TPTP Problem File: SLH0616^1.p
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- 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 : Real_Time_Deque/0022_Common_Proof/prob_00307_009611__6713386_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1437 ( 649 unt; 164 typ; 0 def)
% Number of atoms : 3300 (1335 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 9693 ( 351 ~; 77 |; 144 &;7841 @)
% ( 0 <=>;1280 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 6 avg)
% Number of types : 33 ( 32 usr)
% Number of type conns : 412 ( 412 >; 0 *; 0 +; 0 <<)
% Number of symbols : 135 ( 132 usr; 11 con; 0-4 aty)
% Number of variables : 3216 ( 122 ^;2981 !; 113 ?;3216 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:51:46.872
%------------------------------------------------------------------------------
% Could-be-implicit typings (32)
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J_J,type,
set_Pr765067013931698361st_int: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J_J,type,
set_Pr4048851178543822343list_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J,type,
produc1186641810826059865st_int: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
produc9164743771328383783list_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__Common__Ostate_It__Int__Oint_J_J,type,
produc5934932344166212809te_int: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__Idle__Oidle_It__Int__Oint_J_J,type,
produc400797113818724937le_int: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J_J,type,
list_P3521021558325789923at_int: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
set_Pr958786334691620121nt_int: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J_J,type,
list_P2851791750731487283_nat_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__a_Mt__Common__Ostate_Itf__a_J_J,type,
produc3409137331138395373tate_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__a_Mt__Idle__Oidle_Itf__a_J_J,type,
produc7590564867095724333idle_a: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
set_Product_prod_a_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
product_prod_nat_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J,type,
product_prod_nat_int: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
product_prod_int_int: $tType ).
thf(ty_n_t__List__Olist_I_062_It__Int__Oint_Mt__Nat__Onat_J_J,type,
list_int_nat: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J,type,
product_prod_nat_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
product_prod_a_a: $tType ).
thf(ty_n_t__Current__Ocurrent_It__Int__Oint_J,type,
current_int: $tType ).
thf(ty_n_t__Common__Ostate_It__Int__Oint_J,type,
state_int: $tType ).
thf(ty_n_t__Stack__Ostack_It__Int__Oint_J,type,
stack_int: $tType ).
thf(ty_n_t__List__Olist_It__Int__Oint_J,type,
list_int: $tType ).
thf(ty_n_t__Idle__Oidle_It__Int__Oint_J,type,
idle_int: $tType ).
thf(ty_n_t__Current__Ocurrent_Itf__a_J,type,
current_a: $tType ).
thf(ty_n_t__Common__Ostate_Itf__a_J,type,
state_a: $tType ).
thf(ty_n_t__Stack__Ostack_Itf__a_J,type,
stack_a: $tType ).
thf(ty_n_t__List__Olist_Itf__a_J,type,
list_a: $tType ).
thf(ty_n_t__Idle__Oidle_Itf__a_J,type,
idle_a: $tType ).
thf(ty_n_t__String__Ochar,type,
char: $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 (132)
thf(sy_c_Common_Opop_001t__Int__Oint,type,
pop_int: state_int > produc5934932344166212809te_int ).
thf(sy_c_Common_Opop_001tf__a,type,
pop_a: state_a > produc3409137331138395373tate_a ).
thf(sy_c_Common_Opush_001t__Int__Oint,type,
push_int: int > state_int > state_int ).
thf(sy_c_Common_Opush_001tf__a,type,
push_a: a > state_a > state_a ).
thf(sy_c_Common_Opush__rel_001t__Int__Oint,type,
push_rel_int: produc5934932344166212809te_int > produc5934932344166212809te_int > $o ).
thf(sy_c_Common_Opush__rel_001tf__a,type,
push_rel_a: produc3409137331138395373tate_a > produc3409137331138395373tate_a > $o ).
thf(sy_c_Common_Ostate_OCopy_001t__Int__Oint,type,
copy_int: current_int > list_int > list_int > nat > state_int ).
thf(sy_c_Common_Ostate_OCopy_001tf__a,type,
copy_a: current_a > list_a > list_a > nat > state_a ).
thf(sy_c_Common_Ostate_OIdle_001t__Int__Oint,type,
idle_int2: current_int > idle_int > state_int ).
thf(sy_c_Common_Ostate_OIdle_001tf__a,type,
idle_a2: current_a > idle_a > state_a ).
thf(sy_c_Common__Aux_Olist_001t__Int__Oint,type,
common_list_int: state_int > list_int ).
thf(sy_c_Common__Aux_Olist_001tf__a,type,
common_list_a: state_a > list_a ).
thf(sy_c_Common__Aux_Olist__current_001t__Int__Oint,type,
common8573843843682093467nt_int: state_int > list_int ).
thf(sy_c_Common__Aux_Olist__current_001tf__a,type,
common1102728217005306191rent_a: state_a > list_a ).
thf(sy_c_Common__Aux_Olist__current__rel_001t__Int__Oint,type,
common3364536641936693880el_int: state_int > state_int > $o ).
thf(sy_c_Common__Aux_Olist__current__rel_001tf__a,type,
common1269017819723767218_rel_a: state_a > state_a > $o ).
thf(sy_c_Current_Opush_001t__Int__Oint,type,
push_int2: int > current_int > current_int ).
thf(sy_c_Current_Opush_001tf__a,type,
push_a2: a > current_a > current_a ).
thf(sy_c_Current__Aux_Olist_001t__Int__Oint,type,
current_list_int: current_int > list_int ).
thf(sy_c_Current__Aux_Olist_001tf__a,type,
current_list_a: current_a > list_a ).
thf(sy_c_GCD_Obezw,type,
bezw: nat > nat > product_prod_int_int ).
thf(sy_c_Groups_Oabs__class_Oabs_001t__Int__Oint,type,
abs_abs_int: int > int ).
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_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
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_Idle_Oidle_OIdle_001t__Int__Oint,type,
idle_int3: stack_int > nat > idle_int ).
thf(sy_c_Idle_Oidle_OIdle_001tf__a,type,
idle_a3: stack_a > nat > idle_a ).
thf(sy_c_Idle_Opop_001t__Int__Oint,type,
pop_int2: idle_int > produc400797113818724937le_int ).
thf(sy_c_Idle_Opop_001tf__a,type,
pop_a2: idle_a > produc7590564867095724333idle_a ).
thf(sy_c_Idle_Opop__rel_001tf__a,type,
pop_rel_a: idle_a > idle_a > $o ).
thf(sy_c_Idle_Opush_001t__Int__Oint,type,
push_int3: int > idle_int > idle_int ).
thf(sy_c_Idle_Opush_001tf__a,type,
push_a3: a > idle_a > idle_a ).
thf(sy_c_Idle_Opush__rel_001tf__a,type,
push_rel_a2: produc7590564867095724333idle_a > produc7590564867095724333idle_a > $o ).
thf(sy_c_Idle__Aux_Oinvar__idle__rel_001tf__a,type,
idle_i6200314614184386870_rel_a: idle_a > idle_a > $o ).
thf(sy_c_Idle__Aux_Olist_001t__Int__Oint,type,
idle_list_int: idle_int > list_int ).
thf(sy_c_Idle__Aux_Olist_001tf__a,type,
idle_list_a: idle_a > list_a ).
thf(sy_c_Idle__Aux_Osize__idle__rel_001tf__a,type,
idle_size_idle_rel_a: idle_a > idle_a > $o ).
thf(sy_c_If_001t__Int__Oint,type,
if_int: $o > int > int > int ).
thf(sy_c_If_001t__List__Olist_It__Int__Oint_J,type,
if_list_int: $o > list_int > list_int > list_int ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_Int_Onat,type,
nat2: int > nat ).
thf(sy_c_Int_Oring__1__class_Oof__int_001t__Int__Oint,type,
ring_1_of_int_int: int > int ).
thf(sy_c_List_Oenumerate_001t__Int__Oint,type,
enumerate_int: nat > list_int > list_P3521021558325789923at_int ).
thf(sy_c_List_Oenumerate_001tf__a,type,
enumerate_a: nat > list_a > list_P2851791750731487283_nat_a ).
thf(sy_c_List_Ogen__length_001t__Int__Oint,type,
gen_length_int: nat > list_int > nat ).
thf(sy_c_List_Ogen__length_001tf__a,type,
gen_length_a: nat > list_a > nat ).
thf(sy_c_List_Olenlex_001t__Int__Oint,type,
lenlex_int: set_Pr958786334691620121nt_int > set_Pr765067013931698361st_int ).
thf(sy_c_List_Olenlex_001tf__a,type,
lenlex_a: set_Product_prod_a_a > set_Pr4048851178543822343list_a ).
thf(sy_c_List_Olex_001t__Int__Oint,type,
lex_int: set_Pr958786334691620121nt_int > set_Pr765067013931698361st_int ).
thf(sy_c_List_Olex_001tf__a,type,
lex_a: set_Product_prod_a_a > set_Pr4048851178543822343list_a ).
thf(sy_c_List_Olist_OCons_001_062_It__Int__Oint_Mt__Nat__Onat_J,type,
cons_int_nat: ( int > nat ) > list_int_nat > list_int_nat ).
thf(sy_c_List_Olist_OCons_001t__Int__Oint,type,
cons_int: int > list_int > list_int ).
thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J,type,
cons_P2335045147070616083at_int: product_prod_nat_int > list_P3521021558325789923at_int > list_P3521021558325789923at_int ).
thf(sy_c_List_Olist_OCons_001t__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J,type,
cons_P8443330267410185325_nat_a: product_prod_nat_a > list_P2851791750731487283_nat_a > list_P2851791750731487283_nat_a ).
thf(sy_c_List_Olist_OCons_001tf__a,type,
cons_a: a > list_a > list_a ).
thf(sy_c_List_Olist_Otl_001t__Int__Oint,type,
tl_int: list_int > list_int ).
thf(sy_c_List_Olist_Otl_001tf__a,type,
tl_a: list_a > list_a ).
thf(sy_c_List_Omeasures_001t__Int__Oint,type,
measures_int: list_int_nat > set_Pr958786334691620121nt_int ).
thf(sy_c_List_Oupto__aux,type,
upto_aux: int > int > list_int > list_int ).
thf(sy_c_List_Oupto__rel,type,
upto_rel: product_prod_int_int > product_prod_int_int > $o ).
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_Nat_Osize__class_Osize_001t__Common__Ostate_It__Int__Oint_J,type,
size_size_state_int: state_int > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Common__Ostate_Itf__a_J,type,
size_size_state_a: state_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Idle__Oidle_Itf__a_J,type,
size_size_idle_a: idle_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Int__Oint_J,type,
size_size_list_int: list_int > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__a_J,type,
size_size_list_a: list_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Stack__Ostack_Itf__a_J,type,
size_size_stack_a: stack_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
size_size_char: char > nat ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
neg_nu3811975205180677377ec_int: int > int ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
neg_nu5851722552734809277nc_int: int > int ).
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_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J_J,type,
ord_le1598226405681992910_int_o: ( product_prod_int_int > product_prod_int_int > $o ) > ( product_prod_int_int > product_prod_int_int > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J,type,
ord_le8369615600986905444_int_o: ( product_prod_int_int > $o ) > ( product_prod_int_int > $o ) > $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_Parity_Ounique__euclidean__semiring__with__nat__division__class_Odivides__aux_001t__Int__Oint,type,
unique5329631941980267465ux_int: product_prod_int_int > $o ).
thf(sy_c_Parity_Ounique__euclidean__semiring__with__nat__division__class_Odivides__aux_001t__Nat__Onat,type,
unique5332122412489317741ux_nat: product_prod_nat_nat > $o ).
thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
power_power_int: int > nat > int ).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Common__Ostate_It__Int__Oint_J,type,
produc3637740352169753595te_int: int > state_int > produc5934932344166212809te_int ).
thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Idle__Oidle_It__Int__Oint_J,type,
produc1741743360500042753le_int: int > idle_int > produc400797113818724937le_int ).
thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Int__Oint,type,
product_Pair_int_int: int > int > product_prod_int_int ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Int__Oint_J_001t__List__Olist_It__Int__Oint_J,type,
produc364263696895485585st_int: list_int > list_int > produc1186641810826059865st_int ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
produc6837034575241423639list_a: list_a > list_a > produc9164743771328383783list_a ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Int__Oint,type,
product_Pair_nat_int: nat > int > product_prod_nat_int ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001tf__a,type,
product_Pair_nat_a: nat > a > product_prod_nat_a ).
thf(sy_c_Product__Type_OPair_001tf__a_001t__Common__Ostate_Itf__a_J,type,
produc8263595898873874535tate_a: a > state_a > produc3409137331138395373tate_a ).
thf(sy_c_Product__Type_OPair_001tf__a_001t__Idle__Oidle_Itf__a_J,type,
produc1265230069547855005idle_a: a > idle_a > produc7590564867095724333idle_a ).
thf(sy_c_Product__Type_OPair_001tf__a_001tf__a,type,
product_Pair_a_a: a > a > product_prod_a_a ).
thf(sy_c_Stack_Ofirst_001tf__a,type,
first_a: stack_a > a ).
thf(sy_c_Stack_Opop_001tf__a,type,
pop_a3: stack_a > stack_a ).
thf(sy_c_Stack_Opush_001t__Int__Oint,type,
push_int4: int > stack_int > stack_int ).
thf(sy_c_Stack_Opush_001tf__a,type,
push_a4: a > stack_a > stack_a ).
thf(sy_c_String_Ochar_Osize__char,type,
size_char: char > nat ).
thf(sy_c_Type__Classes_Oinvar__class_Oinvar_001t__Common__Ostate_It__Int__Oint_J,type,
type_i1727439852002712835te_int: state_int > $o ).
thf(sy_c_Type__Classes_Oinvar__class_Oinvar_001t__Common__Ostate_Itf__a_J,type,
type_i4669920168676019581tate_a: state_a > $o ).
thf(sy_c_Type__Classes_Oinvar__class_Oinvar_001t__Idle__Oidle_Itf__a_J,type,
type_i8151583586401621767idle_a: idle_a > $o ).
thf(sy_c_Type__Classes_Ois__empty__class_Ois__empty_001t__Idle__Oidle_It__Int__Oint_J,type,
type_i6790807573071424059le_int: idle_int > $o ).
thf(sy_c_Type__Classes_Ois__empty__class_Ois__empty_001t__Idle__Oidle_Itf__a_J,type,
type_i7304311975391125061idle_a: idle_a > $o ).
thf(sy_c_Type__Classes_Oremaining__steps__class_Oremaining__steps_001t__Common__Ostate_Itf__a_J,type,
type_r2212416260012024137tate_a: state_a > nat ).
thf(sy_c_Type__Classes_Ostep__class_Ostep_001t__Common__Ostate_It__Int__Oint_J,type,
type_s2983202171344810359te_int: state_int > state_int ).
thf(sy_c_Type__Classes_Ostep__class_Ostep_001t__Common__Ostate_Itf__a_J,type,
type_s889635741254954505tate_a: state_a > state_a ).
thf(sy_c_Wellfounded_Oaccp_001t__Common__Ostate_It__Int__Oint_J,type,
accp_state_int: ( state_int > state_int > $o ) > state_int > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Common__Ostate_Itf__a_J,type,
accp_state_a: ( state_a > state_a > $o ) > state_a > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Idle__Oidle_Itf__a_J,type,
accp_idle_a: ( idle_a > idle_a > $o ) > idle_a > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Common__Ostate_It__Int__Oint_J_J,type,
accp_P673987829705199744te_int: ( produc5934932344166212809te_int > produc5934932344166212809te_int > $o ) > produc5934932344166212809te_int > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
accp_P1096762738010456898nt_int: ( product_prod_int_int > product_prod_int_int > $o ) > product_prod_int_int > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_Itf__a_Mt__Common__Ostate_Itf__a_J_J,type,
accp_P3541398268521339172tate_a: ( produc3409137331138395373tate_a > produc3409137331138395373tate_a > $o ) > produc3409137331138395373tate_a > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_Itf__a_Mt__Idle__Oidle_Itf__a_J_J,type,
accp_P2118310398061403062idle_a: ( produc7590564867095724333idle_a > produc7590564867095724333idle_a > $o ) > produc7590564867095724333idle_a > $o ).
thf(sy_c_Wellfounded_Omeasure_001t__Int__Oint,type,
measure_int: ( int > nat ) > set_Pr958786334691620121nt_int ).
thf(sy_c_Wellfounded_Omlex__prod_001t__Int__Oint,type,
mlex_prod_int: ( int > nat ) > set_Pr958786334691620121nt_int > set_Pr958786334691620121nt_int ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
member5262025264175285858nt_int: product_prod_int_int > set_Pr958786334691620121nt_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Int__Oint_J_Mt__List__Olist_It__Int__Oint_J_J,type,
member6698963635872716290st_int: produc1186641810826059865st_int > set_Pr765067013931698361st_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
member8191768239178080336list_a: produc9164743771328383783list_a > set_Pr4048851178543822343list_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
member1426531477525435216od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_v_common_H,type,
common: state_a ).
thf(sy_v_current____,type,
current: current_a ).
thf(sy_v_stackSize____,type,
stackSize: nat ).
thf(sy_v_stack____,type,
stack: stack_a ).
thf(sy_v_xa____,type,
xa: a ).
% Relevant facts (1265)
thf(fact_0__C1_Oprems_C_I1_J,axiom,
type_i4669920168676019581tate_a @ ( idle_a2 @ current @ ( idle_a3 @ stack @ stackSize ) ) ).
% "1.prems"(1)
thf(fact_1__C1_Oprems_C_I2_J,axiom,
ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ ( idle_a2 @ current @ ( idle_a3 @ stack @ stackSize ) ) ) ).
% "1.prems"(2)
thf(fact_2__C1_Oprems_C_I3_J,axiom,
( ( pop_a @ ( idle_a2 @ current @ ( idle_a3 @ stack @ stackSize ) ) )
= ( produc8263595898873874535tate_a @ xa @ common ) ) ).
% "1.prems"(3)
thf(fact_3_idle_Oinject,axiom,
! [X1: stack_a,X2: nat,Y1: stack_a,Y2: nat] :
( ( ( idle_a3 @ X1 @ X2 )
= ( idle_a3 @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% idle.inject
thf(fact_4_state_Oinject_I2_J,axiom,
! [X21: current_a,X22: idle_a,Y21: current_a,Y22: idle_a] :
( ( ( idle_a2 @ X21 @ X22 )
= ( idle_a2 @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% state.inject(2)
thf(fact_5_list_Oinject,axiom,
! [X21: a,X22: list_a,Y21: a,Y22: list_a] :
( ( ( cons_a @ X21 @ X22 )
= ( cons_a @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_6_list_Oinject,axiom,
! [X21: int,X22: list_int,Y21: int,Y22: list_int] :
( ( ( cons_int @ X21 @ X22 )
= ( cons_int @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_7_push__list__current,axiom,
! [X: int,Left: state_int] :
( ( common8573843843682093467nt_int @ ( push_int @ X @ Left ) )
= ( cons_int @ X @ ( common8573843843682093467nt_int @ Left ) ) ) ).
% push_list_current
thf(fact_8_push__list__current,axiom,
! [X: a,Left: state_a] :
( ( common1102728217005306191rent_a @ ( push_a @ X @ Left ) )
= ( cons_a @ X @ ( common1102728217005306191rent_a @ Left ) ) ) ).
% push_list_current
thf(fact_9_idle_Oexhaust,axiom,
! [Y: idle_a] :
~ ! [X12: stack_a,X23: nat] :
( Y
!= ( idle_a3 @ X12 @ X23 ) ) ).
% idle.exhaust
thf(fact_10_is__empty__idle_Ocases,axiom,
! [X: idle_a] :
~ ! [Stack: stack_a,Uu: nat] :
( X
!= ( idle_a3 @ Stack @ Uu ) ) ).
% is_empty_idle.cases
thf(fact_11_step__list__current,axiom,
! [Common: state_int] :
( ( type_i1727439852002712835te_int @ Common )
=> ( ( common8573843843682093467nt_int @ ( type_s2983202171344810359te_int @ Common ) )
= ( common8573843843682093467nt_int @ Common ) ) ) ).
% step_list_current
thf(fact_12_step__list__current,axiom,
! [Common: state_a] :
( ( type_i4669920168676019581tate_a @ Common )
=> ( ( common1102728217005306191rent_a @ ( type_s889635741254954505tate_a @ Common ) )
= ( common1102728217005306191rent_a @ Common ) ) ) ).
% step_list_current
thf(fact_13_not__Cons__self2,axiom,
! [X: a,Xs: list_a] :
( ( cons_a @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_14_not__Cons__self2,axiom,
! [X: int,Xs: list_int] :
( ( cons_int @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_15_list__current_Osimps_I1_J,axiom,
! [Current: current_a,Uu2: idle_a] :
( ( common1102728217005306191rent_a @ ( idle_a2 @ Current @ Uu2 ) )
= ( current_list_a @ Current ) ) ).
% list_current.simps(1)
thf(fact_16_list__current_Osimps_I1_J,axiom,
! [Current: current_int,Uu2: idle_int] :
( ( common8573843843682093467nt_int @ ( idle_int2 @ Current @ Uu2 ) )
= ( current_list_int @ Current ) ) ).
% list_current.simps(1)
thf(fact_17_Common__Proof_Oinvar__pop,axiom,
! [Common: state_a,X: a,Common2: state_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ Common ) )
=> ( ( type_i4669920168676019581tate_a @ Common )
=> ( ( ( pop_a @ Common )
= ( produc8263595898873874535tate_a @ X @ Common2 ) )
=> ( type_i4669920168676019581tate_a @ Common2 ) ) ) ) ).
