% 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_176__3251158_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:12:30.557

% Could-be-implicit typings (9)
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_J_J, type,
    set_se152467259iple_a : $tType).
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__Set__Oset_It__Set__Oset_It__Com__Opname_J_J, type,
    set_set_pname : $tType).
thf(ty_n_t__Set__Oset_It__Com__Opname_J, type,
    set_pname : $tType).
thf(ty_n_t__Com__Ostate, type,
    state : $tType).
thf(ty_n_t__Com__Opname, type,
    pname : $tType).
thf(ty_n_t__Com__Ocom, type,
    com : $tType).
thf(ty_n_tf__a, type,
    a : $tType).

% Explicit typings (39)
thf(sy_c_Com_Ocom_OBODY, type,
    body : pname > com).
thf(sy_c_Com_Ocom_OSKIP, type,
    skip : com).
thf(sy_c_Com_Ocom_OSemi, type,
    semi : com > com > com).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Com__Opname_J, type,
    comple1674572307_pname : set_set_pname > set_pname).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_J, type,
    comple1228397400iple_a : set_se152467259iple_a > set_Ho137910533iple_a).
thf(sy_c_Fun_Oswap_001t__Com__Opname_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    swap_p1758894497iple_a : pname > pname > (pname > hoare_1678595023iple_a) > pname > hoare_1678595023iple_a).
thf(sy_c_HOL_OThe_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    the_Ho1602500360iple_a : (hoare_1678595023iple_a > $o) > hoare_1678595023iple_a).
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_Otriple_Otriple_001tf__a, type,
    hoare_719046530iple_a : (a > state > $o) > com > (a > state > $o) > hoare_1678595023iple_a).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Com__Opname_M_Eo_J, type,
    bot_bot_pname_o : pname > $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__Set__Oset_It__Com__Opname_J, type,
    bot_bot_set_pname : set_pname).
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_Otop__class_Otop_001_062_It__Com__Opname_M_Eo_J, type,
    top_top_pname_o : pname > $o).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Com__Opname_J, type,
    top_top_set_pname : set_pname).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_J, type,
    top_to743720885iple_a : set_Ho137910533iple_a).
thf(sy_c_Set_OCollect_001t__Com__Opname, type,
    collect_pname : (pname > $o) > set_pname).
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_Obind_001t__Com__Opname_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    bind_p599804078iple_a : set_pname > (pname > set_Ho137910533iple_a) > set_Ho137910533iple_a).
thf(sy_c_Set_Obind_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    bind_H209762450iple_a : set_Ho137910533iple_a > (hoare_1678595023iple_a > set_Ho137910533iple_a) > set_Ho137910533iple_a).
thf(sy_c_Set_Oimage_001t__Com__Opname_001t__Com__Opname, type,
    image_pname_pname : (pname > pname) > set_pname > set_pname).
thf(sy_c_Set_Oimage_001t__Com__Opname_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    image_2025567968iple_a : (pname > hoare_1678595023iple_a) > set_pname > set_Ho137910533iple_a).
thf(sy_c_Set_Oimage_001t__Com__Opname_001t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_J, type,
    image_478803990iple_a : (pname > set_Ho137910533iple_a) > set_pname > set_se152467259iple_a).
thf(sy_c_Set_Oimage_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    image_1577193952iple_a : (hoare_1678595023iple_a > hoare_1678595023iple_a) > set_Ho137910533iple_a > set_Ho137910533iple_a).
thf(sy_c_Set_Oinsert_001t__Com__Opname, type,
    insert_pname : pname > set_pname > set_pname).
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__Com__Opname, type,
    is_singleton_pname : set_pname > $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_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__Com__Opname, type,
    member_pname : pname > set_pname > $o).
thf(sy_c_member_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    member1332298086iple_a : hoare_1678595023iple_a > set_Ho137910533iple_a > $o).
thf(sy_c_member_001t__Set__Oset_It__Com__Opname_J, type,
    member_set_pname : set_pname > set_set_pname > $o).
thf(sy_c_member_001t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J_J, type,
    member521824924iple_a : set_Ho137910533iple_a > set_se152467259iple_a > $o).
thf(sy_v_G, type,
    g : set_Ho137910533iple_a).
thf(sy_v_P, type,
    p : pname > a > state > $o).
thf(sy_v_Procs, type,
    procs : set_pname).
thf(sy_v_Q, type,
    q : pname > a > state > $o).
thf(sy_v_pn, type,
    pn : pname).

