TPTP Problem File: DAT119^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : DAT119^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Coinductive list 1279
% 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_list__1279.p [Bla16]
% Status : Theorem
% Rating : 0.33 v8.1.0, 0.25 v7.5.0, 0.33 v7.2.0, 0.25 v7.1.0
% Syntax : Number of formulae : 355 ( 169 unt; 67 typ; 0 def)
% Number of atoms : 673 ( 393 equ; 0 cnn)
% Maximal formula atoms : 24 ( 2 avg)
% Number of connectives : 2810 ( 94 ~; 31 |; 45 &;2378 @)
% ( 0 <=>; 262 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 6 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 211 ( 211 >; 0 *; 0 +; 0 <<)
% Number of symbols : 67 ( 64 usr; 6 con; 0-8 aty)
% Number of variables : 802 ( 32 ^; 660 !; 57 ?; 802 :)
% ( 53 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:49:09.100
%------------------------------------------------------------------------------
%----Could-be-implicit typings (6)
thf(ty_t_Coinductive__List__Mirabelle__kmikjhschf_Ollist,type,
coindu1593790203_llist: $tType > $tType ).
thf(ty_t_Extended__Nat_Oenat,type,
extended_enat: $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 (61)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Omult__zero,type,
mult_zero:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Osemiring__1,type,
semiring_1:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Oordered__ring,type,
ordered_ring:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Osemigroup__mult,type,
semigroup_mult:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Ocomm__semiring__1,type,
comm_semiring_1:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Olinordered__ring,type,
linordered_ring:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Oordered__semiring,type,
ordered_semiring:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocomm__monoid__diff,type,
comm_monoid_diff:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oab__semigroup__mult,type,
ab_semigroup_mult:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Oordered__semiring__0,type,
ordered_semiring_0:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Olinordered__ring__strict,type,
linord581940658strict:
!>[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_Ocancel__ab__semigroup__add,type,
cancel146912293up_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Osemiring__no__zero__divisors,type,
semiri1193490041visors:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni770627133id_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Osemiring__no__zero__divisors__cancel,type,
semiri1923998003cancel:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ogen__lset,type,
coindu1928975208n_lset:
!>[A: $tType] : ( ( set @ A ) > ( coindu1593790203_llist @ A ) > ( set @ A ) ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Oiterates,type,
coindu1529956631erates:
!>[A: $tType] : ( ( A > A ) > A > ( coindu1593790203_llist @ A ) ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olappend,type,
coindu268472904append:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olfinite,type,
coindu1213758845finite:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollength,type,
coindu1018505716length:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > extended_enat ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollexord,type,
coindu300403952lexord:
!>[A: $tType] : ( ( A > A > $o ) > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_OLCons,type,
coindu1121789889_LCons:
!>[A: $tType] : ( A > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_OLNil,type,
coindu1598213697e_LNil:
!>[A: $tType] : ( coindu1593790203_llist @ A ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Ocase__llist,type,
coindu882539134_llist:
!>[B: $tType,A: $tType] : ( B > ( A > ( coindu1593790203_llist @ A ) > B ) > ( coindu1593790203_llist @ A ) > B ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Ocorec__llist,type,
coindu1244876290_llist:
!>[C: $tType,A: $tType] : ( ( C > $o ) > ( C > A ) > ( C > $o ) > ( C > ( coindu1593790203_llist @ A ) ) > ( C > C ) > C > ( coindu1593790203_llist @ A ) ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Opred__llist,type,
coindu44415597_llist:
!>[A: $tType] : ( ( A > $o ) > ( coindu1593790203_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olmember,type,
coindu567634248member:
!>[A: $tType] : ( A > ( coindu1593790203_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olsetp,type,
coindu1055736764_lsetp:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > A > $o ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ounfold__llist,type,
coindu1599971794_llist:
!>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > B ) > ( A > A ) > A > ( coindu1593790203_llist @ B ) ) ).
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_Extended__Nat_OeSuc,type,
extended_eSuc: extended_enat > extended_enat ).
thf(sy_c_Extended__Nat_Oenat,type,
extended_enat2: nat > 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_Oinfinity__class_Oinfinity,type,
extend1396239628finity:
!>[A: $tType] : A ).
thf(sy_c_Extended__Nat_Othe__enat,type,
extended_the_enat: extended_enat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Otimes__class_Otimes,type,
times_times:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux,type,
semiri532925092at_aux:
!>[A: $tType] : ( ( A > A ) > nat > A > A ) ).
thf(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : ( A > nat ) ).
thf(sy_c_Nat__Bijection_Otriangle,type,
nat_triangle: nat > nat ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
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_na____,type,
na: nat ).
thf(sy_v_thesis____,type,
thesis: $o ).
thf(sy_v_xsa____,type,
xsa: coindu1593790203_llist @ a ).
%----Relevant facts (252)
thf(fact_0_len,axiom,
( ( coindu1018505716length @ a @ xsa )
= ( extended_enat2 @ ( suc @ na ) ) ) ).
% len
thf(fact_1_llist_Oinject,axiom,
! [A: $tType,X21: A,X22: coindu1593790203_llist @ A,Y21: A,Y22: coindu1593790203_llist @ A] :
( ( ( coindu1121789889_LCons @ A @ X21 @ X22 )
= ( coindu1121789889_LCons @ A @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% llist.inject
thf(fact_2_lmember__code_I2_J,axiom,
! [A: $tType,X: A,Y: A,Ys: coindu1593790203_llist @ A] :
( ( coindu567634248member @ A @ X @ ( coindu1121789889_LCons @ A @ Y @ Ys ) )
= ( ( X = Y )
| ( coindu567634248member @ A @ X @ Ys ) ) ) ).
% lmember_code(2)
thf(fact_3_iterates_Ocode,axiom,
! [A: $tType] :
( ( coindu1529956631erates @ A )
= ( ^ [F: A > A,X2: A] : ( coindu1121789889_LCons @ A @ X2 @ ( coindu1529956631erates @ A @ F @ ( F @ X2 ) ) ) ) ) ).
% iterates.code
thf(fact_4_llist_Opred__inject_I2_J,axiom,
! [A: $tType,P: A > $o,A2: A,Aa: coindu1593790203_llist @ A] :
( ( coindu44415597_llist @ A @ P @ ( coindu1121789889_LCons @ A @ A2 @ Aa ) )
= ( ( P @ A2 )
& ( coindu44415597_llist @ A @ P @ Aa ) ) ) ).
% llist.pred_inject(2)
thf(fact_5_llexord__LCons__eq,axiom,
! [A: $tType,R: A > A > $o,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,X: A] :
( ( coindu300403952lexord @ A @ R @ Xs @ Ys )
=> ( coindu300403952lexord @ A @ R @ ( coindu1121789889_LCons @ A @ X @ Xs ) @ ( coindu1121789889_LCons @ A @ X @ Ys ) ) ) ).
% llexord_LCons_eq
thf(fact_6_llexord__LCons__less,axiom,
! [A: $tType,R: A > A > $o,X: A,Y: A,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( R @ X @ Y )
=> ( coindu300403952lexord @ A @ R @ ( coindu1121789889_LCons @ A @ X @ Xs ) @ ( coindu1121789889_LCons @ A @ Y @ Ys ) ) ) ).
% llexord_LCons_less
thf(fact_7_unfold__llist__eq__LCons,axiom,
! [A: $tType,B: $tType,IS_LNIL: B > $o,LHD: B > A,LTL: B > B,B2: B,X: A,Xs: coindu1593790203_llist @ A] :
( ( ( coindu1599971794_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ B2 )
= ( coindu1121789889_LCons @ A @ X @ Xs ) )
= ( ~ ( IS_LNIL @ B2 )
& ( X
= ( LHD @ B2 ) )
& ( Xs
= ( coindu1599971794_llist @ B @ A @ IS_LNIL @ LHD @ LTL @ ( LTL @ B2 ) ) ) ) ) ).
% unfold_llist_eq_LCons
thf(fact_8_lsetp_Ointros_I2_J,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,X: A,X3: A] :
( ( coindu1055736764_lsetp @ A @ Xs @ X )
=> ( coindu1055736764_lsetp @ A @ ( coindu1121789889_LCons @ A @ X3 @ Xs ) @ X ) ) ).
% lsetp.intros(2)
thf(fact_9_lsetp_Ointros_I1_J,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A] : ( coindu1055736764_lsetp @ A @ ( coindu1121789889_LCons @ A @ X @ Xs ) @ X ) ).
% lsetp.intros(1)
thf(fact_10_lsetp_Ocases,axiom,
! [A: $tType,A1: coindu1593790203_llist @ A,A22: A] :
( ( coindu1055736764_lsetp @ A @ A1 @ A22 )
=> ( ! [X4: A] :
( ? [Xs2: coindu1593790203_llist @ A] :
( A1
= ( coindu1121789889_LCons @ A @ X4 @ Xs2 ) )
=> ( A22 != X4 ) )
=> ~ ! [Xs2: coindu1593790203_llist @ A] :
( ? [X5: A] :
( A1
= ( coindu1121789889_LCons @ A @ X5 @ Xs2 ) )
=> ~ ( coindu1055736764_lsetp @ A @ Xs2 @ A22 ) ) ) ) ).
