% 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_462__3254902_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:15:11.775

% Could-be-implicit typings (9)
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__Ostate, type,
    state : $tType).
thf(ty_n_t__Com__Opname, type,
    pname : $tType).
thf(ty_n_t__Com__Ocom, type,
    com : $tType).

% Explicit typings (44)
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_Finite__Set_OFpow_001t__Com__Opname, type,
    finite_Fpow_pname : set_pname > set_set_pname).
thf(sy_c_Finite__Set_OFpow_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J, type,
    finite11235755_state : set_Ho840737317_state > set_se421025541_state).
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_Hoare__Mirabelle__raqjowkjvm_Ostate__not__singleton, type,
    hoare_405891322gleton : $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_001_062_It__Com__Opname_M_Eo_J, type,
    ord_less_eq_pname_o : (pname > $o) > (pname > $o) > $o).
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_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J_J_J, type,
    ord_le1363676837_state : set_se421025541_state > set_se421025541_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_Oimage_001t__Set__Oset_It__Com__Opname_J_001t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J_J, type,
    image_434380702_state : (set_pname > set_Ho840737317_state) > set_set_pname > set_se421025541_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_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_c, type,
    c : com).
thf(sy_v_pn, type,
    pn : pname).
thf(sy_v_y, type,
    y : com).

% Relevant facts (200)
thf(fact_0_WT__bodiesD, axiom,
    ((![Pn : pname, B : com]: (wT_bodies => (((body @ Pn) = (some_com @ B)) => (wt @ B)))))). % WT_bodiesD
thf(fact_1_option_Oinject, axiom,
    ((![X2 : com, Y2 : com]: (((some_com @ X2) = (some_com @ Y2)) = (X2 = Y2))))). % option.inject
thf(fact_2_subsetI, axiom,
    ((![A : set_pname, B2 : set_pname]: ((![X : pname]: ((member_pname @ X @ A) => (member_pname @ X @ B2))) => (ord_le865024672_pname @ A @ B2))))). % subsetI
thf(fact_3_subsetI, axiom,
    ((![A : set_Ho840737317_state, B2 : set_Ho840737317_state]: ((![X : hoare_958474565_state]: ((member109514606_state @ X @ A) => (member109514606_state @ X @ B2))) => (ord_le1945819589_state @ A @ B2))))). % subsetI
thf(fact_4_subset__antisym, axiom,
    ((![A : set_Ho840737317_state, B2 : set_Ho840737317_state]: ((ord_le1945819589_state @ A @ B2) => ((ord_le1945819589_state @ B2 @ A) => (A = B2)))))). % subset_antisym
thf(fact_5_image__eqI, axiom,
    ((![B : hoare_958474565_state, F : hoare_958474565_state > hoare_958474565_state, X3 : hoare_958474565_state, A : set_Ho840737317_state]: ((B = (F @ X3)) => ((member109514606_state @ X3 @ A) => (member109514606_state @ B @ (image_1849792670_state @ F @ A))))))). % image_eqI
thf(fact_6_image__eqI, axiom,
    ((![B : pname, F : hoare_958474565_state > pname, X3 : hoare_958474565_state, A : set_Ho840737317_state]: ((B = (F @ X3)) => ((member109514606_state @ X3 @ A) => (member_pname @ B @ (image_871391498_pname @ F @ A))))))). % image_eqI
thf(fact_7_image__eqI, axiom,
    ((![B : pname, F : pname > pname, X3 : pname, A : set_pname]: ((B = (F @ X3)) => ((member_pname @ X3 @ A) => (member_pname @ B @ (image_pname_pname @ F @ A))))))). % image_eqI
thf(fact_8_image__eqI, axiom,
    ((![B : hoare_958474565_state, F : pname > hoare_958474565_state, X3 : pname, A : set_pname]: ((B = (F @ X3)) => ((member_pname @ X3 @ A) => (member109514606_state @ B @ (image_2144627828_state @ F @ A))))))). % image_eqI
thf(fact_9_order__refl, axiom,
    ((![X3 : set_Ho840737317_state]: (ord_le1945819589_state @ X3 @ X3)))). % order_refl
thf(fact_10_image__Collect__subsetI, axiom,
    ((![P : pname > $o, F : pname > pname, B2 : set_pname]: ((![X : pname]: ((P @ X) => (member_pname @ (F @ X) @ B2))) => (ord_le865024672_pname @ (image_pname_pname @ F @ (collect_pname @ P)) @ B2))))). % image_Collect_subsetI
thf(fact_11_image__Collect__subsetI, axiom,
    ((![P : pname > $o, F : pname > hoare_958474565_state, B2 : set_Ho840737317_state]: ((![X : pname]: ((P @ X) => (member109514606_state @ (F @ X) @ B2))) => (ord_le1945819589_state @ (image_2144627828_state @ F @ (collect_pname @ P)) @ B2))))). % image_Collect_subsetI
thf(fact_12_option_Osel, axiom,
    ((![X2 : com]: ((the_com @ (some_com @ X2)) = X2)))). % option.sel
thf(fact_13_domD, axiom,
    ((![A2 : hoare_958474565_state, M : hoare_958474565_state > option_com]: ((member109514606_state @ A2 @ (dom_Ho256784515te_com @ M)) => (?[B3 : com]: ((M @ A2) = (some_com @ B3))))))). % domD
thf(fact_14_domD, axiom,
    ((![A2 : pname, M : pname > option_com]: ((member_pname @ A2 @ (dom_pname_com @ M)) => (?[B3 : com]: ((M @ A2) = (some_com @ B3))))))). % domD
thf(fact_15_domI, axiom,
    ((![M : hoare_958474565_state > option_com, A2 : hoare_958474565_state, B : com]: (((M @ A2) = (some_com @ B)) => (member109514606_state @ A2 @ (dom_Ho256784515te_com @ M)))))). % domI
thf(fact_16_domI, axiom,
    ((![M : pname > option_com, A2 : pname, B : com]: (((M @ A2) = (some_com @ B)) => (member_pname @ A2 @ (dom_pname_com @ M)))))). % domI
thf(fact_17_image__mono, axiom,
    ((![A : set_pname, B2 : set_pname, F : pname > hoare_958474565_state]: ((ord_le865024672_pname @ A @ B2) => (ord_le1945819589_state @ (image_2144627828_state @ F @ A) @ (image_2144627828_state @ F @ B2)))))). % image_mono
thf(fact_18_image__mono, axiom,
    ((![A : set_Ho840737317_state, B2 : set_Ho840737317_state, F : hoare_958474565_state > hoare_958474565_state]: ((ord_le1945819589_state @ A @ B2) => (ord_le1945819589_state @ (image_1849792670_state @ F @ A) @ (image_1849792670_state @ F @ B2)))))). % image_mono
thf(fact_19_less__eq__set__def, axiom,
    ((ord_le865024672_pname = (^[A3 : set_pname]: (^[B4 : set_pname]: (ord_less_eq_pname_o @ (^[X4 : pname]: (member_pname @ X4 @ A3)) @ (^[X4 : pname]: (member_pname @ X4 @ B4)))))))). % less_eq_set_def
thf(fact_20_less__eq__set__def, axiom,
    ((ord_le1945819589_state = (^[A3 : set_Ho840737317_state]: (^[B4 : set_Ho840737317_state]: (ord_le1483368280tate_o @ (^[X4 : hoare_958474565_state]: (member109514606_state @ X4 @ A3)) @ (^[X4 : hoare_958474565_state]: (member109514606_state @ X4 @ B4)))))))). % less_eq_set_def
thf(fact_21_dual__order_Oantisym, axiom,
    ((![B : set_Ho840737317_state, A2 : set_Ho840737317_state]: ((ord_le1945819589_state @ B @ A2) => ((ord_le1945819589_state @ A2 @ B) => (A2 = B)))))). % dual_order.antisym
thf(fact_22_dual__order_Oeq__iff, axiom,
