TPTP Problem File: SLH0941^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Real_Time_Deque/0022_Common_Proof/prob_00377_011549__6743238_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1524 ( 671 unt; 253 typ; 0 def)
% Number of atoms : 3210 (1662 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 9851 ( 478 ~; 86 |; 184 &;7792 @)
% ( 0 <=>;1311 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 6 avg)
% Number of types : 53 ( 52 usr)
% Number of type conns : 564 ( 564 >; 0 *; 0 +; 0 <<)
% Number of symbols : 204 ( 201 usr; 17 con; 0-4 aty)
% Number of variables : 3263 ( 112 ^;3010 !; 141 ?;3263 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:51:52.085
%------------------------------------------------------------------------------
% Could-be-implicit typings (52)
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Int__Oint_J_J_Mt__List__Olist_It__List__Olist_It__Int__Oint_J_J_J_J,type,
set_Pr8140719670968651609st_int: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J_J,type,
set_Pr5382606609415531783list_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Int__Oint_J_J_Mt__List__Olist_It__List__Olist_It__Int__Oint_J_J_J,type,
produc8452586669324383225st_int: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
produc7709606177366032167list_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Int__Oint_J_Mt__Common__Ostate_It__List__Olist_It__Int__Oint_J_J_J,type,
produc2176888055510159817st_int: $tType ).
thf(ty_n_t__Product____Type__Oprod_I_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J_Mt__List__Olist_It__Int__Oint_J_J,type,
produc5834231552977413017st_int: $tType ).
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,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_It__Int__Oint_J_J_J,type,
set_Pr232763997632103205st_int: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Int__Oint_J_Mt__List__Olist_Itf__a_J_J_J,type,
set_Pr8977947362417044851list_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__Common__Ostate_It__List__Olist_Itf__a_J_J_J,type,
produc5879181297811355641list_a: $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,
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thf(ty_n_t__Product____Type__Oprod_I_062_Itf__a_M_062_Itf__a_M_Eo_J_J_Mt__List__Olist_Itf__a_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_It__Int__Oint_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Int__Oint_J_Mt__List__Olist_Itf__a_J_J,type,
produc7853204735038008893list_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__Current__Ocurrent_It__Int__Oint_J_J,type,
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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,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
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thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_It__Int__Oint_J_J_J,type,
list_list_list_int: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__a_Mt__Current__Ocurrent_Itf__a_J_J,type,
produc7805042584321970905rent_a: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mt__Int__Oint_J_J,type,
set_Pr2444352267426396943_a_int: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mtf__a_J_J,type,
set_Pr4265292341059111293_int_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__List__Olist_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
list_list_list_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__Int__Oint_Mt__Int__Oint_J,type,
product_prod_int_int: $tType ).
thf(ty_n_t__Current__Ocurrent_It__List__Olist_It__Int__Oint_J_J,type,
current_list_int: $tType ).
thf(ty_n_t__Common__Ostate_It__List__Olist_It__Int__Oint_J_J,type,
state_list_int: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__a_Mt__Int__Oint_J,type,
product_prod_a_int: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mtf__a_J,type,
product_prod_int_a: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Int__Oint_J_J,type,
list_list_int: $tType ).
thf(ty_n_t__Current__Ocurrent_It__List__Olist_Itf__a_J_J,type,
current_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Int__Oint_J_J,type,
set_list_int: $tType ).
thf(ty_n_t__List__Olist_I_062_Itf__a_Mt__Nat__Onat_J_J,type,
list_a_nat: $tType ).
thf(ty_n_t__Common__Ostate_It__List__Olist_Itf__a_J_J,type,
state_list_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
product_prod_a_a: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_Itf__a_J_J,type,
list_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
set_list_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__List__Olist_It__Int__Oint_J,type,
list_int: $tType ).
thf(ty_n_t__Current__Ocurrent_Itf__a_J,type,
current_a: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__Common__Ostate_Itf__a_J,type,
state_a: $tType ).
thf(ty_n_t__List__Olist_Itf__a_J,type,
list_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_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 (201)
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__Int__Oint,type,
bNF_Gr1870224194279859149ft_int: set_list_int > int > set_list_int ).
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001tf__a,type,
bNF_Greatest_Shift_a: set_list_a > a > set_list_a ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Int__Oint,type,
bNF_Gr6350390219475566417cc_int: set_list_int > list_int > set_int ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001tf__a,type,
bNF_Greatest_Succ_a: set_list_a > list_a > set_a ).
thf(sy_c_Common_Onormalize_001tf__a,type,
normalize_a: state_a > state_a ).
thf(sy_c_Common_Opop_001t__Int__Oint,type,
pop_int: state_int > produc5934932344166212809te_int ).
thf(sy_c_Common_Opop_001t__List__Olist_It__Int__Oint_J,type,
pop_list_int: state_list_int > produc2176888055510159817st_int ).
thf(sy_c_Common_Opop_001t__List__Olist_Itf__a_J,type,
pop_list_a: state_list_a > produc5879181297811355641list_a ).
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_001t__List__Olist_It__Int__Oint_J,type,
push_list_int: list_int > state_list_int > state_list_int ).
thf(sy_c_Common_Opush_001t__List__Olist_Itf__a_J,type,
push_list_a: list_a > state_list_a > state_list_a ).
thf(sy_c_Common_Opush_001tf__a,type,
push_a: a > state_a > state_a ).
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__Aux_Olist_001t__Int__Oint,type,
common_list_int: state_int > list_int ).
thf(sy_c_Common__Aux_Olist_001t__List__Olist_It__Int__Oint_J,type,
common_list_list_int: state_list_int > list_list_int ).
thf(sy_c_Common__Aux_Olist_001t__List__Olist_Itf__a_J,type,
common_list_list_a: state_list_a > list_list_a ).
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_001t__List__Olist_It__Int__Oint_J,type,
common7659038223531881259st_int: state_list_int > list_list_int ).
thf(sy_c_Common__Aux_Olist__current_001t__List__Olist_Itf__a_J,type,
common3845368160371929429list_a: state_list_a > list_list_a ).
thf(sy_c_Common__Aux_Olist__current_001tf__a,type,
common1102728217005306191rent_a: state_a > list_a ).
thf(sy_c_Common__Aux_Otake__rev_001t__Int__Oint,type,
common_take_rev_int: nat > list_int > list_int ).
thf(sy_c_Common__Aux_Otake__rev_001tf__a,type,
common_take_rev_a: nat > list_a > list_a ).
thf(sy_c_Current_Ofirst_001tf__a,type,
first_a: current_a > a ).
thf(sy_c_Current_Opop_001t__Int__Oint,type,
pop_int2: current_int > produc7978559663978235293nt_int ).
thf(sy_c_Current_Opop_001tf__a,type,
pop_a2: current_a > produc7805042584321970905rent_a ).
thf(sy_c_Current_Opush_001t__Int__Oint,type,
push_int2: int > current_int > current_int ).
thf(sy_c_Current_Opush_001t__List__Olist_It__Int__Oint_J,type,
push_list_int2: list_int > current_list_int > current_list_int ).
thf(sy_c_Current_Opush_001t__List__Olist_Itf__a_J,type,
push_list_a2: list_a > current_list_a > current_list_a ).
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_int2: current_int > list_int ).
thf(sy_c_Current__Aux_Olist_001t__List__Olist_It__Int__Oint_J,type,
curren8299908649409640978st_int: current_list_int > list_list_int ).
thf(sy_c_Current__Aux_Olist_001t__List__Olist_Itf__a_J,type,
current_list_list_a: current_list_a > list_list_a ).
thf(sy_c_Current__Aux_Olist_001tf__a,type,
current_list_a2: current_a > list_a ).
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_Ouminus__class_Ouminus_001t__Int__Oint,type,
uminus_uminus_int: int > int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_If_001t__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_Oappend_001t__Int__Oint,type,
append_int: list_int > list_int > list_int ).
thf(sy_c_List_Oappend_001tf__a,type,
append_a: list_a > list_a > list_a ).
thf(sy_c_List_Odrop_001t__Int__Oint,type,
drop_int: nat > list_int > list_int ).
thf(sy_c_List_Odrop_001tf__a,type,
drop_a: nat > list_a > list_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_Oinsert_001t__Int__Oint,type,
insert_int: int > list_int > list_int ).
thf(sy_c_List_Oinsert_001t__List__Olist_It__Int__Oint_J,type,
insert_list_int: list_int > list_list_int > list_list_int ).
thf(sy_c_List_Oinsert_001t__List__Olist_Itf__a_J,type,
insert_list_a: list_a > list_list_a > list_list_a ).
thf(sy_c_List_Oinsert_001tf__a,type,
insert_a: a > list_a > list_a ).
thf(sy_c_List_Olast_001t__Int__Oint,type,
last_int: list_int > int ).
thf(sy_c_List_Olast_001tf__a,type,
last_a: list_a > a ).
thf(sy_c_List_Olenlex_001t__Int__Oint,type,
lenlex_int: set_Pr958786334691620121nt_int > set_Pr765067013931698361st_int ).
thf(sy_c_List_Olenlex_001t__List__Olist_It__Int__Oint_J,type,
lenlex_list_int: set_Pr765067013931698361st_int > set_Pr8140719670968651609st_int ).
thf(sy_c_List_Olenlex_001t__List__Olist_Itf__a_J,type,
lenlex_list_a: set_Pr4048851178543822343list_a > set_Pr5382606609415531783list_a ).
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_Itf__a_Mt__Nat__Onat_J,type,
cons_a_nat: ( a > nat ) > list_a_nat > list_a_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__List__Olist_It__Int__Oint_J,type,
cons_list_int: list_int > list_list_int > list_list_int ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__List__Olist_It__Int__Oint_J_J,type,
cons_list_list_int: list_list_int > list_list_list_int > list_list_list_int ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
cons_list_list_a: list_list_a > list_list_list_a > list_list_list_a ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_Itf__a_J,type,
cons_list_a: list_a > list_list_a > list_list_a ).
thf(sy_c_List_Olist_OCons_001tf__a,type,
cons_a: a > list_a > list_a ).
thf(sy_c_List_Olist_ONil_001_062_Itf__a_Mt__Nat__Onat_J,type,
nil_a_nat: list_a_nat ).
thf(sy_c_List_Olist_ONil_001t__Int__Oint,type,
nil_int: list_int ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Int__Oint_J,type,
nil_list_int: list_list_int ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__List__Olist_It__Int__Oint_J_J,type,
nil_list_list_int: list_list_list_int ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
nil_list_list_a: list_list_list_a ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_Itf__a_J,type,
nil_list_a: list_list_a ).
thf(sy_c_List_Olist_ONil_001tf__a,type,
nil_a: list_a ).
thf(sy_c_List_Olist_Ohd_001t__Int__Oint,type,
hd_int: list_int > int ).
thf(sy_c_List_Olist_Ohd_001tf__a,type,
hd_a: list_a > 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_Olist__ex1_001t__Int__Oint,type,
list_ex1_int: ( int > $o ) > list_int > $o ).
thf(sy_c_List_Olist__ex1_001t__List__Olist_It__Int__Oint_J,type,
list_ex1_list_int: ( list_int > $o ) > list_list_int > $o ).
thf(sy_c_List_Olist__ex1_001t__List__Olist_Itf__a_J,type,
list_ex1_list_a: ( list_a > $o ) > list_list_a > $o ).
thf(sy_c_List_Olist__ex1_001tf__a,type,
list_ex1_a: ( a > $o ) > list_a > $o ).
thf(sy_c_List_Olist__update_001t__Int__Oint,type,
list_update_int: list_int > nat > int > list_int ).
thf(sy_c_List_Olist__update_001tf__a,type,
list_update_a: list_a > nat > a > list_a ).
thf(sy_c_List_Olistrel1_001t__Int__Oint,type,
listrel1_int: set_Pr958786334691620121nt_int > set_Pr765067013931698361st_int ).
thf(sy_c_List_Olistrel1_001tf__a,type,
listrel1_a: set_Product_prod_a_a > set_Pr4048851178543822343list_a ).
thf(sy_c_List_Olistrel_001t__Int__Oint_001t__Int__Oint,type,
listrel_int_int: set_Pr958786334691620121nt_int > set_Pr765067013931698361st_int ).
thf(sy_c_List_Olistrel_001t__Int__Oint_001tf__a,type,
listrel_int_a: set_Pr4265292341059111293_int_a > set_Pr8977947362417044851list_a ).
thf(sy_c_List_Olistrel_001tf__a_001t__Int__Oint,type,
listrel_a_int: set_Pr2444352267426396943_a_int > set_Pr232763997632103205st_int ).
thf(sy_c_List_Olistrel_001tf__a_001tf__a,type,
listrel_a_a: set_Product_prod_a_a > set_Pr4048851178543822343list_a ).
thf(sy_c_List_Omeasures_001tf__a,type,
measures_a: list_a_nat > set_Product_prod_a_a ).
thf(sy_c_List_On__lists_001t__Int__Oint,type,
n_lists_int: nat > list_int > list_list_int ).
thf(sy_c_List_On__lists_001t__List__Olist_It__Int__Oint_J,type,
n_lists_list_int: nat > list_list_int > list_list_list_int ).
thf(sy_c_List_On__lists_001t__List__Olist_Itf__a_J,type,
n_lists_list_a: nat > list_list_a > list_list_list_a ).
thf(sy_c_List_On__lists_001tf__a,type,
n_lists_a: nat > list_a > list_list_a ).
thf(sy_c_List_Onth_001t__Int__Oint,type,
nth_int: list_int > nat > int ).
thf(sy_c_List_Onth_001tf__a,type,
nth_a: list_a > nat > a ).
thf(sy_c_List_Onths_001t__Int__Oint,type,
nths_int: list_int > set_nat > list_int ).
thf(sy_c_List_Onths_001t__List__Olist_It__Int__Oint_J,type,
nths_list_int: list_list_int > set_nat > list_list_int ).
thf(sy_c_List_Onths_001t__List__Olist_Itf__a_J,type,
nths_list_a: list_list_a > set_nat > list_list_a ).
thf(sy_c_List_Onths_001tf__a,type,
nths_a: list_a > set_nat > list_a ).
thf(sy_c_List_Oproduct__lists_001t__Int__Oint,type,
product_lists_int: list_list_int > list_list_int ).
thf(sy_c_List_Oproduct__lists_001t__List__Olist_It__Int__Oint_J,type,
produc2606055431807726861st_int: list_list_list_int > list_list_list_int ).
thf(sy_c_List_Oproduct__lists_001t__List__Olist_Itf__a_J,type,
product_lists_list_a: list_list_list_a > list_list_list_a ).
thf(sy_c_List_Oproduct__lists_001tf__a,type,
product_lists_a: list_list_a > list_list_a ).
thf(sy_c_List_Orev_001t__Int__Oint,type,
rev_int: list_int > list_int ).
thf(sy_c_List_Orev_001tf__a,type,
rev_a: list_a > list_a ).
thf(sy_c_List_Orotate1_001t__Int__Oint,type,
rotate1_int: list_int > list_int ).
thf(sy_c_List_Orotate1_001tf__a,type,
rotate1_a: list_a > list_a ).
thf(sy_c_List_Osubseqs_001tf__a,type,
subseqs_a: list_a > list_list_a ).
thf(sy_c_List_Otake_001t__Int__Oint,type,
take_int: nat > list_int > list_int ).
thf(sy_c_List_Otake_001tf__a,type,
take_a: nat > list_a > list_a ).
thf(sy_c_List_Oupto__aux,type,
upto_aux: int > int > list_int > list_int ).
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_It__List__Olist_It__Int__Oint_J_J,type,
size_s6508937059447799164st_int: state_list_int > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Common__Ostate_It__List__Olist_Itf__a_J_J,type,
size_s2742192117134447578list_a: state_list_a > 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__Current__Ocurrent_It__Int__Oint_J,type,
size_s825146100745393828nt_int: current_int > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Current__Ocurrent_It__List__Olist_It__Int__Oint_J_J,type,
size_s5602925888181301556st_int: current_list_int > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Current__Ocurrent_It__List__Olist_Itf__a_J_J,type,
size_s5124551957227789858list_a: current_list_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Current__Ocurrent_Itf__a_J,type,
size_size_current_a: current_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_It__List__Olist_It__Int__Oint_J_J,type,
size_s533118279054570080st_int: list_list_int > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
size_s349497388124573686list_a: list_list_a > 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__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_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_Product__Type_OPair_001_062_It__Int__Oint_M_062_It__Int__Oint_M_Eo_J_J_001t__List__Olist_It__Int__Oint_J,type,
produc8618682346314911123st_int: ( int > int > $o ) > list_int > produc5834231552977413017st_int ).
thf(sy_c_Product__Type_OPair_001_062_Itf__a_M_062_Itf__a_M_Eo_J_J_001t__List__Olist_Itf__a_J,type,
produc8111569692950616493list_a: ( a > a > $o ) > list_a > produc5032551385658279741list_a ).
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__Int__Oint,type,
product_Pair_int_int: int > int > product_prod_int_int ).
thf(sy_c_Product__Type_OPair_001t__Int__Oint_001tf__a,type,
product_Pair_int_a: int > a > product_prod_int_a ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Int__Oint_J_001t__Common__Ostate_It__List__Olist_It__Int__Oint_J_J,type,
produc9042490306465745403st_int: list_int > state_list_int > produc2176888055510159817st_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_It__Int__Oint_J_001t__List__Olist_Itf__a_J,type,
produc4840137957254204271list_a: list_int > list_a > produc7853204735038008893list_a ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__List__Olist_It__Int__Oint_J_J_001t__List__Olist_It__List__Olist_It__Int__Oint_J_J,type,
produc4355665770860423473st_int: list_list_int > list_list_int > produc8452586669324383225st_int ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
produc8696003437204565271list_a: list_list_a > list_list_a > produc7709606177366032167list_a ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_Itf__a_J_001t__Common__Ostate_It__List__Olist_Itf__a_J_J,type,
produc2216564154414103411list_a: list_a > state_list_a > produc5879181297811355641list_a ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_Itf__a_J_001t__List__Olist_It__Int__Oint_J,type,
produc615098764691696873st_int: list_a > list_int > produc7165688024466062575st_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_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__Current__Ocurrent_Itf__a_J,type,
produc8503237746132909001rent_a: a > current_a > produc7805042584321970905rent_a ).
thf(sy_c_Product__Type_OPair_001tf__a_001t__Int__Oint,type,
product_Pair_a_int: a > int > product_prod_a_int ).
thf(sy_c_Product__Type_OPair_001tf__a_001tf__a,type,
product_Pair_a_a: a > a > product_prod_a_a ).
thf(sy_c_Product__Type_Oprod_Ofst_001t__Int__Oint_001t__Current__Ocurrent_It__Int__Oint_J,type,
produc5847638466952691905nt_int: produc7978559663978235293nt_int > int ).
thf(sy_c_Product__Type_Oprod_Ofst_001tf__a_001t__Current__Ocurrent_Itf__a_J,type,
produc4952273589686483381rent_a: produc7805042584321970905rent_a > a ).
thf(sy_c_Product__Type_Oprod_Osnd_001tf__a_001t__Current__Ocurrent_Itf__a_J,type,
produc4695312889421393143rent_a: produc7805042584321970905rent_a > current_a ).
thf(sy_c_Set_OCollect_001t__Int__Oint,type,
collect_int: ( int > $o ) > set_int ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Int__Oint_J,type,
collect_list_int: ( list_int > $o ) > set_list_int ).
thf(sy_c_Set_OCollect_001t__List__Olist_Itf__a_J,type,
collect_list_a: ( list_a > $o ) > set_list_a ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_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_It__List__Olist_It__Int__Oint_J_J,type,
type_i5837482050862709779st_int: state_list_int > $o ).
thf(sy_c_Type__Classes_Oinvar__class_Oinvar_001t__Common__Ostate_It__List__Olist_Itf__a_J_J,type,
type_i5563415093623196931list_a: state_list_a > $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__Current__Ocurrent_It__Int__Oint_J,type,
type_i7078759447853081037nt_int: current_int > $o ).
thf(sy_c_Type__Classes_Oinvar__class_Oinvar_001t__Current__Ocurrent_It__List__Olist_It__Int__Oint_J_J,type,
type_i4621526582633534429st_int: current_list_int > $o ).
thf(sy_c_Type__Classes_Oinvar__class_Oinvar_001t__Current__Ocurrent_It__List__Olist_Itf__a_J_J,type,
type_i8419033571676804025list_a: current_list_a > $o ).
thf(sy_c_Type__Classes_Oinvar__class_Oinvar_001t__Current__Ocurrent_Itf__a_J,type,
type_i6141643110573041459rent_a: current_a > $o ).
thf(sy_c_Type__Classes_Osize__new__class_Osize__new_001t__Common__Ostate_Itf__a_J,type,
type_s8424385952999958455tate_a: state_a > nat ).
thf(sy_c_Type__Classes_Osize__new__class_Osize__new_001t__Current__Ocurrent_Itf__a_J,type,
type_s933026853152659577rent_a: current_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_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__List__Olist_It__Int__Oint_J,type,
member_list_int: list_int > set_list_int > $o ).
thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
member_list_a: list_a > set_list_a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
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__Int__Oint_Mtf__a_J,type,
member6717207084439649758_int_a: product_prod_int_a > set_Pr4265292341059111293_int_a > $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_It__Int__Oint_J_Mt__List__Olist_Itf__a_J_J,type,
member3048572340474253524list_a: produc7853204735038008893list_a > set_Pr8977947362417044851list_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Int__Oint_J_J_Mt__List__Olist_It__List__Olist_It__Int__Oint_J_J_J,type,
member3583055517200259234st_int: produc8452586669324383225st_int > set_Pr8140719670968651609st_int > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
member1318342207407915856list_a: produc7709606177366032167list_a > set_Pr5382606609415531783list_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_It__Int__Oint_J_J,type,
member2361055629902307206st_int: produc7165688024466062575st_int > set_Pr232763997632103205st_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_Mt__Int__Oint_J,type,
member1546337568877222000_a_int: product_prod_a_int > set_Pr2444352267426396943_a_int > $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_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_aux____,type,
aux: list_a ).
thf(sy_v_current____,type,
current: current_a ).
thf(sy_v_moved____,type,
moved: nat ).
thf(sy_v_new____,type,
new: list_a ).
% Relevant facts (1263)
thf(fact_0__C2_Oprems_C_I3_J,axiom,
type_i4669920168676019581tate_a @ ( copy_a @ current @ aux @ new @ moved ) ).
% "2.prems"(3)
thf(fact_1__C2_Oprems_C_I2_J,axiom,
( ( common_list_a @ ( copy_a @ current @ aux @ new @ moved ) )
= nil_a ) ).
% "2.prems"(2)
thf(fact_2__C2_Oprems_C_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ ( copy_a @ current @ aux @ new @ moved ) ) ).
% "2.prems"(1)
thf(fact_3_Current__Proof_Oinvar__push,axiom,
! [Current: current_int,X: int] :
( ( type_i7078759447853081037nt_int @ Current )
=> ( type_i7078759447853081037nt_int @ ( push_int2 @ X @ Current ) ) ) ).
% Current_Proof.invar_push
thf(fact_4_Current__Proof_Oinvar__push,axiom,
! [Current: current_a,X: a] :
( ( type_i6141643110573041459rent_a @ Current )
=> ( type_i6141643110573041459rent_a @ ( push_a2 @ X @ Current ) ) ) ).
% Current_Proof.invar_push
thf(fact_5_step__list,axiom,
! [Common: state_int] :
( ( type_i1727439852002712835te_int @ Common )
=> ( ( common_list_int @ ( type_s2983202171344810359te_int @ Common ) )
= ( common_list_int @ Common ) ) ) ).
% step_list
thf(fact_6_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_7_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_8_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_9_invar__step,axiom,
! [Common: state_int] :
( ( type_i1727439852002712835te_int @ Common )
=> ( type_i1727439852002712835te_int @ ( type_s2983202171344810359te_int @ Common ) ) ) ).
% invar_step
thf(fact_10_invar__step,axiom,
! [Common: state_a] :
( ( type_i4669920168676019581tate_a @ Common )
=> ( type_i4669920168676019581tate_a @ ( type_s889635741254954505tate_a @ Common ) ) ) ).
