% 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_413__3254546_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:14:40.781

% Could-be-implicit typings (11)
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__Option__Ooption_It__Com__Ocom_J_J, type,
    set_option_com : $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__Set__Oset_It__Com__Ocom_J, type,
    set_com : $tType).
thf(ty_n_t__Com__Ovname, type,
    vname : $tType).
thf(ty_n_t__Com__Ostate, type,
    state : $tType).
thf(ty_n_t__Com__Opname, type,
    pname : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).
thf(ty_n_t__Com__Ocom, type,
    com : $tType).

% Explicit typings (39)
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_Com_Ocom_OCall, type,
    call : vname > pname > (state > nat) > 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_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_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_ONone_001t__Com__Ocom, type,
    none_com : option_com).
thf(sy_c_Option_Ooption_OSome_001t__Com__Ocom, type,
    some_com : com > option_com).
thf(sy_c_Option_Ooption_Othe_001t__Com__Ocom, type,
    the_com : option_com > com).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Com__Opname_M_Eo_J, type,
    bot_bot_pname_o : pname > $o).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J_M_Eo_J, type,
    bot_bo1428770700tate_o : hoare_958474565_state > $o).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Com__Opname_J, type,
    bot_bot_set_pname : set_pname).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J_J, type,
    bot_bo105666705_state : set_Ho840737317_state).
thf(sy_c_Set_OCollect_001t__Com__Opname, type,
    collect_pname : (pname > $o) > set_pname).
thf(sy_c_Set_OCollect_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J, type,
    collec305460656_state : (hoare_958474565_state > $o) > set_Ho840737317_state).
thf(sy_c_Set_Oimage_001t__Com__Ocom_001t__Option__Ooption_It__Com__Ocom_J, type,
    image_com_option_com : (com > option_com) > set_com > set_option_com).
thf(sy_c_Set_Oimage_001t__Com__Opname_001t__Com__Opname, type,
    image_pname_pname : (pname > pname) > set_pname > set_pname).
thf(sy_c_Set_Oimage_001t__Com__Opname_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J, type,
    image_2144627828_state : (pname > hoare_958474565_state) > set_pname > set_Ho840737317_state).
thf(sy_c_Set_Oimage_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J_001t__Com__Opname, type,
    image_871391498_pname : (hoare_958474565_state > pname) > set_Ho840737317_state > set_pname).
thf(sy_c_Set_Oimage_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J, type,
    image_1849792670_state : (hoare_958474565_state > hoare_958474565_state) > set_Ho840737317_state > set_Ho840737317_state).
thf(sy_c_Set_Oinsert_001t__Com__Opname, type,
    insert_pname : pname > set_pname > set_pname).
thf(sy_c_Set_Oinsert_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J, type,
    insert776267541_state : hoare_958474565_state > set_Ho840737317_state > set_Ho840737317_state).
thf(sy_c_Set_Ois__singleton_001t__Com__Opname, type,
    is_singleton_pname : set_pname > $o).
thf(sy_c_Set_Ois__singleton_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J, type,
    is_sin536631353_state : set_Ho840737317_state > $o).
thf(sy_c_Set_Opairwise_001t__Com__Opname, type,
    pairwise_pname : (pname > pname > $o) > set_pname > $o).
thf(sy_c_Set_Opairwise_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J, type,
    pairwi17519280_state : (hoare_958474565_state > hoare_958474565_state > $o) > set_Ho840737317_state > $o).
thf(sy_c_Set_Othe__elem_001t__Com__Opname, type,
    the_elem_pname : set_pname > pname).
thf(sy_c_Set_Othe__elem_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J, type,
    the_el1300254266_state : set_Ho840737317_state > hoare_958474565_state).
thf(sy_c_member_001t__Com__Opname, type,
    member_pname : pname > set_pname > $o).
thf(sy_c_member_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J, type,
    member109514606_state : hoare_958474565_state > set_Ho840737317_state > $o).
thf(sy_c_member_001t__Option__Ooption_It__Com__Ocom_J, type,
    member_option_com : option_com > set_option_com > $o).
thf(sy_v_c, type,
    c : com).
thf(sy_v_uG__0_058ATP, type,
    uG_0_ATP : set_Ho840737317_state).

% Relevant facts (247)
thf(fact_0_single__stateE, axiom,
    ((hoare_405891322gleton => (![T : state]: (~ ((![S : state]: (S = T)))))))). % single_stateE
thf(fact_1_state__not__singleton__def, axiom,
    ((hoare_405891322gleton = (?[S2 : state]: (?[T2 : state]: (~ ((S2 = T2)))))))). % state_not_singleton_def
thf(fact_2_image__ident, axiom,
    ((![Y : set_pname]: ((image_pname_pname @ (^[X : pname]: X) @ Y) = Y)))). % image_ident
thf(fact_3_image__ident, axiom,
    ((![Y : set_Ho840737317_state]: ((image_1849792670_state @ (^[X : hoare_958474565_state]: X) @ Y) = Y)))). % image_ident
thf(fact_4_com_Oinject_I6_J, axiom,
    ((![X7 : pname, Y7 : pname]: (((body2 @ X7) = (body2 @ Y7)) = (X7 = Y7))))). % com.inject(6)
thf(fact_5_image__eqI, axiom,
    ((![B : pname, F : pname > pname, X2 : pname, A : set_pname]: ((B = (F @ X2)) => ((member_pname @ X2 @ A) => (member_pname @ B @ (image_pname_pname @ F @ A))))))). % image_eqI
thf(fact_6_image__eqI, axiom,
    ((![B : pname, F : hoare_958474565_state > pname, X2 : hoare_958474565_state, A : set_Ho840737317_state]: ((B = (F @ X2)) => ((member109514606_state @ X2 @ A) => (member_pname @ B @ (image_871391498_pname @ F @ A))))))). % image_eqI
thf(fact_7_image__eqI, axiom,
    ((![B : hoare_958474565_state, F : hoare_958474565_state > hoare_958474565_state, X2 : hoare_958474565_state, A : set_Ho840737317_state]: ((B = (F @ X2)) => ((member109514606_state @ X2 @ A) => (member109514606_state @ B @ (image_1849792670_state @ F @ A))))))). % image_eqI
thf(fact_8_image__eqI, axiom,
    ((![B : hoare_958474565_state, F : pname > hoare_958474565_state, X2 : pname, A : set_pname]: ((B = (F @ X2)) => ((member_pname @ X2 @ A) => (member109514606_state @ B @ (image_2144627828_state @ F @ A))))))). % image_eqI
thf(fact_9_imageE, axiom,
    ((![B : pname, F : pname > pname, A : set_pname]: ((member_pname @ B @ (image_pname_pname @ F @ A)) => (~ ((![X3 : pname]: ((B = (F @ X3)) => (~ ((member_pname @ X3 @ A))))))))))). % imageE
thf(fact_10_imageE, axiom,
    ((![B : pname, F : hoare_958474565_state > pname, A : set_Ho840737317_state]: ((member_pname @ B @ (image_871391498_pname @ F @ A)) => (~ ((![X3 : hoare_958474565_state]: ((B = (F @ X3)) => (~ ((member109514606_state @ X3 @ A))))))))))). % imageE
thf(fact_11_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)) => (~ ((![X3 : hoare_958474565_state]: ((B = (F @ X3)) => (~ ((member109514606_state @ X3 @ A))))))))))). % imageE
thf(fact_12_imageE, axiom,
    ((![B : hoare_958474565_state, F : pname > hoare_958474565_state, A : set_pname]: ((member109514606_state @ B @ (image_2144627828_state @ F @ A)) => (~ ((![X3 : pname]: ((B = (F @ X3)) => (~ ((member_pname @ X3 @ A))))))))))). % imageE
thf(fact_13_image__image, axiom,
    ((![F : hoare_958474565_state > pname, G : pname > hoare_958474565_state, A : set_pname]: ((image_871391498_pname @ F @ (image_2144627828_state @ G @ A)) = (image_pname_pname @ (^[X : pname]: (F @ (G @ X))) @ A))))). % image_image
thf(fact_14_image__image, axiom,
