% 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/StrongNorm/prob_53__5209776_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:37:04.820

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

% Explicit typings (13)
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat, type,
    one_one_nat : nat).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat, type,
    plus_plus_nat : nat > nat > nat).
thf(sy_c_InductTermi_OIT, type,
    it : dB > $o).
thf(sy_c_Lambda_OdB_OAbs, type,
    abs : dB > dB).
thf(sy_c_Lambda_OdB_OVar, type,
    var : nat > dB).
thf(sy_c_Lambda_Olift, type,
    lift : dB > nat > dB).
thf(sy_c_Lambda_Oliftn, type,
    liftn : nat > dB > nat > dB).
thf(sy_c_Lambda_Osubst, type,
    subst : dB > dB > nat > dB).
thf(sy_c_Nat_OSuc, type,
    suc : nat > nat).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat, type,
    ord_less_nat : nat > nat > $o).
thf(sy_v_ia, type,
    ia : nat).
thf(sy_v_ja, type,
    ja : nat).
thf(sy_v_ra, type,
    ra : dB).

% Relevant facts (134)
thf(fact_0_subst__eq, axiom,
    ((![K : nat, U : dB]: ((subst @ (var @ K) @ U @ K) = U)))). % subst_eq
thf(fact_1_dB_Oinject_I1_J, axiom,
    ((![X1 : nat, Y1 : nat]: (((var @ X1) = (var @ Y1)) = (X1 = Y1))))). % dB.inject(1)
thf(fact_2_dB_Oinject_I3_J, axiom,
    ((![X3 : dB, Y3 : dB]: (((abs @ X3) = (abs @ Y3)) = (X3 = Y3))))). % dB.inject(3)
thf(fact_3_nat_Oinject, axiom,
    ((![X2 : nat, Y2 : nat]: (((suc @ X2) = (suc @ Y2)) = (X2 = Y2))))). % nat.inject
thf(fact_4_old_Onat_Oinject, axiom,
    ((![Nat : nat, Nat2 : nat]: (((suc @ Nat) = (suc @ Nat2)) = (Nat = Nat2))))). % old.nat.inject
thf(fact_5_Lambda, axiom,
    ((![R : dB]: ((it @ R) => (it @ (abs @ R)))))). % Lambda
thf(fact_6_dB_Odistinct_I3_J, axiom,
    ((![X1 : nat, X3 : dB]: (~ (((var @ X1) = (abs @ X3))))))). % dB.distinct(3)
thf(fact_7_Suc__inject, axiom,
    ((![X : nat, Y : nat]: (((suc @ X) = (suc @ Y)) => (X = Y))))). % Suc_inject
thf(fact_8_n__not__Suc__n, axiom,
    ((![N : nat]: (~ ((N = (suc @ N))))))). % n_not_Suc_n
thf(fact_9_lift__IT, axiom,
    ((![T : dB, I : nat]: ((it @ T) => (it @ (lift @ T @ I)))))). % lift_IT
thf(fact_10_subst__lt, axiom,
    ((![J : nat, I : nat, U : dB]: ((ord_less_nat @ J @ I) => ((subst @ (var @ J) @ U @ I) = (var @ J)))))). % subst_lt
thf(fact_11_not__less__eq, axiom,
    ((![M : nat, N : nat]: ((~ ((ord_less_nat @ M @ N))) = (ord_less_nat @ N @ (suc @ M)))))). % not_less_eq
thf(fact_12_Suc__less__eq, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ (suc @ M) @ (suc @ N)) = (ord_less_nat @ M @ N))))). % Suc_less_eq
thf(fact_13_Suc__mono, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (ord_less_nat @ (suc @ M) @ (suc @ N)))))). % Suc_mono
thf(fact_14_lessI, axiom,
    ((![N : nat]: (ord_less_nat @ N @ (suc @ N))))). % lessI
thf(fact_15_subst__lift, axiom,
    ((![T : dB, K : nat, S : dB]: ((subst @ (lift @ T @ K) @ S @ K) = T)))). % subst_lift
