TPTP Problem File: DAT207^1.p
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%------------------------------------------------------------------------------
% File : DAT207^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Sorted list operations 243
% Version : [Bla16] axioms : Especial.
% English :
% Refs : [Lam09] Lammich (2009), Collections Framework
% : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% Source : [Bla16]
% Names : sorted_list_operations__243.p [Bla16]
% Status : Theorem
% Rating : 0.00 v7.3.0, 0.33 v7.2.0, 0.50 v7.1.0
% Syntax : Number of formulae : 348 ( 123 unt; 56 typ; 0 def)
% Number of atoms : 844 ( 453 equ; 0 cnn)
% Maximal formula atoms : 16 ( 2 avg)
% Number of connectives : 4897 ( 170 ~; 14 |; 82 &;4153 @)
% ( 0 <=>; 478 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 8 avg)
% Number of types : 2 ( 1 usr)
% Number of type conns : 277 ( 277 >; 0 *; 0 +; 0 <<)
% Number of symbols : 56 ( 55 usr; 3 con; 0-5 aty)
% Number of variables : 1273 ( 34 ^;1134 !; 49 ?;1273 :)
% ( 56 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:41:25.989
%------------------------------------------------------------------------------
%----Could-be-implicit typings (4)
thf(ty_t_List_Olist,type,
list: $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
thf(ty_tf_a,type,
a: $tType ).
%----Explicit typings (52)
thf(sy_cl_Lattices_Obounded__lattice,type,
bounded_lattice:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Obot,type,
bot:
!>[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_Lattices_Olattice,type,
lattice:
!>[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_Orderings_Oorder__bot,type,
order_bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Owellorder,type,
wellorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Odense__order,type,
dense_order:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Osemilattice__sup,type,
semilattice_sup:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Obounded__lattice__bot,type,
bounded_lattice_bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Lattices_Obounded__semilattice__sup__bot,type,
bounde1808546759up_bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_BNF__Greatest__Fixpoint_Oshift,type,
bNF_Greatest_shift:
!>[A: $tType,B: $tType] : ( ( ( list @ A ) > B ) > A > ( list @ A ) > B ) ).
thf(sy_c_Lattices_Osup__class_Osup,type,
sup_sup:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_List_Oappend,type,
append:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Obind,type,
bind:
!>[A: $tType,B: $tType] : ( ( list @ A ) > ( A > ( list @ B ) ) > ( list @ B ) ) ).
thf(sy_c_List_Oinsert,type,
insert:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olinorder__class_Osorted__list__of__set,type,
linord467138063of_set:
!>[A: $tType] : ( ( set @ 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_Olist_Omap,type,
map:
!>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( list @ A ) > ( list @ Aa ) ) ).
thf(sy_c_List_Olist__ex1,type,
list_ex1:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Olistrelp,type,
listrelp:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( list @ A ) > ( list @ B ) > $o ) ).
thf(sy_c_List_Omap__tailrec,type,
map_tailrec:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( list @ A ) > ( list @ B ) ) ).
thf(sy_c_List_Omap__tailrec__rev,type,
map_tailrec_rev:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( list @ A ) > ( list @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Oord_Olexordp__eq,type,
lexordp_eq:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Oord__class_Olexordp__eq,type,
ord_lexordp_eq:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Oproduct__lists,type,
product_lists:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Orev,type,
rev:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Osplice,type,
splice:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Osublists,type,
sublists:
!>[A: $tType] : ( ( list @ A ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List__More_Ocombinatorial__product,type,
list_c659805718roduct:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_Misc_Olist__collect__set,type,
list_collect_set:
!>[B: $tType,A: $tType] : ( ( B > ( set @ A ) ) > ( list @ B ) > ( set @ A ) ) ).
thf(sy_c_Misc_Omerge,type,
merge:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Misc_Omerge__list,type,
merge_list:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) > ( list @ A ) ) ).
thf(sy_c_Misc_Orevg,type,
revg:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_Ois__empty,type,
is_empty:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Sorted__List__Operations__Mirabelle__fineeiboro_Odiff__sorted,type,
sorted1267110213sorted:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Sorted__List__Operations__Mirabelle__fineeiboro_Ointer__sorted,type,
sorted2037043510sorted:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Sorted__List__Operations__Mirabelle__fineeiboro_Osubset__sorted,type,
sorted1061247458sorted:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_l1_H____,type,
l1: list @ a ).
thf(sy_v_l2a____,type,
l2a: list @ a ).
thf(sy_v_x1____,type,
x1: a ).
%----Relevant facts (255)
thf(fact_0_local_ONil,axiom,
( l2a
= ( nil @ a ) ) ).
% local.Nil
thf(fact_1_ind__hyp,axiom,
! [L2: list @ a] :
( ( l1 = L2 )
= ( ( sorted1061247458sorted @ a @ l1 @ L2 )
& ( sorted1061247458sorted @ a @ L2 @ l1 ) ) ) ).
% ind_hyp
thf(fact_2_subset__sorted_Osimps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X1: A,L1: list @ A] :
~ ( sorted1061247458sorted @ A @ ( cons @ A @ X1 @ L1 ) @ ( nil @ A ) ) ) ).
% subset_sorted.simps(2)
thf(fact_3_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_4_subset__sorted_Osimps_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [L2: list @ A] : ( sorted1061247458sorted @ A @ ( nil @ A ) @ L2 ) ) ).
