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% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:22:45.159

% Could-be-implicit typings (28)
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thf(ty_n_t__Option__Ooption_It__Value__Oval_J, type,
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thf(ty_n_t__List__Olist_It__String__Ochar_J, type,
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thf(ty_n_t__Set__Oset_It__Value__Oval_J, type,
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thf(ty_n_t__Value__Oval, type,
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thf(ty_n_t__Nat__Onat, type,
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% Explicit typings (72)
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thf(sy_c_DefAss_Ohyper__subset_001t__List__Olist_It__String__Ochar_J, type,
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thf(sy_c_JWellForm_Owf__J__mdecl, type,
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thf(sy_c_Map_Omap__add_001t__List__Olist_It__String__Ochar_J_001t__Value__Oval, type,
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thf(sy_c_Objects_Ohext, type,
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thf(sy_c_Option_Ocombine__options_001t__Set__Oset_It__List__Olist_It__String__Ochar_J_J, type,
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thf(sy_c_Option_Ocombine__options_001t__Value__Oval, type,
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thf(sy_c_Option_Ooption_OSome_001t__List__Olist_It__String__Ochar_J, type,
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thf(sy_c_Option_Ooption_OSome_001t__Set__Oset_It__List__Olist_It__String__Ochar_J_J, type,
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thf(sy_c_Option_Ooption_OSome_001t__Value__Oval, type,
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thf(sy_c_Option_Othese_001t__List__Olist_It__String__Ochar_J, type,
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thf(sy_c_Option_Othese_001t__Set__Oset_It__List__Olist_It__String__Ochar_J_J, type,
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thf(sy_c_Option_Othese_001t__Value__Oval, type,
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thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_It__String__Ochar_J_J, type,
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thf(sy_c_Set_Oinsert_001t__Option__Ooption_It__List__Olist_It__String__Ochar_J_J, type,
    insert523167969t_char : option_list_char > set_option_list_char > set_option_list_char).
thf(sy_c_Set_Oinsert_001t__Option__Ooption_It__Set__Oset_It__List__Olist_It__String__Ochar_J_J_J, type,
    insert433234071t_char : option_set_list_char > set_op501702045t_char > set_op501702045t_char).
thf(sy_c_Set_Oinsert_001t__Option__Ooption_It__Value__Oval_J, type,
    insert_option_val : option_val > set_option_val > set_option_val).
thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Expr__Oexp_It__List__Olist_It__String__Ochar_J_J_Mt__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Option__Ooption_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_M_062_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Value__Oval_J_J_J_J_J_M_062_It__List__Olist_It__String__Ochar_J_Mt__Option__Ooption_It__Value__Oval_J_J_J_J_Mt__Product____Type__Oprod_It__Expr__Oexp_It__List__Olist_It__String__Ochar_J_J_Mt__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Option__Ooption_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_M_062_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Value__Oval_J_J_J_J_J_M_062_It__List__Olist_It__String__Ochar_J_Mt__Option__Ooption_It__Value__Oval_J_J_J_J_J, type,
    insert1971494647on_val : produc1172664615on_val > set_Pr1189492743on_val > set_Pr1189492743on_val).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__List__Olist_It__String__Ochar_J_J, type,
    insert_set_list_char : set_list_char > set_set_list_char > set_set_list_char).
thf(sy_c_Set_Oinsert_001t__Value__Oval, type,
    insert_val : val > set_val > set_val).
thf(sy_c_SmallStep_Ored, type,
    red : list_P1234162913t_char > set_Pr1189492743on_val).
thf(sy_c_State_Olcl, type,
    lcl : produc515445032on_val > list_char > option_val).
thf(sy_c_Transitive__Closure_Ortrancl_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Option__Ooption_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_M_062_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Value__Oval_J_J_J_J_J_M_062_It__List__Olist_It__String__Ochar_J_Mt__Option__Ooption_It__Value__Oval_J_J_J, type,
    transi432893227on_val : set_Pr148564103on_val > set_Pr148564103on_val).
thf(sy_c_Transitive__Closure_Ortrancl_001t__Product____Type__Oprod_It__Expr__Oexp_It__List__Olist_It__String__Ochar_J_J_Mt__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Option__Ooption_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_M_062_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Value__Oval_J_J_J_J_J_M_062_It__List__Olist_It__String__Ochar_J_Mt__Option__Ooption_It__Value__Oval_J_J_J_J, type,
    transi111096263on_val : set_Pr1189492743on_val > set_Pr1189492743on_val).
thf(sy_c_Transitive__Closure_Ortrancl_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Expr__Oexp_It__List__Olist_It__String__Ochar_J_J_Mt__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Option__Ooption_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_M_062_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Value__Oval_J_J_J_J_J_M_062_It__List__Olist_It__String__Ochar_J_Mt__Option__Ooption_It__Value__Oval_J_J_J_J_Mt__Product____Type__Oprod_It__Expr__Oexp_It__List__Olist_It__String__Ochar_J_J_Mt__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Option__Ooption_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_M_062_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Value__Oval_J_J_J_J_J_M_062_It__List__Olist_It__String__Ochar_J_Mt__Option__Ooption_It__Value__Oval_J_J_J_J_J, type,
    transi100378922on_val : set_Pr1368603847on_val > set_Pr1368603847on_val).
thf(sy_c_WellForm_Owf__prog_001t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__String__Ochar_J_J_Mt__Expr__Oexp_It__List__Olist_It__String__Ochar_J_J_J, type,
    wf_pro1229751607t_char : (list_P1234162913t_char > list_char > produc1259892823t_char > $o) > list_P1234162913t_char > $o).
thf(sy_c_member_001t__List__Olist_It__String__Ochar_J, type,
    member_list_char : list_char > set_list_char > $o).
thf(sy_c_member_001t__Option__Ooption_It__List__Olist_It__String__Ochar_J_J, type,
    member246349384t_char : option_list_char > set_option_list_char > $o).
thf(sy_c_member_001t__Option__Ooption_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Expr__Oexp_It__List__Olist_It__String__Ochar_J_J_Mt__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Option__Ooption_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_M_062_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Value__Oval_J_J_J_J_J_M_062_It__List__Olist_It__String__Ochar_J_Mt__Option__Ooption_It__Value__Oval_J_J_J_J_Mt__Product____Type__Oprod_It__Expr__Oexp_It__List__Olist_It__String__Ochar_J_J_Mt__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Option__Ooption_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_M_062_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Value__Oval_J_J_J_J_J_M_062_It__List__Olist_It__String__Ochar_J_Mt__Option__Ooption_It__Value__Oval_J_J_J_J_J_J, type,
    member1869742230on_val : option723652845on_val > set_op787718477on_val > $o).
thf(sy_c_member_001t__Option__Ooption_It__Set__Oset_It__List__Olist_It__String__Ochar_J_J_J, type,
    member1976670718t_char : option_set_list_char > set_op501702045t_char > $o).
thf(sy_c_member_001t__Option__Ooption_It__Value__Oval_J, type,
    member_option_val : option_val > set_option_val > $o).
thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Option__Ooption_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_M_062_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Value__Oval_J_J_J_J_J_M_062_It__List__Olist_It__String__Ochar_J_Mt__Option__Ooption_It__Value__Oval_J_J_J, type,
    member652041809on_val : produc515445032on_val > set_Pr723149704on_val > $o).
thf(sy_c_member_001t__Product____Type__Oprod_It__Expr__Oexp_It__List__Olist_It__String__Ochar_J_J_Mt__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Option__Ooption_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_M_062_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Value__Oval_J_J_J_J_J_M_062_It__List__Olist_It__String__Ochar_J_Mt__Option__Ooption_It__Value__Oval_J_J_J_J, type,
    member1888679917on_val : produc1097884484on_val > set_Pr556513572on_val > $o).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Option__Ooption_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_M_062_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Value__Oval_J_J_J_J_J_M_062_It__List__Olist_It__String__Ochar_J_Mt__Option__Ooption_It__Value__Oval_J_J_J_Mt__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Option__Ooption_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_M_062_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Value__Oval_J_J_J_J_J_M_062_It__List__Olist_It__String__Ochar_J_Mt__Option__Ooption_It__Value__Oval_J_J_J_J, type,
    member2027247312on_val : produc744932519on_val > set_Pr148564103on_val > $o).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Expr__Oexp_It__List__Olist_It__String__Ochar_J_J_Mt__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Option__Ooption_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_M_062_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Value__Oval_J_J_J_J_J_M_062_It__List__Olist_It__String__Ochar_J_Mt__Option__Ooption_It__Value__Oval_J_J_J_J_Mt__Product____Type__Oprod_It__Expr__Oexp_It__List__Olist_It__String__Ochar_J_J_Mt__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Option__Ooption_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_M_062_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Value__Oval_J_J_J_J_J_M_062_It__List__Olist_It__String__Ochar_J_Mt__Option__Ooption_It__Value__Oval_J_J_J_J_J, type,
    member354538064on_val : produc1172664615on_val > set_Pr1189492743on_val > $o).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Expr__Oexp_It__List__Olist_It__String__Ochar_J_J_Mt__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Option__Ooption_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_M_062_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Value__Oval_J_J_J_J_J_M_062_It__List__Olist_It__String__Ochar_J_Mt__Option__Ooption_It__Value__Oval_J_J_J_J_Mt__Product____Type__Oprod_It__Expr__Oexp_It__List__Olist_It__String__Ochar_J_J_Mt__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Option__Ooption_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_M_062_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Value__Oval_J_J_J_J_J_M_062_It__List__Olist_It__String__Ochar_J_Mt__Option__Ooption_It__Value__Oval_J_J_J_J_J_Mt__Product____Type__Oprod_It__Product____Type__Oprod_It__Expr__Oexp_It__List__Olist_It__String__Ochar_J_J_Mt__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Option__Ooption_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_M_062_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Value__Oval_J_J_J_J_J_M_062_It__List__Olist_It__String__Ochar_J_Mt__Option__Ooption_It__Value__Oval_J_J_J_J_Mt__Product____Type__Oprod_It__Expr__Oexp_It__List__Olist_It__String__Ochar_J_J_Mt__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Option__Ooption_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_M_062_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_It__String__Ochar_J_J_Mt__Option__Ooption_It__Value__Oval_J_J_J_J_J_M_062_It__List__Olist_It__String__Ochar_J_Mt__Option__Ooption_It__Value__Oval_J_J_J_J_J_J, type,
    member208988944on_val : produc1161703655on_val > set_Pr1368603847on_val > $o).
thf(sy_c_member_001t__Set__Oset_It__List__Olist_It__String__Ochar_J_J, type,
    member_set_list_char : set_list_char > set_set_list_char > $o).
thf(sy_c_member_001t__Value__Oval, type,
    member_val : val > set_val > $o).
thf(sy_v_P, type,
    p : list_P1234162913t_char).
thf(sy_v_e, type,
    e : exp_list_char).
thf(sy_v_e_H, type,
    e2 : exp_list_char).
thf(sy_v_e_Ha____, type,
    e_a : exp_list_char).
thf(sy_v_ea____, type,
    ea : exp_list_char).
thf(sy_v_s, type,
    s : produc515445032on_val).
thf(sy_v_s_H, type,
    s2 : produc515445032on_val).
thf(sy_v_s_Ha____, type,
    s_a : produc515445032on_val).
thf(sy_v_sa____, type,
    sa : produc515445032on_val).