    ((![F : pname > hoare_958474565_state, G : hoare_958474565_state > pname, A : set_Ho840737317_state]: ((image_2144627828_state @ F @ (image_871391498_pname @ G @ A)) = (image_1849792670_state @ (^[X : hoare_958474565_state]: (F @ (G @ X))) @ A))))). % image_image
thf(fact_15_image__image, axiom,
    ((![F : pname > pname, G : pname > pname, A : set_pname]: ((image_pname_pname @ F @ (image_pname_pname @ G @ A)) = (image_pname_pname @ (^[X : pname]: (F @ (G @ X))) @ A))))). % image_image
thf(fact_16_image__image, axiom,
    ((![F : hoare_958474565_state > hoare_958474565_state, G : hoare_958474565_state > hoare_958474565_state, A : set_Ho840737317_state]: ((image_1849792670_state @ F @ (image_1849792670_state @ G @ A)) = (image_1849792670_state @ (^[X : hoare_958474565_state]: (F @ (G @ X))) @ A))))). % image_image
thf(fact_17_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 @ (^[X : pname]: (F @ (G @ X))) @ A))))). % image_image
thf(fact_18_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 @ (^[X : pname]: (F @ (G @ X))) @ A))))). % image_image
thf(fact_19_Compr__image__eq, axiom,
    ((![F : hoare_958474565_state > hoare_958474565_state, A : set_Ho840737317_state, P : hoare_958474565_state > $o]: ((collec305460656_state @ (^[X : hoare_958474565_state]: (((member109514606_state @ X @ (image_1849792670_state @ F @ A))) & ((P @ X))))) = (image_1849792670_state @ F @ (collec305460656_state @ (^[X : hoare_958474565_state]: (((member109514606_state @ X @ A)) & ((P @ (F @ X))))))))))). % Compr_image_eq
thf(fact_20_Compr__image__eq, axiom,
    ((![F : hoare_958474565_state > pname, A : set_Ho840737317_state, P : pname > $o]: ((collect_pname @ (^[X : pname]: (((member_pname @ X @ (image_871391498_pname @ F @ A))) & ((P @ X))))) = (image_871391498_pname @ F @ (collec305460656_state @ (^[X : hoare_958474565_state]: (((member109514606_state @ X @ A)) & ((P @ (F @ X))))))))))). % Compr_image_eq
thf(fact_21_Compr__image__eq, axiom,
    ((![F : pname > pname, A : set_pname, P : pname > $o]: ((collect_pname @ (^[X : pname]: (((member_pname @ X @ (image_pname_pname @ F @ A))) & ((P @ X))))) = (image_pname_pname @ F @ (collect_pname @ (^[X : pname]: (((member_pname @ X @ A)) & ((P @ (F @ X))))))))))). % Compr_image_eq
thf(fact_22_Compr__image__eq, axiom,
    ((![F : pname > hoare_958474565_state, A : set_pname, P : hoare_958474565_state > $o]: ((collec305460656_state @ (^[X : hoare_958474565_state]: (((member109514606_state @ X @ (image_2144627828_state @ F @ A))) & ((P @ X))))) = (image_2144627828_state @ F @ (collect_pname @ (^[X : pname]: (((member_pname @ X @ A)) & ((P @ (F @ X))))))))))). % Compr_image_eq
thf(fact_23_Inf_OINF__identity__eq, axiom,
    ((![Inf : set_pname > pname, A : set_pname]: ((Inf @ (image_pname_pname @ (^[X : pname]: X) @ A)) = (Inf @ A))))). % Inf.INF_identity_eq
thf(fact_24_Inf_OINF__identity__eq, axiom,
    ((![Inf : set_Ho840737317_state > hoare_958474565_state, A : set_Ho840737317_state]: ((Inf @ (image_1849792670_state @ (^[X : hoare_958474565_state]: X) @ A)) = (Inf @ A))))). % Inf.INF_identity_eq
thf(fact_25_Sup_OSUP__identity__eq, axiom,
    ((![Sup : set_pname > pname, A : set_pname]: ((Sup @ (image_pname_pname @ (^[X : pname]: X) @ A)) = (Sup @ A))))). % Sup.SUP_identity_eq
thf(fact_26_Sup_OSUP__identity__eq, axiom,
    ((![Sup : set_Ho840737317_state > hoare_958474565_state, A : set_Ho840737317_state]: ((Sup @ (image_1849792670_state @ (^[X : hoare_958474565_state]: X) @ A)) = (Sup @ A))))). % Sup.SUP_identity_eq
thf(fact_27_imageI, axiom,
    ((![X2 : pname, A : set_pname, F : pname > pname]: ((member_pname @ X2 @ A) => (member_pname @ (F @ X2) @ (image_pname_pname @ F @ A)))))). % imageI
thf(fact_28_imageI, axiom,
    ((![X2 : pname, A : set_pname, F : pname > hoare_958474565_state]: ((member_pname @ X2 @ A) => (member109514606_state @ (F @ X2) @ (image_2144627828_state @ F @ A)))))). % imageI
thf(fact_29_imageI, axiom,
    ((![X2 : hoare_958474565_state, A : set_Ho840737317_state, F : hoare_958474565_state > pname]: ((member109514606_state @ X2 @ A) => (member_pname @ (F @ X2) @ (image_871391498_pname @ F @ A)))))). % imageI
thf(fact_30_imageI, axiom,
    ((![X2 : hoare_958474565_state, A : set_Ho840737317_state, F : hoare_958474565_state > hoare_958474565_state]: ((member109514606_state @ X2 @ A) => (member109514606_state @ (F @ X2) @ (image_1849792670_state @ F @ A)))))). % imageI
thf(fact_31_image__iff, axiom,
    ((![Z : pname, F : pname > pname, A : set_pname]: ((member_pname @ Z @ (image_pname_pname @ F @ A)) = (?[X : pname]: (((member_pname @ X @ A)) & ((Z = (F @ X))))))))). % image_iff
thf(fact_32_image__iff, axiom,
    ((![Z : hoare_958474565_state, F : pname > hoare_958474565_state, A : set_pname]: ((member109514606_state @ Z @ (image_2144627828_state @ F @ A)) = (?[X : pname]: (((member_pname @ X @ A)) & ((Z = (F @ X))))))))). % image_iff
thf(fact_33_image__iff, axiom,
    ((![Z : hoare_958474565_state, F : hoare_958474565_state > hoare_958474565_state, A : set_Ho840737317_state]: ((member109514606_state @ Z @ (image_1849792670_state @ F @ A)) = (?[X : hoare_958474565_state]: (((member109514606_state @ X @ A)) & ((Z = (F @ X))))))))). % image_iff
thf(fact_34_bex__imageD, axiom,
    ((![F : pname > hoare_958474565_state, A : set_pname, P : hoare_958474565_state > $o]: ((?[X4 : hoare_958474565_state]: ((member109514606_state @ X4 @ (image_2144627828_state @ F @ A)) & (P @ X4))) => (?[X3 : pname]: ((member_pname @ X3 @ A) & (P @ (F @ X3)))))))). % bex_imageD
thf(fact_35_bex__imageD, axiom,
    ((![F : pname > pname, A : set_pname, P : pname > $o]: ((?[X4 : pname]: ((member_pname @ X4 @ (image_pname_pname @ F @ A)) & (P @ X4))) => (?[X3 : pname]: ((member_pname @ X3 @ A) & (P @ (F @ X3)))))))). % bex_imageD
thf(fact_36_bex__imageD, axiom,
    ((![F : hoare_958474565_state > hoare_958474565_state, A : set_Ho840737317_state, P : hoare_958474565_state > $o]: ((?[X4 : hoare_958474565_state]: ((member109514606_state @ X4 @ (image_1849792670_state @ F @ A)) & (P @ X4))) => (?[X3 : hoare_958474565_state]: ((member109514606_state @ X3 @ A) & (P @ (F @ X3)))))))). % bex_imageD
thf(fact_37_image__cong, axiom,
    ((![M : set_pname, N : set_pname, F : pname > hoare_958474565_state, G : pname > hoare_958474565_state]: ((M = N) => ((![X3 : pname]: ((member_pname @ X3 @ N) => ((F @ X3) = (G @ X3)))) => ((image_2144627828_state @ F @ M) = (image_2144627828_state @ G @ N))))))). % image_cong
thf(fact_38_image__cong, axiom,
    ((![M : set_pname, N : set_pname, F : pname > pname, G : pname > pname]: ((M = N) => ((![X3 : pname]: ((member_pname @ X3 @ N) => ((F @ X3) = (G @ X3)))) => ((image_pname_pname @ F @ M) = (image_pname_pname @ G @ N))))))). % image_cong
thf(fact_39_image__cong, axiom,
    ((![M : set_Ho840737317_state, N : set_Ho840737317_state, F : hoare_958474565_state > hoare_958474565_state, G : hoare_958474565_state > hoare_958474565_state]: ((M = N) => ((![X3 : hoare_958474565_state]: ((member109514606_state @ X3 @ N) => ((F @ X3) = (G @ X3)))) => ((image_1849792670_state @ F @ M) = (image_1849792670_state @ G @ N))))))). % image_cong
thf(fact_40_Sup_OSUP__cong, axiom,
    ((![A : set_pname, B2 : set_pname, C : pname > hoare_958474565_state, D : pname > hoare_958474565_state, Sup : set_Ho840737317_state > hoare_958474565_state]: ((A = B2) => ((![X3 : pname]: ((member_pname @ X3 @ B2) => ((C @ X3) = (D @ X3)))) => ((Sup @ (image_2144627828_state @ C @ A)) = (Sup @ (image_2144627828_state @ D @ B2)))))))). % Sup.SUP_cong
thf(fact_41_Sup_OSUP__cong, axiom,