% subset_sorted.simps(1)
thf(fact_5_not__Cons__self2,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( cons @ A @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_6_list__tail__coinc,axiom,
! [A: $tType,N1: A,R1: list @ A,N2: A,R2: list @ A] :
( ( ( cons @ A @ N1 @ R1 )
= ( cons @ A @ N2 @ R2 ) )
=> ( ( N1 = N2 )
& ( R1 = R2 ) ) ) ).
% list_tail_coinc
thf(fact_7_subset__sorted_Osimps_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X1: A,L1: list @ A,X2: A,L2: list @ A] :
( ( sorted1061247458sorted @ A @ ( cons @ A @ X1 @ L1 ) @ ( cons @ A @ X2 @ L2 ) )
= ( ~ ( ord_less @ A @ X1 @ X2 )
& ( ~ ( ord_less @ A @ X1 @ X2 )
=> ( ( ( X1 = X2 )
=> ( sorted1061247458sorted @ A @ L1 @ L2 ) )
& ( ( X1 != X2 )
=> ( sorted1061247458sorted @ A @ ( cons @ A @ X1 @ L1 ) @ L2 ) ) ) ) ) ) ) ).
% subset_sorted.simps(3)
thf(fact_8_shift__def,axiom,
! [B: $tType,A: $tType] :
( ( bNF_Greatest_shift @ A @ B )
= ( ^ [Lab: ( list @ A ) > B,K: A,Kl: list @ A] : ( Lab @ ( cons @ A @ K @ Kl ) ) ) ) ).
% shift_def
thf(fact_9_subset__sorted_Oelims_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: list @ A,Xa: list @ A] :
( ~ ( sorted1061247458sorted @ A @ X @ Xa )
=> ( ( ? [X12: A,L12: list @ A] :
( X
= ( cons @ A @ X12 @ L12 ) )
=> ( Xa
!= ( nil @ A ) ) )
=> ~ ! [X12: A,L12: list @ A] :
( ( X
= ( cons @ A @ X12 @ L12 ) )
=> ! [X23: A,L22: list @ A] :
( ( Xa
= ( cons @ A @ X23 @ L22 ) )
=> ( ~ ( ord_less @ A @ X12 @ X23 )
& ( ~ ( ord_less @ A @ X12 @ X23 )
=> ( ( ( X12 = X23 )
=> ( sorted1061247458sorted @ A @ L12 @ L22 ) )
& ( ( X12 != X23 )
=> ( sorted1061247458sorted @ A @ ( cons @ A @ X12 @ L12 ) @ L22 ) ) ) ) ) ) ) ) ) ) ).
% subset_sorted.elims(3)
thf(fact_10_subset__sorted_Oelims_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: list @ A,Xa: list @ A] :
( ( sorted1061247458sorted @ A @ X @ Xa )
=> ( ( ( X
= ( nil @ A ) )
=> ! [L22: list @ A] : Xa != L22 )
=> ~ ! [X12: A,L12: list @ A] :
( ( X
= ( cons @ A @ X12 @ L12 ) )
=> ! [X23: A,L22: list @ A] :
( ( Xa
= ( cons @ A @ X23 @ L22 ) )
=> ~ ( ~ ( ord_less @ A @ X12 @ X23 )
& ( ~ ( ord_less @ A @ X12 @ X23 )
=> ( ( ( X12 = X23 )
=> ( sorted1061247458sorted @ A @ L12 @ L22 ) )
& ( ( X12 != X23 )
=> ( sorted1061247458sorted @ A @ ( cons @ A @ X12 @ L12 ) @ L22 ) ) ) ) ) ) ) ) ) ) ).
% subset_sorted.elims(2)
thf(fact_11_subset__sorted_Oelims_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: list @ A,Xa: list @ A,Y: $o] :
( ( ( sorted1061247458sorted @ A @ X @ Xa )
= Y )
=> ( ( ( X
= ( nil @ A ) )
=> ~ Y )
=> ( ( ? [X12: A,L12: list @ A] :
( X
= ( cons @ A @ X12 @ L12 ) )
=> ( ( Xa
= ( nil @ A ) )
=> Y ) )
=> ~ ! [X12: A,L12: list @ A] :
( ( X
= ( cons @ A @ X12 @ L12 ) )
=> ! [X23: A,L22: list @ A] :
( ( Xa
= ( cons @ A @ X23 @ L22 ) )
=> ( Y
= ( ~ ( ~ ( ord_less @ A @ X12 @ X23 )
& ( ~ ( ord_less @ A @ X12 @ X23 )
=> ( ( ( X12 = X23 )
=> ( sorted1061247458sorted @ A @ L12 @ L22 ) )
& ( ( X12 != X23 )
=> ( sorted1061247458sorted @ A @ ( cons @ A @ X12 @ L12 ) @ L22 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% subset_sorted.elims(1)
thf(fact_12_list_Odistinct_I1_J,axiom,
! [A: $tType,X21: A,X22: list @ A] :
( ( nil @ A )
!= ( cons @ A @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_13_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_14_transpose_Ocases,axiom,
! [A: $tType,X: list @ ( list @ A )] :
( ( X
!= ( nil @ ( list @ A ) ) )
=> ( ! [Xss: list @ ( list @ A )] :
( X
!= ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
=> ~ ! [X3: A,Xs2: list @ A,Xss: list @ ( list @ A )] :
( X
!= ( cons @ ( list @ A ) @ ( cons @ A @ X3 @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_15_subset__sorted_Oinduct,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
( ! [X12: list @ A] : ( P @ ( nil @ A ) @ X12 )
=> ( ! [X12: A,L12: list @ A] : ( P @ ( cons @ A @ X12 @ L12 ) @ ( nil @ A ) )
=> ( ! [X12: A,L12: list @ A,X23: A,L22: list @ A] :
( ( ~ ( ord_less @ A @ X12 @ X23 )
=> ( ( X12 = X23 )
=> ( P @ L12 @ L22 ) ) )
=> ( ( ~ ( ord_less @ A @ X12 @ X23 )
=> ( ( X12 != X23 )
=> ( P @ ( cons @ A @ X12 @ L12 ) @ L22 ) ) )
=> ( P @ ( cons @ A @ X12 @ L12 ) @ ( cons @ A @ X23 @ L22 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ) ).
