% 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_116__3250420_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:12:10.940

% Could-be-implicit typings (2)
thf(ty_n_t__Com__Ostate, type,
    state : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).

% Explicit typings (9)
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat, type,
    times_times_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_Ostate__not__singleton, type,
    hoare_405891322gleton : $o).
thf(sy_c_Nat_OSuc, type,
    suc : nat > nat).
thf(sy_c_Orderings_Oord__class_OLeast_001t__Nat__Onat, type,
    ord_Least_nat : (nat > $o) > nat).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat, type,
    ord_less_nat : nat > nat > $o).
thf(sy_v_s, type,
    s : state).
thf(sy_v_t, type,
    t : state).
thf(sy_v_ta, type,
    ta : state).

% Relevant facts (117)
thf(fact_0_state__not__singleton__def, axiom,
    ((hoare_405891322gleton = (?[S : state]: (?[T : state]: (~ ((S = T)))))))). % state_not_singleton_def
thf(fact_1_old_Onat_Oinject, axiom,
    ((![Nat : nat, Nat2 : nat]: (((suc @ Nat) = (suc @ Nat2)) = (Nat = Nat2))))). % old.nat.inject
thf(fact_2_nat_Oinject, axiom,
    ((![X2 : nat, Y2 : nat]: (((suc @ X2) = (suc @ Y2)) = (X2 = Y2))))). % nat.inject
thf(fact_3_not0__implies__Suc, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (?[M : nat]: (N = (suc @ M))))))). % not0_implies_Suc
thf(fact_4_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_5_old_Onat_Oexhaust, axiom,
    ((![Y : nat]: ((~ ((Y = zero_zero_nat))) => (~ ((![Nat3 : nat]: (~ ((Y = (suc @ Nat3))))))))))). % old.nat.exhaust
thf(fact_6_Zero__not__Suc, axiom,
    ((![M2 : nat]: (~ ((zero_zero_nat = (suc @ M2))))))). % Zero_not_Suc
thf(fact_7_Zero__neq__Suc, axiom,
    ((![M2 : nat]: (~ ((zero_zero_nat = (suc @ M2))))))). % Zero_neq_Suc
thf(fact_8_Suc__neq__Zero, axiom,
    ((![M2 : nat]: (~ (((suc @ M2) = zero_zero_nat)))))). % Suc_neq_Zero
thf(fact_9_zero__reorient, axiom,
    ((![X : nat]: ((zero_zero_nat = X) = (X = zero_zero_nat))))). % zero_reorient
thf(fact_10_Suc__inject, axiom,
    ((![X : nat, Y : nat]: (((suc @ X) = (suc @ Y)) => (X = Y))))). % Suc_inject
thf(fact_11_n__not__Suc__n, axiom,
    ((![N : nat]: (~ ((N = (suc @ N))))))). % n_not_Suc_n
thf(fact_12_nat_Odistinct_I1_J, axiom,
    ((![X2 : nat]: (~ ((zero_zero_nat = (suc @ X2))))))). % nat.distinct(1)
thf(fact_13_old_Onat_Odistinct_I2_J, axiom,
    ((![Nat2 : nat]: (~ (((suc @ Nat2) = zero_zero_nat)))))). % old.nat.distinct(2)
thf(fact_14_old_Onat_Odistinct_I1_J, axiom,
    ((![Nat2 : nat]: (~ ((zero_zero_nat = (suc @ Nat2))))))). % old.nat.distinct(1)
thf(fact_15_nat_OdiscI, axiom,
    ((![Nat : nat, X2 : nat]: ((Nat = (suc @ X2)) => (~ ((Nat = zero_zero_nat))))))). % nat.discI
thf(fact_16_nat__induct, axiom,
    ((![P : nat > $o, N : nat]: ((P @ zero_zero_nat) => ((![N2 : nat]: ((P @ N2) => (P @ (suc @ N2)))) => (P @ N)))))). % nat_induct
thf(fact_17_diff__induct, axiom,
