% 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/Hoare/prob_284__3252294_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:13:27.377

% Could-be-implicit typings (6)
thf(ty_n_t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_J, type,
    set_Ho137910533iple_a : $tType).
thf(ty_n_t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    hoare_1678595023iple_a : $tType).
thf(ty_n_t__Com__Ostate, type,
    state : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).
thf(ty_n_t__Com__Ocom, type,
    com : $tType).
thf(ty_n_tf__a, type,
    a : $tType).

% Explicit typings (24)
thf(sy_c_Com_Ocom_OSKIP, type,
    skip : com).
thf(sy_c_Com_Ocom_OSemi, type,
    semi : com > com > com).
thf(sy_c_Hoare__Mirabelle__raqjowkjvm_Ohoare__derivs_001tf__a, type,
    hoare_129598474rivs_a : set_Ho137910533iple_a > set_Ho137910533iple_a > $o).
thf(sy_c_Hoare__Mirabelle__raqjowkjvm_Ohoare__valids_001tf__a, type,
    hoare_1775499016lids_a : set_Ho137910533iple_a > set_Ho137910533iple_a > $o).
thf(sy_c_Hoare__Mirabelle__raqjowkjvm_Otriple_Otriple_001tf__a, type,
    hoare_719046530iple_a : (a > state > $o) > com > (a > state > $o) > hoare_1678595023iple_a).
thf(sy_c_Hoare__Mirabelle__raqjowkjvm_Otriple__valid_001tf__a, type,
    hoare_1926814542alid_a : nat > hoare_1678595023iple_a > $o).
thf(sy_c_Nat_OSuc, type,
    suc : nat > nat).
thf(sy_c_Natural_Oevaln, type,
    evaln : com > state > nat > state > $o).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_M_Eo_J, type,
    bot_bo431311916le_a_o : hoare_1678595023iple_a > $o).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat, type,
    bot_bot_nat : nat).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_J, type,
    bot_bo1298296729iple_a : set_Ho137910533iple_a).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat, type,
    ord_less_eq_nat : nat > nat > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_J, type,
    ord_le1221261669iple_a : set_Ho137910533iple_a > set_Ho137910533iple_a > $o).
thf(sy_c_Set_OBall_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    ball_H465710501iple_a : set_Ho137910533iple_a > (hoare_1678595023iple_a > $o) > $o).
thf(sy_c_Set_OCollect_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    collec1600235172iple_a : (hoare_1678595023iple_a > $o) > set_Ho137910533iple_a).
thf(sy_c_Set_Oinsert_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    insert1477804543iple_a : hoare_1678595023iple_a > set_Ho137910533iple_a > set_Ho137910533iple_a).
thf(sy_c_Set_Ois__empty_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    is_emp901906557iple_a : set_Ho137910533iple_a > $o).
thf(sy_c_Set_Ois__singleton_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    is_sin1784037339iple_a : set_Ho137910533iple_a > $o).
thf(sy_c_Set_Opairwise_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    pairwi531237284iple_a : (hoare_1678595023iple_a > hoare_1678595023iple_a > $o) > set_Ho137910533iple_a > $o).
thf(sy_c_Set_Othe__elem_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    the_el434698138iple_a : set_Ho137910533iple_a > hoare_1678595023iple_a).
thf(sy_c_member_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    member1332298086iple_a : hoare_1678595023iple_a > set_Ho137910533iple_a > $o).
thf(sy_v_Ga, type,
    ga : set_Ho137910533iple_a).
thf(sy_v_t, type,
    t : hoare_1678595023iple_a).
thf(sy_v_tsa, type,
    tsa : set_Ho137910533iple_a).

% Relevant facts (125)
thf(fact_0_derivs__insertD, axiom,
    ((![G : set_Ho137910533iple_a, T : hoare_1678595023iple_a, Ts : set_Ho137910533iple_a]: ((hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ T @ Ts)) => ((hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ T @ bot_bo1298296729iple_a)) & (hoare_129598474rivs_a @ G @ Ts)))))). % derivs_insertD
thf(fact_1_cut, axiom,
    ((![G2 : set_Ho137910533iple_a, Ts : set_Ho137910533iple_a, G : set_Ho137910533iple_a]: ((hoare_129598474rivs_a @ G2 @ Ts) => ((hoare_129598474rivs_a @ G @ G2) => (hoare_129598474rivs_a @ G @ Ts)))))). % cut
thf(fact_2_empty, axiom,
    ((![G : set_Ho137910533iple_a]: (hoare_129598474rivs_a @ G @ bot_bo1298296729iple_a)))). % empty
thf(fact_3_hoare__derivs_Oinsert, axiom,
