% 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/FFT/prob_417__3226732_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:11:11.570

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

% Explicit typings (14)
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint, type,
    minus_minus_int : int > int > int).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat, type,
    minus_minus_nat : nat > nat > nat).
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_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_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint, type,
    ord_less_eq_int : int > int > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat, type,
    ord_less_eq_nat : nat > nat > $o).
thf(sy_v_i, type,
    i : nat).
thf(sy_v_k, type,
    k : nat).
thf(sy_v_n, type,
    n : nat).

% Relevant facts (199)
thf(fact_0_i__less, axiom,
    ((ord_less_nat @ i @ n))). % i_less
thf(fact_1_diff__less__mono2, axiom,
    ((![M : nat, N : nat, L : nat]: ((ord_less_nat @ M @ N) => ((ord_less_nat @ M @ L) => (ord_less_nat @ (minus_minus_nat @ L @ N) @ (minus_minus_nat @ L @ M))))))). % diff_less_mono2
thf(fact_2_less__imp__diff__less, axiom,
    ((![J : nat, K : nat, N : nat]: ((ord_less_nat @ J @ K) => (ord_less_nat @ (minus_minus_nat @ J @ N) @ K))))). % less_imp_diff_less
thf(fact_3_diff__strict__mono, axiom,
    ((![A : int, B : int, D : int, C : int]: ((ord_less_int @ A @ B) => ((ord_less_int @ D @ C) => (ord_less_int @ (minus_minus_int @ A @ C) @ (minus_minus_int @ B @ D))))))). % diff_strict_mono
thf(fact_4_diff__eq__diff__less, axiom,
    ((![A : int, B : int, C : int, D : int]: (((minus_minus_int @ A @ B) = (minus_minus_int @ C @ D)) => ((ord_less_int @ A @ B) = (ord_less_int @ C @ D)))))). % diff_eq_diff_less
thf(fact_5_diff__strict__left__mono, axiom,
    ((![B : int, A : int, C : int]: ((ord_less_int @ B @ A) => (ord_less_int @ (minus_minus_int @ C @ A) @ (minus_minus_int @ C @ B)))))). % diff_strict_left_mono
thf(fact_6_diff__strict__right__mono, axiom,
    ((![A : int, B : int, C : int]: ((ord_less_int @ A @ B) => (ord_less_int @ (minus_minus_int @ A @ C) @ (minus_minus_int @ B @ C)))))). % diff_strict_right_mono
thf(fact_7_diff__commute, axiom,
    ((![I : nat, J : nat, K : nat]: ((minus_minus_nat @ (minus_minus_nat @ I @ J) @ K) = (minus_minus_nat @ (minus_minus_nat @ I @ K) @ J))))). % diff_commute
thf(fact_8_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_9_less__not__refl, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_not_refl
thf(fact_10_less__not__refl2, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ N @ M) => (~ ((M = N))))))). % less_not_refl2
thf(fact_11_less__not__refl3, axiom,
    ((![S : nat, T : nat]: ((ord_less_nat @ S @ T) => (~ ((S = T))))))). % less_not_refl3
thf(fact_12_diff__right__commute, axiom,
    ((![A : nat, C : nat, B : nat]: ((minus_minus_nat @ (minus_minus_nat @ A @ C) @ B) = (minus_minus_nat @ (minus_minus_nat @ A @ B) @ C))))). % diff_right_commute
thf(fact_13_diff__right__commute, axiom,
    ((![A : int, C : int, B : int]: ((minus_minus_int @ (minus_minus_int @ A @ C) @ B) = (minus_minus_int @ (minus_minus_int @ A @ B) @ C))))). % diff_right_commute
thf(fact_14_diff__eq__diff__eq, axiom,
    ((![A : int, B : int, C : int, D : int]: (((minus_minus_int @ A @ B) = (minus_minus_int @ C @ D)) => ((A = B) = (C = D)))))). % diff_eq_diff_eq
thf(fact_15_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_16_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_17_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_18_less__irrefl__nat, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_irrefl_nat
thf(fact_19_linorder__neqE__linordered__idom, axiom,
    ((![X : int, Y : int]: ((~ ((X = Y))) => ((~ ((ord_less_int @ X @ Y))) => (ord_less_int @ Y @ X)))))). % linorder_neqE_linordered_idom
