TPTP Problem File: SLH0276^1.p
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%------------------------------------------------------------------------------
% 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/0026_RealTimeDeque_Dequeue_Proof/prob_00272_009482__7185286_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1098 ( 492 unt; 98 typ; 0 def)
% Number of atoms : 2556 ( 995 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 10360 ( 288 ~; 53 |; 107 &;8794 @)
% ( 0 <=>;1118 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 7 avg)
% Number of types : 20 ( 19 usr)
% Number of type conns : 231 ( 231 >; 0 *; 0 +; 0 <<)
% Number of symbols : 82 ( 79 usr; 14 con; 0-4 aty)
% Number of variables : 2689 ( 91 ^;2571 !; 27 ?;2689 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:53:06.244
%------------------------------------------------------------------------------
% Could-be-implicit typings (19)
thf(ty_n_t__Product____Type__Oprod_It__States__Ostates_Itf__a_J_Mt__Nat__Onat_J,type,
produc1571854377283420419_a_nat: $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__Product____Type__Oprod_Itf__a_Mt__Small__Ostate_Itf__a_J_J,type,
produc7589950997499123219tate_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__a_Mt__Idle__Oidle_Itf__a_J_J,type,
produc7590564867095724333idle_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_Itf__a_Mt__Big__Ostate_Itf__a_J_J,type,
produc6972303929186420058tate_a: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
product_prod_num_num: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
product_prod_nat_nat: $tType ).
thf(ty_n_t__Current__Ocurrent_Itf__a_J,type,
current_a: $tType ).
thf(ty_n_t__States__Ostates_Itf__a_J,type,
states_a: $tType ).
thf(ty_n_t__Stack__Ostack_Itf__a_J,type,
stack_a: $tType ).
thf(ty_n_t__Small__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__Idle__Oidle_Itf__a_J,type,
idle_a: $tType ).
thf(ty_n_t__Big__Ostate_Itf__a_J,type,
state_a2: $tType ).
thf(ty_n_t__States__Odirection,type,
direction: $tType ).
thf(ty_n_t__String__Ochar,type,
char: $tType ).
thf(ty_n_t__Num__Onum,type,
num: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (79)
thf(sy_c_Big_Opop_001tf__a,type,
pop_a: state_a2 > produc6972303929186420058tate_a ).
thf(sy_c_Big_Opush_001tf__a,type,
push_a: a > state_a2 > state_a2 ).
thf(sy_c_Big_Ostate_OReverse_001tf__a,type,
reverse_a: current_a > stack_a > list_a > nat > state_a2 ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand_001t__Nat__Onat,type,
bit_se727722235901077358nd_nat: nat > nat > nat ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Nat__Onat,type,
bit_se2002935070580805687sk_nat: nat > nat ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Nat__Onat,type,
bit_se1412395901928357646or_nat: nat > nat > nat ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Nat__Onat,type,
bit_se6528837805403552850or_nat: nat > nat > nat ).
thf(sy_c_Current_Ocurrent_OCurrent_001tf__a,type,
current_a2: list_a > nat > stack_a > nat > current_a ).
thf(sy_c_Euclidean__Division_Oeuclidean__semiring__class_Oeuclidean__size_001t__Nat__Onat,type,
euclid4777050414544973029ze_nat: nat > nat ).
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__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Num__Onum,type,
plus_plus_num: num > num > num ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Num__Onum,type,
times_times_num: num > num > num ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_Idle_Opop_001tf__a,type,
pop_a2: idle_a > produc7590564867095724333idle_a ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
if_Pro6206227464963214023at_nat: $o > product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ).
thf(sy_c_List_Olist_ONil_001tf__a,type,
nil_a: list_a ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Ocompow_001_062_It__Nat__Onat_Mt__Nat__Onat_J,type,
compow_nat_nat: nat > ( nat > nat ) > nat > nat ).
thf(sy_c_Nat_Ocompow_001_062_It__States__Ostates_Itf__a_J_Mt__States__Ostates_Itf__a_J_J,type,
compow495008222514391794ates_a: nat > ( states_a > states_a ) > states_a > states_a ).
thf(sy_c_Nat_Ofunpow_001t__Nat__Onat,type,
funpow_nat: nat > ( nat > nat ) > nat > nat ).
thf(sy_c_Nat_Ofunpow_001t__States__Ostates_Itf__a_J,type,
funpow_states_a: nat > ( states_a > states_a ) > states_a > states_a ).
thf(sy_c_Nat_Osize__class_Osize_001t__Big__Ostate_Itf__a_J,type,
size_size_state_a: state_a2 > 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__Idle__Oidle_Itf__a_J,type,
size_size_idle_a: idle_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__Num__Onum,type,
size_size_num: num > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Small__Ostate_Itf__a_J,type,
size_size_state_a2: state_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Stack__Ostack_Itf__a_J,type,
size_size_stack_a: stack_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__States__Odirection,type,
size_size_direction: direction > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__States__Ostates_Itf__a_J,type,
size_size_states_a: states_a > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
size_size_char: char > nat ).
thf(sy_c_Num_Onum_OBit0,type,
bit0: num > num ).
thf(sy_c_Num_Onum_OBit1,type,
bit1: num > num ).
thf(sy_c_Num_Onum_OOne,type,
one: num ).
thf(sy_c_Num_Onum_Osize__num,type,
size_num: num > nat ).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
numeral_numeral_nat: num > nat ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum,type,
ord_less_num: num > num > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum,type,
ord_less_eq_num: num > num > $o ).
thf(sy_c_Parity_Ounique__euclidean__semiring__with__nat__division__class_Odivides__aux_001t__Nat__Onat,type,
unique5332122412489317741ux_nat: product_prod_nat_nat > $o ).
thf(sy_c_Parity_Ounique__euclidean__semiring__with__nat__division__class_Odivmod_001t__Nat__Onat,type,
unique5405566460079783412od_nat: num > num > product_prod_nat_nat ).
thf(sy_c_Parity_Ounique__euclidean__semiring__with__nat__division__class_Odivmod__step_001t__Nat__Onat,type,
unique4036640087844771520ep_nat: nat > product_prod_nat_nat > product_prod_nat_nat ).
thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).
thf(sy_c_Product__Type_OPair_001t__Num__Onum_001t__Num__Onum,type,
product_Pair_num_num: num > num > product_prod_num_num ).
thf(sy_c_Product__Type_OPair_001t__States__Ostates_Itf__a_J_001t__Nat__Onat,type,
produc1877401315875745917_a_nat: states_a > nat > produc1571854377283420419_a_nat ).
thf(sy_c_Product__Type_OPair_001tf__a_001t__Big__Ostate_Itf__a_J,type,
produc8641956578966763338tate_a: a > state_a2 > produc6972303929186420058tate_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__Idle__Oidle_Itf__a_J,type,
produc1265230069547855005idle_a: a > idle_a > produc7590564867095724333idle_a ).
thf(sy_c_Product__Type_OPair_001tf__a_001t__Small__Ostate_Itf__a_J,type,
produc1224139502141355779tate_a: a > state_a > produc7589950997499123219tate_a ).
thf(sy_c_Small_Opop_001tf__a,type,
pop_a3: state_a > produc7589950997499123219tate_a ).
thf(sy_c_Small_Opush_001tf__a,type,
push_a2: a > state_a > state_a ).
thf(sy_c_Small_Ostate_OReverse1_001tf__a,type,
reverse1_a: current_a > stack_a > list_a > state_a ).
thf(sy_c_States_Odirection_OLeft,type,
left: direction ).
thf(sy_c_States_Odirection_Osize__direction,type,
size_direction: direction > nat ).
thf(sy_c_States_Ostates_OStates_001tf__a,type,
states_a2: direction > state_a2 > state_a > states_a ).
thf(sy_c_States_Ostates_Osize__states_001tf__a,type,
size_states_a: ( a > nat ) > states_a > nat ).
thf(sy_c_States__Aux_OlistL_001tf__a,type,
states_listL_a: states_a > list_a ).
thf(sy_c_States__Aux_Osize__ok_H_001tf__a,type,
states_size_ok_a: states_a > nat > $o ).
thf(sy_c_States__Aux_Osize__ok_H__rel_001tf__a,type,
states_size_ok_rel_a: produc1571854377283420419_a_nat > produc1571854377283420419_a_nat > $o ).
thf(sy_c_String_Ochar_Osize__char,type,
size_char: char > nat ).
thf(sy_c_Type__Classes_Oinvar__class_Oinvar_001t__Idle__Oidle_Itf__a_J,type,
type_i8151583586401621767idle_a: idle_a > $o ).
thf(sy_c_Type__Classes_Oinvar__class_Oinvar_001t__States__Ostates_Itf__a_J,type,
type_i8221491762852169479ates_a: states_a > $o ).
thf(sy_c_Type__Classes_Oremaining__steps__class_Oremaining__steps_001t__States__Ostates_Itf__a_J,type,
type_r4519047461186610747ates_a: states_a > nat ).
thf(sy_c_Type__Classes_Osize__new__class_Osize__new_001t__Big__Ostate_Itf__a_J,type,
type_s6530235180886170618tate_a: state_a2 > nat ).
thf(sy_c_Type__Classes_Osize__new__class_Osize__new_001t__Small__Ostate_Itf__a_J,type,
type_s6404775287138157491tate_a: state_a > nat ).
thf(sy_c_Type__Classes_Ostep__class_Ostep_001t__States__Ostates_Itf__a_J,type,
type_s4923920245906622843ates_a: states_a > states_a ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__States__Ostates_Itf__a_J_Mt__Nat__Onat_J,type,
accp_P8670043419270033210_a_nat: ( produc1571854377283420419_a_nat > produc1571854377283420419_a_nat > $o ) > produc1571854377283420419_a_nat > $o ).
thf(sy_v_left_H____,type,
left2: idle_a ).
thf(sy_v_left____,type,
left3: idle_a ).
thf(sy_v_length__left_H____,type,
length_left: nat ).
thf(sy_v_length__right____,type,
length_right: nat ).
thf(sy_v_right____,type,
right: stack_a ).
thf(sy_v_stack__left_H____,type,
stack_left: stack_a ).
thf(sy_v_x____,type,
x: a ).
% Relevant facts (994)
thf(fact_0_size__ok_H,axiom,
states_size_ok_a @ ( states_a2 @ left @ ( reverse_a @ ( current_a2 @ nil_a @ zero_zero_nat @ right @ ( minus_minus_nat @ ( size_size_stack_a @ right ) @ ( suc @ length_left ) ) ) @ right @ nil_a @ ( minus_minus_nat @ ( size_size_stack_a @ right ) @ ( suc @ length_left ) ) ) @ ( reverse1_a @ ( current_a2 @ nil_a @ zero_zero_nat @ stack_left @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ length_left ) ) ) @ stack_left @ nil_a ) ) @ ( minus_minus_nat @ ( type_r4519047461186610747ates_a @ ( states_a2 @ left @ ( reverse_a @ ( current_a2 @ nil_a @ zero_zero_nat @ right @ ( minus_minus_nat @ ( size_size_stack_a @ right ) @ ( suc @ length_left ) ) ) @ right @ nil_a @ ( minus_minus_nat @ ( size_size_stack_a @ right ) @ ( suc @ length_left ) ) ) @ ( reverse1_a @ ( current_a2 @ nil_a @ zero_zero_nat @ stack_left @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ length_left ) ) ) @ stack_left @ nil_a ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ one ) ) ) ) ).
% size_ok'
thf(fact_1_remaining__steps__end,axiom,
ord_less_nat @ zero_zero_nat @ ( type_r4519047461186610747ates_a @ ( compow495008222514391794ates_a @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ one ) ) ) @ type_s4923920245906622843ates_a @ ( states_a2 @ left @ ( reverse_a @ ( current_a2 @ nil_a @ zero_zero_nat @ right @ ( minus_minus_nat @ ( size_size_stack_a @ right ) @ ( suc @ length_left ) ) ) @ right @ nil_a @ ( minus_minus_nat @ ( size_size_stack_a @ right ) @ ( suc @ length_left ) ) ) @ ( reverse1_a @ ( current_a2 @ nil_a @ zero_zero_nat @ stack_left @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ length_left ) ) ) @ stack_left @ nil_a ) ) ) ) ).
% remaining_steps_end
thf(fact_2_remaining__steps,axiom,
ord_less_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ one ) ) ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ left @ ( reverse_a @ ( current_a2 @ nil_a @ zero_zero_nat @ right @ ( minus_minus_nat @ ( size_size_stack_a @ right ) @ ( suc @ length_left ) ) ) @ right @ nil_a @ ( minus_minus_nat @ ( size_size_stack_a @ right ) @ ( suc @ length_left ) ) ) @ ( reverse1_a @ ( current_a2 @ nil_a @ zero_zero_nat @ stack_left @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ length_left ) ) ) @ stack_left @ nil_a ) ) ) ).
% remaining_steps
thf(fact_3_invar__stepped,axiom,
type_i8221491762852169479ates_a @ ( compow495008222514391794ates_a @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ one ) ) ) @ type_s4923920245906622843ates_a @ ( states_a2 @ left @ ( reverse_a @ ( current_a2 @ nil_a @ zero_zero_nat @ right @ ( minus_minus_nat @ ( size_size_stack_a @ right ) @ ( suc @ length_left ) ) ) @ right @ nil_a @ ( minus_minus_nat @ ( size_size_stack_a @ right ) @ ( suc @ length_left ) ) ) @ ( reverse1_a @ ( current_a2 @ nil_a @ zero_zero_nat @ stack_left @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ length_left ) ) ) @ stack_left @ nil_a ) ) ) ).
% invar_stepped
thf(fact_4_invar,axiom,
type_i8221491762852169479ates_a @ ( states_a2 @ left @ ( reverse_a @ ( current_a2 @ nil_a @ zero_zero_nat @ right @ ( minus_minus_nat @ ( size_size_stack_a @ right ) @ ( suc @ length_left ) ) ) @ right @ nil_a @ ( minus_minus_nat @ ( size_size_stack_a @ right ) @ ( suc @ length_left ) ) ) @ ( reverse1_a @ ( current_a2 @ nil_a @ zero_zero_nat @ stack_left @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ length_left ) ) ) @ stack_left @ nil_a ) ) ).
% invar
thf(fact_5_mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_6_one__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M @ N ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_7_funpow__0,axiom,
! [F: states_a > states_a,X: states_a] :
( ( compow495008222514391794ates_a @ zero_zero_nat @ F @ X )
= X ) ).
% funpow_0
thf(fact_8_funpow__0,axiom,
! [F: nat > nat,X: nat] :
( ( compow_nat_nat @ zero_zero_nat @ F @ X )
= X ) ).
% funpow_0
thf(fact_9_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_10_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_11_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_12_numeral__eq__iff,axiom,
! [M: num,N: num] :
( ( ( numeral_numeral_nat @ M )
= ( numeral_numeral_nat @ N ) )
= ( M = N ) ) ).
% numeral_eq_iff
thf(fact_13_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_14_nat_Oinject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
= ( X2 = Y2 ) ) ).
% nat.inject
thf(fact_15_numeral__less__iff,axiom,
! [M: num,N: num] :
( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% numeral_less_iff
thf(fact_16_mult__numeral__left__semiring__numeral,axiom,
! [V: num,W: num,Z: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( times_times_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
= ( times_times_nat @ ( numeral_numeral_nat @ ( times_times_num @ V @ W ) ) @ Z ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_17_numeral__times__numeral,axiom,
! [M: num,N: num] :
( ( times_times_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( times_times_num @ M @ N ) ) ) ).
% numeral_times_numeral
thf(fact_18_num__double,axiom,
! [N: num] :
( ( times_times_num @ ( bit0 @ one ) @ N )
= ( bit0 @ N ) ) ).
% num_double
thf(fact_19_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_20_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_21_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_22_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_23_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_24_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_25_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_26_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_27_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_28_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_29_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_30_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_31_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_32_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_33_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_34_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_35_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_36_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N2 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_37_lift__Suc__mono__less,axiom,
! [F: nat > num,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N2 )
=> ( ord_less_num @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_38_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_39_lift__Suc__mono__less__iff,axiom,
! [F: nat > num,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_num @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_40_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_41_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_42_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_43_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_44_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_45_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
=> ( P @ M2 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_46_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_47_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_48_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_49_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N3 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_50_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_51_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_52_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_53_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_54_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_55_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_56_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_57_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_58_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P @ I2 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I2 @ K2 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_59_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_60_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_61_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_62_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M3: nat] :
( ( M
= ( suc @ M3 ) )
& ( ord_less_nat @ N @ M3 ) ) ) ) ).
% Suc_less_eq2
thf(fact_63_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ I3 ) ) ) ) ).
% All_less_Suc
thf(fact_64_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_65_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_66_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ N )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P @ I3 ) ) ) ) ).
% Ex_less_Suc
thf(fact_67_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_68_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_69_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_70_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_71_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_72_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_73_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_74_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_75_not__numeral__less__zero,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_less_zero
thf(fact_76_zero__less__numeral,axiom,
! [N: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_less_numeral
thf(fact_77_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J3: nat] :
( ( M
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_78_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% gr0_implies_Suc
thf(fact_79_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ ( suc @ I3 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_80_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_81_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P @ ( suc @ I3 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_82_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_83_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_84_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
= ( minus_minus_nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_85_mult__less__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_86_mult__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_87_Suc__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_88_diff__Suc__less,axiom,
! [N: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_89_n__less__n__mult__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
% n_less_n_mult_m
thf(fact_90_n__less__m__mult__n,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
% n_less_m_mult_n
thf(fact_91_one__less__mult,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).
% one_less_mult
thf(fact_92_size__neq__size__imp__neq,axiom,
! [X: state_a2,Y: state_a2] :
( ( ( size_size_state_a @ X )
!= ( size_size_state_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_93_size__neq__size__imp__neq,axiom,
! [X: state_a,Y: state_a] :
( ( ( size_size_state_a2 @ X )
!= ( size_size_state_a2 @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_94_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_95_size__neq__size__imp__neq,axiom,
! [X: states_a,Y: states_a] :
( ( ( size_size_states_a @ X )
!= ( size_size_states_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_96_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_97_size__neq__size__imp__neq,axiom,
! [X: stack_a,Y: stack_a] :
( ( ( size_size_stack_a @ X )
!= ( size_size_stack_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_98_size__neq__size__imp__neq,axiom,
! [X: num,Y: num] :
( ( ( size_size_num @ X )
!= ( size_size_num @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_99_size__neq__size__imp__neq,axiom,
! [X: direction,Y: direction] :
( ( ( size_size_direction @ X )
!= ( size_size_direction @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_100_size__neq__size__imp__neq,axiom,
! [X: idle_a,Y: idle_a] :
( ( ( size_size_idle_a @ X )
!= ( size_size_idle_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_101_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_102_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_103_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_104_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_105_funpow__swap1,axiom,
! [F: states_a > states_a,N: nat,X: states_a] :
( ( F @ ( compow495008222514391794ates_a @ N @ F @ X ) )
= ( compow495008222514391794ates_a @ N @ F @ ( F @ X ) ) ) ).