    ((![G : set_Ho137910533iple_a, T : hoare_1678595023iple_a, Ts : set_Ho137910533iple_a]: ((hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ T @ bot_bo1298296729iple_a)) => ((hoare_129598474rivs_a @ G @ Ts) => (hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ T @ Ts))))))). % hoare_derivs.insert
thf(fact_4_ball__empty, axiom,
    ((![P : hoare_1678595023iple_a > $o, X : hoare_1678595023iple_a]: ((member1332298086iple_a @ X @ bot_bo1298296729iple_a) => (P @ X))))). % ball_empty
thf(fact_5_singletonI, axiom,
    ((![A : hoare_1678595023iple_a]: (member1332298086iple_a @ A @ (insert1477804543iple_a @ A @ bot_bo1298296729iple_a))))). % singletonI
thf(fact_6_insertCI, axiom,
    ((![A : hoare_1678595023iple_a, B : set_Ho137910533iple_a, B2 : hoare_1678595023iple_a]: (((~ ((member1332298086iple_a @ A @ B))) => (A = B2)) => (member1332298086iple_a @ A @ (insert1477804543iple_a @ B2 @ B)))))). % insertCI
thf(fact_7_insert__iff, axiom,
    ((![A : hoare_1678595023iple_a, B2 : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((member1332298086iple_a @ A @ (insert1477804543iple_a @ B2 @ A2)) = (((A = B2)) | ((member1332298086iple_a @ A @ A2))))))). % insert_iff
thf(fact_8_insert__absorb2, axiom,
    ((![X2 : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((insert1477804543iple_a @ X2 @ (insert1477804543iple_a @ X2 @ A2)) = (insert1477804543iple_a @ X2 @ A2))))). % insert_absorb2
thf(fact_9_empty__iff, axiom,
    ((![C : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ C @ bot_bo1298296729iple_a)))))). % empty_iff
thf(fact_10_all__not__in__conv, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((![X3 : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ X3 @ A2)))) = (A2 = bot_bo1298296729iple_a))))). % all_not_in_conv
thf(fact_11_Collect__empty__eq, axiom,
    ((![P : hoare_1678595023iple_a > $o]: (((collec1600235172iple_a @ P) = bot_bo1298296729iple_a) = (![X3 : hoare_1678595023iple_a]: (~ ((P @ X3)))))))). % Collect_empty_eq
thf(fact_12_empty__Collect__eq, axiom,
    ((![P : hoare_1678595023iple_a > $o]: ((bot_bo1298296729iple_a = (collec1600235172iple_a @ P)) = (![X3 : hoare_1678595023iple_a]: (~ ((P @ X3)))))))). % empty_Collect_eq
thf(fact_13_hoare__valids__def, axiom,
    ((hoare_1775499016lids_a = (^[G3 : set_Ho137910533iple_a]: (^[Ts2 : set_Ho137910533iple_a]: (![N : nat]: (((![X3 : hoare_1678595023iple_a]: (((member1332298086iple_a @ X3 @ G3)) => ((hoare_1926814542alid_a @ N @ X3))))) => ((![X3 : hoare_1678595023iple_a]: (((member1332298086iple_a @ X3 @ Ts2)) => ((hoare_1926814542alid_a @ N @ X3)))))))))))). % hoare_valids_def
thf(fact_14_singletonD, axiom,
    ((![B2 : hoare_1678595023iple_a, A : hoare_1678595023iple_a]: ((member1332298086iple_a @ B2 @ (insert1477804543iple_a @ A @ bot_bo1298296729iple_a)) => (B2 = A))))). % singletonD
thf(fact_15_bot__set__def, axiom,
    ((bot_bo1298296729iple_a = (collec1600235172iple_a @ bot_bo431311916le_a_o)))). % bot_set_def
thf(fact_16_ex__in__conv, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((?[X3 : hoare_1678595023iple_a]: (member1332298086iple_a @ X3 @ A2)) = (~ ((A2 = bot_bo1298296729iple_a))))))). % ex_in_conv
thf(fact_17_equals0I, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((![Y : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ Y @ A2)))) => (A2 = bot_bo1298296729iple_a))))). % equals0I
thf(fact_18_equals0D, axiom,
    ((![A2 : set_Ho137910533iple_a, A : hoare_1678595023iple_a]: ((A2 = bot_bo1298296729iple_a) => (~ ((member1332298086iple_a @ A @ A2))))))). % equals0D
thf(fact_19_emptyE, axiom,
    ((![A : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ A @ bot_bo1298296729iple_a)))))). % emptyE
thf(fact_20_mk__disjoint__insert, axiom,
    ((![A : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((member1332298086iple_a @ A @ A2) => (?[B3 : set_Ho137910533iple_a]: ((A2 = (insert1477804543iple_a @ A @ B3)) & (~ ((member1332298086iple_a @ A @ B3))))))))). % mk_disjoint_insert
thf(fact_21_insert__commute, axiom,
    ((![X2 : hoare_1678595023iple_a, Y2 : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((insert1477804543iple_a @ X2 @ (insert1477804543iple_a @ Y2 @ A2)) = (insert1477804543iple_a @ Y2 @ (insert1477804543iple_a @ X2 @ A2)))))). % insert_commute
thf(fact_22_insert__eq__iff, axiom,
