TPTP Problem File: DAT230^1.p
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%------------------------------------------------------------------------------
% File : DAT230^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Red-black trees 205
% Version : [Bla16] axioms : Especial.
% English :
% Refs : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% Source : [Bla16]
% Names : rbt_impl__205.p [Bla16]
% Status : Theorem
% Rating : 0.33 v8.1.0, 0.25 v7.5.0, 0.00 v7.2.0, 0.25 v7.1.0
% Syntax : Number of formulae : 326 ( 115 unt; 60 typ; 0 def)
% Number of atoms : 850 ( 338 equ; 0 cnn)
% Maximal formula atoms : 13 ( 3 avg)
% Number of connectives : 4324 ( 159 ~; 29 |; 106 &;3713 @)
% ( 0 <=>; 317 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 8 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 239 ( 239 >; 0 *; 0 +; 0 <<)
% Number of symbols : 60 ( 57 usr; 7 con; 0-7 aty)
% Number of variables : 1179 ( 22 ^; 988 !; 107 ?;1179 :)
% ( 62 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:39:23.898
%------------------------------------------------------------------------------
%----Could-be-implicit typings (9)
thf(ty_t_RBT__Impl__Mirabelle__msmaddcmtr_Ocolor,type,
rBT_Im1923302023_color: $tType ).
thf(ty_t_RBT__Impl__Mirabelle__msmaddcmtr_Orbt,type,
rBT_Im246033960le_rbt: $tType > $tType > $tType ).
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_Option_Ooption,type,
option: $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_b,type,
b: $tType ).
thf(ty_tf_a,type,
a: $tType ).
%----Explicit typings (51)
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_c_Fun_Oinj__on,type,
inj_on:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_List_Oappend,type,
append:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Obind,type,
bind:
!>[A: $tType,B: $tType] : ( ( list @ A ) > ( A > ( list @ B ) ) > ( list @ B ) ) ).
thf(sy_c_List_Obutlast,type,
butlast:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Odistinct,type,
distinct:
!>[A: $tType] : ( ( list @ A ) > $o ) ).
thf(sy_c_List_Oinsert,type,
insert:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olast,type,
last:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_List_Olist_OCons,type,
cons:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist_ONil,type,
nil:
!>[A: $tType] : ( list @ A ) ).
thf(sy_c_List_Olist_Oset,type,
set2:
!>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
thf(sy_c_List_Omaps,type,
maps:
!>[A: $tType,B: $tType] : ( ( A > ( list @ B ) ) > ( list @ A ) > ( list @ B ) ) ).
thf(sy_c_List_Oord_Olexordp,type,
lexordp:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Oord_Olexordp__eq,type,
lexordp_eq:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Oproduct__lists,type,
product_lists:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Orotate1,type,
rotate1:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Osublists,type,
sublists:
!>[A: $tType] : ( ( list @ A ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Ounion,type,
union:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Map_Odom,type,
dom:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ A ) ) ).
thf(sy_c_Map_Omap__of,type,
map_of:
!>[A: $tType,B: $tType] : ( ( list @ ( product_prod @ A @ B ) ) > A > ( option @ B ) ) ).
thf(sy_c_Option_Ooption_ONone,type,
none:
!>[A: $tType] : ( option @ A ) ).
thf(sy_c_Option_Ooption_OSome,type,
some:
!>[A: $tType] : ( A > ( option @ A ) ) ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oorder_Ostrict__mono,type,
strict_mono:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > $o ) ).
thf(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Oentries,type,
rBT_Im954575269ntries:
!>[A: $tType,B: $tType] : ( ( rBT_Im246033960le_rbt @ A @ B ) > ( list @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Okeys,type,
rBT_Im380146495e_keys:
!>[A: $tType,B: $tType] : ( ( rBT_Im246033960le_rbt @ A @ B ) > ( list @ A ) ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Oord_Orbt__greater,type,
rBT_Im1259024060reater:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > A > ( rBT_Im246033960le_rbt @ A @ B ) > $o ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Oord_Orbt__less,type,
rBT_Im2075124343t_less:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > A > ( rBT_Im246033960le_rbt @ A @ B ) > $o ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Oord_Orbt__lookup,type,
rBT_Im35466040lookup:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( rBT_Im246033960le_rbt @ A @ B ) > A > ( option @ B ) ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Oord_Orbt__sorted,type,
rBT_Im759614907sorted:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( rBT_Im246033960le_rbt @ A @ B ) > $o ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Oord__class_Orbt__greater,type,
rBT_Im1332541288reater:
!>[A: $tType,B: $tType] : ( A > ( rBT_Im246033960le_rbt @ A @ B ) > $o ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Oord__class_Orbt__less,type,
rBT_Im903026891t_less:
!>[A: $tType,B: $tType] : ( A > ( rBT_Im246033960le_rbt @ A @ B ) > $o ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Oord__class_Orbt__sorted,type,
rBT_Im1183586831sorted:
!>[A: $tType,B: $tType] : ( ( rBT_Im246033960le_rbt @ A @ B ) > $o ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Orbt_OBranch,type,
rBT_Im480247531Branch:
!>[A: $tType,B: $tType] : ( rBT_Im1923302023_color > ( rBT_Im246033960le_rbt @ A @ B ) > A > B > ( rBT_Im246033960le_rbt @ A @ B ) > ( rBT_Im246033960le_rbt @ A @ B ) ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Orbt_OEmpty,type,
rBT_Im418718756_Empty:
!>[A: $tType,B: $tType] : ( rBT_Im246033960le_rbt @ A @ B ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_Othe__elem,type,
the_elem:
!>[A: $tType] : ( ( set @ A ) > 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_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_c____,type,
c: rBT_Im1923302023_color ).
thf(sy_v_k____,type,
k: a ).
thf(sy_v_less,type,
less: a > a > $o ).
thf(sy_v_t1____,type,
t1: rBT_Im246033960le_rbt @ a @ b ).
thf(sy_v_t2____,type,
t2: rBT_Im246033960le_rbt @ a @ b ).
thf(sy_v_v____,type,
v: b ).
%----Relevant facts (255)
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_local_Orbt__greater__prop,axiom,
! [B: $tType,K: a,T: rBT_Im246033960le_rbt @ a @ B] :
( ( rBT_Im1259024060reater @ a @ B @ less @ K @ T )
= ( ! [X2: a] :
( ( member @ a @ X2 @ ( set2 @ a @ ( rBT_Im380146495e_keys @ a @ B @ T ) ) )
=> ( less @ K @ X2 ) ) ) ) ).
% local.rbt_greater_prop
thf(fact_26_local_Orbt__greater__trans,axiom,
! [C2: $tType,X: a,Y: a,T: rBT_Im246033960le_rbt @ a @ C2] :
( ( less @ X @ Y )
=> ( ( rBT_Im1259024060reater @ a @ C2 @ less @ Y @ T )
=> ( rBT_Im1259024060reater @ a @ C2 @ less @ X @ T ) ) ) ).
% local.rbt_greater_trans
thf(fact_27__092_060open_062k_A_092_060guillemotleft_062_124_At2_092_060close_062,axiom,
rBT_Im1259024060reater @ a @ b @ less @ k @ t2 ).
% \<open>k \<guillemotleft>| t2\<close>
thf(fact_28_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_29_local_Orbt__less__prop,axiom,
! [B: $tType,K: a,T: rBT_Im246033960le_rbt @ a @ B] :
( ( rBT_Im2075124343t_less @ a @ B @ less @ K @ T )
= ( ! [X2: a] :
( ( member @ a @ X2 @ ( set2 @ a @ ( rBT_Im380146495e_keys @ a @ B @ T ) ) )
=> ( less @ X2 @ K ) ) ) ) ).
% local.rbt_less_prop
thf(fact_30_ord_Orbt__greater__prop,axiom,
! [B: $tType,A: $tType] :
( ( rBT_Im1259024060reater @ A @ B )
= ( ^ [Less: A > A > $o,K2: A,T2: rBT_Im246033960le_rbt @ A @ B] :
! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ ( rBT_Im380146495e_keys @ A @ B @ T2 ) ) )
=> ( Less @ K2 @ X2 ) ) ) ) ).
% ord.rbt_greater_prop
thf(fact_31_local_Orbt__less__trans,axiom,
! [C2: $tType,X: a,T: rBT_Im246033960le_rbt @ a @ C2,Y: a] :
( ( rBT_Im2075124343t_less @ a @ C2 @ less @ X @ T )
=> ( ( less @ X @ Y )
=> ( rBT_Im2075124343t_less @ a @ C2 @ less @ Y @ T ) ) ) ).
% local.rbt_less_trans
thf(fact_32_local_Olexordp__eq__antisym,axiom,
! [Xs: list @ a,Ys: list @ a] :
( ( lexordp_eq @ a @ less @ Xs @ Ys )
=> ( ( lexordp_eq @ a @ less @ Ys @ Xs )
=> ( Xs = Ys ) ) ) ).
% local.lexordp_eq_antisym
thf(fact_33_local_Olexordp__eq__linear,axiom,
! [Xs: list @ a,Ys: list @ a] :
( ( lexordp_eq @ a @ less @ Xs @ Ys )
| ( lexordp_eq @ a @ less @ Ys @ Xs ) ) ).
% local.lexordp_eq_linear
thf(fact_34_local_Olexordp__eq__refl,axiom,
! [Xs: list @ a] : ( lexordp_eq @ a @ less @ Xs @ Xs ) ).
% local.lexordp_eq_refl
thf(fact_35_local_Olexordp__eq__trans,axiom,
! [Xs: list @ a,Ys: list @ a,Zs: list @ a] :
( ( lexordp_eq @ a @ less @ Xs @ Ys )
=> ( ( lexordp_eq @ a @ less @ Ys @ Zs )
=> ( lexordp_eq @ a @ less @ Xs @ Zs ) ) ) ).
% local.lexordp_eq_trans
thf(fact_36_ord_Orbt__less__prop,axiom,
! [B: $tType,A: $tType] :
( ( rBT_Im2075124343t_less @ A @ B )
= ( ^ [Less: A > A > $o,K2: A,T2: rBT_Im246033960le_rbt @ A @ B] :
! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ ( rBT_Im380146495e_keys @ A @ B @ T2 ) ) )
=> ( Less @ X2 @ K2 ) ) ) ) ).
% ord.rbt_less_prop
thf(fact_37_local_Olexordp__antisym,axiom,
! [Xs: list @ a,Ys: list @ a] :
( ( lexordp @ a @ less @ Xs @ Ys )
=> ~ ( lexordp @ a @ less @ Ys @ Xs ) ) ).
% local.lexordp_antisym
thf(fact_38_local_Olexordp__irreflexive,axiom,
! [Xs: list @ a] :
( ! [X3: a] :
~ ( less @ X3 @ X3 )
=> ~ ( lexordp @ a @ less @ Xs @ Xs ) ) ).
% local.lexordp_irreflexive
thf(fact_39_local_Olexordp__irreflexive_H,axiom,
! [Xs: list @ a] :
~ ( lexordp @ a @ less @ Xs @ Xs ) ).
% local.lexordp_irreflexive'
thf(fact_40_local_Olexordp__linear,axiom,
! [Xs: list @ a,Ys: list @ a] :
( ( lexordp @ a @ less @ Xs @ Ys )
| ( Xs = Ys )
| ( lexordp @ a @ less @ Ys @ Xs ) ) ).
% local.lexordp_linear
thf(fact_41_local_Olexordp__trans,axiom,
! [Xs: list @ a,Ys: list @ a,Zs: list @ a] :
( ( lexordp @ a @ less @ Xs @ Ys )
=> ( ( lexordp @ a @ less @ Ys @ Zs )
=> ( lexordp @ a @ less @ Xs @ Zs ) ) ) ).
% local.lexordp_trans
thf(fact_42_local_Olexordp__into__lexordp__eq,axiom,
! [Xs: list @ a,Ys: list @ a] :
( ( lexordp @ a @ less @ Xs @ Ys )
=> ( lexordp_eq @ a @ less @ Xs @ Ys ) ) ).
