TPTP Problem File: SLH0947^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Real_Time_Deque/0025_States_Proof/prob_00491_016108__6968764_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1147 ( 396 unt; 258 typ; 0 def)
% Number of atoms : 2350 (1176 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 8047 ( 372 ~; 65 |; 161 &;6450 @)
% ( 0 <=>; 999 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 6 avg)
% Number of types : 45 ( 44 usr)
% Number of type conns : 573 ( 573 >; 0 *; 0 +; 0 <<)
% Number of symbols : 217 ( 214 usr; 23 con; 0-3 aty)
% Number of variables : 2643 ( 154 ^;2378 !; 111 ?;2643 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 15:52:34.833
%------------------------------------------------------------------------------
% Could-be-implicit typings (44)
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% Explicit typings (214)
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hd_state_a: list_state_a2 > state_a2 ).
thf(sy_c_List_Olist_Ohd_001t__List__Olist_Itf__a_J,type,
hd_list_a: list_list_a > list_a ).
thf(sy_c_List_Olist_Ohd_001t__Nat__Onat,type,
hd_nat: list_nat > nat ).
thf(sy_c_List_Olist_Ohd_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
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thf(sy_c_List_Olist_Ohd_001t__Product____Type__Oprod_It__States__Ostates_Itf__a_J_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Olist_Ohd_001t__Product____Type__Oprod_Itf__a_Mt__Big__Ostate_Itf__a_J_J,type,
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thf(sy_c_List_Olist_Ohd_001t__Product____Type__Oprod_Itf__a_Mt__Small__Ostate_Itf__a_J_J,type,
hd_Pro2196137027883835646tate_a: list_P4482599289824689689tate_a > produc7589950997499123219tate_a ).
thf(sy_c_List_Olist_Ohd_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
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thf(sy_c_List_Olist_Ohd_001t__Small__Ostate_Itf__a_J,type,
hd_state_a2: list_state_a > state_a ).
thf(sy_c_List_Olist_Ohd_001t__States__Ostates_Itf__a_J,type,
hd_states_a: list_states_a > states_a ).
thf(sy_c_List_Olist_Ohd_001tf__a,type,
hd_a: list_a > a ).
thf(sy_c_List_Olist_Osize__list_001tf__a,type,
size_list_a: ( a > nat ) > list_a > nat ).
thf(sy_c_List_Olist_Otl_001tf__a,type,
tl_a: list_a > list_a ).
thf(sy_c_List_Olist__update_001t__Big__Ostate_Itf__a_J,type,
list_update_state_a: list_state_a2 > nat > state_a2 > list_state_a2 ).
thf(sy_c_List_Olist__update_001t__List__Olist_Itf__a_J,type,
list_update_list_a: list_list_a > nat > list_a > list_list_a ).
thf(sy_c_List_Olist__update_001t__Nat__Onat,type,
list_update_nat: list_nat > nat > nat > list_nat ).
thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
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thf(sy_c_List_Olist__update_001t__Product____Type__Oprod_Itf__a_Mt__Small__Ostate_Itf__a_J_J,type,
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thf(sy_c_List_Olist__update_001t__Small__Ostate_Itf__a_J,type,
list_update_state_a2: list_state_a > nat > state_a > list_state_a ).
thf(sy_c_List_Olist__update_001t__States__Ostates_Itf__a_J,type,
list_update_states_a: list_states_a > nat > states_a > list_states_a ).
thf(sy_c_List_Olist__update_001tf__a,type,
list_update_a: list_a > nat > a > list_a ).
thf(sy_c_List_Olistrel1_001t__List__Olist_Itf__a_J,type,
listrel1_list_a: set_Pr4048851178543822343list_a > set_Pr5382606609415531783list_a ).
thf(sy_c_List_Olistrel1_001tf__a,type,
listrel1_a: set_Product_prod_a_a > set_Pr4048851178543822343list_a ).
thf(sy_c_List_Olistrel_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
listre6772471554020304241list_a: set_Pr4048851178543822343list_a > set_Pr5382606609415531783list_a ).
thf(sy_c_List_Olistrel_001t__States__Ostates_Itf__a_J_001t__Nat__Onat,type,
listrel_states_a_nat: set_Pr1464008215722202041_a_nat > set_Pr1437833252353886031st_nat ).
thf(sy_c_List_Olistrel_001tf__a_001t__Big__Ostate_Itf__a_J,type,
listrel_a_state_a: set_Pr4275752383657305402tate_a > set_Pr8989213357517205050tate_a ).
thf(sy_c_List_Olistrel_001tf__a_001t__Small__Ostate_Itf__a_J,type,
listrel_a_state_a2: set_Pr6306228930610421491tate_a > set_Pr6052505092368253171tate_a ).
thf(sy_c_List_Olistrel_001tf__a_001tf__a,type,
listrel_a_a: set_Product_prod_a_a > set_Pr4048851178543822343list_a ).
thf(sy_c_List_Olistrelp_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
listre657891920412899263list_a: ( list_a > list_a > $o ) > list_list_a > list_list_a > $o ).
thf(sy_c_List_Olistrelp_001t__States__Ostates_Itf__a_J_001t__Nat__Onat,type,
listre1698544965105253845_a_nat: ( states_a > nat > $o ) > list_states_a > list_nat > $o ).
thf(sy_c_List_Olistrelp_001tf__a_001t__Big__Ostate_Itf__a_J,type,
listrelp_a_state_a: ( a > state_a2 > $o ) > list_a > list_state_a2 > $o ).
thf(sy_c_List_Olistrelp_001tf__a_001t__Small__Ostate_Itf__a_J,type,
listrelp_a_state_a2: ( a > state_a > $o ) > list_a > list_state_a > $o ).
thf(sy_c_List_Olistrelp_001tf__a_001tf__a,type,
listrelp_a_a: ( a > a > $o ) > list_a > list_a > $o ).
thf(sy_c_List_Omeasures_001t__List__Olist_Itf__a_J,type,
measures_list_a: list_list_a_nat > set_Pr4048851178543822343list_a ).
thf(sy_c_List_On__lists_001tf__a,type,
n_lists_a: nat > list_a > list_list_a ).
thf(sy_c_List_Onth_001t__Big__Ostate_Itf__a_J,type,
nth_state_a: list_state_a2 > nat > state_a2 ).
thf(sy_c_List_Onth_001t__List__Olist_Itf__a_J,type,
nth_list_a: list_list_a > nat > list_a ).
thf(sy_c_List_Onth_001t__Nat__Onat,type,
nth_nat: list_nat > nat > nat ).
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nth_Pr5917933638979213230list_a: list_P321204300973800749list_a > nat > produc9164743771328383783list_a ).
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thf(sy_c_List_Onth_001t__Small__Ostate_Itf__a_J,type,
nth_state_a2: list_state_a > nat > state_a ).
thf(sy_c_List_Onth_001t__States__Ostates_Itf__a_J,type,
nth_states_a: list_states_a > nat > states_a ).
thf(sy_c_List_Onth_001tf__a,type,
nth_a: list_a > nat > a ).
thf(sy_c_List_Onths_001tf__a,type,
nths_a: list_a > set_nat > list_a ).
thf(sy_c_List_Oproduct__lists_001tf__a,type,
product_lists_a: list_list_a > list_list_a ).
thf(sy_c_List_Orev_001tf__a,type,
rev_a: list_a > list_a ).
thf(sy_c_List_Orotate1_001tf__a,type,
rotate1_a: list_a > list_a ).
thf(sy_c_List_Osubseqs_001tf__a,type,
subseqs_a: list_a > list_list_a ).
thf(sy_c_List_Otake_001t__List__Olist_Itf__a_J,type,
take_list_a: nat > list_list_a > list_list_a ).
thf(sy_c_List_Otake_001tf__a,type,
take_a: nat > list_a > list_a ).
thf(sy_c_List_Ozip_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
zip_list_a_list_a: list_list_a > list_list_a > list_P321204300973800749list_a ).
thf(sy_c_List_Ozip_001t__States__Ostates_Itf__a_J_001t__Nat__Onat,type,
zip_states_a_nat: list_states_a > list_nat > list_P4144771487928076819_a_nat ).
thf(sy_c_List_Ozip_001tf__a_001t__Big__Ostate_Itf__a_J,type,
zip_a_state_a: list_a > list_state_a2 > list_P6747491281921308512tate_a ).
thf(sy_c_List_Ozip_001tf__a_001t__Small__Ostate_Itf__a_J,type,
zip_a_state_a2: list_a > list_state_a > list_P4482599289824689689tate_a ).
thf(sy_c_List_Ozip_001tf__a_001tf__a,type,
zip_a_a: list_a > list_a > list_P1396940483166286381od_a_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__List__Olist_It__Big__Ostate_Itf__a_J_J,type,
size_s7859192958365828515tate_a: list_state_a2 > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
size_s349497388124573686list_a: list_list_a > nat ).
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size_size_list_nat: list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Small__Ostate_Itf__a_J_J,type,
size_s8463391772401140188tate_a: list_state_a > nat ).
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size_s3891197933023997302ates_a: list_states_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__Small__Ostate_Itf__a_J,type,
size_size_state_a2: state_a > 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_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__List__Olist_Itf__a_J_M_062_It__List__Olist_Itf__a_J_M_Eo_J_J,type,
ord_le5542992221119063950st_a_o: ( list_a > list_a > $o ) > ( list_a > list_a > $o ) > $o ).
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ord_le4636867679518884800_nat_o: ( states_a > nat > $o ) > ( states_a > nat > $o ) > $o ).
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ord_le5391092903712282779te_a_o: ( a > state_a2 > $o ) > ( a > state_a2 > $o ) > $o ).
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ord_le3878161006114389794te_a_o: ( a > state_a > $o ) > ( a > state_a > $o ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
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thf(sy_c_Product__Type_OPair_001_062_Itf__a_M_062_Itf__a_M_Eo_J_J_001t__List__Olist_Itf__a_J,type,
produc8111569692950616493list_a: ( a > a > $o ) > list_a > produc5032551385658279741list_a ).
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thf(sy_c_Product__Type_OPair_001t__List__Olist_It__States__Ostates_Itf__a_J_J_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_Product__Type_OPair_001t__List__Olist_Itf__a_J_001t__List__Olist_It__Small__Ostate_Itf__a_J_J,type,
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thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001tf__a,type,
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small_list_current_a: state_a > list_a ).
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thf(sy_c_States__Aux_Olists_001tf__a,type,
states_lists_a: states_a > produc9164743771328383783list_a ).
thf(sy_c_States__Aux_Olists__current_001tf__a,type,
states7719277857994474499rent_a: states_a > produc9164743771328383783list_a ).
thf(sy_c_String_Ochar_Osize__char,type,
size_char: char > nat ).
thf(sy_c_Type__Classes_Oinvar__class_Oinvar_001t__Big__Ostate_Itf__a_J,type,
type_i6304938058965754292tate_a: state_a2 > $o ).
thf(sy_c_Type__Classes_Oinvar__class_Oinvar_001t__Small__Ostate_Itf__a_J,type,
type_i464410347872898157tate_a: state_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__Big__Ostate_Itf__a_J,type,
type_r2494999336194962664tate_a: state_a2 > nat ).
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__Big__Ostate_Itf__a_J,type,
type_s3593206172722485288tate_a: state_a2 > state_a2 ).
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_member_001t__List__Olist_Itf__a_J,type,
member_list_a: list_a > set_list_a > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
member1318342207407915856list_a: produc7709606177366032167list_a > set_Pr5382606609415531783list_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__States__Ostates_Itf__a_J_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
member6795120827800458928st_nat: produc4094504297408651929st_nat > set_Pr1437833252353886031st_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_It__Big__Ostate_Itf__a_J_J_J,type,
member6123284207288203267tate_a: produc17304319345593178tate_a > set_Pr8989213357517205050tate_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_It__Small__Ostate_Itf__a_J_J_J,type,
member4112945611203173692tate_a: produc7959480069840336147tate_a > set_Pr6052505092368253171tate_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
member8191768239178080336list_a: produc9164743771328383783list_a > set_Pr4048851178543822343list_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__States__Ostates_Itf__a_J_Mt__Nat__Onat_J,type,
member6483677129516672026_a_nat: produc1571854377283420419_a_nat > set_Pr1464008215722202041_a_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Big__Ostate_Itf__a_J_J,type,
member3175992478928454403tate_a: produc6972303929186420058tate_a > set_Pr4275752383657305402tate_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Small__Ostate_Itf__a_J_J,type,
member8547378267715833660tate_a: produc7589950997499123219tate_a > set_Pr6306228930610421491tate_a > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
member1426531477525435216od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_big,type,
big: state_a2 ).
thf(sy_v_big_H,type,
big2: state_a2 ).
thf(sy_v_dir,type,
dir: direction ).
thf(sy_v_dir_H,type,
dir2: direction ).
thf(sy_v_small,type,
small: state_a ).
thf(sy_v_small_H,type,
small2: state_a ).
% Relevant facts (881)
thf(fact_0_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_1_states_Oinject,axiom,
! [X1: direction,X2: state_a2,X3: state_a,Y1: direction,Y2: state_a2,Y3: state_a] :
( ( ( states_a2 @ X1 @ X2 @ X3 )
= ( states_a2 @ Y1 @ Y2 @ Y3 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 )
& ( X3 = Y3 ) ) ) ).
% states.inject
thf(fact_2_step__lists__current,axiom,
! [States: states_a] :
( ( type_i8221491762852169479ates_a @ States )
=> ( ( states7719277857994474499rent_a @ ( type_s4923920245906622843ates_a @ States ) )
= ( states7719277857994474499rent_a @ States ) ) ) ).
% step_lists_current
thf(fact_3_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_4_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_5_step__lists,axiom,
! [States: states_a] :
( ( type_i8221491762852169479ates_a @ States )
=> ( ( states_lists_a @ ( type_s4923920245906622843ates_a @ States ) )
= ( states_lists_a @ States ) ) ) ).
% step_lists
thf(fact_6_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_7_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_8_states_Oexhaust,axiom,
! [Y: states_a] :
~ ! [X12: direction,X22: state_a2,X32: state_a] :
( Y
!= ( states_a2 @ X12 @ X22 @ X32 ) ) ).
% states.exhaust
thf(fact_9_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_10_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_11_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_12_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_13_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_14_states_Ocase,axiom,
! [F: direction > state_a2 > state_a > nat,X1: direction,X2: state_a2,X3: state_a] :
( ( case_states_a_nat @ F @ ( states_a2 @ X1 @ X2 @ X3 ) )
= ( F @ X1 @ X2 @ X3 ) ) ).
% states.case
thf(fact_15_push__small__lists,axiom,
! [Dir: direction,Big: state_a2,Small: state_a,Big2: list_a,Small2: list_a,X: a] :
( ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( ( states_lists_a @ ( states_a2 @ Dir @ Big @ Small ) )
= ( produc6837034575241423639list_a @ Big2 @ Small2 ) )
=> ( ( states_lists_a @ ( states_a2 @ Dir @ Big @ ( push_a2 @ X @ Small ) ) )
= ( produc6837034575241423639list_a @ Big2 @ ( cons_a @ X @ Small2 ) ) ) ) ) ).
% push_small_lists
thf(fact_16_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_17_step__invars,axiom,
! [States: states_a,Dir: direction,Big: state_a2,Small: state_a] :
( ( type_i8221491762852169479ates_a @ States )
=> ( ( ( type_s4923920245906622843ates_a @ States )
= ( states_a2 @ Dir @ Big @ Small ) )
=> ( ( type_i6304938058965754292tate_a @ Big )
& ( type_i464410347872898157tate_a @ Small ) ) ) ) ).
% step_invars
thf(fact_18_push__big,axiom,
! [Dir: direction,Big: state_a2,Small: state_a,Big2: list_a,Small2: list_a,X: a] :
( ( ( states_lists_a @ ( states_a2 @ Dir @ Big @ Small ) )
= ( produc6837034575241423639list_a @ Big2 @ Small2 ) )
=> ( ( states_lists_a @ ( states_a2 @ Dir @ ( push_a @ X @ Big ) @ Small ) )
= ( produc6837034575241423639list_a @ ( cons_a @ X @ Big2 ) @ Small2 ) ) ) ).
% push_big
thf(fact_19_step__lists__small__first,axiom,
! [States: states_a] :
( ( type_i8221491762852169479ates_a @ States )
=> ( ( states1596304293096088672irst_a @ ( type_s4923920245906622843ates_a @ States ) )
= ( states7886008410469471791irst_a @ ( type_s4923920245906622843ates_a @ States ) ) ) ) ).
% step_lists_small_first
thf(fact_20_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_21_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_22_step__size__new__big__2,axiom,
! [States: states_a] :
( ( type_i8221491762852169479ates_a @ States )
=> ( ( case_states_a_nat
@ ^ [X4: direction,Big4: state_a2,Xa: state_a] : ( type_s6530235180886170618tate_a @ Big4 )
@ ( type_s4923920245906622843ates_a @ States ) )
= ( case_states_a_nat
@ ^ [X4: direction,Big4: state_a2,Xa: state_a] : ( type_s6530235180886170618tate_a @ Big4 )
@ States ) ) ) ).
% step_size_new_big_2
thf(fact_23_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_24_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_25_funpow__0,axiom,
! [F: states_a > states_a,X: states_a] :
( ( compow495008222514391794ates_a @ zero_zero_nat @ F @ X )
= X ) ).
% funpow_0
thf(fact_26_funpow__0,axiom,
! [F: nat > nat,X: nat] :
( ( compow_nat_nat @ zero_zero_nat @ F @ X )
= X ) ).
% funpow_0
thf(fact_27_step__size__new__small__2,axiom,
! [States: states_a] :
( ( type_i8221491762852169479ates_a @ States )
=> ( ( case_states_a_nat
@ ^ [X4: direction,Xa: state_a2] : type_s6404775287138157491tate_a
@ ( type_s4923920245906622843ates_a @ States ) )
= ( case_states_a_nat
@ ^ [X4: direction,Xa: state_a2] : type_s6404775287138157491tate_a
@ States ) ) ) ).
% step_size_new_small_2
thf(fact_28_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_29_states_Osize__neq,axiom,
! [X: states_a] :
( ( size_size_states_a @ X )
!= zero_zero_nat ) ).
% states.size_neq
thf(fact_30_states_Ocase__distrib,axiom,
! [H: nat > nat,F: direction > state_a2 > state_a > nat,States: states_a] :
( ( H @ ( case_states_a_nat @ F @ States ) )
= ( case_states_a_nat
@ ^ [X13: direction,X23: state_a2,X33: state_a] : ( H @ ( F @ X13 @ X23 @ X33 ) )
@ States ) ) ).
% states.case_distrib
thf(fact_31_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ B ) )
=> ? [X5: nat] :
( ( P @ X5 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X5 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_32_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_33_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_34_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_35_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_36_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_37_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_38_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_39_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_40_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_41_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_42_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_43_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_44_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_45_Small__Proof_Oinvar__push,axiom,
! [Small: state_a,X: a] :
( ( type_i464410347872898157tate_a @ Small )
=> ( type_i464410347872898157tate_a @ ( push_a2 @ X @ Small ) ) ) ).
% Small_Proof.invar_push
thf(fact_46_Big__Proof_Oinvar__push,axiom,
! [Big: state_a2,X: a] :
( ( type_i6304938058965754292tate_a @ Big )
=> ( type_i6304938058965754292tate_a @ ( push_a @ X @ Big ) ) ) ).
% Big_Proof.invar_push
thf(fact_47_list_Oinject,axiom,
! [X21: a,X222: list_a,Y21: a,Y22: list_a] :
( ( ( cons_a @ X21 @ X222 )
= ( cons_a @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X222 = Y22 ) ) ) ).
% list.inject
thf(fact_48_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_49_old_Oprod_Oinject,axiom,
! [A: list_a,B: list_a,A2: list_a,B2: list_a] :
( ( ( produc6837034575241423639list_a @ A @ B )
= ( produc6837034575241423639list_a @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_50_old_Oprod_Oinject,axiom,
! [A: states_a,B: nat,A2: states_a,B2: nat] :
( ( ( produc1877401315875745917_a_nat @ A @ B )
= ( produc1877401315875745917_a_nat @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_51_old_Oprod_Oinject,axiom,
! [A: a,B: state_a,A2: a,B2: state_a] :
( ( ( produc1224139502141355779tate_a @ A @ B )
= ( produc1224139502141355779tate_a @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_52_old_Oprod_Oinject,axiom,
! [A: a,B: state_a2,A2: a,B2: state_a2] :
( ( ( produc8641956578966763338tate_a @ A @ B )
= ( produc8641956578966763338tate_a @ A2 @ B2 ) )
= ( ( A = A2 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_53_prod_Oinject,axiom,
! [X1: list_a,X2: list_a,Y1: list_a,Y2: list_a] :
( ( ( produc6837034575241423639list_a @ X1 @ X2 )
= ( produc6837034575241423639list_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_54_prod_Oinject,axiom,
! [X1: states_a,X2: nat,Y1: states_a,Y2: nat] :
( ( ( produc1877401315875745917_a_nat @ X1 @ X2 )
= ( produc1877401315875745917_a_nat @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_55_prod_Oinject,axiom,
! [X1: a,X2: state_a,Y1: a,Y2: state_a] :
( ( ( produc1224139502141355779tate_a @ X1 @ X2 )
= ( produc1224139502141355779tate_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_56_prod_Oinject,axiom,
! [X1: a,X2: state_a2,Y1: a,Y2: state_a2] :
( ( ( produc8641956578966763338tate_a @ X1 @ X2 )
= ( produc8641956578966763338tate_a @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_57_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_58_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_59_Nat_Ofunpow__code__def,axiom,
funpow_states_a = compow495008222514391794ates_a ).
% Nat.funpow_code_def
thf(fact_60_Nat_Ofunpow__code__def,axiom,
funpow_nat = compow_nat_nat ).
% Nat.funpow_code_def
thf(fact_61_size__ok_H_Ocases,axiom,
! [X: produc1571854377283420419_a_nat] :
~ ! [Uu: direction,Big3: state_a2,Small3: state_a,Steps: nat] :
( X
!= ( produc1877401315875745917_a_nat @ ( states_a2 @ Uu @ Big3 @ Small3 ) @ Steps ) ) ).
% size_ok'.cases
thf(fact_62_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_63_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_64_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: nat,Z2: nat] : ( Y6 = Z2 ) )
= ( ^ [X4: nat,Y7: nat] :
( ( ord_less_eq_nat @ X4 @ Y7 )
& ( ord_less_eq_nat @ Y7 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_65_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_66_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_67_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_68_order_Otrans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% order.trans
thf(fact_69_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_70_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: nat,B3: nat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_71_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: nat,Z2: nat] : ( Y6 = Z2 ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_72_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_73_dual__order_Otrans,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_eq_nat @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_74_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_75_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: nat,Z2: nat] : ( Y6 = Z2 ) )
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_76_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_eq_nat @ X5 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_77_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_eq_nat @ X5 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_78_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_79_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_80_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_eq_nat @ X5 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_81_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_eq_nat @ X5 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_82_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_83_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_84_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_85_old_Oprod_Oexhaust,axiom,
! [Y: produc9164743771328383783list_a] :
~ ! [A3: list_a,B3: list_a] :
( Y
!= ( produc6837034575241423639list_a @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_86_old_Oprod_Oexhaust,axiom,
! [Y: produc1571854377283420419_a_nat] :
~ ! [A3: states_a,B3: nat] :
( Y
!= ( produc1877401315875745917_a_nat @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_87_old_Oprod_Oexhaust,axiom,
! [Y: produc7589950997499123219tate_a] :
~ ! [A3: a,B3: state_a] :
( Y
!= ( produc1224139502141355779tate_a @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_88_old_Oprod_Oexhaust,axiom,
! [Y: produc6972303929186420058tate_a] :
~ ! [A3: a,B3: state_a2] :
( Y
!= ( produc8641956578966763338tate_a @ A3 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_89_surj__pair,axiom,
! [P2: produc9164743771328383783list_a] :
? [X5: list_a,Y4: list_a] :
( P2
= ( produc6837034575241423639list_a @ X5 @ Y4 ) ) ).
