TPTP Problem File: DAT235^1.p
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%------------------------------------------------------------------------------
% File : DAT235^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Red-black trees 858
% 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__858.p [Bla16]
% Status : Theorem
% Rating : 1.00 v7.1.0
% Syntax : Number of formulae : 304 ( 135 unt; 43 typ; 0 def)
% Number of atoms : 869 ( 304 equ; 0 cnn)
% Maximal formula atoms : 13 ( 3 avg)
% Number of connectives : 7014 ( 146 ~; 24 |; 85 &;6422 @)
% ( 0 <=>; 337 =>; 0 <=; 0 <~>)
% Maximal formula depth : 30 ( 10 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 329 ( 329 >; 0 *; 0 +; 0 <<)
% Number of symbols : 41 ( 40 usr; 8 con; 0-8 aty)
% Number of variables : 1699 ( 11 ^;1581 !; 60 ?;1699 :)
% ( 47 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:43:34.378
%------------------------------------------------------------------------------
%----Could-be-implicit typings (6)
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_List_Olist,type,
list: $tType > $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
thf(ty_tf_b,type,
b: $tType ).
thf(ty_tf_a,type,
a: $tType ).
%----Explicit typings (37)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_List_Oappend,type,
append:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist_OCons,type,
cons:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist_ONil,type,
nil:
!>[A: $tType] : ( list @ A ) ).
thf(sy_c_List_Oord_Olexordp,type,
lexordp:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Oord_Olexordp__eq,type,
lexordp_eq:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oorder_Ostrict__mono,type,
strict_mono:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > $o ) ).
thf(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Obalance,type,
rBT_Im1648453169alance:
!>[A: $tType,B: $tType] : ( ( rBT_Im246033960le_rbt @ A @ B ) > A > B > ( rBT_Im246033960le_rbt @ A @ B ) > ( rBT_Im246033960le_rbt @ A @ B ) ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Obalance__left,type,
rBT_Im1278323832e_left:
!>[A: $tType,B: $tType] : ( ( rBT_Im246033960le_rbt @ A @ B ) > A > B > ( rBT_Im246033960le_rbt @ A @ B ) > ( rBT_Im246033960le_rbt @ A @ B ) ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Obalance__right,type,
rBT_Im1053358987_right:
!>[A: $tType,B: $tType] : ( ( rBT_Im246033960le_rbt @ A @ B ) > A > B > ( rBT_Im246033960le_rbt @ A @ B ) > ( rBT_Im246033960le_rbt @ A @ B ) ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_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_Ocombine,type,
rBT_Im1591839508ombine:
!>[A: $tType,B: $tType] : ( ( rBT_Im246033960le_rbt @ A @ B ) > ( rBT_Im246033960le_rbt @ A @ B ) > ( rBT_Im246033960le_rbt @ A @ B ) ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Oord_Ois__rbt,type,
rBT_Im862805236is_rbt:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( rBT_Im246033960le_rbt @ A @ B ) > $o ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Oord_Orbt__del,type,
rBT_Im1714610413bt_del:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > A > ( rBT_Im246033960le_rbt @ A @ B ) > ( rBT_Im246033960le_rbt @ A @ B ) ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Oord_Orbt__del__from__left,type,
rBT_Im1152561598m_left:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > A > ( rBT_Im246033960le_rbt @ A @ B ) > A > B > ( rBT_Im246033960le_rbt @ A @ B ) > ( rBT_Im246033960le_rbt @ A @ B ) ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Oord_Orbt__del__from__right,type,
rBT_Im1808746437_right:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > A > ( rBT_Im246033960le_rbt @ A @ B ) > A > B > ( rBT_Im246033960le_rbt @ A @ B ) > ( rBT_Im246033960le_rbt @ A @ B ) ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Oord_Orbt__greater,type,
rBT_Im1259024060reater:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > A > ( rBT_Im246033960le_rbt @ A @ B ) > $o ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Oord_Orbt__ins,type,
rBT_Im1756508336bt_ins:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B > B > B ) > A > B > ( rBT_Im246033960le_rbt @ A @ B ) > ( rBT_Im246033960le_rbt @ A @ B ) ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Oord_Orbt__insert,type,
rBT_Im852463031insert:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > A > B > ( rBT_Im246033960le_rbt @ A @ B ) > ( rBT_Im246033960le_rbt @ A @ B ) ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Oord_Orbt__insert__with,type,
rBT_Im241132701t_with:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > B > B ) > A > B > ( rBT_Im246033960le_rbt @ A @ B ) > ( rBT_Im246033960le_rbt @ A @ B ) ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Oord_Orbt__insert__with__key,type,
rBT_Im627674146th_key:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B > B > B ) > A > B > ( rBT_Im246033960le_rbt @ A @ B ) > ( rBT_Im246033960le_rbt @ A @ B ) ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Oord_Orbt__less,type,
rBT_Im2075124343t_less:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > A > ( rBT_Im246033960le_rbt @ A @ B ) > $o ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Oord_Orbt__sorted,type,
rBT_Im759614907sorted:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( rBT_Im246033960le_rbt @ A @ B ) > $o ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Opaint,type,
rBT_Im945281971_paint:
!>[A: $tType,B: $tType] : ( rBT_Im1923302023_color > ( rBT_Im246033960le_rbt @ A @ B ) > ( rBT_Im246033960le_rbt @ A @ B ) ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Orbt_OBranch,type,
rBT_Im480247531Branch:
!>[A: $tType,B: $tType] : ( rBT_Im1923302023_color > ( rBT_Im246033960le_rbt @ A @ B ) > A > B > ( rBT_Im246033960le_rbt @ A @ B ) > ( rBT_Im246033960le_rbt @ A @ B ) ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Orbt_OEmpty,type,
rBT_Im418718756_Empty:
!>[A: $tType,B: $tType] : ( rBT_Im246033960le_rbt @ A @ B ) ).
thf(sy_v_k,type,
k: a ).
thf(sy_v_less,type,
less: a > a > $o ).
thf(sy_v_lt,type,
lt: rBT_Im246033960le_rbt @ a @ b ).
thf(sy_v_rt,type,
rt: rBT_Im246033960le_rbt @ a @ b ).
thf(sy_v_v,type,
v: a ).
thf(sy_v_x,type,
x: a ).
thf(sy_v_y,type,
y: b ).
%----Relevant facts (255)
thf(fact_0_local_Oantisym__conv3,axiom,
! [Y: a,X: a] :
( ~ ( less @ Y @ X )
=> ( ( ~ ( less @ X @ Y ) )
= ( X = Y ) ) ) ).
% local.antisym_conv3
thf(fact_1_local_Odual__order_Oasym,axiom,
! [B2: a,A2: a] :
( ( less @ B2 @ A2 )
=> ~ ( less @ A2 @ B2 ) ) ).
% local.dual_order.asym
thf(fact_2_local_Odual__order_Ostrict__implies__not__eq,axiom,
! [B2: a,A2: a] :
( ( less @ B2 @ A2 )
=> ( A2 != B2 ) ) ).
% local.dual_order.strict_implies_not_eq
thf(fact_3_local_Odual__order_Ostrict__trans,axiom,
! [B2: a,A2: a,C: a] :
( ( less @ B2 @ A2 )
=> ( ( less @ C @ B2 )
=> ( less @ C @ A2 ) ) ) ).
% local.dual_order.strict_trans
thf(fact_4_local_Oless__asym,axiom,
! [X: a,Y: a] :
( ( less @ X @ Y )
=> ~ ( less @ Y @ X ) ) ).
% local.less_asym
thf(fact_5_local_Oless__asym_H,axiom,
! [A2: a,B2: a] :
( ( less @ A2 @ B2 )
=> ~ ( less @ B2 @ A2 ) ) ).
% local.less_asym'
thf(fact_6_local_Oless__imp__neq,axiom,
! [X: a,Y: a] :
( ( less @ X @ Y )
=> ( X != Y ) ) ).
% local.less_imp_neq
thf(fact_7_local_Oless__imp__not__eq,axiom,
! [X: a,Y: a] :
( ( less @ X @ Y )
=> ( X != Y ) ) ).
% local.less_imp_not_eq
thf(fact_8_local_Oless__imp__not__eq2,axiom,
! [X: a,Y: a] :
( ( less @ X @ Y )
=> ( Y != X ) ) ).
% local.less_imp_not_eq2
thf(fact_9_local_Oless__imp__not__less,axiom,
! [X: a,Y: a] :
( ( less @ X @ Y )
=> ~ ( less @ Y @ X ) ) ).
% local.less_imp_not_less
thf(fact_10_local_Oless__imp__triv,axiom,
! [X: a,Y: a,P: $o] :
( ( less @ X @ Y )
=> ( ( less @ Y @ X )
=> P ) ) ).
% local.less_imp_triv
thf(fact_11_local_Oless__irrefl,axiom,
! [X: a] :
~ ( less @ X @ X ) ).
% local.less_irrefl
thf(fact_12_local_Oless__linear,axiom,
! [X: a,Y: a] :
( ( less @ X @ Y )
| ( X = Y )
| ( less @ Y @ X ) ) ).
% local.less_linear
thf(fact_13_local_Oless__not__sym,axiom,
! [X: a,Y: a] :
( ( less @ X @ Y )
=> ~ ( less @ Y @ X ) ) ).
% local.less_not_sym
thf(fact_14_local_Oless__trans,axiom,
! [X: a,Y: a,Z: a] :
( ( less @ X @ Y )
=> ( ( less @ Y @ Z )
=> ( less @ X @ Z ) ) ) ).
% local.less_trans
thf(fact_15_local_Olinorder__cases,axiom,
! [X: a,Y: a] :
( ~ ( less @ X @ Y )
=> ( ( X != Y )
=> ( less @ Y @ X ) ) ) ).
% local.linorder_cases
thf(fact_16_local_OneqE,axiom,
! [X: a,Y: a] :
( ( X != Y )
=> ( ~ ( less @ X @ Y )
=> ( less @ Y @ X ) ) ) ).
% local.neqE
thf(fact_17_local_Oneq__iff,axiom,
! [X: a,Y: a] :
( ( X != Y )
= ( ( less @ X @ Y )
| ( less @ Y @ X ) ) ) ).
% local.neq_iff
thf(fact_18_local_Onot__less__iff__gr__or__eq,axiom,
! [X: a,Y: a] :
( ( ~ ( less @ X @ Y ) )
= ( ( less @ Y @ X )
| ( X = Y ) ) ) ).
% local.not_less_iff_gr_or_eq
thf(fact_19_local_Oord__eq__less__trans,axiom,
! [A2: a,B2: a,C: a] :
( ( A2 = B2 )
=> ( ( less @ B2 @ C )
=> ( less @ A2 @ C ) ) ) ).
% local.ord_eq_less_trans
thf(fact_20_local_Oord__less__eq__trans,axiom,
! [A2: a,B2: a,C: a] :
( ( less @ A2 @ B2 )
=> ( ( B2 = C )
=> ( less @ A2 @ C ) ) ) ).
% local.ord_less_eq_trans
thf(fact_21_local_Oorder_Oasym,axiom,
! [A2: a,B2: a] :
( ( less @ A2 @ B2 )
=> ~ ( less @ B2 @ A2 ) ) ).
% local.order.asym
thf(fact_22_local_Oorder_Oirrefl,axiom,
! [A2: a] :
~ ( less @ A2 @ A2 ) ).
% local.order.irrefl
thf(fact_23_local_Oorder_Ostrict__implies__not__eq,axiom,
! [A2: a,B2: a] :
( ( less @ A2 @ B2 )
=> ( A2 != B2 ) ) ).
% local.order.strict_implies_not_eq
thf(fact_24_local_Oorder_Ostrict__trans,axiom,
! [A2: a,B2: a,C: a] :
( ( less @ A2 @ B2 )
=> ( ( less @ B2 @ C )
=> ( less @ A2 @ C ) ) ) ).
% local.order.strict_trans
thf(fact_25_local_Orbt__less__trans,axiom,
! [C2: $tType,X: a,T: rBT_Im246033960le_rbt @ a @ C2,Y: a] :
( ( rBT_Im2075124343t_less @ a @ C2 @ less @ X @ T )
=> ( ( less @ X @ Y )
=> ( rBT_Im2075124343t_less @ a @ C2 @ less @ Y @ T ) ) ) ).
% local.rbt_less_trans
thf(fact_26_local_Ostrict__mono__eq,axiom,
! [B: $tType] :
( ( order @ B @ ( type2 @ B ) )
=> ! [F: a > B,X: a,Y: a] :
( ( strict_mono @ a @ B @ less @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ) ).
% local.strict_mono_eq
thf(fact_27_local_Obalance__right__rbt__less,axiom,
! [B: $tType,K: a,A2: rBT_Im246033960le_rbt @ a @ B,B2: rBT_Im246033960le_rbt @ a @ B,X: a,T: B] :
( ( rBT_Im2075124343t_less @ a @ B @ less @ K @ A2 )
=> ( ( rBT_Im2075124343t_less @ a @ B @ less @ K @ B2 )
=> ( ( less @ X @ K )
=> ( rBT_Im2075124343t_less @ a @ B @ less @ K @ ( rBT_Im1053358987_right @ a @ B @ A2 @ X @ T @ B2 ) ) ) ) ) ).
% local.balance_right_rbt_less
thf(fact_28_local_Ocombine__rbt__less,axiom,
! [B: $tType,K: a,L: rBT_Im246033960le_rbt @ a @ B,R: rBT_Im246033960le_rbt @ a @ B] :
( ( rBT_Im2075124343t_less @ a @ B @ less @ K @ L )
=> ( ( rBT_Im2075124343t_less @ a @ B @ less @ K @ R )
=> ( rBT_Im2075124343t_less @ a @ B @ less @ K @ ( rBT_Im1591839508ombine @ a @ B @ L @ R ) ) ) ) ).
% local.combine_rbt_less
thf(fact_29_local_Obalance__left__rbt__less,axiom,
! [B: $tType,K: a,A2: rBT_Im246033960le_rbt @ a @ B,B2: rBT_Im246033960le_rbt @ a @ B,X: a,T: B] :
( ( rBT_Im2075124343t_less @ a @ B @ less @ K @ A2 )
=> ( ( rBT_Im2075124343t_less @ a @ B @ less @ K @ B2 )
=> ( ( less @ X @ K )
=> ( rBT_Im2075124343t_less @ a @ B @ less @ K @ ( rBT_Im1278323832e_left @ a @ B @ A2 @ X @ T @ B2 ) ) ) ) ) ).
