TPTP Problem File: DAT127^1.p
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%------------------------------------------------------------------------------
% File : DAT127^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Coinductive list 2896
% 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__2896.p [Bla16]
% Status : Theorem
% Rating : 1.00 v7.2.0, 0.75 v7.1.0
% Syntax : Number of formulae : 322 ( 89 unt; 43 typ; 0 def)
% Number of atoms : 813 ( 193 equ; 0 cnn)
% Maximal formula atoms : 13 ( 2 avg)
% Number of connectives : 3324 ( 83 ~; 21 |; 31 &;2737 @)
% ( 0 <=>; 452 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 8 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 155 ( 155 >; 0 *; 0 +; 0 <<)
% Number of symbols : 43 ( 40 usr; 6 con; 0-4 aty)
% Number of variables : 932 ( 34 ^; 826 !; 42 ?; 932 :)
% ( 30 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:58:51.939
%------------------------------------------------------------------------------
%----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 (37)
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_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_Olappend,type,
coindu268472904append:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollength,type,
coindu1018505716length:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > extended_enat ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Olmap,type,
coindu1062782156e_lmap:
!>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ Aa ) ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Olset,type,
coindu1112613586e_lset:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > ( set @ A ) ) ).
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_Olprefix,type,
coindu1696667936prefix:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Olstrict__prefix,type,
coindu574146665prefix:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_OltakeWhile,type,
coindu721411036eWhile:
!>[A: $tType] : ( ( A > $o ) > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
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_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_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_P,type,
p: a > $o ).
thf(sy_v_len____,type,
len: nat ).
thf(sy_v_lena____,type,
lena: nat ).
thf(sy_v_xs,type,
xs: coindu1593790203_llist @ a ).
thf(sy_v_xsa____,type,
xsa: coindu1593790203_llist @ a ).
%----Relevant facts (255)
thf(fact_0_Suc_Oprems_I2_J,axiom,
ord_less @ extended_enat @ ( extended_enat2 @ ( suc @ lena ) ) @ ( coindu1018505716length @ a @ xsa ) ).
% Suc.prems(2)
thf(fact_1_len,axiom,
ord_less @ extended_enat @ ( coindu1018505716length @ a @ ( coindu721411036eWhile @ a @ p @ xs ) ) @ ( coindu1018505716length @ a @ xs ) ).
% len
thf(fact_2_llength__ltakeWhile__all,axiom,
! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] :
( ( ( coindu1018505716length @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) )
= ( coindu1018505716length @ A @ Xs ) )
= ( ( coindu721411036eWhile @ A @ P @ Xs )
= Xs ) ) ).
% llength_ltakeWhile_all
thf(fact_3_llist__less__induct,axiom,
! [A: $tType,P: ( coindu1593790203_llist @ A ) > $o,Xs: coindu1593790203_llist @ A] :
( ! [Xs2: coindu1593790203_llist @ A] :
( ! [Ys: coindu1593790203_llist @ A] :
( ( coindu574146665prefix @ A @ Ys @ Xs2 )
=> ( P @ Ys ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% llist_less_induct
thf(fact_4_Suc_Oprems_I1_J,axiom,
p @ ( coindu749330388e_lnth @ a @ xsa @ ( suc @ lena ) ) ).
% Suc.prems(1)
thf(fact_5__092_060open_062llength_A_IltakeWhile_AP_Axs_J_A_061_Aenat_Alen_____092_060close_062,axiom,
( ( coindu1018505716length @ a @ ( coindu721411036eWhile @ a @ p @ xs ) )
= ( extended_enat2 @ len ) ) ).
% \<open>llength (ltakeWhile P xs) = enat len__\<close>
thf(fact_6__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062len_O_Allength_A_IltakeWhile_AP_Axs_J_A_061_Aenat_Alen_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Len: nat] :
( ( coindu1018505716length @ a @ ( coindu721411036eWhile @ a @ p @ xs ) )
!= ( extended_enat2 @ Len ) ) ).
% \<open>\<And>thesis. (\<And>len. llength (ltakeWhile P xs) = enat len \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_7_P,axiom,
p @ ( coindu749330388e_lnth @ a @ xs @ ( extended_the_enat @ ( coindu1018505716length @ a @ ( coindu721411036eWhile @ a @ p @ xs ) ) ) ) ).
% P
thf(fact_8_lstrict__prefix__llength__less,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys2: coindu1593790203_llist @ A] :
( ( coindu574146665prefix @ A @ Xs @ Ys2 )
=> ( ord_less @ extended_enat @ ( coindu1018505716length @ A @ Xs ) @ ( coindu1018505716length @ A @ Ys2 ) ) ) ).
% lstrict_prefix_llength_less
thf(fact_9_Suc_Oprems_I3_J,axiom,
( ( coindu1018505716length @ a @ ( coindu721411036eWhile @ a @ p @ xsa ) )
= ( extended_enat2 @ ( suc @ lena ) ) ) ).
% Suc.prems(3)
thf(fact_10_ltakeWhile__nth,axiom,
! [A: $tType,I: nat,P: A > $o,Xs: coindu1593790203_llist @ A] :
( ( ord_less @ extended_enat @ ( extended_enat2 @ I ) @ ( coindu1018505716length @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) ) )
=> ( ( coindu749330388e_lnth @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) @ I )
= ( coindu749330388e_lnth @ A @ Xs @ I ) ) ) ).
% ltakeWhile_nth
thf(fact_11_Suc_Ohyps,axiom,
! [Xs: coindu1593790203_llist @ a] :
( ( p @ ( coindu749330388e_lnth @ a @ Xs @ lena ) )
=> ( ( ord_less @ extended_enat @ ( extended_enat2 @ lena ) @ ( coindu1018505716length @ a @ Xs ) )
=> ( ( coindu1018505716length @ a @ ( coindu721411036eWhile @ a @ p @ Xs ) )
!= ( extended_enat2 @ lena ) ) ) ) ).
% Suc.hyps
thf(fact_12_enat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( extended_enat2 @ Nat )
= ( extended_enat2 @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% enat.inject
thf(fact_13_nat_Oinject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
= ( X2 = Y2 ) ) ).
% nat.inject
thf(fact_14_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_15_chain__incr,axiom,
! [A: $tType,Y: A > extended_enat,K: nat] :
( ! [I2: A] :
? [J: A] : ( ord_less @ extended_enat @ ( Y @ I2 ) @ ( Y @ J ) )
=> ? [J2: A] : ( ord_less @ extended_enat @ ( extended_enat2 @ K ) @ ( Y @ J2 ) ) ) ).
% chain_incr
thf(fact_16_enat__iless,axiom,
! [N: extended_enat,M: nat] :
( ( ord_less @ extended_enat @ N @ ( extended_enat2 @ M ) )
=> ? [K2: nat] :
( N
= ( extended_enat2 @ K2 ) ) ) ).
% enat_iless
thf(fact_17_lift__Suc__mono__less,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [F: nat > A,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less @ A @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less @ nat @ N @ N2 )
=> ( ord_less @ A @ ( F @ N ) @ ( F @ N2 ) ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_18_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_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_ltakeWhile__K__True,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu721411036eWhile @ A
@ ^ [Uu: A] : $true
@ Xs )
= Xs ) ).
