% 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_237__3251858_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:12:57.153

% Could-be-implicit typings (10)
thf(ty_n_t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__b_J_J, type,
    set_Ho1277474822iple_b : $tType).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J, type,
    set_Product_prod_a_a : $tType).
thf(ty_n_t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__b_J, type,
    hoare_1686856528iple_b : $tType).
thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mtf__a_J_J, type,
    set_Sum_sum_a_a : $tType).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J, type,
    set_set_a : $tType).
thf(ty_n_t__Set__Oset_Itf__a_J, type,
    set_a : $tType).
thf(ty_n_t__Com__Ostate, type,
    state : $tType).
thf(ty_n_t__Com__Ocom, type,
    com : $tType).
thf(ty_n_tf__b, type,
    b : $tType).
thf(ty_n_tf__a, type,
    a : $tType).

% Explicit typings (42)
thf(sy_c_Com_Ocom_OSKIP, type,
    skip : com).
thf(sy_c_Com_Ocom_OSemi, type,
    semi : com > com > com).
thf(sy_c_Com_Ocom_OWhile, type,
    while : (state > $o) > com > com).
thf(sy_c_Finite__Set_Ofinite_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__b_J, type,
    finite1300473319iple_b : set_Ho1277474822iple_b > $o).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_Itf__a_Mtf__a_J, type,
    finite179568208od_a_a : set_Product_prod_a_a > $o).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_Itf__a_J, type,
    finite_finite_set_a : set_set_a > $o).
thf(sy_c_Finite__Set_Ofinite_001t__Sum____Type__Osum_Itf__a_Mtf__a_J, type,
    finite935078844um_a_a : set_Sum_sum_a_a > $o).
thf(sy_c_Finite__Set_Ofinite_001tf__a, type,
    finite_finite_a : set_a > $o).
thf(sy_c_Finite__Set_Ofold_001tf__a_001t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__b_J_J, type,
    finite393365052iple_b : (a > set_Ho1277474822iple_b > set_Ho1277474822iple_b) > set_Ho1277474822iple_b > set_a > set_Ho1277474822iple_b).
thf(sy_c_HOL_OThe_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__b_J, type,
    the_Ho1610761865iple_b : (hoare_1686856528iple_b > $o) > hoare_1686856528iple_b).
thf(sy_c_Hoare__Mirabelle__raqjowkjvm_Ohoare__derivs_001tf__b, type,
    hoare_129598475rivs_b : set_Ho1277474822iple_b > set_Ho1277474822iple_b > $o).
thf(sy_c_Hoare__Mirabelle__raqjowkjvm_Otriple_Otriple_001tf__b, type,
    hoare_719046531iple_b : (b > state > $o) > com > (b > state > $o) > hoare_1686856528iple_b).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__b_J_M_Eo_J, type,
    bot_bo1259081323le_b_o : hoare_1686856528iple_b > $o).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__b_J_J, type,
    bot_bo290377370iple_b : set_Ho1277474822iple_b).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J, type,
    bot_bot_set_a : set_a).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__b_J_J, type,
    top_to1883285174iple_b : set_Ho1277474822iple_b).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J, type,
    top_to398455383od_a_a : set_Product_prod_a_a).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_Itf__a_J_J, type,
    top_top_set_set_a : set_set_a).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mtf__a_J_J, type,
    top_to1089105419um_a_a : set_Sum_sum_a_a).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__a_J, type,
    top_top_set_a : set_a).
thf(sy_c_Set_OCollect_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__b_J, type,
    collec1608496677iple_b : (hoare_1686856528iple_b > $o) > set_Ho1277474822iple_b).
thf(sy_c_Set_OCollect_001tf__a, type,
    collect_a : (a > $o) > set_a).
thf(sy_c_Set_Obind_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__b_J_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__b_J, type,
    bind_H1262675666iple_b : set_Ho1277474822iple_b > (hoare_1686856528iple_b > set_Ho1277474822iple_b) > set_Ho1277474822iple_b).
thf(sy_c_Set_Obind_001tf__a_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__b_J, type,
    bind_a1120646331iple_b : set_a > (a > set_Ho1277474822iple_b) > set_Ho1277474822iple_b).
thf(sy_c_Set_Obind_001tf__a_001tf__a, type,
    bind_a_a : set_a > (a > set_a) > set_a).
thf(sy_c_Set_Oimage_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__b_J_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__b_J, type,
    image_482623520iple_b : (hoare_1686856528iple_b > hoare_1686856528iple_b) > set_Ho1277474822iple_b > set_Ho1277474822iple_b).
thf(sy_c_Set_Oimage_001tf__a_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__b_J, type,
    image_646034953iple_b : (a > hoare_1686856528iple_b) > set_a > set_Ho1277474822iple_b).
thf(sy_c_Set_Oimage_001tf__a_001tf__a, type,
    image_a_a : (a > a) > set_a > set_a).
thf(sy_c_Set_Oinsert_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__b_J, type,
    insert1486066048iple_b : hoare_1686856528iple_b > set_Ho1277474822iple_b > set_Ho1277474822iple_b).
thf(sy_c_Set_Oinsert_001tf__a, type,
    insert_a : a > set_a > set_a).
thf(sy_c_Set_Ois__empty_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__b_J, type,
    is_emp910168062iple_b : set_Ho1277474822iple_b > $o).
thf(sy_c_Set_Ois__singleton_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__b_J, type,
    is_sin1792298844iple_b : set_Ho1277474822iple_b > $o).
thf(sy_c_Set_Othe__elem_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__b_J, type,
    the_el442959643iple_b : set_Ho1277474822iple_b > hoare_1686856528iple_b).
thf(sy_c_member_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__b_J, type,
    member1340559591iple_b : hoare_1686856528iple_b > set_Ho1277474822iple_b > $o).
thf(sy_c_member_001tf__a, type,
    member_a : a > set_a > $o).
thf(sy_v_G, type,
    g : set_Ho1277474822iple_b).
thf(sy_v_P, type,
    p : a > b > state > $o).
thf(sy_v_P_H, type,
    p2 : a > b > state > $o).
thf(sy_v_Q, type,
    q : a > b > state > $o).
thf(sy_v_Q_H, type,
    q2 : a > b > state > $o).
thf(sy_v_U, type,
    u : set_a).
thf(sy_v_c0, type,
    c0 : a > com).