% subset_sorted.induct
thf(fact_16_inter__sorted_Oinduct,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
( ! [X12: list @ A] : ( P @ ( nil @ A ) @ X12 )
=> ( ! [V: A,Va: list @ A] : ( P @ ( cons @ A @ V @ Va ) @ ( nil @ A ) )
=> ( ! [X12: A,L12: list @ A,X23: A,L22: list @ A] :
( ( ( ord_less @ A @ X12 @ X23 )
=> ( P @ L12 @ ( cons @ A @ X23 @ L22 ) ) )
=> ( ( ~ ( ord_less @ A @ X12 @ X23 )
=> ( ( X12 = X23 )
=> ( P @ L12 @ L22 ) ) )
=> ( ( ~ ( ord_less @ A @ X12 @ X23 )
=> ( ( X12 != X23 )
=> ( P @ ( cons @ A @ X12 @ L12 ) @ L22 ) ) )
=> ( P @ ( cons @ A @ X12 @ L12 ) @ ( cons @ A @ X23 @ L22 ) ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ) ).
% inter_sorted.induct
thf(fact_17_mergesort__by__rel__merge__induct,axiom,
! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,R: A > B > $o,Xs: list @ A,Ys: list @ B] :
( ! [Xs2: list @ A] : ( P @ Xs2 @ ( nil @ B ) )
=> ( ! [X12: list @ B] : ( P @ ( nil @ A ) @ X12 )
=> ( ! [X3: A,Xs2: list @ A,Y2: B,Ys2: list @ B] :
( ( R @ X3 @ Y2 )
=> ( ( P @ Xs2 @ ( cons @ B @ Y2 @ Ys2 ) )
=> ( P @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ B @ Y2 @ Ys2 ) ) ) )
=> ( ! [X3: A,Xs2: list @ A,Y2: B,Ys2: list @ B] :
( ~ ( R @ X3 @ Y2 )
=> ( ( P @ ( cons @ A @ X3 @ Xs2 ) @ Ys2 )
=> ( P @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ B @ Y2 @ Ys2 ) ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% mergesort_by_rel_merge_induct
thf(fact_18_mergesort__by__rel__merge_Oinduct,axiom,
! [A: $tType,P: ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o,A0: A > A > $o,A1: list @ A,A2: list @ A] :
( ! [R3: A > A > $o,X3: A,Xs2: list @ A,Y2: A,Ys2: list @ A] :
( ( ( R3 @ X3 @ Y2 )
=> ( P @ R3 @ Xs2 @ ( cons @ A @ Y2 @ Ys2 ) ) )
=> ( ( ~ ( R3 @ X3 @ Y2 )
=> ( P @ R3 @ ( cons @ A @ X3 @ Xs2 ) @ Ys2 ) )
=> ( P @ R3 @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ A @ Y2 @ Ys2 ) ) ) )
=> ( ! [R3: A > A > $o,Xs2: list @ A] : ( P @ R3 @ Xs2 @ ( nil @ A ) )
=> ( ! [R3: A > A > $o,V: A,Va: list @ A] : ( P @ R3 @ ( nil @ A ) @ ( cons @ A @ V @ Va ) )
=> ( P @ A0 @ A1 @ A2 ) ) ) ) ).
% mergesort_by_rel_merge.induct
thf(fact_19_map__tailrec__rev_Oinduct,axiom,
! [A: $tType,B: $tType,P: ( A > B ) > ( list @ A ) > ( list @ B ) > $o,A0: A > B,A1: list @ A,A2: list @ B] :
( ! [F: A > B,X12: list @ B] : ( P @ F @ ( nil @ A ) @ X12 )
=> ( ! [F: A > B,A3: A,As: list @ A,Bs: list @ B] :
( ( P @ F @ As @ ( cons @ B @ ( F @ A3 ) @ Bs ) )
=> ( P @ F @ ( cons @ A @ A3 @ As ) @ Bs ) )
=> ( P @ A0 @ A1 @ A2 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_20_list__nonempty__induct,axiom,
! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X3: A] : ( P @ ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ( ! [X3: A,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( P @ Xs2 )
=> ( P @ ( cons @ A @ X3 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_21_list__all__zip_Oinduct,axiom,
! [A: $tType,B: $tType,P: ( A > B > $o ) > ( list @ A ) > ( list @ B ) > $o,A0: A > B > $o,A1: list @ A,A2: list @ B] :
( ! [P2: A > B > $o] : ( P @ P2 @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [P2: A > B > $o,A3: A,As: list @ A,B2: B,Bs: list @ B] :
( ( P @ P2 @ As @ Bs )
=> ( P @ P2 @ ( cons @ A @ A3 @ As ) @ ( cons @ B @ B2 @ Bs ) ) )
=> ( ! [P2: A > B > $o,V: A,Va: list @ A] : ( P @ P2 @ ( cons @ A @ V @ Va ) @ ( nil @ B ) )
=> ( ! [P2: A > B > $o,V: B,Va: list @ B] : ( P @ P2 @ ( nil @ A ) @ ( cons @ B @ V @ Va ) )
=> ( P @ A0 @ A1 @ A2 ) ) ) ) ) ).
