TPTP Problem File: DAT231^1.p
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%------------------------------------------------------------------------------
% File : DAT231^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Red-black trees 312
% 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__312.p [Bla16]
% Status : Theorem
% Rating : 0.33 v8.1.0, 0.25 v7.5.0, 0.67 v7.2.0, 0.75 v7.1.0
% Syntax : Number of formulae : 368 ( 76 unt; 79 typ; 0 def)
% Number of atoms : 940 ( 215 equ; 0 cnn)
% Maximal formula atoms : 9 ( 3 avg)
% Number of connectives : 4378 ( 85 ~; 19 |; 66 &;3734 @)
% ( 0 <=>; 474 =>; 0 <=; 0 <~>)
% Maximal formula depth : 31 ( 9 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 305 ( 305 >; 0 *; 0 +; 0 <<)
% Number of symbols : 78 ( 75 usr; 5 con; 0-8 aty)
% Number of variables : 1254 ( 31 ^;1091 !; 41 ?;1254 :)
% ( 91 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:40:07.754
%------------------------------------------------------------------------------
%----Could-be-implicit typings (7)
thf(ty_t_RBT__Impl__Mirabelle__msmaddcmtr_Ocolor,type,
rBT_Im1923302023_color: $tType ).
thf(ty_t_RBT__Impl__Mirabelle__msmaddcmtr_Orbt,type,
rBT_Im246033960le_rbt: $tType > $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
thf(ty_tf_b,type,
b: $tType ).
thf(ty_tf_a,type,
a: $tType ).
%----Explicit typings (72)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Ono__bot,type,
no_bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Ono__top,type,
no_top:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Ozero__neq__one,type,
zero_neq_one:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Owellorder,type,
wellorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Ozero__less__one,type,
zero_less_one:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Odense__order,type,
dense_order:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Fields_Olinordered__field,type,
linordered_field:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Odense__linorder,type,
dense_linorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Olinordered__semidom,type,
linordered_semidom:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Olinordered__ab__group__add,type,
linord219039673up_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__comm__monoid__add,type,
ordere216010020id_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__ab__group__add__abs,type,
ordere142940540dd_abs:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
linord1659791738miring:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni770627133id_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ostrict__ordered__comm__monoid__add,type,
strict797366125id_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
ordere236663937imp_le:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
ordere223160158up_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add,type,
strict2144017051up_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type,
ordere516151231imp_le:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
condit1656338222tinuum:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
condit1037483654norder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_BNF__Def_Oeq__onp,type,
bNF_eq_onp:
!>[A: $tType] : ( ( A > $o ) > A > A > $o ) ).
thf(sy_c_Groups_Oabs__class_Oabs,type,
abs_abs:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Groups_Oone__class_Oone,type,
one_one:
!>[A: $tType] : A ).
thf(sy_c_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Osgn__class_Osgn,type,
sgn_sgn:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Groups_Ouminus__class_Ouminus,type,
uminus_uminus:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
semiring_1_of_nat:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : ( A > nat ) ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oorder__class_Ostrict__mono,type,
order_strict_mono:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Power_Opower__class_Opower,type,
power_power:
!>[A: $tType] : ( A > nat > A ) ).
thf(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Obheight,type,
rBT_Im315265182height:
!>[A: $tType,B: $tType] : ( ( rBT_Im246033960le_rbt @ A @ B ) > nat ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Ocolor_OB,type,
rBT_Impl_Mirabelle_B: rBT_Im1923302023_color ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Ocolor_OR,type,
rBT_Impl_Mirabelle_R: rBT_Im1923302023_color ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Ocolor_Ocase__color,type,
rBT_Im1975547051_color:
!>[A: $tType] : ( A > A > rBT_Im1923302023_color > A ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Ocolor_Orec__color,type,
rBT_Im1059957627_color:
!>[A: $tType] : ( A > A > rBT_Im1923302023_color > A ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Ocolor_Osize__color,type,
rBT_Im800559290_color: rBT_Im1923302023_color > nat ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Ocolor__of,type,
rBT_Im1805589839lor_of:
!>[A: $tType,B: $tType] : ( ( rBT_Im246033960le_rbt @ A @ B ) > rBT_Im1923302023_color ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Oinv1,type,
rBT_Im1040612427e_inv1:
!>[A: $tType,B: $tType] : ( ( rBT_Im246033960le_rbt @ A @ B ) > $o ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Oinv1l,type,
rBT_Im1071879905_inv1l:
!>[A: $tType,B: $tType] : ( ( rBT_Im246033960le_rbt @ A @ B ) > $o ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Oord_Orbt__greater,type,
rBT_Im1259024060reater:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > A > ( rBT_Im246033960le_rbt @ A @ B ) > $o ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Oord_Orbt__less,type,
rBT_Im2075124343t_less:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > A > ( rBT_Im246033960le_rbt @ A @ B ) > $o ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Oord_Orbt__sorted,type,
rBT_Im759614907sorted:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( rBT_Im246033960le_rbt @ A @ B ) > $o ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Oord__class_Orbt__greater,type,
rBT_Im1332541288reater:
!>[A: $tType,B: $tType] : ( A > ( rBT_Im246033960le_rbt @ A @ B ) > $o ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Oord__class_Orbt__less,type,
rBT_Im903026891t_less:
!>[A: $tType,B: $tType] : ( A > ( rBT_Im246033960le_rbt @ A @ B ) > $o ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Oord__class_Orbt__sorted,type,
rBT_Im1183586831sorted:
!>[A: $tType,B: $tType] : ( ( rBT_Im246033960le_rbt @ A @ B ) > $o ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Orbt_OBranch,type,
rBT_Im480247531Branch:
!>[A: $tType,B: $tType] : ( rBT_Im1923302023_color > ( rBT_Im246033960le_rbt @ A @ B ) > A > B > ( rBT_Im246033960le_rbt @ A @ B ) > ( rBT_Im246033960le_rbt @ A @ B ) ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Orbt_OEmpty,type,
rBT_Im418718756_Empty:
!>[A: $tType,B: $tType] : ( rBT_Im246033960le_rbt @ A @ B ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Orbt_Ocase__rbt,type,
rBT_Im858806507se_rbt:
!>[C: $tType,A: $tType,B: $tType] : ( C > ( rBT_Im1923302023_color > ( rBT_Im246033960le_rbt @ A @ B ) > A > B > ( rBT_Im246033960le_rbt @ A @ B ) > C ) > ( rBT_Im246033960le_rbt @ A @ B ) > C ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Orbt_Opred__rbt,type,
rBT_Im1931894874ed_rbt:
!>[A: $tType,B: $tType] : ( ( A > $o ) > ( B > $o ) > ( rBT_Im246033960le_rbt @ A @ B ) > $o ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Orbt_Orec__rbt,type,
rBT_Im1947144893ec_rbt:
!>[E: $tType,A: $tType,B: $tType] : ( E > ( rBT_Im1923302023_color > ( rBT_Im246033960le_rbt @ A @ B ) > A > B > ( rBT_Im246033960le_rbt @ A @ B ) > E > E > E ) > ( rBT_Im246033960le_rbt @ A @ B ) > E ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Orbt_Orel__rbt,type,
rBT_Im1000242676el_rbt:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C > $o ) > ( B > D > $o ) > ( rBT_Im246033960le_rbt @ A @ B ) > ( rBT_Im246033960le_rbt @ C @ D ) > $o ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Orbt_Oset1__rbt,type,
rBT_Im1178217900t1_rbt:
!>[A: $tType,B: $tType] : ( ( rBT_Im246033960le_rbt @ A @ B ) > ( set @ A ) ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Orbt_Oset2__rbt,type,
rBT_Im1550225131t2_rbt:
!>[A: $tType,B: $tType] : ( ( rBT_Im246033960le_rbt @ A @ B ) > ( set @ B ) ) ).
thf(sy_c_RBT__Impl__Mirabelle__msmaddcmtr_Orbt_Osize__rbt,type,
rBT_Im1485196410ze_rbt:
!>[A: $tType,B: $tType] : ( ( A > nat ) > ( B > nat ) > ( rBT_Im246033960le_rbt @ A @ B ) > nat ) ).
thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool,type,
zero_neq_one_of_bool:
!>[A: $tType] : ( $o > A ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_t,type,
t: rBT_Im246033960le_rbt @ a @ b ).
%----Relevant facts (256)
thf(fact_0_inv1l_Osimps_I2_J,axiom,
! [B: $tType,A: $tType,C2: rBT_Im1923302023_color,L: rBT_Im246033960le_rbt @ A @ B,K: A,V: B,R: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1071879905_inv1l @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ C2 @ L @ K @ V @ R ) )
= ( ( rBT_Im1040612427e_inv1 @ A @ B @ L )
& ( rBT_Im1040612427e_inv1 @ A @ B @ R ) ) ) ).
% inv1l.simps(2)
thf(fact_1_inv1l_Osimps_I1_J,axiom,
! [B: $tType,A: $tType] : ( rBT_Im1071879905_inv1l @ A @ B @ ( rBT_Im418718756_Empty @ A @ B ) ) ).
% inv1l.simps(1)
thf(fact_2_inv1_Osimps_I1_J,axiom,
! [B: $tType,A: $tType] : ( rBT_Im1040612427e_inv1 @ A @ B @ ( rBT_Im418718756_Empty @ A @ B ) ) ).
% inv1.simps(1)
thf(fact_3_inv1_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,C2: rBT_Im1923302023_color,Lt: rBT_Im246033960le_rbt @ A @ B,K: A,V: B,Rt: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1040612427e_inv1 @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ C2 @ Lt @ K @ V @ Rt ) )
= ( ( rBT_Im1040612427e_inv1 @ A @ B @ Lt )
& ( rBT_Im1040612427e_inv1 @ A @ B @ Rt )
& ( ( C2 = rBT_Impl_Mirabelle_B )
| ( ( ( rBT_Im1805589839lor_of @ A @ B @ Lt )
= rBT_Impl_Mirabelle_B )
& ( ( rBT_Im1805589839lor_of @ A @ B @ Rt )
= rBT_Impl_Mirabelle_B ) ) ) ) ) ).
% inv1.simps(2)
thf(fact_4_color_Ocase__distrib,axiom,
! [A: $tType,B: $tType,H: A > B,F1: A,F2: A,Color: rBT_Im1923302023_color] :
( ( H @ ( rBT_Im1975547051_color @ A @ F1 @ F2 @ Color ) )
= ( rBT_Im1975547051_color @ B @ ( H @ F1 ) @ ( H @ F2 ) @ Color ) ) ).
% color.case_distrib
thf(fact_5_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_6_color_Osimps_I4_J,axiom,
! [A: $tType,F1: A,F2: A] :
( ( rBT_Im1975547051_color @ A @ F1 @ F2 @ rBT_Impl_Mirabelle_B )
= F2 ) ).
% color.simps(4)
thf(fact_7_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_8_color__of_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,C2: rBT_Im1923302023_color,Uu: rBT_Im246033960le_rbt @ A @ B,Uv: A,Uw: B,Ux: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1805589839lor_of @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ C2 @ Uu @ Uv @ Uw @ Ux ) )
= C2 ) ).
