% 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_52__5209770_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:37:03.461

% Could-be-implicit typings (4)
thf(ty_n_t__List__Olist_It__Lambda__OdB_J, type,
    list_dB : $tType).
thf(ty_n_t__Lambda__OdB, type,
    dB : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).
thf(ty_n_t__Int__Oint, type,
    int : $tType).

% Explicit typings (21)
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint, type,
    uminus_uminus_int : int > int).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint, type,
    zero_zero_int : int).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat, type,
    zero_zero_nat : nat).
thf(sy_c_If_001t__Nat__Onat, type,
    if_nat : $o > 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_OApp, type,
    app : dB > 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_Lambda_Osubstn, type,
    substn : dB > dB > nat > dB).
thf(sy_c_List_Ofoldl_001t__Lambda__OdB_001t__Lambda__OdB, type,
    foldl_dB_dB : (dB > dB > dB) > dB > list_dB > dB).
thf(sy_c_List_Olistsp_001t__Lambda__OdB, type,
    listsp_dB : (dB > $o) > list_dB > $o).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint, type,
    semiri2019852685at_int : nat > int).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat, type,
    semiri1382578993at_nat : nat > nat).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint, type,
    ord_less_int : int > int > $o).
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 (192)
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_Lambda, axiom,
    ((![R : dB]: ((it @ R) => (it @ (abs @ R)))))). % Lambda
thf(fact_4_dB_Odistinct_I3_J, axiom,
    ((![X1 : nat, X3 : dB]: (~ (((var @ X1) = (abs @ X3))))))). % dB.distinct(3)
thf(fact_5_lift__IT, axiom,
    ((![T : dB, I : nat]: ((it @ T) => (it @ (lift @ T @ I)))))). % lift_IT
thf(fact_6_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_7_dB_Oinduct, axiom,
    ((![P : dB > $o, DB : dB]: ((![X : nat]: (P @ (var @ X))) => ((![X1a : dB, X2 : dB]: ((P @ X1a) => ((P @ X2) => (P @ (app @ X1a @ X2))))) => ((![X : dB]: ((P @ X) => (P @ (abs @ X)))) => (P @ DB))))))). % dB.induct
thf(fact_8_dB_Oexhaust, axiom,
    ((![Y : dB]: ((![X12 : nat]: (~ ((Y = (var @ X12))))) => ((![X21 : dB, X22 : dB]: (~ ((Y = (app @ X21 @ X22))))) => (~ ((![X32 : dB]: (~ ((Y = (abs @ X32)))))))))))). % dB.exhaust
thf(fact_9_dB_Oinject_I2_J, axiom,
    ((![X212 : dB, X222 : dB, Y21 : dB, Y22 : dB]: (((app @ X212 @ X222) = (app @ Y21 @ Y22)) = (((X212 = Y21)) & ((X222 = Y22))))))). % dB.inject(2)
thf(fact_10_subst__lift, axiom,
    ((![T : dB, K : nat, S : dB]: ((subst @ (lift @ T @ K) @ S @ K) = T)))). % subst_lift
thf(fact_11_lift_Osimps_I2_J, axiom,
    ((![S : dB, T : dB, K : nat]: ((lift @ (app @ S @ T) @ K) = (app @ (lift @ S @ K) @ (lift @ T @ K)))))). % lift.simps(2)
thf(fact_12_dB_Odistinct_I5_J, axiom,
    ((![X212 : dB, X222 : dB, X3 : dB]: (~ (((app @ X212 @ X222) = (abs @ X3))))))). % dB.distinct(5)
thf(fact_13_dB_Odistinct_I1_J, axiom,
    ((![X1 : nat, X212 : dB, X222 : dB]: (~ (((var @ X1) = (app @ X212 @ X222))))))). % dB.distinct(1)
thf(fact_14_subst__App, axiom,
    ((![T : dB, U : dB, S : dB, K : nat]: ((subst @ (app @ T @ U) @ S @ K) = (app @ (subst @ T @ S @ K) @ (subst @ U @ S @ K)))))). % subst_App
thf(fact_15_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_16_less__not__refl, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_not_refl
thf(fact_17_less__not__refl2, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ N @ M) => (~ ((M = N))))))). % less_not_refl2
thf(fact_18_less__not__refl3, axiom,
    ((![S : nat, T : nat]: ((ord_less_nat @ S @ T) => (~ ((S = T))))))). % less_not_refl3
thf(fact_19_less__irrefl__nat, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_irrefl_nat
thf(fact_20_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_21_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_22_linorder__neqE__nat, axiom,
    ((![X4 : nat, Y : nat]: ((~ ((X4 = Y))) => ((~ ((ord_less_nat @ X4 @ Y))) => (ord_less_nat @ Y @ X4)))))). % linorder_neqE_nat
thf(fact_23_pinf_I1_J, axiom,
    ((![P : nat > $o, P2 : nat > $o, Q : nat > $o, Q2 : nat > $o]: ((?[Z : nat]: (![X : nat]: ((ord_less_nat @ Z @ X) => ((P @ X) = (P2 @ X))))) => ((?[Z : nat]: (![X : nat]: ((ord_less_nat @ Z @ X) => ((Q @ X) = (Q2 @ X))))) => (?[Z2 : nat]: (![X5 : nat]: ((ord_less_nat @ Z2 @ X5) => ((((P @ X5)) & ((Q @ X5))) = (((P2 @ X5)) & ((Q2 @ X5)))))))))))). % pinf(1)