    (((^[Y : set_Ho840737317_state]: (^[Z : set_Ho840737317_state]: (Y = Z))) = (^[A4 : set_Ho840737317_state]: (^[B5 : set_Ho840737317_state]: (((ord_le1945819589_state @ B5 @ A4)) & ((ord_le1945819589_state @ A4 @ B5)))))))). % dual_order.eq_iff
thf(fact_23_dual__order_Otrans, axiom,
    ((![B : set_Ho840737317_state, A2 : set_Ho840737317_state, C : set_Ho840737317_state]: ((ord_le1945819589_state @ B @ A2) => ((ord_le1945819589_state @ C @ B) => (ord_le1945819589_state @ C @ A2)))))). % dual_order.trans
thf(fact_24_dual__order_Orefl, axiom,
    ((![A2 : set_Ho840737317_state]: (ord_le1945819589_state @ A2 @ A2)))). % dual_order.refl
thf(fact_25_order__trans, axiom,
    ((![X3 : set_Ho840737317_state, Y3 : set_Ho840737317_state, Z2 : set_Ho840737317_state]: ((ord_le1945819589_state @ X3 @ Y3) => ((ord_le1945819589_state @ Y3 @ Z2) => (ord_le1945819589_state @ X3 @ Z2)))))). % order_trans
thf(fact_26_order__class_Oorder_Oantisym, axiom,
    ((![A2 : set_Ho840737317_state, B : set_Ho840737317_state]: ((ord_le1945819589_state @ A2 @ B) => ((ord_le1945819589_state @ B @ A2) => (A2 = B)))))). % order_class.order.antisym
thf(fact_27_ord__le__eq__trans, axiom,
    ((![A2 : set_Ho840737317_state, B : set_Ho840737317_state, C : set_Ho840737317_state]: ((ord_le1945819589_state @ A2 @ B) => ((B = C) => (ord_le1945819589_state @ A2 @ C)))))). % ord_le_eq_trans
thf(fact_28_ord__eq__le__trans, axiom,
    ((![A2 : set_Ho840737317_state, B : set_Ho840737317_state, C : set_Ho840737317_state]: ((A2 = B) => ((ord_le1945819589_state @ B @ C) => (ord_le1945819589_state @ A2 @ C)))))). % ord_eq_le_trans
thf(fact_29_order__class_Oorder_Oeq__iff, axiom,
    (((^[Y : set_Ho840737317_state]: (^[Z : set_Ho840737317_state]: (Y = Z))) = (^[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_30_antisym__conv, axiom,
    ((![Y3 : set_Ho840737317_state, X3 : set_Ho840737317_state]: ((ord_le1945819589_state @ Y3 @ X3) => ((ord_le1945819589_state @ X3 @ Y3) = (X3 = Y3)))))). % antisym_conv
thf(fact_31_order_Otrans, axiom,
    ((![A2 : set_Ho840737317_state, B : set_Ho840737317_state, C : set_Ho840737317_state]: ((ord_le1945819589_state @ A2 @ B) => ((ord_le1945819589_state @ B @ C) => (ord_le1945819589_state @ A2 @ C)))))). % order.trans
thf(fact_32_eq__refl, axiom,
    ((![X3 : set_Ho840737317_state, Y3 : set_Ho840737317_state]: ((X3 = Y3) => (ord_le1945819589_state @ X3 @ Y3))))). % eq_refl
thf(fact_33_antisym, axiom,
    ((![X3 : set_Ho840737317_state, Y3 : set_Ho840737317_state]: ((ord_le1945819589_state @ X3 @ Y3) => ((ord_le1945819589_state @ Y3 @ X3) => (X3 = Y3)))))). % antisym
thf(fact_34_eq__iff, axiom,
    (((^[Y : set_Ho840737317_state]: (^[Z : set_Ho840737317_state]: (Y = Z))) = (^[X4 : set_Ho840737317_state]: (^[Y4 : set_Ho840737317_state]: (((ord_le1945819589_state @ X4 @ Y4)) & ((ord_le1945819589_state @ Y4 @ X4)))))))). % eq_iff
thf(fact_35_ord__le__eq__subst, axiom,
    ((![A2 : set_Ho840737317_state, B : set_Ho840737317_state, F : set_Ho840737317_state > set_Ho840737317_state, C : set_Ho840737317_state]: ((ord_le1945819589_state @ A2 @ B) => (((F @ B) = C) => ((![X : set_Ho840737317_state, Y5 : set_Ho840737317_state]: ((ord_le1945819589_state @ X @ Y5) => (ord_le1945819589_state @ (F @ X) @ (F @ Y5)))) => (ord_le1945819589_state @ (F @ A2) @ C))))))). % ord_le_eq_subst
thf(fact_36_ord__eq__le__subst, axiom,
    ((![A2 : set_Ho840737317_state, F : set_Ho840737317_state > set_Ho840737317_state, B : set_Ho840737317_state, C : set_Ho840737317_state]: ((A2 = (F @ B)) => ((ord_le1945819589_state @ B @ C) => ((![X : set_Ho840737317_state, Y5 : set_Ho840737317_state]: ((ord_le1945819589_state @ X @ Y5) => (ord_le1945819589_state @ (F @ X) @ (F @ Y5)))) => (ord_le1945819589_state @ A2 @ (F @ C)))))))). % ord_eq_le_subst
thf(fact_37_order__subst2, axiom,
    ((![A2 : set_Ho840737317_state, B : set_Ho840737317_state, F : set_Ho840737317_state > set_Ho840737317_state, C : set_Ho840737317_state]: ((ord_le1945819589_state @ A2 @ B) => ((ord_le1945819589_state @ (F @ B) @ C) => ((![X : set_Ho840737317_state, Y5 : set_Ho840737317_state]: ((ord_le1945819589_state @ X @ Y5) => (ord_le1945819589_state @ (F @ X) @ (F @ Y5)))) => (ord_le1945819589_state @ (F @ A2) @ C))))))). % order_subst2
thf(fact_38_order__subst1, axiom,
    ((![A2 : set_Ho840737317_state, F : set_Ho840737317_state > set_Ho840737317_state, B : set_Ho840737317_state, C : set_Ho840737317_state]: ((ord_le1945819589_state @ A2 @ (F @ B)) => ((ord_le1945819589_state @ B @ C) => ((![X : set_Ho840737317_state, Y5 : set_Ho840737317_state]: ((ord_le1945819589_state @ X @ Y5) => (ord_le1945819589_state @ (F @ X) @ (F @ Y5)))) => (ord_le1945819589_state @ A2 @ (F @ C)))))))). % order_subst1
thf(fact_39_rev__image__eqI, axiom,
    ((![X3 : hoare_958474565_state, A : set_Ho840737317_state, B : hoare_958474565_state, F : hoare_958474565_state > hoare_958474565_state]: ((member109514606_state @ X3 @ A) => ((B = (F @ X3)) => (member109514606_state @ B @ (image_1849792670_state @ F @ A))))))). % rev_image_eqI
thf(fact_40_rev__image__eqI, axiom,
    ((![X3 : hoare_958474565_state, A : set_Ho840737317_state, B : pname, F : hoare_958474565_state > pname]: ((member109514606_state @ X3 @ A) => ((B = (F @ X3)) => (member_pname @ B @ (image_871391498_pname @ F @ A))))))). % rev_image_eqI
thf(fact_41_rev__image__eqI, axiom,
    ((![X3 : pname, A : set_pname, B : hoare_958474565_state, F : pname > hoare_958474565_state]: ((member_pname @ X3 @ A) => ((B = (F @ X3)) => (member109514606_state @ B @ (image_2144627828_state @ F @ A))))))). % rev_image_eqI
thf(fact_42_rev__image__eqI, axiom,
    ((![X3 : pname, A : set_pname, B : pname, F : pname > pname]: ((member_pname @ X3 @ A) => ((B = (F @ X3)) => (member_pname @ B @ (image_pname_pname @ F @ A))))))). % rev_image_eqI
thf(fact_43_ball__imageD, axiom,
    ((![F : pname > hoare_958474565_state, A : set_pname, P : hoare_958474565_state > $o]: ((![X : hoare_958474565_state]: ((member109514606_state @ X @ (image_2144627828_state @ F @ A)) => (P @ X))) => (![X5 : pname]: ((member_pname @ X5 @ A) => (P @ (F @ X5)))))))). % ball_imageD
thf(fact_44_image__cong, axiom,
    ((![M2 : set_pname, N : set_pname, F : pname > hoare_958474565_state, G : pname > hoare_958474565_state]: ((M2 = N) => ((![X : pname]: ((member_pname @ X @ N) => ((F @ X) = (G @ X)))) => ((image_2144627828_state @ F @ M2) = (image_2144627828_state @ G @ N))))))). % image_cong
thf(fact_45_bex__imageD, axiom,
    ((![F : pname > hoare_958474565_state, A : set_pname, P : hoare_958474565_state > $o]: ((?[X5 : hoare_958474565_state]: ((member109514606_state @ X5 @ (image_2144627828_state @ F @ A)) & (P @ X5))) => (?[X : pname]: ((member_pname @ X @ A) & (P @ (F @ X)))))))). % bex_imageD
