% 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_424__3254610_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:14:42.573

% Could-be-implicit typings (7)
thf(ty_n_t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J_J, type,
    set_Ho840737317_state : $tType).
thf(ty_n_t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J, type,
    hoare_958474565_state : $tType).
thf(ty_n_t__Option__Ooption_It__Com__Ocom_J, type,
    option_com : $tType).
thf(ty_n_t__Set__Oset_It__Com__Opname_J, type,
    set_pname : $tType).
thf(ty_n_t__Com__Ostate, type,
    state : $tType).
thf(ty_n_t__Com__Opname, type,
    pname : $tType).
thf(ty_n_t__Com__Ocom, type,
    com : $tType).

% Explicit typings (29)
thf(sy_c_Com_Obody, type,
    body : pname > option_com).
thf(sy_c_Com_Ocom_OBODY, type,
    body2 : pname > com).
thf(sy_c_Finite__Set_Ofinite_001t__Com__Opname, type,
    finite_finite_pname : set_pname > $o).
thf(sy_c_Finite__Set_Ofinite_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J, type,
    finite1986656878_state : set_Ho840737317_state > $o).
thf(sy_c_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_Ohoare__valids_001t__Com__Ostate, type,
    hoare_318887606_state : set_Ho840737317_state > set_Ho840737317_state > $o).
thf(sy_c_Hoare__Mirabelle__raqjowkjvm_Otriple_Otriple_001t__Com__Ostate, type,
    hoare_1659279548_state : (state > state > $o) > com > (state > state > $o) > hoare_958474565_state).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Com__Opname_M_Eo_J, type,
    sup_sup_pname_o : (pname > $o) > (pname > $o) > pname > $o).
thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J_M_Eo_J, type,
    sup_su2044190244tate_o : (hoare_958474565_state > $o) > (hoare_958474565_state > $o) > hoare_958474565_state > $o).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Com__Opname_J, type,
    sup_sup_set_pname : set_pname > set_pname > set_pname).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J_J, type,
    sup_su737426425_state : set_Ho840737317_state > set_Ho840737317_state > set_Ho840737317_state).
thf(sy_c_Map_Odom_001t__Com__Opname_001t__Com__Ocom, type,
    dom_pname_com : (pname > option_com) > set_pname).
thf(sy_c_Natural_Oevalc, type,
    evalc : com > state > state > $o).
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_Set_OCollect_001t__Com__Opname, type,
    collect_pname : (pname > $o) > set_pname).
thf(sy_c_Set_OCollect_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J, type,
    collec305460656_state : (hoare_958474565_state > $o) > set_Ho840737317_state).
thf(sy_c_Set_Oimage_001t__Com__Opname_001t__Com__Opname, type,
    image_pname_pname : (pname > pname) > set_pname > set_pname).
thf(sy_c_Set_Oimage_001t__Com__Opname_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J, type,
    image_2144627828_state : (pname > hoare_958474565_state) > set_pname > set_Ho840737317_state).
thf(sy_c_Set_Oimage_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J_001t__Com__Opname, type,
    image_871391498_pname : (hoare_958474565_state > pname) > set_Ho840737317_state > set_pname).
thf(sy_c_Set_Oimage_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J, type,
    image_1849792670_state : (hoare_958474565_state > hoare_958474565_state) > set_Ho840737317_state > set_Ho840737317_state).
thf(sy_c_Set_Oinsert_001t__Com__Opname, type,
    insert_pname : pname > set_pname > set_pname).
thf(sy_c_Set_Oinsert_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J, type,
    insert776267541_state : hoare_958474565_state > set_Ho840737317_state > set_Ho840737317_state).
thf(sy_c_member_001t__Com__Opname, type,
    member_pname : pname > set_pname > $o).
thf(sy_c_member_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J, type,
    member109514606_state : hoare_958474565_state > set_Ho840737317_state > $o).
thf(sy_v_G, type,
    g : set_Ho840737317_state).
thf(sy_v_Procs, type,
    procs : set_pname).

% Relevant facts (197)
thf(fact_0_triple_Oinject, axiom,
    ((![X1 : state > state > $o, X2 : com, X3 : state > state > $o, Y1 : state > state > $o, Y2 : com, Y3 : state > state > $o]: (((hoare_1659279548_state @ X1 @ X2 @ X3) = (hoare_1659279548_state @ Y1 @ Y2 @ Y3)) = (((X1 = Y1)) & ((((X2 = Y2)) & ((X3 = Y3))))))))). % triple.inject
thf(fact_1_triple_Oinduct, axiom,
    ((![P : hoare_958474565_state > $o, Triple : hoare_958474565_state]: ((![X1a : state > state > $o, X2a : com, X3a : state > state > $o]: (P @ (hoare_1659279548_state @ X1a @ X2a @ X3a))) => (P @ Triple))))). % triple.induct
thf(fact_2_triple_Oexhaust, axiom,
    ((![Y : hoare_958474565_state]: (~ ((![X12 : state > state > $o, X22 : com, X32 : state > state > $o]: (~ ((Y = (hoare_1659279548_state @ X12 @ X22 @ X32)))))))))). % triple.exhaust
thf(fact_3_cut, axiom,
    ((![G : set_Ho840737317_state, Ts : set_Ho840737317_state, G2 : set_Ho840737317_state]: ((hoare_604442164_state @ G @ Ts) => ((hoare_604442164_state @ G2 @ G) => (hoare_604442164_state @ G2 @ Ts)))))). % cut
thf(fact_4_hoare__derivs_OBody, axiom,
    ((![G2 : set_Ho840737317_state, P : pname > state > state > $o, Q : pname > state > state > $o, Procs : set_pname]: ((hoare_604442164_state @ (sup_su737426425_state @ G2 @ (image_2144627828_state @ (^[P2 : pname]: (hoare_1659279548_state @ (P @ P2) @ (body2 @ P2) @ (Q @ P2))) @ Procs)) @ (image_2144627828_state @ (^[P2 : pname]: (hoare_1659279548_state @ (P @ P2) @ (the_com @ (body @ P2)) @ (Q @ P2))) @ Procs)) => (hoare_604442164_state @ G2 @ (image_2144627828_state @ (^[P2 : pname]: (hoare_1659279548_state @ (P @ P2) @ (body2 @ P2) @ (Q @ P2))) @ Procs)))))). % hoare_derivs.Body
thf(fact_5_Body__sound__lemma, axiom,
    ((![G2 : set_Ho840737317_state, P : pname > state > state > $o, Q : pname > state > state > $o, Procs : set_pname]: ((hoare_318887606_state @ (sup_su737426425_state @ G2 @ (image_2144627828_state @ (^[Pn : pname]: (hoare_1659279548_state @ (P @ Pn) @ (body2 @ Pn) @ (Q @ Pn))) @ Procs)) @ (image_2144627828_state @ (^[Pn : pname]: (hoare_1659279548_state @ (P @ Pn) @ (the_com @ (body @ Pn)) @ (Q @ Pn))) @ Procs)) => (hoare_318887606_state @ G2 @ (image_2144627828_state @ (^[Pn : pname]: (hoare_1659279548_state @ (P @ Pn) @ (body2 @ Pn) @ (Q @ Pn))) @ Procs)))))). % Body_sound_lemma
thf(fact_6_finite__Un, axiom,
    ((![F : set_pname, G2 : set_pname]: ((finite_finite_pname @ (sup_sup_set_pname @ F @ G2)) = (((finite_finite_pname @ F)) & ((finite_finite_pname @ G2))))))). % finite_Un
thf(fact_7_finite__Un, axiom,
    ((![F : set_Ho840737317_state, G2 : set_Ho840737317_state]: ((finite1986656878_state @ (sup_su737426425_state @ F @ G2)) = (((finite1986656878_state @ F)) & ((finite1986656878_state @ G2))))))). % finite_Un
thf(fact_8_evalc__elim__cases_I6_J, axiom,
    ((![P : pname, S : state, S1 : state]: ((evalc @ (body2 @ P) @ S @ S1) => (evalc @ (the_com @ (body @ P)) @ S @ S1))))). % evalc_elim_cases(6)
thf(fact_9_evalc_OBody, axiom,
