TPTP Problem File: SLH0426^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : FSM_Tests/0042_Observability/prob_00050_001697__19586310_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1334 ( 620 unt; 59 typ; 0 def)
% Number of atoms : 3276 (1169 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 9394 ( 338 ~; 69 |; 147 &;7484 @)
% ( 0 <=>;1356 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 6 avg)
% Number of types : 7 ( 6 usr)
% Number of type conns : 273 ( 273 >; 0 *; 0 +; 0 <<)
% Number of symbols : 56 ( 53 usr; 14 con; 0-3 aty)
% Number of variables : 3279 ( 208 ^;2987 !; 84 ?;3279 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 11:28:09.782
%------------------------------------------------------------------------------
% Could-be-implicit typings (6)
thf(ty_n_t__FSet__Ofset_It__Nat__Onat_J,type,
fset_nat: $tType ).
thf(ty_n_t__FSet__Ofset_It__Int__Oint_J,type,
fset_int: $tType ).
thf(ty_n_t__FSet__Ofset_Itf__a_J,type,
fset_a: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (53)
thf(sy_c_FSet_Ofcard_001tf__a,type,
fcard_a: fset_a > nat ).
thf(sy_c_FSet_Ofinsert_001t__Int__Oint,type,
finsert_int: int > fset_int > fset_int ).
thf(sy_c_FSet_Ofinsert_001t__Nat__Onat,type,
finsert_nat: nat > fset_nat > fset_nat ).
thf(sy_c_FSet_Ofinsert_001tf__a,type,
finsert_a: a > fset_a > fset_a ).
thf(sy_c_FSet_Ofmember_001t__Int__Oint,type,
fmember_int: int > fset_int > $o ).
thf(sy_c_FSet_Ofmember_001t__Nat__Onat,type,
fmember_nat: nat > fset_nat > $o ).
thf(sy_c_FSet_Ofmember_001tf__a,type,
fmember_a: a > fset_a > $o ).
thf(sy_c_FSet_Ofthe__elem_001tf__a,type,
fthe_elem_a: fset_a > a ).
thf(sy_c_FSet_Olinorder__class_OfMax_001t__Int__Oint,type,
linorder_fMax_int: fset_int > int ).
thf(sy_c_FSet_Olinorder__class_OfMax_001t__Nat__Onat,type,
linorder_fMax_nat: fset_nat > nat ).
thf(sy_c_FSet_Olinorder__class_OfMin_001t__Int__Oint,type,
linorder_fMin_int: fset_int > int ).
thf(sy_c_FSet_Olinorder__class_OfMin_001t__Nat__Onat,type,
linorder_fMin_nat: fset_nat > nat ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
abs_abs_int: int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__FSet__Ofset_Itf__a_J,type,
minus_minus_fset_a: fset_a > fset_a > fset_a ).
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_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
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__FSet__Ofset_Itf__a_J,type,
if_fset_a: $o > fset_a > fset_a > fset_a ).
thf(sy_c_If_001t__Int__Oint,type,
if_int: $o > int > int > int ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001tf__a,type,
if_a: $o > a > a > a ).
thf(sy_c_Int_Onat,type,
nat2: int > nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__FSet__Ofset_Itf__a_J,type,
sup_sup_fset_a: fset_a > fset_a > fset_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Int__Oint,type,
sup_sup_int: int > int > int ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
sup_sup_nat: nat > nat > nat ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
neg_nu5851722552734809277nc_int: int > int ).
thf(sy_c_Orderings_Obot__class_Obot_001t__FSet__Ofset_It__Int__Oint_J,type,
bot_bot_fset_int: fset_int ).
thf(sy_c_Orderings_Obot__class_Obot_001t__FSet__Ofset_It__Nat__Onat_J,type,
bot_bot_fset_nat: fset_nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__FSet__Ofset_Itf__a_J,type,
bot_bot_fset_a: fset_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Oord__class_Oless_001t__FSet__Ofset_Itf__a_J,type,
ord_less_fset_a: fset_a > fset_a > $o ).
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__FSet__Ofset_Itf__a_J,type,
ord_less_eq_fset_a: fset_a > fset_a > $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_c_Orderings_Oord__class_Omax_001t__FSet__Ofset_Itf__a_J,type,
ord_max_fset_a: fset_a > fset_a > fset_a ).
thf(sy_c_Orderings_Oord__class_Omax_001t__Int__Oint,type,
ord_max_int: int > int > int ).
thf(sy_c_Orderings_Oord__class_Omax_001t__Nat__Onat,type,
ord_max_nat: nat > nat > nat ).
thf(sy_v_A,type,
a2: fset_a ).
thf(sy_v_B,type,
b: fset_a ).
thf(sy_v_Ca____,type,
ca: fset_a ).
thf(sy_v_x____,type,
x: a ).
% Relevant facts (1265)
thf(fact_0_insert_Ohyps,axiom,
~ ( fmember_a @ x @ ca ) ).
% insert.hyps
thf(fact_1_a2,axiom,
( ( fmember_a @ x @ a2 )
& ( ord_less_eq_fset_a @ ca @ a2 ) ) ).
% a2
thf(fact_2_insert_Oprems_I2_J,axiom,
ord_less_eq_fset_a @ ( finsert_a @ x @ ca ) @ a2 ).
% insert.prems(2)
thf(fact_3__092_060open_062A_A_124_N_124_A_IC_A_124_092_060union_062_124_Afinsert_Ax_AB_J_A_061_AA_A_124_N_124_AB_A_092_060or_062_A_092_060not_062_AA_A_124_N_124_A_IC_A_124_092_060union_062_124_Afinsert_Ax_AB_J_A_124_092_060subseteq_062_124_AA_A_124_N_124_AB_092_060close_062,axiom,
( ( ( minus_minus_fset_a @ a2 @ ( sup_sup_fset_a @ ca @ ( finsert_a @ x @ b ) ) )
= ( minus_minus_fset_a @ a2 @ b ) )
| ~ ( ord_less_eq_fset_a @ ( minus_minus_fset_a @ a2 @ ( sup_sup_fset_a @ ca @ ( finsert_a @ x @ b ) ) ) @ ( minus_minus_fset_a @ a2 @ b ) ) ) ).
% \<open>A |-| (C |\<union>| finsert x B) = A |-| B \<or> \<not> A |-| (C |\<union>| finsert x B) |\<subseteq>| A |-| B\<close>
thf(fact_4_fsubsetI,axiom,
! [A: fset_a,B: fset_a] :
( ! [X: a] :
( ( fmember_a @ X @ A )
=> ( fmember_a @ X @ B ) )
=> ( ord_less_eq_fset_a @ A @ B ) ) ).
% fsubsetI
thf(fact_5_fsubset__antisym,axiom,
! [A: fset_a,B: fset_a] :
( ( ord_less_eq_fset_a @ A @ B )
=> ( ( ord_less_eq_fset_a @ B @ A )
=> ( A = B ) ) ) ).
% fsubset_antisym
thf(fact_6_order__refl,axiom,
! [X2: fset_a] : ( ord_less_eq_fset_a @ X2 @ X2 ) ).
% order_refl
thf(fact_7_order__refl,axiom,
! [X2: nat] : ( ord_less_eq_nat @ X2 @ X2 ) ).
% order_refl
thf(fact_8_order__refl,axiom,
! [X2: int] : ( ord_less_eq_int @ X2 @ X2 ) ).
% order_refl
thf(fact_9_dual__order_Orefl,axiom,
! [A2: fset_a] : ( ord_less_eq_fset_a @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_10_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_11_dual__order_Orefl,axiom,
! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_12_fin__mono,axiom,
! [A: fset_a,B: fset_a,X2: a] :
( ( ord_less_eq_fset_a @ A @ B )
=> ( ( fmember_a @ X2 @ A )
=> ( fmember_a @ X2 @ B ) ) ) ).
% fin_mono
thf(fact_13_fsubsetD,axiom,
! [A: fset_a,B: fset_a,C: a] :
( ( ord_less_eq_fset_a @ A @ B )
=> ( ( fmember_a @ C @ A )
=> ( fmember_a @ C @ B ) ) ) ).
% fsubsetD
thf(fact_14_finsert__fsubset,axiom,
! [X2: a,A: fset_a,B: fset_a] :
( ( ord_less_eq_fset_a @ ( finsert_a @ X2 @ A ) @ B )
= ( ( fmember_a @ X2 @ B )
& ( ord_less_eq_fset_a @ A @ B ) ) ) ).
% finsert_fsubset
thf(fact_15_fequalityE,axiom,
! [A: fset_a,B: fset_a] :
( ( A = B )
=> ~ ( ( ord_less_eq_fset_a @ A @ B )
=> ~ ( ord_less_eq_fset_a @ B @ A ) ) ) ).
% fequalityE
thf(fact_16_fequalityD1,axiom,
! [A: fset_a,B: fset_a] :
( ( A = B )
=> ( ord_less_eq_fset_a @ A @ B ) ) ).
% fequalityD1
thf(fact_17_fequalityD2,axiom,
! [A: fset_a,B: fset_a] :
( ( A = B )
=> ( ord_less_eq_fset_a @ B @ A ) ) ).
% fequalityD2
thf(fact_18_fsubset__refl,axiom,
! [A: fset_a] : ( ord_less_eq_fset_a @ A @ A ) ).
% fsubset_refl
thf(fact_19_fsubset__trans,axiom,
! [A: fset_a,B: fset_a,C2: fset_a] :
( ( ord_less_eq_fset_a @ A @ B )
=> ( ( ord_less_eq_fset_a @ B @ C2 )
=> ( ord_less_eq_fset_a @ A @ C2 ) ) ) ).
% fsubset_trans
thf(fact_20_finsert__absorb2,axiom,
! [X2: a,A: fset_a] :
( ( finsert_a @ X2 @ ( finsert_a @ X2 @ A ) )
= ( finsert_a @ X2 @ A ) ) ).
% finsert_absorb2
thf(fact_21_fminus__idemp,axiom,
! [A: fset_a,B: fset_a] :
( ( minus_minus_fset_a @ ( minus_minus_fset_a @ A @ B ) @ B )
= ( minus_minus_fset_a @ A @ B ) ) ).
% fminus_idemp
thf(fact_22_finsert__iff,axiom,
! [A2: a,B2: a,A: fset_a] :
( ( fmember_a @ A2 @ ( finsert_a @ B2 @ A ) )
= ( ( A2 = B2 )
| ( fmember_a @ A2 @ A ) ) ) ).
% finsert_iff
thf(fact_23_finsertCI,axiom,
! [A2: a,B: fset_a,B2: a] :
( ( ~ ( fmember_a @ A2 @ B )
=> ( A2 = B2 ) )
=> ( fmember_a @ A2 @ ( finsert_a @ B2 @ B ) ) ) ).
% finsertCI
thf(fact_24_funion__iff,axiom,
! [C: a,A: fset_a,B: fset_a] :
( ( fmember_a @ C @ ( sup_sup_fset_a @ A @ B ) )
= ( ( fmember_a @ C @ A )
| ( fmember_a @ C @ B ) ) ) ).
% funion_iff
thf(fact_25_funionCI,axiom,
! [C: a,B: fset_a,A: fset_a] :
( ( ~ ( fmember_a @ C @ B )
=> ( fmember_a @ C @ A ) )
=> ( fmember_a @ C @ ( sup_sup_fset_a @ A @ B ) ) ) ).
% funionCI
thf(fact_26_fminus__iff,axiom,
! [C: a,A: fset_a,B: fset_a] :
( ( fmember_a @ C @ ( minus_minus_fset_a @ A @ B ) )
= ( ( fmember_a @ C @ A )
& ~ ( fmember_a @ C @ B ) ) ) ).
% fminus_iff
thf(fact_27_fminusI,axiom,
! [C: a,A: fset_a,B: fset_a] :
( ( fmember_a @ C @ A )
=> ( ~ ( fmember_a @ C @ B )
=> ( fmember_a @ C @ ( minus_minus_fset_a @ A @ B ) ) ) ) ).
% fminusI
thf(fact_28_funion__finsert__left,axiom,
! [A2: a,B: fset_a,C2: fset_a] :
( ( sup_sup_fset_a @ ( finsert_a @ A2 @ B ) @ C2 )
= ( finsert_a @ A2 @ ( sup_sup_fset_a @ B @ C2 ) ) ) ).
% funion_finsert_left
thf(fact_29_funion__finsert__right,axiom,
! [A: fset_a,A2: a,B: fset_a] :
( ( sup_sup_fset_a @ A @ ( finsert_a @ A2 @ B ) )
= ( finsert_a @ A2 @ ( sup_sup_fset_a @ A @ B ) ) ) ).
% funion_finsert_right
thf(fact_30_funion__fminus__cancel,axiom,
! [A: fset_a,B: fset_a] :
( ( sup_sup_fset_a @ A @ ( minus_minus_fset_a @ B @ A ) )
= ( sup_sup_fset_a @ A @ B ) ) ).
% funion_fminus_cancel
thf(fact_31_funion__fminus__cancel2,axiom,
! [B: fset_a,A: fset_a] :
( ( sup_sup_fset_a @ ( minus_minus_fset_a @ B @ A ) @ A )
= ( sup_sup_fset_a @ B @ A ) ) ).
% funion_fminus_cancel2
thf(fact_32_finsert__fminus1,axiom,
! [X2: a,B: fset_a,A: fset_a] :
( ( fmember_a @ X2 @ B )
=> ( ( minus_minus_fset_a @ ( finsert_a @ X2 @ A ) @ B )
= ( minus_minus_fset_a @ A @ B ) ) ) ).
% finsert_fminus1
thf(fact_33__092_060open_062_092_060lbrakk_062fcard_A_IA_A_124_N_124_AB_J_A_092_060le_062_Afcard_A_IA_A_124_N_124_A_IB_A_124_092_060union_062_124_AC_J_J_059_AC_A_124_092_060subseteq_062_124_AA_092_060rbrakk_062_A_092_060Longrightarrow_062_AC_A_124_092_060subseteq_062_124_AB_092_060close_062,axiom,
( ( ord_less_eq_nat @ ( fcard_a @ ( minus_minus_fset_a @ a2 @ b ) ) @ ( fcard_a @ ( minus_minus_fset_a @ a2 @ ( sup_sup_fset_a @ b @ ca ) ) ) )
=> ( ( ord_less_eq_fset_a @ ca @ a2 )
=> ( ord_less_eq_fset_a @ ca @ b ) ) ) ).
% \<open>\<lbrakk>fcard (A |-| B) \<le> fcard (A |-| (B |\<union>| C)); C |\<subseteq>| A\<rbrakk> \<Longrightarrow> C |\<subseteq>| B\<close>
thf(fact_34_a1,axiom,
ord_less_eq_nat @ ( fcard_a @ ( minus_minus_fset_a @ a2 @ b ) ) @ ( fcard_a @ ( minus_minus_fset_a @ a2 @ ( finsert_a @ x @ ( sup_sup_fset_a @ b @ ca ) ) ) ) ).
% a1
thf(fact_35_funion__assoc,axiom,
! [A: fset_a,B: fset_a,C2: fset_a] :
( ( sup_sup_fset_a @ ( sup_sup_fset_a @ A @ B ) @ C2 )
= ( sup_sup_fset_a @ A @ ( sup_sup_fset_a @ B @ C2 ) ) ) ).
% funion_assoc
thf(fact_36_funion__absorb,axiom,
! [A: fset_a] :
( ( sup_sup_fset_a @ A @ A )
= A ) ).
% funion_absorb
thf(fact_37_funion__fminus,axiom,
! [A: fset_a,B: fset_a,C2: fset_a] :
( ( minus_minus_fset_a @ ( sup_sup_fset_a @ A @ B ) @ C2 )
= ( sup_sup_fset_a @ ( minus_minus_fset_a @ A @ C2 ) @ ( minus_minus_fset_a @ B @ C2 ) ) ) ).
% funion_fminus
thf(fact_38_funion__commute,axiom,
( sup_sup_fset_a
= ( ^ [A3: fset_a,B3: fset_a] : ( sup_sup_fset_a @ B3 @ A3 ) ) ) ).
% funion_commute
thf(fact_39_finsert__commute,axiom,
! [X2: a,Y: a,A: fset_a] :
( ( finsert_a @ X2 @ ( finsert_a @ Y @ A ) )
= ( finsert_a @ Y @ ( finsert_a @ X2 @ A ) ) ) ).
% finsert_commute
thf(fact_40_funion__left__absorb,axiom,
! [A: fset_a,B: fset_a] :
( ( sup_sup_fset_a @ A @ ( sup_sup_fset_a @ A @ B ) )
= ( sup_sup_fset_a @ A @ B ) ) ).
% funion_left_absorb
thf(fact_41_funion__left__commute,axiom,
! [A: fset_a,B: fset_a,C2: fset_a] :
( ( sup_sup_fset_a @ A @ ( sup_sup_fset_a @ B @ C2 ) )
= ( sup_sup_fset_a @ B @ ( sup_sup_fset_a @ A @ C2 ) ) ) ).
% funion_left_commute
thf(fact_42_finsert__fminus__if,axiom,
! [X2: a,B: fset_a,A: fset_a] :
( ( ( fmember_a @ X2 @ B )
=> ( ( minus_minus_fset_a @ ( finsert_a @ X2 @ A ) @ B )
= ( minus_minus_fset_a @ A @ B ) ) )
& ( ~ ( fmember_a @ X2 @ B )
=> ( ( minus_minus_fset_a @ ( finsert_a @ X2 @ A ) @ B )
= ( finsert_a @ X2 @ ( minus_minus_fset_a @ A @ B ) ) ) ) ) ).
% finsert_fminus_if
thf(fact_43_fminus__fsubset__conv,axiom,
! [A: fset_a,B: fset_a,C2: fset_a] :
( ( ord_less_eq_fset_a @ ( minus_minus_fset_a @ A @ B ) @ C2 )
= ( ord_less_eq_fset_a @ A @ ( sup_sup_fset_a @ B @ C2 ) ) ) ).
% fminus_fsubset_conv
thf(fact_44_fminus__partition,axiom,
! [A: fset_a,B: fset_a] :
( ( ord_less_eq_fset_a @ A @ B )
=> ( ( sup_sup_fset_a @ A @ ( minus_minus_fset_a @ B @ A ) )
= B ) ) ).
% fminus_partition
thf(fact_45_mk__disjoint__finsert,axiom,
! [A2: a,A: fset_a] :
( ( fmember_a @ A2 @ A )
=> ? [B4: fset_a] :
( ( A
= ( finsert_a @ A2 @ B4 ) )
& ~ ( fmember_a @ A2 @ B4 ) ) ) ).
% mk_disjoint_finsert
thf(fact_46_finsert__eq__iff,axiom,
! [A2: a,A: fset_a,B2: a,B: fset_a] :
( ~ ( fmember_a @ A2 @ A )
=> ( ~ ( fmember_a @ B2 @ B )
=> ( ( ( finsert_a @ A2 @ A )
= ( finsert_a @ B2 @ B ) )
= ( ( ( A2 = B2 )
=> ( A = B ) )
& ( ( A2 != B2 )
=> ? [C3: fset_a] :
( ( A
= ( finsert_a @ B2 @ C3 ) )
& ~ ( fmember_a @ B2 @ C3 )
& ( B
= ( finsert_a @ A2 @ C3 ) )
& ~ ( fmember_a @ A2 @ C3 ) ) ) ) ) ) ) ).
% finsert_eq_iff
thf(fact_47_finsert__absorb,axiom,
! [A2: a,A: fset_a] :
( ( fmember_a @ A2 @ A )
=> ( ( finsert_a @ A2 @ A )
= A ) ) ).
% finsert_absorb
thf(fact_48_finsert__ident,axiom,
! [X2: a,A: fset_a,B: fset_a] :
( ~ ( fmember_a @ X2 @ A )
=> ( ~ ( fmember_a @ X2 @ B )
=> ( ( ( finsert_a @ X2 @ A )
= ( finsert_a @ X2 @ B ) )
= ( A = B ) ) ) ) ).
% finsert_ident
thf(fact_49_set__finsert,axiom,
! [X2: a,A: fset_a] :
( ( fmember_a @ X2 @ A )
=> ~ ! [B4: fset_a] :
( ( A
= ( finsert_a @ X2 @ B4 ) )
=> ( fmember_a @ X2 @ B4 ) ) ) ).
% set_finsert
thf(fact_50_finsertI2,axiom,
! [A2: a,B: fset_a,B2: a] :
( ( fmember_a @ A2 @ B )
=> ( fmember_a @ A2 @ ( finsert_a @ B2 @ B ) ) ) ).
% finsertI2
thf(fact_51_finsertI1,axiom,
! [A2: a,B: fset_a] : ( fmember_a @ A2 @ ( finsert_a @ A2 @ B ) ) ).
% finsertI1
thf(fact_52_finsertE,axiom,
! [A2: a,B2: a,A: fset_a] :
( ( fmember_a @ A2 @ ( finsert_a @ B2 @ A ) )
=> ( ( A2 != B2 )
=> ( fmember_a @ A2 @ A ) ) ) ).
% finsertE
thf(fact_53_funionI2,axiom,
! [C: a,B: fset_a,A: fset_a] :
( ( fmember_a @ C @ B )
=> ( fmember_a @ C @ ( sup_sup_fset_a @ A @ B ) ) ) ).
% funionI2
thf(fact_54_funionI1,axiom,
! [C: a,A: fset_a,B: fset_a] :
( ( fmember_a @ C @ A )
=> ( fmember_a @ C @ ( sup_sup_fset_a @ A @ B ) ) ) ).
% funionI1
thf(fact_55_funionE,axiom,
! [C: a,A: fset_a,B: fset_a] :
( ( fmember_a @ C @ ( sup_sup_fset_a @ A @ B ) )
=> ( ~ ( fmember_a @ C @ A )
=> ( fmember_a @ C @ B ) ) ) ).
% funionE
thf(fact_56_fminusD2,axiom,
! [C: a,A: fset_a,B: fset_a] :
( ( fmember_a @ C @ ( minus_minus_fset_a @ A @ B ) )
=> ~ ( fmember_a @ C @ B ) ) ).
% fminusD2
thf(fact_57_fminusD1,axiom,
! [C: a,A: fset_a,B: fset_a] :
( ( fmember_a @ C @ ( minus_minus_fset_a @ A @ B ) )
=> ( fmember_a @ C @ A ) ) ).
% fminusD1
thf(fact_58_fminusE,axiom,
! [C: a,A: fset_a,B: fset_a] :
( ( fmember_a @ C @ ( minus_minus_fset_a @ A @ B ) )
=> ~ ( ( fmember_a @ C @ A )
=> ( fmember_a @ C @ B ) ) ) ).
% fminusE
thf(fact_59_fsubset__finsertI2,axiom,
! [A: fset_a,B: fset_a,B2: a] :
( ( ord_less_eq_fset_a @ A @ B )
=> ( ord_less_eq_fset_a @ A @ ( finsert_a @ B2 @ B ) ) ) ).
% fsubset_finsertI2
thf(fact_60_fsubset__finsertI,axiom,
! [B: fset_a,A2: a] : ( ord_less_eq_fset_a @ B @ ( finsert_a @ A2 @ B ) ) ).
% fsubset_finsertI
thf(fact_61_finsert__mono,axiom,
! [C2: fset_a,D: fset_a,A2: a] :
( ( ord_less_eq_fset_a @ C2 @ D )
=> ( ord_less_eq_fset_a @ ( finsert_a @ A2 @ C2 ) @ ( finsert_a @ A2 @ D ) ) ) ).
% finsert_mono
thf(fact_62_fsubset__funion__eq,axiom,
( ord_less_eq_fset_a
= ( ^ [A3: fset_a,B3: fset_a] :
( ( sup_sup_fset_a @ A3 @ B3 )
= B3 ) ) ) ).
% fsubset_funion_eq
thf(fact_63_funion__absorb2,axiom,
! [B: fset_a,A: fset_a] :
( ( ord_less_eq_fset_a @ B @ A )
=> ( ( sup_sup_fset_a @ A @ B )
= A ) ) ).
% funion_absorb2
thf(fact_64_funion__absorb1,axiom,
! [A: fset_a,B: fset_a] :
( ( ord_less_eq_fset_a @ A @ B )
=> ( ( sup_sup_fset_a @ A @ B )
= B ) ) ).
% funion_absorb1
thf(fact_65_funion__upper2,axiom,
! [B: fset_a,A: fset_a] : ( ord_less_eq_fset_a @ B @ ( sup_sup_fset_a @ A @ B ) ) ).
% funion_upper2
thf(fact_66_funion__upper1,axiom,
! [A: fset_a,B: fset_a] : ( ord_less_eq_fset_a @ A @ ( sup_sup_fset_a @ A @ B ) ) ).
% funion_upper1
thf(fact_67_funion__least,axiom,
! [A: fset_a,C2: fset_a,B: fset_a] :
( ( ord_less_eq_fset_a @ A @ C2 )
=> ( ( ord_less_eq_fset_a @ B @ C2 )
=> ( ord_less_eq_fset_a @ ( sup_sup_fset_a @ A @ B ) @ C2 ) ) ) ).
% funion_least
thf(fact_68_funion__mono,axiom,
! [A: fset_a,C2: fset_a,B: fset_a,D: fset_a] :
( ( ord_less_eq_fset_a @ A @ C2 )
=> ( ( ord_less_eq_fset_a @ B @ D )
=> ( ord_less_eq_fset_a @ ( sup_sup_fset_a @ A @ B ) @ ( sup_sup_fset_a @ C2 @ D ) ) ) ) ).
% funion_mono
thf(fact_69_fminus__fsubset,axiom,
! [A: fset_a,B: fset_a] : ( ord_less_eq_fset_a @ ( minus_minus_fset_a @ A @ B ) @ A ) ).
% fminus_fsubset
thf(fact_70_double__fminus,axiom,
! [A: fset_a,B: fset_a,C2: fset_a] :
( ( ord_less_eq_fset_a @ A @ B )
=> ( ( ord_less_eq_fset_a @ B @ C2 )
=> ( ( minus_minus_fset_a @ B @ ( minus_minus_fset_a @ C2 @ A ) )
= A ) ) ) ).
% double_fminus
thf(fact_71_fminus__mono,axiom,
! [A: fset_a,C2: fset_a,D: fset_a,B: fset_a] :
( ( ord_less_eq_fset_a @ A @ C2 )
=> ( ( ord_less_eq_fset_a @ D @ B )
=> ( ord_less_eq_fset_a @ ( minus_minus_fset_a @ A @ B ) @ ( minus_minus_fset_a @ C2 @ D ) ) ) ) ).
% fminus_mono
thf(fact_72_order__antisym__conv,axiom,
! [Y: fset_a,X2: fset_a] :
( ( ord_less_eq_fset_a @ Y @ X2 )
=> ( ( ord_less_eq_fset_a @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_73_order__antisym__conv,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ( ( ord_less_eq_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_74_order__antisym__conv,axiom,
! [Y: int,X2: int] :
( ( ord_less_eq_int @ Y @ X2 )
=> ( ( ord_less_eq_int @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% order_antisym_conv
thf(fact_75_linorder__le__cases,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_le_cases
thf(fact_76_linorder__le__cases,axiom,
! [X2: int,Y: int] :
( ~ ( ord_less_eq_int @ X2 @ Y )
=> ( ord_less_eq_int @ Y @ X2 ) ) ).
% linorder_le_cases
thf(fact_77_ord__le__eq__subst,axiom,
! [A2: fset_a,B2: fset_a,F: fset_a > fset_a,C: fset_a] :
( ( ord_less_eq_fset_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: fset_a,Y2: fset_a] :
( ( ord_less_eq_fset_a @ X @ Y2 )
=> ( ord_less_eq_fset_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_fset_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_78_ord__le__eq__subst,axiom,
! [A2: fset_a,B2: fset_a,F: fset_a > nat,C: nat] :
( ( ord_less_eq_fset_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: fset_a,Y2: fset_a] :
( ( ord_less_eq_fset_a @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_79_ord__le__eq__subst,axiom,
! [A2: fset_a,B2: fset_a,F: fset_a > int,C: int] :
( ( ord_less_eq_fset_a @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: fset_a,Y2: fset_a] :
( ( ord_less_eq_fset_a @ X @ Y2 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_80_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > fset_a,C: fset_a] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_fset_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_fset_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_81_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_82_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_83_ord__le__eq__subst,axiom,
! [A2: int,B2: int,F: int > fset_a,C: fset_a] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ord_less_eq_fset_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_fset_a @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_84_ord__le__eq__subst,axiom,
! [A2: int,B2: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_85_ord__le__eq__subst,axiom,
! [A2: int,B2: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_86_ord__eq__le__subst,axiom,
! [A2: fset_a,F: fset_a > fset_a,B2: fset_a,C: fset_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_fset_a @ B2 @ C )
=> ( ! [X: fset_a,Y2: fset_a] :
( ( ord_less_eq_fset_a @ X @ Y2 )
=> ( ord_less_eq_fset_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_fset_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_87_ord__eq__le__subst,axiom,
! [A2: nat,F: fset_a > nat,B2: fset_a,C: fset_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_fset_a @ B2 @ C )
=> ( ! [X: fset_a,Y2: fset_a] :
( ( ord_less_eq_fset_a @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_88_ord__eq__le__subst,axiom,
! [A2: int,F: fset_a > int,B2: fset_a,C: fset_a] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_fset_a @ B2 @ C )
=> ( ! [X: fset_a,Y2: fset_a] :
( ( ord_less_eq_fset_a @ X @ Y2 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_89_ord__eq__le__subst,axiom,
! [A2: fset_a,F: nat > fset_a,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_fset_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_fset_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_90_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_91_ord__eq__le__subst,axiom,
! [A2: int,F: nat > int,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_92_ord__eq__le__subst,axiom,
! [A2: fset_a,F: int > fset_a,B2: int,C: int] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ord_less_eq_fset_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_fset_a @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_93_ord__eq__le__subst,axiom,
! [A2: nat,F: int > nat,B2: int,C: int] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_94_ord__eq__le__subst,axiom,
! [A2: int,F: int > int,B2: int,C: int] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_95_linorder__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
| ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_linear
thf(fact_96_linorder__linear,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
| ( ord_less_eq_int @ Y @ X2 ) ) ).
