% 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_445__3254776_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:14:58.908

% Could-be-implicit typings (8)
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J_J_J, type,
    set_se421025541_state : $tType).
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__Set__Oset_It__Set__Oset_It__Com__Opname_J_J, type,
    set_set_pname : $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__Opname, type,
    pname : $tType).
thf(ty_n_t__Com__Ocom, type,
    com : $tType).

% Explicit typings (37)
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_Finite__Set_Ofinite_001t__Com__Opname, type,
    finite_finite_pname : set_pname > $o).
thf(sy_c_Finite__Set_Ofinite_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J, type,
    finite1986656878_state : set_Ho840737317_state > $o).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Com__Opname_J, type,
    finite505202775_pname : set_set_pname > $o).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J_J, type,
    finite1266962254_state : set_se421025541_state > $o).
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_Lattices_Osup__class_Osup_001t__Set__Oset_It__Com__Opname_J, type,
    sup_sup_set_pname : set_pname > set_pname > set_pname).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J_J, type,
    sup_su737426425_state : set_Ho840737317_state > set_Ho840737317_state > set_Ho840737317_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_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_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J_J_J, type,
    bot_bo249060721_state : set_se421025541_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_OCollect_001t__Set__Oset_It__Com__Opname_J, type,
    collect_set_pname : (set_pname > $o) > set_set_pname).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J_J, type,
    collec305218192_state : (set_Ho840737317_state > $o) > set_se421025541_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_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_c_member_001t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J_J, type,
    member1959617614_state : set_Ho840737317_state > set_se421025541_state > $o).
thf(sy_v_Fa, type,
    fa : set_Ho840737317_state).
thf(sy_v_pn, type,
    pn : pname).
thf(sy_v_y, type,
    y : com).

% Relevant facts (202)
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_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_2_derivs__insertD, axiom,
    ((![G2 : set_Ho840737317_state, T : hoare_958474565_state, Ts : set_Ho840737317_state]: ((hoare_604442164_state @ G2 @ (insert776267541_state @ T @ Ts)) => ((hoare_604442164_state @ G2 @ (insert776267541_state @ T @ bot_bo105666705_state)) & (hoare_604442164_state @ G2 @ Ts)))))). % derivs_insertD
thf(fact_3_asm, axiom,
    ((![Ts : set_Ho840737317_state, G2 : set_Ho840737317_state]: ((ord_le1945819589_state @ Ts @ G2) => (hoare_604442164_state @ G2 @ Ts))))). % asm
thf(fact_4_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_5_empty, axiom,
    ((![G2 : set_Ho840737317_state]: (hoare_604442164_state @ G2 @ bot_bo105666705_state)))). % empty
thf(fact_6_hoare__derivs_Oinsert, axiom,
    ((![G2 : set_Ho840737317_state, T : hoare_958474565_state, Ts : set_Ho840737317_state]: ((hoare_604442164_state @ G2 @ (insert776267541_state @ T @ bot_bo105666705_state)) => ((hoare_604442164_state @ G2 @ Ts) => (hoare_604442164_state @ G2 @ (insert776267541_state @ T @ Ts))))))). % hoare_derivs.insert
thf(fact_7_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_8_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_9_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_10_finite__Collect__subsets, axiom,
    ((![A2 : set_pname]: ((finite_finite_pname @ A2) => (finite505202775_pname @ (collect_set_pname @ (^[B2 : set_pname]: (ord_le865024672_pname @ B2 @ A2)))))))). % finite_Collect_subsets
thf(fact_11_finite__Collect__subsets, axiom,
    ((![A2 : set_Ho840737317_state]: ((finite1986656878_state @ A2) => (finite1266962254_state @ (collec305218192_state @ (^[B2 : set_Ho840737317_state]: (ord_le1945819589_state @ B2 @ A2)))))))). % finite_Collect_subsets
thf(fact_12_singleton__conv, axiom,
    ((![A : hoare_958474565_state]: ((collec305460656_state @ (^[X : hoare_958474565_state]: (X = A))) = (insert776267541_state @ A @ bot_bo105666705_state))))). % singleton_conv
thf(fact_13_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_14_insert__subset, axiom,
    ((![X2 : hoare_958474565_state, A2 : set_Ho840737317_state, B3 : set_Ho840737317_state]: ((ord_le1945819589_state @ (insert776267541_state @ X2 @ A2) @ B3) = (((member109514606_state @ X2 @ B3)) & ((ord_le1945819589_state @ A2 @ B3))))))). % insert_subset
thf(fact_15_finite__insert, axiom,
    ((![A : hoare_958474565_state, A2 : set_Ho840737317_state]: ((finite1986656878_state @ (insert776267541_state @ A @ A2)) = (finite1986656878_state @ A2))))). % finite_insert
thf(fact_16_finite__insert, axiom,
    ((![A : pname, A2 : set_pname]: ((finite_finite_pname @ (insert_pname @ A @ A2)) = (finite_finite_pname @ A2))))). % finite_insert
thf(fact_17_singletonI, axiom,
    ((![A : hoare_958474565_state]: (member109514606_state @ A @ (insert776267541_state @ A @ bot_bo105666705_state))))). % singletonI
thf(fact_18_subset__empty, axiom,
    ((![A2 : set_Ho840737317_state]: ((ord_le1945819589_state @ A2 @ bot_bo105666705_state) = (A2 = bot_bo105666705_state))))). % subset_empty
thf(fact_19_empty__subsetI, axiom,
    ((![A2 : set_Ho840737317_state]: (ord_le1945819589_state @ bot_bo105666705_state @ A2)))). % empty_subsetI
thf(fact_20_image__insert, axiom,
    ((![F : pname > hoare_958474565_state, A : pname, B3 : set_pname]: ((image_2144627828_state @ F @ (insert_pname @ A @ B3)) = (insert776267541_state @ (F @ A) @ (image_2144627828_state @ F @ B3)))))). % image_insert
thf(fact_21_image__insert, axiom,
    ((![F : hoare_958474565_state > hoare_958474565_state, A : hoare_958474565_state, B3 : set_Ho840737317_state]: ((image_1849792670_state @ F @ (insert776267541_state @ A @ B3)) = (insert776267541_state @ (F @ A) @ (image_1849792670_state @ F @ B3)))))). % image_insert
thf(fact_22_insert__image, axiom,
    ((![X2 : pname, A2 : set_pname, F : pname > hoare_958474565_state]: ((member_pname @ X2 @ A2) => ((insert776267541_state @ (F @ X2) @ (image_2144627828_state @ F @ A2)) = (image_2144627828_state @ F @ A2)))))). % insert_image
thf(fact_23_insert__image, axiom,
    ((![X2 : hoare_958474565_state, A2 : set_Ho840737317_state, F : hoare_958474565_state > hoare_958474565_state]: ((member109514606_state @ X2 @ A2) => ((insert776267541_state @ (F @ X2) @ (image_1849792670_state @ F @ A2)) = (image_1849792670_state @ F @ A2)))))). % insert_image
thf(fact_24_image__eqI, axiom,
    ((![B : hoare_958474565_state, F : pname > hoare_958474565_state, X2 : pname, A2 : set_pname]: ((B = (F @ X2)) => ((member_pname @ X2 @ A2) => (member109514606_state @ B @ (image_2144627828_state @ F @ A2))))))). % image_eqI
thf(fact_25_image__eqI, axiom,
    ((![B : hoare_958474565_state, F : hoare_958474565_state > hoare_958474565_state, X2 : hoare_958474565_state, A2 : set_Ho840737317_state]: ((B = (F @ X2)) => ((member109514606_state @ X2 @ A2) => (member109514606_state @ B @ (image_1849792670_state @ F @ A2))))))). % image_eqI
