% 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_77__5209994_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:37:22.006

% Could-be-implicit typings (5)
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__Lambda__OdB, type,
    dB : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).
thf(ty_n_t__Int__Oint, type,
    int : $tType).

% Explicit typings (30)
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint, type,
    zero_zero_int : int).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat, type,
    zero_zero_nat : nat).
thf(sy_c_If_001t__Nat__Onat, type,
    if_nat : $o > nat > nat > nat).
thf(sy_c_InductTermi_OIT, type,
    it : dB > $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_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_Ocan__select_001t__Lambda__OdB, type,
    can_select_dB : (dB > $o) > set_dB > $o).
thf(sy_c_List_Ocount__list_001t__Lambda__OdB, type,
    count_list_dB : list_dB > dB > nat).
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_Olist_OCons_001t__Lambda__OdB, type,
    cons_dB : dB > list_dB > list_dB).
thf(sy_c_List_Olist_ONil_001t__Lambda__OdB, type,
    nil_dB : list_dB).
thf(sy_c_List_Olist_Oset_001t__Lambda__OdB, type,
    set_dB2 : list_dB > set_dB).
thf(sy_c_List_Olist__ex1_001t__Lambda__OdB, type,
    list_ex1_dB : (dB > $o) > list_dB > $o).
thf(sy_c_List_Olistsp_001t__Lambda__OdB, type,
    listsp_dB : (dB > $o) > list_dB > $o).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint, type,
    semiri2019852685at_int : nat > int).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat, type,
    semiri1382578993at_nat : nat > nat).
thf(sy_c_Nat_Osize__class_Osize_001t__Lambda__OdB, type,
    size_size_dB : dB > nat).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Lambda__OdB_J, type,
    size_size_list_dB : list_dB > nat).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint, type,
    ord_less_int : int > int > $o).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat, type,
    ord_less_nat : nat > nat > $o).
thf(sy_c_member_001t__Lambda__OdB, type,
    member_dB : dB > set_dB > $o).
thf(sy_v_i, type,
    i : nat).
thf(sy_v_r, type,
    r : dB).

% Relevant facts (142)
thf(fact_0_Var__IT, axiom,
    ((![N : nat]: (it @ (var @ N))))). % Var_IT
thf(fact_1_dB_Oinduct, axiom,
    ((![P : dB > $o, DB : dB]: ((![X : nat]: (P @ (var @ X))) => ((![X1a : dB, X2 : dB]: ((P @ X1a) => ((P @ X2) => (P @ (app @ X1a @ X2))))) => ((![X : dB]: ((P @ X) => (P @ (abs @ X)))) => (P @ DB))))))). % dB.induct
thf(fact_2_dB_Oexhaust, axiom,
    ((![Y : dB]: ((![X1 : nat]: (~ ((Y = (var @ X1))))) => ((![X21 : dB, X22 : dB]: (~ ((Y = (app @ X21 @ X22))))) => (~ ((![X3 : dB]: (~ ((Y = (abs @ X3)))))))))))). % dB.exhaust
thf(fact_3_dB_Oinject_I1_J, axiom,
    ((![X12 : nat, Y1 : nat]: (((var @ X12) = (var @ Y1)) = (X12 = Y1))))). % dB.inject(1)
thf(fact_4_dB_Oinject_I3_J, axiom,
    ((![X32 : dB, Y3 : dB]: (((abs @ X32) = (abs @ Y3)) = (X32 = Y3))))). % dB.inject(3)
thf(fact_5_dB_Oinject_I2_J, axiom,
    ((![X212 : dB, X222 : dB, Y21 : dB, Y22 : dB]: (((app @ X212 @ X222) = (app @ Y21 @ Y22)) = (((X212 = Y21)) & ((X222 = Y22))))))). % dB.inject(2)
thf(fact_6_Lambda, axiom,
    ((![R : dB]: ((it @ R) => (it @ (abs @ R)))))). % Lambda
thf(fact_7_dB_Odistinct_I3_J, axiom,
    ((![X12 : nat, X32 : dB]: (~ (((var @ X12) = (abs @ X32))))))). % dB.distinct(3)
thf(fact_8_dB_Odistinct_I1_J, axiom,
    ((![X12 : nat, X212 : dB, X222 : dB]: (~ (((var @ X12) = (app @ X212 @ X222))))))). % dB.distinct(1)
thf(fact_9_dB_Odistinct_I5_J, axiom,