    ((![Pn2 : pname, S0 : state, S1 : state]: ((evalc @ (the_com @ (body @ Pn2)) @ S0 @ S1) => (evalc @ (body2 @ Pn2) @ S0 @ S1))))). % evalc.Body
thf(fact_10_finite__imageI, axiom,
    ((![F : set_Ho840737317_state, H : hoare_958474565_state > pname]: ((finite1986656878_state @ F) => (finite_finite_pname @ (image_871391498_pname @ H @ F)))))). % finite_imageI
thf(fact_11_finite__imageI, axiom,
    ((![F : set_Ho840737317_state, H : hoare_958474565_state > hoare_958474565_state]: ((finite1986656878_state @ F) => (finite1986656878_state @ (image_1849792670_state @ H @ F)))))). % finite_imageI
thf(fact_12_finite__imageI, axiom,
    ((![F : set_pname, H : pname > hoare_958474565_state]: ((finite_finite_pname @ F) => (finite1986656878_state @ (image_2144627828_state @ H @ F)))))). % finite_imageI
thf(fact_13_finite__imageI, axiom,
    ((![F : set_pname, H : pname > pname]: ((finite_finite_pname @ F) => (finite_finite_pname @ (image_pname_pname @ H @ F)))))). % finite_imageI
thf(fact_14_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_15_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_16_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_17_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_18_image__ident, axiom,
    ((![Y4 : set_pname]: ((image_pname_pname @ (^[X : pname]: X) @ Y4) = Y4)))). % image_ident
thf(fact_19_com_Oinject_I6_J, axiom,
    ((![X7 : pname, Y7 : pname]: (((body2 @ X7) = (body2 @ Y7)) = (X7 = Y7))))). % com.inject(6)
thf(fact_20_UnCI, axiom,
    ((![C : hoare_958474565_state, B : set_Ho840737317_state, A : set_Ho840737317_state]: (((~ ((member109514606_state @ C @ B))) => (member109514606_state @ C @ A)) => (member109514606_state @ C @ (sup_su737426425_state @ A @ B)))))). % UnCI
thf(fact_21_UnCI, axiom,
    ((![C : pname, B : set_pname, A : set_pname]: (((~ ((member_pname @ C @ B))) => (member_pname @ C @ A)) => (member_pname @ C @ (sup_sup_set_pname @ A @ B)))))). % UnCI
thf(fact_22_Un__iff, axiom,
    ((![C : hoare_958474565_state, A : set_Ho840737317_state, B : set_Ho840737317_state]: ((member109514606_state @ C @ (sup_su737426425_state @ A @ B)) = (((member109514606_state @ C @ A)) | ((member109514606_state @ C @ B))))))). % Un_iff
thf(fact_23_Un__iff, axiom,
    ((![C : pname, A : set_pname, B : set_pname]: ((member_pname @ C @ (sup_sup_set_pname @ A @ B)) = (((member_pname @ C @ A)) | ((member_pname @ C @ B))))))). % Un_iff
thf(fact_24_image__eqI, axiom,
    ((![B2 : hoare_958474565_state, F2 : pname > hoare_958474565_state, X4 : pname, A : set_pname]: ((B2 = (F2 @ X4)) => ((member_pname @ X4 @ A) => (member109514606_state @ B2 @ (image_2144627828_state @ F2 @ A))))))). % image_eqI
thf(fact_25_image__eqI, axiom,
    ((![B2 : pname, F2 : pname > pname, X4 : pname, A : set_pname]: ((B2 = (F2 @ X4)) => ((member_pname @ X4 @ A) => (member_pname @ B2 @ (image_pname_pname @ F2 @ A))))))). % image_eqI
thf(fact_26_sup_Oright__idem, axiom,
    ((![A2 : set_Ho840737317_state, B2 : set_Ho840737317_state]: ((sup_su737426425_state @ (sup_su737426425_state @ A2 @ B2) @ B2) = (sup_su737426425_state @ A2 @ B2))))). % sup.right_idem
thf(fact_27_sup_Oright__idem, axiom,
    ((![A2 : set_pname, B2 : set_pname]: ((sup_sup_set_pname @ (sup_sup_set_pname @ A2 @ B2) @ B2) = (sup_sup_set_pname @ A2 @ B2))))). % sup.right_idem
thf(fact_28_sup__left__idem, axiom,
    ((![X4 : set_Ho840737317_state, Y : set_Ho840737317_state]: ((sup_su737426425_state @ X4 @ (sup_su737426425_state @ X4 @ Y)) = (sup_su737426425_state @ X4 @ Y))))). % sup_left_idem
thf(fact_29_sup__left__idem, axiom,
    ((![X4 : set_pname, Y : set_pname]: ((sup_sup_set_pname @ X4 @ (sup_sup_set_pname @ X4 @ Y)) = (sup_sup_set_pname @ X4 @ Y))))). % sup_left_idem
thf(fact_30_sup_Oleft__idem, axiom,
    ((![A2 : set_Ho840737317_state, B2 : set_Ho840737317_state]: ((sup_su737426425_state @ A2 @ (sup_su737426425_state @ A2 @ B2)) = (sup_su737426425_state @ A2 @ B2))))). % sup.left_idem
thf(fact_31_sup_Oleft__idem, axiom,
    ((![A2 : set_pname, B2 : set_pname]: ((sup_sup_set_pname @ A2 @ (sup_sup_set_pname @ A2 @ B2)) = (sup_sup_set_pname @ A2 @ B2))))). % sup.left_idem
thf(fact_32_sup__idem, axiom,
    ((![X4 : set_Ho840737317_state]: ((sup_su737426425_state @ X4 @ X4) = X4)))). % sup_idem
thf(fact_33_sup__idem, axiom,
    ((![X4 : set_pname]: ((sup_sup_set_pname @ X4 @ X4) = X4)))). % sup_idem
thf(fact_34_sup_Oidem, axiom,
    ((![A2 : set_Ho840737317_state]: ((sup_su737426425_state @ A2 @ A2) = A2)))). % sup.idem
thf(fact_35_sup_Oidem, axiom,
    ((![A2 : set_pname]: ((sup_sup_set_pname @ A2 @ A2) = A2)))). % sup.idem
thf(fact_36_sup__set__def, axiom,
    ((sup_su737426425_state = (^[A3 : set_Ho840737317_state]: (^[B3 : set_Ho840737317_state]: (collec305460656_state @ (sup_su2044190244tate_o @ (^[X : hoare_958474565_state]: (member109514606_state @ X @ A3)) @ (^[X : hoare_958474565_state]: (member109514606_state @ X @ B3))))))))). % sup_set_def
thf(fact_37_sup__set__def, axiom,
    ((sup_sup_set_pname = (^[A3 : set_pname]: (^[B3 : set_pname]: (collect_pname @ (sup_sup_pname_o @ (^[X : pname]: (member_pname @ X @ A3)) @ (^[X : pname]: (member_pname @ X @ B3))))))))). % sup_set_def
thf(fact_38_hoare__sound, axiom,
    ((![G2 : set_Ho840737317_state, Ts : set_Ho840737317_state]: ((hoare_604442164_state @ G2 @ Ts) => (hoare_318887606_state @ G2 @ Ts))))). % hoare_sound
thf(fact_39_rev__image__eqI, axiom,
    ((![X4 : pname, A : set_pname, B2 : hoare_958474565_state, F2 : pname > hoare_958474565_state]: ((member_pname @ X4 @ A) => ((B2 = (F2 @ X4)) => (member109514606_state @ B2 @ (image_2144627828_state @ F2 @ A))))))). % rev_image_eqI
thf(fact_40_rev__image__eqI, axiom,
    ((![X4 : pname, A : set_pname, B2 : pname, F2 : pname > pname]: ((member_pname @ X4 @ A) => ((B2 = (F2 @ X4)) => (member_pname @ B2 @ (image_pname_pname @ F2 @ A))))))). % rev_image_eqI
thf(fact_41_ball__imageD, axiom,
    ((![F2 : pname > hoare_958474565_state, A : set_pname, P : hoare_958474565_state > $o]: ((![X5 : hoare_958474565_state]: ((member109514606_state @ X5 @ (image_2144627828_state @ F2 @ A)) => (P @ X5))) => (![X6 : pname]: ((member_pname @ X6 @ A) => (P @ (F2 @ X6)))))))). % ball_imageD
thf(fact_42_ball__imageD, axiom,
    ((![F2 : pname > pname, A : set_pname, P : pname > $o]: ((![X5 : pname]: ((member_pname @ X5 @ (image_pname_pname @ F2 @ A)) => (P @ X5))) => (![X6 : pname]: ((member_pname @ X6 @ A) => (P @ (F2 @ X6)))))))). % ball_imageD
thf(fact_43_image__cong, axiom,
    ((![M : set_pname, N : set_pname, F2 : pname > hoare_958474565_state, G3 : pname > hoare_958474565_state]: ((M = N) => ((![X5 : pname]: ((member_pname @ X5 @ N) => ((F2 @ X5) = (G3 @ X5)))) => ((image_2144627828_state @ F2 @ M) = (image_2144627828_state @ G3 @ N))))))). % image_cong
thf(fact_44_image__cong, axiom,
