
    ! [Xs: list_a,F: a > a] :
      ( ! [X: a] :
          ( ( member_a @ X @ ( set_a2 @ Xs ) )
         => ( ( F @ X )
            = X ) )
     => ( ( map_a_a @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_815_map__idI,axiom,
    ! [Xs: list_P6011104703257516679at_nat,F: product_prod_nat_nat > product_prod_nat_nat] :
      ( ! [X: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
         => ( ( F @ X )
            = X ) )
     => ( ( map_Pr8058819605623181956at_nat @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_816_map__idI,axiom,
    ! [Xs: list_nat,F: nat > nat] :
      ( ! [X: nat] :
          ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
         => ( ( F @ X )
            = X ) )
     => ( ( map_nat_nat @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_817_map__idI,axiom,
    ! [Xs: list_formula_a_t,F: formula_a_t > formula_a_t] :
      ( ! [X: formula_a_t] :
          ( ( member_formula_a_t @ X @ ( set_formula_a_t2 @ Xs ) )
         => ( ( F @ X )
            = X ) )
     => ( ( map_fo8328018262729589004la_a_t @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_818_map__cong,axiom,
    ! [Xs: list_P1095827921439705018_set_a,Ys: list_P1095827921439705018_set_a,F: product_prod_t_set_a > t,G: product_prod_t_set_a > t] :
      ( ( Xs = Ys )
     => ( ! [X: product_prod_t_set_a] :
            ( ( member2646690471337469789_set_a @ X @ ( set_Pr15456023412717001_set_a @ Ys ) )
           => ( ( F @ X )
              = ( G @ X ) ) )
       => ( ( map_Pr119846518281778721et_a_t @ F @ Xs )
          = ( map_Pr119846518281778721et_a_t @ G @ Ys ) ) ) ) ).

% map_cong
thf(fact_819_map__cong,axiom,
    ! [Xs: list_formula_a_t,Ys: list_formula_a_t,F: formula_a_t > $o,G: formula_a_t > $o] :
      ( ( Xs = Ys )
     => ( ! [X: formula_a_t] :
            ( ( member_formula_a_t @ X @ ( set_formula_a_t2 @ Ys ) )
           => ( ( F @ X )
              = ( G @ X ) ) )
       => ( ( map_formula_a_t_o @ F @ Xs )
          = ( map_formula_a_t_o @ G @ Ys ) ) ) ) ).

% map_cong
thf(fact_820_map__cong,axiom,
    ! [Xs: list_formula_a_t,Ys: list_formula_a_t,F: formula_a_t > vydra_aux_a_t_h,G: formula_a_t > vydra_aux_a_t_h] :
      ( ( Xs = Ys )
     => ( ! [X: formula_a_t] :
            ( ( member_formula_a_t @ X @ ( set_formula_a_t2 @ Ys ) )
           => ( ( F @ X )
              = ( G @ X ) ) )
       => ( ( map_fo5746072578778194433_a_t_h @ F @ Xs )
          = ( map_fo5746072578778194433_a_t_h @ G @ Ys ) ) ) ) ).

% map_cong
thf(fact_821_ex__map__conv,axiom,
    ! [Ys: list_t,F: product_prod_t_set_a > t] :
      ( ( ? [Xs2: list_P1095827921439705018_set_a] :
            ( Ys
            = ( map_Pr119846518281778721et_a_t @ F @ Xs2 ) ) )
      = ( ! [X5: t] :
            ( ( member_t @ X5 @ ( set_t2 @ Ys ) )
           => ? [Y5: product_prod_t_set_a] :
                ( X5
                = ( F @ Y5 ) ) ) ) ) ).

% ex_map_conv
thf(fact_822_ex__map__conv,axiom,
    ! [Ys: list_o,F: formula_a_t > $o] :
      ( ( ? [Xs2: list_formula_a_t] :
            ( Ys
            = ( map_formula_a_t_o @ F @ Xs2 ) ) )
      = ( ! [X5: $o] :
            ( ( member_o @ X5 @ ( set_o2 @ Ys ) )
           => ? [Y5: formula_a_t] :
                ( X5
                = ( F @ Y5 ) ) ) ) ) ).

% ex_map_conv
thf(fact_823_ex__map__conv,axiom,
    ! [Ys: list_vydra_aux_a_t_h,F: formula_a_t > vydra_aux_a_t_h] :
      ( ( ? [Xs2: list_formula_a_t] :
            ( Ys
            = ( map_fo5746072578778194433_a_t_h @ F @ Xs2 ) ) )
      = ( ! [X5: vydra_aux_a_t_h] :
            ( ( member5760284237610626351_a_t_h @ X5 @ ( set_vydra_aux_a_t_h2 @ Ys ) )
           => ? [Y5: formula_a_t] :
                ( X5
                = ( F @ Y5 ) ) ) ) ) ).

% ex_map_conv
thf(fact_824_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_nat,Ys: list_nat,Y: nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_nat @ Y @ ( set_nat2 @ Ys ) )
       => ~ ! [X: nat] :
              ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_825_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_h,Ys: list_t,Y: t] :
      ( ( ( size_size_list_h @ Xs )
        = ( size_size_list_t @ Ys ) )
     => ( ( member_t @ Y @ ( set_t2 @ Ys ) )
       => ~ ! [X: h] :
              ~ ( member361086559187601820od_h_t @ ( product_Pair_h_t @ X @ Y ) @ ( set_Product_prod_h_t2 @ ( zip_h_t @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_826_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_t,Ys: list_a,Y: a] :
      ( ( ( size_size_list_t @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( member_a @ Y @ ( set_a2 @ Ys ) )
       => ~ ! [X: t] :
              ~ ( member3805107641173245437od_t_a @ ( product_Pair_t_a @ X @ Y ) @ ( set_Product_prod_t_a2 @ ( zip_t_a @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_827_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_t,Ys: list_nat,Y: nat] :
      ( ( ( size_size_list_t @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_nat @ Y @ ( set_nat2 @ Ys ) )
       => ~ ! [X: t] :
              ~ ( member1527514727993586727_t_nat @ ( product_Pair_t_nat @ X @ Y ) @ ( set_Pr5951681550964978363_t_nat @ ( zip_t_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_828_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_t,Ys: list_o,Y: $o] :
      ( ( ( size_size_list_t @ Xs )
        = ( size_size_list_o @ Ys ) )
     => ( ( member_o @ Y @ ( set_o2 @ Ys ) )
       => ~ ! [X: t] :
              ~ ( member3252940038344094615od_t_o @ ( product_Pair_t_o @ X @ Y ) @ ( set_Product_prod_t_o2 @ ( zip_t_o @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_829_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_t,Ys: list_t,Y: t] :
      ( ( ( size_size_list_t @ Xs )
        = ( size_size_list_t @ Ys ) )
     => ( ( member_t @ Y @ ( set_t2 @ Ys ) )
       => ~ ! [X: t] :
              ~ ( member3805107722934592656od_t_t @ ( product_Pair_t_t @ X @ Y ) @ ( set_Product_prod_t_t2 @ ( zip_t_t @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_830_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_iarray_o,Ys: list_a,Y: a] :
      ( ( ( size_s3243743627286111600rray_o @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( member_a @ Y @ ( set_a2 @ Ys ) )
       => ~ ! [X: iarray_o] :
              ~ ( member7622367487059232464ay_o_a @ ( produc1379057907826328727ay_o_a @ X @ Y ) @ ( set_Pr383680499513403708ay_o_a @ ( zip_iarray_o_a @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_831_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_iarray_o,Ys: list_nat,Y: nat] :
      ( ( ( size_s3243743627286111600rray_o @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_nat @ Y @ ( set_nat2 @ Ys ) )
       => ~ ! [X: iarray_o] :
              ~ ( member3397611872157576340_o_nat @ ( produc7777896477091606007_o_nat @ X @ Y ) @ ( set_Pr7255896997083088680_o_nat @ ( zip_iarray_o_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_832_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_iarray_o,Ys: list_t,Y: t] :
      ( ( ( size_s3243743627286111600rray_o @ Xs )
        = ( size_size_list_t @ Ys ) )
     => ( ( member_t @ Y @ ( set_t2 @ Ys ) )
       => ~ ! [X: iarray_o] :
              ~ ( member7622367568820579683ay_o_t @ ( produc1379057907826328746ay_o_t @ X @ Y ) @ ( set_Pr383680581274750927ay_o_t @ ( zip_iarray_o_t @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_833_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_t,Ys: list_set_a,Y: set_a] :
      ( ( ( size_size_list_t @ Xs )
        = ( size_size_list_set_a @ Ys ) )
     => ( ( member_set_a @ Y @ ( set_set_a2 @ Ys ) )
       => ~ ! [X: t] :
              ~ ( member2646690471337469789_set_a @ ( product_Pair_t_set_a @ X @ Y ) @ ( set_Pr15456023412717001_set_a @ ( zip_t_set_a @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_834_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_nat,Ys: list_nat,X3: nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
       => ~ ! [Y3: nat] :
              ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_835_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_a,Ys: list_t,X3: a] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_t @ Ys ) )
     => ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
       => ~ ! [Y3: t] :
              ~ ( member1426531559286782435od_a_t @ ( product_Pair_a_t @ X3 @ Y3 ) @ ( set_Product_prod_a_t2 @ ( zip_a_t @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_836_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_nat,Ys: list_t,X3: nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_t @ Ys ) )
     => ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
       => ~ ! [Y3: t] :
              ~ ( member8962352133871442893_nat_t @ ( product_Pair_nat_t @ X3 @ Y3 ) @ ( set_Pr4163146919988058721_nat_t @ ( zip_nat_t @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_837_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_h,Ys: list_t,X3: h] :
      ( ( ( size_size_list_h @ Xs )
        = ( size_size_list_t @ Ys ) )
     => ( ( member_h @ X3 @ ( set_h2 @ Xs ) )
       => ~ ! [Y3: t] :
              ~ ( member361086559187601820od_h_t @ ( product_Pair_h_t @ X3 @ Y3 ) @ ( set_Product_prod_h_t2 @ ( zip_h_t @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_838_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_t,Ys: list_o,X3: t] :
      ( ( ( size_size_list_t @ Xs )
        = ( size_size_list_o @ Ys ) )
     => ( ( member_t @ X3 @ ( set_t2 @ Xs ) )
       => ~ ! [Y3: $o] :
              ~ ( member3252940038344094615od_t_o @ ( product_Pair_t_o @ X3 @ Y3 ) @ ( set_Product_prod_t_o2 @ ( zip_t_o @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_839_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_t,Ys: list_t,X3: t] :
      ( ( ( size_size_list_t @ Xs )
        = ( size_size_list_t @ Ys ) )
     => ( ( member_t @ X3 @ ( set_t2 @ Xs ) )
       => ~ ! [Y3: t] :
              ~ ( member3805107722934592656od_t_t @ ( product_Pair_t_t @ X3 @ Y3 ) @ ( set_Product_prod_t_t2 @ ( zip_t_t @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_840_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_a,Ys: list_iarray_o,X3: a] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_s3243743627286111600rray_o @ Ys ) )
     => ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
       => ~ ! [Y3: iarray_o] :
              ~ ( member3327730122600805072rray_o @ ( produc3634157122453939351rray_o @ X3 @ Y3 ) @ ( set_Pr5312415171909752124rray_o @ ( zip_a_iarray_o @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_841_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_nat,Ys: list_iarray_o,X3: nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s3243743627286111600rray_o @ Ys ) )
     => ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
       => ~ ! [Y3: iarray_o] :
              ~ ( member3347103566020543162rray_o @ ( produc5847520750640444501rray_o @ X3 @ Y3 ) @ ( set_Pr7205388690946055502rray_o @ ( zip_nat_iarray_o @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_842_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_iarray_o,Ys: list_t,X3: iarray_o] :
      ( ( ( size_s3243743627286111600rray_o @ Xs )
        = ( size_size_list_t @ Ys ) )
     => ( ( member_iarray_o @ X3 @ ( set_iarray_o2 @ Xs ) )
       => ~ ! [Y3: t] :
              ~ ( member7622367568820579683ay_o_t @ ( produc1379057907826328746ay_o_t @ X3 @ Y3 ) @ ( set_Pr383680581274750927ay_o_t @ ( zip_iarray_o_t @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_843_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_t,Ys: list_set_a,X3: t] :
      ( ( ( size_size_list_t @ Xs )
        = ( size_size_list_set_a @ Ys ) )
     => ( ( member_t @ X3 @ ( set_t2 @ Xs ) )
       => ~ ! [Y3: set_a] :
              ~ ( member2646690471337469789_set_a @ ( product_Pair_t_set_a @ X3 @ Y3 ) @ ( set_Pr15456023412717001_set_a @ ( zip_t_set_a @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_844_find__cong,axiom,
    ! [Xs: list_a,Ys: list_a,P: a > $o,Q: a > $o] :
      ( ( Xs = Ys )
     => ( ! [X: a] :
            ( ( member_a @ X @ ( set_a2 @ Ys ) )
           => ( ( P @ X )
              = ( Q @ X ) ) )
       => ( ( find_a @ P @ Xs )
          = ( find_a @ Q @ Ys ) ) ) ) ).

% find_cong
thf(fact_845_find__cong,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ( Xs = Ys )
     => ( ! [X: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Ys ) )
           => ( ( P @ X )
              = ( Q @ X ) ) )
       => ( ( find_P8199882355184865565at_nat @ P @ Xs )
          = ( find_P8199882355184865565at_nat @ Q @ Ys ) ) ) ) ).

% find_cong
thf(fact_846_find__cong,axiom,
    ! [Xs: list_nat,Ys: list_nat,P: nat > $o,Q: nat > $o] :
      ( ( Xs = Ys )
     => ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ Ys ) )
           => ( ( P @ X )
              = ( Q @ X ) ) )
       => ( ( find_nat @ P @ Xs )
          = ( find_nat @ Q @ Ys ) ) ) ) ).

% find_cong
thf(fact_847_find__cong,axiom,
    ! [Xs: list_formula_a_t,Ys: list_formula_a_t,P: formula_a_t > $o,Q: formula_a_t > $o] :
      ( ( Xs = Ys )
     => ( ! [X: formula_a_t] :
            ( ( member_formula_a_t @ X @ ( set_formula_a_t2 @ Ys ) )
           => ( ( P @ X )
              = ( Q @ X ) ) )
       => ( ( find_formula_a_t @ P @ Xs )
          = ( find_formula_a_t @ Q @ Ys ) ) ) ) ).

% find_cong
thf(fact_848_set__take__subset__set__take,axiom,
    ! [M: nat,N: nat,Xs: list_formula_a_t] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_le7457455060544393785la_a_t @ ( set_formula_a_t2 @ ( take_formula_a_t @ M @ Xs ) ) @ ( set_formula_a_t2 @ ( take_formula_a_t @ N @ Xs ) ) ) ) ).

% set_take_subset_set_take
thf(fact_849_set__drop__subset__set__drop,axiom,
    ! [N: nat,M: nat,Xs: list_formula_a_t] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ord_le7457455060544393785la_a_t @ ( set_formula_a_t2 @ ( drop_formula_a_t @ M @ Xs ) ) @ ( set_formula_a_t2 @ ( drop_formula_a_t @ N @ Xs ) ) ) ) ).

% set_drop_subset_set_drop
thf(fact_850_length__pos__if__in__set,axiom,
    ! [X3: a,Xs: list_a] :
      ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_851_length__pos__if__in__set,axiom,
    ! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s5460976970255530739at_nat @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_852_length__pos__if__in__set,axiom,
    ! [X3: nat,Xs: list_nat] :
      ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_853_length__pos__if__in__set,axiom,
    ! [X3: product_prod_t_set_a,Xs: list_P1095827921439705018_set_a] :
      ( ( member2646690471337469789_set_a @ X3 @ ( set_Pr15456023412717001_set_a @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s5362121731612881446_set_a @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_854_length__pos__if__in__set,axiom,
    ! [X3: vydra_aux_a_t_h,Xs: list_vydra_aux_a_t_h] :
      ( ( member5760284237610626351_a_t_h @ X3 @ ( set_vydra_aux_a_t_h2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s8993897887676352572_a_t_h @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_855_length__pos__if__in__set,axiom,
    ! [X3: iarray_o,Xs: list_iarray_o] :
      ( ( member_iarray_o @ X3 @ ( set_iarray_o2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s3243743627286111600rray_o @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_856_length__pos__if__in__set,axiom,
    ! [X3: formula_a_t,Xs: list_formula_a_t] :
      ( ( member_formula_a_t @ X3 @ ( set_formula_a_t2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s8846756101701226951la_a_t @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_857_length__pos__if__in__set,axiom,
    ! [X3: t,Xs: list_t] :
      ( ( member_t @ X3 @ ( set_t2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_t @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_858_all__set__conv__all__nth,axiom,
    ! [Xs: list_P1095827921439705018_set_a,P: product_prod_t_set_a > $o] :
      ( ( ! [X5: product_prod_t_set_a] :
            ( ( member2646690471337469789_set_a @ X5 @ ( set_Pr15456023412717001_set_a @ Xs ) )
           => ( P @ X5 ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s5362121731612881446_set_a @ Xs ) )
           => ( P @ ( nth_Pr633927046780761275_set_a @ Xs @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_859_all__set__conv__all__nth,axiom,
    ! [Xs: list_vydra_aux_a_t_h,P: vydra_aux_a_t_h > $o] :
      ( ( ! [X5: vydra_aux_a_t_h] :
            ( ( member5760284237610626351_a_t_h @ X5 @ ( set_vydra_aux_a_t_h2 @ Xs ) )
           => ( P @ X5 ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s8993897887676352572_a_t_h @ Xs ) )
           => ( P @ ( nth_vydra_aux_a_t_h @ Xs @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_860_all__set__conv__all__nth,axiom,
    ! [Xs: list_iarray_o,P: iarray_o > $o] :
      ( ( ! [X5: iarray_o] :
            ( ( member_iarray_o @ X5 @ ( set_iarray_o2 @ Xs ) )
           => ( P @ X5 ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s3243743627286111600rray_o @ Xs ) )
           => ( P @ ( nth_iarray_o @ Xs @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_861_all__set__conv__all__nth,axiom,
    ! [Xs: list_formula_a_t,P: formula_a_t > $o] :
      ( ( ! [X5: formula_a_t] :
            ( ( member_formula_a_t @ X5 @ ( set_formula_a_t2 @ Xs ) )
           => ( P @ X5 ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s8846756101701226951la_a_t @ Xs ) )
           => ( P @ ( nth_formula_a_t @ Xs @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_862_all__set__conv__all__nth,axiom,
    ! [Xs: list_t,P: t > $o] :
      ( ( ! [X5: t] :
            ( ( member_t @ X5 @ ( set_t2 @ Xs ) )
           => ( P @ X5 ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_t @ Xs ) )
           => ( P @ ( nth_t @ Xs @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_863_all__nth__imp__all__set,axiom,
    ! [Xs: list_a,P: a > $o,X3: a] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
         => ( P @ ( nth_a @ Xs @ I2 ) ) )
     => ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
       => ( P @ X3 ) ) ) ).

% all_nth_imp_all_set
thf(fact_864_all__nth__imp__all__set,axiom,
    ! [Xs: list_P6011104703257516679at_nat,P: product_prod_nat_nat > $o,X3: product_prod_nat_nat] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s5460976970255530739at_nat @ Xs ) )
         => ( P @ ( nth_Pr7617993195940197384at_nat @ Xs @ I2 ) ) )
     => ( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
       => ( P @ X3 ) ) ) ).

% all_nth_imp_all_set
thf(fact_865_all__nth__imp__all__set,axiom,
    ! [Xs: list_nat,P: nat > $o,X3: nat] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
         => ( P @ ( nth_nat @ Xs @ I2 ) ) )
     => ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
       => ( P @ X3 ) ) ) ).

% all_nth_imp_all_set
thf(fact_866_all__nth__imp__all__set,axiom,
    ! [Xs: list_P1095827921439705018_set_a,P: product_prod_t_set_a > $o,X3: product_prod_t_set_a] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s5362121731612881446_set_a @ Xs ) )
         => ( P @ ( nth_Pr633927046780761275_set_a @ Xs @ I2 ) ) )
     => ( ( member2646690471337469789_set_a @ X3 @ ( set_Pr15456023412717001_set_a @ Xs ) )
       => ( P @ X3 ) ) ) ).

% all_nth_imp_all_set
thf(fact_867_all__nth__imp__all__set,axiom,
    ! [Xs: list_vydra_aux_a_t_h,P: vydra_aux_a_t_h > $o,X3: vydra_aux_a_t_h] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s8993897887676352572_a_t_h @ Xs ) )
         => ( P @ ( nth_vydra_aux_a_t_h @ Xs @ I2 ) ) )
     => ( ( member5760284237610626351_a_t_h @ X3 @ ( set_vydra_aux_a_t_h2 @ Xs ) )
       => ( P @ X3 ) ) ) ).

% all_nth_imp_all_set
thf(fact_868_all__nth__imp__all__set,axiom,
    ! [Xs: list_iarray_o,P: iarray_o > $o,X3: iarray_o] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s3243743627286111600rray_o @ Xs ) )
         => ( P @ ( nth_iarray_o @ Xs @ I2 ) ) )
     => ( ( member_iarray_o @ X3 @ ( set_iarray_o2 @ Xs ) )
       => ( P @ X3 ) ) ) ).

% all_nth_imp_all_set
thf(fact_869_all__nth__imp__all__set,axiom,
    ! [Xs: list_formula_a_t,P: formula_a_t > $o,X3: formula_a_t] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s8846756101701226951la_a_t @ Xs ) )
         => ( P @ ( nth_formula_a_t @ Xs @ I2 ) ) )
     => ( ( member_formula_a_t @ X3 @ ( set_formula_a_t2 @ Xs ) )
       => ( P @ X3 ) ) ) ).

% all_nth_imp_all_set
thf(fact_870_all__nth__imp__all__set,axiom,
    ! [Xs: list_t,P: t > $o,X3: t] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_size_list_t @ Xs ) )
         => ( P @ ( nth_t @ Xs @ I2 ) ) )
     => ( ( member_t @ X3 @ ( set_t2 @ Xs ) )
       => ( P @ X3 ) ) ) ).

% all_nth_imp_all_set
thf(fact_871_in__set__conv__nth,axiom,
    ! [X3: a,Xs: list_a] :
      ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
            & ( ( nth_a @ Xs @ I3 )
              = X3 ) ) ) ) ).

% in_set_conv_nth
thf(fact_872_in__set__conv__nth,axiom,
    ! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s5460976970255530739at_nat @ Xs ) )
            & ( ( nth_Pr7617993195940197384at_nat @ Xs @ I3 )
              = X3 ) ) ) ) ).

% in_set_conv_nth
thf(fact_873_in__set__conv__nth,axiom,
    ! [X3: nat,Xs: list_nat] :
      ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
            & ( ( nth_nat @ Xs @ I3 )
              = X3 ) ) ) ) ).

% in_set_conv_nth
thf(fact_874_in__set__conv__nth,axiom,
    ! [X3: product_prod_t_set_a,Xs: list_P1095827921439705018_set_a] :
      ( ( member2646690471337469789_set_a @ X3 @ ( set_Pr15456023412717001_set_a @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s5362121731612881446_set_a @ Xs ) )
            & ( ( nth_Pr633927046780761275_set_a @ Xs @ I3 )
              = X3 ) ) ) ) ).

% in_set_conv_nth
thf(fact_875_in__set__conv__nth,axiom,
    ! [X3: vydra_aux_a_t_h,Xs: list_vydra_aux_a_t_h] :
      ( ( member5760284237610626351_a_t_h @ X3 @ ( set_vydra_aux_a_t_h2 @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s8993897887676352572_a_t_h @ Xs ) )
            & ( ( nth_vydra_aux_a_t_h @ Xs @ I3 )
              = X3 ) ) ) ) ).

% in_set_conv_nth
thf(fact_876_in__set__conv__nth,axiom,
    ! [X3: iarray_o,Xs: list_iarray_o] :
      ( ( member_iarray_o @ X3 @ ( set_iarray_o2 @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s3243743627286111600rray_o @ Xs ) )
            & ( ( nth_iarray_o @ Xs @ I3 )
              = X3 ) ) ) ) ).

% in_set_conv_nth
thf(fact_877_in__set__conv__nth,axiom,
    ! [X3: formula_a_t,Xs: list_formula_a_t] :
      ( ( member_formula_a_t @ X3 @ ( set_formula_a_t2 @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s8846756101701226951la_a_t @ Xs ) )
            & ( ( nth_formula_a_t @ Xs @ I3 )
              = X3 ) ) ) ) ).

% in_set_conv_nth
thf(fact_878_in__set__conv__nth,axiom,
    ! [X3: t,Xs: list_t] :
      ( ( member_t @ X3 @ ( set_t2 @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_t @ Xs ) )
            & ( ( nth_t @ Xs @ I3 )
              = X3 ) ) ) ) ).

