% 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_441__3254740_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:14:54.161

% 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_Option_Ooption_Othe_001t__Com__Ocom, type,
    the_com : option_com > com).
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_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_001_062_It__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J_M_Eo_J, type,
    ord_le1483368280tate_o : (hoare_958474565_state > $o) > (hoare_958474565_state > $o) > $o).
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__Com__Opname, type,
    image_871391498_pname : (hoare_958474565_state > pname) > set_Ho840737317_state > set_pname).
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__Com__Opname, type,
    is_singleton_pname : set_pname > $o).
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__Com__Opname, type,
    the_elem_pname : set_pname > pname).
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).

% Relevant facts (198)
thf(fact_0_thin, axiom,
    ((![G : set_Ho840737317_state, Ts : set_Ho840737317_state, G2 : set_Ho840737317_state]: ((hoare_604442164_state @ G @ Ts) => ((ord_le1945819589_state @ G @ G2) => (hoare_604442164_state @ G2 @ Ts)))))). % thin
thf(fact_1_single__stateE, axiom,
    ((hoare_405891322gleton => (![T : state]: (~ ((![S : state]: (S = T)))))))). % single_stateE
thf(fact_2_asm, axiom,
    ((![Ts : set_Ho840737317_state, G2 : set_Ho840737317_state]: ((ord_le1945819589_state @ Ts @ G2) => (hoare_604442164_state @ G2 @ Ts))))). % asm
thf(fact_3_cut, axiom,
    ((![G : set_Ho840737317_state, Ts : set_Ho840737317_state, G2 : set_Ho840737317_state]: ((hoare_604442164_state @ G @ Ts) => ((hoare_604442164_state @ G2 @ G) => (hoare_604442164_state @ G2 @ Ts)))))). % cut
thf(fact_4_empty, axiom,
    ((![G2 : set_Ho840737317_state]: (hoare_604442164_state @ G2 @ bot_bo105666705_state)))). % empty
thf(fact_5_weaken, axiom,
    ((![G2 : set_Ho840737317_state, Ts2 : set_Ho840737317_state, Ts : set_Ho840737317_state]: ((hoare_604442164_state @ G2 @ Ts2) => ((ord_le1945819589_state @ Ts @ Ts2) => (hoare_604442164_state @ G2 @ Ts)))))). % weaken
thf(fact_6_state__not__singleton__def, axiom,
    ((hoare_405891322gleton = (?[S2 : state]: (?[T2 : state]: (~ ((S2 = T2)))))))). % state_not_singleton_def
thf(fact_7_subset__empty, axiom,
    ((![A : set_pname]: ((ord_le865024672_pname @ A @ bot_bot_set_pname) = (A = bot_bot_set_pname))))). % subset_empty
thf(fact_8_subset__empty, axiom,
    ((![A : set_Ho840737317_state]: ((ord_le1945819589_state @ A @ bot_bo105666705_state) = (A = bot_bo105666705_state))))). % subset_empty
thf(fact_9_empty__subsetI, axiom,
    ((![A : set_pname]: (ord_le865024672_pname @ bot_bot_set_pname @ A)))). % empty_subsetI
thf(fact_10_empty__subsetI, axiom,
    ((![A : set_Ho840737317_state]: (ord_le1945819589_state @ bot_bo105666705_state @ A)))). % empty_subsetI
thf(fact_11_image__empty, axiom,
    ((![F : hoare_958474565_state > pname]: ((image_871391498_pname @ F @ bot_bo105666705_state) = bot_bot_set_pname)))). % image_empty
thf(fact_12_image__empty, axiom,
    ((![F : pname > pname]: ((image_pname_pname @ F @ bot_bot_set_pname) = bot_bot_set_pname)))). % image_empty
thf(fact_13_image__empty, axiom,
    ((![F : pname > hoare_958474565_state]: ((image_2144627828_state @ F @ bot_bot_set_pname) = bot_bo105666705_state)))). % image_empty
thf(fact_14_image__empty, axiom,
    ((![F : hoare_958474565_state > hoare_958474565_state]: ((image_1849792670_state @ F @ bot_bo105666705_state) = bot_bo105666705_state)))). % image_empty
thf(fact_15_empty__is__image, axiom,
    ((![F : hoare_958474565_state > hoare_958474565_state, A : set_Ho840737317_state]: ((bot_bo105666705_state = (image_1849792670_state @ F @ A)) = (A = bot_bo105666705_state))))). % empty_is_image
thf(fact_16_empty__is__image, axiom,
    ((![F : pname > hoare_958474565_state, A : set_pname]: ((bot_bo105666705_state = (image_2144627828_state @ F @ A)) = (A = bot_bot_set_pname))))). % empty_is_image
thf(fact_17_empty__is__image, axiom,
    ((![F : hoare_958474565_state > pname, A : set_Ho840737317_state]: ((bot_bot_set_pname = (image_871391498_pname @ F @ A)) = (A = bot_bo105666705_state))))). % empty_is_image
thf(fact_18_empty__is__image, axiom,
    ((![F : pname > pname, A : set_pname]: ((bot_bot_set_pname = (image_pname_pname @ F @ A)) = (A = bot_bot_set_pname))))). % empty_is_image
thf(fact_19_image__is__empty, axiom,
    ((![F : hoare_958474565_state > hoare_958474565_state, A : set_Ho840737317_state]: (((image_1849792670_state @ F @ A) = bot_bo105666705_state) = (A = bot_bo105666705_state))))). % image_is_empty
thf(fact_20_image__is__empty, axiom,
    ((![F : pname > hoare_958474565_state, A : set_pname]: (((image_2144627828_state @ F @ A) = bot_bo105666705_state) = (A = bot_bot_set_pname))))). % image_is_empty
thf(fact_21_image__is__empty, axiom,
    ((![F : hoare_958474565_state > pname, A : set_Ho840737317_state]: (((image_871391498_pname @ F @ A) = bot_bot_set_pname) = (A = bot_bo105666705_state))))). % image_is_empty
thf(fact_22_image__is__empty, axiom,
    ((![F : pname > pname, A : set_pname]: (((image_pname_pname @ F @ A) = bot_bot_set_pname) = (A = bot_bot_set_pname))))). % image_is_empty
thf(fact_23_image__ident, axiom,
    ((![Y : set_Ho840737317_state]: ((image_1849792670_state @ (^[X : hoare_958474565_state]: X) @ Y) = Y)))). % image_ident
thf(fact_24_com_Oinject_I6_J, axiom,
    ((![X7 : pname, Y7 : pname]: (((body2 @ X7) = (body2 @ Y7)) = (X7 = Y7))))). % com.inject(6)
thf(fact_25_subsetI, axiom,
    ((![A : set_Ho840737317_state, B : set_Ho840737317_state]: ((![X2 : hoare_958474565_state]: ((member109514606_state @ X2 @ A) => (member109514606_state @ X2 @ B))) => (ord_le1945819589_state @ A @ B))))). % subsetI
thf(fact_26_subset__antisym, axiom,
    ((![A : set_Ho840737317_state, B : set_Ho840737317_state]: ((ord_le1945819589_state @ A @ B) => ((ord_le1945819589_state @ B @ A) => (A = B)))))). % subset_antisym
thf(fact_27_empty__iff, axiom,
    ((![C : hoare_958474565_state]: (~ ((member109514606_state @ C @ bot_bo105666705_state)))))). % empty_iff
thf(fact_28_empty__iff, axiom,
    ((![C : pname]: (~ ((member_pname @ C @ bot_bot_set_pname)))))). % empty_iff
thf(fact_29_all__not__in__conv, axiom,
    ((![A : set_Ho840737317_state]: ((![X : hoare_958474565_state]: (~ ((member109514606_state @ X @ A)))) = (A = bot_bo105666705_state))))). % all_not_in_conv
