TPTP Problem File: DAT157^1.p
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%------------------------------------------------------------------------------
% File : DAT157^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Hamming stream 157
% 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__157.p [Bla16]
% Status : Theorem
% Rating : 0.00 v7.1.0
% Syntax : Number of formulae : 310 ( 100 unt; 48 typ; 0 def)
% Number of atoms : 897 ( 173 equ; 0 cnn)
% Maximal formula atoms : 12 ( 3 avg)
% Number of connectives : 3016 ( 83 ~; 23 |; 37 &;2599 @)
% ( 0 <=>; 274 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 7 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 196 ( 196 >; 0 *; 0 +; 0 <<)
% Number of symbols : 48 ( 47 usr; 5 con; 0-6 aty)
% Number of variables : 767 ( 34 ^; 688 !; 4 ?; 767 :)
% ( 41 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:41:50.421
%------------------------------------------------------------------------------
%----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 (43)
thf(sy_cl_HOL_Otype,type,
type:
!>[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_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_Olset,type,
coinductive_lset:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( set @ 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_Finite__Set_Ofolding,type,
finite_folding:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > $o ) ).
thf(sy_c_Groups_Osemigroup,type,
semigroup:
!>[A: $tType] : ( ( A > A > 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_Oinf__class_Oinf,type,
inf_inf:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Lattices_Osemilattice,type,
semilattice:
!>[A: $tType] : ( ( A > A > A ) > $o ) ).
thf(sy_c_Lattices__Big_Osemilattice__order__set,type,
lattic1693879045er_set:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > $o ) > ( A > A > $o ) > $o ) ).
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_Omin,type,
min:
!>[A: $tType] : ( ( A > A > $o ) > A > A > 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_Relation_Otransp,type,
transp:
!>[A: $tType] : ( ( A > A > $o ) > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord_OatLeast,type,
set_atLeast:
!>[A: $tType] : ( ( A > A > $o ) > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord_OatLeastAtMost,type,
set_atLeastAtMost:
!>[A: $tType] : ( ( A > A > $o ) > A > A > ( 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_OatMost,type,
set_atMost:
!>[A: $tType] : ( ( A > A > $o ) > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord_OgreaterThan,type,
set_greaterThan:
!>[A: $tType] : ( ( A > A > $o ) > 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_Set__Interval_Oord_OlessThan,type,
set_lessThan:
!>[A: $tType] : ( ( A > A > $o ) > 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_thesis____,type,
thesis: $o ).
thf(sy_v_x,type,
x: a ).
thf(sy_v_x_H____,type,
x2: a ).
thf(sy_v_xsa____,type,
xsa: coinductive_llist @ a ).
thf(sy_v_ysa____,type,
ysa: coinductive_llist @ a ).
%----Relevant facts (256)
thf(fact_0_local_Oantisym__conv3,axiom,
! [Y: a,X: a] :
( ~ ( less @ Y @ X )
=> ( ( ~ ( less @ X @ Y ) )
= ( X = Y ) ) ) ).
% local.antisym_conv3
thf(fact_1_local_Odual__order_Oasym,axiom,
! [B2: a,A2: a] :
( ( less @ B2 @ A2 )
=> ~ ( less @ A2 @ B2 ) ) ).
% local.dual_order.asym
thf(fact_2_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_3_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_4_local_Oless__asym,axiom,
! [X: a,Y: a] :
( ( less @ X @ Y )
=> ~ ( less @ Y @ X ) ) ).
% local.less_asym
thf(fact_5_local_Oless__asym_H,axiom,
! [A2: a,B2: a] :
( ( less @ A2 @ B2 )
=> ~ ( less @ B2 @ A2 ) ) ).
% local.less_asym'
thf(fact_6_local_Oless__imp__neq,axiom,
! [X: a,Y: a] :
( ( less @ X @ Y )
=> ( X != Y ) ) ).
% local.less_imp_neq
thf(fact_7_local_Oless__imp__not__eq,axiom,
! [X: a,Y: a] :
( ( less @ X @ Y )
=> ( X != Y ) ) ).
% local.less_imp_not_eq
thf(fact_8_local_Oless__imp__not__eq2,axiom,
! [X: a,Y: a] :
( ( less @ X @ Y )
=> ( Y != X ) ) ).
% local.less_imp_not_eq2
thf(fact_9_local_Oless__imp__not__less,axiom,
! [X: a,Y: a] :
( ( less @ X @ Y )
=> ~ ( less @ Y @ X ) ) ).
% local.less_imp_not_less
thf(fact_10_local_Oless__imp__triv,axiom,
! [X: a,Y: a,P: $o] :
( ( less @ X @ Y )
=> ( ( less @ Y @ X )
=> P ) ) ).
% local.less_imp_triv
thf(fact_11_local_Oless__irrefl,axiom,
! [X: a] :
~ ( less @ X @ X ) ).
% local.less_irrefl
thf(fact_12_local_Oless__linear,axiom,
! [X: a,Y: a] :
( ( less @ X @ Y )
| ( X = Y )
| ( less @ Y @ X ) ) ).
% local.less_linear
thf(fact_13_local_Oless__not__sym,axiom,
! [X: a,Y: a] :
( ( less @ X @ Y )
=> ~ ( less @ Y @ X ) ) ).
% local.less_not_sym
thf(fact_14_local_Oless__trans,axiom,
! [X: a,Y: a,Z: a] :
( ( less @ X @ Y )
=> ( ( less @ Y @ Z )
=> ( less @ X @ Z ) ) ) ).
% local.less_trans
thf(fact_15_local_Olinorder__cases,axiom,
! [X: a,Y: a] :
( ~ ( less @ X @ Y )
=> ( ( X != Y )
=> ( less @ Y @ X ) ) ) ).
% local.linorder_cases
thf(fact_16_local_OneqE,axiom,
! [X: a,Y: a] :
( ( X != Y )
=> ( ~ ( less @ X @ Y )
=> ( less @ Y @ X ) ) ) ).
% local.neqE
thf(fact_17_local_Oneq__iff,axiom,
! [X: a,Y: a] :
( ( X != Y )
= ( ( less @ X @ Y )
| ( less @ Y @ X ) ) ) ).