% local.lexordp_into_lexordp_eq
thf(fact_43_local_Olexordp__eq__conv__lexord,axiom,
! [Xs: list @ a,Ys: list @ a] :
( ( lexordp_eq @ a @ less @ Xs @ Ys )
= ( ( Xs = Ys )
| ( lexordp @ a @ less @ Xs @ Ys ) ) ) ).
% local.lexordp_eq_conv_lexord
thf(fact_44_local_Olexordp__conv__lexordp__eq,axiom,
! [Xs: list @ a,Ys: list @ a] :
( ( lexordp @ a @ less @ Xs @ Ys )
= ( ( lexordp_eq @ a @ less @ Xs @ Ys )
& ~ ( lexordp_eq @ a @ less @ Ys @ Xs ) ) ) ).
% local.lexordp_conv_lexordp_eq
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,A3: set @ A] :
( ( collect @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A3 ) )
= A3 ) ).
% 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_Ostrict__mono__less,axiom,
! [B: $tType] :
( ( order @ B @ ( type2 @ B ) )
=> ! [F: a > B,X: a,Y: a] :
( ( strict_mono @ a @ B @ less @ F )
=> ( ( ord_less @ B @ ( F @ X ) @ ( F @ Y ) )
= ( less @ X @ Y ) ) ) ) ).
% local.strict_mono_less
thf(fact_50_local_Ostrict__mono__def,axiom,
! [B: $tType] :
( ( order @ B @ ( type2 @ B ) )
=> ! [F: a > B] :
( ( strict_mono @ a @ B @ less @ F )
= ( ! [X2: a,Y2: a] :
( ( less @ X2 @ Y2 )
=> ( ord_less @ B @ ( F @ X2 ) @ ( F @ Y2 ) ) ) ) ) ) ).
% local.strict_mono_def
thf(fact_51_local_Ostrict__monoI,axiom,
! [B: $tType] :
( ( order @ B @ ( type2 @ B ) )
=> ! [F: a > B] :
( ! [X3: a,Y3: a] :
( ( less @ X3 @ Y3 )
=> ( ord_less @ B @ ( F @ X3 ) @ ( F @ Y3 ) ) )
=> ( strict_mono @ a @ B @ less @ F ) ) ) ).
% local.strict_monoI
thf(fact_52_local_Ostrict__monoD,axiom,
! [B: $tType] :
( ( order @ B @ ( type2 @ B ) )
=> ! [F: a > B,X: a,Y: a] :
( ( strict_mono @ a @ B @ less @ F )
=> ( ( less @ X @ Y )
=> ( ord_less @ B @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ).
% local.strict_monoD
thf(fact_53_local_Olexordp__eq_ONil,axiom,
! [Ys: list @ a] : ( lexordp_eq @ a @ less @ ( nil @ a ) @ Ys ) ).
% local.lexordp_eq.Nil
thf(fact_54_ord_Olexordp__into__lexordp__eq,axiom,
! [A: $tType,Less2: A > A > $o,Xs: list @ A,Ys: list @ A] :
( ( lexordp @ A @ Less2 @ Xs @ Ys )
=> ( lexordp_eq @ A @ Less2 @ Xs @ Ys ) ) ).
% ord.lexordp_into_lexordp_eq
thf(fact_55_local_Olexordp__eq_OCons__eq,axiom,
! [X: a,Y: a,Xs: list @ a,Ys: list @ a] :
( ~ ( less @ X @ Y )
=> ( ~ ( less @ Y @ X )
=> ( ( lexordp_eq @ a @ less @ Xs @ Ys )
=> ( lexordp_eq @ a @ less @ ( cons @ a @ X @ Xs ) @ ( cons @ a @ Y @ Ys ) ) ) ) ) ).
% local.lexordp_eq.Cons_eq
thf(fact_56_local_Olexordp__eq_OCons,axiom,
! [X: a,Y: a,Xs: list @ a,Ys: list @ a] :
( ( less @ X @ Y )
=> ( lexordp_eq @ a @ less @ ( cons @ a @ X @ Xs ) @ ( cons @ a @ Y @ Ys ) ) ) ).
% local.lexordp_eq.Cons
thf(fact_57_local_Olexordp_OCons,axiom,
! [X: a,Y: a,Xs: list @ a,Ys: list @ a] :
( ( less @ X @ Y )
=> ( lexordp @ a @ less @ ( cons @ a @ X @ Xs ) @ ( cons @ a @ Y @ Ys ) ) ) ).
% local.lexordp.Cons
thf(fact_58_local_Olexordp_OCons__eq,axiom,
! [X: a,Y: a,Xs: list @ a,Ys: list @ a] :
( ~ ( less @ X @ Y )
=> ( ~ ( less @ Y @ X )
=> ( ( lexordp @ a @ less @ Xs @ Ys )
=> ( lexordp @ a @ less @ ( cons @ a @ X @ Xs ) @ ( cons @ a @ Y @ Ys ) ) ) ) ) ).
% local.lexordp.Cons_eq
thf(fact_59_list_Oinject,axiom,
! [A: $tType,X21: A,X22: list @ A,Y21: A,Y22: list @ A] :
( ( ( cons @ A @ X21 @ X22 )
= ( cons @ A @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_60_local_Olexordp_ONil,axiom,
! [Y: a,Ys: list @ a] : ( lexordp @ a @ less @ ( nil @ a ) @ ( cons @ a @ Y @ Ys ) ) ).
% local.lexordp.Nil
thf(fact_61_local_Olexordp_Ocases,axiom,
! [A1: list @ a,A22: list @ a] :
( ( lexordp @ a @ less @ A1 @ A22 )
=> ( ( ( A1
= ( nil @ a ) )
=> ! [Y3: a,Ys2: list @ a] :
( A22
!= ( cons @ a @ Y3 @ Ys2 ) ) )
=> ( ! [X3: a] :
( ? [Xs2: list @ a] :
( A1
= ( cons @ a @ X3 @ Xs2 ) )
=> ! [Y3: a] :
( ? [Ys2: list @ a] :
( A22
= ( cons @ a @ Y3 @ Ys2 ) )
=> ~ ( less @ X3 @ Y3 ) ) )
=> ~ ! [X3: a,Y3: a,Xs2: list @ a] :
( ( A1
= ( cons @ a @ X3 @ Xs2 ) )
=> ! [Ys2: list @ a] :
( ( A22
= ( cons @ a @ Y3 @ Ys2 ) )
=> ( ~ ( less @ X3 @ Y3 )
=> ( ~ ( less @ Y3 @ X3 )
=> ~ ( lexordp @ a @ less @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ).
% local.lexordp.cases
thf(fact_62_local_Olexordp_Oinducts,axiom,
! [X1: list @ a,X23: list @ a,P: ( list @ a ) > ( list @ a ) > $o] :
( ( lexordp @ a @ less @ X1 @ X23 )
=> ( ! [Y3: a,Ys2: list @ a] : ( P @ ( nil @ a ) @ ( cons @ a @ Y3 @ Ys2 ) )
=> ( ! [X3: a,Y3: a,Xs2: list @ a,Ys2: list @ a] :
( ( less @ X3 @ Y3 )
=> ( P @ ( cons @ a @ X3 @ Xs2 ) @ ( cons @ a @ Y3 @ Ys2 ) ) )
=> ( ! [X3: a,Y3: a,Xs2: list @ a,Ys2: list @ a] :
( ~ ( less @ X3 @ Y3 )
=> ( ~ ( less @ Y3 @ X3 )
=> ( ( lexordp @ a @ less @ Xs2 @ Ys2 )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons @ a @ X3 @ Xs2 ) @ ( cons @ a @ Y3 @ Ys2 ) ) ) ) ) )
=> ( P @ X1 @ X23 ) ) ) ) ) ).
% local.lexordp.inducts
thf(fact_63_local_Olexordp_Osimps,axiom,
! [A1: list @ a,A22: list @ a] :
( ( lexordp @ a @ less @ A1 @ A22 )
= ( ? [Y2: a,Ys3: list @ a] :
( ( A1
= ( nil @ a ) )
& ( A22
= ( cons @ a @ Y2 @ Ys3 ) ) )
| ? [X2: a,Y2: a,Xs3: list @ a,Ys3: list @ a] :
( ( A1
= ( cons @ a @ X2 @ Xs3 ) )
& ( A22
= ( cons @ a @ Y2 @ Ys3 ) )
& ( less @ X2 @ Y2 ) )
| ? [X2: a,Y2: a,Xs3: list @ a,Ys3: list @ a] :
( ( A1
= ( cons @ a @ X2 @ Xs3 ) )
& ( A22
= ( cons @ a @ Y2 @ Ys3 ) )
& ~ ( less @ X2 @ Y2 )
& ~ ( less @ Y2 @ X2 )
& ( lexordp @ a @ less @ Xs3 @ Ys3 ) ) ) ) ).
% local.lexordp.simps
thf(fact_64_local_Olexordp__cases,axiom,
! [Xs: list @ a,Ys: list @ a] :
( ( lexordp @ a @ less @ Xs @ Ys )
=> ( ( ( Xs
= ( nil @ a ) )
=> ! [Y3: a,Ys4: list @ a] :
( Ys
!= ( cons @ a @ Y3 @ Ys4 ) ) )
=> ( ! [X3: a] :
( ? [Xs4: list @ a] :
( Xs
= ( cons @ a @ X3 @ Xs4 ) )
=> ! [Y3: a] :
( ? [Ys4: list @ a] :
( Ys
= ( cons @ a @ Y3 @ Ys4 ) )
=> ~ ( less @ X3 @ Y3 ) ) )
=> ~ ! [X3: a,Xs4: list @ a] :
( ( Xs
= ( cons @ a @ X3 @ Xs4 ) )
=> ! [Ys4: list @ a] :
( ( Ys
= ( cons @ a @ X3 @ Ys4 ) )
=> ~ ( lexordp @ a @ less @ Xs4 @ Ys4 ) ) ) ) ) ) ).
% local.lexordp_cases
thf(fact_65_local_Olexordp__induct,axiom,
! [Xs: list @ a,Ys: list @ a,P: ( list @ a ) > ( list @ a ) > $o] :
( ( lexordp @ a @ less @ Xs @ Ys )
=> ( ! [Y3: a,Ys2: list @ a] : ( P @ ( nil @ a ) @ ( cons @ a @ Y3 @ Ys2 ) )
=> ( ! [X3: a,Xs2: list @ a,Y3: a,Ys2: list @ a] :
( ( less @ X3 @ Y3 )
=> ( P @ ( cons @ a @ X3 @ Xs2 ) @ ( cons @ a @ Y3 @ Ys2 ) ) )
=> ( ! [X3: a,Xs2: list @ a,Ys2: list @ a] :
( ( lexordp @ a @ less @ Xs2 @ Ys2 )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons @ a @ X3 @ Xs2 ) @ ( cons @ a @ X3 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% local.lexordp_induct
thf(fact_66_local_Olexordp__eq_Ocases,axiom,
! [A1: list @ a,A22: list @ a] :
( ( lexordp_eq @ a @ less @ A1 @ A22 )
=> ( ( ( A1
= ( nil @ a ) )
=> ! [Ys2: list @ a] : A22 != Ys2 )
=> ( ! [X3: a] :
( ? [Xs2: list @ a] :
( A1
= ( cons @ a @ X3 @ Xs2 ) )
=> ! [Y3: a] :
( ? [Ys2: list @ a] :
( A22
= ( cons @ a @ Y3 @ Ys2 ) )
=> ~ ( less @ X3 @ Y3 ) ) )
=> ~ ! [X3: a,Y3: a,Xs2: list @ a] :
( ( A1
= ( cons @ a @ X3 @ Xs2 ) )
=> ! [Ys2: list @ a] :
( ( A22
= ( cons @ a @ Y3 @ Ys2 ) )
=> ( ~ ( less @ X3 @ Y3 )
=> ( ~ ( less @ Y3 @ X3 )
=> ~ ( lexordp_eq @ a @ less @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ).