% Common_Proof.invar_pop
thf(fact_18_invar__step,axiom,
! [Common: state_a] :
( ( type_i4669920168676019581tate_a @ Common )
=> ( type_i4669920168676019581tate_a @ ( type_s889635741254954505tate_a @ Common ) ) ) ).
% invar_step
thf(fact_19_Common__Proof_Oinvar__push,axiom,
! [Common: state_int,X: int] :
( ( type_i1727439852002712835te_int @ Common )
=> ( type_i1727439852002712835te_int @ ( push_int @ X @ Common ) ) ) ).
% Common_Proof.invar_push
thf(fact_20_Common__Proof_Oinvar__push,axiom,
! [Common: state_a,X: a] :
( ( type_i4669920168676019581tate_a @ Common )
=> ( type_i4669920168676019581tate_a @ ( push_a @ X @ Common ) ) ) ).
% Common_Proof.invar_push
thf(fact_21_step__state_Osimps_I1_J,axiom,
! [Current: current_a,Idle: idle_a] :
( ( type_s889635741254954505tate_a @ ( idle_a2 @ Current @ Idle ) )
= ( idle_a2 @ Current @ Idle ) ) ).
% step_state.simps(1)
thf(fact_22_Common__Proof_Opop__list,axiom,
! [Common: state_int,X: int,Common2: state_int] :
( ( type_i1727439852002712835te_int @ Common )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_int @ Common ) )
=> ( ( ( pop_int @ Common )
= ( produc3637740352169753595te_int @ X @ Common2 ) )
=> ( ( cons_int @ X @ ( common_list_int @ Common2 ) )
= ( common_list_int @ Common ) ) ) ) ) ).
% Common_Proof.pop_list
thf(fact_23_Common__Proof_Opop__list,axiom,
! [Common: state_a,X: a,Common2: state_a] :
( ( type_i4669920168676019581tate_a @ Common )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ Common ) )
=> ( ( ( pop_a @ Common )
= ( produc8263595898873874535tate_a @ X @ Common2 ) )
=> ( ( cons_a @ X @ ( common_list_a @ Common2 ) )
= ( common_list_a @ Common ) ) ) ) ) ).
% Common_Proof.pop_list
thf(fact_24_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_25_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_26_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_27_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_28_prod_Oinject,axiom,
! [X1: a,X2: state_a,Y1: a,Y2: state_a] :
( ( ( produc8263595898873874535tate_a @ X1 @ X2 )
= ( produc8263595898873874535tate_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_29_prod_Oinject,axiom,
! [X1: int,X2: int,Y1: int,Y2: int] :
( ( ( product_Pair_int_int @ X1 @ X2 )
= ( product_Pair_int_int @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_30_old_Oprod_Oinject,axiom,
! [A: a,B: state_a,A2: a,B2: state_a] :
( ( ( produc8263595898873874535tate_a @ A @ B )
= ( produc8263595898873874535tate_a @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_31_old_Oprod_Oinject,axiom,
! [A: int,B: int,A2: int,B2: int] :
( ( ( product_Pair_int_int @ A @ B )
= ( product_Pair_int_int @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_32_step__list,axiom,
! [Common: state_a] :
( ( type_i4669920168676019581tate_a @ Common )
=> ( ( common_list_a @ ( type_s889635741254954505tate_a @ Common ) )
= ( common_list_a @ Common ) ) ) ).
% step_list
thf(fact_33_Common__Proof_Opush__list,axiom,
! [X: a,Common: state_a] :
( ( common_list_a @ ( push_a @ X @ Common ) )
= ( cons_a @ X @ ( common_list_a @ Common ) ) ) ).
% Common_Proof.push_list
thf(fact_34_Common__Proof_Opush__list,axiom,
! [X: int,Common: state_int] :
( ( common_list_int @ ( push_int @ X @ Common ) )
= ( cons_int @ X @ ( common_list_int @ Common ) ) ) ).
% Common_Proof.push_list
thf(fact_35_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_36_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_37_Idle_Opush_Ocases,axiom,
! [X: produc7590564867095724333idle_a] :
~ ! [X3: a,Stack: stack_a,StackSize: nat] :
( X
!= ( produc1265230069547855005idle_a @ X3 @ ( idle_a3 @ Stack @ StackSize ) ) ) ).
% Idle.push.cases
thf(fact_38_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_39_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_40_Pair__inject,axiom,
! [A: a,B: state_a,A2: a,B2: state_a] :
( ( ( produc8263595898873874535tate_a @ A @ B )
= ( produc8263595898873874535tate_a @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_41_Pair__inject,axiom,
! [A: int,B: int,A2: int,B2: int] :
( ( ( product_Pair_int_int @ A @ B )
= ( product_Pair_int_int @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_42_prod__cases,axiom,
! [P: produc3409137331138395373tate_a > $o,P2: produc3409137331138395373tate_a] :
( ! [A3: a,B3: state_a] : ( P @ ( produc8263595898873874535tate_a @ A3 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_43_prod__cases,axiom,
! [P: product_prod_int_int > $o,P2: product_prod_int_int] :
( ! [A3: int,B3: int] : ( P @ ( product_Pair_int_int @ A3 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_44_surj__pair,axiom,
! [P2: produc3409137331138395373tate_a] :
? [X3: a,Y3: state_a] :
( P2
= ( produc8263595898873874535tate_a @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_45_surj__pair,axiom,
! [P2: product_prod_int_int] :
? [X3: int,Y3: int] :
( P2
= ( product_Pair_int_int @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_46_old_Oprod_Oexhaust,axiom,
! [Y: produc3409137331138395373tate_a] :
~ ! [A3: a,B3: state_a] :
( Y
!= ( produc8263595898873874535tate_a @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_47_old_Oprod_Oexhaust,axiom,
! [Y: product_prod_int_int] :
~ ! [A3: int,B3: int] :
( Y
!= ( product_Pair_int_int @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_48_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_49_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M: nat] :
( ( ord_less_nat @ M @ N2 )
& ~ ( P @ M ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_50_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( P @ M ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_51_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_52_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_53_less__not__refl2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( M2 != N ) ) ).
% less_not_refl2
thf(fact_54_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_55_nat__neq__iff,axiom,
! [M2: nat,N: nat] :
( ( M2 != N )
= ( ( ord_less_nat @ M2 @ N )
| ( ord_less_nat @ N @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_56_size__neq__size__imp__neq,axiom,
! [X: state_a,Y: state_a] :
( ( ( size_size_state_a @ X )
!= ( size_size_state_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_57_size__neq__size__imp__neq,axiom,
! [X: char,Y: char] :
( ( ( size_size_char @ X )
!= ( size_size_char @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_58_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_59_gr__implies__not__zero,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_60_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_61_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_62_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M: nat] :
( ( ord_less_nat @ M @ N2 )
& ~ ( P @ M ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_63_gr__implies__not0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_64_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_65_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_66_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_67_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_68_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_69_remaining__steps__state_Osimps_I1_J,axiom,
! [Uu2: current_a,Uv: idle_a] :
( ( type_r2212416260012024137tate_a @ ( idle_a2 @ Uu2 @ Uv ) )
= zero_zero_nat ) ).
% remaining_steps_state.simps(1)
thf(fact_70_in__measures_I2_J,axiom,
! [X: int,Y: int,F: int > nat,Fs: list_int_nat] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( measures_int @ ( cons_int_nat @ F @ Fs ) ) )
= ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
| ( ( ( F @ X )
= ( F @ Y ) )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( measures_int @ Fs ) ) ) ) ) ).
% in_measures(2)
thf(fact_71_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_72_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_73_divides__aux__eq,axiom,
! [Q: nat,R: nat] :
( ( unique5332122412489317741ux_nat @ ( product_Pair_nat_nat @ Q @ R ) )
= ( R = zero_zero_nat ) ) ).
% divides_aux_eq
thf(fact_74_divides__aux__eq,axiom,
! [Q: int,R: int] :
( ( unique5329631941980267465ux_int @ ( product_Pair_int_int @ Q @ R ) )
= ( R = zero_zero_int ) ) ).
% divides_aux_eq
thf(fact_75_in__measure,axiom,
! [X: int,Y: int,F: int > nat] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( measure_int @ F ) )
= ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) ) ) ).
% in_measure
thf(fact_76_measures__less,axiom,
! [F: int > nat,X: int,Y: int,Fs: list_int_nat] :
( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( measures_int @ ( cons_int_nat @ F @ Fs ) ) ) ) ).
% measures_less
thf(fact_77_list__current_Oelims,axiom,
! [X: state_a,Y: list_a] :
( ( ( common1102728217005306191rent_a @ X )
= Y )
=> ( ! [Current2: current_a] :
( ? [Uu: idle_a] :
( X
= ( idle_a2 @ Current2 @ Uu ) )
=> ( Y
!= ( current_list_a @ Current2 ) ) )
=> ~ ! [Current2: current_a] :
( ? [Uv2: list_a,Uw: list_a,Ux: nat] :
( X
= ( copy_a @ Current2 @ Uv2 @ Uw @ Ux ) )
=> ( Y
!= ( current_list_a @ Current2 ) ) ) ) ) ).
% list_current.elims
thf(fact_78_list__current_Oelims,axiom,
! [X: state_int,Y: list_int] :
( ( ( common8573843843682093467nt_int @ X )
= Y )
=> ( ! [Current2: current_int] :
( ? [Uu: idle_int] :
( X
= ( idle_int2 @ Current2 @ Uu ) )
=> ( Y
!= ( current_list_int @ Current2 ) ) )
=> ~ ! [Current2: current_int] :
( ? [Uv2: list_int,Uw: list_int,Ux: nat] :
( X
= ( copy_int @ Current2 @ Uv2 @ Uw @ Ux ) )
=> ( Y
!= ( current_list_int @ Current2 ) ) ) ) ) ).
% list_current.elims
thf(fact_79_Common__Aux_Olist_Osimps_I1_J,axiom,
! [Uu2: current_a,Idle: idle_a] :
( ( common_list_a @ ( idle_a2 @ Uu2 @ Idle ) )
= ( idle_list_a @ Idle ) ) ).
% Common_Aux.list.simps(1)
thf(fact_80_of__nat__eq__iff,axiom,
! [M2: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= ( semiri1314217659103216013at_int @ N ) )
= ( M2 = N ) ) ).
% of_nat_eq_iff
thf(fact_81_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_82_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_83_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_84_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_85_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_86_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_87_of__nat__less__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_iff
thf(fact_88_of__nat__less__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_iff
thf(fact_89_size__idle_Osimps,axiom,
! [Stack2: stack_a,Uu2: nat] :
( ( size_size_idle_a @ ( idle_a3 @ Stack2 @ Uu2 ) )
= ( size_size_stack_a @ Stack2 ) ) ).
% size_idle.simps
thf(fact_90_size__idle_Oelims,axiom,
! [X: idle_a,Y: nat] :
( ( ( size_size_idle_a @ X )
= Y )
=> ~ ! [Stack: stack_a] :
( ? [Uu: nat] :
( X
= ( idle_a3 @ Stack @ Uu ) )
=> ( Y
!= ( size_size_stack_a @ Stack ) ) ) ) ).
% size_idle.elims
thf(fact_91_size__new__state_Ocases,axiom,
! [X: state_a] :
( ! [Current2: current_a,Uu: idle_a] :
( X
!= ( idle_a2 @ Current2 @ Uu ) )
=> ~ ! [Current2: current_a,Uv2: list_a,Uw: list_a,Ux: nat] :
( X
!= ( copy_a @ Current2 @ Uv2 @ Uw @ Ux ) ) ) ).
% size_new_state.cases
thf(fact_92_normalize_Ocases,axiom,
! [X: state_a] :
( ! [Current2: current_a,Old: list_a,New: list_a,Moved: nat] :
( X
!= ( copy_a @ Current2 @ Old @ New @ Moved ) )
=> ~ ! [V: current_a,Va: idle_a] :
( X
!= ( idle_a2 @ V @ Va ) ) ) ).
% normalize.cases
thf(fact_93_state_Oexhaust,axiom,
! [Y: state_a] :
( ! [X11: current_a,X122: list_a,X13: list_a,X14: nat] :
( Y
!= ( copy_a @ X11 @ X122 @ X13 @ X14 ) )
=> ~ ! [X212: current_a,X222: idle_a] :
( Y
!= ( idle_a2 @ X212 @ X222 ) ) ) ).
% state.exhaust
thf(fact_94_state_Odistinct_I1_J,axiom,
! [X112: current_a,X123: list_a,X132: list_a,X142: nat,X21: current_a,X22: idle_a] :
( ( copy_a @ X112 @ X123 @ X132 @ X142 )
!= ( idle_a2 @ X21 @ X22 ) ) ).
% state.distinct(1)
thf(fact_95_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_96_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_97_less__imp__of__nat__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_98_less__imp__of__nat__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_99_of__nat__less__imp__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_100_of__nat__less__imp__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% of_nat_less_imp_less
thf(fact_101_list__current_Osimps_I2_J,axiom,
! [Current: current_a,Uv: list_a,Uw2: list_a,Ux2: nat] :
( ( common1102728217005306191rent_a @ ( copy_a @ Current @ Uv @ Uw2 @ Ux2 ) )
= ( current_list_a @ Current ) ) ).
% list_current.simps(2)
thf(fact_102_list__current_Osimps_I2_J,axiom,
! [Current: current_int,Uv: list_int,Uw2: list_int,Ux2: nat] :
( ( common8573843843682093467nt_int @ ( copy_int @ Current @ Uv @ Uw2 @ Ux2 ) )
= ( current_list_int @ Current ) ) ).
% list_current.simps(2)
thf(fact_103_Common_Opush_Ocases,axiom,
! [X: produc3409137331138395373tate_a] :
( ! [X3: a,Current2: current_a,Stack: stack_a,StackSize: nat] :
( X
!= ( produc8263595898873874535tate_a @ X3 @ ( idle_a2 @ Current2 @ ( idle_a3 @ Stack @ StackSize ) ) ) )
=> ~ ! [X3: a,Current2: current_a,Aux: list_a,New: list_a,Moved: nat] :
( X
!= ( produc8263595898873874535tate_a @ X3 @ ( copy_a @ Current2 @ Aux @ New @ Moved ) ) ) ) ).
% Common.push.cases
thf(fact_104_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_105_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N2: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% pos_int_cases
thf(fact_106_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
& ( K
= ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_107_measures__lesseq,axiom,
! [F: int > nat,X: int,Y: int,Fs: list_int_nat] :
( ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( measures_int @ Fs ) )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( measures_int @ ( cons_int_nat @ F @ Fs ) ) ) ) ) ).
% measures_lesseq
thf(fact_108_mlex__iff,axiom,
! [X: int,Y: int,F: int > nat,R2: set_Pr958786334691620121nt_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( mlex_prod_int @ F @ R2 ) )
= ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
| ( ( ( F @ X )
= ( F @ Y ) )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 ) ) ) ) ).
% mlex_iff
thf(fact_109_mlex__less,axiom,
! [F: int > nat,X: int,Y: int,R2: set_Pr958786334691620121nt_int] :
( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( mlex_prod_int @ F @ R2 ) ) ) ).
% mlex_less
thf(fact_110_Common_Opush_Osimps_I2_J,axiom,
! [X: a,Current: current_a,Aux2: list_a,New2: list_a,Moved2: nat] :
( ( push_a @ X @ ( copy_a @ Current @ Aux2 @ New2 @ Moved2 ) )
= ( copy_a @ ( push_a2 @ X @ Current ) @ Aux2 @ New2 @ Moved2 ) ) ).
% Common.push.simps(2)
thf(fact_111_Common_Opush_Osimps_I2_J,axiom,
! [X: int,Current: current_int,Aux2: list_int,New2: list_int,Moved2: nat] :
( ( push_int @ X @ ( copy_int @ Current @ Aux2 @ New2 @ Moved2 ) )
= ( copy_int @ ( push_int2 @ X @ Current ) @ Aux2 @ New2 @ Moved2 ) ) ).
% Common.push.simps(2)
thf(fact_112_Cons__lenlex__iff,axiom,
! [M2: a,Ms: list_a,N: a,Ns: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ M2 @ Ms ) @ ( cons_a @ N @ Ns ) ) @ ( lenlex_a @ R ) )
= ( ( ord_less_nat @ ( size_size_list_a @ Ms ) @ ( size_size_list_a @ Ns ) )
| ( ( ( size_size_list_a @ Ms )
= ( size_size_list_a @ Ns ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ M2 @ N ) @ R ) )
| ( ( M2 = N )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ms @ Ns ) @ ( lenlex_a @ R ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_113_Cons__lenlex__iff,axiom,
! [M2: int,Ms: list_int,N: int,Ns: list_int,R: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ M2 @ Ms ) @ ( cons_int @ N @ Ns ) ) @ ( lenlex_int @ R ) )
= ( ( ord_less_nat @ ( size_size_list_int @ Ms ) @ ( size_size_list_int @ Ns ) )
| ( ( ( size_size_list_int @ Ms )
= ( size_size_list_int @ Ns ) )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ M2 @ N ) @ R ) )
| ( ( M2 = N )
& ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Ms @ Ns ) @ ( lenlex_int @ R ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_114_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_115_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_116_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_117_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_118_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_119_nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_120_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_121_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_122_of__nat__power,axiom,
! [M2: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( power_power_nat @ M2 @ N ) )
= ( power_power_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ N ) ) ).
% of_nat_power
thf(fact_123_of__nat__power,axiom,
! [M2: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( power_power_nat @ M2 @ N ) )
= ( power_power_int @ ( semiri1314217659103216013at_int @ M2 ) @ N ) ) ).
% of_nat_power
thf(fact_124_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W )
= ( semiri1316708129612266289at_nat @ X ) )
= ( ( power_power_nat @ B @ W )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_125_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W )
= ( semiri1314217659103216013at_int @ X ) )
= ( ( power_power_nat @ B @ W )
= X ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_126_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ( semiri1316708129612266289at_nat @ X )
= ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
= ( X
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_127_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ( semiri1314217659103216013at_int @ X )
= ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
= ( X
= ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_128_of__nat__le__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% of_nat_le_iff
thf(fact_129_of__nat__le__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% of_nat_le_iff
thf(fact_130_power__eq__0__iff,axiom,
! [A: nat,N: nat] :
( ( ( power_power_nat @ A @ N )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_131_power__eq__0__iff,axiom,
! [A: int,N: nat] :
( ( ( power_power_int @ A @ N )
= zero_zero_int )
= ( ( A = zero_zero_int )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_132_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_133_of__nat__le__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X ) )
= ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_134_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_135_of__nat__power__le__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
= ( ord_less_eq_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_136_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X ) )
= ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_137_of__nat__less__of__nat__power__cancel__iff,axiom,
! [B: nat,W: nat,X: nat] :
( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X ) )
= ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_138_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
= ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_139_of__nat__power__less__of__nat__cancel__iff,axiom,
! [X: nat,B: nat,W: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ X ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
= ( ord_less_nat @ X @ ( power_power_nat @ B @ W ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_140_power__mono__iff,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_141_power__mono__iff,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
= ( ord_less_eq_int @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_142_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N2: nat] :
( K
!= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% nonneg_int_cases
thf(fact_143_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N2: nat] :
( K
= ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_144_zle__int,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% zle_int
thf(fact_145_int__int__eq,axiom,
! [M2: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= ( semiri1314217659103216013at_int @ N ) )
= ( M2 = N ) ) ).
% int_int_eq
thf(fact_146_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_147_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_148_eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( M2 = N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% eq_imp_le
thf(fact_149_le__antisym,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_eq_nat @ N @ M2 )
=> ( M2 = N ) ) ) ).
% le_antisym
thf(fact_150_nat__le__linear,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
| ( ord_less_eq_nat @ N @ M2 ) ) ).
% nat_le_linear
thf(fact_151_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_152_zero__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% zero_le_power
thf(fact_153_zero__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% zero_le_power
thf(fact_154_power__mono,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).
% power_mono
thf(fact_155_power__mono,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% power_mono
thf(fact_156_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_157_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_158_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_159_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_160_power__not__zero,axiom,
! [A: nat,N: nat] :
( ( A != zero_zero_nat )
=> ( ( power_power_nat @ A @ N )
!= zero_zero_nat ) ) ).
% power_not_zero
thf(fact_161_power__not__zero,axiom,
! [A: int,N: nat] :
( ( A != zero_zero_int )
=> ( ( power_power_int @ A @ N )
!= zero_zero_int ) ) ).
% power_not_zero
thf(fact_162_power__less__imp__less__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_163_power__less__imp__less__base,axiom,
! [A: int,N: nat,B: int] :
( ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_164_mlex__leq,axiom,
! [F: int > nat,X: int,Y: int,R2: set_Pr958786334691620121nt_int] :
( ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R2 )
=> ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ ( mlex_prod_int @ F @ R2 ) ) ) ) ).
% mlex_leq
thf(fact_165_power__eq__imp__eq__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ( power_power_nat @ A @ N )
= ( power_power_nat @ B @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_166_power__eq__imp__eq__base,axiom,
! [A: int,N: nat,B: int] :
( ( ( power_power_int @ A @ N )
= ( power_power_int @ B @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_167_power__eq__iff__eq__base,axiom,
! [N: nat,A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ( power_power_nat @ A @ N )
= ( power_power_nat @ B @ N ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_168_power__eq__iff__eq__base,axiom,
! [N: nat,A: int,B: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ( power_power_int @ A @ N )
= ( power_power_int @ B @ N ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_169_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_170_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_171_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_172_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_173_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_174_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_175_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_176_impossible__Cons,axiom,
! [Xs: list_a,Ys: list_a,X: a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys ) )
=> ( Xs
!= ( cons_a @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_177_impossible__Cons,axiom,
! [Xs: list_int,Ys: list_int,X: int] :
( ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ ( size_size_list_int @ Ys ) )
=> ( Xs
!= ( cons_int @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_178_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_179_le__neq__implies__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( M2 != N )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_180_less__or__eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_181_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
| ( M3 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_182_less__imp__le__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_imp_le_nat
thf(fact_183_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
& ( M3 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_184_zero__less__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% zero_less_power
thf(fact_185_zero__less__power,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% zero_less_power
thf(fact_186_nat__power__less__imp__less,axiom,
! [I: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ I )
=> ( ( ord_less_nat @ ( power_power_nat @ I @ M2 ) @ ( power_power_nat @ I @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_187_power__strict__mono,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_188_power__strict__mono,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_189_lenlex__irreflexive,axiom,
! [R: set_Pr958786334691620121nt_int,Xs: list_int] :
( ! [X3: int] :
~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ X3 ) @ R )
=> ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Xs ) @ ( lenlex_int @ R ) ) ) ).