thf(fact_16_nat__neq__iff, axiom,
    ((![M : nat, N : nat]: ((~ ((M = N))) = (((ord_less_nat @ M @ N)) | ((ord_less_nat @ N @ M))))))). % nat_neq_iff
thf(fact_17_less__not__refl, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_not_refl
thf(fact_18_less__not__refl2, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ N @ M) => (~ ((M = N))))))). % less_not_refl2
thf(fact_19_less__not__refl3, axiom,
    ((![S : nat, T : nat]: ((ord_less_nat @ S @ T) => (~ ((S = T))))))). % less_not_refl3
thf(fact_20_less__irrefl__nat, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_irrefl_nat
thf(fact_21_nat__less__induct, axiom,
    ((![P : nat > $o, N : nat]: ((![N2 : nat]: ((![M2 : nat]: ((ord_less_nat @ M2 @ N2) => (P @ M2))) => (P @ N2))) => (P @ N))))). % nat_less_induct
thf(fact_22_infinite__descent, axiom,
    ((![P : nat > $o, N : nat]: ((![N2 : nat]: ((~ ((P @ N2))) => (?[M2 : nat]: ((ord_less_nat @ M2 @ N2) & (~ ((P @ M2))))))) => (P @ N))))). % infinite_descent
thf(fact_23_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_24_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_25_lift__Suc__mono__less__iff, axiom,
    ((![F : nat > nat, N : nat, M : nat]: ((![N2 : nat]: (ord_less_nat @ (F @ N2) @ (F @ (suc @ N2)))) => ((ord_less_nat @ (F @ N) @ (F @ M)) = (ord_less_nat @ N @ M)))))). % lift_Suc_mono_less_iff
thf(fact_26_not__less__less__Suc__eq, axiom,
    ((![N : nat, M : nat]: ((~ ((ord_less_nat @ N @ M))) => ((ord_less_nat @ N @ (suc @ M)) = (N = M)))))). % not_less_less_Suc_eq
thf(fact_27_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_28_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_29_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_30_Suc__less__SucD, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ (suc @ M) @ (suc @ N)) => (ord_less_nat @ M @ N))))). % Suc_less_SucD
thf(fact_31_less__antisym, axiom,
    ((![N : nat, M : nat]: ((~ ((ord_less_nat @ N @ M))) => ((ord_less_nat @ N @ (suc @ M)) => (M = N)))))). % less_antisym
thf(fact_32_Suc__less__eq2, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ (suc @ N) @ M) = (?[M3 : nat]: (((M = (suc @ M3))) & ((ord_less_nat @ N @ M3)))))))). % Suc_less_eq2
thf(fact_33_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_34_less__Suc__eq, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ (suc @ N)) = (((ord_less_nat @ M @ N)) | ((M = N))))))). % less_Suc_eq
thf(fact_35_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_36_less__SucI, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (ord_less_nat @ M @ (suc @ N)))))). % less_SucI
thf(fact_37_less__SucE, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ (suc @ N)) => ((~ ((ord_less_nat @ M @ N))) => (M = N)))))). % less_SucE
thf(fact_38_Suc__lessI, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => ((~ (((suc @ M) = N))) => (ord_less_nat @ (suc @ M) @ N)))))). % Suc_lessI
