TPTP Problem File: DAT158^1.p
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%------------------------------------------------------------------------------
% File : DAT158^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Hamming stream 184
% 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 : hamming_stream__184.p [Bla16]
% Status : Theorem
% Rating : 1.00 v8.2.0, 0.67 v8.1.0, 1.00 v7.1.0
% Syntax : Number of formulae : 305 ( 80 unt; 40 typ; 0 def)
% Number of atoms : 914 ( 205 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 3641 ( 168 ~; 51 |; 60 &;2977 @)
% ( 0 <=>; 385 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 8 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 211 ( 211 >; 0 *; 0 +; 0 <<)
% Number of symbols : 40 ( 39 usr; 2 con; 0-6 aty)
% Number of variables : 813 ( 13 ^; 734 !; 31 ?; 813 :)
% ( 35 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:42:10.581
%------------------------------------------------------------------------------
%----Could-be-implicit typings (5)
thf(ty_t_Coinductive__List_Ollist,type,
coinductive_llist: $tType > $tType ).
thf(ty_t_List_Olist,type,
list: $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
thf(ty_tf_a,type,
a: $tType ).
%----Explicit typings (35)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Osemilattice__sup,type,
semilattice_sup:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Oldistinct,type,
coindu351974385stinct:
!>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Olfinite,type,
coinductive_lfinite:
!>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Ollist_OLCons,type,
coinductive_LCons:
!>[A: $tType] : ( A > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_Ollist_OLNil,type,
coinductive_LNil:
!>[A: $tType] : ( coinductive_llist @ A ) ).
thf(sy_c_Coinductive__List_Ollist_Olhd,type,
coinductive_lhd:
!>[A: $tType] : ( ( coinductive_llist @ A ) > A ) ).
thf(sy_c_Coinductive__List_Ollist_Olnull,type,
coinductive_lnull:
!>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Ollist_Olset,type,
coinductive_lset:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( set @ A ) ) ).
thf(sy_c_Coinductive__List_Ollist_Oltl,type,
coinductive_ltl:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_Oord_Olsorted,type,
coinductive_lsorted:
!>[A: $tType] : ( ( A > A > $o ) > ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder_Obdd__above,type,
condit2040224947_above:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Conditionally__Complete__Lattices_Opreorder_Obdd__below,type,
condit1201339847_below:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Hamming__Stream__Mirabelle__rwekfkwckg_Oord_Olmerge,type,
hammin1328233080lmerge:
!>[A: $tType] : ( ( A > A > $o ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Lattices_Osup__class_Osup,type,
sup_sup:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_List_Olinorder_Oinsort__key,type,
insort_key:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > A ) > B > ( list @ B ) > ( list @ B ) ) ).
thf(sy_c_Orderings_Oord_OLeast,type,
least:
!>[A: $tType] : ( ( A > A > $o ) > ( A > $o ) > A ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oorder_Oantimono,type,
antimono:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > $o ) ).
thf(sy_c_Orderings_Oorder_Omono,type,
mono:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > $o ) ).
thf(sy_c_Orderings_Oorder_Ostrict__mono,type,
strict_mono:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > $o ) ).
thf(sy_c_Orderings_Oordering,type,
ordering:
!>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > $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_Set__Interval_Oord_OatLeastLessThan,type,
set_atLeastLessThan:
!>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord_OgreaterThanAtMost,type,
set_gr323396891AtMost:
!>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord_OgreaterThanLessThan,type,
set_gr1161524159ssThan:
!>[A: $tType] : ( ( A > A > $o ) > A > A > ( set @ A ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_less,type,
less: a > a > $o ).
thf(sy_v_less__eq,type,
less_eq: a > a > $o ).
thf(sy_v_xsa____,type,
xsa: coinductive_llist @ a ).
thf(sy_v_ysa____,type,
ysa: coinductive_llist @ a ).
%----Relevant facts (254)
thf(fact_0_local_Olmerge_Oexhaust,axiom,
! [Xs: coinductive_llist @ a,Ys: coinductive_llist @ a] :
( ~ ( ( coinductive_lnull @ a @ Xs )
| ( coinductive_lnull @ a @ Ys ) )
=> ~ ( ~ ( coinductive_lnull @ a @ Xs )
=> ( coinductive_lnull @ a @ Ys ) ) ) ).
% local.lmerge.exhaust
thf(fact_1_local_Oantisym,axiom,
! [X: a,Y: a] :
( ( less_eq @ X @ Y )
=> ( ( less_eq @ Y @ X )
=> ( X = Y ) ) ) ).
% local.antisym
thf(fact_2_local_Oantisym__conv,axiom,
! [Y: a,X: a] :
( ( less_eq @ Y @ X )
=> ( ( less_eq @ X @ Y )
= ( X = Y ) ) ) ).
% local.antisym_conv
thf(fact_3_local_Odual__order_Oantisym,axiom,
! [B2: a,A2: a] :
( ( less_eq @ B2 @ A2 )
=> ( ( less_eq @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% local.dual_order.antisym
thf(fact_4_local_Odual__order_Otrans,axiom,
! [B2: a,A2: a,C: a] :
( ( less_eq @ B2 @ A2 )
=> ( ( less_eq @ C @ B2 )
=> ( less_eq @ C @ A2 ) ) ) ).
% local.dual_order.trans
thf(fact_5_local_Oeq__iff,axiom,
( ( ^ [Y2: a,Z: a] : Y2 = Z )
= ( ^ [X2: a,Y3: a] :
( ( less_eq @ X2 @ Y3 )
& ( less_eq @ Y3 @ X2 ) ) ) ) ).
% local.eq_iff
thf(fact_6_local_Oeq__refl,axiom,
! [X: a,Y: a] :
( ( X = Y )
=> ( less_eq @ X @ Y ) ) ).
% local.eq_refl
thf(fact_7_local_Ole__cases,axiom,
! [X: a,Y: a] :
( ~ ( less_eq @ X @ Y )
=> ( less_eq @ Y @ X ) ) ).
% local.le_cases
thf(fact_8_local_Ole__cases3,axiom,
! [X: a,Y: a,Z2: a] :
( ( ( less_eq @ X @ Y )
=> ~ ( less_eq @ Y @ Z2 ) )
=> ( ( ( less_eq @ Y @ X )
=> ~ ( less_eq @ X @ Z2 ) )
=> ( ( ( less_eq @ X @ Z2 )
=> ~ ( less_eq @ Z2 @ Y ) )
=> ( ( ( less_eq @ Z2 @ Y )
=> ~ ( less_eq @ Y @ X ) )
=> ( ( ( less_eq @ Y @ Z2 )
=> ~ ( less_eq @ Z2 @ X ) )
=> ~ ( ( less_eq @ Z2 @ X )
=> ~ ( less_eq @ X @ Y ) ) ) ) ) ) ) ).
% local.le_cases3
thf(fact_9_local_Olinear,axiom,
! [X: a,Y: a] :
( ( less_eq @ X @ Y )
| ( less_eq @ Y @ X ) ) ).
% local.linear
thf(fact_10_local_Olinorder__wlog,axiom,
! [P: a > a > $o,A2: a,B2: a] :
( ! [A3: a,B3: a] :
( ( less_eq @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: a,B3: a] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A2 @ B2 ) ) ) ).
% local.linorder_wlog
thf(fact_11_local_Oord__eq__le__trans,axiom,
! [A2: a,B2: a,C: a] :
( ( A2 = B2 )
=> ( ( less_eq @ B2 @ C )
=> ( less_eq @ A2 @ C ) ) ) ).
% local.ord_eq_le_trans
thf(fact_12_local_Oord__le__eq__trans,axiom,
! [A2: a,B2: a,C: a] :
( ( less_eq @ A2 @ B2 )
=> ( ( B2 = C )
=> ( less_eq @ A2 @ C ) ) ) ).
% local.ord_le_eq_trans
thf(fact_13_local_Oorder_Oantisym,axiom,
! [A2: a,B2: a] :
( ( less_eq @ A2 @ B2 )
=> ( ( less_eq @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% local.order.antisym
thf(fact_14_local_Oorder_Otrans,axiom,
! [A2: a,B2: a,C: a] :
( ( less_eq @ A2 @ B2 )
=> ( ( less_eq @ B2 @ C )
=> ( less_eq @ A2 @ C ) ) ) ).
% local.order.trans
thf(fact_15_local_Oorder__trans,axiom,
! [X: a,Y: a,Z2: a] :
( ( less_eq @ X @ Y )
=> ( ( less_eq @ Y @ Z2 )
=> ( less_eq @ X @ Z2 ) ) ) ).
% local.order_trans
thf(fact_16_local_Oantisym__conv3,axiom,
! [Y: a,X: a] :
( ~ ( less @ Y @ X )
=> ( ( ~ ( less @ X @ Y ) )
= ( X = Y ) ) ) ).
% local.antisym_conv3
thf(fact_17_local_Odual__order_Oasym,axiom,
! [B2: a,A2: a] :
( ( less @ B2 @ A2 )
=> ~ ( less @ A2 @ B2 ) ) ).
% local.dual_order.asym
thf(fact_18_local_Odual__order_Ostrict__implies__not__eq,axiom,
! [B2: a,A2: a] :
( ( less @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% local.dual_order.strict_implies_not_eq
thf(fact_19_local_Odual__order_Ostrict__trans,axiom,
! [B2: a,A2: a,C: a] :
( ( less @ B2 @ A2 )
=> ( ( less @ C @ B2 )
=> ( less @ C @ A2 ) ) ) ).
% local.dual_order.strict_trans
thf(fact_20_local_Oless__asym,axiom,
! [X: a,Y: a] :
( ( less @ X @ Y )
=> ~ ( less @ Y @ X ) ) ).
% local.less_asym
thf(fact_21_local_Oless__asym_H,axiom,
! [A2: a,B2: a] :
( ( less @ A2 @ B2 )
=> ~ ( less @ B2 @ A2 ) ) ).
% local.less_asym'
thf(fact_22_local_Oless__imp__neq,axiom,
! [X: a,Y: a] :
( ( less @ X @ Y )
=> ( X != Y ) ) ).
% local.less_imp_neq
thf(fact_23_local_Oless__imp__not__eq,axiom,
! [X: a,Y: a] :
( ( less @ X @ Y )
=> ( X != Y ) ) ).
% local.less_imp_not_eq
thf(fact_24_local_Oless__imp__not__eq2,axiom,
! [X: a,Y: a] :
( ( less @ X @ Y )
=> ( Y != X ) ) ).
% local.less_imp_not_eq2
thf(fact_25_local_Oless__imp__not__less,axiom,
! [X: a,Y: a] :
( ( less @ X @ Y )
=> ~ ( less @ Y @ X ) ) ).
% local.less_imp_not_less
thf(fact_26_local_Oless__imp__triv,axiom,
! [X: a,Y: a,P: $o] :
( ( less @ X @ Y )
=> ( ( less @ Y @ X )
=> P ) ) ).
% local.less_imp_triv
thf(fact_27_local_Oless__irrefl,axiom,
! [X: a] :
~ ( less @ X @ X ) ).
% local.less_irrefl
thf(fact_28_local_Oless__linear,axiom,
! [X: a,Y: a] :
( ( less @ X @ Y )
| ( X = Y )
| ( less @ Y @ X ) ) ).
% local.less_linear
thf(fact_29_local_Oless__not__sym,axiom,
! [X: a,Y: a] :
( ( less @ X @ Y )
=> ~ ( less @ Y @ X ) ) ).
% local.less_not_sym
thf(fact_30_local_Oless__trans,axiom,
! [X: a,Y: a,Z2: a] :
( ( less @ X @ Y )
=> ( ( less @ Y @ Z2 )
=> ( less @ X @ Z2 ) ) ) ).
% local.less_trans
thf(fact_31_local_Olinorder__cases,axiom,
! [X: a,Y: a] :
( ~ ( less @ X @ Y )
=> ( ( X != Y )
=> ( less @ Y @ X ) ) ) ).
% local.linorder_cases
thf(fact_32_local_OneqE,axiom,
! [X: a,Y: a] :
( ( X != Y )
=> ( ~ ( less @ X @ Y )
=> ( less @ Y @ X ) ) ) ).