% local.neq_iff
thf(fact_18_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_19_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_20_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_21_local_Oorder_Oasym,axiom,
! [A2: a,B2: a] :
( ( less @ A2 @ B2 )
=> ~ ( less @ B2 @ A2 ) ) ).
% local.order.asym
thf(fact_22_local_Oorder_Oirrefl,axiom,
! [A2: a] :
~ ( less @ A2 @ A2 ) ).
% local.order.irrefl
thf(fact_23_local_Oorder_Ostrict__implies__not__eq,axiom,
! [A2: a,B2: a] :
( ( less @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% local.order.strict_implies_not_eq
thf(fact_24_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_25_step_Ohyps_I2_J,axiom,
x != x2 ).
% step.hyps(2)
thf(fact_26_step_Oprems_I2_J,axiom,
~ ( coinductive_lfinite @ a @ ysa ) ).
% step.prems(2)
thf(fact_27_step_Ohyps_I1_J,axiom,
member @ a @ x @ ( coinductive_lset @ a @ xsa ) ).
% step.hyps(1)
thf(fact_28_local_Oantisym,axiom,
! [X: a,Y: a] :
( ( less_eq @ X @ Y )
=> ( ( less_eq @ Y @ X )
=> ( X = Y ) ) ) ).
% local.antisym
thf(fact_29_local_Oantisym__conv,axiom,
! [Y: a,X: a] :
( ( less_eq @ Y @ X )
=> ( ( less_eq @ X @ Y )
= ( X = Y ) ) ) ).
% local.antisym_conv
thf(fact_30_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_31_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_32_local_Oeq__iff,axiom,
( ( ^ [Y2: a,Z2: a] : Y2 = Z2 )
= ( ^ [X2: a,Y3: a] :
( ( less_eq @ X2 @ Y3 )
& ( less_eq @ Y3 @ X2 ) ) ) ) ).
% local.eq_iff
thf(fact_33_local_Oeq__refl,axiom,
! [X: a,Y: a] :
( ( X = Y )
=> ( less_eq @ X @ Y ) ) ).
% local.eq_refl
thf(fact_34_local_Ole__cases,axiom,
! [X: a,Y: a] :
( ~ ( less_eq @ X @ Y )
=> ( less_eq @ Y @ X ) ) ).
% local.le_cases
thf(fact_35_local_Ole__cases3,axiom,
! [X: a,Y: a,Z: a] :
( ( ( less_eq @ X @ Y )
=> ~ ( less_eq @ Y @ Z ) )
=> ( ( ( less_eq @ Y @ X )
=> ~ ( less_eq @ X @ Z ) )
=> ( ( ( less_eq @ X @ Z )
=> ~ ( less_eq @ Z @ Y ) )
=> ( ( ( less_eq @ Z @ Y )
=> ~ ( less_eq @ Y @ X ) )
=> ( ( ( less_eq @ Y @ Z )
=> ~ ( less_eq @ Z @ X ) )
=> ~ ( ( less_eq @ Z @ X )
=> ~ ( less_eq @ X @ Y ) ) ) ) ) ) ) ).
% local.le_cases3
thf(fact_36_local_Olinear,axiom,
! [X: a,Y: a] :
( ( less_eq @ X @ Y )
| ( less_eq @ Y @ X ) ) ).
% local.linear
thf(fact_37_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_38_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_39_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_40_local_Oorder_Oantisym,axiom,
! [A2: a,B2: a] :
( ( less_eq @ A2 @ B2 )
=> ( ( less_eq @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% local.order.antisym
thf(fact_41_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_42_local_Oorder__trans,axiom,
! [X: a,Y: a,Z: a] :
( ( less_eq @ X @ Y )
=> ( ( less_eq @ Y @ Z )
=> ( less_eq @ X @ Z ) ) ) ).
% local.order_trans
thf(fact_43_step_Oprems_I3_J,axiom,
? [X3: a] :
( ( member @ a @ X3 @ ( coinductive_lset @ a @ ysa ) )
& ( less_eq @ x @ X3 ) ) ).
% step.prems(3)
thf(fact_44_local_Oantisym__conv1,axiom,
! [X: a,Y: a] :
( ~ ( less @ X @ Y )
=> ( ( less_eq @ X @ Y )
= ( X = Y ) ) ) ).
% local.antisym_conv1
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_Oantisym__conv2,axiom,
! [X: a,Y: a] :
( ( less_eq @ X @ Y )
=> ( ( ~ ( less @ X @ Y ) )
= ( X = Y ) ) ) ).
% local.antisym_conv2
thf(fact_50_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_51_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_52_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_53_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_54_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_55_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_56_local_OleD,axiom,
! [Y: a,X: a] :
( ( less_eq @ Y @ X )
=> ~ ( less @ X @ Y ) ) ).
% local.leD
thf(fact_57_local_OleI,axiom,
! [X: a,Y: a] :
( ~ ( less @ X @ Y )
=> ( less_eq @ Y @ X ) ) ).
% local.leI
thf(fact_58_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_59_local_Ole__less,axiom,
! [X: a,Y: a] :
( ( less_eq @ X @ Y )
= ( ( less @ X @ Y )
| ( X = Y ) ) ) ).
% local.le_less
thf(fact_60_local_Ole__less__linear,axiom,
! [X: a,Y: a] :
( ( less_eq @ X @ Y )
| ( less @ Y @ X ) ) ).
% local.le_less_linear
thf(fact_61_local_Ole__less__trans,axiom,
! [X: a,Y: a,Z: a] :
( ( less_eq @ X @ Y )
=> ( ( less @ Y @ Z )
=> ( less @ X @ Z ) ) ) ).
% local.le_less_trans
thf(fact_62_local_Ole__neq__trans,axiom,
! [A2: a,B2: a] :
( ( less_eq @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( less @ A2 @ B2 ) ) ) ).
% local.le_neq_trans
thf(fact_63_local_Oless__imp__le,axiom,
! [X: a,Y: a] :
( ( less @ X @ Y )
=> ( less_eq @ X @ Y ) ) ).