% invar_step
thf(fact_11_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_12_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_13_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_14_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_15_Common_Opush_Osimps_I2_J,axiom,
! [X: int,Current: current_int,Aux: list_int,New: list_int,Moved: nat] :
( ( push_int @ X @ ( copy_int @ Current @ Aux @ New @ Moved ) )
= ( copy_int @ ( push_int2 @ X @ Current ) @ Aux @ New @ Moved ) ) ).
% Common.push.simps(2)
thf(fact_16_Common_Opush_Osimps_I2_J,axiom,
! [X: a,Current: current_a,Aux: list_a,New: list_a,Moved: nat] :
( ( push_a @ X @ ( copy_a @ Current @ Aux @ New @ Moved ) )
= ( copy_a @ ( push_a2 @ X @ Current ) @ Aux @ New @ Moved ) ) ).
% Common.push.simps(2)
thf(fact_17_list__current__size,axiom,
! [Common: state_list_int] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s6508937059447799164st_int @ Common ) )
=> ( ( ( common7659038223531881259st_int @ Common )
= nil_list_int )
=> ~ ( type_i5837482050862709779st_int @ Common ) ) ) ).
% list_current_size
thf(fact_18_list__current__size,axiom,
! [Common: state_list_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s2742192117134447578list_a @ Common ) )
=> ( ( ( common3845368160371929429list_a @ Common )
= nil_list_a )
=> ~ ( type_i5563415093623196931list_a @ Common ) ) ) ).
% list_current_size
thf(fact_19_list__current__size,axiom,
! [Common: state_int] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_int @ Common ) )
=> ( ( ( common8573843843682093467nt_int @ Common )
= nil_int )
=> ~ ( type_i1727439852002712835te_int @ Common ) ) ) ).
% list_current_size
thf(fact_20_list__current__size,axiom,
! [Common: state_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ Common ) )
=> ( ( ( common1102728217005306191rent_a @ Common )
= nil_a )
=> ~ ( type_i4669920168676019581tate_a @ Common ) ) ) ).
% list_current_size
thf(fact_21_state_Oinject_I1_J,axiom,
! [X11: current_a,X12: list_a,X13: list_a,X14: nat,Y11: current_a,Y12: list_a,Y13: list_a,Y14: nat] :
( ( ( copy_a @ X11 @ X12 @ X13 @ X14 )
= ( copy_a @ Y11 @ Y12 @ Y13 @ Y14 ) )
= ( ( X11 = Y11 )
& ( X12 = Y12 )
& ( X13 = Y13 )
& ( X14 = Y14 ) ) ) ).
% state.inject(1)
thf(fact_22_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_23_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_24_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_25_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_26_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_27_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_28_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_29_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_30_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_31_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_32_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_33_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_34_less__not__refl2,axiom,
! [N: nat,M2: nat] :
( ( ord_less_nat @ N @ M2 )
=> ( M2 != N ) ) ).
% less_not_refl2
thf(fact_35_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_36_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_37_size__neq__size__imp__neq,axiom,
! [X: list_a,Y: list_a] :
( ( ( size_size_list_a @ X )
!= ( size_size_list_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_38_size__neq__size__imp__neq,axiom,
! [X: list_int,Y: list_int] :
( ( ( size_size_list_int @ X )
!= ( size_size_list_int @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_39_size__neq__size__imp__neq,axiom,
! [X: current_a,Y: current_a] :
( ( ( size_size_current_a @ X )
!= ( size_size_current_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_40_size__neq__size__imp__neq,axiom,
! [X: state_int,Y: state_int] :
( ( ( size_size_state_int @ X )
!= ( size_size_state_int @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_41_size__neq__size__imp__neq,axiom,
! [X: current_int,Y: current_int] :
( ( ( size_s825146100745393828nt_int @ X )
!= ( size_s825146100745393828nt_int @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_42_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_43_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_44_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_45_gr__implies__not__zero,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_46_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_47_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_48_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_49_gr__implies__not0,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_50_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_51_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_52_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_53_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_54_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_55_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_56_Common__Proof_Opush__list,axiom,
! [X: list_int,Common: state_list_int] :
( ( common_list_list_int @ ( push_list_int @ X @ Common ) )
= ( cons_list_int @ X @ ( common_list_list_int @ Common ) ) ) ).
% Common_Proof.push_list
thf(fact_57_Common__Proof_Opush__list,axiom,
! [X: list_a,Common: state_list_a] :
( ( common_list_list_a @ ( push_list_a @ X @ Common ) )
= ( cons_list_a @ X @ ( common_list_list_a @ Common ) ) ) ).
% Common_Proof.push_list
thf(fact_58_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_59_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_60_mem__Collect__eq,axiom,
! [A: list_int,P: list_int > $o] :
( ( member_list_int @ A @ ( collect_list_int @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_61_mem__Collect__eq,axiom,
! [A: list_a,P: list_a > $o] :
( ( member_list_a @ A @ ( collect_list_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_62_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_63_mem__Collect__eq,axiom,
! [A: int,P: int > $o] :
( ( member_int @ A @ ( collect_int @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_64_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_65_Collect__mem__eq,axiom,
! [A2: set_list_int] :
( ( collect_list_int
@ ^ [X2: list_int] : ( member_list_int @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_66_Collect__mem__eq,axiom,
! [A2: set_list_a] :
( ( collect_list_a
@ ^ [X2: list_a] : ( member_list_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_67_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_68_Collect__mem__eq,axiom,
! [A2: set_int] :
( ( collect_int
@ ^ [X2: int] : ( member_int @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_69_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_70_push__list__current,axiom,
! [X: list_int,Left: state_list_int] :
( ( common7659038223531881259st_int @ ( push_list_int @ X @ Left ) )
= ( cons_list_int @ X @ ( common7659038223531881259st_int @ Left ) ) ) ).
% push_list_current
thf(fact_71_push__list__current,axiom,
! [X: list_a,Left: state_list_a] :
( ( common3845368160371929429list_a @ ( push_list_a @ X @ Left ) )
= ( cons_list_a @ X @ ( common3845368160371929429list_a @ Left ) ) ) ).
% push_list_current
thf(fact_72_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_73_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_74_zero__natural_Orsp,axiom,
zero_zero_nat = zero_zero_nat ).
% zero_natural.rsp
thf(fact_75_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_76_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_77_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_78_order__less__imp__not__eq2,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_79_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_80_order__less__imp__not__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_81_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_82_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_83_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_84_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_85_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_86_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_87_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_88_list_Oinject,axiom,
! [X21: list_int,X22: list_list_int,Y21: list_int,Y22: list_list_int] :
( ( ( cons_list_int @ X21 @ X22 )
= ( cons_list_int @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_89_list_Oinject,axiom,
! [X21: list_a,X22: list_list_a,Y21: list_a,Y22: list_list_a] :
( ( ( cons_list_a @ X21 @ X22 )
= ( cons_list_a @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_90_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_91_of__nat__eq__iff,axiom,
! [M2: nat,N: nat] :
( ( ( semiri1316708129612266289at_nat @ M2 )
= ( semiri1316708129612266289at_nat @ N ) )
= ( M2 = N ) ) ).
% of_nat_eq_iff
thf(fact_92_of__nat__eq__iff,axiom,
! [M2: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= ( semiri1314217659103216013at_int @ N ) )
= ( M2 = N ) ) ).
% of_nat_eq_iff
thf(fact_93_length__0__conv,axiom,
! [Xs: list_list_int] :
( ( ( size_s533118279054570080st_int @ Xs )
= zero_zero_nat )
= ( Xs = nil_list_int ) ) ).
% length_0_conv
thf(fact_94_length__0__conv,axiom,
! [Xs: list_list_a] :
( ( ( size_s349497388124573686list_a @ Xs )
= zero_zero_nat )
= ( Xs = nil_list_a ) ) ).
% length_0_conv
thf(fact_95_length__0__conv,axiom,
! [Xs: list_int] :
( ( ( size_size_list_int @ Xs )
= zero_zero_nat )
= ( Xs = nil_int ) ) ).
% length_0_conv
thf(fact_96_length__0__conv,axiom,
! [Xs: list_a] :
( ( ( size_size_list_a @ Xs )
= zero_zero_nat )
= ( Xs = nil_a ) ) ).
% length_0_conv
thf(fact_97_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_98_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_99_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_100_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_101_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_102_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_103_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_104_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_105_length__greater__0__conv,axiom,
! [Xs: list_list_int] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s533118279054570080st_int @ Xs ) )
= ( Xs != nil_list_int ) ) ).
% length_greater_0_conv
thf(fact_106_length__greater__0__conv,axiom,
! [Xs: list_list_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s349497388124573686list_a @ Xs ) )
= ( Xs != nil_list_a ) ) ).
% length_greater_0_conv
thf(fact_107_length__greater__0__conv,axiom,
! [Xs: list_int] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs ) )
= ( Xs != nil_int ) ) ).
% length_greater_0_conv
thf(fact_108_length__greater__0__conv,axiom,
! [Xs: list_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ Xs ) )
= ( Xs != nil_a ) ) ).
% length_greater_0_conv
thf(fact_109_transpose_Ocases,axiom,
! [X: list_list_list_int] :
( ( X != nil_list_list_int )
=> ( ! [Xss: list_list_list_int] :
( X
!= ( cons_list_list_int @ nil_list_int @ Xss ) )
=> ~ ! [X3: list_int,Xs2: list_list_int,Xss: list_list_list_int] :
( X
!= ( cons_list_list_int @ ( cons_list_int @ X3 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_110_transpose_Ocases,axiom,
! [X: list_list_list_a] :
( ( X != nil_list_list_a )
=> ( ! [Xss: list_list_list_a] :
( X
!= ( cons_list_list_a @ nil_list_a @ Xss ) )
=> ~ ! [X3: list_a,Xs2: list_list_a,Xss: list_list_list_a] :
( X
!= ( cons_list_list_a @ ( cons_list_a @ X3 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_111_transpose_Ocases,axiom,
! [X: list_list_a] :
( ( X != nil_list_a )
=> ( ! [Xss: list_list_a] :
( X
!= ( cons_list_a @ nil_a @ Xss ) )
=> ~ ! [X3: a,Xs2: list_a,Xss: list_list_a] :
( X
!= ( cons_list_a @ ( cons_a @ X3 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_112_transpose_Ocases,axiom,
! [X: list_list_int] :
( ( X != nil_list_int )
=> ( ! [Xss: list_list_int] :
( X
!= ( cons_list_int @ nil_int @ Xss ) )
=> ~ ! [X3: int,Xs2: list_int,Xss: list_list_int] :
( X
!= ( cons_list_int @ ( cons_int @ X3 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_113_list__induct2,axiom,
! [Xs: list_a,Ys: list_a,P: list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( P @ nil_a @ nil_a )
=> ( ! [X3: a,Xs2: list_a,Y2: a,Ys2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_114_list__induct2,axiom,
! [Xs: list_a,Ys: list_int,P: list_a > list_int > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( P @ nil_a @ nil_int )
=> ( ! [X3: a,Xs2: list_a,Y2: int,Ys2: list_int] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_int @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_int @ Y2 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_115_list__induct2,axiom,
! [Xs: list_int,Ys: list_a,P: list_int > list_a > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( P @ nil_int @ nil_a )
=> ( ! [X3: int,Xs2: list_int,Y2: a,Ys2: list_a] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_int @ X3 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_116_list__induct2,axiom,
! [Xs: list_int,Ys: list_int,P: list_int > list_int > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( P @ nil_int @ nil_int )
=> ( ! [X3: int,Xs2: list_int,Y2: int,Ys2: list_int] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_int @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_int @ X3 @ Xs2 ) @ ( cons_int @ Y2 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_117_list__induct2,axiom,
! [Xs: list_list_int,Ys: list_a,P: list_list_int > list_a > $o] :
( ( ( size_s533118279054570080st_int @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( P @ nil_list_int @ nil_a )
=> ( ! [X3: list_int,Xs2: list_list_int,Y2: a,Ys2: list_a] :
( ( ( size_s533118279054570080st_int @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_list_int @ X3 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_118_list__induct2,axiom,
! [Xs: list_list_a,Ys: list_a,P: list_list_a > list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( P @ nil_list_a @ nil_a )
=> ( ! [X3: list_a,Xs2: list_list_a,Y2: a,Ys2: list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_list_a @ X3 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_119_list__induct2,axiom,
! [Xs: list_list_int,Ys: list_int,P: list_list_int > list_int > $o] :
( ( ( size_s533118279054570080st_int @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( P @ nil_list_int @ nil_int )
=> ( ! [X3: list_int,Xs2: list_list_int,Y2: int,Ys2: list_int] :
( ( ( size_s533118279054570080st_int @ Xs2 )
= ( size_size_list_int @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_list_int @ X3 @ Xs2 ) @ ( cons_int @ Y2 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_120_list__induct2,axiom,
! [Xs: list_list_a,Ys: list_int,P: list_list_a > list_int > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( P @ nil_list_a @ nil_int )
=> ( ! [X3: list_a,Xs2: list_list_a,Y2: int,Ys2: list_int] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_size_list_int @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_list_a @ X3 @ Xs2 ) @ ( cons_int @ Y2 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_121_list__induct2,axiom,
! [Xs: list_a,Ys: list_list_int,P: list_a > list_list_int > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_s533118279054570080st_int @ Ys ) )
=> ( ( P @ nil_a @ nil_list_int )
=> ( ! [X3: a,Xs2: list_a,Y2: list_int,Ys2: list_list_int] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s533118279054570080st_int @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_list_int @ Y2 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_122_list__induct2,axiom,
! [Xs: list_a,Ys: list_list_a,P: list_a > list_list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( P @ nil_a @ nil_list_a )
=> ( ! [X3: a,Xs2: list_a,Y2: list_a,Ys2: list_list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_s349497388124573686list_a @ Ys2 ) )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_list_a @ Y2 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_123_list__induct3,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,P: list_a > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ nil_a @ nil_a @ nil_a )
=> ( ! [X3: a,Xs2: list_a,Y2: a,Ys2: list_a,Z: a,Zs2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) @ ( cons_a @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_124_list__induct3,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_int,P: list_a > list_a > list_int > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_int @ Zs ) )
=> ( ( P @ nil_a @ nil_a @ nil_int )
=> ( ! [X3: a,Xs2: list_a,Y2: a,Ys2: list_a,Z: int,Zs2: list_int] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_int @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) @ ( cons_int @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_125_list__induct3,axiom,
! [Xs: list_a,Ys: list_int,Zs: list_a,P: list_a > list_int > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( ( size_size_list_int @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ nil_a @ nil_int @ nil_a )
=> ( ! [X3: a,Xs2: list_a,Y2: int,Ys2: list_int,Z: a,Zs2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_int @ Ys2 ) )
=> ( ( ( size_size_list_int @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_int @ Y2 @ Ys2 ) @ ( cons_a @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_126_list__induct3,axiom,
! [Xs: list_a,Ys: list_int,Zs: list_int,P: list_a > list_int > list_int > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( ( size_size_list_int @ Ys )
= ( size_size_list_int @ Zs ) )
=> ( ( P @ nil_a @ nil_int @ nil_int )
=> ( ! [X3: a,Xs2: list_a,Y2: int,Ys2: list_int,Z: int,Zs2: list_int] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_int @ Ys2 ) )
=> ( ( ( size_size_list_int @ Ys2 )
= ( size_size_list_int @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_int @ Y2 @ Ys2 ) @ ( cons_int @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_127_list__induct3,axiom,
! [Xs: list_int,Ys: list_a,Zs: list_a,P: list_int > list_a > list_a > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ nil_int @ nil_a @ nil_a )
=> ( ! [X3: int,Xs2: list_int,Y2: a,Ys2: list_a,Z: a,Zs2: list_a] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 )
=> ( P @ ( cons_int @ X3 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) @ ( cons_a @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_128_list__induct3,axiom,
! [Xs: list_int,Ys: list_a,Zs: list_int,P: list_int > list_a > list_int > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_int @ Zs ) )
=> ( ( P @ nil_int @ nil_a @ nil_int )
=> ( ! [X3: int,Xs2: list_int,Y2: a,Ys2: list_a,Z: int,Zs2: list_int] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_int @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 )
=> ( P @ ( cons_int @ X3 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) @ ( cons_int @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_129_list__induct3,axiom,
! [Xs: list_int,Ys: list_int,Zs: list_a,P: list_int > list_int > list_a > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( ( size_size_list_int @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ nil_int @ nil_int @ nil_a )
=> ( ! [X3: int,Xs2: list_int,Y2: int,Ys2: list_int,Z: a,Zs2: list_a] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_int @ Ys2 ) )
=> ( ( ( size_size_list_int @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 )
=> ( P @ ( cons_int @ X3 @ Xs2 ) @ ( cons_int @ Y2 @ Ys2 ) @ ( cons_a @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_130_list__induct3,axiom,
! [Xs: list_int,Ys: list_int,Zs: list_int,P: list_int > list_int > list_int > $o] :
( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) )
=> ( ( ( size_size_list_int @ Ys )
= ( size_size_list_int @ Zs ) )
=> ( ( P @ nil_int @ nil_int @ nil_int )
=> ( ! [X3: int,Xs2: list_int,Y2: int,Ys2: list_int,Z: int,Zs2: list_int] :
( ( ( size_size_list_int @ Xs2 )
= ( size_size_list_int @ Ys2 ) )
=> ( ( ( size_size_list_int @ Ys2 )
= ( size_size_list_int @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 )
=> ( P @ ( cons_int @ X3 @ Xs2 ) @ ( cons_int @ Y2 @ Ys2 ) @ ( cons_int @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_131_list__induct3,axiom,
! [Xs: list_list_int,Ys: list_a,Zs: list_a,P: list_list_int > list_a > list_a > $o] :
( ( ( size_s533118279054570080st_int @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ nil_list_int @ nil_a @ nil_a )
=> ( ! [X3: list_int,Xs2: list_list_int,Y2: a,Ys2: list_a,Z: a,Zs2: list_a] :
( ( ( size_s533118279054570080st_int @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 )
=> ( P @ ( cons_list_int @ X3 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) @ ( cons_a @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_132_list__induct3,axiom,
! [Xs: list_list_a,Ys: list_a,Zs: list_a,P: list_list_a > list_a > list_a > $o] :
( ( ( size_s349497388124573686list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ nil_list_a @ nil_a @ nil_a )
=> ( ! [X3: list_a,Xs2: list_list_a,Y2: a,Ys2: list_a,Z: a,Zs2: list_a] :
( ( ( size_s349497388124573686list_a @ Xs2 )
= ( size_size_list_a @ Ys2 ) )
=> ( ( ( size_size_list_a @ Ys2 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys2 @ Zs2 )
=> ( P @ ( cons_list_a @ X3 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) @ ( cons_a @ Z @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_133_not__Cons__self2,axiom,
! [X: a,Xs: list_a] :
( ( cons_a @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_134_not__Cons__self2,axiom,
! [X: list_int,Xs: list_list_int] :
( ( cons_list_int @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_135_not__Cons__self2,axiom,
! [X: list_a,Xs: list_list_a] :
( ( cons_list_a @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_136_not__Cons__self2,axiom,
! [X: int,Xs: list_int] :
( ( cons_int @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_137_length__induct,axiom,
! [P: list_a > $o,Xs: list_a] :
( ! [Xs2: list_a] :
( ! [Ys3: list_a] :
( ( ord_less_nat @ ( size_size_list_a @ Ys3 ) @ ( size_size_list_a @ Xs2 ) )
=> ( P @ Ys3 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_138_length__induct,axiom,
! [P: list_int > $o,Xs: list_int] :
( ! [Xs2: list_int] :
( ! [Ys3: list_int] :
( ( ord_less_nat @ ( size_size_list_int @ Ys3 ) @ ( size_size_list_int @ Xs2 ) )
=> ( P @ Ys3 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_139_list_Odistinct_I1_J,axiom,
! [X21: list_int,X22: list_list_int] :
( nil_list_int
!= ( cons_list_int @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_140_list_Odistinct_I1_J,axiom,
! [X21: list_a,X22: list_list_a] :
( nil_list_a
!= ( cons_list_a @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_141_list_Odistinct_I1_J,axiom,
! [X21: a,X22: list_a] :
( nil_a
!= ( cons_a @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_142_list_Odistinct_I1_J,axiom,
! [X21: int,X22: list_int] :
( nil_int
!= ( cons_int @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_143_list_OdiscI,axiom,
! [List: list_list_int,X21: list_int,X22: list_list_int] :
( ( List
= ( cons_list_int @ X21 @ X22 ) )
=> ( List != nil_list_int ) ) ).
% list.discI
thf(fact_144_list_OdiscI,axiom,
! [List: list_list_a,X21: list_a,X22: list_list_a] :
( ( List
= ( cons_list_a @ X21 @ X22 ) )
=> ( List != nil_list_a ) ) ).
% list.discI
thf(fact_145_list_OdiscI,axiom,
! [List: list_a,X21: a,X22: list_a] :
( ( List
= ( cons_a @ X21 @ X22 ) )
=> ( List != nil_a ) ) ).
% list.discI
thf(fact_146_list_OdiscI,axiom,
! [List: list_int,X21: int,X22: list_int] :
( ( List
= ( cons_int @ X21 @ X22 ) )
=> ( List != nil_int ) ) ).
% list.discI
thf(fact_147_list_Oexhaust,axiom,
! [Y: list_list_int] :
( ( Y != nil_list_int )
=> ~ ! [X212: list_int,X222: list_list_int] :
( Y
!= ( cons_list_int @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_148_list_Oexhaust,axiom,
! [Y: list_list_a] :
( ( Y != nil_list_a )
=> ~ ! [X212: list_a,X222: list_list_a] :
( Y
!= ( cons_list_a @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_149_list_Oexhaust,axiom,
! [Y: list_a] :
( ( Y != nil_a )
=> ~ ! [X212: a,X222: list_a] :
( Y
!= ( cons_a @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_150_list_Oexhaust,axiom,
! [Y: list_int] :
( ( Y != nil_int )
=> ~ ! [X212: int,X222: list_int] :
( Y
!= ( cons_int @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_151_min__list_Ocases,axiom,
! [X: list_int] :
( ! [X3: int,Xs2: list_int] :
( X
!= ( cons_int @ X3 @ Xs2 ) )
=> ( X = nil_int ) ) ).
% min_list.cases
thf(fact_152_remdups__adj_Ocases,axiom,
! [X: list_list_int] :
( ( X != nil_list_int )
=> ( ! [X3: list_int] :
( X
!= ( cons_list_int @ X3 @ nil_list_int ) )
=> ~ ! [X3: list_int,Y2: list_int,Xs2: list_list_int] :
( X
!= ( cons_list_int @ X3 @ ( cons_list_int @ Y2 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_153_remdups__adj_Ocases,axiom,
! [X: list_list_a] :
( ( X != nil_list_a )
=> ( ! [X3: list_a] :
( X
!= ( cons_list_a @ X3 @ nil_list_a ) )
=> ~ ! [X3: list_a,Y2: list_a,Xs2: list_list_a] :
( X
!= ( cons_list_a @ X3 @ ( cons_list_a @ Y2 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_154_remdups__adj_Ocases,axiom,
! [X: list_a] :
( ( X != nil_a )
=> ( ! [X3: a] :
( X
!= ( cons_a @ X3 @ nil_a ) )
=> ~ ! [X3: a,Y2: a,Xs2: list_a] :
( X
!= ( cons_a @ X3 @ ( cons_a @ Y2 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_155_remdups__adj_Ocases,axiom,
! [X: list_int] :
( ( X != nil_int )
=> ( ! [X3: int] :
( X
!= ( cons_int @ X3 @ nil_int ) )
=> ~ ! [X3: int,Y2: int,Xs2: list_int] :
( X
!= ( cons_int @ X3 @ ( cons_int @ Y2 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_156_neq__Nil__conv,axiom,
! [Xs: list_list_int] :
( ( Xs != nil_list_int )
= ( ? [Y3: list_int,Ys4: list_list_int] :
( Xs
= ( cons_list_int @ Y3 @ Ys4 ) ) ) ) ).
% neq_Nil_conv
thf(fact_157_neq__Nil__conv,axiom,
! [Xs: list_list_a] :
( ( Xs != nil_list_a )
= ( ? [Y3: list_a,Ys4: list_list_a] :
( Xs
= ( cons_list_a @ Y3 @ Ys4 ) ) ) ) ).