% local.neqE
thf(fact_33_local_Oneq__iff,axiom,
! [X: a,Y: a] :
( ( X != Y )
= ( ( less @ X @ Y )
| ( less @ Y @ X ) ) ) ).
% local.neq_iff
thf(fact_34_local_Onot__less__iff__gr__or__eq,axiom,
! [X: a,Y: a] :
( ( ~ ( less @ X @ Y ) )
= ( ( less @ Y @ X )
| ( X = Y ) ) ) ).
% local.not_less_iff_gr_or_eq
thf(fact_35_local_Oord__eq__less__trans,axiom,
! [A2: a,B2: a,C: a] :
( ( A2 = B2 )
=> ( ( less @ B2 @ C )
=> ( less @ A2 @ C ) ) ) ).
% local.ord_eq_less_trans
thf(fact_36_local_Oord__less__eq__trans,axiom,
! [A2: a,B2: a,C: a] :
( ( less @ A2 @ B2 )
=> ( ( B2 = C )
=> ( less @ A2 @ C ) ) ) ).
% local.ord_less_eq_trans
thf(fact_37_local_Oorder_Oasym,axiom,
! [A2: a,B2: a] :
( ( less @ A2 @ B2 )
=> ~ ( less @ B2 @ A2 ) ) ).
% local.order.asym
thf(fact_38_local_Oorder_Oirrefl,axiom,
! [A2: a] :
~ ( less @ A2 @ A2 ) ).
% local.order.irrefl
thf(fact_39_local_Oorder_Ostrict__implies__not__eq,axiom,
! [A2: a,B2: a] :
( ( less @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% local.order.strict_implies_not_eq
thf(fact_40_local_Oorder_Ostrict__trans,axiom,
! [A2: a,B2: a,C: a] :
( ( less @ A2 @ B2 )
=> ( ( less @ B2 @ C )
=> ( less @ A2 @ C ) ) ) ).
% local.order.strict_trans
thf(fact_41_lsorted_I2_J,axiom,
coinductive_lsorted @ a @ less_eq @ ysa ).
% lsorted(2)
thf(fact_42_lsorted_I1_J,axiom,
coinductive_lsorted @ a @ less_eq @ xsa ).
% lsorted(1)
thf(fact_43_local_Oantisym__conv1,axiom,
! [X: a,Y: a] :
( ~ ( less @ X @ Y )
=> ( ( less_eq @ X @ Y )
= ( X = Y ) ) ) ).
% local.antisym_conv1
thf(fact_44_local_Oantisym__conv2,axiom,
! [X: a,Y: a] :
( ( less_eq @ X @ Y )
=> ( ( ~ ( less @ X @ Y ) )
= ( X = Y ) ) ) ).
% local.antisym_conv2
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( collect @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X3: A] :
( ( F @ X3 )
= ( G @ X3 ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_local_Odual__order_Onot__eq__order__implies__strict,axiom,
! [A2: a,B2: a] :
( ( A2 != B2 )
=> ( ( less_eq @ B2 @ A2 )
=> ( less @ B2 @ A2 ) ) ) ).
% local.dual_order.not_eq_order_implies_strict
thf(fact_50_local_Odual__order_Oorder__iff__strict,axiom,
! [B2: a,A2: a] :
( ( less_eq @ B2 @ A2 )
= ( ( less @ B2 @ A2 )
| ( A2 = B2 ) ) ) ).
% local.dual_order.order_iff_strict
thf(fact_51_local_Odual__order_Ostrict__iff__order,axiom,
! [B2: a,A2: a] :
( ( less @ B2 @ A2 )
= ( ( less_eq @ B2 @ A2 )
& ( A2 != B2 ) ) ) ).
% local.dual_order.strict_iff_order
thf(fact_52_local_Odual__order_Ostrict__implies__order,axiom,
! [B2: a,A2: a] :
( ( less @ B2 @ A2 )
=> ( less_eq @ B2 @ A2 ) ) ).
% local.dual_order.strict_implies_order
thf(fact_53_local_Odual__order_Ostrict__trans1,axiom,
! [B2: a,A2: a,C: a] :
( ( less_eq @ B2 @ A2 )
=> ( ( less @ C @ B2 )
=> ( less @ C @ A2 ) ) ) ).
% local.dual_order.strict_trans1
thf(fact_54_local_Odual__order_Ostrict__trans2,axiom,
! [B2: a,A2: a,C: a] :
( ( less @ B2 @ A2 )
=> ( ( less_eq @ C @ B2 )
=> ( less @ C @ A2 ) ) ) ).
% local.dual_order.strict_trans2
thf(fact_55_local_OleD,axiom,
! [Y: a,X: a] :
( ( less_eq @ Y @ X )
=> ~ ( less @ X @ Y ) ) ).
% local.leD
thf(fact_56_local_OleI,axiom,
! [X: a,Y: a] :
( ~ ( less @ X @ Y )
=> ( less_eq @ Y @ X ) ) ).
% local.leI
thf(fact_57_local_Ole__imp__less__or__eq,axiom,
! [X: a,Y: a] :
( ( less_eq @ X @ Y )
=> ( ( less @ X @ Y )
| ( X = Y ) ) ) ).
% local.le_imp_less_or_eq
thf(fact_58_local_Ole__less,axiom,
! [X: a,Y: a] :
( ( less_eq @ X @ Y )
= ( ( less @ X @ Y )
| ( X = Y ) ) ) ).
% local.le_less
thf(fact_59_local_Ole__less__linear,axiom,
! [X: a,Y: a] :
( ( less_eq @ X @ Y )
| ( less @ Y @ X ) ) ).
% local.le_less_linear
thf(fact_60_local_Ole__less__trans,axiom,
! [X: a,Y: a,Z2: a] :
( ( less_eq @ X @ Y )
=> ( ( less @ Y @ Z2 )
=> ( less @ X @ Z2 ) ) ) ).
% local.le_less_trans
thf(fact_61_local_Ole__neq__trans,axiom,
! [A2: a,B2: a] :
( ( less_eq @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( less @ A2 @ B2 ) ) ) ).
% local.le_neq_trans
thf(fact_62_local_Oless__imp__le,axiom,
! [X: a,Y: a] :
( ( less @ X @ Y )
=> ( less_eq @ X @ Y ) ) ).
% local.less_imp_le
thf(fact_63_local_Oless__le,axiom,
! [X: a,Y: a] :
( ( less @ X @ Y )
= ( ( less_eq @ X @ Y )
& ( X != Y ) ) ) ).
% local.less_le
thf(fact_64_local_Oless__le__not__le,axiom,
! [X: a,Y: a] :
( ( less @ X @ Y )
= ( ( less_eq @ X @ Y )
& ~ ( less_eq @ Y @ X ) ) ) ).
% local.less_le_not_le
thf(fact_65_local_Oless__le__trans,axiom,
! [X: a,Y: a,Z2: a] :
( ( less @ X @ Y )
=> ( ( less_eq @ Y @ Z2 )
=> ( less @ X @ Z2 ) ) ) ).
% local.less_le_trans
thf(fact_66_local_Onot__le,axiom,
! [X: a,Y: a] :
( ( ~ ( less_eq @ X @ Y ) )
= ( less @ Y @ X ) ) ).
% local.not_le
thf(fact_67_local_Onot__le__imp__less,axiom,
! [Y: a,X: a] :
( ~ ( less_eq @ Y @ X )
=> ( less @ X @ Y ) ) ).
% local.not_le_imp_less
thf(fact_68_local_Onot__less,axiom,
! [X: a,Y: a] :
( ( ~ ( less @ X @ Y ) )
= ( less_eq @ Y @ X ) ) ).
% local.not_less
thf(fact_69_local_Oorder_Onot__eq__order__implies__strict,axiom,
! [A2: a,B2: a] :
( ( A2 != B2 )
=> ( ( less_eq @ A2 @ B2 )
=> ( less @ A2 @ B2 ) ) ) ).
% local.order.not_eq_order_implies_strict
thf(fact_70_local_Oorder_Oorder__iff__strict,axiom,
! [A2: a,B2: a] :
( ( less_eq @ A2 @ B2 )
= ( ( less @ A2 @ B2 )
| ( A2 = B2 ) ) ) ).
% local.order.order_iff_strict
thf(fact_71_local_Oorder_Ostrict__iff__order,axiom,
! [A2: a,B2: a] :
( ( less @ A2 @ B2 )
= ( ( less_eq @ A2 @ B2 )
& ( A2 != B2 ) ) ) ).
% local.order.strict_iff_order
thf(fact_72_local_Oorder_Ostrict__implies__order,axiom,
! [A2: a,B2: a] :
( ( less @ A2 @ B2 )
=> ( less_eq @ A2 @ B2 ) ) ).
% local.order.strict_implies_order
thf(fact_73_local_Oorder_Ostrict__trans1,axiom,
! [A2: a,B2: a,C: a] :
( ( less_eq @ A2 @ B2 )
=> ( ( less @ B2 @ C )
=> ( less @ A2 @ C ) ) ) ).
% local.order.strict_trans1
thf(fact_74_local_Oorder_Ostrict__trans2,axiom,
! [A2: a,B2: a,C: a] :
( ( less @ A2 @ B2 )
=> ( ( less_eq @ B2 @ C )
=> ( less @ A2 @ C ) ) ) ).
% local.order.strict_trans2
thf(fact_75_lsorted_I3_J,axiom,
~ ( coinductive_lnull @ a @ ( hammin1328233080lmerge @ a @ less @ xsa @ ysa ) ) ).
% lsorted(3)
thf(fact_76_local_Oin__lset__lmergeD,axiom,
! [X: a,Xs: coinductive_llist @ a,Ys: coinductive_llist @ a] :
( ( member @ a @ X @ ( coinductive_lset @ a @ ( hammin1328233080lmerge @ a @ less @ Xs @ Ys ) ) )
=> ( ( member @ a @ X @ ( coinductive_lset @ a @ Xs ) )
| ( member @ a @ X @ ( coinductive_lset @ a @ Ys ) ) ) ) ).
% local.in_lset_lmergeD
thf(fact_77_local_Olmerge_Odisc_I2_J,axiom,
! [Xs: coinductive_llist @ a,Ys: coinductive_llist @ a] :
( ~ ( coinductive_lnull @ a @ Xs )
=> ( ~ ( coinductive_lnull @ a @ Ys )
=> ~ ( coinductive_lnull @ a @ ( hammin1328233080lmerge @ a @ less @ Xs @ Ys ) ) ) ) ).
% local.lmerge.disc(2)
thf(fact_78_local_Olmerge_Odisc_I1_J,axiom,
! [Xs: coinductive_llist @ a,Ys: coinductive_llist @ a] :
( ( ( coinductive_lnull @ a @ Xs )
| ( coinductive_lnull @ a @ Ys ) )
=> ( coinductive_lnull @ a @ ( hammin1328233080lmerge @ a @ less @ Xs @ Ys ) ) ) ).
% local.lmerge.disc(1)
thf(fact_79_local_Olsorted__ltlI,axiom,
! [Xs: coinductive_llist @ a] :
( ( coinductive_lsorted @ a @ less_eq @ Xs )
=> ( coinductive_lsorted @ a @ less_eq @ ( coinductive_ltl @ a @ Xs ) ) ) ).
% local.lsorted_ltlI
thf(fact_80_local_Oorder_Orefl,axiom,
! [A2: a] : ( less_eq @ A2 @ A2 ) ).
% local.order.refl
thf(fact_81_local_Oorder__refl,axiom,
! [X: a] : ( less_eq @ X @ X ) ).
% local.order_refl
thf(fact_82_local_OLeastI2__order,axiom,
! [P: a > $o,X: a,Q: a > $o] :
( ( P @ X )
=> ( ! [Y4: a] :
( ( P @ Y4 )
=> ( less_eq @ X @ Y4 ) )
=> ( ! [X3: a] :
( ( P @ X3 )
=> ( ! [Y5: a] :
( ( P @ Y5 )
=> ( less_eq @ X3 @ Y5 ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( least @ a @ less_eq @ P ) ) ) ) ) ).
% local.LeastI2_order
thf(fact_83_local_OLeast__equality,axiom,
! [P: a > $o,X: a] :
( ( P @ X )
=> ( ! [Y4: a] :
( ( P @ Y4 )
=> ( less_eq @ X @ Y4 ) )
=> ( ( least @ a @ less_eq @ P )
= X ) ) ) ).