% linorder_linear
thf(fact_97_order__eq__refl,axiom,
! [X2: fset_a,Y: fset_a] :
( ( X2 = Y )
=> ( ord_less_eq_fset_a @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_98_order__eq__refl,axiom,
! [X2: nat,Y: nat] :
( ( X2 = Y )
=> ( ord_less_eq_nat @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_99_order__eq__refl,axiom,
! [X2: int,Y: int] :
( ( X2 = Y )
=> ( ord_less_eq_int @ X2 @ Y ) ) ).
% order_eq_refl
thf(fact_100_order__subst2,axiom,
! [A2: fset_a,B2: fset_a,F: fset_a > fset_a,C: fset_a] :
( ( ord_less_eq_fset_a @ A2 @ B2 )
=> ( ( ord_less_eq_fset_a @ ( F @ B2 ) @ C )
=> ( ! [X: fset_a,Y2: fset_a] :
( ( ord_less_eq_fset_a @ X @ Y2 )
=> ( ord_less_eq_fset_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_fset_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_101_order__subst2,axiom,
! [A2: fset_a,B2: fset_a,F: fset_a > nat,C: nat] :
( ( ord_less_eq_fset_a @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X: fset_a,Y2: fset_a] :
( ( ord_less_eq_fset_a @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_102_order__subst2,axiom,
! [A2: fset_a,B2: fset_a,F: fset_a > int,C: int] :
( ( ord_less_eq_fset_a @ A2 @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
=> ( ! [X: fset_a,Y2: fset_a] :
( ( ord_less_eq_fset_a @ X @ Y2 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_103_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > fset_a,C: fset_a] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_fset_a @ ( F @ B2 ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_fset_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_fset_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_104_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_105_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_106_order__subst2,axiom,
! [A2: int,B2: int,F: int > fset_a,C: fset_a] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_fset_a @ ( F @ B2 ) @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ord_less_eq_fset_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_fset_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_107_order__subst2,axiom,
! [A2: int,B2: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_108_order__subst2,axiom,
! [A2: int,B2: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_109_order__subst1,axiom,
! [A2: fset_a,F: fset_a > fset_a,B2: fset_a,C: fset_a] :
( ( ord_less_eq_fset_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_fset_a @ B2 @ C )
=> ( ! [X: fset_a,Y2: fset_a] :
( ( ord_less_eq_fset_a @ X @ Y2 )
=> ( ord_less_eq_fset_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_fset_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_110_order__subst1,axiom,
! [A2: fset_a,F: nat > fset_a,B2: nat,C: nat] :
( ( ord_less_eq_fset_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_fset_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_fset_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_111_order__subst1,axiom,
! [A2: fset_a,F: int > fset_a,B2: int,C: int] :
( ( ord_less_eq_fset_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ord_less_eq_fset_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_fset_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_112_order__subst1,axiom,
! [A2: nat,F: fset_a > nat,B2: fset_a,C: fset_a] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_fset_a @ B2 @ C )
=> ( ! [X: fset_a,Y2: fset_a] :
( ( ord_less_eq_fset_a @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_113_order__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_114_order__subst1,axiom,
! [A2: nat,F: int > nat,B2: int,C: int] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_115_order__subst1,axiom,
! [A2: int,F: fset_a > int,B2: fset_a,C: fset_a] :
( ( ord_less_eq_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_fset_a @ B2 @ C )
=> ( ! [X: fset_a,Y2: fset_a] :
( ( ord_less_eq_fset_a @ X @ Y2 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_116_order__subst1,axiom,
! [A2: int,F: nat > int,B2: nat,C: nat] :
( ( ord_less_eq_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_117_order__subst1,axiom,
! [A2: int,F: int > int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_118_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: fset_a,Z: fset_a] : ( Y3 = Z ) )
= ( ^ [A4: fset_a,B5: fset_a] :
( ( ord_less_eq_fset_a @ A4 @ B5 )
& ( ord_less_eq_fset_a @ B5 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_119_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [A4: nat,B5: nat] :
( ( ord_less_eq_nat @ A4 @ B5 )
& ( ord_less_eq_nat @ B5 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_120_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y3: int,Z: int] : ( Y3 = Z ) )
= ( ^ [A4: int,B5: int] :
( ( ord_less_eq_int @ A4 @ B5 )
& ( ord_less_eq_int @ B5 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_121_antisym,axiom,
! [A2: fset_a,B2: fset_a] :
( ( ord_less_eq_fset_a @ A2 @ B2 )
=> ( ( ord_less_eq_fset_a @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_122_antisym,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_123_antisym,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_124_dual__order_Otrans,axiom,
! [B2: fset_a,A2: fset_a,C: fset_a] :
( ( ord_less_eq_fset_a @ B2 @ A2 )
=> ( ( ord_less_eq_fset_a @ C @ B2 )
=> ( ord_less_eq_fset_a @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_125_dual__order_Otrans,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_126_dual__order_Otrans,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ( ord_less_eq_int @ C @ B2 )
=> ( ord_less_eq_int @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_127_dual__order_Oantisym,axiom,
! [B2: fset_a,A2: fset_a] :
( ( ord_less_eq_fset_a @ B2 @ A2 )
=> ( ( ord_less_eq_fset_a @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_128_dual__order_Oantisym,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_129_dual__order_Oantisym,axiom,
! [B2: int,A2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ( ord_less_eq_int @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_130_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: fset_a,Z: fset_a] : ( Y3 = Z ) )
= ( ^ [A4: fset_a,B5: fset_a] :
( ( ord_less_eq_fset_a @ B5 @ A4 )
& ( ord_less_eq_fset_a @ A4 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_131_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [A4: nat,B5: nat] :
( ( ord_less_eq_nat @ B5 @ A4 )
& ( ord_less_eq_nat @ A4 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_132_dual__order_Oeq__iff,axiom,
( ( ^ [Y3: int,Z: int] : ( Y3 = Z ) )
= ( ^ [A4: int,B5: int] :
( ( ord_less_eq_int @ B5 @ A4 )
& ( ord_less_eq_int @ A4 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_133_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A5: nat,B6: nat] :
( ( ord_less_eq_nat @ A5 @ B6 )
=> ( P @ A5 @ B6 ) )
=> ( ! [A5: nat,B6: nat] :
( ( P @ B6 @ A5 )
=> ( P @ A5 @ B6 ) )
=> ( P @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_134_linorder__wlog,axiom,
! [P: int > int > $o,A2: int,B2: int] :
( ! [A5: int,B6: int] :
( ( ord_less_eq_int @ A5 @ B6 )
=> ( P @ A5 @ B6 ) )
=> ( ! [A5: int,B6: int] :
( ( P @ B6 @ A5 )
=> ( P @ A5 @ B6 ) )
=> ( P @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_135_order__trans,axiom,
! [X2: fset_a,Y: fset_a,Z2: fset_a] :
( ( ord_less_eq_fset_a @ X2 @ Y )
=> ( ( ord_less_eq_fset_a @ Y @ Z2 )
=> ( ord_less_eq_fset_a @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_136_order__trans,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_eq_nat @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_137_order__trans,axiom,
! [X2: int,Y: int,Z2: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ( ord_less_eq_int @ Y @ Z2 )
=> ( ord_less_eq_int @ X2 @ Z2 ) ) ) ).
% order_trans
thf(fact_138_order_Otrans,axiom,
! [A2: fset_a,B2: fset_a,C: fset_a] :
( ( ord_less_eq_fset_a @ A2 @ B2 )
=> ( ( ord_less_eq_fset_a @ B2 @ C )
=> ( ord_less_eq_fset_a @ A2 @ C ) ) ) ).
% order.trans
thf(fact_139_order_Otrans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_140_order_Otrans,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% order.trans
thf(fact_141_order__antisym,axiom,
! [X2: fset_a,Y: fset_a] :
( ( ord_less_eq_fset_a @ X2 @ Y )
=> ( ( ord_less_eq_fset_a @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_142_order__antisym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_143_order__antisym,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ( ord_less_eq_int @ Y @ X2 )
=> ( X2 = Y ) ) ) ).
% order_antisym
thf(fact_144_ord__le__eq__trans,axiom,
! [A2: fset_a,B2: fset_a,C: fset_a] :
( ( ord_less_eq_fset_a @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_fset_a @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_145_ord__le__eq__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_146_ord__le__eq__trans,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_147_ord__eq__le__trans,axiom,
! [A2: fset_a,B2: fset_a,C: fset_a] :
( ( A2 = B2 )
=> ( ( ord_less_eq_fset_a @ B2 @ C )
=> ( ord_less_eq_fset_a @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_148_ord__eq__le__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( A2 = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_149_ord__eq__le__trans,axiom,
! [A2: int,B2: int,C: int] :
( ( A2 = B2 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_eq_int @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_150_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: fset_a,Z: fset_a] : ( Y3 = Z ) )
= ( ^ [X3: fset_a,Y4: fset_a] :
( ( ord_less_eq_fset_a @ X3 @ Y4 )
& ( ord_less_eq_fset_a @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_151_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: nat,Z: nat] : ( Y3 = Z ) )
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_152_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y3: int,Z: int] : ( Y3 = Z ) )
= ( ^ [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
& ( ord_less_eq_int @ Y4 @ X3 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_153_le__cases3,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X2 @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X2 )
=> ~ ( ord_less_eq_nat @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_154_le__cases3,axiom,
! [X2: int,Y: int,Z2: int] :
( ( ( ord_less_eq_int @ X2 @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_int @ Y @ X2 )
=> ~ ( ord_less_eq_int @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq_int @ X2 @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_int @ Z2 @ Y )
=> ~ ( ord_less_eq_int @ Y @ X2 ) )
=> ( ( ( ord_less_eq_int @ Y @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq_int @ Z2 @ X2 )
=> ~ ( ord_less_eq_int @ X2 @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_155_nle__le,axiom,
! [A2: nat,B2: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B2 ) )
= ( ( ord_less_eq_nat @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_156_nle__le,axiom,
! [A2: int,B2: int] :
( ( ~ ( ord_less_eq_int @ A2 @ B2 ) )
= ( ( ord_less_eq_int @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_157_eqfelem__imp__iff,axiom,
! [X2: a,Y: a,A: fset_a] :
( ( X2 = Y )
=> ( ( fmember_a @ X2 @ A )
= ( fmember_a @ Y @ A ) ) ) ).
% eqfelem_imp_iff
thf(fact_158_if__split__fmem2,axiom,
! [A2: a,Q: $o,X2: fset_a,Y: fset_a] :
( ( fmember_a @ A2 @ ( if_fset_a @ Q @ X2 @ Y ) )
= ( ( Q
=> ( fmember_a @ A2 @ X2 ) )
& ( ~ Q
=> ( fmember_a @ A2 @ Y ) ) ) ) ).
% if_split_fmem2
thf(fact_159_if__split__fmem1,axiom,
! [Q: $o,X2: a,Y: a,B2: fset_a] :
( ( fmember_a @ ( if_a @ Q @ X2 @ Y ) @ B2 )
= ( ( Q
=> ( fmember_a @ X2 @ B2 ) )
& ( ~ Q
=> ( fmember_a @ Y @ B2 ) ) ) ) ).
% if_split_fmem1
thf(fact_160_eqfset__imp__iff,axiom,
! [A: fset_a,B: fset_a,X2: a] :
( ( A = B )
=> ( ( fmember_a @ X2 @ A )
= ( fmember_a @ X2 @ B ) ) ) ).
% eqfset_imp_iff
thf(fact_161_eq__fmem__trans,axiom,
! [A2: a,B2: a,A: fset_a] :
( ( A2 = B2 )
=> ( ( fmember_a @ B2 @ A )
=> ( fmember_a @ A2 @ A ) ) ) ).
% eq_fmem_trans
thf(fact_162_fequalityCE,axiom,
! [A: fset_a,B: fset_a,C: a] :
( ( A = B )
=> ( ( ( fmember_a @ C @ A )
=> ~ ( fmember_a @ C @ B ) )
=> ~ ( ~ ( fmember_a @ C @ A )
=> ( fmember_a @ C @ B ) ) ) ) ).
% fequalityCE
thf(fact_163_fset__eqI,axiom,
! [A: fset_a,B: fset_a] :
( ! [X: a] :
( ( fmember_a @ X @ A )
= ( fmember_a @ X @ B ) )
=> ( A = B ) ) ).
% fset_eqI
thf(fact_164_fsubset__finsert,axiom,
! [X2: a,A: fset_a,B: fset_a] :
( ~ ( fmember_a @ X2 @ A )
=> ( ( ord_less_eq_fset_a @ A @ ( finsert_a @ X2 @ B ) )
= ( ord_less_eq_fset_a @ A @ B ) ) ) ).
% fsubset_finsert
thf(fact_165_fset__eq__fsubset,axiom,
( ( ^ [Y3: fset_a,Z: fset_a] : ( Y3 = Z ) )
= ( ^ [A3: fset_a,B3: fset_a] :
( ( ord_less_eq_fset_a @ A3 @ B3 )
& ( ord_less_eq_fset_a @ B3 @ A3 ) ) ) ) ).
% fset_eq_fsubset
thf(fact_166_sup_Obounded__iff,axiom,
! [B2: fset_a,C: fset_a,A2: fset_a] :
( ( ord_less_eq_fset_a @ ( sup_sup_fset_a @ B2 @ C ) @ A2 )
= ( ( ord_less_eq_fset_a @ B2 @ A2 )
& ( ord_less_eq_fset_a @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_167_sup_Obounded__iff,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 )
= ( ( ord_less_eq_nat @ B2 @ A2 )
& ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_168_sup_Obounded__iff,axiom,
! [B2: int,C: int,A2: int] :
( ( ord_less_eq_int @ ( sup_sup_int @ B2 @ C ) @ A2 )
= ( ( ord_less_eq_int @ B2 @ A2 )
& ( ord_less_eq_int @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_169_le__sup__iff,axiom,
! [X2: fset_a,Y: fset_a,Z2: fset_a] :
( ( ord_less_eq_fset_a @ ( sup_sup_fset_a @ X2 @ Y ) @ Z2 )
= ( ( ord_less_eq_fset_a @ X2 @ Z2 )
& ( ord_less_eq_fset_a @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_170_le__sup__iff,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X2 @ Y ) @ Z2 )
= ( ( ord_less_eq_nat @ X2 @ Z2 )
& ( ord_less_eq_nat @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_171_le__sup__iff,axiom,
! [X2: int,Y: int,Z2: int] :
( ( ord_less_eq_int @ ( sup_sup_int @ X2 @ Y ) @ Z2 )
= ( ( ord_less_eq_int @ X2 @ Z2 )
& ( ord_less_eq_int @ Y @ Z2 ) ) ) ).
% le_sup_iff
thf(fact_172_insert_Oprems_I1_J,axiom,
~ ( ord_less_nat @ ( fcard_a @ ( minus_minus_fset_a @ a2 @ ( sup_sup_fset_a @ b @ ( finsert_a @ x @ ca ) ) ) ) @ ( fcard_a @ ( minus_minus_fset_a @ a2 @ b ) ) ) ).
% insert.prems(1)
thf(fact_173_insert_OIH,axiom,
( ~ ( ord_less_nat @ ( fcard_a @ ( minus_minus_fset_a @ a2 @ ( sup_sup_fset_a @ b @ ca ) ) ) @ ( fcard_a @ ( minus_minus_fset_a @ a2 @ b ) ) )
=> ( ( ord_less_eq_fset_a @ ca @ a2 )
=> ( ord_less_eq_fset_a @ ca @ b ) ) ) ).
% insert.IH
thf(fact_174_sup_Oidem,axiom,
! [A2: fset_a] :
( ( sup_sup_fset_a @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_175_sup_Oidem,axiom,
! [A2: nat] :
( ( sup_sup_nat @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_176_sup__idem,axiom,
! [X2: fset_a] :
( ( sup_sup_fset_a @ X2 @ X2 )
= X2 ) ).
% sup_idem
thf(fact_177_sup__idem,axiom,
! [X2: nat] :
( ( sup_sup_nat @ X2 @ X2 )
= X2 ) ).
% sup_idem
thf(fact_178_sup_Oleft__idem,axiom,
! [A2: fset_a,B2: fset_a] :
( ( sup_sup_fset_a @ A2 @ ( sup_sup_fset_a @ A2 @ B2 ) )
= ( sup_sup_fset_a @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_179_sup_Oleft__idem,axiom,
! [A2: nat,B2: nat] :
( ( sup_sup_nat @ A2 @ ( sup_sup_nat @ A2 @ B2 ) )
= ( sup_sup_nat @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_180_sup__left__idem,axiom,
! [X2: fset_a,Y: fset_a] :
( ( sup_sup_fset_a @ X2 @ ( sup_sup_fset_a @ X2 @ Y ) )
= ( sup_sup_fset_a @ X2 @ Y ) ) ).
% sup_left_idem
thf(fact_181_sup__left__idem,axiom,
! [X2: nat,Y: nat] :
( ( sup_sup_nat @ X2 @ ( sup_sup_nat @ X2 @ Y ) )
= ( sup_sup_nat @ X2 @ Y ) ) ).
% sup_left_idem
thf(fact_182_sup_Oright__idem,axiom,
! [A2: fset_a,B2: fset_a] :
( ( sup_sup_fset_a @ ( sup_sup_fset_a @ A2 @ B2 ) @ B2 )
= ( sup_sup_fset_a @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_183_sup_Oright__idem,axiom,
! [A2: nat,B2: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ A2 @ B2 ) @ B2 )
= ( sup_sup_nat @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_184_sup_OcoboundedI2,axiom,
! [C: fset_a,B2: fset_a,A2: fset_a] :
( ( ord_less_eq_fset_a @ C @ B2 )
=> ( ord_less_eq_fset_a @ C @ ( sup_sup_fset_a @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_185_sup_OcoboundedI2,axiom,
! [C: nat,B2: nat,A2: nat] :
( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_186_sup_OcoboundedI2,axiom,
! [C: int,B2: int,A2: int] :
( ( ord_less_eq_int @ C @ B2 )
=> ( ord_less_eq_int @ C @ ( sup_sup_int @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_187_lt__ex,axiom,
! [X2: int] :
? [Y2: int] : ( ord_less_int @ Y2 @ X2 ) ).
% lt_ex
thf(fact_188_gt__ex,axiom,
! [X2: nat] :
? [X_1: nat] : ( ord_less_nat @ X2 @ X_1 ) ).
% gt_ex
thf(fact_189_gt__ex,axiom,
! [X2: int] :
? [X_1: int] : ( ord_less_int @ X2 @ X_1 ) ).
% gt_ex
thf(fact_190_less__imp__neq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_neq
thf(fact_191_less__imp__neq,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( X2 != Y ) ) ).
% less_imp_neq
thf(fact_192_order_Oasym,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ~ ( ord_less_nat @ B2 @ A2 ) ) ).
% order.asym
thf(fact_193_order_Oasym,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ~ ( ord_less_int @ B2 @ A2 ) ) ).
% order.asym
thf(fact_194_ord__eq__less__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( A2 = B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_195_ord__eq__less__trans,axiom,
! [A2: int,B2: int,C: int] :
( ( A2 = B2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_196_ord__less__eq__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_197_ord__less__eq__trans,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_198_less__induct,axiom,
! [P: nat > $o,A2: nat] :
( ! [X: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X )
=> ( P @ Y5 ) )
=> ( P @ X ) )
=> ( P @ A2 ) ) ).
% less_induct
thf(fact_199_antisym__conv3,axiom,
! [Y: nat,X2: nat] :
( ~ ( ord_less_nat @ Y @ X2 )
=> ( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv3
thf(fact_200_antisym__conv3,axiom,
! [Y: int,X2: int] :
( ~ ( ord_less_int @ Y @ X2 )
=> ( ( ~ ( ord_less_int @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv3
thf(fact_201_linorder__cases,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ( X2 != Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_cases
thf(fact_202_linorder__cases,axiom,
! [X2: int,Y: int] :
( ~ ( ord_less_int @ X2 @ Y )
=> ( ( X2 != Y )
=> ( ord_less_int @ Y @ X2 ) ) ) ).
% linorder_cases
thf(fact_203_dual__order_Oasym,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ~ ( ord_less_nat @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_204_dual__order_Oasym,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ B2 @ A2 )
=> ~ ( ord_less_int @ A2 @ B2 ) ) ).
% dual_order.asym
thf(fact_205_dual__order_Oirrefl,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_206_dual__order_Oirrefl,axiom,
! [A2: int] :
~ ( ord_less_int @ A2 @ A2 ) ).
% dual_order.irrefl
thf(fact_207_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X4: nat] : ( P2 @ X4 ) )
= ( ^ [P3: nat > $o] :
? [N: nat] :
( ( P3 @ N )
& ! [M: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ( P3 @ M ) ) ) ) ) ).
% exists_least_iff
thf(fact_208_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A5: nat,B6: nat] :
( ( ord_less_nat @ A5 @ B6 )
=> ( P @ A5 @ B6 ) )
=> ( ! [A5: nat] : ( P @ A5 @ A5 )
=> ( ! [A5: nat,B6: nat] :
( ( P @ B6 @ A5 )
=> ( P @ A5 @ B6 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_209_linorder__less__wlog,axiom,
! [P: int > int > $o,A2: int,B2: int] :
( ! [A5: int,B6: int] :
( ( ord_less_int @ A5 @ B6 )
=> ( P @ A5 @ B6 ) )
=> ( ! [A5: int] : ( P @ A5 @ A5 )
=> ( ! [A5: int,B6: int] :
( ( P @ B6 @ A5 )
=> ( P @ A5 @ B6 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_less_wlog
thf(fact_210_order_Ostrict__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_211_order_Ostrict__trans,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans
thf(fact_212_not__less__iff__gr__or__eq,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( ( ord_less_nat @ Y @ X2 )
| ( X2 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_213_not__less__iff__gr__or__eq,axiom,
! [X2: int,Y: int] :
( ( ~ ( ord_less_int @ X2 @ Y ) )
= ( ( ord_less_int @ Y @ X2 )
| ( X2 = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_214_dual__order_Ostrict__trans,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_215_dual__order_Ostrict__trans,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( ( ord_less_int @ C @ B2 )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans
thf(fact_216_order_Ostrict__implies__not__eq,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_217_order_Ostrict__implies__not__eq,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% order.strict_implies_not_eq
thf(fact_218_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_219_dual__order_Ostrict__implies__not__eq,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_220_linorder__neqE,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
=> ( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_neqE
thf(fact_221_linorder__neqE,axiom,
! [X2: int,Y: int] :
( ( X2 != Y )
=> ( ~ ( ord_less_int @ X2 @ Y )
=> ( ord_less_int @ Y @ X2 ) ) ) ).
% linorder_neqE
thf(fact_222_order__less__asym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% order_less_asym
thf(fact_223_order__less__asym,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ~ ( ord_less_int @ Y @ X2 ) ) ).
% order_less_asym
thf(fact_224_linorder__neq__iff,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
= ( ( ord_less_nat @ X2 @ Y )
| ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_225_linorder__neq__iff,axiom,
! [X2: int,Y: int] :
( ( X2 != Y )
= ( ( ord_less_int @ X2 @ Y )
| ( ord_less_int @ Y @ X2 ) ) ) ).
% linorder_neq_iff
thf(fact_226_order__less__asym_H,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ~ ( ord_less_nat @ B2 @ A2 ) ) ).
% order_less_asym'
thf(fact_227_order__less__asym_H,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ~ ( ord_less_int @ B2 @ A2 ) ) ).
% order_less_asym'
thf(fact_228_order__less__trans,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_229_order__less__trans,axiom,
! [X2: int,Y: int,Z2: int] :
( ( ord_less_int @ X2 @ Y )
=> ( ( ord_less_int @ Y @ Z2 )
=> ( ord_less_int @ X2 @ Z2 ) ) ) ).
% order_less_trans
thf(fact_230_ord__eq__less__subst,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_231_ord__eq__less__subst,axiom,
! [A2: int,F: nat > int,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_232_ord__eq__less__subst,axiom,
! [A2: nat,F: int > nat,B2: int,C: int] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_233_ord__eq__less__subst,axiom,
! [A2: int,F: int > int,B2: int,C: int] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_234_ord__less__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_235_ord__less__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_236_ord__less__eq__subst,axiom,
! [A2: int,B2: int,F: int > nat,C: nat] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_237_ord__less__eq__subst,axiom,
! [A2: int,B2: int,F: int > int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_238_order__less__irrefl,axiom,
! [X2: nat] :
~ ( ord_less_nat @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_239_order__less__irrefl,axiom,
! [X2: int] :
~ ( ord_less_int @ X2 @ X2 ) ).
% order_less_irrefl
thf(fact_240_order__less__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_241_order__less__subst1,axiom,
! [A2: nat,F: int > nat,B2: int,C: int] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_242_order__less__subst1,axiom,
! [A2: int,F: nat > int,B2: nat,C: nat] :
( ( ord_less_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_243_order__less__subst1,axiom,
! [A2: int,F: int > int,B2: int,C: int] :
( ( ord_less_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_244_order__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_245_order__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_246_order__less__subst2,axiom,
! [A2: int,B2: int,F: int > nat,C: nat] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_247_order__less__subst2,axiom,
! [A2: int,B2: int,F: int > int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_248_order__less__not__sym,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% order_less_not_sym
thf(fact_249_order__less__not__sym,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ~ ( ord_less_int @ Y @ X2 ) ) ).
% order_less_not_sym
thf(fact_250_order__less__imp__triv,axiom,
! [X2: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_251_order__less__imp__triv,axiom,
! [X2: int,Y: int,P: $o] :
( ( ord_less_int @ X2 @ Y )
=> ( ( ord_less_int @ Y @ X2 )
=> P ) ) ).
% order_less_imp_triv
thf(fact_252_linorder__less__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
| ( X2 = Y )
| ( ord_less_nat @ Y @ X2 ) ) ).
% linorder_less_linear
thf(fact_253_linorder__less__linear,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
| ( X2 = Y )
| ( ord_less_int @ Y @ X2 ) ) ).
% linorder_less_linear
thf(fact_254_order__less__imp__not__eq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( X2 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_255_order__less__imp__not__eq,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( X2 != Y ) ) ).
% order_less_imp_not_eq
thf(fact_256_order__less__imp__not__eq2,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( Y != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_257_order__less__imp__not__eq2,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( Y != X2 ) ) ).
% order_less_imp_not_eq2
thf(fact_258_order__less__imp__not__less,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ~ ( ord_less_nat @ Y @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_259_order__less__imp__not__less,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ~ ( ord_less_int @ Y @ X2 ) ) ).
% order_less_imp_not_less
thf(fact_260_diff__fcard__le__fcard__fminus,axiom,
! [A: fset_a,B: fset_a] : ( ord_less_eq_nat @ ( minus_minus_nat @ ( fcard_a @ A ) @ ( fcard_a @ B ) ) @ ( fcard_a @ ( minus_minus_fset_a @ A @ B ) ) ) ).
% diff_fcard_le_fcard_fminus
thf(fact_261_sup_Ostrict__coboundedI2,axiom,
! [C: fset_a,B2: fset_a,A2: fset_a] :
( ( ord_less_fset_a @ C @ B2 )
=> ( ord_less_fset_a @ C @ ( sup_sup_fset_a @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_262_sup_Ostrict__coboundedI2,axiom,
! [C: nat,B2: nat,A2: nat] :
( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_263_sup_Ostrict__coboundedI2,axiom,
! [C: int,B2: int,A2: int] :
( ( ord_less_int @ C @ B2 )
=> ( ord_less_int @ C @ ( sup_sup_int @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI2
thf(fact_264_sup_Ostrict__coboundedI1,axiom,
! [C: fset_a,A2: fset_a,B2: fset_a] :
( ( ord_less_fset_a @ C @ A2 )
=> ( ord_less_fset_a @ C @ ( sup_sup_fset_a @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_265_sup_Ostrict__coboundedI1,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_nat @ C @ A2 )
=> ( ord_less_nat @ C @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_266_sup_Ostrict__coboundedI1,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_int @ C @ A2 )
=> ( ord_less_int @ C @ ( sup_sup_int @ A2 @ B2 ) ) ) ).
% sup.strict_coboundedI1
thf(fact_267_sup_Ostrict__order__iff,axiom,
( ord_less_fset_a
= ( ^ [B5: fset_a,A4: fset_a] :
( ( A4
= ( sup_sup_fset_a @ A4 @ B5 ) )
& ( A4 != B5 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_268_sup_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [B5: nat,A4: nat] :
( ( A4
= ( sup_sup_nat @ A4 @ B5 ) )
& ( A4 != B5 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_269_sup_Ostrict__order__iff,axiom,
( ord_less_int
= ( ^ [B5: int,A4: int] :
( ( A4
= ( sup_sup_int @ A4 @ B5 ) )
& ( A4 != B5 ) ) ) ) ).
% sup.strict_order_iff
thf(fact_270_sup_Ostrict__boundedE,axiom,
! [B2: fset_a,C: fset_a,A2: fset_a] :
( ( ord_less_fset_a @ ( sup_sup_fset_a @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_fset_a @ B2 @ A2 )
=> ~ ( ord_less_fset_a @ C @ A2 ) ) ) ).
% sup.strict_boundedE
thf(fact_271_sup_Ostrict__boundedE,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( ord_less_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_nat @ B2 @ A2 )
=> ~ ( ord_less_nat @ C @ A2 ) ) ) ).
