TPTP Problem File: DAT123^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : DAT123^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Coinductive list 2041
% 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__2041.p [Bla16]
% Status : Theorem
% Rating : 0.33 v8.1.0, 0.25 v7.5.0, 1.00 v7.2.0, 0.75 v7.1.0
% Syntax : Number of formulae : 323 ( 91 unt; 41 typ; 0 def)
% Number of atoms : 857 ( 196 equ; 0 cnn)
% Maximal formula atoms : 13 ( 3 avg)
% Number of connectives : 3148 ( 86 ~; 21 |; 37 &;2541 @)
% ( 0 <=>; 463 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 8 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 146 ( 146 >; 0 *; 0 +; 0 <<)
% Number of symbols : 39 ( 38 usr; 4 con; 0-4 aty)
% Number of variables : 932 ( 31 ^; 828 !; 46 ?; 932 :)
% ( 27 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:54:22.534
%------------------------------------------------------------------------------
%----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 (35)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Ono__bot,type,
no_bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Ono__top,type,
no_top:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Owellorder,type,
wellorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Odense__order,type,
dense_order:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Odense__linorder,type,
dense_linorder:
!>[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_Oordered__ab__semigroup__add__imp__le,type,
ordere236663937imp_le:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
condit1656338222tinuum:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
condit1037483654norder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olnth,type,
coindu749330388e_lnth:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > nat > A ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Oltake,type,
coindu1802687541_ltake:
!>[A: $tType] : ( extended_enat > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
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_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_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
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_ma____,type,
ma: nat ).
thf(sy_v_na____,type,
na: extended_enat ).
thf(sy_v_thesis____,type,
thesis: $o ).
%----Relevant facts (255)
thf(fact_0_Suc_Oprems,axiom,
ord_less @ extended_enat @ ( extended_enat2 @ ( suc @ ma ) ) @ na ).
% Suc.prems
thf(fact_1_eSuc__inject,axiom,
! [M: extended_enat,N: extended_enat] :
( ( ( extended_eSuc @ M )
= ( extended_eSuc @ N ) )
= ( M = N ) ) ).
% eSuc_inject
thf(fact_2_co_Oenat_Oinject,axiom,
! [X2: extended_enat,Y2: extended_enat] :
( ( ( extended_eSuc @ X2 )
= ( extended_eSuc @ Y2 ) )
= ( X2 = Y2 ) ) ).
% co.enat.inject
thf(fact_3_Extended__Nat_OeSuc__mono,axiom,
! [N: extended_enat,M: extended_enat] :
( ( ord_less @ extended_enat @ ( extended_eSuc @ N ) @ ( extended_eSuc @ M ) )
= ( ord_less @ extended_enat @ N @ M ) ) ).
% Extended_Nat.eSuc_mono
thf(fact_4_eSuc__enat,axiom,
! [N: nat] :
( ( extended_eSuc @ ( extended_enat2 @ N ) )
= ( extended_enat2 @ ( suc @ N ) ) ) ).
% eSuc_enat
thf(fact_5_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_6_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_7_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_8_enat__setp_Ointros_I2_J,axiom,
! [N: extended_enat] :
( ( coindu530039314t_setp @ N )
=> ( coindu530039314t_setp @ ( extended_eSuc @ N ) ) ) ).
% enat_setp.intros(2)
thf(fact_9_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_10_co_Oenat_Ocase_I2_J,axiom,
! [A: $tType,F1: A,F2: extended_enat > A,X2: extended_enat] :
( ( coindu440805660e_enat @ A @ F1 @ F2 @ ( extended_eSuc @ X2 ) )
= ( F2 @ X2 ) ) ).
% co.enat.case(2)
thf(fact_11_enat__cocase__eSuc,axiom,
! [A: $tType,Z: A,S: extended_enat > A,N: extended_enat] :
( ( coindu440805660e_enat @ A @ Z @ S @ ( extended_eSuc @ N ) )
= ( S @ N ) ) ).
% enat_cocase_eSuc
thf(fact_12_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_13_enat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( extended_enat2 @ Nat )
= ( extended_enat2 @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% enat.inject
thf(fact_14_iless__Suc__eq,axiom,
! [M: nat,N: extended_enat] :
( ( ord_less @ extended_enat @ ( extended_enat2 @ M ) @ ( extended_eSuc @ N ) )
= ( ord_less_eq @ extended_enat @ ( extended_enat2 @ M ) @ N ) ) ).
% iless_Suc_eq
thf(fact_15_enat__ile,axiom,
! [N: extended_enat,M: nat] :
( ( ord_less_eq @ extended_enat @ N @ ( extended_enat2 @ M ) )
=> ? [K: nat] :
( N
= ( extended_enat2 @ K ) ) ) ).
% enat_ile
thf(fact_16_Suc__ile__eq,axiom,
! [M: nat,N: extended_enat] :
( ( ord_less_eq @ extended_enat @ ( extended_enat2 @ ( suc @ M ) ) @ N )
= ( ord_less @ extended_enat @ ( extended_enat2 @ M ) @ N ) ) ).
% Suc_ile_eq
thf(fact_17_chain__incr,axiom,
! [A: $tType,Y3: A > extended_enat,K2: nat] :
( ! [I: A] :
? [J: A] : ( ord_less @ extended_enat @ ( Y3 @ I ) @ ( Y3 @ J ) )
=> ? [J2: A] : ( ord_less @ extended_enat @ ( extended_enat2 @ K2 ) @ ( Y3 @ J2 ) ) ) ).
% chain_incr
thf(fact_18_enat__iless,axiom,
! [N: extended_enat,M: nat] :
( ( ord_less @ extended_enat @ N @ ( extended_enat2 @ M ) )
=> ? [K: nat] :
( N
= ( extended_enat2 @ K ) ) ) ).
% enat_iless
thf(fact_19_enat__less__induct,axiom,
! [P: extended_enat > $o,N: extended_enat] :
( ! [N3: extended_enat] :
( ! [M2: extended_enat] :
( ( ord_less @ extended_enat @ M2 @ N3 )
=> ( P @ M2 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% enat_less_induct
thf(fact_20_enat__less__imp__le,axiom,
! [N: extended_enat,M: extended_enat] :
( ! [K: nat] :
( ( ord_less @ extended_enat @ N @ ( extended_enat2 @ K ) )
=> ( ord_less @ extended_enat @ M @ ( extended_enat2 @ K ) ) )
=> ( ord_less_eq @ extended_enat @ M @ N ) ) ).
% enat_less_imp_le
thf(fact_21_enat__setp__enat__set__eq,axiom,
( coindu530039314t_setp
= ( ^ [X3: extended_enat] : ( member @ extended_enat @ X3 @ coinductive_enat_set ) ) ) ).