thf(fact_30_all__not__in__conv, axiom,
    ((![A : set_pname]: ((![X : pname]: (~ ((member_pname @ X @ A)))) = (A = bot_bot_set_pname))))). % all_not_in_conv
thf(fact_31_Collect__empty__eq, axiom,
    ((![P : hoare_958474565_state > $o]: (((collec305460656_state @ P) = bot_bo105666705_state) = (![X : hoare_958474565_state]: (~ ((P @ X)))))))). % Collect_empty_eq
thf(fact_32_Collect__empty__eq, axiom,
    ((![P : pname > $o]: (((collect_pname @ P) = bot_bot_set_pname) = (![X : pname]: (~ ((P @ X)))))))). % Collect_empty_eq
thf(fact_33_image__eqI, axiom,
    ((![B2 : hoare_958474565_state, F : pname > hoare_958474565_state, X3 : pname, A : set_pname]: ((B2 = (F @ X3)) => ((member_pname @ X3 @ A) => (member109514606_state @ B2 @ (image_2144627828_state @ F @ A))))))). % image_eqI
thf(fact_34_image__eqI, axiom,
    ((![B2 : hoare_958474565_state, F : hoare_958474565_state > hoare_958474565_state, X3 : hoare_958474565_state, A : set_Ho840737317_state]: ((B2 = (F @ X3)) => ((member109514606_state @ X3 @ A) => (member109514606_state @ B2 @ (image_1849792670_state @ F @ A))))))). % image_eqI
thf(fact_35_empty__Collect__eq, axiom,
    ((![P : hoare_958474565_state > $o]: ((bot_bo105666705_state = (collec305460656_state @ P)) = (![X : hoare_958474565_state]: (~ ((P @ X)))))))). % empty_Collect_eq
thf(fact_36_empty__Collect__eq, axiom,
    ((![P : pname > $o]: ((bot_bot_set_pname = (collect_pname @ P)) = (![X : pname]: (~ ((P @ X)))))))). % empty_Collect_eq
thf(fact_37_rev__image__eqI, axiom,
    ((![X3 : pname, A : set_pname, B2 : hoare_958474565_state, F : pname > hoare_958474565_state]: ((member_pname @ X3 @ A) => ((B2 = (F @ X3)) => (member109514606_state @ B2 @ (image_2144627828_state @ F @ A))))))). % rev_image_eqI
thf(fact_38_rev__image__eqI, axiom,
    ((![X3 : hoare_958474565_state, A : set_Ho840737317_state, B2 : hoare_958474565_state, F : hoare_958474565_state > hoare_958474565_state]: ((member109514606_state @ X3 @ A) => ((B2 = (F @ X3)) => (member109514606_state @ B2 @ (image_1849792670_state @ F @ A))))))). % rev_image_eqI
thf(fact_39_ball__imageD, axiom,
    ((![F : pname > hoare_958474565_state, A : set_pname, P : hoare_958474565_state > $o]: ((![X2 : hoare_958474565_state]: ((member109514606_state @ X2 @ (image_2144627828_state @ F @ A)) => (P @ X2))) => (![X4 : pname]: ((member_pname @ X4 @ A) => (P @ (F @ X4)))))))). % ball_imageD
thf(fact_40_ball__imageD, axiom,
    ((![F : hoare_958474565_state > hoare_958474565_state, A : set_Ho840737317_state, P : hoare_958474565_state > $o]: ((![X2 : hoare_958474565_state]: ((member109514606_state @ X2 @ (image_1849792670_state @ F @ A)) => (P @ X2))) => (![X4 : hoare_958474565_state]: ((member109514606_state @ X4 @ A) => (P @ (F @ X4)))))))). % ball_imageD
thf(fact_41_image__cong, axiom,
    ((![M : set_pname, N : set_pname, F : pname > hoare_958474565_state, G3 : pname > hoare_958474565_state]: ((M = N) => ((![X2 : pname]: ((member_pname @ X2 @ N) => ((F @ X2) = (G3 @ X2)))) => ((image_2144627828_state @ F @ M) = (image_2144627828_state @ G3 @ N))))))). % image_cong
thf(fact_42_image__cong, axiom,
    ((![M : set_Ho840737317_state, N : set_Ho840737317_state, F : hoare_958474565_state > hoare_958474565_state, G3 : hoare_958474565_state > hoare_958474565_state]: ((M = N) => ((![X2 : hoare_958474565_state]: ((member109514606_state @ X2 @ N) => ((F @ X2) = (G3 @ X2)))) => ((image_1849792670_state @ F @ M) = (image_1849792670_state @ G3 @ N))))))). % image_cong
thf(fact_43_bex__imageD, axiom,
    ((![F : pname > hoare_958474565_state, A : set_pname, P : hoare_958474565_state > $o]: ((?[X4 : hoare_958474565_state]: ((member109514606_state @ X4 @ (image_2144627828_state @ F @ A)) & (P @ X4))) => (?[X2 : pname]: ((member_pname @ X2 @ A) & (P @ (F @ X2)))))))). % bex_imageD
thf(fact_44_bex__imageD, axiom,
    ((![F : hoare_958474565_state > hoare_958474565_state, A : set_Ho840737317_state, P : hoare_958474565_state > $o]: ((?[X4 : hoare_958474565_state]: ((member109514606_state @ X4 @ (image_1849792670_state @ F @ A)) & (P @ X4))) => (?[X2 : hoare_958474565_state]: ((member109514606_state @ X2 @ A) & (P @ (F @ X2)))))))). % bex_imageD
thf(fact_45_image__iff, axiom,
    ((![Z : hoare_958474565_state, F : pname > hoare_958474565_state, A : set_pname]: ((member109514606_state @ Z @ (image_2144627828_state @ F @ A)) = (?[X : pname]: (((member_pname @ X @ A)) & ((Z = (F @ X))))))))). % image_iff
thf(fact_46_image__iff, axiom,
    ((![Z : hoare_958474565_state, F : hoare_958474565_state > hoare_958474565_state, A : set_Ho840737317_state]: ((member109514606_state @ Z @ (image_1849792670_state @ F @ A)) = (?[X : hoare_958474565_state]: (((member109514606_state @ X @ A)) & ((Z = (F @ X))))))))). % image_iff
thf(fact_47_imageI, axiom,
    ((![X3 : pname, A : set_pname, F : pname > hoare_958474565_state]: ((member_pname @ X3 @ A) => (member109514606_state @ (F @ X3) @ (image_2144627828_state @ F @ A)))))). % imageI
thf(fact_48_imageI, axiom,
    ((![X3 : hoare_958474565_state, A : set_Ho840737317_state, F : hoare_958474565_state > hoare_958474565_state]: ((member109514606_state @ X3 @ A) => (member109514606_state @ (F @ X3) @ (image_1849792670_state @ F @ A)))))). % imageI
thf(fact_49_ex__in__conv, axiom,
    ((![A : set_Ho840737317_state]: ((?[X : hoare_958474565_state]: (member109514606_state @ X @ A)) = (~ ((A = bot_bo105666705_state))))))). % ex_in_conv
thf(fact_50_ex__in__conv, axiom,
    ((![A : set_pname]: ((?[X : pname]: (member_pname @ X @ A)) = (~ ((A = bot_bot_set_pname))))))). % ex_in_conv
thf(fact_51_equals0I, axiom,
    ((![A : set_Ho840737317_state]: ((![Y2 : hoare_958474565_state]: (~ ((member109514606_state @ Y2 @ A)))) => (A = bot_bo105666705_state))))). % equals0I
thf(fact_52_equals0I, axiom,
    ((![A : set_pname]: ((![Y2 : pname]: (~ ((member_pname @ Y2 @ A)))) => (A = bot_bot_set_pname))))). % equals0I
thf(fact_53_equals0D, axiom,
    ((![A : set_Ho840737317_state, A2 : hoare_958474565_state]: ((A = bot_bo105666705_state) => (~ ((member109514606_state @ A2 @ A))))))). % equals0D
thf(fact_54_equals0D, axiom,
    ((![A : set_pname, A2 : pname]: ((A = bot_bot_set_pname) => (~ ((member_pname @ A2 @ A))))))). % equals0D
thf(fact_55_emptyE, axiom,
    ((![A2 : hoare_958474565_state]: (~ ((member109514606_state @ A2 @ bot_bo105666705_state)))))). % emptyE
thf(fact_56_emptyE, axiom,