% local.lexordp_eq.cases
thf(fact_67_local_Olexordp__eq_Oinducts,axiom,
! [X1: list @ a,X23: list @ a,P: ( list @ a ) > ( list @ a ) > $o] :
( ( lexordp_eq @ a @ less @ X1 @ X23 )
=> ( ! [X12: list @ a] : ( P @ ( nil @ a ) @ X12 )
=> ( ! [X3: a,Y3: a,Xs2: list @ a,Ys2: list @ a] :
( ( less @ X3 @ Y3 )
=> ( P @ ( cons @ a @ X3 @ Xs2 ) @ ( cons @ a @ Y3 @ Ys2 ) ) )
=> ( ! [X3: a,Y3: a,Xs2: list @ a,Ys2: list @ a] :
( ~ ( less @ X3 @ Y3 )
=> ( ~ ( less @ Y3 @ X3 )
=> ( ( lexordp_eq @ a @ less @ Xs2 @ Ys2 )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons @ a @ X3 @ Xs2 ) @ ( cons @ a @ Y3 @ Ys2 ) ) ) ) ) )
=> ( P @ X1 @ X23 ) ) ) ) ) ).
% local.lexordp_eq.inducts
thf(fact_68_local_Olexordp__eq_Osimps,axiom,
! [A1: list @ a,A22: list @ a] :
( ( lexordp_eq @ a @ less @ A1 @ A22 )
= ( ? [Ys3: list @ a] :
( ( A1
= ( nil @ a ) )
& ( A22 = Ys3 ) )
| ? [X2: a,Y2: a,Xs3: list @ a,Ys3: list @ a] :
( ( A1
= ( cons @ a @ X2 @ Xs3 ) )
& ( A22
= ( cons @ a @ Y2 @ Ys3 ) )
& ( less @ X2 @ Y2 ) )
| ? [X2: a,Y2: a,Xs3: list @ a,Ys3: list @ a] :
( ( A1
= ( cons @ a @ X2 @ Xs3 ) )
& ( A22
= ( cons @ a @ Y2 @ Ys3 ) )
& ~ ( less @ X2 @ Y2 )
& ~ ( less @ Y2 @ X2 )
& ( lexordp_eq @ a @ less @ Xs3 @ Ys3 ) ) ) ) ).
% local.lexordp_eq.simps
thf(fact_69_ord_Olexordp__simps_I3_J,axiom,
! [A: $tType,Less2: A > A > $o,X: A,Xs: list @ A,Y: A,Ys: list @ A] :
( ( lexordp @ A @ Less2 @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) )
= ( ( Less2 @ X @ Y )
| ( ~ ( Less2 @ Y @ X )
& ( lexordp @ A @ Less2 @ Xs @ Ys ) ) ) ) ).
% ord.lexordp_simps(3)
thf(fact_70_ord_Olexordp__simps_I1_J,axiom,
! [A: $tType,Less2: A > A > $o,Ys: list @ A] :
( ( lexordp @ A @ Less2 @ ( nil @ A ) @ Ys )
= ( Ys
!= ( nil @ A ) ) ) ).
% ord.lexordp_simps(1)
thf(fact_71_ord_Olexordp__simps_I2_J,axiom,
! [A: $tType,Less2: A > A > $o,Xs: list @ A] :
~ ( lexordp @ A @ Less2 @ Xs @ ( nil @ A ) ) ).
% ord.lexordp_simps(2)
thf(fact_72_ord_Olexordp__eq__simps_I4_J,axiom,
! [A: $tType,Less2: A > A > $o,X: A,Xs: list @ A,Y: A,Ys: list @ A] :
( ( lexordp_eq @ A @ Less2 @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) )
= ( ( Less2 @ X @ Y )
| ( ~ ( Less2 @ Y @ X )
& ( lexordp_eq @ A @ Less2 @ Xs @ Ys ) ) ) ) ).
% ord.lexordp_eq_simps(4)
thf(fact_73_ord_Olexordp__eq__simps_I1_J,axiom,
! [A: $tType,Less2: A > A > $o,Ys: list @ A] : ( lexordp_eq @ A @ Less2 @ ( nil @ A ) @ Ys ) ).
% ord.lexordp_eq_simps(1)
thf(fact_74_ord_Olexordp__eq__simps_I2_J,axiom,
! [A: $tType,Less2: A > A > $o,Xs: list @ A] :
( ( lexordp_eq @ A @ Less2 @ Xs @ ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% ord.lexordp_eq_simps(2)
thf(fact_75_ord_Olexordp__eq__simps_I3_J,axiom,
! [A: $tType,Less2: A > A > $o,X: A,Xs: list @ A] :
~ ( lexordp_eq @ A @ Less2 @ ( cons @ A @ X @ Xs ) @ ( nil @ A ) ) ).
% ord.lexordp_eq_simps(3)
thf(fact_76_local_Olexordp__simps_I3_J,axiom,
! [X: a,Xs: list @ a,Y: a,Ys: list @ a] :
( ( lexordp @ a @ less @ ( cons @ a @ X @ Xs ) @ ( cons @ a @ Y @ Ys ) )
= ( ( less @ X @ Y )
| ( ~ ( less @ Y @ X )
& ( lexordp @ a @ less @ Xs @ Ys ) ) ) ) ).
% local.lexordp_simps(3)
thf(fact_77_local_Olexordp__eq__simps_I4_J,axiom,
! [X: a,Xs: list @ a,Y: a,Ys: list @ a] :
( ( lexordp_eq @ a @ less @ ( cons @ a @ X @ Xs ) @ ( cons @ a @ Y @ Ys ) )
= ( ( less @ X @ Y )
| ( ~ ( less @ Y @ X )
& ( lexordp_eq @ a @ less @ Xs @ Ys ) ) ) ) ).
% local.lexordp_eq_simps(4)
thf(fact_78_local_Olexordp__simps_I2_J,axiom,
! [Xs: list @ a] :
~ ( lexordp @ a @ less @ Xs @ ( nil @ a ) ) ).
% local.lexordp_simps(2)
thf(fact_79_local_Olexordp__simps_I1_J,axiom,
! [Ys: list @ a] :
( ( lexordp @ a @ less @ ( nil @ a ) @ Ys )
= ( Ys
!= ( nil @ a ) ) ) ).
% local.lexordp_simps(1)
thf(fact_80_local_Olexordp__eq__simps_I2_J,axiom,
! [Xs: list @ a] :
( ( lexordp_eq @ a @ less @ Xs @ ( nil @ a ) )
= ( Xs
= ( nil @ a ) ) ) ).
% local.lexordp_eq_simps(2)
thf(fact_81_local_Olexordp__eq__simps_I1_J,axiom,
! [Ys: list @ a] : ( lexordp_eq @ a @ less @ ( nil @ a ) @ Ys ) ).
% local.lexordp_eq_simps(1)
thf(fact_82_local_Olexordp__eq__simps_I3_J,axiom,
! [X: a,Xs: list @ a] :
~ ( lexordp_eq @ a @ less @ ( cons @ a @ X @ Xs ) @ ( nil @ a ) ) ).
% local.lexordp_eq_simps(3)
thf(fact_83_ord_Olexordp_ONil,axiom,
! [A: $tType,Less2: A > A > $o,Y: A,Ys: list @ A] : ( lexordp @ A @ Less2 @ ( nil @ A ) @ ( cons @ A @ Y @ Ys ) ) ).
% ord.lexordp.Nil
thf(fact_84_ord_Olexordp_Ocases,axiom,
! [A: $tType,Less2: A > A > $o,A1: list @ A,A22: list @ A] :
( ( lexordp @ A @ Less2 @ A1 @ A22 )
=> ( ( ( A1
= ( nil @ A ) )
=> ! [Y3: A,Ys2: list @ A] :
( A22
!= ( cons @ A @ Y3 @ Ys2 ) ) )
=> ( ! [X3: A] :
( ? [Xs2: list @ A] :
( A1
= ( cons @ A @ X3 @ Xs2 ) )
=> ! [Y3: A] :
( ? [Ys2: list @ A] :
( A22
= ( cons @ A @ Y3 @ Ys2 ) )
=> ~ ( Less2 @ X3 @ Y3 ) ) )
=> ~ ! [X3: A,Y3: A,Xs2: list @ A] :
( ( A1
= ( cons @ A @ X3 @ Xs2 ) )
=> ! [Ys2: list @ A] :
( ( A22
= ( cons @ A @ Y3 @ Ys2 ) )
=> ( ~ ( Less2 @ X3 @ Y3 )
=> ( ~ ( Less2 @ Y3 @ X3 )
=> ~ ( lexordp @ A @ Less2 @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ).
% ord.lexordp.cases
thf(fact_85_ord_Olexordp_Osimps,axiom,
! [A: $tType] :
( ( lexordp @ A )
= ( ^ [Less: A > A > $o,A12: list @ A,A23: list @ A] :
( ? [Y2: A,Ys3: list @ A] :
( ( A12
= ( nil @ A ) )
& ( A23
= ( cons @ A @ Y2 @ Ys3 ) ) )
| ? [X2: A,Y2: A,Xs3: list @ A,Ys3: list @ A] :
( ( A12
= ( cons @ A @ X2 @ Xs3 ) )
& ( A23
= ( cons @ A @ Y2 @ Ys3 ) )
& ( Less @ X2 @ Y2 ) )
| ? [X2: A,Y2: A,Xs3: list @ A,Ys3: list @ A] :
( ( A12
= ( cons @ A @ X2 @ Xs3 ) )
& ( A23
= ( cons @ A @ Y2 @ Ys3 ) )
& ~ ( Less @ X2 @ Y2 )
& ~ ( Less @ Y2 @ X2 )
& ( lexordp @ A @ Less @ Xs3 @ Ys3 ) ) ) ) ) ).
% ord.lexordp.simps
thf(fact_86_ord_Olexordp_Oinducts,axiom,
! [A: $tType,Less2: A > A > $o,X1: list @ A,X23: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
( ( lexordp @ A @ Less2 @ X1 @ X23 )
=> ( ! [Y3: A,Ys2: list @ A] : ( P @ ( nil @ A ) @ ( cons @ A @ Y3 @ Ys2 ) )
=> ( ! [X3: A,Y3: A,Xs2: list @ A,Ys2: list @ A] :
( ( Less2 @ X3 @ Y3 )
=> ( P @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ A @ Y3 @ Ys2 ) ) )
=> ( ! [X3: A,Y3: A,Xs2: list @ A,Ys2: list @ A] :
( ~ ( Less2 @ X3 @ Y3 )
=> ( ~ ( Less2 @ Y3 @ X3 )
=> ( ( lexordp @ A @ Less2 @ Xs2 @ Ys2 )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ A @ Y3 @ Ys2 ) ) ) ) ) )
=> ( P @ X1 @ X23 ) ) ) ) ) ).
% ord.lexordp.inducts
thf(fact_87_ord_Olexordp__eq_Ocases,axiom,
! [A: $tType,Less2: A > A > $o,A1: list @ A,A22: list @ A] :
( ( lexordp_eq @ A @ Less2 @ A1 @ A22 )
=> ( ( ( A1
= ( nil @ A ) )
=> ! [Ys2: list @ A] : A22 != Ys2 )
=> ( ! [X3: A] :
( ? [Xs2: list @ A] :
( A1
= ( cons @ A @ X3 @ Xs2 ) )
=> ! [Y3: A] :
( ? [Ys2: list @ A] :
( A22
= ( cons @ A @ Y3 @ Ys2 ) )
=> ~ ( Less2 @ X3 @ Y3 ) ) )
=> ~ ! [X3: A,Y3: A,Xs2: list @ A] :
( ( A1
= ( cons @ A @ X3 @ Xs2 ) )
=> ! [Ys2: list @ A] :
( ( A22
= ( cons @ A @ Y3 @ Ys2 ) )
=> ( ~ ( Less2 @ X3 @ Y3 )
=> ( ~ ( Less2 @ Y3 @ X3 )
=> ~ ( lexordp_eq @ A @ Less2 @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ).