% surj_pair
thf(fact_90_surj__pair,axiom,
! [P2: produc1571854377283420419_a_nat] :
? [X5: states_a,Y4: nat] :
( P2
= ( produc1877401315875745917_a_nat @ X5 @ Y4 ) ) ).
% surj_pair
thf(fact_91_surj__pair,axiom,
! [P2: produc7589950997499123219tate_a] :
? [X5: a,Y4: state_a] :
( P2
= ( produc1224139502141355779tate_a @ X5 @ Y4 ) ) ).
% surj_pair
thf(fact_92_surj__pair,axiom,
! [P2: produc6972303929186420058tate_a] :
? [X5: a,Y4: state_a2] :
( P2
= ( produc8641956578966763338tate_a @ X5 @ Y4 ) ) ).
% surj_pair
thf(fact_93_prod__cases,axiom,
! [P: produc9164743771328383783list_a > $o,P2: produc9164743771328383783list_a] :
( ! [A3: list_a,B3: list_a] : ( P @ ( produc6837034575241423639list_a @ A3 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_94_prod__cases,axiom,
! [P: produc1571854377283420419_a_nat > $o,P2: produc1571854377283420419_a_nat] :
( ! [A3: states_a,B3: nat] : ( P @ ( produc1877401315875745917_a_nat @ A3 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_95_prod__cases,axiom,
! [P: produc7589950997499123219tate_a > $o,P2: produc7589950997499123219tate_a] :
( ! [A3: a,B3: state_a] : ( P @ ( produc1224139502141355779tate_a @ A3 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_96_prod__cases,axiom,
! [P: produc6972303929186420058tate_a > $o,P2: produc6972303929186420058tate_a] :
( ! [A3: a,B3: state_a2] : ( P @ ( produc8641956578966763338tate_a @ A3 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_97_Pair__inject,axiom,
! [A: list_a,B: list_a,A2: list_a,B2: list_a] :
( ( ( produc6837034575241423639list_a @ A @ B )
= ( produc6837034575241423639list_a @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_98_Pair__inject,axiom,
! [A: states_a,B: nat,A2: states_a,B2: nat] :
( ( ( produc1877401315875745917_a_nat @ A @ B )
= ( produc1877401315875745917_a_nat @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_99_Pair__inject,axiom,
! [A: a,B: state_a,A2: a,B2: state_a] :
( ( ( produc1224139502141355779tate_a @ A @ B )
= ( produc1224139502141355779tate_a @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_100_Pair__inject,axiom,
! [A: a,B: state_a2,A2: a,B2: state_a2] :
( ( ( produc8641956578966763338tate_a @ A @ B )
= ( produc8641956578966763338tate_a @ A2 @ B2 ) )
=> ~ ( ( A = A2 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_101_not__Cons__self2,axiom,
! [X: a,Xs: list_a] :
( ( cons_a @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_102_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_103_impossible__Cons,axiom,
! [Xs: list_a,Ys: list_a,X: a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys ) )
=> ( Xs
!= ( cons_a @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_104_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_105_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_106_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_107_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_108_Cons__in__lex,axiom,
! [X: list_a,Xs: list_list_a,Y: list_a,Ys: list_list_a,R: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( cons_list_a @ X @ Xs ) @ ( cons_list_a @ Y @ Ys ) ) @ ( lex_list_a @ R ) )
= ( ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ R )
& ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) ) )
| ( ( X = Y )
& ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Ys ) @ ( lex_list_a @ R ) ) ) ) ) ).
% Cons_in_lex
thf(fact_109_Cons__in__lex,axiom,
! [X: a,Xs: list_a,Y: a,Ys: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X @ Xs ) @ ( cons_a @ Y @ Ys ) ) @ ( lex_a @ R ) )
= ( ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
& ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) ) )
| ( ( X = Y )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( lex_a @ R ) ) ) ) ) ).
% Cons_in_lex
thf(fact_110_Big__Proof_Opush__list__current,axiom,
! [X: a,Big: state_a2] :
( ( big_list_current_a @ ( push_a @ X @ Big ) )
= ( cons_a @ X @ ( big_list_current_a @ Big ) ) ) ).
% Big_Proof.push_list_current
thf(fact_111_Small__Proof_Opush__list__current,axiom,
! [X: a,Small: state_a] :
( ( small_list_current_a @ ( push_a2 @ X @ Small ) )
= ( cons_a @ X @ ( small_list_current_a @ Small ) ) ) ).
% Small_Proof.push_list_current
thf(fact_112_Small__Proof_Osize__size__new,axiom,
! [Small: state_a] :
( ( type_i464410347872898157tate_a @ Small )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a2 @ Small ) )
=> ( ord_less_nat @ zero_zero_nat @ ( type_s6404775287138157491tate_a @ Small ) ) ) ) ).
% Small_Proof.size_size_new
thf(fact_113_Small__Proof_Osize__new__push,axiom,
! [Small: state_a,X: a] :
( ( type_i464410347872898157tate_a @ Small )
=> ( ( type_s6404775287138157491tate_a @ ( push_a2 @ X @ Small ) )
= ( suc @ ( type_s6404775287138157491tate_a @ Small ) ) ) ) ).
% Small_Proof.size_new_push
thf(fact_114_Small__Proof_Osize__push,axiom,
! [Small: state_a,X: a] :
( ( type_i464410347872898157tate_a @ Small )
=> ( ( size_size_state_a2 @ ( push_a2 @ X @ Small ) )
= ( suc @ ( size_size_state_a2 @ Small ) ) ) ) ).
% Small_Proof.size_push
thf(fact_115_add__left__cancel,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C2 ) )
= ( B = C2 ) ) ).
% add_left_cancel
thf(fact_116_add__right__cancel,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C2 @ A ) )
= ( B = C2 ) ) ).
% add_right_cancel
thf(fact_117_nat_Oinject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
= ( X2 = Y2 ) ) ).
% nat.inject
thf(fact_118_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_119_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_120_add__le__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_121_add__le__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_122_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_123_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_124_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_125_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_126_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_127_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_128_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_129_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_130_add__less__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_131_add__less__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_132_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_133_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_134_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_135_add__diff__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_136_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_137_add__diff__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_138_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_139_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_140_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_141_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_142_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_143_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_144_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_145_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_146_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_147_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_148_Suc__funpow,axiom,
! [N: nat] :
( ( compow_nat_nat @ N @ suc )
= ( plus_plus_nat @ N ) ) ).
% Suc_funpow
thf(fact_149_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_150_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_151_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_152_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_153_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_154_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_155_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_156_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_157_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_158_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_159_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_160_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_161_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_162_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_163_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_164_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_165_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_166_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_167_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_168_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_169_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_170_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_171_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_172_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_173_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_174_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_175_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_176_Big__Proof_Osize__push,axiom,
! [Big: state_a2,X: a] :
( ( type_i6304938058965754292tate_a @ Big )
=> ( ( size_size_state_a @ ( push_a @ X @ Big ) )
= ( suc @ ( size_size_state_a @ Big ) ) ) ) ).
% Big_Proof.size_push
thf(fact_177_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_178_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_179_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_180_Big__Proof_Osize__new__push,axiom,
! [Big: state_a2,X: a] :
( ( type_i6304938058965754292tate_a @ Big )
=> ( ( type_s6530235180886170618tate_a @ ( push_a @ X @ Big ) )
= ( suc @ ( type_s6530235180886170618tate_a @ Big ) ) ) ) ).
% Big_Proof.size_new_push
thf(fact_181_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 ) )
| ? [D: nat] :
( ( A
= ( plus_plus_nat @ B @ D ) )
& ~ ( P @ D ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_182_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_183_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 ) )
& ! [D: nat] :
( ( A
= ( plus_plus_nat @ B @ D ) )
=> ( P @ D ) ) ) ) ).
% nat_diff_split
thf(fact_184_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_185_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C2 )
= ( B
= ( plus_plus_nat @ C2 @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_186_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_187_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C2 @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C2 @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_188_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_189_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C2 )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_190_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C2 @ B ) @ A )
= ( plus_plus_nat @ C2 @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_191_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C2 @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C2 @ B ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_192_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_193_le__add__diff,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C2 @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C2 ) @ A ) ) ) ).
% le_add_diff
thf(fact_194_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_195_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_196_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_197_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_198_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_199_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_200_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_201_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_202_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_203_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_204_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_205_add_Oassoc,axiom,
! [A: nat,B: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add.assoc
thf(fact_206_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_207_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_208_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X5: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X5 )
=> ( P @ Y5 ) )
=> ( P @ X5 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_209_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_210_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_211_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_212_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A4: nat,B4: nat] : ( plus_plus_nat @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_213_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_214_exists__least__iff,axiom,
( ( ^ [P3: nat > $o] :
? [X6: nat] : ( P3 @ X6 ) )
= ( ^ [P4: nat > $o] :
? [N2: nat] :
( ( P4 @ N2 )
& ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ~ ( P4 @ M2 ) ) ) ) ) ).
% exists_least_iff
thf(fact_215_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: nat] : ( P @ A3 @ A3 )
=> ( ! [A3: nat,B3: nat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_216_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans
thf(fact_217_add_Oleft__commute,axiom,
! [B: nat,A: nat,C2: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C2 ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add.left_commute
thf(fact_218_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_219_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_220_add__left__imp__eq,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C2 ) )
=> ( B = C2 ) ) ).
% add_left_imp_eq
thf(fact_221_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_222_add__right__imp__eq,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C2 @ A ) )
=> ( B = C2 ) ) ).
% add_right_imp_eq
thf(fact_223_add__implies__diff,axiom,
! [C2: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C2 @ B )
= A )
=> ( C2
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_224_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_225_diff__right__commute,axiom,
! [A: nat,C2: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C2 ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C2 ) ) ).
% diff_right_commute
thf(fact_226_add__strict__mono,axiom,
! [A: nat,B: nat,C2: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C2 @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_227_add__strict__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) ) ) ).
% add_strict_left_mono
thf(fact_228_add__strict__right__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add_strict_right_mono
thf(fact_229_add__less__imp__less__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_230_add__less__imp__less__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_231_diff__diff__eq,axiom,
! [A: nat,B: nat,C2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C2 )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% diff_diff_eq
thf(fact_232_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_233_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_234_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_235_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_236_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_237_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_nat @ X5 @ Y4 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_238_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_nat @ X5 @ Y4 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_239_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_240_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_nat @ X5 @ Y4 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_241_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_nat @ X5 @ Y4 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_242_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_243_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_244_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_245_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_246_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_247_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_248_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_249_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_250_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N4: nat] : ( ord_less_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_nat @ N @ N3 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_251_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M: nat] :
( ! [N4: nat] : ( ord_less_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_252_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_253_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_254_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_255_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_256_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_257_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_258_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_259_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_260_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_261_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ N )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( P @ I2 ) ) ) ) ).
% Ex_less_Suc
thf(fact_262_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_263_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_264_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_265_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_266_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_267_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ N )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P @ I2 ) ) ) ) ).
% All_less_Suc
thf(fact_268_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_269_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_270_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_271_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_272_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_273_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_274_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_275_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_276_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_277_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_278_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_279_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_280_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_281_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_282_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_283_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_284_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_285_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_286_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_287_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_288_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P @ I3 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I3 @ K2 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_289_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_290_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N4: nat] :
( ! [M4: nat] :
( ( ord_less_nat @ M4 @ N4 )
=> ( P @ M4 ) )
=> ( P @ N4 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_291_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_292_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_293_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_294_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N4: nat] :
( ~ ( P @ N4 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N4 )
& ~ ( P @ M4 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_295_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_296_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M2: nat,N2: nat] :
? [K3: nat] :
( N2
= ( suc @ ( plus_plus_nat @ M2 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_297_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_298_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_299_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_300_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I3: nat] :
( ( J
= ( suc @ I3 ) )
=> ( P @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( P @ ( suc @ I3 ) )
=> ( P @ I3 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_301_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N4: nat] :
( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_302_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_303_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_304_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_305_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_306_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_307_add__le__less__mono,axiom,
! [A: nat,B: nat,C2: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C2 @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_le_less_mono
thf(fact_308_add__less__le__mono,axiom,
! [A: nat,B: nat,C2: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_less_le_mono
thf(fact_309_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_310_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_311_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_312_pos__add__strict,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% pos_add_strict
thf(fact_313_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_314_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_315_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_316_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_317_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_318_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_319_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ).
% gr0_implies_Suc
thf(fact_320_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P @ ( suc @ I2 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_321_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_322_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( P @ ( suc @ I2 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_323_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_324_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_325_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_326_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_327_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_328_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_329_diff__less__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C2 ) @ ( minus_minus_nat @ B @ C2 ) ) ) ) ).
% diff_less_mono
thf(fact_330_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_331_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_332_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N2: nat] : ( ord_less_eq_nat @ ( suc @ N2 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_333_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_334_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_335_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_336_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ I @ N4 )
=> ( ( ord_less_nat @ N4 @ J )
=> ( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_337_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ I @ N4 )
=> ( ( ord_less_nat @ N4 @ J )
=> ( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_338_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_339_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_340_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M5: nat,N4: nat] :
( ( ord_less_nat @ M5 @ N4 )
=> ( ord_less_nat @ ( F @ M5 ) @ ( F @ N4 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_341_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_342_add__strict__increasing2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% add_strict_increasing2
thf(fact_343_add__strict__increasing,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% add_strict_increasing
thf(fact_344_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_345_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_346_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_347_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_348_list_Osize_I4_J,axiom,
! [X21: a,X222: list_a] :
( ( size_size_list_a @ ( cons_a @ X21 @ X222 ) )
= ( plus_plus_nat @ ( size_size_list_a @ X222 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_349_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_350_add__le__imp__le__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_351_add__le__imp__le__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_352_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
? [C: nat] :
( B4
= ( plus_plus_nat @ A4 @ C ) ) ) ) ).
% le_iff_add
thf(fact_353_add__right__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add_right_mono
thf(fact_354_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_355_add__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) ) ) ).
% add_left_mono
thf(fact_356_add__mono,axiom,
! [A: nat,B: nat,C2: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ D2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_357_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_358_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_359_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_360_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_361_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_362_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_363_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_364_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_nat @ X5 @ Y4 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_365_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_eq_nat @ X5 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_366_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_eq_nat @ X5 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_367_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X5: nat,Y4: nat] :
( ( ord_less_nat @ X5 @ Y4 )
=> ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_368_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_369_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_370_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_371_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_372_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_373_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_374_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_375_order__less__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y7: nat] :
( ( ord_less_eq_nat @ X4 @ Y7 )
& ( X4 != Y7 ) ) ) ) ).
% order_less_le
thf(fact_376_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y7: nat] :
( ( ord_less_nat @ X4 @ Y7 )
| ( X4 = Y7 ) ) ) ) ).
% order_le_less
thf(fact_377_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_378_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_379_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ~ ( ord_less_eq_nat @ A4 @ B4 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_380_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_381_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_382_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_less_eq_nat @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_383_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B4: nat,A4: nat] :
( ( ord_less_nat @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_384_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ~ ( ord_less_eq_nat @ B4 @ A4 ) ) ) ) ).
% order.strict_iff_not
thf(fact_385_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_386_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_387_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% order.strict_iff_order
thf(fact_388_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% order.order_iff_strict
thf(fact_389_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_390_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y7: nat] :
( ( ord_less_eq_nat @ X4 @ Y7 )
& ~ ( ord_less_eq_nat @ Y7 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_391_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_392_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_393_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_394_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_395_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_396_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_397_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_398_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_399_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_400_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_401_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_402_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ).
% not0_implies_Suc
thf(fact_403_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_404_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_405_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_406_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N4: nat] :
( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_407_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X5: nat] : ( P @ X5 @ zero_zero_nat )
=> ( ! [Y4: nat] : ( P @ zero_zero_nat @ ( suc @ Y4 ) )
=> ( ! [X5: nat,Y4: nat] :
( ( P @ X5 @ Y4 )
=> ( P @ ( suc @ X5 ) @ ( suc @ Y4 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_408_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N4: nat] :
( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_409_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_410_nat_OdiscI,axiom,
! [Nat: nat,X2: nat] :
( ( Nat
= ( suc @ X2 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_411_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_412_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_413_nat_Odistinct_I1_J,axiom,
! [X2: nat] :
( zero_zero_nat
!= ( suc @ X2 ) ) ).
% nat.distinct(1)
thf(fact_414_length__Suc__conv,axiom,
! [Xs: list_a,N: nat] :
( ( ( size_size_list_a @ Xs )
= ( suc @ N ) )
= ( ? [Y7: a,Ys2: list_a] :
( ( Xs
= ( cons_a @ Y7 @ Ys2 ) )
& ( ( size_size_list_a @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_415_Suc__length__conv,axiom,
! [N: nat,Xs: list_a] :
( ( ( suc @ N )
= ( size_size_list_a @ Xs ) )
= ( ? [Y7: a,Ys2: list_a] :
( ( Xs
= ( cons_a @ Y7 @ Ys2 ) )
& ( ( size_size_list_a @ Ys2 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_416_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_417_le__diff__iff_H,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C2 )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A ) @ ( minus_minus_nat @ C2 @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_418_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_419_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_420_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_421_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_422_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_423_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_424_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_425_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R2: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X5: nat] : ( R2 @ X5 @ X5 )
=> ( ! [X5: nat,Y4: nat,Z3: nat] :
( ( R2 @ X5 @ Y4 )
=> ( ( R2 @ Y4 @ Z3 )
=> ( R2 @ X5 @ Z3 ) ) )
=> ( ! [N4: nat] : ( R2 @ N4 @ ( suc @ N4 ) )
=> ( R2 @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_426_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ M @ N4 )
=> ( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_427_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N4: nat] :
( ! [M4: nat] :
( ( ord_less_eq_nat @ ( suc @ M4 ) @ N4 )
=> ( P @ M4 ) )
=> ( P @ N4 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_428_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_429_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_430_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_431_Suc__le__D,axiom,
! [N: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
=> ? [M5: nat] :
( M6
= ( suc @ M5 ) ) ) ).
% Suc_le_D
thf(fact_432_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_433_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_434_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_435_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
? [K3: nat] :
( N2
= ( plus_plus_nat @ M2 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_436_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_437_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_438_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_439_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_440_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N4: nat] :
( L
= ( plus_plus_nat @ K @ N4 ) ) ) ).
% le_Suc_ex
thf(fact_441_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_442_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_443_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_444_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_445_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_446_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ~ ( P @ N4 )
=> ? [M4: nat] :
( ( ord_less_nat @ M4 @ N4 )
& ~ ( P @ M4 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_447_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_448_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_449_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_450_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_451_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_452_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_453_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_454_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_455_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_456_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_457_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_458_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
& ( M2 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_459_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_460_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_461_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_462_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_463_add__increasing2,axiom,
! [C2: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% add_increasing2
thf(fact_464_add__decreasing2,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C2 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ B ) ) ) ).
% add_decreasing2
thf(fact_465_add__increasing,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).
% add_increasing
thf(fact_466_add__decreasing,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ B ) ) ) ).
% add_decreasing
thf(fact_467_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N4 ) ) @ ( F @ N4 ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_468_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N4: nat] : ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ ( suc @ N4 ) ) )
=> ( ( ord_less_eq_nat @ N @ N3 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_469_Suc__le__length__iff,axiom,
! [N: nat,Xs: list_a] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( size_size_list_a @ Xs ) )
= ( ? [X4: a,Ys2: list_a] :
( ( Xs
= ( cons_a @ X4 @ Ys2 ) )
& ( ord_less_eq_nat @ N @ ( size_size_list_a @ Ys2 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_470_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_471_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_472_lists__current_Osimps,axiom,
! [Uu2: direction,Big: state_a2,Small: state_a] :
( ( states7719277857994474499rent_a @ ( states_a2 @ Uu2 @ Big @ Small ) )
= ( produc6837034575241423639list_a @ ( big_list_current_a @ Big ) @ ( small_list_current_a @ Small ) ) ) ).
% lists_current.simps
thf(fact_473_lists__current_Oelims,axiom,
! [X: states_a,Y: produc9164743771328383783list_a] :
( ( ( states7719277857994474499rent_a @ X )
= Y )
=> ~ ! [Uu: direction,Big3: state_a2,Small3: state_a] :
( ( X
= ( states_a2 @ Uu @ Big3 @ Small3 ) )
=> ( Y
!= ( produc6837034575241423639list_a @ ( big_list_current_a @ Big3 ) @ ( small_list_current_a @ Small3 ) ) ) ) ) ).
% lists_current.elims
thf(fact_474_Big__Proof_Osize__size__new,axiom,
! [Big: state_a2] :
( ( type_i6304938058965754292tate_a @ Big )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ Big ) )
=> ( ord_less_nat @ zero_zero_nat @ ( type_s6530235180886170618tate_a @ Big ) ) ) ) ).
% Big_Proof.size_size_new
thf(fact_475_states_Osize_I2_J,axiom,
! [X1: direction,X2: state_a2,X3: state_a] :
( ( size_size_states_a @ ( states_a2 @ X1 @ X2 @ X3 ) )
= ( suc @ zero_zero_nat ) ) ).
% states.size(2)
thf(fact_476_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_477_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_478_list__current__small__first__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_a2 @ Small )
= ( produc1224139502141355779tate_a @ X @ Small2 ) )
=> ( ( cons_a @ X @ ( states7886008410469471791irst_a @ ( states_a2 @ Dir @ Big @ Small2 ) ) )
= ( states7886008410469471791irst_a @ ( states_a2 @ Dir @ Big @ Small ) ) ) ) ) ) ).
% list_current_small_first_pop_small
thf(fact_479_list__small__first__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_a2 @ Small )
= ( produc1224139502141355779tate_a @ X @ Small2 ) )
=> ( ( cons_a @ X @ ( states1596304293096088672irst_a @ ( states_a2 @ Dir @ Big @ Small2 ) ) )
= ( states1596304293096088672irst_a @ ( states_a2 @ Dir @ Big @ Small ) ) ) ) ) ) ).
% list_small_first_pop_small
thf(fact_480_Small__Proof_Opop__list__current,axiom,
! [Small: state_a,X: a,Small2: state_a] :
( ( type_i464410347872898157tate_a @ Small )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a2 @ Small ) )
=> ( ( ( pop_a2 @ Small )
= ( produc1224139502141355779tate_a @ X @ Small2 ) )
=> ( ( cons_a @ X @ ( small_list_current_a @ Small2 ) )
= ( small_list_current_a @ Small ) ) ) ) ) ).
% Small_Proof.pop_list_current
thf(fact_481_lists__small__first__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_a2 @ Small )
= ( produc1224139502141355779tate_a @ X @ Small2 ) )
=> ( ( states1596304293096088672irst_a @ ( states_a2 @ Dir @ Big @ Small2 ) )
= ( states7886008410469471791irst_a @ ( states_a2 @ Dir @ Big @ Small2 ) ) ) ) ) ) ).
% lists_small_first_pop_small
thf(fact_482_invars__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_a2 @ Small )
= ( produc1224139502141355779tate_a @ X @ Small2 ) )
=> ( ( type_i6304938058965754292tate_a @ Big )
& ( type_i464410347872898157tate_a @ Small2 ) ) ) ) ) ).