    ((![A : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, B2 : hoare_1678595023iple_a, B : set_Ho137910533iple_a]: ((~ ((member1332298086iple_a @ A @ A2))) => ((~ ((member1332298086iple_a @ B2 @ B))) => (((insert1477804543iple_a @ A @ A2) = (insert1477804543iple_a @ B2 @ B)) = (((((A = B2)) => ((A2 = B)))) & ((((~ ((A = B2)))) => ((?[C2 : set_Ho137910533iple_a]: (((A2 = (insert1477804543iple_a @ B2 @ C2))) & ((((~ ((member1332298086iple_a @ B2 @ C2)))) & ((((B = (insert1477804543iple_a @ A @ C2))) & ((~ ((member1332298086iple_a @ A @ C2)))))))))))))))))))). % insert_eq_iff
thf(fact_23_insert__absorb, axiom,
    ((![A : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((member1332298086iple_a @ A @ A2) => ((insert1477804543iple_a @ A @ A2) = A2))))). % insert_absorb
thf(fact_24_insert__ident, axiom,
    ((![X2 : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((~ ((member1332298086iple_a @ X2 @ A2))) => ((~ ((member1332298086iple_a @ X2 @ B))) => (((insert1477804543iple_a @ X2 @ A2) = (insert1477804543iple_a @ X2 @ B)) = (A2 = B))))))). % insert_ident
thf(fact_25_Set_Oset__insert, axiom,
    ((![X2 : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((member1332298086iple_a @ X2 @ A2) => (~ ((![B3 : set_Ho137910533iple_a]: ((A2 = (insert1477804543iple_a @ X2 @ B3)) => (member1332298086iple_a @ X2 @ B3))))))))). % Set.set_insert
thf(fact_26_insertI2, axiom,
    ((![A : hoare_1678595023iple_a, B : set_Ho137910533iple_a, B2 : hoare_1678595023iple_a]: ((member1332298086iple_a @ A @ B) => (member1332298086iple_a @ A @ (insert1477804543iple_a @ B2 @ B)))))). % insertI2
thf(fact_27_insertI1, axiom,
    ((![A : hoare_1678595023iple_a, B : set_Ho137910533iple_a]: (member1332298086iple_a @ A @ (insert1477804543iple_a @ A @ B))))). % insertI1
thf(fact_28_insertE, axiom,
    ((![A : hoare_1678595023iple_a, B2 : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((member1332298086iple_a @ A @ (insert1477804543iple_a @ B2 @ A2)) => ((~ ((A = B2))) => (member1332298086iple_a @ A @ A2)))))). % insertE
thf(fact_29_Ball__def, axiom,
    ((ball_H465710501iple_a = (^[A3 : set_Ho137910533iple_a]: (^[P2 : hoare_1678595023iple_a > $o]: (![X3 : hoare_1678595023iple_a]: (((member1332298086iple_a @ X3 @ A3)) => ((P2 @ X3))))))))). % Ball_def
thf(fact_30_singleton__inject, axiom,
    ((![A : hoare_1678595023iple_a, B2 : hoare_1678595023iple_a]: (((insert1477804543iple_a @ A @ bot_bo1298296729iple_a) = (insert1477804543iple_a @ B2 @ bot_bo1298296729iple_a)) => (A = B2))))). % singleton_inject
thf(fact_31_insert__not__empty, axiom,
    ((![A : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: (~ (((insert1477804543iple_a @ A @ A2) = bot_bo1298296729iple_a)))))). % insert_not_empty
thf(fact_32_doubleton__eq__iff, axiom,
    ((![A : hoare_1678595023iple_a, B2 : hoare_1678595023iple_a, C : hoare_1678595023iple_a, D : hoare_1678595023iple_a]: (((insert1477804543iple_a @ A @ (insert1477804543iple_a @ B2 @ bot_bo1298296729iple_a)) = (insert1477804543iple_a @ C @ (insert1477804543iple_a @ D @ bot_bo1298296729iple_a))) = (((((A = C)) & ((B2 = D)))) | ((((A = D)) & ((B2 = C))))))))). % doubleton_eq_iff
thf(fact_33_singleton__iff, axiom,
    ((![B2 : hoare_1678595023iple_a, A : hoare_1678595023iple_a]: ((member1332298086iple_a @ B2 @ (insert1477804543iple_a @ A @ bot_bo1298296729iple_a)) = (B2 = A))))). % singleton_iff
thf(fact_34_the__elem__eq, axiom,
    ((![X2 : hoare_1678595023iple_a]: ((the_el434698138iple_a @ (insert1477804543iple_a @ X2 @ bot_bo1298296729iple_a)) = X2)))). % the_elem_eq
thf(fact_35_is__singletonI, axiom,
    ((![X2 : hoare_1678595023iple_a]: (is_sin1784037339iple_a @ (insert1477804543iple_a @ X2 @ bot_bo1298296729iple_a))))). % is_singletonI
thf(fact_36_Set_Ois__empty__def, axiom,
    ((is_emp901906557iple_a = (^[A3 : set_Ho137910533iple_a]: (A3 = bot_bo1298296729iple_a))))). % Set.is_empty_def
thf(fact_37_is__singleton__def, axiom,
    ((is_sin1784037339iple_a = (^[A3 : set_Ho137910533iple_a]: (?[X3 : hoare_1678595023iple_a]: (A3 = (insert1477804543iple_a @ X3 @ bot_bo1298296729iple_a))))))). % is_singleton_def
thf(fact_38_is__singletonE, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((is_sin1784037339iple_a @ A2) => (~ ((![X4 : hoare_1678595023iple_a]: (~ ((A2 = (insert1477804543iple_a @ X4 @ bot_bo1298296729iple_a))))))))))). % is_singletonE
thf(fact_39_conseq, axiom,