% Relevant facts (196)
thf(fact_0_step_Ohyps_I1_J, axiom,
    ((member354538064on_val @ (produc1069526295on_val @ (produc2023458740on_val @ ea @ sa) @ (produc2023458740on_val @ e_a @ s_a)) @ (red @ p)))). % step.hyps(1)
thf(fact_1_step_Oprems, axiom,
    ((d_list_char @ ea @ (some_set_list_char @ (dom_list_char_val @ (lcl @ sa)))))). % step.prems
thf(fact_2_step_Ohyps_I3_J, axiom,
    (((d_list_char @ e_a @ (some_set_list_char @ (dom_list_char_val @ (lcl @ s_a)))) => (d_list_char @ e2 @ (some_set_list_char @ (dom_list_char_val @ (lcl @ s2))))))). % step.hyps(3)
thf(fact_3_step_Ohyps_I2_J, axiom,
    ((member354538064on_val @ (produc1069526295on_val @ (produc2023458740on_val @ e_a @ s_a) @ (produc2023458740on_val @ e2 @ s2)) @ (transi111096263on_val @ (red @ p))))). % step.hyps(2)
thf(fact_4_reds, axiom,
    ((member354538064on_val @ (produc1069526295on_val @ (produc2023458740on_val @ e @ s) @ (produc2023458740on_val @ e2 @ s2)) @ (transi111096263on_val @ (red @ p))))). % reds
thf(fact_5_option_Oinject, axiom,
    ((![X2 : val, Y2 : val]: (((some_val @ X2) = (some_val @ Y2)) = (X2 = Y2))))). % option.inject
thf(fact_6_option_Oinject, axiom,
    ((![X2 : set_list_char, Y2 : set_list_char]: (((some_set_list_char @ X2) = (some_set_list_char @ Y2)) = (X2 = Y2))))). % option.inject
thf(fact_7_domD, axiom,
    ((![A : list_char, M : list_char > option_set_list_char]: ((member_list_char @ A @ (dom_li665594572t_char @ M)) => (?[B : set_list_char]: ((M @ A) = (some_set_list_char @ B))))))). % domD
thf(fact_8_domD, axiom,
    ((![A : produc1172664615on_val, M : produc1172664615on_val > option_val]: ((member354538064on_val @ A @ (dom_Pr162898261al_val @ M)) => (?[B : val]: ((M @ A) = (some_val @ B))))))). % domD
thf(fact_9_domD, axiom,
    ((![A : list_char, M : list_char > option_val]: ((member_list_char @ A @ (dom_list_char_val @ M)) => (?[B : val]: ((M @ A) = (some_val @ B))))))). % domD
thf(fact_10_domD, axiom,
    ((![A : produc1172664615on_val, M : produc1172664615on_val > option_set_list_char]: ((member354538064on_val @ A @ (dom_Pr1335625324t_char @ M)) => (?[B : set_list_char]: ((M @ A) = (some_set_list_char @ B))))))). % domD
thf(fact_11_domI, axiom,
    ((![M : list_char > option_set_list_char, A : list_char, B2 : set_list_char]: (((M @ A) = (some_set_list_char @ B2)) => (member_list_char @ A @ (dom_li665594572t_char @ M)))))). % domI
thf(fact_12_domI, axiom,
    ((![M : produc1172664615on_val > option_val, A : produc1172664615on_val, B2 : val]: (((M @ A) = (some_val @ B2)) => (member354538064on_val @ A @ (dom_Pr162898261al_val @ M)))))). % domI
thf(fact_13_domI, axiom,
    ((![M : list_char > option_val, A : list_char, B2 : val]: (((M @ A) = (some_val @ B2)) => (member_list_char @ A @ (dom_list_char_val @ M)))))). % domI
thf(fact_14_domI, axiom,
    ((![M : produc1172664615on_val > option_set_list_char, A : produc1172664615on_val, B2 : set_list_char]: (((M @ A) = (some_set_list_char @ B2)) => (member354538064on_val @ A @ (dom_Pr1335625324t_char @ M)))))). % domI
thf(fact_15_dom__const, axiom,
    ((![F : produc1172664615on_val > set_list_char]: ((dom_Pr1335625324t_char @ (^[X : produc1172664615on_val]: (some_set_list_char @ (F @ X)))) = top_to1683965271on_val)))). % dom_const
thf(fact_16_dom__const, axiom,
    ((![F : list_char > set_list_char]: ((dom_li665594572t_char @ (^[X : list_char]: (some_set_list_char @ (F @ X)))) = top_to748628679t_char)))). % dom_const
thf(fact_17_dom__const, axiom,
    ((![F : list_char > val]: ((dom_list_char_val @ (^[X : list_char]: (some_val @ (F @ X)))) = top_to748628679t_char)))). % dom_const
thf(fact_18_insert__dom, axiom,
    ((![F : produc1172664615on_val > option_set_list_char, X3 : produc1172664615on_val, Y : set_list_char]: (((F @ X3) = (some_set_list_char @ Y)) => ((insert1971494647on_val @ X3 @ (dom_Pr1335625324t_char @ F)) = (dom_Pr1335625324t_char @ F)))))). % insert_dom
thf(fact_19_insert__dom, axiom,
    ((![F : list_char > option_set_list_char, X3 : list_char, Y : set_list_char]: (((F @ X3) = (some_set_list_char @ Y)) => ((insert_list_char @ X3 @ (dom_li665594572t_char @ F)) = (dom_li665594572t_char @ F)))))). % insert_dom
thf(fact_20_insert__dom, axiom,
    ((![F : list_char > option_val, X3 : list_char, Y : val]: (((F @ X3) = (some_val @ Y)) => ((insert_list_char @ X3 @ (dom_list_char_val @ F)) = (dom_list_char_val @ F)))))). % insert_dom
thf(fact_21_D__mono, axiom,
    ((![A2 : option_set_list_char, A3 : option_set_list_char, E : exp_list_char]: ((hyper_1448788648t_char @ A2 @ A3) => ((d_list_char @ E @ A2) => (d_list_char @ E @ A3)))))). % D_mono
thf(fact_22_D__mono_H, axiom,
    ((![E : exp_list_char, A2 : option_set_list_char, A3 : option_set_list_char]: ((d_list_char @ E @ A2) => ((hyper_1448788648t_char @ A2 @ A3) => (d_list_char @ E @ A3)))))). % D_mono'
thf(fact_23_red__preserves__defass, axiom,
    ((![P : list_P1234162913t_char, E : exp_list_char, H : nat > option2118145160on_val, L : list_char > option_val, E2 : exp_list_char, H2 : nat > option2118145160on_val, L2 : list_char > option_val]: ((wf_pro1229751607t_char @ wf_J_mdecl @ P) => ((member354538064on_val @ (produc1069526295on_val @ (produc2023458740on_val @ E @ (produc52622304on_val @ H @ L)) @ (produc2023458740on_val @ E2 @ (produc52622304on_val @ H2 @ L2))) @ (red @ P)) => ((d_list_char @ E @ (some_set_list_char @ (dom_list_char_val @ L))) => (d_list_char @ E2 @ (some_set_list_char @ (dom_list_char_val @ L2))))))))). % red_preserves_defass
thf(fact_24_combine__options__simps_I3_J, axiom,
    ((![F : set_list_char > set_list_char > set_list_char, A : set_list_char, B2 : set_list_char]: ((combin255359704t_char @ F @ (some_set_list_char @ A) @ (some_set_list_char @ B2)) = (some_set_list_char @ (F @ A @ B2)))))). % combine_options_simps(3)
thf(fact_25_combine__options__simps_I3_J, axiom,
    ((![F : val > val > val, A : val, B2 : val]: ((combine_options_val @ F @ (some_val @ A) @ (some_val @ B2)) = (some_val @ (F @ A @ B2)))))). % combine_options_simps(3)
thf(fact_26_local_Owf, axiom,
    ((wf_pro1229751607t_char @ wf_J_mdecl @ p))). % local.wf
thf(fact_27_converse__rtrancl__induct__red, axiom,
    ((![E : exp_list_char, H : nat > option2118145160on_val, L : list_char > option_val, E2 : exp_list_char, H2 : nat > option2118145160on_val, L2 : list_char > option_val, P : list_P1234162913t_char, R : exp_list_char > (nat > option2118145160on_val) > (list_char > option_val) > exp_list_char > (nat > option2118145160on_val) > (list_char > option_val) > $o]: ((member354538064on_val @ (produc1069526295on_val @ (produc2023458740on_val @ E @ (produc52622304on_val @ H @ L)) @ (produc2023458740on_val @ E2 @ (produc52622304on_val @ H2 @ L2))) @ (transi111096263on_val @ (red @ P))) => ((![E3 : exp_list_char, H3 : nat > option2118145160on_val, L3 : list_char > option_val]: (R @ E3 @ H3 @ L3 @ E3 @ H3 @ L3)) => ((![E_0 : exp_list_char, H_0 : nat > option2118145160on_val, L_0 : list_char > option_val, E_1 : exp_list_char, H_1 : nat > option2118145160on_val, L_1 : list_char > option_val, E4 : exp_list_char, H4 : nat > option2118145160on_val, L4 : list_char > option_val]: ((member354538064on_val @ (produc1069526295on_val @ (produc2023458740on_val @ E_0 @ (produc52622304on_val @ H_0 @ L_0)) @ (produc2023458740on_val @ E_1 @ (produc52622304on_val @ H_1 @ L_1))) @ (red @ P)) => ((R @ E_1 @ H_1 @ L_1 @ E4 @ H4 @ L4) => (R @ E_0 @ H_0 @ L_0 @ E4 @ H4 @ L4)))) => (R @ E @ H @ L @ E2 @ H2 @ L2))))))). % converse_rtrancl_induct_red
thf(fact_28_rtrancl__induct2, axiom,
    ((![Ax : produc1097884484on_val, Ay : produc1097884484on_val, Bx : produc1097884484on_val, By : produc1097884484on_val, R2 : set_Pr1368603847on_val, P : produc1097884484on_val > produc1097884484on_val > $o]: ((member208988944on_val @ (produc564964183on_val @ (produc1069526295on_val @ Ax @ Ay) @ (produc1069526295on_val @ Bx @ By)) @ (transi100378922on_val @ R2)) => ((P @ Ax @ Ay) => ((![A4 : produc1097884484on_val, B : produc1097884484on_val, Aa : produc1097884484on_val, Ba : produc1097884484on_val]: ((member208988944on_val @ (produc564964183on_val @ (produc1069526295on_val @ Ax @ Ay) @ (produc1069526295on_val @ A4 @ B)) @ (transi100378922on_val @ R2)) => ((member208988944on_val @ (produc564964183on_val @ (produc1069526295on_val @ A4 @ B) @ (produc1069526295on_val @ Aa @ Ba)) @ R2) => ((P @ A4 @ B) => (P @ Aa @ Ba))))) => (P @ Bx @ By))))))). % rtrancl_induct2
