% 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_429__3254640_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:14:47.198

% Could-be-implicit typings (8)
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__Nat__Onat, type,
    nat : $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_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_Hoare__Mirabelle__raqjowkjvm_Otriple__valid_001t__Com__Ostate, type,
    hoare_364318704_state : nat > hoare_958474565_state > $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_Nat_OSuc, type,
    suc : nat > nat).
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_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__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__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J, type,
    image_1849792670_state : (hoare_958474565_state > hoare_958474565_state) > set_Ho840737317_state > set_Ho840737317_state).
thf(sy_c_Set_Oinsert_001t__Com__Opname, type,
    insert_pname : pname > set_pname > set_pname).
thf(sy_c_Set_Oinsert_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J, type,
    insert776267541_state : hoare_958474565_state > set_Ho840737317_state > set_Ho840737317_state).
thf(sy_c_Set_Othe__elem_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J, type,
    the_el1300254266_state : set_Ho840737317_state > hoare_958474565_state).
thf(sy_c_member_001t__Com__Opname, type,
    member_pname : pname > set_pname > $o).
thf(sy_c_member_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J, type,
    member109514606_state : hoare_958474565_state > set_Ho840737317_state > $o).
thf(sy_v_G, type,
    g : set_Ho840737317_state).
thf(sy_v_Procs, type,
    procs : set_pname).
thf(sy_v_pn, type,
    pn : pname).

% Relevant facts (141)
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_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_7_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_8_com_Oinject_I6_J, axiom,
    ((![X7 : pname, Y7 : pname]: (((body2 @ X7) = (body2 @ Y7)) = (X7 = Y7))))). % com.inject(6)
thf(fact_9_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_10_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_11_sup_Oidem, axiom,
    ((![A2 : set_Ho840737317_state]: ((sup_su737426425_state @ A2 @ A2) = A2)))). % sup.idem
thf(fact_12_sup__idem, axiom,
    ((![X : set_Ho840737317_state]: ((sup_su737426425_state @ X @ X) = X)))). % sup_idem
thf(fact_13_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_14_sup__left__idem, axiom,
    ((![X : set_Ho840737317_state, Y : set_Ho840737317_state]: ((sup_su737426425_state @ X @ (sup_su737426425_state @ X @ Y)) = (sup_su737426425_state @ X @ Y))))). % sup_left_idem
thf(fact_15_image__eqI, axiom,
    ((![B2 : hoare_958474565_state, F : pname > hoare_958474565_state, X : pname, A : set_pname]: ((B2 = (F @ X)) => ((member_pname @ X @ A) => (member109514606_state @ B2 @ (image_2144627828_state @ F @ A))))))). % image_eqI
thf(fact_16_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_17_sup__set__def, axiom,
    ((sup_su737426425_state = (^[A3 : set_Ho840737317_state]: (^[B3 : set_Ho840737317_state]: (collec305460656_state @ (sup_su2044190244tate_o @ (^[X4 : hoare_958474565_state]: (member109514606_state @ X4 @ A3)) @ (^[X4 : hoare_958474565_state]: (member109514606_state @ X4 @ B3))))))))). % sup_set_def
thf(fact_18_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_19_rev__image__eqI, axiom,
    ((![X : pname, A : set_pname, B2 : hoare_958474565_state, F : pname > hoare_958474565_state]: ((member_pname @ X @ A) => ((B2 = (F @ X)) => (member109514606_state @ B2 @ (image_2144627828_state @ F @ A))))))). % rev_image_eqI
thf(fact_20_ball__imageD, axiom,
    ((![F : pname > hoare_958474565_state, A : set_pname, P : hoare_958474565_state > $o]: ((![X5 : hoare_958474565_state]: ((member109514606_state @ X5 @ (image_2144627828_state @ F @ A)) => (P @ X5))) => (![X6 : pname]: ((member_pname @ X6 @ A) => (P @ (F @ X6)))))))). % ball_imageD
thf(fact_21_image__cong, axiom,