% sup.strict_boundedE
thf(fact_272_sup_Ostrict__boundedE,axiom,
! [B2: int,C: int,A2: int] :
( ( ord_less_int @ ( sup_sup_int @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_int @ B2 @ A2 )
=> ~ ( ord_less_int @ C @ A2 ) ) ) ).
% sup.strict_boundedE
thf(fact_273_sup_Oabsorb4,axiom,
! [A2: fset_a,B2: fset_a] :
( ( ord_less_fset_a @ A2 @ B2 )
=> ( ( sup_sup_fset_a @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb4
thf(fact_274_sup_Oabsorb4,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb4
thf(fact_275_sup_Oabsorb4,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( sup_sup_int @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb4
thf(fact_276_sup_Oabsorb3,axiom,
! [B2: fset_a,A2: fset_a] :
( ( ord_less_fset_a @ B2 @ A2 )
=> ( ( sup_sup_fset_a @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb3
thf(fact_277_sup_Oabsorb3,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb3
thf(fact_278_sup_Oabsorb3,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( ( sup_sup_int @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb3
thf(fact_279_less__supI2,axiom,
! [X2: fset_a,B2: fset_a,A2: fset_a] :
( ( ord_less_fset_a @ X2 @ B2 )
=> ( ord_less_fset_a @ X2 @ ( sup_sup_fset_a @ A2 @ B2 ) ) ) ).
% less_supI2
thf(fact_280_less__supI2,axiom,
! [X2: nat,B2: nat,A2: nat] :
( ( ord_less_nat @ X2 @ B2 )
=> ( ord_less_nat @ X2 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% less_supI2
thf(fact_281_less__supI2,axiom,
! [X2: int,B2: int,A2: int] :
( ( ord_less_int @ X2 @ B2 )
=> ( ord_less_int @ X2 @ ( sup_sup_int @ A2 @ B2 ) ) ) ).
% less_supI2
thf(fact_282_less__supI1,axiom,
! [X2: fset_a,A2: fset_a,B2: fset_a] :
( ( ord_less_fset_a @ X2 @ A2 )
=> ( ord_less_fset_a @ X2 @ ( sup_sup_fset_a @ A2 @ B2 ) ) ) ).
% less_supI1
thf(fact_283_less__supI1,axiom,
! [X2: nat,A2: nat,B2: nat] :
( ( ord_less_nat @ X2 @ A2 )
=> ( ord_less_nat @ X2 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% less_supI1
thf(fact_284_less__supI1,axiom,
! [X2: int,A2: int,B2: int] :
( ( ord_less_int @ X2 @ A2 )
=> ( ord_less_int @ X2 @ ( sup_sup_int @ A2 @ B2 ) ) ) ).
% less_supI1
thf(fact_285_fcard__funion__fsubset,axiom,
! [B: fset_a,A: fset_a] :
( ( ord_less_eq_fset_a @ B @ A )
=> ( ( fcard_a @ ( minus_minus_fset_a @ A @ B ) )
= ( minus_minus_nat @ ( fcard_a @ A ) @ ( fcard_a @ B ) ) ) ) ).
% fcard_funion_fsubset
thf(fact_286_leD,axiom,
! [Y: fset_a,X2: fset_a] :
( ( ord_less_eq_fset_a @ Y @ X2 )
=> ~ ( ord_less_fset_a @ X2 @ Y ) ) ).
% leD
thf(fact_287_leD,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ~ ( ord_less_nat @ X2 @ Y ) ) ).
% leD
thf(fact_288_leD,axiom,
! [Y: int,X2: int] :
( ( ord_less_eq_int @ Y @ X2 )
=> ~ ( ord_less_int @ X2 @ Y ) ) ).
% leD
thf(fact_289_leI,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ Y @ X2 ) ) ).
% leI
thf(fact_290_leI,axiom,
! [X2: int,Y: int] :
( ~ ( ord_less_int @ X2 @ Y )
=> ( ord_less_eq_int @ Y @ X2 ) ) ).
% leI
thf(fact_291_nless__le,axiom,
! [A2: fset_a,B2: fset_a] :
( ( ~ ( ord_less_fset_a @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_fset_a @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_292_nless__le,axiom,
! [A2: nat,B2: nat] :
( ( ~ ( ord_less_nat @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_293_nless__le,axiom,
! [A2: int,B2: int] :
( ( ~ ( ord_less_int @ A2 @ B2 ) )
= ( ~ ( ord_less_eq_int @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% nless_le
thf(fact_294_antisym__conv1,axiom,
! [X2: fset_a,Y: fset_a] :
( ~ ( ord_less_fset_a @ X2 @ Y )
=> ( ( ord_less_eq_fset_a @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_295_antisym__conv1,axiom,
! [X2: nat,Y: nat] :
( ~ ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_296_antisym__conv1,axiom,
! [X2: int,Y: int] :
( ~ ( ord_less_int @ X2 @ Y )
=> ( ( ord_less_eq_int @ X2 @ Y )
= ( X2 = Y ) ) ) ).
% antisym_conv1
thf(fact_297_antisym__conv2,axiom,
! [X2: fset_a,Y: fset_a] :
( ( ord_less_eq_fset_a @ X2 @ Y )
=> ( ( ~ ( ord_less_fset_a @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_298_antisym__conv2,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_299_antisym__conv2,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ( ~ ( ord_less_int @ X2 @ Y ) )
= ( X2 = Y ) ) ) ).
% antisym_conv2
thf(fact_300_less__le__not__le,axiom,
( ord_less_fset_a
= ( ^ [X3: fset_a,Y4: fset_a] :
( ( ord_less_eq_fset_a @ X3 @ Y4 )
& ~ ( ord_less_eq_fset_a @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_301_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ~ ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_302_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
& ~ ( ord_less_eq_int @ Y4 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_303_not__le__imp__less,axiom,
! [Y: nat,X2: nat] :
( ~ ( ord_less_eq_nat @ Y @ X2 )
=> ( ord_less_nat @ X2 @ Y ) ) ).
% not_le_imp_less
thf(fact_304_not__le__imp__less,axiom,
! [Y: int,X2: int] :
( ~ ( ord_less_eq_int @ Y @ X2 )
=> ( ord_less_int @ X2 @ Y ) ) ).
% not_le_imp_less
thf(fact_305_order_Oorder__iff__strict,axiom,
( ord_less_eq_fset_a
= ( ^ [A4: fset_a,B5: fset_a] :
( ( ord_less_fset_a @ A4 @ B5 )
| ( A4 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_306_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B5: nat] :
( ( ord_less_nat @ A4 @ B5 )
| ( A4 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_307_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B5: int] :
( ( ord_less_int @ A4 @ B5 )
| ( A4 = B5 ) ) ) ) ).
% order.order_iff_strict
thf(fact_308_order_Ostrict__iff__order,axiom,
( ord_less_fset_a
= ( ^ [A4: fset_a,B5: fset_a] :
( ( ord_less_eq_fset_a @ A4 @ B5 )
& ( A4 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_309_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A4: nat,B5: nat] :
( ( ord_less_eq_nat @ A4 @ B5 )
& ( A4 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_310_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A4: int,B5: int] :
( ( ord_less_eq_int @ A4 @ B5 )
& ( A4 != B5 ) ) ) ) ).
% order.strict_iff_order
thf(fact_311_order_Ostrict__trans1,axiom,
! [A2: fset_a,B2: fset_a,C: fset_a] :
( ( ord_less_eq_fset_a @ A2 @ B2 )
=> ( ( ord_less_fset_a @ B2 @ C )
=> ( ord_less_fset_a @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_312_order_Ostrict__trans1,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_313_order_Ostrict__trans1,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans1
thf(fact_314_order_Ostrict__trans2,axiom,
! [A2: fset_a,B2: fset_a,C: fset_a] :
( ( ord_less_fset_a @ A2 @ B2 )
=> ( ( ord_less_eq_fset_a @ B2 @ C )
=> ( ord_less_fset_a @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_315_order_Ostrict__trans2,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_nat @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_316_order_Ostrict__trans2,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ord_less_int @ A2 @ C ) ) ) ).
% order.strict_trans2
thf(fact_317_order_Ostrict__iff__not,axiom,
( ord_less_fset_a
= ( ^ [A4: fset_a,B5: fset_a] :
( ( ord_less_eq_fset_a @ A4 @ B5 )
& ~ ( ord_less_eq_fset_a @ B5 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_318_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A4: nat,B5: nat] :
( ( ord_less_eq_nat @ A4 @ B5 )
& ~ ( ord_less_eq_nat @ B5 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_319_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A4: int,B5: int] :
( ( ord_less_eq_int @ A4 @ B5 )
& ~ ( ord_less_eq_int @ B5 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_320_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_fset_a
= ( ^ [B5: fset_a,A4: fset_a] :
( ( ord_less_fset_a @ B5 @ A4 )
| ( A4 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_321_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A4: nat] :
( ( ord_less_nat @ B5 @ A4 )
| ( A4 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_322_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B5: int,A4: int] :
( ( ord_less_int @ B5 @ A4 )
| ( A4 = B5 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_323_dual__order_Ostrict__iff__order,axiom,
( ord_less_fset_a
= ( ^ [B5: fset_a,A4: fset_a] :
( ( ord_less_eq_fset_a @ B5 @ A4 )
& ( A4 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_324_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B5: nat,A4: nat] :
( ( ord_less_eq_nat @ B5 @ A4 )
& ( A4 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_325_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B5: int,A4: int] :
( ( ord_less_eq_int @ B5 @ A4 )
& ( A4 != B5 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_326_dual__order_Ostrict__trans1,axiom,
! [B2: fset_a,A2: fset_a,C: fset_a] :
( ( ord_less_eq_fset_a @ B2 @ A2 )
=> ( ( ord_less_fset_a @ C @ B2 )
=> ( ord_less_fset_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_327_dual__order_Ostrict__trans1,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_328_dual__order_Ostrict__trans1,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ( ord_less_int @ C @ B2 )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans1
thf(fact_329_dual__order_Ostrict__trans2,axiom,
! [B2: fset_a,A2: fset_a,C: fset_a] :
( ( ord_less_fset_a @ B2 @ A2 )
=> ( ( ord_less_eq_fset_a @ C @ B2 )
=> ( ord_less_fset_a @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_330_dual__order_Ostrict__trans2,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_nat @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_331_dual__order_Ostrict__trans2,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( ( ord_less_eq_int @ C @ B2 )
=> ( ord_less_int @ C @ A2 ) ) ) ).
% dual_order.strict_trans2
thf(fact_332_dual__order_Ostrict__iff__not,axiom,
( ord_less_fset_a
= ( ^ [B5: fset_a,A4: fset_a] :
( ( ord_less_eq_fset_a @ B5 @ A4 )
& ~ ( ord_less_eq_fset_a @ A4 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_333_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B5: nat,A4: nat] :
( ( ord_less_eq_nat @ B5 @ A4 )
& ~ ( ord_less_eq_nat @ A4 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_334_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B5: int,A4: int] :
( ( ord_less_eq_int @ B5 @ A4 )
& ~ ( ord_less_eq_int @ A4 @ B5 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_335_order_Ostrict__implies__order,axiom,
! [A2: fset_a,B2: fset_a] :
( ( ord_less_fset_a @ A2 @ B2 )
=> ( ord_less_eq_fset_a @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_336_order_Ostrict__implies__order,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_337_order_Ostrict__implies__order,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ord_less_eq_int @ A2 @ B2 ) ) ).
% order.strict_implies_order
thf(fact_338_dual__order_Ostrict__implies__order,axiom,
! [B2: fset_a,A2: fset_a] :
( ( ord_less_fset_a @ B2 @ A2 )
=> ( ord_less_eq_fset_a @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_339_dual__order_Ostrict__implies__order,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_340_dual__order_Ostrict__implies__order,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( ord_less_eq_int @ B2 @ A2 ) ) ).
% dual_order.strict_implies_order
thf(fact_341_order__le__less,axiom,
( ord_less_eq_fset_a
= ( ^ [X3: fset_a,Y4: fset_a] :
( ( ord_less_fset_a @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_342_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_nat @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_343_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X3: int,Y4: int] :
( ( ord_less_int @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% order_le_less
thf(fact_344_order__less__le,axiom,
( ord_less_fset_a
= ( ^ [X3: fset_a,Y4: fset_a] :
( ( ord_less_eq_fset_a @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_345_order__less__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y4: nat] :
( ( ord_less_eq_nat @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_346_order__less__le,axiom,
( ord_less_int
= ( ^ [X3: int,Y4: int] :
( ( ord_less_eq_int @ X3 @ Y4 )
& ( X3 != Y4 ) ) ) ) ).
% order_less_le
thf(fact_347_linorder__not__le,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X2 @ Y ) )
= ( ord_less_nat @ Y @ X2 ) ) ).
% linorder_not_le
thf(fact_348_linorder__not__le,axiom,
! [X2: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X2 @ Y ) )
= ( ord_less_int @ Y @ X2 ) ) ).
% linorder_not_le
thf(fact_349_linorder__not__less,axiom,
! [X2: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X2 @ Y ) )
= ( ord_less_eq_nat @ Y @ X2 ) ) ).
% linorder_not_less
thf(fact_350_linorder__not__less,axiom,
! [X2: int,Y: int] :
( ( ~ ( ord_less_int @ X2 @ Y ) )
= ( ord_less_eq_int @ Y @ X2 ) ) ).
% linorder_not_less
thf(fact_351_order__less__imp__le,axiom,
! [X2: fset_a,Y: fset_a] :
( ( ord_less_fset_a @ X2 @ Y )
=> ( ord_less_eq_fset_a @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_352_order__less__imp__le,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_353_order__less__imp__le,axiom,
! [X2: int,Y: int] :
( ( ord_less_int @ X2 @ Y )
=> ( ord_less_eq_int @ X2 @ Y ) ) ).
% order_less_imp_le
thf(fact_354_order__le__neq__trans,axiom,
! [A2: fset_a,B2: fset_a] :
( ( ord_less_eq_fset_a @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_fset_a @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_355_order__le__neq__trans,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_nat @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_356_order__le__neq__trans,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less_int @ A2 @ B2 ) ) ) ).
% order_le_neq_trans
thf(fact_357_order__neq__le__trans,axiom,
! [A2: fset_a,B2: fset_a] :
( ( A2 != B2 )
=> ( ( ord_less_eq_fset_a @ A2 @ B2 )
=> ( ord_less_fset_a @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_358_order__neq__le__trans,axiom,
! [A2: nat,B2: nat] :
( ( A2 != B2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_nat @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_359_order__neq__le__trans,axiom,
! [A2: int,B2: int] :
( ( A2 != B2 )
=> ( ( ord_less_eq_int @ A2 @ B2 )
=> ( ord_less_int @ A2 @ B2 ) ) ) ).
% order_neq_le_trans
thf(fact_360_order__le__less__trans,axiom,
! [X2: fset_a,Y: fset_a,Z2: fset_a] :
( ( ord_less_eq_fset_a @ X2 @ Y )
=> ( ( ord_less_fset_a @ Y @ Z2 )
=> ( ord_less_fset_a @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_361_order__le__less__trans,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_362_order__le__less__trans,axiom,
! [X2: int,Y: int,Z2: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ( ord_less_int @ Y @ Z2 )
=> ( ord_less_int @ X2 @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_363_order__less__le__trans,axiom,
! [X2: fset_a,Y: fset_a,Z2: fset_a] :
( ( ord_less_fset_a @ X2 @ Y )
=> ( ( ord_less_eq_fset_a @ Y @ Z2 )
=> ( ord_less_fset_a @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_364_order__less__le__trans,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_nat @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_365_order__less__le__trans,axiom,
! [X2: int,Y: int,Z2: int] :
( ( ord_less_int @ X2 @ Y )
=> ( ( ord_less_eq_int @ Y @ Z2 )
=> ( ord_less_int @ X2 @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_366_order__le__less__subst1,axiom,
! [A2: fset_a,F: nat > fset_a,B2: nat,C: nat] :
( ( ord_less_eq_fset_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_fset_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_fset_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_367_order__le__less__subst1,axiom,
! [A2: fset_a,F: int > fset_a,B2: int,C: int] :
( ( ord_less_eq_fset_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( ord_less_fset_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_fset_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_368_order__le__less__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_369_order__le__less__subst1,axiom,
! [A2: nat,F: int > nat,B2: int,C: int] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_370_order__le__less__subst1,axiom,
! [A2: int,F: nat > int,B2: nat,C: nat] :
( ( ord_less_eq_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_371_order__le__less__subst1,axiom,
! [A2: int,F: int > int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_int @ B2 @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_372_order__le__less__subst2,axiom,
! [A2: fset_a,B2: fset_a,F: fset_a > fset_a,C: fset_a] :
( ( ord_less_eq_fset_a @ A2 @ B2 )
=> ( ( ord_less_fset_a @ ( F @ B2 ) @ C )
=> ( ! [X: fset_a,Y2: fset_a] :
( ( ord_less_eq_fset_a @ X @ Y2 )
=> ( ord_less_eq_fset_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_fset_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_373_order__le__less__subst2,axiom,
! [A2: fset_a,B2: fset_a,F: fset_a > nat,C: nat] :
( ( ord_less_eq_fset_a @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X: fset_a,Y2: fset_a] :
( ( ord_less_eq_fset_a @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_374_order__le__less__subst2,axiom,
! [A2: fset_a,B2: fset_a,F: fset_a > int,C: int] :
( ( ord_less_eq_fset_a @ A2 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X: fset_a,Y2: fset_a] :
( ( ord_less_eq_fset_a @ X @ Y2 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_375_order__le__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > fset_a,C: fset_a] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_fset_a @ ( F @ B2 ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_fset_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_fset_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_376_order__le__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_377_order__le__less__subst2,axiom,
! [A2: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_378_order__le__less__subst2,axiom,
! [A2: int,B2: int,F: int > fset_a,C: fset_a] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_fset_a @ ( F @ B2 ) @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ord_less_eq_fset_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_fset_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_379_order__le__less__subst2,axiom,
! [A2: int,B2: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_nat @ ( F @ B2 ) @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_380_order__le__less__subst2,axiom,
! [A2: int,B2: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_int @ ( F @ B2 ) @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_381_order__less__le__subst1,axiom,
! [A2: fset_a,F: fset_a > fset_a,B2: fset_a,C: fset_a] :
( ( ord_less_fset_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_fset_a @ B2 @ C )
=> ( ! [X: fset_a,Y2: fset_a] :
( ( ord_less_eq_fset_a @ X @ Y2 )
=> ( ord_less_eq_fset_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_fset_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_382_order__less__le__subst1,axiom,
! [A2: nat,F: fset_a > nat,B2: fset_a,C: fset_a] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_fset_a @ B2 @ C )
=> ( ! [X: fset_a,Y2: fset_a] :
( ( ord_less_eq_fset_a @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_383_order__less__le__subst1,axiom,
! [A2: int,F: fset_a > int,B2: fset_a,C: fset_a] :
( ( ord_less_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_fset_a @ B2 @ C )
=> ( ! [X: fset_a,Y2: fset_a] :
( ( ord_less_eq_fset_a @ X @ Y2 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_384_order__less__le__subst1,axiom,
! [A2: fset_a,F: nat > fset_a,B2: nat,C: nat] :
( ( ord_less_fset_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_fset_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_fset_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_385_order__less__le__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_386_order__less__le__subst1,axiom,
! [A2: int,F: nat > int,B2: nat,C: nat] :
( ( ord_less_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_387_order__less__le__subst1,axiom,
! [A2: fset_a,F: int > fset_a,B2: int,C: int] :
( ( ord_less_fset_a @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ord_less_eq_fset_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_fset_a @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_388_order__less__le__subst1,axiom,
! [A2: nat,F: int > nat,B2: int,C: int] :
( ( ord_less_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_389_order__less__le__subst1,axiom,
! [A2: int,F: int > int,B2: int,C: int] :
( ( ord_less_int @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_int @ B2 @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_eq_int @ X @ Y2 )
=> ( ord_less_eq_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A2 @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_390_order__less__le__subst2,axiom,
! [A2: nat,B2: nat,F: nat > fset_a,C: fset_a] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_fset_a @ ( F @ B2 ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_fset_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_fset_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_391_order__less__le__subst2,axiom,
! [A2: int,B2: int,F: int > fset_a,C: fset_a] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_eq_fset_a @ ( F @ B2 ) @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( ord_less_fset_a @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_fset_a @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_392_order__less__le__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_393_order__less__le__subst2,axiom,
! [A2: int,B2: int,F: int > nat,C: nat] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( ord_less_nat @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_394_order__less__le__subst2,axiom,
! [A2: nat,B2: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
=> ( ! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_395_order__less__le__subst2,axiom,
! [A2: int,B2: int,F: int > int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ ( F @ B2 ) @ C )
=> ( ! [X: int,Y2: int] :
( ( ord_less_int @ X @ Y2 )
=> ( ord_less_int @ ( F @ X ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A2 ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_396_linorder__le__less__linear,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
| ( ord_less_nat @ Y @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_397_linorder__le__less__linear,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
| ( ord_less_int @ Y @ X2 ) ) ).
% linorder_le_less_linear
thf(fact_398_order__le__imp__less__or__eq,axiom,
! [X2: fset_a,Y: fset_a] :
( ( ord_less_eq_fset_a @ X2 @ Y )
=> ( ( ord_less_fset_a @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_399_order__le__imp__less__or__eq,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_less_nat @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_400_order__le__imp__less__or__eq,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ( ord_less_int @ X2 @ Y )
| ( X2 = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_401_fcard__finsert__le,axiom,
! [A: fset_a,X2: a] : ( ord_less_eq_nat @ ( fcard_a @ A ) @ ( fcard_a @ ( finsert_a @ X2 @ A ) ) ) ).
% fcard_finsert_le
thf(fact_402_fcard__seteq,axiom,
! [A: fset_a,B: fset_a] :
( ( ord_less_eq_fset_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( fcard_a @ B ) @ ( fcard_a @ A ) )
=> ( A = B ) ) ) ).
% fcard_seteq
thf(fact_403_fcard__mono,axiom,
! [A: fset_a,B: fset_a] :
( ( ord_less_eq_fset_a @ A @ B )
=> ( ord_less_eq_nat @ ( fcard_a @ A ) @ ( fcard_a @ B ) ) ) ).
% fcard_mono
thf(fact_404_sup__left__commute,axiom,
! [X2: fset_a,Y: fset_a,Z2: fset_a] :
( ( sup_sup_fset_a @ X2 @ ( sup_sup_fset_a @ Y @ Z2 ) )
= ( sup_sup_fset_a @ Y @ ( sup_sup_fset_a @ X2 @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_405_sup__left__commute,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( sup_sup_nat @ X2 @ ( sup_sup_nat @ Y @ Z2 ) )
= ( sup_sup_nat @ Y @ ( sup_sup_nat @ X2 @ Z2 ) ) ) ).
% sup_left_commute
thf(fact_406_sup_Oleft__commute,axiom,
! [B2: fset_a,A2: fset_a,C: fset_a] :
( ( sup_sup_fset_a @ B2 @ ( sup_sup_fset_a @ A2 @ C ) )
= ( sup_sup_fset_a @ A2 @ ( sup_sup_fset_a @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_407_sup_Oleft__commute,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( sup_sup_nat @ B2 @ ( sup_sup_nat @ A2 @ C ) )
= ( sup_sup_nat @ A2 @ ( sup_sup_nat @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_408_sup__commute,axiom,
( sup_sup_fset_a
= ( ^ [X3: fset_a,Y4: fset_a] : ( sup_sup_fset_a @ Y4 @ X3 ) ) ) ).
% sup_commute
thf(fact_409_sup__commute,axiom,
( sup_sup_nat
= ( ^ [X3: nat,Y4: nat] : ( sup_sup_nat @ Y4 @ X3 ) ) ) ).
% sup_commute
thf(fact_410_sup_Ocommute,axiom,
( sup_sup_fset_a
= ( ^ [A4: fset_a,B5: fset_a] : ( sup_sup_fset_a @ B5 @ A4 ) ) ) ).
% sup.commute
thf(fact_411_sup_Ocommute,axiom,
( sup_sup_nat
= ( ^ [A4: nat,B5: nat] : ( sup_sup_nat @ B5 @ A4 ) ) ) ).
% sup.commute
thf(fact_412_sup__assoc,axiom,
! [X2: fset_a,Y: fset_a,Z2: fset_a] :
( ( sup_sup_fset_a @ ( sup_sup_fset_a @ X2 @ Y ) @ Z2 )
= ( sup_sup_fset_a @ X2 @ ( sup_sup_fset_a @ Y @ Z2 ) ) ) ).
% sup_assoc
thf(fact_413_sup__assoc,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ X2 @ Y ) @ Z2 )
= ( sup_sup_nat @ X2 @ ( sup_sup_nat @ Y @ Z2 ) ) ) ).
% sup_assoc
thf(fact_414_sup_Oassoc,axiom,
! [A2: fset_a,B2: fset_a,C: fset_a] :
( ( sup_sup_fset_a @ ( sup_sup_fset_a @ A2 @ B2 ) @ C )
= ( sup_sup_fset_a @ A2 @ ( sup_sup_fset_a @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_415_sup_Oassoc,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ A2 @ B2 ) @ C )
= ( sup_sup_nat @ A2 @ ( sup_sup_nat @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_416_inf__sup__aci_I5_J,axiom,
( sup_sup_fset_a
= ( ^ [X3: fset_a,Y4: fset_a] : ( sup_sup_fset_a @ Y4 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_417_inf__sup__aci_I5_J,axiom,
( sup_sup_nat
= ( ^ [X3: nat,Y4: nat] : ( sup_sup_nat @ Y4 @ X3 ) ) ) ).
% inf_sup_aci(5)
thf(fact_418_inf__sup__aci_I6_J,axiom,
! [X2: fset_a,Y: fset_a,Z2: fset_a] :
( ( sup_sup_fset_a @ ( sup_sup_fset_a @ X2 @ Y ) @ Z2 )
= ( sup_sup_fset_a @ X2 @ ( sup_sup_fset_a @ Y @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_419_inf__sup__aci_I6_J,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( sup_sup_nat @ ( sup_sup_nat @ X2 @ Y ) @ Z2 )
= ( sup_sup_nat @ X2 @ ( sup_sup_nat @ Y @ Z2 ) ) ) ).
% inf_sup_aci(6)
thf(fact_420_inf__sup__aci_I7_J,axiom,
! [X2: fset_a,Y: fset_a,Z2: fset_a] :
( ( sup_sup_fset_a @ X2 @ ( sup_sup_fset_a @ Y @ Z2 ) )
= ( sup_sup_fset_a @ Y @ ( sup_sup_fset_a @ X2 @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_421_inf__sup__aci_I7_J,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( sup_sup_nat @ X2 @ ( sup_sup_nat @ Y @ Z2 ) )
= ( sup_sup_nat @ Y @ ( sup_sup_nat @ X2 @ Z2 ) ) ) ).
% inf_sup_aci(7)
thf(fact_422_inf__sup__aci_I8_J,axiom,
! [X2: fset_a,Y: fset_a] :
( ( sup_sup_fset_a @ X2 @ ( sup_sup_fset_a @ X2 @ Y ) )
= ( sup_sup_fset_a @ X2 @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_423_inf__sup__aci_I8_J,axiom,
! [X2: nat,Y: nat] :
( ( sup_sup_nat @ X2 @ ( sup_sup_nat @ X2 @ Y ) )
= ( sup_sup_nat @ X2 @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_424_inf__sup__ord_I4_J,axiom,
! [Y: fset_a,X2: fset_a] : ( ord_less_eq_fset_a @ Y @ ( sup_sup_fset_a @ X2 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_425_inf__sup__ord_I4_J,axiom,
! [Y: nat,X2: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X2 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_426_inf__sup__ord_I4_J,axiom,
! [Y: int,X2: int] : ( ord_less_eq_int @ Y @ ( sup_sup_int @ X2 @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_427_inf__sup__ord_I3_J,axiom,
! [X2: fset_a,Y: fset_a] : ( ord_less_eq_fset_a @ X2 @ ( sup_sup_fset_a @ X2 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_428_inf__sup__ord_I3_J,axiom,
! [X2: nat,Y: nat] : ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ X2 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_429_inf__sup__ord_I3_J,axiom,
! [X2: int,Y: int] : ( ord_less_eq_int @ X2 @ ( sup_sup_int @ X2 @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_430_le__supE,axiom,
! [A2: fset_a,B2: fset_a,X2: fset_a] :
( ( ord_less_eq_fset_a @ ( sup_sup_fset_a @ A2 @ B2 ) @ X2 )
=> ~ ( ( ord_less_eq_fset_a @ A2 @ X2 )
=> ~ ( ord_less_eq_fset_a @ B2 @ X2 ) ) ) ).
% le_supE
thf(fact_431_le__supE,axiom,
! [A2: nat,B2: nat,X2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ X2 )
=> ~ ( ( ord_less_eq_nat @ A2 @ X2 )
=> ~ ( ord_less_eq_nat @ B2 @ X2 ) ) ) ).
% le_supE
thf(fact_432_le__supE,axiom,
! [A2: int,B2: int,X2: int] :
( ( ord_less_eq_int @ ( sup_sup_int @ A2 @ B2 ) @ X2 )
=> ~ ( ( ord_less_eq_int @ A2 @ X2 )
=> ~ ( ord_less_eq_int @ B2 @ X2 ) ) ) ).
% le_supE
thf(fact_433_le__supI,axiom,
! [A2: fset_a,X2: fset_a,B2: fset_a] :
( ( ord_less_eq_fset_a @ A2 @ X2 )
=> ( ( ord_less_eq_fset_a @ B2 @ X2 )
=> ( ord_less_eq_fset_a @ ( sup_sup_fset_a @ A2 @ B2 ) @ X2 ) ) ) ).
% le_supI
thf(fact_434_le__supI,axiom,
! [A2: nat,X2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ X2 )
=> ( ( ord_less_eq_nat @ B2 @ X2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ X2 ) ) ) ).