% invars_pop_small
thf(fact_483_Small__Proof_Osize__pop,axiom,
! [Small: state_a,X: a,Small2: state_a] :
( ( type_i464410347872898157tate_a @ Small )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a2 @ Small ) )
=> ( ( ( pop_a2 @ Small )
= ( produc1224139502141355779tate_a @ X @ Small2 ) )
=> ( ( suc @ ( size_size_state_a2 @ Small2 ) )
= ( size_size_state_a2 @ Small ) ) ) ) ) ).
% Small_Proof.size_pop
thf(fact_484_Small__Proof_Osize__new__pop,axiom,
! [Small: state_a,X: a,Small2: state_a] :
( ( type_i464410347872898157tate_a @ Small )
=> ( ( ord_less_nat @ zero_zero_nat @ ( type_s6404775287138157491tate_a @ Small ) )
=> ( ( ( pop_a2 @ Small )
= ( produc1224139502141355779tate_a @ X @ Small2 ) )
=> ( ( suc @ ( type_s6404775287138157491tate_a @ Small2 ) )
= ( type_s6404775287138157491tate_a @ Small ) ) ) ) ) ).
% Small_Proof.size_new_pop
thf(fact_485_Small__Proof_Oinvar__pop,axiom,
! [Small: state_a,X: a,Small2: state_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a2 @ Small ) )
=> ( ( type_i464410347872898157tate_a @ Small )
=> ( ( ( pop_a2 @ Small )
= ( produc1224139502141355779tate_a @ X @ Small2 ) )
=> ( type_i464410347872898157tate_a @ Small2 ) ) ) ) ).
% Small_Proof.invar_pop
thf(fact_486_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_a2 @ Small )
= ( produc1224139502141355779tate_a @ X @ Small2 ) )
=> ( type_i8221491762852169479ates_a @ ( states_a2 @ Dir @ Big @ Small2 ) ) ) ) ) ).
% invar_pop_small
thf(fact_487_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_488_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_489_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_490_Big__Proof_Opop__list__current,axiom,
! [Big: state_a2,X: a,Big2: state_a2] :
( ( type_i6304938058965754292tate_a @ Big )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ Big ) )
=> ( ( ( pop_a @ Big )
= ( produc8641956578966763338tate_a @ X @ Big2 ) )
=> ( ( cons_a @ X @ ( big_list_current_a @ Big2 ) )
= ( big_list_current_a @ Big ) ) ) ) ) ).
% Big_Proof.pop_list_current
thf(fact_491_lists__small__first__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 ) )
=> ( ( states1596304293096088672irst_a @ ( states_a2 @ Dir @ Big2 @ Small ) )
= ( states7886008410469471791irst_a @ ( states_a2 @ Dir @ Big2 @ Small ) ) ) ) ) ) ).
% lists_small_first_pop_big
thf(fact_492_invars__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_i6304938058965754292tate_a @ Big2 )
& ( type_i464410347872898157tate_a @ Small ) ) ) ) ) ).
% invars_pop_big
thf(fact_493_Big__Proof_Osize__pop,axiom,
! [Big: state_a2,X: a,Big2: state_a2] :
( ( type_i6304938058965754292tate_a @ Big )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ Big ) )
=> ( ( ( pop_a @ Big )
= ( produc8641956578966763338tate_a @ X @ Big2 ) )
=> ( ( suc @ ( size_size_state_a @ Big2 ) )
= ( size_size_state_a @ Big ) ) ) ) ) ).
% Big_Proof.size_pop
thf(fact_494_Big__Proof_Osize__new__pop,axiom,
! [Big: state_a2,X: a,Big2: state_a2] :
( ( type_i6304938058965754292tate_a @ Big )
=> ( ( ord_less_nat @ zero_zero_nat @ ( type_s6530235180886170618tate_a @ Big ) )
=> ( ( ( pop_a @ Big )
= ( produc8641956578966763338tate_a @ X @ Big2 ) )
=> ( ( suc @ ( type_s6530235180886170618tate_a @ Big2 ) )
= ( type_s6530235180886170618tate_a @ Big ) ) ) ) ) ).
% Big_Proof.size_new_pop
thf(fact_495_Big__Proof_Oinvar__pop,axiom,
! [Big: state_a2,X: a,Big2: state_a2] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ Big ) )
=> ( ( type_i6304938058965754292tate_a @ Big )
=> ( ( ( pop_a @ Big )
= ( produc8641956578966763338tate_a @ X @ Big2 ) )
=> ( type_i6304938058965754292tate_a @ Big2 ) ) ) ) ).
% Big_Proof.invar_pop
thf(fact_496_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_497_list__current__big__first__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 ) )
=> ( ( cons_a @ X @ ( states7295096810965389224irst_a @ ( states_a2 @ Dir @ Big2 @ Small ) ) )
= ( states7295096810965389224irst_a @ ( states_a2 @ Dir @ Big @ Small ) ) ) ) ) ) ).
% list_current_big_first_pop_big
thf(fact_498_list__big__first__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 ) )
=> ( ( cons_a @ X @ ( states1888450819780863577irst_a @ ( states_a2 @ Dir @ Big2 @ Small ) ) )
= ( states1888450819780863577irst_a @ ( states_a2 @ Dir @ Big @ Small ) ) ) ) ) ) ).
% list_big_first_pop_big
thf(fact_499_Big__Proof_Opop__list,axiom,
! [Big: state_a2,X: a,Big2: state_a2] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ Big ) )
=> ( ( type_i6304938058965754292tate_a @ Big )
=> ( ( ( pop_a @ Big )
= ( produc8641956578966763338tate_a @ X @ Big2 ) )
=> ( ( cons_a @ X @ ( big_list_a @ Big2 ) )
= ( big_list_a @ Big ) ) ) ) ) ).
% Big_Proof.pop_list
thf(fact_500_Big__Proof_Oremaining__steps__pop,axiom,
! [Big: state_a2,X: a,Big2: state_a2] :
( ( type_i6304938058965754292tate_a @ Big )
=> ( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ Big ) )
=> ( ( ( pop_a @ Big )
= ( produc8641956578966763338tate_a @ X @ Big2 ) )
=> ( ord_less_eq_nat @ ( type_r2494999336194962664tate_a @ Big2 ) @ ( type_r2494999336194962664tate_a @ Big ) ) ) ) ) ).
% Big_Proof.remaining_steps_pop
thf(fact_501_Big__Proof_Opush__list,axiom,
! [X: a,Big: state_a2] :
( ( big_list_a @ ( push_a @ X @ Big ) )
= ( cons_a @ X @ ( big_list_a @ Big ) ) ) ).
% Big_Proof.push_list
thf(fact_502_invar__list__big__first,axiom,
! [States: states_a] :
( ( type_i8221491762852169479ates_a @ States )
=> ( ( states1888450819780863577irst_a @ States )
= ( states7295096810965389224irst_a @ States ) ) ) ).
% invar_list_big_first
thf(fact_503_list__small__big,axiom,
! [Dir: direction,Big: state_a2,Small: state_a] :
( ( ( states1596304293096088672irst_a @ ( states_a2 @ Dir @ Big @ Small ) )
= ( states7886008410469471791irst_a @ ( states_a2 @ Dir @ Big @ Small ) ) )
= ( ( states1888450819780863577irst_a @ ( states_a2 @ Dir @ Big @ Small ) )
= ( states7295096810965389224irst_a @ ( states_a2 @ Dir @ Big @ Small ) ) ) ) ).
% list_small_big
thf(fact_504_Big__Proof_Oremaining__steps__push,axiom,
! [Big: state_a2,X: a] :
( ( type_i6304938058965754292tate_a @ Big )
=> ( ( type_r2494999336194962664tate_a @ ( push_a @ X @ Big ) )
= ( type_r2494999336194962664tate_a @ Big ) ) ) ).
% Big_Proof.remaining_steps_push
thf(fact_505_lists__big__first__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 ) )
=> ( ( states1888450819780863577irst_a @ ( states_a2 @ Dir @ Big2 @ Small ) )
= ( states7295096810965389224irst_a @ ( states_a2 @ Dir @ Big2 @ Small ) ) ) ) ) ) ).
% lists_big_first_pop_big
thf(fact_506_length__Cons,axiom,
! [X: a,Xs: list_a] :
( ( size_size_list_a @ ( cons_a @ X @ Xs ) )
= ( suc @ ( size_size_list_a @ Xs ) ) ) ).
% length_Cons
thf(fact_507_states_Osize__gen,axiom,
! [X: a > nat,X1: direction,X2: state_a2,X3: state_a] :
( ( size_states_a @ X @ ( states_a2 @ X1 @ X2 @ X3 ) )
= ( suc @ zero_zero_nat ) ) ).
% states.size_gen
thf(fact_508_Big__Proof_Oremaining__steps__step,axiom,
! [Big: state_a2] :
( ( type_i6304938058965754292tate_a @ Big )
=> ( ( ord_less_nat @ zero_zero_nat @ ( type_r2494999336194962664tate_a @ Big ) )
=> ( ( suc @ ( type_r2494999336194962664tate_a @ ( type_s3593206172722485288tate_a @ Big ) ) )
= ( type_r2494999336194962664tate_a @ Big ) ) ) ) ).
% Big_Proof.remaining_steps_step
thf(fact_509_Big__Proof_Ostep__list,axiom,
! [Big: state_a2] :
( ( type_i6304938058965754292tate_a @ Big )
=> ( ( big_list_a @ ( type_s3593206172722485288tate_a @ Big ) )
= ( big_list_a @ Big ) ) ) ).
% Big_Proof.step_list
thf(fact_510_Big__Proof_Ostep__list__current,axiom,
! [Big: state_a2] :
( ( type_i6304938058965754292tate_a @ Big )
=> ( ( big_list_current_a @ ( type_s3593206172722485288tate_a @ Big ) )
= ( big_list_current_a @ Big ) ) ) ).
% Big_Proof.step_list_current
thf(fact_511_Big__Proof_Oremaining__steps__step__0,axiom,
! [Big: state_a2] :
( ( type_i6304938058965754292tate_a @ Big )
=> ( ( ( type_r2494999336194962664tate_a @ Big )
= zero_zero_nat )
=> ( ( type_r2494999336194962664tate_a @ ( type_s3593206172722485288tate_a @ Big ) )
= zero_zero_nat ) ) ) ).
% Big_Proof.remaining_steps_step_0
thf(fact_512_Big__Proof_Oinvar__step,axiom,
! [Big: state_a2] :
( ( type_i6304938058965754292tate_a @ Big )
=> ( type_i6304938058965754292tate_a @ ( type_s3593206172722485288tate_a @ Big ) ) ) ).
% Big_Proof.invar_step
thf(fact_513_Big__Proof_Ostep__size,axiom,
! [Big: state_a2] :
( ( type_i6304938058965754292tate_a @ Big )
=> ( ( size_size_state_a @ Big )
= ( size_size_state_a @ ( type_s3593206172722485288tate_a @ Big ) ) ) ) ).
% Big_Proof.step_size
thf(fact_514_Big__Proof_Opop__list__tl,axiom,
! [Big: state_a2,X: a,Big2: state_a2] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ Big ) )
=> ( ( type_i6304938058965754292tate_a @ Big )
=> ( ( ( pop_a @ Big )
= ( produc8641956578966763338tate_a @ X @ Big2 ) )
=> ( ( big_list_a @ Big2 )
= ( tl_a @ ( big_list_a @ Big ) ) ) ) ) ) ).
% Big_Proof.pop_list_tl
thf(fact_515_list_Osel_I3_J,axiom,
! [X21: a,X222: list_a] :
( ( tl_a @ ( cons_a @ X21 @ X222 ) )
= X222 ) ).
% list.sel(3)
thf(fact_516_cons__tl,axiom,
! [X: a,Xs: list_a,Ys: list_a] :
( ( ( cons_a @ X @ Xs )
= Ys )
=> ( Xs
= ( tl_a @ Ys ) ) ) ).
% cons_tl
thf(fact_517_Cons__lenlex__iff,axiom,
! [M: list_a,Ms: list_list_a,N: list_a,Ns: list_list_a,R: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( cons_list_a @ M @ Ms ) @ ( cons_list_a @ N @ Ns ) ) @ ( lenlex_list_a @ R ) )
= ( ( ord_less_nat @ ( size_s349497388124573686list_a @ Ms ) @ ( size_s349497388124573686list_a @ Ns ) )
| ( ( ( size_s349497388124573686list_a @ Ms )
= ( size_s349497388124573686list_a @ Ns ) )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ M @ N ) @ R ) )
| ( ( M = N )
& ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Ms @ Ns ) @ ( lenlex_list_a @ R ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_518_Cons__lenlex__iff,axiom,
! [M: a,Ms: list_a,N: a,Ns: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ M @ Ms ) @ ( cons_a @ N @ Ns ) ) @ ( lenlex_a @ R ) )
= ( ( ord_less_nat @ ( size_size_list_a @ Ms ) @ ( size_size_list_a @ Ns ) )
| ( ( ( size_size_list_a @ Ms )
= ( size_size_list_a @ Ns ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ M @ N ) @ R ) )
| ( ( M = N )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ms @ Ns ) @ ( lenlex_a @ R ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_519_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I4: nat] :
( ( ord_less_nat @ K2 @ I4 )
=> ( P @ I4 ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_520_lenlex__irreflexive,axiom,
! [R: set_Product_prod_a_a,Xs: list_a] :
( ! [X5: a] :
~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ X5 ) @ R )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Xs ) @ ( lenlex_a @ R ) ) ) ).
% lenlex_irreflexive
thf(fact_521_lenlex__irreflexive,axiom,
! [R: set_Pr4048851178543822343list_a,Xs: list_list_a] :
( ! [X5: list_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X5 @ X5 ) @ R )
=> ~ ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Xs ) @ ( lenlex_list_a @ R ) ) ) ).
% lenlex_irreflexive
thf(fact_522_lenlex__length,axiom,
! [Ms: list_a,Ns: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ms @ Ns ) @ ( lenlex_a @ R ) )
=> ( ord_less_eq_nat @ ( size_size_list_a @ Ms ) @ ( size_size_list_a @ Ns ) ) ) ).
% lenlex_length
thf(fact_523_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( P @ A3 @ B3 )
= ( P @ B3 @ A3 ) )
=> ( ! [A3: nat] : ( P @ A3 @ zero_zero_nat )
=> ( ! [A3: nat,B3: nat] :
( ( P @ A3 @ B3 )
=> ( P @ A3 @ ( plus_plus_nat @ A3 @ B3 ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_524_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_525_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_526_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_527_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_528_verit__comp__simplify1_I3_J,axiom,
! [B2: nat,A2: nat] :
( ( ~ ( ord_less_eq_nat @ B2 @ A2 ) )
= ( ord_less_nat @ A2 @ B2 ) ) ).
% verit_comp_simplify1(3)
thf(fact_529_add__0__iff,axiom,
! [B: nat,A: nat] :
( ( B
= ( plus_plus_nat @ B @ A ) )
= ( A = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_530_minf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z3 )
=> ~ ( ord_less_eq_nat @ T @ X7 ) ) ).
% minf(8)
thf(fact_531_minf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z3 )
=> ( ord_less_eq_nat @ X7 @ T ) ) ).
% minf(6)
thf(fact_532_pinf_I1_J,axiom,
! [P: nat > $o,P5: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( ( P @ X5 )
= ( P5 @ X5 ) ) )
=> ( ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( ( Q2 @ X5 )
= ( Q3 @ X5 ) ) )
=> ? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z3 @ X7 )
=> ( ( ( P @ X7 )
& ( Q2 @ X7 ) )
= ( ( P5 @ X7 )
& ( Q3 @ X7 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_533_pinf_I2_J,axiom,
! [P: nat > $o,P5: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( ( P @ X5 )
= ( P5 @ X5 ) ) )
=> ( ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z4 @ X5 )
=> ( ( Q2 @ X5 )
= ( Q3 @ X5 ) ) )
=> ? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z3 @ X7 )
=> ( ( ( P @ X7 )
| ( Q2 @ X7 ) )
= ( ( P5 @ X7 )
| ( Q3 @ X7 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_534_pinf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z3 @ X7 )
=> ( X7 != T ) ) ).
% pinf(3)
thf(fact_535_pinf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z3 @ X7 )
=> ( X7 != T ) ) ).
% pinf(4)
thf(fact_536_pinf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z3 @ X7 )
=> ~ ( ord_less_nat @ X7 @ T ) ) ).
% pinf(5)
thf(fact_537_pinf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z3 @ X7 )
=> ( ord_less_nat @ T @ X7 ) ) ).
% pinf(7)
thf(fact_538_minf_I1_J,axiom,
! [P: nat > $o,P5: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ( P @ X5 )
= ( P5 @ X5 ) ) )
=> ( ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ( Q2 @ X5 )
= ( Q3 @ X5 ) ) )
=> ? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z3 )
=> ( ( ( P @ X7 )
& ( Q2 @ X7 ) )
= ( ( P5 @ X7 )
& ( Q3 @ X7 ) ) ) ) ) ) ).
% minf(1)
thf(fact_539_minf_I2_J,axiom,
! [P: nat > $o,P5: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ( P @ X5 )
= ( P5 @ X5 ) ) )
=> ( ? [Z4: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z4 )
=> ( ( Q2 @ X5 )
= ( Q3 @ X5 ) ) )
=> ? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z3 )
=> ( ( ( P @ X7 )
| ( Q2 @ X7 ) )
= ( ( P5 @ X7 )
| ( Q3 @ X7 ) ) ) ) ) ) ).
% minf(2)
thf(fact_540_minf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z3 )
=> ( X7 != T ) ) ).
% minf(3)
thf(fact_541_minf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z3 )
=> ( X7 != T ) ) ).
% minf(4)
thf(fact_542_minf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z3 )
=> ( ord_less_nat @ X7 @ T ) ) ).
% minf(5)
thf(fact_543_minf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ X7 @ Z3 )
=> ~ ( ord_less_nat @ T @ X7 ) ) ).
% minf(7)
thf(fact_544_pinf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z3 @ X7 )
=> ~ ( ord_less_eq_nat @ X7 @ T ) ) ).
% pinf(6)
thf(fact_545_pinf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X7: nat] :
( ( ord_less_nat @ Z3 @ X7 )
=> ( ord_less_eq_nat @ T @ X7 ) ) ).
% pinf(8)
thf(fact_546_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C3: nat] :
( ( ord_less_eq_nat @ A @ C3 )
& ( ord_less_eq_nat @ C3 @ B )
& ! [X7: nat] :
( ( ( ord_less_eq_nat @ A @ X7 )
& ( ord_less_nat @ X7 @ C3 ) )
=> ( P @ X7 ) )
& ! [D3: nat] :
( ! [X5: nat] :
( ( ( ord_less_eq_nat @ A @ X5 )
& ( ord_less_nat @ X5 @ D3 ) )
=> ( P @ X5 ) )
=> ( ord_less_eq_nat @ D3 @ C3 ) ) ) ) ) ) ).
% complete_interval
thf(fact_547_list_Osize__gen_I2_J,axiom,
! [X: a > nat,X21: a,X222: list_a] :
( ( size_list_a @ X @ ( cons_a @ X21 @ X222 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( X @ X21 ) @ ( size_list_a @ X @ X222 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size_gen(2)
thf(fact_548_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_549_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_550_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_551_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_552_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_553_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_554_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_555_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_556_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_557_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_558_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_559_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_560_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_561_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_562_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_563_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_564_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_565_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_566_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_567_Suc__eq__plus1,axiom,
( suc
= ( ^ [N2: nat] : ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_568_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_569_Suc__sub,axiom,
! [N: nat,M: nat] :
( ( ( suc @ N )
= M )
=> ( N
= ( minus_minus_nat @ M @ one_one_nat ) ) ) ).
% Suc_sub
thf(fact_570_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_571_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N4: nat] :
( ( ord_less_nat @ zero_zero_nat @ N4 )
=> ( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_572_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_573_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_574_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M2: nat,N2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ N2 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N2 ) ) ) ) ) ).
% add_eq_if
thf(fact_575_Small__Proof_Olist__current__size,axiom,
! [Small: state_a] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a2 @ Small ) )
=> ( ( ( small_list_current_a @ Small )
= nil_a )
=> ~ ( type_i464410347872898157tate_a @ Small ) ) ) ).
% Small_Proof.list_current_size
thf(fact_576_length__0__conv,axiom,
! [Xs: list_a] :
( ( ( size_size_list_a @ Xs )
= zero_zero_nat )
= ( Xs = nil_a ) ) ).
% length_0_conv
thf(fact_577_Nil__lenlex__iff1,axiom,
! [Ns: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ nil_a @ Ns ) @ ( lenlex_a @ R ) )
= ( Ns != nil_a ) ) ).
% Nil_lenlex_iff1
thf(fact_578_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_579_list_Osel_I2_J,axiom,
( ( tl_a @ nil_a )
= nil_a ) ).
% list.sel(2)
thf(fact_580_list_Odistinct_I1_J,axiom,
! [X21: a,X222: list_a] :
( nil_a
!= ( cons_a @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_581_list_OdiscI,axiom,
! [List: list_a,X21: a,X222: list_a] :
( ( List
= ( cons_a @ X21 @ X222 ) )
=> ( List != nil_a ) ) ).
% list.discI
thf(fact_582_list_Oexhaust,axiom,
! [Y: list_a] :
( ( Y != nil_a )
=> ~ ! [X212: a,X223: list_a] :
( Y
!= ( cons_a @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_583_remdups__adj_Ocases,axiom,
! [X: list_a] :
( ( X != nil_a )
=> ( ! [X5: a] :
( X
!= ( cons_a @ X5 @ nil_a ) )
=> ~ ! [X5: a,Y4: a,Xs2: list_a] :
( X
!= ( cons_a @ X5 @ ( cons_a @ Y4 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_584_neq__Nil__conv,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
= ( ? [Y7: a,Ys2: list_a] :
( Xs
= ( cons_a @ Y7 @ Ys2 ) ) ) ) ).