% local.balance_left_rbt_less
thf(fact_30_local_Olexordp__eq__antisym,axiom,
! [Xs: list @ a,Ys: list @ a] :
( ( lexordp_eq @ a @ less @ Xs @ Ys )
=> ( ( lexordp_eq @ a @ less @ Ys @ Xs )
=> ( Xs = Ys ) ) ) ).
% local.lexordp_eq_antisym
thf(fact_31_local_Olexordp__eq__linear,axiom,
! [Xs: list @ a,Ys: list @ a] :
( ( lexordp_eq @ a @ less @ Xs @ Ys )
| ( lexordp_eq @ a @ less @ Ys @ Xs ) ) ).
% local.lexordp_eq_linear
thf(fact_32_local_Olexordp__eq__refl,axiom,
! [Xs: list @ a] : ( lexordp_eq @ a @ less @ Xs @ Xs ) ).
% local.lexordp_eq_refl
thf(fact_33_local_Olexordp__eq__trans,axiom,
! [Xs: list @ a,Ys: list @ a,Zs: list @ a] :
( ( lexordp_eq @ a @ less @ Xs @ Ys )
=> ( ( lexordp_eq @ a @ less @ Ys @ Zs )
=> ( lexordp_eq @ a @ less @ Xs @ Zs ) ) ) ).
% local.lexordp_eq_trans
thf(fact_34_local_Orbt__greater__trans,axiom,
! [C2: $tType,X: a,Y: a,T: rBT_Im246033960le_rbt @ a @ C2] :
( ( less @ X @ Y )
=> ( ( rBT_Im1259024060reater @ a @ C2 @ less @ Y @ T )
=> ( rBT_Im1259024060reater @ a @ C2 @ less @ X @ T ) ) ) ).
% local.rbt_greater_trans
thf(fact_35_local_Opaint__rbt__less,axiom,
! [B: $tType,V: a,C: rBT_Im1923302023_color,T: rBT_Im246033960le_rbt @ a @ B] :
( ( rBT_Im2075124343t_less @ a @ B @ less @ V @ ( rBT_Im945281971_paint @ a @ B @ C @ T ) )
= ( rBT_Im2075124343t_less @ a @ B @ less @ V @ T ) ) ).
% local.paint_rbt_less
thf(fact_36_local_Oins__rbt__less,axiom,
! [B: $tType,V: a,F: a > B > B > B,K: a,X: B,T: rBT_Im246033960le_rbt @ a @ B] :
( ( rBT_Im2075124343t_less @ a @ B @ less @ V @ ( rBT_Im1756508336bt_ins @ a @ B @ less @ F @ K @ X @ T ) )
= ( ( rBT_Im2075124343t_less @ a @ B @ less @ V @ T )
& ( less @ K @ V ) ) ) ).
% local.ins_rbt_less
thf(fact_37_local_Olexordp__antisym,axiom,
! [Xs: list @ a,Ys: list @ a] :
( ( lexordp @ a @ less @ Xs @ Ys )
=> ~ ( lexordp @ a @ less @ Ys @ Xs ) ) ).
% local.lexordp_antisym
thf(fact_38_local_Olexordp__irreflexive,axiom,
! [Xs: list @ a] :
( ! [X2: a] :
~ ( less @ X2 @ X2 )
=> ~ ( lexordp @ a @ less @ Xs @ Xs ) ) ).
% local.lexordp_irreflexive
thf(fact_39_local_Olexordp__trans,axiom,
! [Xs: list @ a,Ys: list @ a,Zs: list @ a] :
( ( lexordp @ a @ less @ Xs @ Ys )
=> ( ( lexordp @ a @ less @ Ys @ Zs )
=> ( lexordp @ a @ less @ Xs @ Zs ) ) ) ).
% local.lexordp_trans
thf(fact_40_local_Olexordp__linear,axiom,
! [Xs: list @ a,Ys: list @ a] :
( ( lexordp @ a @ less @ Xs @ Ys )
| ( Xs = Ys )
| ( lexordp @ a @ less @ Ys @ Xs ) ) ).
% local.lexordp_linear
thf(fact_41_local_Olexordp__irreflexive_H,axiom,
! [Xs: list @ a] :
~ ( lexordp @ a @ less @ Xs @ Xs ) ).
% local.lexordp_irreflexive'
thf(fact_42_local_Olexordp__into__lexordp__eq,axiom,
! [Xs: list @ a,Ys: list @ a] :
( ( lexordp @ a @ less @ Xs @ Ys )
=> ( lexordp_eq @ a @ less @ Xs @ Ys ) ) ).
% local.lexordp_into_lexordp_eq
thf(fact_43_local_Olexordp__eq__conv__lexord,axiom,
! [Xs: list @ a,Ys: list @ a] :
( ( lexordp_eq @ a @ less @ Xs @ Ys )
= ( ( Xs = Ys )
| ( lexordp @ a @ less @ Xs @ Ys ) ) ) ).
% local.lexordp_eq_conv_lexord
thf(fact_44_local_Olexordp__conv__lexordp__eq,axiom,
! [Xs: list @ a,Ys: list @ a] :
( ( lexordp @ a @ less @ Xs @ Ys )
= ( ( lexordp_eq @ a @ less @ Xs @ Ys )
& ~ ( lexordp_eq @ a @ less @ Ys @ Xs ) ) ) ).
% local.lexordp_conv_lexordp_eq
thf(fact_45_local_Ocombine__rbt__greater,axiom,
! [B: $tType,K: a,L: rBT_Im246033960le_rbt @ a @ B,R: rBT_Im246033960le_rbt @ a @ B] :
( ( rBT_Im1259024060reater @ a @ B @ less @ K @ L )
=> ( ( rBT_Im1259024060reater @ a @ B @ less @ K @ R )
=> ( rBT_Im1259024060reater @ a @ B @ less @ K @ ( rBT_Im1591839508ombine @ a @ B @ L @ R ) ) ) ) ).
% local.combine_rbt_greater
thf(fact_46_local_Obalance__left__rbt__greater,axiom,
! [B: $tType,K: a,A2: rBT_Im246033960le_rbt @ a @ B,B2: rBT_Im246033960le_rbt @ a @ B,X: a,T: B] :
( ( rBT_Im1259024060reater @ a @ B @ less @ K @ A2 )
=> ( ( rBT_Im1259024060reater @ a @ B @ less @ K @ B2 )
=> ( ( less @ K @ X )
=> ( rBT_Im1259024060reater @ a @ B @ less @ K @ ( rBT_Im1278323832e_left @ a @ B @ A2 @ X @ T @ B2 ) ) ) ) ) ).
% local.balance_left_rbt_greater
thf(fact_47_local_Obalance__right__rbt__greater,axiom,
! [B: $tType,K: a,A2: rBT_Im246033960le_rbt @ a @ B,B2: rBT_Im246033960le_rbt @ a @ B,X: a,T: B] :
( ( rBT_Im1259024060reater @ a @ B @ less @ K @ A2 )
=> ( ( rBT_Im1259024060reater @ a @ B @ less @ K @ B2 )
=> ( ( less @ K @ X )
=> ( rBT_Im1259024060reater @ a @ B @ less @ K @ ( rBT_Im1053358987_right @ a @ B @ A2 @ X @ T @ B2 ) ) ) ) ) ).
% local.balance_right_rbt_greater
thf(fact_48_local_Oins__rbt__greater,axiom,
! [B: $tType,V: a,F: a > B > B > B,K: a,X: B,T: rBT_Im246033960le_rbt @ a @ B] :
( ( rBT_Im1259024060reater @ a @ B @ less @ V @ ( rBT_Im1756508336bt_ins @ a @ B @ less @ F @ K @ X @ T ) )
= ( ( rBT_Im1259024060reater @ a @ B @ less @ V @ T )
& ( less @ V @ K ) ) ) ).
% local.ins_rbt_greater
thf(fact_49_local_Opaint__rbt__greater,axiom,
! [B: $tType,V: a,C: rBT_Im1923302023_color,T: rBT_Im246033960le_rbt @ a @ B] :
( ( rBT_Im1259024060reater @ a @ B @ less @ V @ ( rBT_Im945281971_paint @ a @ B @ C @ T ) )
= ( rBT_Im1259024060reater @ a @ B @ less @ V @ T ) ) ).
% local.paint_rbt_greater
thf(fact_50_ord_Opaint__rbt__greater,axiom,
! [B: $tType,A: $tType,Less: A > A > $o,V: A,C: rBT_Im1923302023_color,T: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1259024060reater @ A @ B @ Less @ V @ ( rBT_Im945281971_paint @ A @ B @ C @ T ) )
= ( rBT_Im1259024060reater @ A @ B @ Less @ V @ T ) ) ).
% ord.paint_rbt_greater
thf(fact_51_ord_Opaint__rbt__less,axiom,
! [B: $tType,A: $tType,Less: A > A > $o,V: A,C: rBT_Im1923302023_color,T: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im2075124343t_less @ A @ B @ Less @ V @ ( rBT_Im945281971_paint @ A @ B @ C @ T ) )
= ( rBT_Im2075124343t_less @ A @ B @ Less @ V @ T ) ) ).
% ord.paint_rbt_less
thf(fact_52_local_Obalance__left__rbt__sorted,axiom,
! [B: $tType,L: rBT_Im246033960le_rbt @ a @ B,R: rBT_Im246033960le_rbt @ a @ B,K: a,V: B] :
( ( rBT_Im759614907sorted @ a @ B @ less @ L )
=> ( ( rBT_Im759614907sorted @ a @ B @ less @ R )
=> ( ( rBT_Im2075124343t_less @ a @ B @ less @ K @ L )
=> ( ( rBT_Im1259024060reater @ a @ B @ less @ K @ R )
=> ( rBT_Im759614907sorted @ a @ B @ less @ ( rBT_Im1278323832e_left @ a @ B @ L @ K @ V @ R ) ) ) ) ) ) ).
% local.balance_left_rbt_sorted
thf(fact_53_local_Obalance__right__rbt__sorted,axiom,
! [B: $tType,L: rBT_Im246033960le_rbt @ a @ B,R: rBT_Im246033960le_rbt @ a @ B,K: a,V: B] :
( ( rBT_Im759614907sorted @ a @ B @ less @ L )
=> ( ( rBT_Im759614907sorted @ a @ B @ less @ R )
=> ( ( rBT_Im2075124343t_less @ a @ B @ less @ K @ L )
=> ( ( rBT_Im1259024060reater @ a @ B @ less @ K @ R )
=> ( rBT_Im759614907sorted @ a @ B @ less @ ( rBT_Im1053358987_right @ a @ B @ L @ K @ V @ R ) ) ) ) ) ) ).
% local.balance_right_rbt_sorted
thf(fact_54_local_Ocombine__rbt__sorted,axiom,
! [B: $tType,L: rBT_Im246033960le_rbt @ a @ B,R: rBT_Im246033960le_rbt @ a @ B,K: a] :
( ( rBT_Im759614907sorted @ a @ B @ less @ L )
=> ( ( rBT_Im759614907sorted @ a @ B @ less @ R )
=> ( ( rBT_Im2075124343t_less @ a @ B @ less @ K @ L )
=> ( ( rBT_Im1259024060reater @ a @ B @ less @ K @ R )
=> ( rBT_Im759614907sorted @ a @ B @ less @ ( rBT_Im1591839508ombine @ a @ B @ L @ R ) ) ) ) ) ) ).
% local.combine_rbt_sorted
thf(fact_55_local_Ostrict__mono__less,axiom,
! [B: $tType] :
( ( order @ B @ ( type2 @ B ) )
=> ! [F: a > B,X: a,Y: a] :
( ( strict_mono @ a @ B @ less @ F )
=> ( ( ord_less @ B @ ( F @ X ) @ ( F @ Y ) )
= ( less @ X @ Y ) ) ) ) ).
% local.strict_mono_less
thf(fact_56_local_Ostrict__mono__def,axiom,
! [B: $tType] :
( ( order @ B @ ( type2 @ B ) )
=> ! [F: a > B] :
( ( strict_mono @ a @ B @ less @ F )
= ( ! [X3: a,Y2: a] :
( ( less @ X3 @ Y2 )
=> ( ord_less @ B @ ( F @ X3 ) @ ( F @ Y2 ) ) ) ) ) ) ).
% local.strict_mono_def
thf(fact_57_local_Ostrict__monoI,axiom,
! [B: $tType] :
( ( order @ B @ ( type2 @ B ) )
=> ! [F: a > B] :
( ! [X2: a,Y3: a] :
( ( less @ X2 @ Y3 )
=> ( ord_less @ B @ ( F @ X2 ) @ ( F @ Y3 ) ) )
=> ( strict_mono @ a @ B @ less @ F ) ) ) ).
% local.strict_monoI
thf(fact_58_local_Ostrict__monoD,axiom,
! [B: $tType] :
( ( order @ B @ ( type2 @ B ) )
=> ! [F: a > B,X: a,Y: a] :
( ( strict_mono @ a @ B @ less @ F )
=> ( ( less @ X @ Y )
=> ( ord_less @ B @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ).
% local.strict_monoD
thf(fact_59_local_Opaint__rbt__sorted,axiom,
! [B: $tType,T: rBT_Im246033960le_rbt @ a @ B,C: rBT_Im1923302023_color] :
( ( rBT_Im759614907sorted @ a @ B @ less @ T )
=> ( rBT_Im759614907sorted @ a @ B @ less @ ( rBT_Im945281971_paint @ a @ B @ C @ T ) ) ) ).
% local.paint_rbt_sorted
thf(fact_60_local_Oins__rbt__sorted,axiom,
! [B: $tType,T: rBT_Im246033960le_rbt @ a @ B,F: a > B > B > B,K: a,X: B] :
( ( rBT_Im759614907sorted @ a @ B @ less @ T )
=> ( rBT_Im759614907sorted @ a @ B @ less @ ( rBT_Im1756508336bt_ins @ a @ B @ less @ F @ K @ X @ T ) ) ) ).
% local.ins_rbt_sorted
thf(fact_61_local_Orbt__insert__rbt__sorted,axiom,
! [B: $tType,T: rBT_Im246033960le_rbt @ a @ B,K: a,V: B] :
( ( rBT_Im759614907sorted @ a @ B @ less @ T )
=> ( rBT_Im759614907sorted @ a @ B @ less @ ( rBT_Im852463031insert @ a @ B @ less @ K @ V @ T ) ) ) ).