% le_supI
thf(fact_435_le__supI,axiom,
! [A2: int,X2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ X2 )
=> ( ( ord_less_eq_int @ B2 @ X2 )
=> ( ord_less_eq_int @ ( sup_sup_int @ A2 @ B2 ) @ X2 ) ) ) ).
% le_supI
thf(fact_436_sup__ge1,axiom,
! [X2: fset_a,Y: fset_a] : ( ord_less_eq_fset_a @ X2 @ ( sup_sup_fset_a @ X2 @ Y ) ) ).
% sup_ge1
thf(fact_437_sup__ge1,axiom,
! [X2: nat,Y: nat] : ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ X2 @ Y ) ) ).
% sup_ge1
thf(fact_438_sup__ge1,axiom,
! [X2: int,Y: int] : ( ord_less_eq_int @ X2 @ ( sup_sup_int @ X2 @ Y ) ) ).
% sup_ge1
thf(fact_439_sup__ge2,axiom,
! [Y: fset_a,X2: fset_a] : ( ord_less_eq_fset_a @ Y @ ( sup_sup_fset_a @ X2 @ Y ) ) ).
% sup_ge2
thf(fact_440_sup__ge2,axiom,
! [Y: nat,X2: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X2 @ Y ) ) ).
% sup_ge2
thf(fact_441_sup__ge2,axiom,
! [Y: int,X2: int] : ( ord_less_eq_int @ Y @ ( sup_sup_int @ X2 @ Y ) ) ).
% sup_ge2
thf(fact_442_le__supI1,axiom,
! [X2: fset_a,A2: fset_a,B2: fset_a] :
( ( ord_less_eq_fset_a @ X2 @ A2 )
=> ( ord_less_eq_fset_a @ X2 @ ( sup_sup_fset_a @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_443_le__supI1,axiom,
! [X2: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ X2 @ A2 )
=> ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_444_le__supI1,axiom,
! [X2: int,A2: int,B2: int] :
( ( ord_less_eq_int @ X2 @ A2 )
=> ( ord_less_eq_int @ X2 @ ( sup_sup_int @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_445_le__supI2,axiom,
! [X2: fset_a,B2: fset_a,A2: fset_a] :
( ( ord_less_eq_fset_a @ X2 @ B2 )
=> ( ord_less_eq_fset_a @ X2 @ ( sup_sup_fset_a @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_446_le__supI2,axiom,
! [X2: nat,B2: nat,A2: nat] :
( ( ord_less_eq_nat @ X2 @ B2 )
=> ( ord_less_eq_nat @ X2 @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_447_le__supI2,axiom,
! [X2: int,B2: int,A2: int] :
( ( ord_less_eq_int @ X2 @ B2 )
=> ( ord_less_eq_int @ X2 @ ( sup_sup_int @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_448_sup_Omono,axiom,
! [C: fset_a,A2: fset_a,D2: fset_a,B2: fset_a] :
( ( ord_less_eq_fset_a @ C @ A2 )
=> ( ( ord_less_eq_fset_a @ D2 @ B2 )
=> ( ord_less_eq_fset_a @ ( sup_sup_fset_a @ C @ D2 ) @ ( sup_sup_fset_a @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_449_sup_Omono,axiom,
! [C: nat,A2: nat,D2: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ( ord_less_eq_nat @ D2 @ B2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D2 ) @ ( sup_sup_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_450_sup_Omono,axiom,
! [C: int,A2: int,D2: int,B2: int] :
( ( ord_less_eq_int @ C @ A2 )
=> ( ( ord_less_eq_int @ D2 @ B2 )
=> ( ord_less_eq_int @ ( sup_sup_int @ C @ D2 ) @ ( sup_sup_int @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_451_sup__mono,axiom,
! [A2: fset_a,C: fset_a,B2: fset_a,D2: fset_a] :
( ( ord_less_eq_fset_a @ A2 @ C )
=> ( ( ord_less_eq_fset_a @ B2 @ D2 )
=> ( ord_less_eq_fset_a @ ( sup_sup_fset_a @ A2 @ B2 ) @ ( sup_sup_fset_a @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_452_sup__mono,axiom,
! [A2: nat,C: nat,B2: nat,D2: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B2 @ D2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ ( sup_sup_nat @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_453_sup__mono,axiom,
! [A2: int,C: int,B2: int,D2: int] :
( ( ord_less_eq_int @ A2 @ C )
=> ( ( ord_less_eq_int @ B2 @ D2 )
=> ( ord_less_eq_int @ ( sup_sup_int @ A2 @ B2 ) @ ( sup_sup_int @ C @ D2 ) ) ) ) ).
% sup_mono
thf(fact_454_sup__least,axiom,
! [Y: fset_a,X2: fset_a,Z2: fset_a] :
( ( ord_less_eq_fset_a @ Y @ X2 )
=> ( ( ord_less_eq_fset_a @ Z2 @ X2 )
=> ( ord_less_eq_fset_a @ ( sup_sup_fset_a @ Y @ Z2 ) @ X2 ) ) ) ).
% sup_least
thf(fact_455_sup__least,axiom,
! [Y: nat,X2: nat,Z2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ( ( ord_less_eq_nat @ Z2 @ X2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z2 ) @ X2 ) ) ) ).
% sup_least
thf(fact_456_sup__least,axiom,
! [Y: int,X2: int,Z2: int] :
( ( ord_less_eq_int @ Y @ X2 )
=> ( ( ord_less_eq_int @ Z2 @ X2 )
=> ( ord_less_eq_int @ ( sup_sup_int @ Y @ Z2 ) @ X2 ) ) ) ).
% sup_least
thf(fact_457_le__iff__sup,axiom,
( ord_less_eq_fset_a
= ( ^ [X3: fset_a,Y4: fset_a] :
( ( sup_sup_fset_a @ X3 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_458_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y4: nat] :
( ( sup_sup_nat @ X3 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_459_le__iff__sup,axiom,
( ord_less_eq_int
= ( ^ [X3: int,Y4: int] :
( ( sup_sup_int @ X3 @ Y4 )
= Y4 ) ) ) ).
% le_iff_sup
thf(fact_460_sup_OorderE,axiom,
! [B2: fset_a,A2: fset_a] :
( ( ord_less_eq_fset_a @ B2 @ A2 )
=> ( A2
= ( sup_sup_fset_a @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_461_sup_OorderE,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2
= ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_462_sup_OorderE,axiom,
! [B2: int,A2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( A2
= ( sup_sup_int @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_463_sup_OorderI,axiom,
! [A2: fset_a,B2: fset_a] :
( ( A2
= ( sup_sup_fset_a @ A2 @ B2 ) )
=> ( ord_less_eq_fset_a @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_464_sup_OorderI,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( sup_sup_nat @ A2 @ B2 ) )
=> ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_465_sup_OorderI,axiom,
! [A2: int,B2: int] :
( ( A2
= ( sup_sup_int @ A2 @ B2 ) )
=> ( ord_less_eq_int @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_466_sup__unique,axiom,
! [F: fset_a > fset_a > fset_a,X2: fset_a,Y: fset_a] :
( ! [X: fset_a,Y2: fset_a] : ( ord_less_eq_fset_a @ X @ ( F @ X @ Y2 ) )
=> ( ! [X: fset_a,Y2: fset_a] : ( ord_less_eq_fset_a @ Y2 @ ( F @ X @ Y2 ) )
=> ( ! [X: fset_a,Y2: fset_a,Z3: fset_a] :
( ( ord_less_eq_fset_a @ Y2 @ X )
=> ( ( ord_less_eq_fset_a @ Z3 @ X )
=> ( ord_less_eq_fset_a @ ( F @ Y2 @ Z3 ) @ X ) ) )
=> ( ( sup_sup_fset_a @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_467_sup__unique,axiom,
! [F: nat > nat > nat,X2: nat,Y: nat] :
( ! [X: nat,Y2: nat] : ( ord_less_eq_nat @ X @ ( F @ X @ Y2 ) )
=> ( ! [X: nat,Y2: nat] : ( ord_less_eq_nat @ Y2 @ ( F @ X @ Y2 ) )
=> ( ! [X: nat,Y2: nat,Z3: nat] :
( ( ord_less_eq_nat @ Y2 @ X )
=> ( ( ord_less_eq_nat @ Z3 @ X )
=> ( ord_less_eq_nat @ ( F @ Y2 @ Z3 ) @ X ) ) )
=> ( ( sup_sup_nat @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_468_sup__unique,axiom,
! [F: int > int > int,X2: int,Y: int] :
( ! [X: int,Y2: int] : ( ord_less_eq_int @ X @ ( F @ X @ Y2 ) )
=> ( ! [X: int,Y2: int] : ( ord_less_eq_int @ Y2 @ ( F @ X @ Y2 ) )
=> ( ! [X: int,Y2: int,Z3: int] :
( ( ord_less_eq_int @ Y2 @ X )
=> ( ( ord_less_eq_int @ Z3 @ X )
=> ( ord_less_eq_int @ ( F @ Y2 @ Z3 ) @ X ) ) )
=> ( ( sup_sup_int @ X2 @ Y )
= ( F @ X2 @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_469_sup_Oabsorb1,axiom,
! [B2: fset_a,A2: fset_a] :
( ( ord_less_eq_fset_a @ B2 @ A2 )
=> ( ( sup_sup_fset_a @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_470_sup_Oabsorb1,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_471_sup_Oabsorb1,axiom,
! [B2: int,A2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ( sup_sup_int @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_472_sup_Oabsorb2,axiom,
! [A2: fset_a,B2: fset_a] :
( ( ord_less_eq_fset_a @ A2 @ B2 )
=> ( ( sup_sup_fset_a @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_473_sup_Oabsorb2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_474_sup_Oabsorb2,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( sup_sup_int @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_475_sup__absorb1,axiom,
! [Y: fset_a,X2: fset_a] :
( ( ord_less_eq_fset_a @ Y @ X2 )
=> ( ( sup_sup_fset_a @ X2 @ Y )
= X2 ) ) ).
% sup_absorb1
thf(fact_476_sup__absorb1,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ( ( sup_sup_nat @ X2 @ Y )
= X2 ) ) ).
% sup_absorb1
thf(fact_477_sup__absorb1,axiom,
! [Y: int,X2: int] :
( ( ord_less_eq_int @ Y @ X2 )
=> ( ( sup_sup_int @ X2 @ Y )
= X2 ) ) ).
% sup_absorb1
thf(fact_478_sup__absorb2,axiom,
! [X2: fset_a,Y: fset_a] :
( ( ord_less_eq_fset_a @ X2 @ Y )
=> ( ( sup_sup_fset_a @ X2 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_479_sup__absorb2,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( sup_sup_nat @ X2 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_480_sup__absorb2,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ( sup_sup_int @ X2 @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_481_sup_OboundedE,axiom,
! [B2: fset_a,C: fset_a,A2: fset_a] :
( ( ord_less_eq_fset_a @ ( sup_sup_fset_a @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_eq_fset_a @ B2 @ A2 )
=> ~ ( ord_less_eq_fset_a @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_482_sup_OboundedE,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_eq_nat @ B2 @ A2 )
=> ~ ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_483_sup_OboundedE,axiom,
! [B2: int,C: int,A2: int] :
( ( ord_less_eq_int @ ( sup_sup_int @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_eq_int @ B2 @ A2 )
=> ~ ( ord_less_eq_int @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_484_sup_OboundedI,axiom,
! [B2: fset_a,A2: fset_a,C: fset_a] :
( ( ord_less_eq_fset_a @ B2 @ A2 )
=> ( ( ord_less_eq_fset_a @ C @ A2 )
=> ( ord_less_eq_fset_a @ ( sup_sup_fset_a @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_485_sup_OboundedI,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_486_sup_OboundedI,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ( ord_less_eq_int @ C @ A2 )
=> ( ord_less_eq_int @ ( sup_sup_int @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_487_sup_Oorder__iff,axiom,
( ord_less_eq_fset_a
= ( ^ [B5: fset_a,A4: fset_a] :
( A4
= ( sup_sup_fset_a @ A4 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_488_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A4: nat] :
( A4
= ( sup_sup_nat @ A4 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_489_sup_Oorder__iff,axiom,
( ord_less_eq_int
= ( ^ [B5: int,A4: int] :
( A4
= ( sup_sup_int @ A4 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_490_sup_Ocobounded1,axiom,
! [A2: fset_a,B2: fset_a] : ( ord_less_eq_fset_a @ A2 @ ( sup_sup_fset_a @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_491_sup_Ocobounded1,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ A2 @ ( sup_sup_nat @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_492_sup_Ocobounded1,axiom,
! [A2: int,B2: int] : ( ord_less_eq_int @ A2 @ ( sup_sup_int @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_493_sup_Ocobounded2,axiom,
! [B2: fset_a,A2: fset_a] : ( ord_less_eq_fset_a @ B2 @ ( sup_sup_fset_a @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_494_sup_Ocobounded2,axiom,
! [B2: nat,A2: nat] : ( ord_less_eq_nat @ B2 @ ( sup_sup_nat @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_495_sup_Ocobounded2,axiom,
! [B2: int,A2: int] : ( ord_less_eq_int @ B2 @ ( sup_sup_int @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_496_sup_Oabsorb__iff1,axiom,
( ord_less_eq_fset_a
= ( ^ [B5: fset_a,A4: fset_a] :
( ( sup_sup_fset_a @ A4 @ B5 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_497_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A4: nat] :
( ( sup_sup_nat @ A4 @ B5 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_498_sup_Oabsorb__iff1,axiom,
( ord_less_eq_int
= ( ^ [B5: int,A4: int] :
( ( sup_sup_int @ A4 @ B5 )
= A4 ) ) ) ).
% sup.absorb_iff1
thf(fact_499_sup_Oabsorb__iff2,axiom,
( ord_less_eq_fset_a
= ( ^ [A4: fset_a,B5: fset_a] :
( ( sup_sup_fset_a @ A4 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_500_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B5: nat] :
( ( sup_sup_nat @ A4 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_501_sup_Oabsorb__iff2,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B5: int] :
( ( sup_sup_int @ A4 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_502_sup_OcoboundedI1,axiom,
! [C: fset_a,A2: fset_a,B2: fset_a] :
( ( ord_less_eq_fset_a @ C @ A2 )
=> ( ord_less_eq_fset_a @ C @ ( sup_sup_fset_a @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_503_sup_OcoboundedI1,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_504_sup_OcoboundedI1,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_eq_int @ C @ A2 )
=> ( ord_less_eq_int @ C @ ( sup_sup_int @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_505_diff__diff__cancel,axiom,
! [I: nat,N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_506_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
& ( M != N ) ) ) ) ).
% nat_less_le
thf(fact_507_less__diff__iff,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( ord_less_nat @ M2 @ N2 ) ) ) ) ).
% less_diff_iff
thf(fact_508_diff__less__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_nat @ ( minus_minus_nat @ A2 @ C ) @ ( minus_minus_nat @ B2 @ C ) ) ) ) ).
% diff_less_mono
thf(fact_509_less__imp__le__nat,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% less_imp_le_nat
thf(fact_510_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_511_less__or__eq__imp__le,axiom,
! [M2: nat,N2: nat] :
( ( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% less_or_eq_imp_le
thf(fact_512_le__neq__implies__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( M2 != N2 )
=> ( ord_less_nat @ M2 @ N2 ) ) ) ).
% le_neq_implies_less
thf(fact_513_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_514_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_515_pfsubsetD,axiom,
! [A: fset_a,B: fset_a,C: a] :
( ( ord_less_fset_a @ A @ B )
=> ( ( fmember_a @ C @ A )
=> ( fmember_a @ C @ B ) ) ) ).
% pfsubsetD
thf(fact_516_fsubset__iff__pfsubset__eq,axiom,
( ord_less_eq_fset_a
= ( ^ [A3: fset_a,B3: fset_a] :
( ( ord_less_fset_a @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% fsubset_iff_pfsubset_eq
thf(fact_517_pfsubset__fsubset__trans,axiom,
! [A: fset_a,B: fset_a,C2: fset_a] :
( ( ord_less_fset_a @ A @ B )
=> ( ( ord_less_eq_fset_a @ B @ C2 )
=> ( ord_less_fset_a @ A @ C2 ) ) ) ).
% pfsubset_fsubset_trans
thf(fact_518_fsubset__pfsubset__trans,axiom,
! [A: fset_a,B: fset_a,C2: fset_a] :
( ( ord_less_eq_fset_a @ A @ B )
=> ( ( ord_less_fset_a @ B @ C2 )
=> ( ord_less_fset_a @ A @ C2 ) ) ) ).
% fsubset_pfsubset_trans
thf(fact_519_fsubset__not__fsubset__eq,axiom,
( ord_less_fset_a
= ( ^ [A3: fset_a,B3: fset_a] :
( ( ord_less_eq_fset_a @ A3 @ B3 )
& ~ ( ord_less_eq_fset_a @ B3 @ A3 ) ) ) ) ).
% fsubset_not_fsubset_eq
thf(fact_520_pfsubset__imp__fsubset,axiom,
! [A: fset_a,B: fset_a] :
( ( ord_less_fset_a @ A @ B )
=> ( ord_less_eq_fset_a @ A @ B ) ) ).
% pfsubset_imp_fsubset
thf(fact_521_less__fset__def,axiom,
( ord_less_fset_a
= ( ^ [Xs: fset_a,Ys: fset_a] :
( ( ord_less_eq_fset_a @ Xs @ Ys )
& ( Xs != Ys ) ) ) ) ).
% less_fset_def
thf(fact_522_pfsubset__eq,axiom,
( ord_less_fset_a
= ( ^ [A3: fset_a,B3: fset_a] :
( ( ord_less_eq_fset_a @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% pfsubset_eq
thf(fact_523_pfsubset__fcard__mono,axiom,
! [A: fset_a,B: fset_a] :
( ( ord_less_fset_a @ A @ B )
=> ( ord_less_nat @ ( fcard_a @ A ) @ ( fcard_a @ B ) ) ) ).
% pfsubset_fcard_mono
thf(fact_524_pfsubset__imp__ex__fmem,axiom,
! [A: fset_a,B: fset_a] :
( ( ord_less_fset_a @ A @ B )
=> ? [B6: a] : ( fmember_a @ B6 @ ( minus_minus_fset_a @ B @ A ) ) ) ).
% pfsubset_imp_ex_fmem
thf(fact_525_nat__neq__iff,axiom,
! [M2: nat,N2: nat] :
( ( M2 != N2 )
= ( ( ord_less_nat @ M2 @ N2 )
| ( ord_less_nat @ N2 @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_526_less__not__refl,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_not_refl
thf(fact_527_less__not__refl2,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ N2 @ M2 )
=> ( M2 != N2 ) ) ).
% less_not_refl2
thf(fact_528_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_529_less__irrefl__nat,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_irrefl_nat
thf(fact_530_nat__less__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N2 ) ) ).
% nat_less_induct
thf(fact_531_infinite__descent,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) )
=> ( P @ N2 ) ) ).
% infinite_descent
thf(fact_532_linorder__neqE__nat,axiom,
! [X2: nat,Y: nat] :
( ( X2 != Y )
=> ( ~ ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ Y @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_533_less__imp__diff__less,axiom,
! [J: nat,K: nat,N2: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N2 ) @ K ) ) ).
% less_imp_diff_less
thf(fact_534_diff__less__mono2,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ( ord_less_nat @ M2 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_535_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B2 ) )
=> ? [X: nat] :
( ( P @ X )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_536_nat__le__linear,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
| ( ord_less_eq_nat @ N2 @ M2 ) ) ).
% nat_le_linear
thf(fact_537_diff__le__mono2,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_538_le__diff__iff_H,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A2 ) @ ( minus_minus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_539_diff__le__self,axiom,
! [M2: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N2 ) @ M2 ) ).
% diff_le_self
thf(fact_540_diff__le__mono,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N2 @ L ) ) ) ).
% diff_le_mono
thf(fact_541_Nat_Odiff__diff__eq,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_542_le__diff__iff,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ) ) ).
% le_diff_iff
thf(fact_543_eq__diff__iff,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ( minus_minus_nat @ M2 @ K )
= ( minus_minus_nat @ N2 @ K ) )
= ( M2 = N2 ) ) ) ) ).
% eq_diff_iff
thf(fact_544_le__antisym,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ M2 )
=> ( M2 = N2 ) ) ) ).
% le_antisym
thf(fact_545_eq__imp__le,axiom,
! [M2: nat,N2: nat] :
( ( M2 = N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% eq_imp_le
thf(fact_546_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_547_le__refl,axiom,
! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% le_refl
thf(fact_548_fcard__pfsubset,axiom,
! [A: fset_a,B: fset_a] :
( ( ord_less_eq_fset_a @ A @ B )
=> ( ( ord_less_nat @ ( fcard_a @ A ) @ ( fcard_a @ B ) )
=> ( ord_less_fset_a @ A @ B ) ) ) ).
% fcard_pfsubset
thf(fact_549_fcard__fminus__finsert,axiom,
! [A2: a,A: fset_a,B: fset_a] :
( ( fmember_a @ A2 @ A )
=> ( ~ ( fmember_a @ A2 @ B )
=> ( ( fcard_a @ ( minus_minus_fset_a @ A @ ( finsert_a @ A2 @ B ) ) )
= ( minus_minus_nat @ ( fcard_a @ ( minus_minus_fset_a @ A @ B ) ) @ one_one_nat ) ) ) ) ).
% fcard_fminus_finsert
thf(fact_550_nat__descend__induct,axiom,
! [N2: nat,P: nat > $o,M2: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N2 @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ! [I3: nat] :
( ( ord_less_nat @ K2 @ I3 )
=> ( P @ I3 ) )
=> ( P @ K2 ) ) )
=> ( P @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_551_diff__strict__mono,axiom,
! [A2: int,B2: int,D2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_less_int @ D2 @ C )
=> ( ord_less_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B2 @ D2 ) ) ) ) ).
% diff_strict_mono
thf(fact_552_diff__eq__diff__less,axiom,
! [A2: int,B2: int,C: int,D2: int] :
( ( ( minus_minus_int @ A2 @ B2 )
= ( minus_minus_int @ C @ D2 ) )
=> ( ( ord_less_int @ A2 @ B2 )
= ( ord_less_int @ C @ D2 ) ) ) ).
% diff_eq_diff_less
thf(fact_553_diff__strict__left__mono,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( ord_less_int @ ( minus_minus_int @ C @ A2 ) @ ( minus_minus_int @ C @ B2 ) ) ) ).
% diff_strict_left_mono
thf(fact_554_diff__strict__right__mono,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ord_less_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B2 @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_555_one__reorient,axiom,
! [X2: nat] :
( ( one_one_nat = X2 )
= ( X2 = one_one_nat ) ) ).
% one_reorient
thf(fact_556_one__reorient,axiom,
! [X2: int] :
( ( one_one_int = X2 )
= ( X2 = one_one_int ) ) ).
% one_reorient
thf(fact_557_diff__right__commute,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ C ) @ B2 )
= ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B2 ) @ C ) ) ).
% diff_right_commute
thf(fact_558_diff__right__commute,axiom,
! [A2: int,C: int,B2: int] :
( ( minus_minus_int @ ( minus_minus_int @ A2 @ C ) @ B2 )
= ( minus_minus_int @ ( minus_minus_int @ A2 @ B2 ) @ C ) ) ).
% diff_right_commute
thf(fact_559_diff__eq__diff__eq,axiom,
! [A2: int,B2: int,C: int,D2: int] :
( ( ( minus_minus_int @ A2 @ B2 )
= ( minus_minus_int @ C @ D2 ) )
=> ( ( A2 = B2 )
= ( C = D2 ) ) ) ).
% diff_eq_diff_eq
thf(fact_560_diff__mono,axiom,
! [A2: int,B2: int,D2: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ D2 @ C )
=> ( ord_less_eq_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B2 @ D2 ) ) ) ) ).
% diff_mono
thf(fact_561_diff__left__mono,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ord_less_eq_int @ ( minus_minus_int @ C @ A2 ) @ ( minus_minus_int @ C @ B2 ) ) ) ).
% diff_left_mono
thf(fact_562_diff__right__mono,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ord_less_eq_int @ ( minus_minus_int @ A2 @ C ) @ ( minus_minus_int @ B2 @ C ) ) ) ).
% diff_right_mono
thf(fact_563_diff__eq__diff__less__eq,axiom,
! [A2: int,B2: int,C: int,D2: int] :
( ( ( minus_minus_int @ A2 @ B2 )
= ( minus_minus_int @ C @ D2 ) )
=> ( ( ord_less_eq_int @ A2 @ B2 )
= ( ord_less_eq_int @ C @ D2 ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_564_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_565_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_566_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_567_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_568_minf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ~ ( ord_less_eq_nat @ T @ X5 ) ) ).
% minf(8)
thf(fact_569_minf_I8_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z3 )
=> ~ ( ord_less_eq_int @ T @ X5 ) ) ).
% minf(8)
thf(fact_570_minf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( ord_less_eq_nat @ X5 @ T ) ) ).
% minf(6)
thf(fact_571_minf_I6_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z3 )
=> ( ord_less_eq_int @ X5 @ T ) ) ).
% minf(6)
thf(fact_572_pinf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( ord_less_eq_nat @ T @ X5 ) ) ).
% pinf(8)
thf(fact_573_pinf_I8_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ Z3 @ X5 )
=> ( ord_less_eq_int @ T @ X5 ) ) ).
% pinf(8)
thf(fact_574_pinf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ~ ( ord_less_eq_nat @ X5 @ T ) ) ).
% pinf(6)
thf(fact_575_pinf_I6_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ Z3 @ X5 )
=> ~ ( ord_less_eq_int @ X5 @ T ) ) ).
% pinf(6)
thf(fact_576_minf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ~ ( ord_less_nat @ T @ X5 ) ) ).
% minf(7)
thf(fact_577_minf_I7_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z3 )
=> ~ ( ord_less_int @ T @ X5 ) ) ).
% minf(7)
thf(fact_578_minf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( ord_less_nat @ X5 @ T ) ) ).
% minf(5)
thf(fact_579_minf_I5_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z3 )
=> ( ord_less_int @ X5 @ T ) ) ).
% minf(5)
thf(fact_580_minf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_581_minf_I4_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z3 )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_582_minf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_583_minf_I3_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z3 )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_584_minf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z4 )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z4 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_585_minf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X: int] :
( ( ord_less_int @ X @ Z4 )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: int] :
! [X: int] :
( ( ord_less_int @ X @ Z4 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z3 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_586_minf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z4 )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ X @ Z4 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_587_minf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X: int] :
( ( ord_less_int @ X @ Z4 )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: int] :
! [X: int] :
( ( ord_less_int @ X @ Z4 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z3 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_588_pinf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( ord_less_nat @ T @ X5 ) ) ).
% pinf(7)
thf(fact_589_pinf_I7_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ Z3 @ X5 )
=> ( ord_less_int @ T @ X5 ) ) ).
% pinf(7)
thf(fact_590_pinf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ~ ( ord_less_nat @ X5 @ T ) ) ).
% pinf(5)
thf(fact_591_pinf_I5_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ Z3 @ X5 )
=> ~ ( ord_less_int @ X5 @ T ) ) ).
% pinf(5)
thf(fact_592_pinf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_593_pinf_I4_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ Z3 @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_594_pinf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_595_pinf_I3_J,axiom,
! [T: int] :
? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ Z3 @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_596_pinf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ Z4 @ X )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ Z4 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_597_pinf_I2_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X: int] :
( ( ord_less_int @ Z4 @ X )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: int] :
! [X: int] :
( ( ord_less_int @ Z4 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ Z3 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_598_pinf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ Z4 @ X )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: nat] :
! [X: nat] :
( ( ord_less_nat @ Z4 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_599_pinf_I1_J,axiom,
! [P: int > $o,P4: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z4: int] :
! [X: int] :
( ( ord_less_int @ Z4 @ X )
=> ( ( P @ X )
= ( P4 @ X ) ) )
=> ( ? [Z4: int] :
! [X: int] :
( ( ord_less_int @ Z4 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z3: int] :
! [X5: int] :
( ( ord_less_int @ Z3 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_600_complete__interval,axiom,
! [A2: nat,B2: nat,P: nat > $o] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( P @ A2 )
=> ( ~ ( P @ B2 )
=> ? [C4: nat] :
( ( ord_less_eq_nat @ A2 @ C4 )
& ( ord_less_eq_nat @ C4 @ B2 )
& ! [X5: nat] :
( ( ( ord_less_eq_nat @ A2 @ X5 )
& ( ord_less_nat @ X5 @ C4 ) )
=> ( P @ X5 ) )
& ! [D3: nat] :
( ! [X: nat] :
( ( ( ord_less_eq_nat @ A2 @ X )
& ( ord_less_nat @ X @ D3 ) )
=> ( P @ X ) )
=> ( ord_less_eq_nat @ D3 @ C4 ) ) ) ) ) ) ).
% complete_interval
thf(fact_601_complete__interval,axiom,
! [A2: int,B2: int,P: int > $o] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( P @ A2 )
=> ( ~ ( P @ B2 )
=> ? [C4: int] :
( ( ord_less_eq_int @ A2 @ C4 )
& ( ord_less_eq_int @ C4 @ B2 )
& ! [X5: int] :
( ( ( ord_less_eq_int @ A2 @ X5 )
& ( ord_less_int @ X5 @ C4 ) )
=> ( P @ X5 ) )
& ! [D3: int] :
( ! [X: int] :
( ( ( ord_less_eq_int @ A2 @ X )
& ( ord_less_int @ X @ D3 ) )
=> ( P @ X ) )
=> ( ord_less_eq_int @ D3 @ C4 ) ) ) ) ) ) ).
% complete_interval
thf(fact_602_verit__comp__simplify1_I3_J,axiom,
! [B7: nat,A6: nat] :
( ( ~ ( ord_less_eq_nat @ B7 @ A6 ) )
= ( ord_less_nat @ A6 @ B7 ) ) ).