% lenlex_irreflexive
thf(fact_190_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ).
% zero_power
thf(fact_191_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ).
% zero_power
thf(fact_192_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1316708129612266289at_nat @ N ) ) ).
% of_nat_0_le_iff
thf(fact_193_of__nat__0__le__iff,axiom,
! [N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1314217659103216013at_int @ N ) ) ).
% of_nat_0_le_iff
thf(fact_194_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_195_Current__Proof_Opush__list,axiom,
! [X: a,Current: current_a] :
( ( current_list_a @ ( push_a2 @ X @ Current ) )
= ( cons_a @ X @ ( current_list_a @ Current ) ) ) ).
% Current_Proof.push_list
thf(fact_196_Current__Proof_Opush__list,axiom,
! [X: int,Current: current_int] :
( ( current_list_int @ ( push_int2 @ X @ Current ) )
= ( cons_int @ X @ ( current_list_int @ Current ) ) ) ).
% Current_Proof.push_list
thf(fact_197_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_198_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_199_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_200_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_201_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_202_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_203_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_204_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_205_Cons__in__lex,axiom,
! [X: a,Xs: list_a,Y: a,Ys: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X @ Xs ) @ ( cons_a @ Y @ Ys ) ) @ ( lex_a @ R ) )
= ( ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
& ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) ) )
| ( ( X = Y )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( lex_a @ R ) ) ) ) ) ).
% Cons_in_lex
thf(fact_206_Cons__in__lex,axiom,
! [X: int,Xs: list_int,Y: int,Ys: list_int,R: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ X @ Xs ) @ ( cons_int @ Y @ Ys ) ) @ ( lex_int @ R ) )
= ( ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R )
& ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) ) )
| ( ( X = Y )
& ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( lex_int @ R ) ) ) ) ) ).
% Cons_in_lex
thf(fact_207_verit__la__generic,axiom,
! [A: int,X: int] :
( ( ord_less_eq_int @ A @ X )
| ( A = X )
| ( ord_less_eq_int @ X @ A ) ) ).
% verit_la_generic
thf(fact_208_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_209_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_210_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_211_nle__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_eq_int @ A @ B ) )
= ( ( ord_less_eq_int @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_212_le__cases3,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z ) )
=> ( ( ( ord_less_eq_nat @ X @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_213_le__cases3,axiom,
! [X: int,Y: int,Z: int] :
( ( ( ord_less_eq_int @ X @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z ) )
=> ( ( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_eq_int @ X @ Z ) )
=> ( ( ( ord_less_eq_int @ X @ Z )
=> ~ ( ord_less_eq_int @ Z @ Y ) )
=> ( ( ( ord_less_eq_int @ Z @ Y )
=> ~ ( ord_less_eq_int @ Y @ X ) )
=> ( ( ( ord_less_eq_int @ Y @ Z )
=> ~ ( ord_less_eq_int @ Z @ X ) )
=> ~ ( ( ord_less_eq_int @ Z @ X )
=> ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_214_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [X4: nat,Y6: nat] :
( ( ord_less_eq_nat @ X4 @ Y6 )
& ( ord_less_eq_nat @ Y6 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_215_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z2: int] : ( Y5 = Z2 ) )
= ( ^ [X4: int,Y6: int] :
( ( ord_less_eq_int @ X4 @ Y6 )
& ( ord_less_eq_int @ Y6 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_216_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_217_ord__eq__le__trans,axiom,
! [A: int,B: int,C2: int] :
( ( A = B )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ord_less_eq_int @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_218_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_219_ord__le__eq__trans,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_eq_int @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_220_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_221_order__antisym,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_222_order_Otrans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% order.trans
thf(fact_223_order_Otrans,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ord_less_eq_int @ A @ C2 ) ) ) ).
% order.trans
thf(fact_224_order__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_225_order__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z )
=> ( ord_less_eq_int @ X @ Z ) ) ) ).
% order_trans
thf(fact_226_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: nat,B3: nat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_227_linorder__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: int,B3: int] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_228_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_229_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: int,Z2: int] : ( Y5 = Z2 ) )
= ( ^ [A4: int,B4: int] :
( ( ord_less_eq_int @ B4 @ A4 )
& ( ord_less_eq_int @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_230_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_231_dual__order_Oantisym,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_232_dual__order_Otrans,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_eq_nat @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_233_dual__order_Otrans,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C2 @ B )
=> ( ord_less_eq_int @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_234_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_235_antisym,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_236_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_237_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z2: int] : ( Y5 = Z2 ) )
= ( ^ [A4: int,B4: int] :
( ( ord_less_eq_int @ A4 @ B4 )
& ( ord_less_eq_int @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_238_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_239_order__subst1,axiom,
! [A: nat,F: int > nat,B: int,C2: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_240_order__subst1,axiom,
! [A: int,F: nat > int,B: nat,C2: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_241_order__subst1,axiom,
! [A: int,F: int > int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_242_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_243_order__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C2: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_244_order__subst2,axiom,
! [A: int,B: int,F: int > nat,C2: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_245_order__subst2,axiom,
! [A: int,B: int,F: int > int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_246_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_247_order__eq__refl,axiom,
! [X: int,Y: int] :
( ( X = Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_eq_refl
thf(fact_248_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_249_verit__la__disequality,axiom,
! [A: int,B: int] :
( ( A = B )
| ~ ( ord_less_eq_int @ A @ B )
| ~ ( ord_less_eq_int @ B @ A ) ) ).
% verit_la_disequality
thf(fact_250_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_251_linorder__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_linear
thf(fact_252_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_253_ord__eq__le__subst,axiom,
! [A: int,F: nat > int,B: nat,C2: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_254_ord__eq__le__subst,axiom,
! [A: nat,F: int > nat,B: int,C2: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_255_ord__eq__le__subst,axiom,
! [A: int,F: int > int,B: int,C2: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_256_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_257_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C2: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_258_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C2: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_259_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_260_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_261_linorder__le__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_262_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_263_order__antisym__conv,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_264_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_265_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_266_lt__ex,axiom,
! [X: int] :
? [Y3: int] : ( ord_less_int @ Y3 @ X ) ).
% lt_ex
thf(fact_267_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_268_gt__ex,axiom,
! [X: int] :
? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% gt_ex
thf(fact_269_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_270_less__imp__neq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_271_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_272_order_Oasym,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order.asym
thf(fact_273_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_274_ord__eq__less__trans,axiom,
! [A: int,B: int,C2: int] :
( ( A = B )
=> ( ( ord_less_int @ B @ C2 )
=> ( ord_less_int @ A @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_275_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_276_ord__less__eq__trans,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_int @ A @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_277_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X3: nat] :
( ! [Y4: nat] :
( ( ord_less_nat @ Y4 @ X3 )
=> ( P @ Y4 ) )
=> ( P @ X3 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_278_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_279_antisym__conv3,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_int @ Y @ X )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_280_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_281_linorder__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_282_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_283_dual__order_Oasym,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ A @ B ) ) ).
% dual_order.asym
thf(fact_284_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_285_dual__order_Oirrefl,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% dual_order.irrefl
thf(fact_286_exists__least__iff,axiom,
( ( ^ [P3: nat > $o] :
? [X5: nat] : ( P3 @ X5 ) )
= ( ^ [P4: nat > $o] :
? [N3: nat] :
( ( P4 @ N3 )
& ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ~ ( P4 @ M3 ) ) ) ) ) ).
% exists_least_iff
thf(fact_287_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: nat] : ( P @ A3 @ A3 )
=> ( ! [A3: nat,B3: nat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_288_linorder__less__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A3: int,B3: int] :
( ( ord_less_int @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: int] : ( P @ A3 @ A3 )
=> ( ! [A3: int,B3: int] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_289_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans
thf(fact_290_order_Ostrict__trans,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ B @ C2 )
=> ( ord_less_int @ A @ C2 ) ) ) ).
% order.strict_trans
thf(fact_291_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_292_not__less__iff__gr__or__eq,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ( ord_less_int @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_293_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_294_dual__order_Ostrict__trans,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C2 @ B )
=> ( ord_less_int @ C2 @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_295_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_296_order_Ostrict__implies__not__eq,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_297_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_298_dual__order_Ostrict__implies__not__eq,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_299_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_300_linorder__neqE,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_301_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_302_order__less__asym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_asym
thf(fact_303_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_304_linorder__neq__iff,axiom,
! [X: int,Y: int] :
( ( X != Y )
= ( ( ord_less_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_305_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_306_order__less__asym_H,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order_less_asym'
thf(fact_307_order__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_308_order__less__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_309_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_310_ord__eq__less__subst,axiom,
! [A: int,F: nat > int,B: nat,C2: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_311_ord__eq__less__subst,axiom,
! [A: nat,F: int > nat,B: int,C2: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_312_ord__eq__less__subst,axiom,
! [A: int,F: int > int,B: int,C2: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_313_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_314_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C2: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_315_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C2: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_316_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_317_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_318_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_319_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_320_order__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C2: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_321_order__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C2: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_322_order__less__subst1,axiom,
! [A: int,F: int > int,B: int,C2: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_323_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_324_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C2: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_325_order__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C2: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_326_order__less__subst2,axiom,
! [A: int,B: int,F: int > int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_327_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_328_order__less__not__sym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_329_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_330_order__less__imp__triv,axiom,
! [X: int,Y: int,P: $o] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_331_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_332_linorder__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
| ( X = Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_333_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_334_order__less__imp__not__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_335_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_336_order__less__imp__not__eq2,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_337_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_338_order__less__imp__not__less,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_339_int__if,axiom,
! [P: $o,A: nat,B: nat] :
( ( P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
= ( semiri1314217659103216013at_int @ A ) ) )
& ( ~ P
=> ( ( semiri1314217659103216013at_int @ ( if_nat @ P @ A @ B ) )
= ( semiri1314217659103216013at_int @ B ) ) ) ) ).
% int_if
thf(fact_340_nat__int__comparison_I1_J,axiom,
( ( ^ [Y5: nat,Z2: nat] : ( Y5 = Z2 ) )
= ( ^ [A4: nat,B4: nat] :
( ( semiri1314217659103216013at_int @ A4 )
= ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_341_verit__comp__simplify1_I3_J,axiom,
! [B2: nat,A2: nat] :
( ( ~ ( ord_less_eq_nat @ B2 @ A2 ) )
= ( ord_less_nat @ A2 @ B2 ) ) ).
% verit_comp_simplify1(3)
thf(fact_342_verit__comp__simplify1_I3_J,axiom,
! [B2: int,A2: int] :
( ( ~ ( ord_less_eq_int @ B2 @ A2 ) )
= ( ord_less_int @ A2 @ B2 ) ) ).
% verit_comp_simplify1(3)
thf(fact_343_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_344_leD,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_int @ X @ Y ) ) ).
% leD
thf(fact_345_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_346_leI,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% leI
thf(fact_347_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_348_nless__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_int @ A @ B ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_349_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_350_antisym__conv1,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_351_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_352_antisym__conv2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_353_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y6: nat] :
( ( ord_less_eq_nat @ X4 @ Y6 )
& ~ ( ord_less_eq_nat @ Y6 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_354_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X4: int,Y6: int] :
( ( ord_less_eq_int @ X4 @ Y6 )
& ~ ( ord_less_eq_int @ Y6 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_355_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_356_not__le__imp__less,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_eq_int @ Y @ X )
=> ( ord_less_int @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_357_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_358_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B4: int] :
( ( ord_less_int @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_359_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_360_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A4: int,B4: int] :
( ( ord_less_eq_int @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_361_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_362_order_Ostrict__trans1,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ B @ C2 )
=> ( ord_less_int @ A @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_363_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_364_order_Ostrict__trans2,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ord_less_int @ A @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_365_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ~ ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_366_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A4: int,B4: int] :
( ( ord_less_eq_int @ A4 @ B4 )
& ~ ( ord_less_eq_int @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_367_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_less_nat @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_368_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B4: int,A4: int] :
( ( ord_less_int @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_369_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_370_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B4: int,A4: int] :
( ( ord_less_eq_int @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_371_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_372_dual__order_Ostrict__trans1,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_int @ C2 @ B )
=> ( ord_less_int @ C2 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_373_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_374_dual__order_Ostrict__trans2,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_eq_int @ C2 @ B )
=> ( ord_less_int @ C2 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_375_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ~ ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_376_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B4: int,A4: int] :
( ( ord_less_eq_int @ B4 @ A4 )
& ~ ( ord_less_eq_int @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_377_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_378_order_Ostrict__implies__order,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_eq_int @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_379_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_380_dual__order_Ostrict__implies__order,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_eq_int @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_381_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y6: nat] :
( ( ord_less_nat @ X4 @ Y6 )
| ( X4 = Y6 ) ) ) ) ).
% order_le_less
thf(fact_382_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X4: int,Y6: int] :
( ( ord_less_int @ X4 @ Y6 )
| ( X4 = Y6 ) ) ) ) ).
% order_le_less
thf(fact_383_order__less__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y6: nat] :
( ( ord_less_eq_nat @ X4 @ Y6 )
& ( X4 != Y6 ) ) ) ) ).
% order_less_le
thf(fact_384_order__less__le,axiom,
( ord_less_int
= ( ^ [X4: int,Y6: int] :
( ( ord_less_eq_int @ X4 @ Y6 )
& ( X4 != Y6 ) ) ) ) ).
% order_less_le
thf(fact_385_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_386_linorder__not__le,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X @ Y ) )
= ( ord_less_int @ Y @ X ) ) ).
% linorder_not_le
thf(fact_387_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_388_linorder__not__less,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_not_less
thf(fact_389_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_390_order__less__imp__le,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_391_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_392_order__le__neq__trans,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( A != B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_393_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_394_order__neq__le__trans,axiom,
! [A: int,B: int] :
( ( A != B )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_395_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_396_order__le__less__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_397_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_398_order__less__le__trans,axiom,
! [X: int,Y: int,Z: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_399_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_400_order__le__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C2: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_401_order__le__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C2: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_402_order__le__less__subst1,axiom,
! [A: int,F: int > int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_403_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_404_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C2: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_405_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C2: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_406_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_407_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_408_order__less__le__subst1,axiom,
! [A: int,F: nat > int,B: nat,C2: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_409_order__less__le__subst1,axiom,
! [A: nat,F: int > nat,B: int,C2: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_410_order__less__le__subst1,axiom,
! [A: int,F: int > int,B: int,C2: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_eq_int @ X3 @ Y3 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_411_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_412_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > nat,C2: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_413_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C2: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C2 )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_414_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C2 )
=> ( ! [X3: int,Y3: int] :
( ( ord_less_int @ X3 @ Y3 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_415_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_416_linorder__le__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_417_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_418_order__le__imp__less__or__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_419_imp__le__cong,axiom,
! [X: int,X6: int,P: $o,P5: $o] :
( ( X = X6 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> ( P = P5 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> P5 ) ) ) ) ).
% imp_le_cong
thf(fact_420_conj__le__cong,axiom,
! [X: int,X6: int,P: $o,P5: $o] :
( ( X = X6 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X6 )
=> ( P = P5 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X6 )
& P5 ) ) ) ) ).
% conj_le_cong
thf(fact_421_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M2: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I3: nat] :
( ( ord_less_nat @ K2 @ I3 )
=> ( P @ I3 ) )
=> ( P @ K2 ) ) )
=> ( P @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_422_minf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z3 )
=> ~ ( ord_less_eq_nat @ T @ X7 ) ) ).
% minf(8)
thf(fact_423_minf_I8_J,axiom,
! [T: int] :
? [Z3: int] :
! [X7: int] :
( ( ord_less_int @ X7 @ Z3 )
=> ~ ( ord_less_eq_int @ T @ X7 ) ) ).
% minf(8)
thf(fact_424_minf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z3 )
=> ( ord_less_eq_nat @ X7 @ T ) ) ).
% minf(6)
thf(fact_425_minf_I6_J,axiom,
! [T: int] :
? [Z3: int] :
! [X7: int] :
( ( ord_less_int @ X7 @ Z3 )
=> ( ord_less_eq_int @ X7 @ T ) ) ).
% minf(6)
thf(fact_426_pinf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z3 @ X7 )
=> ( ord_less_eq_nat @ T @ X7 ) ) ).
% pinf(8)
thf(fact_427_pinf_I8_J,axiom,
! [T: int] :
? [Z3: int] :
! [X7: int] :
( ( ord_less_int @ Z3 @ X7 )
=> ( ord_less_eq_int @ T @ X7 ) ) ).
% pinf(8)
thf(fact_428_pinf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z3 @ X7 )
=> ~ ( ord_less_eq_nat @ X7 @ T ) ) ).
% pinf(6)
thf(fact_429_pinf_I6_J,axiom,
! [T: int] :
? [Z3: int] :
! [X7: int] :
( ( ord_less_int @ Z3 @ X7 )
=> ~ ( ord_less_eq_int @ X7 @ T ) ) ).
% pinf(6)
thf(fact_430_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C3: nat] :
( ( ord_less_eq_nat @ A @ C3 )
& ( ord_less_eq_nat @ C3 @ B )
& ! [X7: nat] :
( ( ( ord_less_eq_nat @ A @ X7 )
& ( ord_less_nat @ X7 @ C3 ) )
=> ( P @ X7 ) )
& ! [D: nat] :
( ! [X3: nat] :
( ( ( ord_less_eq_nat @ A @ X3 )
& ( ord_less_nat @ X3 @ D ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_nat @ D @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_431_complete__interval,axiom,
! [A: int,B: int,P: int > $o] :
( ( ord_less_int @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C3: int] :
( ( ord_less_eq_int @ A @ C3 )
& ( ord_less_eq_int @ C3 @ B )
& ! [X7: int] :
( ( ( ord_less_eq_int @ A @ X7 )
& ( ord_less_int @ X7 @ C3 ) )
=> ( P @ X7 ) )
& ! [D: int] :
( ! [X3: int] :
( ( ( ord_less_eq_int @ A @ X3 )
& ( ord_less_int @ X3 @ D ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_int @ D @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_432_pinf_I1_J,axiom,
! [P: nat > $o,P5: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z3 @ X7 )
=> ( ( ( P @ X7 )
& ( Q2 @ X7 ) )
= ( ( P5 @ X7 )
& ( Q3 @ X7 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_433_pinf_I1_J,axiom,
! [P: int > $o,P5: int > $o,Q2: int > $o,Q3: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z3: int] :
! [X7: int] :
( ( ord_less_int @ Z3 @ X7 )
=> ( ( ( P @ X7 )
& ( Q2 @ X7 ) )
= ( ( P5 @ X7 )
& ( Q3 @ X7 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_434_pinf_I2_J,axiom,
! [P: nat > $o,P5: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z4 @ X3 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z3 @ X7 )
=> ( ( ( P @ X7 )
| ( Q2 @ X7 ) )
= ( ( P5 @ X7 )
| ( Q3 @ X7 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_435_pinf_I2_J,axiom,
! [P: int > $o,P5: int > $o,Q2: int > $o,Q3: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ Z4 @ X3 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z3: int] :
! [X7: int] :
( ( ord_less_int @ Z3 @ X7 )
=> ( ( ( P @ X7 )
| ( Q2 @ X7 ) )
= ( ( P5 @ X7 )
| ( Q3 @ X7 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_436_pinf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z3 @ X7 )
=> ( X7 != T ) ) ).
% pinf(3)
thf(fact_437_pinf_I3_J,axiom,
! [T: int] :
? [Z3: int] :
! [X7: int] :
( ( ord_less_int @ Z3 @ X7 )
=> ( X7 != T ) ) ).
% pinf(3)
thf(fact_438_pinf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z3 @ X7 )
=> ( X7 != T ) ) ).
% pinf(4)
thf(fact_439_pinf_I4_J,axiom,
! [T: int] :
? [Z3: int] :
! [X7: int] :
( ( ord_less_int @ Z3 @ X7 )
=> ( X7 != T ) ) ).
% pinf(4)
thf(fact_440_pinf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z3 @ X7 )
=> ~ ( ord_less_nat @ X7 @ T ) ) ).
% pinf(5)
thf(fact_441_pinf_I5_J,axiom,
! [T: int] :
? [Z3: int] :
! [X7: int] :
( ( ord_less_int @ Z3 @ X7 )
=> ~ ( ord_less_int @ X7 @ T ) ) ).
% pinf(5)
thf(fact_442_pinf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z3 @ X7 )
=> ( ord_less_nat @ T @ X7 ) ) ).
% pinf(7)
thf(fact_443_pinf_I7_J,axiom,
! [T: int] :
? [Z3: int] :
! [X7: int] :
( ( ord_less_int @ Z3 @ X7 )
=> ( ord_less_int @ T @ X7 ) ) ).
% pinf(7)
thf(fact_444_minf_I1_J,axiom,
! [P: nat > $o,P5: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z3 )
=> ( ( ( P @ X7 )
& ( Q2 @ X7 ) )
= ( ( P5 @ X7 )
& ( Q3 @ X7 ) ) ) ) ) ) ).