    ((![M : set_pname, N : set_pname, F : pname > hoare_958474565_state, G3 : pname > hoare_958474565_state]: ((M = N) => ((![X5 : pname]: ((member_pname @ X5 @ N) => ((F @ X5) = (G3 @ X5)))) => ((image_2144627828_state @ F @ M) = (image_2144627828_state @ G3 @ N))))))). % image_cong
thf(fact_22_bex__imageD, axiom,
    ((![F : pname > hoare_958474565_state, A : set_pname, P : hoare_958474565_state > $o]: ((?[X6 : hoare_958474565_state]: ((member109514606_state @ X6 @ (image_2144627828_state @ F @ A)) & (P @ X6))) => (?[X5 : pname]: ((member_pname @ X5 @ A) & (P @ (F @ X5)))))))). % bex_imageD
thf(fact_23_image__iff, axiom,
    ((![Z : hoare_958474565_state, F : pname > hoare_958474565_state, A : set_pname]: ((member109514606_state @ Z @ (image_2144627828_state @ F @ A)) = (?[X4 : pname]: (((member_pname @ X4 @ A)) & ((Z = (F @ X4))))))))). % image_iff
thf(fact_24_imageI, axiom,
    ((![X : pname, A : set_pname, F : pname > hoare_958474565_state]: ((member_pname @ X @ A) => (member109514606_state @ (F @ X) @ (image_2144627828_state @ F @ A)))))). % imageI
thf(fact_25_sup__left__commute, axiom,
    ((![X : set_Ho840737317_state, Y : set_Ho840737317_state, Z : set_Ho840737317_state]: ((sup_su737426425_state @ X @ (sup_su737426425_state @ Y @ Z)) = (sup_su737426425_state @ Y @ (sup_su737426425_state @ X @ Z)))))). % sup_left_commute
thf(fact_26_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_27_sup__commute, axiom,
    ((sup_su737426425_state = (^[X4 : set_Ho840737317_state]: (^[Y4 : set_Ho840737317_state]: (sup_su737426425_state @ Y4 @ X4)))))). % sup_commute
thf(fact_28_sup_Ocommute, axiom,
    ((sup_su737426425_state = (^[A4 : set_Ho840737317_state]: (^[B4 : set_Ho840737317_state]: (sup_su737426425_state @ B4 @ A4)))))). % sup.commute
thf(fact_29_sup__assoc, axiom,
    ((![X : set_Ho840737317_state, Y : set_Ho840737317_state, Z : set_Ho840737317_state]: ((sup_su737426425_state @ (sup_su737426425_state @ X @ Y) @ Z) = (sup_su737426425_state @ X @ (sup_su737426425_state @ Y @ Z)))))). % sup_assoc
thf(fact_30_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_31_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_32_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_33_inf__sup__aci_I5_J, axiom,
    ((sup_su737426425_state = (^[X4 : set_Ho840737317_state]: (^[Y4 : set_Ho840737317_state]: (sup_su737426425_state @ Y4 @ X4)))))). % inf_sup_aci(5)
thf(fact_34_inf__sup__aci_I6_J, axiom,
    ((![X : set_Ho840737317_state, Y : set_Ho840737317_state, Z : set_Ho840737317_state]: ((sup_su737426425_state @ (sup_su737426425_state @ X @ Y) @ Z) = (sup_su737426425_state @ X @ (sup_su737426425_state @ Y @ Z)))))). % inf_sup_aci(6)
thf(fact_35_inf__sup__aci_I7_J, axiom,
    ((![X : set_Ho840737317_state, Y : set_Ho840737317_state, Z : set_Ho840737317_state]: ((sup_su737426425_state @ X @ (sup_su737426425_state @ Y @ Z)) = (sup_su737426425_state @ Y @ (sup_su737426425_state @ X @ Z)))))). % inf_sup_aci(7)
thf(fact_36_inf__sup__aci_I8_J, axiom,
    ((![X : set_Ho840737317_state, Y : set_Ho840737317_state]: ((sup_su737426425_state @ X @ (sup_su737426425_state @ X @ Y)) = (sup_su737426425_state @ X @ Y))))). % inf_sup_aci(8)
thf(fact_37_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_38_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_39_Un__commute, axiom,
    ((sup_su737426425_state = (^[A3 : set_Ho840737317_state]: (^[B3 : set_Ho840737317_state]: (sup_su737426425_state @ B3 @ A3)))))). % Un_commute
thf(fact_40_Un__absorb, axiom,
    ((![A : set_Ho840737317_state]: ((sup_su737426425_state @ A @ A) = A)))). % Un_absorb
thf(fact_41_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_42_ball__Un, axiom,
    ((![A : set_Ho840737317_state, B : set_Ho840737317_state, P : hoare_958474565_state > $o]: ((![X4 : hoare_958474565_state]: (((member109514606_state @ X4 @ (sup_su737426425_state @ A @ B))) => ((P @ X4)))) = (((![X4 : hoare_958474565_state]: (((member109514606_state @ X4 @ A)) => ((P @ X4))))) & ((![X4 : hoare_958474565_state]: (((member109514606_state @ X4 @ B)) => ((P @ X4)))))))))). % ball_Un
thf(fact_43_bex__Un, axiom,