% neq_Nil_conv
thf(fact_585_list__induct2_H,axiom,
! [P: list_a > list_a > $o,Xs: list_a,Ys: list_a] :
( ( P @ nil_a @ nil_a )
=> ( ! [X5: a,Xs2: list_a] : ( P @ ( cons_a @ X5 @ Xs2 ) @ nil_a )
=> ( ! [Y4: a,Ys3: list_a] : ( P @ nil_a @ ( cons_a @ Y4 @ Ys3 ) )
=> ( ! [X5: a,Xs2: list_a,Y4: a,Ys3: list_a] :
( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_a @ X5 @ Xs2 ) @ ( cons_a @ Y4 @ Ys3 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_586_list__nonempty__induct,axiom,
! [Xs: list_a,P: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X5: a] : ( P @ ( cons_a @ X5 @ nil_a ) )
=> ( ! [X5: a,Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( P @ Xs2 )
=> ( P @ ( cons_a @ X5 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_587_transpose_Ocases,axiom,
! [X: list_list_a] :
( ( X != nil_list_a )
=> ( ! [Xss: list_list_a] :
( X
!= ( cons_list_a @ nil_a @ Xss ) )
=> ~ ! [X5: a,Xs2: list_a,Xss: list_list_a] :
( X
!= ( cons_list_a @ ( cons_a @ X5 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_588_sorted__wrt_Ocases,axiom,
! [X: produc5032551385658279741list_a] :
( ! [P6: a > a > $o] :
( X
!= ( produc8111569692950616493list_a @ P6 @ nil_a ) )
=> ~ ! [P6: a > a > $o,X5: a,Ys3: list_a] :
( X
!= ( produc8111569692950616493list_a @ P6 @ ( cons_a @ X5 @ Ys3 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_589_successively_Ocases,axiom,
! [X: produc5032551385658279741list_a] :
( ! [P6: a > a > $o] :
( X
!= ( produc8111569692950616493list_a @ P6 @ nil_a ) )
=> ( ! [P6: a > a > $o,X5: a] :
( X
!= ( produc8111569692950616493list_a @ P6 @ ( cons_a @ X5 @ nil_a ) ) )
=> ~ ! [P6: a > a > $o,X5: a,Y4: a,Xs2: list_a] :
( X
!= ( produc8111569692950616493list_a @ P6 @ ( cons_a @ X5 @ ( cons_a @ Y4 @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_590_list_Osize_I3_J,axiom,
( ( size_size_list_a @ nil_a )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_591_list__induct4,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,Ws: list_a,P: list_a > list_a > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( ( size_size_list_a @ Zs )
= ( size_size_list_a @ Ws ) )
=> ( ( P @ nil_a @ nil_a @ nil_a @ nil_a )
=> ( ! [X5: a,Xs2: list_a,Y4: a,Ys3: list_a,Z3: a,Zs2: list_a,W: a,Ws2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( ( size_size_list_a @ Ys3 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( ( size_size_list_a @ Zs2 )
= ( size_size_list_a @ Ws2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 @ Ws2 )
=> ( P @ ( cons_a @ X5 @ Xs2 ) @ ( cons_a @ Y4 @ Ys3 ) @ ( cons_a @ Z3 @ Zs2 ) @ ( cons_a @ W @ Ws2 ) ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_592_list__induct3,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,P: list_a > list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( size_size_list_a @ Ys )
= ( size_size_list_a @ Zs ) )
=> ( ( P @ nil_a @ nil_a @ nil_a )
=> ( ! [X5: a,Xs2: list_a,Y4: a,Ys3: list_a,Z3: a,Zs2: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( ( size_size_list_a @ Ys3 )
= ( size_size_list_a @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys3 @ Zs2 )
=> ( P @ ( cons_a @ X5 @ Xs2 ) @ ( cons_a @ Y4 @ Ys3 ) @ ( cons_a @ Z3 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_593_list__induct2,axiom,
! [Xs: list_a,Ys: list_a,P: list_a > list_a > $o] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( P @ nil_a @ nil_a )
=> ( ! [X5: a,Xs2: list_a,Y4: a,Ys3: list_a] :
( ( ( size_size_list_a @ Xs2 )
= ( size_size_list_a @ Ys3 ) )
=> ( ( P @ Xs2 @ Ys3 )
=> ( P @ ( cons_a @ X5 @ Xs2 ) @ ( cons_a @ Y4 @ Ys3 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_594_shuffles_Ocases,axiom,
! [X: produc9164743771328383783list_a] :
( ! [Ys3: list_a] :
( X
!= ( produc6837034575241423639list_a @ nil_a @ Ys3 ) )
=> ( ! [Xs2: list_a] :
( X
!= ( produc6837034575241423639list_a @ Xs2 @ nil_a ) )
=> ~ ! [X5: a,Xs2: list_a,Y4: a,Ys3: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ X5 @ Xs2 ) @ ( cons_a @ Y4 @ Ys3 ) ) ) ) ) ).
% shuffles.cases
thf(fact_595_splice_Ocases,axiom,
! [X: produc9164743771328383783list_a] :
( ! [Ys3: list_a] :
( X
!= ( produc6837034575241423639list_a @ nil_a @ Ys3 ) )
=> ~ ! [X5: a,Xs2: list_a,Ys3: list_a] :
( X
!= ( produc6837034575241423639list_a @ ( cons_a @ X5 @ Xs2 ) @ Ys3 ) ) ) ).
% splice.cases
thf(fact_596_map__tailrec__rev_Ocases,axiom,
! [X: produc1473018763691903991list_a] :
( ! [F2: a > a,Bs: list_a] :
( X
!= ( produc8643929849434629545list_a @ F2 @ ( produc6837034575241423639list_a @ nil_a @ Bs ) ) )
=> ~ ! [F2: a > a,A3: a,As: list_a,Bs: list_a] :
( X
!= ( produc8643929849434629545list_a @ F2 @ ( produc6837034575241423639list_a @ ( cons_a @ A3 @ As ) @ Bs ) ) ) ) ).
% map_tailrec_rev.cases
thf(fact_597_tl__Nil,axiom,
! [Xs: list_a] :
( ( ( tl_a @ Xs )
= nil_a )
= ( ( Xs = nil_a )
| ? [X4: a] :
( Xs
= ( cons_a @ X4 @ nil_a ) ) ) ) ).
% tl_Nil
thf(fact_598_Nil__tl,axiom,
! [Xs: list_a] :
( ( nil_a
= ( tl_a @ Xs ) )
= ( ( Xs = nil_a )
| ? [X4: a] :
( Xs
= ( cons_a @ X4 @ nil_a ) ) ) ) ).
% Nil_tl
thf(fact_599_list_Osize__gen_I1_J,axiom,
! [X: a > nat] :
( ( size_list_a @ X @ nil_a )
= zero_zero_nat ) ).
% list.size_gen(1)
thf(fact_600_Nil__notin__lex,axiom,
! [Ys: list_a,R: set_Product_prod_a_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ nil_a @ Ys ) @ ( lex_a @ R ) ) ).
% Nil_notin_lex
thf(fact_601_Nil2__notin__lex,axiom,
! [Xs: list_a,R: set_Product_prod_a_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ nil_a ) @ ( lex_a @ R ) ) ).
% Nil2_notin_lex
thf(fact_602_Nil__lenlex__iff2,axiom,
! [Ns: list_a,R: set_Product_prod_a_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ns @ nil_a ) @ ( lenlex_a @ R ) ) ).
% Nil_lenlex_iff2
thf(fact_603_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M7: nat] :
( ( P @ X )
=> ( ! [X5: nat] :
( ( P @ X5 )
=> ( ord_less_eq_nat @ X5 @ M7 ) )
=> ~ ! [M5: nat] :
( ( P @ M5 )
=> ~ ! [X7: nat] :
( ( P @ X7 )
=> ( ord_less_eq_nat @ X7 @ M5 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_604_size__list,axiom,
! [Big: state_a2] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ Big ) )
=> ( ( type_i6304938058965754292tate_a @ Big )
=> ( ( big_list_a @ Big )
!= nil_a ) ) ) ).
% size_list
thf(fact_605_Big__Proof_Olist__current__size,axiom,
! [Big: state_a2] :
( ( ord_less_nat @ zero_zero_nat @ ( size_size_state_a @ Big ) )
=> ( ( ( big_list_current_a @ Big )
= nil_a )
=> ~ ( type_i6304938058965754292tate_a @ Big ) ) ) ).
% Big_Proof.list_current_size
thf(fact_606_Nitpick_Osize__list__simp_I2_J,axiom,
( size_size_list_a
= ( ^ [Xs3: list_a] : ( if_nat @ ( Xs3 = nil_a ) @ zero_zero_nat @ ( suc @ ( size_size_list_a @ ( tl_a @ Xs3 ) ) ) ) ) ) ).
% Nitpick.size_list_simp(2)
thf(fact_607_n__lists__Nil,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( n_lists_a @ N @ nil_a )
= ( cons_list_a @ nil_a @ nil_list_a ) ) )
& ( ( N != zero_zero_nat )
=> ( ( n_lists_a @ N @ nil_a )
= nil_list_a ) ) ) ).
% n_lists_Nil
thf(fact_608_n__lists_Osimps_I1_J,axiom,
! [Xs: list_a] :
( ( n_lists_a @ zero_zero_nat @ Xs )
= ( cons_list_a @ nil_a @ nil_list_a ) ) ).
% n_lists.simps(1)
thf(fact_609_product__lists_Osimps_I1_J,axiom,
( ( product_lists_a @ nil_list_a )
= ( cons_list_a @ nil_a @ nil_list_a ) ) ).
% product_lists.simps(1)
thf(fact_610_Nitpick_Osize__list__simp_I1_J,axiom,
( size_list_a
= ( ^ [F3: a > nat,Xs3: list_a] : ( if_nat @ ( Xs3 = nil_a ) @ zero_zero_nat @ ( suc @ ( plus_plus_nat @ ( F3 @ ( hd_a @ Xs3 ) ) @ ( size_list_a @ F3 @ ( tl_a @ Xs3 ) ) ) ) ) ) ) ).
% Nitpick.size_list_simp(1)
thf(fact_611_in__measures_I2_J,axiom,
! [X: list_a,Y: list_a,F: list_a > nat,Fs: list_list_a_nat] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ ( measures_list_a @ ( cons_list_a_nat @ F @ Fs ) ) )
= ( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
| ( ( ( F @ X )
= ( F @ Y ) )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ ( measures_list_a @ Fs ) ) ) ) ) ).
% in_measures(2)
thf(fact_612_in__measures_I1_J,axiom,
! [X: list_a,Y: list_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ ( measures_list_a @ nil_list_a_nat ) ) ).
% in_measures(1)
thf(fact_613_list_Ocollapse,axiom,
! [List: list_a] :
( ( List != nil_a )
=> ( ( cons_a @ ( hd_a @ List ) @ ( tl_a @ List ) )
= List ) ) ).
% list.collapse
thf(fact_614_hd__Cons__tl,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( cons_a @ ( hd_a @ Xs ) @ ( tl_a @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_615_cons__hd,axiom,
! [X: a,Xs: list_a,Ys: list_a] :
( ( ( cons_a @ X @ Xs )
= Ys )
=> ( X
= ( hd_a @ Ys ) ) ) ).
% cons_hd
thf(fact_616_list_Osel_I1_J,axiom,
! [X21: a,X222: list_a] :
( ( hd_a @ ( cons_a @ X21 @ X222 ) )
= X21 ) ).
% list.sel(1)
thf(fact_617_list_Oexpand,axiom,
! [List: list_a,List2: list_a] :
( ( ( List = nil_a )
= ( List2 = nil_a ) )
=> ( ( ( List != nil_a )
=> ( ( List2 != nil_a )
=> ( ( ( hd_a @ List )
= ( hd_a @ List2 ) )
& ( ( tl_a @ List )
= ( tl_a @ List2 ) ) ) ) )
=> ( List = List2 ) ) ) ).
% list.expand
thf(fact_618_list_Oexhaust__sel,axiom,
! [List: list_a] :
( ( List != nil_a )
=> ( List
= ( cons_a @ ( hd_a @ List ) @ ( tl_a @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_619_measures__less,axiom,
! [F: list_a > nat,X: list_a,Y: list_a,Fs: list_list_a_nat] :
( ( ord_less_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ ( measures_list_a @ ( cons_list_a_nat @ F @ Fs ) ) ) ) ).
% measures_less
thf(fact_620_measures__lesseq,axiom,
! [F: list_a > nat,X: list_a,Y: list_a,Fs: list_list_a_nat] :
( ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) )
=> ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ ( measures_list_a @ Fs ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ ( measures_list_a @ ( cons_list_a_nat @ F @ Fs ) ) ) ) ) ).
% measures_lesseq
thf(fact_621_length__one__hd,axiom,
! [Xs: list_a] :
( ( ( size_size_list_a @ Xs )
= one_one_nat )
=> ( Xs
= ( cons_a @ ( hd_a @ Xs ) @ nil_a ) ) ) ).
% length_one_hd
thf(fact_622_take__hd,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( take_a @ ( suc @ zero_zero_nat ) @ Xs )
= ( cons_a @ ( hd_a @ Xs ) @ nil_a ) ) ) ).
% take_hd
thf(fact_623_take__Suc__Cons,axiom,
! [N: nat,X: a,Xs: list_a] :
( ( take_a @ ( suc @ N ) @ ( cons_a @ X @ Xs ) )
= ( cons_a @ X @ ( take_a @ N @ Xs ) ) ) ).
% take_Suc_Cons
thf(fact_624_take0,axiom,
( ( take_a @ zero_zero_nat )
= ( ^ [Xs3: list_a] : nil_a ) ) ).
% take0
thf(fact_625_take__eq__Nil,axiom,
! [N: nat,Xs: list_a] :
( ( ( take_a @ N @ Xs )
= nil_a )
= ( ( N = zero_zero_nat )
| ( Xs = nil_a ) ) ) ).
% take_eq_Nil
thf(fact_626_take__eq__Nil2,axiom,
! [N: nat,Xs: list_a] :
( ( nil_a
= ( take_a @ N @ Xs ) )
= ( ( N = zero_zero_nat )
| ( Xs = nil_a ) ) ) ).
% take_eq_Nil2
thf(fact_627_take__0,axiom,
! [Xs: list_a] :
( ( take_a @ zero_zero_nat @ Xs )
= nil_a ) ).
% take_0
thf(fact_628_take__Nil,axiom,
! [N: nat] :
( ( take_a @ N @ nil_a )
= nil_a ) ).
% take_Nil
thf(fact_629_list_Odisc__eq__case_I2_J,axiom,
! [List: list_a] :
( ( List != nil_a )
= ( case_list_o_a @ $false
@ ^ [Uu3: a,Uv: list_a] : $true
@ List ) ) ).
% list.disc_eq_case(2)
thf(fact_630_list_Odisc__eq__case_I1_J,axiom,
! [List: list_a] :
( ( List = nil_a )
= ( case_list_o_a @ $true
@ ^ [Uu3: a,Uv: list_a] : $false
@ List ) ) ).
% list.disc_eq_case(1)
thf(fact_631_tl__def,axiom,
( tl_a
= ( case_list_list_a_a @ nil_a
@ ^ [X213: a,X224: list_a] : X224 ) ) ).
% tl_def
thf(fact_632_take__Cons_H,axiom,
! [N: nat,X: a,Xs: list_a] :
( ( ( N = zero_zero_nat )
=> ( ( take_a @ N @ ( cons_a @ X @ Xs ) )
= nil_a ) )
& ( ( N != zero_zero_nat )
=> ( ( take_a @ N @ ( cons_a @ X @ Xs ) )
= ( cons_a @ X @ ( take_a @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs ) ) ) ) ) ).
% take_Cons'
thf(fact_633_take__hd_H,axiom,
! [Ys: list_a,X: a,Xs: list_a] :
( ( Ys != nil_a )
=> ( ( ( take_a @ ( size_size_list_a @ Ys ) @ ( cons_a @ X @ Xs ) )
= ( take_a @ ( suc @ ( size_size_list_a @ Xs ) ) @ Ys ) )
=> ( ( hd_a @ Ys )
= X ) ) ) ).
% take_hd'
thf(fact_634_take__Suc,axiom,
! [Xs: list_a,N: nat] :
( ( Xs != nil_a )
=> ( ( take_a @ ( suc @ N ) @ Xs )
= ( cons_a @ ( hd_a @ Xs ) @ ( take_a @ N @ ( tl_a @ Xs ) ) ) ) ) ).
% take_Suc
thf(fact_635_lex__take__index,axiom,
! [Xs: list_list_a,Ys: list_list_a,R: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Ys ) @ ( lex_list_a @ R ) )
=> ~ ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( ord_less_nat @ I3 @ ( size_s349497388124573686list_a @ Ys ) )
=> ( ( ( take_list_a @ I3 @ Xs )
= ( take_list_a @ I3 @ Ys ) )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( nth_list_a @ Xs @ I3 ) @ ( nth_list_a @ Ys @ I3 ) ) @ R ) ) ) ) ) ).
% lex_take_index
thf(fact_636_lex__take__index,axiom,
! [Xs: list_a,Ys: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( lex_a @ R ) )
=> ~ ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
=> ( ( ord_less_nat @ I3 @ ( size_size_list_a @ Ys ) )
=> ( ( ( take_a @ I3 @ Xs )
= ( take_a @ I3 @ Ys ) )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Xs @ I3 ) @ ( nth_a @ Ys @ I3 ) ) @ R ) ) ) ) ) ).
% lex_take_index
thf(fact_637_hd__drop,axiom,
! [N: nat,Xs: list_a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( cons_a @ ( hd_a @ ( drop_a @ N @ Xs ) ) @ ( drop_a @ ( suc @ N ) @ Xs ) )
= ( drop_a @ N @ Xs ) ) ) ).
% hd_drop
thf(fact_638_nth__Cons__0,axiom,
! [X: a,Xs: list_a] :
( ( nth_a @ ( cons_a @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_639_nth__Cons__Suc,axiom,
! [X: a,Xs: list_a,N: nat] :
( ( nth_a @ ( cons_a @ X @ Xs ) @ ( suc @ N ) )
= ( nth_a @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_640_drop__Suc__Cons,axiom,
! [N: nat,X: a,Xs: list_a] :
( ( drop_a @ ( suc @ N ) @ ( cons_a @ X @ Xs ) )
= ( drop_a @ N @ Xs ) ) ).
% drop_Suc_Cons
thf(fact_641_drop__eq__Nil2,axiom,
! [N: nat,Xs: list_a] :
( ( nil_a
= ( drop_a @ N @ Xs ) )
= ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N ) ) ).
% drop_eq_Nil2
thf(fact_642_drop__eq__Nil,axiom,
! [N: nat,Xs: list_a] :
( ( ( drop_a @ N @ Xs )
= nil_a )
= ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N ) ) ).
% drop_eq_Nil
thf(fact_643_drop__all,axiom,
! [Xs: list_a,N: nat] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N )
=> ( ( drop_a @ N @ Xs )
= nil_a ) ) ).
% drop_all
thf(fact_644_nth__Cons__pos,axiom,
! [N: nat,X: a,Xs: list_a] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( nth_a @ ( cons_a @ X @ Xs ) @ N )
= ( nth_a @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% nth_Cons_pos
thf(fact_645_hd__drop__1,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( cons_a @ ( hd_a @ Xs ) @ ( drop_a @ ( suc @ zero_zero_nat ) @ Xs ) )
= Xs ) ) ).
% hd_drop_1
thf(fact_646_drop__Nil,axiom,
! [N: nat] :
( ( drop_a @ N @ nil_a )
= nil_a ) ).
% drop_Nil
thf(fact_647_butlast_Osimps_I1_J,axiom,
( ( butlast_a @ nil_a )
= nil_a ) ).
% butlast.simps(1)
thf(fact_648_Cons__nth__drop__Suc,axiom,
! [I: nat,Xs: list_a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( cons_a @ ( nth_a @ Xs @ I ) @ ( drop_a @ ( suc @ I ) @ Xs ) )
= ( drop_a @ I @ Xs ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_649_nth__via__drop,axiom,
! [N: nat,Xs: list_a,Y: a,Ys: list_a] :
( ( ( drop_a @ N @ Xs )
= ( cons_a @ Y @ Ys ) )
=> ( ( nth_a @ Xs @ N )
= Y ) ) ).
% nth_via_drop
thf(fact_650_butlast_Osimps_I2_J,axiom,
! [Xs: list_a,X: a] :
( ( ( Xs = nil_a )
=> ( ( butlast_a @ ( cons_a @ X @ Xs ) )
= nil_a ) )
& ( ( Xs != nil_a )
=> ( ( butlast_a @ ( cons_a @ X @ Xs ) )
= ( cons_a @ X @ ( butlast_a @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_651_hd__conv__nth,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( hd_a @ Xs )
= ( nth_a @ Xs @ zero_zero_nat ) ) ) ).
% hd_conv_nth
thf(fact_652_drop__Cons_H,axiom,
! [N: nat,X: a,Xs: list_a] :
( ( ( N = zero_zero_nat )
=> ( ( drop_a @ N @ ( cons_a @ X @ Xs ) )
= ( cons_a @ X @ Xs ) ) )
& ( ( N != zero_zero_nat )
=> ( ( drop_a @ N @ ( cons_a @ X @ Xs ) )
= ( drop_a @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs ) ) ) ) ).
% drop_Cons'
thf(fact_653_nth__Cons_H,axiom,
! [N: nat,X: a,Xs: list_a] :
( ( ( N = zero_zero_nat )
=> ( ( nth_a @ ( cons_a @ X @ Xs ) @ N )
= X ) )
& ( ( N != zero_zero_nat )
=> ( ( nth_a @ ( cons_a @ X @ Xs ) @ N )
= ( nth_a @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ).
% nth_Cons'
thf(fact_654_nth__non__equal__first__eq,axiom,
! [X: a,Y: a,Xs: list_a,N: nat] :
( ( X != Y )
=> ( ( ( nth_a @ ( cons_a @ X @ Xs ) @ N )
= Y )
= ( ( ( nth_a @ Xs @ ( minus_minus_nat @ N @ one_one_nat ) )
= Y )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_655_take__hd__drop,axiom,
! [N: nat,Xs: list_a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( append_a @ ( take_a @ N @ Xs ) @ ( cons_a @ ( hd_a @ ( drop_a @ N @ Xs ) ) @ nil_a ) )
= ( take_a @ ( suc @ N ) @ Xs ) ) ) ).
% take_hd_drop
thf(fact_656_id__take__nth__drop,axiom,
! [I: nat,Xs: list_a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( Xs
= ( append_a @ ( take_a @ I @ Xs ) @ ( cons_a @ ( nth_a @ Xs @ I ) @ ( drop_a @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_657_append_Oright__neutral,axiom,
! [A: list_a] :
( ( append_a @ A @ nil_a )
= A ) ).
% append.right_neutral
thf(fact_658_append__Nil2,axiom,
! [Xs: list_a] :
( ( append_a @ Xs @ nil_a )
= Xs ) ).
% append_Nil2
thf(fact_659_append__self__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Xs )
= ( Ys = nil_a ) ) ).
% append_self_conv
thf(fact_660_self__append__conv,axiom,
! [Y: list_a,Ys: list_a] :
( ( Y
= ( append_a @ Y @ Ys ) )
= ( Ys = nil_a ) ) ).
% self_append_conv
thf(fact_661_append__self__conv2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= Ys )
= ( Xs = nil_a ) ) ).
% append_self_conv2
thf(fact_662_self__append__conv2,axiom,
! [Y: list_a,Xs: list_a] :
( ( Y
= ( append_a @ Xs @ Y ) )
= ( Xs = nil_a ) ) ).
% self_append_conv2
thf(fact_663_Nil__is__append__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( nil_a
= ( append_a @ Xs @ Ys ) )
= ( ( Xs = nil_a )
& ( Ys = nil_a ) ) ) ).
% Nil_is_append_conv
thf(fact_664_append__is__Nil__conv,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( append_a @ Xs @ Ys )
= nil_a )
= ( ( Xs = nil_a )
& ( Ys = nil_a ) ) ) ).
% append_is_Nil_conv
thf(fact_665_append1__eq__conv,axiom,
! [Xs: list_a,X: a,Ys: list_a,Y: a] :
( ( ( append_a @ Xs @ ( cons_a @ X @ nil_a ) )
= ( append_a @ Ys @ ( cons_a @ Y @ nil_a ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_666_tl__append2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs != nil_a )
=> ( ( tl_a @ ( append_a @ Xs @ Ys ) )
= ( append_a @ ( tl_a @ Xs ) @ Ys ) ) ) ).
% tl_append2
thf(fact_667_hd__append2,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs != nil_a )
=> ( ( hd_a @ ( append_a @ Xs @ Ys ) )
= ( hd_a @ Xs ) ) ) ).
