% TIMEFORMAT='%3R'; { time (exec 2>&1; /home/martin/bin/satallax -E /home/martin/.isabelle/contrib/e-2.5-1/x86_64-linux/eprover -p tstp -t 5 /home/martin/judgement-day/tptp-thf/tptp/StrongNorm/prob_129__5210544_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:37:38.610

% Could-be-implicit typings (11)
thf(ty_n_t__List__Olist_It__List__Olist_It__LambdaType__Otype_J_J, type,
    list_list_type : $tType).
thf(ty_n_t__Set__Oset_It__List__Olist_It__LambdaType__Otype_J_J, type,
    set_list_type : $tType).
thf(ty_n_t__List__Olist_It__List__Olist_It__Lambda__OdB_J_J, type,
    list_list_dB : $tType).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Lambda__OdB_J_J, type,
    set_list_dB : $tType).
thf(ty_n_t__List__Olist_It__LambdaType__Otype_J, type,
    list_type : $tType).
thf(ty_n_t__Set__Oset_It__LambdaType__Otype_J, type,
    set_type : $tType).
thf(ty_n_t__List__Olist_It__Lambda__OdB_J, type,
    list_dB : $tType).
thf(ty_n_t__Set__Oset_It__Lambda__OdB_J, type,
    set_dB : $tType).
thf(ty_n_t__LambdaType__Otype, type,
    type : $tType).
thf(ty_n_t__Lambda__OdB, type,
    dB : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).

% Explicit typings (57)
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__LambdaType__Otype, type,
    bNF_Gr681920326t_type : set_list_type > type > set_list_type).
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__Lambda__OdB, type,
    bNF_Gr1110975684ift_dB : set_list_dB > dB > set_list_dB).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__LambdaType__Otype, type,
    bNF_Gr1527093962c_type : set_list_type > list_type > set_type).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Lambda__OdB, type,
    bNF_Greatest_Succ_dB : set_list_dB > list_dB > set_dB).
thf(sy_c_InductTermi_OIT, type,
    it : dB > $o).
thf(sy_c_LambdaType_Oshift_001t__LambdaType__Otype, type,
    shift_type : (nat > type) > nat > type > nat > type).
thf(sy_c_LambdaType_Otype_OFun, type,
    fun : type > type > type).
thf(sy_c_LambdaType_Otyping, type,
    typing : (nat > type) > dB > type > $o).
thf(sy_c_LambdaType_Otypings, type,
    typings : (nat > type) > list_dB > list_type > $o).
thf(sy_c_Lambda_OdB_OApp, type,
    app : dB > dB > dB).
thf(sy_c_Lambda_OdB_OVar, type,
    var : nat > dB).
thf(sy_c_Lambda_Olift, type,
    lift : dB > nat > dB).
thf(sy_c_Lambda_Osubst, type,
    subst : dB > dB > nat > dB).
thf(sy_c_List_Oappend_001t__LambdaType__Otype, type,
    append_type : list_type > list_type > list_type).
thf(sy_c_List_Oappend_001t__Lambda__OdB, type,
    append_dB : list_dB > list_dB > list_dB).
thf(sy_c_List_Ofoldl_001t__Lambda__OdB_001t__Lambda__OdB, type,
    foldl_dB_dB : (dB > dB > dB) > dB > list_dB > dB).
thf(sy_c_List_Ofoldr_001t__LambdaType__Otype_001t__LambdaType__Otype, type,
    foldr_type_type : (type > type > type) > list_type > type > type).
thf(sy_c_List_Oinsert_001t__LambdaType__Otype, type,
    insert_type : type > list_type > list_type).
thf(sy_c_List_Oinsert_001t__Lambda__OdB, type,
    insert_dB : dB > list_dB > list_dB).
thf(sy_c_List_Olist_OCons_001t__LambdaType__Otype, type,
    cons_type : type > list_type > list_type).
thf(sy_c_List_Olist_OCons_001t__Lambda__OdB, type,
    cons_dB : dB > list_dB > list_dB).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__LambdaType__Otype_J, type,
    cons_list_type : list_type > list_list_type > list_list_type).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Lambda__OdB_J, type,
    cons_list_dB : list_dB > list_list_dB > list_list_dB).
thf(sy_c_List_Olist_ONil_001t__LambdaType__Otype, type,
    nil_type : list_type).
thf(sy_c_List_Olist_ONil_001t__Lambda__OdB, type,
    nil_dB : list_dB).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__LambdaType__Otype_J, type,
    nil_list_type : list_list_type).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Lambda__OdB_J, type,
    nil_list_dB : list_list_dB).
thf(sy_c_List_Olist_Ocase__list_001_Eo_001t__LambdaType__Otype, type,
    case_list_o_type : $o > (type > list_type > $o) > list_type > $o).
thf(sy_c_List_Olist_Ocase__list_001_Eo_001t__Lambda__OdB, type,
    case_list_o_dB : $o > (dB > list_dB > $o) > list_dB > $o).
thf(sy_c_List_Olist_Omap_001t__LambdaType__Otype_001t__LambdaType__Otype, type,
    map_type_type : (type > type) > list_type > list_type).
thf(sy_c_List_Olist_Omap_001t__LambdaType__Otype_001t__Lambda__OdB, type,
    map_type_dB : (type > dB) > list_type > list_dB).
thf(sy_c_List_Olist_Omap_001t__Lambda__OdB_001t__LambdaType__Otype, type,
    map_dB_type : (dB > type) > list_dB > list_type).
thf(sy_c_List_Olist_Omap_001t__Lambda__OdB_001t__Lambda__OdB, type,
    map_dB_dB : (dB > dB) > list_dB > list_dB).
thf(sy_c_List_Olist_Orec__list_001t__List__Olist_It__Lambda__OdB_J_001t__Lambda__OdB, type,
    rec_list_list_dB_dB : list_dB > (dB > list_dB > list_dB > list_dB) > list_dB > list_dB).
thf(sy_c_List_Omap__tailrec_001t__Lambda__OdB_001t__Lambda__OdB, type,
    map_tailrec_dB_dB : (dB > dB) > list_dB > list_dB).
thf(sy_c_Set_OCollect_001t__LambdaType__Otype, type,
    collect_type : (type > $o) > set_type).
thf(sy_c_Set_OCollect_001t__Lambda__OdB, type,
    collect_dB : (dB > $o) > set_dB).
thf(sy_c_member_001t__LambdaType__Otype, type,
    member_type : type > set_type > $o).
thf(sy_c_member_001t__Lambda__OdB, type,
    member_dB : dB > set_dB > $o).
thf(sy_c_member_001t__List__Olist_It__LambdaType__Otype_J, type,
    member_list_type : list_type > set_list_type > $o).
thf(sy_c_member_001t__List__Olist_It__Lambda__OdB_J, type,
    member_list_dB : list_dB > set_list_dB > $o).
thf(sy_v_T_H1____, type,
    t_1 : type).
thf(sy_v_T_H_H____, type,
    t : type).
thf(sy_v_T_H____, type,
    t2 : type).
thf(sy_v_T____, type,
    t3 : type).
thf(sy_v_Ts____, type,
    ts : list_type).
thf(sy_v_a____, type,
    a : dB).
thf(sy_v_as____, type,
    as : list_dB).
thf(sy_v_e1____, type,
    e1 : nat > type).
thf(sy_v_e____, type,
    e : nat > type).
thf(sy_v_i1____, type,
    i1 : nat).
thf(sy_v_i____, type,
    i : nat).
thf(sy_v_n____, type,
    n : nat).
thf(sy_v_rs____, type,
    rs : list_dB).
thf(sy_v_t____, type,
    t4 : dB).
thf(sy_v_u1____, type,
    u1 : dB).
thf(sy_v_u____, type,
    u : dB).

% Relevant facts (228)
thf(fact_0_uT, axiom,
    ((typing @ e @ u @ t3))). % uT
thf(fact_1_varT, axiom,
    ((typing @ (shift_type @ e @ i @ t3) @ (var @ n) @ (fun @ t @ (foldr_type_type @ fun @ ts @ t2))))). % varT
thf(fact_2__092_060open_062IT_At_092_060close_062, axiom,
    ((it @ t4))). % \<open>IT t\<close>
thf(fact_3_argT, axiom,
    ((typing @ (shift_type @ e @ i @ t3) @ a @ t))). % argT
thf(fact_4_type_Oinject_I2_J, axiom,
    ((![X21 : type, X22 : type, Y21 : type, Y22 : type]: (((fun @ X21 @ X22) = (fun @ Y21 @ Y22)) = (((X21 = Y21)) & ((X22 = Y22))))))). % type.inject(2)
thf(fact_5__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062T_H_H_O_A_092_060lbrakk_062e_092_060langle_062i_058T_092_060rangle_062_A_092_060turnstile_062_AVar_An_A_058_AT_H_H_A_092_060Rightarrow_062_ATs_A_092_060Rrightarrow_062_AT_H_059_Ae_092_060langle_062i_058T_092_060rangle_062_A_092_060turnstile_062_Aa_A_058_AT_H_H_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom,