% color_of.simps(2)
thf(fact_9_color__of_Osimps_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( rBT_Im1805589839lor_of @ A @ B @ ( rBT_Im418718756_Empty @ A @ B ) )
= rBT_Impl_Mirabelle_B ) ).
% color_of.simps(1)
thf(fact_10_rbt_Oinduct,axiom,
! [B: $tType,A: $tType,P: ( rBT_Im246033960le_rbt @ A @ B ) > $o,Rbt: rBT_Im246033960le_rbt @ A @ B] :
( ( P @ ( rBT_Im418718756_Empty @ A @ B ) )
=> ( ! [X1: rBT_Im1923302023_color,X2: rBT_Im246033960le_rbt @ A @ B,X3: A,X4: B,X5: rBT_Im246033960le_rbt @ A @ B] :
( ( P @ X2 )
=> ( ( P @ X5 )
=> ( P @ ( rBT_Im480247531Branch @ A @ B @ X1 @ X2 @ X3 @ X4 @ X5 ) ) ) )
=> ( P @ Rbt ) ) ) ).
% rbt.induct
thf(fact_11_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_12_rbt__cases,axiom,
! [B: $tType,A: $tType,T: rBT_Im246033960le_rbt @ A @ B] :
( ( T
!= ( rBT_Im418718756_Empty @ A @ B ) )
=> ( ! [L2: rBT_Im246033960le_rbt @ A @ B,K2: A,V2: B,R2: rBT_Im246033960le_rbt @ A @ B] :
( T
!= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_R @ L2 @ K2 @ V2 @ R2 ) )
=> ~ ! [L2: rBT_Im246033960le_rbt @ A @ B,K2: A,V2: B,R2: rBT_Im246033960le_rbt @ A @ B] :
( T
!= ( rBT_Im480247531Branch @ A @ B @ rBT_Impl_Mirabelle_B @ L2 @ K2 @ V2 @ R2 ) ) ) ) ).
% rbt_cases
thf(fact_13_color_Osimps_I6_J,axiom,
! [A: $tType,F1: A,F2: A] :
( ( rBT_Im1059957627_color @ A @ F1 @ F2 @ rBT_Impl_Mirabelle_B )
= F2 ) ).
% color.simps(6)
thf(fact_14_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_15_color_Osimps_I3_J,axiom,
! [A: $tType,F1: A,F2: A] :
( ( rBT_Im1975547051_color @ A @ F1 @ F2 @ rBT_Impl_Mirabelle_R )
= F1 ) ).
% color.simps(3)
thf(fact_16_rbt__less__simps_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [K: A] : ( rBT_Im903026891t_less @ A @ B @ K @ ( rBT_Im418718756_Empty @ A @ B ) ) ) ).
% rbt_less_simps(1)
thf(fact_17_rbt__greater__simps_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [K: A] : ( rBT_Im1332541288reater @ A @ B @ K @ ( rBT_Im418718756_Empty @ A @ B ) ) ) ).
% rbt_greater_simps(1)
thf(fact_18_color_Odistinct_I1_J,axiom,
rBT_Impl_Mirabelle_R != rBT_Impl_Mirabelle_B ).
% color.distinct(1)
thf(fact_19_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_20_color_Oexhaust,axiom,
! [Y: rBT_Im1923302023_color] :
( ( Y != rBT_Impl_Mirabelle_R )
=> ( Y = rBT_Impl_Mirabelle_B ) ) ).
% color.exhaust
thf(fact_21_rbt_Osimps_I4_J,axiom,
! [A: $tType,B: $tType,C: $tType,F1: C,F2: rBT_Im1923302023_color > ( rBT_Im246033960le_rbt @ A @ B ) > A > B > ( rBT_Im246033960le_rbt @ A @ B ) > C] :
( ( rBT_Im858806507se_rbt @ C @ A @ B @ F1 @ F2 @ ( rBT_Im418718756_Empty @ A @ B ) )
= F1 ) ).
% rbt.simps(4)
thf(fact_22_rbt_Osimps_I5_J,axiom,
! [C: $tType,B: $tType,A: $tType,F1: C,F2: rBT_Im1923302023_color > ( rBT_Im246033960le_rbt @ A @ B ) > A > B > ( rBT_Im246033960le_rbt @ A @ B ) > C,X21: rBT_Im1923302023_color,X22: rBT_Im246033960le_rbt @ A @ B,X23: A,X24: B,X25: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im858806507se_rbt @ C @ A @ B @ F1 @ F2 @ ( rBT_Im480247531Branch @ A @ B @ X21 @ X22 @ X23 @ X24 @ X25 ) )
= ( F2 @ X21 @ X22 @ X23 @ X24 @ X25 ) ) ).
% rbt.simps(5)
thf(fact_23_color_Osimps_I5_J,axiom,
! [A: $tType,F1: A,F2: A] :
( ( rBT_Im1059957627_color @ A @ F1 @ F2 @ rBT_Impl_Mirabelle_R )
= F1 ) ).
% color.simps(5)
thf(fact_24_rbt__sorted_Osimps_I2_J,axiom,
! [B: $tType,A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [C2: rBT_Im1923302023_color,L: rBT_Im246033960le_rbt @ A @ B,K: A,V: B,R: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1183586831sorted @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ C2 @ L @ K @ V @ R ) )
= ( ( rBT_Im903026891t_less @ A @ B @ K @ L )
& ( rBT_Im1332541288reater @ A @ B @ K @ R )
& ( rBT_Im1183586831sorted @ A @ B @ L )
& ( rBT_Im1183586831sorted @ A @ B @ R ) ) ) ) ).
% rbt_sorted.simps(2)
thf(fact_25_rbt__greater__simps_I2_J,axiom,
! [C: $tType,A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [K: A,C2: rBT_Im1923302023_color,Lt: rBT_Im246033960le_rbt @ A @ C,Kt: A,V: C,Rt: rBT_Im246033960le_rbt @ A @ C] :
( ( rBT_Im1332541288reater @ A @ C @ K @ ( rBT_Im480247531Branch @ A @ C @ C2 @ Lt @ Kt @ V @ Rt ) )
= ( ( ord_less @ A @ K @ Kt )
& ( rBT_Im1332541288reater @ A @ C @ K @ Lt )
& ( rBT_Im1332541288reater @ A @ C @ K @ Rt ) ) ) ) ).
% rbt_greater_simps(2)
thf(fact_26_rbt__less__simps_I2_J,axiom,
! [C: $tType,A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [K: A,C2: rBT_Im1923302023_color,Lt: rBT_Im246033960le_rbt @ A @ C,Kt: A,V: C,Rt: rBT_Im246033960le_rbt @ A @ C] :
( ( rBT_Im903026891t_less @ A @ C @ K @ ( rBT_Im480247531Branch @ A @ C @ C2 @ Lt @ Kt @ V @ Rt ) )
= ( ( ord_less @ A @ Kt @ K )
& ( rBT_Im903026891t_less @ A @ C @ K @ Lt )
& ( rBT_Im903026891t_less @ A @ C @ K @ Rt ) ) ) ) ).
% rbt_less_simps(2)
thf(fact_27_ord_Orbt__sorted_Osimps_I2_J,axiom,
! [B: $tType,A: $tType,Less: A > A > $o,C2: 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 @ C2 @ 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_28_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_29_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_30_ord_Orbt__greater__simps_I2_J,axiom,
! [C: $tType,A: $tType,Less: A > A > $o,K: A,C2: rBT_Im1923302023_color,Lt: rBT_Im246033960le_rbt @ A @ C,Kt: A,V: C,Rt: rBT_Im246033960le_rbt @ A @ C] :
( ( rBT_Im1259024060reater @ A @ C @ Less @ K @ ( rBT_Im480247531Branch @ A @ C @ C2 @ Lt @ Kt @ V @ Rt ) )
= ( ( Less @ K @ Kt )
& ( rBT_Im1259024060reater @ A @ C @ Less @ K @ Lt )
& ( rBT_Im1259024060reater @ A @ C @ Less @ K @ Rt ) ) ) ).
% ord.rbt_greater_simps(2)
thf(fact_31_ord_Orbt__less__simps_I2_J,axiom,
! [C: $tType,A: $tType,Less: A > A > $o,K: A,C2: rBT_Im1923302023_color,Lt: rBT_Im246033960le_rbt @ A @ C,Kt: A,V: C,Rt: rBT_Im246033960le_rbt @ A @ C] :
( ( rBT_Im2075124343t_less @ A @ C @ Less @ K @ ( rBT_Im480247531Branch @ A @ C @ C2 @ Lt @ Kt @ V @ Rt ) )
= ( ( Less @ Kt @ K )
& ( rBT_Im2075124343t_less @ A @ C @ Less @ K @ Lt )
& ( rBT_Im2075124343t_less @ A @ C @ Less @ K @ Rt ) ) ) ).
% ord.rbt_less_simps(2)
thf(fact_32_rbt_Osimps_I6_J,axiom,
! [A: $tType,B: $tType,E: $tType,F1: E,F2: rBT_Im1923302023_color > ( rBT_Im246033960le_rbt @ A @ B ) > A > B > ( rBT_Im246033960le_rbt @ A @ B ) > E > E > E] :
( ( rBT_Im1947144893ec_rbt @ E @ A @ B @ F1 @ F2 @ ( rBT_Im418718756_Empty @ A @ B ) )
= F1 ) ).
% rbt.simps(6)
thf(fact_33_rbt_Osimps_I7_J,axiom,
! [E: $tType,B: $tType,A: $tType,F1: E,F2: rBT_Im1923302023_color > ( rBT_Im246033960le_rbt @ A @ B ) > A > B > ( rBT_Im246033960le_rbt @ A @ B ) > E > E > E,X21: rBT_Im1923302023_color,X22: rBT_Im246033960le_rbt @ A @ B,X23: A,X24: B,X25: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1947144893ec_rbt @ E @ A @ B @ F1 @ F2 @ ( rBT_Im480247531Branch @ A @ B @ X21 @ X22 @ X23 @ X24 @ X25 ) )
= ( F2 @ X21 @ X22 @ X23 @ X24 @ X25 @ ( rBT_Im1947144893ec_rbt @ E @ A @ B @ F1 @ F2 @ X22 ) @ ( rBT_Im1947144893ec_rbt @ E @ A @ B @ F1 @ F2 @ X25 ) ) ) ).
% rbt.simps(7)
thf(fact_34_rbt__less__trans,axiom,
! [C: $tType,A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,T: rBT_Im246033960le_rbt @ A @ C,Y: A] :
( ( rBT_Im903026891t_less @ A @ C @ X @ T )
=> ( ( ord_less @ A @ X @ Y )
=> ( rBT_Im903026891t_less @ A @ C @ Y @ T ) ) ) ) ).
% rbt_less_trans
thf(fact_35_rbt__greater__trans,axiom,
! [C: $tType,A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,T: rBT_Im246033960le_rbt @ A @ C] :
( ( ord_less @ A @ X @ Y )
=> ( ( rBT_Im1332541288reater @ A @ C @ Y @ T )
=> ( rBT_Im1332541288reater @ A @ C @ X @ T ) ) ) ) ).
