TPTP Problem File: DAT236^1.p
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%------------------------------------------------------------------------------
% File : DAT236^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Red-black trees 931
% 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__931.p [Bla16]
% Status : Theorem
% Rating : 0.67 v8.1.0, 0.75 v7.5.0, 0.67 v7.2.0, 0.75 v7.1.0
% Syntax : Number of formulae : 329 ( 131 unt; 64 typ; 0 def)
% Number of atoms : 848 ( 262 equ; 0 cnn)
% Maximal formula atoms : 11 ( 3 avg)
% Number of connectives : 7626 ( 111 ~; 7 |; 64 &;7152 @)
% ( 0 <=>; 292 =>; 0 <=; 0 <~>)
% Maximal formula depth : 30 ( 11 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 413 ( 413 >; 0 *; 0 +; 0 <<)
% Number of symbols : 61 ( 60 usr; 13 con; 0-8 aty)
% Number of variables : 1744 ( 20 ^;1652 !; 3 ?;1744 :)
% ( 69 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:44:15.827
%------------------------------------------------------------------------------
%----Could-be-implicit typings (10)
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_Nat_Onat,type,
nat: $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 (54)
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_c_Fun_Ofun__upd,type,
fun_upd:
!>[A: $tType,B: $tType] : ( ( A > B ) > A > B > A > B ) ).
thf(sy_c_List_Olist_Oset,type,
set2:
!>[A: $tType] : ( ( list @ A ) > ( 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_OSome,type,
some:
!>[A: $tType] : ( A > ( option @ A ) ) ).
thf(sy_c_Orderings_Oorder_Ostrict__mono,type,
strict_mono:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > $o ) ).
thf(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
thf(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Obalance,type,
rBT_Im1648453169alance:
!>[A: $tType,B: $tType] : ( ( rBT_Im246033960le_rbt @ A @ B ) > A > B > ( rBT_Im246033960le_rbt @ A @ B ) > ( rBT_Im246033960le_rbt @ A @ B ) ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Obalance__left,type,
rBT_Im1278323832e_left:
!>[A: $tType,B: $tType] : ( ( rBT_Im246033960le_rbt @ A @ B ) > A > B > ( rBT_Im246033960le_rbt @ A @ B ) > ( rBT_Im246033960le_rbt @ A @ B ) ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Obalance__right,type,
rBT_Im1053358987_right:
!>[A: $tType,B: $tType] : ( ( rBT_Im246033960le_rbt @ A @ B ) > A > B > ( rBT_Im246033960le_rbt @ A @ B ) > ( rBT_Im246033960le_rbt @ A @ B ) ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Obheight,type,
rBT_Im315265182height:
!>[A: $tType,B: $tType] : ( ( rBT_Im246033960le_rbt @ A @ B ) > nat ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Ocolor_OB,type,
rBT_Impl_Mirabelle_B: rBT_Im1923302023_color ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Ocolor_OR,type,
rBT_Impl_Mirabelle_R: rBT_Im1923302023_color ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Ocolor__of,type,
rBT_Im1805589839lor_of:
!>[A: $tType,B: $tType] : ( ( rBT_Im246033960le_rbt @ A @ B ) > rBT_Im1923302023_color ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Ocombine,type,
rBT_Im1591839508ombine:
!>[A: $tType,B: $tType] : ( ( rBT_Im246033960le_rbt @ A @ B ) > ( rBT_Im246033960le_rbt @ A @ B ) > ( rBT_Im246033960le_rbt @ A @ B ) ) ).
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_Oinv1,type,
rBT_Im1040612427e_inv1:
!>[A: $tType,B: $tType] : ( ( rBT_Im246033960le_rbt @ A @ B ) > $o ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Oinv1l,type,
rBT_Im1071879905_inv1l:
!>[A: $tType,B: $tType] : ( ( rBT_Im246033960le_rbt @ A @ B ) > $o ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Oinv2,type,
rBT_Im1040612428e_inv2:
!>[A: $tType,B: $tType] : ( ( rBT_Im246033960le_rbt @ A @ B ) > $o ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Oord_Ois__rbt,type,
rBT_Im862805236is_rbt:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( rBT_Im246033960le_rbt @ A @ B ) > $o ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Oord_Orbt__del,type,
rBT_Im1714610413bt_del:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > A > ( rBT_Im246033960le_rbt @ A @ B ) > ( rBT_Im246033960le_rbt @ A @ B ) ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Oord_Orbt__del__from__left,type,
rBT_Im1152561598m_left:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > A > ( rBT_Im246033960le_rbt @ A @ B ) > A > B > ( rBT_Im246033960le_rbt @ A @ B ) > ( rBT_Im246033960le_rbt @ A @ B ) ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Oord_Orbt__del__from__right,type,
rBT_Im1808746437_right:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > A > ( rBT_Im246033960le_rbt @ A @ B ) > A > B > ( rBT_Im246033960le_rbt @ A @ B ) > ( rBT_Im246033960le_rbt @ A @ B ) ) ).
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__ins,type,
rBT_Im1756508336bt_ins:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B > B > B ) > A > B > ( rBT_Im246033960le_rbt @ A @ B ) > ( rBT_Im246033960le_rbt @ A @ B ) ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Oord_Orbt__insert,type,
rBT_Im852463031insert:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > A > B > ( rBT_Im246033960le_rbt @ A @ B ) > ( rBT_Im246033960le_rbt @ A @ B ) ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Oord_Orbt__insert__with,type,
rBT_Im241132701t_with:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > B > B ) > A > B > ( rBT_Im246033960le_rbt @ A @ B ) > ( rBT_Im246033960le_rbt @ A @ B ) ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Oord_Orbt__insert__with__key,type,
rBT_Im627674146th_key:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B > B > B ) > A > B > ( rBT_Im246033960le_rbt @ A @ B ) > ( rBT_Im246033960le_rbt @ A @ B ) ) ).
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_Opaint,type,
rBT_Im945281971_paint:
!>[A: $tType,B: $tType] : ( rBT_Im1923302023_color > ( rBT_Im246033960le_rbt @ A @ B ) > ( rBT_Im246033960le_rbt @ A @ B ) ) ).
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__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_Transitive__Closure_Oacyclic,type,
transitive_acyclic:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_bb____,type,
bb: rBT_Im246033960le_rbt @ a @ b ).
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_ss____,type,
ss: b ).
thf(sy_v_v,type,
v: b ).
thf(sy_v_vaa____,type,
vaa: rBT_Im246033960le_rbt @ a @ b ).
thf(sy_v_vbb____,type,
vbb: a ).
thf(sy_v_vc____,type,
vc: rBT_Im246033960le_rbt @ a @ b ).
thf(sy_v_vdd____,type,
vdd: b ).
thf(sy_v_xx____,type,
xx: a ).
thf(sy_v_yy____,type,
yy: a ).
%----Relevant facts (256)
thf(fact_0__C4__2_Oprems_C_I6_J,axiom,
rBT_Im1040612427e_inv1 @ a @ b @ bb ).
% "4_2.prems"(6)
thf(fact_1__C4__2_Oprems_C_I8_J,axiom,
rBT_Im1040612428e_inv2 @ a @ b @ bb ).
% "4_2.prems"(8)
thf(fact_2_local_Oantisym__conv3,axiom,
! [Y: a,X: a] :
( ~ ( less @ Y @ X )
=> ( ( ~ ( less @ X @ Y ) )
= ( X = Y ) ) ) ).
% local.antisym_conv3
thf(fact_3_local_Odual__order_Oasym,axiom,
! [B2: a,A2: a] :
( ( less @ B2 @ A2 )
=> ~ ( less @ A2 @ B2 ) ) ).
% local.dual_order.asym
thf(fact_4_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_5_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_6_local_Oless__asym,axiom,
! [X: a,Y: a] :
( ( less @ X @ Y )
=> ~ ( less @ Y @ X ) ) ).
% local.less_asym
thf(fact_7_local_Oless__asym_H,axiom,
! [A2: a,B2: a] :
( ( less @ A2 @ B2 )
=> ~ ( less @ B2 @ A2 ) ) ).
% local.less_asym'
thf(fact_8_local_Oless__imp__neq,axiom,
! [X: a,Y: a] :
( ( less @ X @ Y )
=> ( X != Y ) ) ).
% local.less_imp_neq
thf(fact_9_local_Oless__imp__not__eq,axiom,
! [X: a,Y: a] :
( ( less @ X @ Y )
=> ( X != Y ) ) ).
% local.less_imp_not_eq
thf(fact_10_local_Oless__imp__not__eq2,axiom,
! [X: a,Y: a] :
( ( less @ X @ Y )
=> ( Y != X ) ) ).
% local.less_imp_not_eq2
thf(fact_11_local_Oless__imp__not__less,axiom,
! [X: a,Y: a] :
( ( less @ X @ Y )
=> ~ ( less @ Y @ X ) ) ).
% local.less_imp_not_less
thf(fact_12_local_Oless__imp__triv,axiom,
! [X: a,Y: a,P: $o] :
( ( less @ X @ Y )
=> ( ( less @ Y @ X )
=> P ) ) ).
% local.less_imp_triv
thf(fact_13_local_Oless__irrefl,axiom,
! [X: a] :
~ ( less @ X @ X ) ).
% local.less_irrefl
thf(fact_14_local_Oless__linear,axiom,
! [X: a,Y: a] :
( ( less @ X @ Y )
| ( X = Y )
| ( less @ Y @ X ) ) ).
% local.less_linear
thf(fact_15_local_Oless__not__sym,axiom,
! [X: a,Y: a] :
( ( less @ X @ Y )
=> ~ ( less @ Y @ X ) ) ).
% local.less_not_sym
thf(fact_16_local_Oless__trans,axiom,
! [X: a,Y: a,Z: a] :
( ( less @ X @ Y )
=> ( ( less @ Y @ Z )
=> ( less @ X @ Z ) ) ) ).
% local.less_trans
thf(fact_17_local_Olinorder__cases,axiom,
! [X: a,Y: a] :
( ~ ( less @ X @ Y )
=> ( ( X != Y )
=> ( less @ Y @ X ) ) ) ).
% local.linorder_cases
thf(fact_18_local_OneqE,axiom,
! [X: a,Y: a] :
( ( X != Y )
=> ( ~ ( less @ X @ Y )
=> ( less @ Y @ X ) ) ) ).
% local.neqE
thf(fact_19_local_Oneq__iff,axiom,
! [X: a,Y: a] :
( ( X != Y )
= ( ( less @ X @ Y )
| ( less @ Y @ X ) ) ) ).
% local.neq_iff
thf(fact_20_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_21_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_22_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_23_local_Oorder_Oasym,axiom,
! [A2: a,B2: a] :
( ( less @ A2 @ B2 )
=> ~ ( less @ B2 @ A2 ) ) ).
% local.order.asym
thf(fact_24_local_Oorder_Oirrefl,axiom,
! [A2: a] :
~ ( less @ A2 @ A2 ) ).
% local.order.irrefl
thf(fact_25_local_Oorder_Ostrict__implies__not__eq,axiom,
! [A2: a,B2: a] :
( ( less @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% local.order.strict_implies_not_eq
thf(fact_26_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_27__C4__2_Oprems_C_I2_J,axiom,
rBT_Im759614907sorted @ a @ b @ less @ bb ).
% "4_2.prems"(2)
thf(fact_28__C4__2_Oprems_C_I10_J,axiom,
less @ xx @ yy ).
% "4_2.prems"(10)
thf(fact_29_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_30_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_31__C4__2_Oprems_C_I4_J,axiom,
rBT_Im1259024060reater @ a @ b @ less @ yy @ bb ).
% "4_2.prems"(4)
thf(fact_32__C4__2_Oprems_C_I5_J,axiom,
rBT_Im1040612427e_inv1 @ a @ b @ ( rBT_Im480247531Branch @ a @ b @ rBT_Impl_Mirabelle_R @ vaa @ vbb @ vdd @ vc ) ).
% "4_2.prems"(5)
thf(fact_33__C4__2_Oprems_C_I7_J,axiom,
rBT_Im1040612428e_inv2 @ a @ b @ ( rBT_Im480247531Branch @ a @ b @ rBT_Impl_Mirabelle_R @ vaa @ vbb @ vdd @ vc ) ).
% "4_2.prems"(7)
thf(fact_34_rbt_Oinject,axiom,
! [B: $tType,A: $tType,X21: rBT_Im1923302023_color,X22: rBT_Im246033960le_rbt @ A @ B,X23: A,X24: 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 @ X23 @ X24 @ X25 )
= ( rBT_Im480247531Branch @ A @ B @ Y21 @ Y22 @ Y23 @ Y24 @ Y25 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 )
& ( X23 = Y23 )
& ( X24 = Y24 )
& ( X25 = Y25 ) ) ) ).
% rbt.inject
thf(fact_35__C4__2_Oprems_C_I1_J,axiom,
rBT_Im759614907sorted @ a @ b @ less @ ( rBT_Im480247531Branch @ a @ b @ rBT_Impl_Mirabelle_R @ vaa @ vbb @ vdd @ vc ) ).
% "4_2.prems"(1)
thf(fact_36_rbt__del__from__left__rbt__greater,axiom,
! [B: $tType,V: a,Lt: rBT_Im246033960le_rbt @ a @ B,Rt: rBT_Im246033960le_rbt @ a @ B,K: a,X: a,Y: B] :
( ( rBT_Im1259024060reater @ a @ B @ less @ V @ Lt )
=> ( ( rBT_Im1259024060reater @ a @ B @ less @ V @ Rt )
=> ( ( less @ V @ K )
=> ( rBT_Im1259024060reater @ a @ B @ less @ V @ ( rBT_Im1152561598m_left @ a @ B @ less @ X @ Lt @ K @ Y @ Rt ) ) ) ) ) ).
% rbt_del_from_left_rbt_greater
thf(fact_37_rbt__del__from__left__rbt__less,axiom,
! [B: $tType,V: a,Lt: rBT_Im246033960le_rbt @ a @ B,Rt: rBT_Im246033960le_rbt @ a @ B,K: a,X: a,Y: B] :
( ( rBT_Im2075124343t_less @ a @ B @ less @ V @ Lt )
=> ( ( rBT_Im2075124343t_less @ a @ B @ less @ V @ Rt )
=> ( ( less @ K @ V )
=> ( rBT_Im2075124343t_less @ a @ B @ less @ V @ ( rBT_Im1152561598m_left @ a @ B @ less @ X @ Lt @ K @ Y @ Rt ) ) ) ) ) ).
% rbt_del_from_left_rbt_less
thf(fact_38__C4__2_Oprems_C_I9_J,axiom,
( ( rBT_Im315265182height @ a @ b @ ( rBT_Im480247531Branch @ a @ b @ rBT_Impl_Mirabelle_R @ vaa @ vbb @ vdd @ vc ) )
= ( rBT_Im315265182height @ a @ b @ bb ) ) ).
% "4_2.prems"(9)
thf(fact_39__C4__2_Oprems_C_I3_J,axiom,
rBT_Im2075124343t_less @ a @ b @ less @ yy @ ( rBT_Im480247531Branch @ a @ b @ rBT_Impl_Mirabelle_R @ vaa @ vbb @ vdd @ vc ) ).
% "4_2.prems"(3)
thf(fact_40_rbt__del__rbt__greater,axiom,
! [B: $tType,V: a,Lt: rBT_Im246033960le_rbt @ a @ B,X: a] :
( ( rBT_Im1259024060reater @ a @ B @ less @ V @ Lt )
=> ( rBT_Im1259024060reater @ a @ B @ less @ V @ ( rBT_Im1714610413bt_del @ a @ B @ less @ X @ Lt ) ) ) ).
% rbt_del_rbt_greater
thf(fact_41_rbt__del__rbt__less,axiom,
! [B: $tType,V: a,Lt: rBT_Im246033960le_rbt @ a @ B,X: a] :
( ( rBT_Im2075124343t_less @ a @ B @ less @ V @ Lt )
=> ( rBT_Im2075124343t_less @ a @ B @ less @ V @ ( rBT_Im1714610413bt_del @ a @ B @ less @ X @ Lt ) ) ) ).
% rbt_del_rbt_less
thf(fact_42_rbt__del__rbt__sorted,axiom,
! [B: $tType,Lt: rBT_Im246033960le_rbt @ a @ B,X: a] :
( ( rBT_Im759614907sorted @ a @ B @ less @ Lt )
=> ( rBT_Im759614907sorted @ a @ B @ less @ ( rBT_Im1714610413bt_del @ a @ B @ less @ X @ Lt ) ) ) ).
% rbt_del_rbt_sorted
thf(fact_43_local_Oset__entries__inject,axiom,
! [B: $tType,T1: rBT_Im246033960le_rbt @ a @ B,T2: rBT_Im246033960le_rbt @ a @ B] :
( ( rBT_Im759614907sorted @ a @ B @ less @ T1 )
=> ( ( rBT_Im759614907sorted @ a @ B @ less @ T2 )
=> ( ( ( set2 @ ( product_prod @ a @ B ) @ ( rBT_Im954575269ntries @ a @ B @ T1 ) )
= ( set2 @ ( product_prod @ a @ B ) @ ( rBT_Im954575269ntries @ a @ B @ T2 ) ) )
= ( ( rBT_Im954575269ntries @ a @ B @ T1 )
= ( rBT_Im954575269ntries @ a @ B @ T2 ) ) ) ) ) ).
% local.set_entries_inject
thf(fact_44_local_Orbt__del__from__left_Osimps_I3_J,axiom,
! [B: $tType,X: a,Va: rBT_Im246033960le_rbt @ a @ B,Vb: a,Vc: B,Vd: rBT_Im246033960le_rbt @ a @ B,Y: a,S: B,B2: rBT_Im246033960le_rbt @ a @ B] :
( ( rBT_Im1152561598m_left @ a @ B @ less @ X @ ( rBT_Im480247531Branch @ a @ B @ rBT_Impl_Mirabelle_R @ Va @ Vb @ Vc @ Vd ) @ Y @ S @ B2 )
= ( rBT_Im480247531Branch @ a @ B @ rBT_Impl_Mirabelle_R @ ( rBT_Im1714610413bt_del @ a @ B @ less @ X @ ( rBT_Im480247531Branch @ a @ B @ rBT_Impl_Mirabelle_R @ Va @ Vb @ Vc @ Vd ) ) @ Y @ S @ B2 ) ) ).
% local.rbt_del_from_left.simps(3)
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__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_50__C4__2_Ohyps_C,axiom,
( ( rBT_Im759614907sorted @ a @ b @ less @ ( rBT_Im480247531Branch @ a @ b @ rBT_Impl_Mirabelle_R @ vaa @ vbb @ vdd @ vc ) )
=> ( ( rBT_Im1040612427e_inv1 @ a @ b @ ( rBT_Im480247531Branch @ a @ b @ rBT_Impl_Mirabelle_R @ vaa @ vbb @ vdd @ vc ) )
=> ( ( rBT_Im1040612428e_inv2 @ a @ b @ ( rBT_Im480247531Branch @ a @ b @ rBT_Impl_Mirabelle_R @ vaa @ vbb @ vdd @ vc ) )
=> ( ( member @ ( product_prod @ a @ b ) @ ( product_Pair @ a @ b @ k @ v ) @ ( set2 @ ( product_prod @ a @ b ) @ ( rBT_Im954575269ntries @ a @ b @ ( rBT_Im1714610413bt_del @ a @ b @ less @ xx @ ( rBT_Im480247531Branch @ a @ b @ rBT_Impl_Mirabelle_R @ vaa @ vbb @ vdd @ vc ) ) ) ) )
= ( ( xx != k )
& ( member @ ( product_prod @ a @ b ) @ ( product_Pair @ a @ b @ k @ v ) @ ( set2 @ ( product_prod @ a @ b ) @ ( rBT_Im954575269ntries @ a @ b @ ( rBT_Im480247531Branch @ a @ b @ rBT_Impl_Mirabelle_R @ vaa @ vbb @ vdd @ vc ) ) ) ) ) ) ) ) ) ).