    ((![A : set_pname, B2 : set_pname, C : pname > pname, D : pname > pname, Sup : set_pname > pname]: ((A = B2) => ((![X3 : pname]: ((member_pname @ X3 @ B2) => ((C @ X3) = (D @ X3)))) => ((Sup @ (image_pname_pname @ C @ A)) = (Sup @ (image_pname_pname @ D @ B2)))))))). % Sup.SUP_cong
thf(fact_42_Sup_OSUP__cong, axiom,
    ((![A : set_Ho840737317_state, B2 : set_Ho840737317_state, C : hoare_958474565_state > hoare_958474565_state, D : hoare_958474565_state > hoare_958474565_state, Sup : set_Ho840737317_state > hoare_958474565_state]: ((A = B2) => ((![X3 : hoare_958474565_state]: ((member109514606_state @ X3 @ B2) => ((C @ X3) = (D @ X3)))) => ((Sup @ (image_1849792670_state @ C @ A)) = (Sup @ (image_1849792670_state @ D @ B2)))))))). % Sup.SUP_cong
thf(fact_43_Inf_OINF__cong, axiom,
    ((![A : set_pname, B2 : set_pname, C : pname > hoare_958474565_state, D : pname > hoare_958474565_state, Inf : set_Ho840737317_state > hoare_958474565_state]: ((A = B2) => ((![X3 : pname]: ((member_pname @ X3 @ B2) => ((C @ X3) = (D @ X3)))) => ((Inf @ (image_2144627828_state @ C @ A)) = (Inf @ (image_2144627828_state @ D @ B2)))))))). % Inf.INF_cong
thf(fact_44_Inf_OINF__cong, axiom,
    ((![A : set_pname, B2 : set_pname, C : pname > pname, D : pname > pname, Inf : set_pname > pname]: ((A = B2) => ((![X3 : pname]: ((member_pname @ X3 @ B2) => ((C @ X3) = (D @ X3)))) => ((Inf @ (image_pname_pname @ C @ A)) = (Inf @ (image_pname_pname @ D @ B2)))))))). % Inf.INF_cong
thf(fact_45_Inf_OINF__cong, axiom,
    ((![A : set_Ho840737317_state, B2 : set_Ho840737317_state, C : hoare_958474565_state > hoare_958474565_state, D : hoare_958474565_state > hoare_958474565_state, Inf : set_Ho840737317_state > hoare_958474565_state]: ((A = B2) => ((![X3 : hoare_958474565_state]: ((member109514606_state @ X3 @ B2) => ((C @ X3) = (D @ X3)))) => ((Inf @ (image_1849792670_state @ C @ A)) = (Inf @ (image_1849792670_state @ D @ B2)))))))). % Inf.INF_cong
thf(fact_46_rev__image__eqI, axiom,
    ((![X2 : pname, A : set_pname, B : pname, F : pname > pname]: ((member_pname @ X2 @ A) => ((B = (F @ X2)) => (member_pname @ B @ (image_pname_pname @ F @ A))))))). % rev_image_eqI
thf(fact_47_rev__image__eqI, axiom,
    ((![X2 : pname, A : set_pname, B : hoare_958474565_state, F : pname > hoare_958474565_state]: ((member_pname @ X2 @ A) => ((B = (F @ X2)) => (member109514606_state @ B @ (image_2144627828_state @ F @ A))))))). % rev_image_eqI
thf(fact_48_rev__image__eqI, axiom,
    ((![X2 : hoare_958474565_state, A : set_Ho840737317_state, B : pname, F : hoare_958474565_state > pname]: ((member109514606_state @ X2 @ A) => ((B = (F @ X2)) => (member_pname @ B @ (image_871391498_pname @ F @ A))))))). % rev_image_eqI
thf(fact_49_rev__image__eqI, axiom,
    ((![X2 : hoare_958474565_state, A : set_Ho840737317_state, B : hoare_958474565_state, F : hoare_958474565_state > hoare_958474565_state]: ((member109514606_state @ X2 @ A) => ((B = (F @ X2)) => (member109514606_state @ B @ (image_1849792670_state @ F @ A))))))). % rev_image_eqI
thf(fact_50_ball__imageD, axiom,
    ((![F : pname > hoare_958474565_state, A : set_pname, P : hoare_958474565_state > $o]: ((![X3 : hoare_958474565_state]: ((member109514606_state @ X3 @ (image_2144627828_state @ F @ A)) => (P @ X3))) => (![X4 : pname]: ((member_pname @ X4 @ A) => (P @ (F @ X4)))))))). % ball_imageD
thf(fact_51_ball__imageD, axiom,
    ((![F : pname > pname, A : set_pname, P : pname > $o]: ((![X3 : pname]: ((member_pname @ X3 @ (image_pname_pname @ F @ A)) => (P @ X3))) => (![X4 : pname]: ((member_pname @ X4 @ A) => (P @ (F @ X4)))))))). % ball_imageD
thf(fact_52_ball__imageD, axiom,
    ((![F : hoare_958474565_state > hoare_958474565_state, A : set_Ho840737317_state, P : hoare_958474565_state > $o]: ((![X3 : hoare_958474565_state]: ((member109514606_state @ X3 @ (image_1849792670_state @ F @ A)) => (P @ X3))) => (![X4 : hoare_958474565_state]: ((member109514606_state @ X4 @ A) => (P @ (F @ X4)))))))). % ball_imageD
thf(fact_53_WT__bodiesD, axiom,
    ((![Pn : pname, B : com]: (wT_bodies => (((body @ Pn) = (some_com @ B)) => (wt @ B)))))). % WT_bodiesD
thf(fact_54_WT_OBody, axiom,
    ((![Pn : pname]: ((~ (((body @ Pn) = none_com))) => (wt @ (body2 @ Pn)))))). % WT.Body
thf(fact_55_finite__dom__body, axiom,
    ((finite_finite_pname @ (dom_pname_com @ body)))). % finite_dom_body
thf(fact_56_WTs__elim__cases_I7_J, axiom,
    ((![P : pname]: ((wt @ (body2 @ P)) => (~ ((![Y2 : com]: (~ (((body @ P) = (some_com @ Y2))))))))))). % WTs_elim_cases(7)
thf(fact_57_WT_OCall, axiom,
    ((![Pn : pname, X5 : vname, A2 : state > nat]: ((wt @ (body2 @ Pn)) => (wt @ (call @ X5 @ Pn @ A2)))))). % WT.Call
thf(fact_58_WTs__elim__cases_I8_J, axiom,
    ((![X5 : vname, P : pname, A2 : state > nat]: ((wt @ (call @ X5 @ P @ A2)) => (wt @ (body2 @ P)))))). % WTs_elim_cases(8)
thf(fact_59_MGF__lemma1, axiom,
    ((![G2 : set_Ho840737317_state, C2 : com]: (hoare_405891322gleton => ((![X3 : pname]: ((member_pname @ X3 @ (dom_pname_com @ body)) => (hoare_604442164_state @ G2 @ (insert776267541_state @ (hoare_Mirabelle_MGT @ (body2 @ X3)) @ bot_bo105666705_state)))) => ((wt @ C2) => (hoare_604442164_state @ G2 @ (insert776267541_state @ (hoare_Mirabelle_MGT @ C2) @ bot_bo105666705_state)))))))). % MGF_lemma1
thf(fact_60_pairwise__image, axiom,
    ((![R : hoare_958474565_state > hoare_958474565_state > $o, F : pname > hoare_958474565_state, S3 : set_pname]: ((pairwi17519280_state @ R @ (image_2144627828_state @ F @ S3)) = (pairwise_pname @ (^[X : pname]: (^[Y3 : pname]: (((~ (((F @ X) = (F @ Y3))))) => ((R @ (F @ X) @ (F @ Y3)))))) @ S3))))). % pairwise_image
thf(fact_61_pairwise__image, axiom,
    ((![R : pname > pname > $o, F : pname > pname, S3 : set_pname]: ((pairwise_pname @ R @ (image_pname_pname @ F @ S3)) = (pairwise_pname @ (^[X : pname]: (^[Y3 : pname]: (((~ (((F @ X) = (F @ Y3))))) => ((R @ (F @ X) @ (F @ Y3)))))) @ S3))))). % pairwise_image
thf(fact_62_pairwise__image, axiom,
    ((![R : hoare_958474565_state > hoare_958474565_state > $o, F : hoare_958474565_state > hoare_958474565_state, S3 : set_Ho840737317_state]: ((pairwi17519280_state @ R @ (image_1849792670_state @ F @ S3)) = (pairwi17519280_state @ (^[X : hoare_958474565_state]: (^[Y3 : hoare_958474565_state]: (((~ (((F @ X) = (F @ Y3))))) => ((R @ (F @ X) @ (F @ Y3)))))) @ S3))))). % pairwise_image
thf(fact_63_empty__iff, axiom,
    ((![C2 : pname]: (~ ((member_pname @ C2 @ bot_bot_set_pname)))))). % empty_iff
thf(fact_64_empty__iff, axiom,
    ((![C2 : hoare_958474565_state]: (~ ((member109514606_state @ C2 @ bot_bo105666705_state)))))). % empty_iff
thf(fact_65_all__not__in__conv, axiom,
    ((![A : set_pname]: ((![X : pname]: (~ ((member_pname @ X @ A)))) = (A = bot_bot_set_pname))))). % all_not_in_conv
thf(fact_66_all__not__in__conv, axiom,
    ((![A : set_Ho840737317_state]: ((![X : hoare_958474565_state]: (~ ((member109514606_state @ X @ A)))) = (A = bot_bo105666705_state))))). % all_not_in_conv
thf(fact_67_Collect__empty__eq, axiom,
    ((![P : pname > $o]: (((collect_pname @ P) = bot_bot_set_pname) = (![X : pname]: (~ ((P @ X)))))))). % Collect_empty_eq
thf(fact_68_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_69_empty__Collect__eq, axiom,
    ((![P : pname > $o]: ((bot_bot_set_pname = (collect_pname @ P)) = (![X : pname]: (~ ((P @ X)))))))). % empty_Collect_eq
thf(fact_70_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_71_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_72_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_73_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_74_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_75_insert__absorb2, axiom,