% neq_Nil_conv
thf(fact_158_neq__Nil__conv,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
= ( ? [Y3: a,Ys4: list_a] :
( Xs
= ( cons_a @ Y3 @ Ys4 ) ) ) ) ).
% neq_Nil_conv
thf(fact_159_neq__Nil__conv,axiom,
! [Xs: list_int] :
( ( Xs != nil_int )
= ( ? [Y3: int,Ys4: list_int] :
( Xs
= ( cons_int @ Y3 @ Ys4 ) ) ) ) ).
% neq_Nil_conv
thf(fact_160_list__induct2_H,axiom,
! [P: list_a > list_a > $o,Xs: list_a,Ys: list_a] :
( ( P @ nil_a @ nil_a )
=> ( ! [X3: a,Xs2: list_a] : ( P @ ( cons_a @ X3 @ Xs2 ) @ nil_a )
=> ( ! [Y2: a,Ys2: list_a] : ( P @ nil_a @ ( cons_a @ Y2 @ Ys2 ) )
=> ( ! [X3: a,Xs2: list_a,Y2: a,Ys2: list_a] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_161_list__induct2_H,axiom,
! [P: list_a > list_int > $o,Xs: list_a,Ys: list_int] :
( ( P @ nil_a @ nil_int )
=> ( ! [X3: a,Xs2: list_a] : ( P @ ( cons_a @ X3 @ Xs2 ) @ nil_int )
=> ( ! [Y2: int,Ys2: list_int] : ( P @ nil_a @ ( cons_int @ Y2 @ Ys2 ) )
=> ( ! [X3: a,Xs2: list_a,Y2: int,Ys2: list_int] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_int @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_162_list__induct2_H,axiom,
! [P: list_int > list_a > $o,Xs: list_int,Ys: list_a] :
( ( P @ nil_int @ nil_a )
=> ( ! [X3: int,Xs2: list_int] : ( P @ ( cons_int @ X3 @ Xs2 ) @ nil_a )
=> ( ! [Y2: a,Ys2: list_a] : ( P @ nil_int @ ( cons_a @ Y2 @ Ys2 ) )
=> ( ! [X3: int,Xs2: list_int,Y2: a,Ys2: list_a] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_int @ X3 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_163_list__induct2_H,axiom,
! [P: list_int > list_int > $o,Xs: list_int,Ys: list_int] :
( ( P @ nil_int @ nil_int )
=> ( ! [X3: int,Xs2: list_int] : ( P @ ( cons_int @ X3 @ Xs2 ) @ nil_int )
=> ( ! [Y2: int,Ys2: list_int] : ( P @ nil_int @ ( cons_int @ Y2 @ Ys2 ) )
=> ( ! [X3: int,Xs2: list_int,Y2: int,Ys2: list_int] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_int @ X3 @ Xs2 ) @ ( cons_int @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_164_list__induct2_H,axiom,
! [P: list_int > list_list_int > $o,Xs: list_int,Ys: list_list_int] :
( ( P @ nil_int @ nil_list_int )
=> ( ! [X3: int,Xs2: list_int] : ( P @ ( cons_int @ X3 @ Xs2 ) @ nil_list_int )
=> ( ! [Y2: list_int,Ys2: list_list_int] : ( P @ nil_int @ ( cons_list_int @ Y2 @ Ys2 ) )
=> ( ! [X3: int,Xs2: list_int,Y2: list_int,Ys2: list_list_int] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_int @ X3 @ Xs2 ) @ ( cons_list_int @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_165_list__induct2_H,axiom,
! [P: list_int > list_list_a > $o,Xs: list_int,Ys: list_list_a] :
( ( P @ nil_int @ nil_list_a )
=> ( ! [X3: int,Xs2: list_int] : ( P @ ( cons_int @ X3 @ Xs2 ) @ nil_list_a )
=> ( ! [Y2: list_a,Ys2: list_list_a] : ( P @ nil_int @ ( cons_list_a @ Y2 @ Ys2 ) )
=> ( ! [X3: int,Xs2: list_int,Y2: list_a,Ys2: list_list_a] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_int @ X3 @ Xs2 ) @ ( cons_list_a @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_166_list__induct2_H,axiom,
! [P: list_a > list_list_int > $o,Xs: list_a,Ys: list_list_int] :
( ( P @ nil_a @ nil_list_int )
=> ( ! [X3: a,Xs2: list_a] : ( P @ ( cons_a @ X3 @ Xs2 ) @ nil_list_int )
=> ( ! [Y2: list_int,Ys2: list_list_int] : ( P @ nil_a @ ( cons_list_int @ Y2 @ Ys2 ) )
=> ( ! [X3: a,Xs2: list_a,Y2: list_int,Ys2: list_list_int] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_list_int @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_167_list__induct2_H,axiom,
! [P: list_a > list_list_a > $o,Xs: list_a,Ys: list_list_a] :
( ( P @ nil_a @ nil_list_a )
=> ( ! [X3: a,Xs2: list_a] : ( P @ ( cons_a @ X3 @ Xs2 ) @ nil_list_a )
=> ( ! [Y2: list_a,Ys2: list_list_a] : ( P @ nil_a @ ( cons_list_a @ Y2 @ Ys2 ) )
=> ( ! [X3: a,Xs2: list_a,Y2: list_a,Ys2: list_list_a] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) @ ( cons_list_a @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_168_list__induct2_H,axiom,
! [P: list_list_int > list_int > $o,Xs: list_list_int,Ys: list_int] :
( ( P @ nil_list_int @ nil_int )
=> ( ! [X3: list_int,Xs2: list_list_int] : ( P @ ( cons_list_int @ X3 @ Xs2 ) @ nil_int )
=> ( ! [Y2: int,Ys2: list_int] : ( P @ nil_list_int @ ( cons_int @ Y2 @ Ys2 ) )
=> ( ! [X3: list_int,Xs2: list_list_int,Y2: int,Ys2: list_int] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_list_int @ X3 @ Xs2 ) @ ( cons_int @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_169_list__induct2_H,axiom,
! [P: list_list_int > list_a > $o,Xs: list_list_int,Ys: list_a] :
( ( P @ nil_list_int @ nil_a )
=> ( ! [X3: list_int,Xs2: list_list_int] : ( P @ ( cons_list_int @ X3 @ Xs2 ) @ nil_a )
=> ( ! [Y2: a,Ys2: list_a] : ( P @ nil_list_int @ ( cons_a @ Y2 @ Ys2 ) )
=> ( ! [X3: list_int,Xs2: list_list_int,Y2: a,Ys2: list_a] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons_list_int @ X3 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_170_list__nonempty__induct,axiom,
! [Xs: list_list_int,P: list_list_int > $o] :
( ( Xs != nil_list_int )
=> ( ! [X3: list_int] : ( P @ ( cons_list_int @ X3 @ nil_list_int ) )
=> ( ! [X3: list_int,Xs2: list_list_int] :
( ( Xs2 != nil_list_int )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_list_int @ X3 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_171_list__nonempty__induct,axiom,
! [Xs: list_list_a,P: list_list_a > $o] :
( ( Xs != nil_list_a )
=> ( ! [X3: list_a] : ( P @ ( cons_list_a @ X3 @ nil_list_a ) )
=> ( ! [X3: list_a,Xs2: list_list_a] :
( ( Xs2 != nil_list_a )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_list_a @ X3 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_172_list__nonempty__induct,axiom,
! [Xs: list_a,P: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X3: a] : ( P @ ( cons_a @ X3 @ nil_a ) )
=> ( ! [X3: a,Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_a @ X3 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_173_list__nonempty__induct,axiom,
! [Xs: list_int,P: list_int > $o] :
( ( Xs != nil_int )
=> ( ! [X3: int] : ( P @ ( cons_int @ X3 @ nil_int ) )
=> ( ! [X3: int,Xs2: list_int] :
( ( Xs2 != nil_int )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_int @ X3 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_174_list_Osize_I3_J,axiom,
( ( size_s533118279054570080st_int @ nil_list_int )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_175_list_Osize_I3_J,axiom,
( ( size_s349497388124573686list_a @ nil_list_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_176_list_Osize_I3_J,axiom,
( ( size_size_list_int @ nil_int )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_177_list_Osize_I3_J,axiom,
( ( size_size_list_a @ nil_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_178_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_179_of__nat__less__0__iff,axiom,
! [M2: nat] :
~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_180_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_181_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_182_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_183_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_184_lt__ex,axiom,
! [X: int] :
? [Y2: int] : ( ord_less_int @ Y2 @ X ) ).
% lt_ex
thf(fact_185_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_186_gt__ex,axiom,
! [X: int] :
? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% gt_ex
thf(fact_187_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_188_less__imp__neq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_189_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_190_order_Oasym,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order.asym
thf(fact_191_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_192_ord__eq__less__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_193_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_194_ord__less__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_195_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_196_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_197_antisym__conv3,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_int @ Y @ X )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_198_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_199_linorder__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_200_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_201_dual__order_Oasym,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ A @ B ) ) ).
% dual_order.asym
thf(fact_202_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_203_dual__order_Oirrefl,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% dual_order.irrefl
thf(fact_204_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X4: nat] : ( P2 @ X4 ) )
= ( ^ [P3: nat > $o] :
? [N3: nat] :
( ( P3 @ N3 )
& ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ~ ( P3 @ M3 ) ) ) ) ) ).
% exists_least_iff
thf(fact_205_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: nat] : ( P @ A3 @ A3 )
=> ( ! [A3: nat,B2: nat] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_206_linorder__less__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A3: int,B2: int] :
( ( ord_less_int @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: int] : ( P @ A3 @ A3 )
=> ( ! [A3: int,B2: int] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_207_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_208_order_Ostrict__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_209_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_210_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_211_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_212_dual__order_Ostrict__trans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_213_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_214_order_Ostrict__implies__not__eq,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_215_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_216_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_217_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_218_linorder__neqE,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_219_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_220_order__less__asym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_asym
thf(fact_221_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_222_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_223_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_224_order__less__asym_H,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order_less_asym'
thf(fact_225_order__less__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_226_order__less__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_227_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_228_ord__eq__less__subst,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_229_ord__eq__less__subst,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_230_ord__eq__less__subst,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_231_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_232_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_233_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_234_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_235_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_236_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_237_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_238_order__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_239_order__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_240_order__less__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_241_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_242_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_243_order__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_244_order__less__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_245_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C2: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_246_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_247_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_248_list__size,axiom,
! [Current: current_list_int] :
( ( type_i4621526582633534429st_int @ Current )
=> ( ( ( curren8299908649409640978st_int @ Current )
= nil_list_int )
=> ~ ( ord_less_nat @ zero_zero_nat @ ( size_s5602925888181301556st_int @ Current ) ) ) ) ).
% list_size
thf(fact_249_list__size,axiom,
! [Current: current_list_a] :
( ( type_i8419033571676804025list_a @ Current )
=> ( ( ( current_list_list_a @ Current )
= nil_list_a )
=> ~ ( ord_less_nat @ zero_zero_nat @ ( size_s5124551957227789858list_a @ Current ) ) ) ) ).
% list_size
thf(fact_250_list__size,axiom,
! [Current: current_int] :
( ( type_i7078759447853081037nt_int @ Current )
=> ( ( ( current_list_int2 @ Current )
= nil_int )
=> ~ ( ord_less_nat @ zero_zero_nat @ ( size_s825146100745393828nt_int @ Current ) ) ) ) ).
% list_size
thf(fact_251_list__size,axiom,
! [Current: current_a] :
( ( type_i6141643110573041459rent_a @ Current )
=> ( ( ( current_list_a2 @ Current )
= nil_a )
=> ~ ( ord_less_nat @ zero_zero_nat @ ( size_size_current_a @ Current ) ) ) ) ).
% list_size
thf(fact_252_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_lists_list_int @ N @ nil_list_int )
= ( cons_list_list_int @ nil_list_int @ nil_list_list_int ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_lists_list_int @ N @ nil_list_int )
= nil_list_list_int ) ) ) ).
% n_lists_Nil
thf(fact_253_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_lists_list_a @ N @ nil_list_a )
= ( cons_list_list_a @ nil_list_a @ nil_list_list_a ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_lists_list_a @ N @ nil_list_a )
= nil_list_list_a ) ) ) ).
% n_lists_Nil
thf(fact_254_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_lists_int @ N @ nil_int )
= ( cons_list_int @ nil_int @ nil_list_int ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_lists_int @ N @ nil_int )
= nil_list_int ) ) ) ).
% n_lists_Nil
thf(fact_255_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_lists_a @ N @ nil_a )
= ( cons_list_a @ nil_a @ nil_list_a ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_lists_a @ N @ nil_a )
= nil_list_a ) ) ) ).
% n_lists_Nil
thf(fact_256_Common__Proof_Opop__list,axiom,
! [Common: state_list_int,X: list_int,Common2: state_list_int] :
( ( type_i5837482050862709779st_int @ Common )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_s6508937059447799164st_int @ Common ) )
=> ( ( ( pop_list_int @ Common )
= ( produc9042490306465745403st_int @ X @ Common2 ) )
=> ( ( cons_list_int @ X @ ( common_list_list_int @ Common2 ) )
= ( common_list_list_int @ Common ) ) ) ) ) ).
% Common_Proof.pop_list
thf(fact_257_Common__Proof_Opop__list,axiom,
! [Common: state_list_a,X: list_a,Common2: state_list_a] :
( ( type_i5563415093623196931list_a @ Common )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_s2742192117134447578list_a @ Common ) )
=> ( ( ( pop_list_a @ Common )
= ( produc2216564154414103411list_a @ X @ Common2 ) )
=> ( ( cons_list_a @ X @ ( common_list_list_a @ Common2 ) )
= ( common_list_list_a @ Common ) ) ) ) ) ).
% Common_Proof.pop_list
thf(fact_258_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_259_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_260_insert__Nil,axiom,
! [X: list_int] :
( ( insert_list_int @ X @ nil_list_int )
= ( cons_list_int @ X @ nil_list_int ) ) ).
% insert_Nil
thf(fact_261_insert__Nil,axiom,
! [X: list_a] :
( ( insert_list_a @ X @ nil_list_a )
= ( cons_list_a @ X @ nil_list_a ) ) ).
% insert_Nil
thf(fact_262_insert__Nil,axiom,
! [X: a] :
( ( insert_a @ X @ nil_a )
= ( cons_a @ X @ nil_a ) ) ).
% insert_Nil
thf(fact_263_insert__Nil,axiom,
! [X: int] :
( ( insert_int @ X @ nil_int )
= ( cons_int @ X @ nil_int ) ) ).
% insert_Nil
thf(fact_264_Current__Proof_Opush__list,axiom,
! [X: list_int,Current: current_list_int] :
( ( curren8299908649409640978st_int @ ( push_list_int2 @ X @ Current ) )
= ( cons_list_int @ X @ ( curren8299908649409640978st_int @ Current ) ) ) ).
% Current_Proof.push_list
thf(fact_265_Current__Proof_Opush__list,axiom,
! [X: list_a,Current: current_list_a] :
( ( current_list_list_a @ ( push_list_a2 @ X @ Current ) )
= ( cons_list_a @ X @ ( current_list_list_a @ Current ) ) ) ).
% Current_Proof.push_list
thf(fact_266_Current__Proof_Opush__list,axiom,
! [X: a,Current: current_a] :
( ( current_list_a2 @ ( push_a2 @ X @ Current ) )
= ( cons_a @ X @ ( current_list_a2 @ Current ) ) ) ).
% Current_Proof.push_list
thf(fact_267_Current__Proof_Opush__list,axiom,
! [X: int,Current: current_int] :
( ( current_list_int2 @ ( push_int2 @ X @ Current ) )
= ( cons_int @ X @ ( current_list_int2 @ Current ) ) ) ).
% Current_Proof.push_list
thf(fact_268_nths__singleton,axiom,
! [A2: set_nat,X: list_int] :
( ( ( member_nat @ zero_zero_nat @ A2 )
=> ( ( nths_list_int @ ( cons_list_int @ X @ nil_list_int ) @ A2 )
= ( cons_list_int @ X @ nil_list_int ) ) )
& ( ~ ( member_nat @ zero_zero_nat @ A2 )
=> ( ( nths_list_int @ ( cons_list_int @ X @ nil_list_int ) @ A2 )
= nil_list_int ) ) ) ).
% nths_singleton
thf(fact_269_nths__singleton,axiom,
! [A2: set_nat,X: list_a] :
( ( ( member_nat @ zero_zero_nat @ A2 )
=> ( ( nths_list_a @ ( cons_list_a @ X @ nil_list_a ) @ A2 )
= ( cons_list_a @ X @ nil_list_a ) ) )
& ( ~ ( member_nat @ zero_zero_nat @ A2 )
=> ( ( nths_list_a @ ( cons_list_a @ X @ nil_list_a ) @ A2 )
= nil_list_a ) ) ) ).
% nths_singleton
thf(fact_270_nths__singleton,axiom,
! [A2: set_nat,X: a] :
( ( ( member_nat @ zero_zero_nat @ A2 )
=> ( ( nths_a @ ( cons_a @ X @ nil_a ) @ A2 )
= ( cons_a @ X @ nil_a ) ) )
& ( ~ ( member_nat @ zero_zero_nat @ A2 )
=> ( ( nths_a @ ( cons_a @ X @ nil_a ) @ A2 )
= nil_a ) ) ) ).
% nths_singleton
thf(fact_271_nths__singleton,axiom,
! [A2: set_nat,X: int] :
( ( ( member_nat @ zero_zero_nat @ A2 )
=> ( ( nths_int @ ( cons_int @ X @ nil_int ) @ A2 )
= ( cons_int @ X @ nil_int ) ) )
& ( ~ ( member_nat @ zero_zero_nat @ A2 )
=> ( ( nths_int @ ( cons_int @ X @ nil_int ) @ A2 )
= nil_int ) ) ) ).
% nths_singleton
thf(fact_272_pop__list__current,axiom,
! [Common: state_list_int,X: list_int,Common2: state_list_int] :
( ( type_i5837482050862709779st_int @ Common )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_s6508937059447799164st_int @ Common ) )
=> ( ( ( pop_list_int @ Common )
= ( produc9042490306465745403st_int @ X @ Common2 ) )
=> ( ( cons_list_int @ X @ ( common7659038223531881259st_int @ Common2 ) )
= ( common7659038223531881259st_int @ Common ) ) ) ) ) ).
% pop_list_current
thf(fact_273_pop__list__current,axiom,
! [Common: state_list_a,X: list_a,Common2: state_list_a] :
( ( type_i5563415093623196931list_a @ Common )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_s2742192117134447578list_a @ Common ) )
=> ( ( ( pop_list_a @ Common )
= ( produc2216564154414103411list_a @ X @ Common2 ) )
=> ( ( cons_list_a @ X @ ( common3845368160371929429list_a @ Common2 ) )
= ( common3845368160371929429list_a @ Common ) ) ) ) ) ).
% pop_list_current
thf(fact_274_pop__list__current,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 @ ( common8573843843682093467nt_int @ Common2 ) )
= ( common8573843843682093467nt_int @ Common ) ) ) ) ) ).
% pop_list_current
thf(fact_275_pop__list__current,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 @ ( common1102728217005306191rent_a @ Common2 ) )
= ( common1102728217005306191rent_a @ Common ) ) ) ) ) ).
% pop_list_current
thf(fact_276_list__ex1__simps_I1_J,axiom,
! [P: list_int > $o] :
~ ( list_ex1_list_int @ P @ nil_list_int ) ).
% list_ex1_simps(1)
thf(fact_277_list__ex1__simps_I1_J,axiom,
! [P: int > $o] :
~ ( list_ex1_int @ P @ nil_int ) ).
% list_ex1_simps(1)
thf(fact_278_list__ex1__simps_I1_J,axiom,
! [P: list_a > $o] :
~ ( list_ex1_list_a @ P @ nil_list_a ) ).
% list_ex1_simps(1)
thf(fact_279_list__ex1__simps_I1_J,axiom,
! [P: a > $o] :
~ ( list_ex1_a @ P @ nil_a ) ).
% list_ex1_simps(1)
thf(fact_280_nths__nil,axiom,
! [A2: set_nat] :
( ( nths_list_int @ nil_list_int @ A2 )
= nil_list_int ) ).
% nths_nil
thf(fact_281_nths__nil,axiom,
! [A2: set_nat] :
( ( nths_int @ nil_int @ A2 )
= nil_int ) ).
% nths_nil
thf(fact_282_nths__nil,axiom,
! [A2: set_nat] :
( ( nths_list_a @ nil_list_a @ A2 )
= nil_list_a ) ).
% nths_nil
thf(fact_283_nths__nil,axiom,
! [A2: set_nat] :
( ( nths_a @ nil_a @ A2 )
= nil_a ) ).
% nths_nil
thf(fact_284_zero__integer_Orsp,axiom,
zero_zero_int = zero_zero_int ).
% zero_integer.rsp
thf(fact_285_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_a] :
( ( size_size_list_a @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_286_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_int] :
( ( size_size_list_int @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_287_neq__if__length__neq,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( size_size_list_a @ Xs )
!= ( size_size_list_a @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_288_neq__if__length__neq,axiom,
! [Xs: list_int,Ys: list_int] :
( ( ( size_size_list_int @ Xs )
!= ( size_size_list_int @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_289_int__int__eq,axiom,
! [M2: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M2 )
= ( semiri1314217659103216013at_int @ N ) )
= ( M2 = N ) ) ).
% int_int_eq
thf(fact_290_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_291_nths__all,axiom,
! [Xs: list_a,I: set_nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
=> ( member_nat @ I2 @ I ) )
=> ( ( nths_a @ Xs @ I )
= Xs ) ) ).
% nths_all
thf(fact_292_nths__all,axiom,
! [Xs: list_int,I: set_nat] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
=> ( member_nat @ I2 @ I ) )
=> ( ( nths_int @ Xs @ I )
= Xs ) ) ).
% nths_all
thf(fact_293_Common__Proof_Oinvar__pop,axiom,
! [Common: state_int,X: int,Common2: state_int] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_int @ Common ) )
=> ( ( type_i1727439852002712835te_int @ Common )
=> ( ( ( pop_int @ Common )
= ( produc3637740352169753595te_int @ X @ Common2 ) )
=> ( type_i1727439852002712835te_int @ Common2 ) ) ) ) ).
% Common_Proof.invar_pop
thf(fact_294_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_295_n__lists_Osimps_I1_J,axiom,
! [Xs: list_list_int] :
( ( n_lists_list_int @ zero_zero_nat @ Xs )
= ( cons_list_list_int @ nil_list_int @ nil_list_list_int ) ) ).
% n_lists.simps(1)
thf(fact_296_n__lists_Osimps_I1_J,axiom,
! [Xs: list_list_a] :
( ( n_lists_list_a @ zero_zero_nat @ Xs )
= ( cons_list_list_a @ nil_list_a @ nil_list_list_a ) ) ).
% n_lists.simps(1)
thf(fact_297_n__lists_Osimps_I1_J,axiom,
! [Xs: list_int] :
( ( n_lists_int @ zero_zero_nat @ Xs )
= ( cons_list_int @ nil_int @ nil_list_int ) ) ).
% n_lists.simps(1)
thf(fact_298_n__lists_Osimps_I1_J,axiom,
! [Xs: list_a] :
( ( n_lists_a @ zero_zero_nat @ Xs )
= ( cons_list_a @ nil_a @ nil_list_a ) ) ).
% n_lists.simps(1)
thf(fact_299_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C2: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_300_list__current_Osimps_I2_J,axiom,
! [Current: current_int,Uv: list_int,Uw: list_int,Ux: nat] :
( ( common8573843843682093467nt_int @ ( copy_int @ Current @ Uv @ Uw @ Ux ) )
= ( current_list_int2 @ Current ) ) ).
% list_current.simps(2)
thf(fact_301_list__current_Osimps_I2_J,axiom,
! [Current: current_a,Uv: list_a,Uw: list_a,Ux: nat] :
( ( common1102728217005306191rent_a @ ( copy_a @ Current @ Uv @ Uw @ Ux ) )
= ( current_list_a2 @ Current ) ) ).