thf(fact_46_image__iff, axiom,
    ((![Z2 : hoare_958474565_state, F : pname > hoare_958474565_state, A : set_pname]: ((member109514606_state @ Z2 @ (image_2144627828_state @ F @ A)) = (?[X4 : pname]: (((member_pname @ X4 @ A)) & ((Z2 = (F @ X4))))))))). % image_iff
thf(fact_47_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_48_imageI, axiom,
    ((![X3 : hoare_958474565_state, A : set_Ho840737317_state, F : hoare_958474565_state > pname]: ((member109514606_state @ X3 @ A) => (member_pname @ (F @ X3) @ (image_871391498_pname @ F @ A)))))). % imageI
thf(fact_49_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_50_imageI, axiom,
    ((![X3 : pname, A : set_pname, F : pname > pname]: ((member_pname @ X3 @ A) => (member_pname @ (F @ X3) @ (image_pname_pname @ F @ A)))))). % imageI
thf(fact_51_mem__Collect__eq, axiom,
    ((![A2 : hoare_958474565_state, P : hoare_958474565_state > $o]: ((member109514606_state @ A2 @ (collec305460656_state @ P)) = (P @ A2))))). % mem_Collect_eq
thf(fact_52_mem__Collect__eq, axiom,
    ((![A2 : pname, P : pname > $o]: ((member_pname @ A2 @ (collect_pname @ P)) = (P @ A2))))). % mem_Collect_eq
thf(fact_53_Collect__mem__eq, axiom,
    ((![A : set_Ho840737317_state]: ((collec305460656_state @ (^[X4 : hoare_958474565_state]: (member109514606_state @ X4 @ A))) = A)))). % Collect_mem_eq
thf(fact_54_Collect__mem__eq, axiom,
    ((![A : set_pname]: ((collect_pname @ (^[X4 : pname]: (member_pname @ X4 @ A))) = A)))). % Collect_mem_eq
thf(fact_55_Collect__cong, axiom,
    ((![P : pname > $o, Q : pname > $o]: ((![X : pname]: ((P @ X) = (Q @ X))) => ((collect_pname @ P) = (collect_pname @ Q)))))). % Collect_cong
thf(fact_56_Collect__mono__iff, axiom,
    ((![P : pname > $o, Q : pname > $o]: ((ord_le865024672_pname @ (collect_pname @ P) @ (collect_pname @ Q)) = (![X4 : pname]: (((P @ X4)) => ((Q @ X4)))))))). % Collect_mono_iff
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)) = (![X4 : hoare_958474565_state]: (((P @ X4)) => ((Q @ X4)))))))). % Collect_mono_iff
thf(fact_58_set__eq__subset, axiom,
    (((^[Y : set_Ho840737317_state]: (^[Z : set_Ho840737317_state]: (Y = Z))) = (^[A3 : set_Ho840737317_state]: (^[B4 : set_Ho840737317_state]: (((ord_le1945819589_state @ A3 @ B4)) & ((ord_le1945819589_state @ B4 @ A3)))))))). % set_eq_subset
thf(fact_59_subset__trans, axiom,
    ((![A : set_Ho840737317_state, B2 : set_Ho840737317_state, C2 : set_Ho840737317_state]: ((ord_le1945819589_state @ A @ B2) => ((ord_le1945819589_state @ B2 @ C2) => (ord_le1945819589_state @ A @ C2)))))). % subset_trans
thf(fact_60_Collect__mono, axiom,
    ((![P : pname > $o, Q : pname > $o]: ((![X : pname]: ((P @ X) => (Q @ X))) => (ord_le865024672_pname @ (collect_pname @ P) @ (collect_pname @ Q)))))). % Collect_mono
thf(fact_61_Collect__mono, axiom,
    ((![P : hoare_958474565_state > $o, Q : hoare_958474565_state > $o]: ((![X : hoare_958474565_state]: ((P @ X) => (Q @ X))) => (ord_le1945819589_state @ (collec305460656_state @ P) @ (collec305460656_state @ Q)))))). % Collect_mono
thf(fact_62_subset__refl, axiom,
    ((![A : set_Ho840737317_state]: (ord_le1945819589_state @ A @ A)))). % subset_refl
thf(fact_63_subset__iff, axiom,
    ((ord_le865024672_pname = (^[A3 : set_pname]: (^[B4 : set_pname]: (![T : pname]: (((member_pname @ T @ A3)) => ((member_pname @ T @ B4))))))))). % subset_iff
thf(fact_64_subset__iff, axiom,
    ((ord_le1945819589_state = (^[A3 : set_Ho840737317_state]: (^[B4 : set_Ho840737317_state]: (![T : hoare_958474565_state]: (((member109514606_state @ T @ A3)) => ((member109514606_state @ T @ B4))))))))). % subset_iff
thf(fact_65_equalityD2, axiom,
    ((![A : set_Ho840737317_state, B2 : set_Ho840737317_state]: ((A = B2) => (ord_le1945819589_state @ B2 @ A))))). % equalityD2
thf(fact_66_equalityD1, axiom,
    ((![A : set_Ho840737317_state, B2 : set_Ho840737317_state]: ((A = B2) => (ord_le1945819589_state @ A @ B2))))). % equalityD1
thf(fact_67_subset__eq, axiom,
    ((ord_le865024672_pname = (^[A3 : set_pname]: (^[B4 : set_pname]: (![X4 : pname]: (((member_pname @ X4 @ A3)) => ((member_pname @ X4 @ B4))))))))). % subset_eq
thf(fact_68_subset__eq, axiom,
    ((ord_le1945819589_state = (^[A3 : set_Ho840737317_state]: (^[B4 : set_Ho840737317_state]: (![X4 : hoare_958474565_state]: (((member109514606_state @ X4 @ A3)) => ((member109514606_state @ X4 @ B4))))))))). % subset_eq
thf(fact_69_equalityE, axiom,
    ((![A : set_Ho840737317_state, B2 : set_Ho840737317_state]: ((A = B2) => (~ (((ord_le1945819589_state @ A @ B2) => (~ ((ord_le1945819589_state @ B2 @ A)))))))))). % equalityE
thf(fact_70_subsetD, axiom,
    ((![A : set_pname, B2 : set_pname, C : pname]: ((ord_le865024672_pname @ A @ B2) => ((member_pname @ C @ A) => (member_pname @ C @ B2)))))). % subsetD
thf(fact_71_subsetD, axiom,
    ((![A : set_Ho840737317_state, B2 : set_Ho840737317_state, C : hoare_958474565_state]: ((ord_le1945819589_state @ A @ B2) => ((member109514606_state @ C @ A) => (member109514606_state @ C @ B2)))))). % subsetD
thf(fact_72_in__mono, axiom,
    ((![A : set_pname, B2 : set_pname, X3 : pname]: ((ord_le865024672_pname @ A @ B2) => ((member_pname @ X3 @ A) => (member_pname @ X3 @ B2)))))). % in_mono
thf(fact_73_in__mono, axiom,
    ((![A : set_Ho840737317_state, B2 : set_Ho840737317_state, X3 : hoare_958474565_state]: ((ord_le1945819589_state @ A @ B2) => ((member109514606_state @ X3 @ A) => (member109514606_state @ X3 @ B2)))))). % in_mono
thf(fact_74_Compr__image__eq, axiom,
    ((![F : hoare_958474565_state > hoare_958474565_state, A : set_Ho840737317_state, P : hoare_958474565_state > $o]: ((collec305460656_state @ (^[X4 : hoare_958474565_state]: (((member109514606_state @ X4 @ (image_1849792670_state @ F @ A))) & ((P @ X4))))) = (image_1849792670_state @ F @ (collec305460656_state @ (^[X4 : hoare_958474565_state]: (((member109514606_state @ X4 @ A)) & ((P @ (F @ X4))))))))))). % Compr_image_eq
thf(fact_75_Compr__image__eq, axiom,
    ((![F : pname > hoare_958474565_state, A : set_pname, P : hoare_958474565_state > $o]: ((collec305460656_state @ (^[X4 : hoare_958474565_state]: (((member109514606_state @ X4 @ (image_2144627828_state @ F @ A))) & ((P @ X4))))) = (image_2144627828_state @ F @ (collect_pname @ (^[X4 : pname]: (((member_pname @ X4 @ A)) & ((P @ (F @ X4))))))))))). % Compr_image_eq
thf(fact_76_Compr__image__eq, axiom,
    ((![F : hoare_958474565_state > pname, A : set_Ho840737317_state, P : pname > $o]: ((collect_pname @ (^[X4 : pname]: (((member_pname @ X4 @ (image_871391498_pname @ F @ A))) & ((P @ X4))))) = (image_871391498_pname @ F @ (collec305460656_state @ (^[X4 : hoare_958474565_state]: (((member109514606_state @ X4 @ A)) & ((P @ (F @ X4))))))))))). % Compr_image_eq