% verit_comp_simplify1(3)
thf(fact_603_verit__comp__simplify1_I3_J,axiom,
! [B7: int,A6: int] :
( ( ~ ( ord_less_eq_int @ B7 @ A6 ) )
= ( ord_less_int @ A6 @ B7 ) ) ).
% verit_comp_simplify1(3)
thf(fact_604_fcard__fminus__fsingleton,axiom,
! [X2: a,A: fset_a] :
( ( fmember_a @ X2 @ A )
=> ( ( fcard_a @ ( minus_minus_fset_a @ A @ ( finsert_a @ X2 @ bot_bot_fset_a ) ) )
= ( minus_minus_nat @ ( fcard_a @ A ) @ one_one_nat ) ) ) ).
% fcard_fminus_fsingleton
thf(fact_605_fcard__fminus__fsingleton__if,axiom,
! [X2: a,A: fset_a] :
( ( ( fmember_a @ X2 @ A )
=> ( ( fcard_a @ ( minus_minus_fset_a @ A @ ( finsert_a @ X2 @ bot_bot_fset_a ) ) )
= ( minus_minus_nat @ ( fcard_a @ A ) @ one_one_nat ) ) )
& ( ~ ( fmember_a @ X2 @ A )
=> ( ( fcard_a @ ( minus_minus_fset_a @ A @ ( finsert_a @ X2 @ bot_bot_fset_a ) ) )
= ( fcard_a @ A ) ) ) ) ).
% fcard_fminus_fsingleton_if
thf(fact_606_pfsubset__finsert__iff,axiom,
! [A: fset_a,X2: a,B: fset_a] :
( ( ord_less_fset_a @ A @ ( finsert_a @ X2 @ B ) )
= ( ( ( fmember_a @ X2 @ B )
=> ( ord_less_fset_a @ A @ B ) )
& ( ~ ( fmember_a @ X2 @ B )
=> ( ( ( fmember_a @ X2 @ A )
=> ( ord_less_fset_a @ ( minus_minus_fset_a @ A @ ( finsert_a @ X2 @ bot_bot_fset_a ) ) @ B ) )
& ( ~ ( fmember_a @ X2 @ A )
=> ( ord_less_eq_fset_a @ A @ B ) ) ) ) ) ) ).
% pfsubset_finsert_iff
thf(fact_607_fcard__fminus1__less,axiom,
! [X2: a,A: fset_a] :
( ( fmember_a @ X2 @ A )
=> ( ord_less_nat @ ( fcard_a @ ( minus_minus_fset_a @ A @ ( finsert_a @ X2 @ bot_bot_fset_a ) ) ) @ ( fcard_a @ A ) ) ) ).
% fcard_fminus1_less
thf(fact_608_sup__bot_Oright__neutral,axiom,
! [A2: fset_a] :
( ( sup_sup_fset_a @ A2 @ bot_bot_fset_a )
= A2 ) ).
% sup_bot.right_neutral
thf(fact_609_sup__bot_Oneutr__eq__iff,axiom,
! [A2: fset_a,B2: fset_a] :
( ( bot_bot_fset_a
= ( sup_sup_fset_a @ A2 @ B2 ) )
= ( ( A2 = bot_bot_fset_a )
& ( B2 = bot_bot_fset_a ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_610_sup__bot_Oleft__neutral,axiom,
! [A2: fset_a] :
( ( sup_sup_fset_a @ bot_bot_fset_a @ A2 )
= A2 ) ).
% sup_bot.left_neutral
thf(fact_611_sup__bot_Oeq__neutr__iff,axiom,
! [A2: fset_a,B2: fset_a] :
( ( ( sup_sup_fset_a @ A2 @ B2 )
= bot_bot_fset_a )
= ( ( A2 = bot_bot_fset_a )
& ( B2 = bot_bot_fset_a ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_612_sup__eq__bot__iff,axiom,
! [X2: fset_a,Y: fset_a] :
( ( ( sup_sup_fset_a @ X2 @ Y )
= bot_bot_fset_a )
= ( ( X2 = bot_bot_fset_a )
& ( Y = bot_bot_fset_a ) ) ) ).
% sup_eq_bot_iff
thf(fact_613_bot__eq__sup__iff,axiom,
! [X2: fset_a,Y: fset_a] :
( ( bot_bot_fset_a
= ( sup_sup_fset_a @ X2 @ Y ) )
= ( ( X2 = bot_bot_fset_a )
& ( Y = bot_bot_fset_a ) ) ) ).
% bot_eq_sup_iff
thf(fact_614_sup__bot__right,axiom,
! [X2: fset_a] :
( ( sup_sup_fset_a @ X2 @ bot_bot_fset_a )
= X2 ) ).
% sup_bot_right
thf(fact_615_sup__bot__left,axiom,
! [X2: fset_a] :
( ( sup_sup_fset_a @ bot_bot_fset_a @ X2 )
= X2 ) ).
% sup_bot_left
thf(fact_616_fempty__iff,axiom,
! [C: a] :
~ ( fmember_a @ C @ bot_bot_fset_a ) ).
% fempty_iff
thf(fact_617_all__not__fin__conv,axiom,
! [A: fset_a] :
( ( ! [X3: a] :
~ ( fmember_a @ X3 @ A ) )
= ( A = bot_bot_fset_a ) ) ).
% all_not_fin_conv
thf(fact_618_fsubset__fempty,axiom,
! [A: fset_a] :
( ( ord_less_eq_fset_a @ A @ bot_bot_fset_a )
= ( A = bot_bot_fset_a ) ) ).
% fsubset_fempty
thf(fact_619_fempty__fsubsetI,axiom,
! [X2: fset_a] : ( ord_less_eq_fset_a @ bot_bot_fset_a @ X2 ) ).
% fempty_fsubsetI
thf(fact_620_funion__fempty,axiom,
! [A: fset_a,B: fset_a] :
( ( ( sup_sup_fset_a @ A @ B )
= bot_bot_fset_a )
= ( ( A = bot_bot_fset_a )
& ( B = bot_bot_fset_a ) ) ) ).
% funion_fempty
thf(fact_621_fminus__fempty,axiom,
! [A: fset_a] :
( ( minus_minus_fset_a @ A @ bot_bot_fset_a )
= A ) ).
% fminus_fempty
thf(fact_622_fminus__cancel,axiom,
! [A: fset_a] :
( ( minus_minus_fset_a @ A @ A )
= bot_bot_fset_a ) ).
% fminus_cancel
thf(fact_623_fempty__fminus,axiom,
! [A: fset_a] :
( ( minus_minus_fset_a @ bot_bot_fset_a @ A )
= bot_bot_fset_a ) ).
% fempty_fminus
thf(fact_624_finsert__fminus__single,axiom,
! [A2: a,A: fset_a] :
( ( finsert_a @ A2 @ ( minus_minus_fset_a @ A @ ( finsert_a @ A2 @ bot_bot_fset_a ) ) )
= ( finsert_a @ A2 @ A ) ) ).
% finsert_fminus_single
thf(fact_625_bot_Oextremum,axiom,
! [A2: fset_a] : ( ord_less_eq_fset_a @ bot_bot_fset_a @ A2 ) ).
% bot.extremum
thf(fact_626_bot_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A2 ) ).
% bot.extremum
thf(fact_627_bot_Oextremum__unique,axiom,
! [A2: fset_a] :
( ( ord_less_eq_fset_a @ A2 @ bot_bot_fset_a )
= ( A2 = bot_bot_fset_a ) ) ).
% bot.extremum_unique
thf(fact_628_bot_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
= ( A2 = bot_bot_nat ) ) ).
% bot.extremum_unique
thf(fact_629_bot_Oextremum__uniqueI,axiom,
! [A2: fset_a] :
( ( ord_less_eq_fset_a @ A2 @ bot_bot_fset_a )
=> ( A2 = bot_bot_fset_a ) ) ).
% bot.extremum_uniqueI
thf(fact_630_bot_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ bot_bot_nat )
=> ( A2 = bot_bot_nat ) ) ).
% bot.extremum_uniqueI
thf(fact_631_bot_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ bot_bot_nat ) ).
% bot.extremum_strict
thf(fact_632_bot_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != bot_bot_nat )
= ( ord_less_nat @ bot_bot_nat @ A2 ) ) ).
% bot.not_eq_extremum
thf(fact_633_femptyE,axiom,
! [A2: a] :
~ ( fmember_a @ A2 @ bot_bot_fset_a ) ).
% femptyE
thf(fact_634_ex__fin__conv,axiom,
! [A: fset_a] :
( ( ? [X3: a] : ( fmember_a @ X3 @ A ) )
= ( A != bot_bot_fset_a ) ) ).
% ex_fin_conv
thf(fact_635_equalsffemptyD,axiom,
! [A: fset_a,A2: a] :
( ( A = bot_bot_fset_a )
=> ~ ( fmember_a @ A2 @ A ) ) ).
% equalsffemptyD
thf(fact_636_equalsffemptyI,axiom,
! [A: fset_a] :
( ! [Y2: a] :
~ ( fmember_a @ Y2 @ A )
=> ( A = bot_bot_fset_a ) ) ).
% equalsffemptyI
thf(fact_637_finsert__not__fempty,axiom,
! [A2: a,A: fset_a] :
( ( finsert_a @ A2 @ A )
!= bot_bot_fset_a ) ).
% finsert_not_fempty
thf(fact_638_fsingleton__inject,axiom,
! [A2: a,B2: a] :
( ( ( finsert_a @ A2 @ bot_bot_fset_a )
= ( finsert_a @ B2 @ bot_bot_fset_a ) )
=> ( A2 = B2 ) ) ).
% fsingleton_inject
thf(fact_639_fdoubleton__eq__iff,axiom,
! [A2: a,B2: a,C: a,D2: a] :
( ( ( finsert_a @ A2 @ ( finsert_a @ B2 @ bot_bot_fset_a ) )
= ( finsert_a @ C @ ( finsert_a @ D2 @ bot_bot_fset_a ) ) )
= ( ( ( A2 = C )
& ( B2 = D2 ) )
| ( ( A2 = D2 )
& ( B2 = C ) ) ) ) ).
% fdoubleton_eq_iff
thf(fact_640_fset__exhaust,axiom,
! [S2: fset_a] :
( ( S2 != bot_bot_fset_a )
=> ~ ! [X: a,S3: fset_a] :
( S2
!= ( finsert_a @ X @ S3 ) ) ) ).
% fset_exhaust
thf(fact_641_FSet_Ofset__induct,axiom,
! [P: fset_a > $o,S2: fset_a] :
( ( P @ bot_bot_fset_a )
=> ( ! [X: a,S4: fset_a] :
( ( P @ S4 )
=> ( P @ ( finsert_a @ X @ S4 ) ) )
=> ( P @ S2 ) ) ) ).
% FSet.fset_induct
thf(fact_642_funion__fempty__right,axiom,
! [A: fset_a] :
( ( sup_sup_fset_a @ A @ bot_bot_fset_a )
= A ) ).
% funion_fempty_right
thf(fact_643_funion__fempty__left,axiom,
! [B: fset_a] :
( ( sup_sup_fset_a @ bot_bot_fset_a @ B )
= B ) ).
% funion_fempty_left
thf(fact_644_fset__induct2,axiom,
! [P: fset_a > fset_a > $o,Xsa: fset_a,Ysa: fset_a] :
( ( P @ bot_bot_fset_a @ bot_bot_fset_a )
=> ( ! [X: a,Xs2: fset_a] :
( ~ ( fmember_a @ X @ Xs2 )
=> ( P @ ( finsert_a @ X @ Xs2 ) @ bot_bot_fset_a ) )
=> ( ! [Y2: a,Ys2: fset_a] :
( ~ ( fmember_a @ Y2 @ Ys2 )
=> ( P @ bot_bot_fset_a @ ( finsert_a @ Y2 @ Ys2 ) ) )
=> ( ! [X: a,Xs2: fset_a,Y2: a,Ys2: fset_a] :
( ( P @ Xs2 @ Ys2 )
=> ( ~ ( fmember_a @ X @ Xs2 )
=> ( ~ ( fmember_a @ Y2 @ Ys2 )
=> ( P @ ( finsert_a @ X @ Xs2 ) @ ( finsert_a @ Y2 @ Ys2 ) ) ) ) )
=> ( P @ Xsa @ Ysa ) ) ) ) ) ).
% fset_induct2
thf(fact_645_fsingleton__iff,axiom,
! [B2: a,A2: a] :
( ( fmember_a @ B2 @ ( finsert_a @ A2 @ bot_bot_fset_a ) )
= ( B2 = A2 ) ) ).
% fsingleton_iff
thf(fact_646_fset__strong__cases,axiom,
! [Xs3: fset_a] :
( ( Xs3 != bot_bot_fset_a )
=> ~ ! [Ys2: fset_a,X: a] :
( ~ ( fmember_a @ X @ Ys2 )
=> ( Xs3
!= ( finsert_a @ X @ Ys2 ) ) ) ) ).
% fset_strong_cases
thf(fact_647_fset__induct__stronger,axiom,
! [P: fset_a > $o,S2: fset_a] :
( ( P @ bot_bot_fset_a )
=> ( ! [X: a,S4: fset_a] :
( ~ ( fmember_a @ X @ S4 )
=> ( ( P @ S4 )
=> ( P @ ( finsert_a @ X @ S4 ) ) ) )
=> ( P @ S2 ) ) ) ).
% fset_induct_stronger
thf(fact_648_fsubset__fsingletonD,axiom,
! [A: fset_a,X2: a] :
( ( ord_less_eq_fset_a @ A @ ( finsert_a @ X2 @ bot_bot_fset_a ) )
=> ( ( A = bot_bot_fset_a )
| ( A
= ( finsert_a @ X2 @ bot_bot_fset_a ) ) ) ) ).
% fsubset_fsingletonD
thf(fact_649_funion__fsingleton__iff,axiom,
! [A: fset_a,B: fset_a,X2: a] :
( ( ( sup_sup_fset_a @ A @ B )
= ( finsert_a @ X2 @ bot_bot_fset_a ) )
= ( ( ( A = bot_bot_fset_a )
& ( B
= ( finsert_a @ X2 @ bot_bot_fset_a ) ) )
| ( ( A
= ( finsert_a @ X2 @ bot_bot_fset_a ) )
& ( B = bot_bot_fset_a ) )
| ( ( A
= ( finsert_a @ X2 @ bot_bot_fset_a ) )
& ( B
= ( finsert_a @ X2 @ bot_bot_fset_a ) ) ) ) ) ).
% funion_fsingleton_iff
thf(fact_650_fsingleton__funion__iff,axiom,
! [X2: a,A: fset_a,B: fset_a] :
( ( ( finsert_a @ X2 @ bot_bot_fset_a )
= ( sup_sup_fset_a @ A @ B ) )
= ( ( ( A = bot_bot_fset_a )
& ( B
= ( finsert_a @ X2 @ bot_bot_fset_a ) ) )
| ( ( A
= ( finsert_a @ X2 @ bot_bot_fset_a ) )
& ( B = bot_bot_fset_a ) )
| ( ( A
= ( finsert_a @ X2 @ bot_bot_fset_a ) )
& ( B
= ( finsert_a @ X2 @ bot_bot_fset_a ) ) ) ) ) ).
% fsingleton_funion_iff
thf(fact_651_finsert__is__funion,axiom,
( finsert_a
= ( ^ [A4: a] : ( sup_sup_fset_a @ ( finsert_a @ A4 @ bot_bot_fset_a ) ) ) ) ).
% finsert_is_funion
thf(fact_652_fminus__finsert2,axiom,
! [A: fset_a,A2: a,B: fset_a] :
( ( minus_minus_fset_a @ A @ ( finsert_a @ A2 @ B ) )
= ( minus_minus_fset_a @ ( minus_minus_fset_a @ A @ ( finsert_a @ A2 @ bot_bot_fset_a ) ) @ B ) ) ).
% fminus_finsert2
thf(fact_653_fminus__finsert,axiom,
! [A: fset_a,A2: a,B: fset_a] :
( ( minus_minus_fset_a @ A @ ( finsert_a @ A2 @ B ) )
= ( minus_minus_fset_a @ ( minus_minus_fset_a @ A @ B ) @ ( finsert_a @ A2 @ bot_bot_fset_a ) ) ) ).
% fminus_finsert
thf(fact_654_verit__comp__simplify1_I2_J,axiom,
! [A2: fset_a] : ( ord_less_eq_fset_a @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_655_verit__comp__simplify1_I2_J,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_656_verit__comp__simplify1_I2_J,axiom,
! [A2: int] : ( ord_less_eq_int @ A2 @ A2 ) ).
% verit_comp_simplify1(2)
thf(fact_657_verit__la__disequality,axiom,
! [A2: nat,B2: nat] :
( ( A2 = B2 )
| ~ ( ord_less_eq_nat @ A2 @ B2 )
| ~ ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% verit_la_disequality
thf(fact_658_verit__la__disequality,axiom,
! [A2: int,B2: int] :
( ( A2 = B2 )
| ~ ( ord_less_eq_int @ A2 @ B2 )
| ~ ( ord_less_eq_int @ B2 @ A2 ) ) ).
% verit_la_disequality
thf(fact_659_fset__linorder__min__induct,axiom,
! [P: fset_nat > $o,S2: fset_nat] :
( ( P @ bot_bot_fset_nat )
=> ( ! [X: nat,S4: fset_nat] :
( ! [Y5: nat] :
( ( fmember_nat @ Y5 @ S4 )
=> ( ord_less_nat @ X @ Y5 ) )
=> ( ( P @ S4 )
=> ( P @ ( finsert_nat @ X @ S4 ) ) ) )
=> ( P @ S2 ) ) ) ).
% fset_linorder_min_induct
thf(fact_660_fset__linorder__min__induct,axiom,
! [P: fset_int > $o,S2: fset_int] :
( ( P @ bot_bot_fset_int )
=> ( ! [X: int,S4: fset_int] :
( ! [Y5: int] :
( ( fmember_int @ Y5 @ S4 )
=> ( ord_less_int @ X @ Y5 ) )
=> ( ( P @ S4 )
=> ( P @ ( finsert_int @ X @ S4 ) ) ) )
=> ( P @ S2 ) ) ) ).
% fset_linorder_min_induct
thf(fact_661_fset__linorder__max__induct,axiom,
! [P: fset_nat > $o,S2: fset_nat] :
( ( P @ bot_bot_fset_nat )
=> ( ! [X: nat,S4: fset_nat] :
( ! [Y5: nat] :
( ( fmember_nat @ Y5 @ S4 )
=> ( ord_less_nat @ Y5 @ X ) )
=> ( ( P @ S4 )
=> ( P @ ( finsert_nat @ X @ S4 ) ) ) )
=> ( P @ S2 ) ) ) ).
% fset_linorder_max_induct
thf(fact_662_fset__linorder__max__induct,axiom,
! [P: fset_int > $o,S2: fset_int] :
( ( P @ bot_bot_fset_int )
=> ( ! [X: int,S4: fset_int] :
( ! [Y5: int] :
( ( fmember_int @ Y5 @ S4 )
=> ( ord_less_int @ Y5 @ X ) )
=> ( ( P @ S4 )
=> ( P @ ( finsert_int @ X @ S4 ) ) ) )
=> ( P @ S2 ) ) ) ).
% fset_linorder_max_induct
thf(fact_663_verit__comp__simplify1_I1_J,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_664_verit__comp__simplify1_I1_J,axiom,
! [A2: int] :
~ ( ord_less_int @ A2 @ A2 ) ).
% verit_comp_simplify1(1)
thf(fact_665_finsert__fminus,axiom,
! [A2: a,A: fset_a] :
( ( fmember_a @ A2 @ A )
=> ( ( finsert_a @ A2 @ ( minus_minus_fset_a @ A @ ( finsert_a @ A2 @ bot_bot_fset_a ) ) )
= A ) ) ).
% finsert_fminus
thf(fact_666_fminus__finsert__absorb,axiom,
! [X2: a,A: fset_a] :
( ~ ( fmember_a @ X2 @ A )
=> ( ( minus_minus_fset_a @ ( finsert_a @ X2 @ A ) @ ( finsert_a @ X2 @ bot_bot_fset_a ) )
= A ) ) ).
% fminus_finsert_absorb
thf(fact_667_fminus__single__finsert,axiom,
! [A: fset_a,X2: a,B: fset_a] :
( ( ord_less_eq_fset_a @ ( minus_minus_fset_a @ A @ ( finsert_a @ X2 @ bot_bot_fset_a ) ) @ B )
=> ( ord_less_eq_fset_a @ A @ ( finsert_a @ X2 @ B ) ) ) ).
% fminus_single_finsert
thf(fact_668_fcard__fminus1__le,axiom,
! [A: fset_a,X2: a] : ( ord_less_eq_nat @ ( fcard_a @ ( minus_minus_fset_a @ A @ ( finsert_a @ X2 @ bot_bot_fset_a ) ) ) @ ( fcard_a @ A ) ) ).
% fcard_fminus1_le
thf(fact_669_fsubset__finsert__iff,axiom,
! [A: fset_a,X2: a,B: fset_a] :
( ( ord_less_eq_fset_a @ A @ ( finsert_a @ X2 @ B ) )
= ( ( ( fmember_a @ X2 @ A )
=> ( ord_less_eq_fset_a @ ( minus_minus_fset_a @ A @ ( finsert_a @ X2 @ bot_bot_fset_a ) ) @ B ) )
& ( ~ ( fmember_a @ X2 @ A )
=> ( ord_less_eq_fset_a @ A @ B ) ) ) ) ).
% fsubset_finsert_iff
thf(fact_670_fcard__fminus2__less,axiom,
! [X2: a,A: fset_a,Y: a] :
( ( fmember_a @ X2 @ A )
=> ( ( fmember_a @ Y @ A )
=> ( ord_less_nat @ ( fcard_a @ ( minus_minus_fset_a @ ( minus_minus_fset_a @ A @ ( finsert_a @ X2 @ bot_bot_fset_a ) ) @ ( finsert_a @ Y @ bot_bot_fset_a ) ) ) @ ( fcard_a @ A ) ) ) ) ).
% fcard_fminus2_less
thf(fact_671_fthe__felem__eq,axiom,
! [X2: a] :
( ( fthe_elem_a @ ( finsert_a @ X2 @ bot_bot_fset_a ) )
= X2 ) ).
% fthe_felem_eq
thf(fact_672_fcard__Suc__fminus1,axiom,
! [X2: a,A: fset_a] :
( ( fmember_a @ X2 @ A )
=> ( ( suc @ ( fcard_a @ ( minus_minus_fset_a @ A @ ( finsert_a @ X2 @ bot_bot_fset_a ) ) ) )
= ( fcard_a @ A ) ) ) ).
% fcard_Suc_fminus1
thf(fact_673_fcard__finsert,axiom,
! [X2: a,A: fset_a] :
( ( fcard_a @ ( finsert_a @ X2 @ A ) )
= ( suc @ ( fcard_a @ ( minus_minus_fset_a @ A @ ( finsert_a @ X2 @ bot_bot_fset_a ) ) ) ) ) ).
% fcard_finsert
thf(fact_674_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_675_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_676_lessI,axiom,
! [N2: nat] : ( ord_less_nat @ N2 @ ( suc @ N2 ) ) ).
% lessI
thf(fact_677_Suc__mono,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N2 ) ) ) ).
% Suc_mono
thf(fact_678_Suc__less__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_less_eq
thf(fact_679_Suc__le__mono,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( suc @ M2 ) )
= ( ord_less_eq_nat @ N2 @ M2 ) ) ).
% Suc_le_mono
thf(fact_680_diff__Suc__Suc,axiom,
! [M2: nat,N2: nat] :
( ( minus_minus_nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ).
% diff_Suc_Suc
thf(fact_681_Suc__diff__diff,axiom,
! [M2: nat,N2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N2 ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ N2 ) @ K ) ) ).
% Suc_diff_diff
thf(fact_682_diff__Suc__1,axiom,
! [N2: nat] :
( ( minus_minus_nat @ ( suc @ N2 ) @ one_one_nat )
= N2 ) ).
% diff_Suc_1
thf(fact_683_n__not__Suc__n,axiom,
! [N2: nat] :
( N2
!= ( suc @ N2 ) ) ).
% n_not_Suc_n
thf(fact_684_Suc__inject,axiom,
! [X2: nat,Y: nat] :
( ( ( suc @ X2 )
= ( suc @ Y ) )
=> ( X2 = Y ) ) ).
% Suc_inject
thf(fact_685_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_686_Suc__lessD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ N2 )
=> ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_lessD
thf(fact_687_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_688_Suc__lessI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ( ( suc @ M2 )
!= N2 )
=> ( ord_less_nat @ ( suc @ M2 ) @ N2 ) ) ) ).
% Suc_lessI
thf(fact_689_less__SucE,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
=> ( ~ ( ord_less_nat @ M2 @ N2 )
=> ( M2 = N2 ) ) ) ).
% less_SucE
thf(fact_690_less__SucI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_nat @ M2 @ ( suc @ N2 ) ) ) ).
% less_SucI
thf(fact_691_Ex__less__Suc,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N2 ) )
& ( P @ I4 ) ) )
= ( ( P @ N2 )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
& ( P @ I4 ) ) ) ) ).
% Ex_less_Suc
thf(fact_692_less__Suc__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
= ( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) ) ) ).
% less_Suc_eq
thf(fact_693_not__less__eq,axiom,
! [M2: nat,N2: nat] :
( ( ~ ( ord_less_nat @ M2 @ N2 ) )
= ( ord_less_nat @ N2 @ ( suc @ M2 ) ) ) ).
% not_less_eq
thf(fact_694_All__less__Suc,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N2 ) )
=> ( P @ I4 ) ) )
= ( ( P @ N2 )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
=> ( P @ I4 ) ) ) ) ).
% All_less_Suc
thf(fact_695_Suc__less__eq2,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ N2 ) @ M2 )
= ( ? [M4: nat] :
( ( M2
= ( suc @ M4 ) )
& ( ord_less_nat @ N2 @ M4 ) ) ) ) ).
% Suc_less_eq2
thf(fact_696_less__antisym,axiom,
! [N2: nat,M2: nat] :
( ~ ( ord_less_nat @ N2 @ M2 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M2 ) )
=> ( M2 = N2 ) ) ) ).
% less_antisym
thf(fact_697_Suc__less__SucD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_less_SucD
thf(fact_698_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_699_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_700_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_701_not__less__less__Suc__eq,axiom,
! [N2: nat,M2: nat] :
( ~ ( ord_less_nat @ N2 @ M2 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M2 ) )
= ( N2 = M2 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_702_transitive__stepwise__le,axiom,
! [M2: nat,N2: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ! [X: nat] : ( R @ X @ X )
=> ( ! [X: nat,Y2: nat,Z3: nat] :
( ( R @ X @ Y2 )
=> ( ( R @ Y2 @ Z3 )
=> ( R @ X @ Z3 ) ) )
=> ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
=> ( R @ M2 @ N2 ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_703_nat__induct__at__least,axiom,
! [M2: nat,N2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( P @ M2 )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct_at_least
thf(fact_704_full__nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N2 ) ) ).
% full_nat_induct
thf(fact_705_not__less__eq__eq,axiom,
! [M2: nat,N2: nat] :
( ( ~ ( ord_less_eq_nat @ M2 @ N2 ) )
= ( ord_less_eq_nat @ ( suc @ N2 ) @ M2 ) ) ).
% not_less_eq_eq
thf(fact_706_Suc__n__not__le__n,axiom,
! [N2: nat] :
~ ( ord_less_eq_nat @ ( suc @ N2 ) @ N2 ) ).
% Suc_n_not_le_n
thf(fact_707_le__Suc__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) )
= ( ( ord_less_eq_nat @ M2 @ N2 )
| ( M2
= ( suc @ N2 ) ) ) ) ).
% le_Suc_eq
thf(fact_708_Suc__le__D,axiom,
! [N2: nat,M5: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ M5 )
=> ? [M6: nat] :
( M5
= ( suc @ M6 ) ) ) ).
% Suc_le_D
thf(fact_709_le__SucI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) ) ) ).
% le_SucI
thf(fact_710_le__SucE,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) )
=> ( ~ ( ord_less_eq_nat @ M2 @ N2 )
=> ( M2
= ( suc @ N2 ) ) ) ) ).
% le_SucE
thf(fact_711_Suc__leD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% Suc_leD
thf(fact_712_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_713_lift__Suc__antimono__le,axiom,
! [F: nat > fset_a,N2: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_fset_a @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ord_less_eq_fset_a @ ( F @ N4 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_714_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N2: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_715_lift__Suc__antimono__le,axiom,
! [F: nat > int,N2: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ord_less_eq_int @ ( F @ N4 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_716_lift__Suc__mono__le,axiom,
! [F: nat > fset_a,N2: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_fset_a @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ord_less_eq_fset_a @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_717_lift__Suc__mono__le,axiom,
! [F: nat > nat,N2: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_718_lift__Suc__mono__le,axiom,
! [F: nat > int,N2: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_719_lift__Suc__mono__less,axiom,
! [F: nat > nat,N2: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N2 @ N4 )
=> ( ord_less_nat @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_720_lift__Suc__mono__less,axiom,
! [F: nat > int,N2: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N2 @ N4 )
=> ( ord_less_int @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_721_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N2: nat,M2: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ ( F @ N2 ) @ ( F @ M2 ) )
= ( ord_less_nat @ N2 @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_722_lift__Suc__mono__less__iff,axiom,
! [F: nat > int,N2: nat,M2: nat] :
( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_int @ ( F @ N2 ) @ ( F @ M2 ) )
= ( ord_less_nat @ N2 @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_723_le__imp__less__Suc,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_nat @ M2 @ ( suc @ N2 ) ) ) ).