thf(fact_20_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_21_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_22_order_Ostrict__implies__not__eq, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (~ ((A = B))))))). % order.strict_implies_not_eq
thf(fact_23_order_Ostrict__implies__not__eq, axiom,
    ((![A : int, B : int]: ((ord_less_int @ A @ B) => (~ ((A = B))))))). % order.strict_implies_not_eq
thf(fact_24_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_25_not__less__iff__gr__or__eq, axiom,
    ((![X : int, Y : int]: ((~ ((ord_less_int @ X @ Y))) = (((ord_less_int @ Y @ X)) | ((X = Y))))))). % not_less_iff_gr_or_eq
thf(fact_26_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_27_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_28_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_29_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_30_ord__eq__less__subst, axiom,
    ((![A : nat, F : nat > nat, B : nat, C : nat]: ((A = (F @ B)) => ((ord_less_nat @ B @ C) => ((![X2 : nat, Y2 : nat]: ((ord_less_nat @ X2 @ Y2) => (ord_less_nat @ (F @ X2) @ (F @ Y2)))) => (ord_less_nat @ A @ (F @ C)))))))). % ord_eq_less_subst
thf(fact_31_ord__eq__less__subst, axiom,
    ((![A : int, F : nat > int, B : nat, C : nat]: ((A = (F @ B)) => ((ord_less_nat @ B @ C) => ((![X2 : nat, Y2 : nat]: ((ord_less_nat @ X2 @ Y2) => (ord_less_int @ (F @ X2) @ (F @ Y2)))) => (ord_less_int @ A @ (F @ C)))))))). % ord_eq_less_subst
thf(fact_32_ord__eq__less__subst, axiom,
    ((![A : nat, F : int > nat, B : int, C : int]: ((A = (F @ B)) => ((ord_less_int @ B @ C) => ((![X2 : int, Y2 : int]: ((ord_less_int @ X2 @ Y2) => (ord_less_nat @ (F @ X2) @ (F @ Y2)))) => (ord_less_nat @ A @ (F @ C)))))))). % ord_eq_less_subst
thf(fact_33_ord__eq__less__subst, axiom,
    ((![A : int, F : int > int, B : int, C : int]: ((A = (F @ B)) => ((ord_less_int @ B @ C) => ((![X2 : int, Y2 : int]: ((ord_less_int @ X2 @ Y2) => (ord_less_int @ (F @ X2) @ (F @ Y2)))) => (ord_less_int @ A @ (F @ C)))))))). % ord_eq_less_subst
thf(fact_34_ord__less__eq__subst, axiom,
    ((![A : nat, B : nat, F : nat > nat, C : nat]: ((ord_less_nat @ A @ B) => (((F @ B) = C) => ((![X2 : nat, Y2 : nat]: ((ord_less_nat @ X2 @ Y2) => (ord_less_nat @ (F @ X2) @ (F @ Y2)))) => (ord_less_nat @ (F @ A) @ C))))))). % ord_less_eq_subst
thf(fact_35_ord__less__eq__subst, axiom,
    ((![A : nat, B : nat, F : nat > int, C : int]: ((ord_less_nat @ A @ B) => (((F @ B) = C) => ((![X2 : nat, Y2 : nat]: ((ord_less_nat @ X2 @ Y2) => (ord_less_int @ (F @ X2) @ (F @ Y2)))) => (ord_less_int @ (F @ A) @ C))))))). % ord_less_eq_subst
thf(fact_36_ord__less__eq__subst, axiom,
    ((![A : int, B : int, F : int > nat, C : nat]: ((ord_less_int @ A @ B) => (((F @ B) = C) => ((![X2 : int, Y2 : int]: ((ord_less_int @ X2 @ Y2) => (ord_less_nat @ (F @ X2) @ (F @ Y2)))) => (ord_less_nat @ (F @ A) @ C))))))). % ord_less_eq_subst
thf(fact_37_ord__less__eq__subst, axiom,
    ((![A : int, B : int, F : int > int, C : int]: ((ord_less_int @ A @ B) => (((F @ B) = C) => ((![X2 : int, Y2 : int]: ((ord_less_int @ X2 @ Y2) => (ord_less_int @ (F @ X2) @ (F @ Y2)))) => (ord_less_int @ (F @ A) @ C))))))). % ord_less_eq_subst
thf(fact_38_order__less__subst1, axiom,
    ((![A : nat, F : nat > nat, B : nat, C : nat]: ((ord_less_nat @ A @ (F @ B)) => ((ord_less_nat @ B @ C) => ((![X2 : nat, Y2 : nat]: ((ord_less_nat @ X2 @ Y2) => (ord_less_nat @ (F @ X2) @ (F @ Y2)))) => (ord_less_nat @ A @ (F @ C)))))))). % order_less_subst1
thf(fact_39_order__less__subst1, axiom,
    ((![A : nat, F : int > nat, B : int, C : int]: ((ord_less_nat @ A @ (F @ B)) => ((ord_less_int @ B @ C) => ((![X2 : int, Y2 : int]: ((ord_less_int @ X2 @ Y2) => (ord_less_nat @ (F @ X2) @ (F @ Y2)))) => (ord_less_nat @ A @ (F @ C)))))))). % order_less_subst1
thf(fact_40_order__less__subst1, axiom,