% list_current.simps(2)
thf(fact_302_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_303_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_304_old_Oprod_Oinject,axiom,
! [A: a > a > $o,B: list_a,A5: a > a > $o,B4: list_a] :
( ( ( produc8111569692950616493list_a @ A @ B )
= ( produc8111569692950616493list_a @ A5 @ B4 ) )
= ( ( A = A5 )
& ( B = B4 ) ) ) ).
% old.prod.inject
thf(fact_305_old_Oprod_Oinject,axiom,
! [A: a,B: current_a,A5: a,B4: current_a] :
( ( ( produc8503237746132909001rent_a @ A @ B )
= ( produc8503237746132909001rent_a @ A5 @ B4 ) )
= ( ( A = A5 )
& ( B = B4 ) ) ) ).
% old.prod.inject
thf(fact_306_old_Oprod_Oinject,axiom,
! [A: a,B: state_a,A5: a,B4: state_a] :
( ( ( produc8263595898873874535tate_a @ A @ B )
= ( produc8263595898873874535tate_a @ A5 @ B4 ) )
= ( ( A = A5 )
& ( B = B4 ) ) ) ).
% old.prod.inject
thf(fact_307_old_Oprod_Oinject,axiom,
! [A: a,B: int,A5: a,B4: int] :
( ( ( product_Pair_a_int @ A @ B )
= ( product_Pair_a_int @ A5 @ B4 ) )
= ( ( A = A5 )
& ( B = B4 ) ) ) ).
% old.prod.inject
thf(fact_308_old_Oprod_Oinject,axiom,
! [A: a,B: a,A5: a,B4: a] :
( ( ( product_Pair_a_a @ A @ B )
= ( product_Pair_a_a @ A5 @ B4 ) )
= ( ( A = A5 )
& ( B = B4 ) ) ) ).
% old.prod.inject
thf(fact_309_prod_Oinject,axiom,
! [X1: a > a > $o,X23: list_a,Y1: a > a > $o,Y23: list_a] :
( ( ( produc8111569692950616493list_a @ X1 @ X23 )
= ( produc8111569692950616493list_a @ Y1 @ Y23 ) )
= ( ( X1 = Y1 )
& ( X23 = Y23 ) ) ) ).
% prod.inject
thf(fact_310_prod_Oinject,axiom,
! [X1: a,X23: current_a,Y1: a,Y23: current_a] :
( ( ( produc8503237746132909001rent_a @ X1 @ X23 )
= ( produc8503237746132909001rent_a @ Y1 @ Y23 ) )
= ( ( X1 = Y1 )
& ( X23 = Y23 ) ) ) ).
% prod.inject
thf(fact_311_prod_Oinject,axiom,
! [X1: a,X23: state_a,Y1: a,Y23: state_a] :
( ( ( produc8263595898873874535tate_a @ X1 @ X23 )
= ( produc8263595898873874535tate_a @ Y1 @ Y23 ) )
= ( ( X1 = Y1 )
& ( X23 = Y23 ) ) ) ).
% prod.inject
thf(fact_312_prod_Oinject,axiom,
! [X1: a,X23: int,Y1: a,Y23: int] :
( ( ( product_Pair_a_int @ X1 @ X23 )
= ( product_Pair_a_int @ Y1 @ Y23 ) )
= ( ( X1 = Y1 )
& ( X23 = Y23 ) ) ) ).
% prod.inject
thf(fact_313_prod_Oinject,axiom,
! [X1: a,X23: a,Y1: a,Y23: a] :
( ( ( product_Pair_a_a @ X1 @ X23 )
= ( product_Pair_a_a @ Y1 @ Y23 ) )
= ( ( X1 = Y1 )
& ( X23 = Y23 ) ) ) ).
% prod.inject
thf(fact_314_product__lists_Osimps_I1_J,axiom,
( ( produc2606055431807726861st_int @ nil_list_list_int )
= ( cons_list_list_int @ nil_list_int @ nil_list_list_int ) ) ).
% product_lists.simps(1)
thf(fact_315_product__lists_Osimps_I1_J,axiom,
( ( product_lists_list_a @ nil_list_list_a )
= ( cons_list_list_a @ nil_list_a @ nil_list_list_a ) ) ).
% product_lists.simps(1)
thf(fact_316_product__lists_Osimps_I1_J,axiom,
( ( product_lists_int @ nil_list_int )
= ( cons_list_int @ nil_int @ nil_list_int ) ) ).
% product_lists.simps(1)
thf(fact_317_product__lists_Osimps_I1_J,axiom,
( ( product_lists_a @ nil_list_a )
= ( cons_list_a @ nil_a @ nil_list_a ) ) ).
% product_lists.simps(1)
thf(fact_318_in__measures_I2_J,axiom,
! [X: a,Y: a,F: a > nat,Fs: list_a_nat] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( measures_a @ ( cons_a_nat @ F @ Fs ) ) )
= ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
| ( ( ( F @ X )
= ( F @ Y ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( measures_a @ Fs ) ) ) ) ) ).
% in_measures(2)
thf(fact_319_Cons__lenlex__iff,axiom,
! [M2: list_int,Ms: list_list_int,N: list_int,Ns: list_list_int,R: set_Pr765067013931698361st_int] :
( ( member3583055517200259234st_int @ ( produc4355665770860423473st_int @ ( cons_list_int @ M2 @ Ms ) @ ( cons_list_int @ N @ Ns ) ) @ ( lenlex_list_int @ R ) )
= ( ( ord_less_nat @ ( size_s533118279054570080st_int @ Ms ) @ ( size_s533118279054570080st_int @ Ns ) )
| ( ( ( size_s533118279054570080st_int @ Ms )
= ( size_s533118279054570080st_int @ Ns ) )
& ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ M2 @ N ) @ R ) )
| ( ( M2 = N )
& ( member3583055517200259234st_int @ ( produc4355665770860423473st_int @ Ms @ Ns ) @ ( lenlex_list_int @ R ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_320_Cons__lenlex__iff,axiom,
! [M2: list_a,Ms: list_list_a,N: list_a,Ns: list_list_a,R: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( cons_list_a @ M2 @ Ms ) @ ( cons_list_a @ N @ Ns ) ) @ ( lenlex_list_a @ R ) )
= ( ( ord_less_nat @ ( size_s349497388124573686list_a @ Ms ) @ ( size_s349497388124573686list_a @ Ns ) )
| ( ( ( size_s349497388124573686list_a @ Ms )
= ( size_s349497388124573686list_a @ Ns ) )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ M2 @ N ) @ R ) )
| ( ( M2 = N )
& ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Ms @ Ns ) @ ( lenlex_list_a @ R ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_321_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_322_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_323_in__measures_I1_J,axiom,
! [X: a,Y: a] :
~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( measures_a @ nil_a_nat ) ) ).
% in_measures(1)
thf(fact_324_Nil__lenlex__iff1,axiom,
! [Ns: list_list_int,R: set_Pr765067013931698361st_int] :
( ( member3583055517200259234st_int @ ( produc4355665770860423473st_int @ nil_list_int @ Ns ) @ ( lenlex_list_int @ R ) )
= ( Ns != nil_list_int ) ) ).
% Nil_lenlex_iff1
thf(fact_325_Nil__lenlex__iff1,axiom,
! [Ns: list_int,R: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ nil_int @ Ns ) @ ( lenlex_int @ R ) )
= ( Ns != nil_int ) ) ).
% Nil_lenlex_iff1
thf(fact_326_Nil__lenlex__iff1,axiom,
! [Ns: list_list_a,R: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ nil_list_a @ Ns ) @ ( lenlex_list_a @ R ) )
= ( Ns != nil_list_a ) ) ).
% Nil_lenlex_iff1
thf(fact_327_Nil__lenlex__iff1,axiom,
! [Ns: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ nil_a @ Ns ) @ ( lenlex_a @ R ) )
= ( Ns != nil_a ) ) ).
% Nil_lenlex_iff1
thf(fact_328_Nil__lenlex__iff2,axiom,
! [Ns: list_a,R: set_Product_prod_a_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ns @ nil_a ) @ ( lenlex_a @ R ) ) ).
% Nil_lenlex_iff2
thf(fact_329_sorted__wrt_Ocases,axiom,
! [X: produc5032551385658279741list_a] :
( ! [P4: a > a > $o] :
( X
!= ( produc8111569692950616493list_a @ P4 @ nil_a ) )
=> ~ ! [P4: a > a > $o,X3: a,Ys2: list_a] :
( X
!= ( produc8111569692950616493list_a @ P4 @ ( cons_a @ X3 @ Ys2 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_330_sorted__wrt_Ocases,axiom,
! [X: produc5834231552977413017st_int] :
( ! [P4: int > int > $o] :
( X
!= ( produc8618682346314911123st_int @ P4 @ nil_int ) )
=> ~ ! [P4: int > int > $o,X3: int,Ys2: list_int] :
( X
!= ( produc8618682346314911123st_int @ P4 @ ( cons_int @ X3 @ Ys2 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_331_successively_Ocases,axiom,
! [X: produc5032551385658279741list_a] :
( ! [P4: a > a > $o] :
( X
!= ( produc8111569692950616493list_a @ P4 @ nil_a ) )
=> ( ! [P4: a > a > $o,X3: a] :
( X
!= ( produc8111569692950616493list_a @ P4 @ ( cons_a @ X3 @ nil_a ) ) )
=> ~ ! [P4: a > a > $o,X3: a,Y2: a,Xs2: list_a] :
( X
!= ( produc8111569692950616493list_a @ P4 @ ( cons_a @ X3 @ ( cons_a @ Y2 @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_332_successively_Ocases,axiom,
! [X: produc5834231552977413017st_int] :
( ! [P4: int > int > $o] :
( X
!= ( produc8618682346314911123st_int @ P4 @ nil_int ) )
=> ( ! [P4: int > int > $o,X3: int] :
( X
!= ( produc8618682346314911123st_int @ P4 @ ( cons_int @ X3 @ nil_int ) ) )
=> ~ ! [P4: int > int > $o,X3: int,Y2: int,Xs2: list_int] :
( X
!= ( produc8618682346314911123st_int @ P4 @ ( cons_int @ X3 @ ( cons_int @ Y2 @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_333_splice_Ocases,axiom,
! [X: produc9164743771328383783list_a] :
( ! [Ys2: list_a] :
( X
!= ( produc6837034575241423639list_a @ nil_a @ Ys2 ) )
=> ~ ! [X3: a,Xs2: list_a,Ys2: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ X3 @ Xs2 ) @ Ys2 ) ) ) ).
% splice.cases
thf(fact_334_splice_Ocases,axiom,
! [X: produc1186641810826059865st_int] :
( ! [Ys2: list_int] :
( X
!= ( produc364263696895485585st_int @ nil_int @ Ys2 ) )
=> ~ ! [X3: int,Xs2: list_int,Ys2: list_int] :
( X
!= ( produc364263696895485585st_int @ ( cons_int @ X3 @ Xs2 ) @ Ys2 ) ) ) ).
% splice.cases
thf(fact_335_shuffles_Ocases,axiom,
! [X: produc9164743771328383783list_a] :
( ! [Ys2: list_a] :
( X
!= ( produc6837034575241423639list_a @ nil_a @ Ys2 ) )
=> ( ! [Xs2: list_a] :
( X
!= ( produc6837034575241423639list_a @ Xs2 @ nil_a ) )
=> ~ ! [X3: a,Xs2: list_a,Y2: a,Ys2: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ X3 @ Xs2 ) @ ( cons_a @ Y2 @ Ys2 ) ) ) ) ) ).
% shuffles.cases
thf(fact_336_shuffles_Ocases,axiom,
! [X: produc1186641810826059865st_int] :
( ! [Ys2: list_int] :
( X
!= ( produc364263696895485585st_int @ nil_int @ Ys2 ) )
=> ( ! [Xs2: list_int] :
( X
!= ( produc364263696895485585st_int @ Xs2 @ nil_int ) )
=> ~ ! [X3: int,Xs2: list_int,Y2: int,Ys2: list_int] :
( X
!= ( produc364263696895485585st_int @ ( cons_int @ X3 @ Xs2 ) @ ( cons_int @ Y2 @ Ys2 ) ) ) ) ) ).
% shuffles.cases
thf(fact_337_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_338_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
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,Z3: nat] : ( Y5 = Z3 ) )
= ( ^ [A4: nat,B3: nat] :
( ( semiri1314217659103216013at_int @ A4 )
= ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_341_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_342_Current__Proof_Oinvar__pop,axiom,
! [Current: current_a,X: a,Current2: current_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_current_a @ Current ) )
=> ( ( type_i6141643110573041459rent_a @ Current )
=> ( ( ( pop_a2 @ Current )
= ( produc8503237746132909001rent_a @ X @ Current2 ) )
=> ( type_i6141643110573041459rent_a @ Current2 ) ) ) ) ).
% Current_Proof.invar_pop
thf(fact_343_subseqs_Osimps_I1_J,axiom,
( ( subseqs_a @ nil_a )
= ( cons_list_a @ nil_a @ nil_list_a ) ) ).
% subseqs.simps(1)
thf(fact_344_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_345_listrel_Ocases,axiom,
! [A1: list_a,A22: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ A1 @ A22 ) @ ( listrel_a_a @ R ) )
=> ( ( ( A1 = nil_a )
=> ( A22 != nil_a ) )
=> ~ ! [X3: a,Y2: a,Xs2: list_a] :
( ( A1
= ( cons_a @ X3 @ Xs2 ) )
=> ! [Ys2: list_a] :
( ( A22
= ( cons_a @ Y2 @ Ys2 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y2 ) @ R )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs2 @ Ys2 ) @ ( listrel_a_a @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_346_listrel_Ocases,axiom,
! [A1: list_a,A22: list_int,R: set_Pr2444352267426396943_a_int] :
( ( member2361055629902307206st_int @ ( produc615098764691696873st_int @ A1 @ A22 ) @ ( listrel_a_int @ R ) )
=> ( ( ( A1 = nil_a )
=> ( A22 != nil_int ) )
=> ~ ! [X3: a,Y2: int,Xs2: list_a] :
( ( A1
= ( cons_a @ X3 @ Xs2 ) )
=> ! [Ys2: list_int] :
( ( A22
= ( cons_int @ Y2 @ Ys2 ) )
=> ( ( member1546337568877222000_a_int @ ( product_Pair_a_int @ X3 @ Y2 ) @ R )
=> ~ ( member2361055629902307206st_int @ ( produc615098764691696873st_int @ Xs2 @ Ys2 ) @ ( listrel_a_int @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_347_listrel_Ocases,axiom,
! [A1: list_int,A22: list_a,R: set_Pr4265292341059111293_int_a] :
( ( member3048572340474253524list_a @ ( produc4840137957254204271list_a @ A1 @ A22 ) @ ( listrel_int_a @ R ) )
=> ( ( ( A1 = nil_int )
=> ( A22 != nil_a ) )
=> ~ ! [X3: int,Y2: a,Xs2: list_int] :
( ( A1
= ( cons_int @ X3 @ Xs2 ) )
=> ! [Ys2: list_a] :
( ( A22
= ( cons_a @ Y2 @ Ys2 ) )
=> ( ( member6717207084439649758_int_a @ ( product_Pair_int_a @ X3 @ Y2 ) @ R )
=> ~ ( member3048572340474253524list_a @ ( produc4840137957254204271list_a @ Xs2 @ Ys2 ) @ ( listrel_int_a @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_348_listrel_Ocases,axiom,
! [A1: list_int,A22: list_int,R: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ A1 @ A22 ) @ ( listrel_int_int @ R ) )
=> ( ( ( A1 = nil_int )
=> ( A22 != nil_int ) )
=> ~ ! [X3: int,Y2: int,Xs2: list_int] :
( ( A1
= ( cons_int @ X3 @ Xs2 ) )
=> ! [Ys2: list_int] :
( ( A22
= ( cons_int @ Y2 @ Ys2 ) )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y2 ) @ R )
=> ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs2 @ Ys2 ) @ ( listrel_int_int @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_349_listrel_Osimps,axiom,
! [A1: list_a,A22: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ A1 @ A22 ) @ ( listrel_a_a @ R ) )
= ( ( ( A1 = nil_a )
& ( A22 = nil_a ) )
| ? [X2: a,Y3: a,Xs3: list_a,Ys4: list_a] :
( ( A1
= ( cons_a @ X2 @ Xs3 ) )
& ( A22
= ( cons_a @ Y3 @ Ys4 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X2 @ Y3 ) @ R )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs3 @ Ys4 ) @ ( listrel_a_a @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_350_listrel_Osimps,axiom,
! [A1: list_a,A22: list_int,R: set_Pr2444352267426396943_a_int] :
( ( member2361055629902307206st_int @ ( produc615098764691696873st_int @ A1 @ A22 ) @ ( listrel_a_int @ R ) )
= ( ( ( A1 = nil_a )
& ( A22 = nil_int ) )
| ? [X2: a,Y3: int,Xs3: list_a,Ys4: list_int] :
( ( A1
= ( cons_a @ X2 @ Xs3 ) )
& ( A22
= ( cons_int @ Y3 @ Ys4 ) )
& ( member1546337568877222000_a_int @ ( product_Pair_a_int @ X2 @ Y3 ) @ R )
& ( member2361055629902307206st_int @ ( produc615098764691696873st_int @ Xs3 @ Ys4 ) @ ( listrel_a_int @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_351_listrel_Osimps,axiom,
! [A1: list_int,A22: list_a,R: set_Pr4265292341059111293_int_a] :
( ( member3048572340474253524list_a @ ( produc4840137957254204271list_a @ A1 @ A22 ) @ ( listrel_int_a @ R ) )
= ( ( ( A1 = nil_int )
& ( A22 = nil_a ) )
| ? [X2: int,Y3: a,Xs3: list_int,Ys4: list_a] :
( ( A1
= ( cons_int @ X2 @ Xs3 ) )
& ( A22
= ( cons_a @ Y3 @ Ys4 ) )
& ( member6717207084439649758_int_a @ ( product_Pair_int_a @ X2 @ Y3 ) @ R )
& ( member3048572340474253524list_a @ ( produc4840137957254204271list_a @ Xs3 @ Ys4 ) @ ( listrel_int_a @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_352_listrel_Osimps,axiom,
! [A1: list_int,A22: list_int,R: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ A1 @ A22 ) @ ( listrel_int_int @ R ) )
= ( ( ( A1 = nil_int )
& ( A22 = nil_int ) )
| ? [X2: int,Y3: int,Xs3: list_int,Ys4: list_int] :
( ( A1
= ( cons_int @ X2 @ Xs3 ) )
& ( A22
= ( cons_int @ Y3 @ Ys4 ) )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X2 @ Y3 ) @ R )
& ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs3 @ Ys4 ) @ ( listrel_int_int @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_353_verit__minus__simplify_I4_J,axiom,
! [B: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ B ) )
= B ) ).
% verit_minus_simplify(4)
thf(fact_354_add_Oinverse__inverse,axiom,
! [A: int] :
( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
= A ) ).
% add.inverse_inverse
thf(fact_355_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_356_neg__equal__zero,axiom,
! [A: int] :
( ( ( uminus_uminus_int @ A )
= A )
= ( A = zero_zero_int ) ) ).
% neg_equal_zero
thf(fact_357_equal__neg__zero,axiom,
! [A: int] :
( ( A
= ( uminus_uminus_int @ A ) )
= ( A = zero_zero_int ) ) ).
% equal_neg_zero
thf(fact_358_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_359_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_360_add_Oinverse__neutral,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% add.inverse_neutral
thf(fact_361_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_362_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_363_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_364_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_365_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_366_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_367_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_368_equation__minus__iff,axiom,
! [A: int,B: int] :
( ( A
= ( uminus_uminus_int @ B ) )
= ( B
= ( uminus_uminus_int @ A ) ) ) ).
% equation_minus_iff
thf(fact_369_minus__equation__iff,axiom,
! [A: int,B: int] :
( ( ( uminus_uminus_int @ A )
= B )
= ( ( uminus_uminus_int @ B )
= A ) ) ).
% minus_equation_iff
thf(fact_370_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_371_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_372_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_373_uminus__int__code_I1_J,axiom,
( ( uminus_uminus_int @ zero_zero_int )
= zero_zero_int ) ).
% uminus_int_code(1)
thf(fact_374_int__cases2,axiom,
! [Z2: int] :
( ! [N2: nat] :
( Z2
!= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ! [N2: nat] :
( Z2
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).
% int_cases2
thf(fact_375_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_376_listrel_ONil,axiom,
! [R: set_Product_prod_a_a] : ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ nil_a @ nil_a ) @ ( listrel_a_a @ R ) ) ).
% listrel.Nil
thf(fact_377_listrel__Nil1,axiom,
! [Xs: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ nil_a @ Xs ) @ ( listrel_a_a @ R ) )
=> ( Xs = nil_a ) ) ).
% listrel_Nil1
thf(fact_378_listrel__Nil2,axiom,
! [Xs: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ nil_a ) @ ( listrel_a_a @ R ) )
=> ( Xs = nil_a ) ) ).
% listrel_Nil2
thf(fact_379_gen__length__code_I1_J,axiom,
! [N: nat] :
( ( gen_length_a @ N @ nil_a )
= N ) ).
% gen_length_code(1)
thf(fact_380_Nil__notin__lex,axiom,
! [Ys: list_a,R: set_Product_prod_a_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ nil_a @ Ys ) @ ( lex_a @ R ) ) ).
% Nil_notin_lex
thf(fact_381_Nil2__notin__lex,axiom,
! [Xs: list_a,R: set_Product_prod_a_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ nil_a ) @ ( lex_a @ R ) ) ).
% Nil2_notin_lex
thf(fact_382_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_383_listrel_OCons,axiom,
! [X: int,Y: int,R: set_Pr958786334691620121nt_int,Xs: list_int,Ys: list_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R )
=> ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( listrel_int_int @ R ) )
=> ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ X @ Xs ) @ ( cons_int @ Y @ Ys ) ) @ ( listrel_int_int @ R ) ) ) ) ).
% listrel.Cons
thf(fact_384_listrel__Cons1,axiom,
! [Y: int,Ys: list_int,Xs: list_int,R: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ Y @ Ys ) @ Xs ) @ ( listrel_int_int @ R ) )
=> ~ ! [Y2: int,Ys2: list_int] :
( ( Xs
= ( cons_int @ Y2 @ Ys2 ) )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ Y @ Y2 ) @ R )
=> ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Ys @ Ys2 ) @ ( listrel_int_int @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_385_listrel__Cons2,axiom,
! [Xs: list_int,Y: int,Ys: list_int,R: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ ( cons_int @ Y @ Ys ) ) @ ( listrel_int_int @ R ) )
=> ~ ! [X3: int,Xs2: list_int] :
( ( Xs
= ( cons_int @ X3 @ Xs2 ) )
=> ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y ) @ R )
=> ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs2 @ Ys ) @ ( listrel_int_int @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_386_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_387_size__pop__suc,axiom,
! [Current: current_a,X: a,Current2: current_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_current_a @ Current ) )
=> ( ( type_i6141643110573041459rent_a @ Current )
=> ( ( ( pop_a2 @ Current )
= ( produc8503237746132909001rent_a @ X @ Current2 ) )
=> ( ( suc @ ( size_size_current_a @ Current2 ) )
= ( size_size_current_a @ Current ) ) ) ) ) ).
% size_pop_suc
thf(fact_388_invar__drop__first,axiom,
! [Current: current_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_current_a @ Current ) )
=> ( ( type_i6141643110573041459rent_a @ Current )
=> ( type_i6141643110573041459rent_a @ ( produc4695312889421393143rent_a @ ( pop_a2 @ Current ) ) ) ) ) ).
% invar_drop_first
thf(fact_389_zero__less__nat__eq,axiom,
! [Z2: int] :
( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z2 ) )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% zero_less_nat_eq
thf(fact_390_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_391_nat_Oinject,axiom,
! [X23: nat,Y23: nat] :
( ( ( suc @ X23 )
= ( suc @ Y23 ) )
= ( X23 = Y23 ) ) ).
% nat.inject
thf(fact_392_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_393_Suc__mono,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_394_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_395_nat__int,axiom,
! [N: nat] :
( ( nat2 @ ( semiri1314217659103216013at_int @ N ) )
= N ) ).