% ltakeWhile_K_True
thf(fact_21_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_22_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_23_lessI,axiom,
! [N: nat] : ( ord_less @ nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_24_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_25_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_26_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_not_refl
thf(fact_27_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_28_less__not__refl3,axiom,
! [S: nat,T2: nat] :
( ( ord_less @ nat @ S @ T2 )
=> ( S != T2 ) ) ).
% less_not_refl3
thf(fact_29_measure__induct,axiom,
! [A: $tType,F: A > nat,P: A > $o,A2: A] :
( ! [X: A] :
( ! [Y3: A] :
( ( ord_less @ nat @ ( F @ Y3 ) @ ( F @ X ) )
=> ( P @ Y3 ) )
=> ( P @ X ) )
=> ( P @ A2 ) ) ).
% measure_induct
thf(fact_30_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_31_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_32_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_33_linorder__neqE__nat,axiom,
! [X3: nat,Y4: nat] :
( ( X3 != Y4 )
=> ( ~ ( ord_less @ nat @ X3 @ Y4 )
=> ( ord_less @ nat @ Y4 @ X3 ) ) ) ).
% linorder_neqE_nat
thf(fact_34_measure__induct__rule,axiom,
! [A: $tType,F: A > nat,P: A > $o,A2: A] :
( ! [X: A] :
( ! [Y3: A] :
( ( ord_less @ nat @ ( F @ Y3 ) @ ( F @ X ) )
=> ( P @ Y3 ) )
=> ( P @ X ) )
=> ( P @ A2 ) ) ).
% measure_induct_rule
thf(fact_35_infinite__descent__measure,axiom,
! [A: $tType,P: A > $o,V: A > nat,X3: A] :
( ! [X: A] :
( ~ ( P @ X )
=> ? [Y3: A] :
( ( ord_less @ nat @ ( V @ Y3 ) @ ( V @ X ) )
& ~ ( P @ Y3 ) ) )
=> ( P @ X3 ) ) ).
% infinite_descent_measure
thf(fact_36_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_37_strict__inc__induct,axiom,
! [I: nat,J3: nat,P: nat > $o] :
( ( ord_less @ nat @ I @ J3 )
=> ( ! [I2: nat] :
( ( J3
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ J3 )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_38_less__Suc__induct,axiom,
! [I: nat,J3: nat,P: nat > nat > $o] :
( ( ord_less @ nat @ I @ J3 )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K2: nat] :
( ( ord_less @ nat @ I2 @ J2 )
=> ( ( ord_less @ nat @ J2 @ K2 )
=> ( ( P @ I2 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I2 @ K2 ) ) ) ) )
=> ( P @ I @ J3 ) ) ) ) ).
% less_Suc_induct
thf(fact_39_less__trans__Suc,axiom,
! [I: nat,J3: nat,K: nat] :
( ( ord_less @ nat @ I @ J3 )
=> ( ( ord_less @ nat @ J3 @ K )
=> ( ord_less @ nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_40_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_41_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less @ nat @ N @ M )
=> ( ( ord_less @ nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_42_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( suc @ N ) @ M )
= ( ? [M3: nat] :
( ( M
= ( suc @ M3 ) )
& ( ord_less @ nat @ N @ M3 ) ) ) ) ).
% Suc_less_eq2
thf(fact_43_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less @ nat @ M @ N ) )
= ( ord_less @ nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
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,A3: set @ A] :
( ( collect @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_46_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X: A] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_47_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X: A] :
( ( F @ X )
= ( G @ X ) )
=> ( F = G ) ) ).
% ext
thf(fact_48_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_49_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less @ nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_50_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less @ nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_51_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_52_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less @ nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less @ nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_53_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ M ) @ N )
=> ( ord_less @ nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_54_lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less @ nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less @ nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% lessE
thf(fact_55_the__enat_Osimps,axiom,
! [N: nat] :
( ( extended_the_enat @ ( extended_enat2 @ N ) )
= N ) ).
% the_enat.simps
thf(fact_56_less__enatE,axiom,
! [N: extended_enat,M: nat] :
( ( ord_less @ extended_enat @ N @ ( extended_enat2 @ M ) )
=> ~ ! [K2: nat] :
( ( N
= ( extended_enat2 @ K2 ) )
=> ~ ( ord_less @ nat @ K2 @ M ) ) ) ).
% less_enatE
thf(fact_57_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_58_Suc__inject,axiom,
! [X3: nat,Y4: nat] :
( ( ( suc @ X3 )
= ( suc @ Y4 ) )
=> ( X3 = Y4 ) ) ).
% Suc_inject
thf(fact_59_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_60_lnth__lmap,axiom,
! [B: $tType,A: $tType,N: nat,Xs: coindu1593790203_llist @ A,F: A > B] :
( ( ord_less @ extended_enat @ ( extended_enat2 @ N ) @ ( coindu1018505716length @ A @ Xs ) )
=> ( ( coindu749330388e_lnth @ B @ ( coindu1062782156e_lmap @ A @ B @ F @ Xs ) @ N )
= ( F @ ( coindu749330388e_lnth @ A @ Xs @ N ) ) ) ) ).
% lnth_lmap
thf(fact_61_less__enat__def,axiom,
( ( ord_less @ extended_enat )
= ( ^ [M4: extended_enat,N4: extended_enat] :
( extended_case_enat @ $o
@ ^ [M1: nat] : ( extended_case_enat @ $o @ ( ord_less @ nat @ M1 ) @ $true @ N4 )
@ $false
@ M4 ) ) ) ).
% less_enat_def
thf(fact_62_lprefix__lnthD,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys2: coindu1593790203_llist @ A,N: nat] :
( ( coindu1696667936prefix @ A @ Xs @ Ys2 )
=> ( ( ord_less @ extended_enat @ ( extended_enat2 @ N ) @ ( coindu1018505716length @ A @ Xs ) )
=> ( ( coindu749330388e_lnth @ A @ Xs @ N )
= ( coindu749330388e_lnth @ A @ Ys2 @ N ) ) ) ) ).
% lprefix_lnthD
thf(fact_63_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_64_in__lset__conv__lnth,axiom,
! [A: $tType,X3: A,Xs: coindu1593790203_llist @ A] :
( ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ Xs ) )
= ( ? [N4: nat] :
( ( ord_less @ extended_enat @ ( extended_enat2 @ N4 ) @ ( coindu1018505716length @ A @ Xs ) )
& ( ( coindu749330388e_lnth @ A @ Xs @ N4 )
= X3 ) ) ) ) ).
% in_lset_conv_lnth
thf(fact_65_lnth__lappend1,axiom,
! [A: $tType,N: nat,Xs: coindu1593790203_llist @ A,Ys2: coindu1593790203_llist @ A] :
( ( ord_less @ extended_enat @ ( extended_enat2 @ N ) @ ( coindu1018505716length @ A @ Xs ) )
=> ( ( coindu749330388e_lnth @ A @ ( coindu268472904append @ A @ Xs @ Ys2 ) @ N )
= ( coindu749330388e_lnth @ A @ Xs @ N ) ) ) ).