% lsetp.cases
thf(fact_11_lsetp_Osimps,axiom,
! [A: $tType] :
( ( coindu1055736764_lsetp @ A )
= ( ^ [A12: coindu1593790203_llist @ A,A23: A] :
( ? [X2: A,Xs3: coindu1593790203_llist @ A] :
( ( A12
= ( coindu1121789889_LCons @ A @ X2 @ Xs3 ) )
& ( A23 = X2 ) )
| ? [Xs3: coindu1593790203_llist @ A,X2: A,X6: A] :
( ( A12
= ( coindu1121789889_LCons @ A @ X6 @ Xs3 ) )
& ( A23 = X2 )
& ( coindu1055736764_lsetp @ A @ Xs3 @ X2 ) ) ) ) ) ).
% lsetp.simps
thf(fact_12_lsetp_Oinducts,axiom,
! [A: $tType,X1: coindu1593790203_llist @ A,X23: A,P: ( coindu1593790203_llist @ A ) > A > $o] :
( ( coindu1055736764_lsetp @ A @ X1 @ X23 )
=> ( ! [X4: A,Xs2: coindu1593790203_llist @ A] : ( P @ ( coindu1121789889_LCons @ A @ X4 @ Xs2 ) @ X4 )
=> ( ! [Xs2: coindu1593790203_llist @ A,X4: A,X5: A] :
( ( coindu1055736764_lsetp @ A @ Xs2 @ X4 )
=> ( ( P @ Xs2 @ X4 )
=> ( P @ ( coindu1121789889_LCons @ A @ X5 @ Xs2 ) @ X4 ) ) )
=> ( P @ X1 @ X23 ) ) ) ) ).
% lsetp.inducts
thf(fact_13_llist_Osimps_I5_J,axiom,
! [B: $tType,A: $tType,F1: B,F2: A > ( coindu1593790203_llist @ A ) > B,X21: A,X22: coindu1593790203_llist @ A] :
( ( coindu882539134_llist @ B @ A @ F1 @ F2 @ ( coindu1121789889_LCons @ A @ X21 @ X22 ) )
= ( F2 @ X21 @ X22 ) ) ).
% llist.simps(5)
thf(fact_14_unfold__llist_Octr_I2_J,axiom,
! [B: $tType,A: $tType,P2: A > $o,A2: A,G21: A > B,G22: A > A] :
( ~ ( P2 @ A2 )
=> ( ( coindu1599971794_llist @ A @ B @ P2 @ G21 @ G22 @ A2 )
= ( coindu1121789889_LCons @ B @ ( G21 @ A2 ) @ ( coindu1599971794_llist @ A @ B @ P2 @ G21 @ G22 @ ( G22 @ A2 ) ) ) ) ) ).
% unfold_llist.ctr(2)
thf(fact_15_Suc_Ohyps,axiom,
! [Xs: coindu1593790203_llist @ a] :
( ( ( coindu1018505716length @ a @ Xs )
= ( extended_enat2 @ na ) )
=> ( coindu1213758845finite @ a @ Xs ) ) ).
% Suc.hyps
thf(fact_16_enat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( extended_enat2 @ Nat )
= ( extended_enat2 @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% enat.inject
thf(fact_17_nat_Oinject,axiom,
! [X23: nat,Y2: nat] :
( ( ( suc @ X23 )
= ( suc @ Y2 ) )
= ( X23 = Y2 ) ) ).
% nat.inject
thf(fact_18_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_19_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_20_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_21_llength__LCons,axiom,
! [B: $tType,X: B,Xs: coindu1593790203_llist @ B] :
( ( coindu1018505716length @ B @ ( coindu1121789889_LCons @ B @ X @ Xs ) )
= ( extended_eSuc @ ( coindu1018505716length @ B @ Xs ) ) ) ).
% llength_LCons
thf(fact_22_llexord_Ocoinduct,axiom,
! [A: $tType,X7: ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o,X: coindu1593790203_llist @ A,Xa: coindu1593790203_llist @ A,R: A > A > $o] :
( ( X7 @ X @ Xa )
=> ( ! [X4: coindu1593790203_llist @ A,Xa2: coindu1593790203_llist @ A] :
( ( X7 @ X4 @ Xa2 )
=> ( ? [Xs4: coindu1593790203_llist @ A,Ys2: coindu1593790203_llist @ A,Xb: A] :
( ( X4
= ( coindu1121789889_LCons @ A @ Xb @ Xs4 ) )
& ( Xa2
= ( coindu1121789889_LCons @ A @ Xb @ Ys2 ) )
& ( ( X7 @ Xs4 @ Ys2 )
| ( coindu300403952lexord @ A @ R @ Xs4 @ Ys2 ) ) )
| ? [Xb: A,Y3: A,Xs4: coindu1593790203_llist @ A,Ys2: coindu1593790203_llist @ A] :
( ( X4
= ( coindu1121789889_LCons @ A @ Xb @ Xs4 ) )
& ( Xa2
= ( coindu1121789889_LCons @ A @ Y3 @ Ys2 ) )
& ( R @ Xb @ Y3 ) )
| ? [Ys2: coindu1593790203_llist @ A] :
( ( X4
= ( coindu1598213697e_LNil @ A ) )
& ( Xa2 = Ys2 ) ) ) )
=> ( coindu300403952lexord @ A @ R @ X @ Xa ) ) ) ).
% llexord.coinduct
thf(fact_23_llexord_Osimps,axiom,
! [A: $tType] :
( ( coindu300403952lexord @ A )
= ( ^ [R2: A > A > $o,A12: coindu1593790203_llist @ A,A23: coindu1593790203_llist @ A] :
( ? [Xs3: coindu1593790203_llist @ A,Ys3: coindu1593790203_llist @ A,X2: A] :
( ( A12
= ( coindu1121789889_LCons @ A @ X2 @ Xs3 ) )
& ( A23
= ( coindu1121789889_LCons @ A @ X2 @ Ys3 ) )
& ( coindu300403952lexord @ A @ R2 @ Xs3 @ Ys3 ) )
| ? [X2: A,Y4: A,Xs3: coindu1593790203_llist @ A,Ys3: coindu1593790203_llist @ A] :
( ( A12
= ( coindu1121789889_LCons @ A @ X2 @ Xs3 ) )
& ( A23
= ( coindu1121789889_LCons @ A @ Y4 @ Ys3 ) )
& ( R2 @ X2 @ Y4 ) )
| ? [Ys3: coindu1593790203_llist @ A] :
( ( A12
= ( coindu1598213697e_LNil @ A ) )
& ( A23 = Ys3 ) ) ) ) ) ).
% llexord.simps
thf(fact_24_llexord_Ocases,axiom,
! [A: $tType,R: A > A > $o,A1: coindu1593790203_llist @ A,A22: coindu1593790203_llist @ A] :
( ( coindu300403952lexord @ A @ R @ A1 @ A22 )
=> ( ! [Xs2: coindu1593790203_llist @ A,Ys4: coindu1593790203_llist @ A,X4: A] :
( ( A1
= ( coindu1121789889_LCons @ A @ X4 @ Xs2 ) )
=> ( ( A22
= ( coindu1121789889_LCons @ A @ X4 @ Ys4 ) )
=> ~ ( coindu300403952lexord @ A @ R @ Xs2 @ Ys4 ) ) )
=> ( ! [X4: A] :
( ? [Xs2: coindu1593790203_llist @ A] :
( A1
= ( coindu1121789889_LCons @ A @ X4 @ Xs2 ) )
=> ! [Y5: A] :
( ? [Ys4: coindu1593790203_llist @ A] :
( A22
= ( coindu1121789889_LCons @ A @ Y5 @ Ys4 ) )
=> ~ ( R @ X4 @ Y5 ) ) )
=> ~ ( ( A1
= ( coindu1598213697e_LNil @ A ) )
=> ! [Ys4: coindu1593790203_llist @ A] : A22 != Ys4 ) ) ) ) ).
% llexord.cases
thf(fact_25_eSuc__inject,axiom,
! [M: extended_enat,N: extended_enat] :
( ( ( extended_eSuc @ M )
= ( extended_eSuc @ N ) )
= ( M = N ) ) ).
% eSuc_inject
thf(fact_26_lfinite__code_I2_J,axiom,
! [B: $tType,X: B,Xs: coindu1593790203_llist @ B] :
( ( coindu1213758845finite @ B @ ( coindu1121789889_LCons @ B @ X @ Xs ) )
= ( coindu1213758845finite @ B @ Xs ) ) ).
% lfinite_code(2)
thf(fact_27_lfinite__LCons,axiom,
! [A: $tType,X: A,Xs: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ ( coindu1121789889_LCons @ A @ X @ Xs ) )
= ( coindu1213758845finite @ A @ Xs ) ) ).
% lfinite_LCons
thf(fact_28_lfinite__code_I1_J,axiom,
! [A: $tType] : ( coindu1213758845finite @ A @ ( coindu1598213697e_LNil @ A ) ) ).
% lfinite_code(1)
thf(fact_29_lfinite__LNil,axiom,
! [A: $tType] : ( coindu1213758845finite @ A @ ( coindu1598213697e_LNil @ A ) ) ).