% enat_setp_enat_set_eq
thf(fact_22_wlog__linorder__le,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > A > $o,B: A,A2: A] :
( ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ( ( P @ B @ A2 )
=> ( P @ A2 @ B ) )
=> ( P @ A2 @ B ) ) ) ) ).
% wlog_linorder_le
thf(fact_23_ileI1,axiom,
! [M: extended_enat,N: extended_enat] :
( ( ord_less @ extended_enat @ M @ N )
=> ( ord_less_eq @ extended_enat @ ( extended_eSuc @ M ) @ N ) ) ).
% ileI1
thf(fact_24_eSuc__le__iff,axiom,
! [X: extended_enat,Y: extended_enat] :
( ( ord_less_eq @ extended_enat @ ( extended_eSuc @ X ) @ Y )
= ( ? [Y4: extended_enat] :
( ( Y
= ( extended_eSuc @ Y4 ) )
& ( ord_less_eq @ extended_enat @ X @ Y4 ) ) ) ) ).
% eSuc_le_iff
thf(fact_25_ile__eSuc,axiom,
! [N: extended_enat] : ( ord_less_eq @ extended_enat @ N @ ( extended_eSuc @ N ) ) ).
% ile_eSuc
thf(fact_26_nat_Oinject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
= ( X2 = Y2 ) ) ).
% nat.inject
thf(fact_27_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_28_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_29_lift__Suc__mono__less,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [F: nat > A,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less @ A @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less @ nat @ N @ N4 )
=> ( ord_less @ A @ ( F @ N ) @ ( F @ N4 ) ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_30_lift__Suc__mono__less__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [F: nat > A,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less @ A @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less @ A @ ( F @ N ) @ ( F @ M ) )
= ( ord_less @ nat @ N @ M ) ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_31_lift__Suc__mono__le,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [F: nat > A,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq @ nat @ N @ N4 )
=> ( ord_less_eq @ A @ ( F @ N ) @ ( F @ N4 ) ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_32_lift__Suc__antimono__le,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [F: nat > A,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq @ nat @ N @ N4 )
=> ( ord_less_eq @ A @ ( F @ N4 ) @ ( F @ N ) ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_33_minf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z2: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z2 )
=> ~ ( ord_less_eq @ A @ T2 @ X4 ) ) ) ).
% minf(8)
thf(fact_34_minf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z2: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z2 )
=> ( ord_less_eq @ A @ X4 @ T2 ) ) ) ).
% minf(6)
thf(fact_35_pinf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z2: A] :
! [X4: A] :
( ( ord_less @ A @ Z2 @ X4 )
=> ( ord_less_eq @ A @ T2 @ X4 ) ) ) ).
% pinf(8)
thf(fact_36_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less @ nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_37_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_38_lessI,axiom,
! [N: nat] : ( ord_less @ nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_39_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_40_enat__ord__simps_I1_J,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ extended_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
= ( ord_less_eq @ nat @ M @ N ) ) ).
% enat_ord_simps(1)
thf(fact_41_enat__ord__simps_I2_J,axiom,
! [M: nat,N: nat] :
( ( ord_less @ extended_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
= ( ord_less @ nat @ M @ N ) ) ).
% enat_ord_simps(2)
thf(fact_42_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less_eq @ nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_43_le__refl,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% le_refl
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
@ ^ [X3: A] : ( member @ A @ X3 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_46_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X5: A] :
( ( P @ X5 )
= ( Q @ X5 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_47_ext,axiom,
! [B3: $tType,A: $tType,F: A > B3,G: A > B3] :
( ! [X5: A] :
( ( F @ X5 )
= ( G @ X5 ) )
=> ( F = G ) ) ).
% ext
thf(fact_48_le__trans,axiom,
! [I2: nat,J3: nat,K2: nat] :
( ( ord_less_eq @ nat @ I2 @ J3 )
=> ( ( ord_less_eq @ nat @ J3 @ K2 )
=> ( ord_less_eq @ nat @ I2 @ K2 ) ) ) ).
% le_trans
thf(fact_49_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
= ( ord_less @ nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_50_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_51_dec__induct,axiom,
! [I2: nat,J3: nat,P: nat > $o] :
( ( ord_less_eq @ nat @ I2 @ J3 )
=> ( ( P @ I2 )
=> ( ! [N3: nat] :
( ( ord_less_eq @ nat @ I2 @ N3 )
=> ( ( ord_less @ nat @ N3 @ J3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J3 ) ) ) ) ).
% dec_induct
thf(fact_52_inc__induct,axiom,
! [I2: nat,J3: nat,P: nat > $o] :
( ( ord_less_eq @ nat @ I2 @ J3 )
=> ( ( P @ J3 )
=> ( ! [N3: nat] :
( ( ord_less_eq @ nat @ I2 @ N3 )
=> ( ( ord_less @ nat @ N3 @ J3 )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% inc_induct
thf(fact_53_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_54_nat__less__le,axiom,
( ( ord_less @ nat )
= ( ^ [M3: nat,N2: nat] :
( ( ord_less_eq @ nat @ M3 @ N2 )
& ( M3 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_55_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less @ nat @ M @ N )
| ( ord_less @ nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_56_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
=> ( ord_less @ nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_57_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_not_refl
thf(fact_58_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_59_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less @ nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_60_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ ( suc @ N ) )
= ( ord_less_eq @ nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_61_less__eq__Suc__le,axiom,
( ( ord_less @ nat )
= ( ^ [N2: nat] : ( ord_less_eq @ nat @ ( suc @ N2 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_62_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_63_less__not__refl3,axiom,
! [S: nat,T2: nat] :
( ( ord_less @ nat @ S @ T2 )
=> ( S != T2 ) ) ).
% less_not_refl3
thf(fact_64_measure__induct,axiom,
! [A: $tType,F: A > nat,P: A > $o,A2: A] :
( ! [X5: A] :
( ! [Y5: A] :
( ( ord_less @ nat @ ( F @ Y5 ) @ ( F @ X5 ) )
=> ( P @ Y5 ) )
=> ( P @ X5 ) )
=> ( P @ A2 ) ) ).
% measure_induct
thf(fact_65_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less @ nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_66_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_67_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_68_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M2: nat] :
( ( ord_less @ nat @ M2 @ N3 )
=> ( P @ M2 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_69_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M2: nat] :
( ( ord_less @ nat @ M2 @ N3 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_70_le__eq__less__or__eq,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M3: nat,N2: nat] :
( ( ord_less @ nat @ M3 @ N2 )
| ( M3 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_71_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less @ nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_72_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less @ nat @ X @ Y )
=> ( ord_less @ nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_73_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( M != N )
=> ( ord_less @ nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_74_measure__induct__rule,axiom,
! [A: $tType,F: A > nat,P: A > $o,A2: A] :
( ! [X5: A] :
( ! [Y5: A] :
( ( ord_less @ nat @ ( F @ Y5 ) @ ( F @ X5 ) )
=> ( P @ Y5 ) )
=> ( P @ X5 ) )
=> ( P @ A2 ) ) ).
