TPTP Problem File: DAT151^1.p
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%------------------------------------------------------------------------------
% File : DAT151^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Coinductive stream 392
% Version : [Bla16] axioms : Especial.
% English :
% Refs : [Loc10] Lochbihler (2010), Coinductive
% : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% Source : [Bla16]
% Names : coinductive_stream__392.p [Bla16]
% Status : Theorem
% Rating : 1.00 v7.5.0, 0.33 v7.2.0, 0.50 v7.1.0
% Syntax : Number of formulae : 417 ( 191 unt; 68 typ; 0 def)
% Number of atoms : 808 ( 346 equ; 0 cnn)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 3123 ( 81 ~; 23 |; 56 &;2671 @)
% ( 0 <=>; 292 =>; 0 <=; 0 <~>)
% Maximal formula depth : 19 ( 6 avg)
% Number of types : 7 ( 6 usr)
% Number of type conns : 241 ( 241 >; 0 *; 0 +; 0 <<)
% Number of symbols : 65 ( 62 usr; 4 con; 0-4 aty)
% Number of variables : 851 ( 70 ^; 715 !; 21 ?; 851 :)
% ( 45 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 15:14:34.831
%------------------------------------------------------------------------------
%----Could-be-implicit typings (11)
thf(ty_t_Code__Numeral_Onatural,type,
code_natural: $tType ).
thf(ty_t_Code__Evaluation_Oterm,type,
code_term: $tType ).
thf(ty_t_Product__Type_Ounit,type,
product_unit: $tType ).
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_Extended__Nat_Oenat,type,
extended_enat: $tType ).
thf(ty_t_Sum__Type_Osum,type,
sum_sum: $tType > $tType > $tType ).
thf(ty_t_Stream_Ostream,type,
stream: $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
thf(ty_tf_a,type,
a: $tType ).
%----Explicit typings (57)
thf(sy_cl_Enum_Oenum,type,
enum:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Typerep_Otyperep,type,
typerep:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_HOL_Oequal,type,
cl_HOL_Oequal:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Code__Evaluation_Oterm__of,type,
code_term_of:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Nat_Osize,type,
size:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Omonoid__add,type,
monoid_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Osemigroup__add,type,
semigroup_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Ocomm__semiring__1,type,
comm_semiring_1:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocomm__monoid__add,type,
comm_monoid_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oab__semigroup__add,type,
ab_semigroup_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Quickcheck__Random_Orandom,type,
quickcheck_random:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Olinordered__semidom,type,
linordered_semidom:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
cancel1352612707id_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Olinordered__ab__group__add,type,
linord219039673up_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__comm__monoid__add,type,
ordere216010020id_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
ordere779506340up_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni770627133id_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
ordere236663937imp_le:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type,
ordere516151231imp_le:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,type,
semiri456707255roduct:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_Code__Numeral_OSuc,type,
code_Suc: code_natural > code_natural ).
thf(sy_c_Code__Numeral_Onatural_Ocase__natural,type,
code_case_natural:
!>[T: $tType] : ( T > ( code_natural > T ) > code_natural > T ) ).
thf(sy_c_Code__Numeral_Onatural_Orec__natural,type,
code_rec_natural:
!>[T: $tType] : ( T > ( code_natural > T > T ) > code_natural > T ) ).
thf(sy_c_Code__Numeral_Onatural_Osize__natural,type,
code_size_natural: code_natural > nat ).
thf(sy_c_Coinductive__Nat_OLe__enatp,type,
coinductive_Le_enatp: extended_enat > extended_enat > $o ).
thf(sy_c_Coinductive__Nat_Oco_Oenat_Ocase__enat,type,
coindu440805660e_enat:
!>[A: $tType] : ( A > ( extended_enat > A ) > extended_enat > A ) ).
thf(sy_c_Coinductive__Nat_Oco_Oenat_Oepred,type,
coinductive_epred: extended_enat > extended_enat ).
thf(sy_c_Coinductive__Nat_Oenat__set,type,
coinductive_enat_set: set @ extended_enat ).
thf(sy_c_Coinductive__Nat_Oenat__setp,type,
coindu530039314t_setp: extended_enat > $o ).
thf(sy_c_Coinductive__Nat_Oenat__unfold,type,
coindu1491768222unfold:
!>[A: $tType] : ( ( A > $o ) > ( A > A ) > A > extended_enat ) ).
thf(sy_c_Coinductive__Stream__Mirabelle__dydkjoctes_Oscount,type,
coindu1365464361scount:
!>[S: $tType] : ( ( ( stream @ S ) > $o ) > ( stream @ S ) > extended_enat ) ).
thf(sy_c_Extended__Nat_OeSuc,type,
extended_eSuc: extended_enat > extended_enat ).
thf(sy_c_Extended__Nat_Oenat_Ocase__enat,type,
extended_case_enat:
!>[T: $tType] : ( ( nat > T ) > T > extended_enat > T ) ).
thf(sy_c_Extended__Nat_Oenat_Orec__enat,type,
extended_rec_enat:
!>[T: $tType] : ( ( nat > T ) > T > extended_enat > T ) ).
thf(sy_c_Extended__Nat_Oenat_Osize__enat,type,
extended_size_enat: extended_enat > nat ).
thf(sy_c_Extended__Nat_Othe__enat,type,
extended_the_enat: extended_enat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_HOL_Obool_Osize__bool,type,
size_bool: $o > nat ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Linear__Temporal__Logic__on__Streams_OUNTIL,type,
linear1341015445_UNTIL:
!>[A: $tType] : ( ( ( stream @ A ) > $o ) > ( ( stream @ A ) > $o ) > ( stream @ A ) > $o ) ).
thf(sy_c_Linear__Temporal__Logic__on__Streams_Oalw,type,
linear1386806755on_alw:
!>[A: $tType] : ( ( ( stream @ A ) > $o ) > ( stream @ A ) > $o ) ).
thf(sy_c_Linear__Temporal__Logic__on__Streams_Oev,type,
linear505997466_on_ev:
!>[A: $tType] : ( ( ( stream @ A ) > $o ) > ( stream @ A ) > $o ) ).
thf(sy_c_Linear__Temporal__Logic__on__Streams_Onxt,type,
linear1494993505on_nxt:
!>[A: $tType,B: $tType] : ( ( ( stream @ A ) > B ) > ( stream @ A ) > B ) ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : ( A > nat ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Product__Type_Oold_Obool_Orec__bool,type,
product_rec_bool:
!>[T: $tType] : ( T > T > $o > T ) ).
thf(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_P,type,
p: ( stream @ a ) > $o ).
thf(sy_v__092_060omega_062,type,
omega: stream @ a ).
%----Relevant facts (252)
thf(fact_0_alw__alw,axiom,
! [A: $tType,Phi: ( stream @ A ) > $o] :
( ( linear1386806755on_alw @ A @ ( linear1386806755on_alw @ A @ Phi ) )
= ( linear1386806755on_alw @ A @ Phi ) ) ).
% alw_alw
thf(fact_1_alw__mp,axiom,
! [A: $tType,Phi: ( stream @ A ) > $o,Xs: stream @ A,Psi: ( stream @ A ) > $o] :
( ( linear1386806755on_alw @ A @ Phi @ Xs )
=> ( ( linear1386806755on_alw @ A
@ ^ [Xs2: stream @ A] :
( ( Phi @ Xs2 )
=> ( Psi @ Xs2 ) )
@ Xs )
=> ( linear1386806755on_alw @ A @ Psi @ Xs ) ) ) ).
% alw_mp
thf(fact_2_alw__aand,axiom,
! [A: $tType,Phi: ( stream @ A ) > $o,Psi: ( stream @ A ) > $o] :
( ( linear1386806755on_alw @ A
@ ^ [Xs2: stream @ A] :
( ( Phi @ Xs2 )
& ( Psi @ Xs2 ) ) )
= ( ^ [Xs2: stream @ A] :
( ( linear1386806755on_alw @ A @ Phi @ Xs2 )
& ( linear1386806755on_alw @ A @ Psi @ Xs2 ) ) ) ) ).
% alw_aand
thf(fact_3_alw__False,axiom,
! [A: $tType,Omega: stream @ A] :
~ ( linear1386806755on_alw @ A
@ ^ [X: stream @ A] : $false
@ Omega ) ).
% alw_False
thf(fact_4_fun__cong__unused__0,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( zero @ B @ ( type2 @ B ) )
=> ! [F: ( A > B ) > C,G: C] :
( ( F
= ( ^ [X: A > B] : G ) )
=> ( ( F
@ ^ [X: A] : ( zero_zero @ B ) )
= G ) ) ) ).
% fun_cong_unused_0
thf(fact_5_alwD,axiom,
! [A: $tType,P: ( stream @ A ) > $o,X2: stream @ A] :
( ( linear1386806755on_alw @ A @ P @ X2 )
=> ( P @ X2 ) ) ).
% alwD
thf(fact_6_alw__alwD,axiom,
! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A] :
( ( linear1386806755on_alw @ A @ P @ Omega )
=> ( linear1386806755on_alw @ A @ ( linear1386806755on_alw @ A @ P ) @ Omega ) ) ).
% alw_alwD
thf(fact_7_alw__cong,axiom,
! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A,Q1: ( stream @ A ) > $o,Q2: ( stream @ A ) > $o] :
( ( linear1386806755on_alw @ A @ P @ Omega )
=> ( ! [Omega2: stream @ A] :
( ( P @ Omega2 )
=> ( ( Q1 @ Omega2 )
= ( Q2 @ Omega2 ) ) )
=> ( ( linear1386806755on_alw @ A @ Q1 @ Omega )
= ( linear1386806755on_alw @ A @ Q2 @ Omega ) ) ) ) ).
% alw_cong
thf(fact_8_alw__mono,axiom,
! [A: $tType,Phi: ( stream @ A ) > $o,Xs: stream @ A,Psi: ( stream @ A ) > $o] :
( ( linear1386806755on_alw @ A @ Phi @ Xs )
=> ( ! [Xs3: stream @ A] :
( ( Phi @ Xs3 )
=> ( Psi @ Xs3 ) )
=> ( linear1386806755on_alw @ A @ Psi @ Xs ) ) ) ).
% alw_mono
thf(fact_9_all__imp__alw,axiom,
! [A: $tType,Phi: ( stream @ A ) > $o,Xs: stream @ A] :
( ! [X1: stream @ A] : ( Phi @ X1 )
=> ( linear1386806755on_alw @ A @ Phi @ Xs ) ) ).
% all_imp_alw
thf(fact_10_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A @ ( type2 @ A ) )
=> ! [X2: A] :
( ( ( zero_zero @ A )
= X2 )
= ( X2
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_11_case__enat__0,axiom,
! [A: $tType,F: nat > A,I: A] :
( ( extended_case_enat @ A @ F @ I @ ( zero_zero @ extended_enat ) )
= ( F @ ( zero_zero @ nat ) ) ) ).