% funpow_swap1
thf(fact_106_funpow__swap1,axiom,
! [F: nat > nat,N: nat,X: nat] :
( ( F @ ( compow_nat_nat @ N @ F @ X ) )
= ( compow_nat_nat @ N @ F @ ( F @ X ) ) ) ).
% funpow_swap1
thf(fact_107_less__2__cases__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( ( N = zero_zero_nat )
| ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% less_2_cases_iff
thf(fact_108_less__2__cases,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
=> ( ( N = zero_zero_nat )
| ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% less_2_cases
thf(fact_109_zero__neq__numeral,axiom,
! [N: num] :
( zero_zero_nat
!= ( numeral_numeral_nat @ N ) ) ).
% zero_neq_numeral
thf(fact_110_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% not0_implies_Suc
thf(fact_111_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_112_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_113_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_114_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_115_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X3: nat,Y3: nat] :
( ( P @ X3 @ Y3 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y3 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_116_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_117_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_118_nat_OdiscI,axiom,
! [Nat: nat,X2: nat] :
( ( Nat
= ( suc @ X2 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_119_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_120_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_121_nat_Odistinct_I1_J,axiom,
! [X2: nat] :
( zero_zero_nat
!= ( suc @ X2 ) ) ).
% nat.distinct(1)
thf(fact_122_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_123_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_124_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_125_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_126_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M )
= ( times_times_nat @ ( suc @ K ) @ N ) )
= ( M = N ) ) ).
% Suc_mult_cancel1
thf(fact_127_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_128_diff__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_129_funpow__mult,axiom,
! [N: nat,M: nat,F: states_a > states_a] :
( ( compow495008222514391794ates_a @ N @ ( compow495008222514391794ates_a @ M @ F ) )
= ( compow495008222514391794ates_a @ ( times_times_nat @ M @ N ) @ F ) ) ).
% funpow_mult
thf(fact_130_funpow__mult,axiom,
! [N: nat,M: nat,F: nat > nat] :
( ( compow_nat_nat @ N @ ( compow_nat_nat @ M @ F ) )
= ( compow_nat_nat @ ( times_times_nat @ M @ N ) @ F ) ) ).
% funpow_mult
thf(fact_131_mult__numeral__1__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ ( numeral_numeral_nat @ one ) )
= A ) ).
% mult_numeral_1_right
thf(fact_132_mult__numeral__1,axiom,
! [A: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A )
= A ) ).
% mult_numeral_1
thf(fact_133_num_Oexhaust,axiom,
! [Y: num] :
( ( Y != one )
=> ( ! [X22: num] :
( Y
!= ( bit0 @ X22 ) )
=> ~ ! [X32: num] :
( Y
!= ( bit1 @ X32 ) ) ) ) ).
% num.exhaust
thf(fact_134_numeral__1__eq__Suc__0,axiom,
( ( numeral_numeral_nat @ one )
= ( suc @ zero_zero_nat ) ) ).
% numeral_1_eq_Suc_0
thf(fact_135_eval__nat__numeral_I3_J,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit1 @ N ) )
= ( suc @ ( numeral_numeral_nat @ ( bit0 @ N ) ) ) ) ).
% eval_nat_numeral(3)
thf(fact_136_numeral__2__eq__2,axiom,
( ( numeral_numeral_nat @ ( bit0 @ one ) )
= ( suc @ ( suc @ zero_zero_nat ) ) ) ).
% numeral_2_eq_2
thf(fact_137_numeral__3__eq__3,axiom,
( ( numeral_numeral_nat @ ( bit1 @ one ) )
= ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).
% numeral_3_eq_3
thf(fact_138_remaining__steps__n__steps__sub,axiom,
! [States: states_a,N: nat] :
( ( type_i8221491762852169479ates_a @ States )
=> ( ( type_r4519047461186610747ates_a @ ( compow495008222514391794ates_a @ N @ type_s4923920245906622843ates_a @ States ) )
= ( minus_minus_nat @ ( type_r4519047461186610747ates_a @ States ) @ N ) ) ) ).
% remaining_steps_n_steps_sub
thf(fact_139_remaining__steps__0,axiom,
! [States: states_a] :
( ( type_i8221491762852169479ates_a @ States )
=> ( ( ( type_r4519047461186610747ates_a @ States )
= zero_zero_nat )
=> ( ( type_r4519047461186610747ates_a @ ( type_s4923920245906622843ates_a @ States ) )
= zero_zero_nat ) ) ) ).
% remaining_steps_0
thf(fact_140_size__ok__steps,axiom,
! [States: states_a,N: nat] :
( ( type_i8221491762852169479ates_a @ States )
=> ( ( ord_less_nat @ N @ ( type_r4519047461186610747ates_a @ States ) )
=> ( ( states_size_ok_a @ States @ ( minus_minus_nat @ ( type_r4519047461186610747ates_a @ States ) @ N ) )
=> ( states_size_ok_a @ ( compow495008222514391794ates_a @ N @ type_s4923920245906622843ates_a @ States ) @ ( type_r4519047461186610747ates_a @ ( compow495008222514391794ates_a @ N @ type_s4923920245906622843ates_a @ States ) ) ) ) ) ) ).
% size_ok_steps
thf(fact_141_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_142_nat__bit__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( P @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_bit_induct
thf(fact_143_remaining__steps__decline__Suc,axiom,
! [States: states_a] :
( ( type_i8221491762852169479ates_a @ States )
=> ( ( ord_less_nat @ zero_zero_nat @ ( type_r4519047461186610747ates_a @ States ) )
=> ( ( suc @ ( type_r4519047461186610747ates_a @ ( type_s4923920245906622843ates_a @ States ) ) )
= ( type_r4519047461186610747ates_a @ States ) ) ) ) ).
% remaining_steps_decline_Suc
thf(fact_144_step__n__size__ok,axiom,
! [States: states_a,N: nat] :
( ( type_i8221491762852169479ates_a @ States )
=> ( ( states_size_ok_a @ States @ ( type_r4519047461186610747ates_a @ States ) )
=> ( states_size_ok_a @ ( compow495008222514391794ates_a @ N @ type_s4923920245906622843ates_a @ States ) @ ( type_r4519047461186610747ates_a @ ( compow495008222514391794ates_a @ N @ type_s4923920245906622843ates_a @ States ) ) ) ) ) ).
% step_n_size_ok
thf(fact_145_remaining__steps__0_H,axiom,
! [States: states_a,N: nat] :
( ( type_i8221491762852169479ates_a @ States )
=> ( ( ( type_r4519047461186610747ates_a @ States )
= zero_zero_nat )
=> ( ( type_r4519047461186610747ates_a @ ( compow495008222514391794ates_a @ N @ type_s4923920245906622843ates_a @ States ) )
= zero_zero_nat ) ) ) ).
% remaining_steps_0'
thf(fact_146_double__not__eq__Suc__double,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
!= ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ).
% double_not_eq_Suc_double
thf(fact_147_Suc__double__not__eq__double,axiom,
! [M: nat,N: nat] :
( ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
!= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ).
% Suc_double_not_eq_double
thf(fact_148_semiring__norm_I84_J,axiom,
! [N: num] :
( one
!= ( bit1 @ N ) ) ).
% semiring_norm(84)
thf(fact_149_semiring__norm_I87_J,axiom,
! [M: num,N: num] :
( ( ( bit0 @ M )
= ( bit0 @ N ) )
= ( M = N ) ) ).
% semiring_norm(87)
thf(fact_150_semiring__norm_I90_J,axiom,
! [M: num,N: num] :
( ( ( bit1 @ M )
= ( bit1 @ N ) )
= ( M = N ) ) ).
% semiring_norm(90)
thf(fact_151_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_152_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_153_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_154_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_155_semiring__norm_I85_J,axiom,
! [M: num] :
( ( bit0 @ M )
!= one ) ).
% semiring_norm(85)
thf(fact_156_semiring__norm_I83_J,axiom,
! [N: num] :
( one
!= ( bit0 @ N ) ) ).
% semiring_norm(83)
thf(fact_157_semiring__norm_I89_J,axiom,
! [M: num,N: num] :
( ( bit1 @ M )
!= ( bit0 @ N ) ) ).
% semiring_norm(89)
thf(fact_158_semiring__norm_I88_J,axiom,
! [M: num,N: num] :
( ( bit0 @ M )
!= ( bit1 @ N ) ) ).
% semiring_norm(88)
thf(fact_159_semiring__norm_I86_J,axiom,
! [M: num] :
( ( bit1 @ M )
!= one ) ).
% semiring_norm(86)
thf(fact_160_semiring__norm_I13_J,axiom,
! [M: num,N: num] :
( ( times_times_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( bit0 @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ).
% semiring_norm(13)
thf(fact_161_semiring__norm_I12_J,axiom,
! [N: num] :
( ( times_times_num @ one @ N )
= N ) ).
% semiring_norm(12)
thf(fact_162_semiring__norm_I11_J,axiom,
! [M: num] :
( ( times_times_num @ M @ one )
= M ) ).
% semiring_norm(11)
thf(fact_163_semiring__norm_I78_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% semiring_norm(78)
thf(fact_164_semiring__norm_I75_J,axiom,
! [M: num] :
~ ( ord_less_num @ M @ one ) ).
% semiring_norm(75)
thf(fact_165_semiring__norm_I80_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% semiring_norm(80)
thf(fact_166_semiring__norm_I15_J,axiom,
! [M: num,N: num] :
( ( times_times_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( bit0 @ ( times_times_num @ ( bit1 @ M ) @ N ) ) ) ).
% semiring_norm(15)
thf(fact_167_semiring__norm_I14_J,axiom,
! [M: num,N: num] :
( ( times_times_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( bit0 @ ( times_times_num @ M @ ( bit1 @ N ) ) ) ) ).
% semiring_norm(14)
thf(fact_168_step__size__big,axiom,
! [Dir: direction,Big: state_a2,Small: state_a,Dir2: direction,Big2: state_a2,Small2: state_a] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ( type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
= ( states_a2 @ Dir2 @ Big2 @ Small2 ) )
=> ( ( size_size_state_a @ Big2 )
= ( size_size_state_a @ Big ) ) ) ) ).
% step_size_big
thf(fact_169_step__size__small,axiom,
! [Dir: direction,Big: state_a2,Small: state_a,Dir2: direction,Big2: state_a2,Small2: state_a] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ( type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
= ( states_a2 @ Dir2 @ Big2 @ Small2 ) )
=> ( ( size_size_state_a2 @ Small2 )
= ( size_size_state_a2 @ Small ) ) ) ) ).
% step_size_small
thf(fact_170_semiring__norm_I76_J,axiom,
! [N: num] : ( ord_less_num @ one @ ( bit0 @ N ) ) ).
% semiring_norm(76)
thf(fact_171_semiring__norm_I81_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% semiring_norm(81)
thf(fact_172_semiring__norm_I77_J,axiom,
! [N: num] : ( ord_less_num @ one @ ( bit1 @ N ) ) ).
% semiring_norm(77)
thf(fact_173_step__n__size__big,axiom,
! [Dir: direction,Big: state_a2,Small: state_a,N: nat,Dir2: direction,Big2: state_a2,Small2: state_a] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ( compow495008222514391794ates_a @ N @ type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
= ( states_a2 @ Dir2 @ Big2 @ Small2 ) )
=> ( ( size_size_state_a @ Big2 )
= ( size_size_state_a @ Big ) ) ) ) ).
% step_n_size_big
thf(fact_174_step__n__size__small,axiom,
! [Dir: direction,Big: state_a2,Small: state_a,N: nat,Dir2: direction,Big2: state_a2,Small2: state_a] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ( compow495008222514391794ates_a @ N @ type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
= ( states_a2 @ Dir2 @ Big2 @ Small2 ) )
=> ( ( size_size_state_a2 @ Small2 )
= ( size_size_state_a2 @ Small ) ) ) ) ).
% step_n_size_small
thf(fact_175_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_176_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_177_mult_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_178_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A2: nat,B2: nat] : ( times_times_nat @ B2 @ A2 ) ) ) ).
% mult.commute
thf(fact_179_mult_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_180_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_181_size__ok__size__big,axiom,
! [Dir: direction,Big: state_a2,Small: state_a] :
( ( states_size_ok_a @ ( states_a2 @ Dir @ Big @ Small ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ Big ) ) ) ).
% size_ok_size_big
thf(fact_182_size__ok__size__small,axiom,
! [Dir: direction,Big: state_a2,Small: state_a] :
( ( states_size_ok_a @ ( states_a2 @ Dir @ Big @ Small ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a2 @ Small ) ) ) ).
% size_ok_size_small
thf(fact_183_num_Osize_I4_J,axiom,
( ( size_size_num @ one )
= zero_zero_nat ) ).
% num.size(4)
thf(fact_184_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_185_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_186_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_187_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_188_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_189_step__same,axiom,
! [Dir: direction,Big: state_a2,Small: state_a,Dir2: direction,Big2: state_a2,Small2: state_a] :
( ( ( type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
= ( states_a2 @ Dir2 @ Big2 @ Small2 ) )
=> ( Dir = Dir2 ) ) ).
% step_same
thf(fact_190_size__ok_H__Suc,axiom,
! [States: states_a,Steps: nat] :
( ( states_size_ok_a @ States @ ( suc @ Steps ) )
=> ( states_size_ok_a @ States @ Steps ) ) ).
% size_ok'_Suc
thf(fact_191_States__Proof_Oinvar__step,axiom,
! [States: states_a] :
( ( type_i8221491762852169479ates_a @ States )
=> ( type_i8221491762852169479ates_a @ ( type_s4923920245906622843ates_a @ States ) ) ) ).
% States_Proof.invar_step
thf(fact_192_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_193_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_194_step__n__same,axiom,
! [N: nat,Dir: direction,Big: state_a2,Small: state_a,Dir2: direction,Big2: state_a2,Small2: state_a] :
( ( ( compow495008222514391794ates_a @ N @ type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
= ( states_a2 @ Dir2 @ Big2 @ Small2 ) )
=> ( Dir = Dir2 ) ) ).
% step_n_same
thf(fact_195_invar__step__n,axiom,
! [States: states_a,N: nat] :
( ( type_i8221491762852169479ates_a @ States )
=> ( type_i8221491762852169479ates_a @ ( compow495008222514391794ates_a @ N @ type_s4923920245906622843ates_a @ States ) ) ) ).
% invar_step_n
thf(fact_196_step__consistent__2,axiom,
! [P: states_a > $o,States: states_a,N: nat] :
( ! [States2: states_a] :
( ( type_i8221491762852169479ates_a @ States2 )
=> ( ( P @ States2 )
=> ( P @ ( type_s4923920245906622843ates_a @ States2 ) ) ) )
=> ( ( type_i8221491762852169479ates_a @ States )
=> ( ( P @ States )
=> ( P @ ( compow495008222514391794ates_a @ N @ type_s4923920245906622843ates_a @ States ) ) ) ) ) ).
% step_consistent_2
thf(fact_197_step__size__ok_H,axiom,
! [States: states_a,N: nat] :
( ( type_i8221491762852169479ates_a @ States )
=> ( ( states_size_ok_a @ States @ N )
=> ( states_size_ok_a @ ( type_s4923920245906622843ates_a @ States ) @ N ) ) ) ).
% step_size_ok'
thf(fact_198_step__size__ok,axiom,
! [States: states_a] :
( ( type_i8221491762852169479ates_a @ States )
=> ( ( states_size_ok_a @ States @ ( type_r4519047461186610747ates_a @ States ) )
=> ( states_size_ok_a @ ( type_s4923920245906622843ates_a @ States ) @ ( type_r4519047461186610747ates_a @ ( type_s4923920245906622843ates_a @ States ) ) ) ) ) ).
% step_size_ok
thf(fact_199_step__n__size__ok_H,axiom,
! [States: states_a,X: nat,N: nat] :
( ( type_i8221491762852169479ates_a @ States )
=> ( ( states_size_ok_a @ States @ X )
=> ( states_size_ok_a @ ( compow495008222514391794ates_a @ N @ type_s4923920245906622843ates_a @ States ) @ X ) ) ) ).
% step_n_size_ok'
thf(fact_200_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_201_length__0__conv,axiom,
! [Xs: list_a] :
( ( ( size_size_list_a @ Xs )
= zero_zero_nat )
= ( Xs = nil_a ) ) ).
% length_0_conv
thf(fact_202_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_203_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_204_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_205_mult__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_206_mult__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_207_listL__remaining__steps,axiom,
! [States: states_a] :
( ( ( states_listL_a @ States )
= nil_a )
=> ( ( ord_less_nat @ zero_zero_nat @ ( type_r4519047461186610747ates_a @ States ) )
=> ( ( type_i8221491762852169479ates_a @ States )
=> ~ ( states_size_ok_a @ States @ ( type_r4519047461186610747ates_a @ States ) ) ) ) ) ).
% listL_remaining_steps
thf(fact_208_states_Osize_I2_J,axiom,
! [X1: direction,X2: state_a2,X33: state_a] :
( ( size_size_states_a @ ( states_a2 @ X1 @ X2 @ X33 ) )
= ( suc @ zero_zero_nat ) ) ).
% states.size(2)
thf(fact_209_states_Oinject,axiom,
! [X1: direction,X2: state_a2,X33: state_a,Y1: direction,Y2: state_a2,Y32: state_a] :
( ( ( states_a2 @ X1 @ X2 @ X33 )
= ( states_a2 @ Y1 @ Y2 @ Y32 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 )
& ( X33 = Y32 ) ) ) ).
% states.inject
thf(fact_210_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_211_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_a] :
( ( size_size_list_a @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_212_states_Osize__neq,axiom,
! [X: states_a] :
( ( size_size_states_a @ X )
!= zero_zero_nat ) ).
% states.size_neq
thf(fact_213_length__induct,axiom,
! [P: list_a > $o,Xs: list_a] :
( ! [Xs2: list_a] :
( ! [Ys2: list_a] :
( ( ord_less_nat @ ( size_size_list_a @ Ys2 ) @ ( size_size_list_a @ Xs2 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_214_states_Oexhaust,axiom,
! [Y: states_a] :
~ ! [X12: direction,X22: state_a2,X32: state_a] :
( Y
!= ( states_a2 @ X12 @ X22 @ X32 ) ) ).
% states.exhaust
thf(fact_215_step__listL,axiom,
! [States: states_a] :
( ( type_i8221491762852169479ates_a @ States )
=> ( ( states_listL_a @ ( type_s4923920245906622843ates_a @ States ) )
= ( states_listL_a @ States ) ) ) ).
% step_listL
thf(fact_216_step__n__listL,axiom,
! [States: states_a,N: nat] :
( ( type_i8221491762852169479ates_a @ States )
=> ( ( states_listL_a @ ( compow495008222514391794ates_a @ N @ type_s4923920245906622843ates_a @ States ) )
= ( states_listL_a @ States ) ) ) ).