    ((![A : set_Ho840737317_state, B : set_Ho840737317_state, P : hoare_958474565_state > $o]: ((?[X4 : hoare_958474565_state]: (((member109514606_state @ X4 @ (sup_su737426425_state @ A @ B))) & ((P @ X4)))) = (((?[X4 : hoare_958474565_state]: (((member109514606_state @ X4 @ A)) & ((P @ X4))))) | ((?[X4 : hoare_958474565_state]: (((member109514606_state @ X4 @ B)) & ((P @ X4)))))))))). % bex_Un
thf(fact_44_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_45_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_46_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_47_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_48_Compr__image__eq, axiom,
    ((![F : pname > hoare_958474565_state, A : set_pname, P : hoare_958474565_state > $o]: ((collec305460656_state @ (^[X4 : hoare_958474565_state]: (((member109514606_state @ X4 @ (image_2144627828_state @ F @ A))) & ((P @ X4))))) = (image_2144627828_state @ F @ (collect_pname @ (^[X4 : pname]: (((member_pname @ X4 @ A)) & ((P @ (F @ X4))))))))))). % Compr_image_eq
thf(fact_49_image__image, axiom,
    ((![F : hoare_958474565_state > hoare_958474565_state, G3 : pname > hoare_958474565_state, A : set_pname]: ((image_1849792670_state @ F @ (image_2144627828_state @ G3 @ A)) = (image_2144627828_state @ (^[X4 : pname]: (F @ (G3 @ X4))) @ A))))). % image_image
thf(fact_50_image__image, axiom,
    ((![F : pname > hoare_958474565_state, G3 : pname > pname, A : set_pname]: ((image_2144627828_state @ F @ (image_pname_pname @ G3 @ A)) = (image_2144627828_state @ (^[X4 : pname]: (F @ (G3 @ X4))) @ A))))). % image_image
thf(fact_51_imageE, axiom,
    ((![B2 : hoare_958474565_state, F : pname > hoare_958474565_state, A : set_pname]: ((member109514606_state @ B2 @ (image_2144627828_state @ F @ A)) => (~ ((![X5 : pname]: ((B2 = (F @ X5)) => (~ ((member_pname @ X5 @ A))))))))))). % imageE
thf(fact_52_Collect__disj__eq, axiom,
    ((![P : hoare_958474565_state > $o, Q : hoare_958474565_state > $o]: ((collec305460656_state @ (^[X4 : hoare_958474565_state]: (((P @ X4)) | ((Q @ X4))))) = (sup_su737426425_state @ (collec305460656_state @ P) @ (collec305460656_state @ Q)))))). % Collect_disj_eq
thf(fact_53_Un__def, axiom,
    ((sup_su737426425_state = (^[A3 : set_Ho840737317_state]: (^[B3 : set_Ho840737317_state]: (collec305460656_state @ (^[X4 : hoare_958474565_state]: (((member109514606_state @ X4 @ A3)) | ((member109514606_state @ X4 @ B3)))))))))). % Un_def
thf(fact_54_image__Un, axiom,
    ((![F : pname > hoare_958474565_state, A : set_pname, B : set_pname]: ((image_2144627828_state @ F @ (sup_sup_set_pname @ A @ B)) = (sup_su737426425_state @ (image_2144627828_state @ F @ A) @ (image_2144627828_state @ F @ B)))))). % image_Un
thf(fact_55_image__Un, axiom,
    ((![F : hoare_958474565_state > hoare_958474565_state, A : set_Ho840737317_state, B : set_Ho840737317_state]: ((image_1849792670_state @ F @ (sup_su737426425_state @ A @ B)) = (sup_su737426425_state @ (image_1849792670_state @ F @ A) @ (image_1849792670_state @ F @ B)))))). % image_Un
thf(fact_56_MGT__def, axiom,
    ((hoare_Mirabelle_MGT = (^[C3 : com]: (hoare_1659279548_state @ (^[Y5 : state]: (^[Z2 : state]: (Y5 = Z2))) @ C3 @ (evalc @ C3)))))). % MGT_def
thf(fact_57_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_58_sup__Un__eq, axiom,
    ((![R : set_Ho840737317_state, S2 : set_Ho840737317_state]: ((sup_su2044190244tate_o @ (^[X4 : hoare_958474565_state]: (member109514606_state @ X4 @ R)) @ (^[X4 : hoare_958474565_state]: (member109514606_state @ X4 @ S2))) = (^[X4 : hoare_958474565_state]: (member109514606_state @ X4 @ (sup_su737426425_state @ R @ S2))))))). % sup_Un_eq
thf(fact_59_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_60_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_61_Body__triple__valid__Suc, axiom,
    ((![N2 : nat, P : state > state > $o, Pn2 : pname, Q : state > state > $o]: ((hoare_364318704_state @ N2 @ (hoare_1659279548_state @ P @ (the_com @ (body @ Pn2)) @ Q)) = (hoare_364318704_state @ (suc @ N2) @ (hoare_1659279548_state @ P @ (body2 @ Pn2) @ Q)))))). % Body_triple_valid_Suc