% lnth_lappend1
thf(fact_66_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_67_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_68_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type2 @ A ) )
=> ! [X3: A,Y4: A] :
( ( X3 != Y4 )
=> ( ~ ( ord_less @ A @ X3 @ Y4 )
=> ( ord_less @ A @ Y4 @ X3 ) ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_69_lprefix__refl,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ Xs @ Xs ) ).
% lprefix_refl
thf(fact_70_llist_Oleq__refl,axiom,
! [A: $tType,X3: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ X3 @ X3 ) ).
% llist.leq_refl
thf(fact_71_llist_Omap__ident,axiom,
! [A: $tType,T2: coindu1593790203_llist @ A] :
( ( coindu1062782156e_lmap @ A @ A
@ ^ [X4: A] : X4
@ T2 )
= T2 ) ).
% llist.map_ident
thf(fact_72_llength__lmap,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: coindu1593790203_llist @ B] :
( ( coindu1018505716length @ A @ ( coindu1062782156e_lmap @ B @ A @ F @ Xs ) )
= ( coindu1018505716length @ B @ Xs ) ) ).
% llength_lmap
thf(fact_73_lift__Suc__mono__le,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [F: nat > A,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq @ nat @ N @ N2 )
=> ( ord_less_eq @ A @ ( F @ N ) @ ( F @ N2 ) ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_74_lift__Suc__antimono__le,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [F: nat > A,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq @ nat @ N @ N2 )
=> ( ord_less_eq @ A @ ( F @ N2 ) @ ( F @ N ) ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_75_ltakeWhile__lappend1,axiom,
! [A: $tType,X3: A,Xs: coindu1593790203_llist @ A,P: A > $o,Ys2: coindu1593790203_llist @ A] :
( ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ Xs ) )
=> ( ~ ( P @ X3 )
=> ( ( coindu721411036eWhile @ A @ P @ ( coindu268472904append @ A @ Xs @ Ys2 ) )
= ( coindu721411036eWhile @ A @ P @ Xs ) ) ) ) ).
% ltakeWhile_lappend1
thf(fact_76_lmap__lprefix,axiom,
! [B: $tType,A: $tType,Xs: coindu1593790203_llist @ A,Ys2: coindu1593790203_llist @ A,F: A > B] :
( ( coindu1696667936prefix @ A @ Xs @ Ys2 )
=> ( coindu1696667936prefix @ B @ ( coindu1062782156e_lmap @ A @ B @ F @ Xs ) @ ( coindu1062782156e_lmap @ A @ B @ F @ Ys2 ) ) ) ).
% lmap_lprefix
thf(fact_77_lappend__assoc,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys2: coindu1593790203_llist @ A,Zs: coindu1593790203_llist @ A] :
( ( coindu268472904append @ A @ ( coindu268472904append @ A @ Xs @ Ys2 ) @ Zs )
= ( coindu268472904append @ A @ Xs @ ( coindu268472904append @ A @ Ys2 @ Zs ) ) ) ).
% lappend_assoc
thf(fact_78_lprefix__trans,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys2: coindu1593790203_llist @ A,Zs: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ Ys2 )
=> ( ( coindu1696667936prefix @ A @ Ys2 @ Zs )
=> ( coindu1696667936prefix @ A @ Xs @ Zs ) ) ) ).
% lprefix_trans
thf(fact_79_llist_Oleq__trans,axiom,
! [A: $tType,X3: coindu1593790203_llist @ A,Y4: coindu1593790203_llist @ A,Z: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ X3 @ Y4 )
=> ( ( coindu1696667936prefix @ A @ Y4 @ Z )
=> ( coindu1696667936prefix @ A @ X3 @ Z ) ) ) ).
% llist.leq_trans
thf(fact_80_lprefix__antisym,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys2: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ Ys2 )
=> ( ( coindu1696667936prefix @ A @ Ys2 @ Xs )
=> ( Xs = Ys2 ) ) ) ).
% lprefix_antisym
thf(fact_81_lprefix__lappend,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys2: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ Xs @ ( coindu268472904append @ A @ Xs @ Ys2 ) ) ).
% lprefix_lappend
thf(fact_82_lappend__lprefixE,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys2: coindu1593790203_llist @ A,Zs: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ ( coindu268472904append @ A @ Xs @ Ys2 ) @ Zs )
=> ~ ! [Zs2: coindu1593790203_llist @ A] :
( Zs
!= ( coindu268472904append @ A @ Xs @ Zs2 ) ) ) ).
% lappend_lprefixE
thf(fact_83_lprefix__lappendD,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys2: coindu1593790203_llist @ A,Zs: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ ( coindu268472904append @ A @ Ys2 @ Zs ) )
=> ( ( coindu1696667936prefix @ A @ Xs @ Ys2 )
| ( coindu1696667936prefix @ A @ Ys2 @ Xs ) ) ) ).
% lprefix_lappendD
thf(fact_84_wlog__linorder__le,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > A > $o,B3: A,A2: A] :
( ! [A4: A,B2: A] :
( ( ord_less_eq @ A @ A4 @ B2 )
=> ( P @ A4 @ B2 ) )
=> ( ( ( P @ B3 @ A2 )
=> ( P @ A2 @ B3 ) )
=> ( P @ A2 @ B3 ) ) ) ) ).
% wlog_linorder_le
thf(fact_85_llist_Oleq__antisym,axiom,
! [A: $tType,X3: coindu1593790203_llist @ A,Y4: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ X3 @ Y4 )
=> ( ( coindu1696667936prefix @ A @ Y4 @ X3 )
=> ( X3 = Y4 ) ) ) ).
% llist.leq_antisym
thf(fact_86_lprefix__down__linear,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Zs: coindu1593790203_llist @ A,Ys2: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ Zs )
=> ( ( coindu1696667936prefix @ A @ Ys2 @ Zs )
=> ( ( coindu1696667936prefix @ A @ Xs @ Ys2 )
| ( coindu1696667936prefix @ A @ Ys2 @ Xs ) ) ) ) ).
% lprefix_down_linear
thf(fact_87_lmap__lappend__distrib,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: coindu1593790203_llist @ B,Ys2: coindu1593790203_llist @ B] :
( ( coindu1062782156e_lmap @ B @ A @ F @ ( coindu268472904append @ B @ Xs @ Ys2 ) )
= ( coindu268472904append @ A @ ( coindu1062782156e_lmap @ B @ A @ F @ Xs ) @ ( coindu1062782156e_lmap @ B @ A @ F @ Ys2 ) ) ) ).
% lmap_lappend_distrib
thf(fact_88_lprefix__conv__lappend,axiom,
! [A: $tType] :
( ( coindu1696667936prefix @ A )
= ( ^ [Xs3: coindu1593790203_llist @ A,Ys3: coindu1593790203_llist @ A] :
? [Zs3: coindu1593790203_llist @ A] :
( Ys3
= ( coindu268472904append @ A @ Xs3 @ Zs3 ) ) ) ) ).
% lprefix_conv_lappend
thf(fact_89_lprefix__lappend__sameI,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys2: coindu1593790203_llist @ A,Zs: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ Ys2 )
=> ( coindu1696667936prefix @ A @ ( coindu268472904append @ A @ Zs @ Xs ) @ ( coindu268472904append @ A @ Zs @ Ys2 ) ) ) ).