    ((![X2 : hoare_958474565_state, A : set_Ho840737317_state]: ((insert776267541_state @ X2 @ (insert776267541_state @ X2 @ A)) = (insert776267541_state @ X2 @ A))))). % insert_absorb2
thf(fact_76_com_Oinject_I7_J, axiom,
    ((![X81 : vname, X82 : pname, X83 : state > nat, Y81 : vname, Y82 : pname, Y83 : state > nat]: (((call @ X81 @ X82 @ X83) = (call @ Y81 @ Y82 @ Y83)) = (((X81 = Y81)) & ((((X82 = Y82)) & ((X83 = Y83))))))))). % com.inject(7)
thf(fact_77_image__empty, axiom,
    ((![F : pname > pname]: ((image_pname_pname @ F @ bot_bot_set_pname) = bot_bot_set_pname)))). % image_empty
thf(fact_78_image__empty, axiom,
    ((![F : pname > hoare_958474565_state]: ((image_2144627828_state @ F @ bot_bot_set_pname) = bot_bo105666705_state)))). % image_empty
thf(fact_79_image__empty, axiom,
    ((![F : hoare_958474565_state > hoare_958474565_state]: ((image_1849792670_state @ F @ bot_bo105666705_state) = bot_bo105666705_state)))). % image_empty
thf(fact_80_empty__is__image, axiom,
    ((![F : pname > pname, A : set_pname]: ((bot_bot_set_pname = (image_pname_pname @ F @ A)) = (A = bot_bot_set_pname))))). % empty_is_image
thf(fact_81_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_82_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_83_image__is__empty, axiom,
    ((![F : pname > pname, A : set_pname]: (((image_pname_pname @ F @ A) = bot_bot_set_pname) = (A = bot_bot_set_pname))))). % image_is_empty
thf(fact_84_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_85_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_86_image__insert, axiom,
    ((![F : pname > pname, A2 : pname, B2 : set_pname]: ((image_pname_pname @ F @ (insert_pname @ A2 @ B2)) = (insert_pname @ (F @ A2) @ (image_pname_pname @ F @ B2)))))). % image_insert
thf(fact_87_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_88_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_89_insert__image, axiom,
    ((![X2 : pname, A : set_pname, F : pname > pname]: ((member_pname @ X2 @ A) => ((insert_pname @ (F @ X2) @ (image_pname_pname @ F @ A)) = (image_pname_pname @ F @ A)))))). % insert_image
thf(fact_90_insert__image, axiom,
    ((![X2 : pname, A : set_pname, F : pname > hoare_958474565_state]: ((member_pname @ X2 @ A) => ((insert776267541_state @ (F @ X2) @ (image_2144627828_state @ F @ A)) = (image_2144627828_state @ F @ A)))))). % insert_image
thf(fact_91_insert__image, axiom,
    ((![X2 : hoare_958474565_state, A : set_Ho840737317_state, F : hoare_958474565_state > hoare_958474565_state]: ((member109514606_state @ X2 @ A) => ((insert776267541_state @ (F @ X2) @ (image_1849792670_state @ F @ A)) = (image_1849792670_state @ F @ A)))))). % insert_image
thf(fact_92_singletonI, axiom,
    ((![A2 : pname]: (member_pname @ A2 @ (insert_pname @ A2 @ bot_bot_set_pname))))). % singletonI
thf(fact_93_singletonI, axiom,
    ((![A2 : hoare_958474565_state]: (member109514606_state @ A2 @ (insert776267541_state @ A2 @ bot_bo105666705_state))))). % singletonI
thf(fact_94_singleton__conv, axiom,
    ((![A2 : pname]: ((collect_pname @ (^[X : pname]: (X = A2))) = (insert_pname @ A2 @ bot_bot_set_pname))))). % singleton_conv
thf(fact_95_singleton__conv, axiom,
    ((![A2 : hoare_958474565_state]: ((collec305460656_state @ (^[X : hoare_958474565_state]: (X = A2))) = (insert776267541_state @ A2 @ bot_bo105666705_state))))). % singleton_conv
thf(fact_96_singleton__conv2, axiom,
    ((![A2 : pname]: ((collect_pname @ ((^[Y4 : pname]: (^[Z2 : pname]: (Y4 = Z2))) @ A2)) = (insert_pname @ A2 @ bot_bot_set_pname))))). % singleton_conv2
thf(fact_97_singleton__conv2, axiom,
    ((![A2 : hoare_958474565_state]: ((collec305460656_state @ ((^[Y4 : hoare_958474565_state]: (^[Z2 : hoare_958474565_state]: (Y4 = Z2))) @ A2)) = (insert776267541_state @ A2 @ bot_bo105666705_state))))). % singleton_conv2
thf(fact_98_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_99_mem__Collect__eq, axiom,
    ((![A2 : pname, P : pname > $o]: ((member_pname @ A2 @ (collect_pname @ P)) = (P @ A2))))). % mem_Collect_eq
thf(fact_100_Collect__mem__eq, axiom,
    ((![A : set_Ho840737317_state]: ((collec305460656_state @ (^[X : hoare_958474565_state]: (member109514606_state @ X @ A))) = A)))). % Collect_mem_eq
thf(fact_101_Collect__mem__eq, axiom,
    ((![A : set_pname]: ((collect_pname @ (^[X : pname]: (member_pname @ X @ A))) = A)))). % Collect_mem_eq
thf(fact_102_Collect__cong, axiom,
    ((![P : hoare_958474565_state > $o, Q : hoare_958474565_state > $o]: ((![X3 : hoare_958474565_state]: ((P @ X3) = (Q @ X3))) => ((collec305460656_state @ P) = (collec305460656_state @ Q)))))). % Collect_cong
thf(fact_103_Collect__cong, axiom,
    ((![P : pname > $o, Q : pname > $o]: ((![X3 : pname]: ((P @ X3) = (Q @ X3))) => ((collect_pname @ P) = (collect_pname @ Q)))))). % Collect_cong
thf(fact_104_emptyE, axiom,
    ((![A2 : pname]: (~ ((member_pname @ A2 @ bot_bot_set_pname)))))). % emptyE
thf(fact_105_emptyE, axiom,
    ((![A2 : hoare_958474565_state]: (~ ((member109514606_state @ A2 @ bot_bo105666705_state)))))). % emptyE
thf(fact_106_insertE, axiom,
    ((![A2 : pname, B : pname, A : set_pname]: ((member_pname @ A2 @ (insert_pname @ B @ A)) => ((~ ((A2 = B))) => (member_pname @ A2 @ A)))))). % insertE
thf(fact_107_insertE, 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)))))). % insertE
thf(fact_108_equals0D, axiom,
    ((![A : set_pname, A2 : pname]: ((A = bot_bot_set_pname) => (~ ((member_pname @ A2 @ A))))))). % equals0D
thf(fact_109_equals0D, axiom,
    ((![A : set_Ho840737317_state, A2 : hoare_958474565_state]: ((A = bot_bo105666705_state) => (~ ((member109514606_state @ A2 @ A))))))). % equals0D
thf(fact_110_equals0I, axiom,
    ((![A : set_pname]: ((![Y2 : pname]: (~ ((member_pname @ Y2 @ A)))) => (A = bot_bot_set_pname))))). % equals0I
thf(fact_111_equals0I, axiom,
    ((![A : set_Ho840737317_state]: ((![Y2 : hoare_958474565_state]: (~ ((member109514606_state @ Y2 @ A)))) => (A = bot_bo105666705_state))))). % equals0I
thf(fact_112_insertI1, axiom,
    ((![A2 : pname, B2 : set_pname]: (member_pname @ A2 @ (insert_pname @ A2 @ B2))))). % insertI1
thf(fact_113_insertI1, axiom,
    ((![A2 : hoare_958474565_state, B2 : set_Ho840737317_state]: (member109514606_state @ A2 @ (insert776267541_state @ A2 @ B2))))). % insertI1
thf(fact_114_insertI2, axiom,
    ((![A2 : pname, B2 : set_pname, B : pname]: ((member_pname @ A2 @ B2) => (member_pname @ A2 @ (insert_pname @ B @ B2)))))). % insertI2
thf(fact_115_insertI2, axiom,
    ((![A2 : hoare_958474565_state, B2 : set_Ho840737317_state, B : hoare_958474565_state]: ((member109514606_state @ A2 @ B2) => (member109514606_state @ A2 @ (insert776267541_state @ B @ B2)))))). % insertI2
thf(fact_116_pairwiseD, axiom,
    ((![R2 : pname > pname > $o, S4 : set_pname, X2 : pname, Y5 : pname]: ((pairwise_pname @ R2 @ S4) => ((member_pname @ X2 @ S4) => ((member_pname @ Y5 @ S4) => ((~ ((X2 = Y5))) => (R2 @ X2 @ Y5)))))))). % pairwiseD
thf(fact_117_pairwiseD, axiom,
    ((![R2 : hoare_958474565_state > hoare_958474565_state > $o, S4 : set_Ho840737317_state, X2 : hoare_958474565_state, Y5 : hoare_958474565_state]: ((pairwi17519280_state @ R2 @ S4) => ((member109514606_state @ X2 @ S4) => ((member109514606_state @ Y5 @ S4) => ((~ ((X2 = Y5))) => (R2 @ X2 @ Y5)))))))). % pairwiseD
thf(fact_118_pairwiseI, axiom,