    ((![P : nat > nat > $o, M2 : nat, N : nat]: ((![X3 : nat]: (P @ X3 @ zero_zero_nat)) => ((![Y3 : nat]: (P @ zero_zero_nat @ (suc @ Y3))) => ((![X3 : nat, Y3 : nat]: ((P @ X3 @ Y3) => (P @ (suc @ X3) @ (suc @ Y3)))) => (P @ M2 @ N))))))). % diff_induct
thf(fact_18_zero__induct, axiom,
    ((![P : nat > $o, K : nat]: ((P @ K) => ((![N2 : nat]: ((P @ (suc @ N2)) => (P @ N2))) => (P @ zero_zero_nat)))))). % zero_induct
thf(fact_19_zero__natural_Orsp, axiom,
    ((zero_zero_nat = zero_zero_nat))). % zero_natural.rsp
thf(fact_20_less__Suc0, axiom,
    ((![N : nat]: ((ord_less_nat @ N @ (suc @ zero_zero_nat)) = (N = zero_zero_nat))))). % less_Suc0
thf(fact_21_zero__less__Suc, axiom,
    ((![N : nat]: (ord_less_nat @ zero_zero_nat @ (suc @ N))))). % zero_less_Suc
thf(fact_22_Least__Suc2, axiom,
    ((![P : nat > $o, N : nat, Q : nat > $o, M2 : nat]: ((P @ N) => ((Q @ M2) => ((~ ((P @ zero_zero_nat))) => ((![K2 : nat]: ((P @ (suc @ K2)) = (Q @ K2))) => ((ord_Least_nat @ P) = (suc @ (ord_Least_nat @ Q)))))))))). % Least_Suc2
thf(fact_23_mult__eq__1__iff, axiom,
    ((![M2 : nat, N : nat]: (((times_times_nat @ M2 @ N) = (suc @ zero_zero_nat)) = (((M2 = (suc @ zero_zero_nat))) & ((N = (suc @ zero_zero_nat)))))))). % mult_eq_1_iff
thf(fact_24_one__eq__mult__iff, axiom,
    ((![M2 : nat, N : nat]: (((suc @ zero_zero_nat) = (times_times_nat @ M2 @ N)) = (((M2 = (suc @ zero_zero_nat))) & ((N = (suc @ zero_zero_nat)))))))). % one_eq_mult_iff
thf(fact_25_not__gr__zero, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr_zero
thf(fact_26_Suc__less__eq, axiom,
    ((![M2 : nat, N : nat]: ((ord_less_nat @ (suc @ M2) @ (suc @ N)) = (ord_less_nat @ M2 @ N))))). % Suc_less_eq
thf(fact_27_Suc__mono, axiom,
    ((![M2 : nat, N : nat]: ((ord_less_nat @ M2 @ N) => (ord_less_nat @ (suc @ M2) @ (suc @ N)))))). % Suc_mono
thf(fact_28_lessI, axiom,
    ((![N : nat]: (ord_less_nat @ N @ (suc @ N))))). % lessI
thf(fact_29_bot__nat__0_Onot__eq__extremum, axiom,
    ((![A : nat]: ((~ ((A = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ A))))). % bot_nat_0.not_eq_extremum
thf(fact_30_less__nat__zero__code, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_nat_zero_code
thf(fact_31_neq0__conv, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ N))))). % neq0_conv
thf(fact_32_mult__cancel2, axiom,
    ((![M2 : nat, K : nat, N : nat]: (((times_times_nat @ M2 @ K) = (times_times_nat @ N @ K)) = (((M2 = N)) | ((K = zero_zero_nat))))))). % mult_cancel2
thf(fact_33_mult__cancel1, axiom,
    ((![K : nat, M2 : nat, N : nat]: (((times_times_nat @ K @ M2) = (times_times_nat @ K @ N)) = (((M2 = N)) | ((K = zero_zero_nat))))))). % mult_cancel1
thf(fact_34_mult__0__right, axiom,
    ((![M2 : nat]: ((times_times_nat @ M2 @ zero_zero_nat) = zero_zero_nat)))). % mult_0_right
thf(fact_35_mult__is__0, axiom,
    ((![M2 : nat, N : nat]: (((times_times_nat @ M2 @ N) = zero_zero_nat) = (((M2 = zero_zero_nat)) | ((N = zero_zero_nat))))))). % mult_is_0
thf(fact_36_Least__eq__0, axiom,
    ((![P : nat > $o]: ((P @ zero_zero_nat) => ((ord_Least_nat @ P) = zero_zero_nat))))). % Least_eq_0
thf(fact_37_nat__0__less__mult__iff, axiom,