    ((![M : set_pname, N : set_pname, F2 : pname > pname, G3 : pname > pname]: ((M = N) => ((![X5 : pname]: ((member_pname @ X5 @ N) => ((F2 @ X5) = (G3 @ X5)))) => ((image_pname_pname @ F2 @ M) = (image_pname_pname @ G3 @ N))))))). % image_cong
thf(fact_45_bex__imageD, axiom,
    ((![F2 : pname > hoare_958474565_state, A : set_pname, P : hoare_958474565_state > $o]: ((?[X6 : hoare_958474565_state]: ((member109514606_state @ X6 @ (image_2144627828_state @ F2 @ A)) & (P @ X6))) => (?[X5 : pname]: ((member_pname @ X5 @ A) & (P @ (F2 @ X5)))))))). % bex_imageD
thf(fact_46_bex__imageD, axiom,
    ((![F2 : pname > pname, A : set_pname, P : pname > $o]: ((?[X6 : pname]: ((member_pname @ X6 @ (image_pname_pname @ F2 @ A)) & (P @ X6))) => (?[X5 : pname]: ((member_pname @ X5 @ A) & (P @ (F2 @ X5)))))))). % bex_imageD
thf(fact_47_image__iff, axiom,
    ((![Z : hoare_958474565_state, F2 : pname > hoare_958474565_state, A : set_pname]: ((member109514606_state @ Z @ (image_2144627828_state @ F2 @ A)) = (?[X : pname]: (((member_pname @ X @ A)) & ((Z = (F2 @ X))))))))). % image_iff
thf(fact_48_image__iff, axiom,
    ((![Z : pname, F2 : pname > pname, A : set_pname]: ((member_pname @ Z @ (image_pname_pname @ F2 @ A)) = (?[X : pname]: (((member_pname @ X @ A)) & ((Z = (F2 @ X))))))))). % image_iff
thf(fact_49_imageI, axiom,
    ((![X4 : pname, A : set_pname, F2 : pname > hoare_958474565_state]: ((member_pname @ X4 @ A) => (member109514606_state @ (F2 @ X4) @ (image_2144627828_state @ F2 @ A)))))). % imageI
thf(fact_50_imageI, axiom,
    ((![X4 : pname, A : set_pname, F2 : pname > pname]: ((member_pname @ X4 @ A) => (member_pname @ (F2 @ X4) @ (image_pname_pname @ F2 @ A)))))). % imageI
thf(fact_51_sup__left__commute, axiom,
    ((![X4 : set_Ho840737317_state, Y : set_Ho840737317_state, Z : set_Ho840737317_state]: ((sup_su737426425_state @ X4 @ (sup_su737426425_state @ Y @ Z)) = (sup_su737426425_state @ Y @ (sup_su737426425_state @ X4 @ Z)))))). % sup_left_commute
thf(fact_52_sup__left__commute, axiom,
    ((![X4 : set_pname, Y : set_pname, Z : set_pname]: ((sup_sup_set_pname @ X4 @ (sup_sup_set_pname @ Y @ Z)) = (sup_sup_set_pname @ Y @ (sup_sup_set_pname @ X4 @ Z)))))). % sup_left_commute
thf(fact_53_sup_Oleft__commute, axiom,
    ((![B2 : set_Ho840737317_state, A2 : set_Ho840737317_state, C : set_Ho840737317_state]: ((sup_su737426425_state @ B2 @ (sup_su737426425_state @ A2 @ C)) = (sup_su737426425_state @ A2 @ (sup_su737426425_state @ B2 @ C)))))). % sup.left_commute
thf(fact_54_sup_Oleft__commute, axiom,
    ((![B2 : set_pname, A2 : set_pname, C : set_pname]: ((sup_sup_set_pname @ B2 @ (sup_sup_set_pname @ A2 @ C)) = (sup_sup_set_pname @ A2 @ (sup_sup_set_pname @ B2 @ C)))))). % sup.left_commute
thf(fact_55_sup__commute, axiom,
    ((sup_su737426425_state = (^[X : set_Ho840737317_state]: (^[Y5 : set_Ho840737317_state]: (sup_su737426425_state @ Y5 @ X)))))). % sup_commute
thf(fact_56_sup__commute, axiom,
    ((sup_sup_set_pname = (^[X : set_pname]: (^[Y5 : set_pname]: (sup_sup_set_pname @ Y5 @ X)))))). % sup_commute
thf(fact_57_sup_Ocommute, axiom,
    ((sup_su737426425_state = (^[A4 : set_Ho840737317_state]: (^[B4 : set_Ho840737317_state]: (sup_su737426425_state @ B4 @ A4)))))). % sup.commute
thf(fact_58_sup_Ocommute, axiom,
    ((sup_sup_set_pname = (^[A4 : set_pname]: (^[B4 : set_pname]: (sup_sup_set_pname @ B4 @ A4)))))). % sup.commute
thf(fact_59_sup__assoc, axiom,
    ((![X4 : set_Ho840737317_state, Y : set_Ho840737317_state, Z : set_Ho840737317_state]: ((sup_su737426425_state @ (sup_su737426425_state @ X4 @ Y) @ Z) = (sup_su737426425_state @ X4 @ (sup_su737426425_state @ Y @ Z)))))). % sup_assoc
thf(fact_60_sup__assoc, axiom,
    ((![X4 : set_pname, Y : set_pname, Z : set_pname]: ((sup_sup_set_pname @ (sup_sup_set_pname @ X4 @ Y) @ Z) = (sup_sup_set_pname @ X4 @ (sup_sup_set_pname @ Y @ Z)))))). % sup_assoc
thf(fact_61_sup_Oassoc, axiom,
    ((![A2 : set_Ho840737317_state, B2 : set_Ho840737317_state, C : set_Ho840737317_state]: ((sup_su737426425_state @ (sup_su737426425_state @ A2 @ B2) @ C) = (sup_su737426425_state @ A2 @ (sup_su737426425_state @ B2 @ C)))))). % sup.assoc
thf(fact_62_sup_Oassoc, axiom,
    ((![A2 : set_pname, B2 : set_pname, C : set_pname]: ((sup_sup_set_pname @ (sup_sup_set_pname @ A2 @ B2) @ C) = (sup_sup_set_pname @ A2 @ (sup_sup_set_pname @ B2 @ C)))))). % sup.assoc
thf(fact_63_boolean__algebra__cancel_Osup2, axiom,
    ((![B : set_Ho840737317_state, K : set_Ho840737317_state, B2 : set_Ho840737317_state, A2 : set_Ho840737317_state]: ((B = (sup_su737426425_state @ K @ B2)) => ((sup_su737426425_state @ A2 @ B) = (sup_su737426425_state @ K @ (sup_su737426425_state @ A2 @ B2))))))). % boolean_algebra_cancel.sup2
thf(fact_64_boolean__algebra__cancel_Osup2, axiom,
    ((![B : set_pname, K : set_pname, B2 : set_pname, A2 : set_pname]: ((B = (sup_sup_set_pname @ K @ B2)) => ((sup_sup_set_pname @ A2 @ B) = (sup_sup_set_pname @ K @ (sup_sup_set_pname @ A2 @ B2))))))). % boolean_algebra_cancel.sup2
thf(fact_65_boolean__algebra__cancel_Osup1, axiom,
    ((![A : set_Ho840737317_state, K : set_Ho840737317_state, A2 : set_Ho840737317_state, B2 : set_Ho840737317_state]: ((A = (sup_su737426425_state @ K @ A2)) => ((sup_su737426425_state @ A @ B2) = (sup_su737426425_state @ K @ (sup_su737426425_state @ A2 @ B2))))))). % boolean_algebra_cancel.sup1
thf(fact_66_boolean__algebra__cancel_Osup1, axiom,
    ((![A : set_pname, K : set_pname, A2 : set_pname, B2 : set_pname]: ((A = (sup_sup_set_pname @ K @ A2)) => ((sup_sup_set_pname @ A @ B2) = (sup_sup_set_pname @ K @ (sup_sup_set_pname @ A2 @ B2))))))). % boolean_algebra_cancel.sup1
thf(fact_67_inf__sup__aci_I5_J, axiom,
    ((sup_su737426425_state = (^[X : set_Ho840737317_state]: (^[Y5 : set_Ho840737317_state]: (sup_su737426425_state @ Y5 @ X)))))). % inf_sup_aci(5)
thf(fact_68_inf__sup__aci_I5_J, axiom,
    ((sup_sup_set_pname = (^[X : set_pname]: (^[Y5 : set_pname]: (sup_sup_set_pname @ Y5 @ X)))))). % inf_sup_aci(5)
thf(fact_69_inf__sup__aci_I6_J, axiom,
    ((![X4 : set_Ho840737317_state, Y : set_Ho840737317_state, Z : set_Ho840737317_state]: ((sup_su737426425_state @ (sup_su737426425_state @ X4 @ Y) @ Z) = (sup_su737426425_state @ X4 @ (sup_su737426425_state @ Y @ Z)))))). % inf_sup_aci(6)
thf(fact_70_inf__sup__aci_I6_J, axiom,
    ((![X4 : set_pname, Y : set_pname, Z : set_pname]: ((sup_sup_set_pname @ (sup_sup_set_pname @ X4 @ Y) @ Z) = (sup_sup_set_pname @ X4 @ (sup_sup_set_pname @ Y @ Z)))))). % inf_sup_aci(6)
thf(fact_71_inf__sup__aci_I7_J, axiom,
    ((![X4 : set_Ho840737317_state, Y : set_Ho840737317_state, Z : set_Ho840737317_state]: ((sup_su737426425_state @ X4 @ (sup_su737426425_state @ Y @ Z)) = (sup_su737426425_state @ Y @ (sup_su737426425_state @ X4 @ Z)))))). % inf_sup_aci(7)
thf(fact_72_inf__sup__aci_I7_J, axiom,
    ((![X4 : set_pname, Y : set_pname, Z : set_pname]: ((sup_sup_set_pname @ X4 @ (sup_sup_set_pname @ Y @ Z)) = (sup_sup_set_pname @ Y @ (sup_sup_set_pname @ X4 @ Z)))))). % inf_sup_aci(7)