% in_set_conv_nth
thf(fact_879_list__ball__nth,axiom,
    ! [N: nat,Xs: list_P1095827921439705018_set_a,P: product_prod_t_set_a > $o] :
      ( ( ord_less_nat @ N @ ( size_s5362121731612881446_set_a @ Xs ) )
     => ( ! [X: product_prod_t_set_a] :
            ( ( member2646690471337469789_set_a @ X @ ( set_Pr15456023412717001_set_a @ Xs ) )
           => ( P @ X ) )
       => ( P @ ( nth_Pr633927046780761275_set_a @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_880_list__ball__nth,axiom,
    ! [N: nat,Xs: list_vydra_aux_a_t_h,P: vydra_aux_a_t_h > $o] :
      ( ( ord_less_nat @ N @ ( size_s8993897887676352572_a_t_h @ Xs ) )
     => ( ! [X: vydra_aux_a_t_h] :
            ( ( member5760284237610626351_a_t_h @ X @ ( set_vydra_aux_a_t_h2 @ Xs ) )
           => ( P @ X ) )
       => ( P @ ( nth_vydra_aux_a_t_h @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_881_list__ball__nth,axiom,
    ! [N: nat,Xs: list_iarray_o,P: iarray_o > $o] :
      ( ( ord_less_nat @ N @ ( size_s3243743627286111600rray_o @ Xs ) )
     => ( ! [X: iarray_o] :
            ( ( member_iarray_o @ X @ ( set_iarray_o2 @ Xs ) )
           => ( P @ X ) )
       => ( P @ ( nth_iarray_o @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_882_list__ball__nth,axiom,
    ! [N: nat,Xs: list_formula_a_t,P: formula_a_t > $o] :
      ( ( ord_less_nat @ N @ ( size_s8846756101701226951la_a_t @ Xs ) )
     => ( ! [X: formula_a_t] :
            ( ( member_formula_a_t @ X @ ( set_formula_a_t2 @ Xs ) )
           => ( P @ X ) )
       => ( P @ ( nth_formula_a_t @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_883_list__ball__nth,axiom,
    ! [N: nat,Xs: list_t,P: t > $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_t @ Xs ) )
     => ( ! [X: t] :
            ( ( member_t @ X @ ( set_t2 @ Xs ) )
           => ( P @ X ) )
       => ( P @ ( nth_t @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_884_nth__mem,axiom,
    ! [N: nat,Xs: list_a] :
      ( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
     => ( member_a @ ( nth_a @ Xs @ N ) @ ( set_a2 @ Xs ) ) ) ).

% nth_mem
thf(fact_885_nth__mem,axiom,
    ! [N: nat,Xs: list_P6011104703257516679at_nat] :
      ( ( ord_less_nat @ N @ ( size_s5460976970255530739at_nat @ Xs ) )
     => ( member8440522571783428010at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs @ N ) @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ) ).

% nth_mem
thf(fact_886_nth__mem,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( member_nat @ ( nth_nat @ Xs @ N ) @ ( set_nat2 @ Xs ) ) ) ).

% nth_mem
thf(fact_887_nth__mem,axiom,
    ! [N: nat,Xs: list_P1095827921439705018_set_a] :
      ( ( ord_less_nat @ N @ ( size_s5362121731612881446_set_a @ Xs ) )
     => ( member2646690471337469789_set_a @ ( nth_Pr633927046780761275_set_a @ Xs @ N ) @ ( set_Pr15456023412717001_set_a @ Xs ) ) ) ).

% nth_mem
thf(fact_888_nth__mem,axiom,
    ! [N: nat,Xs: list_vydra_aux_a_t_h] :
      ( ( ord_less_nat @ N @ ( size_s8993897887676352572_a_t_h @ Xs ) )
     => ( member5760284237610626351_a_t_h @ ( nth_vydra_aux_a_t_h @ Xs @ N ) @ ( set_vydra_aux_a_t_h2 @ Xs ) ) ) ).

% nth_mem
thf(fact_889_nth__mem,axiom,
    ! [N: nat,Xs: list_iarray_o] :
      ( ( ord_less_nat @ N @ ( size_s3243743627286111600rray_o @ Xs ) )
     => ( member_iarray_o @ ( nth_iarray_o @ Xs @ N ) @ ( set_iarray_o2 @ Xs ) ) ) ).

% nth_mem
thf(fact_890_nth__mem,axiom,
    ! [N: nat,Xs: list_formula_a_t] :
      ( ( ord_less_nat @ N @ ( size_s8846756101701226951la_a_t @ Xs ) )
     => ( member_formula_a_t @ ( nth_formula_a_t @ Xs @ N ) @ ( set_formula_a_t2 @ Xs ) ) ) ).

% nth_mem
thf(fact_891_nth__mem,axiom,
    ! [N: nat,Xs: list_t] :
      ( ( ord_less_nat @ N @ ( size_size_list_t @ Xs ) )
     => ( member_t @ ( nth_t @ Xs @ N ) @ ( set_t2 @ Xs ) ) ) ).

% nth_mem
thf(fact_892_VYDRA__MDL_Olast__before_Osimps_I2_J,axiom,
    ! [Sigma: trace_a_t,Init_hd: h,Run_hd: h > option5775791656947114332_set_a,P: nat > $o,N: nat] :
      ( ( vYDRA_MDL_a_t_h @ Sigma @ Init_hd @ Run_hd )
     => ( ( ( P @ N )
         => ( ( vYDRA_last_before @ P @ ( suc @ N ) )
            = ( some_nat @ N ) ) )
        & ( ~ ( P @ N )
         => ( ( vYDRA_last_before @ P @ ( suc @ N ) )
            = ( vYDRA_last_before @ P @ N ) ) ) ) ) ).

% VYDRA_MDL.last_before.simps(2)
thf(fact_893_last__before_Oelims,axiom,
    ! [X3: nat > $o,Xa: nat,Y: option_nat] :
      ( ( ( vYDRA_last_before @ X3 @ Xa )
        = Y )
     => ( ( ( Xa = zero_zero_nat )
         => ( Y != none_nat ) )
       => ~ ! [N3: nat] :
              ( ( Xa
                = ( suc @ N3 ) )
             => ~ ( ( ( X3 @ N3 )
                   => ( Y
                      = ( some_nat @ N3 ) ) )
                  & ( ~ ( X3 @ N3 )
                   => ( Y
                      = ( vYDRA_last_before @ X3 @ N3 ) ) ) ) ) ) ) ).

% last_before.elims
thf(fact_894_last__before__Some,axiom,
    ! [P: nat > $o,N: nat,M: nat] :
      ( ( ( vYDRA_last_before @ P @ N )
        = ( some_nat @ M ) )
     => ( ( ord_less_nat @ M @ N )
        & ( P @ M )
        & ! [X10: nat] :
            ( ( member_nat @ X10 @ ( set_or5834768355832116004an_nat @ M @ N ) )
           => ~ ( P @ X10 ) ) ) ) ).

% last_before_Some
thf(fact_895_pred__subset__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S11: set_Pr1261947904930325089at_nat] :
      ( ( ord_le2646555220125990790_nat_o
        @ ^ [X5: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y5 ) @ R )
        @ ^ [X5: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y5 ) @ S11 ) )
      = ( ord_le3146513528884898305at_nat @ R @ S11 ) ) ).

% pred_subset_eq2
thf(fact_896_pred__subset__eq2,axiom,
    ! [R: set_Pr4354077987950702516_o_nat,S11: set_Pr4354077987950702516_o_nat] :
      ( ( ord_le8770491129411382917_nat_o
        @ ^ [X5: nat > $o,Y5: nat] : ( member2897238310647649173_o_nat @ ( produc7277522915581678840_o_nat @ X5 @ Y5 ) @ R )
        @ ^ [X5: nat > $o,Y5: nat] : ( member2897238310647649173_o_nat @ ( produc7277522915581678840_o_nat @ X5 @ Y5 ) @ S11 ) )
      = ( ord_le1487873752969080852_o_nat @ R @ S11 ) ) ).

% pred_subset_eq2
thf(fact_897_pred__subset__eq2,axiom,
    ! [R: set_Product_prod_t_o,S11: set_Product_prod_t_o] :
      ( ( ord_less_eq_t_o_o
        @ ^ [X5: t,Y5: $o] : ( member3252940038344094615od_t_o @ ( product_Pair_t_o @ X5 @ Y5 ) @ R )
        @ ^ [X5: t,Y5: $o] : ( member3252940038344094615od_t_o @ ( product_Pair_t_o @ X5 @ Y5 ) @ S11 ) )
      = ( ord_le1768922996218826606od_t_o @ R @ S11 ) ) ).

% pred_subset_eq2
thf(fact_898_pred__subset__eq2,axiom,
    ! [R: set_Pr6994796776334525076_set_a,S11: set_Pr6994796776334525076_set_a] :
      ( ( ord_le2522191030168684097et_a_o
        @ ^ [X5: t,Y5: set_a] : ( member2646690471337469789_set_a @ ( product_Pair_t_set_a @ X5 @ Y5 ) @ R )
        @ ^ [X5: t,Y5: set_a] : ( member2646690471337469789_set_a @ ( product_Pair_t_set_a @ X5 @ Y5 ) @ S11 ) )
      = ( ord_le3254057565153606196_set_a @ R @ S11 ) ) ).

% pred_subset_eq2
thf(fact_899_pred__subset__eq2,axiom,
    ! [R: set_Pr4350348695907903606_set_a,S11: set_Pr4350348695907903606_set_a] :
      ( ( ord_le1221091523068727007et_a_o
        @ ^ [X5: h,Y5: product_prod_t_set_a] : ( member6770953551192623679_set_a @ ( produc3432067189782488838_set_a @ X5 @ Y5 ) @ R )
        @ ^ [X5: h,Y5: product_prod_t_set_a] : ( member6770953551192623679_set_a @ ( produc3432067189782488838_set_a @ X5 @ Y5 ) @ S11 ) )
      = ( ord_le919132016096821270_set_a @ R @ S11 ) ) ).

% pred_subset_eq2
thf(fact_900_pred__subset__eq2,axiom,
    ! [R: set_Product_prod_h_t,S11: set_Product_prod_h_t] :
      ( ( ord_less_eq_h_t_o
        @ ^ [X5: h,Y5: t] : ( member361086559187601820od_h_t @ ( product_Pair_h_t @ X5 @ Y5 ) @ R )
        @ ^ [X5: h,Y5: t] : ( member361086559187601820od_h_t @ ( product_Pair_h_t @ X5 @ Y5 ) @ S11 ) )
      = ( ord_le6870178751910059251od_h_t @ R @ S11 ) ) ).

% pred_subset_eq2
thf(fact_901_last__before_Osimps_I1_J,axiom,
    ! [P: nat > $o] :
      ( ( vYDRA_last_before @ P @ zero_zero_nat )
      = none_nat ) ).

% last_before.simps(1)
thf(fact_902_last__before__None,axiom,
    ! [P: nat > $o,N: nat,M: nat] :
      ( ( ( vYDRA_last_before @ P @ N )
        = none_nat )
     => ( ( ord_less_nat @ M @ N )
       => ~ ( P @ M ) ) ) ).

% last_before_None
thf(fact_903_not__Some__eq,axiom,
    ! [X3: option4580262698044125979od_t_o] :
      ( ( ! [Y5: produc530307916048688075od_t_o] :
            ( X3
           != ( some_P9127227083975130150od_t_o @ Y5 ) ) )
      = ( X3 = none_P3452577240289636010od_t_o ) ) ).

% not_Some_eq
thf(fact_904_not__Some__eq,axiom,
    ! [X3: option8553266494716013595rray_o] :
      ( ( ! [Y5: produc8228095596854190539rray_o] :
            ( X3
           != ( some_P7279874174034738982rray_o @ Y5 ) ) )
      = ( X3 = none_P945442321178488234rray_o ) ) ).

% not_Some_eq
thf(fact_905_not__Some__eq,axiom,
    ! [X3: option8291060355354214836od_t_o] :
      ( ( ! [Y5: product_prod_t_o] :
            ( X3
           != ( some_P1805796590424124243od_t_o @ Y5 ) ) )
      = ( X3 = none_P7032077165169035727od_t_o ) ) ).

% not_Some_eq
thf(fact_906_not__Some__eq,axiom,
    ! [X3: option5775791656947114332_set_a] :
      ( ( ! [Y5: produc8090089739038349590_set_a] :
            ( X3
           != ( some_P2245991037696317435_set_a @ Y5 ) ) )
      = ( X3 = none_P2868508613186038903_set_a ) ) ).

% not_Some_eq
thf(fact_907_not__Some__eq,axiom,
    ! [X3: option2110264179601050681od_h_t] :
      ( ( ! [Y5: product_prod_h_t] :
            ( X3
           != ( some_P2526622376857543512od_h_t @ Y5 ) ) )
      = ( X3 = none_P849260865442546644od_h_t ) ) ).

% not_Some_eq
thf(fact_908_not__Some__eq,axiom,
    ! [X3: option_nat] :
      ( ( ! [Y5: nat] :
            ( X3
           != ( some_nat @ Y5 ) ) )
      = ( X3 = none_nat ) ) ).

% not_Some_eq
thf(fact_909_not__None__eq,axiom,
    ! [X3: option4580262698044125979od_t_o] :
      ( ( X3 != none_P3452577240289636010od_t_o )
      = ( ? [Y5: produc530307916048688075od_t_o] :
            ( X3
            = ( some_P9127227083975130150od_t_o @ Y5 ) ) ) ) ).

% not_None_eq
thf(fact_910_not__None__eq,axiom,
    ! [X3: option8553266494716013595rray_o] :
      ( ( X3 != none_P945442321178488234rray_o )
      = ( ? [Y5: produc8228095596854190539rray_o] :
            ( X3
            = ( some_P7279874174034738982rray_o @ Y5 ) ) ) ) ).

% not_None_eq
thf(fact_911_not__None__eq,axiom,
    ! [X3: option8291060355354214836od_t_o] :
      ( ( X3 != none_P7032077165169035727od_t_o )
      = ( ? [Y5: product_prod_t_o] :
            ( X3
            = ( some_P1805796590424124243od_t_o @ Y5 ) ) ) ) ).

% not_None_eq
thf(fact_912_not__None__eq,axiom,
    ! [X3: option5775791656947114332_set_a] :
      ( ( X3 != none_P2868508613186038903_set_a )
      = ( ? [Y5: produc8090089739038349590_set_a] :
            ( X3
            = ( some_P2245991037696317435_set_a @ Y5 ) ) ) ) ).

% not_None_eq
thf(fact_913_not__None__eq,axiom,
    ! [X3: option2110264179601050681od_h_t] :
      ( ( X3 != none_P849260865442546644od_h_t )
      = ( ? [Y5: product_prod_h_t] :
            ( X3
            = ( some_P2526622376857543512od_h_t @ Y5 ) ) ) ) ).

% not_None_eq
thf(fact_914_not__None__eq,axiom,
    ! [X3: option_nat] :
      ( ( X3 != none_nat )
      = ( ? [Y5: nat] :
            ( X3
            = ( some_nat @ Y5 ) ) ) ) ).

% not_None_eq
thf(fact_915_psubsetD,axiom,
    ! [A4: set_a,B5: set_a,C2: a] :
      ( ( ord_less_set_a @ A4 @ B5 )
     => ( ( member_a @ C2 @ A4 )
       => ( member_a @ C2 @ B5 ) ) ) ).

% psubsetD
thf(fact_916_psubsetD,axiom,
    ! [A4: set_formula_a_t,B5: set_formula_a_t,C2: formula_a_t] :
      ( ( ord_le2961433037666676781la_a_t @ A4 @ B5 )
     => ( ( member_formula_a_t @ C2 @ A4 )
       => ( member_formula_a_t @ C2 @ B5 ) ) ) ).

% psubsetD
thf(fact_917_psubsetD,axiom,
    ! [A4: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,C2: product_prod_nat_nat] :
      ( ( ord_le7866589430770878221at_nat @ A4 @ B5 )
     => ( ( member8440522571783428010at_nat @ C2 @ A4 )
       => ( member8440522571783428010at_nat @ C2 @ B5 ) ) ) ).

% psubsetD
thf(fact_918_psubsetD,axiom,
    ! [A4: set_nat,B5: set_nat,C2: nat] :
      ( ( ord_less_set_nat @ A4 @ B5 )
     => ( ( member_nat @ C2 @ A4 )
       => ( member_nat @ C2 @ B5 ) ) ) ).

% psubsetD
thf(fact_919_less__set__def,axiom,
    ( ord_less_set_a
    = ( ^ [A6: set_a,B6: set_a] :
          ( ord_less_a_o
          @ ^ [X5: a] : ( member_a @ X5 @ A6 )
          @ ^ [X5: a] : ( member_a @ X5 @ B6 ) ) ) ) ).

% less_set_def
thf(fact_920_less__set__def,axiom,
    ( ord_le2961433037666676781la_a_t
    = ( ^ [A6: set_formula_a_t,B6: set_formula_a_t] :
          ( ord_le2871786250512499864_a_t_o
          @ ^ [X5: formula_a_t] : ( member_formula_a_t @ X5 @ A6 )
          @ ^ [X5: formula_a_t] : ( member_formula_a_t @ X5 @ B6 ) ) ) ) ).

% less_set_def
thf(fact_921_less__set__def,axiom,
    ( ord_le7866589430770878221at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
          ( ord_le549003669493604880_nat_o
          @ ^ [X5: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X5 @ A6 )
          @ ^ [X5: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X5 @ B6 ) ) ) ) ).

% less_set_def
thf(fact_922_less__set__def,axiom,
    ( ord_less_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
          ( ord_less_nat_o
          @ ^ [X5: nat] : ( member_nat @ X5 @ A6 )
          @ ^ [X5: nat] : ( member_nat @ X5 @ B6 ) ) ) ) ).

% less_set_def
thf(fact_923_combine__options__cases,axiom,
    ! [X3: option_nat,P: option_nat > option_nat > $o,Y: option_nat] :
      ( ( ( X3 = none_nat )
       => ( P @ X3 @ Y ) )
     => ( ( ( Y = none_nat )
         => ( P @ X3 @ Y ) )
       => ( ! [A3: nat,B3: nat] :
              ( ( X3
                = ( some_nat @ A3 ) )
             => ( ( Y
                  = ( some_nat @ B3 ) )
               => ( P @ X3 @ Y ) ) )
         => ( P @ X3 @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_924_combine__options__cases,axiom,
    ! [X3: option8291060355354214836od_t_o,P: option8291060355354214836od_t_o > option_nat > $o,Y: option_nat] :
      ( ( ( X3 = none_P7032077165169035727od_t_o )
       => ( P @ X3 @ Y ) )
     => ( ( ( Y = none_nat )
         => ( P @ X3 @ Y ) )
       => ( ! [A3: product_prod_t_o,B3: nat] :
              ( ( X3
                = ( some_P1805796590424124243od_t_o @ A3 ) )
             => ( ( Y
                  = ( some_nat @ B3 ) )
               => ( P @ X3 @ Y ) ) )
         => ( P @ X3 @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_925_combine__options__cases,axiom,
    ! [X3: option2110264179601050681od_h_t,P: option2110264179601050681od_h_t > option_nat > $o,Y: option_nat] :
      ( ( ( X3 = none_P849260865442546644od_h_t )
       => ( P @ X3 @ Y ) )
     => ( ( ( Y = none_nat )
         => ( P @ X3 @ Y ) )
       => ( ! [A3: product_prod_h_t,B3: nat] :
              ( ( X3
                = ( some_P2526622376857543512od_h_t @ A3 ) )
             => ( ( Y
                  = ( some_nat @ B3 ) )
               => ( P @ X3 @ Y ) ) )
         => ( P @ X3 @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_926_combine__options__cases,axiom,
    ! [X3: option_nat,P: option_nat > option8291060355354214836od_t_o > $o,Y: option8291060355354214836od_t_o] :
      ( ( ( X3 = none_nat )
       => ( P @ X3 @ Y ) )
     => ( ( ( Y = none_P7032077165169035727od_t_o )
         => ( P @ X3 @ Y ) )
       => ( ! [A3: nat,B3: product_prod_t_o] :
              ( ( X3
                = ( some_nat @ A3 ) )
             => ( ( Y
                  = ( some_P1805796590424124243od_t_o @ B3 ) )
               => ( P @ X3 @ Y ) ) )
         => ( P @ X3 @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_927_combine__options__cases,axiom,
    ! [X3: option_nat,P: option_nat > option2110264179601050681od_h_t > $o,Y: option2110264179601050681od_h_t] :
      ( ( ( X3 = none_nat )
       => ( P @ X3 @ Y ) )
     => ( ( ( Y = none_P849260865442546644od_h_t )
         => ( P @ X3 @ Y ) )
       => ( ! [A3: nat,B3: product_prod_h_t] :
              ( ( X3
                = ( some_nat @ A3 ) )
             => ( ( Y
                  = ( some_P2526622376857543512od_h_t @ B3 ) )
               => ( P @ X3 @ Y ) ) )
         => ( P @ X3 @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_928_combine__options__cases,axiom,
    ! [X3: option8291060355354214836od_t_o,P: option8291060355354214836od_t_o > option8291060355354214836od_t_o > $o,Y: option8291060355354214836od_t_o] :
      ( ( ( X3 = none_P7032077165169035727od_t_o )
       => ( P @ X3 @ Y ) )
     => ( ( ( Y = none_P7032077165169035727od_t_o )
         => ( P @ X3 @ Y ) )
       => ( ! [A3: product_prod_t_o,B3: product_prod_t_o] :
              ( ( X3
                = ( some_P1805796590424124243od_t_o @ A3 ) )
             => ( ( Y
                  = ( some_P1805796590424124243od_t_o @ B3 ) )
               => ( P @ X3 @ Y ) ) )
         => ( P @ X3 @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_929_combine__options__cases,axiom,
    ! [X3: option8291060355354214836od_t_o,P: option8291060355354214836od_t_o > option2110264179601050681od_h_t > $o,Y: option2110264179601050681od_h_t] :
      ( ( ( X3 = none_P7032077165169035727od_t_o )
       => ( P @ X3 @ Y ) )
     => ( ( ( Y = none_P849260865442546644od_h_t )
         => ( P @ X3 @ Y ) )
       => ( ! [A3: product_prod_t_o,B3: product_prod_h_t] :
              ( ( X3
                = ( some_P1805796590424124243od_t_o @ A3 ) )
             => ( ( Y
                  = ( some_P2526622376857543512od_h_t @ B3 ) )
               => ( P @ X3 @ Y ) ) )
         => ( P @ X3 @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_930_combine__options__cases,axiom,
    ! [X3: option2110264179601050681od_h_t,P: option2110264179601050681od_h_t > option8291060355354214836od_t_o > $o,Y: option8291060355354214836od_t_o] :
      ( ( ( X3 = none_P849260865442546644od_h_t )
       => ( P @ X3 @ Y ) )
     => ( ( ( Y = none_P7032077165169035727od_t_o )
         => ( P @ X3 @ Y ) )
       => ( ! [A3: product_prod_h_t,B3: product_prod_t_o] :
              ( ( X3
                = ( some_P2526622376857543512od_h_t @ A3 ) )
             => ( ( Y
                  = ( some_P1805796590424124243od_t_o @ B3 ) )
               => ( P @ X3 @ Y ) ) )
         => ( P @ X3 @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_931_combine__options__cases,axiom,
    ! [X3: option2110264179601050681od_h_t,P: option2110264179601050681od_h_t > option2110264179601050681od_h_t > $o,Y: option2110264179601050681od_h_t] :
      ( ( ( X3 = none_P849260865442546644od_h_t )
       => ( P @ X3 @ Y ) )
     => ( ( ( Y = none_P849260865442546644od_h_t )
         => ( P @ X3 @ Y ) )
       => ( ! [A3: product_prod_h_t,B3: product_prod_h_t] :
              ( ( X3
                = ( some_P2526622376857543512od_h_t @ A3 ) )
             => ( ( Y
                  = ( some_P2526622376857543512od_h_t @ B3 ) )
               => ( P @ X3 @ Y ) ) )
         => ( P @ X3 @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_932_combine__options__cases,axiom,
    ! [X3: option5775791656947114332_set_a,P: option5775791656947114332_set_a > option_nat > $o,Y: option_nat] :
      ( ( ( X3 = none_P2868508613186038903_set_a )
       => ( P @ X3 @ Y ) )
     => ( ( ( Y = none_nat )
         => ( P @ X3 @ Y ) )
       => ( ! [A3: produc8090089739038349590_set_a,B3: nat] :
              ( ( X3
                = ( some_P2245991037696317435_set_a @ A3 ) )
             => ( ( Y
                  = ( some_nat @ B3 ) )
               => ( P @ X3 @ Y ) ) )
         => ( P @ X3 @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_933_split__option__all,axiom,
    ( ( ^ [P3: option4580262698044125979od_t_o > $o] :
        ! [X8: option4580262698044125979od_t_o] : ( P3 @ X8 ) )
    = ( ^ [P4: option4580262698044125979od_t_o > $o] :
          ( ( P4 @ none_P3452577240289636010od_t_o )
          & ! [X5: produc530307916048688075od_t_o] : ( P4 @ ( some_P9127227083975130150od_t_o @ X5 ) ) ) ) ) ).