% local.Least_equality
thf(fact_84_local_Olhd__lmerge,axiom,
! [Xs: coinductive_llist @ a,Ys: coinductive_llist @ a] :
( ~ ( coinductive_lnull @ a @ Xs )
=> ( ~ ( coinductive_lnull @ a @ Ys )
=> ( ( ( less @ ( coinductive_lhd @ a @ Xs ) @ ( coinductive_lhd @ a @ Ys ) )
=> ( ( coinductive_lhd @ a @ ( hammin1328233080lmerge @ a @ less @ Xs @ Ys ) )
= ( coinductive_lhd @ a @ Xs ) ) )
& ( ~ ( less @ ( coinductive_lhd @ a @ Xs ) @ ( coinductive_lhd @ a @ Ys ) )
=> ( ( coinductive_lhd @ a @ ( hammin1328233080lmerge @ a @ less @ Xs @ Ys ) )
= ( coinductive_lhd @ a @ Ys ) ) ) ) ) ) ).
% local.lhd_lmerge
thf(fact_85_local_Oltl__lmerge,axiom,
! [Xs: coinductive_llist @ a,Ys: coinductive_llist @ a] :
( ~ ( coinductive_lnull @ a @ Xs )
=> ( ~ ( coinductive_lnull @ a @ Ys )
=> ( ( ( less @ ( coinductive_lhd @ a @ Xs ) @ ( coinductive_lhd @ a @ Ys ) )
=> ( ( coinductive_ltl @ a @ ( hammin1328233080lmerge @ a @ less @ Xs @ Ys ) )
= ( hammin1328233080lmerge @ a @ less @ ( coinductive_ltl @ a @ Xs ) @ Ys ) ) )
& ( ~ ( less @ ( coinductive_lhd @ a @ Xs ) @ ( coinductive_lhd @ a @ Ys ) )
=> ( ( ( less @ ( coinductive_lhd @ a @ Ys ) @ ( coinductive_lhd @ a @ Xs ) )
=> ( ( coinductive_ltl @ a @ ( hammin1328233080lmerge @ a @ less @ Xs @ Ys ) )
= ( hammin1328233080lmerge @ a @ less @ Xs @ ( coinductive_ltl @ a @ Ys ) ) ) )
& ( ~ ( less @ ( coinductive_lhd @ a @ Ys ) @ ( coinductive_lhd @ a @ Xs ) )
=> ( ( coinductive_ltl @ a @ ( hammin1328233080lmerge @ a @ less @ Xs @ Ys ) )
= ( hammin1328233080lmerge @ a @ less @ ( coinductive_ltl @ a @ Xs ) @ ( coinductive_ltl @ a @ Ys ) ) ) ) ) ) ) ) ) ).
% local.ltl_lmerge
thf(fact_86_local_Ostrict__mono__eq,axiom,
! [B: $tType] :
( ( order @ B @ ( type2 @ B ) )
=> ! [F: a > B,X: a,Y: a] :
( ( strict_mono @ a @ B @ less @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ) ).
% local.strict_mono_eq
thf(fact_87_local_Olsorted__coinduct_H,axiom,
! [X4: ( coinductive_llist @ a ) > $o,Xs: coinductive_llist @ a] :
( ( X4 @ Xs )
=> ( ! [Xs2: coinductive_llist @ a] :
( ( X4 @ Xs2 )
=> ( ~ ( coinductive_lnull @ a @ Xs2 )
=> ( ~ ( coinductive_lnull @ a @ ( coinductive_ltl @ a @ Xs2 ) )
=> ( ( less_eq @ ( coinductive_lhd @ a @ Xs2 ) @ ( coinductive_lhd @ a @ ( coinductive_ltl @ a @ Xs2 ) ) )
& ( ( X4 @ ( coinductive_ltl @ a @ Xs2 ) )
| ( coinductive_lsorted @ a @ less_eq @ ( coinductive_ltl @ a @ Xs2 ) ) ) ) ) ) )
=> ( coinductive_lsorted @ a @ less_eq @ Xs ) ) ) ).
% local.lsorted_coinduct'
thf(fact_88_local_Olsorted__lhdD,axiom,
! [Xs: coinductive_llist @ a] :
( ( coinductive_lsorted @ a @ less_eq @ Xs )
=> ( ~ ( coinductive_lnull @ a @ Xs )
=> ( ~ ( coinductive_lnull @ a @ ( coinductive_ltl @ a @ Xs ) )
=> ( less_eq @ ( coinductive_lhd @ a @ Xs ) @ ( coinductive_lhd @ a @ ( coinductive_ltl @ a @ Xs ) ) ) ) ) ) ).
% local.lsorted_lhdD
thf(fact_89_local_OlsortedD,axiom,
! [Xs: coinductive_llist @ a,Y: a] :
( ( coinductive_lsorted @ a @ less_eq @ Xs )
=> ( ~ ( coinductive_lnull @ a @ Xs )
=> ( ( member @ a @ Y @ ( coinductive_lset @ a @ ( coinductive_ltl @ a @ Xs ) ) )
=> ( less_eq @ ( coinductive_lhd @ a @ Xs ) @ Y ) ) ) ) ).
% local.lsortedD
thf(fact_90_local_Olsorted__coinduct,axiom,
! [X4: ( coinductive_llist @ a ) > $o,Xs: coinductive_llist @ a] :
( ( X4 @ Xs )
=> ( ! [Xs2: coinductive_llist @ a] :
( ( X4 @ Xs2 )
=> ( ~ ( coinductive_lnull @ a @ Xs2 )
=> ( ! [X3: a] :
( ( member @ a @ X3 @ ( coinductive_lset @ a @ ( coinductive_ltl @ a @ Xs2 ) ) )
=> ( less_eq @ ( coinductive_lhd @ a @ Xs2 ) @ X3 ) )
& ( ( X4 @ ( coinductive_ltl @ a @ Xs2 ) )
| ( coinductive_lsorted @ a @ less_eq @ ( coinductive_ltl @ a @ Xs2 ) ) ) ) ) )
=> ( coinductive_lsorted @ a @ less_eq @ Xs ) ) ) ).
% local.lsorted_coinduct
thf(fact_91_ord_Oin__lset__lmergeD,axiom,
! [A: $tType,X: A,Less: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys ) ) )
=> ( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
| ( member @ A @ X @ ( coinductive_lset @ A @ Ys ) ) ) ) ).
% ord.in_lset_lmergeD
thf(fact_92_local_Olmerge_Osimps_I2_J,axiom,
! [Xs: coinductive_llist @ a,Ys: coinductive_llist @ a] :
( ( ~ ( coinductive_lnull @ a @ ( hammin1328233080lmerge @ a @ less @ Xs @ Ys ) ) )
= ( ~ ( coinductive_lnull @ a @ Xs )
& ~ ( coinductive_lnull @ a @ Ys ) ) ) ).
% local.lmerge.simps(2)
thf(fact_93_local_Olnull__lmerge,axiom,
! [Xs: coinductive_llist @ a,Ys: coinductive_llist @ a] :
( ( coinductive_lnull @ a @ ( hammin1328233080lmerge @ a @ less @ Xs @ Ys ) )
= ( ( coinductive_lnull @ a @ Xs )
| ( coinductive_lnull @ a @ Ys ) ) ) ).
% local.lnull_lmerge
thf(fact_94_local_OIoc__inj,axiom,
! [A2: a,B2: a,C: a,D: a] :
( ( ( set_gr323396891AtMost @ a @ less_eq @ less @ A2 @ B2 )
= ( set_gr323396891AtMost @ a @ less_eq @ less @ C @ D ) )
= ( ( ( less_eq @ B2 @ A2 )
& ( less_eq @ D @ C ) )
| ( ( A2 = C )
& ( B2 = D ) ) ) ) ).
% local.Ioc_inj
thf(fact_95_local_Oorder_Oordering__axioms,axiom,
ordering @ a @ less_eq @ less ).
% local.order.ordering_axioms
thf(fact_96_local_Obdd__above__def,axiom,
! [A4: set @ a] :
( ( condit2040224947_above @ a @ less_eq @ A4 )
= ( ? [M: a] :
! [X2: a] :
( ( member @ a @ X2 @ A4 )
=> ( less_eq @ X2 @ M ) ) ) ) ).
% local.bdd_above_def
thf(fact_97_local_Obdd__below__def,axiom,
! [A4: set @ a] :
( ( condit1201339847_below @ a @ less_eq @ A4 )
= ( ? [M2: a] :
! [X2: a] :
( ( member @ a @ X2 @ A4 )
=> ( less_eq @ M2 @ X2 ) ) ) ) ).
% local.bdd_below_def
thf(fact_98_ord_Oltl__lmerge,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Less: A > A > $o] :
( ~ ( coinductive_lnull @ A @ Xs )
=> ( ~ ( coinductive_lnull @ A @ Ys )
=> ( ( ( Less @ ( coinductive_lhd @ A @ Xs ) @ ( coinductive_lhd @ A @ Ys ) )
=> ( ( coinductive_ltl @ A @ ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys ) )
= ( hammin1328233080lmerge @ A @ Less @ ( coinductive_ltl @ A @ Xs ) @ Ys ) ) )
& ( ~ ( Less @ ( coinductive_lhd @ A @ Xs ) @ ( coinductive_lhd @ A @ Ys ) )
=> ( ( ( Less @ ( coinductive_lhd @ A @ Ys ) @ ( coinductive_lhd @ A @ Xs ) )
=> ( ( coinductive_ltl @ A @ ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys ) )
= ( hammin1328233080lmerge @ A @ Less @ Xs @ ( coinductive_ltl @ A @ Ys ) ) ) )
& ( ~ ( Less @ ( coinductive_lhd @ A @ Ys ) @ ( coinductive_lhd @ A @ Xs ) )
=> ( ( coinductive_ltl @ A @ ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys ) )
= ( hammin1328233080lmerge @ A @ Less @ ( coinductive_ltl @ A @ Xs ) @ ( coinductive_ltl @ A @ Ys ) ) ) ) ) ) ) ) ) ).
% ord.ltl_lmerge
thf(fact_99_local_Oinsort__key__left__comm,axiom,
! [B: $tType,F: B > a,X: B,Y: B,Xs: list @ B] :
( ( ( F @ X )
!= ( F @ Y ) )
=> ( ( insort_key @ a @ B @ less_eq @ F @ Y @ ( insort_key @ a @ B @ less_eq @ F @ X @ Xs ) )
= ( insort_key @ a @ B @ less_eq @ F @ X @ ( insort_key @ a @ B @ less_eq @ F @ Y @ Xs ) ) ) ) ).
% local.insort_key_left_comm
thf(fact_100_ord_Olhd__lmerge,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Less: A > A > $o] :
( ~ ( coinductive_lnull @ A @ Xs )
=> ( ~ ( coinductive_lnull @ A @ Ys )
=> ( ( ( Less @ ( coinductive_lhd @ A @ Xs ) @ ( coinductive_lhd @ A @ Ys ) )
=> ( ( coinductive_lhd @ A @ ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys ) )
= ( coinductive_lhd @ A @ Xs ) ) )
& ( ~ ( Less @ ( coinductive_lhd @ A @ Xs ) @ ( coinductive_lhd @ A @ Ys ) )
=> ( ( coinductive_lhd @ A @ ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys ) )
= ( coinductive_lhd @ A @ Ys ) ) ) ) ) ) ).
% ord.lhd_lmerge
thf(fact_101_local_Olmerge__eq__LNil__iff,axiom,
! [Xs: coinductive_llist @ a,Ys: coinductive_llist @ a] :
( ( ( hammin1328233080lmerge @ a @ less @ Xs @ Ys )
= ( coinductive_LNil @ a ) )
= ( ( Xs
= ( coinductive_LNil @ a ) )
| ( Ys
= ( coinductive_LNil @ a ) ) ) ) ).