% case_enat_0
thf(fact_12_enat__unfold__stop,axiom,
! [A: $tType,Stop: A > $o,A2: A,Next: A > A] :
( ( Stop @ A2 )
=> ( ( coindu1491768222unfold @ A @ Stop @ Next @ A2 )
= ( zero_zero @ extended_enat ) ) ) ).
% enat_unfold_stop
thf(fact_13_enat__unfold__eq__0,axiom,
! [A: $tType,Stop: A > $o,Next: A > A,A2: A] :
( ( ( coindu1491768222unfold @ A @ Stop @ Next @ A2 )
= ( zero_zero @ extended_enat ) )
= ( Stop @ A2 ) ) ).
% enat_unfold_eq_0
thf(fact_14_the__enat__0,axiom,
( ( extended_the_enat @ ( zero_zero @ extended_enat ) )
= ( zero_zero @ nat ) ) ).
% the_enat_0
thf(fact_15_until__false,axiom,
! [A: $tType,Phi: ( stream @ A ) > $o] :
( ( linear1341015445_UNTIL @ A @ Phi
@ ^ [Xs2: stream @ A] : $false )
= ( linear1386806755on_alw @ A @ Phi ) ) ).
% until_false
thf(fact_16_alw__invar,axiom,
! [A: $tType,Phi: ( stream @ A ) > $o,Xs: stream @ A] :
( ( Phi @ Xs )
=> ( ( linear1386806755on_alw @ A
@ ^ [Xs2: stream @ A] :
( ( Phi @ Xs2 )
=> ( linear1494993505on_nxt @ A @ $o @ Phi @ Xs2 ) )
@ Xs )
=> ( linear1386806755on_alw @ A @ Phi @ Xs ) ) ) ).
% alw_invar
thf(fact_17_enat__set_Ointros_I1_J,axiom,
member @ extended_enat @ ( zero_zero @ extended_enat ) @ coinductive_enat_set ).
% enat_set.intros(1)
thf(fact_18_Le__enatp_OLe__enat__zero,axiom,
! [N: extended_enat] : ( coinductive_Le_enatp @ ( zero_zero @ extended_enat ) @ N ) ).
% Le_enatp.Le_enat_zero
thf(fact_19_enat__setp_Ointros_I1_J,axiom,
coindu530039314t_setp @ ( zero_zero @ extended_enat ) ).
% enat_setp.intros(1)
thf(fact_20_co_Oenat_Odisc__eq__case_I1_J,axiom,
! [Enat: extended_enat] :
( ( Enat
= ( zero_zero @ extended_enat ) )
= ( coindu440805660e_enat @ $o @ $true
@ ^ [Uu: extended_enat] : $false
@ Enat ) ) ).
% co.enat.disc_eq_case(1)
thf(fact_21_co_Oenat_Odisc__eq__case_I2_J,axiom,
! [Enat: extended_enat] :
( ( Enat
!= ( zero_zero @ extended_enat ) )
= ( coindu440805660e_enat @ $o @ $false
@ ^ [Uu: extended_enat] : $true
@ Enat ) ) ).
% co.enat.disc_eq_case(2)
thf(fact_22_enat__setp__enat__set__eq,axiom,
( coindu530039314t_setp
= ( ^ [X: extended_enat] : ( member @ extended_enat @ X @ coinductive_enat_set ) ) ) ).
% enat_setp_enat_set_eq
thf(fact_23_nxt__mono,axiom,
! [A: $tType,Phi: ( stream @ A ) > $o,Xs: stream @ A,Psi: ( stream @ A ) > $o] :
( ( linear1494993505on_nxt @ A @ $o @ Phi @ Xs )
=> ( ! [Xs3: stream @ A] :
( ( Phi @ Xs3 )
=> ( Psi @ Xs3 ) )
=> ( linear1494993505on_nxt @ A @ $o @ Psi @ Xs ) ) ) ).
% nxt_mono
thf(fact_24_UNTIL_Obase,axiom,
! [A: $tType,Psi: ( stream @ A ) > $o,Xs: stream @ A,Phi: ( stream @ A ) > $o] :
( ( Psi @ Xs )
=> ( linear1341015445_UNTIL @ A @ Phi @ Psi @ Xs ) ) ).
% UNTIL.base
thf(fact_25_until__mono,axiom,
! [A: $tType,Phi_1: ( stream @ A ) > $o,Psi_1: ( stream @ A ) > $o,Xs: stream @ A,Phi_2: ( stream @ A ) > $o,Psi_2: ( stream @ A ) > $o] :
( ( linear1341015445_UNTIL @ A @ Phi_1 @ Psi_1 @ Xs )
=> ( ! [Xs3: stream @ A] :
( ( Phi_1 @ Xs3 )
=> ( Phi_2 @ Xs3 ) )
=> ( ! [Xs3: stream @ A] :
( ( Psi_1 @ Xs3 )
=> ( Psi_2 @ Xs3 ) )
=> ( linear1341015445_UNTIL @ A @ Phi_2 @ Psi_2 @ Xs ) ) ) ) ).
% until_mono
thf(fact_26_until__monoL,axiom,
! [A: $tType,Phi_1: ( stream @ A ) > $o,Psi: ( stream @ A ) > $o,Xs: stream @ A,Phi_2: ( stream @ A ) > $o] :
( ( linear1341015445_UNTIL @ A @ Phi_1 @ Psi @ Xs )
=> ( ! [Xs3: stream @ A] :
( ( Phi_1 @ Xs3 )
=> ( Phi_2 @ Xs3 ) )
=> ( linear1341015445_UNTIL @ A @ Phi_2 @ Psi @ Xs ) ) ) ).
% until_monoL
thf(fact_27_until__monoR,axiom,
! [A: $tType,Phi: ( stream @ A ) > $o,Psi_1: ( stream @ A ) > $o,Xs: stream @ A,Psi_2: ( stream @ A ) > $o] :
( ( linear1341015445_UNTIL @ A @ Phi @ Psi_1 @ Xs )
=> ( ! [Xs3: stream @ A] :
( ( Psi_1 @ Xs3 )
=> ( Psi_2 @ Xs3 ) )
=> ( linear1341015445_UNTIL @ A @ Phi @ Psi_2 @ Xs ) ) ) ).
% until_monoR
thf(fact_28_enat__set__def,axiom,
( coinductive_enat_set
= ( collect @ extended_enat @ coindu530039314t_setp ) ) ).
% enat_set_def
thf(fact_29_co_Oenat_Ocase__distrib,axiom,
! [B: $tType,A: $tType,H: A > B,F1: A,F2: extended_enat > A,Enat: extended_enat] :
( ( H @ ( coindu440805660e_enat @ A @ F1 @ F2 @ Enat ) )
= ( coindu440805660e_enat @ B @ ( H @ F1 )
@ ^ [X: extended_enat] : ( H @ ( F2 @ X ) )
@ Enat ) ) ).
% co.enat.case_distrib
thf(fact_30_co_Oenat_Ocase_I1_J,axiom,
! [A: $tType,F1: A,F2: extended_enat > A] :
( ( coindu440805660e_enat @ A @ F1 @ F2 @ ( zero_zero @ extended_enat ) )
= F1 ) ).
% co.enat.case(1)
thf(fact_31_enat__cocase__0,axiom,
! [A: $tType,Z: A,S2: extended_enat > A] :
( ( coindu440805660e_enat @ A @ Z @ S2 @ ( zero_zero @ extended_enat ) )
= Z ) ).
% enat_cocase_0
thf(fact_32_alw__nxt,axiom,
! [A: $tType] :
( ( linear1386806755on_alw @ A )
= ( ^ [Phi2: ( stream @ A ) > $o,Xs2: stream @ A] :
( ( Phi2 @ Xs2 )
& ( linear1494993505on_nxt @ A @ $o @ ( linear1386806755on_alw @ A @ Phi2 ) @ Xs2 ) ) ) ) ).
% alw_nxt
thf(fact_33_zero__natural_Orsp,axiom,
( ( zero_zero @ nat )
= ( zero_zero @ nat ) ) ).
% zero_natural.rsp
thf(fact_34_epred__enat__unfold,axiom,
! [A: $tType,Stop: A > $o,A2: A,Next: A > A] :
( ( ( Stop @ A2 )
=> ( ( coinductive_epred @ ( coindu1491768222unfold @ A @ Stop @ Next @ A2 ) )
= ( zero_zero @ extended_enat ) ) )
& ( ~ ( Stop @ A2 )
=> ( ( coinductive_epred @ ( coindu1491768222unfold @ A @ Stop @ Next @ A2 ) )
= ( coindu1491768222unfold @ A @ Stop @ Next @ ( Next @ A2 ) ) ) ) ) ).
% epred_enat_unfold
thf(fact_35_variance,axiom,
! [A: $tType,Phi: ( stream @ A ) > $o,Xs: stream @ A,Psi: ( stream @ A ) > $o] :
( ( Phi @ Xs )
=> ( ( linear1386806755on_alw @ A
@ ^ [Xs2: stream @ A] :
( ( Phi @ Xs2 )
=> ( ( Psi @ Xs2 )
| ( linear1494993505on_nxt @ A @ $o @ Phi @ Xs2 ) ) )
@ Xs )
=> ( ( linear1386806755on_alw @ A @ Phi @ Xs )
| ( linear505997466_on_ev @ A @ Psi @ Xs ) ) ) ) ).
% variance
thf(fact_36_ev__alw__imp__nxt,axiom,
! [A: $tType,Phi: ( stream @ A ) > $o,Xs: stream @ A] :
( ( linear505997466_on_ev @ A @ Phi @ Xs )
=> ( ( linear1386806755on_alw @ A
@ ^ [Xs2: stream @ A] :
( ( Phi @ Xs2 )
=> ( linear1494993505on_nxt @ A @ $o @ Phi @ Xs2 ) )
@ Xs )
=> ( linear505997466_on_ev @ A @ ( linear1386806755on_alw @ A @ Phi ) @ Xs ) ) ) ).
% ev_alw_imp_nxt
thf(fact_37_co_Oenat_Ocase__eq__if,axiom,
! [A: $tType] :
( ( coindu440805660e_enat @ A )
= ( ^ [F12: A,F22: extended_enat > A,Enat2: extended_enat] :
( if @ A
@ ( Enat2
= ( zero_zero @ extended_enat ) )
@ F12
@ ( F22 @ ( coinductive_epred @ Enat2 ) ) ) ) ) ).
% co.enat.case_eq_if
thf(fact_38_enat__setp_Ocoinduct,axiom,
! [X3: extended_enat > $o,X2: extended_enat] :
( ( X3 @ X2 )
=> ( ! [X4: extended_enat] :
( ( X3 @ X4 )
=> ( ( X4
= ( zero_zero @ extended_enat ) )
| ? [N2: extended_enat] :
( ( X4
= ( extended_eSuc @ N2 ) )
& ( ( X3 @ N2 )
| ( coindu530039314t_setp @ N2 ) ) ) ) )
=> ( coindu530039314t_setp @ X2 ) ) ) ).