    ((~ ((![T : type]: ((typing @ (shift_type @ e @ i @ t3) @ (var @ n) @ (fun @ T @ (foldr_type_type @ fun @ ts @ t2))) => (~ ((typing @ (shift_type @ e @ i @ t3) @ a @ T))))))))). % \<open>\<And>thesis. (\<And>T''. \<lbrakk>e\<langle>i:T\<rangle> \<turnstile> Var n : T'' \<Rightarrow> Ts \<Rrightarrow> T'; e\<langle>i:T\<rangle> \<turnstile> a : T''\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_6_argsT, axiom,
    ((typings @ (shift_type @ e @ i @ t3) @ as @ ts))). % argsT
thf(fact_7_headT, axiom,
    ((typing @ (shift_type @ e @ i @ t3) @ (app @ (var @ n) @ a) @ (foldr_type_type @ fun @ ts @ t2)))). % headT
thf(fact_8_type__induct, axiom,
    ((![P : type > $o, T2 : type]: ((![T3 : type]: ((![T1 : type, T22 : type]: ((T3 = (fun @ T1 @ T22)) => (P @ T1))) => ((![T1 : type, T22 : type]: ((T3 = (fun @ T1 @ T22)) => (P @ T22))) => (P @ T3)))) => (P @ T2))))). % type_induct
thf(fact_9_Var_Oprems_I2_J, axiom,
    ((it @ u1))). % Var.prems(2)
thf(fact_10_uIT, axiom,
    ((it @ u))). % uIT
thf(fact_11_MI1, axiom,
    ((![T12 : type, T23 : type, T4 : dB, E : nat > type, I : nat, T2 : type, U : dB]: ((t3 = (fun @ T12 @ T23)) => ((it @ T4) => ((typing @ (shift_type @ E @ I @ T12) @ T4 @ T2) => ((it @ U) => ((typing @ E @ U @ T12) => (it @ (subst @ T4 @ U @ I)))))))))). % MI1
thf(fact_12_MI2, axiom,
    ((![T12 : type, T23 : type, T4 : dB, E : nat > type, I : nat, T2 : type, U : dB]: ((t3 = (fun @ T12 @ T23)) => ((it @ T4) => ((typing @ (shift_type @ E @ I @ T23) @ T4 @ T2) => ((it @ U) => ((typing @ E @ U @ T23) => (it @ (subst @ T4 @ U @ I)))))))))). % MI2
thf(fact_13__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062Ts_O_A_092_060lbrakk_062e_092_060langle_062i_058T_092_060rangle_062_A_092_060turnstile_062_AVar_An_A_092_060degree_062_Aa_A_058_ATs_A_092_060Rrightarrow_062_AT_H_059_Ae_092_060langle_062i_058T_092_060rangle_062_A_092_060tturnstile_062_Aas_A_058_ATs_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom,
    ((~ ((![Ts : list_type]: ((typing @ (shift_type @ e @ i @ t3) @ (app @ (var @ n) @ a) @ (foldr_type_type @ fun @ Ts @ t2)) => (~ ((typings @ (shift_type @ e @ i @ t3) @ as @ Ts))))))))). % \<open>\<And>thesis. (\<And>Ts. \<lbrakk>e\<langle>i:T\<rangle> \<turnstile> Var n \<degree> a : Ts \<Rrightarrow> T'; e\<langle>i:T\<rangle> \<tturnstile> as : Ts\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_14_nT, axiom,
    ((typing @ (shift_type @ e @ i @ t3) @ (foldl_dB_dB @ app @ (var @ n) @ rs) @ t2))). % nT
thf(fact_15_True, axiom,
    ((n = i))). % True
thf(fact_16_shift__eq, axiom,
    ((![I : nat, J : nat, E : nat > type, T2 : type]: ((I = J) => ((shift_type @ E @ I @ T2 @ J) = T2))))). % shift_eq
thf(fact_17__092_060open_062e_092_060langle_062i_058T_092_060rangle_062_A_092_060turnstile_062_AVar_An_A_092_060degree_062_Aa_A_092_060degree_062_092_060degree_062_Aas_A_058_AT_H_092_060close_062, axiom,
    ((typing @ (shift_type @ e @ i @ t3) @ (foldl_dB_dB @ app @ (app @ (var @ n) @ a) @ as) @ t2))). % \<open>e\<langle>i:T\<rangle> \<turnstile> Var n \<degree> a \<degree>\<degree> as : T'\<close>
thf(fact_18_local_OCons, axiom,
    ((rs = (cons_dB @ a @ as)))). % local.Cons
thf(fact_19_typing__elims_I1_J, axiom,
    ((![E : nat > type, I : nat, T2 : type]: ((typing @ E @ (var @ I) @ T2) => ((E @ I) = T2))))). % typing_elims(1)
thf(fact_20_typing_OVar, axiom,
    ((![Env : nat > type, X : nat, T2 : type]: (((Env @ X) = T2) => (typing @ Env @ (var @ X) @ T2))))). % typing.Var
thf(fact_21_subst__lemma, axiom,
    ((![E : nat > type, T4 : dB, T2 : type, E2 : nat > type, U : dB, U2 : type, I : nat]: ((typing @ E @ T4 @ T2) => ((typing @ E2 @ U @ U2) => ((E = (shift_type @ E2 @ I @ U2)) => (typing @ E2 @ (subst @ T4 @ U @ I) @ T2))))))). % subst_lemma
thf(fact_22_var__app__type__eq, axiom,
    ((![E : nat > type, I : nat, Ts2 : list_dB, T2 : type, U2 : type]: ((typing @ E @ (foldl_dB_dB @ app @ (var @ I) @ Ts2) @ T2) => ((typing @ E @ (foldl_dB_dB @ app @ (var @ I) @ Ts2) @ U2) => (T2 = U2)))))). % var_app_type_eq
thf(fact_23_Var__IT, axiom,
    ((![N : nat]: (it @ (var @ N))))). % Var_IT
thf(fact_24_app__Var__IT, axiom,
    ((![T4 : dB, I : nat]: ((it @ T4) => (it @ (app @ T4 @ (var @ I))))))). % app_Var_IT
thf(fact_25_subst__Var__IT, axiom,
    ((![R : dB, I : nat, J : nat]: ((it @ R) => (it @ (subst @ R @ (var @ I) @ J)))))). % subst_Var_IT
thf(fact_26_var__app__typesE, axiom,
    ((![E : nat > type, I : nat, Ts2 : list_dB, T2 : type]: ((typing @ E @ (foldl_dB_dB @ app @ (var @ I) @ Ts2) @ T2) => (~ ((![Ts : list_type]: ((typing @ E @ (var @ I) @ (foldr_type_type @ fun @ Ts @ T2)) => (~ ((typings @ E @ Ts2 @ Ts))))))))))). % var_app_typesE
thf(fact_27_var__app__types, axiom,
    ((![E : nat > type, I : nat, Ts2 : list_dB, Us : list_dB, T2 : type, Ts3 : list_type, U2 : type]: ((typing @ E @ (foldl_dB_dB @ app @ (foldl_dB_dB @ app @ (var @ I) @ Ts2) @ Us) @ T2) => ((typings @ E @ Ts2 @ Ts3) => ((typing @ E @ (foldl_dB_dB @ app @ (var @ I) @ Ts2) @ U2) => (?[Us2 : list_type]: ((U2 = (foldr_type_type @ fun @ Us2 @ T2)) & (typings @ E @ Us @ Us2))))))))). % var_app_types
thf(fact_28_App, axiom,
    ((![Env : nat > type, S : dB, T2 : type, U2 : type, T4 : dB]: ((typing @ Env @ S @ (fun @ T2 @ U2)) => ((typing @ Env @ T4 @ T2) => (typing @ Env @ (app @ S @ T4) @ U2)))))). % App
thf(fact_29_typing__elims_I2_J, axiom,
    ((![E : nat > type, T4 : dB, U : dB, T2 : type]: ((typing @ E @ (app @ T4 @ U) @ T2) => (~ ((![T3 : type]: ((typing @ E @ T4 @ (fun @ T3 @ T2)) => (~ ((typing @ E @ U @ T3))))))))))). % typing_elims(2)
thf(fact_30_list__app__typeI, axiom,
    ((![E : nat > type, T4 : dB, Ts3 : list_type, T2 : type, Ts2 : list_dB]: ((typing @ E @ T4 @ (foldr_type_type @ fun @ Ts3 @ T2)) => ((typings @ E @ Ts2 @ Ts3) => (typing @ E @ (foldl_dB_dB @ app @ T4 @ Ts2) @ T2)))))). % list_app_typeI
thf(fact_31_list__app__typeE, axiom,
    ((![E : nat > type, T4 : dB, Ts2 : list_dB, T2 : type]: ((typing @ E @ (foldl_dB_dB @ app @ T4 @ Ts2) @ T2) => (~ ((![Ts : list_type]: ((typing @ E @ T4 @ (foldr_type_type @ fun @ Ts @ T2)) => (~ ((typings @ E @ Ts2 @ Ts))))))))))). % list_app_typeE
thf(fact_32_list__app__typeD, axiom,
    ((![E : nat > type, T4 : dB, Ts2 : list_dB, T2 : type]: ((typing @ E @ (foldl_dB_dB @ app @ T4 @ Ts2) @ T2) => (?[Ts : list_type]: ((typing @ E @ T4 @ (foldr_type_type @ fun @ Ts @ T2)) & (typings @ E @ Ts2 @ Ts))))))). % list_app_typeD
thf(fact_33_Var__apps__eq__Var__apps__conv, axiom,
    ((![M : nat, Rs : list_dB, N : nat, Ss : list_dB]: (((foldl_dB_dB @ app @ (var @ M) @ Rs) = (foldl_dB_dB @ app @ (var @ N) @ Ss)) = (((M = N)) & ((Rs = Ss))))))). % Var_apps_eq_Var_apps_conv
thf(fact_34_subst__eq, axiom,
    ((![K : nat, U : dB]: ((subst @ (var @ K) @ U @ K) = U)))). % subst_eq
thf(fact_35_apps__eq__tail__conv, axiom,
    ((![R : dB, Ts2 : list_dB, S : dB]: (((foldl_dB_dB @ app @ R @ Ts2) = (foldl_dB_dB @ app @ S @ Ts2)) = (R = S))))). % apps_eq_tail_conv
thf(fact_36_dB_Oinject_I1_J, axiom,
    ((![X1 : nat, Y1 : nat]: (((var @ X1) = (var @ Y1)) = (X1 = Y1))))). % dB.inject(1)
thf(fact_37_dB_Oinject_I2_J, axiom,
    ((![X21 : dB, X22 : dB, Y21 : dB, Y22 : dB]: (((app @ X21 @ X22) = (app @ Y21 @ Y22)) = (((X21 = Y21)) & ((X22 = Y22))))))). % dB.inject(2)
thf(fact_38_subst__App, axiom,
    ((![T4 : dB, U : dB, S : dB, K : nat]: ((subst @ (app @ T4 @ U) @ S @ K) = (app @ (subst @ T4 @ S @ K) @ (subst @ U @ S @ K)))))). % subst_App
thf(fact_39_dB_Odistinct_I1_J, axiom,
    ((![X1 : nat, X21 : dB, X22 : dB]: (~ (((var @ X1) = (app @ X21 @ X22))))))). % dB.distinct(1)
thf(fact_40_Var_Oprems_I1_J, axiom,