% local.rbt_insert_rbt_sorted
thf(fact_62_local_Orbt__insertw__rbt__sorted,axiom,
! [B: $tType,T: rBT_Im246033960le_rbt @ a @ B,F: B > B > B,K: a,V: B] :
( ( rBT_Im759614907sorted @ a @ B @ less @ T )
=> ( rBT_Im759614907sorted @ a @ B @ less @ ( rBT_Im241132701t_with @ a @ B @ less @ F @ K @ V @ T ) ) ) ).
% local.rbt_insertw_rbt_sorted
thf(fact_63_local_Orbt__insertwk__rbt__sorted,axiom,
! [B: $tType,T: rBT_Im246033960le_rbt @ a @ B,F: a > B > B > B,K: a,X: B] :
( ( rBT_Im759614907sorted @ a @ B @ less @ T )
=> ( rBT_Im759614907sorted @ a @ B @ less @ ( rBT_Im627674146th_key @ a @ B @ less @ F @ K @ X @ T ) ) ) ).
% local.rbt_insertwk_rbt_sorted
thf(fact_64_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_65_local_Ois__rbt__rbt__sorted,axiom,
! [B: $tType,T: rBT_Im246033960le_rbt @ a @ B] :
( ( rBT_Im862805236is_rbt @ a @ B @ less @ T )
=> ( rBT_Im759614907sorted @ a @ B @ less @ T ) ) ).
% local.is_rbt_rbt_sorted
thf(fact_66_local_Orbt__sorted_Osimps_I2_J,axiom,
! [B: $tType,C: rBT_Im1923302023_color,L: rBT_Im246033960le_rbt @ a @ B,K: a,V: B,R: rBT_Im246033960le_rbt @ a @ B] :
( ( rBT_Im759614907sorted @ a @ B @ less @ ( rBT_Im480247531Branch @ a @ B @ C @ L @ K @ V @ R ) )
= ( ( rBT_Im2075124343t_less @ a @ B @ less @ K @ L )
& ( rBT_Im1259024060reater @ a @ B @ less @ K @ R )
& ( rBT_Im759614907sorted @ a @ B @ less @ L )
& ( rBT_Im759614907sorted @ a @ B @ less @ R ) ) ) ).
% local.rbt_sorted.simps(2)
thf(fact_67_local_Obalance__rbt__sorted,axiom,
! [B: $tType,L: rBT_Im246033960le_rbt @ a @ B,R: rBT_Im246033960le_rbt @ a @ B,K: a,V: B] :
( ( rBT_Im759614907sorted @ a @ B @ less @ L )
=> ( ( rBT_Im759614907sorted @ a @ B @ less @ R )
=> ( ( rBT_Im2075124343t_less @ a @ B @ less @ K @ L )
=> ( ( rBT_Im1259024060reater @ a @ B @ less @ K @ R )
=> ( rBT_Im759614907sorted @ a @ B @ less @ ( rBT_Im1648453169alance @ a @ B @ L @ K @ V @ R ) ) ) ) ) ) ).
% local.balance_rbt_sorted
thf(fact_68_local_Olexordp__eq_ONil,axiom,
! [Ys: list @ a] : ( lexordp_eq @ a @ less @ ( nil @ a ) @ Ys ) ).
% local.lexordp_eq.Nil
thf(fact_69_ord_Olexordp__into__lexordp__eq,axiom,
! [A: $tType,Less: A > A > $o,Xs: list @ A,Ys: list @ A] :
( ( lexordp @ A @ Less @ Xs @ Ys )
=> ( lexordp_eq @ A @ Less @ Xs @ Ys ) ) ).
% ord.lexordp_into_lexordp_eq
thf(fact_70_local_Olexordp__eq_OCons,axiom,
! [X: a,Y: a,Xs: list @ a,Ys: list @ a] :
( ( less @ X @ Y )
=> ( lexordp_eq @ a @ less @ ( cons @ a @ X @ Xs ) @ ( cons @ a @ Y @ Ys ) ) ) ).
% local.lexordp_eq.Cons
thf(fact_71_local_Olexordp__eq_OCons__eq,axiom,
! [X: a,Y: a,Xs: list @ a,Ys: list @ a] :
( ~ ( less @ X @ Y )
=> ( ~ ( less @ Y @ X )
=> ( ( lexordp_eq @ a @ less @ Xs @ Ys )
=> ( lexordp_eq @ a @ less @ ( cons @ a @ X @ Xs ) @ ( cons @ a @ Y @ Ys ) ) ) ) ) ).
% local.lexordp_eq.Cons_eq
thf(fact_72_local_Olexordp_OCons__eq,axiom,
! [X: a,Y: a,Xs: list @ a,Ys: list @ a] :
( ~ ( less @ X @ Y )
=> ( ~ ( less @ Y @ X )
=> ( ( lexordp @ a @ less @ Xs @ Ys )
=> ( lexordp @ a @ less @ ( cons @ a @ X @ Xs ) @ ( cons @ a @ Y @ Ys ) ) ) ) ) ).
% local.lexordp.Cons_eq
thf(fact_73_local_Olexordp_OCons,axiom,
! [X: a,Y: a,Xs: list @ a,Ys: list @ a] :
( ( less @ X @ Y )
=> ( lexordp @ a @ less @ ( cons @ a @ X @ Xs ) @ ( cons @ a @ Y @ Ys ) ) ) ).
% local.lexordp.Cons
thf(fact_74_local_Orbt__insertwk__is__rbt,axiom,
! [B: $tType,T: rBT_Im246033960le_rbt @ a @ B,F: a > B > B > B,K: a,X: B] :
( ( rBT_Im862805236is_rbt @ a @ B @ less @ T )
=> ( rBT_Im862805236is_rbt @ a @ B @ less @ ( rBT_Im627674146th_key @ a @ B @ less @ F @ K @ X @ T ) ) ) ).
% local.rbt_insertwk_is_rbt
thf(fact_75_local_Orbt__insertw__is__rbt,axiom,
! [B: $tType,T: rBT_Im246033960le_rbt @ a @ B,F: B > B > B,K: a,V: B] :
( ( rBT_Im862805236is_rbt @ a @ B @ less @ T )
=> ( rBT_Im862805236is_rbt @ a @ B @ less @ ( rBT_Im241132701t_with @ a @ B @ less @ F @ K @ V @ T ) ) ) ).
% local.rbt_insertw_is_rbt
thf(fact_76_local_Orbt__insert__is__rbt,axiom,
! [B: $tType,T: rBT_Im246033960le_rbt @ a @ B,K: a,V: B] :
( ( rBT_Im862805236is_rbt @ a @ B @ less @ T )
=> ( rBT_Im862805236is_rbt @ a @ B @ less @ ( rBT_Im852463031insert @ a @ B @ less @ K @ V @ T ) ) ) ).
% local.rbt_insert_is_rbt
thf(fact_77_list_Oinject,axiom,
! [A: $tType,X21: A,X22: list @ A,Y21: A,Y22: list @ A] :
( ( ( cons @ A @ X21 @ X22 )
= ( cons @ A @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_78_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_79_local_Olexordp__induct,axiom,
! [Xs: list @ a,Ys: list @ a,P: ( list @ a ) > ( list @ a ) > $o] :
( ( lexordp @ a @ less @ Xs @ Ys )
=> ( ! [Y3: a,Ys2: list @ a] : ( P @ ( nil @ a ) @ ( cons @ a @ Y3 @ Ys2 ) )
=> ( ! [X2: a,Xs2: list @ a,Y3: a,Ys2: list @ a] :
( ( less @ X2 @ Y3 )
=> ( P @ ( cons @ a @ X2 @ Xs2 ) @ ( cons @ a @ Y3 @ Ys2 ) ) )
=> ( ! [X2: a,Xs2: list @ a,Ys2: list @ a] :
( ( lexordp @ a @ less @ Xs2 @ Ys2 )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons @ a @ X2 @ Xs2 ) @ ( cons @ a @ X2 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% local.lexordp_induct
thf(fact_80_local_Olexordp__cases,axiom,
! [Xs: list @ a,Ys: list @ a] :
( ( lexordp @ a @ less @ Xs @ Ys )
=> ( ( ( Xs
= ( nil @ a ) )
=> ! [Y3: a,Ys3: list @ a] :
( Ys
!= ( cons @ a @ Y3 @ Ys3 ) ) )
=> ( ! [X2: a] :
( ? [Xs3: list @ a] :
( Xs
= ( cons @ a @ X2 @ Xs3 ) )
=> ! [Y3: a] :
( ? [Ys3: list @ a] :
( Ys
= ( cons @ a @ Y3 @ Ys3 ) )
=> ~ ( less @ X2 @ Y3 ) ) )
=> ~ ! [X2: a,Xs3: list @ a] :
( ( Xs
= ( cons @ a @ X2 @ Xs3 ) )
=> ! [Ys3: list @ a] :
( ( Ys
= ( cons @ a @ X2 @ Ys3 ) )
=> ~ ( lexordp @ a @ less @ Xs3 @ Ys3 ) ) ) ) ) ) ).
% local.lexordp_cases
thf(fact_81_local_Olexordp_Osimps,axiom,
! [A1: list @ a,A22: list @ a] :
( ( lexordp @ a @ less @ A1 @ A22 )
= ( ? [Y2: a,Ys4: list @ a] :
( ( A1
= ( nil @ a ) )
& ( A22
= ( cons @ a @ Y2 @ Ys4 ) ) )
| ? [X3: a,Y2: a,Xs4: list @ a,Ys4: list @ a] :
( ( A1
= ( cons @ a @ X3 @ Xs4 ) )
& ( A22
= ( cons @ a @ Y2 @ Ys4 ) )
& ( less @ X3 @ Y2 ) )
| ? [X3: a,Y2: a,Xs4: list @ a,Ys4: list @ a] :
( ( A1
= ( cons @ a @ X3 @ Xs4 ) )
& ( A22
= ( cons @ a @ Y2 @ Ys4 ) )
& ~ ( less @ X3 @ Y2 )
& ~ ( less @ Y2 @ X3 )
& ( lexordp @ a @ less @ Xs4 @ Ys4 ) ) ) ) ).
% local.lexordp.simps
thf(fact_82_local_Olexordp_Oinducts,axiom,
! [X1: list @ a,X26: list @ a,P: ( list @ a ) > ( list @ a ) > $o] :
( ( lexordp @ a @ less @ X1 @ X26 )
=> ( ! [Y3: a,Ys2: list @ a] : ( P @ ( nil @ a ) @ ( cons @ a @ Y3 @ Ys2 ) )
=> ( ! [X2: a,Y3: a,Xs2: list @ a,Ys2: list @ a] :
( ( less @ X2 @ Y3 )
=> ( P @ ( cons @ a @ X2 @ Xs2 ) @ ( cons @ a @ Y3 @ Ys2 ) ) )
=> ( ! [X2: a,Y3: a,Xs2: list @ a,Ys2: list @ a] :
( ~ ( less @ X2 @ Y3 )
=> ( ~ ( less @ Y3 @ X2 )
=> ( ( lexordp @ a @ less @ Xs2 @ Ys2 )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons @ a @ X2 @ Xs2 ) @ ( cons @ a @ Y3 @ Ys2 ) ) ) ) ) )
=> ( P @ X1 @ X26 ) ) ) ) ) ).
% local.lexordp.inducts
thf(fact_83_local_Olexordp_Ocases,axiom,
! [A1: list @ a,A22: list @ a] :
( ( lexordp @ a @ less @ A1 @ A22 )
=> ( ( ( A1
= ( nil @ a ) )
=> ! [Y3: a,Ys2: list @ a] :
( A22
!= ( cons @ a @ Y3 @ Ys2 ) ) )
=> ( ! [X2: a] :
( ? [Xs2: list @ a] :
( A1
= ( cons @ a @ X2 @ Xs2 ) )
=> ! [Y3: a] :
( ? [Ys2: list @ a] :
( A22
= ( cons @ a @ Y3 @ Ys2 ) )
=> ~ ( less @ X2 @ Y3 ) ) )
=> ~ ! [X2: a,Y3: a,Xs2: list @ a] :
( ( A1
= ( cons @ a @ X2 @ Xs2 ) )
=> ! [Ys2: list @ a] :
( ( A22
= ( cons @ a @ Y3 @ Ys2 ) )
=> ( ~ ( less @ X2 @ Y3 )
=> ( ~ ( less @ Y3 @ X2 )
=> ~ ( lexordp @ a @ less @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ).
% local.lexordp.cases
thf(fact_84_local_Olexordp_ONil,axiom,
! [Y: a,Ys: list @ a] : ( lexordp @ a @ less @ ( nil @ a ) @ ( cons @ a @ Y @ Ys ) ) ).
% local.lexordp.Nil
thf(fact_85_local_Olexordp__eq_Osimps,axiom,
! [A1: list @ a,A22: list @ a] :
( ( lexordp_eq @ a @ less @ A1 @ A22 )
= ( ? [Ys4: list @ a] :
( ( A1
= ( nil @ a ) )
& ( A22 = Ys4 ) )
| ? [X3: a,Y2: a,Xs4: list @ a,Ys4: list @ a] :
( ( A1
= ( cons @ a @ X3 @ Xs4 ) )
& ( A22
= ( cons @ a @ Y2 @ Ys4 ) )
& ( less @ X3 @ Y2 ) )
| ? [X3: a,Y2: a,Xs4: list @ a,Ys4: list @ a] :
( ( A1
= ( cons @ a @ X3 @ Xs4 ) )
& ( A22
= ( cons @ a @ Y2 @ Ys4 ) )
& ~ ( less @ X3 @ Y2 )
& ~ ( less @ Y2 @ X3 )
& ( lexordp_eq @ a @ less @ Xs4 @ Ys4 ) ) ) ) ).
% local.lexordp_eq.simps
thf(fact_86_local_Olexordp__eq_Oinducts,axiom,
! [X1: list @ a,X26: list @ a,P: ( list @ a ) > ( list @ a ) > $o] :
( ( lexordp_eq @ a @ less @ X1 @ X26 )
=> ( ! [X12: list @ a] : ( P @ ( nil @ a ) @ X12 )
=> ( ! [X2: a,Y3: a,Xs2: list @ a,Ys2: list @ a] :
( ( less @ X2 @ Y3 )
=> ( P @ ( cons @ a @ X2 @ Xs2 ) @ ( cons @ a @ Y3 @ Ys2 ) ) )
=> ( ! [X2: a,Y3: a,Xs2: list @ a,Ys2: list @ a] :
( ~ ( less @ X2 @ Y3 )
=> ( ~ ( less @ Y3 @ X2 )
=> ( ( lexordp_eq @ a @ less @ Xs2 @ Ys2 )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons @ a @ X2 @ Xs2 ) @ ( cons @ a @ Y3 @ Ys2 ) ) ) ) ) )
=> ( P @ X1 @ X26 ) ) ) ) ) ).