% "4_2.hyps"
thf(fact_51_local_Orbt__greater__simps_I2_J,axiom,
! [C2: $tType,K: a,C: rBT_Im1923302023_color,Lt: rBT_Im246033960le_rbt @ a @ C2,Kt: a,V: C2,Rt: rBT_Im246033960le_rbt @ a @ C2] :
( ( rBT_Im1259024060reater @ a @ C2 @ less @ K @ ( rBT_Im480247531Branch @ a @ C2 @ C @ Lt @ Kt @ V @ 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_52_local_Orbt__less__simps_I2_J,axiom,
! [C2: $tType,K: a,C: rBT_Im1923302023_color,Lt: rBT_Im246033960le_rbt @ a @ C2,Kt: a,V: C2,Rt: rBT_Im246033960le_rbt @ a @ C2] :
( ( rBT_Im2075124343t_less @ a @ C2 @ less @ K @ ( rBT_Im480247531Branch @ a @ C2 @ C @ Lt @ Kt @ V @ 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_53_ord_Orbt__del__from__left_Osimps_I3_J,axiom,
! [B: $tType,A: $tType,Less: A > A > $o,X: A,Va: rBT_Im246033960le_rbt @ A @ B,Vb: A,Vc: B,Vd: rBT_Im246033960le_rbt @ A @ B,Y: A,S: B,B2: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1152561598m_left @ A @ B @ Less @ X @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ Va @ Vb @ Vc @ Vd ) @ Y @ S @ B2 )
= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ ( rBT_Im1714610413bt_del @ A @ B @ Less @ X @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ Va @ Vb @ Vc @ Vd ) ) @ Y @ S @ B2 ) ) ).
% ord.rbt_del_from_left.simps(3)
thf(fact_54_local_OacyclicI__order,axiom,
! [B: $tType,R: set @ ( product_prod @ B @ B ),F: B > a] :
( ! [A4: B,B3: B] :
( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A4 @ B3 ) @ R )
=> ( less @ ( F @ B3 ) @ ( F @ A4 ) ) )
=> ( transitive_acyclic @ B @ R ) ) ).
% local.acyclicI_order
thf(fact_55_local_OgreaterThan__def,axiom,
! [L: a] :
( ( set_greaterThan @ a @ less @ L )
= ( collect @ a @ ( less @ L ) ) ) ).
% local.greaterThan_def
thf(fact_56_local_Orbt__sorted_Osimps_I2_J,axiom,
! [B: $tType,C: rBT_Im1923302023_color,L: rBT_Im246033960le_rbt @ a @ B,K: a,V: B,R: rBT_Im246033960le_rbt @ a @ B] :
( ( rBT_Im759614907sorted @ a @ B @ less @ ( rBT_Im480247531Branch @ a @ B @ C @ L @ K @ V @ 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_57_local_Orbt__insert__rbt__sorted,axiom,
! [B: $tType,T: rBT_Im246033960le_rbt @ a @ B,K: a,V: B] :
( ( rBT_Im759614907sorted @ a @ B @ less @ T )
=> ( rBT_Im759614907sorted @ a @ B @ less @ ( rBT_Im852463031insert @ a @ B @ less @ K @ V @ T ) ) ) ).
% local.rbt_insert_rbt_sorted
thf(fact_58_local_Orbt__insertw__rbt__sorted,axiom,
! [B: $tType,T: rBT_Im246033960le_rbt @ a @ B,F: B > B > B,K: a,V: B] :
( ( rBT_Im759614907sorted @ a @ B @ less @ T )
=> ( rBT_Im759614907sorted @ a @ B @ less @ ( rBT_Im241132701t_with @ a @ B @ less @ F @ K @ V @ T ) ) ) ).
% local.rbt_insertw_rbt_sorted
thf(fact_59_local_OlessThan__def,axiom,
! [U: a] :
( ( set_lessThan @ a @ less @ U )
= ( collect @ a
@ ^ [X2: a] : ( less @ X2 @ U ) ) ) ).
% local.lessThan_def
thf(fact_60_local_Orbt__del__from__right_Osimps_I3_J,axiom,
! [B: $tType,X: a,A2: rBT_Im246033960le_rbt @ a @ B,Y: a,S: B,Va: rBT_Im246033960le_rbt @ a @ B,Vb: a,Vc: B,Vd: rBT_Im246033960le_rbt @ a @ B] :
( ( rBT_Im1808746437_right @ a @ B @ less @ X @ A2 @ Y @ S @ ( rBT_Im480247531Branch @ a @ B @ rBT_Impl_Mirabelle_R @ Va @ Vb @ Vc @ Vd ) )
= ( rBT_Im480247531Branch @ a @ B @ rBT_Impl_Mirabelle_R @ A2 @ Y @ S @ ( rBT_Im1714610413bt_del @ a @ B @ less @ X @ ( rBT_Im480247531Branch @ a @ B @ rBT_Impl_Mirabelle_R @ Va @ Vb @ Vc @ Vd ) ) ) ) ).
% local.rbt_del_from_right.simps(3)
thf(fact_61_local_Orbt__ins_Osimps_I3_J,axiom,
! [B: $tType,K: a,X: a,F: a > B > B > B,V: B,L: rBT_Im246033960le_rbt @ a @ B,Y: B,R: rBT_Im246033960le_rbt @ a @ B] :
( ( ( less @ K @ X )
=> ( ( rBT_Im1756508336bt_ins @ a @ B @ less @ F @ K @ V @ ( rBT_Im480247531Branch @ a @ B @ rBT_Impl_Mirabelle_R @ L @ X @ Y @ R ) )
= ( rBT_Im480247531Branch @ a @ B @ rBT_Impl_Mirabelle_R @ ( rBT_Im1756508336bt_ins @ a @ B @ less @ F @ K @ V @ L ) @ X @ Y @ R ) ) )
& ( ~ ( less @ K @ X )
=> ( ( ( less @ X @ K )
=> ( ( rBT_Im1756508336bt_ins @ a @ B @ less @ F @ K @ V @ ( rBT_Im480247531Branch @ a @ B @ rBT_Impl_Mirabelle_R @ L @ X @ Y @ R ) )
= ( rBT_Im480247531Branch @ a @ B @ rBT_Impl_Mirabelle_R @ L @ X @ Y @ ( rBT_Im1756508336bt_ins @ a @ B @ less @ F @ K @ V @ R ) ) ) )
& ( ~ ( less @ X @ K )
=> ( ( rBT_Im1756508336bt_ins @ a @ B @ less @ F @ K @ V @ ( rBT_Im480247531Branch @ a @ B @ rBT_Impl_Mirabelle_R @ L @ X @ Y @ R ) )
= ( rBT_Im480247531Branch @ a @ B @ rBT_Impl_Mirabelle_R @ L @ X @ ( F @ K @ Y @ V ) @ R ) ) ) ) ) ) ).
% local.rbt_ins.simps(3)
thf(fact_62_local_Orbt__lookup__from__in__tree,axiom,
! [B: $tType,T1: rBT_Im246033960le_rbt @ a @ B,T2: rBT_Im246033960le_rbt @ a @ B,K: a] :
( ( rBT_Im759614907sorted @ a @ B @ less @ T1 )
=> ( ( rBT_Im759614907sorted @ a @ B @ less @ T2 )
=> ( ! [V2: B] :
( ( member @ ( product_prod @ a @ B ) @ ( product_Pair @ a @ B @ K @ V2 ) @ ( set2 @ ( product_prod @ a @ B ) @ ( rBT_Im954575269ntries @ a @ B @ T1 ) ) )
= ( member @ ( product_prod @ a @ B ) @ ( product_Pair @ a @ B @ K @ V2 ) @ ( set2 @ ( product_prod @ a @ B ) @ ( rBT_Im954575269ntries @ a @ B @ T2 ) ) ) )
=> ( ( rBT_Im35466040lookup @ a @ B @ less @ T1 @ K )
= ( rBT_Im35466040lookup @ a @ B @ less @ T2 @ K ) ) ) ) ) ).
% local.rbt_lookup_from_in_tree
thf(fact_63_local_Oins__rbt__sorted,axiom,
! [B: $tType,T: rBT_Im246033960le_rbt @ a @ B,F: a > B > B > B,K: a,X: B] :
( ( rBT_Im759614907sorted @ a @ B @ less @ T )
=> ( rBT_Im759614907sorted @ a @ B @ less @ ( rBT_Im1756508336bt_ins @ a @ B @ less @ F @ K @ X @ T ) ) ) ).
% local.ins_rbt_sorted
thf(fact_64_rbt__del__from__right__rbt__less,axiom,
! [B: $tType,V: a,Lt: rBT_Im246033960le_rbt @ a @ B,Rt: rBT_Im246033960le_rbt @ a @ B,K: a,X: a,Y: B] :
( ( rBT_Im2075124343t_less @ a @ B @ less @ V @ Lt )
=> ( ( rBT_Im2075124343t_less @ a @ B @ less @ V @ Rt )
=> ( ( less @ K @ V )
=> ( rBT_Im2075124343t_less @ a @ B @ less @ V @ ( rBT_Im1808746437_right @ a @ B @ less @ X @ Lt @ K @ Y @ Rt ) ) ) ) ) ).
% rbt_del_from_right_rbt_less
thf(fact_65_rbt__del__from__right__rbt__greater,axiom,
! [B: $tType,V: a,Lt: rBT_Im246033960le_rbt @ a @ B,Rt: rBT_Im246033960le_rbt @ a @ B,K: a,X: a,Y: B] :
( ( rBT_Im1259024060reater @ a @ B @ less @ V @ Lt )
=> ( ( rBT_Im1259024060reater @ a @ B @ less @ V @ Rt )
=> ( ( less @ V @ K )
=> ( rBT_Im1259024060reater @ a @ B @ less @ V @ ( rBT_Im1808746437_right @ a @ B @ less @ X @ Lt @ K @ Y @ Rt ) ) ) ) ) ).
% rbt_del_from_right_rbt_greater
thf(fact_66_local_Oentries__rbt__lookup,axiom,
! [B: $tType,T1: rBT_Im246033960le_rbt @ a @ B,T2: rBT_Im246033960le_rbt @ a @ B] :
( ( rBT_Im759614907sorted @ a @ B @ less @ T1 )
=> ( ( rBT_Im759614907sorted @ a @ B @ less @ T2 )
=> ( ( ( rBT_Im954575269ntries @ a @ B @ T1 )
= ( rBT_Im954575269ntries @ a @ B @ T2 ) )
= ( ( rBT_Im35466040lookup @ a @ B @ less @ T1 )
= ( rBT_Im35466040lookup @ a @ B @ less @ T2 ) ) ) ) ) ).
% local.entries_rbt_lookup
thf(fact_67_local_Oentries__eqI,axiom,
! [B: $tType,T1: rBT_Im246033960le_rbt @ a @ B,T2: rBT_Im246033960le_rbt @ a @ B] :
( ( rBT_Im759614907sorted @ a @ B @ less @ T1 )
=> ( ( rBT_Im759614907sorted @ a @ B @ less @ T2 )
=> ( ( ( rBT_Im35466040lookup @ a @ B @ less @ T1 )
= ( rBT_Im35466040lookup @ a @ B @ less @ T2 ) )
=> ( ( rBT_Im954575269ntries @ a @ B @ T1 )
= ( rBT_Im954575269ntries @ a @ B @ T2 ) ) ) ) ) ).
% local.entries_eqI
thf(fact_68_local_Oins__inv1__inv2_I1_J,axiom,
! [B: $tType,T: rBT_Im246033960le_rbt @ a @ B,F: a > B > B > B,K: a,X: B] :
( ( rBT_Im1040612427e_inv1 @ a @ B @ T )
=> ( ( rBT_Im1040612428e_inv2 @ a @ B @ T )
=> ( rBT_Im1040612428e_inv2 @ a @ B @ ( rBT_Im1756508336bt_ins @ a @ B @ less @ F @ K @ X @ T ) ) ) ) ).
% local.ins_inv1_inv2(1)
thf(fact_69_local_Oins__inv1__inv2_I2_J,axiom,
! [B: $tType,T: rBT_Im246033960le_rbt @ a @ B,F: a > B > B > B,K: a,X: B] :
( ( rBT_Im1040612427e_inv1 @ a @ B @ T )
=> ( ( rBT_Im1040612428e_inv2 @ a @ B @ T )
=> ( ( rBT_Im315265182height @ a @ B @ ( rBT_Im1756508336bt_ins @ a @ B @ less @ F @ K @ X @ T ) )
= ( rBT_Im315265182height @ a @ B @ T ) ) ) ) ).
% local.ins_inv1_inv2(2)
thf(fact_70_local_OlessThan__iff,axiom,
! [I: a,K: a] :
( ( member @ a @ I @ ( set_lessThan @ a @ less @ K ) )
= ( less @ I @ K ) ) ).
% local.lessThan_iff
thf(fact_71_local_OgreaterThan__iff,axiom,
! [I: a,K: a] :
( ( member @ a @ I @ ( set_greaterThan @ a @ less @ K ) )
= ( less @ K @ I ) ) ).
% local.greaterThan_iff
thf(fact_72_local_Oins__rbt__less,axiom,
! [B: $tType,V: a,F: a > B > B > B,K: a,X: B,T: rBT_Im246033960le_rbt @ a @ B] :
( ( rBT_Im2075124343t_less @ a @ B @ less @ V @ ( rBT_Im1756508336bt_ins @ a @ B @ less @ F @ K @ X @ T ) )
= ( ( rBT_Im2075124343t_less @ a @ B @ less @ V @ T )
& ( less @ K @ V ) ) ) ).
% local.ins_rbt_less
thf(fact_73_local_Oins__rbt__greater,axiom,
! [B: $tType,V: a,F: a > B > B > B,K: a,X: B,T: rBT_Im246033960le_rbt @ a @ B] :
( ( rBT_Im1259024060reater @ a @ B @ less @ V @ ( rBT_Im1756508336bt_ins @ a @ B @ less @ F @ K @ X @ T ) )
= ( ( rBT_Im1259024060reater @ a @ B @ less @ V @ T )
& ( less @ V @ K ) ) ) ).
% local.ins_rbt_greater
thf(fact_74_local_Orbt__insertwk__rbt__sorted,axiom,
! [B: $tType,T: rBT_Im246033960le_rbt @ a @ B,F: a > B > B > B,K: a,X: B] :
( ( rBT_Im759614907sorted @ a @ B @ less @ T )
=> ( rBT_Im759614907sorted @ a @ B @ less @ ( rBT_Im627674146th_key @ a @ B @ less @ F @ K @ X @ T ) ) ) ).
% local.rbt_insertwk_rbt_sorted
thf(fact_75_local_Orbt__insertw__def,axiom,
! [B: $tType,F: B > B > B] :
( ( rBT_Im241132701t_with @ a @ B @ less @ F )
= ( rBT_Im627674146th_key @ a @ B @ less
@ ^ [Uu: a] : F ) ) ).
% local.rbt_insertw_def
thf(fact_76_local_Orbt__insert__def,axiom,
! [B: $tType] :
( ( rBT_Im852463031insert @ a @ B @ less )
= ( rBT_Im627674146th_key @ a @ B @ less
@ ^ [Uu: a,Uv: B,Nv: B] : Nv ) ) ).
% local.rbt_insert_def
thf(fact_77_ord_Oins__inv1__inv2_I2_J,axiom,
! [B: $tType,A: $tType,T: rBT_Im246033960le_rbt @ A @ B,Less: A > A > $o,F: A > B > B > B,K: A,X: B] :
( ( rBT_Im1040612427e_inv1 @ A @ B @ T )
=> ( ( rBT_Im1040612428e_inv2 @ A @ B @ T )
=> ( ( rBT_Im315265182height @ A @ B @ ( rBT_Im1756508336bt_ins @ A @ B @ Less @ F @ K @ X @ T ) )
= ( rBT_Im315265182height @ A @ B @ T ) ) ) ) ).
% ord.ins_inv1_inv2(2)
thf(fact_78_ord_Oins__inv1__inv2_I1_J,axiom,
! [B: $tType,A: $tType,T: rBT_Im246033960le_rbt @ A @ B,Less: A > A > $o,F: A > B > B > B,K: A,X: B] :
( ( rBT_Im1040612427e_inv1 @ A @ B @ T )
=> ( ( rBT_Im1040612428e_inv2 @ A @ B @ T )
=> ( rBT_Im1040612428e_inv2 @ A @ B @ ( rBT_Im1756508336bt_ins @ A @ B @ Less @ F @ K @ X @ T ) ) ) ) ).
% ord.ins_inv1_inv2(1)
thf(fact_79_inv2_Osimps_I2_J,axiom,
! [B: $tType,A: $tType,C: rBT_Im1923302023_color,Lt: rBT_Im246033960le_rbt @ A @ B,K: A,V: B,Rt: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1040612428e_inv2 @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ C @ Lt @ K @ V @ Rt ) )
= ( ( rBT_Im1040612428e_inv2 @ A @ B @ Lt )
& ( rBT_Im1040612428e_inv2 @ A @ B @ Rt )
& ( ( rBT_Im315265182height @ A @ B @ Lt )
= ( rBT_Im315265182height @ A @ B @ Rt ) ) ) ) ).
% inv2.simps(2)
thf(fact_80_ord_Orbt__sorted_Osimps_I2_J,axiom,
! [B: $tType,A: $tType,Less: A > A > $o,C: rBT_Im1923302023_color,L: rBT_Im246033960le_rbt @ A @ B,K: A,V: B,R: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im759614907sorted @ A @ B @ Less @ ( rBT_Im480247531Branch @ A @ B @ C @ L @ K @ V @ 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 ) ) ) ).
% ord.rbt_sorted.simps(2)
thf(fact_81_ord_Orbt__ins_Osimps_I3_J,axiom,
! [B: $tType,A: $tType,Less: A > A > $o,K: A,X: A,F: A > B > B > B,V: B,L: rBT_Im246033960le_rbt @ A @ B,Y: B,R: rBT_Im246033960le_rbt @ A @ B] :
( ( ( Less @ K @ X )
=> ( ( rBT_Im1756508336bt_ins @ A @ B @ Less @ F @ K @ V @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ L @ X @ Y @ R ) )
= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ ( rBT_Im1756508336bt_ins @ A @ B @ Less @ F @ K @ V @ L ) @ X @ Y @ R ) ) )
& ( ~ ( Less @ K @ X )
=> ( ( ( Less @ X @ K )
=> ( ( rBT_Im1756508336bt_ins @ A @ B @ Less @ F @ K @ V @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ L @ X @ Y @ R ) )
= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ L @ X @ Y @ ( rBT_Im1756508336bt_ins @ A @ B @ Less @ F @ K @ V @ R ) ) ) )
& ( ~ ( Less @ X @ K )
=> ( ( rBT_Im1756508336bt_ins @ A @ B @ Less @ F @ K @ V @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ L @ X @ Y @ R ) )
= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ L @ X @ ( F @ K @ Y @ V ) @ R ) ) ) ) ) ) ).
% ord.rbt_ins.simps(3)
thf(fact_82_ord_Orbt__del__from__right_Osimps_I3_J,axiom,
! [B: $tType,A: $tType,Less: A > A > $o,X: A,A2: rBT_Im246033960le_rbt @ A @ B,Y: A,S: B,Va: rBT_Im246033960le_rbt @ A @ B,Vb: A,Vc: B,Vd: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1808746437_right @ A @ B @ Less @ X @ A2 @ Y @ S @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ Va @ Vb @ Vc @ Vd ) )
= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ A2 @ Y @ S @ ( rBT_Im1714610413bt_del @ A @ B @ Less @ X @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ Va @ Vb @ Vc @ Vd ) ) ) ) ).
% ord.rbt_del_from_right.simps(3)
thf(fact_83_ord_Orbt__greater__simps_I2_J,axiom,
! [C2: $tType,A: $tType,Less: A > A > $o,K: A,C: rBT_Im1923302023_color,Lt: rBT_Im246033960le_rbt @ A @ C2,Kt: A,V: C2,Rt: rBT_Im246033960le_rbt @ A @ C2] :
( ( rBT_Im1259024060reater @ A @ C2 @ Less @ K @ ( rBT_Im480247531Branch @ A @ C2 @ C @ Lt @ Kt @ V @ Rt ) )
= ( ( Less @ K @ Kt )
& ( rBT_Im1259024060reater @ A @ C2 @ Less @ K @ Lt )
& ( rBT_Im1259024060reater @ A @ C2 @ Less @ K @ Rt ) ) ) ).
% ord.rbt_greater_simps(2)
thf(fact_84_ord_Orbt__less__simps_I2_J,axiom,
! [C2: $tType,A: $tType,Less: A > A > $o,K: A,C: rBT_Im1923302023_color,Lt: rBT_Im246033960le_rbt @ A @ C2,Kt: A,V: C2,Rt: rBT_Im246033960le_rbt @ A @ C2] :
( ( rBT_Im2075124343t_less @ A @ C2 @ Less @ K @ ( rBT_Im480247531Branch @ A @ C2 @ C @ Lt @ Kt @ V @ Rt ) )
= ( ( Less @ Kt @ K )
& ( rBT_Im2075124343t_less @ A @ C2 @ Less @ K @ Lt )
& ( rBT_Im2075124343t_less @ A @ C2 @ Less @ K @ Rt ) ) ) ).
