TPTP Problem File: DAT124^1.p
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%------------------------------------------------------------------------------
% File : DAT124^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Coinductive list 2275
% 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__2275.p [Bla16]
% Status : Theorem
% Rating : 0.00 v7.5.0, 0.33 v7.2.0, 0.25 v7.1.0
% Syntax : Number of formulae : 322 ( 57 unt; 49 typ; 0 def)
% Number of atoms : 878 ( 199 equ; 0 cnn)
% Maximal formula atoms : 13 ( 3 avg)
% Number of connectives : 3697 ( 79 ~; 22 |; 42 &;3073 @)
% ( 0 <=>; 481 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 9 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 109 ( 109 >; 0 *; 0 +; 0 <<)
% Number of symbols : 48 ( 47 usr; 4 con; 0-3 aty)
% Number of variables : 1022 ( 31 ^; 917 !; 33 ?;1022 :)
% ( 41 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:55:40.237
%------------------------------------------------------------------------------
%----Could-be-implicit typings (4)
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_itself,type,
itself: $tType > $tType ).
thf(ty_tf_a,type,
a: $tType ).
%----Explicit typings (45)
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_Groups_Ogroup__add,type,
group_add:
!>[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_Groups_Oab__group__add,type,
ab_group_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Owellorder,type,
wellorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Osemigroup__add,type,
semigroup_add:
!>[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_Groups_Oab__semigroup__add,type,
ab_semigroup_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Odense__linorder,type,
dense_linorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Olinordered__semidom,type,
linordered_semidom:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
cancel1352612707id_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
cancel146912293up_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
ordere779506340up_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni770627133id_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
ordere236663937imp_le:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
ordere223160158up_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__cancel__comm__monoid__diff,type,
ordere623563068d_diff:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add,type,
strict2144017051up_add:
!>[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_Oldrop,type,
coindu191418589_ldrop:
!>[A: $tType] : ( extended_enat > ( 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_OLNil,type,
coindu1598213697e_LNil:
!>[A: $tType] : ( coindu1593790203_llist @ A ) ).
thf(sy_c_Coinductive__List__Mirabelle__kmikjhschf_Ollist_Olnull,type,
coindu335574135_lnull:
!>[A: $tType] : ( ( coindu1593790203_llist @ A ) > $o ) ).
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_Oltake,type,
coindu1802687541_ltake:
!>[A: $tType] : ( extended_enat > ( coindu1593790203_llist @ A ) > ( coindu1593790203_llist @ A ) ) ).
thf(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : ( A > A > A ) ).
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_v_xs,type,
xs: coindu1593790203_llist @ a ).
thf(sy_v_xs_H____,type,
xs2: coindu1593790203_llist @ a ).
thf(sy_v_ys,type,
ys: coindu1593790203_llist @ a ).
thf(sy_v_zs,type,
zs: coindu1593790203_llist @ a ).
%----Relevant facts (254)
thf(fact_0__092_060open_062_092_060not_062_Allength_Ays_A_092_060le_062_Allength_Axs_092_060close_062,axiom,
~ ( ord_less_eq @ extended_enat @ ( coindu1018505716length @ a @ ys ) @ ( coindu1018505716length @ a @ xs ) ) ).
% \<open>\<not> llength ys \<le> llength xs\<close>
thf(fact_1__092_060open_062_092_060not_062_A_Ixs_A_092_060sqsubseteq_062_Ays_A_092_060or_062_Ays_A_092_060sqsubseteq_062_Axs_J_092_060close_062,axiom,
~ ( ( coindu1696667936prefix @ a @ xs @ ys )
| ( coindu1696667936prefix @ a @ ys @ xs ) ) ).
% \<open>\<not> (xs \<sqsubseteq> ys \<or> ys \<sqsubseteq> xs)\<close>
thf(fact_2__092_060open_062_092_060not_062_Axs_A_092_060sqsubseteq_062_Ays_092_060close_062,axiom,
~ ( coindu1696667936prefix @ a @ xs @ ys ) ).
% \<open>\<not> xs \<sqsubseteq> ys\<close>
thf(fact_3__092_060open_062_092_060not_062_Ays_A_092_060sqsubseteq_062_Axs_092_060close_062,axiom,
~ ( coindu1696667936prefix @ a @ ys @ xs ) ).
% \<open>\<not> ys \<sqsubseteq> xs\<close>
thf(fact_4__092_060open_062lappend_Axs_Axs_H_A_061_Alappend_Ays_Azs_092_060close_062,axiom,
( ( coindu268472904append @ a @ xs @ xs2 )
= ( coindu268472904append @ a @ ys @ zs ) ) ).
% \<open>lappend xs xs' = lappend ys zs\<close>
thf(fact_5_lstrict__prefix__llength__less,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu574146665prefix @ A @ Xs @ Ys )
=> ( ord_less @ extended_enat @ ( coindu1018505716length @ A @ Xs ) @ ( coindu1018505716length @ A @ Ys ) ) ) ).
% lstrict_prefix_llength_less
thf(fact_6__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062xs_H_O_Alappend_Axs_Axs_H_A_061_Alappend_Ays_Azs_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Xs2: coindu1593790203_llist @ a] :
( ( coindu268472904append @ a @ xs @ Xs2 )
!= ( coindu268472904append @ a @ ys @ zs ) ) ).
% \<open>\<And>thesis. (\<And>xs'. lappend xs xs' = lappend ys zs \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_7_enat__less__induct,axiom,
! [P: extended_enat > $o,N: extended_enat] :
( ! [N2: extended_enat] :
( ! [M: extended_enat] :
( ( ord_less @ extended_enat @ M @ N2 )
=> ( P @ M ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% enat_less_induct
thf(fact_8_minf_I11_J,axiom,
! [C: $tType,D: $tType] :
( ( ord @ C @ ( type2 @ C ) )
=> ! [F: D] :
? [Z: C] :
! [X: C] :
( ( ord_less @ C @ X @ Z )
=> ( F = F ) ) ) ).
% minf(11)
thf(fact_9_minf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z: A] :
! [X: A] :
( ( ord_less @ A @ X @ Z )
=> ~ ( ord_less @ A @ T @ X ) ) ) ).
% minf(7)
thf(fact_10_minf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z: A] :
! [X: A] :
( ( ord_less @ A @ X @ Z )
=> ( ord_less @ A @ X @ T ) ) ) ).
% minf(5)
thf(fact_11_minf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z: A] :
! [X: A] :
( ( ord_less @ A @ X @ Z )
=> ( X != T ) ) ) ).
% minf(4)
thf(fact_12_minf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z: A] :
! [X: A] :
( ( ord_less @ A @ X @ Z )
=> ( X != T ) ) ) ).
% minf(3)
thf(fact_13_minf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z2: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z2: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: A] :
! [X: A] :
( ( ord_less @ A @ X @ Z )
=> ( ( ( P @ X )
| ( Q @ X ) )
= ( ( P2 @ X )
| ( Q2 @ X ) ) ) ) ) ) ) ).
% minf(2)
thf(fact_14_lprefix__refl,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ Xs @ Xs ) ).
% lprefix_refl
thf(fact_15_llist_Oleq__refl,axiom,
! [A: $tType,X3: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ X3 @ X3 ) ).
% llist.leq_refl
thf(fact_16_assms,axiom,
coindu1696667936prefix @ a @ xs @ ( coindu268472904append @ a @ ys @ zs ) ).
% assms
thf(fact_17_lappend__assoc,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,Zs: coindu1593790203_llist @ A] :
( ( coindu268472904append @ A @ ( coindu268472904append @ A @ Xs @ Ys ) @ Zs )
= ( coindu268472904append @ A @ Xs @ ( coindu268472904append @ A @ Ys @ Zs ) ) ) ).
% lappend_assoc
thf(fact_18_lprefix__trans,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,Zs: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ Ys )
=> ( ( coindu1696667936prefix @ A @ Ys @ Zs )
=> ( coindu1696667936prefix @ A @ Xs @ Zs ) ) ) ).
% lprefix_trans
thf(fact_19_llist_Oleq__trans,axiom,
! [A: $tType,X3: coindu1593790203_llist @ A,Y: coindu1593790203_llist @ A,Z3: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ X3 @ Y )
=> ( ( coindu1696667936prefix @ A @ Y @ Z3 )
=> ( coindu1696667936prefix @ A @ X3 @ Z3 ) ) ) ).
% llist.leq_trans
thf(fact_20_lprefix__antisym,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ Ys )
=> ( ( coindu1696667936prefix @ A @ Ys @ Xs )
=> ( Xs = Ys ) ) ) ).