    ((![A : int, F : nat > int, B : nat, C : nat]: ((ord_less_int @ A @ (F @ B)) => ((ord_less_nat @ B @ C) => ((![X2 : nat, Y2 : nat]: ((ord_less_nat @ X2 @ Y2) => (ord_less_int @ (F @ X2) @ (F @ Y2)))) => (ord_less_int @ A @ (F @ C)))))))). % order_less_subst1
thf(fact_41_order__less__subst1, axiom,
    ((![A : int, F : int > int, B : int, C : int]: ((ord_less_int @ A @ (F @ B)) => ((ord_less_int @ B @ C) => ((![X2 : int, Y2 : int]: ((ord_less_int @ X2 @ Y2) => (ord_less_int @ (F @ X2) @ (F @ Y2)))) => (ord_less_int @ A @ (F @ C)))))))). % order_less_subst1
thf(fact_42_order__less__subst2, axiom,
    ((![A : nat, B : nat, F : nat > nat, C : nat]: ((ord_less_nat @ A @ B) => ((ord_less_nat @ (F @ B) @ C) => ((![X2 : nat, Y2 : nat]: ((ord_less_nat @ X2 @ Y2) => (ord_less_nat @ (F @ X2) @ (F @ Y2)))) => (ord_less_nat @ (F @ A) @ C))))))). % order_less_subst2
thf(fact_43_order__less__subst2, axiom,
    ((![A : nat, B : nat, F : nat > int, C : int]: ((ord_less_nat @ A @ B) => ((ord_less_int @ (F @ B) @ C) => ((![X2 : nat, Y2 : nat]: ((ord_less_nat @ X2 @ Y2) => (ord_less_int @ (F @ X2) @ (F @ Y2)))) => (ord_less_int @ (F @ A) @ C))))))). % order_less_subst2
thf(fact_44_order__less__subst2, axiom,
    ((![A : int, B : int, F : int > nat, C : nat]: ((ord_less_int @ A @ B) => ((ord_less_nat @ (F @ B) @ C) => ((![X2 : int, Y2 : int]: ((ord_less_int @ X2 @ Y2) => (ord_less_nat @ (F @ X2) @ (F @ Y2)))) => (ord_less_nat @ (F @ A) @ C))))))). % order_less_subst2
thf(fact_45_order__less__subst2, axiom,
    ((![A : int, B : int, F : int > int, C : int]: ((ord_less_int @ A @ B) => ((ord_less_int @ (F @ B) @ C) => ((![X2 : int, Y2 : int]: ((ord_less_int @ X2 @ Y2) => (ord_less_int @ (F @ X2) @ (F @ Y2)))) => (ord_less_int @ (F @ A) @ C))))))). % order_less_subst2
thf(fact_46_lt__ex, axiom,
    ((![X : int]: (?[Y2 : int]: (ord_less_int @ Y2 @ X))))). % lt_ex
thf(fact_47_gt__ex, axiom,
    ((![X : nat]: (?[X_1 : nat]: (ord_less_nat @ X @ X_1))))). % gt_ex
thf(fact_48_gt__ex, axiom,
    ((![X : int]: (?[X_1 : int]: (ord_less_int @ X @ X_1))))). % gt_ex
thf(fact_49_neqE, axiom,
    ((![X : nat, Y : nat]: ((~ ((X = Y))) => ((~ ((ord_less_nat @ X @ Y))) => (ord_less_nat @ Y @ X)))))). % neqE
thf(fact_50_neqE, axiom,
    ((![X : int, Y : int]: ((~ ((X = Y))) => ((~ ((ord_less_int @ X @ Y))) => (ord_less_int @ Y @ X)))))). % neqE
thf(fact_51_neq__iff, axiom,
    ((![X : nat, Y : nat]: ((~ ((X = Y))) = (((ord_less_nat @ X @ Y)) | ((ord_less_nat @ Y @ X))))))). % neq_iff
thf(fact_52_neq__iff, axiom,
    ((![X : int, Y : int]: ((~ ((X = Y))) = (((ord_less_int @ X @ Y)) | ((ord_less_int @ Y @ X))))))). % neq_iff
thf(fact_53_order_Oasym, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (~ ((ord_less_nat @ B @ A))))))). % order.asym
thf(fact_54_order_Oasym, axiom,
    ((![A : int, B : int]: ((ord_less_int @ A @ B) => (~ ((ord_less_int @ B @ A))))))). % order.asym
thf(fact_55_less__imp__neq, axiom,
    ((![X : nat, Y : nat]: ((ord_less_nat @ X @ Y) => (~ ((X = Y))))))). % less_imp_neq
thf(fact_56_less__imp__neq, axiom,
    ((![X : int, Y : int]: ((ord_less_int @ X @ Y) => (~ ((X = Y))))))). % less_imp_neq
thf(fact_57_less__asym, axiom,
    ((![X : nat, Y : nat]: ((ord_less_nat @ X @ Y) => (~ ((ord_less_nat @ Y @ X))))))). % less_asym
thf(fact_58_less__asym, axiom,
    ((![X : int, Y : int]: ((ord_less_int @ X @ Y) => (~ ((ord_less_int @ Y @ X))))))). % less_asym
thf(fact_59_less__asym_H, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (~ ((ord_less_nat @ B @ A))))))). % less_asym'
thf(fact_60_less__asym_H, axiom,
    ((![A : int, B : int]: ((ord_less_int @ A @ B) => (~ ((ord_less_int @ B @ A))))))). % less_asym'
thf(fact_61_less__trans, axiom,
    ((![X : nat, Y : nat, Z : nat]: ((ord_less_nat @ X @ Y) => ((ord_less_nat @ Y @ Z) => (ord_less_nat @ X @ Z)))))). % less_trans
thf(fact_62_less__trans, axiom,
    ((![X : int, Y : int, Z : int]: ((ord_less_int @ X @ Y) => ((ord_less_int @ Y @ Z) => (ord_less_int @ X @ Z)))))). % less_trans
thf(fact_63_less__linear, axiom,