% step_n_listL
thf(fact_217_mult__right__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_218_mult__left__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_219_no__zero__divisors,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_220_divisors__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_221_mult__not__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_222_right__diff__distrib_H,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
= ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_223_left__diff__distrib_H,axiom,
! [B: nat,C: nat,A: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A )
= ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_224_list_Osize_I3_J,axiom,
( ( size_size_list_a @ nil_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_225_direction_Osize_I3_J,axiom,
( ( size_size_direction @ left )
= zero_zero_nat ) ).
% direction.size(3)
thf(fact_226_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_227_mult__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_228_mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_229_zero__less__mult__pos2,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_230_zero__less__mult__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_231_mult__pos__neg2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg2
thf(fact_232_mult__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_233_mult__pos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg
thf(fact_234_mult__neg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_neg_pos
thf(fact_235_step__n__push__size__small,axiom,
! [Dir: direction,Big: state_a2,Small: state_a,N: nat,X: a,Dir2: direction,Big2: state_a2,Small2: state_a] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ( compow495008222514391794ates_a @ N @ type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ Big @ ( push_a2 @ X @ Small ) ) )
= ( states_a2 @ Dir2 @ Big2 @ Small2 ) )
=> ( ( size_size_state_a2 @ Small2 )
= ( suc @ ( size_size_state_a2 @ Small ) ) ) ) ) ).
% step_n_push_size_small
thf(fact_236_step__n__push__size__big,axiom,
! [Dir: direction,Big: state_a2,Small: state_a,N: nat,X: a,Dir2: direction,Big2: state_a2,Small2: state_a] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ( compow495008222514391794ates_a @ N @ type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ ( push_a @ X @ Big ) @ Small ) )
= ( states_a2 @ Dir2 @ Big2 @ Small2 ) )
=> ( ( size_size_state_a @ Big2 )
= ( suc @ ( size_size_state_a @ Big ) ) ) ) ) ).
% step_n_push_size_big
thf(fact_237_Small_Ostate_Oinject_I1_J,axiom,
! [X11: current_a,X122: stack_a,X13: list_a,Y11: current_a,Y12: stack_a,Y13: list_a] :
( ( ( reverse1_a @ X11 @ X122 @ X13 )
= ( reverse1_a @ Y11 @ Y12 @ Y13 ) )
= ( ( X11 = Y11 )
& ( X122 = Y12 )
& ( X13 = Y13 ) ) ) ).
% Small.state.inject(1)
thf(fact_238_current_Oinject,axiom,
! [X1: list_a,X2: nat,X33: stack_a,X4: nat,Y1: list_a,Y2: nat,Y32: stack_a,Y4: nat] :
( ( ( current_a2 @ X1 @ X2 @ X33 @ X4 )
= ( current_a2 @ Y1 @ Y2 @ Y32 @ Y4 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 )
& ( X33 = Y32 )
& ( X4 = Y4 ) ) ) ).
% current.inject
thf(fact_239_Big_Ostate_Oinject_I1_J,axiom,
! [X11: current_a,X122: stack_a,X13: list_a,X14: nat,Y11: current_a,Y12: stack_a,Y13: list_a,Y14: nat] :
( ( ( reverse_a @ X11 @ X122 @ X13 @ X14 )
= ( reverse_a @ Y11 @ Y12 @ Y13 @ Y14 ) )
= ( ( X11 = Y11 )
& ( X122 = Y12 )
& ( X13 = Y13 )
& ( X14 = Y14 ) ) ) ).
% Big.state.inject(1)
thf(fact_240_remaining__steps__push__big,axiom,
! [Dir: direction,Big: state_a2,Small: state_a,X: a] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ ( push_a @ X @ Big ) @ Small ) )
= ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) ) ) ).
% remaining_steps_push_big
thf(fact_241_step__push__size__big,axiom,
! [Dir: direction,Big: state_a2,Small: state_a,X: a,Dir2: direction,Big2: state_a2,Small2: state_a] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ( type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ ( push_a @ X @ Big ) @ Small ) )
= ( states_a2 @ Dir2 @ Big2 @ Small2 ) )
=> ( ( size_size_state_a @ Big2 )
= ( suc @ ( size_size_state_a @ Big ) ) ) ) ) ).
% step_push_size_big
thf(fact_242_step__push__size__small,axiom,
! [Dir: direction,Big: state_a2,Small: state_a,X: a,Dir2: direction,Big2: state_a2,Small2: state_a] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ( type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ Big @ ( push_a2 @ X @ Small ) ) )
= ( states_a2 @ Dir2 @ Big2 @ Small2 ) )
=> ( ( size_size_state_a2 @ Small2 )
= ( suc @ ( size_size_state_a2 @ Small ) ) ) ) ) ).
% step_push_size_small
thf(fact_243_invar__push__small,axiom,
! [Dir: direction,Big: state_a2,Small: state_a,X: a] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ ( push_a2 @ X @ Small ) ) ) ) ).
% invar_push_small
thf(fact_244_invar__push__big,axiom,
! [Dir: direction,Big: state_a2,Small: state_a,X: a] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ ( push_a @ X @ Big ) @ Small ) ) ) ).
% invar_push_big
thf(fact_245_current_Oexhaust,axiom,
! [Y: current_a] :
~ ! [X12: list_a,X22: nat,X32: stack_a,X42: nat] :
( Y
!= ( current_a2 @ X12 @ X22 @ X32 @ X42 ) ) ).
% current.exhaust
thf(fact_246_remaining__steps__push__small,axiom,
! [Dir: direction,Big: state_a2,Small: state_a,X: a] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
= ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ ( push_a2 @ X @ Small ) ) ) ) ) ).
% remaining_steps_push_small
thf(fact_247_step__4__remaining__steps__push__big,axiom,
! [Dir: direction,Big: state_a2,Small: state_a,X: a,Dir2: direction,Big2: state_a2,Small2: state_a] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) )
=> ( ( ( compow495008222514391794ates_a @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ ( push_a @ X @ Big ) @ Small ) )
= ( states_a2 @ Dir2 @ Big2 @ Small2 ) )
=> ( ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir2 @ Big2 @ Small2 ) )
= ( minus_minus_nat @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% step_4_remaining_steps_push_big
thf(fact_248_step__4__remaining__steps__push__small,axiom,
! [Dir: direction,Big: state_a2,Small: state_a,X: a,Dir2: direction,Big2: state_a2,Small2: state_a] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) )
=> ( ( ( compow495008222514391794ates_a @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ Big @ ( push_a2 @ X @ Small ) ) )
= ( states_a2 @ Dir2 @ Big2 @ Small2 ) )
=> ( ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir2 @ Big2 @ Small2 ) )
= ( minus_minus_nat @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ).
% step_4_remaining_steps_push_small
thf(fact_249_step__4__push__big__size__ok,axiom,
! [Dir: direction,Big: state_a2,Small: state_a,X: a] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) )
=> ( ( states_size_ok_a @ ( states_a2 @ Dir @ Big @ Small ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) )
=> ( states_size_ok_a @ ( compow495008222514391794ates_a @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ ( push_a @ X @ Big ) @ Small ) ) @ ( type_r4519047461186610747ates_a @ ( compow495008222514391794ates_a @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ ( push_a @ X @ Big ) @ Small ) ) ) ) ) ) ) ).
% step_4_push_big_size_ok
thf(fact_250_step__4__push__small__size__ok,axiom,
! [Dir: direction,Big: state_a2,Small: state_a,X: a] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) )
=> ( ( states_size_ok_a @ ( states_a2 @ Dir @ Big @ Small ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) )
=> ( states_size_ok_a @ ( compow495008222514391794ates_a @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ Big @ ( push_a2 @ X @ Small ) ) ) @ ( type_r4519047461186610747ates_a @ ( compow495008222514391794ates_a @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ Big @ ( push_a2 @ X @ Small ) ) ) ) ) ) ) ) ).
% step_4_push_small_size_ok
thf(fact_251_direction_Osize__gen_I1_J,axiom,
( ( size_direction @ left )
= zero_zero_nat ) ).
% direction.size_gen(1)
thf(fact_252_verit__eq__simplify_I9_J,axiom,
! [X33: num,Y32: num] :
( ( ( bit1 @ X33 )
= ( bit1 @ Y32 ) )
= ( X33 = Y32 ) ) ).
% verit_eq_simplify(9)
thf(fact_253_verit__eq__simplify_I8_J,axiom,
! [X2: num,Y2: num] :
( ( ( bit0 @ X2 )
= ( bit0 @ Y2 ) )
= ( X2 = Y2 ) ) ).
% verit_eq_simplify(8)
thf(fact_254_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_255_numeral__le__iff,axiom,
! [M: num,N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% numeral_le_iff
thf(fact_256_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_257_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_258_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_259_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_260_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_261_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_262_one__le__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_263_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_264_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_265_remaining__steps__decline__n__steps,axiom,
! [States: states_a,N: nat] :
( ( type_i8221491762852169479ates_a @ States )
=> ( ( ord_less_eq_nat @ ( type_r4519047461186610747ates_a @ States ) @ N )
=> ( ( type_r4519047461186610747ates_a @ ( compow495008222514391794ates_a @ N @ type_s4923920245906622843ates_a @ States ) )
= zero_zero_nat ) ) ) ).
% remaining_steps_decline_n_steps
thf(fact_266_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_267_lift__Suc__antimono__le,axiom,
! [F: nat > num,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_num @ ( F @ N2 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_268_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_269_lift__Suc__mono__le,axiom,
! [F: nat > num,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_num @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_270_verit__comp__simplify1_I3_J,axiom,
! [B3: nat,A3: nat] :
( ( ~ ( ord_less_eq_nat @ B3 @ A3 ) )
= ( ord_less_nat @ A3 @ B3 ) ) ).
% verit_comp_simplify1(3)
thf(fact_271_verit__comp__simplify1_I3_J,axiom,
! [B3: num,A3: num] :
( ( ~ ( ord_less_eq_num @ B3 @ A3 ) )
= ( ord_less_num @ A3 @ B3 ) ) ).
% verit_comp_simplify1(3)
thf(fact_272_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_273_verit__comp__simplify1_I2_J,axiom,
! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_274_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_275_verit__la__disequality,axiom,
! [A: num,B: num] :
( ( A = B )
| ~ ( ord_less_eq_num @ A @ B )
| ~ ( ord_less_eq_num @ B @ A ) ) ).
% verit_la_disequality
thf(fact_276_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_277_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_278_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_279_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_280_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_281_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_282_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_283_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_284_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_285_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_286_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_287_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_288_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X3: nat] : ( R @ X3 @ X3 )
=> ( ! [X3: nat,Y3: nat,Z2: nat] :
( ( R @ X3 @ Y3 )
=> ( ( R @ Y3 @ Z2 )
=> ( R @ X3 @ Z2 ) ) )
=> ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_289_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_290_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N3 )
=> ( P @ M2 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_291_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_292_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_293_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_294_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_295_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_296_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_297_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_298_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M5: nat,N4: nat] :
( ( ord_less_eq_nat @ M5 @ N4 )
& ( M5 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_299_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_300_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N4: nat] :
( ( ord_less_nat @ M5 @ N4 )
| ( M5 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_301_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_302_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_303_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_304_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_305_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_306_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_307_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_308_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_309_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_310_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_311_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_312_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_313_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_314_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_315_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_316_size__ok_H__decline,axiom,
! [States: states_a,X: nat,Y: nat] :
( ( states_size_ok_a @ States @ X )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( states_size_ok_a @ States @ Y ) ) ) ).
% size_ok'_decline
thf(fact_317_not__numeral__le__zero,axiom,
! [N: num] :
~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ zero_zero_nat ) ).
% not_numeral_le_zero
thf(fact_318_zero__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N ) ) ).
% zero_le_numeral
thf(fact_319_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_320_mult__nonneg__nonpos2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_321_mult__nonpos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_322_mult__nonneg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_323_mult__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_324_split__mult__neg__le,axiom,
! [A: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_325_mult__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_326_mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_327_mult__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_328_mult__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_329_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_330_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_331_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_332_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_333_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_334_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_335_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_336_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_337_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N4: nat] : ( ord_less_eq_nat @ ( suc @ N4 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_338_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_339_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_340_Suc__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_341_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_342_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_343_mult__less__le__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_344_mult__le__less__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_345_mult__right__le__imp__le,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_346_mult__left__le__imp__le,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_347_mult__strict__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_348_mult__right__less__imp__less,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_349_mult__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_350_mult__left__less__imp__less,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_351_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_352_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_353_verit__comp__simplify1_I1_J,axiom,
! [A: num] :
~ ( ord_less_num @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_354_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_355_remaining__steps__decline,axiom,
! [States: states_a] :
( ( type_i8221491762852169479ates_a @ States )
=> ( ord_less_eq_nat @ ( type_r4519047461186610747ates_a @ ( type_s4923920245906622843ates_a @ States ) ) @ ( type_r4519047461186610747ates_a @ States ) ) ) ).
% remaining_steps_decline
thf(fact_356_verit__eq__simplify_I10_J,axiom,
! [X2: num] :
( one
!= ( bit0 @ X2 ) ) ).
% verit_eq_simplify(10)
thf(fact_357_verit__eq__simplify_I14_J,axiom,
! [X2: num,X33: num] :
( ( bit0 @ X2 )
!= ( bit1 @ X33 ) ) ).
% verit_eq_simplify(14)
thf(fact_358_verit__eq__simplify_I12_J,axiom,
! [X33: num] :
( one
!= ( bit1 @ X33 ) ) ).
% verit_eq_simplify(12)
thf(fact_359_Start__Transformation,axiom,
~ ( ord_less_eq_nat @ length_right @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ ( size_size_idle_a @ left2 ) ) ) ).
% Start_Transformation
thf(fact_360_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C2: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_361_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_362_order__refl,axiom,
! [X: num] : ( ord_less_eq_num @ X @ X ) ).
% order_refl
thf(fact_363_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_364_dual__order_Orefl,axiom,
! [A: num] : ( ord_less_eq_num @ A @ A ) ).
% dual_order.refl
thf(fact_365_states_Osize__gen,axiom,
! [X: a > nat,X1: direction,X2: state_a2,X33: state_a] :
( ( size_states_a @ X @ ( states_a2 @ X1 @ X2 @ X33 ) )
= ( suc @ zero_zero_nat ) ) ).
% states.size_gen
thf(fact_366_semiring__norm_I71_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(71)
thf(fact_367_semiring__norm_I68_J,axiom,
! [N: num] : ( ord_less_eq_num @ one @ N ) ).
% semiring_norm(68)
thf(fact_368_semiring__norm_I73_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(73)
thf(fact_369_semiring__norm_I69_J,axiom,
! [M: num] :
~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).
% semiring_norm(69)
thf(fact_370_semiring__norm_I72_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(72)
thf(fact_371_semiring__norm_I70_J,axiom,
! [M: num] :
~ ( ord_less_eq_num @ ( bit1 @ M ) @ one ) ).
% semiring_norm(70)
thf(fact_372_True,axiom,
ord_less_eq_nat @ one_one_nat @ ( size_size_idle_a @ left2 ) ).
% True
thf(fact_373_semiring__norm_I79_J,axiom,
! [M: num,N: num] :
( ( ord_less_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( ord_less_eq_num @ M @ N ) ) ).
% semiring_norm(79)
thf(fact_374_semiring__norm_I74_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( ord_less_num @ M @ N ) ) ).
% semiring_norm(74)
thf(fact_375_le__num__One__iff,axiom,
! [X: num] :
( ( ord_less_eq_num @ X @ one )
= ( X = one ) ) ).
% le_num_One_iff
thf(fact_376_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_377_order__antisym__conv,axiom,
! [Y: num,X: num] :
( ( ord_less_eq_num @ Y @ X )
=> ( ( ord_less_eq_num @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_378_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_379_linorder__le__cases,axiom,
! [X: num,Y: num] :
( ~ ( ord_less_eq_num @ X @ Y )
=> ( ord_less_eq_num @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_380_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,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_381_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_382_ord__le__eq__subst,axiom,
! [A: num,B: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_383_ord__le__eq__subst,axiom,
! [A: num,B: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_384_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,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_385_ord__eq__le__subst,axiom,
! [A: num,F: nat > num,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_386_ord__eq__le__subst,axiom,
! [A: nat,F: num > nat,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_387_ord__eq__le__subst,axiom,
! [A: num,F: num > num,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_388_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_389_linorder__linear,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
| ( ord_less_eq_num @ Y @ X ) ) ).
% linorder_linear
thf(fact_390_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_391_order__eq__refl,axiom,
! [X: num,Y: num] :
( ( X = Y )
=> ( ord_less_eq_num @ X @ Y ) ) ).
% order_eq_refl
thf(fact_392_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,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_393_order__subst2,axiom,
! [A: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_394_order__subst2,axiom,
! [A: num,B: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_395_order__subst2,axiom,
! [A: num,B: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_396_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,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_397_order__subst1,axiom,
! [A: nat,F: num > nat,B: num,C: num] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_398_order__subst1,axiom,
! [A: num,F: nat > num,B: nat,C: nat] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_399_order__subst1,axiom,
! [A: num,F: num > num,B: num,C: num] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_400_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 ) )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_401_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: num,Z3: num] : ( Y6 = Z3 ) )
= ( ^ [A2: num,B2: num] :
( ( ord_less_eq_num @ A2 @ B2 )
& ( ord_less_eq_num @ B2 @ A2 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_402_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_403_antisym,axiom,
! [A: num,B: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_num @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_404_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_405_dual__order_Otrans,axiom,
! [B: num,A: num,C: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_less_eq_num @ C @ B )
=> ( ord_less_eq_num @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_406_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_407_dual__order_Oantisym,axiom,
! [B: num,A: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_less_eq_num @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_408_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 ) )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_409_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: num,Z3: num] : ( Y6 = Z3 ) )
= ( ^ [A2: num,B2: num] :
( ( ord_less_eq_num @ B2 @ A2 )
& ( ord_less_eq_num @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_410_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_411_linorder__wlog,axiom,
! [P: num > num > $o,A: num,B: num] :
( ! [A4: num,B4: num] :
( ( ord_less_eq_num @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: num,B4: num] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_412_order__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_413_order__trans,axiom,
! [X: num,Y: num,Z: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_less_eq_num @ Y @ Z )
=> ( ord_less_eq_num @ X @ Z ) ) ) ).
% order_trans
thf(fact_414_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_415_order_Otrans,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ord_less_eq_num @ A @ C ) ) ) ).