    ((![M2 : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ (times_times_nat @ M2 @ N)) = (((ord_less_nat @ zero_zero_nat @ M2)) & ((ord_less_nat @ zero_zero_nat @ N))))))). % nat_0_less_mult_iff
thf(fact_38_mult__less__cancel2, axiom,
    ((![M2 : nat, K : nat, N : nat]: ((ord_less_nat @ (times_times_nat @ M2 @ K) @ (times_times_nat @ N @ K)) = (((ord_less_nat @ zero_zero_nat @ K)) & ((ord_less_nat @ M2 @ N))))))). % mult_less_cancel2
thf(fact_39_lift__Suc__mono__less__iff, axiom,
    ((![F : nat > nat, N : nat, M2 : nat]: ((![N2 : nat]: (ord_less_nat @ (F @ N2) @ (F @ (suc @ N2)))) => ((ord_less_nat @ (F @ N) @ (F @ M2)) = (ord_less_nat @ N @ M2)))))). % lift_Suc_mono_less_iff
thf(fact_40_lift__Suc__mono__less, axiom,
    ((![F : nat > nat, N : nat, N3 : nat]: ((![N2 : nat]: (ord_less_nat @ (F @ N2) @ (F @ (suc @ N2)))) => ((ord_less_nat @ N @ N3) => (ord_less_nat @ (F @ N) @ (F @ N3))))))). % lift_Suc_mono_less
thf(fact_41_Suc__mult__less__cancel1, axiom,
    ((![K : nat, M2 : nat, N : nat]: ((ord_less_nat @ (times_times_nat @ (suc @ K) @ M2) @ (times_times_nat @ (suc @ K) @ N)) = (ord_less_nat @ M2 @ N))))). % Suc_mult_less_cancel1
thf(fact_42_mult__less__mono2, axiom,
    ((![I : nat, J : nat, K : nat]: ((ord_less_nat @ I @ J) => ((ord_less_nat @ zero_zero_nat @ K) => (ord_less_nat @ (times_times_nat @ K @ I) @ (times_times_nat @ K @ J))))))). % mult_less_mono2
thf(fact_43_mult__less__mono1, axiom,
    ((![I : nat, J : nat, K : nat]: ((ord_less_nat @ I @ J) => ((ord_less_nat @ zero_zero_nat @ K) => (ord_less_nat @ (times_times_nat @ I @ K) @ (times_times_nat @ J @ K))))))). % mult_less_mono1
thf(fact_44_ab__semigroup__mult__class_Omult__ac_I1_J, axiom,
    ((![A : nat, B : nat, C : nat]: ((times_times_nat @ (times_times_nat @ A @ B) @ C) = (times_times_nat @ A @ (times_times_nat @ B @ C)))))). % ab_semigroup_mult_class.mult_ac(1)
thf(fact_45_nat__neq__iff, axiom,
    ((![M2 : nat, N : nat]: ((~ ((M2 = N))) = (((ord_less_nat @ M2 @ N)) | ((ord_less_nat @ N @ M2))))))). % nat_neq_iff
thf(fact_46_less__not__refl, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_not_refl
thf(fact_47_less__not__refl2, axiom,
    ((![N : nat, M2 : nat]: ((ord_less_nat @ N @ M2) => (~ ((M2 = N))))))). % less_not_refl2
thf(fact_48_less__not__refl3, axiom,
    ((![S2 : nat, T2 : nat]: ((ord_less_nat @ S2 @ T2) => (~ ((S2 = T2))))))). % less_not_refl3
thf(fact_49_less__irrefl__nat, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_irrefl_nat
thf(fact_50_nat__less__induct, axiom,
    ((![P : nat > $o, N : nat]: ((![N2 : nat]: ((![M3 : nat]: ((ord_less_nat @ M3 @ N2) => (P @ M3))) => (P @ N2))) => (P @ N))))). % nat_less_induct
thf(fact_51_infinite__descent, axiom,
    ((![P : nat > $o, N : nat]: ((![N2 : nat]: ((~ ((P @ N2))) => (?[M3 : nat]: ((ord_less_nat @ M3 @ N2) & (~ ((P @ M3))))))) => (P @ N))))). % infinite_descent
thf(fact_52_linorder__neqE__nat, axiom,
    ((![X : nat, Y : nat]: ((~ ((X = Y))) => ((~ ((ord_less_nat @ X @ Y))) => (ord_less_nat @ Y @ X)))))). % linorder_neqE_nat