    ((typing @ (shift_type @ e1 @ i1 @ t3) @ (foldl_dB_dB @ app @ (var @ n) @ rs) @ t_1))). % Var.prems(1)
thf(fact_41_Var_Oprems_I3_J, axiom,
    ((typing @ e1 @ u1 @ t3))). % Var.prems(3)
thf(fact_42_lift__type, axiom,
    ((![E : nat > type, T4 : dB, T2 : type, I : nat, U2 : type]: ((typing @ E @ T4 @ T2) => (typing @ (shift_type @ E @ I @ U2) @ (lift @ T4 @ I) @ T2))))). % lift_type
thf(fact_43_subst__lift, axiom,
    ((![T4 : dB, K : nat, S : dB]: ((subst @ (lift @ T4 @ K) @ S @ K) = T4)))). % subst_lift
thf(fact_44_lift__IT, axiom,
    ((![T4 : dB, I : nat]: ((it @ T4) => (it @ (lift @ T4 @ I)))))). % lift_IT
thf(fact_45_lift_Osimps_I2_J, axiom,
    ((![S : dB, T4 : dB, K : nat]: ((lift @ (app @ S @ T4) @ K) = (app @ (lift @ S @ K) @ (lift @ T4 @ K)))))). % lift.simps(2)
thf(fact_46_list_Oinject, axiom,
    ((![X21 : dB, X22 : list_dB, Y21 : dB, Y22 : list_dB]: (((cons_dB @ X21 @ X22) = (cons_dB @ Y21 @ Y22)) = (((X21 = Y21)) & ((X22 = Y22))))))). % list.inject
thf(fact_47_list_Oinject, axiom,
    ((![X21 : type, X22 : list_type, Y21 : type, Y22 : list_type]: (((cons_type @ X21 @ X22) = (cons_type @ Y21 @ Y22)) = (((X21 = Y21)) & ((X22 = Y22))))))). % list.inject
thf(fact_48_foldl__Cons, axiom,
    ((![F : dB > dB > dB, A : dB, X : dB, Xs : list_dB]: ((foldl_dB_dB @ F @ A @ (cons_dB @ X @ Xs)) = (foldl_dB_dB @ F @ (F @ A @ X) @ Xs))))). % foldl_Cons
thf(fact_49_typings_Osimps_I2_J, axiom,
    ((![E : nat > type, T4 : dB, Ts2 : list_dB, Ts3 : list_type]: ((typings @ E @ (cons_dB @ T4 @ Ts2) @ Ts3) = (case_list_o_type @ $false @ (^[T5 : type]: (^[Ts4 : list_type]: (((typing @ E @ T4 @ T5)) & ((typings @ E @ Ts2 @ Ts4))))) @ Ts3))))). % typings.simps(2)
thf(fact_50_substs__lemma, axiom,
    ((![E : nat > type, U : dB, T2 : type, I : nat, Ts2 : list_dB, Ts3 : list_type]: ((typing @ E @ U @ T2) => ((typings @ (shift_type @ E @ I @ T2) @ Ts2 @ Ts3) => (typings @ E @ (map_dB_dB @ (^[T6 : dB]: (subst @ T6 @ U @ I)) @ Ts2) @ Ts3)))))). % substs_lemma
thf(fact_51_Var__eq__apps__conv, axiom,
    ((![M : nat, S : dB, Ss : list_dB]: (((var @ M) = (foldl_dB_dB @ app @ S @ Ss)) = ((((var @ M) = S)) & ((Ss = nil_dB))))))). % Var_eq_apps_conv
thf(fact_52_lift__map, axiom,
    ((![T4 : dB, Ts2 : list_dB, I : nat]: ((lift @ (foldl_dB_dB @ app @ T4 @ Ts2) @ I) = (foldl_dB_dB @ app @ (lift @ T4 @ I) @ (map_dB_dB @ (^[T6 : dB]: (lift @ T6 @ I)) @ Ts2)))))). % lift_map
thf(fact_53_map__ident, axiom,
    (((map_dB_dB @ (^[X2 : dB]: X2)) = (^[Xs2 : list_dB]: Xs2)))). % map_ident
thf(fact_54_Nil__is__map__conv, axiom,
    ((![F : type > dB, Xs : list_type]: ((nil_dB = (map_type_dB @ F @ Xs)) = (Xs = nil_type))))). % Nil_is_map_conv
thf(fact_55_Nil__is__map__conv, axiom,
    ((![F : dB > type, Xs : list_dB]: ((nil_type = (map_dB_type @ F @ Xs)) = (Xs = nil_dB))))). % Nil_is_map_conv
thf(fact_56_Nil__is__map__conv, axiom,
    ((![F : type > type, Xs : list_type]: ((nil_type = (map_type_type @ F @ Xs)) = (Xs = nil_type))))). % Nil_is_map_conv
thf(fact_57_Nil__is__map__conv, axiom,
    ((![F : dB > dB, Xs : list_dB]: ((nil_dB = (map_dB_dB @ F @ Xs)) = (Xs = nil_dB))))). % Nil_is_map_conv
thf(fact_58_map__is__Nil__conv, axiom,
    ((![F : type > dB, Xs : list_type]: (((map_type_dB @ F @ Xs) = nil_dB) = (Xs = nil_type))))). % map_is_Nil_conv
thf(fact_59_map__is__Nil__conv, axiom,
    ((![F : dB > type, Xs : list_dB]: (((map_dB_type @ F @ Xs) = nil_type) = (Xs = nil_dB))))). % map_is_Nil_conv
thf(fact_60_map__is__Nil__conv, axiom,
    ((![F : type > type, Xs : list_type]: (((map_type_type @ F @ Xs) = nil_type) = (Xs = nil_type))))). % map_is_Nil_conv
thf(fact_61_map__is__Nil__conv, axiom,
    ((![F : dB > dB, Xs : list_dB]: (((map_dB_dB @ F @ Xs) = nil_dB) = (Xs = nil_dB))))). % map_is_Nil_conv
thf(fact_62_list_Omap__disc__iff, axiom,
    ((![F : type > dB, A : list_type]: (((map_type_dB @ F @ A) = nil_dB) = (A = nil_type))))). % list.map_disc_iff
thf(fact_63_list_Omap__disc__iff, axiom,
    ((![F : dB > type, A : list_dB]: (((map_dB_type @ F @ A) = nil_type) = (A = nil_dB))))). % list.map_disc_iff
thf(fact_64_list_Omap__disc__iff, axiom,
    ((![F : type > type, A : list_type]: (((map_type_type @ F @ A) = nil_type) = (A = nil_type))))). % list.map_disc_iff
thf(fact_65_list_Omap__disc__iff, axiom,
    ((![F : dB > dB, A : list_dB]: (((map_dB_dB @ F @ A) = nil_dB) = (A = nil_dB))))). % list.map_disc_iff
thf(fact_66_subst__map, axiom,
    ((![T4 : dB, Ts2 : list_dB, U : dB, I : nat]: ((subst @ (foldl_dB_dB @ app @ T4 @ Ts2) @ U @ I) = (foldl_dB_dB @ app @ (subst @ T4 @ U @ I) @ (map_dB_dB @ (^[T6 : dB]: (subst @ T6 @ U @ I)) @ Ts2)))))). % subst_map
thf(fact_67_foldl__map, axiom,
    ((![G : dB > dB > dB, A : dB, F : dB > dB, Xs : list_dB]: ((foldl_dB_dB @ G @ A @ (map_dB_dB @ F @ Xs)) = (foldl_dB_dB @ (^[A2 : dB]: (^[X2 : dB]: (G @ A2 @ (F @ X2)))) @ A @ Xs))))). % foldl_map
thf(fact_68_map__tailrec__rev_Oinduct, axiom,
    ((![P : (dB > dB) > list_dB > list_dB > $o, A0 : dB > dB, A1 : list_dB, A22 : list_dB]: ((![F2 : dB > dB, X_1 : list_dB]: (P @ F2 @ nil_dB @ X_1)) => ((![F2 : dB > dB, A3 : dB, As : list_dB, Bs : list_dB]: ((P @ F2 @ As @ (cons_dB @ (F2 @ A3) @ Bs)) => (P @ F2 @ (cons_dB @ A3 @ As) @ Bs))) => (P @ A0 @ A1 @ A22)))))). % map_tailrec_rev.induct
thf(fact_69_map__tailrec__rev_Oinduct, axiom,
    ((![P : (type > dB) > list_type > list_dB > $o, A0 : type > dB, A1 : list_type, A22 : list_dB]: ((![F2 : type > dB, X_1 : list_dB]: (P @ F2 @ nil_type @ X_1)) => ((![F2 : type > dB, A3 : type, As : list_type, Bs : list_dB]: ((P @ F2 @ As @ (cons_dB @ (F2 @ A3) @ Bs)) => (P @ F2 @ (cons_type @ A3 @ As) @ Bs))) => (P @ A0 @ A1 @ A22)))))). % map_tailrec_rev.induct
thf(fact_70_map__tailrec__rev_Oinduct, axiom,
    ((![P : (dB > type) > list_dB > list_type > $o, A0 : dB > type, A1 : list_dB, A22 : list_type]: ((![F2 : dB > type, X_1 : list_type]: (P @ F2 @ nil_dB @ X_1)) => ((![F2 : dB > type, A3 : dB, As : list_dB, Bs : list_type]: ((P @ F2 @ As @ (cons_type @ (F2 @ A3) @ Bs)) => (P @ F2 @ (cons_dB @ A3 @ As) @ Bs))) => (P @ A0 @ A1 @ A22)))))). % map_tailrec_rev.induct
thf(fact_71_map__tailrec__rev_Oinduct, axiom,
    ((![P : (type > type) > list_type > list_type > $o, A0 : type > type, A1 : list_type, A22 : list_type]: ((![F2 : type > type, X_1 : list_type]: (P @ F2 @ nil_type @ X_1)) => ((![F2 : type > type, A3 : type, As : list_type, Bs : list_type]: ((P @ F2 @ As @ (cons_type @ (F2 @ A3) @ Bs)) => (P @ F2 @ (cons_type @ A3 @ As) @ Bs))) => (P @ A0 @ A1 @ A22)))))). % map_tailrec_rev.induct
thf(fact_72_list__nonempty__induct, axiom,
    ((![Xs : list_dB, P : list_dB > $o]: ((~ ((Xs = nil_dB))) => ((![X3 : dB]: (P @ (cons_dB @ X3 @ nil_dB))) => ((![X3 : dB, Xs3 : list_dB]: ((~ ((Xs3 = nil_dB))) => ((P @ Xs3) => (P @ (cons_dB @ X3 @ Xs3))))) => (P @ Xs))))))). % list_nonempty_induct
thf(fact_73_list__nonempty__induct, axiom,
    ((![Xs : list_type, P : list_type > $o]: ((~ ((Xs = nil_type))) => ((![X3 : type]: (P @ (cons_type @ X3 @ nil_type))) => ((![X3 : type, Xs3 : list_type]: ((~ ((Xs3 = nil_type))) => ((P @ Xs3) => (P @ (cons_type @ X3 @ Xs3))))) => (P @ Xs))))))). % list_nonempty_induct
thf(fact_74_successively_Oinduct, axiom,
    ((![P : (dB > dB > $o) > list_dB > $o, A0 : dB > dB > $o, A1 : list_dB]: ((![P2 : dB > dB > $o]: (P @ P2 @ nil_dB)) => ((![P2 : dB > dB > $o, X3 : dB]: (P @ P2 @ (cons_dB @ X3 @ nil_dB))) => ((![P2 : dB > dB > $o, X3 : dB, Y : dB, Xs3 : list_dB]: ((P @ P2 @ (cons_dB @ Y @ Xs3)) => (P @ P2 @ (cons_dB @ X3 @ (cons_dB @ Y @ Xs3))))) => (P @ A0 @ A1))))))). % successively.induct