% Relevant facts (145)
thf(fact_0_triple_Oinject, axiom,
    ((![X1 : a > state > $o, X2 : com, X3 : a > state > $o, Y1 : a > state > $o, Y2 : com, Y3 : a > state > $o]: (((hoare_719046530iple_a @ X1 @ X2 @ X3) = (hoare_719046530iple_a @ Y1 @ Y2 @ Y3)) = (((X1 = Y1)) & ((((X2 = Y2)) & ((X3 = Y3))))))))). % triple.inject
thf(fact_1_conseq1, axiom,
    ((![G : set_Ho137910533iple_a, P : a > state > $o, C : com, Q : a > state > $o, P2 : a > state > $o]: ((hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P @ C @ Q) @ bot_bo1298296729iple_a)) => ((![Z : a, S : state]: ((P2 @ Z @ S) => (P @ Z @ S))) => (hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P2 @ C @ Q) @ bot_bo1298296729iple_a))))))). % conseq1
thf(fact_2_conseq2, axiom,
    ((![G : set_Ho137910533iple_a, P2 : a > state > $o, C : com, Q2 : a > state > $o, Q : a > state > $o]: ((hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P2 @ C @ Q2) @ bot_bo1298296729iple_a)) => ((![Z : a, S : state]: ((Q2 @ Z @ S) => (Q @ Z @ S))) => (hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P2 @ C @ Q) @ bot_bo1298296729iple_a))))))). % conseq2
thf(fact_3_conseq12, axiom,
    ((![G : set_Ho137910533iple_a, P : a > state > $o, C : com, Q2 : a > state > $o, P2 : a > state > $o, Q : a > state > $o]: ((hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P @ C @ Q2) @ bot_bo1298296729iple_a)) => ((![Z : a, S : state]: ((P2 @ Z @ S) => (![S2 : state]: ((![Z2 : a]: ((P @ Z2 @ S) => (Q2 @ Z2 @ S2))) => (Q @ Z @ S2))))) => (hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P2 @ C @ Q) @ bot_bo1298296729iple_a))))))). % conseq12
thf(fact_4_triple_Oinduct, axiom,
    ((![P2 : hoare_1678595023iple_a > $o, Triple : hoare_1678595023iple_a]: ((![X1a : a > state > $o, X2a : com, X3a : a > state > $o]: (P2 @ (hoare_719046530iple_a @ X1a @ X2a @ X3a))) => (P2 @ Triple))))). % triple.induct
thf(fact_5_triple_Oexhaust, axiom,
    ((![Y : hoare_1678595023iple_a]: (~ ((![X12 : a > state > $o, X22 : com, X32 : a > state > $o]: (~ ((Y = (hoare_719046530iple_a @ X12 @ X22 @ X32)))))))))). % triple.exhaust
thf(fact_6_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_7_empty, axiom,
    ((![G : set_Ho137910533iple_a]: (hoare_129598474rivs_a @ G @ bot_bo1298296729iple_a)))). % empty
thf(fact_8_conseq, axiom,
    ((![P2 : a > state > $o, G : set_Ho137910533iple_a, C : com, Q : a > state > $o]: ((![Z : a, S : state]: ((P2 @ Z @ S) => (?[P3 : a > state > $o, Q3 : a > state > $o]: ((hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P3 @ C @ Q3) @ bot_bo1298296729iple_a)) & (![S2 : state]: ((![Z2 : a]: ((P3 @ Z2 @ S) => (Q3 @ Z2 @ S2))) => (Q @ Z @ S2))))))) => (hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P2 @ C @ Q) @ bot_bo1298296729iple_a)))))). % conseq
thf(fact_9_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_10_singleton__conv, axiom,
    ((![A : hoare_1678595023iple_a]: ((collec1600235172iple_a @ (^[X : hoare_1678595023iple_a]: (X = A))) = (insert1477804543iple_a @ A @ bot_bo1298296729iple_a))))). % singleton_conv
thf(fact_11_singleton__conv2, axiom,
    ((![A : hoare_1678595023iple_a]: ((collec1600235172iple_a @ ((^[Y4 : hoare_1678595023iple_a]: (^[Z3 : hoare_1678595023iple_a]: (Y4 = Z3))) @ A)) = (insert1477804543iple_a @ A @ bot_bo1298296729iple_a))))). % singleton_conv2
thf(fact_12_singletonI, axiom,
    ((![A : pname]: (member_pname @ A @ (insert_pname @ A @ bot_bot_set_pname))))). % singletonI
thf(fact_13_singletonI, axiom,
    ((![A : hoare_1678595023iple_a]: (member1332298086iple_a @ A @ (insert1477804543iple_a @ A @ bot_bo1298296729iple_a))))). % singletonI
thf(fact_14_image__insert, axiom,
    ((![F : pname > hoare_1678595023iple_a, A : pname, B : set_pname]: ((image_2025567968iple_a @ F @ (insert_pname @ A @ B)) = (insert1477804543iple_a @ (F @ A) @ (image_2025567968iple_a @ F @ B)))))). % image_insert
thf(fact_15_image__insert, axiom,
    ((![F : hoare_1678595023iple_a > hoare_1678595023iple_a, A : hoare_1678595023iple_a, B : set_Ho137910533iple_a]: ((image_1577193952iple_a @ F @ (insert1477804543iple_a @ A @ B)) = (insert1477804543iple_a @ (F @ A) @ (image_1577193952iple_a @ F @ B)))))). % image_insert
thf(fact_16_insert__image, axiom,
    ((![X4 : pname, A2 : set_pname, F : pname > hoare_1678595023iple_a]: ((member_pname @ X4 @ A2) => ((insert1477804543iple_a @ (F @ X4) @ (image_2025567968iple_a @ F @ A2)) = (image_2025567968iple_a @ F @ A2)))))). % insert_image
thf(fact_17_image__empty, axiom,
    ((![F : pname > hoare_1678595023iple_a]: ((image_2025567968iple_a @ F @ bot_bot_set_pname) = bot_bo1298296729iple_a)))). % image_empty
thf(fact_18_image__empty, axiom,