    ((![X212 : dB, X222 : dB, X32 : dB]: (~ (((app @ X212 @ X222) = (abs @ X32))))))). % dB.distinct(5)
thf(fact_10_subst__Var__IT, axiom,
    ((![R : dB, I : nat, J : nat]: ((it @ R) => (it @ (subst @ R @ (var @ I) @ J)))))). % subst_Var_IT
thf(fact_11_ex__head__tail, axiom,
    ((![T : dB]: (?[Ts : list_dB, H : dB]: ((T = (foldl_dB_dB @ app @ H @ Ts)) & ((?[N2 : nat]: (H = (var @ N2))) | (?[U : dB]: (H = (abs @ U))))))))). % ex_head_tail
thf(fact_12_Abs__App__neq__Var__apps, axiom,
    ((![S : dB, T : dB, N : nat, Ss : list_dB]: (~ (((app @ (abs @ S) @ T) = (foldl_dB_dB @ app @ (var @ N) @ Ss))))))). % Abs_App_neq_Var_apps
thf(fact_13_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_14_subst__eq, axiom,
    ((![K : nat, U2 : dB]: ((subst @ (var @ K) @ U2 @ K) = U2)))). % subst_eq
thf(fact_15_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_16_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_17_subst__App, axiom,
    ((![T : dB, U2 : dB, S : dB, K : nat]: ((subst @ (app @ T @ U2) @ S @ K) = (app @ (subst @ T @ S @ K) @ (subst @ U2 @ S @ K)))))). % subst_App
thf(fact_18_Var__apps__neq__Abs__apps, axiom,
    ((![N : nat, Ts2 : list_dB, R : dB, Ss : list_dB]: (~ (((foldl_dB_dB @ app @ (var @ N) @ Ts2) = (foldl_dB_dB @ app @ (abs @ R) @ Ss))))))). % Var_apps_neq_Abs_apps
thf(fact_19_Beta, 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))))))). % Beta
thf(fact_20_IT_OVar, axiom,
    ((![Rs : list_dB, N : nat]: ((listsp_dB @ it @ Rs) => (it @ (foldl_dB_dB @ app @ (var @ N) @ Rs)))))). % IT.Var
thf(fact_21_Apps__dB__induct, axiom,
    ((![P : dB > $o, T : dB]: ((![N2 : nat, Ts : list_dB]: ((![X4 : dB]: ((member_dB @ X4 @ (set_dB2 @ Ts)) => (P @ X4))) => (P @ (foldl_dB_dB @ app @ (var @ N2) @ Ts)))) => ((![U : dB]: ((P @ U) => (![Ts : list_dB]: ((![X4 : dB]: ((member_dB @ X4 @ (set_dB2 @ Ts)) => (P @ X4))) => (P @ (foldl_dB_dB @ app @ (abs @ U) @ Ts)))))) => (P @ T)))))). % Apps_dB_induct
thf(fact_22_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_23_apps__eq__Abs__conv, axiom,
    ((![S : dB, Ss : list_dB, R : dB]: (((foldl_dB_dB @ app @ S @ Ss) = (abs @ R)) = (((S = (abs @ R))) & ((Ss = nil_dB))))))). % apps_eq_Abs_conv
thf(fact_24_Abs__eq__apps__conv, axiom,
    ((![R : dB, S : dB, Ss : list_dB]: (((abs @ R) = (foldl_dB_dB @ app @ S @ Ss)) = ((((abs @ R) = S)) & ((Ss = nil_dB))))))). % Abs_eq_apps_conv
thf(fact_25_IT_Osimps, axiom,
    ((it = (^[A : dB]: (((?[Rs2 : list_dB]: (?[N3 : nat]: (((A = (foldl_dB_dB @ app @ (var @ N3) @ Rs2))) & ((listsp_dB @ it @ Rs2)))))) | ((((?[R2 : dB]: (((A = (abs @ R2))) & ((it @ R2))))) | ((?[R2 : dB]: (?[S2 : dB]: (?[Ss2 : list_dB]: (((A = (foldl_dB_dB @ app @ (app @ (abs @ R2) @ S2) @ Ss2))) & ((((it @ (foldl_dB_dB @ app @ (subst @ R2 @ S2 @ zero_zero_nat) @ Ss2))) & ((it @ S2)))))))))))))))). % IT.simps
thf(fact_26_IT_Ocases, axiom,
    ((![A2 : dB]: ((it @ A2) => ((![Rs3 : list_dB]: ((?[N2 : nat]: (A2 = (foldl_dB_dB @ app @ (var @ N2) @ Rs3))) => (~ ((listsp_dB @ it @ Rs3))))) => ((![R3 : dB]: ((A2 = (abs @ R3)) => (~ ((it @ R3))))) => (~ ((![R3 : dB, S3 : dB, Ss3 : list_dB]: ((A2 = (foldl_dB_dB @ app @ (app @ (abs @ R3) @ S3) @ Ss3)) => ((it @ (foldl_dB_dB @ app @ (subst @ R3 @ S3 @ zero_zero_nat) @ Ss3)) => (~ ((it @ S3)))))))))))))). % IT.cases
thf(fact_27_subst__lt, axiom,
    ((![J : nat, I : nat, U2 : dB]: ((ord_less_nat @ J @ I) => ((subst @ (var @ J) @ U2 @ I) = (var @ J)))))). % subst_lt
thf(fact_28_lem, axiom,