    ((![S4 : set_pname, R2 : pname > pname > $o]: ((![X3 : pname, Y2 : pname]: ((member_pname @ X3 @ S4) => ((member_pname @ Y2 @ S4) => ((~ ((X3 = Y2))) => (R2 @ X3 @ Y2))))) => (pairwise_pname @ R2 @ S4))))). % pairwiseI
thf(fact_119_pairwiseI, axiom,
    ((![S4 : set_Ho840737317_state, R2 : hoare_958474565_state > hoare_958474565_state > $o]: ((![X3 : hoare_958474565_state, Y2 : hoare_958474565_state]: ((member109514606_state @ X3 @ S4) => ((member109514606_state @ Y2 @ S4) => ((~ ((X3 = Y2))) => (R2 @ X3 @ Y2))))) => (pairwi17519280_state @ R2 @ S4))))). % pairwiseI
thf(fact_120_ex__in__conv, axiom,
    ((![A : set_pname]: ((?[X : pname]: (member_pname @ X @ A)) = (~ ((A = bot_bot_set_pname))))))). % ex_in_conv
thf(fact_121_ex__in__conv, axiom,
    ((![A : set_Ho840737317_state]: ((?[X : hoare_958474565_state]: (member109514606_state @ X @ A)) = (~ ((A = bot_bo105666705_state))))))). % ex_in_conv
thf(fact_122_Set_Oset__insert, axiom,
    ((![X2 : pname, A : set_pname]: ((member_pname @ X2 @ A) => (~ ((![B3 : set_pname]: ((A = (insert_pname @ X2 @ B3)) => (member_pname @ X2 @ B3))))))))). % Set.set_insert
thf(fact_123_Set_Oset__insert, axiom,
    ((![X2 : hoare_958474565_state, A : set_Ho840737317_state]: ((member109514606_state @ X2 @ A) => (~ ((![B3 : set_Ho840737317_state]: ((A = (insert776267541_state @ X2 @ B3)) => (member109514606_state @ X2 @ B3))))))))). % Set.set_insert
thf(fact_124_singletonD, axiom,
    ((![B : pname, A2 : pname]: ((member_pname @ B @ (insert_pname @ A2 @ bot_bot_set_pname)) => (B = A2))))). % singletonD
thf(fact_125_singletonD, axiom,
    ((![B : hoare_958474565_state, A2 : hoare_958474565_state]: ((member109514606_state @ B @ (insert776267541_state @ A2 @ bot_bo105666705_state)) => (B = A2))))). % singletonD
thf(fact_126_insert__ident, axiom,
    ((![X2 : pname, A : set_pname, B2 : set_pname]: ((~ ((member_pname @ X2 @ A))) => ((~ ((member_pname @ X2 @ B2))) => (((insert_pname @ X2 @ A) = (insert_pname @ X2 @ B2)) = (A = B2))))))). % insert_ident
thf(fact_127_insert__ident, axiom,
    ((![X2 : hoare_958474565_state, A : set_Ho840737317_state, B2 : set_Ho840737317_state]: ((~ ((member109514606_state @ X2 @ A))) => ((~ ((member109514606_state @ X2 @ B2))) => (((insert776267541_state @ X2 @ A) = (insert776267541_state @ X2 @ B2)) = (A = B2))))))). % insert_ident
thf(fact_128_insert__absorb, axiom,
    ((![A2 : pname, A : set_pname]: ((member_pname @ A2 @ A) => ((insert_pname @ A2 @ A) = A))))). % insert_absorb
thf(fact_129_insert__absorb, axiom,
    ((![A2 : hoare_958474565_state, A : set_Ho840737317_state]: ((member109514606_state @ A2 @ A) => ((insert776267541_state @ A2 @ A) = A))))). % insert_absorb
thf(fact_130_insert__eq__iff, axiom,
    ((![A2 : pname, A : set_pname, B : pname, B2 : set_pname]: ((~ ((member_pname @ A2 @ A))) => ((~ ((member_pname @ B @ B2))) => (((insert_pname @ A2 @ A) = (insert_pname @ B @ B2)) = (((((A2 = B)) => ((A = B2)))) & ((((~ ((A2 = B)))) => ((?[C3 : set_pname]: (((A = (insert_pname @ B @ C3))) & ((((~ ((member_pname @ B @ C3)))) & ((((B2 = (insert_pname @ A2 @ C3))) & ((~ ((member_pname @ A2 @ C3)))))))))))))))))))). % insert_eq_iff
thf(fact_131_insert__eq__iff, axiom,
    ((![A2 : hoare_958474565_state, A : set_Ho840737317_state, B : hoare_958474565_state, B2 : set_Ho840737317_state]: ((~ ((member109514606_state @ A2 @ A))) => ((~ ((member109514606_state @ B @ B2))) => (((insert776267541_state @ A2 @ A) = (insert776267541_state @ B @ B2)) = (((((A2 = B)) => ((A = B2)))) & ((((~ ((A2 = B)))) => ((?[C3 : set_Ho840737317_state]: (((A = (insert776267541_state @ B @ C3))) & ((((~ ((member109514606_state @ B @ C3)))) & ((((B2 = (insert776267541_state @ A2 @ C3))) & ((~ ((member109514606_state @ A2 @ C3)))))))))))))))))))). % insert_eq_iff
thf(fact_132_singleton__iff, axiom,
    ((![B : pname, A2 : pname]: ((member_pname @ B @ (insert_pname @ A2 @ bot_bot_set_pname)) = (B = A2))))). % singleton_iff
thf(fact_133_singleton__iff, axiom,
    ((![B : hoare_958474565_state, A2 : hoare_958474565_state]: ((member109514606_state @ B @ (insert776267541_state @ A2 @ bot_bo105666705_state)) = (B = A2))))). % singleton_iff
thf(fact_134_insert__commute, axiom,
    ((![X2 : hoare_958474565_state, Y5 : hoare_958474565_state, A : set_Ho840737317_state]: ((insert776267541_state @ X2 @ (insert776267541_state @ Y5 @ A)) = (insert776267541_state @ Y5 @ (insert776267541_state @ X2 @ A)))))). % insert_commute
thf(fact_135_pairwise__empty, axiom,
    ((![P : hoare_958474565_state > hoare_958474565_state > $o]: (pairwi17519280_state @ P @ bot_bo105666705_state)))). % pairwise_empty
thf(fact_136_Collect__conv__if, axiom,
    ((![P : pname > $o, A2 : pname]: (((P @ A2) => ((collect_pname @ (^[X : pname]: (((X = A2)) & ((P @ X))))) = (insert_pname @ A2 @ bot_bot_set_pname))) & ((~ ((P @ A2))) => ((collect_pname @ (^[X : pname]: (((X = A2)) & ((P @ X))))) = bot_bot_set_pname)))))). % Collect_conv_if
thf(fact_137_Collect__conv__if, axiom,
    ((![P : hoare_958474565_state > $o, A2 : hoare_958474565_state]: (((P @ A2) => ((collec305460656_state @ (^[X : hoare_958474565_state]: (((X = A2)) & ((P @ X))))) = (insert776267541_state @ A2 @ bot_bo105666705_state))) & ((~ ((P @ A2))) => ((collec305460656_state @ (^[X : hoare_958474565_state]: (((X = A2)) & ((P @ X))))) = bot_bo105666705_state)))))). % Collect_conv_if
thf(fact_138_pairwise__insert, axiom,
    ((![R : pname > pname > $o, X2 : pname, S3 : set_pname]: ((pairwise_pname @ R @ (insert_pname @ X2 @ S3)) = (((![Y3 : pname]: (((((member_pname @ Y3 @ S3)) & ((~ ((Y3 = X2)))))) => ((((R @ X2 @ Y3)) & ((R @ Y3 @ X2))))))) & ((pairwise_pname @ R @ S3))))))). % pairwise_insert
thf(fact_139_pairwise__insert, axiom,
    ((![R : hoare_958474565_state > hoare_958474565_state > $o, X2 : hoare_958474565_state, S3 : set_Ho840737317_state]: ((pairwi17519280_state @ R @ (insert776267541_state @ X2 @ S3)) = (((![Y3 : hoare_958474565_state]: (((((member109514606_state @ Y3 @ S3)) & ((~ ((Y3 = X2)))))) => ((((R @ X2 @ Y3)) & ((R @ Y3 @ X2))))))) & ((pairwi17519280_state @ R @ S3))))))). % pairwise_insert
thf(fact_140_Collect__conv__if2, axiom,
    ((![P : pname > $o, A2 : pname]: (((P @ A2) => ((collect_pname @ (^[X : pname]: (((A2 = X)) & ((P @ X))))) = (insert_pname @ A2 @ bot_bot_set_pname))) & ((~ ((P @ A2))) => ((collect_pname @ (^[X : pname]: (((A2 = X)) & ((P @ X))))) = bot_bot_set_pname)))))). % Collect_conv_if2
thf(fact_141_Collect__conv__if2, axiom,
    ((![P : hoare_958474565_state > $o, A2 : hoare_958474565_state]: (((P @ A2) => ((collec305460656_state @ (^[X : hoare_958474565_state]: (((A2 = X)) & ((P @ X))))) = (insert776267541_state @ A2 @ bot_bo105666705_state))) & ((~ ((P @ A2))) => ((collec305460656_state @ (^[X : hoare_958474565_state]: (((A2 = X)) & ((P @ X))))) = bot_bo105666705_state)))))). % Collect_conv_if2
thf(fact_142_doubleton__eq__iff, axiom,
    ((![A2 : hoare_958474565_state, B : hoare_958474565_state, C2 : hoare_958474565_state, D2 : hoare_958474565_state]: (((insert776267541_state @ A2 @ (insert776267541_state @ B @ bot_bo105666705_state)) = (insert776267541_state @ C2 @ (insert776267541_state @ D2 @ bot_bo105666705_state))) = (((((A2 = C2)) & ((B = D2)))) | ((((A2 = D2)) & ((B = C2))))))))). % doubleton_eq_iff
thf(fact_143_insert__not__empty, axiom,
    ((![A2 : hoare_958474565_state, A : set_Ho840737317_state]: (~ (((insert776267541_state @ A2 @ A) = bot_bo105666705_state)))))). % insert_not_empty
thf(fact_144_singleton__inject, axiom,