thf(fact_53_mult_Oassoc, axiom,
    ((![A : nat, B : nat, C : nat]: ((times_times_nat @ (times_times_nat @ A @ B) @ C) = (times_times_nat @ A @ (times_times_nat @ B @ C)))))). % mult.assoc
thf(fact_54_mult_Ocommute, axiom,
    ((times_times_nat = (^[A2 : nat]: (^[B2 : nat]: (times_times_nat @ B2 @ A2)))))). % mult.commute
thf(fact_55_mult_Oleft__commute, axiom,
    ((![B : nat, A : nat, C : nat]: ((times_times_nat @ B @ (times_times_nat @ A @ C)) = (times_times_nat @ A @ (times_times_nat @ B @ C)))))). % mult.left_commute
thf(fact_56_n__less__n__mult__m, axiom,
    ((![N : nat, M2 : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((ord_less_nat @ (suc @ zero_zero_nat) @ M2) => (ord_less_nat @ N @ (times_times_nat @ N @ M2))))))). % n_less_n_mult_m
thf(fact_57_n__less__m__mult__n, axiom,
    ((![N : nat, M2 : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((ord_less_nat @ (suc @ zero_zero_nat) @ M2) => (ord_less_nat @ N @ (times_times_nat @ M2 @ N))))))). % n_less_m_mult_n
thf(fact_58_one__less__mult, axiom,
    ((![N : nat, M2 : nat]: ((ord_less_nat @ (suc @ zero_zero_nat) @ N) => ((ord_less_nat @ (suc @ zero_zero_nat) @ M2) => (ord_less_nat @ (suc @ zero_zero_nat) @ (times_times_nat @ M2 @ N))))))). % one_less_mult
thf(fact_59_zero__less__iff__neq__zero, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) = (~ ((N = zero_zero_nat))))))). % zero_less_iff_neq_zero
thf(fact_60_gr__implies__not__zero, axiom,
    ((![M2 : nat, N : nat]: ((ord_less_nat @ M2 @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not_zero
thf(fact_61_not__less__zero, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less_zero
thf(fact_62_gr__zeroI, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr_zeroI
thf(fact_63_Suc__mult__cancel1, axiom,
    ((![K : nat, M2 : nat, N : nat]: (((times_times_nat @ (suc @ K) @ M2) = (times_times_nat @ (suc @ K) @ N)) = (M2 = N))))). % Suc_mult_cancel1
thf(fact_64_mult__0, axiom,
    ((![N : nat]: ((times_times_nat @ zero_zero_nat @ N) = zero_zero_nat)))). % mult_0
thf(fact_65_not__less__less__Suc__eq, axiom,
    ((![N : nat, M2 : nat]: ((~ ((ord_less_nat @ N @ M2))) => ((ord_less_nat @ N @ (suc @ M2)) = (N = M2)))))). % not_less_less_Suc_eq
thf(fact_66_strict__inc__induct, axiom,
    ((![I : nat, J : nat, P : nat > $o]: ((ord_less_nat @ I @ J) => ((![I2 : nat]: ((J = (suc @ I2)) => (P @ I2))) => ((![I2 : nat]: ((ord_less_nat @ I2 @ J) => ((P @ (suc @ I2)) => (P @ I2)))) => (P @ I))))))). % strict_inc_induct
thf(fact_67_less__Suc__induct, axiom,
    ((![I : nat, J : nat, P : nat > nat > $o]: ((ord_less_nat @ I @ J) => ((![I2 : nat]: (P @ I2 @ (suc @ I2))) => ((![I2 : nat, J2 : nat, K2 : nat]: ((ord_less_nat @ I2 @ J2) => ((ord_less_nat @ J2 @ K2) => ((P @ I2 @ J2) => ((P @ J2 @ K2) => (P @ I2 @ K2)))))) => (P @ I @ J))))))). % less_Suc_induct
thf(fact_68_less__trans__Suc, axiom,
    ((![I : nat, J : nat, K : nat]: ((ord_less_nat @ I @ J) => ((ord_less_nat @ J @ K) => (ord_less_nat @ (suc @ I) @ K)))))). % less_trans_Suc
thf(fact_69_Suc__less__SucD, axiom,