thf(fact_77_Compr__image__eq, axiom,
    ((![F : pname > pname, A : set_pname, P : pname > $o]: ((collect_pname @ (^[X4 : pname]: (((member_pname @ X4 @ (image_pname_pname @ F @ A))) & ((P @ X4))))) = (image_pname_pname @ F @ (collect_pname @ (^[X4 : pname]: (((member_pname @ X4 @ A)) & ((P @ (F @ X4))))))))))). % Compr_image_eq
thf(fact_78_image__image, axiom,
    ((![F : hoare_958474565_state > hoare_958474565_state, G : pname > hoare_958474565_state, A : set_pname]: ((image_1849792670_state @ F @ (image_2144627828_state @ G @ A)) = (image_2144627828_state @ (^[X4 : pname]: (F @ (G @ X4))) @ A))))). % image_image
thf(fact_79_image__image, axiom,
    ((![F : pname > hoare_958474565_state, G : pname > pname, A : set_pname]: ((image_2144627828_state @ F @ (image_pname_pname @ G @ A)) = (image_2144627828_state @ (^[X4 : pname]: (F @ (G @ X4))) @ A))))). % image_image
thf(fact_80_imageE, axiom,
    ((![B : hoare_958474565_state, F : hoare_958474565_state > hoare_958474565_state, A : set_Ho840737317_state]: ((member109514606_state @ B @ (image_1849792670_state @ F @ A)) => (~ ((![X : hoare_958474565_state]: ((B = (F @ X)) => (~ ((member109514606_state @ X @ A))))))))))). % imageE
thf(fact_81_imageE, axiom,
    ((![B : hoare_958474565_state, F : pname > hoare_958474565_state, A : set_pname]: ((member109514606_state @ B @ (image_2144627828_state @ F @ A)) => (~ ((![X : pname]: ((B = (F @ X)) => (~ ((member_pname @ X @ A))))))))))). % imageE
thf(fact_82_imageE, axiom,
    ((![B : pname, F : hoare_958474565_state > pname, A : set_Ho840737317_state]: ((member_pname @ B @ (image_871391498_pname @ F @ A)) => (~ ((![X : hoare_958474565_state]: ((B = (F @ X)) => (~ ((member109514606_state @ X @ A))))))))))). % imageE
thf(fact_83_imageE, axiom,
    ((![B : pname, F : pname > pname, A : set_pname]: ((member_pname @ B @ (image_pname_pname @ F @ A)) => (~ ((![X : pname]: ((B = (F @ X)) => (~ ((member_pname @ X @ A))))))))))). % imageE
thf(fact_84_Collect__restrict, axiom,
    ((![X6 : set_pname, P : pname > $o]: (ord_le865024672_pname @ (collect_pname @ (^[X4 : pname]: (((member_pname @ X4 @ X6)) & ((P @ X4))))) @ X6)))). % Collect_restrict
thf(fact_85_Collect__restrict, axiom,
    ((![X6 : set_Ho840737317_state, P : hoare_958474565_state > $o]: (ord_le1945819589_state @ (collec305460656_state @ (^[X4 : hoare_958474565_state]: (((member109514606_state @ X4 @ X6)) & ((P @ X4))))) @ X6)))). % Collect_restrict
thf(fact_86_prop__restrict, axiom,
    ((![X3 : pname, Z3 : set_pname, X6 : set_pname, P : pname > $o]: ((member_pname @ X3 @ Z3) => ((ord_le865024672_pname @ Z3 @ (collect_pname @ (^[X4 : pname]: (((member_pname @ X4 @ X6)) & ((P @ X4)))))) => (P @ X3)))))). % prop_restrict
thf(fact_87_prop__restrict, axiom,
    ((![X3 : hoare_958474565_state, Z3 : set_Ho840737317_state, X6 : set_Ho840737317_state, P : hoare_958474565_state > $o]: ((member109514606_state @ X3 @ Z3) => ((ord_le1945819589_state @ Z3 @ (collec305460656_state @ (^[X4 : hoare_958474565_state]: (((member109514606_state @ X4 @ X6)) & ((P @ X4)))))) => (P @ X3)))))). % prop_restrict
thf(fact_88_Collect__subset, axiom,
    ((![A : set_pname, P : pname > $o]: (ord_le865024672_pname @ (collect_pname @ (^[X4 : pname]: (((member_pname @ X4 @ A)) & ((P @ X4))))) @ A)))). % Collect_subset
thf(fact_89_Collect__subset, axiom,
    ((![A : set_Ho840737317_state, P : hoare_958474565_state > $o]: (ord_le1945819589_state @ (collec305460656_state @ (^[X4 : hoare_958474565_state]: (((member109514606_state @ X4 @ A)) & ((P @ X4))))) @ A)))). % Collect_subset
thf(fact_90_subset__image__iff, axiom,
    ((![B2 : set_Ho840737317_state, F : pname > hoare_958474565_state, A : set_pname]: ((ord_le1945819589_state @ B2 @ (image_2144627828_state @ F @ A)) = (?[AA : set_pname]: (((ord_le865024672_pname @ AA @ A)) & ((B2 = (image_2144627828_state @ F @ AA))))))))). % subset_image_iff
thf(fact_91_subset__image__iff, axiom,
    ((![B2 : set_Ho840737317_state, F : hoare_958474565_state > hoare_958474565_state, A : set_Ho840737317_state]: ((ord_le1945819589_state @ B2 @ (image_1849792670_state @ F @ A)) = (?[AA : set_Ho840737317_state]: (((ord_le1945819589_state @ AA @ A)) & ((B2 = (image_1849792670_state @ F @ AA))))))))). % subset_image_iff
thf(fact_92_image__subset__iff, axiom,
    ((![F : pname > hoare_958474565_state, A : set_pname, B2 : set_Ho840737317_state]: ((ord_le1945819589_state @ (image_2144627828_state @ F @ A) @ B2) = (![X4 : pname]: (((member_pname @ X4 @ A)) => ((member109514606_state @ (F @ X4) @ B2)))))))). % image_subset_iff
thf(fact_93_subset__imageE, axiom,
    ((![B2 : set_Ho840737317_state, F : pname > hoare_958474565_state, A : set_pname]: ((ord_le1945819589_state @ B2 @ (image_2144627828_state @ F @ A)) => (~ ((![C3 : set_pname]: ((ord_le865024672_pname @ C3 @ A) => (~ ((B2 = (image_2144627828_state @ F @ C3)))))))))))). % subset_imageE
thf(fact_94_subset__imageE, axiom,
    ((![B2 : set_Ho840737317_state, F : hoare_958474565_state > hoare_958474565_state, A : set_Ho840737317_state]: ((ord_le1945819589_state @ B2 @ (image_1849792670_state @ F @ A)) => (~ ((![C3 : set_Ho840737317_state]: ((ord_le1945819589_state @ C3 @ A) => (~ ((B2 = (image_1849792670_state @ F @ C3)))))))))))). % subset_imageE
thf(fact_95_image__subsetI, axiom,
    ((![A : set_Ho840737317_state, F : hoare_958474565_state > pname, B2 : set_pname]: ((![X : hoare_958474565_state]: ((member109514606_state @ X @ A) => (member_pname @ (F @ X) @ B2))) => (ord_le865024672_pname @ (image_871391498_pname @ F @ A) @ B2))))). % image_subsetI
thf(fact_96_image__subsetI, axiom,
    ((![A : set_pname, F : pname > pname, B2 : set_pname]: ((![X : pname]: ((member_pname @ X @ A) => (member_pname @ (F @ X) @ B2))) => (ord_le865024672_pname @ (image_pname_pname @ F @ A) @ B2))))). % image_subsetI
thf(fact_97_image__subsetI, axiom,
    ((![A : set_Ho840737317_state, F : hoare_958474565_state > hoare_958474565_state, B2 : set_Ho840737317_state]: ((![X : hoare_958474565_state]: ((member109514606_state @ X @ A) => (member109514606_state @ (F @ X) @ B2))) => (ord_le1945819589_state @ (image_1849792670_state @ F @ A) @ B2))))). % image_subsetI
thf(fact_98_image__subsetI, axiom,
    ((![A : set_pname, F : pname > hoare_958474565_state, B2 : set_Ho840737317_state]: ((![X : pname]: ((member_pname @ X @ A) => (member109514606_state @ (F @ X) @ B2))) => (ord_le1945819589_state @ (image_2144627828_state @ F @ A) @ B2))))). % image_subsetI
thf(fact_99_all__subset__image, axiom,