    ((![A2 : hoare_958474565_state, B : hoare_958474565_state]: (((insert776267541_state @ A2 @ bot_bo105666705_state) = (insert776267541_state @ B @ bot_bo105666705_state)) => (A2 = B))))). % singleton_inject
thf(fact_145_mk__disjoint__insert, axiom,
    ((![A2 : pname, A : set_pname]: ((member_pname @ A2 @ A) => (?[B3 : set_pname]: ((A = (insert_pname @ A2 @ B3)) & (~ ((member_pname @ A2 @ B3))))))))). % mk_disjoint_insert
thf(fact_146_mk__disjoint__insert, axiom,
    ((![A2 : hoare_958474565_state, A : set_Ho840737317_state]: ((member109514606_state @ A2 @ A) => (?[B3 : set_Ho840737317_state]: ((A = (insert776267541_state @ A2 @ B3)) & (~ ((member109514606_state @ A2 @ B3))))))))). % mk_disjoint_insert
thf(fact_147_pairwise__singleton, axiom,
    ((![P : hoare_958474565_state > hoare_958474565_state > $o, A : hoare_958474565_state]: (pairwi17519280_state @ P @ (insert776267541_state @ A @ bot_bo105666705_state))))). % pairwise_singleton
thf(fact_148_hoare__derivs_Oinsert, axiom,
    ((![G2 : set_Ho840737317_state, T3 : hoare_958474565_state, Ts : set_Ho840737317_state]: ((hoare_604442164_state @ G2 @ (insert776267541_state @ T3 @ bot_bo105666705_state)) => ((hoare_604442164_state @ G2 @ Ts) => (hoare_604442164_state @ G2 @ (insert776267541_state @ T3 @ Ts))))))). % hoare_derivs.insert
thf(fact_149_empty, axiom,
    ((![G2 : set_Ho840737317_state]: (hoare_604442164_state @ G2 @ bot_bo105666705_state)))). % empty
thf(fact_150_cut, axiom,
    ((![G3 : set_Ho840737317_state, Ts : set_Ho840737317_state, G2 : set_Ho840737317_state]: ((hoare_604442164_state @ G3 @ Ts) => ((hoare_604442164_state @ G2 @ G3) => (hoare_604442164_state @ G2 @ Ts)))))). % cut
thf(fact_151_derivs__insertD, axiom,
    ((![G2 : set_Ho840737317_state, T3 : hoare_958474565_state, Ts : set_Ho840737317_state]: ((hoare_604442164_state @ G2 @ (insert776267541_state @ T3 @ Ts)) => ((hoare_604442164_state @ G2 @ (insert776267541_state @ T3 @ bot_bo105666705_state)) & (hoare_604442164_state @ G2 @ Ts)))))). % derivs_insertD
thf(fact_152_insert__Collect, axiom,
    ((![A2 : hoare_958474565_state, P : hoare_958474565_state > $o]: ((insert776267541_state @ A2 @ (collec305460656_state @ P)) = (collec305460656_state @ (^[U : hoare_958474565_state]: (((~ ((U = A2)))) => ((P @ U))))))))). % insert_Collect
thf(fact_153_insert__Collect, axiom,
    ((![A2 : pname, P : pname > $o]: ((insert_pname @ A2 @ (collect_pname @ P)) = (collect_pname @ (^[U : pname]: (((~ ((U = A2)))) => ((P @ U))))))))). % insert_Collect
thf(fact_154_insert__compr, axiom,
    ((insert776267541_state = (^[A3 : hoare_958474565_state]: (^[B4 : set_Ho840737317_state]: (collec305460656_state @ (^[X : hoare_958474565_state]: (((X = A3)) | ((member109514606_state @ X @ B4)))))))))). % insert_compr
thf(fact_155_insert__compr, axiom,
    ((insert_pname = (^[A3 : pname]: (^[B4 : set_pname]: (collect_pname @ (^[X : pname]: (((X = A3)) | ((member_pname @ X @ B4)))))))))). % insert_compr
thf(fact_156_empty__def, axiom,
    ((bot_bot_set_pname = (collect_pname @ (^[X : pname]: $false))))). % empty_def
thf(fact_157_empty__def, axiom,
    ((bot_bo105666705_state = (collec305460656_state @ (^[X : hoare_958474565_state]: $false))))). % empty_def
thf(fact_158_image__constant, axiom,
    ((![X2 : pname, A : set_pname, C2 : pname]: ((member_pname @ X2 @ A) => ((image_pname_pname @ (^[X : pname]: C2) @ A) = (insert_pname @ C2 @ bot_bot_set_pname)))))). % image_constant
thf(fact_159_image__constant, axiom,
    ((![X2 : pname, A : set_pname, C2 : hoare_958474565_state]: ((member_pname @ X2 @ A) => ((image_2144627828_state @ (^[X : pname]: C2) @ A) = (insert776267541_state @ C2 @ bot_bo105666705_state)))))). % image_constant
thf(fact_160_image__constant, axiom,
    ((![X2 : hoare_958474565_state, A : set_Ho840737317_state, C2 : hoare_958474565_state]: ((member109514606_state @ X2 @ A) => ((image_1849792670_state @ (^[X : hoare_958474565_state]: C2) @ A) = (insert776267541_state @ C2 @ bot_bo105666705_state)))))). % image_constant
thf(fact_161_image__constant__conv, axiom,
    ((![A : set_pname, C2 : pname]: (((A = bot_bot_set_pname) => ((image_pname_pname @ (^[X : pname]: C2) @ A) = bot_bot_set_pname)) & ((~ ((A = bot_bot_set_pname))) => ((image_pname_pname @ (^[X : pname]: C2) @ A) = (insert_pname @ C2 @ bot_bot_set_pname))))))). % image_constant_conv
thf(fact_162_image__constant__conv, axiom,
    ((![A : set_pname, C2 : hoare_958474565_state]: (((A = bot_bot_set_pname) => ((image_2144627828_state @ (^[X : pname]: C2) @ A) = bot_bo105666705_state)) & ((~ ((A = bot_bot_set_pname))) => ((image_2144627828_state @ (^[X : pname]: C2) @ A) = (insert776267541_state @ C2 @ bot_bo105666705_state))))))). % image_constant_conv
thf(fact_163_image__constant__conv, axiom,
    ((![A : set_Ho840737317_state, C2 : hoare_958474565_state]: (((A = bot_bo105666705_state) => ((image_1849792670_state @ (^[X : hoare_958474565_state]: C2) @ A) = bot_bo105666705_state)) & ((~ ((A = bot_bo105666705_state))) => ((image_1849792670_state @ (^[X : hoare_958474565_state]: C2) @ A) = (insert776267541_state @ C2 @ bot_bo105666705_state))))))). % image_constant_conv
thf(fact_164_pairwise__imageI, axiom,
    ((![A : set_pname, F : pname > hoare_958474565_state, P : hoare_958474565_state > hoare_958474565_state > $o]: ((![X3 : pname, Y2 : pname]: ((member_pname @ X3 @ A) => ((member_pname @ Y2 @ A) => ((~ ((X3 = Y2))) => ((~ (((F @ X3) = (F @ Y2)))) => (P @ (F @ X3) @ (F @ Y2))))))) => (pairwi17519280_state @ P @ (image_2144627828_state @ F @ A)))))). % pairwise_imageI
thf(fact_165_pairwise__imageI, axiom,
    ((![A : set_pname, F : pname > pname, P : pname > pname > $o]: ((![X3 : pname, Y2 : pname]: ((member_pname @ X3 @ A) => ((member_pname @ Y2 @ A) => ((~ ((X3 = Y2))) => ((~ (((F @ X3) = (F @ Y2)))) => (P @ (F @ X3) @ (F @ Y2))))))) => (pairwise_pname @ P @ (image_pname_pname @ F @ A)))))). % pairwise_imageI
thf(fact_166_pairwise__imageI, axiom,
    ((![A : set_Ho840737317_state, F : hoare_958474565_state > hoare_958474565_state, P : hoare_958474565_state > hoare_958474565_state > $o]: ((![X3 : hoare_958474565_state, Y2 : hoare_958474565_state]: ((member109514606_state @ X3 @ A) => ((member109514606_state @ Y2 @ A) => ((~ ((X3 = Y2))) => ((~ (((F @ X3) = (F @ Y2)))) => (P @ (F @ X3) @ (F @ Y2))))))) => (pairwi17519280_state @ P @ (image_1849792670_state @ F @ A)))))). % pairwise_imageI
thf(fact_167_com_Odistinct_I55_J, axiom,
    ((![X7 : pname, X81 : vname, X82 : pname, X83 : state > nat]: (~ (((body2 @ X7) = (call @ X81 @ X82 @ X83))))))). % com.distinct(55)
thf(fact_168_dom__empty, axiom,
    (((dom_pname_com @ (^[X : pname]: none_com)) = bot_bot_set_pname))). % dom_empty
thf(fact_169_dom__empty, axiom,
    (((dom_Ho256784515te_com @ (^[X : hoare_958474565_state]: none_com)) = bot_bo105666705_state))). % dom_empty
thf(fact_170_dom__eq__empty__conv, axiom,
    ((![F : pname > option_com]: (((dom_pname_com @ F) = bot_bot_set_pname) = (F = (^[X : pname]: none_com)))))). % dom_eq_empty_conv
thf(fact_171_dom__eq__empty__conv, axiom,
    ((![F : hoare_958474565_state > option_com]: (((dom_Ho256784515te_com @ F) = bot_bo105666705_state) = (F = (^[X : hoare_958474565_state]: none_com)))))). % dom_eq_empty_conv
thf(fact_172_not__Some__eq, axiom,
    ((![X2 : option_com]: ((![Y3 : com]: (~ ((X2 = (some_com @ Y3))))) = (X2 = none_com))))). % not_Some_eq
thf(fact_173_not__None__eq, axiom,
    ((![X2 : option_com]: ((~ ((X2 = none_com))) = (?[Y3 : com]: (X2 = (some_com @ Y3))))))). % not_None_eq
thf(fact_174_finite__insert, axiom,