thf(fact_39_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_40_Suc__lessD, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ (suc @ M) @ N) => (ord_less_nat @ M @ N))))). % Suc_lessD
thf(fact_41_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_42_liftn__lift, axiom,
    ((![N : nat, T : dB, K : nat]: ((liftn @ (suc @ N) @ T @ K) = (lift @ (liftn @ N @ T @ K) @ K))))). % liftn_lift
thf(fact_43_minf_I7_J, axiom,
    ((![T : nat]: (?[Z : nat]: (![X4 : nat]: ((ord_less_nat @ X4 @ Z) => (~ ((ord_less_nat @ T @ X4))))))))). % minf(7)
thf(fact_44_minf_I5_J, axiom,
    ((![T : nat]: (?[Z : nat]: (![X4 : nat]: ((ord_less_nat @ X4 @ Z) => (ord_less_nat @ X4 @ T))))))). % minf(5)
thf(fact_45_minf_I4_J, axiom,
    ((![T : nat]: (?[Z : nat]: (![X4 : nat]: ((ord_less_nat @ X4 @ Z) => (~ ((X4 = T))))))))). % minf(4)
thf(fact_46_minf_I3_J, axiom,
    ((![T : nat]: (?[Z : nat]: (![X4 : nat]: ((ord_less_nat @ X4 @ Z) => (~ ((X4 = T))))))))). % minf(3)
thf(fact_47_minf_I2_J, axiom,
    ((![P : nat > $o, P2 : nat > $o, Q : nat > $o, Q2 : nat > $o]: ((?[Z2 : nat]: (![X5 : nat]: ((ord_less_nat @ X5 @ Z2) => ((P @ X5) = (P2 @ X5))))) => ((?[Z2 : nat]: (![X5 : nat]: ((ord_less_nat @ X5 @ Z2) => ((Q @ X5) = (Q2 @ X5))))) => (?[Z : nat]: (![X4 : nat]: ((ord_less_nat @ X4 @ Z) => ((((P @ X4)) | ((Q @ X4))) = (((P2 @ X4)) | ((Q2 @ X4)))))))))))). % minf(2)
thf(fact_48_minf_I1_J, axiom,
    ((![P : nat > $o, P2 : nat > $o, Q : nat > $o, Q2 : nat > $o]: ((?[Z2 : nat]: (![X5 : nat]: ((ord_less_nat @ X5 @ Z2) => ((P @ X5) = (P2 @ X5))))) => ((?[Z2 : nat]: (![X5 : nat]: ((ord_less_nat @ X5 @ Z2) => ((Q @ X5) = (Q2 @ X5))))) => (?[Z : nat]: (![X4 : nat]: ((ord_less_nat @ X4 @ Z) => ((((P @ X4)) & ((Q @ X4))) = (((P2 @ X4)) & ((Q2 @ X4)))))))))))). % minf(1)
thf(fact_49_pinf_I7_J, axiom,
    ((![T : nat]: (?[Z : nat]: (![X4 : nat]: ((ord_less_nat @ Z @ X4) => (ord_less_nat @ T @ X4))))))). % pinf(7)
thf(fact_50_pinf_I1_J, axiom,
    ((![P : nat > $o, P2 : nat > $o, Q : nat > $o, Q2 : nat > $o]: ((?[Z2 : nat]: (![X5 : nat]: ((ord_less_nat @ Z2 @ X5) => ((P @ X5) = (P2 @ X5))))) => ((?[Z2 : nat]: (![X5 : nat]: ((ord_less_nat @ Z2 @ X5) => ((Q @ X5) = (Q2 @ X5))))) => (?[Z : nat]: (![X4 : nat]: ((ord_less_nat @ Z @ X4) => ((((P @ X4)) & ((Q @ X4))) = (((P2 @ X4)) & ((Q2 @ X4)))))))))))). % pinf(1)
thf(fact_51_pinf_I2_J, axiom,
    ((![P : nat > $o, P2 : nat > $o, Q : nat > $o, Q2 : nat > $o]: ((?[Z2 : nat]: (![X5 : nat]: ((ord_less_nat @ Z2 @ X5) => ((P @ X5) = (P2 @ X5))))) => ((?[Z2 : nat]: (![X5 : nat]: ((ord_less_nat @ Z2 @ X5) => ((Q @ X5) = (Q2 @ X5))))) => (?[Z : nat]: (![X4 : nat]: ((ord_less_nat @ Z @ X4) => ((((P @ X4)) | ((Q @ X4))) = (((P2 @ X4)) | ((Q2 @ X4)))))))))))). % pinf(2)
thf(fact_52_pinf_I3_J, axiom,
    ((![T : nat]: (?[Z : nat]: (![X4 : nat]: ((ord_less_nat @ Z @ X4) => (~ ((X4 = T))))))))). % pinf(3)