thf(fact_73_inf__sup__aci_I8_J, axiom,
    ((![X4 : set_Ho840737317_state, Y : set_Ho840737317_state]: ((sup_su737426425_state @ X4 @ (sup_su737426425_state @ X4 @ Y)) = (sup_su737426425_state @ X4 @ Y))))). % inf_sup_aci(8)
thf(fact_74_inf__sup__aci_I8_J, axiom,
    ((![X4 : set_pname, Y : set_pname]: ((sup_sup_set_pname @ X4 @ (sup_sup_set_pname @ X4 @ Y)) = (sup_sup_set_pname @ X4 @ Y))))). % inf_sup_aci(8)
thf(fact_75_Un__left__commute, axiom,
    ((![A : set_Ho840737317_state, B : set_Ho840737317_state, C2 : set_Ho840737317_state]: ((sup_su737426425_state @ A @ (sup_su737426425_state @ B @ C2)) = (sup_su737426425_state @ B @ (sup_su737426425_state @ A @ C2)))))). % Un_left_commute
thf(fact_76_Un__left__commute, axiom,
    ((![A : set_pname, B : set_pname, C2 : set_pname]: ((sup_sup_set_pname @ A @ (sup_sup_set_pname @ B @ C2)) = (sup_sup_set_pname @ B @ (sup_sup_set_pname @ A @ C2)))))). % Un_left_commute
thf(fact_77_Un__left__absorb, axiom,
    ((![A : set_Ho840737317_state, B : set_Ho840737317_state]: ((sup_su737426425_state @ A @ (sup_su737426425_state @ A @ B)) = (sup_su737426425_state @ A @ B))))). % Un_left_absorb
thf(fact_78_Un__left__absorb, axiom,
    ((![A : set_pname, B : set_pname]: ((sup_sup_set_pname @ A @ (sup_sup_set_pname @ A @ B)) = (sup_sup_set_pname @ A @ B))))). % Un_left_absorb
thf(fact_79_Un__commute, axiom,
    ((sup_su737426425_state = (^[A3 : set_Ho840737317_state]: (^[B3 : set_Ho840737317_state]: (sup_su737426425_state @ B3 @ A3)))))). % Un_commute
thf(fact_80_Un__commute, axiom,
    ((sup_sup_set_pname = (^[A3 : set_pname]: (^[B3 : set_pname]: (sup_sup_set_pname @ B3 @ A3)))))). % Un_commute
thf(fact_81_Un__absorb, axiom,
    ((![A : set_Ho840737317_state]: ((sup_su737426425_state @ A @ A) = A)))). % Un_absorb
thf(fact_82_Un__absorb, axiom,
    ((![A : set_pname]: ((sup_sup_set_pname @ A @ A) = A)))). % Un_absorb
thf(fact_83_Un__assoc, axiom,
    ((![A : set_Ho840737317_state, B : set_Ho840737317_state, C2 : set_Ho840737317_state]: ((sup_su737426425_state @ (sup_su737426425_state @ A @ B) @ C2) = (sup_su737426425_state @ A @ (sup_su737426425_state @ B @ C2)))))). % Un_assoc
thf(fact_84_Un__assoc, axiom,
    ((![A : set_pname, B : set_pname, C2 : set_pname]: ((sup_sup_set_pname @ (sup_sup_set_pname @ A @ B) @ C2) = (sup_sup_set_pname @ A @ (sup_sup_set_pname @ B @ C2)))))). % Un_assoc
thf(fact_85_ball__Un, axiom,
    ((![A : set_Ho840737317_state, B : set_Ho840737317_state, P : hoare_958474565_state > $o]: ((![X : hoare_958474565_state]: (((member109514606_state @ X @ (sup_su737426425_state @ A @ B))) => ((P @ X)))) = (((![X : hoare_958474565_state]: (((member109514606_state @ X @ A)) => ((P @ X))))) & ((![X : hoare_958474565_state]: (((member109514606_state @ X @ B)) => ((P @ X)))))))))). % ball_Un
thf(fact_86_ball__Un, axiom,
    ((![A : set_pname, B : set_pname, P : pname > $o]: ((![X : pname]: (((member_pname @ X @ (sup_sup_set_pname @ A @ B))) => ((P @ X)))) = (((![X : pname]: (((member_pname @ X @ A)) => ((P @ X))))) & ((![X : pname]: (((member_pname @ X @ B)) => ((P @ X)))))))))). % ball_Un
thf(fact_87_bex__Un, axiom,
    ((![A : set_Ho840737317_state, B : set_Ho840737317_state, P : hoare_958474565_state > $o]: ((?[X : hoare_958474565_state]: (((member109514606_state @ X @ (sup_su737426425_state @ A @ B))) & ((P @ X)))) = (((?[X : hoare_958474565_state]: (((member109514606_state @ X @ A)) & ((P @ X))))) | ((?[X : hoare_958474565_state]: (((member109514606_state @ X @ B)) & ((P @ X)))))))))). % bex_Un
thf(fact_88_bex__Un, axiom,
    ((![A : set_pname, B : set_pname, P : pname > $o]: ((?[X : pname]: (((member_pname @ X @ (sup_sup_set_pname @ A @ B))) & ((P @ X)))) = (((?[X : pname]: (((member_pname @ X @ A)) & ((P @ X))))) | ((?[X : pname]: (((member_pname @ X @ B)) & ((P @ X)))))))))). % bex_Un
thf(fact_89_UnI2, axiom,
    ((![C : hoare_958474565_state, B : set_Ho840737317_state, A : set_Ho840737317_state]: ((member109514606_state @ C @ B) => (member109514606_state @ C @ (sup_su737426425_state @ A @ B)))))). % UnI2
thf(fact_90_UnI2, axiom,
    ((![C : pname, B : set_pname, A : set_pname]: ((member_pname @ C @ B) => (member_pname @ C @ (sup_sup_set_pname @ A @ B)))))). % UnI2
thf(fact_91_UnI1, axiom,
    ((![C : hoare_958474565_state, A : set_Ho840737317_state, B : set_Ho840737317_state]: ((member109514606_state @ C @ A) => (member109514606_state @ C @ (sup_su737426425_state @ A @ B)))))). % UnI1
thf(fact_92_UnI1, axiom,
    ((![C : pname, A : set_pname, B : set_pname]: ((member_pname @ C @ A) => (member_pname @ C @ (sup_sup_set_pname @ A @ B)))))). % UnI1
thf(fact_93_UnE, axiom,
    ((![C : hoare_958474565_state, A : set_Ho840737317_state, B : set_Ho840737317_state]: ((member109514606_state @ C @ (sup_su737426425_state @ A @ B)) => ((~ ((member109514606_state @ C @ A))) => (member109514606_state @ C @ B)))))). % UnE
thf(fact_94_UnE, axiom,
    ((![C : pname, A : set_pname, B : set_pname]: ((member_pname @ C @ (sup_sup_set_pname @ A @ B)) => ((~ ((member_pname @ C @ A))) => (member_pname @ C @ B)))))). % UnE
thf(fact_95_com__det, axiom,
    ((![C : com, S : state, T : state, U : state]: ((evalc @ C @ S @ T) => ((evalc @ C @ S @ U) => (U = T)))))). % com_det
thf(fact_96_Compr__image__eq, axiom,
    ((![F2 : pname > hoare_958474565_state, A : set_pname, P : hoare_958474565_state > $o]: ((collec305460656_state @ (^[X : hoare_958474565_state]: (((member109514606_state @ X @ (image_2144627828_state @ F2 @ A))) & ((P @ X))))) = (image_2144627828_state @ F2 @ (collect_pname @ (^[X : pname]: (((member_pname @ X @ A)) & ((P @ (F2 @ X))))))))))). % Compr_image_eq
thf(fact_97_Compr__image__eq, axiom,
    ((![F2 : pname > pname, A : set_pname, P : pname > $o]: ((collect_pname @ (^[X : pname]: (((member_pname @ X @ (image_pname_pname @ F2 @ A))) & ((P @ X))))) = (image_pname_pname @ F2 @ (collect_pname @ (^[X : pname]: (((member_pname @ X @ A)) & ((P @ (F2 @ X))))))))))). % Compr_image_eq
thf(fact_98_image__image, axiom,
    ((![F2 : hoare_958474565_state > hoare_958474565_state, G3 : pname > hoare_958474565_state, A : set_pname]: ((image_1849792670_state @ F2 @ (image_2144627828_state @ G3 @ A)) = (image_2144627828_state @ (^[X : pname]: (F2 @ (G3 @ X))) @ A))))). % image_image
thf(fact_99_image__image, axiom,
    ((![F2 : hoare_958474565_state > pname, G3 : pname > hoare_958474565_state, A : set_pname]: ((image_871391498_pname @ F2 @ (image_2144627828_state @ G3 @ A)) = (image_pname_pname @ (^[X : pname]: (F2 @ (G3 @ X))) @ A))))). % image_image
thf(fact_100_image__image, axiom,
    ((![F2 : pname > hoare_958474565_state, G3 : pname > pname, A : set_pname]: ((image_2144627828_state @ F2 @ (image_pname_pname @ G3 @ A)) = (image_2144627828_state @ (^[X : pname]: (F2 @ (G3 @ X))) @ A))))). % image_image
thf(fact_101_image__image, axiom,
