TPTP Problem File: DAT233^1.p
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%------------------------------------------------------------------------------
% File : DAT233^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Red-black trees 585
% 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__585.p [Bla16]
% Status : Theorem
% Rating : 1.00 v7.2.0, 0.75 v7.1.0
% Syntax : Number of formulae : 337 ( 144 unt; 61 typ; 0 def)
% Number of atoms : 642 ( 239 equ; 0 cnn)
% Maximal formula atoms : 8 ( 2 avg)
% Number of connectives : 6312 ( 54 ~; 5 |; 45 &;5925 @)
% ( 0 <=>; 283 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 9 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 270 ( 270 >; 0 *; 0 +; 0 <<)
% Number of symbols : 60 ( 57 usr; 8 con; 0-7 aty)
% Number of variables : 1545 ( 17 ^;1450 !; 8 ?;1545 :)
% ( 70 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:41:54.023
%------------------------------------------------------------------------------
%----Could-be-implicit typings (7)
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_Option_Ooption,type,
option: $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_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Fields_Olinordered__field,type,
linordered_field:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Olinordered__semidom,type,
linordered_semidom:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Olinordered__ab__group__add,type,
linord219039673up_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni770627133id_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
ordere236663937imp_le:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
ordere223160158up_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add,type,
strict2144017051up_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type,
ordere516151231imp_le:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
condit1656338222tinuum:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Oone__class_Oone,type,
one_one:
!>[A: $tType] : A ).
thf(sy_c_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
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_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_Ocase__color,type,
rBT_Im1975547051_color:
!>[A: $tType] : ( A > A > rBT_Im1923302023_color > A ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Ocolor_Orec__color,type,
rBT_Im1059957627_color:
!>[A: $tType] : ( A > A > rBT_Im1923302023_color > A ) ).
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_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__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__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__sorted,type,
rBT_Im759614907sorted:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( rBT_Im246033960le_rbt @ A @ B ) > $o ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Oord__class_Ois__rbt,type,
rBT_Im222526792is_rbt:
!>[A: $tType,B: $tType] : ( ( rBT_Im246033960le_rbt @ A @ B ) > $o ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Oord__class_Orbt__greater,type,
rBT_Im1332541288reater:
!>[A: $tType,B: $tType] : ( A > ( rBT_Im246033960le_rbt @ A @ B ) > $o ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Oord__class_Orbt__ins,type,
rBT_Im616047964bt_ins:
!>[A: $tType,B: $tType] : ( ( A > B > B > B ) > A > B > ( rBT_Im246033960le_rbt @ A @ B ) > ( rBT_Im246033960le_rbt @ A @ B ) ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Oord__class_Orbt__insert,type,
rBT_Im1276434955insert:
!>[A: $tType,B: $tType] : ( A > B > ( rBT_Im246033960le_rbt @ A @ B ) > ( rBT_Im246033960le_rbt @ A @ B ) ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Oord__class_Orbt__insert__with,type,
rBT_Im1963520753t_with:
!>[B: $tType,A: $tType] : ( ( B > B > B ) > A > B > ( rBT_Im246033960le_rbt @ A @ B ) > ( rBT_Im246033960le_rbt @ A @ B ) ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Oord__class_Orbt__insert__with__key,type,
rBT_Im1451321038th_key:
!>[A: $tType,B: $tType] : ( ( A > B > B > B ) > A > B > ( rBT_Im246033960le_rbt @ A @ B ) > ( rBT_Im246033960le_rbt @ A @ B ) ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Oord__class_Orbt__less,type,
rBT_Im903026891t_less:
!>[A: $tType,B: $tType] : ( A > ( rBT_Im246033960le_rbt @ A @ B ) > $o ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Oord__class_Orbt__lookup,type,
rBT_Im459437964lookup:
!>[A: $tType,B: $tType] : ( ( rBT_Im246033960le_rbt @ A @ B ) > A > ( option @ B ) ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Oord__class_Orbt__sorted,type,
rBT_Im1183586831sorted:
!>[A: $tType,B: $tType] : ( ( rBT_Im246033960le_rbt @ A @ B ) > $o ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_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_v_a,type,
a2: rBT_Im246033960le_rbt @ a @ b ).
thf(sy_v_b,type,
b2: rBT_Im246033960le_rbt @ a @ b ).
thf(sy_v_k,type,
k: a ).
thf(sy_v_x,type,
x: b ).
%----Relevant facts (255)
thf(fact_0_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_1_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_2_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_3_color_Osimps_I6_J,axiom,
! [A: $tType,F1: A,F2: A] :
( ( rBT_Im1059957627_color @ A @ F1 @ F2 @ rBT_Impl_Mirabelle_B )
= F2 ) ).
% color.simps(6)
thf(fact_4_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_5_color_Osimps_I4_J,axiom,
! [A: $tType,F1: A,F2: A] :
( ( rBT_Im1975547051_color @ A @ F1 @ F2 @ rBT_Impl_Mirabelle_B )
= F2 ) ).
% color.simps(4)
thf(fact_6_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_7_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_8_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_9_ins__inv1__inv2_I3_J,axiom,
! [B: $tType,A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [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_Im616047964bt_ins @ A @ B @ F @ K @ X @ T ) ) ) ) ) ) ).
% ins_inv1_inv2(3)
thf(fact_10_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_11_ord_Ois__rbt__def,axiom,
! [B: $tType,A: $tType] :
( ( rBT_Im862805236is_rbt @ A @ B )
= ( ^ [Less2: A > A > $o,T2: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1040612427e_inv1 @ A @ B @ T2 )
& ( rBT_Im1040612428e_inv2 @ A @ B @ T2 )
& ( ( rBT_Im1805589839lor_of @ A @ B @ T2 )
= rBT_Impl_Mirabelle_B )
& ( rBT_Im759614907sorted @ A @ B @ Less2 @ T2 ) ) ) ) ).
% ord.is_rbt_def
thf(fact_12_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_13_inv2_Osimps_I1_J,axiom,
! [B: $tType,A: $tType] : ( rBT_Im1040612428e_inv2 @ A @ B @ ( rBT_Im418718756_Empty @ A @ B ) ) ).
% inv2.simps(1)
thf(fact_14_paint_Osimps_I2_J,axiom,
! [B: $tType,A: $tType,C: rBT_Im1923302023_color,Uu: 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 @ Uu @ L @ K @ V @ R ) )
= ( rBT_Im480247531Branch @ A @ B @ C @ L @ K @ V @ R ) ) ).
% paint.simps(2)
thf(fact_15_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_16_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_17_balance_Osimps_I44_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,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_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_Im480247531Branch @ A @ B @ V @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Vj @ Vk @ Vl @ Vm ) @ Vf @ Vg @ ( rBT_Im418718756_Empty @ A @ B ) ) @ S @ T @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va @ Vb @ Vc @ Vd ) ) ) ).
% balance.simps(44)
thf(fact_18_balance_Osimps_I42_J,axiom,
! [B: $tType,A: $tType,V: rBT_Im1923302023_color,Vf: A,Vg: B,Vi: rBT_Im246033960le_rbt @ A @ B,Vj: A,Vk: B,Vl: 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_Im418718756_Empty @ A @ B ) @ Vf @ Vg @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Vi @ Vj @ Vk @ Vl ) ) @ 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_Im418718756_Empty @ A @ B ) @ Vf @ Vg @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Vi @ Vj @ Vk @ Vl ) ) @ S @ T @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va @ Vb @ Vc @ Vd ) ) ) ).
% balance.simps(42)
thf(fact_19_balance_Osimps_I41_J,axiom,
! [B: $tType,A: $tType,V: rBT_Im1923302023_color,Vf: A,Vg: 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_Im418718756_Empty @ A @ B ) @ Vf @ Vg @ ( 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_Im480247531Branch @ A @ B @ V @ ( rBT_Im418718756_Empty @ A @ B ) @ Vf @ Vg @ ( rBT_Im418718756_Empty @ A @ B ) ) @ S @ T @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va @ Vb @ Vc @ Vd ) ) ) ).
% balance.simps(41)
thf(fact_20_balance_Osimps_I40_J,axiom,
! [B: $tType,A: $tType,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_Im418718756_Empty @ A @ B ) @ 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_Im418718756_Empty @ A @ B ) @ Vf @ Vg @ Vh ) @ S @ T @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va @ Vb @ Vc @ Vd ) ) ) ).
