% 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_453__3254842_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:15:02.712

% Could-be-implicit typings (14)
thf(ty_n_t__Set__Oset_It__Option__Ooption_It__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J_J_J, type,
    set_op1790085995_state : $tType).
thf(ty_n_t__Option__Ooption_It__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J_J, type,
    option1481361675_state : $tType).
thf(ty_n_t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J_J, type,
    set_Ho840737317_state : $tType).
thf(ty_n_t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J, type,
    hoare_958474565_state : $tType).
thf(ty_n_t__Set__Oset_It__Option__Ooption_It__Com__Ocom_J_J, type,
    set_option_com : $tType).
thf(ty_n_t__Option__Ooption_It__Com__Ocom_J, type,
    option_com : $tType).
thf(ty_n_t__Set__Oset_It__Com__Opname_J, type,
    set_pname : $tType).
thf(ty_n_t__Set__Oset_It__Com__Ocom_J, type,
    set_com : $tType).
thf(ty_n_t__Com__Ovname, type,
    vname : $tType).
thf(ty_n_t__Com__Ostate, type,
    state : $tType).
thf(ty_n_t__Com__Opname, type,
    pname : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).
thf(ty_n_t__Com__Oloc, type,
    loc : $tType).
thf(ty_n_t__Com__Ocom, type,
    com : $tType).

% Explicit typings (45)
thf(sy_c_Com_OWT, type,
    wt : com > $o).
thf(sy_c_Com_OWT__bodies, type,
    wT_bodies : $o).
thf(sy_c_Com_Obody, type,
    body : pname > option_com).
thf(sy_c_Com_Ocom_OAss, type,
    ass : vname > (state > nat) > com).
thf(sy_c_Com_Ocom_OBODY, type,
    body2 : pname > com).
thf(sy_c_Com_Ocom_OCall, type,
    call : vname > pname > (state > nat) > com).
thf(sy_c_Com_Ocom_OCond, type,
    cond : (state > $o) > com > com > com).
thf(sy_c_Com_Ocom_OLocal, type,
    local : loc > (state > nat) > com > com).
thf(sy_c_Com_Ocom_OSKIP, type,
    skip : com).
thf(sy_c_Com_Ocom_OSemi, type,
    semi : com > com > com).
thf(sy_c_Com_Ocom_OWhile, type,
    while : (state > $o) > com > 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_Ostate__not__singleton, type,
    hoare_405891322gleton : $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_Map_Odom_001t__Com__Opname_001t__Com__Ocom, type,
    dom_pname_com : (pname > option_com) > set_pname).
thf(sy_c_Map_Odom_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J_001t__Com__Ocom, type,
    dom_Ho256784515te_com : (hoare_958474565_state > option_com) > set_Ho840737317_state).
thf(sy_c_Option_Ooption_ONone_001t__Com__Ocom, type,
    none_com : option_com).
thf(sy_c_Option_Ooption_ONone_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J, type,
    none_H2001963430_state : option1481361675_state).
thf(sy_c_Option_Ooption_OSome_001t__Com__Ocom, type,
    some_com : com > option_com).
thf(sy_c_Option_Ooption_OSome_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J, type,
    some_H1327573290_state : hoare_958474565_state > option1481361675_state).
thf(sy_c_Option_Ooption_Othe_001t__Com__Ocom, type,
    the_com : option_com > com).
thf(sy_c_Option_Othese_001t__Com__Ocom, type,
    these_com : set_option_com > set_com).
thf(sy_c_Option_Othese_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J, type,
    these_1282564994_state : set_op1790085995_state > set_Ho840737317_state).
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__Ocom_J, type,
    bot_bot_set_com : set_com).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Com__Opname_J, type,
    bot_bot_set_pname : set_pname).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J_J, type,
    bot_bo105666705_state : set_Ho840737317_state).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Option__Ooption_It__Com__Ocom_J_J, type,
    bot_bo697186986on_com : set_option_com).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Option__Ooption_It__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J_J_J, type,
    bot_bo472264663_state : set_op1790085995_state).
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_Oinsert_001t__Com__Ocom, type,
    insert_com : com > set_com > set_com).
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_Oinsert_001t__Option__Ooption_It__Com__Ocom_J, type,
    insert_option_com : option_com > set_option_com > set_option_com).
thf(sy_c_Set_Oinsert_001t__Option__Ooption_It__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J_J, type,
    insert677809883_state : option1481361675_state > set_op1790085995_state > set_op1790085995_state).
thf(sy_c_Set_Ois__empty_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J, type,
    is_emp266830871_state : set_Ho840737317_state > $o).
thf(sy_c_Set_Ois__singleton_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J, type,
    is_sin536631353_state : set_Ho840737317_state > $o).
thf(sy_c_Set_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__Ocom, type,
    member_com : com > set_com > $o).
thf(sy_c_member_001t__Com__Opname, type,
    member_pname : pname > set_pname > $o).
thf(sy_c_member_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J, type,
    member109514606_state : hoare_958474565_state > set_Ho840737317_state > $o).
thf(sy_c_member_001t__Option__Ooption_It__Com__Ocom_J, type,
    member_option_com : option_com > set_option_com > $o).
thf(sy_v_c, type,
    c : com).