    ((![P : dB > $o, T : dB, N : nat]: ((![N2 : nat, Ts : list_dB]: ((![X4 : dB]: ((member_dB @ X4 @ (set_dB2 @ Ts)) => (P @ X4))) => (P @ (foldl_dB_dB @ app @ (var @ N2) @ Ts)))) => ((![U : dB]: ((P @ U) => (![Ts : list_dB]: ((![X4 : dB]: ((member_dB @ X4 @ (set_dB2 @ Ts)) => (P @ X4))) => (P @ (foldl_dB_dB @ app @ (abs @ U) @ Ts)))))) => (((size_size_dB @ T) = N) => (P @ T))))))). % lem
thf(fact_29_dB_Osize_I4_J, axiom,
    ((![X12 : nat]: ((size_size_dB @ (var @ X12)) = zero_zero_nat)))). % dB.size(4)
thf(fact_30_in__listspI, axiom,
    ((![Xs : list_dB, A3 : dB > $o]: ((![X : dB]: ((member_dB @ X @ (set_dB2 @ Xs)) => (A3 @ X))) => (listsp_dB @ A3 @ Xs))))). % in_listspI
thf(fact_31_listsp__simps_I1_J, axiom,
    ((![A3 : dB > $o]: (listsp_dB @ A3 @ nil_dB)))). % listsp_simps(1)
thf(fact_32_bot__nat__0_Onot__eq__extremum, axiom,
    ((![A2 : nat]: ((~ ((A2 = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ A2))))). % bot_nat_0.not_eq_extremum
thf(fact_33_less__nat__zero__code, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_nat_zero_code
thf(fact_34_neq0__conv, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ N))))). % neq0_conv
thf(fact_35_not__gr__zero, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr_zero
thf(fact_36_in__listsp__conv__set, axiom,
    ((listsp_dB = (^[A4 : dB > $o]: (^[Xs2 : list_dB]: (![X5 : dB]: (((member_dB @ X5 @ (set_dB2 @ Xs2))) => ((A4 @ X5))))))))). % in_listsp_conv_set
thf(fact_37_zero__reorient, axiom,
    ((![X6 : nat]: ((zero_zero_nat = X6) = (X6 = zero_zero_nat))))). % zero_reorient
thf(fact_38_zero__reorient, axiom,
    ((![X6 : int]: ((zero_zero_int = X6) = (X6 = zero_zero_int))))). % zero_reorient
thf(fact_39_nat__neq__iff, axiom,
    ((![M : nat, N : nat]: ((~ ((M = N))) = (((ord_less_nat @ M @ N)) | ((ord_less_nat @ N @ M))))))). % nat_neq_iff
thf(fact_40_less__not__refl, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_not_refl
thf(fact_41_less__not__refl2, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ N @ M) => (~ ((M = N))))))). % less_not_refl2
thf(fact_42_less__not__refl3, axiom,
    ((![S : nat, T : nat]: ((ord_less_nat @ S @ T) => (~ ((S = T))))))). % less_not_refl3
thf(fact_43_less__irrefl__nat, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_irrefl_nat
thf(fact_44_nat__less__induct, axiom,
    ((![P : nat > $o, N : nat]: ((![N2 : nat]: ((![M2 : nat]: ((ord_less_nat @ M2 @ N2) => (P @ M2))) => (P @ N2))) => (P @ N))))). % nat_less_induct
thf(fact_45_infinite__descent, axiom,
    ((![P : nat > $o, N : nat]: ((![N2 : nat]: ((~ ((P @ N2))) => (?[M2 : nat]: ((ord_less_nat @ M2 @ N2) & (~ ((P @ M2))))))) => (P @ N))))). % infinite_descent
thf(fact_46_linorder__neqE__nat, axiom,
    ((![X6 : nat, Y : nat]: ((~ ((X6 = Y))) => ((~ ((ord_less_nat @ X6 @ Y))) => (ord_less_nat @ Y @ X6)))))). % linorder_neqE_nat
thf(fact_47_size__neq__size__imp__neq, axiom,
    ((![X6 : dB, Y : dB]: ((~ (((size_size_dB @ X6) = (size_size_dB @ Y)))) => (~ ((X6 = Y))))))). % size_neq_size_imp_neq
thf(fact_48_gr__zeroI, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr_zeroI
thf(fact_49_not__less__zero, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less_zero
thf(fact_50_gr__implies__not__zero, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not_zero
thf(fact_51_zero__less__iff__neq__zero, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) = (~ ((N = zero_zero_nat))))))). % zero_less_iff_neq_zero
thf(fact_52_gr0I, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr0I
thf(fact_53_not__gr0, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr0
thf(fact_54_not__less0, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less0
thf(fact_55_less__zeroE, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_zeroE