% split_option_all
thf(fact_934_split__option__all,axiom,
    ( ( ^ [P3: option8553266494716013595rray_o > $o] :
        ! [X8: option8553266494716013595rray_o] : ( P3 @ X8 ) )
    = ( ^ [P4: option8553266494716013595rray_o > $o] :
          ( ( P4 @ none_P945442321178488234rray_o )
          & ! [X5: produc8228095596854190539rray_o] : ( P4 @ ( some_P7279874174034738982rray_o @ X5 ) ) ) ) ) ).

% split_option_all
thf(fact_935_split__option__all,axiom,
    ( ( ^ [P3: option8291060355354214836od_t_o > $o] :
        ! [X8: option8291060355354214836od_t_o] : ( P3 @ X8 ) )
    = ( ^ [P4: option8291060355354214836od_t_o > $o] :
          ( ( P4 @ none_P7032077165169035727od_t_o )
          & ! [X5: product_prod_t_o] : ( P4 @ ( some_P1805796590424124243od_t_o @ X5 ) ) ) ) ) ).

% split_option_all
thf(fact_936_split__option__all,axiom,
    ( ( ^ [P3: option5775791656947114332_set_a > $o] :
        ! [X8: option5775791656947114332_set_a] : ( P3 @ X8 ) )
    = ( ^ [P4: option5775791656947114332_set_a > $o] :
          ( ( P4 @ none_P2868508613186038903_set_a )
          & ! [X5: produc8090089739038349590_set_a] : ( P4 @ ( some_P2245991037696317435_set_a @ X5 ) ) ) ) ) ).

% split_option_all
thf(fact_937_split__option__all,axiom,
    ( ( ^ [P3: option2110264179601050681od_h_t > $o] :
        ! [X8: option2110264179601050681od_h_t] : ( P3 @ X8 ) )
    = ( ^ [P4: option2110264179601050681od_h_t > $o] :
          ( ( P4 @ none_P849260865442546644od_h_t )
          & ! [X5: product_prod_h_t] : ( P4 @ ( some_P2526622376857543512od_h_t @ X5 ) ) ) ) ) ).

% split_option_all
thf(fact_938_split__option__all,axiom,
    ( ( ^ [P3: option_nat > $o] :
        ! [X8: option_nat] : ( P3 @ X8 ) )
    = ( ^ [P4: option_nat > $o] :
          ( ( P4 @ none_nat )
          & ! [X5: nat] : ( P4 @ ( some_nat @ X5 ) ) ) ) ) ).

% split_option_all
thf(fact_939_split__option__ex,axiom,
    ( ( ^ [P3: option4580262698044125979od_t_o > $o] :
        ? [X8: option4580262698044125979od_t_o] : ( P3 @ X8 ) )
    = ( ^ [P4: option4580262698044125979od_t_o > $o] :
          ( ( P4 @ none_P3452577240289636010od_t_o )
          | ? [X5: produc530307916048688075od_t_o] : ( P4 @ ( some_P9127227083975130150od_t_o @ X5 ) ) ) ) ) ).

% split_option_ex
thf(fact_940_split__option__ex,axiom,
    ( ( ^ [P3: option8553266494716013595rray_o > $o] :
        ? [X8: option8553266494716013595rray_o] : ( P3 @ X8 ) )
    = ( ^ [P4: option8553266494716013595rray_o > $o] :
          ( ( P4 @ none_P945442321178488234rray_o )
          | ? [X5: produc8228095596854190539rray_o] : ( P4 @ ( some_P7279874174034738982rray_o @ X5 ) ) ) ) ) ).

% split_option_ex
thf(fact_941_split__option__ex,axiom,
    ( ( ^ [P3: option8291060355354214836od_t_o > $o] :
        ? [X8: option8291060355354214836od_t_o] : ( P3 @ X8 ) )
    = ( ^ [P4: option8291060355354214836od_t_o > $o] :
          ( ( P4 @ none_P7032077165169035727od_t_o )
          | ? [X5: product_prod_t_o] : ( P4 @ ( some_P1805796590424124243od_t_o @ X5 ) ) ) ) ) ).

% split_option_ex
thf(fact_942_split__option__ex,axiom,
    ( ( ^ [P3: option5775791656947114332_set_a > $o] :
        ? [X8: option5775791656947114332_set_a] : ( P3 @ X8 ) )
    = ( ^ [P4: option5775791656947114332_set_a > $o] :
          ( ( P4 @ none_P2868508613186038903_set_a )
          | ? [X5: produc8090089739038349590_set_a] : ( P4 @ ( some_P2245991037696317435_set_a @ X5 ) ) ) ) ) ).

% split_option_ex
thf(fact_943_split__option__ex,axiom,
    ( ( ^ [P3: option2110264179601050681od_h_t > $o] :
        ? [X8: option2110264179601050681od_h_t] : ( P3 @ X8 ) )
    = ( ^ [P4: option2110264179601050681od_h_t > $o] :
          ( ( P4 @ none_P849260865442546644od_h_t )
          | ? [X5: product_prod_h_t] : ( P4 @ ( some_P2526622376857543512od_h_t @ X5 ) ) ) ) ) ).

% split_option_ex
thf(fact_944_split__option__ex,axiom,
    ( ( ^ [P3: option_nat > $o] :
        ? [X8: option_nat] : ( P3 @ X8 ) )
    = ( ^ [P4: option_nat > $o] :
          ( ( P4 @ none_nat )
          | ? [X5: nat] : ( P4 @ ( some_nat @ X5 ) ) ) ) ) ).

% split_option_ex
thf(fact_945_option_Oexhaust,axiom,
    ! [Y: option4580262698044125979od_t_o] :
      ( ( Y != none_P3452577240289636010od_t_o )
     => ~ ! [X23: produc530307916048688075od_t_o] :
            ( Y
           != ( some_P9127227083975130150od_t_o @ X23 ) ) ) ).

% option.exhaust
thf(fact_946_option_Oexhaust,axiom,
    ! [Y: option8553266494716013595rray_o] :
      ( ( Y != none_P945442321178488234rray_o )
     => ~ ! [X23: produc8228095596854190539rray_o] :
            ( Y
           != ( some_P7279874174034738982rray_o @ X23 ) ) ) ).

% option.exhaust
thf(fact_947_option_Oexhaust,axiom,
    ! [Y: option8291060355354214836od_t_o] :
      ( ( Y != none_P7032077165169035727od_t_o )
     => ~ ! [X23: product_prod_t_o] :
            ( Y
           != ( some_P1805796590424124243od_t_o @ X23 ) ) ) ).

% option.exhaust
thf(fact_948_option_Oexhaust,axiom,
    ! [Y: option5775791656947114332_set_a] :
      ( ( Y != none_P2868508613186038903_set_a )
     => ~ ! [X23: produc8090089739038349590_set_a] :
            ( Y
           != ( some_P2245991037696317435_set_a @ X23 ) ) ) ).

% option.exhaust
thf(fact_949_option_Oexhaust,axiom,
    ! [Y: option2110264179601050681od_h_t] :
      ( ( Y != none_P849260865442546644od_h_t )
     => ~ ! [X23: product_prod_h_t] :
            ( Y
           != ( some_P2526622376857543512od_h_t @ X23 ) ) ) ).

% option.exhaust
thf(fact_950_option_Oexhaust,axiom,
    ! [Y: option_nat] :
      ( ( Y != none_nat )
     => ~ ! [X23: nat] :
            ( Y
           != ( some_nat @ X23 ) ) ) ).

% option.exhaust
thf(fact_951_option_OdiscI,axiom,
    ! [Option: option4580262698044125979od_t_o,X2: produc530307916048688075od_t_o] :
      ( ( Option
        = ( some_P9127227083975130150od_t_o @ X2 ) )
     => ( Option != none_P3452577240289636010od_t_o ) ) ).

% option.discI
thf(fact_952_option_OdiscI,axiom,
    ! [Option: option8553266494716013595rray_o,X2: produc8228095596854190539rray_o] :
      ( ( Option
        = ( some_P7279874174034738982rray_o @ X2 ) )
     => ( Option != none_P945442321178488234rray_o ) ) ).

% option.discI
thf(fact_953_option_OdiscI,axiom,
    ! [Option: option8291060355354214836od_t_o,X2: product_prod_t_o] :
      ( ( Option
        = ( some_P1805796590424124243od_t_o @ X2 ) )
     => ( Option != none_P7032077165169035727od_t_o ) ) ).

% option.discI
thf(fact_954_option_OdiscI,axiom,
    ! [Option: option5775791656947114332_set_a,X2: produc8090089739038349590_set_a] :
      ( ( Option
        = ( some_P2245991037696317435_set_a @ X2 ) )
     => ( Option != none_P2868508613186038903_set_a ) ) ).

% option.discI
thf(fact_955_option_OdiscI,axiom,
    ! [Option: option2110264179601050681od_h_t,X2: product_prod_h_t] :
      ( ( Option
        = ( some_P2526622376857543512od_h_t @ X2 ) )
     => ( Option != none_P849260865442546644od_h_t ) ) ).

% option.discI
thf(fact_956_option_OdiscI,axiom,
    ! [Option: option_nat,X2: nat] :
      ( ( Option
        = ( some_nat @ X2 ) )
     => ( Option != none_nat ) ) ).

% option.discI
thf(fact_957_option_Odistinct_I1_J,axiom,
    ! [X2: produc530307916048688075od_t_o] :
      ( none_P3452577240289636010od_t_o
     != ( some_P9127227083975130150od_t_o @ X2 ) ) ).

% option.distinct(1)
thf(fact_958_option_Odistinct_I1_J,axiom,
    ! [X2: produc8228095596854190539rray_o] :
      ( none_P945442321178488234rray_o
     != ( some_P7279874174034738982rray_o @ X2 ) ) ).

% option.distinct(1)
thf(fact_959_option_Odistinct_I1_J,axiom,
    ! [X2: product_prod_t_o] :
      ( none_P7032077165169035727od_t_o
     != ( some_P1805796590424124243od_t_o @ X2 ) ) ).

% option.distinct(1)
thf(fact_960_option_Odistinct_I1_J,axiom,
    ! [X2: produc8090089739038349590_set_a] :
      ( none_P2868508613186038903_set_a
     != ( some_P2245991037696317435_set_a @ X2 ) ) ).

% option.distinct(1)
thf(fact_961_option_Odistinct_I1_J,axiom,
    ! [X2: product_prod_h_t] :
      ( none_P849260865442546644od_h_t
     != ( some_P2526622376857543512od_h_t @ X2 ) ) ).

% option.distinct(1)
thf(fact_962_option_Odistinct_I1_J,axiom,
    ! [X2: nat] :
      ( none_nat
     != ( some_nat @ X2 ) ) ).

% option.distinct(1)
thf(fact_963_find__None__iff,axiom,
    ! [P: a > $o,Xs: list_a] :
      ( ( ( find_a @ P @ Xs )
        = none_a )
      = ( ~ ? [X5: a] :
              ( ( member_a @ X5 @ ( set_a2 @ Xs ) )
              & ( P @ X5 ) ) ) ) ).

% find_None_iff
thf(fact_964_find__None__iff,axiom,
    ! [P: product_prod_nat_nat > $o,Xs: list_P6011104703257516679at_nat] :
      ( ( ( find_P8199882355184865565at_nat @ P @ Xs )
        = none_P5556105721700978146at_nat )
      = ( ~ ? [X5: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X5 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
              & ( P @ X5 ) ) ) ) ).

% find_None_iff
thf(fact_965_find__None__iff,axiom,
    ! [P: formula_a_t > $o,Xs: list_formula_a_t] :
      ( ( ( find_formula_a_t @ P @ Xs )
        = none_formula_a_t )
      = ( ~ ? [X5: formula_a_t] :
              ( ( member_formula_a_t @ X5 @ ( set_formula_a_t2 @ Xs ) )
              & ( P @ X5 ) ) ) ) ).

% find_None_iff
thf(fact_966_find__None__iff,axiom,
    ! [P: nat > $o,Xs: list_nat] :
      ( ( ( find_nat @ P @ Xs )
        = none_nat )
      = ( ~ ? [X5: nat] :
              ( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
              & ( P @ X5 ) ) ) ) ).

% find_None_iff
thf(fact_967_find__None__iff,axiom,
    ! [P: produc530307916048688075od_t_o > $o,Xs: list_P7167060751680638171od_t_o] :
      ( ( ( find_P8232462763046280367od_t_o @ P @ Xs )
        = none_P3452577240289636010od_t_o )
      = ( ~ ? [X5: produc530307916048688075od_t_o] :
              ( ( member3775657527989285602od_t_o @ X5 @ ( set_Pr513341922006242166od_t_o @ Xs ) )
              & ( P @ X5 ) ) ) ) ).

% find_None_iff
thf(fact_968_find__None__iff,axiom,
    ! [P: product_prod_t_o > $o,Xs: list_P5575445741472356340od_t_o] :
      ( ( ( find_P3690432262111017226od_t_o @ P @ Xs )
        = none_P7032077165169035727od_t_o )
      = ( ~ ? [X5: product_prod_t_o] :
              ( ( member3252940038344094615od_t_o @ X5 @ ( set_Product_prod_t_o2 @ Xs ) )
              & ( P @ X5 ) ) ) ) ).

% find_None_iff
thf(fact_969_find__None__iff2,axiom,
    ! [P: a > $o,Xs: list_a] :
      ( ( none_a
        = ( find_a @ P @ Xs ) )
      = ( ~ ? [X5: a] :
              ( ( member_a @ X5 @ ( set_a2 @ Xs ) )
              & ( P @ X5 ) ) ) ) ).

% find_None_iff2
thf(fact_970_find__None__iff2,axiom,
    ! [P: product_prod_nat_nat > $o,Xs: list_P6011104703257516679at_nat] :
      ( ( none_P5556105721700978146at_nat
        = ( find_P8199882355184865565at_nat @ P @ Xs ) )
      = ( ~ ? [X5: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X5 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
              & ( P @ X5 ) ) ) ) ).

% find_None_iff2
thf(fact_971_find__None__iff2,axiom,
    ! [P: formula_a_t > $o,Xs: list_formula_a_t] :
      ( ( none_formula_a_t
        = ( find_formula_a_t @ P @ Xs ) )
      = ( ~ ? [X5: formula_a_t] :
              ( ( member_formula_a_t @ X5 @ ( set_formula_a_t2 @ Xs ) )
              & ( P @ X5 ) ) ) ) ).

% find_None_iff2
thf(fact_972_find__None__iff2,axiom,
    ! [P: nat > $o,Xs: list_nat] :
      ( ( none_nat
        = ( find_nat @ P @ Xs ) )
      = ( ~ ? [X5: nat] :
              ( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
              & ( P @ X5 ) ) ) ) ).

% find_None_iff2
thf(fact_973_find__None__iff2,axiom,
    ! [P: produc530307916048688075od_t_o > $o,Xs: list_P7167060751680638171od_t_o] :
      ( ( none_P3452577240289636010od_t_o
        = ( find_P8232462763046280367od_t_o @ P @ Xs ) )
      = ( ~ ? [X5: produc530307916048688075od_t_o] :
              ( ( member3775657527989285602od_t_o @ X5 @ ( set_Pr513341922006242166od_t_o @ Xs ) )
              & ( P @ X5 ) ) ) ) ).

% find_None_iff2
thf(fact_974_find__None__iff2,axiom,
    ! [P: product_prod_t_o > $o,Xs: list_P5575445741472356340od_t_o] :
      ( ( none_P7032077165169035727od_t_o
        = ( find_P3690432262111017226od_t_o @ P @ Xs ) )
      = ( ~ ? [X5: product_prod_t_o] :
              ( ( member3252940038344094615od_t_o @ X5 @ ( set_Product_prod_t_o2 @ Xs ) )
              & ( P @ X5 ) ) ) ) ).

% find_None_iff2
thf(fact_975_VYDRA__MDL_Olast__before_Osimps_I1_J,axiom,
    ! [Sigma: trace_a_t,Init_hd: h,Run_hd: h > option5775791656947114332_set_a,P: nat > $o] :
      ( ( vYDRA_MDL_a_t_h @ Sigma @ Init_hd @ Run_hd )
     => ( ( vYDRA_last_before @ P @ zero_zero_nat )
        = none_nat ) ) ).

% VYDRA_MDL.last_before.simps(1)
thf(fact_976_VYDRA__MDL_Olast__before__None,axiom,
    ! [Sigma: trace_a_t,Init_hd: h,Run_hd: h > option5775791656947114332_set_a,P: nat > $o,N: nat,M: nat] :
      ( ( vYDRA_MDL_a_t_h @ Sigma @ Init_hd @ Run_hd )
     => ( ( ( vYDRA_last_before @ P @ N )
          = none_nat )
       => ( ( ord_less_nat @ M @ N )
         => ~ ( P @ M ) ) ) ) ).

% VYDRA_MDL.last_before_None
thf(fact_977_option_Osize_I3_J,axiom,
    ( ( size_size_option_nat @ none_nat )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_978_option_Osize_I3_J,axiom,
    ( ( size_s8621177079540999087od_t_o @ none_P3452577240289636010od_t_o )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_979_option_Osize_I3_J,axiom,
    ( ( size_s4900030450070283808od_t_o @ none_P7032077165169035727od_t_o )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_980_pred__equals__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S11: set_Pr1261947904930325089at_nat] :
      ( ( ( ^ [X5: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y5 ) @ R ) )
        = ( ^ [X5: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y5 ) @ S11 ) ) )
      = ( R = S11 ) ) ).

% pred_equals_eq2
thf(fact_981_pred__equals__eq2,axiom,
    ! [R: set_Pr4354077987950702516_o_nat,S11: set_Pr4354077987950702516_o_nat] :
      ( ( ( ^ [X5: nat > $o,Y5: nat] : ( member2897238310647649173_o_nat @ ( produc7277522915581678840_o_nat @ X5 @ Y5 ) @ R ) )
        = ( ^ [X5: nat > $o,Y5: nat] : ( member2897238310647649173_o_nat @ ( produc7277522915581678840_o_nat @ X5 @ Y5 ) @ S11 ) ) )
      = ( R = S11 ) ) ).

% pred_equals_eq2
thf(fact_982_pred__equals__eq2,axiom,
    ! [R: set_Product_prod_t_o,S11: set_Product_prod_t_o] :
      ( ( ( ^ [X5: t,Y5: $o] : ( member3252940038344094615od_t_o @ ( product_Pair_t_o @ X5 @ Y5 ) @ R ) )
        = ( ^ [X5: t,Y5: $o] : ( member3252940038344094615od_t_o @ ( product_Pair_t_o @ X5 @ Y5 ) @ S11 ) ) )
      = ( R = S11 ) ) ).

% pred_equals_eq2
thf(fact_983_pred__equals__eq2,axiom,
    ! [R: set_Pr6994796776334525076_set_a,S11: set_Pr6994796776334525076_set_a] :
      ( ( ( ^ [X5: t,Y5: set_a] : ( member2646690471337469789_set_a @ ( product_Pair_t_set_a @ X5 @ Y5 ) @ R ) )
        = ( ^ [X5: t,Y5: set_a] : ( member2646690471337469789_set_a @ ( product_Pair_t_set_a @ X5 @ Y5 ) @ S11 ) ) )
      = ( R = S11 ) ) ).

% pred_equals_eq2
thf(fact_984_pred__equals__eq2,axiom,
    ! [R: set_Pr4350348695907903606_set_a,S11: set_Pr4350348695907903606_set_a] :
      ( ( ( ^ [X5: h,Y5: product_prod_t_set_a] : ( member6770953551192623679_set_a @ ( produc3432067189782488838_set_a @ X5 @ Y5 ) @ R ) )
        = ( ^ [X5: h,Y5: product_prod_t_set_a] : ( member6770953551192623679_set_a @ ( produc3432067189782488838_set_a @ X5 @ Y5 ) @ S11 ) ) )
      = ( R = S11 ) ) ).

% pred_equals_eq2
thf(fact_985_pred__equals__eq2,axiom,
    ! [R: set_Product_prod_h_t,S11: set_Product_prod_h_t] :
      ( ( ( ^ [X5: h,Y5: t] : ( member361086559187601820od_h_t @ ( product_Pair_h_t @ X5 @ Y5 ) @ R ) )
        = ( ^ [X5: h,Y5: t] : ( member361086559187601820od_h_t @ ( product_Pair_h_t @ X5 @ Y5 ) @ S11 ) ) )
      = ( R = S11 ) ) ).

% pred_equals_eq2
thf(fact_986_VYDRA__MDL_Olast__before__Some,axiom,
    ! [Sigma: trace_a_t,Init_hd: h,Run_hd: h > option5775791656947114332_set_a,P: nat > $o,N: nat,M: nat] :
      ( ( vYDRA_MDL_a_t_h @ Sigma @ Init_hd @ Run_hd )
     => ( ( ( vYDRA_last_before @ P @ N )
          = ( some_nat @ M ) )
       => ( ( ord_less_nat @ M @ N )
          & ( P @ M )
          & ! [X10: nat] :
              ( ( member_nat @ X10 @ ( set_or5834768355832116004an_nat @ M @ N ) )
             => ~ ( P @ X10 ) ) ) ) ) ).

% VYDRA_MDL.last_before_Some
thf(fact_987_VYDRA__MDL_Olast__before_Oelims,axiom,
    ! [Sigma: trace_a_t,Init_hd: h,Run_hd: h > option5775791656947114332_set_a,X3: nat > $o,Xa: nat,Y: option_nat] :
      ( ( vYDRA_MDL_a_t_h @ Sigma @ Init_hd @ Run_hd )
     => ( ( ( vYDRA_last_before @ X3 @ Xa )
          = Y )
       => ( ( ( Xa = zero_zero_nat )
           => ( Y != none_nat ) )
         => ~ ! [N3: nat] :
                ( ( Xa
                  = ( suc @ N3 ) )
               => ~ ( ( ( X3 @ N3 )
                     => ( Y
                        = ( some_nat @ N3 ) ) )
                    & ( ~ ( X3 @ N3 )
                     => ( Y
                        = ( vYDRA_last_before @ X3 @ N3 ) ) ) ) ) ) ) ) ).

% VYDRA_MDL.last_before.elims
thf(fact_988_subrelI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ! [X: nat,Y3: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ R2 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ S ) )
     => ( ord_le3146513528884898305at_nat @ R2 @ S ) ) ).

% subrelI
thf(fact_989_subrelI,axiom,
    ! [R2: set_Pr4354077987950702516_o_nat,S: set_Pr4354077987950702516_o_nat] :
      ( ! [X: nat > $o,Y3: nat] :
          ( ( member2897238310647649173_o_nat @ ( produc7277522915581678840_o_nat @ X @ Y3 ) @ R2 )
         => ( member2897238310647649173_o_nat @ ( produc7277522915581678840_o_nat @ X @ Y3 ) @ S ) )
     => ( ord_le1487873752969080852_o_nat @ R2 @ S ) ) ).

% subrelI
thf(fact_990_subrelI,axiom,
    ! [R2: set_Product_prod_t_o,S: set_Product_prod_t_o] :
      ( ! [X: t,Y3: $o] :
          ( ( member3252940038344094615od_t_o @ ( product_Pair_t_o @ X @ Y3 ) @ R2 )
         => ( member3252940038344094615od_t_o @ ( product_Pair_t_o @ X @ Y3 ) @ S ) )
     => ( ord_le1768922996218826606od_t_o @ R2 @ S ) ) ).

% subrelI
thf(fact_991_subrelI,axiom,
    ! [R2: set_Pr6994796776334525076_set_a,S: set_Pr6994796776334525076_set_a] :
      ( ! [X: t,Y3: set_a] :
          ( ( member2646690471337469789_set_a @ ( product_Pair_t_set_a @ X @ Y3 ) @ R2 )
         => ( member2646690471337469789_set_a @ ( product_Pair_t_set_a @ X @ Y3 ) @ S ) )
     => ( ord_le3254057565153606196_set_a @ R2 @ S ) ) ).

% subrelI
thf(fact_992_subrelI,axiom,
    ! [R2: set_Pr4350348695907903606_set_a,S: set_Pr4350348695907903606_set_a] :
      ( ! [X: h,Y3: product_prod_t_set_a] :
          ( ( member6770953551192623679_set_a @ ( produc3432067189782488838_set_a @ X @ Y3 ) @ R2 )
         => ( member6770953551192623679_set_a @ ( produc3432067189782488838_set_a @ X @ Y3 ) @ S ) )
     => ( ord_le919132016096821270_set_a @ R2 @ S ) ) ).