thf(fact_29_rtrancl__induct2, axiom,
    ((![Ax : nat > option2118145160on_val, Ay : list_char > option_val, Bx : nat > option2118145160on_val, By : list_char > option_val, R2 : set_Pr148564103on_val, P : (nat > option2118145160on_val) > (list_char > option_val) > $o]: ((member2027247312on_val @ (produc2106877847on_val @ (produc52622304on_val @ Ax @ Ay) @ (produc52622304on_val @ Bx @ By)) @ (transi432893227on_val @ R2)) => ((P @ Ax @ Ay) => ((![A4 : nat > option2118145160on_val, B : list_char > option_val, Aa : nat > option2118145160on_val, Ba : list_char > option_val]: ((member2027247312on_val @ (produc2106877847on_val @ (produc52622304on_val @ Ax @ Ay) @ (produc52622304on_val @ A4 @ B)) @ (transi432893227on_val @ R2)) => ((member2027247312on_val @ (produc2106877847on_val @ (produc52622304on_val @ A4 @ B) @ (produc52622304on_val @ Aa @ Ba)) @ R2) => ((P @ A4 @ B) => (P @ Aa @ Ba))))) => (P @ Bx @ By))))))). % rtrancl_induct2
thf(fact_30_rtrancl__induct2, axiom,
    ((![Ax : exp_list_char, Ay : produc515445032on_val, Bx : exp_list_char, By : produc515445032on_val, R2 : set_Pr1189492743on_val, P : exp_list_char > produc515445032on_val > $o]: ((member354538064on_val @ (produc1069526295on_val @ (produc2023458740on_val @ Ax @ Ay) @ (produc2023458740on_val @ Bx @ By)) @ (transi111096263on_val @ R2)) => ((P @ Ax @ Ay) => ((![A4 : exp_list_char, B : produc515445032on_val, Aa : exp_list_char, Ba : produc515445032on_val]: ((member354538064on_val @ (produc1069526295on_val @ (produc2023458740on_val @ Ax @ Ay) @ (produc2023458740on_val @ A4 @ B)) @ (transi111096263on_val @ R2)) => ((member354538064on_val @ (produc1069526295on_val @ (produc2023458740on_val @ A4 @ B) @ (produc2023458740on_val @ Aa @ Ba)) @ R2) => ((P @ A4 @ B) => (P @ Aa @ Ba))))) => (P @ Bx @ By))))))). % rtrancl_induct2
thf(fact_31_converse__rtranclE2, axiom,
    ((![Xa : produc1097884484on_val, Xb : produc1097884484on_val, Za : produc1097884484on_val, Zb : produc1097884484on_val, R2 : set_Pr1368603847on_val]: ((member208988944on_val @ (produc564964183on_val @ (produc1069526295on_val @ Xa @ Xb) @ (produc1069526295on_val @ Za @ Zb)) @ (transi100378922on_val @ R2)) => ((~ (((produc1069526295on_val @ Xa @ Xb) = (produc1069526295on_val @ Za @ Zb)))) => (~ ((![A4 : produc1097884484on_val, B : produc1097884484on_val]: ((member208988944on_val @ (produc564964183on_val @ (produc1069526295on_val @ Xa @ Xb) @ (produc1069526295on_val @ A4 @ B)) @ R2) => (~ ((member208988944on_val @ (produc564964183on_val @ (produc1069526295on_val @ A4 @ B) @ (produc1069526295on_val @ Za @ Zb)) @ (transi100378922on_val @ R2))))))))))))). % converse_rtranclE2
thf(fact_32_converse__rtranclE2, axiom,
    ((![Xa : nat > option2118145160on_val, Xb : list_char > option_val, Za : nat > option2118145160on_val, Zb : list_char > option_val, R2 : set_Pr148564103on_val]: ((member2027247312on_val @ (produc2106877847on_val @ (produc52622304on_val @ Xa @ Xb) @ (produc52622304on_val @ Za @ Zb)) @ (transi432893227on_val @ R2)) => ((~ (((produc52622304on_val @ Xa @ Xb) = (produc52622304on_val @ Za @ Zb)))) => (~ ((![A4 : nat > option2118145160on_val, B : list_char > option_val]: ((member2027247312on_val @ (produc2106877847on_val @ (produc52622304on_val @ Xa @ Xb) @ (produc52622304on_val @ A4 @ B)) @ R2) => (~ ((member2027247312on_val @ (produc2106877847on_val @ (produc52622304on_val @ A4 @ B) @ (produc52622304on_val @ Za @ Zb)) @ (transi432893227on_val @ R2))))))))))))). % converse_rtranclE2
thf(fact_33_converse__rtranclE2, axiom,
    ((![Xa : exp_list_char, Xb : produc515445032on_val, Za : exp_list_char, Zb : produc515445032on_val, R2 : set_Pr1189492743on_val]: ((member354538064on_val @ (produc1069526295on_val @ (produc2023458740on_val @ Xa @ Xb) @ (produc2023458740on_val @ Za @ Zb)) @ (transi111096263on_val @ R2)) => ((~ (((produc2023458740on_val @ Xa @ Xb) = (produc2023458740on_val @ Za @ Zb)))) => (~ ((![A4 : exp_list_char, B : produc515445032on_val]: ((member354538064on_val @ (produc1069526295on_val @ (produc2023458740on_val @ Xa @ Xb) @ (produc2023458740on_val @ A4 @ B)) @ R2) => (~ ((member354538064on_val @ (produc1069526295on_val @ (produc2023458740on_val @ A4 @ B) @ (produc2023458740on_val @ Za @ Zb)) @ (transi111096263on_val @ R2))))))))))))). % converse_rtranclE2
thf(fact_34_converse__rtrancl__induct2, axiom,
    ((![Ax : produc1097884484on_val, Ay : produc1097884484on_val, Bx : produc1097884484on_val, By : produc1097884484on_val, R2 : set_Pr1368603847on_val, P : produc1097884484on_val > produc1097884484on_val > $o]: ((member208988944on_val @ (produc564964183on_val @ (produc1069526295on_val @ Ax @ Ay) @ (produc1069526295on_val @ Bx @ By)) @ (transi100378922on_val @ R2)) => ((P @ Bx @ By) => ((![A4 : produc1097884484on_val, B : produc1097884484on_val, Aa : produc1097884484on_val, Ba : produc1097884484on_val]: ((member208988944on_val @ (produc564964183on_val @ (produc1069526295on_val @ A4 @ B) @ (produc1069526295on_val @ Aa @ Ba)) @ R2) => ((member208988944on_val @ (produc564964183on_val @ (produc1069526295on_val @ Aa @ Ba) @ (produc1069526295on_val @ Bx @ By)) @ (transi100378922on_val @ R2)) => ((P @ Aa @ Ba) => (P @ A4 @ B))))) => (P @ Ax @ Ay))))))). % converse_rtrancl_induct2
thf(fact_35_converse__rtrancl__induct2, axiom,
    ((![Ax : nat > option2118145160on_val, Ay : list_char > option_val, Bx : nat > option2118145160on_val, By : list_char > option_val, R2 : set_Pr148564103on_val, P : (nat > option2118145160on_val) > (list_char > option_val) > $o]: ((member2027247312on_val @ (produc2106877847on_val @ (produc52622304on_val @ Ax @ Ay) @ (produc52622304on_val @ Bx @ By)) @ (transi432893227on_val @ R2)) => ((P @ Bx @ By) => ((![A4 : nat > option2118145160on_val, B : list_char > option_val, Aa : nat > option2118145160on_val, Ba : list_char > option_val]: ((member2027247312on_val @ (produc2106877847on_val @ (produc52622304on_val @ A4 @ B) @ (produc52622304on_val @ Aa @ Ba)) @ R2) => ((member2027247312on_val @ (produc2106877847on_val @ (produc52622304on_val @ Aa @ Ba) @ (produc52622304on_val @ Bx @ By)) @ (transi432893227on_val @ R2)) => ((P @ Aa @ Ba) => (P @ A4 @ B))))) => (P @ Ax @ Ay))))))). % converse_rtrancl_induct2
thf(fact_36_converse__rtrancl__induct2, axiom,
    ((![Ax : exp_list_char, Ay : produc515445032on_val, Bx : exp_list_char, By : produc515445032on_val, R2 : set_Pr1189492743on_val, P : exp_list_char > produc515445032on_val > $o]: ((member354538064on_val @ (produc1069526295on_val @ (produc2023458740on_val @ Ax @ Ay) @ (produc2023458740on_val @ Bx @ By)) @ (transi111096263on_val @ R2)) => ((P @ Bx @ By) => ((![A4 : exp_list_char, B : produc515445032on_val, Aa : exp_list_char, Ba : produc515445032on_val]: ((member354538064on_val @ (produc1069526295on_val @ (produc2023458740on_val @ A4 @ B) @ (produc2023458740on_val @ Aa @ Ba)) @ R2) => ((member354538064on_val @ (produc1069526295on_val @ (produc2023458740on_val @ Aa @ Ba) @ (produc2023458740on_val @ Bx @ By)) @ (transi111096263on_val @ R2)) => ((P @ Aa @ Ba) => (P @ A4 @ B))))) => (P @ Ax @ Ay))))))). % converse_rtrancl_induct2
thf(fact_37_prod__cases4, axiom,
    ((![Y : produc1172664615on_val]: (~ ((![A4 : produc1097884484on_val, B : exp_list_char, C : nat > option2118145160on_val, D : list_char > option_val]: (~ ((Y = (produc1069526295on_val @ A4 @ (produc2023458740on_val @ B @ (produc52622304on_val @ C @ D)))))))))))). % prod_cases4
thf(fact_38_prod__induct4, axiom,
    ((![P : produc1172664615on_val > $o, X3 : produc1172664615on_val]: ((![A4 : produc1097884484on_val, B : exp_list_char, C : nat > option2118145160on_val, D : list_char > option_val]: (P @ (produc1069526295on_val @ A4 @ (produc2023458740on_val @ B @ (produc52622304on_val @ C @ D))))) => (P @ X3))))). % prod_induct4
thf(fact_39_old_Oprod_Oinject, axiom,
    ((![A : produc1097884484on_val, B2 : produc1097884484on_val, A5 : produc1097884484on_val, B3 : produc1097884484on_val]: (((produc1069526295on_val @ A @ B2) = (produc1069526295on_val @ A5 @ B3)) = (((A = A5)) & ((B2 = B3))))))). % old.prod.inject
thf(fact_40_old_Oprod_Oinject, axiom,