    ((![P : a > state > $o, G : set_Ho137910533iple_a, C : com, Q : a > state > $o]: ((![Z : a, S : state]: ((P @ Z @ S) => (?[P3 : a > state > $o, Q2 : a > state > $o]: ((hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P3 @ C @ Q2) @ bot_bo1298296729iple_a)) & (![S2 : state]: ((![Z2 : a]: ((P3 @ Z2 @ S) => (Q2 @ Z2 @ S2))) => (Q @ Z @ S2))))))) => (hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P @ C @ Q) @ bot_bo1298296729iple_a)))))). % conseq
thf(fact_40_conseq12, axiom,
    ((![G : set_Ho137910533iple_a, P4 : a > state > $o, C : com, Q3 : a > state > $o, P : a > state > $o, Q : a > state > $o]: ((hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P4 @ C @ Q3) @ bot_bo1298296729iple_a)) => ((![Z : a, S : state]: ((P @ Z @ S) => (![S2 : state]: ((![Z2 : a]: ((P4 @ Z2 @ S) => (Q3 @ Z2 @ S2))) => (Q @ Z @ S2))))) => (hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P @ C @ Q) @ bot_bo1298296729iple_a))))))). % conseq12
thf(fact_41_conseq2, axiom,
    ((![G : set_Ho137910533iple_a, P : a > state > $o, C : com, Q3 : a > state > $o, Q : a > state > $o]: ((hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P @ C @ Q3) @ bot_bo1298296729iple_a)) => ((![Z : a, S : state]: ((Q3 @ Z @ S) => (Q @ Z @ S))) => (hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P @ C @ Q) @ bot_bo1298296729iple_a))))))). % conseq2
thf(fact_42_conseq1, axiom,
    ((![G : set_Ho137910533iple_a, P4 : a > state > $o, C : com, Q : a > state > $o, P : a > state > $o]: ((hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P4 @ C @ Q) @ bot_bo1298296729iple_a)) => ((![Z : a, S : state]: ((P @ Z @ S) => (P4 @ Z @ S))) => (hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P @ C @ Q) @ bot_bo1298296729iple_a))))))). % conseq1
thf(fact_43_mem__Collect__eq, axiom,
    ((![A : hoare_1678595023iple_a, P : hoare_1678595023iple_a > $o]: ((member1332298086iple_a @ A @ (collec1600235172iple_a @ P)) = (P @ A))))). % mem_Collect_eq
thf(fact_44_Collect__mem__eq, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((collec1600235172iple_a @ (^[X3 : hoare_1678595023iple_a]: (member1332298086iple_a @ X3 @ A2))) = A2)))). % Collect_mem_eq
thf(fact_45_is__singleton__the__elem, axiom,
    ((is_sin1784037339iple_a = (^[A3 : set_Ho137910533iple_a]: (A3 = (insert1477804543iple_a @ (the_el434698138iple_a @ A3) @ bot_bo1298296729iple_a)))))). % is_singleton_the_elem
thf(fact_46_is__singletonI_H, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((~ ((A2 = bot_bo1298296729iple_a))) => ((![X4 : hoare_1678595023iple_a, Y : hoare_1678595023iple_a]: ((member1332298086iple_a @ X4 @ A2) => ((member1332298086iple_a @ Y @ A2) => (X4 = Y)))) => (is_sin1784037339iple_a @ A2)))))). % is_singletonI'
thf(fact_47_bot__empty__eq, axiom,
    ((bot_bo431311916le_a_o = (^[X3 : hoare_1678595023iple_a]: (member1332298086iple_a @ X3 @ bot_bo1298296729iple_a))))). % bot_empty_eq
thf(fact_48_Collect__empty__eq__bot, axiom,
    ((![P : hoare_1678595023iple_a > $o]: (((collec1600235172iple_a @ P) = bot_bo1298296729iple_a) = (P = bot_bo431311916le_a_o))))). % Collect_empty_eq_bot
thf(fact_49_hoare__derivs_OSkip, axiom,
    ((![G : set_Ho137910533iple_a, P : a > state > $o]: (hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P @ skip @ P) @ bot_bo1298296729iple_a))))). % hoare_derivs.Skip
thf(fact_50_Comp, axiom,
    ((![G : set_Ho137910533iple_a, P : a > state > $o, C : com, Q : a > state > $o, D : com, R : a > state > $o]: ((hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P @ C @ Q) @ bot_bo1298296729iple_a)) => ((hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ Q @ D @ R) @ bot_bo1298296729iple_a)) => (hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P @ (semi @ C @ D) @ R) @ bot_bo1298296729iple_a))))))). % Comp
thf(fact_51_ball__reg, axiom,
    ((![R : set_Ho137910533iple_a, P : hoare_1678595023iple_a > $o, Q : hoare_1678595023iple_a > $o]: ((![X4 : hoare_1678595023iple_a]: ((member1332298086iple_a @ X4 @ R) => ((P @ X4) => (Q @ X4)))) => ((![X4 : hoare_1678595023iple_a]: ((member1332298086iple_a @ X4 @ R) => (P @ X4))) => (![X : hoare_1678595023iple_a]: ((member1332298086iple_a @ X @ R) => (Q @ X)))))))). % ball_reg
thf(fact_52_com_Oinject_I3_J, axiom,