% order.trans
thf(fact_416_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_417_order__antisym,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_less_eq_num @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_418_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_419_ord__le__eq__trans,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_num @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_420_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_421_ord__eq__le__trans,axiom,
! [A: num,B: num,C: num] :
( ( A = B )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ord_less_eq_num @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_422_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 ) )
= ( ^ [X5: nat,Y7: nat] :
( ( ord_less_eq_nat @ X5 @ Y7 )
& ( ord_less_eq_nat @ Y7 @ X5 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_423_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: num,Z3: num] : ( Y6 = Z3 ) )
= ( ^ [X5: num,Y7: num] :
( ( ord_less_eq_num @ X5 @ Y7 )
& ( ord_less_eq_num @ Y7 @ X5 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_424_le__cases3,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z ) )
=> ( ( ( ord_less_eq_nat @ X @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_425_le__cases3,axiom,
! [X: num,Y: num,Z: num] :
( ( ( ord_less_eq_num @ X @ Y )
=> ~ ( ord_less_eq_num @ Y @ Z ) )
=> ( ( ( ord_less_eq_num @ Y @ X )
=> ~ ( ord_less_eq_num @ X @ Z ) )
=> ( ( ( ord_less_eq_num @ X @ Z )
=> ~ ( ord_less_eq_num @ Z @ Y ) )
=> ( ( ( ord_less_eq_num @ Z @ Y )
=> ~ ( ord_less_eq_num @ Y @ X ) )
=> ( ( ( ord_less_eq_num @ Y @ Z )
=> ~ ( ord_less_eq_num @ Z @ X ) )
=> ~ ( ( ord_less_eq_num @ Z @ X )
=> ~ ( ord_less_eq_num @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_426_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_427_nle__le,axiom,
! [A: num,B: num] :
( ( ~ ( ord_less_eq_num @ A @ B ) )
= ( ( ord_less_eq_num @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_428_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_429_order__less__imp__not__less,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ~ ( ord_less_num @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_430_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_431_order__less__imp__not__eq2,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_432_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_433_order__less__imp__not__eq,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_434_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_435_linorder__less__linear,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
| ( X = Y )
| ( ord_less_num @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_436_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_437_order__less__imp__triv,axiom,
! [X: num,Y: num,P: $o] :
( ( ord_less_num @ X @ Y )
=> ( ( ord_less_num @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_438_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_439_order__less__not__sym,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ~ ( ord_less_num @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_440_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,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_441_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_442_order__less__subst2,axiom,
! [A: num,B: num,F: num > nat,C: nat] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_443_order__less__subst2,axiom,
! [A: num,B: num,F: num > num,C: num] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_444_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,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_445_order__less__subst1,axiom,
! [A: nat,F: num > nat,B: num,C: num] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_446_order__less__subst1,axiom,
! [A: num,F: nat > num,B: nat,C: nat] :
( ( ord_less_num @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_447_order__less__subst1,axiom,
! [A: num,F: num > num,B: num,C: num] :
( ( ord_less_num @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_448_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_449_order__less__irrefl,axiom,
! [X: num] :
~ ( ord_less_num @ X @ X ) ).
% order_less_irrefl
thf(fact_450_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_451_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_452_ord__less__eq__subst,axiom,
! [A: num,B: num,F: num > nat,C: nat] :
( ( ord_less_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_453_ord__less__eq__subst,axiom,
! [A: num,B: num,F: num > num,C: num] :
( ( ord_less_num @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_454_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_455_ord__eq__less__subst,axiom,
! [A: num,F: nat > num,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_456_ord__eq__less__subst,axiom,
! [A: nat,F: num > nat,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_457_ord__eq__less__subst,axiom,
! [A: num,F: num > num,B: num,C: num] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_458_order__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_459_order__less__trans,axiom,
! [X: num,Y: num,Z: num] :
( ( ord_less_num @ X @ Y )
=> ( ( ord_less_num @ Y @ Z )
=> ( ord_less_num @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_460_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_461_order__less__asym_H,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ~ ( ord_less_num @ B @ A ) ) ).
% order_less_asym'
thf(fact_462_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_463_linorder__neq__iff,axiom,
! [X: num,Y: num] :
( ( X != Y )
= ( ( ord_less_num @ X @ Y )
| ( ord_less_num @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_464_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_465_order__less__asym,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ~ ( ord_less_num @ Y @ X ) ) ).
% order_less_asym
thf(fact_466_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_467_linorder__neqE,axiom,
! [X: num,Y: num] :
( ( X != Y )
=> ( ~ ( ord_less_num @ X @ Y )
=> ( ord_less_num @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_468_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_469_dual__order_Ostrict__implies__not__eq,axiom,
! [B: num,A: num] :
( ( ord_less_num @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_470_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_471_order_Ostrict__implies__not__eq,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_472_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_473_dual__order_Ostrict__trans,axiom,
! [B: num,A: num,C: num] :
( ( ord_less_num @ B @ A )
=> ( ( ord_less_num @ C @ B )
=> ( ord_less_num @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_474_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_475_not__less__iff__gr__or__eq,axiom,
! [X: num,Y: num] :
( ( ~ ( ord_less_num @ X @ Y ) )
= ( ( ord_less_num @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_476_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_477_order_Ostrict__trans,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_num @ B @ C )
=> ( ord_less_num @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_478_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ A4 )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_479_linorder__less__wlog,axiom,
! [P: num > num > $o,A: num,B: num] :
( ! [A4: num,B4: num] :
( ( ord_less_num @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: num] : ( P @ A4 @ A4 )
=> ( ! [A4: num,B4: num] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_480_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X6: nat] : ( P2 @ X6 ) )
= ( ^ [P3: nat > $o] :
? [N4: nat] :
( ( P3 @ N4 )
& ! [M5: nat] :
( ( ord_less_nat @ M5 @ N4 )
=> ~ ( P3 @ M5 ) ) ) ) ) ).
% exists_least_iff
thf(fact_481_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_482_dual__order_Oirrefl,axiom,
! [A: num] :
~ ( ord_less_num @ A @ A ) ).
% dual_order.irrefl
thf(fact_483_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_484_dual__order_Oasym,axiom,
! [B: num,A: num] :
( ( ord_less_num @ B @ A )
=> ~ ( ord_less_num @ A @ B ) ) ).
% dual_order.asym
thf(fact_485_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_486_linorder__cases,axiom,
! [X: num,Y: num] :
( ~ ( ord_less_num @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_num @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_487_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_488_antisym__conv3,axiom,
! [Y: num,X: num] :
( ~ ( ord_less_num @ Y @ X )
=> ( ( ~ ( ord_less_num @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_489_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X3: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X3 )
=> ( P @ Y5 ) )
=> ( P @ X3 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_490_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_491_ord__less__eq__trans,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_num @ A @ B )
=> ( ( B = C )
=> ( ord_less_num @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_492_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_493_ord__eq__less__trans,axiom,
! [A: num,B: num,C: num] :
( ( A = B )
=> ( ( ord_less_num @ B @ C )
=> ( ord_less_num @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_494_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_495_order_Oasym,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ~ ( ord_less_num @ B @ A ) ) ).
% order.asym
thf(fact_496_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_497_less__imp__neq,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_498_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_499_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_500_leD,axiom,
! [Y: num,X: num] :
( ( ord_less_eq_num @ Y @ X )
=> ~ ( ord_less_num @ X @ Y ) ) ).
% leD
thf(fact_501_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_502_leI,axiom,
! [X: num,Y: num] :
( ~ ( ord_less_num @ X @ Y )
=> ( ord_less_eq_num @ Y @ X ) ) ).
% leI
thf(fact_503_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_504_nless__le,axiom,
! [A: num,B: num] :
( ( ~ ( ord_less_num @ A @ B ) )
= ( ~ ( ord_less_eq_num @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_505_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_506_antisym__conv1,axiom,
! [X: num,Y: num] :
( ~ ( ord_less_num @ X @ Y )
=> ( ( ord_less_eq_num @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_507_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_508_antisym__conv2,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ~ ( ord_less_num @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_509_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X5: nat,Y7: nat] :
( ( ord_less_eq_nat @ X5 @ Y7 )
& ~ ( ord_less_eq_nat @ Y7 @ X5 ) ) ) ) ).
% less_le_not_le
thf(fact_510_less__le__not__le,axiom,
( ord_less_num
= ( ^ [X5: num,Y7: num] :
( ( ord_less_eq_num @ X5 @ Y7 )
& ~ ( ord_less_eq_num @ Y7 @ X5 ) ) ) ) ).
% less_le_not_le
thf(fact_511_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_512_not__le__imp__less,axiom,
! [Y: num,X: num] :
( ~ ( ord_less_eq_num @ Y @ X )
=> ( ord_less_num @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_513_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_514_order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [A2: num,B2: num] :
( ( ord_less_num @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_515_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_516_order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [A2: num,B2: num] :
( ( ord_less_eq_num @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_517_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_518_order_Ostrict__trans1,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_num @ B @ C )
=> ( ord_less_num @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_519_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_520_order_Ostrict__trans2,axiom,
! [A: num,B: num,C: num] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ord_less_num @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_521_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ~ ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_522_order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [A2: num,B2: num] :
( ( ord_less_eq_num @ A2 @ B2 )
& ~ ( ord_less_eq_num @ B2 @ A2 ) ) ) ) ).
% order.strict_iff_not
thf(fact_523_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_524_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_num
= ( ^ [B2: num,A2: num] :
( ( ord_less_num @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_525_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_526_dual__order_Ostrict__iff__order,axiom,
( ord_less_num
= ( ^ [B2: num,A2: num] :
( ( ord_less_eq_num @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_527_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_528_dual__order_Ostrict__trans1,axiom,
! [B: num,A: num,C: num] :
( ( ord_less_eq_num @ B @ A )
=> ( ( ord_less_num @ C @ B )
=> ( ord_less_num @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_529_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_530_dual__order_Ostrict__trans2,axiom,
! [B: num,A: num,C: num] :
( ( ord_less_num @ B @ A )
=> ( ( ord_less_eq_num @ C @ B )
=> ( ord_less_num @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_531_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ~ ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_532_dual__order_Ostrict__iff__not,axiom,
( ord_less_num
= ( ^ [B2: num,A2: num] :
( ( ord_less_eq_num @ B2 @ A2 )
& ~ ( ord_less_eq_num @ A2 @ B2 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_533_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_534_order_Ostrict__implies__order,axiom,
! [A: num,B: num] :
( ( ord_less_num @ A @ B )
=> ( ord_less_eq_num @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_535_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_536_dual__order_Ostrict__implies__order,axiom,
! [B: num,A: num] :
( ( ord_less_num @ B @ A )
=> ( ord_less_eq_num @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_537_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X5: nat,Y7: nat] :
( ( ord_less_nat @ X5 @ Y7 )
| ( X5 = Y7 ) ) ) ) ).
% order_le_less
thf(fact_538_order__le__less,axiom,
( ord_less_eq_num
= ( ^ [X5: num,Y7: num] :
( ( ord_less_num @ X5 @ Y7 )
| ( X5 = Y7 ) ) ) ) ).
% order_le_less
thf(fact_539_order__less__le,axiom,
( ord_less_nat
= ( ^ [X5: nat,Y7: nat] :
( ( ord_less_eq_nat @ X5 @ Y7 )
& ( X5 != Y7 ) ) ) ) ).
% order_less_le
thf(fact_540_order__less__le,axiom,
( ord_less_num
= ( ^ [X5: num,Y7: num] :
( ( ord_less_eq_num @ X5 @ Y7 )
& ( X5 != Y7 ) ) ) ) ).
% order_less_le
thf(fact_541_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_542_linorder__not__le,axiom,
! [X: num,Y: num] :
( ( ~ ( ord_less_eq_num @ X @ Y ) )
= ( ord_less_num @ Y @ X ) ) ).
% linorder_not_le
thf(fact_543_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_544_linorder__not__less,axiom,
! [X: num,Y: num] :
( ( ~ ( ord_less_num @ X @ Y ) )
= ( ord_less_eq_num @ Y @ X ) ) ).
% linorder_not_less
thf(fact_545_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_546_order__less__imp__le,axiom,
! [X: num,Y: num] :
( ( ord_less_num @ X @ Y )
=> ( ord_less_eq_num @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_547_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_548_order__le__neq__trans,axiom,
! [A: num,B: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( A != B )
=> ( ord_less_num @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_549_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_550_order__neq__le__trans,axiom,
! [A: num,B: num] :
( ( A != B )
=> ( ( ord_less_eq_num @ A @ B )
=> ( ord_less_num @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_551_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_552_order__le__less__trans,axiom,
! [X: num,Y: num,Z: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_less_num @ Y @ Z )
=> ( ord_less_num @ X @ Z ) ) ) ).
% order_le_less_trans
thf(fact_553_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_554_order__less__le__trans,axiom,
! [X: num,Y: num,Z: num] :
( ( ord_less_num @ X @ Y )
=> ( ( ord_less_eq_num @ Y @ Z )
=> ( ord_less_num @ X @ Z ) ) ) ).
% order_less_le_trans
thf(fact_555_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,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_556_order__le__less__subst1,axiom,
! [A: nat,F: num > nat,B: num,C: num] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_557_order__le__less__subst1,axiom,
! [A: num,F: nat > num,B: nat,C: nat] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_558_order__le__less__subst1,axiom,
! [A: num,F: num > num,B: num,C: num] :
( ( ord_less_eq_num @ A @ ( F @ B ) )
=> ( ( ord_less_num @ B @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_559_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,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_560_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_561_order__le__less__subst2,axiom,
! [A: num,B: num,F: num > nat,C: nat] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_562_order__le__less__subst2,axiom,
! [A: num,B: num,F: num > num,C: num] :
( ( ord_less_eq_num @ A @ B )
=> ( ( ord_less_num @ ( F @ B ) @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_563_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,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_564_order__less__le__subst1,axiom,
! [A: num,F: nat > num,B: nat,C: nat] :
( ( ord_less_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_eq_nat @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_565_order__less__le__subst1,axiom,
! [A: nat,F: num > nat,B: num,C: num] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_566_order__less__le__subst1,axiom,
! [A: num,F: num > num,B: num,C: num] :
( ( ord_less_num @ A @ ( F @ B ) )
=> ( ( ord_less_eq_num @ B @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_eq_num @ X3 @ Y3 )
=> ( ord_less_eq_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_567_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,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_568_order__less__le__subst2,axiom,
! [A: num,B: num,F: num > nat,C: nat] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_nat @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_569_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > num,C: num] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X3: nat,Y3: nat] :
( ( ord_less_nat @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_570_order__less__le__subst2,axiom,
! [A: num,B: num,F: num > num,C: num] :
( ( ord_less_num @ A @ B )
=> ( ( ord_less_eq_num @ ( F @ B ) @ C )
=> ( ! [X3: num,Y3: num] :
( ( ord_less_num @ X3 @ Y3 )
=> ( ord_less_num @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_571_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_572_linorder__le__less__linear,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
| ( ord_less_num @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_573_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_574_order__le__imp__less__or__eq,axiom,
! [X: num,Y: num] :
( ( ord_less_eq_num @ X @ Y )
=> ( ( ord_less_num @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_575_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C2: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_576_invar__left_H_I1_J,axiom,
type_i8151583586401621767idle_a @ left2 ).
% invar_left'(1)
thf(fact_577_Nat_Ofunpow__code__def,axiom,
funpow_states_a = compow495008222514391794ates_a ).
% Nat.funpow_code_def
thf(fact_578_Nat_Ofunpow__code__def,axiom,
funpow_nat = compow_nat_nat ).
% Nat.funpow_code_def
thf(fact_579_step__n__push__size__new__small,axiom,
! [Dir: direction,Big: state_a2,Small: state_a,N: nat,X: a,Dir2: direction,Big2: state_a2,Small2: state_a] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ( compow495008222514391794ates_a @ N @ type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ Big @ ( push_a2 @ X @ Small ) ) )
= ( states_a2 @ Dir2 @ Big2 @ Small2 ) )
=> ( ( type_s6404775287138157491tate_a @ Small2 )
= ( suc @ ( type_s6404775287138157491tate_a @ Small ) ) ) ) ) ).
% step_n_push_size_new_small
thf(fact_580_step__n__push__size__new__big,axiom,
! [Dir: direction,Big: state_a2,Small: state_a,N: nat,X: a,Dir2: direction,Big2: state_a2,Small2: state_a] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ( compow495008222514391794ates_a @ N @ type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ ( push_a @ X @ Big ) @ Small ) )
= ( states_a2 @ Dir2 @ Big2 @ Small2 ) )
=> ( ( type_s6530235180886170618tate_a @ Big2 )
= ( suc @ ( type_s6530235180886170618tate_a @ Big ) ) ) ) ) ).
% step_n_push_size_new_big
thf(fact_581_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_582_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_583_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_584_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_585_one__eq__numeral__iff,axiom,
! [N: num] :
( ( one_one_nat
= ( numeral_numeral_nat @ N ) )
= ( one = N ) ) ).
% one_eq_numeral_iff
thf(fact_586_numeral__eq__one__iff,axiom,
! [N: num] :
( ( ( numeral_numeral_nat @ N )
= one_one_nat )
= ( N = one ) ) ).
% numeral_eq_one_iff
thf(fact_587_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_588_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_589_step__size__new__big,axiom,
! [Dir: direction,Big: state_a2,Small: state_a,Dir2: direction,Big2: state_a2,Small2: state_a] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ( type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
= ( states_a2 @ Dir2 @ Big2 @ Small2 ) )
=> ( ( type_s6530235180886170618tate_a @ Big2 )
= ( type_s6530235180886170618tate_a @ Big ) ) ) ) ).
% step_size_new_big
thf(fact_590_step__size__new__small,axiom,
! [Dir: direction,Big: state_a2,Small: state_a,Dir2: direction,Big2: state_a2,Small2: state_a] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ( type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
= ( states_a2 @ Dir2 @ Big2 @ Small2 ) )
=> ( ( type_s6404775287138157491tate_a @ Small2 )
= ( type_s6404775287138157491tate_a @ Small ) ) ) ) ).
% step_size_new_small
thf(fact_591_Suc__1,axiom,
( ( suc @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% Suc_1
thf(fact_592_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_593_numeral__le__one__iff,axiom,
! [N: num] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
= ( ord_less_eq_num @ N @ one ) ) ).
% numeral_le_one_iff
thf(fact_594_one__less__numeral__iff,axiom,
! [N: num] :
( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
= ( ord_less_num @ one @ N ) ) ).
% one_less_numeral_iff
thf(fact_595_remaining__steps__decline__sub,axiom,
! [States: states_a] :
( ( type_i8221491762852169479ates_a @ States )
=> ( ( type_r4519047461186610747ates_a @ ( type_s4923920245906622843ates_a @ States ) )
= ( minus_minus_nat @ ( type_r4519047461186610747ates_a @ States ) @ one_one_nat ) ) ) ).
% remaining_steps_decline_sub
thf(fact_596_step__n__size__new__big,axiom,
! [Dir: direction,Big: state_a2,Small: state_a,N: nat,Dir2: direction,Big2: state_a2,Small2: state_a] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ( compow495008222514391794ates_a @ N @ type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
= ( states_a2 @ Dir2 @ Big2 @ Small2 ) )
=> ( ( type_s6530235180886170618tate_a @ Big2 )
= ( type_s6530235180886170618tate_a @ Big ) ) ) ) ).
% step_n_size_new_big
thf(fact_597_step__n__size__new__small,axiom,
! [Dir: direction,Big: state_a2,Small: state_a,N: nat,Dir2: direction,Big2: state_a2,Small2: state_a] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ( compow495008222514391794ates_a @ N @ type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
= ( states_a2 @ Dir2 @ Big2 @ Small2 ) )
=> ( ( type_s6404775287138157491tate_a @ Small2 )
= ( type_s6404775287138157491tate_a @ Small ) ) ) ) ).