% measure_induct_rule
thf(fact_75_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I2: nat,J3: nat] :
( ! [I: nat,J2: nat] :
( ( ord_less @ nat @ I @ J2 )
=> ( ord_less @ nat @ ( F @ I ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq @ nat @ I2 @ J3 )
=> ( ord_less_eq @ nat @ ( F @ I2 ) @ ( F @ J3 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_76_infinite__descent__measure,axiom,
! [A: $tType,P: A > $o,V: A > nat,X: A] :
( ! [X5: A] :
( ~ ( P @ X5 )
=> ? [Y5: A] :
( ( ord_less @ nat @ ( V @ Y5 ) @ ( V @ X5 ) )
& ~ ( P @ Y5 ) ) )
=> ( P @ X ) ) ).
% infinite_descent_measure
thf(fact_77_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less @ nat @ N @ M )
=> ( ( ord_less @ nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_78_strict__inc__induct,axiom,
! [I2: nat,J3: nat,P: nat > $o] :
( ( ord_less @ nat @ I2 @ J3 )
=> ( ! [I: nat] :
( ( J3
= ( suc @ I ) )
=> ( P @ I ) )
=> ( ! [I: nat] :
( ( ord_less @ nat @ I @ J3 )
=> ( ( P @ ( suc @ I ) )
=> ( P @ I ) ) )
=> ( P @ I2 ) ) ) ) ).
% strict_inc_induct
thf(fact_79_less__Suc__induct,axiom,
! [I2: nat,J3: nat,P: nat > nat > $o] :
( ( ord_less @ nat @ I2 @ J3 )
=> ( ! [I: nat] : ( P @ I @ ( suc @ I ) )
=> ( ! [I: nat,J2: nat,K: nat] :
( ( ord_less @ nat @ I @ J2 )
=> ( ( ord_less @ nat @ J2 @ K )
=> ( ( P @ I @ J2 )
=> ( ( P @ J2 @ K )
=> ( P @ I @ K ) ) ) ) )
=> ( P @ I2 @ J3 ) ) ) ) ).
% less_Suc_induct
thf(fact_80_less__trans__Suc,axiom,
! [I2: nat,J3: nat,K2: nat] :
( ( ord_less @ nat @ I2 @ J3 )
=> ( ( ord_less @ nat @ J3 @ K2 )
=> ( ord_less @ nat @ ( suc @ I2 ) @ K2 ) ) ) ).
% less_trans_Suc
thf(fact_81_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less @ nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_82_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less @ nat @ N @ M )
=> ( ( ord_less @ nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_83_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( suc @ N ) @ M )
= ( ? [M4: nat] :
( ( M
= ( suc @ M4 ) )
& ( ord_less @ nat @ N @ M4 ) ) ) ) ).
% Suc_less_eq2
thf(fact_84_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less @ nat @ M @ N ) )
= ( ord_less @ nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_85_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ ( suc @ N ) )
= ( ( ord_less @ nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_86_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less @ nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_87_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less @ nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_88_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less @ nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_89_Suc__lessE,axiom,
! [I2: nat,K2: nat] :
( ( ord_less @ nat @ ( suc @ I2 ) @ K2 )
=> ~ ! [J2: nat] :
( ( ord_less @ nat @ I2 @ J2 )
=> ( K2
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_90_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ M ) @ N )
=> ( ord_less @ nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_91_lessE,axiom,
! [I2: nat,K2: nat] :
( ( ord_less @ nat @ I2 @ K2 )
=> ( ( K2
!= ( suc @ I2 ) )
=> ~ ! [J2: nat] :
( ( ord_less @ nat @ I2 @ J2 )
=> ( K2
!= ( suc @ J2 ) ) ) ) ) ).
% lessE
thf(fact_92_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M2: nat] :
( ( ord_less_eq @ nat @ ( suc @ M2 ) @ N3 )
=> ( P @ M2 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_93_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_94_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq @ nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_95_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_96_Suc__le__D,axiom,
! [N: nat,M5: nat] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ M5 )
=> ? [M6: nat] :
( M5
= ( suc @ M6 ) ) ) ).
% Suc_le_D
thf(fact_97_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_98_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_99_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% Suc_leD
thf(fact_100_less__enatE,axiom,
! [N: extended_enat,M: nat] :
( ( ord_less @ extended_enat @ N @ ( extended_enat2 @ M ) )
=> ~ ! [K: nat] :
( ( N
= ( extended_enat2 @ K ) )
=> ~ ( ord_less @ nat @ K @ M ) ) ) ).
% less_enatE
thf(fact_101_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B: A,A2: A] :
( ( ord_less_eq @ A @ B @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ B )
=> ( A2 = B ) ) ) ) ).
% dual_order.antisym
thf(fact_102_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B: A,A2: A,C: A] :
( ( ord_less_eq @ A @ B @ A2 )
=> ( ( ord_less_eq @ A @ C @ B )
=> ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% dual_order.trans
thf(fact_103_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > A > $o,A2: A,B: A] :
( ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( P @ A3 @ B2 ) )
=> ( ! [A3: A,B2: A] :
( ( P @ B2 @ A3 )
=> ( P @ A3 @ B2 ) )
=> ( P @ A2 @ B ) ) ) ) ).
% linorder_wlog
thf(fact_104_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_105_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z )
=> ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% order_trans
thf(fact_106_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( ord_less_eq @ A @ B @ A2 )
=> ( A2 = B ) ) ) ) ).
% order_class.order.antisym
thf(fact_107_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% ord_le_eq_trans
thf(fact_108_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C: A] :
( ( A2 = B )
=> ( ( ord_less_eq @ A @ B @ C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% ord_eq_le_trans
thf(fact_109_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_less_eq @ A @ X @ Y )
= ( X = Y ) ) ) ) ).
% antisym_conv
thf(fact_110_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( ( ord_less_eq @ A @ X @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z ) )
=> ( ( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less_eq @ A @ X @ Z ) )
=> ( ( ( ord_less_eq @ A @ X @ Z )
=> ~ ( ord_less_eq @ A @ Z @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z )
=> ~ ( ord_less_eq @ A @ Z @ X ) )
=> ~ ( ( ord_less_eq @ A @ Z @ X )
=> ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_111_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( ord_less_eq @ A @ B @ C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% order.trans
thf(fact_112_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ~ ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% le_cases
thf(fact_113_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( X = Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% eq_refl
thf(fact_114_linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
| ( ord_less_eq @ A @ Y @ X ) ) ) ).