% rbt_greater_trans
thf(fact_36_rbt__sorted_Osimps_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ( rBT_Im1183586831sorted @ A @ B @ ( rBT_Im418718756_Empty @ A @ B ) ) ) ).
% rbt_sorted.simps(1)
thf(fact_37_minf_I11_J,axiom,
! [C: $tType,D: $tType] :
( ( ord @ C @ ( type2 @ C ) )
=> ! [F: D] :
? [Z: C] :
! [X6: C] :
( ( ord_less @ C @ X6 @ Z )
=> ( F = F ) ) ) ).
% minf(11)
thf(fact_38_minf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z: A] :
! [X6: A] :
( ( ord_less @ A @ X6 @ Z )
=> ~ ( ord_less @ A @ T @ X6 ) ) ) ).
% minf(7)
thf(fact_39_minf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z: A] :
! [X6: A] :
( ( ord_less @ A @ X6 @ Z )
=> ( ord_less @ A @ X6 @ T ) ) ) ).
% minf(5)
thf(fact_40_minf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z: A] :
! [X6: A] :
( ( ord_less @ A @ X6 @ Z )
=> ( X6 != T ) ) ) ).
% minf(4)
thf(fact_41_minf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z: A] :
! [X6: A] :
( ( ord_less @ A @ X6 @ Z )
=> ( X6 != T ) ) ) ).
% minf(3)
thf(fact_42_minf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z2: A] :
! [X7: A] :
( ( ord_less @ A @ X7 @ Z2 )
=> ( ( P @ X7 )
= ( P2 @ X7 ) ) )
=> ( ? [Z2: A] :
! [X7: A] :
( ( ord_less @ A @ X7 @ Z2 )
=> ( ( Q @ X7 )
= ( Q2 @ X7 ) ) )
=> ? [Z: A] :
! [X6: A] :
( ( ord_less @ A @ X6 @ Z )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P2 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ) ).
% minf(2)
thf(fact_43_minf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z2: A] :
! [X7: A] :
( ( ord_less @ A @ X7 @ Z2 )
=> ( ( P @ X7 )
= ( P2 @ X7 ) ) )
=> ( ? [Z2: A] :
! [X7: A] :
( ( ord_less @ A @ X7 @ Z2 )
=> ( ( Q @ X7 )
= ( Q2 @ X7 ) ) )
=> ? [Z: A] :
! [X6: A] :
( ( ord_less @ A @ X6 @ Z )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P2 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ) ).
% minf(1)
thf(fact_44_pinf_I11_J,axiom,
! [C: $tType,D: $tType] :
( ( ord @ C @ ( type2 @ C ) )
=> ! [F: D] :
? [Z: C] :
! [X6: C] :
( ( ord_less @ C @ Z @ X6 )
=> ( F = F ) ) ) ).
% pinf(11)
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( collect @ A
@ ^ [X8: A] : ( member @ A @ X8 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X7: A] :
( ( P @ X7 )
= ( Q @ X7 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F3: A > B,G: A > B] :
( ! [X7: A] :
( ( F3 @ X7 )
= ( G @ X7 ) )
=> ( F3 = G ) ) ).
% ext
thf(fact_49_pinf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z: A] :
! [X6: A] :
( ( ord_less @ A @ Z @ X6 )
=> ( ord_less @ A @ T @ X6 ) ) ) ).
% pinf(7)
thf(fact_50_pinf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z2: A] :
! [X7: A] :
( ( ord_less @ A @ Z2 @ X7 )
=> ( ( P @ X7 )
= ( P2 @ X7 ) ) )
=> ( ? [Z2: A] :
! [X7: A] :
( ( ord_less @ A @ Z2 @ X7 )
=> ( ( Q @ X7 )
= ( Q2 @ X7 ) ) )
=> ? [Z: A] :
! [X6: A] :
( ( ord_less @ A @ Z @ X6 )
=> ( ( ( P @ X6 )
& ( Q @ X6 ) )
= ( ( P2 @ X6 )
& ( Q2 @ X6 ) ) ) ) ) ) ) ).
% pinf(1)
thf(fact_51_pinf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z2: A] :
! [X7: A] :
( ( ord_less @ A @ Z2 @ X7 )
=> ( ( P @ X7 )
= ( P2 @ X7 ) ) )
=> ( ? [Z2: A] :
! [X7: A] :
( ( ord_less @ A @ Z2 @ X7 )
=> ( ( Q @ X7 )
= ( Q2 @ X7 ) ) )
=> ? [Z: A] :
! [X6: A] :
( ( ord_less @ A @ Z @ X6 )
=> ( ( ( P @ X6 )
| ( Q @ X6 ) )
= ( ( P2 @ X6 )
| ( Q2 @ X6 ) ) ) ) ) ) ) ).
% pinf(2)
thf(fact_52_pinf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z: A] :
! [X6: A] :
( ( ord_less @ A @ Z @ X6 )
=> ( X6 != T ) ) ) ).
% pinf(3)
thf(fact_53_pinf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z: A] :
! [X6: A] :
( ( ord_less @ A @ Z @ X6 )
=> ( X6 != T ) ) ) ).
% pinf(4)
thf(fact_54_pinf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z: A] :
! [X6: A] :
( ( ord_less @ A @ Z @ X6 )
=> ~ ( ord_less @ A @ X6 @ T ) ) ) ).
% pinf(5)
thf(fact_55_ex__gt__or__lt,axiom,
! [A: $tType] :
( ( condit1656338222tinuum @ A @ ( type2 @ A ) )
=> ! [A2: A] :
? [B2: A] :
( ( ord_less @ A @ A2 @ B2 )
| ( ord_less @ A @ B2 @ A2 ) ) ) ).
% ex_gt_or_lt
thf(fact_56_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( X != Y )
=> ( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_57_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A] :
( ( ord_less @ A @ B3 @ A2 )
=> ( A2 != B3 ) ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_58_linordered__field__no__ub,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [X6: A] :
? [X1: A] : ( ord_less @ A @ X6 @ X1 ) ) ).
% linordered_field_no_ub
thf(fact_59_linordered__field__no__lb,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [X6: A] :
? [Y2: A] : ( ord_less @ A @ Y2 @ X6 ) ) ).
% linordered_field_no_lb
thf(fact_60_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( ord_less @ A @ A2 @ B3 )
=> ( A2 != B3 ) ) ) ).
% order.strict_implies_not_eq
thf(fact_61_dependent__wellorder__choice,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ A @ ( type2 @ A ) )
=> ! [P: ( A > B ) > A > B > $o] :
( ! [R2: B,F4: A > B,G2: A > B,X7: A] :
( ! [Y3: A] :
( ( ord_less @ A @ Y3 @ X7 )
=> ( ( F4 @ Y3 )
= ( G2 @ Y3 ) ) )
=> ( ( P @ F4 @ X7 @ R2 )
= ( P @ G2 @ X7 @ R2 ) ) )
=> ( ! [X7: A,F4: A > B] :
( ! [Y3: A] :
( ( ord_less @ A @ Y3 @ X7 )
=> ( P @ F4 @ Y3 @ ( F4 @ Y3 ) ) )
=> ? [X12: B] : ( P @ F4 @ X7 @ X12 ) )
=> ? [F4: A > B] :
! [X6: A] : ( P @ F4 @ X6 @ ( F4 @ X6 ) ) ) ) ) ).
% dependent_wellorder_choice
thf(fact_62_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less @ A @ X @ Y ) )
= ( ( ord_less @ A @ Y @ X )
| ( X = Y ) ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_63_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F3: B > A,B3: B,C2: B] :
( ( A2
= ( F3 @ B3 ) )
=> ( ( ord_less @ B @ B3 @ C2 )
=> ( ! [X7: B,Y2: B] :
( ( ord_less @ B @ X7 @ Y2 )
=> ( ord_less @ A @ ( F3 @ X7 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less @ A @ A2 @ ( F3 @ C2 ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_64_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B3: A,F3: A > B,C2: B] :
( ( ord_less @ A @ A2 @ B3 )
=> ( ( ( F3 @ B3 )
= C2 )
=> ( ! [X7: A,Y2: A] :
( ( ord_less @ A @ X7 @ Y2 )
=> ( ord_less @ B @ ( F3 @ X7 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less @ B @ ( F3 @ A2 ) @ C2 ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_65_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F3: B > A,B3: B,C2: B] :
( ( ord_less @ A @ A2 @ ( F3 @ B3 ) )
=> ( ( ord_less @ B @ B3 @ C2 )
=> ( ! [X7: B,Y2: B] :
( ( ord_less @ B @ X7 @ Y2 )
=> ( ord_less @ A @ ( F3 @ X7 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less @ A @ A2 @ ( F3 @ C2 ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_66_order__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C @ ( type2 @ C ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B3: A,F3: A > C,C2: C] :
( ( ord_less @ A @ A2 @ B3 )
=> ( ( ord_less @ C @ ( F3 @ B3 ) @ C2 )
=> ( ! [X7: A,Y2: A] :
( ( ord_less @ A @ X7 @ Y2 )
=> ( ord_less @ C @ ( F3 @ X7 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less @ C @ ( F3 @ A2 ) @ C2 ) ) ) ) ) ).
% order_less_subst2
thf(fact_67_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A @ ( type2 @ A ) )
=> ! [X: A] :
? [Y2: A] : ( ord_less @ A @ Y2 @ X ) ) ).
% lt_ex
thf(fact_68_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A @ ( type2 @ A ) )
=> ! [X: A] :
? [X1: A] : ( ord_less @ A @ X @ X1 ) ) ).
% gt_ex
thf(fact_69_neqE,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( X != Y )
=> ( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% neqE
thf(fact_70_neq__iff,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( X != Y )
= ( ( ord_less @ A @ X @ Y )
| ( ord_less @ A @ Y @ X ) ) ) ) ).
% neq_iff
thf(fact_71_order_Oasym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( ord_less @ A @ A2 @ B3 )
=> ~ ( ord_less @ A @ B3 @ A2 ) ) ) ).
% order.asym
thf(fact_72_dense,axiom,
! [A: $tType] :
( ( dense_order @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ? [Z: A] :
( ( ord_less @ A @ X @ Z )
& ( ord_less @ A @ Z @ Y ) ) ) ) ).
% dense
thf(fact_73_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( X != Y ) ) ) ).
% less_imp_neq
thf(fact_74_less__asym,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% less_asym
thf(fact_75_less__asym_H,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( ord_less @ A @ A2 @ B3 )
=> ~ ( ord_less @ A @ B3 @ A2 ) ) ) ).
% less_asym'
thf(fact_76_less__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z3: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ Z3 )
=> ( ord_less @ A @ X @ Z3 ) ) ) ) ).
% less_trans
thf(fact_77_less__linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
| ( X = Y )
| ( ord_less @ A @ Y @ X ) ) ) ).
% less_linear
thf(fact_78_less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A] :
~ ( ord_less @ A @ X @ X ) ) ).