% lprefix_antisym
thf(fact_21_lprefix__lappend,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ Xs @ ( coindu268472904append @ A @ Xs @ Ys ) ) ).
% lprefix_lappend
thf(fact_22_lappend__lprefixE,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,Zs: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ ( coindu268472904append @ A @ Xs @ Ys ) @ Zs )
=> ~ ! [Zs2: coindu1593790203_llist @ A] :
( Zs
!= ( coindu268472904append @ A @ Xs @ Zs2 ) ) ) ).
% lappend_lprefixE
thf(fact_23_llist_Oleq__antisym,axiom,
! [A: $tType,X3: coindu1593790203_llist @ A,Y: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ X3 @ Y )
=> ( ( coindu1696667936prefix @ A @ Y @ X3 )
=> ( X3 = Y ) ) ) ).
% llist.leq_antisym
thf(fact_24_lstrict__prefix__def,axiom,
! [A: $tType] :
( ( coindu574146665prefix @ A )
= ( ^ [Xs3: coindu1593790203_llist @ A,Ys2: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs3 @ Ys2 )
& ( Xs3 != Ys2 ) ) ) ) ).
% lstrict_prefix_def
thf(fact_25_lprefix__down__linear,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Zs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ Zs )
=> ( ( coindu1696667936prefix @ A @ Ys @ Zs )
=> ( ( coindu1696667936prefix @ A @ Xs @ Ys )
| ( coindu1696667936prefix @ A @ Ys @ Xs ) ) ) ) ).
% lprefix_down_linear
thf(fact_26_lprefix__conv__lappend,axiom,
! [A: $tType] :
( ( coindu1696667936prefix @ A )
= ( ^ [Xs3: coindu1593790203_llist @ A,Ys2: coindu1593790203_llist @ A] :
? [Zs3: coindu1593790203_llist @ A] :
( Ys2
= ( coindu268472904append @ A @ Xs3 @ Zs3 ) ) ) ) ).
% lprefix_conv_lappend
thf(fact_27_lprefix__lappend__sameI,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,Zs: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ Ys )
=> ( coindu1696667936prefix @ A @ ( coindu268472904append @ A @ Zs @ Xs ) @ ( coindu268472904append @ A @ Zs @ Ys ) ) ) ).
% lprefix_lappend_sameI
thf(fact_28_llist__less__induct,axiom,
! [A: $tType,P: ( coindu1593790203_llist @ A ) > $o,Xs: coindu1593790203_llist @ A] :
( ! [Xs4: coindu1593790203_llist @ A] :
( ! [Ys3: coindu1593790203_llist @ A] :
( ( coindu574146665prefix @ A @ Ys3 @ Xs4 )
=> ( P @ Ys3 ) )
=> ( P @ Xs4 ) )
=> ( P @ Xs ) ) ).
% llist_less_induct
thf(fact_29_lprefix__llength__le,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ Ys )
=> ( ord_less_eq @ extended_enat @ ( coindu1018505716length @ A @ Xs ) @ ( coindu1018505716length @ A @ Ys ) ) ) ).
% lprefix_llength_le
thf(fact_30_lprefix__llength__eq__imp__eq,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ Ys )
=> ( ( ( coindu1018505716length @ A @ Xs )
= ( coindu1018505716length @ A @ Ys ) )
=> ( Xs = Ys ) ) ) ).
% lprefix_llength_eq_imp_eq
thf(fact_31_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_32_pinf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z: A] :
! [X: A] :
( ( ord_less @ A @ Z @ X )
=> ~ ( ord_less_eq @ A @ X @ T ) ) ) ).
% pinf(6)
thf(fact_33_pinf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z: A] :
! [X: A] :
( ( ord_less @ A @ Z @ X )
=> ( ord_less_eq @ A @ T @ X ) ) ) ).
% pinf(8)
thf(fact_34_minf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z: A] :
! [X: A] :
( ( ord_less @ A @ X @ Z )
=> ( ord_less_eq @ A @ X @ T ) ) ) ).
% minf(6)
thf(fact_35_minf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z: A] :
! [X: A] :
( ( ord_less @ A @ X @ Z )
=> ~ ( ord_less_eq @ A @ T @ X ) ) ) ).
% minf(8)
thf(fact_36_pinf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z2: A] :
! [X2: A] :
( ( ord_less @ A @ Z2 @ X2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z2: A] :
! [X2: A] :
( ( ord_less @ A @ Z2 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: A] :
! [X: A] :
( ( ord_less @ A @ Z @ X )
=> ( ( ( P @ X )
& ( Q @ X ) )
= ( ( P2 @ X )
& ( Q2 @ X ) ) ) ) ) ) ) ).
% pinf(1)
thf(fact_37_pinf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z2: A] :
! [X2: A] :
( ( ord_less @ A @ Z2 @ X2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z2: A] :
! [X2: A] :
( ( ord_less @ A @ Z2 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: A] :
! [X: A] :
( ( ord_less @ A @ Z @ X )
=> ( ( ( P @ X )
| ( Q @ X ) )
= ( ( P2 @ X )
| ( Q2 @ X ) ) ) ) ) ) ) ).
% pinf(2)
thf(fact_38_pinf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z: A] :
! [X: A] :
( ( ord_less @ A @ Z @ X )
=> ( X != T ) ) ) ).
% pinf(3)
thf(fact_39_pinf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z: A] :
! [X: A] :
( ( ord_less @ A @ Z @ X )
=> ( X != T ) ) ) ).
% pinf(4)
thf(fact_40_pinf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z: A] :
! [X: A] :
( ( ord_less @ A @ Z @ X )
=> ~ ( ord_less @ A @ X @ T ) ) ) ).
% pinf(5)
thf(fact_41_pinf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z: A] :
! [X: A] :
( ( ord_less @ A @ Z @ X )
=> ( ord_less @ A @ T @ X ) ) ) ).
% pinf(7)
thf(fact_42_pinf_I11_J,axiom,
! [C: $tType,D: $tType] :
( ( ord @ C @ ( type2 @ C ) )
=> ! [F: D] :
? [Z: C] :
! [X: C] :
( ( ord_less @ C @ Z @ X )
=> ( F = F ) ) ) ).
% pinf(11)
thf(fact_43_minf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z2: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z2: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: A] :
! [X: A] :
( ( ord_less @ A @ X @ Z )
=> ( ( ( P @ X )
& ( Q @ X ) )
= ( ( P2 @ X )
& ( Q2 @ X ) ) ) ) ) ) ) ).
% minf(1)
thf(fact_44_eq,axiom,
( ( coindu268472904append @ a @ ( coindu1802687541_ltake @ a @ ( coindu1018505716length @ a @ ys ) @ xs ) @ ( coindu268472904append @ a @ ( coindu191418589_ldrop @ a @ ( coindu1018505716length @ a @ ys ) @ xs ) @ xs2 ) )
= ( coindu268472904append @ a @ ys @ zs ) ) ).
% eq
thf(fact_45_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A] : ( ord_less_eq @ A @ X3 @ X3 ) ) ).
% order_refl
thf(fact_46_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 )
=> ? [C2: A] :
( ( ord_less_eq @ A @ A2 @ C2 )
& ( ord_less_eq @ A @ C2 @ B )
& ! [X: A] :
( ( ( ord_less_eq @ A @ A2 @ X )
& ( ord_less @ A @ X @ C2 ) )
=> ( P @ X ) )
& ! [D2: A] :
( ! [X2: A] :
( ( ( ord_less_eq @ A @ A2 @ X2 )
& ( ord_less @ A @ X2 @ D2 ) )
=> ( P @ X2 ) )
=> ( ord_less_eq @ A @ D2 @ C2 ) ) ) ) ) ) ) ).
% complete_interval
thf(fact_47_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_48_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_49_dual__order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [B3: A,A4: A] :
( ( ord_less_eq @ A @ B3 @ A4 )
& ( A4 != B3 ) ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_50_dual__order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [B3: A,A4: A] :
( ( ord_less @ A @ B3 @ A4 )
| ( A4 = B3 ) ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_51_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_52_dense__le__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y: A,Z3: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ! [W: A] :
( ( ord_less @ A @ X3 @ W )
=> ( ( ord_less @ A @ W @ Y )
=> ( ord_less_eq @ A @ W @ Z3 ) ) )
=> ( ord_less_eq @ A @ Y @ Z3 ) ) ) ) ).
% dense_le_bounded
thf(fact_53_dense__ge__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Z3: A,X3: A,Y: A] :
( ( ord_less @ A @ Z3 @ X3 )
=> ( ! [W: A] :
( ( ord_less @ A @ Z3 @ W )
=> ( ( ord_less @ A @ W @ X3 )
=> ( ord_less_eq @ A @ Y @ W ) ) )
=> ( ord_less_eq @ A @ Y @ Z3 ) ) ) ) ).