% step_n_size_new_small
thf(fact_598_step__push__size__new__big,axiom,
! [Dir: direction,Big: state_a2,Small: state_a,X: a,Dir2: direction,Big2: state_a2,Small2: state_a] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ( type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ ( push_a @ X @ Big ) @ Small ) )
= ( states_a2 @ Dir2 @ Big2 @ Small2 ) )
=> ( ( type_s6530235180886170618tate_a @ Big2 )
= ( suc @ ( type_s6530235180886170618tate_a @ Big ) ) ) ) ) ).
% step_push_size_new_big
thf(fact_599_step__push__size__new__small,axiom,
! [Dir: direction,Big: state_a2,Small: state_a,X: a,Dir2: direction,Big2: state_a2,Small2: state_a] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ( type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ Big @ ( push_a2 @ X @ Small ) ) )
= ( states_a2 @ Dir2 @ Big2 @ Small2 ) )
=> ( ( type_s6404775287138157491tate_a @ Small2 )
= ( suc @ ( type_s6404775287138157491tate_a @ Small ) ) ) ) ) ).
% step_push_size_new_small
thf(fact_600_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_601_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_602_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_603_comm__monoid__mult__class_Omult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_604_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_605_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_606_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_607_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_608_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_609_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_610_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_611_one__le__numeral,axiom,
! [N: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) ) ).
% one_le_numeral
thf(fact_612_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_613_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_614_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_615_not__numeral__less__one,axiom,
! [N: num] :
~ ( ord_less_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat ) ).
% not_numeral_less_one
thf(fact_616_less__1__mult,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ M )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_617_numeral__One,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numeral_One
thf(fact_618_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_619_numerals_I1_J,axiom,
( ( numeral_numeral_nat @ one )
= one_one_nat ) ).
% numerals(1)
thf(fact_620_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_621_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_622_mult__left__le,axiom,
! [C: nat,A: nat] :
( ( ord_less_eq_nat @ C @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_623_mult__le__one,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_624_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_625_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_626_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_627_size__ok__size__new__big,axiom,
! [Dir: direction,Big: state_a2,Small: state_a] :
( ( states_size_ok_a @ ( states_a2 @ Dir @ Big @ Small ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( type_s6530235180886170618tate_a @ Big ) ) ) ).
% size_ok_size_new_big
thf(fact_628_size__ok__size__new__small,axiom,
! [Dir: direction,Big: state_a2,Small: state_a] :
( ( states_size_ok_a @ ( states_a2 @ Dir @ Big @ Small ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) )
=> ( ord_less_nat @ zero_zero_nat @ ( type_s6404775287138157491tate_a @ Small ) ) ) ).
% size_ok_size_new_small
thf(fact_629_invar__left_H_I2_J,axiom,
type_i8151583586401621767idle_a @ left3 ).
% invar_left'(2)
thf(fact_630_step__4__pop__small__size__ok__4__aux,axiom,
! [Small: state_a,Dir: direction,Big: state_a2] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a2 @ Small ) )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( type_s6530235180886170618tate_a @ Big ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ ( type_s6404775287138157491tate_a @ Small ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( type_s6530235180886170618tate_a @ Big ) @ ( minus_minus_nat @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ ( minus_minus_nat @ ( type_s6404775287138157491tate_a @ Small ) @ one_one_nat ) ) ) ) ) ) ).
% step_4_pop_small_size_ok_4_aux
thf(fact_631_step__4__pop__big__size__ok__4__aux,axiom,
! [Big: state_a2,Dir: direction,Small: state_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ Big ) )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( type_s6530235180886170618tate_a @ Big ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ ( type_s6404775287138157491tate_a @ Small ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( minus_minus_nat @ ( type_s6530235180886170618tate_a @ Big ) @ one_one_nat ) @ ( minus_minus_nat @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ ( type_s6404775287138157491tate_a @ Small ) ) ) ) ) ) ).
% step_4_pop_big_size_ok_4_aux
thf(fact_632_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_633_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_634_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_635_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_636_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_637_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_638_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_639_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_640_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_641_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_642_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_643_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_644_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_645_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_646_numeral__plus__numeral,axiom,
! [M: num,N: num] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ M @ N ) ) ) ).
% numeral_plus_numeral
thf(fact_647_add__numeral__left,axiom,
! [V: num,W: num,Z: nat] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).
% add_numeral_left
thf(fact_648_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_649_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_650_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_651_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_652_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_653_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_654_Suc__funpow,axiom,
! [N: nat] :
( ( compow_nat_nat @ N @ suc )
= ( plus_plus_nat @ N ) ) ).
% Suc_funpow
thf(fact_655_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_656_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_657_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_658_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_659_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_660_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_661_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_662_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_663_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_664_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_665_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_666_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_667_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_668_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_669_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_670_distrib__left__numeral,axiom,
! [V: num,B: nat,C: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ B ) @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ C ) ) ) ).
% distrib_left_numeral
thf(fact_671_distrib__right__numeral,axiom,
! [A: nat,B: nat,V: num] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ ( numeral_numeral_nat @ V ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ ( numeral_numeral_nat @ V ) ) @ ( times_times_nat @ B @ ( numeral_numeral_nat @ V ) ) ) ) ).
% distrib_right_numeral
thf(fact_672_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_673_mult__Suc__right,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ M @ ( suc @ N ) )
= ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc_right
thf(fact_674_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_675_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_676_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_677_one__plus__numeral,axiom,
! [N: num] :
( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ one @ N ) ) ) ).
% one_plus_numeral
thf(fact_678_numeral__plus__one,axiom,
! [N: num] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ one_one_nat )
= ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% numeral_plus_one
thf(fact_679_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_680_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_681_one__add__one,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% one_add_one
thf(fact_682_add__2__eq__Suc,axiom,
! [N: nat] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
= ( suc @ ( suc @ N ) ) ) ).
% add_2_eq_Suc
thf(fact_683_add__2__eq__Suc_H,axiom,
! [N: nat] :
( ( plus_plus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( suc @ ( suc @ N ) ) ) ).
% add_2_eq_Suc'
thf(fact_684_remaining__steps__n__steps__plus,axiom,
! [N: nat,States: states_a] :
( ( ord_less_eq_nat @ N @ ( type_r4519047461186610747ates_a @ States ) )
=> ( ( type_i8221491762852169479ates_a @ States )
=> ( ( plus_plus_nat @ ( type_r4519047461186610747ates_a @ ( compow495008222514391794ates_a @ N @ type_s4923920245906622843ates_a @ States ) ) @ N )
= ( type_r4519047461186610747ates_a @ States ) ) ) ) ).
% remaining_steps_n_steps_plus
thf(fact_685__092_060open_062length__left_H_A_061_Asize_Aleft_A_N_A1_092_060close_062,axiom,
( length_left
= ( minus_minus_nat @ ( size_size_idle_a @ left3 ) @ one_one_nat ) ) ).
% \<open>length_left' = size left - 1\<close>
thf(fact_686__092_060open_062size_Astack__left_H_A_061_Asize_Aleft_A_N_A1_092_060close_062,axiom,
( ( size_size_stack_a @ stack_left )
= ( minus_minus_nat @ ( size_size_idle_a @ left3 ) @ one_one_nat ) ) ).
% \<open>size stack_left' = size left - 1\<close>
thf(fact_687_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_688_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_689_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_690_add__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_691_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_692_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C3: nat] :
( B
!= ( plus_plus_nat @ A @ C3 ) ) ) ).
% less_eqE
thf(fact_693_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_694_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
? [C2: nat] :
( B2
= ( plus_plus_nat @ A2 @ C2 ) ) ) ) ).
% le_iff_add
thf(fact_695_add__le__imp__le__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_imp_le_left
thf(fact_696_add__le__imp__le__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_imp_le_right
thf(fact_697_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_698_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_699_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_700_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_701_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_702_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_703_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_704_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_705_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_706_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_707_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N4: nat] :
? [K3: nat] :
( N4
= ( plus_plus_nat @ M5 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_708_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_709_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_710_group__cancel_Oadd1,axiom,
! [A5: nat,K: nat,A: nat,B: nat] :
( ( A5
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A5 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_711_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_712_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_713_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A2: nat,B2: nat] : ( plus_plus_nat @ B2 @ A2 ) ) ) ).
% add.commute
thf(fact_714_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_715_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_716_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_717_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_718_add__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_719_left__add__mult__distrib,axiom,
! [I: nat,U: nat,J: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I @ J ) @ U ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_720_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_721_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_722_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_723_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_724_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_725_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_726_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_727_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_728_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_729_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_730_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_731_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_732_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_733_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_734_nat__arith_Osuc1,axiom,
! [A5: nat,K: nat,A: nat] :
( ( A5
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A5 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_735_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_736_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_737_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_738_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_739_comm__semiring__class_Odistrib,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_740_distrib__left,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% distrib_left
thf(fact_741_distrib__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% distrib_right
thf(fact_742_combine__common__factor,axiom,
! [A: nat,E: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E ) @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_743_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_744_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_745_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_746_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_747_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_748_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_749_add__less__imp__less__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_imp_less_left
thf(fact_750_add__less__imp__less__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_imp_less_right
thf(fact_751_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_752_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_753_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_754_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_755_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_756_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_757_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_758_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_759_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_760_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_761_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_762_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_763_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_764_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_765_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_766_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_767_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C3: nat] :
( ( B
= ( plus_plus_nat @ A @ C3 ) )
=> ( C3 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_768_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_769_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_770_add__le__add__imp__diff__le,axiom,
! [I: nat,K: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_771_add__le__imp__le__diff,axiom,
! [I: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ N )
=> ( ord_less_eq_nat @ I @ ( minus_minus_nat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_772_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
thf(fact_773_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_774_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_775_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_776_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_777_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_778_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_779_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_780_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_781_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_782_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_783_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_784_numeral__Bit0,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit0 @ N ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) ) ).
% numeral_Bit0
thf(fact_785_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_786_one__plus__numeral__commute,axiom,
! [X: num] :
( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ X ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ X ) @ one_one_nat ) ) ).
% one_plus_numeral_commute
thf(fact_787_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_788_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_789_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_790_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K2: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_791_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M5: nat,N4: nat] :
? [K3: nat] :
( N4
= ( suc @ ( plus_plus_nat @ M5 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_792_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_793_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_794_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q ) ) ) ) ).
% less_natE
thf(fact_795_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M4: nat,N3: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ( ord_less_nat @ ( F @ M4 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_796_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_797_mult__Suc,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc
thf(fact_798_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_799_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_800_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_801_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_802_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_803_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_804_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_805_Suc__eq__plus1,axiom,
( suc
= ( ^ [N4: nat] : ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_806_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_807_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_808_add__strict__increasing2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_809_add__strict__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_810_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_811_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_812_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_813_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_814_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_815_numeral__Bit1,axiom,
! [N: num] :
( ( numeral_numeral_nat @ ( bit1 @ N ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ N ) @ ( numeral_numeral_nat @ N ) ) @ one_one_nat ) ) ).
% numeral_Bit1
thf(fact_816_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D2: nat] :
( ( A
= ( plus_plus_nat @ B @ D2 ) )
=> ( P @ D2 ) ) ) ) ).
% nat_diff_split
thf(fact_817_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D2: nat] :
( ( A
= ( plus_plus_nat @ B @ D2 ) )
& ~ ( P @ D2 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_818_less__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_819_nat__eq__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M )
= N ) ) ) ).
% nat_eq_add_iff1
thf(fact_820_nat__eq__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( M
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_821_nat__le__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_le_add_iff1
thf(fact_822_nat__le__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_le_add_iff2
thf(fact_823_nat__diff__add__eq1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_diff_add_eq1
thf(fact_824_nat__diff__add__eq2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_825_mult__2,axiom,
! [Z: nat] :
( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z )
= ( plus_plus_nat @ Z @ Z ) ) ).
% mult_2
thf(fact_826_mult__2__right,axiom,
! [Z: nat] :
( ( times_times_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
= ( plus_plus_nat @ Z @ Z ) ) ).
% mult_2_right
thf(fact_827_left__add__twice,axiom,
! [A: nat,B: nat] :
( ( plus_plus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).
% left_add_twice
thf(fact_828_numeral__add__unfold__funpow,axiom,
! [K: num,A: nat] :
( ( plus_plus_nat @ ( numeral_numeral_nat @ K ) @ A )
= ( compow_nat_nat @ ( numeral_numeral_nat @ K ) @ ( plus_plus_nat @ one_one_nat ) @ A ) ) ).
% numeral_add_unfold_funpow
thf(fact_829_Suc3__eq__add__3,axiom,
! [N: nat] :
( ( suc @ ( suc @ ( suc @ N ) ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N ) ) ).
% Suc3_eq_add_3
thf(fact_830_nat__1__add__1,axiom,
( ( plus_plus_nat @ one_one_nat @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).
% nat_1_add_1
thf(fact_831_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M5: nat,N4: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ N4 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M5 @ one_one_nat ) @ N4 ) ) ) ) ) ).
% add_eq_if
thf(fact_832_nat__less__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_less_add_iff1
thf(fact_833_nat__less__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N ) ) ) ) ).
% nat_less_add_iff2
thf(fact_834_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M5: nat,N4: nat] : ( if_nat @ ( M5 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N4 @ ( times_times_nat @ ( minus_minus_nat @ M5 @ one_one_nat ) @ N4 ) ) ) ) ) ).
% mult_eq_if
thf(fact_835_num_Osize_I5_J,axiom,
! [X2: num] :
( ( size_size_num @ ( bit0 @ X2 ) )
= ( plus_plus_nat @ ( size_size_num @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).
% num.size(5)
thf(fact_836_num_Osize_I6_J,axiom,
! [X33: num] :
( ( size_size_num @ ( bit1 @ X33 ) )
= ( plus_plus_nat @ ( size_size_num @ X33 ) @ ( suc @ zero_zero_nat ) ) ) ).
% num.size(6)
thf(fact_837_numeral__unfold__funpow,axiom,
( numeral_numeral_nat
= ( ^ [K3: num] : ( compow_nat_nat @ ( numeral_numeral_nat @ K3 ) @ ( plus_plus_nat @ one_one_nat ) @ zero_zero_nat ) ) ) ).
% numeral_unfold_funpow
thf(fact_838_nat__induct2,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( plus_plus_nat @ N3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct2
thf(fact_839_step__size__ok__4,axiom,
! [Dir: direction,Big: state_a2,Small: state_a,Dir2: direction,Big2: state_a2,Small2: state_a] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ( type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
= ( states_a2 @ Dir2 @ Big2 @ Small2 ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) @ one_one_nat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( size_size_state_a @ Big ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir2 @ Big2 @ Small2 ) ) @ one_one_nat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( size_size_state_a @ Big2 ) ) ) ) ) ) ).
% step_size_ok_4
thf(fact_840_step__size__ok__3,axiom,
! [Dir: direction,Big: state_a2,Small: state_a,Dir2: direction,Big2: state_a2,Small2: state_a] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ( type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
= ( states_a2 @ Dir2 @ Big2 @ Small2 ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) @ one_one_nat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( size_size_state_a2 @ Small ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir2 @ Big2 @ Small2 ) ) @ one_one_nat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( size_size_state_a2 @ Small2 ) ) ) ) ) ) ).
% step_size_ok_3
thf(fact_841_step__4__push__big__size__ok__3__aux,axiom,
! [Dir: direction,Big: state_a2,Small: state_a] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( type_s6404775287138157491tate_a @ Small ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ ( type_s6530235180886170618tate_a @ Big ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( type_s6404775287138157491tate_a @ Small ) @ ( minus_minus_nat @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ ( suc @ ( type_s6530235180886170618tate_a @ Big ) ) ) ) ) ) ).
% step_4_push_big_size_ok_3_aux
thf(fact_842_step__4__push__big__size__ok__4__aux,axiom,
! [Dir: direction,Big: state_a2,Small: state_a] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( type_s6530235180886170618tate_a @ Big ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ ( type_s6404775287138157491tate_a @ Small ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( suc @ ( type_s6530235180886170618tate_a @ Big ) ) @ ( minus_minus_nat @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ ( type_s6404775287138157491tate_a @ Small ) ) ) ) ) ).
% step_4_push_big_size_ok_4_aux
thf(fact_843_step__4__push__small__size__ok__3__aux,axiom,
! [Dir: direction,Big: state_a2,Small: state_a] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( type_s6404775287138157491tate_a @ Small ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ ( type_s6530235180886170618tate_a @ Big ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( suc @ ( type_s6404775287138157491tate_a @ Small ) ) @ ( minus_minus_nat @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ ( type_s6530235180886170618tate_a @ Big ) ) ) ) ) ).
% step_4_push_small_size_ok_3_aux
thf(fact_844_step__4__push__small__size__ok__4__aux,axiom,
! [Dir: direction,Big: state_a2,Small: state_a] :
( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( type_s6530235180886170618tate_a @ Big ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ ( type_s6404775287138157491tate_a @ Small ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( type_s6530235180886170618tate_a @ Big ) @ ( minus_minus_nat @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ ( suc @ ( type_s6404775287138157491tate_a @ Small ) ) ) ) ) ) ).
% step_4_push_small_size_ok_4_aux
thf(fact_845_step__size__ok__1,axiom,
! [Dir: direction,Big: state_a2,Small: state_a,Dir2: direction,Big2: state_a2,Small2: state_a] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ( type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
= ( states_a2 @ Dir2 @ Big2 @ Small2 ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( type_s6530235180886170618tate_a @ Big ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ ( type_s6404775287138157491tate_a @ Small ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( type_s6530235180886170618tate_a @ Big2 ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir2 @ Big2 @ Small2 ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ ( type_s6404775287138157491tate_a @ Small2 ) ) ) ) ) ) ).
% step_size_ok_1
thf(fact_846_step__size__ok__2,axiom,
! [Dir: direction,Big: state_a2,Small: state_a,Dir2: direction,Big2: state_a2,Small2: state_a] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ( type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
= ( states_a2 @ Dir2 @ Big2 @ Small2 ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( type_s6404775287138157491tate_a @ Small ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ ( type_s6530235180886170618tate_a @ Big ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( type_s6404775287138157491tate_a @ Small2 ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir2 @ Big2 @ Small2 ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ ( type_s6530235180886170618tate_a @ Big2 ) ) ) ) ) ) ).
% step_size_ok_2
thf(fact_847_step__4__push__big__size__ok__2,axiom,
! [Dir: direction,Big: state_a2,Small: state_a,X: a,Dir2: direction,Big2: state_a2,Small2: state_a] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) )
=> ( ( ( compow495008222514391794ates_a @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ ( push_a @ X @ Big ) @ Small ) )
= ( states_a2 @ Dir2 @ Big2 @ Small2 ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) @ one_one_nat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( size_size_state_a @ Big ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir2 @ Big2 @ Small2 ) ) @ one_one_nat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( size_size_state_a @ Big2 ) ) ) ) ) ) ) ).