% balance.simps(40)
thf(fact_21_balance_Osimps_I37_J,axiom,
! [B: $tType,A: $tType,V: rBT_Im1923302023_color,Vf: rBT_Im246033960le_rbt @ A @ B,Vg: A,Vh: B,Vi: rBT_Im246033960le_rbt @ A @ B,Vb: A,Vc: B,Ve: rBT_Im246033960le_rbt @ A @ B,Vj: A,Vk: B,Vl: rBT_Im246033960le_rbt @ A @ B,S: A,T: B] :
( ( rBT_Im1648453169alance @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ V @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Vf @ Vg @ Vh @ Vi ) @ Vb @ Vc @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Ve @ Vj @ Vk @ Vl ) ) @ S @ T @ ( rBT_Im418718756_Empty @ A @ B ) )
= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ ( rBT_Im480247531Branch @ A @ B @ V @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Vf @ Vg @ Vh @ Vi ) @ Vb @ Vc @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Ve @ Vj @ Vk @ Vl ) ) @ S @ T @ ( rBT_Im418718756_Empty @ A @ B ) ) ) ).
% balance.simps(37)
thf(fact_22_balance_Osimps_I36_J,axiom,
! [B: $tType,A: $tType,V: rBT_Im1923302023_color,Vf: rBT_Im246033960le_rbt @ A @ B,Vg: A,Vh: B,Vi: rBT_Im246033960le_rbt @ A @ B,Vb: A,Vc: B,S: A,T: B] :
( ( rBT_Im1648453169alance @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ V @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Vf @ Vg @ Vh @ Vi ) @ Vb @ Vc @ ( rBT_Im418718756_Empty @ A @ B ) ) @ S @ T @ ( rBT_Im418718756_Empty @ A @ B ) )
= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ ( rBT_Im480247531Branch @ A @ B @ V @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Vf @ Vg @ Vh @ Vi ) @ Vb @ Vc @ ( rBT_Im418718756_Empty @ A @ B ) ) @ S @ T @ ( rBT_Im418718756_Empty @ A @ B ) ) ) ).
% balance.simps(36)
thf(fact_23_balance_Osimps_I35_J,axiom,
! [B: $tType,A: $tType,Vf: rBT_Im246033960le_rbt @ A @ B,Vg: A,Vh: B,Vi: 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 @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Vf @ Vg @ Vh @ Vi ) @ 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 @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Vf @ Vg @ Vh @ Vi ) @ Vb @ Vc @ Vd ) @ S @ T @ ( rBT_Im418718756_Empty @ A @ B ) ) ) ).
% balance.simps(35)
thf(fact_24_balance_Osimps_I34_J,axiom,
! [B: $tType,A: $tType,V: rBT_Im1923302023_color,Vb: A,Vc: B,Ve: rBT_Im246033960le_rbt @ A @ B,Vf: A,Vg: B,Vh: rBT_Im246033960le_rbt @ A @ B,S: A,T: B] :
( ( rBT_Im1648453169alance @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ V @ ( rBT_Im418718756_Empty @ A @ B ) @ Vb @ Vc @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Ve @ Vf @ Vg @ Vh ) ) @ S @ T @ ( rBT_Im418718756_Empty @ A @ B ) )
= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ ( rBT_Im480247531Branch @ A @ B @ V @ ( rBT_Im418718756_Empty @ A @ B ) @ Vb @ Vc @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Ve @ Vf @ Vg @ Vh ) ) @ S @ T @ ( rBT_Im418718756_Empty @ A @ B ) ) ) ).
% balance.simps(34)
thf(fact_25_balance_Osimps_I33_J,axiom,
! [B: $tType,A: $tType,V: rBT_Im1923302023_color,Vb: A,Vc: B,S: A,T: B] :
( ( rBT_Im1648453169alance @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ V @ ( rBT_Im418718756_Empty @ A @ B ) @ Vb @ Vc @ ( rBT_Im418718756_Empty @ A @ B ) ) @ S @ T @ ( rBT_Im418718756_Empty @ A @ B ) )
= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ ( rBT_Im480247531Branch @ A @ B @ V @ ( rBT_Im418718756_Empty @ A @ B ) @ Vb @ Vc @ ( rBT_Im418718756_Empty @ A @ B ) ) @ S @ T @ ( rBT_Im418718756_Empty @ A @ B ) ) ) ).
% balance.simps(33)
thf(fact_26_balance_Osimps_I32_J,axiom,
! [B: $tType,A: $tType,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 @ ( rBT_Im418718756_Empty @ A @ B ) @ 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 @ ( rBT_Im418718756_Empty @ A @ B ) @ Vb @ Vc @ Vd ) @ S @ T @ ( rBT_Im418718756_Empty @ A @ B ) ) ) ).
% balance.simps(32)
thf(fact_27_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)
thf(fact_28_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_29_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_30_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_31_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_32_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_33_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_34_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_35_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_36_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_37_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_38_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_39_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_40_ord_Orbt__ins_Osimps_I2_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_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 ) ) ) ) ) ) ).
% ord.rbt_ins.simps(2)
thf(fact_41_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_42_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_43_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,X2: rBT_Im246033960le_rbt @ A @ B,X3: A,X4: B,X5: rBT_Im246033960le_rbt @ A @ B] :
( ( P @ X2 )
=> ( ( P @ X5 )
=> ( P @ ( rBT_Im480247531Branch @ A @ B @ X1 @ X2 @ X3 @ X4 @ X5 ) ) ) )
=> ( P @ Rbt ) ) ) ).
% rbt.induct
thf(fact_44_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_45_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X6: A] :
( ( F @ X6 )
= ( G @ X6 ) )
=> ( F = G ) ) ).
% ext
thf(fact_46_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_47_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_48_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_49_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_50_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_51_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_52_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_53_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_54_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_55_color_Ocase__distrib,axiom,
! [A: $tType,B: $tType,H: A > B,F1: A,F2: A,Color: rBT_Im1923302023_color] :
( ( H @ ( rBT_Im1975547051_color @ A @ F1 @ F2 @ Color ) )
= ( rBT_Im1975547051_color @ B @ ( H @ F1 ) @ ( H @ F2 ) @ Color ) ) ).
% color.case_distrib
thf(fact_56_ins__inv1__inv2_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [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_Im616047964bt_ins @ A @ B @ F @ K @ X @ T ) ) ) ) ) ).
% ins_inv1_inv2(1)
thf(fact_57_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_58_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_59_inv1_Osimps_I1_J,axiom,
! [B: $tType,A: $tType] : ( rBT_Im1040612427e_inv1 @ A @ B @ ( rBT_Im418718756_Empty @ A @ B ) ) ).
% inv1.simps(1)
thf(fact_60_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_61_color__of_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,C: rBT_Im1923302023_color,Uu: rBT_Im246033960le_rbt @ A @ B,Uv: A,Uw: B,Ux: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1805589839lor_of @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ C @ Uu @ Uv @ Uw @ Ux ) )
= C ) ).
% color_of.simps(2)
thf(fact_62_inv1l_Osimps_I1_J,axiom,
! [B: $tType,A: $tType] : ( rBT_Im1071879905_inv1l @ A @ B @ ( rBT_Im418718756_Empty @ A @ B ) ) ).
% inv1l.simps(1)
thf(fact_63_ins__inv1__inv2_I4_J,axiom,
! [B: $tType,A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [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_Im616047964bt_ins @ A @ B @ F @ K @ X @ T ) ) ) ) ) ).
% ins_inv1_inv2(4)
thf(fact_64_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_65_ord_Orbt__insert__with__key__def,axiom,
! [B: $tType,A: $tType] :
( ( rBT_Im627674146th_key @ A @ B )
= ( ^ [Less2: A > A > $o,F3: A > B > B > B,K2: A,V2: B,T2: rBT_Im246033960le_rbt @ A @ B] : ( rBT_Im945281971_paint @ A @ B @ rBT_Impl_Mirabelle_B @ ( rBT_Im1756508336bt_ins @ A @ B @ Less2 @ F3 @ K2 @ V2 @ T2 ) ) ) ) ).
% ord.rbt_insert_with_key_def
thf(fact_66_rbt__insert__with__key__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ( ( rBT_Im1451321038th_key @ A @ B )
= ( ^ [F3: A > B > B > B,K2: A,V2: B,T2: rBT_Im246033960le_rbt @ A @ B] : ( rBT_Im945281971_paint @ A @ B @ rBT_Impl_Mirabelle_B @ ( rBT_Im616047964bt_ins @ A @ B @ F3 @ K2 @ V2 @ T2 ) ) ) ) ) ).
% rbt_insert_with_key_def
thf(fact_67_balance__left_Osimps_I4_J,axiom,
! [B: $tType,A: $tType,K: A,X: B,A2: rBT_Im246033960le_rbt @ A @ B,S: A,Y: B,B2: rBT_Im246033960le_rbt @ A @ B,T: A,Z: B,C: 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_B @ A2 @ S @ Y @ B2 ) @ T @ Z @ C ) )
= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ ( rBT_Im418718756_Empty @ A @ B ) @ K @ X @ A2 ) @ S @ Y @ ( rBT_Im1648453169alance @ A @ B @ B2 @ T @ Z @ ( rBT_Im945281971_paint @ A @ B @ rBT_Impl_Mirabelle_R @ C ) ) ) ) ).