thf(fact_62_empty__iff, axiom,
    ((![C : hoare_958474565_state]: (~ ((member109514606_state @ C @ bot_bo105666705_state)))))). % empty_iff
thf(fact_63_all__not__in__conv, axiom,
    ((![A : set_Ho840737317_state]: ((![X4 : hoare_958474565_state]: (~ ((member109514606_state @ X4 @ A)))) = (A = bot_bo105666705_state))))). % all_not_in_conv
thf(fact_64_Collect__empty__eq, axiom,
    ((![P : hoare_958474565_state > $o]: (((collec305460656_state @ P) = bot_bo105666705_state) = (![X4 : hoare_958474565_state]: (~ ((P @ X4)))))))). % Collect_empty_eq
thf(fact_65_empty__Collect__eq, axiom,
    ((![P : hoare_958474565_state > $o]: ((bot_bo105666705_state = (collec305460656_state @ P)) = (![X4 : hoare_958474565_state]: (~ ((P @ X4)))))))). % empty_Collect_eq
thf(fact_66_insertCI, axiom,
    ((![A2 : hoare_958474565_state, B : set_Ho840737317_state, B2 : hoare_958474565_state]: (((~ ((member109514606_state @ A2 @ B))) => (A2 = B2)) => (member109514606_state @ A2 @ (insert776267541_state @ B2 @ B)))))). % insertCI
thf(fact_67_insert__iff, axiom,
    ((![A2 : hoare_958474565_state, B2 : hoare_958474565_state, A : set_Ho840737317_state]: ((member109514606_state @ A2 @ (insert776267541_state @ B2 @ A)) = (((A2 = B2)) | ((member109514606_state @ A2 @ A))))))). % insert_iff
thf(fact_68_insert__absorb2, axiom,
    ((![X : hoare_958474565_state, A : set_Ho840737317_state]: ((insert776267541_state @ X @ (insert776267541_state @ X @ A)) = (insert776267541_state @ X @ A))))). % insert_absorb2
thf(fact_69_image__empty, axiom,
    ((![F : pname > hoare_958474565_state]: ((image_2144627828_state @ F @ bot_bot_set_pname) = bot_bo105666705_state)))). % image_empty
thf(fact_70_image__empty, axiom,
    ((![F : hoare_958474565_state > hoare_958474565_state]: ((image_1849792670_state @ F @ bot_bo105666705_state) = bot_bo105666705_state)))). % image_empty
thf(fact_71_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_72_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_73_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_74_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_75_image__insert, axiom,
    ((![F : pname > hoare_958474565_state, A2 : pname, B : set_pname]: ((image_2144627828_state @ F @ (insert_pname @ A2 @ B)) = (insert776267541_state @ (F @ A2) @ (image_2144627828_state @ F @ B)))))). % image_insert
thf(fact_76_image__insert, axiom,
    ((![F : hoare_958474565_state > hoare_958474565_state, A2 : hoare_958474565_state, B : set_Ho840737317_state]: ((image_1849792670_state @ F @ (insert776267541_state @ A2 @ B)) = (insert776267541_state @ (F @ A2) @ (image_1849792670_state @ F @ B)))))). % image_insert
thf(fact_77_insert__image, axiom,
    ((![X : pname, A : set_pname, F : pname > hoare_958474565_state]: ((member_pname @ X @ A) => ((insert776267541_state @ (F @ X) @ (image_2144627828_state @ F @ A)) = (image_2144627828_state @ F @ A)))))). % insert_image
thf(fact_78_sup__bot__left, axiom,
    ((![X : set_Ho840737317_state]: ((sup_su737426425_state @ bot_bo105666705_state @ X) = X)))). % sup_bot_left
thf(fact_79_sup__bot__right, axiom,
    ((![X : set_Ho840737317_state]: ((sup_su737426425_state @ X @ bot_bo105666705_state) = X)))). % sup_bot_right
thf(fact_80_bot__eq__sup__iff, axiom,
    ((![X : set_Ho840737317_state, Y : set_Ho840737317_state]: ((bot_bo105666705_state = (sup_su737426425_state @ X @ Y)) = (((X = bot_bo105666705_state)) & ((Y = bot_bo105666705_state))))))). % bot_eq_sup_iff
thf(fact_81_sup__eq__bot__iff, axiom,
    ((![X : set_Ho840737317_state, Y : set_Ho840737317_state]: (((sup_su737426425_state @ X @ Y) = bot_bo105666705_state) = (((X = bot_bo105666705_state)) & ((Y = bot_bo105666705_state))))))). % sup_eq_bot_iff
thf(fact_82_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_83_sup__bot_Oleft__neutral, axiom,
    ((![A2 : set_Ho840737317_state]: ((sup_su737426425_state @ bot_bo105666705_state @ A2) = A2)))). % sup_bot.left_neutral
thf(fact_84_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_85_sup__bot_Oright__neutral, axiom,