    ((![A2 : pname]: (~ ((member_pname @ A2 @ bot_bot_set_pname)))))). % emptyE
thf(fact_57_Collect__mono__iff, axiom,
    ((![P : hoare_958474565_state > $o, Q : hoare_958474565_state > $o]: ((ord_le1945819589_state @ (collec305460656_state @ P) @ (collec305460656_state @ Q)) = (![X : hoare_958474565_state]: (((P @ X)) => ((Q @ X)))))))). % Collect_mono_iff
thf(fact_58_set__eq__subset, axiom,
    (((^[Y3 : set_Ho840737317_state]: (^[Z2 : set_Ho840737317_state]: (Y3 = Z2))) = (^[A3 : set_Ho840737317_state]: (^[B3 : set_Ho840737317_state]: (((ord_le1945819589_state @ A3 @ B3)) & ((ord_le1945819589_state @ B3 @ A3)))))))). % set_eq_subset
thf(fact_59_subset__trans, axiom,
    ((![A : set_Ho840737317_state, B : set_Ho840737317_state, C2 : set_Ho840737317_state]: ((ord_le1945819589_state @ A @ B) => ((ord_le1945819589_state @ B @ C2) => (ord_le1945819589_state @ A @ C2)))))). % subset_trans
thf(fact_60_Collect__mono, axiom,
    ((![P : hoare_958474565_state > $o, Q : hoare_958474565_state > $o]: ((![X2 : hoare_958474565_state]: ((P @ X2) => (Q @ X2))) => (ord_le1945819589_state @ (collec305460656_state @ P) @ (collec305460656_state @ Q)))))). % Collect_mono
thf(fact_61_subset__refl, axiom,
    ((![A : set_Ho840737317_state]: (ord_le1945819589_state @ A @ A)))). % subset_refl
thf(fact_62_subset__iff, axiom,
    ((ord_le1945819589_state = (^[A3 : set_Ho840737317_state]: (^[B3 : set_Ho840737317_state]: (![T2 : hoare_958474565_state]: (((member109514606_state @ T2 @ A3)) => ((member109514606_state @ T2 @ B3))))))))). % subset_iff
thf(fact_63_equalityD2, axiom,
    ((![A : set_Ho840737317_state, B : set_Ho840737317_state]: ((A = B) => (ord_le1945819589_state @ B @ A))))). % equalityD2
thf(fact_64_equalityD1, axiom,
    ((![A : set_Ho840737317_state, B : set_Ho840737317_state]: ((A = B) => (ord_le1945819589_state @ A @ B))))). % equalityD1
thf(fact_65_subset__eq, axiom,
    ((ord_le1945819589_state = (^[A3 : set_Ho840737317_state]: (^[B3 : set_Ho840737317_state]: (![X : hoare_958474565_state]: (((member109514606_state @ X @ A3)) => ((member109514606_state @ X @ B3))))))))). % subset_eq
thf(fact_66_equalityE, axiom,
    ((![A : set_Ho840737317_state, B : set_Ho840737317_state]: ((A = B) => (~ (((ord_le1945819589_state @ A @ B) => (~ ((ord_le1945819589_state @ B @ A)))))))))). % equalityE
thf(fact_67_subsetD, axiom,
    ((![A : set_Ho840737317_state, B : set_Ho840737317_state, C : hoare_958474565_state]: ((ord_le1945819589_state @ A @ B) => ((member109514606_state @ C @ A) => (member109514606_state @ C @ B)))))). % subsetD
thf(fact_68_in__mono, axiom,
    ((![A : set_Ho840737317_state, B : set_Ho840737317_state, X3 : hoare_958474565_state]: ((ord_le1945819589_state @ A @ B) => ((member109514606_state @ X3 @ A) => (member109514606_state @ X3 @ B)))))). % in_mono
thf(fact_69_Compr__image__eq, axiom,
    ((![F : pname > hoare_958474565_state, A : set_pname, P : hoare_958474565_state > $o]: ((collec305460656_state @ (^[X : hoare_958474565_state]: (((member109514606_state @ X @ (image_2144627828_state @ F @ A))) & ((P @ X))))) = (image_2144627828_state @ F @ (collect_pname @ (^[X : pname]: (((member_pname @ X @ A)) & ((P @ (F @ X))))))))))). % Compr_image_eq
thf(fact_70_Compr__image__eq, axiom,
    ((![F : hoare_958474565_state > hoare_958474565_state, A : set_Ho840737317_state, P : hoare_958474565_state > $o]: ((collec305460656_state @ (^[X : hoare_958474565_state]: (((member109514606_state @ X @ (image_1849792670_state @ F @ A))) & ((P @ X))))) = (image_1849792670_state @ F @ (collec305460656_state @ (^[X : hoare_958474565_state]: (((member109514606_state @ X @ A)) & ((P @ (F @ X))))))))))). % Compr_image_eq
thf(fact_71_image__image, axiom,
    ((![F : pname > hoare_958474565_state, G3 : pname > pname, A : set_pname]: ((image_2144627828_state @ F @ (image_pname_pname @ G3 @ A)) = (image_2144627828_state @ (^[X : pname]: (F @ (G3 @ X))) @ A))))). % image_image
thf(fact_72_image__image, axiom,
    ((![F : pname > hoare_958474565_state, G3 : hoare_958474565_state > pname, A : set_Ho840737317_state]: ((image_2144627828_state @ F @ (image_871391498_pname @ G3 @ A)) = (image_1849792670_state @ (^[X : hoare_958474565_state]: (F @ (G3 @ X))) @ A))))). % image_image
thf(fact_73_image__image, axiom,
    ((![F : hoare_958474565_state > hoare_958474565_state, G3 : pname > hoare_958474565_state, A : set_pname]: ((image_1849792670_state @ F @ (image_2144627828_state @ G3 @ A)) = (image_2144627828_state @ (^[X : pname]: (F @ (G3 @ X))) @ A))))). % image_image
thf(fact_74_image__image, axiom,
    ((![F : hoare_958474565_state > hoare_958474565_state, G3 : hoare_958474565_state > hoare_958474565_state, A : set_Ho840737317_state]: ((image_1849792670_state @ F @ (image_1849792670_state @ G3 @ A)) = (image_1849792670_state @ (^[X : hoare_958474565_state]: (F @ (G3 @ X))) @ A))))). % image_image
thf(fact_75_imageE, axiom,
    ((![B2 : hoare_958474565_state, F : pname > hoare_958474565_state, A : set_pname]: ((member109514606_state @ B2 @ (image_2144627828_state @ F @ A)) => (~ ((![X2 : pname]: ((B2 = (F @ X2)) => (~ ((member_pname @ X2 @ A))))))))))). % imageE
thf(fact_76_imageE, axiom,
    ((![B2 : hoare_958474565_state, F : hoare_958474565_state > hoare_958474565_state, A : set_Ho840737317_state]: ((member109514606_state @ B2 @ (image_1849792670_state @ F @ A)) => (~ ((![X2 : hoare_958474565_state]: ((B2 = (F @ X2)) => (~ ((member109514606_state @ X2 @ A))))))))))). % imageE
thf(fact_77_empty__def, axiom,
    ((bot_bo105666705_state = (collec305460656_state @ (^[X : hoare_958474565_state]: $false))))). % empty_def
thf(fact_78_empty__def, axiom,
    ((bot_bot_set_pname = (collect_pname @ (^[X : pname]: $false))))). % empty_def
thf(fact_79_Collect__subset, axiom,
    ((![A : set_Ho840737317_state, P : hoare_958474565_state > $o]: (ord_le1945819589_state @ (collec305460656_state @ (^[X : hoare_958474565_state]: (((member109514606_state @ X @ A)) & ((P @ X))))) @ A)))). % Collect_subset
thf(fact_80_subset__image__iff, axiom,
    ((![B : set_Ho840737317_state, F : pname > hoare_958474565_state, A : set_pname]: ((ord_le1945819589_state @ B @ (image_2144627828_state @ F @ A)) = (?[AA : set_pname]: (((ord_le865024672_pname @ AA @ A)) & ((B = (image_2144627828_state @ F @ AA))))))))). % subset_image_iff
thf(fact_81_subset__image__iff, axiom,