% balance_left.simps(4)
thf(fact_68_balance__left_Osimps_I5_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,A2: rBT_Im246033960le_rbt @ A @ B,S: A,Y: B,B2: rBT_Im246033960le_rbt @ A @ B,T: A,Z: B,C: 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_B @ A2 @ S @ Y @ B2 ) @ T @ Z @ C ) )
= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va @ Vb @ Vc @ Vd ) @ K @ X @ A2 ) @ S @ Y @ ( rBT_Im1648453169alance @ A @ B @ B2 @ T @ Z @ ( rBT_Im945281971_paint @ A @ B @ rBT_Impl_Mirabelle_R @ C ) ) ) ) ).
% balance_left.simps(5)
thf(fact_69_balance__left_Osimps_I2_J,axiom,
! [B: $tType,A: $tType,K: A,X: B,A2: rBT_Im246033960le_rbt @ A @ B,S: A,Y: B,B2: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1278323832e_left @ A @ B @ ( rBT_Im418718756_Empty @ A @ B ) @ K @ X @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ A2 @ S @ Y @ B2 ) )
= ( rBT_Im1648453169alance @ A @ B @ ( rBT_Im418718756_Empty @ A @ B ) @ K @ X @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ A2 @ S @ Y @ B2 ) ) ) ).
% balance_left.simps(2)
thf(fact_70_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_71_ins__inv1__inv2_I2_J,axiom,
! [B: $tType,A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [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_Im616047964bt_ins @ A @ B @ F @ K @ X @ T ) )
= ( rBT_Im315265182height @ A @ B @ T ) ) ) ) ) ).
% ins_inv1_inv2(2)
thf(fact_72_rbt__ins_Osimps_I2_J,axiom,
! [B: $tType,A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [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] :
( ( ( ord_less @ A @ K @ X )
=> ( ( rBT_Im616047964bt_ins @ A @ B @ F @ K @ V @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ L @ X @ Y @ R ) )
= ( rBT_Im1648453169alance @ A @ B @ ( rBT_Im616047964bt_ins @ A @ B @ F @ K @ V @ L ) @ X @ Y @ R ) ) )
& ( ~ ( ord_less @ A @ K @ X )
=> ( ( ( ord_less @ A @ X @ K )
=> ( ( rBT_Im616047964bt_ins @ A @ B @ F @ K @ V @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ L @ X @ Y @ R ) )
= ( rBT_Im1648453169alance @ A @ B @ L @ X @ Y @ ( rBT_Im616047964bt_ins @ A @ B @ F @ K @ V @ R ) ) ) )
& ( ~ ( ord_less @ A @ X @ K )
=> ( ( rBT_Im616047964bt_ins @ A @ B @ 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 ) ) ) ) ) ) ) ).
% rbt_ins.simps(2)
thf(fact_73_balance__left_Osimps_I3_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,A2: rBT_Im246033960le_rbt @ A @ B,S: A,Y: B,B2: 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_B @ A2 @ S @ Y @ B2 ) )
= ( rBT_Im1648453169alance @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va @ Vb @ Vc @ Vd ) @ K @ X @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ A2 @ S @ Y @ B2 ) ) ) ).
% balance_left.simps(3)
thf(fact_74_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_75_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_76_rbt__ins_Osimps_I3_J,axiom,
! [B: $tType,A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [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] :
( ( ( ord_less @ A @ K @ X )
=> ( ( rBT_Im616047964bt_ins @ A @ B @ F @ K @ V @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ L @ X @ Y @ R ) )
= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ ( rBT_Im616047964bt_ins @ A @ B @ F @ K @ V @ L ) @ X @ Y @ R ) ) )
& ( ~ ( ord_less @ A @ K @ X )
=> ( ( ( ord_less @ A @ X @ K )
=> ( ( rBT_Im616047964bt_ins @ A @ B @ 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_Im616047964bt_ins @ A @ B @ F @ K @ V @ R ) ) ) )
& ( ~ ( ord_less @ A @ X @ K )
=> ( ( rBT_Im616047964bt_ins @ A @ B @ 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 ) ) ) ) ) ) ) ).
% rbt_ins.simps(3)
thf(fact_77_color_Odistinct_I1_J,axiom,
rBT_Impl_Mirabelle_R != rBT_Impl_Mirabelle_B ).
% color.distinct(1)
thf(fact_78_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_79_color_Oexhaust,axiom,
! [Y: rBT_Im1923302023_color] :
( ( Y != rBT_Impl_Mirabelle_R )
=> ( Y = rBT_Impl_Mirabelle_B ) ) ).
% color.exhaust
thf(fact_80_rbt__ins_Oinduct,axiom,
! [B: $tType,A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [P: ( A > B > B > B ) > A > B > ( rBT_Im246033960le_rbt @ A @ B ) > $o,A0: A > B > B > B,A1: A,A22: B,A3: rBT_Im246033960le_rbt @ A @ B] :
( ! [F4: A > B > B > B,K3: A,V3: B] : ( P @ F4 @ K3 @ V3 @ ( rBT_Im418718756_Empty @ A @ B ) )
=> ( ! [F4: A > B > B > B,K3: A,V3: B,L2: rBT_Im246033960le_rbt @ A @ B,X6: A,Y2: B,R2: rBT_Im246033960le_rbt @ A @ B] :
( ( ( ord_less @ A @ K3 @ X6 )
=> ( P @ F4 @ K3 @ V3 @ L2 ) )
=> ( ( ~ ( ord_less @ A @ K3 @ X6 )
=> ( ( ord_less @ A @ X6 @ K3 )
=> ( P @ F4 @ K3 @ V3 @ R2 ) ) )
=> ( P @ F4 @ K3 @ V3 @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ L2 @ X6 @ Y2 @ R2 ) ) ) )
=> ( ! [F4: A > B > B > B,K3: A,V3: B,L2: rBT_Im246033960le_rbt @ A @ B,X6: A,Y2: B,R2: rBT_Im246033960le_rbt @ A @ B] :
( ( ( ord_less @ A @ K3 @ X6 )
=> ( P @ F4 @ K3 @ V3 @ L2 ) )
=> ( ( ~ ( ord_less @ A @ K3 @ X6 )
=> ( ( ord_less @ A @ X6 @ K3 )
=> ( P @ F4 @ K3 @ V3 @ R2 ) ) )
=> ( P @ F4 @ K3 @ V3 @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ L2 @ X6 @ Y2 @ R2 ) ) ) )
=> ( P @ A0 @ A1 @ A22 @ A3 ) ) ) ) ) ).
% rbt_ins.induct
thf(fact_81_color_Osimps_I3_J,axiom,
! [A: $tType,F1: A,F2: A] :
( ( rBT_Im1975547051_color @ A @ F1 @ F2 @ rBT_Impl_Mirabelle_R )
= F1 ) ).
% color.simps(3)
thf(fact_82_color_Osimps_I5_J,axiom,
! [A: $tType,F1: A,F2: A] :
( ( rBT_Im1059957627_color @ A @ F1 @ F2 @ rBT_Impl_Mirabelle_R )
= F1 ) ).
% color.simps(5)
thf(fact_83_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_84_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_85_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_86_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,A3: rBT_Im246033960le_rbt @ A @ B] :
( ! [F4: A > B > B > B,K3: A,V3: B] : ( P @ F4 @ K3 @ V3 @ ( rBT_Im418718756_Empty @ A @ B ) )
=> ( ! [F4: A > B > B > B,K3: A,V3: B,L2: rBT_Im246033960le_rbt @ A @ B,X6: A,Y2: B,R2: rBT_Im246033960le_rbt @ A @ B] :
( ( ( Less @ K3 @ X6 )
=> ( P @ F4 @ K3 @ V3 @ L2 ) )
=> ( ( ~ ( Less @ K3 @ X6 )
=> ( ( Less @ X6 @ K3 )
=> ( P @ F4 @ K3 @ V3 @ R2 ) ) )
=> ( P @ F4 @ K3 @ V3 @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ L2 @ X6 @ Y2 @ R2 ) ) ) )
=> ( ! [F4: A > B > B > B,K3: A,V3: B,L2: rBT_Im246033960le_rbt @ A @ B,X6: A,Y2: B,R2: rBT_Im246033960le_rbt @ A @ B] :
( ( ( Less @ K3 @ X6 )
=> ( P @ F4 @ K3 @ V3 @ L2 ) )
=> ( ( ~ ( Less @ K3 @ X6 )
=> ( ( Less @ X6 @ K3 )
=> ( P @ F4 @ K3 @ V3 @ R2 ) ) )
=> ( P @ F4 @ K3 @ V3 @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ L2 @ X6 @ Y2 @ R2 ) ) ) )
=> ( P @ A0 @ A1 @ A22 @ A3 ) ) ) ) ).