% enat_setp.coinduct
thf(fact_39_enat__setp_Osimps,axiom,
( coindu530039314t_setp
= ( ^ [A3: extended_enat] :
( ( A3
= ( zero_zero @ extended_enat ) )
| ? [N3: extended_enat] :
( ( A3
= ( extended_eSuc @ N3 ) )
& ( coindu530039314t_setp @ N3 ) ) ) ) ) ).
% enat_setp.simps
thf(fact_40_enat__setp_Ocases,axiom,
! [A2: extended_enat] :
( ( coindu530039314t_setp @ A2 )
=> ( ( A2
!= ( zero_zero @ extended_enat ) )
=> ~ ! [N4: extended_enat] :
( ( A2
= ( extended_eSuc @ N4 ) )
=> ~ ( coindu530039314t_setp @ N4 ) ) ) ) ).
% enat_setp.cases
thf(fact_41_enat__unfold__unique,axiom,
! [A: $tType,Stop: A > $o,H: A > extended_enat,Next: A > A,X2: A] :
( ! [X4: A] :
( ( ( Stop @ X4 )
=> ( ( H @ X4 )
= ( zero_zero @ extended_enat ) ) )
& ( ~ ( Stop @ X4 )
=> ( ( H @ X4 )
= ( extended_eSuc @ ( H @ ( Next @ X4 ) ) ) ) ) )
=> ( ( H @ X2 )
= ( coindu1491768222unfold @ A @ Stop @ Next @ X2 ) ) ) ).
% enat_unfold_unique
thf(fact_42_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_43_Collect__mem__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( collect @ A
@ ^ [X: A] : ( member @ A @ X @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_44_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X4: A] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_45_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X4: A] :
( ( F @ X4 )
= ( G @ X4 ) )
=> ( F = G ) ) ).
% ext
thf(fact_46_co_Oenat_Oinject,axiom,
! [X22: extended_enat,Y2: extended_enat] :
( ( ( extended_eSuc @ X22 )
= ( extended_eSuc @ Y2 ) )
= ( X22 = Y2 ) ) ).
% co.enat.inject
thf(fact_47_ev__ev,axiom,
! [A: $tType,Phi: ( stream @ A ) > $o] :
( ( linear505997466_on_ev @ A @ ( linear505997466_on_ev @ A @ Phi ) )
= ( linear505997466_on_ev @ A @ Phi ) ) ).
% ev_ev
thf(fact_48_not__alw__not,axiom,
! [A: $tType,Phi: ( stream @ A ) > $o] :
( ( ^ [Xs2: stream @ A] :
~ ( linear1386806755on_alw @ A
@ ^ [Xt: stream @ A] :
~ ( Phi @ Xt )
@ Xs2 ) )
= ( linear505997466_on_ev @ A @ Phi ) ) ).
% not_alw_not
thf(fact_49_not__ev__not,axiom,
! [A: $tType,Phi: ( stream @ A ) > $o] :
( ( ^ [Xs2: stream @ A] :
~ ( linear505997466_on_ev @ A
@ ^ [Xt: stream @ A] :
~ ( Phi @ Xt )
@ Xs2 ) )
= ( linear1386806755on_alw @ A @ Phi ) ) ).
% not_ev_not
thf(fact_50_eSuc__epred,axiom,
! [N: extended_enat] :
( ( N
!= ( zero_zero @ extended_enat ) )
=> ( ( extended_eSuc @ ( coinductive_epred @ N ) )
= N ) ) ).
% eSuc_epred
thf(fact_51_co_Oenat_Ocollapse,axiom,
! [Enat: extended_enat] :
( ( Enat
!= ( zero_zero @ extended_enat ) )
=> ( ( extended_eSuc @ ( coinductive_epred @ Enat ) )
= Enat ) ) ).
% co.enat.collapse
thf(fact_52_ev__False,axiom,
! [A: $tType,Omega: stream @ A] :
~ ( linear505997466_on_ev @ A
@ ^ [X: stream @ A] : $false
@ Omega ) ).
% ev_False
thf(fact_53_ev__or,axiom,
! [A: $tType,Phi: ( stream @ A ) > $o,Psi: ( stream @ A ) > $o] :
( ( linear505997466_on_ev @ A
@ ^ [Xs2: stream @ A] :
( ( Phi @ Xs2 )
| ( Psi @ Xs2 ) ) )
= ( ^ [Xs2: stream @ A] :
( ( linear505997466_on_ev @ A @ Phi @ Xs2 )
| ( linear505997466_on_ev @ A @ Psi @ Xs2 ) ) ) ) ).
% ev_or
thf(fact_54_co_Oenat_Oexhaust__sel,axiom,
! [Enat: extended_enat] :
( ( Enat
!= ( zero_zero @ extended_enat ) )
=> ( Enat
= ( extended_eSuc @ ( coinductive_epred @ Enat ) ) ) ) ).
% co.enat.exhaust_sel
thf(fact_55_ev__mono,axiom,
! [A: $tType,Phi: ( stream @ A ) > $o,Xs: stream @ A,Psi: ( stream @ A ) > $o] :
( ( linear505997466_on_ev @ A @ Phi @ Xs )
=> ( ! [Xs3: stream @ A] :
( ( Phi @ Xs3 )
=> ( Psi @ Xs3 ) )
=> ( linear505997466_on_ev @ A @ Psi @ Xs ) ) ) ).
% ev_mono
thf(fact_56_ev_Obase,axiom,
! [A: $tType,Phi: ( stream @ A ) > $o,Xs: stream @ A] :
( ( Phi @ Xs )
=> ( linear505997466_on_ev @ A @ Phi @ Xs ) ) ).
% ev.base
thf(fact_57_epred__eSuc,axiom,
! [N: extended_enat] :
( ( coinductive_epred @ ( extended_eSuc @ N ) )
= N ) ).
% epred_eSuc
thf(fact_58_co_Oenat_Osel_I2_J,axiom,
! [X22: extended_enat] :
( ( coinductive_epred @ ( extended_eSuc @ X22 ) )
= X22 ) ).
% co.enat.sel(2)
thf(fact_59_co_Oepred__def,axiom,
( coinductive_epred
= ( coindu440805660e_enat @ extended_enat @ ( zero_zero @ extended_enat )
@ ^ [X23: extended_enat] : X23 ) ) ).
% co.epred_def
thf(fact_60_co_Oenat_Osplit__sel__asm,axiom,
! [A: $tType,P: A > $o,F1: A,F2: extended_enat > A,Enat: extended_enat] :
( ( P @ ( coindu440805660e_enat @ A @ F1 @ F2 @ Enat ) )
= ( ~ ( ( ( Enat
= ( zero_zero @ extended_enat ) )
& ~ ( P @ F1 ) )
| ( ( Enat
= ( extended_eSuc @ ( coinductive_epred @ Enat ) ) )
& ~ ( P @ ( F2 @ ( coinductive_epred @ Enat ) ) ) ) ) ) ) ).
% co.enat.split_sel_asm
thf(fact_61_co_Oenat_Osplit__sel,axiom,
! [A: $tType,P: A > $o,F1: A,F2: extended_enat > A,Enat: extended_enat] :
( ( P @ ( coindu440805660e_enat @ A @ F1 @ F2 @ Enat ) )
= ( ( ( Enat
= ( zero_zero @ extended_enat ) )
=> ( P @ F1 ) )
& ( ( Enat
= ( extended_eSuc @ ( coinductive_epred @ Enat ) ) )
=> ( P @ ( F2 @ ( coinductive_epred @ Enat ) ) ) ) ) ) ).
% co.enat.split_sel
thf(fact_62_ev__alw__imp__alw__ev,axiom,
! [A: $tType,Phi: ( stream @ A ) > $o,Xs: stream @ A] :
( ( linear505997466_on_ev @ A @ ( linear1386806755on_alw @ A @ Phi ) @ Xs )
=> ( linear1386806755on_alw @ A @ ( linear505997466_on_ev @ A @ Phi ) @ Xs ) ) ).
% ev_alw_imp_alw_ev
thf(fact_63_ev__cong,axiom,
! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A,Q1: ( stream @ A ) > $o,Q2: ( stream @ A ) > $o] :
( ( linear1386806755on_alw @ A @ P @ Omega )
=> ( ! [Omega2: stream @ A] :
( ( P @ Omega2 )
=> ( ( Q1 @ Omega2 )
= ( Q2 @ Omega2 ) ) )
=> ( ( linear505997466_on_ev @ A @ Q1 @ Omega )
= ( linear505997466_on_ev @ A @ Q2 @ Omega ) ) ) ) ).
% ev_cong
thf(fact_64_neq__zero__conv__eSuc,axiom,
! [N: extended_enat] :
( ( N
!= ( zero_zero @ extended_enat ) )
= ( ? [N5: extended_enat] :
( N
= ( extended_eSuc @ N5 ) ) ) ) ).
% neq_zero_conv_eSuc
thf(fact_65_co_Oenat_Oexhaust,axiom,
! [Y: extended_enat] :
( ( Y
!= ( zero_zero @ extended_enat ) )
=> ~ ! [X24: extended_enat] :
( Y
!= ( extended_eSuc @ X24 ) ) ) ).
% co.enat.exhaust
thf(fact_66_enat__coexhaust,axiom,
! [N: extended_enat] :
( ( N
!= ( zero_zero @ extended_enat ) )
=> ~ ! [N6: extended_enat] :
( N
!= ( extended_eSuc @ N6 ) ) ) ).
% enat_coexhaust
thf(fact_67_co_Oenat_OdiscI,axiom,
! [Enat: extended_enat,X22: extended_enat] :
( ( Enat
= ( extended_eSuc @ X22 ) )
=> ( Enat
!= ( zero_zero @ extended_enat ) ) ) ).
% co.enat.discI
thf(fact_68_co_Oenat_Odistinct_I1_J,axiom,
! [X22: extended_enat] :
( ( zero_zero @ extended_enat )
!= ( extended_eSuc @ X22 ) ) ).
% co.enat.distinct(1)
thf(fact_69_enat__coinduct2,axiom,
! [P: extended_enat > extended_enat > $o,M: extended_enat,N: extended_enat] :
( ( P @ M @ N )
=> ( ! [M2: extended_enat,N4: extended_enat] :
( ( P @ M2 @ N4 )
=> ( ( M2
= ( zero_zero @ extended_enat ) )
= ( N4
= ( zero_zero @ extended_enat ) ) ) )
=> ( ! [M2: extended_enat,N4: extended_enat] :
( ( P @ M2 @ N4 )
=> ( ( M2
!= ( zero_zero @ extended_enat ) )
=> ( ( N4
!= ( zero_zero @ extended_enat ) )
=> ( ( P @ ( coinductive_epred @ M2 ) @ ( coinductive_epred @ N4 ) )
| ( ( coinductive_epred @ M2 )
= ( coinductive_epred @ N4 ) ) ) ) ) )
=> ( M = N ) ) ) ) ).