% nat_int
thf(fact_396_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_397_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_398_nth__Cons__Suc,axiom,
! [X: int,Xs: list_int,N: nat] :
( ( nth_int @ ( cons_int @ X @ Xs ) @ ( suc @ N ) )
= ( nth_int @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_399_nth__Cons__0,axiom,
! [X: int,Xs: list_int] :
( ( nth_int @ ( cons_int @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_400_Current__Proof_Osize__push,axiom,
! [X: a,Current: current_a] :
( ( size_size_current_a @ ( push_a2 @ X @ Current ) )
= ( suc @ ( size_size_current_a @ Current ) ) ) ).
% Current_Proof.size_push
thf(fact_401_zless__nat__conj,axiom,
! [W: int,Z2: int] :
( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
= ( ( ord_less_int @ zero_zero_int @ Z2 )
& ( ord_less_int @ W @ Z2 ) ) ) ).
% zless_nat_conj
thf(fact_402_negative__zless,axiom,
! [N: nat,M2: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ ( semiri1314217659103216013at_int @ M2 ) ) ).
% negative_zless
thf(fact_403_nat__zminus__int,axiom,
! [N: nat] :
( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) )
= zero_zero_nat ) ).
% nat_zminus_int
thf(fact_404_Current__Proof_Osize__pop,axiom,
! [Current: current_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_current_a @ Current ) )
=> ( ( type_i6141643110573041459rent_a @ Current )
=> ( ( suc @ ( size_size_current_a @ ( produc4695312889421393143rent_a @ ( pop_a2 @ Current ) ) ) )
= ( size_size_current_a @ Current ) ) ) ) ).
% Current_Proof.size_pop
thf(fact_405_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_406_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_407_nat_Odistinct_I1_J,axiom,
! [X23: nat] :
( zero_zero_nat
!= ( suc @ X23 ) ) ).
% nat.distinct(1)
thf(fact_408_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_409_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_410_nat_OdiscI,axiom,
! [Nat: nat,X23: nat] :
( ( Nat
= ( suc @ X23 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_411_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_412_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_413_diff__induct,axiom,
! [P: nat > nat > $o,M2: nat,N: nat] :
( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
=> ( ! [Y2: nat] : ( P @ zero_zero_nat @ ( suc @ Y2 ) )
=> ( ! [X3: nat,Y2: nat] :
( ( P @ X3 @ Y2 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y2 ) ) )
=> ( P @ M2 @ N ) ) ) ) ).
% diff_induct
thf(fact_414_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_415_Suc__neq__Zero,axiom,
! [M2: nat] :
( ( suc @ M2 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_416_Zero__neq__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_neq_Suc
thf(fact_417_Zero__not__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_not_Suc
thf(fact_418_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% not0_implies_Suc
thf(fact_419_Nat_OlessE,axiom,
! [I3: nat,K: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( ( K
!= ( suc @ I3 ) )
=> ~ ! [J: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( K
!= ( suc @ J ) ) ) ) ) ).
% Nat.lessE
thf(fact_420_Suc__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_nat @ M2 @ N ) ) ).
% Suc_lessD
thf(fact_421_Suc__lessE,axiom,
! [I3: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I3 ) @ K )
=> ~ ! [J: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( K
!= ( suc @ J ) ) ) ) ).
% Suc_lessE
thf(fact_422_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_423_less__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M2 @ N )
=> ( M2 = N ) ) ) ).
% less_SucE
thf(fact_424_less__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_425_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_426_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_427_not__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_nat @ M2 @ N ) )
= ( ord_less_nat @ N @ ( suc @ M2 ) ) ) ).
% not_less_eq
thf(fact_428_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_429_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_430_less__antisym,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less_nat @ N @ M2 )
=> ( ( ord_less_nat @ N @ ( suc @ M2 ) )
=> ( M2 = N ) ) ) ).
% less_antisym
thf(fact_431_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_432_less__trans__Suc,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ K )
=> ( ord_less_nat @ ( suc @ I3 ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_433_less__Suc__induct,axiom,
! [I3: nat,J2: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J: nat,K2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ J @ K2 )
=> ( ( P @ I2 @ J )
=> ( ( P @ J @ K2 )
=> ( P @ I2 @ K2 ) ) ) ) )
=> ( P @ I3 @ J2 ) ) ) ) ).
% less_Suc_induct
thf(fact_434_strict__inc__induct,axiom,
! [I3: nat,J2: nat,P: nat > $o] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ! [I2: nat] :
( ( J2
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I3 ) ) ) ) ).
% strict_inc_induct
thf(fact_435_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_436_nat__zero__as__int,axiom,
( zero_zero_nat
= ( nat2 @ zero_zero_int ) ) ).
% nat_zero_as_int
thf(fact_437_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_438_lift__Suc__mono__less,axiom,
! [F: nat > int,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_439_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_440_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_441_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ N ) )
!= zero_zero_nat ) ).
% of_nat_neq_0
thf(fact_442_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
!= zero_zero_int ) ).
% of_nat_neq_0
thf(fact_443_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_444_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_445_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_446_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% gr0_implies_Suc
thf(fact_447_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_448_Suc__length__conv,axiom,
! [N: nat,Xs: list_int] :
( ( ( suc @ N )
= ( size_size_list_int @ Xs ) )
= ( ? [Y3: int,Ys4: list_int] :
( ( Xs
= ( cons_int @ Y3 @ Ys4 ) )
& ( ( size_size_list_int @ Ys4 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_449_length__Suc__conv,axiom,
! [Xs: list_int,N: nat] :
( ( ( size_size_list_int @ Xs )
= ( suc @ N ) )
= ( ? [Y3: int,Ys4: list_int] :
( ( Xs
= ( cons_int @ Y3 @ Ys4 ) )
& ( ( size_size_list_int @ Ys4 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_450_int__cases,axiom,
! [Z2: int] :
( ! [N2: nat] :
( Z2
!= ( semiri1314217659103216013at_int @ N2 ) )
=> ~ ! [N2: nat] :
( Z2
!= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).
% int_cases
thf(fact_451_int__of__nat__induct,axiom,
! [P: int > $o,Z2: int] :
( ! [N2: nat] : ( P @ ( semiri1314217659103216013at_int @ N2 ) )
=> ( ! [N2: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) )
=> ( P @ Z2 ) ) ) ).
% int_of_nat_induct
thf(fact_452_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_453_nat__mono__iff,axiom,
! [Z2: int,W: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
= ( ord_less_int @ W @ Z2 ) ) ) ).
% nat_mono_iff
thf(fact_454_zless__nat__eq__int__zless,axiom,
! [M2: nat,Z2: int] :
( ( ord_less_nat @ M2 @ ( nat2 @ Z2 ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ M2 ) @ Z2 ) ) ).
% zless_nat_eq_int_zless
thf(fact_455_split__nat,axiom,
! [P: nat > $o,I3: int] :
( ( P @ ( nat2 @ I3 ) )
= ( ! [N3: nat] :
( ( I3
= ( semiri1314217659103216013at_int @ N3 ) )
=> ( P @ N3 ) )
& ( ( ord_less_int @ I3 @ zero_zero_int )
=> ( P @ zero_zero_nat ) ) ) ) ).
% split_nat
thf(fact_456_negative__zless__0,axiom,
! [N: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) @ zero_zero_int ) ).
% negative_zless_0
thf(fact_457_negD,axiom,
! [X: int] :
( ( ord_less_int @ X @ zero_zero_int )
=> ? [N2: nat] :
( X
= ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) ) ) ).
% negD
thf(fact_458_size__new__pop,axiom,
! [Current: current_a] :
( ( ord_less_nat @ zero_zero_nat @ ( type_s933026853152659577rent_a @ Current ) )
=> ( ( type_i6141643110573041459rent_a @ Current )
=> ( ( suc @ ( type_s933026853152659577rent_a @ ( produc4695312889421393143rent_a @ ( pop_a2 @ Current ) ) ) )
= ( type_s933026853152659577rent_a @ Current ) ) ) ) ).
% size_new_pop
thf(fact_459_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_460_drop__first__list,axiom,
! [Current: current_a] :
( ( type_i6141643110573041459rent_a @ Current )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_current_a @ Current ) )
=> ( ( current_list_a2 @ ( produc4695312889421393143rent_a @ ( pop_a2 @ Current ) ) )
= ( tl_a @ ( current_list_a2 @ Current ) ) ) ) ) ).
% drop_first_list
thf(fact_461_size__new__push,axiom,
! [Current: current_a,X: a] :
( ( type_i6141643110573041459rent_a @ Current )
=> ( ( type_s933026853152659577rent_a @ ( push_a2 @ X @ Current ) )
= ( suc @ ( type_s933026853152659577rent_a @ Current ) ) ) ) ).
% size_new_push
thf(fact_462_list_Osel_I3_J,axiom,
! [X21: int,X22: list_int] :
( ( tl_int @ ( cons_int @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_463_list_Osel_I2_J,axiom,
( ( tl_a @ nil_a )
= nil_a ) ).
% list.sel(2)
thf(fact_464_tl__Nil,axiom,
! [Xs: list_a] :
( ( ( tl_a @ Xs )
= nil_a )
= ( ( Xs = nil_a )
| ? [X2: a] :
( Xs
= ( cons_a @ X2 @ nil_a ) ) ) ) ).
% tl_Nil
thf(fact_465_tl__Nil,axiom,
! [Xs: list_int] :
( ( ( tl_int @ Xs )
= nil_int )
= ( ( Xs = nil_int )
| ? [X2: int] :
( Xs
= ( cons_int @ X2 @ nil_int ) ) ) ) ).
% tl_Nil
thf(fact_466_Nil__tl,axiom,
! [Xs: list_a] :
( ( nil_a
= ( tl_a @ Xs ) )
= ( ( Xs = nil_a )
| ? [X2: a] :
( Xs
= ( cons_a @ X2 @ nil_a ) ) ) ) ).
% Nil_tl
thf(fact_467_Nil__tl,axiom,
! [Xs: list_int] :
( ( nil_int
= ( tl_int @ Xs ) )
= ( ( Xs = nil_int )
| ? [X2: int] :
( Xs
= ( cons_int @ X2 @ nil_int ) ) ) ) ).
% Nil_tl
thf(fact_468_Nitpick_Osize__list__simp_I2_J,axiom,
( size_size_list_a
= ( ^ [Xs3: list_a] : ( if_nat @ ( Xs3 = nil_a ) @ zero_zero_nat @ ( suc @ ( size_size_list_a @ ( tl_a @ Xs3 ) ) ) ) ) ) ).
% Nitpick.size_list_simp(2)
thf(fact_469_Current__Proof_Opop__list,axiom,
! [Current: current_int] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s825146100745393828nt_int @ Current ) )
=> ( ( type_i7078759447853081037nt_int @ Current )
=> ( ( cons_int @ ( produc5847638466952691905nt_int @ ( pop_int2 @ Current ) ) @ ( tl_int @ ( current_list_int2 @ Current ) ) )
= ( current_list_int2 @ Current ) ) ) ) ).
% Current_Proof.pop_list
thf(fact_470_Current__Proof_Opop__list,axiom,
! [Current: current_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_current_a @ Current ) )
=> ( ( type_i6141643110573041459rent_a @ Current )
=> ( ( cons_a @ ( produc4952273589686483381rent_a @ ( pop_a2 @ Current ) ) @ ( tl_a @ ( current_list_a2 @ Current ) ) )
= ( current_list_a2 @ Current ) ) ) ) ).
% Current_Proof.pop_list
thf(fact_471_cons__tl,axiom,
! [X: int,Xs: list_int,Ys: list_int] :
( ( ( cons_int @ X @ Xs )
= Ys )
=> ( Xs
= ( tl_int @ Ys ) ) ) ).
% cons_tl
thf(fact_472_size__new__state_Osimps_I2_J,axiom,
! [Current: current_a,Uv: list_a,Uw: list_a,Ux: nat] :
( ( type_s8424385952999958455tate_a @ ( copy_a @ Current @ Uv @ Uw @ Ux ) )
= ( type_s933026853152659577rent_a @ Current ) ) ).
% size_new_state.simps(2)
thf(fact_473_one__less__nat__eq,axiom,
! [Z2: int] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z2 ) )
= ( ord_less_int @ one_one_int @ Z2 ) ) ).
% one_less_nat_eq
thf(fact_474_Common_Opop_Osimps_I2_J,axiom,
! [Current: current_a,Aux: list_a,New: list_a,Moved: nat] :
( ( pop_a @ ( copy_a @ Current @ Aux @ New @ Moved ) )
= ( produc8263595898873874535tate_a @ ( first_a @ Current ) @ ( normalize_a @ ( copy_a @ ( produc4695312889421393143rent_a @ ( pop_a2 @ Current ) ) @ Aux @ New @ Moved ) ) ) ) ).
% Common.pop.simps(2)
thf(fact_475_of__nat__1,axiom,
( ( semiri1316708129612266289at_nat @ one_one_nat )
= one_one_nat ) ).
% of_nat_1
thf(fact_476_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_477_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_478_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_479_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_480_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_481_nat__1,axiom,
( ( nat2 @ one_one_int )
= ( suc @ zero_zero_nat ) ) ).
% nat_1
thf(fact_482_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_483_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_484_one__integer_Orsp,axiom,
one_one_int = one_one_int ).
% one_integer.rsp
thf(fact_485_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_486_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_487_one__neq__neg__one,axiom,
( one_one_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% one_neq_neg_one
thf(fact_488_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_489_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_490_zero__neq__neg__one,axiom,
( zero_zero_int
!= ( uminus_uminus_int @ one_one_int ) ) ).
% zero_neq_neg_one
thf(fact_491_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_492_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_493_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_494_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_495_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_496_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_497_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_498_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_499_dbl__dec__simps_I2_J,axiom,
( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
= ( uminus_uminus_int @ one_one_int ) ) ).
% dbl_dec_simps(2)
thf(fact_500_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_501_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_502_dbl__dec__simps_I3_J,axiom,
( ( neg_nu3811975205180677377ec_int @ one_one_int )
= one_one_int ) ).
% dbl_dec_simps(3)
thf(fact_503_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_504_one__natural_Orsp,axiom,
one_one_nat = one_one_nat ).
% one_natural.rsp
thf(fact_505_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_506_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_507_nat__one__as__int,axiom,
( one_one_nat
= ( nat2 @ one_one_int ) ) ).
% nat_one_as_int
thf(fact_508_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_509_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_510_size__drop__first__sub,axiom,
! [Current: current_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_current_a @ Current ) )
=> ( ( type_i6141643110573041459rent_a @ Current )
=> ( ( size_size_current_a @ ( produc4695312889421393143rent_a @ ( pop_a2 @ Current ) ) )
= ( minus_minus_nat @ ( size_size_current_a @ Current ) @ one_one_nat ) ) ) ) ).
% size_drop_first_sub
thf(fact_511_Current__Proof_Osize__pop__sub,axiom,
! [Current: current_a,X: a,Current2: current_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_current_a @ Current ) )
=> ( ( type_i6141643110573041459rent_a @ Current )
=> ( ( ( pop_a2 @ Current )
= ( produc8503237746132909001rent_a @ X @ Current2 ) )
=> ( ( size_size_current_a @ Current2 )
= ( minus_minus_nat @ ( size_size_current_a @ Current ) @ one_one_nat ) ) ) ) ) ).
% Current_Proof.size_pop_sub
thf(fact_512_nth__Cons__pos,axiom,
! [N: nat,X: int,Xs: list_int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_int @ ( cons_int @ X @ Xs ) @ N )
= ( nth_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_513_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_514_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
= one_one_int ) ).
% dbl_inc_simps(2)
thf(fact_515_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_516_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_517_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_518_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_519_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_520_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_521_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_522_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_523_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_524_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_525_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_526_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_527_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_528_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_529_diff__0,axiom,
! [A: int] :
( ( minus_minus_int @ zero_zero_int @ A )
= ( uminus_uminus_int @ A ) ) ).
% diff_0
thf(fact_530_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_531_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_532_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_533_take__Suc__Cons,axiom,
! [N: nat,X: int,Xs: list_int] :
( ( take_int @ ( suc @ N ) @ ( cons_int @ X @ Xs ) )
= ( cons_int @ X @ ( take_int @ N @ Xs ) ) ) ).
% take_Suc_Cons
thf(fact_534_take0,axiom,
( ( take_a @ zero_zero_nat )
= ( ^ [Xs3: list_a] : nil_a ) ) ).
% take0
thf(fact_535_take__eq__Nil,axiom,
! [N: nat,Xs: list_a] :
( ( ( take_a @ N @ Xs )
= nil_a )
= ( ( N = zero_zero_nat )
| ( Xs = nil_a ) ) ) ).
% take_eq_Nil
thf(fact_536_take__eq__Nil2,axiom,
! [N: nat,Xs: list_a] :
( ( nil_a
= ( take_a @ N @ Xs ) )
= ( ( N = zero_zero_nat )
| ( Xs = nil_a ) ) ) ).
% take_eq_Nil2
thf(fact_537_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_538_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_539_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_540_diff__commute,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J2 ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I3 @ K ) @ J2 ) ) ).
% diff_commute
thf(fact_541_diff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_542_diff__right__commute,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
= ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_543_diff__eq__diff__eq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_544_take__Nil,axiom,
! [N: nat] :
( ( take_a @ N @ nil_a )
= nil_a ) ).
% take_Nil
thf(fact_545_eq__iff__diff__eq__0,axiom,
( ( ^ [Y5: int,Z3: int] : ( Y5 = Z3 ) )
= ( ^ [A4: int,B3: int] :
( ( minus_minus_int @ A4 @ B3 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_546_diff__strict__mono,axiom,
! [A: int,B: int,D: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ D @ C )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_547_diff__eq__diff__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_int @ A @ B )
= ( ord_less_int @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_548_diff__strict__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_549_diff__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_550_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_551_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I3: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus_nat @ K @ I3 ) ) ) ) ).
% zero_induct_lemma
thf(fact_552_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% minus_nat.diff_0
thf(fact_553_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_554_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_555_less__imp__diff__less,axiom,
! [J2: nat,K: nat,N: nat] :
( ( ord_less_nat @ J2 @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J2 @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_556_take__Cons_H,axiom,
! [N: nat,X: a,Xs: list_a] :
( ( ( N = zero_zero_nat )
=> ( ( take_a @ N @ ( cons_a @ X @ Xs ) )
= nil_a ) )
& ( ( N != zero_zero_nat )
=> ( ( take_a @ N @ ( cons_a @ X @ Xs ) )
= ( cons_a @ X @ ( take_a @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs ) ) ) ) ) ).
% take_Cons'
thf(fact_557_take__Cons_H,axiom,
! [N: nat,X: int,Xs: list_int] :
( ( ( N = zero_zero_nat )
=> ( ( take_int @ N @ ( cons_int @ X @ Xs ) )
= nil_int ) )
& ( ( N != zero_zero_nat )
=> ( ( take_int @ N @ ( cons_int @ X @ Xs ) )
= ( cons_int @ X @ ( take_int @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs ) ) ) ) ) ).
% take_Cons'
thf(fact_558_take__0,axiom,
! [Xs: list_a] :
( ( take_a @ zero_zero_nat @ Xs )
= nil_a ) ).
% take_0
thf(fact_559_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A4: int,B3: int] : ( ord_less_int @ ( minus_minus_int @ A4 @ B3 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_560_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_561_diff__less__Suc,axiom,
! [M2: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ ( suc @ M2 ) ) ).
% diff_less_Suc
thf(fact_562_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_563_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_564_diff__Suc__less,axiom,
! [N: nat,I3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I3 ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_565_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_566_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_567_nth__Cons_H,axiom,
! [N: nat,X: int,Xs: list_int] :
( ( ( N = zero_zero_nat )
=> ( ( nth_int @ ( cons_int @ X @ Xs ) @ N )
= X ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_int @ ( cons_int @ X @ Xs ) @ N )
= ( nth_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_568_nth__non__equal__first__eq,axiom,
! [X: int,Y: int,Xs: list_int,N: nat] :
( ( X != Y )
=> ( ( ( nth_int @ ( cons_int @ X @ Xs ) @ N )
= Y )
= ( ( ( nth_int @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_569_take__Suc__conv__app__nth,axiom,
! [I3: nat,Xs: list_a] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
=> ( ( take_a @ ( suc @ I3 ) @ Xs )
= ( append_a @ ( take_a @ I3 @ Xs ) @ ( cons_a @ ( nth_a @ Xs @ I3 ) @ nil_a ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_570_take__Suc__conv__app__nth,axiom,
! [I3: nat,Xs: list_int] :
( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
=> ( ( take_int @ ( suc @ I3 ) @ Xs )
= ( append_int @ ( take_int @ I3 @ Xs ) @ ( cons_int @ ( nth_int @ Xs @ I3 ) @ nil_int ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_571_take__hd,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( take_a @ ( suc @ zero_zero_nat ) @ Xs )
= ( cons_a @ ( hd_a @ Xs ) @ nil_a ) ) ) ).
% take_hd
thf(fact_572_take__hd,axiom,
! [Xs: list_int] :
( ( Xs != nil_int )
=> ( ( take_int @ ( suc @ zero_zero_nat ) @ Xs )
= ( cons_int @ ( hd_int @ Xs ) @ nil_int ) ) ) ).
% take_hd
thf(fact_573_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_574_last__conv__nth,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( last_a @ Xs )
= ( nth_a @ Xs @ ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) ) ) ) ).
% last_conv_nth
thf(fact_575_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_576_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_577_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_578_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_579_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_580_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_581_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_582_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_583_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_584_append__is__Nil__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= nil_a )
= ( ( Xs = nil_a )
& ( Ys = nil_a ) ) ) ).
% append_is_Nil_conv
thf(fact_585_Nil__is__append__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( nil_a
= ( append_a @ Xs @ Ys ) )
= ( ( Xs = nil_a )
& ( Ys = nil_a ) ) ) ).
% Nil_is_append_conv
thf(fact_586_self__append__conv2,axiom,
! [Y: list_a,Xs: list_a] :
( ( Y
= ( append_a @ Xs @ Y ) )
= ( Xs = nil_a ) ) ).
% self_append_conv2
thf(fact_587_append__self__conv2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Ys )
= ( Xs = nil_a ) ) ).
% append_self_conv2
thf(fact_588_self__append__conv,axiom,
! [Y: list_a,Ys: list_a] :
( ( Y
= ( append_a @ Y @ Ys ) )
= ( Ys = nil_a ) ) ).
% self_append_conv
thf(fact_589_append__self__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Xs )
= ( Ys = nil_a ) ) ).
% append_self_conv
thf(fact_590_append__Nil2,axiom,
! [Xs: list_a] :
( ( append_a @ Xs @ nil_a )
= Xs ) ).
% append_Nil2
thf(fact_591_append_Oright__neutral,axiom,
! [A: list_a] :
( ( append_a @ A @ nil_a )
= A ) ).
% append.right_neutral
thf(fact_592_of__int__diff,axiom,
! [W: int,Z2: int] :
( ( ring_1_of_int_int @ ( minus_minus_int @ W @ Z2 ) )
= ( minus_minus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z2 ) ) ) ).
% of_int_diff
thf(fact_593_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_594_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_595_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_596_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_597_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_598_append1__eq__conv,axiom,
! [Xs: list_a,X: a,Ys: list_a,Y: a] :
( ( ( append_a @ Xs @ ( cons_a @ X @ nil_a ) )
= ( append_a @ Ys @ ( cons_a @ Y @ nil_a ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_599_append1__eq__conv,axiom,
! [Xs: list_int,X: int,Ys: list_int,Y: int] :
( ( ( append_int @ Xs @ ( cons_int @ X @ nil_int ) )
= ( append_int @ Ys @ ( cons_int @ Y @ nil_int ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_600_of__int__eq__0__iff,axiom,
! [Z2: int] :
( ( ( ring_1_of_int_int @ Z2 )
= zero_zero_int )
= ( Z2 = zero_zero_int ) ) ).
% of_int_eq_0_iff
thf(fact_601_of__int__0__eq__iff,axiom,
! [Z2: int] :
( ( zero_zero_int
= ( ring_1_of_int_int @ Z2 ) )
= ( Z2 = zero_zero_int ) ) ).
% of_int_0_eq_iff
thf(fact_602_of__int__0,axiom,
( ( ring_1_of_int_int @ zero_zero_int )
= zero_zero_int ) ).