% ord.rbt_ins.induct
thf(fact_87_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,K3: A,V3: B,R2: rBT_Im246033960le_rbt @ A @ B] :
( T
!= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ L2 @ K3 @ V3 @ R2 ) )
=> ~ ! [L2: rBT_Im246033960le_rbt @ A @ B,K3: A,V3: B,R2: rBT_Im246033960le_rbt @ A @ B] :
( T
!= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ L2 @ K3 @ V3 @ R2 ) ) ) ) ).
% rbt_cases
thf(fact_88_balance_Osimps_I1_J,axiom,
! [B: $tType,A: $tType,A2: rBT_Im246033960le_rbt @ A @ B,W: A,X: B,B2: rBT_Im246033960le_rbt @ A @ B,S: A,T: B,C: rBT_Im246033960le_rbt @ A @ B,Y: A,Z: B,D: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1648453169alance @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ A2 @ W @ X @ B2 ) @ S @ T @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ C @ Y @ Z @ D ) )
= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ A2 @ W @ X @ B2 ) @ S @ T @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ C @ Y @ Z @ D ) ) ) ).
% balance.simps(1)
thf(fact_89_balance_Osimps_I3_J,axiom,
! [B: $tType,A: $tType,A2: rBT_Im246033960le_rbt @ A @ B,W: A,X: B,B2: rBT_Im246033960le_rbt @ A @ B,S: A,T: B,C: rBT_Im246033960le_rbt @ A @ B,Y: A,Z: 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_R @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ A2 @ W @ X @ B2 ) @ S @ T @ C ) @ Y @ Z @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va @ Vb @ Vc @ Vd ) )
= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ A2 @ W @ X @ B2 ) @ S @ T @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ C @ Y @ Z @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va @ Vb @ Vc @ Vd ) ) ) ) ).
% balance.simps(3)
thf(fact_90_balance_Osimps_I7_J,axiom,
! [B: $tType,A: $tType,Ve: rBT_Im246033960le_rbt @ A @ B,Vf: A,Vg: B,Vh: rBT_Im246033960le_rbt @ A @ B,W: A,X: B,B2: rBT_Im246033960le_rbt @ A @ B,S: A,T: B,C: rBT_Im246033960le_rbt @ A @ B,Y: A,Z: 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_R @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Ve @ Vf @ Vg @ Vh ) @ W @ X @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ B2 @ S @ T @ C ) ) @ Y @ Z @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va @ Vb @ Vc @ Vd ) )
= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Ve @ Vf @ Vg @ Vh ) @ W @ X @ B2 ) @ S @ T @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ C @ Y @ Z @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va @ Vb @ Vc @ Vd ) ) ) ) ).
% balance.simps(7)
thf(fact_91_balance_Osimps_I9_J,axiom,
! [B: $tType,A: $tType,Va: rBT_Im246033960le_rbt @ A @ B,Vb: A,Vc: B,Vd: rBT_Im246033960le_rbt @ A @ B,W: A,X: B,B2: rBT_Im246033960le_rbt @ A @ B,S: A,T: B,C: rBT_Im246033960le_rbt @ A @ B,Y: A,Z: B,D: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1648453169alance @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va @ Vb @ Vc @ Vd ) @ W @ X @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ B2 @ S @ T @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ C @ Y @ Z @ D ) ) )
= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va @ Vb @ Vc @ Vd ) @ W @ X @ B2 ) @ S @ T @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ C @ Y @ Z @ D ) ) ) ).
% balance.simps(9)
thf(fact_92_balance_Osimps_I13_J,axiom,
! [B: $tType,A: $tType,Va: rBT_Im246033960le_rbt @ A @ B,Vb: A,Vc: B,Vd: rBT_Im246033960le_rbt @ A @ B,W: A,X: B,B2: rBT_Im246033960le_rbt @ A @ B,S: A,T: B,C: rBT_Im246033960le_rbt @ A @ B,Y: A,Z: 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 ) @ W @ X @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ B2 @ S @ T @ C ) @ Y @ Z @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Ve @ Vf @ Vg @ Vh ) ) )
= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va @ Vb @ Vc @ Vd ) @ W @ X @ B2 ) @ S @ T @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ C @ Y @ Z @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Ve @ Vf @ Vg @ Vh ) ) ) ) ).
% balance.simps(13)
thf(fact_93_rbt__ins_Osimps_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [F: A > B > B > B,K: A,V: B] :
( ( rBT_Im616047964bt_ins @ A @ B @ 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 ) ) ) ) ).
% rbt_ins.simps(1)
thf(fact_94_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_95_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_96_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_97_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_98_balance_Osimps_I2_J,axiom,
! [B: $tType,A: $tType,A2: rBT_Im246033960le_rbt @ A @ B,W: A,X: B,B2: rBT_Im246033960le_rbt @ A @ B,S: A,T: B,C: rBT_Im246033960le_rbt @ A @ B,Y: A,Z: B] :
( ( rBT_Im1648453169alance @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ A2 @ W @ X @ B2 ) @ S @ T @ C ) @ Y @ Z @ ( rBT_Im418718756_Empty @ A @ B ) )
= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ A2 @ W @ X @ B2 ) @ S @ T @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ C @ Y @ Z @ ( rBT_Im418718756_Empty @ A @ B ) ) ) ) ).
% balance.simps(2)
thf(fact_99_balance_Osimps_I4_J,axiom,
! [B: $tType,A: $tType,W: A,X: B,B2: rBT_Im246033960le_rbt @ A @ B,S: A,T: B,C: rBT_Im246033960le_rbt @ A @ B,Y: A,Z: B] :
( ( rBT_Im1648453169alance @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ ( rBT_Im418718756_Empty @ A @ B ) @ W @ X @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ B2 @ S @ T @ C ) ) @ Y @ Z @ ( rBT_Im418718756_Empty @ A @ B ) )
= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ ( rBT_Im418718756_Empty @ A @ B ) @ W @ X @ B2 ) @ S @ T @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ C @ Y @ Z @ ( rBT_Im418718756_Empty @ A @ B ) ) ) ) ).
% balance.simps(4)
thf(fact_100_balance_Osimps_I5_J,axiom,
! [B: $tType,A: $tType,Va: rBT_Im246033960le_rbt @ A @ B,Vb: A,Vc: B,Vd: rBT_Im246033960le_rbt @ A @ B,W: A,X: B,B2: rBT_Im246033960le_rbt @ A @ B,S: A,T: B,C: rBT_Im246033960le_rbt @ A @ B,Y: A,Z: B] :
( ( rBT_Im1648453169alance @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va @ Vb @ Vc @ Vd ) @ W @ X @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ B2 @ S @ T @ C ) ) @ Y @ Z @ ( rBT_Im418718756_Empty @ A @ B ) )
= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va @ Vb @ Vc @ Vd ) @ W @ X @ B2 ) @ S @ T @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ C @ Y @ Z @ ( rBT_Im418718756_Empty @ A @ B ) ) ) ) ).
% balance.simps(5)
thf(fact_101_balance_Osimps_I6_J,axiom,
! [B: $tType,A: $tType,W: A,X: B,B2: rBT_Im246033960le_rbt @ A @ B,S: A,T: B,C: rBT_Im246033960le_rbt @ A @ B,Y: A,Z: 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_R @ ( rBT_Im418718756_Empty @ A @ B ) @ W @ X @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ B2 @ S @ T @ C ) ) @ Y @ Z @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va @ Vb @ Vc @ Vd ) )
= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ ( rBT_Im418718756_Empty @ A @ B ) @ W @ X @ B2 ) @ S @ T @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ C @ Y @ Z @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va @ Vb @ Vc @ Vd ) ) ) ) ).
% balance.simps(6)
thf(fact_102_balance_Osimps_I8_J,axiom,
! [B: $tType,A: $tType,W: A,X: B,B2: rBT_Im246033960le_rbt @ A @ B,S: A,T: B,C: rBT_Im246033960le_rbt @ A @ B,Y: A,Z: B,D: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1648453169alance @ A @ B @ ( rBT_Im418718756_Empty @ A @ B ) @ W @ X @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ B2 @ S @ T @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ C @ Y @ Z @ D ) ) )
= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ ( rBT_Im418718756_Empty @ A @ B ) @ W @ X @ B2 ) @ S @ T @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ C @ Y @ Z @ D ) ) ) ).
% balance.simps(8)
thf(fact_103_balance_Osimps_I10_J,axiom,
! [B: $tType,A: $tType,W: A,X: B,B2: rBT_Im246033960le_rbt @ A @ B,S: A,T: B,C: rBT_Im246033960le_rbt @ A @ B,Y: A,Z: B] :
( ( rBT_Im1648453169alance @ A @ B @ ( rBT_Im418718756_Empty @ A @ B ) @ W @ X @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ B2 @ S @ T @ C ) @ Y @ Z @ ( rBT_Im418718756_Empty @ A @ B ) ) )
= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ ( rBT_Im418718756_Empty @ A @ B ) @ W @ X @ B2 ) @ S @ T @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ C @ Y @ Z @ ( rBT_Im418718756_Empty @ A @ B ) ) ) ) ).