    ((![A : exp_list_char, B2 : produc515445032on_val, A5 : exp_list_char, B3 : produc515445032on_val]: (((produc2023458740on_val @ A @ B2) = (produc2023458740on_val @ A5 @ B3)) = (((A = A5)) & ((B2 = B3))))))). % old.prod.inject
thf(fact_41_old_Oprod_Oinject, axiom,
    ((![A : nat > option2118145160on_val, B2 : list_char > option_val, A5 : nat > option2118145160on_val, B3 : list_char > option_val]: (((produc52622304on_val @ A @ B2) = (produc52622304on_val @ A5 @ B3)) = (((A = A5)) & ((B2 = B3))))))). % old.prod.inject
thf(fact_42_prod_Oinject, axiom,
    ((![X1 : produc1097884484on_val, X2 : produc1097884484on_val, Y1 : produc1097884484on_val, Y2 : produc1097884484on_val]: (((produc1069526295on_val @ X1 @ X2) = (produc1069526295on_val @ Y1 @ Y2)) = (((X1 = Y1)) & ((X2 = Y2))))))). % prod.inject
thf(fact_43_prod_Oinject, axiom,
    ((![X1 : exp_list_char, X2 : produc515445032on_val, Y1 : exp_list_char, Y2 : produc515445032on_val]: (((produc2023458740on_val @ X1 @ X2) = (produc2023458740on_val @ Y1 @ Y2)) = (((X1 = Y1)) & ((X2 = Y2))))))). % prod.inject
thf(fact_44_prod_Oinject, axiom,
    ((![X1 : nat > option2118145160on_val, X2 : list_char > option_val, Y1 : nat > option2118145160on_val, Y2 : list_char > option_val]: (((produc52622304on_val @ X1 @ X2) = (produc52622304on_val @ Y1 @ Y2)) = (((X1 = Y1)) & ((X2 = Y2))))))). % prod.inject
thf(fact_45_r__into__rtrancl, axiom,
    ((![P2 : produc1172664615on_val, R2 : set_Pr1189492743on_val]: ((member354538064on_val @ P2 @ R2) => (member354538064on_val @ P2 @ (transi111096263on_val @ R2)))))). % r_into_rtrancl
thf(fact_46_rtrancl__idemp, axiom,
    ((![R2 : set_Pr1189492743on_val]: ((transi111096263on_val @ (transi111096263on_val @ R2)) = (transi111096263on_val @ R2))))). % rtrancl_idemp
thf(fact_47_old_Oprod_Oinducts, axiom,
    ((![P : produc1172664615on_val > $o, Prod : produc1172664615on_val]: ((![A4 : produc1097884484on_val, B : produc1097884484on_val]: (P @ (produc1069526295on_val @ A4 @ B))) => (P @ Prod))))). % old.prod.inducts
thf(fact_48_old_Oprod_Oinducts, axiom,
    ((![P : produc1097884484on_val > $o, Prod : produc1097884484on_val]: ((![A4 : exp_list_char, B : produc515445032on_val]: (P @ (produc2023458740on_val @ A4 @ B))) => (P @ Prod))))). % old.prod.inducts
thf(fact_49_old_Oprod_Oinducts, axiom,
    ((![P : produc515445032on_val > $o, Prod : produc515445032on_val]: ((![A4 : nat > option2118145160on_val, B : list_char > option_val]: (P @ (produc52622304on_val @ A4 @ B))) => (P @ Prod))))). % old.prod.inducts
thf(fact_50_old_Oprod_Oexhaust, axiom,
    ((![Y : produc1172664615on_val]: (~ ((![A4 : produc1097884484on_val, B : produc1097884484on_val]: (~ ((Y = (produc1069526295on_val @ A4 @ B)))))))))). % old.prod.exhaust
thf(fact_51_old_Oprod_Oexhaust, axiom,
    ((![Y : produc1097884484on_val]: (~ ((![A4 : exp_list_char, B : produc515445032on_val]: (~ ((Y = (produc2023458740on_val @ A4 @ B)))))))))). % old.prod.exhaust
thf(fact_52_old_Oprod_Oexhaust, axiom,
    ((![Y : produc515445032on_val]: (~ ((![A4 : nat > option2118145160on_val, B : list_char > option_val]: (~ ((Y = (produc52622304on_val @ A4 @ B)))))))))). % old.prod.exhaust
thf(fact_53_Pair__inject, axiom,
    ((![A : produc1097884484on_val, B2 : produc1097884484on_val, A5 : produc1097884484on_val, B3 : produc1097884484on_val]: (((produc1069526295on_val @ A @ B2) = (produc1069526295on_val @ A5 @ B3)) => (~ (((A = A5) => (~ ((B2 = B3)))))))))). % Pair_inject
thf(fact_54_Pair__inject, axiom,
    ((![A : exp_list_char, B2 : produc515445032on_val, A5 : exp_list_char, B3 : produc515445032on_val]: (((produc2023458740on_val @ A @ B2) = (produc2023458740on_val @ A5 @ B3)) => (~ (((A = A5) => (~ ((B2 = B3)))))))))). % Pair_inject
thf(fact_55_Pair__inject, axiom,
    ((![A : nat > option2118145160on_val, B2 : list_char > option_val, A5 : nat > option2118145160on_val, B3 : list_char > option_val]: (((produc52622304on_val @ A @ B2) = (produc52622304on_val @ A5 @ B3)) => (~ (((A = A5) => (~ ((B2 = B3)))))))))). % Pair_inject
thf(fact_56_prod__cases, axiom,
    ((![P : produc1172664615on_val > $o, P2 : produc1172664615on_val]: ((![A4 : produc1097884484on_val, B : produc1097884484on_val]: (P @ (produc1069526295on_val @ A4 @ B))) => (P @ P2))))). % prod_cases
thf(fact_57_prod__cases, axiom,
    ((![P : produc1097884484on_val > $o, P2 : produc1097884484on_val]: ((![A4 : exp_list_char, B : produc515445032on_val]: (P @ (produc2023458740on_val @ A4 @ B))) => (P @ P2))))). % prod_cases
thf(fact_58_prod__cases, axiom,
    ((![P : produc515445032on_val > $o, P2 : produc515445032on_val]: ((![A4 : nat > option2118145160on_val, B : list_char > option_val]: (P @ (produc52622304on_val @ A4 @ B))) => (P @ P2))))). % prod_cases
thf(fact_59_surj__pair, axiom,
    ((![P2 : produc1172664615on_val]: (?[X4 : produc1097884484on_val, Y3 : produc1097884484on_val]: (P2 = (produc1069526295on_val @ X4 @ Y3)))))). % surj_pair
thf(fact_60_surj__pair, axiom,
    ((![P2 : produc1097884484on_val]: (?[X4 : exp_list_char, Y3 : produc515445032on_val]: (P2 = (produc2023458740on_val @ X4 @ Y3)))))). % surj_pair
thf(fact_61_surj__pair, axiom,
    ((![P2 : produc515445032on_val]: (?[X4 : nat > option2118145160on_val, Y3 : list_char > option_val]: (P2 = (produc52622304on_val @ X4 @ Y3)))))). % surj_pair
thf(fact_62_prod__induct3, axiom,
    ((![P : produc1172664615on_val > $o, X3 : produc1172664615on_val]: ((![A4 : produc1097884484on_val, B : exp_list_char, C : produc515445032on_val]: (P @ (produc1069526295on_val @ A4 @ (produc2023458740on_val @ B @ C)))) => (P @ X3))))). % prod_induct3
thf(fact_63_prod__induct3, axiom,
    ((![P : produc1097884484on_val > $o, X3 : produc1097884484on_val]: ((![A4 : exp_list_char, B : nat > option2118145160on_val, C : list_char > option_val]: (P @ (produc2023458740on_val @ A4 @ (produc52622304on_val @ B @ C)))) => (P @ X3))))). % prod_induct3
thf(fact_64_prod__cases3, axiom,
    ((![Y : produc1172664615on_val]: (~ ((![A4 : produc1097884484on_val, B : exp_list_char, C : produc515445032on_val]: (~ ((Y = (produc1069526295on_val @ A4 @ (produc2023458740on_val @ B @ C))))))))))). % prod_cases3
thf(fact_65_prod__cases3, axiom,
    ((![Y : produc1097884484on_val]: (~ ((![A4 : exp_list_char, B : nat > option2118145160on_val, C : list_char > option_val]: (~ ((Y = (produc2023458740on_val @ A4 @ (produc52622304on_val @ B @ C))))))))))). % prod_cases3
thf(fact_66_converse__rtrancl__into__rtrancl, axiom,
    ((![A : produc1097884484on_val, B2 : produc1097884484on_val, R2 : set_Pr1189492743on_val, C2 : produc1097884484on_val]: ((member354538064on_val @ (produc1069526295on_val @ A @ B2) @ R2) => ((member354538064on_val @ (produc1069526295on_val @ B2 @ C2) @ (transi111096263on_val @ R2)) => (member354538064on_val @ (produc1069526295on_val @ A @ C2) @ (transi111096263on_val @ R2))))))). % converse_rtrancl_into_rtrancl
thf(fact_67_rtrancl_Ortrancl__into__rtrancl, axiom,
    ((![A : produc1097884484on_val, B2 : produc1097884484on_val, R2 : set_Pr1189492743on_val, C2 : produc1097884484on_val]: ((member354538064on_val @ (produc1069526295on_val @ A @ B2) @ (transi111096263on_val @ R2)) => ((member354538064on_val @ (produc1069526295on_val @ B2 @ C2) @ R2) => (member354538064on_val @ (produc1069526295on_val @ A @ C2) @ (transi111096263on_val @ R2))))))). % rtrancl.rtrancl_into_rtrancl
thf(fact_68_converse__rtrancl__induct, axiom,
    ((![A : produc1097884484on_val, B2 : produc1097884484on_val, R2 : set_Pr1189492743on_val, P : produc1097884484on_val > $o]: ((member354538064on_val @ (produc1069526295on_val @ A @ B2) @ (transi111096263on_val @ R2)) => ((P @ B2) => ((![Y3 : produc1097884484on_val, Z : produc1097884484on_val]: ((member354538064on_val @ (produc1069526295on_val @ Y3 @ Z) @ R2) => ((member354538064on_val @ (produc1069526295on_val @ Z @ B2) @ (transi111096263on_val @ R2)) => ((P @ Z) => (P @ Y3))))) => (P @ A))))))). % converse_rtrancl_induct
thf(fact_69_rtrancl_Ortrancl__refl, axiom,
    ((![A : produc1097884484on_val, R2 : set_Pr1189492743on_val]: (member354538064on_val @ (produc1069526295on_val @ A @ A) @ (transi111096263on_val @ R2))))). % rtrancl.rtrancl_refl
thf(fact_70_mem__Collect__eq, axiom,
    ((![A : produc1172664615on_val, P : produc1172664615on_val > $o]: ((member354538064on_val @ A @ (collec2055896722on_val @ P)) = (P @ A))))). % mem_Collect_eq
thf(fact_71_mem__Collect__eq, axiom,