% hd_append2
thf(fact_668_nth__append__length,axiom,
! [Xs: list_a,X: a,Ys: list_a] :
( ( nth_a @ ( append_a @ Xs @ ( cons_a @ X @ Ys ) ) @ ( size_size_list_a @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_669_butlast__snoc,axiom,
! [Xs: list_a,X: a] :
( ( butlast_a @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_670_butlast__append,axiom,
! [Ys: list_a,Xs: list_a] :
( ( ( Ys = nil_a )
=> ( ( butlast_a @ ( append_a @ Xs @ Ys ) )
= ( butlast_a @ Xs ) ) )
& ( ( Ys != nil_a )
=> ( ( butlast_a @ ( append_a @ Xs @ Ys ) )
= ( append_a @ Xs @ ( butlast_a @ Ys ) ) ) ) ) ).
% butlast_append
thf(fact_671_hd__append,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( Xs = nil_a )
=> ( ( hd_a @ ( append_a @ Xs @ Ys ) )
= ( hd_a @ Ys ) ) )
& ( ( Xs != nil_a )
=> ( ( hd_a @ ( append_a @ Xs @ Ys ) )
= ( hd_a @ Xs ) ) ) ) ).
% hd_append
thf(fact_672_longest__common__prefix,axiom,
! [Xs: list_a,Ys: list_a] :
? [Ps: list_a,Xs4: list_a,Ys4: list_a] :
( ( Xs
= ( append_a @ Ps @ Xs4 ) )
& ( Ys
= ( append_a @ Ps @ Ys4 ) )
& ( ( Xs4 = nil_a )
| ( Ys4 = nil_a )
| ( ( hd_a @ Xs4 )
!= ( hd_a @ Ys4 ) ) ) ) ).
% longest_common_prefix
thf(fact_673_tl__append__if,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( Xs = nil_a )
=> ( ( tl_a @ ( append_a @ Xs @ Ys ) )
= ( tl_a @ Ys ) ) )
& ( ( Xs != nil_a )
=> ( ( tl_a @ ( append_a @ Xs @ Ys ) )
= ( append_a @ ( tl_a @ Xs ) @ Ys ) ) ) ) ).
% tl_append_if
thf(fact_674_Cons__eq__appendI,axiom,
! [X: a,Xs1: list_a,Ys: list_a,Xs: list_a,Zs: list_a] :
( ( ( cons_a @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append_a @ Xs1 @ Zs ) )
=> ( ( cons_a @ X @ Xs )
= ( append_a @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_675_append__Cons,axiom,
! [X: a,Xs: list_a,Ys: list_a] :
( ( append_a @ ( cons_a @ X @ Xs ) @ Ys )
= ( cons_a @ X @ ( append_a @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_676_append__Nil,axiom,
! [Ys: list_a] :
( ( append_a @ nil_a @ Ys )
= Ys ) ).
% append_Nil
thf(fact_677_append_Oleft__neutral,axiom,
! [A: list_a] :
( ( append_a @ nil_a @ A )
= A ) ).
% append.left_neutral
thf(fact_678_eq__Nil__appendI,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs = Ys )
=> ( Xs
= ( append_a @ nil_a @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_679_rev__nonempty__induct,axiom,
! [Xs: list_a,P: list_a > $o] :
( ( Xs != nil_a )
=> ( ! [X5: a] : ( P @ ( cons_a @ X5 @ nil_a ) )
=> ( ! [X5: a,Xs2: list_a] :
( ( Xs2 != nil_a )
=> ( ( P @ Xs2 )
=> ( P @ ( append_a @ Xs2 @ ( cons_a @ X5 @ nil_a ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_680_append__eq__Cons__conv,axiom,
! [Ys: list_a,Zs: list_a,X: a,Xs: list_a] :
( ( ( append_a @ Ys @ Zs )
= ( cons_a @ X @ Xs ) )
= ( ( ( Ys = nil_a )
& ( Zs
= ( cons_a @ X @ Xs ) ) )
| ? [Ys5: list_a] :
( ( Ys
= ( cons_a @ X @ Ys5 ) )
& ( ( append_a @ Ys5 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_681_Cons__eq__append__conv,axiom,
! [X: a,Xs: list_a,Ys: list_a,Zs: list_a] :
( ( ( cons_a @ X @ Xs )
= ( append_a @ Ys @ Zs ) )
= ( ( ( Ys = nil_a )
& ( ( cons_a @ X @ Xs )
= Zs ) )
| ? [Ys5: list_a] :
( ( ( cons_a @ X @ Ys5 )
= Ys )
& ( Xs
= ( append_a @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_682_rev__exhaust,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ~ ! [Ys3: list_a,Y4: a] :
( Xs
!= ( append_a @ Ys3 @ ( cons_a @ Y4 @ nil_a ) ) ) ) ).
% rev_exhaust
thf(fact_683_rev__induct,axiom,
! [P: list_a > $o,Xs: list_a] :
( ( P @ nil_a )
=> ( ! [X5: a,Xs2: list_a] :
( ( P @ Xs2 )
=> ( P @ ( append_a @ Xs2 @ ( cons_a @ X5 @ nil_a ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_684_lex__append__leftI,axiom,
! [Ys: list_a,Zs: list_a,R: set_Product_prod_a_a,Xs: list_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ys @ Zs ) @ ( lex_a @ R ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Xs @ Ys ) @ ( append_a @ Xs @ Zs ) ) @ ( lex_a @ R ) ) ) ).
% lex_append_leftI
thf(fact_685_same__length__different,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs != Ys )
=> ( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ? [Pre: list_a,X5: a,Xs4: list_a,Y4: a,Ys4: list_a] :
( ( X5 != Y4 )
& ( Xs
= ( append_a @ Pre @ ( append_a @ ( cons_a @ X5 @ nil_a ) @ Xs4 ) ) )
& ( Ys
= ( append_a @ Pre @ ( append_a @ ( cons_a @ Y4 @ nil_a ) @ Ys4 ) ) ) ) ) ) ).
% same_length_different
thf(fact_686_lex__append__leftD,axiom,
! [R: set_Product_prod_a_a,Xs: list_a,Ys: list_a,Zs: list_a] :
( ! [X5: a] :
~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ X5 ) @ R )
=> ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Xs @ Ys ) @ ( append_a @ Xs @ Zs ) ) @ ( lex_a @ R ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ys @ Zs ) @ ( lex_a @ R ) ) ) ) ).
% lex_append_leftD
thf(fact_687_lex__append__leftD,axiom,
! [R: set_Pr4048851178543822343list_a,Xs: list_list_a,Ys: list_list_a,Zs: list_list_a] :
( ! [X5: list_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X5 @ X5 ) @ R )
=> ( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( append_list_a @ Xs @ Ys ) @ ( append_list_a @ Xs @ Zs ) ) @ ( lex_list_a @ R ) )
=> ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Ys @ Zs ) @ ( lex_list_a @ R ) ) ) ) ).
% lex_append_leftD
thf(fact_688_lex__append__left__iff,axiom,
! [R: set_Product_prod_a_a,Xs: list_a,Ys: list_a,Zs: list_a] :
( ! [X5: a] :
~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ X5 ) @ R )
=> ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Xs @ Ys ) @ ( append_a @ Xs @ Zs ) ) @ ( lex_a @ R ) )
= ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ys @ Zs ) @ ( lex_a @ R ) ) ) ) ).
% lex_append_left_iff
thf(fact_689_lex__append__left__iff,axiom,
! [R: set_Pr4048851178543822343list_a,Xs: list_list_a,Ys: list_list_a,Zs: list_list_a] :
( ! [X5: list_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X5 @ X5 ) @ R )
=> ( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( append_list_a @ Xs @ Ys ) @ ( append_list_a @ Xs @ Zs ) ) @ ( lex_list_a @ R ) )
= ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Ys @ Zs ) @ ( lex_list_a @ R ) ) ) ) ).
% lex_append_left_iff
thf(fact_690_lex__append__rightI,axiom,
! [Xs: list_a,Ys: list_a,R: set_Product_prod_a_a,Vs: list_a,Us: list_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( lex_a @ R ) )
=> ( ( ( size_size_list_a @ Vs )
= ( size_size_list_a @ Us ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Xs @ Us ) @ ( append_a @ Ys @ Vs ) ) @ ( lex_a @ R ) ) ) ) ).
% lex_append_rightI
thf(fact_691_lenlex__append1,axiom,
! [Us: list_a,Xs: list_a,R2: set_Product_prod_a_a,Vs: list_a,Ys: list_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Us @ Xs ) @ ( lenlex_a @ R2 ) )
=> ( ( ( size_size_list_a @ Vs )
= ( size_size_list_a @ Ys ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Us @ Vs ) @ ( append_a @ Xs @ Ys ) ) @ ( lenlex_a @ R2 ) ) ) ) ).
% lenlex_append1
thf(fact_692_length__append__singleton,axiom,
! [Xs: list_a,X: a] :
( ( size_size_list_a @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) )
= ( suc @ ( size_size_list_a @ Xs ) ) ) ).
% length_append_singleton
thf(fact_693_length__Suc__conv__rev,axiom,
! [Xs: list_a,N: nat] :
( ( ( size_size_list_a @ Xs )
= ( suc @ N ) )
= ( ? [Y7: a,Ys2: list_a] :
( ( Xs
= ( append_a @ Ys2 @ ( cons_a @ Y7 @ nil_a ) ) )
& ( ( size_size_list_a @ Ys2 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_694_take__Suc__conv__app__nth,axiom,
! [I: nat,Xs: list_a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( take_a @ ( suc @ I ) @ Xs )
= ( append_a @ ( take_a @ I @ Xs ) @ ( cons_a @ ( nth_a @ Xs @ I ) @ nil_a ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_695_Succ__def,axiom,
( bNF_Greatest_Succ_a
= ( ^ [Kl: set_list_a,Kl2: list_a] :
( collect_a
@ ^ [K3: a] : ( member_list_a @ ( append_a @ Kl2 @ ( cons_a @ K3 @ nil_a ) ) @ Kl ) ) ) ) ).
% Succ_def
thf(fact_696_take__rev__tl__hd,axiom,
! [N: nat,Xs: list_a,Ys: list_a] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( Xs != nil_a )
=> ( ( append_a @ ( common_take_rev_a @ N @ Xs ) @ Ys )
= ( append_a @ ( common_take_rev_a @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) @ ( tl_a @ Xs ) ) @ ( cons_a @ ( hd_a @ Xs ) @ Ys ) ) ) ) ) ).
% take_rev_tl_hd
thf(fact_697_take__rev__nth,axiom,
! [N: nat,Xs: list_a,X: a,Ys: list_a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( X
= ( nth_a @ Xs @ N ) )
=> ( ( cons_a @ X @ ( append_a @ ( common_take_rev_a @ N @ Xs ) @ Ys ) )
= ( append_a @ ( common_take_rev_a @ ( suc @ N ) @ Xs ) @ Ys ) ) ) ) ).
% take_rev_nth
thf(fact_698_take__rev__step,axiom,
! [Xs: list_a,N: nat,Acc: list_a] :
( ( Xs != nil_a )
=> ( ( append_a @ ( common_take_rev_a @ N @ ( tl_a @ Xs ) ) @ ( cons_a @ ( hd_a @ Xs ) @ Acc ) )
= ( append_a @ ( common_take_rev_a @ ( suc @ N ) @ Xs ) @ Acc ) ) ) ).
% take_rev_step
thf(fact_699_SuccI,axiom,
! [Kl3: list_a,K: a,Kl4: set_list_a] :
( ( member_list_a @ ( append_a @ Kl3 @ ( cons_a @ K @ nil_a ) ) @ Kl4 )
=> ( member_a @ K @ ( bNF_Greatest_Succ_a @ Kl4 @ Kl3 ) ) ) ).
% SuccI
thf(fact_700_SuccD,axiom,
! [K: a,Kl4: set_list_a,Kl3: list_a] :
( ( member_a @ K @ ( bNF_Greatest_Succ_a @ Kl4 @ Kl3 ) )
=> ( member_list_a @ ( append_a @ Kl3 @ ( cons_a @ K @ nil_a ) ) @ Kl4 ) ) ).
% SuccD
thf(fact_701_upd__conv__take__nth__drop,axiom,
! [I: nat,Xs: list_a,A: a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( list_update_a @ Xs @ I @ A )
= ( append_a @ ( take_a @ I @ Xs ) @ ( cons_a @ A @ ( drop_a @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% upd_conv_take_nth_drop
thf(fact_702_Succ__Shift,axiom,
! [Kl4: set_list_a,K: a,Kl3: list_a] :
( ( bNF_Greatest_Succ_a @ ( bNF_Greatest_Shift_a @ Kl4 @ K ) @ Kl3 )
= ( bNF_Greatest_Succ_a @ Kl4 @ ( cons_a @ K @ Kl3 ) ) ) ).
% Succ_Shift
thf(fact_703_list__update__nonempty,axiom,
! [Xs: list_a,K: nat,X: a] :
( ( ( list_update_a @ Xs @ K @ X )
= nil_a )
= ( Xs = nil_a ) ) ).
% list_update_nonempty
thf(fact_704_list__update__length,axiom,
! [Xs: list_a,X: a,Ys: list_a,Y: a] :
( ( list_update_a @ ( append_a @ Xs @ ( cons_a @ X @ Ys ) ) @ ( size_size_list_a @ Xs ) @ Y )
= ( append_a @ Xs @ ( cons_a @ Y @ Ys ) ) ) ).
% list_update_length
thf(fact_705_list__update__code_I3_J,axiom,
! [X: a,Xs: list_a,I: nat,Y: a] :
( ( list_update_a @ ( cons_a @ X @ Xs ) @ ( suc @ I ) @ Y )
= ( cons_a @ X @ ( list_update_a @ Xs @ I @ Y ) ) ) ).
% list_update_code(3)
thf(fact_706_list__update__code_I2_J,axiom,
! [X: a,Xs: list_a,Y: a] :
( ( list_update_a @ ( cons_a @ X @ Xs ) @ zero_zero_nat @ Y )
= ( cons_a @ Y @ Xs ) ) ).
% list_update_code(2)
thf(fact_707_list__update__code_I1_J,axiom,
! [I: nat,Y: a] :
( ( list_update_a @ nil_a @ I @ Y )
= nil_a ) ).
% list_update_code(1)
thf(fact_708_list__update_Osimps_I1_J,axiom,
! [I: nat,V: a] :
( ( list_update_a @ nil_a @ I @ V )
= nil_a ) ).
% list_update.simps(1)
thf(fact_709_ShiftD,axiom,
! [Kl3: list_a,Kl4: set_list_a,K: a] :
( ( member_list_a @ Kl3 @ ( bNF_Greatest_Shift_a @ Kl4 @ K ) )
=> ( member_list_a @ ( cons_a @ K @ Kl3 ) @ Kl4 ) ) ).
% ShiftD
thf(fact_710_Shift__def,axiom,
( bNF_Greatest_Shift_a
= ( ^ [Kl: set_list_a,K3: a] :
( collect_list_a
@ ^ [Kl2: list_a] : ( member_list_a @ ( cons_a @ K3 @ Kl2 ) @ Kl ) ) ) ) ).
% Shift_def
thf(fact_711_rotate1__hd__tl,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( rotate1_a @ Xs )
= ( append_a @ ( tl_a @ Xs ) @ ( cons_a @ ( hd_a @ Xs ) @ nil_a ) ) ) ) ).
% rotate1_hd_tl
thf(fact_712_listrel1__iff__update,axiom,
! [Xs: list_list_a,Ys: list_list_a,R: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Ys ) @ ( listrel1_list_a @ R ) )
= ( ? [Y7: list_a,N2: nat] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( nth_list_a @ Xs @ N2 ) @ Y7 ) @ R )
& ( ord_less_nat @ N2 @ ( size_s349497388124573686list_a @ Xs ) )
& ( Ys
= ( list_update_list_a @ Xs @ N2 @ Y7 ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_713_listrel1__iff__update,axiom,
! [Xs: list_a,Ys: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel1_a @ R ) )
= ( ? [Y7: a,N2: nat] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Xs @ N2 ) @ Y7 ) @ R )
& ( ord_less_nat @ N2 @ ( size_size_list_a @ Xs ) )
& ( Ys
= ( list_update_a @ Xs @ N2 @ Y7 ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_714_rotate1__is__Nil__conv,axiom,
! [Xs: list_a] :
( ( ( rotate1_a @ Xs )
= nil_a )
= ( Xs = nil_a ) ) ).
% rotate1_is_Nil_conv
thf(fact_715_Cons__listrel1__Cons,axiom,
! [X: list_a,Xs: list_list_a,Y: list_a,Ys: list_list_a,R: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( cons_list_a @ X @ Xs ) @ ( cons_list_a @ Y @ Ys ) ) @ ( listrel1_list_a @ R ) )
= ( ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ R )
& ( Xs = Ys ) )
| ( ( X = Y )
& ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Ys ) @ ( listrel1_list_a @ R ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_716_Cons__listrel1__Cons,axiom,
! [X: a,Xs: list_a,Y: a,Ys: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X @ Xs ) @ ( cons_a @ Y @ Ys ) ) @ ( listrel1_a @ R ) )
= ( ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
& ( Xs = Ys ) )
| ( ( X = Y )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel1_a @ R ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_717_append__listrel1I,axiom,
! [Xs: list_a,Ys: list_a,R: set_Product_prod_a_a,Us: list_a,Vs: list_a] :
( ( ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel1_a @ R ) )
& ( Us = Vs ) )
| ( ( Xs = Ys )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Us @ Vs ) @ ( listrel1_a @ R ) ) ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Xs @ Us ) @ ( append_a @ Ys @ Vs ) ) @ ( listrel1_a @ R ) ) ) ).
% append_listrel1I
thf(fact_718_listrel1I2,axiom,
! [Xs: list_a,Ys: list_a,R: set_Product_prod_a_a,X: a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel1_a @ R ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X @ Xs ) @ ( cons_a @ X @ Ys ) ) @ ( listrel1_a @ R ) ) ) ).
% listrel1I2
thf(fact_719_not__listrel1__Nil,axiom,
! [Xs: list_a,R: set_Product_prod_a_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ nil_a ) @ ( listrel1_a @ R ) ) ).
% not_listrel1_Nil
thf(fact_720_not__Nil__listrel1,axiom,
! [Xs: list_a,R: set_Product_prod_a_a] :
~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ nil_a @ Xs ) @ ( listrel1_a @ R ) ) ).
% not_Nil_listrel1
thf(fact_721_listrel1__eq__len,axiom,
! [Xs: list_a,Ys: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel1_a @ R ) )
=> ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) ) ) ).
% listrel1_eq_len
thf(fact_722_rotate1_Osimps_I1_J,axiom,
( ( rotate1_a @ nil_a )
= nil_a ) ).
% rotate1.simps(1)
thf(fact_723_Cons__listrel1E2,axiom,
! [Xs: list_list_a,Y: list_a,Ys: list_list_a,R: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ ( cons_list_a @ Y @ Ys ) ) @ ( listrel1_list_a @ R ) )
=> ( ! [X5: list_a] :
( ( Xs
= ( cons_list_a @ X5 @ Ys ) )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X5 @ Y ) @ R ) )
=> ~ ! [Zs2: list_list_a] :
( ( Xs
= ( cons_list_a @ Y @ Zs2 ) )
=> ~ ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Zs2 @ Ys ) @ ( listrel1_list_a @ R ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_724_Cons__listrel1E2,axiom,
! [Xs: list_a,Y: a,Ys: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ ( cons_a @ Y @ Ys ) ) @ ( listrel1_a @ R ) )
=> ( ! [X5: a] :
( ( Xs
= ( cons_a @ X5 @ Ys ) )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Y ) @ R ) )
=> ~ ! [Zs2: list_a] :
( ( Xs
= ( cons_a @ Y @ Zs2 ) )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Zs2 @ Ys ) @ ( listrel1_a @ R ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_725_Cons__listrel1E1,axiom,
! [X: list_a,Xs: list_list_a,Ys: list_list_a,R: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( cons_list_a @ X @ Xs ) @ Ys ) @ ( listrel1_list_a @ R ) )
=> ( ! [Y4: list_a] :
( ( Ys
= ( cons_list_a @ Y4 @ Xs ) )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y4 ) @ R ) )
=> ~ ! [Zs2: list_list_a] :
( ( Ys
= ( cons_list_a @ X @ Zs2 ) )
=> ~ ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Zs2 ) @ ( listrel1_list_a @ R ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_726_Cons__listrel1E1,axiom,
! [X: a,Xs: list_a,Ys: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X @ Xs ) @ Ys ) @ ( listrel1_a @ R ) )
=> ( ! [Y4: a] :
( ( Ys
= ( cons_a @ Y4 @ Xs ) )
=> ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y4 ) @ R ) )
=> ~ ! [Zs2: list_a] :
( ( Ys
= ( cons_a @ X @ Zs2 ) )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Zs2 ) @ ( listrel1_a @ R ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_727_listrel1I1,axiom,
! [X: a,Y: a,R: set_Product_prod_a_a,Xs: list_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X @ Xs ) @ ( cons_a @ Y @ Xs ) ) @ ( listrel1_a @ R ) ) ) ).
% listrel1I1
thf(fact_728_listrel1I1,axiom,
! [X: list_a,Y: list_a,R: set_Pr4048851178543822343list_a,Xs: list_list_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ R )
=> ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( cons_list_a @ X @ Xs ) @ ( cons_list_a @ Y @ Xs ) ) @ ( listrel1_list_a @ R ) ) ) ).
% listrel1I1
thf(fact_729_rotate1_Osimps_I2_J,axiom,
! [X: a,Xs: list_a] :
( ( rotate1_a @ ( cons_a @ X @ Xs ) )
= ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) ) ).
% rotate1.simps(2)
thf(fact_730_listrel1I,axiom,
! [X: a,Y: a,R: set_Product_prod_a_a,Xs: list_a,Us: list_a,Vs: list_a,Ys: list_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
=> ( ( Xs
= ( append_a @ Us @ ( cons_a @ X @ Vs ) ) )
=> ( ( Ys
= ( append_a @ Us @ ( cons_a @ Y @ Vs ) ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel1_a @ R ) ) ) ) ) ).
% listrel1I
thf(fact_731_listrel1I,axiom,
! [X: list_a,Y: list_a,R: set_Pr4048851178543822343list_a,Xs: list_list_a,Us: list_list_a,Vs: list_list_a,Ys: list_list_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ R )
=> ( ( Xs
= ( append_list_a @ Us @ ( cons_list_a @ X @ Vs ) ) )
=> ( ( Ys
= ( append_list_a @ Us @ ( cons_list_a @ Y @ Vs ) ) )
=> ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Ys ) @ ( listrel1_list_a @ R ) ) ) ) ) ).