thf(fact_75_successively_Oinduct, axiom,
    ((![P : (type > type > $o) > list_type > $o, A0 : type > type > $o, A1 : list_type]: ((![P2 : type > type > $o]: (P @ P2 @ nil_type)) => ((![P2 : type > type > $o, X3 : type]: (P @ P2 @ (cons_type @ X3 @ nil_type))) => ((![P2 : type > type > $o, X3 : type, Y : type, Xs3 : list_type]: ((P @ P2 @ (cons_type @ Y @ Xs3)) => (P @ P2 @ (cons_type @ X3 @ (cons_type @ Y @ Xs3))))) => (P @ A0 @ A1))))))). % successively.induct
thf(fact_76_list_Omap__ident, axiom,
    ((![T4 : list_dB]: ((map_dB_dB @ (^[X2 : dB]: X2) @ T4) = T4)))). % list.map_ident
thf(fact_77_remdups__adj_Oinduct, axiom,
    ((![P : list_dB > $o, A0 : list_dB]: ((P @ nil_dB) => ((![X3 : dB]: (P @ (cons_dB @ X3 @ nil_dB))) => ((![X3 : dB, Y : dB, Xs3 : list_dB]: (((X3 = Y) => (P @ (cons_dB @ X3 @ Xs3))) => (((~ ((X3 = Y))) => (P @ (cons_dB @ Y @ Xs3))) => (P @ (cons_dB @ X3 @ (cons_dB @ Y @ Xs3)))))) => (P @ A0))))))). % remdups_adj.induct
thf(fact_78_remdups__adj_Oinduct, axiom,
    ((![P : list_type > $o, A0 : list_type]: ((P @ nil_type) => ((![X3 : type]: (P @ (cons_type @ X3 @ nil_type))) => ((![X3 : type, Y : type, Xs3 : list_type]: (((X3 = Y) => (P @ (cons_type @ X3 @ Xs3))) => (((~ ((X3 = Y))) => (P @ (cons_type @ Y @ Xs3))) => (P @ (cons_type @ X3 @ (cons_type @ Y @ Xs3)))))) => (P @ A0))))))). % remdups_adj.induct
thf(fact_79_sorted__wrt_Oinduct, axiom,
    ((![P : (dB > dB > $o) > list_dB > $o, A0 : dB > dB > $o, A1 : list_dB]: ((![P2 : dB > dB > $o]: (P @ P2 @ nil_dB)) => ((![P2 : dB > dB > $o, X3 : dB, Ys : list_dB]: ((P @ P2 @ Ys) => (P @ P2 @ (cons_dB @ X3 @ Ys)))) => (P @ A0 @ A1)))))). % sorted_wrt.induct
thf(fact_80_sorted__wrt_Oinduct, axiom,
    ((![P : (type > type > $o) > list_type > $o, A0 : type > type > $o, A1 : list_type]: ((![P2 : type > type > $o]: (P @ P2 @ nil_type)) => ((![P2 : type > type > $o, X3 : type, Ys : list_type]: ((P @ P2 @ Ys) => (P @ P2 @ (cons_type @ X3 @ Ys)))) => (P @ A0 @ A1)))))). % sorted_wrt.induct
thf(fact_81_remdups__adj_Ocases, axiom,
    ((![X : list_dB]: ((~ ((X = nil_dB))) => ((![X3 : dB]: (~ ((X = (cons_dB @ X3 @ nil_dB))))) => (~ ((![X3 : dB, Y : dB, Xs3 : list_dB]: (~ ((X = (cons_dB @ X3 @ (cons_dB @ Y @ Xs3))))))))))))). % remdups_adj.cases
thf(fact_82_remdups__adj_Ocases, axiom,
    ((![X : list_type]: ((~ ((X = nil_type))) => ((![X3 : type]: (~ ((X = (cons_type @ X3 @ nil_type))))) => (~ ((![X3 : type, Y : type, Xs3 : list_type]: (~ ((X = (cons_type @ X3 @ (cons_type @ Y @ Xs3))))))))))))). % remdups_adj.cases
thf(fact_83_list_Ocase__distrib, axiom,
    ((![H : $o > $o, F1 : $o, F22 : type > list_type > $o, List : list_type]: ((H @ (case_list_o_type @ F1 @ F22 @ List)) = (case_list_o_type @ (H @ F1) @ (^[X12 : type]: (^[X23 : list_type]: (H @ (F22 @ X12 @ X23)))) @ List))))). % list.case_distrib
thf(fact_84_map__eq__Cons__conv, axiom,
    ((![F : dB > dB, Xs : list_dB, Y2 : dB, Ys2 : list_dB]: (((map_dB_dB @ F @ Xs) = (cons_dB @ Y2 @ Ys2)) = (?[Z : dB]: (?[Zs : list_dB]: (((Xs = (cons_dB @ Z @ Zs))) & (((((F @ Z) = Y2)) & (((map_dB_dB @ F @ Zs) = Ys2))))))))))). % map_eq_Cons_conv
thf(fact_85_map__eq__Cons__conv, axiom,
    ((![F : type > dB, Xs : list_type, Y2 : dB, Ys2 : list_dB]: (((map_type_dB @ F @ Xs) = (cons_dB @ Y2 @ Ys2)) = (?[Z : type]: (?[Zs : list_type]: (((Xs = (cons_type @ Z @ Zs))) & (((((F @ Z) = Y2)) & (((map_type_dB @ F @ Zs) = Ys2))))))))))). % map_eq_Cons_conv
thf(fact_86_map__eq__Cons__conv, axiom,
    ((![F : dB > type, Xs : list_dB, Y2 : type, Ys2 : list_type]: (((map_dB_type @ F @ Xs) = (cons_type @ Y2 @ Ys2)) = (?[Z : dB]: (?[Zs : list_dB]: (((Xs = (cons_dB @ Z @ Zs))) & (((((F @ Z) = Y2)) & (((map_dB_type @ F @ Zs) = Ys2))))))))))). % map_eq_Cons_conv
thf(fact_87_map__eq__Cons__conv, axiom,
    ((![F : type > type, Xs : list_type, Y2 : type, Ys2 : list_type]: (((map_type_type @ F @ Xs) = (cons_type @ Y2 @ Ys2)) = (?[Z : type]: (?[Zs : list_type]: (((Xs = (cons_type @ Z @ Zs))) & (((((F @ Z) = Y2)) & (((map_type_type @ F @ Zs) = Ys2))))))))))). % map_eq_Cons_conv
thf(fact_88_Cons__eq__map__conv, axiom,
    ((![X : dB, Xs : list_dB, F : dB > dB, Ys2 : list_dB]: (((cons_dB @ X @ Xs) = (map_dB_dB @ F @ Ys2)) = (?[Z : dB]: (?[Zs : list_dB]: (((Ys2 = (cons_dB @ Z @ Zs))) & ((((X = (F @ Z))) & ((Xs = (map_dB_dB @ F @ Zs)))))))))))). % Cons_eq_map_conv
thf(fact_89_Cons__eq__map__conv, axiom,
    ((![X : dB, Xs : list_dB, F : type > dB, Ys2 : list_type]: (((cons_dB @ X @ Xs) = (map_type_dB @ F @ Ys2)) = (?[Z : type]: (?[Zs : list_type]: (((Ys2 = (cons_type @ Z @ Zs))) & ((((X = (F @ Z))) & ((Xs = (map_type_dB @ F @ Zs)))))))))))). % Cons_eq_map_conv
thf(fact_90_Cons__eq__map__conv, axiom,
    ((![X : type, Xs : list_type, F : dB > type, Ys2 : list_dB]: (((cons_type @ X @ Xs) = (map_dB_type @ F @ Ys2)) = (?[Z : dB]: (?[Zs : list_dB]: (((Ys2 = (cons_dB @ Z @ Zs))) & ((((X = (F @ Z))) & ((Xs = (map_dB_type @ F @ Zs)))))))))))). % Cons_eq_map_conv
thf(fact_91_Cons__eq__map__conv, axiom,
    ((![X : type, Xs : list_type, F : type > type, Ys2 : list_type]: (((cons_type @ X @ Xs) = (map_type_type @ F @ Ys2)) = (?[Z : type]: (?[Zs : list_type]: (((Ys2 = (cons_type @ Z @ Zs))) & ((((X = (F @ Z))) & ((Xs = (map_type_type @ F @ Zs)))))))))))). % Cons_eq_map_conv
thf(fact_92_transpose_Ocases, axiom,
    ((![X : list_list_dB]: ((~ ((X = nil_list_dB))) => ((![Xss : list_list_dB]: (~ ((X = (cons_list_dB @ nil_dB @ Xss))))) => (~ ((![X3 : dB, Xs3 : list_dB, Xss : list_list_dB]: (~ ((X = (cons_list_dB @ (cons_dB @ X3 @ Xs3) @ Xss)))))))))))). % transpose.cases
thf(fact_93_transpose_Ocases, axiom,
    ((![X : list_list_type]: ((~ ((X = nil_list_type))) => ((![Xss : list_list_type]: (~ ((X = (cons_list_type @ nil_type @ Xss))))) => (~ ((![X3 : type, Xs3 : list_type, Xss : list_list_type]: (~ ((X = (cons_list_type @ (cons_type @ X3 @ Xs3) @ Xss)))))))))))). % transpose.cases
thf(fact_94_shuffles_Oinduct, axiom,
    ((![P : list_dB > list_dB > $o, A0 : list_dB, A1 : list_dB]: ((![X_1 : list_dB]: (P @ nil_dB @ X_1)) => ((![Xs3 : list_dB]: (P @ Xs3 @ nil_dB)) => ((![X3 : dB, Xs3 : list_dB, Y : dB, Ys : list_dB]: ((P @ Xs3 @ (cons_dB @ Y @ Ys)) => ((P @ (cons_dB @ X3 @ Xs3) @ Ys) => (P @ (cons_dB @ X3 @ Xs3) @ (cons_dB @ Y @ Ys))))) => (P @ A0 @ A1))))))). % shuffles.induct
thf(fact_95_shuffles_Oinduct, axiom,
    ((![P : list_type > list_type > $o, A0 : list_type, A1 : list_type]: ((![X_1 : list_type]: (P @ nil_type @ X_1)) => ((![Xs3 : list_type]: (P @ Xs3 @ nil_type)) => ((![X3 : type, Xs3 : list_type, Y : type, Ys : list_type]: ((P @ Xs3 @ (cons_type @ Y @ Ys)) => ((P @ (cons_type @ X3 @ Xs3) @ Ys) => (P @ (cons_type @ X3 @ Xs3) @ (cons_type @ Y @ Ys))))) => (P @ A0 @ A1))))))). % shuffles.induct
thf(fact_96_induct__list012, axiom,
    ((![P : list_dB > $o, Xs : list_dB]: ((P @ nil_dB) => ((![X3 : dB]: (P @ (cons_dB @ X3 @ nil_dB))) => ((![X3 : dB, Y : dB, Zs2 : list_dB]: ((P @ Zs2) => ((P @ (cons_dB @ Y @ Zs2)) => (P @ (cons_dB @ X3 @ (cons_dB @ Y @ Zs2)))))) => (P @ Xs))))))). % induct_list012
thf(fact_97_induct__list012, axiom,
    ((![P : list_type > $o, Xs : list_type]: ((P @ nil_type) => ((![X3 : type]: (P @ (cons_type @ X3 @ nil_type))) => ((![X3 : type, Y : type, Zs2 : list_type]: ((P @ Zs2) => ((P @ (cons_type @ Y @ Zs2)) => (P @ (cons_type @ X3 @ (cons_type @ Y @ Zs2)))))) => (P @ Xs))))))). % induct_list012