thf(fact_26_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_27_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_28_all__not__in__conv, axiom,
    ((![A2 : set_Ho840737317_state]: ((![X : hoare_958474565_state]: (~ ((member109514606_state @ X @ A2)))) = (A2 = bot_bo105666705_state))))). % all_not_in_conv
thf(fact_29_empty__iff, axiom,
    ((![C : hoare_958474565_state]: (~ ((member109514606_state @ C @ bot_bo105666705_state)))))). % empty_iff
thf(fact_30_subset__antisym, axiom,
    ((![A2 : set_Ho840737317_state, B3 : set_Ho840737317_state]: ((ord_le1945819589_state @ A2 @ B3) => ((ord_le1945819589_state @ B3 @ A2) => (A2 = B3)))))). % subset_antisym
thf(fact_31_subsetI, axiom,
    ((![A2 : set_Ho840737317_state, B3 : set_Ho840737317_state]: ((![X3 : hoare_958474565_state]: ((member109514606_state @ X3 @ A2) => (member109514606_state @ X3 @ B3))) => (ord_le1945819589_state @ A2 @ B3))))). % subsetI
thf(fact_32_insert__absorb2, axiom,
    ((![X2 : hoare_958474565_state, A2 : set_Ho840737317_state]: ((insert776267541_state @ X2 @ (insert776267541_state @ X2 @ A2)) = (insert776267541_state @ X2 @ A2))))). % insert_absorb2
thf(fact_33_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_34_insertCI, axiom,
    ((![A : hoare_958474565_state, B3 : set_Ho840737317_state, B : hoare_958474565_state]: (((~ ((member109514606_state @ A @ B3))) => (A = B)) => (member109514606_state @ A @ (insert776267541_state @ B @ B3)))))). % insertCI
thf(fact_35_finite__Collect__disjI, axiom,
    ((![P : hoare_958474565_state > $o, Q : hoare_958474565_state > $o]: ((finite1986656878_state @ (collec305460656_state @ (^[X : hoare_958474565_state]: (((P @ X)) | ((Q @ X)))))) = (((finite1986656878_state @ (collec305460656_state @ P))) & ((finite1986656878_state @ (collec305460656_state @ Q)))))))). % finite_Collect_disjI
thf(fact_36_finite__Collect__disjI, axiom,
    ((![P : pname > $o, Q : pname > $o]: ((finite_finite_pname @ (collect_pname @ (^[X : pname]: (((P @ X)) | ((Q @ X)))))) = (((finite_finite_pname @ (collect_pname @ P))) & ((finite_finite_pname @ (collect_pname @ Q)))))))). % finite_Collect_disjI
thf(fact_37_finite__Collect__conjI, axiom,
    ((![P : hoare_958474565_state > $o, Q : hoare_958474565_state > $o]: (((finite1986656878_state @ (collec305460656_state @ P)) | (finite1986656878_state @ (collec305460656_state @ Q))) => (finite1986656878_state @ (collec305460656_state @ (^[X : hoare_958474565_state]: (((P @ X)) & ((Q @ X)))))))))). % finite_Collect_conjI
thf(fact_38_finite__Collect__conjI, axiom,
    ((![P : pname > $o, Q : pname > $o]: (((finite_finite_pname @ (collect_pname @ P)) | (finite_finite_pname @ (collect_pname @ Q))) => (finite_finite_pname @ (collect_pname @ (^[X : pname]: (((P @ X)) & ((Q @ X)))))))))). % finite_Collect_conjI
thf(fact_39_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_40_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_41_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_42_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_43_image__empty, axiom,
    ((![F : pname > hoare_958474565_state]: ((image_2144627828_state @ F @ bot_bot_set_pname) = bot_bo105666705_state)))). % image_empty
thf(fact_44_image__empty, axiom,
    ((![F : hoare_958474565_state > hoare_958474565_state]: ((image_1849792670_state @ F @ bot_bo105666705_state) = bot_bo105666705_state)))). % image_empty
thf(fact_45_finite__imageI, axiom,
    ((![F2 : set_Ho840737317_state, H : hoare_958474565_state > hoare_958474565_state]: ((finite1986656878_state @ F2) => (finite1986656878_state @ (image_1849792670_state @ H @ F2)))))). % finite_imageI
thf(fact_46_finite__imageI, axiom,
    ((![F2 : set_Ho840737317_state, H : hoare_958474565_state > pname]: ((finite1986656878_state @ F2) => (finite_finite_pname @ (image_871391498_pname @ H @ F2)))))). % finite_imageI
thf(fact_47_finite__imageI, axiom,
    ((![F2 : set_pname, H : pname > hoare_958474565_state]: ((finite_finite_pname @ F2) => (finite1986656878_state @ (image_2144627828_state @ H @ F2)))))). % finite_imageI
thf(fact_48_finite__imageI, axiom,
    ((![F2 : set_pname, H : pname > pname]: ((finite_finite_pname @ F2) => (finite_finite_pname @ (image_pname_pname @ H @ F2)))))). % finite_imageI
thf(fact_49_rev__image__eqI, axiom,
    ((![X2 : pname, A2 : set_pname, B : hoare_958474565_state, F : pname > hoare_958474565_state]: ((member_pname @ X2 @ A2) => ((B = (F @ X2)) => (member109514606_state @ B @ (image_2144627828_state @ F @ A2))))))). % rev_image_eqI
thf(fact_50_rev__image__eqI, axiom,
    ((![X2 : hoare_958474565_state, A2 : set_Ho840737317_state, B : hoare_958474565_state, F : hoare_958474565_state > hoare_958474565_state]: ((member109514606_state @ X2 @ A2) => ((B = (F @ X2)) => (member109514606_state @ B @ (image_1849792670_state @ F @ A2))))))). % rev_image_eqI
thf(fact_51_ball__imageD, axiom,
    ((![F : pname > hoare_958474565_state, A2 : set_pname, P : hoare_958474565_state > $o]: ((![X3 : hoare_958474565_state]: ((member109514606_state @ X3 @ (image_2144627828_state @ F @ A2)) => (P @ X3))) => (![X4 : pname]: ((member_pname @ X4 @ A2) => (P @ (F @ X4)))))))). % ball_imageD
thf(fact_52_image__cong, axiom,
    ((![M : set_pname, N : set_pname, F : pname > hoare_958474565_state, G3 : pname > hoare_958474565_state]: ((M = N) => ((![X3 : pname]: ((member_pname @ X3 @ N) => ((F @ X3) = (G3 @ X3)))) => ((image_2144627828_state @ F @ M) = (image_2144627828_state @ G3 @ N))))))). % image_cong
thf(fact_53_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))) => (?[X3 : pname]: ((member_pname @ X3 @ A2) & (P @ (F @ X3)))))))). % bex_imageD
thf(fact_54_image__iff, axiom,
    ((![Z2 : hoare_958474565_state, F : pname > hoare_958474565_state, A2 : set_pname]: ((member109514606_state @ Z2 @ (image_2144627828_state @ F @ A2)) = (?[X : pname]: (((member_pname @ X @ A2)) & ((Z2 = (F @ X))))))))). % image_iff
thf(fact_55_imageI, axiom,
    ((![X2 : pname, A2 : set_pname, F : pname > hoare_958474565_state]: ((member_pname @ X2 @ A2) => (member109514606_state @ (F @ X2) @ (image_2144627828_state @ F @ A2)))))). % imageI
thf(fact_56_imageI, axiom,
    ((![X2 : hoare_958474565_state, A2 : set_Ho840737317_state, F : hoare_958474565_state > hoare_958474565_state]: ((member109514606_state @ X2 @ A2) => (member109514606_state @ (F @ X2) @ (image_1849792670_state @ F @ A2)))))). % imageI
thf(fact_57_ex__in__conv, axiom,
    ((![A2 : set_Ho840737317_state]: ((?[X : hoare_958474565_state]: (member109514606_state @ X @ A2)) = (~ ((A2 = bot_bo105666705_state))))))). % ex_in_conv
thf(fact_58_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_59_Collect__mem__eq, axiom,
    ((![A2 : set_Ho840737317_state]: ((collec305460656_state @ (^[X : hoare_958474565_state]: (member109514606_state @ X @ A2))) = A2)))). % Collect_mem_eq
thf(fact_60_equals0I, axiom,
    ((![A2 : set_Ho840737317_state]: ((![Y2 : hoare_958474565_state]: (~ ((member109514606_state @ Y2 @ A2)))) => (A2 = bot_bo105666705_state))))). % equals0I
thf(fact_61_equals0D, axiom,