% subrelI
thf(fact_993_subrelI,axiom,
    ! [R2: set_Product_prod_h_t,S: set_Product_prod_h_t] :
      ( ! [X: h,Y3: t] :
          ( ( member361086559187601820od_h_t @ ( product_Pair_h_t @ X @ Y3 ) @ R2 )
         => ( member361086559187601820od_h_t @ ( product_Pair_h_t @ X @ Y3 ) @ S ) )
     => ( ord_le6870178751910059251od_h_t @ R2 @ S ) ) ).

% subrelI
thf(fact_994_greaterThanLessThan__iff,axiom,
    ! [I: product_prod_nat_nat,L: product_prod_nat_nat,U: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ I @ ( set_or3577856640977352877at_nat @ L @ U ) )
      = ( ( ord_le1203424502768444845at_nat @ L @ I )
        & ( ord_le1203424502768444845at_nat @ I @ U ) ) ) ).

% greaterThanLessThan_iff
thf(fact_995_greaterThanLessThan__iff,axiom,
    ! [I: nat,L: nat,U: nat] :
      ( ( member_nat @ I @ ( set_or5834768355832116004an_nat @ L @ U ) )
      = ( ( ord_less_nat @ L @ I )
        & ( ord_less_nat @ I @ U ) ) ) ).

% greaterThanLessThan_iff
thf(fact_996_last__before_Opelims,axiom,
    ! [X3: nat > $o,Xa: nat,Y: option_nat] :
      ( ( ( vYDRA_last_before @ X3 @ Xa )
        = Y )
     => ( ( accp_P7902722568424861365_o_nat @ vYDRA_5175910486691664417re_rel @ ( produc7277522915581678840_o_nat @ X3 @ Xa ) )
       => ( ( ( Xa = zero_zero_nat )
           => ( ( Y = none_nat )
             => ~ ( accp_P7902722568424861365_o_nat @ vYDRA_5175910486691664417re_rel @ ( produc7277522915581678840_o_nat @ X3 @ zero_zero_nat ) ) ) )
         => ~ ! [N3: nat] :
                ( ( Xa
                  = ( suc @ N3 ) )
               => ( ( ( ( X3 @ N3 )
                     => ( Y
                        = ( some_nat @ N3 ) ) )
                    & ( ~ ( X3 @ N3 )
                     => ( Y
                        = ( vYDRA_last_before @ X3 @ N3 ) ) ) )
                 => ~ ( accp_P7902722568424861365_o_nat @ vYDRA_5175910486691664417re_rel @ ( produc7277522915581678840_o_nat @ X3 @ ( suc @ N3 ) ) ) ) ) ) ) ) ).

% last_before.pelims
thf(fact_997_bs__sat,axiom,
    ! [R2: regex_a_t,N: nat,I: nat,Vs: list_vydra_aux_a_t_h,Vs2: list_vydra_aux_a_t_h,Bs: iarray_o] :
      ( ! [Phi4: formula_a_t,I2: nat,V3: vydra_aux_a_t_h,V5: vydra_aux_a_t_h,B3: product_prod_t_o] :
          ( ( member_formula_a_t @ Phi4 @ ( set_formula_a_t2 @ ( collect_subfmlas_a_t @ R2 @ nil_formula_a_t ) ) )
         => ( ( vYDRA_wf_vydra_a_t_h @ sigma @ init_hd @ run_hd @ Phi4 @ I2 @ N @ V3 )
           => ( ( ( run_h_t_a @ run_hd @ N @ V3 )
                = ( some_P9127227083975130150od_t_o @ ( produc5956389175556509053od_t_o @ V5 @ B3 ) ) )
             => ( ( product_snd_t_o @ B3 )
                = ( sat_a_t @ sigma @ Phi4 @ I2 ) ) ) ) )
     => ( ! [J2: nat] :
            ( ( ord_less_nat @ J2 @ ( size_s8846756101701226951la_a_t @ ( collect_subfmlas_a_t @ R2 @ nil_formula_a_t ) ) )
           => ( vYDRA_wf_vydra_a_t_h @ sigma @ init_hd @ run_hd @ ( nth_formula_a_t @ ( collect_subfmlas_a_t @ R2 @ nil_formula_a_t ) @ J2 ) @ I @ N @ ( nth_vydra_aux_a_t_h @ Vs @ J2 ) ) )
       => ( ( ( run_su3116026553151384105_h_t_o @ ( run_h_t_a @ run_hd @ N ) @ Vs )
            = ( some_P7279874174034738982rray_o @ ( produc8472968102330193021rray_o @ Vs2 @ Bs ) ) )
         => ( ( ( size_s8993897887676352572_a_t_h @ Vs )
              = ( size_s8846756101701226951la_a_t @ ( collect_subfmlas_a_t @ R2 @ nil_formula_a_t ) ) )
           => ( Bs
              = ( iarray_of_list_o
                @ ( map_formula_a_t_o
                  @ ^ [Phi5: formula_a_t] : ( sat_a_t @ sigma @ Phi5 @ I )
                  @ ( collect_subfmlas_a_t @ R2 @ nil_formula_a_t ) ) ) ) ) ) ) ) ).

% bs_sat
thf(fact_998_map__is__Nil__conv,axiom,
    ! [F: formula_a_t > formula_a_t,Xs: list_formula_a_t] :
      ( ( ( map_fo8328018262729589004la_a_t @ F @ Xs )
        = nil_formula_a_t )
      = ( Xs = nil_formula_a_t ) ) ).

% map_is_Nil_conv
thf(fact_999_map__is__Nil__conv,axiom,
    ! [F: product_prod_t_set_a > t,Xs: list_P1095827921439705018_set_a] :
      ( ( ( map_Pr119846518281778721et_a_t @ F @ Xs )
        = nil_t )
      = ( Xs = nil_Pr7982011819685168954_set_a ) ) ).

% map_is_Nil_conv
thf(fact_1000_map__is__Nil__conv,axiom,
    ! [F: formula_a_t > $o,Xs: list_formula_a_t] :
      ( ( ( map_formula_a_t_o @ F @ Xs )
        = nil_o )
      = ( Xs = nil_formula_a_t ) ) ).

% map_is_Nil_conv
thf(fact_1001_map__is__Nil__conv,axiom,
    ! [F: formula_a_t > vydra_aux_a_t_h,Xs: list_formula_a_t] :
      ( ( ( map_fo5746072578778194433_a_t_h @ F @ Xs )
        = nil_vydra_aux_a_t_h )
      = ( Xs = nil_formula_a_t ) ) ).

% map_is_Nil_conv
thf(fact_1002_Nil__is__map__conv,axiom,
    ! [F: formula_a_t > formula_a_t,Xs: list_formula_a_t] :
      ( ( nil_formula_a_t
        = ( map_fo8328018262729589004la_a_t @ F @ Xs ) )
      = ( Xs = nil_formula_a_t ) ) ).

% Nil_is_map_conv
thf(fact_1003_Nil__is__map__conv,axiom,
    ! [F: product_prod_t_set_a > t,Xs: list_P1095827921439705018_set_a] :
      ( ( nil_t
        = ( map_Pr119846518281778721et_a_t @ F @ Xs ) )
      = ( Xs = nil_Pr7982011819685168954_set_a ) ) ).

% Nil_is_map_conv
thf(fact_1004_Nil__is__map__conv,axiom,
    ! [F: formula_a_t > $o,Xs: list_formula_a_t] :
      ( ( nil_o
        = ( map_formula_a_t_o @ F @ Xs ) )
      = ( Xs = nil_formula_a_t ) ) ).

% Nil_is_map_conv
thf(fact_1005_Nil__is__map__conv,axiom,
    ! [F: formula_a_t > vydra_aux_a_t_h,Xs: list_formula_a_t] :
      ( ( nil_vydra_aux_a_t_h
        = ( map_fo5746072578778194433_a_t_h @ F @ Xs ) )
      = ( Xs = nil_formula_a_t ) ) ).

% Nil_is_map_conv
thf(fact_1006_list_Omap__disc__iff,axiom,
    ! [F: formula_a_t > formula_a_t,A: list_formula_a_t] :
      ( ( ( map_fo8328018262729589004la_a_t @ F @ A )
        = nil_formula_a_t )
      = ( A = nil_formula_a_t ) ) ).

% list.map_disc_iff
thf(fact_1007_list_Omap__disc__iff,axiom,
    ! [F: product_prod_t_set_a > t,A: list_P1095827921439705018_set_a] :
      ( ( ( map_Pr119846518281778721et_a_t @ F @ A )
        = nil_t )
      = ( A = nil_Pr7982011819685168954_set_a ) ) ).

% list.map_disc_iff
thf(fact_1008_list_Omap__disc__iff,axiom,
    ! [F: formula_a_t > $o,A: list_formula_a_t] :
      ( ( ( map_formula_a_t_o @ F @ A )
        = nil_o )
      = ( A = nil_formula_a_t ) ) ).

% list.map_disc_iff
thf(fact_1009_list_Omap__disc__iff,axiom,
    ! [F: formula_a_t > vydra_aux_a_t_h,A: list_formula_a_t] :
      ( ( ( map_fo5746072578778194433_a_t_h @ F @ A )
        = nil_vydra_aux_a_t_h )
      = ( A = nil_formula_a_t ) ) ).

% list.map_disc_iff
thf(fact_1010_zip__eq__Nil__iff,axiom,
    ! [Xs: list_formula_a_t,Ys: list_formula_a_t] :
      ( ( ( zip_fo8638544115243153151la_a_t @ Xs @ Ys )
        = nil_Pr7564245829727862829la_a_t )
      = ( ( Xs = nil_formula_a_t )
        | ( Ys = nil_formula_a_t ) ) ) ).

% zip_eq_Nil_iff
thf(fact_1011_Nil__eq__zip__iff,axiom,
    ! [Xs: list_formula_a_t,Ys: list_formula_a_t] :
      ( ( nil_Pr7564245829727862829la_a_t
        = ( zip_fo8638544115243153151la_a_t @ Xs @ Ys ) )
      = ( ( Xs = nil_formula_a_t )
        | ( Ys = nil_formula_a_t ) ) ) ).

% Nil_eq_zip_iff
thf(fact_1012_length__0__conv,axiom,
    ! [Xs: list_P1095827921439705018_set_a] :
      ( ( ( size_s5362121731612881446_set_a @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_Pr7982011819685168954_set_a ) ) ).

% length_0_conv
thf(fact_1013_length__0__conv,axiom,
    ! [Xs: list_vydra_aux_a_t_h] :
      ( ( ( size_s8993897887676352572_a_t_h @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_vydra_aux_a_t_h ) ) ).

% length_0_conv
thf(fact_1014_length__0__conv,axiom,
    ! [Xs: list_iarray_o] :
      ( ( ( size_s3243743627286111600rray_o @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_iarray_o ) ) ).

% length_0_conv
thf(fact_1015_length__0__conv,axiom,
    ! [Xs: list_formula_a_t] :
      ( ( ( size_s8846756101701226951la_a_t @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_formula_a_t ) ) ).

% length_0_conv
thf(fact_1016_length__0__conv,axiom,
    ! [Xs: list_t] :
      ( ( ( size_size_list_t @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_t ) ) ).

% length_0_conv
thf(fact_1017_take0,axiom,
    ( ( take_formula_a_t @ zero_zero_nat )
    = ( ^ [Xs2: list_formula_a_t] : nil_formula_a_t ) ) ).

% take0
thf(fact_1018_take__eq__Nil,axiom,
    ! [N: nat,Xs: list_formula_a_t] :
      ( ( ( take_formula_a_t @ N @ Xs )
        = nil_formula_a_t )
      = ( ( N = zero_zero_nat )
        | ( Xs = nil_formula_a_t ) ) ) ).

% take_eq_Nil
thf(fact_1019_take__eq__Nil2,axiom,
    ! [N: nat,Xs: list_formula_a_t] :
      ( ( nil_formula_a_t
        = ( take_formula_a_t @ N @ Xs ) )
      = ( ( N = zero_zero_nat )
        | ( Xs = nil_formula_a_t ) ) ) ).

% take_eq_Nil2
thf(fact_1020_reach__run__subs__len,axiom,
    ! [N: nat,R2: regex_a_t,Rho3: list_iarray_o,Vs: list_vydra_aux_a_t_h] :
      ( ( reache2401979074735462992rray_o @ ( run_su3116026553151384105_h_t_o @ ( run_h_t_a @ run_hd @ N ) ) @ ( map_fo5746072578778194433_a_t_h @ ( sub_h_t_a @ init_hd @ run_hd @ N ) @ ( collect_subfmlas_a_t @ R2 @ nil_formula_a_t ) ) @ Rho3 @ Vs )
     => ( ( size_s8993897887676352572_a_t_h @ Vs )
        = ( size_s8846756101701226951la_a_t @ ( collect_subfmlas_a_t @ R2 @ nil_formula_a_t ) ) ) ) ).

% reach_run_subs_len
thf(fact_1021_length__greater__0__conv,axiom,
    ! [Xs: list_P1095827921439705018_set_a] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_s5362121731612881446_set_a @ Xs ) )
      = ( Xs != nil_Pr7982011819685168954_set_a ) ) ).

% length_greater_0_conv
thf(fact_1022_length__greater__0__conv,axiom,
    ! [Xs: list_vydra_aux_a_t_h] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_s8993897887676352572_a_t_h @ Xs ) )
      = ( Xs != nil_vydra_aux_a_t_h ) ) ).

% length_greater_0_conv
thf(fact_1023_length__greater__0__conv,axiom,
    ! [Xs: list_iarray_o] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_s3243743627286111600rray_o @ Xs ) )
      = ( Xs != nil_iarray_o ) ) ).

% length_greater_0_conv
thf(fact_1024_length__greater__0__conv,axiom,
    ! [Xs: list_formula_a_t] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_s8846756101701226951la_a_t @ Xs ) )
      = ( Xs != nil_formula_a_t ) ) ).

% length_greater_0_conv
thf(fact_1025_length__greater__0__conv,axiom,
    ! [Xs: list_t] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_t @ Xs ) )
      = ( Xs != nil_t ) ) ).

% length_greater_0_conv
thf(fact_1026_drop__all,axiom,
    ! [Xs: list_P1095827921439705018_set_a,N: nat] :
      ( ( ord_less_eq_nat @ ( size_s5362121731612881446_set_a @ Xs ) @ N )
     => ( ( drop_P283403782220237946_set_a @ N @ Xs )
        = nil_Pr7982011819685168954_set_a ) ) ).

% drop_all
thf(fact_1027_drop__all,axiom,
    ! [Xs: list_vydra_aux_a_t_h,N: nat] :
      ( ( ord_less_eq_nat @ ( size_s8993897887676352572_a_t_h @ Xs ) @ N )
     => ( ( drop_vydra_aux_a_t_h @ N @ Xs )
        = nil_vydra_aux_a_t_h ) ) ).

% drop_all
thf(fact_1028_drop__all,axiom,
    ! [Xs: list_iarray_o,N: nat] :
      ( ( ord_less_eq_nat @ ( size_s3243743627286111600rray_o @ Xs ) @ N )
     => ( ( drop_iarray_o @ N @ Xs )
        = nil_iarray_o ) ) ).

% drop_all
thf(fact_1029_drop__all,axiom,
    ! [Xs: list_formula_a_t,N: nat] :
      ( ( ord_less_eq_nat @ ( size_s8846756101701226951la_a_t @ Xs ) @ N )
     => ( ( drop_formula_a_t @ N @ Xs )
        = nil_formula_a_t ) ) ).

% drop_all
thf(fact_1030_drop__all,axiom,
    ! [Xs: list_t,N: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_t @ Xs ) @ N )
     => ( ( drop_t @ N @ Xs )
        = nil_t ) ) ).

% drop_all
thf(fact_1031_drop__eq__Nil,axiom,
    ! [N: nat,Xs: list_P1095827921439705018_set_a] :
      ( ( ( drop_P283403782220237946_set_a @ N @ Xs )
        = nil_Pr7982011819685168954_set_a )
      = ( ord_less_eq_nat @ ( size_s5362121731612881446_set_a @ Xs ) @ N ) ) ).

% drop_eq_Nil
thf(fact_1032_drop__eq__Nil,axiom,
    ! [N: nat,Xs: list_vydra_aux_a_t_h] :
      ( ( ( drop_vydra_aux_a_t_h @ N @ Xs )
        = nil_vydra_aux_a_t_h )
      = ( ord_less_eq_nat @ ( size_s8993897887676352572_a_t_h @ Xs ) @ N ) ) ).

% drop_eq_Nil
thf(fact_1033_drop__eq__Nil,axiom,
    ! [N: nat,Xs: list_iarray_o] :
      ( ( ( drop_iarray_o @ N @ Xs )
        = nil_iarray_o )
      = ( ord_less_eq_nat @ ( size_s3243743627286111600rray_o @ Xs ) @ N ) ) ).

% drop_eq_Nil
thf(fact_1034_drop__eq__Nil,axiom,
    ! [N: nat,Xs: list_formula_a_t] :
      ( ( ( drop_formula_a_t @ N @ Xs )
        = nil_formula_a_t )
      = ( ord_less_eq_nat @ ( size_s8846756101701226951la_a_t @ Xs ) @ N ) ) ).

% drop_eq_Nil
thf(fact_1035_drop__eq__Nil,axiom,
    ! [N: nat,Xs: list_t] :
      ( ( ( drop_t @ N @ Xs )
        = nil_t )
      = ( ord_less_eq_nat @ ( size_size_list_t @ Xs ) @ N ) ) ).

% drop_eq_Nil
thf(fact_1036_drop__eq__Nil2,axiom,
    ! [N: nat,Xs: list_P1095827921439705018_set_a] :
      ( ( nil_Pr7982011819685168954_set_a
        = ( drop_P283403782220237946_set_a @ N @ Xs ) )
      = ( ord_less_eq_nat @ ( size_s5362121731612881446_set_a @ Xs ) @ N ) ) ).

% drop_eq_Nil2
thf(fact_1037_drop__eq__Nil2,axiom,
    ! [N: nat,Xs: list_vydra_aux_a_t_h] :
      ( ( nil_vydra_aux_a_t_h
        = ( drop_vydra_aux_a_t_h @ N @ Xs ) )
      = ( ord_less_eq_nat @ ( size_s8993897887676352572_a_t_h @ Xs ) @ N ) ) ).

% drop_eq_Nil2
thf(fact_1038_drop__eq__Nil2,axiom,
    ! [N: nat,Xs: list_iarray_o] :
      ( ( nil_iarray_o
        = ( drop_iarray_o @ N @ Xs ) )
      = ( ord_less_eq_nat @ ( size_s3243743627286111600rray_o @ Xs ) @ N ) ) ).

% drop_eq_Nil2
thf(fact_1039_drop__eq__Nil2,axiom,
    ! [N: nat,Xs: list_formula_a_t] :
      ( ( nil_formula_a_t
        = ( drop_formula_a_t @ N @ Xs ) )
      = ( ord_less_eq_nat @ ( size_s8846756101701226951la_a_t @ Xs ) @ N ) ) ).

% drop_eq_Nil2
thf(fact_1040_drop__eq__Nil2,axiom,
    ! [N: nat,Xs: list_t] :
      ( ( nil_t
        = ( drop_t @ N @ Xs ) )
      = ( ord_less_eq_nat @ ( size_size_list_t @ Xs ) @ N ) ) ).

% drop_eq_Nil2
thf(fact_1041_reach__run__subs__run,axiom,
    ! [N: nat,R2: regex_a_t,Rho3: list_iarray_o,Vs: list_vydra_aux_a_t_h,J: nat,Phi2: formula_a_t] :
      ( ( reache2401979074735462992rray_o @ ( run_su3116026553151384105_h_t_o @ ( run_h_t_a @ run_hd @ N ) ) @ ( map_fo5746072578778194433_a_t_h @ ( sub_h_t_a @ init_hd @ run_hd @ N ) @ ( collect_subfmlas_a_t @ R2 @ nil_formula_a_t ) ) @ Rho3 @ Vs )
     => ( ( ord_less_nat @ J @ ( size_s8846756101701226951la_a_t @ ( collect_subfmlas_a_t @ R2 @ nil_formula_a_t ) ) )
       => ( ( Phi2
            = ( nth_formula_a_t @ ( collect_subfmlas_a_t @ R2 @ nil_formula_a_t ) @ J ) )
         => ? [Rho4: list_P5575445741472356340od_t_o] :
              ( ( reache8691153155344877136od_t_o @ ( run_h_t_a @ run_hd @ N ) @ ( sub_h_t_a @ init_hd @ run_hd @ N @ Phi2 ) @ Rho4 @ ( nth_vydra_aux_a_t_h @ Vs @ J ) )
              & ( ( size_s4106575409306904672od_t_o @ Rho4 )
                = ( size_s3243743627286111600rray_o @ Rho3 ) ) ) ) ) ) ).

% reach_run_subs_run
thf(fact_1042_length__collect__subfmlas,axiom,
    ! [Phis: list_formula_a_t,R2: regex_a_t] : ( ord_less_eq_nat @ ( size_s8846756101701226951la_a_t @ Phis ) @ ( size_s8846756101701226951la_a_t @ ( collect_subfmlas_a_t @ R2 @ Phis ) ) ) ).

% length_collect_subfmlas
thf(fact_1043_run__t_Osimps_I1_J,axiom,
    ! [Run_hd: h > option5775791656947114332_set_a] :
      ( ( run_t_h_t_a @ Run_hd @ none_P849260865442546644od_h_t )
      = none_P797220180637652640_h_t_t ) ).

% run_t.simps(1)
thf(fact_1044_reaches__on__init__Some,axiom,
    ! [R2: h > option5775791656947114332_set_a,S: h,Xs: list_P1095827921439705018_set_a,S2: h] :
      ( ( reache3397181163414509427_set_a @ R2 @ S @ Xs @ S2 )
     => ( ( ( R2 @ S2 )
         != none_P2868508613186038903_set_a )
       => ( ( R2 @ S )
         != none_P2868508613186038903_set_a ) ) ) ).

% reaches_on_init_Some
thf(fact_1045_reaches__on__init__Some,axiom,
    ! [R2: option2110264179601050681od_h_t > option4241632866121027589_h_t_t,S: option2110264179601050681od_h_t,Xs: list_t,S2: option2110264179601050681od_h_t] :
      ( ( reache645946503560514460_h_t_t @ R2 @ S @ Xs @ S2 )
     => ( ( ( R2 @ S2 )
         != none_P797220180637652640_h_t_t )
       => ( ( R2 @ S )
         != none_P797220180637652640_h_t_t ) ) ) ).

% reaches_on_init_Some
thf(fact_1046_reaches__on__init__Some,axiom,
    ! [R2: list_vydra_aux_a_t_h > option8553266494716013595rray_o,S: list_vydra_aux_a_t_h,Xs: list_iarray_o,S2: list_vydra_aux_a_t_h] :
      ( ( reache2401979074735462992rray_o @ R2 @ S @ Xs @ S2 )
     => ( ( ( R2 @ S2 )
         != none_P945442321178488234rray_o )
       => ( ( R2 @ S )
         != none_P945442321178488234rray_o ) ) ) ).

% reaches_on_init_Some
thf(fact_1047_reaches__on__init__Some,axiom,
    ! [R2: vydra_aux_a_t_h > option4580262698044125979od_t_o,S: vydra_aux_a_t_h,Xs: list_P5575445741472356340od_t_o,S2: vydra_aux_a_t_h] :
      ( ( reache8691153155344877136od_t_o @ R2 @ S @ Xs @ S2 )
     => ( ( ( R2 @ S2 )
         != none_P3452577240289636010od_t_o )
       => ( ( R2 @ S )
         != none_P3452577240289636010od_t_o ) ) ) ).

% reaches_on_init_Some
thf(fact_1048_reaches__on__init__Some,axiom,
    ! [R2: t > option8291060355354214836od_t_o,S: t,Xs: list_o,S2: t] :
      ( ( reaches_on_t_o @ R2 @ S @ Xs @ S2 )
     => ( ( ( R2 @ S2 )
         != none_P7032077165169035727od_t_o )
       => ( ( R2 @ S )
         != none_P7032077165169035727od_t_o ) ) ) ).

% reaches_on_init_Some
thf(fact_1049_reaches__on__NilD,axiom,
    ! [Run: h > option5775791656947114332_set_a,Q3: h,Q4: h] :
      ( ( reache3397181163414509427_set_a @ Run @ Q3 @ nil_Pr7982011819685168954_set_a @ Q4 )
     => ( Q3 = Q4 ) ) ).

% reaches_on_NilD
thf(fact_1050_reaches__on__NilD,axiom,
    ! [Run: option2110264179601050681od_h_t > option4241632866121027589_h_t_t,Q3: option2110264179601050681od_h_t,Q4: option2110264179601050681od_h_t] :
      ( ( reache645946503560514460_h_t_t @ Run @ Q3 @ nil_t @ Q4 )
     => ( Q3 = Q4 ) ) ).