% local.lexordp_eq.inducts
thf(fact_87_local_Olexordp__eq_Ocases,axiom,
! [A1: list @ a,A22: list @ a] :
( ( lexordp_eq @ a @ less @ A1 @ A22 )
=> ( ( ( A1
= ( nil @ a ) )
=> ! [Ys2: list @ a] : A22 != Ys2 )
=> ( ! [X2: a] :
( ? [Xs2: list @ a] :
( A1
= ( cons @ a @ X2 @ Xs2 ) )
=> ! [Y3: a] :
( ? [Ys2: list @ a] :
( A22
= ( cons @ a @ Y3 @ Ys2 ) )
=> ~ ( less @ X2 @ Y3 ) ) )
=> ~ ! [X2: a,Y3: a,Xs2: list @ a] :
( ( A1
= ( cons @ a @ X2 @ Xs2 ) )
=> ! [Ys2: list @ a] :
( ( A22
= ( cons @ a @ Y3 @ Ys2 ) )
=> ( ~ ( less @ X2 @ Y3 )
=> ( ~ ( less @ Y3 @ X2 )
=> ~ ( lexordp_eq @ a @ less @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ).
% local.lexordp_eq.cases
thf(fact_88_ord_Olexordp__simps_I3_J,axiom,
! [A: $tType,Less: A > A > $o,X: A,Xs: list @ A,Y: A,Ys: list @ A] :
( ( lexordp @ A @ Less @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) )
= ( ( Less @ X @ Y )
| ( ~ ( Less @ Y @ X )
& ( lexordp @ A @ Less @ Xs @ Ys ) ) ) ) ).
% ord.lexordp_simps(3)
thf(fact_89_ord_Olexordp__simps_I2_J,axiom,
! [A: $tType,Less: A > A > $o,Xs: list @ A] :
~ ( lexordp @ A @ Less @ Xs @ ( nil @ A ) ) ).
% ord.lexordp_simps(2)
thf(fact_90_ord_Olexordp__simps_I1_J,axiom,
! [A: $tType,Less: A > A > $o,Ys: list @ A] :
( ( lexordp @ A @ Less @ ( nil @ A ) @ Ys )
= ( Ys
!= ( nil @ A ) ) ) ).
% ord.lexordp_simps(1)
thf(fact_91_ord_Olexordp__eq__simps_I4_J,axiom,
! [A: $tType,Less: A > A > $o,X: A,Xs: list @ A,Y: A,Ys: list @ A] :
( ( lexordp_eq @ A @ Less @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) )
= ( ( Less @ X @ Y )
| ( ~ ( Less @ Y @ X )
& ( lexordp_eq @ A @ Less @ Xs @ Ys ) ) ) ) ).
% ord.lexordp_eq_simps(4)
thf(fact_92_ord_Olexordp__eq__simps_I2_J,axiom,
! [A: $tType,Less: A > A > $o,Xs: list @ A] :
( ( lexordp_eq @ A @ Less @ Xs @ ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% ord.lexordp_eq_simps(2)
thf(fact_93_ord_Olexordp__eq__simps_I1_J,axiom,
! [A: $tType,Less: A > A > $o,Ys: list @ A] : ( lexordp_eq @ A @ Less @ ( nil @ A ) @ Ys ) ).
% ord.lexordp_eq_simps(1)
thf(fact_94_ord_Olexordp__eq__simps_I3_J,axiom,
! [A: $tType,Less: A > A > $o,X: A,Xs: list @ A] :
~ ( lexordp_eq @ A @ Less @ ( cons @ A @ X @ Xs ) @ ( nil @ A ) ) ).
% ord.lexordp_eq_simps(3)
thf(fact_95_local_Olexordp__simps_I3_J,axiom,
! [X: a,Xs: list @ a,Y: a,Ys: list @ a] :
( ( lexordp @ a @ less @ ( cons @ a @ X @ Xs ) @ ( cons @ a @ Y @ Ys ) )
= ( ( less @ X @ Y )
| ( ~ ( less @ Y @ X )
& ( lexordp @ a @ less @ Xs @ Ys ) ) ) ) ).
% local.lexordp_simps(3)
thf(fact_96_local_Olexordp__eq__simps_I4_J,axiom,
! [X: a,Xs: list @ a,Y: a,Ys: list @ a] :
( ( lexordp_eq @ a @ less @ ( cons @ a @ X @ Xs ) @ ( cons @ a @ Y @ Ys ) )
= ( ( less @ X @ Y )
| ( ~ ( less @ Y @ X )
& ( lexordp_eq @ a @ less @ Xs @ Ys ) ) ) ) ).
% local.lexordp_eq_simps(4)
thf(fact_97_local_Olexordp__simps_I1_J,axiom,
! [Ys: list @ a] :
( ( lexordp @ a @ less @ ( nil @ a ) @ Ys )
= ( Ys
!= ( nil @ a ) ) ) ).
% local.lexordp_simps(1)
thf(fact_98_local_Olexordp__simps_I2_J,axiom,
! [Xs: list @ a] :
~ ( lexordp @ a @ less @ Xs @ ( nil @ a ) ) ).
% local.lexordp_simps(2)
thf(fact_99_local_Orbt__greater__simps_I2_J,axiom,
! [C2: $tType,K: a,C: rBT_Im1923302023_color,Lt: rBT_Im246033960le_rbt @ a @ C2,Kt: a,V: C2,Rt: rBT_Im246033960le_rbt @ a @ C2] :
( ( rBT_Im1259024060reater @ a @ C2 @ less @ K @ ( rBT_Im480247531Branch @ a @ C2 @ C @ Lt @ Kt @ V @ Rt ) )
= ( ( less @ K @ Kt )
& ( rBT_Im1259024060reater @ a @ C2 @ less @ K @ Lt )
& ( rBT_Im1259024060reater @ a @ C2 @ less @ K @ Rt ) ) ) ).
% local.rbt_greater_simps(2)
thf(fact_100_local_Orbt__less__simps_I2_J,axiom,
! [C2: $tType,K: a,C: rBT_Im1923302023_color,Lt: rBT_Im246033960le_rbt @ a @ C2,Kt: a,V: C2,Rt: rBT_Im246033960le_rbt @ a @ C2] :
( ( rBT_Im2075124343t_less @ a @ C2 @ less @ K @ ( rBT_Im480247531Branch @ a @ C2 @ C @ Lt @ Kt @ V @ Rt ) )
= ( ( less @ Kt @ K )
& ( rBT_Im2075124343t_less @ a @ C2 @ less @ K @ Lt )
& ( rBT_Im2075124343t_less @ a @ C2 @ less @ K @ Rt ) ) ) ).
% local.rbt_less_simps(2)
thf(fact_101_local_Olexordp__eq__simps_I1_J,axiom,
! [Ys: list @ a] : ( lexordp_eq @ a @ less @ ( nil @ a ) @ Ys ) ).
% local.lexordp_eq_simps(1)
thf(fact_102_local_Olexordp__eq__simps_I2_J,axiom,
! [Xs: list @ a] :
( ( lexordp_eq @ a @ less @ Xs @ ( nil @ a ) )
= ( Xs
= ( nil @ a ) ) ) ).
% local.lexordp_eq_simps(2)
thf(fact_103_local_Obalance__rbt__greater,axiom,
! [B: $tType,V: a,A2: rBT_Im246033960le_rbt @ a @ B,K: a,X: B,B2: rBT_Im246033960le_rbt @ a @ B] :
( ( rBT_Im1259024060reater @ a @ B @ less @ V @ ( rBT_Im1648453169alance @ a @ B @ A2 @ K @ X @ B2 ) )
= ( ( rBT_Im1259024060reater @ a @ B @ less @ V @ A2 )
& ( rBT_Im1259024060reater @ a @ B @ less @ V @ B2 )
& ( less @ V @ K ) ) ) ).
% local.balance_rbt_greater
thf(fact_104_local_Obalance__rbt__less,axiom,
! [B: $tType,V: a,A2: rBT_Im246033960le_rbt @ a @ B,K: a,X: B,B2: rBT_Im246033960le_rbt @ a @ B] :
( ( rBT_Im2075124343t_less @ a @ B @ less @ V @ ( rBT_Im1648453169alance @ a @ B @ A2 @ K @ X @ B2 ) )
= ( ( rBT_Im2075124343t_less @ a @ B @ less @ V @ A2 )
& ( rBT_Im2075124343t_less @ a @ B @ less @ V @ B2 )
& ( less @ K @ V ) ) ) ).
% local.balance_rbt_less
thf(fact_105_local_Olexordp__eq__simps_I3_J,axiom,
! [X: a,Xs: list @ a] :
~ ( lexordp_eq @ a @ less @ ( cons @ a @ X @ Xs ) @ ( nil @ a ) ) ).
% local.lexordp_eq_simps(3)
thf(fact_106_ord_Olexordp_OCons__eq,axiom,
! [A: $tType,Less: A > A > $o,X: A,Y: A,Xs: list @ A,Ys: list @ A] :
( ~ ( Less @ X @ Y )
=> ( ~ ( Less @ Y @ X )
=> ( ( lexordp @ A @ Less @ Xs @ Ys )
=> ( lexordp @ A @ Less @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) ) ) ).
% ord.lexordp.Cons_eq
thf(fact_107_ord_Olexordp_OCons,axiom,
! [A: $tType,Less: A > A > $o,X: A,Y: A,Xs: list @ A,Ys: list @ A] :
( ( Less @ X @ Y )
=> ( lexordp @ A @ Less @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) ).
% ord.lexordp.Cons
thf(fact_108_ord_Olexordp__eq_OCons__eq,axiom,
! [A: $tType,Less: A > A > $o,X: A,Y: A,Xs: list @ A,Ys: list @ A] :
( ~ ( Less @ X @ Y )
=> ( ~ ( Less @ Y @ X )
=> ( ( lexordp_eq @ A @ Less @ Xs @ Ys )
=> ( lexordp_eq @ A @ Less @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) ) ) ).
% ord.lexordp_eq.Cons_eq
thf(fact_109_ord_Olexordp__eq_OCons,axiom,
! [A: $tType,Less: A > A > $o,X: A,Y: A,Xs: list @ A,Ys: list @ A] :
( ( Less @ X @ Y )
=> ( lexordp_eq @ A @ Less @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) ).
% ord.lexordp_eq.Cons
thf(fact_110_ord_Olexordp__eq_ONil,axiom,
! [A: $tType,Less: A > A > $o,Ys: list @ A] : ( lexordp_eq @ A @ Less @ ( nil @ A ) @ Ys ) ).
% ord.lexordp_eq.Nil
thf(fact_111_list_Odistinct_I1_J,axiom,
! [A: $tType,X21: A,X22: list @ A] :
( ( nil @ A )
!= ( cons @ A @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_112_list_OdiscI,axiom,
! [A: $tType,List: list @ A,X21: A,X22: list @ A] :
( ( List
= ( cons @ A @ X21 @ X22 ) )
=> ( List
!= ( nil @ A ) ) ) ).
% list.discI
thf(fact_113_list_Oexhaust,axiom,
! [A: $tType,Y: list @ A] :
( ( Y
!= ( nil @ A ) )
=> ~ ! [X212: A,X222: list @ A] :
( Y
!= ( cons @ A @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_114_list_Oinducts,axiom,
! [A: $tType,P: ( list @ A ) > $o,List: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X12: A,X27: list @ A] :
( ( P @ X27 )
=> ( P @ ( cons @ A @ X12 @ X27 ) ) )
=> ( P @ List ) ) ) ).
% list.inducts
thf(fact_115_neq__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
= ( ? [Y2: A,Ys4: list @ A] :
( Xs
= ( cons @ A @ Y2 @ Ys4 ) ) ) ) ).
% neq_Nil_conv
thf(fact_116_list__induct2_H,axiom,
! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,Xs: list @ A,Ys: list @ B] :
( ( P @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X2: A,Xs2: list @ A] : ( P @ ( cons @ A @ X2 @ Xs2 ) @ ( nil @ B ) )
=> ( ! [Y3: B,Ys2: list @ B] : ( P @ ( nil @ A ) @ ( cons @ B @ Y3 @ Ys2 ) )
=> ( ! [X2: A,Xs2: list @ A,Y3: B,Ys2: list @ B] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ B @ Y3 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_117_splice_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
( ! [X12: list @ A] : ( P @ ( nil @ A ) @ X12 )
=> ( ! [V2: A,Va: list @ A] : ( P @ ( cons @ A @ V2 @ Va ) @ ( nil @ A ) )
=> ( ! [X2: A,Xs2: list @ A,Y3: A,Ys2: list @ A] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ A @ Y3 @ Ys2 ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% splice.induct
thf(fact_118_not__Cons__self2,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( cons @ A @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_119_transpose_Ocases,axiom,
! [A: $tType,X: list @ ( list @ A )] :
( ( X
!= ( nil @ ( list @ A ) ) )
=> ( ! [Xss: list @ ( list @ A )] :
( X
!= ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
=> ~ ! [X2: A,Xs2: list @ A,Xss: list @ ( list @ A )] :
( X
!= ( cons @ ( list @ A ) @ ( cons @ A @ X2 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_120_remdups__adj_Ocases,axiom,
! [A: $tType,X: list @ A] :
( ( X
!= ( nil @ A ) )
=> ( ! [X2: A] :
( X
!= ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ~ ! [X2: A,Y3: A,Xs2: list @ A] :
( X
!= ( cons @ A @ X2 @ ( cons @ A @ Y3 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_121_remdups__adj_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > $o,A0: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ( ! [X2: A,Y3: A,Xs2: list @ A] :
( ( ( X2 = Y3 )
=> ( P @ ( cons @ A @ X2 @ Xs2 ) ) )
=> ( ( ( X2 != Y3 )
=> ( P @ ( cons @ A @ Y3 @ Xs2 ) ) )
=> ( P @ ( cons @ A @ X2 @ ( cons @ A @ Y3 @ Xs2 ) ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% remdups_adj.induct
thf(fact_122_list__nonempty__induct,axiom,
! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ( ! [X2: A,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( P @ Xs2 )
=> ( P @ ( cons @ A @ X2 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_123_map__tailrec__rev_Oinduct,axiom,
! [A: $tType,B: $tType,P: ( A > B ) > ( list @ A ) > ( list @ B ) > $o,A0: A > B,A1: list @ A,A22: list @ B] :
( ! [F2: A > B,X12: list @ B] : ( P @ F2 @ ( nil @ A ) @ X12 )
=> ( ! [F2: A > B,A3: A,As: list @ A,Bs: list @ B] :
( ( P @ F2 @ As @ ( cons @ B @ ( F2 @ A3 ) @ Bs ) )
=> ( P @ F2 @ ( cons @ A @ A3 @ As ) @ Bs ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_124_ord_Olexordp__eq_Oinducts,axiom,
! [A: $tType,Less: A > A > $o,X1: list @ A,X26: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
( ( lexordp_eq @ A @ Less @ X1 @ X26 )
=> ( ! [X12: list @ A] : ( P @ ( nil @ A ) @ X12 )
=> ( ! [X2: A,Y3: A,Xs2: list @ A,Ys2: list @ A] :
( ( Less @ X2 @ Y3 )
=> ( P @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ A @ Y3 @ Ys2 ) ) )
=> ( ! [X2: A,Y3: A,Xs2: list @ A,Ys2: list @ A] :
( ~ ( Less @ X2 @ Y3 )
=> ( ~ ( Less @ Y3 @ X2 )
=> ( ( lexordp_eq @ A @ Less @ Xs2 @ Ys2 )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ A @ Y3 @ Ys2 ) ) ) ) ) )
=> ( P @ X1 @ X26 ) ) ) ) ) ).