thf(fact_56_gr__implies__not0, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not0
thf(fact_57_infinite__descent0, axiom,
    ((![P : nat > $o, N : nat]: ((P @ zero_zero_nat) => ((![N2 : nat]: ((ord_less_nat @ zero_zero_nat @ N2) => ((~ ((P @ N2))) => (?[M2 : nat]: ((ord_less_nat @ M2 @ N2) & (~ ((P @ M2)))))))) => (P @ N)))))). % infinite_descent0
thf(fact_58_bot__nat__0_Oextremum__strict, axiom,
    ((![A2 : nat]: (~ ((ord_less_nat @ A2 @ zero_zero_nat)))))). % bot_nat_0.extremum_strict
thf(fact_59_foldl__Nil, axiom,
    ((![F : dB > dB > dB, A2 : dB]: ((foldl_dB_dB @ F @ A2 @ nil_dB) = A2)))). % foldl_Nil
thf(fact_60_foldl__cong, axiom,
    ((![A2 : dB, B : dB, L : list_dB, K : list_dB, F : dB > dB > dB, G : dB > dB > dB]: ((A2 = B) => ((L = K) => ((![A5 : dB, X : dB]: ((member_dB @ X @ (set_dB2 @ L)) => ((F @ A5 @ X) = (G @ A5 @ X)))) => ((foldl_dB_dB @ F @ A2 @ L) = (foldl_dB_dB @ G @ B @ K)))))))). % foldl_cong
thf(fact_61_listsp_ONil, axiom,
    ((![A3 : dB > $o]: (listsp_dB @ A3 @ nil_dB)))). % listsp.Nil
thf(fact_62_in__listspD, axiom,
    ((![A3 : dB > $o, Xs : list_dB]: ((listsp_dB @ A3 @ Xs) => (![X4 : dB]: ((member_dB @ X4 @ (set_dB2 @ Xs)) => (A3 @ X4))))))). % in_listspD
thf(fact_63_less__numeral__extra_I3_J, axiom,
    ((~ ((ord_less_nat @ zero_zero_nat @ zero_zero_nat))))). % less_numeral_extra(3)
thf(fact_64_less__numeral__extra_I3_J, axiom,
    ((~ ((ord_less_int @ zero_zero_int @ zero_zero_int))))). % less_numeral_extra(3)
thf(fact_65_count__notin, axiom,
    ((![X6 : dB, Xs : list_dB]: ((~ ((member_dB @ X6 @ (set_dB2 @ Xs)))) => ((count_list_dB @ Xs @ X6) = zero_zero_nat))))). % count_notin
thf(fact_66_dB_Osize__gen_I1_J, axiom,
    ((![X12 : nat]: ((size_dB @ (var @ X12)) = zero_zero_nat)))). % dB.size_gen(1)
thf(fact_67_of__nat__0__less__iff, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ (semiri1382578993at_nat @ N)) = (ord_less_nat @ zero_zero_nat @ N))))). % of_nat_0_less_iff
thf(fact_68_of__nat__0__less__iff, axiom,
    ((![N : nat]: ((ord_less_int @ zero_zero_int @ (semiri2019852685at_int @ N)) = (ord_less_nat @ zero_zero_nat @ N))))). % of_nat_0_less_iff
thf(fact_69_list__ex1__simps_I1_J, axiom,
    ((![P : dB > $o]: (~ ((list_ex1_dB @ P @ nil_dB)))))). % list_ex1_simps(1)
thf(fact_70_substn__subst__0, axiom,
    ((![T : dB, S : dB]: ((substn @ T @ S @ zero_zero_nat) = (subst @ T @ S @ zero_zero_nat))))). % substn_subst_0
thf(fact_71_beta_Oinducts, axiom,
    ((![X12 : dB, X23 : dB, P : dB > dB > $o]: ((beta @ X12 @ X23) => ((![S3 : dB, T2 : dB]: (P @ (app @ (abs @ S3) @ T2) @ (subst @ S3 @ T2 @ zero_zero_nat))) => ((![S3 : dB, T2 : dB, U : dB]: ((beta @ S3 @ T2) => ((P @ S3 @ T2) => (P @ (app @ S3 @ U) @ (app @ T2 @ U))))) => ((![S3 : dB, T2 : dB, U : dB]: ((beta @ S3 @ T2) => ((P @ S3 @ T2) => (P @ (app @ U @ S3) @ (app @ U @ T2))))) => ((![S3 : dB, T2 : dB]: ((beta @ S3 @ T2) => ((P @ S3 @ T2) => (P @ (abs @ S3) @ (abs @ T2))))) => (P @ X12 @ X23))))))))). % beta.inducts
thf(fact_72_of__nat__eq__iff, axiom,
    ((![M : nat, N : nat]: (((semiri2019852685at_int @ M) = (semiri2019852685at_int @ N)) = (M = N))))). % of_nat_eq_iff
thf(fact_73_length__0__conv, axiom,
    ((![Xs : list_dB]: (((size_size_list_dB @ Xs) = zero_zero_nat) = (Xs = nil_dB))))). % length_0_conv
thf(fact_74_of__nat__eq__0__iff, axiom,
    ((![M : nat]: (((semiri1382578993at_nat @ M) = zero_zero_nat) = (M = zero_zero_nat))))). % of_nat_eq_0_iff
thf(fact_75_of__nat__eq__0__iff, axiom,
    ((![M : nat]: (((semiri2019852685at_int @ M) = zero_zero_int) = (M = zero_zero_nat))))). % of_nat_eq_0_iff
thf(fact_76_of__nat__0__eq__iff, axiom,