% list_all_zip.induct
thf(fact_22_list__induct__first2,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X3: A] : ( P @ ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ( ! [X12: A,X23: A,Xs2: list @ A] :
( ( P @ Xs2 )
=> ( P @ ( cons @ A @ X12 @ ( cons @ A @ X23 @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_induct_first2
thf(fact_23_remdups__adj_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > $o,A0: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X3: A] : ( P @ ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ( ! [X3: A,Y2: A,Xs2: list @ A] :
( ( ( X3 = Y2 )
=> ( P @ ( cons @ A @ X3 @ Xs2 ) ) )
=> ( ( ( X3 != Y2 )
=> ( P @ ( cons @ A @ Y2 @ Xs2 ) ) )
=> ( P @ ( cons @ A @ X3 @ ( cons @ A @ Y2 @ Xs2 ) ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% remdups_adj.induct
thf(fact_24_remdups__adj_Ocases,axiom,
! [A: $tType,X: list @ A] :
( ( X
!= ( nil @ A ) )
=> ( ! [X3: A] :
( X
!= ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ~ ! [X3: A,Y2: A,Xs2: list @ A] :
( X
!= ( cons @ A @ X3 @ ( cons @ A @ Y2 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_25_list__2pre__induct,axiom,
! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,W1: list @ A,W2: list @ B] :
( ( P @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [E: A,W12: list @ A,W22: list @ B] :
( ( P @ W12 @ W22 )
=> ( P @ ( cons @ A @ E @ W12 ) @ W22 ) )
=> ( ! [E: B,W13: list @ A,W23: list @ B] :
( ( P @ W13 @ W23 )
=> ( P @ W13 @ ( cons @ B @ E @ W23 ) ) )
=> ( P @ W1 @ W2 ) ) ) ) ).
% list_2pre_induct
thf(fact_26_splice_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
( ! [X12: list @ A] : ( P @ ( nil @ A ) @ X12 )
=> ( ! [V: A,Va: list @ A] : ( P @ ( cons @ A @ V @ Va ) @ ( nil @ A ) )
=> ( ! [X3: A,Xs2: list @ A,Y2: A,Ys2: list @ A] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ A @ Y2 @ Ys2 ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% splice.induct
thf(fact_27_list__induct2_H,axiom,
! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,Xs: list @ A,Ys: list @ B] :
( ( P @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X3: A,Xs2: list @ A] : ( P @ ( cons @ A @ X3 @ Xs2 ) @ ( nil @ B ) )
=> ( ! [Y2: B,Ys2: list @ B] : ( P @ ( nil @ A ) @ ( cons @ B @ Y2 @ Ys2 ) )
=> ( ! [X3: A,Xs2: list @ A,Y2: B,Ys2: list @ B] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ B @ Y2 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_28_neq__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
= ( ? [Y3: A,Ys3: list @ A] :
( Xs
= ( cons @ A @ Y3 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_29_list_Oinducts,axiom,
! [A: $tType,P: ( list @ A ) > $o,List: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X12: A,X23: list @ A] :
( ( P @ X23 )
=> ( P @ ( cons @ A @ X12 @ X23 ) ) )
=> ( P @ List ) ) ) ).
% list.inducts
thf(fact_30_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_31_zipf_Oinduct,axiom,
! [A: $tType,C: $tType,B: $tType,P: ( A > B > C ) > ( list @ A ) > ( list @ B ) > $o,A0: A > B > C,A1: list @ A,A2: list @ B] :
( ! [F: A > B > C] : ( P @ F @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [F: A > B > C,A3: A,As: list @ A,B2: B,Bs: list @ B] :
( ( P @ F @ As @ Bs )
=> ( P @ F @ ( cons @ A @ A3 @ As ) @ ( cons @ B @ B2 @ Bs ) ) )
=> ( ! [A3: A > B > C,V: A,Va: list @ A] : ( P @ A3 @ ( cons @ A @ V @ Va ) @ ( nil @ B ) )
=> ( ! [A3: A > B > C,V: B,Va: list @ B] : ( P @ A3 @ ( nil @ A ) @ ( cons @ B @ V @ Va ) )
=> ( P @ A0 @ A1 @ A2 ) ) ) ) ) ).
% zipf.induct
thf(fact_32_revg_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
( ! [X12: list @ A] : ( P @ ( nil @ A ) @ X12 )
=> ( ! [A3: A,As: list @ A,B2: list @ A] :
( ( P @ As @ ( cons @ A @ A3 @ B2 ) )
=> ( P @ ( cons @ A @ A3 @ As ) @ B2 ) )
=> ( P @ A0 @ A1 ) ) ) ).
% revg.induct
thf(fact_33_quicksort_Ocases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: list @ A] :
( ( X
!= ( nil @ A ) )
=> ~ ! [X3: A,Xs2: list @ A] :
( X
!= ( cons @ A @ X3 @ Xs2 ) ) ) ) ).
% quicksort.cases
thf(fact_34_insert__Nil,axiom,
! [A: $tType,X: A] :
( ( insert @ A @ X @ ( nil @ A ) )
= ( cons @ A @ X @ ( nil @ A ) ) ) ).
% insert_Nil
thf(fact_35_list__ex1__simps_I1_J,axiom,
! [A: $tType,P: A > $o] :
~ ( list_ex1 @ A @ P @ ( nil @ A ) ) ).