% linear
thf(fact_115_antisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ X )
=> ( X = Y ) ) ) ) ).
% antisym
thf(fact_116_eq__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ^ [Y6: A,Z3: A] : Y6 = Z3 )
= ( ^ [X3: A,Y7: A] :
( ( ord_less_eq @ A @ X3 @ Y7 )
& ( ord_less_eq @ A @ Y7 @ X3 ) ) ) ) ) ).
% eq_iff
thf(fact_117_ord__le__eq__subst,axiom,
! [A: $tType,B3: $tType] :
( ( ( ord @ B3 @ ( type2 @ B3 ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B: A,F: A > B3,C: B3] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X5: A,Y8: A] :
( ( ord_less_eq @ A @ X5 @ Y8 )
=> ( ord_less_eq @ B3 @ ( F @ X5 ) @ ( F @ Y8 ) ) )
=> ( ord_less_eq @ B3 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_118_ord__eq__le__subst,axiom,
! [A: $tType,B3: $tType] :
( ( ( ord @ B3 @ ( type2 @ B3 ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B3 > A,B: B3,C: B3] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq @ B3 @ B @ C )
=> ( ! [X5: B3,Y8: B3] :
( ( ord_less_eq @ B3 @ X5 @ Y8 )
=> ( ord_less_eq @ A @ ( F @ X5 ) @ ( F @ Y8 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_119_order__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 @ ( type2 @ C2 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B: A,F: A > C2,C: C2] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( ord_less_eq @ C2 @ ( F @ B ) @ C )
=> ( ! [X5: A,Y8: A] :
( ( ord_less_eq @ A @ X5 @ Y8 )
=> ( ord_less_eq @ C2 @ ( F @ X5 ) @ ( F @ Y8 ) ) )
=> ( ord_less_eq @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_subst2
thf(fact_120_order__subst1,axiom,
! [A: $tType,B3: $tType] :
( ( ( order @ B3 @ ( type2 @ B3 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B3 > A,B: B3,C: B3] :
( ( ord_less_eq @ A @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq @ B3 @ B @ C )
=> ( ! [X5: B3,Y8: B3] :
( ( ord_less_eq @ B3 @ X5 @ Y8 )
=> ( ord_less_eq @ A @ ( F @ X5 ) @ ( F @ Y8 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_subst1
thf(fact_121_le__fun__def,axiom,
! [B3: $tType,A: $tType] :
( ( ord @ B3 @ ( type2 @ B3 ) )
=> ( ( ord_less_eq @ ( A > B3 ) )
= ( ^ [F3: A > B3,G2: A > B3] :
! [X3: A] : ( ord_less_eq @ B3 @ ( F3 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% le_fun_def
thf(fact_122_le__funI,axiom,
! [B3: $tType,A: $tType] :
( ( ord @ B3 @ ( type2 @ B3 ) )
=> ! [F: A > B3,G: A > B3] :
( ! [X5: A] : ( ord_less_eq @ B3 @ ( F @ X5 ) @ ( G @ X5 ) )
=> ( ord_less_eq @ ( A > B3 ) @ F @ G ) ) ) ).
% le_funI
thf(fact_123_le__funE,axiom,
! [B3: $tType,A: $tType] :
( ( ord @ B3 @ ( type2 @ B3 ) )
=> ! [F: A > B3,G: A > B3,X: A] :
( ( ord_less_eq @ ( A > B3 ) @ F @ G )
=> ( ord_less_eq @ B3 @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% le_funE
thf(fact_124_le__funD,axiom,
! [B3: $tType,A: $tType] :
( ( ord @ B3 @ ( type2 @ B3 ) )
=> ! [F: A > B3,G: A > B3,X: A] :
( ( ord_less_eq @ ( A > B3 ) @ F @ G )
=> ( ord_less_eq @ B3 @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% le_funD
thf(fact_125_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B: A,A2: A] :
( ( ord_less @ A @ B @ A2 )
=> ( A2 != B ) ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_126_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ A2 @ B )
=> ( A2 != B ) ) ) ).
% order.strict_implies_not_eq
thf(fact_127_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less @ A @ X @ Y ) )
= ( ( ord_less @ A @ Y @ X )
| ( X = Y ) ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_128_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B: A,A2: A,C: A] :
( ( ord_less @ A @ B @ A2 )
=> ( ( ord_less @ A @ C @ B )
=> ( ord_less @ A @ C @ A2 ) ) ) ) ).
% dual_order.strict_trans
thf(fact_129_less__imp__not__less,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% less_imp_not_less
thf(fact_130_order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C: A] :
( ( ord_less @ A @ A2 @ B )
=> ( ( ord_less @ A @ B @ C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% order.strict_trans
thf(fact_131_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ A2 ) ) ).
% dual_order.irrefl
thf(fact_132_linorder__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ( X != Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% linorder_cases
thf(fact_133_less__imp__triv,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,P: $o] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ X )
=> P ) ) ) ).
% less_imp_triv
thf(fact_134_less__imp__not__eq2,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( Y != X ) ) ) ).
% less_imp_not_eq2
thf(fact_135_antisym__conv3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A] :
( ~ ( ord_less @ A @ Y @ X )
=> ( ( ~ ( ord_less @ A @ X @ Y ) )
= ( X = Y ) ) ) ) ).
% antisym_conv3
thf(fact_136_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,A2: A] :
( ! [X5: A] :
( ! [Y5: A] :
( ( ord_less @ A @ Y5 @ X5 )
=> ( P @ Y5 ) )
=> ( P @ X5 ) )
=> ( P @ A2 ) ) ) ).
% less_induct
thf(fact_137_less__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% less_not_sym
thf(fact_138_less__imp__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( X != Y ) ) ) ).
% less_imp_not_eq
thf(fact_139_dual__order_Oasym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B: A,A2: A] :
( ( ord_less @ A @ B @ A2 )
=> ~ ( ord_less @ A @ A2 @ B ) ) ) ).
% dual_order.asym
thf(fact_140_ord__less__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C: A] :
( ( ord_less @ A @ A2 @ B )
=> ( ( B = C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% ord_less_eq_trans
thf(fact_141_ord__eq__less__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C: A] :
( ( A2 = B )
=> ( ( ord_less @ A @ B @ C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% ord_eq_less_trans
thf(fact_142_less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A] :
~ ( ord_less @ A @ X @ X ) ) ).
% less_irrefl
thf(fact_143_less__linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
| ( X = Y )
| ( ord_less @ A @ Y @ X ) ) ) ).
% less_linear
thf(fact_144_less__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ Z )
=> ( ord_less @ A @ X @ Z ) ) ) ) ).
% less_trans
thf(fact_145_less__asym_H,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ A2 @ B )
=> ~ ( ord_less @ A @ B @ A2 ) ) ) ).