% Relevant facts (138)
thf(fact_0_triple_Oinject, axiom,
    ((![X1 : b > state > $o, X2 : com, X3 : b > state > $o, Y1 : b > state > $o, Y2 : com, Y3 : b > state > $o]: (((hoare_719046531iple_b @ X1 @ X2 @ X3) = (hoare_719046531iple_b @ Y1 @ Y2 @ Y3)) = (((X1 = Y1)) & ((((X2 = Y2)) & ((X3 = Y3))))))))). % triple.inject
thf(fact_1_escape, axiom,
    ((![P : b > state > $o, G : set_Ho1277474822iple_b, C : com, Q : b > state > $o]: ((![Z : b, S : state]: ((P @ Z @ S) => (hoare_129598475rivs_b @ G @ (insert1486066048iple_b @ (hoare_719046531iple_b @ (^[Za : b]: (^[S2 : state]: (S2 = S))) @ C @ (^[Z2 : b]: (Q @ Z))) @ bot_bo290377370iple_b)))) => (hoare_129598475rivs_b @ G @ (insert1486066048iple_b @ (hoare_719046531iple_b @ P @ C @ Q) @ bot_bo290377370iple_b)))))). % escape
thf(fact_2_conseq1, axiom,
    ((![G : set_Ho1277474822iple_b, P2 : b > state > $o, C : com, Q : b > state > $o, P : b > state > $o]: ((hoare_129598475rivs_b @ G @ (insert1486066048iple_b @ (hoare_719046531iple_b @ P2 @ C @ Q) @ bot_bo290377370iple_b)) => ((![Z : b, S : state]: ((P @ Z @ S) => (P2 @ Z @ S))) => (hoare_129598475rivs_b @ G @ (insert1486066048iple_b @ (hoare_719046531iple_b @ P @ C @ Q) @ bot_bo290377370iple_b))))))). % conseq1
thf(fact_3_conseq2, axiom,
    ((![G : set_Ho1277474822iple_b, P : b > state > $o, C : com, Q2 : b > state > $o, Q : b > state > $o]: ((hoare_129598475rivs_b @ G @ (insert1486066048iple_b @ (hoare_719046531iple_b @ P @ C @ Q2) @ bot_bo290377370iple_b)) => ((![Z : b, S : state]: ((Q2 @ Z @ S) => (Q @ Z @ S))) => (hoare_129598475rivs_b @ G @ (insert1486066048iple_b @ (hoare_719046531iple_b @ P @ C @ Q) @ bot_bo290377370iple_b))))))). % conseq2
thf(fact_4_conseq12, axiom,
    ((![G : set_Ho1277474822iple_b, P2 : b > state > $o, C : com, Q2 : b > state > $o, P : b > state > $o, Q : b > state > $o]: ((hoare_129598475rivs_b @ G @ (insert1486066048iple_b @ (hoare_719046531iple_b @ P2 @ C @ Q2) @ bot_bo290377370iple_b)) => ((![Z : b, S : state]: ((P @ Z @ S) => (![S3 : state]: ((![Z3 : b]: ((P2 @ Z3 @ S) => (Q2 @ Z3 @ S3))) => (Q @ Z @ S3))))) => (hoare_129598475rivs_b @ G @ (insert1486066048iple_b @ (hoare_719046531iple_b @ P @ C @ Q) @ bot_bo290377370iple_b))))))). % conseq12
thf(fact_5__Cconstant_C, axiom,
    ((![C2 : $o, G : set_Ho1277474822iple_b, P : b > state > $o, C : com, Q : b > state > $o]: ((C2 => (hoare_129598475rivs_b @ G @ (insert1486066048iple_b @ (hoare_719046531iple_b @ P @ C @ Q) @ bot_bo290377370iple_b))) => (hoare_129598475rivs_b @ G @ (insert1486066048iple_b @ (hoare_719046531iple_b @ (^[Z4 : b]: (^[S4 : state]: (((P @ Z4 @ S4)) & (C2)))) @ C @ Q) @ bot_bo290377370iple_b)))))). % "constant"
thf(fact_6_triple_Oinduct, axiom,
    ((![P : hoare_1686856528iple_b > $o, Triple : hoare_1686856528iple_b]: ((![X1a : b > state > $o, X2a : com, X3a : b > state > $o]: (P @ (hoare_719046531iple_b @ X1a @ X2a @ X3a))) => (P @ Triple))))). % triple.induct
thf(fact_7_derivs__insertD, axiom,
    ((![G : set_Ho1277474822iple_b, T : hoare_1686856528iple_b, Ts : set_Ho1277474822iple_b]: ((hoare_129598475rivs_b @ G @ (insert1486066048iple_b @ T @ Ts)) => ((hoare_129598475rivs_b @ G @ (insert1486066048iple_b @ T @ bot_bo290377370iple_b)) & (hoare_129598475rivs_b @ G @ Ts)))))). % derivs_insertD
thf(fact_8_triple_Oexhaust, axiom,
    ((![Y : hoare_1686856528iple_b]: (~ ((![X12 : b > state > $o, X22 : com, X32 : b > state > $o]: (~ ((Y = (hoare_719046531iple_b @ X12 @ X22 @ X32)))))))))). % triple.exhaust
thf(fact_9_cut, axiom,
    ((![G2 : set_Ho1277474822iple_b, Ts : set_Ho1277474822iple_b, G : set_Ho1277474822iple_b]: ((hoare_129598475rivs_b @ G2 @ Ts) => ((hoare_129598475rivs_b @ G @ G2) => (hoare_129598475rivs_b @ G @ Ts)))))). % cut
thf(fact_10_empty, axiom,
    ((![G : set_Ho1277474822iple_b]: (hoare_129598475rivs_b @ G @ bot_bo290377370iple_b)))). % empty
thf(fact_11_conseq, axiom,
    ((![P : b > state > $o, G : set_Ho1277474822iple_b, C : com, Q : b > state > $o]: ((![Z : b, S : state]: ((P @ Z @ S) => (?[P3 : b > state > $o, Q3 : b > state > $o]: ((hoare_129598475rivs_b @ G @ (insert1486066048iple_b @ (hoare_719046531iple_b @ P3 @ C @ Q3) @ bot_bo290377370iple_b)) & (![S3 : state]: ((![Z3 : b]: ((P3 @ Z3 @ S) => (Q3 @ Z3 @ S3))) => (Q @ Z @ S3))))))) => (hoare_129598475rivs_b @ G @ (insert1486066048iple_b @ (hoare_719046531iple_b @ P @ C @ Q) @ bot_bo290377370iple_b)))))). % conseq
thf(fact_12_hoare__derivs_Oinsert, axiom,
    ((![G : set_Ho1277474822iple_b, T : hoare_1686856528iple_b, Ts : set_Ho1277474822iple_b]: ((hoare_129598475rivs_b @ G @ (insert1486066048iple_b @ T @ bot_bo290377370iple_b)) => ((hoare_129598475rivs_b @ G @ Ts) => (hoare_129598475rivs_b @ G @ (insert1486066048iple_b @ T @ Ts))))))). % hoare_derivs.insert
thf(fact_13_singleton__conv, axiom,
    ((![A : hoare_1686856528iple_b]: ((collec1608496677iple_b @ (^[X : hoare_1686856528iple_b]: (X = A))) = (insert1486066048iple_b @ A @ bot_bo290377370iple_b))))). % singleton_conv
thf(fact_14_singleton__conv2, axiom,
    ((![A : hoare_1686856528iple_b]: ((collec1608496677iple_b @ ((^[Y4 : hoare_1686856528iple_b]: (^[Z5 : hoare_1686856528iple_b]: (Y4 = Z5))) @ A)) = (insert1486066048iple_b @ A @ bot_bo290377370iple_b))))). % singleton_conv2
thf(fact_15_finite__insert, axiom,