% reaches_on_NilD
thf(fact_1051_reaches__on__NilD,axiom,
    ! [Run: list_vydra_aux_a_t_h > option8553266494716013595rray_o,Q3: list_vydra_aux_a_t_h,Q4: list_vydra_aux_a_t_h] :
      ( ( reache2401979074735462992rray_o @ Run @ Q3 @ nil_iarray_o @ Q4 )
     => ( Q3 = Q4 ) ) ).

% reaches_on_NilD
thf(fact_1052_reaches__on__NilD,axiom,
    ! [Run: vydra_aux_a_t_h > option4580262698044125979od_t_o,Q3: vydra_aux_a_t_h,Q4: vydra_aux_a_t_h] :
      ( ( reache8691153155344877136od_t_o @ Run @ Q3 @ nil_Product_prod_t_o @ Q4 )
     => ( Q3 = Q4 ) ) ).

% reaches_on_NilD
thf(fact_1053_reaches__on_Ointros_I1_J,axiom,
    ! [Run: h > option5775791656947114332_set_a,S: h] : ( reache3397181163414509427_set_a @ Run @ S @ nil_Pr7982011819685168954_set_a @ S ) ).

% reaches_on.intros(1)
thf(fact_1054_reaches__on_Ointros_I1_J,axiom,
    ! [Run: option2110264179601050681od_h_t > option4241632866121027589_h_t_t,S: option2110264179601050681od_h_t] : ( reache645946503560514460_h_t_t @ Run @ S @ nil_t @ S ) ).

% reaches_on.intros(1)
thf(fact_1055_reaches__on_Ointros_I1_J,axiom,
    ! [Run: list_vydra_aux_a_t_h > option8553266494716013595rray_o,S: list_vydra_aux_a_t_h] : ( reache2401979074735462992rray_o @ Run @ S @ nil_iarray_o @ S ) ).

% reaches_on.intros(1)
thf(fact_1056_reaches__on_Ointros_I1_J,axiom,
    ! [Run: vydra_aux_a_t_h > option4580262698044125979od_t_o,S: vydra_aux_a_t_h] : ( reache8691153155344877136od_t_o @ Run @ S @ nil_Product_prod_t_o @ S ) ).

% reaches_on.intros(1)
thf(fact_1057_drop__Nil,axiom,
    ! [N: nat] :
      ( ( drop_formula_a_t @ N @ nil_formula_a_t )
      = nil_formula_a_t ) ).

% drop_Nil
thf(fact_1058_list_Osimps_I8_J,axiom,
    ! [F: formula_a_t > formula_a_t] :
      ( ( map_fo8328018262729589004la_a_t @ F @ nil_formula_a_t )
      = nil_formula_a_t ) ).

% list.simps(8)
thf(fact_1059_list_Osimps_I8_J,axiom,
    ! [F: product_prod_t_set_a > t] :
      ( ( map_Pr119846518281778721et_a_t @ F @ nil_Pr7982011819685168954_set_a )
      = nil_t ) ).

% list.simps(8)
thf(fact_1060_list_Osimps_I8_J,axiom,
    ! [F: formula_a_t > $o] :
      ( ( map_formula_a_t_o @ F @ nil_formula_a_t )
      = nil_o ) ).

% list.simps(8)
thf(fact_1061_list_Osimps_I8_J,axiom,
    ! [F: formula_a_t > vydra_aux_a_t_h] :
      ( ( map_fo5746072578778194433_a_t_h @ F @ nil_formula_a_t )
      = nil_vydra_aux_a_t_h ) ).

% list.simps(8)
thf(fact_1062_take__Nil,axiom,
    ! [N: nat] :
      ( ( take_formula_a_t @ N @ nil_formula_a_t )
      = nil_formula_a_t ) ).

% take_Nil
thf(fact_1063_listrelp_ONil,axiom,
    ! [R2: formula_a_t > formula_a_t > $o] : ( listre5601473105921689735la_a_t @ R2 @ nil_formula_a_t @ nil_formula_a_t ) ).

% listrelp.Nil
thf(fact_1064_VYDRA__MDL_Oreach__run__subs__len,axiom,
    ! [Sigma: trace_a_t,Init_hd: h,Run_hd: h > option5775791656947114332_set_a,N: nat,R2: regex_a_t,Rho3: list_iarray_o,Vs: list_vydra_aux_a_t_h] :
      ( ( vYDRA_MDL_a_t_h @ Sigma @ Init_hd @ Run_hd )
     => ( ( reache2401979074735462992rray_o @ ( run_su3116026553151384105_h_t_o @ ( run_h_t_a @ Run_hd @ N ) ) @ ( map_fo5746072578778194433_a_t_h @ ( sub_h_t_a @ Init_hd @ Run_hd @ N ) @ ( collect_subfmlas_a_t @ R2 @ nil_formula_a_t ) ) @ Rho3 @ Vs )
       => ( ( size_s8993897887676352572_a_t_h @ Vs )
          = ( size_s8846756101701226951la_a_t @ ( collect_subfmlas_a_t @ R2 @ nil_formula_a_t ) ) ) ) ) ).

% VYDRA_MDL.reach_run_subs_len
thf(fact_1065_collect__subfmlas__size,axiom,
    ! [X3: formula_a_t,R2: regex_a_t] :
      ( ( member_formula_a_t @ X3 @ ( set_formula_a_t2 @ ( collect_subfmlas_a_t @ R2 @ nil_formula_a_t ) ) )
     => ( ord_less_nat @ ( size_s4016968051272393527la_a_t @ X3 ) @ ( size_size_regex_a_t @ R2 ) ) ) ).

% collect_subfmlas_size
thf(fact_1066_VYDRA__MDL_Oreach__run__subs__run,axiom,
    ! [Sigma: trace_a_t,Init_hd: h,Run_hd: h > option5775791656947114332_set_a,N: nat,R2: regex_a_t,Rho3: list_iarray_o,Vs: list_vydra_aux_a_t_h,J: nat,Phi2: formula_a_t] :
      ( ( vYDRA_MDL_a_t_h @ Sigma @ Init_hd @ Run_hd )
     => ( ( reache2401979074735462992rray_o @ ( run_su3116026553151384105_h_t_o @ ( run_h_t_a @ Run_hd @ N ) ) @ ( map_fo5746072578778194433_a_t_h @ ( sub_h_t_a @ Init_hd @ Run_hd @ N ) @ ( collect_subfmlas_a_t @ R2 @ nil_formula_a_t ) ) @ Rho3 @ Vs )
       => ( ( ord_less_nat @ J @ ( size_s8846756101701226951la_a_t @ ( collect_subfmlas_a_t @ R2 @ nil_formula_a_t ) ) )
         => ( ( Phi2
              = ( nth_formula_a_t @ ( collect_subfmlas_a_t @ R2 @ nil_formula_a_t ) @ J ) )
           => ? [Rho4: list_P5575445741472356340od_t_o] :
                ( ( reache8691153155344877136od_t_o @ ( run_h_t_a @ Run_hd @ N ) @ ( sub_h_t_a @ Init_hd @ Run_hd @ N @ Phi2 ) @ Rho4 @ ( nth_vydra_aux_a_t_h @ Vs @ J ) )
                & ( ( size_s4106575409306904672od_t_o @ Rho4 )
                  = ( size_s3243743627286111600rray_o @ Rho3 ) ) ) ) ) ) ) ).

% VYDRA_MDL.reach_run_subs_run
thf(fact_1067_read__t_Ocases,axiom,
    ! [X3: option7421465107808516430_o_nat] :
      ( ( X3 != none_P4770284390835442525_o_nat )
     => ~ ! [E: nat > $o,T5: nat] :
            ( X3
           != ( some_P2630223654056210137_o_nat @ ( produc7277522915581678840_o_nat @ E @ T5 ) ) ) ) ).

% read_t.cases
thf(fact_1068_read__t_Ocases,axiom,
    ! [X3: option2110264179601050681od_h_t] :
      ( ( X3 != none_P849260865442546644od_h_t )
     => ~ ! [E: h,T5: t] :
            ( X3
           != ( some_P2526622376857543512od_h_t @ ( product_Pair_h_t @ E @ T5 ) ) ) ) ).

% read_t.cases
thf(fact_1069_list_Osize_I3_J,axiom,
    ( ( size_s5362121731612881446_set_a @ nil_Pr7982011819685168954_set_a )
    = zero_zero_nat ) ).

% list.size(3)
thf(fact_1070_list_Osize_I3_J,axiom,
    ( ( size_s8993897887676352572_a_t_h @ nil_vydra_aux_a_t_h )
    = zero_zero_nat ) ).

% list.size(3)
thf(fact_1071_list_Osize_I3_J,axiom,
    ( ( size_s3243743627286111600rray_o @ nil_iarray_o )
    = zero_zero_nat ) ).

% list.size(3)
thf(fact_1072_list_Osize_I3_J,axiom,
    ( ( size_s8846756101701226951la_a_t @ nil_formula_a_t )
    = zero_zero_nat ) ).

% list.size(3)
thf(fact_1073_list_Osize_I3_J,axiom,
    ( ( size_size_list_t @ nil_t )
    = zero_zero_nat ) ).

% list.size(3)
thf(fact_1074_reaches__on__len,axiom,
    ! [Run: vydra_aux_a_t_h > option4580262698044125979od_t_o,S: vydra_aux_a_t_h,N: nat] :
    ? [Vs3: list_P5575445741472356340od_t_o,S6: vydra_aux_a_t_h] :
      ( ( reache8691153155344877136od_t_o @ Run @ S @ Vs3 @ S6 )
      & ( ( ( size_s4106575409306904672od_t_o @ Vs3 )
          = N )
        | ( ( Run @ S6 )
          = none_P3452577240289636010od_t_o ) ) ) ).

% reaches_on_len
thf(fact_1075_reaches__on__len,axiom,
    ! [Run: t > option8291060355354214836od_t_o,S: t,N: nat] :
    ? [Vs3: list_o,S6: t] :
      ( ( reaches_on_t_o @ Run @ S @ Vs3 @ S6 )
      & ( ( ( size_size_list_o @ Vs3 )
          = N )
        | ( ( Run @ S6 )
          = none_P7032077165169035727od_t_o ) ) ) ).

% reaches_on_len
thf(fact_1076_reaches__on__len,axiom,
    ! [Run: h > option5775791656947114332_set_a,S: h,N: nat] :
    ? [Vs3: list_P1095827921439705018_set_a,S6: h] :
      ( ( reache3397181163414509427_set_a @ Run @ S @ Vs3 @ S6 )
      & ( ( ( size_s5362121731612881446_set_a @ Vs3 )
          = N )
        | ( ( Run @ S6 )
          = none_P2868508613186038903_set_a ) ) ) ).

% reaches_on_len
thf(fact_1077_reaches__on__len,axiom,
    ! [Run: list_vydra_aux_a_t_h > option8553266494716013595rray_o,S: list_vydra_aux_a_t_h,N: nat] :
    ? [Vs3: list_iarray_o,S6: list_vydra_aux_a_t_h] :
      ( ( reache2401979074735462992rray_o @ Run @ S @ Vs3 @ S6 )
      & ( ( ( size_s3243743627286111600rray_o @ Vs3 )
          = N )
        | ( ( Run @ S6 )
          = none_P945442321178488234rray_o ) ) ) ).

% reaches_on_len
thf(fact_1078_reaches__on__len,axiom,
    ! [Run: option2110264179601050681od_h_t > option4241632866121027589_h_t_t,S: option2110264179601050681od_h_t,N: nat] :
    ? [Vs3: list_t,S6: option2110264179601050681od_h_t] :
      ( ( reache645946503560514460_h_t_t @ Run @ S @ Vs3 @ S6 )
      & ( ( ( size_size_list_t @ Vs3 )
          = N )
        | ( ( Run @ S6 )
          = none_P797220180637652640_h_t_t ) ) ) ).

% reaches_on_len
thf(fact_1079_take__0,axiom,
    ! [Xs: list_formula_a_t] :
      ( ( take_formula_a_t @ zero_zero_nat @ Xs )
      = nil_formula_a_t ) ).

% take_0
thf(fact_1080_listrel__Nil2,axiom,
    ! [Xs: list_formula_a_t,R2: set_Pr3703585533673849735la_a_t] :
      ( ( member5181786688746205936la_a_t @ ( produc7251354486509092607la_a_t @ Xs @ nil_formula_a_t ) @ ( listre5244141508577473849la_a_t @ R2 ) )
     => ( Xs = nil_formula_a_t ) ) ).

% listrel_Nil2
thf(fact_1081_listrel__Nil1,axiom,
    ! [Xs: list_formula_a_t,R2: set_Pr3703585533673849735la_a_t] :
      ( ( member5181786688746205936la_a_t @ ( produc7251354486509092607la_a_t @ nil_formula_a_t @ Xs ) @ ( listre5244141508577473849la_a_t @ R2 ) )
     => ( Xs = nil_formula_a_t ) ) ).

% listrel_Nil1
thf(fact_1082_listrel_ONil,axiom,
    ! [R2: set_Pr3703585533673849735la_a_t] : ( member5181786688746205936la_a_t @ ( produc7251354486509092607la_a_t @ nil_formula_a_t @ nil_formula_a_t ) @ ( listre5244141508577473849la_a_t @ R2 ) ) ).

% listrel.Nil
thf(fact_1083_Nil__notin__lex,axiom,
    ! [Ys: list_formula_a_t,R2: set_Pr3703585533673849735la_a_t] :
      ~ ( member5181786688746205936la_a_t @ ( produc7251354486509092607la_a_t @ nil_formula_a_t @ Ys ) @ ( lex_formula_a_t @ R2 ) ) ).

% Nil_notin_lex
thf(fact_1084_Nil2__notin__lex,axiom,
    ! [Xs: list_formula_a_t,R2: set_Pr3703585533673849735la_a_t] :
      ~ ( member5181786688746205936la_a_t @ ( produc7251354486509092607la_a_t @ Xs @ nil_formula_a_t ) @ ( lex_formula_a_t @ R2 ) ) ).

% Nil2_notin_lex
thf(fact_1085_find_Osimps_I1_J,axiom,
    ! [Uu: formula_a_t > $o] :
      ( ( find_formula_a_t @ Uu @ nil_formula_a_t )
      = none_formula_a_t ) ).

% find.simps(1)
thf(fact_1086_find_Osimps_I1_J,axiom,
    ! [Uu: nat > $o] :
      ( ( find_nat @ Uu @ nil_nat )
      = none_nat ) ).

% find.simps(1)
thf(fact_1087_find_Osimps_I1_J,axiom,
    ! [Uu: produc530307916048688075od_t_o > $o] :
      ( ( find_P8232462763046280367od_t_o @ Uu @ nil_Pr3088597731192347973od_t_o )
      = none_P3452577240289636010od_t_o ) ).

% find.simps(1)
thf(fact_1088_find_Osimps_I1_J,axiom,
    ! [Uu: product_prod_t_o > $o] :
      ( ( find_P3690432262111017226od_t_o @ Uu @ nil_Product_prod_t_o )
      = none_P7032077165169035727od_t_o ) ).

% find.simps(1)
thf(fact_1089_VYDRA__MDL_Obs__sat,axiom,
    ! [Sigma: trace_a_t,Init_hd: h,Run_hd: h > option5775791656947114332_set_a,R2: regex_a_t,N: nat,I: nat,Vs: list_vydra_aux_a_t_h,Vs2: list_vydra_aux_a_t_h,Bs: iarray_o] :
      ( ( vYDRA_MDL_a_t_h @ Sigma @ Init_hd @ Run_hd )
     => ( ! [Phi4: formula_a_t,I2: nat,V3: vydra_aux_a_t_h,V5: vydra_aux_a_t_h,B3: product_prod_t_o] :
            ( ( member_formula_a_t @ Phi4 @ ( set_formula_a_t2 @ ( collect_subfmlas_a_t @ R2 @ nil_formula_a_t ) ) )
           => ( ( vYDRA_wf_vydra_a_t_h @ Sigma @ Init_hd @ Run_hd @ Phi4 @ I2 @ N @ V3 )
             => ( ( ( run_h_t_a @ Run_hd @ N @ V3 )
                  = ( some_P9127227083975130150od_t_o @ ( produc5956389175556509053od_t_o @ V5 @ B3 ) ) )
               => ( ( product_snd_t_o @ B3 )
                  = ( sat_a_t @ Sigma @ Phi4 @ I2 ) ) ) ) )
       => ( ! [J2: nat] :
              ( ( ord_less_nat @ J2 @ ( size_s8846756101701226951la_a_t @ ( collect_subfmlas_a_t @ R2 @ nil_formula_a_t ) ) )
             => ( vYDRA_wf_vydra_a_t_h @ Sigma @ Init_hd @ Run_hd @ ( nth_formula_a_t @ ( collect_subfmlas_a_t @ R2 @ nil_formula_a_t ) @ J2 ) @ I @ N @ ( nth_vydra_aux_a_t_h @ Vs @ J2 ) ) )
         => ( ( ( run_su3116026553151384105_h_t_o @ ( run_h_t_a @ Run_hd @ N ) @ Vs )
              = ( some_P7279874174034738982rray_o @ ( produc8472968102330193021rray_o @ Vs2 @ Bs ) ) )
           => ( ( ( size_s8993897887676352572_a_t_h @ Vs )
                = ( size_s8846756101701226951la_a_t @ ( collect_subfmlas_a_t @ R2 @ nil_formula_a_t ) ) )
             => ( Bs
                = ( iarray_of_list_o
                  @ ( map_formula_a_t_o
                    @ ^ [Phi5: formula_a_t] : ( sat_a_t @ Sigma @ Phi5 @ I )
                    @ ( collect_subfmlas_a_t @ R2 @ nil_formula_a_t ) ) ) ) ) ) ) ) ) ).

% VYDRA_MDL.bs_sat
thf(fact_1090_bounded__Max__nat,axiom,
    ! [P: nat > $o,X3: nat,M7: nat] :
      ( ( P @ X3 )
     => ( ! [X: nat] :
            ( ( P @ X )
           => ( ord_less_eq_nat @ X @ M7 ) )
       => ~ ! [M3: nat] :
              ( ( P @ M3 )
             => ~ ! [X10: nat] :
                    ( ( P @ X10 )
                   => ( ord_less_eq_nat @ X10 @ M3 ) ) ) ) ) ).

% bounded_Max_nat
thf(fact_1091_VYDRA__MDL_Olast__before_Opelims,axiom,
    ! [Sigma: trace_a_t,Init_hd: h,Run_hd: h > option5775791656947114332_set_a,X3: nat > $o,Xa: nat,Y: option_nat] :
      ( ( vYDRA_MDL_a_t_h @ Sigma @ Init_hd @ Run_hd )
     => ( ( ( vYDRA_last_before @ X3 @ Xa )
          = Y )
       => ( ( accp_P7902722568424861365_o_nat @ vYDRA_5175910486691664417re_rel @ ( produc7277522915581678840_o_nat @ X3 @ Xa ) )
         => ( ( ( Xa = zero_zero_nat )
             => ( ( Y = none_nat )
               => ~ ( accp_P7902722568424861365_o_nat @ vYDRA_5175910486691664417re_rel @ ( produc7277522915581678840_o_nat @ X3 @ zero_zero_nat ) ) ) )
           => ~ ! [N3: nat] :
                  ( ( Xa
                    = ( suc @ N3 ) )
                 => ( ( ( ( X3 @ N3 )
                       => ( Y
                          = ( some_nat @ N3 ) ) )
                      & ( ~ ( X3 @ N3 )
                       => ( Y
                          = ( vYDRA_last_before @ X3 @ N3 ) ) ) )
                   => ~ ( accp_P7902722568424861365_o_nat @ vYDRA_5175910486691664417re_rel @ ( produc7277522915581678840_o_nat @ X3 @ ( suc @ N3 ) ) ) ) ) ) ) ) ) ).

% VYDRA_MDL.last_before.pelims
thf(fact_1092_set__n__lists,axiom,
    ! [N: nat,Xs: list_P1095827921439705018_set_a] :
      ( ( set_li5089069168345991631_set_a @ ( n_list7540261771967413058_set_a @ N @ Xs ) )
      = ( collec5706338785728018085_set_a
        @ ^ [Ys2: list_P1095827921439705018_set_a] :
            ( ( ( size_s5362121731612881446_set_a @ Ys2 )
              = N )
            & ( ord_le3254057565153606196_set_a @ ( set_Pr15456023412717001_set_a @ Ys2 ) @ ( set_Pr15456023412717001_set_a @ Xs ) ) ) ) ) ).

% set_n_lists
thf(fact_1093_set__n__lists,axiom,
    ! [N: nat,Xs: list_vydra_aux_a_t_h] :
      ( ( set_li3104226997595317331_a_t_h @ ( n_list420448431600190986_a_t_h @ N @ Xs ) )
      = ( collec4460019958060512765_a_t_h
        @ ^ [Ys2: list_vydra_aux_a_t_h] :
            ( ( ( size_s8993897887676352572_a_t_h @ Ys2 )
              = N )
            & ( ord_le7451535499628331182_a_t_h @ ( set_vydra_aux_a_t_h2 @ Ys2 ) @ ( set_vydra_aux_a_t_h2 @ Xs ) ) ) ) ) ).

% set_n_lists
thf(fact_1094_set__n__lists,axiom,
    ! [N: nat,Xs: list_iarray_o] :
      ( ( set_list_iarray_o2 @ ( n_lists_iarray_o @ N @ Xs ) )
      = ( collec7735512679378346479rray_o
        @ ^ [Ys2: list_iarray_o] :
            ( ( ( size_s3243743627286111600rray_o @ Ys2 )
              = N )
            & ( ord_le6505580897047731582rray_o @ ( set_iarray_o2 @ Ys2 ) @ ( set_iarray_o2 @ Xs ) ) ) ) ) ).

% set_n_lists
thf(fact_1095_set__n__lists,axiom,
    ! [N: nat,Xs: list_formula_a_t] :
      ( ( set_list_formula_a_t2 @ ( n_lists_formula_a_t @ N @ Xs ) )
      = ( collec2153012532608959368la_a_t
        @ ^ [Ys2: list_formula_a_t] :
            ( ( ( size_s8846756101701226951la_a_t @ Ys2 )
              = N )
            & ( ord_le7457455060544393785la_a_t @ ( set_formula_a_t2 @ Ys2 ) @ ( set_formula_a_t2 @ Xs ) ) ) ) ) ).

% set_n_lists
thf(fact_1096_set__n__lists,axiom,
    ! [N: nat,Xs: list_t] :
      ( ( set_list_t2 @ ( n_lists_t @ N @ Xs ) )
      = ( collect_list_t
        @ ^ [Ys2: list_t] :
            ( ( ( size_size_list_t @ Ys2 )
              = N )
            & ( ord_less_eq_set_t @ ( set_t2 @ Ys2 ) @ ( set_t2 @ Xs ) ) ) ) ) ).

% set_n_lists
thf(fact_1097_collect__subfmlas__msize,axiom,
    ! [X3: formula_a_t,R2: regex_a_t] :
      ( ( member_formula_a_t @ X3 @ ( set_formula_a_t2 @ ( collect_subfmlas_a_t @ R2 @ nil_formula_a_t ) ) )
     => ( ord_less_eq_nat @ ( msize_fmla_a_t @ X3 ) @ ( msize_regex_a_t @ R2 ) ) ) ).

% collect_subfmlas_msize
thf(fact_1098_option_Osize__gen_I1_J,axiom,
    ! [X3: nat > nat] :
      ( ( size_option_nat @ X3 @ none_nat )
      = ( suc @ zero_zero_nat ) ) ).

% option.size_gen(1)
thf(fact_1099_option_Osize__gen_I1_J,axiom,
    ! [X3: produc530307916048688075od_t_o > nat] :
      ( ( size_o2629515801175699432od_t_o @ X3 @ none_P3452577240289636010od_t_o )
      = ( suc @ zero_zero_nat ) ) ).

% option.size_gen(1)
thf(fact_1100_option_Osize__gen_I1_J,axiom,
    ! [X3: product_prod_t_o > nat] :
      ( ( size_o5219344772683989009od_t_o @ X3 @ none_P7032077165169035727od_t_o )
      = ( suc @ zero_zero_nat ) ) ).

% option.size_gen(1)
thf(fact_1101_Id__on__set,axiom,
    ! [Xs: list_formula_a_t] :
      ( ( id_on_formula_a_t @ ( set_formula_a_t2 @ Xs ) )
      = ( set_Pr507822682862612540la_a_t
        @ ( map_fo1213494298497741438la_a_t
          @ ^ [X5: formula_a_t] : ( produc8281305675666354143la_a_t @ X5 @ X5 )
          @ Xs ) ) ) ).