% ord.rbt_less_simps(2)
thf(fact_85_local_Orbt__insertw__is__rbt,axiom,
! [B: $tType,T: rBT_Im246033960le_rbt @ a @ B,F: B > B > B,K: a,V: B] :
( ( rBT_Im862805236is_rbt @ a @ B @ less @ T )
=> ( rBT_Im862805236is_rbt @ a @ B @ less @ ( rBT_Im241132701t_with @ a @ B @ less @ F @ K @ V @ T ) ) ) ).
% local.rbt_insertw_is_rbt
thf(fact_86_local_Orbt__insert__is__rbt,axiom,
! [B: $tType,T: rBT_Im246033960le_rbt @ a @ B,K: a,V: B] :
( ( rBT_Im862805236is_rbt @ a @ B @ less @ T )
=> ( rBT_Im862805236is_rbt @ a @ B @ less @ ( rBT_Im852463031insert @ a @ B @ less @ K @ V @ T ) ) ) ).
% local.rbt_insert_is_rbt
thf(fact_87_local_Oins__inv1__inv2_I4_J,axiom,
! [B: $tType,T: rBT_Im246033960le_rbt @ a @ B,F: a > B > B > B,K: a,X: B] :
( ( rBT_Im1040612427e_inv1 @ a @ B @ T )
=> ( ( rBT_Im1040612428e_inv2 @ a @ B @ T )
=> ( rBT_Im1071879905_inv1l @ a @ B @ ( rBT_Im1756508336bt_ins @ a @ B @ less @ F @ K @ X @ T ) ) ) ) ).
% local.ins_inv1_inv2(4)
thf(fact_88_local_Ois__rbt__rbt__sorted,axiom,
! [B: $tType,T: rBT_Im246033960le_rbt @ a @ B] :
( ( rBT_Im862805236is_rbt @ a @ B @ less @ T )
=> ( rBT_Im759614907sorted @ a @ B @ less @ T ) ) ).
% local.is_rbt_rbt_sorted
thf(fact_89_local_Orbt__del_Osimps_I2_J,axiom,
! [B: $tType,X: a,Y: a,C: rBT_Im1923302023_color,A2: rBT_Im246033960le_rbt @ a @ B,S: B,B2: rBT_Im246033960le_rbt @ a @ B] :
( ( ( less @ X @ Y )
=> ( ( rBT_Im1714610413bt_del @ a @ B @ less @ X @ ( rBT_Im480247531Branch @ a @ B @ C @ A2 @ Y @ S @ B2 ) )
= ( rBT_Im1152561598m_left @ a @ B @ less @ X @ A2 @ Y @ S @ B2 ) ) )
& ( ~ ( less @ X @ Y )
=> ( ( ( less @ Y @ X )
=> ( ( rBT_Im1714610413bt_del @ a @ B @ less @ X @ ( rBT_Im480247531Branch @ a @ B @ C @ A2 @ Y @ S @ B2 ) )
= ( rBT_Im1808746437_right @ a @ B @ less @ X @ A2 @ Y @ S @ B2 ) ) )
& ( ~ ( less @ Y @ X )
=> ( ( rBT_Im1714610413bt_del @ a @ B @ less @ X @ ( rBT_Im480247531Branch @ a @ B @ C @ A2 @ Y @ S @ B2 ) )
= ( rBT_Im1591839508ombine @ a @ B @ A2 @ B2 ) ) ) ) ) ) ).
% local.rbt_del.simps(2)
thf(fact_90_local_Obalance__right__rbt__sorted,axiom,
! [B: $tType,L: rBT_Im246033960le_rbt @ a @ B,R: rBT_Im246033960le_rbt @ a @ B,K: a,V: B] :
( ( rBT_Im759614907sorted @ a @ B @ less @ L )
=> ( ( rBT_Im759614907sorted @ a @ B @ less @ R )
=> ( ( rBT_Im2075124343t_less @ a @ B @ less @ K @ L )
=> ( ( rBT_Im1259024060reater @ a @ B @ less @ K @ R )
=> ( rBT_Im759614907sorted @ a @ B @ less @ ( rBT_Im1053358987_right @ a @ B @ L @ K @ V @ R ) ) ) ) ) ) ).
% local.balance_right_rbt_sorted
thf(fact_91_local_Ocombine__rbt__sorted,axiom,
! [B: $tType,L: rBT_Im246033960le_rbt @ a @ B,R: rBT_Im246033960le_rbt @ a @ B,K: a] :
( ( rBT_Im759614907sorted @ a @ B @ less @ L )
=> ( ( rBT_Im759614907sorted @ a @ B @ less @ R )
=> ( ( rBT_Im2075124343t_less @ a @ B @ less @ K @ L )
=> ( ( rBT_Im1259024060reater @ a @ B @ less @ K @ R )
=> ( rBT_Im759614907sorted @ a @ B @ less @ ( rBT_Im1591839508ombine @ a @ B @ L @ R ) ) ) ) ) ) ).
% local.combine_rbt_sorted
thf(fact_92_local_Ocombine__rbt__greater,axiom,
! [B: $tType,K: a,L: rBT_Im246033960le_rbt @ a @ B,R: rBT_Im246033960le_rbt @ a @ B] :
( ( rBT_Im1259024060reater @ a @ B @ less @ K @ L )
=> ( ( rBT_Im1259024060reater @ a @ B @ less @ K @ R )
=> ( rBT_Im1259024060reater @ a @ B @ less @ K @ ( rBT_Im1591839508ombine @ a @ B @ L @ R ) ) ) ) ).
% local.combine_rbt_greater
thf(fact_93_local_Ocombine__rbt__less,axiom,
! [B: $tType,K: a,L: rBT_Im246033960le_rbt @ a @ B,R: rBT_Im246033960le_rbt @ a @ B] :
( ( rBT_Im2075124343t_less @ a @ B @ less @ K @ L )
=> ( ( rBT_Im2075124343t_less @ a @ B @ less @ K @ R )
=> ( rBT_Im2075124343t_less @ a @ B @ less @ K @ ( rBT_Im1591839508ombine @ a @ B @ L @ R ) ) ) ) ).
% local.combine_rbt_less
thf(fact_94_local_Obalance__right__rbt__greater,axiom,
! [B: $tType,K: a,A2: rBT_Im246033960le_rbt @ a @ B,B2: rBT_Im246033960le_rbt @ a @ B,X: a,T: B] :
( ( rBT_Im1259024060reater @ a @ B @ less @ K @ A2 )
=> ( ( rBT_Im1259024060reater @ a @ B @ less @ K @ B2 )
=> ( ( less @ K @ X )
=> ( rBT_Im1259024060reater @ a @ B @ less @ K @ ( rBT_Im1053358987_right @ a @ B @ A2 @ X @ T @ B2 ) ) ) ) ) ).
% local.balance_right_rbt_greater
thf(fact_95_local_Obalance__right__rbt__less,axiom,
! [B: $tType,K: a,A2: rBT_Im246033960le_rbt @ a @ B,B2: rBT_Im246033960le_rbt @ a @ B,X: a,T: B] :
( ( rBT_Im2075124343t_less @ a @ B @ less @ K @ A2 )
=> ( ( rBT_Im2075124343t_less @ a @ B @ less @ K @ B2 )
=> ( ( less @ X @ K )
=> ( rBT_Im2075124343t_less @ a @ B @ less @ K @ ( rBT_Im1053358987_right @ a @ B @ A2 @ X @ T @ B2 ) ) ) ) ) ).
% local.balance_right_rbt_less
thf(fact_96_local_Orbt__insertwk__is__rbt,axiom,
! [B: $tType,T: rBT_Im246033960le_rbt @ a @ B,F: a > B > B > B,K: a,X: B] :
( ( rBT_Im862805236is_rbt @ a @ B @ less @ T )
=> ( rBT_Im862805236is_rbt @ a @ B @ less @ ( rBT_Im627674146th_key @ a @ B @ less @ F @ K @ X @ T ) ) ) ).
% local.rbt_insertwk_is_rbt
thf(fact_97_combine__inv1_I2_J,axiom,
! [B: $tType,A: $tType,Lt: rBT_Im246033960le_rbt @ A @ B,Rt: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1040612427e_inv1 @ A @ B @ Lt )
=> ( ( rBT_Im1040612427e_inv1 @ A @ B @ Rt )
=> ( rBT_Im1071879905_inv1l @ A @ B @ ( rBT_Im1591839508ombine @ A @ B @ Lt @ Rt ) ) ) ) ).
% combine_inv1(2)
thf(fact_98_balance__right__inv1l,axiom,
! [B: $tType,A: $tType,Lt: rBT_Im246033960le_rbt @ A @ B,Rt: rBT_Im246033960le_rbt @ A @ B,K: A,X: B] :
( ( rBT_Im1040612427e_inv1 @ A @ B @ Lt )
=> ( ( rBT_Im1071879905_inv1l @ A @ B @ Rt )
=> ( rBT_Im1071879905_inv1l @ A @ B @ ( rBT_Im1053358987_right @ A @ B @ Lt @ K @ X @ Rt ) ) ) ) ).
% balance_right_inv1l
thf(fact_99_inv1l_Osimps_I2_J,axiom,
! [B: $tType,A: $tType,C: rBT_Im1923302023_color,L: rBT_Im246033960le_rbt @ A @ B,K: A,V: B,R: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1071879905_inv1l @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ C @ L @ K @ V @ R ) )
= ( ( rBT_Im1040612427e_inv1 @ A @ B @ L )
& ( rBT_Im1040612427e_inv1 @ A @ B @ R ) ) ) ).
% inv1l.simps(2)
thf(fact_100_combine__inv2_I1_J,axiom,
! [B: $tType,A: $tType,Lt: rBT_Im246033960le_rbt @ A @ B,Rt: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1040612428e_inv2 @ A @ B @ Lt )
=> ( ( rBT_Im1040612428e_inv2 @ A @ B @ Rt )
=> ( ( ( rBT_Im315265182height @ A @ B @ Lt )
= ( rBT_Im315265182height @ A @ B @ Rt ) )
=> ( ( rBT_Im315265182height @ A @ B @ ( rBT_Im1591839508ombine @ A @ B @ Lt @ Rt ) )
= ( rBT_Im315265182height @ A @ B @ Lt ) ) ) ) ) ).
% combine_inv2(1)
thf(fact_101_combine__inv2_I2_J,axiom,
! [B: $tType,A: $tType,Lt: rBT_Im246033960le_rbt @ A @ B,Rt: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1040612428e_inv2 @ A @ B @ Lt )
=> ( ( rBT_Im1040612428e_inv2 @ A @ B @ Rt )
=> ( ( ( rBT_Im315265182height @ A @ B @ Lt )
= ( rBT_Im315265182height @ A @ B @ Rt ) )
=> ( rBT_Im1040612428e_inv2 @ A @ B @ ( rBT_Im1591839508ombine @ A @ B @ Lt @ Rt ) ) ) ) ) ).
% combine_inv2(2)
thf(fact_102_ord_Ois__rbt__rbt__sorted,axiom,
! [B: $tType,A: $tType,Less: A > A > $o,T: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im862805236is_rbt @ A @ B @ Less @ T )
=> ( rBT_Im759614907sorted @ A @ B @ Less @ T ) ) ).
% ord.is_rbt_rbt_sorted
thf(fact_103_ord_Oins__inv1__inv2_I4_J,axiom,
! [B: $tType,A: $tType,T: rBT_Im246033960le_rbt @ A @ B,Less: A > A > $o,F: A > B > B > B,K: A,X: B] :
( ( rBT_Im1040612427e_inv1 @ A @ B @ T )
=> ( ( rBT_Im1040612428e_inv2 @ A @ B @ T )
=> ( rBT_Im1071879905_inv1l @ A @ B @ ( rBT_Im1756508336bt_ins @ A @ B @ Less @ F @ K @ X @ T ) ) ) ) ).
% ord.ins_inv1_inv2(4)
thf(fact_104_ord_Orbt__insert__def,axiom,
! [B: $tType,A: $tType] :
( ( rBT_Im852463031insert @ A @ B )
= ( ^ [Less2: A > A > $o] :
( rBT_Im627674146th_key @ A @ B @ Less2
@ ^ [Uu: A,Uv: B,Nv: B] : Nv ) ) ) ).
% ord.rbt_insert_def
thf(fact_105_ord_Orbt__insertw__def,axiom,
! [B: $tType,A: $tType] :
( ( rBT_Im241132701t_with @ A @ B )
= ( ^ [Less2: A > A > $o,F2: B > B > B] :
( rBT_Im627674146th_key @ A @ B @ Less2
@ ^ [Uu: A] : F2 ) ) ) ).
% ord.rbt_insertw_def
thf(fact_106_ord_Orbt__del_Osimps_I2_J,axiom,
! [B: $tType,A: $tType,Less: A > A > $o,X: A,Y: A,C: rBT_Im1923302023_color,A2: rBT_Im246033960le_rbt @ A @ B,S: B,B2: rBT_Im246033960le_rbt @ A @ B] :
( ( ( Less @ X @ Y )
=> ( ( rBT_Im1714610413bt_del @ A @ B @ Less @ X @ ( rBT_Im480247531Branch @ A @ B @ C @ A2 @ Y @ S @ B2 ) )
= ( rBT_Im1152561598m_left @ A @ B @ Less @ X @ A2 @ Y @ S @ B2 ) ) )
& ( ~ ( Less @ X @ Y )
=> ( ( ( Less @ Y @ X )
=> ( ( rBT_Im1714610413bt_del @ A @ B @ Less @ X @ ( rBT_Im480247531Branch @ A @ B @ C @ A2 @ Y @ S @ B2 ) )
= ( rBT_Im1808746437_right @ A @ B @ Less @ X @ A2 @ Y @ S @ B2 ) ) )
& ( ~ ( Less @ Y @ X )
=> ( ( rBT_Im1714610413bt_del @ A @ B @ Less @ X @ ( rBT_Im480247531Branch @ A @ B @ C @ A2 @ Y @ S @ B2 ) )
= ( rBT_Im1591839508ombine @ A @ B @ A2 @ B2 ) ) ) ) ) ) ).
% ord.rbt_del.simps(2)
thf(fact_107_combine__in__tree,axiom,
! [B: $tType,A: $tType,L: rBT_Im246033960le_rbt @ A @ B,R: rBT_Im246033960le_rbt @ A @ B,K: A,V: B] :
( ( rBT_Im1040612428e_inv2 @ A @ B @ L )
=> ( ( rBT_Im1040612428e_inv2 @ A @ B @ R )
=> ( ( ( rBT_Im315265182height @ A @ B @ L )
= ( rBT_Im315265182height @ A @ B @ R ) )
=> ( ( rBT_Im1040612427e_inv1 @ A @ B @ L )
=> ( ( rBT_Im1040612427e_inv1 @ A @ B @ R )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V ) @ ( set2 @ ( product_prod @ A @ B ) @ ( rBT_Im954575269ntries @ A @ B @ ( rBT_Im1591839508ombine @ A @ B @ L @ R ) ) ) )
= ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V ) @ ( set2 @ ( product_prod @ A @ B ) @ ( rBT_Im954575269ntries @ A @ B @ L ) ) )
| ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V ) @ ( set2 @ ( product_prod @ A @ B ) @ ( rBT_Im954575269ntries @ A @ B @ R ) ) ) ) ) ) ) ) ) ) ).
% combine_in_tree
thf(fact_108_local_Obalance__left__rbt__sorted,axiom,
! [B: $tType,L: rBT_Im246033960le_rbt @ a @ B,R: rBT_Im246033960le_rbt @ a @ B,K: a,V: B] :
( ( rBT_Im759614907sorted @ a @ B @ less @ L )
=> ( ( rBT_Im759614907sorted @ a @ B @ less @ R )
=> ( ( rBT_Im2075124343t_less @ a @ B @ less @ K @ L )
=> ( ( rBT_Im1259024060reater @ a @ B @ less @ K @ R )
=> ( rBT_Im759614907sorted @ a @ B @ less @ ( rBT_Im1278323832e_left @ a @ B @ L @ K @ V @ R ) ) ) ) ) ) ).
% local.balance_left_rbt_sorted
thf(fact_109_local_Orbt__del_Oelims,axiom,
! [B: $tType,X: a,Xa: rBT_Im246033960le_rbt @ a @ B,Y: rBT_Im246033960le_rbt @ a @ B] :
( ( ( rBT_Im1714610413bt_del @ a @ B @ less @ X @ Xa )
= Y )
=> ( ( ( Xa
= ( rBT_Im418718756_Empty @ a @ B ) )
=> ( Y
!= ( rBT_Im418718756_Empty @ a @ B ) ) )
=> ~ ! [C3: rBT_Im1923302023_color,A4: rBT_Im246033960le_rbt @ a @ B,Y2: a,S2: B,B3: rBT_Im246033960le_rbt @ a @ B] :
( ( Xa
= ( rBT_Im480247531Branch @ a @ B @ C3 @ A4 @ Y2 @ S2 @ B3 ) )
=> ~ ( ( ( less @ X @ Y2 )
=> ( Y
= ( rBT_Im1152561598m_left @ a @ B @ less @ X @ A4 @ Y2 @ S2 @ B3 ) ) )
& ( ~ ( less @ X @ Y2 )
=> ( ( ( less @ Y2 @ X )
=> ( Y
= ( rBT_Im1808746437_right @ a @ B @ less @ X @ A4 @ Y2 @ S2 @ B3 ) ) )
& ( ~ ( less @ Y2 @ X )
=> ( Y
= ( rBT_Im1591839508ombine @ a @ B @ A4 @ B3 ) ) ) ) ) ) ) ) ) ).
% local.rbt_del.elims
thf(fact_110_local_Orbt__del__from__right_Osimps_I1_J,axiom,
! [B: $tType,X: a,A2: rBT_Im246033960le_rbt @ a @ B,Y: a,S: B,Lt: rBT_Im246033960le_rbt @ a @ B,Z: a,V: B,Rt: rBT_Im246033960le_rbt @ a @ B] :
( ( rBT_Im1808746437_right @ a @ B @ less @ X @ A2 @ Y @ S @ ( rBT_Im480247531Branch @ a @ B @ rBT_Impl_Mirabelle_B @ Lt @ Z @ V @ Rt ) )
= ( rBT_Im1053358987_right @ a @ B @ A2 @ Y @ S @ ( rBT_Im1714610413bt_del @ a @ B @ less @ X @ ( rBT_Im480247531Branch @ a @ B @ rBT_Impl_Mirabelle_B @ Lt @ Z @ V @ Rt ) ) ) ) ).
% local.rbt_del_from_right.simps(1)
thf(fact_111_local_Orbt__lookup__in__tree,axiom,
! [B: $tType,T: rBT_Im246033960le_rbt @ a @ B,K: a,V: B] :
( ( rBT_Im759614907sorted @ a @ B @ less @ T )
=> ( ( ( rBT_Im35466040lookup @ a @ B @ less @ T @ K )
= ( some @ B @ V ) )
= ( member @ ( product_prod @ a @ B ) @ ( product_Pair @ a @ B @ K @ V ) @ ( set2 @ ( product_prod @ a @ B ) @ ( rBT_Im954575269ntries @ a @ B @ T ) ) ) ) ) ).
% local.rbt_lookup_in_tree
thf(fact_112_local_Orbt__del__from__left_Osimps_I2_J,axiom,
! [B: $tType,X: a,Y: a,S: B,B2: rBT_Im246033960le_rbt @ a @ B] :
( ( rBT_Im1152561598m_left @ a @ B @ less @ X @ ( rBT_Im418718756_Empty @ a @ B ) @ Y @ S @ B2 )
= ( rBT_Im480247531Branch @ a @ B @ rBT_Impl_Mirabelle_R @ ( rBT_Im1714610413bt_del @ a @ B @ less @ X @ ( rBT_Im418718756_Empty @ a @ B ) ) @ Y @ S @ B2 ) ) ).
% local.rbt_del_from_left.simps(2)
thf(fact_113_local_Orbt__del_Ocases,axiom,
! [B: $tType,X: product_prod @ a @ ( rBT_Im246033960le_rbt @ a @ B )] :
( ! [X3: a] :
( X
!= ( product_Pair @ a @ ( rBT_Im246033960le_rbt @ a @ B ) @ X3 @ ( rBT_Im418718756_Empty @ a @ B ) ) )
=> ~ ! [X3: a,C3: rBT_Im1923302023_color,A4: rBT_Im246033960le_rbt @ a @ B,Y2: a,S2: B,B3: rBT_Im246033960le_rbt @ a @ B] :
( X
!= ( product_Pair @ a @ ( rBT_Im246033960le_rbt @ a @ B ) @ X3 @ ( rBT_Im480247531Branch @ a @ B @ C3 @ A4 @ Y2 @ S2 @ B3 ) ) ) ) ).