% local.lmerge_eq_LNil_iff
thf(fact_102_in__lset__lmergeI1,axiom,
! [X: a,Xs: coinductive_llist @ a,Ys: coinductive_llist @ a] :
( ( member @ a @ X @ ( coinductive_lset @ a @ Xs ) )
=> ( ( coinductive_lsorted @ a @ less_eq @ Xs )
=> ( ~ ( coinductive_lfinite @ a @ Ys )
=> ( ? [X5: a] :
( ( member @ a @ X5 @ ( coinductive_lset @ a @ Ys ) )
& ( less_eq @ X @ X5 ) )
=> ( member @ a @ X @ ( coinductive_lset @ a @ ( hammin1328233080lmerge @ a @ less @ Xs @ Ys ) ) ) ) ) ) ) ).
% in_lset_lmergeI1
thf(fact_103_in__lset__lmergeI2,axiom,
! [X: a,Ys: coinductive_llist @ a,Xs: coinductive_llist @ a] :
( ( member @ a @ X @ ( coinductive_lset @ a @ Ys ) )
=> ( ( coinductive_lsorted @ a @ less_eq @ Ys )
=> ( ~ ( coinductive_lfinite @ a @ Xs )
=> ( ? [X5: a] :
( ( member @ a @ X5 @ ( coinductive_lset @ a @ Xs ) )
& ( less_eq @ X @ X5 ) )
=> ( member @ a @ X @ ( coinductive_lset @ a @ ( hammin1328233080lmerge @ a @ less @ Xs @ Ys ) ) ) ) ) ) ) ).
% in_lset_lmergeI2
thf(fact_104_local_Olsorted_OLNil,axiom,
coinductive_lsorted @ a @ less_eq @ ( coinductive_LNil @ a ) ).
% local.lsorted.LNil
thf(fact_105_local_Olinfinite__lmerge,axiom,
! [Xs: coinductive_llist @ a,Ys: coinductive_llist @ a] :
( ~ ( coinductive_lfinite @ a @ Xs )
=> ( ~ ( coinductive_lfinite @ a @ Ys )
=> ~ ( coinductive_lfinite @ a @ ( hammin1328233080lmerge @ a @ less @ Xs @ Ys ) ) ) ) ).
% local.linfinite_lmerge
thf(fact_106_local_Olfinite__lmergeI,axiom,
! [Xs: coinductive_llist @ a,Ys: coinductive_llist @ a] :
( ( coinductive_lfinite @ a @ Xs )
=> ( ( coinductive_lfinite @ a @ Ys )
=> ( coinductive_lfinite @ a @ ( hammin1328233080lmerge @ a @ less @ Xs @ Ys ) ) ) ) ).
% local.lfinite_lmergeI
thf(fact_107_local_Olfinite__lmergeD,axiom,
! [Xs: coinductive_llist @ a,Ys: coinductive_llist @ a] :
( ( coinductive_lfinite @ a @ ( hammin1328233080lmerge @ a @ less @ Xs @ Ys ) )
=> ( ( coinductive_lfinite @ a @ Xs )
| ( coinductive_lfinite @ a @ Ys ) ) ) ).
% local.lfinite_lmergeD
thf(fact_108_local_Olmerge_Octr_I1_J,axiom,
! [Xs: coinductive_llist @ a,Ys: coinductive_llist @ a] :
( ( ( coinductive_lnull @ a @ Xs )
| ( coinductive_lnull @ a @ Ys ) )
=> ( ( hammin1328233080lmerge @ a @ less @ Xs @ Ys )
= ( coinductive_LNil @ a ) ) ) ).
% local.lmerge.ctr(1)
thf(fact_109_local_Obdd__belowI,axiom,
! [A4: set @ a,M3: a] :
( ! [X3: a] :
( ( member @ a @ X3 @ A4 )
=> ( less_eq @ M3 @ X3 ) )
=> ( condit1201339847_below @ a @ less_eq @ A4 ) ) ).
% local.bdd_belowI
thf(fact_110_local_Obdd__aboveI,axiom,
! [A4: set @ a,M4: a] :
( ! [X3: a] :
( ( member @ a @ X3 @ A4 )
=> ( less_eq @ X3 @ M4 ) )
=> ( condit2040224947_above @ a @ less_eq @ A4 ) ) ).
% local.bdd_aboveI
thf(fact_111_local_Olsorted__code_I1_J,axiom,
coinductive_lsorted @ a @ less_eq @ ( coinductive_LNil @ a ) ).
% local.lsorted_code(1)
thf(fact_112_local_Olmerge__LNil_I1_J,axiom,
! [Ys: coinductive_llist @ a] :
( ( hammin1328233080lmerge @ a @ less @ ( coinductive_LNil @ a ) @ Ys )
= ( coinductive_LNil @ a ) ) ).
% local.lmerge_LNil(1)
thf(fact_113_local_Olmerge__LNil_I2_J,axiom,
! [Xs: coinductive_llist @ a] :
( ( hammin1328233080lmerge @ a @ less @ Xs @ ( coinductive_LNil @ a ) )
= ( coinductive_LNil @ a ) ) ).
% local.lmerge_LNil(2)
thf(fact_114_local_OgreaterThanAtMost__iff,axiom,
! [I: a,L: a,U: a] :
( ( member @ a @ I @ ( set_gr323396891AtMost @ a @ less_eq @ less @ L @ U ) )
= ( ( less @ L @ I )
& ( less_eq @ I @ U ) ) ) ).
% local.greaterThanAtMost_iff
thf(fact_115_local_Obdd__below__Ioc,axiom,
! [A2: a,B2: a] : ( condit1201339847_below @ a @ less_eq @ ( set_gr323396891AtMost @ a @ less_eq @ less @ A2 @ B2 ) ) ).
% local.bdd_below_Ioc
thf(fact_116_local_Obdd__above__Ioc,axiom,
! [A2: a,B2: a] : ( condit2040224947_above @ a @ less_eq @ ( set_gr323396891AtMost @ a @ less_eq @ less @ A2 @ B2 ) ) ).
% local.bdd_above_Ioc
thf(fact_117_local_Ostrict__mono__mono,axiom,
! [B: $tType] :
( ( order @ B @ ( type2 @ B ) )
=> ! [F: a > B] :
( ( strict_mono @ a @ B @ less @ F )
=> ( mono @ a @ B @ less_eq @ F ) ) ) ).
% local.strict_mono_mono
thf(fact_118_ord_Olmerge_Octr_I1_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Less: A > A > $o] :
( ( ( coinductive_lnull @ A @ Xs )
| ( coinductive_lnull @ A @ Ys ) )
=> ( ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys )
= ( coinductive_LNil @ A ) ) ) ).
% ord.lmerge.ctr(1)
thf(fact_119_ord_Olinfinite__lmerge,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Less: A > A > $o] :
( ~ ( coinductive_lfinite @ A @ Xs )
=> ( ~ ( coinductive_lfinite @ A @ Ys )
=> ~ ( coinductive_lfinite @ A @ ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys ) ) ) ) ).
% ord.linfinite_lmerge
thf(fact_120_ord_Olfinite__lmergeI,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Less: A > A > $o] :
( ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_lfinite @ A @ Ys )
=> ( coinductive_lfinite @ A @ ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys ) ) ) ) ).
% ord.lfinite_lmergeI
thf(fact_121_ord_Olfinite__lmergeD,axiom,
! [A: $tType,Less: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys ) )
=> ( ( coinductive_lfinite @ A @ Xs )
| ( coinductive_lfinite @ A @ Ys ) ) ) ).
% ord.lfinite_lmergeD
thf(fact_122_ord_Olmerge__eq__LNil__iff,axiom,
! [A: $tType,Less: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys )
= ( coinductive_LNil @ A ) )
= ( ( Xs
= ( coinductive_LNil @ A ) )
| ( Ys
= ( coinductive_LNil @ A ) ) ) ) ).
% ord.lmerge_eq_LNil_iff
thf(fact_123_ord_Olmerge__LNil_I1_J,axiom,
! [A: $tType,Less: A > A > $o,Ys: coinductive_llist @ A] :
( ( hammin1328233080lmerge @ A @ Less @ ( coinductive_LNil @ A ) @ Ys )
= ( coinductive_LNil @ A ) ) ).
% ord.lmerge_LNil(1)
thf(fact_124_ord_Olmerge__LNil_I2_J,axiom,
! [A: $tType,Less: A > A > $o,Xs: coinductive_llist @ A] :
( ( hammin1328233080lmerge @ A @ Less @ Xs @ ( coinductive_LNil @ A ) )
= ( coinductive_LNil @ A ) ) ).
% ord.lmerge_LNil(2)
thf(fact_125_ord__class_Olmerge_Oexhaust,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ~ ( ( coinductive_lnull @ A @ Xs )
| ( coinductive_lnull @ A @ Ys ) )
=> ~ ( ~ ( coinductive_lnull @ A @ Xs )
=> ( coinductive_lnull @ A @ Ys ) ) ) ) ).
% ord_class.lmerge.exhaust
thf(fact_126_ord_Olmerge_Oexhaust,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ~ ( ( coinductive_lnull @ A @ Xs )
| ( coinductive_lnull @ A @ Ys ) )
=> ~ ( ~ ( coinductive_lnull @ A @ Xs )
=> ( coinductive_lnull @ A @ Ys ) ) ) ).
% ord.lmerge.exhaust
thf(fact_127_ord_Olnull__lmerge,axiom,
! [A: $tType,Less: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys ) )
= ( ( coinductive_lnull @ A @ Xs )
| ( coinductive_lnull @ A @ Ys ) ) ) ).
% ord.lnull_lmerge
thf(fact_128_ord_Olmerge_Odisc__iff_I2_J,axiom,
! [A: $tType,Less: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ~ ( coinductive_lnull @ A @ ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys ) ) )
= ( ~ ( coinductive_lnull @ A @ Xs )
& ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% ord.lmerge.disc_iff(2)
thf(fact_129_ord_Olmerge_Odisc_I1_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Less: A > A > $o] :
( ( ( coinductive_lnull @ A @ Xs )
| ( coinductive_lnull @ A @ Ys ) )
=> ( coinductive_lnull @ A @ ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys ) ) ) ).
% ord.lmerge.disc(1)
thf(fact_130_ord_Olmerge_Odisc_I2_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Less: A > A > $o] :
( ~ ( coinductive_lnull @ A @ Xs )
=> ( ~ ( coinductive_lnull @ A @ Ys )
=> ~ ( coinductive_lnull @ A @ ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys ) ) ) ) ).
% ord.lmerge.disc(2)
thf(fact_131_lfinite__ltl,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ ( coinductive_ltl @ A @ Xs ) )
= ( coinductive_lfinite @ A @ Xs ) ) ).
% lfinite_ltl
thf(fact_132_lfinite__code_I1_J,axiom,
! [A: $tType] : ( coinductive_lfinite @ A @ ( coinductive_LNil @ A ) ) ).
% lfinite_code(1)
thf(fact_133_ord_Olsorted__coinduct_H,axiom,
! [A: $tType,X4: ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A,Less_eq: A > A > $o] :
( ( X4 @ Xs )
=> ( ! [Xs2: coinductive_llist @ A] :
( ( X4 @ Xs2 )
=> ( ~ ( coinductive_lnull @ A @ Xs2 )
=> ( ~ ( coinductive_lnull @ A @ ( coinductive_ltl @ A @ Xs2 ) )
=> ( ( Less_eq @ ( coinductive_lhd @ A @ Xs2 ) @ ( coinductive_lhd @ A @ ( coinductive_ltl @ A @ Xs2 ) ) )
& ( ( X4 @ ( coinductive_ltl @ A @ Xs2 ) )
| ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_ltl @ A @ Xs2 ) ) ) ) ) ) )
=> ( coinductive_lsorted @ A @ Less_eq @ Xs ) ) ) ).
% ord.lsorted_coinduct'
thf(fact_134_ord_Olsorted__lhdD,axiom,
! [A: $tType,Less_eq: A > A > $o,Xs: coinductive_llist @ A] :
( ( coinductive_lsorted @ A @ Less_eq @ Xs )
=> ( ~ ( coinductive_lnull @ A @ Xs )
=> ( ~ ( coinductive_lnull @ A @ ( coinductive_ltl @ A @ Xs ) )
=> ( Less_eq @ ( coinductive_lhd @ A @ Xs ) @ ( coinductive_lhd @ A @ ( coinductive_ltl @ A @ Xs ) ) ) ) ) ) ).