% dense_ge_bounded
thf(fact_54_dual__order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B: A,A2: A,C3: A] :
( ( ord_less @ A @ B @ A2 )
=> ( ( ord_less_eq @ A @ C3 @ B )
=> ( ord_less @ A @ C3 @ A2 ) ) ) ) ).
% dual_order.strict_trans2
thf(fact_55_ltake__is__lprefix,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ ( coindu1802687541_ltake @ A @ N @ Xs ) @ Xs ) ).
% ltake_is_lprefix
thf(fact_56_lprefix__ltake__same,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A,M2: extended_enat] :
( ( coindu1696667936prefix @ A @ ( coindu1802687541_ltake @ A @ N @ Xs ) @ ( coindu1802687541_ltake @ A @ M2 @ Xs ) )
= ( ( ord_less_eq @ extended_enat @ N @ M2 )
| ( ord_less_eq @ extended_enat @ ( coindu1018505716length @ A @ Xs ) @ M2 ) ) ) ).
% lprefix_ltake_same
thf(fact_57_lappend__ltake__ldrop,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
( ( coindu268472904append @ A @ ( coindu1802687541_ltake @ A @ N @ Xs ) @ ( coindu191418589_ldrop @ A @ N @ Xs ) )
= Xs ) ).
% lappend_ltake_ldrop
thf(fact_58_ltake__eq__ltake__antimono,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A,M2: extended_enat] :
( ( ( coindu1802687541_ltake @ A @ N @ Xs )
= ( coindu1802687541_ltake @ A @ N @ Ys ) )
=> ( ( ord_less_eq @ extended_enat @ M2 @ N )
=> ( ( coindu1802687541_ltake @ A @ M2 @ Xs )
= ( coindu1802687541_ltake @ A @ M2 @ Ys ) ) ) ) ).
% ltake_eq_ltake_antimono
thf(fact_59_ltake__all,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,M2: extended_enat] :
( ( ord_less_eq @ extended_enat @ ( coindu1018505716length @ A @ Xs ) @ M2 )
=> ( ( coindu1802687541_ltake @ A @ M2 @ Xs )
= Xs ) ) ).
% ltake_all
thf(fact_60_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_61_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B: A,A2: A,C3: A] :
( ( ord_less_eq @ A @ B @ A2 )
=> ( ( ord_less_eq @ A @ C3 @ B )
=> ( ord_less_eq @ A @ C3 @ A2 ) ) ) ) ).
% dual_order.trans
thf(fact_62_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_63_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_64_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y: A,Z3: A] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z3 )
=> ( ord_less_eq @ A @ X3 @ Z3 ) ) ) ) ).
% order_trans
thf(fact_65_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_66_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( B = C3 )
=> ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_67_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C3: A] :
( ( A2 = B )
=> ( ( ord_less_eq @ A @ B @ C3 )
=> ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_68_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [Y: A,X3: A] :
( ( ord_less_eq @ A @ Y @ X3 )
=> ( ( ord_less_eq @ A @ X3 @ Y )
= ( X3 = Y ) ) ) ) ).
% antisym_conv
thf(fact_69_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y: A,Z3: A] :
( ( ( ord_less_eq @ A @ X3 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z3 ) )
=> ( ( ( ord_less_eq @ A @ Y @ X3 )
=> ~ ( ord_less_eq @ A @ X3 @ Z3 ) )
=> ( ( ( ord_less_eq @ A @ X3 @ Z3 )
=> ~ ( ord_less_eq @ A @ Z3 @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z3 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X3 ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z3 )
=> ~ ( ord_less_eq @ A @ Z3 @ X3 ) )
=> ~ ( ( ord_less_eq @ A @ Z3 @ X3 )
=> ~ ( ord_less_eq @ A @ X3 @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_70_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( ord_less_eq @ A @ B @ C3 )
=> ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% order.trans
thf(fact_71_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y: A] :
( ~ ( ord_less_eq @ A @ X3 @ Y )
=> ( ord_less_eq @ A @ Y @ X3 ) ) ) ).
% le_cases
thf(fact_72_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y: A] :
( ( X3 = Y )
=> ( ord_less_eq @ A @ X3 @ Y ) ) ) ).
% eq_refl
thf(fact_73_linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
| ( ord_less_eq @ A @ Y @ X3 ) ) ) ).
% linear
thf(fact_74_antisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( ord_less_eq @ A @ Y @ X3 )
=> ( X3 = Y ) ) ) ) ).
% antisym
thf(fact_75_eq__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ^ [Y2: A,Z4: A] : Y2 = Z4 )
= ( ^ [X4: A,Y3: A] :
( ( ord_less_eq @ A @ X4 @ Y3 )
& ( ord_less_eq @ A @ Y3 @ X4 ) ) ) ) ) ).
% eq_iff
thf(fact_76_ord__le__eq__subst,axiom,
! [A: $tType,B4: $tType] :
( ( ( ord @ B4 @ ( type2 @ B4 ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B: A,F2: A > B4,C3: B4] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( ( F2 @ B )
= C3 )
=> ( ! [X2: A,Y4: A] :
( ( ord_less_eq @ A @ X2 @ Y4 )
=> ( ord_less_eq @ B4 @ ( F2 @ X2 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq @ B4 @ ( F2 @ A2 ) @ C3 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_77_ord__eq__le__subst,axiom,
! [A: $tType,B4: $tType] :
( ( ( ord @ B4 @ ( type2 @ B4 ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F2: B4 > A,B: B4,C3: B4] :
( ( A2
= ( F2 @ B ) )
=> ( ( ord_less_eq @ B4 @ B @ C3 )
=> ( ! [X2: B4,Y4: B4] :
( ( ord_less_eq @ B4 @ X2 @ Y4 )
=> ( ord_less_eq @ A @ ( F2 @ X2 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F2 @ C3 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_78_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C @ ( type2 @ C ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B: A,F2: A > C,C3: C] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( ord_less_eq @ C @ ( F2 @ B ) @ C3 )
=> ( ! [X2: A,Y4: A] :
( ( ord_less_eq @ A @ X2 @ Y4 )
=> ( ord_less_eq @ C @ ( F2 @ X2 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq @ C @ ( F2 @ A2 ) @ C3 ) ) ) ) ) ).
% order_subst2
thf(fact_79_order__subst1,axiom,
! [A: $tType,B4: $tType] :
( ( ( order @ B4 @ ( type2 @ B4 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F2: B4 > A,B: B4,C3: B4] :
( ( ord_less_eq @ A @ A2 @ ( F2 @ B ) )
=> ( ( ord_less_eq @ B4 @ B @ C3 )
=> ( ! [X2: B4,Y4: B4] :
( ( ord_less_eq @ B4 @ X2 @ Y4 )
=> ( ord_less_eq @ A @ ( F2 @ X2 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F2 @ C3 ) ) ) ) ) ) ).
% order_subst1
thf(fact_80_le__fun__def,axiom,
! [B4: $tType,A: $tType] :
( ( ord @ B4 @ ( type2 @ B4 ) )
=> ( ( ord_less_eq @ ( A > B4 ) )
= ( ^ [F3: A > B4,G: A > B4] :
! [X4: A] : ( ord_less_eq @ B4 @ ( F3 @ X4 ) @ ( G @ X4 ) ) ) ) ) ).
% le_fun_def
thf(fact_81_le__funI,axiom,
! [B4: $tType,A: $tType] :
( ( ord @ B4 @ ( type2 @ B4 ) )
=> ! [F2: A > B4,G2: A > B4] :
( ! [X2: A] : ( ord_less_eq @ B4 @ ( F2 @ X2 ) @ ( G2 @ X2 ) )
=> ( ord_less_eq @ ( A > B4 ) @ F2 @ G2 ) ) ) ).
% le_funI
thf(fact_82_le__funE,axiom,
! [B4: $tType,A: $tType] :
( ( ord @ B4 @ ( type2 @ B4 ) )
=> ! [F2: A > B4,G2: A > B4,X3: A] :
( ( ord_less_eq @ ( A > B4 ) @ F2 @ G2 )
=> ( ord_less_eq @ B4 @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ).
% le_funE
thf(fact_83_le__funD,axiom,
! [B4: $tType,A: $tType] :
( ( ord @ B4 @ ( type2 @ B4 ) )
=> ! [F2: A > B4,G2: A > B4,X3: A] :
( ( ord_less_eq @ ( A > B4 ) @ F2 @ G2 )
=> ( ord_less_eq @ B4 @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ).