    ((![N : nat]: ((zero_zero_nat = (semiri1382578993at_nat @ N)) = (zero_zero_nat = N))))). % of_nat_0_eq_iff
thf(fact_77_of__nat__0__eq__iff, axiom,
    ((![N : nat]: ((zero_zero_int = (semiri2019852685at_int @ N)) = (zero_zero_nat = N))))). % of_nat_0_eq_iff
thf(fact_78_of__nat__0, axiom,
    (((semiri1382578993at_nat @ zero_zero_nat) = zero_zero_nat))). % of_nat_0
thf(fact_79_of__nat__0, axiom,
    (((semiri2019852685at_int @ zero_zero_nat) = zero_zero_int))). % of_nat_0
thf(fact_80_of__nat__less__iff, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ (semiri1382578993at_nat @ M) @ (semiri1382578993at_nat @ N)) = (ord_less_nat @ M @ N))))). % of_nat_less_iff
thf(fact_81_of__nat__less__iff, axiom,
    ((![M : nat, N : nat]: ((ord_less_int @ (semiri2019852685at_int @ M) @ (semiri2019852685at_int @ N)) = (ord_less_nat @ M @ N))))). % of_nat_less_iff
thf(fact_82_length__greater__0__conv, axiom,
    ((![Xs : list_dB]: ((ord_less_nat @ zero_zero_nat @ (size_size_list_dB @ Xs)) = (~ ((Xs = nil_dB))))))). % length_greater_0_conv
thf(fact_83_appL, axiom,
    ((![S : dB, T : dB, U2 : dB]: ((beta @ S @ T) => (beta @ (app @ S @ U2) @ (app @ T @ U2)))))). % appL
thf(fact_84_appR, axiom,
    ((![S : dB, T : dB, U2 : dB]: ((beta @ S @ T) => (beta @ (app @ U2 @ S) @ (app @ U2 @ T)))))). % appR
thf(fact_85_beta__cases_I2_J, axiom,
    ((![R : dB, S : dB]: ((beta @ (abs @ R) @ S) => (~ ((![T2 : dB]: ((S = (abs @ T2)) => (~ ((beta @ R @ T2))))))))))). % beta_cases(2)
thf(fact_86_abs, axiom,
    ((![S : dB, T : dB]: ((beta @ S @ T) => (beta @ (abs @ S) @ (abs @ T)))))). % abs
thf(fact_87_beta__cases_I1_J, axiom,
    ((![I : nat, T : dB]: (~ ((beta @ (var @ I) @ T)))))). % beta_cases(1)
thf(fact_88_subst__preserves__beta, axiom,
    ((![R : dB, S : dB, T : dB, I : nat]: ((beta @ R @ S) => (beta @ (subst @ R @ T @ I) @ (subst @ S @ T @ I)))))). % subst_preserves_beta
thf(fact_89_of__nat__less__0__iff, axiom,
    ((![M : nat]: (~ ((ord_less_nat @ (semiri1382578993at_nat @ M) @ zero_zero_nat)))))). % of_nat_less_0_iff
thf(fact_90_of__nat__less__0__iff, axiom,
    ((![M : nat]: (~ ((ord_less_int @ (semiri2019852685at_int @ M) @ zero_zero_int)))))). % of_nat_less_0_iff
thf(fact_91_of__nat__less__imp__less, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ (semiri1382578993at_nat @ M) @ (semiri1382578993at_nat @ N)) => (ord_less_nat @ M @ N))))). % of_nat_less_imp_less
thf(fact_92_of__nat__less__imp__less, axiom,
    ((![M : nat, N : nat]: ((ord_less_int @ (semiri2019852685at_int @ M) @ (semiri2019852685at_int @ N)) => (ord_less_nat @ M @ N))))). % of_nat_less_imp_less
thf(fact_93_less__imp__of__nat__less, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (ord_less_nat @ (semiri1382578993at_nat @ M) @ (semiri1382578993at_nat @ N)))))). % less_imp_of_nat_less
thf(fact_94_less__imp__of__nat__less, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (ord_less_int @ (semiri2019852685at_int @ M) @ (semiri2019852685at_int @ N)))))). % less_imp_of_nat_less
thf(fact_95_list_Osize_I3_J, axiom,
    (((size_size_list_dB @ nil_dB) = zero_zero_nat))). % list.size(3)
thf(fact_96_substn_Osimps_I2_J, axiom,
    ((![T : dB, U2 : dB, S : dB, K : nat]: ((substn @ (app @ T @ U2) @ S @ K) = (app @ (substn @ T @ S @ K) @ (substn @ U2 @ S @ K)))))). % substn.simps(2)
thf(fact_97_list__ex1__iff, axiom,
    ((list_ex1_dB = (^[P2 : dB > $o]: (^[Xs2 : list_dB]: (?[X5 : dB]: (((((member_dB @ X5 @ (set_dB2 @ Xs2))) & ((P2 @ X5)))) & ((![Y2 : dB]: (((((member_dB @ Y2 @ (set_dB2 @ Xs2))) & ((P2 @ Y2)))) => ((Y2 = X5)))))))))))). % list_ex1_iff