% local.rbt_del.cases
thf(fact_114_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_115_local_Orbt__del_Osimps_I1_J,axiom,
! [B: $tType,X: a] :
( ( rBT_Im1714610413bt_del @ a @ B @ less @ X @ ( rBT_Im418718756_Empty @ a @ B ) )
= ( rBT_Im418718756_Empty @ a @ B ) ) ).
% local.rbt_del.simps(1)
thf(fact_116_local_OEmpty__is__rbt,axiom,
! [B: $tType] : ( rBT_Im862805236is_rbt @ a @ B @ less @ ( rBT_Im418718756_Empty @ a @ B ) ) ).
% local.Empty_is_rbt
thf(fact_117_local_Obalance__left__rbt__less,axiom,
! [B: $tType,K: a,A2: rBT_Im246033960le_rbt @ a @ B,B2: rBT_Im246033960le_rbt @ a @ B,X: a,T: B] :
( ( rBT_Im2075124343t_less @ a @ B @ less @ K @ A2 )
=> ( ( rBT_Im2075124343t_less @ a @ B @ less @ K @ B2 )
=> ( ( less @ X @ K )
=> ( rBT_Im2075124343t_less @ a @ B @ less @ K @ ( rBT_Im1278323832e_left @ a @ B @ A2 @ X @ T @ B2 ) ) ) ) ) ).
% local.balance_left_rbt_less
thf(fact_118_local_Obalance__left__rbt__greater,axiom,
! [B: $tType,K: a,A2: rBT_Im246033960le_rbt @ a @ B,B2: rBT_Im246033960le_rbt @ a @ B,X: a,T: B] :
( ( rBT_Im1259024060reater @ a @ B @ less @ K @ A2 )
=> ( ( rBT_Im1259024060reater @ a @ B @ less @ K @ B2 )
=> ( ( less @ K @ X )
=> ( rBT_Im1259024060reater @ a @ B @ less @ K @ ( rBT_Im1278323832e_left @ a @ B @ A2 @ X @ T @ B2 ) ) ) ) ) ).
% local.balance_left_rbt_greater
thf(fact_119_local_Orbt__del__from__left_Ocases,axiom,
! [B: $tType,X: product_prod @ a @ ( product_prod @ ( rBT_Im246033960le_rbt @ a @ B ) @ ( product_prod @ a @ ( product_prod @ B @ ( rBT_Im246033960le_rbt @ a @ B ) ) ) )] :
( ! [X3: a,Lt2: rBT_Im246033960le_rbt @ a @ B,Z2: a,V2: B,Rt2: rBT_Im246033960le_rbt @ a @ B,Y2: a,S2: B,B3: rBT_Im246033960le_rbt @ a @ B] :
( X
!= ( product_Pair @ a @ ( product_prod @ ( rBT_Im246033960le_rbt @ a @ B ) @ ( product_prod @ a @ ( product_prod @ B @ ( rBT_Im246033960le_rbt @ a @ B ) ) ) ) @ X3 @ ( product_Pair @ ( rBT_Im246033960le_rbt @ a @ B ) @ ( product_prod @ a @ ( product_prod @ B @ ( rBT_Im246033960le_rbt @ a @ B ) ) ) @ ( rBT_Im480247531Branch @ a @ B @ rBT_Impl_Mirabelle_B @ Lt2 @ Z2 @ V2 @ Rt2 ) @ ( product_Pair @ a @ ( product_prod @ B @ ( rBT_Im246033960le_rbt @ a @ B ) ) @ Y2 @ ( product_Pair @ B @ ( rBT_Im246033960le_rbt @ a @ B ) @ S2 @ B3 ) ) ) ) )
=> ( ! [X3: a,Y2: a,S2: B,B3: rBT_Im246033960le_rbt @ a @ B] :
( X
!= ( product_Pair @ a @ ( product_prod @ ( rBT_Im246033960le_rbt @ a @ B ) @ ( product_prod @ a @ ( product_prod @ B @ ( rBT_Im246033960le_rbt @ a @ B ) ) ) ) @ X3 @ ( product_Pair @ ( rBT_Im246033960le_rbt @ a @ B ) @ ( product_prod @ a @ ( product_prod @ B @ ( rBT_Im246033960le_rbt @ a @ B ) ) ) @ ( rBT_Im418718756_Empty @ a @ B ) @ ( product_Pair @ a @ ( product_prod @ B @ ( rBT_Im246033960le_rbt @ a @ B ) ) @ Y2 @ ( product_Pair @ B @ ( rBT_Im246033960le_rbt @ a @ B ) @ S2 @ B3 ) ) ) ) )
=> ~ ! [X3: a,Va2: rBT_Im246033960le_rbt @ a @ B,Vb2: a,Vc2: B,Vd2: rBT_Im246033960le_rbt @ a @ B,Y2: a,S2: B,B3: rBT_Im246033960le_rbt @ a @ B] :
( X
!= ( product_Pair @ a @ ( product_prod @ ( rBT_Im246033960le_rbt @ a @ B ) @ ( product_prod @ a @ ( product_prod @ B @ ( rBT_Im246033960le_rbt @ a @ B ) ) ) ) @ X3 @ ( product_Pair @ ( rBT_Im246033960le_rbt @ a @ B ) @ ( product_prod @ a @ ( product_prod @ B @ ( rBT_Im246033960le_rbt @ a @ B ) ) ) @ ( rBT_Im480247531Branch @ a @ B @ rBT_Impl_Mirabelle_R @ Va2 @ Vb2 @ Vc2 @ Vd2 ) @ ( product_Pair @ a @ ( product_prod @ B @ ( rBT_Im246033960le_rbt @ a @ B ) ) @ Y2 @ ( product_Pair @ B @ ( rBT_Im246033960le_rbt @ a @ B ) @ S2 @ B3 ) ) ) ) ) ) ) ).
% local.rbt_del_from_left.cases
thf(fact_120_local_Orbt__ins_Ocases,axiom,
! [B: $tType,X: product_prod @ ( a > B > B > B ) @ ( product_prod @ a @ ( product_prod @ B @ ( rBT_Im246033960le_rbt @ a @ B ) ) )] :
( ! [F3: a > B > B > B,K2: a,V2: B] :
( X
!= ( product_Pair @ ( a > B > B > B ) @ ( product_prod @ a @ ( product_prod @ B @ ( rBT_Im246033960le_rbt @ a @ B ) ) ) @ F3 @ ( product_Pair @ a @ ( product_prod @ B @ ( rBT_Im246033960le_rbt @ a @ B ) ) @ K2 @ ( product_Pair @ B @ ( rBT_Im246033960le_rbt @ a @ B ) @ V2 @ ( rBT_Im418718756_Empty @ a @ B ) ) ) ) )
=> ( ! [F3: a > B > B > B,K2: a,V2: B,L2: rBT_Im246033960le_rbt @ a @ B,X3: a,Y2: B,R2: rBT_Im246033960le_rbt @ a @ B] :
( X
!= ( product_Pair @ ( a > B > B > B ) @ ( product_prod @ a @ ( product_prod @ B @ ( rBT_Im246033960le_rbt @ a @ B ) ) ) @ F3 @ ( product_Pair @ a @ ( product_prod @ B @ ( rBT_Im246033960le_rbt @ a @ B ) ) @ K2 @ ( product_Pair @ B @ ( rBT_Im246033960le_rbt @ a @ B ) @ V2 @ ( rBT_Im480247531Branch @ a @ B @ rBT_Impl_Mirabelle_B @ L2 @ X3 @ Y2 @ R2 ) ) ) ) )
=> ~ ! [F3: a > B > B > B,K2: a,V2: B,L2: rBT_Im246033960le_rbt @ a @ B,X3: a,Y2: B,R2: rBT_Im246033960le_rbt @ a @ B] :
( X
!= ( product_Pair @ ( a > B > B > B ) @ ( product_prod @ a @ ( product_prod @ B @ ( rBT_Im246033960le_rbt @ a @ B ) ) ) @ F3 @ ( product_Pair @ a @ ( product_prod @ B @ ( rBT_Im246033960le_rbt @ a @ B ) ) @ K2 @ ( product_Pair @ B @ ( rBT_Im246033960le_rbt @ a @ B ) @ V2 @ ( rBT_Im480247531Branch @ a @ B @ rBT_Impl_Mirabelle_R @ L2 @ X3 @ Y2 @ R2 ) ) ) ) ) ) ) ).
% local.rbt_ins.cases
thf(fact_121_local_Orbt__lookup_Osimps_I2_J,axiom,
! [B: $tType,K: a,X: a,Uu2: 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 @ Uu2 @ 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 @ Uu2 @ L @ X @ Y @ R ) @ K )
= ( rBT_Im35466040lookup @ a @ B @ less @ R @ K ) ) )
& ( ~ ( less @ X @ K )
=> ( ( rBT_Im35466040lookup @ a @ B @ less @ ( rBT_Im480247531Branch @ a @ B @ Uu2 @ L @ X @ Y @ R ) @ K )
= ( some @ B @ Y ) ) ) ) ) ) ).
% local.rbt_lookup.simps(2)
thf(fact_122_local_Orbt__ins_Oinduct,axiom,
! [B: $tType,P: ( a > B > B > B ) > a > B > ( rBT_Im246033960le_rbt @ a @ B ) > $o,A0: a > B > B > B,A1: a,A22: B,A32: rBT_Im246033960le_rbt @ a @ B] :
( ! [F3: a > B > B > B,K2: a,V2: B] : ( P @ F3 @ K2 @ V2 @ ( rBT_Im418718756_Empty @ a @ B ) )
=> ( ! [F3: a > B > B > B,K2: a,V2: B,L2: rBT_Im246033960le_rbt @ a @ B,X3: a,Y2: B,R2: rBT_Im246033960le_rbt @ a @ B] :
( ( ( less @ K2 @ X3 )
=> ( P @ F3 @ K2 @ V2 @ L2 ) )
=> ( ( ~ ( less @ K2 @ X3 )
=> ( ( less @ X3 @ K2 )
=> ( P @ F3 @ K2 @ V2 @ R2 ) ) )
=> ( P @ F3 @ K2 @ V2 @ ( rBT_Im480247531Branch @ a @ B @ rBT_Impl_Mirabelle_B @ L2 @ X3 @ Y2 @ R2 ) ) ) )
=> ( ! [F3: a > B > B > B,K2: a,V2: B,L2: rBT_Im246033960le_rbt @ a @ B,X3: a,Y2: B,R2: rBT_Im246033960le_rbt @ a @ B] :
( ( ( less @ K2 @ X3 )
=> ( P @ F3 @ K2 @ V2 @ L2 ) )
=> ( ( ~ ( less @ K2 @ X3 )
=> ( ( less @ X3 @ K2 )
=> ( P @ F3 @ K2 @ V2 @ R2 ) ) )
=> ( P @ F3 @ K2 @ V2 @ ( rBT_Im480247531Branch @ a @ B @ rBT_Impl_Mirabelle_R @ L2 @ X3 @ Y2 @ R2 ) ) ) )
=> ( P @ A0 @ A1 @ A22 @ A32 ) ) ) ) ).
% local.rbt_ins.induct
thf(fact_123_local_Orbt__ins_Osimps_I1_J,axiom,
! [B: $tType,F: a > B > B > B,K: a,V: B] :
( ( rBT_Im1756508336bt_ins @ a @ B @ less @ F @ K @ V @ ( rBT_Im418718756_Empty @ a @ B ) )
= ( rBT_Im480247531Branch @ a @ B @ rBT_Impl_Mirabelle_R @ ( rBT_Im418718756_Empty @ a @ B ) @ K @ V @ ( rBT_Im418718756_Empty @ a @ B ) ) ) ).
% local.rbt_ins.simps(1)
thf(fact_124_local_Orbt__del__from__right_Osimps_I2_J,axiom,
! [B: $tType,X: a,A2: rBT_Im246033960le_rbt @ a @ B,Y: a,S: B] :
( ( rBT_Im1808746437_right @ a @ B @ less @ X @ A2 @ Y @ S @ ( rBT_Im418718756_Empty @ a @ B ) )
= ( rBT_Im480247531Branch @ a @ B @ rBT_Impl_Mirabelle_R @ A2 @ Y @ S @ ( rBT_Im1714610413bt_del @ a @ B @ less @ X @ ( rBT_Im418718756_Empty @ a @ B ) ) ) ) ).
% local.rbt_del_from_right.simps(2)
thf(fact_125_local_Orbt__del__from__left_Osimps_I1_J,axiom,
! [B: $tType,X: a,Lt: rBT_Im246033960le_rbt @ a @ B,Z: a,V: B,Rt: rBT_Im246033960le_rbt @ a @ B,Y: a,S: B,B2: rBT_Im246033960le_rbt @ a @ B] :
( ( rBT_Im1152561598m_left @ a @ B @ less @ X @ ( rBT_Im480247531Branch @ a @ B @ rBT_Impl_Mirabelle_B @ Lt @ Z @ V @ Rt ) @ Y @ S @ B2 )
= ( rBT_Im1278323832e_left @ a @ B @ ( rBT_Im1714610413bt_del @ a @ B @ less @ X @ ( rBT_Im480247531Branch @ a @ B @ rBT_Impl_Mirabelle_B @ Lt @ Z @ V @ Rt ) ) @ Y @ S @ B2 ) ) ).
% local.rbt_del_from_left.simps(1)
thf(fact_126_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_127_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_128_rbt__cases,axiom,
! [B: $tType,A: $tType,T: rBT_Im246033960le_rbt @ A @ B] :
( ( T
!= ( rBT_Im418718756_Empty @ A @ B ) )
=> ( ! [L2: rBT_Im246033960le_rbt @ A @ B,K2: A,V2: B,R2: rBT_Im246033960le_rbt @ A @ B] :
( T
!= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ L2 @ K2 @ V2 @ R2 ) )
=> ~ ! [L2: rBT_Im246033960le_rbt @ A @ B,K2: A,V2: B,R2: rBT_Im246033960le_rbt @ A @ B] :
( T
!= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ L2 @ K2 @ V2 @ R2 ) ) ) ) ).
% rbt_cases
thf(fact_129_combine_Oinduct,axiom,
! [B: $tType,A: $tType,P: ( rBT_Im246033960le_rbt @ A @ B ) > ( rBT_Im246033960le_rbt @ A @ B ) > $o,A0: rBT_Im246033960le_rbt @ A @ B,A1: rBT_Im246033960le_rbt @ A @ B] :
( ! [X1: rBT_Im246033960le_rbt @ A @ B] : ( P @ ( rBT_Im418718756_Empty @ A @ B ) @ X1 )
=> ( ! [V2: rBT_Im1923302023_color,Va2: rBT_Im246033960le_rbt @ A @ B,Vb2: A,Vc2: B,Vd2: rBT_Im246033960le_rbt @ A @ B] : ( P @ ( rBT_Im480247531Branch @ A @ B @ V2 @ Va2 @ Vb2 @ Vc2 @ Vd2 ) @ ( rBT_Im418718756_Empty @ A @ B ) )
=> ( ! [A4: rBT_Im246033960le_rbt @ A @ B,K2: A,X3: B,B3: rBT_Im246033960le_rbt @ A @ B,C3: rBT_Im246033960le_rbt @ A @ B,S2: A,Y2: B,D: rBT_Im246033960le_rbt @ A @ B] :
( ( P @ B3 @ C3 )
=> ( P @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ A4 @ K2 @ X3 @ B3 ) @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ C3 @ S2 @ Y2 @ D ) ) )
=> ( ! [A4: rBT_Im246033960le_rbt @ A @ B,K2: A,X3: B,B3: rBT_Im246033960le_rbt @ A @ B,C3: rBT_Im246033960le_rbt @ A @ B,S2: A,Y2: B,D: rBT_Im246033960le_rbt @ A @ B] :
( ( P @ B3 @ C3 )
=> ( P @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ A4 @ K2 @ X3 @ B3 ) @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ C3 @ S2 @ Y2 @ D ) ) )
=> ( ! [Va2: rBT_Im246033960le_rbt @ A @ B,Vb2: A,Vc2: B,Vd2: rBT_Im246033960le_rbt @ A @ B,B3: rBT_Im246033960le_rbt @ A @ B,K2: A,X3: B,C3: rBT_Im246033960le_rbt @ A @ B] :
( ( P @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va2 @ Vb2 @ Vc2 @ Vd2 ) @ B3 )
=> ( P @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va2 @ Vb2 @ Vc2 @ Vd2 ) @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ B3 @ K2 @ X3 @ C3 ) ) )
=> ( ! [A4: rBT_Im246033960le_rbt @ A @ B,K2: A,X3: B,B3: rBT_Im246033960le_rbt @ A @ B,Va2: rBT_Im246033960le_rbt @ A @ B,Vb2: A,Vc2: B,Vd2: rBT_Im246033960le_rbt @ A @ B] :
( ( P @ B3 @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va2 @ Vb2 @ Vc2 @ Vd2 ) )
=> ( P @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ A4 @ K2 @ X3 @ B3 ) @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va2 @ Vb2 @ Vc2 @ Vd2 ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ) ) ) ).
% combine.induct
thf(fact_130_ord_Orbt__ins_Oinduct,axiom,
! [B: $tType,A: $tType,P: ( A > B > B > B ) > A > B > ( rBT_Im246033960le_rbt @ A @ B ) > $o,Less: A > A > $o,A0: A > B > B > B,A1: A,A22: B,A32: rBT_Im246033960le_rbt @ A @ B] :
( ! [F3: A > B > B > B,K2: A,V2: B] : ( P @ F3 @ K2 @ V2 @ ( rBT_Im418718756_Empty @ A @ B ) )
=> ( ! [F3: A > B > B > B,K2: A,V2: B,L2: rBT_Im246033960le_rbt @ A @ B,X3: A,Y2: B,R2: rBT_Im246033960le_rbt @ A @ B] :
( ( ( Less @ K2 @ X3 )
=> ( P @ F3 @ K2 @ V2 @ L2 ) )
=> ( ( ~ ( Less @ K2 @ X3 )
=> ( ( Less @ X3 @ K2 )
=> ( P @ F3 @ K2 @ V2 @ R2 ) ) )
=> ( P @ F3 @ K2 @ V2 @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ L2 @ X3 @ Y2 @ R2 ) ) ) )
=> ( ! [F3: A > B > B > B,K2: A,V2: B,L2: rBT_Im246033960le_rbt @ A @ B,X3: A,Y2: B,R2: rBT_Im246033960le_rbt @ A @ B] :
( ( ( Less @ K2 @ X3 )
=> ( P @ F3 @ K2 @ V2 @ L2 ) )
=> ( ( ~ ( Less @ K2 @ X3 )
=> ( ( Less @ X3 @ K2 )
=> ( P @ F3 @ K2 @ V2 @ R2 ) ) )
=> ( P @ F3 @ K2 @ V2 @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ L2 @ X3 @ Y2 @ R2 ) ) ) )
=> ( P @ A0 @ A1 @ A22 @ A32 ) ) ) ) ).