% less_irrefl
thf(fact_79_ord__eq__less__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A] :
( ( A2 = B3 )
=> ( ( ord_less @ A @ B3 @ C2 )
=> ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% ord_eq_less_trans
thf(fact_80_ord__less__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A] :
( ( ord_less @ A @ A2 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% ord_less_eq_trans
thf(fact_81_dual__order_Oasym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A] :
( ( ord_less @ A @ B3 @ A2 )
=> ~ ( ord_less @ A @ A2 @ B3 ) ) ) ).
% dual_order.asym
thf(fact_82_less__imp__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( X != Y ) ) ) ).
% less_imp_not_eq
thf(fact_83_less__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% less_not_sym
thf(fact_84_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,A2: A] :
( ! [X7: A] :
( ! [Y3: A] :
( ( ord_less @ A @ Y3 @ X7 )
=> ( P @ Y3 ) )
=> ( P @ X7 ) )
=> ( P @ A2 ) ) ) ).
% less_induct
thf(fact_85_antisym__conv3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A] :
( ~ ( ord_less @ A @ Y @ X )
=> ( ( ~ ( ord_less @ A @ X @ Y ) )
= ( X = Y ) ) ) ) ).
% antisym_conv3
thf(fact_86_less__imp__not__eq2,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( Y != X ) ) ) ).
% less_imp_not_eq2
thf(fact_87_less__imp__triv,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,P: $o] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ X )
=> P ) ) ) ).
% less_imp_triv
thf(fact_88_linorder__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ( X != Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% linorder_cases
thf(fact_89_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ A2 ) ) ).
% dual_order.irrefl
thf(fact_90_order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A] :
( ( ord_less @ A @ A2 @ B3 )
=> ( ( ord_less @ A @ B3 @ C2 )
=> ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% order.strict_trans
thf(fact_91_less__imp__not__less,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% less_imp_not_less
thf(fact_92_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A,C2: A] :
( ( ord_less @ A @ B3 @ A2 )
=> ( ( ord_less @ A @ C2 @ B3 )
=> ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% dual_order.strict_trans
thf(fact_93_rbt_Opred__inject_I2_J,axiom,
! [B: $tType,A: $tType,P1: A > $o,P22: B > $o,A2: rBT_Im1923302023_color,Aa: rBT_Im246033960le_rbt @ A @ B,Ab: A,Ac: B,Ad: rBT_Im246033960le_rbt @ A @ B] :
( ( rBT_Im1931894874ed_rbt @ A @ B @ P1 @ P22 @ ( rBT_Im480247531Branch @ A @ B @ A2 @ Aa @ Ab @ Ac @ Ad ) )
= ( ( rBT_Im1931894874ed_rbt @ A @ B @ P1 @ P22 @ Aa )
& ( P1 @ Ab )
& ( P22 @ Ac )
& ( rBT_Im1931894874ed_rbt @ A @ B @ P1 @ P22 @ Ad ) ) ) ).
% rbt.pred_inject(2)
thf(fact_94_rbt_Orel__distinct_I2_J,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,R1: A > C > $o,R22: B > D > $o,Y21: rBT_Im1923302023_color,Y22: rBT_Im246033960le_rbt @ A @ B,Y23: A,Y24: B,Y25: rBT_Im246033960le_rbt @ A @ B] :
~ ( rBT_Im1000242676el_rbt @ A @ C @ B @ D @ R1 @ R22 @ ( rBT_Im480247531Branch @ A @ B @ Y21 @ Y22 @ Y23 @ Y24 @ Y25 ) @ ( rBT_Im418718756_Empty @ C @ D ) ) ).
% rbt.rel_distinct(2)
thf(fact_95_rbt_Orel__distinct_I1_J,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,R1: A > C > $o,R22: B > D > $o,Y21: rBT_Im1923302023_color,Y22: rBT_Im246033960le_rbt @ C @ D,Y23: C,Y24: D,Y25: rBT_Im246033960le_rbt @ C @ D] :
~ ( rBT_Im1000242676el_rbt @ A @ C @ B @ D @ R1 @ R22 @ ( rBT_Im418718756_Empty @ A @ B ) @ ( rBT_Im480247531Branch @ C @ D @ Y21 @ Y22 @ Y23 @ Y24 @ Y25 ) ) ).
% rbt.rel_distinct(1)
thf(fact_96_strict__monoD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A @ ( type2 @ A ) )
& ( order @ B @ ( type2 @ B ) ) )
=> ! [F3: A > B,X: A,Y: A] :
( ( order_strict_mono @ A @ B @ F3 )
=> ( ( ord_less @ A @ X @ Y )
=> ( ord_less @ B @ ( F3 @ X ) @ ( F3 @ Y ) ) ) ) ) ).
% strict_monoD
thf(fact_97_strict__monoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A @ ( type2 @ A ) )
& ( order @ B @ ( type2 @ B ) ) )
=> ! [F3: A > B] :
( ! [X7: A,Y2: A] :
( ( ord_less @ A @ X7 @ Y2 )
=> ( ord_less @ B @ ( F3 @ X7 ) @ ( F3 @ Y2 ) ) )
=> ( order_strict_mono @ A @ B @ F3 ) ) ) ).
% strict_monoI
thf(fact_98_strict__mono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A @ ( type2 @ A ) )
& ( order @ B @ ( type2 @ B ) ) )
=> ( ( order_strict_mono @ A @ B )
= ( ^ [F5: A > B] :
! [X8: A,Y4: A] :
( ( ord_less @ A @ X8 @ Y4 )
=> ( ord_less @ B @ ( F5 @ X8 ) @ ( F5 @ Y4 ) ) ) ) ) ) ).
% strict_mono_def
thf(fact_99_strict__mono__less,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A @ ( type2 @ A ) )
& ( order @ B @ ( type2 @ B ) ) )
=> ! [F3: A > B,X: A,Y: A] :
( ( order_strict_mono @ A @ B @ F3 )
=> ( ( ord_less @ B @ ( F3 @ X ) @ ( F3 @ Y ) )
= ( ord_less @ A @ X @ Y ) ) ) ) ).
% strict_mono_less
thf(fact_100_strict__mono__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A @ ( type2 @ A ) )
& ( order @ B @ ( type2 @ B ) ) )
=> ! [F3: A > B,X: A,Y: A] :
( ( order_strict_mono @ A @ B @ F3 )
=> ( ( ( F3 @ X )
= ( F3 @ Y ) )
= ( X = Y ) ) ) ) ).
% strict_mono_eq
thf(fact_101_rbt_Orel__refl,axiom,
! [D: $tType,C: $tType,R1a: C > C > $o,R2a: D > D > $o,X: rBT_Im246033960le_rbt @ C @ D] :
( ! [X7: C] : ( R1a @ X7 @ X7 )
=> ( ! [X7: D] : ( R2a @ X7 @ X7 )
=> ( rBT_Im1000242676el_rbt @ C @ C @ D @ D @ R1a @ R2a @ X @ X ) ) ) ).
% rbt.rel_refl
thf(fact_102_rbt_Orel__eq,axiom,
! [B: $tType,A: $tType] :
( ( rBT_Im1000242676el_rbt @ A @ A @ B @ B
@ ^ [Y5: A,Z4: A] : Y5 = Z4
@ ^ [Y5: B,Z4: B] : Y5 = Z4 )
= ( ^ [Y5: rBT_Im246033960le_rbt @ A @ B,Z4: rBT_Im246033960le_rbt @ A @ B] : Y5 = Z4 ) ) ).
% rbt.rel_eq
thf(fact_103_rbt_Orel__intros_I2_J,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,X21: rBT_Im1923302023_color,Y21: rBT_Im1923302023_color,R1: A > C > $o,R22: B > D > $o,X22: rBT_Im246033960le_rbt @ A @ B,Y22: rBT_Im246033960le_rbt @ C @ D,X23: A,Y23: C,X24: B,Y24: D,X25: rBT_Im246033960le_rbt @ A @ B,Y25: rBT_Im246033960le_rbt @ C @ D] :
( ( X21 = Y21 )
=> ( ( rBT_Im1000242676el_rbt @ A @ C @ B @ D @ R1 @ R22 @ X22 @ Y22 )
=> ( ( R1 @ X23 @ Y23 )
=> ( ( R22 @ X24 @ Y24 )
=> ( ( rBT_Im1000242676el_rbt @ A @ C @ B @ D @ R1 @ R22 @ X25 @ Y25 )
=> ( rBT_Im1000242676el_rbt @ A @ C @ B @ D @ R1 @ R22 @ ( rBT_Im480247531Branch @ A @ B @ X21 @ X22 @ X23 @ X24 @ X25 ) @ ( rBT_Im480247531Branch @ C @ D @ Y21 @ Y22 @ Y23 @ Y24 @ Y25 ) ) ) ) ) ) ) ).
% rbt.rel_intros(2)
thf(fact_104_rbt_Orel__inject_I2_J,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,R1: A > C > $o,R22: B > D > $o,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 @ C @ D,Y23: C,Y24: D,Y25: rBT_Im246033960le_rbt @ C @ D] :
( ( rBT_Im1000242676el_rbt @ A @ C @ B @ D @ R1 @ R22 @ ( rBT_Im480247531Branch @ A @ B @ X21 @ X22 @ X23 @ X24 @ X25 ) @ ( rBT_Im480247531Branch @ C @ D @ Y21 @ Y22 @ Y23 @ Y24 @ Y25 ) )
= ( ( X21 = Y21 )
& ( rBT_Im1000242676el_rbt @ A @ C @ B @ D @ R1 @ R22 @ X22 @ Y22 )
& ( R1 @ X23 @ Y23 )
& ( R22 @ X24 @ Y24 )
& ( rBT_Im1000242676el_rbt @ A @ C @ B @ D @ R1 @ R22 @ X25 @ Y25 ) ) ) ).
% rbt.rel_inject(2)
thf(fact_105_rbt_Octr__transfer_I1_J,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,R1: A > C > $o,R22: B > D > $o] : ( rBT_Im1000242676el_rbt @ A @ C @ B @ D @ R1 @ R22 @ ( rBT_Im418718756_Empty @ A @ B ) @ ( rBT_Im418718756_Empty @ C @ D ) ) ).
% rbt.ctr_transfer(1)
thf(fact_106_rbt_Opred__inject_I1_J,axiom,
! [B: $tType,A: $tType,P1: A > $o,P22: B > $o] : ( rBT_Im1931894874ed_rbt @ A @ B @ P1 @ P22 @ ( rBT_Im418718756_Empty @ A @ B ) ) ).
% rbt.pred_inject(1)
thf(fact_107_rbt_Opred__rel,axiom,
! [B: $tType,A: $tType] :
( ( rBT_Im1931894874ed_rbt @ A @ B )
= ( ^ [P12: A > $o,P23: B > $o,X8: rBT_Im246033960le_rbt @ A @ B] : ( rBT_Im1000242676el_rbt @ A @ A @ B @ B @ ( bNF_eq_onp @ A @ P12 ) @ ( bNF_eq_onp @ B @ P23 ) @ X8 @ X8 ) ) ) ).
