% 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_236__5212014_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:38:35.630

% Could-be-implicit typings (5)
thf(ty_n_t__List__Olist_It__LambdaType__Otype_J, type,
    list_type : $tType).
thf(ty_n_t__List__Olist_It__Lambda__OdB_J, type,
    list_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 (42)
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat, type,
    plus_plus_nat : nat > nat > nat).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat, type,
    zero_zero_nat : nat).
thf(sy_c_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_Obeta, type,
    beta : dB > dB > $o).
thf(sy_c_Lambda_OdB_OAbs, type,
    abs : dB > dB).
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_OdB_Osize__dB, type,
    size_dB : dB > nat).
thf(sy_c_Lambda_Olift, type,
    lift : dB > nat > dB).
thf(sy_c_Lambda_Oliftn, type,
    liftn : nat > dB > nat > dB).
thf(sy_c_Lambda_Osubst, type,
    subst : dB > dB > nat > dB).
thf(sy_c_Lambda_Osubstn, type,
    substn : dB > dB > nat > dB).
thf(sy_c_ListOrder_Ostep1_001t__Lambda__OdB, type,
    step1_dB : (dB > dB > $o) > list_dB > list_dB > $o).
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_Ogen__length_001t__Lambda__OdB, type,
    gen_length_dB : nat > list_dB > nat).
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_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_Omap_001t__Lambda__OdB_001t__Lambda__OdB, type,
    map_dB_dB : (dB > dB) > list_dB > list_dB).
thf(sy_c_List_Olistsp_001t__Lambda__OdB, type,
    listsp_dB : (dB > $o) > list_dB > $o).
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_Nat_OSuc, type,
    suc : nat > nat).
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_Ua____, type,
    ua : 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_r____, type,
    r : dB).
thf(sy_v_t____, type,
    t4 : dB).
thf(sy_v_u1____, type,
    u1 : dB).
thf(sy_v_u____, type,
    u : dB).

% Relevant facts (144)
thf(fact_0__092_060open_062IT_At_092_060close_062, axiom,
    ((it @ t4))). % \<open>IT t\<close>
thf(fact_1_Beta_Oprems_I2_J, axiom,
    ((it @ u1))). % Beta.prems(2)
thf(fact_2_uIT, axiom,
    ((it @ u))). % uIT
thf(fact_3_uT, axiom,
    ((typing @ e @ u @ t3))). % uT
thf(fact_4_Beta_Ohyps_I3_J, axiom,
    ((it @ a))). % Beta.hyps(3)
thf(fact_5__092_060open_062e_092_060langle_062i_058T_092_060rangle_062_A_092_060turnstile_062_Aa_A_058_AT_H_H_092_060close_062, axiom,
    ((typing @ (shift_type @ e @ i @ t3) @ a @ t))). % \<open>e\<langle>i:T\<rangle> \<turnstile> a : T''\<close>
thf(fact_6__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062T_H_H_O_Ae_092_060langle_062i_058T_092_060rangle_062_A_092_060turnstile_062_Aa_A_058_AT_H_H_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom,
    ((~ ((![T : type]: (~ ((typing @ (shift_type @ e @ i @ t3) @ a @ T)))))))). % \<open>\<And>thesis. (\<And>T''. e\<langle>i:T\<rangle> \<turnstile> a : T'' \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_7_MI1, axiom,
    ((![T1 : type, T2 : type, T3 : dB, E : nat > type, I : nat, T4 : type, U : dB]: ((t3 = (fun @ T1 @ T2)) => ((it @ T3) => ((typing @ (shift_type @ E @ I @ T1) @ T3 @ T4) => ((it @ U) => ((typing @ E @ U @ T1) => (it @ (subst @ T3 @ U @ I)))))))))). % MI1
thf(fact_8_MI2, axiom,
    ((![T1 : type, T2 : type, T3 : dB, E : nat > type, I : nat, T4 : type, U : dB]: ((t3 = (fun @ T1 @ T2)) => ((it @ T3) => ((typing @ (shift_type @ E @ I @ T2) @ T3 @ T4) => ((it @ U) => ((typing @ E @ U @ T2) => (it @ (subst @ T3 @ U @ I)))))))))). % MI2
thf(fact_9_subst__Var__IT, axiom,
    ((![R : dB, I : nat, J : nat]: ((it @ R) => (it @ (subst @ R @ (var @ I) @ J)))))). % subst_Var_IT
thf(fact_10_lift__IT, axiom,
    ((![T3 : dB, I : nat]: ((it @ T3) => (it @ (lift @ T3 @ I)))))). % lift_IT
thf(fact_11__092_060open_062IT_A_I_Ir_091a_P0_093_A_092_060degree_062_092_060degree_062_Aas_J_091u_Pi_093_J_092_060close_062, axiom,
    ((it @ (subst @ (foldl_dB_dB @ app @ (subst @ r @ a @ zero_zero_nat) @ as) @ u @ i)))). % \<open>IT ((r[a/0] \<degree>\<degree> as)[u/i])\<close>
thf(fact_12_SI2, axiom,