% Relevant facts (143)
thf(fact_0_single__stateE, axiom,
    ((hoare_405891322gleton => (![T : state]: (~ ((![S : state]: (S = T)))))))). % single_stateE
thf(fact_1_state__not__singleton__def, axiom,
    ((hoare_405891322gleton = (?[S2 : state]: (?[T2 : state]: (~ ((S2 = T2)))))))). % state_not_singleton_def
thf(fact_2_WT__bodiesD, axiom,
    ((![Pn : pname, B : com]: (wT_bodies => (((body @ Pn) = (some_com @ B)) => (wt @ B)))))). % WT_bodiesD
thf(fact_3_MGF, axiom,
    ((![C : com]: (hoare_405891322gleton => (wT_bodies => ((wt @ C) => (hoare_604442164_state @ bot_bo105666705_state @ (insert776267541_state @ (hoare_Mirabelle_MGT @ C) @ bot_bo105666705_state)))))))). % MGF
thf(fact_4_WT_OSkip, axiom,
    ((wt @ skip))). % WT.Skip
thf(fact_5_WTs__elim__cases_I4_J, axiom,
    ((![C1 : com, C2 : com]: ((wt @ (semi @ C1 @ C2)) => (~ (((wt @ C1) => (~ ((wt @ C2)))))))))). % WTs_elim_cases(4)
thf(fact_6_WT_OSemi, axiom,
    ((![C0 : com, C1 : com]: ((wt @ C0) => ((wt @ C1) => (wt @ (semi @ C0 @ C1))))))). % WT.Semi
thf(fact_7_WTs__elim__cases_I3_J, axiom,
    ((![Y : loc, A : state > nat, C : com]: ((wt @ (local @ Y @ A @ C)) => (wt @ C))))). % WTs_elim_cases(3)
thf(fact_8_WT_OLocal, axiom,
    ((![C : com, Y : loc, A : state > nat]: ((wt @ C) => (wt @ (local @ Y @ A @ C)))))). % WT.Local
thf(fact_9_WT_OAssign, axiom,
    ((![X : vname, A : state > nat]: (wt @ (ass @ X @ A))))). % WT.Assign
thf(fact_10_WTs__elim__cases_I5_J, axiom,
    ((![B : state > $o, C1 : com, C2 : com]: ((wt @ (cond @ B @ C1 @ C2)) => (~ (((wt @ C1) => (~ ((wt @ C2)))))))))). % WTs_elim_cases(5)
thf(fact_11_WT_OIf, axiom,
    ((![C0 : com, C1 : com, B : state > $o]: ((wt @ C0) => ((wt @ C1) => (wt @ (cond @ B @ C0 @ C1))))))). % WT.If
thf(fact_12_com_Oinject_I4_J, axiom,
    ((![X51 : state > $o, X52 : com, X53 : com, Y51 : state > $o, Y52 : com, Y53 : com]: (((cond @ X51 @ X52 @ X53) = (cond @ Y51 @ Y52 @ Y53)) = (((X51 = Y51)) & ((((X52 = Y52)) & ((X53 = Y53))))))))). % com.inject(4)
thf(fact_13_com_Oinject_I3_J, axiom,
    ((![X41 : com, X42 : com, Y41 : com, Y42 : com]: (((semi @ X41 @ X42) = (semi @ Y41 @ Y42)) = (((X41 = Y41)) & ((X42 = Y42))))))). % com.inject(3)
thf(fact_14_com_Oinject_I2_J, axiom,
    ((![X31 : loc, X32 : state > nat, X33 : com, Y31 : loc, Y32 : state > nat, Y33 : com]: (((local @ X31 @ X32 @ X33) = (local @ Y31 @ Y32 @ Y33)) = (((X31 = Y31)) & ((((X32 = Y32)) & ((X33 = Y33))))))))). % com.inject(2)
thf(fact_15_com_Oinject_I1_J, axiom,
    ((![X21 : vname, X22 : state > nat, Y21 : vname, Y22 : state > nat]: (((ass @ X21 @ X22) = (ass @ Y21 @ Y22)) = (((X21 = Y21)) & ((X22 = Y22))))))). % com.inject(1)
thf(fact_16_com_Odistinct_I37_J, axiom,
    ((![X41 : com, X42 : com, X51 : state > $o, X52 : com, X53 : com]: (~ (((semi @ X41 @ X42) = (cond @ X51 @ X52 @ X53))))))). % com.distinct(37)
thf(fact_17_com_Odistinct_I29_J, axiom,
    ((![X31 : loc, X32 : state > nat, X33 : com, X51 : state > $o, X52 : com, X53 : com]: (~ (((local @ X31 @ X32 @ X33) = (cond @ X51 @ X52 @ X53))))))). % com.distinct(29)
thf(fact_18_com_Odistinct_I27_J, axiom,
    ((![X31 : loc, X32 : state > nat, X33 : com, X41 : com, X42 : com]: (~ (((local @ X31 @ X32 @ X33) = (semi @ X41 @ X42))))))). % com.distinct(27)
thf(fact_19_com_Odistinct_I19_J, axiom,
    ((![X21 : vname, X22 : state > nat, X51 : state > $o, X52 : com, X53 : com]: (~ (((ass @ X21 @ X22) = (cond @ X51 @ X52 @ X53))))))). % com.distinct(19)
thf(fact_20_com_Odistinct_I17_J, axiom,
    ((![X21 : vname, X22 : state > nat, X41 : com, X42 : com]: (~ (((ass @ X21 @ X22) = (semi @ X41 @ X42))))))). % com.distinct(17)
thf(fact_21_com_Odistinct_I15_J, axiom,
    ((![X21 : vname, X22 : state > nat, X31 : loc, X32 : state > nat, X33 : com]: (~ (((ass @ X21 @ X22) = (local @ X31 @ X32 @ X33))))))). % com.distinct(15)
thf(fact_22_com_Odistinct_I7_J, axiom,
    ((![X51 : state > $o, X52 : com, X53 : com]: (~ ((skip = (cond @ X51 @ X52 @ X53))))))). % com.distinct(7)
thf(fact_23_com_Odistinct_I5_J, axiom,
    ((![X41 : com, X42 : com]: (~ ((skip = (semi @ X41 @ X42))))))). % com.distinct(5)
thf(fact_24_com_Odistinct_I3_J, axiom,
    ((![X31 : loc, X32 : state > nat, X33 : com]: (~ ((skip = (local @ X31 @ X32 @ X33))))))). % com.distinct(3)
thf(fact_25_com_Odistinct_I1_J, axiom,
    ((![X21 : vname, X22 : state > nat]: (~ ((skip = (ass @ X21 @ X22))))))). % com.distinct(1)
thf(fact_26_derivs__insertD, axiom,
    ((![G : set_Ho840737317_state, T3 : hoare_958474565_state, Ts : set_Ho840737317_state]: ((hoare_604442164_state @ G @ (insert776267541_state @ T3 @ Ts)) => ((hoare_604442164_state @ G @ (insert776267541_state @ T3 @ bot_bo105666705_state)) & (hoare_604442164_state @ G @ Ts)))))). % derivs_insertD