% rbt.pred_rel
thf(fact_108_bheight_Osimps_I2_J,axiom,
! [B: $tType,A: $tType,C2: rBT_Im1923302023_color,Lt: rBT_Im246033960le_rbt @ A @ B,K: A,V: B,Rt: rBT_Im246033960le_rbt @ A @ B] :
( ( ( C2 = rBT_Impl_Mirabelle_B )
=> ( ( rBT_Im315265182height @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ C2 @ Lt @ K @ V @ Rt ) )
= ( suc @ ( rBT_Im315265182height @ A @ B @ Lt ) ) ) )
& ( ( C2 != rBT_Impl_Mirabelle_B )
=> ( ( rBT_Im315265182height @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ C2 @ Lt @ K @ V @ Rt ) )
= ( rBT_Im315265182height @ A @ B @ Lt ) ) ) ) ).
% bheight.simps(2)
thf(fact_109_rbt_Oset__intros_I1_J,axiom,
! [B: $tType,A: $tType,X: A,A22: rBT_Im246033960le_rbt @ A @ B,A1: rBT_Im1923302023_color,A32: A,A4: B,A5: rBT_Im246033960le_rbt @ A @ B] :
( ( member @ A @ X @ ( rBT_Im1178217900t1_rbt @ A @ B @ A22 ) )
=> ( member @ A @ X @ ( rBT_Im1178217900t1_rbt @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ A1 @ A22 @ A32 @ A4 @ A5 ) ) ) ) ).
% rbt.set_intros(1)
thf(fact_110_rbt_Oset__intros_I2_J,axiom,
! [B: $tType,A: $tType,A32: A,A1: rBT_Im1923302023_color,A22: rBT_Im246033960le_rbt @ A @ B,A4: B,A5: rBT_Im246033960le_rbt @ A @ B] : ( member @ A @ A32 @ ( rBT_Im1178217900t1_rbt @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ A1 @ A22 @ A32 @ A4 @ A5 ) ) ) ).
% rbt.set_intros(2)
thf(fact_111_rbt_Oset__intros_I3_J,axiom,
! [B: $tType,A: $tType,Xb: A,A5: rBT_Im246033960le_rbt @ A @ B,A1: rBT_Im1923302023_color,A22: rBT_Im246033960le_rbt @ A @ B,A32: A,A4: B] :
( ( member @ A @ Xb @ ( rBT_Im1178217900t1_rbt @ A @ B @ A5 ) )
=> ( member @ A @ Xb @ ( rBT_Im1178217900t1_rbt @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ A1 @ A22 @ A32 @ A4 @ A5 ) ) ) ) ).
% rbt.set_intros(3)
thf(fact_112_rbt_Orel__eq__onp,axiom,
! [A: $tType,B: $tType,P1: A > $o,P22: B > $o] :
( ( rBT_Im1000242676el_rbt @ A @ A @ B @ B @ ( bNF_eq_onp @ A @ P1 ) @ ( bNF_eq_onp @ B @ P22 ) )
= ( bNF_eq_onp @ ( rBT_Im246033960le_rbt @ A @ B ) @ ( rBT_Im1931894874ed_rbt @ A @ B @ P1 @ P22 ) ) ) ).
% rbt.rel_eq_onp
thf(fact_113_rbt_Oset__cases_I1_J,axiom,
! [B: $tType,A: $tType,E2: A,A2: rBT_Im246033960le_rbt @ A @ B] :
( ( member @ A @ E2 @ ( rBT_Im1178217900t1_rbt @ A @ B @ A2 ) )
=> ( ! [Z1: rBT_Im1923302023_color,Z22: rBT_Im246033960le_rbt @ A @ B] :
( ? [Z32: A,Z42: B,Z5: rBT_Im246033960le_rbt @ A @ B] :
( A2
= ( rBT_Im480247531Branch @ A @ B @ Z1 @ Z22 @ Z32 @ Z42 @ Z5 ) )
=> ~ ( member @ A @ E2 @ ( rBT_Im1178217900t1_rbt @ A @ B @ Z22 ) ) )
=> ( ! [Z1: rBT_Im1923302023_color,Z22: rBT_Im246033960le_rbt @ A @ B,Z42: B,Z5: rBT_Im246033960le_rbt @ A @ B] :
( A2
!= ( rBT_Im480247531Branch @ A @ B @ Z1 @ Z22 @ E2 @ Z42 @ Z5 ) )
=> ~ ! [Z1: rBT_Im1923302023_color,Z22: rBT_Im246033960le_rbt @ A @ B,Z32: A,Z42: B,Z5: rBT_Im246033960le_rbt @ A @ B] :
( ( A2
= ( rBT_Im480247531Branch @ A @ B @ Z1 @ Z22 @ Z32 @ Z42 @ Z5 ) )
=> ~ ( member @ A @ E2 @ ( rBT_Im1178217900t1_rbt @ A @ B @ Z5 ) ) ) ) ) ) ).
% rbt.set_cases(1)
thf(fact_114_lift__Suc__mono__less__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [F3: nat > A,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less @ A @ ( F3 @ N2 ) @ ( F3 @ ( suc @ N2 ) ) )
=> ( ( ord_less @ A @ ( F3 @ N ) @ ( F3 @ M ) )
= ( ord_less @ nat @ N @ M ) ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_115_lift__Suc__mono__less,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [F3: nat > A,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less @ A @ ( F3 @ N2 ) @ ( F3 @ ( suc @ N2 ) ) )
=> ( ( ord_less @ nat @ N @ N3 )
=> ( ord_less @ A @ ( F3 @ N ) @ ( F3 @ N3 ) ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_116_bheight_Osimps_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( rBT_Im315265182height @ A @ B @ ( rBT_Im418718756_Empty @ A @ B ) )
= ( zero_zero @ nat ) ) ).
% bheight.simps(1)
thf(fact_117_rbt_Opred__mono__strong,axiom,
! [B: $tType,A: $tType,P1: A > $o,P22: B > $o,X: rBT_Im246033960le_rbt @ A @ B,P1a: A > $o,P2a: B > $o] :
( ( rBT_Im1931894874ed_rbt @ A @ B @ P1 @ P22 @ X )
=> ( ! [Z1: A] :
( ( member @ A @ Z1 @ ( rBT_Im1178217900t1_rbt @ A @ B @ X ) )
=> ( ( P1 @ Z1 )
=> ( P1a @ Z1 ) ) )
=> ( ! [Z22: B] :
( ( member @ B @ Z22 @ ( rBT_Im1550225131t2_rbt @ A @ B @ X ) )
=> ( ( P22 @ Z22 )
=> ( P2a @ Z22 ) ) )
=> ( rBT_Im1931894874ed_rbt @ A @ B @ P1a @ P2a @ X ) ) ) ) ).
% rbt.pred_mono_strong
thf(fact_118_rbt_Opred__cong,axiom,
! [B: $tType,A: $tType,X: rBT_Im246033960le_rbt @ A @ B,Ya: rBT_Im246033960le_rbt @ A @ B,P1: A > $o,P1a: A > $o,P22: B > $o,P2a: B > $o] :
( ( X = Ya )
=> ( ! [Z1: A] :
( ( member @ A @ Z1 @ ( rBT_Im1178217900t1_rbt @ A @ B @ Ya ) )
=> ( ( P1 @ Z1 )
= ( P1a @ Z1 ) ) )
=> ( ! [Z22: B] :
( ( member @ B @ Z22 @ ( rBT_Im1550225131t2_rbt @ A @ B @ Ya ) )
=> ( ( P22 @ Z22 )
= ( P2a @ Z22 ) ) )
=> ( ( rBT_Im1931894874ed_rbt @ A @ B @ P1 @ P22 @ X )
= ( rBT_Im1931894874ed_rbt @ A @ B @ P1a @ P2a @ Ya ) ) ) ) ) ).
% rbt.pred_cong
thf(fact_119_rbt_Oset__intros_I4_J,axiom,
! [B: $tType,A: $tType,Xe: B,A2a: rBT_Im246033960le_rbt @ A @ B,A1a: rBT_Im1923302023_color,A3a: A,A4a: B,A5a: rBT_Im246033960le_rbt @ A @ B] :
( ( member @ B @ Xe @ ( rBT_Im1550225131t2_rbt @ A @ B @ A2a ) )
=> ( member @ B @ Xe @ ( rBT_Im1550225131t2_rbt @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ A1a @ A2a @ A3a @ A4a @ A5a ) ) ) ) ).
% rbt.set_intros(4)
thf(fact_120_rbt_Oset__intros_I5_J,axiom,
! [B: $tType,A: $tType,A4a: B,A1a: rBT_Im1923302023_color,A2a: rBT_Im246033960le_rbt @ A @ B,A3a: A,A5a: rBT_Im246033960le_rbt @ A @ B] : ( member @ B @ A4a @ ( rBT_Im1550225131t2_rbt @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ A1a @ A2a @ A3a @ A4a @ A5a ) ) ) ).
% rbt.set_intros(5)
thf(fact_121_rbt_Oset__intros_I6_J,axiom,
! [B: $tType,A: $tType,Xg: B,A5a: rBT_Im246033960le_rbt @ A @ B,A1a: rBT_Im1923302023_color,A2a: rBT_Im246033960le_rbt @ A @ B,A3a: A,A4a: B] :
( ( member @ B @ Xg @ ( rBT_Im1550225131t2_rbt @ A @ B @ A5a ) )
=> ( member @ B @ Xg @ ( rBT_Im1550225131t2_rbt @ A @ B @ ( rBT_Im480247531Branch @ A @ B @ A1a @ A2a @ A3a @ A4a @ A5a ) ) ) ) ).
% rbt.set_intros(6)
thf(fact_122_rbt_Oset__cases_I2_J,axiom,
! [B: $tType,A: $tType,E2: B,A2: rBT_Im246033960le_rbt @ A @ B] :
( ( member @ B @ E2 @ ( rBT_Im1550225131t2_rbt @ A @ B @ A2 ) )
=> ( ! [Z1: rBT_Im1923302023_color,Z22: rBT_Im246033960le_rbt @ A @ B] :
( ? [Z32: A,Z42: B,Z5: rBT_Im246033960le_rbt @ A @ B] :
( A2
= ( rBT_Im480247531Branch @ A @ B @ Z1 @ Z22 @ Z32 @ Z42 @ Z5 ) )
=> ~ ( member @ B @ E2 @ ( rBT_Im1550225131t2_rbt @ A @ B @ Z22 ) ) )
=> ( ! [Z1: rBT_Im1923302023_color,Z22: rBT_Im246033960le_rbt @ A @ B,Z32: A,Z5: rBT_Im246033960le_rbt @ A @ B] :
( A2
!= ( rBT_Im480247531Branch @ A @ B @ Z1 @ Z22 @ Z32 @ E2 @ Z5 ) )
=> ~ ! [Z1: rBT_Im1923302023_color,Z22: rBT_Im246033960le_rbt @ A @ B,Z32: A,Z42: B,Z5: rBT_Im246033960le_rbt @ A @ B] :
( ( A2
= ( rBT_Im480247531Branch @ A @ B @ Z1 @ Z22 @ Z32 @ Z42 @ Z5 ) )
=> ~ ( member @ B @ E2 @ ( rBT_Im1550225131t2_rbt @ A @ B @ Z5 ) ) ) ) ) ) ).