    ((![A : hoare_1686856528iple_b, A2 : set_Ho1277474822iple_b]: ((finite1300473319iple_b @ (insert1486066048iple_b @ A @ A2)) = (finite1300473319iple_b @ A2))))). % finite_insert
thf(fact_16_finite__insert, axiom,
    ((![A : a, A2 : set_a]: ((finite_finite_a @ (insert_a @ A @ A2)) = (finite_finite_a @ A2))))). % finite_insert
thf(fact_17_singletonI, axiom,
    ((![A : hoare_1686856528iple_b]: (member1340559591iple_b @ A @ (insert1486066048iple_b @ A @ bot_bo290377370iple_b))))). % singletonI
thf(fact_18_image__insert, axiom,
    ((![F : a > hoare_1686856528iple_b, A : a, B : set_a]: ((image_646034953iple_b @ F @ (insert_a @ A @ B)) = (insert1486066048iple_b @ (F @ A) @ (image_646034953iple_b @ F @ B)))))). % image_insert
thf(fact_19_image__insert, axiom,
    ((![F : hoare_1686856528iple_b > hoare_1686856528iple_b, A : hoare_1686856528iple_b, B : set_Ho1277474822iple_b]: ((image_482623520iple_b @ F @ (insert1486066048iple_b @ A @ B)) = (insert1486066048iple_b @ (F @ A) @ (image_482623520iple_b @ F @ B)))))). % image_insert
thf(fact_20_insert__image, axiom,
    ((![X4 : a, A2 : set_a, F : a > hoare_1686856528iple_b]: ((member_a @ X4 @ A2) => ((insert1486066048iple_b @ (F @ X4) @ (image_646034953iple_b @ F @ A2)) = (image_646034953iple_b @ F @ A2)))))). % insert_image
thf(fact_21_finite__imageI, axiom,
    ((![F2 : set_a, H : a > hoare_1686856528iple_b]: ((finite_finite_a @ F2) => (finite1300473319iple_b @ (image_646034953iple_b @ H @ F2)))))). % finite_imageI
thf(fact_22_finite__imageI, axiom,
    ((![F2 : set_a, H : a > a]: ((finite_finite_a @ F2) => (finite_finite_a @ (image_a_a @ H @ F2)))))). % finite_imageI
thf(fact_23_image__empty, axiom,
    ((![F : a > hoare_1686856528iple_b]: ((image_646034953iple_b @ F @ bot_bot_set_a) = bot_bo290377370iple_b)))). % image_empty
thf(fact_24_image__empty, axiom,
    ((![F : hoare_1686856528iple_b > hoare_1686856528iple_b]: ((image_482623520iple_b @ F @ bot_bo290377370iple_b) = bot_bo290377370iple_b)))). % image_empty
thf(fact_25_empty__is__image, axiom,
    ((![F : a > hoare_1686856528iple_b, A2 : set_a]: ((bot_bo290377370iple_b = (image_646034953iple_b @ F @ A2)) = (A2 = bot_bot_set_a))))). % empty_is_image
thf(fact_26_empty__is__image, axiom,
    ((![F : hoare_1686856528iple_b > hoare_1686856528iple_b, A2 : set_Ho1277474822iple_b]: ((bot_bo290377370iple_b = (image_482623520iple_b @ F @ A2)) = (A2 = bot_bo290377370iple_b))))). % empty_is_image
thf(fact_27_image__is__empty, axiom,
    ((![F : a > hoare_1686856528iple_b, A2 : set_a]: (((image_646034953iple_b @ F @ A2) = bot_bo290377370iple_b) = (A2 = bot_bot_set_a))))). % image_is_empty
thf(fact_28_image__is__empty, axiom,
    ((![F : hoare_1686856528iple_b > hoare_1686856528iple_b, A2 : set_Ho1277474822iple_b]: (((image_482623520iple_b @ F @ A2) = bot_bo290377370iple_b) = (A2 = bot_bo290377370iple_b))))). % image_is_empty
thf(fact_29_finite__Collect__conjI, axiom,
    ((![P : a > $o, Q : a > $o]: (((finite_finite_a @ (collect_a @ P)) | (finite_finite_a @ (collect_a @ Q))) => (finite_finite_a @ (collect_a @ (^[X : a]: (((P @ X)) & ((Q @ X)))))))))). % finite_Collect_conjI
thf(fact_30_finite__Collect__disjI, axiom,
    ((![P : a > $o, Q : a > $o]: ((finite_finite_a @ (collect_a @ (^[X : a]: (((P @ X)) | ((Q @ X)))))) = (((finite_finite_a @ (collect_a @ P))) & ((finite_finite_a @ (collect_a @ Q)))))))). % finite_Collect_disjI
thf(fact_31_image__eqI, axiom,
    ((![B2 : hoare_1686856528iple_b, F : a > hoare_1686856528iple_b, X4 : a, A2 : set_a]: ((B2 = (F @ X4)) => ((member_a @ X4 @ A2) => (member1340559591iple_b @ B2 @ (image_646034953iple_b @ F @ A2))))))). % image_eqI
thf(fact_32_empty__Collect__eq, axiom,
    ((![P : hoare_1686856528iple_b > $o]: ((bot_bo290377370iple_b = (collec1608496677iple_b @ P)) = (![X : hoare_1686856528iple_b]: (~ ((P @ X)))))))). % empty_Collect_eq
thf(fact_33_Collect__empty__eq, axiom,
    ((![P : hoare_1686856528iple_b > $o]: (((collec1608496677iple_b @ P) = bot_bo290377370iple_b) = (![X : hoare_1686856528iple_b]: (~ ((P @ X)))))))). % Collect_empty_eq
thf(fact_34_all__not__in__conv, axiom,
    ((![A2 : set_Ho1277474822iple_b]: ((![X : hoare_1686856528iple_b]: (~ ((member1340559591iple_b @ X @ A2)))) = (A2 = bot_bo290377370iple_b))))). % all_not_in_conv
thf(fact_35_empty__iff, axiom,
    ((![C : hoare_1686856528iple_b]: (~ ((member1340559591iple_b @ C @ bot_bo290377370iple_b)))))). % empty_iff
thf(fact_36_insert__absorb2, axiom,
    ((![X4 : hoare_1686856528iple_b, A2 : set_Ho1277474822iple_b]: ((insert1486066048iple_b @ X4 @ (insert1486066048iple_b @ X4 @ A2)) = (insert1486066048iple_b @ X4 @ A2))))). % insert_absorb2
thf(fact_37_insert__iff, axiom,
    ((![A : hoare_1686856528iple_b, B2 : hoare_1686856528iple_b, A2 : set_Ho1277474822iple_b]: ((member1340559591iple_b @ A @ (insert1486066048iple_b @ B2 @ A2)) = (((A = B2)) | ((member1340559591iple_b @ A @ A2))))))). % insert_iff
thf(fact_38_insertCI, axiom,
    ((![A : hoare_1686856528iple_b, B : set_Ho1277474822iple_b, B2 : hoare_1686856528iple_b]: (((~ ((member1340559591iple_b @ A @ B))) => (A = B2)) => (member1340559591iple_b @ A @ (insert1486066048iple_b @ B2 @ B)))))). % insertCI
thf(fact_39_rev__image__eqI, axiom,