thf(fact_24_pinf_I1_J, axiom,
    ((![P : int > $o, P2 : int > $o, Q : int > $o, Q2 : int > $o]: ((?[Z : int]: (![X : int]: ((ord_less_int @ Z @ X) => ((P @ X) = (P2 @ X))))) => ((?[Z : int]: (![X : int]: ((ord_less_int @ Z @ X) => ((Q @ X) = (Q2 @ X))))) => (?[Z2 : int]: (![X5 : int]: ((ord_less_int @ Z2 @ X5) => ((((P @ X5)) & ((Q @ X5))) = (((P2 @ X5)) & ((Q2 @ X5)))))))))))). % pinf(1)
thf(fact_25_pinf_I2_J, axiom,
    ((![P : nat > $o, P2 : nat > $o, Q : nat > $o, Q2 : nat > $o]: ((?[Z : nat]: (![X : nat]: ((ord_less_nat @ Z @ X) => ((P @ X) = (P2 @ X))))) => ((?[Z : nat]: (![X : nat]: ((ord_less_nat @ Z @ X) => ((Q @ X) = (Q2 @ X))))) => (?[Z2 : nat]: (![X5 : nat]: ((ord_less_nat @ Z2 @ X5) => ((((P @ X5)) | ((Q @ X5))) = (((P2 @ X5)) | ((Q2 @ X5)))))))))))). % pinf(2)
thf(fact_26_pinf_I2_J, axiom,
    ((![P : int > $o, P2 : int > $o, Q : int > $o, Q2 : int > $o]: ((?[Z : int]: (![X : int]: ((ord_less_int @ Z @ X) => ((P @ X) = (P2 @ X))))) => ((?[Z : int]: (![X : int]: ((ord_less_int @ Z @ X) => ((Q @ X) = (Q2 @ X))))) => (?[Z2 : int]: (![X5 : int]: ((ord_less_int @ Z2 @ X5) => ((((P @ X5)) | ((Q @ X5))) = (((P2 @ X5)) | ((Q2 @ X5)))))))))))). % pinf(2)
thf(fact_27_pinf_I3_J, axiom,
    ((![T : nat]: (?[Z2 : nat]: (![X5 : nat]: ((ord_less_nat @ Z2 @ X5) => (~ ((X5 = T))))))))). % pinf(3)
thf(fact_28_pinf_I3_J, axiom,
    ((![T : int]: (?[Z2 : int]: (![X5 : int]: ((ord_less_int @ Z2 @ X5) => (~ ((X5 = T))))))))). % pinf(3)
thf(fact_29_pinf_I4_J, axiom,
    ((![T : nat]: (?[Z2 : nat]: (![X5 : nat]: ((ord_less_nat @ Z2 @ X5) => (~ ((X5 = T))))))))). % pinf(4)
thf(fact_30_pinf_I4_J, axiom,
    ((![T : int]: (?[Z2 : int]: (![X5 : int]: ((ord_less_int @ Z2 @ X5) => (~ ((X5 = T))))))))). % pinf(4)
thf(fact_31_pinf_I5_J, axiom,
    ((![T : nat]: (?[Z2 : nat]: (![X5 : nat]: ((ord_less_nat @ Z2 @ X5) => (~ ((ord_less_nat @ X5 @ T))))))))). % pinf(5)
thf(fact_32_pinf_I5_J, axiom,
    ((![T : int]: (?[Z2 : int]: (![X5 : int]: ((ord_less_int @ Z2 @ X5) => (~ ((ord_less_int @ X5 @ T))))))))). % pinf(5)
thf(fact_33_pinf_I7_J, axiom,
    ((![T : nat]: (?[Z2 : nat]: (![X5 : nat]: ((ord_less_nat @ Z2 @ X5) => (ord_less_nat @ T @ X5))))))). % pinf(7)
thf(fact_34_pinf_I7_J, axiom,
    ((![T : int]: (?[Z2 : int]: (![X5 : int]: ((ord_less_int @ Z2 @ X5) => (ord_less_int @ T @ X5))))))). % pinf(7)
thf(fact_35_minf_I1_J, axiom,
    ((![P : nat > $o, P2 : nat > $o, Q : nat > $o, Q2 : nat > $o]: ((?[Z : nat]: (![X : nat]: ((ord_less_nat @ X @ Z) => ((P @ X) = (P2 @ X))))) => ((?[Z : nat]: (![X : nat]: ((ord_less_nat @ X @ Z) => ((Q @ X) = (Q2 @ X))))) => (?[Z2 : nat]: (![X5 : nat]: ((ord_less_nat @ X5 @ Z2) => ((((P @ X5)) & ((Q @ X5))) = (((P2 @ X5)) & ((Q2 @ X5)))))))))))). % minf(1)
thf(fact_36_minf_I1_J, axiom,
    ((![P : int > $o, P2 : int > $o, Q : int > $o, Q2 : int > $o]: ((?[Z : int]: (![X : int]: ((ord_less_int @ X @ Z) => ((P @ X) = (P2 @ X))))) => ((?[Z : int]: (![X : int]: ((ord_less_int @ X @ Z) => ((Q @ X) = (Q2 @ X))))) => (?[Z2 : int]: (![X5 : int]: ((ord_less_int @ X5 @ Z2) => ((((P @ X5)) & ((Q @ X5))) = (((P2 @ X5)) & ((Q2 @ X5)))))))))))). % minf(1)
thf(fact_37_minf_I2_J, axiom,
    ((![P : nat > $o, P2 : nat > $o, Q : nat > $o, Q2 : nat > $o]: ((?[Z : nat]: (![X : nat]: ((ord_less_nat @ X @ Z) => ((P @ X) = (P2 @ X))))) => ((?[Z : nat]: (![X : nat]: ((ord_less_nat @ X @ Z) => ((Q @ X) = (Q2 @ X))))) => (?[Z2 : nat]: (![X5 : nat]: ((ord_less_nat @ X5 @ Z2) => ((((P @ X5)) | ((Q @ X5))) = (((P2 @ X5)) | ((Q2 @ X5)))))))))))). % minf(2)
thf(fact_38_minf_I2_J, axiom,
    ((![P : int > $o, P2 : int > $o, Q : int > $o, Q2 : int > $o]: ((?[Z : int]: (![X : int]: ((ord_less_int @ X @ Z) => ((P @ X) = (P2 @ X))))) => ((?[Z : int]: (![X : int]: ((ord_less_int @ X @ Z) => ((Q @ X) = (Q2 @ X))))) => (?[Z2 : int]: (![X5 : int]: ((ord_less_int @ X5 @ Z2) => ((((P @ X5)) | ((Q @ X5))) = (((P2 @ X5)) | ((Q2 @ X5)))))))))))). % minf(2)
thf(fact_39_minf_I3_J, axiom,
    ((![T : nat]: (?[Z2 : nat]: (![X5 : nat]: ((ord_less_nat @ X5 @ Z2) => (~ ((X5 = T))))))))). % minf(3)
thf(fact_40_minf_I3_J, axiom,
    ((![T : int]: (?[Z2 : int]: (![X5 : int]: ((ord_less_int @ X5 @ Z2) => (~ ((X5 = T))))))))). % minf(3)
thf(fact_41_minf_I4_J, axiom,