% ord.lsorted_lhdD
thf(fact_135_llist__set__induct,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A,P: A > ( coinductive_llist @ A ) > $o] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
=> ( ! [Xs2: coinductive_llist @ A] :
( ~ ( coinductive_lnull @ A @ Xs2 )
=> ( P @ ( coinductive_lhd @ A @ Xs2 ) @ Xs2 ) )
=> ( ! [Xs2: coinductive_llist @ A,Y4: A] :
( ~ ( coinductive_lnull @ A @ Xs2 )
=> ( ( member @ A @ Y4 @ ( coinductive_lset @ A @ ( coinductive_ltl @ A @ Xs2 ) ) )
=> ( ( P @ Y4 @ ( coinductive_ltl @ A @ Xs2 ) )
=> ( P @ Y4 @ Xs2 ) ) ) )
=> ( P @ X @ Xs ) ) ) ) ).
% llist_set_induct
thf(fact_136_lappend_Oexhaust,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ( coinductive_lnull @ A @ Xs )
=> ~ ( coinductive_lnull @ A @ Ys ) )
=> ( ~ ( coinductive_lnull @ A @ Xs )
| ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% lappend.exhaust
thf(fact_137_lzip_Oexhaust,axiom,
! [A: $tType,B: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
( ~ ( ( coinductive_lnull @ A @ Xs )
| ( coinductive_lnull @ B @ Ys ) )
=> ~ ( ~ ( coinductive_lnull @ A @ Xs )
=> ( coinductive_lnull @ B @ Ys ) ) ) ).
% lzip.exhaust
thf(fact_138_lnull__def,axiom,
! [A: $tType] :
( ( coinductive_lnull @ A )
= ( ^ [Llist: coinductive_llist @ A] :
( Llist
= ( coinductive_LNil @ A ) ) ) ) ).
% lnull_def
thf(fact_139_llist_Ocollapse_I1_J,axiom,
! [A: $tType,Llist2: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Llist2 )
=> ( Llist2
= ( coinductive_LNil @ A ) ) ) ).
% llist.collapse(1)
thf(fact_140_llist_OdiscI_I1_J,axiom,
! [A: $tType,Llist2: coinductive_llist @ A] :
( ( Llist2
= ( coinductive_LNil @ A ) )
=> ( coinductive_lnull @ A @ Llist2 ) ) ).
% llist.discI(1)
thf(fact_141_llist_Odisc_I1_J,axiom,
! [A: $tType] : ( coinductive_lnull @ A @ ( coinductive_LNil @ A ) ) ).
% llist.disc(1)
thf(fact_142_ltakeWhile_Oexhaust,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P: A > $o] :
( ~ ( ( coinductive_lnull @ A @ Xs )
| ~ ( P @ ( coinductive_lhd @ A @ Xs ) ) )
=> ~ ( ~ ( coinductive_lnull @ A @ Xs )
=> ~ ( P @ ( coinductive_lhd @ A @ Xs ) ) ) ) ).
% ltakeWhile.exhaust
thf(fact_143_lnull__ltlI,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( coinductive_lnull @ A @ ( coinductive_ltl @ A @ Xs ) ) ) ).
% lnull_ltlI
thf(fact_144_lnull__imp__lfinite,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( coinductive_lfinite @ A @ Xs ) ) ).
% lnull_imp_lfinite
thf(fact_145_ltl__simps_I1_J,axiom,
! [A: $tType] :
( ( coinductive_ltl @ A @ ( coinductive_LNil @ A ) )
= ( coinductive_LNil @ A ) ) ).
% ltl_simps(1)
thf(fact_146_lfinite__LNil,axiom,
! [A: $tType] : ( coinductive_lfinite @ A @ ( coinductive_LNil @ A ) ) ).
% lfinite_LNil
thf(fact_147_in__lset__ltlD,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_ltl @ A @ Xs ) ) )
=> ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) ) ) ).
% in_lset_ltlD
thf(fact_148_ord_OLNil,axiom,
! [A: $tType,Less_eq: A > A > $o] : ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LNil @ A ) ) ).
% ord.LNil
thf(fact_149_ord_Olsorted__code_I1_J,axiom,
! [A: $tType,Less_eq: A > A > $o] : ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LNil @ A ) ) ).
% ord.lsorted_code(1)
thf(fact_150_ord_Olsorted__ltlI,axiom,
! [A: $tType,Less_eq: A > A > $o,Xs: coinductive_llist @ A] :
( ( coinductive_lsorted @ A @ Less_eq @ Xs )
=> ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_ltl @ A @ Xs ) ) ) ).
% ord.lsorted_ltlI
thf(fact_151_llist_Oset__sel_I1_J,axiom,
! [A: $tType,A2: coinductive_llist @ A] :
( ~ ( coinductive_lnull @ A @ A2 )
=> ( member @ A @ ( coinductive_lhd @ A @ A2 ) @ ( coinductive_lset @ A @ A2 ) ) ) ).
% llist.set_sel(1)
thf(fact_152_llist_Oexpand,axiom,
! [A: $tType,Llist2: coinductive_llist @ A,Llist3: coinductive_llist @ A] :
( ( ( coinductive_lnull @ A @ Llist2 )
= ( coinductive_lnull @ A @ Llist3 ) )
=> ( ( ~ ( coinductive_lnull @ A @ Llist2 )
=> ( ~ ( coinductive_lnull @ A @ Llist3 )
=> ( ( ( coinductive_lhd @ A @ Llist2 )
= ( coinductive_lhd @ A @ Llist3 ) )
& ( ( coinductive_ltl @ A @ Llist2 )
= ( coinductive_ltl @ A @ Llist3 ) ) ) ) )
=> ( Llist2 = Llist3 ) ) ) ).
% llist.expand
thf(fact_153_llist_Ocoinduct,axiom,
! [A: $tType,R: ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o,Llist2: coinductive_llist @ A,Llist3: coinductive_llist @ A] :
( ( R @ Llist2 @ Llist3 )
=> ( ! [Llist4: coinductive_llist @ A,Llist5: coinductive_llist @ A] :
( ( R @ Llist4 @ Llist5 )
=> ( ( ( coinductive_lnull @ A @ Llist4 )
= ( coinductive_lnull @ A @ Llist5 ) )
& ( ~ ( coinductive_lnull @ A @ Llist4 )
=> ( ~ ( coinductive_lnull @ A @ Llist5 )
=> ( ( ( coinductive_lhd @ A @ Llist4 )
= ( coinductive_lhd @ A @ Llist5 ) )
& ( R @ ( coinductive_ltl @ A @ Llist4 ) @ ( coinductive_ltl @ A @ Llist5 ) ) ) ) ) ) )
=> ( Llist2 = Llist3 ) ) ) ).
% llist.coinduct
thf(fact_154_llist_Ocoinduct__strong,axiom,
! [A: $tType,R: ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o,Llist2: coinductive_llist @ A,Llist3: coinductive_llist @ A] :
( ( R @ Llist2 @ Llist3 )
=> ( ! [Llist4: coinductive_llist @ A,Llist5: coinductive_llist @ A] :
( ( R @ Llist4 @ Llist5 )
=> ( ( ( coinductive_lnull @ A @ Llist4 )
= ( coinductive_lnull @ A @ Llist5 ) )
& ( ~ ( coinductive_lnull @ A @ Llist4 )
=> ( ~ ( coinductive_lnull @ A @ Llist5 )
=> ( ( ( coinductive_lhd @ A @ Llist4 )
= ( coinductive_lhd @ A @ Llist5 ) )
& ( ( R @ ( coinductive_ltl @ A @ Llist4 ) @ ( coinductive_ltl @ A @ Llist5 ) )
| ( ( coinductive_ltl @ A @ Llist4 )
= ( coinductive_ltl @ A @ Llist5 ) ) ) ) ) ) ) )
=> ( Llist2 = Llist3 ) ) ) ).
% llist.coinduct_strong
thf(fact_155_llist_Oset__sel_I2_J,axiom,
! [A: $tType,A2: coinductive_llist @ A,X: A] :
( ~ ( coinductive_lnull @ A @ A2 )
=> ( ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_ltl @ A @ A2 ) ) )
=> ( member @ A @ X @ ( coinductive_lset @ A @ A2 ) ) ) ) ).
% llist.set_sel(2)
thf(fact_156_lfinite__induct,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
( ( coinductive_lfinite @ A @ Xs )
=> ( ! [Xs2: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs2 )
=> ( P @ Xs2 ) )
=> ( ! [Xs2: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs2 )
=> ( ~ ( coinductive_lnull @ A @ Xs2 )
=> ( ( P @ ( coinductive_ltl @ A @ Xs2 ) )
=> ( P @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% lfinite_induct
thf(fact_157_local_Olsorted__ldistinct__lset__unique,axiom,
! [Xs: coinductive_llist @ a,Ys: coinductive_llist @ a] :
( ( coinductive_lsorted @ a @ less_eq @ Xs )
=> ( ( coindu351974385stinct @ a @ Xs )
=> ( ( coinductive_lsorted @ a @ less_eq @ Ys )
=> ( ( coindu351974385stinct @ a @ Ys )
=> ( ( ( coinductive_lset @ a @ Xs )
= ( coinductive_lset @ a @ Ys ) )
=> ( Xs = Ys ) ) ) ) ) ) ).
% local.lsorted_ldistinct_lset_unique
thf(fact_158_local_Ostrict__mono__less__eq,axiom,
! [B: $tType] :
( ( order @ B @ ( type2 @ B ) )
=> ! [F: a > B,X: a,Y: a] :
( ( strict_mono @ a @ B @ less @ F )
=> ( ( ord_less_eq @ B @ ( F @ X ) @ ( F @ Y ) )
= ( less_eq @ X @ Y ) ) ) ) ).
% local.strict_mono_less_eq
thf(fact_159_local_OIoc__subset__iff,axiom,
! [A2: a,B2: a,C: a,D: a] :
( ( ord_less_eq @ ( set @ a ) @ ( set_gr323396891AtMost @ a @ less_eq @ less @ A2 @ B2 ) @ ( set_gr323396891AtMost @ a @ less_eq @ less @ C @ D ) )
= ( ( less_eq @ B2 @ A2 )
| ( ( less_eq @ C @ A2 )
& ( less_eq @ B2 @ D ) ) ) ) ).
% local.Ioc_subset_iff
thf(fact_160_local_Olsorted__LCons_H,axiom,
! [X: a,Xs: coinductive_llist @ a] :
( ( coinductive_lsorted @ a @ less_eq @ ( coinductive_LCons @ a @ X @ Xs ) )
= ( ~ ( coinductive_lnull @ a @ Xs )
=> ( ( less_eq @ X @ ( coinductive_lhd @ a @ Xs ) )
& ( coinductive_lsorted @ a @ less_eq @ Xs ) ) ) ) ).
% local.lsorted_LCons'
thf(fact_161_local_Olsorted_OLCons__LCons,axiom,
! [X: a,Y: a,Xs: coinductive_llist @ a] :
( ( less_eq @ X @ Y )
=> ( ( coinductive_lsorted @ a @ less_eq @ ( coinductive_LCons @ a @ Y @ Xs ) )
=> ( coinductive_lsorted @ a @ less_eq @ ( coinductive_LCons @ a @ X @ ( coinductive_LCons @ a @ Y @ Xs ) ) ) ) ) ).
% local.lsorted.LCons_LCons
thf(fact_162_local_Omono__def,axiom,
! [B: $tType] :
( ( order @ B @ ( type2 @ B ) )
=> ! [F: a > B] :
( ( mono @ a @ B @ less_eq @ F )
= ( ! [X2: a,Y3: a] :
( ( less_eq @ X2 @ Y3 )
=> ( ord_less_eq @ B @ ( F @ X2 ) @ ( F @ Y3 ) ) ) ) ) ) ).
% local.mono_def
thf(fact_163_local_OmonoI,axiom,
! [B: $tType] :
( ( order @ B @ ( type2 @ B ) )
=> ! [F: a > B] :
( ! [X3: a,Y4: a] :
( ( less_eq @ X3 @ Y4 )
=> ( ord_less_eq @ B @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( mono @ a @ B @ less_eq @ F ) ) ) ).
% local.monoI
thf(fact_164_local_OmonoE,axiom,
! [B: $tType] :
( ( order @ B @ ( type2 @ B ) )
=> ! [F: a > B,X: a,Y: a] :
( ( mono @ a @ B @ less_eq @ F )
=> ( ( less_eq @ X @ Y )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ).
