% 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_144__3250778_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:12:20.587

% Could-be-implicit typings (9)
thf(ty_n_t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J, type,
    hoare_958474565_state : $tType).
thf(ty_n_t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    hoare_1678595023iple_a : $tType).
thf(ty_n_t__Option__Ooption_It__Com__Ocom_J, type,
    option_com : $tType).
thf(ty_n_t__Typerep__Otyperep, type,
    typerep : $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).
thf(ty_n_tf__a, type,
    a : $tType).

% Explicit typings (28)
thf(sy_c_Com_Obody, type,
    body : pname > option_com).
thf(sy_c_Com_Ocom_OBODY, type,
    body2 : pname > com).
thf(sy_c_Com_Ocom_Osize__com, type,
    size_com : com > nat).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat, type,
    plus_plus_nat : nat > nat > nat).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat, type,
    zero_zero_nat : nat).
thf(sy_c_Hoare__Mirabelle__raqjowkjvm_OMGT, type,
    hoare_Mirabelle_MGT : com > hoare_958474565_state).
thf(sy_c_Hoare__Mirabelle__raqjowkjvm_Otriple_Ocase__triple_001tf__a_001_Eo, type,
    hoare_1012081509le_a_o : ((a > state > $o) > com > (a > state > $o) > $o) > hoare_1678595023iple_a > $o).
thf(sy_c_Hoare__Mirabelle__raqjowkjvm_Otriple_Osize__triple_001t__Com__Ostate, type,
    hoare_103296669_state : (state > nat) > hoare_958474565_state > nat).
thf(sy_c_Hoare__Mirabelle__raqjowkjvm_Otriple_Osize__triple_001tf__a, type,
    hoare_201533281iple_a : (a > nat) > hoare_1678595023iple_a > nat).
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_Otriple_001tf__a, type,
    hoare_719046530iple_a : (a > state > $o) > com > (a > state > $o) > hoare_1678595023iple_a).
thf(sy_c_Hoare__Mirabelle__raqjowkjvm_Otriple__valid_001t__Com__Ostate, type,
    hoare_364318704_state : nat > hoare_958474565_state > $o).
thf(sy_c_Hoare__Mirabelle__raqjowkjvm_Otriple__valid_001tf__a, type,
    hoare_1926814542alid_a : nat > hoare_1678595023iple_a > $o).
thf(sy_c_Nat_OSuc, type,
    suc : nat > nat).
thf(sy_c_Nat_Osize__class_Osize_001t__Com__Ocom, type,
    size_size_com : com > nat).
thf(sy_c_Nat_Osize__class_Osize_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_It__Com__Ostate_J, type,
    size_s1457777073_state : hoare_958474565_state > nat).
thf(sy_c_Nat_Osize__class_Osize_001t__Hoare____Mirabelle____raqjowkjvm__Otriple_Itf__a_J, type,
    size_s648929379iple_a : hoare_1678595023iple_a > nat).
thf(sy_c_Nat_Osize__class_Osize_001t__Typerep__Otyperep, type,
    size_size_typerep : typerep > nat).
thf(sy_c_Natural_Oevalc, type,
    evalc : com > state > state > $o).
thf(sy_c_Natural_Oevaln, type,
    evaln : com > state > nat > state > $o).
thf(sy_c_Option_Ooption_ONone_001t__Com__Ocom, type,
    none_com : option_com).
thf(sy_c_Option_Ooption_OSome_001t__Com__Ocom, type,
    some_com : com > option_com).
thf(sy_c_Option_Ooption_Othe_001t__Com__Ocom, type,
    the_com : option_com > com).
thf(sy_c_Orderings_Oord__class_OLeast_001t__Nat__Onat, type,
    ord_Least_nat : (nat > $o) > nat).
thf(sy_v_n, type,
    n : nat).
thf(sy_v_x1a, type,
    x1a : a > state > $o).
thf(sy_v_x2a, type,
    x2a : com).
thf(sy_v_x3a, type,
    x3a : a > state > $o).