% rbt.set_cases(2)
thf(fact_123_rbt_Orel__cong,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,X: rBT_Im246033960le_rbt @ A @ B,Ya: rBT_Im246033960le_rbt @ A @ B,Y: rBT_Im246033960le_rbt @ C @ D,Xa: rBT_Im246033960le_rbt @ C @ D,R1: A > C > $o,R1a: A > C > $o,R22: B > D > $o,R2a: B > D > $o] :
( ( X = Ya )
=> ( ( Y = Xa )
=> ( ! [Z1: A,Y1: C] :
( ( member @ A @ Z1 @ ( rBT_Im1178217900t1_rbt @ A @ B @ Ya ) )
=> ( ( member @ C @ Y1 @ ( rBT_Im1178217900t1_rbt @ C @ D @ Xa ) )
=> ( ( R1 @ Z1 @ Y1 )
= ( R1a @ Z1 @ Y1 ) ) ) )
=> ( ! [Z22: B,Y26: D] :
( ( member @ B @ Z22 @ ( rBT_Im1550225131t2_rbt @ A @ B @ Ya ) )
=> ( ( member @ D @ Y26 @ ( rBT_Im1550225131t2_rbt @ C @ D @ Xa ) )
=> ( ( R22 @ Z22 @ Y26 )
= ( R2a @ Z22 @ Y26 ) ) ) )
=> ( ( rBT_Im1000242676el_rbt @ A @ C @ B @ D @ R1 @ R22 @ X @ Y )
= ( rBT_Im1000242676el_rbt @ A @ C @ B @ D @ R1a @ R2a @ Ya @ Xa ) ) ) ) ) ) ).
% rbt.rel_cong
thf(fact_124_rbt_Orel__mono__strong,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,R1: A > C > $o,R22: B > D > $o,X: rBT_Im246033960le_rbt @ A @ B,Y: rBT_Im246033960le_rbt @ C @ D,R1a: A > C > $o,R2a: B > D > $o] :
( ( rBT_Im1000242676el_rbt @ A @ C @ B @ D @ R1 @ R22 @ X @ Y )
=> ( ! [Z1: A,Y1: C] :
( ( member @ A @ Z1 @ ( rBT_Im1178217900t1_rbt @ A @ B @ X ) )
=> ( ( member @ C @ Y1 @ ( rBT_Im1178217900t1_rbt @ C @ D @ Y ) )
=> ( ( R1 @ Z1 @ Y1 )
=> ( R1a @ Z1 @ Y1 ) ) ) )
=> ( ! [Z22: B,Y26: D] :
( ( member @ B @ Z22 @ ( rBT_Im1550225131t2_rbt @ A @ B @ X ) )
=> ( ( member @ D @ Y26 @ ( rBT_Im1550225131t2_rbt @ C @ D @ Y ) )
=> ( ( R22 @ Z22 @ Y26 )
=> ( R2a @ Z22 @ Y26 ) ) ) )
=> ( rBT_Im1000242676el_rbt @ A @ C @ B @ D @ R1a @ R2a @ X @ Y ) ) ) ) ).
% rbt.rel_mono_strong
thf(fact_125_rbt_Orel__refl__strong,axiom,
! [B: $tType,A: $tType,X: rBT_Im246033960le_rbt @ A @ B,R1a: A > A > $o,R2a: B > B > $o] :
( ! [Z1: A] :
( ( member @ A @ Z1 @ ( rBT_Im1178217900t1_rbt @ A @ B @ X ) )
=> ( R1a @ Z1 @ Z1 ) )
=> ( ! [Z22: B] :
( ( member @ B @ Z22 @ ( rBT_Im1550225131t2_rbt @ A @ B @ X ) )
=> ( R2a @ Z22 @ Z22 ) )
=> ( rBT_Im1000242676el_rbt @ A @ A @ B @ B @ R1a @ R2a @ X @ X ) ) ) ).
% rbt.rel_refl_strong
thf(fact_126_not__gr__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [N: A] :
( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N ) )
= ( N
= ( zero_zero @ A ) ) ) ) ).
% not_gr_zero
thf(fact_127_color_Osize__gen_I1_J,axiom,
( ( rBT_Im800559290_color @ rBT_Impl_Mirabelle_R )
= ( zero_zero @ nat ) ) ).
% color.size_gen(1)
thf(fact_128_color_Osize__gen_I2_J,axiom,
( ( rBT_Im800559290_color @ rBT_Impl_Mirabelle_B )
= ( zero_zero @ nat ) ) ).
% color.size_gen(2)
thf(fact_129_less__numeral__extra_I3_J,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% less_numeral_extra(3)
thf(fact_130_zero__less__iff__neq__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [N: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ N )
= ( N
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_iff_neq_zero
thf(fact_131_gr__implies__not__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [M: A,N: A] :
( ( ord_less @ A @ M @ N )
=> ( N
!= ( zero_zero @ A ) ) ) ) ).
% gr_implies_not_zero
thf(fact_132_not__less__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [N: A] :
~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% not_less_zero
thf(fact_133_gr__zeroI,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [N: A] :
( ( N
!= ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ N ) ) ) ).
% gr_zeroI
thf(fact_134_color_Osize_I3_J,axiom,
( ( size_size @ rBT_Im1923302023_color @ rBT_Impl_Mirabelle_R )
= ( zero_zero @ nat ) ) ).
% color.size(3)
thf(fact_135_color_Osize_I4_J,axiom,
( ( size_size @ rBT_Im1923302023_color @ rBT_Impl_Mirabelle_B )
= ( zero_zero @ nat ) ) ).
% color.size(4)
thf(fact_136_rbt_Osize__gen_I1_J,axiom,
! [A: $tType,B: $tType,Xa: A > nat,X: B > nat] :
( ( rBT_Im1485196410ze_rbt @ A @ B @ Xa @ X @ ( rBT_Im418718756_Empty @ A @ B ) )
= ( zero_zero @ nat ) ) ).
% rbt.size_gen(1)
thf(fact_137_rbt_Osize_I3_J,axiom,
! [A: $tType,B: $tType] :
( ( size_size @ ( rBT_Im246033960le_rbt @ A @ B ) @ ( rBT_Im418718756_Empty @ A @ B ) )
= ( zero_zero @ nat ) ) ).
% rbt.size(3)
thf(fact_138_rbt_Osize_I4_J,axiom,
! [A: $tType,B: $tType,X21: rBT_Im1923302023_color,X22: rBT_Im246033960le_rbt @ A @ B,X23: A,X24: B,X25: rBT_Im246033960le_rbt @ A @ B] :
( ( size_size @ ( rBT_Im246033960le_rbt @ A @ B ) @ ( rBT_Im480247531Branch @ A @ B @ X21 @ X22 @ X23 @ X24 @ X25 ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( size_size @ ( rBT_Im246033960le_rbt @ A @ B ) @ X22 ) @ ( size_size @ ( rBT_Im246033960le_rbt @ A @ B ) @ X25 ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% rbt.size(4)
thf(fact_139_of__nat__0__less__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% of_nat_0_less_iff
thf(fact_140_sgn__greater,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( sgn_sgn @ A @ A2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% sgn_greater
thf(fact_141_sgn__less,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less @ A @ ( sgn_sgn @ A @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% sgn_less
thf(fact_142_sgn__sgn,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( sgn_sgn @ A @ ( sgn_sgn @ A @ A2 ) )
= ( sgn_sgn @ A @ A2 ) ) ) ).
% sgn_sgn
thf(fact_143_add__less__cancel__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,C2: A,B3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B3 @ C2 ) )
= ( ord_less @ A @ A2 @ B3 ) ) ) ).
% add_less_cancel_right
thf(fact_144_add__less__cancel__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [C2: A,A2: A,B3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B3 ) )
= ( ord_less @ A @ A2 @ B3 ) ) ) ).
% add_less_cancel_left
thf(fact_145_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_146_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_147_less__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( ord_less @ A @ A2 @ ( plus_plus @ A @ B3 @ A2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B3 ) ) ) ).
% less_add_same_cancel2
thf(fact_148_less__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( ord_less @ A @ A2 @ ( plus_plus @ A @ A2 @ B3 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B3 ) ) ) ).
% less_add_same_cancel1
thf(fact_149_add__less__same__cancel2,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ B3 ) @ B3 )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel2
thf(fact_150_add__less__same__cancel1,axiom,
! [A: $tType] :
( ( ordere516151231imp_le @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ B3 @ A2 ) @ B3 )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel1
thf(fact_151_of__nat__less__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [M: nat,N: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
= ( ord_less @ nat @ M @ N ) ) ) ).
% of_nat_less_iff
thf(fact_152_linordered__field__class_Osign__simps_I26_J,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B3 ) @ C2 )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B3 @ C2 ) ) ) ) ).
% linordered_field_class.sign_simps(26)
thf(fact_153_linordered__field__class_Osign__simps_I27_J,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ( ( plus_plus @ A )
= ( ^ [A6: A,B4: A] : ( plus_plus @ A @ B4 @ A6 ) ) ) ) ).
% linordered_field_class.sign_simps(27)
thf(fact_154_linordered__field__class_Osign__simps_I28_J,axiom,
! [A: $tType] :
( ( linordered_field @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A,C2: A] :
( ( plus_plus @ A @ B3 @ ( plus_plus @ A @ A2 @ C2 ) )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B3 @ C2 ) ) ) ) ).
% linordered_field_class.sign_simps(28)
thf(fact_155_add__less__imp__less__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [A2: A,C2: A,B3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B3 @ C2 ) )
=> ( ord_less @ A @ A2 @ B3 ) ) ) ).
% add_less_imp_less_right
thf(fact_156_add__less__imp__less__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A @ ( type2 @ A ) )
=> ! [C2: A,A2: A,B3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B3 ) )
=> ( ord_less @ A @ A2 @ B3 ) ) ) ).
% add_less_imp_less_left
thf(fact_157_add__strict__right__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A] :
( ( ord_less @ A @ A2 @ B3 )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B3 @ C2 ) ) ) ) ).
% add_strict_right_mono
thf(fact_158_add__strict__left__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A] :
( ( ord_less @ A @ A2 @ B3 )
=> ( ord_less @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B3 ) ) ) ) ).
% add_strict_left_mono
thf(fact_159_add__strict__mono,axiom,
! [A: $tType] :
( ( strict2144017051up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A,D2: A] :
( ( ord_less @ A @ A2 @ B3 )
=> ( ( ord_less @ A @ C2 @ D2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B3 @ D2 ) ) ) ) ) ).
% add_strict_mono
thf(fact_160_add__mono__thms__linordered__field_I1_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( K = L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_161_add__mono__thms__linordered__field_I2_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_162_add__mono__thms__linordered__field_I5_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A @ ( type2 @ A ) )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_163_sgn__0__0,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ( sgn_sgn @ A @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% sgn_0_0
thf(fact_164_of__nat__less__0__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [M: nat] :
~ ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( zero_zero @ A ) ) ) ).