% local.monoE
thf(fact_165_local_OmonoD,axiom,
! [B: $tType] :
( ( order @ B @ ( type2 @ B ) )
=> ! [F: a > B,X: a,Y: a] :
( ( mono @ a @ B @ less_eq @ F )
=> ( ( less_eq @ X @ Y )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ).
% local.monoD
thf(fact_166_local_Olmerge__simps,axiom,
! [X: a,Y: a,Xs: coinductive_llist @ a,Ys: coinductive_llist @ a] :
( ( ( less @ X @ Y )
=> ( ( hammin1328233080lmerge @ a @ less @ ( coinductive_LCons @ a @ X @ Xs ) @ ( coinductive_LCons @ a @ Y @ Ys ) )
= ( coinductive_LCons @ a @ X @ ( hammin1328233080lmerge @ a @ less @ Xs @ ( coinductive_LCons @ a @ Y @ Ys ) ) ) ) )
& ( ~ ( less @ X @ Y )
=> ( ( ( less @ Y @ X )
=> ( ( hammin1328233080lmerge @ a @ less @ ( coinductive_LCons @ a @ X @ Xs ) @ ( coinductive_LCons @ a @ Y @ Ys ) )
= ( coinductive_LCons @ a @ Y @ ( hammin1328233080lmerge @ a @ less @ ( coinductive_LCons @ a @ X @ Xs ) @ Ys ) ) ) )
& ( ~ ( less @ Y @ X )
=> ( ( hammin1328233080lmerge @ a @ less @ ( coinductive_LCons @ a @ X @ Xs ) @ ( coinductive_LCons @ a @ Y @ Ys ) )
= ( coinductive_LCons @ a @ Y @ ( hammin1328233080lmerge @ a @ less @ Xs @ Ys ) ) ) ) ) ) ) ).
% local.lmerge_simps
thf(fact_167_local_Obdd__below__mono,axiom,
! [B4: set @ a,A4: set @ a] :
( ( condit1201339847_below @ a @ less_eq @ B4 )
=> ( ( ord_less_eq @ ( set @ a ) @ A4 @ B4 )
=> ( condit1201339847_below @ a @ less_eq @ A4 ) ) ) ).
% local.bdd_below_mono
thf(fact_168_local_Obdd__above__mono,axiom,
! [B4: set @ a,A4: set @ a] :
( ( condit2040224947_above @ a @ less_eq @ B4 )
=> ( ( ord_less_eq @ ( set @ a ) @ A4 @ B4 )
=> ( condit2040224947_above @ a @ less_eq @ A4 ) ) ) ).
% local.bdd_above_mono
thf(fact_169_llist_Oinject,axiom,
! [A: $tType,X21: A,X22: coinductive_llist @ A,Y21: A,Y22: coinductive_llist @ A] :
( ( ( coinductive_LCons @ A @ X21 @ X22 )
= ( coinductive_LCons @ A @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% llist.inject
thf(fact_170_local_Olsorted_Osimps,axiom,
! [A2: coinductive_llist @ a] :
( ( coinductive_lsorted @ a @ less_eq @ A2 )
= ( ( A2
= ( coinductive_LNil @ a ) )
| ? [X2: a] :
( A2
= ( coinductive_LCons @ a @ X2 @ ( coinductive_LNil @ a ) ) )
| ? [X2: a,Y3: a,Xs3: coinductive_llist @ a] :
( ( A2
= ( coinductive_LCons @ a @ X2 @ ( coinductive_LCons @ a @ Y3 @ Xs3 ) ) )
& ( less_eq @ X2 @ Y3 )
& ( coinductive_lsorted @ a @ less_eq @ ( coinductive_LCons @ a @ Y3 @ Xs3 ) ) ) ) ) ).
% local.lsorted.simps
thf(fact_171_local_Olsorted_Ocoinduct,axiom,
! [X4: ( coinductive_llist @ a ) > $o,X: coinductive_llist @ a] :
( ( X4 @ X )
=> ( ! [X3: coinductive_llist @ a] :
( ( X4 @ X3 )
=> ( ( X3
= ( coinductive_LNil @ a ) )
| ? [Xa: a] :
( X3
= ( coinductive_LCons @ a @ Xa @ ( coinductive_LNil @ a ) ) )
| ? [Xa: a,Y5: a,Xs4: coinductive_llist @ a] :
( ( X3
= ( coinductive_LCons @ a @ Xa @ ( coinductive_LCons @ a @ Y5 @ Xs4 ) ) )
& ( less_eq @ Xa @ Y5 )
& ( ( X4 @ ( coinductive_LCons @ a @ Y5 @ Xs4 ) )
| ( coinductive_lsorted @ a @ less_eq @ ( coinductive_LCons @ a @ Y5 @ Xs4 ) ) ) ) ) )
=> ( coinductive_lsorted @ a @ less_eq @ X ) ) ) ).
% local.lsorted.coinduct
thf(fact_172_local_Olsorted_Ocases,axiom,
! [A2: coinductive_llist @ a] :
( ( coinductive_lsorted @ a @ less_eq @ A2 )
=> ( ( A2
!= ( coinductive_LNil @ a ) )
=> ( ! [X3: a] :
( A2
!= ( coinductive_LCons @ a @ X3 @ ( coinductive_LNil @ a ) ) )
=> ~ ! [X3: a,Y4: a,Xs2: coinductive_llist @ a] :
( ( A2
= ( coinductive_LCons @ a @ X3 @ ( coinductive_LCons @ a @ Y4 @ Xs2 ) ) )
=> ( ( less_eq @ X3 @ Y4 )
=> ~ ( coinductive_lsorted @ a @ less_eq @ ( coinductive_LCons @ a @ Y4 @ Xs2 ) ) ) ) ) ) ) ).
% local.lsorted.cases
thf(fact_173_local_Olsorted_OSingleton,axiom,
! [X: a] : ( coinductive_lsorted @ a @ less_eq @ ( coinductive_LCons @ a @ X @ ( coinductive_LNil @ a ) ) ) ).
% local.lsorted.Singleton
thf(fact_174_local_Olsorted__LCons,axiom,
! [X: a,Xs: coinductive_llist @ a] :
( ( coinductive_lsorted @ a @ less_eq @ ( coinductive_LCons @ a @ X @ Xs ) )
= ( ( coinductive_lsorted @ a @ less_eq @ Xs )
& ! [X2: a] :
( ( member @ a @ X2 @ ( coinductive_lset @ a @ Xs ) )
=> ( less_eq @ X @ X2 ) ) ) ) ).
% local.lsorted_LCons
thf(fact_175_local_Oantimono__def,axiom,
! [B: $tType] :
( ( order @ B @ ( type2 @ B ) )
=> ! [F: a > B] :
( ( antimono @ a @ B @ less_eq @ F )
= ( ! [X2: a,Y3: a] :
( ( less_eq @ X2 @ Y3 )
=> ( ord_less_eq @ B @ ( F @ Y3 ) @ ( F @ X2 ) ) ) ) ) ) ).
% local.antimono_def
thf(fact_176_local_OantimonoI,axiom,
! [B: $tType] :
( ( order @ B @ ( type2 @ B ) )
=> ! [F: a > B] :
( ! [X3: a,Y4: a] :
( ( less_eq @ X3 @ Y4 )
=> ( ord_less_eq @ B @ ( F @ Y4 ) @ ( F @ X3 ) ) )
=> ( antimono @ a @ B @ less_eq @ F ) ) ) ).
% local.antimonoI
thf(fact_177_local_OantimonoE,axiom,
! [B: $tType] :
( ( order @ B @ ( type2 @ B ) )
=> ! [F: a > B,X: a,Y: a] :
( ( antimono @ a @ B @ less_eq @ F )
=> ( ( less_eq @ X @ Y )
=> ( ord_less_eq @ B @ ( F @ Y ) @ ( F @ X ) ) ) ) ) ).
% local.antimonoE
thf(fact_178_local_OantimonoD,axiom,
! [B: $tType] :
( ( order @ B @ ( type2 @ B ) )
=> ! [F: a > B,X: a,Y: a] :
( ( antimono @ a @ B @ less_eq @ F )
=> ( ( less_eq @ X @ Y )
=> ( ord_less_eq @ B @ ( F @ Y ) @ ( F @ X ) ) ) ) ) ).
% local.antimonoD
thf(fact_179_lfinite__code_I2_J,axiom,
! [B: $tType,X: B,Xs: coinductive_llist @ B] :
( ( coinductive_lfinite @ B @ ( coinductive_LCons @ B @ X @ Xs ) )
= ( coinductive_lfinite @ B @ Xs ) ) ).
% lfinite_code(2)
thf(fact_180_lfinite__LCons,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
= ( coinductive_lfinite @ A @ Xs ) ) ).
% lfinite_LCons
thf(fact_181_ldistinct__LNil__code,axiom,
! [A: $tType] : ( coindu351974385stinct @ A @ ( coinductive_LNil @ A ) ) ).
% ldistinct_LNil_code
thf(fact_182_ldistinct__LCons,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A] :
( ( coindu351974385stinct @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
= ( ~ ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
& ( coindu351974385stinct @ A @ Xs ) ) ) ).
% ldistinct_LCons
thf(fact_183_local_Olsorted__LCons__LCons,axiom,
! [X: a,Y: a,Xs: coinductive_llist @ a] :
( ( coinductive_lsorted @ a @ less_eq @ ( coinductive_LCons @ a @ X @ ( coinductive_LCons @ a @ Y @ Xs ) ) )
= ( ( less_eq @ X @ Y )
& ( coinductive_lsorted @ a @ less_eq @ ( coinductive_LCons @ a @ Y @ Xs ) ) ) ) ).
% local.lsorted_LCons_LCons
thf(fact_184_lhd__LCons__ltl,axiom,
! [A: $tType,Llist2: coinductive_llist @ A] :
( ~ ( coinductive_lnull @ A @ Llist2 )
=> ( ( coinductive_LCons @ A @ ( coinductive_lhd @ A @ Llist2 ) @ ( coinductive_ltl @ A @ Llist2 ) )
= Llist2 ) ) ).
% lhd_LCons_ltl
thf(fact_185_local_Olsorted__code_I2_J,axiom,
! [X: a] : ( coinductive_lsorted @ a @ less_eq @ ( coinductive_LCons @ a @ X @ ( coinductive_LNil @ a ) ) ) ).
% local.lsorted_code(2)
thf(fact_186_llist_Odisc_I2_J,axiom,
! [A: $tType,X21: A,X22: coinductive_llist @ A] :
~ ( coinductive_lnull @ A @ ( coinductive_LCons @ A @ X21 @ X22 ) ) ).
% llist.disc(2)
thf(fact_187_llist_OdiscI_I2_J,axiom,
! [A: $tType,Llist2: coinductive_llist @ A,X21: A,X22: coinductive_llist @ A] :
( ( Llist2
= ( coinductive_LCons @ A @ X21 @ X22 ) )
=> ~ ( coinductive_lnull @ A @ Llist2 ) ) ).
% llist.discI(2)
thf(fact_188_llist_Odistinct_I1_J,axiom,
! [A: $tType,X21: A,X22: coinductive_llist @ A] :
( ( coinductive_LNil @ A )
!= ( coinductive_LCons @ A @ X21 @ X22 ) ) ).
% llist.distinct(1)
thf(fact_189_llist_Oexhaust,axiom,
! [A: $tType,Y: coinductive_llist @ A] :
( ( Y
!= ( coinductive_LNil @ A ) )
=> ~ ! [X212: A,X222: coinductive_llist @ A] :
( Y
!= ( coinductive_LCons @ A @ X212 @ X222 ) ) ) ).
% llist.exhaust
thf(fact_190_neq__LNil__conv,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( Xs
!= ( coinductive_LNil @ A ) )
= ( ? [X2: A,Xs5: coinductive_llist @ A] :
( Xs
= ( coinductive_LCons @ A @ X2 @ Xs5 ) ) ) ) ).
% neq_LNil_conv
thf(fact_191_ldistinct_OLNil,axiom,
! [A: $tType] : ( coindu351974385stinct @ A @ ( coinductive_LNil @ A ) ) ).