% step_4_push_big_size_ok_2
thf(fact_848_step__4__push__small__size__ok__2,axiom,
! [Dir: direction,Big: state_a2,Small: state_a,X: a,Dir2: direction,Big2: state_a2,Small2: state_a] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) )
=> ( ( ( compow495008222514391794ates_a @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ Big @ ( push_a2 @ X @ Small ) ) )
= ( states_a2 @ Dir2 @ Big2 @ Small2 ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) @ one_one_nat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( size_size_state_a @ Big ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir2 @ Big2 @ Small2 ) ) @ one_one_nat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( size_size_state_a @ Big2 ) ) ) ) ) ) ) ).
% step_4_push_small_size_ok_2
thf(fact_849_step__4__push__big__size__ok__1,axiom,
! [Dir: direction,Big: state_a2,Small: state_a,X: a,Dir2: direction,Big2: state_a2,Small2: state_a] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) )
=> ( ( ( compow495008222514391794ates_a @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ ( push_a @ X @ Big ) @ Small ) )
= ( states_a2 @ Dir2 @ Big2 @ Small2 ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) @ one_one_nat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( size_size_state_a2 @ Small ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir2 @ Big2 @ Small2 ) ) @ one_one_nat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( size_size_state_a2 @ Small2 ) ) ) ) ) ) ) ).
% step_4_push_big_size_ok_1
thf(fact_850_step__4__push__small__size__ok__1,axiom,
! [Dir: direction,Big: state_a2,Small: state_a,X: a,Dir2: direction,Big2: state_a2,Small2: state_a] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) )
=> ( ( ( compow495008222514391794ates_a @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ Big @ ( push_a2 @ X @ Small ) ) )
= ( states_a2 @ Dir2 @ Big2 @ Small2 ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) @ one_one_nat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( size_size_state_a2 @ Small ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir2 @ Big2 @ Small2 ) ) @ one_one_nat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( size_size_state_a2 @ Small2 ) ) ) ) ) ) ) ).
% step_4_push_small_size_ok_1
thf(fact_851_step__4__push__big__size__ok__3,axiom,
! [Dir: direction,Big: state_a2,Small: state_a,X: a,Dir2: direction,Big2: state_a2,Small2: state_a] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) )
=> ( ( ( compow495008222514391794ates_a @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ ( push_a @ X @ Big ) @ Small ) )
= ( states_a2 @ Dir2 @ Big2 @ Small2 ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( type_s6404775287138157491tate_a @ Small ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ ( type_s6530235180886170618tate_a @ Big ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( type_s6404775287138157491tate_a @ Small2 ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir2 @ Big2 @ Small2 ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ ( type_s6530235180886170618tate_a @ Big2 ) ) ) ) ) ) ) ).
% step_4_push_big_size_ok_3
thf(fact_852_step__4__push__big__size__ok__4,axiom,
! [Dir: direction,Big: state_a2,Small: state_a,X: a,Dir2: direction,Big2: state_a2,Small2: state_a] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) )
=> ( ( ( compow495008222514391794ates_a @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ ( push_a @ X @ Big ) @ Small ) )
= ( states_a2 @ Dir2 @ Big2 @ Small2 ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( type_s6530235180886170618tate_a @ Big ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ ( type_s6404775287138157491tate_a @ Small ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( type_s6530235180886170618tate_a @ Big2 ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir2 @ Big2 @ Small2 ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ ( type_s6404775287138157491tate_a @ Small2 ) ) ) ) ) ) ) ).
% step_4_push_big_size_ok_4
thf(fact_853_step__4__push__small__size__ok__3,axiom,
! [Dir: direction,Big: state_a2,Small: state_a,X: a,Dir2: direction,Big2: state_a2,Small2: state_a] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) )
=> ( ( ( compow495008222514391794ates_a @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ Big @ ( push_a2 @ X @ Small ) ) )
= ( states_a2 @ Dir2 @ Big2 @ Small2 ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( type_s6404775287138157491tate_a @ Small ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ ( type_s6530235180886170618tate_a @ Big ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( type_s6404775287138157491tate_a @ Small2 ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir2 @ Big2 @ Small2 ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ ( type_s6530235180886170618tate_a @ Big2 ) ) ) ) ) ) ) ).
% step_4_push_small_size_ok_3
thf(fact_854_step__4__push__small__size__ok__4,axiom,
! [Dir: direction,Big: state_a2,Small: state_a,X: a,Dir2: direction,Big2: state_a2,Small2: state_a] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) )
=> ( ( ( compow495008222514391794ates_a @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ Big @ ( push_a2 @ X @ Small ) ) )
= ( states_a2 @ Dir2 @ Big2 @ Small2 ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( type_s6530235180886170618tate_a @ Big ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ ( type_s6404775287138157491tate_a @ Small ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( type_s6530235180886170618tate_a @ Big2 ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir2 @ Big2 @ Small2 ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ ( type_s6404775287138157491tate_a @ Small2 ) ) ) ) ) ) ) ).
% step_4_push_small_size_ok_4
thf(fact_855_step__4__pop__big__size__ok__3__aux,axiom,
! [Big: state_a2,Dir: direction,Small: state_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ Big ) )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( type_s6404775287138157491tate_a @ Small ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ ( type_s6530235180886170618tate_a @ Big ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( type_s6404775287138157491tate_a @ Small ) @ ( minus_minus_nat @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ ( minus_minus_nat @ ( type_s6530235180886170618tate_a @ Big ) @ one_one_nat ) ) ) ) ) ) ).
% step_4_pop_big_size_ok_3_aux
thf(fact_856_size__ok_H_Oelims_I3_J,axiom,
! [X: states_a,Xa: nat] :
( ~ ( states_size_ok_a @ X @ Xa )
=> ~ ! [Uu: direction,Big3: state_a2,Small3: state_a] :
( ( X
= ( states_a2 @ Uu @ Big3 @ Small3 ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( type_s6404775287138157491tate_a @ Small3 ) @ Xa ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ ( type_s6530235180886170618tate_a @ Big3 ) ) )
& ( ord_less_eq_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( type_s6530235180886170618tate_a @ Big3 ) @ Xa ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ ( type_s6404775287138157491tate_a @ Small3 ) ) )
& ( ord_less_eq_nat @ ( plus_plus_nat @ Xa @ one_one_nat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( size_size_state_a2 @ Small3 ) ) )
& ( ord_less_eq_nat @ ( plus_plus_nat @ Xa @ one_one_nat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( size_size_state_a @ Big3 ) ) ) ) ) ) ).
% size_ok'.elims(3)
thf(fact_857_size__ok_H_Oelims_I2_J,axiom,
! [X: states_a,Xa: nat] :
( ( states_size_ok_a @ X @ Xa )
=> ~ ! [Uu: direction,Big3: state_a2,Small3: state_a] :
( ( X
= ( states_a2 @ Uu @ Big3 @ Small3 ) )
=> ~ ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( type_s6404775287138157491tate_a @ Small3 ) @ Xa ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ ( type_s6530235180886170618tate_a @ Big3 ) ) )
& ( ord_less_eq_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( type_s6530235180886170618tate_a @ Big3 ) @ Xa ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ ( type_s6404775287138157491tate_a @ Small3 ) ) )
& ( ord_less_eq_nat @ ( plus_plus_nat @ Xa @ one_one_nat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( size_size_state_a2 @ Small3 ) ) )
& ( ord_less_eq_nat @ ( plus_plus_nat @ Xa @ one_one_nat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( size_size_state_a @ Big3 ) ) ) ) ) ) ).
% size_ok'.elims(2)
thf(fact_858_size__ok_H_Oelims_I1_J,axiom,
! [X: states_a,Xa: nat,Y: $o] :
( ( ( states_size_ok_a @ X @ Xa )
= Y )
=> ~ ! [Uu: direction,Big3: state_a2,Small3: state_a] :
( ( X
= ( states_a2 @ Uu @ Big3 @ Small3 ) )
=> ( Y
= ( ~ ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( type_s6404775287138157491tate_a @ Small3 ) @ Xa ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ ( type_s6530235180886170618tate_a @ Big3 ) ) )
& ( ord_less_eq_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( type_s6530235180886170618tate_a @ Big3 ) @ Xa ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ ( type_s6404775287138157491tate_a @ Small3 ) ) )
& ( ord_less_eq_nat @ ( plus_plus_nat @ Xa @ one_one_nat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( size_size_state_a2 @ Small3 ) ) )
& ( ord_less_eq_nat @ ( plus_plus_nat @ Xa @ one_one_nat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( size_size_state_a @ Big3 ) ) ) ) ) ) ) ) ).
% size_ok'.elims(1)
thf(fact_859_semiring__norm_I6_J,axiom,
! [M: num,N: num] :
( ( plus_plus_num @ ( bit0 @ M ) @ ( bit0 @ N ) )
= ( bit0 @ ( plus_plus_num @ M @ N ) ) ) ).
% semiring_norm(6)
thf(fact_860_semiring__norm_I2_J,axiom,
( ( plus_plus_num @ one @ one )
= ( bit0 @ one ) ) ).
% semiring_norm(2)
thf(fact_861_semiring__norm_I7_J,axiom,
! [M: num,N: num] :
( ( plus_plus_num @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).
% semiring_norm(7)
thf(fact_862_semiring__norm_I9_J,axiom,
! [M: num,N: num] :
( ( plus_plus_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( bit1 @ ( plus_plus_num @ M @ N ) ) ) ).
% semiring_norm(9)
thf(fact_863_semiring__norm_I3_J,axiom,
! [N: num] :
( ( plus_plus_num @ one @ ( bit0 @ N ) )
= ( bit1 @ N ) ) ).
% semiring_norm(3)
thf(fact_864_semiring__norm_I4_J,axiom,
! [N: num] :
( ( plus_plus_num @ one @ ( bit1 @ N ) )
= ( bit0 @ ( plus_plus_num @ N @ one ) ) ) ).
% semiring_norm(4)
thf(fact_865_semiring__norm_I5_J,axiom,
! [M: num] :
( ( plus_plus_num @ ( bit0 @ M ) @ one )
= ( bit1 @ M ) ) ).
% semiring_norm(5)
thf(fact_866_semiring__norm_I8_J,axiom,
! [M: num] :
( ( plus_plus_num @ ( bit1 @ M ) @ one )
= ( bit0 @ ( plus_plus_num @ M @ one ) ) ) ).
% semiring_norm(8)
thf(fact_867_semiring__norm_I10_J,axiom,
! [M: num,N: num] :
( ( plus_plus_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( bit0 @ ( plus_plus_num @ ( plus_plus_num @ M @ N ) @ one ) ) ) ).
% semiring_norm(10)
thf(fact_868_Suc__numeral,axiom,
! [N: num] :
( ( suc @ ( numeral_numeral_nat @ N ) )
= ( numeral_numeral_nat @ ( plus_plus_num @ N @ one ) ) ) ).
% Suc_numeral
thf(fact_869_semiring__norm_I16_J,axiom,
! [M: num,N: num] :
( ( times_times_num @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( bit1 @ ( plus_plus_num @ ( plus_plus_num @ M @ N ) @ ( bit0 @ ( times_times_num @ M @ N ) ) ) ) ) ).
% semiring_norm(16)
thf(fact_870_add__One__commute,axiom,
! [N: num] :
( ( plus_plus_num @ one @ N )
= ( plus_plus_num @ N @ one ) ) ).
% add_One_commute
thf(fact_871_remaining__steps__states_Ocases,axiom,
! [X: states_a] :
~ ! [Uu: direction,Big3: state_a2,Small3: state_a] :
( X
!= ( states_a2 @ Uu @ Big3 @ Small3 ) ) ).
% remaining_steps_states.cases
thf(fact_872_Suc__nat__number__of__add,axiom,
! [V: num,N: nat] :
( ( suc @ ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ N ) )
= ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ one ) ) @ N ) ) ).
% Suc_nat_number_of_add
thf(fact_873_size__ok_H_Osimps,axiom,
! [Uu2: direction,Big: state_a2,Small: state_a,Steps: nat] :
( ( states_size_ok_a @ ( states_a2 @ Uu2 @ Big @ Small ) @ Steps )
= ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( type_s6404775287138157491tate_a @ Small ) @ Steps ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ ( type_s6530235180886170618tate_a @ Big ) ) )
& ( ord_less_eq_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( type_s6530235180886170618tate_a @ Big ) @ Steps ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ ( type_s6404775287138157491tate_a @ Small ) ) )
& ( ord_less_eq_nat @ ( plus_plus_nat @ Steps @ one_one_nat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( size_size_state_a2 @ Small ) ) )
& ( ord_less_eq_nat @ ( plus_plus_nat @ Steps @ one_one_nat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( size_size_state_a @ Big ) ) ) ) ) ).
% size_ok'.simps
thf(fact_874_size__current_Osimps,axiom,
! [Uu2: list_a,Added: nat,Old: stack_a,Uv: nat] :
( ( size_size_current_a @ ( current_a2 @ Uu2 @ Added @ Old @ Uv ) )
= ( plus_plus_nat @ Added @ ( size_size_stack_a @ Old ) ) ) ).
% size_current.simps
thf(fact_875_size__new__current_Ocases,axiom,
! [X: current_a] :
~ ! [Uu: list_a,Added2: nat,Uv2: stack_a,Remained: nat] :
( X
!= ( current_a2 @ Uu @ Added2 @ Uv2 @ Remained ) ) ).
% size_new_current.cases
thf(fact_876_size__current_Oelims,axiom,
! [X: current_a,Y: nat] :
( ( ( size_size_current_a @ X )
= Y )
=> ~ ! [Uu: list_a,Added2: nat,Old2: stack_a] :
( ? [Uv2: nat] :
( X
= ( current_a2 @ Uu @ Added2 @ Old2 @ Uv2 ) )
=> ( Y
!= ( plus_plus_nat @ Added2 @ ( size_size_stack_a @ Old2 ) ) ) ) ) ).
% size_current.elims
thf(fact_877_Suc__sub,axiom,
! [N: nat,M: nat] :
( ( ( suc @ N )
= M )
=> ( N
= ( minus_minus_nat @ M @ one_one_nat ) ) ) ).
% Suc_sub
thf(fact_878_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M7: nat] :
( ( P @ X )
=> ( ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M7 ) )
=> ~ ! [M4: nat] :
( ( P @ M4 )
=> ~ ! [X7: nat] :
( ( P @ X7 )
=> ( ord_less_eq_nat @ X7 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_879_add__scale__eq__noteq,axiom,
! [R2: nat,A: nat,B: nat,C: nat,D: nat] :
( ( R2 != zero_zero_nat )
=> ( ( ( A = B )
& ( C != D ) )
=> ( ( plus_plus_nat @ A @ ( times_times_nat @ R2 @ C ) )
!= ( plus_plus_nat @ B @ ( times_times_nat @ R2 @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_880_num_Osize__gen_I3_J,axiom,
! [X33: num] :
( ( size_num @ ( bit1 @ X33 ) )
= ( plus_plus_nat @ ( size_num @ X33 ) @ ( suc @ zero_zero_nat ) ) ) ).
% num.size_gen(3)
thf(fact_881_num_Osize__gen_I1_J,axiom,
( ( size_num @ one )
= zero_zero_nat ) ).
% num.size_gen(1)
thf(fact_882_add__0__iff,axiom,
! [B: nat,A: nat] :
( ( B
= ( plus_plus_nat @ B @ A ) )
= ( A = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_883_crossproduct__noteq,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( A != B )
& ( C != D ) )
= ( ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) )
!= ( plus_plus_nat @ ( times_times_nat @ A @ D ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_884_crossproduct__eq,axiom,
! [W: nat,Y: nat,X: nat,Z: nat] :
( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y ) @ ( times_times_nat @ X @ Z ) )
= ( plus_plus_nat @ ( times_times_nat @ W @ Z ) @ ( times_times_nat @ X @ Y ) ) )
= ( ( W = X )
| ( Y = Z ) ) ) ).
% crossproduct_eq
thf(fact_885_num_Osize__gen_I2_J,axiom,
! [X2: num] :
( ( size_num @ ( bit0 @ X2 ) )
= ( plus_plus_nat @ ( size_num @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).
% num.size_gen(2)
thf(fact_886_step__4__pop__small__size__ok__3,axiom,
! [Dir: direction,Big: state_a2,Small: state_a,X: a,SmallP: state_a,Dir2: direction,Big2: state_a2,Small2: state_a] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a2 @ Small ) )
=> ( ( ( pop_a3 @ Small )
= ( produc1224139502141355779tate_a @ X @ SmallP ) )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ SmallP ) ) )
=> ( ( ( compow495008222514391794ates_a @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ Big @ SmallP ) )
= ( states_a2 @ Dir2 @ Big2 @ Small2 ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( type_s6404775287138157491tate_a @ Small ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ ( type_s6530235180886170618tate_a @ Big ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( type_s6404775287138157491tate_a @ Small2 ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir2 @ Big2 @ Small2 ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ ( type_s6530235180886170618tate_a @ Big2 ) ) ) ) ) ) ) ) ) ).
% step_4_pop_small_size_ok_3
thf(fact_887_step__4__pop__small__size__ok__4,axiom,
! [Dir: direction,Big: state_a2,Small: state_a,X: a,SmallP: state_a,Dir2: direction,Big2: state_a2,Small2: state_a] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a2 @ Small ) )
=> ( ( ( pop_a3 @ Small )
= ( produc1224139502141355779tate_a @ X @ SmallP ) )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ SmallP ) ) )
=> ( ( ( compow495008222514391794ates_a @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ Big @ SmallP ) )
= ( states_a2 @ Dir2 @ Big2 @ Small2 ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( type_s6530235180886170618tate_a @ Big ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ ( type_s6404775287138157491tate_a @ Small ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( type_s6530235180886170618tate_a @ Big2 ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir2 @ Big2 @ Small2 ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ ( type_s6404775287138157491tate_a @ Small2 ) ) ) ) ) ) ) ) ) ).
% step_4_pop_small_size_ok_4
thf(fact_888_step__n__pop__size__small,axiom,
! [Dir: direction,Big: state_a2,Small: state_a,X: a,SmallP: state_a,N: nat,Dir2: direction,Big2: state_a2,Small2: state_a] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a2 @ Small ) )
=> ( ( ( pop_a3 @ Small )
= ( produc1224139502141355779tate_a @ X @ SmallP ) )
=> ( ( ( compow495008222514391794ates_a @ N @ type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ Big @ SmallP ) )
= ( states_a2 @ Dir2 @ Big2 @ Small2 ) )
=> ( ( suc @ ( size_size_state_a2 @ Small2 ) )
= ( size_size_state_a2 @ Small ) ) ) ) ) ) ).
% step_n_pop_size_small
thf(fact_889_step__n__pop__size__new__small,axiom,
! [Dir: direction,Big: state_a2,Small: state_a,X: a,SmallP: state_a,N: nat,Dir2: direction,Big2: state_a2,Small2: state_a] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a2 @ Small ) )
=> ( ( ( pop_a3 @ Small )
= ( produc1224139502141355779tate_a @ X @ SmallP ) )
=> ( ( ( compow495008222514391794ates_a @ N @ type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ Big @ SmallP ) )
= ( states_a2 @ Dir2 @ Big2 @ Small2 ) )
=> ( ( suc @ ( type_s6404775287138157491tate_a @ Small2 ) )
= ( type_s6404775287138157491tate_a @ Small ) ) ) ) ) ) ).