    ((![X : nat, Y : nat]: ((ord_less_nat @ X @ Y) | ((X = Y) | (ord_less_nat @ Y @ X)))))). % less_linear
thf(fact_64_less__linear, axiom,
    ((![X : int, Y : int]: ((ord_less_int @ X @ Y) | ((X = Y) | (ord_less_int @ Y @ X)))))). % less_linear
thf(fact_65_less__irrefl, axiom,
    ((![X : nat]: (~ ((ord_less_nat @ X @ X)))))). % less_irrefl
thf(fact_66_less__irrefl, axiom,
    ((![X : int]: (~ ((ord_less_int @ X @ X)))))). % less_irrefl
thf(fact_67_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_68_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_69_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_70_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_71_dual__order_Oasym, axiom,
    ((![B : nat, A : nat]: ((ord_less_nat @ B @ A) => (~ ((ord_less_nat @ A @ B))))))). % dual_order.asym
thf(fact_72_dual__order_Oasym, axiom,
    ((![B : int, A : int]: ((ord_less_int @ B @ A) => (~ ((ord_less_int @ A @ B))))))). % dual_order.asym
thf(fact_73_less__imp__not__eq, axiom,
    ((![X : nat, Y : nat]: ((ord_less_nat @ X @ Y) => (~ ((X = Y))))))). % less_imp_not_eq
thf(fact_74_less__imp__not__eq, axiom,
    ((![X : int, Y : int]: ((ord_less_int @ X @ Y) => (~ ((X = Y))))))). % less_imp_not_eq
thf(fact_75_less__not__sym, axiom,
    ((![X : nat, Y : nat]: ((ord_less_nat @ X @ Y) => (~ ((ord_less_nat @ Y @ X))))))). % less_not_sym
thf(fact_76_less__not__sym, axiom,
    ((![X : int, Y : int]: ((ord_less_int @ X @ Y) => (~ ((ord_less_int @ Y @ X))))))). % less_not_sym
thf(fact_77_less__induct, axiom,
    ((![P : nat > $o, A : nat]: ((![X2 : nat]: ((![Y3 : nat]: ((ord_less_nat @ Y3 @ X2) => (P @ Y3))) => (P @ X2))) => (P @ A))))). % less_induct
thf(fact_78_antisym__conv3, axiom,
    ((![Y : nat, X : nat]: ((~ ((ord_less_nat @ Y @ X))) => ((~ ((ord_less_nat @ X @ Y))) = (X = Y)))))). % antisym_conv3
thf(fact_79_antisym__conv3, axiom,
    ((![Y : int, X : int]: ((~ ((ord_less_int @ Y @ X))) => ((~ ((ord_less_int @ X @ Y))) = (X = Y)))))). % antisym_conv3
thf(fact_80_less__imp__not__eq2, axiom,
    ((![X : nat, Y : nat]: ((ord_less_nat @ X @ Y) => (~ ((Y = X))))))). % less_imp_not_eq2
thf(fact_81_less__imp__not__eq2, axiom,
    ((![X : int, Y : int]: ((ord_less_int @ X @ Y) => (~ ((Y = X))))))). % less_imp_not_eq2
thf(fact_82_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_83_less__imp__triv, axiom,
    ((![X : int, Y : int, P : $o]: ((ord_less_int @ X @ Y) => ((ord_less_int @ Y @ X) => P))))). % less_imp_triv
thf(fact_84_linorder__cases, axiom,
    ((![X : nat, Y : nat]: ((~ ((ord_less_nat @ X @ Y))) => ((~ ((X = Y))) => (ord_less_nat @ Y @ X)))))). % linorder_cases
thf(fact_85_linorder__cases, axiom,
    ((![X : int, Y : int]: ((~ ((ord_less_int @ X @ Y))) => ((~ ((X = Y))) => (ord_less_int @ Y @ X)))))). % linorder_cases
thf(fact_86_dual__order_Oirrefl, axiom,
    ((![A : nat]: (~ ((ord_less_nat @ A @ A)))))). % dual_order.irrefl
thf(fact_87_dual__order_Oirrefl, axiom,
    ((![A : int]: (~ ((ord_less_int @ A @ A)))))). % dual_order.irrefl
thf(fact_88_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_89_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_90_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_91_less__imp__not__less, axiom,
    ((![X : int, Y : int]: ((ord_less_int @ X @ Y) => (~ ((ord_less_int @ Y @ X))))))). % less_imp_not_less
thf(fact_92_exists__least__iff, axiom,
    (((^[P2 : nat > $o]: (?[X3 : nat]: (P2 @ X3))) = (^[P3 : nat > $o]: (?[N3 : nat]: (((P3 @ N3)) & ((![M3 : nat]: (((ord_less_nat @ M3 @ N3)) => ((~ ((P3 @ M3))))))))))))). % exists_least_iff
thf(fact_93_minf_I7_J, axiom,
    ((![T : nat]: (?[Z2 : nat]: (![X4 : nat]: ((ord_less_nat @ X4 @ Z2) => (~ ((ord_less_nat @ T @ X4))))))))). % minf(7)
thf(fact_94_minf_I7_J, axiom,
    ((![T : int]: (?[Z2 : int]: (![X4 : int]: ((ord_less_int @ X4 @ Z2) => (~ ((ord_less_int @ T @ X4))))))))). % minf(7)
thf(fact_95_minf_I5_J, axiom,