    ((![A : list_char, P : list_char > $o]: ((member_list_char @ A @ (collect_list_char @ P)) = (P @ A))))). % mem_Collect_eq
thf(fact_72_Collect__mem__eq, axiom,
    ((![A2 : set_Pr1189492743on_val]: ((collec2055896722on_val @ (^[X : produc1172664615on_val]: (member354538064on_val @ X @ A2))) = A2)))). % Collect_mem_eq
thf(fact_73_Collect__mem__eq, axiom,
    ((![A2 : set_list_char]: ((collect_list_char @ (^[X : list_char]: (member_list_char @ X @ A2))) = A2)))). % Collect_mem_eq
thf(fact_74_converse__rtranclE, axiom,
    ((![X3 : produc1097884484on_val, Z2 : produc1097884484on_val, R2 : set_Pr1189492743on_val]: ((member354538064on_val @ (produc1069526295on_val @ X3 @ Z2) @ (transi111096263on_val @ R2)) => ((~ ((X3 = Z2))) => (~ ((![Y3 : produc1097884484on_val]: ((member354538064on_val @ (produc1069526295on_val @ X3 @ Y3) @ R2) => (~ ((member354538064on_val @ (produc1069526295on_val @ Y3 @ Z2) @ (transi111096263on_val @ R2))))))))))))). % converse_rtranclE
thf(fact_75_rtrancl_Oinducts, axiom,
    ((![X1 : produc1097884484on_val, X2 : produc1097884484on_val, R2 : set_Pr1189492743on_val, P : produc1097884484on_val > produc1097884484on_val > $o]: ((member354538064on_val @ (produc1069526295on_val @ X1 @ X2) @ (transi111096263on_val @ R2)) => ((![A4 : produc1097884484on_val]: (P @ A4 @ A4)) => ((![A4 : produc1097884484on_val, B : produc1097884484on_val, C : produc1097884484on_val]: ((member354538064on_val @ (produc1069526295on_val @ A4 @ B) @ (transi111096263on_val @ R2)) => ((P @ A4 @ B) => ((member354538064on_val @ (produc1069526295on_val @ B @ C) @ R2) => (P @ A4 @ C))))) => (P @ X1 @ X2))))))). % rtrancl.inducts
thf(fact_76_rtrancl__induct, axiom,
    ((![A : produc1097884484on_val, B2 : produc1097884484on_val, R2 : set_Pr1189492743on_val, P : produc1097884484on_val > $o]: ((member354538064on_val @ (produc1069526295on_val @ A @ B2) @ (transi111096263on_val @ R2)) => ((P @ A) => ((![Y3 : produc1097884484on_val, Z : produc1097884484on_val]: ((member354538064on_val @ (produc1069526295on_val @ A @ Y3) @ (transi111096263on_val @ R2)) => ((member354538064on_val @ (produc1069526295on_val @ Y3 @ Z) @ R2) => ((P @ Y3) => (P @ Z))))) => (P @ B2))))))). % rtrancl_induct
thf(fact_77_rtrancl__trans, axiom,
    ((![X3 : produc1097884484on_val, Y : produc1097884484on_val, R2 : set_Pr1189492743on_val, Z2 : produc1097884484on_val]: ((member354538064on_val @ (produc1069526295on_val @ X3 @ Y) @ (transi111096263on_val @ R2)) => ((member354538064on_val @ (produc1069526295on_val @ Y @ Z2) @ (transi111096263on_val @ R2)) => (member354538064on_val @ (produc1069526295on_val @ X3 @ Z2) @ (transi111096263on_val @ R2))))))). % rtrancl_trans
thf(fact_78_rtrancl_Osimps, axiom,
    ((![A1 : produc1097884484on_val, A22 : produc1097884484on_val, R2 : set_Pr1189492743on_val]: ((member354538064on_val @ (produc1069526295on_val @ A1 @ A22) @ (transi111096263on_val @ R2)) = (((?[A6 : produc1097884484on_val]: (((A1 = A6)) & ((A22 = A6))))) | ((?[A6 : produc1097884484on_val]: (?[B4 : produc1097884484on_val]: (?[C3 : produc1097884484on_val]: (((A1 = A6)) & ((((A22 = C3)) & ((((member354538064on_val @ (produc1069526295on_val @ A6 @ B4) @ (transi111096263on_val @ R2))) & ((member354538064on_val @ (produc1069526295on_val @ B4 @ C3) @ R2)))))))))))))))). % rtrancl.simps
thf(fact_79_rtrancl_Ocases, axiom,
    ((![A1 : produc1097884484on_val, A22 : produc1097884484on_val, R2 : set_Pr1189492743on_val]: ((member354538064on_val @ (produc1069526295on_val @ A1 @ A22) @ (transi111096263on_val @ R2)) => ((~ ((A22 = A1))) => (~ ((![B : produc1097884484on_val]: ((member354538064on_val @ (produc1069526295on_val @ A1 @ B) @ (transi111096263on_val @ R2)) => (~ ((member354538064on_val @ (produc1069526295on_val @ B @ A22) @ R2)))))))))))). % rtrancl.cases
thf(fact_80_rtranclE, axiom,
    ((![A : produc1097884484on_val, B2 : produc1097884484on_val, R2 : set_Pr1189492743on_val]: ((member354538064on_val @ (produc1069526295on_val @ A @ B2) @ (transi111096263on_val @ R2)) => ((~ ((A = B2))) => (~ ((![Y3 : produc1097884484on_val]: ((member354538064on_val @ (produc1069526295on_val @ A @ Y3) @ (transi111096263on_val @ R2)) => (~ ((member354538064on_val @ (produc1069526295on_val @ Y3 @ B2) @ R2)))))))))))). % rtranclE
thf(fact_81_iso__tuple__UNIV__I, axiom,
    ((![X3 : produc1172664615on_val]: (member354538064on_val @ X3 @ top_to1683965271on_val)))). % iso_tuple_UNIV_I
thf(fact_82_iso__tuple__UNIV__I, axiom,
    ((![X3 : list_char]: (member_list_char @ X3 @ top_to748628679t_char)))). % iso_tuple_UNIV_I
thf(fact_83_UNIV__I, axiom,
    ((![X3 : produc1172664615on_val]: (member354538064on_val @ X3 @ top_to1683965271on_val)))). % UNIV_I
thf(fact_84_UNIV__I, axiom,
    ((![X3 : list_char]: (member_list_char @ X3 @ top_to748628679t_char)))). % UNIV_I
thf(fact_85_insert__absorb2, axiom,
    ((![X3 : list_char, A2 : set_list_char]: ((insert_list_char @ X3 @ (insert_list_char @ X3 @ A2)) = (insert_list_char @ X3 @ A2))))). % insert_absorb2
thf(fact_86_insert__iff, axiom,
    ((![A : produc1172664615on_val, B2 : produc1172664615on_val, A2 : set_Pr1189492743on_val]: ((member354538064on_val @ A @ (insert1971494647on_val @ B2 @ A2)) = (((A = B2)) | ((member354538064on_val @ A @ A2))))))). % insert_iff
thf(fact_87_insert__iff, axiom,
    ((![A : list_char, B2 : list_char, A2 : set_list_char]: ((member_list_char @ A @ (insert_list_char @ B2 @ A2)) = (((A = B2)) | ((member_list_char @ A @ A2))))))). % insert_iff
thf(fact_88_insertCI, axiom,
    ((![A : produc1172664615on_val, B5 : set_Pr1189492743on_val, B2 : produc1172664615on_val]: (((~ ((member354538064on_val @ A @ B5))) => (A = B2)) => (member354538064on_val @ A @ (insert1971494647on_val @ B2 @ B5)))))). % insertCI
thf(fact_89_insertCI, axiom,
    ((![A : list_char, B5 : set_list_char, B2 : list_char]: (((~ ((member_list_char @ A @ B5))) => (A = B2)) => (member_list_char @ A @ (insert_list_char @ B2 @ B5)))))). % insertCI
thf(fact_90_insert__UNIV, axiom,
    ((![X3 : list_char]: ((insert_list_char @ X3 @ top_to748628679t_char) = top_to748628679t_char)))). % insert_UNIV
thf(fact_91_top__set__def, axiom,
    ((top_to748628679t_char = (collect_list_char @ top_top_list_char_o)))). % top_set_def
thf(fact_92_insertE, axiom,
    ((![A : produc1172664615on_val, B2 : produc1172664615on_val, A2 : set_Pr1189492743on_val]: ((member354538064on_val @ A @ (insert1971494647on_val @ B2 @ A2)) => ((~ ((A = B2))) => (member354538064on_val @ A @ A2)))))). % insertE
thf(fact_93_insertE, axiom,
    ((![A : list_char, B2 : list_char, A2 : set_list_char]: ((member_list_char @ A @ (insert_list_char @ B2 @ A2)) => ((~ ((A = B2))) => (member_list_char @ A @ A2)))))). % insertE
thf(fact_94_insertI1, axiom,
    ((![A : produc1172664615on_val, B5 : set_Pr1189492743on_val]: (member354538064on_val @ A @ (insert1971494647on_val @ A @ B5))))). % insertI1
thf(fact_95_insertI1, axiom,
    ((![A : list_char, B5 : set_list_char]: (member_list_char @ A @ (insert_list_char @ A @ B5))))). % insertI1
thf(fact_96_insertI2, axiom,
    ((![A : produc1172664615on_val, B5 : set_Pr1189492743on_val, B2 : produc1172664615on_val]: ((member354538064on_val @ A @ B5) => (member354538064on_val @ A @ (insert1971494647on_val @ B2 @ B5)))))). % insertI2
thf(fact_97_insertI2, axiom,
    ((![A : list_char, B5 : set_list_char, B2 : list_char]: ((member_list_char @ A @ B5) => (member_list_char @ A @ (insert_list_char @ B2 @ B5)))))). % insertI2
thf(fact_98_Set_Oset__insert, axiom,
    ((![X3 : produc1172664615on_val, A2 : set_Pr1189492743on_val]: ((member354538064on_val @ X3 @ A2) => (~ ((![B6 : set_Pr1189492743on_val]: ((A2 = (insert1971494647on_val @ X3 @ B6)) => (member354538064on_val @ X3 @ B6))))))))). % Set.set_insert
thf(fact_99_Set_Oset__insert, axiom,
    ((![X3 : list_char, A2 : set_list_char]: ((member_list_char @ X3 @ A2) => (~ ((![B6 : set_list_char]: ((A2 = (insert_list_char @ X3 @ B6)) => (member_list_char @ X3 @ B6))))))))). % Set.set_insert
thf(fact_100_insert__ident, axiom,
    ((![X3 : produc1172664615on_val, A2 : set_Pr1189492743on_val, B5 : set_Pr1189492743on_val]: ((~ ((member354538064on_val @ X3 @ A2))) => ((~ ((member354538064on_val @ X3 @ B5))) => (((insert1971494647on_val @ X3 @ A2) = (insert1971494647on_val @ X3 @ B5)) = (A2 = B5))))))). % insert_ident