    ((![X41 : com, X42 : com, Y41 : com, Y42 : com]: (((semi @ X41 @ X42) = (semi @ Y41 @ Y42)) = (((X41 = Y41)) & ((X42 = Y42))))))). % com.inject(3)
thf(fact_53_com_Odistinct_I5_J, axiom,
    ((![X41 : com, X42 : com]: (~ ((skip = (semi @ X41 @ X42))))))). % com.distinct(5)
thf(fact_54_triples__valid__Suc, axiom,
    ((![Ts : set_Ho137910533iple_a, N2 : nat]: ((![X4 : hoare_1678595023iple_a]: ((member1332298086iple_a @ X4 @ Ts) => (hoare_1926814542alid_a @ (suc @ N2) @ X4))) => (![X : hoare_1678595023iple_a]: ((member1332298086iple_a @ X @ Ts) => (hoare_1926814542alid_a @ N2 @ X))))))). % triples_valid_Suc
thf(fact_55_triple__valid__Suc, axiom,
    ((![N2 : nat, T : hoare_1678595023iple_a]: ((hoare_1926814542alid_a @ (suc @ N2) @ T) => (hoare_1926814542alid_a @ N2 @ T))))). % triple_valid_Suc
thf(fact_56_old_Onat_Oinject, axiom,
    ((![Nat : nat, Nat2 : nat]: (((suc @ Nat) = (suc @ Nat2)) = (Nat = Nat2))))). % old.nat.inject
thf(fact_57_nat_Oinject, axiom,
    ((![X22 : nat, Y22 : nat]: (((suc @ X22) = (suc @ Y22)) = (X22 = Y22))))). % nat.inject
thf(fact_58_n__not__Suc__n, axiom,
    ((![N2 : nat]: (~ ((N2 = (suc @ N2))))))). % n_not_Suc_n
thf(fact_59_Suc__inject, axiom,
    ((![X2 : nat, Y2 : nat]: (((suc @ X2) = (suc @ Y2)) => (X2 = Y2))))). % Suc_inject
thf(fact_60_pairwise__singleton, axiom,
    ((![P : hoare_1678595023iple_a > hoare_1678595023iple_a > $o, A2 : hoare_1678595023iple_a]: (pairwi531237284iple_a @ P @ (insert1477804543iple_a @ A2 @ bot_bo1298296729iple_a))))). % pairwise_singleton
thf(fact_61_triple__valid__def2, axiom,
    ((![N2 : nat, P : a > state > $o, C : com, Q : a > state > $o]: ((hoare_1926814542alid_a @ N2 @ (hoare_719046530iple_a @ P @ C @ Q)) = (![Z3 : a]: (![S3 : state]: (((P @ Z3 @ S3)) => ((![S4 : state]: (((evaln @ C @ S3 @ N2 @ S4)) => ((Q @ Z3 @ S4)))))))))))). % triple_valid_def2
thf(fact_62_singleton__insert__inj__eq_H, axiom,
    ((![A : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, B2 : hoare_1678595023iple_a]: (((insert1477804543iple_a @ A @ A2) = (insert1477804543iple_a @ B2 @ bot_bo1298296729iple_a)) = (((A = B2)) & ((ord_le1221261669iple_a @ A2 @ (insert1477804543iple_a @ B2 @ bot_bo1298296729iple_a)))))))). % singleton_insert_inj_eq'
thf(fact_63_order__refl, axiom,
    ((![X2 : nat]: (ord_less_eq_nat @ X2 @ X2)))). % order_refl
thf(fact_64_subsetI, axiom,
    ((![A2 : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((![X4 : hoare_1678595023iple_a]: ((member1332298086iple_a @ X4 @ A2) => (member1332298086iple_a @ X4 @ B))) => (ord_le1221261669iple_a @ A2 @ B))))). % subsetI
thf(fact_65_subset__empty, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A2 @ bot_bo1298296729iple_a) = (A2 = bot_bo1298296729iple_a))))). % subset_empty
thf(fact_66_empty__subsetI, axiom,
    ((![A2 : set_Ho137910533iple_a]: (ord_le1221261669iple_a @ bot_bo1298296729iple_a @ A2)))). % empty_subsetI
thf(fact_67_insert__subset, axiom,
    ((![X2 : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ (insert1477804543iple_a @ X2 @ A2) @ B) = (((member1332298086iple_a @ X2 @ B)) & ((ord_le1221261669iple_a @ A2 @ B))))))). % insert_subset
thf(fact_68_singleton__insert__inj__eq, axiom,
    ((![B2 : hoare_1678595023iple_a, A : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: (((insert1477804543iple_a @ B2 @ bot_bo1298296729iple_a) = (insert1477804543iple_a @ A @ A2)) = (((A = B2)) & ((ord_le1221261669iple_a @ A2 @ (insert1477804543iple_a @ B2 @ bot_bo1298296729iple_a)))))))). % singleton_insert_inj_eq
thf(fact_69_lift__Suc__antimono__le, axiom,
    ((![F : nat > nat, N2 : nat, N3 : nat]: ((![N4 : nat]: (ord_less_eq_nat @ (F @ (suc @ N4)) @ (F @ N4))) => ((ord_less_eq_nat @ N2 @ N3) => (ord_less_eq_nat @ (F @ N3) @ (F @ N2))))))). % lift_Suc_antimono_le
thf(fact_70_lift__Suc__mono__le, axiom,
    ((![F : nat > nat, N2 : nat, N3 : nat]: ((![N4 : nat]: (ord_less_eq_nat @ (F @ N4) @ (F @ (suc @ N4)))) => ((ord_less_eq_nat @ N2 @ N3) => (ord_less_eq_nat @ (F @ N2) @ (F @ N3))))))). % lift_Suc_mono_le
thf(fact_71_in__mono, axiom,
    ((![A2 : set_Ho137910533iple_a, B : set_Ho137910533iple_a, X2 : hoare_1678595023iple_a]: ((ord_le1221261669iple_a @ A2 @ B) => ((member1332298086iple_a @ X2 @ A2) => (member1332298086iple_a @ X2 @ B)))))). % in_mono
thf(fact_72_subsetD, axiom,