% ord.lexordp_eq.cases
thf(fact_88_ord_Olexordp__eq_Osimps,axiom,
! [A: $tType] :
( ( lexordp_eq @ A )
= ( ^ [Less: A > A > $o,A12: list @ A,A23: list @ A] :
( ? [Ys3: list @ A] :
( ( A12
= ( nil @ A ) )
& ( A23 = Ys3 ) )
| ? [X2: A,Y2: A,Xs3: list @ A,Ys3: list @ A] :
( ( A12
= ( cons @ A @ X2 @ Xs3 ) )
& ( A23
= ( cons @ A @ Y2 @ Ys3 ) )
& ( Less @ X2 @ Y2 ) )
| ? [X2: A,Y2: A,Xs3: list @ A,Ys3: list @ A] :
( ( A12
= ( cons @ A @ X2 @ Xs3 ) )
& ( A23
= ( cons @ A @ Y2 @ Ys3 ) )
& ~ ( Less @ X2 @ Y2 )
& ~ ( Less @ Y2 @ X2 )
& ( lexordp_eq @ A @ Less @ Xs3 @ Ys3 ) ) ) ) ) ).
% ord.lexordp_eq.simps
thf(fact_89_ord_Olexordp__eq_Oinducts,axiom,
! [A: $tType,Less2: A > A > $o,X1: list @ A,X23: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
( ( lexordp_eq @ A @ Less2 @ X1 @ X23 )
=> ( ! [X12: list @ A] : ( P @ ( nil @ A ) @ X12 )
=> ( ! [X3: A,Y3: A,Xs2: list @ A,Ys2: list @ A] :
( ( Less2 @ X3 @ Y3 )
=> ( P @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ A @ Y3 @ Ys2 ) ) )
=> ( ! [X3: A,Y3: A,Xs2: list @ A,Ys2: list @ A] :
( ~ ( Less2 @ X3 @ Y3 )
=> ( ~ ( Less2 @ Y3 @ X3 )
=> ( ( lexordp_eq @ A @ Less2 @ Xs2 @ Ys2 )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ A @ Y3 @ Ys2 ) ) ) ) ) )
=> ( P @ X1 @ X23 ) ) ) ) ) ).
% ord.lexordp_eq.inducts
thf(fact_90_list_Oset__cases,axiom,
! [A: $tType,E: A,A2: list @ A] :
( ( member @ A @ E @ ( set2 @ A @ A2 ) )
=> ( ! [Z2: list @ A] :
( A2
!= ( cons @ A @ E @ Z2 ) )
=> ~ ! [Z1: A,Z2: list @ A] :
( ( A2
= ( cons @ A @ Z1 @ Z2 ) )
=> ~ ( member @ A @ E @ ( set2 @ A @ Z2 ) ) ) ) ) ).
% list.set_cases
thf(fact_91_set__ConsD,axiom,
! [A: $tType,Y: A,X: A,Xs: list @ A] :
( ( member @ A @ Y @ ( set2 @ A @ ( cons @ A @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member @ A @ Y @ ( set2 @ A @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_92_list_Oset__intros_I1_J,axiom,
! [A: $tType,A1: A,A22: list @ A] : ( member @ A @ A1 @ ( set2 @ A @ ( cons @ A @ A1 @ A22 ) ) ) ).
% list.set_intros(1)
thf(fact_93_list_Oset__intros_I2_J,axiom,
! [A: $tType,X: A,A22: list @ A,A1: A] :
( ( member @ A @ X @ ( set2 @ A @ A22 ) )
=> ( member @ A @ X @ ( set2 @ A @ ( cons @ A @ A1 @ A22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_94_map__tailrec__rev_Oinduct,axiom,
! [A: $tType,B: $tType,P: ( A > B ) > ( list @ A ) > ( list @ B ) > $o,A0: A > B,A1: list @ A,A22: list @ B] :
( ! [F2: A > B,X12: list @ B] : ( P @ F2 @ ( nil @ A ) @ X12 )
=> ( ! [F2: A > B,A4: A,As: list @ A,Bs: list @ B] :
( ( P @ F2 @ As @ ( cons @ B @ ( F2 @ A4 ) @ Bs ) )
=> ( P @ F2 @ ( cons @ A @ A4 @ As ) @ Bs ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_95_list__nonempty__induct,axiom,
! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X3: A] : ( P @ ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ( ! [X3: A,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( P @ Xs2 )
=> ( P @ ( cons @ A @ X3 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_96_remdups__adj_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > $o,A0: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X3: A] : ( P @ ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ( ! [X3: A,Y3: A,Xs2: list @ A] :
( ( ( X3 = Y3 )
=> ( P @ ( cons @ A @ X3 @ Xs2 ) ) )
=> ( ( ( X3 != Y3 )
=> ( P @ ( cons @ A @ Y3 @ Xs2 ) ) )
=> ( P @ ( cons @ A @ X3 @ ( cons @ A @ Y3 @ Xs2 ) ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% remdups_adj.induct
thf(fact_97_remdups__adj_Ocases,axiom,
! [A: $tType,X: list @ A] :
( ( X
!= ( nil @ A ) )
=> ( ! [X3: A] :
( X
!= ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ~ ! [X3: A,Y3: A,Xs2: list @ A] :
( X
!= ( cons @ A @ X3 @ ( cons @ A @ Y3 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_98_transpose_Ocases,axiom,
! [A: $tType,X: list @ ( list @ A )] :
( ( X
!= ( nil @ ( list @ A ) ) )
=> ( ! [Xss: list @ ( list @ A )] :
( X
!= ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
=> ~ ! [X3: A,Xs2: list @ A,Xss: list @ ( list @ A )] :
( X
!= ( cons @ ( list @ A ) @ ( cons @ A @ X3 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_99_not__Cons__self2,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( cons @ A @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_100_splice_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
( ! [X12: list @ A] : ( P @ ( nil @ A ) @ X12 )
=> ( ! [V: A,Va: list @ A] : ( P @ ( cons @ A @ V @ Va ) @ ( nil @ A ) )
=> ( ! [X3: A,Xs2: list @ A,Y3: A,Ys2: list @ A] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ A @ Y3 @ Ys2 ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% splice.induct
thf(fact_101_list__induct2_H,axiom,
! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,Xs: list @ A,Ys: list @ B] :
( ( P @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X3: A,Xs2: list @ A] : ( P @ ( cons @ A @ X3 @ Xs2 ) @ ( nil @ B ) )
=> ( ! [Y3: B,Ys2: list @ B] : ( P @ ( nil @ A ) @ ( cons @ B @ Y3 @ Ys2 ) )
=> ( ! [X3: A,Xs2: list @ A,Y3: B,Ys2: list @ B] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ B @ Y3 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_102_neq__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
= ( ? [Y2: A,Ys3: list @ A] :
( Xs
= ( cons @ A @ Y2 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_103_list_Oinducts,axiom,
! [A: $tType,P: ( list @ A ) > $o,List: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X12: A,X24: list @ A] :
( ( P @ X24 )
=> ( P @ ( cons @ A @ X12 @ X24 ) ) )
=> ( P @ List ) ) ) ).
% list.inducts
thf(fact_104_list_Oexhaust,axiom,
! [A: $tType,Y: list @ A] :
( ( Y
!= ( nil @ A ) )
=> ~ ! [X212: A,X222: list @ A] :
( Y
!= ( cons @ A @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_105_list_OdiscI,axiom,
! [A: $tType,List: list @ A,X21: A,X22: list @ A] :
( ( List
= ( cons @ A @ X21 @ X22 ) )
=> ( List
!= ( nil @ A ) ) ) ).
% list.discI
thf(fact_106_list_Odistinct_I1_J,axiom,
! [A: $tType,X21: A,X22: list @ A] :
( ( nil @ A )
!= ( cons @ A @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_107_ord_Olexordp_OCons__eq,axiom,
! [A: $tType,Less2: A > A > $o,X: A,Y: A,Xs: list @ A,Ys: list @ A] :
( ~ ( Less2 @ X @ Y )
=> ( ~ ( Less2 @ Y @ X )
=> ( ( lexordp @ A @ Less2 @ Xs @ Ys )
=> ( lexordp @ A @ Less2 @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) ) ) ).
% ord.lexordp.Cons_eq
thf(fact_108_ord_Olexordp_OCons,axiom,
! [A: $tType,Less2: A > A > $o,X: A,Y: A,Xs: list @ A,Ys: list @ A] :
( ( Less2 @ X @ Y )
=> ( lexordp @ A @ Less2 @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) ).
% ord.lexordp.Cons
thf(fact_109_ord_Olexordp__eq_OCons__eq,axiom,
! [A: $tType,Less2: A > A > $o,X: A,Y: A,Xs: list @ A,Ys: list @ A] :
( ~ ( Less2 @ X @ Y )
=> ( ~ ( Less2 @ Y @ X )
=> ( ( lexordp_eq @ A @ Less2 @ Xs @ Ys )
=> ( lexordp_eq @ A @ Less2 @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) ) ) ).
% ord.lexordp_eq.Cons_eq
thf(fact_110_ord_Olexordp__eq_OCons,axiom,
! [A: $tType,Less2: A > A > $o,X: A,Y: A,Xs: list @ A,Ys: list @ A] :
( ( Less2 @ X @ Y )
=> ( lexordp_eq @ A @ Less2 @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) ).
% ord.lexordp_eq.Cons
thf(fact_111_ord_Olexordp__eq_ONil,axiom,
! [A: $tType,Less2: A > A > $o,Ys: list @ A] : ( lexordp_eq @ A @ Less2 @ ( nil @ A ) @ Ys ) ).
% ord.lexordp_eq.Nil
thf(fact_112_ord_Olexordp__irreflexive,axiom,
! [A: $tType,Less2: A > A > $o,Xs: list @ A] :
( ! [X3: A] :
~ ( Less2 @ X3 @ X3 )
=> ~ ( lexordp @ A @ Less2 @ Xs @ Xs ) ) ).
% ord.lexordp_irreflexive
thf(fact_113_ord_Olexordp__eq__refl,axiom,
! [A: $tType,Less2: A > A > $o,Xs: list @ A] : ( lexordp_eq @ A @ Less2 @ Xs @ Xs ) ).
% ord.lexordp_eq_refl
thf(fact_114_local_Olexordp__append__rightI,axiom,
! [Ys: list @ a,Xs: list @ a] :
( ( Ys
!= ( nil @ a ) )
=> ( lexordp @ a @ less @ Xs @ ( append @ a @ Xs @ Ys ) ) ) ).
% local.lexordp_append_rightI
thf(fact_115_local_Olexordp__append__left__rightI,axiom,
! [X: a,Y: a,Us: list @ a,Xs: list @ a,Ys: list @ a] :
( ( less @ X @ Y )
=> ( lexordp @ a @ less @ ( append @ a @ Us @ ( cons @ a @ X @ Xs ) ) @ ( append @ a @ Us @ ( cons @ a @ Y @ Ys ) ) ) ) ).
% local.lexordp_append_left_rightI
thf(fact_116_local_Olexordp__iff,axiom,
! [Xs: list @ a,Ys: list @ a] :
( ( lexordp @ a @ less @ Xs @ Ys )
= ( ? [X2: a,Vs: list @ a] :
( Ys
= ( append @ a @ Xs @ ( cons @ a @ X2 @ Vs ) ) )
| ? [Us2: list @ a,A5: a,B3: a,Vs: list @ a,Ws: list @ a] :
( ( less @ A5 @ B3 )
& ( Xs
= ( append @ a @ Us2 @ ( cons @ a @ A5 @ Vs ) ) )
& ( Ys
= ( append @ a @ Us2 @ ( cons @ a @ B3 @ Ws ) ) ) ) ) ) ).