thf(fact_101_insert__ident, axiom,
    ((![X3 : list_char, A2 : set_list_char, B5 : set_list_char]: ((~ ((member_list_char @ X3 @ A2))) => ((~ ((member_list_char @ X3 @ B5))) => (((insert_list_char @ X3 @ A2) = (insert_list_char @ X3 @ B5)) = (A2 = B5))))))). % insert_ident
thf(fact_102_insert__absorb, axiom,
    ((![A : produc1172664615on_val, A2 : set_Pr1189492743on_val]: ((member354538064on_val @ A @ A2) => ((insert1971494647on_val @ A @ A2) = A2))))). % insert_absorb
thf(fact_103_insert__absorb, axiom,
    ((![A : list_char, A2 : set_list_char]: ((member_list_char @ A @ A2) => ((insert_list_char @ A @ A2) = A2))))). % insert_absorb
thf(fact_104_insert__eq__iff, axiom,
    ((![A : produc1172664615on_val, A2 : set_Pr1189492743on_val, B2 : produc1172664615on_val, B5 : set_Pr1189492743on_val]: ((~ ((member354538064on_val @ A @ A2))) => ((~ ((member354538064on_val @ B2 @ B5))) => (((insert1971494647on_val @ A @ A2) = (insert1971494647on_val @ B2 @ B5)) = (((((A = B2)) => ((A2 = B5)))) & ((((~ ((A = B2)))) => ((?[C4 : set_Pr1189492743on_val]: (((A2 = (insert1971494647on_val @ B2 @ C4))) & ((((~ ((member354538064on_val @ B2 @ C4)))) & ((((B5 = (insert1971494647on_val @ A @ C4))) & ((~ ((member354538064on_val @ A @ C4)))))))))))))))))))). % insert_eq_iff
thf(fact_105_insert__eq__iff, axiom,
    ((![A : list_char, A2 : set_list_char, B2 : list_char, B5 : set_list_char]: ((~ ((member_list_char @ A @ A2))) => ((~ ((member_list_char @ B2 @ B5))) => (((insert_list_char @ A @ A2) = (insert_list_char @ B2 @ B5)) = (((((A = B2)) => ((A2 = B5)))) & ((((~ ((A = B2)))) => ((?[C4 : set_list_char]: (((A2 = (insert_list_char @ B2 @ C4))) & ((((~ ((member_list_char @ B2 @ C4)))) & ((((B5 = (insert_list_char @ A @ C4))) & ((~ ((member_list_char @ A @ C4)))))))))))))))))))). % insert_eq_iff
thf(fact_106_insert__commute, axiom,
    ((![X3 : list_char, Y : list_char, A2 : set_list_char]: ((insert_list_char @ X3 @ (insert_list_char @ Y @ A2)) = (insert_list_char @ Y @ (insert_list_char @ X3 @ A2)))))). % insert_commute
thf(fact_107_mk__disjoint__insert, axiom,
    ((![A : produc1172664615on_val, A2 : set_Pr1189492743on_val]: ((member354538064on_val @ A @ A2) => (?[B6 : set_Pr1189492743on_val]: ((A2 = (insert1971494647on_val @ A @ B6)) & (~ ((member354538064on_val @ A @ B6))))))))). % mk_disjoint_insert
thf(fact_108_mk__disjoint__insert, axiom,
    ((![A : list_char, A2 : set_list_char]: ((member_list_char @ A @ A2) => (?[B6 : set_list_char]: ((A2 = (insert_list_char @ A @ B6)) & (~ ((member_list_char @ A @ B6))))))))). % mk_disjoint_insert
thf(fact_109_UNIV__eq__I, axiom,
    ((![A2 : set_Pr1189492743on_val]: ((![X4 : produc1172664615on_val]: (member354538064on_val @ X4 @ A2)) => (top_to1683965271on_val = A2))))). % UNIV_eq_I
thf(fact_110_UNIV__eq__I, axiom,
    ((![A2 : set_list_char]: ((![X4 : list_char]: (member_list_char @ X4 @ A2)) => (top_to748628679t_char = A2))))). % UNIV_eq_I
thf(fact_111_UNIV__witness, axiom,
    ((?[X4 : produc1172664615on_val]: (member354538064on_val @ X4 @ top_to1683965271on_val)))). % UNIV_witness
thf(fact_112_UNIV__witness, axiom,
    ((?[X4 : list_char]: (member_list_char @ X4 @ top_to748628679t_char)))). % UNIV_witness
thf(fact_113_insert__compr, axiom,
    ((insert1971494647on_val = (^[A6 : produc1172664615on_val]: (^[B7 : set_Pr1189492743on_val]: (collec2055896722on_val @ (^[X : produc1172664615on_val]: (((X = A6)) | ((member354538064on_val @ X @ B7)))))))))). % insert_compr
thf(fact_114_insert__compr, axiom,
    ((insert_list_char = (^[A6 : list_char]: (^[B7 : set_list_char]: (collect_list_char @ (^[X : list_char]: (((X = A6)) | ((member_list_char @ X @ B7)))))))))). % insert_compr
thf(fact_115_insert__Collect, axiom,
    ((![A : list_char, P : list_char > $o]: ((insert_list_char @ A @ (collect_list_char @ P)) = (collect_list_char @ (^[U : list_char]: (((~ ((U = A)))) => ((P @ U))))))))). % insert_Collect
thf(fact_116_UNIV__def, axiom,
    ((top_to748628679t_char = (collect_list_char @ (^[X : list_char]: $true))))). % UNIV_def
thf(fact_117_red__lcl__incr, axiom,
    ((![E : exp_list_char, H_02 : nat > option2118145160on_val, L_02 : list_char > option_val, E2 : exp_list_char, H_12 : nat > option2118145160on_val, L_12 : list_char > option_val, P : list_P1234162913t_char]: ((member354538064on_val @ (produc1069526295on_val @ (produc2023458740on_val @ E @ (produc52622304on_val @ H_02 @ L_02)) @ (produc2023458740on_val @ E2 @ (produc52622304on_val @ H_12 @ L_12))) @ (red @ P)) => (ord_le1836582007t_char @ (dom_list_char_val @ L_02) @ (dom_list_char_val @ L_12)))))). % red_lcl_incr
thf(fact_118_Red__lcl__add, axiom,
    ((![E : exp_list_char, H : nat > option2118145160on_val, L : list_char > option_val, E2 : exp_list_char, H2 : nat > option2118145160on_val, L2 : list_char > option_val, P : list_P1234162913t_char, L_02 : list_char > option_val]: ((member354538064on_val @ (produc1069526295on_val @ (produc2023458740on_val @ E @ (produc52622304on_val @ H @ L)) @ (produc2023458740on_val @ E2 @ (produc52622304on_val @ H2 @ L2))) @ (transi111096263on_val @ (red @ P))) => (member354538064on_val @ (produc1069526295on_val @ (produc2023458740on_val @ E @ (produc52622304on_val @ H @ (map_ad1615113112ar_val @ L_02 @ L))) @ (produc2023458740on_val @ E2 @ (produc52622304on_val @ H2 @ (map_ad1615113112ar_val @ L_02 @ L2)))) @ (transi111096263on_val @ (red @ P))))))). % Red_lcl_add
thf(fact_119_red__hext__incr, axiom,
    ((![E : exp_list_char, H : nat > option2118145160on_val, L : list_char > option_val, E2 : exp_list_char, H2 : nat > option2118145160on_val, L2 : list_char > option_val, P : list_P1234162913t_char]: ((member354538064on_val @ (produc1069526295on_val @ (produc2023458740on_val @ E @ (produc52622304on_val @ H @ L)) @ (produc2023458740on_val @ E2 @ (produc52622304on_val @ H2 @ L2))) @ (red @ P)) => (hext @ H @ H2))))). % red_hext_incr
thf(fact_120_these__insert__Some, axiom,
    ((![X3 : list_char, A2 : set_option_list_char]: ((these_list_char @ (insert523167969t_char @ (some_list_char @ X3) @ A2)) = (insert_list_char @ X3 @ (these_list_char @ A2)))))). % these_insert_Some
thf(fact_121_these__insert__Some, axiom,
    ((![X3 : set_list_char, A2 : set_op501702045t_char]: ((these_set_list_char @ (insert433234071t_char @ (some_set_list_char @ X3) @ A2)) = (insert_set_list_char @ X3 @ (these_set_list_char @ A2)))))). % these_insert_Some
thf(fact_122_these__insert__Some, axiom,
    ((![X3 : val, A2 : set_option_val]: ((these_val @ (insert_option_val @ (some_val @ X3) @ A2)) = (insert_val @ X3 @ (these_val @ A2)))))). % these_insert_Some
thf(fact_123_top__empty__eq2, axiom,
    ((top_to1666156862_val_o = (^[X : produc1097884484on_val]: (^[Y4 : produc1097884484on_val]: (member354538064on_val @ (produc1069526295on_val @ X @ Y4) @ top_to1683965271on_val)))))). % top_empty_eq2
thf(fact_124_top__empty__eq2, axiom,
    ((top_to2035973601_val_o = (^[X : exp_list_char]: (^[Y4 : produc515445032on_val]: (member1888679917on_val @ (produc2023458740on_val @ X @ Y4) @ top_to796958324on_val)))))). % top_empty_eq2
thf(fact_125_top__empty__eq2, axiom,
    ((top_to1645522895_val_o = (^[X : nat > option2118145160on_val]: (^[Y4 : list_char > option_val]: (member652041809on_val @ (produc52622304on_val @ X @ Y4) @ top_to374550744on_val)))))). % top_empty_eq2
thf(fact_126_order__refl, axiom,
    ((![X3 : set_list_char]: (ord_le1836582007t_char @ X3 @ X3)))). % order_refl
thf(fact_127_subsetI, axiom,
    ((![A2 : set_Pr1189492743on_val, B5 : set_Pr1189492743on_val]: ((![X4 : produc1172664615on_val]: ((member354538064on_val @ X4 @ A2) => (member354538064on_val @ X4 @ B5))) => (ord_le837504935on_val @ A2 @ B5))))). % subsetI
thf(fact_128_subsetI, axiom,
    ((![A2 : set_list_char, B5 : set_list_char]: ((![X4 : list_char]: ((member_list_char @ X4 @ A2) => (member_list_char @ X4 @ B5))) => (ord_le1836582007t_char @ A2 @ B5))))). % subsetI
thf(fact_129_subset__antisym, axiom,
    ((![A2 : set_list_char, B5 : set_list_char]: ((ord_le1836582007t_char @ A2 @ B5) => ((ord_le1836582007t_char @ B5 @ A2) => (A2 = B5)))))). % subset_antisym
thf(fact_130_map__add__assoc, axiom,
    ((![M1 : list_char > option_val, M2 : list_char > option_val, M3 : list_char > option_val]: ((map_ad1615113112ar_val @ M1 @ (map_ad1615113112ar_val @ M2 @ M3)) = (map_ad1615113112ar_val @ (map_ad1615113112ar_val @ M1 @ M2) @ M3))))). % map_add_assoc