% Relevant facts (97)
thf(fact_0_triple_Oinject, axiom,
    ((![X1 : a > state > $o, X2 : com, X3 : a > state > $o, Y1 : a > state > $o, Y2 : com, Y3 : a > state > $o]: (((hoare_719046530iple_a @ X1 @ X2 @ X3) = (hoare_719046530iple_a @ Y1 @ Y2 @ Y3)) = (((X1 = Y1)) & ((((X2 = Y2)) & ((X3 = Y3))))))))). % triple.inject
thf(fact_1_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_2_triple_Ocase, axiom,
    ((![F : (a > state > $o) > com > (a > state > $o) > $o, X1 : a > state > $o, X2 : com, X3 : a > state > $o]: ((hoare_1012081509le_a_o @ F @ (hoare_719046530iple_a @ X1 @ X2 @ X3)) = (F @ X1 @ X2 @ X3))))). % triple.case
thf(fact_3_triple_Oinduct, axiom,
    ((![P : hoare_1678595023iple_a > $o, Triple : hoare_1678595023iple_a]: ((![X1a : a > state > $o, X2a : com, X3a : a > state > $o]: (P @ (hoare_719046530iple_a @ X1a @ X2a @ X3a))) => (P @ Triple))))). % triple.induct
thf(fact_4_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_5_triple_Oexhaust, axiom,
    ((![Y : hoare_1678595023iple_a]: (~ ((![X12 : a > state > $o, X22 : com, X32 : a > state > $o]: (~ ((Y = (hoare_719046530iple_a @ X12 @ X22 @ X32)))))))))). % triple.exhaust
thf(fact_6_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_7_triple_Ocase__distrib, axiom,
    ((![H : $o > $o, F : (a > state > $o) > com > (a > state > $o) > $o, Triple : hoare_1678595023iple_a]: ((H @ (hoare_1012081509le_a_o @ F @ Triple)) = (hoare_1012081509le_a_o @ (^[X13 : a > state > $o]: (^[X23 : com]: (^[X33 : a > state > $o]: (H @ (F @ X13 @ X23 @ X33))))) @ Triple))))). % triple.case_distrib
thf(fact_8_triple__valid__def, axiom,
    ((hoare_1926814542alid_a = (^[N : nat]: (hoare_1012081509le_a_o @ (^[P2 : a > state > $o]: (^[C : com]: (^[Q : a > state > $o]: (![Z : a]: (![S : state]: (((P2 @ Z @ S)) => ((![S2 : state]: (((evaln @ C @ S @ N @ S2)) => ((Q @ Z @ S2)))))))))))))))). % triple_valid_def
thf(fact_9_nat_Oinject, axiom,
    ((![X2 : nat, Y2 : nat]: (((suc @ X2) = (suc @ Y2)) = (X2 = Y2))))). % nat.inject
thf(fact_10_old_Onat_Oinject, axiom,
    ((![Nat : nat, Nat2 : nat]: (((suc @ Nat) = (suc @ Nat2)) = (Nat = Nat2))))). % old.nat.inject
thf(fact_11_evaln__Suc, axiom,
    ((![C2 : com, S3 : state, N2 : nat, S4 : state]: ((evaln @ C2 @ S3 @ N2 @ S4) => (evaln @ C2 @ S3 @ (suc @ N2) @ S4))))). % evaln_Suc
thf(fact_12_triple__valid__def2, axiom,
    ((![N2 : nat, P : a > state > $o, C2 : com, Q2 : a > state > $o]: ((hoare_1926814542alid_a @ N2 @ (hoare_719046530iple_a @ P @ C2 @ Q2)) = (![Z : a]: (![S : state]: (((P @ Z @ S)) => ((![S2 : state]: (((evaln @ C2 @ S @ N2 @ S2)) => ((Q2 @ Z @ S2)))))))))))). % triple_valid_def2