% lprefix_lappend_sameI
thf(fact_90_llist_Omap__cong,axiom,
! [B: $tType,A: $tType,X3: coindu1593790203_llist @ A,Ya: coindu1593790203_llist @ A,F: A > B,G: A > B] :
( ( X3 = Ya )
=> ( ! [Z2: A] :
( ( member @ A @ Z2 @ ( coindu1112613586e_lset @ A @ Ya ) )
=> ( ( F @ Z2 )
= ( G @ Z2 ) ) )
=> ( ( coindu1062782156e_lmap @ A @ B @ F @ X3 )
= ( coindu1062782156e_lmap @ A @ B @ G @ Ya ) ) ) ) ).
% llist.map_cong
thf(fact_91_llist_Omap__cong0,axiom,
! [B: $tType,A: $tType,X3: coindu1593790203_llist @ A,F: A > B,G: A > B] :
( ! [Z2: A] :
( ( member @ A @ Z2 @ ( coindu1112613586e_lset @ A @ X3 ) )
=> ( ( F @ Z2 )
= ( G @ Z2 ) ) )
=> ( ( coindu1062782156e_lmap @ A @ B @ F @ X3 )
= ( coindu1062782156e_lmap @ A @ B @ G @ X3 ) ) ) ).
% llist.map_cong0
thf(fact_92_llist_Oinj__map__strong,axiom,
! [B: $tType,A: $tType,X3: coindu1593790203_llist @ A,Xa: coindu1593790203_llist @ A,F: A > B,Fa: A > B] :
( ! [Z2: A,Za: A] :
( ( member @ A @ Z2 @ ( coindu1112613586e_lset @ A @ X3 ) )
=> ( ( member @ A @ Za @ ( coindu1112613586e_lset @ A @ Xa ) )
=> ( ( ( F @ Z2 )
= ( Fa @ Za ) )
=> ( Z2 = Za ) ) ) )
=> ( ( ( coindu1062782156e_lmap @ A @ B @ F @ X3 )
= ( coindu1062782156e_lmap @ A @ B @ Fa @ Xa ) )
=> ( X3 = Xa ) ) ) ).
% llist.inj_map_strong
thf(fact_93_lprefix__llength__le,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys2: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ Ys2 )
=> ( ord_less_eq @ extended_enat @ ( coindu1018505716length @ A @ Xs ) @ ( coindu1018505716length @ A @ Ys2 ) ) ) ).
% lprefix_llength_le
thf(fact_94_complete__interval,axiom,
! [A: $tType] :
( ( condit1037483654norder @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,P: A > $o] :
( ( ord_less @ A @ A2 @ B3 )
=> ( ( P @ A2 )
=> ( ~ ( P @ B3 )
=> ? [C: A] :
( ( ord_less_eq @ A @ A2 @ C )
& ( ord_less_eq @ A @ C @ B3 )
& ! [X5: A] :
( ( ( ord_less_eq @ A @ A2 @ X5 )
& ( ord_less @ A @ X5 @ C ) )
=> ( P @ X5 ) )
& ! [D: A] :
( ! [X: A] :
( ( ( ord_less_eq @ A @ A2 @ X )
& ( ord_less @ A @ X @ D ) )
=> ( P @ X ) )
=> ( ord_less_eq @ A @ D @ C ) ) ) ) ) ) ) ).
% complete_interval
thf(fact_95_case__enat__def,axiom,
! [T: $tType] :
( ( extended_case_enat @ T )
= ( extended_rec_enat @ T ) ) ).
% case_enat_def
thf(fact_96_ltakeWhile__all,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o] :
( ! [X: A] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Xs ) )
=> ( P @ X ) )
=> ( ( coindu721411036eWhile @ A @ P @ Xs )
= Xs ) ) ).
% ltakeWhile_all
thf(fact_97_ltakeWhile__cong,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys2: coindu1593790203_llist @ A,P: A > $o,Q: A > $o] :
( ( Xs = Ys2 )
=> ( ! [X: A] :
( ( member @ A @ X @ ( coindu1112613586e_lset @ A @ Ys2 ) )
=> ( ( P @ X )
= ( Q @ X ) ) )
=> ( ( coindu721411036eWhile @ A @ P @ Xs )
= ( coindu721411036eWhile @ A @ Q @ Ys2 ) ) ) ) ).
% ltakeWhile_cong
thf(fact_98_lset__ltakeWhileD,axiom,
! [A: $tType,X3: A,P: A > $o,Xs: coindu1593790203_llist @ A] :
( ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) ) )
=> ( ( member @ A @ X3 @ ( coindu1112613586e_lset @ A @ Xs ) )
& ( P @ X3 ) ) ) ).
% lset_ltakeWhileD
thf(fact_99_lprefix__llength__eq__imp__eq,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys2: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ Ys2 )
=> ( ( ( coindu1018505716length @ A @ Xs )
= ( coindu1018505716length @ A @ Ys2 ) )
=> ( Xs = Ys2 ) ) ) ).
% lprefix_llength_eq_imp_eq
thf(fact_100_lprefix__ltakeWhile,axiom,
! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) @ Xs ) ).
% lprefix_ltakeWhile
thf(fact_101_enat__ile,axiom,
! [N: extended_enat,M: nat] :
( ( ord_less_eq @ extended_enat @ N @ ( extended_enat2 @ M ) )
=> ? [K2: nat] :
( N
= ( extended_enat2 @ K2 ) ) ) ).
% enat_ile
thf(fact_102_lmap__lstrict__prefix,axiom,
! [B: $tType,A: $tType,Xs: coindu1593790203_llist @ A,Ys2: coindu1593790203_llist @ A,F: A > B] :
( ( coindu574146665prefix @ A @ Xs @ Ys2 )
=> ( coindu574146665prefix @ B @ ( coindu1062782156e_lmap @ A @ B @ F @ Xs ) @ ( coindu1062782156e_lmap @ A @ B @ F @ Ys2 ) ) ) ).
% lmap_lstrict_prefix
thf(fact_103_lstrict__prefix__def,axiom,
! [A: $tType] :
( ( coindu574146665prefix @ A )
= ( ^ [Xs3: coindu1593790203_llist @ A,Ys3: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs3 @ Ys3 )
& ( Xs3 != Ys3 ) ) ) ) ).
% lstrict_prefix_def
thf(fact_104_llength__ltakeWhile__le,axiom,
! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] : ( ord_less_eq @ extended_enat @ ( coindu1018505716length @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) ) @ ( coindu1018505716length @ A @ Xs ) ) ).
% llength_ltakeWhile_le
thf(fact_105_llength__ltakeWhile__lt__iff,axiom,
! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] :
( ( ord_less @ extended_enat @ ( coindu1018505716length @ A @ ( coindu721411036eWhile @ A @ P @ Xs ) ) @ ( coindu1018505716length @ A @ Xs ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ ( coindu1112613586e_lset @ A @ Xs ) )
& ~ ( P @ X4 ) ) ) ) ).