% enat_coinduct2
thf(fact_70_co_Oenat_Oexpand,axiom,
! [Enat: extended_enat,Enat3: extended_enat] :
( ( ( Enat
= ( zero_zero @ extended_enat ) )
= ( Enat3
= ( zero_zero @ extended_enat ) ) )
=> ( ( ( Enat
!= ( zero_zero @ extended_enat ) )
=> ( ( Enat3
!= ( zero_zero @ extended_enat ) )
=> ( ( coinductive_epred @ Enat )
= ( coinductive_epred @ Enat3 ) ) ) )
=> ( Enat = Enat3 ) ) ) ).
% co.enat.expand
thf(fact_71_enat__coinduct,axiom,
! [P: extended_enat > extended_enat > $o,M: extended_enat,N: extended_enat] :
( ( P @ M @ N )
=> ( ! [M2: extended_enat,N4: extended_enat] :
( ( P @ M2 @ N4 )
=> ( ( ( M2
= ( zero_zero @ extended_enat ) )
= ( N4
= ( zero_zero @ extended_enat ) ) )
& ( ( M2
!= ( zero_zero @ extended_enat ) )
=> ( ( N4
!= ( zero_zero @ extended_enat ) )
=> ( ( P @ ( coinductive_epred @ M2 ) @ ( coinductive_epred @ N4 ) )
| ( ( coinductive_epred @ M2 )
= ( coinductive_epred @ N4 ) ) ) ) ) ) )
=> ( M = N ) ) ) ).
% enat_coinduct
thf(fact_72_epred__inject,axiom,
! [X2: extended_enat,Y: extended_enat] :
( ( X2
!= ( zero_zero @ extended_enat ) )
=> ( ( Y
!= ( zero_zero @ extended_enat ) )
=> ( ( ( coinductive_epred @ X2 )
= ( coinductive_epred @ Y ) )
= ( X2 = Y ) ) ) ) ).
% epred_inject
thf(fact_73_epred__0,axiom,
( ( coinductive_epred @ ( zero_zero @ extended_enat ) )
= ( zero_zero @ extended_enat ) ) ).
% epred_0
thf(fact_74_ev__nxt,axiom,
! [A: $tType] :
( ( linear505997466_on_ev @ A )
= ( ^ [Phi2: ( stream @ A ) > $o,Xs2: stream @ A] :
( ( Phi2 @ Xs2 )
| ( linear1494993505on_nxt @ A @ $o @ ( linear505997466_on_ev @ A @ Phi2 ) @ Xs2 ) ) ) ) ).
% ev_nxt
thf(fact_75_enat__cocase__eSuc,axiom,
! [A: $tType,Z: A,S2: extended_enat > A,N: extended_enat] :
( ( coindu440805660e_enat @ A @ Z @ S2 @ ( extended_eSuc @ N ) )
= ( S2 @ N ) ) ).
% enat_cocase_eSuc
thf(fact_76_co_Oenat_Ocase_I2_J,axiom,
! [A: $tType,F1: A,F2: extended_enat > A,X22: extended_enat] :
( ( coindu440805660e_enat @ A @ F1 @ F2 @ ( extended_eSuc @ X22 ) )
= ( F2 @ X22 ) ) ).
% co.enat.case(2)
thf(fact_77_enat__unfold__next,axiom,
! [A: $tType,Stop: A > $o,A2: A,Next: A > A] :
( ~ ( Stop @ A2 )
=> ( ( coindu1491768222unfold @ A @ Stop @ Next @ A2 )
= ( extended_eSuc @ ( coindu1491768222unfold @ A @ Stop @ Next @ ( Next @ A2 ) ) ) ) ) ).
% enat_unfold_next
thf(fact_78_enat__set_Ointros_I2_J,axiom,
! [N: extended_enat] :
( ( member @ extended_enat @ N @ coinductive_enat_set )
=> ( member @ extended_enat @ ( extended_eSuc @ N ) @ coinductive_enat_set ) ) ).
% enat_set.intros(2)
thf(fact_79_alw__ev__imp__ev__alw,axiom,
! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A] :
( ( linear1386806755on_alw @ A @ ( linear505997466_on_ev @ A @ P ) @ Omega )
=> ( linear505997466_on_ev @ A
@ ^ [Xs2: stream @ A] :
( ( P @ Xs2 )
& ( linear1386806755on_alw @ A @ ( linear505997466_on_ev @ A @ P ) @ Xs2 ) )
@ Omega ) ) ).
% alw_ev_imp_ev_alw
thf(fact_80_ev__alw__alw__impl,axiom,
! [A: $tType,Phi: ( stream @ A ) > $o,Xs: stream @ A,Psi: ( stream @ A ) > $o] :
( ( linear505997466_on_ev @ A @ ( linear1386806755on_alw @ A @ Phi ) @ Xs )
=> ( ( linear1386806755on_alw @ A
@ ^ [Xs2: stream @ A] :
( ( linear1386806755on_alw @ A @ Phi @ Xs2 )
=> ( linear505997466_on_ev @ A @ Psi @ Xs2 ) )
@ Xs )
=> ( linear505997466_on_ev @ A @ Psi @ Xs ) ) ) ).
% ev_alw_alw_impl
thf(fact_81_alw__impl__ev__alw,axiom,
! [A: $tType,Phi: ( stream @ A ) > $o,Psi: ( stream @ A ) > $o,Xs: stream @ A] :
( ( linear1386806755on_alw @ A
@ ^ [Xs2: stream @ A] :
( ( Phi @ Xs2 )
=> ( linear505997466_on_ev @ A @ Psi @ Xs2 ) )
@ Xs )
=> ( linear1386806755on_alw @ A
@ ^ [Xs2: stream @ A] :
( ( linear505997466_on_ev @ A @ Phi @ Xs2 )
=> ( linear505997466_on_ev @ A @ Psi @ Xs2 ) )
@ Xs ) ) ).
% alw_impl_ev_alw
thf(fact_82_alw__alw__impl__ev,axiom,
! [A: $tType,Phi: ( stream @ A ) > $o,Psi: ( stream @ A ) > $o] :
( ( linear1386806755on_alw @ A
@ ^ [Xs2: stream @ A] :
( ( linear1386806755on_alw @ A @ Phi @ Xs2 )
=> ( linear505997466_on_ev @ A @ Psi @ Xs2 ) ) )
= ( ^ [Xs2: stream @ A] :
( ( linear505997466_on_ev @ A @ ( linear1386806755on_alw @ A @ Phi ) @ Xs2 )
=> ( linear1386806755on_alw @ A @ ( linear505997466_on_ev @ A @ Psi ) @ Xs2 ) ) ) ) ).
% alw_alw_impl_ev
thf(fact_83_ev__alw__impl__ev,axiom,
! [A: $tType,Phi: ( stream @ A ) > $o,Xs: stream @ A,Psi: ( stream @ A ) > $o] :
( ( linear505997466_on_ev @ A @ Phi @ Xs )
=> ( ( linear1386806755on_alw @ A
@ ^ [Xs2: stream @ A] :
( ( Phi @ Xs2 )
=> ( linear505997466_on_ev @ A @ Psi @ Xs2 ) )
@ Xs )
=> ( linear505997466_on_ev @ A @ Psi @ Xs ) ) ) ).
% ev_alw_impl_ev
thf(fact_84_not__alw__iff,axiom,
! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A] :
( ( ~ ( linear1386806755on_alw @ A @ P @ Omega ) )
= ( linear505997466_on_ev @ A
@ ^ [Xs2: stream @ A] :
~ ( P @ Xs2 )
@ Omega ) ) ).
% not_alw_iff
thf(fact_85_ev__alw__impl,axiom,
! [A: $tType,Phi: ( stream @ A ) > $o,Xs: stream @ A,Psi: ( stream @ A ) > $o] :
( ( linear505997466_on_ev @ A @ Phi @ Xs )
=> ( ( linear1386806755on_alw @ A
@ ^ [Xs2: stream @ A] :
( ( Phi @ Xs2 )
=> ( Psi @ Xs2 ) )
@ Xs )
=> ( linear505997466_on_ev @ A @ Psi @ Xs ) ) ) ).
% ev_alw_impl
thf(fact_86_ev__alw__aand,axiom,
! [A: $tType,Phi: ( stream @ A ) > $o,Xs: stream @ A,Psi: ( stream @ A ) > $o] :
( ( linear505997466_on_ev @ A @ ( linear1386806755on_alw @ A @ Phi ) @ Xs )
=> ( ( linear505997466_on_ev @ A @ ( linear1386806755on_alw @ A @ Psi ) @ Xs )
=> ( linear505997466_on_ev @ A
@ ( linear1386806755on_alw @ A
@ ^ [Xs2: stream @ A] :
( ( Phi @ Xs2 )
& ( Psi @ Xs2 ) ) )
@ Xs ) ) ) ).
% ev_alw_aand
thf(fact_87_not__ev__iff,axiom,
! [A: $tType,P: ( stream @ A ) > $o,Omega: stream @ A] :
( ( ~ ( linear505997466_on_ev @ A @ P @ Omega ) )
= ( linear1386806755on_alw @ A
@ ^ [Xs2: stream @ A] :
~ ( P @ Xs2 )
@ Omega ) ) ).
% not_ev_iff
thf(fact_88_not__alw,axiom,
! [A: $tType,Phi: ( stream @ A ) > $o] :
( ( ^ [Xs2: stream @ A] :
~ ( linear1386806755on_alw @ A @ Phi @ Xs2 ) )
= ( linear505997466_on_ev @ A
@ ^ [Xs2: stream @ A] :
~ ( Phi @ Xs2 ) ) ) ).
% not_alw
thf(fact_89_not__ev,axiom,
! [A: $tType,Phi: ( stream @ A ) > $o] :
( ( ^ [Xs2: stream @ A] :
~ ( linear505997466_on_ev @ A @ Phi @ Xs2 ) )
= ( linear1386806755on_alw @ A
@ ^ [Xs2: stream @ A] :
~ ( Phi @ Xs2 ) ) ) ).
% not_ev
thf(fact_90_enat__setp_Ointros_I2_J,axiom,
! [N: extended_enat] :
( ( coindu530039314t_setp @ N )
=> ( coindu530039314t_setp @ ( extended_eSuc @ N ) ) ) ).
% enat_setp.intros(2)
thf(fact_91_enat__cosplit,axiom,
! [A: $tType,P: A > $o,C2: A,D: extended_enat > A,N: extended_enat] :
( ( P @ ( coindu440805660e_enat @ A @ C2 @ D @ N ) )
= ( ( ( N
= ( zero_zero @ extended_enat ) )
=> ( P @ C2 ) )
& ! [M3: extended_enat] :
( ( N
= ( extended_eSuc @ M3 ) )
=> ( P @ ( D @ M3 ) ) ) ) ) ).