thf(fact_53_pinf_I4_J, axiom,
    ((![T : nat]: (?[Z : nat]: (![X4 : nat]: ((ord_less_nat @ Z @ X4) => (~ ((X4 = T))))))))). % pinf(4)
thf(fact_54_pinf_I5_J, axiom,
    ((![T : nat]: (?[Z : nat]: (![X4 : nat]: ((ord_less_nat @ Z @ X4) => (~ ((ord_less_nat @ X4 @ T))))))))). % pinf(5)
thf(fact_55_liftn_Osimps_I1_J, axiom,
    ((![I : nat, K : nat, N : nat]: (((ord_less_nat @ I @ K) => ((liftn @ N @ (var @ I) @ K) = (var @ I))) & ((~ ((ord_less_nat @ I @ K))) => ((liftn @ N @ (var @ I) @ K) = (var @ (plus_plus_nat @ I @ N)))))))). % liftn.simps(1)
thf(fact_56_dual__order_Ostrict__implies__not__eq, axiom,
    ((![B : nat, A : nat]: ((ord_less_nat @ B @ A) => (~ ((A = B))))))). % dual_order.strict_implies_not_eq
thf(fact_57_add__Suc__right, axiom,
    ((![M : nat, N : nat]: ((plus_plus_nat @ M @ (suc @ N)) = (suc @ (plus_plus_nat @ M @ N)))))). % add_Suc_right
thf(fact_58_nat__add__left__cancel__less, axiom,
    ((![K : nat, M : nat, N : nat]: ((ord_less_nat @ (plus_plus_nat @ K @ M) @ (plus_plus_nat @ K @ N)) = (ord_less_nat @ M @ N))))). % nat_add_left_cancel_less
thf(fact_59_add__Suc, axiom,
    ((![M : nat, N : nat]: ((plus_plus_nat @ (suc @ M) @ N) = (suc @ (plus_plus_nat @ M @ N)))))). % add_Suc
thf(fact_60_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_61_add__Suc__shift, axiom,
    ((![M : nat, N : nat]: ((plus_plus_nat @ (suc @ M) @ N) = (plus_plus_nat @ M @ (suc @ N)))))). % add_Suc_shift
thf(fact_62_add__lessD1, axiom,
    ((![I : nat, J : nat, K : nat]: ((ord_less_nat @ (plus_plus_nat @ I @ J) @ K) => (ord_less_nat @ I @ K))))). % add_lessD1
thf(fact_63_add__less__mono, axiom,
    ((![I : nat, J : nat, K : nat, L : nat]: ((ord_less_nat @ I @ J) => ((ord_less_nat @ K @ L) => (ord_less_nat @ (plus_plus_nat @ I @ K) @ (plus_plus_nat @ J @ L))))))). % add_less_mono
thf(fact_64_not__add__less1, axiom,
    ((![I : nat, J : nat]: (~ ((ord_less_nat @ (plus_plus_nat @ I @ J) @ I)))))). % not_add_less1
thf(fact_65_not__add__less2, axiom,
    ((![J : nat, I : nat]: (~ ((ord_less_nat @ (plus_plus_nat @ J @ I) @ I)))))). % not_add_less2
thf(fact_66_add__less__mono1, axiom,
    ((![I : nat, J : nat, K : nat]: ((ord_less_nat @ I @ J) => (ord_less_nat @ (plus_plus_nat @ I @ K) @ (plus_plus_nat @ J @ K)))))). % add_less_mono1
thf(fact_67_trans__less__add1, axiom,
    ((![I : nat, J : nat, M : nat]: ((ord_less_nat @ I @ J) => (ord_less_nat @ I @ (plus_plus_nat @ J @ M)))))). % trans_less_add1
thf(fact_68_trans__less__add2, axiom,
    ((![I : nat, J : nat, M : nat]: ((ord_less_nat @ I @ J) => (ord_less_nat @ I @ (plus_plus_nat @ M @ J)))))). % trans_less_add2
thf(fact_69_less__add__eq__less, axiom,
    ((![K : nat, L : nat, M : nat, N : nat]: ((ord_less_nat @ K @ L) => (((plus_plus_nat @ M @ L) = (plus_plus_nat @ K @ N)) => (ord_less_nat @ M @ N)))))). % less_add_eq_less