% ord.lexordp_eq.inducts
thf(fact_125_ord_Olexordp__eq_Osimps,axiom,
! [A: $tType] :
( ( lexordp_eq @ A )
= ( ^ [Less2: A > A > $o,A12: list @ A,A23: list @ A] :
( ? [Ys4: list @ A] :
( ( A12
= ( nil @ A ) )
& ( A23 = Ys4 ) )
| ? [X3: A,Y2: A,Xs4: list @ A,Ys4: list @ A] :
( ( A12
= ( cons @ A @ X3 @ Xs4 ) )
& ( A23
= ( cons @ A @ Y2 @ Ys4 ) )
& ( Less2 @ X3 @ Y2 ) )
| ? [X3: A,Y2: A,Xs4: list @ A,Ys4: list @ A] :
( ( A12
= ( cons @ A @ X3 @ Xs4 ) )
& ( A23
= ( cons @ A @ Y2 @ Ys4 ) )
& ~ ( Less2 @ X3 @ Y2 )
& ~ ( Less2 @ Y2 @ X3 )
& ( lexordp_eq @ A @ Less2 @ Xs4 @ Ys4 ) ) ) ) ) ).
% ord.lexordp_eq.simps
thf(fact_126_ord_Olexordp__eq_Ocases,axiom,
! [A: $tType,Less: A > A > $o,A1: list @ A,A22: list @ A] :
( ( lexordp_eq @ A @ Less @ A1 @ A22 )
=> ( ( ( A1
= ( nil @ A ) )
=> ! [Ys2: list @ A] : A22 != Ys2 )
=> ( ! [X2: A] :
( ? [Xs2: list @ A] :
( A1
= ( cons @ A @ X2 @ Xs2 ) )
=> ! [Y3: A] :
( ? [Ys2: list @ A] :
( A22
= ( cons @ A @ Y3 @ Ys2 ) )
=> ~ ( Less @ X2 @ Y3 ) ) )
=> ~ ! [X2: A,Y3: A,Xs2: list @ A] :
( ( A1
= ( cons @ A @ X2 @ Xs2 ) )
=> ! [Ys2: list @ A] :
( ( A22
= ( cons @ A @ Y3 @ Ys2 ) )
=> ( ~ ( Less @ X2 @ Y3 )
=> ( ~ ( Less @ Y3 @ X2 )
=> ~ ( lexordp_eq @ A @ Less @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ).
% ord.lexordp_eq.cases
thf(fact_127_ord_Olexordp_Oinducts,axiom,
! [A: $tType,Less: A > A > $o,X1: list @ A,X26: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
( ( lexordp @ A @ Less @ X1 @ X26 )
=> ( ! [Y3: A,Ys2: list @ A] : ( P @ ( nil @ A ) @ ( cons @ A @ Y3 @ Ys2 ) )
=> ( ! [X2: A,Y3: A,Xs2: list @ A,Ys2: list @ A] :
( ( Less @ X2 @ Y3 )
=> ( P @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ A @ Y3 @ Ys2 ) ) )
=> ( ! [X2: A,Y3: A,Xs2: list @ A,Ys2: list @ A] :
( ~ ( Less @ X2 @ Y3 )
=> ( ~ ( Less @ Y3 @ X2 )
=> ( ( lexordp @ A @ Less @ Xs2 @ Ys2 )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons @ A @ X2 @ Xs2 ) @ ( cons @ A @ Y3 @ Ys2 ) ) ) ) ) )
=> ( P @ X1 @ X26 ) ) ) ) ) ).
% ord.lexordp.inducts
thf(fact_128_ord_Olexordp_Osimps,axiom,
! [A: $tType] :
( ( lexordp @ A )
= ( ^ [Less2: A > A > $o,A12: list @ A,A23: list @ A] :
( ? [Y2: A,Ys4: list @ A] :
( ( A12
= ( nil @ A ) )
& ( A23
= ( cons @ A @ Y2 @ Ys4 ) ) )
| ? [X3: A,Y2: A,Xs4: list @ A,Ys4: list @ A] :
( ( A12
= ( cons @ A @ X3 @ Xs4 ) )
& ( A23
= ( cons @ A @ Y2 @ Ys4 ) )
& ( Less2 @ X3 @ Y2 ) )
| ? [X3: A,Y2: A,Xs4: list @ A,Ys4: list @ A] :
( ( A12
= ( cons @ A @ X3 @ Xs4 ) )
& ( A23
= ( cons @ A @ Y2 @ Ys4 ) )
& ~ ( Less2 @ X3 @ Y2 )
& ~ ( Less2 @ Y2 @ X3 )
& ( lexordp @ A @ Less2 @ Xs4 @ Ys4 ) ) ) ) ) ).
% ord.lexordp.simps
thf(fact_129_ord_Olexordp_Ocases,axiom,
! [A: $tType,Less: A > A > $o,A1: list @ A,A22: list @ A] :
( ( lexordp @ A @ Less @ A1 @ A22 )
=> ( ( ( A1
= ( nil @ A ) )
=> ! [Y3: A,Ys2: list @ A] :
( A22
!= ( cons @ A @ Y3 @ Ys2 ) ) )
=> ( ! [X2: A] :
( ? [Xs2: list @ A] :
( A1
= ( cons @ A @ X2 @ Xs2 ) )
=> ! [Y3: A] :
( ? [Ys2: list @ A] :
( A22
= ( cons @ A @ Y3 @ Ys2 ) )
=> ~ ( Less @ X2 @ Y3 ) ) )
=> ~ ! [X2: A,Y3: A,Xs2: list @ A] :
( ( A1
= ( cons @ A @ X2 @ Xs2 ) )
=> ! [Ys2: list @ A] :
( ( A22
= ( cons @ A @ Y3 @ Ys2 ) )
=> ( ~ ( Less @ X2 @ Y3 )
=> ( ~ ( Less @ Y3 @ X2 )
=> ~ ( lexordp @ A @ Less @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ).
% ord.lexordp.cases
thf(fact_130_ord_Olexordp_ONil,axiom,
! [A: $tType,Less: A > A > $o,Y: A,Ys: list @ A] : ( lexordp @ A @ Less @ ( nil @ A ) @ ( cons @ A @ Y @ Ys ) ) ).
% ord.lexordp.Nil
thf(fact_131_ord_Orbt__greater__simps_I2_J,axiom,
! [C2: $tType,A: $tType,Less: A > A > $o,K: A,C: rBT_Im1923302023_color,Lt: rBT_Im246033960le_rbt @ A @ C2,Kt: A,V: C2,Rt: rBT_Im246033960le_rbt @ A @ C2] :
( ( rBT_Im1259024060reater @ A @ C2 @ Less @ K @ ( rBT_Im480247531Branch @ A @ C2 @ C @ Lt @ Kt @ V @ Rt ) )
= ( ( Less @ K @ Kt )
& ( rBT_Im1259024060reater @ A @ C2 @ Less @ K @ Lt )
& ( rBT_Im1259024060reater @ A @ C2 @ Less @ K @ Rt ) ) ) ).
% ord.rbt_greater_simps(2)
thf(fact_132_ord_Orbt__less__simps_I2_J,axiom,
! [C2: $tType,A: $tType,Less: A > A > $o,K: A,C: rBT_Im1923302023_color,Lt: rBT_Im246033960le_rbt @ A @ C2,Kt: A,V: C2,Rt: rBT_Im246033960le_rbt @ A @ C2] :
( ( rBT_Im2075124343t_less @ A @ C2 @ Less @ K @ ( rBT_Im480247531Branch @ A @ C2 @ C @ Lt @ Kt @ V @ Rt ) )
= ( ( Less @ Kt @ K )
& ( rBT_Im2075124343t_less @ A @ C2 @ Less @ K @ Lt )
& ( rBT_Im2075124343t_less @ A @ C2 @ Less @ K @ Rt ) ) ) ).
% ord.rbt_less_simps(2)
thf(fact_133_ord_Obalance__rbt__greater,axiom,
! [B: $tType,A: $tType,Less: A > A > $o,V: A,A2: rBT_Im246033960le_rbt @ A @ B,K: A,X: B,B2: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1259024060reater @ A @ B @ Less @ V @ ( rBT_Im1648453169alance @ A @ B @ A2 @ K @ X @ B2 ) )
= ( ( rBT_Im1259024060reater @ A @ B @ Less @ V @ A2 )
& ( rBT_Im1259024060reater @ A @ B @ Less @ V @ B2 )
& ( Less @ V @ K ) ) ) ).
% ord.balance_rbt_greater
thf(fact_134_ord_Obalance__rbt__less,axiom,
! [B: $tType,A: $tType,Less: A > A > $o,V: A,A2: rBT_Im246033960le_rbt @ A @ B,K: A,X: B,B2: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im2075124343t_less @ A @ B @ Less @ V @ ( rBT_Im1648453169alance @ A @ B @ A2 @ K @ X @ B2 ) )
= ( ( rBT_Im2075124343t_less @ A @ B @ Less @ V @ A2 )
& ( rBT_Im2075124343t_less @ A @ B @ Less @ V @ B2 )
& ( Less @ K @ V ) ) ) ).
% ord.balance_rbt_less
thf(fact_135_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_136_ord_Orbt__sorted_Osimps_I2_J,axiom,
! [B: $tType,A: $tType,Less: A > A > $o,C: rBT_Im1923302023_color,L: rBT_Im246033960le_rbt @ A @ B,K: A,V: B,R: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im759614907sorted @ A @ B @ Less @ ( rBT_Im480247531Branch @ A @ B @ C @ L @ K @ V @ R ) )
= ( ( rBT_Im2075124343t_less @ A @ B @ Less @ K @ L )
& ( rBT_Im1259024060reater @ A @ B @ Less @ K @ R )
& ( rBT_Im759614907sorted @ A @ B @ Less @ L )
& ( rBT_Im759614907sorted @ A @ B @ Less @ R ) ) ) ).
% ord.rbt_sorted.simps(2)
thf(fact_137_ord_Olexordp__irreflexive,axiom,
! [A: $tType,Less: A > A > $o,Xs: list @ A] :
( ! [X2: A] :
~ ( Less @ X2 @ X2 )
=> ~ ( lexordp @ A @ Less @ Xs @ Xs ) ) ).
% ord.lexordp_irreflexive
thf(fact_138_ord_Olexordp__eq__refl,axiom,
! [A: $tType,Less: A > A > $o,Xs: list @ A] : ( lexordp_eq @ A @ Less @ Xs @ Xs ) ).
% ord.lexordp_eq_refl
thf(fact_139_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_140_local_Orbt__insert__with__key__def,axiom,
! [B: $tType,F: a > B > B > B,K: a,V: B,T: rBT_Im246033960le_rbt @ a @ B] :
( ( rBT_Im627674146th_key @ a @ B @ less @ F @ K @ V @ T )
= ( rBT_Im945281971_paint @ a @ B @ rBT_Impl_Mirabelle_B @ ( rBT_Im1756508336bt_ins @ a @ B @ less @ F @ K @ V @ T ) ) ) ).
% local.rbt_insert_with_key_def
thf(fact_141_local_Orbt__ins_Osimps_I2_J,axiom,
! [B: $tType,K: a,X: a,F: a > B > B > B,V: B,L: rBT_Im246033960le_rbt @ a @ B,Y: B,R: rBT_Im246033960le_rbt @ a @ B] :
( ( ( less @ K @ X )
=> ( ( rBT_Im1756508336bt_ins @ a @ B @ less @ F @ K @ V @ ( rBT_Im480247531Branch @ a @ B @ rBT_Impl_Mirabelle_B @ L @ X @ Y @ R ) )
= ( rBT_Im1648453169alance @ a @ B @ ( rBT_Im1756508336bt_ins @ a @ B @ less @ F @ K @ V @ L ) @ X @ Y @ R ) ) )
& ( ~ ( less @ K @ X )
=> ( ( ( less @ X @ K )
=> ( ( rBT_Im1756508336bt_ins @ a @ B @ less @ F @ K @ V @ ( rBT_Im480247531Branch @ a @ B @ rBT_Impl_Mirabelle_B @ L @ X @ Y @ R ) )
= ( rBT_Im1648453169alance @ a @ B @ L @ X @ Y @ ( rBT_Im1756508336bt_ins @ a @ B @ less @ F @ K @ V @ R ) ) ) )
& ( ~ ( less @ X @ K )
=> ( ( rBT_Im1756508336bt_ins @ a @ B @ less @ F @ K @ V @ ( rBT_Im480247531Branch @ a @ B @ rBT_Impl_Mirabelle_B @ L @ X @ Y @ R ) )
= ( rBT_Im480247531Branch @ a @ B @ rBT_Impl_Mirabelle_B @ L @ X @ ( F @ K @ Y @ V ) @ R ) ) ) ) ) ) ).