% lfinite_LNil
thf(fact_30_lfinite_Ocases,axiom,
! [A: $tType,A2: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ A2 )
=> ( ( A2
!= ( coindu1598213697e_LNil @ A ) )
=> ~ ! [Xs2: coindu1593790203_llist @ A] :
( ? [X4: A] :
( A2
= ( coindu1121789889_LCons @ A @ X4 @ Xs2 ) )
=> ~ ( coindu1213758845finite @ A @ Xs2 ) ) ) ) ).
% lfinite.cases
thf(fact_31_lfinite_Osimps,axiom,
! [A: $tType] :
( ( coindu1213758845finite @ A )
= ( ^ [A3: coindu1593790203_llist @ A] :
( ( A3
= ( coindu1598213697e_LNil @ A ) )
| ? [Xs3: coindu1593790203_llist @ A,X2: A] :
( ( A3
= ( coindu1121789889_LCons @ A @ X2 @ Xs3 ) )
& ( coindu1213758845finite @ A @ Xs3 ) ) ) ) ) ).
% lfinite.simps
thf(fact_32_lfinite_Oinducts,axiom,
! [A: $tType,X: coindu1593790203_llist @ A,P: ( coindu1593790203_llist @ A ) > $o] :
( ( coindu1213758845finite @ A @ X )
=> ( ( P @ ( coindu1598213697e_LNil @ A ) )
=> ( ! [Xs2: coindu1593790203_llist @ A,X4: A] :
( ( coindu1213758845finite @ A @ Xs2 )
=> ( ( P @ Xs2 )
=> ( P @ ( coindu1121789889_LCons @ A @ X4 @ Xs2 ) ) ) )
=> ( P @ X ) ) ) ) ).
% lfinite.inducts
thf(fact_33_llist_Odistinct_I1_J,axiom,
! [A: $tType,X21: A,X22: coindu1593790203_llist @ A] :
( ( coindu1598213697e_LNil @ A )
!= ( coindu1121789889_LCons @ A @ X21 @ X22 ) ) ).
% llist.distinct(1)
thf(fact_34_llist_Oexhaust,axiom,
! [A: $tType,Y: coindu1593790203_llist @ A] :
( ( Y
!= ( coindu1598213697e_LNil @ A ) )
=> ~ ! [X212: A,X222: coindu1593790203_llist @ A] :
( Y
!= ( coindu1121789889_LCons @ A @ X212 @ X222 ) ) ) ).
% llist.exhaust
thf(fact_35_neq__LNil__conv,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( Xs
!= ( coindu1598213697e_LNil @ A ) )
= ( ? [X2: A,Xs5: coindu1593790203_llist @ A] :
( Xs
= ( coindu1121789889_LCons @ A @ X2 @ Xs5 ) ) ) ) ).
% neq_LNil_conv
thf(fact_36_lfinite__LConsI,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,X: A] :
( ( coindu1213758845finite @ A @ Xs )
=> ( coindu1213758845finite @ A @ ( coindu1121789889_LCons @ A @ X @ Xs ) ) ) ).
% lfinite_LConsI
thf(fact_37_llist_Osimps_I4_J,axiom,
! [A: $tType,B: $tType,F1: B,F2: A > ( coindu1593790203_llist @ A ) > B] :
( ( coindu882539134_llist @ B @ A @ F1 @ F2 @ ( coindu1598213697e_LNil @ A ) )
= F1 ) ).
% llist.simps(4)
thf(fact_38_enat__eSuc__iff,axiom,
! [Y: nat,X: extended_enat] :
( ( ( extended_enat2 @ Y )
= ( extended_eSuc @ X ) )
= ( ? [N2: nat] :
( ( Y
= ( suc @ N2 ) )
& ( ( extended_enat2 @ N2 )
= X ) ) ) ) ).
% enat_eSuc_iff
thf(fact_39_eSuc__enat__iff,axiom,
! [X: extended_enat,Y: nat] :
( ( ( extended_eSuc @ X )
= ( extended_enat2 @ Y ) )
= ( ? [N2: nat] :
( ( Y
= ( suc @ N2 ) )
& ( X
= ( extended_enat2 @ N2 ) ) ) ) ) ).
% eSuc_enat_iff
thf(fact_40_eSuc__enat,axiom,
! [N: nat] :
( ( extended_eSuc @ ( extended_enat2 @ N ) )
= ( extended_enat2 @ ( suc @ N ) ) ) ).
% eSuc_enat
thf(fact_41_unfold__llist_Octr_I1_J,axiom,
! [A: $tType,B: $tType,P2: A > $o,A2: A,G21: A > B,G22: A > A] :
( ( P2 @ A2 )
=> ( ( coindu1599971794_llist @ A @ B @ P2 @ G21 @ G22 @ A2 )
= ( coindu1598213697e_LNil @ B ) ) ) ).
% unfold_llist.ctr(1)
thf(fact_42_llist_Opred__inject_I1_J,axiom,
! [A: $tType,P: A > $o] : ( coindu44415597_llist @ A @ P @ ( coindu1598213697e_LNil @ A ) ) ).
% llist.pred_inject(1)
thf(fact_43_llexord__LNil,axiom,
! [A: $tType,R: A > A > $o,Ys: coindu1593790203_llist @ A] : ( coindu300403952lexord @ A @ R @ ( coindu1598213697e_LNil @ A ) @ Ys ) ).
% llexord_LNil
thf(fact_44_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( collect @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_46_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_47_ext,axiom,
! [B: $tType,A: $tType,F3: A > B,G: A > B] :
( ! [X4: A] :
( ( F3 @ X4 )
= ( G @ X4 ) )
=> ( F3 = G ) ) ).
% ext
thf(fact_48_lmember__code_I1_J,axiom,
! [A: $tType,X: A] :
~ ( coindu567634248member @ A @ X @ ( coindu1598213697e_LNil @ A ) ) ).
% lmember_code(1)
thf(fact_49_unfold__llist_Ocode,axiom,
! [B: $tType,A: $tType] :
( ( coindu1599971794_llist @ A @ B )
= ( ^ [P3: A > $o,G212: A > B,G222: A > A,A3: A] : ( if @ ( coindu1593790203_llist @ B ) @ ( P3 @ A3 ) @ ( coindu1598213697e_LNil @ B ) @ ( coindu1121789889_LCons @ B @ ( G212 @ A3 ) @ ( coindu1599971794_llist @ A @ B @ P3 @ G212 @ G222 @ ( G222 @ A3 ) ) ) ) ) ) ).
% unfold_llist.code
thf(fact_50_co_Oenat_Oinject,axiom,
! [X23: extended_enat,Y2: extended_enat] :
( ( ( extended_eSuc @ X23 )
= ( extended_eSuc @ Y2 ) )
= ( X23 = Y2 ) ) ).
% co.enat.inject
thf(fact_51_enat_Osimps_I6_J,axiom,
! [T: $tType,F1: nat > T,F2: T,Nat: nat] :
( ( extended_rec_enat @ T @ F1 @ F2 @ ( extended_enat2 @ Nat ) )
= ( F1 @ Nat ) ) ).
% enat.simps(6)
thf(fact_52_enat_Osimps_I4_J,axiom,
! [T: $tType,F1: nat > T,F2: T,Nat: nat] :
( ( extended_case_enat @ T @ F1 @ F2 @ ( extended_enat2 @ Nat ) )
= ( F1 @ Nat ) ) ).
% enat.simps(4)
thf(fact_53_lfinite__rev__induct,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,P: ( coindu1593790203_llist @ A ) > $o] :
( ( coindu1213758845finite @ A @ Xs )
=> ( ( P @ ( coindu1598213697e_LNil @ A ) )
=> ( ! [X4: A,Xs2: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ Xs2 )
=> ( ( P @ Xs2 )
=> ( P @ ( coindu268472904append @ A @ Xs2 @ ( coindu1121789889_LCons @ A @ X4 @ ( coindu1598213697e_LNil @ A ) ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% lfinite_rev_induct
thf(fact_54_llength__LNil,axiom,
! [A: $tType] :
( ( coindu1018505716length @ A @ ( coindu1598213697e_LNil @ A ) )
= ( zero_zero @ extended_enat ) ) ).
% llength_LNil
thf(fact_55_gen__lset__code_I1_J,axiom,
! [A: $tType,A4: set @ A] :
( ( coindu1928975208n_lset @ A @ A4 @ ( coindu1598213697e_LNil @ A ) )
= A4 ) ).
% gen_lset_code(1)
thf(fact_56_of__nat__aux_Osimps_I2_J,axiom,
! [A: $tType] :
( ( semiring_1 @ A @ ( type2 @ A ) )
=> ! [Inc: A > A,N: nat,I: A] :
( ( semiri532925092at_aux @ A @ Inc @ ( suc @ N ) @ I )
= ( semiri532925092at_aux @ A @ Inc @ N @ ( Inc @ I ) ) ) ) ).
% of_nat_aux.simps(2)
thf(fact_57_llist_Ocorec__code,axiom,
! [A: $tType,C: $tType] :
( ( coindu1244876290_llist @ C @ A )
= ( ^ [P3: C > $o,G212: C > A,Q22: C > $o,G221: C > ( coindu1593790203_llist @ A ),G2222: C > C,A3: C] : ( if @ ( coindu1593790203_llist @ A ) @ ( P3 @ A3 ) @ ( coindu1598213697e_LNil @ A ) @ ( coindu1121789889_LCons @ A @ ( G212 @ A3 ) @ ( if @ ( coindu1593790203_llist @ A ) @ ( Q22 @ A3 ) @ ( G221 @ A3 ) @ ( coindu1244876290_llist @ C @ A @ P3 @ G212 @ Q22 @ G221 @ G2222 @ ( G2222 @ A3 ) ) ) ) ) ) ) ).