    ((![A2 : set_Ho137910533iple_a, B : set_Ho137910533iple_a, C : hoare_1678595023iple_a]: ((ord_le1221261669iple_a @ A2 @ B) => ((member1332298086iple_a @ C @ A2) => (member1332298086iple_a @ C @ B)))))). % subsetD
thf(fact_73_pairwiseD, axiom,
    ((![R : hoare_1678595023iple_a > hoare_1678595023iple_a > $o, S5 : set_Ho137910533iple_a, X2 : hoare_1678595023iple_a, Y2 : hoare_1678595023iple_a]: ((pairwi531237284iple_a @ R @ S5) => ((member1332298086iple_a @ X2 @ S5) => ((member1332298086iple_a @ Y2 @ S5) => ((~ ((X2 = Y2))) => (R @ X2 @ Y2)))))))). % pairwiseD
thf(fact_74_pairwiseI, axiom,
    ((![S5 : set_Ho137910533iple_a, R : hoare_1678595023iple_a > hoare_1678595023iple_a > $o]: ((![X4 : hoare_1678595023iple_a, Y : hoare_1678595023iple_a]: ((member1332298086iple_a @ X4 @ S5) => ((member1332298086iple_a @ Y @ S5) => ((~ ((X4 = Y))) => (R @ X4 @ Y))))) => (pairwi531237284iple_a @ R @ S5))))). % pairwiseI
thf(fact_75_subset__eq, axiom,
    ((ord_le1221261669iple_a = (^[A3 : set_Ho137910533iple_a]: (^[B4 : set_Ho137910533iple_a]: (![X3 : hoare_1678595023iple_a]: (((member1332298086iple_a @ X3 @ A3)) => ((member1332298086iple_a @ X3 @ B4))))))))). % subset_eq
thf(fact_76_subset__iff, axiom,
    ((ord_le1221261669iple_a = (^[A3 : set_Ho137910533iple_a]: (^[B4 : set_Ho137910533iple_a]: (![T2 : hoare_1678595023iple_a]: (((member1332298086iple_a @ T2 @ A3)) => ((member1332298086iple_a @ T2 @ B4))))))))). % subset_iff
thf(fact_77_order__subst1, axiom,
    ((![A : nat, F : nat > nat, B2 : nat, C : nat]: ((ord_less_eq_nat @ A @ (F @ B2)) => ((ord_less_eq_nat @ B2 @ C) => ((![X4 : nat, Y : nat]: ((ord_less_eq_nat @ X4 @ Y) => (ord_less_eq_nat @ (F @ X4) @ (F @ Y)))) => (ord_less_eq_nat @ A @ (F @ C)))))))). % order_subst1
thf(fact_78_order__subst2, axiom,
    ((![A : nat, B2 : nat, F : nat > nat, C : nat]: ((ord_less_eq_nat @ A @ B2) => ((ord_less_eq_nat @ (F @ B2) @ C) => ((![X4 : nat, Y : nat]: ((ord_less_eq_nat @ X4 @ Y) => (ord_less_eq_nat @ (F @ X4) @ (F @ Y)))) => (ord_less_eq_nat @ (F @ A) @ C))))))). % order_subst2
thf(fact_79_ord__eq__le__subst, axiom,
    ((![A : nat, F : nat > nat, B2 : nat, C : nat]: ((A = (F @ B2)) => ((ord_less_eq_nat @ B2 @ C) => ((![X4 : nat, Y : nat]: ((ord_less_eq_nat @ X4 @ Y) => (ord_less_eq_nat @ (F @ X4) @ (F @ Y)))) => (ord_less_eq_nat @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_80_ord__le__eq__subst, axiom,
    ((![A : nat, B2 : nat, F : nat > nat, C : nat]: ((ord_less_eq_nat @ A @ B2) => (((F @ B2) = C) => ((![X4 : nat, Y : nat]: ((ord_less_eq_nat @ X4 @ Y) => (ord_less_eq_nat @ (F @ X4) @ (F @ Y)))) => (ord_less_eq_nat @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_81_eq__iff, axiom,
    (((^[Y3 : nat]: (^[Z4 : nat]: (Y3 = Z4))) = (^[X3 : nat]: (^[Y4 : nat]: (((ord_less_eq_nat @ X3 @ Y4)) & ((ord_less_eq_nat @ Y4 @ X3)))))))). % eq_iff
thf(fact_82_antisym, axiom,
    ((![X2 : nat, Y2 : nat]: ((ord_less_eq_nat @ X2 @ Y2) => ((ord_less_eq_nat @ Y2 @ X2) => (X2 = Y2)))))). % antisym
thf(fact_83_linear, axiom,
    ((![X2 : nat, Y2 : nat]: ((ord_less_eq_nat @ X2 @ Y2) | (ord_less_eq_nat @ Y2 @ X2))))). % linear
thf(fact_84_eq__refl, axiom,
    ((![X2 : nat, Y2 : nat]: ((X2 = Y2) => (ord_less_eq_nat @ X2 @ Y2))))). % eq_refl
thf(fact_85_le__cases, axiom,
    ((![X2 : nat, Y2 : nat]: ((~ ((ord_less_eq_nat @ X2 @ Y2))) => (ord_less_eq_nat @ Y2 @ X2))))). % le_cases
thf(fact_86_order_Otrans, axiom,
    ((![A : nat, B2 : nat, C : nat]: ((ord_less_eq_nat @ A @ B2) => ((ord_less_eq_nat @ B2 @ C) => (ord_less_eq_nat @ A @ C)))))). % order.trans
thf(fact_87_le__cases3, axiom,
    ((![X2 : nat, Y2 : nat, Z5 : nat]: (((ord_less_eq_nat @ X2 @ Y2) => (~ ((ord_less_eq_nat @ Y2 @ Z5)))) => (((ord_less_eq_nat @ Y2 @ X2) => (~ ((ord_less_eq_nat @ X2 @ Z5)))) => (((ord_less_eq_nat @ X2 @ Z5) => (~ ((ord_less_eq_nat @ Z5 @ Y2)))) => (((ord_less_eq_nat @ Z5 @ Y2) => (~ ((ord_less_eq_nat @ Y2 @ X2)))) => (((ord_less_eq_nat @ Y2 @ Z5) => (~ ((ord_less_eq_nat @ Z5 @ X2)))) => (~ (((ord_less_eq_nat @ Z5 @ X2) => (~ ((ord_less_eq_nat @ X2 @ Y2)))))))))))))). % le_cases3
thf(fact_88_antisym__conv, axiom,
    ((![Y2 : nat, X2 : nat]: ((ord_less_eq_nat @ Y2 @ X2) => ((ord_less_eq_nat @ X2 @ Y2) = (X2 = Y2)))))). % antisym_conv
thf(fact_89_order__class_Oorder_Oeq__iff, axiom,