    ((![M2 : nat, N : nat]: ((ord_less_nat @ (suc @ M2) @ (suc @ N)) => (ord_less_nat @ M2 @ N))))). % Suc_less_SucD
thf(fact_70_less__antisym, axiom,
    ((![N : nat, M2 : nat]: ((~ ((ord_less_nat @ N @ M2))) => ((ord_less_nat @ N @ (suc @ M2)) => (M2 = N)))))). % less_antisym
thf(fact_71_Suc__less__eq2, axiom,
    ((![N : nat, M2 : nat]: ((ord_less_nat @ (suc @ N) @ M2) = (?[M4 : nat]: (((M2 = (suc @ M4))) & ((ord_less_nat @ N @ M4)))))))). % Suc_less_eq2
thf(fact_72_All__less__Suc, axiom,
    ((![N : nat, P : nat > $o]: ((![I3 : nat]: (((ord_less_nat @ I3 @ (suc @ N))) => ((P @ I3)))) = (((P @ N)) & ((![I3 : nat]: (((ord_less_nat @ I3 @ N)) => ((P @ I3)))))))))). % All_less_Suc
thf(fact_73_not__less__eq, axiom,
    ((![M2 : nat, N : nat]: ((~ ((ord_less_nat @ M2 @ N))) = (ord_less_nat @ N @ (suc @ M2)))))). % not_less_eq
thf(fact_74_less__Suc__eq, axiom,
    ((![M2 : nat, N : nat]: ((ord_less_nat @ M2 @ (suc @ N)) = (((ord_less_nat @ M2 @ N)) | ((M2 = N))))))). % less_Suc_eq
thf(fact_75_Ex__less__Suc, axiom,
    ((![N : nat, P : nat > $o]: ((?[I3 : nat]: (((ord_less_nat @ I3 @ (suc @ N))) & ((P @ I3)))) = (((P @ N)) | ((?[I3 : nat]: (((ord_less_nat @ I3 @ N)) & ((P @ I3)))))))))). % Ex_less_Suc
thf(fact_76_less__SucI, axiom,
    ((![M2 : nat, N : nat]: ((ord_less_nat @ M2 @ N) => (ord_less_nat @ M2 @ (suc @ N)))))). % less_SucI
thf(fact_77_less__SucE, axiom,
    ((![M2 : nat, N : nat]: ((ord_less_nat @ M2 @ (suc @ N)) => ((~ ((ord_less_nat @ M2 @ N))) => (M2 = N)))))). % less_SucE
thf(fact_78_Suc__lessI, axiom,
    ((![M2 : nat, N : nat]: ((ord_less_nat @ M2 @ N) => ((~ (((suc @ M2) = N))) => (ord_less_nat @ (suc @ M2) @ N)))))). % Suc_lessI
thf(fact_79_Suc__lessE, axiom,
    ((![I : nat, K : nat]: ((ord_less_nat @ (suc @ I) @ K) => (~ ((![J2 : nat]: ((ord_less_nat @ I @ J2) => (~ ((K = (suc @ J2)))))))))))). % Suc_lessE
thf(fact_80_Suc__lessD, axiom,
    ((![M2 : nat, N : nat]: ((ord_less_nat @ (suc @ M2) @ N) => (ord_less_nat @ M2 @ N))))). % Suc_lessD
thf(fact_81_Nat_OlessE, axiom,
    ((![I : nat, K : nat]: ((ord_less_nat @ I @ K) => ((~ ((K = (suc @ I)))) => (~ ((![J2 : nat]: ((ord_less_nat @ I @ J2) => (~ ((K = (suc @ J2))))))))))))). % Nat.lessE
thf(fact_82_bot__nat__0_Oextremum__strict, axiom,
    ((![A : nat]: (~ ((ord_less_nat @ A @ zero_zero_nat)))))). % bot_nat_0.extremum_strict
thf(fact_83_infinite__descent0, axiom,
    ((![P : nat > $o, N : nat]: ((P @ zero_zero_nat) => ((![N2 : nat]: ((ord_less_nat @ zero_zero_nat @ N2) => ((~ ((P @ N2))) => (?[M3 : nat]: ((ord_less_nat @ M3 @ N2) & (~ ((P @ M3)))))))) => (P @ N)))))). % infinite_descent0
thf(fact_84_gr__implies__not0, axiom,
    ((![M2 : nat, N : nat]: ((ord_less_nat @ M2 @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not0
thf(fact_85_less__zeroE, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_zeroE
thf(fact_86_not__less0, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less0
thf(fact_87_not__gr0, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr0
thf(fact_88_gr0I, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr0I