    ((![F : pname > hoare_958474565_state, A : set_pname, P : set_Ho840737317_state > $o]: ((![B4 : set_Ho840737317_state]: (((ord_le1945819589_state @ B4 @ (image_2144627828_state @ F @ A))) => ((P @ B4)))) = (![B4 : set_pname]: (((ord_le865024672_pname @ B4 @ A)) => ((P @ (image_2144627828_state @ F @ B4))))))))). % all_subset_image
thf(fact_100_all__subset__image, axiom,
    ((![F : hoare_958474565_state > hoare_958474565_state, A : set_Ho840737317_state, P : set_Ho840737317_state > $o]: ((![B4 : set_Ho840737317_state]: (((ord_le1945819589_state @ B4 @ (image_1849792670_state @ F @ A))) => ((P @ B4)))) = (![B4 : set_Ho840737317_state]: (((ord_le1945819589_state @ B4 @ A)) => ((P @ (image_1849792670_state @ F @ B4))))))))). % all_subset_image
thf(fact_101_MGF__lemma2__simult, axiom,
    ((![F2 : set_Ho840737317_state]: (hoare_405891322gleton => (wT_bodies => ((ord_le1945819589_state @ F2 @ (image_2144627828_state @ (^[Pn2 : pname]: (hoare_Mirabelle_MGT @ (the_com @ (body @ Pn2)))) @ (dom_pname_com @ body))) => (hoare_604442164_state @ (image_2144627828_state @ (^[Pn2 : pname]: (hoare_Mirabelle_MGT @ (body2 @ Pn2))) @ (dom_pname_com @ body)) @ F2))))))). % MGF_lemma2_simult
thf(fact_102_WTs__elim__cases_I7_J, axiom,
    ((![P : pname]: ((wt @ (body2 @ P)) => (~ ((![Y5 : com]: (~ (((body @ P) = (some_com @ Y5))))))))))). % WTs_elim_cases(7)
thf(fact_103_pred__subset__eq, axiom,
    ((![R : set_pname, S : set_pname]: ((ord_less_eq_pname_o @ (^[X4 : pname]: (member_pname @ X4 @ R)) @ (^[X4 : pname]: (member_pname @ X4 @ S))) = (ord_le865024672_pname @ R @ S))))). % pred_subset_eq
thf(fact_104_pred__subset__eq, axiom,
    ((![R : set_Ho840737317_state, S : set_Ho840737317_state]: ((ord_le1483368280tate_o @ (^[X4 : hoare_958474565_state]: (member109514606_state @ X4 @ R)) @ (^[X4 : hoare_958474565_state]: (member109514606_state @ X4 @ S))) = (ord_le1945819589_state @ R @ S))))). % pred_subset_eq
thf(fact_105_subset__Collect__iff, axiom,
    ((![B2 : set_pname, A : set_pname, P : pname > $o]: ((ord_le865024672_pname @ B2 @ A) => ((ord_le865024672_pname @ B2 @ (collect_pname @ (^[X4 : pname]: (((member_pname @ X4 @ A)) & ((P @ X4)))))) = (![X4 : pname]: (((member_pname @ X4 @ B2)) => ((P @ X4))))))))). % subset_Collect_iff
thf(fact_106_subset__Collect__iff, axiom,
    ((![B2 : set_Ho840737317_state, A : set_Ho840737317_state, P : hoare_958474565_state > $o]: ((ord_le1945819589_state @ B2 @ A) => ((ord_le1945819589_state @ B2 @ (collec305460656_state @ (^[X4 : hoare_958474565_state]: (((member109514606_state @ X4 @ A)) & ((P @ X4)))))) = (![X4 : hoare_958474565_state]: (((member109514606_state @ X4 @ B2)) => ((P @ X4))))))))). % subset_Collect_iff
thf(fact_107_subset__CollectI, axiom,
    ((![B2 : set_pname, A : set_pname, Q : pname > $o, P : pname > $o]: ((ord_le865024672_pname @ B2 @ A) => ((![X : pname]: ((member_pname @ X @ B2) => ((Q @ X) => (P @ X)))) => (ord_le865024672_pname @ (collect_pname @ (^[X4 : pname]: (((member_pname @ X4 @ B2)) & ((Q @ X4))))) @ (collect_pname @ (^[X4 : pname]: (((member_pname @ X4 @ A)) & ((P @ X4))))))))))). % subset_CollectI
thf(fact_108_subset__CollectI, axiom,
    ((![B2 : set_Ho840737317_state, A : set_Ho840737317_state, Q : hoare_958474565_state > $o, P : hoare_958474565_state > $o]: ((ord_le1945819589_state @ B2 @ A) => ((![X : hoare_958474565_state]: ((member109514606_state @ X @ B2) => ((Q @ X) => (P @ X)))) => (ord_le1945819589_state @ (collec305460656_state @ (^[X4 : hoare_958474565_state]: (((member109514606_state @ X4 @ B2)) & ((Q @ X4))))) @ (collec305460656_state @ (^[X4 : hoare_958474565_state]: (((member109514606_state @ X4 @ A)) & ((P @ X4))))))))))). % subset_CollectI
thf(fact_109_conj__subset__def, axiom,
    ((![A : set_pname, P : pname > $o, Q : pname > $o]: ((ord_le865024672_pname @ A @ (collect_pname @ (^[X4 : pname]: (((P @ X4)) & ((Q @ X4)))))) = (((ord_le865024672_pname @ A @ (collect_pname @ P))) & ((ord_le865024672_pname @ A @ (collect_pname @ Q)))))))). % conj_subset_def
thf(fact_110_conj__subset__def, axiom,
    ((![A : set_Ho840737317_state, P : hoare_958474565_state > $o, Q : hoare_958474565_state > $o]: ((ord_le1945819589_state @ A @ (collec305460656_state @ (^[X4 : hoare_958474565_state]: (((P @ X4)) & ((Q @ X4)))))) = (((ord_le1945819589_state @ A @ (collec305460656_state @ P))) & ((ord_le1945819589_state @ A @ (collec305460656_state @ Q)))))))). % conj_subset_def
thf(fact_111_com_Oinject_I6_J, axiom,
    ((![X7 : pname, Y7 : pname]: (((body2 @ X7) = (body2 @ Y7)) = (X7 = Y7))))). % com.inject(6)
thf(fact_112_cut, axiom,
    ((![G2 : set_Ho840737317_state, Ts : set_Ho840737317_state, G3 : set_Ho840737317_state]: ((hoare_604442164_state @ G2 @ Ts) => ((hoare_604442164_state @ G3 @ G2) => (hoare_604442164_state @ G3 @ Ts)))))). % cut
thf(fact_113_single__stateE, axiom,
    ((hoare_405891322gleton => (![T2 : state]: (~ ((![S2 : state]: (S2 = T2)))))))). % single_stateE
thf(fact_114_state__not__singleton__def, axiom,
    ((hoare_405891322gleton = (?[S3 : state]: (?[T : state]: (~ ((S3 = T)))))))). % state_not_singleton_def
thf(fact_115_thin, axiom,
    ((![G2 : set_Ho840737317_state, Ts : set_Ho840737317_state, G3 : set_Ho840737317_state]: ((hoare_604442164_state @ G2 @ Ts) => ((ord_le1945819589_state @ G2 @ G3) => (hoare_604442164_state @ G3 @ Ts)))))). % thin
thf(fact_116_asm, axiom,
    ((![Ts : set_Ho840737317_state, G3 : set_Ho840737317_state]: ((ord_le1945819589_state @ Ts @ G3) => (hoare_604442164_state @ G3 @ Ts))))). % asm
thf(fact_117_weaken, axiom,
    ((![G3 : set_Ho840737317_state, Ts2 : set_Ho840737317_state, Ts : set_Ho840737317_state]: ((hoare_604442164_state @ G3 @ Ts2) => ((ord_le1945819589_state @ Ts @ Ts2) => (hoare_604442164_state @ G3 @ Ts)))))). % weaken
thf(fact_118_Inf_OINF__cong, axiom,
    ((![A : set_pname, B2 : set_pname, C2 : pname > hoare_958474565_state, D : pname > hoare_958474565_state, Inf : set_Ho840737317_state > hoare_958474565_state]: ((A = B2) => ((![X : pname]: ((member_pname @ X @ B2) => ((C2 @ X) = (D @ X)))) => ((Inf @ (image_2144627828_state @ C2 @ A)) = (Inf @ (image_2144627828_state @ D @ B2)))))))). % Inf.INF_cong
thf(fact_119_Sup_OSUP__cong, axiom,
    ((![A : set_pname, B2 : set_pname, C2 : pname > hoare_958474565_state, D : pname > hoare_958474565_state, Sup : set_Ho840737317_state > hoare_958474565_state]: ((A = B2) => ((![X : pname]: ((member_pname @ X @ B2) => ((C2 @ X) = (D @ X)))) => ((Sup @ (image_2144627828_state @ C2 @ A)) = (Sup @ (image_2144627828_state @ D @ B2)))))))). % Sup.SUP_cong
thf(fact_120_MGF__lemma1, axiom,