thf(fact_27_cut, axiom,
    ((![G2 : set_Ho840737317_state, Ts : set_Ho840737317_state, G : set_Ho840737317_state]: ((hoare_604442164_state @ G2 @ Ts) => ((hoare_604442164_state @ G @ G2) => (hoare_604442164_state @ G @ Ts)))))). % cut
thf(fact_28_empty, axiom,
    ((![G : set_Ho840737317_state]: (hoare_604442164_state @ G @ bot_bo105666705_state)))). % empty
thf(fact_29_hoare__derivs_Oinsert, axiom,
    ((![G : set_Ho840737317_state, T3 : hoare_958474565_state, Ts : set_Ho840737317_state]: ((hoare_604442164_state @ G @ (insert776267541_state @ T3 @ bot_bo105666705_state)) => ((hoare_604442164_state @ G @ Ts) => (hoare_604442164_state @ G @ (insert776267541_state @ T3 @ Ts))))))). % hoare_derivs.insert
thf(fact_30_singletonI, axiom,
    ((![A : hoare_958474565_state]: (member109514606_state @ A @ (insert776267541_state @ A @ bot_bo105666705_state))))). % singletonI
thf(fact_31_option_Oinject, axiom,
    ((![X2 : com, Y2 : com]: (((some_com @ X2) = (some_com @ Y2)) = (X2 = Y2))))). % option.inject
thf(fact_32_insertCI, axiom,
    ((![A : hoare_958474565_state, B2 : set_Ho840737317_state, B : hoare_958474565_state]: (((~ ((member109514606_state @ A @ B2))) => (A = B)) => (member109514606_state @ A @ (insert776267541_state @ B @ B2)))))). % insertCI
thf(fact_33_insert__iff, axiom,
    ((![A : hoare_958474565_state, B : hoare_958474565_state, A2 : set_Ho840737317_state]: ((member109514606_state @ A @ (insert776267541_state @ B @ A2)) = (((A = B)) | ((member109514606_state @ A @ A2))))))). % insert_iff
thf(fact_34_insert__absorb2, axiom,
    ((![X3 : hoare_958474565_state, A2 : set_Ho840737317_state]: ((insert776267541_state @ X3 @ (insert776267541_state @ X3 @ A2)) = (insert776267541_state @ X3 @ A2))))). % insert_absorb2
thf(fact_35_empty__iff, axiom,
    ((![C : hoare_958474565_state]: (~ ((member109514606_state @ C @ bot_bo105666705_state)))))). % empty_iff
thf(fact_36_all__not__in__conv, axiom,
    ((![A2 : set_Ho840737317_state]: ((![X4 : hoare_958474565_state]: (~ ((member109514606_state @ X4 @ A2)))) = (A2 = bot_bo105666705_state))))). % all_not_in_conv
thf(fact_37_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_38_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_39_bot__set__def, axiom,
    ((bot_bo105666705_state = (collec305460656_state @ bot_bo1428770700tate_o)))). % bot_set_def
thf(fact_40_ex__in__conv, axiom,
    ((![A2 : set_Ho840737317_state]: ((?[X4 : hoare_958474565_state]: (member109514606_state @ X4 @ A2)) = (~ ((A2 = bot_bo105666705_state))))))). % ex_in_conv
thf(fact_41_equals0I, axiom,
    ((![A2 : set_Ho840737317_state]: ((![Y3 : hoare_958474565_state]: (~ ((member109514606_state @ Y3 @ A2)))) => (A2 = bot_bo105666705_state))))). % equals0I
thf(fact_42_equals0D, axiom,
    ((![A2 : set_Ho840737317_state, A : hoare_958474565_state]: ((A2 = bot_bo105666705_state) => (~ ((member109514606_state @ A @ A2))))))). % equals0D
thf(fact_43_emptyE, axiom,
    ((![A : hoare_958474565_state]: (~ ((member109514606_state @ A @ bot_bo105666705_state)))))). % emptyE
thf(fact_44_mk__disjoint__insert, axiom,
    ((![A : hoare_958474565_state, A2 : set_Ho840737317_state]: ((member109514606_state @ A @ A2) => (?[B3 : set_Ho840737317_state]: ((A2 = (insert776267541_state @ A @ B3)) & (~ ((member109514606_state @ A @ B3))))))))). % mk_disjoint_insert
thf(fact_45_insert__commute, axiom,
    ((![X3 : hoare_958474565_state, Y4 : hoare_958474565_state, A2 : set_Ho840737317_state]: ((insert776267541_state @ X3 @ (insert776267541_state @ Y4 @ A2)) = (insert776267541_state @ Y4 @ (insert776267541_state @ X3 @ A2)))))). % insert_commute
thf(fact_46_insert__eq__iff, axiom,
    ((![A : hoare_958474565_state, A2 : set_Ho840737317_state, B : hoare_958474565_state, B2 : set_Ho840737317_state]: ((~ ((member109514606_state @ A @ A2))) => ((~ ((member109514606_state @ B @ B2))) => (((insert776267541_state @ A @ A2) = (insert776267541_state @ B @ B2)) = (((((A = B)) => ((A2 = B2)))) & ((((~ ((A = B)))) => ((?[C3 : set_Ho840737317_state]: (((A2 = (insert776267541_state @ B @ C3))) & ((((~ ((member109514606_state @ B @ C3)))) & ((((B2 = (insert776267541_state @ A @ C3))) & ((~ ((member109514606_state @ A @ C3)))))))))))))))))))). % insert_eq_iff
thf(fact_47_insert__absorb, axiom,
    ((![A : hoare_958474565_state, A2 : set_Ho840737317_state]: ((member109514606_state @ A @ A2) => ((insert776267541_state @ A @ A2) = A2))))). % insert_absorb
thf(fact_48_insert__ident, axiom,
    ((![X3 : hoare_958474565_state, A2 : set_Ho840737317_state, B2 : set_Ho840737317_state]: ((~ ((member109514606_state @ X3 @ A2))) => ((~ ((member109514606_state @ X3 @ B2))) => (((insert776267541_state @ X3 @ A2) = (insert776267541_state @ X3 @ B2)) = (A2 = B2))))))). % insert_ident