% balance.simps(10)
thf(fact_104_balance_Osimps_I11_J,axiom,
! [B: $tType,A: $tType,W: A,X: B,B2: rBT_Im246033960le_rbt @ A @ B,S: A,T: B,C: rBT_Im246033960le_rbt @ A @ B,Y: A,Z: 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 ) @ W @ X @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ B2 @ S @ T @ C ) @ Y @ Z @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va @ Vb @ Vc @ Vd ) ) )
= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ ( rBT_Im418718756_Empty @ A @ B ) @ W @ X @ B2 ) @ S @ T @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ C @ Y @ Z @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va @ Vb @ Vc @ Vd ) ) ) ) ).
% balance.simps(11)
thf(fact_105_balance_Osimps_I12_J,axiom,
! [B: $tType,A: $tType,Va: rBT_Im246033960le_rbt @ A @ B,Vb: A,Vc: B,Vd: rBT_Im246033960le_rbt @ A @ B,W: A,X: B,B2: rBT_Im246033960le_rbt @ A @ B,S: A,T: B,C: rBT_Im246033960le_rbt @ A @ B,Y: A,Z: B] :
( ( rBT_Im1648453169alance @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va @ Vb @ Vc @ Vd ) @ W @ X @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ B2 @ S @ T @ C ) @ Y @ Z @ ( rBT_Im418718756_Empty @ A @ B ) ) )
= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va @ Vb @ Vc @ Vd ) @ W @ X @ B2 ) @ S @ T @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ C @ Y @ Z @ ( rBT_Im418718756_Empty @ A @ B ) ) ) ) ).
% balance.simps(12)
thf(fact_106_ins__rbt__greater,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [V: A,F: A > B > B > B,K: A,X: B,T: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1332541288reater @ A @ B @ V @ ( rBT_Im616047964bt_ins @ A @ B @ F @ K @ X @ T ) )
= ( ( rBT_Im1332541288reater @ A @ B @ V @ T )
& ( ord_less @ A @ V @ K ) ) ) ) ).
% ins_rbt_greater
thf(fact_107_ins__rbt__less,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [V: A,F: A > B > B > B,K: A,X: B,T: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im903026891t_less @ A @ B @ V @ ( rBT_Im616047964bt_ins @ A @ B @ F @ K @ X @ T ) )
= ( ( rBT_Im903026891t_less @ A @ B @ V @ T )
& ( ord_less @ A @ K @ V ) ) ) ) ).
% ins_rbt_less
thf(fact_108_balance__rbt__greater,axiom,
! [B: $tType,A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [V: A,A2: rBT_Im246033960le_rbt @ A @ B,K: A,X: B,B2: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1332541288reater @ A @ B @ V @ ( rBT_Im1648453169alance @ A @ B @ A2 @ K @ X @ B2 ) )
= ( ( rBT_Im1332541288reater @ A @ B @ V @ A2 )
& ( rBT_Im1332541288reater @ A @ B @ V @ B2 )
& ( ord_less @ A @ V @ K ) ) ) ) ).
% balance_rbt_greater
thf(fact_109_balance__rbt__less,axiom,
! [B: $tType,A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [V: A,A2: rBT_Im246033960le_rbt @ A @ B,K: A,X: B,B2: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im903026891t_less @ A @ B @ V @ ( rBT_Im1648453169alance @ A @ B @ A2 @ K @ X @ B2 ) )
= ( ( rBT_Im903026891t_less @ A @ B @ V @ A2 )
& ( rBT_Im903026891t_less @ A @ B @ V @ B2 )
& ( ord_less @ A @ K @ V ) ) ) ) ).
% balance_rbt_less
thf(fact_110_rbt__greater__simps_I2_J,axiom,
! [C2: $tType,A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [K: A,C: rBT_Im1923302023_color,Lt: rBT_Im246033960le_rbt @ A @ C2,Kt: A,V: C2,Rt: rBT_Im246033960le_rbt @ A @ C2] :
( ( rBT_Im1332541288reater @ A @ C2 @ K @ ( rBT_Im480247531Branch @ A @ C2 @ C @ Lt @ Kt @ V @ Rt ) )
= ( ( ord_less @ A @ K @ Kt )
& ( rBT_Im1332541288reater @ A @ C2 @ K @ Lt )
& ( rBT_Im1332541288reater @ A @ C2 @ K @ Rt ) ) ) ) ).
% rbt_greater_simps(2)
thf(fact_111_rbt__less__simps_I2_J,axiom,
! [C2: $tType,A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [K: A,C: rBT_Im1923302023_color,Lt: rBT_Im246033960le_rbt @ A @ C2,Kt: A,V: C2,Rt: rBT_Im246033960le_rbt @ A @ C2] :
( ( rBT_Im903026891t_less @ A @ C2 @ K @ ( rBT_Im480247531Branch @ A @ C2 @ C @ Lt @ Kt @ V @ Rt ) )
= ( ( ord_less @ A @ Kt @ K )
& ( rBT_Im903026891t_less @ A @ C2 @ K @ Lt )
& ( rBT_Im903026891t_less @ A @ C2 @ K @ Rt ) ) ) ) ).
% rbt_less_simps(2)
thf(fact_112_rbt__less__simps_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [K: A] : ( rBT_Im903026891t_less @ A @ B @ K @ ( rBT_Im418718756_Empty @ A @ B ) ) ) ).
% rbt_less_simps(1)
thf(fact_113_rbt__greater__simps_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [K: A] : ( rBT_Im1332541288reater @ A @ B @ K @ ( rBT_Im418718756_Empty @ A @ B ) ) ) ).
% rbt_greater_simps(1)
thf(fact_114_paint__rbt__less,axiom,
! [B: $tType,A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [V: A,C: rBT_Im1923302023_color,T: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im903026891t_less @ A @ B @ V @ ( rBT_Im945281971_paint @ A @ B @ C @ T ) )
= ( rBT_Im903026891t_less @ A @ B @ V @ T ) ) ) ).
% paint_rbt_less
thf(fact_115_paint__rbt__greater,axiom,
! [B: $tType,A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [V: A,C: rBT_Im1923302023_color,T: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1332541288reater @ A @ B @ V @ ( rBT_Im945281971_paint @ A @ B @ C @ T ) )
= ( rBT_Im1332541288reater @ A @ B @ V @ T ) ) ) ).
% paint_rbt_greater
thf(fact_116_rbt__less__trans,axiom,
! [C2: $tType,A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,T: rBT_Im246033960le_rbt @ A @ C2,Y: A] :
( ( rBT_Im903026891t_less @ A @ C2 @ X @ T )
=> ( ( ord_less @ A @ X @ Y )
=> ( rBT_Im903026891t_less @ A @ C2 @ Y @ T ) ) ) ) ).
% rbt_less_trans
thf(fact_117_rbt__greater__trans,axiom,
! [C2: $tType,A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,T: rBT_Im246033960le_rbt @ A @ C2] :
( ( ord_less @ A @ X @ Y )
=> ( ( rBT_Im1332541288reater @ A @ C2 @ Y @ T )
=> ( rBT_Im1332541288reater @ A @ C2 @ X @ T ) ) ) ) ).
% rbt_greater_trans
thf(fact_118_balance__rbt__sorted,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [L: rBT_Im246033960le_rbt @ A @ B,R: rBT_Im246033960le_rbt @ A @ B,K: A,V: B] :
( ( rBT_Im1183586831sorted @ A @ B @ L )
=> ( ( rBT_Im1183586831sorted @ A @ B @ R )
=> ( ( rBT_Im903026891t_less @ A @ B @ K @ L )
=> ( ( rBT_Im1332541288reater @ A @ B @ K @ R )
=> ( rBT_Im1183586831sorted @ A @ B @ ( rBT_Im1648453169alance @ A @ B @ L @ K @ V @ R ) ) ) ) ) ) ) ).
% balance_rbt_sorted
thf(fact_119_rbt__sorted_Osimps_I2_J,axiom,
! [B: $tType,A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [C: rBT_Im1923302023_color,L: rBT_Im246033960le_rbt @ A @ B,K: A,V: B,R: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1183586831sorted @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ C @ L @ K @ V @ R ) )
= ( ( rBT_Im903026891t_less @ A @ B @ K @ L )
& ( rBT_Im1332541288reater @ A @ B @ K @ R )
& ( rBT_Im1183586831sorted @ A @ B @ L )
& ( rBT_Im1183586831sorted @ A @ B @ R ) ) ) ) ).