% ord.rbt_ins.induct
thf(fact_131_ord__class_Orbt__del__from__left_Ocases,axiom,
! [B: $tType,A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X: product_prod @ A @ ( product_prod @ ( rBT_Im246033960le_rbt @ A @ B ) @ ( product_prod @ A @ ( product_prod @ B @ ( rBT_Im246033960le_rbt @ A @ B ) ) ) )] :
( ! [X3: A,Lt2: rBT_Im246033960le_rbt @ A @ B,Z2: A,V2: B,Rt2: rBT_Im246033960le_rbt @ A @ B,Y2: A,S2: B,B3: rBT_Im246033960le_rbt @ A @ B] :
( X
!= ( product_Pair @ A @ ( product_prod @ ( rBT_Im246033960le_rbt @ A @ B ) @ ( product_prod @ A @ ( product_prod @ B @ ( rBT_Im246033960le_rbt @ A @ B ) ) ) ) @ X3 @ ( product_Pair @ ( rBT_Im246033960le_rbt @ A @ B ) @ ( product_prod @ A @ ( product_prod @ B @ ( rBT_Im246033960le_rbt @ A @ B ) ) ) @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Lt2 @ Z2 @ V2 @ Rt2 ) @ ( product_Pair @ A @ ( product_prod @ B @ ( rBT_Im246033960le_rbt @ A @ B ) ) @ Y2 @ ( product_Pair @ B @ ( rBT_Im246033960le_rbt @ A @ B ) @ S2 @ B3 ) ) ) ) )
=> ( ! [X3: A,Y2: A,S2: B,B3: rBT_Im246033960le_rbt @ A @ B] :
( X
!= ( product_Pair @ A @ ( product_prod @ ( rBT_Im246033960le_rbt @ A @ B ) @ ( product_prod @ A @ ( product_prod @ B @ ( rBT_Im246033960le_rbt @ A @ B ) ) ) ) @ X3 @ ( product_Pair @ ( rBT_Im246033960le_rbt @ A @ B ) @ ( product_prod @ A @ ( product_prod @ B @ ( rBT_Im246033960le_rbt @ A @ B ) ) ) @ ( rBT_Im418718756_Empty @ A @ B ) @ ( product_Pair @ A @ ( product_prod @ B @ ( rBT_Im246033960le_rbt @ A @ B ) ) @ Y2 @ ( product_Pair @ B @ ( rBT_Im246033960le_rbt @ A @ B ) @ S2 @ B3 ) ) ) ) )
=> ~ ! [X3: A,Va2: rBT_Im246033960le_rbt @ A @ B,Vb2: A,Vc2: B,Vd2: rBT_Im246033960le_rbt @ A @ B,Y2: A,S2: B,B3: rBT_Im246033960le_rbt @ A @ B] :
( X
!= ( product_Pair @ A @ ( product_prod @ ( rBT_Im246033960le_rbt @ A @ B ) @ ( product_prod @ A @ ( product_prod @ B @ ( rBT_Im246033960le_rbt @ A @ B ) ) ) ) @ X3 @ ( product_Pair @ ( rBT_Im246033960le_rbt @ A @ B ) @ ( product_prod @ A @ ( product_prod @ B @ ( rBT_Im246033960le_rbt @ A @ B ) ) ) @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ Va2 @ Vb2 @ Vc2 @ Vd2 ) @ ( product_Pair @ A @ ( product_prod @ B @ ( rBT_Im246033960le_rbt @ A @ B ) ) @ Y2 @ ( product_Pair @ B @ ( rBT_Im246033960le_rbt @ A @ B ) @ S2 @ B3 ) ) ) ) ) ) ) ) ).
% ord_class.rbt_del_from_left.cases
thf(fact_132_ord_Orbt__del__from__left_Ocases,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ ( product_prod @ ( rBT_Im246033960le_rbt @ A @ B ) @ ( product_prod @ A @ ( product_prod @ B @ ( rBT_Im246033960le_rbt @ A @ B ) ) ) )] :
( ! [X3: A,Lt2: rBT_Im246033960le_rbt @ A @ B,Z2: A,V2: B,Rt2: rBT_Im246033960le_rbt @ A @ B,Y2: A,S2: B,B3: rBT_Im246033960le_rbt @ A @ B] :
( X
!= ( product_Pair @ A @ ( product_prod @ ( rBT_Im246033960le_rbt @ A @ B ) @ ( product_prod @ A @ ( product_prod @ B @ ( rBT_Im246033960le_rbt @ A @ B ) ) ) ) @ X3 @ ( product_Pair @ ( rBT_Im246033960le_rbt @ A @ B ) @ ( product_prod @ A @ ( product_prod @ B @ ( rBT_Im246033960le_rbt @ A @ B ) ) ) @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Lt2 @ Z2 @ V2 @ Rt2 ) @ ( product_Pair @ A @ ( product_prod @ B @ ( rBT_Im246033960le_rbt @ A @ B ) ) @ Y2 @ ( product_Pair @ B @ ( rBT_Im246033960le_rbt @ A @ B ) @ S2 @ B3 ) ) ) ) )
=> ( ! [X3: A,Y2: A,S2: B,B3: rBT_Im246033960le_rbt @ A @ B] :
( X
!= ( product_Pair @ A @ ( product_prod @ ( rBT_Im246033960le_rbt @ A @ B ) @ ( product_prod @ A @ ( product_prod @ B @ ( rBT_Im246033960le_rbt @ A @ B ) ) ) ) @ X3 @ ( product_Pair @ ( rBT_Im246033960le_rbt @ A @ B ) @ ( product_prod @ A @ ( product_prod @ B @ ( rBT_Im246033960le_rbt @ A @ B ) ) ) @ ( rBT_Im418718756_Empty @ A @ B ) @ ( product_Pair @ A @ ( product_prod @ B @ ( rBT_Im246033960le_rbt @ A @ B ) ) @ Y2 @ ( product_Pair @ B @ ( rBT_Im246033960le_rbt @ A @ B ) @ S2 @ B3 ) ) ) ) )
=> ~ ! [X3: A,Va2: rBT_Im246033960le_rbt @ A @ B,Vb2: A,Vc2: B,Vd2: rBT_Im246033960le_rbt @ A @ B,Y2: A,S2: B,B3: rBT_Im246033960le_rbt @ A @ B] :
( X
!= ( product_Pair @ A @ ( product_prod @ ( rBT_Im246033960le_rbt @ A @ B ) @ ( product_prod @ A @ ( product_prod @ B @ ( rBT_Im246033960le_rbt @ A @ B ) ) ) ) @ X3 @ ( product_Pair @ ( rBT_Im246033960le_rbt @ A @ B ) @ ( product_prod @ A @ ( product_prod @ B @ ( rBT_Im246033960le_rbt @ A @ B ) ) ) @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ Va2 @ Vb2 @ Vc2 @ Vd2 ) @ ( product_Pair @ A @ ( product_prod @ B @ ( rBT_Im246033960le_rbt @ A @ B ) ) @ Y2 @ ( product_Pair @ B @ ( rBT_Im246033960le_rbt @ A @ B ) @ S2 @ B3 ) ) ) ) ) ) ) ).
% ord.rbt_del_from_left.cases
thf(fact_133_ord__class_Orbt__ins_Ocases,axiom,
! [B: $tType,A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X: product_prod @ ( A > B > B > B ) @ ( product_prod @ A @ ( product_prod @ B @ ( rBT_Im246033960le_rbt @ A @ B ) ) )] :
( ! [F3: A > B > B > B,K2: A,V2: B] :
( X
!= ( product_Pair @ ( A > B > B > B ) @ ( product_prod @ A @ ( product_prod @ B @ ( rBT_Im246033960le_rbt @ A @ B ) ) ) @ F3 @ ( product_Pair @ A @ ( product_prod @ B @ ( rBT_Im246033960le_rbt @ A @ B ) ) @ K2 @ ( product_Pair @ B @ ( rBT_Im246033960le_rbt @ A @ B ) @ V2 @ ( rBT_Im418718756_Empty @ A @ B ) ) ) ) )
=> ( ! [F3: A > B > B > B,K2: A,V2: B,L2: rBT_Im246033960le_rbt @ A @ B,X3: A,Y2: B,R2: rBT_Im246033960le_rbt @ A @ B] :
( X
!= ( product_Pair @ ( A > B > B > B ) @ ( product_prod @ A @ ( product_prod @ B @ ( rBT_Im246033960le_rbt @ A @ B ) ) ) @ F3 @ ( product_Pair @ A @ ( product_prod @ B @ ( rBT_Im246033960le_rbt @ A @ B ) ) @ K2 @ ( product_Pair @ B @ ( rBT_Im246033960le_rbt @ A @ B ) @ V2 @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ L2 @ X3 @ Y2 @ R2 ) ) ) ) )
=> ~ ! [F3: A > B > B > B,K2: A,V2: B,L2: rBT_Im246033960le_rbt @ A @ B,X3: A,Y2: B,R2: rBT_Im246033960le_rbt @ A @ B] :
( X
!= ( product_Pair @ ( A > B > B > B ) @ ( product_prod @ A @ ( product_prod @ B @ ( rBT_Im246033960le_rbt @ A @ B ) ) ) @ F3 @ ( product_Pair @ A @ ( product_prod @ B @ ( rBT_Im246033960le_rbt @ A @ B ) ) @ K2 @ ( product_Pair @ B @ ( rBT_Im246033960le_rbt @ A @ B ) @ V2 @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ L2 @ X3 @ Y2 @ R2 ) ) ) ) ) ) ) ) ).
% ord_class.rbt_ins.cases
thf(fact_134_ord_Orbt__ins_Ocases,axiom,
! [B: $tType,A: $tType,X: product_prod @ ( A > B > B > B ) @ ( product_prod @ A @ ( product_prod @ B @ ( rBT_Im246033960le_rbt @ A @ B ) ) )] :
( ! [F3: A > B > B > B,K2: A,V2: B] :
( X
!= ( product_Pair @ ( A > B > B > B ) @ ( product_prod @ A @ ( product_prod @ B @ ( rBT_Im246033960le_rbt @ A @ B ) ) ) @ F3 @ ( product_Pair @ A @ ( product_prod @ B @ ( rBT_Im246033960le_rbt @ A @ B ) ) @ K2 @ ( product_Pair @ B @ ( rBT_Im246033960le_rbt @ A @ B ) @ V2 @ ( rBT_Im418718756_Empty @ A @ B ) ) ) ) )
=> ( ! [F3: A > B > B > B,K2: A,V2: B,L2: rBT_Im246033960le_rbt @ A @ B,X3: A,Y2: B,R2: rBT_Im246033960le_rbt @ A @ B] :
( X
!= ( product_Pair @ ( A > B > B > B ) @ ( product_prod @ A @ ( product_prod @ B @ ( rBT_Im246033960le_rbt @ A @ B ) ) ) @ F3 @ ( product_Pair @ A @ ( product_prod @ B @ ( rBT_Im246033960le_rbt @ A @ B ) ) @ K2 @ ( product_Pair @ B @ ( rBT_Im246033960le_rbt @ A @ B ) @ V2 @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ L2 @ X3 @ Y2 @ R2 ) ) ) ) )
=> ~ ! [F3: A > B > B > B,K2: A,V2: B,L2: rBT_Im246033960le_rbt @ A @ B,X3: A,Y2: B,R2: rBT_Im246033960le_rbt @ A @ B] :
( X
!= ( product_Pair @ ( A > B > B > B ) @ ( product_prod @ A @ ( product_prod @ B @ ( rBT_Im246033960le_rbt @ A @ B ) ) ) @ F3 @ ( product_Pair @ A @ ( product_prod @ B @ ( rBT_Im246033960le_rbt @ A @ B ) ) @ K2 @ ( product_Pair @ B @ ( rBT_Im246033960le_rbt @ A @ B ) @ V2 @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ L2 @ X3 @ Y2 @ R2 ) ) ) ) ) ) ) ).
% ord.rbt_ins.cases
thf(fact_135_balance__left_Osimps_I6_J,axiom,
! [B: $tType,A: $tType,K: A,X: B] :
( ( rBT_Im1278323832e_left @ A @ B @ ( rBT_Im418718756_Empty @ A @ B ) @ K @ X @ ( rBT_Im418718756_Empty @ A @ B ) )
= ( rBT_Im418718756_Empty @ A @ B ) ) ).
% balance_left.simps(6)
thf(fact_136_balance__left_Osimps_I11_J,axiom,
! [B: $tType,A: $tType,Va: rBT_Im246033960le_rbt @ A @ B,Vb: A,Vc: B,Vd: rBT_Im246033960le_rbt @ A @ B,K: A,X: B,Vi: rBT_Im246033960le_rbt @ A @ B,Vj: A,Vk: B,Vl: rBT_Im246033960le_rbt @ A @ B,Vf: A,Vg: B,Vh: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1278323832e_left @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va @ Vb @ Vc @ Vd ) @ K @ X @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ Vi @ Vj @ Vk @ Vl ) @ Vf @ Vg @ Vh ) )
= ( rBT_Im418718756_Empty @ A @ B ) ) ).
% balance_left.simps(11)
thf(fact_137_balance__left_Osimps_I10_J,axiom,
! [B: $tType,A: $tType,Va: rBT_Im246033960le_rbt @ A @ B,Vb: A,Vc: B,Vd: rBT_Im246033960le_rbt @ A @ B,K: A,X: B,Vf: A,Vg: B,Vh: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1278323832e_left @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va @ Vb @ Vc @ Vd ) @ K @ X @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ ( rBT_Im418718756_Empty @ A @ B ) @ Vf @ Vg @ Vh ) )
= ( rBT_Im418718756_Empty @ A @ B ) ) ).
% balance_left.simps(10)
thf(fact_138_balance__left_Osimps_I9_J,axiom,
! [B: $tType,A: $tType,Va: rBT_Im246033960le_rbt @ A @ B,Vb: A,Vc: B,Vd: rBT_Im246033960le_rbt @ A @ B,K: A,X: B] :
( ( rBT_Im1278323832e_left @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va @ Vb @ Vc @ Vd ) @ K @ X @ ( rBT_Im418718756_Empty @ A @ B ) )
= ( rBT_Im418718756_Empty @ A @ B ) ) ).
% balance_left.simps(9)
thf(fact_139_balance__right_Osimps_I9_J,axiom,
! [B: $tType,A: $tType,K: A,X: B,Va: rBT_Im246033960le_rbt @ A @ B,Vb: A,Vc: B,Vd: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1053358987_right @ A @ B @ ( rBT_Im418718756_Empty @ A @ B ) @ K @ X @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va @ Vb @ Vc @ Vd ) )
= ( rBT_Im418718756_Empty @ A @ B ) ) ).
% balance_right.simps(9)
thf(fact_140_balance__left_Osimps_I7_J,axiom,
! [B: $tType,A: $tType,K: A,X: B,Vb: A,Vc: B,Vd: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1278323832e_left @ A @ B @ ( rBT_Im418718756_Empty @ A @ B ) @ K @ X @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ ( rBT_Im418718756_Empty @ A @ B ) @ Vb @ Vc @ Vd ) )
= ( rBT_Im418718756_Empty @ A @ B ) ) ).
% balance_left.simps(7)
thf(fact_141_balance__left_Osimps_I8_J,axiom,
! [B: $tType,A: $tType,K: A,X: B,Ve: rBT_Im246033960le_rbt @ A @ B,Vf: A,Vg: B,Vh: rBT_Im246033960le_rbt @ A @ B,Vb: A,Vc: B,Vd: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1278323832e_left @ A @ B @ ( rBT_Im418718756_Empty @ A @ B ) @ K @ X @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ Ve @ Vf @ Vg @ Vh ) @ Vb @ Vc @ Vd ) )
= ( rBT_Im418718756_Empty @ A @ B ) ) ).
% balance_left.simps(8)
thf(fact_142_ord__class_Orbt__del_Ocases,axiom,
! [B: $tType,A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X: product_prod @ A @ ( rBT_Im246033960le_rbt @ A @ B )] :
( ! [X3: A] :
( X
!= ( product_Pair @ A @ ( rBT_Im246033960le_rbt @ A @ B ) @ X3 @ ( rBT_Im418718756_Empty @ A @ B ) ) )
=> ~ ! [X3: A,C3: rBT_Im1923302023_color,A4: rBT_Im246033960le_rbt @ A @ B,Y2: A,S2: B,B3: rBT_Im246033960le_rbt @ A @ B] :
( X
!= ( product_Pair @ A @ ( rBT_Im246033960le_rbt @ A @ B ) @ X3 @ ( rBT_Im480247531Branch @ A @ B @ C3 @ A4 @ Y2 @ S2 @ B3 ) ) ) ) ) ).
% ord_class.rbt_del.cases
thf(fact_143_ord_Orbt__del_Ocases,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ ( rBT_Im246033960le_rbt @ A @ B )] :
( ! [X3: A] :
( X
!= ( product_Pair @ A @ ( rBT_Im246033960le_rbt @ A @ B ) @ X3 @ ( rBT_Im418718756_Empty @ A @ B ) ) )
=> ~ ! [X3: A,C3: rBT_Im1923302023_color,A4: rBT_Im246033960le_rbt @ A @ B,Y2: A,S2: B,B3: rBT_Im246033960le_rbt @ A @ B] :
( X
!= ( product_Pair @ A @ ( rBT_Im246033960le_rbt @ A @ B ) @ X3 @ ( rBT_Im480247531Branch @ A @ B @ C3 @ A4 @ Y2 @ S2 @ B3 ) ) ) ) ).
% ord.rbt_del.cases
thf(fact_144_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,X232: A,X242: B,X252: rBT_Im246033960le_rbt @ A @ B] :
( Y
!= ( rBT_Im480247531Branch @ A @ B @ X212 @ X222 @ X232 @ X242 @ X252 ) ) ) ).
% rbt.exhaust
thf(fact_145_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 ) )
=> ( ! [X1: rBT_Im1923302023_color,X26: rBT_Im246033960le_rbt @ A @ B,X32: A,X4: B,X5: rBT_Im246033960le_rbt @ A @ B] :
( ( P @ X26 )
=> ( ( P @ X5 )
=> ( P @ ( rBT_Im480247531Branch @ A @ B @ X1 @ X26 @ X32 @ X4 @ X5 ) ) ) )
=> ( P @ Rbt ) ) ) ).
% rbt.induct
thf(fact_146_rbt_Odistinct_I1_J,axiom,
! [B: $tType,A: $tType,X21: rBT_Im1923302023_color,X22: rBT_Im246033960le_rbt @ A @ B,X23: A,X24: B,X25: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im418718756_Empty @ A @ B )
!= ( rBT_Im480247531Branch @ A @ B @ X21 @ X22 @ X23 @ X24 @ X25 ) ) ).
% rbt.distinct(1)
thf(fact_147_balance__left_Osimps_I1_J,axiom,
! [B: $tType,A: $tType,A2: rBT_Im246033960le_rbt @ A @ B,K: A,X: B,B2: rBT_Im246033960le_rbt @ A @ B,S: A,Y: B,C: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1278323832e_left @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ A2 @ K @ X @ B2 ) @ S @ Y @ C )
= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ A2 @ K @ X @ B2 ) @ S @ Y @ C ) ) ).
% balance_left.simps(1)
thf(fact_148_color_Oexhaust,axiom,
! [Y: rBT_Im1923302023_color] :
( ( Y != rBT_Impl_Mirabelle_R )
=> ( Y = rBT_Impl_Mirabelle_B ) ) ).
% color.exhaust
thf(fact_149_color_Oinduct,axiom,
! [P: rBT_Im1923302023_color > $o,Color: rBT_Im1923302023_color] :
( ( P @ rBT_Impl_Mirabelle_R )
=> ( ( P @ rBT_Impl_Mirabelle_B )
=> ( P @ Color ) ) ) ).
% color.induct
thf(fact_150_color_Odistinct_I1_J,axiom,
rBT_Impl_Mirabelle_R != rBT_Impl_Mirabelle_B ).
% color.distinct(1)
thf(fact_151_ord_Orbt__sorted_Osimps_I1_J,axiom,
! [B: $tType,A: $tType,Less: A > A > $o] : ( rBT_Im759614907sorted @ A @ B @ Less @ ( rBT_Im418718756_Empty @ A @ B ) ) ).
% ord.rbt_sorted.simps(1)
thf(fact_152_inv2_Osimps_I1_J,axiom,
! [B: $tType,A: $tType] : ( rBT_Im1040612428e_inv2 @ A @ B @ ( rBT_Im418718756_Empty @ A @ B ) ) ).
% inv2.simps(1)
thf(fact_153_ord_Orbt__less__simps_I1_J,axiom,
! [B: $tType,A: $tType,Less: A > A > $o,K: A] : ( rBT_Im2075124343t_less @ A @ B @ Less @ K @ ( rBT_Im418718756_Empty @ A @ B ) ) ).
% ord.rbt_less_simps(1)
thf(fact_154_ord_Orbt__greater__simps_I1_J,axiom,
! [B: $tType,A: $tType,Less: A > A > $o,K: A] : ( rBT_Im1259024060reater @ A @ B @ Less @ K @ ( rBT_Im418718756_Empty @ A @ B ) ) ).
% ord.rbt_greater_simps(1)
thf(fact_155_inv1_Osimps_I1_J,axiom,
! [B: $tType,A: $tType] : ( rBT_Im1040612427e_inv1 @ A @ B @ ( rBT_Im418718756_Empty @ A @ B ) ) ).
% inv1.simps(1)
thf(fact_156_ord_Orbt__del_Osimps_I1_J,axiom,
! [B: $tType,A: $tType,Less: A > A > $o,X: A] :
( ( rBT_Im1714610413bt_del @ A @ B @ Less @ X @ ( rBT_Im418718756_Empty @ A @ B ) )
= ( rBT_Im418718756_Empty @ A @ B ) ) ).
% ord.rbt_del.simps(1)
thf(fact_157_combine_Osimps_I1_J,axiom,
! [B: $tType,A: $tType,X: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1591839508ombine @ A @ B @ ( rBT_Im418718756_Empty @ A @ B ) @ X )
= X ) ).