thf(fact_13_triple__valid__def2, axiom,
    ((![N2 : nat, P : state > state > $o, C2 : com, Q2 : state > state > $o]: ((hoare_364318704_state @ N2 @ (hoare_1659279548_state @ P @ C2 @ Q2)) = (![Z : state]: (![S : state]: (((P @ Z @ S)) => ((![S2 : state]: (((evaln @ C2 @ S @ N2 @ S2)) => ((Q2 @ Z @ S2)))))))))))). % triple_valid_def2
thf(fact_14_evaln__max2, axiom,
    ((![C1 : com, S1 : state, N1 : nat, T1 : state, C22 : com, S22 : state, N22 : nat, T2 : state]: ((evaln @ C1 @ S1 @ N1 @ T1) => ((evaln @ C22 @ S22 @ N22 @ T2) => (?[N3 : nat]: ((evaln @ C1 @ S1 @ N3 @ T1) & (evaln @ C22 @ S22 @ N3 @ T2)))))))). % evaln_max2
thf(fact_15_Suc__inject, axiom,
    ((![X : nat, Y : nat]: (((suc @ X) = (suc @ Y)) => (X = Y))))). % Suc_inject
thf(fact_16_n__not__Suc__n, axiom,
    ((![N2 : nat]: (~ ((N2 = (suc @ N2))))))). % n_not_Suc_n
thf(fact_17_triple_Osize__gen, axiom,
    ((![X : a > nat, X1 : a > state > $o, X2 : com, X3 : a > state > $o]: ((hoare_201533281iple_a @ X @ (hoare_719046530iple_a @ X1 @ X2 @ X3)) = (suc @ zero_zero_nat))))). % triple.size_gen
thf(fact_18_triple_Osize__gen, axiom,
    ((![X : state > nat, X1 : state > state > $o, X2 : com, X3 : state > state > $o]: ((hoare_103296669_state @ X @ (hoare_1659279548_state @ X1 @ X2 @ X3)) = (suc @ zero_zero_nat))))). % triple.size_gen
thf(fact_19_not0__implies__Suc, axiom,
    ((![N2 : nat]: ((~ ((N2 = zero_zero_nat))) => (?[M : nat]: (N2 = (suc @ M))))))). % not0_implies_Suc
thf(fact_20_old_Onat_Oinducts, axiom,
    ((![P : nat > $o, Nat : nat]: ((P @ zero_zero_nat) => ((![Nat3 : nat]: ((P @ Nat3) => (P @ (suc @ Nat3)))) => (P @ Nat)))))). % old.nat.inducts
thf(fact_21_old_Onat_Oexhaust, axiom,
    ((![Y : nat]: ((~ ((Y = zero_zero_nat))) => (~ ((![Nat3 : nat]: (~ ((Y = (suc @ Nat3))))))))))). % old.nat.exhaust
thf(fact_22_Zero__not__Suc, axiom,
    ((![M2 : nat]: (~ ((zero_zero_nat = (suc @ M2))))))). % Zero_not_Suc
thf(fact_23_Zero__neq__Suc, axiom,
    ((![M2 : nat]: (~ ((zero_zero_nat = (suc @ M2))))))). % Zero_neq_Suc
thf(fact_24_Suc__neq__Zero, axiom,
    ((![M2 : nat]: (~ (((suc @ M2) = zero_zero_nat)))))). % Suc_neq_Zero
thf(fact_25_zero__induct, axiom,
    ((![P : nat > $o, K : nat]: ((P @ K) => ((![N3 : nat]: ((P @ (suc @ N3)) => (P @ N3))) => (P @ zero_zero_nat)))))). % zero_induct
thf(fact_26_diff__induct, axiom,