% less_asym'
thf(fact_146_less__asym,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% less_asym
thf(fact_147_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( X != Y ) ) ) ).
% less_imp_neq
thf(fact_148_dense,axiom,
! [A: $tType] :
( ( dense_order @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ? [Z2: A] :
( ( ord_less @ A @ X @ Z2 )
& ( ord_less @ A @ Z2 @ Y ) ) ) ) ).
% dense
thf(fact_149_order_Oasym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ A2 @ B )
=> ~ ( ord_less @ A @ B @ A2 ) ) ) ).
% order.asym
thf(fact_150_neq__iff,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( X != Y )
= ( ( ord_less @ A @ X @ Y )
| ( ord_less @ A @ Y @ X ) ) ) ) ).
% neq_iff
thf(fact_151_neqE,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( X != Y )
=> ( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% neqE
thf(fact_152_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A @ ( type2 @ A ) )
=> ! [X: A] :
? [X1: A] : ( ord_less @ A @ X @ X1 ) ) ).
% gt_ex
thf(fact_153_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A @ ( type2 @ A ) )
=> ! [X: A] :
? [Y8: A] : ( ord_less @ A @ Y8 @ X ) ) ).
% lt_ex
thf(fact_154_order__less__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 @ ( type2 @ C2 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B: A,F: A > C2,C: C2] :
( ( ord_less @ A @ A2 @ B )
=> ( ( ord_less @ C2 @ ( F @ B ) @ C )
=> ( ! [X5: A,Y8: A] :
( ( ord_less @ A @ X5 @ Y8 )
=> ( ord_less @ C2 @ ( F @ X5 ) @ ( F @ Y8 ) ) )
=> ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_less_subst2
thf(fact_155_order__less__subst1,axiom,
! [A: $tType,B3: $tType] :
( ( ( order @ B3 @ ( type2 @ B3 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B3 > A,B: B3,C: B3] :
( ( ord_less @ A @ A2 @ ( F @ B ) )
=> ( ( ord_less @ B3 @ B @ C )
=> ( ! [X5: B3,Y8: B3] :
( ( ord_less @ B3 @ X5 @ Y8 )
=> ( ord_less @ A @ ( F @ X5 ) @ ( F @ Y8 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_156_ord__less__eq__subst,axiom,
! [A: $tType,B3: $tType] :
( ( ( ord @ B3 @ ( type2 @ B3 ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B: A,F: A > B3,C: B3] :
( ( ord_less @ A @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X5: A,Y8: A] :
( ( ord_less @ A @ X5 @ Y8 )
=> ( ord_less @ B3 @ ( F @ X5 ) @ ( F @ Y8 ) ) )
=> ( ord_less @ B3 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_157_ord__eq__less__subst,axiom,
! [A: $tType,B3: $tType] :
( ( ( ord @ B3 @ ( type2 @ B3 ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B3 > A,B: B3,C: B3] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less @ B3 @ B @ C )
=> ( ! [X5: B3,Y8: B3] :
( ( ord_less @ B3 @ X5 @ Y8 )
=> ( ord_less @ A @ ( F @ X5 ) @ ( F @ Y8 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_158_pinf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ Z4 @ X5 )
=> ( ( P @ X5 )
= ( P2 @ X5 ) ) )
=> ( ? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ Z4 @ X5 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z2: A] :
! [X4: A] :
( ( ord_less @ A @ Z2 @ X4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P2 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ) ).
% pinf(1)
thf(fact_159_pinf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ Z4 @ X5 )
=> ( ( P @ X5 )
= ( P2 @ X5 ) ) )
=> ( ? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ Z4 @ X5 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z2: A] :
! [X4: A] :
( ( ord_less @ A @ Z2 @ X4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P2 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ) ).
% pinf(2)
thf(fact_160_pinf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z2: A] :
! [X4: A] :
( ( ord_less @ A @ Z2 @ X4 )
=> ( X4 != T2 ) ) ) ).
% pinf(3)
thf(fact_161_pinf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z2: A] :
! [X4: A] :
( ( ord_less @ A @ Z2 @ X4 )
=> ( X4 != T2 ) ) ) ).
% pinf(4)
thf(fact_162_pinf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z2: A] :
! [X4: A] :
( ( ord_less @ A @ Z2 @ X4 )
=> ~ ( ord_less @ A @ X4 @ T2 ) ) ) ).
% pinf(5)
thf(fact_163_pinf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z2: A] :
! [X4: A] :
( ( ord_less @ A @ Z2 @ X4 )
=> ( ord_less @ A @ T2 @ X4 ) ) ) ).
% pinf(7)
thf(fact_164_pinf_I11_J,axiom,
! [C2: $tType,D: $tType] :
( ( ord @ C2 @ ( type2 @ C2 ) )
=> ! [F4: D] :
? [Z2: C2] :
! [X4: C2] :
( ( ord_less @ C2 @ Z2 @ X4 )
=> ( F4 = F4 ) ) ) ).
% pinf(11)
thf(fact_165_minf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z4 )
=> ( ( P @ X5 )
= ( P2 @ X5 ) ) )
=> ( ? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z4 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z2: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z2 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P2 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ) ).
% minf(1)
thf(fact_166_minf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z4 )
=> ( ( P @ X5 )
= ( P2 @ X5 ) ) )
=> ( ? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z4 )
=> ( ( Q @ X5 )
= ( Q2 @ X5 ) ) )
=> ? [Z2: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z2 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P2 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ) ).
% minf(2)
thf(fact_167_minf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z2: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z2 )
=> ( X4 != T2 ) ) ) ).
% minf(3)
thf(fact_168_minf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z2: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z2 )
=> ( X4 != T2 ) ) ) ).
% minf(4)
thf(fact_169_minf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z2: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z2 )
=> ( ord_less @ A @ X4 @ T2 ) ) ) ).
% minf(5)
thf(fact_170_minf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z2: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z2 )
=> ~ ( ord_less @ A @ T2 @ X4 ) ) ) ).
% minf(7)
thf(fact_171_minf_I11_J,axiom,
! [C2: $tType,D: $tType] :
( ( ord @ C2 @ ( type2 @ C2 ) )
=> ! [F4: D] :
? [Z2: C2] :
! [X4: C2] :
( ( ord_less @ C2 @ X4 @ Z2 )
=> ( F4 = F4 ) ) ) ).
% minf(11)
thf(fact_172_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_173_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_174_order_Onot__eq__order__implies__strict,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A] :
( ( A2 != B )
=> ( ( ord_less_eq @ A @ A2 @ B )
=> ( ord_less @ A @ A2 @ B ) ) ) ) ).
% order.not_eq_order_implies_strict
thf(fact_175_dual__order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B: A,A2: A] :
( ( ord_less @ A @ B @ A2 )
=> ( ord_less_eq @ A @ B @ A2 ) ) ) ).