% llist.corec_code
thf(fact_58_the__enat_Osimps,axiom,
! [N: nat] :
( ( extended_the_enat @ ( extended_enat2 @ N ) )
= N ) ).
% the_enat.simps
thf(fact_59_lappend__code_I2_J,axiom,
! [A: $tType,Xa: A,X: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu268472904append @ A @ ( coindu1121789889_LCons @ A @ Xa @ X ) @ Ys )
= ( coindu1121789889_LCons @ A @ Xa @ ( coindu268472904append @ A @ X @ Ys ) ) ) ).
% lappend_code(2)
thf(fact_60_lappend__code_I1_J,axiom,
! [A: $tType,Ys: coindu1593790203_llist @ A] :
( ( coindu268472904append @ A @ ( coindu1598213697e_LNil @ A ) @ Ys )
= Ys ) ).
% lappend_code(1)
thf(fact_61_lappend__LNil2,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu268472904append @ A @ Xs @ ( coindu1598213697e_LNil @ A ) )
= Xs ) ).
% lappend_LNil2
thf(fact_62_lfinite__lappend,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1213758845finite @ A @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( ( coindu1213758845finite @ A @ Xs )
& ( coindu1213758845finite @ A @ Ys ) ) ) ).
% lfinite_lappend
thf(fact_63_lappend__assoc,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,Zs: coindu1593790203_llist @ A] :
( ( coindu268472904append @ A @ ( coindu268472904append @ A @ Xs @ Ys ) @ Zs )
= ( coindu268472904append @ A @ Xs @ ( coindu268472904append @ A @ Ys @ Zs ) ) ) ).
% lappend_assoc
thf(fact_64_co_Oenat_Odistinct_I1_J,axiom,
! [X23: extended_enat] :
( ( zero_zero @ extended_enat )
!= ( extended_eSuc @ X23 ) ) ).
% co.enat.distinct(1)
thf(fact_65_co_Oenat_OdiscI,axiom,
! [Enat: extended_enat,X23: extended_enat] :
( ( Enat
= ( extended_eSuc @ X23 ) )
=> ( Enat
!= ( zero_zero @ extended_enat ) ) ) ).
% co.enat.discI
thf(fact_66_enat__coexhaust,axiom,
! [N: extended_enat] :
( ( N
!= ( zero_zero @ extended_enat ) )
=> ~ ! [N3: extended_enat] :
( N
!= ( extended_eSuc @ N3 ) ) ) ).
% enat_coexhaust
thf(fact_67_co_Oenat_Oexhaust,axiom,
! [Y: extended_enat] :
( ( Y
!= ( zero_zero @ extended_enat ) )
=> ~ ! [X24: extended_enat] :
( Y
!= ( extended_eSuc @ X24 ) ) ) ).
% co.enat.exhaust
thf(fact_68_neq__zero__conv__eSuc,axiom,
! [N: extended_enat] :
( ( N
!= ( zero_zero @ extended_enat ) )
= ( ? [N4: extended_enat] :
( N
= ( extended_eSuc @ N4 ) ) ) ) ).
% neq_zero_conv_eSuc
thf(fact_69_lappend__LNil__LNil,axiom,
! [A: $tType] :
( ( coindu268472904append @ A @ ( coindu1598213697e_LNil @ A ) @ ( coindu1598213697e_LNil @ A ) )
= ( coindu1598213697e_LNil @ A ) ) ).
% lappend_LNil_LNil
thf(fact_70_LNil__eq__lappend__iff,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( ( coindu1598213697e_LNil @ A )
= ( coindu268472904append @ A @ Xs @ Ys ) )
= ( ( Xs
= ( coindu1598213697e_LNil @ A ) )
& ( Ys
= ( coindu1598213697e_LNil @ A ) ) ) ) ).
% LNil_eq_lappend_iff
thf(fact_71_lappend__eq__LNil__iff,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( ( coindu268472904append @ A @ Xs @ Ys )
= ( coindu1598213697e_LNil @ A ) )
= ( ( Xs
= ( coindu1598213697e_LNil @ A ) )
& ( Ys
= ( coindu1598213697e_LNil @ A ) ) ) ) ).
% lappend_eq_LNil_iff
thf(fact_72_lappend__inf,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ~ ( coindu1213758845finite @ A @ Xs )
=> ( ( coindu268472904append @ A @ Xs @ Ys )
= Xs ) ) ).
% lappend_inf
thf(fact_73_zero__ne__eSuc,axiom,
! [N: extended_enat] :
( ( zero_zero @ extended_enat )
!= ( extended_eSuc @ N ) ) ).
% zero_ne_eSuc
thf(fact_74_llist_Ocorec_I2_J,axiom,
! [A: $tType,C: $tType,P2: C > $o,A2: C,G21: C > A,Q222: C > $o,G2212: C > ( coindu1593790203_llist @ A ),G2223: C > C] :
( ~ ( P2 @ A2 )
=> ( ( coindu1244876290_llist @ C @ A @ P2 @ G21 @ Q222 @ G2212 @ G2223 @ A2 )
= ( coindu1121789889_LCons @ A @ ( G21 @ A2 ) @ ( if @ ( coindu1593790203_llist @ A ) @ ( Q222 @ A2 ) @ ( G2212 @ A2 ) @ ( coindu1244876290_llist @ C @ A @ P2 @ G21 @ Q222 @ G2212 @ G2223 @ ( G2223 @ A2 ) ) ) ) ) ) ).
% llist.corec(2)
thf(fact_75_llist_Ocorec_I1_J,axiom,
! [C: $tType,A: $tType,P2: C > $o,A2: C,G21: C > A,Q222: C > $o,G2212: C > ( coindu1593790203_llist @ A ),G2223: C > C] :
( ( P2 @ A2 )
=> ( ( coindu1244876290_llist @ C @ A @ P2 @ G21 @ Q222 @ G2212 @ G2223 @ A2 )
= ( coindu1598213697e_LNil @ A ) ) ) ).
% llist.corec(1)
thf(fact_76_lappend__snocL1__conv__LCons2,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Y: A,Ys: coindu1593790203_llist @ A] :
( ( coindu268472904append @ A @ ( coindu268472904append @ A @ Xs @ ( coindu1121789889_LCons @ A @ Y @ ( coindu1598213697e_LNil @ A ) ) ) @ Ys )
= ( coindu268472904append @ A @ Xs @ ( coindu1121789889_LCons @ A @ Y @ Ys ) ) ) ).
% lappend_snocL1_conv_LCons2
thf(fact_77_enat__set_Ocoinduct,axiom,
! [X7: extended_enat > $o,X: extended_enat] :
( ( X7 @ X )
=> ( ! [X4: extended_enat] :
( ( X7 @ X4 )
=> ( ( X4
= ( zero_zero @ extended_enat ) )
| ? [N5: extended_enat] :
( ( X4
= ( extended_eSuc @ N5 ) )
& ( ( X7 @ N5 )
| ( member @ extended_enat @ N5 @ coinductive_enat_set ) ) ) ) )
=> ( member @ extended_enat @ X @ coinductive_enat_set ) ) ) ).
% enat_set.coinduct
thf(fact_78_enat__set_Osimps,axiom,
! [A2: extended_enat] :
( ( member @ extended_enat @ A2 @ coinductive_enat_set )
= ( ( A2
= ( zero_zero @ extended_enat ) )
| ? [N2: extended_enat] :
( ( A2
= ( extended_eSuc @ N2 ) )
& ( member @ extended_enat @ N2 @ coinductive_enat_set ) ) ) ) ).
% enat_set.simps
thf(fact_79_enat__set_Ocases,axiom,
! [A2: extended_enat] :
( ( member @ extended_enat @ A2 @ coinductive_enat_set )
=> ( ( A2
!= ( zero_zero @ extended_enat ) )
=> ~ ! [N6: extended_enat] :
( ( A2
= ( extended_eSuc @ N6 ) )
=> ~ ( member @ extended_enat @ N6 @ coinductive_enat_set ) ) ) ) ).
% enat_set.cases
thf(fact_80_enat__setp_Ocoinduct,axiom,
! [X7: extended_enat > $o,X: extended_enat] :
( ( X7 @ X )
=> ( ! [X4: extended_enat] :
( ( X7 @ X4 )
=> ( ( X4
= ( zero_zero @ extended_enat ) )
| ? [N5: extended_enat] :
( ( X4
= ( extended_eSuc @ N5 ) )
& ( ( X7 @ N5 )
| ( coindu530039314t_setp @ N5 ) ) ) ) )
=> ( coindu530039314t_setp @ X ) ) ) ).
% enat_setp.coinduct
thf(fact_81_enat__setp_Osimps,axiom,
( coindu530039314t_setp
= ( ^ [A3: extended_enat] :
( ( A3
= ( zero_zero @ extended_enat ) )
| ? [N2: extended_enat] :
( ( A3
= ( extended_eSuc @ N2 ) )
& ( coindu530039314t_setp @ N2 ) ) ) ) ) ).
