% 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_457__3254872_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:15:06.283

% Could-be-implicit typings (7)
thf(ty_n_t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J_J, type,
    set_Ho840737317_state : $tType).
thf(ty_n_t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J, type,
    hoare_958474565_state : $tType).
thf(ty_n_t__Option__Ooption_It__Com__Ocom_J, type,
    option_com : $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).

% Explicit typings (31)
thf(sy_c_Com_OWT, type,
    wt : com > $o).
thf(sy_c_Com_OWT__bodies, type,
    wT_bodies : $o).
thf(sy_c_Com_Obody, type,
    body : pname > option_com).
thf(sy_c_Com_Ocom_OBODY, type,
    body2 : pname > com).
thf(sy_c_Hoare__Mirabelle__raqjowkjvm_OMGT, type,
    hoare_Mirabelle_MGT : com > hoare_958474565_state).
thf(sy_c_Hoare__Mirabelle__raqjowkjvm_Ohoare__derivs_001t__Com__Ostate, type,
    hoare_604442164_state : set_Ho840737317_state > set_Ho840737317_state > $o).
thf(sy_c_Hoare__Mirabelle__raqjowkjvm_Ostate__not__singleton, type,
    hoare_405891322gleton : $o).
thf(sy_c_Hoare__Mirabelle__raqjowkjvm_Otriple_Otriple_001t__Com__Ostate, type,
    hoare_1659279548_state : (state > state > $o) > com > (state > state > $o) > hoare_958474565_state).
thf(sy_c_Map_Odom_001t__Com__Opname_001t__Com__Ocom, type,
    dom_pname_com : (pname > option_com) > set_pname).
thf(sy_c_Map_Odom_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J_001t__Com__Ocom, type,
    dom_Ho256784515te_com : (hoare_958474565_state > option_com) > set_Ho840737317_state).
thf(sy_c_Option_Ooption_OSome_001t__Com__Ocom, type,
    some_com : com > option_com).
thf(sy_c_Option_Ooption_Othe_001t__Com__Ocom, type,
    the_com : option_com > com).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J_M_Eo_J, type,
    bot_bo1428770700tate_o : hoare_958474565_state > $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_It__Com__Ostate_J_J, type,
    bot_bo105666705_state : set_Ho840737317_state).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Com__Opname_J, type,
    ord_le865024672_pname : set_pname > set_pname > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J_J, type,
    ord_le1945819589_state : set_Ho840737317_state > set_Ho840737317_state > $o).
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_It__Com__Ostate_J, type,
    collec305460656_state : (hoare_958474565_state > $o) > set_Ho840737317_state).
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_It__Com__Ostate_J, type,
    image_2144627828_state : (pname > hoare_958474565_state) > set_pname > set_Ho840737317_state).
thf(sy_c_Set_Oimage_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J, type,
    image_1849792670_state : (hoare_958474565_state > hoare_958474565_state) > set_Ho840737317_state > set_Ho840737317_state).
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_It__Com__Ostate_J, type,
    insert776267541_state : hoare_958474565_state > set_Ho840737317_state > set_Ho840737317_state).
thf(sy_c_Set_Ois__singleton_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J, type,
    is_sin536631353_state : set_Ho840737317_state > $o).
thf(sy_c_Set_Othe__elem_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J, type,
    the_el1300254266_state : set_Ho840737317_state > hoare_958474565_state).
thf(sy_c_member_001t__Com__Opname, type,
    member_pname : pname > set_pname > $o).
thf(sy_c_member_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J, type,
    member109514606_state : hoare_958474565_state > set_Ho840737317_state > $o).
thf(sy_v_c, type,
    c : com).
thf(sy_v_pn, type,
    pn : pname).
thf(sy_v_y, type,
    y : com).

% Relevant facts (150)
thf(fact_0_MGF, axiom,
    ((![C : com]: (hoare_405891322gleton => (wT_bodies => ((wt @ C) => (hoare_604442164_state @ bot_bo105666705_state @ (insert776267541_state @ (hoare_Mirabelle_MGT @ C) @ bot_bo105666705_state)))))))). % MGF
thf(fact_1_MGF__lemma1, axiom,
    ((![G : set_Ho840737317_state, C : com]: (hoare_405891322gleton => ((![X : pname]: ((member_pname @ X @ (dom_pname_com @ body)) => (hoare_604442164_state @ G @ (insert776267541_state @ (hoare_Mirabelle_MGT @ (body2 @ X)) @ bot_bo105666705_state)))) => ((wt @ C) => (hoare_604442164_state @ G @ (insert776267541_state @ (hoare_Mirabelle_MGT @ C) @ bot_bo105666705_state)))))))). % MGF_lemma1
thf(fact_2_single__stateE, axiom,
    ((hoare_405891322gleton => (![T : state]: (~ ((![S : state]: (S = T)))))))). % single_stateE
thf(fact_3_derivs__insertD, axiom,
    ((![G : set_Ho840737317_state, T2 : hoare_958474565_state, Ts : set_Ho840737317_state]: ((hoare_604442164_state @ G @ (insert776267541_state @ T2 @ Ts)) => ((hoare_604442164_state @ G @ (insert776267541_state @ T2 @ bot_bo105666705_state)) & (hoare_604442164_state @ G @ Ts)))))). % derivs_insertD
thf(fact_4_cut, axiom,
    ((![G2 : set_Ho840737317_state, Ts : set_Ho840737317_state, G : set_Ho840737317_state]: ((hoare_604442164_state @ G2 @ Ts) => ((hoare_604442164_state @ G @ G2) => (hoare_604442164_state @ G @ Ts)))))). % cut
thf(fact_5_empty, axiom,
    ((![G : set_Ho840737317_state]: (hoare_604442164_state @ G @ bot_bo105666705_state)))). % empty
thf(fact_6_hoare__derivs_Oinsert, axiom,