% dual_order.strict_implies_order
thf(fact_176_dual__order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [B4: A,A5: A] :
( ( ord_less_eq @ A @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_177_dual__order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [B4: A,A5: A] :
( ( ord_less @ A @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_178_order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ A2 @ B )
=> ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% order.strict_implies_order
thf(fact_179_dense__le__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less @ A @ X @ Y )
=> ( ! [W: A] :
( ( ord_less @ A @ X @ W )
=> ( ( ord_less @ A @ W @ Y )
=> ( ord_less_eq @ A @ W @ Z ) ) )
=> ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% dense_le_bounded
thf(fact_180_dense__ge__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Z: A,X: A,Y: A] :
( ( ord_less @ A @ Z @ X )
=> ( ! [W: A] :
( ( ord_less @ A @ Z @ W )
=> ( ( ord_less @ A @ W @ X )
=> ( ord_less_eq @ A @ Y @ W ) ) )
=> ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% dense_ge_bounded
thf(fact_181_dual__order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B: A,A2: A,C: A] :
( ( ord_less @ A @ B @ A2 )
=> ( ( ord_less_eq @ A @ C @ B )
=> ( ord_less @ A @ C @ A2 ) ) ) ) ).
% dual_order.strict_trans2
thf(fact_182_dual__order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B: A,A2: A,C: A] :
( ( ord_less_eq @ A @ B @ A2 )
=> ( ( ord_less @ A @ C @ B )
=> ( ord_less @ A @ C @ A2 ) ) ) ) ).
% dual_order.strict_trans1
thf(fact_183_order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [A5: A,B4: A] :
( ( ord_less_eq @ A @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ) ).
% order.strict_iff_order
thf(fact_184_order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B4: A] :
( ( ord_less @ A @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ) ).
% order.order_iff_strict
thf(fact_185_order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C: A] :
( ( ord_less @ A @ A2 @ B )
=> ( ( ord_less_eq @ A @ B @ C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% order.strict_trans2
thf(fact_186_order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( ord_less @ A @ B @ C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% order.strict_trans1
thf(fact_187_not__le__imp__less,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A] :
( ~ ( ord_less_eq @ A @ Y @ X )
=> ( ord_less @ A @ X @ Y ) ) ) ).
% not_le_imp_less
thf(fact_188_less__le__not__le,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [X3: A,Y7: A] :
( ( ord_less_eq @ A @ X3 @ Y7 )
& ~ ( ord_less_eq @ A @ Y7 @ X3 ) ) ) ) ) ).
% less_le_not_le
thf(fact_189_le__imp__less__or__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ X @ Y )
| ( X = Y ) ) ) ) ).
% le_imp_less_or_eq
thf(fact_190_le__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
| ( ord_less @ A @ Y @ X ) ) ) ).
% le_less_linear
thf(fact_191_dense__le,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Y: A,Z: A] :
( ! [X5: A] :
( ( ord_less @ A @ X5 @ Y )
=> ( ord_less_eq @ A @ X5 @ Z ) )
=> ( ord_less_eq @ A @ Y @ Z ) ) ) ).
% dense_le
thf(fact_192_dense__ge,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Z: A,Y: A] :
( ! [X5: A] :
( ( ord_less @ A @ Z @ X5 )
=> ( ord_less_eq @ A @ Y @ X5 ) )
=> ( ord_less_eq @ A @ Y @ Z ) ) ) ).
% dense_ge
thf(fact_193_less__le__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z )
=> ( ord_less @ A @ X @ Z ) ) ) ) ).
% less_le_trans
thf(fact_194_le__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ Z )
=> ( ord_less @ A @ X @ Z ) ) ) ) ).
% le_less_trans
thf(fact_195_antisym__conv2,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ~ ( ord_less @ A @ X @ Y ) )
= ( X = Y ) ) ) ) ).
% antisym_conv2
thf(fact_196_antisym__conv1,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ X @ Y )
= ( X = Y ) ) ) ) ).
% antisym_conv1
thf(fact_197_less__imp__le,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% less_imp_le
thf(fact_198_le__neq__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less @ A @ A2 @ B ) ) ) ) ).
% le_neq_trans
thf(fact_199_not__less,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less @ A @ X @ Y ) )
= ( ord_less_eq @ A @ Y @ X ) ) ) ).
% not_less
thf(fact_200_not__le,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less_eq @ A @ X @ Y ) )
= ( ord_less @ A @ Y @ X ) ) ) ).
% not_le
thf(fact_201_order__less__le__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 @ ( type2 @ C2 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B: A,F: A > C2,C: C2] :
( ( ord_less @ A @ A2 @ B )
=> ( ( ord_less_eq @ C2 @ ( F @ B ) @ C )
=> ( ! [X5: A,Y8: A] :
( ( ord_less @ A @ X5 @ Y8 )
=> ( ord_less @ C2 @ ( F @ X5 ) @ ( F @ Y8 ) ) )
=> ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_less_le_subst2
thf(fact_202_order__less__le__subst1,axiom,
! [A: $tType,B3: $tType] :
( ( ( order @ B3 @ ( type2 @ B3 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B3 > A,B: B3,C: B3] :
( ( ord_less @ A @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq @ B3 @ B @ C )
=> ( ! [X5: B3,Y8: B3] :
( ( ord_less_eq @ B3 @ X5 @ Y8 )
=> ( ord_less_eq @ A @ ( F @ X5 ) @ ( F @ Y8 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_203_order__le__less__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 @ ( type2 @ C2 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B: A,F: A > C2,C: C2] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( ord_less @ C2 @ ( F @ B ) @ C )
=> ( ! [X5: A,Y8: A] :
( ( ord_less_eq @ A @ X5 @ Y8 )
=> ( ord_less_eq @ C2 @ ( F @ X5 ) @ ( F @ Y8 ) ) )
=> ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_le_less_subst2
thf(fact_204_order__le__less__subst1,axiom,
! [A: $tType,B3: $tType] :
( ( ( order @ B3 @ ( type2 @ B3 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B3 > A,B: B3,C: B3] :
( ( ord_less_eq @ A @ A2 @ ( F @ B ) )
=> ( ( ord_less @ B3 @ B @ C )
=> ( ! [X5: B3,Y8: B3] :
( ( ord_less @ B3 @ X5 @ Y8 )
=> ( ord_less @ A @ ( F @ X5 ) @ ( F @ Y8 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_205_less__le,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [X3: A,Y7: A] :
( ( ord_less_eq @ A @ X3 @ Y7 )
& ( X3 != Y7 ) ) ) ) ) ).
% less_le
thf(fact_206_le__less,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [X3: A,Y7: A] :
( ( ord_less @ A @ X3 @ Y7 )
| ( X3 = Y7 ) ) ) ) ) ).