% local.rbt_ins.simps(2)
thf(fact_142_local_Orbt__ins_Osimps_I3_J,axiom,
! [B: $tType,K: a,X: a,F: a > B > B > B,V: B,L: rBT_Im246033960le_rbt @ a @ B,Y: B,R: rBT_Im246033960le_rbt @ a @ B] :
( ( ( less @ K @ X )
=> ( ( rBT_Im1756508336bt_ins @ a @ B @ less @ F @ K @ V @ ( rBT_Im480247531Branch @ a @ B @ rBT_Impl_Mirabelle_R @ L @ X @ Y @ R ) )
= ( rBT_Im480247531Branch @ a @ B @ rBT_Impl_Mirabelle_R @ ( rBT_Im1756508336bt_ins @ a @ B @ less @ F @ K @ V @ L ) @ X @ Y @ R ) ) )
& ( ~ ( less @ K @ X )
=> ( ( ( less @ X @ K )
=> ( ( rBT_Im1756508336bt_ins @ a @ B @ less @ F @ K @ V @ ( rBT_Im480247531Branch @ a @ B @ rBT_Impl_Mirabelle_R @ L @ X @ Y @ R ) )
= ( rBT_Im480247531Branch @ a @ B @ rBT_Impl_Mirabelle_R @ L @ X @ Y @ ( rBT_Im1756508336bt_ins @ a @ B @ less @ F @ K @ V @ R ) ) ) )
& ( ~ ( less @ X @ K )
=> ( ( rBT_Im1756508336bt_ins @ a @ B @ less @ F @ K @ V @ ( rBT_Im480247531Branch @ a @ B @ rBT_Impl_Mirabelle_R @ L @ X @ Y @ R ) )
= ( rBT_Im480247531Branch @ a @ B @ rBT_Impl_Mirabelle_R @ L @ X @ ( F @ K @ Y @ V ) @ R ) ) ) ) ) ) ).
% local.rbt_ins.simps(3)
thf(fact_143_local_Olexordp__append__rightI,axiom,
! [Ys: list @ a,Xs: list @ a] :
( ( Ys
!= ( nil @ a ) )
=> ( lexordp @ a @ less @ Xs @ ( append @ a @ Xs @ Ys ) ) ) ).
% local.lexordp_append_rightI
thf(fact_144_local_Olexordp__iff,axiom,
! [Xs: list @ a,Ys: list @ a] :
( ( lexordp @ a @ less @ Xs @ Ys )
= ( ? [X3: a,Vs: list @ a] :
( Ys
= ( append @ a @ Xs @ ( cons @ a @ X3 @ Vs ) ) )
| ? [Us: list @ a,A4: a,B3: a,Vs: list @ a,Ws: list @ a] :
( ( less @ A4 @ B3 )
& ( Xs
= ( append @ a @ Us @ ( cons @ a @ A4 @ Vs ) ) )
& ( Ys
= ( append @ a @ Us @ ( cons @ a @ B3 @ Ws ) ) ) ) ) ) ).
% local.lexordp_iff
thf(fact_145_local_Olexordp__append__left__rightI,axiom,
! [X: a,Y: a,Us2: list @ a,Xs: list @ a,Ys: list @ a] :
( ( less @ X @ Y )
=> ( lexordp @ a @ less @ ( append @ a @ Us2 @ ( cons @ a @ X @ Xs ) ) @ ( append @ a @ Us2 @ ( cons @ a @ Y @ Ys ) ) ) ) ).
% local.lexordp_append_left_rightI
thf(fact_146_local_Olexordp__append__leftD,axiom,
! [Xs: list @ a,Us2: list @ a,Vs2: list @ a] :
( ( lexordp @ a @ less @ ( append @ a @ Xs @ Us2 ) @ ( append @ a @ Xs @ Vs2 ) )
=> ( ! [A3: a] :
~ ( less @ A3 @ A3 )
=> ( lexordp @ a @ less @ Us2 @ Vs2 ) ) ) ).
% local.lexordp_append_leftD
thf(fact_147_local_Olexordp__append__leftI,axiom,
! [Us2: list @ a,Vs2: list @ a,Xs: list @ a] :
( ( lexordp @ a @ less @ Us2 @ Vs2 )
=> ( lexordp @ a @ less @ ( append @ a @ Xs @ Us2 ) @ ( append @ a @ Xs @ Vs2 ) ) ) ).
% local.lexordp_append_leftI
thf(fact_148_append__assoc,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( append @ A @ ( append @ A @ Xs @ Ys ) @ Zs )
= ( append @ A @ Xs @ ( append @ A @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_149_append__same__eq,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A,Zs: list @ A] :
( ( ( append @ A @ Ys @ Xs )
= ( append @ A @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_150_same__append__eq,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_151_append__is__Nil__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= ( nil @ A ) )
= ( ( Xs
= ( nil @ A ) )
& ( Ys
= ( nil @ A ) ) ) ) ).
% append_is_Nil_conv
thf(fact_152_Nil__is__append__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( nil @ A )
= ( append @ A @ Xs @ Ys ) )
= ( ( Xs
= ( nil @ A ) )
& ( Ys
= ( nil @ A ) ) ) ) ).
% Nil_is_append_conv
thf(fact_153_self__append__conv2,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A] :
( ( Ys
= ( append @ A @ Xs @ Ys ) )
= ( Xs
= ( nil @ A ) ) ) ).
% self_append_conv2
thf(fact_154_append__self__conv2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= Ys )
= ( Xs
= ( nil @ A ) ) ) ).
% append_self_conv2
thf(fact_155_self__append__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs
= ( append @ A @ Xs @ Ys ) )
= ( Ys
= ( nil @ A ) ) ) ).
% self_append_conv
thf(fact_156_append__self__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= Xs )
= ( Ys
= ( nil @ A ) ) ) ).
% append_self_conv
thf(fact_157_append__Nil2,axiom,
! [A: $tType,Xs: list @ A] :
( ( append @ A @ Xs @ ( nil @ A ) )
= Xs ) ).
% append_Nil2
thf(fact_158_append1__eq__conv,axiom,
! [A: $tType,Xs: list @ A,X: A,Ys: list @ A,Y: A] :
( ( ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) )
= ( append @ A @ Ys @ ( cons @ A @ Y @ ( nil @ A ) ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_159_append__eq__appendI,axiom,
! [A: $tType,Xs: list @ A,Xs1: list @ A,Zs: list @ A,Ys: list @ A,Us2: list @ A] :
( ( ( append @ A @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append @ A @ Xs1 @ Us2 ) )
=> ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Zs @ Us2 ) ) ) ) ).
% append_eq_appendI
thf(fact_160_append__eq__append__conv2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A,Ts: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Zs @ Ts ) )
= ( ? [Us: list @ A] :
( ( ( Xs
= ( append @ A @ Zs @ Us ) )
& ( ( append @ A @ Us @ Ys )
= Ts ) )
| ( ( ( append @ A @ Xs @ Us )
= Zs )
& ( Ys
= ( append @ A @ Us @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_161_color_Oexhaust,axiom,
! [Y: rBT_Im1923302023_color] :
( ( Y != rBT_Impl_Mirabelle_R )
=> ( Y = rBT_Impl_Mirabelle_B ) ) ).
% color.exhaust
thf(fact_162_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_163_color_Odistinct_I1_J,axiom,
rBT_Impl_Mirabelle_R != rBT_Impl_Mirabelle_B ).
% color.distinct(1)
thf(fact_164_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_165_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,Va2: 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 @ Va2 @ 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 @ Va2 @ Vb @ Vc @ Vd ) ) ) ) ).
% balance.simps(3)
thf(fact_166_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,Va2: 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 @ Va2 @ 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 @ Va2 @ Vb @ Vc @ Vd ) ) ) ) ).
% balance.simps(7)
thf(fact_167_balance_Osimps_I9_J,axiom,
! [B: $tType,A: $tType,Va2: 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 @ Va2 @ 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 @ Va2 @ 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_168_balance_Osimps_I13_J,axiom,
! [B: $tType,A: $tType,Va2: 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 @ Va2 @ 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 @ Va2 @ 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_169_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_170_combine_Osimps_I6_J,axiom,
! [B: $tType,A: $tType,A2: rBT_Im246033960le_rbt @ A @ B,K: A,X: B,B2: rBT_Im246033960le_rbt @ A @ B,Va2: rBT_Im246033960le_rbt @ A @ B,Vb: A,Vc: B,Vd: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1591839508ombine @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ A2 @ K @ X @ B2 ) @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va2 @ Vb @ Vc @ Vd ) )
= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ A2 @ K @ X @ ( rBT_Im1591839508ombine @ A @ B @ B2 @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va2 @ Vb @ Vc @ Vd ) ) ) ) ).
% combine.simps(6)
thf(fact_171_combine_Osimps_I5_J,axiom,
! [B: $tType,A: $tType,Va2: rBT_Im246033960le_rbt @ A @ B,Vb: A,Vc: B,Vd: rBT_Im246033960le_rbt @ A @ B,B2: rBT_Im246033960le_rbt @ A @ B,K: A,X: B,C: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1591839508ombine @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va2 @ Vb @ Vc @ Vd ) @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ B2 @ K @ X @ C ) )
= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ ( rBT_Im1591839508ombine @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va2 @ Vb @ Vc @ Vd ) @ B2 ) @ K @ X @ C ) ) ).
% combine.simps(5)
thf(fact_172_balance__right_Osimps_I1_J,axiom,
! [B: $tType,A: $tType,A2: rBT_Im246033960le_rbt @ A @ B,K: A,X: B,B2: rBT_Im246033960le_rbt @ A @ B,S: A,Y: B,C: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1053358987_right @ A @ B @ A2 @ K @ X @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ B2 @ S @ Y @ C ) )
= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ A2 @ K @ X @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ B2 @ S @ Y @ C ) ) ) ).
% balance_right.simps(1)
thf(fact_173_append__Cons,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A] :
( ( append @ A @ ( cons @ A @ X @ Xs ) @ Ys )
= ( cons @ A @ X @ ( append @ A @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_174_Cons__eq__appendI,axiom,
! [A: $tType,X: A,Xs1: list @ A,Ys: list @ A,Xs: list @ A,Zs: list @ A] :
( ( ( cons @ A @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append @ A @ Xs1 @ Zs ) )
=> ( ( cons @ A @ X @ Xs )
= ( append @ A @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_175_append__Nil,axiom,
! [A: $tType,Ys: list @ A] :
( ( append @ A @ ( nil @ A ) @ Ys )
= Ys ) ).
% append_Nil
thf(fact_176_eq__Nil__appendI,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs = Ys )
=> ( Xs
= ( append @ A @ ( nil @ A ) @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_177_ord_Olexordp__append__leftD,axiom,
! [A: $tType,Less: A > A > $o,Xs: list @ A,Us2: list @ A,Vs2: list @ A] :
( ( lexordp @ A @ Less @ ( append @ A @ Xs @ Us2 ) @ ( append @ A @ Xs @ Vs2 ) )
=> ( ! [A3: A] :
~ ( Less @ A3 @ A3 )
=> ( lexordp @ A @ Less @ Us2 @ Vs2 ) ) ) ).
% ord.lexordp_append_leftD
thf(fact_178_ord_Olexordp__append__leftI,axiom,
! [A: $tType,Less: A > A > $o,Us2: list @ A,Vs2: list @ A,Xs: list @ A] :
( ( lexordp @ A @ Less @ Us2 @ Vs2 )
=> ( lexordp @ A @ Less @ ( append @ A @ Xs @ Us2 ) @ ( append @ A @ Xs @ Vs2 ) ) ) ).
% ord.lexordp_append_leftI
thf(fact_179_balance__left_Osimps_I3_J,axiom,
! [B: $tType,A: $tType,Va2: 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 @ Va2 @ 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 @ Va2 @ Vb @ Vc @ Vd ) @ K @ X @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ A2 @ S @ Y @ B2 ) ) ) ).
% balance_left.simps(3)
thf(fact_180_balance__right_Osimps_I3_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,Va2: rBT_Im246033960le_rbt @ A @ B,Vb: A,Vc: B,Vd: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1053358987_right @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ A2 @ K @ X @ B2 ) @ S @ Y @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va2 @ Vb @ Vc @ Vd ) )
= ( rBT_Im1648453169alance @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ A2 @ K @ X @ B2 ) @ S @ Y @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va2 @ Vb @ Vc @ Vd ) ) ) ).
% balance_right.simps(3)
thf(fact_181_rev__nonempty__induct,axiom,
! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X2: A] : ( P @ ( cons @ A @ X2 @ ( nil @ A ) ) )
=> ( ! [X2: A,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( P @ Xs2 )
=> ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_182_append__eq__Cons__conv,axiom,
! [A: $tType,Ys: list @ A,Zs: list @ A,X: A,Xs: list @ A] :
( ( ( append @ A @ Ys @ Zs )
= ( cons @ A @ X @ Xs ) )
= ( ( ( Ys
= ( nil @ A ) )
& ( Zs
= ( cons @ A @ X @ Xs ) ) )
| ? [Ys5: list @ A] :
( ( Ys
= ( cons @ A @ X @ Ys5 ) )
& ( ( append @ A @ Ys5 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_183_Cons__eq__append__conv,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( cons @ A @ X @ Xs )
= ( append @ A @ Ys @ Zs ) )
= ( ( ( Ys
= ( nil @ A ) )
& ( ( cons @ A @ X @ Xs )
= Zs ) )
| ? [Ys5: list @ A] :
( ( ( cons @ A @ X @ Ys5 )
= Ys )
& ( Xs
= ( append @ A @ Ys5 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_184_rev__exhaust,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ~ ! [Ys2: list @ A,Y3: A] :
( Xs
!= ( append @ A @ Ys2 @ ( cons @ A @ Y3 @ ( nil @ A ) ) ) ) ) ).
% rev_exhaust
thf(fact_185_rev__induct,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X2: A,Xs2: list @ A] :
( ( P @ Xs2 )
=> ( P @ ( append @ A @ Xs2 @ ( cons @ A @ X2 @ ( nil @ A ) ) ) ) )
=> ( P @ Xs ) ) ) ).
% rev_induct
thf(fact_186_ord_Olexordp__append__left__rightI,axiom,
! [A: $tType,Less: A > A > $o,X: A,Y: A,Us2: list @ A,Xs: list @ A,Ys: list @ A] :
( ( Less @ X @ Y )
=> ( lexordp @ A @ Less @ ( append @ A @ Us2 @ ( cons @ A @ X @ Xs ) ) @ ( append @ A @ Us2 @ ( cons @ A @ Y @ Ys ) ) ) ) ).