% local.less_imp_le
thf(fact_64_local_Oless__le,axiom,
! [X: a,Y: a] :
( ( less @ X @ Y )
= ( ( less_eq @ X @ Y )
& ( X != Y ) ) ) ).
% local.less_le
thf(fact_65_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_66_local_Oless__le__trans,axiom,
! [X: a,Y: a,Z: a] :
( ( less @ X @ Y )
=> ( ( less_eq @ Y @ Z )
=> ( less @ X @ Z ) ) ) ).
% local.less_le_trans
thf(fact_67_local_Onot__le,axiom,
! [X: a,Y: a] :
( ( ~ ( less_eq @ X @ Y ) )
= ( less @ Y @ X ) ) ).
% local.not_le
thf(fact_68_local_Onot__le__imp__less,axiom,
! [Y: a,X: a] :
( ~ ( less_eq @ Y @ X )
=> ( less @ X @ Y ) ) ).
% local.not_le_imp_less
thf(fact_69_local_Onot__less,axiom,
! [X: a,Y: a] :
( ( ~ ( less @ X @ Y ) )
= ( less_eq @ Y @ X ) ) ).
% local.not_less
thf(fact_70_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_71_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_72_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_73_local_Oorder_Ostrict__implies__order,axiom,
! [A2: a,B2: a] :
( ( less @ A2 @ B2 )
=> ( less_eq @ A2 @ B2 ) ) ).
% local.order.strict_implies_order
thf(fact_74_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_75_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_76_local_Oorder_Orefl,axiom,
! [A2: a] : ( less_eq @ A2 @ A2 ) ).
% local.order.refl
thf(fact_77_local_Oorder__refl,axiom,
! [X: a] : ( less_eq @ X @ X ) ).
% local.order_refl
thf(fact_78_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_79_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_80_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_81_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_82_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_83_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_84_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_85_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_86_local_Otransp__le,axiom,
transp @ a @ less_eq ).
% local.transp_le
thf(fact_87_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_88_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_89_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_90_local_Omin_Oabsorb1,axiom,
! [A2: a,B2: a] :
( ( less_eq @ A2 @ B2 )
=> ( ( min @ a @ less_eq @ A2 @ B2 )
= A2 ) ) ).
% local.min.absorb1
thf(fact_91_local_Omin_Oabsorb2,axiom,
! [B2: a,A2: a] :
( ( less_eq @ B2 @ A2 )
=> ( ( min @ a @ less_eq @ A2 @ B2 )
= B2 ) ) ).
% local.min.absorb2
thf(fact_92_local_Omin_Oabsorb__iff1,axiom,
! [A2: a,B2: a] :
( ( less_eq @ A2 @ B2 )
= ( ( min @ a @ less_eq @ A2 @ B2 )
= A2 ) ) ).
% local.min.absorb_iff1
thf(fact_93_local_Omin_Oabsorb__iff2,axiom,
! [B2: a,A2: a] :
( ( less_eq @ B2 @ A2 )
= ( ( min @ a @ less_eq @ A2 @ B2 )
= B2 ) ) ).
% local.min.absorb_iff2
thf(fact_94_local_Omin__le__iff__disj,axiom,
! [X: a,Y: a,Z: a] :
( ( less_eq @ ( min @ a @ less_eq @ X @ Y ) @ Z )
= ( ( less_eq @ X @ Z )
| ( less_eq @ Y @ Z ) ) ) ).
% local.min_le_iff_disj
thf(fact_95_local_Omin__def,axiom,
! [A2: a,B2: a] :
( ( ( less_eq @ A2 @ B2 )
=> ( ( min @ a @ less_eq @ A2 @ B2 )
= A2 ) )
& ( ~ ( less_eq @ A2 @ B2 )
=> ( ( min @ a @ less_eq @ A2 @ B2 )
= B2 ) ) ) ).
% local.min_def
thf(fact_96_local_Omin_Oorder__iff,axiom,
! [A2: a,B2: a] :
( ( less_eq @ A2 @ B2 )
= ( A2
= ( min @ a @ less_eq @ A2 @ B2 ) ) ) ).
% local.min.order_iff
thf(fact_97_local_Omin_OorderI,axiom,
! [A2: a,B2: a] :
( ( A2
= ( min @ a @ less_eq @ A2 @ B2 ) )
=> ( less_eq @ A2 @ B2 ) ) ).
% local.min.orderI
thf(fact_98_local_Omin_OorderE,axiom,
! [A2: a,B2: a] :
( ( less_eq @ A2 @ B2 )
=> ( A2
= ( min @ a @ less_eq @ A2 @ B2 ) ) ) ).
% local.min.orderE
thf(fact_99_local_Omin_Omono,axiom,
! [A2: a,C: a,B2: a,D: a] :
( ( less_eq @ A2 @ C )
=> ( ( less_eq @ B2 @ D )
=> ( less_eq @ ( min @ a @ less_eq @ A2 @ B2 ) @ ( min @ a @ less_eq @ C @ D ) ) ) ) ).
% local.min.mono
thf(fact_100_local_Omin_Oleft__commute,axiom,
! [B2: a,A2: a,C: a] :
( ( min @ a @ less_eq @ B2 @ ( min @ a @ less_eq @ A2 @ C ) )
= ( min @ a @ less_eq @ A2 @ ( min @ a @ less_eq @ B2 @ C ) ) ) ).
% local.min.left_commute
thf(fact_101_local_Omin_Ocommute,axiom,
! [A2: a,B2: a] :
( ( min @ a @ less_eq @ A2 @ B2 )
= ( min @ a @ less_eq @ B2 @ A2 ) ) ).
% local.min.commute
thf(fact_102_local_Omin_OcoboundedI2,axiom,
! [B2: a,C: a,A2: a] :
( ( less_eq @ B2 @ C )
=> ( less_eq @ ( min @ a @ less_eq @ A2 @ B2 ) @ C ) ) ).
% local.min.coboundedI2
thf(fact_103_local_Omin_OcoboundedI1,axiom,
! [A2: a,C: a,B2: a] :
( ( less_eq @ A2 @ C )
=> ( less_eq @ ( min @ a @ less_eq @ A2 @ B2 ) @ C ) ) ).