thf(fact_49_Set_Oset__insert, axiom,
    ((![X3 : hoare_958474565_state, A2 : set_Ho840737317_state]: ((member109514606_state @ X3 @ A2) => (~ ((![B3 : set_Ho840737317_state]: ((A2 = (insert776267541_state @ X3 @ B3)) => (member109514606_state @ X3 @ B3))))))))). % Set.set_insert
thf(fact_50_insertI2, axiom,
    ((![A : hoare_958474565_state, B2 : set_Ho840737317_state, B : hoare_958474565_state]: ((member109514606_state @ A @ B2) => (member109514606_state @ A @ (insert776267541_state @ B @ B2)))))). % insertI2
thf(fact_51_insertI1, axiom,
    ((![A : hoare_958474565_state, B2 : set_Ho840737317_state]: (member109514606_state @ A @ (insert776267541_state @ A @ B2))))). % insertI1
thf(fact_52_insertE, axiom,
    ((![A : hoare_958474565_state, B : hoare_958474565_state, A2 : set_Ho840737317_state]: ((member109514606_state @ A @ (insert776267541_state @ B @ A2)) => ((~ ((A = B))) => (member109514606_state @ A @ A2)))))). % insertE
thf(fact_53_singleton__inject, axiom,
    ((![A : hoare_958474565_state, B : hoare_958474565_state]: (((insert776267541_state @ A @ bot_bo105666705_state) = (insert776267541_state @ B @ bot_bo105666705_state)) => (A = B))))). % singleton_inject
thf(fact_54_insert__not__empty, axiom,
    ((![A : hoare_958474565_state, A2 : set_Ho840737317_state]: (~ (((insert776267541_state @ A @ A2) = bot_bo105666705_state)))))). % insert_not_empty
thf(fact_55_doubleton__eq__iff, axiom,
    ((![A : hoare_958474565_state, B : hoare_958474565_state, C : hoare_958474565_state, D : hoare_958474565_state]: (((insert776267541_state @ A @ (insert776267541_state @ B @ bot_bo105666705_state)) = (insert776267541_state @ C @ (insert776267541_state @ D @ bot_bo105666705_state))) = (((((A = C)) & ((B = D)))) | ((((A = D)) & ((B = C))))))))). % doubleton_eq_iff
thf(fact_56_singleton__iff, axiom,
    ((![B : hoare_958474565_state, A : hoare_958474565_state]: ((member109514606_state @ B @ (insert776267541_state @ A @ bot_bo105666705_state)) = (B = A))))). % singleton_iff
thf(fact_57_singletonD, axiom,
    ((![B : hoare_958474565_state, A : hoare_958474565_state]: ((member109514606_state @ B @ (insert776267541_state @ A @ bot_bo105666705_state)) => (B = A))))). % singletonD
thf(fact_58_the__elem__eq, axiom,
    ((![X3 : hoare_958474565_state]: ((the_el1300254266_state @ (insert776267541_state @ X3 @ bot_bo105666705_state)) = X3)))). % the_elem_eq
thf(fact_59_is__singletonI, axiom,
    ((![X3 : hoare_958474565_state]: (is_sin536631353_state @ (insert776267541_state @ X3 @ bot_bo105666705_state))))). % is_singletonI
thf(fact_60_these__insert__Some, axiom,
    ((![X3 : hoare_958474565_state, A2 : set_op1790085995_state]: ((these_1282564994_state @ (insert677809883_state @ (some_H1327573290_state @ X3) @ A2)) = (insert776267541_state @ X3 @ (these_1282564994_state @ A2)))))). % these_insert_Some
thf(fact_61_these__insert__Some, axiom,
    ((![X3 : com, A2 : set_option_com]: ((these_com @ (insert_option_com @ (some_com @ X3) @ A2)) = (insert_com @ X3 @ (these_com @ A2)))))). % these_insert_Some
thf(fact_62_Set_Ois__empty__def, axiom,
    ((is_emp266830871_state = (^[A3 : set_Ho840737317_state]: (A3 = bot_bo105666705_state))))). % Set.is_empty_def
thf(fact_63_WTs__elim__cases_I7_J, axiom,
    ((![P : pname]: ((wt @ (body2 @ P)) => (~ ((![Y3 : com]: (~ (((body @ P) = (some_com @ Y3))))))))))). % WTs_elim_cases(7)
thf(fact_64_is__singleton__def, axiom,
    ((is_sin536631353_state = (^[A3 : set_Ho840737317_state]: (?[X4 : hoare_958474565_state]: (A3 = (insert776267541_state @ X4 @ bot_bo105666705_state))))))). % is_singleton_def
thf(fact_65_com_Oinject_I6_J, axiom,
    ((![X7 : pname, Y7 : pname]: (((body2 @ X7) = (body2 @ Y7)) = (X7 = Y7))))). % com.inject(6)
thf(fact_66_these__empty, axiom,
    (((these_1282564994_state @ bot_bo472264663_state) = bot_bo105666705_state))). % these_empty
thf(fact_67_com_Odistinct_I47_J, axiom,
    ((![X51 : state > $o, X52 : com, X53 : com, X7 : pname]: (~ (((cond @ X51 @ X52 @ X53) = (body2 @ X7))))))). % com.distinct(47)
thf(fact_68_com_Odistinct_I41_J, axiom,
    ((![X41 : com, X42 : com, X7 : pname]: (~ (((semi @ X41 @ X42) = (body2 @ X7))))))). % com.distinct(41)
thf(fact_69_com_Odistinct_I33_J, axiom,
    ((![X31 : loc, X32 : state > nat, X33 : com, X7 : pname]: (~ (((local @ X31 @ X32 @ X33) = (body2 @ X7))))))). % com.distinct(33)
thf(fact_70_com_Odistinct_I23_J, axiom,
    ((![X21 : vname, X22 : state > nat, X7 : pname]: (~ (((ass @ X21 @ X22) = (body2 @ X7))))))). % com.distinct(23)
thf(fact_71_com_Odistinct_I11_J, axiom,
    ((![X7 : pname]: (~ ((skip = (body2 @ X7))))))). % com.distinct(11)
thf(fact_72_in__these__eq, axiom,