% step_n_pop_size_new_small
thf(fact_890_invar__pop__small,axiom,
! [Dir: direction,Big: state_a2,Small: state_a,X: a,Small2: state_a] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a2 @ Small ) )
=> ( ( ( pop_a3 @ Small )
= ( produc1224139502141355779tate_a @ X @ Small2 ) )
=> ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small2 ) ) ) ) ) ).
% invar_pop_small
thf(fact_891_remaining__steps__pop__small,axiom,
! [Dir: direction,Big: state_a2,Small: state_a,X: a,SmallP: state_a] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a2 @ Small ) )
=> ( ( ( pop_a3 @ Small )
= ( produc1224139502141355779tate_a @ X @ SmallP ) )
=> ( ord_less_eq_nat @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ SmallP ) ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) ) ) ) ) ).
% remaining_steps_pop_small
thf(fact_892_step__4__remaining__steps__pop__small,axiom,
! [Dir: direction,Big: state_a2,Small: state_a,X: a,SmallP: state_a,Dir2: direction,Big2: state_a2,Small2: state_a] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a2 @ Small ) )
=> ( ( ( pop_a3 @ Small )
= ( produc1224139502141355779tate_a @ X @ SmallP ) )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ SmallP ) ) )
=> ( ( ( compow495008222514391794ates_a @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ Big @ SmallP ) )
= ( states_a2 @ Dir2 @ Big2 @ Small2 ) )
=> ( ord_less_eq_nat @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir2 @ Big2 @ Small2 ) ) @ ( minus_minus_nat @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% step_4_remaining_steps_pop_small
thf(fact_893_step__4__pop__small__size__ok,axiom,
! [Dir: direction,Big: state_a2,Small: state_a,X: a,SmallP: state_a] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a2 @ Small ) )
=> ( ( ( pop_a3 @ Small )
= ( produc1224139502141355779tate_a @ X @ SmallP ) )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ SmallP ) ) )
=> ( ( states_size_ok_a @ ( states_a2 @ Dir @ Big @ Small ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) )
=> ( states_size_ok_a @ ( compow495008222514391794ates_a @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ Big @ SmallP ) ) @ ( type_r4519047461186610747ates_a @ ( compow495008222514391794ates_a @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ Big @ SmallP ) ) ) ) ) ) ) ) ) ).
% step_4_pop_small_size_ok
thf(fact_894_step__4__pop__small__size__ok__1,axiom,
! [Dir: direction,Big: state_a2,Small: state_a,X: a,SmallP: state_a,Dir2: direction,Big2: state_a2,Small2: state_a] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a2 @ Small ) )
=> ( ( ( pop_a3 @ Small )
= ( produc1224139502141355779tate_a @ X @ SmallP ) )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ SmallP ) ) )
=> ( ( ( compow495008222514391794ates_a @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ Big @ SmallP ) )
= ( states_a2 @ Dir2 @ Big2 @ Small2 ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) @ one_one_nat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( size_size_state_a2 @ Small ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir2 @ Big2 @ Small2 ) ) @ one_one_nat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( size_size_state_a2 @ Small2 ) ) ) ) ) ) ) ) ) ).
% step_4_pop_small_size_ok_1
thf(fact_895_step__4__pop__small__size__ok__2,axiom,
! [Dir: direction,Big: state_a2,Small: state_a,X: a,SmallP: state_a,Dir2: direction,Big2: state_a2,Small2: state_a] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a2 @ Small ) )
=> ( ( ( pop_a3 @ Small )
= ( produc1224139502141355779tate_a @ X @ SmallP ) )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ SmallP ) ) )
=> ( ( ( compow495008222514391794ates_a @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ Big @ SmallP ) )
= ( states_a2 @ Dir2 @ Big2 @ Small2 ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) @ one_one_nat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( size_size_state_a @ Big ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir2 @ Big2 @ Small2 ) ) @ one_one_nat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( size_size_state_a @ Big2 ) ) ) ) ) ) ) ) ) ).
% step_4_pop_small_size_ok_2
thf(fact_896_divides__aux__eq,axiom,
! [Q2: nat,R2: nat] :
( ( unique5332122412489317741ux_nat @ ( product_Pair_nat_nat @ Q2 @ R2 ) )
= ( R2 = zero_zero_nat ) ) ).
% divides_aux_eq
thf(fact_897_step__4__pop__big__size__ok__4,axiom,
! [Dir: direction,Big: state_a2,Small: state_a,X: a,BigP: state_a2,Dir2: direction,Big2: state_a2,Small2: state_a] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ Big ) )
=> ( ( ( pop_a @ Big )
= ( produc8641956578966763338tate_a @ X @ BigP ) )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ BigP @ Small ) ) )
=> ( ( ( compow495008222514391794ates_a @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ BigP @ Small ) )
= ( states_a2 @ Dir2 @ Big2 @ Small2 ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( type_s6530235180886170618tate_a @ Big ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ ( type_s6404775287138157491tate_a @ Small ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( type_s6530235180886170618tate_a @ Big2 ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir2 @ Big2 @ Small2 ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ ( type_s6404775287138157491tate_a @ Small2 ) ) ) ) ) ) ) ) ) ).
% step_4_pop_big_size_ok_4
thf(fact_898_step__pop__size__big,axiom,
! [Dir: direction,Big: state_a2,Small: state_a,X: a,BigP: state_a2,Dir2: direction,Big2: state_a2,Small2: state_a] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ Big ) )
=> ( ( ( pop_a @ Big )
= ( produc8641956578966763338tate_a @ X @ BigP ) )
=> ( ( ( type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ BigP @ Small ) )
= ( states_a2 @ Dir2 @ Big2 @ Small2 ) )
=> ( ( suc @ ( size_size_state_a @ Big2 ) )
= ( size_size_state_a @ Big ) ) ) ) ) ) ).
% step_pop_size_big
thf(fact_899_step__n__pop__size__big,axiom,
! [Dir: direction,Big: state_a2,Small: state_a,X: a,BigP: state_a2,N: nat,Dir2: direction,Big2: state_a2,Small2: state_a] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ Big ) )
=> ( ( ( pop_a @ Big )
= ( produc8641956578966763338tate_a @ X @ BigP ) )
=> ( ( ( compow495008222514391794ates_a @ N @ type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ BigP @ Small ) )
= ( states_a2 @ Dir2 @ Big2 @ Small2 ) )
=> ( ( suc @ ( size_size_state_a @ Big2 ) )
= ( size_size_state_a @ Big ) ) ) ) ) ) ).
% step_n_pop_size_big
thf(fact_900_step__pop__size__new__big,axiom,
! [Dir: direction,Big: state_a2,Small: state_a,X: a,BigP: state_a2,Dir2: direction,Big2: state_a2,Small2: state_a] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ Big ) )
=> ( ( ( pop_a @ Big )
= ( produc8641956578966763338tate_a @ X @ BigP ) )
=> ( ( ( type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ BigP @ Small ) )
= ( states_a2 @ Dir2 @ Big2 @ Small2 ) )
=> ( ( suc @ ( type_s6530235180886170618tate_a @ Big2 ) )
= ( type_s6530235180886170618tate_a @ Big ) ) ) ) ) ) ).
% step_pop_size_new_big
thf(fact_901_Current_Opush_Ocases,axiom,
! [X: produc7805042584321970905rent_a] :
~ ! [X3: a,Extra: list_a,Added2: nat,Old2: stack_a,Remained: nat] :
( X
!= ( produc8503237746132909001rent_a @ X3 @ ( current_a2 @ Extra @ Added2 @ Old2 @ Remained ) ) ) ).
% Current.push.cases
thf(fact_902_size__ok_H_Ocases,axiom,
! [X: produc1571854377283420419_a_nat] :
~ ! [Uu: direction,Big3: state_a2,Small3: state_a,Steps2: nat] :
( X
!= ( produc1877401315875745917_a_nat @ ( states_a2 @ Uu @ Big3 @ Small3 ) @ Steps2 ) ) ).
% size_ok'.cases
thf(fact_903_invar__pop__big,axiom,
! [Dir: direction,Big: state_a2,Small: state_a,X: a,Big2: state_a2] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ Big ) )
=> ( ( ( pop_a @ Big )
= ( produc8641956578966763338tate_a @ X @ Big2 ) )
=> ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big2 @ Small ) ) ) ) ) ).
% invar_pop_big
thf(fact_904_xor__num_Ocases,axiom,
! [X: product_prod_num_num] :
( ( X
!= ( product_Pair_num_num @ one @ one ) )
=> ( ! [N3: num] :
( X
!= ( product_Pair_num_num @ one @ ( bit0 @ N3 ) ) )
=> ( ! [N3: num] :
( X
!= ( product_Pair_num_num @ one @ ( bit1 @ N3 ) ) )
=> ( ! [M4: num] :
( X
!= ( product_Pair_num_num @ ( bit0 @ M4 ) @ one ) )
=> ( ! [M4: num,N3: num] :
( X
!= ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit0 @ N3 ) ) )
=> ( ! [M4: num,N3: num] :
( X
!= ( product_Pair_num_num @ ( bit0 @ M4 ) @ ( bit1 @ N3 ) ) )
=> ( ! [M4: num] :
( X
!= ( product_Pair_num_num @ ( bit1 @ M4 ) @ one ) )
=> ( ! [M4: num,N3: num] :
( X
!= ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit0 @ N3 ) ) )
=> ~ ! [M4: num,N3: num] :
( X
!= ( product_Pair_num_num @ ( bit1 @ M4 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ).
% xor_num.cases
thf(fact_905_remaining__steps__pop__big,axiom,
! [Dir: direction,Big: state_a2,Small: state_a,X: a,BigP: state_a2] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ Big ) )
=> ( ( ( pop_a @ Big )
= ( produc8641956578966763338tate_a @ X @ BigP ) )
=> ( ord_less_eq_nat @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ BigP @ Small ) ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) ) ) ) ) ).
% remaining_steps_pop_big
thf(fact_906_step__n__pop__size__new__big,axiom,
! [Dir: direction,Big: state_a2,Small: state_a,X: a,BigP: state_a2,N: nat,Dir2: direction,Big2: state_a2,Small2: state_a] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ Big ) )
=> ( ( ( pop_a @ Big )
= ( produc8641956578966763338tate_a @ X @ BigP ) )
=> ( ( ( compow495008222514391794ates_a @ N @ type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ BigP @ Small ) )
= ( states_a2 @ Dir2 @ Big2 @ Small2 ) )
=> ( ( suc @ ( type_s6530235180886170618tate_a @ Big2 ) )
= ( type_s6530235180886170618tate_a @ Big ) ) ) ) ) ) ).
% step_n_pop_size_new_big
thf(fact_907_step__4__remaining__steps__pop__big,axiom,
! [Dir: direction,Big: state_a2,Small: state_a,X: a,BigP: state_a2,Dir2: direction,Big2: state_a2,Small2: state_a] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ Big ) )
=> ( ( ( pop_a @ Big )
= ( produc8641956578966763338tate_a @ X @ BigP ) )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ BigP @ Small ) ) )
=> ( ( ( compow495008222514391794ates_a @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ BigP @ Small ) )
= ( states_a2 @ Dir2 @ Big2 @ Small2 ) )
=> ( ord_less_eq_nat @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir2 @ Big2 @ Small2 ) ) @ ( minus_minus_nat @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).
% step_4_remaining_steps_pop_big
thf(fact_908_step__4__pop__big__size__ok,axiom,
! [Dir: direction,Big: state_a2,Small: state_a,X: a,BigP: state_a2] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ Big ) )
=> ( ( ( pop_a @ Big )
= ( produc8641956578966763338tate_a @ X @ BigP ) )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ BigP @ Small ) ) )
=> ( ( states_size_ok_a @ ( states_a2 @ Dir @ Big @ Small ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) )
=> ( states_size_ok_a @ ( compow495008222514391794ates_a @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ BigP @ Small ) ) @ ( type_r4519047461186610747ates_a @ ( compow495008222514391794ates_a @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ BigP @ Small ) ) ) ) ) ) ) ) ) ).
% step_4_pop_big_size_ok
thf(fact_909_step__4__pop__big__size__ok__2,axiom,
! [Dir: direction,Big: state_a2,Small: state_a,X: a,BigP: state_a2,Dir2: direction,Big2: state_a2,Small2: state_a] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ Big ) )
=> ( ( ( pop_a @ Big )
= ( produc8641956578966763338tate_a @ X @ BigP ) )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ BigP @ Small ) ) )
=> ( ( ( compow495008222514391794ates_a @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ BigP @ Small ) )
= ( states_a2 @ Dir2 @ Big2 @ Small2 ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) @ one_one_nat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( size_size_state_a @ Big ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir2 @ Big2 @ Small2 ) ) @ one_one_nat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( size_size_state_a @ Big2 ) ) ) ) ) ) ) ) ) ).
% step_4_pop_big_size_ok_2
thf(fact_910_step__4__pop__big__size__ok__1,axiom,
! [Dir: direction,Big: state_a2,Small: state_a,X: a,BigP: state_a2,Dir2: direction,Big2: state_a2,Small2: state_a] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ Big ) )
=> ( ( ( pop_a @ Big )
= ( produc8641956578966763338tate_a @ X @ BigP ) )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ BigP @ Small ) ) )
=> ( ( ( compow495008222514391794ates_a @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ BigP @ Small ) )
= ( states_a2 @ Dir2 @ Big2 @ Small2 ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) @ one_one_nat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( size_size_state_a2 @ Small ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir2 @ Big2 @ Small2 ) ) @ one_one_nat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( size_size_state_a2 @ Small2 ) ) ) ) ) ) ) ) ) ).
% step_4_pop_big_size_ok_1
thf(fact_911_step__4__pop__big__size__ok__3,axiom,
! [Dir: direction,Big: state_a2,Small: state_a,X: a,BigP: state_a2,Dir2: direction,Big2: state_a2,Small2: state_a] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ Big ) )
=> ( ( ( pop_a @ Big )
= ( produc8641956578966763338tate_a @ X @ BigP ) )
=> ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ BigP @ Small ) ) )
=> ( ( ( compow495008222514391794ates_a @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ type_s4923920245906622843ates_a @ ( states_a2 @ Dir @ BigP @ Small ) )
= ( states_a2 @ Dir2 @ Big2 @ Small2 ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( type_s6404775287138157491tate_a @ Small ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir @ Big @ Small ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ ( type_s6530235180886170618tate_a @ Big ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( type_s6404775287138157491tate_a @ Small2 ) @ ( type_r4519047461186610747ates_a @ ( states_a2 @ Dir2 @ Big2 @ Small2 ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ ( type_s6530235180886170618tate_a @ Big2 ) ) ) ) ) ) ) ) ) ).
% step_4_pop_big_size_ok_3
thf(fact_912__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062x_Aleft_H_O_AIdle_Opop_Aleft_A_061_A_Ix_M_Aleft_H_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [X3: a,Left: idle_a] :
( ( pop_a2 @ left3 )
!= ( produc1265230069547855005idle_a @ X3 @ Left ) ) ).
% \<open>\<And>thesis. (\<And>x left'. Idle.pop left = (x, left') \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_913_size__ok_H_Opelims_I3_J,axiom,
! [X: states_a,Xa: nat] :
( ~ ( states_size_ok_a @ X @ Xa )
=> ( ( accp_P8670043419270033210_a_nat @ states_size_ok_rel_a @ ( produc1877401315875745917_a_nat @ X @ Xa ) )
=> ~ ! [Uu: direction,Big3: state_a2,Small3: state_a] :
( ( X
= ( states_a2 @ Uu @ Big3 @ Small3 ) )
=> ( ( accp_P8670043419270033210_a_nat @ states_size_ok_rel_a @ ( produc1877401315875745917_a_nat @ ( states_a2 @ Uu @ Big3 @ Small3 ) @ Xa ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( type_s6404775287138157491tate_a @ Small3 ) @ Xa ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ ( type_s6530235180886170618tate_a @ Big3 ) ) )
& ( ord_less_eq_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( type_s6530235180886170618tate_a @ Big3 ) @ Xa ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ ( type_s6404775287138157491tate_a @ Small3 ) ) )
& ( ord_less_eq_nat @ ( plus_plus_nat @ Xa @ one_one_nat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( size_size_state_a2 @ Small3 ) ) )
& ( ord_less_eq_nat @ ( plus_plus_nat @ Xa @ one_one_nat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( size_size_state_a @ Big3 ) ) ) ) ) ) ) ) ).
% size_ok'.pelims(3)
thf(fact_914_size__ok_H_Opelims_I1_J,axiom,
! [X: states_a,Xa: nat,Y: $o] :
( ( ( states_size_ok_a @ X @ Xa )
= Y )
=> ( ( accp_P8670043419270033210_a_nat @ states_size_ok_rel_a @ ( produc1877401315875745917_a_nat @ X @ Xa ) )
=> ~ ! [Uu: direction,Big3: state_a2,Small3: state_a] :
( ( X
= ( states_a2 @ Uu @ Big3 @ Small3 ) )
=> ( ( Y
= ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( type_s6404775287138157491tate_a @ Small3 ) @ Xa ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ ( type_s6530235180886170618tate_a @ Big3 ) ) )
& ( ord_less_eq_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( type_s6530235180886170618tate_a @ Big3 ) @ Xa ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ ( type_s6404775287138157491tate_a @ Small3 ) ) )
& ( ord_less_eq_nat @ ( plus_plus_nat @ Xa @ one_one_nat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( size_size_state_a2 @ Small3 ) ) )
& ( ord_less_eq_nat @ ( plus_plus_nat @ Xa @ one_one_nat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( size_size_state_a @ Big3 ) ) ) ) )
=> ~ ( accp_P8670043419270033210_a_nat @ states_size_ok_rel_a @ ( produc1877401315875745917_a_nat @ ( states_a2 @ Uu @ Big3 @ Small3 ) @ Xa ) ) ) ) ) ) ).
% size_ok'.pelims(1)
thf(fact_915_size__ok_H_Opelims_I2_J,axiom,
! [X: states_a,Xa: nat] :
( ( states_size_ok_a @ X @ Xa )
=> ( ( accp_P8670043419270033210_a_nat @ states_size_ok_rel_a @ ( produc1877401315875745917_a_nat @ X @ Xa ) )
=> ~ ! [Uu: direction,Big3: state_a2,Small3: state_a] :
( ( X
= ( states_a2 @ Uu @ Big3 @ Small3 ) )
=> ( ( accp_P8670043419270033210_a_nat @ states_size_ok_rel_a @ ( produc1877401315875745917_a_nat @ ( states_a2 @ Uu @ Big3 @ Small3 ) @ Xa ) )
=> ~ ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( type_s6404775287138157491tate_a @ Small3 ) @ Xa ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ ( type_s6530235180886170618tate_a @ Big3 ) ) )
& ( ord_less_eq_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( type_s6530235180886170618tate_a @ Big3 ) @ Xa ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ ( type_s6404775287138157491tate_a @ Small3 ) ) )
& ( ord_less_eq_nat @ ( plus_plus_nat @ Xa @ one_one_nat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( size_size_state_a2 @ Small3 ) ) )
& ( ord_less_eq_nat @ ( plus_plus_nat @ Xa @ one_one_nat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) @ ( size_size_state_a @ Big3 ) ) ) ) ) ) ) ) ).