    ((![G : set_Ho840737317_state, T2 : hoare_958474565_state, Ts : set_Ho840737317_state]: ((hoare_604442164_state @ G @ (insert776267541_state @ T2 @ bot_bo105666705_state)) => ((hoare_604442164_state @ G @ Ts) => (hoare_604442164_state @ G @ (insert776267541_state @ T2 @ Ts))))))). % hoare_derivs.insert
thf(fact_7_state__not__singleton__def, axiom,
    ((hoare_405891322gleton = (?[S2 : state]: (?[T3 : state]: (~ ((S2 = T3)))))))). % state_not_singleton_def
thf(fact_8_WT__bodiesD, axiom,
    ((![Pn : pname, B : com]: (wT_bodies => (((body @ Pn) = (some_com @ B)) => (wt @ B)))))). % WT_bodiesD
thf(fact_9_singleton__conv, axiom,
    ((![A : hoare_958474565_state]: ((collec305460656_state @ (^[X2 : hoare_958474565_state]: (X2 = A))) = (insert776267541_state @ A @ bot_bo105666705_state))))). % singleton_conv
thf(fact_10_singleton__conv2, axiom,
    ((![A : hoare_958474565_state]: ((collec305460656_state @ ((^[Y : hoare_958474565_state]: (^[Z : hoare_958474565_state]: (Y = Z))) @ A)) = (insert776267541_state @ A @ bot_bo105666705_state))))). % singleton_conv2
thf(fact_11_WTs__elim__cases_I7_J, axiom,
    ((![P : pname]: ((wt @ (body2 @ P)) => (~ ((![Y2 : com]: (~ (((body @ P) = (some_com @ Y2))))))))))). % WTs_elim_cases(7)
thf(fact_12_singletonI, axiom,
    ((![A : hoare_958474565_state]: (member109514606_state @ A @ (insert776267541_state @ A @ bot_bo105666705_state))))). % singletonI
thf(fact_13_image__insert, axiom,
    ((![F : pname > hoare_958474565_state, A : pname, B2 : set_pname]: ((image_2144627828_state @ F @ (insert_pname @ A @ B2)) = (insert776267541_state @ (F @ A) @ (image_2144627828_state @ F @ B2)))))). % image_insert
thf(fact_14_image__insert, axiom,
    ((![F : hoare_958474565_state > hoare_958474565_state, A : hoare_958474565_state, B2 : set_Ho840737317_state]: ((image_1849792670_state @ F @ (insert776267541_state @ A @ B2)) = (insert776267541_state @ (F @ A) @ (image_1849792670_state @ F @ B2)))))). % image_insert
thf(fact_15_insert__image, axiom,
    ((![X3 : pname, A2 : set_pname, F : pname > hoare_958474565_state]: ((member_pname @ X3 @ A2) => ((insert776267541_state @ (F @ X3) @ (image_2144627828_state @ F @ A2)) = (image_2144627828_state @ F @ A2)))))). % insert_image
thf(fact_16_image__empty, axiom,
    ((![F : pname > hoare_958474565_state]: ((image_2144627828_state @ F @ bot_bot_set_pname) = bot_bo105666705_state)))). % image_empty
thf(fact_17_image__empty, axiom,
    ((![F : hoare_958474565_state > hoare_958474565_state]: ((image_1849792670_state @ F @ bot_bo105666705_state) = bot_bo105666705_state)))). % image_empty
thf(fact_18_empty__is__image, axiom,
    ((![F : pname > hoare_958474565_state, A2 : set_pname]: ((bot_bo105666705_state = (image_2144627828_state @ F @ A2)) = (A2 = bot_bot_set_pname))))). % empty_is_image
thf(fact_19_empty__is__image, axiom,
    ((![F : hoare_958474565_state > hoare_958474565_state, A2 : set_Ho840737317_state]: ((bot_bo105666705_state = (image_1849792670_state @ F @ A2)) = (A2 = bot_bo105666705_state))))). % empty_is_image
thf(fact_20_image__is__empty, axiom,
    ((![F : pname > hoare_958474565_state, A2 : set_pname]: (((image_2144627828_state @ F @ A2) = bot_bo105666705_state) = (A2 = bot_bot_set_pname))))). % image_is_empty
thf(fact_21_image__is__empty, axiom,
    ((![F : hoare_958474565_state > hoare_958474565_state, A2 : set_Ho840737317_state]: (((image_1849792670_state @ F @ A2) = bot_bo105666705_state) = (A2 = bot_bo105666705_state))))). % image_is_empty
thf(fact_22_image__constant, axiom,
    ((![X3 : pname, A2 : set_pname, C : hoare_958474565_state]: ((member_pname @ X3 @ A2) => ((image_2144627828_state @ (^[X2 : pname]: C) @ A2) = (insert776267541_state @ C @ bot_bo105666705_state)))))). % image_constant
thf(fact_23_image__eqI, axiom,
    ((![B : hoare_958474565_state, F : pname > hoare_958474565_state, X3 : pname, A2 : set_pname]: ((B = (F @ X3)) => ((member_pname @ X3 @ A2) => (member109514606_state @ B @ (image_2144627828_state @ F @ A2))))))). % image_eqI
thf(fact_24_empty__Collect__eq, axiom,
    ((![P : hoare_958474565_state > $o]: ((bot_bo105666705_state = (collec305460656_state @ P)) = (![X2 : hoare_958474565_state]: (~ ((P @ X2)))))))). % empty_Collect_eq
thf(fact_25_Collect__empty__eq, axiom,
    ((![P : hoare_958474565_state > $o]: (((collec305460656_state @ P) = bot_bo105666705_state) = (![X2 : hoare_958474565_state]: (~ ((P @ X2)))))))). % Collect_empty_eq
thf(fact_26_all__not__in__conv, axiom,
    ((![A2 : set_Ho840737317_state]: ((![X2 : hoare_958474565_state]: (~ ((member109514606_state @ X2 @ A2)))) = (A2 = bot_bo105666705_state))))). % all_not_in_conv
thf(fact_27_empty__iff, axiom,
    ((![C : hoare_958474565_state]: (~ ((member109514606_state @ C @ bot_bo105666705_state)))))). % empty_iff
thf(fact_28_insert__absorb2, axiom,
    ((![X3 : hoare_958474565_state, A2 : set_Ho840737317_state]: ((insert776267541_state @ X3 @ (insert776267541_state @ X3 @ A2)) = (insert776267541_state @ X3 @ A2))))). % insert_absorb2
thf(fact_29_insert__iff, axiom,
    ((![A : hoare_958474565_state, B : hoare_958474565_state, A2 : set_Ho840737317_state]: ((member109514606_state @ A @ (insert776267541_state @ B @ A2)) = (((A = B)) | ((member109514606_state @ A @ A2))))))). % insert_iff
thf(fact_30_insertCI, axiom,
    ((![A : hoare_958474565_state, B2 : set_Ho840737317_state, B : hoare_958474565_state]: (((~ ((member109514606_state @ A @ B2))) => (A = B)) => (member109514606_state @ A @ (insert776267541_state @ B @ B2)))))). % insertCI
thf(fact_31_com_Oinject_I6_J, axiom,