    ((![B : set_Ho840737317_state, F : hoare_958474565_state > hoare_958474565_state, A : set_Ho840737317_state]: ((ord_le1945819589_state @ B @ (image_1849792670_state @ F @ A)) = (?[AA : set_Ho840737317_state]: (((ord_le1945819589_state @ AA @ A)) & ((B = (image_1849792670_state @ F @ AA))))))))). % subset_image_iff
thf(fact_82_image__subset__iff, axiom,
    ((![F : pname > hoare_958474565_state, A : set_pname, B : set_Ho840737317_state]: ((ord_le1945819589_state @ (image_2144627828_state @ F @ A) @ B) = (![X : pname]: (((member_pname @ X @ A)) => ((member109514606_state @ (F @ X) @ B)))))))). % image_subset_iff
thf(fact_83_image__subset__iff, axiom,
    ((![F : hoare_958474565_state > hoare_958474565_state, A : set_Ho840737317_state, B : set_Ho840737317_state]: ((ord_le1945819589_state @ (image_1849792670_state @ F @ A) @ B) = (![X : hoare_958474565_state]: (((member109514606_state @ X @ A)) => ((member109514606_state @ (F @ X) @ B)))))))). % image_subset_iff
thf(fact_84_subset__imageE, axiom,
    ((![B : set_Ho840737317_state, F : pname > hoare_958474565_state, A : set_pname]: ((ord_le1945819589_state @ B @ (image_2144627828_state @ F @ A)) => (~ ((![C3 : set_pname]: ((ord_le865024672_pname @ C3 @ A) => (~ ((B = (image_2144627828_state @ F @ C3)))))))))))). % subset_imageE
thf(fact_85_subset__imageE, axiom,
    ((![B : set_Ho840737317_state, F : hoare_958474565_state > hoare_958474565_state, A : set_Ho840737317_state]: ((ord_le1945819589_state @ B @ (image_1849792670_state @ F @ A)) => (~ ((![C3 : set_Ho840737317_state]: ((ord_le1945819589_state @ C3 @ A) => (~ ((B = (image_1849792670_state @ F @ C3)))))))))))). % subset_imageE
thf(fact_86_image__subsetI, axiom,
    ((![A : set_pname, F : pname > hoare_958474565_state, B : set_Ho840737317_state]: ((![X2 : pname]: ((member_pname @ X2 @ A) => (member109514606_state @ (F @ X2) @ B))) => (ord_le1945819589_state @ (image_2144627828_state @ F @ A) @ B))))). % image_subsetI
thf(fact_87_image__subsetI, axiom,
    ((![A : set_Ho840737317_state, F : hoare_958474565_state > hoare_958474565_state, B : set_Ho840737317_state]: ((![X2 : hoare_958474565_state]: ((member109514606_state @ X2 @ A) => (member109514606_state @ (F @ X2) @ B))) => (ord_le1945819589_state @ (image_1849792670_state @ F @ A) @ B))))). % image_subsetI
thf(fact_88_image__mono, axiom,
    ((![A : set_pname, B : set_pname, F : pname > hoare_958474565_state]: ((ord_le865024672_pname @ A @ B) => (ord_le1945819589_state @ (image_2144627828_state @ F @ A) @ (image_2144627828_state @ F @ B)))))). % image_mono
thf(fact_89_image__mono, axiom,
    ((![A : set_Ho840737317_state, B : set_Ho840737317_state, F : hoare_958474565_state > hoare_958474565_state]: ((ord_le1945819589_state @ A @ B) => (ord_le1945819589_state @ (image_1849792670_state @ F @ A) @ (image_1849792670_state @ F @ B)))))). % image_mono
thf(fact_90_order__refl, axiom,
    ((![X3 : set_Ho840737317_state]: (ord_le1945819589_state @ X3 @ X3)))). % order_refl
thf(fact_91_image__Collect__subsetI, axiom,
    ((![P : pname > $o, F : pname > hoare_958474565_state, B : set_Ho840737317_state]: ((![X2 : pname]: ((P @ X2) => (member109514606_state @ (F @ X2) @ B))) => (ord_le1945819589_state @ (image_2144627828_state @ F @ (collect_pname @ P)) @ B))))). % image_Collect_subsetI
thf(fact_92_image__Collect__subsetI, axiom,
    ((![P : hoare_958474565_state > $o, F : hoare_958474565_state > hoare_958474565_state, B : set_Ho840737317_state]: ((![X2 : hoare_958474565_state]: ((P @ X2) => (member109514606_state @ (F @ X2) @ B))) => (ord_le1945819589_state @ (image_1849792670_state @ F @ (collec305460656_state @ P)) @ B))))). % image_Collect_subsetI
thf(fact_93_MGT__BodyN, axiom,
    ((![Pn : pname, G2 : set_Ho840737317_state]: ((hoare_604442164_state @ (insert776267541_state @ (hoare_Mirabelle_MGT @ (body2 @ Pn)) @ G2) @ (insert776267541_state @ (hoare_Mirabelle_MGT @ (the_com @ (body @ Pn))) @ bot_bo105666705_state)) => (hoare_604442164_state @ G2 @ (insert776267541_state @ (hoare_Mirabelle_MGT @ (body2 @ Pn)) @ bot_bo105666705_state)))))). % MGT_BodyN
thf(fact_94_subset__emptyI, axiom,
    ((![A : set_pname]: ((![X2 : pname]: (~ ((member_pname @ X2 @ A)))) => (ord_le865024672_pname @ A @ bot_bot_set_pname))))). % subset_emptyI
thf(fact_95_subset__emptyI, axiom,
    ((![A : set_Ho840737317_state]: ((![X2 : hoare_958474565_state]: (~ ((member109514606_state @ X2 @ A)))) => (ord_le1945819589_state @ A @ bot_bo105666705_state))))). % subset_emptyI
thf(fact_96_all__subset__image, axiom,
    ((![F : pname > hoare_958474565_state, A : set_pname, P : set_Ho840737317_state > $o]: ((![B3 : set_Ho840737317_state]: (((ord_le1945819589_state @ B3 @ (image_2144627828_state @ F @ A))) => ((P @ B3)))) = (![B3 : set_pname]: (((ord_le865024672_pname @ B3 @ A)) => ((P @ (image_2144627828_state @ F @ B3))))))))). % all_subset_image
thf(fact_97_all__subset__image, axiom,
    ((![F : hoare_958474565_state > hoare_958474565_state, A : set_Ho840737317_state, P : set_Ho840737317_state > $o]: ((![B3 : set_Ho840737317_state]: (((ord_le1945819589_state @ B3 @ (image_1849792670_state @ F @ A))) => ((P @ B3)))) = (![B3 : set_Ho840737317_state]: (((ord_le1945819589_state @ B3 @ A)) => ((P @ (image_1849792670_state @ F @ B3))))))))). % all_subset_image
thf(fact_98_bot_Oextremum__uniqueI, axiom,
    ((![A2 : set_pname]: ((ord_le865024672_pname @ A2 @ bot_bot_set_pname) => (A2 = bot_bot_set_pname))))). % bot.extremum_uniqueI
thf(fact_99_bot_Oextremum__uniqueI, axiom,
    ((![A2 : set_Ho840737317_state]: ((ord_le1945819589_state @ A2 @ bot_bo105666705_state) => (A2 = bot_bo105666705_state))))). % bot.extremum_uniqueI
thf(fact_100_bot_Oextremum__unique, axiom,
    ((![A2 : set_pname]: ((ord_le865024672_pname @ A2 @ bot_bot_set_pname) = (A2 = bot_bot_set_pname))))). % bot.extremum_unique
thf(fact_101_bot_Oextremum__unique, axiom,
    ((![A2 : set_Ho840737317_state]: ((ord_le1945819589_state @ A2 @ bot_bo105666705_state) = (A2 = bot_bo105666705_state))))). % bot.extremum_unique
thf(fact_102_bot_Oextremum, axiom,
    ((![A2 : set_pname]: (ord_le865024672_pname @ bot_bot_set_pname @ A2)))). % bot.extremum
thf(fact_103_bot_Oextremum, axiom,
    ((![A2 : set_Ho840737317_state]: (ord_le1945819589_state @ bot_bo105666705_state @ A2)))). % bot.extremum
thf(fact_104_insertCI, axiom,
    ((![A2 : hoare_958474565_state, B : set_Ho840737317_state, B2 : hoare_958474565_state]: (((~ ((member109514606_state @ A2 @ B))) => (A2 = B2)) => (member109514606_state @ A2 @ (insert776267541_state @ B2 @ B)))))). % insertCI
thf(fact_105_insert__iff, axiom,