    ((![X4 : a, A2 : set_a, B2 : hoare_1686856528iple_b, F : a > hoare_1686856528iple_b]: ((member_a @ X4 @ A2) => ((B2 = (F @ X4)) => (member1340559591iple_b @ B2 @ (image_646034953iple_b @ F @ A2))))))). % rev_image_eqI
thf(fact_40_ball__imageD, axiom,
    ((![F : a > hoare_1686856528iple_b, A2 : set_a, P : hoare_1686856528iple_b > $o]: ((![X5 : hoare_1686856528iple_b]: ((member1340559591iple_b @ X5 @ (image_646034953iple_b @ F @ A2)) => (P @ X5))) => (![X6 : a]: ((member_a @ X6 @ A2) => (P @ (F @ X6)))))))). % ball_imageD
thf(fact_41_image__cong, axiom,
    ((![M : set_a, N : set_a, F : a > hoare_1686856528iple_b, G3 : a > hoare_1686856528iple_b]: ((M = N) => ((![X5 : a]: ((member_a @ X5 @ N) => ((F @ X5) = (G3 @ X5)))) => ((image_646034953iple_b @ F @ M) = (image_646034953iple_b @ G3 @ N))))))). % image_cong
thf(fact_42_bex__imageD, axiom,
    ((![F : a > hoare_1686856528iple_b, A2 : set_a, P : hoare_1686856528iple_b > $o]: ((?[X6 : hoare_1686856528iple_b]: ((member1340559591iple_b @ X6 @ (image_646034953iple_b @ F @ A2)) & (P @ X6))) => (?[X5 : a]: ((member_a @ X5 @ A2) & (P @ (F @ X5)))))))). % bex_imageD
thf(fact_43_image__iff, axiom,
    ((![Z6 : hoare_1686856528iple_b, F : a > hoare_1686856528iple_b, A2 : set_a]: ((member1340559591iple_b @ Z6 @ (image_646034953iple_b @ F @ A2)) = (?[X : a]: (((member_a @ X @ A2)) & ((Z6 = (F @ X))))))))). % image_iff
thf(fact_44_imageI, axiom,
    ((![X4 : a, A2 : set_a, F : a > hoare_1686856528iple_b]: ((member_a @ X4 @ A2) => (member1340559591iple_b @ (F @ X4) @ (image_646034953iple_b @ F @ A2)))))). % imageI
thf(fact_45_ex__in__conv, axiom,
    ((![A2 : set_Ho1277474822iple_b]: ((?[X : hoare_1686856528iple_b]: (member1340559591iple_b @ X @ A2)) = (~ ((A2 = bot_bo290377370iple_b))))))). % ex_in_conv
thf(fact_46_equals0I, axiom,
    ((![A2 : set_Ho1277474822iple_b]: ((![Y5 : hoare_1686856528iple_b]: (~ ((member1340559591iple_b @ Y5 @ A2)))) => (A2 = bot_bo290377370iple_b))))). % equals0I
thf(fact_47_equals0D, axiom,
    ((![A2 : set_Ho1277474822iple_b, A : hoare_1686856528iple_b]: ((A2 = bot_bo290377370iple_b) => (~ ((member1340559591iple_b @ A @ A2))))))). % equals0D
thf(fact_48_emptyE, axiom,
    ((![A : hoare_1686856528iple_b]: (~ ((member1340559591iple_b @ A @ bot_bo290377370iple_b)))))). % emptyE
thf(fact_49_mk__disjoint__insert, axiom,
    ((![A : hoare_1686856528iple_b, A2 : set_Ho1277474822iple_b]: ((member1340559591iple_b @ A @ A2) => (?[B3 : set_Ho1277474822iple_b]: ((A2 = (insert1486066048iple_b @ A @ B3)) & (~ ((member1340559591iple_b @ A @ B3))))))))). % mk_disjoint_insert
thf(fact_50_insert__commute, axiom,
    ((![X4 : hoare_1686856528iple_b, Y : hoare_1686856528iple_b, A2 : set_Ho1277474822iple_b]: ((insert1486066048iple_b @ X4 @ (insert1486066048iple_b @ Y @ A2)) = (insert1486066048iple_b @ Y @ (insert1486066048iple_b @ X4 @ A2)))))). % insert_commute
thf(fact_51_insert__eq__iff, axiom,
    ((![A : hoare_1686856528iple_b, A2 : set_Ho1277474822iple_b, B2 : hoare_1686856528iple_b, B : set_Ho1277474822iple_b]: ((~ ((member1340559591iple_b @ A @ A2))) => ((~ ((member1340559591iple_b @ B2 @ B))) => (((insert1486066048iple_b @ A @ A2) = (insert1486066048iple_b @ B2 @ B)) = (((((A = B2)) => ((A2 = B)))) & ((((~ ((A = B2)))) => ((?[C3 : set_Ho1277474822iple_b]: (((A2 = (insert1486066048iple_b @ B2 @ C3))) & ((((~ ((member1340559591iple_b @ B2 @ C3)))) & ((((B = (insert1486066048iple_b @ A @ C3))) & ((~ ((member1340559591iple_b @ A @ C3)))))))))))))))))))). % insert_eq_iff
thf(fact_52_insert__absorb, axiom,
    ((![A : hoare_1686856528iple_b, A2 : set_Ho1277474822iple_b]: ((member1340559591iple_b @ A @ A2) => ((insert1486066048iple_b @ A @ A2) = A2))))). % insert_absorb
thf(fact_53_insert__ident, axiom,
    ((![X4 : hoare_1686856528iple_b, A2 : set_Ho1277474822iple_b, B : set_Ho1277474822iple_b]: ((~ ((member1340559591iple_b @ X4 @ A2))) => ((~ ((member1340559591iple_b @ X4 @ B))) => (((insert1486066048iple_b @ X4 @ A2) = (insert1486066048iple_b @ X4 @ B)) = (A2 = B))))))). % insert_ident
thf(fact_54_Set_Oset__insert, axiom,
    ((![X4 : hoare_1686856528iple_b, A2 : set_Ho1277474822iple_b]: ((member1340559591iple_b @ X4 @ A2) => (~ ((![B3 : set_Ho1277474822iple_b]: ((A2 = (insert1486066048iple_b @ X4 @ B3)) => (member1340559591iple_b @ X4 @ B3))))))))). % Set.set_insert
thf(fact_55_insertI2, axiom,
    ((![A : hoare_1686856528iple_b, B : set_Ho1277474822iple_b, B2 : hoare_1686856528iple_b]: ((member1340559591iple_b @ A @ B) => (member1340559591iple_b @ A @ (insert1486066048iple_b @ B2 @ B)))))). % insertI2
thf(fact_56_insertI1, axiom,
    ((![A : hoare_1686856528iple_b, B : set_Ho1277474822iple_b]: (member1340559591iple_b @ A @ (insert1486066048iple_b @ A @ B))))). % insertI1
thf(fact_57_insertE, axiom,
    ((![A : hoare_1686856528iple_b, B2 : hoare_1686856528iple_b, A2 : set_Ho1277474822iple_b]: ((member1340559591iple_b @ A @ (insert1486066048iple_b @ B2 @ A2)) => ((~ ((A = B2))) => (member1340559591iple_b @ A @ A2)))))). % insertE
thf(fact_58_Compr__image__eq, axiom,
    ((![F : a > hoare_1686856528iple_b, A2 : set_a, P : hoare_1686856528iple_b > $o]: ((collec1608496677iple_b @ (^[X : hoare_1686856528iple_b]: (((member1340559591iple_b @ X @ (image_646034953iple_b @ F @ A2))) & ((P @ X))))) = (image_646034953iple_b @ F @ (collect_a @ (^[X : a]: (((member_a @ X @ A2)) & ((P @ (F @ X))))))))))). % Compr_image_eq