% listrel1I
thf(fact_732_listrel1E,axiom,
! [Xs: list_list_a,Ys: list_list_a,R: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Ys ) @ ( listrel1_list_a @ R ) )
=> ~ ! [X5: list_a,Y4: list_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X5 @ Y4 ) @ R )
=> ! [Us2: list_list_a,Vs2: list_list_a] :
( ( Xs
= ( append_list_a @ Us2 @ ( cons_list_a @ X5 @ Vs2 ) ) )
=> ( Ys
!= ( append_list_a @ Us2 @ ( cons_list_a @ Y4 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_733_listrel1E,axiom,
! [Xs: list_a,Ys: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel1_a @ R ) )
=> ~ ! [X5: a,Y4: a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Y4 ) @ R )
=> ! [Us2: list_a,Vs2: list_a] :
( ( Xs
= ( append_a @ Us2 @ ( cons_a @ X5 @ Vs2 ) ) )
=> ( Ys
!= ( append_a @ Us2 @ ( cons_a @ Y4 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_734_snoc__listrel1__snoc__iff,axiom,
! [Xs: list_list_a,X: list_a,Ys: list_list_a,Y: list_a,R: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( append_list_a @ Xs @ ( cons_list_a @ X @ nil_list_a ) ) @ ( append_list_a @ Ys @ ( cons_list_a @ Y @ nil_list_a ) ) ) @ ( listrel1_list_a @ R ) )
= ( ( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Ys ) @ ( listrel1_list_a @ R ) )
& ( X = Y ) )
| ( ( Xs = Ys )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ R ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_735_snoc__listrel1__snoc__iff,axiom,
! [Xs: list_a,X: a,Ys: list_a,Y: a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) @ ( append_a @ Ys @ ( cons_a @ Y @ nil_a ) ) ) @ ( listrel1_a @ R ) )
= ( ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel1_a @ R ) )
& ( X = Y ) )
| ( ( Xs = Ys )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_736_nths__Cons,axiom,
! [X: a,L: list_a,A5: set_nat] :
( ( nths_a @ ( cons_a @ X @ L ) @ A5 )
= ( append_a @ ( if_list_a @ ( member_nat @ zero_zero_nat @ A5 ) @ ( cons_a @ X @ nil_a ) @ nil_a )
@ ( nths_a @ L
@ ( collect_nat
@ ^ [J3: nat] : ( member_nat @ ( suc @ J3 ) @ A5 ) ) ) ) ) ).
% nths_Cons
thf(fact_737_nths__nil,axiom,
! [A5: set_nat] :
( ( nths_a @ nil_a @ A5 )
= nil_a ) ).
% nths_nil
thf(fact_738_bind__simps_I1_J,axiom,
! [F: a > list_a] :
( ( bind_a_a @ nil_a @ F )
= nil_a ) ).
% bind_simps(1)
thf(fact_739_nths__singleton,axiom,
! [A5: set_nat,X: a] :
( ( ( member_nat @ zero_zero_nat @ A5 )
=> ( ( nths_a @ ( cons_a @ X @ nil_a ) @ A5 )
= ( cons_a @ X @ nil_a ) ) )
& ( ~ ( member_nat @ zero_zero_nat @ A5 )
=> ( ( nths_a @ ( cons_a @ X @ nil_a ) @ A5 )
= nil_a ) ) ) ).
% nths_singleton
thf(fact_740_pred__subset__eq2,axiom,
! [R2: set_Pr4048851178543822343list_a,S2: set_Pr4048851178543822343list_a] :
( ( ord_le5542992221119063950st_a_o
@ ^ [X4: list_a,Y7: list_a] : ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X4 @ Y7 ) @ R2 )
@ ^ [X4: list_a,Y7: list_a] : ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X4 @ Y7 ) @ S2 ) )
= ( ord_le7857023143581076903list_a @ R2 @ S2 ) ) ).
% pred_subset_eq2
thf(fact_741_pred__subset__eq2,axiom,
! [R2: set_Pr1464008215722202041_a_nat,S2: set_Pr1464008215722202041_a_nat] :
( ( ord_le4636867679518884800_nat_o
@ ^ [X4: states_a,Y7: nat] : ( member6483677129516672026_a_nat @ ( produc1877401315875745917_a_nat @ X4 @ Y7 ) @ R2 )
@ ^ [X4: states_a,Y7: nat] : ( member6483677129516672026_a_nat @ ( produc1877401315875745917_a_nat @ X4 @ Y7 ) @ S2 ) )
= ( ord_le6196172653455811609_a_nat @ R2 @ S2 ) ) ).
% pred_subset_eq2
thf(fact_742_pred__subset__eq2,axiom,
! [R2: set_Pr6306228930610421491tate_a,S2: set_Pr6306228930610421491tate_a] :
( ( ord_le3878161006114389794te_a_o
@ ^ [X4: a,Y7: state_a] : ( member8547378267715833660tate_a @ ( produc1224139502141355779tate_a @ X4 @ Y7 ) @ R2 )
@ ^ [X4: a,Y7: state_a] : ( member8547378267715833660tate_a @ ( produc1224139502141355779tate_a @ X4 @ Y7 ) @ S2 ) )
= ( ord_le926351553812985491tate_a @ R2 @ S2 ) ) ).
% pred_subset_eq2
thf(fact_743_pred__subset__eq2,axiom,
! [R2: set_Pr4275752383657305402tate_a,S2: set_Pr4275752383657305402tate_a] :
( ( ord_le5391092903712282779te_a_o
@ ^ [X4: a,Y7: state_a2] : ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ X4 @ Y7 ) @ R2 )
@ ^ [X4: a,Y7: state_a2] : ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ X4 @ Y7 ) @ S2 ) )
= ( ord_le7345504482307493082tate_a @ R2 @ S2 ) ) ).
% pred_subset_eq2
thf(fact_744_pred__equals__eq2,axiom,
! [R2: set_Pr4048851178543822343list_a,S2: set_Pr4048851178543822343list_a] :
( ( ( ^ [X4: list_a,Y7: list_a] : ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X4 @ Y7 ) @ R2 ) )
= ( ^ [X4: list_a,Y7: list_a] : ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X4 @ Y7 ) @ S2 ) ) )
= ( R2 = S2 ) ) ).
% pred_equals_eq2
thf(fact_745_pred__equals__eq2,axiom,
! [R2: set_Pr1464008215722202041_a_nat,S2: set_Pr1464008215722202041_a_nat] :
( ( ( ^ [X4: states_a,Y7: nat] : ( member6483677129516672026_a_nat @ ( produc1877401315875745917_a_nat @ X4 @ Y7 ) @ R2 ) )
= ( ^ [X4: states_a,Y7: nat] : ( member6483677129516672026_a_nat @ ( produc1877401315875745917_a_nat @ X4 @ Y7 ) @ S2 ) ) )
= ( R2 = S2 ) ) ).
% pred_equals_eq2
thf(fact_746_pred__equals__eq2,axiom,
! [R2: set_Pr6306228930610421491tate_a,S2: set_Pr6306228930610421491tate_a] :
( ( ( ^ [X4: a,Y7: state_a] : ( member8547378267715833660tate_a @ ( produc1224139502141355779tate_a @ X4 @ Y7 ) @ R2 ) )
= ( ^ [X4: a,Y7: state_a] : ( member8547378267715833660tate_a @ ( produc1224139502141355779tate_a @ X4 @ Y7 ) @ S2 ) ) )
= ( R2 = S2 ) ) ).
% pred_equals_eq2
thf(fact_747_pred__equals__eq2,axiom,
! [R2: set_Pr4275752383657305402tate_a,S2: set_Pr4275752383657305402tate_a] :
( ( ( ^ [X4: a,Y7: state_a2] : ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ X4 @ Y7 ) @ R2 ) )
= ( ^ [X4: a,Y7: state_a2] : ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ X4 @ Y7 ) @ S2 ) ) )
= ( R2 = S2 ) ) ).
% pred_equals_eq2
thf(fact_748_subrelI,axiom,
! [R: set_Pr4048851178543822343list_a,S: set_Pr4048851178543822343list_a] :
( ! [X5: list_a,Y4: list_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X5 @ Y4 ) @ R )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X5 @ Y4 ) @ S ) )
=> ( ord_le7857023143581076903list_a @ R @ S ) ) ).
% subrelI
thf(fact_749_subrelI,axiom,
! [R: set_Pr1464008215722202041_a_nat,S: set_Pr1464008215722202041_a_nat] :
( ! [X5: states_a,Y4: nat] :
( ( member6483677129516672026_a_nat @ ( produc1877401315875745917_a_nat @ X5 @ Y4 ) @ R )
=> ( member6483677129516672026_a_nat @ ( produc1877401315875745917_a_nat @ X5 @ Y4 ) @ S ) )
=> ( ord_le6196172653455811609_a_nat @ R @ S ) ) ).
% subrelI
thf(fact_750_subrelI,axiom,
! [R: set_Pr6306228930610421491tate_a,S: set_Pr6306228930610421491tate_a] :
( ! [X5: a,Y4: state_a] :
( ( member8547378267715833660tate_a @ ( produc1224139502141355779tate_a @ X5 @ Y4 ) @ R )
=> ( member8547378267715833660tate_a @ ( produc1224139502141355779tate_a @ X5 @ Y4 ) @ S ) )
=> ( ord_le926351553812985491tate_a @ R @ S ) ) ).
% subrelI
thf(fact_751_subrelI,axiom,
! [R: set_Pr4275752383657305402tate_a,S: set_Pr4275752383657305402tate_a] :
( ! [X5: a,Y4: state_a2] :
( ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ X5 @ Y4 ) @ R )
=> ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ X5 @ Y4 ) @ S ) )
=> ( ord_le7345504482307493082tate_a @ R @ S ) ) ).
% subrelI
thf(fact_752_listrel__iff__nth,axiom,
! [Xs: list_list_a,Ys: list_list_a,R: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Ys ) @ ( listre6772471554020304241list_a @ R ) )
= ( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
& ! [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_s349497388124573686list_a @ Xs ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( nth_list_a @ Xs @ N2 ) @ ( nth_list_a @ Ys @ N2 ) ) @ R ) ) ) ) ).
% listrel_iff_nth
thf(fact_753_listrel__iff__nth,axiom,
! [Xs: list_states_a,Ys: list_nat,R: set_Pr1464008215722202041_a_nat] :
( ( member6795120827800458928st_nat @ ( produc327714349733490451st_nat @ Xs @ Ys ) @ ( listrel_states_a_nat @ R ) )
= ( ( ( size_s3891197933023997302ates_a @ Xs )
= ( size_size_list_nat @ Ys ) )
& ! [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_s3891197933023997302ates_a @ Xs ) )
=> ( member6483677129516672026_a_nat @ ( produc1877401315875745917_a_nat @ ( nth_states_a @ Xs @ N2 ) @ ( nth_nat @ Ys @ N2 ) ) @ R ) ) ) ) ).
% listrel_iff_nth
thf(fact_754_listrel__iff__nth,axiom,
! [Xs: list_a,Ys: list_state_a,R: set_Pr6306228930610421491tate_a] :
( ( member4112945611203173692tate_a @ ( produc1997082749353321475tate_a @ Xs @ Ys ) @ ( listrel_a_state_a2 @ R ) )
= ( ( ( size_size_list_a @ Xs )
= ( size_s8463391772401140188tate_a @ Ys ) )
& ! [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_a @ Xs ) )
=> ( member8547378267715833660tate_a @ ( produc1224139502141355779tate_a @ ( nth_a @ Xs @ N2 ) @ ( nth_state_a2 @ Ys @ N2 ) ) @ R ) ) ) ) ).
% listrel_iff_nth
thf(fact_755_listrel__iff__nth,axiom,
! [Xs: list_a,Ys: list_state_a2,R: set_Pr4275752383657305402tate_a] :
( ( member6123284207288203267tate_a @ ( produc3968967176812022602tate_a @ Xs @ Ys ) @ ( listrel_a_state_a @ R ) )
= ( ( ( size_size_list_a @ Xs )
= ( size_s7859192958365828515tate_a @ Ys ) )
& ! [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_a @ Xs ) )
=> ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ ( nth_a @ Xs @ N2 ) @ ( nth_state_a @ Ys @ N2 ) ) @ R ) ) ) ) ).
% listrel_iff_nth
thf(fact_756_listrel__iff__nth,axiom,
! [Xs: list_a,Ys: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel_a_a @ R ) )
= ( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
& ! [N2: nat] :
( ( ord_less_nat @ N2 @ ( size_size_list_a @ Xs ) )
=> ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Xs @ N2 ) @ ( nth_a @ Ys @ N2 ) ) @ R ) ) ) ) ).
% listrel_iff_nth
thf(fact_757_enumerate__simps_I1_J,axiom,
! [N: nat] :
( ( enumerate_a @ N @ nil_a )
= nil_Pr1417316670369895453_nat_a ) ).
% enumerate_simps(1)
thf(fact_758_enumerate__simps_I2_J,axiom,
! [N: nat,X: a,Xs: list_a] :
( ( enumerate_a @ N @ ( cons_a @ X @ Xs ) )
= ( cons_P8443330267410185325_nat_a @ ( product_Pair_nat_a @ N @ X ) @ ( enumerate_a @ ( suc @ N ) @ Xs ) ) ) ).
% enumerate_simps(2)
thf(fact_759_listrel_ONil,axiom,
! [R: set_Product_prod_a_a] : ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ nil_a @ nil_a ) @ ( listrel_a_a @ R ) ) ).
% listrel.Nil
thf(fact_760_listrel__Nil1,axiom,
! [Xs: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ nil_a @ Xs ) @ ( listrel_a_a @ R ) )
=> ( Xs = nil_a ) ) ).
% listrel_Nil1
thf(fact_761_listrel__Nil2,axiom,
! [Xs: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ nil_a ) @ ( listrel_a_a @ R ) )
=> ( Xs = nil_a ) ) ).
% listrel_Nil2
thf(fact_762_listrel__eq__len,axiom,
! [Xs: list_a,Ys: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel_a_a @ R ) )
=> ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) ) ) ).
% listrel_eq_len
thf(fact_763_listrel_OCons,axiom,
! [X: a,Y: a,R: set_Product_prod_a_a,Xs: list_a,Ys: list_a] :
( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R )
=> ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ Ys ) @ ( listrel_a_a @ R ) )
=> ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ X @ Xs ) @ ( cons_a @ Y @ Ys ) ) @ ( listrel_a_a @ R ) ) ) ) ).
% listrel.Cons
thf(fact_764_listrel_OCons,axiom,
! [X: list_a,Y: list_a,R: set_Pr4048851178543822343list_a,Xs: list_list_a,Ys: list_list_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ R )
=> ( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ Ys ) @ ( listre6772471554020304241list_a @ R ) )
=> ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( cons_list_a @ X @ Xs ) @ ( cons_list_a @ Y @ Ys ) ) @ ( listre6772471554020304241list_a @ R ) ) ) ) ).
% listrel.Cons
thf(fact_765_listrel_OCons,axiom,
! [X: states_a,Y: nat,R: set_Pr1464008215722202041_a_nat,Xs: list_states_a,Ys: list_nat] :
( ( member6483677129516672026_a_nat @ ( produc1877401315875745917_a_nat @ X @ Y ) @ R )
=> ( ( member6795120827800458928st_nat @ ( produc327714349733490451st_nat @ Xs @ Ys ) @ ( listrel_states_a_nat @ R ) )
=> ( member6795120827800458928st_nat @ ( produc327714349733490451st_nat @ ( cons_states_a @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) ) @ ( listrel_states_a_nat @ R ) ) ) ) ).
% listrel.Cons
thf(fact_766_listrel_OCons,axiom,
! [X: a,Y: state_a,R: set_Pr6306228930610421491tate_a,Xs: list_a,Ys: list_state_a] :
( ( member8547378267715833660tate_a @ ( produc1224139502141355779tate_a @ X @ Y ) @ R )
=> ( ( member4112945611203173692tate_a @ ( produc1997082749353321475tate_a @ Xs @ Ys ) @ ( listrel_a_state_a2 @ R ) )
=> ( member4112945611203173692tate_a @ ( produc1997082749353321475tate_a @ ( cons_a @ X @ Xs ) @ ( cons_state_a2 @ Y @ Ys ) ) @ ( listrel_a_state_a2 @ R ) ) ) ) ).
% listrel.Cons
thf(fact_767_listrel_OCons,axiom,
! [X: a,Y: state_a2,R: set_Pr4275752383657305402tate_a,Xs: list_a,Ys: list_state_a2] :
( ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ X @ Y ) @ R )
=> ( ( member6123284207288203267tate_a @ ( produc3968967176812022602tate_a @ Xs @ Ys ) @ ( listrel_a_state_a @ R ) )
=> ( member6123284207288203267tate_a @ ( produc3968967176812022602tate_a @ ( cons_a @ X @ Xs ) @ ( cons_state_a @ Y @ Ys ) ) @ ( listrel_a_state_a @ R ) ) ) ) ).
% listrel.Cons
thf(fact_768_listrel__Cons1,axiom,
! [Y: list_a,Ys: list_list_a,Xs: list_list_a,R: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ ( cons_list_a @ Y @ Ys ) @ Xs ) @ ( listre6772471554020304241list_a @ R ) )
=> ~ ! [Y4: list_a,Ys3: list_list_a] :
( ( Xs
= ( cons_list_a @ Y4 @ Ys3 ) )
=> ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Y @ Y4 ) @ R )
=> ~ ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Ys @ Ys3 ) @ ( listre6772471554020304241list_a @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_769_listrel__Cons1,axiom,
! [Y: states_a,Ys: list_states_a,Xs: list_nat,R: set_Pr1464008215722202041_a_nat] :
( ( member6795120827800458928st_nat @ ( produc327714349733490451st_nat @ ( cons_states_a @ Y @ Ys ) @ Xs ) @ ( listrel_states_a_nat @ R ) )
=> ~ ! [Y4: nat,Ys3: list_nat] :
( ( Xs
= ( cons_nat @ Y4 @ Ys3 ) )
=> ( ( member6483677129516672026_a_nat @ ( produc1877401315875745917_a_nat @ Y @ Y4 ) @ R )
=> ~ ( member6795120827800458928st_nat @ ( produc327714349733490451st_nat @ Ys @ Ys3 ) @ ( listrel_states_a_nat @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_770_listrel__Cons1,axiom,
! [Y: a,Ys: list_a,Xs: list_state_a,R: set_Pr6306228930610421491tate_a] :
( ( member4112945611203173692tate_a @ ( produc1997082749353321475tate_a @ ( cons_a @ Y @ Ys ) @ Xs ) @ ( listrel_a_state_a2 @ R ) )
=> ~ ! [Y4: state_a,Ys3: list_state_a] :
( ( Xs
= ( cons_state_a2 @ Y4 @ Ys3 ) )
=> ( ( member8547378267715833660tate_a @ ( produc1224139502141355779tate_a @ Y @ Y4 ) @ R )
=> ~ ( member4112945611203173692tate_a @ ( produc1997082749353321475tate_a @ Ys @ Ys3 ) @ ( listrel_a_state_a2 @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_771_listrel__Cons1,axiom,
! [Y: a,Ys: list_a,Xs: list_state_a2,R: set_Pr4275752383657305402tate_a] :
( ( member6123284207288203267tate_a @ ( produc3968967176812022602tate_a @ ( cons_a @ Y @ Ys ) @ Xs ) @ ( listrel_a_state_a @ R ) )
=> ~ ! [Y4: state_a2,Ys3: list_state_a2] :
( ( Xs
= ( cons_state_a @ Y4 @ Ys3 ) )
=> ( ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ Y @ Y4 ) @ R )
=> ~ ( member6123284207288203267tate_a @ ( produc3968967176812022602tate_a @ Ys @ Ys3 ) @ ( listrel_a_state_a @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_772_listrel__Cons1,axiom,
! [Y: a,Ys: list_a,Xs: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( cons_a @ Y @ Ys ) @ Xs ) @ ( listrel_a_a @ R ) )
=> ~ ! [Y4: a,Ys3: list_a] :
( ( Xs
= ( cons_a @ Y4 @ Ys3 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ Y4 ) @ R )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ys @ Ys3 ) @ ( listrel_a_a @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_773_listrel__Cons2,axiom,
! [Xs: list_list_a,Y: list_a,Ys: list_list_a,R: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs @ ( cons_list_a @ Y @ Ys ) ) @ ( listre6772471554020304241list_a @ R ) )
=> ~ ! [X5: list_a,Xs2: list_list_a] :
( ( Xs
= ( cons_list_a @ X5 @ Xs2 ) )
=> ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X5 @ Y ) @ R )
=> ~ ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs2 @ Ys ) @ ( listre6772471554020304241list_a @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_774_listrel__Cons2,axiom,
! [Xs: list_states_a,Y: nat,Ys: list_nat,R: set_Pr1464008215722202041_a_nat] :
( ( member6795120827800458928st_nat @ ( produc327714349733490451st_nat @ Xs @ ( cons_nat @ Y @ Ys ) ) @ ( listrel_states_a_nat @ R ) )
=> ~ ! [X5: states_a,Xs2: list_states_a] :
( ( Xs
= ( cons_states_a @ X5 @ Xs2 ) )
=> ( ( member6483677129516672026_a_nat @ ( produc1877401315875745917_a_nat @ X5 @ Y ) @ R )
=> ~ ( member6795120827800458928st_nat @ ( produc327714349733490451st_nat @ Xs2 @ Ys ) @ ( listrel_states_a_nat @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_775_listrel__Cons2,axiom,
! [Xs: list_a,Y: state_a,Ys: list_state_a,R: set_Pr6306228930610421491tate_a] :
( ( member4112945611203173692tate_a @ ( produc1997082749353321475tate_a @ Xs @ ( cons_state_a2 @ Y @ Ys ) ) @ ( listrel_a_state_a2 @ R ) )
=> ~ ! [X5: a,Xs2: list_a] :
( ( Xs
= ( cons_a @ X5 @ Xs2 ) )
=> ( ( member8547378267715833660tate_a @ ( produc1224139502141355779tate_a @ X5 @ Y ) @ R )
=> ~ ( member4112945611203173692tate_a @ ( produc1997082749353321475tate_a @ Xs2 @ Ys ) @ ( listrel_a_state_a2 @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_776_listrel__Cons2,axiom,
! [Xs: list_a,Y: state_a2,Ys: list_state_a2,R: set_Pr4275752383657305402tate_a] :
( ( member6123284207288203267tate_a @ ( produc3968967176812022602tate_a @ Xs @ ( cons_state_a @ Y @ Ys ) ) @ ( listrel_a_state_a @ R ) )
=> ~ ! [X5: a,Xs2: list_a] :
( ( Xs
= ( cons_a @ X5 @ Xs2 ) )
=> ( ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ X5 @ Y ) @ R )