    ((![A2 : set_Ho840737317_state, A : hoare_958474565_state]: ((A2 = bot_bo105666705_state) => (~ ((member109514606_state @ A @ A2))))))). % equals0D
thf(fact_62_emptyE, axiom,
    ((![A : hoare_958474565_state]: (~ ((member109514606_state @ A @ bot_bo105666705_state)))))). % emptyE
thf(fact_63_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_64_set__eq__subset, axiom,
    (((^[Y : set_Ho840737317_state]: (^[Z : set_Ho840737317_state]: (Y = Z))) = (^[A3 : set_Ho840737317_state]: (^[B2 : set_Ho840737317_state]: (((ord_le1945819589_state @ A3 @ B2)) & ((ord_le1945819589_state @ B2 @ A3)))))))). % set_eq_subset
thf(fact_65_subset__trans, axiom,
    ((![A2 : set_Ho840737317_state, B3 : set_Ho840737317_state, C2 : set_Ho840737317_state]: ((ord_le1945819589_state @ A2 @ B3) => ((ord_le1945819589_state @ B3 @ C2) => (ord_le1945819589_state @ A2 @ C2)))))). % subset_trans
thf(fact_66_Collect__mono, axiom,
    ((![P : hoare_958474565_state > $o, Q : hoare_958474565_state > $o]: ((![X3 : hoare_958474565_state]: ((P @ X3) => (Q @ X3))) => (ord_le1945819589_state @ (collec305460656_state @ P) @ (collec305460656_state @ Q)))))). % Collect_mono
thf(fact_67_subset__refl, axiom,
    ((![A2 : set_Ho840737317_state]: (ord_le1945819589_state @ A2 @ A2)))). % subset_refl
thf(fact_68_subset__iff, axiom,
    ((ord_le1945819589_state = (^[A3 : set_Ho840737317_state]: (^[B2 : set_Ho840737317_state]: (![T2 : hoare_958474565_state]: (((member109514606_state @ T2 @ A3)) => ((member109514606_state @ T2 @ B2))))))))). % subset_iff
thf(fact_69_equalityD2, axiom,
    ((![A2 : set_Ho840737317_state, B3 : set_Ho840737317_state]: ((A2 = B3) => (ord_le1945819589_state @ B3 @ A2))))). % equalityD2
thf(fact_70_equalityD1, axiom,
    ((![A2 : set_Ho840737317_state, B3 : set_Ho840737317_state]: ((A2 = B3) => (ord_le1945819589_state @ A2 @ B3))))). % equalityD1
thf(fact_71_subset__eq, axiom,
    ((ord_le1945819589_state = (^[A3 : set_Ho840737317_state]: (^[B2 : set_Ho840737317_state]: (![X : hoare_958474565_state]: (((member109514606_state @ X @ A3)) => ((member109514606_state @ X @ B2))))))))). % subset_eq
thf(fact_72_equalityE, axiom,
    ((![A2 : set_Ho840737317_state, B3 : set_Ho840737317_state]: ((A2 = B3) => (~ (((ord_le1945819589_state @ A2 @ B3) => (~ ((ord_le1945819589_state @ B3 @ A2)))))))))). % equalityE
thf(fact_73_subsetD, axiom,
    ((![A2 : set_Ho840737317_state, B3 : set_Ho840737317_state, C : hoare_958474565_state]: ((ord_le1945819589_state @ A2 @ B3) => ((member109514606_state @ C @ A2) => (member109514606_state @ C @ B3)))))). % subsetD
thf(fact_74_in__mono, axiom,
    ((![A2 : set_Ho840737317_state, B3 : set_Ho840737317_state, X2 : hoare_958474565_state]: ((ord_le1945819589_state @ A2 @ B3) => ((member109514606_state @ X2 @ A2) => (member109514606_state @ X2 @ B3)))))). % in_mono
thf(fact_75_mk__disjoint__insert, axiom,
    ((![A : hoare_958474565_state, A2 : set_Ho840737317_state]: ((member109514606_state @ A @ A2) => (?[B4 : set_Ho840737317_state]: ((A2 = (insert776267541_state @ A @ B4)) & (~ ((member109514606_state @ A @ B4))))))))). % mk_disjoint_insert
thf(fact_76_insert__commute, axiom,
    ((![X2 : hoare_958474565_state, Y3 : hoare_958474565_state, A2 : set_Ho840737317_state]: ((insert776267541_state @ X2 @ (insert776267541_state @ Y3 @ A2)) = (insert776267541_state @ Y3 @ (insert776267541_state @ X2 @ A2)))))). % insert_commute
thf(fact_77_insert__eq__iff, axiom,
    ((![A : hoare_958474565_state, A2 : set_Ho840737317_state, B : hoare_958474565_state, B3 : set_Ho840737317_state]: ((~ ((member109514606_state @ A @ A2))) => ((~ ((member109514606_state @ B @ B3))) => (((insert776267541_state @ A @ A2) = (insert776267541_state @ B @ B3)) = (((((A = B)) => ((A2 = B3)))) & ((((~ ((A = B)))) => ((?[C3 : set_Ho840737317_state]: (((A2 = (insert776267541_state @ B @ C3))) & ((((~ ((member109514606_state @ B @ C3)))) & ((((B3 = (insert776267541_state @ A @ C3))) & ((~ ((member109514606_state @ A @ C3)))))))))))))))))))). % insert_eq_iff
thf(fact_78_insert__absorb, axiom,
    ((![A : hoare_958474565_state, A2 : set_Ho840737317_state]: ((member109514606_state @ A @ A2) => ((insert776267541_state @ A @ A2) = A2))))). % insert_absorb
thf(fact_79_insert__ident, axiom,
    ((![X2 : hoare_958474565_state, A2 : set_Ho840737317_state, B3 : set_Ho840737317_state]: ((~ ((member109514606_state @ X2 @ A2))) => ((~ ((member109514606_state @ X2 @ B3))) => (((insert776267541_state @ X2 @ A2) = (insert776267541_state @ X2 @ B3)) = (A2 = B3))))))). % insert_ident
thf(fact_80_Set_Oset__insert, axiom,
    ((![X2 : hoare_958474565_state, A2 : set_Ho840737317_state]: ((member109514606_state @ X2 @ A2) => (~ ((![B4 : set_Ho840737317_state]: ((A2 = (insert776267541_state @ X2 @ B4)) => (member109514606_state @ X2 @ B4))))))))). % Set.set_insert
thf(fact_81_insertI2, axiom,
    ((![A : hoare_958474565_state, B3 : set_Ho840737317_state, B : hoare_958474565_state]: ((member109514606_state @ A @ B3) => (member109514606_state @ A @ (insert776267541_state @ B @ B3)))))). % insertI2
thf(fact_82_insertI1, axiom,
    ((![A : hoare_958474565_state, B3 : set_Ho840737317_state]: (member109514606_state @ A @ (insert776267541_state @ A @ B3))))). % insertI1
thf(fact_83_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_84_Compr__image__eq, axiom,
    ((![F : pname > hoare_958474565_state, A2 : set_pname, P : hoare_958474565_state > $o]: ((collec305460656_state @ (^[X : hoare_958474565_state]: (((member109514606_state @ X @ (image_2144627828_state @ F @ A2))) & ((P @ X))))) = (image_2144627828_state @ F @ (collect_pname @ (^[X : pname]: (((member_pname @ X @ A2)) & ((P @ (F @ X))))))))))). % Compr_image_eq
thf(fact_85_Compr__image__eq, axiom,
    ((![F : hoare_958474565_state > hoare_958474565_state, A2 : set_Ho840737317_state, P : hoare_958474565_state > $o]: ((collec305460656_state @ (^[X : hoare_958474565_state]: (((member109514606_state @ X @ (image_1849792670_state @ F @ A2))) & ((P @ X))))) = (image_1849792670_state @ F @ (collec305460656_state @ (^[X : hoare_958474565_state]: (((member109514606_state @ X @ A2)) & ((P @ (F @ X))))))))))). % Compr_image_eq
thf(fact_86_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 @ (^[X : pname]: (F @ (G3 @ X))) @ A2))))). % image_image
thf(fact_87_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 @ (^[X : pname]: (F @ (G3 @ X))) @ A2))))). % image_image
thf(fact_88_imageE, axiom,
    ((![B : hoare_958474565_state, F : pname > hoare_958474565_state, A2 : set_pname]: ((member109514606_state @ B @ (image_2144627828_state @ F @ A2)) => (~ ((![X3 : pname]: ((B = (F @ X3)) => (~ ((member_pname @ X3 @ A2))))))))))). % imageE
thf(fact_89_imageE, axiom,
    ((![B : hoare_958474565_state, F : hoare_958474565_state > hoare_958474565_state, A2 : set_Ho840737317_state]: ((member109514606_state @ B @ (image_1849792670_state @ F @ A2)) => (~ ((![X3 : hoare_958474565_state]: ((B = (F @ X3)) => (~ ((member109514606_state @ X3 @ A2))))))))))). % imageE
thf(fact_90_empty__def, axiom,
    ((bot_bo105666705_state = (collec305460656_state @ (^[X : hoare_958474565_state]: $false))))). % empty_def
thf(fact_91_pigeonhole__infinite__rel, axiom,