    ((![A2 : hoare_958474565_state, A : set_Ho840737317_state]: ((finite1986656878_state @ (insert776267541_state @ A2 @ A)) = (finite1986656878_state @ A))))). % finite_insert
thf(fact_175_finite__insert, axiom,
    ((![A2 : pname, A : set_pname]: ((finite_finite_pname @ (insert_pname @ A2 @ A)) = (finite_finite_pname @ A))))). % finite_insert
thf(fact_176_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_177_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_178_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_179_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_180_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_181_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_182_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_183_option_Oinject, axiom,
    ((![X22 : com, Y22 : com]: (((some_com @ X22) = (some_com @ Y22)) = (X22 = Y22))))). % option.inject
thf(fact_184_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_185_not__finite__existsD, axiom,
    ((![P : pname > $o]: ((~ ((finite_finite_pname @ (collect_pname @ P)))) => (?[X_1 : pname]: (P @ X_1)))))). % not_finite_existsD
thf(fact_186_pigeonhole__infinite__rel, axiom,
    ((![A : set_Ho840737317_state, B2 : set_pname, R2 : hoare_958474565_state > pname > $o]: ((~ ((finite1986656878_state @ A))) => ((finite_finite_pname @ B2) => ((![X3 : hoare_958474565_state]: ((member109514606_state @ X3 @ A) => (?[Xa : pname]: ((member_pname @ Xa @ B2) & (R2 @ X3 @ Xa))))) => (?[X3 : pname]: ((member_pname @ X3 @ B2) & (~ ((finite1986656878_state @ (collec305460656_state @ (^[A3 : hoare_958474565_state]: (((member109514606_state @ A3 @ A)) & ((R2 @ A3 @ X3)))))))))))))))). % pigeonhole_infinite_rel
thf(fact_187_pigeonhole__infinite__rel, axiom,
    ((![A : set_pname, B2 : set_pname, R2 : pname > pname > $o]: ((~ ((finite_finite_pname @ A))) => ((finite_finite_pname @ B2) => ((![X3 : pname]: ((member_pname @ X3 @ A) => (?[Xa : pname]: ((member_pname @ Xa @ B2) & (R2 @ X3 @ Xa))))) => (?[X3 : pname]: ((member_pname @ X3 @ B2) & (~ ((finite_finite_pname @ (collect_pname @ (^[A3 : pname]: (((member_pname @ A3 @ A)) & ((R2 @ A3 @ X3)))))))))))))))). % pigeonhole_infinite_rel
thf(fact_188_finite_OemptyI, axiom,
    ((finite_finite_pname @ bot_bot_set_pname))). % finite.emptyI
thf(fact_189_finite_OemptyI, axiom,
    ((finite1986656878_state @ bot_bo105666705_state))). % finite.emptyI
thf(fact_190_infinite__imp__nonempty, axiom,
    ((![S4 : set_pname]: ((~ ((finite_finite_pname @ S4))) => (~ ((S4 = bot_bot_set_pname))))))). % infinite_imp_nonempty
thf(fact_191_infinite__imp__nonempty, axiom,
    ((![S4 : set_Ho840737317_state]: ((~ ((finite1986656878_state @ S4))) => (~ ((S4 = bot_bo105666705_state))))))). % infinite_imp_nonempty
thf(fact_192_finite_OinsertI, axiom,
    ((![A : set_Ho840737317_state, A2 : hoare_958474565_state]: ((finite1986656878_state @ A) => (finite1986656878_state @ (insert776267541_state @ A2 @ A)))))). % finite.insertI
thf(fact_193_finite_OinsertI, axiom,
    ((![A : set_pname, A2 : pname]: ((finite_finite_pname @ A) => (finite_finite_pname @ (insert_pname @ A2 @ A)))))). % finite.insertI
thf(fact_194_option_Odistinct_I1_J, axiom,
    ((![X22 : com]: (~ ((none_com = (some_com @ X22))))))). % option.distinct(1)
thf(fact_195_option_OdiscI, axiom,
    ((![Option : option_com, X22 : com]: ((Option = (some_com @ X22)) => (~ ((Option = none_com))))))). % option.discI
thf(fact_196_option_Oexhaust, axiom,
    ((![Y5 : option_com]: ((~ ((Y5 = none_com))) => (~ ((![X23 : com]: (~ ((Y5 = (some_com @ X23))))))))))). % option.exhaust
thf(fact_197_option_Oinducts, axiom,
    ((![P : option_com > $o, Option : option_com]: ((P @ none_com) => ((![X3 : com]: (P @ (some_com @ X3))) => (P @ Option)))))). % option.inducts
thf(fact_198_split__option__ex, axiom,
    (((^[P2 : option_com > $o]: (?[X6 : option_com]: (P2 @ X6))) = (^[P3 : option_com > $o]: (((P3 @ none_com)) | ((?[X : com]: (P3 @ (some_com @ X))))))))). % split_option_ex
thf(fact_199_split__option__all, axiom,
    (((^[P2 : option_com > $o]: (![X6 : option_com]: (P2 @ X6))) = (^[P3 : option_com > $o]: (((P3 @ none_com)) & ((![X : com]: (P3 @ (some_com @ X))))))))). % split_option_all
thf(fact_200_combine__options__cases, axiom,
    ((![X2 : option_com, P : option_com > option_com > $o, Y5 : option_com]: (((X2 = none_com) => (P @ X2 @ Y5)) => (((Y5 = none_com) => (P @ X2 @ Y5)) => ((![A4 : com, B5 : com]: ((X2 = (some_com @ A4)) => ((Y5 = (some_com @ B5)) => (P @ X2 @ Y5)))) => (P @ X2 @ Y5))))))). % combine_options_cases
thf(fact_201_None__notin__image__Some, axiom,
    ((![A : set_com]: (~ ((member_option_com @ none_com @ (image_com_option_com @ some_com @ A))))))). % None_notin_image_Some
thf(fact_202_domIff, axiom,
    ((![A2 : pname, M2 : pname > option_com]: ((member_pname @ A2 @ (dom_pname_com @ M2)) = (~ (((M2 @ A2) = none_com))))))). % domIff
thf(fact_203_domIff, axiom,
    ((![A2 : hoare_958474565_state, M2 : hoare_958474565_state > option_com]: ((member109514606_state @ A2 @ (dom_Ho256784515te_com @ M2)) = (~ (((M2 @ A2) = none_com))))))). % domIff
thf(fact_204_domI, axiom,
    ((![M2 : pname > option_com, A2 : pname, B : com]: (((M2 @ A2) = (some_com @ B)) => (member_pname @ A2 @ (dom_pname_com @ M2)))))). % domI
thf(fact_205_domI, axiom,
    ((![M2 : hoare_958474565_state > option_com, A2 : hoare_958474565_state, B : com]: (((M2 @ A2) = (some_com @ B)) => (member109514606_state @ A2 @ (dom_Ho256784515te_com @ M2)))))). % domI
thf(fact_206_domD, axiom,
    ((![A2 : pname, M2 : pname > option_com]: ((member_pname @ A2 @ (dom_pname_com @ M2)) => (?[B5 : com]: ((M2 @ A2) = (some_com @ B5))))))). % domD
thf(fact_207_domD, axiom,
    ((![A2 : hoare_958474565_state, M2 : hoare_958474565_state > option_com]: ((member109514606_state @ A2 @ (dom_Ho256784515te_com @ M2)) => (?[B5 : com]: ((M2 @ A2) = (some_com @ B5))))))). % domD
thf(fact_208_pigeonhole__infinite, axiom,
    ((![A : set_Ho840737317_state, F : hoare_958474565_state > hoare_958474565_state]: ((~ ((finite1986656878_state @ A))) => ((finite1986656878_state @ (image_1849792670_state @ F @ A)) => (?[X3 : hoare_958474565_state]: ((member109514606_state @ X3 @ A) & (~ ((finite1986656878_state @ (collec305460656_state @ (^[A3 : hoare_958474565_state]: (((member109514606_state @ A3 @ A)) & (((F @ A3) = (F @ X3)))))))))))))))). % pigeonhole_infinite
thf(fact_209_pigeonhole__infinite, axiom,
    ((![A : set_Ho840737317_state, F : hoare_958474565_state > pname]: ((~ ((finite1986656878_state @ A))) => ((finite_finite_pname @ (image_871391498_pname @ F @ A)) => (?[X3 : hoare_958474565_state]: ((member109514606_state @ X3 @ A) & (~ ((finite1986656878_state @ (collec305460656_state @ (^[A3 : hoare_958474565_state]: (((member109514606_state @ A3 @ A)) & (((F @ A3) = (F @ X3)))))))))))))))). % pigeonhole_infinite
thf(fact_210_pigeonhole__infinite, axiom,
    ((![A : set_pname, F : pname > hoare_958474565_state]: ((~ ((finite_finite_pname @ A))) => ((finite1986656878_state @ (image_2144627828_state @ F @ A)) => (?[X3 : pname]: ((member_pname @ X3 @ A) & (~ ((finite_finite_pname @ (collect_pname @ (^[A3 : pname]: (((member_pname @ A3 @ A)) & (((F @ A3) = (F @ X3)))))))))))))))). % pigeonhole_infinite
thf(fact_211_pigeonhole__infinite, axiom,
    ((![A : set_pname, F : pname > pname]: ((~ ((finite_finite_pname @ A))) => ((finite_finite_pname @ (image_pname_pname @ F @ A)) => (?[X3 : pname]: ((member_pname @ X3 @ A) & (~ ((finite_finite_pname @ (collect_pname @ (^[A3 : pname]: (((member_pname @ A3 @ A)) & (((F @ A3) = (F @ X3)))))))))))))))). % pigeonhole_infinite