% le_imp_less_Suc
thf(fact_724_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N: nat] : ( ord_less_eq_nat @ ( suc @ N ) ) ) ) ).
% less_eq_Suc_le
thf(fact_725_less__Suc__eq__le,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% less_Suc_eq_le
thf(fact_726_le__less__Suc__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M2 ) )
= ( N2 = M2 ) ) ) ).
% le_less_Suc_eq
thf(fact_727_Suc__le__lessD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
=> ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_le_lessD
thf(fact_728_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_729_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_730_Suc__le__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_le_eq
thf(fact_731_Suc__leI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 ) ) ).
% Suc_leI
thf(fact_732_diff__less__Suc,axiom,
! [M2: nat,N2: nat] : ( ord_less_nat @ ( minus_minus_nat @ M2 @ N2 ) @ ( suc @ M2 ) ) ).
% diff_less_Suc
thf(fact_733_Suc__diff__Suc,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ N2 @ M2 )
=> ( ( suc @ ( minus_minus_nat @ M2 @ ( suc @ N2 ) ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ) ).
% Suc_diff_Suc
thf(fact_734_Suc__diff__le,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_eq_nat @ N2 @ M2 )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N2 )
= ( suc @ ( minus_minus_nat @ M2 @ N2 ) ) ) ) ).
% Suc_diff_le
thf(fact_735_diff__Suc__eq__diff__pred,axiom,
! [M2: nat,N2: nat] :
( ( minus_minus_nat @ M2 @ ( suc @ N2 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N2 ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_736_fset__card__induct,axiom,
! [P: fset_a > $o,S2: fset_a] :
( ( P @ bot_bot_fset_a )
=> ( ! [S4: fset_a,T2: fset_a] :
( ( ( suc @ ( fcard_a @ S4 ) )
= ( fcard_a @ T2 ) )
=> ( ( P @ S4 )
=> ( P @ T2 ) ) )
=> ( P @ S2 ) ) ) ).
% fset_card_induct
thf(fact_737_fMax__eqI,axiom,
! [A: fset_nat,X2: nat] :
( ! [Y2: nat] :
( ( fmember_nat @ Y2 @ A )
=> ( ord_less_eq_nat @ Y2 @ X2 ) )
=> ( ( fmember_nat @ X2 @ A )
=> ( ( linorder_fMax_nat @ A )
= X2 ) ) ) ).
% fMax_eqI
thf(fact_738_fMax__eqI,axiom,
! [A: fset_int,X2: int] :
( ! [Y2: int] :
( ( fmember_int @ Y2 @ A )
=> ( ord_less_eq_int @ Y2 @ X2 ) )
=> ( ( fmember_int @ X2 @ A )
=> ( ( linorder_fMax_int @ A )
= X2 ) ) ) ).
% fMax_eqI
thf(fact_739_fMax__ge,axiom,
! [X2: nat,A: fset_nat] :
( ( fmember_nat @ X2 @ A )
=> ( ord_less_eq_nat @ X2 @ ( linorder_fMax_nat @ A ) ) ) ).
% fMax_ge
thf(fact_740_fMax__ge,axiom,
! [X2: int,A: fset_int] :
( ( fmember_int @ X2 @ A )
=> ( ord_less_eq_int @ X2 @ ( linorder_fMax_int @ A ) ) ) ).
% fMax_ge
thf(fact_741_boolean__algebra__cancel_Osup1,axiom,
! [A: fset_a,K: fset_a,A2: fset_a,B2: fset_a] :
( ( A
= ( sup_sup_fset_a @ K @ A2 ) )
=> ( ( sup_sup_fset_a @ A @ B2 )
= ( sup_sup_fset_a @ K @ ( sup_sup_fset_a @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_742_boolean__algebra__cancel_Osup1,axiom,
! [A: nat,K: nat,A2: nat,B2: nat] :
( ( A
= ( sup_sup_nat @ K @ A2 ) )
=> ( ( sup_sup_nat @ A @ B2 )
= ( sup_sup_nat @ K @ ( sup_sup_nat @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_743_boolean__algebra__cancel_Osup2,axiom,
! [B: fset_a,K: fset_a,B2: fset_a,A2: fset_a] :
( ( B
= ( sup_sup_fset_a @ K @ B2 ) )
=> ( ( sup_sup_fset_a @ A2 @ B )
= ( sup_sup_fset_a @ K @ ( sup_sup_fset_a @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_744_boolean__algebra__cancel_Osup2,axiom,
! [B: nat,K: nat,B2: nat,A2: nat] :
( ( B
= ( sup_sup_nat @ K @ B2 ) )
=> ( ( sup_sup_nat @ A2 @ B )
= ( sup_sup_nat @ K @ ( sup_sup_nat @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_745_fcard__finsert__if,axiom,
! [X2: a,A: fset_a] :
( ( ( fmember_a @ X2 @ A )
=> ( ( fcard_a @ ( finsert_a @ X2 @ A ) )
= ( fcard_a @ A ) ) )
& ( ~ ( fmember_a @ X2 @ A )
=> ( ( fcard_a @ ( finsert_a @ X2 @ A ) )
= ( suc @ ( fcard_a @ A ) ) ) ) ) ).
% fcard_finsert_if
thf(fact_746_fcard__finsert__disjoint,axiom,
! [X2: a,A: fset_a] :
( ~ ( fmember_a @ X2 @ A )
=> ( ( fcard_a @ ( finsert_a @ X2 @ A ) )
= ( suc @ ( fcard_a @ A ) ) ) ) ).
% fcard_finsert_disjoint
thf(fact_747_minimal__fixpoint__helper_I2_J,axiom,
! [F: nat > nat,P: nat > $o,K: nat,X2: nat,X6: nat] :
( ( F
= ( ^ [X3: nat] : ( if_nat @ ( P @ X3 ) @ X3 @ ( F @ ( suc @ X3 ) ) ) ) )
=> ( ! [X: nat] :
( ( ord_less_eq_nat @ K @ X )
=> ( P @ X ) )
=> ( ( ord_less_eq_nat @ X2 @ X6 )
=> ( ( ord_less_nat @ X6 @ ( F @ X2 ) )
=> ~ ( P @ X6 ) ) ) ) ) ).
% minimal_fixpoint_helper(2)
thf(fact_748_minimal__fixpoint__helper_I1_J,axiom,
! [F: nat > nat,P: nat > $o,K: nat,X2: nat] :
( ( F
= ( ^ [X3: nat] : ( if_nat @ ( P @ X3 ) @ X3 @ ( F @ ( suc @ X3 ) ) ) ) )
=> ( ! [X: nat] :
( ( ord_less_eq_nat @ K @ X )
=> ( P @ X ) )
=> ( P @ ( F @ X2 ) ) ) ) ).
% minimal_fixpoint_helper(1)
thf(fact_749_recursion__renaming__helper,axiom,
! [F1: nat > nat,P: nat > $o,F2: nat > nat,K: nat] :
( ( F1
= ( ^ [X3: nat] : ( if_nat @ ( P @ X3 ) @ X3 @ ( F1 @ ( suc @ X3 ) ) ) ) )
=> ( ( F2
= ( ^ [X3: nat] : ( if_nat @ ( P @ X3 ) @ X3 @ ( F2 @ ( suc @ X3 ) ) ) ) )
=> ( ! [X: nat] :
( ( ord_less_eq_nat @ K @ X )
=> ( P @ X ) )
=> ( F1 = F2 ) ) ) ) ).
% recursion_renaming_helper
thf(fact_750_Suc__diff__1,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) )
= N2 ) ) ).
% Suc_diff_1
thf(fact_751_le__zero__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_752_not__gr__zero,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_753_diff__self,axiom,
! [A2: int] :
( ( minus_minus_int @ A2 @ A2 )
= zero_zero_int ) ).
% diff_self
thf(fact_754_diff__0__right,axiom,
! [A2: int] :
( ( minus_minus_int @ A2 @ zero_zero_int )
= A2 ) ).
% diff_0_right
thf(fact_755_zero__diff,axiom,
! [A2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A2 )
= zero_zero_nat ) ).
% zero_diff
thf(fact_756_diff__zero,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% diff_zero
thf(fact_757_diff__zero,axiom,
! [A2: int] :
( ( minus_minus_int @ A2 @ zero_zero_int )
= A2 ) ).
% diff_zero
thf(fact_758_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: nat] :
( ( minus_minus_nat @ A2 @ A2 )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_759_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A2: int] :
( ( minus_minus_int @ A2 @ A2 )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_760_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_761_neq0__conv,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% neq0_conv
thf(fact_762_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_763_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_764_le0,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% le0
thf(fact_765_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_766_diff__0__eq__0,axiom,
! [N2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_767_diff__ge__0__iff__ge,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A2 @ B2 ) )
= ( ord_less_eq_int @ B2 @ A2 ) ) ).
% diff_ge_0_iff_ge
thf(fact_768_diff__gt__0__iff__gt,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A2 @ B2 ) )
= ( ord_less_int @ B2 @ A2 ) ) ).
% diff_gt_0_iff_gt
thf(fact_769_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_770_less__Suc0,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ ( suc @ zero_zero_nat ) )
= ( N2 = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_771_zero__less__Suc,axiom,
! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N2 ) ) ).
% zero_less_Suc
thf(fact_772_zero__less__diff,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% zero_less_diff
thf(fact_773_diff__is__0__eq,axiom,
! [M2: nat,N2: nat] :
( ( ( minus_minus_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% diff_is_0_eq
thf(fact_774_diff__is__0__eq_H,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( minus_minus_nat @ M2 @ N2 )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_775_less__one,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ one_one_nat )
= ( N2 = zero_zero_nat ) ) ).
% less_one
thf(fact_776_Suc__pred,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( suc @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) )
= N2 ) ) ).
% Suc_pred
thf(fact_777_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_778_zero__reorient,axiom,
! [X2: nat] :
( ( zero_zero_nat = X2 )
= ( X2 = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_779_zero__reorient,axiom,
! [X2: int] :
( ( zero_zero_int = X2 )
= ( X2 = zero_zero_int ) ) ).
% zero_reorient
thf(fact_780_zero__le,axiom,
! [X2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X2 ) ).
% zero_le
thf(fact_781_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_782_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_783_gr__zeroI,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr_zeroI
thf(fact_784_not__less__zero,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less_zero
thf(fact_785_gr__implies__not__zero,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_786_zero__less__iff__neq__zero,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( N2 != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_787_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_788_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_789_eq__iff__diff__eq__0,axiom,
( ( ^ [Y3: int,Z: int] : ( Y3 = Z ) )
= ( ^ [A4: int,B5: int] :
( ( minus_minus_int @ A4 @ B5 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_790_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_791_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_792_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_793_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_794_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_795_nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N2 ) ) ) ).
% nat_induct
thf(fact_796_diff__induct,axiom,
! [P: nat > nat > $o,M2: nat,N2: nat] :
( ! [X: nat] : ( P @ X @ zero_zero_nat )
=> ( ! [Y2: nat] : ( P @ zero_zero_nat @ ( suc @ Y2 ) )
=> ( ! [X: nat,Y2: nat] :
( ( P @ X @ Y2 )
=> ( P @ ( suc @ X ) @ ( suc @ Y2 ) ) )
=> ( P @ M2 @ N2 ) ) ) ) ).
% diff_induct
thf(fact_797_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_798_Suc__neq__Zero,axiom,
! [M2: nat] :
( ( suc @ M2 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_799_Zero__neq__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_neq_Suc
thf(fact_800_Zero__not__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_not_Suc
thf(fact_801_not0__implies__Suc,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ? [M6: nat] :
( N2
= ( suc @ M6 ) ) ) ).
% not0_implies_Suc
thf(fact_802_infinite__descent0,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N2 ) ) ) ).
% infinite_descent0
thf(fact_803_gr__implies__not0,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_804_less__zeroE,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_805_not__less0,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less0
thf(fact_806_not__gr0,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_807_gr0I,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr0I
thf(fact_808_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_809_less__eq__nat_Osimps_I1_J,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% less_eq_nat.simps(1)
thf(fact_810_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_811_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_812_le__0__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_813_diffs0__imp__equal,axiom,
! [M2: nat,N2: nat] :
( ( ( minus_minus_nat @ M2 @ N2 )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N2 @ M2 )
= zero_zero_nat )
=> ( M2 = N2 ) ) ) ).
% diffs0_imp_equal
thf(fact_814_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% minus_nat.diff_0
thf(fact_815_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B5: int] : ( ord_less_eq_int @ ( minus_minus_int @ A4 @ B5 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_816_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_817_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_818_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A4: int,B5: int] : ( ord_less_int @ ( minus_minus_int @ A4 @ B5 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_819_Ex__less__Suc2,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N2 ) )
& ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
& ( P @ ( suc @ I4 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_820_gr0__conv__Suc,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( ? [M: nat] :
( N2
= ( suc @ M ) ) ) ) ).
% gr0_conv_Suc
thf(fact_821_All__less__Suc2,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N2 ) )
=> ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
=> ( P @ ( suc @ I4 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_822_gr0__implies__Suc,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ? [M6: nat] :
( N2
= ( suc @ M6 ) ) ) ).
% gr0_implies_Suc
thf(fact_823_less__Suc__eq__0__disj,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
= ( ( M2 = zero_zero_nat )
| ? [J3: nat] :
( ( M2
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N2 ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_824_ex__least__nat__le,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N2 )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_825_diff__less,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( minus_minus_nat @ M2 @ N2 ) @ M2 ) ) ) ).
% diff_less
thf(fact_826_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_827_fcard__fempty,axiom,
( ( fcard_a @ bot_bot_fset_a )
= zero_zero_nat ) ).
% fcard_fempty
thf(fact_828_monotone__function__with__limit__witness__helper,axiom,
! [F: nat > nat,K: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_eq_nat @ I2 @ J2 )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ( F @ I2 )
= ( F @ J2 ) )
=> ! [M6: nat] :
( ( ord_less_eq_nat @ J2 @ M6 )
=> ( ( F @ I2 )
= ( F @ M6 ) ) ) ) )
=> ( ! [I2: nat] : ( ord_less_eq_nat @ ( F @ I2 ) @ K )
=> ~ ! [X: nat] :
( ( ( F @ ( suc @ X ) )
= ( F @ X ) )
=> ~ ( ord_less_eq_nat @ X @ ( minus_minus_nat @ K @ ( F @ zero_zero_nat ) ) ) ) ) ) ) ).
% monotone_function_with_limit_witness_helper
thf(fact_829_ex__least__nat__less,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N2 )
& ! [I3: nat] :
( ( ord_less_eq_nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_830_diff__Suc__less,axiom,
! [N2: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ ( minus_minus_nat @ N2 @ ( suc @ I ) ) @ N2 ) ) ).
% diff_Suc_less
thf(fact_831_nat__induct__non__zero,axiom,
! [N2: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct_non_zero
thf(fact_832_Suc__pred_H,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( N2
= ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_833_Suc__diff__eq__diff__pred,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N2 )
= ( minus_minus_nat @ M2 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_834_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_835_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_836_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_837_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_838_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_839_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_840_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_841_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% linordered_nonzero_semiring_class.zero_le_one
thf(fact_842_linorder__neqE__linordered__idom,axiom,
! [X2: int,Y: int] :
( ( X2 != Y )
=> ( ~ ( ord_less_int @ X2 @ Y )
=> ( ord_less_int @ Y @ X2 ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_843_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_844_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_845_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_846_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_le_one
thf(fact_847_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
= one_one_int ) ).
% dbl_inc_simps(2)
thf(fact_848_of__nat__0__less__iff,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N2 ) )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% of_nat_0_less_iff
thf(fact_849_of__nat__0__less__iff,axiom,
! [N2: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% of_nat_0_less_iff
thf(fact_850_fMax_Oinsert__not__elem,axiom,
! [X2: nat,A: fset_nat] :
( ~ ( fmember_nat @ X2 @ A )
=> ( ( A != bot_bot_fset_nat )
=> ( ( linorder_fMax_nat @ ( finsert_nat @ X2 @ A ) )
= ( ord_max_nat @ X2 @ ( linorder_fMax_nat @ A ) ) ) ) ) ).
% fMax.insert_not_elem
thf(fact_851_max__Suc__Suc,axiom,
! [M2: nat,N2: nat] :
( ( ord_max_nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
= ( suc @ ( ord_max_nat @ M2 @ N2 ) ) ) ).
% max_Suc_Suc
thf(fact_852_max__nat_Oeq__neutr__iff,axiom,
! [A2: nat,B2: nat] :
( ( ( ord_max_nat @ A2 @ B2 )
= zero_zero_nat )
= ( ( A2 = zero_zero_nat )
& ( B2 = zero_zero_nat ) ) ) ).
% max_nat.eq_neutr_iff
thf(fact_853_max__nat_Oleft__neutral,axiom,
! [A2: nat] :
( ( ord_max_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% max_nat.left_neutral
thf(fact_854_max__nat_Oneutr__eq__iff,axiom,
! [A2: nat,B2: nat] :
( ( zero_zero_nat
= ( ord_max_nat @ A2 @ B2 ) )
= ( ( A2 = zero_zero_nat )
& ( B2 = zero_zero_nat ) ) ) ).
% max_nat.neutr_eq_iff
thf(fact_855_max__nat_Oright__neutral,axiom,
! [A2: nat] :
( ( ord_max_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% max_nat.right_neutral
thf(fact_856_max__0L,axiom,
! [N2: nat] :
( ( ord_max_nat @ zero_zero_nat @ N2 )
= N2 ) ).
% max_0L
thf(fact_857_max__0R,axiom,
! [N2: nat] :
( ( ord_max_nat @ N2 @ zero_zero_nat )
= N2 ) ).
% max_0R
thf(fact_858_of__nat__eq__iff,axiom,
! [M2: nat,N2: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= ( semiri1314217659103216013at_int @ N2 ) )
= ( M2 = N2 ) ) ).
% of_nat_eq_iff
thf(fact_859_max_Oidem,axiom,
! [A2: nat] :
( ( ord_max_nat @ A2 @ A2 )
= A2 ) ).
% max.idem
thf(fact_860_max_Oleft__idem,axiom,
! [A2: nat,B2: nat] :
( ( ord_max_nat @ A2 @ ( ord_max_nat @ A2 @ B2 ) )
= ( ord_max_nat @ A2 @ B2 ) ) ).
% max.left_idem
thf(fact_861_max_Oright__idem,axiom,
! [A2: nat,B2: nat] :
( ( ord_max_nat @ ( ord_max_nat @ A2 @ B2 ) @ B2 )
= ( ord_max_nat @ A2 @ B2 ) ) ).
% max.right_idem
thf(fact_862_max_Obounded__iff,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( ord_max_nat @ B2 @ C ) @ A2 )
= ( ( ord_less_eq_nat @ B2 @ A2 )
& ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% max.bounded_iff
thf(fact_863_max_Obounded__iff,axiom,
! [B2: int,C: int,A2: int] :
( ( ord_less_eq_int @ ( ord_max_int @ B2 @ C ) @ A2 )
= ( ( ord_less_eq_int @ B2 @ A2 )
& ( ord_less_eq_int @ C @ A2 ) ) ) ).
% max.bounded_iff
thf(fact_864_max_Oabsorb2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_max_nat @ A2 @ B2 )
= B2 ) ) ).
% max.absorb2
thf(fact_865_max_Oabsorb2,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_max_int @ A2 @ B2 )
= B2 ) ) ).
% max.absorb2
thf(fact_866_max_Oabsorb1,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_max_nat @ A2 @ B2 )
= A2 ) ) ).
% max.absorb1
thf(fact_867_max_Oabsorb1,axiom,
! [B2: int,A2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ( ord_max_int @ A2 @ B2 )
= A2 ) ) ).
% max.absorb1
thf(fact_868_max_Oabsorb3,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
=> ( ( ord_max_nat @ A2 @ B2 )
= A2 ) ) ).
% max.absorb3
thf(fact_869_max_Oabsorb3,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ B2 @ A2 )
=> ( ( ord_max_int @ A2 @ B2 )
= A2 ) ) ).
% max.absorb3
thf(fact_870_max_Oabsorb4,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
=> ( ( ord_max_nat @ A2 @ B2 )
= B2 ) ) ).
% max.absorb4
thf(fact_871_max_Oabsorb4,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ B2 )
=> ( ( ord_max_int @ A2 @ B2 )
= B2 ) ) ).
% max.absorb4
thf(fact_872_max__less__iff__conj,axiom,
! [X2: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ ( ord_max_nat @ X2 @ Y ) @ Z2 )
= ( ( ord_less_nat @ X2 @ Z2 )
& ( ord_less_nat @ Y @ Z2 ) ) ) ).
% max_less_iff_conj
thf(fact_873_max__less__iff__conj,axiom,
! [X2: int,Y: int,Z2: int] :
( ( ord_less_int @ ( ord_max_int @ X2 @ Y ) @ Z2 )
= ( ( ord_less_int @ X2 @ Z2 )
& ( ord_less_int @ Y @ Z2 ) ) ) ).
% max_less_iff_conj
thf(fact_874_max__bot,axiom,
! [X2: nat] :
( ( ord_max_nat @ bot_bot_nat @ X2 )
= X2 ) ).
% max_bot
thf(fact_875_max__bot2,axiom,
! [X2: nat] :
( ( ord_max_nat @ X2 @ bot_bot_nat )
= X2 ) ).
% max_bot2
thf(fact_876_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_877_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_878_of__nat__0__eq__iff,axiom,
! [N2: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N2 ) )
= ( zero_zero_nat = N2 ) ) ).
% of_nat_0_eq_iff
thf(fact_879_of__nat__0__eq__iff,axiom,
! [N2: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N2 ) )
= ( zero_zero_nat = N2 ) ) ).
% of_nat_0_eq_iff
thf(fact_880_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri1316708129612266289at_nat @ M2 )
= zero_zero_nat )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_881_of__nat__eq__0__iff,axiom,
! [M2: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= zero_zero_int )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_882_of__nat__le__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% of_nat_le_iff
thf(fact_883_of__nat__le__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% of_nat_le_iff
thf(fact_884_of__nat__less__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% of_nat_less_iff
thf(fact_885_of__nat__less__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% of_nat_less_iff
thf(fact_886_max__0__1_I1_J,axiom,
( ( ord_max_int @ zero_zero_int @ one_one_int )
= one_one_int ) ).
% max_0_1(1)
thf(fact_887_max__0__1_I1_J,axiom,
( ( ord_max_nat @ zero_zero_nat @ one_one_nat )
= one_one_nat ) ).
% max_0_1(1)
thf(fact_888_max__0__1_I2_J,axiom,
( ( ord_max_int @ one_one_int @ zero_zero_int )
= one_one_int ) ).
% max_0_1(2)
thf(fact_889_max__0__1_I2_J,axiom,
( ( ord_max_nat @ one_one_nat @ zero_zero_nat )
= one_one_nat ) ).
% max_0_1(2)
thf(fact_890_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_891_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_892_of__nat__1__eq__iff,axiom,
! [N2: nat] :
( ( one_one_nat
= ( semiri1316708129612266289at_nat @ N2 ) )
= ( N2 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_893_of__nat__1__eq__iff,axiom,
! [N2: nat] :
( ( one_one_int
= ( semiri1314217659103216013at_int @ N2 ) )
= ( N2 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_894_of__nat__eq__1__iff,axiom,
! [N2: nat] :
( ( ( semiri1316708129612266289at_nat @ N2 )
= one_one_nat )
= ( N2 = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_895_of__nat__eq__1__iff,axiom,
! [N2: nat] :
( ( ( semiri1314217659103216013at_int @ N2 )
= one_one_int )
= ( N2 = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_896_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_897_of__nat__le__0__iff,axiom,
! [M2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int )
= ( M2 = zero_zero_nat ) ) ).
% of_nat_le_0_iff
thf(fact_898_fMax_Oinsert,axiom,
! [A: fset_nat,X2: nat] :
( ( A != bot_bot_fset_nat )
=> ( ( linorder_fMax_nat @ ( finsert_nat @ X2 @ A ) )
= ( ord_max_nat @ X2 @ ( linorder_fMax_nat @ A ) ) ) ) ).
% fMax.insert
thf(fact_899_fMax__finsert,axiom,
! [A: fset_nat,X2: nat] :
( ( ( A = bot_bot_fset_nat )
=> ( ( linorder_fMax_nat @ ( finsert_nat @ X2 @ A ) )
= X2 ) )
& ( ( A != bot_bot_fset_nat )
=> ( ( linorder_fMax_nat @ ( finsert_nat @ X2 @ A ) )
= ( ord_max_nat @ X2 @ ( linorder_fMax_nat @ A ) ) ) ) ) ).
% fMax_finsert
thf(fact_900_int__ops_I6_J,axiom,
! [A2: nat,B2: nat] :
( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A2 @ B2 ) )
= zero_zero_int ) )
& ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A2 @ B2 ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).
% int_ops(6)
thf(fact_901_max__absorb2,axiom,
! [X2: fset_a,Y: fset_a] :
( ( ord_less_eq_fset_a @ X2 @ Y )
=> ( ( ord_max_fset_a @ X2 @ Y )
= Y ) ) ).
% max_absorb2
thf(fact_902_max__absorb2,axiom,
! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ( ord_max_nat @ X2 @ Y )
= Y ) ) ).
% max_absorb2
thf(fact_903_max__absorb2,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ( ord_max_int @ X2 @ Y )
= Y ) ) ).
% max_absorb2
thf(fact_904_max__absorb1,axiom,
! [Y: fset_a,X2: fset_a] :
( ( ord_less_eq_fset_a @ Y @ X2 )
=> ( ( ord_max_fset_a @ X2 @ Y )
= X2 ) ) ).
% max_absorb1
thf(fact_905_max__absorb1,axiom,
! [Y: nat,X2: nat] :
( ( ord_less_eq_nat @ Y @ X2 )
=> ( ( ord_max_nat @ X2 @ Y )
= X2 ) ) ).
% max_absorb1
thf(fact_906_max__absorb1,axiom,
! [Y: int,X2: int] :
( ( ord_less_eq_int @ Y @ X2 )
=> ( ( ord_max_int @ X2 @ Y )
= X2 ) ) ).
% max_absorb1
thf(fact_907_max__def,axiom,
( ord_max_fset_a
= ( ^ [A4: fset_a,B5: fset_a] : ( if_fset_a @ ( ord_less_eq_fset_a @ A4 @ B5 ) @ B5 @ A4 ) ) ) ).
% max_def
thf(fact_908_max__def,axiom,
( ord_max_nat
= ( ^ [A4: nat,B5: nat] : ( if_nat @ ( ord_less_eq_nat @ A4 @ B5 ) @ B5 @ A4 ) ) ) ).
% max_def
thf(fact_909_max__def,axiom,
( ord_max_int
= ( ^ [A4: int,B5: int] : ( if_int @ ( ord_less_eq_int @ A4 @ B5 ) @ B5 @ A4 ) ) ) ).
% max_def
thf(fact_910_max_OcoboundedI2,axiom,
! [C: nat,B2: nat,A2: nat] :
( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ ( ord_max_nat @ A2 @ B2 ) ) ) ).
% max.coboundedI2
thf(fact_911_max_OcoboundedI2,axiom,
! [C: int,B2: int,A2: int] :
( ( ord_less_eq_int @ C @ B2 )
=> ( ord_less_eq_int @ C @ ( ord_max_int @ A2 @ B2 ) ) ) ).
% max.coboundedI2
thf(fact_912_max_OcoboundedI1,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ C @ ( ord_max_nat @ A2 @ B2 ) ) ) ).
% max.coboundedI1
thf(fact_913_max_OcoboundedI1,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_eq_int @ C @ A2 )
=> ( ord_less_eq_int @ C @ ( ord_max_int @ A2 @ B2 ) ) ) ).
% max.coboundedI1
thf(fact_914_max_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B5: nat] :
( ( ord_max_nat @ A4 @ B5 )
= B5 ) ) ) ).
% max.absorb_iff2
thf(fact_915_max_Oabsorb__iff2,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B5: int] :
( ( ord_max_int @ A4 @ B5 )
= B5 ) ) ) ).
% max.absorb_iff2
thf(fact_916_max_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A4: nat] :
( ( ord_max_nat @ A4 @ B5 )
= A4 ) ) ) ).
% max.absorb_iff1
thf(fact_917_max_Oabsorb__iff1,axiom,
( ord_less_eq_int
= ( ^ [B5: int,A4: int] :
( ( ord_max_int @ A4 @ B5 )
= A4 ) ) ) ).
% max.absorb_iff1
thf(fact_918_le__max__iff__disj,axiom,
! [Z2: nat,X2: nat,Y: nat] :
( ( ord_less_eq_nat @ Z2 @ ( ord_max_nat @ X2 @ Y ) )
= ( ( ord_less_eq_nat @ Z2 @ X2 )
| ( ord_less_eq_nat @ Z2 @ Y ) ) ) ).
% le_max_iff_disj
thf(fact_919_le__max__iff__disj,axiom,
! [Z2: int,X2: int,Y: int] :
( ( ord_less_eq_int @ Z2 @ ( ord_max_int @ X2 @ Y ) )
= ( ( ord_less_eq_int @ Z2 @ X2 )
| ( ord_less_eq_int @ Z2 @ Y ) ) ) ).
% le_max_iff_disj
thf(fact_920_max_Ocobounded2,axiom,
! [B2: nat,A2: nat] : ( ord_less_eq_nat @ B2 @ ( ord_max_nat @ A2 @ B2 ) ) ).
% max.cobounded2
thf(fact_921_max_Ocobounded2,axiom,
! [B2: int,A2: int] : ( ord_less_eq_int @ B2 @ ( ord_max_int @ A2 @ B2 ) ) ).
% max.cobounded2
thf(fact_922_max_Ocobounded1,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ A2 @ ( ord_max_nat @ A2 @ B2 ) ) ).