    ((![A2 : set_Ho840737317_state, B3 : set_Ho840737317_state, R : hoare_958474565_state > hoare_958474565_state > $o]: ((~ ((finite1986656878_state @ A2))) => ((finite1986656878_state @ B3) => ((![X3 : hoare_958474565_state]: ((member109514606_state @ X3 @ A2) => (?[Xa : hoare_958474565_state]: ((member109514606_state @ Xa @ B3) & (R @ X3 @ Xa))))) => (?[X3 : hoare_958474565_state]: ((member109514606_state @ X3 @ B3) & (~ ((finite1986656878_state @ (collec305460656_state @ (^[A4 : hoare_958474565_state]: (((member109514606_state @ A4 @ A2)) & ((R @ A4 @ X3)))))))))))))))). % pigeonhole_infinite_rel
thf(fact_92_pigeonhole__infinite__rel, axiom,
    ((![A2 : set_Ho840737317_state, B3 : set_pname, R : hoare_958474565_state > pname > $o]: ((~ ((finite1986656878_state @ A2))) => ((finite_finite_pname @ B3) => ((![X3 : hoare_958474565_state]: ((member109514606_state @ X3 @ A2) => (?[Xa : pname]: ((member_pname @ Xa @ B3) & (R @ X3 @ Xa))))) => (?[X3 : pname]: ((member_pname @ X3 @ B3) & (~ ((finite1986656878_state @ (collec305460656_state @ (^[A4 : hoare_958474565_state]: (((member109514606_state @ A4 @ A2)) & ((R @ A4 @ X3)))))))))))))))). % pigeonhole_infinite_rel
thf(fact_93_pigeonhole__infinite__rel, axiom,
    ((![A2 : set_pname, B3 : set_Ho840737317_state, R : pname > hoare_958474565_state > $o]: ((~ ((finite_finite_pname @ A2))) => ((finite1986656878_state @ B3) => ((![X3 : pname]: ((member_pname @ X3 @ A2) => (?[Xa : hoare_958474565_state]: ((member109514606_state @ Xa @ B3) & (R @ X3 @ Xa))))) => (?[X3 : hoare_958474565_state]: ((member109514606_state @ X3 @ B3) & (~ ((finite_finite_pname @ (collect_pname @ (^[A4 : pname]: (((member_pname @ A4 @ A2)) & ((R @ A4 @ X3)))))))))))))))). % pigeonhole_infinite_rel
thf(fact_94_pigeonhole__infinite__rel, axiom,
    ((![A2 : set_pname, B3 : set_pname, R : pname > pname > $o]: ((~ ((finite_finite_pname @ A2))) => ((finite_finite_pname @ B3) => ((![X3 : pname]: ((member_pname @ X3 @ A2) => (?[Xa : pname]: ((member_pname @ Xa @ B3) & (R @ X3 @ Xa))))) => (?[X3 : pname]: ((member_pname @ X3 @ B3) & (~ ((finite_finite_pname @ (collect_pname @ (^[A4 : pname]: (((member_pname @ A4 @ A2)) & ((R @ A4 @ X3)))))))))))))))). % pigeonhole_infinite_rel
thf(fact_95_not__finite__existsD, axiom,
    ((![P : hoare_958474565_state > $o]: ((~ ((finite1986656878_state @ (collec305460656_state @ P)))) => (?[X_1 : hoare_958474565_state]: (P @ X_1)))))). % not_finite_existsD
thf(fact_96_not__finite__existsD, axiom,
    ((![P : pname > $o]: ((~ ((finite_finite_pname @ (collect_pname @ P)))) => (?[X_1 : pname]: (P @ X_1)))))). % not_finite_existsD
thf(fact_97_Collect__subset, axiom,
    ((![A2 : set_Ho840737317_state, P : hoare_958474565_state > $o]: (ord_le1945819589_state @ (collec305460656_state @ (^[X : hoare_958474565_state]: (((member109514606_state @ X @ A2)) & ((P @ X))))) @ A2)))). % Collect_subset
thf(fact_98_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_99_insert__compr, axiom,
    ((insert776267541_state = (^[A4 : hoare_958474565_state]: (^[B2 : set_Ho840737317_state]: (collec305460656_state @ (^[X : hoare_958474565_state]: (((X = A4)) | ((member109514606_state @ X @ B2)))))))))). % insert_compr
thf(fact_100_finite__has__minimal2, axiom,
    ((![A2 : set_se421025541_state, A : set_Ho840737317_state]: ((finite1266962254_state @ A2) => ((member1959617614_state @ A @ A2) => (?[X3 : set_Ho840737317_state]: ((member1959617614_state @ X3 @ A2) & ((ord_le1945819589_state @ X3 @ A) & (![Xa : set_Ho840737317_state]: ((member1959617614_state @ Xa @ A2) => ((ord_le1945819589_state @ Xa @ X3) => (X3 = Xa)))))))))))). % finite_has_minimal2
thf(fact_101_finite__has__maximal2, axiom,
    ((![A2 : set_se421025541_state, A : set_Ho840737317_state]: ((finite1266962254_state @ A2) => ((member1959617614_state @ A @ A2) => (?[X3 : set_Ho840737317_state]: ((member1959617614_state @ X3 @ A2) & ((ord_le1945819589_state @ A @ X3) & (![Xa : set_Ho840737317_state]: ((member1959617614_state @ Xa @ A2) => ((ord_le1945819589_state @ X3 @ Xa) => (X3 = Xa)))))))))))). % finite_has_maximal2
thf(fact_102_infinite__imp__nonempty, axiom,
    ((![S : set_pname]: ((~ ((finite_finite_pname @ S))) => (~ ((S = bot_bot_set_pname))))))). % infinite_imp_nonempty
thf(fact_103_infinite__imp__nonempty, axiom,
    ((![S : set_Ho840737317_state]: ((~ ((finite1986656878_state @ S))) => (~ ((S = bot_bo105666705_state))))))). % infinite_imp_nonempty
thf(fact_104_finite_OemptyI, axiom,
    ((finite_finite_pname @ bot_bot_set_pname))). % finite.emptyI
thf(fact_105_finite_OemptyI, axiom,
    ((finite1986656878_state @ bot_bo105666705_state))). % finite.emptyI
thf(fact_106_all__subset__image, axiom,
    ((![F : pname > hoare_958474565_state, A2 : set_pname, P : set_Ho840737317_state > $o]: ((![B2 : set_Ho840737317_state]: (((ord_le1945819589_state @ B2 @ (image_2144627828_state @ F @ A2))) => ((P @ B2)))) = (![B2 : set_pname]: (((ord_le865024672_pname @ B2 @ A2)) => ((P @ (image_2144627828_state @ F @ B2))))))))). % all_subset_image
thf(fact_107_all__subset__image, axiom,
    ((![F : hoare_958474565_state > hoare_958474565_state, A2 : set_Ho840737317_state, P : set_Ho840737317_state > $o]: ((![B2 : set_Ho840737317_state]: (((ord_le1945819589_state @ B2 @ (image_1849792670_state @ F @ A2))) => ((P @ B2)))) = (![B2 : set_Ho840737317_state]: (((ord_le1945819589_state @ B2 @ A2)) => ((P @ (image_1849792670_state @ F @ B2))))))))). % all_subset_image
thf(fact_108_subset__image__iff, axiom,
    ((![B3 : set_Ho840737317_state, F : pname > hoare_958474565_state, A2 : set_pname]: ((ord_le1945819589_state @ B3 @ (image_2144627828_state @ F @ A2)) = (?[AA : set_pname]: (((ord_le865024672_pname @ AA @ A2)) & ((B3 = (image_2144627828_state @ F @ AA))))))))). % subset_image_iff
thf(fact_109_subset__image__iff, axiom,
    ((![B3 : set_Ho840737317_state, F : hoare_958474565_state > hoare_958474565_state, A2 : set_Ho840737317_state]: ((ord_le1945819589_state @ B3 @ (image_1849792670_state @ F @ A2)) = (?[AA : set_Ho840737317_state]: (((ord_le1945819589_state @ AA @ A2)) & ((B3 = (image_1849792670_state @ F @ AA))))))))). % subset_image_iff
thf(fact_110_image__subset__iff, axiom,
    ((![F : pname > hoare_958474565_state, A2 : set_pname, B3 : set_Ho840737317_state]: ((ord_le1945819589_state @ (image_2144627828_state @ F @ A2) @ B3) = (![X : pname]: (((member_pname @ X @ A2)) => ((member109514606_state @ (F @ X) @ B3)))))))). % image_subset_iff
thf(fact_111_subset__imageE, axiom,
    ((![B3 : set_Ho840737317_state, F : pname > hoare_958474565_state, A2 : set_pname]: ((ord_le1945819589_state @ B3 @ (image_2144627828_state @ F @ A2)) => (~ ((![C4 : set_pname]: ((ord_le865024672_pname @ C4 @ A2) => (~ ((B3 = (image_2144627828_state @ F @ C4)))))))))))). % subset_imageE
thf(fact_112_subset__imageE, axiom,
    ((![B3 : set_Ho840737317_state, F : hoare_958474565_state > hoare_958474565_state, A2 : set_Ho840737317_state]: ((ord_le1945819589_state @ B3 @ (image_1849792670_state @ F @ A2)) => (~ ((![C4 : set_Ho840737317_state]: ((ord_le1945819589_state @ C4 @ A2) => (~ ((B3 = (image_1849792670_state @ F @ C4)))))))))))). % subset_imageE
thf(fact_113_image__subsetI, axiom,