    ((![T : nat]: (?[Z2 : nat]: (![X4 : nat]: ((ord_less_nat @ X4 @ Z2) => (ord_less_nat @ X4 @ T))))))). % minf(5)
thf(fact_96_minf_I5_J, axiom,
    ((![T : int]: (?[Z2 : int]: (![X4 : int]: ((ord_less_int @ X4 @ Z2) => (ord_less_int @ X4 @ T))))))). % minf(5)
thf(fact_97_minf_I4_J, axiom,
    ((![T : nat]: (?[Z2 : nat]: (![X4 : nat]: ((ord_less_nat @ X4 @ Z2) => (~ ((X4 = T))))))))). % minf(4)
thf(fact_98_minf_I4_J, axiom,
    ((![T : int]: (?[Z2 : int]: (![X4 : int]: ((ord_less_int @ X4 @ Z2) => (~ ((X4 = T))))))))). % minf(4)
thf(fact_99_minf_I3_J, axiom,
    ((![T : nat]: (?[Z2 : nat]: (![X4 : nat]: ((ord_less_nat @ X4 @ Z2) => (~ ((X4 = T))))))))). % minf(3)
thf(fact_100_minf_I3_J, axiom,
    ((![T : int]: (?[Z2 : int]: (![X4 : int]: ((ord_less_int @ X4 @ Z2) => (~ ((X4 = T))))))))). % minf(3)
thf(fact_101_minf_I2_J, axiom,
    ((![P : nat > $o, P4 : nat > $o, Q : nat > $o, Q2 : nat > $o]: ((?[Z3 : nat]: (![X2 : nat]: ((ord_less_nat @ X2 @ Z3) => ((P @ X2) = (P4 @ X2))))) => ((?[Z3 : nat]: (![X2 : nat]: ((ord_less_nat @ X2 @ Z3) => ((Q @ X2) = (Q2 @ X2))))) => (?[Z2 : nat]: (![X4 : nat]: ((ord_less_nat @ X4 @ Z2) => ((((P @ X4)) | ((Q @ X4))) = (((P4 @ X4)) | ((Q2 @ X4)))))))))))). % minf(2)
thf(fact_102_minf_I2_J, axiom,
    ((![P : int > $o, P4 : int > $o, Q : int > $o, Q2 : int > $o]: ((?[Z3 : int]: (![X2 : int]: ((ord_less_int @ X2 @ Z3) => ((P @ X2) = (P4 @ X2))))) => ((?[Z3 : int]: (![X2 : int]: ((ord_less_int @ X2 @ Z3) => ((Q @ X2) = (Q2 @ X2))))) => (?[Z2 : int]: (![X4 : int]: ((ord_less_int @ X4 @ Z2) => ((((P @ X4)) | ((Q @ X4))) = (((P4 @ X4)) | ((Q2 @ X4)))))))))))). % minf(2)
thf(fact_103_minf_I1_J, axiom,
    ((![P : nat > $o, P4 : nat > $o, Q : nat > $o, Q2 : nat > $o]: ((?[Z3 : nat]: (![X2 : nat]: ((ord_less_nat @ X2 @ Z3) => ((P @ X2) = (P4 @ X2))))) => ((?[Z3 : nat]: (![X2 : nat]: ((ord_less_nat @ X2 @ Z3) => ((Q @ X2) = (Q2 @ X2))))) => (?[Z2 : nat]: (![X4 : nat]: ((ord_less_nat @ X4 @ Z2) => ((((P @ X4)) & ((Q @ X4))) = (((P4 @ X4)) & ((Q2 @ X4)))))))))))). % minf(1)
thf(fact_104_minf_I1_J, axiom,
    ((![P : int > $o, P4 : int > $o, Q : int > $o, Q2 : int > $o]: ((?[Z3 : int]: (![X2 : int]: ((ord_less_int @ X2 @ Z3) => ((P @ X2) = (P4 @ X2))))) => ((?[Z3 : int]: (![X2 : int]: ((ord_less_int @ X2 @ Z3) => ((Q @ X2) = (Q2 @ X2))))) => (?[Z2 : int]: (![X4 : int]: ((ord_less_int @ X4 @ Z2) => ((((P @ X4)) & ((Q @ X4))) = (((P4 @ X4)) & ((Q2 @ X4)))))))))))). % minf(1)
thf(fact_105_pinf_I1_J, axiom,
    ((![P : nat > $o, P4 : nat > $o, Q : nat > $o, Q2 : nat > $o]: ((?[Z3 : nat]: (![X2 : nat]: ((ord_less_nat @ Z3 @ X2) => ((P @ X2) = (P4 @ X2))))) => ((?[Z3 : nat]: (![X2 : nat]: ((ord_less_nat @ Z3 @ X2) => ((Q @ X2) = (Q2 @ X2))))) => (?[Z2 : nat]: (![X4 : nat]: ((ord_less_nat @ Z2 @ X4) => ((((P @ X4)) & ((Q @ X4))) = (((P4 @ X4)) & ((Q2 @ X4)))))))))))). % pinf(1)
thf(fact_106_pinf_I1_J, axiom,
    ((![P : int > $o, P4 : int > $o, Q : int > $o, Q2 : int > $o]: ((?[Z3 : int]: (![X2 : int]: ((ord_less_int @ Z3 @ X2) => ((P @ X2) = (P4 @ X2))))) => ((?[Z3 : int]: (![X2 : int]: ((ord_less_int @ Z3 @ X2) => ((Q @ X2) = (Q2 @ X2))))) => (?[Z2 : int]: (![X4 : int]: ((ord_less_int @ Z2 @ X4) => ((((P @ X4)) & ((Q @ X4))) = (((P4 @ X4)) & ((Q2 @ X4)))))))))))). % pinf(1)
thf(fact_107_pinf_I2_J, axiom,
    ((![P : nat > $o, P4 : nat > $o, Q : nat > $o, Q2 : nat > $o]: ((?[Z3 : nat]: (![X2 : nat]: ((ord_less_nat @ Z3 @ X2) => ((P @ X2) = (P4 @ X2))))) => ((?[Z3 : nat]: (![X2 : nat]: ((ord_less_nat @ Z3 @ X2) => ((Q @ X2) = (Q2 @ X2))))) => (?[Z2 : nat]: (![X4 : nat]: ((ord_less_nat @ Z2 @ X4) => ((((P @ X4)) | ((Q @ X4))) = (((P4 @ X4)) | ((Q2 @ X4)))))))))))). % pinf(2)
thf(fact_108_pinf_I2_J, axiom,
    ((![P : int > $o, P4 : int > $o, Q : int > $o, Q2 : int > $o]: ((?[Z3 : int]: (![X2 : int]: ((ord_less_int @ Z3 @ X2) => ((P @ X2) = (P4 @ X2))))) => ((?[Z3 : int]: (![X2 : int]: ((ord_less_int @ Z3 @ X2) => ((Q @ X2) = (Q2 @ X2))))) => (?[Z2 : int]: (![X4 : int]: ((ord_less_int @ Z2 @ X4) => ((((P @ X4)) | ((Q @ X4))) = (((P4 @ X4)) | ((Q2 @ X4)))))))))))). % pinf(2)