% list_ex1_simps(1)
thf(fact_36_inter__sorted_Oelims,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: list @ A,Xa: list @ A,Y: list @ A] :
( ( ( sorted2037043510sorted @ A @ X @ Xa )
= Y )
=> ( ( ( X
= ( nil @ A ) )
=> ( Y
!= ( nil @ A ) ) )
=> ( ( ? [V: A,Va: list @ A] :
( X
= ( cons @ A @ V @ Va ) )
=> ( ( Xa
= ( nil @ A ) )
=> ( Y
!= ( nil @ A ) ) ) )
=> ~ ! [X12: A,L12: list @ A] :
( ( X
= ( cons @ A @ X12 @ L12 ) )
=> ! [X23: A,L22: list @ A] :
( ( Xa
= ( cons @ A @ X23 @ L22 ) )
=> ~ ( ( ( ord_less @ A @ X12 @ X23 )
=> ( Y
= ( sorted2037043510sorted @ A @ L12 @ ( cons @ A @ X23 @ L22 ) ) ) )
& ( ~ ( ord_less @ A @ X12 @ X23 )
=> ( ( ( X12 = X23 )
=> ( Y
= ( cons @ A @ X12 @ ( sorted2037043510sorted @ A @ L12 @ L22 ) ) ) )
& ( ( X12 != X23 )
=> ( Y
= ( sorted2037043510sorted @ A @ ( cons @ A @ X12 @ L12 ) @ L22 ) ) ) ) ) ) ) ) ) ) ) ) ).
% inter_sorted.elims
thf(fact_37_diff__sorted_Oelims,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: list @ A,Xa: list @ A,Y: list @ A] :
( ( ( sorted1267110213sorted @ A @ X @ Xa )
= Y )
=> ( ( ( X
= ( nil @ A ) )
=> ( Y
!= ( nil @ A ) ) )
=> ( ! [V: A,Va: list @ A] :
( ( X
= ( cons @ A @ V @ Va ) )
=> ( ( Xa
= ( nil @ A ) )
=> ( Y
!= ( cons @ A @ V @ Va ) ) ) )
=> ~ ! [X12: A,L12: list @ A] :
( ( X
= ( cons @ A @ X12 @ L12 ) )
=> ! [X23: A,L22: list @ A] :
( ( Xa
= ( cons @ A @ X23 @ L22 ) )
=> ~ ( ( ( ord_less @ A @ X12 @ X23 )
=> ( Y
= ( cons @ A @ X12 @ ( sorted1267110213sorted @ A @ L12 @ ( cons @ A @ X23 @ L22 ) ) ) ) )
& ( ~ ( ord_less @ A @ X12 @ X23 )
=> ( ( ( X12 = X23 )
=> ( Y
= ( sorted1267110213sorted @ A @ L12 @ L22 ) ) )
& ( ( X12 != X23 )
=> ( Y
= ( sorted1267110213sorted @ A @ ( cons @ A @ X12 @ L12 ) @ L22 ) ) ) ) ) ) ) ) ) ) ) ) ).
% diff_sorted.elims
thf(fact_38_merge_Oelims,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: list @ A,Xa: list @ A,Y: list @ A] :
( ( ( merge @ A @ X @ Xa )
= Y )
=> ( ( ( X
= ( nil @ A ) )
=> ( Y != Xa ) )
=> ( ! [V: A,Va: list @ A] :
( ( X
= ( cons @ A @ V @ Va ) )
=> ( ( Xa
= ( nil @ A ) )
=> ( Y
!= ( cons @ A @ V @ Va ) ) ) )
=> ~ ! [X12: A,L12: list @ A] :
( ( X
= ( cons @ A @ X12 @ L12 ) )
=> ! [X23: A,L22: list @ A] :
( ( Xa
= ( cons @ A @ X23 @ L22 ) )
=> ~ ( ( ( ord_less @ A @ X12 @ X23 )
=> ( Y
= ( cons @ A @ X12 @ ( merge @ A @ L12 @ ( cons @ A @ X23 @ L22 ) ) ) ) )
& ( ~ ( ord_less @ A @ X12 @ X23 )
=> ( ( ( X12 = X23 )
=> ( Y
= ( cons @ A @ X12 @ ( merge @ A @ L12 @ L22 ) ) ) )
& ( ( X12 != X23 )
=> ( Y
= ( cons @ A @ X23 @ ( merge @ A @ ( cons @ A @ X12 @ L12 ) @ L22 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% merge.elims
thf(fact_39_revg_Oelims,axiom,
! [A: $tType,X: list @ A,Xa: list @ A,Y: list @ A] :
( ( ( revg @ A @ X @ Xa )
= Y )
=> ( ( ( X
= ( nil @ A ) )
=> ( Y != Xa ) )
=> ~ ! [A3: A,As: list @ A] :
( ( X
= ( cons @ A @ A3 @ As ) )
=> ( Y
!= ( revg @ A @ As @ ( cons @ A @ A3 @ Xa ) ) ) ) ) ) ).
% revg.elims
thf(fact_40_list__collect__set__simps_I2_J,axiom,
! [A: $tType,B: $tType,F2: B > ( set @ A ),A4: B] :
( ( list_collect_set @ B @ A @ F2 @ ( cons @ B @ A4 @ ( nil @ B ) ) )
= ( F2 @ A4 ) ) ).