thf(fact_70_less__natE, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((![Q3 : nat]: (~ ((N = (suc @ (plus_plus_nat @ M @ Q3)))))))))))). % less_natE
thf(fact_71_less__add__Suc1, axiom,
    ((![I : nat, M : nat]: (ord_less_nat @ I @ (suc @ (plus_plus_nat @ I @ M)))))). % less_add_Suc1
thf(fact_72_less__add__Suc2, axiom,
    ((![I : nat, M : nat]: (ord_less_nat @ I @ (suc @ (plus_plus_nat @ M @ I)))))). % less_add_Suc2
thf(fact_73_less__iff__Suc__add, axiom,
    ((ord_less_nat = (^[M4 : nat]: (^[N4 : nat]: (?[K3 : nat]: (N4 = (suc @ (plus_plus_nat @ M4 @ K3))))))))). % less_iff_Suc_add
thf(fact_74_less__imp__Suc__add, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (?[K2 : nat]: (N = (suc @ (plus_plus_nat @ M @ K2)))))))). % less_imp_Suc_add
thf(fact_75_ord__eq__less__subst, axiom,
    ((![A : nat, F : nat > nat, B : nat, C : nat]: ((A = (F @ B)) => ((ord_less_nat @ B @ C) => ((![X5 : nat, Y4 : nat]: ((ord_less_nat @ X5 @ Y4) => (ord_less_nat @ (F @ X5) @ (F @ Y4)))) => (ord_less_nat @ A @ (F @ C)))))))). % ord_eq_less_subst
thf(fact_76_ord__less__eq__subst, axiom,
    ((![A : nat, B : nat, F : nat > nat, C : nat]: ((ord_less_nat @ A @ B) => (((F @ B) = C) => ((![X5 : nat, Y4 : nat]: ((ord_less_nat @ X5 @ Y4) => (ord_less_nat @ (F @ X5) @ (F @ Y4)))) => (ord_less_nat @ (F @ A) @ C))))))). % ord_less_eq_subst
thf(fact_77_order__less__subst1, axiom,
    ((![A : nat, F : nat > nat, B : nat, C : nat]: ((ord_less_nat @ A @ (F @ B)) => ((ord_less_nat @ B @ C) => ((![X5 : nat, Y4 : nat]: ((ord_less_nat @ X5 @ Y4) => (ord_less_nat @ (F @ X5) @ (F @ Y4)))) => (ord_less_nat @ A @ (F @ C)))))))). % order_less_subst1
thf(fact_78_order__less__subst2, axiom,
    ((![A : nat, B : nat, F : nat > nat, C : nat]: ((ord_less_nat @ A @ B) => ((ord_less_nat @ (F @ B) @ C) => ((![X5 : nat, Y4 : nat]: ((ord_less_nat @ X5 @ Y4) => (ord_less_nat @ (F @ X5) @ (F @ Y4)))) => (ord_less_nat @ (F @ A) @ C))))))). % order_less_subst2
thf(fact_79_gt__ex, axiom,
    ((![X : nat]: (?[X_1 : nat]: (ord_less_nat @ X @ X_1))))). % gt_ex
thf(fact_80_neqE, axiom,
    ((![X : nat, Y : nat]: ((~ ((X = Y))) => ((~ ((ord_less_nat @ X @ Y))) => (ord_less_nat @ Y @ X)))))). % neqE
thf(fact_81_neq__iff, axiom,
    ((![X : nat, Y : nat]: ((~ ((X = Y))) = (((ord_less_nat @ X @ Y)) | ((ord_less_nat @ Y @ X))))))). % neq_iff
thf(fact_82_order_Oasym, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (~ ((ord_less_nat @ B @ A))))))). % order.asym
thf(fact_83_less__imp__neq, axiom,
    ((![X : nat, Y : nat]: ((ord_less_nat @ X @ Y) => (~ ((X = Y))))))). % less_imp_neq
thf(fact_84_less__asym, axiom,
    ((![X : nat, Y : nat]: ((ord_less_nat @ X @ Y) => (~ ((ord_less_nat @ Y @ X))))))). % less_asym
thf(fact_85_less__asym_H, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (~ ((ord_less_nat @ B @ A))))))). % less_asym'
thf(fact_86_less__trans, axiom,