=> ~ ( member6123284207288203267tate_a @ ( produc3968967176812022602tate_a @ Xs2 @ Ys ) @ ( listrel_a_state_a @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_777_listrel__Cons2,axiom,
! [Xs: list_a,Y: a,Ys: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs @ ( cons_a @ Y @ Ys ) ) @ ( listrel_a_a @ R ) )
=> ~ ! [X5: a,Xs2: list_a] :
( ( Xs
= ( cons_a @ X5 @ Xs2 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Y ) @ R )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs2 @ Ys ) @ ( listrel_a_a @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_778_listrel_Osimps,axiom,
! [A1: list_list_a,A22: list_list_a,R: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ A1 @ A22 ) @ ( listre6772471554020304241list_a @ R ) )
= ( ( ( A1 = nil_list_a )
& ( A22 = nil_list_a ) )
| ? [X4: list_a,Y7: list_a,Xs3: list_list_a,Ys2: list_list_a] :
( ( A1
= ( cons_list_a @ X4 @ Xs3 ) )
& ( A22
= ( cons_list_a @ Y7 @ Ys2 ) )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X4 @ Y7 ) @ R )
& ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs3 @ Ys2 ) @ ( listre6772471554020304241list_a @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_779_listrel_Osimps,axiom,
! [A1: list_states_a,A22: list_nat,R: set_Pr1464008215722202041_a_nat] :
( ( member6795120827800458928st_nat @ ( produc327714349733490451st_nat @ A1 @ A22 ) @ ( listrel_states_a_nat @ R ) )
= ( ( ( A1 = nil_states_a )
& ( A22 = nil_nat ) )
| ? [X4: states_a,Y7: nat,Xs3: list_states_a,Ys2: list_nat] :
( ( A1
= ( cons_states_a @ X4 @ Xs3 ) )
& ( A22
= ( cons_nat @ Y7 @ Ys2 ) )
& ( member6483677129516672026_a_nat @ ( produc1877401315875745917_a_nat @ X4 @ Y7 ) @ R )
& ( member6795120827800458928st_nat @ ( produc327714349733490451st_nat @ Xs3 @ Ys2 ) @ ( listrel_states_a_nat @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_780_listrel_Osimps,axiom,
! [A1: list_a,A22: list_state_a,R: set_Pr6306228930610421491tate_a] :
( ( member4112945611203173692tate_a @ ( produc1997082749353321475tate_a @ A1 @ A22 ) @ ( listrel_a_state_a2 @ R ) )
= ( ( ( A1 = nil_a )
& ( A22 = nil_state_a2 ) )
| ? [X4: a,Y7: state_a,Xs3: list_a,Ys2: list_state_a] :
( ( A1
= ( cons_a @ X4 @ Xs3 ) )
& ( A22
= ( cons_state_a2 @ Y7 @ Ys2 ) )
& ( member8547378267715833660tate_a @ ( produc1224139502141355779tate_a @ X4 @ Y7 ) @ R )
& ( member4112945611203173692tate_a @ ( produc1997082749353321475tate_a @ Xs3 @ Ys2 ) @ ( listrel_a_state_a2 @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_781_listrel_Osimps,axiom,
! [A1: list_a,A22: list_state_a2,R: set_Pr4275752383657305402tate_a] :
( ( member6123284207288203267tate_a @ ( produc3968967176812022602tate_a @ A1 @ A22 ) @ ( listrel_a_state_a @ R ) )
= ( ( ( A1 = nil_a )
& ( A22 = nil_state_a ) )
| ? [X4: a,Y7: state_a2,Xs3: list_a,Ys2: list_state_a2] :
( ( A1
= ( cons_a @ X4 @ Xs3 ) )
& ( A22
= ( cons_state_a @ Y7 @ Ys2 ) )
& ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ X4 @ Y7 ) @ R )
& ( member6123284207288203267tate_a @ ( produc3968967176812022602tate_a @ Xs3 @ Ys2 ) @ ( listrel_a_state_a @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_782_listrel_Osimps,axiom,
! [A1: list_a,A22: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ A1 @ A22 ) @ ( listrel_a_a @ R ) )
= ( ( ( A1 = nil_a )
& ( A22 = nil_a ) )
| ? [X4: a,Y7: a,Xs3: list_a,Ys2: list_a] :
( ( A1
= ( cons_a @ X4 @ Xs3 ) )
& ( A22
= ( cons_a @ Y7 @ Ys2 ) )
& ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y7 ) @ R )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs3 @ Ys2 ) @ ( listrel_a_a @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_783_listrel_Ocases,axiom,
! [A1: list_list_a,A22: list_list_a,R: set_Pr4048851178543822343list_a] :
( ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ A1 @ A22 ) @ ( listre6772471554020304241list_a @ R ) )
=> ( ( ( A1 = nil_list_a )
=> ( A22 != nil_list_a ) )
=> ~ ! [X5: list_a,Y4: list_a,Xs2: list_list_a] :
( ( A1
= ( cons_list_a @ X5 @ Xs2 ) )
=> ! [Ys3: list_list_a] :
( ( A22
= ( cons_list_a @ Y4 @ Ys3 ) )
=> ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X5 @ Y4 ) @ R )
=> ~ ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ Xs2 @ Ys3 ) @ ( listre6772471554020304241list_a @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_784_listrel_Ocases,axiom,
! [A1: list_states_a,A22: list_nat,R: set_Pr1464008215722202041_a_nat] :
( ( member6795120827800458928st_nat @ ( produc327714349733490451st_nat @ A1 @ A22 ) @ ( listrel_states_a_nat @ R ) )
=> ( ( ( A1 = nil_states_a )
=> ( A22 != nil_nat ) )
=> ~ ! [X5: states_a,Y4: nat,Xs2: list_states_a] :
( ( A1
= ( cons_states_a @ X5 @ Xs2 ) )
=> ! [Ys3: list_nat] :
( ( A22
= ( cons_nat @ Y4 @ Ys3 ) )
=> ( ( member6483677129516672026_a_nat @ ( produc1877401315875745917_a_nat @ X5 @ Y4 ) @ R )
=> ~ ( member6795120827800458928st_nat @ ( produc327714349733490451st_nat @ Xs2 @ Ys3 ) @ ( listrel_states_a_nat @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_785_listrel_Ocases,axiom,
! [A1: list_a,A22: list_state_a,R: set_Pr6306228930610421491tate_a] :
( ( member4112945611203173692tate_a @ ( produc1997082749353321475tate_a @ A1 @ A22 ) @ ( listrel_a_state_a2 @ R ) )
=> ( ( ( A1 = nil_a )
=> ( A22 != nil_state_a2 ) )
=> ~ ! [X5: a,Y4: state_a,Xs2: list_a] :
( ( A1
= ( cons_a @ X5 @ Xs2 ) )
=> ! [Ys3: list_state_a] :
( ( A22
= ( cons_state_a2 @ Y4 @ Ys3 ) )
=> ( ( member8547378267715833660tate_a @ ( produc1224139502141355779tate_a @ X5 @ Y4 ) @ R )
=> ~ ( member4112945611203173692tate_a @ ( produc1997082749353321475tate_a @ Xs2 @ Ys3 ) @ ( listrel_a_state_a2 @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_786_listrel_Ocases,axiom,
! [A1: list_a,A22: list_state_a2,R: set_Pr4275752383657305402tate_a] :
( ( member6123284207288203267tate_a @ ( produc3968967176812022602tate_a @ A1 @ A22 ) @ ( listrel_a_state_a @ R ) )
=> ( ( ( A1 = nil_a )
=> ( A22 != nil_state_a ) )
=> ~ ! [X5: a,Y4: state_a2,Xs2: list_a] :
( ( A1
= ( cons_a @ X5 @ Xs2 ) )
=> ! [Ys3: list_state_a2] :
( ( A22
= ( cons_state_a @ Y4 @ Ys3 ) )
=> ( ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ X5 @ Y4 ) @ R )
=> ~ ( member6123284207288203267tate_a @ ( produc3968967176812022602tate_a @ Xs2 @ Ys3 ) @ ( listrel_a_state_a @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_787_listrel_Ocases,axiom,
! [A1: list_a,A22: list_a,R: set_Product_prod_a_a] :
( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ A1 @ A22 ) @ ( listrel_a_a @ R ) )
=> ( ( ( A1 = nil_a )
=> ( A22 != nil_a ) )
=> ~ ! [X5: a,Y4: a,Xs2: list_a] :
( ( A1
= ( cons_a @ X5 @ Xs2 ) )
=> ! [Ys3: list_a] :
( ( A22
= ( cons_a @ Y4 @ Ys3 ) )
=> ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X5 @ Y4 ) @ R )
=> ~ ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs2 @ Ys3 ) @ ( listrel_a_a @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_788_nth__zip,axiom,
! [I: nat,Xs: list_list_a,Ys: list_list_a] :
( ( ord_less_nat @ I @ ( size_s349497388124573686list_a @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_s349497388124573686list_a @ Ys ) )
=> ( ( nth_Pr5917933638979213230list_a @ ( zip_list_a_list_a @ Xs @ Ys ) @ I )
= ( produc6837034575241423639list_a @ ( nth_list_a @ Xs @ I ) @ ( nth_list_a @ Ys @ I ) ) ) ) ) ).
% nth_zip
thf(fact_789_nth__zip,axiom,
! [I: nat,Xs: list_states_a,Ys: list_nat] :
( ( ord_less_nat @ I @ ( size_s3891197933023997302ates_a @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
=> ( ( nth_Pr6783065760942753852_a_nat @ ( zip_states_a_nat @ Xs @ Ys ) @ I )
= ( produc1877401315875745917_a_nat @ ( nth_states_a @ Xs @ I ) @ ( nth_nat @ Ys @ I ) ) ) ) ) ).
% nth_zip
thf(fact_790_nth__zip,axiom,
! [I: nat,Xs: list_a,Ys: list_state_a] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_s8463391772401140188tate_a @ Ys ) )
=> ( ( nth_Pr7598562053491293850tate_a @ ( zip_a_state_a2 @ Xs @ Ys ) @ I )
= ( produc1224139502141355779tate_a @ ( nth_a @ Xs @ I ) @ ( nth_state_a2 @ Ys @ I ) ) ) ) ) ).
% nth_zip
thf(fact_791_nth__zip,axiom,
! [I: nat,Xs: list_a,Ys: list_state_a2] :
( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
=> ( ( ord_less_nat @ I @ ( size_s7859192958365828515tate_a @ Ys ) )
=> ( ( nth_Pr2816242342134088033tate_a @ ( zip_a_state_a @ Xs @ Ys ) @ I )
= ( produc8641956578966763338tate_a @ ( nth_a @ Xs @ I ) @ ( nth_state_a @ Ys @ I ) ) ) ) ) ).
% nth_zip
thf(fact_792_listrelp__listrel__eq,axiom,
! [R: set_Product_prod_a_a] :
( ( listrelp_a_a
@ ^ [X4: a,Y7: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y7 ) @ R ) )
= ( ^ [X4: list_a,Y7: list_a] : ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X4 @ Y7 ) @ ( listrel_a_a @ R ) ) ) ) ).
% listrelp_listrel_eq
thf(fact_793_listrelp__listrel__eq,axiom,
! [R: set_Pr4048851178543822343list_a] :
( ( listre657891920412899263list_a
@ ^ [X4: list_a,Y7: list_a] : ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ X4 @ Y7 ) @ R ) )
= ( ^ [X4: list_list_a,Y7: list_list_a] : ( member1318342207407915856list_a @ ( produc8696003437204565271list_a @ X4 @ Y7 ) @ ( listre6772471554020304241list_a @ R ) ) ) ) ).
% listrelp_listrel_eq
thf(fact_794_listrelp__listrel__eq,axiom,
! [R: set_Pr1464008215722202041_a_nat] :
( ( listre1698544965105253845_a_nat
@ ^ [X4: states_a,Y7: nat] : ( member6483677129516672026_a_nat @ ( produc1877401315875745917_a_nat @ X4 @ Y7 ) @ R ) )
= ( ^ [X4: list_states_a,Y7: list_nat] : ( member6795120827800458928st_nat @ ( produc327714349733490451st_nat @ X4 @ Y7 ) @ ( listrel_states_a_nat @ R ) ) ) ) ).
% listrelp_listrel_eq
thf(fact_795_listrelp__listrel__eq,axiom,
! [R: set_Pr6306228930610421491tate_a] :
( ( listrelp_a_state_a2
@ ^ [X4: a,Y7: state_a] : ( member8547378267715833660tate_a @ ( produc1224139502141355779tate_a @ X4 @ Y7 ) @ R ) )
= ( ^ [X4: list_a,Y7: list_state_a] : ( member4112945611203173692tate_a @ ( produc1997082749353321475tate_a @ X4 @ Y7 ) @ ( listrel_a_state_a2 @ R ) ) ) ) ).
% listrelp_listrel_eq
thf(fact_796_listrelp__listrel__eq,axiom,
! [R: set_Pr4275752383657305402tate_a] :
( ( listrelp_a_state_a
@ ^ [X4: a,Y7: state_a2] : ( member3175992478928454403tate_a @ ( produc8641956578966763338tate_a @ X4 @ Y7 ) @ R ) )
= ( ^ [X4: list_a,Y7: list_state_a2] : ( member6123284207288203267tate_a @ ( produc3968967176812022602tate_a @ X4 @ Y7 ) @ ( listrel_a_state_a @ R ) ) ) ) ).
% listrelp_listrel_eq
thf(fact_797_zip__eq__Nil__iff,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( zip_a_a @ Xs @ Ys )
= nil_Product_prod_a_a )
= ( ( Xs = nil_a )
| ( Ys = nil_a ) ) ) ).
% zip_eq_Nil_iff
thf(fact_798_Nil__eq__zip__iff,axiom,
! [Xs: list_a,Ys: list_a] :
( ( nil_Product_prod_a_a
= ( zip_a_a @ Xs @ Ys ) )
= ( ( Xs = nil_a )
| ( Ys = nil_a ) ) ) ).
% Nil_eq_zip_iff
thf(fact_799_zip__Cons__Cons,axiom,
! [X: a,Xs: list_a,Y: a,Ys: list_a] :
( ( zip_a_a @ ( cons_a @ X @ Xs ) @ ( cons_a @ Y @ Ys ) )
= ( cons_P7316939126706565853od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( zip_a_a @ Xs @ Ys ) ) ) ).
% zip_Cons_Cons
thf(fact_800_zip__Cons__Cons,axiom,
! [X: list_a,Xs: list_list_a,Y: list_a,Ys: list_list_a] :
( ( zip_list_a_list_a @ ( cons_list_a @ X @ Xs ) @ ( cons_list_a @ Y @ Ys ) )
= ( cons_P5184657343811988189list_a @ ( produc6837034575241423639list_a @ X @ Y ) @ ( zip_list_a_list_a @ Xs @ Ys ) ) ) ).
% zip_Cons_Cons
thf(fact_801_zip__Cons__Cons,axiom,
! [X: states_a,Xs: list_states_a,Y: nat,Ys: list_nat] :
( ( zip_states_a_nat @ ( cons_states_a @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) )
= ( cons_P1204124355475727053_a_nat @ ( produc1877401315875745917_a_nat @ X @ Y ) @ ( zip_states_a_nat @ Xs @ Ys ) ) ) ).
% zip_Cons_Cons
thf(fact_802_zip__Cons__Cons,axiom,
! [X: a,Xs: list_a,Y: state_a,Ys: list_state_a] :
( ( zip_a_state_a2 @ ( cons_a @ X @ Xs ) @ ( cons_state_a2 @ Y @ Ys ) )
= ( cons_P5520042648066362825tate_a @ ( produc1224139502141355779tate_a @ X @ Y ) @ ( zip_a_state_a2 @ Xs @ Ys ) ) ) ).
% zip_Cons_Cons
thf(fact_803_zip__Cons__Cons,axiom,
! [X: a,Xs: list_a,Y: state_a2,Ys: list_state_a2] :
( ( zip_a_state_a @ ( cons_a @ X @ Xs ) @ ( cons_state_a @ Y @ Ys ) )
= ( cons_P6411715633656068112tate_a @ ( produc8641956578966763338tate_a @ X @ Y ) @ ( zip_a_state_a @ Xs @ Ys ) ) ) ).
% zip_Cons_Cons
thf(fact_804_zip__Cons,axiom,
! [Xs: list_list_a,Y: list_a,Ys: list_list_a] :
( ( zip_list_a_list_a @ Xs @ ( cons_list_a @ Y @ Ys ) )
= ( case_l5846627494734287741list_a @ nil_Pr3188421586756112173list_a
@ ^ [Z5: list_a,Zs3: list_list_a] : ( cons_P5184657343811988189list_a @ ( produc6837034575241423639list_a @ Z5 @ Y ) @ ( zip_list_a_list_a @ Zs3 @ Ys ) )
@ Xs ) ) ).
% zip_Cons
thf(fact_805_zip__Cons,axiom,
! [Xs: list_states_a,Y: nat,Ys: list_nat] :
( ( zip_states_a_nat @ Xs @ ( cons_nat @ Y @ Ys ) )
= ( case_l3969150313965745097ates_a @ nil_Pr5685302264115950205_a_nat
@ ^ [Z5: states_a,Zs3: list_states_a] : ( cons_P1204124355475727053_a_nat @ ( produc1877401315875745917_a_nat @ Z5 @ Y ) @ ( zip_states_a_nat @ Zs3 @ Ys ) )
@ Xs ) ) ).
% zip_Cons
thf(fact_806_zip__Cons,axiom,
! [Xs: list_a,Y: state_a,Ys: list_state_a] :
( ( zip_a_state_a2 @ Xs @ ( cons_state_a2 @ Y @ Ys ) )
= ( case_l6193896632244483467te_a_a @ nil_Pr821976538969420313tate_a
@ ^ [Z5: a,Zs3: list_a] : ( cons_P5520042648066362825tate_a @ ( produc1224139502141355779tate_a @ Z5 @ Y ) @ ( zip_a_state_a2 @ Zs3 @ Ys ) )
@ Xs ) ) ).
% zip_Cons
thf(fact_807_zip__Cons,axiom,
! [Xs: list_a,Y: state_a2,Ys: list_state_a2] :
( ( zip_a_state_a @ Xs @ ( cons_state_a @ Y @ Ys ) )
= ( case_l5203556057060921220te_a_a @ nil_Pr4916735315485279328tate_a
@ ^ [Z5: a,Zs3: list_a] : ( cons_P6411715633656068112tate_a @ ( produc8641956578966763338tate_a @ Z5 @ Y ) @ ( zip_a_state_a @ Zs3 @ Ys ) )
@ Xs ) ) ).
% zip_Cons
thf(fact_808_zip__Cons1,axiom,
! [X: list_a,Xs: list_list_a,Ys: list_list_a] :
( ( zip_list_a_list_a @ ( cons_list_a @ X @ Xs ) @ Ys )
= ( case_l5846627494734287741list_a @ nil_Pr3188421586756112173list_a
@ ^ [Y7: list_a,Ys2: list_list_a] : ( cons_P5184657343811988189list_a @ ( produc6837034575241423639list_a @ X @ Y7 ) @ ( zip_list_a_list_a @ Xs @ Ys2 ) )
@ Ys ) ) ).
% zip_Cons1
thf(fact_809_zip__Cons1,axiom,
! [X: states_a,Xs: list_states_a,Ys: list_nat] :
( ( zip_states_a_nat @ ( cons_states_a @ X @ Xs ) @ Ys )
= ( case_l5899030235993850187at_nat @ nil_Pr5685302264115950205_a_nat
@ ^ [Y7: nat,Ys2: list_nat] : ( cons_P1204124355475727053_a_nat @ ( produc1877401315875745917_a_nat @ X @ Y7 ) @ ( zip_states_a_nat @ Xs @ Ys2 ) )
@ Ys ) ) ).
% zip_Cons1
thf(fact_810_zip__Cons1,axiom,
! [X: a,Xs: list_a,Ys: list_state_a] :
( ( zip_a_state_a2 @ ( cons_a @ X @ Xs ) @ Ys )
= ( case_l4389968015973919095tate_a @ nil_Pr821976538969420313tate_a
@ ^ [Y7: state_a,Ys2: list_state_a] : ( cons_P5520042648066362825tate_a @ ( produc1224139502141355779tate_a @ X @ Y7 ) @ ( zip_a_state_a2 @ Xs @ Ys2 ) )
@ Ys ) ) ).
% zip_Cons1
thf(fact_811_zip__Cons1,axiom,
! [X: a,Xs: list_a,Ys: list_state_a2] :
( ( zip_a_state_a @ ( cons_a @ X @ Xs ) @ Ys )
= ( case_l8234815367818063159tate_a @ nil_Pr4916735315485279328tate_a
@ ^ [Y7: state_a2,Ys2: list_state_a2] : ( cons_P6411715633656068112tate_a @ ( produc8641956578966763338tate_a @ X @ Y7 ) @ ( zip_a_state_a @ Xs @ Ys2 ) )
@ Ys ) ) ).
% zip_Cons1
thf(fact_812_zip__eq__ConsE,axiom,
! [Xs: list_a,Ys: list_a,Xy: product_prod_a_a,Xys: list_P1396940483166286381od_a_a] :
( ( ( zip_a_a @ Xs @ Ys )
= ( cons_P7316939126706565853od_a_a @ Xy @ Xys ) )
=> ~ ! [X5: a,Xs4: list_a] :
( ( Xs
= ( cons_a @ X5 @ Xs4 ) )
=> ! [Y4: a,Ys4: list_a] :
( ( Ys
= ( cons_a @ Y4 @ Ys4 ) )
=> ( ( Xy
= ( product_Pair_a_a @ X5 @ Y4 ) )
=> ( Xys
!= ( zip_a_a @ Xs4 @ Ys4 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_813_zip__eq__ConsE,axiom,
! [Xs: list_list_a,Ys: list_list_a,Xy: produc9164743771328383783list_a,Xys: list_P321204300973800749list_a] :
( ( ( zip_list_a_list_a @ Xs @ Ys )
= ( cons_P5184657343811988189list_a @ Xy @ Xys ) )
=> ~ ! [X5: list_a,Xs4: list_list_a] :
( ( Xs
= ( cons_list_a @ X5 @ Xs4 ) )
=> ! [Y4: list_a,Ys4: list_list_a] :
( ( Ys
= ( cons_list_a @ Y4 @ Ys4 ) )
=> ( ( Xy
= ( produc6837034575241423639list_a @ X5 @ Y4 ) )
=> ( Xys
!= ( zip_list_a_list_a @ Xs4 @ Ys4 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_814_zip__eq__ConsE,axiom,
! [Xs: list_states_a,Ys: list_nat,Xy: produc1571854377283420419_a_nat,Xys: list_P4144771487928076819_a_nat] :
( ( ( zip_states_a_nat @ Xs @ Ys )
= ( cons_P1204124355475727053_a_nat @ Xy @ Xys ) )
=> ~ ! [X5: states_a,Xs4: list_states_a] :
( ( Xs
= ( cons_states_a @ X5 @ Xs4 ) )
=> ! [Y4: nat,Ys4: list_nat] :
( ( Ys
= ( cons_nat @ Y4 @ Ys4 ) )
=> ( ( Xy
= ( produc1877401315875745917_a_nat @ X5 @ Y4 ) )
=> ( Xys
!= ( zip_states_a_nat @ Xs4 @ Ys4 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_815_zip__eq__ConsE,axiom,
! [Xs: list_a,Ys: list_state_a,Xy: produc7589950997499123219tate_a,Xys: list_P4482599289824689689tate_a] :
( ( ( zip_a_state_a2 @ Xs @ Ys )
= ( cons_P5520042648066362825tate_a @ Xy @ Xys ) )
=> ~ ! [X5: a,Xs4: list_a] :
( ( Xs
= ( cons_a @ X5 @ Xs4 ) )
=> ! [Y4: state_a,Ys4: list_state_a] :
( ( Ys
= ( cons_state_a2 @ Y4 @ Ys4 ) )
=> ( ( Xy
= ( produc1224139502141355779tate_a @ X5 @ Y4 ) )
=> ( Xys
!= ( zip_a_state_a2 @ Xs4 @ Ys4 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_816_zip__eq__ConsE,axiom,
! [Xs: list_a,Ys: list_state_a2,Xy: produc6972303929186420058tate_a,Xys: list_P6747491281921308512tate_a] :
( ( ( zip_a_state_a @ Xs @ Ys )
= ( cons_P6411715633656068112tate_a @ Xy @ Xys ) )
=> ~ ! [X5: a,Xs4: list_a] :
( ( Xs
= ( cons_a @ X5 @ Xs4 ) )
=> ! [Y4: state_a2,Ys4: list_state_a2] :
( ( Ys
= ( cons_state_a @ Y4 @ Ys4 ) )
=> ( ( Xy
= ( produc8641956578966763338tate_a @ X5 @ Y4 ) )
=> ( Xys
!= ( zip_a_state_a @ Xs4 @ Ys4 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_817_zip__update,axiom,
! [Xs: list_list_a,I: nat,X: list_a,Ys: list_list_a,Y: list_a] :
( ( zip_list_a_list_a @ ( list_update_list_a @ Xs @ I @ X ) @ ( list_update_list_a @ Ys @ I @ Y ) )
= ( list_u6458906768619699605list_a @ ( zip_list_a_list_a @ Xs @ Ys ) @ I @ ( produc6837034575241423639list_a @ X @ Y ) ) ) ).