% local.min.coboundedI1
thf(fact_104_local_Omin_Ocobounded2,axiom,
! [A2: a,B2: a] : ( less_eq @ ( min @ a @ less_eq @ A2 @ B2 ) @ B2 ) ).
% local.min.cobounded2
thf(fact_105_local_Omin_Ocobounded1,axiom,
! [A2: a,B2: a] : ( less_eq @ ( min @ a @ less_eq @ A2 @ B2 ) @ A2 ) ).
% local.min.cobounded1
thf(fact_106_local_Omin_OboundedI,axiom,
! [A2: a,B2: a,C: a] :
( ( less_eq @ A2 @ B2 )
=> ( ( less_eq @ A2 @ C )
=> ( less_eq @ A2 @ ( min @ a @ less_eq @ B2 @ C ) ) ) ) ).
% local.min.boundedI
thf(fact_107_local_Omin_OboundedE,axiom,
! [A2: a,B2: a,C: a] :
( ( less_eq @ A2 @ ( min @ a @ less_eq @ B2 @ C ) )
=> ~ ( ( less_eq @ A2 @ B2 )
=> ~ ( less_eq @ A2 @ C ) ) ) ).
% local.min.boundedE
thf(fact_108_local_Omin_Oassoc,axiom,
! [A2: a,B2: a,C: a] :
( ( min @ a @ less_eq @ ( min @ a @ less_eq @ A2 @ B2 ) @ C )
= ( min @ a @ less_eq @ A2 @ ( min @ a @ less_eq @ B2 @ C ) ) ) ).
% local.min.assoc
thf(fact_109_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_110_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_111_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_112_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_113_local_Omin__less__iff__disj,axiom,
! [X: a,Y: a,Z: a] :
( ( less @ ( min @ a @ less_eq @ X @ Y ) @ Z )
= ( ( less @ X @ Z )
| ( less @ Y @ Z ) ) ) ).
% local.min_less_iff_disj
thf(fact_114_local_Omin_Ostrict__order__iff,axiom,
! [A2: a,B2: a] :
( ( less @ A2 @ B2 )
= ( ( A2
= ( min @ a @ less_eq @ A2 @ B2 ) )
& ( A2 != B2 ) ) ) ).
% local.min.strict_order_iff
thf(fact_115_local_Omin_Ostrict__coboundedI2,axiom,
! [B2: a,C: a,A2: a] :
( ( less @ B2 @ C )
=> ( less @ ( min @ a @ less_eq @ A2 @ B2 ) @ C ) ) ).
% local.min.strict_coboundedI2
thf(fact_116_local_Omin_Ostrict__coboundedI1,axiom,
! [A2: a,C: a,B2: a] :
( ( less @ A2 @ C )
=> ( less @ ( min @ a @ less_eq @ A2 @ B2 ) @ C ) ) ).
% local.min.strict_coboundedI1
thf(fact_117_local_Omin_Ostrict__boundedE,axiom,
! [A2: a,B2: a,C: a] :
( ( less @ A2 @ ( min @ a @ less_eq @ B2 @ C ) )
=> ~ ( ( less @ A2 @ B2 )
=> ~ ( less @ A2 @ C ) ) ) ).
% local.min.strict_boundedE
thf(fact_118_local_Omin_Oright__idem,axiom,
! [A2: a,B2: a] :
( ( min @ a @ less_eq @ ( min @ a @ less_eq @ A2 @ B2 ) @ B2 )
= ( min @ a @ less_eq @ A2 @ B2 ) ) ).
% local.min.right_idem
thf(fact_119_local_Omin_Oleft__idem,axiom,
! [A2: a,B2: a] :
( ( min @ a @ less_eq @ A2 @ ( min @ a @ less_eq @ A2 @ B2 ) )
= ( min @ a @ less_eq @ A2 @ B2 ) ) ).
% local.min.left_idem
thf(fact_120_local_Omin_Oidem,axiom,
! [A2: a] :
( ( min @ a @ less_eq @ A2 @ A2 )
= A2 ) ).
% local.min.idem
thf(fact_121_local_Omin_Obounded__iff,axiom,
! [A2: a,B2: a,C: a] :
( ( less_eq @ A2 @ ( min @ a @ less_eq @ B2 @ C ) )
= ( ( less_eq @ A2 @ B2 )
& ( less_eq @ A2 @ C ) ) ) ).
% local.min.bounded_iff
thf(fact_122_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_123_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_124_local_Omin__less__iff__conj,axiom,
! [Z: a,X: a,Y: a] :
( ( less @ Z @ ( min @ a @ less_eq @ X @ Y ) )
= ( ( less @ Z @ X )
& ( less @ Z @ Y ) ) ) ).
% local.min_less_iff_conj
thf(fact_125_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_126_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_127_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_128_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_129_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_130_local_OMin_Osemilattice__order__set__axioms,axiom,
lattic1693879045er_set @ a @ ( min @ a @ less_eq ) @ less_eq @ less ).
% local.Min.semilattice_order_set_axioms
thf(fact_131_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_132_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_133_step_OIH,axiom,
! [Ys: coinductive_llist @ a] :
( ( coinductive_lsorted @ a @ less_eq @ xsa )
=> ( ~ ( coinductive_lfinite @ a @ Ys )
=> ( ? [X4: a] :
( ( member @ a @ X4 @ ( coinductive_lset @ a @ Ys ) )
& ( less_eq @ x @ X4 ) )
=> ( member @ a @ x @ ( coinductive_lset @ a @ ( hammin1328233080lmerge @ a @ less @ xsa @ Ys ) ) ) ) ) ) ).
% step.IH
thf(fact_134_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_135_local_Oinsort__left__comm,axiom,
! [X: a,Y: a,Xs: list @ a] :
( ( insort_key @ a @ a @ less_eq
@ ^ [X2: a] : X2
@ X
@ ( insort_key @ a @ a @ less_eq
@ ^ [X2: a] : X2
@ Y
@ Xs ) )
= ( insort_key @ a @ a @ less_eq
@ ^ [X2: a] : X2
@ Y
@ ( insort_key @ a @ a @ less_eq
@ ^ [X2: a] : X2
@ X
@ Xs ) ) ) ).