% local.lexordp_iff
thf(fact_117_the__elem__set,axiom,
! [A: $tType,X: A] :
( ( the_elem @ A @ ( set2 @ A @ ( cons @ A @ X @ ( nil @ A ) ) ) )
= X ) ).
% the_elem_set
thf(fact_118_local_Olexordp__append__leftD,axiom,
! [Xs: list @ a,Us: list @ a,Vs2: list @ a] :
( ( lexordp @ a @ less @ ( append @ a @ Xs @ Us ) @ ( append @ a @ Xs @ Vs2 ) )
=> ( ! [A4: a] :
~ ( less @ A4 @ A4 )
=> ( lexordp @ a @ less @ Us @ Vs2 ) ) ) ).
% local.lexordp_append_leftD
thf(fact_119_local_Olexordp__append__leftI,axiom,
! [Us: list @ a,Vs2: list @ a,Xs: list @ a] :
( ( lexordp @ a @ less @ Us @ Vs2 )
=> ( lexordp @ a @ less @ ( append @ a @ Xs @ Us ) @ ( append @ a @ Xs @ Vs2 ) ) ) ).
% local.lexordp_append_leftI
thf(fact_120_local_Ostrict__mono__imp__inj__on,axiom,
! [B: $tType] :
( ( order @ B @ ( type2 @ B ) )
=> ! [F: a > B,A3: set @ a] :
( ( strict_mono @ a @ B @ less @ F )
=> ( inj_on @ a @ B @ F @ A3 ) ) ) ).
% local.strict_mono_imp_inj_on
thf(fact_121_same__append__eq,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_122_append__same__eq,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A,Zs: list @ A] :
( ( ( append @ A @ Ys @ Xs )
= ( append @ A @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_123_append__assoc,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( append @ A @ ( append @ A @ Xs @ Ys ) @ Zs )
= ( append @ A @ Xs @ ( append @ A @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_124_append__Nil2,axiom,
! [A: $tType,Xs: list @ A] :
( ( append @ A @ Xs @ ( nil @ A ) )
= Xs ) ).
% append_Nil2
thf(fact_125_append__self__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= Xs )
= ( Ys
= ( nil @ A ) ) ) ).
% append_self_conv
thf(fact_126_self__append__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs
= ( append @ A @ Xs @ Ys ) )
= ( Ys
= ( nil @ A ) ) ) ).
% self_append_conv
thf(fact_127_append__self__conv2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= Ys )
= ( Xs
= ( nil @ A ) ) ) ).
% append_self_conv2
thf(fact_128_self__append__conv2,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A] :
( ( Ys
= ( append @ A @ Xs @ Ys ) )
= ( Xs
= ( nil @ A ) ) ) ).
% self_append_conv2
thf(fact_129_Nil__is__append__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( nil @ A )
= ( append @ A @ Xs @ Ys ) )
= ( ( Xs
= ( nil @ A ) )
& ( Ys
= ( nil @ A ) ) ) ) ).
% Nil_is_append_conv
thf(fact_130_append__is__Nil__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= ( nil @ A ) )
= ( ( Xs
= ( nil @ A ) )
& ( Ys
= ( nil @ A ) ) ) ) ).
% append_is_Nil_conv
thf(fact_131_append1__eq__conv,axiom,
! [A: $tType,Xs: list @ A,X: A,Ys: list @ A,Y: A] :
( ( ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) )
= ( append @ A @ Ys @ ( cons @ A @ Y @ ( nil @ A ) ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_132_append__eq__append__conv2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A,Ts: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Zs @ Ts ) )
= ( ? [Us2: list @ A] :
( ( ( Xs
= ( append @ A @ Zs @ Us2 ) )
& ( ( append @ A @ Us2 @ Ys )
= Ts ) )
| ( ( ( append @ A @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append @ A @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_133_append__eq__appendI,axiom,
! [A: $tType,Xs: list @ A,Xs1: list @ A,Zs: list @ A,Ys: list @ A,Us: list @ A] :
( ( ( append @ A @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append @ A @ Xs1 @ Us ) )
=> ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_134_Cons__eq__appendI,axiom,
! [A: $tType,X: A,Xs1: list @ A,Ys: list @ A,Xs: list @ A,Zs: list @ A] :
( ( ( cons @ A @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append @ A @ Xs1 @ Zs ) )
=> ( ( cons @ A @ X @ Xs )
= ( append @ A @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_135_append__Cons,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A] :
( ( append @ A @ ( cons @ A @ X @ Xs ) @ Ys )
= ( cons @ A @ X @ ( append @ A @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_136_eq__Nil__appendI,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs = Ys )
=> ( Xs
= ( append @ A @ ( nil @ A ) @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_137_append__Nil,axiom,
! [A: $tType,Ys: list @ A] :
( ( append @ A @ ( nil @ A ) @ Ys )
= Ys ) ).
% append_Nil
thf(fact_138_ord_Olexordp__append__leftI,axiom,
! [A: $tType,Less2: A > A > $o,Us: list @ A,Vs2: list @ A,Xs: list @ A] :
( ( lexordp @ A @ Less2 @ Us @ Vs2 )
=> ( lexordp @ A @ Less2 @ ( append @ A @ Xs @ Us ) @ ( append @ A @ Xs @ Vs2 ) ) ) ).
% ord.lexordp_append_leftI
thf(fact_139_ord_Olexordp__append__leftD,axiom,
! [A: $tType,Less2: A > A > $o,Xs: list @ A,Us: list @ A,Vs2: list @ A] :
( ( lexordp @ A @ Less2 @ ( append @ A @ Xs @ Us ) @ ( append @ A @ Xs @ Vs2 ) )
=> ( ! [A4: A] :
~ ( Less2 @ A4 @ A4 )
=> ( lexordp @ A @ Less2 @ Us @ Vs2 ) ) ) ).
% ord.lexordp_append_leftD
thf(fact_140_rev__induct,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X3: A,Xs2: list @ A] :
( ( P @ Xs2 )
=> ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_141_rev__exhaust,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ~ ! [Ys2: list @ A,Y3: A] :
( Xs
!= ( append @ A @ Ys2 @ ( cons @ A @ Y3 @ ( nil @ A ) ) ) ) ) ).
% rev_exhaust
thf(fact_142_Cons__eq__append__conv,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( cons @ A @ X @ Xs )
= ( append @ A @ Ys @ Zs ) )
= ( ( ( Ys
= ( nil @ A ) )
& ( ( cons @ A @ X @ Xs )
= Zs ) )
| ? [Ys5: list @ A] :
( ( ( cons @ A @ X @ Ys5 )
= Ys )
& ( Xs
= ( append @ A @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_143_append__eq__Cons__conv,axiom,
! [A: $tType,Ys: list @ A,Zs: list @ A,X: A,Xs: list @ A] :
( ( ( append @ A @ Ys @ Zs )
= ( cons @ A @ X @ Xs ) )
= ( ( ( Ys
= ( nil @ A ) )
& ( Zs
= ( cons @ A @ X @ Xs ) ) )
| ? [Ys5: list @ A] :
( ( Ys
= ( cons @ A @ X @ Ys5 ) )
& ( ( append @ A @ Ys5 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_144_rev__nonempty__induct,axiom,
! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X3: A] : ( P @ ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ( ! [X3: A,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( P @ Xs2 )
=> ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_145_split__list__first__prop__iff,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ( ? [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
& ( P @ X2 ) ) )
= ( ? [Ys3: list @ A,X2: A] :
( ? [Zs2: list @ A] :
( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X2 @ Zs2 ) ) )
& ( P @ X2 )
& ! [Y2: A] :
( ( member @ A @ Y2 @ ( set2 @ A @ Ys3 ) )
=> ~ ( P @ Y2 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_146_split__list__last__prop__iff,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ( ? [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
& ( P @ X2 ) ) )
= ( ? [Ys3: list @ A,X2: A,Zs2: list @ A] :
( ( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X2 @ Zs2 ) ) )
& ( P @ X2 )
& ! [Y2: A] :
( ( member @ A @ Y2 @ ( set2 @ A @ Zs2 ) )
=> ~ ( P @ Y2 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_147_in__set__conv__decomp__first,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
= ( ? [Ys3: list @ A,Zs2: list @ A] :
( ( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X @ Zs2 ) ) )
& ~ ( member @ A @ X @ ( set2 @ A @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_148_in__set__conv__decomp__last,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
= ( ? [Ys3: list @ A,Zs2: list @ A] :
( ( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X @ Zs2 ) ) )
& ~ ( member @ A @ X @ ( set2 @ A @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_149_split__list__first__propE,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ? [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys2: list @ A,X3: A] :
( ? [Zs3: list @ A] :
( Xs
= ( append @ A @ Ys2 @ ( cons @ A @ X3 @ Zs3 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa: A] :
( ( member @ A @ Xa @ ( set2 @ A @ Ys2 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_150_split__list__last__propE,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ? [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys2: list @ A,X3: A,Zs3: list @ A] :
( ( Xs
= ( append @ A @ Ys2 @ ( cons @ A @ X3 @ Zs3 ) ) )
=> ( ( P @ X3 )
=> ~ ! [Xa: A] :
( ( member @ A @ Xa @ ( set2 @ A @ Zs3 ) )
=> ~ ( P @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_151_split__list__first__prop,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ? [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys2: list @ A,X3: A] :
( ? [Zs3: list @ A] :
( Xs
= ( append @ A @ Ys2 @ ( cons @ A @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Xa: A] :
( ( member @ A @ Xa @ ( set2 @ A @ Ys2 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_152_split__list__last__prop,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ? [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys2: list @ A,X3: A,Zs3: list @ A] :
( ( Xs
= ( append @ A @ Ys2 @ ( cons @ A @ X3 @ Zs3 ) ) )
& ( P @ X3 )
& ! [Xa: A] :
( ( member @ A @ Xa @ ( set2 @ A @ Zs3 ) )
=> ~ ( P @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_153_in__set__conv__decomp,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
= ( ? [Ys3: list @ A,Zs2: list @ A] :
( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X @ Zs2 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_154_split__list__propE,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ? [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
& ( P @ X4 ) )
=> ~ ! [Ys2: list @ A,X3: A] :
( ? [Zs3: list @ A] :
( Xs
= ( append @ A @ Ys2 @ ( cons @ A @ X3 @ Zs3 ) ) )
=> ~ ( P @ X3 ) ) ) ).
% split_list_propE
thf(fact_155_split__list__first,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ? [Ys2: list @ A,Zs3: list @ A] :
( ( Xs
= ( append @ A @ Ys2 @ ( cons @ A @ X @ Zs3 ) ) )
& ~ ( member @ A @ X @ ( set2 @ A @ Ys2 ) ) ) ) ).
% split_list_first
thf(fact_156_split__list__prop,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ? [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
& ( P @ X4 ) )
=> ? [Ys2: list @ A,X3: A] :
( ? [Zs3: list @ A] :
( Xs
= ( append @ A @ Ys2 @ ( cons @ A @ X3 @ Zs3 ) ) )
& ( P @ X3 ) ) ) ).
% split_list_prop
thf(fact_157_split__list__last,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ? [Ys2: list @ A,Zs3: list @ A] :
( ( Xs
= ( append @ A @ Ys2 @ ( cons @ A @ X @ Zs3 ) ) )
& ~ ( member @ A @ X @ ( set2 @ A @ Zs3 ) ) ) ) ).
% split_list_last
thf(fact_158_split__list,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ? [Ys2: list @ A,Zs3: list @ A] :
( Xs
= ( append @ A @ Ys2 @ ( cons @ A @ X @ Zs3 ) ) ) ) ).
% split_list
thf(fact_159_ord_Olexordp__append__left__rightI,axiom,
! [A: $tType,Less2: A > A > $o,X: A,Y: A,Us: list @ A,Xs: list @ A,Ys: list @ A] :
( ( Less2 @ X @ Y )
=> ( lexordp @ A @ Less2 @ ( append @ A @ Us @ ( cons @ A @ X @ Xs ) ) @ ( append @ A @ Us @ ( cons @ A @ Y @ Ys ) ) ) ) ).