    ((![A2 : set_Ho840737317_state]: ((sup_su737426425_state @ A2 @ bot_bo105666705_state) = A2)))). % sup_bot.right_neutral
thf(fact_86_singletonI, axiom,
    ((![A2 : hoare_958474565_state]: (member109514606_state @ A2 @ (insert776267541_state @ A2 @ bot_bo105666705_state))))). % singletonI
thf(fact_87_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_88_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_89_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_90_singleton__conv, axiom,
    ((![A2 : hoare_958474565_state]: ((collec305460656_state @ (^[X4 : hoare_958474565_state]: (X4 = A2))) = (insert776267541_state @ A2 @ bot_bo105666705_state))))). % singleton_conv
thf(fact_91_singleton__conv2, axiom,
    ((![A2 : hoare_958474565_state]: ((collec305460656_state @ ((^[Y5 : hoare_958474565_state]: (^[Z2 : hoare_958474565_state]: (Y5 = Z2))) @ A2)) = (insert776267541_state @ A2 @ bot_bo105666705_state))))). % singleton_conv2
thf(fact_92_emptyE, axiom,
    ((![A2 : hoare_958474565_state]: (~ ((member109514606_state @ A2 @ bot_bo105666705_state)))))). % emptyE
thf(fact_93_insertE, axiom,
    ((![A2 : hoare_958474565_state, B2 : hoare_958474565_state, A : set_Ho840737317_state]: ((member109514606_state @ A2 @ (insert776267541_state @ B2 @ A)) => ((~ ((A2 = B2))) => (member109514606_state @ A2 @ A)))))). % insertE
thf(fact_94_equals0D, axiom,
    ((![A : set_Ho840737317_state, A2 : hoare_958474565_state]: ((A = bot_bo105666705_state) => (~ ((member109514606_state @ A2 @ A))))))). % equals0D
thf(fact_95_equals0I, axiom,
    ((![A : set_Ho840737317_state]: ((![Y6 : hoare_958474565_state]: (~ ((member109514606_state @ Y6 @ A)))) => (A = bot_bo105666705_state))))). % equals0I
thf(fact_96_insertI1, axiom,
    ((![A2 : hoare_958474565_state, B : set_Ho840737317_state]: (member109514606_state @ A2 @ (insert776267541_state @ A2 @ B))))). % insertI1
thf(fact_97_insertI2, axiom,
    ((![A2 : hoare_958474565_state, B : set_Ho840737317_state, B2 : hoare_958474565_state]: ((member109514606_state @ A2 @ B) => (member109514606_state @ A2 @ (insert776267541_state @ B2 @ B)))))). % insertI2
thf(fact_98_ex__in__conv, axiom,
    ((![A : set_Ho840737317_state]: ((?[X4 : hoare_958474565_state]: (member109514606_state @ X4 @ A)) = (~ ((A = bot_bo105666705_state))))))). % ex_in_conv
thf(fact_99_Set_Oset__insert, axiom,
    ((![X : hoare_958474565_state, A : set_Ho840737317_state]: ((member109514606_state @ X @ A) => (~ ((![B5 : set_Ho840737317_state]: ((A = (insert776267541_state @ X @ B5)) => (member109514606_state @ X @ B5))))))))). % Set.set_insert
thf(fact_100_singletonD, axiom,
    ((![B2 : hoare_958474565_state, A2 : hoare_958474565_state]: ((member109514606_state @ B2 @ (insert776267541_state @ A2 @ bot_bo105666705_state)) => (B2 = A2))))). % singletonD
thf(fact_101_insert__compr, axiom,
    ((insert776267541_state = (^[A4 : hoare_958474565_state]: (^[B3 : set_Ho840737317_state]: (collec305460656_state @ (^[X4 : hoare_958474565_state]: (((X4 = A4)) | ((member109514606_state @ X4 @ B3)))))))))). % insert_compr
thf(fact_102_insert__ident, axiom,
    ((![X : hoare_958474565_state, A : set_Ho840737317_state, B : set_Ho840737317_state]: ((~ ((member109514606_state @ X @ A))) => ((~ ((member109514606_state @ X @ B))) => (((insert776267541_state @ X @ A) = (insert776267541_state @ X @ B)) = (A = B))))))). % insert_ident
thf(fact_103_insert__is__Un, axiom,
    ((insert776267541_state = (^[A4 : hoare_958474565_state]: (sup_su737426425_state @ (insert776267541_state @ A4 @ bot_bo105666705_state)))))). % insert_is_Un
thf(fact_104_insert__absorb, axiom,
    ((![A2 : hoare_958474565_state, A : set_Ho840737317_state]: ((member109514606_state @ A2 @ A) => ((insert776267541_state @ A2 @ A) = A))))). % insert_absorb
thf(fact_105_insert__eq__iff, axiom,
    ((![A2 : hoare_958474565_state, A : set_Ho840737317_state, B2 : hoare_958474565_state, B : set_Ho840737317_state]: ((~ ((member109514606_state @ A2 @ A))) => ((~ ((member109514606_state @ B2 @ B))) => (((insert776267541_state @ A2 @ A) = (insert776267541_state @ B2 @ B)) = (((((A2 = B2)) => ((A = B)))) & ((((~ ((A2 = B2)))) => ((?[C4 : set_Ho840737317_state]: (((A = (insert776267541_state @ B2 @ C4))) & ((((~ ((member109514606_state @ B2 @ C4)))) & ((((B = (insert776267541_state @ A2 @ C4))) & ((~ ((member109514606_state @ A2 @ C4)))))))))))))))))))). % insert_eq_iff