% max.cobounded1
thf(fact_923_max_Ocobounded1,axiom,
! [A2: int,B2: int] : ( ord_less_eq_int @ A2 @ ( ord_max_int @ A2 @ B2 ) ) ).
% max.cobounded1
thf(fact_924_max_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A4: nat] :
( A4
= ( ord_max_nat @ A4 @ B5 ) ) ) ) ).
% max.order_iff
thf(fact_925_max_Oorder__iff,axiom,
( ord_less_eq_int
= ( ^ [B5: int,A4: int] :
( A4
= ( ord_max_int @ A4 @ B5 ) ) ) ) ).
% max.order_iff
thf(fact_926_max_OboundedI,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ ( ord_max_nat @ B2 @ C ) @ A2 ) ) ) ).
% max.boundedI
thf(fact_927_max_OboundedI,axiom,
! [B2: int,A2: int,C: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ( ord_less_eq_int @ C @ A2 )
=> ( ord_less_eq_int @ ( ord_max_int @ B2 @ C ) @ A2 ) ) ) ).
% max.boundedI
thf(fact_928_max_OboundedE,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( ord_max_nat @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_eq_nat @ B2 @ A2 )
=> ~ ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% max.boundedE
thf(fact_929_max_OboundedE,axiom,
! [B2: int,C: int,A2: int] :
( ( ord_less_eq_int @ ( ord_max_int @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_eq_int @ B2 @ A2 )
=> ~ ( ord_less_eq_int @ C @ A2 ) ) ) ).
% max.boundedE
thf(fact_930_max_OorderI,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( ord_max_nat @ A2 @ B2 ) )
=> ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% max.orderI
thf(fact_931_max_OorderI,axiom,
! [A2: int,B2: int] :
( ( A2
= ( ord_max_int @ A2 @ B2 ) )
=> ( ord_less_eq_int @ B2 @ A2 ) ) ).
% max.orderI
thf(fact_932_max_OorderE,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2
= ( ord_max_nat @ A2 @ B2 ) ) ) ).
% max.orderE
thf(fact_933_max_OorderE,axiom,
! [B2: int,A2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( A2
= ( ord_max_int @ A2 @ B2 ) ) ) ).
% max.orderE
thf(fact_934_max_Omono,axiom,
! [C: nat,A2: nat,D2: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ( ord_less_eq_nat @ D2 @ B2 )
=> ( ord_less_eq_nat @ ( ord_max_nat @ C @ D2 ) @ ( ord_max_nat @ A2 @ B2 ) ) ) ) ).
% max.mono
thf(fact_935_max_Omono,axiom,
! [C: int,A2: int,D2: int,B2: int] :
( ( ord_less_eq_int @ C @ A2 )
=> ( ( ord_less_eq_int @ D2 @ B2 )
=> ( ord_less_eq_int @ ( ord_max_int @ C @ D2 ) @ ( ord_max_int @ A2 @ B2 ) ) ) ) ).
% max.mono
thf(fact_936_less__max__iff__disj,axiom,
! [Z2: nat,X2: nat,Y: nat] :
( ( ord_less_nat @ Z2 @ ( ord_max_nat @ X2 @ Y ) )
= ( ( ord_less_nat @ Z2 @ X2 )
| ( ord_less_nat @ Z2 @ Y ) ) ) ).
% less_max_iff_disj
thf(fact_937_less__max__iff__disj,axiom,
! [Z2: int,X2: int,Y: int] :
( ( ord_less_int @ Z2 @ ( ord_max_int @ X2 @ Y ) )
= ( ( ord_less_int @ Z2 @ X2 )
| ( ord_less_int @ Z2 @ Y ) ) ) ).
% less_max_iff_disj
thf(fact_938_max_Ostrict__boundedE,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( ord_less_nat @ ( ord_max_nat @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_nat @ B2 @ A2 )
=> ~ ( ord_less_nat @ C @ A2 ) ) ) ).
% max.strict_boundedE
thf(fact_939_max_Ostrict__boundedE,axiom,
! [B2: int,C: int,A2: int] :
( ( ord_less_int @ ( ord_max_int @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_int @ B2 @ A2 )
=> ~ ( ord_less_int @ C @ A2 ) ) ) ).
% max.strict_boundedE
thf(fact_940_max_Ostrict__order__iff,axiom,
( ord_less_nat
= ( ^ [B5: nat,A4: nat] :
( ( A4
= ( ord_max_nat @ A4 @ B5 ) )
& ( A4 != B5 ) ) ) ) ).
% max.strict_order_iff
thf(fact_941_max_Ostrict__order__iff,axiom,
( ord_less_int
= ( ^ [B5: int,A4: int] :
( ( A4
= ( ord_max_int @ A4 @ B5 ) )
& ( A4 != B5 ) ) ) ) ).
% max.strict_order_iff
thf(fact_942_max_Ostrict__coboundedI1,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_nat @ C @ A2 )
=> ( ord_less_nat @ C @ ( ord_max_nat @ A2 @ B2 ) ) ) ).
% max.strict_coboundedI1
thf(fact_943_max_Ostrict__coboundedI1,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_int @ C @ A2 )
=> ( ord_less_int @ C @ ( ord_max_int @ A2 @ B2 ) ) ) ).
% max.strict_coboundedI1
thf(fact_944_max_Ostrict__coboundedI2,axiom,
! [C: nat,B2: nat,A2: nat] :
( ( ord_less_nat @ C @ B2 )
=> ( ord_less_nat @ C @ ( ord_max_nat @ A2 @ B2 ) ) ) ).
% max.strict_coboundedI2
thf(fact_945_max_Ostrict__coboundedI2,axiom,
! [C: int,B2: int,A2: int] :
( ( ord_less_int @ C @ B2 )
=> ( ord_less_int @ C @ ( ord_max_int @ A2 @ B2 ) ) ) ).
% max.strict_coboundedI2
thf(fact_946_max__diff__distrib__left,axiom,
! [X2: int,Y: int,Z2: int] :
( ( minus_minus_int @ ( ord_max_int @ X2 @ Y ) @ Z2 )
= ( ord_max_int @ ( minus_minus_int @ X2 @ Z2 ) @ ( minus_minus_int @ Y @ Z2 ) ) ) ).
% max_diff_distrib_left
thf(fact_947_sup__max,axiom,
sup_sup_nat = ord_max_nat ).
% sup_max
thf(fact_948_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A4: nat,B5: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_949_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B5: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B5 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_950_of__nat__max,axiom,
! [X2: nat,Y: nat] :
( ( semiri1316708129612266289at_nat @ ( ord_max_nat @ X2 @ Y ) )
= ( ord_max_nat @ ( semiri1316708129612266289at_nat @ X2 ) @ ( semiri1316708129612266289at_nat @ Y ) ) ) ).
% of_nat_max
thf(fact_951_of__nat__max,axiom,
! [X2: nat,Y: nat] :
( ( semiri1314217659103216013at_int @ ( ord_max_nat @ X2 @ Y ) )
= ( ord_max_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( semiri1314217659103216013at_int @ Y ) ) ) ).
% of_nat_max
thf(fact_952_max_Oassoc,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_max_nat @ ( ord_max_nat @ A2 @ B2 ) @ C )
= ( ord_max_nat @ A2 @ ( ord_max_nat @ B2 @ C ) ) ) ).
% max.assoc
thf(fact_953_max_Ocommute,axiom,
( ord_max_nat
= ( ^ [A4: nat,B5: nat] : ( ord_max_nat @ B5 @ A4 ) ) ) ).
% max.commute
thf(fact_954_max_Oleft__commute,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_max_nat @ B2 @ ( ord_max_nat @ A2 @ C ) )
= ( ord_max_nat @ A2 @ ( ord_max_nat @ B2 @ C ) ) ) ).
% max.left_commute
thf(fact_955_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_956_of__nat__0__le__iff,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N2 ) ) ).
% of_nat_0_le_iff
thf(fact_957_of__nat__0__le__iff,axiom,
! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N2 ) ) ).
% of_nat_0_le_iff
thf(fact_958_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_959_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_960_of__nat__neq__0,axiom,
! [N2: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ N2 ) )
!= zero_zero_nat ) ).
% of_nat_neq_0
thf(fact_961_of__nat__neq__0,axiom,
! [N2: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N2 ) )
!= zero_zero_int ) ).
% of_nat_neq_0
thf(fact_962_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).
% of_nat_mono
thf(fact_963_of__nat__mono,axiom,
! [I: nat,J: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).
% of_nat_mono
thf(fact_964_of__nat__less__imp__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ).
% of_nat_less_imp_less
thf(fact_965_of__nat__less__imp__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ).
% of_nat_less_imp_less
thf(fact_966_less__imp__of__nat__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% less_imp_of_nat_less
thf(fact_967_less__imp__of__nat__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% less_imp_of_nat_less
thf(fact_968_fMax_Oin__idem,axiom,
! [X2: nat,A: fset_nat] :
( ( fmember_nat @ X2 @ A )
=> ( ( ord_max_nat @ X2 @ ( linorder_fMax_nat @ A ) )
= ( linorder_fMax_nat @ A ) ) ) ).
% fMax.in_idem
thf(fact_969_of__nat__diff,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_eq_nat @ N2 @ M2 )
=> ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M2 @ N2 ) )
= ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ) ).
% of_nat_diff
thf(fact_970_of__nat__diff,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_eq_nat @ N2 @ M2 )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M2 @ N2 ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% of_nat_diff
thf(fact_971_fMin__le,axiom,
! [X2: nat,A: fset_nat] :
( ( fmember_nat @ X2 @ A )
=> ( ord_less_eq_nat @ ( linorder_fMin_nat @ A ) @ X2 ) ) ).
% fMin_le
thf(fact_972_fMin__le,axiom,
! [X2: int,A: fset_int] :
( ( fmember_int @ X2 @ A )
=> ( ord_less_eq_int @ ( linorder_fMin_int @ A ) @ X2 ) ) ).
% fMin_le
thf(fact_973_fMin__eqI,axiom,
! [A: fset_nat,X2: nat] :
( ! [Y2: nat] :
( ( fmember_nat @ Y2 @ A )
=> ( ord_less_eq_nat @ X2 @ Y2 ) )
=> ( ( fmember_nat @ X2 @ A )
=> ( ( linorder_fMin_nat @ A )
= X2 ) ) ) ).
% fMin_eqI
thf(fact_974_fMin__eqI,axiom,
! [A: fset_int,X2: int] :
( ! [Y2: int] :
( ( fmember_int @ Y2 @ A )
=> ( ord_less_eq_int @ X2 @ Y2 ) )
=> ( ( fmember_int @ X2 @ A )
=> ( ( linorder_fMin_int @ A )
= X2 ) ) ) ).
% fMin_eqI
thf(fact_975_zdiff__int__split,axiom,
! [P: int > $o,X2: nat,Y: nat] :
( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X2 @ Y ) ) )
= ( ( ( ord_less_eq_nat @ Y @ X2 )
=> ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X2 ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
& ( ( ord_less_nat @ X2 @ Y )
=> ( P @ zero_zero_int ) ) ) ) ).
% zdiff_int_split
thf(fact_976_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% pos_int_cases
thf(fact_977_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
& ( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_978_zle__diff1__eq,axiom,
! [W: int,Z2: int] :
( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z2 @ one_one_int ) )
= ( ord_less_int @ W @ Z2 ) ) ).
% zle_diff1_eq
thf(fact_979_sup__nat__def,axiom,
sup_sup_nat = ord_max_nat ).
% sup_nat_def
thf(fact_980_int__less__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_int @ I @ K )
=> ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_less_induct
thf(fact_981_int__le__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_eq_int @ I @ K )
=> ( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_le_induct
thf(fact_982_conj__le__cong,axiom,
! [X2: int,X6: int,P: $o,P4: $o] :
( ( X2 = X6 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X6 )
& P4 ) ) ) ) ).
% conj_le_cong
thf(fact_983_imp__le__cong,axiom,
! [X2: int,X6: int,P: $o,P4: $o] :
( ( X2 = X6 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> ( P = P4 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> P4 ) ) ) ) ).
% imp_le_cong
thf(fact_984_int__one__le__iff__zero__less,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ one_one_int @ Z2 )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% int_one_le_iff_zero_less
thf(fact_985_zle__int,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% zle_int
thf(fact_986_nat__ivt__aux,axiom,
! [N2: nat,F: nat > int,K: int] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N2 )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N2 ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ I2 @ N2 )
& ( ( F @ I2 )
= K ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_987_nat__intermed__int__val,axiom,
! [M2: nat,N2: nat,F: nat > int,K: int] :
( ! [I2: nat] :
( ( ( ord_less_eq_nat @ M2 @ I2 )
& ( ord_less_nat @ I2 @ N2 ) )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( ord_less_eq_int @ ( F @ M2 ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N2 ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ M2 @ I2 )
& ( ord_less_eq_nat @ I2 @ N2 )
& ( ( F @ I2 )
= K ) ) ) ) ) ) ).
% nat_intermed_int_val
thf(fact_988_abs__idempotent,axiom,
! [A2: int] :
( ( abs_abs_int @ ( abs_abs_int @ A2 ) )
= ( abs_abs_int @ A2 ) ) ).
% abs_idempotent
thf(fact_989_abs__zero,axiom,
( ( abs_abs_int @ zero_zero_int )
= zero_zero_int ) ).
% abs_zero
thf(fact_990_abs__eq__0,axiom,
! [A2: int] :
( ( ( abs_abs_int @ A2 )
= zero_zero_int )
= ( A2 = zero_zero_int ) ) ).
% abs_eq_0
thf(fact_991_abs__0__eq,axiom,
! [A2: int] :
( ( zero_zero_int
= ( abs_abs_int @ A2 ) )
= ( A2 = zero_zero_int ) ) ).
% abs_0_eq
thf(fact_992_abs__1,axiom,
( ( abs_abs_int @ one_one_int )
= one_one_int ) ).
% abs_1
thf(fact_993_abs__of__nat,axiom,
! [N2: nat] :
( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N2 ) )
= ( semiri1314217659103216013at_int @ N2 ) ) ).
% abs_of_nat
thf(fact_994_abs__of__nonneg,axiom,
! [A2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A2 )
=> ( ( abs_abs_int @ A2 )
= A2 ) ) ).
% abs_of_nonneg
thf(fact_995_abs__le__self__iff,axiom,
! [A2: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A2 ) @ A2 )
= ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).
% abs_le_self_iff
thf(fact_996_abs__le__zero__iff,axiom,
! [A2: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A2 ) @ zero_zero_int )
= ( A2 = zero_zero_int ) ) ).
% abs_le_zero_iff
thf(fact_997_zero__less__abs__iff,axiom,
! [A2: int] :
( ( ord_less_int @ zero_zero_int @ ( abs_abs_int @ A2 ) )
= ( A2 != zero_zero_int ) ) ).
% zero_less_abs_iff
thf(fact_998_zabs__less__one__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ ( abs_abs_int @ Z2 ) @ one_one_int )
= ( Z2 = zero_zero_int ) ) ).
% zabs_less_one_iff
thf(fact_999_abs__minus__commute,axiom,
! [A2: int,B2: int] :
( ( abs_abs_int @ ( minus_minus_int @ A2 @ B2 ) )
= ( abs_abs_int @ ( minus_minus_int @ B2 @ A2 ) ) ) ).
% abs_minus_commute
thf(fact_1000_abs__ge__self,axiom,
! [A2: int] : ( ord_less_eq_int @ A2 @ ( abs_abs_int @ A2 ) ) ).
% abs_ge_self
thf(fact_1001_abs__le__D1,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A2 ) @ B2 )
=> ( ord_less_eq_int @ A2 @ B2 ) ) ).
% abs_le_D1
thf(fact_1002_abs__one,axiom,
( ( abs_abs_int @ one_one_int )
= one_one_int ) ).
% abs_one
thf(fact_1003_abs__ge__zero,axiom,
! [A2: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A2 ) ) ).
% abs_ge_zero
thf(fact_1004_abs__of__pos,axiom,
! [A2: int] :
( ( ord_less_int @ zero_zero_int @ A2 )
=> ( ( abs_abs_int @ A2 )
= A2 ) ) ).
% abs_of_pos
thf(fact_1005_abs__not__less__zero,axiom,
! [A2: int] :
~ ( ord_less_int @ ( abs_abs_int @ A2 ) @ zero_zero_int ) ).
% abs_not_less_zero
thf(fact_1006_abs__triangle__ineq2,axiom,
! [A2: int,B2: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B2 ) ) @ ( abs_abs_int @ ( minus_minus_int @ A2 @ B2 ) ) ) ).
% abs_triangle_ineq2
thf(fact_1007_abs__triangle__ineq3,axiom,
! [A2: int,B2: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B2 ) ) ) @ ( abs_abs_int @ ( minus_minus_int @ A2 @ B2 ) ) ) ).
% abs_triangle_ineq3
thf(fact_1008_abs__triangle__ineq2__sym,axiom,
! [A2: int,B2: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B2 ) ) @ ( abs_abs_int @ ( minus_minus_int @ B2 @ A2 ) ) ) ).
% abs_triangle_ineq2_sym
thf(fact_1009_nat0__intermed__int__val,axiom,
! [N2: nat,F: nat > int,K: int] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N2 )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I2 @ one_one_nat ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N2 ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ I2 @ N2 )
& ( ( F @ I2 )
= K ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_1010_one__less__nat__eq,axiom,
! [Z2: int] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z2 ) )
= ( ord_less_int @ one_one_int @ Z2 ) ) ).
% one_less_nat_eq
thf(fact_1011_neg__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ K @ zero_zero_int )
=> ~ ! [N3: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% neg_int_cases
thf(fact_1012_add__right__cancel,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ( plus_plus_nat @ B2 @ A2 )
= ( plus_plus_nat @ C @ A2 ) )
= ( B2 = C ) ) ).
% add_right_cancel
thf(fact_1013_add__right__cancel,axiom,
! [B2: int,A2: int,C: int] :
( ( ( plus_plus_int @ B2 @ A2 )
= ( plus_plus_int @ C @ A2 ) )
= ( B2 = C ) ) ).
% add_right_cancel
thf(fact_1014_add__left__cancel,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ( plus_plus_nat @ A2 @ B2 )
= ( plus_plus_nat @ A2 @ C ) )
= ( B2 = C ) ) ).
% add_left_cancel
thf(fact_1015_add__left__cancel,axiom,
! [A2: int,B2: int,C: int] :
( ( ( plus_plus_int @ A2 @ B2 )
= ( plus_plus_int @ A2 @ C ) )
= ( B2 = C ) ) ).
% add_left_cancel
thf(fact_1016_neg__equal__iff__equal,axiom,
! [A2: int,B2: int] :
( ( ( uminus_uminus_int @ A2 )
= ( uminus_uminus_int @ B2 ) )
= ( A2 = B2 ) ) ).
% neg_equal_iff_equal
thf(fact_1017_add_Oinverse__inverse,axiom,
! [A2: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A2 ) )
= A2 ) ).
% add.inverse_inverse
thf(fact_1018_add__le__cancel__left,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% add_le_cancel_left
thf(fact_1019_add__le__cancel__left,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B2 ) )
= ( ord_less_eq_int @ A2 @ B2 ) ) ).
% add_le_cancel_left
thf(fact_1020_add__le__cancel__right,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% add_le_cancel_right
thf(fact_1021_add__le__cancel__right,axiom,
! [A2: int,C: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ C ) )
= ( ord_less_eq_int @ A2 @ B2 ) ) ).
% add_le_cancel_right
thf(fact_1022_add__0,axiom,
! [A2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% add_0
thf(fact_1023_add__0,axiom,
! [A2: int] :
( ( plus_plus_int @ zero_zero_int @ A2 )
= A2 ) ).
% add_0
thf(fact_1024_zero__eq__add__iff__both__eq__0,axiom,
! [X2: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X2 @ Y ) )
= ( ( X2 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_1025_add__eq__0__iff__both__eq__0,axiom,
! [X2: nat,Y: nat] :
( ( ( plus_plus_nat @ X2 @ Y )
= zero_zero_nat )
= ( ( X2 = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_1026_add__cancel__right__right,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( plus_plus_nat @ A2 @ B2 ) )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_1027_add__cancel__right__right,axiom,
! [A2: int,B2: int] :
( ( A2
= ( plus_plus_int @ A2 @ B2 ) )
= ( B2 = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_1028_add__cancel__right__left,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( plus_plus_nat @ B2 @ A2 ) )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_1029_add__cancel__right__left,axiom,
! [A2: int,B2: int] :
( ( A2
= ( plus_plus_int @ B2 @ A2 ) )
= ( B2 = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_1030_add__cancel__left__right,axiom,
! [A2: nat,B2: nat] :
( ( ( plus_plus_nat @ A2 @ B2 )
= A2 )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_1031_add__cancel__left__right,axiom,
! [A2: int,B2: int] :
( ( ( plus_plus_int @ A2 @ B2 )
= A2 )
= ( B2 = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_1032_add__cancel__left__left,axiom,
! [B2: nat,A2: nat] :
( ( ( plus_plus_nat @ B2 @ A2 )
= A2 )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_1033_add__cancel__left__left,axiom,
! [B2: int,A2: int] :
( ( ( plus_plus_int @ B2 @ A2 )
= A2 )
= ( B2 = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_1034_double__zero__sym,axiom,
! [A2: int] :
( ( zero_zero_int
= ( plus_plus_int @ A2 @ A2 ) )
= ( A2 = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_1035_add_Oright__neutral,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% add.right_neutral
thf(fact_1036_add_Oright__neutral,axiom,
! [A2: int] :
( ( plus_plus_int @ A2 @ zero_zero_int )
= A2 ) ).
% add.right_neutral
thf(fact_1037_add__less__cancel__right,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( ord_less_nat @ A2 @ B2 ) ) ).
% add_less_cancel_right
thf(fact_1038_add__less__cancel__right,axiom,
! [A2: int,C: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ C ) )
= ( ord_less_int @ A2 @ B2 ) ) ).
% add_less_cancel_right
thf(fact_1039_add__less__cancel__left,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
= ( ord_less_nat @ A2 @ B2 ) ) ).
% add_less_cancel_left
thf(fact_1040_add__less__cancel__left,axiom,
! [C: int,A2: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B2 ) )
= ( ord_less_int @ A2 @ B2 ) ) ).
% add_less_cancel_left
thf(fact_1041_neg__le__iff__le,axiom,
! [B2: int,A2: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_eq_int @ A2 @ B2 ) ) ).
% neg_le_iff_le
thf(fact_1042_neg__equal__zero,axiom,
! [A2: int] :
( ( ( uminus_uminus_int @ A2 )
= A2 )
= ( A2 = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_1043_equal__neg__zero,axiom,
! [A2: int] :
( ( A2
= ( uminus_uminus_int @ A2 ) )
= ( A2 = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_1044_neg__equal__0__iff__equal,axiom,
! [A2: int] :
( ( ( uminus_uminus_int @ A2 )
= zero_zero_int )
= ( A2 = zero_zero_int ) ) ).
% neg_equal_0_iff_equal
thf(fact_1045_neg__0__equal__iff__equal,axiom,
! [A2: int] :
( ( zero_zero_int
= ( uminus_uminus_int @ A2 ) )
= ( zero_zero_int = A2 ) ) ).
% neg_0_equal_iff_equal
thf(fact_1046_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_1047_neg__less__iff__less,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_int @ A2 @ B2 ) ) ).
% neg_less_iff_less
thf(fact_1048_add__diff__cancel__right_H,axiom,
! [A2: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B2 ) @ B2 )
= A2 ) ).
% add_diff_cancel_right'
thf(fact_1049_add__diff__cancel__right_H,axiom,
! [A2: int,B2: int] :
( ( minus_minus_int @ ( plus_plus_int @ A2 @ B2 ) @ B2 )
= A2 ) ).
% add_diff_cancel_right'
thf(fact_1050_add__diff__cancel__right,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( minus_minus_nat @ A2 @ B2 ) ) ).
% add_diff_cancel_right
thf(fact_1051_add__diff__cancel__right,axiom,
! [A2: int,C: int,B2: int] :
( ( minus_minus_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ C ) )
= ( minus_minus_int @ A2 @ B2 ) ) ).
% add_diff_cancel_right
thf(fact_1052_add__diff__cancel__left_H,axiom,
! [A2: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B2 ) @ A2 )
= B2 ) ).
% add_diff_cancel_left'
thf(fact_1053_add__diff__cancel__left_H,axiom,
! [A2: int,B2: int] :
( ( minus_minus_int @ ( plus_plus_int @ A2 @ B2 ) @ A2 )
= B2 ) ).
% add_diff_cancel_left'
thf(fact_1054_add__diff__cancel__left,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
= ( minus_minus_nat @ A2 @ B2 ) ) ).
% add_diff_cancel_left
thf(fact_1055_add__diff__cancel__left,axiom,
! [C: int,A2: int,B2: int] :
( ( minus_minus_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B2 ) )
= ( minus_minus_int @ A2 @ B2 ) ) ).
% add_diff_cancel_left
thf(fact_1056_diff__add__cancel,axiom,
! [A2: int,B2: int] :
( ( plus_plus_int @ ( minus_minus_int @ A2 @ B2 ) @ B2 )
= A2 ) ).
% diff_add_cancel
thf(fact_1057_add__diff__cancel,axiom,
! [A2: int,B2: int] :
( ( minus_minus_int @ ( plus_plus_int @ A2 @ B2 ) @ B2 )
= A2 ) ).
% add_diff_cancel
thf(fact_1058_minus__add__distrib,axiom,
! [A2: int,B2: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A2 @ B2 ) )
= ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ ( uminus_uminus_int @ B2 ) ) ) ).
% minus_add_distrib
thf(fact_1059_minus__add__cancel,axiom,
! [A2: int,B2: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ ( plus_plus_int @ A2 @ B2 ) )
= B2 ) ).
% minus_add_cancel
thf(fact_1060_add__minus__cancel,axiom,
! [A2: int,B2: int] :
( ( plus_plus_int @ A2 @ ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ B2 ) )
= B2 ) ).
% add_minus_cancel
thf(fact_1061_minus__diff__eq,axiom,
! [A2: int,B2: int] :
( ( uminus_uminus_int @ ( minus_minus_int @ A2 @ B2 ) )
= ( minus_minus_int @ B2 @ A2 ) ) ).
% minus_diff_eq
thf(fact_1062_add__Suc__right,axiom,
! [M2: nat,N2: nat] :
( ( plus_plus_nat @ M2 @ ( suc @ N2 ) )
= ( suc @ ( plus_plus_nat @ M2 @ N2 ) ) ) ).
% add_Suc_right
thf(fact_1063_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% Nat.add_0_right
thf(fact_1064_add__is__0,axiom,
! [M2: nat,N2: nat] :
( ( ( plus_plus_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
& ( N2 = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1065_abs__add__abs,axiom,
! [A2: int,B2: int] :
( ( abs_abs_int @ ( plus_plus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B2 ) ) )
= ( plus_plus_int @ ( abs_abs_int @ A2 ) @ ( abs_abs_int @ B2 ) ) ) ).
% abs_add_abs
thf(fact_1066_nat__add__left__cancel__less,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% nat_add_left_cancel_less
thf(fact_1067_abs__minus__cancel,axiom,
! [A2: int] :
( ( abs_abs_int @ ( uminus_uminus_int @ A2 ) )
= ( abs_abs_int @ A2 ) ) ).
% abs_minus_cancel
thf(fact_1068_nat__add__left__cancel__le,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% nat_add_left_cancel_le
thf(fact_1069_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_1070_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A2 @ A2 ) )
= ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_1071_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ A2 ) @ zero_zero_int )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_1072_le__add__same__cancel2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ B2 @ A2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% le_add_same_cancel2
thf(fact_1073_le__add__same__cancel2,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ ( plus_plus_int @ B2 @ A2 ) )
= ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ).
% le_add_same_cancel2
thf(fact_1074_le__add__same__cancel1,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ ( plus_plus_nat @ A2 @ B2 ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B2 ) ) ).
% le_add_same_cancel1
thf(fact_1075_le__add__same__cancel1,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ ( plus_plus_int @ A2 @ B2 ) )
= ( ord_less_eq_int @ zero_zero_int @ B2 ) ) ).
% le_add_same_cancel1
thf(fact_1076_add__le__same__cancel2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ B2 ) @ B2 )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_1077_add__le__same__cancel2,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A2 @ B2 ) @ B2 )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_1078_add__le__same__cancel1,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B2 @ A2 ) @ B2 )
= ( ord_less_eq_nat @ A2 @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_1079_add__le__same__cancel1,axiom,
! [B2: int,A2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B2 @ A2 ) @ B2 )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_1080_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A2: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A2 @ A2 ) )
= ( ord_less_int @ zero_zero_int @ A2 ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_1081_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A2: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ A2 ) @ zero_zero_int )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_1082_less__add__same__cancel2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ ( plus_plus_nat @ B2 @ A2 ) )
= ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).
% less_add_same_cancel2
thf(fact_1083_less__add__same__cancel2,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ ( plus_plus_int @ B2 @ A2 ) )
= ( ord_less_int @ zero_zero_int @ B2 ) ) ).
% less_add_same_cancel2
thf(fact_1084_less__add__same__cancel1,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ B2 ) )
= ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).
% less_add_same_cancel1
thf(fact_1085_less__add__same__cancel1,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ A2 @ ( plus_plus_int @ A2 @ B2 ) )
= ( ord_less_int @ zero_zero_int @ B2 ) ) ).
% less_add_same_cancel1
thf(fact_1086_add__less__same__cancel2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A2 @ B2 ) @ B2 )
= ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_1087_add__less__same__cancel2,axiom,
! [A2: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ A2 @ B2 ) @ B2 )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_1088_add__less__same__cancel1,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B2 @ A2 ) @ B2 )
= ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_1089_add__less__same__cancel1,axiom,
! [B2: int,A2: int] :
( ( ord_less_int @ ( plus_plus_int @ B2 @ A2 ) @ B2 )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_1090_neg__less__eq__nonneg,axiom,
! [A2: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ A2 )
= ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).