    ((![X7 : pname, Y7 : pname]: (((body2 @ X7) = (body2 @ Y7)) = (X7 = Y7))))). % com.inject(6)
thf(fact_32_rev__image__eqI, axiom,
    ((![X3 : pname, A2 : set_pname, B : hoare_958474565_state, F : pname > hoare_958474565_state]: ((member_pname @ X3 @ A2) => ((B = (F @ X3)) => (member109514606_state @ B @ (image_2144627828_state @ F @ A2))))))). % rev_image_eqI
thf(fact_33_ball__imageD, axiom,
    ((![F : pname > hoare_958474565_state, A2 : set_pname, P : hoare_958474565_state > $o]: ((![X : hoare_958474565_state]: ((member109514606_state @ X @ (image_2144627828_state @ F @ A2)) => (P @ X))) => (![X4 : pname]: ((member_pname @ X4 @ A2) => (P @ (F @ X4)))))))). % ball_imageD
thf(fact_34_image__cong, axiom,
    ((![M : set_pname, N : set_pname, F : pname > hoare_958474565_state, G3 : pname > hoare_958474565_state]: ((M = N) => ((![X : pname]: ((member_pname @ X @ N) => ((F @ X) = (G3 @ X)))) => ((image_2144627828_state @ F @ M) = (image_2144627828_state @ G3 @ N))))))). % image_cong
thf(fact_35_bex__imageD, axiom,
    ((![F : pname > hoare_958474565_state, A2 : set_pname, P : hoare_958474565_state > $o]: ((?[X4 : hoare_958474565_state]: ((member109514606_state @ X4 @ (image_2144627828_state @ F @ A2)) & (P @ X4))) => (?[X : pname]: ((member_pname @ X @ A2) & (P @ (F @ X)))))))). % bex_imageD
thf(fact_36_image__iff, axiom,
    ((![Z2 : hoare_958474565_state, F : pname > hoare_958474565_state, A2 : set_pname]: ((member109514606_state @ Z2 @ (image_2144627828_state @ F @ A2)) = (?[X2 : pname]: (((member_pname @ X2 @ A2)) & ((Z2 = (F @ X2))))))))). % image_iff
thf(fact_37_imageI, axiom,
    ((![X3 : pname, A2 : set_pname, F : pname > hoare_958474565_state]: ((member_pname @ X3 @ A2) => (member109514606_state @ (F @ X3) @ (image_2144627828_state @ F @ A2)))))). % imageI
thf(fact_38_ex__in__conv, axiom,
    ((![A2 : set_Ho840737317_state]: ((?[X2 : hoare_958474565_state]: (member109514606_state @ X2 @ A2)) = (~ ((A2 = bot_bo105666705_state))))))). % ex_in_conv
thf(fact_39_equals0I, axiom,
    ((![A2 : set_Ho840737317_state]: ((![Y2 : hoare_958474565_state]: (~ ((member109514606_state @ Y2 @ A2)))) => (A2 = bot_bo105666705_state))))). % equals0I
thf(fact_40_equals0D, axiom,
    ((![A2 : set_Ho840737317_state, A : hoare_958474565_state]: ((A2 = bot_bo105666705_state) => (~ ((member109514606_state @ A @ A2))))))). % equals0D
thf(fact_41_emptyE, axiom,
    ((![A : hoare_958474565_state]: (~ ((member109514606_state @ A @ bot_bo105666705_state)))))). % emptyE
thf(fact_42_mk__disjoint__insert, axiom,
    ((![A : hoare_958474565_state, A2 : set_Ho840737317_state]: ((member109514606_state @ A @ A2) => (?[B3 : set_Ho840737317_state]: ((A2 = (insert776267541_state @ A @ B3)) & (~ ((member109514606_state @ A @ B3))))))))). % mk_disjoint_insert
thf(fact_43_insert__commute, axiom,
    ((![X3 : hoare_958474565_state, Y3 : hoare_958474565_state, A2 : set_Ho840737317_state]: ((insert776267541_state @ X3 @ (insert776267541_state @ Y3 @ A2)) = (insert776267541_state @ Y3 @ (insert776267541_state @ X3 @ A2)))))). % insert_commute
thf(fact_44_insert__eq__iff, axiom,
    ((![A : hoare_958474565_state, A2 : set_Ho840737317_state, B : hoare_958474565_state, B2 : set_Ho840737317_state]: ((~ ((member109514606_state @ A @ A2))) => ((~ ((member109514606_state @ B @ B2))) => (((insert776267541_state @ A @ A2) = (insert776267541_state @ B @ B2)) = (((((A = B)) => ((A2 = B2)))) & ((((~ ((A = B)))) => ((?[C2 : set_Ho840737317_state]: (((A2 = (insert776267541_state @ B @ C2))) & ((((~ ((member109514606_state @ B @ C2)))) & ((((B2 = (insert776267541_state @ A @ C2))) & ((~ ((member109514606_state @ A @ C2)))))))))))))))))))). % insert_eq_iff
thf(fact_45_insert__absorb, axiom,
    ((![A : hoare_958474565_state, A2 : set_Ho840737317_state]: ((member109514606_state @ A @ A2) => ((insert776267541_state @ A @ A2) = A2))))). % insert_absorb
thf(fact_46_insert__ident, axiom,
    ((![X3 : hoare_958474565_state, A2 : set_Ho840737317_state, B2 : set_Ho840737317_state]: ((~ ((member109514606_state @ X3 @ A2))) => ((~ ((member109514606_state @ X3 @ B2))) => (((insert776267541_state @ X3 @ A2) = (insert776267541_state @ X3 @ B2)) = (A2 = B2))))))). % insert_ident
thf(fact_47_Set_Oset__insert, axiom,
    ((![X3 : hoare_958474565_state, A2 : set_Ho840737317_state]: ((member109514606_state @ X3 @ A2) => (~ ((![B3 : set_Ho840737317_state]: ((A2 = (insert776267541_state @ X3 @ B3)) => (member109514606_state @ X3 @ B3))))))))). % Set.set_insert
thf(fact_48_mem__Collect__eq, axiom,
    ((![A : hoare_958474565_state, P : hoare_958474565_state > $o]: ((member109514606_state @ A @ (collec305460656_state @ P)) = (P @ A))))). % mem_Collect_eq
thf(fact_49_Collect__mem__eq, axiom,
    ((![A2 : set_Ho840737317_state]: ((collec305460656_state @ (^[X2 : hoare_958474565_state]: (member109514606_state @ X2 @ A2))) = A2)))). % Collect_mem_eq
thf(fact_50_Collect__cong, axiom,
    ((![P : hoare_958474565_state > $o, Q : hoare_958474565_state > $o]: ((![X : hoare_958474565_state]: ((P @ X) = (Q @ X))) => ((collec305460656_state @ P) = (collec305460656_state @ Q)))))). % Collect_cong
thf(fact_51_insertI2, axiom,
    ((![A : hoare_958474565_state, B2 : set_Ho840737317_state, B : hoare_958474565_state]: ((member109514606_state @ A @ B2) => (member109514606_state @ A @ (insert776267541_state @ B @ B2)))))). % insertI2
thf(fact_52_insertI1, axiom,