thf(fact_98_length__pos__if__in__set, axiom,
    ((![X6 : dB, Xs : list_dB]: ((member_dB @ X6 @ (set_dB2 @ Xs)) => (ord_less_nat @ zero_zero_nat @ (size_size_list_dB @ Xs)))))). % length_pos_if_in_set
thf(fact_99_count__list_Osimps_I1_J, axiom,
    ((![Y : dB]: ((count_list_dB @ nil_dB @ Y) = zero_zero_nat)))). % count_list.simps(1)
thf(fact_100_beta__cases_I3_J, axiom,
    ((![S : dB, T : dB, U2 : dB]: ((beta @ (app @ S @ T) @ U2) => ((![S3 : dB]: ((S = (abs @ S3)) => (~ ((U2 = (subst @ S3 @ T @ zero_zero_nat)))))) => ((![T2 : dB]: ((U2 = (app @ T2 @ T)) => (~ ((beta @ S @ T2))))) => (~ ((![T2 : dB]: ((U2 = (app @ S @ T2)) => (~ ((beta @ T @ T2))))))))))))). % beta_cases(3)
thf(fact_101_beta, axiom,
    ((![S : dB, T : dB]: (beta @ (app @ (abs @ S) @ T) @ (subst @ S @ T @ zero_zero_nat))))). % beta
thf(fact_102_beta_Ocases, axiom,
    ((![A1 : dB, A22 : dB]: ((beta @ A1 @ A22) => ((![S3 : dB, T2 : dB]: ((A1 = (app @ (abs @ S3) @ T2)) => (~ ((A22 = (subst @ S3 @ T2 @ zero_zero_nat)))))) => ((![S3 : dB, T2 : dB, U : dB]: ((A1 = (app @ S3 @ U)) => ((A22 = (app @ T2 @ U)) => (~ ((beta @ S3 @ T2)))))) => ((![S3 : dB, T2 : dB, U : dB]: ((A1 = (app @ U @ S3)) => ((A22 = (app @ U @ T2)) => (~ ((beta @ S3 @ T2)))))) => (~ ((![S3 : dB]: ((A1 = (abs @ S3)) => (![T2 : dB]: ((A22 = (abs @ T2)) => (~ ((beta @ S3 @ T2)))))))))))))))). % beta.cases
thf(fact_103_beta_Osimps, axiom,
    ((beta = (^[A12 : dB]: (^[A23 : dB]: (((?[S2 : dB]: (?[T3 : dB]: (((A12 = (app @ (abs @ S2) @ T3))) & ((A23 = (subst @ S2 @ T3 @ zero_zero_nat))))))) | ((((?[S2 : dB]: (?[T3 : dB]: (?[U3 : dB]: (((A12 = (app @ S2 @ U3))) & ((((A23 = (app @ T3 @ U3))) & ((beta @ S2 @ T3))))))))) | ((((?[S2 : dB]: (?[T3 : dB]: (?[U3 : dB]: (((A12 = (app @ U3 @ S2))) & ((((A23 = (app @ U3 @ T3))) & ((beta @ S2 @ T3))))))))) | ((?[S2 : dB]: (?[T3 : dB]: (((A12 = (abs @ S2))) & ((((A23 = (abs @ T3))) & ((beta @ S2 @ T3)))))))))))))))))). % beta.simps
thf(fact_104_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_105_can__select__set__list__ex1, axiom,
    ((![P : dB > $o, A3 : list_dB]: ((can_select_dB @ P @ (set_dB2 @ A3)) = (list_ex1_dB @ P @ A3))))). % can_select_set_list_ex1
thf(fact_106_substn__subst__n, axiom,
    ((substn = (^[T3 : dB]: (^[S2 : dB]: (^[N3 : nat]: (subst @ T3 @ (liftn @ N3 @ S2 @ zero_zero_nat) @ N3))))))). % substn_subst_n
thf(fact_107_pos__int__cases, axiom,
    ((![K : int]: ((ord_less_int @ zero_zero_int @ K) => (~ ((![N2 : nat]: ((K = (semiri2019852685at_int @ N2)) => (~ ((ord_less_nat @ zero_zero_nat @ N2))))))))))). % pos_int_cases
thf(fact_108_liftn__0, axiom,
    ((![T : dB, K : nat]: ((liftn @ zero_zero_nat @ T @ K) = T)))). % liftn_0
thf(fact_109_int__int__eq, axiom,
    ((![M : nat, N : nat]: (((semiri2019852685at_int @ M) = (semiri2019852685at_int @ N)) = (M = N))))). % int_int_eq
thf(fact_110_less__int__code_I1_J, axiom,
    ((~ ((ord_less_int @ zero_zero_int @ zero_zero_int))))). % less_int_code(1)
thf(fact_111_liftn_Osimps_I2_J, axiom,
    ((![N : nat, S : dB, T : dB, K : nat]: ((liftn @ N @ (app @ S @ T) @ K) = (app @ (liftn @ N @ S @ K) @ (liftn @ N @ T @ K)))))). % liftn.simps(2)
thf(fact_112_zero__less__imp__eq__int, axiom,
    ((![K : int]: ((ord_less_int @ zero_zero_int @ K) => (?[N2 : nat]: ((ord_less_nat @ zero_zero_nat @ N2) & (K = (semiri2019852685at_int @ N2)))))))). % zero_less_imp_eq_int
thf(fact_113_nat__int__comparison_I2_J, axiom,
    ((ord_less_nat = (^[A : nat]: (^[B2 : nat]: (ord_less_int @ (semiri2019852685at_int @ A) @ (semiri2019852685at_int @ B2))))))). % nat_int_comparison(2)
thf(fact_114_int__ops_I1_J, axiom,