    (((^[Y3 : nat]: (^[Z4 : nat]: (Y3 = Z4))) = (^[A4 : nat]: (^[B5 : nat]: (((ord_less_eq_nat @ A4 @ B5)) & ((ord_less_eq_nat @ B5 @ A4)))))))). % order_class.order.eq_iff
thf(fact_90_ord__eq__le__trans, axiom,
    ((![A : nat, B2 : nat, C : nat]: ((A = B2) => ((ord_less_eq_nat @ B2 @ C) => (ord_less_eq_nat @ A @ C)))))). % ord_eq_le_trans
thf(fact_91_ord__le__eq__trans, axiom,
    ((![A : nat, B2 : nat, C : nat]: ((ord_less_eq_nat @ A @ B2) => ((B2 = C) => (ord_less_eq_nat @ A @ C)))))). % ord_le_eq_trans
thf(fact_92_order__class_Oorder_Oantisym, axiom,
    ((![A : nat, B2 : nat]: ((ord_less_eq_nat @ A @ B2) => ((ord_less_eq_nat @ B2 @ A) => (A = B2)))))). % order_class.order.antisym
thf(fact_93_order__trans, axiom,
    ((![X2 : nat, Y2 : nat, Z5 : nat]: ((ord_less_eq_nat @ X2 @ Y2) => ((ord_less_eq_nat @ Y2 @ Z5) => (ord_less_eq_nat @ X2 @ Z5)))))). % order_trans
thf(fact_94_dual__order_Orefl, axiom,
    ((![A : nat]: (ord_less_eq_nat @ A @ A)))). % dual_order.refl
thf(fact_95_linorder__wlog, axiom,
    ((![P : nat > nat > $o, A : nat, B2 : nat]: ((![A5 : nat, B6 : nat]: ((ord_less_eq_nat @ A5 @ B6) => (P @ A5 @ B6))) => ((![A5 : nat, B6 : nat]: ((P @ B6 @ A5) => (P @ A5 @ B6))) => (P @ A @ B2)))))). % linorder_wlog
thf(fact_96_dual__order_Otrans, axiom,
    ((![B2 : nat, A : nat, C : nat]: ((ord_less_eq_nat @ B2 @ A) => ((ord_less_eq_nat @ C @ B2) => (ord_less_eq_nat @ C @ A)))))). % dual_order.trans
thf(fact_97_dual__order_Oeq__iff, axiom,
    (((^[Y3 : nat]: (^[Z4 : nat]: (Y3 = Z4))) = (^[A4 : nat]: (^[B5 : nat]: (((ord_less_eq_nat @ B5 @ A4)) & ((ord_less_eq_nat @ A4 @ B5)))))))). % dual_order.eq_iff
thf(fact_98_dual__order_Oantisym, axiom,
    ((![B2 : nat, A : nat]: ((ord_less_eq_nat @ B2 @ A) => ((ord_less_eq_nat @ A @ B2) => (A = B2)))))). % dual_order.antisym
thf(fact_99_subset__insertI2, axiom,
    ((![A2 : set_Ho137910533iple_a, B : set_Ho137910533iple_a, B2 : hoare_1678595023iple_a]: ((ord_le1221261669iple_a @ A2 @ B) => (ord_le1221261669iple_a @ A2 @ (insert1477804543iple_a @ B2 @ B)))))). % subset_insertI2
thf(fact_100_subset__insertI, axiom,
    ((![B : set_Ho137910533iple_a, A : hoare_1678595023iple_a]: (ord_le1221261669iple_a @ B @ (insert1477804543iple_a @ A @ B))))). % subset_insertI
thf(fact_101_subset__insert, axiom,
    ((![X2 : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((~ ((member1332298086iple_a @ X2 @ A2))) => ((ord_le1221261669iple_a @ A2 @ (insert1477804543iple_a @ X2 @ B)) = (ord_le1221261669iple_a @ A2 @ B)))))). % subset_insert
thf(fact_102_insert__mono, axiom,
    ((![C3 : set_Ho137910533iple_a, D2 : set_Ho137910533iple_a, A : hoare_1678595023iple_a]: ((ord_le1221261669iple_a @ C3 @ D2) => (ord_le1221261669iple_a @ (insert1477804543iple_a @ A @ C3) @ (insert1477804543iple_a @ A @ D2)))))). % insert_mono
thf(fact_103_bot_Oextremum__uniqueI, axiom,
    ((![A : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A @ bot_bo1298296729iple_a) => (A = bot_bo1298296729iple_a))))). % bot.extremum_uniqueI
thf(fact_104_bot_Oextremum__uniqueI, axiom,
    ((![A : nat]: ((ord_less_eq_nat @ A @ bot_bot_nat) => (A = bot_bot_nat))))). % bot.extremum_uniqueI
thf(fact_105_bot_Oextremum__unique, axiom,
    ((![A : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ A @ bot_bo1298296729iple_a) = (A = bot_bo1298296729iple_a))))). % bot.extremum_unique
thf(fact_106_bot_Oextremum__unique, axiom,
    ((![A : nat]: ((ord_less_eq_nat @ A @ bot_bot_nat) = (A = bot_bot_nat))))). % bot.extremum_unique
thf(fact_107_bot_Oextremum, axiom,
    ((![A : set_Ho137910533iple_a]: (ord_le1221261669iple_a @ bot_bo1298296729iple_a @ A)))). % bot.extremum
thf(fact_108_bot_Oextremum, axiom,
    ((![A : nat]: (ord_less_eq_nat @ bot_bot_nat @ A)))). % bot.extremum
thf(fact_109_thin, axiom,
    ((![G2 : set_Ho137910533iple_a, Ts : set_Ho137910533iple_a, G : set_Ho137910533iple_a]: ((hoare_129598474rivs_a @ G2 @ Ts) => ((ord_le1221261669iple_a @ G2 @ G) => (hoare_129598474rivs_a @ G @ Ts)))))). % thin
thf(fact_110_asm, axiom,
    ((![Ts : set_Ho137910533iple_a, G : set_Ho137910533iple_a]: ((ord_le1221261669iple_a @ Ts @ G) => (hoare_129598474rivs_a @ G @ Ts))))). % asm
thf(fact_111_weaken, axiom,
    ((![G : set_Ho137910533iple_a, Ts3 : set_Ho137910533iple_a, Ts : set_Ho137910533iple_a]: ((hoare_129598474rivs_a @ G @ Ts3) => ((ord_le1221261669iple_a @ Ts @ Ts3) => (hoare_129598474rivs_a @ G @ Ts)))))). % weaken