% of_int_0
thf(fact_603_of__int__le__iff,axiom,
! [W: int,Z2: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z2 ) )
= ( ord_less_eq_int @ W @ Z2 ) ) ).
% of_int_le_iff
thf(fact_604_of__int__less__iff,axiom,
! [W: int,Z2: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z2 ) )
= ( ord_less_int @ W @ Z2 ) ) ).
% of_int_less_iff
thf(fact_605_of__int__1,axiom,
( ( ring_1_of_int_int @ one_one_int )
= one_one_int ) ).
% of_int_1
thf(fact_606_of__int__eq__1__iff,axiom,
! [Z2: int] :
( ( ( ring_1_of_int_int @ Z2 )
= one_one_int )
= ( Z2 = one_one_int ) ) ).
% of_int_eq_1_iff
thf(fact_607_zle__diff1__eq,axiom,
! [W: int,Z2: int] :
( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z2 @ one_one_int ) )
= ( ord_less_int @ W @ Z2 ) ) ).
% zle_diff1_eq
thf(fact_608_of__int__minus,axiom,
! [Z2: int] :
( ( ring_1_of_int_int @ ( uminus_uminus_int @ Z2 ) )
= ( uminus_uminus_int @ ( ring_1_of_int_int @ Z2 ) ) ) ).
% of_int_minus
thf(fact_609_negative__zle,axiom,
! [N: nat,M2: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ ( semiri1314217659103216013at_int @ M2 ) ) ).
% negative_zle
thf(fact_610_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_611_hd__append2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs != nil_a )
=> ( ( hd_a @ ( append_a @ Xs @ Ys ) )
= ( hd_a @ Xs ) ) ) ).
% hd_append2
thf(fact_612_tl__append2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs != nil_a )
=> ( ( tl_a @ ( append_a @ Xs @ Ys ) )
= ( append_a @ ( tl_a @ Xs ) @ Ys ) ) ) ).
% tl_append2
thf(fact_613_last__appendR,axiom,
! [Ys: list_a,Xs: list_a] :
( ( Ys != nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys ) )
= ( last_a @ Ys ) ) ) ).
% last_appendR
thf(fact_614_last__appendL,axiom,
! [Ys: list_a,Xs: list_a] :
( ( Ys = nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys ) )
= ( last_a @ Xs ) ) ) ).
% last_appendL
thf(fact_615_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_616_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_617_nth__append__length,axiom,
! [Xs: list_int,X: int,Ys: list_int] :
( ( nth_int @ ( append_int @ Xs @ ( cons_int @ X @ Ys ) ) @ ( size_size_list_int @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_618_nat__0__iff,axiom,
! [I3: int] :
( ( ( nat2 @ I3 )
= zero_zero_nat )
= ( ord_less_eq_int @ I3 @ zero_zero_int ) ) ).
% nat_0_iff
thf(fact_619_nat__le__0,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ Z2 @ zero_zero_int )
=> ( ( nat2 @ Z2 )
= zero_zero_nat ) ) ).
% nat_le_0
thf(fact_620_int__nat__eq,axiom,
! [Z2: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
= Z2 ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
= zero_zero_int ) ) ) ).
% int_nat_eq
thf(fact_621_last__snoc,axiom,
! [Xs: list_a,X: a] :
( ( last_a @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) )
= X ) ).
% last_snoc
thf(fact_622_last__snoc,axiom,
! [Xs: list_int,X: int] :
( ( last_int @ ( append_int @ Xs @ ( cons_int @ X @ nil_int ) ) )
= X ) ).
% last_snoc
thf(fact_623_hd__Cons__tl,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( cons_a @ ( hd_a @ Xs ) @ ( tl_a @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_624_hd__Cons__tl,axiom,
! [Xs: list_int] :
( ( Xs != nil_int )
=> ( ( cons_int @ ( hd_int @ Xs ) @ ( tl_int @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_625_list_Ocollapse,axiom,
! [List: list_a] :
( ( List != nil_a )
=> ( ( cons_a @ ( hd_a @ List ) @ ( tl_a @ List ) )
= List ) ) ).
% list.collapse
thf(fact_626_list_Ocollapse,axiom,
! [List: list_int] :
( ( List != nil_int )
=> ( ( cons_int @ ( hd_int @ List ) @ ( tl_int @ List ) )
= List ) ) ).
% list.collapse
thf(fact_627_of__int__0__le__iff,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) )
= ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ).
% of_int_0_le_iff
thf(fact_628_of__int__le__0__iff,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z2 ) @ zero_zero_int )
= ( ord_less_eq_int @ Z2 @ zero_zero_int ) ) ).
% of_int_le_0_iff
thf(fact_629_of__int__less__0__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ Z2 ) @ zero_zero_int )
= ( ord_less_int @ Z2 @ zero_zero_int ) ) ).
% of_int_less_0_iff
thf(fact_630_of__int__0__less__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% of_int_0_less_iff
thf(fact_631_of__int__1__le__iff,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ one_one_int @ ( ring_1_of_int_int @ Z2 ) )
= ( ord_less_eq_int @ one_one_int @ Z2 ) ) ).
% of_int_1_le_iff
thf(fact_632_of__int__le__1__iff,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z2 ) @ one_one_int )
= ( ord_less_eq_int @ Z2 @ one_one_int ) ) ).
% of_int_le_1_iff
thf(fact_633_of__int__1__less__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ one_one_int @ ( ring_1_of_int_int @ Z2 ) )
= ( ord_less_int @ one_one_int @ Z2 ) ) ).
% of_int_1_less_iff
thf(fact_634_of__int__less__1__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ ( ring_1_of_int_int @ Z2 ) @ one_one_int )
= ( ord_less_int @ Z2 @ one_one_int ) ) ).
% of_int_less_1_iff
thf(fact_635_of__nat__nat,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
= ( ring_1_of_int_int @ Z2 ) ) ) ).
% of_nat_nat
thf(fact_636_int__le__induct,axiom,
! [I3: int,K: int,P: int > $o] :
( ( ord_less_eq_int @ I3 @ K )
=> ( ( P @ K )
=> ( ! [I2: int] :
( ( ord_less_eq_int @ I2 @ K )
=> ( ( P @ I2 )
=> ( P @ ( minus_minus_int @ I2 @ one_one_int ) ) ) )
=> ( P @ I3 ) ) ) ) ).
% int_le_induct
thf(fact_637_lift__Suc__antimono__le,axiom,
! [F: nat > int,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_int @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_638_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_639_lift__Suc__mono__le,axiom,
! [F: nat > int,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_int @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_640_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_641_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_642_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_643_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_644_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_645_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_646_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_647_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_648_le__cases3,axiom,
! [X: int,Y: int,Z2: int] :
( ( ( ord_less_eq_int @ X @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_eq_int @ X @ Z2 ) )
=> ( ( ( ord_less_eq_int @ X @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_int @ Z2 @ Y )
=> ~ ( ord_less_eq_int @ Y @ X ) )
=> ( ( ( ord_less_eq_int @ Y @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_int @ Z2 @ X )
=> ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_649_le__cases3,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_650_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z3: int] : ( Y5 = Z3 ) )
= ( ^ [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
& ( ord_less_eq_int @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_651_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
& ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_652_ord__eq__le__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_653_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_654_ord__le__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_655_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_656_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_657_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_658_order_Otrans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% order.trans
thf(fact_659_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_660_order__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z2 )
=> ( ord_less_eq_int @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_661_order__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_662_linorder__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A3: int,B2: int] :
( ( ord_less_eq_int @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: int,B2: int] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_663_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: nat,B2: nat] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_664_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: int,Z3: int] : ( Y5 = Z3 ) )
= ( ^ [A4: int,B3: int] :
( ( ord_less_eq_int @ B3 @ A4 )
& ( ord_less_eq_int @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_665_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_666_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_667_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_668_dual__order_Otrans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_669_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_670_antisym,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_671_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_672_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: int,Z3: int] : ( Y5 = Z3 ) )
= ( ^ [A4: int,B3: int] :
( ( ord_less_eq_int @ A4 @ B3 )
& ( ord_less_eq_int @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_673_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y5: nat,Z3: nat] : ( Y5 = Z3 ) )
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_674_order__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_675_order__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_676_order__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_677_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_678_order__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_679_order__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_680_order__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_681_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_682_order__eq__refl,axiom,
! [X: int,Y: int] :
( ( X = Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_eq_refl
thf(fact_683_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_684_linorder__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_linear
thf(fact_685_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_686_ord__eq__le__subst,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_687_ord__eq__le__subst,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_688_ord__eq__le__subst,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_689_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_690_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_691_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_692_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_693_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_694_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_695_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_696_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_697_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_698_of__nat__mono,axiom,
! [I3: nat,J2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I3 ) @ ( semiri1314217659103216013at_int @ J2 ) ) ) ).
% of_nat_mono
thf(fact_699_of__nat__mono,axiom,
! [I3: nat,J2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I3 ) @ ( semiri1316708129612266289at_nat @ J2 ) ) ) ).
% of_nat_mono
thf(fact_700_of__int__nonneg,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) ) ) ).
% of_int_nonneg
thf(fact_701_longest__common__prefix,axiom,
! [Xs: list_a,Ys: list_a] :
? [Ps: list_a,Xs4: list_a,Ys5: list_a] :
( ( Xs
= ( append_a @ Ps @ Xs4 ) )
& ( Ys
= ( append_a @ Ps @ Ys5 ) )
& ( ( Xs4 = nil_a )
| ( Ys5 = nil_a )
| ( ( hd_a @ Xs4 )
!= ( hd_a @ Ys5 ) ) ) ) ).
% longest_common_prefix
thf(fact_702_hd__append,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( Xs = nil_a )
=> ( ( hd_a @ ( append_a @ Xs @ Ys ) )
= ( hd_a @ Ys ) ) )
& ( ( Xs != nil_a )
=> ( ( hd_a @ ( append_a @ Xs @ Ys ) )
= ( hd_a @ Xs ) ) ) ) ).
% hd_append
thf(fact_703_hd__Nil__eq__last,axiom,
( ( hd_a @ nil_a )
= ( last_a @ nil_a ) ) ).
% hd_Nil_eq_last
thf(fact_704_longest__common__suffix,axiom,
! [Xs: list_a,Ys: list_a] :
? [Ss: list_a,Xs4: list_a,Ys5: list_a] :
( ( Xs
= ( append_a @ Xs4 @ Ss ) )
& ( Ys
= ( append_a @ Ys5 @ Ss ) )
& ( ( Xs4 = nil_a )
| ( Ys5 = nil_a )
| ( ( last_a @ Xs4 )
!= ( last_a @ Ys5 ) ) ) ) ).
% longest_common_suffix
thf(fact_705_last__append,axiom,
! [Ys: list_a,Xs: list_a] :
( ( ( Ys = nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys ) )
= ( last_a @ Xs ) ) )
& ( ( Ys != nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys ) )
= ( last_a @ Ys ) ) ) ) ).
% last_append
thf(fact_706_list_Osel_I1_J,axiom,
! [X21: int,X22: list_int] :
( ( hd_int @ ( cons_int @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_707_cons__hd,axiom,
! [X: int,Xs: list_int,Ys: list_int] :
( ( ( cons_int @ X @ Xs )
= Ys )
=> ( X
= ( hd_int @ Ys ) ) ) ).
% cons_hd
thf(fact_708_Cons__eq__appendI,axiom,
! [X: int,Xs1: list_int,Ys: list_int,Xs: list_int,Zs: list_int] :
( ( ( cons_int @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_int @ Xs1 @ Zs ) )
=> ( ( cons_int @ X @ Xs )
= ( append_int @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_709_append__Cons,axiom,
! [X: int,Xs: list_int,Ys: list_int] :
( ( append_int @ ( cons_int @ X @ Xs ) @ Ys )
= ( cons_int @ X @ ( append_int @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_710_eq__Nil__appendI,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs = Ys )
=> ( Xs
= ( append_a @ nil_a @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_711_append_Oleft__neutral,axiom,
! [A: list_a] :
( ( append_a @ nil_a @ A )
= A ) ).
% append.left_neutral
thf(fact_712_append__Nil,axiom,
! [Ys: list_a] :
( ( append_a @ nil_a @ Ys )
= Ys ) ).
% append_Nil
thf(fact_713_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_714_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_715_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_716_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus_int @ K @ zero_zero_int )
= K ) ).
% minus_int_code(1)
thf(fact_717_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_718_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_719_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_720_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_721_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_722_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_723_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_724_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_725_order__less__le__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_726_order__less__le__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_727_order__less__le__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_728_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_729_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_730_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_eq_int @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_731_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_732_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_eq_nat @ X3 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_733_order__le__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_734_order__le__less__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_int @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_735_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y2: nat] :
( ( ord_less_nat @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_736_order__le__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y2: int] :
( ( ord_less_int @ X3 @ Y2 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_737_order__less__le__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_738_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_739_order__le__less__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_740_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_741_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_742_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_743_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_744_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_745_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_746_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_747_linorder__not__less,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_not_less
thf(fact_748_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_749_linorder__not__le,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X @ Y ) )
= ( ord_less_int @ Y @ X ) ) ).
% linorder_not_le
thf(fact_750_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_751_order__less__le,axiom,
( ord_less_int
= ( ^ [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
& ( X2 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_752_order__less__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
& ( X2 != Y3 ) ) ) ) ).
% order_less_le
thf(fact_753_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X2: int,Y3: int] :
( ( ord_less_int @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_754_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_nat @ X2 @ Y3 )
| ( X2 = Y3 ) ) ) ) ).
% order_le_less
thf(fact_755_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_756_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_757_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_758_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_759_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B3: int,A4: int] :
( ( ord_less_eq_int @ B3 @ A4 )
& ~ ( ord_less_eq_int @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_760_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ~ ( ord_less_eq_nat @ A4 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_761_dual__order_Ostrict__trans2,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_762_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_763_dual__order_Ostrict__trans1,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_764_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_765_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B3: int,A4: int] :
( ( ord_less_eq_int @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_766_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_eq_nat @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_767_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B3: int,A4: int] :
( ( ord_less_int @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_768_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A4: nat] :
( ( ord_less_nat @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_769_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A4: int,B3: int] :
( ( ord_less_eq_int @ A4 @ B3 )
& ~ ( ord_less_eq_int @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_770_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ~ ( ord_less_eq_nat @ B3 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_771_order_Ostrict__trans2,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_772_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_773_order_Ostrict__trans1,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_774_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_775_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A4: int,B3: int] :
( ( ord_less_eq_int @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_776_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_777_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B3: int] :
( ( ord_less_int @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_778_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] :
( ( ord_less_nat @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_779_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_780_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_781_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X2: int,Y3: int] :
( ( ord_less_eq_int @ X2 @ Y3 )
& ~ ( ord_less_eq_int @ Y3 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_782_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y3: nat] :
( ( ord_less_eq_nat @ X2 @ Y3 )
& ~ ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_783_antisym__conv2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_784_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_785_antisym__conv1,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_786_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_787_nless__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_int @ A @ B ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_788_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_789_leI,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% leI
thf(fact_790_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_791_leD,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_int @ X @ Y ) ) ).
% leD
thf(fact_792_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_793_verit__comp__simplify1_I3_J,axiom,
! [B4: int,A5: int] :
( ( ~ ( ord_less_eq_int @ B4 @ A5 ) )
= ( ord_less_int @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_794_verit__comp__simplify1_I3_J,axiom,
! [B4: nat,A5: nat] :
( ( ~ ( ord_less_eq_nat @ B4 @ A5 ) )
= ( ord_less_nat @ A5 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_795_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_796_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_797_diff__mono,axiom,
! [A: int,B: int,D: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ D @ C )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_798_diff__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_799_diff__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_800_diff__eq__diff__less__eq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_eq_int @ A @ B )
= ( ord_less_eq_int @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_801_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_802_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_803_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_804_int__diff__cases,axiom,
! [Z2: int] :
~ ! [M4: nat,N2: nat] :
( Z2
!= ( minus_minus_int @ ( semiri1314217659103216013at_int @ M4 ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% int_diff_cases
thf(fact_805_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_806_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_807_nat__diff__distrib,axiom,
! [Z4: int,Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z4 )
=> ( ( ord_less_eq_int @ Z4 @ Z2 )
=> ( ( nat2 @ ( minus_minus_int @ Z2 @ Z4 ) )
= ( minus_minus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z4 ) ) ) ) ) ).
% nat_diff_distrib
thf(fact_808_rev__nonempty__induct,axiom,
! [Xs: list_a,P: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X3: a] : ( P @ ( cons_a @ X3 @ nil_a ) )
=> ( ! [X3: a,Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( P @ Xs2 )
=> ( P @ ( append_a @ Xs2 @ ( cons_a @ X3 @ nil_a ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_809_rev__nonempty__induct,axiom,
! [Xs: list_int,P: list_int > $o] :
( ( Xs != nil_int )
=> ( ! [X3: int] : ( P @ ( cons_int @ X3 @ nil_int ) )
=> ( ! [X3: int,Xs2: list_int] :
( ( Xs2 != nil_int )
=> ( ( P @ Xs2 )
=> ( P @ ( append_int @ Xs2 @ ( cons_int @ X3 @ nil_int ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_810_append__eq__Cons__conv,axiom,
! [Ys: list_a,Zs: list_a,X: a,Xs: list_a] :
( ( ( append_a @ Ys @ Zs )
= ( cons_a @ X @ Xs ) )
= ( ( ( Ys = nil_a )
& ( Zs
= ( cons_a @ X @ Xs ) ) )
| ? [Ys6: list_a] :
( ( Ys
= ( cons_a @ X @ Ys6 ) )
& ( ( append_a @ Ys6 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_811_append__eq__Cons__conv,axiom,
! [Ys: list_int,Zs: list_int,X: int,Xs: list_int] :
( ( ( append_int @ Ys @ Zs )
= ( cons_int @ X @ Xs ) )
= ( ( ( Ys = nil_int )
& ( Zs
= ( cons_int @ X @ Xs ) ) )
| ? [Ys6: list_int] :
( ( Ys
= ( cons_int @ X @ Ys6 ) )
& ( ( append_int @ Ys6 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_812_Cons__eq__append__conv,axiom,
! [X: a,Xs: list_a,Ys: list_a,Zs: list_a] :
( ( ( cons_a @ X @ Xs )
= ( append_a @ Ys @ Zs ) )
= ( ( ( Ys = nil_a )
& ( ( cons_a @ X @ Xs )
= Zs ) )
| ? [Ys6: list_a] :
( ( ( cons_a @ X @ Ys6 )
= Ys )
& ( Xs
= ( append_a @ Ys6 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_813_Cons__eq__append__conv,axiom,
! [X: int,Xs: list_int,Ys: list_int,Zs: list_int] :
( ( ( cons_int @ X @ Xs )
= ( append_int @ Ys @ Zs ) )
= ( ( ( Ys = nil_int )
& ( ( cons_int @ X @ Xs )
= Zs ) )
| ? [Ys6: list_int] :
( ( ( cons_int @ X @ Ys6 )
= Ys )
& ( Xs
= ( append_int @ Ys6 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_814_rev__exhaust,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ~ ! [Ys2: list_a,Y2: a] :
( Xs
!= ( append_a @ Ys2 @ ( cons_a @ Y2 @ nil_a ) ) ) ) ).
% rev_exhaust
thf(fact_815_rev__exhaust,axiom,
! [Xs: list_int] :
( ( Xs != nil_int )
=> ~ ! [Ys2: list_int,Y2: int] :
( Xs
!= ( append_int @ Ys2 @ ( cons_int @ Y2 @ nil_int ) ) ) ) ).
% rev_exhaust
thf(fact_816_rev__induct,axiom,
! [P: list_a > $o,Xs: list_a] :
( ( P @ nil_a )
=> ( ! [X3: a,Xs2: list_a] :
( ( P @ Xs2 )
=> ( P @ ( append_a @ Xs2 @ ( cons_a @ X3 @ nil_a ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_817_rev__induct,axiom,
! [P: list_int > $o,Xs: list_int] :
( ( P @ nil_int )
=> ( ! [X3: int,Xs2: list_int] :
( ( P @ Xs2 )
=> ( P @ ( append_int @ Xs2 @ ( cons_int @ X3 @ nil_int ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_818_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_819_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_820_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_821_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_822_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_823_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_824_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A4: int,B3: int] : ( ord_less_eq_int @ ( minus_minus_int @ A4 @ B3 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_825_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_826_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_827_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_828_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_829_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_830_list_Oexpand,axiom,
! [List: list_a,List2: list_a] :
( ( ( List = nil_a )
= ( List2 = nil_a ) )
=> ( ( ( List != nil_a )
=> ( ( List2 != nil_a )
=> ( ( ( hd_a @ List )
= ( hd_a @ List2 ) )
& ( ( tl_a @ List )
= ( tl_a @ List2 ) ) ) ) )
=> ( List = List2 ) ) ) ).
% list.expand
thf(fact_831_tl__append__if,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( Xs = nil_a )
=> ( ( tl_a @ ( append_a @ Xs @ Ys ) )
= ( tl_a @ Ys ) ) )
& ( ( Xs != nil_a )
=> ( ( tl_a @ ( append_a @ Xs @ Ys ) )
= ( append_a @ ( tl_a @ Xs ) @ Ys ) ) ) ) ).
% tl_append_if
thf(fact_832_int__less__induct,axiom,
! [I3: int,K: int,P: int > $o] :
( ( ord_less_int @ I3 @ 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 @ I3 ) ) ) ) ).
% int_less_induct
thf(fact_833_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_834_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_835_ex__nat,axiom,
( ( ^ [P2: nat > $o] :
? [X4: nat] : ( P2 @ X4 ) )
= ( ^ [P3: nat > $o] :
? [X2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
& ( P3 @ ( nat2 @ X2 ) ) ) ) ) ).
% ex_nat
thf(fact_836_all__nat,axiom,
( ( ^ [P2: nat > $o] :
! [X4: nat] : ( P2 @ X4 ) )
= ( ^ [P3: nat > $o] :
! [X2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> ( P3 @ ( nat2 @ X2 ) ) ) ) ) ).
% all_nat
thf(fact_837_eq__nat__nat__iff,axiom,
! [Z2: int,Z4: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z4 )
=> ( ( ( nat2 @ Z2 )
= ( nat2 @ Z4 ) )
= ( Z2 = Z4 ) ) ) ) ).
% eq_nat_nat_iff
thf(fact_838_last__ConsR,axiom,
! [Xs: list_a,X: a] :
( ( Xs != nil_a )
=> ( ( last_a @ ( cons_a @ X @ Xs ) )
= ( last_a @ Xs ) ) ) ).
% last_ConsR
thf(fact_839_last__ConsR,axiom,
! [Xs: list_int,X: int] :
( ( Xs != nil_int )
=> ( ( last_int @ ( cons_int @ X @ Xs ) )
= ( last_int @ Xs ) ) ) ).
% last_ConsR
thf(fact_840_last__ConsL,axiom,
! [Xs: list_a,X: a] :
( ( Xs = nil_a )
=> ( ( last_a @ ( cons_a @ X @ Xs ) )
= X ) ) ).
% last_ConsL
thf(fact_841_last__ConsL,axiom,
! [Xs: list_int,X: int] :
( ( Xs = nil_int )
=> ( ( last_int @ ( cons_int @ X @ Xs ) )
= X ) ) ).
% last_ConsL
thf(fact_842_last_Osimps,axiom,
! [Xs: list_a,X: a] :
( ( ( Xs = nil_a )
=> ( ( last_a @ ( cons_a @ X @ Xs ) )
= X ) )
& ( ( Xs != nil_a )
=> ( ( last_a @ ( cons_a @ X @ Xs ) )
= ( last_a @ Xs ) ) ) ) ).
% last.simps
thf(fact_843_last_Osimps,axiom,
! [Xs: list_int,X: int] :
( ( ( Xs = nil_int )
=> ( ( last_int @ ( cons_int @ X @ Xs ) )
= X ) )
& ( ( Xs != nil_int )
=> ( ( last_int @ ( cons_int @ X @ Xs ) )
= ( last_int @ Xs ) ) ) ) ).
% last.simps
thf(fact_844_last__tl,axiom,
! [Xs: list_a] :
( ( ( Xs = nil_a )
| ( ( tl_a @ Xs )
!= nil_a ) )
=> ( ( last_a @ ( tl_a @ Xs ) )
= ( last_a @ Xs ) ) ) ).