    ((![F : hoare_1678595023iple_a > hoare_1678595023iple_a]: ((image_1577193952iple_a @ F @ bot_bo1298296729iple_a) = bot_bo1298296729iple_a)))). % image_empty
thf(fact_19_empty__is__image, axiom,
    ((![F : pname > hoare_1678595023iple_a, A2 : set_pname]: ((bot_bo1298296729iple_a = (image_2025567968iple_a @ F @ A2)) = (A2 = bot_bot_set_pname))))). % empty_is_image
thf(fact_20_empty__is__image, axiom,
    ((![F : hoare_1678595023iple_a > hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((bot_bo1298296729iple_a = (image_1577193952iple_a @ F @ A2)) = (A2 = bot_bo1298296729iple_a))))). % empty_is_image
thf(fact_21_image__is__empty, axiom,
    ((![F : pname > hoare_1678595023iple_a, A2 : set_pname]: (((image_2025567968iple_a @ F @ A2) = bot_bo1298296729iple_a) = (A2 = bot_bot_set_pname))))). % image_is_empty
thf(fact_22_image__is__empty, axiom,
    ((![F : hoare_1678595023iple_a > hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: (((image_1577193952iple_a @ F @ A2) = bot_bo1298296729iple_a) = (A2 = bot_bo1298296729iple_a))))). % image_is_empty
thf(fact_23_image__constant, axiom,
    ((![X4 : pname, A2 : set_pname, C : hoare_1678595023iple_a]: ((member_pname @ X4 @ A2) => ((image_2025567968iple_a @ (^[X : pname]: C) @ A2) = (insert1477804543iple_a @ C @ bot_bo1298296729iple_a)))))). % image_constant
thf(fact_24_image__constant__conv, axiom,
    ((![A2 : set_pname, C : hoare_1678595023iple_a]: (((A2 = bot_bot_set_pname) => ((image_2025567968iple_a @ (^[X : pname]: C) @ A2) = bot_bo1298296729iple_a)) & ((~ ((A2 = bot_bot_set_pname))) => ((image_2025567968iple_a @ (^[X : pname]: C) @ A2) = (insert1477804543iple_a @ C @ bot_bo1298296729iple_a))))))). % image_constant_conv
thf(fact_25_image__constant__conv, axiom,
    ((![A2 : set_Ho137910533iple_a, C : hoare_1678595023iple_a]: (((A2 = bot_bo1298296729iple_a) => ((image_1577193952iple_a @ (^[X : hoare_1678595023iple_a]: C) @ A2) = bot_bo1298296729iple_a)) & ((~ ((A2 = bot_bo1298296729iple_a))) => ((image_1577193952iple_a @ (^[X : hoare_1678595023iple_a]: C) @ A2) = (insert1477804543iple_a @ C @ bot_bo1298296729iple_a))))))). % image_constant_conv
thf(fact_26_com_Oinject_I6_J, axiom,
    ((![X7 : pname, Y7 : pname]: (((body @ X7) = (body @ Y7)) = (X7 = Y7))))). % com.inject(6)
thf(fact_27_image__eqI, axiom,
    ((![B2 : hoare_1678595023iple_a, F : pname > hoare_1678595023iple_a, X4 : pname, A2 : set_pname]: ((B2 = (F @ X4)) => ((member_pname @ X4 @ A2) => (member1332298086iple_a @ B2 @ (image_2025567968iple_a @ F @ A2))))))). % image_eqI
thf(fact_28_image__eqI, axiom,
    ((![B2 : pname, F : pname > pname, X4 : pname, A2 : set_pname]: ((B2 = (F @ X4)) => ((member_pname @ X4 @ A2) => (member_pname @ B2 @ (image_pname_pname @ F @ A2))))))). % image_eqI
thf(fact_29_empty__Collect__eq, axiom,
    ((![P2 : hoare_1678595023iple_a > $o]: ((bot_bo1298296729iple_a = (collec1600235172iple_a @ P2)) = (![X : hoare_1678595023iple_a]: (~ ((P2 @ X)))))))). % empty_Collect_eq
thf(fact_30_Collect__empty__eq, axiom,
    ((![P2 : hoare_1678595023iple_a > $o]: (((collec1600235172iple_a @ P2) = bot_bo1298296729iple_a) = (![X : hoare_1678595023iple_a]: (~ ((P2 @ X)))))))). % Collect_empty_eq
thf(fact_31_all__not__in__conv, axiom,
    ((![A2 : set_pname]: ((![X : pname]: (~ ((member_pname @ X @ A2)))) = (A2 = bot_bot_set_pname))))). % all_not_in_conv
thf(fact_32_all__not__in__conv, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((![X : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ X @ A2)))) = (A2 = bot_bo1298296729iple_a))))). % all_not_in_conv
thf(fact_33_empty__iff, axiom,
    ((![C : pname]: (~ ((member_pname @ C @ bot_bot_set_pname)))))). % empty_iff
thf(fact_34_empty__iff, axiom,
    ((![C : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ C @ bot_bo1298296729iple_a)))))). % empty_iff
thf(fact_35_insert__absorb2, axiom,
    ((![X4 : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((insert1477804543iple_a @ X4 @ (insert1477804543iple_a @ X4 @ A2)) = (insert1477804543iple_a @ X4 @ A2))))). % insert_absorb2
thf(fact_36_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_37_insert__iff, axiom,
    ((![A : pname, B2 : pname, A2 : set_pname]: ((member_pname @ A @ (insert_pname @ B2 @ A2)) = (((A = B2)) | ((member_pname @ A @ A2))))))). % insert_iff
thf(fact_38_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_39_insertCI, axiom,
    ((![A : pname, B : set_pname, B2 : pname]: (((~ ((member_pname @ A @ B))) => (A = B2)) => (member_pname @ A @ (insert_pname @ B2 @ B)))))). % insertCI
thf(fact_40_rev__image__eqI, axiom,