    ((![A2 : set_pname, F : pname > hoare_958474565_state, B3 : set_Ho840737317_state]: ((![X3 : pname]: ((member_pname @ X3 @ A2) => (member109514606_state @ (F @ X3) @ B3))) => (ord_le1945819589_state @ (image_2144627828_state @ F @ A2) @ B3))))). % image_subsetI
thf(fact_114_image__subsetI, axiom,
    ((![A2 : set_Ho840737317_state, F : hoare_958474565_state > hoare_958474565_state, B3 : set_Ho840737317_state]: ((![X3 : hoare_958474565_state]: ((member109514606_state @ X3 @ A2) => (member109514606_state @ (F @ X3) @ B3))) => (ord_le1945819589_state @ (image_1849792670_state @ F @ A2) @ B3))))). % image_subsetI
thf(fact_115_image__mono, axiom,
    ((![A2 : set_pname, B3 : set_pname, F : pname > hoare_958474565_state]: ((ord_le865024672_pname @ A2 @ B3) => (ord_le1945819589_state @ (image_2144627828_state @ F @ A2) @ (image_2144627828_state @ F @ B3)))))). % image_mono
thf(fact_116_image__mono, axiom,
    ((![A2 : set_Ho840737317_state, B3 : set_Ho840737317_state, F : hoare_958474565_state > hoare_958474565_state]: ((ord_le1945819589_state @ A2 @ B3) => (ord_le1945819589_state @ (image_1849792670_state @ F @ A2) @ (image_1849792670_state @ F @ B3)))))). % image_mono
thf(fact_117_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_118_insert__not__empty, axiom,
    ((![A : hoare_958474565_state, A2 : set_Ho840737317_state]: (~ (((insert776267541_state @ A @ A2) = bot_bo105666705_state)))))). % insert_not_empty
thf(fact_119_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_120_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_121_singletonD, axiom,
    ((![B : hoare_958474565_state, A : hoare_958474565_state]: ((member109514606_state @ B @ (insert776267541_state @ A @ bot_bo105666705_state)) => (B = A))))). % singletonD
thf(fact_122_rev__finite__subset, axiom,
    ((![B3 : set_pname, A2 : set_pname]: ((finite_finite_pname @ B3) => ((ord_le865024672_pname @ A2 @ B3) => (finite_finite_pname @ A2)))))). % rev_finite_subset
thf(fact_123_rev__finite__subset, axiom,
    ((![B3 : set_Ho840737317_state, A2 : set_Ho840737317_state]: ((finite1986656878_state @ B3) => ((ord_le1945819589_state @ A2 @ B3) => (finite1986656878_state @ A2)))))). % rev_finite_subset
thf(fact_124_infinite__super, axiom,
    ((![S : set_pname, T3 : set_pname]: ((ord_le865024672_pname @ S @ T3) => ((~ ((finite_finite_pname @ S))) => (~ ((finite_finite_pname @ T3)))))))). % infinite_super
thf(fact_125_infinite__super, axiom,
    ((![S : set_Ho840737317_state, T3 : set_Ho840737317_state]: ((ord_le1945819589_state @ S @ T3) => ((~ ((finite1986656878_state @ S))) => (~ ((finite1986656878_state @ T3)))))))). % infinite_super
thf(fact_126_finite__subset, axiom,
    ((![A2 : set_pname, B3 : set_pname]: ((ord_le865024672_pname @ A2 @ B3) => ((finite_finite_pname @ B3) => (finite_finite_pname @ A2)))))). % finite_subset
thf(fact_127_finite__subset, axiom,
    ((![A2 : set_Ho840737317_state, B3 : set_Ho840737317_state]: ((ord_le1945819589_state @ A2 @ B3) => ((finite1986656878_state @ B3) => (finite1986656878_state @ A2)))))). % finite_subset
thf(fact_128_finite_OinsertI, axiom,
    ((![A2 : set_Ho840737317_state, A : hoare_958474565_state]: ((finite1986656878_state @ A2) => (finite1986656878_state @ (insert776267541_state @ A @ A2)))))). % finite.insertI
thf(fact_129_finite_OinsertI, axiom,
    ((![A2 : set_pname, A : pname]: ((finite_finite_pname @ A2) => (finite_finite_pname @ (insert_pname @ A @ A2)))))). % finite.insertI
thf(fact_130_subset__insertI2, axiom,
    ((![A2 : set_Ho840737317_state, B3 : set_Ho840737317_state, B : hoare_958474565_state]: ((ord_le1945819589_state @ A2 @ B3) => (ord_le1945819589_state @ A2 @ (insert776267541_state @ B @ B3)))))). % subset_insertI2
thf(fact_131_subset__insertI, axiom,
    ((![B3 : set_Ho840737317_state, A : hoare_958474565_state]: (ord_le1945819589_state @ B3 @ (insert776267541_state @ A @ B3))))). % subset_insertI
thf(fact_132_subset__insert, axiom,
    ((![X2 : hoare_958474565_state, A2 : set_Ho840737317_state, B3 : set_Ho840737317_state]: ((~ ((member109514606_state @ X2 @ A2))) => ((ord_le1945819589_state @ A2 @ (insert776267541_state @ X2 @ B3)) = (ord_le1945819589_state @ A2 @ B3)))))). % subset_insert
thf(fact_133_insert__mono, axiom,
    ((![C2 : set_Ho840737317_state, D2 : set_Ho840737317_state, A : hoare_958474565_state]: ((ord_le1945819589_state @ C2 @ D2) => (ord_le1945819589_state @ (insert776267541_state @ A @ C2) @ (insert776267541_state @ A @ D2)))))). % insert_mono
thf(fact_134_pigeonhole__infinite, axiom,
    ((![A2 : set_Ho840737317_state, F : hoare_958474565_state > hoare_958474565_state]: ((~ ((finite1986656878_state @ A2))) => ((finite1986656878_state @ (image_1849792670_state @ F @ A2)) => (?[X3 : hoare_958474565_state]: ((member109514606_state @ X3 @ A2) & (~ ((finite1986656878_state @ (collec305460656_state @ (^[A4 : hoare_958474565_state]: (((member109514606_state @ A4 @ A2)) & (((F @ A4) = (F @ X3)))))))))))))))). % pigeonhole_infinite
thf(fact_135_pigeonhole__infinite, axiom,
    ((![A2 : set_Ho840737317_state, F : hoare_958474565_state > pname]: ((~ ((finite1986656878_state @ A2))) => ((finite_finite_pname @ (image_871391498_pname @ F @ A2)) => (?[X3 : hoare_958474565_state]: ((member109514606_state @ X3 @ A2) & (~ ((finite1986656878_state @ (collec305460656_state @ (^[A4 : hoare_958474565_state]: (((member109514606_state @ A4 @ A2)) & (((F @ A4) = (F @ X3)))))))))))))))). % pigeonhole_infinite
thf(fact_136_pigeonhole__infinite, axiom,
    ((![A2 : set_pname, F : pname > hoare_958474565_state]: ((~ ((finite_finite_pname @ A2))) => ((finite1986656878_state @ (image_2144627828_state @ F @ A2)) => (?[X3 : pname]: ((member_pname @ X3 @ A2) & (~ ((finite_finite_pname @ (collect_pname @ (^[A4 : pname]: (((member_pname @ A4 @ A2)) & (((F @ A4) = (F @ X3)))))))))))))))). % pigeonhole_infinite
thf(fact_137_pigeonhole__infinite, axiom,
    ((![A2 : set_pname, F : pname > pname]: ((~ ((finite_finite_pname @ A2))) => ((finite_finite_pname @ (image_pname_pname @ F @ A2)) => (?[X3 : pname]: ((member_pname @ X3 @ A2) & (~ ((finite_finite_pname @ (collect_pname @ (^[A4 : pname]: (((member_pname @ A4 @ A2)) & (((F @ A4) = (F @ X3)))))))))))))))). % pigeonhole_infinite
thf(fact_138_Collect__conv__if2, axiom,
    ((![P : hoare_958474565_state > $o, A : hoare_958474565_state]: (((P @ A) => ((collec305460656_state @ (^[X : hoare_958474565_state]: (((A = X)) & ((P @ X))))) = (insert776267541_state @ A @ bot_bo105666705_state))) & ((~ ((P @ A))) => ((collec305460656_state @ (^[X : hoare_958474565_state]: (((A = X)) & ((P @ X))))) = bot_bo105666705_state)))))). % Collect_conv_if2
thf(fact_139_Collect__conv__if, axiom,
    ((![P : hoare_958474565_state > $o, A : hoare_958474565_state]: (((P @ A) => ((collec305460656_state @ (^[X : hoare_958474565_state]: (((X = A)) & ((P @ X))))) = (insert776267541_state @ A @ bot_bo105666705_state))) & ((~ ((P @ A))) => ((collec305460656_state @ (^[X : hoare_958474565_state]: (((X = A)) & ((P @ X))))) = bot_bo105666705_state)))))). % Collect_conv_if
thf(fact_140_finite__has__minimal, axiom,
    ((![A2 : set_se421025541_state]: ((finite1266962254_state @ A2) => ((~ ((A2 = bot_bo249060721_state))) => (?[X3 : set_Ho840737317_state]: ((member1959617614_state @ X3 @ A2) & (![Xa : set_Ho840737317_state]: ((member1959617614_state @ Xa @ A2) => ((ord_le1945819589_state @ Xa @ X3) => (X3 = Xa))))))))))). % finite_has_minimal