thf(fact_112_pairwise__empty, axiom,
    ((![P : hoare_1678595023iple_a > hoare_1678595023iple_a > $o]: (pairwi531237284iple_a @ P @ bot_bo1298296729iple_a)))). % pairwise_empty
thf(fact_113_pairwise__insert, axiom,
    ((![R2 : hoare_1678595023iple_a > hoare_1678595023iple_a > $o, X2 : hoare_1678595023iple_a, S6 : set_Ho137910533iple_a]: ((pairwi531237284iple_a @ R2 @ (insert1477804543iple_a @ X2 @ S6)) = (((![Y4 : hoare_1678595023iple_a]: (((((member1332298086iple_a @ Y4 @ S6)) & ((~ ((Y4 = X2)))))) => ((((R2 @ X2 @ Y4)) & ((R2 @ Y4 @ X2))))))) & ((pairwi531237284iple_a @ R2 @ S6))))))). % pairwise_insert
thf(fact_114_subset__singletonD, axiom,
    ((![A2 : set_Ho137910533iple_a, X2 : hoare_1678595023iple_a]: ((ord_le1221261669iple_a @ A2 @ (insert1477804543iple_a @ X2 @ bot_bo1298296729iple_a)) => ((A2 = bot_bo1298296729iple_a) | (A2 = (insert1477804543iple_a @ X2 @ bot_bo1298296729iple_a))))))). % subset_singletonD
thf(fact_115_subset__singleton__iff, axiom,
    ((![X5 : set_Ho137910533iple_a, A : hoare_1678595023iple_a]: ((ord_le1221261669iple_a @ X5 @ (insert1477804543iple_a @ A @ bot_bo1298296729iple_a)) = (((X5 = bot_bo1298296729iple_a)) | ((X5 = (insert1477804543iple_a @ A @ bot_bo1298296729iple_a)))))))). % subset_singleton_iff
thf(fact_116_evaln__elim__cases_I1_J, axiom,
    ((![S6 : state, N2 : nat, T : state]: ((evaln @ skip @ S6 @ N2 @ T) => (T = S6))))). % evaln_elim_cases(1)
thf(fact_117_evaln_OSkip, axiom,
    ((![S6 : state, N2 : nat]: (evaln @ skip @ S6 @ N2 @ S6)))). % evaln.Skip
thf(fact_118_evaln__elim__cases_I4_J, axiom,
    ((![C1 : com, C22 : com, S6 : state, N2 : nat, T : state]: ((evaln @ (semi @ C1 @ C22) @ S6 @ N2 @ T) => (~ ((![S1 : state]: ((evaln @ C1 @ S6 @ N2 @ S1) => (~ ((evaln @ C22 @ S1 @ N2 @ T))))))))))). % evaln_elim_cases(4)
thf(fact_119_evaln_OSemi, axiom,
    ((![C0 : com, S0 : state, N2 : nat, S12 : state, C1 : com, S22 : state]: ((evaln @ C0 @ S0 @ N2 @ S12) => ((evaln @ C1 @ S12 @ N2 @ S22) => (evaln @ (semi @ C0 @ C1) @ S0 @ N2 @ S22)))))). % evaln.Semi
thf(fact_120_Suc__le__mono, axiom,
    ((![N2 : nat, M : nat]: ((ord_less_eq_nat @ (suc @ N2) @ (suc @ M)) = (ord_less_eq_nat @ N2 @ M))))). % Suc_le_mono
thf(fact_121_transitive__stepwise__le, axiom,
    ((![M : nat, N2 : nat, R : nat > nat > $o]: ((ord_less_eq_nat @ M @ N2) => ((![X4 : nat]: (R @ X4 @ X4)) => ((![X4 : nat, Y : nat, Z6 : nat]: ((R @ X4 @ Y) => ((R @ Y @ Z6) => (R @ X4 @ Z6)))) => ((![N4 : nat]: (R @ N4 @ (suc @ N4))) => (R @ M @ N2)))))))). % transitive_stepwise_le
thf(fact_122_nat__induct__at__least, axiom,
    ((![M : nat, N2 : nat, P : nat > $o]: ((ord_less_eq_nat @ M @ N2) => ((P @ M) => ((![N4 : nat]: ((ord_less_eq_nat @ M @ N4) => ((P @ N4) => (P @ (suc @ N4))))) => (P @ N2))))))). % nat_induct_at_least
thf(fact_123_full__nat__induct, axiom,
    ((![P : nat > $o, N2 : nat]: ((![N4 : nat]: ((![M2 : nat]: ((ord_less_eq_nat @ (suc @ M2) @ N4) => (P @ M2))) => (P @ N4))) => (P @ N2))))). % full_nat_induct
thf(fact_124_not__less__eq__eq, axiom,
    ((![M : nat, N2 : nat]: ((~ ((ord_less_eq_nat @ M @ N2))) = (ord_less_eq_nat @ (suc @ N2) @ M))))). % not_less_eq_eq

% Conjectures (5)
thf(conj_0, hypothesis,
    ((hoare_129598474rivs_a @ ga @ (insert1477804543iple_a @ t @ bot_bo1298296729iple_a)))).
thf(conj_1, hypothesis,
    ((![N5 : nat]: ((![X4 : hoare_1678595023iple_a]: ((member1332298086iple_a @ X4 @ ga) => (hoare_1926814542alid_a @ N5 @ X4))) => (![X : hoare_1678595023iple_a]: ((member1332298086iple_a @ X @ (insert1477804543iple_a @ t @ bot_bo1298296729iple_a)) => (hoare_1926814542alid_a @ N5 @ X))))))).
thf(conj_2, hypothesis,
    ((hoare_129598474rivs_a @ ga @ tsa))).
thf(conj_3, hypothesis,
    ((![N5 : nat]: ((![X4 : hoare_1678595023iple_a]: ((member1332298086iple_a @ X4 @ ga) => (hoare_1926814542alid_a @ N5 @ X4))) => (![X : hoare_1678595023iple_a]: ((member1332298086iple_a @ X @ tsa) => (hoare_1926814542alid_a @ N5 @ X))))))).
thf(conj_4, conjecture,
    ((![N4 : nat]: ((?[X : hoare_1678595023iple_a]: ((member1332298086iple_a @ X @ ga) & (~ ((hoare_1926814542alid_a @ N4 @ X))))) | (![X4 : hoare_1678595023iple_a]: ((member1332298086iple_a @ X4 @ (insert1477804543iple_a @ t @ tsa)) => (hoare_1926814542alid_a @ N4 @ X4))))))).