    ((![X : nat, Y : nat, Z3 : nat]: ((ord_less_nat @ X @ Y) => ((ord_less_nat @ Y @ Z3) => (ord_less_nat @ X @ Z3)))))). % less_trans
thf(fact_87_less__linear, axiom,
    ((![X : nat, Y : nat]: ((ord_less_nat @ X @ Y) | ((X = Y) | (ord_less_nat @ Y @ X)))))). % less_linear
thf(fact_88_less__irrefl, axiom,
    ((![X : nat]: (~ ((ord_less_nat @ X @ X)))))). % less_irrefl
thf(fact_89_ord__eq__less__trans, axiom,
    ((![A : nat, B : nat, C : nat]: ((A = B) => ((ord_less_nat @ B @ C) => (ord_less_nat @ A @ C)))))). % ord_eq_less_trans
thf(fact_90_ord__less__eq__trans, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ A @ B) => ((B = C) => (ord_less_nat @ A @ C)))))). % ord_less_eq_trans
thf(fact_91_dual__order_Oasym, axiom,
    ((![B : nat, A : nat]: ((ord_less_nat @ B @ A) => (~ ((ord_less_nat @ A @ B))))))). % dual_order.asym
thf(fact_92_less__imp__not__eq, axiom,
    ((![X : nat, Y : nat]: ((ord_less_nat @ X @ Y) => (~ ((X = Y))))))). % less_imp_not_eq
thf(fact_93_less__not__sym, axiom,
    ((![X : nat, Y : nat]: ((ord_less_nat @ X @ Y) => (~ ((ord_less_nat @ Y @ X))))))). % less_not_sym
thf(fact_94_less__induct, axiom,
    ((![P : nat > $o, A : nat]: ((![X5 : nat]: ((![Y5 : nat]: ((ord_less_nat @ Y5 @ X5) => (P @ Y5))) => (P @ X5))) => (P @ A))))). % less_induct
thf(fact_95_antisym__conv3, axiom,
    ((![Y : nat, X : nat]: ((~ ((ord_less_nat @ Y @ X))) => ((~ ((ord_less_nat @ X @ Y))) = (X = Y)))))). % antisym_conv3
thf(fact_96_less__imp__not__eq2, axiom,
    ((![X : nat, Y : nat]: ((ord_less_nat @ X @ Y) => (~ ((Y = X))))))). % less_imp_not_eq2
thf(fact_97_less__imp__triv, axiom,
    ((![X : nat, Y : nat, P : $o]: ((ord_less_nat @ X @ Y) => ((ord_less_nat @ Y @ X) => P))))). % less_imp_triv
thf(fact_98_linorder__cases, axiom,
    ((![X : nat, Y : nat]: ((~ ((ord_less_nat @ X @ Y))) => ((~ ((X = Y))) => (ord_less_nat @ Y @ X)))))). % linorder_cases
thf(fact_99_dual__order_Oirrefl, axiom,
    ((![A : nat]: (~ ((ord_less_nat @ A @ A)))))). % dual_order.irrefl
thf(fact_100_order_Ostrict__trans, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ A @ B) => ((ord_less_nat @ B @ C) => (ord_less_nat @ A @ C)))))). % order.strict_trans
thf(fact_101_less__imp__not__less, axiom,
    ((![X : nat, Y : nat]: ((ord_less_nat @ X @ Y) => (~ ((ord_less_nat @ Y @ X))))))). % less_imp_not_less
thf(fact_102_exists__least__iff, axiom,
    (((^[P3 : nat > $o]: (?[X6 : nat]: (P3 @ X6))) = (^[P4 : nat > $o]: (?[N4 : nat]: (((P4 @ N4)) & ((![M4 : nat]: (((ord_less_nat @ M4 @ N4)) => ((~ ((P4 @ M4))))))))))))). % exists_least_iff
thf(fact_103_linorder__less__wlog, axiom,
    ((![P : nat > nat > $o, A : nat, B : nat]: ((![A3 : nat, B2 : nat]: ((ord_less_nat @ A3 @ B2) => (P @ A3 @ B2))) => ((![A3 : nat]: (P @ A3 @ A3)) => ((![A3 : nat, B2 : nat]: ((P @ B2 @ A3) => (P @ A3 @ B2))) => (P @ A @ B))))))). % linorder_less_wlog
thf(fact_104_dual__order_Ostrict__trans, axiom,