    ((![X3 : com, A2 : set_option_com]: ((member_com @ X3 @ (these_com @ A2)) = (member_option_com @ (some_com @ X3) @ A2))))). % in_these_eq
thf(fact_73_is__singleton__the__elem, axiom,
    ((is_sin536631353_state = (^[A3 : set_Ho840737317_state]: (A3 = (insert776267541_state @ (the_el1300254266_state @ A3) @ bot_bo105666705_state)))))). % is_singleton_the_elem
thf(fact_74_is__singletonI_H, axiom,
    ((![A2 : set_Ho840737317_state]: ((~ ((A2 = bot_bo105666705_state))) => ((![X5 : hoare_958474565_state, Y3 : hoare_958474565_state]: ((member109514606_state @ X5 @ A2) => ((member109514606_state @ Y3 @ A2) => (X5 = Y3)))) => (is_sin536631353_state @ A2)))))). % is_singletonI'
thf(fact_75_is__singletonE, axiom,
    ((![A2 : set_Ho840737317_state]: ((is_sin536631353_state @ A2) => (~ ((![X5 : hoare_958474565_state]: (~ ((A2 = (insert776267541_state @ X5 @ bot_bo105666705_state))))))))))). % is_singletonE
thf(fact_76_MGF__lemma1, axiom,
    ((![G : set_Ho840737317_state, C : com]: (hoare_405891322gleton => ((![X5 : pname]: ((member_pname @ X5 @ (dom_pname_com @ body)) => (hoare_604442164_state @ G @ (insert776267541_state @ (hoare_Mirabelle_MGT @ (body2 @ X5)) @ bot_bo105666705_state)))) => ((wt @ C) => (hoare_604442164_state @ G @ (insert776267541_state @ (hoare_Mirabelle_MGT @ C) @ bot_bo105666705_state)))))))). % MGF_lemma1
thf(fact_77_Collect__empty__eq__bot, axiom,
    ((![P : hoare_958474565_state > $o]: (((collec305460656_state @ P) = bot_bo105666705_state) = (P = bot_bo1428770700tate_o))))). % Collect_empty_eq_bot
thf(fact_78_bot__empty__eq, axiom,
    ((bot_bo1428770700tate_o = (^[X4 : hoare_958474565_state]: (member109514606_state @ X4 @ bot_bo105666705_state))))). % bot_empty_eq
thf(fact_79_MGT__BodyN, axiom,
    ((![Pn : pname, G : set_Ho840737317_state]: ((hoare_604442164_state @ (insert776267541_state @ (hoare_Mirabelle_MGT @ (body2 @ Pn)) @ G) @ (insert776267541_state @ (hoare_Mirabelle_MGT @ (the_com @ (body @ Pn))) @ bot_bo105666705_state)) => (hoare_604442164_state @ G @ (insert776267541_state @ (hoare_Mirabelle_MGT @ (body2 @ Pn)) @ bot_bo105666705_state)))))). % MGT_BodyN
thf(fact_80_option_Osel, axiom,
    ((![X2 : com]: ((the_com @ (some_com @ X2)) = X2)))). % option.sel
thf(fact_81_insert__dom, axiom,
    ((![F : pname > option_com, X3 : pname, Y4 : com]: (((F @ X3) = (some_com @ Y4)) => ((insert_pname @ X3 @ (dom_pname_com @ F)) = (dom_pname_com @ F)))))). % insert_dom
thf(fact_82_insert__dom, axiom,
    ((![F : hoare_958474565_state > option_com, X3 : hoare_958474565_state, Y4 : com]: (((F @ X3) = (some_com @ Y4)) => ((insert776267541_state @ X3 @ (dom_Ho256784515te_com @ F)) = (dom_Ho256784515te_com @ F)))))). % insert_dom
thf(fact_83_domD, axiom,
    ((![A : pname, M : pname > option_com]: ((member_pname @ A @ (dom_pname_com @ M)) => (?[B4 : com]: ((M @ A) = (some_com @ B4))))))). % domD
thf(fact_84_domI, axiom,
    ((![M : pname > option_com, A : pname, B : com]: (((M @ A) = (some_com @ B)) => (member_pname @ A @ (dom_pname_com @ M)))))). % domI
thf(fact_85_BodyN, axiom,
    ((![P : state > state > $o, Pn : pname, Q : state > state > $o, G : set_Ho840737317_state]: ((hoare_604442164_state @ (insert776267541_state @ (hoare_1659279548_state @ P @ (body2 @ Pn) @ Q) @ G) @ (insert776267541_state @ (hoare_1659279548_state @ P @ (the_com @ (body @ Pn)) @ Q) @ bot_bo105666705_state)) => (hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ P @ (body2 @ Pn) @ Q) @ bot_bo105666705_state)))))). % BodyN
thf(fact_86_weak__Body, axiom,
    ((![G : set_Ho840737317_state, P : state > state > $o, Pn : pname, Q : state > state > $o]: ((hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ P @ (the_com @ (body @ Pn)) @ Q) @ bot_bo105666705_state)) => (hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ P @ (body2 @ Pn) @ Q) @ bot_bo105666705_state)))))). % weak_Body
thf(fact_87_these__empty__eq, axiom,
    ((![B2 : set_option_com]: (((these_com @ B2) = bot_bot_set_com) = (((B2 = bot_bo697186986on_com)) | ((B2 = (insert_option_com @ none_com @ bot_bo697186986on_com)))))))). % these_empty_eq
thf(fact_88_these__empty__eq, axiom,
    ((![B2 : set_op1790085995_state]: (((these_1282564994_state @ B2) = bot_bo105666705_state) = (((B2 = bot_bo472264663_state)) | ((B2 = (insert677809883_state @ none_H2001963430_state @ bot_bo472264663_state)))))))). % these_empty_eq
thf(fact_89_triple_Oinject, axiom,
    ((![X1 : state > state > $o, X2 : com, X34 : state > state > $o, Y1 : state > state > $o, Y2 : com, Y34 : state > state > $o]: (((hoare_1659279548_state @ X1 @ X2 @ X34) = (hoare_1659279548_state @ Y1 @ Y2 @ Y34)) = (((X1 = Y1)) & ((((X2 = Y2)) & ((X34 = Y34))))))))). % triple.inject
thf(fact_90_not__None__eq, axiom,