    ((![T : nat]: (?[Z2 : nat]: (![X5 : nat]: ((ord_less_nat @ X5 @ Z2) => (~ ((X5 = T))))))))). % minf(4)
thf(fact_42_minf_I4_J, axiom,
    ((![T : int]: (?[Z2 : int]: (![X5 : int]: ((ord_less_int @ X5 @ Z2) => (~ ((X5 = T))))))))). % minf(4)
thf(fact_43_minf_I5_J, axiom,
    ((![T : nat]: (?[Z2 : nat]: (![X5 : nat]: ((ord_less_nat @ X5 @ Z2) => (ord_less_nat @ X5 @ T))))))). % minf(5)
thf(fact_44_minf_I5_J, axiom,
    ((![T : int]: (?[Z2 : int]: (![X5 : int]: ((ord_less_int @ X5 @ Z2) => (ord_less_int @ X5 @ T))))))). % minf(5)
thf(fact_45_minf_I7_J, axiom,
    ((![T : nat]: (?[Z2 : nat]: (![X5 : nat]: ((ord_less_nat @ X5 @ Z2) => (~ ((ord_less_nat @ T @ X5))))))))). % minf(7)
thf(fact_46_minf_I7_J, axiom,
    ((![T : int]: (?[Z2 : int]: (![X5 : int]: ((ord_less_int @ X5 @ Z2) => (~ ((ord_less_int @ T @ X5))))))))). % minf(7)
thf(fact_47_linorder__neqE__linordered__idom, axiom,
    ((![X4 : int, Y : int]: ((~ ((X4 = Y))) => ((~ ((ord_less_int @ X4 @ Y))) => (ord_less_int @ Y @ X4)))))). % linorder_neqE_linordered_idom
thf(fact_48_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_49_dual__order_Ostrict__implies__not__eq, axiom,
    ((![B : int, A : int]: ((ord_less_int @ B @ A) => (~ ((A = B))))))). % dual_order.strict_implies_not_eq
thf(fact_50_order_Ostrict__implies__not__eq, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (~ ((A = B))))))). % order.strict_implies_not_eq
thf(fact_51_order_Ostrict__implies__not__eq, axiom,
    ((![A : int, B : int]: ((ord_less_int @ A @ B) => (~ ((A = B))))))). % order.strict_implies_not_eq
thf(fact_52_ord__eq__less__subst, axiom,
    ((![A : nat, F : nat > nat, B : nat, C : nat]: ((A = (F @ B)) => ((ord_less_nat @ B @ C) => ((![X : nat, Y2 : nat]: ((ord_less_nat @ X @ Y2) => (ord_less_nat @ (F @ X) @ (F @ Y2)))) => (ord_less_nat @ A @ (F @ C)))))))). % ord_eq_less_subst
thf(fact_53_ord__eq__less__subst, axiom,
    ((![A : int, F : nat > int, B : nat, C : nat]: ((A = (F @ B)) => ((ord_less_nat @ B @ C) => ((![X : nat, Y2 : nat]: ((ord_less_nat @ X @ Y2) => (ord_less_int @ (F @ X) @ (F @ Y2)))) => (ord_less_int @ A @ (F @ C)))))))). % ord_eq_less_subst
thf(fact_54_ord__eq__less__subst, axiom,
    ((![A : nat, F : int > nat, B : int, C : int]: ((A = (F @ B)) => ((ord_less_int @ B @ C) => ((![X : int, Y2 : int]: ((ord_less_int @ X @ Y2) => (ord_less_nat @ (F @ X) @ (F @ Y2)))) => (ord_less_nat @ A @ (F @ C)))))))). % ord_eq_less_subst
thf(fact_55_ord__eq__less__subst, axiom,
    ((![A : int, F : int > int, B : int, C : int]: ((A = (F @ B)) => ((ord_less_int @ B @ C) => ((![X : int, Y2 : int]: ((ord_less_int @ X @ Y2) => (ord_less_int @ (F @ X) @ (F @ Y2)))) => (ord_less_int @ A @ (F @ C)))))))). % ord_eq_less_subst
thf(fact_56_ord__less__eq__subst, axiom,
    ((![A : nat, B : nat, F : nat > nat, C : nat]: ((ord_less_nat @ A @ B) => (((F @ B) = C) => ((![X : nat, Y2 : nat]: ((ord_less_nat @ X @ Y2) => (ord_less_nat @ (F @ X) @ (F @ Y2)))) => (ord_less_nat @ (F @ A) @ C))))))). % ord_less_eq_subst
thf(fact_57_ord__less__eq__subst, axiom,
    ((![A : nat, B : nat, F : nat > int, C : int]: ((ord_less_nat @ A @ B) => (((F @ B) = C) => ((![X : nat, Y2 : nat]: ((ord_less_nat @ X @ Y2) => (ord_less_int @ (F @ X) @ (F @ Y2)))) => (ord_less_int @ (F @ A) @ C))))))). % ord_less_eq_subst
thf(fact_58_ord__less__eq__subst, axiom,
    ((![A : int, B : int, F : int > nat, C : nat]: ((ord_less_int @ A @ B) => (((F @ B) = C) => ((![X : int, Y2 : int]: ((ord_less_int @ X @ Y2) => (ord_less_nat @ (F @ X) @ (F @ Y2)))) => (ord_less_nat @ (F @ A) @ C))))))). % ord_less_eq_subst
thf(fact_59_ord__less__eq__subst, axiom,
    ((![A : int, B : int, F : int > int, C : int]: ((ord_less_int @ A @ B) => (((F @ B) = C) => ((![X : int, Y2 : int]: ((ord_less_int @ X @ Y2) => (ord_less_int @ (F @ X) @ (F @ Y2)))) => (ord_less_int @ (F @ A) @ C))))))). % ord_less_eq_subst
thf(fact_60_order__less__subst1, axiom,
    ((![A : nat, F : nat > nat, B : nat, C : nat]: ((ord_less_nat @ A @ (F @ B)) => ((ord_less_nat @ B @ C) => ((![X : nat, Y2 : nat]: ((ord_less_nat @ X @ Y2) => (ord_less_nat @ (F @ X) @ (F @ Y2)))) => (ord_less_nat @ A @ (F @ C)))))))). % order_less_subst1
thf(fact_61_order__less__subst1, axiom,
    ((![A : nat, F : int > nat, B : int, C : int]: ((ord_less_nat @ A @ (F @ B)) => ((ord_less_int @ B @ C) => ((![X : int, Y2 : int]: ((ord_less_int @ X @ Y2) => (ord_less_nat @ (F @ X) @ (F @ Y2)))) => (ord_less_nat @ A @ (F @ C)))))))). % order_less_subst1