    ((![A : hoare_958474565_state, B2 : set_Ho840737317_state]: (member109514606_state @ A @ (insert776267541_state @ A @ B2))))). % insertI1
thf(fact_53_insertE, axiom,
    ((![A : hoare_958474565_state, B : hoare_958474565_state, A2 : set_Ho840737317_state]: ((member109514606_state @ A @ (insert776267541_state @ B @ A2)) => ((~ ((A = B))) => (member109514606_state @ A @ A2)))))). % insertE
thf(fact_54_Compr__image__eq, axiom,
    ((![F : pname > hoare_958474565_state, A2 : set_pname, P : hoare_958474565_state > $o]: ((collec305460656_state @ (^[X2 : hoare_958474565_state]: (((member109514606_state @ X2 @ (image_2144627828_state @ F @ A2))) & ((P @ X2))))) = (image_2144627828_state @ F @ (collect_pname @ (^[X2 : pname]: (((member_pname @ X2 @ A2)) & ((P @ (F @ X2))))))))))). % Compr_image_eq
thf(fact_55_Compr__image__eq, axiom,
    ((![F : hoare_958474565_state > hoare_958474565_state, A2 : set_Ho840737317_state, P : hoare_958474565_state > $o]: ((collec305460656_state @ (^[X2 : hoare_958474565_state]: (((member109514606_state @ X2 @ (image_1849792670_state @ F @ A2))) & ((P @ X2))))) = (image_1849792670_state @ F @ (collec305460656_state @ (^[X2 : hoare_958474565_state]: (((member109514606_state @ X2 @ A2)) & ((P @ (F @ X2))))))))))). % Compr_image_eq
thf(fact_56_image__image, axiom,
    ((![F : hoare_958474565_state > hoare_958474565_state, G3 : pname > hoare_958474565_state, A2 : set_pname]: ((image_1849792670_state @ F @ (image_2144627828_state @ G3 @ A2)) = (image_2144627828_state @ (^[X2 : pname]: (F @ (G3 @ X2))) @ A2))))). % image_image
thf(fact_57_image__image, axiom,
    ((![F : pname > hoare_958474565_state, G3 : pname > pname, A2 : set_pname]: ((image_2144627828_state @ F @ (image_pname_pname @ G3 @ A2)) = (image_2144627828_state @ (^[X2 : pname]: (F @ (G3 @ X2))) @ A2))))). % image_image
thf(fact_58_imageE, axiom,
    ((![B : hoare_958474565_state, F : pname > hoare_958474565_state, A2 : set_pname]: ((member109514606_state @ B @ (image_2144627828_state @ F @ A2)) => (~ ((![X : pname]: ((B = (F @ X)) => (~ ((member_pname @ X @ A2))))))))))). % imageE
thf(fact_59_empty__def, axiom,
    ((bot_bo105666705_state = (collec305460656_state @ (^[X2 : hoare_958474565_state]: $false))))). % empty_def
thf(fact_60_insert__Collect, axiom,
    ((![A : hoare_958474565_state, P : hoare_958474565_state > $o]: ((insert776267541_state @ A @ (collec305460656_state @ P)) = (collec305460656_state @ (^[U : hoare_958474565_state]: (((~ ((U = A)))) => ((P @ U))))))))). % insert_Collect
thf(fact_61_insert__compr, axiom,
    ((insert776267541_state = (^[A3 : hoare_958474565_state]: (^[B4 : set_Ho840737317_state]: (collec305460656_state @ (^[X2 : hoare_958474565_state]: (((X2 = A3)) | ((member109514606_state @ X2 @ B4)))))))))). % insert_compr
thf(fact_62_singleton__inject, axiom,
    ((![A : hoare_958474565_state, B : hoare_958474565_state]: (((insert776267541_state @ A @ bot_bo105666705_state) = (insert776267541_state @ B @ bot_bo105666705_state)) => (A = B))))). % singleton_inject
thf(fact_63_insert__not__empty, axiom,
    ((![A : hoare_958474565_state, A2 : set_Ho840737317_state]: (~ (((insert776267541_state @ A @ A2) = bot_bo105666705_state)))))). % insert_not_empty
thf(fact_64_doubleton__eq__iff, axiom,
    ((![A : hoare_958474565_state, B : hoare_958474565_state, C : hoare_958474565_state, D : hoare_958474565_state]: (((insert776267541_state @ A @ (insert776267541_state @ B @ bot_bo105666705_state)) = (insert776267541_state @ C @ (insert776267541_state @ D @ bot_bo105666705_state))) = (((((A = C)) & ((B = D)))) | ((((A = D)) & ((B = C))))))))). % doubleton_eq_iff
thf(fact_65_singleton__iff, axiom,
    ((![B : hoare_958474565_state, A : hoare_958474565_state]: ((member109514606_state @ B @ (insert776267541_state @ A @ bot_bo105666705_state)) = (B = A))))). % singleton_iff
thf(fact_66_singletonD, axiom,
    ((![B : hoare_958474565_state, A : hoare_958474565_state]: ((member109514606_state @ B @ (insert776267541_state @ A @ bot_bo105666705_state)) => (B = A))))). % singletonD
thf(fact_67_Collect__conv__if2, axiom,
    ((![P : hoare_958474565_state > $o, A : hoare_958474565_state]: (((P @ A) => ((collec305460656_state @ (^[X2 : hoare_958474565_state]: (((A = X2)) & ((P @ X2))))) = (insert776267541_state @ A @ bot_bo105666705_state))) & ((~ ((P @ A))) => ((collec305460656_state @ (^[X2 : hoare_958474565_state]: (((A = X2)) & ((P @ X2))))) = bot_bo105666705_state)))))). % Collect_conv_if2
thf(fact_68_Collect__conv__if, axiom,
    ((![P : hoare_958474565_state > $o, A : hoare_958474565_state]: (((P @ A) => ((collec305460656_state @ (^[X2 : hoare_958474565_state]: (((X2 = A)) & ((P @ X2))))) = (insert776267541_state @ A @ bot_bo105666705_state))) & ((~ ((P @ A))) => ((collec305460656_state @ (^[X2 : hoare_958474565_state]: (((X2 = A)) & ((P @ X2))))) = bot_bo105666705_state)))))). % Collect_conv_if
thf(fact_69_image__constant__conv, axiom,
    ((![A2 : set_pname, C : hoare_958474565_state]: (((A2 = bot_bot_set_pname) => ((image_2144627828_state @ (^[X2 : pname]: C) @ A2) = bot_bo105666705_state)) & ((~ ((A2 = bot_bot_set_pname))) => ((image_2144627828_state @ (^[X2 : pname]: C) @ A2) = (insert776267541_state @ C @ bot_bo105666705_state))))))). % image_constant_conv
thf(fact_70_image__constant__conv, axiom,