    ((![E : nat > type, I : nat, T5 : type, U : dB]: ((typing @ (shift_type @ E @ I @ t3) @ a @ T5) => ((it @ U) => ((typing @ E @ U @ t3) => (it @ (subst @ a @ U @ I)))))))). % SI2
thf(fact_13_Var__IT, axiom,
    ((![N : nat]: (it @ (var @ N))))). % Var_IT
thf(fact_14__092_060open_062e_092_060langle_062i_058T_092_060rangle_062_A_092_060turnstile_062_AAbs_Ar_A_092_060degree_062_Aa_A_058_AU_092_060close_062, axiom,
    ((typing @ (shift_type @ e @ i @ t3) @ (app @ (abs @ r) @ a) @ ua))). % \<open>e\<langle>i:T\<rangle> \<turnstile> Abs r \<degree> a : U\<close>
thf(fact_15_subst__lemma, axiom,
    ((![E : nat > type, T3 : dB, T4 : type, E2 : nat > type, U : dB, U2 : type, I : nat]: ((typing @ E @ T3 @ T4) => ((typing @ E2 @ U @ U2) => ((E = (shift_type @ E2 @ I @ U2)) => (typing @ E2 @ (subst @ T3 @ U @ I) @ T4))))))). % subst_lemma
thf(fact_16__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062U_O_Ae_092_060langle_062i_058T_092_060rangle_062_A_092_060turnstile_062_AAbs_Ar_A_092_060degree_062_Aa_A_058_AU_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom,
    ((~ ((![U3 : type]: (~ ((typing @ (shift_type @ e @ i @ t3) @ (app @ (abs @ r) @ a) @ U3)))))))). % \<open>\<And>thesis. (\<And>U. e\<langle>i:T\<rangle> \<turnstile> Abs r \<degree> a : U \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_17_Beta_Ohyps_I1_J, axiom,
    ((it @ (foldl_dB_dB @ app @ (subst @ r @ a @ zero_zero_nat) @ as)))). % Beta.hyps(1)
thf(fact_18_Beta_Oprems_I3_J, axiom,
    ((typing @ e1 @ u1 @ t3))). % Beta.prems(3)
thf(fact_19_shift__eq, axiom,
    ((![I : nat, J : nat, E : nat > type, T4 : type]: ((I = J) => ((shift_type @ E @ I @ T4 @ J) = T4))))). % shift_eq
thf(fact_20_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_21_T, axiom,
    ((typing @ (shift_type @ e @ i @ t3) @ (foldl_dB_dB @ app @ (app @ (abs @ r) @ a) @ as) @ t2))). % T
thf(fact_22_lift__type, axiom,
    ((![E : nat > type, T3 : dB, T4 : type, I : nat, U2 : type]: ((typing @ E @ T3 @ T4) => (typing @ (shift_type @ E @ I @ U2) @ (lift @ T3 @ I) @ T4))))). % lift_type
thf(fact_23_SI1, axiom,
    ((![E : nat > type, I : nat, T5 : type, U : dB]: ((typing @ (shift_type @ E @ I @ t3) @ (foldl_dB_dB @ app @ (subst @ r @ a @ zero_zero_nat) @ as) @ T5) => ((it @ U) => ((typing @ E @ U @ t3) => (it @ (subst @ (foldl_dB_dB @ app @ (subst @ r @ a @ zero_zero_nat) @ as) @ U @ I)))))))). % SI1
thf(fact_24_Beta_Oprems_I1_J, axiom,
    ((typing @ (shift_type @ e1 @ i1 @ t3) @ (foldl_dB_dB @ app @ (app @ (abs @ r) @ a) @ as) @ t_1))). % Beta.prems(1)
thf(fact_25__092_060open_062e_092_060langle_062i_058T_092_060rangle_062_A_092_060turnstile_062_Ar_091a_P0_093_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 @ (subst @ r @ a @ zero_zero_nat) @ as) @ t2))). % \<open>e\<langle>i:T\<rangle> \<turnstile> r[a/0] \<degree>\<degree> as : T'\<close>
thf(fact_26_typing__elims_I3_J, axiom,
    ((![E : nat > type, T3 : dB, T4 : type]: ((typing @ E @ (abs @ T3) @ T4) => (~ ((![T6 : type, U3 : type]: ((T4 = (fun @ T6 @ U3)) => (~ ((typing @ (shift_type @ E @ zero_zero_nat @ T6) @ T3 @ U3))))))))))). % typing_elims(3)
thf(fact_27_typing__elims_I2_J, axiom,
    ((![E : nat > type, T3 : dB, U : dB, T4 : type]: ((typing @ E @ (app @ T3 @ U) @ T4) => (~ ((![T6 : type]: ((typing @ E @ T3 @ (fun @ T6 @ T4)) => (~ ((typing @ E @ U @ T6))))))))))). % typing_elims(2)
thf(fact_28_typing__elims_I1_J, axiom,
    ((![E : nat > type, I : nat, T4 : type]: ((typing @ E @ (var @ I) @ T4) => ((E @ I) = T4))))). % typing_elims(1)
thf(fact_29_abs__typeE, axiom,
    ((![E : nat > type, T3 : dB, T4 : type]: ((typing @ E @ (abs @ T3) @ T4) => (~ ((![U3 : type, V : type]: (~ ((typing @ (shift_type @ E @ zero_zero_nat @ U3) @ T3 @ V)))))))))). % abs_typeE
thf(fact_30_Abs, axiom,
    ((![Env : nat > type, T4 : type, T3 : dB, U2 : type]: ((typing @ (shift_type @ Env @ zero_zero_nat @ T4) @ T3 @ U2) => (typing @ Env @ (abs @ T3) @ (fun @ T4 @ U2)))))). % Abs
thf(fact_31_App, axiom,
    ((![Env : nat > type, S : dB, T4 : type, U2 : type, T3 : dB]: ((typing @ Env @ S @ (fun @ T4 @ U2)) => ((typing @ Env @ T3 @ T4) => (typing @ Env @ (app @ S @ T3) @ U2)))))). % App