    ((![P : nat > nat > $o, M2 : nat, N2 : nat]: ((![X4 : nat]: (P @ X4 @ zero_zero_nat)) => ((![Y4 : nat]: (P @ zero_zero_nat @ (suc @ Y4))) => ((![X4 : nat, Y4 : nat]: ((P @ X4 @ Y4) => (P @ (suc @ X4) @ (suc @ Y4)))) => (P @ M2 @ N2))))))). % diff_induct
thf(fact_27_nat__induct, axiom,
    ((![P : nat > $o, N2 : nat]: ((P @ zero_zero_nat) => ((![N3 : nat]: ((P @ N3) => (P @ (suc @ N3)))) => (P @ N2)))))). % nat_induct
thf(fact_28_nat_OdiscI, axiom,
    ((![Nat : nat, X2 : nat]: ((Nat = (suc @ X2)) => (~ ((Nat = zero_zero_nat))))))). % nat.discI
thf(fact_29_old_Onat_Odistinct_I1_J, axiom,
    ((![Nat2 : nat]: (~ ((zero_zero_nat = (suc @ Nat2))))))). % old.nat.distinct(1)
thf(fact_30_old_Onat_Odistinct_I2_J, axiom,
    ((![Nat2 : nat]: (~ (((suc @ Nat2) = zero_zero_nat)))))). % old.nat.distinct(2)
thf(fact_31_nat_Odistinct_I1_J, axiom,
    ((![X2 : nat]: (~ ((zero_zero_nat = (suc @ X2))))))). % nat.distinct(1)
thf(fact_32_triple_Osize_I2_J, axiom,
    ((![X1 : a > state > $o, X2 : com, X3 : a > state > $o]: ((size_s648929379iple_a @ (hoare_719046530iple_a @ X1 @ X2 @ X3)) = (suc @ zero_zero_nat))))). % triple.size(2)
thf(fact_33_triple_Osize_I2_J, axiom,
    ((![X1 : state > state > $o, X2 : com, X3 : state > state > $o]: ((size_s1457777073_state @ (hoare_1659279548_state @ X1 @ X2 @ X3)) = (suc @ zero_zero_nat))))). % triple.size(2)
thf(fact_34_Body__triple__valid__0, axiom,
    ((![P : a > state > $o, Pn : pname, Q2 : a > state > $o]: (hoare_1926814542alid_a @ zero_zero_nat @ (hoare_719046530iple_a @ P @ (body2 @ Pn) @ Q2))))). % Body_triple_valid_0
thf(fact_35_Body__triple__valid__0, axiom,
    ((![P : state > state > $o, Pn : pname, Q2 : state > state > $o]: (hoare_364318704_state @ zero_zero_nat @ (hoare_1659279548_state @ P @ (body2 @ Pn) @ Q2))))). % Body_triple_valid_0
thf(fact_36_zero__natural_Orsp, axiom,
    ((zero_zero_nat = zero_zero_nat))). % zero_natural.rsp
thf(fact_37_zero__reorient, axiom,
    ((![X : nat]: ((zero_zero_nat = X) = (X = zero_zero_nat))))). % zero_reorient
thf(fact_38_Least__Suc, axiom,
    ((![P : nat > $o, N2 : nat]: ((P @ N2) => ((~ ((P @ zero_zero_nat))) => ((ord_Least_nat @ P) = (suc @ (ord_Least_nat @ (^[M3 : nat]: (P @ (suc @ M3))))))))))). % Least_Suc
thf(fact_39_Least__eq__0, axiom,
    ((![P : nat > $o]: ((P @ zero_zero_nat) => ((ord_Least_nat @ P) = zero_zero_nat))))). % Least_eq_0
thf(fact_40_size__neq__size__imp__neq, axiom,
    ((![X : com, Y : com]: ((~ (((size_size_com @ X) = (size_size_com @ Y)))) => (~ ((X = Y))))))). % size_neq_size_imp_neq
thf(fact_41_size__neq__size__imp__neq, axiom,
    ((![X : typerep, Y : typerep]: ((~ (((size_size_typerep @ X) = (size_size_typerep @ Y)))) => (~ ((X = Y))))))). % size_neq_size_imp_neq