    ((![B : nat, A : nat, C : nat]: ((ord_less_nat @ B @ A) => ((ord_less_nat @ C @ B) => (ord_less_nat @ C @ A)))))). % dual_order.strict_trans
thf(fact_105_not__less__iff__gr__or__eq, axiom,
    ((![X : nat, Y : nat]: ((~ ((ord_less_nat @ X @ Y))) = (((ord_less_nat @ Y @ X)) | ((X = Y))))))). % not_less_iff_gr_or_eq
thf(fact_106_order_Ostrict__implies__not__eq, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (~ ((A = B))))))). % order.strict_implies_not_eq
thf(fact_107_add__less__cancel__right, axiom,
    ((![A : nat, C : nat, B : nat]: ((ord_less_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ C)) = (ord_less_nat @ A @ B))))). % add_less_cancel_right
thf(fact_108_add__less__cancel__left, axiom,
    ((![C : nat, A : nat, B : nat]: ((ord_less_nat @ (plus_plus_nat @ C @ A) @ (plus_plus_nat @ C @ B)) = (ord_less_nat @ A @ B))))). % add_less_cancel_left
thf(fact_109_add__left__cancel, axiom,
    ((![A : nat, B : nat, C : nat]: (((plus_plus_nat @ A @ B) = (plus_plus_nat @ A @ C)) = (B = C))))). % add_left_cancel
thf(fact_110_add__right__cancel, axiom,
    ((![B : nat, A : nat, C : nat]: (((plus_plus_nat @ B @ A) = (plus_plus_nat @ C @ A)) = (B = C))))). % add_right_cancel
thf(fact_111_add__right__imp__eq, axiom,
    ((![B : nat, A : nat, C : nat]: (((plus_plus_nat @ B @ A) = (plus_plus_nat @ C @ A)) => (B = C))))). % add_right_imp_eq
thf(fact_112_add__left__imp__eq, axiom,
    ((![A : nat, B : nat, C : nat]: (((plus_plus_nat @ A @ B) = (plus_plus_nat @ A @ C)) => (B = C))))). % add_left_imp_eq
thf(fact_113_add_Oleft__commute, axiom,
    ((![B : nat, A : nat, C : nat]: ((plus_plus_nat @ B @ (plus_plus_nat @ A @ C)) = (plus_plus_nat @ A @ (plus_plus_nat @ B @ C)))))). % add.left_commute
thf(fact_114_add_Ocommute, axiom,
    ((plus_plus_nat = (^[A4 : nat]: (^[B3 : nat]: (plus_plus_nat @ B3 @ A4)))))). % add.commute
thf(fact_115_add_Oassoc, axiom,
    ((![A : nat, B : nat, C : nat]: ((plus_plus_nat @ (plus_plus_nat @ A @ B) @ C) = (plus_plus_nat @ A @ (plus_plus_nat @ B @ C)))))). % add.assoc
thf(fact_116_group__cancel_Oadd2, axiom,
    ((![B4 : nat, K : nat, B : nat, A : nat]: ((B4 = (plus_plus_nat @ K @ B)) => ((plus_plus_nat @ A @ B4) = (plus_plus_nat @ K @ (plus_plus_nat @ A @ B))))))). % group_cancel.add2
thf(fact_117_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_118_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_119_ab__semigroup__add__class_Oadd__ac_I1_J, axiom,
    ((![A : nat, B : nat, C : nat]: ((plus_plus_nat @ (plus_plus_nat @ A @ B) @ C) = (plus_plus_nat @ A @ (plus_plus_nat @ B @ C)))))). % ab_semigroup_add_class.add_ac(1)
thf(fact_120_add__mono__thms__linordered__field_I5_J, axiom,
    ((![I : nat, J : nat, K : nat, L : nat]: (((ord_less_nat @ I @ J) & (ord_less_nat @ K @ L)) => (ord_less_nat @ (plus_plus_nat @ I @ K) @ (plus_plus_nat @ J @ L)))))). % add_mono_thms_linordered_field(5)