thf(fact_59_image__image, axiom,
    ((![F : hoare_1686856528iple_b > hoare_1686856528iple_b, G3 : a > hoare_1686856528iple_b, A2 : set_a]: ((image_482623520iple_b @ F @ (image_646034953iple_b @ G3 @ A2)) = (image_646034953iple_b @ (^[X : a]: (F @ (G3 @ X))) @ A2))))). % image_image
thf(fact_60_image__image, axiom,
    ((![F : a > hoare_1686856528iple_b, G3 : a > a, A2 : set_a]: ((image_646034953iple_b @ F @ (image_a_a @ G3 @ A2)) = (image_646034953iple_b @ (^[X : a]: (F @ (G3 @ X))) @ A2))))). % image_image
thf(fact_61_imageE, axiom,
    ((![B2 : hoare_1686856528iple_b, F : a > hoare_1686856528iple_b, A2 : set_a]: ((member1340559591iple_b @ B2 @ (image_646034953iple_b @ F @ A2)) => (~ ((![X5 : a]: ((B2 = (F @ X5)) => (~ ((member_a @ X5 @ A2))))))))))). % imageE
thf(fact_62_empty__def, axiom,
    ((bot_bo290377370iple_b = (collec1608496677iple_b @ (^[X : hoare_1686856528iple_b]: $false))))). % empty_def
thf(fact_63_pigeonhole__infinite__rel, axiom,
    ((![A2 : set_a, B : set_a, R : a > a > $o]: ((~ ((finite_finite_a @ A2))) => ((finite_finite_a @ B) => ((![X5 : a]: ((member_a @ X5 @ A2) => (?[Xa : a]: ((member_a @ Xa @ B) & (R @ X5 @ Xa))))) => (?[X5 : a]: ((member_a @ X5 @ B) & (~ ((finite_finite_a @ (collect_a @ (^[A3 : a]: (((member_a @ A3 @ A2)) & ((R @ A3 @ X5)))))))))))))))). % pigeonhole_infinite_rel
thf(fact_64_not__finite__existsD, axiom,
    ((![P : a > $o]: ((~ ((finite_finite_a @ (collect_a @ P)))) => (?[X_1 : a]: (P @ X_1)))))). % not_finite_existsD
thf(fact_65_insert__Collect, axiom,
    ((![A : hoare_1686856528iple_b, P : hoare_1686856528iple_b > $o]: ((insert1486066048iple_b @ A @ (collec1608496677iple_b @ P)) = (collec1608496677iple_b @ (^[U : hoare_1686856528iple_b]: (((~ ((U = A)))) => ((P @ U))))))))). % insert_Collect
thf(fact_66_insert__compr, axiom,
    ((insert1486066048iple_b = (^[A3 : hoare_1686856528iple_b]: (^[B4 : set_Ho1277474822iple_b]: (collec1608496677iple_b @ (^[X : hoare_1686856528iple_b]: (((X = A3)) | ((member1340559591iple_b @ X @ B4)))))))))). % insert_compr
thf(fact_67_infinite__imp__nonempty, axiom,
    ((![S5 : set_a]: ((~ ((finite_finite_a @ S5))) => (~ ((S5 = bot_bot_set_a))))))). % infinite_imp_nonempty
thf(fact_68_infinite__imp__nonempty, axiom,
    ((![S5 : set_Ho1277474822iple_b]: ((~ ((finite1300473319iple_b @ S5))) => (~ ((S5 = bot_bo290377370iple_b))))))). % infinite_imp_nonempty
thf(fact_69_finite_OemptyI, axiom,
    ((finite_finite_a @ bot_bot_set_a))). % finite.emptyI
thf(fact_70_finite_OemptyI, axiom,
    ((finite1300473319iple_b @ bot_bo290377370iple_b))). % finite.emptyI
thf(fact_71_singleton__inject, axiom,
    ((![A : hoare_1686856528iple_b, B2 : hoare_1686856528iple_b]: (((insert1486066048iple_b @ A @ bot_bo290377370iple_b) = (insert1486066048iple_b @ B2 @ bot_bo290377370iple_b)) => (A = B2))))). % singleton_inject
thf(fact_72_insert__not__empty, axiom,
    ((![A : hoare_1686856528iple_b, A2 : set_Ho1277474822iple_b]: (~ (((insert1486066048iple_b @ A @ A2) = bot_bo290377370iple_b)))))). % insert_not_empty
thf(fact_73_doubleton__eq__iff, axiom,
    ((![A : hoare_1686856528iple_b, B2 : hoare_1686856528iple_b, C : hoare_1686856528iple_b, D : hoare_1686856528iple_b]: (((insert1486066048iple_b @ A @ (insert1486066048iple_b @ B2 @ bot_bo290377370iple_b)) = (insert1486066048iple_b @ C @ (insert1486066048iple_b @ D @ bot_bo290377370iple_b))) = (((((A = C)) & ((B2 = D)))) | ((((A = D)) & ((B2 = C))))))))). % doubleton_eq_iff
thf(fact_74_singleton__iff, axiom,
    ((![B2 : hoare_1686856528iple_b, A : hoare_1686856528iple_b]: ((member1340559591iple_b @ B2 @ (insert1486066048iple_b @ A @ bot_bo290377370iple_b)) = (B2 = A))))). % singleton_iff
thf(fact_75_singletonD, axiom,
    ((![B2 : hoare_1686856528iple_b, A : hoare_1686856528iple_b]: ((member1340559591iple_b @ B2 @ (insert1486066048iple_b @ A @ bot_bo290377370iple_b)) => (B2 = A))))). % singletonD
thf(fact_76_finite_OinsertI, axiom,
    ((![A2 : set_Ho1277474822iple_b, A : hoare_1686856528iple_b]: ((finite1300473319iple_b @ A2) => (finite1300473319iple_b @ (insert1486066048iple_b @ A @ A2)))))). % finite.insertI
thf(fact_77_finite_OinsertI, axiom,
    ((![A2 : set_a, A : a]: ((finite_finite_a @ A2) => (finite_finite_a @ (insert_a @ A @ A2)))))). % finite.insertI
thf(fact_78_pigeonhole__infinite, axiom,
    ((![A2 : set_a, F : a > hoare_1686856528iple_b]: ((~ ((finite_finite_a @ A2))) => ((finite1300473319iple_b @ (image_646034953iple_b @ F @ A2)) => (?[X5 : a]: ((member_a @ X5 @ A2) & (~ ((finite_finite_a @ (collect_a @ (^[A3 : a]: (((member_a @ A3 @ A2)) & (((F @ A3) = (F @ X5)))))))))))))))). % pigeonhole_infinite
thf(fact_79_pigeonhole__infinite, axiom,
    ((![A2 : set_a, F : a > a]: ((~ ((finite_finite_a @ A2))) => ((finite_finite_a @ (image_a_a @ F @ A2)) => (?[X5 : a]: ((member_a @ X5 @ A2) & (~ ((finite_finite_a @ (collect_a @ (^[A3 : a]: (((member_a @ A3 @ A2)) & (((F @ A3) = (F @ X5)))))))))))))))). % pigeonhole_infinite
thf(fact_80_Collect__conv__if2, axiom,
    ((![P : hoare_1686856528iple_b > $o, A : hoare_1686856528iple_b]: (((P @ A) => ((collec1608496677iple_b @ (^[X : hoare_1686856528iple_b]: (((A = X)) & ((P @ X))))) = (insert1486066048iple_b @ A @ bot_bo290377370iple_b))) & ((~ ((P @ A))) => ((collec1608496677iple_b @ (^[X : hoare_1686856528iple_b]: (((A = X)) & ((P @ X))))) = bot_bo290377370iple_b)))))). % Collect_conv_if2