% ord.lexordp_append_left_rightI
thf(fact_160_ord_Olexordp__append__rightI,axiom,
! [A: $tType,Ys: list @ A,Less2: A > A > $o,Xs: list @ A] :
( ( Ys
!= ( nil @ A ) )
=> ( lexordp @ A @ Less2 @ Xs @ ( append @ A @ Xs @ Ys ) ) ) ).
% ord.lexordp_append_rightI
thf(fact_161_sublists_Osimps_I1_J,axiom,
! [A: $tType] :
( ( sublists @ A @ ( nil @ A ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% sublists.simps(1)
thf(fact_162_product__lists_Osimps_I1_J,axiom,
! [A: $tType] :
( ( product_lists @ A @ ( nil @ ( list @ A ) ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% product_lists.simps(1)
thf(fact_163_bind__simps_I2_J,axiom,
! [A: $tType,B: $tType,X: B,Xs: list @ B,F: B > ( list @ A )] :
( ( bind @ B @ A @ ( cons @ B @ X @ Xs ) @ F )
= ( append @ A @ ( F @ X ) @ ( bind @ B @ A @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_164_ord__class_Orbt__greater__prop,axiom,
! [B: $tType,A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ( ( rBT_Im1332541288reater @ A @ B )
= ( ^ [K2: A,T2: rBT_Im246033960le_rbt @ A @ B] :
! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ ( rBT_Im380146495e_keys @ A @ B @ T2 ) ) )
=> ( ord_less @ A @ K2 @ X2 ) ) ) ) ) ).
% ord_class.rbt_greater_prop
thf(fact_165_bind__simps_I1_J,axiom,
! [B: $tType,A: $tType,F: B > ( list @ A )] :
( ( bind @ B @ A @ ( nil @ B ) @ F )
= ( nil @ A ) ) ).
% bind_simps(1)
thf(fact_166_order__class_Orbt__greater__trans,axiom,
! [C2: $tType,A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,T: rBT_Im246033960le_rbt @ A @ C2] :
( ( ord_less @ A @ X @ Y )
=> ( ( rBT_Im1332541288reater @ A @ C2 @ Y @ T )
=> ( rBT_Im1332541288reater @ A @ C2 @ X @ T ) ) ) ) ).
% order_class.rbt_greater_trans
thf(fact_167_list__bind__cong,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ A,F: A > ( list @ B ),G: A > ( list @ B )] :
( ( Xs = Ys )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ( F @ X3 )
= ( G @ X3 ) ) )
=> ( ( bind @ A @ B @ Xs @ F )
= ( bind @ A @ B @ Ys @ G ) ) ) ) ).
% list_bind_cong
thf(fact_168_ord__class_Orbt__less__prop,axiom,
! [B: $tType,A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ( ( rBT_Im903026891t_less @ A @ B )
= ( ^ [K2: A,T2: rBT_Im246033960le_rbt @ A @ B] :
! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ ( rBT_Im380146495e_keys @ A @ B @ T2 ) ) )
=> ( ord_less @ A @ X2 @ K2 ) ) ) ) ) ).
% ord_class.rbt_less_prop
thf(fact_169_maps__simps_I1_J,axiom,
! [A: $tType,B: $tType,F: B > ( list @ A ),X: B,Xs: list @ B] :
( ( maps @ B @ A @ F @ ( cons @ B @ X @ Xs ) )
= ( append @ A @ ( F @ X ) @ ( maps @ B @ A @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_170_not__in__set__insert,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( insert @ A @ X @ Xs )
= ( cons @ A @ X @ Xs ) ) ) ).
% not_in_set_insert
thf(fact_171_in__set__insert,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( insert @ A @ X @ Xs )
= Xs ) ) ).
% in_set_insert
thf(fact_172_insert__Nil,axiom,
! [A: $tType,X: A] :
( ( insert @ A @ X @ ( nil @ A ) )
= ( cons @ A @ X @ ( nil @ A ) ) ) ).
% insert_Nil
thf(fact_173_order__class_Orbt__less__trans,axiom,
! [C2: $tType,A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,T: rBT_Im246033960le_rbt @ A @ C2,Y: A] :
( ( rBT_Im903026891t_less @ A @ C2 @ X @ T )
=> ( ( ord_less @ A @ X @ Y )
=> ( rBT_Im903026891t_less @ A @ C2 @ Y @ T ) ) ) ) ).
% order_class.rbt_less_trans
thf(fact_174_maps__simps_I2_J,axiom,
! [B: $tType,A: $tType,F: B > ( list @ A )] :
( ( maps @ B @ A @ F @ ( nil @ B ) )
= ( nil @ A ) ) ).
% maps_simps(2)
thf(fact_175_List_Oinsert__def,axiom,
! [A: $tType] :
( ( insert @ A )
= ( ^ [X2: A,Xs3: list @ A] : ( if @ ( list @ A ) @ ( member @ A @ X2 @ ( set2 @ A @ Xs3 ) ) @ Xs3 @ ( cons @ A @ X2 @ Xs3 ) ) ) ) ).
% List.insert_def
thf(fact_176_rotate1_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( rotate1 @ A @ ( cons @ A @ X @ Xs ) )
= ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ).
% rotate1.simps(2)
thf(fact_177_local_Orbt__less__simps_I1_J,axiom,
! [B: $tType,K: a] : ( rBT_Im2075124343t_less @ a @ B @ less @ K @ ( rBT_Im418718756_Empty @ a @ B ) ) ).
% local.rbt_less_simps(1)
thf(fact_178_last__snoc,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( last @ A @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) )
= X ) ).
% last_snoc
thf(fact_179_rotate1__is__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( rotate1 @ A @ Xs )
= ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% rotate1_is_Nil_conv
thf(fact_180_set__rotate1,axiom,
! [A: $tType,Xs: list @ A] :
( ( set2 @ A @ ( rotate1 @ A @ Xs ) )
= ( set2 @ A @ Xs ) ) ).
% set_rotate1
thf(fact_181_ord__class_Orbt__greater__simps_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [K: A] : ( rBT_Im1332541288reater @ A @ B @ K @ ( rBT_Im418718756_Empty @ A @ B ) ) ) ).
% ord_class.rbt_greater_simps(1)
thf(fact_182_ord__class_Orbt__less__simps_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [K: A] : ( rBT_Im903026891t_less @ A @ B @ K @ ( rBT_Im418718756_Empty @ A @ B ) ) ) ).
% ord_class.rbt_less_simps(1)
thf(fact_183_last__appendL,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A] :
( ( Ys
= ( nil @ A ) )
=> ( ( last @ A @ ( append @ A @ Xs @ Ys ) )
= ( last @ A @ Xs ) ) ) ).
% last_appendL
thf(fact_184_last__appendR,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A] :
( ( Ys
!= ( nil @ A ) )
=> ( ( last @ A @ ( append @ A @ Xs @ Ys ) )
= ( last @ A @ Ys ) ) ) ).
% last_appendR
thf(fact_185_keys__simps_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( rBT_Im380146495e_keys @ A @ B @ ( rBT_Im418718756_Empty @ A @ B ) )
= ( nil @ A ) ) ).
% keys_simps(1)
thf(fact_186_local_Orbt__greater__simps_I1_J,axiom,
! [B: $tType,K: a] : ( rBT_Im1259024060reater @ a @ B @ less @ K @ ( rBT_Im418718756_Empty @ a @ B ) ) ).
% local.rbt_greater_simps(1)
thf(fact_187_rotate1_Osimps_I1_J,axiom,
! [A: $tType] :
( ( rotate1 @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% rotate1.simps(1)
thf(fact_188_ord_Orbt__less__simps_I1_J,axiom,
! [B: $tType,A: $tType,Less2: A > A > $o,K: A] : ( rBT_Im2075124343t_less @ A @ B @ Less2 @ K @ ( rBT_Im418718756_Empty @ A @ B ) ) ).
% ord.rbt_less_simps(1)
thf(fact_189_ord_Orbt__greater__simps_I1_J,axiom,
! [B: $tType,A: $tType,Less2: A > A > $o,K: A] : ( rBT_Im1259024060reater @ A @ B @ Less2 @ K @ ( rBT_Im418718756_Empty @ A @ B ) ) ).
% ord.rbt_greater_simps(1)
thf(fact_190_last__ConsR,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X @ Xs ) )
= ( last @ A @ Xs ) ) ) ).
% last_ConsR
thf(fact_191_last__ConsL,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( Xs
= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X @ Xs ) )
= X ) ) ).
% last_ConsL
thf(fact_192_last_Osimps,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( ( Xs
= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X @ Xs ) )
= X ) )
& ( ( Xs
!= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X @ Xs ) )
= ( last @ A @ Xs ) ) ) ) ).
% last.simps
thf(fact_193_last__in__set,axiom,
! [A: $tType,As2: list @ A] :
( ( As2
!= ( nil @ A ) )
=> ( member @ A @ ( last @ A @ As2 ) @ ( set2 @ A @ As2 ) ) ) ).
% last_in_set
thf(fact_194_last__append,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A] :
( ( ( Ys
= ( nil @ A ) )
=> ( ( last @ A @ ( append @ A @ Xs @ Ys ) )
= ( last @ A @ Xs ) ) )
& ( ( Ys
!= ( nil @ A ) )
=> ( ( last @ A @ ( append @ A @ Xs @ Ys ) )
= ( last @ A @ Ys ) ) ) ) ).
% last_append
thf(fact_195_longest__common__suffix,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
? [Ss: list @ A,Xs4: list @ A,Ys4: list @ A] :
( ( Xs
= ( append @ A @ Xs4 @ Ss ) )
& ( Ys
= ( append @ A @ Ys4 @ Ss ) )
& ( ( Xs4
= ( nil @ A ) )
| ( Ys4
= ( nil @ A ) )
| ( ( last @ A @ Xs4 )
!= ( last @ A @ Ys4 ) ) ) ) ).
% longest_common_suffix
thf(fact_196_non__empty__rbt__keys,axiom,
! [B: $tType,A: $tType,T: rBT_Im246033960le_rbt @ A @ B] :
( ( T
!= ( rBT_Im418718756_Empty @ A @ B ) )
=> ( ( rBT_Im380146495e_keys @ A @ B @ T )
!= ( nil @ A ) ) ) ).
% non_empty_rbt_keys
thf(fact_197_local_Orbt__sorted_Osimps_I1_J,axiom,
! [B: $tType] : ( rBT_Im759614907sorted @ a @ B @ less @ ( rBT_Im418718756_Empty @ a @ B ) ) ).
% local.rbt_sorted.simps(1)
thf(fact_198_append__butlast__last__id,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( append @ A @ ( butlast @ A @ Xs ) @ ( cons @ A @ ( last @ A @ Xs ) @ ( nil @ A ) ) )
= Xs ) ) ).
% append_butlast_last_id
thf(fact_199_snoc__eq__iff__butlast,axiom,
! [A: $tType,Xs: list @ A,X: A,Ys: list @ A] :
( ( ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) )
= Ys )
= ( ( Ys
!= ( nil @ A ) )
& ( ( butlast @ A @ Ys )
= Xs )
& ( ( last @ A @ Ys )
= X ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_200_butlast__snoc,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( butlast @ A @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_201_butlast_Osimps_I1_J,axiom,
! [A: $tType] :
( ( butlast @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% butlast.simps(1)
thf(fact_202_in__set__butlastD,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ ( butlast @ A @ Xs ) ) )
=> ( member @ A @ X @ ( set2 @ A @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_203_butlast_Osimps_I2_J,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( ( Xs
= ( nil @ A ) )
=> ( ( butlast @ A @ ( cons @ A @ X @ Xs ) )
= ( nil @ A ) ) )
& ( ( Xs
!= ( nil @ A ) )
=> ( ( butlast @ A @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X @ ( butlast @ A @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_204_butlast__append,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A] :
( ( ( Ys
= ( nil @ A ) )
=> ( ( butlast @ A @ ( append @ A @ Xs @ Ys ) )
= ( butlast @ A @ Xs ) ) )
& ( ( Ys
!= ( nil @ A ) )
=> ( ( butlast @ A @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ Xs @ ( butlast @ A @ Ys ) ) ) ) ) ).