% le_less
thf(fact_207_leI,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% leI
thf(fact_208_leD,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less @ A @ X @ Y ) ) ) ).
% leD
thf(fact_209_pinf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z2: A] :
! [X4: A] :
( ( ord_less @ A @ Z2 @ X4 )
=> ~ ( ord_less_eq @ A @ X4 @ T2 ) ) ) ).
% pinf(6)
thf(fact_210_ex__has__greatest__nat,axiom,
! [A: $tType,P: A > $o,K2: A,M: A > nat,B: nat] :
( ( P @ K2 )
=> ( ! [Y8: A] :
( ( P @ Y8 )
=> ( ord_less @ nat @ ( M @ Y8 ) @ B ) )
=> ? [X5: A] :
( ( P @ X5 )
& ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ nat @ ( M @ Y5 ) @ ( M @ X5 ) ) ) ) ) ) ).
% ex_has_greatest_nat
thf(fact_211_complete__interval,axiom,
! [A: $tType] :
( ( condit1037483654norder @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,P: A > $o] :
( ( ord_less @ A @ A2 @ B )
=> ( ( P @ A2 )
=> ( ~ ( P @ B )
=> ? [C3: A] :
( ( ord_less_eq @ A @ A2 @ C3 )
& ( ord_less_eq @ A @ C3 @ B )
& ! [X4: A] :
( ( ( ord_less_eq @ A @ A2 @ X4 )
& ( ord_less @ A @ X4 @ C3 ) )
=> ( P @ X4 ) )
& ! [D2: A] :
( ! [X5: A] :
( ( ( ord_less_eq @ A @ A2 @ X5 )
& ( ord_less @ A @ X5 @ D2 ) )
=> ( P @ X5 ) )
=> ( ord_less_eq @ A @ D2 @ C3 ) ) ) ) ) ) ) ).
% complete_interval
thf(fact_212_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_213_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_214_Suc_Ohyps,axiom,
! [N: extended_enat,Xs: coindu1593790203_llist @ a] :
( ( ord_less @ extended_enat @ ( extended_enat2 @ ma ) @ N )
=> ( ( coindu749330388e_lnth @ a @ ( coindu1802687541_ltake @ a @ N @ Xs ) @ ma )
= ( coindu749330388e_lnth @ a @ Xs @ ma ) ) ) ).
% Suc.hyps
thf(fact_215_ex__has__least__nat,axiom,
! [A: $tType,P: A > $o,K2: A,M: A > nat] :
( ( P @ K2 )
=> ? [X5: A] :
( ( P @ X5 )
& ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ nat @ ( M @ X5 ) @ ( M @ Y5 ) ) ) ) ) ).
% ex_has_least_nat
thf(fact_216_enat__less__enat__plusI,axiom,
! [X: nat,Y: nat,Z: extended_enat] :
( ( ord_less @ nat @ X @ Y )
=> ( ord_less @ extended_enat @ ( extended_enat2 @ X ) @ ( plus_plus @ extended_enat @ ( extended_enat2 @ Y ) @ Z ) ) ) ).
% enat_less_enat_plusI
thf(fact_217_ex__gt__or__lt,axiom,
! [A: $tType] :
( ( condit1656338222tinuum @ A @ ( type2 @ A ) )
=> ! [A2: A] :
? [B2: A] :
( ( ord_less @ A @ A2 @ B2 )
| ( ord_less @ A @ B2 @ A2 ) ) ) ).
% ex_gt_or_lt
thf(fact_218_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( X != Y )
=> ( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_219_enat__add1__eq,axiom,
! [X: nat,Y: extended_enat,Z: extended_enat] :
( ( ( plus_plus @ extended_enat @ ( extended_enat2 @ X ) @ Y )
= ( plus_plus @ extended_enat @ ( extended_enat2 @ X ) @ Z ) )
= ( Y = Z ) ) ).
% enat_add1_eq
thf(fact_220_enat__add2__eq,axiom,
! [Y: extended_enat,X: nat,Z: extended_enat] :
( ( ( plus_plus @ extended_enat @ Y @ ( extended_enat2 @ X ) )
= ( plus_plus @ extended_enat @ Z @ ( extended_enat2 @ X ) ) )
= ( Y = Z ) ) ).
% enat_add2_eq
thf(fact_221_enat__add__mono,axiom,
! [X: nat,Y: extended_enat,Z: extended_enat] :
( ( ord_less @ extended_enat @ ( plus_plus @ extended_enat @ ( extended_enat2 @ X ) @ Y ) @ ( plus_plus @ extended_enat @ ( extended_enat2 @ X ) @ Z ) )
= ( ord_less @ extended_enat @ Y @ Z ) ) ).
% enat_add_mono
thf(fact_222_less__fun__def,axiom,
! [B3: $tType,A: $tType] :
( ( ord @ B3 @ ( type2 @ B3 ) )
=> ( ( ord_less @ ( A > B3 ) )
= ( ^ [F3: A > B3,G2: A > B3] :
( ( ord_less_eq @ ( A > B3 ) @ F3 @ G2 )
& ~ ( ord_less_eq @ ( A > B3 ) @ G2 @ F3 ) ) ) ) ) ).
% less_fun_def
thf(fact_223_ltake__eq__ltake__antimono,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,M: extended_enat] :
( ( ( coindu1802687541_ltake @ A @ N @ Xs )
= ( coindu1802687541_ltake @ A @ N @ Ys ) )
=> ( ( ord_less_eq @ extended_enat @ M @ N )
=> ( ( coindu1802687541_ltake @ A @ M @ Xs )
= ( coindu1802687541_ltake @ A @ M @ Ys ) ) ) ) ).
% ltake_eq_ltake_antimono
thf(fact_224_enat__le__plus__same_I2_J,axiom,
! [X: extended_enat,Y: extended_enat] : ( ord_less_eq @ extended_enat @ X @ ( plus_plus @ extended_enat @ Y @ X ) ) ).
% enat_le_plus_same(2)
thf(fact_225_enat__le__plus__same_I1_J,axiom,
! [X: extended_enat,Y: extended_enat] : ( ord_less_eq @ extended_enat @ X @ ( plus_plus @ extended_enat @ X @ Y ) ) ).
% enat_le_plus_same(1)
thf(fact_226_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_227_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_228_add__less__cancel__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [C: A,A2: A,B: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) )
= ( ord_less @ A @ A2 @ B ) ) ) ).
% add_less_cancel_left
thf(fact_229_add__less__cancel__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,C: A,B: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) )
= ( ord_less @ A @ A2 @ B ) ) ) ).
% add_less_cancel_right
thf(fact_230_add__right__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
=> ! [B: A,A2: A,C: A] :
( ( ( plus_plus @ A @ B @ A2 )
= ( plus_plus @ A @ C @ A2 ) )
= ( B = C ) ) ) ).