    (((semiri2019852685at_int @ zero_zero_nat) = zero_zero_int))). % int_ops(1)
thf(fact_115_head__Var__reduction, axiom,
    ((![N : nat, Rs : list_dB, V : dB]: ((beta @ (foldl_dB_dB @ app @ (var @ N) @ Rs) @ V) => (?[Ss3 : list_dB]: ((step1_dB @ beta @ Rs @ Ss3) & (V = (foldl_dB_dB @ app @ (var @ N) @ Ss3)))))))). % head_Var_reduction
thf(fact_116_verit__comp__simplify1_I1_J, axiom,
    ((![A2 : nat]: (~ ((ord_less_nat @ A2 @ A2)))))). % verit_comp_simplify1(1)
thf(fact_117_verit__comp__simplify1_I1_J, axiom,
    ((![A2 : int]: (~ ((ord_less_int @ A2 @ A2)))))). % verit_comp_simplify1(1)
thf(fact_118_nat__int__comparison_I1_J, axiom,
    (((^[Y4 : nat]: (^[Z : nat]: (Y4 = Z))) = (^[A : nat]: (^[B2 : nat]: ((semiri2019852685at_int @ A) = (semiri2019852685at_int @ B2))))))). % nat_int_comparison(1)
thf(fact_119_int__if, axiom,
    ((![P : $o, A2 : nat, B : nat]: ((P => ((semiri2019852685at_int @ (if_nat @ P @ A2 @ B)) = (semiri2019852685at_int @ A2))) & ((~ (P)) => ((semiri2019852685at_int @ (if_nat @ P @ A2 @ B)) = (semiri2019852685at_int @ B))))))). % int_if
thf(fact_120_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_121_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)))))) => (~ ((![T2 : dB]: ((R = (abs @ T2)) => (![U : dB, Us : list_dB]: ((Rs = (cons_dB @ U @ Us)) => (~ ((S = (foldl_dB_dB @ app @ (subst @ T2 @ U @ zero_zero_nat) @ Us)))))))))))))))). % apps_betasE
thf(fact_122_ex__step1I, axiom,
    ((![X6 : dB, Xs : list_dB, R : dB > dB > $o, Y : dB]: ((member_dB @ X6 @ (set_dB2 @ Xs)) => ((R @ Y @ X6) => (?[Ys : list_dB]: ((step1_dB @ R @ Ys @ Xs) & (member_dB @ Y @ (set_dB2 @ Ys))))))))). % ex_step1I
thf(fact_123_list_Oinject, axiom,
    ((![X212 : dB, X222 : list_dB, Y21 : dB, Y22 : list_dB]: (((cons_dB @ X212 @ X222) = (cons_dB @ Y21 @ Y22)) = (((X212 = Y21)) & ((X222 = Y22))))))). % list.inject
thf(fact_124_listsp_Oinducts, axiom,
    ((![A3 : dB > $o, X6 : list_dB, P : list_dB > $o]: ((listsp_dB @ A3 @ X6) => ((P @ nil_dB) => ((![A5 : dB, L2 : list_dB]: ((A3 @ A5) => ((listsp_dB @ A3 @ L2) => ((P @ L2) => (P @ (cons_dB @ A5 @ L2)))))) => (P @ X6))))))). % listsp.inducts
thf(fact_125_listsp_Osimps, axiom,
    ((listsp_dB = (^[A4 : dB > $o]: (^[A : list_dB]: (((A = nil_dB)) | ((?[B2 : dB]: (?[L3 : list_dB]: (((A = (cons_dB @ B2 @ L3))) & ((((A4 @ B2)) & ((listsp_dB @ A4 @ L3)))))))))))))). % listsp.simps
thf(fact_126_listsp_Ocases, axiom,
    ((![A3 : dB > $o, A2 : list_dB]: ((listsp_dB @ A3 @ A2) => ((~ ((A2 = nil_dB))) => (~ ((![A5 : dB, L2 : list_dB]: ((A2 = (cons_dB @ A5 @ L2)) => ((A3 @ A5) => (~ ((listsp_dB @ A3 @ L2))))))))))))). % listsp.cases
thf(fact_127_list_Odistinct_I1_J, axiom,
    ((![X212 : dB, X222 : list_dB]: (~ ((nil_dB = (cons_dB @ X212 @ X222))))))). % list.distinct(1)
thf(fact_128_list_OdiscI, axiom,
    ((![List : list_dB, X212 : dB, X222 : list_dB]: ((List = (cons_dB @ X212 @ X222)) => (~ ((List = nil_dB))))))). % list.discI
thf(fact_129_list_Oexhaust, axiom,
    ((![Y : list_dB]: ((~ ((Y = nil_dB))) => (~ ((![X21 : dB, X22 : list_dB]: (~ ((Y = (cons_dB @ X21 @ X22))))))))))). % list.exhaust
thf(fact_130_list_Oinducts, axiom,
    ((![P : list_dB > $o, List : list_dB]: ((P @ nil_dB) => ((![X1 : dB, X2 : list_dB]: ((P @ X2) => (P @ (cons_dB @ X1 @ X2)))) => (P @ List)))))). % list.inducts
thf(fact_131_neq__Nil__conv, axiom,
    ((![Xs : list_dB]: ((~ ((Xs = nil_dB))) = (?[Y2 : dB]: (?[Ys2 : list_dB]: (Xs = (cons_dB @ Y2 @ Ys2)))))))). % neq_Nil_conv
thf(fact_132_list__induct2_H, axiom,