thf(fact_42_Least__Suc2, axiom,
    ((![P : nat > $o, N2 : nat, Q2 : nat > $o, M2 : nat]: ((P @ N2) => ((Q2 @ M2) => ((~ ((P @ zero_zero_nat))) => ((![K2 : nat]: ((P @ (suc @ K2)) = (Q2 @ K2))) => ((ord_Least_nat @ P) = (suc @ (ord_Least_nat @ Q2)))))))))). % Least_Suc2
thf(fact_43_com_Oinject_I6_J, axiom,
    ((![X7 : pname, Y7 : pname]: (((body2 @ X7) = (body2 @ Y7)) = (X7 = Y7))))). % com.inject(6)
thf(fact_44_LeastI, axiom,
    ((![P : nat > $o, K : nat]: ((P @ K) => (P @ (ord_Least_nat @ P)))))). % LeastI
thf(fact_45_LeastI2__ex, axiom,
    ((![P : nat > $o, Q2 : nat > $o]: ((?[X_1 : nat]: (P @ X_1)) => ((![X4 : nat]: ((P @ X4) => (Q2 @ X4))) => (Q2 @ (ord_Least_nat @ P))))))). % LeastI2_ex
thf(fact_46_LeastI__ex, axiom,
    ((![P : nat > $o]: ((?[X_1 : nat]: (P @ X_1)) => (P @ (ord_Least_nat @ P)))))). % LeastI_ex
thf(fact_47_LeastI2, axiom,
    ((![P : nat > $o, A : nat, Q2 : nat > $o]: ((P @ A) => ((![X4 : nat]: ((P @ X4) => (Q2 @ X4))) => (Q2 @ (ord_Least_nat @ P))))))). % LeastI2
thf(fact_48_Body__triple__valid__Suc, axiom,
    ((![N2 : nat, P : a > state > $o, Pn : pname, Q2 : a > state > $o]: ((hoare_1926814542alid_a @ N2 @ (hoare_719046530iple_a @ P @ (the_com @ (body @ Pn)) @ Q2)) = (hoare_1926814542alid_a @ (suc @ N2) @ (hoare_719046530iple_a @ P @ (body2 @ Pn) @ Q2)))))). % Body_triple_valid_Suc
thf(fact_49_Body__triple__valid__Suc, axiom,
    ((![N2 : nat, P : state > state > $o, Pn : pname, Q2 : state > state > $o]: ((hoare_364318704_state @ N2 @ (hoare_1659279548_state @ P @ (the_com @ (body @ Pn)) @ Q2)) = (hoare_364318704_state @ (suc @ N2) @ (hoare_1659279548_state @ P @ (body2 @ Pn) @ Q2)))))). % Body_triple_valid_Suc
thf(fact_50_com_Osize_I15_J, axiom,
    ((![X7 : pname]: ((size_size_com @ (body2 @ X7)) = zero_zero_nat)))). % com.size(15)
thf(fact_51_evaln_OBody, axiom,
    ((![Pn : pname, S0 : state, N2 : nat, S1 : state]: ((evaln @ (the_com @ (body @ Pn)) @ S0 @ N2 @ S1) => (evaln @ (body2 @ Pn) @ S0 @ (suc @ N2) @ S1))))). % evaln.Body
thf(fact_52_evaln__elim__cases_I6_J, axiom,
    ((![P : pname, S3 : state, N2 : nat, S1 : state]: ((evaln @ (body2 @ P) @ S3 @ N2 @ S1) => (~ ((![N3 : nat]: ((N2 = (suc @ N3)) => (~ ((evaln @ (the_com @ (body @ P)) @ S3 @ N3 @ S1))))))))))). % evaln_elim_cases(6)
thf(fact_53_evalc__elim__cases_I6_J, axiom,
    ((![P : pname, S3 : state, S1 : state]: ((evalc @ (body2 @ P) @ S3 @ S1) => (evalc @ (the_com @ (body @ P)) @ S3 @ S1))))). % evalc_elim_cases(6)