% enat_cosplit
thf(fact_92_enat__cosplit__asm,axiom,
! [A: $tType,P: A > $o,C2: A,D: extended_enat > A,N: extended_enat] :
( ( P @ ( coindu440805660e_enat @ A @ C2 @ D @ N ) )
= ( ~ ( ( ( N
= ( zero_zero @ extended_enat ) )
& ~ ( P @ C2 ) )
| ? [M3: extended_enat] :
( ( N
= ( extended_eSuc @ M3 ) )
& ~ ( P @ ( D @ M3 ) ) ) ) ) ) ).
% enat_cosplit_asm
thf(fact_93_enat__set_Ocases,axiom,
! [A2: extended_enat] :
( ( member @ extended_enat @ A2 @ coinductive_enat_set )
=> ( ( A2
!= ( zero_zero @ extended_enat ) )
=> ~ ! [N4: extended_enat] :
( ( A2
= ( extended_eSuc @ N4 ) )
=> ~ ( member @ extended_enat @ N4 @ coinductive_enat_set ) ) ) ) ).
% enat_set.cases
thf(fact_94_enat__set_Osimps,axiom,
! [A2: extended_enat] :
( ( member @ extended_enat @ A2 @ coinductive_enat_set )
= ( ( A2
= ( zero_zero @ extended_enat ) )
| ? [N3: extended_enat] :
( ( A2
= ( extended_eSuc @ N3 ) )
& ( member @ extended_enat @ N3 @ coinductive_enat_set ) ) ) ) ).
% enat_set.simps
thf(fact_95_enat__set_Ocoinduct,axiom,
! [X3: extended_enat > $o,X2: extended_enat] :
( ( X3 @ X2 )
=> ( ! [X4: extended_enat] :
( ( X3 @ X4 )
=> ( ( X4
= ( zero_zero @ extended_enat ) )
| ? [N2: extended_enat] :
( ( X4
= ( extended_eSuc @ N2 ) )
& ( ( X3 @ N2 )
| ( member @ extended_enat @ N2 @ coinductive_enat_set ) ) ) ) )
=> ( member @ extended_enat @ X2 @ coinductive_enat_set ) ) ) ).
% enat_set.coinduct
thf(fact_96_enat__unfold_Osimps,axiom,
! [A: $tType] :
( ( coindu1491768222unfold @ A )
= ( ^ [Stop2: A > $o,Next2: A > A,A3: A] : ( if @ extended_enat @ ( Stop2 @ A3 ) @ ( zero_zero @ extended_enat ) @ ( extended_eSuc @ ( coindu1491768222unfold @ A @ Stop2 @ Next2 @ ( Next2 @ A3 ) ) ) ) ) ) ).
% enat_unfold.simps
thf(fact_97_eSuc__inject,axiom,
! [M: extended_enat,N: extended_enat] :
( ( ( extended_eSuc @ M )
= ( extended_eSuc @ N ) )
= ( M = N ) ) ).
% eSuc_inject
thf(fact_98_zero__ne__eSuc,axiom,
! [N: extended_enat] :
( ( zero_zero @ extended_enat )
!= ( extended_eSuc @ N ) ) ).
% zero_ne_eSuc
thf(fact_99_natural_Osize_I1_J,axiom,
( ( code_size_natural @ ( zero_zero @ code_natural ) )
= ( zero_zero @ nat ) ) ).
% natural.size(1)
thf(fact_100_natural_Osize_I3_J,axiom,
( ( size_size @ code_natural @ ( zero_zero @ code_natural ) )
= ( zero_zero @ nat ) ) ).
% natural.size(3)
thf(fact_101_Le__enatp_OLe__enat__add,axiom,
! [M: extended_enat,N: extended_enat,K: extended_enat] :
( ( coinductive_Le_enatp @ M @ N )
=> ( ( K
!= ( zero_zero @ extended_enat ) )
=> ( coinductive_Le_enatp @ ( extended_eSuc @ M ) @ ( plus_plus @ extended_enat @ N @ K ) ) ) ) ).
% Le_enatp.Le_enat_add
thf(fact_102_Le__enatp_Ocoinduct,axiom,
! [X3: extended_enat > extended_enat > $o,X2: extended_enat,Xa: extended_enat] :
( ( X3 @ X2 @ Xa )
=> ( ! [X4: extended_enat,Xa2: extended_enat] :
( ( X3 @ X4 @ Xa2 )
=> ( ? [N2: extended_enat] :
( ( X4
= ( zero_zero @ extended_enat ) )
& ( Xa2 = N2 ) )
| ? [M4: extended_enat,N2: extended_enat,K2: extended_enat] :
( ( X4
= ( extended_eSuc @ M4 ) )
& ( Xa2
= ( plus_plus @ extended_enat @ N2 @ K2 ) )
& ( ( X3 @ M4 @ N2 )
| ( coinductive_Le_enatp @ M4 @ N2 ) )
& ( K2
!= ( zero_zero @ extended_enat ) ) ) ) )
=> ( coinductive_Le_enatp @ X2 @ Xa ) ) ) ).
% Le_enatp.coinduct
thf(fact_103_Le__enatp_Osimps,axiom,
( coinductive_Le_enatp
= ( ^ [A1: extended_enat,A22: extended_enat] :
( ? [N3: extended_enat] :
( ( A1
= ( zero_zero @ extended_enat ) )
& ( A22 = N3 ) )
| ? [M3: extended_enat,N3: extended_enat,K3: extended_enat] :
( ( A1
= ( extended_eSuc @ M3 ) )
& ( A22
= ( plus_plus @ extended_enat @ N3 @ K3 ) )
& ( coinductive_Le_enatp @ M3 @ N3 )
& ( K3
!= ( zero_zero @ extended_enat ) ) ) ) ) ) ).
% Le_enatp.simps
thf(fact_104_Le__enatp_Ocases,axiom,
! [A12: extended_enat,A23: extended_enat] :
( ( coinductive_Le_enatp @ A12 @ A23 )
=> ( ( ( A12
= ( zero_zero @ extended_enat ) )
=> ! [N4: extended_enat] : A23 != N4 )
=> ~ ! [M2: extended_enat] :
( ( A12
= ( extended_eSuc @ M2 ) )
=> ! [N4: extended_enat,K4: extended_enat] :
( ( A23
= ( plus_plus @ extended_enat @ N4 @ K4 ) )
=> ( ( coinductive_Le_enatp @ M2 @ N4 )
=> ( K4
= ( zero_zero @ extended_enat ) ) ) ) ) ) ) ).
% Le_enatp.cases
thf(fact_105_add__left__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C2 ) )
= ( B2 = C2 ) ) ) ).
% add_left_cancel
thf(fact_106_add__right__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C2: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C2 @ A2 ) )
= ( B2 = C2 ) ) ) ).
% add_right_cancel
thf(fact_107_add_Oleft__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% add.left_neutral
thf(fact_108_add_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% add.right_neutral
thf(fact_109_double__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ( plus_plus @ A @ A2 @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% double_zero
thf(fact_110_double__zero__sym,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ( zero_zero @ A )
= ( plus_plus @ A @ A2 @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% double_zero_sym
thf(fact_111_add__cancel__left__left,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= A2 )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_left
thf(fact_112_add__cancel__left__right,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= A2 )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_right
thf(fact_113_add__cancel__right__left,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( A2
= ( plus_plus @ A @ B2 @ A2 ) )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_left
thf(fact_114_add__cancel__right__right,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( A2
= ( plus_plus @ A @ A2 @ B2 ) )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_right
thf(fact_115_iadd__Suc__right,axiom,
! [M: extended_enat,N: extended_enat] :
( ( plus_plus @ extended_enat @ M @ ( extended_eSuc @ N ) )
= ( extended_eSuc @ ( plus_plus @ extended_enat @ M @ N ) ) ) ).
% iadd_Suc_right
thf(fact_116_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_117_add__mono__thms__linordered__semiring_I4_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A @ ( type2 @ A ) )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus @ A @ I @ K )
= ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_118_add_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% add.assoc
thf(fact_119_add_Oleft__cancel,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C2 ) )
= ( B2 = C2 ) ) ) ).
% add.left_cancel
thf(fact_120_add_Oright__cancel,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C2: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C2 @ A2 ) )
= ( B2 = C2 ) ) ) ).
% add.right_cancel
thf(fact_121_add_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A @ ( type2 @ A ) )
=> ( ( plus_plus @ A )
= ( ^ [A3: A,B3: A] : ( plus_plus @ A @ B3 @ A3 ) ) ) ) ).
% add.commute
thf(fact_122_add_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C2: A] :
( ( plus_plus @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% add.left_commute
thf(fact_123_add__left__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C2 ) )
=> ( B2 = C2 ) ) ) ).
% add_left_imp_eq
thf(fact_124_add__right__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C2: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C2 @ A2 ) )
=> ( B2 = C2 ) ) ) ).
% add_right_imp_eq
thf(fact_125_comm__monoid__add__class_Oadd__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% comm_monoid_add_class.add_0
thf(fact_126_add_Ocomm__neutral,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% add.comm_neutral
thf(fact_127_add_Ogroup__left__neutral,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% add.group_left_neutral
thf(fact_128_add__eq__0__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [X2: A,Y: A] :
( ( ( plus_plus @ A @ X2 @ Y )
= ( zero_zero @ A ) )
= ( ( X2
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_129_plus__enat__eq__0__conv,axiom,
! [M: extended_enat,N: extended_enat] :
( ( ( plus_plus @ extended_enat @ M @ N )
= ( zero_zero @ extended_enat ) )
= ( ( M
= ( zero_zero @ extended_enat ) )
& ( N
= ( zero_zero @ extended_enat ) ) ) ) ).
% plus_enat_eq_0_conv
thf(fact_130_eSuc__plus,axiom,
! [M: extended_enat,N: extended_enat] :
( ( plus_plus @ extended_enat @ ( extended_eSuc @ M ) @ N )
= ( extended_eSuc @ ( plus_plus @ extended_enat @ M @ N ) ) ) ).
% eSuc_plus
thf(fact_131_epred__iadd1,axiom,
! [A2: extended_enat,B2: extended_enat] :
( ( A2
!= ( zero_zero @ extended_enat ) )
=> ( ( coinductive_epred @ ( plus_plus @ extended_enat @ A2 @ B2 ) )
= ( plus_plus @ extended_enat @ ( coinductive_epred @ A2 ) @ B2 ) ) ) ).
% epred_iadd1
thf(fact_132_natural_Osimps_I4_J,axiom,
! [T: $tType,F1: T,F2: code_natural > T] :
( ( code_case_natural @ T @ F1 @ F2 @ ( zero_zero @ code_natural ) )
= F1 ) ).
% natural.simps(4)
thf(fact_133_add__0__iff,axiom,
! [A: $tType] :
( ( semiri456707255roduct @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A] :
( ( B2
= ( plus_plus @ A @ B2 @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% add_0_iff
thf(fact_134_semiring__normalization__rules_I5_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% semiring_normalization_rules(5)
thf(fact_135_semiring__normalization__rules_I6_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% semiring_normalization_rules(6)
thf(fact_136_natural_Osimps_I6_J,axiom,
! [T: $tType,F1: T,F2: code_natural > T > T] :
( ( code_rec_natural @ T @ F1 @ F2 @ ( zero_zero @ code_natural ) )
= F1 ) ).