    ((![F2 : pname > pname, G3 : pname > pname, A : set_pname]: ((image_pname_pname @ F2 @ (image_pname_pname @ G3 @ A)) = (image_pname_pname @ (^[X : pname]: (F2 @ (G3 @ X))) @ A))))). % image_image
thf(fact_102_imageE, axiom,
    ((![B2 : hoare_958474565_state, F2 : pname > hoare_958474565_state, A : set_pname]: ((member109514606_state @ B2 @ (image_2144627828_state @ F2 @ A)) => (~ ((![X5 : pname]: ((B2 = (F2 @ X5)) => (~ ((member_pname @ X5 @ A))))))))))). % imageE
thf(fact_103_imageE, axiom,
    ((![B2 : pname, F2 : pname > pname, A : set_pname]: ((member_pname @ B2 @ (image_pname_pname @ F2 @ A)) => (~ ((![X5 : pname]: ((B2 = (F2 @ X5)) => (~ ((member_pname @ X5 @ A))))))))))). % imageE
thf(fact_104_pigeonhole__infinite__rel, axiom,
    ((![A : set_pname, B : set_pname, R : pname > pname > $o]: ((~ ((finite_finite_pname @ A))) => ((finite_finite_pname @ B) => ((![X5 : pname]: ((member_pname @ X5 @ A) => (?[Xa : pname]: ((member_pname @ Xa @ B) & (R @ X5 @ Xa))))) => (?[X5 : pname]: ((member_pname @ X5 @ B) & (~ ((finite_finite_pname @ (collect_pname @ (^[A4 : pname]: (((member_pname @ A4 @ A)) & ((R @ A4 @ X5)))))))))))))))). % pigeonhole_infinite_rel
thf(fact_105_pigeonhole__infinite__rel, axiom,
    ((![A : set_pname, B : set_Ho840737317_state, R : pname > hoare_958474565_state > $o]: ((~ ((finite_finite_pname @ A))) => ((finite1986656878_state @ B) => ((![X5 : pname]: ((member_pname @ X5 @ A) => (?[Xa : hoare_958474565_state]: ((member109514606_state @ Xa @ B) & (R @ X5 @ Xa))))) => (?[X5 : hoare_958474565_state]: ((member109514606_state @ X5 @ B) & (~ ((finite_finite_pname @ (collect_pname @ (^[A4 : pname]: (((member_pname @ A4 @ A)) & ((R @ A4 @ X5)))))))))))))))). % pigeonhole_infinite_rel
thf(fact_106_pigeonhole__infinite__rel, axiom,
    ((![A : set_Ho840737317_state, B : set_pname, R : hoare_958474565_state > pname > $o]: ((~ ((finite1986656878_state @ A))) => ((finite_finite_pname @ B) => ((![X5 : hoare_958474565_state]: ((member109514606_state @ X5 @ A) => (?[Xa : pname]: ((member_pname @ Xa @ B) & (R @ X5 @ Xa))))) => (?[X5 : pname]: ((member_pname @ X5 @ B) & (~ ((finite1986656878_state @ (collec305460656_state @ (^[A4 : hoare_958474565_state]: (((member109514606_state @ A4 @ A)) & ((R @ A4 @ X5)))))))))))))))). % pigeonhole_infinite_rel
thf(fact_107_pigeonhole__infinite__rel, axiom,
    ((![A : set_Ho840737317_state, B : set_Ho840737317_state, R : hoare_958474565_state > hoare_958474565_state > $o]: ((~ ((finite1986656878_state @ A))) => ((finite1986656878_state @ B) => ((![X5 : hoare_958474565_state]: ((member109514606_state @ X5 @ A) => (?[Xa : hoare_958474565_state]: ((member109514606_state @ Xa @ B) & (R @ X5 @ Xa))))) => (?[X5 : hoare_958474565_state]: ((member109514606_state @ X5 @ B) & (~ ((finite1986656878_state @ (collec305460656_state @ (^[A4 : hoare_958474565_state]: (((member109514606_state @ A4 @ A)) & ((R @ A4 @ X5)))))))))))))))). % pigeonhole_infinite_rel
thf(fact_108_not__finite__existsD, axiom,
    ((![P : pname > $o]: ((~ ((finite_finite_pname @ (collect_pname @ P)))) => (?[X_1 : pname]: (P @ X_1)))))). % not_finite_existsD
thf(fact_109_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_110_Collect__disj__eq, axiom,
    ((![P : hoare_958474565_state > $o, Q : hoare_958474565_state > $o]: ((collec305460656_state @ (^[X : hoare_958474565_state]: (((P @ X)) | ((Q @ X))))) = (sup_su737426425_state @ (collec305460656_state @ P) @ (collec305460656_state @ Q)))))). % Collect_disj_eq
thf(fact_111_Collect__disj__eq, axiom,
    ((![P : pname > $o, Q : pname > $o]: ((collect_pname @ (^[X : pname]: (((P @ X)) | ((Q @ X))))) = (sup_sup_set_pname @ (collect_pname @ P) @ (collect_pname @ Q)))))). % Collect_disj_eq
thf(fact_112_Un__def, axiom,
    ((sup_su737426425_state = (^[A3 : set_Ho840737317_state]: (^[B3 : set_Ho840737317_state]: (collec305460656_state @ (^[X : hoare_958474565_state]: (((member109514606_state @ X @ A3)) | ((member109514606_state @ X @ B3)))))))))). % Un_def
thf(fact_113_Un__def, axiom,
    ((sup_sup_set_pname = (^[A3 : set_pname]: (^[B3 : set_pname]: (collect_pname @ (^[X : pname]: (((member_pname @ X @ A3)) | ((member_pname @ X @ B3)))))))))). % Un_def
thf(fact_114_image__Un, axiom,
    ((![F2 : hoare_958474565_state > hoare_958474565_state, A : set_Ho840737317_state, B : set_Ho840737317_state]: ((image_1849792670_state @ F2 @ (sup_su737426425_state @ A @ B)) = (sup_su737426425_state @ (image_1849792670_state @ F2 @ A) @ (image_1849792670_state @ F2 @ B)))))). % image_Un
thf(fact_115_image__Un, axiom,
    ((![F2 : hoare_958474565_state > pname, A : set_Ho840737317_state, B : set_Ho840737317_state]: ((image_871391498_pname @ F2 @ (sup_su737426425_state @ A @ B)) = (sup_sup_set_pname @ (image_871391498_pname @ F2 @ A) @ (image_871391498_pname @ F2 @ B)))))). % image_Un
thf(fact_116_image__Un, axiom,
    ((![F2 : pname > hoare_958474565_state, A : set_pname, B : set_pname]: ((image_2144627828_state @ F2 @ (sup_sup_set_pname @ A @ B)) = (sup_su737426425_state @ (image_2144627828_state @ F2 @ A) @ (image_2144627828_state @ F2 @ B)))))). % image_Un
thf(fact_117_image__Un, axiom,
    ((![F2 : pname > pname, A : set_pname, B : set_pname]: ((image_pname_pname @ F2 @ (sup_sup_set_pname @ A @ B)) = (sup_sup_set_pname @ (image_pname_pname @ F2 @ A) @ (image_pname_pname @ F2 @ B)))))). % image_Un
thf(fact_118_infinite__Un, axiom,
    ((![S2 : set_Ho840737317_state, T2 : set_Ho840737317_state]: ((~ ((finite1986656878_state @ (sup_su737426425_state @ S2 @ T2)))) = (((~ ((finite1986656878_state @ S2)))) | ((~ ((finite1986656878_state @ T2))))))))). % infinite_Un
thf(fact_119_infinite__Un, axiom,
    ((![S2 : set_pname, T2 : set_pname]: ((~ ((finite_finite_pname @ (sup_sup_set_pname @ S2 @ T2)))) = (((~ ((finite_finite_pname @ S2)))) | ((~ ((finite_finite_pname @ T2))))))))). % infinite_Un
thf(fact_120_Un__infinite, axiom,
    ((![S2 : set_Ho840737317_state, T2 : set_Ho840737317_state]: ((~ ((finite1986656878_state @ S2))) => (~ ((finite1986656878_state @ (sup_su737426425_state @ S2 @ T2)))))))). % Un_infinite
thf(fact_121_Un__infinite, axiom,
    ((![S2 : set_pname, T2 : set_pname]: ((~ ((finite_finite_pname @ S2))) => (~ ((finite_finite_pname @ (sup_sup_set_pname @ S2 @ T2)))))))). % Un_infinite
thf(fact_122_finite__UnI, axiom,
    ((![F : set_Ho840737317_state, G2 : set_Ho840737317_state]: ((finite1986656878_state @ F) => ((finite1986656878_state @ G2) => (finite1986656878_state @ (sup_su737426425_state @ F @ G2))))))). % finite_UnI
thf(fact_123_finite__UnI, axiom,
    ((![F : set_pname, G2 : set_pname]: ((finite_finite_pname @ F) => ((finite_finite_pname @ G2) => (finite_finite_pname @ (sup_sup_set_pname @ F @ G2))))))). % finite_UnI
thf(fact_124_pigeonhole__infinite, axiom,
    ((![A : set_pname, F2 : pname > pname]: ((~ ((finite_finite_pname @ A))) => ((finite_finite_pname @ (image_pname_pname @ F2 @ A)) => (?[X5 : pname]: ((member_pname @ X5 @ A) & (~ ((finite_finite_pname @ (collect_pname @ (^[A4 : pname]: (((member_pname @ A4 @ A)) & (((F2 @ A4) = (F2 @ X5)))))))))))))))). % pigeonhole_infinite