% minf(1)
thf(fact_445_minf_I1_J,axiom,
! [P: int > $o,P5: int > $o,Q2: int > $o,Q3: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z3: int] :
! [X7: int] :
( ( ord_less_int @ X7 @ Z3 )
=> ( ( ( P @ X7 )
& ( Q2 @ X7 ) )
= ( ( P5 @ X7 )
& ( Q3 @ X7 ) ) ) ) ) ) ).
% minf(1)
thf(fact_446_minf_I2_J,axiom,
! [P: nat > $o,P5: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z4: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z4 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z3 )
=> ( ( ( P @ X7 )
| ( Q2 @ X7 ) )
= ( ( P5 @ X7 )
| ( Q3 @ X7 ) ) ) ) ) ) ).
% minf(2)
thf(fact_447_minf_I2_J,axiom,
! [P: int > $o,P5: int > $o,Q2: int > $o,Q3: int > $o] :
( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z4: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z4 )
=> ( ( Q2 @ X3 )
= ( Q3 @ X3 ) ) )
=> ? [Z3: int] :
! [X7: int] :
( ( ord_less_int @ X7 @ Z3 )
=> ( ( ( P @ X7 )
| ( Q2 @ X7 ) )
= ( ( P5 @ X7 )
| ( Q3 @ X7 ) ) ) ) ) ) ).
% minf(2)
thf(fact_448_minf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z3 )
=> ( X7 != T ) ) ).
% minf(3)
thf(fact_449_minf_I3_J,axiom,
! [T: int] :
? [Z3: int] :
! [X7: int] :
( ( ord_less_int @ X7 @ Z3 )
=> ( X7 != T ) ) ).
% minf(3)
thf(fact_450_minf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z3 )
=> ( X7 != T ) ) ).
% minf(4)
thf(fact_451_minf_I4_J,axiom,
! [T: int] :
? [Z3: int] :
! [X7: int] :
( ( ord_less_int @ X7 @ Z3 )
=> ( X7 != T ) ) ).
% minf(4)
thf(fact_452_minf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z3 )
=> ( ord_less_nat @ X7 @ T ) ) ).
% minf(5)
thf(fact_453_minf_I5_J,axiom,
! [T: int] :
? [Z3: int] :
! [X7: int] :
( ( ord_less_int @ X7 @ Z3 )
=> ( ord_less_int @ X7 @ T ) ) ).
% minf(5)
thf(fact_454_minf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z3 )
=> ~ ( ord_less_nat @ T @ X7 ) ) ).
% minf(7)
thf(fact_455_minf_I7_J,axiom,
! [T: int] :
? [Z3: int] :
! [X7: int] :
( ( ord_less_int @ X7 @ Z3 )
=> ~ ( ord_less_int @ T @ X7 ) ) ).
% minf(7)
thf(fact_456_power__decreasing__iff,axiom,
! [B: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M2 ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_eq_nat @ N @ M2 ) ) ) ) ).
% power_decreasing_iff
thf(fact_457_power__decreasing__iff,axiom,
! [B: int,M2: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ one_one_int )
=> ( ( ord_less_eq_int @ ( power_power_int @ B @ M2 ) @ ( power_power_int @ B @ N ) )
= ( ord_less_eq_nat @ N @ M2 ) ) ) ) ).
% power_decreasing_iff
thf(fact_458_power__increasing__iff,axiom,
! [B: nat,X: nat,Y: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power_increasing_iff
thf(fact_459_power__increasing__iff,axiom,
! [B: int,X: nat,Y: nat] :
( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_eq_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power_increasing_iff
thf(fact_460_neg__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ K @ zero_zero_int )
=> ~ ! [N2: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% neg_int_cases
thf(fact_461_power__strict__decreasing__iff,axiom,
! [B: nat,M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ M2 ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_nat @ N @ M2 ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_462_power__strict__decreasing__iff,axiom,
! [B: int,M2: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ one_one_int )
=> ( ( ord_less_int @ ( power_power_int @ B @ M2 ) @ ( power_power_int @ B @ N ) )
= ( ord_less_nat @ N @ M2 ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_463_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_464_neg__equal__iff__equal,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= ( uminus_uminus_int @ B ) )
= ( A = B ) ) ).
% neg_equal_iff_equal
thf(fact_465_add_Oinverse__inverse,axiom,
! [A: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_466_verit__minus__simplify_I4_J,axiom,
! [B: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_467_power__one__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_468_power__one__right,axiom,
! [A: int] :
( ( power_power_int @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_469_neg__le__iff__le,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% neg_le_iff_le
thf(fact_470_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_471_neg__0__equal__iff__equal,axiom,
! [A: int] :
( ( zero_zero_int
= ( uminus_uminus_int @ A ) )
= ( zero_zero_int = A ) ) ).
% neg_0_equal_iff_equal
thf(fact_472_neg__equal__0__iff__equal,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% neg_equal_0_iff_equal
thf(fact_473_equal__neg__zero,axiom,
! [A: int] :
( ( A
= ( uminus_uminus_int @ A ) )
= ( A = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_474_neg__equal__zero,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= A )
= ( A = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_475_neg__less__iff__less,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ B ) ) ).
% neg_less_iff_less
thf(fact_476_power__one,axiom,
! [N: nat] :
( ( power_power_nat @ one_one_nat @ N )
= one_one_nat ) ).
% power_one
thf(fact_477_power__one,axiom,
! [N: nat] :
( ( power_power_int @ one_one_int @ N )
= one_one_int ) ).
% power_one
thf(fact_478_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_479_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_480_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_nat
= ( semiri1316708129612266289at_nat @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_481_of__nat__1__eq__iff,axiom,
! [N: nat] :
( ( one_one_int
= ( semiri1314217659103216013at_int @ N ) )
= ( N = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_482_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1316708129612266289at_nat @ N )
= one_one_nat )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_483_of__nat__eq__1__iff,axiom,
! [N: nat] :
( ( ( semiri1314217659103216013at_int @ N )
= one_one_int )
= ( N = one_one_nat ) ) ).
% of_nat_eq_1_iff
thf(fact_484_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_485_nat__int,axiom,
! [N: nat] :
( ( nat2 @ ( semiri1314217659103216013at_int @ N ) )
= N ) ).
% nat_int
thf(fact_486_neg__less__eq__nonneg,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_less_eq_nonneg
thf(fact_487_less__eq__neg__nonpos,axiom,
! [A: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% less_eq_neg_nonpos
thf(fact_488_neg__le__0__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% neg_le_0_iff_le
thf(fact_489_neg__0__le__iff__le,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% neg_0_le_iff_le
thf(fact_490_less__neg__neg,axiom,
! [A: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% less_neg_neg
thf(fact_491_neg__less__pos,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ A )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_pos
thf(fact_492_neg__0__less__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ A ) )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% neg_0_less_iff_less
thf(fact_493_neg__less__0__iff__less,axiom,
! [A: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ zero_zero_int )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% neg_less_0_iff_less
thf(fact_494_power__inject__exp,axiom,
! [A: nat,M2: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ( power_power_nat @ A @ M2 )
= ( power_power_nat @ A @ N ) )
= ( M2 = N ) ) ) ).
% power_inject_exp
thf(fact_495_power__inject__exp,axiom,
! [A: int,M2: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ( power_power_int @ A @ M2 )
= ( power_power_int @ A @ N ) )
= ( M2 = N ) ) ) ).
% power_inject_exp
thf(fact_496_negative__eq__positive,axiom,
! [N: nat,M2: nat] :
( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ M2 ) )
= ( ( N = zero_zero_nat )
& ( M2 = zero_zero_nat ) ) ) ).
% negative_eq_positive
thf(fact_497_negative__zle,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M2 ) ) ).
% negative_zle
thf(fact_498_power__strict__increasing__iff,axiom,
! [B: nat,X: nat,Y: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_499_power__strict__increasing__iff,axiom,
! [B: int,X: nat,Y: nat] :
( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_500_nat__le__0,axiom,
! [Z: int] :
( ( ord_less_eq_int @ Z @ zero_zero_int )
=> ( ( nat2 @ Z )
= zero_zero_nat ) ) ).
% nat_le_0
thf(fact_501_nat__0__iff,axiom,
! [I: int] :
( ( ( nat2 @ I )
= zero_zero_nat )
= ( ord_less_eq_int @ I @ zero_zero_int ) ) ).
% nat_0_iff
thf(fact_502_zless__nat__conj,axiom,
! [W: int,Z: int] :
( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
= ( ( ord_less_int @ zero_zero_int @ Z )
& ( ord_less_int @ W @ Z ) ) ) ).
% zless_nat_conj
thf(fact_503_nat__zminus__int,axiom,
! [N: nat] :
( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
= zero_zero_nat ) ).
% nat_zminus_int
thf(fact_504_int__nat__eq,axiom,
! [Z: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
= Z ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
= zero_zero_int ) ) ) ).
% int_nat_eq
thf(fact_505_zero__less__nat__eq,axiom,
! [Z: int] :
( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z ) )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% zero_less_nat_eq
thf(fact_506_le__minus__one__simps_I4_J,axiom,
~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% le_minus_one_simps(4)
thf(fact_507_le__minus__one__simps_I2_J,axiom,
ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% le_minus_one_simps(2)
thf(fact_508_less__minus__one__simps_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% less_minus_one_simps(4)
thf(fact_509_less__minus__one__simps_I2_J,axiom,
ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ).
% less_minus_one_simps(2)
thf(fact_510_zero__neq__neg__one,axiom,
( zero_zero_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% zero_neq_neg_one
thf(fact_511_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_512_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_513_minus__equation__iff,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( uminus_uminus_int @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_514_equation__minus__iff,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% equation_minus_iff
thf(fact_515_one__neq__neg__one,axiom,
( one_one_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% one_neq_neg_one
thf(fact_516_nat__one__as__int,axiom,
( one_one_nat
= ( nat2 @ one_one_int ) ) ).
% nat_one_as_int
thf(fact_517_verit__negate__coefficient_I3_J,axiom,
! [A: int,B: int] :
( ( A = B )
=> ( ( uminus_uminus_int @ A )
= ( uminus_uminus_int @ B ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_518_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_519_le__minus__one__simps_I1_J,axiom,
ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% le_minus_one_simps(1)
thf(fact_520_le__minus__one__simps_I3_J,axiom,
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% le_minus_one_simps(3)
thf(fact_521_less__minus__one__simps_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ one_one_int ) ) ).
% less_minus_one_simps(3)
thf(fact_522_less__minus__one__simps_I1_J,axiom,
ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ zero_zero_int ).
% less_minus_one_simps(1)
thf(fact_523_le__imp__neg__le,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% le_imp_neg_le
thf(fact_524_minus__le__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
= ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% minus_le_iff
thf(fact_525_le__minus__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
= ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% le_minus_iff
thf(fact_526_verit__negate__coefficient_I2_J,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_527_less__minus__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( uminus_uminus_int @ B ) )
= ( ord_less_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% less_minus_iff
thf(fact_528_minus__less__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B )
= ( ord_less_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% minus_less_iff
thf(fact_529_uminus__int__code_I1_J,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% uminus_int_code(1)
thf(fact_530_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_531_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_532_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_533_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_534_int__cases2,axiom,
! [Z: int] :
( ! [N2: nat] :
( Z
!= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ! [N2: nat] :
( Z
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% int_cases2
thf(fact_535_nat__zero__as__int,axiom,
( zero_zero_nat
= ( nat2 @ zero_zero_int ) ) ).
% nat_zero_as_int
thf(fact_536_nat__mono,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ).
% nat_mono
thf(fact_537_ex__nat,axiom,
( ( ^ [P3: nat > $o] :
? [X5: nat] : ( P3 @ X5 ) )
= ( ^ [P4: nat > $o] :
? [X4: int] :
( ( ord_less_eq_int @ zero_zero_int @ X4 )
& ( P4 @ ( nat2 @ X4 ) ) ) ) ) ).
% ex_nat
thf(fact_538_all__nat,axiom,
( ( ^ [P3: nat > $o] :
! [X5: nat] : ( P3 @ X5 ) )
= ( ^ [P4: nat > $o] :
! [X4: int] :
( ( ord_less_eq_int @ zero_zero_int @ X4 )
=> ( P4 @ ( nat2 @ X4 ) ) ) ) ) ).
% all_nat
thf(fact_539_eq__nat__nat__iff,axiom,
! [Z: int,Z5: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z5 )
=> ( ( ( nat2 @ Z )
= ( nat2 @ Z5 ) )
= ( Z = Z5 ) ) ) ) ).
% eq_nat_nat_iff
thf(fact_540_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_541_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_542_one__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ).
% one_le_power
thf(fact_543_one__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ).
% one_le_power
thf(fact_544_not__int__zless__negative,axiom,
! [N: nat,M2: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M2 ) ) ) ).
% not_int_zless_negative
thf(fact_545_power__0,axiom,
! [A: nat] :
( ( power_power_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% power_0
thf(fact_546_power__0,axiom,
! [A: int] :
( ( power_power_int @ A @ zero_zero_nat )
= one_one_int ) ).
% power_0
thf(fact_547_nat__mono__iff,axiom,
! [Z: int,W: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
= ( ord_less_int @ W @ Z ) ) ) ).
% nat_mono_iff
thf(fact_548_zless__nat__eq__int__zless,axiom,
! [M2: nat,Z: int] :
( ( ord_less_nat @ M2 @ ( nat2 @ Z ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ Z ) ) ).
% zless_nat_eq_int_zless
thf(fact_549_nat__le__iff,axiom,
! [X: int,N: nat] :
( ( ord_less_eq_nat @ ( nat2 @ X ) @ N )
= ( ord_less_eq_int @ X @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% nat_le_iff
thf(fact_550_int__eq__iff,axiom,
! [M2: nat,Z: int] :
( ( ( semiri1314217659103216013at_int @ M2 )
= Z )
= ( ( M2
= ( nat2 @ Z ) )
& ( ord_less_eq_int @ zero_zero_int @ Z ) ) ) ).
% int_eq_iff
thf(fact_551_nat__0__le,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
= Z ) ) ).
% nat_0_le
thf(fact_552_nat__power__eq,axiom,
! [Z: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( nat2 @ ( power_power_int @ Z @ N ) )
= ( power_power_nat @ ( nat2 @ Z ) @ N ) ) ) ).
% nat_power_eq
thf(fact_553_int__cases4,axiom,
! [M2: int] :
( ! [N2: nat] :
( M2
!= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( M2
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).
% int_cases4
thf(fact_554_int__zle__neg,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M2 ) ) )
= ( ( N = zero_zero_nat )
& ( M2 = zero_zero_nat ) ) ) ).
% int_zle_neg
thf(fact_555_power__le__one,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ one_one_nat ) ) ) ).
% power_le_one
thf(fact_556_power__le__one,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ one_one_int ) ) ) ).
% power_le_one
thf(fact_557_nonpos__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ K @ zero_zero_int )
=> ~ ! [N2: nat] :
( K
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% nonpos_int_cases
thf(fact_558_negative__zle__0,axiom,
! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% negative_zle_0
thf(fact_559_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= one_one_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ) ).
% power_0_left
thf(fact_560_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= one_one_int ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ) ).
% power_0_left
thf(fact_561_power__increasing,axiom,
! [N: nat,N4: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N4 )
=> ( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N4 ) ) ) ) ).
% power_increasing
thf(fact_562_power__increasing,axiom,
! [N: nat,N4: nat,A: int] :
( ( ord_less_eq_nat @ N @ N4 )
=> ( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N4 ) ) ) ) ).
% power_increasing
thf(fact_563_power__less__imp__less__exp,axiom,
! [A: nat,M2: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_564_power__less__imp__less__exp,axiom,
! [A: int,M2: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_565_power__strict__increasing,axiom,
! [N: nat,N4: nat,A: nat] :
( ( ord_less_nat @ N @ N4 )
=> ( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N4 ) ) ) ) ).
% power_strict_increasing
thf(fact_566_power__strict__increasing,axiom,
! [N: nat,N4: nat,A: int] :
( ( ord_less_nat @ N @ N4 )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N4 ) ) ) ) ).
% power_strict_increasing
thf(fact_567_int__one__le__iff__zero__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ one_one_int @ Z )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% int_one_le_iff_zero_less
thf(fact_568_nat__eq__iff2,axiom,
! [M2: nat,W: int] :
( ( M2
= ( nat2 @ W ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( W
= ( semiri1314217659103216013at_int @ M2 ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
=> ( M2 = zero_zero_nat ) ) ) ) ).
% nat_eq_iff2
thf(fact_569_nat__eq__iff,axiom,
! [W: int,M2: nat] :
( ( ( nat2 @ W )
= M2 )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( W
= ( semiri1314217659103216013at_int @ M2 ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
=> ( M2 = zero_zero_nat ) ) ) ) ).
% nat_eq_iff
thf(fact_570_nat__less__eq__zless,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
= ( ord_less_int @ W @ Z ) ) ) ).
% nat_less_eq_zless
thf(fact_571_split__nat,axiom,
! [P: nat > $o,I: int] :
( ( P @ ( nat2 @ I ) )
= ( ! [N3: nat] :
( ( I
= ( semiri1314217659103216013at_int @ N3 ) )
=> ( P @ N3 ) )
& ( ( ord_less_int @ I @ zero_zero_int )
=> ( P @ zero_zero_nat ) ) ) ) ).
% split_nat
thf(fact_572_nat__le__eq__zle,axiom,
! [W: int,Z: int] :
( ( ( ord_less_int @ zero_zero_int @ W )
| ( ord_less_eq_int @ zero_zero_int @ Z ) )
=> ( ( ord_less_eq_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
= ( ord_less_eq_int @ W @ Z ) ) ) ).
% nat_le_eq_zle
thf(fact_573_le__nat__iff,axiom,
! [K: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_eq_nat @ N @ ( nat2 @ K ) )
= ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N ) @ K ) ) ) ).
% le_nat_iff
thf(fact_574_int__cases3,axiom,
! [K: int] :
( ( K != zero_zero_int )
=> ( ! [N2: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
=> ~ ! [N2: nat] :
( ( K
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ) ).
% int_cases3
thf(fact_575_power__decreasing,axiom,
! [N: nat,N4: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N4 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N4 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_576_power__decreasing,axiom,
! [N: nat,N4: nat,A: int] :
( ( ord_less_eq_nat @ N @ N4 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N4 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_577_power__strict__decreasing,axiom,
! [N: nat,N4: nat,A: nat] :
( ( ord_less_nat @ N @ N4 )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A @ N4 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_578_power__strict__decreasing,axiom,
! [N: nat,N4: nat,A: int] :
( ( ord_less_nat @ N @ N4 )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A @ N4 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_579_power__le__imp__le__exp,axiom,
! [A: nat,M2: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M2 ) @ ( power_power_nat @ A @ N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_580_power__le__imp__le__exp,axiom,
! [A: int,M2: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_eq_int @ ( power_power_int @ A @ M2 ) @ ( power_power_int @ A @ N ) )
=> ( ord_less_eq_nat @ M2 @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_581_self__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% self_le_power
thf(fact_582_self__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ one_one_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% self_le_power
thf(fact_583_one__less__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_584_one__less__power,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_585_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_586_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_587_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_588_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_589_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_590_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_591_linorder__neqE__linordered__idom,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_592_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_593_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_594_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_595_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_596_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_597_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_598_dbl__dec__simps_I2_J,axiom,
( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_dec_simps(2)
thf(fact_599_one__less__nat__eq,axiom,
! [Z: int] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z ) )
= ( ord_less_int @ one_one_int @ Z ) ) ).
% one_less_nat_eq
thf(fact_600_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_601_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
= one_one_int ) ).
% dbl_inc_simps(2)
thf(fact_602_of__int__of__nat,axiom,
( ring_1_of_int_int
= ( ^ [K3: int] : ( if_int @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( nat2 @ ( uminus_uminus_int @ K3 ) ) ) ) @ ( semiri1314217659103216013at_int @ ( nat2 @ K3 ) ) ) ) ) ).
% of_int_of_nat
thf(fact_603_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_604_nat_Oinject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
= ( X2 = Y2 ) ) ).
% nat.inject
thf(fact_605_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_606_Suc__mono,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_607_Suc__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_less_eq
thf(fact_608_Suc__le__mono,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M2 ) )
= ( ord_less_eq_nat @ N @ M2 ) ) ).
% Suc_le_mono
thf(fact_609_dbl__dec__simps_I3_J,axiom,
( ( neg_nu3811975205180677377ec_int @ one_one_int )
= one_one_int ) ).
% dbl_dec_simps(3)
thf(fact_610_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
= zero_zero_nat ) ).
% power_0_Suc
thf(fact_611_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
= zero_zero_int ) ).
% power_0_Suc
thf(fact_612_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_613_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_614_power__Suc0__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_615_power__Suc0__right,axiom,
! [A: int] :
( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_616_of__int__0,axiom,
( ( ring_1_of_int_int @ zero_zero_int )
= zero_zero_int ) ).
% of_int_0
thf(fact_617_of__int__0__eq__iff,axiom,
! [Z: int] :
( ( zero_zero_int
= ( ring_1_of_int_int @ Z ) )
= ( Z = zero_zero_int ) ) ).
% of_int_0_eq_iff
thf(fact_618_of__int__eq__0__iff,axiom,
! [Z: int] :
( ( ( ring_1_of_int_int @ Z )
= zero_zero_int )
= ( Z = zero_zero_int ) ) ).
% of_int_eq_0_iff
thf(fact_619_of__int__le__iff,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_eq_int @ W @ Z ) ) ).
% of_int_le_iff
thf(fact_620_of__int__less__iff,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_int @ W @ Z ) ) ).