% natural.simps(6)
thf(fact_137_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus @ nat @ M @ ( zero_zero @ nat ) )
= M ) ).
% Nat.add_0_right
thf(fact_138_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_139_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus @ nat @ ( zero_zero @ nat ) @ N )
= N ) ).
% plus_nat.add_0
thf(fact_140_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_141_case__natural__def,axiom,
! [T: $tType] :
( ( code_case_natural @ T )
= ( ^ [F12: T,F22: code_natural > T] :
( code_rec_natural @ T @ F12
@ ^ [X12: code_natural,X23: T] : ( F22 @ X12 ) ) ) ) ).
% case_natural_def
thf(fact_142_semiring__normalization__rules_I25_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A2: A,C2: A,D: A] :
( ( plus_plus @ A @ A2 @ ( plus_plus @ A @ C2 @ D ) )
= ( plus_plus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ D ) ) ) ).
% semiring_normalization_rules(25)
thf(fact_143_semiring__normalization__rules_I24_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ( ( plus_plus @ A )
= ( ^ [A3: A,C3: A] : ( plus_plus @ A @ C3 @ A3 ) ) ) ) ).
% semiring_normalization_rules(24)
thf(fact_144_semiring__normalization__rules_I23_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B2 ) ) ) ).
% semiring_normalization_rules(23)
thf(fact_145_semiring__normalization__rules_I22_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A2: A,C2: A,D: A] :
( ( plus_plus @ A @ A2 @ ( plus_plus @ A @ C2 @ D ) )
= ( plus_plus @ A @ C2 @ ( plus_plus @ A @ A2 @ D ) ) ) ) ).
% semiring_normalization_rules(22)
thf(fact_146_semiring__normalization__rules_I21_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% semiring_normalization_rules(21)
thf(fact_147_semiring__normalization__rules_I20_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C2: A,D: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( plus_plus @ A @ C2 @ D ) )
= ( plus_plus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D ) ) ) ) ).
% semiring_normalization_rules(20)
thf(fact_148_case__enat__def,axiom,
! [T: $tType] :
( ( extended_case_enat @ T )
= ( extended_rec_enat @ T ) ) ).
% case_enat_def
thf(fact_149_natural_Osimps_I5_J,axiom,
! [T: $tType,F1: T,F2: code_natural > T,Natural: code_natural] :
( ( code_case_natural @ T @ F1 @ F2 @ ( code_Suc @ Natural ) )
= ( F2 @ Natural ) ) ).
% natural.simps(5)
thf(fact_150_enat__le__coinduct,axiom,
! [P: extended_enat > extended_enat > $o,M: extended_enat,N: extended_enat] :
( ( P @ M @ N )
=> ( ! [M2: extended_enat,N4: extended_enat] :
( ( P @ M2 @ N4 )
=> ( ( ( N4
= ( zero_zero @ extended_enat ) )
=> ( M2
= ( zero_zero @ extended_enat ) ) )
& ( ( M2
!= ( zero_zero @ extended_enat ) )
=> ( ( N4
!= ( zero_zero @ extended_enat ) )
=> ( ? [K2: extended_enat,N7: extended_enat] :
( ( P @ ( coinductive_epred @ M2 ) @ N7 )
& ( ( coinductive_epred @ N4 )
= ( plus_plus @ extended_enat @ N7 @ K2 ) ) )
| ( ord_less_eq @ extended_enat @ ( coinductive_epred @ M2 ) @ ( coinductive_epred @ N4 ) ) ) ) ) ) )
=> ( ord_less_eq @ extended_enat @ M @ N ) ) ) ).
% enat_le_coinduct
thf(fact_151_size__ne__size__imp__ne,axiom,
! [A: $tType] :
( ( size @ A @ ( type2 @ A ) )
=> ! [X2: A,Y: A] :
( ( ( size_size @ A @ X2 )
!= ( size_size @ A @ Y ) )
=> ( X2 != Y ) ) ) ).
% size_ne_size_imp_ne
thf(fact_152_natural_Oinject,axiom,
! [Natural: code_natural,Natural2: code_natural] :
( ( ( code_Suc @ Natural )
= ( code_Suc @ Natural2 ) )
= ( Natural = Natural2 ) ) ).
% natural.inject
thf(fact_153_le__zero__eq,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [N: A] :
( ( ord_less_eq @ A @ N @ ( zero_zero @ A ) )
= ( N
= ( zero_zero @ A ) ) ) ) ).
% le_zero_eq
thf(fact_154_add__le__cancel__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_cancel_right
thf(fact_155_add__le__cancel__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_cancel_left
thf(fact_156_eSuc__ile__mono,axiom,
! [N: extended_enat,M: extended_enat] :
( ( ord_less_eq @ extended_enat @ ( extended_eSuc @ N ) @ ( extended_eSuc @ M ) )
= ( ord_less_eq @ extended_enat @ N @ M ) ) ).
% eSuc_ile_mono
thf(fact_157_natural_Osimps_I7_J,axiom,
! [T: $tType,F1: T,F2: code_natural > T > T,Natural: code_natural] :
( ( code_rec_natural @ T @ F1 @ F2 @ ( code_Suc @ Natural ) )
= ( F2 @ Natural @ ( code_rec_natural @ T @ F1 @ F2 @ Natural ) ) ) ).
% natural.simps(7)
thf(fact_158_add__le__same__cancel1,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_le_same_cancel1
thf(fact_159_add__le__same__cancel2,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_le_same_cancel2
thf(fact_160_le__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% le_add_same_cancel1
thf(fact_161_le__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ B2 @ A2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% le_add_same_cancel2
thf(fact_162_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_163_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_164_nat__add__left__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( plus_plus @ nat @ K @ M )
= ( plus_plus @ nat @ K @ N ) )
= ( M = N ) ) ).
% nat_add_left_cancel
thf(fact_165_nat__add__right__cancel,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( plus_plus @ nat @ M @ K )
= ( plus_plus @ nat @ N @ K ) )
= ( M = N ) ) ).
% nat_add_right_cancel
thf(fact_166_add__mono__thms__linordered__semiring_I3_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A @ ( type2 @ A ) )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( K = L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_167_add__mono__thms__linordered__semiring_I2_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A @ ( type2 @ A ) )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_168_add__mono__thms__linordered__semiring_I1_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A @ ( type2 @ A ) )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_169_add__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C2: A,D: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D ) ) ) ) ) ).
% add_mono
thf(fact_170_add__left__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% add_left_mono
thf(fact_171_add__right__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% add_right_mono
thf(fact_172_le__iff__add,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [A3: A,B3: A] :
? [C3: A] :
( B3
= ( plus_plus @ A @ A3 @ C3 ) ) ) ) ) ).
% le_iff_add
thf(fact_173_add__le__imp__le__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_imp_le_left
thf(fact_174_add__le__imp__le__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_imp_le_right
thf(fact_175_i0__lb,axiom,
! [N: extended_enat] : ( ord_less_eq @ extended_enat @ ( zero_zero @ extended_enat ) @ N ) ).
% i0_lb
thf(fact_176_ile0__eq,axiom,
! [N: extended_enat] :
( ( ord_less_eq @ extended_enat @ N @ ( zero_zero @ extended_enat ) )
= ( N
= ( zero_zero @ extended_enat ) ) ) ).
% ile0_eq
thf(fact_177_eSuc__le__iff,axiom,
! [X2: extended_enat,Y: extended_enat] :
( ( ord_less_eq @ extended_enat @ ( extended_eSuc @ X2 ) @ Y )
= ( ? [Y3: extended_enat] :
( ( Y
= ( extended_eSuc @ Y3 ) )
& ( ord_less_eq @ extended_enat @ X2 @ Y3 ) ) ) ) ).
% eSuc_le_iff
thf(fact_178_ile__eSuc,axiom,
! [N: extended_enat] : ( ord_less_eq @ extended_enat @ N @ ( extended_eSuc @ N ) ) ).
% ile_eSuc
thf(fact_179_enat__le__plus__same_I1_J,axiom,
! [X2: extended_enat,Y: extended_enat] : ( ord_less_eq @ extended_enat @ X2 @ ( plus_plus @ extended_enat @ X2 @ Y ) ) ).
% enat_le_plus_same(1)
thf(fact_180_enat__le__plus__same_I2_J,axiom,
! [X2: extended_enat,Y: extended_enat] : ( ord_less_eq @ extended_enat @ X2 @ ( plus_plus @ extended_enat @ Y @ X2 ) ) ).
% enat_le_plus_same(2)
thf(fact_181_epred__le__epredI,axiom,
! [N: extended_enat,M: extended_enat] :
( ( ord_less_eq @ extended_enat @ N @ M )
=> ( ord_less_eq @ extended_enat @ ( coinductive_epred @ N ) @ ( coinductive_epred @ M ) ) ) ).
% epred_le_epredI
thf(fact_182_zero__le,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [X2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 ) ) ).
% zero_le
thf(fact_183_size__enat__overloaded__def,axiom,
( ( size_size @ extended_enat )
= ( extended_rec_enat @ nat
@ ^ [X: nat] : ( zero_zero @ nat )
@ ( zero_zero @ nat ) ) ) ).
% size_enat_overloaded_def
thf(fact_184_natural_Odistinct_I2_J,axiom,
! [Natural3: code_natural] :
( ( code_Suc @ Natural3 )
!= ( zero_zero @ code_natural ) ) ).
% natural.distinct(2)
thf(fact_185_natural_Odistinct_I1_J,axiom,
! [Natural2: code_natural] :
( ( zero_zero @ code_natural )
!= ( code_Suc @ Natural2 ) ) ).
% natural.distinct(1)
thf(fact_186_natural_Oexhaust,axiom,
! [Y: code_natural] :
( ( Y
!= ( zero_zero @ code_natural ) )
=> ~ ! [Natural4: code_natural] :
( Y
!= ( code_Suc @ Natural4 ) ) ) ).
% natural.exhaust
thf(fact_187_natural_Oinducts,axiom,
! [P: code_natural > $o,Natural: code_natural] :
( ( P @ ( zero_zero @ code_natural ) )
=> ( ! [Natural4: code_natural] :
( ( P @ Natural4 )
=> ( P @ ( code_Suc @ Natural4 ) ) )
=> ( P @ Natural ) ) ) ).
% natural.inducts
thf(fact_188_add__decreasing,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B2 ) ) ) ) ).
% add_decreasing
thf(fact_189_add__increasing,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) ) ) ) ) ).
% add_increasing
thf(fact_190_add__decreasing2,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A @ ( type2 @ A ) )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B2 ) ) ) ) ).
% add_decreasing2
thf(fact_191_add__increasing2,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A @ ( type2 @ A ) )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) ) ) ) ) ).