% le_funD
thf(fact_84_ltake__lappend1,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( ord_less_eq @ extended_enat @ N @ ( coindu1018505716length @ A @ Xs ) )
=> ( ( coindu1802687541_ltake @ A @ N @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( coindu1802687541_ltake @ A @ N @ Xs ) ) ) ).
% ltake_lappend1
thf(fact_85_ord__eq__less__subst,axiom,
! [A: $tType,B4: $tType] :
( ( ( ord @ B4 @ ( type2 @ B4 ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F2: B4 > A,B: B4,C3: B4] :
( ( A2
= ( F2 @ B ) )
=> ( ( ord_less @ B4 @ B @ C3 )
=> ( ! [X2: B4,Y4: B4] :
( ( ord_less @ B4 @ X2 @ Y4 )
=> ( ord_less @ A @ ( F2 @ X2 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less @ A @ A2 @ ( F2 @ C3 ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_86_ord__less__eq__subst,axiom,
! [A: $tType,B4: $tType] :
( ( ( ord @ B4 @ ( type2 @ B4 ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B: A,F2: A > B4,C3: B4] :
( ( ord_less @ A @ A2 @ B )
=> ( ( ( F2 @ B )
= C3 )
=> ( ! [X2: A,Y4: A] :
( ( ord_less @ A @ X2 @ Y4 )
=> ( ord_less @ B4 @ ( F2 @ X2 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less @ B4 @ ( F2 @ A2 ) @ C3 ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_87_order__less__subst1,axiom,
! [A: $tType,B4: $tType] :
( ( ( order @ B4 @ ( type2 @ B4 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F2: B4 > A,B: B4,C3: B4] :
( ( ord_less @ A @ A2 @ ( F2 @ B ) )
=> ( ( ord_less @ B4 @ B @ C3 )
=> ( ! [X2: B4,Y4: B4] :
( ( ord_less @ B4 @ X2 @ Y4 )
=> ( ord_less @ A @ ( F2 @ X2 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less @ A @ A2 @ ( F2 @ C3 ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_88_order__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C @ ( type2 @ C ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B: A,F2: A > C,C3: C] :
( ( ord_less @ A @ A2 @ B )
=> ( ( ord_less @ C @ ( F2 @ B ) @ C3 )
=> ( ! [X2: A,Y4: A] :
( ( ord_less @ A @ X2 @ Y4 )
=> ( ord_less @ C @ ( F2 @ X2 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less @ C @ ( F2 @ A2 ) @ C3 ) ) ) ) ) ).
% order_less_subst2
thf(fact_89_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A @ ( type2 @ A ) )
=> ! [X3: A] :
? [Y4: A] : ( ord_less @ A @ Y4 @ X3 ) ) ).
% lt_ex
thf(fact_90_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A @ ( type2 @ A ) )
=> ! [X3: A] :
? [X1: A] : ( ord_less @ A @ X3 @ X1 ) ) ).
% gt_ex
thf(fact_91_neqE,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y: A] :
( ( X3 != Y )
=> ( ~ ( ord_less @ A @ X3 @ Y )
=> ( ord_less @ A @ Y @ X3 ) ) ) ) ).
% neqE
thf(fact_92_neq__iff,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y: A] :
( ( X3 != Y )
= ( ( ord_less @ A @ X3 @ Y )
| ( ord_less @ A @ Y @ X3 ) ) ) ) ).
% neq_iff
thf(fact_93_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_94_dense,axiom,
! [A: $tType] :
( ( dense_order @ A @ ( type2 @ A ) )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ? [Z: A] :
( ( ord_less @ A @ X3 @ Z )
& ( ord_less @ A @ Z @ Y ) ) ) ) ).
% dense
thf(fact_95_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( X3 != Y ) ) ) ).
% less_imp_neq
thf(fact_96_less__asym,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ~ ( ord_less @ A @ Y @ X3 ) ) ) ).
% less_asym
thf(fact_97_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_98_less__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y: A,Z3: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ( ord_less @ A @ Y @ Z3 )
=> ( ord_less @ A @ X3 @ Z3 ) ) ) ) ).
% less_trans
thf(fact_99_less__linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
| ( X3 = Y )
| ( ord_less @ A @ Y @ X3 ) ) ) ).
% less_linear
thf(fact_100_less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A] :
~ ( ord_less @ A @ X3 @ X3 ) ) ).
% less_irrefl
thf(fact_101_ord__eq__less__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C3: A] :
( ( A2 = B )
=> ( ( ord_less @ A @ B @ C3 )
=> ( ord_less @ A @ A2 @ C3 ) ) ) ) ).
% ord_eq_less_trans
thf(fact_102_ord__less__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C3: A] :
( ( ord_less @ A @ A2 @ B )
=> ( ( B = C3 )
=> ( ord_less @ A @ A2 @ C3 ) ) ) ) ).
% ord_less_eq_trans
thf(fact_103_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_104_less__imp__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( X3 != Y ) ) ) ).
% less_imp_not_eq
thf(fact_105_less__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ~ ( ord_less @ A @ Y @ X3 ) ) ) ).
% less_not_sym
thf(fact_106_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,A2: A] :
( ! [X2: A] :
( ! [Y5: A] :
( ( ord_less @ A @ Y5 @ X2 )
=> ( P @ Y5 ) )
=> ( P @ X2 ) )
=> ( P @ A2 ) ) ) ).
% less_induct
thf(fact_107_antisym__conv3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y: A,X3: A] :
( ~ ( ord_less @ A @ Y @ X3 )
=> ( ( ~ ( ord_less @ A @ X3 @ Y ) )
= ( X3 = Y ) ) ) ) ).
% antisym_conv3
thf(fact_108_less__imp__not__eq2,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( Y != X3 ) ) ) ).
% less_imp_not_eq2
thf(fact_109_less__imp__triv,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y: A,P: $o] :
( ( ord_less @ A @ X3 @ Y )
=> ( ( ord_less @ A @ Y @ X3 )
=> P ) ) ) ).
% less_imp_triv
thf(fact_110_linorder__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y: A] :
( ~ ( ord_less @ A @ X3 @ Y )
=> ( ( X3 != Y )
=> ( ord_less @ A @ Y @ X3 ) ) ) ) ).
% linorder_cases
thf(fact_111_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ A2 ) ) ).
% dual_order.irrefl
thf(fact_112_order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C3: A] :
( ( ord_less @ A @ A2 @ B )
=> ( ( ord_less @ A @ B @ C3 )
=> ( ord_less @ A @ A2 @ C3 ) ) ) ) ).
% order.strict_trans
thf(fact_113_less__imp__not__less,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ~ ( ord_less @ A @ Y @ X3 ) ) ) ).
% less_imp_not_less
thf(fact_114_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B: A,A2: A,C3: A] :
( ( ord_less @ A @ B @ A2 )
=> ( ( ord_less @ A @ C3 @ B )
=> ( ord_less @ A @ C3 @ A2 ) ) ) ) ).
% dual_order.strict_trans
thf(fact_115_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y: A] :
( ( ~ ( ord_less @ A @ X3 @ Y ) )
= ( ( ord_less @ A @ Y @ X3 )
| ( X3 = Y ) ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_116_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_117_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_118_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_119_leD,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y: A,X3: A] :
( ( ord_less_eq @ A @ Y @ X3 )
=> ~ ( ord_less @ A @ X3 @ Y ) ) ) ).
% leD
thf(fact_120_leI,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y: A] :
( ~ ( ord_less @ A @ X3 @ Y )
=> ( ord_less_eq @ A @ Y @ X3 ) ) ) ).
% leI
thf(fact_121_le__less,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [X4: A,Y3: A] :
( ( ord_less @ A @ X4 @ Y3 )
| ( X4 = Y3 ) ) ) ) ) ).
% le_less
thf(fact_122_less__le,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [X4: A,Y3: A] :
( ( ord_less_eq @ A @ X4 @ Y3 )
& ( X4 != Y3 ) ) ) ) ) ).
% less_le
thf(fact_123_order__le__less__subst1,axiom,
! [A: $tType,B4: $tType] :
( ( ( order @ B4 @ ( type2 @ B4 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F2: B4 > A,B: B4,C3: B4] :
( ( ord_less_eq @ A @ A2 @ ( F2 @ B ) )
=> ( ( ord_less @ B4 @ B @ C3 )
=> ( ! [X2: B4,Y4: B4] :
( ( ord_less @ B4 @ X2 @ Y4 )
=> ( ord_less @ A @ ( F2 @ X2 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less @ A @ A2 @ ( F2 @ C3 ) ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_124_order__le__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C @ ( type2 @ C ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B: A,F2: A > C,C3: C] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( ord_less @ C @ ( F2 @ B ) @ C3 )
=> ( ! [X2: A,Y4: A] :
( ( ord_less_eq @ A @ X2 @ Y4 )
=> ( ord_less_eq @ C @ ( F2 @ X2 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less @ C @ ( F2 @ A2 ) @ C3 ) ) ) ) ) ).