% last_tl
thf(fact_845_same__length__different,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs != Ys )
=> ( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ? [Pre: list_a,X3: a,Xs4: list_a,Y2: a,Ys5: list_a] :
( ( X3 != Y2 )
& ( Xs
= ( append_a @ Pre @ ( append_a @ ( cons_a @ X3 @ nil_a ) @ Xs4 ) ) )
& ( Ys
= ( append_a @ Pre @ ( append_a @ ( cons_a @ Y2 @ nil_a ) @ Ys5 ) ) ) ) ) ) ).
% same_length_different
thf(fact_846_same__length__different,axiom,
! [Xs: list_int,Ys: list_int] :
( ( Xs != Ys )
=> ( ( ( size_size_list_int @ Xs )
= ( size_size_list_int @ Ys ) )
=> ? [Pre: list_int,X3: int,Xs4: list_int,Y2: int,Ys5: list_int] :
( ( X3 != Y2 )
& ( Xs
= ( append_int @ Pre @ ( append_int @ ( cons_int @ X3 @ nil_int ) @ Xs4 ) ) )
& ( Ys
= ( append_int @ Pre @ ( append_int @ ( cons_int @ Y2 @ nil_int ) @ Ys5 ) ) ) ) ) ) ).
% same_length_different
thf(fact_847_hd__conv__nth,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( hd_a @ Xs )
= ( nth_a @ Xs @ zero_zero_nat ) ) ) ).
% hd_conv_nth
thf(fact_848_list_Oexhaust__sel,axiom,
! [List: list_a] :
( ( List != nil_a )
=> ( List
= ( cons_a @ ( hd_a @ List ) @ ( tl_a @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_849_list_Oexhaust__sel,axiom,
! [List: list_int] :
( ( List != nil_int )
=> ( List
= ( cons_int @ ( hd_int @ List ) @ ( tl_int @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_850_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_851_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_852_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_853_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_854_int__one__le__iff__zero__less,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ one_one_int @ Z2 )
= ( ord_less_int @ zero_zero_int @ Z2 ) ) ).
% int_one_le_iff_zero_less
thf(fact_855_negative__zle__0,axiom,
! [N: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N ) ) @ zero_zero_int ) ).
% negative_zle_0
thf(fact_856_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_857_nat__0__le,axiom,
! [Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z2 )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
= Z2 ) ) ).
% nat_0_le
thf(fact_858_int__eq__iff,axiom,
! [M2: nat,Z2: int] :
( ( ( semiri1314217659103216013at_int @ M2 )
= Z2 )
= ( ( M2
= ( nat2 @ Z2 ) )
& ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ) ).
% int_eq_iff
thf(fact_859_of__int__pos,axiom,
! [Z2: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) ) ) ).
% of_int_pos
thf(fact_860_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_861_length__append__singleton,axiom,
! [Xs: list_a,X: a] :
( ( size_size_list_a @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) )
= ( suc @ ( size_size_list_a @ Xs ) ) ) ).
% length_append_singleton
thf(fact_862_length__append__singleton,axiom,
! [Xs: list_int,X: int] :
( ( size_size_list_int @ ( append_int @ Xs @ ( cons_int @ X @ nil_int ) ) )
= ( suc @ ( size_size_list_int @ Xs ) ) ) ).
% length_append_singleton
thf(fact_863_length__Suc__conv__rev,axiom,
! [Xs: list_a,N: nat] :
( ( ( size_size_list_a @ Xs )
= ( suc @ N ) )
= ( ? [Y3: a,Ys4: list_a] :
( ( Xs
= ( append_a @ Ys4 @ ( cons_a @ Y3 @ nil_a ) ) )
& ( ( size_size_list_a @ Ys4 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_864_length__Suc__conv__rev,axiom,
! [Xs: list_int,N: nat] :
( ( ( size_size_list_int @ Xs )
= ( suc @ N ) )
= ( ? [Y3: int,Ys4: list_int] :
( ( Xs
= ( append_int @ Ys4 @ ( cons_int @ Y3 @ nil_int ) ) )
& ( ( size_size_list_int @ Ys4 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_865_length__one__hd,axiom,
! [Xs: list_a] :
( ( ( size_size_list_a @ Xs )
= one_one_nat )
=> ( Xs
= ( cons_a @ ( hd_a @ Xs ) @ nil_a ) ) ) ).
% length_one_hd
thf(fact_866_length__one__hd,axiom,
! [Xs: list_int] :
( ( ( size_size_list_int @ Xs )
= one_one_nat )
=> ( Xs
= ( cons_int @ ( hd_int @ Xs ) @ nil_int ) ) ) ).
% length_one_hd
thf(fact_867_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_868_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_869_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_870_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_871_nat__less__eq__zless,axiom,
! [W: int,Z2: int] :
( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
= ( ord_less_int @ W @ Z2 ) ) ) ).
% nat_less_eq_zless
thf(fact_872_take__hd_H,axiom,
! [Ys: list_a,X: a,Xs: list_a] :
( ( Ys != nil_a )
=> ( ( ( take_a @ ( size_size_list_a @ Ys ) @ ( cons_a @ X @ Xs ) )
= ( take_a @ ( suc @ ( size_size_list_a @ Xs ) ) @ Ys ) )
=> ( ( hd_a @ Ys )
= X ) ) ) ).
% take_hd'
thf(fact_873_take__hd_H,axiom,
! [Ys: list_int,X: int,Xs: list_int] :
( ( Ys != nil_int )
=> ( ( ( take_int @ ( size_size_list_int @ Ys ) @ ( cons_int @ X @ Xs ) )
= ( take_int @ ( suc @ ( size_size_list_int @ Xs ) ) @ Ys ) )
=> ( ( hd_int @ Ys )
= X ) ) ) ).
% take_hd'
thf(fact_874_take__Suc,axiom,
! [Xs: list_a,N: nat] :
( ( Xs != nil_a )
=> ( ( take_a @ ( suc @ N ) @ Xs )
= ( cons_a @ ( hd_a @ Xs ) @ ( take_a @ N @ ( tl_a @ Xs ) ) ) ) ) ).
% take_Suc
thf(fact_875_take__Suc,axiom,
! [Xs: list_int,N: nat] :
( ( Xs != nil_int )
=> ( ( take_int @ ( suc @ N ) @ Xs )
= ( cons_int @ ( hd_int @ Xs ) @ ( take_int @ N @ ( tl_int @ Xs ) ) ) ) ) ).
% take_Suc
thf(fact_876_take__rev__tl__hd,axiom,
! [N: nat,Xs: list_a,Ys: list_a] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( Xs != nil_a )
=> ( ( append_a @ ( common_take_rev_a @ N @ Xs ) @ Ys )
= ( append_a @ ( common_take_rev_a @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( tl_a @ Xs ) ) @ ( cons_a @ ( hd_a @ Xs ) @ Ys ) ) ) ) ) ).
% take_rev_tl_hd
thf(fact_877_take__rev__tl__hd,axiom,
! [N: nat,Xs: list_int,Ys: list_int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( Xs != nil_int )
=> ( ( append_int @ ( common_take_rev_int @ N @ Xs ) @ Ys )
= ( append_int @ ( common_take_rev_int @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( tl_int @ Xs ) ) @ ( cons_int @ ( hd_int @ Xs ) @ Ys ) ) ) ) ) ).
% take_rev_tl_hd
thf(fact_878_take__rev__step,axiom,
! [Xs: list_a,N: nat,Acc: list_a] :
( ( Xs != nil_a )
=> ( ( append_a @ ( common_take_rev_a @ N @ ( tl_a @ Xs ) ) @ ( cons_a @ ( hd_a @ Xs ) @ Acc ) )
= ( append_a @ ( common_take_rev_a @ ( suc @ N ) @ Xs ) @ Acc ) ) ) ).
% take_rev_step
thf(fact_879_take__rev__step,axiom,
! [Xs: list_int,N: nat,Acc: list_int] :
( ( Xs != nil_int )
=> ( ( append_int @ ( common_take_rev_int @ N @ ( tl_int @ Xs ) ) @ ( cons_int @ ( hd_int @ Xs ) @ Acc ) )
= ( append_int @ ( common_take_rev_int @ ( suc @ N ) @ Xs ) @ Acc ) ) ) ).
% take_rev_step
thf(fact_880_take__rev__nth,axiom,
! [N: nat,Xs: list_int,X: int,Ys: list_int] :
( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
=> ( ( X
= ( nth_int @ Xs @ N ) )
=> ( ( cons_int @ X @ ( append_int @ ( common_take_rev_int @ N @ Xs ) @ Ys ) )
= ( append_int @ ( common_take_rev_int @ ( suc @ N ) @ Xs ) @ Ys ) ) ) ) ).
% take_rev_nth
thf(fact_881_take__hd__drop,axiom,
! [N: nat,Xs: list_a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( append_a @ ( take_a @ N @ Xs ) @ ( cons_a @ ( hd_a @ ( drop_a @ N @ Xs ) ) @ nil_a ) )
= ( take_a @ ( suc @ N ) @ Xs ) ) ) ).
% take_hd_drop
thf(fact_882_take__hd__drop,axiom,
! [N: nat,Xs: list_int] :
( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
=> ( ( append_int @ ( take_int @ N @ Xs ) @ ( cons_int @ ( hd_int @ ( drop_int @ N @ Xs ) ) @ nil_int ) )
= ( take_int @ ( suc @ N ) @ Xs ) ) ) ).
% take_hd_drop
thf(fact_883_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_884_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_885_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_886_diff__diff__cancel,axiom,
! [I3: nat,N: nat] :
( ( ord_less_eq_nat @ I3 @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I3 ) )
= I3 ) ) ).
% diff_diff_cancel
thf(fact_887_take__rev__empty,axiom,
! [N: nat] :
( ( common_take_rev_a @ N @ nil_a )
= nil_a ) ).
% take_rev_empty
thf(fact_888_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_889_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_890_drop__Suc__Cons,axiom,
! [N: nat,X: int,Xs: list_int] :
( ( drop_int @ ( suc @ N ) @ ( cons_int @ X @ Xs ) )
= ( drop_int @ N @ Xs ) ) ).
% drop_Suc_Cons
thf(fact_891_drop__all,axiom,
! [Xs: list_a,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N )
=> ( ( drop_a @ N @ Xs )
= nil_a ) ) ).
% drop_all
thf(fact_892_drop__eq__Nil,axiom,
! [N: nat,Xs: list_a] :
( ( ( drop_a @ N @ Xs )
= nil_a )
= ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N ) ) ).
% drop_eq_Nil
thf(fact_893_drop__eq__Nil2,axiom,
! [N: nat,Xs: list_a] :
( ( nil_a
= ( drop_a @ N @ Xs ) )
= ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N ) ) ).
% drop_eq_Nil2
thf(fact_894_hd__drop__1,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( cons_a @ ( hd_a @ Xs ) @ ( drop_a @ ( suc @ zero_zero_nat ) @ Xs ) )
= Xs ) ) ).
% hd_drop_1
thf(fact_895_hd__drop__1,axiom,
! [Xs: list_int] :
( ( Xs != nil_int )
=> ( ( cons_int @ ( hd_int @ Xs ) @ ( drop_int @ ( suc @ zero_zero_nat ) @ Xs ) )
= Xs ) ) ).
% hd_drop_1
thf(fact_896_hd__drop,axiom,
! [N: nat,Xs: list_int] :
( ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
=> ( ( cons_int @ ( hd_int @ ( drop_int @ N @ Xs ) ) @ ( drop_int @ ( suc @ N ) @ Xs ) )
= ( drop_int @ N @ Xs ) ) ) ).
% hd_drop
thf(fact_897_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_898_le__trans,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_eq_nat @ J2 @ K )
=> ( ord_less_eq_nat @ I3 @ K ) ) ) ).
% le_trans
thf(fact_899_eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( M2 = N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% eq_imp_le
thf(fact_900_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_901_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_902_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_903_Suc__leD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
=> ( ord_less_eq_nat @ M2 @ N ) ) ).
% Suc_leD
thf(fact_904_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_905_le__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq_nat @ M2 @ N )
=> ( ord_less_eq_nat @ M2 @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_906_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_907_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_908_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_909_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_910_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_911_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_912_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,Y2: nat,Z: nat] :
( ( R2 @ X3 @ Y2 )
=> ( ( R2 @ Y2 @ Z )
=> ( R2 @ X3 @ Z ) ) )
=> ( ! [N2: nat] : ( R2 @ N2 @ ( suc @ N2 ) )
=> ( R2 @ M2 @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_913_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_914_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_915_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_916_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_917_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I3: nat,J2: nat] :
( ! [I2: nat,J: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J ) ) )
=> ( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( F @ J2 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_918_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_919_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_920_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_921_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_922_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
& ( M3 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_923_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_924_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_925_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_926_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_927_diff__le__self,axiom,
! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ).
% diff_le_self
thf(fact_928_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_929_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_930_drop__Nil,axiom,
! [N: nat] :
( ( drop_a @ N @ nil_a )
= nil_a ) ).
% drop_Nil
thf(fact_931_nth__via__drop,axiom,
! [N: nat,Xs: list_int,Y: int,Ys: list_int] :
( ( ( drop_int @ N @ Xs )
= ( cons_int @ Y @ Ys ) )
=> ( ( nth_int @ Xs @ N )
= Y ) ) ).
% nth_via_drop
thf(fact_932_Suc__leI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( ord_less_eq_nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_leI
thf(fact_933_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_934_dec__induct,axiom,
! [I3: nat,J2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( P @ I3 )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I3 @ N2 )
=> ( ( ord_less_nat @ N2 @ J2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J2 ) ) ) ) ).
% dec_induct
thf(fact_935_inc__induct,axiom,
! [I3: nat,J2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( P @ J2 )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I3 @ N2 )
=> ( ( ord_less_nat @ N2 @ J2 )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I3 ) ) ) ) ).
% inc_induct
thf(fact_936_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_937_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_938_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_939_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_940_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_941_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ K2 )
=> ~ ( P @ I5 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_942_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_943_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_944_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_945_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_946_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_947_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B3: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A4 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_948_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_949_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I5: nat] :
( ( ord_less_eq_nat @ I5 @ K2 )
=> ~ ( P @ I5 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_950_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_int] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_int @ Xs ) )
= ( ? [X2: int,Ys4: list_int] :
( ( Xs
= ( cons_int @ X2 @ Ys4 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_int @ Ys4 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_951_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_952_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_953_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_954_drop__Cons_H,axiom,
! [N: nat,X: int,Xs: list_int] :
( ( ( N = zero_zero_nat )
=> ( ( drop_int @ N @ ( cons_int @ X @ Xs ) )
= ( cons_int @ X @ Xs ) ) )
& ( ( N != zero_zero_nat )
=> ( ( drop_int @ N @ ( cons_int @ X @ Xs ) )
= ( drop_int @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs ) ) ) ) ).
% drop_Cons'
thf(fact_955_nat__le__eq__zle,axiom,
! [W: int,Z2: int] :
( ( ( ord_less_int @ zero_zero_int @ W )
| ( ord_less_eq_int @ zero_zero_int @ Z2 ) )
=> ( ( ord_less_eq_nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
= ( ord_less_eq_int @ W @ Z2 ) ) ) ).
% nat_le_eq_zle
thf(fact_956_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_957_Cons__nth__drop__Suc,axiom,
! [I3: nat,Xs: list_int] :
( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
=> ( ( cons_int @ ( nth_int @ Xs @ I3 ) @ ( drop_int @ ( suc @ I3 ) @ Xs ) )
= ( drop_int @ I3 @ Xs ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_958_id__take__nth__drop,axiom,
! [I3: nat,Xs: list_int] :
( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
=> ( Xs
= ( append_int @ ( take_int @ I3 @ Xs ) @ ( cons_int @ ( nth_int @ Xs @ I3 ) @ ( drop_int @ ( suc @ I3 ) @ Xs ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_959_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_960_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_961_take__last__length,axiom,
! [Xs: list_a] :
( ( ( take_a @ ( suc @ zero_zero_nat ) @ ( rev_a @ Xs ) )
= ( cons_a @ ( last_a @ Xs ) @ nil_a ) )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( size_size_list_a @ Xs ) ) ) ).
% take_last_length
thf(fact_962_take__last__length,axiom,
! [Xs: list_int] :
( ( ( take_int @ ( suc @ zero_zero_nat ) @ ( rev_int @ Xs ) )
= ( cons_int @ ( last_int @ Xs ) @ nil_int ) )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( size_size_list_int @ Xs ) ) ) ).
% take_last_length
thf(fact_963_Nil__is__rev__conv,axiom,
! [Xs: list_a] :
( ( nil_a
= ( rev_a @ Xs ) )
= ( Xs = nil_a ) ) ).
% Nil_is_rev_conv
thf(fact_964_rev__is__Nil__conv,axiom,
! [Xs: list_a] :
( ( ( rev_a @ Xs )
= nil_a )
= ( Xs = nil_a ) ) ).
% rev_is_Nil_conv
thf(fact_965_singleton__rev__conv,axiom,
! [X: a,Xs: list_a] :
( ( ( cons_a @ X @ nil_a )
= ( rev_a @ Xs ) )
= ( ( cons_a @ X @ nil_a )
= Xs ) ) ).
% singleton_rev_conv
thf(fact_966_singleton__rev__conv,axiom,
! [X: int,Xs: list_int] :
( ( ( cons_int @ X @ nil_int )
= ( rev_int @ Xs ) )
= ( ( cons_int @ X @ nil_int )
= Xs ) ) ).
% singleton_rev_conv
thf(fact_967_rev__singleton__conv,axiom,
! [Xs: list_a,X: a] :
( ( ( rev_a @ Xs )
= ( cons_a @ X @ nil_a ) )
= ( Xs
= ( cons_a @ X @ nil_a ) ) ) ).
% rev_singleton_conv
thf(fact_968_rev__singleton__conv,axiom,
! [Xs: list_int,X: int] :
( ( ( rev_int @ Xs )
= ( cons_int @ X @ nil_int ) )
= ( Xs
= ( cons_int @ X @ nil_int ) ) ) ).
% rev_singleton_conv
thf(fact_969_rev__eq__Cons__iff,axiom,
! [Xs: list_a,Y: a,Ys: list_a] :
( ( ( rev_a @ Xs )
= ( cons_a @ Y @ Ys ) )
= ( Xs
= ( append_a @ ( rev_a @ Ys ) @ ( cons_a @ Y @ nil_a ) ) ) ) ).
% rev_eq_Cons_iff
thf(fact_970_rev__eq__Cons__iff,axiom,
! [Xs: list_int,Y: int,Ys: list_int] :
( ( ( rev_int @ Xs )
= ( cons_int @ Y @ Ys ) )
= ( Xs
= ( append_int @ ( rev_int @ Ys ) @ ( cons_int @ Y @ nil_int ) ) ) ) ).
% rev_eq_Cons_iff
thf(fact_971_rev_Osimps_I1_J,axiom,
( ( rev_a @ nil_a )
= nil_a ) ).
% rev.simps(1)
thf(fact_972_rev__app__single,axiom,
! [Xs: list_a,X: a] :
( ( append_a @ ( rev_a @ Xs ) @ ( cons_a @ X @ nil_a ) )
= ( rev_a @ ( cons_a @ X @ Xs ) ) ) ).
% rev_app_single
thf(fact_973_rev__app__single,axiom,
! [Xs: list_int,X: int] :
( ( append_int @ ( rev_int @ Xs ) @ ( cons_int @ X @ nil_int ) )
= ( rev_int @ ( cons_int @ X @ Xs ) ) ) ).
% rev_app_single
thf(fact_974_pinf_I1_J,axiom,
! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P5 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_975_pinf_I1_J,axiom,
! [P: int > $o,P5: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: int] :
! [X5: int] :
( ( ord_less_int @ Z @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P5 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_976_pinf_I2_J,axiom,
! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P5 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_977_pinf_I2_J,axiom,
! [P: int > $o,P5: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: int] :
! [X5: int] :
( ( ord_less_int @ Z @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P5 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_978_pinf_I3_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_979_pinf_I3_J,axiom,
! [T: int] :
? [Z: int] :
! [X5: int] :
( ( ord_less_int @ Z @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_980_pinf_I4_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_981_pinf_I4_J,axiom,
! [T: int] :
? [Z: int] :
! [X5: int] :
( ( ord_less_int @ Z @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_982_pinf_I5_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z @ X5 )
=> ~ ( ord_less_nat @ X5 @ T ) ) ).
% pinf(5)
thf(fact_983_pinf_I5_J,axiom,
! [T: int] :
? [Z: int] :
! [X5: int] :
( ( ord_less_int @ Z @ X5 )
=> ~ ( ord_less_int @ X5 @ T ) ) ).
% pinf(5)
thf(fact_984_pinf_I7_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z @ X5 )
=> ( ord_less_nat @ T @ X5 ) ) ).
% pinf(7)
thf(fact_985_pinf_I7_J,axiom,
! [T: int] :
? [Z: int] :
! [X5: int] :
( ( ord_less_int @ Z @ X5 )
=> ( ord_less_int @ T @ X5 ) ) ).
% pinf(7)
thf(fact_986_minf_I1_J,axiom,
! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P5 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_987_minf_I1_J,axiom,
! [P: int > $o,P5: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P5 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_988_minf_I2_J,axiom,
! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z5: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P5 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_989_minf_I2_J,axiom,
! [P: int > $o,P5: int > $o,Q: int > $o,Q2: int > $o] :
( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less_int @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P5 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_990_minf_I3_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_991_minf_I3_J,axiom,
! [T: int] :
? [Z: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_992_minf_I4_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_993_minf_I4_J,axiom,
! [T: int] :
? [Z: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_994_minf_I5_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z )
=> ( ord_less_nat @ X5 @ T ) ) ).
% minf(5)
thf(fact_995_minf_I5_J,axiom,
! [T: int] :
? [Z: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z )
=> ( ord_less_int @ X5 @ T ) ) ).
% minf(5)
thf(fact_996_minf_I7_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z )
=> ~ ( ord_less_nat @ T @ X5 ) ) ).
% minf(7)
thf(fact_997_minf_I7_J,axiom,
! [T: int] :
? [Z: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z )
=> ~ ( ord_less_int @ T @ X5 ) ) ).
% minf(7)
thf(fact_998_rev__tl__hd,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( append_a @ ( rev_a @ ( tl_a @ Xs ) ) @ ( cons_a @ ( hd_a @ Xs ) @ nil_a ) )
= ( rev_a @ Xs ) ) ) ).
% rev_tl_hd
thf(fact_999_rev__tl__hd,axiom,
! [Xs: list_int] :
( ( Xs != nil_int )
=> ( ( append_int @ ( rev_int @ ( tl_int @ Xs ) ) @ ( cons_int @ ( hd_int @ Xs ) @ nil_int ) )
= ( rev_int @ Xs ) ) ) ).
% rev_tl_hd
thf(fact_1000_take__last,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( take_a @ one_one_nat @ ( rev_a @ Xs ) )
= ( cons_a @ ( last_a @ Xs ) @ nil_a ) ) ) ).
% take_last
thf(fact_1001_take__last,axiom,
! [Xs: list_int] :
( ( Xs != nil_int )
=> ( ( take_int @ one_one_nat @ ( rev_int @ Xs ) )
= ( cons_int @ ( last_int @ Xs ) @ nil_int ) ) ) ).
% take_last
thf(fact_1002_last__drop__rev,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( cons_a @ ( last_a @ Xs ) @ ( drop_a @ one_one_nat @ ( rev_a @ Xs ) ) )
= ( rev_a @ Xs ) ) ) ).
% last_drop_rev
thf(fact_1003_last__drop__rev,axiom,
! [Xs: list_int] :
( ( Xs != nil_int )
=> ( ( cons_int @ ( last_int @ Xs ) @ ( drop_int @ one_one_nat @ ( rev_int @ Xs ) ) )
= ( rev_int @ Xs ) ) ) ).
% last_drop_rev
thf(fact_1004_minf_I8_J,axiom,
! [T: int] :
? [Z: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z )
=> ~ ( ord_less_eq_int @ T @ X5 ) ) ).