thf(fact_62_order__less__subst1, axiom,
    ((![A : int, F : nat > int, B : nat, C : nat]: ((ord_less_int @ A @ (F @ B)) => ((ord_less_nat @ B @ C) => ((![X : nat, Y2 : nat]: ((ord_less_nat @ X @ Y2) => (ord_less_int @ (F @ X) @ (F @ Y2)))) => (ord_less_int @ A @ (F @ C)))))))). % order_less_subst1
thf(fact_63_order__less__subst1, axiom,
    ((![A : int, F : int > int, B : int, C : int]: ((ord_less_int @ A @ (F @ B)) => ((ord_less_int @ B @ C) => ((![X : int, Y2 : int]: ((ord_less_int @ X @ Y2) => (ord_less_int @ (F @ X) @ (F @ Y2)))) => (ord_less_int @ A @ (F @ C)))))))). % order_less_subst1
thf(fact_64_order__less__subst2, axiom,
    ((![A : nat, B : nat, F : nat > nat, C : nat]: ((ord_less_nat @ A @ B) => ((ord_less_nat @ (F @ B) @ C) => ((![X : nat, Y2 : nat]: ((ord_less_nat @ X @ Y2) => (ord_less_nat @ (F @ X) @ (F @ Y2)))) => (ord_less_nat @ (F @ A) @ C))))))). % order_less_subst2
thf(fact_65_order__less__subst2, axiom,
    ((![A : nat, B : nat, F : nat > int, C : int]: ((ord_less_nat @ A @ B) => ((ord_less_int @ (F @ B) @ C) => ((![X : nat, Y2 : nat]: ((ord_less_nat @ X @ Y2) => (ord_less_int @ (F @ X) @ (F @ Y2)))) => (ord_less_int @ (F @ A) @ C))))))). % order_less_subst2
thf(fact_66_order__less__subst2, axiom,
    ((![A : int, B : int, F : int > nat, C : nat]: ((ord_less_int @ A @ B) => ((ord_less_nat @ (F @ B) @ C) => ((![X : int, Y2 : int]: ((ord_less_int @ X @ Y2) => (ord_less_nat @ (F @ X) @ (F @ Y2)))) => (ord_less_nat @ (F @ A) @ C))))))). % order_less_subst2
thf(fact_67_order__less__subst2, axiom,
    ((![A : int, B : int, F : int > int, C : int]: ((ord_less_int @ A @ B) => ((ord_less_int @ (F @ B) @ C) => ((![X : int, Y2 : int]: ((ord_less_int @ X @ Y2) => (ord_less_int @ (F @ X) @ (F @ Y2)))) => (ord_less_int @ (F @ A) @ C))))))). % order_less_subst2
thf(fact_68_lt__ex, axiom,
    ((![X4 : int]: (?[Y2 : int]: (ord_less_int @ Y2 @ X4))))). % lt_ex
thf(fact_69_gt__ex, axiom,
    ((![X4 : nat]: (?[X_1 : nat]: (ord_less_nat @ X4 @ X_1))))). % gt_ex
thf(fact_70_gt__ex, axiom,
    ((![X4 : int]: (?[X_1 : int]: (ord_less_int @ X4 @ X_1))))). % gt_ex
thf(fact_71_neqE, axiom,
    ((![X4 : nat, Y : nat]: ((~ ((X4 = Y))) => ((~ ((ord_less_nat @ X4 @ Y))) => (ord_less_nat @ Y @ X4)))))). % neqE
thf(fact_72_neqE, axiom,
    ((![X4 : int, Y : int]: ((~ ((X4 = Y))) => ((~ ((ord_less_int @ X4 @ Y))) => (ord_less_int @ Y @ X4)))))). % neqE
thf(fact_73_neq__iff, axiom,
    ((![X4 : nat, Y : nat]: ((~ ((X4 = Y))) = (((ord_less_nat @ X4 @ Y)) | ((ord_less_nat @ Y @ X4))))))). % neq_iff
thf(fact_74_neq__iff, axiom,
    ((![X4 : int, Y : int]: ((~ ((X4 = Y))) = (((ord_less_int @ X4 @ Y)) | ((ord_less_int @ Y @ X4))))))). % neq_iff
thf(fact_75_order_Oasym, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (~ ((ord_less_nat @ B @ A))))))). % order.asym
thf(fact_76_order_Oasym, axiom,
    ((![A : int, B : int]: ((ord_less_int @ A @ B) => (~ ((ord_less_int @ B @ A))))))). % order.asym
thf(fact_77_less__imp__neq, axiom,
    ((![X4 : nat, Y : nat]: ((ord_less_nat @ X4 @ Y) => (~ ((X4 = Y))))))). % less_imp_neq
thf(fact_78_less__imp__neq, axiom,
    ((![X4 : int, Y : int]: ((ord_less_int @ X4 @ Y) => (~ ((X4 = Y))))))). % less_imp_neq
thf(fact_79_less__asym, axiom,
    ((![X4 : nat, Y : nat]: ((ord_less_nat @ X4 @ Y) => (~ ((ord_less_nat @ Y @ X4))))))). % less_asym
thf(fact_80_less__asym, axiom,
    ((![X4 : int, Y : int]: ((ord_less_int @ X4 @ Y) => (~ ((ord_less_int @ Y @ X4))))))). % less_asym
thf(fact_81_less__asym_H, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (~ ((ord_less_nat @ B @ A))))))). % less_asym'
thf(fact_82_less__asym_H, axiom,
    ((![A : int, B : int]: ((ord_less_int @ A @ B) => (~ ((ord_less_int @ B @ A))))))). % less_asym'
thf(fact_83_less__trans, axiom,
    ((![X4 : nat, Y : nat, Z3 : nat]: ((ord_less_nat @ X4 @ Y) => ((ord_less_nat @ Y @ Z3) => (ord_less_nat @ X4 @ Z3)))))). % less_trans
thf(fact_84_less__trans, axiom,
    ((![X4 : int, Y : int, Z3 : int]: ((ord_less_int @ X4 @ Y) => ((ord_less_int @ Y @ Z3) => (ord_less_int @ X4 @ Z3)))))). % less_trans
thf(fact_85_less__linear, axiom,
    ((![X4 : nat, Y : nat]: ((ord_less_nat @ X4 @ Y) | ((X4 = Y) | (ord_less_nat @ Y @ X4)))))). % less_linear
thf(fact_86_less__linear, axiom,