% Id_on_set
thf(fact_1102_Id__onI,axiom,
    ! [A: a,A4: set_a] :
      ( ( member_a @ A @ A4 )
     => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ A ) @ ( id_on_a @ A4 ) ) ) ).

% Id_onI
thf(fact_1103_Id__onI,axiom,
    ! [A: formula_a_t,A4: set_formula_a_t] :
      ( ( member_formula_a_t @ A @ A4 )
     => ( member9065953759106212816la_a_t @ ( produc8281305675666354143la_a_t @ A @ A ) @ ( id_on_formula_a_t @ A4 ) ) ) ).

% Id_onI
thf(fact_1104_Id__onI,axiom,
    ! [A: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A @ A4 )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ A ) @ ( id_on_2554058798563519774at_nat @ A4 ) ) ) ).

% Id_onI
thf(fact_1105_Id__onI,axiom,
    ! [A: nat,A4: set_nat] :
      ( ( member_nat @ A @ A4 )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A ) @ ( id_on_nat @ A4 ) ) ) ).

% Id_onI
thf(fact_1106_Id__onE,axiom,
    ! [C2: product_prod_a_a,A4: set_a] :
      ( ( member1426531477525435216od_a_a @ C2 @ ( id_on_a @ A4 ) )
     => ~ ! [X: a] :
            ( ( member_a @ X @ A4 )
           => ( C2
             != ( product_Pair_a_a @ X @ X ) ) ) ) ).

% Id_onE
thf(fact_1107_Id__onE,axiom,
    ! [C2: produc2959973871163602727la_a_t,A4: set_formula_a_t] :
      ( ( member9065953759106212816la_a_t @ C2 @ ( id_on_formula_a_t @ A4 ) )
     => ~ ! [X: formula_a_t] :
            ( ( member_formula_a_t @ X @ A4 )
           => ( C2
             != ( produc8281305675666354143la_a_t @ X @ X ) ) ) ) ).

% Id_onE
thf(fact_1108_Id__onE,axiom,
    ! [C2: produc859450856879609959at_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ C2 @ ( id_on_2554058798563519774at_nat @ A4 ) )
     => ~ ! [X: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X @ A4 )
           => ( C2
             != ( produc6161850002892822231at_nat @ X @ X ) ) ) ) ).

% Id_onE
thf(fact_1109_Id__onE,axiom,
    ! [C2: product_prod_nat_nat,A4: set_nat] :
      ( ( member8440522571783428010at_nat @ C2 @ ( id_on_nat @ A4 ) )
     => ~ ! [X: nat] :
            ( ( member_nat @ X @ A4 )
           => ( C2
             != ( product_Pair_nat_nat @ X @ X ) ) ) ) ).

% Id_onE
thf(fact_1110_Id__on__eqI,axiom,
    ! [A: a,B: a,A4: set_a] :
      ( ( A = B )
     => ( ( member_a @ A @ A4 )
       => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ ( id_on_a @ A4 ) ) ) ) ).

% Id_on_eqI
thf(fact_1111_Id__on__eqI,axiom,
    ! [A: formula_a_t,B: formula_a_t,A4: set_formula_a_t] :
      ( ( A = B )
     => ( ( member_formula_a_t @ A @ A4 )
       => ( member9065953759106212816la_a_t @ ( produc8281305675666354143la_a_t @ A @ B ) @ ( id_on_formula_a_t @ A4 ) ) ) ) ).

% Id_on_eqI
thf(fact_1112_Id__on__eqI,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( A = B )
     => ( ( member8440522571783428010at_nat @ A @ A4 )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( id_on_2554058798563519774at_nat @ A4 ) ) ) ) ).

% Id_on_eqI
thf(fact_1113_Id__on__eqI,axiom,
    ! [A: nat,B: nat,A4: set_nat] :
      ( ( A = B )
     => ( ( member_nat @ A @ A4 )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( id_on_nat @ A4 ) ) ) ) ).

% Id_on_eqI
thf(fact_1114_Id__on__iff,axiom,
    ! [X3: a,Y: a,A4: set_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ ( id_on_a @ A4 ) )
      = ( ( X3 = Y )
        & ( member_a @ X3 @ A4 ) ) ) ).

% Id_on_iff
thf(fact_1115_Id__on__iff,axiom,
    ! [X3: formula_a_t,Y: formula_a_t,A4: set_formula_a_t] :
      ( ( member9065953759106212816la_a_t @ ( produc8281305675666354143la_a_t @ X3 @ Y ) @ ( id_on_formula_a_t @ A4 ) )
      = ( ( X3 = Y )
        & ( member_formula_a_t @ X3 @ A4 ) ) ) ).

% Id_on_iff
thf(fact_1116_Id__on__iff,axiom,
    ! [X3: product_prod_nat_nat,Y: product_prod_nat_nat,A4: set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y ) @ ( id_on_2554058798563519774at_nat @ A4 ) )
      = ( ( X3 = Y )
        & ( member8440522571783428010at_nat @ X3 @ A4 ) ) ) ).

% Id_on_iff
thf(fact_1117_Id__on__iff,axiom,
    ! [X3: nat,Y: nat,A4: set_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( id_on_nat @ A4 ) )
      = ( ( X3 = Y )
        & ( member_nat @ X3 @ A4 ) ) ) ).

% Id_on_iff
thf(fact_1118_length__n__lists__elem,axiom,
    ! [Ys: list_P1095827921439705018_set_a,N: nat,Xs: list_P1095827921439705018_set_a] :
      ( ( member1645050908308627811_set_a @ Ys @ ( set_li5089069168345991631_set_a @ ( n_list7540261771967413058_set_a @ N @ Xs ) ) )
     => ( ( size_s5362121731612881446_set_a @ Ys )
        = N ) ) ).

% length_n_lists_elem
thf(fact_1119_length__n__lists__elem,axiom,
    ! [Ys: list_vydra_aux_a_t_h,N: nat,Xs: list_vydra_aux_a_t_h] :
      ( ( member2827466511716433855_a_t_h @ Ys @ ( set_li3104226997595317331_a_t_h @ ( n_list420448431600190986_a_t_h @ N @ Xs ) ) )
     => ( ( size_s8993897887676352572_a_t_h @ Ys )
        = N ) ) ).

% length_n_lists_elem
thf(fact_1120_length__n__lists__elem,axiom,
    ! [Ys: list_iarray_o,N: nat,Xs: list_iarray_o] :
      ( ( member_list_iarray_o @ Ys @ ( set_list_iarray_o2 @ ( n_lists_iarray_o @ N @ Xs ) ) )
     => ( ( size_s3243743627286111600rray_o @ Ys )
        = N ) ) ).

% length_n_lists_elem
thf(fact_1121_length__n__lists__elem,axiom,
    ! [Ys: list_formula_a_t,N: nat,Xs: list_formula_a_t] :
      ( ( member8982599720545899850la_a_t @ Ys @ ( set_list_formula_a_t2 @ ( n_lists_formula_a_t @ N @ Xs ) ) )
     => ( ( size_s8846756101701226951la_a_t @ Ys )
        = N ) ) ).

% length_n_lists_elem
thf(fact_1122_length__n__lists__elem,axiom,
    ! [Ys: list_t,N: nat,Xs: list_t] :
      ( ( member_list_t @ Ys @ ( set_list_t2 @ ( n_lists_t @ N @ Xs ) ) )
     => ( ( size_size_list_t @ Ys )
        = N ) ) ).

% length_n_lists_elem
thf(fact_1123_read__t_Oelims,axiom,
    ! [X3: option7421465107808516430_o_nat,Y: option_nat] :
      ( ( ( read_t_nat_o_nat @ X3 )
        = Y )
     => ( ( ( X3 = none_P4770284390835442525_o_nat )
         => ( Y != none_nat ) )
       => ~ ! [E: nat > $o,T5: nat] :
              ( ( X3
                = ( some_P2630223654056210137_o_nat @ ( produc7277522915581678840_o_nat @ E @ T5 ) ) )
             => ( Y
               != ( some_nat @ T5 ) ) ) ) ) ).

% read_t.elims
thf(fact_1124_read__t_Oelims,axiom,
    ! [X3: option2110264179601050681od_h_t,Y: option_t] :
      ( ( ( read_t_h_t @ X3 )
        = Y )
     => ( ( ( X3 = none_P849260865442546644od_h_t )
         => ( Y != none_t ) )
       => ~ ! [E: h,T5: t] :
              ( ( X3
                = ( some_P2526622376857543512od_h_t @ ( product_Pair_h_t @ E @ T5 ) ) )
             => ( Y
               != ( some_t @ T5 ) ) ) ) ) ).

% read_t.elims
thf(fact_1125_wf__vydra_Ointros_I1_J,axiom,
    ! [Phi2: formula_a_t,I: nat,N: nat,W: vydra_aux_a_t_h,I5: i_t] :
      ( ( vYDRA_wf_vydra_a_t_h @ sigma @ init_hd @ run_hd @ Phi2 @ I @ N @ W )
     => ( ( ( run_h_t_a @ run_hd @ N @ W )
          = none_P3452577240289636010od_t_o )
       => ( vYDRA_wf_vydra_a_t_h @ sigma @ init_hd @ run_hd @ ( prev_t_a @ I5 @ Phi2 ) @ ( suc @ I ) @ ( suc @ N ) @ vydra_4195442553028166619_a_t_h ) ) ) ).

% wf_vydra.intros(1)
thf(fact_1126_formula_Oinject_I5_J,axiom,
    ! [X51: i_t,X52: formula_a_t,Y51: i_t,Y52: formula_a_t] :
      ( ( ( prev_t_a @ X51 @ X52 )
        = ( prev_t_a @ Y51 @ Y52 ) )
      = ( ( X51 = Y51 )
        & ( X52 = Y52 ) ) ) ).

% formula.inject(5)
thf(fact_1127_append_Oright__neutral,axiom,
    ! [A: list_formula_a_t] :
      ( ( append_formula_a_t @ A @ nil_formula_a_t )
      = A ) ).

% append.right_neutral
thf(fact_1128_append__Nil2,axiom,
    ! [Xs: list_formula_a_t] :
      ( ( append_formula_a_t @ Xs @ nil_formula_a_t )
      = Xs ) ).

% append_Nil2
thf(fact_1129_append__self__conv,axiom,
    ! [Xs: list_formula_a_t,Ys: list_formula_a_t] :
      ( ( ( append_formula_a_t @ Xs @ Ys )
        = Xs )
      = ( Ys = nil_formula_a_t ) ) ).

% append_self_conv
thf(fact_1130_self__append__conv,axiom,
    ! [Y: list_formula_a_t,Ys: list_formula_a_t] :
      ( ( Y
        = ( append_formula_a_t @ Y @ Ys ) )
      = ( Ys = nil_formula_a_t ) ) ).

% self_append_conv
thf(fact_1131_append__self__conv2,axiom,
    ! [Xs: list_formula_a_t,Ys: list_formula_a_t] :
      ( ( ( append_formula_a_t @ Xs @ Ys )
        = Ys )
      = ( Xs = nil_formula_a_t ) ) ).

% append_self_conv2
thf(fact_1132_self__append__conv2,axiom,
    ! [Y: list_formula_a_t,Xs: list_formula_a_t] :
      ( ( Y
        = ( append_formula_a_t @ Xs @ Y ) )
      = ( Xs = nil_formula_a_t ) ) ).

% self_append_conv2
thf(fact_1133_Nil__is__append__conv,axiom,
    ! [Xs: list_formula_a_t,Ys: list_formula_a_t] :
      ( ( nil_formula_a_t
        = ( append_formula_a_t @ Xs @ Ys ) )
      = ( ( Xs = nil_formula_a_t )
        & ( Ys = nil_formula_a_t ) ) ) ).

% Nil_is_append_conv
thf(fact_1134_append__is__Nil__conv,axiom,
    ! [Xs: list_formula_a_t,Ys: list_formula_a_t] :
      ( ( ( append_formula_a_t @ Xs @ Ys )
        = nil_formula_a_t )
      = ( ( Xs = nil_formula_a_t )
        & ( Ys = nil_formula_a_t ) ) ) ).

% append_is_Nil_conv
thf(fact_1135_append__eq__append__conv,axiom,
    ! [Xs: list_P1095827921439705018_set_a,Ys: list_P1095827921439705018_set_a,Us: list_P1095827921439705018_set_a,Vs: list_P1095827921439705018_set_a] :
      ( ( ( ( size_s5362121731612881446_set_a @ Xs )
          = ( size_s5362121731612881446_set_a @ Ys ) )
        | ( ( size_s5362121731612881446_set_a @ Us )
          = ( size_s5362121731612881446_set_a @ Vs ) ) )
     => ( ( ( append8768163766229313743_set_a @ Xs @ Us )
          = ( append8768163766229313743_set_a @ Ys @ Vs ) )
        = ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_1136_append__eq__append__conv,axiom,
    ! [Xs: list_vydra_aux_a_t_h,Ys: list_vydra_aux_a_t_h,Us: list_vydra_aux_a_t_h,Vs: list_vydra_aux_a_t_h] :
      ( ( ( ( size_s8993897887676352572_a_t_h @ Xs )
          = ( size_s8993897887676352572_a_t_h @ Ys ) )
        | ( ( size_s8993897887676352572_a_t_h @ Us )
          = ( size_s8993897887676352572_a_t_h @ Vs ) ) )
     => ( ( ( append7715335228782332733_a_t_h @ Xs @ Us )
          = ( append7715335228782332733_a_t_h @ Ys @ Vs ) )
        = ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_1137_append__eq__append__conv,axiom,
    ! [Xs: list_iarray_o,Ys: list_iarray_o,Us: list_iarray_o,Vs: list_iarray_o] :
      ( ( ( ( size_s3243743627286111600rray_o @ Xs )
          = ( size_s3243743627286111600rray_o @ Ys ) )
        | ( ( size_s3243743627286111600rray_o @ Us )
          = ( size_s3243743627286111600rray_o @ Vs ) ) )
     => ( ( ( append_iarray_o @ Xs @ Us )
          = ( append_iarray_o @ Ys @ Vs ) )
        = ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_1138_append__eq__append__conv,axiom,
    ! [Xs: list_formula_a_t,Ys: list_formula_a_t,Us: list_formula_a_t,Vs: list_formula_a_t] :
      ( ( ( ( size_s8846756101701226951la_a_t @ Xs )
          = ( size_s8846756101701226951la_a_t @ Ys ) )
        | ( ( size_s8846756101701226951la_a_t @ Us )
          = ( size_s8846756101701226951la_a_t @ Vs ) ) )
     => ( ( ( append_formula_a_t @ Xs @ Us )
          = ( append_formula_a_t @ Ys @ Vs ) )
        = ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_1139_append__eq__append__conv,axiom,
    ! [Xs: list_t,Ys: list_t,Us: list_t,Vs: list_t] :
      ( ( ( ( size_size_list_t @ Xs )
          = ( size_size_list_t @ Ys ) )
        | ( ( size_size_list_t @ Us )
          = ( size_size_list_t @ Vs ) ) )
     => ( ( ( append_t @ Xs @ Us )
          = ( append_t @ Ys @ Vs ) )
        = ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_1140_map__append,axiom,
    ! [F: product_prod_t_set_a > t,Xs: list_P1095827921439705018_set_a,Ys: list_P1095827921439705018_set_a] :
      ( ( map_Pr119846518281778721et_a_t @ F @ ( append8768163766229313743_set_a @ Xs @ Ys ) )
      = ( append_t @ ( map_Pr119846518281778721et_a_t @ F @ Xs ) @ ( map_Pr119846518281778721et_a_t @ F @ Ys ) ) ) ).

% map_append
thf(fact_1141_map__append,axiom,
    ! [F: formula_a_t > $o,Xs: list_formula_a_t,Ys: list_formula_a_t] :
      ( ( map_formula_a_t_o @ F @ ( append_formula_a_t @ Xs @ Ys ) )
      = ( append_o @ ( map_formula_a_t_o @ F @ Xs ) @ ( map_formula_a_t_o @ F @ Ys ) ) ) ).

% map_append
thf(fact_1142_map__append,axiom,
    ! [F: formula_a_t > vydra_aux_a_t_h,Xs: list_formula_a_t,Ys: list_formula_a_t] :
      ( ( map_fo5746072578778194433_a_t_h @ F @ ( append_formula_a_t @ Xs @ Ys ) )
      = ( append7715335228782332733_a_t_h @ ( map_fo5746072578778194433_a_t_h @ F @ Xs ) @ ( map_fo5746072578778194433_a_t_h @ F @ Ys ) ) ) ).

% map_append
thf(fact_1143_append1__eq__conv,axiom,
    ! [Xs: list_formula_a_t,X3: formula_a_t,Ys: list_formula_a_t,Y: formula_a_t] :
      ( ( ( append_formula_a_t @ Xs @ ( cons_formula_a_t @ X3 @ nil_formula_a_t ) )
        = ( append_formula_a_t @ Ys @ ( cons_formula_a_t @ Y @ nil_formula_a_t ) ) )
      = ( ( Xs = Ys )
        & ( X3 = Y ) ) ) ).

% append1_eq_conv
thf(fact_1144_nth__Cons__Suc,axiom,
    ! [X3: t,Xs: list_t,N: nat] :
      ( ( nth_t @ ( cons_t @ X3 @ Xs ) @ ( suc @ N ) )
      = ( nth_t @ Xs @ N ) ) ).

% nth_Cons_Suc
thf(fact_1145_nth__Cons__Suc,axiom,
    ! [X3: formula_a_t,Xs: list_formula_a_t,N: nat] :
      ( ( nth_formula_a_t @ ( cons_formula_a_t @ X3 @ Xs ) @ ( suc @ N ) )
      = ( nth_formula_a_t @ Xs @ N ) ) ).

% nth_Cons_Suc
thf(fact_1146_nth__Cons__Suc,axiom,
    ! [X3: vydra_aux_a_t_h,Xs: list_vydra_aux_a_t_h,N: nat] :
      ( ( nth_vydra_aux_a_t_h @ ( cons_vydra_aux_a_t_h @ X3 @ Xs ) @ ( suc @ N ) )
      = ( nth_vydra_aux_a_t_h @ Xs @ N ) ) ).

% nth_Cons_Suc
thf(fact_1147_nth__Cons__0,axiom,
    ! [X3: t,Xs: list_t] :
      ( ( nth_t @ ( cons_t @ X3 @ Xs ) @ zero_zero_nat )
      = X3 ) ).

% nth_Cons_0
thf(fact_1148_nth__Cons__0,axiom,
    ! [X3: formula_a_t,Xs: list_formula_a_t] :
      ( ( nth_formula_a_t @ ( cons_formula_a_t @ X3 @ Xs ) @ zero_zero_nat )
      = X3 ) ).

% nth_Cons_0
thf(fact_1149_nth__Cons__0,axiom,
    ! [X3: vydra_aux_a_t_h,Xs: list_vydra_aux_a_t_h] :
      ( ( nth_vydra_aux_a_t_h @ ( cons_vydra_aux_a_t_h @ X3 @ Xs ) @ zero_zero_nat )
      = X3 ) ).

% nth_Cons_0
thf(fact_1150_nth__append__length,axiom,
    ! [Xs: list_P1095827921439705018_set_a,X3: product_prod_t_set_a,Ys: list_P1095827921439705018_set_a] :
      ( ( nth_Pr633927046780761275_set_a @ ( append8768163766229313743_set_a @ Xs @ ( cons_P6565363215658504938_set_a @ X3 @ Ys ) ) @ ( size_s5362121731612881446_set_a @ Xs ) )
      = X3 ) ).

% nth_append_length
thf(fact_1151_nth__append__length,axiom,
    ! [Xs: list_vydra_aux_a_t_h,X3: vydra_aux_a_t_h,Ys: list_vydra_aux_a_t_h] :
      ( ( nth_vydra_aux_a_t_h @ ( append7715335228782332733_a_t_h @ Xs @ ( cons_vydra_aux_a_t_h @ X3 @ Ys ) ) @ ( size_s8993897887676352572_a_t_h @ Xs ) )
      = X3 ) ).

% nth_append_length
thf(fact_1152_nth__append__length,axiom,
    ! [Xs: list_iarray_o,X3: iarray_o,Ys: list_iarray_o] :
      ( ( nth_iarray_o @ ( append_iarray_o @ Xs @ ( cons_iarray_o @ X3 @ Ys ) ) @ ( size_s3243743627286111600rray_o @ Xs ) )
      = X3 ) ).

% nth_append_length
thf(fact_1153_nth__append__length,axiom,
    ! [Xs: list_formula_a_t,X3: formula_a_t,Ys: list_formula_a_t] :
      ( ( nth_formula_a_t @ ( append_formula_a_t @ Xs @ ( cons_formula_a_t @ X3 @ Ys ) ) @ ( size_s8846756101701226951la_a_t @ Xs ) )
      = X3 ) ).

% nth_append_length
thf(fact_1154_nth__append__length,axiom,
    ! [Xs: list_t,X3: t,Ys: list_t] :
      ( ( nth_t @ ( append_t @ Xs @ ( cons_t @ X3 @ Ys ) ) @ ( size_size_list_t @ Xs ) )
      = X3 ) ).