% local.insort_left_comm
thf(fact_136_step_Oprems_I1_J,axiom,
coinductive_lsorted @ a @ less_eq @ ( coinductive_LCons @ a @ x2 @ xsa ) ).
% step.prems(1)
thf(fact_137_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_138_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_139_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_140_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_141_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_142_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_143_lset__intros_I2_J,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A,X5: A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
=> ( member @ A @ X @ ( coinductive_lset @ A @ ( coinductive_LCons @ A @ X5 @ Xs ) ) ) ) ).
% lset_intros(2)
thf(fact_144_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_145_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_146_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_147_lset__cases,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
=> ( ! [Xs2: coinductive_llist @ A] :
( Xs
!= ( coinductive_LCons @ A @ X @ Xs2 ) )
=> ~ ! [X6: A,Xs2: coinductive_llist @ A] :
( ( Xs
= ( coinductive_LCons @ A @ X6 @ Xs2 ) )
=> ~ ( member @ A @ X @ ( coinductive_lset @ A @ Xs2 ) ) ) ) ) ).
% lset_cases
thf(fact_148_lset__induct,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
=> ( ! [Xs3: coinductive_llist @ A] : ( P @ ( coinductive_LCons @ A @ X @ Xs3 ) )
=> ( ! [X6: A,Xs3: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs3 ) )
=> ( ( X != X6 )
=> ( ( P @ Xs3 )
=> ( P @ ( coinductive_LCons @ A @ X6 @ Xs3 ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% lset_induct
thf(fact_149_lset__induct_H,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs ) )
=> ( ! [Xs3: coinductive_llist @ A] : ( P @ ( coinductive_LCons @ A @ X @ Xs3 ) )
=> ( ! [X6: A,Xs3: coinductive_llist @ A] :
( ( member @ A @ X @ ( coinductive_lset @ A @ Xs3 ) )
=> ( ( P @ Xs3 )
=> ( P @ ( coinductive_LCons @ A @ X6 @ Xs3 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% lset_induct'
thf(fact_150_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_151_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,Xa: A] :
( ( member @ A @ Xa @ ( coinductive_lset @ A @ Z22 ) )
=> ( ( P @ Xa @ Z22 )
=> ( P @ Xa @ ( coinductive_LCons @ A @ Z1 @ Z22 ) ) ) )
=> ( P @ X @ A2 ) ) ) ) ).
% llist.set_induct
thf(fact_152_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_153_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_154_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_155_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_156_local_Osorted__list__of__set_Ofolding__axioms,axiom,
( finite_folding @ a @ ( list @ a )
@ ( insort_key @ a @ a @ less_eq
@ ^ [X2: a] : X2 ) ) ).
% local.sorted_list_of_set.folding_axioms
thf(fact_157_local_OatLeast__def,axiom,
! [L: a] :
( ( set_atLeast @ a @ less_eq @ L )
= ( collect @ a @ ( less_eq @ L ) ) ) ).
% local.atLeast_def
thf(fact_158_local_Odual__order_Oordering__axioms,axiom,
( ordering @ a
@ ^ [X2: a,Y3: a] : ( less_eq @ Y3 @ X2 )
@ ^ [X2: a,Y3: a] : ( less @ Y3 @ X2 ) ) ).
% local.dual_order.ordering_axioms
thf(fact_159_local_OatMost__def,axiom,
! [U: a] :
( ( set_atMost @ a @ less_eq @ U )
= ( collect @ a
@ ^ [X2: a] : ( less_eq @ X2 @ U ) ) ) ).
% local.atMost_def
thf(fact_160_local_Oorder_Oordering__axioms,axiom,
ordering @ a @ less_eq @ less ).
% local.order.ordering_axioms
thf(fact_161_local_OatMost__iff,axiom,
! [I: a,K: a] :
( ( member @ a @ I @ ( set_atMost @ a @ less_eq @ K ) )
= ( less_eq @ I @ K ) ) ).
% local.atMost_iff
thf(fact_162_local_OatLeast__iff,axiom,
! [I: a,K: a] :
( ( member @ a @ I @ ( set_atLeast @ a @ less_eq @ K ) )
= ( less_eq @ K @ I ) ) ).
% local.atLeast_iff
thf(fact_163_local_Obdd__above__Iic,axiom,
! [B2: a] : ( condit2040224947_above @ a @ less_eq @ ( set_atMost @ a @ less_eq @ B2 ) ) ).
% local.bdd_above_Iic
thf(fact_164_local_Obdd__below__Ici,axiom,
! [A2: a] : ( condit1201339847_below @ a @ less_eq @ ( set_atLeast @ a @ less_eq @ A2 ) ) ).
% local.bdd_below_Ici
thf(fact_165_local_OgreaterThan__def,axiom,
! [L: a] :
( ( set_greaterThan @ a @ less @ L )
= ( collect @ a @ ( less @ L ) ) ) ).
% local.greaterThan_def
thf(fact_166_local_Omin_Osemilattice__axioms,axiom,
semilattice @ a @ ( min @ a @ less_eq ) ).
% local.min.semilattice_axioms
thf(fact_167_local_Omin_Osemigroup__axioms,axiom,
semigroup @ a @ ( min @ a @ less_eq ) ).
% local.min.semigroup_axioms
thf(fact_168_local_OgreaterThan__iff,axiom,
! [I: a,K: a] :
( ( member @ a @ I @ ( set_greaterThan @ a @ less @ K ) )
= ( less @ K @ I ) ) ).
% local.greaterThan_iff
thf(fact_169_local_Obdd__below__Ioi,axiom,
! [A2: a] : ( condit1201339847_below @ a @ less_eq @ ( set_greaterThan @ a @ less @ A2 ) ) ).
% local.bdd_below_Ioi
thf(fact_170_local_OgreaterThanAtMost__def,axiom,
! [L: a,U: a] :
( ( set_gr323396891AtMost @ a @ less_eq @ less @ L @ U )
= ( inf_inf @ ( set @ a ) @ ( set_greaterThan @ a @ less @ L ) @ ( set_atMost @ a @ less_eq @ U ) ) ) ).