% list_collect_set_simps(2)
thf(fact_41_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_42_lexordp__eq_Ocases,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A1: list @ A,A2: list @ A] :
( ( ord_lexordp_eq @ A @ A1 @ A2 )
=> ( ( ( A1
= ( nil @ A ) )
=> ! [Ys2: list @ A] : A2 != Ys2 )
=> ( ! [X3: A] :
( ? [Xs2: list @ A] :
( A1
= ( cons @ A @ X3 @ Xs2 ) )
=> ! [Y2: A] :
( ? [Ys2: list @ A] :
( A2
= ( cons @ A @ Y2 @ Ys2 ) )
=> ~ ( ord_less @ A @ X3 @ Y2 ) ) )
=> ~ ! [X3: A,Y2: A,Xs2: list @ A] :
( ( A1
= ( cons @ A @ X3 @ Xs2 ) )
=> ! [Ys2: list @ A] :
( ( A2
= ( cons @ A @ Y2 @ Ys2 ) )
=> ( ~ ( ord_less @ A @ X3 @ Y2 )
=> ( ~ ( ord_less @ A @ Y2 @ X3 )
=> ~ ( ord_lexordp_eq @ A @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ) ).
% lexordp_eq.cases
thf(fact_43_lexordp__eq_Osimps,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ( ( ord_lexordp_eq @ A )
= ( ^ [A12: list @ A,A22: list @ A] :
( ? [Ys3: list @ A] :
( ( A12
= ( nil @ A ) )
& ( A22 = Ys3 ) )
| ? [X4: A,Y3: A,Xs3: list @ A,Ys3: list @ A] :
( ( A12
= ( cons @ A @ X4 @ Xs3 ) )
& ( A22
= ( cons @ A @ Y3 @ Ys3 ) )
& ( ord_less @ A @ X4 @ Y3 ) )
| ? [X4: A,Y3: A,Xs3: list @ A,Ys3: list @ A] :
( ( A12
= ( cons @ A @ X4 @ Xs3 ) )
& ( A22
= ( cons @ A @ Y3 @ Ys3 ) )
& ~ ( ord_less @ A @ X4 @ Y3 )
& ~ ( ord_less @ A @ Y3 @ X4 )
& ( ord_lexordp_eq @ A @ Xs3 @ Ys3 ) ) ) ) ) ) ).
% lexordp_eq.simps
thf(fact_44_mem__Collect__eq,axiom,
! [A: $tType,A4: A,P: A > $o] :
( ( member @ A @ A4 @ ( collect @ A @ P ) )
= ( P @ A4 ) ) ).
% mem_Collect_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A: $tType,A5: set @ A] :
( ( collect @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_46_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_47_ext,axiom,
! [B: $tType,A: $tType,F2: A > B,G: A > B] :
( ! [X3: A] :
( ( F2 @ X3 )
= ( G @ X3 ) )
=> ( F2 = G ) ) ).
% ext
thf(fact_48_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_49_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_50_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_51_lexordp__eq__simps_I1_J,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [Ys: list @ A] : ( ord_lexordp_eq @ A @ ( nil @ A ) @ Ys ) ) ).
% lexordp_eq_simps(1)
thf(fact_52_lexordp__eq__simps_I2_J,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [Xs: list @ A] :
( ( ord_lexordp_eq @ A @ Xs @ ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ) ).
% lexordp_eq_simps(2)
thf(fact_53_lexordp__eq__simps_I4_J,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X: A,Xs: list @ A,Y: A,Ys: list @ A] :
( ( ord_lexordp_eq @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) )
= ( ( ord_less @ A @ X @ Y )
| ( ~ ( ord_less @ A @ Y @ X )
& ( ord_lexordp_eq @ A @ Xs @ Ys ) ) ) ) ) ).
% lexordp_eq_simps(4)
thf(fact_54_lexordp__eq__simps_I3_J,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X: A,Xs: list @ A] :
~ ( ord_lexordp_eq @ A @ ( cons @ A @ X @ Xs ) @ ( nil @ A ) ) ) ).
% lexordp_eq_simps(3)
thf(fact_55_merge__list_Oinduct,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) > $o,A0: list @ ( list @ A ),A1: list @ ( list @ A )] :
( ( P @ ( nil @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) )
=> ( ! [L: list @ A] : ( P @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L @ ( nil @ ( list @ A ) ) ) )
=> ( ! [La: list @ A,Acc2: list @ ( list @ A )] :
( ( P @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La @ Acc2 ) )
=> ( P @ ( cons @ ( list @ A ) @ La @ Acc2 ) @ ( nil @ ( list @ A ) ) ) )
=> ( ! [La: list @ A,Acc2: list @ ( list @ A ),L: list @ A] :
( ( P @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L @ ( cons @ ( list @ A ) @ La @ Acc2 ) ) )
=> ( P @ ( cons @ ( list @ A ) @ La @ Acc2 ) @ ( cons @ ( list @ A ) @ L @ ( nil @ ( list @ A ) ) ) ) )
=> ( ! [Acc2: list @ ( list @ A ),L12: list @ A,L22: list @ A,Ls: list @ ( list @ A )] :
( ( P @ ( cons @ ( list @ A ) @ ( merge @ A @ L12 @ L22 ) @ Acc2 ) @ Ls )
=> ( P @ Acc2 @ ( cons @ ( list @ A ) @ L12 @ ( cons @ ( list @ A ) @ L22 @ Ls ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ) ) ) ).
% merge_list.induct
thf(fact_56_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_57_lexordp__eq__refl,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [Xs: list @ A] : ( ord_lexordp_eq @ A @ Xs @ Xs ) ) ).
% lexordp_eq_refl
thf(fact_58_lexordp__eq__trans,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ord_lexordp_eq @ A @ Xs @ Ys )
=> ( ( ord_lexordp_eq @ A @ Ys @ Zs )
=> ( ord_lexordp_eq @ A @ Xs @ Zs ) ) ) ) ).