thf(fact_54_evalc_OBody, axiom,
    ((![Pn : pname, S0 : state, S1 : state]: ((evalc @ (the_com @ (body @ Pn)) @ S0 @ S1) => (evalc @ (body2 @ Pn) @ S0 @ S1))))). % evalc.Body
thf(fact_55_com_Osize__gen_I7_J, axiom,
    ((![X7 : pname]: ((size_com @ (body2 @ X7)) = zero_zero_nat)))). % com.size_gen(7)
thf(fact_56_typerep_Osize__neq, axiom,
    ((![X : typerep]: (~ (((size_size_typerep @ X) = zero_zero_nat)))))). % typerep.size_neq
thf(fact_57_com__det, axiom,
    ((![C2 : com, S3 : state, T : state, U : state]: ((evalc @ C2 @ S3 @ T) => ((evalc @ C2 @ S3 @ U) => (U = T)))))). % com_det
thf(fact_58_eval__eq, axiom,
    ((evalc = (^[C : com]: (^[S : state]: (^[T3 : state]: (?[N : nat]: (evaln @ C @ S @ N @ T3)))))))). % eval_eq
thf(fact_59_evalc__evaln, axiom,
    ((![C2 : com, S3 : state, T : state]: ((evalc @ C2 @ S3 @ T) => (?[N3 : nat]: (evaln @ C2 @ S3 @ N3 @ T)))))). % evalc_evaln
thf(fact_60_evaln__evalc, axiom,
    ((![C2 : com, S3 : state, N2 : nat, T : state]: ((evaln @ C2 @ S3 @ N2 @ T) => (evalc @ C2 @ S3 @ T))))). % evaln_evalc
thf(fact_61_MGT__def, axiom,
    ((hoare_Mirabelle_MGT = (^[C : com]: (hoare_1659279548_state @ (^[Y5 : state]: (^[Z2 : state]: (Y5 = Z2))) @ C @ (evalc @ C)))))). % MGT_def
thf(fact_62_option_Ocollapse, axiom,
    ((![Option : option_com]: ((~ ((Option = none_com))) => ((some_com @ (the_com @ Option)) = Option))))). % option.collapse
thf(fact_63_option_Oexhaust__sel, axiom,
    ((![Option : option_com]: ((~ ((Option = none_com))) => (Option = (some_com @ (the_com @ Option))))))). % option.exhaust_sel
thf(fact_64_option_Osel, axiom,
    ((![X2 : com]: ((the_com @ (some_com @ X2)) = X2)))). % option.sel
thf(fact_65_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_66_add__left__cancel, axiom,
    ((![A : nat, B : nat, C2 : nat]: (((plus_plus_nat @ A @ B) = (plus_plus_nat @ A @ C2)) = (B = C2))))). % add_left_cancel
thf(fact_67_add__right__cancel, axiom,
    ((![B : nat, A : nat, C2 : nat]: (((plus_plus_nat @ B @ A) = (plus_plus_nat @ C2 @ A)) = (B = C2))))). % add_right_cancel
thf(fact_68_add_Oleft__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ zero_zero_nat @ A) = A)))). % add.left_neutral
thf(fact_69_add_Oright__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ A @ zero_zero_nat) = A)))). % add.right_neutral
thf(fact_70_add__cancel__left__left, axiom,
    ((![B : nat, A : nat]: (((plus_plus_nat @ B @ A) = A) = (B = zero_zero_nat))))). % add_cancel_left_left
thf(fact_71_add__cancel__left__right, axiom,
    ((![A : nat, B : nat]: (((plus_plus_nat @ A @ B) = A) = (B = zero_zero_nat))))). % add_cancel_left_right