thf(fact_141_finite__has__maximal, axiom,
    ((![A2 : set_se421025541_state]: ((finite1266962254_state @ A2) => ((~ ((A2 = bot_bo249060721_state))) => (?[X3 : set_Ho840737317_state]: ((member1959617614_state @ X3 @ A2) & (![Xa : set_Ho840737317_state]: ((member1959617614_state @ Xa @ A2) => ((ord_le1945819589_state @ X3 @ Xa) => (X3 = Xa))))))))))). % finite_has_maximal
thf(fact_142_all__finite__subset__image, axiom,
    ((![F : pname > pname, A2 : set_pname, P : set_pname > $o]: ((![B2 : set_pname]: (((((finite_finite_pname @ B2)) & ((ord_le865024672_pname @ B2 @ (image_pname_pname @ F @ A2))))) => ((P @ B2)))) = (![B2 : set_pname]: (((((finite_finite_pname @ B2)) & ((ord_le865024672_pname @ B2 @ A2)))) => ((P @ (image_pname_pname @ F @ B2))))))))). % all_finite_subset_image
thf(fact_143_all__finite__subset__image, axiom,
    ((![F : hoare_958474565_state > pname, A2 : set_Ho840737317_state, P : set_pname > $o]: ((![B2 : set_pname]: (((((finite_finite_pname @ B2)) & ((ord_le865024672_pname @ B2 @ (image_871391498_pname @ F @ A2))))) => ((P @ B2)))) = (![B2 : set_Ho840737317_state]: (((((finite1986656878_state @ B2)) & ((ord_le1945819589_state @ B2 @ A2)))) => ((P @ (image_871391498_pname @ F @ B2))))))))). % all_finite_subset_image
thf(fact_144_all__finite__subset__image, axiom,
    ((![F : pname > hoare_958474565_state, A2 : set_pname, P : set_Ho840737317_state > $o]: ((![B2 : set_Ho840737317_state]: (((((finite1986656878_state @ B2)) & ((ord_le1945819589_state @ B2 @ (image_2144627828_state @ F @ A2))))) => ((P @ B2)))) = (![B2 : set_pname]: (((((finite_finite_pname @ B2)) & ((ord_le865024672_pname @ B2 @ A2)))) => ((P @ (image_2144627828_state @ F @ B2))))))))). % all_finite_subset_image
thf(fact_145_all__finite__subset__image, axiom,
    ((![F : hoare_958474565_state > hoare_958474565_state, A2 : set_Ho840737317_state, P : set_Ho840737317_state > $o]: ((![B2 : set_Ho840737317_state]: (((((finite1986656878_state @ B2)) & ((ord_le1945819589_state @ B2 @ (image_1849792670_state @ F @ A2))))) => ((P @ B2)))) = (![B2 : set_Ho840737317_state]: (((((finite1986656878_state @ B2)) & ((ord_le1945819589_state @ B2 @ A2)))) => ((P @ (image_1849792670_state @ F @ B2))))))))). % all_finite_subset_image
thf(fact_146_ex__finite__subset__image, axiom,
    ((![F : pname > pname, A2 : set_pname, P : set_pname > $o]: ((?[B2 : set_pname]: (((finite_finite_pname @ B2)) & ((((ord_le865024672_pname @ B2 @ (image_pname_pname @ F @ A2))) & ((P @ B2)))))) = (?[B2 : set_pname]: (((finite_finite_pname @ B2)) & ((((ord_le865024672_pname @ B2 @ A2)) & ((P @ (image_pname_pname @ F @ B2))))))))))). % ex_finite_subset_image
thf(fact_147_ex__finite__subset__image, axiom,
    ((![F : hoare_958474565_state > pname, A2 : set_Ho840737317_state, P : set_pname > $o]: ((?[B2 : set_pname]: (((finite_finite_pname @ B2)) & ((((ord_le865024672_pname @ B2 @ (image_871391498_pname @ F @ A2))) & ((P @ B2)))))) = (?[B2 : set_Ho840737317_state]: (((finite1986656878_state @ B2)) & ((((ord_le1945819589_state @ B2 @ A2)) & ((P @ (image_871391498_pname @ F @ B2))))))))))). % ex_finite_subset_image
thf(fact_148_ex__finite__subset__image, axiom,
    ((![F : pname > hoare_958474565_state, A2 : set_pname, P : set_Ho840737317_state > $o]: ((?[B2 : set_Ho840737317_state]: (((finite1986656878_state @ B2)) & ((((ord_le1945819589_state @ B2 @ (image_2144627828_state @ F @ A2))) & ((P @ B2)))))) = (?[B2 : set_pname]: (((finite_finite_pname @ B2)) & ((((ord_le865024672_pname @ B2 @ A2)) & ((P @ (image_2144627828_state @ F @ B2))))))))))). % ex_finite_subset_image
thf(fact_149_ex__finite__subset__image, axiom,
    ((![F : hoare_958474565_state > hoare_958474565_state, A2 : set_Ho840737317_state, P : set_Ho840737317_state > $o]: ((?[B2 : set_Ho840737317_state]: (((finite1986656878_state @ B2)) & ((((ord_le1945819589_state @ B2 @ (image_1849792670_state @ F @ A2))) & ((P @ B2)))))) = (?[B2 : set_Ho840737317_state]: (((finite1986656878_state @ B2)) & ((((ord_le1945819589_state @ B2 @ A2)) & ((P @ (image_1849792670_state @ F @ B2))))))))))). % ex_finite_subset_image
thf(fact_150_finite__subset__image, axiom,
    ((![B3 : set_pname, F : pname > pname, A2 : set_pname]: ((finite_finite_pname @ B3) => ((ord_le865024672_pname @ B3 @ (image_pname_pname @ F @ A2)) => (?[C4 : set_pname]: ((ord_le865024672_pname @ C4 @ A2) & ((finite_finite_pname @ C4) & (B3 = (image_pname_pname @ F @ C4)))))))))). % finite_subset_image
thf(fact_151_finite__subset__image, axiom,
    ((![B3 : set_pname, F : hoare_958474565_state > pname, A2 : set_Ho840737317_state]: ((finite_finite_pname @ B3) => ((ord_le865024672_pname @ B3 @ (image_871391498_pname @ F @ A2)) => (?[C4 : set_Ho840737317_state]: ((ord_le1945819589_state @ C4 @ A2) & ((finite1986656878_state @ C4) & (B3 = (image_871391498_pname @ F @ C4)))))))))). % finite_subset_image
thf(fact_152_finite__subset__image, axiom,
    ((![B3 : set_Ho840737317_state, F : pname > hoare_958474565_state, A2 : set_pname]: ((finite1986656878_state @ B3) => ((ord_le1945819589_state @ B3 @ (image_2144627828_state @ F @ A2)) => (?[C4 : set_pname]: ((ord_le865024672_pname @ C4 @ A2) & ((finite_finite_pname @ C4) & (B3 = (image_2144627828_state @ F @ C4)))))))))). % finite_subset_image
thf(fact_153_finite__subset__image, axiom,
    ((![B3 : set_Ho840737317_state, F : hoare_958474565_state > hoare_958474565_state, A2 : set_Ho840737317_state]: ((finite1986656878_state @ B3) => ((ord_le1945819589_state @ B3 @ (image_1849792670_state @ F @ A2)) => (?[C4 : set_Ho840737317_state]: ((ord_le1945819589_state @ C4 @ A2) & ((finite1986656878_state @ C4) & (B3 = (image_1849792670_state @ F @ C4)))))))))). % finite_subset_image
thf(fact_154_finite__surj, axiom,
    ((![A2 : set_Ho840737317_state, B3 : set_pname, F : hoare_958474565_state > pname]: ((finite1986656878_state @ A2) => ((ord_le865024672_pname @ B3 @ (image_871391498_pname @ F @ A2)) => (finite_finite_pname @ B3)))))). % finite_surj
thf(fact_155_finite__surj, axiom,
    ((![A2 : set_pname, B3 : set_pname, F : pname > pname]: ((finite_finite_pname @ A2) => ((ord_le865024672_pname @ B3 @ (image_pname_pname @ F @ A2)) => (finite_finite_pname @ B3)))))). % finite_surj
thf(fact_156_finite__surj, axiom,
    ((![A2 : set_Ho840737317_state, B3 : set_Ho840737317_state, F : hoare_958474565_state > hoare_958474565_state]: ((finite1986656878_state @ A2) => ((ord_le1945819589_state @ B3 @ (image_1849792670_state @ F @ A2)) => (finite1986656878_state @ B3)))))). % finite_surj
thf(fact_157_finite__surj, axiom,
    ((![A2 : set_pname, B3 : set_Ho840737317_state, F : pname > hoare_958474565_state]: ((finite_finite_pname @ A2) => ((ord_le1945819589_state @ B3 @ (image_2144627828_state @ F @ A2)) => (finite1986656878_state @ B3)))))). % finite_surj
thf(fact_158_infinite__finite__induct, axiom,
    ((![P : set_pname > $o, A2 : set_pname]: ((![A5 : set_pname]: ((~ ((finite_finite_pname @ A5))) => (P @ A5))) => ((P @ bot_bot_set_pname) => ((![X3 : pname, F3 : set_pname]: ((finite_finite_pname @ F3) => ((~ ((member_pname @ X3 @ F3))) => ((P @ F3) => (P @ (insert_pname @ X3 @ F3)))))) => (P @ A2))))))). % infinite_finite_induct