    ((![X4 : pname, A2 : set_pname, B2 : hoare_1678595023iple_a, F : pname > hoare_1678595023iple_a]: ((member_pname @ X4 @ A2) => ((B2 = (F @ X4)) => (member1332298086iple_a @ B2 @ (image_2025567968iple_a @ F @ A2))))))). % rev_image_eqI
thf(fact_41_rev__image__eqI, axiom,
    ((![X4 : pname, A2 : set_pname, B2 : pname, F : pname > pname]: ((member_pname @ X4 @ A2) => ((B2 = (F @ X4)) => (member_pname @ B2 @ (image_pname_pname @ F @ A2))))))). % rev_image_eqI
thf(fact_42_ball__imageD, axiom,
    ((![F : pname > hoare_1678595023iple_a, A2 : set_pname, P2 : hoare_1678595023iple_a > $o]: ((![X5 : hoare_1678595023iple_a]: ((member1332298086iple_a @ X5 @ (image_2025567968iple_a @ F @ A2)) => (P2 @ X5))) => (![X6 : pname]: ((member_pname @ X6 @ A2) => (P2 @ (F @ X6)))))))). % ball_imageD
thf(fact_43_image__cong, axiom,
    ((![M : set_pname, N : set_pname, F : pname > hoare_1678595023iple_a, G3 : pname > hoare_1678595023iple_a]: ((M = N) => ((![X5 : pname]: ((member_pname @ X5 @ N) => ((F @ X5) = (G3 @ X5)))) => ((image_2025567968iple_a @ F @ M) = (image_2025567968iple_a @ G3 @ N))))))). % image_cong
thf(fact_44_bex__imageD, axiom,
    ((![F : pname > hoare_1678595023iple_a, A2 : set_pname, P2 : hoare_1678595023iple_a > $o]: ((?[X6 : hoare_1678595023iple_a]: ((member1332298086iple_a @ X6 @ (image_2025567968iple_a @ F @ A2)) & (P2 @ X6))) => (?[X5 : pname]: ((member_pname @ X5 @ A2) & (P2 @ (F @ X5)))))))). % bex_imageD
thf(fact_45_image__iff, axiom,
    ((![Z4 : hoare_1678595023iple_a, F : pname > hoare_1678595023iple_a, A2 : set_pname]: ((member1332298086iple_a @ Z4 @ (image_2025567968iple_a @ F @ A2)) = (?[X : pname]: (((member_pname @ X @ A2)) & ((Z4 = (F @ X))))))))). % image_iff
thf(fact_46_imageI, axiom,
    ((![X4 : pname, A2 : set_pname, F : pname > hoare_1678595023iple_a]: ((member_pname @ X4 @ A2) => (member1332298086iple_a @ (F @ X4) @ (image_2025567968iple_a @ F @ A2)))))). % imageI
thf(fact_47_imageI, axiom,
    ((![X4 : pname, A2 : set_pname, F : pname > pname]: ((member_pname @ X4 @ A2) => (member_pname @ (F @ X4) @ (image_pname_pname @ F @ A2)))))). % imageI
thf(fact_48_ex__in__conv, axiom,
    ((![A2 : set_pname]: ((?[X : pname]: (member_pname @ X @ A2)) = (~ ((A2 = bot_bot_set_pname))))))). % ex_in_conv
thf(fact_49_ex__in__conv, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((?[X : hoare_1678595023iple_a]: (member1332298086iple_a @ X @ A2)) = (~ ((A2 = bot_bo1298296729iple_a))))))). % ex_in_conv
thf(fact_50_equals0I, axiom,
    ((![A2 : set_pname]: ((![Y5 : pname]: (~ ((member_pname @ Y5 @ A2)))) => (A2 = bot_bot_set_pname))))). % equals0I
thf(fact_51_equals0I, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((![Y5 : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ Y5 @ A2)))) => (A2 = bot_bo1298296729iple_a))))). % equals0I
thf(fact_52_equals0D, axiom,
    ((![A2 : set_pname, A : pname]: ((A2 = bot_bot_set_pname) => (~ ((member_pname @ A @ A2))))))). % equals0D
thf(fact_53_equals0D, axiom,
    ((![A2 : set_Ho137910533iple_a, A : hoare_1678595023iple_a]: ((A2 = bot_bo1298296729iple_a) => (~ ((member1332298086iple_a @ A @ A2))))))). % equals0D
thf(fact_54_emptyE, axiom,
    ((![A : pname]: (~ ((member_pname @ A @ bot_bot_set_pname)))))). % emptyE
thf(fact_55_emptyE, axiom,
    ((![A : hoare_1678595023iple_a]: (~ ((member1332298086iple_a @ A @ bot_bo1298296729iple_a)))))). % emptyE
thf(fact_56_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_57_mk__disjoint__insert, axiom,
    ((![A : pname, A2 : set_pname]: ((member_pname @ A @ A2) => (?[B3 : set_pname]: ((A2 = (insert_pname @ A @ B3)) & (~ ((member_pname @ A @ B3))))))))). % mk_disjoint_insert
thf(fact_58_insert__commute, axiom,
    ((![X4 : hoare_1678595023iple_a, Y : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((insert1477804543iple_a @ X4 @ (insert1477804543iple_a @ Y @ A2)) = (insert1477804543iple_a @ Y @ (insert1477804543iple_a @ X4 @ A2)))))). % insert_commute
thf(fact_59_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_60_insert__eq__iff, axiom,
    ((![A : pname, A2 : set_pname, B2 : pname, B : set_pname]: ((~ ((member_pname @ A @ A2))) => ((~ ((member_pname @ B2 @ B))) => (((insert_pname @ A @ A2) = (insert_pname @ B2 @ B)) = (((((A = B2)) => ((A2 = B)))) & ((((~ ((A = B2)))) => ((?[C2 : set_pname]: (((A2 = (insert_pname @ B2 @ C2))) & ((((~ ((member_pname @ B2 @ C2)))) & ((((B = (insert_pname @ A @ C2))) & ((~ ((member_pname @ A @ C2)))))))))))))))))))). % insert_eq_iff
thf(fact_61_insert__absorb, axiom,
    ((![A : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((member1332298086iple_a @ A @ A2) => ((insert1477804543iple_a @ A @ A2) = A2))))). % insert_absorb
thf(fact_62_insert__absorb, axiom,