% add_increasing2
thf(fact_192_add__nonneg__nonneg,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_193_add__nonpos__nonpos,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_nonpos_nonpos
thf(fact_194_add__nonneg__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A @ ( type2 @ A ) )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ( plus_plus @ A @ X2 @ Y )
= ( zero_zero @ A ) )
= ( ( X2
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_195_add__nonpos__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A @ ( type2 @ A ) )
=> ! [X2: A,Y: A] :
( ( ord_less_eq @ A @ X2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) )
=> ( ( ( plus_plus @ A @ X2 @ Y )
= ( zero_zero @ A ) )
= ( ( X2
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_196_not__eSuc__ilei0,axiom,
! [N: extended_enat] :
~ ( ord_less_eq @ extended_enat @ ( extended_eSuc @ N ) @ ( zero_zero @ extended_enat ) ) ).
% not_eSuc_ilei0
thf(fact_197_random__aux__rec,axiom,
! [A: $tType,Random_aux: code_natural > A,Rhs: code_natural > A,K: code_natural] :
( ( ( Random_aux @ ( zero_zero @ code_natural ) )
= ( Rhs @ ( zero_zero @ code_natural ) ) )
=> ( ! [K4: code_natural] :
( ( Random_aux @ ( code_Suc @ K4 ) )
= ( Rhs @ ( code_Suc @ K4 ) ) )
=> ( ( Random_aux @ K )
= ( Rhs @ K ) ) ) ) ).
% random_aux_rec
thf(fact_198_random__aux__set_Oinduct,axiom,
! [B: $tType] :
( ( quickcheck_random @ B @ ( type2 @ B ) )
=> ! [P: code_natural > code_natural > $o,A0: code_natural,A12: code_natural] :
( ! [X1: code_natural] : ( P @ ( zero_zero @ code_natural ) @ X1 )
=> ( ! [I2: code_natural,J2: code_natural] :
( ! [X5: product_prod @ B @ ( product_unit > code_term )] : ( P @ I2 @ J2 )
=> ( P @ ( code_Suc @ I2 ) @ J2 ) )
=> ( P @ A0 @ A12 ) ) ) ) ).
% random_aux_set.induct
thf(fact_199_bool_Osize_I3_J,axiom,
( ( size_size @ $o @ $true )
= ( zero_zero @ nat ) ) ).
% bool.size(3)
thf(fact_200_le0,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% le0
thf(fact_201_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_202_bool_Osize_I4_J,axiom,
( ( size_size @ $o @ $false )
= ( zero_zero @ nat ) ) ).
% bool.size(4)
thf(fact_203_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_204_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ N @ M ) ) ).
% le_add1
thf(fact_205_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ M @ N ) ) ).
% le_add2
thf(fact_206_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_207_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_208_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_209_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_210_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_211_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_212_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_213_nat__le__iff__add,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M3: nat,N3: nat] :
? [K3: nat] :
( N3
= ( plus_plus @ nat @ M3 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_214_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% le_0_eq
thf(fact_215_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% less_eq_nat.simps(1)
thf(fact_216_less__eq__enat__def,axiom,
( ( ord_less_eq @ extended_enat )
= ( ^ [M3: extended_enat] :
( extended_case_enat @ $o
@ ^ [N1: nat] :
( extended_case_enat @ $o
@ ^ [M1: nat] : ( ord_less_eq @ nat @ M1 @ N1 )
@ $false
@ M3 )
@ $true ) ) ) ).
% less_eq_enat_def
thf(fact_217_size__bool,axiom,
( ( size_size @ $o )
= ( ^ [B3: $o] : ( zero_zero @ nat ) ) ) ).
% size_bool
thf(fact_218_sum_Osize__neq,axiom,
! [A: $tType,B: $tType,X2: sum_sum @ A @ B] :
( ( size_size @ ( sum_sum @ A @ B ) @ X2 )
!= ( zero_zero @ nat ) ) ).
% sum.size_neq
thf(fact_219_prod_Osize__neq,axiom,
! [A: $tType,B: $tType,X2: product_prod @ A @ B] :
( ( size_size @ ( product_prod @ A @ B ) @ X2 )
!= ( zero_zero @ nat ) ) ).
% prod.size_neq
thf(fact_220_size__enat__def,axiom,
( extended_size_enat
= ( extended_rec_enat @ nat
@ ^ [X: nat] : ( zero_zero @ nat )
@ ( zero_zero @ nat ) ) ) ).
% size_enat_def
thf(fact_221_bool_Osize_I1_J,axiom,
( ( size_bool @ $true )
= ( zero_zero @ nat ) ) ).
% bool.size(1)
thf(fact_222_bool_Osize_I2_J,axiom,
( ( size_bool @ $false )
= ( zero_zero @ nat ) ) ).
% bool.size(2)
thf(fact_223_le__refl,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% le_refl
thf(fact_224_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_225_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_226_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_227_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_228_size__bool__def,axiom,
( size_bool
= ( product_rec_bool @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) ) ).
% size_bool_def
thf(fact_229_ind__euclid,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A5: nat,B4: nat] :
( ( P @ A5 @ B4 )
= ( P @ B4 @ A5 ) )
=> ( ! [A5: nat] : ( P @ A5 @ ( zero_zero @ nat ) )
=> ( ! [A5: nat,B4: nat] :
( ( P @ A5 @ B4 )
=> ( P @ A5 @ ( plus_plus @ nat @ A5 @ B4 ) ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% ind_euclid
thf(fact_230_size__bool__overloaded__def,axiom,
( ( size_size @ $o )
= ( product_rec_bool @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) ) ).
% size_bool_overloaded_def
thf(fact_231_le__numeral__extra_I3_J,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% le_numeral_extra(3)
thf(fact_232_natural_Osize_I4_J,axiom,
! [Natural: code_natural] :
( ( size_size @ code_natural @ ( code_Suc @ Natural ) )
= ( plus_plus @ nat @ ( size_size @ code_natural @ Natural ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% natural.size(4)
thf(fact_233_nat_Oinject,axiom,
! [X22: nat,Y2: nat] :
( ( ( suc @ X22 )
= ( suc @ Y2 ) )
= ( X22 = Y2 ) ) ).
% nat.inject
thf(fact_234_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_235_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_236_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_237_natural_Osize_I2_J,axiom,
! [Natural: code_natural] :
( ( code_size_natural @ ( code_Suc @ Natural ) )
= ( plus_plus @ nat @ ( code_size_natural @ Natural ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% natural.size(2)
thf(fact_238_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_239_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_240_Suc__inject,axiom,
! [X2: nat,Y: nat] :
( ( ( suc @ X2 )
= ( suc @ Y ) )
=> ( X2 = Y ) ) ).
% Suc_inject
thf(fact_241_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_242_not0__implies__Suc,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ).
% not0_implies_Suc
thf(fact_243_old_Onat_Oinducts,axiom,
! [P: nat > $o,Nat: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [Nat3: nat] :
( ( P @ Nat3 )
=> ( P @ ( suc @ Nat3 ) ) )
=> ( P @ Nat ) ) ) ).
% old.nat.inducts
thf(fact_244_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y
!= ( zero_zero @ nat ) )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_245_Zero__not__Suc,axiom,
! [M: nat] :
( ( zero_zero @ nat )
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_246_Zero__neq__Suc,axiom,
! [M: nat] :
( ( zero_zero @ nat )
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_247_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= ( zero_zero @ nat ) ) ).
% Suc_neq_Zero
thf(fact_248_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_249_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X4: nat] : ( P @ X4 @ ( zero_zero @ nat ) )
=> ( ! [Y4: nat] : ( P @ ( zero_zero @ nat ) @ ( suc @ Y4 ) )
=> ( ! [X4: nat,Y4: nat] :
( ( P @ X4 @ Y4 )
=> ( P @ ( suc @ X4 ) @ ( suc @ Y4 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_250_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_251_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat
!= ( zero_zero @ nat ) ) ) ).
% nat.discI
%----Type constructors (92)
thf(tcon_Code__Numeral_Onatural___Code__Evaluation_Oterm__of,axiom,
code_term_of @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___HOL_Oequal,axiom,
cl_HOL_Oequal @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Evaluation_Oterm___Code__Evaluation_Oterm__of_1,axiom,
code_term_of @ code_term @ ( type2 @ code_term ) ).
thf(tcon_Code__Evaluation_Oterm___HOL_Oequal_2,axiom,
cl_HOL_Oequal @ code_term @ ( type2 @ code_term ) ).
thf(tcon_Product__Type_Ounit___Code__Evaluation_Oterm__of_3,axiom,
code_term_of @ product_unit @ ( type2 @ product_unit ) ).
thf(tcon_Product__Type_Ounit___HOL_Oequal_4,axiom,
cl_HOL_Oequal @ product_unit @ ( type2 @ product_unit ) ).
thf(tcon_Product__Type_Oprod___Code__Evaluation_Oterm__of_5,axiom,
! [A6: $tType,A7: $tType] :
( ( ( typerep @ A6 @ ( type2 @ A6 ) )
& ( typerep @ A7 @ ( type2 @ A7 ) ) )
=> ( code_term_of @ ( product_prod @ A6 @ A7 ) @ ( type2 @ ( product_prod @ A6 @ A7 ) ) ) ) ).
thf(tcon_Product__Type_Oprod___HOL_Oequal_6,axiom,
! [A6: $tType,A7: $tType] : ( cl_HOL_Oequal @ ( product_prod @ A6 @ A7 ) @ ( type2 @ ( product_prod @ A6 @ A7 ) ) ) ).
thf(tcon_Extended__Nat_Oenat___Code__Evaluation_Oterm__of_7,axiom,
code_term_of @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___HOL_Oequal_8,axiom,
cl_HOL_Oequal @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Stream_Ostream___Code__Evaluation_Oterm__of_9,axiom,
! [A6: $tType] :
( ( typerep @ A6 @ ( type2 @ A6 ) )
=> ( code_term_of @ ( stream @ A6 ) @ ( type2 @ ( stream @ A6 ) ) ) ) ).
thf(tcon_Stream_Ostream___HOL_Oequal_10,axiom,
! [A6: $tType] : ( cl_HOL_Oequal @ ( stream @ A6 ) @ ( type2 @ ( stream @ A6 ) ) ) ).
thf(tcon_Sum__Type_Osum___Code__Evaluation_Oterm__of_11,axiom,
! [A6: $tType,A7: $tType] :
( ( ( typerep @ A6 @ ( type2 @ A6 ) )
& ( typerep @ A7 @ ( type2 @ A7 ) ) )
=> ( code_term_of @ ( sum_sum @ A6 @ A7 ) @ ( type2 @ ( sum_sum @ A6 @ A7 ) ) ) ) ).
thf(tcon_Sum__Type_Osum___HOL_Oequal_12,axiom,
! [A6: $tType,A7: $tType] : ( cl_HOL_Oequal @ ( sum_sum @ A6 @ A7 ) @ ( type2 @ ( sum_sum @ A6 @ A7 ) ) ) ).