thf(fact_32_typing_OVar, axiom,
    ((![Env : nat > type, X : nat, T4 : type]: (((Env @ X) = T4) => (typing @ Env @ (var @ X) @ T4))))). % typing.Var
thf(fact_33_type__induct, axiom,
    ((![P : type > $o, T4 : type]: ((![T6 : type]: ((![T12 : type, T22 : type]: ((T6 = (fun @ T12 @ T22)) => (P @ T12))) => ((![T12 : type, T22 : type]: ((T6 = (fun @ T12 @ T22)) => (P @ T22))) => (P @ T6)))) => (P @ T4))))). % type_induct
thf(fact_34_typing_Ocases, axiom,
    ((![A1 : nat > type, A2 : dB, A3 : type]: ((typing @ A1 @ A2 @ A3) => ((![X2 : nat]: ((A2 = (var @ X2)) => (~ (((A1 @ X2) = A3))))) => ((![T6 : type, T7 : dB]: ((A2 = (abs @ T7)) => (![U3 : type]: ((A3 = (fun @ T6 @ U3)) => (~ ((typing @ (shift_type @ A1 @ zero_zero_nat @ T6) @ T7 @ U3))))))) => (~ ((![S2 : dB, T6 : type, U3 : type, T7 : dB]: ((A2 = (app @ S2 @ T7)) => ((A3 = U3) => ((typing @ A1 @ S2 @ (fun @ T6 @ U3)) => (~ ((typing @ A1 @ T7 @ T6))))))))))))))). % typing.cases
thf(fact_35_typing_Osimps, axiom,
    ((typing = (^[A12 : nat > type]: (^[A22 : dB]: (^[A32 : type]: (((?[Env2 : nat > type]: (?[X3 : nat]: (?[T8 : type]: (((A12 = Env2)) & ((((A22 = (var @ X3))) & ((((A32 = T8)) & (((Env2 @ X3) = T8))))))))))) | ((((?[Env2 : nat > type]: (?[T8 : type]: (?[T9 : dB]: (?[U4 : type]: (((A12 = Env2)) & ((((A22 = (abs @ T9))) & ((((A32 = (fun @ T8 @ U4))) & ((typing @ (shift_type @ Env2 @ zero_zero_nat @ T8) @ T9 @ U4)))))))))))) | ((?[Env2 : nat > type]: (?[S3 : dB]: (?[T8 : type]: (?[U4 : type]: (?[T9 : dB]: (((A12 = Env2)) & ((((A22 = (app @ S3 @ T9))) & ((((A32 = U4)) & ((((typing @ Env2 @ S3 @ (fun @ T8 @ U4))) & ((typing @ Env2 @ T9 @ T8)))))))))))))))))))))))). % typing.simps
thf(fact_36_typing_Oinducts, axiom,
    ((![X1 : nat > type, X23 : dB, X32 : type, P : (nat > type) > dB > type > $o]: ((typing @ X1 @ X23 @ X32) => ((![Env3 : nat > type, X2 : nat, T6 : type]: (((Env3 @ X2) = T6) => (P @ Env3 @ (var @ X2) @ T6))) => ((![Env3 : nat > type, T6 : type, T7 : dB, U3 : type]: ((typing @ (shift_type @ Env3 @ zero_zero_nat @ T6) @ T7 @ U3) => ((P @ (shift_type @ Env3 @ zero_zero_nat @ T6) @ T7 @ U3) => (P @ Env3 @ (abs @ T7) @ (fun @ T6 @ U3))))) => ((![Env3 : nat > type, S2 : dB, T6 : type, U3 : type, T7 : dB]: ((typing @ Env3 @ S2 @ (fun @ T6 @ U3)) => ((P @ Env3 @ S2 @ (fun @ T6 @ U3)) => ((typing @ Env3 @ T7 @ T6) => ((P @ Env3 @ T7 @ T6) => (P @ Env3 @ (app @ S2 @ T7) @ U3)))))) => (P @ X1 @ X23 @ X32)))))))). % typing.inducts
thf(fact_37_var__app__type__eq, axiom,
    ((![E : nat > type, I : nat, Ts : list_dB, T4 : type, U2 : type]: ((typing @ E @ (foldl_dB_dB @ app @ (var @ I) @ Ts) @ T4) => ((typing @ E @ (foldl_dB_dB @ app @ (var @ I) @ Ts) @ U2) => (T4 = U2)))))). % var_app_type_eq
thf(fact_38_app__Var__IT, axiom,
    ((![T3 : dB, I : nat]: ((it @ T3) => (it @ (app @ T3 @ (var @ I))))))). % app_Var_IT
thf(fact_39_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_40_Abs__apps__eq__Abs__apps__conv, axiom,
    ((![R : dB, Rs : list_dB, S : dB, Ss : list_dB]: (((foldl_dB_dB @ app @ (abs @ R) @ Rs) = (foldl_dB_dB @ app @ (abs @ S) @ Ss)) = (((R = S)) & ((Rs = Ss))))))). % Abs_apps_eq_Abs_apps_conv
thf(fact_41__092_060open_062Abs_Ar_A_092_060degree_062_Aa_A_092_060degree_062_092_060degree_062_Aas_A_092_060rightarrow_062_092_060_094sub_062_092_060beta_062_Ar_091a_P0_093_A_092_060degree_062_092_060degree_062_Aas_092_060close_062, axiom,
    ((beta @ (foldl_dB_dB @ app @ (app @ (abs @ r) @ a) @ as) @ (foldl_dB_dB @ app @ (subst @ r @ a @ zero_zero_nat) @ as)))). % \<open>Abs r \<degree> a \<degree>\<degree> as \<rightarrow>\<^sub>\<beta> r[a/0] \<degree>\<degree> as\<close>
thf(fact_42_IT_OBeta, axiom,
    ((![R : dB, S : dB, Ss : list_dB]: ((it @ (foldl_dB_dB @ app @ (subst @ R @ S @ zero_zero_nat) @ Ss)) => ((it @ S) => (it @ (foldl_dB_dB @ app @ (app @ (abs @ R) @ S) @ Ss))))))). % IT.Beta
thf(fact_43_subst__lift, axiom,
    ((![T3 : dB, K : nat, S : dB]: ((subst @ (lift @ T3 @ K) @ S @ K) = T3)))). % subst_lift
thf(fact_44_subst__eq, axiom,
    ((![K : nat, U : dB]: ((subst @ (var @ K) @ U @ K) = U)))). % subst_eq
thf(fact_45_apps__eq__tail__conv, axiom,