% butlast_append
thf(fact_205_in__set__butlast__appendI,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A] :
( ( ( member @ A @ X @ ( set2 @ A @ ( butlast @ A @ Xs ) ) )
| ( member @ A @ X @ ( set2 @ A @ ( butlast @ A @ Ys ) ) ) )
=> ( member @ A @ X @ ( set2 @ A @ ( butlast @ A @ ( append @ A @ Xs @ Ys ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_206_ord_Orbt__sorted_Osimps_I1_J,axiom,
! [B: $tType,A: $tType,Less2: A > A > $o] : ( rBT_Im759614907sorted @ A @ B @ Less2 @ ( rBT_Im418718756_Empty @ A @ B ) ) ).
% ord.rbt_sorted.simps(1)
thf(fact_207_local_Orbt__sorted_Osimps_I2_J,axiom,
! [B: $tType,C: rBT_Im1923302023_color,L: rBT_Im246033960le_rbt @ a @ B,K: a,V2: B,R: rBT_Im246033960le_rbt @ a @ B] :
( ( rBT_Im759614907sorted @ a @ B @ less @ ( rBT_Im480247531Branch @ a @ B @ C @ L @ K @ V2 @ R ) )
= ( ( rBT_Im2075124343t_less @ a @ B @ less @ K @ L )
& ( rBT_Im1259024060reater @ a @ B @ less @ K @ R )
& ( rBT_Im759614907sorted @ a @ B @ less @ L )
& ( rBT_Im759614907sorted @ a @ B @ less @ R ) ) ) ).
% local.rbt_sorted.simps(2)
thf(fact_208_local_Odistinct__keys,axiom,
! [B: $tType,T: rBT_Im246033960le_rbt @ a @ B] :
( ( rBT_Im759614907sorted @ a @ B @ less @ T )
=> ( distinct @ a @ ( rBT_Im380146495e_keys @ a @ B @ T ) ) ) ).
% local.distinct_keys
thf(fact_209_rbt_Oinject,axiom,
! [B: $tType,A: $tType,X21: rBT_Im1923302023_color,X22: rBT_Im246033960le_rbt @ A @ B,X232: A,X242: B,X25: rBT_Im246033960le_rbt @ A @ B,Y21: rBT_Im1923302023_color,Y22: rBT_Im246033960le_rbt @ A @ B,Y23: A,Y24: B,Y25: rBT_Im246033960le_rbt @ A @ B] :
( ( ( rBT_Im480247531Branch @ A @ B @ X21 @ X22 @ X232 @ X242 @ X25 )
= ( rBT_Im480247531Branch @ A @ B @ Y21 @ Y22 @ Y23 @ Y24 @ Y25 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 )
& ( X232 = Y23 )
& ( X242 = Y24 )
& ( X25 = Y25 ) ) ) ).
% rbt.inject
thf(fact_210_distinct1__rotate,axiom,
! [A: $tType,Xs: list @ A] :
( ( distinct @ A @ ( rotate1 @ A @ Xs ) )
= ( distinct @ A @ Xs ) ) ).
% distinct1_rotate
thf(fact_211_distinct__insert,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( distinct @ A @ ( insert @ A @ X @ Xs ) )
= ( distinct @ A @ Xs ) ) ).
% distinct_insert
thf(fact_212_ord__class_Orbt__greater__simps_I2_J,axiom,
! [C2: $tType,A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [K: A,C: rBT_Im1923302023_color,Lt: rBT_Im246033960le_rbt @ A @ C2,Kt: A,V2: C2,Rt: rBT_Im246033960le_rbt @ A @ C2] :
( ( rBT_Im1332541288reater @ A @ C2 @ K @ ( rBT_Im480247531Branch @ A @ C2 @ C @ Lt @ Kt @ V2 @ Rt ) )
= ( ( ord_less @ A @ K @ Kt )
& ( rBT_Im1332541288reater @ A @ C2 @ K @ Lt )
& ( rBT_Im1332541288reater @ A @ C2 @ K @ Rt ) ) ) ) ).
% ord_class.rbt_greater_simps(2)
thf(fact_213_ord__class_Orbt__less__simps_I2_J,axiom,
! [C2: $tType,A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [K: A,C: rBT_Im1923302023_color,Lt: rBT_Im246033960le_rbt @ A @ C2,Kt: A,V2: C2,Rt: rBT_Im246033960le_rbt @ A @ C2] :
( ( rBT_Im903026891t_less @ A @ C2 @ K @ ( rBT_Im480247531Branch @ A @ C2 @ C @ Lt @ Kt @ V2 @ Rt ) )
= ( ( ord_less @ A @ Kt @ K )
& ( rBT_Im903026891t_less @ A @ C2 @ K @ Lt )
& ( rBT_Im903026891t_less @ A @ C2 @ K @ Rt ) ) ) ) ).
% ord_class.rbt_less_simps(2)
thf(fact_214_local_Orbt__greater__simps_I2_J,axiom,
! [C2: $tType,K: a,C: rBT_Im1923302023_color,Lt: rBT_Im246033960le_rbt @ a @ C2,Kt: a,V2: C2,Rt: rBT_Im246033960le_rbt @ a @ C2] :
( ( rBT_Im1259024060reater @ a @ C2 @ less @ K @ ( rBT_Im480247531Branch @ a @ C2 @ C @ Lt @ Kt @ V2 @ Rt ) )
= ( ( less @ K @ Kt )
& ( rBT_Im1259024060reater @ a @ C2 @ less @ K @ Lt )
& ( rBT_Im1259024060reater @ a @ C2 @ less @ K @ Rt ) ) ) ).
% local.rbt_greater_simps(2)
thf(fact_215_local_Orbt__less__simps_I2_J,axiom,
! [C2: $tType,K: a,C: rBT_Im1923302023_color,Lt: rBT_Im246033960le_rbt @ a @ C2,Kt: a,V2: C2,Rt: rBT_Im246033960le_rbt @ a @ C2] :
( ( rBT_Im2075124343t_less @ a @ C2 @ less @ K @ ( rBT_Im480247531Branch @ a @ C2 @ C @ Lt @ Kt @ V2 @ Rt ) )
= ( ( less @ Kt @ K )
& ( rBT_Im2075124343t_less @ a @ C2 @ less @ K @ Lt )
& ( rBT_Im2075124343t_less @ a @ C2 @ less @ K @ Rt ) ) ) ).
% local.rbt_less_simps(2)
thf(fact_216_keys__simps_I2_J,axiom,
! [D: $tType,C2: $tType,C: rBT_Im1923302023_color,L: rBT_Im246033960le_rbt @ C2 @ D,K: C2,V2: D,R: rBT_Im246033960le_rbt @ C2 @ D] :
( ( rBT_Im380146495e_keys @ C2 @ D @ ( rBT_Im480247531Branch @ C2 @ D @ C @ L @ K @ V2 @ R ) )
= ( append @ C2 @ ( rBT_Im380146495e_keys @ C2 @ D @ L ) @ ( cons @ C2 @ K @ ( rBT_Im380146495e_keys @ C2 @ D @ R ) ) ) ) ).
% keys_simps(2)
thf(fact_217_rbt_Oexhaust,axiom,
! [B: $tType,A: $tType,Y: rBT_Im246033960le_rbt @ A @ B] :
( ( Y
!= ( rBT_Im418718756_Empty @ A @ B ) )
=> ~ ! [X212: rBT_Im1923302023_color,X222: rBT_Im246033960le_rbt @ A @ B,X233: A,X243: B,X252: rBT_Im246033960le_rbt @ A @ B] :
( Y
!= ( rBT_Im480247531Branch @ A @ B @ X212 @ X222 @ X233 @ X243 @ X252 ) ) ) ).
% rbt.exhaust
thf(fact_218_rbt_Oinduct,axiom,
! [B: $tType,A: $tType,P: ( rBT_Im246033960le_rbt @ A @ B ) > $o,Rbt: rBT_Im246033960le_rbt @ A @ B] :
( ( P @ ( rBT_Im418718756_Empty @ A @ B ) )
=> ( ! [X12: rBT_Im1923302023_color,X24: rBT_Im246033960le_rbt @ A @ B,X32: A,X42: B,X5: rBT_Im246033960le_rbt @ A @ B] :
( ( P @ X24 )
=> ( ( P @ X5 )
=> ( P @ ( rBT_Im480247531Branch @ A @ B @ X12 @ X24 @ X32 @ X42 @ X5 ) ) ) )
=> ( P @ Rbt ) ) ) ).
% rbt.induct
thf(fact_219_rbt_Odistinct_I1_J,axiom,
! [B: $tType,A: $tType,X21: rBT_Im1923302023_color,X22: rBT_Im246033960le_rbt @ A @ B,X232: A,X242: B,X25: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im418718756_Empty @ A @ B )
!= ( rBT_Im480247531Branch @ A @ B @ X21 @ X22 @ X232 @ X242 @ X25 ) ) ).
% rbt.distinct(1)
thf(fact_220_distinct__butlast,axiom,
! [A: $tType,Xs: list @ A] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( butlast @ A @ Xs ) ) ) ).
% distinct_butlast
thf(fact_221_distinct__singleton,axiom,
! [A: $tType,X: A] : ( distinct @ A @ ( cons @ A @ X @ ( nil @ A ) ) ) ).
% distinct_singleton
thf(fact_222_distinct_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( distinct @ A @ ( cons @ A @ X @ Xs ) )
= ( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
& ( distinct @ A @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_223_distinct__length__2__or__more,axiom,
! [A: $tType,A2: A,B2: A,Xs: list @ A] :
( ( distinct @ A @ ( cons @ A @ A2 @ ( cons @ A @ B2 @ Xs ) ) )
= ( ( A2 != B2 )
& ( distinct @ A @ ( cons @ A @ A2 @ Xs ) )
& ( distinct @ A @ ( cons @ A @ B2 @ Xs ) ) ) ) ).
% distinct_length_2_or_more
thf(fact_224_distinct_Osimps_I1_J,axiom,
! [A: $tType] : ( distinct @ A @ ( nil @ A ) ) ).
% distinct.simps(1)
thf(fact_225_ord_Orbt__less__simps_I2_J,axiom,
! [C2: $tType,A: $tType,Less2: A > A > $o,K: A,C: rBT_Im1923302023_color,Lt: rBT_Im246033960le_rbt @ A @ C2,Kt: A,V2: C2,Rt: rBT_Im246033960le_rbt @ A @ C2] :
( ( rBT_Im2075124343t_less @ A @ C2 @ Less2 @ K @ ( rBT_Im480247531Branch @ A @ C2 @ C @ Lt @ Kt @ V2 @ Rt ) )
= ( ( Less2 @ Kt @ K )
& ( rBT_Im2075124343t_less @ A @ C2 @ Less2 @ K @ Lt )
& ( rBT_Im2075124343t_less @ A @ C2 @ Less2 @ K @ Rt ) ) ) ).
% ord.rbt_less_simps(2)
thf(fact_226_ord_Orbt__greater__simps_I2_J,axiom,
! [C2: $tType,A: $tType,Less2: A > A > $o,K: A,C: rBT_Im1923302023_color,Lt: rBT_Im246033960le_rbt @ A @ C2,Kt: A,V2: C2,Rt: rBT_Im246033960le_rbt @ A @ C2] :
( ( rBT_Im1259024060reater @ A @ C2 @ Less2 @ K @ ( rBT_Im480247531Branch @ A @ C2 @ C @ Lt @ Kt @ V2 @ Rt ) )
= ( ( Less2 @ K @ Kt )
& ( rBT_Im1259024060reater @ A @ C2 @ Less2 @ K @ Lt )
& ( rBT_Im1259024060reater @ A @ C2 @ Less2 @ K @ Rt ) ) ) ).