thf(fact_106_singleton__iff, axiom,
    ((![B2 : hoare_958474565_state, A2 : hoare_958474565_state]: ((member109514606_state @ B2 @ (insert776267541_state @ A2 @ bot_bo105666705_state)) = (B2 = A2))))). % singleton_iff
thf(fact_107_insert__Collect, axiom,
    ((![A2 : hoare_958474565_state, P : hoare_958474565_state > $o]: ((insert776267541_state @ A2 @ (collec305460656_state @ P)) = (collec305460656_state @ (^[U2 : hoare_958474565_state]: (((~ ((U2 = A2)))) => ((P @ U2))))))))). % insert_Collect
thf(fact_108_insert__commute, axiom,
    ((![X : hoare_958474565_state, Y : hoare_958474565_state, A : set_Ho840737317_state]: ((insert776267541_state @ X @ (insert776267541_state @ Y @ A)) = (insert776267541_state @ Y @ (insert776267541_state @ X @ A)))))). % insert_commute
thf(fact_109_Collect__conv__if, axiom,
    ((![P : hoare_958474565_state > $o, A2 : hoare_958474565_state]: (((P @ A2) => ((collec305460656_state @ (^[X4 : hoare_958474565_state]: (((X4 = A2)) & ((P @ X4))))) = (insert776267541_state @ A2 @ bot_bo105666705_state))) & ((~ ((P @ A2))) => ((collec305460656_state @ (^[X4 : hoare_958474565_state]: (((X4 = A2)) & ((P @ X4))))) = bot_bo105666705_state)))))). % Collect_conv_if
thf(fact_110_Collect__conv__if2, axiom,
    ((![P : hoare_958474565_state > $o, A2 : hoare_958474565_state]: (((P @ A2) => ((collec305460656_state @ (^[X4 : hoare_958474565_state]: (((A2 = X4)) & ((P @ X4))))) = (insert776267541_state @ A2 @ bot_bo105666705_state))) & ((~ ((P @ A2))) => ((collec305460656_state @ (^[X4 : hoare_958474565_state]: (((A2 = X4)) & ((P @ X4))))) = bot_bo105666705_state)))))). % Collect_conv_if2
thf(fact_111_Un__singleton__iff, axiom,
    ((![A : set_Ho840737317_state, B : set_Ho840737317_state, X : hoare_958474565_state]: (((sup_su737426425_state @ A @ B) = (insert776267541_state @ X @ bot_bo105666705_state)) = (((((A = bot_bo105666705_state)) & ((B = (insert776267541_state @ X @ bot_bo105666705_state))))) | ((((((A = (insert776267541_state @ X @ bot_bo105666705_state))) & ((B = bot_bo105666705_state)))) | ((((A = (insert776267541_state @ X @ bot_bo105666705_state))) & ((B = (insert776267541_state @ X @ bot_bo105666705_state)))))))))))). % Un_singleton_iff
thf(fact_112_doubleton__eq__iff, axiom,
    ((![A2 : hoare_958474565_state, B2 : hoare_958474565_state, C : hoare_958474565_state, D2 : hoare_958474565_state]: (((insert776267541_state @ A2 @ (insert776267541_state @ B2 @ bot_bo105666705_state)) = (insert776267541_state @ C @ (insert776267541_state @ D2 @ bot_bo105666705_state))) = (((((A2 = C)) & ((B2 = D2)))) | ((((A2 = D2)) & ((B2 = C))))))))). % doubleton_eq_iff
thf(fact_113_insert__not__empty, axiom,
    ((![A2 : hoare_958474565_state, A : set_Ho840737317_state]: (~ (((insert776267541_state @ A2 @ A) = bot_bo105666705_state)))))). % insert_not_empty
thf(fact_114_singleton__Un__iff, axiom,
    ((![X : hoare_958474565_state, A : set_Ho840737317_state, B : set_Ho840737317_state]: (((insert776267541_state @ X @ bot_bo105666705_state) = (sup_su737426425_state @ A @ B)) = (((((A = bot_bo105666705_state)) & ((B = (insert776267541_state @ X @ bot_bo105666705_state))))) | ((((((A = (insert776267541_state @ X @ bot_bo105666705_state))) & ((B = bot_bo105666705_state)))) | ((((A = (insert776267541_state @ X @ bot_bo105666705_state))) & ((B = (insert776267541_state @ X @ bot_bo105666705_state)))))))))))). % singleton_Un_iff
thf(fact_115_singleton__inject, axiom,
    ((![A2 : hoare_958474565_state, B2 : hoare_958474565_state]: (((insert776267541_state @ A2 @ bot_bo105666705_state) = (insert776267541_state @ B2 @ bot_bo105666705_state)) => (A2 = B2))))). % singleton_inject
thf(fact_116_mk__disjoint__insert, axiom,
    ((![A2 : hoare_958474565_state, A : set_Ho840737317_state]: ((member109514606_state @ A2 @ A) => (?[B5 : set_Ho840737317_state]: ((A = (insert776267541_state @ A2 @ B5)) & (~ ((member109514606_state @ A2 @ B5))))))))). % mk_disjoint_insert