thf(fact_81_Collect__conv__if, axiom,
    ((![P : hoare_1686856528iple_b > $o, A : hoare_1686856528iple_b]: (((P @ A) => ((collec1608496677iple_b @ (^[X : hoare_1686856528iple_b]: (((X = A)) & ((P @ X))))) = (insert1486066048iple_b @ A @ bot_bo290377370iple_b))) & ((~ ((P @ A))) => ((collec1608496677iple_b @ (^[X : hoare_1686856528iple_b]: (((X = A)) & ((P @ X))))) = bot_bo290377370iple_b)))))). % Collect_conv_if
thf(fact_82_infinite__finite__induct, axiom,
    ((![P : set_a > $o, A2 : set_a]: ((![A4 : set_a]: ((~ ((finite_finite_a @ A4))) => (P @ A4))) => ((P @ bot_bot_set_a) => ((![X5 : a, F3 : set_a]: ((finite_finite_a @ F3) => ((~ ((member_a @ X5 @ F3))) => ((P @ F3) => (P @ (insert_a @ X5 @ F3)))))) => (P @ A2))))))). % infinite_finite_induct
thf(fact_83_infinite__finite__induct, axiom,
    ((![P : set_Ho1277474822iple_b > $o, A2 : set_Ho1277474822iple_b]: ((![A4 : set_Ho1277474822iple_b]: ((~ ((finite1300473319iple_b @ A4))) => (P @ A4))) => ((P @ bot_bo290377370iple_b) => ((![X5 : hoare_1686856528iple_b, F3 : set_Ho1277474822iple_b]: ((finite1300473319iple_b @ F3) => ((~ ((member1340559591iple_b @ X5 @ F3))) => ((P @ F3) => (P @ (insert1486066048iple_b @ X5 @ F3)))))) => (P @ A2))))))). % infinite_finite_induct
thf(fact_84_finite__ne__induct, axiom,
    ((![F2 : set_a, P : set_a > $o]: ((finite_finite_a @ F2) => ((~ ((F2 = bot_bot_set_a))) => ((![X5 : a]: (P @ (insert_a @ X5 @ bot_bot_set_a))) => ((![X5 : a, F3 : set_a]: ((finite_finite_a @ F3) => ((~ ((F3 = bot_bot_set_a))) => ((~ ((member_a @ X5 @ F3))) => ((P @ F3) => (P @ (insert_a @ X5 @ F3))))))) => (P @ F2)))))))). % finite_ne_induct
thf(fact_85_finite__ne__induct, axiom,
    ((![F2 : set_Ho1277474822iple_b, P : set_Ho1277474822iple_b > $o]: ((finite1300473319iple_b @ F2) => ((~ ((F2 = bot_bo290377370iple_b))) => ((![X5 : hoare_1686856528iple_b]: (P @ (insert1486066048iple_b @ X5 @ bot_bo290377370iple_b))) => ((![X5 : hoare_1686856528iple_b, F3 : set_Ho1277474822iple_b]: ((finite1300473319iple_b @ F3) => ((~ ((F3 = bot_bo290377370iple_b))) => ((~ ((member1340559591iple_b @ X5 @ F3))) => ((P @ F3) => (P @ (insert1486066048iple_b @ X5 @ F3))))))) => (P @ F2)))))))). % finite_ne_induct
thf(fact_86_finite_Oinducts, axiom,
    ((![X4 : set_a, P : set_a > $o]: ((finite_finite_a @ X4) => ((P @ bot_bot_set_a) => ((![A4 : set_a, A5 : a]: ((finite_finite_a @ A4) => ((P @ A4) => (P @ (insert_a @ A5 @ A4))))) => (P @ X4))))))). % finite.inducts
thf(fact_87_finite_Oinducts, axiom,
    ((![X4 : set_Ho1277474822iple_b, P : set_Ho1277474822iple_b > $o]: ((finite1300473319iple_b @ X4) => ((P @ bot_bo290377370iple_b) => ((![A4 : set_Ho1277474822iple_b, A5 : hoare_1686856528iple_b]: ((finite1300473319iple_b @ A4) => ((P @ A4) => (P @ (insert1486066048iple_b @ A5 @ A4))))) => (P @ X4))))))). % finite.inducts
thf(fact_88_finite__induct, axiom,
    ((![F2 : set_a, P : set_a > $o]: ((finite_finite_a @ F2) => ((P @ bot_bot_set_a) => ((![X5 : a, F3 : set_a]: ((finite_finite_a @ F3) => ((~ ((member_a @ X5 @ F3))) => ((P @ F3) => (P @ (insert_a @ X5 @ F3)))))) => (P @ F2))))))). % finite_induct
thf(fact_89_finite__induct, axiom,
    ((![F2 : set_Ho1277474822iple_b, P : set_Ho1277474822iple_b > $o]: ((finite1300473319iple_b @ F2) => ((P @ bot_bo290377370iple_b) => ((![X5 : hoare_1686856528iple_b, F3 : set_Ho1277474822iple_b]: ((finite1300473319iple_b @ F3) => ((~ ((member1340559591iple_b @ X5 @ F3))) => ((P @ F3) => (P @ (insert1486066048iple_b @ X5 @ F3)))))) => (P @ F2))))))). % finite_induct
thf(fact_90_finite_Osimps, axiom,
    ((finite_finite_a = (^[A3 : set_a]: (((A3 = bot_bot_set_a)) | ((?[A6 : set_a]: (?[B5 : a]: (((A3 = (insert_a @ B5 @ A6))) & ((finite_finite_a @ A6))))))))))). % finite.simps
thf(fact_91_finite_Osimps, axiom,
    ((finite1300473319iple_b = (^[A3 : set_Ho1277474822iple_b]: (((A3 = bot_bo290377370iple_b)) | ((?[A6 : set_Ho1277474822iple_b]: (?[B5 : hoare_1686856528iple_b]: (((A3 = (insert1486066048iple_b @ B5 @ A6))) & ((finite1300473319iple_b @ A6))))))))))). % finite.simps
thf(fact_92_finite_Ocases, axiom,
    ((![A : set_a]: ((finite_finite_a @ A) => ((~ ((A = bot_bot_set_a))) => (~ ((![A4 : set_a]: ((?[A5 : a]: (A = (insert_a @ A5 @ A4))) => (~ ((finite_finite_a @ A4)))))))))))). % finite.cases
thf(fact_93_finite_Ocases, axiom,
    ((![A : set_Ho1277474822iple_b]: ((finite1300473319iple_b @ A) => ((~ ((A = bot_bo290377370iple_b))) => (~ ((![A4 : set_Ho1277474822iple_b]: ((?[A5 : hoare_1686856528iple_b]: (A = (insert1486066048iple_b @ A5 @ A4))) => (~ ((finite1300473319iple_b @ A4)))))))))))). % finite.cases
thf(fact_94_image__constant__conv, axiom,
    ((![A2 : set_a, C : hoare_1686856528iple_b]: (((A2 = bot_bot_set_a) => ((image_646034953iple_b @ (^[X : a]: C) @ A2) = bot_bo290377370iple_b)) & ((~ ((A2 = bot_bot_set_a))) => ((image_646034953iple_b @ (^[X : a]: C) @ A2) = (insert1486066048iple_b @ C @ bot_bo290377370iple_b))))))). % image_constant_conv
thf(fact_95_image__constant__conv, axiom,
    ((![A2 : set_Ho1277474822iple_b, C : hoare_1686856528iple_b]: (((A2 = bot_bo290377370iple_b) => ((image_482623520iple_b @ (^[X : hoare_1686856528iple_b]: C) @ A2) = bot_bo290377370iple_b)) & ((~ ((A2 = bot_bo290377370iple_b))) => ((image_482623520iple_b @ (^[X : hoare_1686856528iple_b]: C) @ A2) = (insert1486066048iple_b @ C @ bot_bo290377370iple_b))))))). % image_constant_conv