    ((![A : pname, A2 : set_pname]: ((member_pname @ A @ A2) => ((insert_pname @ A @ A2) = A2))))). % insert_absorb
thf(fact_63_insert__ident, axiom,
    ((![X4 : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: ((~ ((member1332298086iple_a @ X4 @ A2))) => ((~ ((member1332298086iple_a @ X4 @ B))) => (((insert1477804543iple_a @ X4 @ A2) = (insert1477804543iple_a @ X4 @ B)) = (A2 = B))))))). % insert_ident
thf(fact_64_insert__ident, axiom,
    ((![X4 : pname, A2 : set_pname, B : set_pname]: ((~ ((member_pname @ X4 @ A2))) => ((~ ((member_pname @ X4 @ B))) => (((insert_pname @ X4 @ A2) = (insert_pname @ X4 @ B)) = (A2 = B))))))). % insert_ident
thf(fact_65_mem__Collect__eq, axiom,
    ((![A : pname, P2 : pname > $o]: ((member_pname @ A @ (collect_pname @ P2)) = (P2 @ A))))). % mem_Collect_eq
thf(fact_66_Collect__mem__eq, axiom,
    ((![A2 : set_pname]: ((collect_pname @ (^[X : pname]: (member_pname @ X @ A2))) = A2)))). % Collect_mem_eq
thf(fact_67_Set_Oset__insert, axiom,
    ((![X4 : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: ((member1332298086iple_a @ X4 @ A2) => (~ ((![B3 : set_Ho137910533iple_a]: ((A2 = (insert1477804543iple_a @ X4 @ B3)) => (member1332298086iple_a @ X4 @ B3))))))))). % Set.set_insert
thf(fact_68_Set_Oset__insert, axiom,
    ((![X4 : pname, A2 : set_pname]: ((member_pname @ X4 @ A2) => (~ ((![B3 : set_pname]: ((A2 = (insert_pname @ X4 @ B3)) => (member_pname @ X4 @ B3))))))))). % Set.set_insert
thf(fact_69_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_70_insertI2, axiom,
    ((![A : pname, B : set_pname, B2 : pname]: ((member_pname @ A @ B) => (member_pname @ A @ (insert_pname @ B2 @ B)))))). % insertI2
thf(fact_71_insertI1, axiom,
    ((![A : hoare_1678595023iple_a, B : set_Ho137910533iple_a]: (member1332298086iple_a @ A @ (insert1477804543iple_a @ A @ B))))). % insertI1
thf(fact_72_insertI1, axiom,
    ((![A : pname, B : set_pname]: (member_pname @ A @ (insert_pname @ A @ B))))). % insertI1
thf(fact_73_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_74_insertE, axiom,
    ((![A : pname, B2 : pname, A2 : set_pname]: ((member_pname @ A @ (insert_pname @ B2 @ A2)) => ((~ ((A = B2))) => (member_pname @ A @ A2)))))). % insertE
thf(fact_75_Compr__image__eq, axiom,
    ((![F : pname > hoare_1678595023iple_a, A2 : set_pname, P2 : hoare_1678595023iple_a > $o]: ((collec1600235172iple_a @ (^[X : hoare_1678595023iple_a]: (((member1332298086iple_a @ X @ (image_2025567968iple_a @ F @ A2))) & ((P2 @ X))))) = (image_2025567968iple_a @ F @ (collect_pname @ (^[X : pname]: (((member_pname @ X @ A2)) & ((P2 @ (F @ X))))))))))). % Compr_image_eq
thf(fact_76_Compr__image__eq, axiom,
    ((![F : pname > pname, A2 : set_pname, P2 : pname > $o]: ((collect_pname @ (^[X : pname]: (((member_pname @ X @ (image_pname_pname @ F @ A2))) & ((P2 @ X))))) = (image_pname_pname @ F @ (collect_pname @ (^[X : pname]: (((member_pname @ X @ A2)) & ((P2 @ (F @ X))))))))))). % Compr_image_eq
thf(fact_77_image__image, axiom,
    ((![F : hoare_1678595023iple_a > hoare_1678595023iple_a, G3 : pname > hoare_1678595023iple_a, A2 : set_pname]: ((image_1577193952iple_a @ F @ (image_2025567968iple_a @ G3 @ A2)) = (image_2025567968iple_a @ (^[X : pname]: (F @ (G3 @ X))) @ A2))))). % image_image
thf(fact_78_image__image, axiom,
    ((![F : pname > hoare_1678595023iple_a, G3 : pname > pname, A2 : set_pname]: ((image_2025567968iple_a @ F @ (image_pname_pname @ G3 @ A2)) = (image_2025567968iple_a @ (^[X : pname]: (F @ (G3 @ X))) @ A2))))). % image_image
thf(fact_79_imageE, axiom,
    ((![B2 : hoare_1678595023iple_a, F : pname > hoare_1678595023iple_a, A2 : set_pname]: ((member1332298086iple_a @ B2 @ (image_2025567968iple_a @ F @ A2)) => (~ ((![X5 : pname]: ((B2 = (F @ X5)) => (~ ((member_pname @ X5 @ A2))))))))))). % imageE
thf(fact_80_imageE, axiom,
    ((![B2 : pname, F : pname > pname, A2 : set_pname]: ((member_pname @ B2 @ (image_pname_pname @ F @ A2)) => (~ ((![X5 : pname]: ((B2 = (F @ X5)) => (~ ((member_pname @ X5 @ A2))))))))))). % imageE
thf(fact_81_empty__def, axiom,
    ((bot_bo1298296729iple_a = (collec1600235172iple_a @ (^[X : hoare_1678595023iple_a]: $false))))). % empty_def
thf(fact_82_insert__Collect, axiom,
    ((![A : hoare_1678595023iple_a, P2 : hoare_1678595023iple_a > $o]: ((insert1477804543iple_a @ A @ (collec1600235172iple_a @ P2)) = (collec1600235172iple_a @ (^[U : hoare_1678595023iple_a]: (((~ ((U = A)))) => ((P2 @ U))))))))). % insert_Collect
thf(fact_83_insert__compr, axiom,
    ((insert1477804543iple_a = (^[A3 : hoare_1678595023iple_a]: (^[B4 : set_Ho137910533iple_a]: (collec1600235172iple_a @ (^[X : hoare_1678595023iple_a]: (((X = A3)) | ((member1332298086iple_a @ X @ B4)))))))))). % insert_compr