thf(fact_117_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_118_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_119_empty__def, axiom,
    ((bot_bo105666705_state = (collec305460656_state @ (^[X4 : hoare_958474565_state]: $false))))). % empty_def
thf(fact_120_image__constant, axiom,
    ((![X : pname, A : set_pname, C : hoare_958474565_state]: ((member_pname @ X @ A) => ((image_2144627828_state @ (^[X4 : pname]: C) @ A) = (insert776267541_state @ C @ bot_bo105666705_state)))))). % image_constant
thf(fact_121_image__constant__conv, axiom,
    ((![A : set_pname, C : hoare_958474565_state]: (((A = bot_bot_set_pname) => ((image_2144627828_state @ (^[X4 : pname]: C) @ A) = bot_bo105666705_state)) & ((~ ((A = bot_bot_set_pname))) => ((image_2144627828_state @ (^[X4 : pname]: C) @ A) = (insert776267541_state @ C @ bot_bo105666705_state))))))). % image_constant_conv
thf(fact_122_image__constant__conv, axiom,
    ((![A : set_Ho840737317_state, C : hoare_958474565_state]: (((A = bot_bo105666705_state) => ((image_1849792670_state @ (^[X4 : hoare_958474565_state]: C) @ A) = bot_bo105666705_state)) & ((~ ((A = bot_bo105666705_state))) => ((image_1849792670_state @ (^[X4 : hoare_958474565_state]: C) @ A) = (insert776267541_state @ C @ bot_bo105666705_state))))))). % image_constant_conv
thf(fact_123_MGF__complete, axiom,
    ((![C : com, P : state > state > $o, Q : state > state > $o]: ((hoare_604442164_state @ bot_bo105666705_state @ (insert776267541_state @ (hoare_Mirabelle_MGT @ C) @ bot_bo105666705_state)) => ((hoare_318887606_state @ bot_bo105666705_state @ (insert776267541_state @ (hoare_1659279548_state @ P @ C @ Q) @ bot_bo105666705_state)) => (hoare_604442164_state @ bot_bo105666705_state @ (insert776267541_state @ (hoare_1659279548_state @ P @ C @ Q) @ bot_bo105666705_state))))))). % MGF_complete
thf(fact_124_conseq1, axiom,
    ((![G2 : set_Ho840737317_state, P3 : state > state > $o, C : com, Q : state > state > $o, P : state > state > $o]: ((hoare_604442164_state @ G2 @ (insert776267541_state @ (hoare_1659279548_state @ P3 @ C @ Q) @ bot_bo105666705_state)) => ((![Z3 : state, S3 : state]: ((P @ Z3 @ S3) => (P3 @ Z3 @ S3))) => (hoare_604442164_state @ G2 @ (insert776267541_state @ (hoare_1659279548_state @ P @ C @ Q) @ bot_bo105666705_state))))))). % conseq1
thf(fact_125_conseq2, axiom,
    ((![G2 : set_Ho840737317_state, P : state > state > $o, C : com, Q2 : state > state > $o, Q : state > state > $o]: ((hoare_604442164_state @ G2 @ (insert776267541_state @ (hoare_1659279548_state @ P @ C @ Q2) @ bot_bo105666705_state)) => ((![Z3 : state, S3 : state]: ((Q2 @ Z3 @ S3) => (Q @ Z3 @ S3))) => (hoare_604442164_state @ G2 @ (insert776267541_state @ (hoare_1659279548_state @ P @ C @ Q) @ bot_bo105666705_state))))))). % conseq2
thf(fact_126_conseq12, axiom,
    ((![G2 : set_Ho840737317_state, P3 : state > state > $o, C : com, Q2 : state > state > $o, P : state > state > $o, Q : state > state > $o]: ((hoare_604442164_state @ G2 @ (insert776267541_state @ (hoare_1659279548_state @ P3 @ C @ Q2) @ bot_bo105666705_state)) => ((![Z3 : state, S3 : state]: ((P @ Z3 @ S3) => (![S4 : state]: ((![Z4 : state]: ((P3 @ Z4 @ S3) => (Q2 @ Z4 @ S4))) => (Q @ Z3 @ S4))))) => (hoare_604442164_state @ G2 @ (insert776267541_state @ (hoare_1659279548_state @ P @ C @ Q) @ bot_bo105666705_state))))))). % conseq12
thf(fact_127_conseq, axiom,
    ((![P : state > state > $o, G2 : set_Ho840737317_state, C : com, Q : state > state > $o]: ((![Z3 : state, S3 : state]: ((P @ Z3 @ S3) => (?[P4 : state > state > $o, Q3 : state > state > $o]: ((hoare_604442164_state @ G2 @ (insert776267541_state @ (hoare_1659279548_state @ P4 @ C @ Q3) @ bot_bo105666705_state)) & (![S4 : state]: ((![Z4 : state]: ((P4 @ Z4 @ S3) => (Q3 @ Z4 @ S4))) => (Q @ Z3 @ S4))))))) => (hoare_604442164_state @ G2 @ (insert776267541_state @ (hoare_1659279548_state @ P @ C @ Q) @ bot_bo105666705_state)))))). % conseq
thf(fact_128_Un__empty__left, axiom,