    ((![X3 : option_com]: ((~ ((X3 = none_com))) = (?[Y5 : com]: (X3 = (some_com @ Y5))))))). % not_None_eq
thf(fact_91_not__Some__eq, axiom,
    ((![X3 : option_com]: ((![Y5 : com]: (~ ((X3 = (some_com @ Y5))))) = (X3 = none_com))))). % not_Some_eq
thf(fact_92_dom__eq__empty__conv, axiom,
    ((![F : pname > option_com]: (((dom_pname_com @ F) = bot_bot_set_pname) = (F = (^[X4 : pname]: none_com)))))). % dom_eq_empty_conv
thf(fact_93_dom__eq__empty__conv, axiom,
    ((![F : hoare_958474565_state > option_com]: (((dom_Ho256784515te_com @ F) = bot_bo105666705_state) = (F = (^[X4 : hoare_958474565_state]: none_com)))))). % dom_eq_empty_conv
thf(fact_94_option_Ocollapse, axiom,
    ((![Option : option_com]: ((~ ((Option = none_com))) => ((some_com @ (the_com @ Option)) = Option))))). % option.collapse
thf(fact_95_these__insert__None, axiom,
    ((![A2 : set_option_com]: ((these_com @ (insert_option_com @ none_com @ A2)) = (these_com @ A2))))). % these_insert_None
thf(fact_96_domIff, axiom,
    ((![A : pname, M : pname > option_com]: ((member_pname @ A @ (dom_pname_com @ M)) = (~ (((M @ A) = none_com))))))). % domIff
thf(fact_97_option_Oexpand, axiom,
    ((![Option : option_com, Option2 : option_com]: (((Option = none_com) = (Option2 = none_com)) => (((~ ((Option = none_com))) => ((~ ((Option2 = none_com))) => ((the_com @ Option) = (the_com @ Option2)))) => (Option = Option2)))))). % option.expand
thf(fact_98_option_Odistinct_I1_J, axiom,
    ((![X2 : com]: (~ ((none_com = (some_com @ X2))))))). % option.distinct(1)
thf(fact_99_option_OdiscI, axiom,
    ((![Option : option_com, X2 : com]: ((Option = (some_com @ X2)) => (~ ((Option = none_com))))))). % option.discI
thf(fact_100_option_Oexhaust, axiom,
    ((![Y4 : option_com]: ((~ ((Y4 = none_com))) => (~ ((![X23 : com]: (~ ((Y4 = (some_com @ X23))))))))))). % option.exhaust
thf(fact_101_option_Oinducts, axiom,
    ((![P : option_com > $o, Option : option_com]: ((P @ none_com) => ((![X5 : com]: (P @ (some_com @ X5))) => (P @ Option)))))). % option.inducts
thf(fact_102_split__option__ex, axiom,
    (((^[P2 : option_com > $o]: (?[X6 : option_com]: (P2 @ X6))) = (^[P3 : option_com > $o]: (((P3 @ none_com)) | ((?[X4 : com]: (P3 @ (some_com @ X4))))))))). % split_option_ex
thf(fact_103_split__option__all, axiom,
    (((^[P2 : option_com > $o]: (![X6 : option_com]: (P2 @ X6))) = (^[P3 : option_com > $o]: (((P3 @ none_com)) & ((![X4 : com]: (P3 @ (some_com @ X4))))))))). % split_option_all
thf(fact_104_combine__options__cases, axiom,
    ((![X3 : option_com, P : option_com > option_com > $o, Y4 : option_com]: (((X3 = none_com) => (P @ X3 @ Y4)) => (((Y4 = none_com) => (P @ X3 @ Y4)) => ((![A4 : com, B4 : com]: ((X3 = (some_com @ A4)) => ((Y4 = (some_com @ B4)) => (P @ X3 @ Y4)))) => (P @ X3 @ Y4))))))). % combine_options_cases
thf(fact_105_triple_Oexhaust, axiom,
    ((![Y4 : hoare_958474565_state]: (~ ((![X12 : state > state > $o, X23 : com, X35 : state > state > $o]: (~ ((Y4 = (hoare_1659279548_state @ X12 @ X23 @ X35)))))))))). % triple.exhaust
thf(fact_106_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_107_option_Oexhaust__sel, axiom,
    ((![Option : option_com]: ((~ ((Option = none_com))) => (Option = (some_com @ (the_com @ Option))))))). % option.exhaust_sel
thf(fact_108_conseq, axiom,
    ((![P : state > state > $o, G : set_Ho840737317_state, C : com, Q : state > state > $o]: ((![Z : state, S : state]: ((P @ Z @ S) => (?[P4 : state > state > $o, Q2 : state > state > $o]: ((hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ P4 @ C @ Q2) @ bot_bo105666705_state)) & (![S3 : state]: ((![Z2 : state]: ((P4 @ Z2 @ S) => (Q2 @ Z2 @ S3))) => (Q @ Z @ S3))))))) => (hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ P @ C @ Q) @ bot_bo105666705_state)))))). % conseq
thf(fact_109_conseq12, axiom,
    ((![G : set_Ho840737317_state, P5 : state > state > $o, C : com, Q3 : state > state > $o, P : state > state > $o, Q : state > state > $o]: ((hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ P5 @ C @ Q3) @ bot_bo105666705_state)) => ((![Z : state, S : state]: ((P @ Z @ S) => (![S3 : state]: ((![Z2 : state]: ((P5 @ Z2 @ S) => (Q3 @ Z2 @ S3))) => (Q @ Z @ S3))))) => (hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ P @ C @ Q) @ bot_bo105666705_state))))))). % conseq12
thf(fact_110_conseq2, axiom,
    ((![G : set_Ho840737317_state, P : state > state > $o, C : com, Q3 : state > state > $o, Q : state > state > $o]: ((hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ P @ C @ Q3) @ bot_bo105666705_state)) => ((![Z : state, S : state]: ((Q3 @ Z @ S) => (Q @ Z @ S))) => (hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ P @ C @ Q) @ bot_bo105666705_state))))))). % conseq2
thf(fact_111_conseq1, axiom,