% local.greaterThanAtMost_def
thf(fact_171_local_OIci__subset__Ioi__iff,axiom,
! [A2: a,B2: a] :
( ( ord_less_eq @ ( set @ a ) @ ( set_atLeast @ a @ less_eq @ A2 ) @ ( set_greaterThan @ a @ less @ B2 ) )
= ( less @ B2 @ A2 ) ) ).
% local.Ici_subset_Ioi_iff
thf(fact_172_local_OIoi__le__Ico,axiom,
! [A2: a] : ( ord_less_eq @ ( set @ a ) @ ( set_greaterThan @ a @ less @ A2 ) @ ( set_atLeast @ a @ less_eq @ A2 ) ) ).
% local.Ioi_le_Ico
thf(fact_173_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_174_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_175_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_176_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_177_local_Obdd__below__Int2,axiom,
! [B4: set @ a,A4: set @ a] :
( ( condit1201339847_below @ a @ less_eq @ B4 )
=> ( condit1201339847_below @ a @ less_eq @ ( inf_inf @ ( set @ a ) @ A4 @ B4 ) ) ) ).
% local.bdd_below_Int2
thf(fact_178_local_Obdd__below__Int1,axiom,
! [A4: set @ a,B4: set @ a] :
( ( condit1201339847_below @ a @ less_eq @ A4 )
=> ( condit1201339847_below @ a @ less_eq @ ( inf_inf @ ( set @ a ) @ A4 @ B4 ) ) ) ).
% local.bdd_below_Int1
thf(fact_179_local_Obdd__above__Int2,axiom,
! [B4: set @ a,A4: set @ a] :
( ( condit2040224947_above @ a @ less_eq @ B4 )
=> ( condit2040224947_above @ a @ less_eq @ ( inf_inf @ ( set @ a ) @ A4 @ B4 ) ) ) ).
% local.bdd_above_Int2
thf(fact_180_local_Obdd__above__Int1,axiom,
! [A4: set @ a,B4: set @ a] :
( ( condit2040224947_above @ a @ less_eq @ A4 )
=> ( condit2040224947_above @ a @ less_eq @ ( inf_inf @ ( set @ a ) @ A4 @ B4 ) ) ) ).
% local.bdd_above_Int1
thf(fact_181_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_182_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_183_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_184_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_185_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_186_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_187_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_188_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_189_local_OInt__atMost,axiom,
! [A2: a,B2: a] :
( ( inf_inf @ ( set @ a ) @ ( set_atMost @ a @ less_eq @ A2 ) @ ( set_atMost @ a @ less_eq @ B2 ) )
= ( set_atMost @ a @ less_eq @ ( min @ a @ less_eq @ A2 @ B2 ) ) ) ).
% local.Int_atMost
thf(fact_190_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_191_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_192_local_OIic__subset__Iio__iff,axiom,
! [A2: a,B2: a] :
( ( ord_less_eq @ ( set @ a ) @ ( set_atMost @ a @ less_eq @ A2 ) @ ( set_lessThan @ a @ less @ B2 ) )
= ( less @ A2 @ B2 ) ) ).
% local.Iic_subset_Iio_iff
thf(fact_193_local_OgreaterThan__Int__greaterThan,axiom,
! [A2: a,B2: a] :
( ( inf_inf @ ( set @ a ) @ ( set_lessThan @ a @ less @ A2 ) @ ( set_lessThan @ a @ less @ B2 ) )
= ( set_lessThan @ a @ less @ ( min @ a @ less_eq @ A2 @ B2 ) ) ) ).
% local.greaterThan_Int_greaterThan
thf(fact_194_local_OlessThan__def,axiom,
! [U: a] :
( ( set_lessThan @ a @ less @ U )
= ( collect @ a
@ ^ [X2: a] : ( less @ X2 @ U ) ) ) ).
% local.lessThan_def
thf(fact_195_local_OatLeastLessThan__def,axiom,
! [L: a,U: a] :
( ( set_atLeastLessThan @ a @ less_eq @ less @ L @ U )
= ( inf_inf @ ( set @ a ) @ ( set_atLeast @ a @ less_eq @ L ) @ ( set_lessThan @ a @ less @ U ) ) ) ).
% local.atLeastLessThan_def
thf(fact_196_local_OlessThan__iff,axiom,
! [I: a,K: a] :
( ( member @ a @ I @ ( set_lessThan @ a @ less @ K ) )
= ( less @ I @ K ) ) ).
% local.lessThan_iff
thf(fact_197_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_198_local_Obdd__above__Iio,axiom,
! [B2: a] : ( condit2040224947_above @ a @ less_eq @ ( set_lessThan @ a @ less @ B2 ) ) ).
% local.bdd_above_Iio
thf(fact_199_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_200_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_201_local_OgreaterThanLessThan__eq,axiom,
! [A2: a,B2: a] :
( ( set_gr1161524159ssThan @ a @ less @ A2 @ B2 )
= ( inf_inf @ ( set @ a ) @ ( set_greaterThan @ a @ less @ A2 ) @ ( set_lessThan @ a @ less @ B2 ) ) ) ).
% local.greaterThanLessThan_eq
thf(fact_202_local_OgreaterThanLessThan__def,axiom,
! [L: a,U: a] :
( ( set_gr1161524159ssThan @ a @ less @ L @ U )
= ( inf_inf @ ( set @ a ) @ ( set_greaterThan @ a @ less @ L ) @ ( set_lessThan @ a @ less @ U ) ) ) ).
% local.greaterThanLessThan_def
thf(fact_203_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_204_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_205_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_206_local_OatLeastAtMost__def,axiom,
! [L: a,U: a] :
( ( set_atLeastAtMost @ a @ less_eq @ L @ U )
= ( inf_inf @ ( set @ a ) @ ( set_atLeast @ a @ less_eq @ L ) @ ( set_atMost @ a @ less_eq @ U ) ) ) ).