    ((![A2 : hoare_958474565_state, B2 : hoare_958474565_state, A : set_Ho840737317_state]: ((member109514606_state @ A2 @ (insert776267541_state @ B2 @ A)) = (((A2 = B2)) | ((member109514606_state @ A2 @ A))))))). % insert_iff
thf(fact_106_insert__absorb2, axiom,
    ((![X3 : hoare_958474565_state, A : set_Ho840737317_state]: ((insert776267541_state @ X3 @ (insert776267541_state @ X3 @ A)) = (insert776267541_state @ X3 @ A))))). % insert_absorb2
thf(fact_107_image__insert, axiom,
    ((![F : pname > hoare_958474565_state, A2 : pname, B : set_pname]: ((image_2144627828_state @ F @ (insert_pname @ A2 @ B)) = (insert776267541_state @ (F @ A2) @ (image_2144627828_state @ F @ B)))))). % image_insert
thf(fact_108_image__insert, axiom,
    ((![F : hoare_958474565_state > hoare_958474565_state, A2 : hoare_958474565_state, B : set_Ho840737317_state]: ((image_1849792670_state @ F @ (insert776267541_state @ A2 @ B)) = (insert776267541_state @ (F @ A2) @ (image_1849792670_state @ F @ B)))))). % image_insert
thf(fact_109_insert__image, axiom,
    ((![X3 : pname, A : set_pname, F : pname > hoare_958474565_state]: ((member_pname @ X3 @ A) => ((insert776267541_state @ (F @ X3) @ (image_2144627828_state @ F @ A)) = (image_2144627828_state @ F @ A)))))). % insert_image
thf(fact_110_insert__image, axiom,
    ((![X3 : hoare_958474565_state, A : set_Ho840737317_state, F : hoare_958474565_state > hoare_958474565_state]: ((member109514606_state @ X3 @ A) => ((insert776267541_state @ (F @ X3) @ (image_1849792670_state @ F @ A)) = (image_1849792670_state @ F @ A)))))). % insert_image
thf(fact_111_singletonI, axiom,
    ((![A2 : hoare_958474565_state]: (member109514606_state @ A2 @ (insert776267541_state @ A2 @ bot_bo105666705_state))))). % singletonI
thf(fact_112_singletonI, axiom,
    ((![A2 : pname]: (member_pname @ A2 @ (insert_pname @ A2 @ bot_bot_set_pname))))). % singletonI
thf(fact_113_insert__subset, axiom,
    ((![X3 : hoare_958474565_state, A : set_Ho840737317_state, B : set_Ho840737317_state]: ((ord_le1945819589_state @ (insert776267541_state @ X3 @ A) @ B) = (((member109514606_state @ X3 @ B)) & ((ord_le1945819589_state @ A @ B))))))). % insert_subset
thf(fact_114_singleton__conv, axiom,
    ((![A2 : hoare_958474565_state]: ((collec305460656_state @ (^[X : hoare_958474565_state]: (X = A2))) = (insert776267541_state @ A2 @ bot_bo105666705_state))))). % singleton_conv
thf(fact_115_singleton__conv, axiom,
    ((![A2 : pname]: ((collect_pname @ (^[X : pname]: (X = A2))) = (insert_pname @ A2 @ bot_bot_set_pname))))). % singleton_conv
thf(fact_116_singleton__conv2, axiom,
    ((![A2 : hoare_958474565_state]: ((collec305460656_state @ ((^[Y3 : hoare_958474565_state]: (^[Z2 : hoare_958474565_state]: (Y3 = Z2))) @ A2)) = (insert776267541_state @ A2 @ bot_bo105666705_state))))). % singleton_conv2
thf(fact_117_singleton__conv2, axiom,
    ((![A2 : pname]: ((collect_pname @ ((^[Y3 : pname]: (^[Z2 : pname]: (Y3 = Z2))) @ A2)) = (insert_pname @ A2 @ bot_bot_set_pname))))). % singleton_conv2
thf(fact_118_singleton__insert__inj__eq, axiom,
    ((![B2 : pname, A2 : pname, A : set_pname]: (((insert_pname @ B2 @ bot_bot_set_pname) = (insert_pname @ A2 @ A)) = (((A2 = B2)) & ((ord_le865024672_pname @ A @ (insert_pname @ B2 @ bot_bot_set_pname)))))))). % singleton_insert_inj_eq
thf(fact_119_singleton__insert__inj__eq, axiom,
    ((![B2 : hoare_958474565_state, A2 : hoare_958474565_state, A : set_Ho840737317_state]: (((insert776267541_state @ B2 @ bot_bo105666705_state) = (insert776267541_state @ A2 @ A)) = (((A2 = B2)) & ((ord_le1945819589_state @ A @ (insert776267541_state @ B2 @ bot_bo105666705_state)))))))). % singleton_insert_inj_eq
thf(fact_120_singleton__insert__inj__eq_H, axiom,
    ((![A2 : pname, A : set_pname, B2 : pname]: (((insert_pname @ A2 @ A) = (insert_pname @ B2 @ bot_bot_set_pname)) = (((A2 = B2)) & ((ord_le865024672_pname @ A @ (insert_pname @ B2 @ bot_bot_set_pname)))))))). % singleton_insert_inj_eq'
thf(fact_121_singleton__insert__inj__eq_H, axiom,
    ((![A2 : hoare_958474565_state, A : set_Ho840737317_state, B2 : hoare_958474565_state]: (((insert776267541_state @ A2 @ A) = (insert776267541_state @ B2 @ bot_bo105666705_state)) = (((A2 = B2)) & ((ord_le1945819589_state @ A @ (insert776267541_state @ B2 @ bot_bo105666705_state)))))))). % singleton_insert_inj_eq'
thf(fact_122_insertE, axiom,
    ((![A2 : hoare_958474565_state, B2 : hoare_958474565_state, A : set_Ho840737317_state]: ((member109514606_state @ A2 @ (insert776267541_state @ B2 @ A)) => ((~ ((A2 = B2))) => (member109514606_state @ A2 @ A)))))). % insertE
thf(fact_123_insertI1, axiom,
    ((![A2 : hoare_958474565_state, B : set_Ho840737317_state]: (member109514606_state @ A2 @ (insert776267541_state @ A2 @ B))))). % insertI1
thf(fact_124_insertI2, axiom,
    ((![A2 : hoare_958474565_state, B : set_Ho840737317_state, B2 : hoare_958474565_state]: ((member109514606_state @ A2 @ B) => (member109514606_state @ A2 @ (insert776267541_state @ B2 @ B)))))). % insertI2
thf(fact_125_Set_Oset__insert, axiom,
    ((![X3 : hoare_958474565_state, A : set_Ho840737317_state]: ((member109514606_state @ X3 @ A) => (~ ((![B4 : set_Ho840737317_state]: ((A = (insert776267541_state @ X3 @ B4)) => (member109514606_state @ X3 @ B4))))))))). % Set.set_insert
thf(fact_126_insert__compr, axiom,
    ((insert776267541_state = (^[A4 : hoare_958474565_state]: (^[B3 : set_Ho840737317_state]: (collec305460656_state @ (^[X : hoare_958474565_state]: (((X = A4)) | ((member109514606_state @ X @ B3)))))))))). % insert_compr
thf(fact_127_insert__ident, axiom,
    ((![X3 : hoare_958474565_state, A : set_Ho840737317_state, B : set_Ho840737317_state]: ((~ ((member109514606_state @ X3 @ A))) => ((~ ((member109514606_state @ X3 @ B))) => (((insert776267541_state @ X3 @ A) = (insert776267541_state @ X3 @ B)) = (A = B))))))). % insert_ident
thf(fact_128_insert__absorb, axiom,
    ((![A2 : hoare_958474565_state, A : set_Ho840737317_state]: ((member109514606_state @ A2 @ A) => ((insert776267541_state @ A2 @ A) = A))))). % insert_absorb
thf(fact_129_insert__eq__iff, axiom,
    ((![A2 : hoare_958474565_state, A : set_Ho840737317_state, B2 : hoare_958474565_state, B : set_Ho840737317_state]: ((~ ((member109514606_state @ A2 @ A))) => ((~ ((member109514606_state @ B2 @ B))) => (((insert776267541_state @ A2 @ A) = (insert776267541_state @ B2 @ B)) = (((((A2 = B2)) => ((A = B)))) & ((((~ ((A2 = B2)))) => ((?[C4 : set_Ho840737317_state]: (((A = (insert776267541_state @ B2 @ C4))) & ((((~ ((member109514606_state @ B2 @ C4)))) & ((((B = (insert776267541_state @ A2 @ C4))) & ((~ ((member109514606_state @ A2 @ C4)))))))))))))))))))). % insert_eq_iff