% rbt_sorted.simps(2)
thf(fact_120_rbt__lookup__balance,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [L: rBT_Im246033960le_rbt @ A @ B,R: rBT_Im246033960le_rbt @ A @ B,K: A,V: B,X: A] :
( ( rBT_Im1183586831sorted @ A @ B @ L )
=> ( ( rBT_Im1183586831sorted @ A @ B @ R )
=> ( ( rBT_Im903026891t_less @ A @ B @ K @ L )
=> ( ( rBT_Im1332541288reater @ A @ B @ K @ R )
=> ( ( rBT_Im459437964lookup @ A @ B @ ( rBT_Im1648453169alance @ A @ B @ L @ K @ V @ R ) @ X )
= ( rBT_Im459437964lookup @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ L @ K @ V @ R ) @ X ) ) ) ) ) ) ) ).
% rbt_lookup_balance
thf(fact_121_bheight_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] :
( ( ( C = rBT_Impl_Mirabelle_B )
=> ( ( rBT_Im315265182height @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ C @ Lt @ K @ V @ Rt ) )
= ( suc @ ( rBT_Im315265182height @ A @ B @ Lt ) ) ) )
& ( ( C != rBT_Impl_Mirabelle_B )
=> ( ( rBT_Im315265182height @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ C @ Lt @ K @ V @ Rt ) )
= ( rBT_Im315265182height @ A @ B @ Lt ) ) ) ) ).
% bheight.simps(2)
thf(fact_122_paint__rbt__lookup,axiom,
! [B: $tType,A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [C: rBT_Im1923302023_color,T: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im459437964lookup @ A @ B @ ( rBT_Im945281971_paint @ A @ B @ C @ T ) )
= ( rBT_Im459437964lookup @ A @ B @ T ) ) ) ).
% paint_rbt_lookup
thf(fact_123_rbt__sorted_Osimps_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ( rBT_Im1183586831sorted @ A @ B @ ( rBT_Im418718756_Empty @ A @ B ) ) ) ).
% rbt_sorted.simps(1)
thf(fact_124_paint__rbt__sorted,axiom,
! [B: $tType,A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [T: rBT_Im246033960le_rbt @ A @ B,C: rBT_Im1923302023_color] :
( ( rBT_Im1183586831sorted @ A @ B @ T )
=> ( rBT_Im1183586831sorted @ A @ B @ ( rBT_Im945281971_paint @ A @ B @ C @ T ) ) ) ) ).
% paint_rbt_sorted
thf(fact_125_ins__rbt__sorted,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: rBT_Im246033960le_rbt @ A @ B,F: A > B > B > B,K: A,X: B] :
( ( rBT_Im1183586831sorted @ A @ B @ T )
=> ( rBT_Im1183586831sorted @ A @ B @ ( rBT_Im616047964bt_ins @ A @ B @ F @ K @ X @ T ) ) ) ) ).
% ins_rbt_sorted
thf(fact_126_rbt__insertwk__rbt__sorted,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: rBT_Im246033960le_rbt @ A @ B,F: A > B > B > B,K: A,X: B] :
( ( rBT_Im1183586831sorted @ A @ B @ T )
=> ( rBT_Im1183586831sorted @ A @ B @ ( rBT_Im1451321038th_key @ A @ B @ F @ K @ X @ T ) ) ) ) ).
% rbt_insertwk_rbt_sorted
thf(fact_127_balance__bheight,axiom,
! [B: $tType,A: $tType,L: rBT_Im246033960le_rbt @ A @ B,R: rBT_Im246033960le_rbt @ A @ B,K: A,V: B] :
( ( ( rBT_Im315265182height @ A @ B @ L )
= ( rBT_Im315265182height @ A @ B @ R ) )
=> ( ( rBT_Im315265182height @ A @ B @ ( rBT_Im1648453169alance @ A @ B @ L @ K @ V @ R ) )
= ( suc @ ( rBT_Im315265182height @ A @ B @ L ) ) ) ) ).
% balance_bheight
thf(fact_128_is__rbt__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ( ( rBT_Im222526792is_rbt @ A @ B )
= ( ^ [T2: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1040612427e_inv1 @ A @ B @ T2 )
& ( rBT_Im1040612428e_inv2 @ A @ B @ T2 )
& ( ( rBT_Im1805589839lor_of @ A @ B @ T2 )
= rBT_Impl_Mirabelle_B )
& ( rBT_Im1183586831sorted @ A @ B @ T2 ) ) ) ) ) ).
% is_rbt_def
thf(fact_129_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_130_nat_Oinject,axiom,
! [X26: nat,Y26: nat] :
( ( ( suc @ X26 )
= ( suc @ Y26 ) )
= ( X26 = Y26 ) ) ).
% nat.inject
thf(fact_131_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less @ nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_132_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_133_lessI,axiom,
! [N: nat] : ( ord_less @ nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_134_Empty__is__rbt,axiom,
! [B: $tType,A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ( rBT_Im222526792is_rbt @ A @ B @ ( rBT_Im418718756_Empty @ A @ B ) ) ) ).
% Empty_is_rbt
thf(fact_135_is__rbt__rbt__sorted,axiom,
! [B: $tType,A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [T: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im222526792is_rbt @ A @ B @ T )
=> ( rBT_Im1183586831sorted @ A @ B @ T ) ) ) ).
% is_rbt_rbt_sorted
thf(fact_136_rbt__insertwk__is__rbt,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: rBT_Im246033960le_rbt @ A @ B,F: A > B > B > B,K: A,X: B] :
( ( rBT_Im222526792is_rbt @ A @ B @ T )
=> ( rBT_Im222526792is_rbt @ A @ B @ ( rBT_Im1451321038th_key @ A @ B @ F @ K @ X @ T ) ) ) ) ).
% rbt_insertwk_is_rbt
thf(fact_137_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less @ nat @ N @ M )
=> ( ( ord_less @ nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_138_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less @ nat @ I @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ J )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_139_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less @ nat @ I @ J )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K3: nat] :
( ( ord_less @ nat @ I2 @ J2 )
=> ( ( ord_less @ nat @ J2 @ K3 )
=> ( ( P @ I2 @ J2 )
=> ( ( P @ J2 @ K3 )
=> ( P @ I2 @ K3 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_140_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( ord_less @ nat @ J @ K )
=> ( ord_less @ nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_141_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less @ nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_142_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less @ nat @ N @ M )
=> ( ( ord_less @ nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_143_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( suc @ N ) @ M )
= ( ? [M2: nat] :
( ( M
= ( suc @ M2 ) )
& ( ord_less @ nat @ N @ M2 ) ) ) ) ).
% Suc_less_eq2
thf(fact_144_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less @ nat @ M @ N ) )
= ( ord_less @ nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_145_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ ( suc @ N ) )
= ( ( ord_less @ nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_146_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less @ nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_147_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less @ nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_148_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less @ nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_149_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less @ nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less @ nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_150_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ M ) @ N )
=> ( ord_less @ nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_151_lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less @ nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less @ nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% lessE
thf(fact_152_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_153_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_154_lift__Suc__mono__less__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [F: nat > A,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less @ A @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less @ A @ ( F @ N ) @ ( F @ M ) )
= ( ord_less @ nat @ N @ M ) ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_155_lift__Suc__mono__less,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [F: nat > A,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less @ A @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less @ nat @ N @ N3 )
=> ( ord_less @ A @ ( F @ N ) @ ( F @ N3 ) ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_156_rbt__insert__is__rbt,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: rBT_Im246033960le_rbt @ A @ B,K: A,V: B] :
( ( rBT_Im222526792is_rbt @ A @ B @ T )
=> ( rBT_Im222526792is_rbt @ A @ B @ ( rBT_Im1276434955insert @ A @ B @ K @ V @ T ) ) ) ) ).
% rbt_insert_is_rbt
thf(fact_157_rbt__insertw__is__rbt,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: rBT_Im246033960le_rbt @ A @ B,F: B > B > B,K: A,V: B] :
( ( rBT_Im222526792is_rbt @ A @ B @ T )
=> ( rBT_Im222526792is_rbt @ A @ B @ ( rBT_Im1963520753t_with @ B @ A @ F @ K @ V @ T ) ) ) ) ).
% rbt_insertw_is_rbt
thf(fact_158_rbt__insert__rbt__sorted,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: rBT_Im246033960le_rbt @ A @ B,K: A,V: B] :
( ( rBT_Im1183586831sorted @ A @ B @ T )
=> ( rBT_Im1183586831sorted @ A @ B @ ( rBT_Im1276434955insert @ A @ B @ K @ V @ T ) ) ) ) ).
% rbt_insert_rbt_sorted
thf(fact_159_rbt__insertw__rbt__sorted,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: rBT_Im246033960le_rbt @ A @ B,F: B > B > B,K: A,V: B] :
( ( rBT_Im1183586831sorted @ A @ B @ T )
=> ( rBT_Im1183586831sorted @ A @ B @ ( rBT_Im1963520753t_with @ B @ A @ F @ K @ V @ T ) ) ) ) ).