% local.atLeastAtMost_def
thf(fact_207_Int__subset__iff,axiom,
! [A: $tType,C2: set @ A,A4: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C2 @ ( inf_inf @ ( set @ A ) @ A4 @ B4 ) )
= ( ( ord_less_eq @ ( set @ A ) @ C2 @ A4 )
& ( ord_less_eq @ ( set @ A ) @ C2 @ B4 ) ) ) ).
% Int_subset_iff
thf(fact_208_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_209_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_210_Int__iff,axiom,
! [A: $tType,C: A,A4: set @ A,B4: set @ A] :
( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A4 @ B4 ) )
= ( ( member @ A @ C @ A4 )
& ( member @ A @ C @ B4 ) ) ) ).
% Int_iff
thf(fact_211_IntI,axiom,
! [A: $tType,C: A,A4: set @ A,B4: set @ A] :
( ( member @ A @ C @ A4 )
=> ( ( member @ A @ C @ B4 )
=> ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A4 @ B4 ) ) ) ) ).
% IntI
thf(fact_212_local_OatLeastAtMost__iff,axiom,
! [I: a,L: a,U: a] :
( ( member @ a @ I @ ( set_atLeastAtMost @ a @ less_eq @ L @ U ) )
= ( ( less_eq @ L @ I )
& ( less_eq @ I @ U ) ) ) ).
% local.atLeastAtMost_iff
thf(fact_213_local_OIcc__eq__Icc,axiom,
! [L: a,H: a,L2: a,H2: a] :
( ( ( set_atLeastAtMost @ a @ less_eq @ L @ H )
= ( set_atLeastAtMost @ a @ less_eq @ L2 @ H2 ) )
= ( ( ( L = L2 )
& ( H = H2 ) )
| ( ~ ( less_eq @ L @ H )
& ~ ( less_eq @ L2 @ H2 ) ) ) ) ).
% local.Icc_eq_Icc
thf(fact_214_local_OatLeastatMost__subset__iff,axiom,
! [A2: a,B2: a,C: a,D: a] :
( ( ord_less_eq @ ( set @ a ) @ ( set_atLeastAtMost @ a @ less_eq @ A2 @ B2 ) @ ( set_atLeastAtMost @ a @ less_eq @ C @ D ) )
= ( ~ ( less_eq @ A2 @ B2 )
| ( ( less_eq @ C @ A2 )
& ( less_eq @ B2 @ D ) ) ) ) ).
% local.atLeastatMost_subset_iff
thf(fact_215_local_Obdd__above__Icc,axiom,
! [A2: a,B2: a] : ( condit2040224947_above @ a @ less_eq @ ( set_atLeastAtMost @ a @ less_eq @ A2 @ B2 ) ) ).
% local.bdd_above_Icc
thf(fact_216_local_Obdd__below__Icc,axiom,
! [A2: a,B2: a] : ( condit1201339847_below @ a @ less_eq @ ( set_atLeastAtMost @ a @ less_eq @ A2 @ B2 ) ) ).
% local.bdd_below_Icc
thf(fact_217_local_OIcc__subset__Iic__iff,axiom,
! [L: a,H: a,H2: a] :
( ( ord_less_eq @ ( set @ a ) @ ( set_atLeastAtMost @ a @ less_eq @ L @ H ) @ ( set_atMost @ a @ less_eq @ H2 ) )
= ( ~ ( less_eq @ L @ H )
| ( less_eq @ H @ H2 ) ) ) ).
% local.Icc_subset_Iic_iff
thf(fact_218_local_OIcc__subset__Ici__iff,axiom,
! [L: a,H: a,L2: a] :
( ( ord_less_eq @ ( set @ a ) @ ( set_atLeastAtMost @ a @ less_eq @ L @ H ) @ ( set_atLeast @ a @ less_eq @ L2 ) )
= ( ~ ( less_eq @ L @ H )
| ( less_eq @ L2 @ L ) ) ) ).
% local.Icc_subset_Ici_iff
thf(fact_219_local_OInt__atLeastAtMostL1,axiom,
! [A2: a,B2: a,D: a] :
( ( inf_inf @ ( set @ a ) @ ( set_atLeastAtMost @ a @ less_eq @ A2 @ B2 ) @ ( set_atMost @ a @ less_eq @ D ) )
= ( set_atLeastAtMost @ a @ less_eq @ A2 @ ( min @ a @ less_eq @ B2 @ D ) ) ) ).
% local.Int_atLeastAtMostL1
thf(fact_220_local_OInt__atLeastAtMostR1,axiom,
! [B2: a,C: a,D: a] :
( ( inf_inf @ ( set @ a ) @ ( set_atMost @ a @ less_eq @ B2 ) @ ( set_atLeastAtMost @ a @ less_eq @ C @ D ) )
= ( set_atLeastAtMost @ a @ less_eq @ C @ ( min @ a @ less_eq @ B2 @ D ) ) ) ).
% local.Int_atLeastAtMostR1
thf(fact_221_Collect__mono__iff,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) )
= ( ! [X2: A] :
( ( P @ X2 )
=> ( Q @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_222_contra__subsetD,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,C: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ~ ( member @ A @ C @ B4 )
=> ~ ( member @ A @ C @ A4 ) ) ) ).
% contra_subsetD
thf(fact_223_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y2: set @ A,Z2: set @ A] : Y2 = Z2 )
= ( ^ [A5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B5 )
& ( ord_less_eq @ ( set @ A ) @ B5 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_224_subset__trans,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,C2: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ( ord_less_eq @ ( set @ A ) @ B4 @ C2 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ C2 ) ) ) ).
% subset_trans
thf(fact_225_Collect__mono,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_mono
thf(fact_226_subset__refl,axiom,
! [A: $tType,A4: set @ A] : ( ord_less_eq @ ( set @ A ) @ A4 @ A4 ) ).
% subset_refl
thf(fact_227_rev__subsetD,axiom,
! [A: $tType,C: A,A4: set @ A,B4: set @ A] :
( ( member @ A @ C @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( member @ A @ C @ B4 ) ) ) ).
% rev_subsetD
thf(fact_228_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A5: set @ A,B5: set @ A] :
! [T: A] :
( ( member @ A @ T @ A5 )
=> ( member @ A @ T @ B5 ) ) ) ) ).