% of_nat_less_0_iff
thf(fact_165_of__nat__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [M: nat,N: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) )
=> ( ord_less @ nat @ M @ N ) ) ) ).
% of_nat_less_imp_less
thf(fact_166_less__imp__of__nat__less,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% less_imp_of_nat_less
thf(fact_167_add__neg__neg,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_neg_neg
thf(fact_168_add__pos__pos,axiom,
! [A: $tType] :
( ( ordere216010020id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B3 ) ) ) ) ) ).
% add_pos_pos
thf(fact_169_pos__add__strict,axiom,
! [A: $tType] :
( ( strict797366125id_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ B3 @ C2 )
=> ( ord_less @ A @ B3 @ ( plus_plus @ A @ A2 @ C2 ) ) ) ) ) ).
% pos_add_strict
thf(fact_170_sgn__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( sgn_sgn @ A @ A2 )
= ( one_one @ A ) ) ) ) ).
% sgn_pos
thf(fact_171_of__nat__zero__less__power__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [X: nat,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ X ) @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ X )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_172_of__bool__eq_I1_J,axiom,
! [A: $tType] :
( ( zero_neq_one @ A @ ( type2 @ A ) )
=> ( ( zero_neq_one_of_bool @ A @ $false )
= ( zero_zero @ A ) ) ) ).
% of_bool_eq(1)
thf(fact_173_rbt_Opred__set,axiom,
! [B: $tType,A: $tType] :
( ( rBT_Im1931894874ed_rbt @ A @ B )
= ( ^ [P12: A > $o,P23: B > $o,X8: rBT_Im246033960le_rbt @ A @ B] :
( ! [Y4: A] :
( ( member @ A @ Y4 @ ( rBT_Im1178217900t1_rbt @ A @ B @ X8 ) )
=> ( P12 @ Y4 ) )
& ! [Y4: B] :
( ( member @ B @ Y4 @ ( rBT_Im1550225131t2_rbt @ A @ B @ X8 ) )
=> ( P23 @ Y4 ) ) ) ) ) ).
% rbt.pred_set
thf(fact_174_of__bool__eq_I2_J,axiom,
! [A: $tType] :
( ( zero_neq_one @ A @ ( type2 @ A ) )
=> ( ( zero_neq_one_of_bool @ A @ $true )
= ( one_one @ A ) ) ) ).
% of_bool_eq(2)
thf(fact_175_power__inject__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [A2: A,M: nat,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ( ( power_power @ A @ A2 @ M )
= ( power_power @ A @ A2 @ N ) )
= ( M = N ) ) ) ) ).
% power_inject_exp
thf(fact_176_power__strict__increasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [B3: A,X: nat,Y: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ B3 )
=> ( ( ord_less @ A @ ( power_power @ A @ B3 @ X ) @ ( power_power @ A @ B3 @ Y ) )
= ( ord_less @ nat @ X @ Y ) ) ) ) ).
% power_strict_increasing_iff
thf(fact_177_power__Suc__less__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [A2: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ A2 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( power_power @ A @ A2 @ ( suc @ N ) ) @ ( one_one @ A ) ) ) ) ) ).
% power_Suc_less_one
thf(fact_178_of__bool__def,axiom,
! [A: $tType] :
( ( zero_neq_one @ A @ ( type2 @ A ) )
=> ( ( zero_neq_one_of_bool @ A )
= ( ^ [P3: $o] : ( if @ A @ P3 @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ) ) ).
% of_bool_def
thf(fact_179_zero__neq__one,axiom,
! [A: $tType] :
( ( zero_neq_one @ A @ ( type2 @ A ) )
=> ( ( zero_zero @ A )
!= ( one_one @ A ) ) ) ).
% zero_neq_one
thf(fact_180_split__of__bool,axiom,
! [A: $tType] :
( ( zero_neq_one @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P4: $o] :
( ( P @ ( zero_neq_one_of_bool @ A @ P4 ) )
= ( ( P4
=> ( P @ ( one_one @ A ) ) )
& ( ~ P4
=> ( P @ ( zero_zero @ A ) ) ) ) ) ) ).
% split_of_bool
thf(fact_181_split__of__bool__asm,axiom,
! [A: $tType] :
( ( zero_neq_one @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P4: $o] :
( ( P @ ( zero_neq_one_of_bool @ A @ P4 ) )
= ( ~ ( ( P4
& ~ ( P @ ( one_one @ A ) ) )
| ( ~ P4
& ~ ( P @ ( zero_zero @ A ) ) ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_182_less__numeral__extra_I4_J,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% less_numeral_extra(4)
thf(fact_183_one__less__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [A2: A,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ A @ ( one_one @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ) ).
% one_less_power
thf(fact_184_zero__less__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [A2: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% zero_less_power
thf(fact_185_power__strict__decreasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [N: nat,N4: nat,A2: A] :
( ( ord_less @ nat @ N @ N4 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ A2 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( power_power @ A @ A2 @ N4 ) @ ( power_power @ A @ A2 @ N ) ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_186_of__bool__eq__iff,axiom,
! [A: $tType] :
( ( zero_neq_one @ A @ ( type2 @ A ) )
=> ! [P4: $o,Q3: $o] :
( ( ( zero_neq_one_of_bool @ A @ P4 )
= ( zero_neq_one_of_bool @ A @ Q3 ) )
= ( P4 = Q3 ) ) ) ).
% of_bool_eq_iff
thf(fact_187_power__strict__increasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [N: nat,N4: nat,A2: A] :
( ( ord_less @ nat @ N @ N4 )
=> ( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ord_less @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ A2 @ N4 ) ) ) ) ) ).
% power_strict_increasing
thf(fact_188_power__less__imp__less__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [A2: A,M: nat,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ( ord_less @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N ) )
=> ( ord_less @ nat @ M @ N ) ) ) ) ).
% power_less_imp_less_exp
thf(fact_189_power__gt1,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [A2: A,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ord_less @ A @ ( one_one @ A ) @ ( power_power @ A @ A2 @ ( suc @ N ) ) ) ) ) ).
% power_gt1
thf(fact_190_not__one__less__zero,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A @ ( type2 @ A ) )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% not_one_less_zero
thf(fact_191_zero__less__one,axiom,
! [A: $tType] :
( ( zero_less_one @ A @ ( type2 @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% zero_less_one
thf(fact_192_less__numeral__extra_I1_J,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% less_numeral_extra(1)
thf(fact_193_less__numeral__extra_I2_J,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% less_numeral_extra(2)
thf(fact_194_less__add__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [A2: A] : ( ord_less @ A @ A2 @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) ) ) ).
% less_add_one
thf(fact_195_zero__less__two,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A @ ( type2 @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ) ).
% zero_less_two
thf(fact_196_sgn__1__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ( sgn_sgn @ A @ A2 )
= ( one_one @ A ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% sgn_1_pos
thf(fact_197_power__strict__mono,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,N: nat] :
( ( ord_less @ A @ A2 @ B3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B3 @ N ) ) ) ) ) ) ).
% power_strict_mono
thf(fact_198_sgn__neg,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( sgn_sgn @ A @ A2 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% sgn_neg
thf(fact_199_zero__less__power__abs__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type2 @ A ) )
=> ! [A2: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( abs_abs @ A @ A2 ) @ N ) )
= ( ( A2
!= ( zero_zero @ A ) )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% zero_less_power_abs_iff
thf(fact_200_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_201_neg__less__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ B3 ) @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less @ A @ A2 @ B3 ) ) ) ).
% neg_less_iff_less
thf(fact_202_abs__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type2 @ A ) )
=> ( ( abs_abs @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% abs_one
thf(fact_203_neg__less__0__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% neg_less_0_iff_less
thf(fact_204_neg__0__less__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% neg_0_less_iff_less
thf(fact_205_neg__less__pos,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% neg_less_pos
thf(fact_206_less__neg__neg,axiom,
! [A: $tType] :
( ( linord219039673up_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% less_neg_neg
thf(fact_207_zero__less__abs__iff,axiom,
! [A: $tType] :
( ( ordere142940540dd_abs @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( abs_abs @ A @ A2 ) )
= ( A2
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_abs_iff
thf(fact_208_power__increasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [B3: A,X: nat,Y: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ B3 )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ B3 @ X ) @ ( power_power @ A @ B3 @ Y ) )
= ( ord_less_eq @ nat @ X @ Y ) ) ) ) ).
% power_increasing_iff
thf(fact_209_power__le__imp__le__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ! [A2: A,M: nat,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N ) )
=> ( ord_less_eq @ nat @ M @ N ) ) ) ) ).
% power_le_imp_le_exp
thf(fact_210_strict__mono__less__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A @ ( type2 @ A ) )
& ( order @ B @ ( type2 @ B ) ) )
=> ! [F3: A > B,X: A,Y: A] :
( ( order_strict_mono @ A @ B @ F3 )
=> ( ( ord_less_eq @ B @ ( F3 @ X ) @ ( F3 @ Y ) )
= ( ord_less_eq @ A @ X @ Y ) ) ) ) ).
% strict_mono_less_eq
thf(fact_211_abs__not__less__zero,axiom,
! [A: $tType] :
( ( ordere142940540dd_abs @ A @ ( type2 @ A ) )
=> ! [A2: A] :
~ ( ord_less @ A @ ( abs_abs @ A @ A2 ) @ ( zero_zero @ A ) ) ) ).
% abs_not_less_zero
thf(fact_212_abs__of__pos,axiom,
! [A: $tType] :
( ( ordere142940540dd_abs @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( abs_abs @ A @ A2 )
= A2 ) ) ) ).
% abs_of_pos
thf(fact_213_complete__interval,axiom,
! [A: $tType] :
( ( condit1037483654norder @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,P: A > $o] :
( ( ord_less @ A @ A2 @ B3 )
=> ( ( P @ A2 )
=> ( ~ ( P @ B3 )
=> ? [C3: A] :
( ( ord_less_eq @ A @ A2 @ C3 )
& ( ord_less_eq @ A @ C3 @ B3 )
& ! [X6: A] :
( ( ( ord_less_eq @ A @ A2 @ X6 )
& ( ord_less @ A @ X6 @ C3 ) )
=> ( P @ X6 ) )
& ! [D3: A] :
( ! [X7: A] :
( ( ( ord_less_eq @ A @ A2 @ X7 )
& ( ord_less @ A @ X7 @ D3 ) )
=> ( P @ X7 ) )
=> ( ord_less_eq @ A @ D3 @ C3 ) ) ) ) ) ) ) ).
% complete_interval
thf(fact_214_order_Onot__eq__order__implies__strict,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( A2 != B3 )
=> ( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ord_less @ A @ A2 @ B3 ) ) ) ) ).
% order.not_eq_order_implies_strict
thf(fact_215_dual__order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A] :
( ( ord_less @ A @ B3 @ A2 )
=> ( ord_less_eq @ A @ B3 @ A2 ) ) ) ).
% dual_order.strict_implies_order
thf(fact_216_dual__order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [B4: A,A6: A] :
( ( ord_less_eq @ A @ B4 @ A6 )
& ( A6 != B4 ) ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_217_dual__order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [B4: A,A6: A] :
( ( ord_less @ A @ B4 @ A6 )
| ( A6 = B4 ) ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_218_order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( ord_less @ A @ A2 @ B3 )
=> ( ord_less_eq @ A @ A2 @ B3 ) ) ) ).