    ((![A2 : set_Ho840737317_state, C : hoare_958474565_state]: (((A2 = bot_bo105666705_state) => ((image_1849792670_state @ (^[X2 : hoare_958474565_state]: C) @ A2) = bot_bo105666705_state)) & ((~ ((A2 = bot_bo105666705_state))) => ((image_1849792670_state @ (^[X2 : hoare_958474565_state]: C) @ A2) = (insert776267541_state @ C @ bot_bo105666705_state))))))). % image_constant_conv
thf(fact_71_insert__dom, axiom,
    ((![F : pname > option_com, X3 : pname, Y3 : com]: (((F @ X3) = (some_com @ Y3)) => ((insert_pname @ X3 @ (dom_pname_com @ F)) = (dom_pname_com @ F)))))). % insert_dom
thf(fact_72_insert__dom, axiom,
    ((![F : hoare_958474565_state > option_com, X3 : hoare_958474565_state, Y3 : com]: (((F @ X3) = (some_com @ Y3)) => ((insert776267541_state @ X3 @ (dom_Ho256784515te_com @ F)) = (dom_Ho256784515te_com @ F)))))). % insert_dom
thf(fact_73_option_Oinject, axiom,
    ((![X22 : com, Y22 : com]: (((some_com @ X22) = (some_com @ Y22)) = (X22 = Y22))))). % option.inject
thf(fact_74_the__elem__eq, axiom,
    ((![X3 : hoare_958474565_state]: ((the_el1300254266_state @ (insert776267541_state @ X3 @ bot_bo105666705_state)) = X3)))). % the_elem_eq
thf(fact_75_MGT__BodyN, axiom,
    ((![Pn : pname, G : set_Ho840737317_state]: ((hoare_604442164_state @ (insert776267541_state @ (hoare_Mirabelle_MGT @ (body2 @ Pn)) @ G) @ (insert776267541_state @ (hoare_Mirabelle_MGT @ (the_com @ (body @ Pn))) @ bot_bo105666705_state)) => (hoare_604442164_state @ G @ (insert776267541_state @ (hoare_Mirabelle_MGT @ (body2 @ Pn)) @ bot_bo105666705_state)))))). % MGT_BodyN
thf(fact_76_domI, axiom,
    ((![M2 : pname > option_com, A : pname, B : com]: (((M2 @ A) = (some_com @ B)) => (member_pname @ A @ (dom_pname_com @ M2)))))). % domI
thf(fact_77_domD, axiom,
    ((![A : pname, M2 : pname > option_com]: ((member_pname @ A @ (dom_pname_com @ M2)) => (?[B5 : com]: ((M2 @ A) = (some_com @ B5))))))). % domD
thf(fact_78_is__singletonI, axiom,
    ((![X3 : hoare_958474565_state]: (is_sin536631353_state @ (insert776267541_state @ X3 @ bot_bo105666705_state))))). % is_singletonI
thf(fact_79_MGF__lemma2__simult, axiom,
    ((![F2 : set_Ho840737317_state]: (hoare_405891322gleton => (wT_bodies => ((ord_le1945819589_state @ F2 @ (image_2144627828_state @ (^[Pn2 : pname]: (hoare_Mirabelle_MGT @ (the_com @ (body @ Pn2)))) @ (dom_pname_com @ body))) => (hoare_604442164_state @ (image_2144627828_state @ (^[Pn2 : pname]: (hoare_Mirabelle_MGT @ (body2 @ Pn2))) @ (dom_pname_com @ body)) @ F2))))))). % MGF_lemma2_simult
thf(fact_80_order__refl, axiom,
    ((![X3 : set_Ho840737317_state]: (ord_le1945819589_state @ X3 @ X3)))). % order_refl
thf(fact_81_subsetI, axiom,
    ((![A2 : set_Ho840737317_state, B2 : set_Ho840737317_state]: ((![X : hoare_958474565_state]: ((member109514606_state @ X @ A2) => (member109514606_state @ X @ B2))) => (ord_le1945819589_state @ A2 @ B2))))). % subsetI
thf(fact_82_subset__antisym, axiom,
    ((![A2 : set_Ho840737317_state, B2 : set_Ho840737317_state]: ((ord_le1945819589_state @ A2 @ B2) => ((ord_le1945819589_state @ B2 @ A2) => (A2 = B2)))))). % subset_antisym
thf(fact_83_subset__empty, axiom,
    ((![A2 : set_Ho840737317_state]: ((ord_le1945819589_state @ A2 @ bot_bo105666705_state) = (A2 = bot_bo105666705_state))))). % subset_empty
thf(fact_84_empty__subsetI, axiom,
    ((![A2 : set_Ho840737317_state]: (ord_le1945819589_state @ bot_bo105666705_state @ A2)))). % empty_subsetI
thf(fact_85_insert__subset, axiom,
    ((![X3 : hoare_958474565_state, A2 : set_Ho840737317_state, B2 : set_Ho840737317_state]: ((ord_le1945819589_state @ (insert776267541_state @ X3 @ A2) @ B2) = (((member109514606_state @ X3 @ B2)) & ((ord_le1945819589_state @ A2 @ B2))))))). % insert_subset
thf(fact_86_singleton__insert__inj__eq, axiom,
    ((![B : hoare_958474565_state, A : hoare_958474565_state, A2 : set_Ho840737317_state]: (((insert776267541_state @ B @ bot_bo105666705_state) = (insert776267541_state @ A @ A2)) = (((A = B)) & ((ord_le1945819589_state @ A2 @ (insert776267541_state @ B @ bot_bo105666705_state)))))))). % singleton_insert_inj_eq
thf(fact_87_singleton__insert__inj__eq_H, axiom,
    ((![A : hoare_958474565_state, A2 : set_Ho840737317_state, B : hoare_958474565_state]: (((insert776267541_state @ A @ A2) = (insert776267541_state @ B @ bot_bo105666705_state)) = (((A = B)) & ((ord_le1945819589_state @ A2 @ (insert776267541_state @ B @ bot_bo105666705_state)))))))). % singleton_insert_inj_eq'
thf(fact_88_in__mono, axiom,
    ((![A2 : set_Ho840737317_state, B2 : set_Ho840737317_state, X3 : hoare_958474565_state]: ((ord_le1945819589_state @ A2 @ B2) => ((member109514606_state @ X3 @ A2) => (member109514606_state @ X3 @ B2)))))). % in_mono
thf(fact_89_subsetD, axiom,
    ((![A2 : set_Ho840737317_state, B2 : set_Ho840737317_state, C : hoare_958474565_state]: ((ord_le1945819589_state @ A2 @ B2) => ((member109514606_state @ C @ A2) => (member109514606_state @ C @ B2)))))). % subsetD
thf(fact_90_equalityE, axiom,
    ((![A2 : set_Ho840737317_state, B2 : set_Ho840737317_state]: ((A2 = B2) => (~ (((ord_le1945819589_state @ A2 @ B2) => (~ ((ord_le1945819589_state @ B2 @ A2)))))))))). % equalityE
thf(fact_91_subset__eq, axiom,
    ((ord_le1945819589_state = (^[A4 : set_Ho840737317_state]: (^[B4 : set_Ho840737317_state]: (![X2 : hoare_958474565_state]: (((member109514606_state @ X2 @ A4)) => ((member109514606_state @ X2 @ B4))))))))). % subset_eq
thf(fact_92_equalityD1, axiom,