thf(fact_84_insert__compr, axiom,
    ((insert_pname = (^[A3 : pname]: (^[B4 : set_pname]: (collect_pname @ (^[X : pname]: (((X = A3)) | ((member_pname @ X @ B4)))))))))). % insert_compr
thf(fact_85_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_86_insert__not__empty, axiom,
    ((![A : hoare_1678595023iple_a, A2 : set_Ho137910533iple_a]: (~ (((insert1477804543iple_a @ A @ A2) = bot_bo1298296729iple_a)))))). % insert_not_empty
thf(fact_87_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_88_singleton__iff, axiom,
    ((![B2 : pname, A : pname]: ((member_pname @ B2 @ (insert_pname @ A @ bot_bot_set_pname)) = (B2 = A))))). % singleton_iff
thf(fact_89_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_90_singletonD, axiom,
    ((![B2 : pname, A : pname]: ((member_pname @ B2 @ (insert_pname @ A @ bot_bot_set_pname)) => (B2 = A))))). % singletonD
thf(fact_91_singletonD, axiom,
    ((![B2 : hoare_1678595023iple_a, A : hoare_1678595023iple_a]: ((member1332298086iple_a @ B2 @ (insert1477804543iple_a @ A @ bot_bo1298296729iple_a)) => (B2 = A))))). % singletonD
thf(fact_92_Collect__conv__if2, axiom,
    ((![P2 : hoare_1678595023iple_a > $o, A : hoare_1678595023iple_a]: (((P2 @ A) => ((collec1600235172iple_a @ (^[X : hoare_1678595023iple_a]: (((A = X)) & ((P2 @ X))))) = (insert1477804543iple_a @ A @ bot_bo1298296729iple_a))) & ((~ ((P2 @ A))) => ((collec1600235172iple_a @ (^[X : hoare_1678595023iple_a]: (((A = X)) & ((P2 @ X))))) = bot_bo1298296729iple_a)))))). % Collect_conv_if2
thf(fact_93_Collect__conv__if, axiom,
    ((![P2 : hoare_1678595023iple_a > $o, A : hoare_1678595023iple_a]: (((P2 @ A) => ((collec1600235172iple_a @ (^[X : hoare_1678595023iple_a]: (((X = A)) & ((P2 @ X))))) = (insert1477804543iple_a @ A @ bot_bo1298296729iple_a))) & ((~ ((P2 @ A))) => ((collec1600235172iple_a @ (^[X : hoare_1678595023iple_a]: (((X = A)) & ((P2 @ X))))) = bot_bo1298296729iple_a)))))). % Collect_conv_if
thf(fact_94_the__elem__eq, axiom,
    ((![X4 : hoare_1678595023iple_a]: ((the_el434698138iple_a @ (insert1477804543iple_a @ X4 @ bot_bo1298296729iple_a)) = X4)))). % the_elem_eq
thf(fact_95_is__singletonI, axiom,
    ((![X4 : hoare_1678595023iple_a]: (is_sin1784037339iple_a @ (insert1477804543iple_a @ X4 @ bot_bo1298296729iple_a))))). % is_singletonI
thf(fact_96_hoare__derivs_OSkip, axiom,
    ((![G : set_Ho137910533iple_a, P2 : a > state > $o]: (hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P2 @ skip @ P2) @ bot_bo1298296729iple_a))))). % hoare_derivs.Skip
thf(fact_97_Comp, axiom,
    ((![G : set_Ho137910533iple_a, P2 : a > state > $o, C : com, Q : a > state > $o, D : com, R : a > state > $o]: ((hoare_129598474rivs_a @ G @ (insert1477804543iple_a @ (hoare_719046530iple_a @ P2 @ 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 @ P2 @ (semi @ C @ D) @ R) @ bot_bo1298296729iple_a))))))). % Comp
thf(fact_98_the__elem__image__unique, axiom,
    ((![A2 : set_pname, F : pname > hoare_1678595023iple_a, X4 : pname]: ((~ ((A2 = bot_bot_set_pname))) => ((![Y5 : pname]: ((member_pname @ Y5 @ A2) => ((F @ Y5) = (F @ X4)))) => ((the_el434698138iple_a @ (image_2025567968iple_a @ F @ A2)) = (F @ X4))))))). % the_elem_image_unique
thf(fact_99_Set_Ois__empty__def, axiom,
    ((is_emp901906557iple_a = (^[A4 : set_Ho137910533iple_a]: (A4 = bot_bo1298296729iple_a))))). % Set.is_empty_def
thf(fact_100_is__singleton__def, axiom,
    ((is_sin1784037339iple_a = (^[A4 : set_Ho137910533iple_a]: (?[X : hoare_1678595023iple_a]: (A4 = (insert1477804543iple_a @ X @ bot_bo1298296729iple_a))))))). % is_singleton_def
thf(fact_101_is__singletonE, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((is_sin1784037339iple_a @ A2) => (~ ((![X5 : hoare_1678595023iple_a]: (~ ((A2 = (insert1477804543iple_a @ X5 @ bot_bo1298296729iple_a))))))))))). % is_singletonE
thf(fact_102_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_103_com_Odistinct_I5_J, axiom,
    ((![X41 : com, X42 : com]: (~ ((skip = (semi @ X41 @ X42))))))). % com.distinct(5)
thf(fact_104_bot__set__def, axiom,
    ((bot_bo1298296729iple_a = (collec1600235172iple_a @ bot_bo431311916le_a_o)))). % bot_set_def
thf(fact_105_com_Odistinct_I41_J, axiom,
    ((![X41 : com, X42 : com, X7 : pname]: (~ (((semi @ X41 @ X42) = (body @ X7))))))). % com.distinct(41)
thf(fact_106_com_Odistinct_I11_J, axiom,
    ((![X7 : pname]: (~ ((skip = (body @ X7))))))). % com.distinct(11)
thf(fact_107_is__singleton__the__elem, axiom,
    ((is_sin1784037339iple_a = (^[A4 : set_Ho137910533iple_a]: (A4 = (insert1477804543iple_a @ (the_el434698138iple_a @ A4) @ bot_bo1298296729iple_a)))))). % is_singleton_the_elem