% order.strict_implies_order
thf(fact_219_dense__le__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z3: A] :
( ( ord_less @ A @ X @ Y )
=> ( ! [W: A] :
( ( ord_less @ A @ X @ W )
=> ( ( ord_less @ A @ W @ Y )
=> ( ord_less_eq @ A @ W @ Z3 ) ) )
=> ( ord_less_eq @ A @ Y @ Z3 ) ) ) ) ).
% dense_le_bounded
thf(fact_220_dense__ge__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Z3: A,X: A,Y: A] :
( ( ord_less @ A @ Z3 @ X )
=> ( ! [W: A] :
( ( ord_less @ A @ Z3 @ W )
=> ( ( ord_less @ A @ W @ X )
=> ( ord_less_eq @ A @ Y @ W ) ) )
=> ( ord_less_eq @ A @ Y @ Z3 ) ) ) ) ).
% dense_ge_bounded
thf(fact_221_dual__order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A,C2: A] :
( ( ord_less @ A @ B3 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ B3 )
=> ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% dual_order.strict_trans2
thf(fact_222_dual__order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B3 @ A2 )
=> ( ( ord_less @ A @ C2 @ B3 )
=> ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% dual_order.strict_trans1
thf(fact_223_order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [A6: A,B4: A] :
( ( ord_less_eq @ A @ A6 @ B4 )
& ( A6 != B4 ) ) ) ) ) ).
% order.strict_iff_order
thf(fact_224_order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [A6: A,B4: A] :
( ( ord_less @ A @ A6 @ B4 )
| ( A6 = B4 ) ) ) ) ) ).
% order.order_iff_strict
thf(fact_225_order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A] :
( ( ord_less @ A @ A2 @ B3 )
=> ( ( ord_less_eq @ A @ B3 @ C2 )
=> ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% order.strict_trans2
thf(fact_226_order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( ord_less @ A @ B3 @ C2 )
=> ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% order.strict_trans1
thf(fact_227_not__le__imp__less,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A] :
( ~ ( ord_less_eq @ A @ Y @ X )
=> ( ord_less @ A @ X @ Y ) ) ) ).
% not_le_imp_less
thf(fact_228_less__le__not__le,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [X8: A,Y4: A] :
( ( ord_less_eq @ A @ X8 @ Y4 )
& ~ ( ord_less_eq @ A @ Y4 @ X8 ) ) ) ) ) ).
% less_le_not_le
thf(fact_229_le__imp__less__or__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ X @ Y )
| ( X = Y ) ) ) ) ).
% le_imp_less_or_eq
thf(fact_230_le__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
| ( ord_less @ A @ Y @ X ) ) ) ).
% le_less_linear
thf(fact_231_dense__le,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Y: A,Z3: A] :
( ! [X7: A] :
( ( ord_less @ A @ X7 @ Y )
=> ( ord_less_eq @ A @ X7 @ Z3 ) )
=> ( ord_less_eq @ A @ Y @ Z3 ) ) ) ).
% dense_le
thf(fact_232_dense__ge,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Z3: A,Y: A] :
( ! [X7: A] :
( ( ord_less @ A @ Z3 @ X7 )
=> ( ord_less_eq @ A @ Y @ X7 ) )
=> ( ord_less_eq @ A @ Y @ Z3 ) ) ) ).
% dense_ge
thf(fact_233_less__le__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z3: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z3 )
=> ( ord_less @ A @ X @ Z3 ) ) ) ) ).
% less_le_trans
thf(fact_234_le__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z3: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ Z3 )
=> ( ord_less @ A @ X @ Z3 ) ) ) ) ).
% le_less_trans
thf(fact_235_antisym__conv2,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ~ ( ord_less @ A @ X @ Y ) )
= ( X = Y ) ) ) ) ).
% antisym_conv2
thf(fact_236_antisym__conv1,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ X @ Y )
= ( X = Y ) ) ) ) ).
% antisym_conv1
thf(fact_237_less__imp__le,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% less_imp_le
thf(fact_238_le__neq__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( A2 != B3 )
=> ( ord_less @ A @ A2 @ B3 ) ) ) ) ).
% le_neq_trans
thf(fact_239_not__less,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less @ A @ X @ Y ) )
= ( ord_less_eq @ A @ Y @ X ) ) ) ).
% not_less
thf(fact_240_not__le,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less_eq @ A @ X @ Y ) )
= ( ord_less @ A @ Y @ X ) ) ) ).
% not_le
thf(fact_241_order__less__le__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C @ ( type2 @ C ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B3: A,F3: A > C,C2: C] :
( ( ord_less @ A @ A2 @ B3 )
=> ( ( ord_less_eq @ C @ ( F3 @ B3 ) @ C2 )
=> ( ! [X7: A,Y2: A] :
( ( ord_less @ A @ X7 @ Y2 )
=> ( ord_less @ C @ ( F3 @ X7 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less @ C @ ( F3 @ A2 ) @ C2 ) ) ) ) ) ).
% order_less_le_subst2
thf(fact_242_order__less__le__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F3: B > A,B3: B,C2: B] :
( ( ord_less @ A @ A2 @ ( F3 @ B3 ) )
=> ( ( ord_less_eq @ B @ B3 @ C2 )
=> ( ! [X7: B,Y2: B] :
( ( ord_less_eq @ B @ X7 @ Y2 )
=> ( ord_less_eq @ A @ ( F3 @ X7 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less @ A @ A2 @ ( F3 @ C2 ) ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_243_order__le__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C @ ( type2 @ C ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B3: A,F3: A > C,C2: C] :
( ( ord_less_eq @ A @ A2 @ B3 )
=> ( ( ord_less @ C @ ( F3 @ B3 ) @ C2 )
=> ( ! [X7: A,Y2: A] :
( ( ord_less_eq @ A @ X7 @ Y2 )
=> ( ord_less_eq @ C @ ( F3 @ X7 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less @ C @ ( F3 @ A2 ) @ C2 ) ) ) ) ) ).
% order_le_less_subst2
thf(fact_244_order__le__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F3: B > A,B3: B,C2: B] :
( ( ord_less_eq @ A @ A2 @ ( F3 @ B3 ) )
=> ( ( ord_less @ B @ B3 @ C2 )
=> ( ! [X7: B,Y2: B] :
( ( ord_less @ B @ X7 @ Y2 )
=> ( ord_less @ A @ ( F3 @ X7 ) @ ( F3 @ Y2 ) ) )
=> ( ord_less @ A @ A2 @ ( F3 @ C2 ) ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_245_less__le,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [X8: A,Y4: A] :
( ( ord_less_eq @ A @ X8 @ Y4 )
& ( X8 != Y4 ) ) ) ) ) ).
% less_le
thf(fact_246_le__less,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [X8: A,Y4: A] :
( ( ord_less @ A @ X8 @ Y4 )
| ( X8 = Y4 ) ) ) ) ) ).
% le_less
thf(fact_247_leI,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% leI
thf(fact_248_leD,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less @ A @ X @ Y ) ) ) ).
% leD
thf(fact_249_pinf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z: A] :
! [X6: A] :
( ( ord_less @ A @ Z @ X6 )
=> ~ ( ord_less_eq @ A @ X6 @ T ) ) ) ).
% pinf(6)
thf(fact_250_pinf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z: A] :
! [X6: A] :
( ( ord_less @ A @ Z @ X6 )
=> ( ord_less_eq @ A @ T @ X6 ) ) ) ).
% pinf(8)
thf(fact_251_minf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z: A] :
! [X6: A] :
( ( ord_less @ A @ X6 @ Z )
=> ( ord_less_eq @ A @ X6 @ T ) ) ) ).
% minf(6)
thf(fact_252_minf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z: A] :
! [X6: A] :
( ( ord_less @ A @ X6 @ Z )
=> ~ ( ord_less_eq @ A @ T @ X6 ) ) ) ).
% minf(8)
thf(fact_253_abs__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type2 @ A ) )
=> ! [A2: A,B3: A] :
( ( ord_less @ A @ ( abs_abs @ A @ A2 ) @ B3 )
= ( ( ord_less @ A @ A2 @ B3 )
& ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ B3 ) ) ) ) ).
% abs_less_iff
thf(fact_254_rbt_Orel__mono,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,R1: A > C > $o,R1a: A > C > $o,R22: B > D > $o,R2a: B > D > $o] :
( ( ord_less_eq @ ( A > C > $o ) @ R1 @ R1a )
=> ( ( ord_less_eq @ ( B > D > $o ) @ R22 @ R2a )
=> ( ord_less_eq @ ( ( rBT_Im246033960le_rbt @ A @ B ) > ( rBT_Im246033960le_rbt @ C @ D ) > $o ) @ ( rBT_Im1000242676el_rbt @ A @ C @ B @ D @ R1 @ R22 ) @ ( rBT_Im1000242676el_rbt @ A @ C @ B @ D @ R1a @ R2a ) ) ) ) ).
% rbt.rel_mono
thf(fact_255_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A2: A] :
( ( ord_less_eq @ A @ B3 @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ B3 )
=> ( A2 = B3 ) ) ) ) ).
% dual_order.antisym
%----Type constructors (28)
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A7: $tType,A8: $tType] :
( ( preorder @ A8 @ ( type2 @ A8 ) )
=> ( preorder @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A7: $tType,A8: $tType] :
( ( order @ A8 @ ( type2 @ A8 ) )
=> ( order @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A7: $tType,A8: $tType] :
( ( ord @ A8 @ ( type2 @ A8 ) )
=> ( ord @ ( A7 > A8 ) @ ( type2 @ ( A7 > A8 ) ) ) ) ).
thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
condit1037483654norder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
ordere516151231imp_le @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
strict2144017051up_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add,axiom,
ordere223160158up_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere236663937imp_le @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add,axiom,
strict797366125id_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
canoni770627133id_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring,axiom,
linord1659791738miring @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add,axiom,
ordere216010020id_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Olinordered__semidom,axiom,
linordered_semidom @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Ozero__less__one,axiom,
zero_less_one @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
wellorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Ozero__neq__one,axiom,
zero_neq_one @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
preorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
linorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Ono__top,axiom,
no_top @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
order @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
ord @ nat @ ( type2 @ nat ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_4,axiom,
! [A7: $tType] : ( preorder @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_5,axiom,
! [A7: $tType] : ( order @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_6,axiom,
! [A7: $tType] : ( ord @ ( set @ A7 ) @ ( type2 @ ( set @ A7 ) ) ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_7,axiom,
preorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder_8,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_9,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_10,axiom,
ord @ $o @ ( type2 @ $o ) ).
%----Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $true @ X @ Y )
= X ) ).
%----Conjectures (2)
thf(conj_0,hypothesis,
rBT_Im1040612427e_inv1 @ a @ b @ t ).
thf(conj_1,conjecture,
rBT_Im1071879905_inv1l @ a @ b @ t ).
%------------------------------------------------------------------------------