thf(fact_212_dom__def, axiom,
    ((dom_Ho256784515te_com = (^[M3 : hoare_958474565_state > option_com]: (collec305460656_state @ (^[A3 : hoare_958474565_state]: (~ (((M3 @ A3) = none_com))))))))). % dom_def
thf(fact_213_dom__def, axiom,
    ((dom_pname_com = (^[M3 : pname > option_com]: (collect_pname @ (^[A3 : pname]: (~ (((M3 @ A3) = none_com))))))))). % dom_def
thf(fact_214_finite_Ocases, axiom,
    ((![A2 : set_pname]: ((finite_finite_pname @ A2) => ((~ ((A2 = bot_bot_set_pname))) => (~ ((![A5 : set_pname]: ((?[A4 : pname]: (A2 = (insert_pname @ A4 @ A5))) => (~ ((finite_finite_pname @ A5)))))))))))). % finite.cases
thf(fact_215_finite_Ocases, axiom,
    ((![A2 : set_Ho840737317_state]: ((finite1986656878_state @ A2) => ((~ ((A2 = bot_bo105666705_state))) => (~ ((![A5 : set_Ho840737317_state]: ((?[A4 : hoare_958474565_state]: (A2 = (insert776267541_state @ A4 @ A5))) => (~ ((finite1986656878_state @ A5)))))))))))). % finite.cases
thf(fact_216_finite_Osimps, axiom,
    ((finite_finite_pname = (^[A3 : set_pname]: (((A3 = bot_bot_set_pname)) | ((?[A6 : set_pname]: (?[B6 : pname]: (((A3 = (insert_pname @ B6 @ A6))) & ((finite_finite_pname @ A6))))))))))). % finite.simps
thf(fact_217_finite_Osimps, axiom,
    ((finite1986656878_state = (^[A3 : set_Ho840737317_state]: (((A3 = bot_bo105666705_state)) | ((?[A6 : set_Ho840737317_state]: (?[B6 : hoare_958474565_state]: (((A3 = (insert776267541_state @ B6 @ A6))) & ((finite1986656878_state @ A6))))))))))). % finite.simps
thf(fact_218_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_219_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_220_finite_Oinducts, axiom,
    ((![X2 : set_pname, P : set_pname > $o]: ((finite_finite_pname @ X2) => ((P @ bot_bot_set_pname) => ((![A5 : set_pname, A4 : pname]: ((finite_finite_pname @ A5) => ((P @ A5) => (P @ (insert_pname @ A4 @ A5))))) => (P @ X2))))))). % finite.inducts
thf(fact_221_finite_Oinducts, axiom,
    ((![X2 : set_Ho840737317_state, P : set_Ho840737317_state > $o]: ((finite1986656878_state @ X2) => ((P @ bot_bo105666705_state) => ((![A5 : set_Ho840737317_state, A4 : hoare_958474565_state]: ((finite1986656878_state @ A5) => ((P @ A5) => (P @ (insert776267541_state @ A4 @ A5))))) => (P @ X2))))))). % finite.inducts
thf(fact_222_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_223_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_224_infinite__finite__induct, axiom,
    ((![P : set_pname > $o, A : 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 @ A))))))). % infinite_finite_induct
thf(fact_225_infinite__finite__induct, axiom,
    ((![P : set_Ho840737317_state > $o, A : 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 @ A))))))). % infinite_finite_induct
thf(fact_226_insert__dom, axiom,
    ((![F : pname > option_com, X2 : pname, Y5 : com]: (((F @ X2) = (some_com @ Y5)) => ((insert_pname @ X2 @ (dom_pname_com @ F)) = (dom_pname_com @ F)))))). % insert_dom
thf(fact_227_insert__dom, axiom,
    ((![F : hoare_958474565_state > option_com, X2 : hoare_958474565_state, Y5 : com]: (((F @ X2) = (some_com @ Y5)) => ((insert776267541_state @ X2 @ (dom_Ho256784515te_com @ F)) = (dom_Ho256784515te_com @ F)))))). % insert_dom
thf(fact_228_the__elem__eq, axiom,
    ((![X2 : hoare_958474565_state]: ((the_el1300254266_state @ (insert776267541_state @ X2 @ bot_bo105666705_state)) = X2)))). % the_elem_eq
thf(fact_229_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_230_option_Ocollapse, axiom,
    ((![Option : option_com]: ((~ ((Option = none_com))) => ((some_com @ (the_com @ Option)) = Option))))). % option.collapse
thf(fact_231_option_Oexpand, axiom,
    ((![Option : option_com, Option2 : option_com]: (((Option = none_com) = (Option2 = none_com)) => (((~ ((Option = none_com))) => ((~ ((Option2 = none_com))) => ((the_com @ Option) = (the_com @ Option2)))) => (Option = Option2)))))). % option.expand
thf(fact_232_option_Osel, axiom,
    ((![X22 : com]: ((the_com @ (some_com @ X22)) = X22)))). % option.sel
thf(fact_233_bot__set__def, axiom,
    ((bot_bot_set_pname = (collect_pname @ bot_bot_pname_o)))). % bot_set_def
thf(fact_234_bot__set__def, axiom,
    ((bot_bo105666705_state = (collec305460656_state @ bot_bo1428770700tate_o)))). % bot_set_def
thf(fact_235_option_Oexhaust__sel, axiom,
    ((![Option : option_com]: ((~ ((Option = none_com))) => (Option = (some_com @ (the_com @ Option))))))). % option.exhaust_sel
thf(fact_236_the__elem__image__unique, axiom,
    ((![A : set_pname, F : pname > hoare_958474565_state, X2 : pname]: ((~ ((A = bot_bot_set_pname))) => ((![Y2 : pname]: ((member_pname @ Y2 @ A) => ((F @ Y2) = (F @ X2)))) => ((the_el1300254266_state @ (image_2144627828_state @ F @ A)) = (F @ X2))))))). % the_elem_image_unique
thf(fact_237_the__elem__image__unique, axiom,
    ((![A : set_pname, F : pname > pname, X2 : pname]: ((~ ((A = bot_bot_set_pname))) => ((![Y2 : pname]: ((member_pname @ Y2 @ A) => ((F @ Y2) = (F @ X2)))) => ((the_elem_pname @ (image_pname_pname @ F @ A)) = (F @ X2))))))). % the_elem_image_unique
thf(fact_238_the__elem__image__unique, axiom,
    ((![A : set_Ho840737317_state, F : hoare_958474565_state > hoare_958474565_state, X2 : hoare_958474565_state]: ((~ ((A = bot_bo105666705_state))) => ((![Y2 : hoare_958474565_state]: ((member109514606_state @ Y2 @ A) => ((F @ Y2) = (F @ X2)))) => ((the_el1300254266_state @ (image_1849792670_state @ F @ A)) = (F @ X2))))))). % the_elem_image_unique
thf(fact_239_is__singleton__the__elem, axiom,
    ((is_sin536631353_state = (^[A6 : set_Ho840737317_state]: (A6 = (insert776267541_state @ (the_el1300254266_state @ A6) @ bot_bo105666705_state)))))). % is_singleton_the_elem
thf(fact_240_is__singletonI, axiom,
    ((![X2 : hoare_958474565_state]: (is_sin536631353_state @ (insert776267541_state @ X2 @ bot_bo105666705_state))))). % is_singletonI
thf(fact_241_is__singletonI_H, axiom,
    ((![A : set_pname]: ((~ ((A = bot_bot_set_pname))) => ((![X3 : pname, Y2 : pname]: ((member_pname @ X3 @ A) => ((member_pname @ Y2 @ A) => (X3 = Y2)))) => (is_singleton_pname @ A)))))). % is_singletonI'
thf(fact_242_is__singletonI_H, axiom,
    ((![A : set_Ho840737317_state]: ((~ ((A = bot_bo105666705_state))) => ((![X3 : hoare_958474565_state, Y2 : hoare_958474565_state]: ((member109514606_state @ X3 @ A) => ((member109514606_state @ Y2 @ A) => (X3 = Y2)))) => (is_sin536631353_state @ A)))))). % is_singletonI'
thf(fact_243_is__singleton__def, axiom,
    ((is_sin536631353_state = (^[A6 : set_Ho840737317_state]: (?[X : hoare_958474565_state]: (A6 = (insert776267541_state @ X @ bot_bo105666705_state))))))). % is_singleton_def
thf(fact_244_is__singletonE, axiom,
    ((![A : set_Ho840737317_state]: ((is_sin536631353_state @ A) => (~ ((![X3 : hoare_958474565_state]: (~ ((A = (insert776267541_state @ X3 @ bot_bo105666705_state))))))))))). % is_singletonE
thf(fact_245_bot__empty__eq, axiom,
    ((bot_bot_pname_o = (^[X : pname]: (member_pname @ X @ bot_bot_set_pname))))). % bot_empty_eq
thf(fact_246_bot__empty__eq, axiom,
    ((bot_bo1428770700tate_o = (^[X : hoare_958474565_state]: (member109514606_state @ X @ bot_bo105666705_state))))). % bot_empty_eq

% Conjectures (4)
thf(conj_0, hypothesis,
    (hoare_405891322gleton)).
thf(conj_1, hypothesis,
    (wT_bodies)).
thf(conj_2, hypothesis,
    ((wt @ c))).
thf(conj_3, conjecture,
    ((uG_0_ATP = (image_2144627828_state @ (^[Pn2 : pname]: (hoare_Mirabelle_MGT @ (body2 @ Pn2))) @ (dom_pname_com @ body))))).