% order_le_less_subst2
thf(fact_125_order__less__le__subst1,axiom,
! [A: $tType,B4: $tType] :
( ( ( order @ B4 @ ( type2 @ B4 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F2: B4 > A,B: B4,C3: B4] :
( ( ord_less @ A @ A2 @ ( F2 @ B ) )
=> ( ( ord_less_eq @ B4 @ B @ C3 )
=> ( ! [X2: B4,Y4: B4] :
( ( ord_less_eq @ B4 @ X2 @ Y4 )
=> ( ord_less_eq @ A @ ( F2 @ X2 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less @ A @ A2 @ ( F2 @ C3 ) ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_126_order__less__le__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C @ ( type2 @ C ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B: A,F2: A > C,C3: C] :
( ( ord_less @ A @ A2 @ B )
=> ( ( ord_less_eq @ C @ ( F2 @ B ) @ C3 )
=> ( ! [X2: A,Y4: A] :
( ( ord_less @ A @ X2 @ Y4 )
=> ( ord_less @ C @ ( F2 @ X2 ) @ ( F2 @ Y4 ) ) )
=> ( ord_less @ C @ ( F2 @ A2 ) @ C3 ) ) ) ) ) ).
% order_less_le_subst2
thf(fact_127_not__le,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y: A] :
( ( ~ ( ord_less_eq @ A @ X3 @ Y ) )
= ( ord_less @ A @ Y @ X3 ) ) ) ).
% not_le
thf(fact_128_not__less,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y: A] :
( ( ~ ( ord_less @ A @ X3 @ Y ) )
= ( ord_less_eq @ A @ Y @ X3 ) ) ) ).
% not_less
thf(fact_129_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_130_less__imp__le,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ord_less_eq @ A @ X3 @ Y ) ) ) ).
% less_imp_le
thf(fact_131_antisym__conv1,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y: A] :
( ~ ( ord_less @ A @ X3 @ Y )
=> ( ( ord_less_eq @ A @ X3 @ Y )
= ( X3 = Y ) ) ) ) ).
% antisym_conv1
thf(fact_132_antisym__conv2,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( ~ ( ord_less @ A @ X3 @ Y ) )
= ( X3 = Y ) ) ) ) ).
% antisym_conv2
thf(fact_133_le__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y: A,Z3: A] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( ord_less @ A @ Y @ Z3 )
=> ( ord_less @ A @ X3 @ Z3 ) ) ) ) ).
% le_less_trans
thf(fact_134_less__le__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y: A,Z3: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z3 )
=> ( ord_less @ A @ X3 @ Z3 ) ) ) ) ).
% less_le_trans
thf(fact_135_dense__ge,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Z3: A,Y: A] :
( ! [X2: A] :
( ( ord_less @ A @ Z3 @ X2 )
=> ( ord_less_eq @ A @ Y @ X2 ) )
=> ( ord_less_eq @ A @ Y @ Z3 ) ) ) ).
% dense_ge
thf(fact_136_dense__le,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Y: A,Z3: A] :
( ! [X2: A] :
( ( ord_less @ A @ X2 @ Y )
=> ( ord_less_eq @ A @ X2 @ Z3 ) )
=> ( ord_less_eq @ A @ Y @ Z3 ) ) ) ).
% dense_le
thf(fact_137_le__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
| ( ord_less @ A @ Y @ X3 ) ) ) ).
% le_less_linear
thf(fact_138_le__imp__less__or__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X3: A,Y: A] :
( ( ord_less_eq @ A @ X3 @ Y )
=> ( ( ord_less @ A @ X3 @ Y )
| ( X3 = Y ) ) ) ) ).
% le_imp_less_or_eq
thf(fact_139_less__le__not__le,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [X4: A,Y3: A] :
( ( ord_less_eq @ A @ X4 @ Y3 )
& ~ ( ord_less_eq @ A @ Y3 @ X4 ) ) ) ) ) ).
% less_le_not_le
thf(fact_140_not__le__imp__less,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y: A,X3: A] :
( ~ ( ord_less_eq @ A @ Y @ X3 )
=> ( ord_less @ A @ X3 @ Y ) ) ) ).
% not_le_imp_less
thf(fact_141_order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( ord_less @ A @ B @ C3 )
=> ( ord_less @ A @ A2 @ C3 ) ) ) ) ).
% order.strict_trans1
thf(fact_142_order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C3: A] :
( ( ord_less @ A @ A2 @ B )
=> ( ( ord_less_eq @ A @ B @ C3 )
=> ( ord_less @ A @ A2 @ C3 ) ) ) ) ).
% order.strict_trans2
thf(fact_143_order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ B3 )
| ( A4 = B3 ) ) ) ) ) ).
% order.order_iff_strict
thf(fact_144_order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
& ( A4 != B3 ) ) ) ) ) ).
% order.strict_iff_order
thf(fact_145_dual__order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B: A,A2: A,C3: A] :
( ( ord_less_eq @ A @ B @ A2 )
=> ( ( ord_less @ A @ C3 @ B )
=> ( ord_less @ A @ C3 @ A2 ) ) ) ) ).
% dual_order.strict_trans1
thf(fact_146_ldrop__lappend,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( ( ord_less @ extended_enat @ N @ ( coindu1018505716length @ A @ Xs ) )
=> ( ( coindu191418589_ldrop @ A @ N @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( coindu268472904append @ A @ ( coindu191418589_ldrop @ A @ N @ Xs ) @ Ys ) ) )
& ( ~ ( ord_less @ extended_enat @ N @ ( coindu1018505716length @ A @ Xs ) )
=> ( ( coindu191418589_ldrop @ A @ N @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( coindu191418589_ldrop @ A @ ( minus_minus @ extended_enat @ N @ ( coindu1018505716length @ A @ Xs ) ) @ Ys ) ) ) ) ).
% ldrop_lappend
thf(fact_147_ltake__lappend2,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,N: extended_enat,Ys: coindu1593790203_llist @ A] :
( ( ord_less_eq @ extended_enat @ ( coindu1018505716length @ A @ Xs ) @ N )
=> ( ( coindu1802687541_ltake @ A @ N @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( coindu268472904append @ A @ Xs @ ( coindu1802687541_ltake @ A @ ( minus_minus @ extended_enat @ N @ ( coindu1018505716length @ A @ Xs ) ) @ Ys ) ) ) ) ).
% ltake_lappend2
thf(fact_148_ldrop__eq__LNil,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
( ( ( coindu191418589_ldrop @ A @ N @ Xs )
= ( coindu1598213697e_LNil @ A ) )
= ( ord_less_eq @ extended_enat @ ( coindu1018505716length @ A @ Xs ) @ N ) ) ).
% ldrop_eq_LNil
thf(fact_149_lnull__ldrop,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ ( coindu191418589_ldrop @ A @ N @ Xs ) )
= ( ord_less_eq @ extended_enat @ ( coindu1018505716length @ A @ Xs ) @ N ) ) ).
% lnull_ldrop
thf(fact_150_ltake__plus__conv__lappend,axiom,
! [A: $tType,N: extended_enat,M2: extended_enat,Xs: coindu1593790203_llist @ A] :
( ( coindu1802687541_ltake @ A @ ( plus_plus @ extended_enat @ N @ M2 ) @ Xs )
= ( coindu268472904append @ A @ ( coindu1802687541_ltake @ A @ N @ Xs ) @ ( coindu1802687541_ltake @ A @ M2 @ ( coindu191418589_ldrop @ A @ N @ Xs ) ) ) ) ).
% ltake_plus_conv_lappend
thf(fact_151_ltake__lappend,axiom,
! [A: $tType,N: extended_enat,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1802687541_ltake @ A @ N @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( coindu268472904append @ A @ ( coindu1802687541_ltake @ A @ N @ Xs ) @ ( coindu1802687541_ltake @ A @ ( minus_minus @ extended_enat @ N @ ( coindu1018505716length @ A @ Xs ) ) @ Ys ) ) ) ).
% ltake_lappend
thf(fact_152_ldrop__all,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,M2: extended_enat] :
( ( ord_less_eq @ extended_enat @ ( coindu1018505716length @ A @ Xs ) @ M2 )
=> ( ( coindu191418589_ldrop @ A @ M2 @ Xs )
= ( coindu1598213697e_LNil @ A ) ) ) ).
% ldrop_all
thf(fact_153_lprefix__code_I1_J,axiom,
! [A: $tType,Ys: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ ( coindu1598213697e_LNil @ A ) @ Ys ) ).