thf(fact_98_splice_Oinduct, axiom,
    ((![P : list_dB > list_dB > $o, A0 : list_dB, A1 : list_dB]: ((![X_1 : list_dB]: (P @ nil_dB @ X_1)) => ((![X3 : dB, Xs3 : list_dB, Ys : list_dB]: ((P @ Ys @ Xs3) => (P @ (cons_dB @ X3 @ Xs3) @ Ys))) => (P @ A0 @ A1)))))). % splice.induct
thf(fact_99_splice_Oinduct, axiom,
    ((![P : list_type > list_type > $o, A0 : list_type, A1 : list_type]: ((![X_1 : list_type]: (P @ nil_type @ X_1)) => ((![X3 : type, Xs3 : list_type, Ys : list_type]: ((P @ Ys @ Xs3) => (P @ (cons_type @ X3 @ Xs3) @ Ys))) => (P @ A0 @ A1)))))). % splice.induct
thf(fact_100_map__eq__Cons__D, axiom,
    ((![F : dB > dB, Xs : list_dB, Y2 : dB, Ys2 : list_dB]: (((map_dB_dB @ F @ Xs) = (cons_dB @ Y2 @ Ys2)) => (?[Z2 : dB, Zs2 : list_dB]: ((Xs = (cons_dB @ Z2 @ Zs2)) & (((F @ Z2) = Y2) & ((map_dB_dB @ F @ Zs2) = Ys2)))))))). % map_eq_Cons_D
thf(fact_101_map__eq__Cons__D, axiom,
    ((![F : type > dB, Xs : list_type, Y2 : dB, Ys2 : list_dB]: (((map_type_dB @ F @ Xs) = (cons_dB @ Y2 @ Ys2)) => (?[Z2 : type, Zs2 : list_type]: ((Xs = (cons_type @ Z2 @ Zs2)) & (((F @ Z2) = Y2) & ((map_type_dB @ F @ Zs2) = Ys2)))))))). % map_eq_Cons_D
thf(fact_102_map__eq__Cons__D, axiom,
    ((![F : dB > type, Xs : list_dB, Y2 : type, Ys2 : list_type]: (((map_dB_type @ F @ Xs) = (cons_type @ Y2 @ Ys2)) => (?[Z2 : dB, Zs2 : list_dB]: ((Xs = (cons_dB @ Z2 @ Zs2)) & (((F @ Z2) = Y2) & ((map_dB_type @ F @ Zs2) = Ys2)))))))). % map_eq_Cons_D
thf(fact_103_map__eq__Cons__D, axiom,
    ((![F : type > type, Xs : list_type, Y2 : type, Ys2 : list_type]: (((map_type_type @ F @ Xs) = (cons_type @ Y2 @ Ys2)) => (?[Z2 : type, Zs2 : list_type]: ((Xs = (cons_type @ Z2 @ Zs2)) & (((F @ Z2) = Y2) & ((map_type_type @ F @ Zs2) = Ys2)))))))). % map_eq_Cons_D
thf(fact_104_list__induct2_H, axiom,
    ((![P : list_dB > list_dB > $o, Xs : list_dB, Ys2 : list_dB]: ((P @ nil_dB @ nil_dB) => ((![X3 : dB, Xs3 : list_dB]: (P @ (cons_dB @ X3 @ Xs3) @ nil_dB)) => ((![Y : dB, Ys : list_dB]: (P @ nil_dB @ (cons_dB @ Y @ Ys))) => ((![X3 : dB, Xs3 : list_dB, Y : dB, Ys : list_dB]: ((P @ Xs3 @ Ys) => (P @ (cons_dB @ X3 @ Xs3) @ (cons_dB @ Y @ Ys)))) => (P @ Xs @ Ys2)))))))). % list_induct2'
thf(fact_105_list__induct2_H, axiom,
    ((![P : list_dB > list_type > $o, Xs : list_dB, Ys2 : list_type]: ((P @ nil_dB @ nil_type) => ((![X3 : dB, Xs3 : list_dB]: (P @ (cons_dB @ X3 @ Xs3) @ nil_type)) => ((![Y : type, Ys : list_type]: (P @ nil_dB @ (cons_type @ Y @ Ys))) => ((![X3 : dB, Xs3 : list_dB, Y : type, Ys : list_type]: ((P @ Xs3 @ Ys) => (P @ (cons_dB @ X3 @ Xs3) @ (cons_type @ Y @ Ys)))) => (P @ Xs @ Ys2)))))))). % list_induct2'
thf(fact_106_list__induct2_H, axiom,
    ((![P : list_type > list_dB > $o, Xs : list_type, Ys2 : list_dB]: ((P @ nil_type @ nil_dB) => ((![X3 : type, Xs3 : list_type]: (P @ (cons_type @ X3 @ Xs3) @ nil_dB)) => ((![Y : dB, Ys : list_dB]: (P @ nil_type @ (cons_dB @ Y @ Ys))) => ((![X3 : type, Xs3 : list_type, Y : dB, Ys : list_dB]: ((P @ Xs3 @ Ys) => (P @ (cons_type @ X3 @ Xs3) @ (cons_dB @ Y @ Ys)))) => (P @ Xs @ Ys2)))))))). % list_induct2'
thf(fact_107_list__induct2_H, axiom,
    ((![P : list_type > list_type > $o, Xs : list_type, Ys2 : list_type]: ((P @ nil_type @ nil_type) => ((![X3 : type, Xs3 : list_type]: (P @ (cons_type @ X3 @ Xs3) @ nil_type)) => ((![Y : type, Ys : list_type]: (P @ nil_type @ (cons_type @ Y @ Ys))) => ((![X3 : type, Xs3 : list_type, Y : type, Ys : list_type]: ((P @ Xs3 @ Ys) => (P @ (cons_type @ X3 @ Xs3) @ (cons_type @ Y @ Ys)))) => (P @ Xs @ Ys2)))))))). % list_induct2'
thf(fact_108_Cons__eq__map__D, axiom,
    ((![X : dB, Xs : list_dB, F : dB > dB, Ys2 : list_dB]: (((cons_dB @ X @ Xs) = (map_dB_dB @ F @ Ys2)) => (?[Z2 : dB, Zs2 : list_dB]: ((Ys2 = (cons_dB @ Z2 @ Zs2)) & ((X = (F @ Z2)) & (Xs = (map_dB_dB @ F @ Zs2))))))))). % Cons_eq_map_D
thf(fact_109_Cons__eq__map__D, axiom,
    ((![X : dB, Xs : list_dB, F : type > dB, Ys2 : list_type]: (((cons_dB @ X @ Xs) = (map_type_dB @ F @ Ys2)) => (?[Z2 : type, Zs2 : list_type]: ((Ys2 = (cons_type @ Z2 @ Zs2)) & ((X = (F @ Z2)) & (Xs = (map_type_dB @ F @ Zs2))))))))). % Cons_eq_map_D
thf(fact_110_Cons__eq__map__D, axiom,
    ((![X : type, Xs : list_type, F : dB > type, Ys2 : list_dB]: (((cons_type @ X @ Xs) = (map_dB_type @ F @ Ys2)) => (?[Z2 : dB, Zs2 : list_dB]: ((Ys2 = (cons_dB @ Z2 @ Zs2)) & ((X = (F @ Z2)) & (Xs = (map_dB_type @ F @ Zs2))))))))). % Cons_eq_map_D
thf(fact_111_Cons__eq__map__D, axiom,
    ((![X : type, Xs : list_type, F : type > type, Ys2 : list_type]: (((cons_type @ X @ Xs) = (map_type_type @ F @ Ys2)) => (?[Z2 : type, Zs2 : list_type]: ((Ys2 = (cons_type @ Z2 @ Zs2)) & ((X = (F @ Z2)) & (Xs = (map_type_type @ F @ Zs2))))))))). % Cons_eq_map_D
thf(fact_112_neq__Nil__conv, axiom,
    ((![Xs : list_dB]: ((~ ((Xs = nil_dB))) = (?[Y3 : dB]: (?[Ys3 : list_dB]: (Xs = (cons_dB @ Y3 @ Ys3)))))))). % neq_Nil_conv
thf(fact_113_neq__Nil__conv, axiom,
    ((![Xs : list_type]: ((~ ((Xs = nil_type))) = (?[Y3 : type]: (?[Ys3 : list_type]: (Xs = (cons_type @ Y3 @ Ys3)))))))). % neq_Nil_conv
thf(fact_114_list_Oinducts, axiom,
    ((![P : list_dB > $o, List : list_dB]: ((P @ nil_dB) => ((![X13 : dB, X24 : list_dB]: ((P @ X24) => (P @ (cons_dB @ X13 @ X24)))) => (P @ List)))))). % list.inducts
thf(fact_115_list_Oinducts, axiom,
    ((![P : list_type > $o, List : list_type]: ((P @ nil_type) => ((![X13 : type, X24 : list_type]: ((P @ X24) => (P @ (cons_type @ X13 @ X24)))) => (P @ List)))))). % list.inducts
thf(fact_116_list_Oexhaust, axiom,
    ((![Y2 : list_dB]: ((~ ((Y2 = nil_dB))) => (~ ((![X212 : dB, X222 : list_dB]: (~ ((Y2 = (cons_dB @ X212 @ X222))))))))))). % list.exhaust
thf(fact_117_list_Oexhaust, axiom,
    ((![Y2 : list_type]: ((~ ((Y2 = nil_type))) => (~ ((![X212 : type, X222 : list_type]: (~ ((Y2 = (cons_type @ X212 @ X222))))))))))). % list.exhaust
thf(fact_118_list_OdiscI, axiom,
    ((![List : list_dB, X21 : dB, X22 : list_dB]: ((List = (cons_dB @ X21 @ X22)) => (~ ((List = nil_dB))))))). % list.discI
thf(fact_119_list_OdiscI, axiom,
    ((![List : list_type, X21 : type, X22 : list_type]: ((List = (cons_type @ X21 @ X22)) => (~ ((List = nil_type))))))). % list.discI
thf(fact_120_list_Odisc__eq__case_I1_J, axiom,
    ((![List : list_dB]: ((List = nil_dB) = (case_list_o_dB @ $true @ (^[Uu : dB]: (^[Uv : list_dB]: $false)) @ List))))). % list.disc_eq_case(1)
thf(fact_121_list_Odisc__eq__case_I1_J, axiom,
    ((![List : list_type]: ((List = nil_type) = (case_list_o_type @ $true @ (^[Uu : type]: (^[Uv : list_type]: $false)) @ List))))). % list.disc_eq_case(1)
thf(fact_122_list_Odisc__eq__case_I2_J, axiom,
    ((![List : list_dB]: ((~ ((List = nil_dB))) = (case_list_o_dB @ $false @ (^[Uu : dB]: (^[Uv : list_dB]: $true)) @ List))))). % list.disc_eq_case(2)
thf(fact_123_list_Odisc__eq__case_I2_J, axiom,
    ((![List : list_type]: ((~ ((List = nil_type))) = (case_list_o_type @ $false @ (^[Uu : type]: (^[Uv : list_type]: $true)) @ List))))). % list.disc_eq_case(2)
thf(fact_124_list_Odistinct_I1_J, axiom,
    ((![X21 : dB, X22 : list_dB]: (~ ((nil_dB = (cons_dB @ X21 @ X22))))))). % list.distinct(1)
thf(fact_125_list_Odistinct_I1_J, axiom,
    ((![X21 : type, X22 : list_type]: (~ ((nil_type = (cons_type @ X21 @ X22))))))). % list.distinct(1)
thf(fact_126_list_Osimps_I5_J, axiom,