thf(fact_89_less__Suc__eq__0__disj, axiom,
    ((![M2 : nat, N : nat]: ((ord_less_nat @ M2 @ (suc @ N)) = (((M2 = zero_zero_nat)) | ((?[J3 : nat]: (((M2 = (suc @ J3))) & ((ord_less_nat @ J3 @ N)))))))))). % less_Suc_eq_0_disj
thf(fact_90_gr0__implies__Suc, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) => (?[M : nat]: (N = (suc @ M))))))). % gr0_implies_Suc
thf(fact_91_All__less__Suc2, axiom,
    ((![N : nat, P : nat > $o]: ((![I3 : nat]: (((ord_less_nat @ I3 @ (suc @ N))) => ((P @ I3)))) = (((P @ zero_zero_nat)) & ((![I3 : nat]: (((ord_less_nat @ I3 @ N)) => ((P @ (suc @ I3))))))))))). % All_less_Suc2
thf(fact_92_gr0__conv__Suc, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) = (?[M5 : nat]: (N = (suc @ M5))))))). % gr0_conv_Suc
thf(fact_93_Ex__less__Suc2, axiom,
    ((![N : nat, P : nat > $o]: ((?[I3 : nat]: (((ord_less_nat @ I3 @ (suc @ N))) & ((P @ I3)))) = (((P @ zero_zero_nat)) | ((?[I3 : nat]: (((ord_less_nat @ I3 @ N)) & ((P @ (suc @ I3))))))))))). % Ex_less_Suc2
thf(fact_94_nat__mult__less__cancel__disj, axiom,
    ((![K : nat, M2 : nat, N : nat]: ((ord_less_nat @ (times_times_nat @ K @ M2) @ (times_times_nat @ K @ N)) = (((ord_less_nat @ zero_zero_nat @ K)) & ((ord_less_nat @ M2 @ N))))))). % nat_mult_less_cancel_disj
thf(fact_95_mult__cancel__right, axiom,
    ((![A : nat, C : nat, B : nat]: (((times_times_nat @ A @ C) = (times_times_nat @ B @ C)) = (((C = zero_zero_nat)) | ((A = B))))))). % mult_cancel_right
thf(fact_96_mult__cancel__left, axiom,
    ((![C : nat, A : nat, B : nat]: (((times_times_nat @ C @ A) = (times_times_nat @ C @ B)) = (((C = zero_zero_nat)) | ((A = B))))))). % mult_cancel_left
thf(fact_97_mult__eq__0__iff, axiom,
    ((![A : nat, B : nat]: (((times_times_nat @ A @ B) = zero_zero_nat) = (((A = zero_zero_nat)) | ((B = zero_zero_nat))))))). % mult_eq_0_iff
thf(fact_98_mult__zero__right, axiom,
    ((![A : nat]: ((times_times_nat @ A @ zero_zero_nat) = zero_zero_nat)))). % mult_zero_right
thf(fact_99_mult__zero__left, axiom,
    ((![A : nat]: ((times_times_nat @ zero_zero_nat @ A) = zero_zero_nat)))). % mult_zero_left
thf(fact_100_mult__right__cancel, axiom,
    ((![C : nat, A : nat, B : nat]: ((~ ((C = zero_zero_nat))) => (((times_times_nat @ A @ C) = (times_times_nat @ B @ C)) = (A = B)))))). % mult_right_cancel
thf(fact_101_mult__left__cancel, axiom,
    ((![C : nat, A : nat, B : nat]: ((~ ((C = zero_zero_nat))) => (((times_times_nat @ C @ A) = (times_times_nat @ C @ B)) = (A = B)))))). % mult_left_cancel
thf(fact_102_no__zero__divisors, axiom,
    ((![A : nat, B : nat]: ((~ ((A = zero_zero_nat))) => ((~ ((B = zero_zero_nat))) => (~ (((times_times_nat @ A @ B) = zero_zero_nat)))))))). % no_zero_divisors
thf(fact_103_divisors__zero, axiom,
    ((![A : nat, B : nat]: (((times_times_nat @ A @ B) = zero_zero_nat) => ((A = zero_zero_nat) | (B = zero_zero_nat)))))). % divisors_zero
thf(fact_104_mult__not__zero, axiom,
    ((![A : nat, B : nat]: ((~ (((times_times_nat @ A @ B) = zero_zero_nat))) => ((~ ((A = zero_zero_nat))) & (~ ((B = zero_zero_nat)))))))). % mult_not_zero