% llength_ltakeWhile_lt_iff
thf(fact_106_enat__less__imp__le,axiom,
! [N: extended_enat,M: extended_enat] :
( ! [K2: nat] :
( ( ord_less @ extended_enat @ N @ ( extended_enat2 @ K2 ) )
=> ( ord_less @ extended_enat @ M @ ( extended_enat2 @ K2 ) ) )
=> ( ord_less_eq @ extended_enat @ M @ N ) ) ).
% enat_less_imp_le
thf(fact_107_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A] : ( ord_less_eq @ A @ X3 @ X3 ) ) ).
% order_refl
thf(fact_108_lnth__lappend,axiom,
! [A: $tType,N: nat,Xs: coindu1593790203_llist @ A,Ys2: coindu1593790203_llist @ A] :
( ( ( ord_less @ extended_enat @ ( extended_enat2 @ N ) @ ( coindu1018505716length @ A @ Xs ) )
=> ( ( coindu749330388e_lnth @ A @ ( coindu268472904append @ A @ Xs @ Ys2 ) @ N )
= ( coindu749330388e_lnth @ A @ Xs @ N ) ) )
& ( ~ ( ord_less @ extended_enat @ ( extended_enat2 @ N ) @ ( coindu1018505716length @ A @ Xs ) )
=> ( ( coindu749330388e_lnth @ A @ ( coindu268472904append @ A @ Xs @ Ys2 ) @ N )
= ( coindu749330388e_lnth @ A @ Ys2 @ ( minus_minus @ nat @ N @ ( extended_the_enat @ ( coindu1018505716length @ A @ Xs ) ) ) ) ) ) ) ).
% lnth_lappend
thf(fact_109_minf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z2: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z2 )
=> ~ ( ord_less_eq @ A @ T2 @ X5 ) ) ) ).
% minf(8)
thf(fact_110_minf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z2: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z2 )
=> ( ord_less_eq @ A @ X5 @ T2 ) ) ) ).
% minf(6)
thf(fact_111_pinf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z2: A] :
! [X5: A] :
( ( ord_less @ A @ Z2 @ X5 )
=> ( ord_less_eq @ A @ T2 @ X5 ) ) ) ).
% pinf(8)
thf(fact_112_pinf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z2: A] :
! [X5: A] :
( ( ord_less @ A @ Z2 @ X5 )
=> ~ ( ord_less_eq @ A @ X5 @ T2 ) ) ) ).
% pinf(6)
thf(fact_113_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_114_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_115_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_116_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_117_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_118_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_119_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ M )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( ( ord_less @ nat @ ( minus_minus @ nat @ M @ K ) @ ( minus_minus @ nat @ N @ K ) )
= ( ord_less @ nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_120_diff__less__mono,axiom,
! [A2: nat,B3: nat,C2: nat] :
( ( ord_less @ nat @ A2 @ B3 )
=> ( ( ord_less_eq @ nat @ C2 @ A2 )
=> ( ord_less @ nat @ ( minus_minus @ nat @ A2 @ C2 ) @ ( minus_minus @ nat @ B3 @ C2 ) ) ) ) ).
% diff_less_mono
thf(fact_121_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_122_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_123_le__diff__iff_H,axiom,
! [A2: nat,C2: nat,B3: nat] :
( ( ord_less_eq @ nat @ A2 @ C2 )
=> ( ( ord_less_eq @ nat @ B3 @ C2 )
=> ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ C2 @ A2 ) @ ( minus_minus @ nat @ C2 @ B3 ) )
= ( ord_less_eq @ nat @ B3 @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_124_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_125_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_126_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_127_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_128_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_129_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_130_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_131_le__trans,axiom,
! [I: nat,J3: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J3 )
=> ( ( ord_less_eq @ nat @ J3 @ K )
=> ( ord_less_eq @ nat @ I @ K ) ) ) ).
% le_trans
thf(fact_132_le__refl,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% le_refl
thf(fact_133_diff__commute,axiom,
! [I: nat,J3: nat,K: nat] :
( ( minus_minus @ nat @ ( minus_minus @ nat @ I @ J3 ) @ K )
= ( minus_minus @ nat @ ( minus_minus @ nat @ I @ K ) @ J3 ) ) ).
% diff_commute
thf(fact_134_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ ( minus_minus @ nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_135_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ( ord_less @ nat @ M @ L )
=> ( ord_less @ nat @ ( minus_minus @ nat @ L @ N ) @ ( minus_minus @ nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_136_less__imp__diff__less,axiom,
! [J3: nat,K: nat,N: nat] :
( ( ord_less @ nat @ J3 @ K )
=> ( ord_less @ nat @ ( minus_minus @ nat @ J3 @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_137_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_138_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_139_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq @ nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_140_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_141_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_142_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_143_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_144_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% Suc_leD
thf(fact_145_nat__less__le,axiom,
( ( ord_less @ nat )
= ( ^ [M4: nat,N4: nat] :
( ( ord_less_eq @ nat @ M4 @ N4 )
& ( M4 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_146_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_147_le__eq__less__or__eq,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M4: nat,N4: nat] :
( ( ord_less @ nat @ M4 @ N4 )
| ( M4 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_148_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_149_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_150_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J3: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less @ nat @ I2 @ J2 )
=> ( ord_less @ nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq @ nat @ I @ J3 )
=> ( ord_less_eq @ nat @ ( F @ I ) @ ( F @ J3 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_151_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ N @ M )
=> ( ( suc @ ( minus_minus @ nat @ M @ ( suc @ N ) ) )
= ( minus_minus @ nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_152_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less @ nat @ ( minus_minus @ nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_153_lnth__lappend2,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,K: nat,N: nat,Ys2: coindu1593790203_llist @ A] :
( ( ( coindu1018505716length @ A @ Xs )
= ( extended_enat2 @ K ) )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( ( coindu749330388e_lnth @ A @ ( coindu268472904append @ A @ Xs @ Ys2 ) @ N )
= ( coindu749330388e_lnth @ A @ Ys2 @ ( minus_minus @ nat @ N @ K ) ) ) ) ) ).
% lnth_lappend2
thf(fact_154_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less_eq @ nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_155_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_156_dec__induct,axiom,
! [I: nat,J3: nat,P: nat > $o] :
( ( ord_less_eq @ nat @ I @ J3 )
=> ( ( P @ I )
=> ( ! [N3: nat] :
( ( ord_less_eq @ nat @ I @ N3 )
=> ( ( ord_less @ nat @ N3 @ J3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J3 ) ) ) ) ).
% dec_induct
thf(fact_157_inc__induct,axiom,
! [I: nat,J3: nat,P: nat > $o] :
( ( ord_less_eq @ nat @ I @ J3 )
=> ( ( P @ J3 )
=> ( ! [N3: nat] :
( ( ord_less_eq @ nat @ I @ N3 )
=> ( ( ord_less @ nat @ N3 @ J3 )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_158_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_159_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_160_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_161_less__eq__Suc__le,axiom,
( ( ord_less @ nat )
= ( ^ [N4: nat] : ( ord_less_eq @ nat @ ( suc @ N4 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_162_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_163_lprefix__lsetD,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys2: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ Ys2 )
=> ( ord_less_eq @ ( set @ A ) @ ( coindu1112613586e_lset @ A @ Xs ) @ ( coindu1112613586e_lset @ A @ Ys2 ) ) ) ).