    ((![F1 : $o, F22 : type > list_type > $o, X21 : type, X22 : list_type]: ((case_list_o_type @ F1 @ F22 @ (cons_type @ X21 @ X22)) = (F22 @ X21 @ X22))))). % list.simps(5)
thf(fact_127_list_Osimps_I9_J, axiom,
    ((![F : dB > dB, X21 : dB, X22 : list_dB]: ((map_dB_dB @ F @ (cons_dB @ X21 @ X22)) = (cons_dB @ (F @ X21) @ (map_dB_dB @ F @ X22)))))). % list.simps(9)
thf(fact_128_list_Osimps_I9_J, axiom,
    ((![F : dB > type, X21 : dB, X22 : list_dB]: ((map_dB_type @ F @ (cons_dB @ X21 @ X22)) = (cons_type @ (F @ X21) @ (map_dB_type @ F @ X22)))))). % list.simps(9)
thf(fact_129_list_Osimps_I9_J, axiom,
    ((![F : type > dB, X21 : type, X22 : list_type]: ((map_type_dB @ F @ (cons_type @ X21 @ X22)) = (cons_dB @ (F @ X21) @ (map_type_dB @ F @ X22)))))). % list.simps(9)
thf(fact_130_list_Osimps_I9_J, axiom,
    ((![F : type > type, X21 : type, X22 : list_type]: ((map_type_type @ F @ (cons_type @ X21 @ X22)) = (cons_type @ (F @ X21) @ (map_type_type @ F @ X22)))))). % list.simps(9)
thf(fact_131_foldl__Nil, axiom,
    ((![F : dB > dB > dB, A : dB]: ((foldl_dB_dB @ F @ A @ nil_dB) = A)))). % foldl_Nil
thf(fact_132_list_Osimps_I8_J, axiom,
    ((![F : dB > type]: ((map_dB_type @ F @ nil_dB) = nil_type)))). % list.simps(8)
thf(fact_133_list_Osimps_I8_J, axiom,
    ((![F : type > dB]: ((map_type_dB @ F @ nil_type) = nil_dB)))). % list.simps(8)
thf(fact_134_list_Osimps_I8_J, axiom,
    ((![F : type > type]: ((map_type_type @ F @ nil_type) = nil_type)))). % list.simps(8)
thf(fact_135_list_Osimps_I8_J, axiom,
    ((![F : dB > dB]: ((map_dB_dB @ F @ nil_dB) = nil_dB)))). % list.simps(8)
thf(fact_136_list_Osimps_I4_J, axiom,
    ((![F1 : $o, F22 : type > list_type > $o]: ((case_list_o_type @ F1 @ F22 @ nil_type) = F1)))). % list.simps(4)
thf(fact_137_not__Cons__self2, axiom,
    ((![X : dB, Xs : list_dB]: (~ (((cons_dB @ X @ Xs) = Xs)))))). % not_Cons_self2
thf(fact_138_not__Cons__self2, axiom,
    ((![X : type, Xs : list_type]: (~ (((cons_type @ X @ Xs) = Xs)))))). % not_Cons_self2
thf(fact_139_lift__types, axiom,
    ((![E : nat > type, Ts2 : list_dB, Ts3 : list_type, I : nat, U2 : type]: ((typings @ E @ Ts2 @ Ts3) => (typings @ (shift_type @ E @ I @ U2) @ (map_dB_dB @ (^[T6 : dB]: (lift @ T6 @ I)) @ Ts2) @ Ts3))))). % lift_types
thf(fact_140_map__rec, axiom,
    ((map_dB_dB = (^[F3 : dB > dB]: (rec_list_list_dB_dB @ nil_dB @ (^[X2 : dB]: (^[Uu : list_dB]: (cons_dB @ (F3 @ X2))))))))). % map_rec
thf(fact_141_insert__Nil, axiom,
    ((![X : dB]: ((insert_dB @ X @ nil_dB) = (cons_dB @ X @ nil_dB))))). % insert_Nil
thf(fact_142_insert__Nil, axiom,
    ((![X : type]: ((insert_type @ X @ nil_type) = (cons_type @ X @ nil_type))))). % insert_Nil
thf(fact_143_App__eq__foldl__conv, axiom,
    ((![R : dB, S : dB, T4 : dB, Ts2 : list_dB]: (((app @ R @ S) = (foldl_dB_dB @ app @ T4 @ Ts2)) = (((((Ts2 = nil_dB)) => (((app @ R @ S) = T4)))) & ((((~ ((Ts2 = nil_dB)))) => ((?[Ss2 : list_dB]: (((Ts2 = (append_dB @ Ss2 @ (cons_dB @ S @ nil_dB)))) & ((R = (foldl_dB_dB @ app @ T4 @ Ss2))))))))))))). % App_eq_foldl_conv
thf(fact_144_app__last, axiom,
    ((![T4 : dB, Ts2 : list_dB, U : dB]: ((app @ (foldl_dB_dB @ app @ T4 @ Ts2) @ U) = (foldl_dB_dB @ app @ T4 @ (append_dB @ Ts2 @ (cons_dB @ U @ nil_dB))))))). % app_last
thf(fact_145_map__eq__map__tailrec, axiom,
    ((map_dB_dB = map_tailrec_dB_dB))). % map_eq_map_tailrec
thf(fact_146_append_Oassoc, axiom,
    ((![A : list_dB, B : list_dB, C : list_dB]: ((append_dB @ (append_dB @ A @ B) @ C) = (append_dB @ A @ (append_dB @ B @ C)))))). % append.assoc
thf(fact_147_append_Oassoc, axiom,
    ((![A : list_type, B : list_type, C : list_type]: ((append_type @ (append_type @ A @ B) @ C) = (append_type @ A @ (append_type @ B @ C)))))). % append.assoc
thf(fact_148_append__assoc, axiom,
    ((![Xs : list_dB, Ys2 : list_dB, Zs3 : list_dB]: ((append_dB @ (append_dB @ Xs @ Ys2) @ Zs3) = (append_dB @ Xs @ (append_dB @ Ys2 @ Zs3)))))). % append_assoc
thf(fact_149_append__assoc, axiom,
    ((![Xs : list_type, Ys2 : list_type, Zs3 : list_type]: ((append_type @ (append_type @ Xs @ Ys2) @ Zs3) = (append_type @ Xs @ (append_type @ Ys2 @ Zs3)))))). % append_assoc
thf(fact_150_append__same__eq, axiom,
    ((![Ys2 : list_dB, Xs : list_dB, Zs3 : list_dB]: (((append_dB @ Ys2 @ Xs) = (append_dB @ Zs3 @ Xs)) = (Ys2 = Zs3))))). % append_same_eq
thf(fact_151_append__same__eq, axiom,
    ((![Ys2 : list_type, Xs : list_type, Zs3 : list_type]: (((append_type @ Ys2 @ Xs) = (append_type @ Zs3 @ Xs)) = (Ys2 = Zs3))))). % append_same_eq
thf(fact_152_same__append__eq, axiom,
    ((![Xs : list_dB, Ys2 : list_dB, Zs3 : list_dB]: (((append_dB @ Xs @ Ys2) = (append_dB @ Xs @ Zs3)) = (Ys2 = Zs3))))). % same_append_eq
thf(fact_153_same__append__eq, axiom,
    ((![Xs : list_type, Ys2 : list_type, Zs3 : list_type]: (((append_type @ Xs @ Ys2) = (append_type @ Xs @ Zs3)) = (Ys2 = Zs3))))). % same_append_eq
thf(fact_154_append_Oright__neutral, axiom,
    ((![A : list_dB]: ((append_dB @ A @ nil_dB) = A)))). % append.right_neutral
thf(fact_155_append_Oright__neutral, axiom,
    ((![A : list_type]: ((append_type @ A @ nil_type) = A)))). % append.right_neutral
thf(fact_156_append__is__Nil__conv, axiom,
    ((![Xs : list_dB, Ys2 : list_dB]: (((append_dB @ Xs @ Ys2) = nil_dB) = (((Xs = nil_dB)) & ((Ys2 = nil_dB))))))). % append_is_Nil_conv
thf(fact_157_append__is__Nil__conv, axiom,
    ((![Xs : list_type, Ys2 : list_type]: (((append_type @ Xs @ Ys2) = nil_type) = (((Xs = nil_type)) & ((Ys2 = nil_type))))))). % append_is_Nil_conv
thf(fact_158_Nil__is__append__conv, axiom,
    ((![Xs : list_dB, Ys2 : list_dB]: ((nil_dB = (append_dB @ Xs @ Ys2)) = (((Xs = nil_dB)) & ((Ys2 = nil_dB))))))). % Nil_is_append_conv
thf(fact_159_Nil__is__append__conv, axiom,
    ((![Xs : list_type, Ys2 : list_type]: ((nil_type = (append_type @ Xs @ Ys2)) = (((Xs = nil_type)) & ((Ys2 = nil_type))))))). % Nil_is_append_conv
thf(fact_160_self__append__conv2, axiom,
    ((![Ys2 : list_dB, Xs : list_dB]: ((Ys2 = (append_dB @ Xs @ Ys2)) = (Xs = nil_dB))))). % self_append_conv2
thf(fact_161_self__append__conv2, axiom,
    ((![Ys2 : list_type, Xs : list_type]: ((Ys2 = (append_type @ Xs @ Ys2)) = (Xs = nil_type))))). % self_append_conv2
thf(fact_162_append__self__conv2, axiom,
    ((![Xs : list_dB, Ys2 : list_dB]: (((append_dB @ Xs @ Ys2) = Ys2) = (Xs = nil_dB))))). % append_self_conv2
thf(fact_163_append__self__conv2, axiom,
    ((![Xs : list_type, Ys2 : list_type]: (((append_type @ Xs @ Ys2) = Ys2) = (Xs = nil_type))))). % append_self_conv2
thf(fact_164_self__append__conv, axiom,
    ((![Xs : list_dB, Ys2 : list_dB]: ((Xs = (append_dB @ Xs @ Ys2)) = (Ys2 = nil_dB))))). % self_append_conv
thf(fact_165_self__append__conv, axiom,
    ((![Xs : list_type, Ys2 : list_type]: ((Xs = (append_type @ Xs @ Ys2)) = (Ys2 = nil_type))))). % self_append_conv
thf(fact_166_append__self__conv, axiom,
    ((![Xs : list_dB, Ys2 : list_dB]: (((append_dB @ Xs @ Ys2) = Xs) = (Ys2 = nil_dB))))). % append_self_conv
thf(fact_167_append__self__conv, axiom,
    ((![Xs : list_type, Ys2 : list_type]: (((append_type @ Xs @ Ys2) = Xs) = (Ys2 = nil_type))))). % append_self_conv
thf(fact_168_append__Nil2, axiom,
    ((![Xs : list_dB]: ((append_dB @ Xs @ nil_dB) = Xs)))). % append_Nil2
thf(fact_169_append__Nil2, axiom,
    ((![Xs : list_type]: ((append_type @ Xs @ nil_type) = Xs)))). % append_Nil2
thf(fact_170_map__append, axiom,