thf(fact_109_pinf_I3_J, axiom,
    ((![T : nat]: (?[Z2 : nat]: (![X4 : nat]: ((ord_less_nat @ Z2 @ X4) => (~ ((X4 = T))))))))). % pinf(3)
thf(fact_110_pinf_I3_J, axiom,
    ((![T : int]: (?[Z2 : int]: (![X4 : int]: ((ord_less_int @ Z2 @ X4) => (~ ((X4 = T))))))))). % pinf(3)
thf(fact_111_pinf_I4_J, axiom,
    ((![T : nat]: (?[Z2 : nat]: (![X4 : nat]: ((ord_less_nat @ Z2 @ X4) => (~ ((X4 = T))))))))). % pinf(4)
thf(fact_112_pinf_I4_J, axiom,
    ((![T : int]: (?[Z2 : int]: (![X4 : int]: ((ord_less_int @ Z2 @ X4) => (~ ((X4 = T))))))))). % pinf(4)
thf(fact_113_pinf_I5_J, axiom,
    ((![T : nat]: (?[Z2 : nat]: (![X4 : nat]: ((ord_less_nat @ Z2 @ X4) => (~ ((ord_less_nat @ X4 @ T))))))))). % pinf(5)
thf(fact_114_pinf_I5_J, axiom,
    ((![T : int]: (?[Z2 : int]: (![X4 : int]: ((ord_less_int @ Z2 @ X4) => (~ ((ord_less_int @ X4 @ T))))))))). % pinf(5)
thf(fact_115_pinf_I7_J, axiom,
    ((![T : nat]: (?[Z2 : nat]: (![X4 : nat]: ((ord_less_nat @ Z2 @ X4) => (ord_less_nat @ T @ X4))))))). % pinf(7)
thf(fact_116_pinf_I7_J, axiom,
    ((![T : int]: (?[Z2 : int]: (![X4 : int]: ((ord_less_int @ Z2 @ X4) => (ord_less_int @ T @ X4))))))). % pinf(7)
thf(fact_117_verit__comp__simplify1_I1_J, axiom,
    ((![A : nat]: (~ ((ord_less_nat @ A @ A)))))). % verit_comp_simplify1(1)
thf(fact_118_verit__comp__simplify1_I1_J, axiom,
    ((![A : int]: (~ ((ord_less_int @ A @ A)))))). % verit_comp_simplify1(1)
thf(fact_119_zero__less__diff, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ zero_zero_nat @ (minus_minus_nat @ N @ M)) = (ord_less_nat @ M @ N))))). % zero_less_diff
thf(fact_120_diff__gt__0__iff__gt, axiom,
    ((![A : int, B : int]: ((ord_less_int @ zero_zero_int @ (minus_minus_int @ A @ B)) = (ord_less_int @ B @ A))))). % diff_gt_0_iff_gt
thf(fact_121_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_122_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_123_of__nat__eq__iff, axiom,
    ((![M : nat, N : nat]: (((semiri2019852685at_int @ M) = (semiri2019852685at_int @ N)) = (M = N))))). % of_nat_eq_iff
thf(fact_124_not__gr__zero, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr_zero
thf(fact_125_diff__self, axiom,
    ((![A : int]: ((minus_minus_int @ A @ A) = zero_zero_int)))). % diff_self
thf(fact_126_diff__0__right, axiom,
    ((![A : int]: ((minus_minus_int @ A @ zero_zero_int) = A)))). % diff_0_right
thf(fact_127_zero__diff, axiom,
    ((![A : nat]: ((minus_minus_nat @ zero_zero_nat @ A) = zero_zero_nat)))). % zero_diff
thf(fact_128_diff__zero, axiom,
    ((![A : nat]: ((minus_minus_nat @ A @ zero_zero_nat) = A)))). % diff_zero
thf(fact_129_diff__zero, axiom,
    ((![A : int]: ((minus_minus_int @ A @ zero_zero_int) = A)))). % diff_zero
thf(fact_130_cancel__comm__monoid__add__class_Odiff__cancel, axiom,
    ((![A : nat]: ((minus_minus_nat @ A @ A) = zero_zero_nat)))). % cancel_comm_monoid_add_class.diff_cancel
thf(fact_131_cancel__comm__monoid__add__class_Odiff__cancel, axiom,
    ((![A : int]: ((minus_minus_int @ A @ A) = zero_zero_int)))). % cancel_comm_monoid_add_class.diff_cancel
thf(fact_132_neq0__conv, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ N))))). % neq0_conv
thf(fact_133_less__nat__zero__code, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_nat_zero_code
thf(fact_134_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_135_diff__0__eq__0, axiom,
    ((![N : nat]: ((minus_minus_nat @ zero_zero_nat @ N) = zero_zero_nat)))). % diff_0_eq_0
thf(fact_136_diff__self__eq__0, axiom,
    ((![M : nat]: ((minus_minus_nat @ M @ M) = zero_zero_nat)))). % diff_self_eq_0
thf(fact_137_of__nat__0, axiom,
    (((semiri1382578993at_nat @ zero_zero_nat) = zero_zero_nat))). % of_nat_0
thf(fact_138_of__nat__0, axiom,