thf(fact_130_insert__Collect, axiom,
    ((![A2 : hoare_958474565_state, P : hoare_958474565_state > $o]: ((insert776267541_state @ A2 @ (collec305460656_state @ P)) = (collec305460656_state @ (^[U : hoare_958474565_state]: (((~ ((U = A2)))) => ((P @ U))))))))). % insert_Collect
thf(fact_131_insert__commute, axiom,
    ((![X3 : hoare_958474565_state, Y4 : hoare_958474565_state, A : set_Ho840737317_state]: ((insert776267541_state @ X3 @ (insert776267541_state @ Y4 @ A)) = (insert776267541_state @ Y4 @ (insert776267541_state @ X3 @ A)))))). % insert_commute
thf(fact_132_mk__disjoint__insert, axiom,
    ((![A2 : hoare_958474565_state, A : set_Ho840737317_state]: ((member109514606_state @ A2 @ A) => (?[B4 : set_Ho840737317_state]: ((A = (insert776267541_state @ A2 @ B4)) & (~ ((member109514606_state @ A2 @ B4))))))))). % mk_disjoint_insert
thf(fact_133_insert__subsetI, axiom,
    ((![X3 : hoare_958474565_state, A : set_Ho840737317_state, X5 : set_Ho840737317_state]: ((member109514606_state @ X3 @ A) => ((ord_le1945819589_state @ X5 @ A) => (ord_le1945819589_state @ (insert776267541_state @ X3 @ X5) @ A)))))). % insert_subsetI
thf(fact_134_bot__set__def, axiom,
    ((bot_bo105666705_state = (collec305460656_state @ bot_bo1428770700tate_o)))). % bot_set_def
thf(fact_135_bot__set__def, axiom,
    ((bot_bot_set_pname = (collect_pname @ bot_bot_pname_o)))). % bot_set_def
thf(fact_136_singletonD, axiom,
    ((![B2 : hoare_958474565_state, A2 : hoare_958474565_state]: ((member109514606_state @ B2 @ (insert776267541_state @ A2 @ bot_bo105666705_state)) => (B2 = A2))))). % singletonD
thf(fact_137_singletonD, axiom,
    ((![B2 : pname, A2 : pname]: ((member_pname @ B2 @ (insert_pname @ A2 @ bot_bot_set_pname)) => (B2 = A2))))). % singletonD
thf(fact_138_singleton__iff, axiom,
    ((![B2 : hoare_958474565_state, A2 : hoare_958474565_state]: ((member109514606_state @ B2 @ (insert776267541_state @ A2 @ bot_bo105666705_state)) = (B2 = A2))))). % singleton_iff
thf(fact_139_singleton__iff, axiom,
    ((![B2 : pname, A2 : pname]: ((member_pname @ B2 @ (insert_pname @ A2 @ bot_bot_set_pname)) = (B2 = A2))))). % singleton_iff
thf(fact_140_doubleton__eq__iff, axiom,
    ((![A2 : hoare_958474565_state, B2 : hoare_958474565_state, C : hoare_958474565_state, D : hoare_958474565_state]: (((insert776267541_state @ A2 @ (insert776267541_state @ B2 @ bot_bo105666705_state)) = (insert776267541_state @ C @ (insert776267541_state @ D @ bot_bo105666705_state))) = (((((A2 = C)) & ((B2 = D)))) | ((((A2 = D)) & ((B2 = C))))))))). % doubleton_eq_iff
thf(fact_141_doubleton__eq__iff, axiom,
    ((![A2 : pname, B2 : pname, C : pname, D : pname]: (((insert_pname @ A2 @ (insert_pname @ B2 @ bot_bot_set_pname)) = (insert_pname @ C @ (insert_pname @ D @ bot_bot_set_pname))) = (((((A2 = C)) & ((B2 = D)))) | ((((A2 = D)) & ((B2 = C))))))))). % doubleton_eq_iff
thf(fact_142_insert__not__empty, axiom,
    ((![A2 : hoare_958474565_state, A : set_Ho840737317_state]: (~ (((insert776267541_state @ A2 @ A) = bot_bo105666705_state)))))). % insert_not_empty
thf(fact_143_insert__not__empty, axiom,
    ((![A2 : pname, A : set_pname]: (~ (((insert_pname @ A2 @ A) = bot_bot_set_pname)))))). % insert_not_empty
thf(fact_144_singleton__inject, axiom,
    ((![A2 : hoare_958474565_state, B2 : hoare_958474565_state]: (((insert776267541_state @ A2 @ bot_bo105666705_state) = (insert776267541_state @ B2 @ bot_bo105666705_state)) => (A2 = B2))))). % singleton_inject
thf(fact_145_singleton__inject, axiom,
    ((![A2 : pname, B2 : pname]: (((insert_pname @ A2 @ bot_bot_set_pname) = (insert_pname @ B2 @ bot_bot_set_pname)) => (A2 = B2))))). % singleton_inject
thf(fact_146_insert__mono, axiom,
    ((![C2 : set_Ho840737317_state, D2 : set_Ho840737317_state, A2 : hoare_958474565_state]: ((ord_le1945819589_state @ C2 @ D2) => (ord_le1945819589_state @ (insert776267541_state @ A2 @ C2) @ (insert776267541_state @ A2 @ D2)))))). % insert_mono
thf(fact_147_subset__insert, axiom,
    ((![X3 : hoare_958474565_state, A : set_Ho840737317_state, B : set_Ho840737317_state]: ((~ ((member109514606_state @ X3 @ A))) => ((ord_le1945819589_state @ A @ (insert776267541_state @ X3 @ B)) = (ord_le1945819589_state @ A @ B)))))). % subset_insert
thf(fact_148_subset__insertI, axiom,
    ((![B : set_Ho840737317_state, A2 : hoare_958474565_state]: (ord_le1945819589_state @ B @ (insert776267541_state @ A2 @ B))))). % subset_insertI
thf(fact_149_subset__insertI2, axiom,
    ((![A : set_Ho840737317_state, B : set_Ho840737317_state, B2 : hoare_958474565_state]: ((ord_le1945819589_state @ A @ B) => (ord_le1945819589_state @ A @ (insert776267541_state @ B2 @ B)))))). % subset_insertI2
thf(fact_150_less__eq__set__def, axiom,
    ((ord_le1945819589_state = (^[A3 : set_Ho840737317_state]: (^[B3 : set_Ho840737317_state]: (ord_le1483368280tate_o @ (^[X : hoare_958474565_state]: (member109514606_state @ X @ A3)) @ (^[X : hoare_958474565_state]: (member109514606_state @ X @ B3)))))))). % less_eq_set_def
thf(fact_151_Collect__conv__if, axiom,
    ((![P : hoare_958474565_state > $o, A2 : hoare_958474565_state]: (((P @ A2) => ((collec305460656_state @ (^[X : hoare_958474565_state]: (((X = A2)) & ((P @ X))))) = (insert776267541_state @ A2 @ bot_bo105666705_state))) & ((~ ((P @ A2))) => ((collec305460656_state @ (^[X : hoare_958474565_state]: (((X = A2)) & ((P @ X))))) = bot_bo105666705_state)))))). % Collect_conv_if
thf(fact_152_Collect__conv__if, axiom,
    ((![P : pname > $o, A2 : pname]: (((P @ A2) => ((collect_pname @ (^[X : pname]: (((X = A2)) & ((P @ X))))) = (insert_pname @ A2 @ bot_bot_set_pname))) & ((~ ((P @ A2))) => ((collect_pname @ (^[X : pname]: (((X = A2)) & ((P @ X))))) = bot_bot_set_pname)))))). % Collect_conv_if
thf(fact_153_Collect__conv__if2, axiom,
    ((![P : hoare_958474565_state > $o, A2 : hoare_958474565_state]: (((P @ A2) => ((collec305460656_state @ (^[X : hoare_958474565_state]: (((A2 = X)) & ((P @ X))))) = (insert776267541_state @ A2 @ bot_bo105666705_state))) & ((~ ((P @ A2))) => ((collec305460656_state @ (^[X : hoare_958474565_state]: (((A2 = X)) & ((P @ X))))) = bot_bo105666705_state)))))). % Collect_conv_if2
thf(fact_154_Collect__conv__if2, axiom,