% of_int_less_iff
thf(fact_621_of__int__eq__1__iff,axiom,
! [Z: int] :
( ( ( ring_1_of_int_int @ Z )
= one_one_int )
= ( Z = one_one_int ) ) ).
% of_int_eq_1_iff
thf(fact_622_of__int__1,axiom,
( ( ring_1_of_int_int @ one_one_int )
= one_one_int ) ).
% of_int_1
thf(fact_623_power__Suc__0,axiom,
! [N: nat] :
( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
= ( suc @ zero_zero_nat ) ) ).
% power_Suc_0
thf(fact_624_nat__power__eq__Suc__0__iff,axiom,
! [X: nat,M2: nat] :
( ( ( power_power_nat @ X @ M2 )
= ( suc @ zero_zero_nat ) )
= ( ( M2 = zero_zero_nat )
| ( X
= ( suc @ zero_zero_nat ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_625_of__int__minus,axiom,
! [Z: int] :
( ( ring_1_of_int_int @ ( uminus_uminus_int @ Z ) )
= ( uminus_uminus_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% of_int_minus
thf(fact_626_of__int__of__nat__eq,axiom,
! [N: nat] :
( ( ring_1_of_int_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ N ) ) ).
% of_int_of_nat_eq
thf(fact_627_of__int__power,axiom,
! [Z: int,N: nat] :
( ( ring_1_of_int_int @ ( power_power_int @ Z @ N ) )
= ( power_power_int @ ( ring_1_of_int_int @ Z ) @ N ) ) ).
% of_int_power
thf(fact_628_of__int__eq__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,X: int] :
( ( ( power_power_int @ ( ring_1_of_int_int @ B ) @ W )
= ( ring_1_of_int_int @ X ) )
= ( ( power_power_int @ B @ W )
= X ) ) ).
% of_int_eq_of_int_power_cancel_iff
thf(fact_629_of__int__power__eq__of__int__cancel__iff,axiom,
! [X: int,B: int,W: nat] :
( ( ( ring_1_of_int_int @ X )
= ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
= ( X
= ( power_power_int @ B @ W ) ) ) ).
% of_int_power_eq_of_int_cancel_iff
thf(fact_630_nat__1,axiom,
( ( nat2 @ one_one_int )
= ( suc @ zero_zero_nat ) ) ).
% nat_1
thf(fact_631_negative__zless,axiom,
! [N: nat,M2: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M2 ) ) ).
% negative_zless
thf(fact_632_of__int__0__le__iff,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).
% of_int_0_le_iff
thf(fact_633_of__int__le__0__iff,axiom,
! [Z: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
= ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).
% of_int_le_0_iff
thf(fact_634_of__int__0__less__iff,axiom,
! [Z: int] :
( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% of_int_0_less_iff
thf(fact_635_of__int__less__0__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
= ( ord_less_int @ Z @ zero_zero_int ) ) ).
% of_int_less_0_iff
thf(fact_636_of__int__le__1__iff,axiom,
! [Z: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
= ( ord_less_eq_int @ Z @ one_one_int ) ) ).
% of_int_le_1_iff
thf(fact_637_of__int__1__le__iff,axiom,
! [Z: int] :
( ( ord_less_eq_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_eq_int @ one_one_int @ Z ) ) ).
% of_int_1_le_iff
thf(fact_638_of__int__less__1__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
= ( ord_less_int @ Z @ one_one_int ) ) ).
% of_int_less_1_iff
thf(fact_639_of__int__1__less__iff,axiom,
! [Z: int] :
( ( ord_less_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
= ( ord_less_int @ one_one_int @ Z ) ) ).
% of_int_1_less_iff
thf(fact_640_of__int__le__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,X: int] :
( ( ord_less_eq_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) @ ( ring_1_of_int_int @ X ) )
= ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% of_int_le_of_int_power_cancel_iff
thf(fact_641_of__int__power__le__of__int__cancel__iff,axiom,
! [X: int,B: int,W: nat] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
= ( ord_less_eq_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% of_int_power_le_of_int_cancel_iff
thf(fact_642_of__nat__nat,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
= ( ring_1_of_int_int @ Z ) ) ) ).
% of_nat_nat
thf(fact_643_of__int__less__of__int__power__cancel__iff,axiom,
! [B: int,W: nat,X: int] :
( ( ord_less_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) @ ( ring_1_of_int_int @ X ) )
= ( ord_less_int @ ( power_power_int @ B @ W ) @ X ) ) ).
% of_int_less_of_int_power_cancel_iff
thf(fact_644_of__int__power__less__of__int__cancel__iff,axiom,
! [X: int,B: int,W: nat] :
( ( ord_less_int @ ( ring_1_of_int_int @ X ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
= ( ord_less_int @ X @ ( power_power_int @ B @ W ) ) ) ).
% of_int_power_less_of_int_cancel_iff
thf(fact_645_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% not0_implies_Suc
thf(fact_646_Zero__not__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_not_Suc
thf(fact_647_Zero__neq__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_neq_Suc
thf(fact_648_Suc__neq__Zero,axiom,
! [M2: nat] :
( ( suc @ M2 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_649_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_650_diff__induct,axiom,
! [P: nat > nat > $o,M2: nat,N: nat] :
( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X3: nat,Y3: nat] :
( ( P @ X3 @ Y3 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y3 ) ) )
=> ( P @ M2 @ N ) ) ) ) ).
% diff_induct
thf(fact_651_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_652_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_653_nat_OdiscI,axiom,
! [Nat: nat,X2: nat] :
( ( Nat
= ( suc @ X2 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_654_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_655_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_656_nat_Odistinct_I1_J,axiom,
! [X2: nat] :
( zero_zero_nat
!= ( suc @ X2 ) ) ).
% nat.distinct(1)
thf(fact_657_not__less__less__Suc__eq,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_658_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_659_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_660_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_661_Suc__less__SucD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_less_SucD
thf(fact_662_less__antisym,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
=> ( M2 = N ) ) ) ).
% less_antisym
thf(fact_663_Suc__less__eq2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M2 )
= ( ? [M5: nat] :
( ( M2
= ( suc @ M5 ) )
& ( ord_less_nat @ N @ M5 ) ) ) ) ).
% Suc_less_eq2
thf(fact_664_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P @ I4 ) ) ) ) ).
% All_less_Suc
thf(fact_665_not__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_nat @ M2 @ N ) )
= ( ord_less_nat @ N @ ( suc @ M2 ) ) ) ).
% not_less_eq
thf(fact_666_less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ( ord_less_nat @ M2 @ N )
| ( M2 = N ) ) ) ).
% less_Suc_eq
thf(fact_667_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ N )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P @ I4 ) ) ) ) ).
% Ex_less_Suc
thf(fact_668_less__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_669_less__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M2 @ N )
=> ( M2 = N ) ) ) ).
% less_SucE
thf(fact_670_Suc__lessI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ( suc @ M2 )
!= N )
=> ( ord_less_nat @ ( suc @ M2 ) @ N ) ) ) ).
% Suc_lessI
thf(fact_671_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_672_Suc__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_lessD
thf(fact_673_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_674_Suc__leD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% Suc_leD
thf(fact_675_le__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M2 @ N )
=> ( M2
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_676_le__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_677_Suc__le__D,axiom,
! [N: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
=> ? [M4: nat] :
( M6
= ( suc @ M4 ) ) ) ).
% Suc_le_D
thf(fact_678_le__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M2 @ N )
| ( M2
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_679_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_680_not__less__eq__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M2 @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M2 ) ) ).
% not_less_eq_eq
thf(fact_681_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
=> ( P @ M ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_682_nat__induct__at__least,axiom,
! [M2: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( P @ M2 )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_683_transitive__stepwise__le,axiom,
! [M2: nat,N: nat,R2: nat > nat > $o] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ! [X3: nat] : ( R2 @ X3 @ X3 )
=> ( ! [X3: nat,Y3: nat,Z3: nat] :
( ( R2 @ X3 @ Y3 )
=> ( ( R2 @ Y3 @ Z3 )
=> ( R2 @ X3 @ Z3 ) ) )
=> ( ! [N2: nat] : ( R2 @ N2 @ ( suc @ N2 ) )
=> ( R2 @ M2 @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_684_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_685_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_686_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N5: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N5 )
=> ( ord_less_eq_nat @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_687_lift__Suc__antimono__le,axiom,
! [F: nat > int,N: nat,N5: nat] :
( ! [N2: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N5 )
=> ( ord_less_eq_int @ ( F @ N5 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_688_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N5: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N5 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_689_lift__Suc__mono__le,axiom,
! [F: nat > int,N: nat,N5: nat] :
( ! [N2: nat] : ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N5 )
=> ( ord_less_eq_int @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_690_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N5: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N5 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_691_lift__Suc__mono__less,axiom,
! [F: nat > int,N: nat,N5: nat] :
( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N5 )
=> ( ord_less_int @ ( F @ N ) @ ( F @ N5 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_692_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M2: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M2 ) )
= ( ord_less_nat @ N @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_693_lift__Suc__mono__less__iff,axiom,
! [F: nat > int,N: nat,M2: nat] :
( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_int @ ( F @ N ) @ ( F @ M2 ) )
= ( ord_less_nat @ N @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_694_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
!= zero_zero_nat ) ).
% of_nat_neq_0
thf(fact_695_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
!= zero_zero_int ) ).
% of_nat_neq_0
thf(fact_696_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P @ ( suc @ I4 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_697_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_698_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P @ ( suc @ I4 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_699_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% gr0_implies_Suc
thf(fact_700_less__Suc__eq__0__disj,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ( M2 = zero_zero_nat )
| ? [J3: nat] :
( ( M2
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_701_Suc__length__conv,axiom,
! [N: nat,Xs: list_a] :
( ( ( suc @ N )
= ( size_size_list_a @ Xs ) )
= ( ? [Y6: a,Ys2: list_a] :
( ( Xs
= ( cons_a @ Y6 @ Ys2 ) )
& ( ( size_size_list_a @ Ys2 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_702_Suc__length__conv,axiom,
! [N: nat,Xs: list_int] :
( ( ( suc @ N )
= ( size_size_list_int @ Xs ) )
= ( ? [Y6: int,Ys2: list_int] :
( ( Xs
= ( cons_int @ Y6 @ Ys2 ) )
& ( ( size_size_list_int @ Ys2 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_703_length__Suc__conv,axiom,
! [Xs: list_a,N: nat] :
( ( ( size_size_list_a @ Xs )
= ( suc @ N ) )
= ( ? [Y6: a,Ys2: list_a] :
( ( Xs
= ( cons_a @ Y6 @ Ys2 ) )
& ( ( size_size_list_a @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_704_length__Suc__conv,axiom,
! [Xs: list_int,N: nat] :
( ( ( size_size_list_int @ Xs )
= ( suc @ N ) )
= ( ? [Y6: int,Ys2: list_int] :
( ( Xs
= ( cons_int @ Y6 @ Ys2 ) )
& ( ( size_size_list_int @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_705_le__imp__less__Suc,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_706_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_707_less__Suc__eq__le,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% less_Suc_eq_le
thf(fact_708_le__less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% le_less_Suc_eq
thf(fact_709_Suc__le__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_le_lessD
thf(fact_710_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_711_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_712_Suc__le__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
= ( ord_less_nat @ M2 @ N ) ) ).
% Suc_le_eq
thf(fact_713_Suc__leI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_leI
thf(fact_714_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_715_int__cases,axiom,
! [Z: int] :
( ! [N2: nat] :
( Z
!= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ! [N2: nat] :
( Z
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).
% int_cases
thf(fact_716_int__of__nat__induct,axiom,
! [P: int > $o,Z: int] :
( ! [N2: nat] : ( P @ ( semiri1314217659103216013at_int @ N2 ) )
=> ( ! [N2: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) )
=> ( P @ Z ) ) ) ).
% int_of_nat_induct
thf(fact_717_gen__length__code_I2_J,axiom,
! [N: nat,X: a,Xs: list_a] :
( ( gen_length_a @ N @ ( cons_a @ X @ Xs ) )
= ( gen_length_a @ ( suc @ N ) @ Xs ) ) ).
% gen_length_code(2)
thf(fact_718_gen__length__code_I2_J,axiom,
! [N: nat,X: int,Xs: list_int] :
( ( gen_length_int @ N @ ( cons_int @ X @ Xs ) )
= ( gen_length_int @ ( suc @ N ) @ Xs ) ) ).
% gen_length_code(2)
thf(fact_719_power__inject__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ( power_power_nat @ A @ ( suc @ N ) )
= ( power_power_nat @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( A = B ) ) ) ) ).
% power_inject_base
thf(fact_720_power__inject__base,axiom,
! [A: int,N: nat,B: int] :
( ( ( power_power_int @ A @ ( suc @ N ) )
= ( power_power_int @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( A = B ) ) ) ) ).
% power_inject_base
thf(fact_721_power__le__imp__le__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ ( power_power_nat @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% power_le_imp_le_base
thf(fact_722_power__le__imp__le__base,axiom,
! [A: int,N: nat,B: int] :
( ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ ( power_power_int @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% power_le_imp_le_base
thf(fact_723_power__gt1,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ ( suc @ N ) ) ) ) ).
% power_gt1
thf(fact_724_power__gt1,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A @ ( suc @ N ) ) ) ) ).
% power_gt1
thf(fact_725_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I3: nat] :
( ( ord_less_eq_nat @ I3 @ K2 )
=> ~ ( P @ I3 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_726_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_a] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_a @ Xs ) )
= ( ? [X4: a,Ys2: list_a] :
( ( Xs
= ( cons_a @ X4 @ Ys2 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_a @ Ys2 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_727_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_int] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_int @ Xs ) )
= ( ? [X4: int,Ys2: list_int] :
( ( Xs
= ( cons_int @ X4 @ Ys2 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_int @ Ys2 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_728_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_729_power__gt__expt,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ord_less_nat @ K @ ( power_power_nat @ N @ K ) ) ) ).
% power_gt_expt
thf(fact_730_nat__one__le__power,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N ) ) ) ).
% nat_one_le_power
thf(fact_731_of__int__nonneg,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% of_int_nonneg
thf(fact_732_of__int__pos,axiom,
! [Z: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).
% of_int_pos
thf(fact_733_of__nat__less__of__int__iff,axiom,
! [N: nat,X: int] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ ( ring_1_of_int_int @ X ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ N ) @ X ) ) ).
% of_nat_less_of_int_iff
thf(fact_734_power__Suc__le__self,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ A ) ) ) ).
% power_Suc_le_self
thf(fact_735_power__Suc__le__self,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ A ) ) ) ).
% power_Suc_le_self
thf(fact_736_power__Suc__less__one,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ one_one_nat ) ) ) ).
% power_Suc_less_one
thf(fact_737_power__Suc__less__one,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A @ ( suc @ N ) ) @ one_one_int ) ) ) ).
% power_Suc_less_one
thf(fact_738_not__zle__0__negative,axiom,
! [N: nat] :
~ ( ord_less_eq_int @ zero_zero_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ).
% not_zle_0_negative
thf(fact_739_negative__zless__0,axiom,
! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).
% negative_zless_0
thf(fact_740_negD,axiom,
! [X: int] :
( ( ord_less_int @ X @ zero_zero_int )
=> ? [N2: nat] :
( X
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).
% negD
thf(fact_741_Common_Opush_Oelims,axiom,
! [X: int,Xa: state_int,Y: state_int] :
( ( ( push_int @ X @ Xa )
= Y )
=> ( ! [Current2: current_int,Stack: stack_int,StackSize: nat] :
( ( Xa
= ( idle_int2 @ Current2 @ ( idle_int3 @ Stack @ StackSize ) ) )
=> ( Y
!= ( idle_int2 @ ( push_int2 @ X @ Current2 ) @ ( idle_int3 @ ( push_int4 @ X @ Stack ) @ ( suc @ StackSize ) ) ) ) )
=> ~ ! [Current2: current_int,Aux: list_int,New: list_int,Moved: nat] :
( ( Xa
= ( copy_int @ Current2 @ Aux @ New @ Moved ) )
=> ( Y
!= ( copy_int @ ( push_int2 @ X @ Current2 ) @ Aux @ New @ Moved ) ) ) ) ) ).
% Common.push.elims
thf(fact_742_Common_Opush_Oelims,axiom,
! [X: a,Xa: state_a,Y: state_a] :
( ( ( push_a @ X @ Xa )
= Y )
=> ( ! [Current2: current_a,Stack: stack_a,StackSize: nat] :
( ( Xa
= ( idle_a2 @ Current2 @ ( idle_a3 @ Stack @ StackSize ) ) )
=> ( Y
!= ( idle_a2 @ ( push_a2 @ X @ Current2 ) @ ( idle_a3 @ ( push_a4 @ X @ Stack ) @ ( suc @ StackSize ) ) ) ) )
=> ~ ! [Current2: current_a,Aux: list_a,New: list_a,Moved: nat] :
( ( Xa
= ( copy_a @ Current2 @ Aux @ New @ Moved ) )
=> ( Y
!= ( copy_a @ ( push_a2 @ X @ Current2 ) @ Aux @ New @ Moved ) ) ) ) ) ).
% Common.push.elims
thf(fact_743_length__Cons,axiom,
! [X: a,Xs: list_a] :
( ( size_size_list_a @ ( cons_a @ X @ Xs ) )
= ( suc @ ( size_size_list_a @ Xs ) ) ) ).
% length_Cons
thf(fact_744_length__Cons,axiom,
! [X: int,Xs: list_int] :
( ( size_size_list_int @ ( cons_int @ X @ Xs ) )
= ( suc @ ( size_size_list_int @ Xs ) ) ) ).
% length_Cons
thf(fact_745_Common_Opush_Osimps_I1_J,axiom,
! [X: int,Current: current_int,Stack2: stack_int,StackSize2: nat] :
( ( push_int @ X @ ( idle_int2 @ Current @ ( idle_int3 @ Stack2 @ StackSize2 ) ) )
= ( idle_int2 @ ( push_int2 @ X @ Current ) @ ( idle_int3 @ ( push_int4 @ X @ Stack2 ) @ ( suc @ StackSize2 ) ) ) ) ).
% Common.push.simps(1)
thf(fact_746_Common_Opush_Osimps_I1_J,axiom,
! [X: a,Current: current_a,Stack2: stack_a,StackSize2: nat] :
( ( push_a @ X @ ( idle_a2 @ Current @ ( idle_a3 @ Stack2 @ StackSize2 ) ) )
= ( idle_a2 @ ( push_a2 @ X @ Current ) @ ( idle_a3 @ ( push_a4 @ X @ Stack2 ) @ ( suc @ StackSize2 ) ) ) ) ).
% Common.push.simps(1)
thf(fact_747_Common_Opush_Opelims,axiom,
! [X: int,Xa: state_int,Y: state_int] :
( ( ( push_int @ X @ Xa )
= Y )
=> ( ( accp_P673987829705199744te_int @ push_rel_int @ ( produc3637740352169753595te_int @ X @ Xa ) )
=> ( ! [Current2: current_int,Stack: stack_int,StackSize: nat] :
( ( Xa
= ( idle_int2 @ Current2 @ ( idle_int3 @ Stack @ StackSize ) ) )
=> ( ( Y
= ( idle_int2 @ ( push_int2 @ X @ Current2 ) @ ( idle_int3 @ ( push_int4 @ X @ Stack ) @ ( suc @ StackSize ) ) ) )
=> ~ ( accp_P673987829705199744te_int @ push_rel_int @ ( produc3637740352169753595te_int @ X @ ( idle_int2 @ Current2 @ ( idle_int3 @ Stack @ StackSize ) ) ) ) ) )
=> ~ ! [Current2: current_int,Aux: list_int,New: list_int,Moved: nat] :
( ( Xa
= ( copy_int @ Current2 @ Aux @ New @ Moved ) )
=> ( ( Y
= ( copy_int @ ( push_int2 @ X @ Current2 ) @ Aux @ New @ Moved ) )
=> ~ ( accp_P673987829705199744te_int @ push_rel_int @ ( produc3637740352169753595te_int @ X @ ( copy_int @ Current2 @ Aux @ New @ Moved ) ) ) ) ) ) ) ) ).
% Common.push.pelims
thf(fact_748_Common_Opush_Opelims,axiom,
! [X: a,Xa: state_a,Y: state_a] :
( ( ( push_a @ X @ Xa )
= Y )
=> ( ( accp_P3541398268521339172tate_a @ push_rel_a @ ( produc8263595898873874535tate_a @ X @ Xa ) )
=> ( ! [Current2: current_a,Stack: stack_a,StackSize: nat] :
( ( Xa
= ( idle_a2 @ Current2 @ ( idle_a3 @ Stack @ StackSize ) ) )
=> ( ( Y
= ( idle_a2 @ ( push_a2 @ X @ Current2 ) @ ( idle_a3 @ ( push_a4 @ X @ Stack ) @ ( suc @ StackSize ) ) ) )
=> ~ ( accp_P3541398268521339172tate_a @ push_rel_a @ ( produc8263595898873874535tate_a @ X @ ( idle_a2 @ Current2 @ ( idle_a3 @ Stack @ StackSize ) ) ) ) ) )
=> ~ ! [Current2: current_a,Aux: list_a,New: list_a,Moved: nat] :
( ( Xa
= ( copy_a @ Current2 @ Aux @ New @ Moved ) )
=> ( ( Y
= ( copy_a @ ( push_a2 @ X @ Current2 ) @ Aux @ New @ Moved ) )
=> ~ ( accp_P3541398268521339172tate_a @ push_rel_a @ ( produc8263595898873874535tate_a @ X @ ( copy_a @ Current2 @ Aux @ New @ Moved ) ) ) ) ) ) ) ) ).