    ((![R : dB, Ts : list_dB, S : dB]: (((foldl_dB_dB @ app @ R @ Ts) = (foldl_dB_dB @ app @ S @ Ts)) = (R = S))))). % apps_eq_tail_conv
thf(fact_46_ex__head__tail, axiom,
    ((![T3 : dB]: (?[Ts2 : list_dB, H : dB]: ((T3 = (foldl_dB_dB @ app @ H @ Ts2)) & ((?[N2 : nat]: (H = (var @ N2))) | (?[U5 : dB]: (H = (abs @ U5))))))))). % ex_head_tail
thf(fact_47_Abs__App__neq__Var__apps, axiom,
    ((![S : dB, T3 : dB, N : nat, Ss : list_dB]: (~ (((app @ (abs @ S) @ T3) = (foldl_dB_dB @ app @ (var @ N) @ Ss))))))). % Abs_App_neq_Var_apps
thf(fact_48_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_49_dB_Oinject_I3_J, axiom,
    ((![X32 : dB, Y3 : dB]: (((abs @ X32) = (abs @ Y3)) = (X32 = Y3))))). % dB.inject(3)
thf(fact_50_dB_Oinject_I1_J, axiom,
    ((![X1 : nat, Y1 : nat]: (((var @ X1) = (var @ Y1)) = (X1 = Y1))))). % dB.inject(1)
thf(fact_51_appR, axiom,
    ((![S : dB, T3 : dB, U : dB]: ((beta @ S @ T3) => (beta @ (app @ U @ S) @ (app @ U @ T3)))))). % appR
thf(fact_52_appL, axiom,
    ((![S : dB, T3 : dB, U : dB]: ((beta @ S @ T3) => (beta @ (app @ S @ U) @ (app @ T3 @ U)))))). % appL
thf(fact_53_abs, axiom,
    ((![S : dB, T3 : dB]: ((beta @ S @ T3) => (beta @ (abs @ S) @ (abs @ T3)))))). % abs
thf(fact_54_beta__cases_I2_J, axiom,
    ((![R : dB, S : dB]: ((beta @ (abs @ R) @ S) => (~ ((![T7 : dB]: ((S = (abs @ T7)) => (~ ((beta @ R @ T7))))))))))). % beta_cases(2)
thf(fact_55_subst__preserves__beta, axiom,
    ((![R : dB, S : dB, T3 : dB, I : nat]: ((beta @ R @ S) => (beta @ (subst @ R @ T3 @ I) @ (subst @ S @ T3 @ I)))))). % subst_preserves_beta
thf(fact_56_beta__cases_I1_J, axiom,
    ((![I : nat, T3 : dB]: (~ ((beta @ (var @ I) @ T3)))))). % beta_cases(1)
thf(fact_57_lift__preserves__beta, axiom,
    ((![R : dB, S : dB, I : nat]: ((beta @ R @ S) => (beta @ (lift @ R @ I) @ (lift @ S @ I)))))). % lift_preserves_beta
thf(fact_58_subject__reduction, axiom,
    ((![E : nat > type, T3 : dB, T4 : type, T10 : dB]: ((typing @ E @ T3 @ T4) => ((beta @ T3 @ T10) => (typing @ E @ T10 @ T4)))))). % subject_reduction
thf(fact_59_beta_Oinducts, axiom,
    ((![X1 : dB, X23 : dB, P : dB > dB > $o]: ((beta @ X1 @ X23) => ((![S2 : dB, T7 : dB]: (P @ (app @ (abs @ S2) @ T7) @ (subst @ S2 @ T7 @ zero_zero_nat))) => ((![S2 : dB, T7 : dB, U5 : dB]: ((beta @ S2 @ T7) => ((P @ S2 @ T7) => (P @ (app @ S2 @ U5) @ (app @ T7 @ U5))))) => ((![S2 : dB, T7 : dB, U5 : dB]: ((beta @ S2 @ T7) => ((P @ S2 @ T7) => (P @ (app @ U5 @ S2) @ (app @ U5 @ T7))))) => ((![S2 : dB, T7 : dB]: ((beta @ S2 @ T7) => ((P @ S2 @ T7) => (P @ (abs @ S2) @ (abs @ T7))))) => (P @ X1 @ X23))))))))). % beta.inducts
thf(fact_60_beta_Osimps, axiom,
    ((beta = (^[A12 : dB]: (^[A22 : dB]: (((?[S3 : dB]: (?[T9 : dB]: (((A12 = (app @ (abs @ S3) @ T9))) & ((A22 = (subst @ S3 @ T9 @ zero_zero_nat))))))) | ((((?[S3 : dB]: (?[T9 : dB]: (?[U6 : dB]: (((A12 = (app @ S3 @ U6))) & ((((A22 = (app @ T9 @ U6))) & ((beta @ S3 @ T9))))))))) | ((((?[S3 : dB]: (?[T9 : dB]: (?[U6 : dB]: (((A12 = (app @ U6 @ S3))) & ((((A22 = (app @ U6 @ T9))) & ((beta @ S3 @ T9))))))))) | ((?[S3 : dB]: (?[T9 : dB]: (((A12 = (abs @ S3))) & ((((A22 = (abs @ T9))) & ((beta @ S3 @ T9)))))))))))))))))). % beta.simps
thf(fact_61_beta_Ocases, axiom,
    ((![A1 : dB, A2 : dB]: ((beta @ A1 @ A2) => ((![S2 : dB, T7 : dB]: ((A1 = (app @ (abs @ S2) @ T7)) => (~ ((A2 = (subst @ S2 @ T7 @ zero_zero_nat)))))) => ((![S2 : dB, T7 : dB, U5 : dB]: ((A1 = (app @ S2 @ U5)) => ((A2 = (app @ T7 @ U5)) => (~ ((beta @ S2 @ T7)))))) => ((![S2 : dB, T7 : dB, U5 : dB]: ((A1 = (app @ U5 @ S2)) => ((A2 = (app @ U5 @ T7)) => (~ ((beta @ S2 @ T7)))))) => (~ ((![S2 : dB]: ((A1 = (abs @ S2)) => (![T7 : dB]: ((A2 = (abs @ T7)) => (~ ((beta @ S2 @ T7)))))))))))))))). % beta.cases
thf(fact_62_beta, axiom,
    ((![S : dB, T3 : dB]: (beta @ (app @ (abs @ S) @ T3) @ (subst @ S @ T3 @ zero_zero_nat))))). % beta
thf(fact_63_beta__cases_I3_J, axiom,
    ((![S : dB, T3 : dB, U : dB]: ((beta @ (app @ S @ T3) @ U) => ((![S2 : dB]: ((S = (abs @ S2)) => (~ ((U = (subst @ S2 @ T3 @ zero_zero_nat)))))) => ((![T7 : dB]: ((U = (app @ T7 @ T3)) => (~ ((beta @ S @ T7))))) => (~ ((![T7 : dB]: ((U = (app @ S @ T7)) => (~ ((beta @ T3 @ T7))))))))))))). % beta_cases(3)