% lexordp_eq_trans
thf(fact_59_lexordp__eq__linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Xs: list @ A,Ys: list @ A] :
( ( ord_lexordp_eq @ A @ Xs @ Ys )
| ( ord_lexordp_eq @ A @ Ys @ Xs ) ) ) ).
% lexordp_eq_linear
thf(fact_60_lexordp__eq__antisym,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Xs: list @ A,Ys: list @ A] :
( ( ord_lexordp_eq @ A @ Xs @ Ys )
=> ( ( ord_lexordp_eq @ A @ Ys @ Xs )
=> ( Xs = Ys ) ) ) ) ).
% lexordp_eq_antisym
thf(fact_61_lexordp__eq_ONil,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [Ys: list @ A] : ( ord_lexordp_eq @ A @ ( nil @ A ) @ Ys ) ) ).
% lexordp_eq.Nil
thf(fact_62_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_63_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_64_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_65_merge_Osimps_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [L2: list @ A] :
( ( merge @ A @ ( nil @ A ) @ L2 )
= L2 ) ) ).
% merge.simps(1)
thf(fact_66_inter__sorted_Osimps_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [L2: list @ A] :
( ( sorted2037043510sorted @ A @ ( nil @ A ) @ L2 )
= ( nil @ A ) ) ) ).
% inter_sorted.simps(1)
thf(fact_67_diff__sorted_Osimps_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [L2: list @ A] :
( ( sorted1267110213sorted @ A @ ( nil @ A ) @ L2 )
= ( nil @ A ) ) ) ).
% diff_sorted.simps(1)
thf(fact_68_revg_Osimps_I2_J,axiom,
! [A: $tType,A4: A,As2: list @ A,B3: list @ A] :
( ( revg @ A @ ( cons @ A @ A4 @ As2 ) @ B3 )
= ( revg @ A @ As2 @ ( cons @ A @ A4 @ B3 ) ) ) ).
% revg.simps(2)
thf(fact_69_revg_Osimps_I1_J,axiom,
! [A: $tType,B3: list @ A] :
( ( revg @ A @ ( nil @ A ) @ B3 )
= B3 ) ).
% revg.simps(1)
thf(fact_70_lexordp__eq_OCons__eq,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Xs: list @ A,Ys: list @ A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ~ ( ord_less @ A @ Y @ X )
=> ( ( ord_lexordp_eq @ A @ Xs @ Ys )
=> ( ord_lexordp_eq @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) ) ) ) ).
% lexordp_eq.Cons_eq
thf(fact_71_lexordp__eq_OCons,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Xs: list @ A,Ys: list @ A] :
( ( ord_less @ A @ X @ Y )
=> ( ord_lexordp_eq @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) ) ).
% lexordp_eq.Cons
thf(fact_72_ord_Olexordp__eq_Ocases,axiom,
! [A: $tType,Less: A > A > $o,A1: list @ A,A2: list @ A] :
( ( lexordp_eq @ A @ Less @ A1 @ A2 )
=> ( ( ( A1
= ( nil @ A ) )
=> ! [Ys2: list @ A] : A2 != Ys2 )
=> ( ! [X3: A] :
( ? [Xs2: list @ A] :
( A1
= ( cons @ A @ X3 @ Xs2 ) )
=> ! [Y2: A] :
( ? [Ys2: list @ A] :
( A2
= ( cons @ A @ Y2 @ Ys2 ) )
=> ~ ( Less @ X3 @ Y2 ) ) )
=> ~ ! [X3: A,Y2: A,Xs2: list @ A] :
( ( A1
= ( cons @ A @ X3 @ Xs2 ) )
=> ! [Ys2: list @ A] :
( ( A2
= ( cons @ A @ Y2 @ Ys2 ) )
=> ( ~ ( Less @ X3 @ Y2 )
=> ( ~ ( Less @ Y2 @ X3 )
=> ~ ( lexordp_eq @ A @ Less @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ).
% ord.lexordp_eq.cases
thf(fact_73_ord_Olexordp__eq_Osimps,axiom,
! [A: $tType] :
( ( lexordp_eq @ A )
= ( ^ [Less2: A > A > $o,A12: list @ A,A22: list @ A] :
( ? [Ys3: list @ A] :
( ( A12
= ( nil @ A ) )
& ( A22 = Ys3 ) )
| ? [X4: A,Y3: A,Xs3: list @ A,Ys3: list @ A] :
( ( A12
= ( cons @ A @ X4 @ Xs3 ) )
& ( A22
= ( cons @ A @ Y3 @ Ys3 ) )
& ( Less2 @ X4 @ Y3 ) )
| ? [X4: A,Y3: A,Xs3: list @ A,Ys3: list @ A] :
( ( A12
= ( cons @ A @ X4 @ Xs3 ) )
& ( A22
= ( cons @ A @ Y3 @ Ys3 ) )
& ~ ( Less2 @ X4 @ Y3 )
& ~ ( Less2 @ Y3 @ X4 )
& ( lexordp_eq @ A @ Less2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% ord.lexordp_eq.simps
thf(fact_74_ord_Olexordp__eq_Oinducts,axiom,
! [A: $tType,Less: A > A > $o,X1: list @ A,X2: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
( ( lexordp_eq @ A @ Less @ X1 @ X2 )
=> ( ! [X12: list @ A] : ( P @ ( nil @ A ) @ X12 )
=> ( ! [X3: A,Y2: A,Xs2: list @ A,Ys2: list @ A] :
( ( Less @ X3 @ Y2 )
=> ( P @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ A @ Y2 @ Ys2 ) ) )
=> ( ! [X3: A,Y2: A,Xs2: list @ A,Ys2: list @ A] :
( ~ ( Less @ X3 @ Y2 )
=> ( ~ ( Less @ Y2 @ X3 )
=> ( ( lexordp_eq @ A @ Less @ Xs2 @ Ys2 )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ A @ Y2 @ Ys2 ) ) ) ) ) )
=> ( P @ X1 @ X2 ) ) ) ) ) ).