% ord.lexordp_append_left_rightI
thf(fact_187_ord_Olexordp__append__rightI,axiom,
! [A: $tType,Ys: list @ A,Less: A > A > $o,Xs: list @ A] :
( ( Ys
!= ( nil @ A ) )
=> ( lexordp @ A @ Less @ Xs @ ( append @ A @ Xs @ Ys ) ) ) ).
% ord.lexordp_append_rightI
thf(fact_188_balance__left_Osimps_I5_J,axiom,
! [B: $tType,A: $tType,Va2: 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 @ Va2 @ 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 @ Va2 @ 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_189_balance__right_Osimps_I5_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,T: A,Z: B,Va2: rBT_Im246033960le_rbt @ A @ B,Vb: A,Vc: B,Vd: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1053358987_right @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ A2 @ K @ X @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ B2 @ S @ Y @ C ) ) @ T @ Z @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va2 @ Vb @ Vc @ Vd ) )
= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ ( rBT_Im1648453169alance @ A @ B @ ( rBT_Im945281971_paint @ A @ B @ rBT_Impl_Mirabelle_R @ A2 ) @ K @ X @ B2 ) @ S @ Y @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ C @ T @ Z @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va2 @ Vb @ Vc @ Vd ) ) ) ) ).
% balance_right.simps(5)
thf(fact_190_balance_Osimps_I23_J,axiom,
! [B: $tType,A: $tType,Va2: 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 @ Va2 @ 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 @ Va2 @ Vb @ Vc @ Vd ) @ S @ T @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Ve @ Vf @ Vg @ Vh ) ) ) ).
% balance.simps(23)
thf(fact_191_balance_Osimps_I27_J,axiom,
! [B: $tType,A: $tType,Va2: 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 @ Va2 @ 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 @ Va2 @ 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_192_balance_Osimps_I29_J,axiom,
! [B: $tType,A: $tType,Va2: 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 @ Va2 @ 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 @ Va2 @ 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_193_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,Va2: 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 @ Va2 @ 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 @ Va2 @ Vb @ Vc @ Vd ) ) ) ).
% balance.simps(39)
thf(fact_194_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,Va2: 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 @ Va2 @ 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 @ Va2 @ Vb @ Vc @ Vd ) ) ) ).
% balance.simps(43)
thf(fact_195_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,Va2: 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 @ Va2 @ 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 @ Va2 @ Vb @ Vc @ Vd ) ) ) ).
% balance.simps(45)
thf(fact_196_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_197_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_198_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,V3: B,T2: rBT_Im246033960le_rbt @ A @ B] : ( rBT_Im945281971_paint @ A @ B @ rBT_Impl_Mirabelle_B @ ( rBT_Im1756508336bt_ins @ A @ B @ Less2 @ F3 @ K2 @ V3 @ T2 ) ) ) ) ).
% ord.rbt_insert_with_key_def
thf(fact_199_local_Orbt__del__from__left_Osimps_I1_J,axiom,
! [B: $tType,X: a,Lt: rBT_Im246033960le_rbt @ a @ B,Z: a,V: B,Rt: rBT_Im246033960le_rbt @ a @ B,Y: a,S: B,B2: rBT_Im246033960le_rbt @ a @ B] :
( ( rBT_Im1152561598m_left @ a @ B @ less @ X @ ( rBT_Im480247531Branch @ a @ B @ rBT_Impl_Mirabelle_B @ Lt @ Z @ V @ Rt ) @ Y @ S @ B2 )
= ( rBT_Im1278323832e_left @ a @ B @ ( rBT_Im1714610413bt_del @ a @ B @ less @ X @ ( rBT_Im480247531Branch @ a @ B @ rBT_Impl_Mirabelle_B @ Lt @ Z @ V @ Rt ) ) @ Y @ S @ B2 ) ) ).
% local.rbt_del_from_left.simps(1)
thf(fact_200_local_Orbt__del__from__left_Osimps_I3_J,axiom,
! [B: $tType,X: a,Va2: rBT_Im246033960le_rbt @ a @ B,Vb: a,Vc: B,Vd: rBT_Im246033960le_rbt @ a @ B,Y: a,S: B,B2: rBT_Im246033960le_rbt @ a @ B] :
( ( rBT_Im1152561598m_left @ a @ B @ less @ X @ ( rBT_Im480247531Branch @ a @ B @ rBT_Impl_Mirabelle_R @ Va2 @ Vb @ Vc @ Vd ) @ Y @ S @ B2 )
= ( rBT_Im480247531Branch @ a @ B @ rBT_Impl_Mirabelle_R @ ( rBT_Im1714610413bt_del @ a @ B @ less @ X @ ( rBT_Im480247531Branch @ a @ B @ rBT_Impl_Mirabelle_R @ Va2 @ Vb @ Vc @ Vd ) ) @ Y @ S @ B2 ) ) ).
% local.rbt_del_from_left.simps(3)
thf(fact_201_local_Orbt__ins_Osimps_I1_J,axiom,
! [B: $tType,F: a > B > B > B,K: a,V: B] :
( ( rBT_Im1756508336bt_ins @ a @ B @ less @ F @ K @ V @ ( rBT_Im418718756_Empty @ a @ B ) )
= ( rBT_Im480247531Branch @ a @ B @ rBT_Impl_Mirabelle_R @ ( rBT_Im418718756_Empty @ a @ B ) @ K @ V @ ( rBT_Im418718756_Empty @ a @ B ) ) ) ).
% local.rbt_ins.simps(1)
thf(fact_202_local_Orbt__sorted_Osimps_I1_J,axiom,
! [B: $tType] : ( rBT_Im759614907sorted @ a @ B @ less @ ( rBT_Im418718756_Empty @ a @ B ) ) ).
% local.rbt_sorted.simps(1)
thf(fact_203_local_Orbt__del_Osimps_I1_J,axiom,
! [B: $tType,X: a] :
( ( rBT_Im1714610413bt_del @ a @ B @ less @ X @ ( rBT_Im418718756_Empty @ a @ B ) )
= ( rBT_Im418718756_Empty @ a @ B ) ) ).
% local.rbt_del.simps(1)
thf(fact_204_local_OEmpty__is__rbt,axiom,
! [B: $tType] : ( rBT_Im862805236is_rbt @ a @ B @ less @ ( rBT_Im418718756_Empty @ a @ B ) ) ).
% local.Empty_is_rbt
thf(fact_205_local_Orbt__ins_Oinduct,axiom,
! [B: $tType,P: ( a > B > B > B ) > a > B > ( rBT_Im246033960le_rbt @ a @ B ) > $o,A0: a > B > B > B,A1: a,A22: B,A32: rBT_Im246033960le_rbt @ a @ B] :
( ! [F2: a > B > B > B,K3: a,V2: B] : ( P @ F2 @ K3 @ V2 @ ( rBT_Im418718756_Empty @ a @ B ) )
=> ( ! [F2: a > B > B > B,K3: a,V2: B,L2: rBT_Im246033960le_rbt @ a @ B,X2: a,Y3: B,R2: rBT_Im246033960le_rbt @ a @ B] :
( ( ( less @ K3 @ X2 )
=> ( P @ F2 @ K3 @ V2 @ L2 ) )
=> ( ( ~ ( less @ K3 @ X2 )
=> ( ( less @ X2 @ K3 )
=> ( P @ F2 @ K3 @ V2 @ R2 ) ) )
=> ( P @ F2 @ K3 @ V2 @ ( rBT_Im480247531Branch @ a @ B @ rBT_Impl_Mirabelle_B @ L2 @ X2 @ Y3 @ R2 ) ) ) )
=> ( ! [F2: a > B > B > B,K3: a,V2: B,L2: rBT_Im246033960le_rbt @ a @ B,X2: a,Y3: B,R2: rBT_Im246033960le_rbt @ a @ B] :
( ( ( less @ K3 @ X2 )
=> ( P @ F2 @ K3 @ V2 @ L2 ) )
=> ( ( ~ ( less @ K3 @ X2 )
=> ( ( less @ X2 @ K3 )
=> ( P @ F2 @ K3 @ V2 @ R2 ) ) )
=> ( P @ F2 @ K3 @ V2 @ ( rBT_Im480247531Branch @ a @ B @ rBT_Impl_Mirabelle_R @ L2 @ X2 @ Y3 @ R2 ) ) ) )
=> ( P @ A0 @ A1 @ A22 @ A32 ) ) ) ) ).
% local.rbt_ins.induct
thf(fact_206_local_Orbt__del__from__left_Osimps_I2_J,axiom,
! [B: $tType,X: a,Y: a,S: B,B2: rBT_Im246033960le_rbt @ a @ B] :
( ( rBT_Im1152561598m_left @ a @ B @ less @ X @ ( rBT_Im418718756_Empty @ a @ B ) @ Y @ S @ B2 )
= ( rBT_Im480247531Branch @ a @ B @ rBT_Impl_Mirabelle_R @ ( rBT_Im1714610413bt_del @ a @ B @ less @ X @ ( rBT_Im418718756_Empty @ a @ B ) ) @ Y @ S @ B2 ) ) ).
% local.rbt_del_from_left.simps(2)
thf(fact_207_local_Orbt__greater__simps_I1_J,axiom,
! [B: $tType,K: a] : ( rBT_Im1259024060reater @ a @ B @ less @ K @ ( rBT_Im418718756_Empty @ a @ B ) ) ).
% local.rbt_greater_simps(1)
thf(fact_208_local_Orbt__less__simps_I1_J,axiom,
! [B: $tType,K: a] : ( rBT_Im2075124343t_less @ a @ B @ less @ K @ ( rBT_Im418718756_Empty @ a @ B ) ) ).
% local.rbt_less_simps(1)
thf(fact_209_local_Orbt__del__from__right_Osimps_I3_J,axiom,
! [B: $tType,X: a,A2: rBT_Im246033960le_rbt @ a @ B,Y: a,S: B,Va2: rBT_Im246033960le_rbt @ a @ B,Vb: a,Vc: B,Vd: rBT_Im246033960le_rbt @ a @ B] :
( ( rBT_Im1808746437_right @ a @ B @ less @ X @ A2 @ Y @ S @ ( rBT_Im480247531Branch @ a @ B @ rBT_Impl_Mirabelle_R @ Va2 @ Vb @ Vc @ Vd ) )
= ( rBT_Im480247531Branch @ a @ B @ rBT_Impl_Mirabelle_R @ A2 @ Y @ S @ ( rBT_Im1714610413bt_del @ a @ B @ less @ X @ ( rBT_Im480247531Branch @ a @ B @ rBT_Impl_Mirabelle_R @ Va2 @ Vb @ Vc @ Vd ) ) ) ) ).
% local.rbt_del_from_right.simps(3)
thf(fact_210_local_Orbt__del__from__right_Osimps_I2_J,axiom,
! [B: $tType,X: a,A2: rBT_Im246033960le_rbt @ a @ B,Y: a,S: B] :
( ( rBT_Im1808746437_right @ a @ B @ less @ X @ A2 @ Y @ S @ ( rBT_Im418718756_Empty @ a @ B ) )
= ( rBT_Im480247531Branch @ a @ B @ rBT_Impl_Mirabelle_R @ A2 @ Y @ S @ ( rBT_Im1714610413bt_del @ a @ B @ less @ X @ ( rBT_Im418718756_Empty @ a @ B ) ) ) ) ).
% local.rbt_del_from_right.simps(2)
thf(fact_211_local_Orbt__del__from__right_Osimps_I1_J,axiom,
! [B: $tType,X: a,A2: rBT_Im246033960le_rbt @ a @ B,Y: a,S: B,Lt: rBT_Im246033960le_rbt @ a @ B,Z: a,V: B,Rt: rBT_Im246033960le_rbt @ a @ B] :
( ( rBT_Im1808746437_right @ a @ B @ less @ X @ A2 @ Y @ S @ ( rBT_Im480247531Branch @ a @ B @ rBT_Impl_Mirabelle_B @ Lt @ Z @ V @ Rt ) )
= ( rBT_Im1053358987_right @ a @ B @ A2 @ Y @ S @ ( rBT_Im1714610413bt_del @ a @ B @ less @ X @ ( rBT_Im480247531Branch @ a @ B @ rBT_Impl_Mirabelle_B @ Lt @ Z @ V @ Rt ) ) ) ) ).
% local.rbt_del_from_right.simps(1)
thf(fact_212_local_Orbt__del_Osimps_I2_J,axiom,
! [B: $tType,X: a,Y: a,C: rBT_Im1923302023_color,A2: rBT_Im246033960le_rbt @ a @ B,S: B,B2: rBT_Im246033960le_rbt @ a @ B] :
( ( ( less @ X @ Y )
=> ( ( rBT_Im1714610413bt_del @ a @ B @ less @ X @ ( rBT_Im480247531Branch @ a @ B @ C @ A2 @ Y @ S @ B2 ) )
= ( rBT_Im1152561598m_left @ a @ B @ less @ X @ A2 @ Y @ S @ B2 ) ) )
& ( ~ ( less @ X @ Y )
=> ( ( ( less @ Y @ X )
=> ( ( rBT_Im1714610413bt_del @ a @ B @ less @ X @ ( rBT_Im480247531Branch @ a @ B @ C @ A2 @ Y @ S @ B2 ) )
= ( rBT_Im1808746437_right @ a @ B @ less @ X @ A2 @ Y @ S @ B2 ) ) )
& ( ~ ( less @ Y @ X )
=> ( ( rBT_Im1714610413bt_del @ a @ B @ less @ X @ ( rBT_Im480247531Branch @ a @ B @ C @ A2 @ Y @ S @ B2 ) )
= ( rBT_Im1591839508ombine @ a @ B @ A2 @ B2 ) ) ) ) ) ) ).