    (((semiri2019852685at_int @ zero_zero_nat) = zero_zero_int))). % of_nat_0
thf(fact_139_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_140_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_141_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_142_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_143_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_144_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_145_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_146_gr__implies__not__zero, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not_zero
thf(fact_147_not__less__zero, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less_zero
thf(fact_148_gr__zeroI, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr_zeroI
thf(fact_149_eq__iff__diff__eq__0, axiom,
    (((^[Y4 : int]: (^[Z4 : int]: (Y4 = Z4))) = (^[A3 : int]: (^[B3 : int]: ((minus_minus_int @ A3 @ B3) = zero_zero_int)))))). % eq_iff_diff_eq_0
thf(fact_150_bot__nat__0_Oextremum__strict, axiom,
    ((![A : nat]: (~ ((ord_less_nat @ A @ zero_zero_nat)))))). % bot_nat_0.extremum_strict
thf(fact_151_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_152_gr__implies__not0, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not0
thf(fact_153_less__zeroE, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_zeroE
thf(fact_154_not__less0, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less0
thf(fact_155_not__gr0, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr0
thf(fact_156_gr0I, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr0I
thf(fact_157_minus__nat_Odiff__0, axiom,
    ((![M : nat]: ((minus_minus_nat @ M @ zero_zero_nat) = M)))). % minus_nat.diff_0
thf(fact_158_diffs0__imp__equal, axiom,
    ((![M : nat, N : nat]: (((minus_minus_nat @ M @ N) = zero_zero_nat) => (((minus_minus_nat @ N @ M) = zero_zero_nat) => (M = N)))))). % diffs0_imp_equal
thf(fact_159_int__ops_I6_J, axiom,
    ((![A : nat, B : nat]: (((ord_less_int @ (semiri2019852685at_int @ A) @ (semiri2019852685at_int @ B)) => ((semiri2019852685at_int @ (minus_minus_nat @ A @ B)) = zero_zero_int)) & ((~ ((ord_less_int @ (semiri2019852685at_int @ A) @ (semiri2019852685at_int @ B)))) => ((semiri2019852685at_int @ (minus_minus_nat @ A @ B)) = (minus_minus_int @ (semiri2019852685at_int @ A) @ (semiri2019852685at_int @ B)))))))). % int_ops(6)
thf(fact_160_zero__reorient, axiom,
    ((![X : nat]: ((zero_zero_nat = X) = (X = zero_zero_nat))))). % zero_reorient
thf(fact_161_zero__reorient, axiom,
    ((![X : int]: ((zero_zero_int = X) = (X = zero_zero_int))))). % zero_reorient
thf(fact_162_of__nat__less__0__iff, axiom,
    ((![M : nat]: (~ ((ord_less_nat @ (semiri1382578993at_nat @ M) @ zero_zero_nat)))))). % of_nat_less_0_iff
thf(fact_163_of__nat__less__0__iff, axiom,
    ((![M : nat]: (~ ((ord_less_int @ (semiri2019852685at_int @ M) @ zero_zero_int)))))). % of_nat_less_0_iff
thf(fact_164_int__ops_I1_J, axiom,
    (((semiri2019852685at_int @ zero_zero_nat) = zero_zero_int))). % int_ops(1)
thf(fact_165_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_166_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_167_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_168_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_169_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_170_less__iff__diff__less__0, axiom,
    ((ord_less_int = (^[A3 : int]: (^[B3 : int]: (ord_less_int @ (minus_minus_int @ A3 @ B3) @ zero_zero_int)))))). % less_iff_diff_less_0
thf(fact_171_diff__less, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((ord_less_nat @ zero_zero_nat @ M) => (ord_less_nat @ (minus_minus_nat @ M @ N) @ M)))))). % diff_less
thf(fact_172_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_173_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_174_less__int__code_I1_J, axiom,
    ((~ ((ord_less_int @ zero_zero_int @ zero_zero_int))))). % less_int_code(1)
thf(fact_175_minus__int__code_I1_J, axiom,