thf(fact_72_add__cancel__right__left, axiom,
    ((![A : nat, B : nat]: ((A = (plus_plus_nat @ B @ A)) = (B = zero_zero_nat))))). % add_cancel_right_left
thf(fact_73_add__cancel__right__right, axiom,
    ((![A : nat, B : nat]: ((A = (plus_plus_nat @ A @ B)) = (B = zero_zero_nat))))). % add_cancel_right_right
thf(fact_74_add__eq__0__iff__both__eq__0, axiom,
    ((![X : nat, Y : nat]: (((plus_plus_nat @ X @ Y) = zero_zero_nat) = (((X = zero_zero_nat)) & ((Y = zero_zero_nat))))))). % add_eq_0_iff_both_eq_0
thf(fact_75_zero__eq__add__iff__both__eq__0, axiom,
    ((![X : nat, Y : nat]: ((zero_zero_nat = (plus_plus_nat @ X @ Y)) = (((X = zero_zero_nat)) & ((Y = zero_zero_nat))))))). % zero_eq_add_iff_both_eq_0
thf(fact_76_add__Suc__right, axiom,
    ((![M2 : nat, N2 : nat]: ((plus_plus_nat @ M2 @ (suc @ N2)) = (suc @ (plus_plus_nat @ M2 @ N2)))))). % add_Suc_right
thf(fact_77_add__is__0, axiom,
    ((![M2 : nat, N2 : nat]: (((plus_plus_nat @ M2 @ N2) = zero_zero_nat) = (((M2 = zero_zero_nat)) & ((N2 = zero_zero_nat))))))). % add_is_0
thf(fact_78_Nat_Oadd__0__right, axiom,
    ((![M2 : nat]: ((plus_plus_nat @ M2 @ zero_zero_nat) = M2)))). % Nat.add_0_right
thf(fact_79_add__is__1, axiom,
    ((![M2 : nat, N2 : nat]: (((plus_plus_nat @ M2 @ N2) = (suc @ zero_zero_nat)) = (((((M2 = (suc @ zero_zero_nat))) & ((N2 = zero_zero_nat)))) | ((((M2 = zero_zero_nat)) & ((N2 = (suc @ zero_zero_nat)))))))))). % add_is_1
thf(fact_80_one__is__add, axiom,
    ((![M2 : nat, N2 : nat]: (((suc @ zero_zero_nat) = (plus_plus_nat @ M2 @ N2)) = (((((M2 = (suc @ zero_zero_nat))) & ((N2 = zero_zero_nat)))) | ((((M2 = zero_zero_nat)) & ((N2 = (suc @ zero_zero_nat)))))))))). % one_is_add
thf(fact_81_ab__semigroup__add__class_Oadd__ac_I1_J, axiom,
    ((![A : nat, B : nat, C2 : nat]: ((plus_plus_nat @ (plus_plus_nat @ A @ B) @ C2) = (plus_plus_nat @ A @ (plus_plus_nat @ B @ C2)))))). % ab_semigroup_add_class.add_ac(1)
thf(fact_82_add__mono__thms__linordered__semiring_I4_J, axiom,
    ((![I : nat, J : nat, K : nat, L : nat]: (((I = J) & (K = L)) => ((plus_plus_nat @ I @ K) = (plus_plus_nat @ J @ L)))))). % add_mono_thms_linordered_semiring(4)
thf(fact_83_group__cancel_Oadd1, axiom,
    ((![A2 : nat, K : nat, A : nat, B : nat]: ((A2 = (plus_plus_nat @ K @ A)) => ((plus_plus_nat @ A2 @ B) = (plus_plus_nat @ K @ (plus_plus_nat @ A @ B))))))). % group_cancel.add1
thf(fact_84_group__cancel_Oadd2, axiom,