    ((![X4 : int, Y : int]: ((ord_less_int @ X4 @ Y) | ((X4 = Y) | (ord_less_int @ Y @ X4)))))). % less_linear
thf(fact_87_less__irrefl, axiom,
    ((![X4 : nat]: (~ ((ord_less_nat @ X4 @ X4)))))). % less_irrefl
thf(fact_88_less__irrefl, axiom,
    ((![X4 : int]: (~ ((ord_less_int @ X4 @ X4)))))). % 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__eq__less__trans, axiom,
    ((![A : int, B : int, C : int]: ((A = B) => ((ord_less_int @ B @ C) => (ord_less_int @ A @ C)))))). % ord_eq_less_trans
thf(fact_91_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_92_ord__less__eq__trans, axiom,
    ((![A : int, B : int, C : int]: ((ord_less_int @ A @ B) => ((B = C) => (ord_less_int @ A @ C)))))). % ord_less_eq_trans
thf(fact_93_dual__order_Oasym, axiom,
    ((![B : nat, A : nat]: ((ord_less_nat @ B @ A) => (~ ((ord_less_nat @ A @ B))))))). % dual_order.asym
thf(fact_94_dual__order_Oasym, axiom,
    ((![B : int, A : int]: ((ord_less_int @ B @ A) => (~ ((ord_less_int @ A @ B))))))). % dual_order.asym
thf(fact_95_less__imp__not__eq, axiom,
    ((![X4 : nat, Y : nat]: ((ord_less_nat @ X4 @ Y) => (~ ((X4 = Y))))))). % less_imp_not_eq
thf(fact_96_less__imp__not__eq, axiom,
    ((![X4 : int, Y : int]: ((ord_less_int @ X4 @ Y) => (~ ((X4 = Y))))))). % less_imp_not_eq
thf(fact_97_less__not__sym, axiom,
    ((![X4 : nat, Y : nat]: ((ord_less_nat @ X4 @ Y) => (~ ((ord_less_nat @ Y @ X4))))))). % less_not_sym
thf(fact_98_less__not__sym, axiom,
    ((![X4 : int, Y : int]: ((ord_less_int @ X4 @ Y) => (~ ((ord_less_int @ Y @ X4))))))). % less_not_sym
thf(fact_99_less__induct, axiom,
    ((![P : nat > $o, A : nat]: ((![X : nat]: ((![Y4 : nat]: ((ord_less_nat @ Y4 @ X) => (P @ Y4))) => (P @ X))) => (P @ A))))). % less_induct
thf(fact_100_antisym__conv3, axiom,
    ((![Y : nat, X4 : nat]: ((~ ((ord_less_nat @ Y @ X4))) => ((~ ((ord_less_nat @ X4 @ Y))) = (X4 = Y)))))). % antisym_conv3
thf(fact_101_antisym__conv3, axiom,
    ((![Y : int, X4 : int]: ((~ ((ord_less_int @ Y @ X4))) => ((~ ((ord_less_int @ X4 @ Y))) = (X4 = Y)))))). % antisym_conv3
thf(fact_102_less__imp__not__eq2, axiom,
    ((![X4 : nat, Y : nat]: ((ord_less_nat @ X4 @ Y) => (~ ((Y = X4))))))). % less_imp_not_eq2
thf(fact_103_less__imp__not__eq2, axiom,
    ((![X4 : int, Y : int]: ((ord_less_int @ X4 @ Y) => (~ ((Y = X4))))))). % less_imp_not_eq2
thf(fact_104_less__imp__triv, axiom,
    ((![X4 : nat, Y : nat, P : $o]: ((ord_less_nat @ X4 @ Y) => ((ord_less_nat @ Y @ X4) => P))))). % less_imp_triv
thf(fact_105_less__imp__triv, axiom,
    ((![X4 : int, Y : int, P : $o]: ((ord_less_int @ X4 @ Y) => ((ord_less_int @ Y @ X4) => P))))). % less_imp_triv
thf(fact_106_linorder__cases, axiom,
    ((![X4 : nat, Y : nat]: ((~ ((ord_less_nat @ X4 @ Y))) => ((~ ((X4 = Y))) => (ord_less_nat @ Y @ X4)))))). % linorder_cases
thf(fact_107_linorder__cases, axiom,
    ((![X4 : int, Y : int]: ((~ ((ord_less_int @ X4 @ Y))) => ((~ ((X4 = Y))) => (ord_less_int @ Y @ X4)))))). % linorder_cases
thf(fact_108_dual__order_Oirrefl, axiom,
    ((![A : nat]: (~ ((ord_less_nat @ A @ A)))))). % dual_order.irrefl
thf(fact_109_dual__order_Oirrefl, axiom,
    ((![A : int]: (~ ((ord_less_int @ A @ A)))))). % dual_order.irrefl
thf(fact_110_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_111_order_Ostrict__trans, axiom,
    ((![A : int, B : int, C : int]: ((ord_less_int @ A @ B) => ((ord_less_int @ B @ C) => (ord_less_int @ A @ C)))))). % order.strict_trans
thf(fact_112_less__imp__not__less, axiom,
    ((![X4 : nat, Y : nat]: ((ord_less_nat @ X4 @ Y) => (~ ((ord_less_nat @ Y @ X4))))))). % less_imp_not_less
thf(fact_113_less__imp__not__less, axiom,
    ((![X4 : int, Y : int]: ((ord_less_int @ X4 @ Y) => (~ ((ord_less_int @ Y @ X4))))))). % less_imp_not_less
thf(fact_114_exists__least__iff, axiom,
    (((^[P3 : nat > $o]: (?[X6 : nat]: (P3 @ X6))) = (^[P4 : nat > $o]: (?[N3 : nat]: (((P4 @ N3)) & ((![M3 : nat]: (((ord_less_nat @ M3 @ N3)) => ((~ ((P4 @ M3))))))))))))). % exists_least_iff
thf(fact_115_linorder__less__wlog, axiom,
    ((![P : nat > nat > $o, A : nat, B : nat]: ((![A2 : nat, B2 : nat]: ((ord_less_nat @ A2 @ B2) => (P @ A2 @ B2))) => ((![A2 : nat]: (P @ A2 @ A2)) => ((![A2 : nat, B2 : nat]: ((P @ B2 @ A2) => (P @ A2 @ B2))) => (P @ A @ B))))))). % linorder_less_wlog
thf(fact_116_linorder__less__wlog, axiom,