thf(fact_159_infinite__finite__induct, axiom,
    ((![P : set_Ho840737317_state > $o, A2 : set_Ho840737317_state]: ((![A5 : set_Ho840737317_state]: ((~ ((finite1986656878_state @ A5))) => (P @ A5))) => ((P @ bot_bo105666705_state) => ((![X3 : hoare_958474565_state, F3 : set_Ho840737317_state]: ((finite1986656878_state @ F3) => ((~ ((member109514606_state @ X3 @ F3))) => ((P @ F3) => (P @ (insert776267541_state @ X3 @ F3)))))) => (P @ A2))))))). % infinite_finite_induct
thf(fact_160_finite__ne__induct, axiom,
    ((![F2 : set_pname, P : set_pname > $o]: ((finite_finite_pname @ F2) => ((~ ((F2 = bot_bot_set_pname))) => ((![X3 : pname]: (P @ (insert_pname @ X3 @ bot_bot_set_pname))) => ((![X3 : pname, F3 : set_pname]: ((finite_finite_pname @ F3) => ((~ ((F3 = bot_bot_set_pname))) => ((~ ((member_pname @ X3 @ F3))) => ((P @ F3) => (P @ (insert_pname @ X3 @ F3))))))) => (P @ F2)))))))). % finite_ne_induct
thf(fact_161_finite__ne__induct, axiom,
    ((![F2 : set_Ho840737317_state, P : set_Ho840737317_state > $o]: ((finite1986656878_state @ F2) => ((~ ((F2 = bot_bo105666705_state))) => ((![X3 : hoare_958474565_state]: (P @ (insert776267541_state @ X3 @ bot_bo105666705_state))) => ((![X3 : hoare_958474565_state, F3 : set_Ho840737317_state]: ((finite1986656878_state @ F3) => ((~ ((F3 = bot_bo105666705_state))) => ((~ ((member109514606_state @ X3 @ F3))) => ((P @ F3) => (P @ (insert776267541_state @ X3 @ F3))))))) => (P @ F2)))))))). % finite_ne_induct
thf(fact_162_finite_Oinducts, axiom,
    ((![X2 : set_pname, P : set_pname > $o]: ((finite_finite_pname @ X2) => ((P @ bot_bot_set_pname) => ((![A5 : set_pname, A6 : pname]: ((finite_finite_pname @ A5) => ((P @ A5) => (P @ (insert_pname @ A6 @ A5))))) => (P @ X2))))))). % finite.inducts
thf(fact_163_finite_Oinducts, axiom,
    ((![X2 : set_Ho840737317_state, P : set_Ho840737317_state > $o]: ((finite1986656878_state @ X2) => ((P @ bot_bo105666705_state) => ((![A5 : set_Ho840737317_state, A6 : hoare_958474565_state]: ((finite1986656878_state @ A5) => ((P @ A5) => (P @ (insert776267541_state @ A6 @ A5))))) => (P @ X2))))))). % finite.inducts
thf(fact_164_finite__induct, axiom,
    ((![F2 : set_pname, P : set_pname > $o]: ((finite_finite_pname @ F2) => ((P @ bot_bot_set_pname) => ((![X3 : pname, F3 : set_pname]: ((finite_finite_pname @ F3) => ((~ ((member_pname @ X3 @ F3))) => ((P @ F3) => (P @ (insert_pname @ X3 @ F3)))))) => (P @ F2))))))). % finite_induct
thf(fact_165_finite__induct, axiom,
    ((![F2 : set_Ho840737317_state, P : set_Ho840737317_state > $o]: ((finite1986656878_state @ F2) => ((P @ bot_bo105666705_state) => ((![X3 : hoare_958474565_state, F3 : set_Ho840737317_state]: ((finite1986656878_state @ F3) => ((~ ((member109514606_state @ X3 @ F3))) => ((P @ F3) => (P @ (insert776267541_state @ X3 @ F3)))))) => (P @ F2))))))). % finite_induct
thf(fact_166_finite_Osimps, axiom,
    ((finite_finite_pname = (^[A4 : set_pname]: (((A4 = bot_bot_set_pname)) | ((?[A3 : set_pname]: (?[B5 : pname]: (((A4 = (insert_pname @ B5 @ A3))) & ((finite_finite_pname @ A3))))))))))). % finite.simps
thf(fact_167_finite_Osimps, axiom,
    ((finite1986656878_state = (^[A4 : set_Ho840737317_state]: (((A4 = bot_bo105666705_state)) | ((?[A3 : set_Ho840737317_state]: (?[B5 : hoare_958474565_state]: (((A4 = (insert776267541_state @ B5 @ A3))) & ((finite1986656878_state @ A3))))))))))). % finite.simps
thf(fact_168_finite_Ocases, axiom,
    ((![A : set_pname]: ((finite_finite_pname @ A) => ((~ ((A = bot_bot_set_pname))) => (~ ((![A5 : set_pname]: ((?[A6 : pname]: (A = (insert_pname @ A6 @ A5))) => (~ ((finite_finite_pname @ A5)))))))))))). % finite.cases
thf(fact_169_finite_Ocases, axiom,
    ((![A : set_Ho840737317_state]: ((finite1986656878_state @ A) => ((~ ((A = bot_bo105666705_state))) => (~ ((![A5 : set_Ho840737317_state]: ((?[A6 : hoare_958474565_state]: (A = (insert776267541_state @ A6 @ A5))) => (~ ((finite1986656878_state @ A5)))))))))))). % finite.cases
thf(fact_170_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_171_subset__singletonD, axiom,
    ((![A2 : set_Ho840737317_state, X2 : hoare_958474565_state]: ((ord_le1945819589_state @ A2 @ (insert776267541_state @ X2 @ bot_bo105666705_state)) => ((A2 = bot_bo105666705_state) | (A2 = (insert776267541_state @ X2 @ bot_bo105666705_state))))))). % subset_singletonD
thf(fact_172_image__constant__conv, axiom,
    ((![A2 : set_pname, C : hoare_958474565_state]: (((A2 = bot_bot_set_pname) => ((image_2144627828_state @ (^[X : pname]: C) @ A2) = bot_bo105666705_state)) & ((~ ((A2 = bot_bot_set_pname))) => ((image_2144627828_state @ (^[X : pname]: C) @ A2) = (insert776267541_state @ C @ bot_bo105666705_state))))))). % image_constant_conv
thf(fact_173_image__constant__conv, axiom,
    ((![A2 : set_Ho840737317_state, C : hoare_958474565_state]: (((A2 = bot_bo105666705_state) => ((image_1849792670_state @ (^[X : hoare_958474565_state]: C) @ A2) = bot_bo105666705_state)) & ((~ ((A2 = bot_bo105666705_state))) => ((image_1849792670_state @ (^[X : hoare_958474565_state]: C) @ A2) = (insert776267541_state @ C @ bot_bo105666705_state))))))). % image_constant_conv
thf(fact_174_image__constant, axiom,
    ((![X2 : pname, A2 : set_pname, C : hoare_958474565_state]: ((member_pname @ X2 @ A2) => ((image_2144627828_state @ (^[X : pname]: C) @ A2) = (insert776267541_state @ C @ bot_bo105666705_state)))))). % image_constant
thf(fact_175_image__constant, axiom,
    ((![X2 : hoare_958474565_state, A2 : set_Ho840737317_state, C : hoare_958474565_state]: ((member109514606_state @ X2 @ A2) => ((image_1849792670_state @ (^[X : hoare_958474565_state]: C) @ A2) = (insert776267541_state @ C @ bot_bo105666705_state)))))). % image_constant
thf(fact_176_finite__subset__induct_H, axiom,
    ((![F2 : set_pname, A2 : set_pname, P : set_pname > $o]: ((finite_finite_pname @ F2) => ((ord_le865024672_pname @ F2 @ A2) => ((P @ bot_bot_set_pname) => ((![A6 : pname, F3 : set_pname]: ((finite_finite_pname @ F3) => ((member_pname @ A6 @ A2) => ((ord_le865024672_pname @ F3 @ A2) => ((~ ((member_pname @ A6 @ F3))) => ((P @ F3) => (P @ (insert_pname @ A6 @ F3)))))))) => (P @ F2)))))))). % finite_subset_induct'
thf(fact_177_finite__subset__induct_H, axiom,
    ((![F2 : set_Ho840737317_state, A2 : set_Ho840737317_state, P : set_Ho840737317_state > $o]: ((finite1986656878_state @ F2) => ((ord_le1945819589_state @ F2 @ A2) => ((P @ bot_bo105666705_state) => ((![A6 : hoare_958474565_state, F3 : set_Ho840737317_state]: ((finite1986656878_state @ F3) => ((member109514606_state @ A6 @ A2) => ((ord_le1945819589_state @ F3 @ A2) => ((~ ((member109514606_state @ A6 @ F3))) => ((P @ F3) => (P @ (insert776267541_state @ A6 @ F3)))))))) => (P @ F2)))))))). % finite_subset_induct'
thf(fact_178_finite__subset__induct, axiom,
    ((![F2 : set_pname, A2 : set_pname, P : set_pname > $o]: ((finite_finite_pname @ F2) => ((ord_le865024672_pname @ F2 @ A2) => ((P @ bot_bot_set_pname) => ((![A6 : pname, F3 : set_pname]: ((finite_finite_pname @ F3) => ((member_pname @ A6 @ A2) => ((~ ((member_pname @ A6 @ F3))) => ((P @ F3) => (P @ (insert_pname @ A6 @ F3))))))) => (P @ F2)))))))). % finite_subset_induct