    ((![P : pname > $o, A2 : pname]: (((P @ A2) => ((collect_pname @ (^[X : pname]: (((A2 = X)) & ((P @ X))))) = (insert_pname @ A2 @ bot_bot_set_pname))) & ((~ ((P @ A2))) => ((collect_pname @ (^[X : pname]: (((A2 = X)) & ((P @ X))))) = bot_bot_set_pname)))))). % Collect_conv_if2
thf(fact_155_subset__singletonD, axiom,
    ((![A : set_pname, X3 : pname]: ((ord_le865024672_pname @ A @ (insert_pname @ X3 @ bot_bot_set_pname)) => ((A = bot_bot_set_pname) | (A = (insert_pname @ X3 @ bot_bot_set_pname))))))). % subset_singletonD
thf(fact_156_subset__singletonD, axiom,
    ((![A : set_Ho840737317_state, X3 : hoare_958474565_state]: ((ord_le1945819589_state @ A @ (insert776267541_state @ X3 @ bot_bo105666705_state)) => ((A = bot_bo105666705_state) | (A = (insert776267541_state @ X3 @ bot_bo105666705_state))))))). % subset_singletonD
thf(fact_157_subset__singleton__iff, axiom,
    ((![X5 : set_pname, A2 : pname]: ((ord_le865024672_pname @ X5 @ (insert_pname @ A2 @ bot_bot_set_pname)) = (((X5 = bot_bot_set_pname)) | ((X5 = (insert_pname @ A2 @ bot_bot_set_pname)))))))). % subset_singleton_iff
thf(fact_158_subset__singleton__iff, axiom,
    ((![X5 : set_Ho840737317_state, A2 : hoare_958474565_state]: ((ord_le1945819589_state @ X5 @ (insert776267541_state @ A2 @ bot_bo105666705_state)) = (((X5 = bot_bo105666705_state)) | ((X5 = (insert776267541_state @ A2 @ bot_bo105666705_state)))))))). % subset_singleton_iff
thf(fact_159_derivs__insertD, axiom,
    ((![G2 : set_Ho840737317_state, T3 : hoare_958474565_state, Ts : set_Ho840737317_state]: ((hoare_604442164_state @ G2 @ (insert776267541_state @ T3 @ Ts)) => ((hoare_604442164_state @ G2 @ (insert776267541_state @ T3 @ bot_bo105666705_state)) & (hoare_604442164_state @ G2 @ Ts)))))). % derivs_insertD
thf(fact_160_hoare__derivs_Oinsert, axiom,
    ((![G2 : set_Ho840737317_state, T3 : hoare_958474565_state, Ts : set_Ho840737317_state]: ((hoare_604442164_state @ G2 @ (insert776267541_state @ T3 @ bot_bo105666705_state)) => ((hoare_604442164_state @ G2 @ Ts) => (hoare_604442164_state @ G2 @ (insert776267541_state @ T3 @ Ts))))))). % hoare_derivs.insert
thf(fact_161_image__constant, axiom,
    ((![X3 : pname, A : set_pname, C : hoare_958474565_state]: ((member_pname @ X3 @ A) => ((image_2144627828_state @ (^[X : pname]: C) @ A) = (insert776267541_state @ C @ bot_bo105666705_state)))))). % image_constant
thf(fact_162_image__constant, axiom,
    ((![X3 : hoare_958474565_state, A : set_Ho840737317_state, C : hoare_958474565_state]: ((member109514606_state @ X3 @ A) => ((image_1849792670_state @ (^[X : hoare_958474565_state]: C) @ A) = (insert776267541_state @ C @ bot_bo105666705_state)))))). % image_constant
thf(fact_163_image__constant__conv, axiom,
    ((![A : set_Ho840737317_state, C : hoare_958474565_state]: (((A = bot_bo105666705_state) => ((image_1849792670_state @ (^[X : hoare_958474565_state]: C) @ A) = bot_bo105666705_state)) & ((~ ((A = bot_bo105666705_state))) => ((image_1849792670_state @ (^[X : hoare_958474565_state]: C) @ A) = (insert776267541_state @ C @ bot_bo105666705_state))))))). % image_constant_conv
thf(fact_164_image__constant__conv, axiom,
    ((![A : set_Ho840737317_state, C : pname]: (((A = bot_bo105666705_state) => ((image_871391498_pname @ (^[X : hoare_958474565_state]: C) @ A) = bot_bot_set_pname)) & ((~ ((A = bot_bo105666705_state))) => ((image_871391498_pname @ (^[X : hoare_958474565_state]: C) @ A) = (insert_pname @ C @ bot_bot_set_pname))))))). % image_constant_conv
thf(fact_165_image__constant__conv, axiom,
    ((![A : set_pname, C : hoare_958474565_state]: (((A = bot_bot_set_pname) => ((image_2144627828_state @ (^[X : pname]: C) @ A) = bot_bo105666705_state)) & ((~ ((A = bot_bot_set_pname))) => ((image_2144627828_state @ (^[X : pname]: C) @ A) = (insert776267541_state @ C @ bot_bo105666705_state))))))). % image_constant_conv
thf(fact_166_image__constant__conv, axiom,
    ((![A : set_pname, C : pname]: (((A = bot_bot_set_pname) => ((image_pname_pname @ (^[X : pname]: C) @ A) = bot_bot_set_pname)) & ((~ ((A = bot_bot_set_pname))) => ((image_pname_pname @ (^[X : pname]: C) @ A) = (insert_pname @ C @ bot_bot_set_pname))))))). % image_constant_conv
thf(fact_167_order__subst1, axiom,
    ((![A2 : set_Ho840737317_state, F : set_Ho840737317_state > set_Ho840737317_state, B2 : set_Ho840737317_state, C : set_Ho840737317_state]: ((ord_le1945819589_state @ A2 @ (F @ B2)) => ((ord_le1945819589_state @ B2 @ C) => ((![X2 : set_Ho840737317_state, Y2 : set_Ho840737317_state]: ((ord_le1945819589_state @ X2 @ Y2) => (ord_le1945819589_state @ (F @ X2) @ (F @ Y2)))) => (ord_le1945819589_state @ A2 @ (F @ C)))))))). % order_subst1
thf(fact_168_order__subst2, axiom,
    ((![A2 : set_Ho840737317_state, B2 : set_Ho840737317_state, F : set_Ho840737317_state > set_Ho840737317_state, C : set_Ho840737317_state]: ((ord_le1945819589_state @ A2 @ B2) => ((ord_le1945819589_state @ (F @ B2) @ C) => ((![X2 : set_Ho840737317_state, Y2 : set_Ho840737317_state]: ((ord_le1945819589_state @ X2 @ Y2) => (ord_le1945819589_state @ (F @ X2) @ (F @ Y2)))) => (ord_le1945819589_state @ (F @ A2) @ C))))))). % order_subst2
thf(fact_169_ord__eq__le__subst, axiom,
    ((![A2 : set_Ho840737317_state, F : set_Ho840737317_state > set_Ho840737317_state, B2 : set_Ho840737317_state, C : set_Ho840737317_state]: ((A2 = (F @ B2)) => ((ord_le1945819589_state @ B2 @ C) => ((![X2 : set_Ho840737317_state, Y2 : set_Ho840737317_state]: ((ord_le1945819589_state @ X2 @ Y2) => (ord_le1945819589_state @ (F @ X2) @ (F @ Y2)))) => (ord_le1945819589_state @ A2 @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_170_ord__le__eq__subst, axiom,
    ((![A2 : set_Ho840737317_state, B2 : set_Ho840737317_state, F : set_Ho840737317_state > set_Ho840737317_state, C : set_Ho840737317_state]: ((ord_le1945819589_state @ A2 @ B2) => (((F @ B2) = C) => ((![X2 : set_Ho840737317_state, Y2 : set_Ho840737317_state]: ((ord_le1945819589_state @ X2 @ Y2) => (ord_le1945819589_state @ (F @ X2) @ (F @ Y2)))) => (ord_le1945819589_state @ (F @ A2) @ C))))))). % ord_le_eq_subst
thf(fact_171_eq__iff, axiom,
    (((^[Y3 : set_Ho840737317_state]: (^[Z2 : set_Ho840737317_state]: (Y3 = Z2))) = (^[X : set_Ho840737317_state]: (^[Y5 : set_Ho840737317_state]: (((ord_le1945819589_state @ X @ Y5)) & ((ord_le1945819589_state @ Y5 @ X)))))))). % eq_iff
thf(fact_172_antisym, axiom,