% zip_update
thf(fact_818_zip__update,axiom,
! [Xs: list_states_a,I: nat,X: states_a,Ys: list_nat,Y: nat] :
( ( zip_states_a_nat @ ( list_update_states_a @ Xs @ I @ X ) @ ( list_update_nat @ Ys @ I @ Y ) )
= ( list_u4184495706060574229_a_nat @ ( zip_states_a_nat @ Xs @ Ys ) @ I @ ( produc1877401315875745917_a_nat @ X @ Y ) ) ) ).
% zip_update
thf(fact_819_zip__update,axiom,
! [Xs: list_a,I: nat,X: a,Ys: list_state_a,Y: state_a] :
( ( zip_a_state_a2 @ ( list_update_a @ Xs @ I @ X ) @ ( list_update_state_a2 @ Ys @ I @ Y ) )
= ( list_u2256020193368096897tate_a @ ( zip_a_state_a2 @ Xs @ Ys ) @ I @ ( produc1224139502141355779tate_a @ X @ Y ) ) ) ).
% zip_update
thf(fact_820_zip__update,axiom,
! [Xs: list_a,I: nat,X: a,Ys: list_state_a2,Y: state_a2] :
( ( zip_a_state_a @ ( list_update_a @ Xs @ I @ X ) @ ( list_update_state_a @ Ys @ I @ Y ) )
= ( list_u2732569323712218824tate_a @ ( zip_a_state_a @ Xs @ Ys ) @ I @ ( produc8641956578966763338tate_a @ X @ Y ) ) ) ).
% zip_update
thf(fact_821_hd__zip,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs != nil_a )
=> ( ( Ys != nil_a )
=> ( ( hd_Product_prod_a_a @ ( zip_a_a @ Xs @ Ys ) )
= ( product_Pair_a_a @ ( hd_a @ Xs ) @ ( hd_a @ Ys ) ) ) ) ) ).
% hd_zip
thf(fact_822_hd__zip,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( Xs != nil_list_a )
=> ( ( Ys != nil_list_a )
=> ( ( hd_Pro6433485837049129490list_a @ ( zip_list_a_list_a @ Xs @ Ys ) )
= ( produc6837034575241423639list_a @ ( hd_list_a @ Xs ) @ ( hd_list_a @ Ys ) ) ) ) ) ).
% hd_zip
thf(fact_823_hd__zip,axiom,
! [Xs: list_states_a,Ys: list_nat] :
( ( Xs != nil_states_a )
=> ( ( Ys != nil_nat )
=> ( ( hd_Pro4761346146552465112_a_nat @ ( zip_states_a_nat @ Xs @ Ys ) )
= ( produc1877401315875745917_a_nat @ ( hd_states_a @ Xs ) @ ( hd_nat @ Ys ) ) ) ) ) ).
% hd_zip
thf(fact_824_hd__zip,axiom,
! [Xs: list_a,Ys: list_state_a] :
( ( Xs != nil_a )
=> ( ( Ys != nil_state_a2 )
=> ( ( hd_Pro2196137027883835646tate_a @ ( zip_a_state_a2 @ Xs @ Ys ) )
= ( produc1224139502141355779tate_a @ ( hd_a @ Xs ) @ ( hd_state_a2 @ Ys ) ) ) ) ) ).
% hd_zip
thf(fact_825_hd__zip,axiom,
! [Xs: list_a,Ys: list_state_a2] :
( ( Xs != nil_a )
=> ( ( Ys != nil_state_a )
=> ( ( hd_Pro8463085911274001861tate_a @ ( zip_a_state_a @ Xs @ Ys ) )
= ( produc8641956578966763338tate_a @ ( hd_a @ Xs ) @ ( hd_state_a @ Ys ) ) ) ) ) ).
% hd_zip
thf(fact_826_listrelp_ONil,axiom,
! [R: a > a > $o] : ( listrelp_a_a @ R @ nil_a @ nil_a ) ).
% listrelp.Nil
thf(fact_827_listrelp_OCons,axiom,
! [R: a > a > $o,X: a,Y: a,Xs: list_a,Ys: list_a] :
( ( R @ X @ Y )
=> ( ( listrelp_a_a @ R @ Xs @ Ys )
=> ( listrelp_a_a @ R @ ( cons_a @ X @ Xs ) @ ( cons_a @ Y @ Ys ) ) ) ) ).
% listrelp.Cons
thf(fact_828_listrelp_Osimps,axiom,
( listrelp_a_a
= ( ^ [R3: a > a > $o,A12: list_a,A23: list_a] :
( ( ( A12 = nil_a )
& ( A23 = nil_a ) )
| ? [X4: a,Y7: a,Xs3: list_a,Ys2: list_a] :
( ( A12
= ( cons_a @ X4 @ Xs3 ) )
& ( A23
= ( cons_a @ Y7 @ Ys2 ) )
& ( R3 @ X4 @ Y7 )
& ( listrelp_a_a @ R3 @ Xs3 @ Ys2 ) ) ) ) ) ).
% listrelp.simps
thf(fact_829_listrelp_Ocases,axiom,
! [R: a > a > $o,A1: list_a,A22: list_a] :
( ( listrelp_a_a @ R @ A1 @ A22 )
=> ( ( ( A1 = nil_a )
=> ( A22 != nil_a ) )
=> ~ ! [X5: a,Y4: a,Xs2: list_a] :
( ( A1
= ( cons_a @ X5 @ Xs2 ) )
=> ! [Ys3: list_a] :
( ( A22
= ( cons_a @ Y4 @ Ys3 ) )
=> ( ( R @ X5 @ Y4 )
=> ~ ( listrelp_a_a @ R @ Xs2 @ Ys3 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_830_append__butlast__last__id,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( append_a @ ( butlast_a @ Xs ) @ ( cons_a @ ( last_a @ Xs ) @ nil_a ) )
= Xs ) ) ).
% append_butlast_last_id
thf(fact_831_last__list__update,axiom,
! [Xs: list_a,K: nat,X: a] :
( ( Xs != nil_a )
=> ( ( ( K
= ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) )
=> ( ( last_a @ ( list_update_a @ Xs @ K @ X ) )
= X ) )
& ( ( K
!= ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) )
=> ( ( last_a @ ( list_update_a @ Xs @ K @ X ) )
= ( last_a @ Xs ) ) ) ) ) ).
% last_list_update
thf(fact_832_last__appendL,axiom,
! [Ys: list_a,Xs: list_a] :
( ( Ys = nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys ) )
= ( last_a @ Xs ) ) ) ).
% last_appendL
thf(fact_833_last__appendR,axiom,
! [Ys: list_a,Xs: list_a] :
( ( Ys != nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys ) )
= ( last_a @ Ys ) ) ) ).
% last_appendR
thf(fact_834_last__snoc,axiom,
! [Xs: list_a,X: a] :
( ( last_a @ ( append_a @ Xs @ ( cons_a @ X @ nil_a ) ) )
= X ) ).
% last_snoc
thf(fact_835_last__zip,axiom,
! [Xs: list_a,Ys: list_a] :
( ( Xs != nil_a )
=> ( ( Ys != nil_a )
=> ( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( last_P8790725268278465478od_a_a @ ( zip_a_a @ Xs @ Ys ) )
= ( product_Pair_a_a @ ( last_a @ Xs ) @ ( last_a @ Ys ) ) ) ) ) ) ).
% last_zip
thf(fact_836_last__zip,axiom,
! [Xs: list_list_a,Ys: list_list_a] :
( ( Xs != nil_list_a )
=> ( ( Ys != nil_list_a )
=> ( ( ( size_s349497388124573686list_a @ Xs )
= ( size_s349497388124573686list_a @ Ys ) )
=> ( ( last_P2616948663241161414list_a @ ( zip_list_a_list_a @ Xs @ Ys ) )
= ( produc6837034575241423639list_a @ ( last_list_a @ Xs ) @ ( last_list_a @ Ys ) ) ) ) ) ) ).
% last_zip
thf(fact_837_last__zip,axiom,
! [Xs: list_states_a,Ys: list_nat] :
( ( Xs != nil_states_a )
=> ( ( Ys != nil_nat )
=> ( ( ( size_s3891197933023997302ates_a @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( last_P1521129688936790820_a_nat @ ( zip_states_a_nat @ Xs @ Ys ) )
= ( produc1877401315875745917_a_nat @ ( last_states_a @ Xs ) @ ( last_nat @ Ys ) ) ) ) ) ) ).
% last_zip
thf(fact_838_last__zip,axiom,
! [Xs: list_a,Ys: list_state_a] :
( ( Xs != nil_a )
=> ( ( Ys != nil_state_a2 )
=> ( ( ( size_size_list_a @ Xs )
= ( size_s8463391772401140188tate_a @ Ys ) )
=> ( ( last_P6377108300570313138tate_a @ ( zip_a_state_a2 @ Xs @ Ys ) )
= ( produc1224139502141355779tate_a @ ( last_a @ Xs ) @ ( last_state_a2 @ Ys ) ) ) ) ) ) ).
% last_zip
thf(fact_839_last__zip,axiom,
! [Xs: list_a,Ys: list_state_a2] :
( ( Xs != nil_a )
=> ( ( Ys != nil_state_a )
=> ( ( ( size_size_list_a @ Xs )
= ( size_s7859192958365828515tate_a @ Ys ) )
=> ( ( last_P7000629880413671545tate_a @ ( zip_a_state_a @ Xs @ Ys ) )
= ( produc8641956578966763338tate_a @ ( last_a @ Xs ) @ ( last_state_a @ Ys ) ) ) ) ) ) ).
% last_zip
thf(fact_840_last__append,axiom,
! [Ys: list_a,Xs: list_a] :
( ( ( Ys = nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys ) )
= ( last_a @ Xs ) ) )
& ( ( Ys != nil_a )
=> ( ( last_a @ ( append_a @ Xs @ Ys ) )
= ( last_a @ Ys ) ) ) ) ).
% last_append
thf(fact_841_longest__common__suffix,axiom,
! [Xs: list_a,Ys: list_a] :
? [Ss: list_a,Xs4: list_a,Ys4: list_a] :
( ( Xs
= ( append_a @ Xs4 @ Ss ) )
& ( Ys
= ( append_a @ Ys4 @ Ss ) )
& ( ( Xs4 = nil_a )
| ( Ys4 = nil_a )
| ( ( last_a @ Xs4 )
!= ( last_a @ Ys4 ) ) ) ) ).
% longest_common_suffix
thf(fact_842_last_Osimps,axiom,
! [Xs: list_a,X: a] :
( ( ( Xs = nil_a )
=> ( ( last_a @ ( cons_a @ X @ Xs ) )
= X ) )
& ( ( Xs != nil_a )
=> ( ( last_a @ ( cons_a @ X @ Xs ) )
= ( last_a @ Xs ) ) ) ) ).
% last.simps
thf(fact_843_last__ConsL,axiom,
! [Xs: list_a,X: a] :
( ( Xs = nil_a )
=> ( ( last_a @ ( cons_a @ X @ Xs ) )
= X ) ) ).
% last_ConsL
thf(fact_844_last__ConsR,axiom,
! [Xs: list_a,X: a] :
( ( Xs != nil_a )
=> ( ( last_a @ ( cons_a @ X @ Xs ) )
= ( last_a @ Xs ) ) ) ).
% last_ConsR
thf(fact_845_last__tl,axiom,
! [Xs: list_a] :
( ( ( Xs = nil_a )
| ( ( tl_a @ Xs )
!= nil_a ) )
=> ( ( last_a @ ( tl_a @ Xs ) )
= ( last_a @ Xs ) ) ) ).
% last_tl
thf(fact_846_hd__Nil__eq__last,axiom,
( ( hd_a @ nil_a )
= ( last_a @ nil_a ) ) ).
% hd_Nil_eq_last
thf(fact_847_snoc__eq__iff__butlast,axiom,
! [Xs: list_a,X: a,Ys: list_a] :
( ( ( append_a @ Xs @ ( cons_a @ X @ nil_a ) )
= Ys )
= ( ( Ys != nil_a )
& ( ( butlast_a @ Ys )
= Xs )
& ( ( last_a @ Ys )
= X ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_848_last__conv__nth,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( last_a @ Xs )
= ( nth_a @ Xs @ ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ one_one_nat ) ) ) ) ).
% last_conv_nth
thf(fact_849_take__last__length,axiom,
! [Xs: list_a] :
( ( ( take_a @ ( suc @ zero_zero_nat ) @ ( rev_a @ Xs ) )
= ( cons_a @ ( last_a @ Xs ) @ nil_a ) )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( size_size_list_a @ Xs ) ) ) ).
% take_last_length
thf(fact_850_subseqs_Osimps_I1_J,axiom,
( ( subseqs_a @ nil_a )
= ( cons_list_a @ nil_a @ nil_list_a ) ) ).
% subseqs.simps(1)
thf(fact_851_rev__is__Nil__conv,axiom,
! [Xs: list_a] :
( ( ( rev_a @ Xs )
= nil_a )
= ( Xs = nil_a ) ) ).
% rev_is_Nil_conv
thf(fact_852_Nil__is__rev__conv,axiom,
! [Xs: list_a] :
( ( nil_a
= ( rev_a @ Xs ) )
= ( Xs = nil_a ) ) ).
% Nil_is_rev_conv
thf(fact_853_rev__singleton__conv,axiom,
! [Xs: list_a,X: a] :
( ( ( rev_a @ Xs )
= ( cons_a @ X @ nil_a ) )
= ( Xs
= ( cons_a @ X @ nil_a ) ) ) ).
% rev_singleton_conv
thf(fact_854_singleton__rev__conv,axiom,
! [X: a,Xs: list_a] :
( ( ( cons_a @ X @ nil_a )
= ( rev_a @ Xs ) )
= ( ( cons_a @ X @ nil_a )
= Xs ) ) ).
% singleton_rev_conv
thf(fact_855_rev__eq__Cons__iff,axiom,
! [Xs: list_a,Y: a,Ys: list_a] :
( ( ( rev_a @ Xs )
= ( cons_a @ Y @ Ys ) )
= ( Xs
= ( append_a @ ( rev_a @ Ys ) @ ( cons_a @ Y @ nil_a ) ) ) ) ).
% rev_eq_Cons_iff
thf(fact_856_rev__app__single,axiom,
! [Xs: list_a,X: a] :
( ( append_a @ ( rev_a @ Xs ) @ ( cons_a @ X @ nil_a ) )
= ( rev_a @ ( cons_a @ X @ Xs ) ) ) ).
% rev_app_single
thf(fact_857_rev_Osimps_I1_J,axiom,
( ( rev_a @ nil_a )
= nil_a ) ).
% rev.simps(1)
thf(fact_858_rev__tl__hd,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( append_a @ ( rev_a @ ( tl_a @ Xs ) ) @ ( cons_a @ ( hd_a @ Xs ) @ nil_a ) )
= ( rev_a @ Xs ) ) ) ).
% rev_tl_hd
thf(fact_859_take__last,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( take_a @ one_one_nat @ ( rev_a @ Xs ) )
= ( cons_a @ ( last_a @ Xs ) @ nil_a ) ) ) ).
% take_last
thf(fact_860_last__drop__rev,axiom,
! [Xs: list_a] :
( ( Xs != nil_a )
=> ( ( cons_a @ ( last_a @ Xs ) @ ( drop_a @ one_one_nat @ ( rev_a @ Xs ) ) )
= ( rev_a @ Xs ) ) ) ).
% last_drop_rev
thf(fact_861_distinct__adj__append__iff,axiom,
! [Xs: list_a,Ys: list_a] :
( ( distinct_adj_a @ ( append_a @ Xs @ Ys ) )
= ( ( distinct_adj_a @ Xs )
& ( distinct_adj_a @ Ys )
& ( ( Xs = nil_a )
| ( Ys = nil_a )
| ( ( last_a @ Xs )
!= ( hd_a @ Ys ) ) ) ) ) ).
% distinct_adj_append_iff
thf(fact_862_lenlex__conv,axiom,
( lenlex_a
= ( ^ [R3: set_Product_prod_a_a] :
( collec943055143889122450list_a
@ ( produc8172378796822260076st_a_o
@ ^ [Xs3: list_a,Ys2: list_a] :
( ( ord_less_nat @ ( size_size_list_a @ Xs3 ) @ ( size_size_list_a @ Ys2 ) )
| ( ( ( size_size_list_a @ Xs3 )
= ( size_size_list_a @ Ys2 ) )
& ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs3 @ Ys2 ) @ ( lex_a @ R3 ) ) ) ) ) ) ) ) ).
% lenlex_conv
thf(fact_863_distinct__adj__Cons__Cons,axiom,
! [X: a,Y: a,Xs: list_a] :
( ( distinct_adj_a @ ( cons_a @ X @ ( cons_a @ Y @ Xs ) ) )
= ( ( X != Y )
& ( distinct_adj_a @ ( cons_a @ Y @ Xs ) ) ) ) ).
% distinct_adj_Cons_Cons
thf(fact_864_case__prodI,axiom,
! [F: list_a > list_a > $o,A: list_a,B: list_a] :
( ( F @ A @ B )
=> ( produc8172378796822260076st_a_o @ F @ ( produc6837034575241423639list_a @ A @ B ) ) ) ).
% case_prodI
thf(fact_865_case__prodI,axiom,
! [F: states_a > nat > $o,A: states_a,B: nat] :
( ( F @ A @ B )
=> ( produc6678156928310571630_nat_o @ F @ ( produc1877401315875745917_a_nat @ A @ B ) ) ) ).
% case_prodI
thf(fact_866_case__prodI,axiom,
! [F: a > state_a > $o,A: a,B: state_a] :
( ( F @ A @ B )
=> ( produc9121072869769030272te_a_o @ F @ ( produc1224139502141355779tate_a @ A @ B ) ) ) ).
% case_prodI
thf(fact_867_case__prodI,axiom,
! [F: a > state_a2 > $o,A: a,B: state_a2] :
( ( F @ A @ B )
=> ( produc3717778009801768057te_a_o @ F @ ( produc8641956578966763338tate_a @ A @ B ) ) ) ).
% case_prodI
thf(fact_868_case__prodI2,axiom,
! [P2: produc9164743771328383783list_a,C2: list_a > list_a > $o] :
( ! [A3: list_a,B3: list_a] :
( ( P2
= ( produc6837034575241423639list_a @ A3 @ B3 ) )
=> ( C2 @ A3 @ B3 ) )
=> ( produc8172378796822260076st_a_o @ C2 @ P2 ) ) ).
% case_prodI2
thf(fact_869_case__prodI2,axiom,
! [P2: produc1571854377283420419_a_nat,C2: states_a > nat > $o] :
( ! [A3: states_a,B3: nat] :
( ( P2
= ( produc1877401315875745917_a_nat @ A3 @ B3 ) )
=> ( C2 @ A3 @ B3 ) )
=> ( produc6678156928310571630_nat_o @ C2 @ P2 ) ) ).
% case_prodI2
thf(fact_870_case__prodI2,axiom,
! [P2: produc7589950997499123219tate_a,C2: a > state_a > $o] :
( ! [A3: a,B3: state_a] :
( ( P2
= ( produc1224139502141355779tate_a @ A3 @ B3 ) )
=> ( C2 @ A3 @ B3 ) )
=> ( produc9121072869769030272te_a_o @ C2 @ P2 ) ) ).
% case_prodI2
thf(fact_871_case__prodI2,axiom,
! [P2: produc6972303929186420058tate_a,C2: a > state_a2 > $o] :
( ! [A3: a,B3: state_a2] :
( ( P2
= ( produc8641956578966763338tate_a @ A3 @ B3 ) )
=> ( C2 @ A3 @ B3 ) )
=> ( produc3717778009801768057te_a_o @ C2 @ P2 ) ) ).
% case_prodI2
thf(fact_872_case__prodE,axiom,
! [C2: list_a > list_a > $o,P2: produc9164743771328383783list_a] :
( ( produc8172378796822260076st_a_o @ C2 @ P2 )
=> ~ ! [X5: list_a,Y4: list_a] :
( ( P2
= ( produc6837034575241423639list_a @ X5 @ Y4 ) )
=> ~ ( C2 @ X5 @ Y4 ) ) ) ).
% case_prodE
thf(fact_873_case__prodE,axiom,
! [C2: states_a > nat > $o,P2: produc1571854377283420419_a_nat] :
( ( produc6678156928310571630_nat_o @ C2 @ P2 )
=> ~ ! [X5: states_a,Y4: nat] :
( ( P2
= ( produc1877401315875745917_a_nat @ X5 @ Y4 ) )
=> ~ ( C2 @ X5 @ Y4 ) ) ) ).
% case_prodE
thf(fact_874_case__prodE,axiom,
! [C2: a > state_a > $o,P2: produc7589950997499123219tate_a] :
( ( produc9121072869769030272te_a_o @ C2 @ P2 )
=> ~ ! [X5: a,Y4: state_a] :
( ( P2
= ( produc1224139502141355779tate_a @ X5 @ Y4 ) )
=> ~ ( C2 @ X5 @ Y4 ) ) ) ).
% case_prodE
thf(fact_875_case__prodE,axiom,
! [C2: a > state_a2 > $o,P2: produc6972303929186420058tate_a] :
( ( produc3717778009801768057te_a_o @ C2 @ P2 )
=> ~ ! [X5: a,Y4: state_a2] :
( ( P2
= ( produc8641956578966763338tate_a @ X5 @ Y4 ) )
=> ~ ( C2 @ X5 @ Y4 ) ) ) ).
% case_prodE
thf(fact_876_case__prodD,axiom,
! [F: list_a > list_a > $o,A: list_a,B: list_a] :
( ( produc8172378796822260076st_a_o @ F @ ( produc6837034575241423639list_a @ A @ B ) )
=> ( F @ A @ B ) ) ).
% case_prodD
thf(fact_877_case__prodD,axiom,
! [F: states_a > nat > $o,A: states_a,B: nat] :
( ( produc6678156928310571630_nat_o @ F @ ( produc1877401315875745917_a_nat @ A @ B ) )
=> ( F @ A @ B ) ) ).
% case_prodD
thf(fact_878_case__prodD,axiom,
! [F: a > state_a > $o,A: a,B: state_a] :
( ( produc9121072869769030272te_a_o @ F @ ( produc1224139502141355779tate_a @ A @ B ) )
=> ( F @ A @ B ) ) ).
% case_prodD
thf(fact_879_case__prodD,axiom,
! [F: a > state_a2 > $o,A: a,B: state_a2] :
( ( produc3717778009801768057te_a_o @ F @ ( produc8641956578966763338tate_a @ A @ B ) )
=> ( F @ A @ B ) ) ).
% case_prodD
thf(fact_880_distinct__adj__singleton,axiom,
! [X: a] : ( distinct_adj_a @ ( cons_a @ X @ nil_a ) ) ).
% distinct_adj_singleton
% 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__List__Olist_Itf__a_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X: list_a,Y: list_a] :
( ( if_list_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__List__Olist_Itf__a_J_T,axiom,
! [X: list_a,Y: list_a] :
( ( if_list_a @ $true @ X @ Y )
= X ) ).
% Conjectures (3)
thf(conj_0,hypothesis,
type_i8221491762852169479ates_a @ ( states_a2 @ dir @ big @ small ) ).
thf(conj_1,hypothesis,
( ( type_s4923920245906622843ates_a @ ( states_a2 @ dir @ big @ small ) )
= ( states_a2 @ dir2 @ big2 @ small2 ) ) ).
thf(conj_2,conjecture,
( ( size_size_state_a2 @ small2 )
= ( size_size_state_a2 @ small ) ) ).
%------------------------------------------------------------------------------