% Common.push.pelims
thf(fact_749_Idle_Opush_Oelims,axiom,
! [X: a,Xa: idle_a,Y: idle_a] :
( ( ( push_a3 @ X @ Xa )
= Y )
=> ~ ! [Stack: stack_a,StackSize: nat] :
( ( Xa
= ( idle_a3 @ Stack @ StackSize ) )
=> ( Y
!= ( idle_a3 @ ( push_a4 @ X @ Stack ) @ ( suc @ StackSize ) ) ) ) ) ).
% Idle.push.elims
thf(fact_750_Idle_Opush_Osimps,axiom,
! [X: a,Stack2: stack_a,StackSize2: nat] :
( ( push_a3 @ X @ ( idle_a3 @ Stack2 @ StackSize2 ) )
= ( idle_a3 @ ( push_a4 @ X @ Stack2 ) @ ( suc @ StackSize2 ) ) ) ).
% Idle.push.simps
thf(fact_751_accp__subset,axiom,
! [R1: product_prod_int_int > product_prod_int_int > $o,R22: product_prod_int_int > product_prod_int_int > $o] :
( ( ord_le1598226405681992910_int_o @ R1 @ R22 )
=> ( ord_le8369615600986905444_int_o @ ( accp_P1096762738010456898nt_int @ R22 ) @ ( accp_P1096762738010456898nt_int @ R1 ) ) ) ).
% accp_subset
thf(fact_752_accp__subset__induct,axiom,
! [D2: product_prod_int_int > $o,R2: product_prod_int_int > product_prod_int_int > $o,X: product_prod_int_int,P: product_prod_int_int > $o] :
( ( ord_le8369615600986905444_int_o @ D2 @ ( accp_P1096762738010456898nt_int @ R2 ) )
=> ( ! [X3: product_prod_int_int,Z3: product_prod_int_int] :
( ( D2 @ X3 )
=> ( ( R2 @ Z3 @ X3 )
=> ( D2 @ Z3 ) ) )
=> ( ( D2 @ X )
=> ( ! [X3: product_prod_int_int] :
( ( D2 @ X3 )
=> ( ! [Z4: product_prod_int_int] :
( ( R2 @ Z4 @ X3 )
=> ( P @ Z4 ) )
=> ( P @ X3 ) ) )
=> ( P @ X ) ) ) ) ) ).
% accp_subset_induct
thf(fact_753_accp__induct__rule,axiom,
! [R: product_prod_int_int > product_prod_int_int > $o,A: product_prod_int_int,P: product_prod_int_int > $o] :
( ( accp_P1096762738010456898nt_int @ R @ A )
=> ( ! [X3: product_prod_int_int] :
( ( accp_P1096762738010456898nt_int @ R @ X3 )
=> ( ! [Y4: product_prod_int_int] :
( ( R @ Y4 @ X3 )
=> ( P @ Y4 ) )
=> ( P @ X3 ) ) )
=> ( P @ A ) ) ) ).
% accp_induct_rule
thf(fact_754_not__accp__down,axiom,
! [R2: product_prod_int_int > product_prod_int_int > $o,X: product_prod_int_int] :
( ~ ( accp_P1096762738010456898nt_int @ R2 @ X )
=> ~ ! [Z3: product_prod_int_int] :
( ( R2 @ Z3 @ X )
=> ( accp_P1096762738010456898nt_int @ R2 @ Z3 ) ) ) ).
% not_accp_down
thf(fact_755_accp__downward,axiom,
! [R: product_prod_int_int > product_prod_int_int > $o,B: product_prod_int_int,A: product_prod_int_int] :
( ( accp_P1096762738010456898nt_int @ R @ B )
=> ( ( R @ A @ B )
=> ( accp_P1096762738010456898nt_int @ R @ A ) ) ) ).
% accp_downward
thf(fact_756_accp__induct,axiom,
! [R: product_prod_int_int > product_prod_int_int > $o,A: product_prod_int_int,P: product_prod_int_int > $o] :
( ( accp_P1096762738010456898nt_int @ R @ A )
=> ( ! [X3: product_prod_int_int] :
( ( accp_P1096762738010456898nt_int @ R @ X3 )
=> ( ! [Y4: product_prod_int_int] :
( ( R @ Y4 @ X3 )
=> ( P @ Y4 ) )
=> ( P @ X3 ) ) )
=> ( P @ A ) ) ) ).
% accp_induct
thf(fact_757_accpI,axiom,
! [R: product_prod_int_int > product_prod_int_int > $o,X: product_prod_int_int] :
( ! [Y3: product_prod_int_int] :
( ( R @ Y3 @ X )
=> ( accp_P1096762738010456898nt_int @ R @ Y3 ) )
=> ( accp_P1096762738010456898nt_int @ R @ X ) ) ).
% accpI
thf(fact_758_accp_Osimps,axiom,
( accp_P1096762738010456898nt_int
= ( ^ [R3: product_prod_int_int > product_prod_int_int > $o,A4: product_prod_int_int] :
? [X4: product_prod_int_int] :
( ( A4 = X4 )
& ! [Y6: product_prod_int_int] :
( ( R3 @ Y6 @ X4 )
=> ( accp_P1096762738010456898nt_int @ R3 @ Y6 ) ) ) ) ) ).
% accp.simps
thf(fact_759_accp_Ocases,axiom,
! [R: product_prod_int_int > product_prod_int_int > $o,A: product_prod_int_int] :
( ( accp_P1096762738010456898nt_int @ R @ A )
=> ! [Y4: product_prod_int_int] :
( ( R @ Y4 @ A )
=> ( accp_P1096762738010456898nt_int @ R @ Y4 ) ) ) ).
% accp.cases
thf(fact_760_Idle__Proof_Opush__list,axiom,
! [X: a,Idle: idle_a] :
( ( idle_list_a @ ( push_a3 @ X @ Idle ) )
= ( cons_a @ X @ ( idle_list_a @ Idle ) ) ) ).
% Idle_Proof.push_list
thf(fact_761_Idle__Proof_Opush__list,axiom,
! [X: int,Idle: idle_int] :
( ( idle_list_int @ ( push_int3 @ X @ Idle ) )
= ( cons_int @ X @ ( idle_list_int @ Idle ) ) ) ).
% Idle_Proof.push_list
thf(fact_762_Idle_Opush_Opelims,axiom,
! [X: a,Xa: idle_a,Y: idle_a] :
( ( ( push_a3 @ X @ Xa )
= Y )
=> ( ( accp_P2118310398061403062idle_a @ push_rel_a2 @ ( produc1265230069547855005idle_a @ X @ Xa ) )
=> ~ ! [Stack: stack_a,StackSize: nat] :
( ( Xa
= ( idle_a3 @ Stack @ StackSize ) )
=> ( ( Y
= ( idle_a3 @ ( push_a4 @ X @ Stack ) @ ( suc @ StackSize ) ) )
=> ~ ( accp_P2118310398061403062idle_a @ push_rel_a2 @ ( produc1265230069547855005idle_a @ X @ ( idle_a3 @ Stack @ StackSize ) ) ) ) ) ) ) ).
% Idle.push.pelims
thf(fact_763_list__current_Opelims,axiom,
! [X: state_a,Y: list_a] :
( ( ( common1102728217005306191rent_a @ X )
= Y )
=> ( ( accp_state_a @ common1269017819723767218_rel_a @ X )
=> ( ! [Current2: current_a,Uu: idle_a] :
( ( X
= ( idle_a2 @ Current2 @ Uu ) )
=> ( ( Y
= ( current_list_a @ Current2 ) )
=> ~ ( accp_state_a @ common1269017819723767218_rel_a @ ( idle_a2 @ Current2 @ Uu ) ) ) )
=> ~ ! [Current2: current_a,Uv2: list_a,Uw: list_a,Ux: nat] :
( ( X
= ( copy_a @ Current2 @ Uv2 @ Uw @ Ux ) )
=> ( ( Y
= ( current_list_a @ Current2 ) )
=> ~ ( accp_state_a @ common1269017819723767218_rel_a @ ( copy_a @ Current2 @ Uv2 @ Uw @ Ux ) ) ) ) ) ) ) ).
% list_current.pelims
thf(fact_764_list__current_Opelims,axiom,
! [X: state_int,Y: list_int] :
( ( ( common8573843843682093467nt_int @ X )
= Y )
=> ( ( accp_state_int @ common3364536641936693880el_int @ X )
=> ( ! [Current2: current_int,Uu: idle_int] :
( ( X
= ( idle_int2 @ Current2 @ Uu ) )
=> ( ( Y
= ( current_list_int @ Current2 ) )
=> ~ ( accp_state_int @ common3364536641936693880el_int @ ( idle_int2 @ Current2 @ Uu ) ) ) )
=> ~ ! [Current2: current_int,Uv2: list_int,Uw: list_int,Ux: nat] :
( ( X
= ( copy_int @ Current2 @ Uv2 @ Uw @ Ux ) )
=> ( ( Y
= ( current_list_int @ Current2 ) )
=> ~ ( accp_state_int @ common3364536641936693880el_int @ ( copy_int @ Current2 @ Uv2 @ Uw @ Ux ) ) ) ) ) ) ) ).
% list_current.pelims
thf(fact_765_size__idle_Opelims,axiom,
! [X: idle_a,Y: nat] :
( ( ( size_size_idle_a @ X )
= Y )
=> ( ( accp_idle_a @ idle_size_idle_rel_a @ X )
=> ~ ! [Stack: stack_a,Uu: nat] :
( ( X
= ( idle_a3 @ Stack @ Uu ) )
=> ( ( Y
= ( size_size_stack_a @ Stack ) )
=> ~ ( accp_idle_a @ idle_size_idle_rel_a @ ( idle_a3 @ Stack @ Uu ) ) ) ) ) ) ).
% size_idle.pelims
thf(fact_766_Suc__nat__eq__nat__zadd1,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( suc @ ( nat2 @ Z ) )
= ( nat2 @ ( plus_plus_int @ one_one_int @ Z ) ) ) ) ).
% Suc_nat_eq_nat_zadd1
thf(fact_767_add__right__cancel,axiom,
! [B: int,A: int,C2: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C2 @ A ) )
= ( B = C2 ) ) ).
% add_right_cancel
thf(fact_768_add__right__cancel,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C2 @ A ) )
= ( B = C2 ) ) ).
% add_right_cancel
thf(fact_769_add__left__cancel,axiom,
! [A: int,B: int,C2: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C2 ) )
= ( B = C2 ) ) ).
% add_left_cancel
thf(fact_770_add__left__cancel,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C2 ) )
= ( B = C2 ) ) ).
% add_left_cancel
thf(fact_771_add__le__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_772_add__le__cancel__right,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_773_add__le__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_774_add__le__cancel__left,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_775_double__eq__0__iff,axiom,
! [A: int] :
( ( ( plus_plus_int @ A @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_776_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_777_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_778_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_779_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_780_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_781_add__cancel__right__right,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ A @ B ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_782_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_783_add__cancel__right__left,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ B @ A ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_784_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_785_add__cancel__left__right,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_786_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_787_add__cancel__left__left,axiom,
! [B: int,A: int] :
( ( ( plus_plus_int @ B @ A )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_788_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_789_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_790_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_791_add__less__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_792_add__less__cancel__right,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_793_add__less__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_794_add__less__cancel__left,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_795_add__minus__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
= B ) ).
% add_minus_cancel
thf(fact_796_minus__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
= B ) ).
% minus_add_cancel
thf(fact_797_minus__add__distrib,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).
% minus_add_distrib
thf(fact_798_of__nat__add,axiom,
! [M2: nat,N: nat] :
( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M2 @ N ) )
= ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ).
% of_nat_add
thf(fact_799_of__nat__add,axiom,
! [M2: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M2 @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_add
thf(fact_800_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_801_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_802_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_803_le__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel2
thf(fact_804_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_805_le__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel1
thf(fact_806_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_807_add__le__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_808_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_809_add__le__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_810_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_811_add__less__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_812_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_813_add__less__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_814_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_815_less__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel1
thf(fact_816_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_817_less__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel2
thf(fact_818_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_819_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_820_add_Oright__inverse,axiom,
! [A: int] :
( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
= zero_zero_int ) ).
% add.right_inverse
thf(fact_821_ab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_left_minus
thf(fact_822_of__int__add,axiom,
! [W: int,Z: int] :
( ( ring_1_of_int_int @ ( plus_plus_int @ W @ Z ) )
= ( plus_plus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% of_int_add
thf(fact_823_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_824_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_825_of__nat__Suc,axiom,
! [M2: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ M2 ) )
= ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M2 ) ) ) ).
% of_nat_Suc
thf(fact_826_of__nat__Suc,axiom,
! [M2: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ M2 ) )
= ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M2 ) ) ) ).
% of_nat_Suc
thf(fact_827_zle__add1__eq__le,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ord_less_eq_int @ W @ Z ) ) ).
% zle_add1_eq_le
thf(fact_828_add__le__imp__le__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_829_add__le__imp__le__right,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_830_add__le__imp__le__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_831_add__le__imp__le__left,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_832_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
? [C: nat] :
( B4
= ( plus_plus_nat @ A4 @ C ) ) ) ) ).
% le_iff_add
thf(fact_833_add__right__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add_right_mono
thf(fact_834_add__right__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) ) ) ).
% add_right_mono
thf(fact_835_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C3: nat] :
( B
!= ( plus_plus_nat @ A @ C3 ) ) ) ).
% less_eqE
thf(fact_836_add__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) ) ) ).
% add_left_mono
thf(fact_837_add__left__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) ) ) ).
% add_left_mono
thf(fact_838_add__mono,axiom,
! [A: nat,B: nat,C2: nat,D3: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ D3 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D3 ) ) ) ) ).
% add_mono
thf(fact_839_add__mono,axiom,
! [A: int,B: int,C2: int,D3: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C2 @ D3 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ D3 ) ) ) ) ).
% add_mono
thf(fact_840_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_841_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_842_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_843_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_844_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_845_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_846_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_847_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_848_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_849_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_850_comm__monoid__add__class_Oadd__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_851_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_852_verit__sum__simplify,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% verit_sum_simplify
thf(fact_853_add__less__imp__less__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_854_add__less__imp__less__right,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_855_add__less__imp__less__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_856_add__less__imp__less__left,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_857_add__strict__right__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add_strict_right_mono
thf(fact_858_add__strict__right__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) ) ) ).
% add_strict_right_mono
thf(fact_859_add__strict__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) ) ) ).
% add_strict_left_mono
thf(fact_860_add__strict__left__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) ) ) ).
% add_strict_left_mono
thf(fact_861_add__strict__mono,axiom,
! [A: nat,B: nat,C2: nat,D3: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C2 @ D3 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D3 ) ) ) ) ).
% add_strict_mono
thf(fact_862_add__strict__mono,axiom,
! [A: int,B: int,C2: int,D3: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C2 @ D3 )
=> ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ D3 ) ) ) ) ).
% add_strict_mono
thf(fact_863_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_864_add__mono__thms__linordered__field_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( K = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_865_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_866_add__mono__thms__linordered__field_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_867_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_868_add__mono__thms__linordered__field_I5_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_869_is__num__normalize_I8_J,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% is_num_normalize(8)
thf(fact_870_group__cancel_Oneg1,axiom,
! [A5: int,K: int,A: int] :
( ( A5
= ( plus_plus_int @ K @ A ) )
=> ( ( uminus_uminus_int @ A5 )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_871_add_Oinverse__distrib__swap,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
= ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% add.inverse_distrib_swap
thf(fact_872_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_873_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_874_add__right__imp__eq,axiom,
! [B: int,A: int,C2: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C2 @ A ) )
=> ( B = C2 ) ) ).
% add_right_imp_eq
thf(fact_875_add__right__imp__eq,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C2 @ A ) )
=> ( B = C2 ) ) ).
% add_right_imp_eq
thf(fact_876_add__left__imp__eq,axiom,
! [A: int,B: int,C2: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C2 ) )
=> ( B = C2 ) ) ).
% add_left_imp_eq
thf(fact_877_add__left__imp__eq,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C2 ) )
=> ( B = C2 ) ) ).
% add_left_imp_eq
thf(fact_878_add_Oleft__commute,axiom,
! [B: int,A: int,C2: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C2 ) )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).
% add.left_commute
thf(fact_879_add_Oleft__commute,axiom,
! [B: nat,A: nat,C2: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C2 ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add.left_commute
thf(fact_880_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A4: int,B4: int] : ( plus_plus_int @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_881_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A4: nat,B4: nat] : ( plus_plus_nat @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_882_add_Oright__cancel,axiom,
! [B: int,A: int,C2: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C2 @ A ) )
= ( B = C2 ) ) ).
% add.right_cancel
thf(fact_883_add_Oleft__cancel,axiom,
! [A: int,B: int,C2: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C2 ) )
= ( B = C2 ) ) ).
% add.left_cancel
thf(fact_884_add_Oassoc,axiom,
! [A: int,B: int,C2: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C2 )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).
% add.assoc
thf(fact_885_add_Oassoc,axiom,
! [A: nat,B: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add.assoc
thf(fact_886_group__cancel_Oadd2,axiom,
! [B5: int,K: int,B: int,A: int] :
( ( B5
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A @ B5 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_887_group__cancel_Oadd2,axiom,
! [B5: nat,K: nat,B: nat,A: nat] :
( ( B5
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B5 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_888_group__cancel_Oadd1,axiom,
! [A5: int,K: int,A: int,B: int] :
( ( A5
= ( plus_plus_int @ K @ A ) )
=> ( ( plus_plus_int @ A5 @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_889_group__cancel_Oadd1,axiom,
! [A5: nat,K: nat,A: nat,B: nat] :
( ( A5
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A5 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_890_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_891_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_892_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: int,B: int,C2: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C2 )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_893_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_894_is__num__normalize_I1_J,axiom,
! [A: int,B: int,C2: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C2 )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).
% is_num_normalize(1)
thf(fact_895_add__decreasing,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ B ) ) ) ).
% add_decreasing
thf(fact_896_add__decreasing,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ C2 @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ B ) ) ) ).
% add_decreasing
thf(fact_897_add__increasing,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% add_increasing
thf(fact_898_add__increasing,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).
% add_increasing
thf(fact_899_add__decreasing2,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ B ) ) ) ).
% add_decreasing2
thf(fact_900_add__decreasing2,axiom,
! [C2: int,A: int,B: int] :
( ( ord_less_eq_int @ C2 @ zero_zero_int )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C2 ) @ B ) ) ) ).
% add_decreasing2
thf(fact_901_add__increasing2,axiom,
! [C2: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% add_increasing2
thf(fact_902_add__increasing2,axiom,
! [C2: int,B: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ C2 )
=> ( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).
% add_increasing2
thf(fact_903_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_904_add__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_905_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_906_add__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_nonpos
thf(fact_907_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_908_add__nonneg__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_909_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_910_add__nonpos__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_911_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_912_add__mono__thms__linordered__field_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_913_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_914_add__mono__thms__linordered__field_I3_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_915_add__le__less__mono,axiom,
! [A: nat,B: nat,C2: nat,D3: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C2 @ D3 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D3 ) ) ) ) ).
% add_le_less_mono
thf(fact_916_add__le__less__mono,axiom,
! [A: int,B: int,C2: int,D3: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C2 @ D3 )
=> ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ D3 ) ) ) ) ).
% add_le_less_mono
thf(fact_917_add__less__le__mono,axiom,
! [A: nat,B: nat,C2: nat,D3: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ D3 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D3 ) ) ) ) ).
% add_less_le_mono
thf(fact_918_add__less__le__mono,axiom,
! [A: int,B: int,C2: int,D3: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C2 @ D3 )
=> ( ord_less_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ D3 ) ) ) ) ).
% add_less_le_mono
thf(fact_919_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_920_add__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_921_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_922_add__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_923_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C3: nat] :
( ( B
= ( plus_plus_nat @ A @ C3 ) )
=> ( C3 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_924_pos__add__strict,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% pos_add_strict
thf(fact_925_pos__add__strict,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C2 )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).
% pos_add_strict
thf(fact_926_add__less__zeroD,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ ( plus_plus_int @ X @ Y ) @ zero_zero_int )
=> ( ( ord_less_int @ X @ zero_zero_int )
| ( ord_less_int @ Y @ zero_zero_int ) ) ) ).
% add_less_zeroD
thf(fact_927_add__mono1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_928_add__mono1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).
% add_mono1
thf(fact_929_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_930_less__add__one,axiom,
! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% less_add_one
thf(fact_931_neg__eq__iff__add__eq__0,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( plus_plus_int @ A @ B )
= zero_zero_int ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_932_eq__neg__iff__add__eq__0,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( ( plus_plus_int @ A @ B )
= zero_zero_int ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_933_add_Oinverse__unique,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= zero_zero_int )
=> ( ( uminus_uminus_int @ A )
= B ) ) ).
% add.inverse_unique
thf(fact_934_ab__group__add__class_Oab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_group_add_class.ab_left_minus
thf(fact_935_add__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= zero_zero_int )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% add_eq_0_iff
thf(fact_936_odd__nonzero,axiom,
! [Z: int] :
( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z )
!= zero_zero_int ) ).
% odd_nonzero
thf(fact_937_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_938_zless__add1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
= ( ( ord_less_int @ W @ Z )
| ( W = Z ) ) ) ).
% zless_add1_eq
thf(fact_939_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_940_zle__iff__zadd,axiom,
( ord_less_eq_int
= ( ^ [W2: int,Z6: int] :
? [N3: nat] :
( Z6
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_941_dbl__inc__def,axiom,
( neg_nu5851722552734809277nc_int
= ( ^ [X4: int] : ( plus_plus_int @ ( plus_plus_int @ X4 @ X4 ) @ one_one_int ) ) ) ).
% dbl_inc_def
thf(fact_942_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_943_add__neg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_nonpos
thf(fact_944_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_945_add__nonneg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_946_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_947_add__nonpos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_neg
thf(fact_948_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_949_add__pos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_950_add__strict__increasing,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% add_strict_increasing
thf(fact_951_add__strict__increasing,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C2 )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).