% minf(8)
thf(fact_1005_minf_I8_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z )
=> ~ ( ord_less_eq_nat @ T @ X5 ) ) ).
% minf(8)
thf(fact_1006_minf_I6_J,axiom,
! [T: int] :
? [Z: int] :
! [X5: int] :
( ( ord_less_int @ X5 @ Z )
=> ( ord_less_eq_int @ X5 @ T ) ) ).
% minf(6)
thf(fact_1007_minf_I6_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z )
=> ( ord_less_eq_nat @ X5 @ T ) ) ).
% minf(6)
thf(fact_1008_pinf_I8_J,axiom,
! [T: int] :
? [Z: int] :
! [X5: int] :
( ( ord_less_int @ Z @ X5 )
=> ( ord_less_eq_int @ T @ X5 ) ) ).
% pinf(8)
thf(fact_1009_pinf_I8_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z @ X5 )
=> ( ord_less_eq_nat @ T @ X5 ) ) ).
% pinf(8)
thf(fact_1010_pinf_I6_J,axiom,
! [T: int] :
? [Z: int] :
! [X5: int] :
( ( ord_less_int @ Z @ X5 )
=> ~ ( ord_less_eq_int @ X5 @ T ) ) ).
% pinf(6)
thf(fact_1011_pinf_I6_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z @ X5 )
=> ~ ( ord_less_eq_nat @ X5 @ T ) ) ).
% pinf(6)
thf(fact_1012_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_1013_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_1014_upd__conv__take__nth__drop,axiom,
! [I3: nat,Xs: list_int,A: int] :
( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
=> ( ( list_update_int @ Xs @ I3 @ A )
= ( append_int @ ( take_int @ I3 @ Xs ) @ ( cons_int @ A @ ( drop_int @ ( suc @ I3 ) @ Xs ) ) ) ) ) ).
% upd_conv_take_nth_drop
thf(fact_1015_rotate1__hd__tl,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( rotate1_a @ Xs )
= ( append_a @ ( tl_a @ Xs ) @ ( cons_a @ ( hd_a @ Xs ) @ nil_a ) ) ) ) ).
% rotate1_hd_tl
thf(fact_1016_rotate1__hd__tl,axiom,
! [Xs: list_int] :
( ( Xs != nil_int )
=> ( ( rotate1_int @ Xs )
= ( append_int @ ( tl_int @ Xs ) @ ( cons_int @ ( hd_int @ Xs ) @ nil_int ) ) ) ) ).
% rotate1_hd_tl
thf(fact_1017_list__update__nonempty,axiom,
! [Xs: list_a,K: nat,X: a] :
( ( ( list_update_a @ Xs @ K @ X )
= nil_a )
= ( Xs = nil_a ) ) ).
% list_update_nonempty
thf(fact_1018_rotate1__is__Nil__conv,axiom,
! [Xs: list_a] :
( ( ( rotate1_a @ Xs )
= nil_a )
= ( Xs = nil_a ) ) ).
% rotate1_is_Nil_conv
thf(fact_1019_list__update__length,axiom,
! [Xs: list_int,X: int,Ys: list_int,Y: int] :
( ( list_update_int @ ( append_int @ Xs @ ( cons_int @ X @ Ys ) ) @ ( size_size_list_int @ Xs ) @ Y )
= ( append_int @ Xs @ ( cons_int @ Y @ Ys ) ) ) ).
% list_update_length
thf(fact_1020_list__update__code_I3_J,axiom,
! [X: int,Xs: list_int,I3: nat,Y: int] :
( ( list_update_int @ ( cons_int @ X @ Xs ) @ ( suc @ I3 ) @ Y )
= ( cons_int @ X @ ( list_update_int @ Xs @ I3 @ Y ) ) ) ).
% list_update_code(3)
thf(fact_1021_list__update__code_I2_J,axiom,
! [X: int,Xs: list_int,Y: int] :
( ( list_update_int @ ( cons_int @ X @ Xs ) @ zero_zero_nat @ Y )
= ( cons_int @ Y @ Xs ) ) ).
% list_update_code(2)
thf(fact_1022_list__update_Osimps_I1_J,axiom,
! [I3: nat,V: a] :
( ( list_update_a @ nil_a @ I3 @ V )
= nil_a ) ).
% list_update.simps(1)
thf(fact_1023_list__update__code_I1_J,axiom,
! [I3: nat,Y: a] :
( ( list_update_a @ nil_a @ I3 @ Y )
= nil_a ) ).
% list_update_code(1)
thf(fact_1024_rotate1_Osimps_I1_J,axiom,
( ( rotate1_a @ nil_a )
= nil_a ) ).
% rotate1.simps(1)
thf(fact_1025_rotate1_Osimps_I2_J,axiom,
! [X: a,Xs: list_a] :
( ( rotate1_a @ ( cons_a @ X @ Xs ) )
= ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) ) ).
% rotate1.simps(2)
thf(fact_1026_rotate1_Osimps_I2_J,axiom,
! [X: int,Xs: list_int] :
( ( rotate1_int @ ( cons_int @ X @ Xs ) )
= ( append_int @ Xs @ ( cons_int @ X @ nil_int ) ) ) ).
% rotate1.simps(2)
thf(fact_1027_last__list__update,axiom,
! [Xs: list_a,K: nat,X: a] :
( ( Xs != nil_a )
=> ( ( ( K
= ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) )
=> ( ( last_a @ ( list_update_a @ Xs @ K @ X ) )
= X ) )
& ( ( K
!= ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) )
=> ( ( last_a @ ( list_update_a @ Xs @ K @ X ) )
= ( last_a @ Xs ) ) ) ) ) ).
% last_list_update
thf(fact_1028_SuccD,axiom,
! [K: a,Kl: set_list_a,Kl2: list_a] :
( ( member_a @ K @ ( bNF_Greatest_Succ_a @ Kl @ Kl2 ) )
=> ( member_list_a @ ( append_a @ Kl2 @ ( cons_a @ K @ nil_a ) ) @ Kl ) ) ).
% SuccD
thf(fact_1029_SuccD,axiom,
! [K: int,Kl: set_list_int,Kl2: list_int] :
( ( member_int @ K @ ( bNF_Gr6350390219475566417cc_int @ Kl @ Kl2 ) )
=> ( member_list_int @ ( append_int @ Kl2 @ ( cons_int @ K @ nil_int ) ) @ Kl ) ) ).
% SuccD
thf(fact_1030_SuccI,axiom,
! [Kl2: list_a,K: a,Kl: set_list_a] :
( ( member_list_a @ ( append_a @ Kl2 @ ( cons_a @ K @ nil_a ) ) @ Kl )
=> ( member_a @ K @ ( bNF_Greatest_Succ_a @ Kl @ Kl2 ) ) ) ).
% SuccI
thf(fact_1031_SuccI,axiom,
! [Kl2: list_int,K: int,Kl: set_list_int] :
( ( member_list_int @ ( append_int @ Kl2 @ ( cons_int @ K @ nil_int ) ) @ Kl )
=> ( member_int @ K @ ( bNF_Gr6350390219475566417cc_int @ Kl @ Kl2 ) ) ) ).
% SuccI
thf(fact_1032_empty__Shift,axiom,
! [Kl: set_list_a,K: a] :
( ( member_list_a @ nil_a @ Kl )
=> ( ( member_a @ K @ ( bNF_Greatest_Succ_a @ Kl @ nil_a ) )
=> ( member_list_a @ nil_a @ ( bNF_Greatest_Shift_a @ Kl @ K ) ) ) ) ).
% empty_Shift
thf(fact_1033_Succ__Shift,axiom,
! [Kl: set_list_int,K: int,Kl2: list_int] :
( ( bNF_Gr6350390219475566417cc_int @ ( bNF_Gr1870224194279859149ft_int @ Kl @ K ) @ Kl2 )
= ( bNF_Gr6350390219475566417cc_int @ Kl @ ( cons_int @ K @ Kl2 ) ) ) ).
% Succ_Shift
thf(fact_1034_ShiftD,axiom,
! [Kl2: list_int,Kl: set_list_int,K: int] :
( ( member_list_int @ Kl2 @ ( bNF_Gr1870224194279859149ft_int @ Kl @ K ) )
=> ( member_list_int @ ( cons_int @ K @ Kl2 ) @ Kl ) ) ).
% ShiftD
thf(fact_1035_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 )
=> ( ! [I5: nat] :
( ( ord_less_nat @ K2 @ I5 )
=> ( P @ I5 ) )
=> ( P @ K2 ) ) )
=> ( P @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_1036_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_1037_abs__idempotent,axiom,
! [A: int] :
( ( abs_abs_int @ ( abs_abs_int @ A ) )
= ( abs_abs_int @ A ) ) ).
% abs_idempotent
thf(fact_1038_abs__0,axiom,
( ( abs_abs_int @ zero_zero_int )
= zero_zero_int ) ).
% abs_0
thf(fact_1039_abs__zero,axiom,
( ( abs_abs_int @ zero_zero_int )
= zero_zero_int ) ).
% abs_zero
thf(fact_1040_abs__eq__0,axiom,
! [A: int] :
( ( ( abs_abs_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% abs_eq_0
thf(fact_1041_abs__0__eq,axiom,
! [A: int] :
( ( zero_zero_int
= ( abs_abs_int @ A ) )
= ( A = zero_zero_int ) ) ).
% abs_0_eq
thf(fact_1042_abs__minus__cancel,axiom,
! [A: int] :
( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
= ( abs_abs_int @ A ) ) ).
% abs_minus_cancel
thf(fact_1043_abs__of__nat,axiom,
! [N: nat] :
( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N ) )
= ( semiri1314217659103216013at_int @ N ) ) ).
% abs_of_nat
thf(fact_1044_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_1045_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_1046_abs__of__nonneg,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( abs_abs_int @ A )
= A ) ) ).
% abs_of_nonneg
thf(fact_1047_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_1048_abs__neg__one,axiom,
( ( abs_abs_int @ ( uminus_uminus_int @ one_one_int ) )
= one_one_int ) ).
% abs_neg_one
thf(fact_1049_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_1050_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_1051_zabs__less__one__iff,axiom,
! [Z2: int] :
( ( ord_less_int @ ( abs_abs_int @ Z2 ) @ one_one_int )
= ( Z2 = zero_zero_int ) ) ).
% zabs_less_one_iff
thf(fact_1052_Cons__listrel1__Cons,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 ) ) @ ( listrel1_int @ R ) )
= ( ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R )
& ( Xs = Ys ) )
| ( ( X = Y )
& ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( listrel1_int @ R ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_1053_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_1054_abs__ge__self,axiom,
! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).
% abs_ge_self
thf(fact_1055_abs__ge__zero,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A ) ) ).
% abs_ge_zero
thf(fact_1056_abs__not__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( abs_abs_int @ A ) @ zero_zero_int ) ).
% abs_not_less_zero
thf(fact_1057_abs__of__pos,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( abs_abs_int @ A )
= A ) ) ).
% abs_of_pos
thf(fact_1058_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_1059_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_1060_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_1061_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_1062_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_1063_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_1064_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_1065_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_1066_abs__eq__0__iff,axiom,
! [A: int] :
( ( ( abs_abs_int @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% abs_eq_0_iff
thf(fact_1067_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_1068_not__Nil__listrel1,axiom,
! [Xs: list_a,R: set_Product_prod_a_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ nil_a @ Xs ) @ ( listrel1_a @ R ) ) ).
% not_Nil_listrel1
thf(fact_1069_not__listrel1__Nil,axiom,
! [Xs: list_a,R: set_Product_prod_a_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ nil_a ) @ ( listrel1_a @ R ) ) ).
% not_listrel1_Nil
thf(fact_1070_listrel1I2,axiom,
! [Xs: list_int,Ys: list_int,R: set_Pr958786334691620121nt_int,X: int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( listrel1_int @ R ) )
=> ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ X @ Xs ) @ ( cons_int @ X @ Ys ) ) @ ( listrel1_int @ R ) ) ) ).
% listrel1I2
thf(fact_1071_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_1072_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_1073_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_1074_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_1075_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_1076_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_1077_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_1078_Cons__listrel1E2,axiom,
! [Xs: list_int,Y: int,Ys: list_int,R: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ ( cons_int @ Y @ Ys ) ) @ ( listrel1_int @ R ) )
=> ( ! [X3: int] :
( ( Xs
= ( cons_int @ X3 @ Ys ) )
=> ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y ) @ R ) )
=> ~ ! [Zs2: list_int] :
( ( Xs
= ( cons_int @ Y @ Zs2 ) )
=> ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Zs2 @ Ys ) @ ( listrel1_int @ R ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_1079_Cons__listrel1E1,axiom,
! [X: int,Xs: list_int,Ys: list_int,R: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ X @ Xs ) @ Ys ) @ ( listrel1_int @ R ) )
=> ( ! [Y2: int] :
( ( Ys
= ( cons_int @ Y2 @ Xs ) )
=> ~ ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y2 ) @ R ) )
=> ~ ! [Zs2: list_int] :
( ( Ys
= ( cons_int @ X @ Zs2 ) )
=> ~ ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Zs2 ) @ ( listrel1_int @ R ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_1080_listrel1I1,axiom,
! [X: int,Y: int,R: set_Pr958786334691620121nt_int,Xs: list_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R )
=> ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( cons_int @ X @ Xs ) @ ( cons_int @ Y @ Xs ) ) @ ( listrel1_int @ R ) ) ) ).
% listrel1I1
thf(fact_1081_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_1082_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_1083_listrel1I,axiom,
! [X: int,Y: int,R: set_Pr958786334691620121nt_int,Xs: list_int,Us: list_int,Vs: list_int,Ys: list_int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R )
=> ( ( Xs
= ( append_int @ Us @ ( cons_int @ X @ Vs ) ) )
=> ( ( Ys
= ( append_int @ Us @ ( cons_int @ Y @ Vs ) ) )
=> ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( listrel1_int @ R ) ) ) ) ) ).
% listrel1I
thf(fact_1084_listrel1E,axiom,
! [Xs: list_int,Ys: list_int,R: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( listrel1_int @ R ) )
=> ~ ! [X3: int,Y2: int] :
( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y2 ) @ R )
=> ! [Us2: list_int,Vs2: list_int] :
( ( Xs
= ( append_int @ Us2 @ ( cons_int @ X3 @ Vs2 ) ) )
=> ( Ys
!= ( append_int @ Us2 @ ( cons_int @ Y2 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_1085_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_1086_snoc__listrel1__snoc__iff,axiom,
! [Xs: list_a,X: a,Ys: list_a,Y: a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) @ ( append_a @ Ys @ ( cons_a @ Y @ nil_a ) ) ) @ ( listrel1_a @ R ) )
= ( ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel1_a @ R ) )
& ( X = Y ) )
| ( ( Xs = Ys )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_1087_snoc__listrel1__snoc__iff,axiom,
! [Xs: list_int,X: int,Ys: list_int,Y: int,R: set_Pr958786334691620121nt_int] :
( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ ( append_int @ Xs @ ( cons_int @ X @ nil_int ) ) @ ( append_int @ Ys @ ( cons_int @ Y @ nil_int ) ) ) @ ( listrel1_int @ R ) )
= ( ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( listrel1_int @ R ) )
& ( X = Y ) )
| ( ( Xs = Ys )
& ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_1088_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_1089_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_1090_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 )
& ! [X5: int] :
( ( ( ord_less_eq_int @ A @ X5 )
& ( ord_less_int @ X5 @ C3 ) )
=> ( P @ X5 ) )
& ! [D2: int] :
( ! [X3: int] :
( ( ( ord_less_eq_int @ A @ X3 )
& ( ord_less_int @ X3 @ D2 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_int @ D2 @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_1091_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 )
& ! [X5: nat] :
( ( ( ord_less_eq_nat @ A @ X5 )
& ( ord_less_nat @ X5 @ C3 ) )
=> ( P @ X5 ) )
& ! [D2: nat] :
( ! [X3: nat] :
( ( ( ord_less_eq_nat @ A @ X3 )
& ( ord_less_nat @ X3 @ D2 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq_nat @ D2 @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_1092_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_1093_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_1094_add__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_1095_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_1096_add__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_1097_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_1098_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_1099_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_1100_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_1101_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_1102_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_1103_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_1104_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_1105_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_1106_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_1107_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_1108_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_1109_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_1110_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_1111_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_1112_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_1113_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_1114_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_1115_add__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_1116_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_1117_add__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_1118_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_1119_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_1120_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_1121_add__diff__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_1122_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_1123_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_1124_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_1125_add__diff__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_1126_diff__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_1127_add__diff__cancel,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_1128_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_1129_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_1130_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_1131_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_1132_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_1133_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% Nat.add_0_right
thf(fact_1134_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_1135_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_1136_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_1137_diff__diff__left,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J2 ) @ K )
= ( minus_minus_nat @ I3 @ ( plus_plus_nat @ J2 @ K ) ) ) ).
% diff_diff_left
thf(fact_1138_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_1139_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_1140_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_1141_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_1142_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_1143_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_1144_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_1145_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_1146_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_1147_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_1148_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_1149_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_1150_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_1151_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_1152_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_1153_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_1154_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_1155_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_1156_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_1157_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_1158_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_1159_add_Oright__inverse,axiom,
! [A: int] :
( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
= zero_zero_int ) ).
% add.right_inverse
thf(fact_1160_ab__left__minus,axiom,
! [A: int] :
( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
= zero_zero_int ) ).
% ab_left_minus
thf(fact_1161_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_1162_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_1163_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_1164_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_1165_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_1166_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( plus_plus_nat @ I3 @ ( minus_minus_nat @ J2 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I3 @ J2 ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1167_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K ) @ I3 )
= ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I3 ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1168_Nat_Odiff__diff__right,axiom,
! [K: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ I3 @ ( minus_minus_nat @ J2 @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I3 @ K ) @ J2 ) ) ) ).
% Nat.diff_diff_right
thf(fact_1169_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_1170_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_1171_of__nat__Suc,axiom,
! [M2: nat] :
( ( semiri1316708129612266289at_nat @ ( suc @ M2 ) )
= ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M2 ) ) ) ).
% of_nat_Suc
thf(fact_1172_of__nat__Suc,axiom,
! [M2: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ M2 ) )
= ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M2 ) ) ) ).
% of_nat_Suc
thf(fact_1173_diff__Suc__diff__eq1,axiom,
! [K: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ I3 @ ( suc @ ( minus_minus_nat @ J2 @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I3 @ K ) @ ( suc @ J2 ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1174_diff__Suc__diff__eq2,axiom,
! [K: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J2 @ K ) ) @ I3 )
= ( minus_minus_nat @ ( suc @ J2 ) @ ( plus_plus_nat @ K @ I3 ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1175_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_1176_group__cancel_Osub1,axiom,
! [A2: int,K: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( minus_minus_int @ A2 @ B )
= ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_1177_diff__eq__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( minus_minus_int @ A @ B )
= C )
= ( A
= ( plus_plus_int @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_1178_eq__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( A
= ( minus_minus_int @ C @ B ) )
= ( ( plus_plus_int @ A @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_1179_add__diff__eq,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_1180_diff__diff__eq2,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_1181_diff__add__eq,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_1182_diff__add__eq__diff__diff__swap,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_1183_add__implies__diff,axiom,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C @ B )
= A )
=> ( C
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_1184_add__implies__diff,axiom,
! [C: int,B: int,A: int] :
( ( ( plus_plus_int @ C @ B )
= A )
=> ( C
= ( minus_minus_int @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_1185_diff__diff__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_1186_diff__diff__eq,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_1187_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_1188_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_1189_diff__add__inverse,axiom,
! [N: nat,M2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M2 ) @ N )
= M2 ) ).
% diff_add_inverse
thf(fact_1190_diff__add__inverse2,axiom,
! [M2: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N ) @ N )
= M2 ) ).
% diff_add_inverse2
thf(fact_1191_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_1192_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I3: nat,J2: nat,K: nat,L: nat] :
( ( ( I3 = J2 )
& ( K = L ) )
=> ( ( plus_plus_nat @ I3 @ K )
= ( plus_plus_nat @ J2 @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_1193_group__cancel_Oadd1,axiom,
! [A2: nat,K: nat,A: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_1194_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_1195_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_1196_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A4: nat,B3: nat] : ( plus_plus_nat @ B3 @ A4 ) ) ) ).
% add.commute
thf(fact_1197_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_1198_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_1199_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_1200_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1201_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_1202_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_1203_add__Suc,axiom,
! [M2: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M2 ) @ N )
= ( suc @ ( plus_plus_nat @ M2 @ N ) ) ) ).
% add_Suc
thf(fact_1204_nat__arith_Osuc1,axiom,
! [A2: nat,K: nat,A: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A2 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_1205_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_1206_group__cancel_Oneg1,axiom,
! [A2: int,K: int,A: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( uminus_uminus_int @ A2 )
= ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).
% group_cancel.neg1
thf(fact_1207_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_1208_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_1209_verit__sum__simplify,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% verit_sum_simplify
thf(fact_1210_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_1211_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_1212_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_1213_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_1214_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_1215_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1216_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1217_Suc__eq__plus1,axiom,
( suc
= ( ^ [N3: nat] : ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1218_Nat_Ole__imp__diff__is__add,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ( minus_minus_nat @ J2 @ I3 )
= K )
= ( J2
= ( plus_plus_nat @ K @ I3 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1219_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I3 ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K ) @ I3 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1220_Nat_Odiff__add__assoc,axiom,
! [K: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I3 @ J2 ) @ K )
= ( plus_plus_nat @ I3 @ ( minus_minus_nat @ J2 @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1221_Nat_Ole__diff__conv2,axiom,
! [K: nat,J2: nat,I3: nat] :
( ( ord_less_eq_nat @ K @ J2 )
=> ( ( ord_less_eq_nat @ I3 @ ( minus_minus_nat @ J2 @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ K ) @ J2 ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1222_le__diff__conv,axiom,
! [J2: nat,K: nat,I3: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J2 @ K ) @ I3 )
= ( ord_less_eq_nat @ J2 @ ( plus_plus_nat @ I3 @ K ) ) ) ).
% le_diff_conv
thf(fact_1223_less__diff__conv,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( ord_less_nat @ I3 @ ( minus_minus_nat @ J2 @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ J2 ) ) ).
% less_diff_conv
thf(fact_1224_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_1225_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_1226_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_1227_less__imp__add__positive,axiom,
! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I3 @ K2 )
= J2 ) ) ) ).
% less_imp_add_positive
thf(fact_1228_less__natE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_nat @ M2 @ N )
=> ~ ! [Q3: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M2 @ Q3 ) ) ) ) ).
% less_natE
thf(fact_1229_less__add__Suc1,axiom,
! [I3: nat,M2: nat] : ( ord_less_nat @ I3 @ ( suc @ ( plus_plus_nat @ I3 @ M2 ) ) ) ).
% less_add_Suc1
thf(fact_1230_less__add__Suc2,axiom,
! [I3: nat,M2: nat] : ( ord_less_nat @ I3 @ ( suc @ ( plus_plus_nat @ M2 @ I3 ) ) ) ).
% less_add_Suc2
thf(fact_1231_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_1232_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_1233_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_1234_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_1235_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A4: int,B3: int] : ( plus_plus_int @ A4 @ ( uminus_uminus_int @ B3 ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_1236_diff__conv__add__uminus,axiom,
( minus_minus_int
= ( ^ [A4: int,B3: int] : ( plus_plus_int @ A4 @ ( uminus_uminus_int @ B3 ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_1237_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_1238_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_1239_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_1240_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_1241_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_1242_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_1243_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_1244_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_1245_diff__less__eq,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C )
= ( ord_less_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% diff_less_eq
thf(fact_1246_less__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ A @ ( minus_minus_int @ C @ B ) )
= ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% less_diff_eq
thf(fact_1247_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_1248_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_1249_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_1250_less__add__one,axiom,
! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% less_add_one
thf(fact_1251_diff__le__eq,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C )
= ( ord_less_eq_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% diff_le_eq
thf(fact_1252_le__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ ( minus_minus_int @ C @ B ) )
= ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% le_diff_eq
thf(fact_1253_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_1254_le__add__diff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% le_add_diff
thf(fact_1255_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1256_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1257_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1258_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1259_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1260_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1261_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_1262_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C )
= ( B
= ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
% 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,
type_i6141643110573041459rent_a @ current ).
%------------------------------------------------------------------------------