    ((![G : set_Ho840737317_state, P5 : state > state > $o, C : com, Q : state > state > $o, P : state > state > $o]: ((hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ P5 @ C @ Q) @ bot_bo105666705_state)) => ((![Z : state, S : state]: ((P @ Z @ S) => (P5 @ Z @ S))) => (hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ P @ C @ Q) @ bot_bo105666705_state))))))). % conseq1
thf(fact_112_WT_OBody, axiom,
    ((![Pn : pname]: ((~ (((body @ Pn) = none_com))) => (wt @ (body2 @ Pn)))))). % WT.Body
thf(fact_113_Comp, axiom,
    ((![G : set_Ho840737317_state, P : state > state > $o, C : com, Q : state > state > $o, D : com, R : state > state > $o]: ((hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ P @ C @ Q) @ bot_bo105666705_state)) => ((hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ Q @ D @ R) @ bot_bo105666705_state)) => (hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ P @ (semi @ C @ D) @ R) @ bot_bo105666705_state))))))). % Comp
thf(fact_114_hoare__derivs_OSkip, axiom,
    ((![G : set_Ho840737317_state, P : state > state > $o]: (hoare_604442164_state @ G @ (insert776267541_state @ (hoare_1659279548_state @ P @ skip @ P) @ bot_bo105666705_state))))). % hoare_derivs.Skip
thf(fact_115_these__not__empty__eq, axiom,
    ((![B2 : set_option_com]: ((~ (((these_com @ B2) = bot_bot_set_com))) = (((~ ((B2 = bot_bo697186986on_com)))) & ((~ ((B2 = (insert_option_com @ none_com @ bot_bo697186986on_com)))))))))). % these_not_empty_eq
thf(fact_116_these__not__empty__eq, axiom,
    ((![B2 : set_op1790085995_state]: ((~ (((these_1282564994_state @ B2) = bot_bo105666705_state))) = (((~ ((B2 = bot_bo472264663_state)))) & ((~ ((B2 = (insert677809883_state @ none_H2001963430_state @ bot_bo472264663_state)))))))))). % these_not_empty_eq
thf(fact_117_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_118_WT_Oinducts, axiom,
    ((![X3 : com, P : com > $o]: ((wt @ X3) => ((P @ skip) => ((![X8 : vname, A4 : state > nat]: (P @ (ass @ X8 @ A4))) => ((![C4 : com, Y6 : loc, A4 : state > nat]: ((wt @ C4) => ((P @ C4) => (P @ (local @ Y6 @ A4 @ C4))))) => ((![C02 : com, C12 : com]: ((wt @ C02) => ((P @ C02) => ((wt @ C12) => ((P @ C12) => (P @ (semi @ C02 @ C12))))))) => ((![C02 : com, C12 : com, B4 : state > $o]: ((wt @ C02) => ((P @ C02) => ((wt @ C12) => ((P @ C12) => (P @ (cond @ B4 @ C02 @ C12))))))) => ((![C4 : com, B4 : state > $o]: ((wt @ C4) => ((P @ C4) => (P @ (while @ B4 @ C4))))) => ((![Pn2 : pname]: ((~ (((body @ Pn2) = none_com))) => (P @ (body2 @ Pn2)))) => ((![Pn2 : pname, X8 : vname, A4 : state > nat]: ((wt @ (body2 @ Pn2)) => ((P @ (body2 @ Pn2)) => (P @ (call @ X8 @ Pn2 @ A4))))) => (P @ X3))))))))))))). % WT.inducts
thf(fact_119_WT_Osimps, axiom,
    ((wt = (^[A5 : com]: (((A5 = skip)) | ((((?[X9 : vname]: (?[B5 : state > nat]: (A5 = (ass @ X9 @ B5))))) | ((((?[C5 : com]: (?[Y8 : loc]: (?[B5 : state > nat]: (((A5 = (local @ Y8 @ B5 @ C5))) & ((wt @ C5))))))) | ((((?[C03 : com]: (?[C13 : com]: (((A5 = (semi @ C03 @ C13))) & ((((wt @ C03)) & ((wt @ C13)))))))) | ((((?[C03 : com]: (?[C13 : com]: (?[B5 : state > $o]: (((A5 = (cond @ B5 @ C03 @ C13))) & ((((wt @ C03)) & ((wt @ C13))))))))) | ((((?[C5 : com]: (?[B5 : state > $o]: (((A5 = (while @ B5 @ C5))) & ((wt @ C5)))))) | ((((?[Pn3 : pname]: (((A5 = (body2 @ Pn3))) & ((~ (((body @ Pn3) = none_com))))))) | ((?[Pn3 : pname]: (?[X9 : vname]: (?[B5 : state > nat]: (((A5 = (call @ X9 @ Pn3 @ B5))) & ((wt @ (body2 @ Pn3))))))))))))))))))))))))). % WT.simps
thf(fact_120_com_Oinject_I5_J, axiom,
    ((![X61 : state > $o, X62 : com, Y61 : state > $o, Y62 : com]: (((while @ X61 @ X62) = (while @ Y61 @ Y62)) = (((X61 = Y61)) & ((X62 = Y62))))))). % com.inject(5)
thf(fact_121_com_Oinject_I7_J, axiom,
    ((![X81 : vname, X82 : pname, X83 : state > nat, Y81 : vname, Y82 : pname, Y83 : state > nat]: (((call @ X81 @ X82 @ X83) = (call @ Y81 @ Y82 @ Y83)) = (((X81 = Y81)) & ((((X82 = Y82)) & ((X83 = Y83))))))))). % com.inject(7)
thf(fact_122_com_Odistinct_I25_J, axiom,
    ((![X21 : vname, X22 : state > nat, X81 : vname, X82 : pname, X83 : state > nat]: (~ (((ass @ X21 @ X22) = (call @ X81 @ X82 @ X83))))))). % com.distinct(25)
thf(fact_123_com_Odistinct_I35_J, axiom,
    ((![X31 : loc, X32 : state > nat, X33 : com, X81 : vname, X82 : pname, X83 : state > nat]: (~ (((local @ X31 @ X32 @ X33) = (call @ X81 @ X82 @ X83))))))). % com.distinct(35)
thf(fact_124_com_Odistinct_I43_J, axiom,
    ((![X41 : com, X42 : com, X81 : vname, X82 : pname, X83 : state > nat]: (~ (((semi @ X41 @ X42) = (call @ X81 @ X82 @ X83))))))). % com.distinct(43)
thf(fact_125_com_Odistinct_I13_J, axiom,
    ((![X81 : vname, X82 : pname, X83 : state > nat]: (~ ((skip = (call @ X81 @ X82 @ X83))))))). % com.distinct(13)