    ((![B : set_Ho840737317_state]: ((sup_su737426425_state @ bot_bo105666705_state @ B) = B)))). % Un_empty_left
thf(fact_129_Un__empty__right, axiom,
    ((![A : set_Ho840737317_state]: ((sup_su737426425_state @ A @ bot_bo105666705_state) = A)))). % Un_empty_right
thf(fact_130_escape, axiom,
    ((![P : state > state > $o, G2 : set_Ho840737317_state, C : com, Q : state > state > $o]: ((![Z3 : state, S3 : state]: ((P @ Z3 @ S3) => (hoare_604442164_state @ G2 @ (insert776267541_state @ (hoare_1659279548_state @ (^[Za : state]: (^[S5 : state]: (S5 = S3))) @ C @ (^[Z5 : state]: (Q @ Z3))) @ bot_bo105666705_state)))) => (hoare_604442164_state @ G2 @ (insert776267541_state @ (hoare_1659279548_state @ P @ C @ Q) @ bot_bo105666705_state)))))). % escape
thf(fact_131__Cconstant_C, axiom,
    ((![C2 : $o, G2 : set_Ho840737317_state, P : state > state > $o, C : com, Q : state > state > $o]: ((C2 => (hoare_604442164_state @ G2 @ (insert776267541_state @ (hoare_1659279548_state @ P @ C @ Q) @ bot_bo105666705_state))) => (hoare_604442164_state @ G2 @ (insert776267541_state @ (hoare_1659279548_state @ (^[Z6 : state]: (^[S6 : state]: (((P @ Z6 @ S6)) & (C2)))) @ C @ Q) @ bot_bo105666705_state)))))). % "constant"
thf(fact_132_empty, axiom,
    ((![G2 : set_Ho840737317_state]: (hoare_604442164_state @ G2 @ bot_bo105666705_state)))). % empty
thf(fact_133_insert__def, axiom,
    ((insert776267541_state = (^[A4 : hoare_958474565_state]: (sup_su737426425_state @ (collec305460656_state @ (^[X4 : hoare_958474565_state]: (X4 = A4)))))))). % insert_def
thf(fact_134_MGT__BodyN, axiom,
    ((![Pn2 : pname, G2 : set_Ho840737317_state]: ((hoare_604442164_state @ (insert776267541_state @ (hoare_Mirabelle_MGT @ (body2 @ Pn2)) @ G2) @ (insert776267541_state @ (hoare_Mirabelle_MGT @ (the_com @ (body @ Pn2))) @ bot_bo105666705_state)) => (hoare_604442164_state @ G2 @ (insert776267541_state @ (hoare_Mirabelle_MGT @ (body2 @ Pn2)) @ bot_bo105666705_state)))))). % MGT_BodyN
thf(fact_135_BodyN, axiom,
    ((![P : state > state > $o, Pn2 : pname, Q : state > state > $o, G2 : set_Ho840737317_state]: ((hoare_604442164_state @ (insert776267541_state @ (hoare_1659279548_state @ P @ (body2 @ Pn2) @ Q) @ G2) @ (insert776267541_state @ (hoare_1659279548_state @ P @ (the_com @ (body @ Pn2)) @ Q) @ bot_bo105666705_state)) => (hoare_604442164_state @ G2 @ (insert776267541_state @ (hoare_1659279548_state @ P @ (body2 @ Pn2) @ Q) @ bot_bo105666705_state)))))). % BodyN
thf(fact_136_weak__Body, axiom,
    ((![G2 : set_Ho840737317_state, P : state > state > $o, Pn2 : pname, Q : state > state > $o]: ((hoare_604442164_state @ G2 @ (insert776267541_state @ (hoare_1659279548_state @ P @ (the_com @ (body @ Pn2)) @ Q) @ bot_bo105666705_state)) => (hoare_604442164_state @ G2 @ (insert776267541_state @ (hoare_1659279548_state @ P @ (body2 @ Pn2) @ Q) @ bot_bo105666705_state)))))). % weak_Body
thf(fact_137_old_Onat_Oinject, axiom,
    ((![Nat : nat, Nat2 : nat]: (((suc @ Nat) = (suc @ Nat2)) = (Nat = Nat2))))). % old.nat.inject
thf(fact_138_nat_Oinject, axiom,
    ((![X2 : nat, Y2 : nat]: (((suc @ X2) = (suc @ Y2)) = (X2 = Y2))))). % nat.inject
thf(fact_139_the__elem__eq, axiom,
    ((![X : hoare_958474565_state]: ((the_el1300254266_state @ (insert776267541_state @ X @ bot_bo105666705_state)) = X)))). % the_elem_eq
thf(fact_140_bot__set__def, axiom,
    ((bot_bo105666705_state = (collec305460656_state @ bot_bo1428770700tate_o)))). % bot_set_def

% Conjectures (2)
thf(conj_0, hypothesis,
    ((hoare_604442164_state @ (sup_su737426425_state @ g @ (image_2144627828_state @ (^[Pn : pname]: (hoare_1659279548_state @ (^[Y5 : state]: (^[Z2 : state]: (Y5 = Z2))) @ (body2 @ Pn) @ (evalc @ (body2 @ Pn)))) @ procs)) @ (image_2144627828_state @ (^[Pn : pname]: (hoare_1659279548_state @ (^[Y5 : state]: (^[Z2 : state]: (Y5 = Z2))) @ (the_com @ (body @ Pn)) @ (evalc @ (the_com @ (body @ Pn))))) @ procs)))).
thf(conj_1, conjecture,
    ((![Z3 : state, S3 : state]: ((~ ((evalc @ (the_com @ (body @ pn)) @ Z3 @ S3))) | (evalc @ (body2 @ pn) @ Z3 @ S3))))).