thf(fact_64_dB_Odistinct_I5_J, axiom,
    ((![X21 : dB, X22 : dB, X32 : dB]: (~ (((app @ X21 @ X22) = (abs @ X32))))))). % dB.distinct(5)
thf(fact_65_subst__App, axiom,
    ((![T3 : dB, U : dB, S : dB, K : nat]: ((subst @ (app @ T3 @ U) @ S @ K) = (app @ (subst @ T3 @ S @ K) @ (subst @ U @ S @ K)))))). % subst_App
thf(fact_66_dB_Odistinct_I1_J, axiom,
    ((![X1 : nat, X21 : dB, X22 : dB]: (~ (((var @ X1) = (app @ X21 @ X22))))))). % dB.distinct(1)
thf(fact_67_dB_Odistinct_I3_J, axiom,
    ((![X1 : nat, X32 : dB]: (~ (((var @ X1) = (abs @ X32))))))). % dB.distinct(3)
thf(fact_68_lift_Osimps_I2_J, axiom,
    ((![S : dB, T3 : dB, K : nat]: ((lift @ (app @ S @ T3) @ K) = (app @ (lift @ S @ K) @ (lift @ T3 @ K)))))). % lift.simps(2)
thf(fact_69_Lambda, axiom,
    ((![R : dB]: ((it @ R) => (it @ (abs @ R)))))). % Lambda
thf(fact_70_dB_Oexhaust, axiom,
    ((![Y : dB]: ((![X12 : nat]: (~ ((Y = (var @ X12))))) => ((![X212 : dB, X222 : dB]: (~ ((Y = (app @ X212 @ X222))))) => (~ ((![X33 : dB]: (~ ((Y = (abs @ X33)))))))))))). % dB.exhaust
thf(fact_71_dB_Oinduct, axiom,
    ((![P : dB > $o, DB : dB]: ((![X2 : nat]: (P @ (var @ X2))) => ((![X1a : dB, X24 : dB]: ((P @ X1a) => ((P @ X24) => (P @ (app @ X1a @ X24))))) => ((![X2 : dB]: ((P @ X2) => (P @ (abs @ X2)))) => (P @ DB))))))). % dB.induct
thf(fact_72_Var__apps__neq__Abs__apps, axiom,
    ((![N : nat, Ts : list_dB, R : dB, Ss : list_dB]: (~ (((foldl_dB_dB @ app @ (var @ N) @ Ts) = (foldl_dB_dB @ app @ (abs @ R) @ Ss))))))). % Var_apps_neq_Abs_apps
thf(fact_73_apps__preserves__beta, axiom,
    ((![R : dB, S : dB, Ss : list_dB]: ((beta @ R @ S) => (beta @ (foldl_dB_dB @ app @ R @ Ss) @ (foldl_dB_dB @ app @ S @ Ss)))))). % apps_preserves_beta
thf(fact_74__092_060open_062IT_A_Ir_091lift_Au_A0_PSuc_Ai_093_091a_091u_Pi_093_P0_093_A_092_060degree_062_092_060degree_062_Amap_A_I_092_060lambda_062t_O_At_091u_Pi_093_J_Aas_J_092_060close_062, axiom,
    ((it @ (foldl_dB_dB @ app @ (subst @ (subst @ r @ (lift @ u @ zero_zero_nat) @ (suc @ i)) @ (subst @ a @ u @ i) @ zero_zero_nat) @ (map_dB_dB @ (^[T9 : dB]: (subst @ T9 @ u @ i)) @ as))))). % \<open>IT (r[lift u 0/Suc i][a[u/i]/0] \<degree>\<degree> map (\<lambda>t. t[u/i]) as)\<close>
thf(fact_75_IT_Oinducts, axiom,
    ((![X : dB, P : dB > $o]: ((it @ X) => ((![Rs2 : list_dB, N2 : nat]: ((listsp_dB @ (^[X3 : dB]: (((it @ X3)) & ((P @ X3)))) @ Rs2) => (P @ (foldl_dB_dB @ app @ (var @ N2) @ Rs2)))) => ((![R2 : dB]: ((it @ R2) => ((P @ R2) => (P @ (abs @ R2))))) => ((![R2 : dB, S2 : dB, Ss2 : list_dB]: ((it @ (foldl_dB_dB @ app @ (subst @ R2 @ S2 @ zero_zero_nat) @ Ss2)) => ((P @ (foldl_dB_dB @ app @ (subst @ R2 @ S2 @ zero_zero_nat) @ Ss2)) => ((it @ S2) => ((P @ S2) => (P @ (foldl_dB_dB @ app @ (app @ (abs @ R2) @ S2) @ Ss2))))))) => (P @ X)))))))). % IT.inducts
thf(fact_76_IT_Osimps, axiom,
    ((it = (^[A : dB]: (((?[Rs3 : list_dB]: (?[N3 : nat]: (((A = (foldl_dB_dB @ app @ (var @ N3) @ Rs3))) & ((listsp_dB @ it @ Rs3)))))) | ((((?[R3 : dB]: (((A = (abs @ R3))) & ((it @ R3))))) | ((?[R3 : dB]: (?[S3 : dB]: (?[Ss3 : list_dB]: (((A = (foldl_dB_dB @ app @ (app @ (abs @ R3) @ S3) @ Ss3))) & ((((it @ (foldl_dB_dB @ app @ (subst @ R3 @ S3 @ zero_zero_nat) @ Ss3))) & ((it @ S3)))))))))))))))). % IT.simps
thf(fact_77_IT_Ocases, axiom,
    ((![A4 : dB]: ((it @ A4) => ((![Rs2 : list_dB]: ((?[N2 : nat]: (A4 = (foldl_dB_dB @ app @ (var @ N2) @ Rs2))) => (~ ((listsp_dB @ it @ Rs2))))) => ((![R2 : dB]: ((A4 = (abs @ R2)) => (~ ((it @ R2))))) => (~ ((![R2 : dB, S2 : dB, Ss2 : list_dB]: ((A4 = (foldl_dB_dB @ app @ (app @ (abs @ R2) @ S2) @ Ss2)) => ((it @ (foldl_dB_dB @ app @ (subst @ R2 @ S2 @ zero_zero_nat) @ Ss2)) => (~ ((it @ S2)))))))))))))). % IT.cases
thf(fact_78_shift__commute, axiom,
    ((![E : nat > type, I : nat, U2 : type, T4 : type]: ((shift_type @ (shift_type @ E @ I @ U2) @ zero_zero_nat @ T4) = (shift_type @ (shift_type @ E @ zero_zero_nat @ T4) @ (suc @ I) @ U2))))). % shift_commute
thf(fact_79_subst__map, axiom,