    ((![G3 : set_Ho840737317_state, C : com]: (hoare_405891322gleton => ((![X : pname]: ((member_pname @ X @ (dom_pname_com @ body)) => (hoare_604442164_state @ G3 @ (insert776267541_state @ (hoare_Mirabelle_MGT @ (body2 @ X)) @ bot_bo105666705_state)))) => ((wt @ C) => (hoare_604442164_state @ G3 @ (insert776267541_state @ (hoare_Mirabelle_MGT @ C) @ bot_bo105666705_state)))))))). % MGF_lemma1
thf(fact_121_MGT__Body, axiom,
    ((![G3 : set_Ho840737317_state, Procs : set_pname]: ((hoare_604442164_state @ (sup_su737426425_state @ G3 @ (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 @ G3 @ (image_2144627828_state @ (^[Pn2 : pname]: (hoare_Mirabelle_MGT @ (body2 @ Pn2))) @ Procs))))))). % MGT_Body
thf(fact_122_image__Fpow__mono, axiom,
    ((![F : pname > hoare_958474565_state, A : set_pname, B2 : set_Ho840737317_state]: ((ord_le1945819589_state @ (image_2144627828_state @ F @ A) @ B2) => (ord_le1363676837_state @ (image_434380702_state @ (image_2144627828_state @ F) @ (finite_Fpow_pname @ A)) @ (finite11235755_state @ B2)))))). % image_Fpow_mono
thf(fact_123_empty__Collect__eq, axiom,
    ((![P : pname > $o]: ((bot_bot_set_pname = (collect_pname @ P)) = (![X4 : pname]: (~ ((P @ X4)))))))). % empty_Collect_eq
thf(fact_124_empty__Collect__eq, axiom,
    ((![P : hoare_958474565_state > $o]: ((bot_bo105666705_state = (collec305460656_state @ P)) = (![X4 : hoare_958474565_state]: (~ ((P @ X4)))))))). % empty_Collect_eq
thf(fact_125_Collect__empty__eq, axiom,
    ((![P : pname > $o]: (((collect_pname @ P) = bot_bot_set_pname) = (![X4 : pname]: (~ ((P @ X4)))))))). % Collect_empty_eq
thf(fact_126_Collect__empty__eq, axiom,
    ((![P : hoare_958474565_state > $o]: (((collec305460656_state @ P) = bot_bo105666705_state) = (![X4 : hoare_958474565_state]: (~ ((P @ X4)))))))). % Collect_empty_eq
thf(fact_127_all__not__in__conv, axiom,
    ((![A : set_pname]: ((![X4 : pname]: (~ ((member_pname @ X4 @ A)))) = (A = bot_bot_set_pname))))). % all_not_in_conv
thf(fact_128_all__not__in__conv, axiom,
    ((![A : set_Ho840737317_state]: ((![X4 : hoare_958474565_state]: (~ ((member109514606_state @ X4 @ A)))) = (A = bot_bo105666705_state))))). % all_not_in_conv
thf(fact_129_empty__iff, axiom,
    ((![C : pname]: (~ ((member_pname @ C @ bot_bot_set_pname)))))). % empty_iff
thf(fact_130_empty__iff, axiom,
    ((![C : hoare_958474565_state]: (~ ((member109514606_state @ C @ bot_bo105666705_state)))))). % empty_iff
thf(fact_131_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_132_insert__iff, axiom,
    ((![A2 : hoare_958474565_state, B : hoare_958474565_state, A : set_Ho840737317_state]: ((member109514606_state @ A2 @ (insert776267541_state @ B @ A)) = (((A2 = B)) | ((member109514606_state @ A2 @ A))))))). % insert_iff
thf(fact_133_insert__iff, axiom,
    ((![A2 : pname, B : pname, A : set_pname]: ((member_pname @ A2 @ (insert_pname @ B @ A)) = (((A2 = B)) | ((member_pname @ A2 @ A))))))). % insert_iff
thf(fact_134_insertCI, axiom,
    ((![A2 : hoare_958474565_state, B2 : set_Ho840737317_state, B : hoare_958474565_state]: (((~ ((member109514606_state @ A2 @ B2))) => (A2 = B)) => (member109514606_state @ A2 @ (insert776267541_state @ B @ B2)))))). % insertCI
thf(fact_135_insertCI, axiom,
    ((![A2 : pname, B2 : set_pname, B : pname]: (((~ ((member_pname @ A2 @ B2))) => (A2 = B)) => (member_pname @ A2 @ (insert_pname @ B @ B2)))))). % insertCI
thf(fact_136_Un__iff, axiom,
    ((![C : pname, A : set_pname, B2 : set_pname]: ((member_pname @ C @ (sup_sup_set_pname @ A @ B2)) = (((member_pname @ C @ A)) | ((member_pname @ C @ B2))))))). % Un_iff
thf(fact_137_Un__iff, axiom,
    ((![C : hoare_958474565_state, A : set_Ho840737317_state, B2 : set_Ho840737317_state]: ((member109514606_state @ C @ (sup_su737426425_state @ A @ B2)) = (((member109514606_state @ C @ A)) | ((member109514606_state @ C @ B2))))))). % Un_iff
thf(fact_138_UnCI, axiom,
    ((![C : pname, B2 : set_pname, A : set_pname]: (((~ ((member_pname @ C @ B2))) => (member_pname @ C @ A)) => (member_pname @ C @ (sup_sup_set_pname @ A @ B2)))))). % UnCI
thf(fact_139_UnCI, axiom,
    ((![C : hoare_958474565_state, B2 : set_Ho840737317_state, A : set_Ho840737317_state]: (((~ ((member109514606_state @ C @ B2))) => (member109514606_state @ C @ A)) => (member109514606_state @ C @ (sup_su737426425_state @ A @ B2)))))). % UnCI
thf(fact_140_finite__Collect__disjI, axiom,
    ((![P : pname > $o, Q : pname > $o]: ((finite_finite_pname @ (collect_pname @ (^[X4 : pname]: (((P @ X4)) | ((Q @ X4)))))) = (((finite_finite_pname @ (collect_pname @ P))) & ((finite_finite_pname @ (collect_pname @ Q)))))))). % finite_Collect_disjI
thf(fact_141_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 @ (^[X4 : pname]: (((P @ X4)) & ((Q @ X4)))))))))). % finite_Collect_conjI
thf(fact_142_image__empty, axiom,
    ((![F : pname > hoare_958474565_state]: ((image_2144627828_state @ F @ bot_bot_set_pname) = bot_bo105666705_state)))). % image_empty
thf(fact_143_image__empty, axiom,
    ((![F : hoare_958474565_state > hoare_958474565_state]: ((image_1849792670_state @ F @ bot_bo105666705_state) = bot_bo105666705_state)))). % image_empty
thf(fact_144_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_145_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_146_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_147_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_148_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_149_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_150_image__insert, axiom,
    ((![F : pname > hoare_958474565_state, A2 : pname, B2 : set_pname]: ((image_2144627828_state @ F @ (insert_pname @ A2 @ B2)) = (insert776267541_state @ (F @ A2) @ (image_2144627828_state @ F @ B2)))))). % image_insert
thf(fact_151_image__insert, axiom,
    ((![F : hoare_958474565_state > hoare_958474565_state, A2 : hoare_958474565_state, B2 : set_Ho840737317_state]: ((image_1849792670_state @ F @ (insert776267541_state @ A2 @ B2)) = (insert776267541_state @ (F @ A2) @ (image_1849792670_state @ F @ B2)))))). % image_insert
thf(fact_152_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_153_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_154_empty__subsetI, axiom,
    ((![A : set_Ho840737317_state]: (ord_le1945819589_state @ bot_bo105666705_state @ A)))). % empty_subsetI
thf(fact_155_subset__empty, axiom,
    ((![A : set_Ho840737317_state]: ((ord_le1945819589_state @ A @ bot_bo105666705_state) = (A = bot_bo105666705_state))))). % subset_empty
thf(fact_156_singletonI, axiom,
    ((![A2 : pname]: (member_pname @ A2 @ (insert_pname @ A2 @ bot_bot_set_pname))))). % singletonI
thf(fact_157_singletonI, axiom,
    ((![A2 : hoare_958474565_state]: (member109514606_state @ A2 @ (insert776267541_state @ A2 @ bot_bo105666705_state))))). % singletonI
thf(fact_158_finite__insert, axiom,