thf(fact_179_finite__subset__induct, axiom,
    ((![F2 : set_Ho840737317_state, A2 : set_Ho840737317_state, P : set_Ho840737317_state > $o]: ((finite1986656878_state @ F2) => ((ord_le1945819589_state @ F2 @ A2) => ((P @ bot_bo105666705_state) => ((![A6 : hoare_958474565_state, F3 : set_Ho840737317_state]: ((finite1986656878_state @ F3) => ((member109514606_state @ A6 @ A2) => ((~ ((member109514606_state @ A6 @ F3))) => ((P @ F3) => (P @ (insert776267541_state @ A6 @ F3))))))) => (P @ F2)))))))). % finite_subset_induct
thf(fact_180_finite__dom__body, axiom,
    ((finite_finite_pname @ (dom_pname_com @ body)))). % finite_dom_body
thf(fact_181_insert__dom, axiom,
    ((![F : pname > option_com, X2 : pname, Y3 : com]: (((F @ X2) = (some_com @ Y3)) => ((insert_pname @ X2 @ (dom_pname_com @ F)) = (dom_pname_com @ F)))))). % insert_dom
thf(fact_182_insert__dom, axiom,
    ((![F : hoare_958474565_state > option_com, X2 : hoare_958474565_state, Y3 : com]: (((F @ X2) = (some_com @ Y3)) => ((insert776267541_state @ X2 @ (dom_Ho256784515te_com @ F)) = (dom_Ho256784515te_com @ F)))))). % insert_dom
thf(fact_183_com_Oinject_I6_J, axiom,
    ((![X7 : pname, Y7 : pname]: (((body2 @ X7) = (body2 @ Y7)) = (X7 = Y7))))). % com.inject(6)
thf(fact_184_option_Oinject, axiom,
    ((![X22 : com, Y22 : com]: (((some_com @ X22) = (some_com @ Y22)) = (X22 = Y22))))). % option.inject
thf(fact_185_the__elem__eq, axiom,
    ((![X2 : hoare_958474565_state]: ((the_el1300254266_state @ (insert776267541_state @ X2 @ bot_bo105666705_state)) = X2)))). % the_elem_eq
thf(fact_186_MGT__Body, axiom,
    ((![G2 : set_Ho840737317_state, Procs : set_pname]: ((hoare_604442164_state @ (sup_su737426425_state @ G2 @ (image_2144627828_state @ (^[Pn2 : pname]: (hoare_Mirabelle_MGT @ (body2 @ Pn2))) @ Procs)) @ (image_2144627828_state @ (^[Pn2 : pname]: (hoare_Mirabelle_MGT @ (the_com @ (body @ Pn2)))) @ Procs)) => ((finite_finite_pname @ Procs) => (hoare_604442164_state @ G2 @ (image_2144627828_state @ (^[Pn2 : pname]: (hoare_Mirabelle_MGT @ (body2 @ Pn2))) @ Procs))))))). % MGT_Body
thf(fact_187_order__refl, axiom,
    ((![X2 : set_Ho840737317_state]: (ord_le1945819589_state @ X2 @ X2)))). % order_refl
thf(fact_188_UnCI, axiom,
    ((![C : hoare_958474565_state, B3 : set_Ho840737317_state, A2 : set_Ho840737317_state]: (((~ ((member109514606_state @ C @ B3))) => (member109514606_state @ C @ A2)) => (member109514606_state @ C @ (sup_su737426425_state @ A2 @ B3)))))). % UnCI
thf(fact_189_Un__iff, axiom,
    ((![C : hoare_958474565_state, A2 : set_Ho840737317_state, B3 : set_Ho840737317_state]: ((member109514606_state @ C @ (sup_su737426425_state @ A2 @ B3)) = (((member109514606_state @ C @ A2)) | ((member109514606_state @ C @ B3))))))). % Un_iff
thf(fact_190_Un__empty, axiom,
    ((![A2 : set_Ho840737317_state, B3 : set_Ho840737317_state]: (((sup_su737426425_state @ A2 @ B3) = bot_bo105666705_state) = (((A2 = bot_bo105666705_state)) & ((B3 = bot_bo105666705_state))))))). % Un_empty
thf(fact_191_finite__Un, axiom,
    ((![F2 : set_pname, G2 : set_pname]: ((finite_finite_pname @ (sup_sup_set_pname @ F2 @ G2)) = (((finite_finite_pname @ F2)) & ((finite_finite_pname @ G2))))))). % finite_Un
thf(fact_192_finite__Un, axiom,
    ((![F2 : set_Ho840737317_state, G2 : set_Ho840737317_state]: ((finite1986656878_state @ (sup_su737426425_state @ F2 @ G2)) = (((finite1986656878_state @ F2)) & ((finite1986656878_state @ G2))))))). % finite_Un
thf(fact_193_Un__subset__iff, axiom,
    ((![A2 : set_Ho840737317_state, B3 : set_Ho840737317_state, C2 : set_Ho840737317_state]: ((ord_le1945819589_state @ (sup_su737426425_state @ A2 @ B3) @ C2) = (((ord_le1945819589_state @ A2 @ C2)) & ((ord_le1945819589_state @ B3 @ C2))))))). % Un_subset_iff
thf(fact_194_Un__insert__left, axiom,
    ((![A : hoare_958474565_state, B3 : set_Ho840737317_state, C2 : set_Ho840737317_state]: ((sup_su737426425_state @ (insert776267541_state @ A @ B3) @ C2) = (insert776267541_state @ A @ (sup_su737426425_state @ B3 @ C2)))))). % Un_insert_left
thf(fact_195_Un__insert__right, axiom,
    ((![A2 : set_Ho840737317_state, A : hoare_958474565_state, B3 : set_Ho840737317_state]: ((sup_su737426425_state @ A2 @ (insert776267541_state @ A @ B3)) = (insert776267541_state @ A @ (sup_su737426425_state @ A2 @ B3)))))). % Un_insert_right
thf(fact_196_UnE, axiom,
    ((![C : hoare_958474565_state, A2 : set_Ho840737317_state, B3 : set_Ho840737317_state]: ((member109514606_state @ C @ (sup_su737426425_state @ A2 @ B3)) => ((~ ((member109514606_state @ C @ A2))) => (member109514606_state @ C @ B3)))))). % UnE
thf(fact_197_UnI1, axiom,
    ((![C : hoare_958474565_state, A2 : set_Ho840737317_state, B3 : set_Ho840737317_state]: ((member109514606_state @ C @ A2) => (member109514606_state @ C @ (sup_su737426425_state @ A2 @ B3)))))). % UnI1
thf(fact_198_UnI2, axiom,
    ((![C : hoare_958474565_state, B3 : set_Ho840737317_state, A2 : set_Ho840737317_state]: ((member109514606_state @ C @ B3) => (member109514606_state @ C @ (sup_su737426425_state @ A2 @ B3)))))). % UnI2
thf(fact_199_Un__def, axiom,
    ((sup_su737426425_state = (^[A3 : set_Ho840737317_state]: (^[B2 : set_Ho840737317_state]: (collec305460656_state @ (^[X : hoare_958474565_state]: (((member109514606_state @ X @ A3)) | ((member109514606_state @ X @ B2)))))))))). % Un_def
thf(fact_200_bex__Un, axiom,
    ((![A2 : set_Ho840737317_state, B3 : set_Ho840737317_state, P : hoare_958474565_state > $o]: ((?[X : hoare_958474565_state]: (((member109514606_state @ X @ (sup_su737426425_state @ A2 @ B3))) & ((P @ X)))) = (((?[X : hoare_958474565_state]: (((member109514606_state @ X @ A2)) & ((P @ X))))) | ((?[X : hoare_958474565_state]: (((member109514606_state @ X @ B3)) & ((P @ X)))))))))). % bex_Un
thf(fact_201_ball__Un, axiom,
    ((![A2 : set_Ho840737317_state, B3 : set_Ho840737317_state, P : hoare_958474565_state > $o]: ((![X : hoare_958474565_state]: (((member109514606_state @ X @ (sup_su737426425_state @ A2 @ B3))) => ((P @ X)))) = (((![X : hoare_958474565_state]: (((member109514606_state @ X @ A2)) => ((P @ X))))) & ((![X : hoare_958474565_state]: (((member109514606_state @ X @ B3)) => ((P @ X)))))))))). % ball_Un

% Conjectures (7)
thf(conj_0, hypothesis,
    (wT_bodies)).
thf(conj_1, hypothesis,
    ((finite1986656878_state @ fa))).
thf(conj_2, hypothesis,
    ((~ ((member109514606_state @ (hoare_Mirabelle_MGT @ y) @ fa))))).
thf(conj_3, hypothesis,
    ((ord_le1945819589_state @ fa @ (image_2144627828_state @ (^[Pn2 : pname]: (hoare_Mirabelle_MGT @ (the_com @ (body @ Pn2)))) @ (dom_pname_com @ body))))).
thf(conj_4, hypothesis,
    (((body @ pn) = (some_com @ y)))).
thf(conj_5, hypothesis,
    ((hoare_604442164_state @ (image_2144627828_state @ (^[Pn2 : pname]: (hoare_Mirabelle_MGT @ (body2 @ Pn2))) @ (dom_pname_com @ body)) @ fa))).
thf(conj_6, conjecture,
    ((![X3 : pname]: ((member_pname @ X3 @ (dom_pname_com @ body)) => (hoare_604442164_state @ (image_2144627828_state @ (^[Pn2 : pname]: (hoare_Mirabelle_MGT @ (body2 @ Pn2))) @ (dom_pname_com @ body)) @ (insert776267541_state @ (hoare_Mirabelle_MGT @ (body2 @ X3)) @ bot_bo105666705_state)))))).