% neg_less_eq_nonneg
thf(fact_1091_less__eq__neg__nonpos,axiom,
! [A2: int] :
( ( ord_less_eq_int @ A2 @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% less_eq_neg_nonpos
thf(fact_1092_neg__le__0__iff__le,axiom,
! [A2: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ zero_zero_int )
= ( ord_less_eq_int @ zero_zero_int @ A2 ) ) ).
% neg_le_0_iff_le
thf(fact_1093_neg__0__le__iff__le,axiom,
! [A2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_eq_int @ A2 @ zero_zero_int ) ) ).
% neg_0_le_iff_le
thf(fact_1094_le__add__diff__inverse2,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A2 @ B2 ) @ B2 )
= A2 ) ) ).
% le_add_diff_inverse2
thf(fact_1095_le__add__diff__inverse2,axiom,
! [B2: int,A2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ( plus_plus_int @ ( minus_minus_int @ A2 @ B2 ) @ B2 )
= A2 ) ) ).
% le_add_diff_inverse2
thf(fact_1096_le__add__diff__inverse,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( plus_plus_nat @ B2 @ ( minus_minus_nat @ A2 @ B2 ) )
= A2 ) ) ).
% le_add_diff_inverse
thf(fact_1097_le__add__diff__inverse,axiom,
! [B2: int,A2: int] :
( ( ord_less_eq_int @ B2 @ A2 )
=> ( ( plus_plus_int @ B2 @ ( minus_minus_int @ A2 @ B2 ) )
= A2 ) ) ).
% le_add_diff_inverse
thf(fact_1098_neg__less__0__iff__less,axiom,
! [A2: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A2 ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A2 ) ) ).
% neg_less_0_iff_less
thf(fact_1099_neg__0__less__iff__less,axiom,
! [A2: int] :
( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% neg_0_less_iff_less
thf(fact_1100_neg__less__pos,axiom,
! [A2: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A2 ) @ A2 )
= ( ord_less_int @ zero_zero_int @ A2 ) ) ).
% neg_less_pos
thf(fact_1101_less__neg__neg,axiom,
! [A2: int] :
( ( ord_less_int @ A2 @ ( uminus_uminus_int @ A2 ) )
= ( ord_less_int @ A2 @ zero_zero_int ) ) ).
% less_neg_neg
thf(fact_1102_diff__add__zero,axiom,
! [A2: nat,B2: nat] :
( ( minus_minus_nat @ A2 @ ( plus_plus_nat @ A2 @ B2 ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_1103_add_Oright__inverse,axiom,
! [A2: int] :
( ( plus_plus_int @ A2 @ ( uminus_uminus_int @ A2 ) )
= zero_zero_int ) ).
% add.right_inverse
thf(fact_1104_ab__left__minus,axiom,
! [A2: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ A2 )
= zero_zero_int ) ).
% ab_left_minus
thf(fact_1105_verit__minus__simplify_I3_J,axiom,
! [B2: int] :
( ( minus_minus_int @ zero_zero_int @ B2 )
= ( uminus_uminus_int @ B2 ) ) ).
% verit_minus_simplify(3)
thf(fact_1106_diff__0,axiom,
! [A2: int] :
( ( minus_minus_int @ zero_zero_int @ A2 )
= ( uminus_uminus_int @ A2 ) ) ).
% diff_0
thf(fact_1107_uminus__add__conv__diff,axiom,
! [A2: int,B2: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ B2 )
= ( minus_minus_int @ B2 @ A2 ) ) ).
% uminus_add_conv_diff
thf(fact_1108_diff__minus__eq__add,axiom,
! [A2: int,B2: int] :
( ( minus_minus_int @ A2 @ ( uminus_uminus_int @ B2 ) )
= ( plus_plus_int @ A2 @ B2 ) ) ).
% diff_minus_eq_add
thf(fact_1109_abs__neg__one,axiom,
( ( abs_abs_int @ ( uminus_uminus_int @ one_one_int ) )
= one_one_int ) ).
% abs_neg_one
thf(fact_1110_of__nat__add,axiom,
! [M2: nat,N2: nat] :
( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M2 @ N2 ) )
= ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).
% of_nat_add
thf(fact_1111_of__nat__add,axiom,
! [M2: nat,N2: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M2 @ N2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% of_nat_add
thf(fact_1112_add__gr__0,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
| ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% add_gr_0
thf(fact_1113_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1114_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1115_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1116_dbl__inc__simps_I4_J,axiom,
( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ one_one_int ) )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_inc_simps(4)
thf(fact_1117_add__neg__numeral__special_I7_J,axiom,
( ( plus_plus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% add_neg_numeral_special(7)
thf(fact_1118_add__neg__numeral__special_I8_J,axiom,
( ( plus_plus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
= zero_zero_int ) ).
% add_neg_numeral_special(8)
thf(fact_1119_diff__numeral__special_I12_J,axiom,
( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ one_one_int ) )
= zero_zero_int ) ).
% diff_numeral_special(12)
thf(fact_1120_abs__of__nonpos,axiom,
! [A2: int] :
( ( ord_less_eq_int @ A2 @ zero_zero_int )
=> ( ( abs_abs_int @ A2 )
= ( uminus_uminus_int @ A2 ) ) ) ).
% abs_of_nonpos
thf(fact_1121_of__nat__Suc,axiom,
! [M2: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ M2 ) )
= ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M2 ) ) ) ).
% of_nat_Suc
thf(fact_1122_of__nat__Suc,axiom,
! [M2: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ M2 ) )
= ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M2 ) ) ) ).
% of_nat_Suc
thf(fact_1123_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1124_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1125_nat__1,axiom,
( ( nat2 @ one_one_int )
= ( suc @ zero_zero_nat ) ) ).
% nat_1
thf(fact_1126_zless__nat__conj,axiom,
! [W: int,Z2: int] :
( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
= ( ( ord_less_int @ zero_zero_int @ Z2 )
& ( ord_less_int @ W @ Z2 ) ) ) ).
% zless_nat_conj
thf(fact_1127_zero__less__nat__eq,axiom,
! [Z2: int] :
( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z2 ) )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% zero_less_nat_eq
thf(fact_1128_nat__arith_Osuc1,axiom,
! [A: nat,K: nat,A2: nat] :
( ( A
= ( plus_plus_nat @ K @ A2 ) )
=> ( ( suc @ A )
= ( plus_plus_nat @ K @ ( suc @ A2 ) ) ) ) ).
% nat_arith.suc1
thf(fact_1129_add__Suc,axiom,
! [M2: nat,N2: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N2 )
= ( suc @ ( plus_plus_nat @ M2 @ N2 ) ) ) ).
% add_Suc
thf(fact_1130_add__Suc__shift,axiom,
! [M2: nat,N2: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N2 )
= ( plus_plus_nat @ M2 @ ( suc @ N2 ) ) ) ).
% add_Suc_shift
thf(fact_1131_add__right__imp__eq,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ( plus_plus_nat @ B2 @ A2 )
= ( plus_plus_nat @ C @ A2 ) )
=> ( B2 = C ) ) ).
% add_right_imp_eq
thf(fact_1132_add__right__imp__eq,axiom,
! [B2: int,A2: int,C: int] :
( ( ( plus_plus_int @ B2 @ A2 )
= ( plus_plus_int @ C @ A2 ) )
=> ( B2 = C ) ) ).
% add_right_imp_eq
thf(fact_1133_add__left__imp__eq,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ( plus_plus_nat @ A2 @ B2 )
= ( plus_plus_nat @ A2 @ C ) )
=> ( B2 = C ) ) ).
% add_left_imp_eq
thf(fact_1134_add__left__imp__eq,axiom,
! [A2: int,B2: int,C: int] :
( ( ( plus_plus_int @ A2 @ B2 )
= ( plus_plus_int @ A2 @ C ) )
=> ( B2 = C ) ) ).
% add_left_imp_eq
thf(fact_1135_add_Oinverse__distrib__swap,axiom,
! [A2: int,B2: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A2 @ B2 ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A2 ) ) ) ).
% add.inverse_distrib_swap
thf(fact_1136_add_Oleft__commute,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( plus_plus_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add.left_commute
thf(fact_1137_add_Oleft__commute,axiom,
! [B2: int,A2: int,C: int] :
( ( plus_plus_int @ B2 @ ( plus_plus_int @ A2 @ C ) )
= ( plus_plus_int @ A2 @ ( plus_plus_int @ B2 @ C ) ) ) ).
% add.left_commute
thf(fact_1138_minus__equation__iff,axiom,
! [A2: int,B2: int] :
( ( ( uminus_uminus_int @ A2 )
= B2 )
= ( ( uminus_uminus_int @ B2 )
= A2 ) ) ).
% minus_equation_iff
thf(fact_1139_equation__minus__iff,axiom,
! [A2: int,B2: int] :
( ( A2
= ( uminus_uminus_int @ B2 ) )
= ( B2
= ( uminus_uminus_int @ A2 ) ) ) ).
% equation_minus_iff
thf(fact_1140_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A4: nat,B5: nat] : ( plus_plus_nat @ B5 @ A4 ) ) ) ).
% add.commute
thf(fact_1141_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A4: int,B5: int] : ( plus_plus_int @ B5 @ A4 ) ) ) ).
% add.commute
thf(fact_1142_add_Oright__cancel,axiom,
! [B2: int,A2: int,C: int] :
( ( ( plus_plus_int @ B2 @ A2 )
= ( plus_plus_int @ C @ A2 ) )
= ( B2 = C ) ) ).
% add.right_cancel
thf(fact_1143_add_Oleft__cancel,axiom,
! [A2: int,B2: int,C: int] :
( ( ( plus_plus_int @ A2 @ B2 )
= ( plus_plus_int @ A2 @ C ) )
= ( B2 = C ) ) ).
% add.left_cancel
thf(fact_1144_add_Oassoc,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add.assoc
thf(fact_1145_add_Oassoc,axiom,
! [A2: int,B2: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A2 @ B2 ) @ C )
= ( plus_plus_int @ A2 @ ( plus_plus_int @ B2 @ C ) ) ) ).
% add.assoc
thf(fact_1146_group__cancel_Oneg1,axiom,
! [A: int,K: int,A2: int] :
( ( A
= ( plus_plus_int @ K @ A2 ) )
=> ( ( uminus_uminus_int @ A )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A2 ) ) ) ) ).
% group_cancel.neg1
thf(fact_1147_group__cancel_Oadd2,axiom,
! [B: nat,K: nat,B2: nat,A2: nat] :
( ( B
= ( plus_plus_nat @ K @ B2 ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).
% group_cancel.add2
thf(fact_1148_group__cancel_Oadd2,axiom,
! [B: int,K: int,B2: int,A2: int] :
( ( B
= ( plus_plus_int @ K @ B2 ) )
=> ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A2 @ B2 ) ) ) ) ).
% group_cancel.add2
thf(fact_1149_group__cancel_Oadd1,axiom,
! [A: nat,K: nat,A2: nat,B2: nat] :
( ( A
= ( plus_plus_nat @ K @ A2 ) )
=> ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).
% group_cancel.add1
thf(fact_1150_group__cancel_Oadd1,axiom,
! [A: int,K: int,A2: int,B2: int] :
( ( A
= ( plus_plus_int @ K @ A2 ) )
=> ( ( plus_plus_int @ A @ B2 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A2 @ B2 ) ) ) ) ).
% group_cancel.add1
thf(fact_1151_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_1152_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_int @ I @ K )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_1153_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C )
= ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_1154_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A2: int,B2: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A2 @ B2 ) @ C )
= ( plus_plus_int @ A2 @ ( plus_plus_int @ B2 @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_1155_is__num__normalize_I1_J,axiom,
! [A2: int,B2: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A2 @ B2 ) @ C )
= ( plus_plus_int @ A2 @ ( plus_plus_int @ B2 @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_1156_is__num__normalize_I8_J,axiom,
! [A2: int,B2: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A2 @ B2 ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A2 ) ) ) ).
% is_num_normalize(8)
thf(fact_1157_plus__nat_Oadd__0,axiom,
! [N2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N2 )
= N2 ) ).
% plus_nat.add_0
thf(fact_1158_add__eq__self__zero,axiom,
! [M2: nat,N2: nat] :
( ( ( plus_plus_nat @ M2 @ N2 )
= M2 )
=> ( N2 = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1159_comm__monoid__add__class_Oadd__0,axiom,
! [A2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A2 )
= A2 ) ).
% comm_monoid_add_class.add_0
thf(fact_1160_comm__monoid__add__class_Oadd__0,axiom,
! [A2: int] :
( ( plus_plus_int @ zero_zero_int @ A2 )
= A2 ) ).
% comm_monoid_add_class.add_0
thf(fact_1161_add__eq__0__iff,axiom,
! [A2: int,B2: int] :
( ( ( plus_plus_int @ A2 @ B2 )
= zero_zero_int )
= ( B2
= ( uminus_uminus_int @ A2 ) ) ) ).
% add_eq_0_iff
thf(fact_1162_ab__group__add__class_Oab__left__minus,axiom,
! [A2: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A2 ) @ A2 )
= zero_zero_int ) ).
% ab_group_add_class.ab_left_minus
thf(fact_1163_add_Oinverse__unique,axiom,
! [A2: int,B2: int] :
( ( ( plus_plus_int @ A2 @ B2 )
= zero_zero_int )
=> ( ( uminus_uminus_int @ A2 )
= B2 ) ) ).
% add.inverse_unique
thf(fact_1164_eq__neg__iff__add__eq__0,axiom,
! [A2: int,B2: int] :
( ( A2
= ( uminus_uminus_int @ B2 ) )
= ( ( plus_plus_int @ A2 @ B2 )
= zero_zero_int ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_1165_neg__eq__iff__add__eq__0,axiom,
! [A2: int,B2: int] :
( ( ( uminus_uminus_int @ A2 )
= B2 )
= ( ( plus_plus_int @ A2 @ B2 )
= zero_zero_int ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_1166_add_Ocomm__neutral,axiom,
! [A2: nat] :
( ( plus_plus_nat @ A2 @ zero_zero_nat )
= A2 ) ).
% add.comm_neutral
thf(fact_1167_add_Ocomm__neutral,axiom,
! [A2: int] :
( ( plus_plus_int @ A2 @ zero_zero_int )
= A2 ) ).
% add.comm_neutral
thf(fact_1168_add_Ogroup__left__neutral,axiom,
! [A2: int] :
( ( plus_plus_int @ zero_zero_int @ A2 )
= A2 ) ).
% add.group_left_neutral
thf(fact_1169_le__minus__iff,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ ( uminus_uminus_int @ B2 ) )
= ( ord_less_eq_int @ B2 @ ( uminus_uminus_int @ A2 ) ) ) ).
% le_minus_iff
thf(fact_1170_minus__le__iff,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A2 ) @ B2 )
= ( ord_less_eq_int @ ( uminus_uminus_int @ B2 ) @ A2 ) ) ).
% minus_le_iff
thf(fact_1171_le__imp__neg__le,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ B2 ) @ ( uminus_uminus_int @ A2 ) ) ) ).
% le_imp_neg_le
thf(fact_1172_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1173_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1174_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1175_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1176_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1177_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1178_add__mono,axiom,
! [A2: nat,B2: nat,C: nat,D2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ C @ D2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ D2 ) ) ) ) ).
% add_mono
thf(fact_1179_add__mono,axiom,
! [A2: int,B2: int,C: int,D2: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ( ord_less_eq_int @ C @ D2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A2 @ C ) @ ( plus_plus_int @ B2 @ D2 ) ) ) ) ).
% add_mono
thf(fact_1180_add__left__mono,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).
% add_left_mono
thf(fact_1181_add__left__mono,axiom,
! [A2: int,B2: int,C: int] :
( ( ord_less_eq_int @ A2 @ B2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ C @ A2 ) @ ( plus_plus_int @ C @ B2 ) ) ) ).
% add_left_mono
thf(fact_1182_less__eqE,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ~ ! [C4: nat] :
( B2
!= ( plus_plus_nat @ A2 @ C4 ) ) ) ).
% less_eqE
thf(fact_1183_Nat_Odiff__cancel,axiom,
! [K: nat,M2: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N2 ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ).
% Nat.diff_cancel
thf(fact_1184_diff__cancel2,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N2 @ K ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ).
% diff_cancel2
thf(fact_1185_diff__add__inverse,axiom,
! [N2: nat,M2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N2 @ M2 ) @ N2 )
= M2 ) ).
% diff_add_inverse
thf(fact_1186_diff__add__inverse2,axiom,
! [M2: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N2 ) @ N2 )
= M2 ) ).
% diff_add_inverse2
thf(fact_1187_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N: nat] :
? [K3: nat] :
( N
= ( plus_plus_nat @ M @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1188_trans__le__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_le_add2
thf(fact_1189_trans__le__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_le_add1
thf(fact_1190_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_1191_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1192_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_1193_add__leD2,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N2 )
=> ( ord_less_eq_nat @ K @ N2 ) ) ).
% add_leD2
thf(fact_1194_add__leD1,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% add_leD1
thf(fact_1195_le__add2,axiom,
! [N2: nat,M2: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ M2 @ N2 ) ) ).
% le_add2
thf(fact_1196_le__add1,axiom,
! [N2: nat,M2: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ N2 @ M2 ) ) ).
% le_add1
thf(fact_1197_add__leE,axiom,
! [M2: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N2 )
=> ~ ( ( ord_less_eq_nat @ M2 @ N2 )
=> ~ ( ord_less_eq_nat @ K @ N2 ) ) ) ).
% add_leE
thf(fact_1198_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_1199_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_1200_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_1201_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_1202_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_1203_trans__less__add1,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).
% trans_less_add1
thf(fact_1204_trans__less__add2,axiom,
! [I: nat,J: nat,M2: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).
% trans_less_add2
thf(fact_1205_less__add__eq__less,axiom,
! [K: nat,L: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M2 @ L )
= ( plus_plus_nat @ K @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ) ).
% less_add_eq_less
thf(fact_1206_nat__add__max__right,axiom,
! [M2: nat,N2: nat,Q3: nat] :
( ( plus_plus_nat @ M2 @ ( ord_max_nat @ N2 @ Q3 ) )
= ( ord_max_nat @ ( plus_plus_nat @ M2 @ N2 ) @ ( plus_plus_nat @ M2 @ Q3 ) ) ) ).
% nat_add_max_right
thf(fact_1207_nat__add__max__left,axiom,
! [M2: nat,N2: nat,Q3: nat] :
( ( plus_plus_nat @ ( ord_max_nat @ M2 @ N2 ) @ Q3 )
= ( ord_max_nat @ ( plus_plus_nat @ M2 @ Q3 ) @ ( plus_plus_nat @ N2 @ Q3 ) ) ) ).
% nat_add_max_left
thf(fact_1208_nat__mono,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ X2 @ Y )
=> ( ord_less_eq_nat @ ( nat2 @ X2 ) @ ( nat2 @ Y ) ) ) ).
% nat_mono
thf(fact_1209_nat__one__as__int,axiom,
( one_one_nat
= ( nat2 @ one_one_int ) ) ).
% nat_one_as_int
thf(fact_1210_add__is__1,axiom,
! [M2: nat,N2: nat] :
( ( ( plus_plus_nat @ M2 @ N2 )
= ( suc @ zero_zero_nat ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N2 = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_1211_one__is__add,axiom,
! [M2: nat,N2: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M2 @ N2 ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N2 = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_1212_less__imp__Suc__add,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ? [K2: nat] :
( N2
= ( suc @ ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1213_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M: nat,N: nat] :
? [K3: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1214_less__add__Suc2,axiom,
! [I: nat,M2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M2 @ I ) ) ) ).
% less_add_Suc2
thf(fact_1215_less__add__Suc1,axiom,
! [I: nat,M2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M2 ) ) ) ).
% less_add_Suc1
thf(fact_1216_less__natE,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ~ ! [Q4: nat] :
( N2
!= ( suc @ ( plus_plus_nat @ M2 @ Q4 ) ) ) ) ).
% less_natE
thf(fact_1217_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1218_mono__nat__linear__lb,axiom,
! [F: nat > nat,M2: nat,K: nat] :
( ! [M6: nat,N3: nat] :
( ( ord_less_nat @ M6 @ N3 )
=> ( ord_less_nat @ ( F @ M6 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M2 ) @ K ) @ ( F @ ( plus_plus_nat @ M2 @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1219_diff__add__0,axiom,
! [N2: nat,M2: nat] :
( ( minus_minus_nat @ N2 @ ( plus_plus_nat @ N2 @ M2 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1220_add__diff__inverse__nat,axiom,
! [M2: nat,N2: nat] :
( ~ ( ord_less_nat @ M2 @ N2 )
=> ( ( plus_plus_nat @ N2 @ ( minus_minus_nat @ M2 @ N2 ) )
= M2 ) ) ).
% add_diff_inverse_nat
thf(fact_1221_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_1222_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1223_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1224_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1225_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1226_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_1227_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1228_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1229_Suc__eq__plus1,axiom,
( suc
= ( ^ [N: nat] : ( plus_plus_nat @ N @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1230_nat__minus__add__max,axiom,
! [N2: nat,M2: nat] :
( ( plus_plus_nat @ ( minus_minus_nat @ N2 @ M2 ) @ M2 )
= ( ord_max_nat @ N2 @ M2 ) ) ).
% nat_minus_add_max
thf(fact_1231_nat__mono__iff,axiom,
! [Z2: int,W: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
= ( ord_less_int @ W @ Z2 ) ) ) ).
% nat_mono_iff
thf(fact_1232_zless__nat__eq__int__zless,axiom,
! [M2: nat,Z2: int] :
( ( ord_less_nat @ M2 @ ( nat2 @ Z2 ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ Z2 ) ) ).
% zless_nat_eq_int_zless
thf(fact_1233_nat__le__iff,axiom,
! [X2: int,N2: nat] :
( ( ord_less_eq_nat @ ( nat2 @ X2 ) @ N2 )
= ( ord_less_eq_int @ X2 @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% nat_le_iff
thf(fact_1234_int__minus,axiom,
! [N2: nat,M2: nat] :
( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N2 @ M2 ) )
= ( semiri1314217659103216013at_int @ ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( semiri1314217659103216013at_int @ M2 ) ) ) ) ) ).
% int_minus
thf(fact_1235_int__cases4,axiom,
! [M2: int] :
( ! [N3: nat] :
( M2
!= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( M2
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% int_cases4
thf(fact_1236_nat__diff__split__asm,axiom,
! [P: nat > $o,A2: nat,B2: nat] :
( ( P @ ( minus_minus_nat @ A2 @ B2 ) )
= ( ~ ( ( ( ord_less_nat @ A2 @ B2 )
& ~ ( P @ zero_zero_nat ) )
| ? [D4: nat] :
( ( A2
= ( plus_plus_nat @ B2 @ D4 ) )
& ~ ( P @ D4 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1237_nat__diff__split,axiom,
! [P: nat > $o,A2: nat,B2: nat] :
( ( P @ ( minus_minus_nat @ A2 @ B2 ) )
= ( ( ( ord_less_nat @ A2 @ B2 )
=> ( P @ zero_zero_nat ) )
& ! [D4: nat] :
( ( A2
= ( plus_plus_nat @ B2 @ D4 ) )
=> ( P @ D4 ) ) ) ) ).
% nat_diff_split
thf(fact_1238_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1239_nat__less__eq__zless,axiom,
! [W: int,Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
= ( ord_less_int @ W @ Z2 ) ) ) ).
% nat_less_eq_zless
thf(fact_1240_nat__le__eq__zle,axiom,
! [W: int,Z2: int] :
( ( ( ord_less_int @ zero_zero_int @ W )
| ( ord_less_eq_int @ zero_zero_int @ Z2 ) )
=> ( ( ord_less_eq_nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
= ( ord_less_eq_int @ W @ Z2 ) ) ) ).
% nat_le_eq_zle
thf(fact_1241_le__nat__iff,axiom,
! [K: int,N2: nat] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_eq_nat @ N2 @ ( nat2 @ K ) )
= ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N2 ) @ K ) ) ) ).
% le_nat_iff
thf(fact_1242_nat__diff__distrib_H,axiom,
! [X2: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( nat2 @ ( minus_minus_int @ X2 @ Y ) )
= ( minus_minus_nat @ ( nat2 @ X2 ) @ ( nat2 @ Y ) ) ) ) ) ).
% nat_diff_distrib'
thf(fact_1243_nat__diff__distrib,axiom,
! [Z5: int,Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z5 )
=> ( ( ord_less_eq_int @ Z5 @ Z2 )
=> ( ( nat2 @ ( minus_minus_int @ Z2 @ Z5 ) )
= ( minus_minus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z5 ) ) ) ) ) ).
% nat_diff_distrib
thf(fact_1244_int__cases3,axiom,
! [K: int] :
( ( K != zero_zero_int )
=> ( ! [N3: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) )
=> ~ ! [N3: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ) ).
% int_cases3
thf(fact_1245_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M: nat,N: nat] : ( if_nat @ ( M = zero_zero_nat ) @ N @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ) ) ) ).
% add_eq_if
thf(fact_1246_nat__less__iff,axiom,
! [W: int,M2: nat] :
( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ M2 )
= ( ord_less_int @ W @ ( semiri1314217659103216013at_int @ M2 ) ) ) ) ).
% nat_less_iff
thf(fact_1247_nat__abs__int__diff,axiom,
! [A2: nat,B2: nat] :
( ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) )
= ( minus_minus_nat @ B2 @ A2 ) ) )
& ( ~ ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A2 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) )
= ( minus_minus_nat @ A2 @ B2 ) ) ) ) ).
% nat_abs_int_diff
thf(fact_1248_zle__add1__eq__le,axiom,
! [W: int,Z2: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z2 @ one_one_int ) )
= ( ord_less_eq_int @ W @ Z2 ) ) ).
% zle_add1_eq_le
thf(fact_1249_odd__nonzero,axiom,
! [Z2: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_1250_int__ge__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_eq_int @ K @ I )
=> ( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_ge_induct
thf(fact_1251_zless__add1__eq,axiom,
! [W: int,Z2: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z2 @ one_one_int ) )
= ( ( ord_less_int @ W @ Z2 )
| ( W = Z2 ) ) ) ).
% zless_add1_eq
thf(fact_1252_int__gr__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_int @ K @ I )
=> ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
=> ( ! [I2: int] :
( ( ord_less_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_gr_induct
thf(fact_1253_int__ops_I4_J,axiom,
! [A2: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ A2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A2 ) @ one_one_int ) ) ).
% int_ops(4)
thf(fact_1254_int__Suc,axiom,
! [N2: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) ).
% int_Suc
thf(fact_1255_odd__less__0__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 ) @ zero_zero_int )
= ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_1256_add1__zle__eq,axiom,
! [W: int,Z2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z2 )
= ( ord_less_int @ W @ Z2 ) ) ).
% add1_zle_eq
thf(fact_1257_zless__imp__add1__zle,axiom,
! [W: int,Z2: int] :
( ( ord_less_int @ W @ Z2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z2 ) ) ).
% zless_imp_add1_zle
thf(fact_1258_int__induct,axiom,
! [P: int > $o,K: int,I: int] :
( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ K @ I2 )
=> ( ( P @ I2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) ) ) )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_induct
thf(fact_1259_bounded__Max__nat,axiom,
! [P: nat > $o,X2: nat,M7: nat] :
( ( P @ X2 )
=> ( ! [X: nat] :
( ( P @ X )
=> ( ord_less_eq_nat @ X @ M7 ) )
=> ~ ! [M6: nat] :
( ( P @ M6 )
=> ~ ! [X5: nat] :
( ( P @ X5 )
=> ( ord_less_eq_nat @ X5 @ M6 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1260_le__imp__0__less,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z2 ) ) ) ).
% le_imp_0_less
thf(fact_1261_Suc__as__int,axiom,
( suc
= ( ^ [A4: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A4 ) @ one_one_int ) ) ) ) ).
% Suc_as_int
thf(fact_1262_nat__abs__triangle__ineq,axiom,
! [K: int,L: int] : ( ord_less_eq_nat @ ( nat2 @ ( abs_abs_int @ ( plus_plus_int @ K @ L ) ) ) @ ( plus_plus_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ).
% nat_abs_triangle_ineq
thf(fact_1263_Suc__nat__eq__nat__zadd1,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( suc @ ( nat2 @ Z2 ) )
= ( nat2 @ ( plus_plus_int @ one_one_int @ Z2 ) ) ) ) ).
% Suc_nat_eq_nat_zadd1
thf(fact_1264_decr__lemma,axiom,
! [D2: int,X2: int,Z2: int] :
( ( ord_less_int @ zero_zero_int @ D2 )
=> ( ord_less_int @ ( minus_minus_int @ X2 @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X2 @ Z2 ) ) @ one_one_int ) @ D2 ) ) @ Z2 ) ) ).
% decr_lemma
% Helper facts (9)
thf(help_If_2_1_If_001tf__a_T,axiom,
! [X2: a,Y: a] :
( ( if_a @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001tf__a_T,axiom,
! [X2: a,Y: a] :
( ( if_a @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X2: int,Y: int] :
( ( if_int @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X2: int,Y: int] :
( ( if_int @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y: nat] :
( ( if_nat @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X2: nat,Y: nat] :
( ( if_nat @ $true @ X2 @ Y )
= X2 ) ).
thf(help_If_3_1_If_001t__FSet__Ofset_Itf__a_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__FSet__Ofset_Itf__a_J_T,axiom,
! [X2: fset_a,Y: fset_a] :
( ( if_fset_a @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__FSet__Ofset_Itf__a_J_T,axiom,
! [X2: fset_a,Y: fset_a] :
( ( if_fset_a @ $true @ X2 @ Y )
= X2 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( fmember_a @ x @ b )
& ( ord_less_eq_fset_a @ ca @ b ) ) ).
%------------------------------------------------------------------------------