    ((![F : dB > type, Xs : list_dB, Ys2 : list_dB]: ((map_dB_type @ F @ (append_dB @ Xs @ Ys2)) = (append_type @ (map_dB_type @ F @ Xs) @ (map_dB_type @ F @ Ys2)))))). % map_append
thf(fact_171_map__append, axiom,
    ((![F : type > dB, Xs : list_type, Ys2 : list_type]: ((map_type_dB @ F @ (append_type @ Xs @ Ys2)) = (append_dB @ (map_type_dB @ F @ Xs) @ (map_type_dB @ F @ Ys2)))))). % map_append
thf(fact_172_map__append, axiom,
    ((![F : type > type, Xs : list_type, Ys2 : list_type]: ((map_type_type @ F @ (append_type @ Xs @ Ys2)) = (append_type @ (map_type_type @ F @ Xs) @ (map_type_type @ F @ Ys2)))))). % map_append
thf(fact_173_map__append, axiom,
    ((![F : dB > dB, Xs : list_dB, Ys2 : list_dB]: ((map_dB_dB @ F @ (append_dB @ Xs @ Ys2)) = (append_dB @ (map_dB_dB @ F @ Xs) @ (map_dB_dB @ F @ Ys2)))))). % map_append
thf(fact_174_foldl__append, axiom,
    ((![F : dB > dB > dB, A : dB, Xs : list_dB, Ys2 : list_dB]: ((foldl_dB_dB @ F @ A @ (append_dB @ Xs @ Ys2)) = (foldl_dB_dB @ F @ (foldl_dB_dB @ F @ A @ Xs) @ Ys2))))). % foldl_append
thf(fact_175_foldr__append, axiom,
    ((![F : type > type > type, Xs : list_type, Ys2 : list_type, A : type]: ((foldr_type_type @ F @ (append_type @ Xs @ Ys2) @ A) = (foldr_type_type @ F @ Xs @ (foldr_type_type @ F @ Ys2 @ A)))))). % foldr_append
thf(fact_176_append1__eq__conv, axiom,
    ((![Xs : list_dB, X : dB, Ys2 : list_dB, Y2 : dB]: (((append_dB @ Xs @ (cons_dB @ X @ nil_dB)) = (append_dB @ Ys2 @ (cons_dB @ Y2 @ nil_dB))) = (((Xs = Ys2)) & ((X = Y2))))))). % append1_eq_conv
thf(fact_177_append1__eq__conv, axiom,
    ((![Xs : list_type, X : type, Ys2 : list_type, Y2 : type]: (((append_type @ Xs @ (cons_type @ X @ nil_type)) = (append_type @ Ys2 @ (cons_type @ Y2 @ nil_type))) = (((Xs = Ys2)) & ((X = Y2))))))). % append1_eq_conv
thf(fact_178_append__eq__appendI, axiom,
    ((![Xs : list_dB, Xs1 : list_dB, Zs3 : list_dB, Ys2 : list_dB, Us : list_dB]: (((append_dB @ Xs @ Xs1) = Zs3) => ((Ys2 = (append_dB @ Xs1 @ Us)) => ((append_dB @ Xs @ Ys2) = (append_dB @ Zs3 @ Us))))))). % append_eq_appendI
thf(fact_179_append__eq__appendI, axiom,
    ((![Xs : list_type, Xs1 : list_type, Zs3 : list_type, Ys2 : list_type, Us : list_type]: (((append_type @ Xs @ Xs1) = Zs3) => ((Ys2 = (append_type @ Xs1 @ Us)) => ((append_type @ Xs @ Ys2) = (append_type @ Zs3 @ Us))))))). % append_eq_appendI
thf(fact_180_append__eq__append__conv2, axiom,
    ((![Xs : list_dB, Ys2 : list_dB, Zs3 : list_dB, Ts2 : list_dB]: (((append_dB @ Xs @ Ys2) = (append_dB @ Zs3 @ Ts2)) = (?[Us3 : list_dB]: (((((Xs = (append_dB @ Zs3 @ Us3))) & (((append_dB @ Us3 @ Ys2) = Ts2)))) | (((((append_dB @ Xs @ Us3) = Zs3)) & ((Ys2 = (append_dB @ Us3 @ Ts2))))))))))). % append_eq_append_conv2
thf(fact_181_append__eq__append__conv2, axiom,
    ((![Xs : list_type, Ys2 : list_type, Zs3 : list_type, Ts2 : list_type]: (((append_type @ Xs @ Ys2) = (append_type @ Zs3 @ Ts2)) = (?[Us3 : list_type]: (((((Xs = (append_type @ Zs3 @ Us3))) & (((append_type @ Us3 @ Ys2) = Ts2)))) | (((((append_type @ Xs @ Us3) = Zs3)) & ((Ys2 = (append_type @ Us3 @ Ts2))))))))))). % append_eq_append_conv2
thf(fact_182_Cons__eq__appendI, axiom,
    ((![X : dB, Xs1 : list_dB, Ys2 : list_dB, Xs : list_dB, Zs3 : list_dB]: (((cons_dB @ X @ Xs1) = Ys2) => ((Xs = (append_dB @ Xs1 @ Zs3)) => ((cons_dB @ X @ Xs) = (append_dB @ Ys2 @ Zs3))))))). % Cons_eq_appendI
thf(fact_183_Cons__eq__appendI, axiom,
    ((![X : type, Xs1 : list_type, Ys2 : list_type, Xs : list_type, Zs3 : list_type]: (((cons_type @ X @ Xs1) = Ys2) => ((Xs = (append_type @ Xs1 @ Zs3)) => ((cons_type @ X @ Xs) = (append_type @ Ys2 @ Zs3))))))). % Cons_eq_appendI
thf(fact_184_append__Cons, axiom,
    ((![X : dB, Xs : list_dB, Ys2 : list_dB]: ((append_dB @ (cons_dB @ X @ Xs) @ Ys2) = (cons_dB @ X @ (append_dB @ Xs @ Ys2)))))). % append_Cons
thf(fact_185_append__Cons, axiom,
    ((![X : type, Xs : list_type, Ys2 : list_type]: ((append_type @ (cons_type @ X @ Xs) @ Ys2) = (cons_type @ X @ (append_type @ Xs @ Ys2)))))). % append_Cons
thf(fact_186_append_Oleft__neutral, axiom,
    ((![A : list_dB]: ((append_dB @ nil_dB @ A) = A)))). % append.left_neutral
thf(fact_187_append_Oleft__neutral, axiom,
    ((![A : list_type]: ((append_type @ nil_type @ A) = A)))). % append.left_neutral
thf(fact_188_append__Nil, axiom,
    ((![Ys2 : list_dB]: ((append_dB @ nil_dB @ Ys2) = Ys2)))). % append_Nil
thf(fact_189_append__Nil, axiom,
    ((![Ys2 : list_type]: ((append_type @ nil_type @ Ys2) = Ys2)))). % append_Nil
thf(fact_190_eq__Nil__appendI, axiom,
    ((![Xs : list_dB, Ys2 : list_dB]: ((Xs = Ys2) => (Xs = (append_dB @ nil_dB @ Ys2)))))). % eq_Nil_appendI
thf(fact_191_eq__Nil__appendI, axiom,
    ((![Xs : list_type, Ys2 : list_type]: ((Xs = Ys2) => (Xs = (append_type @ nil_type @ Ys2)))))). % eq_Nil_appendI
thf(fact_192_map__eq__append__conv, axiom,
    ((![F : type > dB, Xs : list_type, Ys2 : list_dB, Zs3 : list_dB]: (((map_type_dB @ F @ Xs) = (append_dB @ Ys2 @ Zs3)) = (?[Us3 : list_type]: (?[Vs : list_type]: (((Xs = (append_type @ Us3 @ Vs))) & ((((Ys2 = (map_type_dB @ F @ Us3))) & ((Zs3 = (map_type_dB @ F @ Vs)))))))))))). % map_eq_append_conv
thf(fact_193_map__eq__append__conv, axiom,
    ((![F : dB > type, Xs : list_dB, Ys2 : list_type, Zs3 : list_type]: (((map_dB_type @ F @ Xs) = (append_type @ Ys2 @ Zs3)) = (?[Us3 : list_dB]: (?[Vs : list_dB]: (((Xs = (append_dB @ Us3 @ Vs))) & ((((Ys2 = (map_dB_type @ F @ Us3))) & ((Zs3 = (map_dB_type @ F @ Vs)))))))))))). % map_eq_append_conv
thf(fact_194_map__eq__append__conv, axiom,
    ((![F : type > type, Xs : list_type, Ys2 : list_type, Zs3 : list_type]: (((map_type_type @ F @ Xs) = (append_type @ Ys2 @ Zs3)) = (?[Us3 : list_type]: (?[Vs : list_type]: (((Xs = (append_type @ Us3 @ Vs))) & ((((Ys2 = (map_type_type @ F @ Us3))) & ((Zs3 = (map_type_type @ F @ Vs)))))))))))). % map_eq_append_conv
thf(fact_195_map__eq__append__conv, axiom,
    ((![F : dB > dB, Xs : list_dB, Ys2 : list_dB, Zs3 : list_dB]: (((map_dB_dB @ F @ Xs) = (append_dB @ Ys2 @ Zs3)) = (?[Us3 : list_dB]: (?[Vs : list_dB]: (((Xs = (append_dB @ Us3 @ Vs))) & ((((Ys2 = (map_dB_dB @ F @ Us3))) & ((Zs3 = (map_dB_dB @ F @ Vs)))))))))))). % map_eq_append_conv
thf(fact_196_append__eq__map__conv, axiom,
    ((![Ys2 : list_dB, Zs3 : list_dB, F : type > dB, Xs : list_type]: (((append_dB @ Ys2 @ Zs3) = (map_type_dB @ F @ Xs)) = (?[Us3 : list_type]: (?[Vs : list_type]: (((Xs = (append_type @ Us3 @ Vs))) & ((((Ys2 = (map_type_dB @ F @ Us3))) & ((Zs3 = (map_type_dB @ F @ Vs)))))))))))). % append_eq_map_conv
thf(fact_197_append__eq__map__conv, axiom,
    ((![Ys2 : list_type, Zs3 : list_type, F : dB > type, Xs : list_dB]: (((append_type @ Ys2 @ Zs3) = (map_dB_type @ F @ Xs)) = (?[Us3 : list_dB]: (?[Vs : list_dB]: (((Xs = (append_dB @ Us3 @ Vs))) & ((((Ys2 = (map_dB_type @ F @ Us3))) & ((Zs3 = (map_dB_type @ F @ Vs)))))))))))). % append_eq_map_conv
thf(fact_198_append__eq__map__conv, axiom,
    ((![Ys2 : list_type, Zs3 : list_type, F : type > type, Xs : list_type]: (((append_type @ Ys2 @ Zs3) = (map_type_type @ F @ Xs)) = (?[Us3 : list_type]: (?[Vs : list_type]: (((Xs = (append_type @ Us3 @ Vs))) & ((((Ys2 = (map_type_type @ F @ Us3))) & ((Zs3 = (map_type_type @ F @ Vs)))))))))))). % append_eq_map_conv
thf(fact_199_append__eq__map__conv, axiom,