thf(fact_108_is__singletonI_H, axiom,
    ((![A2 : set_pname]: ((~ ((A2 = bot_bot_set_pname))) => ((![X5 : pname, Y5 : pname]: ((member_pname @ X5 @ A2) => ((member_pname @ Y5 @ A2) => (X5 = Y5)))) => (is_singleton_pname @ A2)))))). % is_singletonI'
thf(fact_109_is__singletonI_H, axiom,
    ((![A2 : set_Ho137910533iple_a]: ((~ ((A2 = bot_bo1298296729iple_a))) => ((![X5 : hoare_1678595023iple_a, Y5 : hoare_1678595023iple_a]: ((member1332298086iple_a @ X5 @ A2) => ((member1332298086iple_a @ Y5 @ A2) => (X5 = Y5)))) => (is_sin1784037339iple_a @ A2)))))). % is_singletonI'
thf(fact_110_bind__singleton__conv__image, axiom,
    ((![A2 : set_pname, F : pname > hoare_1678595023iple_a]: ((bind_p599804078iple_a @ A2 @ (^[X : pname]: (insert1477804543iple_a @ (F @ X) @ bot_bo1298296729iple_a))) = (image_2025567968iple_a @ F @ A2))))). % bind_singleton_conv_image
thf(fact_111_the__elem__def, axiom,
    ((the_el434698138iple_a = (^[X8 : set_Ho137910533iple_a]: (the_Ho1602500360iple_a @ (^[X : hoare_1678595023iple_a]: (X8 = (insert1477804543iple_a @ X @ bot_bo1298296729iple_a)))))))). % the_elem_def
thf(fact_112_empty__bind, axiom,
    ((![F : hoare_1678595023iple_a > set_Ho137910533iple_a]: ((bind_H209762450iple_a @ bot_bo1298296729iple_a @ F) = bot_bo1298296729iple_a)))). % empty_bind
thf(fact_113_bind__const, axiom,
    ((![A2 : set_Ho137910533iple_a, B : set_Ho137910533iple_a]: (((A2 = bot_bo1298296729iple_a) => ((bind_H209762450iple_a @ A2 @ (^[Uu : hoare_1678595023iple_a]: B)) = bot_bo1298296729iple_a)) & ((~ ((A2 = bot_bo1298296729iple_a))) => ((bind_H209762450iple_a @ A2 @ (^[Uu : hoare_1678595023iple_a]: B)) = B)))))). % bind_const
thf(fact_114_Sup_OSUP__cong, axiom,
    ((![A2 : set_pname, B : set_pname, C3 : pname > hoare_1678595023iple_a, D2 : pname > hoare_1678595023iple_a, Sup : set_Ho137910533iple_a > hoare_1678595023iple_a]: ((A2 = B) => ((![X5 : pname]: ((member_pname @ X5 @ B) => ((C3 @ X5) = (D2 @ X5)))) => ((Sup @ (image_2025567968iple_a @ C3 @ A2)) = (Sup @ (image_2025567968iple_a @ D2 @ B)))))))). % Sup.SUP_cong
thf(fact_115_Inf_OINF__cong, axiom,
    ((![A2 : set_pname, B : set_pname, C3 : pname > hoare_1678595023iple_a, D2 : pname > hoare_1678595023iple_a, Inf : set_Ho137910533iple_a > hoare_1678595023iple_a]: ((A2 = B) => ((![X5 : pname]: ((member_pname @ X5 @ B) => ((C3 @ X5) = (D2 @ X5)))) => ((Inf @ (image_2025567968iple_a @ C3 @ A2)) = (Inf @ (image_2025567968iple_a @ D2 @ B)))))))). % Inf.INF_cong
thf(fact_116_Collect__empty__eq__bot, axiom,
    ((![P2 : hoare_1678595023iple_a > $o]: (((collec1600235172iple_a @ P2) = bot_bo1298296729iple_a) = (P2 = bot_bo431311916le_a_o))))). % Collect_empty_eq_bot
thf(fact_117_bot__empty__eq, axiom,
    ((bot_bot_pname_o = (^[X : pname]: (member_pname @ X @ bot_bot_set_pname))))). % bot_empty_eq
thf(fact_118_bot__empty__eq, axiom,
    ((bot_bo431311916le_a_o = (^[X : hoare_1678595023iple_a]: (member1332298086iple_a @ X @ bot_bo1298296729iple_a))))). % bot_empty_eq
thf(fact_119_range__constant, axiom,
    ((![X4 : hoare_1678595023iple_a]: ((image_2025567968iple_a @ (^[Uu : pname]: X4) @ top_top_set_pname) = (insert1477804543iple_a @ X4 @ bot_bo1298296729iple_a))))). % range_constant
thf(fact_120_UNIV__I, axiom,
    ((![X4 : pname]: (member_pname @ X4 @ top_top_set_pname)))). % UNIV_I
thf(fact_121_Collect__const, axiom,
    ((![P2 : $o]: ((P2 => ((collec1600235172iple_a @ (^[S3 : hoare_1678595023iple_a]: P2)) = top_to743720885iple_a)) & ((~ (P2)) => ((collec1600235172iple_a @ (^[S3 : hoare_1678595023iple_a]: P2)) = bot_bo1298296729iple_a)))))). % Collect_const
thf(fact_122_rangeI, axiom,
    ((![F : pname > hoare_1678595023iple_a, X4 : pname]: (member1332298086iple_a @ (F @ X4) @ (image_2025567968iple_a @ F @ top_top_set_pname))))). % rangeI
thf(fact_123_range__eqI, axiom,
    ((![B2 : hoare_1678595023iple_a, F : pname > hoare_1678595023iple_a, X4 : pname]: ((B2 = (F @ X4)) => (member1332298086iple_a @ B2 @ (image_2025567968iple_a @ F @ top_top_set_pname)))))). % range_eqI
thf(fact_124_insert__UNIV, axiom,
    ((![X4 : hoare_1678595023iple_a]: ((insert1477804543iple_a @ X4 @ top_to743720885iple_a) = top_to743720885iple_a)))). % insert_UNIV
thf(fact_125_UNIV__witness, axiom,
    ((?[X5 : pname]: (member_pname @ X5 @ top_top_set_pname)))). % UNIV_witness
thf(fact_126_UNIV__eq__I, axiom,
    ((![A2 : set_pname]: ((![X5 : pname]: (member_pname @ X5 @ A2)) => (top_top_set_pname = A2))))). % UNIV_eq_I
thf(fact_127_rangeE, axiom,
    ((![B2 : hoare_1678595023iple_a, F : pname > hoare_1678595023iple_a]: ((member1332298086iple_a @ B2 @ (image_2025567968iple_a @ F @ top_top_set_pname)) => (~ ((![X5 : pname]: (~ ((B2 = (F @ X5))))))))))). % rangeE