    ((![T3 : dB, Ts : list_dB, U : dB, I : nat]: ((subst @ (foldl_dB_dB @ app @ T3 @ Ts) @ U @ I) = (foldl_dB_dB @ app @ (subst @ T3 @ U @ I) @ (map_dB_dB @ (^[T9 : dB]: (subst @ T9 @ U @ I)) @ Ts)))))). % subst_map
thf(fact_80_lift__map, axiom,
    ((![T3 : dB, Ts : list_dB, I : nat]: ((lift @ (foldl_dB_dB @ app @ T3 @ Ts) @ I) = (foldl_dB_dB @ app @ (lift @ T3 @ I) @ (map_dB_dB @ (^[T9 : dB]: (lift @ T9 @ I)) @ Ts)))))). % lift_map
thf(fact_81_lifts__IT, axiom,
    ((![Ts : list_dB]: ((listsp_dB @ it @ Ts) => (listsp_dB @ it @ (map_dB_dB @ (^[T9 : dB]: (lift @ T9 @ zero_zero_nat)) @ Ts)))))). % lifts_IT
thf(fact_82_zero__reorient, axiom,
    ((![X : nat]: ((zero_zero_nat = X) = (X = zero_zero_nat))))). % zero_reorient
thf(fact_83_IT_OVar, axiom,
    ((![Rs : list_dB, N : nat]: ((listsp_dB @ it @ Rs) => (it @ (foldl_dB_dB @ app @ (var @ N) @ Rs)))))). % IT.Var
thf(fact_84_listsp__conj__eq, axiom,
    ((![A5 : dB > $o, B : dB > $o]: ((listsp_dB @ (^[X3 : dB]: (((A5 @ X3)) & ((B @ X3))))) = (^[X3 : list_dB]: (((listsp_dB @ A5 @ X3)) & ((listsp_dB @ B @ X3)))))))). % listsp_conj_eq
thf(fact_85_map__ident, axiom,
    (((map_dB_dB @ (^[X3 : dB]: X3)) = (^[Xs : list_dB]: Xs)))). % map_ident
thf(fact_86_foldl__map, axiom,
    ((![G : dB > dB > dB, A4 : dB, F : dB > dB, Xs2 : list_dB]: ((foldl_dB_dB @ G @ A4 @ (map_dB_dB @ F @ Xs2)) = (foldl_dB_dB @ (^[A : dB]: (^[X3 : dB]: (G @ A @ (F @ X3)))) @ A4 @ Xs2))))). % foldl_map
thf(fact_87_old_Onat_Oinject, axiom,
    ((![Nat : nat, Nat2 : nat]: (((suc @ Nat) = (suc @ Nat2)) = (Nat = Nat2))))). % old.nat.inject
thf(fact_88_nat_Oinject, axiom,
    ((![X23 : nat, Y2 : nat]: (((suc @ X23) = (suc @ Y2)) = (X23 = Y2))))). % nat.inject
thf(fact_89_n__not__Suc__n, axiom,
    ((![N : nat]: (~ ((N = (suc @ N))))))). % n_not_Suc_n
thf(fact_90_Suc__inject, axiom,
    ((![X : nat, Y : nat]: (((suc @ X) = (suc @ Y)) => (X = Y))))). % Suc_inject
thf(fact_91_list_Omap__ident, axiom,
    ((![T3 : list_dB]: ((map_dB_dB @ (^[X3 : dB]: X3) @ T3) = T3)))). % list.map_ident
thf(fact_92_nat_Odistinct_I1_J, axiom,
    ((![X23 : nat]: (~ ((zero_zero_nat = (suc @ X23))))))). % nat.distinct(1)
thf(fact_93_old_Onat_Odistinct_I2_J, axiom,
    ((![Nat2 : nat]: (~ (((suc @ Nat2) = zero_zero_nat)))))). % old.nat.distinct(2)
thf(fact_94_old_Onat_Odistinct_I1_J, axiom,
    ((![Nat2 : nat]: (~ ((zero_zero_nat = (suc @ Nat2))))))). % old.nat.distinct(1)
thf(fact_95_nat_OdiscI, axiom,
    ((![Nat : nat, X23 : nat]: ((Nat = (suc @ X23)) => (~ ((Nat = zero_zero_nat))))))). % nat.discI
thf(fact_96_nat__induct, axiom,
    ((![P : nat > $o, N : nat]: ((P @ zero_zero_nat) => ((![N2 : nat]: ((P @ N2) => (P @ (suc @ N2)))) => (P @ N)))))). % nat_induct
thf(fact_97_diff__induct, axiom,
    ((![P : nat > nat > $o, M : nat, N : nat]: ((![X2 : nat]: (P @ X2 @ zero_zero_nat)) => ((![Y4 : nat]: (P @ zero_zero_nat @ (suc @ Y4))) => ((![X2 : nat, Y4 : nat]: ((P @ X2 @ Y4) => (P @ (suc @ X2) @ (suc @ Y4)))) => (P @ M @ N))))))). % diff_induct
thf(fact_98_zero__induct, axiom,
    ((![P : nat > $o, K : nat]: ((P @ K) => ((![N2 : nat]: ((P @ (suc @ N2)) => (P @ N2))) => (P @ zero_zero_nat)))))). % zero_induct
thf(fact_99_Suc__neq__Zero, axiom,
    ((![M : nat]: (~ (((suc @ M) = zero_zero_nat)))))). % Suc_neq_Zero
thf(fact_100_Zero__neq__Suc, axiom,
    ((![M : nat]: (~ ((zero_zero_nat = (suc @ M))))))). % Zero_neq_Suc
thf(fact_101_Zero__not__Suc, axiom,
    ((![M : nat]: (~ ((zero_zero_nat = (suc @ M))))))). % Zero_not_Suc
thf(fact_102_old_Onat_Oexhaust, axiom,
    ((![Y : nat]: ((~ ((Y = zero_zero_nat))) => (~ ((![Nat3 : nat]: (~ ((Y = (suc @ Nat3))))))))))). % old.nat.exhaust
thf(fact_103_old_Onat_Oinducts, axiom,
    ((![P : nat > $o, Nat : nat]: ((P @ zero_zero_nat) => ((![Nat3 : nat]: ((P @ Nat3) => (P @ (suc @ Nat3)))) => (P @ Nat)))))). % old.nat.inducts
thf(fact_104_not0__implies__Suc, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (?[M2 : nat]: (N = (suc @ M2))))))). % not0_implies_Suc
thf(fact_105_substs__lemma, axiom,
    ((![E : nat > type, U : dB, T4 : type, I : nat, Ts : list_dB, Ts3 : list_type]: ((typing @ E @ U @ T4) => ((typings @ (shift_type @ E @ I @ T4) @ Ts @ Ts3) => (typings @ E @ (map_dB_dB @ (^[T9 : dB]: (subst @ T9 @ U @ I)) @ Ts) @ Ts3)))))). % substs_lemma