    ((![P : int > int > $o, A : int, B : int]: ((![A2 : int, B2 : int]: ((ord_less_int @ A2 @ B2) => (P @ A2 @ B2))) => ((![A2 : int]: (P @ A2 @ A2)) => ((![A2 : int, B2 : int]: ((P @ B2 @ A2) => (P @ A2 @ B2))) => (P @ A @ B))))))). % linorder_less_wlog
thf(fact_117_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_118_dual__order_Ostrict__trans, axiom,
    ((![B : int, A : int, C : int]: ((ord_less_int @ B @ A) => ((ord_less_int @ C @ B) => (ord_less_int @ C @ A)))))). % dual_order.strict_trans
thf(fact_119_not__less__iff__gr__or__eq, axiom,
    ((![X4 : nat, Y : nat]: ((~ ((ord_less_nat @ X4 @ Y))) = (((ord_less_nat @ Y @ X4)) | ((X4 = Y))))))). % not_less_iff_gr_or_eq
thf(fact_120_not__less__iff__gr__or__eq, axiom,
    ((![X4 : int, Y : int]: ((~ ((ord_less_int @ X4 @ Y))) = (((ord_less_int @ Y @ X4)) | ((X4 = Y))))))). % not_less_iff_gr_or_eq
thf(fact_121_verit__comp__simplify1_I1_J, axiom,
    ((![A : nat]: (~ ((ord_less_nat @ A @ A)))))). % verit_comp_simplify1(1)
thf(fact_122_verit__comp__simplify1_I1_J, axiom,
    ((![A : int]: (~ ((ord_less_int @ A @ A)))))). % verit_comp_simplify1(1)
thf(fact_123_substn_Osimps_I2_J, axiom,
    ((![T : dB, U : dB, S : dB, K : nat]: ((substn @ (app @ T @ U) @ S @ K) = (app @ (substn @ T @ S @ K) @ (substn @ U @ S @ K)))))). % substn.simps(2)
thf(fact_124_of__nat__less__iff, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ (semiri1382578993at_nat @ M) @ (semiri1382578993at_nat @ N)) = (ord_less_nat @ M @ N))))). % of_nat_less_iff
thf(fact_125_of__nat__less__iff, axiom,
    ((![M : nat, N : nat]: ((ord_less_int @ (semiri2019852685at_int @ M) @ (semiri2019852685at_int @ N)) = (ord_less_nat @ M @ N))))). % of_nat_less_iff
thf(fact_126_Beta, axiom,
    ((![R : dB, S : dB, Ss : list_dB]: ((it @ (foldl_dB_dB @ app @ (subst @ R @ S @ zero_zero_nat) @ Ss)) => ((it @ S) => (it @ (foldl_dB_dB @ app @ (app @ (abs @ R) @ S) @ Ss))))))). % Beta
thf(fact_127_liftn_Osimps_I2_J, axiom,
    ((![N : nat, S : dB, T : dB, K : nat]: ((liftn @ N @ (app @ S @ T) @ K) = (app @ (liftn @ N @ S @ K) @ (liftn @ N @ T @ K)))))). % liftn.simps(2)
thf(fact_128_of__nat__eq__iff, axiom,
    ((![M : nat, N : nat]: (((semiri2019852685at_int @ M) = (semiri2019852685at_int @ N)) = (M = N))))). % of_nat_eq_iff
thf(fact_129_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_130_less__nat__zero__code, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_nat_zero_code
thf(fact_131_neq0__conv, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ N))))). % neq0_conv
thf(fact_132_liftn__0, axiom,
    ((![T : dB, K : nat]: ((liftn @ zero_zero_nat @ T @ K) = T)))). % liftn_0
thf(fact_133_of__nat__0, axiom,
    (((semiri1382578993at_nat @ zero_zero_nat) = zero_zero_nat))). % of_nat_0
thf(fact_134_of__nat__0, axiom,
    (((semiri2019852685at_int @ zero_zero_nat) = zero_zero_int))). % of_nat_0
thf(fact_135_of__nat__0__eq__iff, axiom,
    ((![N : nat]: ((zero_zero_nat = (semiri1382578993at_nat @ N)) = (zero_zero_nat = N))))). % of_nat_0_eq_iff
thf(fact_136_of__nat__0__eq__iff, axiom,
    ((![N : nat]: ((zero_zero_int = (semiri2019852685at_int @ N)) = (zero_zero_nat = N))))). % of_nat_0_eq_iff
thf(fact_137_of__nat__eq__0__iff, axiom,
    ((![M : nat]: (((semiri1382578993at_nat @ M) = zero_zero_nat) = (M = zero_zero_nat))))). % of_nat_eq_0_iff
thf(fact_138_of__nat__eq__0__iff, axiom,
    ((![M : nat]: (((semiri2019852685at_int @ M) = zero_zero_int) = (M = zero_zero_nat))))). % of_nat_eq_0_iff
thf(fact_139_of__nat__0__less__iff, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ (semiri1382578993at_nat @ N)) = (ord_less_nat @ zero_zero_nat @ N))))). % of_nat_0_less_iff
thf(fact_140_of__nat__0__less__iff, axiom,
    ((![N : nat]: ((ord_less_int @ zero_zero_int @ (semiri2019852685at_int @ N)) = (ord_less_nat @ zero_zero_nat @ N))))). % of_nat_0_less_iff
thf(fact_141_substn__subst__n, axiom,
    ((substn = (^[T2 : dB]: (^[S2 : dB]: (^[N3 : nat]: (subst @ T2 @ (liftn @ N3 @ S2 @ zero_zero_nat) @ N3))))))). % substn_subst_n
thf(fact_142_substn__subst__0, axiom,
    ((![T : dB, S : dB]: ((substn @ T @ S @ zero_zero_nat) = (subst @ T @ S @ zero_zero_nat))))). % substn_subst_0
thf(fact_143_of__nat__less__0__iff, axiom,
    ((![M : nat]: (~ ((ord_less_nat @ (semiri1382578993at_nat @ M) @ zero_zero_nat)))))). % of_nat_less_0_iff
thf(fact_144_of__nat__less__0__iff, axiom,