% enat_setp.simps
thf(fact_82_enat__setp_Ocases,axiom,
! [A2: extended_enat] :
( ( coindu530039314t_setp @ A2 )
=> ( ( A2
!= ( zero_zero @ extended_enat ) )
=> ~ ! [N6: extended_enat] :
( ( A2
= ( extended_eSuc @ N6 ) )
=> ~ ( coindu530039314t_setp @ N6 ) ) ) ) ).
% enat_setp.cases
thf(fact_83_case__enat__0,axiom,
! [A: $tType,F3: nat > A,I: A] :
( ( extended_case_enat @ A @ F3 @ I @ ( zero_zero @ extended_enat ) )
= ( F3 @ ( zero_zero @ nat ) ) ) ).
% case_enat_0
thf(fact_84_the__enat__0,axiom,
( ( extended_the_enat @ ( zero_zero @ extended_enat ) )
= ( zero_zero @ nat ) ) ).
% the_enat_0
thf(fact_85_enat__setp__enat__set__eq,axiom,
( coindu530039314t_setp
= ( ^ [X2: extended_enat] : ( member @ extended_enat @ X2 @ coinductive_enat_set ) ) ) ).
% enat_setp_enat_set_eq
thf(fact_86_not0__implies__Suc,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ).
% not0_implies_Suc
thf(fact_87_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_88_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y
!= ( zero_zero @ nat ) )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_89_Zero__not__Suc,axiom,
! [M: nat] :
( ( zero_zero @ nat )
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_90_Zero__neq__Suc,axiom,
! [M: nat] :
( ( zero_zero @ nat )
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_91_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= ( zero_zero @ nat ) ) ).
% Suc_neq_Zero
thf(fact_92_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N6: nat] :
( ( P @ ( suc @ N6 ) )
=> ( P @ N6 ) )
=> ( P @ ( zero_zero @ nat ) ) ) ) ).
% zero_induct
thf(fact_93_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X4: nat] : ( P @ X4 @ ( zero_zero @ nat ) )
=> ( ! [Y5: nat] : ( P @ ( zero_zero @ nat ) @ ( suc @ Y5 ) )
=> ( ! [X4: nat,Y5: nat] :
( ( P @ X4 @ Y5 )
=> ( P @ ( suc @ X4 ) @ ( suc @ Y5 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_94_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N6: nat] :
( ( P @ N6 )
=> ( P @ ( suc @ N6 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_95_nat_OdiscI,axiom,
! [Nat: nat,X23: nat] :
( ( Nat
= ( suc @ X23 ) )
=> ( Nat
!= ( zero_zero @ nat ) ) ) ).
% nat.discI
thf(fact_96_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( ( zero_zero @ nat )
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_97_old_Onat_Odistinct_I2_J,axiom,
! [Nat4: nat] :
( ( suc @ Nat4 )
!= ( zero_zero @ nat ) ) ).
% old.nat.distinct(2)
thf(fact_98_nat_Odistinct_I1_J,axiom,
! [X23: nat] :
( ( zero_zero @ nat )
!= ( suc @ X23 ) ) ).
% nat.distinct(1)
thf(fact_99_of__nat__aux_Osimps_I1_J,axiom,
! [A: $tType] :
( ( semiring_1 @ A @ ( type2 @ A ) )
=> ! [Inc: A > A,I: A] :
( ( semiri532925092at_aux @ A @ Inc @ ( zero_zero @ nat ) @ I )
= I ) ) ).
% of_nat_aux.simps(1)
thf(fact_100_enat__setp_Ointros_I1_J,axiom,
coindu530039314t_setp @ ( zero_zero @ extended_enat ) ).
% enat_setp.intros(1)
thf(fact_101_enat__setp_Ointros_I2_J,axiom,
! [N: extended_enat] :
( ( coindu530039314t_setp @ N )
=> ( coindu530039314t_setp @ ( extended_eSuc @ N ) ) ) ).
% enat_setp.intros(2)
thf(fact_102_enat__set_Ointros_I1_J,axiom,
member @ extended_enat @ ( zero_zero @ extended_enat ) @ coinductive_enat_set ).
% enat_set.intros(1)
thf(fact_103_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_104_enat__0__iff_I2_J,axiom,
! [X: nat] :
( ( ( zero_zero @ extended_enat )
= ( extended_enat2 @ X ) )
= ( X
= ( zero_zero @ nat ) ) ) ).
% enat_0_iff(2)
thf(fact_105_enat__0__iff_I1_J,axiom,
! [X: nat] :
( ( ( extended_enat2 @ X )
= ( zero_zero @ extended_enat ) )
= ( X
= ( zero_zero @ nat ) ) ) ).
% enat_0_iff(1)
thf(fact_106_zero__enat__def,axiom,
( ( zero_zero @ extended_enat )
= ( extended_enat2 @ ( zero_zero @ nat ) ) ) ).
% zero_enat_def
thf(fact_107_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( ( zero_zero @ A )
= X )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_108_enat_Osize_I1_J,axiom,
! [Nat: nat] :
( ( extended_size_enat @ ( extended_enat2 @ Nat ) )
= ( zero_zero @ nat ) ) ).
% enat.size(1)
thf(fact_109_enat_Osize_I3_J,axiom,
! [Nat: nat] :
( ( size_size @ extended_enat @ ( extended_enat2 @ Nat ) )
= ( zero_zero @ nat ) ) ).
% enat.size(3)
thf(fact_110_enat__unfold_Osimps,axiom,
! [A: $tType] :
( ( coindu1491768222unfold @ A )
= ( ^ [Stop: A > $o,Next: A > A,A3: A] : ( if @ extended_enat @ ( Stop @ A3 ) @ ( zero_zero @ extended_enat ) @ ( extended_eSuc @ ( coindu1491768222unfold @ A @ Stop @ Next @ ( Next @ A3 ) ) ) ) ) ) ).
% enat_unfold.simps
thf(fact_111_enat__unfold__unique,axiom,
! [A: $tType,Stop2: A > $o,H: A > extended_enat,Next2: A > A,X: A] :
( ! [X4: A] :
( ( ( Stop2 @ X4 )
=> ( ( H @ X4 )
= ( zero_zero @ extended_enat ) ) )
& ( ~ ( Stop2 @ X4 )
=> ( ( H @ X4 )
= ( extended_eSuc @ ( H @ ( Next2 @ X4 ) ) ) ) ) )
=> ( ( H @ X )
= ( coindu1491768222unfold @ A @ Stop2 @ Next2 @ X ) ) ) ).
% enat_unfold_unique
thf(fact_112_the__enat__eSuc,axiom,
! [N: extended_enat] :
( ( N
!= ( extend1396239628finity @ extended_enat ) )
=> ( ( extended_the_enat @ ( extended_eSuc @ N ) )
= ( suc @ ( extended_the_enat @ N ) ) ) ) ).
% the_enat_eSuc
thf(fact_113_list__decode_Ocases,axiom,
! [X: nat] :
( ( X
!= ( zero_zero @ nat ) )
=> ~ ! [N6: nat] :
( X
!= ( suc @ N6 ) ) ) ).
% list_decode.cases
thf(fact_114_not__enat__eq,axiom,
! [X: extended_enat] :
( ( ! [Y4: nat] :
( X
!= ( extended_enat2 @ Y4 ) ) )
= ( X
= ( extend1396239628finity @ extended_enat ) ) ) ).
% not_enat_eq
thf(fact_115_not__infinity__eq,axiom,
! [X: extended_enat] :
( ( X
!= ( extend1396239628finity @ extended_enat ) )
= ( ? [I2: nat] :
( X
= ( extended_enat2 @ I2 ) ) ) ) ).
% not_infinity_eq
thf(fact_116_eSuc__infinity,axiom,
( ( extended_eSuc @ ( extend1396239628finity @ extended_enat ) )
= ( extend1396239628finity @ extended_enat ) ) ).
% eSuc_infinity
thf(fact_117_enat_Osimps_I5_J,axiom,
! [T: $tType,F1: nat > T,F2: T] :
( ( extended_case_enat @ T @ F1 @ F2 @ ( extend1396239628finity @ extended_enat ) )
= F2 ) ).
% enat.simps(5)
thf(fact_118_enat__unfold__eq__0,axiom,
! [A: $tType,Stop2: A > $o,Next2: A > A,A2: A] :
( ( ( coindu1491768222unfold @ A @ Stop2 @ Next2 @ A2 )
= ( zero_zero @ extended_enat ) )
= ( Stop2 @ A2 ) ) ).
% enat_unfold_eq_0
thf(fact_119_enat__unfold__stop,axiom,
! [A: $tType,Stop2: A > $o,A2: A,Next2: A > A] :
( ( Stop2 @ A2 )
=> ( ( coindu1491768222unfold @ A @ Stop2 @ Next2 @ A2 )
= ( zero_zero @ extended_enat ) ) ) ).
% enat_unfold_stop
thf(fact_120_enat_Osimps_I7_J,axiom,
! [T: $tType,F1: nat > T,F2: T] :
( ( extended_rec_enat @ T @ F1 @ F2 @ ( extend1396239628finity @ extended_enat ) )
= F2 ) ).
% enat.simps(7)
thf(fact_121_enat_Osize_I2_J,axiom,
( ( extended_size_enat @ ( extend1396239628finity @ extended_enat ) )
= ( zero_zero @ nat ) ) ).