% lprefix_lsetD
thf(fact_164_lset__lappend1,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys2: coindu1593790203_llist @ A] : ( ord_less_eq @ ( set @ A ) @ ( coindu1112613586e_lset @ A @ Xs ) @ ( coindu1112613586e_lset @ A @ ( coindu268472904append @ A @ Xs @ Ys2 ) ) ) ).
% lset_lappend1
thf(fact_165_ltakeWhile__all__conv,axiom,
! [A: $tType,P: A > $o,Xs: coindu1593790203_llist @ A] :
( ( ( coindu721411036eWhile @ A @ P @ Xs )
= Xs )
= ( ord_less_eq @ ( set @ A ) @ ( coindu1112613586e_lset @ A @ Xs ) @ ( collect @ A @ P ) ) ) ).
% ltakeWhile_all_conv
thf(fact_166_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A] :
( ( ord_less_eq @ A @ B3 @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ) ).
% dual_order.antisym
thf(fact_167_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B3 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ B3 )
=> ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% dual_order.trans
thf(fact_168_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > A > $o,A2: A,B3: A] :
( ! [A4: A,B2: A] :
( ( ord_less_eq @ A @ A4 @ B2 )
=> ( P @ A4 @ B2 ) )
=> ( ! [A4: A,B2: A] :
( ( P @ B2 @ A4 )
=> ( P @ A4 @ B2 ) )
=> ( P @ A2 @ B3 ) ) ) ) ).
% linorder_wlog
thf(fact_169_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_170_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y4: A,Z: A] :
( ( ord_less_eq @ A @ X3 @ Y4 )
=> ( ( ord_less_eq @ A @ Y4 @ Z )
=> ( ord_less_eq @ A @ X3 @ Z ) ) ) ) ).
% order_trans
thf(fact_171_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ) ).
% order_class.order.antisym
thf(fact_172_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_173_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A] :
( ( A2 = B3 )
=> ( ( ord_less_eq @ A @ B3 @ C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_174_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [Y4: A,X3: A] :
( ( ord_less_eq @ A @ Y4 @ X3 )
=> ( ( ord_less_eq @ A @ X3 @ Y4 )
= ( X3 = Y4 ) ) ) ) ).
% antisym_conv
thf(fact_175_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y4: A,Z: A] :
( ( ( ord_less_eq @ A @ X3 @ Y4 )
=> ~ ( ord_less_eq @ A @ Y4 @ Z ) )
=> ( ( ( ord_less_eq @ A @ Y4 @ X3 )
=> ~ ( ord_less_eq @ A @ X3 @ Z ) )
=> ( ( ( ord_less_eq @ A @ X3 @ Z )
=> ~ ( ord_less_eq @ A @ Z @ Y4 ) )
=> ( ( ( ord_less_eq @ A @ Z @ Y4 )
=> ~ ( ord_less_eq @ A @ Y4 @ X3 ) )
=> ( ( ( ord_less_eq @ A @ Y4 @ Z )
=> ~ ( ord_less_eq @ A @ Z @ X3 ) )
=> ~ ( ( ord_less_eq @ A @ Z @ X3 )
=> ~ ( ord_less_eq @ A @ X3 @ Y4 ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_176_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% order.trans
thf(fact_177_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y4: A] :
( ~ ( ord_less_eq @ A @ X3 @ Y4 )
=> ( ord_less_eq @ A @ Y4 @ X3 ) ) ) ).
% le_cases
thf(fact_178_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y4: A] :
( ( X3 = Y4 )
=> ( ord_less_eq @ A @ X3 @ Y4 ) ) ) ).
% eq_refl
thf(fact_179_linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y4: A] :
( ( ord_less_eq @ A @ X3 @ Y4 )
| ( ord_less_eq @ A @ Y4 @ X3 ) ) ) ).
% linear
thf(fact_180_antisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X3: A,Y4: A] :
( ( ord_less_eq @ A @ X3 @ Y4 )
=> ( ( ord_less_eq @ A @ Y4 @ X3 )
=> ( X3 = Y4 ) ) ) ) ).
% antisym
thf(fact_181_eq__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ^ [Y5: A,Z3: A] : Y5 = Z3 )
= ( ^ [X4: A,Y6: A] :
( ( ord_less_eq @ A @ X4 @ Y6 )
& ( ord_less_eq @ A @ Y6 @ X4 ) ) ) ) ) ).
% eq_iff
thf(fact_182_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B3: A,F: A > B,C2: B] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X: A,Y7: A] :
( ( ord_less_eq @ A @ X @ Y7 )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( F @ Y7 ) ) )
=> ( ord_less_eq @ B @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_183_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B > A,B3: B,C2: B] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C2 )
=> ( ! [X: B,Y7: B] :
( ( ord_less_eq @ B @ X @ Y7 )
=> ( ord_less_eq @ A @ ( F @ X ) @ ( F @ Y7 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_184_order__subst2,axiom,
! [A: $tType,C3: $tType] :
( ( ( order @ C3 @ ( type2 @ C3 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B3: A,F: A > C3,C2: C3] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( ord_less_eq @ C3 @ ( F @ B3 ) @ C2 )
=> ( ! [X: A,Y7: A] :
( ( ord_less_eq @ A @ X @ Y7 )
=> ( ord_less_eq @ C3 @ ( F @ X ) @ ( F @ Y7 ) ) )
=> ( ord_less_eq @ C3 @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% order_subst2
thf(fact_185_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B > A,B3: B,C2: B] :
( ( ord_less_eq @ A @ A2 @ ( F @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C2 )
=> ( ! [X: B,Y7: B] :
( ( ord_less_eq @ B @ X @ Y7 )
=> ( ord_less_eq @ A @ ( F @ X ) @ ( F @ Y7 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% order_subst1
thf(fact_186_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F3: A > B,G2: A > B] :
! [X4: A] : ( ord_less_eq @ B @ ( F3 @ X4 ) @ ( G2 @ X4 ) ) ) ) ) ).
% le_fun_def
thf(fact_187_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B] :
( ! [X: A] : ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) )
=> ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% le_funI
thf(fact_188_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B,X3: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) ) ) ) ).
% le_funE
thf(fact_189_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B,X3: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X3 ) @ ( G @ X3 ) ) ) ) ).
% le_funD
thf(fact_190_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A] :
( ( ord_less @ A @ B3 @ A2 )
=> ( A2 != B3 ) ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_191_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( ord_less @ A @ A2 @ B3 )
=> ( A2 != B3 ) ) ) ).
% order.strict_implies_not_eq
thf(fact_192_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y4: A] :
( ( ~ ( ord_less @ A @ X3 @ Y4 ) )
= ( ( ord_less @ A @ Y4 @ X3 )
| ( X3 = Y4 ) ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_193_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A,C2: A] :
( ( ord_less @ A @ B3 @ A2 )
=> ( ( ord_less @ A @ C2 @ B3 )
=> ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% dual_order.strict_trans
thf(fact_194_less__imp__not__less,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y4: A] :
( ( ord_less @ A @ X3 @ Y4 )
=> ~ ( ord_less @ A @ Y4 @ X3 ) ) ) ).