% ord.rbt_greater_simps(2)
thf(fact_227_distinct__product__lists,axiom,
! [A: $tType,Xss2: list @ ( list @ A )] :
( ! [X3: list @ A] :
( ( member @ ( list @ A ) @ X3 @ ( set2 @ ( list @ A ) @ Xss2 ) )
=> ( distinct @ A @ X3 ) )
=> ( distinct @ ( list @ A ) @ ( product_lists @ A @ Xss2 ) ) ) ).
% distinct_product_lists
thf(fact_228_not__distinct__decomp,axiom,
! [A: $tType,Ws2: list @ A] :
( ~ ( distinct @ A @ Ws2 )
=> ? [Xs2: list @ A,Ys2: list @ A,Zs3: list @ A,Y3: A] :
( Ws2
= ( append @ A @ Xs2 @ ( append @ A @ ( cons @ A @ Y3 @ ( nil @ A ) ) @ ( append @ A @ Ys2 @ ( append @ A @ ( cons @ A @ Y3 @ ( nil @ A ) ) @ Zs3 ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_229_not__distinct__conv__prefix,axiom,
! [A: $tType,As2: list @ A] :
( ( ~ ( distinct @ A @ As2 ) )
= ( ? [Xs3: list @ A,Y2: A,Ys3: list @ A] :
( ( member @ A @ Y2 @ ( set2 @ A @ Xs3 ) )
& ( distinct @ A @ Xs3 )
& ( As2
= ( append @ A @ Xs3 @ ( cons @ A @ Y2 @ Ys3 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_230_ord_Orbt__sorted_Osimps_I2_J,axiom,
! [B: $tType,A: $tType,Less2: A > A > $o,C: rBT_Im1923302023_color,L: rBT_Im246033960le_rbt @ A @ B,K: A,V2: B,R: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im759614907sorted @ A @ B @ Less2 @ ( rBT_Im480247531Branch @ A @ B @ C @ L @ K @ V2 @ R ) )
= ( ( rBT_Im2075124343t_less @ A @ B @ Less2 @ K @ L )
& ( rBT_Im1259024060reater @ A @ B @ Less2 @ K @ R )
& ( rBT_Im759614907sorted @ A @ B @ Less2 @ L )
& ( rBT_Im759614907sorted @ A @ B @ Less2 @ R ) ) ) ).
% ord.rbt_sorted.simps(2)
thf(fact_231_distinct__union,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( distinct @ A @ ( union @ A @ Xs @ Ys ) )
= ( distinct @ A @ Ys ) ) ).
% distinct_union
thf(fact_232_RBT__SORTED,axiom,
rBT_Im759614907sorted @ a @ b @ less @ ( rBT_Im480247531Branch @ a @ b @ c @ t1 @ k @ v @ t2 ) ).
% RBT_SORTED
thf(fact_233_Branch_Oprems,axiom,
rBT_Im759614907sorted @ a @ b @ less @ ( rBT_Im480247531Branch @ a @ b @ c @ t1 @ k @ v @ t2 ) ).
% Branch.prems
thf(fact_234_ord__class_Orbt__sorted_Osimps_I2_J,axiom,
! [B: $tType,A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [C: rBT_Im1923302023_color,L: rBT_Im246033960le_rbt @ A @ B,K: A,V2: B,R: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1183586831sorted @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ C @ L @ K @ V2 @ R ) )
= ( ( rBT_Im903026891t_less @ A @ B @ K @ L )
& ( rBT_Im1332541288reater @ A @ B @ K @ R )
& ( rBT_Im1183586831sorted @ A @ B @ L )
& ( rBT_Im1183586831sorted @ A @ B @ R ) ) ) ) ).
% ord_class.rbt_sorted.simps(2)
thf(fact_235_ord__class_Orbt__sorted_Osimps_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ( rBT_Im1183586831sorted @ A @ B @ ( rBT_Im418718756_Empty @ A @ B ) ) ) ).
% ord_class.rbt_sorted.simps(1)
thf(fact_236_linorder__class_Odistinct__keys,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1183586831sorted @ A @ B @ T )
=> ( distinct @ A @ ( rBT_Im380146495e_keys @ A @ B @ T ) ) ) ) ).
% linorder_class.distinct_keys
thf(fact_237_local_OgreaterThan__def,axiom,
! [L: a] :
( ( set_greaterThan @ a @ less @ L )
= ( collect @ a @ ( less @ L ) ) ) ).
% local.greaterThan_def
thf(fact_238_local_OlessThan__def,axiom,
! [U: a] :
( ( set_lessThan @ a @ less @ U )
= ( collect @ a
@ ^ [X2: a] : ( less @ X2 @ U ) ) ) ).
% local.lessThan_def
thf(fact_239_local_OlessThan__iff,axiom,
! [I: a,K: a] :
( ( member @ a @ I @ ( set_lessThan @ a @ less @ K ) )
= ( less @ I @ K ) ) ).
% local.lessThan_iff
thf(fact_240_local_OgreaterThan__iff,axiom,
! [I: a,K: a] :
( ( member @ a @ I @ ( set_greaterThan @ a @ less @ K ) )
= ( less @ K @ I ) ) ).
% local.greaterThan_iff
thf(fact_241_DOM__T1,axiom,
! [K3: a] :
( ( member @ a @ K3 @ ( dom @ a @ b @ ( rBT_Im35466040lookup @ a @ b @ less @ t1 ) ) )
=> ( less @ K3 @ k ) ) ).
% DOM_T1
thf(fact_242_local_Orbt__lookup__keys,axiom,
! [B: $tType,T: rBT_Im246033960le_rbt @ a @ B] :
( ( rBT_Im759614907sorted @ a @ B @ less @ T )
=> ( ( dom @ a @ B @ ( rBT_Im35466040lookup @ a @ B @ less @ T ) )
= ( set2 @ a @ ( rBT_Im380146495e_keys @ a @ B @ T ) ) ) ) ).
% local.rbt_lookup_keys
thf(fact_243_Branch_Ohyps_I1_J,axiom,
( ( rBT_Im759614907sorted @ a @ b @ less @ t1 )
=> ( ( map_of @ a @ b @ ( rBT_Im954575269ntries @ a @ b @ t1 ) )
= ( rBT_Im35466040lookup @ a @ b @ less @ t1 ) ) ) ).
% Branch.hyps(1)
thf(fact_244_Branch_Ohyps_I2_J,axiom,
( ( rBT_Im759614907sorted @ a @ b @ less @ t2 )
=> ( ( map_of @ a @ b @ ( rBT_Im954575269ntries @ a @ b @ t2 ) )
= ( rBT_Im35466040lookup @ a @ b @ less @ t2 ) ) ) ).
% Branch.hyps(2)
thf(fact_245_entries_Osimps_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( rBT_Im954575269ntries @ A @ B @ ( rBT_Im418718756_Empty @ A @ B ) )
= ( nil @ ( product_prod @ A @ B ) ) ) ).
% entries.simps(1)
thf(fact_246_local_Orbt__lookup__Empty,axiom,
! [B: $tType] :
( ( rBT_Im35466040lookup @ a @ B @ less @ ( rBT_Im418718756_Empty @ a @ B ) )
= ( ^ [X2: a] : ( none @ B ) ) ) ).
% local.rbt_lookup_Empty
thf(fact_247_local_Orbt__lookup_Osimps_I1_J,axiom,
! [B: $tType,K: a] :
( ( rBT_Im35466040lookup @ a @ B @ less @ ( rBT_Im418718756_Empty @ a @ B ) @ K )
= ( none @ B ) ) ).
% local.rbt_lookup.simps(1)
thf(fact_248_local_Orbt__lookup__rbt__greater,axiom,
! [B: $tType,K: a,T: rBT_Im246033960le_rbt @ a @ B] :
( ( rBT_Im1259024060reater @ a @ B @ less @ K @ T )
=> ( ( rBT_Im35466040lookup @ a @ B @ less @ T @ K )
= ( none @ B ) ) ) ).
% local.rbt_lookup_rbt_greater
thf(fact_249_local_Orbt__lookup__rbt__less,axiom,
! [B: $tType,K: a,T: rBT_Im246033960le_rbt @ a @ B] :
( ( rBT_Im2075124343t_less @ a @ B @ less @ K @ T )
=> ( ( rBT_Im35466040lookup @ a @ B @ less @ T @ K )
= ( none @ B ) ) ) ).
% local.rbt_lookup_rbt_less
thf(fact_250_ord_Orbt__lookup__rbt__less,axiom,
! [A: $tType,B: $tType,Less2: A > A > $o,K: A,T: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im2075124343t_less @ A @ B @ Less2 @ K @ T )
=> ( ( rBT_Im35466040lookup @ A @ B @ Less2 @ T @ K )
= ( none @ B ) ) ) ).
% ord.rbt_lookup_rbt_less
thf(fact_251_ord_Orbt__lookup__rbt__greater,axiom,
! [A: $tType,B: $tType,Less2: A > A > $o,K: A,T: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1259024060reater @ A @ B @ Less2 @ K @ T )
=> ( ( rBT_Im35466040lookup @ A @ B @ Less2 @ T @ K )
= ( none @ B ) ) ) ).
% ord.rbt_lookup_rbt_greater
thf(fact_252_ord_Orbt__lookup_Osimps_I1_J,axiom,
! [A: $tType,B: $tType,Less2: A > A > $o,K: A] :
( ( rBT_Im35466040lookup @ A @ B @ Less2 @ ( rBT_Im418718756_Empty @ A @ B ) @ K )
= ( none @ B ) ) ).
% ord.rbt_lookup.simps(1)
thf(fact_253_ord_Orbt__lookup__Empty,axiom,
! [B: $tType,A: $tType,Less2: A > A > $o] :
( ( rBT_Im35466040lookup @ A @ B @ Less2 @ ( rBT_Im418718756_Empty @ A @ B ) )
= ( ^ [X2: A] : ( none @ B ) ) ) ).
% ord.rbt_lookup_Empty
thf(fact_254_local_Orbt__lookup_Osimps_I2_J,axiom,
! [B: $tType,K: a,X: a,Uu: rBT_Im1923302023_color,L: rBT_Im246033960le_rbt @ a @ B,Y: B,R: rBT_Im246033960le_rbt @ a @ B] :
( ( ( less @ K @ X )
=> ( ( rBT_Im35466040lookup @ a @ B @ less @ ( rBT_Im480247531Branch @ a @ B @ Uu @ L @ X @ Y @ R ) @ K )
= ( rBT_Im35466040lookup @ a @ B @ less @ L @ K ) ) )
& ( ~ ( less @ K @ X )
=> ( ( ( less @ X @ K )
=> ( ( rBT_Im35466040lookup @ a @ B @ less @ ( rBT_Im480247531Branch @ a @ B @ Uu @ L @ X @ Y @ R ) @ K )
= ( rBT_Im35466040lookup @ a @ B @ less @ R @ K ) ) )
& ( ~ ( less @ X @ K )
=> ( ( rBT_Im35466040lookup @ a @ B @ less @ ( rBT_Im480247531Branch @ a @ B @ Uu @ L @ X @ Y @ R ) @ K )
= ( some @ B @ Y ) ) ) ) ) ) ).
% local.rbt_lookup.simps(2)
%----Type constructors (7)
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___Orderings_Oorder_1,axiom,
! [A6: $tType] : ( order @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_2,axiom,
! [A6: $tType] : ( ord @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_3,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_4,axiom,
ord @ $o @ ( type2 @ $o ) ).
%----Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $true @ X @ Y )
= X ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
! [X3: a] :
( ( member @ a @ X3 @ ( set2 @ a @ ( rBT_Im380146495e_keys @ a @ b @ t2 ) ) )
=> ( less @ k @ X3 ) ) ).
%------------------------------------------------------------------------------