% rbt_insertw_rbt_sorted
thf(fact_160_infinite__descent__measure,axiom,
! [A: $tType,P: A > $o,V4: A > nat,X: A] :
( ! [X6: A] :
( ~ ( P @ X6 )
=> ? [Y3: A] :
( ( ord_less @ nat @ ( V4 @ Y3 ) @ ( V4 @ X6 ) )
& ~ ( P @ Y3 ) ) )
=> ( P @ X ) ) ).
% infinite_descent_measure
thf(fact_161_measure__induct__rule,axiom,
! [A: $tType,F: A > nat,P: A > $o,A2: A] :
( ! [X6: A] :
( ! [Y3: A] :
( ( ord_less @ nat @ ( F @ Y3 ) @ ( F @ X6 ) )
=> ( P @ Y3 ) )
=> ( P @ X6 ) )
=> ( P @ A2 ) ) ).
% measure_induct_rule
thf(fact_162_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less @ nat @ X @ Y )
=> ( ord_less @ nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_163_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M3: nat] :
( ( ord_less @ nat @ M3 @ N2 )
& ~ ( P @ M3 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_164_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M3: nat] :
( ( ord_less @ nat @ M3 @ N2 )
=> ( P @ M3 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_165_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_166_measure__induct,axiom,
! [A: $tType,F: A > nat,P: A > $o,A2: A] :
( ! [X6: A] :
( ! [Y3: A] :
( ( ord_less @ nat @ ( F @ Y3 ) @ ( F @ X6 ) )
=> ( P @ Y3 ) )
=> ( P @ X6 ) )
=> ( P @ A2 ) ) ).
% measure_induct
thf(fact_167_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less @ nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_168_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_169_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_not_refl
thf(fact_170_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less @ nat @ M @ N )
| ( ord_less @ nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_171_bheight__paintR_H,axiom,
! [A: $tType,B: $tType,T: rBT_Im246033960le_rbt @ A @ B] :
( ( ( rBT_Im1805589839lor_of @ A @ B @ T )
= rBT_Impl_Mirabelle_B )
=> ( ( rBT_Im315265182height @ A @ B @ ( rBT_Im945281971_paint @ A @ B @ rBT_Impl_Mirabelle_R @ T ) )
= ( minus_minus @ nat @ ( rBT_Im315265182height @ A @ B @ T ) @ ( one_one @ nat ) ) ) ) ).
% bheight_paintR'
thf(fact_172_ex__gt__or__lt,axiom,
! [A: $tType] :
( ( condit1656338222tinuum @ A @ ( type2 @ A ) )
=> ! [A2: A] :
? [B3: A] :
( ( ord_less @ A @ A2 @ B3 )
| ( ord_less @ A @ B3 @ A2 ) ) ) ).
% ex_gt_or_lt
thf(fact_173_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( X != Y )
=> ( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_174_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus @ nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus @ nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_175_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus @ nat @ ( minus_minus @ nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus @ nat @ ( minus_minus @ nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_176_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus @ nat @ ( suc @ N ) @ ( one_one @ nat ) )
= N ) ).
% diff_Suc_1
thf(fact_177_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less @ nat @ J @ K )
=> ( ord_less @ nat @ ( minus_minus @ nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_178_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ( ord_less @ nat @ M @ L )
=> ( ord_less @ nat @ ( minus_minus @ nat @ L @ N ) @ ( minus_minus @ nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_179_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus @ nat @ ( minus_minus @ nat @ I @ J ) @ K )
= ( minus_minus @ nat @ ( minus_minus @ nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_180_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus @ nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_181_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus @ nat @ M @ ( suc @ N ) )
= ( minus_minus @ nat @ ( minus_minus @ nat @ M @ ( one_one @ nat ) ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_182_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less @ nat @ ( minus_minus @ nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_183_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ N @ M )
=> ( ( suc @ ( minus_minus @ nat @ M @ ( suc @ N ) ) )
= ( minus_minus @ nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_184_balance__left__inv2__app_I1_J,axiom,
! [B: $tType,A: $tType,Lt: rBT_Im246033960le_rbt @ A @ B,Rt: rBT_Im246033960le_rbt @ A @ B,K: A,V: B] :
( ( rBT_Im1040612428e_inv2 @ A @ B @ Lt )
=> ( ( rBT_Im1040612428e_inv2 @ A @ B @ Rt )
=> ( ( ( plus_plus @ nat @ ( rBT_Im315265182height @ A @ B @ Lt ) @ ( one_one @ nat ) )
= ( rBT_Im315265182height @ A @ B @ Rt ) )
=> ( ( ( rBT_Im1805589839lor_of @ A @ B @ Rt )
= rBT_Impl_Mirabelle_B )
=> ( rBT_Im1040612428e_inv2 @ A @ B @ ( rBT_Im1278323832e_left @ A @ B @ Lt @ K @ V @ Rt ) ) ) ) ) ) ).
% balance_left_inv2_app(1)
thf(fact_185_balance__left__inv2__app_I2_J,axiom,
! [B: $tType,A: $tType,Lt: rBT_Im246033960le_rbt @ A @ B,Rt: rBT_Im246033960le_rbt @ A @ B,K: A,V: B] :
( ( rBT_Im1040612428e_inv2 @ A @ B @ Lt )
=> ( ( rBT_Im1040612428e_inv2 @ A @ B @ Rt )
=> ( ( ( plus_plus @ nat @ ( rBT_Im315265182height @ A @ B @ Lt ) @ ( one_one @ nat ) )
= ( rBT_Im315265182height @ A @ B @ Rt ) )
=> ( ( ( rBT_Im1805589839lor_of @ A @ B @ Rt )
= rBT_Impl_Mirabelle_B )
=> ( ( rBT_Im315265182height @ A @ B @ ( rBT_Im1278323832e_left @ A @ B @ Lt @ K @ V @ Rt ) )
= ( rBT_Im315265182height @ A @ B @ Rt ) ) ) ) ) ) ).
% balance_left_inv2_app(2)
thf(fact_186_less__numeral__extra_I4_J,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% less_numeral_extra(4)
thf(fact_187_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus @ nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus @ nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_188_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
= ( ord_less @ nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_189_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus @ nat @ ( minus_minus @ nat @ I @ J ) @ K )
= ( minus_minus @ nat @ I @ ( plus_plus @ nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_190_less__add__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [A2: A] : ( ord_less @ A @ A2 @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) ) ) ).
% less_add_one
thf(fact_191_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ~ ( ord_less @ A @ A2 @ B2 )
=> ( ( plus_plus @ A @ B2 @ ( minus_minus @ A @ A2 @ B2 ) )
= A2 ) ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_192_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_193_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_194_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ M @ K ) @ ( plus_plus @ nat @ N @ K ) )
= ( minus_minus @ nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_195_Nat_Odiff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ K @ M ) @ ( plus_plus @ nat @ K @ N ) )
= ( minus_minus @ nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_196_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus @ nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus @ nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_197_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus @ nat @ ( suc @ M ) @ N )
= ( plus_plus @ nat @ M @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_198_nat__add__left__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( plus_plus @ nat @ K @ M )
= ( plus_plus @ nat @ K @ N ) )
= ( M = N ) ) ).
% nat_add_left_cancel
thf(fact_199_nat__add__right__cancel,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( plus_plus @ nat @ M @ K )
= ( plus_plus @ nat @ N @ K ) )
= ( M = N ) ) ).
% nat_add_right_cancel
thf(fact_200_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ K )
=> ( ord_less @ nat @ I @ K ) ) ).
% add_lessD1
thf(fact_201_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ( ord_less @ nat @ K @ L )
=> ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_202_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_203_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_204_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_205_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ord_less @ nat @ I @ ( plus_plus @ nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_206_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less @ nat @ I @ J )
=> ( ord_less @ nat @ I @ ( plus_plus @ nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_207_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less @ nat @ K @ L )
=> ( ( ( plus_plus @ nat @ M @ L )
= ( plus_plus @ nat @ K @ N ) )
=> ( ord_less @ nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_208_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less @ nat @ I @ ( suc @ ( plus_plus @ nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_209_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less @ nat @ I @ ( suc @ ( plus_plus @ nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_210_less__iff__Suc__add,axiom,
( ( ord_less @ nat )
= ( ^ [M4: nat,N4: nat] :
? [K2: nat] :
( N4
= ( suc @ ( plus_plus @ nat @ M4 @ K2 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_211_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ? [K3: nat] :
( N
= ( suc @ ( plus_plus @ nat @ M @ K3 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_212_less__diff__conv,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less @ nat @ I @ ( minus_minus @ nat @ J @ K ) )
= ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_213_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less @ nat @ M @ N )
=> ( ( plus_plus @ nat @ N @ ( minus_minus @ nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_214_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus @ nat @ ( one_one @ nat ) ) ) ).
% Suc_eq_plus1_left
thf(fact_215_Suc__eq__plus1,axiom,
( suc
= ( ^ [N4: nat] : ( plus_plus @ nat @ N4 @ ( one_one @ nat ) ) ) ) ).