    ((![A2 : hoare_958474565_state, A : set_Ho840737317_state]: ((finite1986656878_state @ (insert776267541_state @ A2 @ A)) = (finite1986656878_state @ A))))). % finite_insert
thf(fact_159_finite__insert, axiom,
    ((![A2 : pname, A : set_pname]: ((finite_finite_pname @ (insert_pname @ A2 @ A)) = (finite_finite_pname @ A))))). % finite_insert
thf(fact_160_insert__subset, axiom,
    ((![X3 : pname, A : set_pname, B2 : set_pname]: ((ord_le865024672_pname @ (insert_pname @ X3 @ A) @ B2) = (((member_pname @ X3 @ B2)) & ((ord_le865024672_pname @ A @ B2))))))). % insert_subset
thf(fact_161_insert__subset, axiom,
    ((![X3 : hoare_958474565_state, A : set_Ho840737317_state, B2 : set_Ho840737317_state]: ((ord_le1945819589_state @ (insert776267541_state @ X3 @ A) @ B2) = (((member109514606_state @ X3 @ B2)) & ((ord_le1945819589_state @ A @ B2))))))). % insert_subset
thf(fact_162_Un__empty, axiom,
    ((![A : set_Ho840737317_state, B2 : set_Ho840737317_state]: (((sup_su737426425_state @ A @ B2) = bot_bo105666705_state) = (((A = bot_bo105666705_state)) & ((B2 = bot_bo105666705_state))))))). % Un_empty
thf(fact_163_finite__Un, axiom,
    ((![F2 : set_pname, G3 : set_pname]: ((finite_finite_pname @ (sup_sup_set_pname @ F2 @ G3)) = (((finite_finite_pname @ F2)) & ((finite_finite_pname @ G3))))))). % finite_Un
thf(fact_164_finite__Un, axiom,
    ((![F2 : set_Ho840737317_state, G3 : set_Ho840737317_state]: ((finite1986656878_state @ (sup_su737426425_state @ F2 @ G3)) = (((finite1986656878_state @ F2)) & ((finite1986656878_state @ G3))))))). % finite_Un
thf(fact_165_Un__subset__iff, axiom,
    ((![A : set_Ho840737317_state, B2 : set_Ho840737317_state, C2 : set_Ho840737317_state]: ((ord_le1945819589_state @ (sup_su737426425_state @ A @ B2) @ C2) = (((ord_le1945819589_state @ A @ C2)) & ((ord_le1945819589_state @ B2 @ C2))))))). % Un_subset_iff
thf(fact_166_Un__insert__right, axiom,
    ((![A : set_Ho840737317_state, A2 : hoare_958474565_state, B2 : set_Ho840737317_state]: ((sup_su737426425_state @ A @ (insert776267541_state @ A2 @ B2)) = (insert776267541_state @ A2 @ (sup_su737426425_state @ A @ B2)))))). % Un_insert_right
thf(fact_167_Un__insert__left, axiom,
    ((![A2 : hoare_958474565_state, B2 : set_Ho840737317_state, C2 : set_Ho840737317_state]: ((sup_su737426425_state @ (insert776267541_state @ A2 @ B2) @ C2) = (insert776267541_state @ A2 @ (sup_su737426425_state @ B2 @ C2)))))). % Un_insert_left
thf(fact_168_singleton__conv2, axiom,
    ((![A2 : pname]: ((collect_pname @ ((^[Y : pname]: (^[Z : pname]: (Y = Z))) @ A2)) = (insert_pname @ A2 @ bot_bot_set_pname))))). % singleton_conv2
thf(fact_169_singleton__conv2, axiom,
    ((![A2 : hoare_958474565_state]: ((collec305460656_state @ ((^[Y : hoare_958474565_state]: (^[Z : hoare_958474565_state]: (Y = Z))) @ A2)) = (insert776267541_state @ A2 @ bot_bo105666705_state))))). % singleton_conv2
thf(fact_170_singleton__conv, axiom,
    ((![A2 : pname]: ((collect_pname @ (^[X4 : pname]: (X4 = A2))) = (insert_pname @ A2 @ bot_bot_set_pname))))). % singleton_conv
thf(fact_171_singleton__conv, axiom,
    ((![A2 : hoare_958474565_state]: ((collec305460656_state @ (^[X4 : hoare_958474565_state]: (X4 = A2))) = (insert776267541_state @ A2 @ bot_bo105666705_state))))). % singleton_conv
thf(fact_172_finite__Collect__subsets, axiom,
    ((![A : set_pname]: ((finite_finite_pname @ A) => (finite505202775_pname @ (collect_set_pname @ (^[B4 : set_pname]: (ord_le865024672_pname @ B4 @ A)))))))). % finite_Collect_subsets
thf(fact_173_finite__Collect__subsets, axiom,
    ((![A : set_Ho840737317_state]: ((finite1986656878_state @ A) => (finite1266962254_state @ (collec305218192_state @ (^[B4 : set_Ho840737317_state]: (ord_le1945819589_state @ B4 @ A)))))))). % finite_Collect_subsets
thf(fact_174_singleton__insert__inj__eq_H, axiom,
    ((![A2 : hoare_958474565_state, A : set_Ho840737317_state, B : hoare_958474565_state]: (((insert776267541_state @ A2 @ A) = (insert776267541_state @ B @ bot_bo105666705_state)) = (((A2 = B)) & ((ord_le1945819589_state @ A @ (insert776267541_state @ B @ bot_bo105666705_state)))))))). % singleton_insert_inj_eq'
thf(fact_175_singleton__insert__inj__eq, axiom,
    ((![B : hoare_958474565_state, A2 : hoare_958474565_state, A : set_Ho840737317_state]: (((insert776267541_state @ B @ bot_bo105666705_state) = (insert776267541_state @ A2 @ A)) = (((A2 = B)) & ((ord_le1945819589_state @ A @ (insert776267541_state @ B @ bot_bo105666705_state)))))))). % singleton_insert_inj_eq
thf(fact_176_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_177_subset__singleton__iff, axiom,
    ((![X6 : set_Ho840737317_state, A2 : hoare_958474565_state]: ((ord_le1945819589_state @ X6 @ (insert776267541_state @ A2 @ bot_bo105666705_state)) = (((X6 = bot_bo105666705_state)) | ((X6 = (insert776267541_state @ A2 @ bot_bo105666705_state)))))))). % subset_singleton_iff
thf(fact_178_finite__subset__induct, axiom,
    ((![F2 : set_pname, A : set_pname, P : set_pname > $o]: ((finite_finite_pname @ F2) => ((ord_le865024672_pname @ F2 @ A) => ((P @ bot_bot_set_pname) => ((![A5 : pname, F3 : set_pname]: ((finite_finite_pname @ F3) => ((member_pname @ A5 @ A) => ((~ ((member_pname @ A5 @ F3))) => ((P @ F3) => (P @ (insert_pname @ A5 @ F3))))))) => (P @ F2)))))))). % finite_subset_induct
thf(fact_179_finite__subset__induct, axiom,
    ((![F2 : set_Ho840737317_state, A : set_Ho840737317_state, P : set_Ho840737317_state > $o]: ((finite1986656878_state @ F2) => ((ord_le1945819589_state @ F2 @ A) => ((P @ bot_bo105666705_state) => ((![A5 : hoare_958474565_state, F3 : set_Ho840737317_state]: ((finite1986656878_state @ F3) => ((member109514606_state @ A5 @ A) => ((~ ((member109514606_state @ A5 @ F3))) => ((P @ F3) => (P @ (insert776267541_state @ A5 @ F3))))))) => (P @ F2)))))))). % finite_subset_induct
thf(fact_180_finite__subset__induct_H, axiom,
    ((![F2 : set_pname, A : set_pname, P : set_pname > $o]: ((finite_finite_pname @ F2) => ((ord_le865024672_pname @ F2 @ A) => ((P @ bot_bot_set_pname) => ((![A5 : pname, F3 : set_pname]: ((finite_finite_pname @ F3) => ((member_pname @ A5 @ A) => ((ord_le865024672_pname @ F3 @ A) => ((~ ((member_pname @ A5 @ F3))) => ((P @ F3) => (P @ (insert_pname @ A5 @ F3)))))))) => (P @ F2)))))))). % finite_subset_induct'
thf(fact_181_finite__subset__induct_H, axiom,
    ((![F2 : set_Ho840737317_state, A : set_Ho840737317_state, P : set_Ho840737317_state > $o]: ((finite1986656878_state @ F2) => ((ord_le1945819589_state @ F2 @ A) => ((P @ bot_bo105666705_state) => ((![A5 : hoare_958474565_state, F3 : set_Ho840737317_state]: ((finite1986656878_state @ F3) => ((member109514606_state @ A5 @ A) => ((ord_le1945819589_state @ F3 @ A) => ((~ ((member109514606_state @ A5 @ F3))) => ((P @ F3) => (P @ (insert776267541_state @ A5 @ F3)))))))) => (P @ F2)))))))). % finite_subset_induct'