% add_strict_increasing
thf(fact_952_add__strict__increasing2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% add_strict_increasing2
thf(fact_953_add__strict__increasing2,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C2 )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C2 ) ) ) ) ).
% add_strict_increasing2
thf(fact_954_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_955_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% zero_less_two
thf(fact_956_int__Suc,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).
% int_Suc
thf(fact_957_int__ops_I4_J,axiom,
! [A: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ A ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ one_one_int ) ) ).
% int_ops(4)
thf(fact_958_zless__iff__Suc__zadd,axiom,
( ord_less_int
= ( ^ [W2: int,Z6: int] :
? [N3: nat] :
( Z6
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_959_odd__less__0__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) @ zero_zero_int )
= ( ord_less_int @ Z @ zero_zero_int ) ) ).
% odd_less_0_iff
thf(fact_960_zless__imp__add1__zle,axiom,
! [W: int,Z: int] :
( ( ord_less_int @ W @ Z )
=> ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z ) ) ).
% zless_imp_add1_zle
thf(fact_961_add1__zle__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z )
= ( ord_less_int @ W @ Z ) ) ).
% add1_zle_eq
thf(fact_962_le__imp__0__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).
% le_imp_0_less
thf(fact_963_Suc__as__int,axiom,
( suc
= ( ^ [A4: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A4 ) @ one_one_int ) ) ) ) ).
% Suc_as_int
thf(fact_964_add__Suc__right,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ M2 @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M2 @ N ) ) ) ).
% add_Suc_right
thf(fact_965_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% Nat.add_0_right
thf(fact_966_add__is__0,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_967_nat__add__left__cancel__less,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_968_nat__add__left__cancel__le,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_969_add__gr__0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_970_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N3: nat] :
? [K3: nat] :
( N3
= ( plus_plus_nat @ M3 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_971_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_972_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_973_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_974_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_975_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_976_add__leD2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_977_add__leD1,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% add_leD1
thf(fact_978_le__add2,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M2 @ N ) ) ).
% le_add2
thf(fact_979_le__add1,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M2 ) ) ).
% le_add1
thf(fact_980_add__leE,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M2 @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_981_add__eq__self__zero,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= M2 )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_982_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_983_less__add__eq__less,axiom,
! [K: nat,L: nat,M2: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M2 @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M2 @ N ) ) ) ).
% less_add_eq_less
thf(fact_984_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_985_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_986_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_987_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_988_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_989_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_990_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_991_nat__arith_Osuc1,axiom,
! [A5: nat,K: nat,A: nat] :
( ( A5
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A5 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_992_add__Suc,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N )
= ( suc @ ( plus_plus_nat @ M2 @ N ) ) ) ).
% add_Suc
thf(fact_993_add__Suc__shift,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N )
= ( plus_plus_nat @ M2 @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_994_add__is__1,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus_nat @ M2 @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_995_one__is__add,axiom,
! [M2: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M2 @ N ) )
= ( ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M2 = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_996_less__imp__Suc__add,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ? [K2: nat] :
( N
= ( suc @ ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_997_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M3: nat,N3: nat] :
? [K3: nat] :
( N3
= ( suc @ ( plus_plus_nat @ M3 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_998_less__add__Suc2,axiom,
! [I: nat,M2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M2 @ I ) ) ) ).
% less_add_Suc2
thf(fact_999_less__add__Suc1,axiom,
! [I: nat,M2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M2 ) ) ) ).
% less_add_Suc1
thf(fact_1000_less__natE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ~ ! [Q4: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M2 @ Q4 ) ) ) ) ).
% less_natE
thf(fact_1001_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_1002_mono__nat__linear__lb,axiom,
! [F: nat > nat,M2: nat,K: nat] :
( ! [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( ord_less_nat @ ( F @ M4 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M2 ) @ K ) @ ( F @ ( plus_plus_nat @ M2 @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1003_Suc__eq__plus1,axiom,
( suc
= ( ^ [N3: nat] : ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1004_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1005_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1006_int__ops_I5_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(5)
thf(fact_1007_int__plus,axiom,
! [N: nat,M2: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M2 ) ) ) ).
% int_plus
thf(fact_1008_zadd__int__left,axiom,
! [M2: nat,N: nat,Z: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M2 @ N ) ) @ Z ) ) ).
% zadd_int_left
thf(fact_1009_nat__int__add,axiom,
! [A: nat,B: nat] :
( ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) )
= ( plus_plus_nat @ A @ B ) ) ).
% nat_int_add
thf(fact_1010_list_Osize_I4_J,axiom,
! [X21: a,X22: list_a] :
( ( size_size_list_a @ ( cons_a @ X21 @ X22 ) )
= ( plus_plus_nat @ ( size_size_list_a @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_1011_list_Osize_I4_J,axiom,
! [X21: int,X22: list_int] :
( ( size_size_list_int @ ( cons_int @ X21 @ X22 ) )
= ( plus_plus_nat @ ( size_size_list_int @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_1012_nat__add__distrib,axiom,
! [Z: int,Z5: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z5 )
=> ( ( nat2 @ ( plus_plus_int @ Z @ Z5 ) )
= ( plus_plus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z5 ) ) ) ) ) ).
% nat_add_distrib
thf(fact_1013_upto_Opinduct,axiom,
! [A0: int,A1: int,P: int > int > $o] :
( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A1 ) )
=> ( ! [I2: int,J2: int] :
( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I2 @ J2 ) )
=> ( ( ( ord_less_eq_int @ I2 @ J2 )
=> ( P @ ( plus_plus_int @ I2 @ one_one_int ) @ J2 ) )
=> ( P @ I2 @ J2 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% upto.pinduct
thf(fact_1014_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( P @ A3 @ B3 )
= ( P @ B3 @ A3 ) )
=> ( ! [A3: nat] : ( P @ A3 @ zero_zero_nat )
=> ( ! [A3: nat,B3: nat] :
( ( P @ A3 @ B3 )
=> ( P @ A3 @ ( plus_plus_nat @ A3 @ B3 ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_1015_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M7: nat] :
( ( P @ X )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M7 ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X7: nat] :
( ( P @ X7 )
=> ( ord_less_eq_nat @ X7 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1016_bezw__0,axiom,
! [X: nat] :
( ( bezw @ X @ zero_zero_nat )
= ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) ).
% bezw_0
thf(fact_1017_add__0__iff,axiom,
! [B: nat,A: nat] :
( ( B
= ( plus_plus_nat @ B @ A ) )
= ( A = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_1018_add__0__iff,axiom,
! [B: int,A: int] :
( ( B
= ( plus_plus_int @ B @ A ) )
= ( A = zero_zero_int ) ) ).
% add_0_iff
thf(fact_1019_enumerate__simps_I2_J,axiom,
! [N: nat,X: a,Xs: list_a] :
( ( enumerate_a @ N @ ( cons_a @ X @ Xs ) )
= ( cons_P8443330267410185325_nat_a @ ( product_Pair_nat_a @ N @ X ) @ ( enumerate_a @ ( suc @ N ) @ Xs ) ) ) ).
% enumerate_simps(2)
thf(fact_1020_enumerate__simps_I2_J,axiom,
! [N: nat,X: int,Xs: list_int] :
( ( enumerate_int @ N @ ( cons_int @ X @ Xs ) )
= ( cons_P2335045147070616083at_int @ ( product_Pair_nat_int @ N @ X ) @ ( enumerate_int @ ( suc @ N ) @ Xs ) ) ) ).
% enumerate_simps(2)
thf(fact_1021_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( plus_plus_nat @ N @ K ) )
= ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
thf(fact_1022_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_1023_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1024_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1025_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_1026_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_1027_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_1028_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_1029_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_1030_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_1031_add__diff__cancel__right_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_1032_add__diff__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_1033_add__diff__cancel__right,axiom,
! [A: int,C2: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ C2 ) @ ( plus_plus_int @ B @ C2 ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_1034_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_1035_add__diff__cancel__left_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_1036_add__diff__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_1037_add__diff__cancel__left,axiom,
! [C2: int,A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ C2 @ A ) @ ( plus_plus_int @ C2 @ B ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_1038_diff__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_1039_add__diff__cancel,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_1040_minus__diff__eq,axiom,
! [A: int,B: int] :
( ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) )
= ( minus_minus_int @ B @ A ) ) ).
% minus_diff_eq
thf(fact_1041_diff__Suc__Suc,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% diff_Suc_Suc
thf(fact_1042_Suc__diff__diff,axiom,
! [M2: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_1043_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1044_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1045_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1046_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_1047_of__int__diff,axiom,
! [W: int,Z: int] :
( ( ring_1_of_int_int @ ( minus_minus_int @ W @ Z ) )
= ( minus_minus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).
% of_int_diff
thf(fact_1048_diff__ge__0__iff__ge,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_1049_diff__gt__0__iff__gt,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_int @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_1050_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_1051_le__add__diff__inverse2,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_1052_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_1053_le__add__diff__inverse,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_1054_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_1055_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_1056_diff__0,axiom,
! [A: int] :
( ( minus_minus_int @ zero_zero_int @ A )
= ( uminus_uminus_int @ A ) ) ).
% diff_0
thf(fact_1057_verit__minus__simplify_I3_J,axiom,
! [B: int] :
( ( minus_minus_int @ zero_zero_int @ B )
= ( uminus_uminus_int @ B ) ) ).
% verit_minus_simplify(3)
thf(fact_1058_uminus__add__conv__diff,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B )
= ( minus_minus_int @ B @ A ) ) ).
% uminus_add_conv_diff
thf(fact_1059_diff__minus__eq__add,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ A @ ( uminus_uminus_int @ B ) )
= ( plus_plus_int @ A @ B ) ) ).
% diff_minus_eq_add
thf(fact_1060_zero__less__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M2 ) )
= ( ord_less_nat @ M2 @ N ) ) ).
% zero_less_diff
thf(fact_1061_diff__is__0__eq_H,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1062_diff__is__0__eq,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M2 @ N ) ) ).
% diff_is_0_eq
thf(fact_1063_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_1064_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_1065_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_1066_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_1067_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_1068_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_1069_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_1070_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_1071_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_1072_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_1073_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_1074_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_1075_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_1076_diff__add__0,axiom,
! [N: nat,M2: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M2 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1077_diff__add__inverse2,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N ) @ N )
= M2 ) ).
% diff_add_inverse2
thf(fact_1078_diff__add__inverse,axiom,
! [N: nat,M2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M2 ) @ N )
= M2 ) ).
% diff_add_inverse
thf(fact_1079_diff__diff__eq,axiom,
! [A: nat,B: nat,C2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C2 )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% diff_diff_eq
thf(fact_1080_diff__diff__eq,axiom,
! [A: int,B: int,C2: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C2 )
= ( minus_minus_int @ A @ ( plus_plus_int @ B @ C2 ) ) ) ).
% diff_diff_eq
thf(fact_1081_diff__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% diff_cancel2
thf(fact_1082_Nat_Odiff__cancel,axiom,
! [K: nat,M2: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M2 @ N ) ) ).
% Nat.diff_cancel
thf(fact_1083_add__implies__diff,axiom,
! [C2: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C2 @ B )
= A )
=> ( C2
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_1084_add__implies__diff,axiom,
! [C2: int,B: int,A: int] :
( ( ( plus_plus_int @ C2 @ B )
= A )
=> ( C2
= ( minus_minus_int @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_1085_diff__add__eq__diff__diff__swap,axiom,
! [A: int,B: int,C2: int] :
( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C2 ) )
= ( minus_minus_int @ ( minus_minus_int @ A @ C2 ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_1086_diff__add__eq,axiom,
! [A: int,B: int,C2: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C2 )
= ( minus_minus_int @ ( plus_plus_int @ A @ C2 ) @ B ) ) ).
% diff_add_eq
thf(fact_1087_diff__diff__eq2,axiom,
! [A: int,B: int,C2: int] :
( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C2 ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ C2 ) @ B ) ) ).
% diff_diff_eq2
thf(fact_1088_add__diff__eq,axiom,
! [A: int,B: int,C2: int] :
( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C2 ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C2 ) ) ).
% add_diff_eq
thf(fact_1089_eq__diff__eq,axiom,
! [A: int,C2: int,B: int] :
( ( A
= ( minus_minus_int @ C2 @ B ) )
= ( ( plus_plus_int @ A @ B )
= C2 ) ) ).
% eq_diff_eq
thf(fact_1090_diff__eq__eq,axiom,
! [A: int,B: int,C2: int] :
( ( ( minus_minus_int @ A @ B )
= C2 )
= ( A
= ( plus_plus_int @ C2 @ B ) ) ) ).
% diff_eq_eq
thf(fact_1091_group__cancel_Osub1,axiom,
! [A5: int,K: int,A: int,B: int] :
( ( A5
= ( plus_plus_int @ K @ A ) )
=> ( ( minus_minus_int @ A5 @ B )
= ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_1092_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_1093_add__diff__inverse__nat,axiom,
! [M2: nat,N: nat] :
( ~ ( ord_less_nat @ M2 @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M2 @ N ) )
= M2 ) ) ).
% add_diff_inverse_nat
thf(fact_1094_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B4: int] : ( ord_less_eq_int @ ( minus_minus_int @ A4 @ B4 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_1095_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A4: int,B4: int] : ( ord_less_int @ ( minus_minus_int @ A4 @ B4 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_1096_add__le__imp__le__diff,axiom,
! [I: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_1097_add__le__imp__le__diff,axiom,
! [I: int,K: int,N: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ord_less_eq_int @ I @ ( minus_minus_int @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_1098_add__le__add__imp__diff__le,axiom,
! [I: nat,K: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1099_add__le__add__imp__diff__le,axiom,
! [I: int,K: int,N: int,J: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
=> ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1100_diff__le__eq,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C2 )
= ( ord_less_eq_int @ A @ ( plus_plus_int @ C2 @ B ) ) ) ).
% diff_le_eq
thf(fact_1101_le__diff__eq,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_eq_int @ A @ ( minus_minus_int @ C2 @ B ) )
= ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C2 ) ) ).
% le_diff_eq
thf(fact_1102_diff__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ) ).
% diff_add
thf(fact_1103_le__add__diff,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C2 @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A ) ) ) ).
% le_add_diff
thf(fact_1104_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1105_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C2 @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C2 @ B ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1106_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C2 @ B ) @ A )
= ( plus_plus_nat @ C2 @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1107_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C2 )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1108_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1109_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C2 @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C2 @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1110_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_1111_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C2 )
= ( B
= ( plus_plus_nat @ C2 @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1112_less__diff__eq,axiom,
! [A: int,C2: int,B: int] :
( ( ord_less_int @ A @ ( minus_minus_int @ C2 @ B ) )
= ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C2 ) ) ).
% less_diff_eq
thf(fact_1113_diff__less__eq,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C2 )
= ( ord_less_int @ A @ ( plus_plus_int @ C2 @ B ) ) ) ).
% diff_less_eq
thf(fact_1114_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1115_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: int,B: int] :
( ~ ( ord_less_int @ A @ B )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1116_group__cancel_Osub2,axiom,
! [B5: int,K: int,B: int,A: int] :
( ( B5
= ( plus_plus_int @ K @ B ) )
=> ( ( minus_minus_int @ A @ B5 )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub2
thf(fact_1117_diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A4: int,B4: int] : ( plus_plus_int @ A4 @ ( uminus_uminus_int @ B4 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_1118_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A4: int,B4: int] : ( plus_plus_int @ A4 @ ( uminus_uminus_int @ B4 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_1119_diff__less__Suc,axiom,
! [M2: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ ( suc @ M2 ) ) ).
% diff_less_Suc
thf(fact_1120_Suc__diff__Suc,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( ( suc @ ( minus_minus_nat @ M2 @ ( suc @ N ) ) )
= ( minus_minus_nat @ M2 @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_1121_diff__less,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ) ) ).
% diff_less
thf(fact_1122_Suc__diff__le,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N )
= ( suc @ ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_1123_diff__less__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C2 ) @ ( minus_minus_nat @ B @ C2 ) ) ) ) ).
% diff_less_mono
thf(fact_1124_less__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M2 @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1125_diff__le__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_1126_le__diff__iff_H,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A ) @ ( minus_minus_nat @ C2 @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_1127_diff__le__self,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ).
% diff_le_self
thf(fact_1128_diff__le__mono,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_1129_Nat_Odiff__diff__eq,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M2 @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1130_le__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M2 @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1131_eq__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M2 )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M2 @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M2 = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1132_diff__eq__diff__less__eq,axiom,
! [A: int,B: int,C2: int,D3: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C2 @ D3 ) )
=> ( ( ord_less_eq_int @ A @ B )
= ( ord_less_eq_int @ C2 @ D3 ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_1133_diff__right__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B @ C2 ) ) ) ).
% diff_right_mono
thf(fact_1134_diff__left__mono,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ ( minus_minus_int @ C2 @ A ) @ ( minus_minus_int @ C2 @ B ) ) ) ).
% diff_left_mono
thf(fact_1135_diff__mono,axiom,
! [A: int,B: int,D3: int,C2: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ D3 @ C2 )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B @ D3 ) ) ) ) ).
% diff_mono
thf(fact_1136_of__nat__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M2 @ N ) )
= ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1316708129612266289at_nat @ N ) ) ) ) ).
% of_nat_diff
thf(fact_1137_of__nat__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ N @ M2 )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M2 @ N ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ).
% of_nat_diff
thf(fact_1138_eq__iff__diff__eq__0,axiom,
( ( ^ [Y5: int,Z2: int] : ( Y5 = Z2 ) )
= ( ^ [A4: int,B4: int] :
( ( minus_minus_int @ A4 @ B4 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_1139_diff__strict__right__mono,axiom,
! [A: int,B: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B @ C2 ) ) ) ).
% diff_strict_right_mono
thf(fact_1140_diff__strict__left__mono,axiom,
! [B: int,A: int,C2: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_int @ ( minus_minus_int @ C2 @ A ) @ ( minus_minus_int @ C2 @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_1141_diff__eq__diff__less,axiom,
! [A: int,B: int,C2: int,D3: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C2 @ D3 ) )
=> ( ( ord_less_int @ A @ B )
= ( ord_less_int @ C2 @ D3 ) ) ) ).
% diff_eq_diff_less
thf(fact_1142_diff__strict__mono,axiom,
! [A: int,B: int,D3: int,C2: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ D3 @ C2 )
=> ( ord_less_int @ ( minus_minus_int @ A @ C2 ) @ ( minus_minus_int @ B @ D3 ) ) ) ) ).
% diff_strict_mono
thf(fact_1143_minus__diff__commute,axiom,
! [B: int,A: int] :
( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A )
= ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% minus_diff_commute
thf(fact_1144_diffs0__imp__equal,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus_nat @ M2 @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M2 )
= zero_zero_nat )
=> ( M2 = N ) ) ) ).
% diffs0_imp_equal
thf(fact_1145_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% minus_nat.diff_0
thf(fact_1146_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_1147_diff__less__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ( ord_less_nat @ M2 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_1148_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_1149_diff__right__commute,axiom,
! [A: nat,C2: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C2 ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C2 ) ) ).
% diff_right_commute
thf(fact_1150_diff__right__commute,axiom,
! [A: int,C2: int,B: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ C2 ) @ B )
= ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C2 ) ) ).
% diff_right_commute
thf(fact_1151_diff__eq__diff__eq,axiom,
! [A: int,B: int,C2: int,D3: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C2 @ D3 ) )
=> ( ( A = B )
= ( C2 = D3 ) ) ) ).
% diff_eq_diff_eq
thf(fact_1152_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_1153_diff__Suc__eq__diff__pred,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ M2 @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1154_diff__Suc__less,axiom,
! [N: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_1155_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D4: nat] :
( ( A
= ( plus_plus_nat @ B @ D4 ) )
=> ( P @ D4 ) ) ) ) ).
% nat_diff_split
thf(fact_1156_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D4: nat] :
( ( A
= ( plus_plus_nat @ B @ D4 ) )
& ~ ( P @ D4 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1157_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_1158_dbl__dec__def,axiom,
( neg_nu3811975205180677377ec_int
= ( ^ [X4: int] : ( minus_minus_int @ ( plus_plus_int @ X4 @ X4 ) @ one_one_int ) ) ) ).
% dbl_dec_def
thf(fact_1159_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N )
= ( minus_minus_nat @ M2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_1160_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_1161_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M3: nat,N3: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N3 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N3 ) ) ) ) ) ).
% add_eq_if
thf(fact_1162_zle__diff1__eq,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z @ one_one_int ) )
= ( ord_less_int @ W @ Z ) ) ).
% zle_diff1_eq
thf(fact_1163_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_1164_int__diff__cases,axiom,
! [Z: int] :
~ ! [M4: nat,N2: nat] :
( Z
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% int_diff_cases
thf(fact_1165_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_1166_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_1167_minus__int__code_I2_J,axiom,
! [L: int] :
( ( minus_minus_int @ zero_zero_int @ L )
= ( uminus_uminus_int @ L ) ) ).
% minus_int_code(2)
thf(fact_1168_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_1169_int__minus,axiom,
! [N: nat,M2: nat] :
( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ N @ M2 ) )
= ( semiri1314217659103216013at_int @ ( nat2 @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M2 ) ) ) ) ) ).
% int_minus
thf(fact_1170_int__ops_I6_J,axiom,
! [A: nat,B: nat] :
( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= zero_zero_int ) )
& ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).
% int_ops(6)
thf(fact_1171_nat__diff__distrib,axiom,
! [Z5: int,Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z5 )
=> ( ( ord_less_eq_int @ Z5 @ Z )
=> ( ( nat2 @ ( minus_minus_int @ Z @ Z5 ) )
= ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z5 ) ) ) ) ) ).