% Suc_eq_plus1
thf(fact_216_balance__left__inv2__with__inv1_I2_J,axiom,
! [B: $tType,A: $tType,Lt: rBT_Im246033960le_rbt @ A @ B,Rt: rBT_Im246033960le_rbt @ A @ B,K: A,V: B] :
( ( rBT_Im1040612428e_inv2 @ A @ B @ Lt )
=> ( ( rBT_Im1040612428e_inv2 @ A @ B @ Rt )
=> ( ( ( plus_plus @ nat @ ( rBT_Im315265182height @ A @ B @ Lt ) @ ( one_one @ nat ) )
= ( rBT_Im315265182height @ A @ B @ Rt ) )
=> ( ( rBT_Im1040612427e_inv1 @ A @ B @ Rt )
=> ( rBT_Im1040612428e_inv2 @ A @ B @ ( rBT_Im1278323832e_left @ A @ B @ Lt @ K @ V @ Rt ) ) ) ) ) ) ).
% balance_left_inv2_with_inv1(2)
thf(fact_217_balance__left__inv2__with__inv1_I1_J,axiom,
! [A: $tType,B: $tType,Lt: rBT_Im246033960le_rbt @ A @ B,Rt: rBT_Im246033960le_rbt @ A @ B,K: A,V: B] :
( ( rBT_Im1040612428e_inv2 @ A @ B @ Lt )
=> ( ( rBT_Im1040612428e_inv2 @ A @ B @ Rt )
=> ( ( ( plus_plus @ nat @ ( rBT_Im315265182height @ A @ B @ Lt ) @ ( one_one @ nat ) )
= ( rBT_Im315265182height @ A @ B @ Rt ) )
=> ( ( rBT_Im1040612427e_inv1 @ A @ B @ Rt )
=> ( ( rBT_Im315265182height @ A @ B @ ( rBT_Im1278323832e_left @ A @ B @ Lt @ K @ V @ Rt ) )
= ( plus_plus @ nat @ ( rBT_Im315265182height @ A @ B @ Lt ) @ ( one_one @ nat ) ) ) ) ) ) ) ).
% balance_left_inv2_with_inv1(1)
thf(fact_218_add__less__cancel__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,C: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% add_less_cancel_right
thf(fact_219_add__less__cancel__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [C: A,A2: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% add_less_cancel_left
thf(fact_220_add__mono__thms__linordered__field_I5_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_221_add__mono__thms__linordered__field_I2_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_222_add__mono__thms__linordered__field_I1_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( K = L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_223_add__strict__mono,axiom,
! [A: $tType] :
( ( strict2144017051up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A,D: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C @ D )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ D ) ) ) ) ) ).
% add_strict_mono
thf(fact_224_add__strict__left__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) ) ) ) ).
% add_strict_left_mono
thf(fact_225_add__strict__right__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) ) ) ) ).
% add_strict_right_mono
thf(fact_226_add__less__imp__less__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [C: A,A2: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C @ A2 ) @ ( plus_plus @ A @ C @ B2 ) )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ).
% add_less_imp_less_left
thf(fact_227_add__less__imp__less__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,C: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C ) @ ( plus_plus @ A @ B2 @ C ) )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ).
% add_less_imp_less_right
thf(fact_228_diff__strict__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,D: A,C: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ D @ C )
=> ( ord_less @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B2 @ D ) ) ) ) ) ).
% diff_strict_mono
thf(fact_229_diff__eq__diff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A,D: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= ( minus_minus @ A @ C @ D ) )
=> ( ( ord_less @ A @ A2 @ B2 )
= ( ord_less @ A @ C @ D ) ) ) ) ).
% diff_eq_diff_less
thf(fact_230_diff__strict__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ord_less @ A @ ( minus_minus @ A @ C @ A2 ) @ ( minus_minus @ A @ C @ B2 ) ) ) ) ).
% diff_strict_left_mono
thf(fact_231_diff__strict__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B2 @ C ) ) ) ) ).
% diff_strict_right_mono
thf(fact_232_less__diff__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,C: A,B2: A] :
( ( ord_less @ A @ A2 @ ( minus_minus @ A @ C @ B2 ) )
= ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C ) ) ) ).
% less_diff_eq
thf(fact_233_diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C )
= ( ord_less @ A @ A2 @ ( plus_plus @ A @ C @ B2 ) ) ) ) ).
% diff_less_eq
thf(fact_234_linordered__field__class_Osign__simps_I40_J,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C )
= ( ord_less @ A @ A2 @ ( plus_plus @ A @ C @ B2 ) ) ) ) ).
% linordered_field_class.sign_simps(40)
thf(fact_235_linordered__field__class_Osign__simps_I41_J,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [A2: A,C: A,B2: A] :
( ( ord_less @ A @ A2 @ ( minus_minus @ A @ C @ B2 ) )
= ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C ) ) ) ).
% linordered_field_class.sign_simps(41)
thf(fact_236_linordered__field__no__ub,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [X7: A] :
? [X1: A] : ( ord_less @ A @ X7 @ X1 ) ) ).
% linordered_field_no_ub
thf(fact_237_linordered__field__no__lb,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [X7: A] :
? [Y2: A] : ( ord_less @ A @ Y2 @ X7 ) ) ).
% linordered_field_no_lb
thf(fact_238_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( suc @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
= N ) ) ).
% Suc_diff_1
thf(fact_239_add__eq__if,axiom,
( ( plus_plus @ nat )
= ( ^ [M4: nat,N4: nat] :
( if @ nat
@ ( M4
= ( zero_zero @ nat ) )
@ N4
@ ( suc @ ( plus_plus @ nat @ ( minus_minus @ nat @ M4 @ ( one_one @ nat ) ) @ N4 ) ) ) ) ) ).
% add_eq_if
thf(fact_240_not__gr__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [N: A] :
( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
= ( N
= ( zero_zero @ A ) ) ) ) ).
% not_gr_zero
thf(fact_241_neq0__conv,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% neq0_conv
thf(fact_242_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus @ nat @ M @ ( zero_zero @ nat ) )
= M ) ).
% Nat.add_0_right
thf(fact_243_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus @ nat @ M @ N )
= ( zero_zero @ nat ) )
= ( ( M
= ( zero_zero @ nat ) )
& ( N
= ( zero_zero @ nat ) ) ) ) ).
% add_is_0
thf(fact_244_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus @ nat @ M @ M )
= ( zero_zero @ nat ) ) ).
% diff_self_eq_0
thf(fact_245_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus @ nat @ ( zero_zero @ nat ) @ N )
= ( zero_zero @ nat ) ) ).
% diff_0_eq_0
thf(fact_246_add__less__same__cancel1,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel1
thf(fact_247_add__less__same__cancel2,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel2
thf(fact_248_less__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% less_add_same_cancel1
thf(fact_249_less__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( plus_plus @ A @ B2 @ A2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% less_add_same_cancel2
thf(fact_250_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_251_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_252_diff__gt__0__iff__gt,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B2 ) )
= ( ord_less @ A @ B2 @ A2 ) ) ) ).
% diff_gt_0_iff_gt
thf(fact_253_zero__less__Suc,axiom,
! [N: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_254_less__Suc0,axiom,
! [N: nat] :
( ( ord_less @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% less_Suc0
%----Type constructors (14)
thf(tcon_fun___Orderings_Oorder,axiom,
! [A4: $tType,A5: $tType] :
( ( order @ A5 @ ( type2 @ A5 ) )
=> ( order @ ( A4 > A5 ) @ ( type2 @ ( A4 > A5 ) ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A4: $tType,A5: $tType] :
( ( ord @ A5 @ ( type2 @ A5 ) )
=> ( ord @ ( A4 > A5 ) @ ( type2 @ ( A4 > A5 ) ) ) ) ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
ordere516151231imp_le @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
strict2144017051up_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add,axiom,
ordere223160158up_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere236663937imp_le @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
canoni770627133id_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Olinordered__semidom,axiom,
linordered_semidom @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
linorder @ nat @ ( type2 @ nat ) ).
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_HOL_Obool___Orderings_Olinorder_3,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_4,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_5,axiom,
ord @ $o @ ( type2 @ $o ) ).
%----Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $true @ X @ Y )
= X ) ).
%----Conjectures (4)
thf(conj_0,hypothesis,
rBT_Im1071879905_inv1l @ a @ b @ a2 ).
thf(conj_1,hypothesis,
rBT_Im1040612427e_inv1 @ a @ b @ b2 ).
thf(conj_2,hypothesis,
( ( rBT_Im1805589839lor_of @ a @ b @ b2 )
= rBT_Impl_Mirabelle_B ) ).
thf(conj_3,conjecture,
rBT_Im1040612427e_inv1 @ a @ b @ ( rBT_Im1278323832e_left @ a @ b @ a2 @ k @ x @ b2 ) ).
%------------------------------------------------------------------------------