% combine.simps(1)
thf(fact_158_balance__right_Osimps_I6_J,axiom,
! [B: $tType,A: $tType,K: A,X: B] :
( ( rBT_Im1053358987_right @ A @ B @ ( rBT_Im418718756_Empty @ A @ B ) @ K @ X @ ( rBT_Im418718756_Empty @ A @ B ) )
= ( rBT_Im418718756_Empty @ A @ B ) ) ).
% balance_right.simps(6)
thf(fact_159_inv1l_Osimps_I1_J,axiom,
! [B: $tType,A: $tType] : ( rBT_Im1071879905_inv1l @ A @ B @ ( rBT_Im418718756_Empty @ A @ B ) ) ).
% inv1l.simps(1)
thf(fact_160_ord_OEmpty__is__rbt,axiom,
! [B: $tType,A: $tType,Less: A > A > $o] : ( rBT_Im862805236is_rbt @ A @ B @ Less @ ( rBT_Im418718756_Empty @ A @ B ) ) ).
% ord.Empty_is_rbt
thf(fact_161_balance__right_Osimps_I11_J,axiom,
! [B: $tType,A: $tType,Ve: rBT_Im246033960le_rbt @ A @ B,Vf: A,Vg: B,Vi: rBT_Im246033960le_rbt @ A @ B,Vj: A,Vk: B,Vl: rBT_Im246033960le_rbt @ A @ B,K: A,X: B,Va: rBT_Im246033960le_rbt @ A @ B,Vb: A,Vc: B,Vd: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1053358987_right @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ Ve @ Vf @ Vg @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ Vi @ Vj @ Vk @ Vl ) ) @ K @ X @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va @ Vb @ Vc @ Vd ) )
= ( rBT_Im418718756_Empty @ A @ B ) ) ).
% balance_right.simps(11)
thf(fact_162_balance__right_Osimps_I10_J,axiom,
! [B: $tType,A: $tType,Ve: rBT_Im246033960le_rbt @ A @ B,Vf: A,Vg: B,K: A,X: B,Va: rBT_Im246033960le_rbt @ A @ B,Vb: A,Vc: B,Vd: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1053358987_right @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ Ve @ Vf @ Vg @ ( rBT_Im418718756_Empty @ A @ B ) ) @ K @ X @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va @ Vb @ Vc @ Vd ) )
= ( rBT_Im418718756_Empty @ A @ B ) ) ).
% balance_right.simps(10)
thf(fact_163_ord_Orbt__del__from__left_Osimps_I1_J,axiom,
! [B: $tType,A: $tType,Less: A > A > $o,X: A,Lt: rBT_Im246033960le_rbt @ A @ B,Z: A,V: B,Rt: rBT_Im246033960le_rbt @ A @ B,Y: A,S: B,B2: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1152561598m_left @ A @ B @ Less @ X @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Lt @ Z @ V @ Rt ) @ Y @ S @ B2 )
= ( rBT_Im1278323832e_left @ A @ B @ ( rBT_Im1714610413bt_del @ A @ B @ Less @ X @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Lt @ Z @ V @ Rt ) ) @ Y @ S @ B2 ) ) ).
% ord.rbt_del_from_left.simps(1)
thf(fact_164_ord_Orbt__lookup_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,Less: A > A > $o,K: A,X: A,Uu2: 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 @ Uu2 @ 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 @ Uu2 @ L @ X @ Y @ R ) @ K )
= ( rBT_Im35466040lookup @ A @ B @ Less @ R @ K ) ) )
& ( ~ ( Less @ X @ K )
=> ( ( rBT_Im35466040lookup @ A @ B @ Less @ ( rBT_Im480247531Branch @ A @ B @ Uu2 @ L @ X @ Y @ R ) @ K )
= ( some @ B @ Y ) ) ) ) ) ) ).
% ord.rbt_lookup.simps(2)
thf(fact_165_combine_Osimps_I2_J,axiom,
! [B: $tType,A: $tType,V: rBT_Im1923302023_color,Va: rBT_Im246033960le_rbt @ A @ B,Vb: A,Vc: B,Vd: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1591839508ombine @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ V @ Va @ Vb @ Vc @ Vd ) @ ( rBT_Im418718756_Empty @ A @ B ) )
= ( rBT_Im480247531Branch @ A @ B @ V @ Va @ Vb @ Vc @ Vd ) ) ).
% combine.simps(2)
thf(fact_166_balance__left__inv1l,axiom,
! [B: $tType,A: $tType,Lt: rBT_Im246033960le_rbt @ A @ B,Rt: rBT_Im246033960le_rbt @ A @ B,K: A,X: B] :
( ( rBT_Im1071879905_inv1l @ A @ B @ Lt )
=> ( ( rBT_Im1040612427e_inv1 @ A @ B @ Rt )
=> ( rBT_Im1071879905_inv1l @ A @ B @ ( rBT_Im1278323832e_left @ A @ B @ Lt @ K @ X @ Rt ) ) ) ) ).
% balance_left_inv1l
thf(fact_167_ord_Orbt__ins_Osimps_I1_J,axiom,
! [B: $tType,A: $tType,Less: A > A > $o,F: A > B > B > B,K: A,V: B] :
( ( rBT_Im1756508336bt_ins @ A @ B @ Less @ F @ K @ V @ ( rBT_Im418718756_Empty @ A @ B ) )
= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ ( rBT_Im418718756_Empty @ A @ B ) @ K @ V @ ( rBT_Im418718756_Empty @ A @ B ) ) ) ).
% ord.rbt_ins.simps(1)
thf(fact_168_balance__right_Osimps_I7_J,axiom,
! [B: $tType,A: $tType,Va: rBT_Im246033960le_rbt @ A @ B,Vb: A,Vc: B,K: A,X: B] :
( ( rBT_Im1053358987_right @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ Va @ Vb @ Vc @ ( rBT_Im418718756_Empty @ A @ B ) ) @ K @ X @ ( rBT_Im418718756_Empty @ A @ B ) )
= ( rBT_Im418718756_Empty @ A @ B ) ) ).
% balance_right.simps(7)
thf(fact_169_balance__right_Osimps_I8_J,axiom,
! [B: $tType,A: $tType,Va: rBT_Im246033960le_rbt @ A @ B,Vb: A,Vc: B,Ve: rBT_Im246033960le_rbt @ A @ B,Vf: A,Vg: B,Vh: rBT_Im246033960le_rbt @ A @ B,K: A,X: B] :
( ( rBT_Im1053358987_right @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ Va @ Vb @ Vc @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ Ve @ Vf @ Vg @ Vh ) ) @ K @ X @ ( rBT_Im418718756_Empty @ A @ B ) )
= ( rBT_Im418718756_Empty @ A @ B ) ) ).
% balance_right.simps(8)
thf(fact_170_combine_Osimps_I5_J,axiom,
! [B: $tType,A: $tType,Va: rBT_Im246033960le_rbt @ A @ B,Vb: A,Vc: B,Vd: rBT_Im246033960le_rbt @ A @ B,B2: rBT_Im246033960le_rbt @ A @ B,K: A,X: B,C: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1591839508ombine @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va @ Vb @ Vc @ Vd ) @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ B2 @ K @ X @ C ) )
= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ ( rBT_Im1591839508ombine @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va @ Vb @ Vc @ Vd ) @ B2 ) @ K @ X @ C ) ) ).
% combine.simps(5)
thf(fact_171_combine_Osimps_I6_J,axiom,
! [B: $tType,A: $tType,A2: rBT_Im246033960le_rbt @ A @ B,K: A,X: B,B2: rBT_Im246033960le_rbt @ A @ B,Va: rBT_Im246033960le_rbt @ A @ B,Vb: A,Vc: B,Vd: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1591839508ombine @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ A2 @ K @ X @ B2 ) @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va @ Vb @ Vc @ Vd ) )
= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ A2 @ K @ X @ ( rBT_Im1591839508ombine @ A @ B @ B2 @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va @ Vb @ Vc @ Vd ) ) ) ) ).
% combine.simps(6)
thf(fact_172_balance__right_Osimps_I1_J,axiom,
! [B: $tType,A: $tType,A2: rBT_Im246033960le_rbt @ A @ B,K: A,X: B,B2: rBT_Im246033960le_rbt @ A @ B,S: A,Y: B,C: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1053358987_right @ A @ B @ A2 @ K @ X @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ B2 @ S @ Y @ C ) )
= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ A2 @ K @ X @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ B2 @ S @ Y @ C ) ) ) ).
% balance_right.simps(1)
thf(fact_173_ord_Orbt__del__from__right_Osimps_I2_J,axiom,
! [B: $tType,A: $tType,Less: A > A > $o,X: A,A2: rBT_Im246033960le_rbt @ A @ B,Y: A,S: B] :
( ( rBT_Im1808746437_right @ A @ B @ Less @ X @ A2 @ Y @ S @ ( rBT_Im418718756_Empty @ A @ B ) )
= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ A2 @ Y @ S @ ( rBT_Im1714610413bt_del @ A @ B @ Less @ X @ ( rBT_Im418718756_Empty @ A @ B ) ) ) ) ).
% ord.rbt_del_from_right.simps(2)
thf(fact_174_ord_Orbt__del__from__left_Osimps_I2_J,axiom,
! [B: $tType,A: $tType,Less: A > A > $o,X: A,Y: A,S: B,B2: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1152561598m_left @ A @ B @ Less @ X @ ( rBT_Im418718756_Empty @ A @ B ) @ Y @ S @ B2 )
= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ ( rBT_Im1714610413bt_del @ A @ B @ Less @ X @ ( rBT_Im418718756_Empty @ A @ B ) ) @ Y @ S @ B2 ) ) ).
% ord.rbt_del_from_left.simps(2)
thf(fact_175_ord_Orbt__del__from__right_Osimps_I1_J,axiom,
! [B: $tType,A: $tType,Less: A > A > $o,X: A,A2: rBT_Im246033960le_rbt @ A @ B,Y: A,S: B,Lt: rBT_Im246033960le_rbt @ A @ B,Z: A,V: B,Rt: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1808746437_right @ A @ B @ Less @ X @ A2 @ Y @ S @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Lt @ Z @ V @ Rt ) )
= ( rBT_Im1053358987_right @ A @ B @ A2 @ Y @ S @ ( rBT_Im1714610413bt_del @ A @ B @ Less @ X @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Lt @ Z @ V @ Rt ) ) ) ) ).
% ord.rbt_del_from_right.simps(1)
thf(fact_176_ord_Orbt__del_Oelims,axiom,
! [B: $tType,A: $tType,Less: A > A > $o,X: A,Xa: rBT_Im246033960le_rbt @ A @ B,Y: rBT_Im246033960le_rbt @ A @ B] :
( ( ( rBT_Im1714610413bt_del @ A @ B @ Less @ X @ Xa )
= Y )
=> ( ( ( Xa
= ( rBT_Im418718756_Empty @ A @ B ) )
=> ( Y
!= ( rBT_Im418718756_Empty @ A @ B ) ) )
=> ~ ! [C3: rBT_Im1923302023_color,A4: rBT_Im246033960le_rbt @ A @ B,Y2: A,S2: B,B3: rBT_Im246033960le_rbt @ A @ B] :
( ( Xa
= ( rBT_Im480247531Branch @ A @ B @ C3 @ A4 @ Y2 @ S2 @ B3 ) )
=> ~ ( ( ( Less @ X @ Y2 )
=> ( Y
= ( rBT_Im1152561598m_left @ A @ B @ Less @ X @ A4 @ Y2 @ S2 @ B3 ) ) )
& ( ~ ( Less @ X @ Y2 )
=> ( ( ( Less @ Y2 @ X )
=> ( Y
= ( rBT_Im1808746437_right @ A @ B @ Less @ X @ A4 @ Y2 @ S2 @ B3 ) ) )
& ( ~ ( Less @ Y2 @ X )
=> ( Y
= ( rBT_Im1591839508ombine @ A @ B @ A4 @ B3 ) ) ) ) ) ) ) ) ) ).
% ord.rbt_del.elims
thf(fact_177_local_Ois__rbt__def,axiom,
! [B: $tType,T: rBT_Im246033960le_rbt @ a @ B] :
( ( rBT_Im862805236is_rbt @ a @ B @ less @ T )
= ( ( rBT_Im1040612427e_inv1 @ a @ B @ T )
& ( rBT_Im1040612428e_inv2 @ a @ B @ T )
& ( ( rBT_Im1805589839lor_of @ a @ B @ T )
= rBT_Impl_Mirabelle_B )
& ( rBT_Im759614907sorted @ a @ B @ less @ T ) ) ) ).
% local.is_rbt_def
thf(fact_178_local_Oins__inv1__inv2_I3_J,axiom,
! [B: $tType,T: rBT_Im246033960le_rbt @ a @ B,F: a > B > B > B,K: a,X: B] :
( ( rBT_Im1040612427e_inv1 @ a @ B @ T )
=> ( ( rBT_Im1040612428e_inv2 @ a @ B @ T )
=> ( ( ( rBT_Im1805589839lor_of @ a @ B @ T )
= rBT_Impl_Mirabelle_B )
=> ( rBT_Im1040612427e_inv1 @ a @ B @ ( rBT_Im1756508336bt_ins @ a @ B @ less @ F @ K @ X @ T ) ) ) ) ) ).
% local.ins_inv1_inv2(3)
thf(fact_179_local_Orbt__insert__with__key__def,axiom,
! [B: $tType,F: a > B > B > B,K: a,V: B,T: rBT_Im246033960le_rbt @ a @ B] :
( ( rBT_Im627674146th_key @ a @ B @ less @ F @ K @ V @ T )
= ( rBT_Im945281971_paint @ a @ B @ rBT_Impl_Mirabelle_B @ ( rBT_Im1756508336bt_ins @ a @ B @ less @ F @ K @ V @ T ) ) ) ).
% local.rbt_insert_with_key_def
thf(fact_180_local_Omap__of__entries,axiom,
! [B: $tType,T: rBT_Im246033960le_rbt @ a @ B] :
( ( rBT_Im759614907sorted @ a @ B @ less @ T )
=> ( ( map_of @ a @ B @ ( rBT_Im954575269ntries @ a @ B @ T ) )
= ( rBT_Im35466040lookup @ a @ B @ less @ T ) ) ) ).
% local.map_of_entries
thf(fact_181_local_Opaint__rbt__sorted,axiom,
! [B: $tType,T: rBT_Im246033960le_rbt @ a @ B,C: rBT_Im1923302023_color] :
( ( rBT_Im759614907sorted @ a @ B @ less @ T )
=> ( rBT_Im759614907sorted @ a @ B @ less @ ( rBT_Im945281971_paint @ a @ B @ C @ T ) ) ) ).
% local.paint_rbt_sorted
thf(fact_182_paint__color__of,axiom,
! [A: $tType,B: $tType,T: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1805589839lor_of @ A @ B @ ( rBT_Im945281971_paint @ A @ B @ rBT_Impl_Mirabelle_B @ T ) )
= rBT_Impl_Mirabelle_B ) ).
% paint_color_of
thf(fact_183_local_Opaint__rbt__greater,axiom,
! [B: $tType,V: a,C: rBT_Im1923302023_color,T: rBT_Im246033960le_rbt @ a @ B] :
( ( rBT_Im1259024060reater @ a @ B @ less @ V @ ( rBT_Im945281971_paint @ a @ B @ C @ T ) )
= ( rBT_Im1259024060reater @ a @ B @ less @ V @ T ) ) ).
% local.paint_rbt_greater
thf(fact_184_local_Opaint__rbt__less,axiom,
! [B: $tType,V: a,C: rBT_Im1923302023_color,T: rBT_Im246033960le_rbt @ a @ B] :
( ( rBT_Im2075124343t_less @ a @ B @ less @ V @ ( rBT_Im945281971_paint @ a @ B @ C @ T ) )
= ( rBT_Im2075124343t_less @ a @ B @ less @ V @ T ) ) ).
% local.paint_rbt_less
thf(fact_185_local_Opaint__rbt__lookup,axiom,
! [B: $tType,C: rBT_Im1923302023_color,T: rBT_Im246033960le_rbt @ a @ B] :
( ( rBT_Im35466040lookup @ a @ B @ less @ ( rBT_Im945281971_paint @ a @ B @ C @ T ) )
= ( rBT_Im35466040lookup @ a @ B @ less @ T ) ) ).
% local.paint_rbt_lookup
thf(fact_186_paint__in__tree,axiom,
! [B: $tType,A: $tType,K: A,X: B,C: rBT_Im1923302023_color,T: rBT_Im246033960le_rbt @ A @ B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ X ) @ ( set2 @ ( product_prod @ A @ B ) @ ( rBT_Im954575269ntries @ A @ B @ ( rBT_Im945281971_paint @ A @ B @ C @ T ) ) ) )
= ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ X ) @ ( set2 @ ( product_prod @ A @ B ) @ ( rBT_Im954575269ntries @ A @ B @ T ) ) ) ) ).
% paint_in_tree
thf(fact_187_paint_Osimps_I2_J,axiom,
! [B: $tType,A: $tType,C: rBT_Im1923302023_color,Uu2: rBT_Im1923302023_color,L: rBT_Im246033960le_rbt @ A @ B,K: A,V: B,R: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im945281971_paint @ A @ B @ C @ ( rBT_Im480247531Branch @ A @ B @ Uu2 @ L @ K @ V @ R ) )
= ( rBT_Im480247531Branch @ A @ B @ C @ L @ K @ V @ R ) ) ).
% paint.simps(2)
thf(fact_188_paint_Osimps_I1_J,axiom,
! [B: $tType,A: $tType,C: rBT_Im1923302023_color] :
( ( rBT_Im945281971_paint @ A @ B @ C @ ( rBT_Im418718756_Empty @ A @ B ) )
= ( rBT_Im418718756_Empty @ A @ B ) ) ).
% paint.simps(1)
thf(fact_189_color__of_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,C: rBT_Im1923302023_color,Uu2: rBT_Im246033960le_rbt @ A @ B,Uv2: A,Uw: B,Ux: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1805589839lor_of @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ C @ Uu2 @ Uv2 @ Uw @ Ux ) )
= C ) ).
% color_of.simps(2)
thf(fact_190_ord_Opaint__rbt__sorted,axiom,
! [B: $tType,A: $tType,Less: A > A > $o,T: rBT_Im246033960le_rbt @ A @ B,C: rBT_Im1923302023_color] :
( ( rBT_Im759614907sorted @ A @ B @ Less @ T )
=> ( rBT_Im759614907sorted @ A @ B @ Less @ ( rBT_Im945281971_paint @ A @ B @ C @ T ) ) ) ).
% ord.paint_rbt_sorted
thf(fact_191_paint__inv2,axiom,
! [B: $tType,A: $tType,T: rBT_Im246033960le_rbt @ A @ B,C: rBT_Im1923302023_color] :
( ( rBT_Im1040612428e_inv2 @ A @ B @ T )
=> ( rBT_Im1040612428e_inv2 @ A @ B @ ( rBT_Im945281971_paint @ A @ B @ C @ T ) ) ) ).
% paint_inv2
thf(fact_192_ord_Opaint__rbt__greater,axiom,
! [B: $tType,A: $tType,Less: A > A > $o,V: A,C: rBT_Im1923302023_color,T: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1259024060reater @ A @ B @ Less @ V @ ( rBT_Im945281971_paint @ A @ B @ C @ T ) )
= ( rBT_Im1259024060reater @ A @ B @ Less @ V @ T ) ) ).
% ord.paint_rbt_greater
thf(fact_193_ord_Opaint__rbt__less,axiom,
! [B: $tType,A: $tType,Less: A > A > $o,V: A,C: rBT_Im1923302023_color,T: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im2075124343t_less @ A @ B @ Less @ V @ ( rBT_Im945281971_paint @ A @ B @ C @ T ) )
= ( rBT_Im2075124343t_less @ A @ B @ Less @ V @ T ) ) ).
% ord.paint_rbt_less
thf(fact_194_ord_Opaint__rbt__lookup,axiom,
! [B: $tType,A: $tType,Less: A > A > $o,C: rBT_Im1923302023_color,T: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im35466040lookup @ A @ B @ Less @ ( rBT_Im945281971_paint @ A @ B @ C @ T ) )
= ( rBT_Im35466040lookup @ A @ B @ Less @ T ) ) ).
% ord.paint_rbt_lookup
thf(fact_195_paint__inv1l,axiom,
! [B: $tType,A: $tType,T: rBT_Im246033960le_rbt @ A @ B,C: rBT_Im1923302023_color] :
( ( rBT_Im1071879905_inv1l @ A @ B @ T )
=> ( rBT_Im1071879905_inv1l @ A @ B @ ( rBT_Im945281971_paint @ A @ B @ C @ T ) ) ) ).