% less_imp_not_less
thf(fact_195_order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A] :
( ( ord_less @ A @ A2 @ B3 )
=> ( ( ord_less @ A @ B3 @ C2 )
=> ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% order.strict_trans
thf(fact_196_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ A2 ) ) ).
% dual_order.irrefl
thf(fact_197_linorder__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y4: A] :
( ~ ( ord_less @ A @ X3 @ Y4 )
=> ( ( X3 != Y4 )
=> ( ord_less @ A @ Y4 @ X3 ) ) ) ) ).
% linorder_cases
thf(fact_198_less__imp__triv,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y4: A,P: $o] :
( ( ord_less @ A @ X3 @ Y4 )
=> ( ( ord_less @ A @ Y4 @ X3 )
=> P ) ) ) ).
% less_imp_triv
thf(fact_199_less__imp__not__eq2,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X3: A,Y4: A] :
( ( ord_less @ A @ X3 @ Y4 )
=> ( Y4 != X3 ) ) ) ).
% less_imp_not_eq2
thf(fact_200_antisym__conv3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y4: A,X3: A] :
( ~ ( ord_less @ A @ Y4 @ X3 )
=> ( ( ~ ( ord_less @ A @ X3 @ Y4 ) )
= ( X3 = Y4 ) ) ) ) ).
% antisym_conv3
thf(fact_201_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,A2: A] :
( ! [X: A] :
( ! [Y3: A] :
( ( ord_less @ A @ Y3 @ X )
=> ( P @ Y3 ) )
=> ( P @ X ) )
=> ( P @ A2 ) ) ) ).
% less_induct
thf(fact_202_less__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y4: A] :
( ( ord_less @ A @ X3 @ Y4 )
=> ~ ( ord_less @ A @ Y4 @ X3 ) ) ) ).
% less_not_sym
thf(fact_203_less__imp__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X3: A,Y4: A] :
( ( ord_less @ A @ X3 @ Y4 )
=> ( X3 != Y4 ) ) ) ).
% less_imp_not_eq
thf(fact_204_dual__order_Oasym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A] :
( ( ord_less @ A @ B3 @ A2 )
=> ~ ( ord_less @ A @ A2 @ B3 ) ) ) ).
% dual_order.asym
thf(fact_205_ord__less__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A] :
( ( ord_less @ A @ A2 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% ord_less_eq_trans
thf(fact_206_ord__eq__less__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A] :
( ( A2 = B3 )
=> ( ( ord_less @ A @ B3 @ C2 )
=> ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% ord_eq_less_trans
thf(fact_207_less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A] :
~ ( ord_less @ A @ X3 @ X3 ) ) ).
% less_irrefl
thf(fact_208_less__linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y4: A] :
( ( ord_less @ A @ X3 @ Y4 )
| ( X3 = Y4 )
| ( ord_less @ A @ Y4 @ X3 ) ) ) ).
% less_linear
thf(fact_209_less__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y4: A,Z: A] :
( ( ord_less @ A @ X3 @ Y4 )
=> ( ( ord_less @ A @ Y4 @ Z )
=> ( ord_less @ A @ X3 @ Z ) ) ) ) ).
% less_trans
thf(fact_210_less__asym_H,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( ord_less @ A @ A2 @ B3 )
=> ~ ( ord_less @ A @ B3 @ A2 ) ) ) ).
% less_asym'
thf(fact_211_less__asym,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y4: A] :
( ( ord_less @ A @ X3 @ Y4 )
=> ~ ( ord_less @ A @ Y4 @ X3 ) ) ) ).
% less_asym
thf(fact_212_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X3: A,Y4: A] :
( ( ord_less @ A @ X3 @ Y4 )
=> ( X3 != Y4 ) ) ) ).
% less_imp_neq
thf(fact_213_dense,axiom,
! [A: $tType] :
( ( dense_order @ A @ ( type2 @ A ) )
=> ! [X3: A,Y4: A] :
( ( ord_less @ A @ X3 @ Y4 )
=> ? [Z2: A] :
( ( ord_less @ A @ X3 @ Z2 )
& ( ord_less @ A @ Z2 @ Y4 ) ) ) ) ).
% dense
thf(fact_214_order_Oasym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( ord_less @ A @ A2 @ B3 )
=> ~ ( ord_less @ A @ B3 @ A2 ) ) ) ).
% order.asym
thf(fact_215_neq__iff,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y4: A] :
( ( X3 != Y4 )
= ( ( ord_less @ A @ X3 @ Y4 )
| ( ord_less @ A @ Y4 @ X3 ) ) ) ) ).
% neq_iff
thf(fact_216_neqE,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y4: A] :
( ( X3 != Y4 )
=> ( ~ ( ord_less @ A @ X3 @ Y4 )
=> ( ord_less @ A @ Y4 @ X3 ) ) ) ) ).
% neqE
thf(fact_217_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A @ ( type2 @ A ) )
=> ! [X3: A] :
? [X1: A] : ( ord_less @ A @ X3 @ X1 ) ) ).
% gt_ex
thf(fact_218_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A @ ( type2 @ A ) )
=> ! [X3: A] :
? [Y7: A] : ( ord_less @ A @ Y7 @ X3 ) ) ).
% lt_ex
thf(fact_219_order__less__subst2,axiom,
! [A: $tType,C3: $tType] :
( ( ( order @ C3 @ ( type2 @ C3 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B3: A,F: A > C3,C2: C3] :
( ( ord_less @ A @ A2 @ B3 )
=> ( ( ord_less @ C3 @ ( F @ B3 ) @ C2 )
=> ( ! [X: A,Y7: A] :
( ( ord_less @ A @ X @ Y7 )
=> ( ord_less @ C3 @ ( F @ X ) @ ( F @ Y7 ) ) )
=> ( ord_less @ C3 @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% order_less_subst2
thf(fact_220_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B > A,B3: B,C2: B] :
( ( ord_less @ A @ A2 @ ( F @ B3 ) )
=> ( ( ord_less @ B @ B3 @ C2 )
=> ( ! [X: B,Y7: B] :
( ( ord_less @ B @ X @ Y7 )
=> ( ord_less @ A @ ( F @ X ) @ ( F @ Y7 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_221_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B3: A,F: A > B,C2: B] :
( ( ord_less @ A @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C2 )
=> ( ! [X: A,Y7: A] :
( ( ord_less @ A @ X @ Y7 )
=> ( ord_less @ B @ ( F @ X ) @ ( F @ Y7 ) ) )
=> ( ord_less @ B @ ( F @ A2 ) @ C2 ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_222_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B > A,B3: B,C2: B] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less @ B @ B3 @ C2 )
=> ( ! [X: B,Y7: B] :
( ( ord_less @ B @ X @ Y7 )
=> ( ord_less @ A @ ( F @ X ) @ ( F @ Y7 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C2 ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_223_pinf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X: A] :
( ( ord_less @ A @ Z4 @ X )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z4: A] :
! [X: A] :
( ( ord_less @ A @ Z4 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: A] :
! [X5: A] :
( ( ord_less @ A @ Z2 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P2 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ) ).