% lprefix_code(1)
thf(fact_154_lappend_Odisc__iff_I2_J,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( ~ ( coindu335574135_lnull @ A @ ( coindu268472904append @ A @ Xs @ Ys ) ) )
= ( ~ ( coindu335574135_lnull @ A @ Xs )
| ~ ( coindu335574135_lnull @ A @ Ys ) ) ) ).
% lappend.disc_iff(2)
thf(fact_155_lnull__lappend,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( ( coindu335574135_lnull @ A @ Xs )
& ( coindu335574135_lnull @ A @ Ys ) ) ) ).
% lnull_lappend
thf(fact_156_lappend__code_I1_J,axiom,
! [A: $tType,Ys: coindu1593790203_llist @ A] :
( ( coindu268472904append @ A @ ( coindu1598213697e_LNil @ A ) @ Ys )
= Ys ) ).
% lappend_code(1)
thf(fact_157_lappend__LNil2,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu268472904append @ A @ Xs @ ( coindu1598213697e_LNil @ A ) )
= Xs ) ).
% lappend_LNil2
thf(fact_158_ltake__LNil,axiom,
! [A: $tType,N: extended_enat] :
( ( coindu1802687541_ltake @ A @ N @ ( coindu1598213697e_LNil @ A ) )
= ( coindu1598213697e_LNil @ A ) ) ).
% ltake_LNil
thf(fact_159_ldrop__LNil,axiom,
! [A: $tType,N: extended_enat] :
( ( coindu191418589_ldrop @ A @ N @ ( coindu1598213697e_LNil @ A ) )
= ( coindu1598213697e_LNil @ A ) ) ).
% ldrop_LNil
thf(fact_160_ldrop__ldrop,axiom,
! [A: $tType,N: extended_enat,M2: extended_enat,Xs: coindu1593790203_llist @ A] :
( ( coindu191418589_ldrop @ A @ N @ ( coindu191418589_ldrop @ A @ M2 @ Xs ) )
= ( coindu191418589_ldrop @ A @ ( plus_plus @ extended_enat @ N @ M2 ) @ Xs ) ) ).
% ldrop_ldrop
thf(fact_161_lstrict__prefix__code_I1_J,axiom,
! [A: $tType] :
~ ( coindu574146665prefix @ A @ ( coindu1598213697e_LNil @ A ) @ ( coindu1598213697e_LNil @ A ) ) ).
% lstrict_prefix_code(1)
thf(fact_162_lprefix__LNil,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ ( coindu1598213697e_LNil @ A ) )
= ( coindu335574135_lnull @ A @ Xs ) ) ).
% lprefix_LNil
thf(fact_163_llength__lappend,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1018505716length @ A @ ( coindu268472904append @ A @ Xs @ Ys ) )
= ( plus_plus @ extended_enat @ ( coindu1018505716length @ A @ Xs ) @ ( coindu1018505716length @ A @ Ys ) ) ) ).
% llength_lappend
thf(fact_164_less__fun__def,axiom,
! [B4: $tType,A: $tType] :
( ( ord @ B4 @ ( type2 @ B4 ) )
=> ( ( ord_less @ ( A > B4 ) )
= ( ^ [F3: A > B4,G: A > B4] :
( ( ord_less_eq @ ( A > B4 ) @ F3 @ G )
& ~ ( ord_less_eq @ ( A > B4 ) @ G @ F3 ) ) ) ) ) ).
% less_fun_def
thf(fact_165_ldrop__lnull,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,N: extended_enat] :
( ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu191418589_ldrop @ A @ N @ Xs )
= ( coindu1598213697e_LNil @ A ) ) ) ).
% ldrop_lnull
thf(fact_166_lappend_Octr_I1_J,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu335574135_lnull @ A @ Ys )
=> ( ( coindu268472904append @ A @ Xs @ Ys )
= ( coindu1598213697e_LNil @ A ) ) ) ) ).
% lappend.ctr(1)
thf(fact_167_llist_Odisc_I1_J,axiom,
! [A: $tType] : ( coindu335574135_lnull @ A @ ( coindu1598213697e_LNil @ A ) ) ).
% llist.disc(1)
thf(fact_168_llist_OdiscI_I1_J,axiom,
! [A: $tType,Llist: coindu1593790203_llist @ A] :
( ( Llist
= ( coindu1598213697e_LNil @ A ) )
=> ( coindu335574135_lnull @ A @ Llist ) ) ).
% llist.discI(1)
thf(fact_169_llist_Ocollapse_I1_J,axiom,
! [A: $tType,Llist: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Llist )
=> ( Llist
= ( coindu1598213697e_LNil @ A ) ) ) ).
% llist.collapse(1)
thf(fact_170_lzip_Oexhaust,axiom,
! [A: $tType,B4: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ B4] :
( ~ ( ( coindu335574135_lnull @ A @ Xs )
| ( coindu335574135_lnull @ B4 @ Ys ) )
=> ~ ( ~ ( coindu335574135_lnull @ A @ Xs )
=> ( coindu335574135_lnull @ B4 @ Ys ) ) ) ).
% lzip.exhaust
thf(fact_171_lappend_Oexhaust,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( ( coindu335574135_lnull @ A @ Xs )
=> ~ ( coindu335574135_lnull @ A @ Ys ) )
=> ( ~ ( coindu335574135_lnull @ A @ Xs )
| ~ ( coindu335574135_lnull @ A @ Ys ) ) ) ).
% lappend.exhaust
thf(fact_172_lnull__def,axiom,
! [A: $tType] :
( ( coindu335574135_lnull @ A )
= ( ^ [Llist2: coindu1593790203_llist @ A] :
( Llist2
= ( coindu1598213697e_LNil @ A ) ) ) ) ).
% lnull_def
thf(fact_173_lnull__lprefix,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Xs )
=> ( coindu1696667936prefix @ A @ Xs @ Ys ) ) ).
% lnull_lprefix
thf(fact_174_lprefix__lnull,axiom,
! [A: $tType,Ys: coindu1593790203_llist @ A,Xs: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Ys )
=> ( ( coindu1696667936prefix @ A @ Xs @ Ys )
= ( coindu335574135_lnull @ A @ Xs ) ) ) ).
% lprefix_lnull
thf(fact_175_lprefix__lnullD,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ Ys )
=> ( ( coindu335574135_lnull @ A @ Ys )
=> ( coindu335574135_lnull @ A @ Xs ) ) ) ).
% lprefix_lnullD
thf(fact_176_lprefix__not__lnullD,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu1696667936prefix @ A @ Xs @ Ys )
=> ( ~ ( coindu335574135_lnull @ A @ Xs )
=> ~ ( coindu335574135_lnull @ A @ Ys ) ) ) ).
% lprefix_not_lnullD
thf(fact_177_LNil__lprefix,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A] : ( coindu1696667936prefix @ A @ ( coindu1598213697e_LNil @ A ) @ Xs ) ).
% LNil_lprefix
thf(fact_178_lappend_Odisc_I2_J,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( ~ ( coindu335574135_lnull @ A @ Xs )
| ~ ( coindu335574135_lnull @ A @ Ys ) )
=> ~ ( coindu335574135_lnull @ A @ ( coindu268472904append @ A @ Xs @ Ys ) ) ) ).
% lappend.disc(2)
thf(fact_179_lappend_Odisc_I1_J,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu335574135_lnull @ A @ Ys )
=> ( coindu335574135_lnull @ A @ ( coindu268472904append @ A @ Xs @ Ys ) ) ) ) ).
% lappend.disc(1)
thf(fact_180_lappend__lnull1,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Xs )
=> ( ( coindu268472904append @ A @ Xs @ Ys )
= Ys ) ) ).
% lappend_lnull1
thf(fact_181_lappend__lnull2,axiom,
! [A: $tType,Ys: coindu1593790203_llist @ A,Xs: coindu1593790203_llist @ A] :
( ( coindu335574135_lnull @ A @ Ys )
=> ( ( coindu268472904append @ A @ Xs @ Ys )
= Xs ) ) ).
% lappend_lnull2
thf(fact_182_lappend__LNil__LNil,axiom,
! [A: $tType] :
( ( coindu268472904append @ A @ ( coindu1598213697e_LNil @ A ) @ ( coindu1598213697e_LNil @ A ) )
= ( coindu1598213697e_LNil @ A ) ) ).
% lappend_LNil_LNil
thf(fact_183_LNil__eq__lappend__iff,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( ( coindu1598213697e_LNil @ A )
= ( coindu268472904append @ A @ Xs @ Ys ) )
= ( ( Xs
= ( coindu1598213697e_LNil @ A ) )
& ( Ys
= ( coindu1598213697e_LNil @ A ) ) ) ) ).