% enat.size(2)
thf(fact_122_enat_Osize_I4_J,axiom,
( ( size_size @ extended_enat @ ( extend1396239628finity @ extended_enat ) )
= ( zero_zero @ nat ) ) ).
% enat.size(4)
thf(fact_123_infinity__ne__i0,axiom,
( ( extend1396239628finity @ extended_enat )
!= ( zero_zero @ extended_enat ) ) ).
% infinity_ne_i0
thf(fact_124_enat_Odistinct_I2_J,axiom,
! [Nat5: nat] :
( ( extend1396239628finity @ extended_enat )
!= ( extended_enat2 @ Nat5 ) ) ).
% enat.distinct(2)
thf(fact_125_enat_Odistinct_I1_J,axiom,
! [Nat: nat] :
( ( extended_enat2 @ Nat )
!= ( extend1396239628finity @ extended_enat ) ) ).
% enat.distinct(1)
thf(fact_126_enat2__cases,axiom,
! [Y: extended_enat,Ya: extended_enat] :
( ( ? [Nat3: nat] :
( Y
= ( extended_enat2 @ Nat3 ) )
=> ! [Nata: nat] :
( Ya
!= ( extended_enat2 @ Nata ) ) )
=> ( ( ? [Nat3: nat] :
( Y
= ( extended_enat2 @ Nat3 ) )
=> ( Ya
!= ( extend1396239628finity @ extended_enat ) ) )
=> ( ( ( Y
= ( extend1396239628finity @ extended_enat ) )
=> ! [Nat3: nat] :
( Ya
!= ( extended_enat2 @ Nat3 ) ) )
=> ~ ( ( Y
= ( extend1396239628finity @ extended_enat ) )
=> ( Ya
!= ( extend1396239628finity @ extended_enat ) ) ) ) ) ) ).
% enat2_cases
thf(fact_127_enat3__cases,axiom,
! [Y: extended_enat,Ya: extended_enat,Yb: extended_enat] :
( ( ? [Nat3: nat] :
( Y
= ( extended_enat2 @ Nat3 ) )
=> ( ? [Nata: nat] :
( Ya
= ( extended_enat2 @ Nata ) )
=> ! [Natb: nat] :
( Yb
!= ( extended_enat2 @ Natb ) ) ) )
=> ( ( ? [Nat3: nat] :
( Y
= ( extended_enat2 @ Nat3 ) )
=> ( ? [Nata: nat] :
( Ya
= ( extended_enat2 @ Nata ) )
=> ( Yb
!= ( extend1396239628finity @ extended_enat ) ) ) )
=> ( ( ? [Nat3: nat] :
( Y
= ( extended_enat2 @ Nat3 ) )
=> ( ( Ya
= ( extend1396239628finity @ extended_enat ) )
=> ! [Nata: nat] :
( Yb
!= ( extended_enat2 @ Nata ) ) ) )
=> ( ( ? [Nat3: nat] :
( Y
= ( extended_enat2 @ Nat3 ) )
=> ( ( Ya
= ( extend1396239628finity @ extended_enat ) )
=> ( Yb
!= ( extend1396239628finity @ extended_enat ) ) ) )
=> ( ( ( Y
= ( extend1396239628finity @ extended_enat ) )
=> ( ? [Nat3: nat] :
( Ya
= ( extended_enat2 @ Nat3 ) )
=> ! [Nata: nat] :
( Yb
!= ( extended_enat2 @ Nata ) ) ) )
=> ( ( ( Y
= ( extend1396239628finity @ extended_enat ) )
=> ( ? [Nat3: nat] :
( Ya
= ( extended_enat2 @ Nat3 ) )
=> ( Yb
!= ( extend1396239628finity @ extended_enat ) ) ) )
=> ( ( ( Y
= ( extend1396239628finity @ extended_enat ) )
=> ( ( Ya
= ( extend1396239628finity @ extended_enat ) )
=> ! [Nat3: nat] :
( Yb
!= ( extended_enat2 @ Nat3 ) ) ) )
=> ~ ( ( Y
= ( extend1396239628finity @ extended_enat ) )
=> ( ( Ya
= ( extend1396239628finity @ extended_enat ) )
=> ( Yb
!= ( extend1396239628finity @ extended_enat ) ) ) ) ) ) ) ) ) ) ) ).
% enat3_cases
thf(fact_128_enat_Oexhaust,axiom,
! [Y: extended_enat] :
( ! [Nat3: nat] :
( Y
!= ( extended_enat2 @ Nat3 ) )
=> ( Y
= ( extend1396239628finity @ extended_enat ) ) ) ).
% enat.exhaust
thf(fact_129_enat_Oinducts,axiom,
! [P: extended_enat > $o,Enat: extended_enat] :
( ! [Nat3: nat] : ( P @ ( extended_enat2 @ Nat3 ) )
=> ( ( P @ ( extend1396239628finity @ extended_enat ) )
=> ( P @ Enat ) ) ) ).
% enat.inducts
thf(fact_130_enat__ex__split,axiom,
( ( ^ [P4: extended_enat > $o] :
? [X8: extended_enat] : ( P4 @ X8 ) )
= ( ^ [P5: extended_enat > $o] :
( ( P5 @ ( extend1396239628finity @ extended_enat ) )
| ? [X2: nat] : ( P5 @ ( extended_enat2 @ X2 ) ) ) ) ) ).
% enat_ex_split
thf(fact_131_eSuc__eq__infinity__iff,axiom,
! [N: extended_enat] :
( ( ( extended_eSuc @ N )
= ( extend1396239628finity @ extended_enat ) )
= ( N
= ( extend1396239628finity @ extended_enat ) ) ) ).
% eSuc_eq_infinity_iff
thf(fact_132_infinity__eq__eSuc__iff,axiom,
! [N: extended_enat] :
( ( ( extend1396239628finity @ extended_enat )
= ( extended_eSuc @ N ) )
= ( N
= ( extend1396239628finity @ extended_enat ) ) ) ).
% infinity_eq_eSuc_iff
thf(fact_133_enat__unfold__next,axiom,
! [A: $tType,Stop2: A > $o,A2: A,Next2: A > A] :
( ~ ( Stop2 @ A2 )
=> ( ( coindu1491768222unfold @ A @ Stop2 @ Next2 @ A2 )
= ( extended_eSuc @ ( coindu1491768222unfold @ A @ Stop2 @ Next2 @ ( Next2 @ A2 ) ) ) ) ) ).
% enat_unfold_next
thf(fact_134_enat__the__enat,axiom,
! [N: extended_enat] :
( ( N
!= ( extend1396239628finity @ extended_enat ) )
=> ( ( extended_enat2 @ ( extended_the_enat @ N ) )
= N ) ) ).
% enat_the_enat
thf(fact_135_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ ( zero_zero @ nat ) )
=> ( ? [X12: nat] : ( P @ X12 )
=> ? [N6: nat] :
( ~ ( P @ N6 )
& ( P @ ( suc @ N6 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_136_dependent__nat__choice,axiom,
! [A: $tType,P: nat > A > $o,Q: nat > A > A > $o] :
( ? [X12: A] : ( P @ ( zero_zero @ nat ) @ X12 )
=> ( ! [X4: A,N6: nat] :
( ( P @ N6 @ X4 )
=> ? [Y3: A] :
( ( P @ ( suc @ N6 ) @ Y3 )
& ( Q @ N6 @ X4 @ Y3 ) ) )
=> ? [F4: nat > A] :
! [N5: nat] :
( ( P @ N5 @ ( F4 @ N5 ) )
& ( Q @ N5 @ ( F4 @ N5 ) @ ( F4 @ ( suc @ N5 ) ) ) ) ) ) ).
% dependent_nat_choice
thf(fact_137_times__enat__simps_I3_J,axiom,
! [N: nat] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( times_times @ extended_enat @ ( extend1396239628finity @ extended_enat ) @ ( extended_enat2 @ N ) )
= ( zero_zero @ extended_enat ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( times_times @ extended_enat @ ( extend1396239628finity @ extended_enat ) @ ( extended_enat2 @ N ) )
= ( extend1396239628finity @ extended_enat ) ) ) ) ).
% times_enat_simps(3)
thf(fact_138_times__enat__simps_I4_J,axiom,
! [M: nat] :
( ( ( M
= ( zero_zero @ nat ) )
=> ( ( times_times @ extended_enat @ ( extended_enat2 @ M ) @ ( extend1396239628finity @ extended_enat ) )
= ( zero_zero @ extended_enat ) ) )
& ( ( M
!= ( zero_zero @ nat ) )
=> ( ( times_times @ extended_enat @ ( extended_enat2 @ M ) @ ( extend1396239628finity @ extended_enat ) )
= ( extend1396239628finity @ extended_enat ) ) ) ) ).
% times_enat_simps(4)
thf(fact_139_times__enat__simps_I2_J,axiom,
( ( times_times @ extended_enat @ ( extend1396239628finity @ extended_enat ) @ ( extend1396239628finity @ extended_enat ) )
= ( extend1396239628finity @ extended_enat ) ) ).