thf(fact_182_finite__has__maximal, axiom,
    ((![A : set_se421025541_state]: ((finite1266962254_state @ A) => ((~ ((A = bot_bo249060721_state))) => (?[X : set_Ho840737317_state]: ((member1959617614_state @ X @ A) & (![Xa : set_Ho840737317_state]: ((member1959617614_state @ Xa @ A) => ((ord_le1945819589_state @ X @ Xa) => (X = Xa))))))))))). % finite_has_maximal
thf(fact_183_finite__has__minimal, axiom,
    ((![A : set_se421025541_state]: ((finite1266962254_state @ A) => ((~ ((A = bot_bo249060721_state))) => (?[X : set_Ho840737317_state]: ((member1959617614_state @ X @ A) & (![Xa : set_Ho840737317_state]: ((member1959617614_state @ Xa @ A) => ((ord_le1945819589_state @ Xa @ X) => (X = Xa))))))))))). % finite_has_minimal
thf(fact_184_hoare__derivs_Oinsert, axiom,
    ((![G3 : set_Ho840737317_state, T3 : hoare_958474565_state, Ts : set_Ho840737317_state]: ((hoare_604442164_state @ G3 @ (insert776267541_state @ T3 @ bot_bo105666705_state)) => ((hoare_604442164_state @ G3 @ Ts) => (hoare_604442164_state @ G3 @ (insert776267541_state @ T3 @ Ts))))))). % hoare_derivs.insert
thf(fact_185_empty, axiom,
    ((![G3 : set_Ho840737317_state]: (hoare_604442164_state @ G3 @ bot_bo105666705_state)))). % empty
thf(fact_186_derivs__insertD, axiom,
    ((![G3 : set_Ho840737317_state, T3 : hoare_958474565_state, Ts : set_Ho840737317_state]: ((hoare_604442164_state @ G3 @ (insert776267541_state @ T3 @ Ts)) => ((hoare_604442164_state @ G3 @ (insert776267541_state @ T3 @ bot_bo105666705_state)) & (hoare_604442164_state @ G3 @ Ts)))))). % derivs_insertD
thf(fact_187_pigeonhole__infinite__rel, axiom,
    ((![A : set_Ho840737317_state, B2 : set_pname, R : hoare_958474565_state > pname > $o]: ((~ ((finite1986656878_state @ A))) => ((finite_finite_pname @ B2) => ((![X : hoare_958474565_state]: ((member109514606_state @ X @ A) => (?[Xa : pname]: ((member_pname @ Xa @ B2) & (R @ X @ Xa))))) => (?[X : pname]: ((member_pname @ X @ B2) & (~ ((finite1986656878_state @ (collec305460656_state @ (^[A4 : hoare_958474565_state]: (((member109514606_state @ A4 @ A)) & ((R @ A4 @ X)))))))))))))))). % pigeonhole_infinite_rel
thf(fact_188_pigeonhole__infinite__rel, axiom,
    ((![A : set_pname, B2 : set_pname, R : pname > pname > $o]: ((~ ((finite_finite_pname @ A))) => ((finite_finite_pname @ B2) => ((![X : pname]: ((member_pname @ X @ A) => (?[Xa : pname]: ((member_pname @ Xa @ B2) & (R @ X @ Xa))))) => (?[X : pname]: ((member_pname @ X @ B2) & (~ ((finite_finite_pname @ (collect_pname @ (^[A4 : pname]: (((member_pname @ A4 @ A)) & ((R @ A4 @ X)))))))))))))))). % pigeonhole_infinite_rel
thf(fact_189_infinite__finite__induct, axiom,
    ((![P : set_pname > $o, A : set_pname]: ((![A6 : set_pname]: ((~ ((finite_finite_pname @ A6))) => (P @ A6))) => ((P @ bot_bot_set_pname) => ((![X : pname, F3 : set_pname]: ((finite_finite_pname @ F3) => ((~ ((member_pname @ X @ F3))) => ((P @ F3) => (P @ (insert_pname @ X @ F3)))))) => (P @ A))))))). % infinite_finite_induct
thf(fact_190_infinite__finite__induct, axiom,
    ((![P : set_Ho840737317_state > $o, A : set_Ho840737317_state]: ((![A6 : set_Ho840737317_state]: ((~ ((finite1986656878_state @ A6))) => (P @ A6))) => ((P @ bot_bo105666705_state) => ((![X : hoare_958474565_state, F3 : set_Ho840737317_state]: ((finite1986656878_state @ F3) => ((~ ((member109514606_state @ X @ F3))) => ((P @ F3) => (P @ (insert776267541_state @ X @ F3)))))) => (P @ A))))))). % infinite_finite_induct
thf(fact_191_infinite__imp__nonempty, axiom,
    ((![S : set_pname]: ((~ ((finite_finite_pname @ S))) => (~ ((S = bot_bot_set_pname))))))). % infinite_imp_nonempty
thf(fact_192_infinite__imp__nonempty, axiom,
    ((![S : set_Ho840737317_state]: ((~ ((finite1986656878_state @ S))) => (~ ((S = bot_bo105666705_state))))))). % infinite_imp_nonempty
thf(fact_193_not__finite__existsD, axiom,
    ((![P : pname > $o]: ((~ ((finite_finite_pname @ (collect_pname @ P)))) => (?[X_1 : pname]: (P @ X_1)))))). % not_finite_existsD
thf(fact_194_finite__ne__induct, axiom,
    ((![F2 : set_pname, P : set_pname > $o]: ((finite_finite_pname @ F2) => ((~ ((F2 = bot_bot_set_pname))) => ((![X : pname]: (P @ (insert_pname @ X @ bot_bot_set_pname))) => ((![X : pname, F3 : set_pname]: ((finite_finite_pname @ F3) => ((~ ((F3 = bot_bot_set_pname))) => ((~ ((member_pname @ X @ F3))) => ((P @ F3) => (P @ (insert_pname @ X @ F3))))))) => (P @ F2)))))))). % finite_ne_induct
thf(fact_195_finite__ne__induct, axiom,
    ((![F2 : set_Ho840737317_state, P : set_Ho840737317_state > $o]: ((finite1986656878_state @ F2) => ((~ ((F2 = bot_bo105666705_state))) => ((![X : hoare_958474565_state]: (P @ (insert776267541_state @ X @ bot_bo105666705_state))) => ((![X : hoare_958474565_state, F3 : set_Ho840737317_state]: ((finite1986656878_state @ F3) => ((~ ((F3 = bot_bo105666705_state))) => ((~ ((member109514606_state @ X @ F3))) => ((P @ F3) => (P @ (insert776267541_state @ X @ F3))))))) => (P @ F2)))))))). % finite_ne_induct
thf(fact_196_finite_OinsertI, axiom,
    ((![A : set_Ho840737317_state, A2 : hoare_958474565_state]: ((finite1986656878_state @ A) => (finite1986656878_state @ (insert776267541_state @ A2 @ A)))))). % finite.insertI
thf(fact_197_finite_OinsertI, axiom,
    ((![A : set_pname, A2 : pname]: ((finite_finite_pname @ A) => (finite_finite_pname @ (insert_pname @ A2 @ A)))))). % finite.insertI
thf(fact_198_finite_Oinducts, axiom,
    ((![X3 : set_pname, P : set_pname > $o]: ((finite_finite_pname @ X3) => ((P @ bot_bot_set_pname) => ((![A6 : set_pname, A5 : pname]: ((finite_finite_pname @ A6) => ((P @ A6) => (P @ (insert_pname @ A5 @ A6))))) => (P @ X3))))))). % finite.inducts
thf(fact_199_finite_Oinducts, axiom,
    ((![X3 : set_Ho840737317_state, P : set_Ho840737317_state > $o]: ((finite1986656878_state @ X3) => ((P @ bot_bo105666705_state) => ((![A6 : set_Ho840737317_state, A5 : hoare_958474565_state]: ((finite1986656878_state @ A6) => ((P @ A6) => (P @ (insert776267541_state @ A5 @ A6))))) => (P @ X3))))))). % finite.inducts

% Conjectures (4)
thf(conj_0, hypothesis,
    (wT_bodies)).
thf(conj_1, hypothesis,
    ((wt @ c))).
thf(conj_2, hypothesis,
    (((body @ pn) = (some_com @ y)))).
thf(conj_3, conjecture,
    ((ord_le1945819589_state @ (image_2144627828_state @ (^[Pn2 : pname]: (hoare_Mirabelle_MGT @ (the_com @ (body @ Pn2)))) @ (dom_pname_com @ body)) @ (image_2144627828_state @ (^[Pn2 : pname]: (hoare_Mirabelle_MGT @ (the_com @ (body @ Pn2)))) @ (dom_pname_com @ body))))).