    ((![K : int]: ((minus_minus_int @ K @ zero_zero_int) = K)))). % minus_int_code(1)
thf(fact_176_int__diff__cases, axiom,
    ((![Z : int]: (~ ((![M4 : nat, N2 : nat]: (~ ((Z = (minus_minus_int @ (semiri2019852685at_int @ M4) @ (semiri2019852685at_int @ N2))))))))))). % int_diff_cases
thf(fact_177_int__int__eq, axiom,
    ((![M : nat, N : nat]: (((semiri2019852685at_int @ M) = (semiri2019852685at_int @ N)) = (M = N))))). % int_int_eq
thf(fact_178_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_179_nat__int__comparison_I1_J, axiom,
    (((^[Y4 : nat]: (^[Z4 : nat]: (Y4 = Z4))) = (^[A3 : nat]: (^[B3 : nat]: ((semiri2019852685at_int @ A3) = (semiri2019852685at_int @ B3))))))). % nat_int_comparison(1)
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_zdiff__int__split, axiom,
    ((![P : int > $o, X : nat, Y : nat]: ((P @ (semiri2019852685at_int @ (minus_minus_nat @ X @ Y))) = (((((ord_less_eq_nat @ Y @ X)) => ((P @ (minus_minus_int @ (semiri2019852685at_int @ X) @ (semiri2019852685at_int @ Y)))))) & ((((ord_less_nat @ X @ Y)) => ((P @ zero_zero_int))))))))). % zdiff_int_split
thf(fact_183_order__refl, axiom,
    ((![X : nat]: (ord_less_eq_nat @ X @ X)))). % order_refl
thf(fact_184_le__zero__eq, axiom,
    ((![N : nat]: ((ord_less_eq_nat @ N @ zero_zero_nat) = (N = zero_zero_nat))))). % le_zero_eq
thf(fact_185_le0, axiom,
    ((![N : nat]: (ord_less_eq_nat @ zero_zero_nat @ N)))). % le0
thf(fact_186_bot__nat__0_Oextremum, axiom,
    ((![A : nat]: (ord_less_eq_nat @ zero_zero_nat @ A)))). % bot_nat_0.extremum
thf(fact_187_diff__diff__cancel, axiom,
    ((![I : nat, N : nat]: ((ord_less_eq_nat @ I @ N) => ((minus_minus_nat @ N @ (minus_minus_nat @ N @ I)) = I))))). % diff_diff_cancel
thf(fact_188_diff__ge__0__iff__ge, axiom,
    ((![A : int, B : int]: ((ord_less_eq_int @ zero_zero_int @ (minus_minus_int @ A @ B)) = (ord_less_eq_int @ B @ A))))). % diff_ge_0_iff_ge
thf(fact_189_of__nat__le__iff, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_int @ (semiri2019852685at_int @ M) @ (semiri2019852685at_int @ N)) = (ord_less_eq_nat @ M @ N))))). % of_nat_le_iff
thf(fact_190_of__nat__le__iff, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_nat @ (semiri1382578993at_nat @ M) @ (semiri1382578993at_nat @ N)) = (ord_less_eq_nat @ M @ N))))). % of_nat_le_iff
thf(fact_191_diff__is__0__eq_H, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_nat @ M @ N) => ((minus_minus_nat @ M @ N) = zero_zero_nat))))). % diff_is_0_eq'
thf(fact_192_diff__is__0__eq, axiom,
    ((![M : nat, N : nat]: (((minus_minus_nat @ M @ N) = zero_zero_nat) = (ord_less_eq_nat @ M @ N))))). % diff_is_0_eq
thf(fact_193_of__nat__le__0__iff, axiom,
    ((![M : nat]: ((ord_less_eq_int @ (semiri2019852685at_int @ M) @ zero_zero_int) = (M = zero_zero_nat))))). % of_nat_le_0_iff
thf(fact_194_of__nat__le__0__iff, axiom,
    ((![M : nat]: ((ord_less_eq_nat @ (semiri1382578993at_nat @ M) @ zero_zero_nat) = (M = zero_zero_nat))))). % of_nat_le_0_iff
thf(fact_195_less__eq__nat_Osimps_I1_J, axiom,
    ((![N : nat]: (ord_less_eq_nat @ zero_zero_nat @ N)))). % less_eq_nat.simps(1)
thf(fact_196_le__0__eq, axiom,
    ((![N : nat]: ((ord_less_eq_nat @ N @ zero_zero_nat) = (N = zero_zero_nat))))). % le_0_eq
thf(fact_197_bot__nat__0_Oextremum__unique, axiom,
    ((![A : nat]: ((ord_less_eq_nat @ A @ zero_zero_nat) = (A = zero_zero_nat))))). % bot_nat_0.extremum_unique
thf(fact_198_bot__nat__0_Oextremum__uniqueI, axiom,
    ((![A : nat]: ((ord_less_eq_nat @ A @ zero_zero_nat) => (A = zero_zero_nat))))). % bot_nat_0.extremum_uniqueI

% 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,
    ((![X : nat, Y : nat]: ((if_nat @ $false @ X @ Y) = Y)))).
thf(help_If_1_1_If_001t__Nat__Onat_T, axiom,
    ((![X : nat, Y : nat]: ((if_nat @ $true @ X @ Y) = X)))).

% Conjectures (1)
thf(conj_0, conjecture,
    ((ord_less_nat @ (minus_minus_nat @ i @ k) @ n))).