thf(fact_131_insert__subset, axiom,
    ((![X3 : produc1172664615on_val, A2 : set_Pr1189492743on_val, B5 : set_Pr1189492743on_val]: ((ord_le837504935on_val @ (insert1971494647on_val @ X3 @ A2) @ B5) = (((member354538064on_val @ X3 @ B5)) & ((ord_le837504935on_val @ A2 @ B5))))))). % insert_subset
thf(fact_132_insert__subset, axiom,
    ((![X3 : list_char, A2 : set_list_char, B5 : set_list_char]: ((ord_le1836582007t_char @ (insert_list_char @ X3 @ A2) @ B5) = (((member_list_char @ X3 @ B5)) & ((ord_le1836582007t_char @ A2 @ B5))))))). % insert_subset
thf(fact_133_map__add__find__right, axiom,
    ((![N : list_char > option_val, K : list_char, Xx : val, M : list_char > option_val]: (((N @ K) = (some_val @ Xx)) => ((map_ad1615113112ar_val @ M @ N @ K) = (some_val @ Xx)))))). % map_add_find_right
thf(fact_134_top_Oextremum__uniqueI, axiom,
    ((![A : set_list_char]: ((ord_le1836582007t_char @ top_to748628679t_char @ A) => (A = top_to748628679t_char))))). % top.extremum_uniqueI
thf(fact_135_top_Oextremum__unique, axiom,
    ((![A : set_list_char]: ((ord_le1836582007t_char @ top_to748628679t_char @ A) = (A = top_to748628679t_char))))). % top.extremum_unique
thf(fact_136_top__greatest, axiom,
    ((![A : set_list_char]: (ord_le1836582007t_char @ A @ top_to748628679t_char)))). % top_greatest
thf(fact_137_subset__insertI2, axiom,
    ((![A2 : set_list_char, B5 : set_list_char, B2 : list_char]: ((ord_le1836582007t_char @ A2 @ B5) => (ord_le1836582007t_char @ A2 @ (insert_list_char @ B2 @ B5)))))). % subset_insertI2
thf(fact_138_subset__insertI, axiom,
    ((![B5 : set_list_char, A : list_char]: (ord_le1836582007t_char @ B5 @ (insert_list_char @ A @ B5))))). % subset_insertI
thf(fact_139_subset__insert, axiom,
    ((![X3 : produc1172664615on_val, A2 : set_Pr1189492743on_val, B5 : set_Pr1189492743on_val]: ((~ ((member354538064on_val @ X3 @ A2))) => ((ord_le837504935on_val @ A2 @ (insert1971494647on_val @ X3 @ B5)) = (ord_le837504935on_val @ A2 @ B5)))))). % subset_insert
thf(fact_140_subset__insert, axiom,
    ((![X3 : list_char, A2 : set_list_char, B5 : set_list_char]: ((~ ((member_list_char @ X3 @ A2))) => ((ord_le1836582007t_char @ A2 @ (insert_list_char @ X3 @ B5)) = (ord_le1836582007t_char @ A2 @ B5)))))). % subset_insert
thf(fact_141_insert__mono, axiom,
    ((![C5 : set_list_char, D2 : set_list_char, A : list_char]: ((ord_le1836582007t_char @ C5 @ D2) => (ord_le1836582007t_char @ (insert_list_char @ A @ C5) @ (insert_list_char @ A @ D2)))))). % insert_mono
thf(fact_142_subset__UNIV, axiom,
    ((![A2 : set_list_char]: (ord_le1836582007t_char @ A2 @ top_to748628679t_char)))). % subset_UNIV
thf(fact_143_in__mono, axiom,
    ((![A2 : set_Pr1189492743on_val, B5 : set_Pr1189492743on_val, X3 : produc1172664615on_val]: ((ord_le837504935on_val @ A2 @ B5) => ((member354538064on_val @ X3 @ A2) => (member354538064on_val @ X3 @ B5)))))). % in_mono
thf(fact_144_in__mono, axiom,
    ((![A2 : set_list_char, B5 : set_list_char, X3 : list_char]: ((ord_le1836582007t_char @ A2 @ B5) => ((member_list_char @ X3 @ A2) => (member_list_char @ X3 @ B5)))))). % in_mono
thf(fact_145_subsetD, axiom,
    ((![A2 : set_Pr1189492743on_val, B5 : set_Pr1189492743on_val, C2 : produc1172664615on_val]: ((ord_le837504935on_val @ A2 @ B5) => ((member354538064on_val @ C2 @ A2) => (member354538064on_val @ C2 @ B5)))))). % subsetD
thf(fact_146_subsetD, axiom,
    ((![A2 : set_list_char, B5 : set_list_char, C2 : list_char]: ((ord_le1836582007t_char @ A2 @ B5) => ((member_list_char @ C2 @ A2) => (member_list_char @ C2 @ B5)))))). % subsetD
thf(fact_147_equalityE, axiom,
    ((![A2 : set_list_char, B5 : set_list_char]: ((A2 = B5) => (~ (((ord_le1836582007t_char @ A2 @ B5) => (~ ((ord_le1836582007t_char @ B5 @ A2)))))))))). % equalityE
thf(fact_148_subset__eq, axiom,
    ((ord_le837504935on_val = (^[A7 : set_Pr1189492743on_val]: (^[B7 : set_Pr1189492743on_val]: (![X : produc1172664615on_val]: (((member354538064on_val @ X @ A7)) => ((member354538064on_val @ X @ B7))))))))). % subset_eq
thf(fact_149_subset__eq, axiom,
    ((ord_le1836582007t_char = (^[A7 : set_list_char]: (^[B7 : set_list_char]: (![X : list_char]: (((member_list_char @ X @ A7)) => ((member_list_char @ X @ B7))))))))). % subset_eq
thf(fact_150_equalityD1, axiom,
    ((![A2 : set_list_char, B5 : set_list_char]: ((A2 = B5) => (ord_le1836582007t_char @ A2 @ B5))))). % equalityD1
thf(fact_151_equalityD2, axiom,
    ((![A2 : set_list_char, B5 : set_list_char]: ((A2 = B5) => (ord_le1836582007t_char @ B5 @ A2))))). % equalityD2
thf(fact_152_subset__iff, axiom,
    ((ord_le837504935on_val = (^[A7 : set_Pr1189492743on_val]: (^[B7 : set_Pr1189492743on_val]: (![T : produc1172664615on_val]: (((member354538064on_val @ T @ A7)) => ((member354538064on_val @ T @ B7))))))))). % subset_iff
thf(fact_153_subset__iff, axiom,
    ((ord_le1836582007t_char = (^[A7 : set_list_char]: (^[B7 : set_list_char]: (![T : list_char]: (((member_list_char @ T @ A7)) => ((member_list_char @ T @ B7))))))))). % subset_iff
thf(fact_154_subset__refl, axiom,
    ((![A2 : set_list_char]: (ord_le1836582007t_char @ A2 @ A2)))). % subset_refl
thf(fact_155_Collect__mono, axiom,
    ((![P : list_char > $o, Q : list_char > $o]: ((![X4 : list_char]: ((P @ X4) => (Q @ X4))) => (ord_le1836582007t_char @ (collect_list_char @ P) @ (collect_list_char @ Q)))))). % Collect_mono
thf(fact_156_subset__trans, axiom,
    ((![A2 : set_list_char, B5 : set_list_char, C5 : set_list_char]: ((ord_le1836582007t_char @ A2 @ B5) => ((ord_le1836582007t_char @ B5 @ C5) => (ord_le1836582007t_char @ A2 @ C5)))))). % subset_trans
thf(fact_157_set__eq__subset, axiom,
    (((^[Y5 : set_list_char]: (^[Z3 : set_list_char]: (Y5 = Z3))) = (^[A7 : set_list_char]: (^[B7 : set_list_char]: (((ord_le1836582007t_char @ A7 @ B7)) & ((ord_le1836582007t_char @ B7 @ A7)))))))). % set_eq_subset
thf(fact_158_Collect__subset, axiom,
    ((![A2 : set_Pr1189492743on_val, P : produc1172664615on_val > $o]: (ord_le837504935on_val @ (collec2055896722on_val @ (^[X : produc1172664615on_val]: (((member354538064on_val @ X @ A2)) & ((P @ X))))) @ A2)))). % Collect_subset
thf(fact_159_Collect__subset, axiom,
    ((![A2 : set_list_char, P : list_char > $o]: (ord_le1836582007t_char @ (collect_list_char @ (^[X : list_char]: (((member_list_char @ X @ A2)) & ((P @ X))))) @ A2)))). % Collect_subset
thf(fact_160_Collect__mono__iff, axiom,
    ((![P : list_char > $o, Q : list_char > $o]: ((ord_le1836582007t_char @ (collect_list_char @ P) @ (collect_list_char @ Q)) = (![X : list_char]: (((P @ X)) => ((Q @ X)))))))). % Collect_mono_iff
thf(fact_161_order__subst1, axiom,
    ((![A : set_list_char, F : set_list_char > set_list_char, B2 : set_list_char, C2 : set_list_char]: ((ord_le1836582007t_char @ A @ (F @ B2)) => ((ord_le1836582007t_char @ B2 @ C2) => ((![X4 : set_list_char, Y3 : set_list_char]: ((ord_le1836582007t_char @ X4 @ Y3) => (ord_le1836582007t_char @ (F @ X4) @ (F @ Y3)))) => (ord_le1836582007t_char @ A @ (F @ C2)))))))). % order_subst1
thf(fact_162_order__subst2, axiom,
    ((![A : set_list_char, B2 : set_list_char, F : set_list_char > set_list_char, C2 : set_list_char]: ((ord_le1836582007t_char @ A @ B2) => ((ord_le1836582007t_char @ (F @ B2) @ C2) => ((![X4 : set_list_char, Y3 : set_list_char]: ((ord_le1836582007t_char @ X4 @ Y3) => (ord_le1836582007t_char @ (F @ X4) @ (F @ Y3)))) => (ord_le1836582007t_char @ (F @ A) @ C2))))))). % order_subst2
thf(fact_163_ord__eq__le__subst, axiom,
    ((![A : set_list_char, F : set_list_char > set_list_char, B2 : set_list_char, C2 : set_list_char]: ((A = (F @ B2)) => ((ord_le1836582007t_char @ B2 @ C2) => ((![X4 : set_list_char, Y3 : set_list_char]: ((ord_le1836582007t_char @ X4 @ Y3) => (ord_le1836582007t_char @ (F @ X4) @ (F @ Y3)))) => (ord_le1836582007t_char @ A @ (F @ C2)))))))). % ord_eq_le_subst
thf(fact_164_ord__le__eq__subst, axiom,
    ((![A : set_list_char, B2 : set_list_char, F : set_list_char > set_list_char, C2 : set_list_char]: ((ord_le1836582007t_char @ A @ B2) => (((F @ B2) = C2) => ((![X4 : set_list_char, Y3 : set_list_char]: ((ord_le1836582007t_char @ X4 @ Y3) => (ord_le1836582007t_char @ (F @ X4) @ (F @ Y3)))) => (ord_le1836582007t_char @ (F @ A) @ C2))))))). % ord_le_eq_subst