thf(tcon_HOL_Obool___Code__Evaluation_Oterm__of_13,axiom,
code_term_of @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___HOL_Oequal_14,axiom,
cl_HOL_Oequal @ $o @ ( type2 @ $o ) ).
thf(tcon_Set_Oset___Code__Evaluation_Oterm__of_15,axiom,
! [A6: $tType] :
( ( typerep @ A6 @ ( type2 @ A6 ) )
=> ( code_term_of @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ) ).
thf(tcon_Set_Oset___HOL_Oequal_16,axiom,
! [A6: $tType] :
( ( cl_HOL_Oequal @ A6 @ ( type2 @ A6 ) )
=> ( cl_HOL_Oequal @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ) ).
thf(tcon_Nat_Onat___Code__Evaluation_Oterm__of_17,axiom,
code_term_of @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___HOL_Oequal_18,axiom,
cl_HOL_Oequal @ nat @ ( type2 @ nat ) ).
thf(tcon_fun___Code__Evaluation_Oterm__of_19,axiom,
! [A6: $tType,A7: $tType] :
( ( ( typerep @ A6 @ ( type2 @ A6 ) )
& ( typerep @ A7 @ ( type2 @ A7 ) ) )
=> ( code_term_of @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___HOL_Oequal_20,axiom,
! [A6: $tType,A7: $tType] :
( ( ( enum @ A6 @ ( type2 @ A6 ) )
& ( cl_HOL_Oequal @ A7 @ ( type2 @ A7 ) ) )
=> ( cl_HOL_Oequal @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Typerep_Otyperep,axiom,
! [A6: $tType,A7: $tType] :
( ( ( typerep @ A6 @ ( type2 @ A6 ) )
& ( typerep @ A7 @ ( type2 @ A7 ) ) )
=> ( typerep @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Enum_Oenum,axiom,
! [A6: $tType,A7: $tType] :
( ( ( enum @ A6 @ ( type2 @ A6 ) )
& ( enum @ A7 @ ( type2 @ A7 ) ) )
=> ( enum @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_Nat_Onat___Typerep_Otyperep_21,axiom,
typerep @ nat @ ( type2 @ nat ) ).
thf(tcon_Set_Oset___Typerep_Otyperep_22,axiom,
! [A6: $tType] :
( ( typerep @ A6 @ ( type2 @ A6 ) )
=> ( typerep @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ) ).
thf(tcon_Set_Oset___Enum_Oenum_23,axiom,
! [A6: $tType] :
( ( enum @ A6 @ ( type2 @ A6 ) )
=> ( enum @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ) ).
thf(tcon_HOL_Obool___Typerep_Otyperep_24,axiom,
typerep @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Enum_Oenum_25,axiom,
enum @ $o @ ( type2 @ $o ) ).
thf(tcon_Sum__Type_Osum___Typerep_Otyperep_26,axiom,
! [A6: $tType,A7: $tType] :
( ( ( typerep @ A6 @ ( type2 @ A6 ) )
& ( typerep @ A7 @ ( type2 @ A7 ) ) )
=> ( typerep @ ( sum_sum @ A6 @ A7 ) @ ( type2 @ ( sum_sum @ A6 @ A7 ) ) ) ) ).
thf(tcon_Sum__Type_Osum___Enum_Oenum_27,axiom,
! [A6: $tType,A7: $tType] :
( ( ( enum @ A6 @ ( type2 @ A6 ) )
& ( enum @ A7 @ ( type2 @ A7 ) ) )
=> ( enum @ ( sum_sum @ A6 @ A7 ) @ ( type2 @ ( sum_sum @ A6 @ A7 ) ) ) ) ).
thf(tcon_Stream_Ostream___Typerep_Otyperep_28,axiom,
! [A6: $tType] :
( ( typerep @ A6 @ ( type2 @ A6 ) )
=> ( typerep @ ( stream @ A6 ) @ ( type2 @ ( stream @ A6 ) ) ) ) ).
thf(tcon_Extended__Nat_Oenat___Typerep_Otyperep_29,axiom,
typerep @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Product__Type_Oprod___Typerep_Otyperep_30,axiom,
! [A6: $tType,A7: $tType] :
( ( ( typerep @ A6 @ ( type2 @ A6 ) )
& ( typerep @ A7 @ ( type2 @ A7 ) ) )
=> ( typerep @ ( product_prod @ A6 @ A7 ) @ ( type2 @ ( product_prod @ A6 @ A7 ) ) ) ) ).
thf(tcon_Product__Type_Oprod___Enum_Oenum_31,axiom,
! [A6: $tType,A7: $tType] :
( ( ( enum @ A6 @ ( type2 @ A6 ) )
& ( enum @ A7 @ ( type2 @ A7 ) ) )
=> ( enum @ ( product_prod @ A6 @ A7 ) @ ( type2 @ ( product_prod @ A6 @ A7 ) ) ) ) ).
thf(tcon_Product__Type_Ounit___Typerep_Otyperep_32,axiom,
typerep @ product_unit @ ( type2 @ product_unit ) ).
thf(tcon_Product__Type_Ounit___Enum_Oenum_33,axiom,
enum @ product_unit @ ( type2 @ product_unit ) ).
thf(tcon_Code__Evaluation_Oterm___Typerep_Otyperep_34,axiom,
typerep @ code_term @ ( type2 @ code_term ) ).
thf(tcon_Code__Numeral_Onatural___Typerep_Otyperep_35,axiom,
typerep @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_fun___Quickcheck__Random_Orandom,axiom,
! [A6: $tType,A7: $tType] :
( ( ( code_term_of @ A6 @ ( type2 @ A6 ) )
& ( cl_HOL_Oequal @ A6 @ ( type2 @ A6 ) )
& ( quickcheck_random @ A7 @ ( type2 @ A7 ) ) )
=> ( quickcheck_random @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
semiri456707255roduct @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
ordere516151231imp_le @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere236663937imp_le @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
canoni770627133id_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add,axiom,
ordere779506340up_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add,axiom,
ordere216010020id_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add,axiom,
cancel1352612707id_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Olinordered__semidom,axiom,
linordered_semidom @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Quickcheck__Random_Orandom_36,axiom,
quickcheck_random @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1 @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Osemigroup__add,axiom,
semigroup_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Omonoid__add,axiom,
monoid_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ozero,axiom,
zero @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Nat_Osize,axiom,
size @ nat @ ( type2 @ nat ) ).
thf(tcon_Set_Oset___Quickcheck__Random_Orandom_37,axiom,
! [A6: $tType] :
( ( quickcheck_random @ A6 @ ( type2 @ A6 ) )
=> ( quickcheck_random @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ) ).
thf(tcon_HOL_Obool___Quickcheck__Random_Orandom_38,axiom,
quickcheck_random @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Nat_Osize_39,axiom,
size @ $o @ ( type2 @ $o ) ).
thf(tcon_Sum__Type_Osum___Quickcheck__Random_Orandom_40,axiom,
! [A6: $tType,A7: $tType] :
( ( ( quickcheck_random @ A6 @ ( type2 @ A6 ) )
& ( quickcheck_random @ A7 @ ( type2 @ A7 ) ) )
=> ( quickcheck_random @ ( sum_sum @ A6 @ A7 ) @ ( type2 @ ( sum_sum @ A6 @ A7 ) ) ) ) ).
thf(tcon_Sum__Type_Osum___Nat_Osize_41,axiom,
! [A6: $tType,A7: $tType] : ( size @ ( sum_sum @ A6 @ A7 ) @ ( type2 @ ( sum_sum @ A6 @ A7 ) ) ) ).
thf(tcon_Stream_Ostream___Quickcheck__Random_Orandom_42,axiom,
! [A6: $tType] :
( ( quickcheck_random @ A6 @ ( type2 @ A6 ) )
=> ( quickcheck_random @ ( stream @ A6 ) @ ( type2 @ ( stream @ A6 ) ) ) ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_43,axiom,
canoni770627133id_add @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Oordered__ab__semigroup__add_44,axiom,
ordere779506340up_add @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Oordered__comm__monoid__add_45,axiom,
ordere216010020id_add @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Quickcheck__Random_Orandom_46,axiom,
quickcheck_random @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__add_47,axiom,
ab_semigroup_add @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__add_48,axiom,
comm_monoid_add @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__1_49,axiom,
comm_semiring_1 @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Osemigroup__add_50,axiom,
semigroup_add @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Omonoid__add_51,axiom,
monoid_add @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Ozero_52,axiom,
zero @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Nat_Osize_53,axiom,
size @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Product__Type_Oprod___Quickcheck__Random_Orandom_54,axiom,
! [A6: $tType,A7: $tType] :
( ( ( quickcheck_random @ A6 @ ( type2 @ A6 ) )
& ( quickcheck_random @ A7 @ ( type2 @ A7 ) ) )
=> ( quickcheck_random @ ( product_prod @ A6 @ A7 ) @ ( type2 @ ( product_prod @ A6 @ A7 ) ) ) ) ).
thf(tcon_Product__Type_Oprod___Nat_Osize_55,axiom,
! [A6: $tType,A7: $tType] : ( size @ ( product_prod @ A6 @ A7 ) @ ( type2 @ ( product_prod @ A6 @ A7 ) ) ) ).
thf(tcon_Product__Type_Ounit___Quickcheck__Random_Orandom_56,axiom,
quickcheck_random @ product_unit @ ( type2 @ product_unit ) ).
thf(tcon_Product__Type_Ounit___Nat_Osize_57,axiom,
size @ product_unit @ ( type2 @ product_unit ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__monoid__add__imp__le_58,axiom,
ordere516151231imp_le @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add__imp__le_59,axiom,
ordere236663937imp_le @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add_60,axiom,
ordere779506340up_add @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__comm__monoid__add_61,axiom,
ordere216010020id_add @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__comm__monoid__add_62,axiom,
cancel1352612707id_add @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__semigroup__add_63,axiom,
cancel_semigroup_add @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Quickcheck__Random_Orandom_64,axiom,
quickcheck_random @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__add_65,axiom,
ab_semigroup_add @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__add_66,axiom,
comm_monoid_add @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1_67,axiom,
comm_semiring_1 @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Osemigroup__add_68,axiom,
semigroup_add @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Omonoid__add_69,axiom,
monoid_add @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Groups_Ozero_70,axiom,
zero @ code_natural @ ( type2 @ code_natural ) ).
thf(tcon_Code__Numeral_Onatural___Nat_Osize_71,axiom,
size @ code_natural @ ( type2 @ code_natural ) ).
%----Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X2: A,Y: A] :
( ( if @ A @ $false @ X2 @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X2: A,Y: A] :
( ( if @ A @ $true @ X2 @ Y )
= X2 ) ).
%----Conjectures (2)
thf(conj_0,hypothesis,
( linear1386806755on_alw @ a
@ ^ [Xs2: stream @ a] :
~ ( p @ Xs2 )
@ omega ) ).
thf(conj_1,conjecture,
( ( coindu1365464361scount @ a @ p @ omega )
= ( zero_zero @ extended_enat ) ) ).
%------------------------------------------------------------------------------