    ((![X3 : set_Ho840737317_state, Y4 : set_Ho840737317_state]: ((ord_le1945819589_state @ X3 @ Y4) => ((ord_le1945819589_state @ Y4 @ X3) => (X3 = Y4)))))). % antisym
thf(fact_173_eq__refl, axiom,
    ((![X3 : set_Ho840737317_state, Y4 : set_Ho840737317_state]: ((X3 = Y4) => (ord_le1945819589_state @ X3 @ Y4))))). % eq_refl
thf(fact_174_order_Otrans, axiom,
    ((![A2 : set_Ho840737317_state, B2 : set_Ho840737317_state, C : set_Ho840737317_state]: ((ord_le1945819589_state @ A2 @ B2) => ((ord_le1945819589_state @ B2 @ C) => (ord_le1945819589_state @ A2 @ C)))))). % order.trans
thf(fact_175_antisym__conv, axiom,
    ((![Y4 : set_Ho840737317_state, X3 : set_Ho840737317_state]: ((ord_le1945819589_state @ Y4 @ X3) => ((ord_le1945819589_state @ X3 @ Y4) = (X3 = Y4)))))). % antisym_conv
thf(fact_176_order__class_Oorder_Oeq__iff, axiom,
    (((^[Y3 : set_Ho840737317_state]: (^[Z2 : set_Ho840737317_state]: (Y3 = Z2))) = (^[A4 : set_Ho840737317_state]: (^[B5 : set_Ho840737317_state]: (((ord_le1945819589_state @ A4 @ B5)) & ((ord_le1945819589_state @ B5 @ A4)))))))). % order_class.order.eq_iff
thf(fact_177_ord__eq__le__trans, axiom,
    ((![A2 : set_Ho840737317_state, B2 : set_Ho840737317_state, C : set_Ho840737317_state]: ((A2 = B2) => ((ord_le1945819589_state @ B2 @ C) => (ord_le1945819589_state @ A2 @ C)))))). % ord_eq_le_trans
thf(fact_178_ord__le__eq__trans, axiom,
    ((![A2 : set_Ho840737317_state, B2 : set_Ho840737317_state, C : set_Ho840737317_state]: ((ord_le1945819589_state @ A2 @ B2) => ((B2 = C) => (ord_le1945819589_state @ A2 @ C)))))). % ord_le_eq_trans
thf(fact_179_order__class_Oorder_Oantisym, axiom,
    ((![A2 : set_Ho840737317_state, B2 : set_Ho840737317_state]: ((ord_le1945819589_state @ A2 @ B2) => ((ord_le1945819589_state @ B2 @ A2) => (A2 = B2)))))). % order_class.order.antisym
thf(fact_180_order__trans, axiom,
    ((![X3 : set_Ho840737317_state, Y4 : set_Ho840737317_state, Z : set_Ho840737317_state]: ((ord_le1945819589_state @ X3 @ Y4) => ((ord_le1945819589_state @ Y4 @ Z) => (ord_le1945819589_state @ X3 @ Z)))))). % order_trans
thf(fact_181_dual__order_Orefl, axiom,
    ((![A2 : set_Ho840737317_state]: (ord_le1945819589_state @ A2 @ A2)))). % dual_order.refl
thf(fact_182_dual__order_Otrans, axiom,
    ((![B2 : set_Ho840737317_state, A2 : set_Ho840737317_state, C : set_Ho840737317_state]: ((ord_le1945819589_state @ B2 @ A2) => ((ord_le1945819589_state @ C @ B2) => (ord_le1945819589_state @ C @ A2)))))). % dual_order.trans
thf(fact_183_dual__order_Oeq__iff, axiom,
    (((^[Y3 : set_Ho840737317_state]: (^[Z2 : set_Ho840737317_state]: (Y3 = Z2))) = (^[A4 : set_Ho840737317_state]: (^[B5 : set_Ho840737317_state]: (((ord_le1945819589_state @ B5 @ A4)) & ((ord_le1945819589_state @ A4 @ B5)))))))). % dual_order.eq_iff
thf(fact_184_dual__order_Oantisym, axiom,
    ((![B2 : set_Ho840737317_state, A2 : set_Ho840737317_state]: ((ord_le1945819589_state @ B2 @ A2) => ((ord_le1945819589_state @ A2 @ B2) => (A2 = B2)))))). % dual_order.antisym
thf(fact_185_prop__restrict, axiom,
    ((![X3 : hoare_958474565_state, Z3 : set_Ho840737317_state, X5 : set_Ho840737317_state, P : hoare_958474565_state > $o]: ((member109514606_state @ X3 @ Z3) => ((ord_le1945819589_state @ Z3 @ (collec305460656_state @ (^[X : hoare_958474565_state]: (((member109514606_state @ X @ X5)) & ((P @ X)))))) => (P @ X3)))))). % prop_restrict
thf(fact_186_Collect__restrict, axiom,
    ((![X5 : set_Ho840737317_state, P : hoare_958474565_state > $o]: (ord_le1945819589_state @ (collec305460656_state @ (^[X : hoare_958474565_state]: (((member109514606_state @ X @ X5)) & ((P @ X))))) @ X5)))). % Collect_restrict
thf(fact_187_MGF__lemma1, axiom,
    ((![G2 : set_Ho840737317_state, C : com]: (hoare_405891322gleton => ((![X2 : pname]: ((member_pname @ X2 @ (dom_pname_com @ body)) => (hoare_604442164_state @ G2 @ (insert776267541_state @ (hoare_Mirabelle_MGT @ (body2 @ X2)) @ bot_bo105666705_state)))) => ((wt @ C) => (hoare_604442164_state @ G2 @ (insert776267541_state @ (hoare_Mirabelle_MGT @ C) @ bot_bo105666705_state)))))))). % MGF_lemma1
thf(fact_188_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_189_the__elem__eq, axiom,
    ((![X3 : hoare_958474565_state]: ((the_el1300254266_state @ (insert776267541_state @ X3 @ bot_bo105666705_state)) = X3)))). % the_elem_eq
thf(fact_190_the__elem__eq, axiom,
    ((![X3 : pname]: ((the_elem_pname @ (insert_pname @ X3 @ bot_bot_set_pname)) = X3)))). % the_elem_eq
thf(fact_191_is__singletonI, axiom,
    ((![X3 : hoare_958474565_state]: (is_sin536631353_state @ (insert776267541_state @ X3 @ bot_bo105666705_state))))). % is_singletonI
thf(fact_192_is__singletonI, axiom,
    ((![X3 : pname]: (is_singleton_pname @ (insert_pname @ X3 @ bot_bot_set_pname))))). % is_singletonI
thf(fact_193_BodyN, axiom,
    ((![P : state > state > $o, Pn : pname, Q : state > state > $o, G2 : set_Ho840737317_state]: ((hoare_604442164_state @ (insert776267541_state @ (hoare_1659279548_state @ P @ (body2 @ Pn) @ Q) @ G2) @ (insert776267541_state @ (hoare_1659279548_state @ P @ (the_com @ (body @ Pn)) @ Q) @ bot_bo105666705_state)) => (hoare_604442164_state @ G2 @ (insert776267541_state @ (hoare_1659279548_state @ P @ (body2 @ Pn) @ Q) @ bot_bo105666705_state)))))). % BodyN
thf(fact_194_is__singleton__the__elem, axiom,
    ((is_sin536631353_state = (^[A3 : set_Ho840737317_state]: (A3 = (insert776267541_state @ (the_el1300254266_state @ A3) @ bot_bo105666705_state)))))). % is_singleton_the_elem
thf(fact_195_is__singleton__the__elem, axiom,
    ((is_singleton_pname = (^[A3 : set_pname]: (A3 = (insert_pname @ (the_elem_pname @ A3) @ bot_bot_set_pname)))))). % is_singleton_the_elem
thf(fact_196_is__singletonI_H, axiom,
    ((![A : set_Ho840737317_state]: ((~ ((A = bot_bo105666705_state))) => ((![X2 : hoare_958474565_state, Y2 : hoare_958474565_state]: ((member109514606_state @ X2 @ A) => ((member109514606_state @ Y2 @ A) => (X2 = Y2)))) => (is_sin536631353_state @ A)))))). % is_singletonI'
thf(fact_197_is__singletonI_H, axiom,
    ((![A : set_pname]: ((~ ((A = bot_bot_set_pname))) => ((![X2 : pname, Y2 : pname]: ((member_pname @ X2 @ A) => ((member_pname @ Y2 @ A) => (X2 = Y2)))) => (is_singleton_pname @ A)))))). % is_singletonI'

% Conjectures (3)
thf(conj_0, hypothesis,
    (hoare_405891322gleton)).
thf(conj_1, hypothesis,
    (wT_bodies)).
thf(conj_2, conjecture,
    (((~ ((ord_le1945819589_state @ bot_bo105666705_state @ (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)) @ bot_bo105666705_state)))).