% nat_diff_distrib
thf(fact_1172_nat__diff__distrib_H,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( nat2 @ ( minus_minus_int @ X @ Y ) )
= ( minus_minus_nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ) ).
% nat_diff_distrib'
thf(fact_1173_zdiff__int__split,axiom,
! [P: int > $o,X: nat,Y: nat] :
( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y ) ) )
= ( ( ( ord_less_eq_nat @ Y @ X )
=> ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
& ( ( ord_less_nat @ X @ Y )
=> ( P @ zero_zero_int ) ) ) ) ).
% zdiff_int_split
thf(fact_1174_Idle_Opop_Osimps,axiom,
! [Stack2: stack_a,StackSize2: nat] :
( ( pop_a2 @ ( idle_a3 @ Stack2 @ StackSize2 ) )
= ( produc1265230069547855005idle_a @ ( first_a @ Stack2 ) @ ( idle_a3 @ ( pop_a3 @ Stack2 ) @ ( minus_minus_nat @ StackSize2 @ one_one_nat ) ) ) ) ).
% Idle.pop.simps
thf(fact_1175_Idle_Opop_Oelims,axiom,
! [X: idle_a,Y: produc7590564867095724333idle_a] :
( ( ( pop_a2 @ X )
= Y )
=> ~ ! [Stack: stack_a,StackSize: nat] :
( ( X
= ( idle_a3 @ Stack @ StackSize ) )
=> ( Y
!= ( produc1265230069547855005idle_a @ ( first_a @ Stack ) @ ( idle_a3 @ ( pop_a3 @ Stack ) @ ( minus_minus_nat @ StackSize @ one_one_nat ) ) ) ) ) ) ).
% Idle.pop.elims
thf(fact_1176_Idle_Opop_Opelims,axiom,
! [X: idle_a,Y: produc7590564867095724333idle_a] :
( ( ( pop_a2 @ X )
= Y )
=> ( ( accp_idle_a @ pop_rel_a @ X )
=> ~ ! [Stack: stack_a,StackSize: nat] :
( ( X
= ( idle_a3 @ Stack @ StackSize ) )
=> ( ( Y
= ( produc1265230069547855005idle_a @ ( first_a @ Stack ) @ ( idle_a3 @ ( pop_a3 @ Stack ) @ ( minus_minus_nat @ StackSize @ one_one_nat ) ) ) )
=> ~ ( accp_idle_a @ pop_rel_a @ ( idle_a3 @ Stack @ StackSize ) ) ) ) ) ) ).
% Idle.pop.pelims
thf(fact_1177_Idle__Proof_Opop__list,axiom,
! [Idle: idle_a,X: a,Idle2: idle_a] :
( ~ ( type_i7304311975391125061idle_a @ Idle )
=> ( ( ( pop_a2 @ Idle )
= ( produc1265230069547855005idle_a @ X @ Idle2 ) )
=> ( ( cons_a @ X @ ( idle_list_a @ Idle2 ) )
= ( idle_list_a @ Idle ) ) ) ) ).
% Idle_Proof.pop_list
thf(fact_1178_Idle__Proof_Opop__list,axiom,
! [Idle: idle_int,X: int,Idle2: idle_int] :
( ~ ( type_i6790807573071424059le_int @ Idle )
=> ( ( ( pop_int2 @ Idle )
= ( produc1741743360500042753le_int @ X @ Idle2 ) )
=> ( ( cons_int @ X @ ( idle_list_int @ Idle2 ) )
= ( idle_list_int @ Idle ) ) ) ) ).
% Idle_Proof.pop_list
thf(fact_1179_upto__aux__rec,axiom,
( upto_aux
= ( ^ [I4: int,J3: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J3 @ I4 ) @ Js @ ( upto_aux @ I4 @ ( minus_minus_int @ J3 @ one_one_int ) @ ( cons_int @ J3 @ Js ) ) ) ) ) ).
% upto_aux_rec
thf(fact_1180_nat0__intermed__int__val,axiom,
! [N: nat,F: nat > int,K: int] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( 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 @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ I2 @ N )
& ( ( F @ I2 )
= K ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_1181_abs__idempotent,axiom,
! [A: int] :
( ( abs_abs_int @ ( abs_abs_int @ A ) )
= ( abs_abs_int @ A ) ) ).
% abs_idempotent
thf(fact_1182_abs__0,axiom,
( ( abs_abs_int @ zero_zero_int )
= zero_zero_int ) ).
% abs_0
thf(fact_1183_abs__zero,axiom,
( ( abs_abs_int @ zero_zero_int )
= zero_zero_int ) ).
% abs_zero
thf(fact_1184_abs__eq__0,axiom,
! [A: int] :
( ( ( abs_abs_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% abs_eq_0
thf(fact_1185_abs__0__eq,axiom,
! [A: int] :
( ( zero_zero_int
= ( abs_abs_int @ A ) )
= ( A = zero_zero_int ) ) ).
% abs_0_eq
thf(fact_1186_abs__add__abs,axiom,
! [A: int,B: int] :
( ( abs_abs_int @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) )
= ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% abs_add_abs
thf(fact_1187_abs__minus__cancel,axiom,
! [A: int] :
( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
= ( abs_abs_int @ A ) ) ).
% abs_minus_cancel
thf(fact_1188_abs__of__nat,axiom,
! [N: nat] :
( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ N ) ) ).
% abs_of_nat
thf(fact_1189_abs__of__nonneg,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( abs_abs_int @ A )
= A ) ) ).
% abs_of_nonneg
thf(fact_1190_abs__le__self__iff,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ A )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% abs_le_self_iff
thf(fact_1191_abs__le__zero__iff,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ zero_zero_int )
= ( A = zero_zero_int ) ) ).
% abs_le_zero_iff
thf(fact_1192_zero__less__abs__iff,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( abs_abs_int @ A ) )
= ( A != zero_zero_int ) ) ).
% zero_less_abs_iff
thf(fact_1193_abs__neg__one,axiom,
( ( abs_abs_int @ ( uminus_uminus_int @ one_one_int ) )
= one_one_int ) ).
% abs_neg_one
thf(fact_1194_abs__power__minus,axiom,
! [A: int,N: nat] :
( ( abs_abs_int @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N ) )
= ( abs_abs_int @ ( power_power_int @ A @ N ) ) ) ).
% abs_power_minus
thf(fact_1195_of__int__abs,axiom,
! [X: int] :
( ( ring_1_of_int_int @ ( abs_abs_int @ X ) )
= ( abs_abs_int @ ( ring_1_of_int_int @ X ) ) ) ).
% of_int_abs
thf(fact_1196_abs__of__nonpos,axiom,
! [A: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( abs_abs_int @ A )
= ( uminus_uminus_int @ A ) ) ) ).
% abs_of_nonpos
thf(fact_1197_zabs__less__one__iff,axiom,
! [Z: int] :
( ( ord_less_int @ ( abs_abs_int @ Z ) @ one_one_int )
= ( Z = zero_zero_int ) ) ).
% zabs_less_one_iff
thf(fact_1198_zero__less__power__abs__iff,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N ) )
= ( ( A != zero_zero_int )
| ( N = zero_zero_nat ) ) ) ).
% zero_less_power_abs_iff
thf(fact_1199_abs__triangle__ineq2__sym,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).
% abs_triangle_ineq2_sym
thf(fact_1200_abs__triangle__ineq3,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).
% abs_triangle_ineq3
thf(fact_1201_abs__triangle__ineq2,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).
% abs_triangle_ineq2
thf(fact_1202_abs__minus__commute,axiom,
! [A: int,B: int] :
( ( abs_abs_int @ ( minus_minus_int @ A @ B ) )
= ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).
% abs_minus_commute
thf(fact_1203_eq__abs__iff_H,axiom,
! [A: int,B: int] :
( ( A
= ( abs_abs_int @ B ) )
= ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ( B = A )
| ( B
= ( uminus_uminus_int @ A ) ) ) ) ) ).
% eq_abs_iff'
thf(fact_1204_abs__eq__iff_H,axiom,
! [A: int,B: int] :
( ( ( abs_abs_int @ A )
= B )
= ( ( ord_less_eq_int @ zero_zero_int @ B )
& ( ( A = B )
| ( A
= ( uminus_uminus_int @ B ) ) ) ) ) ).
% abs_eq_iff'
thf(fact_1205_abs__minus__le__zero,axiom,
! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( abs_abs_int @ A ) ) @ zero_zero_int ) ).
% abs_minus_le_zero
thf(fact_1206_abs__diff__le__iff,axiom,
! [X: int,A: int,R: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A ) ) @ R )
= ( ( ord_less_eq_int @ ( minus_minus_int @ A @ R ) @ X )
& ( ord_less_eq_int @ X @ ( plus_plus_int @ A @ R ) ) ) ) ).
% abs_diff_le_iff
thf(fact_1207_abs__triangle__ineq4,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% abs_triangle_ineq4
thf(fact_1208_abs__diff__triangle__ineq,axiom,
! [A: int,B: int,C2: int,D3: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ ( plus_plus_int @ C2 @ D3 ) ) ) @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ A @ C2 ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ D3 ) ) ) ) ).
% abs_diff_triangle_ineq
thf(fact_1209_abs__if,axiom,
( abs_abs_int
= ( ^ [A4: int] : ( if_int @ ( ord_less_int @ A4 @ zero_zero_int ) @ ( uminus_uminus_int @ A4 ) @ A4 ) ) ) ).
% abs_if
thf(fact_1210_abs__if__raw,axiom,
( abs_abs_int
= ( ^ [A4: int] : ( if_int @ ( ord_less_int @ A4 @ zero_zero_int ) @ ( uminus_uminus_int @ A4 ) @ A4 ) ) ) ).
% abs_if_raw
thf(fact_1211_abs__of__neg,axiom,
! [A: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( abs_abs_int @ A )
= ( uminus_uminus_int @ A ) ) ) ).
% abs_of_neg
thf(fact_1212_zero__le__power__abs,axiom,
! [A: int,N: nat] : ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ ( abs_abs_int @ A ) @ N ) ) ).
% zero_le_power_abs
thf(fact_1213_abs__diff__less__iff,axiom,
! [X: int,A: int,R: int] :
( ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X @ A ) ) @ R )
= ( ( ord_less_int @ ( minus_minus_int @ A @ R ) @ X )
& ( ord_less_int @ X @ ( plus_plus_int @ A @ R ) ) ) ) ).
% abs_diff_less_iff
thf(fact_1214_abs__ge__zero,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A ) ) ).
% abs_ge_zero
thf(fact_1215_abs__not__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( abs_abs_int @ A ) @ zero_zero_int ) ).
% abs_not_less_zero
thf(fact_1216_abs__of__pos,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( abs_abs_int @ A )
= A ) ) ).
% abs_of_pos
thf(fact_1217_abs__triangle__ineq,axiom,
! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( plus_plus_int @ A @ B ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).
% abs_triangle_ineq
thf(fact_1218_abs__ge__minus__self,axiom,
! [A: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ ( abs_abs_int @ A ) ) ).
% abs_ge_minus_self
thf(fact_1219_abs__le__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
= ( ( ord_less_eq_int @ A @ B )
& ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% abs_le_iff
thf(fact_1220_abs__le__D2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
=> ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% abs_le_D2
thf(fact_1221_abs__leI,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
=> ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B ) ) ) ).
% abs_leI
thf(fact_1222_abs__less__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( abs_abs_int @ A ) @ B )
= ( ( ord_less_int @ A @ B )
& ( ord_less_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).
% abs_less_iff
thf(fact_1223_abs__ge__self,axiom,
! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).
% abs_ge_self
thf(fact_1224_abs__le__D1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
=> ( ord_less_eq_int @ A @ B ) ) ).
% abs_le_D1
thf(fact_1225_abs__eq__0__iff,axiom,
! [A: int] :
( ( ( abs_abs_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% abs_eq_0_iff
thf(fact_1226_power__abs,axiom,
! [A: int,N: nat] :
( ( abs_abs_int @ ( power_power_int @ A @ N ) )
= ( power_power_int @ ( abs_abs_int @ A ) @ N ) ) ).
% power_abs
thf(fact_1227_zabs__def,axiom,
( abs_abs_int
= ( ^ [I4: int] : ( if_int @ ( ord_less_int @ I4 @ zero_zero_int ) @ ( uminus_uminus_int @ I4 ) @ I4 ) ) ) ).
% zabs_def
thf(fact_1228_abs__add__one__gt__zero,axiom,
! [X: int] : ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ ( abs_abs_int @ X ) ) ) ).
% abs_add_one_gt_zero
thf(fact_1229_of__int__leD,axiom,
! [N: int,X: int] :
( ( ord_less_eq_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X )
=> ( ( N = zero_zero_int )
| ( ord_less_eq_int @ one_one_int @ X ) ) ) ).
% of_int_leD
thf(fact_1230_of__int__lessD,axiom,
! [N: int,X: int] :
( ( ord_less_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N ) ) @ X )
=> ( ( N = zero_zero_int )
| ( ord_less_int @ one_one_int @ X ) ) ) ).
% of_int_lessD
thf(fact_1231_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_1232_nat__abs__int__diff,axiom,
! [A: nat,B: nat] :
( ( ( ord_less_eq_nat @ A @ B )
=> ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
= ( minus_minus_nat @ B @ A ) ) )
& ( ~ ( ord_less_eq_nat @ A @ B )
=> ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
= ( minus_minus_nat @ A @ B ) ) ) ) ).
% nat_abs_int_diff
thf(fact_1233_nat__intermed__int__val,axiom,
! [M2: nat,N: nat,F: nat > int,K: int] :
( ! [I2: nat] :
( ( ( ord_less_eq_nat @ M2 @ I2 )
& ( ord_less_nat @ I2 @ N ) )
=> ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) ) ) @ one_one_int ) )
=> ( ( ord_less_eq_nat @ M2 @ N )
=> ( ( ord_less_eq_int @ ( F @ M2 ) @ K )
=> ( ( ord_less_eq_int @ K @ ( F @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ M2 @ I2 )
& ( ord_less_eq_nat @ I2 @ N )
& ( ( F @ I2 )
= K ) ) ) ) ) ) ).
% nat_intermed_int_val
thf(fact_1234_nat__ivt__aux,axiom,
! [N: nat,F: nat > int,K: int] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( 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 @ N ) )
=> ? [I2: nat] :
( ( ord_less_eq_nat @ I2 @ N )
& ( ( F @ I2 )
= K ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_1235_pop__list__tl,axiom,
! [Idle: idle_a,X: a,Idle2: idle_a] :
( ~ ( type_i7304311975391125061idle_a @ Idle )
=> ( ( ( pop_a2 @ Idle )
= ( produc1265230069547855005idle_a @ X @ Idle2 ) )
=> ( ( cons_a @ X @ ( tl_a @ ( idle_list_a @ Idle ) ) )
= ( idle_list_a @ Idle ) ) ) ) ).
% pop_list_tl
thf(fact_1236_pop__list__tl,axiom,
! [Idle: idle_int,X: int,Idle2: idle_int] :
( ~ ( type_i6790807573071424059le_int @ Idle )
=> ( ( ( pop_int2 @ Idle )
= ( produc1741743360500042753le_int @ X @ Idle2 ) )
=> ( ( cons_int @ X @ ( tl_int @ ( idle_list_int @ Idle ) ) )
= ( idle_list_int @ Idle ) ) ) ) ).
% pop_list_tl
thf(fact_1237_list_Osel_I3_J,axiom,
! [X21: a,X22: list_a] :
( ( tl_a @ ( cons_a @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_1238_list_Osel_I3_J,axiom,
! [X21: int,X22: list_int] :
( ( tl_int @ ( cons_int @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_1239_invar__idle_Oelims_I3_J,axiom,
! [X: idle_a] :
( ~ ( type_i8151583586401621767idle_a @ X )
=> ~ ! [Stack: stack_a,StackSize: nat] :
( ( X
= ( idle_a3 @ Stack @ StackSize ) )
=> ( ( size_size_stack_a @ Stack )
= StackSize ) ) ) ).
% invar_idle.elims(3)
thf(fact_1240_invar__idle_Oelims_I2_J,axiom,
! [X: idle_a] :
( ( type_i8151583586401621767idle_a @ X )
=> ~ ! [Stack: stack_a,StackSize: nat] :
( ( X
= ( idle_a3 @ Stack @ StackSize ) )
=> ( ( size_size_stack_a @ Stack )
!= StackSize ) ) ) ).
% invar_idle.elims(2)
thf(fact_1241_invar__idle_Oelims_I1_J,axiom,
! [X: idle_a,Y: $o] :
( ( ( type_i8151583586401621767idle_a @ X )
= Y )
=> ~ ! [Stack: stack_a,StackSize: nat] :
( ( X
= ( idle_a3 @ Stack @ StackSize ) )
=> ( Y
= ( ( size_size_stack_a @ Stack )
!= StackSize ) ) ) ) ).
% invar_idle.elims(1)
thf(fact_1242_invar__idle_Osimps,axiom,
! [Stack2: stack_a,StackSize2: nat] :
( ( type_i8151583586401621767idle_a @ ( idle_a3 @ Stack2 @ StackSize2 ) )
= ( ( size_size_stack_a @ Stack2 )
= StackSize2 ) ) ).
% invar_idle.simps
thf(fact_1243_invar__idle_Opelims_I3_J,axiom,
! [X: idle_a] :
( ~ ( type_i8151583586401621767idle_a @ X )
=> ( ( accp_idle_a @ idle_i6200314614184386870_rel_a @ X )
=> ~ ! [Stack: stack_a,StackSize: nat] :
( ( X
= ( idle_a3 @ Stack @ StackSize ) )
=> ( ( accp_idle_a @ idle_i6200314614184386870_rel_a @ ( idle_a3 @ Stack @ StackSize ) )
=> ( ( size_size_stack_a @ Stack )
= StackSize ) ) ) ) ) ).
% invar_idle.pelims(3)
thf(fact_1244_invar__idle_Opelims_I2_J,axiom,
! [X: idle_a] :
( ( type_i8151583586401621767idle_a @ X )
=> ( ( accp_idle_a @ idle_i6200314614184386870_rel_a @ X )
=> ~ ! [Stack: stack_a,StackSize: nat] :
( ( X
= ( idle_a3 @ Stack @ StackSize ) )
=> ( ( accp_idle_a @ idle_i6200314614184386870_rel_a @ ( idle_a3 @ Stack @ StackSize ) )
=> ( ( size_size_stack_a @ Stack )
!= StackSize ) ) ) ) ) ).
% invar_idle.pelims(2)
thf(fact_1245_invar__idle_Opelims_I1_J,axiom,
! [X: idle_a,Y: $o] :
( ( ( type_i8151583586401621767idle_a @ X )
= Y )
=> ( ( accp_idle_a @ idle_i6200314614184386870_rel_a @ X )
=> ~ ! [Stack: stack_a,StackSize: nat] :
( ( X
= ( idle_a3 @ Stack @ StackSize ) )
=> ( ( Y
= ( ( size_size_stack_a @ Stack )
= StackSize ) )
=> ~ ( accp_idle_a @ idle_i6200314614184386870_rel_a @ ( idle_a3 @ Stack @ StackSize ) ) ) ) ) ) ).
% invar_idle.pelims(1)
thf(fact_1246_cons__tl,axiom,
! [X: a,Xs: list_a,Ys: list_a] :
( ( ( cons_a @ X @ Xs )
= Ys )
=> ( Xs
= ( tl_a @ Ys ) ) ) ).
% cons_tl
thf(fact_1247_cons__tl,axiom,
! [X: int,Xs: list_int,Ys: list_int] :
( ( ( cons_int @ X @ Xs )
= Ys )
=> ( Xs
= ( tl_int @ Ys ) ) ) ).
% cons_tl
thf(fact_1248_decr__lemma,axiom,
! [D3: int,X: int,Z: int] :
( ( ord_less_int @ zero_zero_int @ D3 )
=> ( ord_less_int @ ( minus_minus_int @ X @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X @ Z ) ) @ one_one_int ) @ D3 ) ) @ Z ) ) ).
% decr_lemma
thf(fact_1249_mult__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ( times_times_nat @ A @ C2 )
= ( times_times_nat @ B @ C2 ) )
= ( ( C2 = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_1250_mult__cancel__right,axiom,
! [A: int,C2: int,B: int] :
( ( ( times_times_int @ A @ C2 )
= ( times_times_int @ B @ C2 ) )
= ( ( C2 = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_1251_mult__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C2 @ A )
= ( times_times_nat @ C2 @ B ) )
= ( ( C2 = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_1252_mult__cancel__left,axiom,
! [C2: int,A: int,B: int] :
( ( ( times_times_int @ C2 @ A )
= ( times_times_int @ C2 @ B ) )
= ( ( C2 = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_1253_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_1254_mult__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
= ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_1255_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_1256_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_1257_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_1258_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_1259_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_1260_mult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% mult_1
thf(fact_1261_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_1262_mult_Oright__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.right_neutral
thf(fact_1263_of__nat__mult,axiom,
! [M2: nat,N: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ M2 @ N ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ).
% of_nat_mult
thf(fact_1264_mult__cancel__right2,axiom,
! [A: int,C2: int] :
( ( ( times_times_int @ A @ C2 )
= C2 )
= ( ( C2 = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_right2
% Helper facts (7)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $true @ X @ Y )
= X ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [X: list_int,Y: list_int] :
( ( if_list_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
! [X: list_int,Y: list_int] :
( ( if_list_int @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( cons_a @ xa @ ( common1102728217005306191rent_a @ common ) )
= ( common1102728217005306191rent_a @ ( idle_a2 @ current @ ( idle_a3 @ stack @ stackSize ) ) ) ) ).
%------------------------------------------------------------------------------