thf(fact_125_pigeonhole__infinite, axiom,
    ((![A : set_pname, F2 : pname > hoare_958474565_state]: ((~ ((finite_finite_pname @ A))) => ((finite1986656878_state @ (image_2144627828_state @ F2 @ A)) => (?[X5 : pname]: ((member_pname @ X5 @ A) & (~ ((finite_finite_pname @ (collect_pname @ (^[A4 : pname]: (((member_pname @ A4 @ A)) & (((F2 @ A4) = (F2 @ X5)))))))))))))))). % pigeonhole_infinite
thf(fact_126_pigeonhole__infinite, axiom,
    ((![A : set_Ho840737317_state, F2 : hoare_958474565_state > pname]: ((~ ((finite1986656878_state @ A))) => ((finite_finite_pname @ (image_871391498_pname @ F2 @ A)) => (?[X5 : hoare_958474565_state]: ((member109514606_state @ X5 @ A) & (~ ((finite1986656878_state @ (collec305460656_state @ (^[A4 : hoare_958474565_state]: (((member109514606_state @ A4 @ A)) & (((F2 @ A4) = (F2 @ X5)))))))))))))))). % pigeonhole_infinite
thf(fact_127_pigeonhole__infinite, axiom,
    ((![A : set_Ho840737317_state, F2 : hoare_958474565_state > hoare_958474565_state]: ((~ ((finite1986656878_state @ A))) => ((finite1986656878_state @ (image_1849792670_state @ F2 @ A)) => (?[X5 : hoare_958474565_state]: ((member109514606_state @ X5 @ A) & (~ ((finite1986656878_state @ (collec305460656_state @ (^[A4 : hoare_958474565_state]: (((member109514606_state @ A4 @ A)) & (((F2 @ A4) = (F2 @ X5)))))))))))))))). % pigeonhole_infinite
thf(fact_128_MGT__def, axiom,
    ((hoare_Mirabelle_MGT = (^[C3 : com]: (hoare_1659279548_state @ (^[Y6 : state]: (^[Z2 : state]: (Y6 = Z2))) @ C3 @ (evalc @ C3)))))). % MGT_def
thf(fact_129_Body1, axiom,
    ((![G2 : set_Ho840737317_state, P : pname > state > state > $o, Q : pname > state > state > $o, Procs : set_pname, Pn2 : pname]: ((hoare_604442164_state @ (sup_su737426425_state @ G2 @ (image_2144627828_state @ (^[P2 : pname]: (hoare_1659279548_state @ (P @ P2) @ (body2 @ P2) @ (Q @ P2))) @ Procs)) @ (image_2144627828_state @ (^[P2 : pname]: (hoare_1659279548_state @ (P @ P2) @ (the_com @ (body @ P2)) @ (Q @ P2))) @ Procs)) => ((member_pname @ Pn2 @ Procs) => (hoare_604442164_state @ G2 @ (insert776267541_state @ (hoare_1659279548_state @ (P @ Pn2) @ (body2 @ Pn2) @ (Q @ Pn2)) @ bot_bo105666705_state))))))). % Body1
thf(fact_130_sup__Un__eq, axiom,
    ((![R : set_Ho840737317_state, S2 : set_Ho840737317_state]: ((sup_su2044190244tate_o @ (^[X : hoare_958474565_state]: (member109514606_state @ X @ R)) @ (^[X : hoare_958474565_state]: (member109514606_state @ X @ S2))) = (^[X : hoare_958474565_state]: (member109514606_state @ X @ (sup_su737426425_state @ R @ S2))))))). % sup_Un_eq
thf(fact_131_sup__Un__eq, axiom,
    ((![R : set_pname, S2 : set_pname]: ((sup_sup_pname_o @ (^[X : pname]: (member_pname @ X @ R)) @ (^[X : pname]: (member_pname @ X @ S2))) = (^[X : pname]: (member_pname @ X @ (sup_sup_set_pname @ R @ S2))))))). % sup_Un_eq
thf(fact_132_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_133_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_134_finite__dom__body, axiom,
    ((finite_finite_pname @ (dom_pname_com @ body)))). % finite_dom_body
thf(fact_135_Sup_OSUP__cong, axiom,
    ((![A : set_pname, B : set_pname, C2 : pname > hoare_958474565_state, D : pname > hoare_958474565_state, Sup : set_Ho840737317_state > hoare_958474565_state]: ((A = B) => ((![X5 : pname]: ((member_pname @ X5 @ B) => ((C2 @ X5) = (D @ X5)))) => ((Sup @ (image_2144627828_state @ C2 @ A)) = (Sup @ (image_2144627828_state @ D @ B)))))))). % Sup.SUP_cong
thf(fact_136_Sup_OSUP__cong, axiom,
    ((![A : set_pname, B : set_pname, C2 : pname > pname, D : pname > pname, Sup : set_pname > pname]: ((A = B) => ((![X5 : pname]: ((member_pname @ X5 @ B) => ((C2 @ X5) = (D @ X5)))) => ((Sup @ (image_pname_pname @ C2 @ A)) = (Sup @ (image_pname_pname @ D @ B)))))))). % Sup.SUP_cong
thf(fact_137_Inf_OINF__cong, axiom,
    ((![A : set_pname, B : set_pname, C2 : pname > hoare_958474565_state, D : pname > hoare_958474565_state, Inf : set_Ho840737317_state > hoare_958474565_state]: ((A = B) => ((![X5 : pname]: ((member_pname @ X5 @ B) => ((C2 @ X5) = (D @ X5)))) => ((Inf @ (image_2144627828_state @ C2 @ A)) = (Inf @ (image_2144627828_state @ D @ B)))))))). % Inf.INF_cong
thf(fact_138_Inf_OINF__cong, axiom,
    ((![A : set_pname, B : set_pname, C2 : pname > pname, D : pname > pname, Inf : set_pname > pname]: ((A = B) => ((![X5 : pname]: ((member_pname @ X5 @ B) => ((C2 @ X5) = (D @ X5)))) => ((Inf @ (image_pname_pname @ C2 @ A)) = (Inf @ (image_pname_pname @ D @ B)))))))). % Inf.INF_cong
thf(fact_139_image__empty, axiom,
    ((![F2 : pname > hoare_958474565_state]: ((image_2144627828_state @ F2 @ bot_bot_set_pname) = bot_bo105666705_state)))). % image_empty
thf(fact_140_image__empty, axiom,
    ((![F2 : pname > pname]: ((image_pname_pname @ F2 @ bot_bot_set_pname) = bot_bot_set_pname)))). % image_empty
thf(fact_141_empty__is__image, axiom,
    ((![F2 : pname > hoare_958474565_state, A : set_pname]: ((bot_bo105666705_state = (image_2144627828_state @ F2 @ A)) = (A = bot_bot_set_pname))))). % empty_is_image
thf(fact_142_empty__is__image, axiom,
    ((![F2 : pname > pname, A : set_pname]: ((bot_bot_set_pname = (image_pname_pname @ F2 @ A)) = (A = bot_bot_set_pname))))). % empty_is_image
thf(fact_143_image__is__empty, axiom,
    ((![F2 : pname > hoare_958474565_state, A : set_pname]: (((image_2144627828_state @ F2 @ A) = bot_bo105666705_state) = (A = bot_bot_set_pname))))). % image_is_empty
thf(fact_144_image__is__empty, axiom,
    ((![F2 : pname > pname, A : set_pname]: (((image_pname_pname @ F2 @ A) = bot_bot_set_pname) = (A = bot_bot_set_pname))))). % image_is_empty
thf(fact_145_image__insert, axiom,
    ((![F2 : pname > hoare_958474565_state, A2 : pname, B : set_pname]: ((image_2144627828_state @ F2 @ (insert_pname @ A2 @ B)) = (insert776267541_state @ (F2 @ A2) @ (image_2144627828_state @ F2 @ B)))))). % image_insert
thf(fact_146_image__insert, axiom,
    ((![F2 : pname > pname, A2 : pname, B : set_pname]: ((image_pname_pname @ F2 @ (insert_pname @ A2 @ B)) = (insert_pname @ (F2 @ A2) @ (image_pname_pname @ F2 @ B)))))). % image_insert
thf(fact_147_insert__image, axiom,
    ((![X4 : pname, A : set_pname, F2 : pname > hoare_958474565_state]: ((member_pname @ X4 @ A) => ((insert776267541_state @ (F2 @ X4) @ (image_2144627828_state @ F2 @ A)) = (image_2144627828_state @ F2 @ A)))))). % insert_image
thf(fact_148_insert__image, axiom,
    ((![X4 : pname, A : set_pname, F2 : pname > pname]: ((member_pname @ X4 @ A) => ((insert_pname @ (F2 @ X4) @ (image_pname_pname @ F2 @ A)) = (image_pname_pname @ F2 @ A)))))). % insert_image
thf(fact_149_sup__bot__left, axiom,
    ((![X4 : set_Ho840737317_state]: ((sup_su737426425_state @ bot_bo105666705_state @ X4) = X4)))). % sup_bot_left
thf(fact_150_sup__bot__left, axiom,
    ((![X4 : set_pname]: ((sup_sup_set_pname @ bot_bot_set_pname @ X4) = X4)))). % sup_bot_left