% subset_iff
thf(fact_229_set__rev__mp,axiom,
! [A: $tType,X: A,A4: set @ A,B4: set @ A] :
( ( member @ A @ X @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( member @ A @ X @ B4 ) ) ) ).
% set_rev_mp
thf(fact_230_equalityD2,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( A4 = B4 )
=> ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ).
% equalityD2
thf(fact_231_equalityD1,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( A4 = B4 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B4 ) ) ).
% equalityD1
thf(fact_232_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A5: set @ A,B5: set @ A] :
! [X2: A] :
( ( member @ A @ X2 @ A5 )
=> ( member @ A @ X2 @ B5 ) ) ) ) ).
% subset_eq
thf(fact_233_equalityE,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( A4 = B4 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B4 @ A4 ) ) ) ).
% equalityE
thf(fact_234_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
thf(fact_235_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_236_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_237_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_238_Int__left__commute,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,C2: set @ A] :
( ( inf_inf @ ( set @ A ) @ A4 @ ( inf_inf @ ( set @ A ) @ B4 @ C2 ) )
= ( inf_inf @ ( set @ A ) @ B4 @ ( inf_inf @ ( set @ A ) @ A4 @ C2 ) ) ) ).
% Int_left_commute
thf(fact_239_Int__left__absorb,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( inf_inf @ ( set @ A ) @ A4 @ ( inf_inf @ ( set @ A ) @ A4 @ B4 ) )
= ( inf_inf @ ( set @ A ) @ A4 @ B4 ) ) ).
% Int_left_absorb
thf(fact_240_Int__commute,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A5: set @ A,B5: set @ A] : ( inf_inf @ ( set @ A ) @ B5 @ A5 ) ) ) ).
% Int_commute
thf(fact_241_Int__absorb,axiom,
! [A: $tType,A4: set @ A] :
( ( inf_inf @ ( set @ A ) @ A4 @ A4 )
= A4 ) ).
% Int_absorb
thf(fact_242_Int__assoc,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,C2: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B4 ) @ C2 )
= ( inf_inf @ ( set @ A ) @ A4 @ ( inf_inf @ ( set @ A ) @ B4 @ C2 ) ) ) ).
% Int_assoc
thf(fact_243_IntD2,axiom,
! [A: $tType,C: A,A4: set @ A,B4: set @ A] :
( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A4 @ B4 ) )
=> ( member @ A @ C @ B4 ) ) ).
% IntD2
thf(fact_244_IntD1,axiom,
! [A: $tType,C: A,A4: set @ A,B4: set @ A] :
( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A4 @ B4 ) )
=> ( member @ A @ C @ A4 ) ) ).
% IntD1
thf(fact_245_IntE,axiom,
! [A: $tType,C: A,A4: set @ A,B4: set @ A] :
( ( member @ A @ C @ ( inf_inf @ ( set @ A ) @ A4 @ B4 ) )
=> ~ ( ( member @ A @ C @ A4 )
=> ~ ( member @ A @ C @ B4 ) ) ) ).
% IntE
thf(fact_246_less__eq__set__def,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A5: set @ A,B5: set @ A] :
( ord_less_eq @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ A5 )
@ ^ [X2: A] : ( member @ A @ X2 @ B5 ) ) ) ) ).
% less_eq_set_def
thf(fact_247_Int__def,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A5: set @ A,B5: set @ A] :
( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ A5 )
& ( member @ A @ X2 @ B5 ) ) ) ) ) ).
% Int_def
thf(fact_248_Int__Collect,axiom,
! [A: $tType,X: A,A4: set @ A,P: A > $o] :
( ( member @ A @ X @ ( inf_inf @ ( set @ A ) @ A4 @ ( collect @ A @ P ) ) )
= ( ( member @ A @ X @ A4 )
& ( P @ X ) ) ) ).
% Int_Collect
thf(fact_249_Collect__conj__eq,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( collect @ A
@ ^ [X2: A] :
( ( P @ X2 )
& ( Q @ X2 ) ) )
= ( inf_inf @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_conj_eq
thf(fact_250_Int__Collect__mono,axiom,
! [A: $tType,A4: set @ A,B4: set @ A,P: A > $o,Q: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( ( P @ X3 )
=> ( Q @ X3 ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ ( collect @ A @ P ) ) @ ( inf_inf @ ( set @ A ) @ B4 @ ( collect @ A @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_251_Int__greatest,axiom,
! [A: $tType,C2: set @ A,A4: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C2 @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ C2 @ B4 )
=> ( ord_less_eq @ ( set @ A ) @ C2 @ ( inf_inf @ ( set @ A ) @ A4 @ B4 ) ) ) ) ).
% Int_greatest
thf(fact_252_Int__absorb2,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B4 )
=> ( ( inf_inf @ ( set @ A ) @ A4 @ B4 )
= A4 ) ) ).
% Int_absorb2
thf(fact_253_Int__absorb1,axiom,
! [A: $tType,B4: set @ A,A4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B4 @ A4 )
=> ( ( inf_inf @ ( set @ A ) @ A4 @ B4 )
= B4 ) ) ).
% Int_absorb1
thf(fact_254_Int__lower2,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B4 ) @ B4 ) ).
% Int_lower2
thf(fact_255_Int__lower1,axiom,
! [A: $tType,A4: set @ A,B4: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B4 ) @ A4 ) ).
% Int_lower1
%----Type constructors (4)
thf(tcon_fun___Orderings_Oorder,axiom,
! [A6: $tType,A7: $tType] :
( ( order @ A7 @ ( type2 @ A7 ) )
=> ( order @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_1,axiom,
! [A6: $tType] : ( order @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_2,axiom,
order @ $o @ ( type2 @ $o ) ).
%----Conjectures (2)
thf(conj_0,hypothesis,
! [Y5: a] :
( ( member @ a @ Y5 @ ( coinductive_lset @ a @ ysa ) )
=> ( ( less_eq @ x @ Y5 )
=> thesis ) ) ).
thf(conj_1,conjecture,
thesis ).
%------------------------------------------------------------------------------