    ((![Ys2 : list_dB, Zs3 : list_dB, F : dB > dB, Xs : list_dB]: (((append_dB @ Ys2 @ Zs3) = (map_dB_dB @ F @ Xs)) = (?[Us3 : list_dB]: (?[Vs : list_dB]: (((Xs = (append_dB @ Us3 @ Vs))) & ((((Ys2 = (map_dB_dB @ F @ Us3))) & ((Zs3 = (map_dB_dB @ F @ Vs)))))))))))). % append_eq_map_conv
thf(fact_200_rev__induct, axiom,
    ((![P : list_dB > $o, Xs : list_dB]: ((P @ nil_dB) => ((![X3 : dB, Xs3 : list_dB]: ((P @ Xs3) => (P @ (append_dB @ Xs3 @ (cons_dB @ X3 @ nil_dB))))) => (P @ Xs)))))). % rev_induct
thf(fact_201_rev__induct, axiom,
    ((![P : list_type > $o, Xs : list_type]: ((P @ nil_type) => ((![X3 : type, Xs3 : list_type]: ((P @ Xs3) => (P @ (append_type @ Xs3 @ (cons_type @ X3 @ nil_type))))) => (P @ Xs)))))). % rev_induct
thf(fact_202_rev__exhaust, axiom,
    ((![Xs : list_dB]: ((~ ((Xs = nil_dB))) => (~ ((![Ys : list_dB, Y : dB]: (~ ((Xs = (append_dB @ Ys @ (cons_dB @ Y @ nil_dB)))))))))))). % rev_exhaust
thf(fact_203_rev__exhaust, axiom,
    ((![Xs : list_type]: ((~ ((Xs = nil_type))) => (~ ((![Ys : list_type, Y : type]: (~ ((Xs = (append_type @ Ys @ (cons_type @ Y @ nil_type)))))))))))). % rev_exhaust
thf(fact_204_Cons__eq__append__conv, axiom,
    ((![X : dB, Xs : list_dB, Ys2 : list_dB, Zs3 : list_dB]: (((cons_dB @ X @ Xs) = (append_dB @ Ys2 @ Zs3)) = (((((Ys2 = nil_dB)) & (((cons_dB @ X @ Xs) = Zs3)))) | ((?[Ys4 : list_dB]: ((((cons_dB @ X @ Ys4) = Ys2)) & ((Xs = (append_dB @ Ys4 @ Zs3))))))))))). % Cons_eq_append_conv
thf(fact_205_Cons__eq__append__conv, axiom,
    ((![X : type, Xs : list_type, Ys2 : list_type, Zs3 : list_type]: (((cons_type @ X @ Xs) = (append_type @ Ys2 @ Zs3)) = (((((Ys2 = nil_type)) & (((cons_type @ X @ Xs) = Zs3)))) | ((?[Ys4 : list_type]: ((((cons_type @ X @ Ys4) = Ys2)) & ((Xs = (append_type @ Ys4 @ Zs3))))))))))). % Cons_eq_append_conv
thf(fact_206_append__eq__Cons__conv, axiom,
    ((![Ys2 : list_dB, Zs3 : list_dB, X : dB, Xs : list_dB]: (((append_dB @ Ys2 @ Zs3) = (cons_dB @ X @ Xs)) = (((((Ys2 = nil_dB)) & ((Zs3 = (cons_dB @ X @ Xs))))) | ((?[Ys4 : list_dB]: (((Ys2 = (cons_dB @ X @ Ys4))) & (((append_dB @ Ys4 @ Zs3) = Xs)))))))))). % append_eq_Cons_conv
thf(fact_207_append__eq__Cons__conv, axiom,
    ((![Ys2 : list_type, Zs3 : list_type, X : type, Xs : list_type]: (((append_type @ Ys2 @ Zs3) = (cons_type @ X @ Xs)) = (((((Ys2 = nil_type)) & ((Zs3 = (cons_type @ X @ Xs))))) | ((?[Ys4 : list_type]: (((Ys2 = (cons_type @ X @ Ys4))) & (((append_type @ Ys4 @ Zs3) = Xs)))))))))). % append_eq_Cons_conv
thf(fact_208_rev__nonempty__induct, axiom,
    ((![Xs : list_dB, P : list_dB > $o]: ((~ ((Xs = nil_dB))) => ((![X3 : dB]: (P @ (cons_dB @ X3 @ nil_dB))) => ((![X3 : dB, Xs3 : list_dB]: ((~ ((Xs3 = nil_dB))) => ((P @ Xs3) => (P @ (append_dB @ Xs3 @ (cons_dB @ X3 @ nil_dB)))))) => (P @ Xs))))))). % rev_nonempty_induct
thf(fact_209_rev__nonempty__induct, axiom,
    ((![Xs : list_type, P : list_type > $o]: ((~ ((Xs = nil_type))) => ((![X3 : type]: (P @ (cons_type @ X3 @ nil_type))) => ((![X3 : type, Xs3 : list_type]: ((~ ((Xs3 = nil_type))) => ((P @ Xs3) => (P @ (append_type @ Xs3 @ (cons_type @ X3 @ nil_type)))))) => (P @ Xs))))))). % rev_nonempty_induct
thf(fact_210_rev__exhaust2, axiom,
    ((![Xs : list_dB]: ((~ ((Xs = nil_dB))) => (~ ((![Ys : list_dB, Y : dB]: (~ ((Xs = (append_dB @ Ys @ (cons_dB @ Y @ nil_dB)))))))))))). % rev_exhaust2
thf(fact_211_rev__exhaust2, axiom,
    ((![Xs : list_type]: ((~ ((Xs = nil_type))) => (~ ((![Ys : list_type, Y : type]: (~ ((Xs = (append_type @ Ys @ (cons_type @ Y @ nil_type)))))))))))). % rev_exhaust2
thf(fact_212_typings_Osimps_I1_J, axiom,
    ((![E : nat > type, Ts3 : list_type]: ((typings @ E @ nil_dB @ Ts3) = (Ts3 = nil_type))))). % typings.simps(1)
thf(fact_213_types__snoc__eq, axiom,
    ((![E : nat > type, Ts2 : list_dB, T4 : dB, Ts3 : list_type, T2 : type]: ((typings @ E @ (append_dB @ Ts2 @ (cons_dB @ T4 @ nil_dB)) @ (append_type @ Ts3 @ (cons_type @ T2 @ nil_type))) = (((typings @ E @ Ts2 @ Ts3)) & ((typing @ E @ T4 @ T2))))))). % types_snoc_eq
thf(fact_214_types__snoc, axiom,
    ((![E : nat > type, Ts2 : list_dB, Ts3 : list_type, T4 : dB, T2 : type]: ((typings @ E @ Ts2 @ Ts3) => ((typing @ E @ T4 @ T2) => (typings @ E @ (append_dB @ Ts2 @ (cons_dB @ T4 @ nil_dB)) @ (append_type @ Ts3 @ (cons_type @ T2 @ nil_type)))))))). % types_snoc
thf(fact_215_types__snocE, axiom,
    ((![E : nat > type, Ts2 : list_dB, T4 : dB, Ts3 : list_type]: ((typings @ E @ (append_dB @ Ts2 @ (cons_dB @ T4 @ nil_dB)) @ Ts3) => (~ ((![Us2 : list_type, U3 : type]: ((Ts3 = (append_type @ Us2 @ (cons_type @ U3 @ nil_type))) => ((typings @ E @ Ts2 @ Us2) => (~ ((typing @ E @ T4 @ U3)))))))))))). % types_snocE
thf(fact_216_Succ__def, axiom,
    ((bNF_Greatest_Succ_dB = (^[Kl : set_list_dB]: (^[Kl2 : list_dB]: (collect_dB @ (^[K2 : dB]: (member_list_dB @ (append_dB @ Kl2 @ (cons_dB @ K2 @ nil_dB)) @ Kl)))))))). % Succ_def
thf(fact_217_Succ__def, axiom,
    ((bNF_Gr1527093962c_type = (^[Kl : set_list_type]: (^[Kl2 : list_type]: (collect_type @ (^[K2 : type]: (member_list_type @ (append_type @ Kl2 @ (cons_type @ K2 @ nil_type)) @ Kl)))))))). % Succ_def
thf(fact_218_SuccI, axiom,
    ((![Kl3 : list_dB, K : dB, Kl4 : set_list_dB]: ((member_list_dB @ (append_dB @ Kl3 @ (cons_dB @ K @ nil_dB)) @ Kl4) => (member_dB @ K @ (bNF_Greatest_Succ_dB @ Kl4 @ Kl3)))))). % SuccI
thf(fact_219_SuccI, axiom,
    ((![Kl3 : list_type, K : type, Kl4 : set_list_type]: ((member_list_type @ (append_type @ Kl3 @ (cons_type @ K @ nil_type)) @ Kl4) => (member_type @ K @ (bNF_Gr1527093962c_type @ Kl4 @ Kl3)))))). % SuccI
thf(fact_220_SuccD, axiom,
    ((![K : dB, Kl4 : set_list_dB, Kl3 : list_dB]: ((member_dB @ K @ (bNF_Greatest_Succ_dB @ Kl4 @ Kl3)) => (member_list_dB @ (append_dB @ Kl3 @ (cons_dB @ K @ nil_dB)) @ Kl4))))). % SuccD
thf(fact_221_SuccD, axiom,
    ((![K : type, Kl4 : set_list_type, Kl3 : list_type]: ((member_type @ K @ (bNF_Gr1527093962c_type @ Kl4 @ Kl3)) => (member_list_type @ (append_type @ Kl3 @ (cons_type @ K @ nil_type)) @ Kl4))))). % SuccD
thf(fact_222_empty__Shift, axiom,
    ((![Kl4 : set_list_dB, K : dB]: ((member_list_dB @ nil_dB @ Kl4) => ((member_dB @ K @ (bNF_Greatest_Succ_dB @ Kl4 @ nil_dB)) => (member_list_dB @ nil_dB @ (bNF_Gr1110975684ift_dB @ Kl4 @ K))))))). % empty_Shift
thf(fact_223_empty__Shift, axiom,
    ((![Kl4 : set_list_type, K : type]: ((member_list_type @ nil_type @ Kl4) => ((member_type @ K @ (bNF_Gr1527093962c_type @ Kl4 @ nil_type)) => (member_list_type @ nil_type @ (bNF_Gr681920326t_type @ Kl4 @ K))))))). % empty_Shift
thf(fact_224_Succ__Shift, axiom,
    ((![Kl4 : set_list_dB, K : dB, Kl3 : list_dB]: ((bNF_Greatest_Succ_dB @ (bNF_Gr1110975684ift_dB @ Kl4 @ K) @ Kl3) = (bNF_Greatest_Succ_dB @ Kl4 @ (cons_dB @ K @ Kl3)))))). % Succ_Shift
thf(fact_225_Succ__Shift, axiom,
    ((![Kl4 : set_list_type, K : type, Kl3 : list_type]: ((bNF_Gr1527093962c_type @ (bNF_Gr681920326t_type @ Kl4 @ K) @ Kl3) = (bNF_Gr1527093962c_type @ Kl4 @ (cons_type @ K @ Kl3)))))). % Succ_Shift
thf(fact_226_ShiftD, axiom,
    ((![Kl3 : list_dB, Kl4 : set_list_dB, K : dB]: ((member_list_dB @ Kl3 @ (bNF_Gr1110975684ift_dB @ Kl4 @ K)) => (member_list_dB @ (cons_dB @ K @ Kl3) @ Kl4))))). % ShiftD
thf(fact_227_ShiftD, axiom,
    ((![Kl3 : list_type, Kl4 : set_list_type, K : type]: ((member_list_type @ Kl3 @ (bNF_Gr681920326t_type @ Kl4 @ K)) => (member_list_type @ (cons_type @ K @ Kl3) @ Kl4))))). % ShiftD

% Conjectures (1)
thf(conj_0, conjecture,
    ((t3 = (fun @ t @ (foldr_type_type @ fun @ ts @ t2))))).