    ((![A2 : set_Ho840737317_state, B2 : set_Ho840737317_state]: ((A2 = B2) => (ord_le1945819589_state @ A2 @ B2))))). % equalityD1
thf(fact_93_equalityD2, axiom,
    ((![A2 : set_Ho840737317_state, B2 : set_Ho840737317_state]: ((A2 = B2) => (ord_le1945819589_state @ B2 @ A2))))). % equalityD2
thf(fact_94_subset__iff, axiom,
    ((ord_le1945819589_state = (^[A4 : set_Ho840737317_state]: (^[B4 : set_Ho840737317_state]: (![T3 : hoare_958474565_state]: (((member109514606_state @ T3 @ A4)) => ((member109514606_state @ T3 @ B4))))))))). % subset_iff
thf(fact_95_subset__refl, axiom,
    ((![A2 : set_Ho840737317_state]: (ord_le1945819589_state @ A2 @ A2)))). % subset_refl
thf(fact_96_Collect__mono, axiom,
    ((![P : hoare_958474565_state > $o, Q : hoare_958474565_state > $o]: ((![X : hoare_958474565_state]: ((P @ X) => (Q @ X))) => (ord_le1945819589_state @ (collec305460656_state @ P) @ (collec305460656_state @ Q)))))). % Collect_mono
thf(fact_97_subset__trans, axiom,
    ((![A2 : set_Ho840737317_state, B2 : set_Ho840737317_state, C3 : set_Ho840737317_state]: ((ord_le1945819589_state @ A2 @ B2) => ((ord_le1945819589_state @ B2 @ C3) => (ord_le1945819589_state @ A2 @ C3)))))). % subset_trans
thf(fact_98_set__eq__subset, axiom,
    (((^[Y : set_Ho840737317_state]: (^[Z : set_Ho840737317_state]: (Y = Z))) = (^[A4 : set_Ho840737317_state]: (^[B4 : set_Ho840737317_state]: (((ord_le1945819589_state @ A4 @ B4)) & ((ord_le1945819589_state @ B4 @ A4)))))))). % set_eq_subset
thf(fact_99_Collect__subset, axiom,
    ((![A2 : set_Ho840737317_state, P : hoare_958474565_state > $o]: (ord_le1945819589_state @ (collec305460656_state @ (^[X2 : hoare_958474565_state]: (((member109514606_state @ X2 @ A2)) & ((P @ X2))))) @ A2)))). % Collect_subset
thf(fact_100_Collect__mono__iff, axiom,
    ((![P : hoare_958474565_state > $o, Q : hoare_958474565_state > $o]: ((ord_le1945819589_state @ (collec305460656_state @ P) @ (collec305460656_state @ Q)) = (![X2 : hoare_958474565_state]: (((P @ X2)) => ((Q @ X2)))))))). % Collect_mono_iff
thf(fact_101_bot_Oextremum__uniqueI, axiom,
    ((![A : set_Ho840737317_state]: ((ord_le1945819589_state @ A @ bot_bo105666705_state) => (A = bot_bo105666705_state))))). % bot.extremum_uniqueI
thf(fact_102_bot_Oextremum__unique, axiom,
    ((![A : set_Ho840737317_state]: ((ord_le1945819589_state @ A @ bot_bo105666705_state) = (A = bot_bo105666705_state))))). % bot.extremum_unique
thf(fact_103_bot_Oextremum, axiom,
    ((![A : set_Ho840737317_state]: (ord_le1945819589_state @ bot_bo105666705_state @ A)))). % bot.extremum
thf(fact_104_dual__order_Oantisym, axiom,
    ((![B : set_Ho840737317_state, A : set_Ho840737317_state]: ((ord_le1945819589_state @ B @ A) => ((ord_le1945819589_state @ A @ B) => (A = B)))))). % dual_order.antisym
thf(fact_105_dual__order_Oeq__iff, axiom,
    (((^[Y : set_Ho840737317_state]: (^[Z : set_Ho840737317_state]: (Y = Z))) = (^[A3 : set_Ho840737317_state]: (^[B6 : set_Ho840737317_state]: (((ord_le1945819589_state @ B6 @ A3)) & ((ord_le1945819589_state @ A3 @ B6)))))))). % dual_order.eq_iff
thf(fact_106_dual__order_Otrans, axiom,
    ((![B : set_Ho840737317_state, A : set_Ho840737317_state, C : set_Ho840737317_state]: ((ord_le1945819589_state @ B @ A) => ((ord_le1945819589_state @ C @ B) => (ord_le1945819589_state @ C @ A)))))). % dual_order.trans
thf(fact_107_dual__order_Orefl, axiom,
    ((![A : set_Ho840737317_state]: (ord_le1945819589_state @ A @ A)))). % dual_order.refl
thf(fact_108_order__trans, axiom,
    ((![X3 : set_Ho840737317_state, Y3 : set_Ho840737317_state, Z2 : set_Ho840737317_state]: ((ord_le1945819589_state @ X3 @ Y3) => ((ord_le1945819589_state @ Y3 @ Z2) => (ord_le1945819589_state @ X3 @ Z2)))))). % order_trans
thf(fact_109_order__class_Oorder_Oantisym, axiom,
    ((![A : set_Ho840737317_state, B : set_Ho840737317_state]: ((ord_le1945819589_state @ A @ B) => ((ord_le1945819589_state @ B @ A) => (A = B)))))). % order_class.order.antisym
thf(fact_110_ord__le__eq__trans, axiom,
    ((![A : set_Ho840737317_state, B : set_Ho840737317_state, C : set_Ho840737317_state]: ((ord_le1945819589_state @ A @ B) => ((B = C) => (ord_le1945819589_state @ A @ C)))))). % ord_le_eq_trans
thf(fact_111_ord__eq__le__trans, axiom,
    ((![A : set_Ho840737317_state, B : set_Ho840737317_state, C : set_Ho840737317_state]: ((A = B) => ((ord_le1945819589_state @ B @ C) => (ord_le1945819589_state @ A @ C)))))). % ord_eq_le_trans
thf(fact_112_order__class_Oorder_Oeq__iff, axiom,
    (((^[Y : set_Ho840737317_state]: (^[Z : set_Ho840737317_state]: (Y = Z))) = (^[A3 : set_Ho840737317_state]: (^[B6 : set_Ho840737317_state]: (((ord_le1945819589_state @ A3 @ B6)) & ((ord_le1945819589_state @ B6 @ A3)))))))). % order_class.order.eq_iff
thf(fact_113_antisym__conv, axiom,
    ((![Y3 : set_Ho840737317_state, X3 : set_Ho840737317_state]: ((ord_le1945819589_state @ Y3 @ X3) => ((ord_le1945819589_state @ X3 @ Y3) = (X3 = Y3)))))). % antisym_conv
thf(fact_114_order_Otrans, axiom,
    ((![A : set_Ho840737317_state, B : set_Ho840737317_state, C : set_Ho840737317_state]: ((ord_le1945819589_state @ A @ B) => ((ord_le1945819589_state @ B @ C) => (ord_le1945819589_state @ A @ C)))))). % order.trans
thf(fact_115_eq__refl, axiom,
    ((![X3 : set_Ho840737317_state, Y3 : set_Ho840737317_state]: ((X3 = Y3) => (ord_le1945819589_state @ X3 @ Y3))))). % eq_refl
thf(fact_116_antisym, axiom,