thf(fact_105_nat__mult__eq__cancel__disj, axiom,
    ((![K : nat, M2 : nat, N : nat]: (((times_times_nat @ K @ M2) = (times_times_nat @ K @ N)) = (((K = zero_zero_nat)) | ((M2 = N))))))). % nat_mult_eq_cancel_disj
thf(fact_106_mult__neg__pos, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ zero_zero_nat) => ((ord_less_nat @ zero_zero_nat @ B) => (ord_less_nat @ (times_times_nat @ A @ B) @ zero_zero_nat)))))). % mult_neg_pos
thf(fact_107_mult__pos__neg, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ zero_zero_nat @ A) => ((ord_less_nat @ B @ zero_zero_nat) => (ord_less_nat @ (times_times_nat @ A @ B) @ zero_zero_nat)))))). % mult_pos_neg
thf(fact_108_mult__pos__pos, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ zero_zero_nat @ A) => ((ord_less_nat @ zero_zero_nat @ B) => (ord_less_nat @ zero_zero_nat @ (times_times_nat @ A @ B))))))). % mult_pos_pos
thf(fact_109_mult__pos__neg2, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ zero_zero_nat @ A) => ((ord_less_nat @ B @ zero_zero_nat) => (ord_less_nat @ (times_times_nat @ B @ A) @ zero_zero_nat)))))). % mult_pos_neg2
thf(fact_110_zero__less__mult__pos, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ zero_zero_nat @ (times_times_nat @ A @ B)) => ((ord_less_nat @ zero_zero_nat @ A) => (ord_less_nat @ zero_zero_nat @ B)))))). % zero_less_mult_pos
thf(fact_111_zero__less__mult__pos2, axiom,
    ((![B : nat, A : nat]: ((ord_less_nat @ zero_zero_nat @ (times_times_nat @ B @ A)) => ((ord_less_nat @ zero_zero_nat @ A) => (ord_less_nat @ zero_zero_nat @ B)))))). % zero_less_mult_pos2
thf(fact_112_mult__strict__left__mono, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ A @ B) => ((ord_less_nat @ zero_zero_nat @ C) => (ord_less_nat @ (times_times_nat @ C @ A) @ (times_times_nat @ C @ B))))))). % mult_strict_left_mono
thf(fact_113_mult__strict__right__mono, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ A @ B) => ((ord_less_nat @ zero_zero_nat @ C) => (ord_less_nat @ (times_times_nat @ A @ C) @ (times_times_nat @ B @ C))))))). % mult_strict_right_mono
thf(fact_114_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ A @ B) => ((ord_less_nat @ zero_zero_nat @ C) => (ord_less_nat @ (times_times_nat @ C @ A) @ (times_times_nat @ C @ B))))))). % linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_115_nat__mult__eq__cancel1, axiom,
    ((![K : nat, M2 : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ K) => (((times_times_nat @ K @ M2) = (times_times_nat @ K @ N)) = (M2 = N)))))). % nat_mult_eq_cancel1
thf(fact_116_nat__mult__less__cancel1, axiom,
    ((![K : nat, M2 : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ K) => ((ord_less_nat @ (times_times_nat @ K @ M2) @ (times_times_nat @ K @ N)) = (ord_less_nat @ M2 @ N)))))). % nat_mult_less_cancel1

% Conjectures (4)
thf(conj_0, hypothesis,
    ((~ ((s = t))))).
thf(conj_1, hypothesis,
    ((![S3 : state]: (S3 = ta)))).
thf(conj_2, hypothesis,
    ((~ ((ta = t))))).
thf(conj_3, conjecture,
    ($false)).