% pinf(1)
thf(fact_224_pinf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X: A] :
( ( ord_less @ A @ Z4 @ X )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z4: A] :
! [X: A] :
( ( ord_less @ A @ Z4 @ X )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: A] :
! [X5: A] :
( ( ord_less @ A @ Z2 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P2 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ) ).
% pinf(2)
thf(fact_225_pinf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z2: A] :
! [X5: A] :
( ( ord_less @ A @ Z2 @ X5 )
=> ( X5 != T2 ) ) ) ).
% pinf(3)
thf(fact_226_pinf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z2: A] :
! [X5: A] :
( ( ord_less @ A @ Z2 @ X5 )
=> ( X5 != T2 ) ) ) ).
% pinf(4)
thf(fact_227_pinf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z2: A] :
! [X5: A] :
( ( ord_less @ A @ Z2 @ X5 )
=> ~ ( ord_less @ A @ X5 @ T2 ) ) ) ).
% pinf(5)
thf(fact_228_pinf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z2: A] :
! [X5: A] :
( ( ord_less @ A @ Z2 @ X5 )
=> ( ord_less @ A @ T2 @ X5 ) ) ) ).
% pinf(7)
thf(fact_229_pinf_I11_J,axiom,
! [C3: $tType,D2: $tType] :
( ( ord @ C3 @ ( type2 @ C3 ) )
=> ! [F4: D2] :
? [Z2: C3] :
! [X5: C3] :
( ( ord_less @ C3 @ Z2 @ X5 )
=> ( F4 = F4 ) ) ) ).
% pinf(11)
thf(fact_230_minf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X: A] :
( ( ord_less @ A @ X @ Z4 )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z4: A] :
! [X: A] :
( ( ord_less @ A @ X @ Z4 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z2 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P2 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ) ).
% minf(1)
thf(fact_231_minf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X: A] :
( ( ord_less @ A @ X @ Z4 )
=> ( ( P @ X )
= ( P2 @ X ) ) )
=> ( ? [Z4: A] :
! [X: A] :
( ( ord_less @ A @ X @ Z4 )
=> ( ( Q @ X )
= ( Q2 @ X ) ) )
=> ? [Z2: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z2 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P2 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ) ).
% minf(2)
thf(fact_232_minf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z2: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z2 )
=> ( X5 != T2 ) ) ) ).
% minf(3)
thf(fact_233_minf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z2: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z2 )
=> ( X5 != T2 ) ) ) ).
% minf(4)
thf(fact_234_minf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z2: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z2 )
=> ( ord_less @ A @ X5 @ T2 ) ) ) ).
% minf(5)
thf(fact_235_minf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T2: A] :
? [Z2: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z2 )
=> ~ ( ord_less @ A @ T2 @ X5 ) ) ) ).
% minf(7)
thf(fact_236_minf_I11_J,axiom,
! [C3: $tType,D2: $tType] :
( ( ord @ C3 @ ( type2 @ C3 ) )
=> ! [F4: D2] :
? [Z2: C3] :
! [X5: C3] :
( ( ord_less @ C3 @ X5 @ Z2 )
=> ( F4 = F4 ) ) ) ).
% minf(11)
thf(fact_237_less__eq__enat__def,axiom,
( ( ord_less_eq @ extended_enat )
= ( ^ [M4: extended_enat] :
( extended_case_enat @ $o
@ ^ [N1: nat] :
( extended_case_enat @ $o
@ ^ [M1: nat] : ( ord_less_eq @ nat @ M1 @ N1 )
@ $false
@ M4 )
@ $true ) ) ) ).
% less_eq_enat_def
thf(fact_238_ltakeWhile__lappend2,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,P: A > $o,Ys2: coindu1593790203_llist @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( coindu1112613586e_lset @ A @ Xs ) @ ( collect @ A @ P ) )
=> ( ( coindu721411036eWhile @ A @ P @ ( coindu268472904append @ A @ Xs @ Ys2 ) )
= ( coindu268472904append @ A @ Xs @ ( coindu721411036eWhile @ A @ P @ Ys2 ) ) ) ) ).
% ltakeWhile_lappend2
thf(fact_239_order_Onot__eq__order__implies__strict,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( A2 != B3 )
=> ( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ord_less @ A @ A2 @ B3 ) ) ) ) ).
% order.not_eq_order_implies_strict
thf(fact_240_dual__order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A] :
( ( ord_less @ A @ B3 @ A2 )
=> ( ord_less_eq @ A @ B3 @ A2 ) ) ) ).
% dual_order.strict_implies_order
thf(fact_241_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_242_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_243_order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( ord_less @ A @ A2 @ B3 )
=> ( ord_less_eq @ A @ A2 @ B3 ) ) ) ).
% order.strict_implies_order
thf(fact_244_dense__le__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y4: A,Z: A] :
( ( ord_less @ A @ X3 @ Y4 )
=> ( ! [W: A] :
( ( ord_less @ A @ X3 @ W )
=> ( ( ord_less @ A @ W @ Y4 )
=> ( ord_less_eq @ A @ W @ Z ) ) )
=> ( ord_less_eq @ A @ Y4 @ Z ) ) ) ) ).
% dense_le_bounded
thf(fact_245_dense__ge__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Z: A,X3: A,Y4: A] :
( ( ord_less @ A @ Z @ X3 )
=> ( ! [W: A] :
( ( ord_less @ A @ Z @ W )
=> ( ( ord_less @ A @ W @ X3 )
=> ( ord_less_eq @ A @ Y4 @ W ) ) )
=> ( ord_less_eq @ A @ Y4 @ Z ) ) ) ) ).
% dense_ge_bounded
thf(fact_246_dual__order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A,C2: A] :
( ( ord_less @ A @ B3 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ B3 )
=> ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% dual_order.strict_trans2
thf(fact_247_dual__order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B3 @ A2 )
=> ( ( ord_less @ A @ C2 @ B3 )
=> ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% dual_order.strict_trans1
thf(fact_248_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_249_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_250_order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A] :
( ( ord_less @ A @ A2 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ C2 )
=> ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% order.strict_trans2
thf(fact_251_order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( ord_less @ A @ B3 @ C2 )
=> ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% order.strict_trans1
thf(fact_252_not__le__imp__less,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y4: A,X3: A] :
( ~ ( ord_less_eq @ A @ Y4 @ X3 )
=> ( ord_less @ A @ X3 @ Y4 ) ) ) ).
% not_le_imp_less
thf(fact_253_less__le__not__le,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [X4: A,Y6: A] :
( ( ord_less_eq @ A @ X4 @ Y6 )
& ~ ( ord_less_eq @ A @ Y6 @ X4 ) ) ) ) ) ).
% less_le_not_le
thf(fact_254_le__imp__less__or__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X3: A,Y4: A] :
( ( ord_less_eq @ A @ X3 @ Y4 )
=> ( ( ord_less @ A @ X3 @ Y4 )
| ( X3 = Y4 ) ) ) ) ).
% le_imp_less_or_eq
%----Type constructors (23)
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___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 (1)
thf(conj_0,conjecture,
$false ).
%------------------------------------------------------------------------------