thf(fact_96_image__constant, axiom,
    ((![X4 : a, A2 : set_a, C : hoare_1686856528iple_b]: ((member_a @ X4 @ A2) => ((image_646034953iple_b @ (^[X : a]: C) @ A2) = (insert1486066048iple_b @ C @ bot_bo290377370iple_b)))))). % image_constant
thf(fact_97_the__elem__eq, axiom,
    ((![X4 : hoare_1686856528iple_b]: ((the_el442959643iple_b @ (insert1486066048iple_b @ X4 @ bot_bo290377370iple_b)) = X4)))). % the_elem_eq
thf(fact_98_is__singletonI, axiom,
    ((![X4 : hoare_1686856528iple_b]: (is_sin1792298844iple_b @ (insert1486066048iple_b @ X4 @ bot_bo290377370iple_b))))). % is_singletonI
thf(fact_99_hoare__derivs_OSkip, axiom,
    ((![G : set_Ho1277474822iple_b, P : b > state > $o]: (hoare_129598475rivs_b @ G @ (insert1486066048iple_b @ (hoare_719046531iple_b @ P @ skip @ P) @ bot_bo290377370iple_b))))). % hoare_derivs.Skip
thf(fact_100_Comp, axiom,
    ((![G : set_Ho1277474822iple_b, P : b > state > $o, C : com, Q : b > state > $o, D : com, R : b > state > $o]: ((hoare_129598475rivs_b @ G @ (insert1486066048iple_b @ (hoare_719046531iple_b @ P @ C @ Q) @ bot_bo290377370iple_b)) => ((hoare_129598475rivs_b @ G @ (insert1486066048iple_b @ (hoare_719046531iple_b @ Q @ D @ R) @ bot_bo290377370iple_b)) => (hoare_129598475rivs_b @ G @ (insert1486066048iple_b @ (hoare_719046531iple_b @ P @ (semi @ C @ D) @ R) @ bot_bo290377370iple_b))))))). % Comp
thf(fact_101_LoopF, axiom,
    ((![G : set_Ho1277474822iple_b, P : b > state > $o, B2 : state > $o, C : com]: (hoare_129598475rivs_b @ G @ (insert1486066048iple_b @ (hoare_719046531iple_b @ (^[Z4 : b]: (^[S4 : state]: (((P @ Z4 @ S4)) & ((~ ((B2 @ S4))))))) @ (while @ B2 @ C) @ P) @ bot_bo290377370iple_b))))). % LoopF
thf(fact_102_the__elem__image__unique, axiom,
    ((![A2 : set_a, F : a > hoare_1686856528iple_b, X4 : a]: ((~ ((A2 = bot_bot_set_a))) => ((![Y5 : a]: ((member_a @ Y5 @ A2) => ((F @ Y5) = (F @ X4)))) => ((the_el442959643iple_b @ (image_646034953iple_b @ F @ A2)) = (F @ X4))))))). % the_elem_image_unique
thf(fact_103_Set_Ois__empty__def, axiom,
    ((is_emp910168062iple_b = (^[A6 : set_Ho1277474822iple_b]: (A6 = bot_bo290377370iple_b))))). % Set.is_empty_def
thf(fact_104_bot__set__def, axiom,
    ((bot_bo290377370iple_b = (collec1608496677iple_b @ bot_bo1259081323le_b_o)))). % bot_set_def
thf(fact_105_is__singleton__the__elem, axiom,
    ((is_sin1792298844iple_b = (^[A6 : set_Ho1277474822iple_b]: (A6 = (insert1486066048iple_b @ (the_el442959643iple_b @ A6) @ bot_bo290377370iple_b)))))). % is_singleton_the_elem
thf(fact_106_is__singletonI_H, axiom,
    ((![A2 : set_Ho1277474822iple_b]: ((~ ((A2 = bot_bo290377370iple_b))) => ((![X5 : hoare_1686856528iple_b, Y5 : hoare_1686856528iple_b]: ((member1340559591iple_b @ X5 @ A2) => ((member1340559591iple_b @ Y5 @ A2) => (X5 = Y5)))) => (is_sin1792298844iple_b @ A2)))))). % is_singletonI'
thf(fact_107_is__singletonE, axiom,
    ((![A2 : set_Ho1277474822iple_b]: ((is_sin1792298844iple_b @ A2) => (~ ((![X5 : hoare_1686856528iple_b]: (~ ((A2 = (insert1486066048iple_b @ X5 @ bot_bo290377370iple_b))))))))))). % is_singletonE
thf(fact_108_is__singleton__def, axiom,
    ((is_sin1792298844iple_b = (^[A6 : set_Ho1277474822iple_b]: (?[X : hoare_1686856528iple_b]: (A6 = (insert1486066048iple_b @ X @ bot_bo290377370iple_b))))))). % is_singleton_def
thf(fact_109_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_110_com_Oinject_I5_J, axiom,
    ((![X61 : state > $o, X62 : com, Y61 : state > $o, Y62 : com]: (((while @ X61 @ X62) = (while @ Y61 @ Y62)) = (((X61 = Y61)) & ((X62 = Y62))))))). % com.inject(5)
thf(fact_111_com_Odistinct_I5_J, axiom,
    ((![X41 : com, X42 : com]: (~ ((skip = (semi @ X41 @ X42))))))). % com.distinct(5)
thf(fact_112_com_Odistinct_I39_J, axiom,
    ((![X41 : com, X42 : com, X61 : state > $o, X62 : com]: (~ (((semi @ X41 @ X42) = (while @ X61 @ X62))))))). % com.distinct(39)
thf(fact_113_com_Odistinct_I9_J, axiom,
    ((![X61 : state > $o, X62 : com]: (~ ((skip = (while @ X61 @ X62))))))). % com.distinct(9)
thf(fact_114_bot__empty__eq, axiom,
    ((bot_bo1259081323le_b_o = (^[X : hoare_1686856528iple_b]: (member1340559591iple_b @ X @ bot_bo290377370iple_b))))). % bot_empty_eq
thf(fact_115_Collect__empty__eq__bot, axiom,
    ((![P : hoare_1686856528iple_b > $o]: (((collec1608496677iple_b @ P) = bot_bo290377370iple_b) = (P = bot_bo1259081323le_b_o))))). % Collect_empty_eq_bot
thf(fact_116_bind__singleton__conv__image, axiom,
    ((![A2 : set_a, F : a > hoare_1686856528iple_b]: ((bind_a1120646331iple_b @ A2 @ (^[X : a]: (insert1486066048iple_b @ (F @ X) @ bot_bo290377370iple_b))) = (image_646034953iple_b @ F @ A2))))). % bind_singleton_conv_image
thf(fact_117_empty__bind, axiom,
    ((![F : hoare_1686856528iple_b > set_Ho1277474822iple_b]: ((bind_H1262675666iple_b @ bot_bo290377370iple_b @ F) = bot_bo290377370iple_b)))). % empty_bind
thf(fact_118_finite__bind, axiom,
    ((![S5 : set_a, F : a > set_a]: ((finite_finite_a @ S5) => ((![X5 : a]: ((member_a @ X5 @ S5) => (finite_finite_a @ (F @ X5)))) => (finite_finite_a @ (bind_a_a @ S5 @ F))))))). % finite_bind
thf(fact_119_bind__const, axiom,