% size_ok'.pelims(2)
thf(fact_916_pop__left,axiom,
( ( pop_a2 @ left3 )
= ( produc1265230069547855005idle_a @ x @ left2 ) ) ).
% pop_left
thf(fact_917_Idle__Proof_Osize__pop__sub,axiom,
! [Idle: idle_a,X: a,Idle2: idle_a] :
( ( ( pop_a2 @ Idle )
= ( produc1265230069547855005idle_a @ X @ Idle2 ) )
=> ( ( size_size_idle_a @ Idle2 )
= ( minus_minus_nat @ ( size_size_idle_a @ Idle ) @ one_one_nat ) ) ) ).
% Idle_Proof.size_pop_sub
thf(fact_918_divmod__algorithm__code_I3_J,axiom,
! [N: num] :
( ( unique5405566460079783412od_nat @ one @ ( bit1 @ N ) )
= ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).
% divmod_algorithm_code(3)
thf(fact_919_divmod__algorithm__code_I2_J,axiom,
! [N: num] :
( ( unique5405566460079783412od_nat @ one @ ( bit0 @ N ) )
= ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ one ) ) ) ).
% divmod_algorithm_code(2)
thf(fact_920_divmod__algorithm__code_I1_J,axiom,
! [M: num] :
( ( unique5405566460079783412od_nat @ M @ one )
= ( product_Pair_nat_nat @ ( numeral_numeral_nat @ M ) @ zero_zero_nat ) ) ).
% divmod_algorithm_code(1)
thf(fact_921_divmod__algorithm__code_I6_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_eq_num @ M @ N )
=> ( ( unique5405566460079783412od_nat @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) ) ) )
& ( ~ ( ord_less_eq_num @ M @ N )
=> ( ( unique5405566460079783412od_nat @ ( bit0 @ M ) @ ( bit1 @ N ) )
= ( unique4036640087844771520ep_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( unique5405566460079783412od_nat @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% divmod_algorithm_code(6)
thf(fact_922_divmod__algorithm__code_I7_J,axiom,
! [M: num,N: num] :
( ( ( ord_less_num @ M @ N )
=> ( ( unique5405566460079783412od_nat @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) ) ) )
& ( ~ ( ord_less_num @ M @ N )
=> ( ( unique5405566460079783412od_nat @ ( bit1 @ M ) @ ( bit1 @ N ) )
= ( unique4036640087844771520ep_nat @ ( numeral_numeral_nat @ ( bit1 @ N ) ) @ ( unique5405566460079783412od_nat @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ).
% divmod_algorithm_code(7)
thf(fact_923_divmod__divmod__step,axiom,
( unique5405566460079783412od_nat
= ( ^ [M5: num,N4: num] : ( if_Pro6206227464963214023at_nat @ ( ord_less_num @ M5 @ N4 ) @ ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ M5 ) ) @ ( unique4036640087844771520ep_nat @ ( numeral_numeral_nat @ N4 ) @ ( unique5405566460079783412od_nat @ M5 @ ( bit0 @ N4 ) ) ) ) ) ) ).
% divmod_divmod_step
thf(fact_924_divmod__step__def,axiom,
! [L: nat,R2: nat,Q2: nat] :
( ( ( ord_less_eq_nat @ ( euclid4777050414544973029ze_nat @ L ) @ ( euclid4777050414544973029ze_nat @ R2 ) )
=> ( ( unique4036640087844771520ep_nat @ L @ ( product_Pair_nat_nat @ Q2 @ R2 ) )
= ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q2 ) @ one_one_nat ) @ ( minus_minus_nat @ R2 @ L ) ) ) )
& ( ~ ( ord_less_eq_nat @ ( euclid4777050414544973029ze_nat @ L ) @ ( euclid4777050414544973029ze_nat @ R2 ) )
=> ( ( unique4036640087844771520ep_nat @ L @ ( product_Pair_nat_nat @ Q2 @ R2 ) )
= ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q2 ) @ R2 ) ) ) ) ).
% divmod_step_def
thf(fact_925_xor__numerals_I6_J,axiom,
! [X: num,Y: num] :
( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
= ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).
% xor_numerals(6)
thf(fact_926_xor__self__eq,axiom,
! [A: nat] :
( ( bit_se6528837805403552850or_nat @ A @ A )
= zero_zero_nat ) ).
% xor_self_eq
thf(fact_927_xor_Oleft__neutral,axiom,
! [A: nat] :
( ( bit_se6528837805403552850or_nat @ zero_zero_nat @ A )
= A ) ).
% xor.left_neutral
thf(fact_928_xor_Oright__neutral,axiom,
! [A: nat] :
( ( bit_se6528837805403552850or_nat @ A @ zero_zero_nat )
= A ) ).
% xor.right_neutral
thf(fact_929_euclidean__size__numeral,axiom,
! [K: num] :
( ( euclid4777050414544973029ze_nat @ ( numeral_numeral_nat @ K ) )
= ( numeral_numeral_nat @ K ) ) ).
% euclidean_size_numeral
thf(fact_930_euclidean__size__1,axiom,
( ( euclid4777050414544973029ze_nat @ one_one_nat )
= one_one_nat ) ).
% euclidean_size_1
thf(fact_931_xor__nat__numerals_I1_J,axiom,
! [Y: num] :
( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
= ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).
% xor_nat_numerals(1)
thf(fact_932_xor__nat__numerals_I2_J,axiom,
! [Y: num] :
( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
= ( numeral_numeral_nat @ ( bit0 @ Y ) ) ) ).
% xor_nat_numerals(2)
thf(fact_933_xor__nat__numerals_I3_J,axiom,
! [X: num] :
( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
= ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% xor_nat_numerals(3)
thf(fact_934_xor__nat__numerals_I4_J,axiom,
! [X: num] :
( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
= ( numeral_numeral_nat @ ( bit0 @ X ) ) ) ).
% xor_nat_numerals(4)
thf(fact_935_xor__numerals_I3_J,axiom,
! [X: num,Y: num] :
( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).
% xor_numerals(3)
thf(fact_936_xor__numerals_I1_J,axiom,
! [Y: num] :
( ( bit_se6528837805403552850or_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
= ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).
% xor_numerals(1)
thf(fact_937_xor__numerals_I2_J,axiom,
! [Y: num] :
( ( bit_se6528837805403552850or_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
= ( numeral_numeral_nat @ ( bit0 @ Y ) ) ) ).
% xor_numerals(2)
thf(fact_938_xor__numerals_I5_J,axiom,
! [X: num] :
( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ one_one_nat )
= ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% xor_numerals(5)
thf(fact_939_xor__numerals_I8_J,axiom,
! [X: num] :
( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ one_one_nat )
= ( numeral_numeral_nat @ ( bit0 @ X ) ) ) ).
% xor_numerals(8)
thf(fact_940_xor__numerals_I7_J,axiom,
! [X: num,Y: num] :
( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).
% xor_numerals(7)
thf(fact_941_xor__numerals_I4_J,axiom,
! [X: num,Y: num] :
( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
= ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).
% xor_numerals(4)
thf(fact_942_xor_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( bit_se6528837805403552850or_nat @ B @ ( bit_se6528837805403552850or_nat @ A @ C ) )
= ( bit_se6528837805403552850or_nat @ A @ ( bit_se6528837805403552850or_nat @ B @ C ) ) ) ).
% xor.left_commute
thf(fact_943_xor_Ocommute,axiom,
( bit_se6528837805403552850or_nat
= ( ^ [A2: nat,B2: nat] : ( bit_se6528837805403552850or_nat @ B2 @ A2 ) ) ) ).
% xor.commute
thf(fact_944_xor_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( bit_se6528837805403552850or_nat @ ( bit_se6528837805403552850or_nat @ A @ B ) @ C )
= ( bit_se6528837805403552850or_nat @ A @ ( bit_se6528837805403552850or_nat @ B @ C ) ) ) ).
% xor.assoc
thf(fact_945_euclidean__size__nat__less__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( euclid4777050414544973029ze_nat @ M ) @ ( euclid4777050414544973029ze_nat @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% euclidean_size_nat_less_eq_iff
thf(fact_946_euclidean__size__greater__0__iff,axiom,
! [B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( euclid4777050414544973029ze_nat @ B ) )
= ( B != zero_zero_nat ) ) ).
% euclidean_size_greater_0_iff
thf(fact_947_size__0,axiom,
( ( euclid4777050414544973029ze_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% size_0
thf(fact_948_euclidean__size__eq__0__iff,axiom,
! [B: nat] :
( ( ( euclid4777050414544973029ze_nat @ B )
= zero_zero_nat )
= ( B = zero_zero_nat ) ) ).
% euclidean_size_eq_0_iff
thf(fact_949_euclidean__size__mult,axiom,
! [A: nat,B: nat] :
( ( euclid4777050414544973029ze_nat @ ( times_times_nat @ A @ B ) )
= ( times_times_nat @ ( euclid4777050414544973029ze_nat @ A ) @ ( euclid4777050414544973029ze_nat @ B ) ) ) ).
% euclidean_size_mult
thf(fact_950_size__mult__mono,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ord_less_eq_nat @ ( euclid4777050414544973029ze_nat @ A ) @ ( euclid4777050414544973029ze_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% size_mult_mono
thf(fact_951_size__mult__mono_H,axiom,
! [B: nat,A: nat] :
( ( B != zero_zero_nat )
=> ( ord_less_eq_nat @ ( euclid4777050414544973029ze_nat @ A ) @ ( euclid4777050414544973029ze_nat @ ( times_times_nat @ B @ A ) ) ) ) ).
% size_mult_mono'
thf(fact_952_or__numerals_I7_J,axiom,
! [X: num,Y: num] :
( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
= ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).
% or_numerals(7)
thf(fact_953_or__numerals_I6_J,axiom,
! [X: num,Y: num] :
( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
= ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).
% or_numerals(6)
thf(fact_954_or_Oright__idem,axiom,
! [A: nat,B: nat] :
( ( bit_se1412395901928357646or_nat @ ( bit_se1412395901928357646or_nat @ A @ B ) @ B )
= ( bit_se1412395901928357646or_nat @ A @ B ) ) ).
% or.right_idem
thf(fact_955_or_Oleft__idem,axiom,
! [A: nat,B: nat] :
( ( bit_se1412395901928357646or_nat @ A @ ( bit_se1412395901928357646or_nat @ A @ B ) )
= ( bit_se1412395901928357646or_nat @ A @ B ) ) ).
% or.left_idem
thf(fact_956_or_Oidem,axiom,
! [A: nat] :
( ( bit_se1412395901928357646or_nat @ A @ A )
= A ) ).
% or.idem
thf(fact_957_or_Oleft__neutral,axiom,
! [A: nat] :
( ( bit_se1412395901928357646or_nat @ zero_zero_nat @ A )
= A ) ).
% or.left_neutral
thf(fact_958_or_Oright__neutral,axiom,
! [A: nat] :
( ( bit_se1412395901928357646or_nat @ A @ zero_zero_nat )
= A ) ).
% or.right_neutral
thf(fact_959_or__nat__numerals_I2_J,axiom,
! [Y: num] :
( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
= ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).
% or_nat_numerals(2)
thf(fact_960_or__nat__numerals_I4_J,axiom,
! [X: num] :
( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
= ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% or_nat_numerals(4)
thf(fact_961_or__numerals_I8_J,axiom,
! [X: num] :
( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ one_one_nat )
= ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% or_numerals(8)
thf(fact_962_or__numerals_I2_J,axiom,
! [Y: num] :
( ( bit_se1412395901928357646or_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
= ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).
% or_numerals(2)
thf(fact_963_or__nat__numerals_I1_J,axiom,
! [Y: num] :
( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
= ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).
% or_nat_numerals(1)
thf(fact_964_or__nat__numerals_I3_J,axiom,
! [X: num] :
( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
= ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% or_nat_numerals(3)
thf(fact_965_or__numerals_I3_J,axiom,
! [X: num,Y: num] :
( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).
% or_numerals(3)
thf(fact_966_or__numerals_I1_J,axiom,
! [Y: num] :
( ( bit_se1412395901928357646or_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
= ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).
% or_numerals(1)
thf(fact_967_or__numerals_I5_J,axiom,
! [X: num] :
( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ one_one_nat )
= ( numeral_numeral_nat @ ( bit1 @ X ) ) ) ).
% or_numerals(5)
thf(fact_968_or__numerals_I4_J,axiom,
! [X: num,Y: num] :
( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
= ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).
% or_numerals(4)
thf(fact_969_or__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( bit_se1412395901928357646or_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
& ( B = zero_zero_nat ) ) ) ).
% or_eq_0_iff
thf(fact_970_or_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( bit_se1412395901928357646or_nat @ B @ ( bit_se1412395901928357646or_nat @ A @ C ) )
= ( bit_se1412395901928357646or_nat @ A @ ( bit_se1412395901928357646or_nat @ B @ C ) ) ) ).
% or.left_commute
thf(fact_971_or_Ocommute,axiom,
( bit_se1412395901928357646or_nat
= ( ^ [A2: nat,B2: nat] : ( bit_se1412395901928357646or_nat @ B2 @ A2 ) ) ) ).
% or.commute
thf(fact_972_or_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( bit_se1412395901928357646or_nat @ ( bit_se1412395901928357646or_nat @ A @ B ) @ C )
= ( bit_se1412395901928357646or_nat @ A @ ( bit_se1412395901928357646or_nat @ B @ C ) ) ) ).
% or.assoc
thf(fact_973_mask__Suc__double,axiom,
! [N: nat] :
( ( bit_se2002935070580805687sk_nat @ ( suc @ N ) )
= ( bit_se1412395901928357646or_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2002935070580805687sk_nat @ N ) ) ) ) ).
% mask_Suc_double
thf(fact_974_and__numerals_I7_J,axiom,
! [X: num,Y: num] :
( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
= ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).
% and_numerals(7)
thf(fact_975_and_Oidem,axiom,
! [A: nat] :
( ( bit_se727722235901077358nd_nat @ A @ A )
= A ) ).
% and.idem
thf(fact_976_and_Oleft__idem,axiom,
! [A: nat,B: nat] :
( ( bit_se727722235901077358nd_nat @ A @ ( bit_se727722235901077358nd_nat @ A @ B ) )
= ( bit_se727722235901077358nd_nat @ A @ B ) ) ).
% and.left_idem
thf(fact_977_and_Oright__idem,axiom,
! [A: nat,B: nat] :
( ( bit_se727722235901077358nd_nat @ ( bit_se727722235901077358nd_nat @ A @ B ) @ B )
= ( bit_se727722235901077358nd_nat @ A @ B ) ) ).
% and.right_idem
thf(fact_978_mask__nat__positive__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( bit_se2002935070580805687sk_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% mask_nat_positive_iff
thf(fact_979_zero__and__eq,axiom,
! [A: nat] :
( ( bit_se727722235901077358nd_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_and_eq
thf(fact_980_and__zero__eq,axiom,
! [A: nat] :
( ( bit_se727722235901077358nd_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% and_zero_eq
thf(fact_981_mask__eq__0__iff,axiom,
! [N: nat] :
( ( ( bit_se2002935070580805687sk_nat @ N )
= zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% mask_eq_0_iff
thf(fact_982_mask__0,axiom,
( ( bit_se2002935070580805687sk_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% mask_0
thf(fact_983_and__nat__numerals_I3_J,axiom,
! [X: num] :
( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( suc @ zero_zero_nat ) )
= zero_zero_nat ) ).
% and_nat_numerals(3)
thf(fact_984_and__nat__numerals_I1_J,axiom,
! [Y: num] :
( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
= zero_zero_nat ) ).
% and_nat_numerals(1)
thf(fact_985_and__numerals_I8_J,axiom,
! [X: num] :
( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ one_one_nat )
= one_one_nat ) ).
% and_numerals(8)
thf(fact_986_and__numerals_I2_J,axiom,
! [Y: num] :
( ( bit_se727722235901077358nd_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
= one_one_nat ) ).
% and_numerals(2)
thf(fact_987_mask__Suc__0,axiom,
( ( bit_se2002935070580805687sk_nat @ ( suc @ zero_zero_nat ) )
= one_one_nat ) ).
% mask_Suc_0
thf(fact_988_and__numerals_I5_J,axiom,
! [X: num] :
( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ one_one_nat )
= zero_zero_nat ) ).
% and_numerals(5)
thf(fact_989_and__numerals_I1_J,axiom,
! [Y: num] :
( ( bit_se727722235901077358nd_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
= zero_zero_nat ) ).
% and_numerals(1)
thf(fact_990_and__numerals_I3_J,axiom,
! [X: num,Y: num] :
( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).
% and_numerals(3)
thf(fact_991_and__nat__numerals_I4_J,axiom,
! [X: num] :
( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( suc @ zero_zero_nat ) )
= one_one_nat ) ).
% and_nat_numerals(4)
thf(fact_992_and__nat__numerals_I2_J,axiom,
! [Y: num] :
( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
= one_one_nat ) ).
% and_nat_numerals(2)
thf(fact_993_and__numerals_I6_J,axiom,
! [X: num,Y: num] :
( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
= ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).
% and_numerals(6)
% Helper facts (5)
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__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( if_Pro6206227464963214023at_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( if_Pro6206227464963214023at_nat @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
states_size_ok_a @ ( compow495008222514391794ates_a @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ one ) ) ) @ type_s4923920245906622843ates_a @ ( states_a2 @ left @ ( reverse_a @ ( current_a2 @ nil_a @ zero_zero_nat @ right @ ( minus_minus_nat @ ( size_size_stack_a @ right ) @ ( suc @ length_left ) ) ) @ right @ nil_a @ ( minus_minus_nat @ ( size_size_stack_a @ right ) @ ( suc @ length_left ) ) ) @ ( reverse1_a @ ( current_a2 @ nil_a @ zero_zero_nat @ stack_left @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ length_left ) ) ) @ stack_left @ nil_a ) ) ) @ ( type_r4519047461186610747ates_a @ ( compow495008222514391794ates_a @ ( numeral_numeral_nat @ ( bit0 @ ( bit1 @ one ) ) ) @ type_s4923920245906622843ates_a @ ( states_a2 @ left @ ( reverse_a @ ( current_a2 @ nil_a @ zero_zero_nat @ right @ ( minus_minus_nat @ ( size_size_stack_a @ right ) @ ( suc @ length_left ) ) ) @ right @ nil_a @ ( minus_minus_nat @ ( size_size_stack_a @ right ) @ ( suc @ length_left ) ) ) @ ( reverse1_a @ ( current_a2 @ nil_a @ zero_zero_nat @ stack_left @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ length_left ) ) ) @ stack_left @ nil_a ) ) ) ) ).
%------------------------------------------------------------------------------