thf(fact_151_sup__bot__right, axiom,
    ((![X4 : set_Ho840737317_state]: ((sup_su737426425_state @ X4 @ bot_bo105666705_state) = X4)))). % sup_bot_right
thf(fact_152_sup__bot__right, axiom,
    ((![X4 : set_pname]: ((sup_sup_set_pname @ X4 @ bot_bot_set_pname) = X4)))). % sup_bot_right
thf(fact_153_bot__eq__sup__iff, axiom,
    ((![X4 : set_Ho840737317_state, Y : set_Ho840737317_state]: ((bot_bo105666705_state = (sup_su737426425_state @ X4 @ Y)) = (((X4 = bot_bo105666705_state)) & ((Y = bot_bo105666705_state))))))). % bot_eq_sup_iff
thf(fact_154_bot__eq__sup__iff, axiom,
    ((![X4 : set_pname, Y : set_pname]: ((bot_bot_set_pname = (sup_sup_set_pname @ X4 @ Y)) = (((X4 = bot_bot_set_pname)) & ((Y = bot_bot_set_pname))))))). % bot_eq_sup_iff
thf(fact_155_sup__eq__bot__iff, axiom,
    ((![X4 : set_Ho840737317_state, Y : set_Ho840737317_state]: (((sup_su737426425_state @ X4 @ Y) = bot_bo105666705_state) = (((X4 = bot_bo105666705_state)) & ((Y = bot_bo105666705_state))))))). % sup_eq_bot_iff
thf(fact_156_sup__eq__bot__iff, axiom,
    ((![X4 : set_pname, Y : set_pname]: (((sup_sup_set_pname @ X4 @ Y) = bot_bot_set_pname) = (((X4 = bot_bot_set_pname)) & ((Y = bot_bot_set_pname))))))). % sup_eq_bot_iff
thf(fact_157_sup__bot_Oeq__neutr__iff, axiom,
    ((![A2 : set_Ho840737317_state, B2 : set_Ho840737317_state]: (((sup_su737426425_state @ A2 @ B2) = bot_bo105666705_state) = (((A2 = bot_bo105666705_state)) & ((B2 = bot_bo105666705_state))))))). % sup_bot.eq_neutr_iff
thf(fact_158_sup__bot_Oeq__neutr__iff, axiom,
    ((![A2 : set_pname, B2 : set_pname]: (((sup_sup_set_pname @ A2 @ B2) = bot_bot_set_pname) = (((A2 = bot_bot_set_pname)) & ((B2 = bot_bot_set_pname))))))). % sup_bot.eq_neutr_iff
thf(fact_159_sup__bot_Oleft__neutral, axiom,
    ((![A2 : set_Ho840737317_state]: ((sup_su737426425_state @ bot_bo105666705_state @ A2) = A2)))). % sup_bot.left_neutral
thf(fact_160_sup__bot_Oleft__neutral, axiom,
    ((![A2 : set_pname]: ((sup_sup_set_pname @ bot_bot_set_pname @ A2) = A2)))). % sup_bot.left_neutral
thf(fact_161_sup__bot_Oneutr__eq__iff, axiom,
    ((![A2 : set_Ho840737317_state, B2 : set_Ho840737317_state]: ((bot_bo105666705_state = (sup_su737426425_state @ A2 @ B2)) = (((A2 = bot_bo105666705_state)) & ((B2 = bot_bo105666705_state))))))). % sup_bot.neutr_eq_iff
thf(fact_162_sup__bot_Oneutr__eq__iff, axiom,
    ((![A2 : set_pname, B2 : set_pname]: ((bot_bot_set_pname = (sup_sup_set_pname @ A2 @ B2)) = (((A2 = bot_bot_set_pname)) & ((B2 = bot_bot_set_pname))))))). % sup_bot.neutr_eq_iff
thf(fact_163_sup__bot_Oright__neutral, axiom,
    ((![A2 : set_Ho840737317_state]: ((sup_su737426425_state @ A2 @ bot_bo105666705_state) = A2)))). % sup_bot.right_neutral
thf(fact_164_sup__bot_Oright__neutral, axiom,
    ((![A2 : set_pname]: ((sup_sup_set_pname @ A2 @ bot_bot_set_pname) = A2)))). % sup_bot.right_neutral
thf(fact_165_finite__insert, axiom,
    ((![A2 : pname, A : set_pname]: ((finite_finite_pname @ (insert_pname @ A2 @ A)) = (finite_finite_pname @ A))))). % finite_insert
thf(fact_166_finite__insert, axiom,
    ((![A2 : hoare_958474565_state, A : set_Ho840737317_state]: ((finite1986656878_state @ (insert776267541_state @ A2 @ A)) = (finite1986656878_state @ A))))). % finite_insert
thf(fact_167_Un__empty, axiom,
    ((![A : set_Ho840737317_state, B : set_Ho840737317_state]: (((sup_su737426425_state @ A @ B) = bot_bo105666705_state) = (((A = bot_bo105666705_state)) & ((B = bot_bo105666705_state))))))). % Un_empty
thf(fact_168_Un__empty, axiom,
    ((![A : set_pname, B : set_pname]: (((sup_sup_set_pname @ A @ B) = bot_bot_set_pname) = (((A = bot_bot_set_pname)) & ((B = bot_bot_set_pname))))))). % Un_empty
thf(fact_169_Un__insert__left, axiom,
    ((![A2 : hoare_958474565_state, B : set_Ho840737317_state, C2 : set_Ho840737317_state]: ((sup_su737426425_state @ (insert776267541_state @ A2 @ B) @ C2) = (insert776267541_state @ A2 @ (sup_su737426425_state @ B @ C2)))))). % Un_insert_left
thf(fact_170_Un__insert__left, axiom,
    ((![A2 : pname, B : set_pname, C2 : set_pname]: ((sup_sup_set_pname @ (insert_pname @ A2 @ B) @ C2) = (insert_pname @ A2 @ (sup_sup_set_pname @ B @ C2)))))). % Un_insert_left
thf(fact_171_Un__insert__right, axiom,
    ((![A : set_Ho840737317_state, A2 : hoare_958474565_state, B : set_Ho840737317_state]: ((sup_su737426425_state @ A @ (insert776267541_state @ A2 @ B)) = (insert776267541_state @ A2 @ (sup_su737426425_state @ A @ B)))))). % Un_insert_right
thf(fact_172_Un__insert__right, axiom,
    ((![A : set_pname, A2 : pname, B : set_pname]: ((sup_sup_set_pname @ A @ (insert_pname @ A2 @ B)) = (insert_pname @ A2 @ (sup_sup_set_pname @ A @ B)))))). % Un_insert_right
thf(fact_173_insert__is__Un, axiom,
    ((insert776267541_state = (^[A4 : hoare_958474565_state]: (sup_su737426425_state @ (insert776267541_state @ A4 @ bot_bo105666705_state)))))). % insert_is_Un
thf(fact_174_insert__is__Un, axiom,
    ((insert_pname = (^[A4 : pname]: (sup_sup_set_pname @ (insert_pname @ A4 @ bot_bot_set_pname)))))). % insert_is_Un
thf(fact_175_Un__singleton__iff, axiom,
    ((![A : set_Ho840737317_state, B : set_Ho840737317_state, X4 : hoare_958474565_state]: (((sup_su737426425_state @ A @ B) = (insert776267541_state @ X4 @ bot_bo105666705_state)) = (((((A = bot_bo105666705_state)) & ((B = (insert776267541_state @ X4 @ bot_bo105666705_state))))) | ((((((A = (insert776267541_state @ X4 @ bot_bo105666705_state))) & ((B = bot_bo105666705_state)))) | ((((A = (insert776267541_state @ X4 @ bot_bo105666705_state))) & ((B = (insert776267541_state @ X4 @ bot_bo105666705_state)))))))))))). % Un_singleton_iff
thf(fact_176_Un__singleton__iff, axiom,
    ((![A : set_pname, B : set_pname, X4 : pname]: (((sup_sup_set_pname @ A @ B) = (insert_pname @ X4 @ bot_bot_set_pname)) = (((((A = bot_bot_set_pname)) & ((B = (insert_pname @ X4 @ bot_bot_set_pname))))) | ((((((A = (insert_pname @ X4 @ bot_bot_set_pname))) & ((B = bot_bot_set_pname)))) | ((((A = (insert_pname @ X4 @ bot_bot_set_pname))) & ((B = (insert_pname @ X4 @ bot_bot_set_pname)))))))))))). % Un_singleton_iff
thf(fact_177_singleton__Un__iff, axiom,
    ((![X4 : hoare_958474565_state, A : set_Ho840737317_state, B : set_Ho840737317_state]: (((insert776267541_state @ X4 @ bot_bo105666705_state) = (sup_su737426425_state @ A @ B)) = (((((A = bot_bo105666705_state)) & ((B = (insert776267541_state @ X4 @ bot_bo105666705_state))))) | ((((((A = (insert776267541_state @ X4 @ bot_bo105666705_state))) & ((B = bot_bo105666705_state)))) | ((((A = (insert776267541_state @ X4 @ bot_bo105666705_state))) & ((B = (insert776267541_state @ X4 @ bot_bo105666705_state)))))))))))). % singleton_Un_iff
thf(fact_178_singleton__Un__iff, axiom,