% nth_append_length
thf(fact_1155_zip__Cons__Cons,axiom,
    ! [X3: nat > $o,Xs: list_nat_o,Y: nat,Ys: list_nat] :
      ( ( zip_nat_o_nat @ ( cons_nat_o @ X3 @ Xs ) @ ( cons_nat @ Y @ Ys ) )
      = ( cons_P1301391208966372296_o_nat @ ( produc7277522915581678840_o_nat @ X3 @ Y ) @ ( zip_nat_o_nat @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_1156_zip__Cons__Cons,axiom,
    ! [X3: t,Xs: list_t,Y: $o,Ys: list_o] :
      ( ( zip_t_o @ ( cons_t @ X3 @ Xs ) @ ( cons_o @ Y @ Ys ) )
      = ( cons_P8726414264574200228od_t_o @ ( product_Pair_t_o @ X3 @ Y ) @ ( zip_t_o @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_1157_zip__Cons__Cons,axiom,
    ! [X3: t,Xs: list_t,Y: set_a,Ys: list_set_a] :
      ( ( zip_t_set_a @ ( cons_t @ X3 @ Xs ) @ ( cons_set_a @ Y @ Ys ) )
      = ( cons_P6565363215658504938_set_a @ ( product_Pair_t_set_a @ X3 @ Y ) @ ( zip_t_set_a @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_1158_zip__Cons__Cons,axiom,
    ! [X3: h,Xs: list_h,Y: product_prod_t_set_a,Ys: list_P1095827921439705018_set_a] :
      ( ( zip_h_8035635146645487142_set_a @ ( cons_h @ X3 @ Xs ) @ ( cons_P6565363215658504938_set_a @ Y @ Ys ) )
      = ( cons_P8657861537117511884_set_a @ ( produc3432067189782488838_set_a @ X3 @ Y ) @ ( zip_h_8035635146645487142_set_a @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_1159_zip__Cons__Cons,axiom,
    ! [X3: h,Xs: list_h,Y: t,Ys: list_t] :
      ( ( zip_h_t @ ( cons_h @ X3 @ Xs ) @ ( cons_t @ Y @ Ys ) )
      = ( cons_P6251494208368732457od_h_t @ ( product_Pair_h_t @ X3 @ Y ) @ ( zip_h_t @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_1160_zip__append,axiom,
    ! [Xs: list_t,Us: list_t,Ys: list_t,Vs: list_t] :
      ( ( ( size_size_list_t @ Xs )
        = ( size_size_list_t @ Us ) )
     => ( ( zip_t_t @ ( append_t @ Xs @ Ys ) @ ( append_t @ Us @ Vs ) )
        = ( append7713785064455990786od_t_t @ ( zip_t_t @ Xs @ Us ) @ ( zip_t_t @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_1161_zip__append,axiom,
    ! [Xs: list_iarray_o,Us: list_t,Ys: list_iarray_o,Vs: list_t] :
      ( ( ( size_s3243743627286111600rray_o @ Xs )
        = ( size_size_list_t @ Us ) )
     => ( ( zip_iarray_o_t @ ( append_iarray_o @ Xs @ Ys ) @ ( append_t @ Us @ Vs ) )
        = ( append3406593766146243797ay_o_t @ ( zip_iarray_o_t @ Xs @ Us ) @ ( zip_iarray_o_t @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_1162_zip__append,axiom,
    ! [Xs: list_t,Us: list_iarray_o,Ys: list_t,Vs: list_iarray_o] :
      ( ( ( size_size_list_t @ Xs )
        = ( size_s3243743627286111600rray_o @ Us ) )
     => ( ( zip_t_iarray_o @ ( append_t @ Xs @ Ys ) @ ( append_iarray_o @ Us @ Vs ) )
        = ( append391921101841443823rray_o @ ( zip_t_iarray_o @ Xs @ Us ) @ ( zip_t_iarray_o @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_1163_zip__append,axiom,
    ! [Xs: list_iarray_o,Us: list_iarray_o,Ys: list_iarray_o,Vs: list_iarray_o] :
      ( ( ( size_s3243743627286111600rray_o @ Xs )
        = ( size_s3243743627286111600rray_o @ Us ) )
     => ( ( zip_ia7658775312918225207rray_o @ ( append_iarray_o @ Xs @ Ys ) @ ( append_iarray_o @ Us @ Vs ) )
        = ( append2067669574830570434rray_o @ ( zip_ia7658775312918225207rray_o @ Xs @ Us ) @ ( zip_ia7658775312918225207rray_o @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_1164_zip__append,axiom,
    ! [Xs: list_formula_a_t,Us: list_t,Ys: list_formula_a_t,Vs: list_t] :
      ( ( ( size_s8846756101701226951la_a_t @ Xs )
        = ( size_size_list_t @ Us ) )
     => ( ( zip_formula_a_t_t @ ( append_formula_a_t @ Xs @ Ys ) @ ( append_t @ Us @ Vs ) )
        = ( append7857587814059881928_a_t_t @ ( zip_formula_a_t_t @ Xs @ Us ) @ ( zip_formula_a_t_t @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_1165_zip__append,axiom,
    ! [Xs: list_t,Us: list_formula_a_t,Ys: list_t,Vs: list_formula_a_t] :
      ( ( ( size_size_list_t @ Xs )
        = ( size_s8846756101701226951la_a_t @ Us ) )
     => ( ( zip_t_formula_a_t @ ( append_t @ Xs @ Ys ) @ ( append_formula_a_t @ Us @ Vs ) )
        = ( append2895453439901546888la_a_t @ ( zip_t_formula_a_t @ Xs @ Us ) @ ( zip_t_formula_a_t @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_1166_zip__append,axiom,
    ! [Xs: list_P1095827921439705018_set_a,Us: list_t,Ys: list_P1095827921439705018_set_a,Vs: list_t] :
      ( ( ( size_s5362121731612881446_set_a @ Xs )
        = ( size_size_list_t @ Us ) )
     => ( ( zip_Pr6246384304659936660et_a_t @ ( append8768163766229313743_set_a @ Xs @ Ys ) @ ( append_t @ Us @ Vs ) )
        = ( append4948748235560564383et_a_t @ ( zip_Pr6246384304659936660et_a_t @ Xs @ Us ) @ ( zip_Pr6246384304659936660et_a_t @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_1167_zip__append,axiom,
    ! [Xs: list_vydra_aux_a_t_h,Us: list_t,Ys: list_vydra_aux_a_t_h,Vs: list_t] :
      ( ( ( size_s8993897887676352572_a_t_h @ Xs )
        = ( size_size_list_t @ Us ) )
     => ( ( zip_vy170765141419497472_t_h_t @ ( append7715335228782332733_a_t_h @ Xs @ Ys ) @ ( append_t @ Us @ Vs ) )
        = ( append2187503897680144403_t_h_t @ ( zip_vy170765141419497472_t_h_t @ Xs @ Us ) @ ( zip_vy170765141419497472_t_h_t @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_1168_zip__append,axiom,
    ! [Xs: list_iarray_o,Us: list_formula_a_t,Ys: list_iarray_o,Vs: list_formula_a_t] :
      ( ( ( size_s3243743627286111600rray_o @ Xs )
        = ( size_s8846756101701226951la_a_t @ Us ) )
     => ( ( zip_ia3729555385294884522la_a_t @ ( append_iarray_o @ Xs @ Ys ) @ ( append_formula_a_t @ Us @ Vs ) )
        = ( append1079506867748072181la_a_t @ ( zip_ia3729555385294884522la_a_t @ Xs @ Us ) @ ( zip_ia3729555385294884522la_a_t @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_1169_zip__append,axiom,
    ! [Xs: list_formula_a_t,Us: list_iarray_o,Ys: list_formula_a_t,Vs: list_iarray_o] :
      ( ( ( size_s8846756101701226951la_a_t @ Xs )
        = ( size_s3243743627286111600rray_o @ Us ) )
     => ( ( zip_fo3361909093787821090rray_o @ ( append_formula_a_t @ Xs @ Ys ) @ ( append_iarray_o @ Us @ Vs ) )
        = ( append1155512193423955381rray_o @ ( zip_fo3361909093787821090rray_o @ Xs @ Us ) @ ( zip_fo3361909093787821090rray_o @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_1170_Cons__in__lex,axiom,
    ! [X3: nat,Xs: list_nat,Y: nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs ) @ ( cons_nat @ Y @ Ys ) ) @ ( lex_nat @ R2 ) )
      = ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ R2 )
          & ( ( size_size_list_nat @ Xs )
            = ( size_size_list_nat @ Ys ) ) )
        | ( ( X3 = Y )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( lex_nat @ R2 ) ) ) ) ) ).

% Cons_in_lex
thf(fact_1171_Cons__in__lex,axiom,
    ! [X3: product_prod_t_set_a,Xs: list_P1095827921439705018_set_a,Y: product_prod_t_set_a,Ys: list_P1095827921439705018_set_a,R2: set_Pr2751536223740281991_set_a] :
      ( ( member7105239521983867088_set_a @ ( produc3154345580163335831_set_a @ ( cons_P6565363215658504938_set_a @ X3 @ Xs ) @ ( cons_P6565363215658504938_set_a @ Y @ Ys ) ) @ ( lex_Pr3546421067709956510_set_a @ R2 ) )
      = ( ( ( member6953777265722981072_set_a @ ( produc4993190531567155607_set_a @ X3 @ Y ) @ R2 )
          & ( ( size_s5362121731612881446_set_a @ Xs )
            = ( size_s5362121731612881446_set_a @ Ys ) ) )
        | ( ( X3 = Y )
          & ( member7105239521983867088_set_a @ ( produc3154345580163335831_set_a @ Xs @ Ys ) @ ( lex_Pr3546421067709956510_set_a @ R2 ) ) ) ) ) ).

% Cons_in_lex
thf(fact_1172_Cons__in__lex,axiom,
    ! [X3: vydra_aux_a_t_h,Xs: list_vydra_aux_a_t_h,Y: vydra_aux_a_t_h,Ys: list_vydra_aux_a_t_h,R2: set_Pr5596025548626726129_a_t_h] :
      ( ( member2805742798364816858_a_t_h @ ( produc7080071857504742505_a_t_h @ ( cons_vydra_aux_a_t_h @ X3 @ Xs ) @ ( cons_vydra_aux_a_t_h @ Y @ Ys ) ) @ ( lex_vydra_aux_a_t_h @ R2 ) )
      = ( ( ( member3443537796814201402_a_t_h @ ( produc6881572889943414985_a_t_h @ X3 @ Y ) @ R2 )
          & ( ( size_s8993897887676352572_a_t_h @ Xs )
            = ( size_s8993897887676352572_a_t_h @ Ys ) ) )
        | ( ( X3 = Y )
          & ( member2805742798364816858_a_t_h @ ( produc7080071857504742505_a_t_h @ Xs @ Ys ) @ ( lex_vydra_aux_a_t_h @ R2 ) ) ) ) ) ).

% Cons_in_lex
thf(fact_1173_Cons__in__lex,axiom,
    ! [X3: iarray_o,Xs: list_iarray_o,Y: iarray_o,Ys: list_iarray_o,R2: set_Pr3159307878933344263rray_o] :
      ( ( member8606389602990657872rray_o @ ( produc2869105288327869463rray_o @ ( cons_iarray_o @ X3 @ Xs ) @ ( cons_iarray_o @ Y @ Ys ) ) @ ( lex_iarray_o @ R2 ) )
      = ( ( ( member3227708950830999120rray_o @ ( produc513795717669825047rray_o @ X3 @ Y ) @ R2 )
          & ( ( size_s3243743627286111600rray_o @ Xs )
            = ( size_s3243743627286111600rray_o @ Ys ) ) )
        | ( ( X3 = Y )
          & ( member8606389602990657872rray_o @ ( produc2869105288327869463rray_o @ Xs @ Ys ) @ ( lex_iarray_o @ R2 ) ) ) ) ) ).

% Cons_in_lex
thf(fact_1174_Cons__in__lex,axiom,
    ! [X3: formula_a_t,Xs: list_formula_a_t,Y: formula_a_t,Ys: list_formula_a_t,R2: set_Pr3703585533673849735la_a_t] :
      ( ( member5181786688746205936la_a_t @ ( produc7251354486509092607la_a_t @ ( cons_formula_a_t @ X3 @ Xs ) @ ( cons_formula_a_t @ Y @ Ys ) ) @ ( lex_formula_a_t @ R2 ) )
      = ( ( ( member9065953759106212816la_a_t @ ( produc8281305675666354143la_a_t @ X3 @ Y ) @ R2 )
          & ( ( size_s8846756101701226951la_a_t @ Xs )
            = ( size_s8846756101701226951la_a_t @ Ys ) ) )
        | ( ( X3 = Y )
          & ( member5181786688746205936la_a_t @ ( produc7251354486509092607la_a_t @ Xs @ Ys ) @ ( lex_formula_a_t @ R2 ) ) ) ) ) ).

% Cons_in_lex
thf(fact_1175_Cons__in__lex,axiom,
    ! [X3: t,Xs: list_t,Y: t,Ys: list_t,R2: set_Product_prod_t_t] :
      ( ( member1598007745118852880list_t @ ( produc8116999438917745495list_t @ ( cons_t @ X3 @ Xs ) @ ( cons_t @ Y @ Ys ) ) @ ( lex_t @ R2 ) )
      = ( ( ( member3805107722934592656od_t_t @ ( product_Pair_t_t @ X3 @ Y ) @ R2 )
          & ( ( size_size_list_t @ Xs )
            = ( size_size_list_t @ Ys ) ) )
        | ( ( X3 = Y )
          & ( member1598007745118852880list_t @ ( produc8116999438917745495list_t @ Xs @ Ys ) @ ( lex_t @ R2 ) ) ) ) ) ).

% Cons_in_lex
thf(fact_1176_reaches__on__setD,axiom,
    ! [Run: h > option5775791656947114332_set_a,S: h,Vs: list_P1095827921439705018_set_a,S2: h,X3: product_prod_t_set_a] :
      ( ( reache3397181163414509427_set_a @ Run @ S @ Vs @ S2 )
     => ( ( member2646690471337469789_set_a @ X3 @ ( set_Pr15456023412717001_set_a @ Vs ) )
       => ? [Vs4: list_P1095827921439705018_set_a,Vs5: list_P1095827921439705018_set_a,S8: h] :
            ( ( reache3397181163414509427_set_a @ Run @ S @ ( append8768163766229313743_set_a @ Vs4 @ ( cons_P6565363215658504938_set_a @ X3 @ nil_Pr7982011819685168954_set_a ) ) @ S8 )
            & ( reache3397181163414509427_set_a @ Run @ S8 @ Vs5 @ S2 )
            & ( Vs
              = ( append8768163766229313743_set_a @ Vs4 @ ( cons_P6565363215658504938_set_a @ X3 @ Vs5 ) ) ) ) ) ) ).

% reaches_on_setD
thf(fact_1177_reaches__on__setD,axiom,
    ! [Run: option2110264179601050681od_h_t > option4241632866121027589_h_t_t,S: option2110264179601050681od_h_t,Vs: list_t,S2: option2110264179601050681od_h_t,X3: t] :
      ( ( reache645946503560514460_h_t_t @ Run @ S @ Vs @ S2 )
     => ( ( member_t @ X3 @ ( set_t2 @ Vs ) )
       => ? [Vs4: list_t,Vs5: list_t,S8: option2110264179601050681od_h_t] :
            ( ( reache645946503560514460_h_t_t @ Run @ S @ ( append_t @ Vs4 @ ( cons_t @ X3 @ nil_t ) ) @ S8 )
            & ( reache645946503560514460_h_t_t @ Run @ S8 @ Vs5 @ S2 )
            & ( Vs
              = ( append_t @ Vs4 @ ( cons_t @ X3 @ Vs5 ) ) ) ) ) ) ).

% reaches_on_setD
thf(fact_1178_reaches__on__setD,axiom,
    ! [Run: list_vydra_aux_a_t_h > option8553266494716013595rray_o,S: list_vydra_aux_a_t_h,Vs: list_iarray_o,S2: list_vydra_aux_a_t_h,X3: iarray_o] :
      ( ( reache2401979074735462992rray_o @ Run @ S @ Vs @ S2 )
     => ( ( member_iarray_o @ X3 @ ( set_iarray_o2 @ Vs ) )
       => ? [Vs4: list_iarray_o,Vs5: list_iarray_o,S8: list_vydra_aux_a_t_h] :
            ( ( reache2401979074735462992rray_o @ Run @ S @ ( append_iarray_o @ Vs4 @ ( cons_iarray_o @ X3 @ nil_iarray_o ) ) @ S8 )
            & ( reache2401979074735462992rray_o @ Run @ S8 @ Vs5 @ S2 )
            & ( Vs
              = ( append_iarray_o @ Vs4 @ ( cons_iarray_o @ X3 @ Vs5 ) ) ) ) ) ) ).

% reaches_on_setD
thf(fact_1179_reaches__on__setD,axiom,
    ! [Run: vydra_aux_a_t_h > option4580262698044125979od_t_o,S: vydra_aux_a_t_h,Vs: list_P5575445741472356340od_t_o,S2: vydra_aux_a_t_h,X3: product_prod_t_o] :
      ( ( reache8691153155344877136od_t_o @ Run @ S @ Vs @ S2 )
     => ( ( member3252940038344094615od_t_o @ X3 @ ( set_Product_prod_t_o2 @ Vs ) )
       => ? [Vs4: list_P5575445741472356340od_t_o,Vs5: list_P5575445741472356340od_t_o,S8: vydra_aux_a_t_h] :
            ( ( reache8691153155344877136od_t_o @ Run @ S @ ( append2717171661090821641od_t_o @ Vs4 @ ( cons_P8726414264574200228od_t_o @ X3 @ nil_Product_prod_t_o ) ) @ S8 )
            & ( reache8691153155344877136od_t_o @ Run @ S8 @ Vs5 @ S2 )
            & ( Vs
              = ( append2717171661090821641od_t_o @ Vs4 @ ( cons_P8726414264574200228od_t_o @ X3 @ Vs5 ) ) ) ) ) ) ).

% reaches_on_setD
thf(fact_1180_list_Oset__intros_I2_J,axiom,
    ! [Y: a,X22: list_a,X21: a] :
      ( ( member_a @ Y @ ( set_a2 @ X22 ) )
     => ( member_a @ Y @ ( set_a2 @ ( cons_a @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_1181_list_Oset__intros_I2_J,axiom,
    ! [Y: product_prod_nat_nat,X22: list_P6011104703257516679at_nat,X21: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ Y @ ( set_Pr5648618587558075414at_nat @ X22 ) )
     => ( member8440522571783428010at_nat @ Y @ ( set_Pr5648618587558075414at_nat @ ( cons_P6512896166579812791at_nat @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_1182_list_Oset__intros_I2_J,axiom,
    ! [Y: nat,X22: list_nat,X21: nat] :
      ( ( member_nat @ Y @ ( set_nat2 @ X22 ) )
     => ( member_nat @ Y @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_1183_list_Oset__intros_I2_J,axiom,
    ! [Y: formula_a_t,X22: list_formula_a_t,X21: formula_a_t] :
      ( ( member_formula_a_t @ Y @ ( set_formula_a_t2 @ X22 ) )
     => ( member_formula_a_t @ Y @ ( set_formula_a_t2 @ ( cons_formula_a_t @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_1184_list_Oset__intros_I1_J,axiom,
    ! [X21: a,X22: list_a] : ( member_a @ X21 @ ( set_a2 @ ( cons_a @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_1185_list_Oset__intros_I1_J,axiom,
    ! [X21: product_prod_nat_nat,X22: list_P6011104703257516679at_nat] : ( member8440522571783428010at_nat @ X21 @ ( set_Pr5648618587558075414at_nat @ ( cons_P6512896166579812791at_nat @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_1186_list_Oset__intros_I1_J,axiom,
    ! [X21: nat,X22: list_nat] : ( member_nat @ X21 @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_1187_list_Oset__intros_I1_J,axiom,
    ! [X21: formula_a_t,X22: list_formula_a_t] : ( member_formula_a_t @ X21 @ ( set_formula_a_t2 @ ( cons_formula_a_t @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_1188_list_Oset__cases,axiom,
    ! [E2: a,A: list_a] :
      ( ( member_a @ E2 @ ( set_a2 @ A ) )
     => ( ! [Z22: list_a] :
            ( A
           != ( cons_a @ E2 @ Z22 ) )
       => ~ ! [Z1: a,Z22: list_a] :
              ( ( A
                = ( cons_a @ Z1 @ Z22 ) )
             => ~ ( member_a @ E2 @ ( set_a2 @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_1189_list_Oset__cases,axiom,
    ! [E2: product_prod_nat_nat,A: list_P6011104703257516679at_nat] :
      ( ( member8440522571783428010at_nat @ E2 @ ( set_Pr5648618587558075414at_nat @ A ) )
     => ( ! [Z22: list_P6011104703257516679at_nat] :
            ( A
           != ( cons_P6512896166579812791at_nat @ E2 @ Z22 ) )
       => ~ ! [Z1: product_prod_nat_nat,Z22: list_P6011104703257516679at_nat] :
              ( ( A
                = ( cons_P6512896166579812791at_nat @ Z1 @ Z22 ) )
             => ~ ( member8440522571783428010at_nat @ E2 @ ( set_Pr5648618587558075414at_nat @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_1190_list_Oset__cases,axiom,
    ! [E2: nat,A: list_nat] :
      ( ( member_nat @ E2 @ ( set_nat2 @ A ) )
     => ( ! [Z22: list_nat] :
            ( A
           != ( cons_nat @ E2 @ Z22 ) )
       => ~ ! [Z1: nat,Z22: list_nat] :
              ( ( A
                = ( cons_nat @ Z1 @ Z22 ) )
             => ~ ( member_nat @ E2 @ ( set_nat2 @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_1191_list_Oset__cases,axiom,
    ! [E2: formula_a_t,A: list_formula_a_t] :
      ( ( member_formula_a_t @ E2 @ ( set_formula_a_t2 @ A ) )
     => ( ! [Z22: list_formula_a_t] :
            ( A
           != ( cons_formula_a_t @ E2 @ Z22 ) )
       => ~ ! [Z1: formula_a_t,Z22: list_formula_a_t] :
              ( ( A
                = ( cons_formula_a_t @ Z1 @ Z22 ) )
             => ~ ( member_formula_a_t @ E2 @ ( set_formula_a_t2 @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_1192_set__ConsD,axiom,
    ! [Y: a,X3: a,Xs: list_a] :
      ( ( member_a @ Y @ ( set_a2 @ ( cons_a @ X3 @ Xs ) ) )
     => ( ( Y = X3 )
        | ( member_a @ Y @ ( set_a2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_1193_set__ConsD,axiom,
    ! [Y: product_prod_nat_nat,X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( member8440522571783428010at_nat @ Y @ ( set_Pr5648618587558075414at_nat @ ( cons_P6512896166579812791at_nat @ X3 @ Xs ) ) )
     => ( ( Y = X3 )
        | ( member8440522571783428010at_nat @ Y @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_1194_set__ConsD,axiom,
    ! [Y: nat,X3: nat,Xs: list_nat] :
      ( ( member_nat @ Y @ ( set_nat2 @ ( cons_nat @ X3 @ Xs ) ) )
     => ( ( Y = X3 )
        | ( member_nat @ Y @ ( set_nat2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_1195_set__ConsD,axiom,
    ! [Y: formula_a_t,X3: formula_a_t,Xs: list_formula_a_t] :
      ( ( member_formula_a_t @ Y @ ( set_formula_a_t2 @ ( cons_formula_a_t @ X3 @ Xs ) ) )
     => ( ( Y = X3 )
        | ( member_formula_a_t @ Y @ ( set_formula_a_t2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_1196_split__list,axiom,
    ! [X3: a,Xs: list_a] :
      ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
     => ? [Ys4: list_a,Zs2: list_a] :
          ( Xs
          = ( append_a @ Ys4 @ ( cons_a @ X3 @ Zs2 ) ) ) ) ).

% split_list
thf(fact_1197_split__list,axiom,
    ! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
     => ? [Ys4: list_P6011104703257516679at_nat,Zs2: list_P6011104703257516679at_nat] :
          ( Xs
          = ( append985823374593552924at_nat @ Ys4 @ ( cons_P6512896166579812791at_nat @ X3 @ Zs2 ) ) ) ) ).

% split_list
thf(fact_1198_split__list,axiom,
    ! [X3: nat,Xs: list_nat] :
      ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
     => ? [Ys4: list_nat,Zs2: list_nat] :
          ( Xs
          = ( append_nat @ Ys4 @ ( cons_nat @ X3 @ Zs2 ) ) ) ) ).

% split_list
thf(fact_1199_split__list,axiom,
    ! [X3: formula_a_t,Xs: list_formula_a_t] :
      ( ( member_formula_a_t @ X3 @ ( set_formula_a_t2 @ Xs ) )
     => ? [Ys4: list_formula_a_t,Zs2: list_formula_a_t] :
          ( Xs
          = ( append_formula_a_t @ Ys4 @ ( cons_formula_a_t @ X3 @ Zs2 ) ) ) ) ).

% split_list
thf(fact_1200_split__list__last,axiom,
    ! [X3: a,Xs: list_a] :
      ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
     => ? [Ys4: list_a,Zs2: list_a] :
          ( ( Xs
            = ( append_a @ Ys4 @ ( cons_a @ X3 @ Zs2 ) ) )
          & ~ ( member_a @ X3 @ ( set_a2 @ Zs2 ) ) ) ) ).

% split_list_last
thf(fact_1201_split__list__last,axiom,
    ! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
     => ? [Ys4: list_P6011104703257516679at_nat,Zs2: list_P6011104703257516679at_nat] :
          ( ( Xs
            = ( append985823374593552924at_nat @ Ys4 @ ( cons_P6512896166579812791at_nat @ X3 @ Zs2 ) ) )
          & ~ ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Zs2 ) ) ) ) ).

% split_list_last
thf(fact_1202_split__list__last,axiom,
    ! [X3: nat,Xs: list_nat] :
      ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
     => ? [Ys4: list_nat,Zs2: list_nat] :
          ( ( Xs
            = ( append_nat @ Ys4 @ ( cons_nat @ X3 @ Zs2 ) ) )
          & ~ ( member_nat @ X3 @ ( set_nat2 @ Zs2 ) ) ) ) ).

% split_list_last
thf(fact_1203_split__list__last,axiom,
    ! [X3: formula_a_t,Xs: list_formula_a_t] :
      ( ( member_formula_a_t @ X3 @ ( set_formula_a_t2 @ Xs ) )
     => ? [Ys4: list_formula_a_t,Zs2: list_formula_a_t] :
          ( ( Xs
            = ( append_formula_a_t @ Ys4 @ ( cons_formula_a_t @ X3 @ Zs2 ) ) )
          & ~ ( member_formula_a_t @ X3 @ ( set_formula_a_t2 @ Zs2 ) ) ) ) ).

% split_list_last
thf(fact_1204_split__list__prop,axiom,
    ! [Xs: list_formula_a_t,P: formula_a_t > $o] :
      ( ? [X10: formula_a_t] :
          ( ( member_formula_a_t @ X10 @ ( set_formula_a_t2 @ Xs ) )
          & ( P @ X10 ) )
     => ? [Ys4: list_formula_a_t,X: formula_a_t] :
          ( ? [Zs2: list_formula_a_t] :
              ( Xs
              = ( append_formula_a_t @ Ys4 @ ( cons_formula_a_t @ X @ Zs2 ) ) )
          & ( P @ X ) ) ) ).

% split_list_prop
thf(fact_1205_split__list__first,axiom,
    ! [X3: a,Xs: list_a] :
      ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
     => ? [Ys4: list_a,Zs2: list_a] :
          ( ( Xs
            = ( append_a @ Ys4 @ ( cons_a @ X3 @ Zs2 ) ) )
          & ~ ( member_a @ X3 @ ( set_a2 @ Ys4 ) ) ) ) ).

% split_list_first
thf(fact_1206_split__list__first,axiom,
    ! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
     => ? [Ys4: list_P6011104703257516679at_nat,Zs2: list_P6011104703257516679at_nat] :
          ( ( Xs
            = ( append985823374593552924at_nat @ Ys4 @ ( cons_P6512896166579812791at_nat @ X3 @ Zs2 ) ) )
          & ~ ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Ys4 ) ) ) ) ).

% split_list_first
thf(fact_1207_split__list__first,axiom,
    ! [X3: nat,Xs: list_nat] :
      ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
     => ? [Ys4: list_nat,Zs2: list_nat] :
          ( ( Xs
            = ( append_nat @ Ys4 @ ( cons_nat @ X3 @ Zs2 ) ) )
          & ~ ( member_nat @ X3 @ ( set_nat2 @ Ys4 ) ) ) ) ).