% times_enat_simps(2)
thf(fact_140_imult__is__0,axiom,
! [M: extended_enat,N: extended_enat] :
( ( ( times_times @ extended_enat @ M @ N )
= ( zero_zero @ extended_enat ) )
= ( ( M
= ( zero_zero @ extended_enat ) )
| ( N
= ( zero_zero @ extended_enat ) ) ) ) ).
% imult_is_0
thf(fact_141_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
= ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_142_mult_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_mult @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
= ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% mult.assoc
thf(fact_143_mult_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A @ ( type2 @ A ) )
=> ( ( times_times @ A )
= ( ^ [A3: A,B3: A] : ( times_times @ A @ B3 @ A3 ) ) ) ) ).
% mult.commute
thf(fact_144_mult_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C2: A] :
( ( times_times @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) )
= ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% mult.left_commute
thf(fact_145_imult__is__infinity,axiom,
! [A2: extended_enat,B2: extended_enat] :
( ( ( times_times @ extended_enat @ A2 @ B2 )
= ( extend1396239628finity @ extended_enat ) )
= ( ( ( A2
= ( extend1396239628finity @ extended_enat ) )
& ( B2
!= ( zero_zero @ extended_enat ) ) )
| ( ( B2
= ( extend1396239628finity @ extended_enat ) )
& ( A2
!= ( zero_zero @ extended_enat ) ) ) ) ) ).
% imult_is_infinity
thf(fact_146_mult__zero__left,axiom,
! [A: $tType] :
( ( mult_zero @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( times_times @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% mult_zero_left
thf(fact_147_mult__zero__right,axiom,
! [A: $tType] :
( ( mult_zero @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( times_times @ A @ A2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% mult_zero_right
thf(fact_148_mult__eq__0__iff,axiom,
! [A: $tType] :
( ( semiri1193490041visors @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ( times_times @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
= ( ( A2
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% mult_eq_0_iff
thf(fact_149_mult__cancel__left,axiom,
! [A: $tType] :
( ( semiri1923998003cancel @ A @ ( type2 @ A ) )
=> ! [C2: A,A2: A,B2: A] :
( ( ( times_times @ A @ C2 @ A2 )
= ( times_times @ A @ C2 @ B2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( A2 = B2 ) ) ) ) ).
% mult_cancel_left
thf(fact_150_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times @ nat @ M @ N )
= ( zero_zero @ nat ) )
= ( ( M
= ( zero_zero @ nat ) )
| ( N
= ( zero_zero @ nat ) ) ) ) ).
% mult_is_0
thf(fact_151_mult__0__right,axiom,
! [M: nat] :
( ( times_times @ nat @ M @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% mult_0_right
thf(fact_152_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times @ nat @ K @ M )
= ( times_times @ nat @ K @ N ) )
= ( ( M = N )
| ( K
= ( zero_zero @ nat ) ) ) ) ).
% mult_cancel1
thf(fact_153_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times @ nat @ M @ K )
= ( times_times @ nat @ N @ K ) )
= ( ( M = N )
| ( K
= ( zero_zero @ nat ) ) ) ) ).
% mult_cancel2
thf(fact_154_mult__cancel__right,axiom,
! [A: $tType] :
( ( semiri1923998003cancel @ A @ ( type2 @ A ) )
=> ! [A2: A,C2: A,B2: A] :
( ( ( times_times @ A @ A2 @ C2 )
= ( times_times @ A @ B2 @ C2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( A2 = B2 ) ) ) ) ).
% mult_cancel_right
thf(fact_155_mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times @ nat @ M @ N )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ( M
= ( suc @ ( zero_zero @ nat ) ) )
& ( N
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% mult_eq_1_iff
thf(fact_156_one__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ( suc @ ( zero_zero @ nat ) )
= ( times_times @ nat @ M @ N ) )
= ( ( M
= ( suc @ ( zero_zero @ nat ) ) )
& ( N
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% one_eq_mult_iff
thf(fact_157_times__enat__simps_I1_J,axiom,
! [M: nat,N: nat] :
( ( times_times @ extended_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
= ( extended_enat2 @ ( times_times @ nat @ M @ N ) ) ) ).
% times_enat_simps(1)
thf(fact_158_mult__0,axiom,
! [N: nat] :
( ( times_times @ nat @ ( zero_zero @ nat ) @ N )
= ( zero_zero @ nat ) ) ).
% mult_0
thf(fact_159_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times @ nat @ ( suc @ K ) @ M )
= ( times_times @ nat @ ( suc @ K ) @ N ) )
= ( M = N ) ) ).
% Suc_mult_cancel1
thf(fact_160_mult__right__cancel,axiom,
! [A: $tType] :
( ( semiri1923998003cancel @ A @ ( type2 @ A ) )
=> ! [C2: A,A2: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ A2 @ C2 )
= ( times_times @ A @ B2 @ C2 ) )
= ( A2 = B2 ) ) ) ) ).
% mult_right_cancel
thf(fact_161_mult__left__cancel,axiom,
! [A: $tType] :
( ( semiri1923998003cancel @ A @ ( type2 @ A ) )
=> ! [C2: A,A2: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ C2 @ A2 )
= ( times_times @ A @ C2 @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% mult_left_cancel
thf(fact_162_no__zero__divisors,axiom,
! [A: $tType] :
( ( semiri1193490041visors @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A2 @ B2 )
!= ( zero_zero @ A ) ) ) ) ) ).
% no_zero_divisors
thf(fact_163_divisors__zero,axiom,
! [A: $tType] :
( ( semiri1193490041visors @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ( times_times @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
=> ( ( A2
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% divisors_zero
thf(fact_164_mult__not__zero,axiom,
! [A: $tType] :
( ( mult_zero @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ( times_times @ A @ A2 @ B2 )
!= ( zero_zero @ A ) )
=> ( ( A2
!= ( zero_zero @ A ) )
& ( B2
!= ( zero_zero @ A ) ) ) ) ) ).
% mult_not_zero
thf(fact_165_semiring__normalization__rules_I9_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( times_times @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% semiring_normalization_rules(9)
thf(fact_166_semiring__normalization__rules_I10_J,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( times_times @ A @ A2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% semiring_normalization_rules(10)
thf(fact_167_idiff__infinity__right,axiom,
! [A2: nat] :
( ( minus_minus @ extended_enat @ ( extended_enat2 @ A2 ) @ ( extend1396239628finity @ extended_enat ) )
= ( zero_zero @ extended_enat ) ) ).
% idiff_infinity_right
thf(fact_168_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ A2 )
= ( zero_zero @ A ) ) ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_169_diff__zero,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% diff_zero
thf(fact_170_zero__diff,axiom,
! [A: $tType] :
( ( comm_monoid_diff @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( minus_minus @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% zero_diff
thf(fact_171_diff__0__right,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% diff_0_right
thf(fact_172_diff__self,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ A2 )
= ( zero_zero @ A ) ) ) ).
% diff_self
thf(fact_173_idiff__0__right,axiom,
! [N: extended_enat] :
( ( minus_minus @ extended_enat @ N @ ( zero_zero @ extended_enat ) )
= N ) ).
% idiff_0_right
thf(fact_174_idiff__0,axiom,
! [N: extended_enat] :
( ( minus_minus @ extended_enat @ ( zero_zero @ extended_enat ) @ N )
= ( zero_zero @ extended_enat ) ) ).
% idiff_0
thf(fact_175_eSuc__minus__eSuc,axiom,
! [N: extended_enat,M: extended_enat] :
( ( minus_minus @ extended_enat @ ( extended_eSuc @ N ) @ ( extended_eSuc @ M ) )
= ( minus_minus @ extended_enat @ N @ M ) ) ).
% eSuc_minus_eSuc
thf(fact_176_idiff__infinity,axiom,
! [N: extended_enat] :
( ( minus_minus @ extended_enat @ ( extend1396239628finity @ extended_enat ) @ N )
= ( extend1396239628finity @ extended_enat ) ) ).
% idiff_infinity
thf(fact_177_idiff__enat__0__right,axiom,
! [N: extended_enat] :
( ( minus_minus @ extended_enat @ N @ ( extended_enat2 @ ( zero_zero @ nat ) ) )
= N ) ).
% idiff_enat_0_right
thf(fact_178_idiff__enat__0,axiom,
! [N: extended_enat] :
( ( minus_minus @ extended_enat @ ( extended_enat2 @ ( zero_zero @ nat ) ) @ N )
= ( extended_enat2 @ ( zero_zero @ nat ) ) ) ).
% idiff_enat_0
thf(fact_179_idiff__self,axiom,
! [N: extended_enat] :
( ( N
!= ( extend1396239628finity @ extended_enat ) )
=> ( ( minus_minus @ extended_enat @ N @ N )
= ( zero_zero @ extended_enat ) ) ) ).
% idiff_self
thf(fact_180_eq__iff__diff__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ( ( ^ [Y6: A,Z: A] : Y6 = Z )
= ( ^ [A3: A,B3: A] :
( ( minus_minus @ A @ A3 @ B3 )
= ( zero_zero @ A ) ) ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_181_diff__eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C2: A,D: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= ( minus_minus @ A @ C2 @ D ) )
=> ( ( A2 = B2 )
= ( C2 = D ) ) ) ) ).