    ((![B2 : nat, K : nat, B : nat, A : nat]: ((B2 = (plus_plus_nat @ K @ B)) => ((plus_plus_nat @ A @ B2) = (plus_plus_nat @ K @ (plus_plus_nat @ A @ B))))))). % group_cancel.add2
thf(fact_85_add_Oassoc, axiom,
    ((![A : nat, B : nat, C2 : nat]: ((plus_plus_nat @ (plus_plus_nat @ A @ B) @ C2) = (plus_plus_nat @ A @ (plus_plus_nat @ B @ C2)))))). % add.assoc
thf(fact_86_add_Ocommute, axiom,
    ((plus_plus_nat = (^[A3 : nat]: (^[B3 : nat]: (plus_plus_nat @ B3 @ A3)))))). % add.commute
thf(fact_87_add_Oleft__commute, axiom,
    ((![B : nat, A : nat, C2 : nat]: ((plus_plus_nat @ B @ (plus_plus_nat @ A @ C2)) = (plus_plus_nat @ A @ (plus_plus_nat @ B @ C2)))))). % add.left_commute
thf(fact_88_add__left__imp__eq, axiom,
    ((![A : nat, B : nat, C2 : nat]: (((plus_plus_nat @ A @ B) = (plus_plus_nat @ A @ C2)) => (B = C2))))). % add_left_imp_eq
thf(fact_89_add__right__imp__eq, axiom,
    ((![B : nat, A : nat, C2 : nat]: (((plus_plus_nat @ B @ A) = (plus_plus_nat @ C2 @ A)) => (B = C2))))). % add_right_imp_eq
thf(fact_90_plus__nat_Oadd__0, axiom,
    ((![N2 : nat]: ((plus_plus_nat @ zero_zero_nat @ N2) = N2)))). % plus_nat.add_0
thf(fact_91_add__eq__self__zero, axiom,
    ((![M2 : nat, N2 : nat]: (((plus_plus_nat @ M2 @ N2) = M2) => (N2 = zero_zero_nat))))). % add_eq_self_zero
thf(fact_92_add__Suc, axiom,
    ((![M2 : nat, N2 : nat]: ((plus_plus_nat @ (suc @ M2) @ N2) = (suc @ (plus_plus_nat @ M2 @ N2)))))). % add_Suc
thf(fact_93_nat__arith_Osuc1, axiom,
    ((![A2 : nat, K : nat, A : nat]: ((A2 = (plus_plus_nat @ K @ A)) => ((suc @ A2) = (plus_plus_nat @ K @ (suc @ A))))))). % nat_arith.suc1
thf(fact_94_add__Suc__shift, axiom,
    ((![M2 : nat, N2 : nat]: ((plus_plus_nat @ (suc @ M2) @ N2) = (plus_plus_nat @ M2 @ (suc @ N2)))))). % add_Suc_shift
thf(fact_95_add_Ocomm__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ A @ zero_zero_nat) = A)))). % add.comm_neutral
thf(fact_96_comm__monoid__add__class_Oadd__0, axiom,
    ((![A : nat]: ((plus_plus_nat @ zero_zero_nat @ A) = A)))). % comm_monoid_add_class.add_0

% Conjectures (1)
thf(conj_0, conjecture,
    (((~ ((hoare_1012081509le_a_o @ (^[P2 : a > state > $o]: (^[C : com]: (^[Q : a > state > $o]: (![Z : a]: (![S : state]: (((P2 @ Z @ S)) => ((![S2 : state]: (((evaln @ C @ S @ (suc @ n) @ S2)) => ((Q @ Z @ S2))))))))))) @ (hoare_719046530iple_a @ x1a @ x2a @ x3a)))) | (hoare_1012081509le_a_o @ (^[P2 : a > state > $o]: (^[C : com]: (^[Q : a > state > $o]: (![Z : a]: (![S : state]: (((P2 @ Z @ S)) => ((![S2 : state]: (((evaln @ C @ S @ n @ S2)) => ((Q @ Z @ S2))))))))))) @ (hoare_719046530iple_a @ x1a @ x2a @ x3a))))).