% LNil_eq_lappend_iff
thf(fact_184_lappend__eq__LNil__iff,axiom,
! [A: $tType,Xs: coindu1593790203_llist @ A,Ys: coindu1593790203_llist @ A] :
( ( ( coindu268472904append @ A @ Xs @ Ys )
= ( coindu1598213697e_LNil @ A ) )
= ( ( Xs
= ( coindu1598213697e_LNil @ A ) )
& ( Ys
= ( coindu1598213697e_LNil @ A ) ) ) ) ).
% lappend_eq_LNil_iff
thf(fact_185_ldrop__ltake,axiom,
! [A: $tType,N: extended_enat,M2: extended_enat,Xs: coindu1593790203_llist @ A] :
( ( coindu191418589_ldrop @ A @ N @ ( coindu1802687541_ltake @ A @ M2 @ Xs ) )
= ( coindu1802687541_ltake @ A @ ( minus_minus @ extended_enat @ M2 @ N ) @ ( coindu191418589_ldrop @ A @ N @ Xs ) ) ) ).
% ldrop_ltake
thf(fact_186_ltake__ldrop,axiom,
! [A: $tType,N: extended_enat,M2: extended_enat,Xs: coindu1593790203_llist @ A] :
( ( coindu1802687541_ltake @ A @ N @ ( coindu191418589_ldrop @ A @ M2 @ Xs ) )
= ( coindu191418589_ldrop @ A @ M2 @ ( coindu1802687541_ltake @ A @ ( plus_plus @ extended_enat @ N @ M2 ) @ Xs ) ) ) ).
% ltake_ldrop
thf(fact_187_le__add__diff__inverse,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [B: A,A2: A] :
( ( ord_less_eq @ A @ B @ A2 )
=> ( ( plus_plus @ A @ B @ ( minus_minus @ A @ A2 @ B ) )
= A2 ) ) ) ).
% le_add_diff_inverse
thf(fact_188_le__add__diff__inverse2,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [B: A,A2: A] :
( ( ord_less_eq @ A @ B @ A2 )
=> ( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B ) @ B )
= A2 ) ) ) ).
% le_add_diff_inverse2
thf(fact_189_add__diff__cancel,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B ) @ B )
= A2 ) ) ).
% add_diff_cancel
thf(fact_190_add__right__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
=> ! [B: A,A2: A,C3: A] :
( ( ( plus_plus @ A @ B @ A2 )
= ( plus_plus @ A @ C3 @ A2 ) )
= ( B = C3 ) ) ) ).
% add_right_cancel
thf(fact_191_add__left__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C3: A] :
( ( ( plus_plus @ A @ A2 @ B )
= ( plus_plus @ A @ A2 @ C3 ) )
= ( B = C3 ) ) ) ).
% add_left_cancel
thf(fact_192_add__le__cancel__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [C3: A,A2: A,B: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C3 @ A2 ) @ ( plus_plus @ A @ C3 @ B ) )
= ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% add_le_cancel_left
thf(fact_193_add__le__cancel__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,C3: A,B: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C3 ) @ ( plus_plus @ A @ B @ C3 ) )
= ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% add_le_cancel_right
thf(fact_194_add__less__cancel__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,C3: A,B: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C3 ) @ ( plus_plus @ A @ B @ C3 ) )
= ( ord_less @ A @ A2 @ B ) ) ) ).
% add_less_cancel_right
thf(fact_195_add__less__cancel__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [C3: A,A2: A,B: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C3 @ A2 ) @ ( plus_plus @ A @ C3 @ B ) )
= ( ord_less @ A @ A2 @ B ) ) ) ).
% add_less_cancel_left
thf(fact_196_add__diff__cancel__right_H,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B ) @ B )
= A2 ) ) ).
% add_diff_cancel_right'
thf(fact_197_add__diff__cancel__right,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,C3: A,B: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C3 ) @ ( plus_plus @ A @ B @ C3 ) )
= ( minus_minus @ A @ A2 @ B ) ) ) ).
% add_diff_cancel_right
thf(fact_198_add__diff__cancel__left_H,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B ) @ A2 )
= B ) ) ).
% add_diff_cancel_left'
thf(fact_199_add__diff__cancel__left,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A @ ( type2 @ A ) )
=> ! [C3: A,A2: A,B: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ C3 @ A2 ) @ ( plus_plus @ A @ C3 @ B ) )
= ( minus_minus @ A @ A2 @ B ) ) ) ).
% add_diff_cancel_left
thf(fact_200_diff__add__cancel,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A] :
( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B ) @ B )
= A2 ) ) ).
% diff_add_cancel
thf(fact_201_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type2 @ A ) )
=> ! [X3: A,Y: A] :
( ( X3 != Y )
=> ( ~ ( ord_less @ A @ X3 @ Y )
=> ( ord_less @ A @ Y @ X3 ) ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_202_add__right__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
=> ! [B: A,A2: A,C3: A] :
( ( ( plus_plus @ A @ B @ A2 )
= ( plus_plus @ A @ C3 @ A2 ) )
=> ( B = C3 ) ) ) ).
% add_right_imp_eq
thf(fact_203_add__left__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C3: A] :
( ( ( plus_plus @ A @ A2 @ B )
= ( plus_plus @ A @ A2 @ C3 ) )
=> ( B = C3 ) ) ) ).
% add_left_imp_eq
thf(fact_204_add_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A @ ( type2 @ A ) )
=> ! [B: A,A2: A,C3: A] :
( ( plus_plus @ A @ B @ ( plus_plus @ A @ A2 @ C3 ) )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B @ C3 ) ) ) ) ).
% add.left_commute
thf(fact_205_add_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A @ ( type2 @ A ) )
=> ( ( plus_plus @ A )
= ( ^ [A4: A,B3: A] : ( plus_plus @ A @ B3 @ A4 ) ) ) ) ).
% add.commute
thf(fact_206_add_Oright__cancel,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [B: A,A2: A,C3: A] :
( ( ( plus_plus @ A @ B @ A2 )
= ( plus_plus @ A @ C3 @ A2 ) )
= ( B = C3 ) ) ) ).
% add.right_cancel
thf(fact_207_add_Oleft__cancel,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C3: A] :
( ( ( plus_plus @ A @ A2 @ B )
= ( plus_plus @ A @ A2 @ C3 ) )
= ( B = C3 ) ) ) ).
% add.left_cancel
thf(fact_208_add_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C3: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B ) @ C3 )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B @ C3 ) ) ) ) ).
% add.assoc
thf(fact_209_add__mono__thms__linordered__semiring_I4_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A @ ( type2 @ A ) )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus @ A @ I @ K )
= ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_210_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C3: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B ) @ C3 )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B @ C3 ) ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_211_diff__right__commute,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,C3: A,B: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C3 ) @ B )
= ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B ) @ C3 ) ) ) ).
% diff_right_commute
thf(fact_212_diff__eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C3: A,D3: A] :
( ( ( minus_minus @ A @ A2 @ B )
= ( minus_minus @ A @ C3 @ D3 ) )
=> ( ( A2 = B )
= ( C3 = D3 ) ) ) ) ).
% diff_eq_diff_eq
thf(fact_213_add__mono__thms__linordered__semiring_I3_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A @ ( type2 @ A ) )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( K = L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_214_add__mono__thms__linordered__semiring_I2_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A @ ( type2 @ A ) )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_215_add__mono__thms__linordered__semiring_I1_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A @ ( type2 @ A ) )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_216_add__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C3: A,D3: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( ord_less_eq @ A @ C3 @ D3 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C3 ) @ ( plus_plus @ A @ B @ D3 ) ) ) ) ) ).
% add_mono
thf(fact_217_add__left__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ C3 @ A2 ) @ ( plus_plus @ A @ C3 @ B ) ) ) ) ).
% add_left_mono
thf(fact_218_add__right__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C3 ) @ ( plus_plus @ A @ B @ C3 ) ) ) ) ).
% add_right_mono
thf(fact_219_le__iff__add,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [A4: A,B3: A] :
? [C4: A] :
( B3
= ( plus_plus @ A @ A4 @ C4 ) ) ) ) ) ).
% le_iff_add
thf(fact_220_add__le__imp__le__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [C3: A,A2: A,B: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C3 @ A2 ) @ ( plus_plus @ A @ C3 @ B ) )
=> ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% add_le_imp_le_left
thf(fact_221_add__le__imp__le__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,C3: A,B: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C3 ) @ ( plus_plus @ A @ B @ C3 ) )
=> ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% add_le_imp_le_right
thf(fact_222_diff__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,D3: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( ord_less_eq @ A @ D3 @ C3 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C3 ) @ ( minus_minus @ A @ B @ D3 ) ) ) ) ) ).