% ldistinct.LNil
thf(fact_192_not__lnull__conv,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( ~ ( coinductive_lnull @ A @ Xs ) )
= ( ? [X2: A,Xs5: coinductive_llist @ A] :
( Xs
= ( coinductive_LCons @ A @ X2 @ Xs5 ) ) ) ) ).
% not_lnull_conv
thf(fact_193_ldistinct_OLCons,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A] :
( ~ ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
=> ( ( coindu351974385stinct @ A @ Xs )
=> ( coindu351974385stinct @ A @ ( coinductive_LCons @ A @ X @ Xs ) ) ) ) ).
% ldistinct.LCons
thf(fact_194_ldistinct_Ocases,axiom,
! [A: $tType,A2: coinductive_llist @ A] :
( ( coindu351974385stinct @ A @ A2 )
=> ( ( A2
!= ( coinductive_LNil @ A ) )
=> ~ ! [X3: A,Xs2: coinductive_llist @ A] :
( ( A2
= ( coinductive_LCons @ A @ X3 @ Xs2 ) )
=> ( ~ ( member @ A @ X3 @ ( coinductive_lset @ A @ Xs2 ) )
=> ~ ( coindu351974385stinct @ A @ Xs2 ) ) ) ) ) ).
% ldistinct.cases
thf(fact_195_ldistinct_Osimps,axiom,
! [A: $tType] :
( ( coindu351974385stinct @ A )
= ( ^ [A5: coinductive_llist @ A] :
( ( A5
= ( coinductive_LNil @ A ) )
| ? [X2: A,Xs3: coinductive_llist @ A] :
( ( A5
= ( coinductive_LCons @ A @ X2 @ Xs3 ) )
& ~ ( member @ A @ X2 @ ( coinductive_lset @ A @ Xs3 ) )
& ( coindu351974385stinct @ A @ Xs3 ) ) ) ) ) ).
% ldistinct.simps
thf(fact_196_ldistinct_Ocoinduct,axiom,
! [A: $tType,X4: ( coinductive_llist @ A ) > $o,X: coinductive_llist @ A] :
( ( X4 @ X )
=> ( ! [X3: coinductive_llist @ A] :
( ( X4 @ X3 )
=> ( ( X3
= ( coinductive_LNil @ A ) )
| ? [Xa: A,Xs4: coinductive_llist @ A] :
( ( X3
= ( coinductive_LCons @ A @ Xa @ Xs4 ) )
& ~ ( member @ A @ Xa @ ( coinductive_lset @ A @ Xs4 ) )
& ( ( X4 @ Xs4 )
| ( coindu351974385stinct @ A @ Xs4 ) ) ) ) )
=> ( coindu351974385stinct @ A @ X ) ) ) ).
% ldistinct.coinduct
thf(fact_197_lfinite__LConsI,axiom,
! [A: $tType,Xs: coinductive_llist @ A,X: A] :
( ( coinductive_lfinite @ A @ Xs )
=> ( coinductive_lfinite @ A @ ( coinductive_LCons @ A @ X @ Xs ) ) ) ).
% lfinite_LConsI
thf(fact_198_ord_OLCons__LCons,axiom,
! [A: $tType,Less_eq: A > A > $o,X: A,Y: A,Xs: coinductive_llist @ A] :
( ( Less_eq @ X @ Y )
=> ( ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LCons @ A @ Y @ Xs ) )
=> ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LCons @ A @ X @ ( coinductive_LCons @ A @ Y @ Xs ) ) ) ) ) ).
% ord.LCons_LCons
thf(fact_199_ord_Olsorted__LCons__LCons,axiom,
! [A: $tType,Less_eq: A > A > $o,X: A,Y: A,Xs: coinductive_llist @ A] :
( ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LCons @ A @ X @ ( coinductive_LCons @ A @ Y @ Xs ) ) )
= ( ( Less_eq @ X @ Y )
& ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LCons @ A @ Y @ Xs ) ) ) ) ).
% ord.lsorted_LCons_LCons
thf(fact_200_lhd__LCons,axiom,
! [A: $tType,X21: A,X22: coinductive_llist @ A] :
( ( coinductive_lhd @ A @ ( coinductive_LCons @ A @ X21 @ X22 ) )
= X21 ) ).
% lhd_LCons
thf(fact_201_llist_Oset__induct,axiom,
! [A: $tType,X: A,A2: coinductive_llist @ A,P: A > ( coinductive_llist @ A ) > $o] :
( ( member @ A @ X @ ( coinductive_lset @ A @ A2 ) )
=> ( ! [Z1: A,Z22: coinductive_llist @ A] : ( P @ Z1 @ ( coinductive_LCons @ A @ Z1 @ Z22 ) )
=> ( ! [Z1: A,Z22: coinductive_llist @ A,Xa2: A] :
( ( member @ A @ Xa2 @ ( coinductive_lset @ A @ Z22 ) )
=> ( ( P @ Xa2 @ Z22 )
=> ( P @ Xa2 @ ( coinductive_LCons @ A @ Z1 @ Z22 ) ) ) )
=> ( P @ X @ A2 ) ) ) ) ).
% llist.set_induct
thf(fact_202_llist_Oset__cases,axiom,
! [A: $tType,E: A,A2: coinductive_llist @ A] :
( ( member @ A @ E @ ( coinductive_lset @ A @ A2 ) )
=> ( ! [Z22: coinductive_llist @ A] :
( A2
!= ( coinductive_LCons @ A @ E @ Z22 ) )
=> ~ ! [Z1: A,Z22: coinductive_llist @ A] :
( ( A2
= ( coinductive_LCons @ A @ Z1 @ Z22 ) )
=> ~ ( member @ A @ E @ ( coinductive_lset @ A @ Z22 ) ) ) ) ) ).
% llist.set_cases
thf(fact_203_lset__induct_H,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
=> ( ! [Xs2: coinductive_llist @ A] : ( P @ ( coinductive_LCons @ A @ X @ Xs2 ) )
=> ( ! [X6: A,Xs2: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs2 ) )
=> ( ( P @ Xs2 )
=> ( P @ ( coinductive_LCons @ A @ X6 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% lset_induct'
thf(fact_204_lset__induct,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
=> ( ! [Xs2: coinductive_llist @ A] : ( P @ ( coinductive_LCons @ A @ X @ Xs2 ) )
=> ( ! [X6: A,Xs2: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs2 ) )
=> ( ( X != X6 )
=> ( ( P @ Xs2 )
=> ( P @ ( coinductive_LCons @ A @ X6 @ Xs2 ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% lset_induct
thf(fact_205_lset__cases,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
=> ( ! [Xs6: coinductive_llist @ A] :
( Xs
!= ( coinductive_LCons @ A @ X @ Xs6 ) )
=> ~ ! [X6: A,Xs6: coinductive_llist @ A] :
( ( Xs
= ( coinductive_LCons @ A @ X6 @ Xs6 ) )
=> ~ ( member @ A @ X @ ( coinductive_lset @ A @ Xs6 ) ) ) ) ) ).
% lset_cases
thf(fact_206_llist_Oset__intros_I1_J,axiom,
! [A: $tType,A1: A,A22: coinductive_llist @ A] : ( member @ A @ A1 @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ A1 @ A22 ) ) ) ).
% llist.set_intros(1)
thf(fact_207_llist_Oset__intros_I2_J,axiom,
! [A: $tType,X: A,A22: coinductive_llist @ A,A1: A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ A22 ) )
=> ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ A1 @ A22 ) ) ) ) ).
% llist.set_intros(2)
thf(fact_208_lset__intros_I1_J,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A] : ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ X @ Xs ) ) ) ).
% lset_intros(1)
thf(fact_209_lset__intros_I2_J,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A,X7: A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
=> ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ X7 @ Xs ) ) ) ) ).
% lset_intros(2)
thf(fact_210_ltl__simps_I2_J,axiom,
! [A: $tType,X21: A,X22: coinductive_llist @ A] :
( ( coinductive_ltl @ A @ ( coinductive_LCons @ A @ X21 @ X22 ) )
= X22 ) ).
% ltl_simps(2)
thf(fact_211_ldistinct__ltlI,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coindu351974385stinct @ A @ Xs )
=> ( coindu351974385stinct @ A @ ( coinductive_ltl @ A @ Xs ) ) ) ).
% ldistinct_ltlI
thf(fact_212_wlog__linorder__le,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > A > $o,B2: A,A2: A] :
( ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% wlog_linorder_le
thf(fact_213_ord_Olmerge__simps,axiom,
! [A: $tType,Less: A > A > $o,X: A,Y: A,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ( Less @ X @ Y )
=> ( ( hammin1328233080lmerge @ A @ Less @ ( coinductive_LCons @ A @ X @ Xs ) @ ( coinductive_LCons @ A @ Y @ Ys ) )
= ( coinductive_LCons @ A @ X @ ( hammin1328233080lmerge @ A @ Less @ Xs @ ( coinductive_LCons @ A @ Y @ Ys ) ) ) ) )
& ( ~ ( Less @ X @ Y )
=> ( ( ( Less @ Y @ X )
=> ( ( hammin1328233080lmerge @ A @ Less @ ( coinductive_LCons @ A @ X @ Xs ) @ ( coinductive_LCons @ A @ Y @ Ys ) )
= ( coinductive_LCons @ A @ Y @ ( hammin1328233080lmerge @ A @ Less @ ( coinductive_LCons @ A @ X @ Xs ) @ Ys ) ) ) )
& ( ~ ( Less @ Y @ X )
=> ( ( hammin1328233080lmerge @ A @ Less @ ( coinductive_LCons @ A @ X @ Xs ) @ ( coinductive_LCons @ A @ Y @ Ys ) )
= ( coinductive_LCons @ A @ Y @ ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys ) ) ) ) ) ) ) ).
% ord.lmerge_simps
thf(fact_214_lfinite_Ocases,axiom,
! [A: $tType,A2: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ A2 )
=> ( ( A2
!= ( coinductive_LNil @ A ) )
=> ~ ! [Xs2: coinductive_llist @ A] :
( ? [X3: A] :
( A2
= ( coinductive_LCons @ A @ X3 @ Xs2 ) )
=> ~ ( coinductive_lfinite @ A @ Xs2 ) ) ) ) ).
% lfinite.cases
thf(fact_215_lfinite_Osimps,axiom,
! [A: $tType] :
( ( coinductive_lfinite @ A )
= ( ^ [A5: coinductive_llist @ A] :
( ( A5
= ( coinductive_LNil @ A ) )
| ? [Xs3: coinductive_llist @ A,X2: A] :
( ( A5
= ( coinductive_LCons @ A @ X2 @ Xs3 ) )
& ( coinductive_lfinite @ A @ Xs3 ) ) ) ) ) ).
% lfinite.simps
thf(fact_216_lfinite_Oinducts,axiom,
! [A: $tType,X: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
( ( coinductive_lfinite @ A @ X )
=> ( ( P @ ( coinductive_LNil @ A ) )
=> ( ! [Xs2: coinductive_llist @ A,X3: A] :
( ( coinductive_lfinite @ A @ Xs2 )
=> ( ( P @ Xs2 )
=> ( P @ ( coinductive_LCons @ A @ X3 @ Xs2 ) ) ) )
=> ( P @ X ) ) ) ) ).
% lfinite.inducts
thf(fact_217_ord_Olsorted__code_I2_J,axiom,
! [A: $tType,Less_eq: A > A > $o,X: A] : ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LCons @ A @ X @ ( coinductive_LNil @ A ) ) ) ).
% ord.lsorted_code(2)
thf(fact_218_ord_Olsorted_Ocases,axiom,
! [A: $tType,Less_eq: A > A > $o,A2: coinductive_llist @ A] :
( ( coinductive_lsorted @ A @ Less_eq @ A2 )
=> ( ( A2
!= ( coinductive_LNil @ A ) )
=> ( ! [X3: A] :
( A2
!= ( coinductive_LCons @ A @ X3 @ ( coinductive_LNil @ A ) ) )
=> ~ ! [X3: A,Y4: A,Xs2: coinductive_llist @ A] :
( ( A2
= ( coinductive_LCons @ A @ X3 @ ( coinductive_LCons @ A @ Y4 @ Xs2 ) ) )
=> ( ( Less_eq @ X3 @ Y4 )
=> ~ ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LCons @ A @ Y4 @ Xs2 ) ) ) ) ) ) ) ).