    ((![P : list_dB > list_dB > $o, Xs : list_dB, Ys3 : list_dB]: ((P @ nil_dB @ nil_dB) => ((![X : dB, Xs3 : list_dB]: (P @ (cons_dB @ X @ Xs3) @ nil_dB)) => ((![Y5 : dB, Ys : list_dB]: (P @ nil_dB @ (cons_dB @ Y5 @ Ys))) => ((![X : dB, Xs3 : list_dB, Y5 : dB, Ys : list_dB]: ((P @ Xs3 @ Ys) => (P @ (cons_dB @ X @ Xs3) @ (cons_dB @ Y5 @ Ys)))) => (P @ Xs @ Ys3)))))))). % list_induct2'
thf(fact_133_splice_Oinduct, axiom,
    ((![P : list_dB > list_dB > $o, A0 : list_dB, A1 : list_dB]: ((![X_1 : list_dB]: (P @ nil_dB @ X_1)) => ((![X : dB, Xs3 : list_dB, Ys : list_dB]: ((P @ Ys @ Xs3) => (P @ (cons_dB @ X @ Xs3) @ Ys))) => (P @ A0 @ A1)))))). % splice.induct
thf(fact_134_induct__list012, axiom,
    ((![P : list_dB > $o, Xs : list_dB]: ((P @ nil_dB) => ((![X : dB]: (P @ (cons_dB @ X @ nil_dB))) => ((![X : dB, Y5 : dB, Zs : list_dB]: ((P @ Zs) => ((P @ (cons_dB @ Y5 @ Zs)) => (P @ (cons_dB @ X @ (cons_dB @ Y5 @ Zs)))))) => (P @ Xs))))))). % induct_list012
thf(fact_135_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)) => ((![X : dB, Xs3 : list_dB, Y5 : dB, Ys : list_dB]: ((P @ Xs3 @ (cons_dB @ Y5 @ Ys)) => ((P @ (cons_dB @ X @ Xs3) @ Ys) => (P @ (cons_dB @ X @ Xs3) @ (cons_dB @ Y5 @ Ys))))) => (P @ A0 @ A1))))))). % shuffles.induct
thf(fact_136_remdups__adj_Ocases, axiom,
    ((![X6 : list_dB]: ((~ ((X6 = nil_dB))) => ((![X : dB]: (~ ((X6 = (cons_dB @ X @ nil_dB))))) => (~ ((![X : dB, Y5 : dB, Xs3 : list_dB]: (~ ((X6 = (cons_dB @ X @ (cons_dB @ Y5 @ Xs3))))))))))))). % remdups_adj.cases
thf(fact_137_sorted__wrt_Oinduct, axiom,
    ((![P : (dB > dB > $o) > list_dB > $o, A0 : dB > dB > $o, A1 : list_dB]: ((![P3 : dB > dB > $o]: (P @ P3 @ nil_dB)) => ((![P3 : dB > dB > $o, X : dB, Ys : list_dB]: ((P @ P3 @ Ys) => (P @ P3 @ (cons_dB @ X @ Ys)))) => (P @ A0 @ A1)))))). % sorted_wrt.induct
thf(fact_138_remdups__adj_Oinduct, axiom,
    ((![P : list_dB > $o, A0 : list_dB]: ((P @ nil_dB) => ((![X : dB]: (P @ (cons_dB @ X @ nil_dB))) => ((![X : dB, Y5 : dB, Xs3 : list_dB]: (((X = Y5) => (P @ (cons_dB @ X @ Xs3))) => (((~ ((X = Y5))) => (P @ (cons_dB @ Y5 @ Xs3))) => (P @ (cons_dB @ X @ (cons_dB @ Y5 @ Xs3)))))) => (P @ A0))))))). % remdups_adj.induct
thf(fact_139_successively_Oinduct, axiom,
    ((![P : (dB > dB > $o) > list_dB > $o, A0 : dB > dB > $o, A1 : list_dB]: ((![P3 : dB > dB > $o]: (P @ P3 @ nil_dB)) => ((![P3 : dB > dB > $o, X : dB]: (P @ P3 @ (cons_dB @ X @ nil_dB))) => ((![P3 : dB > dB > $o, X : dB, Y5 : dB, Xs3 : list_dB]: ((P @ P3 @ (cons_dB @ Y5 @ Xs3)) => (P @ P3 @ (cons_dB @ X @ (cons_dB @ Y5 @ Xs3))))) => (P @ A0 @ A1))))))). % successively.induct
thf(fact_140_list__nonempty__induct, axiom,
    ((![Xs : list_dB, P : list_dB > $o]: ((~ ((Xs = nil_dB))) => ((![X : dB]: (P @ (cons_dB @ X @ nil_dB))) => ((![X : dB, Xs3 : list_dB]: ((~ ((Xs3 = nil_dB))) => ((P @ Xs3) => (P @ (cons_dB @ X @ Xs3))))) => (P @ Xs))))))). % list_nonempty_induct
thf(fact_141_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, A5 : dB, As : list_dB, Bs : list_dB]: ((P @ F2 @ As @ (cons_dB @ (F2 @ A5) @ Bs)) => (P @ F2 @ (cons_dB @ A5 @ As) @ Bs))) => (P @ A0 @ A1 @ A22)))))). % map_tailrec_rev.induct

% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T, axiom,
    ((![P : $o]: ((P = $true) | (P = $false))))).
thf(help_If_2_1_If_001t__Nat__Onat_T, axiom,
    ((![X6 : nat, Y : nat]: ((if_nat @ $false @ X6 @ Y) = Y)))).
thf(help_If_1_1_If_001t__Nat__Onat_T, axiom,
    ((![X6 : nat, Y : nat]: ((if_nat @ $true @ X6 @ Y) = X6)))).

% Conjectures (3)
thf(conj_0, hypothesis,
    ((it @ r))).
thf(conj_1, hypothesis,
    ((it @ (app @ r @ (var @ i))))).
thf(conj_2, conjecture,
    ((it @ (app @ (abs @ r) @ (var @ i))))).