% add_right_cancel
thf(fact_231_add__left__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C: A] :
( ( ( plus_plus @ A @ A2 @ B )
= ( plus_plus @ A @ A2 @ C ) )
= ( B = C ) ) ) ).
% add_left_cancel
thf(fact_232_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_233_nat__add__left__cancel__less,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ K2 @ M ) @ ( plus_plus @ nat @ K2 @ N ) )
= ( ord_less @ nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_234_nat__add__left__cancel__le,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ K2 @ M ) @ ( plus_plus @ nat @ K2 @ N ) )
= ( ord_less_eq @ nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_235_add__le__cancel__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,C: A,B: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B @ C ) )
= ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% add_le_cancel_right
thf(fact_236_add__le__cancel__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [C: A,A2: A,B: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B ) )
= ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% add_le_cancel_left
thf(fact_237_plus__enat__simps_I1_J,axiom,
! [M: nat,N: nat] :
( ( plus_plus @ extended_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
= ( extended_enat2 @ ( plus_plus @ nat @ M @ N ) ) ) ).
% plus_enat_simps(1)
thf(fact_238_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus @ nat @ ( suc @ M ) @ N )
= ( plus_plus @ nat @ M @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_239_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus @ nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus @ nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_240_less__add__eq__less,axiom,
! [K2: nat,L: nat,M: nat,N: nat] :
( ( ord_less @ nat @ K2 @ L )
=> ( ( ( plus_plus @ nat @ M @ L )
= ( plus_plus @ nat @ K2 @ N ) )
=> ( ord_less @ nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_241_trans__less__add2,axiom,
! [I2: nat,J3: nat,M: nat] :
( ( ord_less @ nat @ I2 @ J3 )
=> ( ord_less @ nat @ I2 @ ( plus_plus @ nat @ M @ J3 ) ) ) ).
% trans_less_add2
thf(fact_242_trans__less__add1,axiom,
! [I2: nat,J3: nat,M: nat] :
( ( ord_less @ nat @ I2 @ J3 )
=> ( ord_less @ nat @ I2 @ ( plus_plus @ nat @ J3 @ M ) ) ) ).
% trans_less_add1
thf(fact_243_add__less__mono1,axiom,
! [I2: nat,J3: nat,K2: nat] :
( ( ord_less @ nat @ I2 @ J3 )
=> ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ K2 ) @ ( plus_plus @ nat @ J3 @ K2 ) ) ) ).
% add_less_mono1
thf(fact_244_not__add__less2,axiom,
! [J3: nat,I2: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ J3 @ I2 ) @ I2 ) ).
% not_add_less2
thf(fact_245_not__add__less1,axiom,
! [I2: nat,J3: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ J3 ) @ I2 ) ).
% not_add_less1
thf(fact_246_add__less__mono,axiom,
! [I2: nat,J3: nat,K2: nat,L: nat] :
( ( ord_less @ nat @ I2 @ J3 )
=> ( ( ord_less @ nat @ K2 @ L )
=> ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ K2 ) @ ( plus_plus @ nat @ J3 @ L ) ) ) ) ).
% add_less_mono
thf(fact_247_add__lessD1,axiom,
! [I2: nat,J3: nat,K2: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ J3 ) @ K2 )
=> ( ord_less @ nat @ I2 @ K2 ) ) ).
% add_lessD1
thf(fact_248_nat__le__iff__add,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M3: nat,N2: nat] :
? [K3: nat] :
( N2
= ( plus_plus @ nat @ M3 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_249_trans__le__add2,axiom,
! [I2: nat,J3: nat,M: nat] :
( ( ord_less_eq @ nat @ I2 @ J3 )
=> ( ord_less_eq @ nat @ I2 @ ( plus_plus @ nat @ M @ J3 ) ) ) ).
% trans_le_add2
thf(fact_250_trans__le__add1,axiom,
! [I2: nat,J3: nat,M: nat] :
( ( ord_less_eq @ nat @ I2 @ J3 )
=> ( ord_less_eq @ nat @ I2 @ ( plus_plus @ nat @ J3 @ M ) ) ) ).
% trans_le_add1
thf(fact_251_add__le__mono1,axiom,
! [I2: nat,J3: nat,K2: nat] :
( ( ord_less_eq @ nat @ I2 @ J3 )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ I2 @ K2 ) @ ( plus_plus @ nat @ J3 @ K2 ) ) ) ).
% add_le_mono1
thf(fact_252_add__le__mono,axiom,
! [I2: nat,J3: nat,K2: nat,L: nat] :
( ( ord_less_eq @ nat @ I2 @ J3 )
=> ( ( ord_less_eq @ nat @ K2 @ L )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ I2 @ K2 ) @ ( plus_plus @ nat @ J3 @ L ) ) ) ) ).
% add_le_mono
thf(fact_253_le__Suc__ex,axiom,
! [K2: nat,L: nat] :
( ( ord_less_eq @ nat @ K2 @ L )
=> ? [N3: nat] :
( L
= ( plus_plus @ nat @ K2 @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_254_add__leD2,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K2 ) @ N )
=> ( ord_less_eq @ nat @ K2 @ N ) ) ).
% add_leD2
%----Type constructors (25)
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A6: $tType,A7: $tType] :
( ( preorder @ A7 @ ( type2 @ A7 ) )
=> ( preorder @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A6: $tType,A7: $tType] :
( ( order @ A7 @ ( type2 @ A7 ) )
=> ( order @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A6: $tType,A7: $tType] :
( ( ord @ A7 @ ( type2 @ A7 ) )
=> ( ord @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
condit1037483654norder @ 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_Ocancel__semigroup__add,axiom,
cancel_semigroup_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
wellorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
preorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
linorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Ono__top,axiom,
no_top @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
order @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
ord @ nat @ ( type2 @ nat ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_4,axiom,
! [A6: $tType] : ( preorder @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_5,axiom,
! [A6: $tType] : ( order @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_6,axiom,
! [A6: $tType] : ( ord @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_7,axiom,
preorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder_8,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_9,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_10,axiom,
ord @ $o @ ( type2 @ $o ) ).
thf(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_11,axiom,
condit1037483654norder @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Orderings_Owellorder_12,axiom,
wellorder @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Orderings_Opreorder_13,axiom,
preorder @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Orderings_Olinorder_14,axiom,
linorder @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oorder_15,axiom,
order @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oord_16,axiom,
ord @ extended_enat @ ( type2 @ extended_enat ) ).
%----Conjectures (2)
thf(conj_0,hypothesis,
! [N5: extended_enat] :
( ( na
= ( extended_eSuc @ N5 ) )
=> thesis ) ).
thf(conj_1,conjecture,
thesis ).
%------------------------------------------------------------------------------