    ((![X4 : pname, A : set_pname, B : set_pname]: (((insert_pname @ X4 @ bot_bot_set_pname) = (sup_sup_set_pname @ A @ B)) = (((((A = bot_bot_set_pname)) & ((B = (insert_pname @ X4 @ bot_bot_set_pname))))) | ((((((A = (insert_pname @ X4 @ bot_bot_set_pname))) & ((B = bot_bot_set_pname)))) | ((((A = (insert_pname @ X4 @ bot_bot_set_pname))) & ((B = (insert_pname @ X4 @ bot_bot_set_pname)))))))))))). % singleton_Un_iff
thf(fact_179_finite_Ocases, axiom,
    ((![A2 : set_pname]: ((finite_finite_pname @ A2) => ((~ ((A2 = bot_bot_set_pname))) => (~ ((![A5 : set_pname]: ((?[A6 : pname]: (A2 = (insert_pname @ A6 @ A5))) => (~ ((finite_finite_pname @ A5)))))))))))). % finite.cases
thf(fact_180_finite_Ocases, axiom,
    ((![A2 : set_Ho840737317_state]: ((finite1986656878_state @ A2) => ((~ ((A2 = bot_bo105666705_state))) => (~ ((![A5 : set_Ho840737317_state]: ((?[A6 : hoare_958474565_state]: (A2 = (insert776267541_state @ A6 @ A5))) => (~ ((finite1986656878_state @ A5)))))))))))). % finite.cases
thf(fact_181_finite_Osimps, axiom,
    ((finite_finite_pname = (^[A4 : set_pname]: (((A4 = bot_bot_set_pname)) | ((?[A3 : set_pname]: (?[B4 : pname]: (((A4 = (insert_pname @ B4 @ A3))) & ((finite_finite_pname @ A3))))))))))). % finite.simps
thf(fact_182_finite_Osimps, axiom,
    ((finite1986656878_state = (^[A4 : set_Ho840737317_state]: (((A4 = bot_bo105666705_state)) | ((?[A3 : set_Ho840737317_state]: (?[B4 : hoare_958474565_state]: (((A4 = (insert776267541_state @ B4 @ A3))) & ((finite1986656878_state @ A3))))))))))). % finite.simps
thf(fact_183_finite__induct, axiom,
    ((![F : set_pname, P : set_pname > $o]: ((finite_finite_pname @ F) => ((P @ bot_bot_set_pname) => ((![X5 : pname, F3 : set_pname]: ((finite_finite_pname @ F3) => ((~ ((member_pname @ X5 @ F3))) => ((P @ F3) => (P @ (insert_pname @ X5 @ F3)))))) => (P @ F))))))). % finite_induct
thf(fact_184_finite__induct, axiom,
    ((![F : set_Ho840737317_state, P : set_Ho840737317_state > $o]: ((finite1986656878_state @ F) => ((P @ bot_bo105666705_state) => ((![X5 : hoare_958474565_state, F3 : set_Ho840737317_state]: ((finite1986656878_state @ F3) => ((~ ((member109514606_state @ X5 @ F3))) => ((P @ F3) => (P @ (insert776267541_state @ X5 @ F3)))))) => (P @ F))))))). % finite_induct
thf(fact_185_finite_Oinducts, axiom,
    ((![X4 : set_pname, P : set_pname > $o]: ((finite_finite_pname @ X4) => ((P @ bot_bot_set_pname) => ((![A5 : set_pname, A6 : pname]: ((finite_finite_pname @ A5) => ((P @ A5) => (P @ (insert_pname @ A6 @ A5))))) => (P @ X4))))))). % finite.inducts
thf(fact_186_finite_Oinducts, axiom,
    ((![X4 : set_Ho840737317_state, P : set_Ho840737317_state > $o]: ((finite1986656878_state @ X4) => ((P @ bot_bo105666705_state) => ((![A5 : set_Ho840737317_state, A6 : hoare_958474565_state]: ((finite1986656878_state @ A5) => ((P @ A5) => (P @ (insert776267541_state @ A6 @ A5))))) => (P @ X4))))))). % finite.inducts
thf(fact_187_finite__ne__induct, axiom,
    ((![F : set_pname, P : set_pname > $o]: ((finite_finite_pname @ F) => ((~ ((F = bot_bot_set_pname))) => ((![X5 : pname]: (P @ (insert_pname @ X5 @ bot_bot_set_pname))) => ((![X5 : pname, F3 : set_pname]: ((finite_finite_pname @ F3) => ((~ ((F3 = bot_bot_set_pname))) => ((~ ((member_pname @ X5 @ F3))) => ((P @ F3) => (P @ (insert_pname @ X5 @ F3))))))) => (P @ F)))))))). % finite_ne_induct
thf(fact_188_finite__ne__induct, axiom,
    ((![F : set_Ho840737317_state, P : set_Ho840737317_state > $o]: ((finite1986656878_state @ F) => ((~ ((F = bot_bo105666705_state))) => ((![X5 : hoare_958474565_state]: (P @ (insert776267541_state @ X5 @ bot_bo105666705_state))) => ((![X5 : hoare_958474565_state, F3 : set_Ho840737317_state]: ((finite1986656878_state @ F3) => ((~ ((F3 = bot_bo105666705_state))) => ((~ ((member109514606_state @ X5 @ F3))) => ((P @ F3) => (P @ (insert776267541_state @ X5 @ F3))))))) => (P @ F)))))))). % finite_ne_induct
thf(fact_189_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) => ((![X5 : pname, F3 : set_pname]: ((finite_finite_pname @ F3) => ((~ ((member_pname @ X5 @ F3))) => ((P @ F3) => (P @ (insert_pname @ X5 @ F3)))))) => (P @ A))))))). % infinite_finite_induct
thf(fact_190_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) => ((![X5 : hoare_958474565_state, F3 : set_Ho840737317_state]: ((finite1986656878_state @ F3) => ((~ ((member109514606_state @ X5 @ F3))) => ((P @ F3) => (P @ (insert776267541_state @ X5 @ F3)))))) => (P @ A))))))). % infinite_finite_induct
thf(fact_191_derivs__insertD, axiom,
    ((![G2 : set_Ho840737317_state, T : hoare_958474565_state, Ts : set_Ho840737317_state]: ((hoare_604442164_state @ G2 @ (insert776267541_state @ T @ Ts)) => ((hoare_604442164_state @ G2 @ (insert776267541_state @ T @ bot_bo105666705_state)) & (hoare_604442164_state @ G2 @ Ts)))))). % derivs_insertD
thf(fact_192_hoare__derivs_Oinsert, axiom,
    ((![G2 : set_Ho840737317_state, T : hoare_958474565_state, Ts : set_Ho840737317_state]: ((hoare_604442164_state @ G2 @ (insert776267541_state @ T @ bot_bo105666705_state)) => ((hoare_604442164_state @ G2 @ Ts) => (hoare_604442164_state @ G2 @ (insert776267541_state @ T @ Ts))))))). % hoare_derivs.insert
thf(fact_193_image__constant, axiom,
    ((![X4 : pname, A : set_pname, C : hoare_958474565_state]: ((member_pname @ X4 @ A) => ((image_2144627828_state @ (^[X : pname]: C) @ A) = (insert776267541_state @ C @ bot_bo105666705_state)))))). % image_constant
thf(fact_194_image__constant, axiom,
    ((![X4 : pname, A : set_pname, C : pname]: ((member_pname @ X4 @ A) => ((image_pname_pname @ (^[X : pname]: C) @ A) = (insert_pname @ C @ bot_bot_set_pname)))))). % image_constant
thf(fact_195_image__constant__conv, axiom,
    ((![A : set_pname, C : hoare_958474565_state]: (((A = bot_bot_set_pname) => ((image_2144627828_state @ (^[X : pname]: C) @ A) = bot_bo105666705_state)) & ((~ ((A = bot_bot_set_pname))) => ((image_2144627828_state @ (^[X : pname]: C) @ A) = (insert776267541_state @ C @ bot_bo105666705_state))))))). % image_constant_conv
thf(fact_196_image__constant__conv, axiom,
    ((![A : set_pname, C : pname]: (((A = bot_bot_set_pname) => ((image_pname_pname @ (^[X : pname]: C) @ A) = bot_bot_set_pname)) & ((~ ((A = bot_bot_set_pname))) => ((image_pname_pname @ (^[X : pname]: C) @ A) = (insert_pname @ C @ bot_bot_set_pname))))))). % image_constant_conv

% Conjectures (3)
thf(conj_0, hypothesis,
    ((hoare_604442164_state @ (sup_su737426425_state @ g @ (image_2144627828_state @ (^[Pn : pname]: (hoare_1659279548_state @ (^[Y6 : state]: (^[Z2 : state]: (Y6 = Z2))) @ (body2 @ Pn) @ (evalc @ (body2 @ Pn)))) @ procs)) @ (image_2144627828_state @ (^[Pn : pname]: (hoare_1659279548_state @ (^[Y6 : state]: (^[Z2 : state]: (Y6 = Z2))) @ (the_com @ (body @ Pn)) @ (evalc @ (the_com @ (body @ Pn))))) @ procs)))).
thf(conj_1, hypothesis,
    ((finite_finite_pname @ procs))).
thf(conj_2, conjecture,
    ((hoare_604442164_state @ g @ (image_2144627828_state @ (^[Pn : pname]: (hoare_1659279548_state @ (^[Y6 : state]: (^[Z2 : state]: (Y6 = Z2))) @ (body2 @ Pn) @ (evalc @ (body2 @ Pn)))) @ procs)))).