thf(fact_106_lift__types, axiom,
    ((![E : nat > type, Ts : list_dB, Ts3 : list_type, I : nat, U2 : type]: ((typings @ E @ Ts @ Ts3) => (typings @ (shift_type @ E @ I @ U2) @ (map_dB_dB @ (^[T9 : dB]: (lift @ T9 @ I)) @ Ts) @ Ts3))))). % lift_types
thf(fact_107_var__app__types, axiom,
    ((![E : nat > type, I : nat, Ts : list_dB, Us : list_dB, T4 : type, Ts3 : list_type, U2 : type]: ((typing @ E @ (foldl_dB_dB @ app @ (foldl_dB_dB @ app @ (var @ I) @ Ts) @ Us) @ T4) => ((typings @ E @ Ts @ Ts3) => ((typing @ E @ (foldl_dB_dB @ app @ (var @ I) @ Ts) @ U2) => (?[Us2 : list_type]: ((U2 = (foldr_type_type @ fun @ Us2 @ T4)) & (typings @ E @ Us @ Us2))))))))). % var_app_types
thf(fact_108_var__app__typesE, axiom,
    ((![E : nat > type, I : nat, Ts : list_dB, T4 : type]: ((typing @ E @ (foldl_dB_dB @ app @ (var @ I) @ Ts) @ T4) => (~ ((![Ts4 : list_type]: ((typing @ E @ (var @ I) @ (foldr_type_type @ fun @ Ts4 @ T4)) => (~ ((typings @ E @ Ts @ Ts4))))))))))). % var_app_typesE
thf(fact_109_list__app__typeI, axiom,
    ((![E : nat > type, T3 : dB, Ts3 : list_type, T4 : type, Ts : list_dB]: ((typing @ E @ T3 @ (foldr_type_type @ fun @ Ts3 @ T4)) => ((typings @ E @ Ts @ Ts3) => (typing @ E @ (foldl_dB_dB @ app @ T3 @ Ts) @ T4)))))). % list_app_typeI
thf(fact_110_list__app__typeE, axiom,
    ((![E : nat > type, T3 : dB, Ts : list_dB, T4 : type]: ((typing @ E @ (foldl_dB_dB @ app @ T3 @ Ts) @ T4) => (~ ((![Ts4 : list_type]: ((typing @ E @ T3 @ (foldr_type_type @ fun @ Ts4 @ T4)) => (~ ((typings @ E @ Ts @ Ts4))))))))))). % list_app_typeE
thf(fact_111_list__app__typeD, axiom,
    ((![E : nat > type, T3 : dB, Ts : list_dB, T4 : type]: ((typing @ E @ (foldl_dB_dB @ app @ T3 @ Ts) @ T4) => (?[Ts4 : list_type]: ((typing @ E @ T3 @ (foldr_type_type @ fun @ Ts4 @ T4)) & (typings @ E @ Ts @ Ts4))))))). % list_app_typeD
thf(fact_112_liftn__lift, axiom,
    ((![N : nat, T3 : dB, K : nat]: ((liftn @ (suc @ N) @ T3 @ K) = (lift @ (liftn @ N @ T3 @ K) @ K))))). % liftn_lift
thf(fact_113_head__Var__reduction, axiom,
    ((![N : nat, Rs : list_dB, V2 : dB]: ((beta @ (foldl_dB_dB @ app @ (var @ N) @ Rs) @ V2) => (?[Ss2 : list_dB]: ((step1_dB @ beta @ Rs @ Ss2) & (V2 = (foldl_dB_dB @ app @ (var @ N) @ Ss2)))))))). % head_Var_reduction
thf(fact_114_liftn__0, axiom,
    ((![T3 : dB, K : nat]: ((liftn @ zero_zero_nat @ T3 @ K) = T3)))). % liftn_0
thf(fact_115_liftn_Osimps_I2_J, axiom,
    ((![N : nat, S : dB, T3 : dB, K : nat]: ((liftn @ N @ (app @ S @ T3) @ K) = (app @ (liftn @ N @ S @ K) @ (liftn @ N @ T3 @ K)))))). % liftn.simps(2)
thf(fact_116_apps__preserves__betas, axiom,
    ((![Rs : list_dB, Ss : list_dB, R : dB]: ((step1_dB @ beta @ Rs @ Ss) => (beta @ (foldl_dB_dB @ app @ R @ Rs) @ (foldl_dB_dB @ app @ R @ Ss)))))). % apps_preserves_betas
thf(fact_117_substn__subst__n, axiom,
    ((substn = (^[T9 : dB]: (^[S3 : dB]: (^[N3 : nat]: (subst @ T9 @ (liftn @ N3 @ S3 @ zero_zero_nat) @ N3))))))). % substn_subst_n
thf(fact_118_apps__betasE, axiom,
    ((![R : dB, Rs : list_dB, S : dB]: ((beta @ (foldl_dB_dB @ app @ R @ Rs) @ S) => ((![R4 : dB]: ((beta @ R @ R4) => (~ ((S = (foldl_dB_dB @ app @ R4 @ Rs)))))) => ((![Rs4 : list_dB]: ((step1_dB @ beta @ Rs @ Rs4) => (~ ((S = (foldl_dB_dB @ app @ R @ Rs4)))))) => (~ ((![T7 : dB]: ((R = (abs @ T7)) => (![U5 : dB, Us3 : list_dB]: ((Rs = (cons_dB @ U5 @ Us3)) => (~ ((S = (foldl_dB_dB @ app @ (subst @ T7 @ U5 @ zero_zero_nat) @ Us3)))))))))))))))). % apps_betasE
thf(fact_119_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_120_substn_Osimps_I2_J, axiom,
    ((![T3 : dB, U : dB, S : dB, K : nat]: ((substn @ (app @ T3 @ U) @ S @ K) = (app @ (substn @ T3 @ S @ K) @ (substn @ U @ S @ K)))))). % substn.simps(2)
thf(fact_121_foldl__Cons, axiom,
    ((![F : dB > dB > dB, A4 : dB, X : dB, Xs2 : list_dB]: ((foldl_dB_dB @ F @ A4 @ (cons_dB @ X @ Xs2)) = (foldl_dB_dB @ F @ (F @ A4 @ X) @ Xs2))))). % foldl_Cons
thf(fact_122_not__Cons__self2, axiom,
    ((![X : dB, Xs2 : list_dB]: (~ (((cons_dB @ X @ Xs2) = Xs2)))))). % not_Cons_self2
thf(fact_123_map__eq__Cons__conv, axiom,