    ((![X3 : set_Ho840737317_state, Y3 : set_Ho840737317_state]: ((ord_le1945819589_state @ X3 @ Y3) => ((ord_le1945819589_state @ Y3 @ X3) => (X3 = Y3)))))). % antisym
thf(fact_117_eq__iff, axiom,
    (((^[Y : set_Ho840737317_state]: (^[Z : set_Ho840737317_state]: (Y = Z))) = (^[X2 : set_Ho840737317_state]: (^[Y4 : set_Ho840737317_state]: (((ord_le1945819589_state @ X2 @ Y4)) & ((ord_le1945819589_state @ Y4 @ X2)))))))). % eq_iff
thf(fact_118_ord__le__eq__subst, axiom,
    ((![A : set_Ho840737317_state, B : set_Ho840737317_state, F : set_Ho840737317_state > set_Ho840737317_state, C : set_Ho840737317_state]: ((ord_le1945819589_state @ A @ B) => (((F @ B) = C) => ((![X : set_Ho840737317_state, Y2 : set_Ho840737317_state]: ((ord_le1945819589_state @ X @ Y2) => (ord_le1945819589_state @ (F @ X) @ (F @ Y2)))) => (ord_le1945819589_state @ (F @ A) @ C))))))). % ord_le_eq_subst
thf(fact_119_ord__eq__le__subst, axiom,
    ((![A : set_Ho840737317_state, F : set_Ho840737317_state > set_Ho840737317_state, B : set_Ho840737317_state, C : set_Ho840737317_state]: ((A = (F @ B)) => ((ord_le1945819589_state @ B @ C) => ((![X : set_Ho840737317_state, Y2 : set_Ho840737317_state]: ((ord_le1945819589_state @ X @ Y2) => (ord_le1945819589_state @ (F @ X) @ (F @ Y2)))) => (ord_le1945819589_state @ A @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_120_order__subst2, axiom,
    ((![A : set_Ho840737317_state, B : set_Ho840737317_state, F : set_Ho840737317_state > set_Ho840737317_state, C : set_Ho840737317_state]: ((ord_le1945819589_state @ A @ B) => ((ord_le1945819589_state @ (F @ B) @ C) => ((![X : set_Ho840737317_state, Y2 : set_Ho840737317_state]: ((ord_le1945819589_state @ X @ Y2) => (ord_le1945819589_state @ (F @ X) @ (F @ Y2)))) => (ord_le1945819589_state @ (F @ A) @ C))))))). % order_subst2
thf(fact_121_order__subst1, axiom,
    ((![A : set_Ho840737317_state, F : set_Ho840737317_state > set_Ho840737317_state, B : set_Ho840737317_state, C : set_Ho840737317_state]: ((ord_le1945819589_state @ A @ (F @ B)) => ((ord_le1945819589_state @ B @ C) => ((![X : set_Ho840737317_state, Y2 : set_Ho840737317_state]: ((ord_le1945819589_state @ X @ Y2) => (ord_le1945819589_state @ (F @ X) @ (F @ Y2)))) => (ord_le1945819589_state @ A @ (F @ C)))))))). % order_subst1
thf(fact_122_option_Osel, axiom,
    ((![X22 : com]: ((the_com @ (some_com @ X22)) = X22)))). % option.sel
thf(fact_123_bot__set__def, axiom,
    ((bot_bo105666705_state = (collec305460656_state @ bot_bo1428770700tate_o)))). % bot_set_def
thf(fact_124_image__mono, axiom,
    ((![A2 : set_pname, B2 : set_pname, F : pname > hoare_958474565_state]: ((ord_le865024672_pname @ A2 @ B2) => (ord_le1945819589_state @ (image_2144627828_state @ F @ A2) @ (image_2144627828_state @ F @ B2)))))). % image_mono
thf(fact_125_image__mono, axiom,
    ((![A2 : set_Ho840737317_state, B2 : set_Ho840737317_state, F : hoare_958474565_state > hoare_958474565_state]: ((ord_le1945819589_state @ A2 @ B2) => (ord_le1945819589_state @ (image_1849792670_state @ F @ A2) @ (image_1849792670_state @ F @ B2)))))). % image_mono
thf(fact_126_image__subsetI, axiom,
    ((![A2 : set_pname, F : pname > hoare_958474565_state, B2 : set_Ho840737317_state]: ((![X : pname]: ((member_pname @ X @ A2) => (member109514606_state @ (F @ X) @ B2))) => (ord_le1945819589_state @ (image_2144627828_state @ F @ A2) @ B2))))). % image_subsetI
thf(fact_127_subset__imageE, axiom,
    ((![B2 : set_Ho840737317_state, F : pname > hoare_958474565_state, A2 : set_pname]: ((ord_le1945819589_state @ B2 @ (image_2144627828_state @ F @ A2)) => (~ ((![C4 : set_pname]: ((ord_le865024672_pname @ C4 @ A2) => (~ ((B2 = (image_2144627828_state @ F @ C4)))))))))))). % subset_imageE
thf(fact_128_subset__imageE, axiom,
    ((![B2 : set_Ho840737317_state, F : hoare_958474565_state > hoare_958474565_state, A2 : set_Ho840737317_state]: ((ord_le1945819589_state @ B2 @ (image_1849792670_state @ F @ A2)) => (~ ((![C4 : set_Ho840737317_state]: ((ord_le1945819589_state @ C4 @ A2) => (~ ((B2 = (image_1849792670_state @ F @ C4)))))))))))). % subset_imageE
thf(fact_129_image__subset__iff, axiom,
    ((![F : pname > hoare_958474565_state, A2 : set_pname, B2 : set_Ho840737317_state]: ((ord_le1945819589_state @ (image_2144627828_state @ F @ A2) @ B2) = (![X2 : pname]: (((member_pname @ X2 @ A2)) => ((member109514606_state @ (F @ X2) @ B2)))))))). % image_subset_iff
thf(fact_130_subset__image__iff, axiom,
    ((![B2 : set_Ho840737317_state, F : pname > hoare_958474565_state, A2 : set_pname]: ((ord_le1945819589_state @ B2 @ (image_2144627828_state @ F @ A2)) = (?[AA : set_pname]: (((ord_le865024672_pname @ AA @ A2)) & ((B2 = (image_2144627828_state @ F @ AA))))))))). % subset_image_iff
thf(fact_131_subset__image__iff, axiom,
    ((![B2 : set_Ho840737317_state, F : hoare_958474565_state > hoare_958474565_state, A2 : set_Ho840737317_state]: ((ord_le1945819589_state @ B2 @ (image_1849792670_state @ F @ A2)) = (?[AA : set_Ho840737317_state]: (((ord_le1945819589_state @ AA @ A2)) & ((B2 = (image_1849792670_state @ F @ AA))))))))). % subset_image_iff
thf(fact_132_insert__mono, axiom,
    ((![C3 : set_Ho840737317_state, D2 : set_Ho840737317_state, A : hoare_958474565_state]: ((ord_le1945819589_state @ C3 @ D2) => (ord_le1945819589_state @ (insert776267541_state @ A @ C3) @ (insert776267541_state @ A @ D2)))))). % insert_mono
thf(fact_133_subset__insert, axiom,