    ((![M : nat]: (~ ((ord_less_int @ (semiri2019852685at_int @ M) @ zero_zero_int)))))). % of_nat_less_0_iff
thf(fact_145_nat__int__comparison_I2_J, axiom,
    ((ord_less_nat = (^[A3 : nat]: (^[B3 : nat]: (ord_less_int @ (semiri2019852685at_int @ A3) @ (semiri2019852685at_int @ B3))))))). % nat_int_comparison(2)
thf(fact_146_bot__nat__0_Oextremum__strict, axiom,
    ((![A : nat]: (~ ((ord_less_nat @ A @ zero_zero_nat)))))). % bot_nat_0.extremum_strict
thf(fact_147_infinite__descent0, axiom,
    ((![P : nat > $o, N : nat]: ((P @ zero_zero_nat) => ((![N2 : nat]: ((ord_less_nat @ zero_zero_nat @ N2) => ((~ ((P @ N2))) => (?[M2 : nat]: ((ord_less_nat @ M2 @ N2) & (~ ((P @ M2)))))))) => (P @ N)))))). % infinite_descent0
thf(fact_148_gr__implies__not0, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not0
thf(fact_149_less__zeroE, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_zeroE
thf(fact_150_not__less0, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less0
thf(fact_151_not__gr0, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr0
thf(fact_152_gr0I, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr0I
thf(fact_153_of__nat__less__imp__less, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ (semiri1382578993at_nat @ M) @ (semiri1382578993at_nat @ N)) => (ord_less_nat @ M @ N))))). % of_nat_less_imp_less
thf(fact_154_of__nat__less__imp__less, axiom,
    ((![M : nat, N : nat]: ((ord_less_int @ (semiri2019852685at_int @ M) @ (semiri2019852685at_int @ N)) => (ord_less_nat @ M @ N))))). % of_nat_less_imp_less
thf(fact_155_less__imp__of__nat__less, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (ord_less_nat @ (semiri1382578993at_nat @ M) @ (semiri1382578993at_nat @ N)))))). % less_imp_of_nat_less
thf(fact_156_less__imp__of__nat__less, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (ord_less_int @ (semiri2019852685at_int @ M) @ (semiri2019852685at_int @ N)))))). % less_imp_of_nat_less
thf(fact_157_Var__apps__eq__Var__apps__conv, axiom,
    ((![M : nat, Rs : list_dB, N : nat, Ss : list_dB]: (((foldl_dB_dB @ app @ (var @ M) @ Rs) = (foldl_dB_dB @ app @ (var @ N) @ Ss)) = (((M = N)) & ((Rs = Ss))))))). % Var_apps_eq_Var_apps_conv
thf(fact_158_Abs__apps__eq__Abs__apps__conv, axiom,
    ((![R : dB, Rs : list_dB, S : dB, Ss : list_dB]: (((foldl_dB_dB @ app @ (abs @ R) @ Rs) = (foldl_dB_dB @ app @ (abs @ S) @ Ss)) = (((R = S)) & ((Rs = Ss))))))). % Abs_apps_eq_Abs_apps_conv
thf(fact_159_apps__eq__tail__conv, axiom,
    ((![R : dB, Ts : list_dB, S : dB]: (((foldl_dB_dB @ app @ R @ Ts) = (foldl_dB_dB @ app @ S @ Ts)) = (R = S))))). % apps_eq_tail_conv
thf(fact_160_int__if, axiom,
    ((![P : $o, A : nat, B : nat]: ((P => ((semiri2019852685at_int @ (if_nat @ P @ A @ B)) = (semiri2019852685at_int @ A))) & ((~ (P)) => ((semiri2019852685at_int @ (if_nat @ P @ A @ B)) = (semiri2019852685at_int @ B))))))). % int_if
thf(fact_161_nat__int__comparison_I1_J, axiom,
    (((^[Y5 : nat]: (^[Z4 : nat]: (Y5 = Z4))) = (^[A3 : nat]: (^[B3 : nat]: ((semiri2019852685at_int @ A3) = (semiri2019852685at_int @ B3))))))). % nat_int_comparison(1)
thf(fact_162_int__ops_I1_J, axiom,
    (((semiri2019852685at_int @ zero_zero_nat) = zero_zero_int))). % int_ops(1)
thf(fact_163_Var__apps__neq__Abs__apps, axiom,
    ((![N : nat, Ts : list_dB, R : dB, Ss : list_dB]: (~ (((foldl_dB_dB @ app @ (var @ N) @ Ts) = (foldl_dB_dB @ app @ (abs @ R) @ Ss))))))). % Var_apps_neq_Abs_apps
thf(fact_164_Abs__App__neq__Var__apps, axiom,
    ((![S : dB, T : dB, N : nat, Ss : list_dB]: (~ (((app @ (abs @ S) @ T) = (foldl_dB_dB @ app @ (var @ N) @ Ss))))))). % Abs_App_neq_Var_apps
thf(fact_165_ex__head__tail, axiom,
    ((![T : dB]: (?[Ts2 : list_dB, H : dB]: ((T = (foldl_dB_dB @ app @ H @ Ts2)) & ((?[N2 : nat]: (H = (var @ N2))) | (?[U2 : dB]: (H = (abs @ U2))))))))). % ex_head_tail
thf(fact_166_not__gr__zero, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr_zero
thf(fact_167_zero__less__imp__eq__int, axiom,
    ((![K : int]: ((ord_less_int @ zero_zero_int @ K) => (?[N2 : nat]: ((ord_less_nat @ zero_zero_nat @ N2) & (K = (semiri2019852685at_int @ N2)))))))). % zero_less_imp_eq_int
thf(fact_168_zero__reorient, axiom,
    ((![X4 : nat]: ((zero_zero_nat = X4) = (X4 = zero_zero_nat))))). % zero_reorient
thf(fact_169_zero__reorient, axiom,
    ((![X4 : int]: ((zero_zero_int = X4) = (X4 = zero_zero_int))))). % zero_reorient