    ((![F : dB > dB, Xs2 : list_dB, Y : dB, Ys : list_dB]: (((map_dB_dB @ F @ Xs2) = (cons_dB @ Y @ Ys)) = (?[Z : dB]: (?[Zs : list_dB]: (((Xs2 = (cons_dB @ Z @ Zs))) & (((((F @ Z) = Y)) & (((map_dB_dB @ F @ Zs) = Ys))))))))))). % map_eq_Cons_conv
thf(fact_124_Cons__eq__map__conv, axiom,
    ((![X : dB, Xs2 : list_dB, F : dB > dB, Ys : list_dB]: (((cons_dB @ X @ Xs2) = (map_dB_dB @ F @ Ys)) = (?[Z : dB]: (?[Zs : list_dB]: (((Ys = (cons_dB @ Z @ Zs))) & ((((X = (F @ Z))) & ((Xs2 = (map_dB_dB @ F @ Zs)))))))))))). % Cons_eq_map_conv
thf(fact_125_map__eq__Cons__D, axiom,
    ((![F : dB > dB, Xs2 : list_dB, Y : dB, Ys : list_dB]: (((map_dB_dB @ F @ Xs2) = (cons_dB @ Y @ Ys)) => (?[Z2 : dB, Zs2 : list_dB]: ((Xs2 = (cons_dB @ Z2 @ Zs2)) & (((F @ Z2) = Y) & ((map_dB_dB @ F @ Zs2) = Ys)))))))). % map_eq_Cons_D
thf(fact_126_Cons__eq__map__D, axiom,
    ((![X : dB, Xs2 : list_dB, F : dB > dB, Ys : list_dB]: (((cons_dB @ X @ Xs2) = (map_dB_dB @ F @ Ys)) => (?[Z2 : dB, Zs2 : list_dB]: ((Ys = (cons_dB @ Z2 @ Zs2)) & ((X = (F @ Z2)) & (Xs2 = (map_dB_dB @ F @ Zs2))))))))). % Cons_eq_map_D
thf(fact_127_listsp_OCons, axiom,
    ((![A5 : dB > $o, A4 : dB, L : list_dB]: ((A5 @ A4) => ((listsp_dB @ A5 @ L) => (listsp_dB @ A5 @ (cons_dB @ A4 @ L))))))). % listsp.Cons
thf(fact_128_listspE, axiom,
    ((![A5 : dB > $o, X : dB, L : list_dB]: ((listsp_dB @ A5 @ (cons_dB @ X @ L)) => (~ (((A5 @ X) => (~ ((listsp_dB @ A5 @ L)))))))))). % listspE
thf(fact_129_listsp__simps_I2_J, axiom,
    ((![A5 : dB > $o, X : dB, Xs2 : list_dB]: ((listsp_dB @ A5 @ (cons_dB @ X @ Xs2)) = (((A5 @ X)) & ((listsp_dB @ A5 @ Xs2))))))). % listsp_simps(2)
thf(fact_130_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_131_substn__subst__0, axiom,
    ((![T3 : dB, S : dB]: ((substn @ T3 @ S @ zero_zero_nat) = (subst @ T3 @ S @ zero_zero_nat))))). % substn_subst_0
thf(fact_132_typings_Osimps_I2_J, axiom,
    ((![E : nat > type, T3 : dB, Ts : list_dB, Ts3 : list_type]: ((typings @ E @ (cons_dB @ T3 @ Ts) @ Ts3) = (case_list_o_type @ $false @ (^[T8 : type]: (^[Ts5 : list_type]: (((typing @ E @ T3 @ T8)) & ((typings @ E @ Ts @ Ts5))))) @ Ts3))))). % typings.simps(2)
thf(fact_133_gen__length__code_I2_J, axiom,
    ((![N : nat, X : dB, Xs2 : list_dB]: ((gen_length_dB @ N @ (cons_dB @ X @ Xs2)) = (gen_length_dB @ (suc @ N) @ Xs2))))). % gen_length_code(2)
thf(fact_134_list_Osimps_I5_J, axiom,
    ((![F1 : $o, F2 : type > list_type > $o, X21 : type, X22 : list_type]: ((case_list_o_type @ F1 @ F2 @ (cons_type @ X21 @ X22)) = (F2 @ X21 @ X22))))). % list.simps(5)
thf(fact_135_list_Ocase__distrib, axiom,
    ((![H2 : $o > $o, F1 : $o, F2 : type > list_type > $o, List : list_type]: ((H2 @ (case_list_o_type @ F1 @ F2 @ List)) = (case_list_o_type @ (H2 @ F1) @ (^[X13 : type]: (^[X25 : list_type]: (H2 @ (F2 @ X13 @ X25)))) @ List))))). % list.case_distrib
thf(fact_136_dB_Osize__gen_I1_J, axiom,
    ((![X1 : nat]: ((size_dB @ (var @ X1)) = zero_zero_nat)))). % dB.size_gen(1)
thf(fact_137_map__eq__map__tailrec, axiom,
    ((map_dB_dB = map_tailrec_dB_dB))). % map_eq_map_tailrec
thf(fact_138_dB_Osize__gen_I3_J, axiom,
    ((![X32 : dB]: ((size_dB @ (abs @ X32)) = (plus_plus_nat @ (size_dB @ X32) @ (suc @ zero_zero_nat)))))). % dB.size_gen(3)
thf(fact_139_dB_Osize__gen_I2_J, axiom,
    ((![X21 : dB, X22 : dB]: ((size_dB @ (app @ X21 @ X22)) = (plus_plus_nat @ (plus_plus_nat @ (size_dB @ X21) @ (size_dB @ X22)) @ (suc @ zero_zero_nat)))))). % dB.size_gen(2)
thf(fact_140_add__right__cancel, axiom,
    ((![B2 : nat, A4 : nat, C : nat]: (((plus_plus_nat @ B2 @ A4) = (plus_plus_nat @ C @ A4)) = (B2 = C))))). % add_right_cancel
thf(fact_141_add__left__cancel, axiom,
    ((![A4 : nat, B2 : nat, C : nat]: (((plus_plus_nat @ A4 @ B2) = (plus_plus_nat @ A4 @ C)) = (B2 = C))))). % add_left_cancel
thf(fact_142_add_Oleft__neutral, axiom,
    ((![A4 : nat]: ((plus_plus_nat @ zero_zero_nat @ A4) = A4)))). % add.left_neutral
thf(fact_143_add_Oright__neutral, axiom,
    ((![A4 : nat]: ((plus_plus_nat @ A4 @ zero_zero_nat) = A4)))). % add.right_neutral

% Conjectures (1)
thf(conj_0, conjecture,
    ((it @ (subst @ a @ u @ i)))).