% ord.lsorted.cases
thf(fact_219_ord_Olsorted_Osimps,axiom,
! [A: $tType] :
( ( coinductive_lsorted @ A )
= ( ^ [Less_eq2: A > A > $o,A5: coinductive_llist @ A] :
( ( A5
= ( coinductive_LNil @ A ) )
| ? [X2: A] :
( A5
= ( coinductive_LCons @ A @ X2 @ ( coinductive_LNil @ A ) ) )
| ? [X2: A,Y3: A,Xs3: coinductive_llist @ A] :
( ( A5
= ( coinductive_LCons @ A @ X2 @ ( coinductive_LCons @ A @ Y3 @ Xs3 ) ) )
& ( Less_eq2 @ X2 @ Y3 )
& ( coinductive_lsorted @ A @ Less_eq2 @ ( coinductive_LCons @ A @ Y3 @ Xs3 ) ) ) ) ) ) ).
% ord.lsorted.simps
thf(fact_220_ord_Olsorted_Ocoinduct,axiom,
! [A: $tType,X4: ( coinductive_llist @ A ) > $o,X: coinductive_llist @ A,Less_eq: A > A > $o] :
( ( X4 @ X )
=> ( ! [X3: coinductive_llist @ A] :
( ( X4 @ X3 )
=> ( ( X3
= ( coinductive_LNil @ A ) )
| ? [Xa: A] :
( X3
= ( coinductive_LCons @ A @ Xa @ ( coinductive_LNil @ A ) ) )
| ? [Xa: A,Y5: A,Xs4: coinductive_llist @ A] :
( ( X3
= ( coinductive_LCons @ A @ Xa @ ( coinductive_LCons @ A @ Y5 @ Xs4 ) ) )
& ( Less_eq @ Xa @ Y5 )
& ( ( X4 @ ( coinductive_LCons @ A @ Y5 @ Xs4 ) )
| ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LCons @ A @ Y5 @ Xs4 ) ) ) ) ) )
=> ( coinductive_lsorted @ A @ Less_eq @ X ) ) ) ).
% ord.lsorted.coinduct
thf(fact_221_ord_OSingleton,axiom,
! [A: $tType,Less_eq: A > A > $o,X: A] : ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LCons @ A @ X @ ( coinductive_LNil @ A ) ) ) ).
% ord.Singleton
thf(fact_222_lset__ltl,axiom,
! [A: $tType,Xs: coinductive_llist @ A] : ( ord_less_eq @ ( set @ A ) @ ( coinductive_lset @ A @ ( coinductive_ltl @ A @ Xs ) ) @ ( coinductive_lset @ A @ Xs ) ) ).
% lset_ltl
thf(fact_223_llist_Oexhaust__sel,axiom,
! [A: $tType,Llist2: coinductive_llist @ A] :
( ( Llist2
!= ( coinductive_LNil @ A ) )
=> ( Llist2
= ( coinductive_LCons @ A @ ( coinductive_lhd @ A @ Llist2 ) @ ( coinductive_ltl @ A @ Llist2 ) ) ) ) ).
% llist.exhaust_sel
thf(fact_224_eq__LConsD,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
( ( Xs
= ( coinductive_LCons @ A @ Y @ Ys ) )
=> ( ( Xs
!= ( coinductive_LNil @ A ) )
& ( ( coinductive_lhd @ A @ Xs )
= Y )
& ( ( coinductive_ltl @ A @ Xs )
= Ys ) ) ) ).
% eq_LConsD
thf(fact_225_ord_Olsorted__LCons_H,axiom,
! [A: $tType,Less_eq: A > A > $o,X: A,Xs: coinductive_llist @ A] :
( ( coinductive_lsorted @ A @ Less_eq @ ( coinductive_LCons @ A @ X @ Xs ) )
= ( ~ ( coinductive_lnull @ A @ Xs )
=> ( ( Less_eq @ X @ ( coinductive_lhd @ A @ Xs ) )
& ( coinductive_lsorted @ A @ Less_eq @ Xs ) ) ) ) ).
% ord.lsorted_LCons'
thf(fact_226_ldistinct__coinduct,axiom,
! [A: $tType,X4: ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A] :
( ( X4 @ Xs )
=> ( ! [Xs2: coinductive_llist @ A] :
( ( X4 @ Xs2 )
=> ( ~ ( coinductive_lnull @ A @ Xs2 )
=> ( ~ ( member @ A @ ( coinductive_lhd @ A @ Xs2 ) @ ( coinductive_lset @ A @ ( coinductive_ltl @ A @ Xs2 ) ) )
& ( ( X4 @ ( coinductive_ltl @ A @ Xs2 ) )
| ( coindu351974385stinct @ A @ ( coinductive_ltl @ A @ Xs2 ) ) ) ) ) )
=> ( coindu351974385stinct @ A @ Xs ) ) ) ).
% ldistinct_coinduct
thf(fact_227_ldistinct__lhdD,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coindu351974385stinct @ A @ Xs )
=> ( ~ ( coinductive_lnull @ A @ Xs )
=> ~ ( member @ A @ ( coinductive_lhd @ A @ Xs ) @ ( coinductive_lset @ A @ ( coinductive_ltl @ A @ Xs ) ) ) ) ) ).
% ldistinct_lhdD
thf(fact_228_local_OatLeastLessThan__subset__iff,axiom,
! [A2: a,B2: a,C: a,D: a] :
( ( ord_less_eq @ ( set @ a ) @ ( set_atLeastLessThan @ a @ less_eq @ less @ A2 @ B2 ) @ ( set_atLeastLessThan @ a @ less_eq @ less @ C @ D ) )
=> ( ( less_eq @ B2 @ A2 )
| ( ( less_eq @ C @ A2 )
& ( less_eq @ B2 @ D ) ) ) ) ).
% local.atLeastLessThan_subset_iff
thf(fact_229_local_Olset__lmerge,axiom,
! [Xs: coinductive_llist @ a,Ys: coinductive_llist @ a] : ( ord_less_eq @ ( set @ a ) @ ( coinductive_lset @ a @ ( hammin1328233080lmerge @ a @ less @ Xs @ Ys ) ) @ ( sup_sup @ ( set @ a ) @ ( coinductive_lset @ a @ Xs ) @ ( coinductive_lset @ a @ Ys ) ) ) ).
% local.lset_lmerge
thf(fact_230_local_Obdd__below__Ioo,axiom,
! [A2: a,B2: a] : ( condit1201339847_below @ a @ less_eq @ ( set_gr1161524159ssThan @ a @ less @ A2 @ B2 ) ) ).
% local.bdd_below_Ioo
thf(fact_231_local_OgreaterThanLessThan__iff,axiom,
! [I: a,L: a,U: a] :
( ( member @ a @ I @ ( set_gr1161524159ssThan @ a @ less @ L @ U ) )
= ( ( less @ L @ I )
& ( less @ I @ U ) ) ) ).
% local.greaterThanLessThan_iff
thf(fact_232_local_OatLeastLessThan__iff,axiom,
! [I: a,L: a,U: a] :
( ( member @ a @ I @ ( set_atLeastLessThan @ a @ less_eq @ less @ L @ U ) )
= ( ( less_eq @ L @ I )
& ( less @ I @ U ) ) ) ).
% local.atLeastLessThan_iff
thf(fact_233_local_Obdd__above__Ico,axiom,
! [A2: a,B2: a] : ( condit2040224947_above @ a @ less_eq @ ( set_atLeastLessThan @ a @ less_eq @ less @ A2 @ B2 ) ) ).
% local.bdd_above_Ico
thf(fact_234_local_Obdd__below__Ico,axiom,
! [A2: a,B2: a] : ( condit1201339847_below @ a @ less_eq @ ( set_atLeastLessThan @ a @ less_eq @ less @ A2 @ B2 ) ) ).
% local.bdd_below_Ico
thf(fact_235_local_Obdd__above__Ioo,axiom,
! [A2: a,B2: a] : ( condit2040224947_above @ a @ less_eq @ ( set_gr1161524159ssThan @ a @ less @ A2 @ B2 ) ) ).
% local.bdd_above_Ioo
thf(fact_236_ord_Olset__lmerge,axiom,
! [A: $tType,Less: A > A > $o,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] : ( ord_less_eq @ ( set @ A ) @ ( coinductive_lset @ A @ ( hammin1328233080lmerge @ A @ Less @ Xs @ Ys ) ) @ ( sup_sup @ ( set @ A ) @ ( coinductive_lset @ A @ Xs ) @ ( coinductive_lset @ A @ Ys ) ) ) ).
% ord.lset_lmerge
thf(fact_237_Un__subset__iff,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,C2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) @ C2 )
= ( ( ord_less_eq @ ( set @ A ) @ A4 @ C2 )
& ( ord_less_eq @ ( set @ A ) @ B4 @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_238_le__sup__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ X @ Y ) @ Z2 )
= ( ( ord_less_eq @ A @ X @ Z2 )
& ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% le_sup_iff
thf(fact_239_subsetI,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( member @ A @ X3 @ B4 ) )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ).
% subsetI
thf(fact_240_subset__antisym,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ A4 )
=> ( A4 = B4 ) ) ) ).
% subset_antisym
thf(fact_241_sup__apply,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B @ ( type2 @ B ) )
=> ( ( sup_sup @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B,X2: A] : ( sup_sup @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% sup_apply
thf(fact_242_sup_Oidem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( sup_sup @ A @ A2 @ A2 )
= A2 ) ) ).
% sup.idem
thf(fact_243_sup__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( sup_sup @ A @ X @ X )
= X ) ) ).
% sup_idem
thf(fact_244_sup_Oleft__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( sup_sup @ A @ A2 @ ( sup_sup @ A @ A2 @ B2 ) )
= ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% sup.left_idem
thf(fact_245_sup__left__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
= ( sup_sup @ A @ X @ Y ) ) ) ).
% sup_left_idem
thf(fact_246_sup_Oright__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B2 ) @ B2 )
= ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% sup.right_idem
thf(fact_247_UnCI,axiom,
! [A: $tType,C: A,B4: set @ A,A4: set @ A] :
( ( ~ ( member @ A @ C @ B4 )
=> ( member @ A @ C @ A4 ) )
=> ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) ) ) ).
% UnCI
thf(fact_248_Un__iff,axiom,
! [A: $tType,C: A,A4: set @ A,B4: set @ A] :
( ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A4 @ B4 ) )
= ( ( member @ A @ C @ A4 )
| ( member @ A @ C @ B4 ) ) ) ).
% Un_iff
thf(fact_249_sup_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B2: A,C: A,A2: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C ) @ A2 )
= ( ( ord_less_eq @ A @ B2 @ A2 )
& ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% sup.bounded_iff
thf(fact_250_set__mp,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ( member @ A @ X @ A4 )
=> ( member @ A @ X @ B4 ) ) ) ).
% set_mp
thf(fact_251_in__mono,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ( member @ A @ X @ A4 )
=> ( member @ A @ X @ B4 ) ) ) ).
% in_mono
thf(fact_252_subsetD,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,C: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ( member @ A @ C @ A4 )
=> ( member @ A @ C @ B4 ) ) ) ).
% subsetD
thf(fact_253_subsetCE,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,C: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ( member @ A @ C @ A4 )
=> ( member @ A @ C @ B4 ) ) ) ).
% subsetCE
%----Type constructors (10)
thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A6: $tType,A7: $tType] :
( ( semilattice_sup @ A7 @ ( type2 @ A7 ) )
=> ( semilattice_sup @ ( 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_Set_Oset___Lattices_Osemilattice__sup_1,axiom,
! [A6: $tType] : ( semilattice_sup @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_2,axiom,
! [A6: $tType] : ( order @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_3,axiom,
! [A6: $tType] : ( ord @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_4,axiom,
semilattice_sup @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_5,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_6,axiom,
ord @ $o @ ( type2 @ $o ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
! [X3: a] :
( ( member @ a @ X3 @ ( coinductive_lset @ a @ ( coinductive_ltl @ a @ ( hammin1328233080lmerge @ a @ less @ xsa @ ysa ) ) ) )
=> ( less_eq @ ( coinductive_lhd @ a @ ( hammin1328233080lmerge @ a @ less @ xsa @ ysa ) ) @ X3 ) ) ).
%------------------------------------------------------------------------------