thf(fact_128_range__composition, axiom,
    ((![F : hoare_1678595023iple_a > hoare_1678595023iple_a, G3 : pname > hoare_1678595023iple_a]: ((image_2025567968iple_a @ (^[X : pname]: (F @ (G3 @ X))) @ top_top_set_pname) = (image_1577193952iple_a @ F @ (image_2025567968iple_a @ G3 @ top_top_set_pname)))))). % range_composition
thf(fact_129_range__composition, axiom,
    ((![F : pname > hoare_1678595023iple_a, G3 : pname > pname]: ((image_2025567968iple_a @ (^[X : pname]: (F @ (G3 @ X))) @ top_top_set_pname) = (image_2025567968iple_a @ F @ (image_pname_pname @ G3 @ top_top_set_pname)))))). % range_composition
thf(fact_130_empty__not__UNIV, axiom,
    ((~ ((bot_bo1298296729iple_a = top_to743720885iple_a))))). % empty_not_UNIV
thf(fact_131_range__eq__singletonD, axiom,
    ((![F : pname > hoare_1678595023iple_a, A : hoare_1678595023iple_a, X4 : pname]: (((image_2025567968iple_a @ F @ top_top_set_pname) = (insert1477804543iple_a @ A @ bot_bo1298296729iple_a)) => ((F @ X4) = A))))). % range_eq_singletonD
thf(fact_132_iso__tuple__UNIV__I, axiom,
    ((![X4 : pname]: (member_pname @ X4 @ top_top_set_pname)))). % iso_tuple_UNIV_I
thf(fact_133_surjD, axiom,
    ((![F : pname > hoare_1678595023iple_a, Y : hoare_1678595023iple_a]: (((image_2025567968iple_a @ F @ top_top_set_pname) = top_to743720885iple_a) => (?[X5 : pname]: (Y = (F @ X5))))))). % surjD
thf(fact_134_surjE, axiom,
    ((![F : pname > hoare_1678595023iple_a, Y : hoare_1678595023iple_a]: (((image_2025567968iple_a @ F @ top_top_set_pname) = top_to743720885iple_a) => (~ ((![X5 : pname]: (~ ((Y = (F @ X5))))))))))). % surjE
thf(fact_135_top__empty__eq, axiom,
    ((top_top_pname_o = (^[X : pname]: (member_pname @ X @ top_top_set_pname))))). % top_empty_eq
thf(fact_136_surj__def, axiom,
    ((![F : pname > hoare_1678595023iple_a]: (((image_2025567968iple_a @ F @ top_top_set_pname) = top_to743720885iple_a) = (![Y6 : hoare_1678595023iple_a]: (?[X : pname]: (Y6 = (F @ X)))))))). % surj_def
thf(fact_137_surjI, axiom,
    ((![G3 : pname > hoare_1678595023iple_a, F : hoare_1678595023iple_a > pname]: ((![X5 : hoare_1678595023iple_a]: ((G3 @ (F @ X5)) = X5)) => ((image_2025567968iple_a @ G3 @ top_top_set_pname) = top_to743720885iple_a))))). % surjI
thf(fact_138_surj__swap__iff, axiom,
    ((![A : pname, B2 : pname, F : pname > hoare_1678595023iple_a]: (((image_2025567968iple_a @ (swap_p1758894497iple_a @ A @ B2 @ F) @ top_top_set_pname) = top_to743720885iple_a) = ((image_2025567968iple_a @ F @ top_top_set_pname) = top_to743720885iple_a))))). % surj_swap_iff
thf(fact_139_UNION__singleton__eq__range, axiom,
    ((![F : pname > hoare_1678595023iple_a, A2 : set_pname]: ((comple1228397400iple_a @ (image_478803990iple_a @ (^[X : pname]: (insert1477804543iple_a @ (F @ X) @ bot_bo1298296729iple_a)) @ A2)) = (image_2025567968iple_a @ F @ A2))))). % UNION_singleton_eq_range
thf(fact_140_UnionI, axiom,
    ((![X9 : set_pname, C3 : set_set_pname, A2 : pname]: ((member_set_pname @ X9 @ C3) => ((member_pname @ A2 @ X9) => (member_pname @ A2 @ (comple1674572307_pname @ C3))))))). % UnionI
thf(fact_141_Union__iff, axiom,
    ((![A2 : pname, C3 : set_set_pname]: ((member_pname @ A2 @ (comple1674572307_pname @ C3)) = (?[X : set_pname]: (((member_set_pname @ X @ C3)) & ((member_pname @ A2 @ X)))))))). % Union_iff
thf(fact_142_Sup__bot__conv_I2_J, axiom,
    ((![A2 : set_se152467259iple_a]: ((bot_bo1298296729iple_a = (comple1228397400iple_a @ A2)) = (![X : set_Ho137910533iple_a]: (((member521824924iple_a @ X @ A2)) => ((X = bot_bo1298296729iple_a)))))))). % Sup_bot_conv(2)
thf(fact_143_Sup__bot__conv_I1_J, axiom,
    ((![A2 : set_se152467259iple_a]: (((comple1228397400iple_a @ A2) = bot_bo1298296729iple_a) = (![X : set_Ho137910533iple_a]: (((member521824924iple_a @ X @ A2)) => ((X = bot_bo1298296729iple_a)))))))). % Sup_bot_conv(1)
thf(fact_144_swap__image__eq, axiom,
    ((![A : pname, A2 : set_pname, B2 : pname, F : pname > hoare_1678595023iple_a]: ((member_pname @ A @ A2) => ((member_pname @ B2 @ A2) => ((image_2025567968iple_a @ (swap_p1758894497iple_a @ A @ B2 @ F) @ A2) = (image_2025567968iple_a @ F @ A2))))))). % swap_image_eq

% Conjectures (3)
thf(conj_0, hypothesis,
    ((member_pname @ pn @ procs))).
thf(conj_1, hypothesis,
    ((hoare_129598474rivs_a @ g @ (image_2025567968iple_a @ (^[P4 : pname]: (hoare_719046530iple_a @ (p @ P4) @ (body @ P4) @ (q @ P4))) @ procs)))).
thf(conj_2, conjecture,
    ((hoare_129598474rivs_a @ g @ (insert1477804543iple_a @ (hoare_719046530iple_a @ (p @ pn) @ (body @ pn) @ (q @ pn)) @ bot_bo1298296729iple_a)))).