    ((![A2 : set_Ho1277474822iple_b, B : set_Ho1277474822iple_b]: (((A2 = bot_bo290377370iple_b) => ((bind_H1262675666iple_b @ A2 @ (^[Uu : hoare_1686856528iple_b]: B)) = bot_bo290377370iple_b)) & ((~ ((A2 = bot_bo290377370iple_b))) => ((bind_H1262675666iple_b @ A2 @ (^[Uu : hoare_1686856528iple_b]: B)) = B)))))). % bind_const
thf(fact_120_the__elem__def, axiom,
    ((the_el442959643iple_b = (^[X7 : set_Ho1277474822iple_b]: (the_Ho1610761865iple_b @ (^[X : hoare_1686856528iple_b]: (X7 = (insert1486066048iple_b @ X @ bot_bo290377370iple_b)))))))). % the_elem_def
thf(fact_121_Sup_OSUP__cong, axiom,
    ((![A2 : set_a, B : set_a, C2 : a > hoare_1686856528iple_b, D2 : a > hoare_1686856528iple_b, Sup : set_Ho1277474822iple_b > hoare_1686856528iple_b]: ((A2 = B) => ((![X5 : a]: ((member_a @ X5 @ B) => ((C2 @ X5) = (D2 @ X5)))) => ((Sup @ (image_646034953iple_b @ C2 @ A2)) = (Sup @ (image_646034953iple_b @ D2 @ B)))))))). % Sup.SUP_cong
thf(fact_122_Inf_OINF__cong, axiom,
    ((![A2 : set_a, B : set_a, C2 : a > hoare_1686856528iple_b, D2 : a > hoare_1686856528iple_b, Inf : set_Ho1277474822iple_b > hoare_1686856528iple_b]: ((A2 = B) => ((![X5 : a]: ((member_a @ X5 @ B) => ((C2 @ X5) = (D2 @ X5)))) => ((Inf @ (image_646034953iple_b @ C2 @ A2)) = (Inf @ (image_646034953iple_b @ D2 @ B)))))))). % Inf.INF_cong
thf(fact_123_range__constant, axiom,
    ((![X4 : hoare_1686856528iple_b]: ((image_646034953iple_b @ (^[Uu : a]: X4) @ top_top_set_a) = (insert1486066048iple_b @ X4 @ bot_bo290377370iple_b))))). % range_constant
thf(fact_124_image__fold__insert, axiom,
    ((![A2 : set_a, F : a > hoare_1686856528iple_b]: ((finite_finite_a @ A2) => ((image_646034953iple_b @ F @ A2) = (finite393365052iple_b @ (^[K : a]: (insert1486066048iple_b @ (F @ K))) @ bot_bo290377370iple_b @ A2)))))). % image_fold_insert
thf(fact_125_finite__Plus__UNIV__iff, axiom,
    (((finite935078844um_a_a @ top_to1089105419um_a_a) = (((finite_finite_a @ top_top_set_a)) & ((finite_finite_a @ top_top_set_a)))))). % finite_Plus_UNIV_iff
thf(fact_126_Collect__const, axiom,
    ((![P : $o]: ((P => ((collec1608496677iple_b @ (^[S4 : hoare_1686856528iple_b]: P)) = top_to1883285174iple_b)) & ((~ (P)) => ((collec1608496677iple_b @ (^[S4 : hoare_1686856528iple_b]: P)) = bot_bo290377370iple_b)))))). % Collect_const
thf(fact_127_finite__Collect__not, axiom,
    ((![P : a > $o]: ((finite_finite_a @ (collect_a @ P)) => ((finite_finite_a @ (collect_a @ (^[X : a]: (~ ((P @ X)))))) = (finite_finite_a @ top_top_set_a)))))). % finite_Collect_not
thf(fact_128_rangeI, axiom,
    ((![F : a > hoare_1686856528iple_b, X4 : a]: (member1340559591iple_b @ (F @ X4) @ (image_646034953iple_b @ F @ top_top_set_a))))). % rangeI
thf(fact_129_range__eqI, axiom,
    ((![B2 : hoare_1686856528iple_b, F : a > hoare_1686856528iple_b, X4 : a]: ((B2 = (F @ X4)) => (member1340559591iple_b @ B2 @ (image_646034953iple_b @ F @ top_top_set_a)))))). % range_eqI
thf(fact_130_ex__new__if__finite, axiom,
    ((![A2 : set_a]: ((~ ((finite_finite_a @ top_top_set_a))) => ((finite_finite_a @ A2) => (?[A5 : a]: (~ ((member_a @ A5 @ A2))))))))). % ex_new_if_finite
thf(fact_131_insert__UNIV, axiom,
    ((![X4 : hoare_1686856528iple_b]: ((insert1486066048iple_b @ X4 @ top_to1883285174iple_b) = top_to1883285174iple_b)))). % insert_UNIV
thf(fact_132_rangeE, axiom,
    ((![B2 : hoare_1686856528iple_b, F : a > hoare_1686856528iple_b]: ((member1340559591iple_b @ B2 @ (image_646034953iple_b @ F @ top_top_set_a)) => (~ ((![X5 : a]: (~ ((B2 = (F @ X5))))))))))). % rangeE
thf(fact_133_range__composition, axiom,
    ((![F : hoare_1686856528iple_b > hoare_1686856528iple_b, G3 : a > hoare_1686856528iple_b]: ((image_646034953iple_b @ (^[X : a]: (F @ (G3 @ X))) @ top_top_set_a) = (image_482623520iple_b @ F @ (image_646034953iple_b @ G3 @ top_top_set_a)))))). % range_composition
thf(fact_134_range__composition, axiom,
    ((![F : a > hoare_1686856528iple_b, G3 : a > a]: ((image_646034953iple_b @ (^[X : a]: (F @ (G3 @ X))) @ top_top_set_a) = (image_646034953iple_b @ F @ (image_a_a @ G3 @ top_top_set_a)))))). % range_composition
thf(fact_135_finite__Prod__UNIV, axiom,
    (((finite_finite_a @ top_top_set_a) => ((finite_finite_a @ top_top_set_a) => (finite179568208od_a_a @ top_to398455383od_a_a))))). % finite_Prod_UNIV
thf(fact_136_finite__prod, axiom,
    (((finite179568208od_a_a @ top_to398455383od_a_a) = (((finite_finite_a @ top_top_set_a)) & ((finite_finite_a @ top_top_set_a)))))). % finite_prod
thf(fact_137_Finite__Set_Ofinite__set, axiom,
    (((finite_finite_set_a @ top_top_set_set_a) = (finite_finite_a @ top_top_set_a)))). % Finite_Set.finite_set

% Conjectures (3)
thf(conj_0, hypothesis,
    ((finite_finite_a @ u))).
thf(conj_1, hypothesis,
    ((![P4 : a]: ((hoare_129598475rivs_b @ g @ (insert1486066048iple_b @ (hoare_719046531iple_b @ (p2 @ P4) @ (c0 @ P4) @ (q2 @ P4)) @ bot_bo290377370iple_b)) => (hoare_129598475rivs_b @ g @ (insert1486066048iple_b @ (hoare_719046531iple_b @ (p @ P4) @ (c0 @ P4) @ (q @ P4)) @ bot_bo290377370iple_b)))))).
thf(conj_2, conjecture,
    (((~ ((hoare_129598475rivs_b @ g @ (image_646034953iple_b @ (^[P5 : a]: (hoare_719046531iple_b @ (p2 @ P5) @ (c0 @ P5) @ (q2 @ P5))) @ u)))) | (hoare_129598475rivs_b @ g @ (image_646034953iple_b @ (^[P5 : a]: (hoare_719046531iple_b @ (p @ P5) @ (c0 @ P5) @ (q @ P5))) @ u))))).