thf(fact_126_WTs__elim__cases_I6_J, axiom,
    ((![B : state > $o, C : com]: ((wt @ (while @ B @ C)) => (wt @ C))))). % WTs_elim_cases(6)
thf(fact_127_While, axiom,
    ((![C : com, B : state > $o]: ((wt @ C) => (wt @ (while @ B @ C)))))). % While
thf(fact_128_com_Odistinct_I51_J, axiom,
    ((![X61 : state > $o, X62 : com, X7 : pname]: (~ (((while @ X61 @ X62) = (body2 @ X7))))))). % com.distinct(51)
thf(fact_129_com_Odistinct_I45_J, axiom,
    ((![X51 : state > $o, X52 : com, X53 : com, X61 : state > $o, X62 : com]: (~ (((cond @ X51 @ X52 @ X53) = (while @ X61 @ X62))))))). % com.distinct(45)
thf(fact_130_com_Odistinct_I21_J, axiom,
    ((![X21 : vname, X22 : state > nat, X61 : state > $o, X62 : com]: (~ (((ass @ X21 @ X22) = (while @ X61 @ X62))))))). % com.distinct(21)
thf(fact_131_com_Odistinct_I31_J, axiom,
    ((![X31 : loc, X32 : state > nat, X33 : com, X61 : state > $o, X62 : com]: (~ (((local @ X31 @ X32 @ X33) = (while @ X61 @ X62))))))). % com.distinct(31)
thf(fact_132_com_Odistinct_I39_J, axiom,
    ((![X41 : com, X42 : com, X61 : state > $o, X62 : com]: (~ (((semi @ X41 @ X42) = (while @ X61 @ X62))))))). % com.distinct(39)
thf(fact_133_com_Odistinct_I9_J, axiom,
    ((![X61 : state > $o, X62 : com]: (~ ((skip = (while @ X61 @ X62))))))). % com.distinct(9)
thf(fact_134_com_Odistinct_I55_J, axiom,
    ((![X7 : pname, X81 : vname, X82 : pname, X83 : state > nat]: (~ (((body2 @ X7) = (call @ X81 @ X82 @ X83))))))). % com.distinct(55)
thf(fact_135_com_Odistinct_I53_J, axiom,
    ((![X61 : state > $o, X62 : com, X81 : vname, X82 : pname, X83 : state > nat]: (~ (((while @ X61 @ X62) = (call @ X81 @ X82 @ X83))))))). % com.distinct(53)
thf(fact_136_com_Odistinct_I49_J, axiom,
    ((![X51 : state > $o, X52 : com, X53 : com, X81 : vname, X82 : pname, X83 : state > nat]: (~ (((cond @ X51 @ X52 @ X53) = (call @ X81 @ X82 @ X83))))))). % com.distinct(49)
thf(fact_137_hoare__sound, axiom,
    ((![G : set_Ho840737317_state, Ts : set_Ho840737317_state]: ((hoare_604442164_state @ G @ Ts) => (hoare_318887606_state @ G @ Ts))))). % hoare_sound
thf(fact_138_WT_OCall, axiom,
    ((![Pn : pname, X : vname, A : state > nat]: ((wt @ (body2 @ Pn)) => (wt @ (call @ X @ Pn @ A)))))). % WT.Call
thf(fact_139_WTs__elim__cases_I8_J, axiom,
    ((![X : vname, P : pname, A : state > nat]: ((wt @ (call @ X @ P @ A)) => (wt @ (body2 @ P)))))). % WTs_elim_cases(8)
thf(fact_140_com_Oinduct, axiom,
    ((![P : com > $o, Com : com]: ((P @ skip) => ((![X12 : vname, X23 : state > nat]: (P @ (ass @ X12 @ X23))) => ((![X12 : loc, X23 : state > nat, X35 : com]: ((P @ X35) => (P @ (local @ X12 @ X23 @ X35)))) => ((![X12 : com, X23 : com]: ((P @ X12) => ((P @ X23) => (P @ (semi @ X12 @ X23))))) => ((![X12 : state > $o, X23 : com, X35 : com]: ((P @ X23) => ((P @ X35) => (P @ (cond @ X12 @ X23 @ X35))))) => ((![X12 : state > $o, X23 : com]: ((P @ X23) => (P @ (while @ X12 @ X23)))) => ((![X5 : pname]: (P @ (body2 @ X5))) => ((![X12 : vname, X23 : pname, X35 : state > nat]: (P @ (call @ X12 @ X23 @ X35))) => (P @ Com)))))))))))). % com.induct
thf(fact_141_com_Oexhaust, axiom,
    ((![Y4 : com]: ((~ ((Y4 = skip))) => ((![X212 : vname, X222 : state > nat]: (~ ((Y4 = (ass @ X212 @ X222))))) => ((![X312 : loc, X322 : state > nat, X332 : com]: (~ ((Y4 = (local @ X312 @ X322 @ X332))))) => ((![X412 : com, X422 : com]: (~ ((Y4 = (semi @ X412 @ X422))))) => ((![X512 : state > $o, X522 : com, X532 : com]: (~ ((Y4 = (cond @ X512 @ X522 @ X532))))) => ((![X612 : state > $o, X622 : com]: (~ ((Y4 = (while @ X612 @ X622))))) => ((![X72 : pname]: (~ ((Y4 = (body2 @ X72))))) => (~ ((![X812 : vname, X822 : pname, X832 : state > nat]: (~ ((Y4 = (call @ X812 @ X822 @ X832))))))))))))))))). % com.exhaust
thf(fact_142_WT_Ocases, axiom,
    ((![A : com]: ((wt @ A) => ((~ ((A = skip))) => ((![X8 : vname, A4 : state > nat]: (~ ((A = (ass @ X8 @ A4))))) => ((![C4 : com]: ((?[Y6 : loc, A4 : state > nat]: (A = (local @ Y6 @ A4 @ C4))) => (~ ((wt @ C4))))) => ((![C02 : com, C12 : com]: ((A = (semi @ C02 @ C12)) => ((wt @ C02) => (~ ((wt @ C12)))))) => ((![C02 : com, C12 : com]: ((?[B4 : state > $o]: (A = (cond @ B4 @ C02 @ C12))) => ((wt @ C02) => (~ ((wt @ C12)))))) => ((![C4 : com]: ((?[B4 : state > $o]: (A = (while @ B4 @ C4))) => (~ ((wt @ C4))))) => ((![Pn2 : pname]: ((A = (body2 @ Pn2)) => ((body @ Pn2) = none_com))) => (~ ((![Pn2 : pname]: ((?[X8 : vname, A4 : state > nat]: (A = (call @ X8 @ Pn2 @ A4))) => (~ ((wt @ (body2 @ Pn2))))))))))))))))))). % WT.cases

% Conjectures (4)
thf(conj_0, hypothesis,
    (hoare_405891322gleton)).
thf(conj_1, hypothesis,
    (wT_bodies)).
thf(conj_2, hypothesis,
    ((wt @ c))).
thf(conj_3, conjecture,
    (hoare_405891322gleton)).