% paint_inv1l
thf(fact_196_color__of_Osimps_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( rBT_Im1805589839lor_of @ A @ B @ ( rBT_Im418718756_Empty @ A @ B ) )
= rBT_Impl_Mirabelle_B ) ).
% color_of.simps(1)
thf(fact_197_inv1_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,C: rBT_Im1923302023_color,Lt: rBT_Im246033960le_rbt @ A @ B,K: A,V: B,Rt: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1040612427e_inv1 @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ C @ Lt @ K @ V @ Rt ) )
= ( ( rBT_Im1040612427e_inv1 @ A @ B @ Lt )
& ( rBT_Im1040612427e_inv1 @ A @ B @ Rt )
& ( ( C = rBT_Impl_Mirabelle_B )
| ( ( ( rBT_Im1805589839lor_of @ A @ B @ Lt )
= rBT_Impl_Mirabelle_B )
& ( ( rBT_Im1805589839lor_of @ A @ B @ Rt )
= rBT_Impl_Mirabelle_B ) ) ) ) ) ).
% inv1.simps(2)
thf(fact_198_paint__inv1,axiom,
! [B: $tType,A: $tType,T: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1071879905_inv1l @ A @ B @ T )
=> ( rBT_Im1040612427e_inv1 @ A @ B @ ( rBT_Im945281971_paint @ A @ B @ rBT_Impl_Mirabelle_B @ T ) ) ) ).
% paint_inv1
thf(fact_199_combine__inv1_I1_J,axiom,
! [B: $tType,A: $tType,Lt: rBT_Im246033960le_rbt @ A @ B,Rt: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1040612427e_inv1 @ A @ B @ Lt )
=> ( ( rBT_Im1040612427e_inv1 @ A @ B @ Rt )
=> ( ( ( rBT_Im1805589839lor_of @ A @ B @ Lt )
= rBT_Impl_Mirabelle_B )
=> ( ( ( rBT_Im1805589839lor_of @ A @ B @ Rt )
= rBT_Impl_Mirabelle_B )
=> ( rBT_Im1040612427e_inv1 @ A @ B @ ( rBT_Im1591839508ombine @ A @ B @ Lt @ Rt ) ) ) ) ) ) ).
% combine_inv1(1)
thf(fact_200_ord_Orbt__insert__with__key__def,axiom,
! [B: $tType,A: $tType] :
( ( rBT_Im627674146th_key @ A @ B )
= ( ^ [Less2: A > A > $o,F2: A > B > B > B,K3: A,V3: B,T3: rBT_Im246033960le_rbt @ A @ B] : ( rBT_Im945281971_paint @ A @ B @ rBT_Impl_Mirabelle_B @ ( rBT_Im1756508336bt_ins @ A @ B @ Less2 @ F2 @ K3 @ V3 @ T3 ) ) ) ) ).
% ord.rbt_insert_with_key_def
thf(fact_201_ord_Oins__inv1__inv2_I3_J,axiom,
! [B: $tType,A: $tType,T: rBT_Im246033960le_rbt @ A @ B,Less: A > A > $o,F: A > B > B > B,K: A,X: B] :
( ( rBT_Im1040612427e_inv1 @ A @ B @ T )
=> ( ( rBT_Im1040612428e_inv2 @ A @ B @ T )
=> ( ( ( rBT_Im1805589839lor_of @ A @ B @ T )
= rBT_Impl_Mirabelle_B )
=> ( rBT_Im1040612427e_inv1 @ A @ B @ ( rBT_Im1756508336bt_ins @ A @ B @ Less @ F @ K @ X @ T ) ) ) ) ) ).
% ord.ins_inv1_inv2(3)
thf(fact_202_balance__left__inv1,axiom,
! [B: $tType,A: $tType,A2: rBT_Im246033960le_rbt @ A @ B,B2: rBT_Im246033960le_rbt @ A @ B,K: A,X: B] :
( ( rBT_Im1071879905_inv1l @ A @ B @ A2 )
=> ( ( rBT_Im1040612427e_inv1 @ A @ B @ B2 )
=> ( ( ( rBT_Im1805589839lor_of @ A @ B @ B2 )
= rBT_Impl_Mirabelle_B )
=> ( rBT_Im1040612427e_inv1 @ A @ B @ ( rBT_Im1278323832e_left @ A @ B @ A2 @ K @ X @ B2 ) ) ) ) ) ).
% balance_left_inv1
thf(fact_203_balance__right__inv1,axiom,
! [B: $tType,A: $tType,A2: rBT_Im246033960le_rbt @ A @ B,B2: rBT_Im246033960le_rbt @ A @ B,K: A,X: B] :
( ( rBT_Im1040612427e_inv1 @ A @ B @ A2 )
=> ( ( rBT_Im1071879905_inv1l @ A @ B @ B2 )
=> ( ( ( rBT_Im1805589839lor_of @ A @ B @ A2 )
= rBT_Impl_Mirabelle_B )
=> ( rBT_Im1040612427e_inv1 @ A @ B @ ( rBT_Im1053358987_right @ A @ B @ A2 @ K @ X @ B2 ) ) ) ) ) ).
% balance_right_inv1
thf(fact_204_ord_Ois__rbt__def,axiom,
! [B: $tType,A: $tType] :
( ( rBT_Im862805236is_rbt @ A @ B )
= ( ^ [Less2: A > A > $o,T3: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1040612427e_inv1 @ A @ B @ T3 )
& ( rBT_Im1040612428e_inv2 @ A @ B @ T3 )
& ( ( rBT_Im1805589839lor_of @ A @ B @ T3 )
= rBT_Impl_Mirabelle_B )
& ( rBT_Im759614907sorted @ A @ B @ Less2 @ T3 ) ) ) ) ).
% ord.is_rbt_def
thf(fact_205_prod__cases5,axiom,
! [A: $tType,B: $tType,C2: $tType,D2: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C2 @ ( product_prod @ D2 @ E ) ) )] :
~ ! [A4: A,B3: B,C3: C2,D: D2,E2: E] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C2 @ ( product_prod @ D2 @ E ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C2 @ ( product_prod @ D2 @ E ) ) @ B3 @ ( product_Pair @ C2 @ ( product_prod @ D2 @ E ) @ C3 @ ( product_Pair @ D2 @ E @ D @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_206_prod__cases6,axiom,
! [A: $tType,B: $tType,C2: $tType,D2: $tType,E: $tType,F4: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C2 @ ( product_prod @ D2 @ ( product_prod @ E @ F4 ) ) ) )] :
~ ! [A4: A,B3: B,C3: C2,D: D2,E2: E,F3: F4] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C2 @ ( product_prod @ D2 @ ( product_prod @ E @ F4 ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C2 @ ( product_prod @ D2 @ ( product_prod @ E @ F4 ) ) ) @ B3 @ ( product_Pair @ C2 @ ( product_prod @ D2 @ ( product_prod @ E @ F4 ) ) @ C3 @ ( product_Pair @ D2 @ ( product_prod @ E @ F4 ) @ D @ ( product_Pair @ E @ F4 @ E2 @ F3 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_207_prod__cases7,axiom,
! [A: $tType,B: $tType,C2: $tType,D2: $tType,E: $tType,F4: $tType,G2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C2 @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F4 @ G2 ) ) ) ) )] :
~ ! [A4: A,B3: B,C3: C2,D: D2,E2: E,F3: F4,G3: G2] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C2 @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F4 @ G2 ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C2 @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F4 @ G2 ) ) ) ) @ B3 @ ( product_Pair @ C2 @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F4 @ G2 ) ) ) @ C3 @ ( product_Pair @ D2 @ ( product_prod @ E @ ( product_prod @ F4 @ G2 ) ) @ D @ ( product_Pair @ E @ ( product_prod @ F4 @ G2 ) @ E2 @ ( product_Pair @ F4 @ G2 @ F3 @ G3 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_208_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A5: A,B4: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A5 @ B4 ) )
= ( ( A2 = A5 )
& ( B2 = B4 ) ) ) ).
% old.prod.inject
thf(fact_209_prod_Oinject,axiom,
! [A: $tType,B: $tType,X12: A,X27: B,Y1: A,Y26: B] :
( ( ( product_Pair @ A @ B @ X12 @ X27 )
= ( product_Pair @ A @ B @ Y1 @ Y26 ) )
= ( ( X12 = Y1 )
& ( X27 = Y26 ) ) ) ).
% prod.inject
thf(fact_210_old_Oprod_Oinducts,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,Prod: product_prod @ A @ B] :
( ! [A4: A,B3: B] : ( P @ ( product_Pair @ A @ B @ A4 @ B3 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_211_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
~ ! [A4: A,B3: B] :
( Y
!= ( product_Pair @ A @ B @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_212_Pair__inject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A5: A,B4: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A5 @ B4 ) )
=> ~ ( ( A2 = A5 )
=> ( B2 != B4 ) ) ) ).
% Pair_inject
thf(fact_213_prod__cases,axiom,
! [B: $tType,A: $tType,P: ( product_prod @ A @ B ) > $o,P2: product_prod @ A @ B] :
( ! [A4: A,B3: B] : ( P @ ( product_Pair @ A @ B @ A4 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_214_surj__pair,axiom,
! [A: $tType,B: $tType,P2: product_prod @ A @ B] :
? [X3: A,Y2: B] :
( P2
= ( product_Pair @ A @ B @ X3 @ Y2 ) ) ).
% surj_pair
thf(fact_215_prod__induct3,axiom,
! [C2: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ C2 ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C2 )] :
( ! [A4: A,B3: B,C3: C2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ C2 ) @ A4 @ ( product_Pair @ B @ C2 @ B3 @ C3 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_216_prod__cases3,axiom,
! [A: $tType,B: $tType,C2: $tType,Y: product_prod @ A @ ( product_prod @ B @ C2 )] :
~ ! [A4: A,B3: B,C3: C2] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ C2 ) @ A4 @ ( product_Pair @ B @ C2 @ B3 @ C3 ) ) ) ).
% prod_cases3
thf(fact_217_prod__induct4,axiom,
! [D2: $tType,C2: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C2 @ D2 ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C2 @ D2 ) )] :
( ! [A4: A,B3: B,C3: C2,D: D2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C2 @ D2 ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C2 @ D2 ) @ B3 @ ( product_Pair @ C2 @ D2 @ C3 @ D ) ) ) )
=> ( P @ X ) ) ).
% prod_induct4
thf(fact_218_prod__cases4,axiom,
! [A: $tType,B: $tType,C2: $tType,D2: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C2 @ D2 ) )] :
~ ! [A4: A,B3: B,C3: C2,D: D2] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C2 @ D2 ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C2 @ D2 ) @ B3 @ ( product_Pair @ C2 @ D2 @ C3 @ D ) ) ) ) ).
% prod_cases4
thf(fact_219_prod__induct7,axiom,
! [G2: $tType,F4: $tType,E: $tType,D2: $tType,C2: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C2 @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F4 @ G2 ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C2 @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F4 @ G2 ) ) ) ) )] :
( ! [A4: A,B3: B,C3: C2,D: D2,E2: E,F3: F4,G3: G2] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C2 @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F4 @ G2 ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C2 @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F4 @ G2 ) ) ) ) @ B3 @ ( product_Pair @ C2 @ ( product_prod @ D2 @ ( product_prod @ E @ ( product_prod @ F4 @ G2 ) ) ) @ C3 @ ( product_Pair @ D2 @ ( product_prod @ E @ ( product_prod @ F4 @ G2 ) ) @ D @ ( product_Pair @ E @ ( product_prod @ F4 @ G2 ) @ E2 @ ( product_Pair @ F4 @ G2 @ F3 @ G3 ) ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct7
thf(fact_220_prod__induct6,axiom,
! [F4: $tType,E: $tType,D2: $tType,C2: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C2 @ ( product_prod @ D2 @ ( product_prod @ E @ F4 ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C2 @ ( product_prod @ D2 @ ( product_prod @ E @ F4 ) ) ) )] :
( ! [A4: A,B3: B,C3: C2,D: D2,E2: E,F3: F4] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C2 @ ( product_prod @ D2 @ ( product_prod @ E @ F4 ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C2 @ ( product_prod @ D2 @ ( product_prod @ E @ F4 ) ) ) @ B3 @ ( product_Pair @ C2 @ ( product_prod @ D2 @ ( product_prod @ E @ F4 ) ) @ C3 @ ( product_Pair @ D2 @ ( product_prod @ E @ F4 ) @ D @ ( product_Pair @ E @ F4 @ E2 @ F3 ) ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct6
thf(fact_221_prod__induct5,axiom,
! [E: $tType,D2: $tType,C2: $tType,B: $tType,A: $tType,P: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C2 @ ( product_prod @ D2 @ E ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C2 @ ( product_prod @ D2 @ E ) ) )] :
( ! [A4: A,B3: B,C3: C2,D: D2,E2: E] : ( P @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C2 @ ( product_prod @ D2 @ E ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C2 @ ( product_prod @ D2 @ E ) ) @ B3 @ ( product_Pair @ C2 @ ( product_prod @ D2 @ E ) @ C3 @ ( product_Pair @ D2 @ E @ D @ E2 ) ) ) ) )
=> ( P @ X ) ) ).
% prod_induct5
thf(fact_222_map__of__SomeD,axiom,
! [A: $tType,B: $tType,Xs: list @ ( product_prod @ B @ A ),K: B,Y: A] :
( ( ( map_of @ B @ A @ Xs @ K )
= ( some @ A @ Y ) )
=> ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ K @ Y ) @ ( set2 @ ( product_prod @ B @ A ) @ Xs ) ) ) ).
% map_of_SomeD
thf(fact_223_weak__map__of__SomeI,axiom,
! [A: $tType,B: $tType,K: A,X: B,L: list @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ X ) @ ( set2 @ ( product_prod @ A @ B ) @ L ) )
=> ? [X3: B] :
( ( map_of @ A @ B @ L @ K )
= ( some @ B @ X3 ) ) ) ).
% weak_map_of_SomeI
thf(fact_224_local_Orbt__lookup__rbt__insert,axiom,
! [B: $tType,T: rBT_Im246033960le_rbt @ a @ B,K: a,V: B] :
( ( rBT_Im862805236is_rbt @ a @ B @ less @ T )
=> ( ( rBT_Im35466040lookup @ a @ B @ less @ ( rBT_Im852463031insert @ a @ B @ less @ K @ V @ T ) )
= ( fun_upd @ a @ ( option @ B ) @ ( rBT_Im35466040lookup @ a @ B @ less @ T ) @ K @ ( some @ B @ V ) ) ) ) ).
% local.rbt_lookup_rbt_insert
thf(fact_225_local_Obalance__rbt__sorted,axiom,
! [B: $tType,L: rBT_Im246033960le_rbt @ a @ B,R: rBT_Im246033960le_rbt @ a @ B,K: a,V: B] :
( ( rBT_Im759614907sorted @ a @ B @ less @ L )
=> ( ( rBT_Im759614907sorted @ a @ B @ less @ R )
=> ( ( rBT_Im2075124343t_less @ a @ B @ less @ K @ L )
=> ( ( rBT_Im1259024060reater @ a @ B @ less @ K @ R )
=> ( rBT_Im759614907sorted @ a @ B @ less @ ( rBT_Im1648453169alance @ a @ B @ L @ K @ V @ R ) ) ) ) ) ) ).
% local.balance_rbt_sorted
thf(fact_226_local_Orbt__ins_Osimps_I2_J,axiom,
! [B: $tType,K: a,X: a,F: a > B > B > B,V: B,L: rBT_Im246033960le_rbt @ a @ B,Y: B,R: rBT_Im246033960le_rbt @ a @ B] :
( ( ( less @ K @ X )
=> ( ( rBT_Im1756508336bt_ins @ a @ B @ less @ F @ K @ V @ ( rBT_Im480247531Branch @ a @ B @ rBT_Impl_Mirabelle_B @ L @ X @ Y @ R ) )
= ( rBT_Im1648453169alance @ a @ B @ ( rBT_Im1756508336bt_ins @ a @ B @ less @ F @ K @ V @ L ) @ X @ Y @ R ) ) )
& ( ~ ( less @ K @ X )
=> ( ( ( less @ X @ K )
=> ( ( rBT_Im1756508336bt_ins @ a @ B @ less @ F @ K @ V @ ( rBT_Im480247531Branch @ a @ B @ rBT_Impl_Mirabelle_B @ L @ X @ Y @ R ) )
= ( rBT_Im1648453169alance @ a @ B @ L @ X @ Y @ ( rBT_Im1756508336bt_ins @ a @ B @ less @ F @ K @ V @ R ) ) ) )
& ( ~ ( less @ X @ K )
=> ( ( rBT_Im1756508336bt_ins @ a @ B @ less @ F @ K @ V @ ( rBT_Im480247531Branch @ a @ B @ rBT_Impl_Mirabelle_B @ L @ X @ Y @ R ) )
= ( rBT_Im480247531Branch @ a @ B @ rBT_Impl_Mirabelle_B @ L @ X @ ( F @ K @ Y @ V ) @ R ) ) ) ) ) ) ).
% local.rbt_ins.simps(2)
thf(fact_227_local_Obalance__rbt__less,axiom,
! [B: $tType,V: a,A2: rBT_Im246033960le_rbt @ a @ B,K: a,X: B,B2: rBT_Im246033960le_rbt @ a @ B] :
( ( rBT_Im2075124343t_less @ a @ B @ less @ V @ ( rBT_Im1648453169alance @ a @ B @ A2 @ K @ X @ B2 ) )
= ( ( rBT_Im2075124343t_less @ a @ B @ less @ V @ A2 )
& ( rBT_Im2075124343t_less @ a @ B @ less @ V @ B2 )
& ( less @ K @ V ) ) ) ).
% local.balance_rbt_less
thf(fact_228_local_Obalance__rbt__greater,axiom,
! [B: $tType,V: a,A2: rBT_Im246033960le_rbt @ a @ B,K: a,X: B,B2: rBT_Im246033960le_rbt @ a @ B] :
( ( rBT_Im1259024060reater @ a @ B @ less @ V @ ( rBT_Im1648453169alance @ a @ B @ A2 @ K @ X @ B2 ) )
= ( ( rBT_Im1259024060reater @ a @ B @ less @ V @ A2 )
& ( rBT_Im1259024060reater @ a @ B @ less @ V @ B2 )
& ( less @ V @ K ) ) ) ).
% local.balance_rbt_greater
thf(fact_229_local_Orbt__lookup__balance,axiom,
! [B: $tType,L: rBT_Im246033960le_rbt @ a @ B,R: rBT_Im246033960le_rbt @ a @ B,K: a,V: B,X: a] :
( ( rBT_Im759614907sorted @ a @ B @ less @ L )
=> ( ( rBT_Im759614907sorted @ a @ B @ less @ R )
=> ( ( rBT_Im2075124343t_less @ a @ B @ less @ K @ L )
=> ( ( rBT_Im1259024060reater @ a @ B @ less @ K @ R )
=> ( ( rBT_Im35466040lookup @ a @ B @ less @ ( rBT_Im1648453169alance @ a @ B @ L @ K @ V @ R ) @ X )
= ( rBT_Im35466040lookup @ a @ B @ less @ ( rBT_Im480247531Branch @ a @ B @ rBT_Impl_Mirabelle_B @ L @ K @ V @ R ) @ X ) ) ) ) ) ) ).
% local.rbt_lookup_balance
thf(fact_230_map__upd__eqD1,axiom,
! [A: $tType,B: $tType,M: A > ( option @ B ),A2: A,X: B,N: A > ( option @ B ),Y: B] :
( ( ( fun_upd @ A @ ( option @ B ) @ M @ A2 @ ( some @ B @ X ) )
= ( fun_upd @ A @ ( option @ B ) @ N @ A2 @ ( some @ B @ Y ) ) )
=> ( X = Y ) ) ).
% map_upd_eqD1
thf(fact_231_map__upd__triv,axiom,
! [A: $tType,B: $tType,T: B > ( option @ A ),K: B,X: A] :
( ( ( T @ K )
= ( some @ A @ X ) )
=> ( ( fun_upd @ B @ ( option @ A ) @ T @ K @ ( some @ A @ X ) )
= T ) ) ).
% map_upd_triv
thf(fact_232_map__upd__Some__unfold,axiom,
! [B: $tType,A: $tType,M: B > ( option @ A ),A2: B,B2: A,X: B,Y: A] :
( ( ( fun_upd @ B @ ( option @ A ) @ M @ A2 @ ( some @ A @ B2 ) @ X )
= ( some @ A @ Y ) )
= ( ( ( X = A2 )
& ( B2 = Y ) )
| ( ( X != A2 )
& ( ( M @ X )
= ( some @ A @ Y ) ) ) ) ) ).