thf(fact_170_less__int__code_I1_J, axiom,
    ((~ ((ord_less_int @ zero_zero_int @ zero_zero_int))))). % less_int_code(1)
thf(fact_171_int__int__eq, axiom,
    ((![M : nat, N : nat]: (((semiri2019852685at_int @ M) = (semiri2019852685at_int @ N)) = (M = N))))). % int_int_eq
thf(fact_172_gr__zeroI, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr_zeroI
thf(fact_173_not__less__zero, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less_zero
thf(fact_174_gr__implies__not__zero, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not_zero
thf(fact_175_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_176_pos__int__cases, axiom,
    ((![K : int]: ((ord_less_int @ zero_zero_int @ K) => (~ ((![N2 : nat]: ((K = (semiri2019852685at_int @ N2)) => (~ ((ord_less_nat @ zero_zero_nat @ N2))))))))))). % pos_int_cases
thf(fact_177_IT_Ocases, axiom,
    ((![A : dB]: ((it @ A) => ((![Rs2 : list_dB]: ((?[N2 : nat]: (A = (foldl_dB_dB @ app @ (var @ N2) @ Rs2))) => (~ ((listsp_dB @ it @ Rs2))))) => ((![R2 : dB]: ((A = (abs @ R2)) => (~ ((it @ R2))))) => (~ ((![R2 : dB, S3 : dB, Ss2 : list_dB]: ((A = (foldl_dB_dB @ app @ (app @ (abs @ R2) @ S3) @ Ss2)) => ((it @ (foldl_dB_dB @ app @ (subst @ R2 @ S3 @ zero_zero_nat) @ Ss2)) => (~ ((it @ S3)))))))))))))). % IT.cases
thf(fact_178_IT_Osimps, axiom,
    ((it = (^[A3 : dB]: (((?[Rs3 : list_dB]: (?[N3 : nat]: (((A3 = (foldl_dB_dB @ app @ (var @ N3) @ Rs3))) & ((listsp_dB @ it @ Rs3)))))) | ((((?[R3 : dB]: (((A3 = (abs @ R3))) & ((it @ R3))))) | ((?[R3 : dB]: (?[S2 : dB]: (?[Ss3 : list_dB]: (((A3 = (foldl_dB_dB @ app @ (app @ (abs @ R3) @ S2) @ Ss3))) & ((((it @ (foldl_dB_dB @ app @ (subst @ R3 @ S2 @ zero_zero_nat) @ Ss3))) & ((it @ S2)))))))))))))))). % IT.simps
thf(fact_179_IT_OVar, axiom,
    ((![Rs : list_dB, N : nat]: ((listsp_dB @ it @ Rs) => (it @ (foldl_dB_dB @ app @ (var @ N) @ Rs)))))). % IT.Var
thf(fact_180_less__numeral__extra_I3_J, axiom,
    ((~ ((ord_less_nat @ zero_zero_nat @ zero_zero_nat))))). % less_numeral_extra(3)
thf(fact_181_less__numeral__extra_I3_J, axiom,
    ((~ ((ord_less_int @ zero_zero_int @ zero_zero_int))))). % less_numeral_extra(3)
thf(fact_182_neg__int__cases, axiom,
    ((![K : int]: ((ord_less_int @ K @ zero_zero_int) => (~ ((![N2 : nat]: ((K = (uminus_uminus_int @ (semiri2019852685at_int @ N2))) => (~ ((ord_less_nat @ zero_zero_nat @ N2))))))))))). % neg_int_cases
thf(fact_183_verit__minus__simplify_I4_J, axiom,
    ((![B : int]: ((uminus_uminus_int @ (uminus_uminus_int @ B)) = B)))). % verit_minus_simplify(4)
thf(fact_184_add_Oinverse__neutral, axiom,
    (((uminus_uminus_int @ zero_zero_int) = zero_zero_int))). % add.inverse_neutral
thf(fact_185_neg__0__equal__iff__equal, axiom,
    ((![A : int]: ((zero_zero_int = (uminus_uminus_int @ A)) = (zero_zero_int = A))))). % neg_0_equal_iff_equal
thf(fact_186_neg__equal__0__iff__equal, axiom,
    ((![A : int]: (((uminus_uminus_int @ A) = zero_zero_int) = (A = zero_zero_int))))). % neg_equal_0_iff_equal
thf(fact_187_equal__neg__zero, axiom,
    ((![A : int]: ((A = (uminus_uminus_int @ A)) = (A = zero_zero_int))))). % equal_neg_zero
thf(fact_188_neg__equal__zero, axiom,
    ((![A : int]: (((uminus_uminus_int @ A) = A) = (A = zero_zero_int))))). % neg_equal_zero
thf(fact_189_neg__less__iff__less, axiom,
    ((![B : int, A : int]: ((ord_less_int @ (uminus_uminus_int @ B) @ (uminus_uminus_int @ A)) = (ord_less_int @ A @ B))))). % neg_less_iff_less
thf(fact_190_less__neg__neg, axiom,
    ((![A : int]: ((ord_less_int @ A @ (uminus_uminus_int @ A)) = (ord_less_int @ A @ zero_zero_int))))). % less_neg_neg
thf(fact_191_neg__less__pos, axiom,
    ((![A : int]: ((ord_less_int @ (uminus_uminus_int @ A) @ A) = (ord_less_int @ zero_zero_int @ A))))). % neg_less_pos

% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T, axiom,
    ((![P : $o]: ((P = $true) | (P = $false))))).
thf(help_If_2_1_If_001t__Nat__Onat_T, axiom,
    ((![X4 : nat, Y : nat]: ((if_nat @ $false @ X4 @ Y) = Y)))).
thf(help_If_1_1_If_001t__Nat__Onat_T, axiom,
    ((![X4 : nat, Y : nat]: ((if_nat @ $true @ X4 @ Y) = X4)))).

% Conjectures (3)
thf(conj_0, hypothesis,
    ((it @ ra))).
thf(conj_1, hypothesis,
    ((![I2 : nat, J2 : nat]: (it @ (subst @ ra @ (var @ I2) @ J2))))).
thf(conj_2, conjecture,
    ((it @ (subst @ (abs @ ra) @ (var @ ia) @ ja)))).