% split_list_first
thf(fact_1208_split__list__first,axiom,
    ! [X3: formula_a_t,Xs: list_formula_a_t] :
      ( ( member_formula_a_t @ X3 @ ( set_formula_a_t2 @ Xs ) )
     => ? [Ys4: list_formula_a_t,Zs2: list_formula_a_t] :
          ( ( Xs
            = ( append_formula_a_t @ Ys4 @ ( cons_formula_a_t @ X3 @ Zs2 ) ) )
          & ~ ( member_formula_a_t @ X3 @ ( set_formula_a_t2 @ Ys4 ) ) ) ) ).

% split_list_first
thf(fact_1209_split__list__propE,axiom,
    ! [Xs: list_formula_a_t,P: formula_a_t > $o] :
      ( ? [X10: formula_a_t] :
          ( ( member_formula_a_t @ X10 @ ( set_formula_a_t2 @ Xs ) )
          & ( P @ X10 ) )
     => ~ ! [Ys4: list_formula_a_t,X: formula_a_t] :
            ( ? [Zs2: list_formula_a_t] :
                ( Xs
                = ( append_formula_a_t @ Ys4 @ ( cons_formula_a_t @ X @ Zs2 ) ) )
           => ~ ( P @ X ) ) ) ).

% split_list_propE
thf(fact_1210_append__Cons__eq__iff,axiom,
    ! [X3: a,Xs: list_a,Ys: list_a,Xs4: list_a,Ys5: list_a] :
      ( ~ ( member_a @ X3 @ ( set_a2 @ Xs ) )
     => ( ~ ( member_a @ X3 @ ( set_a2 @ Ys ) )
       => ( ( ( append_a @ Xs @ ( cons_a @ X3 @ Ys ) )
            = ( append_a @ Xs4 @ ( cons_a @ X3 @ Ys5 ) ) )
          = ( ( Xs = Xs4 )
            & ( Ys = Ys5 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_1211_append__Cons__eq__iff,axiom,
    ! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,Xs4: list_P6011104703257516679at_nat,Ys5: list_P6011104703257516679at_nat] :
      ( ~ ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
     => ( ~ ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Ys ) )
       => ( ( ( append985823374593552924at_nat @ Xs @ ( cons_P6512896166579812791at_nat @ X3 @ Ys ) )
            = ( append985823374593552924at_nat @ Xs4 @ ( cons_P6512896166579812791at_nat @ X3 @ Ys5 ) ) )
          = ( ( Xs = Xs4 )
            & ( Ys = Ys5 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_1212_append__Cons__eq__iff,axiom,
    ! [X3: nat,Xs: list_nat,Ys: list_nat,Xs4: list_nat,Ys5: list_nat] :
      ( ~ ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
     => ( ~ ( member_nat @ X3 @ ( set_nat2 @ Ys ) )
       => ( ( ( append_nat @ Xs @ ( cons_nat @ X3 @ Ys ) )
            = ( append_nat @ Xs4 @ ( cons_nat @ X3 @ Ys5 ) ) )
          = ( ( Xs = Xs4 )
            & ( Ys = Ys5 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_1213_append__Cons__eq__iff,axiom,
    ! [X3: formula_a_t,Xs: list_formula_a_t,Ys: list_formula_a_t,Xs4: list_formula_a_t,Ys5: list_formula_a_t] :
      ( ~ ( member_formula_a_t @ X3 @ ( set_formula_a_t2 @ Xs ) )
     => ( ~ ( member_formula_a_t @ X3 @ ( set_formula_a_t2 @ Ys ) )
       => ( ( ( append_formula_a_t @ Xs @ ( cons_formula_a_t @ X3 @ Ys ) )
            = ( append_formula_a_t @ Xs4 @ ( cons_formula_a_t @ X3 @ Ys5 ) ) )
          = ( ( Xs = Xs4 )
            & ( Ys = Ys5 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_1214_in__set__conv__decomp,axiom,
    ! [X3: a,Xs: list_a] :
      ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
      = ( ? [Ys2: list_a,Zs3: list_a] :
            ( Xs
            = ( append_a @ Ys2 @ ( cons_a @ X3 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_1215_in__set__conv__decomp,axiom,
    ! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
      = ( ? [Ys2: list_P6011104703257516679at_nat,Zs3: list_P6011104703257516679at_nat] :
            ( Xs
            = ( append985823374593552924at_nat @ Ys2 @ ( cons_P6512896166579812791at_nat @ X3 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_1216_in__set__conv__decomp,axiom,
    ! [X3: nat,Xs: list_nat] :
      ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
      = ( ? [Ys2: list_nat,Zs3: list_nat] :
            ( Xs
            = ( append_nat @ Ys2 @ ( cons_nat @ X3 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_1217_in__set__conv__decomp,axiom,
    ! [X3: formula_a_t,Xs: list_formula_a_t] :
      ( ( member_formula_a_t @ X3 @ ( set_formula_a_t2 @ Xs ) )
      = ( ? [Ys2: list_formula_a_t,Zs3: list_formula_a_t] :
            ( Xs
            = ( append_formula_a_t @ Ys2 @ ( cons_formula_a_t @ X3 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_1218_split__list__last__prop,axiom,
    ! [Xs: list_formula_a_t,P: formula_a_t > $o] :
      ( ? [X10: formula_a_t] :
          ( ( member_formula_a_t @ X10 @ ( set_formula_a_t2 @ Xs ) )
          & ( P @ X10 ) )
     => ? [Ys4: list_formula_a_t,X: formula_a_t,Zs2: list_formula_a_t] :
          ( ( Xs
            = ( append_formula_a_t @ Ys4 @ ( cons_formula_a_t @ X @ Zs2 ) ) )
          & ( P @ X )
          & ! [Xa2: formula_a_t] :
              ( ( member_formula_a_t @ Xa2 @ ( set_formula_a_t2 @ Zs2 ) )
             => ~ ( P @ Xa2 ) ) ) ) ).

% split_list_last_prop
thf(fact_1219_split__list__first__prop,axiom,
    ! [Xs: list_formula_a_t,P: formula_a_t > $o] :
      ( ? [X10: formula_a_t] :
          ( ( member_formula_a_t @ X10 @ ( set_formula_a_t2 @ Xs ) )
          & ( P @ X10 ) )
     => ? [Ys4: list_formula_a_t,X: formula_a_t] :
          ( ? [Zs2: list_formula_a_t] :
              ( Xs
              = ( append_formula_a_t @ Ys4 @ ( cons_formula_a_t @ X @ Zs2 ) ) )
          & ( P @ X )
          & ! [Xa2: formula_a_t] :
              ( ( member_formula_a_t @ Xa2 @ ( set_formula_a_t2 @ Ys4 ) )
             => ~ ( P @ Xa2 ) ) ) ) ).

% split_list_first_prop
thf(fact_1220_split__list__last__propE,axiom,
    ! [Xs: list_formula_a_t,P: formula_a_t > $o] :
      ( ? [X10: formula_a_t] :
          ( ( member_formula_a_t @ X10 @ ( set_formula_a_t2 @ Xs ) )
          & ( P @ X10 ) )
     => ~ ! [Ys4: list_formula_a_t,X: formula_a_t,Zs2: list_formula_a_t] :
            ( ( Xs
              = ( append_formula_a_t @ Ys4 @ ( cons_formula_a_t @ X @ Zs2 ) ) )
           => ( ( P @ X )
             => ~ ! [Xa2: formula_a_t] :
                    ( ( member_formula_a_t @ Xa2 @ ( set_formula_a_t2 @ Zs2 ) )
                   => ~ ( P @ Xa2 ) ) ) ) ) ).

% split_list_last_propE
thf(fact_1221_split__list__first__propE,axiom,
    ! [Xs: list_formula_a_t,P: formula_a_t > $o] :
      ( ? [X10: formula_a_t] :
          ( ( member_formula_a_t @ X10 @ ( set_formula_a_t2 @ Xs ) )
          & ( P @ X10 ) )
     => ~ ! [Ys4: list_formula_a_t,X: formula_a_t] :
            ( ? [Zs2: list_formula_a_t] :
                ( Xs
                = ( append_formula_a_t @ Ys4 @ ( cons_formula_a_t @ X @ Zs2 ) ) )
           => ( ( P @ X )
             => ~ ! [Xa2: formula_a_t] :
                    ( ( member_formula_a_t @ Xa2 @ ( set_formula_a_t2 @ Ys4 ) )
                   => ~ ( P @ Xa2 ) ) ) ) ) ).

% split_list_first_propE
thf(fact_1222_in__set__conv__decomp__last,axiom,
    ! [X3: a,Xs: list_a] :
      ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
      = ( ? [Ys2: list_a,Zs3: list_a] :
            ( ( Xs
              = ( append_a @ Ys2 @ ( cons_a @ X3 @ Zs3 ) ) )
            & ~ ( member_a @ X3 @ ( set_a2 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_1223_in__set__conv__decomp__last,axiom,
    ! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
      = ( ? [Ys2: list_P6011104703257516679at_nat,Zs3: list_P6011104703257516679at_nat] :
            ( ( Xs
              = ( append985823374593552924at_nat @ Ys2 @ ( cons_P6512896166579812791at_nat @ X3 @ Zs3 ) ) )
            & ~ ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_1224_in__set__conv__decomp__last,axiom,
    ! [X3: nat,Xs: list_nat] :
      ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
      = ( ? [Ys2: list_nat,Zs3: list_nat] :
            ( ( Xs
              = ( append_nat @ Ys2 @ ( cons_nat @ X3 @ Zs3 ) ) )
            & ~ ( member_nat @ X3 @ ( set_nat2 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_1225_in__set__conv__decomp__last,axiom,
    ! [X3: formula_a_t,Xs: list_formula_a_t] :
      ( ( member_formula_a_t @ X3 @ ( set_formula_a_t2 @ Xs ) )
      = ( ? [Ys2: list_formula_a_t,Zs3: list_formula_a_t] :
            ( ( Xs
              = ( append_formula_a_t @ Ys2 @ ( cons_formula_a_t @ X3 @ Zs3 ) ) )
            & ~ ( member_formula_a_t @ X3 @ ( set_formula_a_t2 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_1226_in__set__conv__decomp__first,axiom,
    ! [X3: a,Xs: list_a] :
      ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
      = ( ? [Ys2: list_a,Zs3: list_a] :
            ( ( Xs
              = ( append_a @ Ys2 @ ( cons_a @ X3 @ Zs3 ) ) )
            & ~ ( member_a @ X3 @ ( set_a2 @ Ys2 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_1227_in__set__conv__decomp__first,axiom,
    ! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
      = ( ? [Ys2: list_P6011104703257516679at_nat,Zs3: list_P6011104703257516679at_nat] :
            ( ( Xs
              = ( append985823374593552924at_nat @ Ys2 @ ( cons_P6512896166579812791at_nat @ X3 @ Zs3 ) ) )
            & ~ ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Ys2 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_1228_in__set__conv__decomp__first,axiom,
    ! [X3: nat,Xs: list_nat] :
      ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
      = ( ? [Ys2: list_nat,Zs3: list_nat] :
            ( ( Xs
              = ( append_nat @ Ys2 @ ( cons_nat @ X3 @ Zs3 ) ) )
            & ~ ( member_nat @ X3 @ ( set_nat2 @ Ys2 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_1229_in__set__conv__decomp__first,axiom,
    ! [X3: formula_a_t,Xs: list_formula_a_t] :
      ( ( member_formula_a_t @ X3 @ ( set_formula_a_t2 @ Xs ) )
      = ( ? [Ys2: list_formula_a_t,Zs3: list_formula_a_t] :
            ( ( Xs
              = ( append_formula_a_t @ Ys2 @ ( cons_formula_a_t @ X3 @ Zs3 ) ) )
            & ~ ( member_formula_a_t @ X3 @ ( set_formula_a_t2 @ Ys2 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_1230_split__list__last__prop__iff,axiom,
    ! [Xs: list_formula_a_t,P: formula_a_t > $o] :
      ( ( ? [X5: formula_a_t] :
            ( ( member_formula_a_t @ X5 @ ( set_formula_a_t2 @ Xs ) )
            & ( P @ X5 ) ) )
      = ( ? [Ys2: list_formula_a_t,X5: formula_a_t,Zs3: list_formula_a_t] :
            ( ( Xs
              = ( append_formula_a_t @ Ys2 @ ( cons_formula_a_t @ X5 @ Zs3 ) ) )
            & ( P @ X5 )
            & ! [Y5: formula_a_t] :
                ( ( member_formula_a_t @ Y5 @ ( set_formula_a_t2 @ Zs3 ) )
               => ~ ( P @ Y5 ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_1231_split__list__first__prop__iff,axiom,
    ! [Xs: list_formula_a_t,P: formula_a_t > $o] :
      ( ( ? [X5: formula_a_t] :
            ( ( member_formula_a_t @ X5 @ ( set_formula_a_t2 @ Xs ) )
            & ( P @ X5 ) ) )
      = ( ? [Ys2: list_formula_a_t,X5: formula_a_t] :
            ( ? [Zs3: list_formula_a_t] :
                ( Xs
                = ( append_formula_a_t @ Ys2 @ ( cons_formula_a_t @ X5 @ Zs3 ) ) )
            & ( P @ X5 )
            & ! [Y5: formula_a_t] :
                ( ( member_formula_a_t @ Y5 @ ( set_formula_a_t2 @ Ys2 ) )
               => ~ ( P @ Y5 ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_1232_chain__le__app,axiom,
    ! [Zs: list_nat,Z3: nat,W: nat] :
      ( ( chain_le_nat @ ( append_nat @ Zs @ ( cons_nat @ Z3 @ nil_nat ) ) )
     => ( ( ord_less_eq_nat @ Z3 @ W )
       => ( chain_le_nat @ ( append_nat @ ( append_nat @ Zs @ ( cons_nat @ Z3 @ nil_nat ) ) @ ( cons_nat @ W @ nil_nat ) ) ) ) ) ).

% chain_le_app
thf(fact_1233_run_Osimps_I1_J,axiom,
    ! [Run_hd: h > option5775791656947114332_set_a,N: nat] :
      ( ( run_h_t_a @ Run_hd @ N @ vydra_4195442553028166619_a_t_h )
      = none_P3452577240289636010od_t_o ) ).

% run.simps(1)
thf(fact_1234_sorted__wrt_Ocases,axiom,
    ! [X3: produc4771377321613502370la_a_t] :
      ( ! [P6: formula_a_t > formula_a_t > $o] :
          ( X3
         != ( produc2584558968105417244la_a_t @ P6 @ nil_formula_a_t ) )
     => ~ ! [P6: formula_a_t > formula_a_t > $o,X: formula_a_t,Ys4: list_formula_a_t] :
            ( X3
           != ( produc2584558968105417244la_a_t @ P6 @ ( cons_formula_a_t @ X @ Ys4 ) ) ) ) ).

% sorted_wrt.cases
thf(fact_1235_wf__vydra_Ointros_I2_J,axiom,
    ! [Phi2: formula_a_t,I: nat,N: nat,W: vydra_aux_a_t_h,I5: i_t] :
      ( ( vYDRA_wf_vydra_a_t_h @ sigma @ init_hd @ run_hd @ Phi2 @ I @ N @ W )
     => ( ( ( run_h_t_a @ run_hd @ N @ W )
          = none_P3452577240289636010od_t_o )
       => ( vYDRA_wf_vydra_a_t_h @ sigma @ init_hd @ run_hd @ ( next_t_a @ I5 @ Phi2 ) @ I @ ( suc @ N ) @ vydra_4195442553028166619_a_t_h ) ) ) ).

% wf_vydra.intros(2)
thf(fact_1236_wf__vydra_Ointros_I7_J,axiom,
    ! [Phi2: formula_a_t,I: nat,N: nat,V: vydra_aux_a_t_h,Es: list_P1095827921439705018_set_a,Sub: h,I5: i_t] :
      ( ( vYDRA_wf_vydra_a_t_h @ sigma @ init_hd @ run_hd @ Phi2 @ I @ N @ V )
     => ( ( reache3397181163414509427_set_a @ run_hd @ init_hd @ Es @ Sub )
       => ( ( ( size_s5362121731612881446_set_a @ Es )
            = I )
         => ( vYDRA_wf_vydra_a_t_h @ sigma @ init_hd @ run_hd @ ( prev_t_a @ I5 @ Phi2 ) @ I @ ( suc @ N )
            @ ( vydra_3189297434896079910_t_a_h @ I5 @ V @ Sub
              @ ( case_n7548220105206534174od_t_o @ none_P7032077165169035727od_t_o
                @ ^ [J3: nat] : ( some_P1805796590424124243od_t_o @ ( product_Pair_t_o @ ( tau_a_t @ sigma @ J3 ) @ ( sat_a_t @ sigma @ Phi2 @ J3 ) ) )
                @ I ) ) ) ) ) ) ).

% wf_vydra.intros(7)
thf(fact_1237_sat_Osimps_I6_J,axiom,
    ! [I5: i_t,Phi: formula_a_t,I: nat] :
      ( ( sat_a_t @ sigma @ ( next_t_a @ I5 @ Phi ) @ I )
      = ( ( mem_t @ ( tau_a_t @ sigma @ I ) @ ( tau_a_t @ sigma @ ( suc @ I ) ) @ I5 )
        & ( sat_a_t @ sigma @ Phi @ ( suc @ I ) ) ) ) ).

% sat.simps(6)
thf(fact_1238_sat_Osimps_I5_J,axiom,
    ! [I5: i_t,Phi: formula_a_t,I: nat] :
      ( ( sat_a_t @ sigma @ ( prev_t_a @ I5 @ Phi ) @ I )
      = ( case_nat_o @ $false
        @ ^ [J3: nat] :
            ( ( mem_t @ ( tau_a_t @ sigma @ J3 ) @ ( tau_a_t @ sigma @ I ) @ I5 )
            & ( sat_a_t @ sigma @ Phi @ J3 ) )
        @ I ) ) ).

% sat.simps(5)
thf(fact_1239_nat_Odisc__eq__case_I1_J,axiom,
    ! [Nat: nat] :
      ( ( Nat = zero_zero_nat )
      = ( case_nat_o @ $true
        @ ^ [Uu2: nat] : $false
        @ Nat ) ) ).

% nat.disc_eq_case(1)
thf(fact_1240_nat_Odisc__eq__case_I2_J,axiom,
    ! [Nat: nat] :
      ( ( Nat != zero_zero_nat )
      = ( case_nat_o @ $false
        @ ^ [Uu2: nat] : $true
        @ Nat ) ) ).

% nat.disc_eq_case(2)
thf(fact_1241_less__eq__nat_Osimps_I2_J,axiom,
    ! [M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N )
      = ( case_nat_o @ $false @ ( ord_less_eq_nat @ M ) @ N ) ) ).

% less_eq_nat.simps(2)
thf(fact_1242_match__le,axiom,
    ! [I: nat,J: nat,R2: regex_a_t] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ J ) @ ( match_a_t @ sigma @ R2 ) )
     => ( ord_less_eq_nat @ I @ J ) ) ).

% match_le
thf(fact_1243_sat_Osimps_I10_J,axiom,
    ! [I5: i_t,R2: regex_a_t,I: nat] :
      ( ( sat_a_t @ sigma @ ( matchF_t_a @ I5 @ R2 ) @ I )
      = ( ? [J3: nat] :
            ( ( ord_less_eq_nat @ I @ J3 )
            & ( mem_t @ ( tau_a_t @ sigma @ I ) @ ( tau_a_t @ sigma @ J3 ) @ I5 )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ ( suc @ J3 ) ) @ ( match_a_t @ sigma @ R2 ) ) ) ) ) ).

% sat.simps(10)
thf(fact_1244_sat_Osimps_I9_J,axiom,
    ! [I5: i_t,R2: regex_a_t,I: nat] :
      ( ( sat_a_t @ sigma @ ( matchP_t_a @ I5 @ R2 ) @ I )
      = ( ? [J3: nat] :
            ( ( ord_less_eq_nat @ J3 @ I )
            & ( mem_t @ ( tau_a_t @ sigma @ J3 ) @ ( tau_a_t @ sigma @ I ) @ I5 )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ J3 @ ( suc @ I ) ) @ ( match_a_t @ sigma @ R2 ) ) ) ) ) ).

% sat.simps(9)
thf(fact_1245_match__refl__eps,axiom,
    ! [I: nat,R2: regex_a_t] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ I ) @ ( match_a_t @ sigma @ R2 ) )
     => ( eps_a_t @ R2 ) ) ).

% match_refl_eps
thf(fact_1246_wf__regex__eps__match,axiom,
    ! [R2: regex_a_t,I: nat] :
      ( ( wf_regex_a_t @ R2 )
     => ( ( eps_a_t @ R2 )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ I ) @ ( match_a_t @ sigma @ R2 ) ) ) ) ).

% wf_regex_eps_match
thf(fact_1247_pred__def,axiom,
    ( pred
    = ( case_nat_nat @ zero_zero_nat
      @ ^ [X24: nat] : X24 ) ) ).

% pred_def
thf(fact_1248_match__rderive,axiom,
    ! [R2: regex_a_t,I: nat,J: nat] :
      ( ( wf_regex_a_t @ R2 )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ ( suc @ J ) ) @ ( match_a_t @ sigma @ R2 ) )
          = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ J ) @ ( match_a_t @ sigma @ ( rderive_a_t @ R2 ) ) ) ) ) ) ).

% match_rderive
thf(fact_1249_all__nat__less__eq,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [M5: nat] :
            ( ( ord_less_nat @ M5 @ N )
           => ( P @ M5 ) ) )
      = ( ! [X5: nat] :
            ( ( member_nat @ X5 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
           => ( P @ X5 ) ) ) ) ).

% all_nat_less_eq
thf(fact_1250_ex__nat__less__eq,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [M5: nat] :
            ( ( ord_less_nat @ M5 @ N )
            & ( P @ M5 ) ) )
      = ( ? [X5: nat] :
            ( ( member_nat @ X5 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
            & ( P @ X5 ) ) ) ) ).

% ex_nat_less_eq
thf(fact_1251_atLeastSucLessThan__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] :
      ( ( set_or4665077453230672383an_nat @ ( suc @ L ) @ U )
      = ( set_or5834768355832116004an_nat @ L @ U ) ) ).

% atLeastSucLessThan_greaterThanLessThan
thf(fact_1252_match__Star__unfold,axiom,
    ! [I: nat,J: nat,R2: regex_a_t] :
      ( ( ord_less_nat @ I @ J )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ J ) @ ( match_a_t @ sigma @ ( star_a_t @ R2 ) ) )
       => ? [X: nat] :
            ( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ I @ J ) )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ X ) @ ( match_a_t @ sigma @ ( star_a_t @ R2 ) ) )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ J ) @ ( match_a_t @ sigma @ R2 ) ) ) ) ) ).

% match_Star_unfold
thf(fact_1253_match_Osimps_I5_J,axiom,
    ! [R2: regex_a_t] :
      ( ( match_a_t @ sigma @ ( star_a_t @ R2 ) )
      = ( transi2905341329935302413cl_nat @ ( match_a_t @ sigma @ R2 ) ) ) ).

% match.simps(5)
thf(fact_1254_image__Suc__atLeastLessThan,axiom,
    ! [I: nat,J: nat] :
      ( ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ I @ J ) )
      = ( set_or4665077453230672383an_nat @ ( suc @ I ) @ ( suc @ J ) ) ) ).

% image_Suc_atLeastLessThan
thf(fact_1255_zero__notin__Suc__image,axiom,
    ! [A4: set_nat] :
      ~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A4 ) ) ).

% zero_notin_Suc_image

% Conjectures (3)
thf(conj_0,hypothesis,
    vYDRA_wf_vydra_a_t_h @ sigma @ init_hd @ run_hd @ x @ ia @ na @ vb ).

thf(conj_1,hypothesis,
    ( ( run_h_t_a @ run_hd @ na @ vb )
    = ( some_P9127227083975130150od_t_o @ ( produc5956389175556509053od_t_o @ v_b @ ( product_Pair_t_o @ tb @ bb ) ) ) ) ).

thf(conj_2,conjecture,
    ( ( vYDRA_wf_vydra_a_t_h @ sigma @ init_hd @ run_hd @ x @ ( suc @ ia ) @ na @ v_b )
    & ? [Es3: list_P1095827921439705018_set_a,E7: h] :
        ( ( reache3397181163414509427_set_a @ run_hd @ init_hd @ Es3 @ E7 )
        & ( ( size_s5362121731612881446_set_a @ Es3 )
          = ( suc @ ia ) ) )
    & ( tb
      = ( tau_a_t @ sigma @ ia ) )
    & ( bb
      = ( sat_a_t @ sigma @ x @ ia ) ) ) ).

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