% diff_eq_diff_eq
thf(fact_182_diff__right__commute,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,C2: A,B2: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C2 ) @ B2 )
= ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% diff_right_commute
thf(fact_183_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times @ nat @ K @ M )
= ( times_times @ nat @ K @ N ) )
= ( ( K
= ( zero_zero @ nat ) )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_184_one__le__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( times_times @ nat @ M @ N ) )
= ( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M )
& ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_185_triangle__0,axiom,
( ( nat_triangle @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% triangle_0
thf(fact_186_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus @ nat @ M @ M )
= ( zero_zero @ nat ) ) ).
% diff_self_eq_0
thf(fact_187_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus @ nat @ ( zero_zero @ nat ) @ N )
= ( zero_zero @ nat ) ) ).
% diff_0_eq_0
thf(fact_188_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_189_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_190_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_191_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_192_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_193_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_194_le0,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% le0
thf(fact_195_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_196_idiff__enat__enat,axiom,
! [A2: nat,B2: nat] :
( ( minus_minus @ extended_enat @ ( extended_enat2 @ A2 ) @ ( extended_enat2 @ B2 ) )
= ( extended_enat2 @ ( minus_minus @ nat @ A2 @ B2 ) ) ) ).
% idiff_enat_enat
thf(fact_197_diff__ge__0__iff__ge,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% diff_ge_0_iff_ge
thf(fact_198_le__cube,axiom,
! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ ( times_times @ nat @ M @ M ) ) ) ).
% le_cube
thf(fact_199_le__square,axiom,
! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ M ) ) ).
% le_square
thf(fact_200_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ( ord_less_eq @ nat @ K @ L )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ I @ K ) @ ( times_times @ nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_201_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ I @ K ) @ ( times_times @ nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_202_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ K @ I ) @ ( times_times @ nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_203_diff__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times @ nat @ ( minus_minus @ nat @ M @ N ) @ K )
= ( minus_minus @ nat @ ( times_times @ nat @ M @ K ) @ ( times_times @ nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_204_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times @ nat @ K @ ( minus_minus @ nat @ M @ N ) )
= ( minus_minus @ nat @ ( times_times @ nat @ K @ M ) @ ( times_times @ nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_205_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus @ nat @ M @ ( zero_zero @ nat ) )
= M ) ).
% minus_nat.diff_0
thf(fact_206_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_207_prod__decode__aux_Oinduct,axiom,
! [P: nat > nat > $o,A0: nat,A1: nat] :
( ! [K2: nat,M2: nat] :
( ( ~ ( ord_less_eq @ nat @ M2 @ K2 )
=> ( P @ ( suc @ K2 ) @ ( minus_minus @ nat @ M2 @ ( suc @ K2 ) ) ) )
=> ( P @ K2 @ M2 ) )
=> ( P @ A0 @ A1 ) ) ).
% prod_decode_aux.induct
thf(fact_208_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N6: nat] :
( ( P @ ( suc @ N6 ) )
=> ( P @ N6 ) )
=> ( P @ ( minus_minus @ nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_209_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_210_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_211_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_212_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_213_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_214_le__diff__iff_H,axiom,
! [A2: nat,C2: nat,B2: nat] :
( ( ord_less_eq @ nat @ A2 @ C2 )
=> ( ( ord_less_eq @ nat @ B2 @ C2 )
=> ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ C2 @ A2 ) @ ( minus_minus @ nat @ C2 @ B2 ) )
= ( ord_less_eq @ nat @ B2 @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_215_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_216_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_217_diff__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,D: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ D @ C2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B2 @ D ) ) ) ) ) ).
% diff_mono
thf(fact_218_diff__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ C2 @ A2 ) @ ( minus_minus @ A @ C2 @ B2 ) ) ) ) ).
% diff_left_mono
thf(fact_219_diff__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B2 @ C2 ) ) ) ) ).
% diff_right_mono
thf(fact_220_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C2: A,D: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= ( minus_minus @ A @ C2 @ D ) )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
= ( ord_less_eq @ A @ C2 @ D ) ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_221_lift__Suc__mono__le,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [F3: nat > A,N: nat,N7: nat] :
( ! [N6: nat] : ( ord_less_eq @ A @ ( F3 @ N6 ) @ ( F3 @ ( suc @ N6 ) ) )
=> ( ( ord_less_eq @ nat @ N @ N7 )
=> ( ord_less_eq @ A @ ( F3 @ N ) @ ( F3 @ N7 ) ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_222_lift__Suc__antimono__le,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [F3: nat > A,N: nat,N7: nat] :
( ! [N6: nat] : ( ord_less_eq @ A @ ( F3 @ ( suc @ N6 ) ) @ ( F3 @ N6 ) )
=> ( ( ord_less_eq @ nat @ N @ N7 )
=> ( ord_less_eq @ A @ ( F3 @ N7 ) @ ( F3 @ N ) ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_223_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_224_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_225_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_226_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_227_le__refl,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% le_refl
thf(fact_228_wlog__linorder__le,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > A > $o,B2: A,A2: A] :
( ! [A5: A,B4: A] :
( ( ord_less_eq @ A @ A5 @ B4 )
=> ( P @ A5 @ B4 ) )
=> ( ( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% wlog_linorder_le
thf(fact_229_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N6: nat] :
( ! [M3: nat] :
( ( ord_less_eq @ nat @ ( suc @ M3 ) @ N6 )
=> ( P @ M3 ) )
=> ( P @ N6 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_230_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_231_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq @ nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_232_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_233_Suc__le__D,axiom,
! [N: nat,M4: nat] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ M4 )
=> ? [M2: nat] :
( M4
= ( suc @ M2 ) ) ) ).
% Suc_le_D
thf(fact_234_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_235_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_236_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% Suc_leD
thf(fact_237_zero__le,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% zero_le
thf(fact_238_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% less_eq_nat.simps(1)
thf(fact_239_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% le_0_eq
thf(fact_240_mult__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ 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 @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D ) ) ) ) ) ) ) ).
% mult_mono
thf(fact_241_mult__mono_H,axiom,
! [A: $tType] :
( ( ordered_semiring @ 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 @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D ) ) ) ) ) ) ) ).
% mult_mono'
thf(fact_242_zero__le__square,axiom,
! [A: $tType] :
( ( linordered_ring @ A @ ( type2 @ A ) )
=> ! [A2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ A2 ) ) ) ).
% zero_le_square
thf(fact_243_split__mult__pos__le,axiom,
! [A: $tType] :
( ( ordered_ring @ 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 @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) ) ) ).
% split_mult_pos_le
thf(fact_244_mult__left__mono__neg,axiom,
! [A: $tType] :
( ( ordered_ring @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% mult_left_mono_neg
thf(fact_245_mult__nonpos__nonpos,axiom,
! [A: $tType] :
( ( ordered_ring @ 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 @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_246_mult__left__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% mult_left_mono
thf(fact_247_mult__right__mono__neg,axiom,
! [A: $tType] :
( ( ordered_ring @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% mult_right_mono_neg
thf(fact_248_mult__right__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% mult_right_mono
thf(fact_249_mult__le__0__iff,axiom,
! [A: $tType] :
( ( linord581940658strict @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).
% mult_le_0_iff
thf(fact_250_split__mult__neg__le,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ).
% split_mult_neg_le
thf(fact_251_mult__nonneg__nonneg,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ 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 ) @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ).
% mult_nonneg_nonneg
%----Type constructors (31)
thf(tcon_fun___Orderings_Oorder,axiom,
! [A6: $tType,A7: $tType] :
( ( order @ A7 @ ( type2 @ A7 ) )
=> ( order @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel,axiom,
semiri1923998003cancel @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
canoni770627133id_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors,axiom,
semiri1193490041visors @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add,axiom,
cancel1352612707id_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Oordered__semiring__0,axiom,
ordered_semiring_0 @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oab__semigroup__mult,axiom,
ab_semigroup_mult @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__diff,axiom,
comm_monoid_diff @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Oordered__semiring,axiom,
ordered_semiring @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1 @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Osemigroup__mult,axiom,
semigroup_mult @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
linorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Osemiring__1,axiom,
semiring_1 @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Omult__zero,axiom,
mult_zero @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oorder_1,axiom,
order @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ozero,axiom,
zero @ nat @ ( type2 @ nat ) ).
thf(tcon_Set_Oset___Orderings_Oorder_2,axiom,
! [A6: $tType] : ( order @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder_3,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_4,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_5,axiom,
canoni770627133id_add @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__no__zero__divisors_6,axiom,
semiri1193490041visors @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__mult_7,axiom,
ab_semigroup_mult @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Rings_Oordered__semiring_8,axiom,
ordered_semiring @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__1_9,axiom,
comm_semiring_1 @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Osemigroup__mult_10,axiom,
semigroup_mult @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Orderings_Olinorder_11,axiom,
linorder @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__1_12,axiom,
semiring_1 @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Rings_Omult__zero_13,axiom,
mult_zero @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oorder_14,axiom,
order @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Ozero_15,axiom,
zero @ extended_enat @ ( type2 @ extended_enat ) ).
%----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,X: A,Y: A] :
( ( if @ A @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $true @ X @ Y )
= X ) ).
%----Conjectures (2)
thf(conj_0,hypothesis,
! [X9: a,Xs6: coindu1593790203_llist @ a] :
( ( xsa
= ( coindu1121789889_LCons @ a @ X9 @ Xs6 ) )
=> thesis ) ).
thf(conj_1,conjecture,
thesis ).
%------------------------------------------------------------------------------