thf(fact_121_add__mono__thms__linordered__field_I2_J, axiom,
    ((![I : nat, J : nat, K : nat, L : nat]: (((I = J) & (ord_less_nat @ K @ L)) => (ord_less_nat @ (plus_plus_nat @ I @ K) @ (plus_plus_nat @ J @ L)))))). % add_mono_thms_linordered_field(2)
thf(fact_122_add__mono__thms__linordered__field_I1_J, axiom,
    ((![I : nat, J : nat, K : nat, L : nat]: (((ord_less_nat @ I @ J) & (K = L)) => (ord_less_nat @ (plus_plus_nat @ I @ K) @ (plus_plus_nat @ J @ L)))))). % add_mono_thms_linordered_field(1)
thf(fact_123_add__strict__mono, axiom,
    ((![A : nat, B : nat, C : nat, D : nat]: ((ord_less_nat @ A @ B) => ((ord_less_nat @ C @ D) => (ord_less_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ D))))))). % add_strict_mono
thf(fact_124_add__strict__left__mono, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ A @ B) => (ord_less_nat @ (plus_plus_nat @ C @ A) @ (plus_plus_nat @ C @ B)))))). % add_strict_left_mono
thf(fact_125_add__strict__right__mono, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ A @ B) => (ord_less_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ C)))))). % add_strict_right_mono
thf(fact_126_add__less__imp__less__left, axiom,
    ((![C : nat, A : nat, B : nat]: ((ord_less_nat @ (plus_plus_nat @ C @ A) @ (plus_plus_nat @ C @ B)) => (ord_less_nat @ A @ B))))). % add_less_imp_less_left
thf(fact_127_add__less__imp__less__right, axiom,
    ((![A : nat, C : nat, B : nat]: ((ord_less_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ C)) => (ord_less_nat @ A @ B))))). % add_less_imp_less_right
thf(fact_128_lift__subst, axiom,
    ((![J : nat, I : nat, T : dB, S : dB]: ((ord_less_nat @ J @ (plus_plus_nat @ I @ one_one_nat)) => ((lift @ (subst @ T @ S @ J) @ I) = (subst @ (lift @ T @ (plus_plus_nat @ I @ one_one_nat)) @ (lift @ S @ I) @ J)))))). % lift_subst
thf(fact_129_subst__subst, axiom,
    ((![I : nat, J : nat, T : dB, V : dB, U : dB]: ((ord_less_nat @ I @ (plus_plus_nat @ J @ one_one_nat)) => ((subst @ (subst @ T @ (lift @ V @ I) @ (suc @ J)) @ (subst @ U @ V @ J) @ I) = (subst @ (subst @ T @ U @ I) @ V @ J)))))). % subst_subst
thf(fact_130_one__reorient, axiom,
    ((![X : nat]: ((one_one_nat = X) = (X = one_one_nat))))). % one_reorient
thf(fact_131_add__mono1, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (ord_less_nat @ (plus_plus_nat @ A @ one_one_nat) @ (plus_plus_nat @ B @ one_one_nat)))))). % add_mono1
thf(fact_132_less__add__one, axiom,
    ((![A : nat]: (ord_less_nat @ A @ (plus_plus_nat @ A @ one_one_nat))))). % less_add_one
thf(fact_133_Suc__eq__plus1, axiom,
    ((suc = (^[N4 : nat]: (plus_plus_nat @ N4 @ one_one_nat))))). % Suc_eq_plus1

% Conjectures (3)
thf(conj_0, hypothesis,
    ((it @ ra))).
thf(conj_1, hypothesis,
    ((![I4 : nat, J3 : nat]: (it @ (subst @ ra @ (var @ I4) @ J3))))).
thf(conj_2, conjecture,
    ((it @ (abs @ (subst @ ra @ (var @ (suc @ ia)) @ (suc @ ja)))))).