% map_upd_Some_unfold
thf(fact_233_ord_Obalance__rbt__greater,axiom,
! [B: $tType,A: $tType,Less: A > A > $o,V: A,A2: rBT_Im246033960le_rbt @ A @ B,K: A,X: B,B2: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1259024060reater @ A @ B @ Less @ V @ ( rBT_Im1648453169alance @ A @ B @ A2 @ K @ X @ B2 ) )
= ( ( rBT_Im1259024060reater @ A @ B @ Less @ V @ A2 )
& ( rBT_Im1259024060reater @ A @ B @ Less @ V @ B2 )
& ( Less @ V @ K ) ) ) ).
% ord.balance_rbt_greater
thf(fact_234_ord_Obalance__rbt__less,axiom,
! [B: $tType,A: $tType,Less: A > A > $o,V: A,A2: rBT_Im246033960le_rbt @ A @ B,K: A,X: B,B2: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im2075124343t_less @ A @ B @ Less @ V @ ( rBT_Im1648453169alance @ A @ B @ A2 @ K @ X @ B2 ) )
= ( ( rBT_Im2075124343t_less @ A @ B @ Less @ V @ A2 )
& ( rBT_Im2075124343t_less @ A @ B @ Less @ V @ B2 )
& ( Less @ K @ V ) ) ) ).
% ord.balance_rbt_less
thf(fact_235_balance__inv1,axiom,
! [B: $tType,A: $tType,L: rBT_Im246033960le_rbt @ A @ B,R: rBT_Im246033960le_rbt @ A @ B,K: A,V: B] :
( ( rBT_Im1071879905_inv1l @ A @ B @ L )
=> ( ( rBT_Im1071879905_inv1l @ A @ B @ R )
=> ( rBT_Im1040612427e_inv1 @ A @ B @ ( rBT_Im1648453169alance @ A @ B @ L @ K @ V @ R ) ) ) ) ).
% balance_inv1
thf(fact_236_balance__inv2,axiom,
! [B: $tType,A: $tType,L: rBT_Im246033960le_rbt @ A @ B,R: rBT_Im246033960le_rbt @ A @ B,K: A,V: B] :
( ( rBT_Im1040612428e_inv2 @ A @ B @ L )
=> ( ( rBT_Im1040612428e_inv2 @ A @ B @ R )
=> ( ( ( rBT_Im315265182height @ A @ B @ L )
= ( rBT_Im315265182height @ A @ B @ R ) )
=> ( rBT_Im1040612428e_inv2 @ A @ B @ ( rBT_Im1648453169alance @ A @ B @ L @ K @ V @ R ) ) ) ) ) ).
% balance_inv2
thf(fact_237_balance_Osimps_I23_J,axiom,
! [B: $tType,A: $tType,Va: rBT_Im246033960le_rbt @ A @ B,Vb: A,Vc: B,Vd: rBT_Im246033960le_rbt @ A @ B,S: A,T: B,Ve: rBT_Im246033960le_rbt @ A @ B,Vf: A,Vg: B,Vh: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1648453169alance @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va @ Vb @ Vc @ Vd ) @ S @ T @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Ve @ Vf @ Vg @ Vh ) )
= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va @ Vb @ Vc @ Vd ) @ S @ T @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Ve @ Vf @ Vg @ Vh ) ) ) ).
% balance.simps(23)
thf(fact_238_balance_Osimps_I27_J,axiom,
! [B: $tType,A: $tType,Va: rBT_Im246033960le_rbt @ A @ B,Vb: A,Vc: B,Vd: rBT_Im246033960le_rbt @ A @ B,S: A,T: B,Ve: rBT_Im246033960le_rbt @ A @ B,Vf: A,Vg: B,Vj: rBT_Im246033960le_rbt @ A @ B,Vk: A,Vl: B,Vm: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1648453169alance @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va @ Vb @ Vc @ Vd ) @ S @ T @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Ve @ Vf @ Vg @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Vj @ Vk @ Vl @ Vm ) ) )
= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va @ Vb @ Vc @ Vd ) @ S @ T @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Ve @ Vf @ Vg @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Vj @ Vk @ Vl @ Vm ) ) ) ) ).
% balance.simps(27)
thf(fact_239_balance_Osimps_I29_J,axiom,
! [B: $tType,A: $tType,Va: rBT_Im246033960le_rbt @ A @ B,Vb: A,Vc: B,Vd: rBT_Im246033960le_rbt @ A @ B,S: A,T: B,V: rBT_Im1923302023_color,Vi: rBT_Im246033960le_rbt @ A @ B,Vn: A,Vo: B,Vp: rBT_Im246033960le_rbt @ A @ B,Vf: A,Vg: B,Vj: rBT_Im246033960le_rbt @ A @ B,Vk: A,Vl: B,Vm: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1648453169alance @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va @ Vb @ Vc @ Vd ) @ S @ T @ ( rBT_Im480247531Branch @ A @ B @ V @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Vi @ Vn @ Vo @ Vp ) @ Vf @ Vg @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Vj @ Vk @ Vl @ Vm ) ) )
= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va @ Vb @ Vc @ Vd ) @ S @ T @ ( rBT_Im480247531Branch @ A @ B @ V @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Vi @ Vn @ Vo @ Vp ) @ Vf @ Vg @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Vj @ Vk @ Vl @ Vm ) ) ) ) ).
% balance.simps(29)
thf(fact_240_balance_Osimps_I39_J,axiom,
! [B: $tType,A: $tType,Ve: rBT_Im246033960le_rbt @ A @ B,Vf: A,Vg: B,Vh: rBT_Im246033960le_rbt @ A @ B,S: A,T: B,Va: rBT_Im246033960le_rbt @ A @ B,Vb: A,Vc: B,Vd: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1648453169alance @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Ve @ Vf @ Vg @ Vh ) @ S @ T @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va @ Vb @ Vc @ Vd ) )
= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Ve @ Vf @ Vg @ Vh ) @ S @ T @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va @ Vb @ Vc @ Vd ) ) ) ).
% balance.simps(39)
thf(fact_241_balance_Osimps_I43_J,axiom,
! [B: $tType,A: $tType,Vj: rBT_Im246033960le_rbt @ A @ B,Vk: A,Vl: B,Vm: rBT_Im246033960le_rbt @ A @ B,Vf: A,Vg: B,Vh: rBT_Im246033960le_rbt @ A @ B,S: A,T: B,Va: rBT_Im246033960le_rbt @ A @ B,Vb: A,Vc: B,Vd: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1648453169alance @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Vj @ Vk @ Vl @ Vm ) @ Vf @ Vg @ Vh ) @ S @ T @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va @ Vb @ Vc @ Vd ) )
= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Vj @ Vk @ Vl @ Vm ) @ Vf @ Vg @ Vh ) @ S @ T @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va @ Vb @ Vc @ Vd ) ) ) ).
% balance.simps(43)
thf(fact_242_balance_Osimps_I45_J,axiom,
! [B: $tType,A: $tType,V: rBT_Im1923302023_color,Vj: rBT_Im246033960le_rbt @ A @ B,Vk: A,Vl: B,Vm: rBT_Im246033960le_rbt @ A @ B,Vf: A,Vg: B,Vi: rBT_Im246033960le_rbt @ A @ B,Vn: A,Vo: B,Vp: rBT_Im246033960le_rbt @ A @ B,S: A,T: B,Va: rBT_Im246033960le_rbt @ A @ B,Vb: A,Vc: B,Vd: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1648453169alance @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ V @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Vj @ Vk @ Vl @ Vm ) @ Vf @ Vg @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Vi @ Vn @ Vo @ Vp ) ) @ S @ T @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va @ Vb @ Vc @ Vd ) )
= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ ( rBT_Im480247531Branch @ A @ B @ V @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Vj @ Vk @ Vl @ Vm ) @ Vf @ Vg @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Vi @ Vn @ Vo @ Vp ) ) @ S @ T @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va @ Vb @ Vc @ Vd ) ) ) ).
% balance.simps(45)
thf(fact_243_balance_Osimps_I14_J,axiom,
! [B: $tType,A: $tType,S: A,T: B] :
( ( rBT_Im1648453169alance @ A @ B @ ( rBT_Im418718756_Empty @ A @ B ) @ S @ T @ ( rBT_Im418718756_Empty @ A @ B ) )
= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ ( rBT_Im418718756_Empty @ A @ B ) @ S @ T @ ( rBT_Im418718756_Empty @ A @ B ) ) ) ).
% balance.simps(14)
thf(fact_244_balance_Osimps_I15_J,axiom,
! [B: $tType,A: $tType,S: A,T: B,Va: rBT_Im246033960le_rbt @ A @ B,Vb: A,Vc: B,Vd: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1648453169alance @ A @ B @ ( rBT_Im418718756_Empty @ A @ B ) @ S @ T @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va @ Vb @ Vc @ Vd ) )
= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ ( rBT_Im418718756_Empty @ A @ B ) @ S @ T @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va @ Vb @ Vc @ Vd ) ) ) ).
% balance.simps(15)
thf(fact_245_balance_Osimps_I16_J,axiom,
! [B: $tType,A: $tType,S: A,T: B,Va: rBT_Im246033960le_rbt @ A @ B,Vb: A,Vc: B] :
( ( rBT_Im1648453169alance @ A @ B @ ( rBT_Im418718756_Empty @ A @ B ) @ S @ T @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va @ Vb @ Vc @ ( rBT_Im418718756_Empty @ A @ B ) ) )
= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ ( rBT_Im418718756_Empty @ A @ B ) @ S @ T @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va @ Vb @ Vc @ ( rBT_Im418718756_Empty @ A @ B ) ) ) ) ).
% balance.simps(16)
thf(fact_246_balance_Osimps_I17_J,axiom,
! [B: $tType,A: $tType,S: A,T: B,V: rBT_Im1923302023_color,Vb: A,Vc: B] :
( ( rBT_Im1648453169alance @ A @ B @ ( rBT_Im418718756_Empty @ A @ B ) @ S @ T @ ( rBT_Im480247531Branch @ A @ B @ V @ ( rBT_Im418718756_Empty @ A @ B ) @ Vb @ Vc @ ( rBT_Im418718756_Empty @ A @ B ) ) )
= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ ( rBT_Im418718756_Empty @ A @ B ) @ S @ T @ ( rBT_Im480247531Branch @ A @ B @ V @ ( rBT_Im418718756_Empty @ A @ B ) @ Vb @ Vc @ ( rBT_Im418718756_Empty @ A @ B ) ) ) ) ).
% balance.simps(17)
thf(fact_247_balance_Osimps_I18_J,axiom,
! [B: $tType,A: $tType,S: A,T: B,V: rBT_Im1923302023_color,Ve: rBT_Im246033960le_rbt @ A @ B,Vf: A,Vg: B,Vh: rBT_Im246033960le_rbt @ A @ B,Vb: A,Vc: B] :
( ( rBT_Im1648453169alance @ A @ B @ ( rBT_Im418718756_Empty @ A @ B ) @ S @ T @ ( rBT_Im480247531Branch @ A @ B @ V @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Ve @ Vf @ Vg @ Vh ) @ Vb @ Vc @ ( rBT_Im418718756_Empty @ A @ B ) ) )
= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ ( rBT_Im418718756_Empty @ A @ B ) @ S @ T @ ( rBT_Im480247531Branch @ A @ B @ V @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Ve @ Vf @ Vg @ Vh ) @ Vb @ Vc @ ( rBT_Im418718756_Empty @ A @ B ) ) ) ) ).
% balance.simps(18)
thf(fact_248_balance_Osimps_I19_J,axiom,
! [B: $tType,A: $tType,S: A,T: B,Va: rBT_Im246033960le_rbt @ A @ B,Vb: A,Vc: B,Vf: rBT_Im246033960le_rbt @ A @ B,Vg: A,Vh: B,Vi: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1648453169alance @ A @ B @ ( rBT_Im418718756_Empty @ A @ B ) @ S @ T @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va @ Vb @ Vc @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Vf @ Vg @ Vh @ Vi ) ) )
= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ ( rBT_Im418718756_Empty @ A @ B ) @ S @ T @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va @ Vb @ Vc @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Vf @ Vg @ Vh @ Vi ) ) ) ) ).
% balance.simps(19)
thf(fact_249_balance_Osimps_I20_J,axiom,
! [B: $tType,A: $tType,S: A,T: B,V: rBT_Im1923302023_color,Vb: A,Vc: B,Vf: rBT_Im246033960le_rbt @ A @ B,Vg: A,Vh: B,Vi: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1648453169alance @ A @ B @ ( rBT_Im418718756_Empty @ A @ B ) @ S @ T @ ( rBT_Im480247531Branch @ A @ B @ V @ ( rBT_Im418718756_Empty @ A @ B ) @ Vb @ Vc @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Vf @ Vg @ Vh @ Vi ) ) )
= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ ( rBT_Im418718756_Empty @ A @ B ) @ S @ T @ ( rBT_Im480247531Branch @ A @ B @ V @ ( rBT_Im418718756_Empty @ A @ B ) @ Vb @ Vc @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Vf @ Vg @ Vh @ Vi ) ) ) ) ).
% balance.simps(20)
thf(fact_250_balance_Osimps_I21_J,axiom,
! [B: $tType,A: $tType,S: A,T: B,V: rBT_Im1923302023_color,Ve: rBT_Im246033960le_rbt @ A @ B,Vj: A,Vk: B,Vl: rBT_Im246033960le_rbt @ A @ B,Vb: A,Vc: B,Vf: rBT_Im246033960le_rbt @ A @ B,Vg: A,Vh: B,Vi: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1648453169alance @ A @ B @ ( rBT_Im418718756_Empty @ A @ B ) @ S @ T @ ( rBT_Im480247531Branch @ A @ B @ V @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Ve @ Vj @ Vk @ Vl ) @ Vb @ Vc @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Vf @ Vg @ Vh @ Vi ) ) )
= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ ( rBT_Im418718756_Empty @ A @ B ) @ S @ T @ ( rBT_Im480247531Branch @ A @ B @ V @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Ve @ Vj @ Vk @ Vl ) @ Vb @ Vc @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Vf @ Vg @ Vh @ Vi ) ) ) ) ).
% balance.simps(21)
thf(fact_251_balance_Osimps_I22_J,axiom,
! [B: $tType,A: $tType,Va: rBT_Im246033960le_rbt @ A @ B,Vb: A,Vc: B,Vd: rBT_Im246033960le_rbt @ A @ B,S: A,T: B] :
( ( rBT_Im1648453169alance @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va @ Vb @ Vc @ Vd ) @ S @ T @ ( rBT_Im418718756_Empty @ A @ B ) )
= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va @ Vb @ Vc @ Vd ) @ S @ T @ ( rBT_Im418718756_Empty @ A @ B ) ) ) ).
% balance.simps(22)
thf(fact_252_balance_Osimps_I24_J,axiom,
! [B: $tType,A: $tType,Va: rBT_Im246033960le_rbt @ A @ B,Vb: A,Vc: B,Vd: rBT_Im246033960le_rbt @ A @ B,S: A,T: B,Ve: rBT_Im246033960le_rbt @ A @ B,Vf: A,Vg: B] :
( ( rBT_Im1648453169alance @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va @ Vb @ Vc @ Vd ) @ S @ T @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Ve @ Vf @ Vg @ ( rBT_Im418718756_Empty @ A @ B ) ) )
= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va @ Vb @ Vc @ Vd ) @ S @ T @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Ve @ Vf @ Vg @ ( rBT_Im418718756_Empty @ A @ B ) ) ) ) ).
% balance.simps(24)
thf(fact_253_balance_Osimps_I25_J,axiom,
! [B: $tType,A: $tType,Va: rBT_Im246033960le_rbt @ A @ B,Vb: A,Vc: B,Vd: rBT_Im246033960le_rbt @ A @ B,S: A,T: B,V: rBT_Im1923302023_color,Vf: A,Vg: B] :
( ( rBT_Im1648453169alance @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va @ Vb @ Vc @ Vd ) @ S @ T @ ( rBT_Im480247531Branch @ A @ B @ V @ ( rBT_Im418718756_Empty @ A @ B ) @ Vf @ Vg @ ( rBT_Im418718756_Empty @ A @ B ) ) )
= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va @ Vb @ Vc @ Vd ) @ S @ T @ ( rBT_Im480247531Branch @ A @ B @ V @ ( rBT_Im418718756_Empty @ A @ B ) @ Vf @ Vg @ ( rBT_Im418718756_Empty @ A @ B ) ) ) ) ).
% balance.simps(25)
thf(fact_254_balance_Osimps_I26_J,axiom,
! [B: $tType,A: $tType,Va: rBT_Im246033960le_rbt @ A @ B,Vb: A,Vc: B,Vd: rBT_Im246033960le_rbt @ A @ B,S: A,T: B,V: rBT_Im1923302023_color,Vi: rBT_Im246033960le_rbt @ A @ B,Vj: A,Vk: B,Vl: rBT_Im246033960le_rbt @ A @ B,Vf: A,Vg: B] :
( ( rBT_Im1648453169alance @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va @ Vb @ Vc @ Vd ) @ S @ T @ ( rBT_Im480247531Branch @ A @ B @ V @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Vi @ Vj @ Vk @ Vl ) @ Vf @ Vg @ ( rBT_Im418718756_Empty @ A @ B ) ) )
= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va @ Vb @ Vc @ Vd ) @ S @ T @ ( rBT_Im480247531Branch @ A @ B @ V @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Vi @ Vj @ Vk @ Vl ) @ Vf @ Vg @ ( rBT_Im418718756_Empty @ A @ B ) ) ) ) ).
% balance.simps(26)
thf(fact_255_balance_Osimps_I28_J,axiom,
! [B: $tType,A: $tType,Va: rBT_Im246033960le_rbt @ A @ B,Vb: A,Vc: B,Vd: rBT_Im246033960le_rbt @ A @ B,S: A,T: B,V: rBT_Im1923302023_color,Vf: A,Vg: B,Vj: rBT_Im246033960le_rbt @ A @ B,Vk: A,Vl: B,Vm: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1648453169alance @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va @ Vb @ Vc @ Vd ) @ S @ T @ ( rBT_Im480247531Branch @ A @ B @ V @ ( rBT_Im418718756_Empty @ A @ B ) @ Vf @ Vg @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Vj @ Vk @ Vl @ Vm ) ) )
= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va @ Vb @ Vc @ Vd ) @ S @ T @ ( rBT_Im480247531Branch @ A @ B @ V @ ( rBT_Im418718756_Empty @ A @ B ) @ Vf @ Vg @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Vj @ Vk @ Vl @ Vm ) ) ) ) ).
% balance.simps(28)
%----Type constructors (8)
thf(tcon_fun___Orderings_Oorder,axiom,
! [A6: $tType,A7: $tType] :
( ( order @ A7 @ ( type2 @ A7 ) )
=> ( order @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A6: $tType,A7: $tType] :
( ( ord @ A7 @ ( type2 @ A7 ) )
=> ( ord @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_Nat_Onat___Orderings_Oorder_1,axiom,
order @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oord_2,axiom,
ord @ nat @ ( type2 @ nat ) ).
thf(tcon_Set_Oset___Orderings_Oorder_3,axiom,
! [A6: $tType] : ( order @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_4,axiom,
! [A6: $tType] : ( ord @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_5,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_6,axiom,
ord @ $o @ ( type2 @ $o ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
( ( member @ ( product_prod @ a @ b ) @ ( product_Pair @ a @ b @ k @ v ) @ ( set2 @ ( product_prod @ a @ b ) @ ( rBT_Im954575269ntries @ a @ b @ ( rBT_Im1152561598m_left @ a @ b @ less @ xx @ ( rBT_Im480247531Branch @ a @ b @ rBT_Impl_Mirabelle_R @ vaa @ vbb @ vdd @ vc ) @ yy @ ss @ bb ) ) ) )
!= ( ~ ( ( xx != k )
& ( member @ ( product_prod @ a @ b ) @ ( product_Pair @ a @ b @ k @ v ) @ ( set2 @ ( product_prod @ a @ b ) @ ( rBT_Im954575269ntries @ a @ b @ ( rBT_Im480247531Branch @ a @ b @ c @ ( rBT_Im480247531Branch @ a @ b @ rBT_Impl_Mirabelle_R @ vaa @ vbb @ vdd @ vc ) @ yy @ ss @ bb ) ) ) ) ) ) ) ).
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