    ((![X3 : hoare_958474565_state, A2 : set_Ho840737317_state, B2 : set_Ho840737317_state]: ((~ ((member109514606_state @ X3 @ A2))) => ((ord_le1945819589_state @ A2 @ (insert776267541_state @ X3 @ B2)) = (ord_le1945819589_state @ A2 @ B2)))))). % subset_insert
thf(fact_134_subset__insertI, axiom,
    ((![B2 : set_Ho840737317_state, A : hoare_958474565_state]: (ord_le1945819589_state @ B2 @ (insert776267541_state @ A @ B2))))). % subset_insertI
thf(fact_135_subset__insertI2, axiom,
    ((![A2 : set_Ho840737317_state, B2 : set_Ho840737317_state, B : hoare_958474565_state]: ((ord_le1945819589_state @ A2 @ B2) => (ord_le1945819589_state @ A2 @ (insert776267541_state @ B @ B2)))))). % subset_insertI2
thf(fact_136_thin, axiom,
    ((![G2 : set_Ho840737317_state, Ts : set_Ho840737317_state, G : set_Ho840737317_state]: ((hoare_604442164_state @ G2 @ Ts) => ((ord_le1945819589_state @ G2 @ G) => (hoare_604442164_state @ G @ Ts)))))). % thin
thf(fact_137_asm, axiom,
    ((![Ts : set_Ho840737317_state, G : set_Ho840737317_state]: ((ord_le1945819589_state @ Ts @ G) => (hoare_604442164_state @ G @ Ts))))). % asm
thf(fact_138_weaken, axiom,
    ((![G : set_Ho840737317_state, Ts2 : set_Ho840737317_state, Ts : set_Ho840737317_state]: ((hoare_604442164_state @ G @ Ts2) => ((ord_le1945819589_state @ Ts @ Ts2) => (hoare_604442164_state @ G @ Ts)))))). % weaken
thf(fact_139_subset__singletonD, axiom,
    ((![A2 : set_Ho840737317_state, X3 : hoare_958474565_state]: ((ord_le1945819589_state @ A2 @ (insert776267541_state @ X3 @ bot_bo105666705_state)) => ((A2 = bot_bo105666705_state) | (A2 = (insert776267541_state @ X3 @ bot_bo105666705_state))))))). % subset_singletonD
thf(fact_140_subset__singleton__iff, axiom,
    ((![X5 : set_Ho840737317_state, A : hoare_958474565_state]: ((ord_le1945819589_state @ X5 @ (insert776267541_state @ A @ bot_bo105666705_state)) = (((X5 = bot_bo105666705_state)) | ((X5 = (insert776267541_state @ A @ bot_bo105666705_state)))))))). % subset_singleton_iff
thf(fact_141_is__singleton__the__elem, axiom,
    ((is_sin536631353_state = (^[A4 : set_Ho840737317_state]: (A4 = (insert776267541_state @ (the_el1300254266_state @ A4) @ bot_bo105666705_state)))))). % is_singleton_the_elem
thf(fact_142_is__singletonI_H, axiom,
    ((![A2 : set_Ho840737317_state]: ((~ ((A2 = bot_bo105666705_state))) => ((![X : hoare_958474565_state, Y2 : hoare_958474565_state]: ((member109514606_state @ X @ A2) => ((member109514606_state @ Y2 @ A2) => (X = Y2)))) => (is_sin536631353_state @ A2)))))). % is_singletonI'
thf(fact_143_is__singletonE, axiom,
    ((![A2 : set_Ho840737317_state]: ((is_sin536631353_state @ A2) => (~ ((![X : hoare_958474565_state]: (~ ((A2 = (insert776267541_state @ X @ bot_bo105666705_state))))))))))). % is_singletonE
thf(fact_144_is__singleton__def, axiom,
    ((is_sin536631353_state = (^[A4 : set_Ho840737317_state]: (?[X2 : hoare_958474565_state]: (A4 = (insert776267541_state @ X2 @ bot_bo105666705_state))))))). % is_singleton_def
thf(fact_145_the__elem__image__unique, axiom,
    ((![A2 : set_pname, F : pname > hoare_958474565_state, X3 : pname]: ((~ ((A2 = bot_bot_set_pname))) => ((![Y2 : pname]: ((member_pname @ Y2 @ A2) => ((F @ Y2) = (F @ X3)))) => ((the_el1300254266_state @ (image_2144627828_state @ F @ A2)) = (F @ X3))))))). % the_elem_image_unique
thf(fact_146_image__Collect__subsetI, axiom,
    ((![P : pname > $o, F : pname > hoare_958474565_state, B2 : set_Ho840737317_state]: ((![X : pname]: ((P @ X) => (member109514606_state @ (F @ X) @ B2))) => (ord_le1945819589_state @ (image_2144627828_state @ F @ (collect_pname @ P)) @ B2))))). % image_Collect_subsetI
thf(fact_147_image__Collect__subsetI, axiom,
    ((![P : hoare_958474565_state > $o, F : hoare_958474565_state > hoare_958474565_state, B2 : set_Ho840737317_state]: ((![X : hoare_958474565_state]: ((P @ X) => (member109514606_state @ (F @ X) @ B2))) => (ord_le1945819589_state @ (image_1849792670_state @ F @ (collec305460656_state @ P)) @ B2))))). % image_Collect_subsetI
thf(fact_148_BodyN, axiom,
    ((![P : state > state > $o, Pn : pname, Q : state > state > $o, G : set_Ho840737317_state]: ((hoare_604442164_state @ (insert776267541_state @ (hoare_1659279548_state @ P @ (body2 @ Pn) @ Q) @ G) @ (insert776267541_state @ (hoare_1659279548_state @ P @ (the_com @ (body @ Pn)) @ Q) @ bot_bo105666705_state)) => (hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ P @ (body2 @ Pn) @ Q) @ bot_bo105666705_state)))))). % BodyN
thf(fact_149_weak__Body, axiom,
    ((![G : set_Ho840737317_state, P : state > state > $o, Pn : pname, Q : state > state > $o]: ((hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ P @ (the_com @ (body @ Pn)) @ Q) @ bot_bo105666705_state)) => (hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ P @ (body2 @ Pn) @ Q) @ bot_bo105666705_state)))))). % weak_Body

% Conjectures (6)
thf(conj_0, hypothesis,
    (hoare_405891322gleton)).
thf(conj_1, hypothesis,
    (wT_bodies)).
thf(conj_2, hypothesis,
    ((wt @ c))).
thf(conj_3, hypothesis,
    (((body @ pn) = (some_com @ y)))).
thf(conj_4, hypothesis,
    ((hoare_604442164_state @ bot_bo105666705_state @ (image_2144627828_state @ (^[Pn2 : pname]: (hoare_Mirabelle_MGT @ (body2 @ Pn2))) @ (dom_pname_com @ body))))).
thf(conj_5, conjecture,
    ((hoare_604442164_state @ bot_bo105666705_state @ (insert776267541_state @ (hoare_Mirabelle_MGT @ (body2 @ pn)) @ bot_bo105666705_state)))).