thf(fact_165_eq__iff, axiom,
    (((^[Y5 : set_list_char]: (^[Z3 : set_list_char]: (Y5 = Z3))) = (^[X : set_list_char]: (^[Y4 : set_list_char]: (((ord_le1836582007t_char @ X @ Y4)) & ((ord_le1836582007t_char @ Y4 @ X)))))))). % eq_iff
thf(fact_166_antisym, axiom,
    ((![X3 : set_list_char, Y : set_list_char]: ((ord_le1836582007t_char @ X3 @ Y) => ((ord_le1836582007t_char @ Y @ X3) => (X3 = Y)))))). % antisym
thf(fact_167_eq__refl, axiom,
    ((![X3 : set_list_char, Y : set_list_char]: ((X3 = Y) => (ord_le1836582007t_char @ X3 @ Y))))). % eq_refl
thf(fact_168_order_Otrans, axiom,
    ((![A : set_list_char, B2 : set_list_char, C2 : set_list_char]: ((ord_le1836582007t_char @ A @ B2) => ((ord_le1836582007t_char @ B2 @ C2) => (ord_le1836582007t_char @ A @ C2)))))). % order.trans
thf(fact_169_antisym__conv, axiom,
    ((![Y : set_list_char, X3 : set_list_char]: ((ord_le1836582007t_char @ Y @ X3) => ((ord_le1836582007t_char @ X3 @ Y) = (X3 = Y)))))). % antisym_conv
thf(fact_170_order__class_Oorder_Oeq__iff, axiom,
    (((^[Y5 : set_list_char]: (^[Z3 : set_list_char]: (Y5 = Z3))) = (^[A6 : set_list_char]: (^[B4 : set_list_char]: (((ord_le1836582007t_char @ A6 @ B4)) & ((ord_le1836582007t_char @ B4 @ A6)))))))). % order_class.order.eq_iff
thf(fact_171_ord__eq__le__trans, axiom,
    ((![A : set_list_char, B2 : set_list_char, C2 : set_list_char]: ((A = B2) => ((ord_le1836582007t_char @ B2 @ C2) => (ord_le1836582007t_char @ A @ C2)))))). % ord_eq_le_trans
thf(fact_172_ord__le__eq__trans, axiom,
    ((![A : set_list_char, B2 : set_list_char, C2 : set_list_char]: ((ord_le1836582007t_char @ A @ B2) => ((B2 = C2) => (ord_le1836582007t_char @ A @ C2)))))). % ord_le_eq_trans
thf(fact_173_order__class_Oorder_Oantisym, axiom,
    ((![A : set_list_char, B2 : set_list_char]: ((ord_le1836582007t_char @ A @ B2) => ((ord_le1836582007t_char @ B2 @ A) => (A = B2)))))). % order_class.order.antisym
thf(fact_174_order__trans, axiom,
    ((![X3 : set_list_char, Y : set_list_char, Z2 : set_list_char]: ((ord_le1836582007t_char @ X3 @ Y) => ((ord_le1836582007t_char @ Y @ Z2) => (ord_le1836582007t_char @ X3 @ Z2)))))). % order_trans
thf(fact_175_dual__order_Orefl, axiom,
    ((![A : set_list_char]: (ord_le1836582007t_char @ A @ A)))). % dual_order.refl
thf(fact_176_dual__order_Otrans, axiom,
    ((![B2 : set_list_char, A : set_list_char, C2 : set_list_char]: ((ord_le1836582007t_char @ B2 @ A) => ((ord_le1836582007t_char @ C2 @ B2) => (ord_le1836582007t_char @ C2 @ A)))))). % dual_order.trans
thf(fact_177_dual__order_Oeq__iff, axiom,
    (((^[Y5 : set_list_char]: (^[Z3 : set_list_char]: (Y5 = Z3))) = (^[A6 : set_list_char]: (^[B4 : set_list_char]: (((ord_le1836582007t_char @ B4 @ A6)) & ((ord_le1836582007t_char @ A6 @ B4)))))))). % dual_order.eq_iff
thf(fact_178_dual__order_Oantisym, axiom,
    ((![B2 : set_list_char, A : set_list_char]: ((ord_le1836582007t_char @ B2 @ A) => ((ord_le1836582007t_char @ A @ B2) => (A = B2)))))). % dual_order.antisym
thf(fact_179_map__add__dom__app__simps_I3_J, axiom,
    ((![M : produc1172664615on_val, L22 : produc1172664615on_val > option_set_list_char, L1 : produc1172664615on_val > option_set_list_char]: ((~ ((member354538064on_val @ M @ (dom_Pr1335625324t_char @ L22)))) => ((map_ad1871482825t_char @ L1 @ L22 @ M) = (L1 @ M)))))). % map_add_dom_app_simps(3)
thf(fact_180_map__add__dom__app__simps_I3_J, axiom,
    ((![M : list_char, L22 : list_char > option_val, L1 : list_char > option_val]: ((~ ((member_list_char @ M @ (dom_list_char_val @ L22)))) => ((map_ad1615113112ar_val @ L1 @ L22 @ M) = (L1 @ M)))))). % map_add_dom_app_simps(3)
thf(fact_181_map__add__dom__app__simps_I2_J, axiom,
    ((![M : produc1172664615on_val, L1 : produc1172664615on_val > option_set_list_char, L22 : produc1172664615on_val > option_set_list_char]: ((~ ((member354538064on_val @ M @ (dom_Pr1335625324t_char @ L1)))) => ((map_ad1871482825t_char @ L1 @ L22 @ M) = (L22 @ M)))))). % map_add_dom_app_simps(2)
thf(fact_182_map__add__dom__app__simps_I2_J, axiom,
    ((![M : list_char, L1 : list_char > option_val, L22 : list_char > option_val]: ((~ ((member_list_char @ M @ (dom_list_char_val @ L1)))) => ((map_ad1615113112ar_val @ L1 @ L22 @ M) = (L22 @ M)))))). % map_add_dom_app_simps(2)
thf(fact_183_map__add__dom__app__simps_I1_J, axiom,
    ((![M : produc1172664615on_val, L22 : produc1172664615on_val > option_set_list_char, L1 : produc1172664615on_val > option_set_list_char]: ((member354538064on_val @ M @ (dom_Pr1335625324t_char @ L22)) => ((map_ad1871482825t_char @ L1 @ L22 @ M) = (L22 @ M)))))). % map_add_dom_app_simps(1)
thf(fact_184_map__add__dom__app__simps_I1_J, axiom,
    ((![M : list_char, L22 : list_char > option_val, L1 : list_char > option_val]: ((member_list_char @ M @ (dom_list_char_val @ L22)) => ((map_ad1615113112ar_val @ L1 @ L22 @ M) = (L22 @ M)))))). % map_add_dom_app_simps(1)
thf(fact_185_in__these__eq, axiom,
    ((![X3 : produc1172664615on_val, A2 : set_op787718477on_val]: ((member354538064on_val @ X3 @ (these_1398366820on_val @ A2)) = (member1869742230on_val @ (some_P83435532on_val @ X3) @ A2))))). % in_these_eq
thf(fact_186_in__these__eq, axiom,
    ((![X3 : list_char, A2 : set_option_list_char]: ((member_list_char @ X3 @ (these_list_char @ A2)) = (member246349384t_char @ (some_list_char @ X3) @ A2))))). % in_these_eq
thf(fact_187_in__these__eq, axiom,
    ((![X3 : set_list_char, A2 : set_op501702045t_char]: ((member_set_list_char @ X3 @ (these_set_list_char @ A2)) = (member1976670718t_char @ (some_set_list_char @ X3) @ A2))))). % in_these_eq
thf(fact_188_in__these__eq, axiom,
    ((![X3 : val, A2 : set_option_val]: ((member_val @ X3 @ (these_val @ A2)) = (member_option_val @ (some_val @ X3) @ A2))))). % in_these_eq
thf(fact_189_top__empty__eq, axiom,
    ((top_to1455478342_val_o = (^[X : produc1172664615on_val]: (member354538064on_val @ X @ top_to1683965271on_val))))). % top_empty_eq
thf(fact_190_top__empty__eq, axiom,
    ((top_top_list_char_o = (^[X : list_char]: (member_list_char @ X @ top_to748628679t_char))))). % top_empty_eq
thf(fact_191_red__lcl__add, axiom,
    ((![E : exp_list_char, H : nat > option2118145160on_val, L : list_char > option_val, E2 : exp_list_char, H2 : nat > option2118145160on_val, L2 : list_char > option_val, P : list_P1234162913t_char, L_02 : list_char > option_val]: ((member354538064on_val @ (produc1069526295on_val @ (produc2023458740on_val @ E @ (produc52622304on_val @ H @ L)) @ (produc2023458740on_val @ E2 @ (produc52622304on_val @ H2 @ L2))) @ (red @ P)) => (member354538064on_val @ (produc1069526295on_val @ (produc2023458740on_val @ E @ (produc52622304on_val @ H @ (map_ad1615113112ar_val @ L_02 @ L))) @ (produc2023458740on_val @ E2 @ (produc52622304on_val @ H2 @ (map_ad1615113112ar_val @ L_02 @ L2)))) @ (red @ P)))))). % red_lcl_add
thf(fact_192_pred__equals__eq2, axiom,
    ((![R : set_Pr1189492743on_val, S : set_Pr1189492743on_val]: (((^[X : produc1097884484on_val]: (^[Y4 : produc1097884484on_val]: (member354538064on_val @ (produc1069526295on_val @ X @ Y4) @ R))) = (^[X : produc1097884484on_val]: (^[Y4 : produc1097884484on_val]: (member354538064on_val @ (produc1069526295on_val @ X @ Y4) @ S)))) = (R = S))))). % pred_equals_eq2
thf(fact_193_pred__equals__eq2, axiom,
    ((![R : set_Pr556513572on_val, S : set_Pr556513572on_val]: (((^[X : exp_list_char]: (^[Y4 : produc515445032on_val]: (member1888679917on_val @ (produc2023458740on_val @ X @ Y4) @ R))) = (^[X : exp_list_char]: (^[Y4 : produc515445032on_val]: (member1888679917on_val @ (produc2023458740on_val @ X @ Y4) @ S)))) = (R = S))))). % pred_equals_eq2
thf(fact_194_pred__equals__eq2, axiom,
    ((![R : set_Pr723149704on_val, S : set_Pr723149704on_val]: (((^[X : nat > option2118145160on_val]: (^[Y4 : list_char > option_val]: (member652041809on_val @ (produc52622304on_val @ X @ Y4) @ R))) = (^[X : nat > option2118145160on_val]: (^[Y4 : list_char > option_val]: (member652041809on_val @ (produc52622304on_val @ X @ Y4) @ S)))) = (R = S))))). % pred_equals_eq2
thf(fact_195_hext__refl, axiom,
    ((![H : nat > option2118145160on_val]: (hext @ H @ H)))). % hext_refl

% Conjectures (1)
thf(conj_0, conjecture,
    ((d_list_char @ e2 @ (some_set_list_char @ (dom_list_char_val @ (lcl @ s2)))))).