% diff_mono
thf(fact_223_diff__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [B: A,A2: A,C3: A] :
( ( ord_less_eq @ A @ B @ A2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ C3 @ A2 ) @ ( minus_minus @ A @ C3 @ B ) ) ) ) ).
% diff_left_mono
thf(fact_224_diff__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C3 ) @ ( minus_minus @ A @ B @ C3 ) ) ) ) ).
% diff_right_mono
thf(fact_225_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C3: A,D3: A] :
( ( ( minus_minus @ A @ A2 @ B )
= ( minus_minus @ A @ C3 @ D3 ) )
=> ( ( ord_less_eq @ A @ A2 @ B )
= ( ord_less_eq @ A @ C3 @ D3 ) ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_226_add__less__imp__less__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,C3: A,B: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C3 ) @ ( plus_plus @ A @ B @ C3 ) )
=> ( ord_less @ A @ A2 @ B ) ) ) ).
% add_less_imp_less_right
thf(fact_227_add__less__imp__less__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [C3: A,A2: A,B: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C3 @ A2 ) @ ( plus_plus @ A @ C3 @ B ) )
=> ( ord_less @ A @ A2 @ B ) ) ) ).
% add_less_imp_less_left
thf(fact_228_add__strict__right__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C3: A] :
( ( ord_less @ A @ A2 @ B )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C3 ) @ ( plus_plus @ A @ B @ C3 ) ) ) ) ).
% add_strict_right_mono
thf(fact_229_add__strict__left__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C3: A] :
( ( ord_less @ A @ A2 @ B )
=> ( ord_less @ A @ ( plus_plus @ A @ C3 @ A2 ) @ ( plus_plus @ A @ C3 @ B ) ) ) ) ).
% add_strict_left_mono
thf(fact_230_add__strict__mono,axiom,
! [A: $tType] :
( ( strict2144017051up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C3: A,D3: A] :
( ( ord_less @ A @ A2 @ B )
=> ( ( ord_less @ A @ C3 @ D3 )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C3 ) @ ( plus_plus @ A @ B @ D3 ) ) ) ) ) ).
% add_strict_mono
thf(fact_231_add__mono__thms__linordered__field_I1_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( K = L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_232_add__mono__thms__linordered__field_I2_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_233_add__mono__thms__linordered__field_I5_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_234_diff__strict__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C3: A] :
( ( ord_less @ A @ A2 @ B )
=> ( ord_less @ A @ ( minus_minus @ A @ A2 @ C3 ) @ ( minus_minus @ A @ B @ C3 ) ) ) ) ).
% diff_strict_right_mono
thf(fact_235_diff__strict__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [B: A,A2: A,C3: A] :
( ( ord_less @ A @ B @ A2 )
=> ( ord_less @ A @ ( minus_minus @ A @ C3 @ A2 ) @ ( minus_minus @ A @ C3 @ B ) ) ) ) ).
% diff_strict_left_mono
thf(fact_236_diff__eq__diff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C3: A,D3: A] :
( ( ( minus_minus @ A @ A2 @ B )
= ( minus_minus @ A @ C3 @ D3 ) )
=> ( ( ord_less @ A @ A2 @ B )
= ( ord_less @ A @ C3 @ D3 ) ) ) ) ).
% diff_eq_diff_less
thf(fact_237_diff__strict__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,D3: A,C3: A] :
( ( ord_less @ A @ A2 @ B )
=> ( ( ord_less @ A @ D3 @ C3 )
=> ( ord_less @ A @ ( minus_minus @ A @ A2 @ C3 ) @ ( minus_minus @ A @ B @ D3 ) ) ) ) ) ).
% diff_strict_mono
thf(fact_238_add__implies__diff,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
=> ! [C3: A,B: A,A2: A] :
( ( ( plus_plus @ A @ C3 @ B )
= A2 )
=> ( C3
= ( minus_minus @ A @ A2 @ B ) ) ) ) ).
% add_implies_diff
thf(fact_239_diff__diff__add,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C3: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B ) @ C3 )
= ( minus_minus @ A @ A2 @ ( plus_plus @ A @ B @ C3 ) ) ) ) ).
% diff_diff_add
thf(fact_240_diff__add__eq__diff__diff__swap,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C3: A] :
( ( minus_minus @ A @ A2 @ ( plus_plus @ A @ B @ C3 ) )
= ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C3 ) @ B ) ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_241_diff__add__eq,axiom,
! [A: $tType] :
( ( ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C3: A] :
( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B ) @ C3 )
= ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C3 ) @ B ) ) ) ).
% diff_add_eq
thf(fact_242_diff__diff__eq2,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C3: A] :
( ( minus_minus @ A @ A2 @ ( minus_minus @ A @ B @ C3 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C3 ) @ B ) ) ) ).
% diff_diff_eq2
thf(fact_243_add__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C3: A] :
( ( plus_plus @ A @ A2 @ ( minus_minus @ A @ B @ C3 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B ) @ C3 ) ) ) ).
% add_diff_eq
thf(fact_244_eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,C3: A,B: A] :
( ( A2
= ( minus_minus @ A @ C3 @ B ) )
= ( ( plus_plus @ A @ A2 @ B )
= C3 ) ) ) ).
% eq_diff_eq
thf(fact_245_diff__eq__eq,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C3: A] :
( ( ( minus_minus @ A @ A2 @ B )
= C3 )
= ( A2
= ( plus_plus @ A @ C3 @ B ) ) ) ) ).
% diff_eq_eq
thf(fact_246_add__mono__thms__linordered__field_I4_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_247_add__mono__thms__linordered__field_I3_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_248_add__le__less__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C3: A,D3: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( ord_less @ A @ C3 @ D3 )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C3 ) @ ( plus_plus @ A @ B @ D3 ) ) ) ) ) ).
% add_le_less_mono
thf(fact_249_add__less__le__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C3: A,D3: A] :
( ( ord_less @ A @ A2 @ B )
=> ( ( ord_less_eq @ A @ C3 @ D3 )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C3 ) @ ( plus_plus @ A @ B @ D3 ) ) ) ) ) ).
% add_less_le_mono
thf(fact_250_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: $tType] :
( ( ordere623563068d_diff @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( ord_less_eq @ A @ A2 @ B )
=> ( ( ( minus_minus @ A @ B @ A2 )
= C3 )
= ( B
= ( plus_plus @ A @ C3 @ A2 ) ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_251_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: $tType] :
( ( ordere623563068d_diff @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( plus_plus @ A @ A2 @ ( minus_minus @ A @ B @ A2 ) )
= B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_252_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: $tType] :
( ( ordere623563068d_diff @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( minus_minus @ A @ C3 @ ( minus_minus @ A @ B @ A2 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ C3 @ A2 ) @ B ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_253_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: $tType] :
( ( ordere623563068d_diff @ A @ ( type2 @ A ) )
=> ! [A2: A,B: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( minus_minus @ A @ ( plus_plus @ A @ B @ C3 ) @ A2 )
= ( plus_plus @ A @ ( minus_minus @ A @ B @ A2 ) @ C3 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
%----Type constructors (18)
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A5: $tType,A6: $tType] :
( ( preorder @ A6 @ ( type2 @ A6 ) )
=> ( preorder @ ( A5 > A6 ) @ ( type2 @ ( A5 > A6 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A5: $tType,A6: $tType] :
( ( order @ A6 @ ( type2 @ A6 ) )
=> ( order @ ( A5 > A6 ) @ ( type2 @ ( A5 > A6 ) ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A5: $tType,A6: $tType] :
( ( ord @ A6 @ ( type2 @ A6 ) )
=> ( ord @ ( A5 > A6 ) @ ( type2 @ ( A5 > A6 ) ) ) ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_1,axiom,
preorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_2,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_3,axiom,
ord @ $o @ ( type2 @ $o ) ).
thf(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
condit1037483654norder @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
strict2144017051up_add @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add,axiom,
canoni770627133id_add @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Oordered__ab__semigroup__add,axiom,
ordere779506340up_add @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Groups_Osemigroup__add,axiom,
semigroup_add @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Orderings_Owellorder,axiom,
wellorder @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Orderings_Opreorder_4,axiom,
preorder @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Orderings_Olinorder_5,axiom,
linorder @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oorder_6,axiom,
order @ extended_enat @ ( type2 @ extended_enat ) ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oord_7,axiom,
ord @ extended_enat @ ( type2 @ extended_enat ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
ord_less @ extended_enat @ ( coindu1018505716length @ a @ xs ) @ ( coindu1018505716length @ a @ ys ) ).
%------------------------------------------------------------------------------