% local.rbt_del.simps(2)
thf(fact_213_local_Orbt__del_Oelims,axiom,
! [B: $tType,X: a,Xa: rBT_Im246033960le_rbt @ a @ B,Y: rBT_Im246033960le_rbt @ a @ B] :
( ( ( rBT_Im1714610413bt_del @ a @ B @ less @ X @ Xa )
= Y )
=> ( ( ( Xa
= ( rBT_Im418718756_Empty @ a @ B ) )
=> ( Y
!= ( rBT_Im418718756_Empty @ a @ B ) ) )
=> ~ ! [C3: rBT_Im1923302023_color,A3: rBT_Im246033960le_rbt @ a @ B,Y3: a,S2: B,B4: rBT_Im246033960le_rbt @ a @ B] :
( ( Xa
= ( rBT_Im480247531Branch @ a @ B @ C3 @ A3 @ Y3 @ S2 @ B4 ) )
=> ~ ( ( ( less @ X @ Y3 )
=> ( Y
= ( rBT_Im1152561598m_left @ a @ B @ less @ X @ A3 @ Y3 @ S2 @ B4 ) ) )
& ( ~ ( less @ X @ Y3 )
=> ( ( ( less @ Y3 @ X )
=> ( Y
= ( rBT_Im1808746437_right @ a @ B @ less @ X @ A3 @ Y3 @ S2 @ B4 ) ) )
& ( ~ ( less @ Y3 @ X )
=> ( Y
= ( rBT_Im1591839508ombine @ a @ B @ A3 @ B4 ) ) ) ) ) ) ) ) ) ).
% local.rbt_del.elims
thf(fact_214_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_215_rbt_Oinduct,axiom,
! [B: $tType,A: $tType,P: ( rBT_Im246033960le_rbt @ A @ B ) > $o,Rbt: rBT_Im246033960le_rbt @ A @ B] :
( ( P @ ( rBT_Im418718756_Empty @ A @ B ) )
=> ( ! [X12: rBT_Im1923302023_color,X27: rBT_Im246033960le_rbt @ A @ B,X32: A,X4: B,X5: rBT_Im246033960le_rbt @ A @ B] :
( ( P @ X27 )
=> ( ( P @ X5 )
=> ( P @ ( rBT_Im480247531Branch @ A @ B @ X12 @ X27 @ X32 @ X4 @ X5 ) ) ) )
=> ( P @ Rbt ) ) ) ).
% rbt.induct
thf(fact_216_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_217_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_218_ord_Orbt__less__simps_I1_J,axiom,
! [B: $tType,A: $tType,Less: A > A > $o,K: A] : ( rBT_Im2075124343t_less @ A @ B @ Less @ K @ ( rBT_Im418718756_Empty @ A @ B ) ) ).
% ord.rbt_less_simps(1)
thf(fact_219_ord_Orbt__greater__simps_I1_J,axiom,
! [B: $tType,A: $tType,Less: A > A > $o,K: A] : ( rBT_Im1259024060reater @ A @ B @ Less @ K @ ( rBT_Im418718756_Empty @ A @ B ) ) ).
% ord.rbt_greater_simps(1)
thf(fact_220_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_221_combine_Osimps_I1_J,axiom,
! [B: $tType,A: $tType,X: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1591839508ombine @ A @ B @ ( rBT_Im418718756_Empty @ A @ B ) @ X )
= X ) ).
% combine.simps(1)
thf(fact_222_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_223_balance__right_Osimps_I6_J,axiom,
! [B: $tType,A: $tType,K: A,X: B] :
( ( rBT_Im1053358987_right @ A @ B @ ( rBT_Im418718756_Empty @ A @ B ) @ K @ X @ ( rBT_Im418718756_Empty @ A @ B ) )
= ( rBT_Im418718756_Empty @ A @ B ) ) ).
% balance_right.simps(6)
thf(fact_224_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_225_ord_Orbt__del_Osimps_I1_J,axiom,
! [B: $tType,A: $tType,Less: A > A > $o,X: A] :
( ( rBT_Im1714610413bt_del @ A @ B @ Less @ X @ ( rBT_Im418718756_Empty @ A @ B ) )
= ( rBT_Im418718756_Empty @ A @ B ) ) ).
% ord.rbt_del.simps(1)
thf(fact_226_ord_Orbt__del__from__left_Osimps_I2_J,axiom,
! [B: $tType,A: $tType,Less: A > A > $o,X: A,Y: A,S: B,B2: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1152561598m_left @ A @ B @ Less @ X @ ( rBT_Im418718756_Empty @ A @ B ) @ Y @ S @ B2 )
= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ ( rBT_Im1714610413bt_del @ A @ B @ Less @ X @ ( rBT_Im418718756_Empty @ A @ B ) ) @ Y @ S @ B2 ) ) ).
% ord.rbt_del_from_left.simps(2)
thf(fact_227_combine_Osimps_I2_J,axiom,
! [B: $tType,A: $tType,V: rBT_Im1923302023_color,Va2: rBT_Im246033960le_rbt @ A @ B,Vb: A,Vc: B,Vd: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1591839508ombine @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ V @ Va2 @ Vb @ Vc @ Vd ) @ ( rBT_Im418718756_Empty @ A @ B ) )
= ( rBT_Im480247531Branch @ A @ B @ V @ Va2 @ Vb @ Vc @ Vd ) ) ).
% combine.simps(2)
thf(fact_228_ord_Orbt__ins_Oinduct,axiom,
! [B: $tType,A: $tType,P: ( A > B > B > B ) > A > B > ( rBT_Im246033960le_rbt @ A @ B ) > $o,Less: A > A > $o,A0: A > B > B > B,A1: A,A22: B,A32: rBT_Im246033960le_rbt @ A @ B] :
( ! [F2: A > B > B > B,K3: A,V2: B] : ( P @ F2 @ K3 @ V2 @ ( rBT_Im418718756_Empty @ A @ B ) )
=> ( ! [F2: A > B > B > B,K3: A,V2: B,L2: rBT_Im246033960le_rbt @ A @ B,X2: A,Y3: B,R2: rBT_Im246033960le_rbt @ A @ B] :
( ( ( Less @ K3 @ X2 )
=> ( P @ F2 @ K3 @ V2 @ L2 ) )
=> ( ( ~ ( Less @ K3 @ X2 )
=> ( ( Less @ X2 @ K3 )
=> ( P @ F2 @ K3 @ V2 @ R2 ) ) )
=> ( P @ F2 @ K3 @ V2 @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ L2 @ X2 @ Y3 @ R2 ) ) ) )
=> ( ! [F2: A > B > B > B,K3: A,V2: B,L2: rBT_Im246033960le_rbt @ A @ B,X2: A,Y3: B,R2: rBT_Im246033960le_rbt @ A @ B] :
( ( ( Less @ K3 @ X2 )
=> ( P @ F2 @ K3 @ V2 @ L2 ) )
=> ( ( ~ ( Less @ K3 @ X2 )
=> ( ( Less @ X2 @ K3 )
=> ( P @ F2 @ K3 @ V2 @ R2 ) ) )
=> ( P @ F2 @ K3 @ V2 @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ L2 @ X2 @ Y3 @ R2 ) ) ) )
=> ( P @ A0 @ A1 @ A22 @ A32 ) ) ) ) ).
% ord.rbt_ins.induct
thf(fact_229_combine_Oinduct,axiom,
! [B: $tType,A: $tType,P: ( rBT_Im246033960le_rbt @ A @ B ) > ( rBT_Im246033960le_rbt @ A @ B ) > $o,A0: rBT_Im246033960le_rbt @ A @ B,A1: rBT_Im246033960le_rbt @ A @ B] :
( ! [X12: rBT_Im246033960le_rbt @ A @ B] : ( P @ ( rBT_Im418718756_Empty @ A @ B ) @ X12 )
=> ( ! [V2: rBT_Im1923302023_color,Va: rBT_Im246033960le_rbt @ A @ B,Vb2: A,Vc2: B,Vd2: rBT_Im246033960le_rbt @ A @ B] : ( P @ ( rBT_Im480247531Branch @ A @ B @ V2 @ Va @ Vb2 @ Vc2 @ Vd2 ) @ ( rBT_Im418718756_Empty @ A @ B ) )
=> ( ! [A3: rBT_Im246033960le_rbt @ A @ B,K3: A,X2: B,B4: rBT_Im246033960le_rbt @ A @ B,C3: rBT_Im246033960le_rbt @ A @ B,S2: A,Y3: B,D2: rBT_Im246033960le_rbt @ A @ B] :
( ( P @ B4 @ C3 )
=> ( P @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ A3 @ K3 @ X2 @ B4 ) @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ C3 @ S2 @ Y3 @ D2 ) ) )
=> ( ! [A3: rBT_Im246033960le_rbt @ A @ B,K3: A,X2: B,B4: rBT_Im246033960le_rbt @ A @ B,C3: rBT_Im246033960le_rbt @ A @ B,S2: A,Y3: B,D2: rBT_Im246033960le_rbt @ A @ B] :
( ( P @ B4 @ C3 )
=> ( P @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ A3 @ K3 @ X2 @ B4 ) @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ C3 @ S2 @ Y3 @ D2 ) ) )
=> ( ! [Va: rBT_Im246033960le_rbt @ A @ B,Vb2: A,Vc2: B,Vd2: rBT_Im246033960le_rbt @ A @ B,B4: rBT_Im246033960le_rbt @ A @ B,K3: A,X2: B,C3: rBT_Im246033960le_rbt @ A @ B] :
( ( P @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va @ Vb2 @ Vc2 @ Vd2 ) @ B4 )
=> ( P @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va @ Vb2 @ Vc2 @ Vd2 ) @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ B4 @ K3 @ X2 @ C3 ) ) )
=> ( ! [A3: rBT_Im246033960le_rbt @ A @ B,K3: A,X2: B,B4: rBT_Im246033960le_rbt @ A @ B,Va: rBT_Im246033960le_rbt @ A @ B,Vb2: A,Vc2: B,Vd2: rBT_Im246033960le_rbt @ A @ B] :
( ( P @ B4 @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va @ Vb2 @ Vc2 @ Vd2 ) )
=> ( P @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ A3 @ K3 @ X2 @ B4 ) @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Va @ Vb2 @ Vc2 @ Vd2 ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ) ) ) ).
% combine.induct
thf(fact_230_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,V2: B,R2: rBT_Im246033960le_rbt @ A @ B] :
( T
!= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ L2 @ K3 @ V2 @ R2 ) )
=> ~ ! [L2: rBT_Im246033960le_rbt @ A @ B,K3: A,V2: B,R2: rBT_Im246033960le_rbt @ A @ B] :
( T
!= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ L2 @ K3 @ V2 @ R2 ) ) ) ) ).
% rbt_cases
thf(fact_231_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,Va2: 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 @ Va2 @ 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 @ Va2 @ Vb @ Vc @ Vd ) ) ) ).
% balance.simps(44)
thf(fact_232_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,Va2: 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 @ Va2 @ 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 @ Va2 @ Vb @ Vc @ Vd ) ) ) ).
% balance.simps(42)
thf(fact_233_balance_Osimps_I41_J,axiom,
! [B: $tType,A: $tType,V: rBT_Im1923302023_color,Vf: A,Vg: B,S: A,T: B,Va2: 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 @ Va2 @ 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 @ Va2 @ Vb @ Vc @ Vd ) ) ) ).
% balance.simps(41)
thf(fact_234_balance_Osimps_I40_J,axiom,
! [B: $tType,A: $tType,Vf: A,Vg: B,Vh: rBT_Im246033960le_rbt @ A @ B,S: A,T: B,Va2: 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 @ Va2 @ 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 @ Va2 @ Vb @ Vc @ Vd ) ) ) ).
% balance.simps(40)
thf(fact_235_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_236_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_237_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_238_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_239_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_240_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_241_balance_Osimps_I28_J,axiom,
! [B: $tType,A: $tType,Va2: 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 @ Va2 @ 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 @ Va2 @ 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_242_balance_Osimps_I26_J,axiom,
! [B: $tType,A: $tType,Va2: 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 @ Va2 @ 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 @ Va2 @ 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_243_balance_Osimps_I25_J,axiom,
! [B: $tType,A: $tType,Va2: 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 @ Va2 @ 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 @ Va2 @ 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_244_balance_Osimps_I24_J,axiom,
! [B: $tType,A: $tType,Va2: 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 @ Va2 @ 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 @ Va2 @ 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_245_balance_Osimps_I22_J,axiom,
! [B: $tType,A: $tType,Va2: 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 @ Va2 @ 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 @ Va2 @ Vb @ Vc @ Vd ) @ S @ T @ ( rBT_Im418718756_Empty @ A @ B ) ) ) ).
% balance.simps(22)
thf(fact_246_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_247_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_248_balance_Osimps_I19_J,axiom,
! [B: $tType,A: $tType,S: A,T: B,Va2: 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 @ Va2 @ 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 @ Va2 @ Vb @ Vc @ ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ Vf @ Vg @ Vh @ Vi ) ) ) ) ).
% balance.simps(19)
thf(fact_249_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_250_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_251_balance_Osimps_I16_J,axiom,
! [B: $tType,A: $tType,S: A,T: B,Va2: 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 @ Va2 @ 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 @ Va2 @ Vb @ Vc @ ( rBT_Im418718756_Empty @ A @ B ) ) ) ) ).
% balance.simps(16)
thf(fact_252_balance_Osimps_I15_J,axiom,
! [B: $tType,A: $tType,S: A,T: B,Va2: 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 @ Va2 @ 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 @ Va2 @ Vb @ Vc @ Vd ) ) ) ).
% balance.simps(15)
thf(fact_253_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_254_balance__left_Osimps_I9_J,axiom,
! [B: $tType,A: $tType,Va2: 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 @ Va2 @ Vb @ Vc @ Vd ) @ K @ X @ ( rBT_Im418718756_Empty @ A @ B ) )
= ( rBT_Im418718756_Empty @ A @ B ) ) ).
% balance_left.simps(9)
%----Type constructors (2)
thf(tcon_fun___Orderings_Oorder,axiom,
! [A5: $tType,A6: $tType] :
( ( order @ A6 @ ( type2 @ A6 ) )
=> ( order @ ( A5 > A6 ) @ ( type2 @ ( A5 > A6 ) ) ) ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_1,axiom,
order @ $o @ ( type2 @ $o ) ).
%----Conjectures (4)
thf(conj_0,hypothesis,
rBT_Im2075124343t_less @ a @ b @ less @ v @ lt ).
thf(conj_1,hypothesis,
rBT_Im2075124343t_less @ a @ b @ less @ v @ rt ).
thf(conj_2,hypothesis,
less @ k @ v ).
thf(conj_3,conjecture,
rBT_Im2075124343t_less @ a @ b @ less @ v @ ( rBT_Im1152561598m_left @ a @ b @ less @ x @ lt @ k @ y @ rt ) ).
%------------------------------------------------------------------------------