% ord.lexordp_eq.inducts
thf(fact_75_merge_Osimps_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X1: A,X2: A,L1: list @ A,L2: list @ A] :
( ( ( ord_less @ A @ X1 @ X2 )
=> ( ( merge @ A @ ( cons @ A @ X1 @ L1 ) @ ( cons @ A @ X2 @ L2 ) )
= ( cons @ A @ X1 @ ( merge @ A @ L1 @ ( cons @ A @ X2 @ L2 ) ) ) ) )
& ( ~ ( ord_less @ A @ X1 @ X2 )
=> ( ( ( X1 = X2 )
=> ( ( merge @ A @ ( cons @ A @ X1 @ L1 ) @ ( cons @ A @ X2 @ L2 ) )
= ( cons @ A @ X1 @ ( merge @ A @ L1 @ L2 ) ) ) )
& ( ( X1 != X2 )
=> ( ( merge @ A @ ( cons @ A @ X1 @ L1 ) @ ( cons @ A @ X2 @ L2 ) )
= ( cons @ A @ X2 @ ( merge @ A @ ( cons @ A @ X1 @ L1 ) @ L2 ) ) ) ) ) ) ) ) ).
% merge.simps(3)
thf(fact_76_merge_Osimps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [V2: A,Va2: list @ A] :
( ( merge @ A @ ( cons @ A @ V2 @ Va2 ) @ ( nil @ A ) )
= ( cons @ A @ V2 @ Va2 ) ) ) ).
% merge.simps(2)
thf(fact_77_diff__sorted_Osimps_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X1: A,X2: A,L1: list @ A,L2: list @ A] :
( ( ( ord_less @ A @ X1 @ X2 )
=> ( ( sorted1267110213sorted @ A @ ( cons @ A @ X1 @ L1 ) @ ( cons @ A @ X2 @ L2 ) )
= ( cons @ A @ X1 @ ( sorted1267110213sorted @ A @ L1 @ ( cons @ A @ X2 @ L2 ) ) ) ) )
& ( ~ ( ord_less @ A @ X1 @ X2 )
=> ( ( ( X1 = X2 )
=> ( ( sorted1267110213sorted @ A @ ( cons @ A @ X1 @ L1 ) @ ( cons @ A @ X2 @ L2 ) )
= ( sorted1267110213sorted @ A @ L1 @ L2 ) ) )
& ( ( X1 != X2 )
=> ( ( sorted1267110213sorted @ A @ ( cons @ A @ X1 @ L1 ) @ ( cons @ A @ X2 @ L2 ) )
= ( sorted1267110213sorted @ A @ ( cons @ A @ X1 @ L1 ) @ L2 ) ) ) ) ) ) ) ).
% diff_sorted.simps(3)
thf(fact_78_inter__sorted_Osimps_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X1: A,X2: A,L1: list @ A,L2: list @ A] :
( ( ( ord_less @ A @ X1 @ X2 )
=> ( ( sorted2037043510sorted @ A @ ( cons @ A @ X1 @ L1 ) @ ( cons @ A @ X2 @ L2 ) )
= ( sorted2037043510sorted @ A @ L1 @ ( cons @ A @ X2 @ L2 ) ) ) )
& ( ~ ( ord_less @ A @ X1 @ X2 )
=> ( ( ( X1 = X2 )
=> ( ( sorted2037043510sorted @ A @ ( cons @ A @ X1 @ L1 ) @ ( cons @ A @ X2 @ L2 ) )
= ( cons @ A @ X1 @ ( sorted2037043510sorted @ A @ L1 @ L2 ) ) ) )
& ( ( X1 != X2 )
=> ( ( sorted2037043510sorted @ A @ ( cons @ A @ X1 @ L1 ) @ ( cons @ A @ X2 @ L2 ) )
= ( sorted2037043510sorted @ A @ ( cons @ A @ X1 @ L1 ) @ L2 ) ) ) ) ) ) ) ).
% inter_sorted.simps(3)
thf(fact_79_diff__sorted_Osimps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [V2: A,Va2: list @ A] :
( ( sorted1267110213sorted @ A @ ( cons @ A @ V2 @ Va2 ) @ ( nil @ A ) )
= ( cons @ A @ V2 @ Va2 ) ) ) ).
% diff_sorted.simps(2)
thf(fact_80_inter__sorted_Osimps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [V2: A,Va2: list @ A] :
( ( sorted2037043510sorted @ A @ ( cons @ A @ V2 @ Va2 ) @ ( nil @ A ) )
= ( nil @ A ) ) ) ).
% inter_sorted.simps(2)
thf(fact_81_lexordp__eq_Oinducts,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X1: list @ A,X2: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
( ( ord_lexordp_eq @ A @ X1 @ X2 )
=> ( ! [X12: list @ A] : ( P @ ( nil @ A ) @ X12 )
=> ( ! [X3: A,Y2: A,Xs2: list @ A,Ys2: list @ A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( P @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ A @ Y2 @ Ys2 ) ) )
=> ( ! [X3: A,Y2: A,Xs2: list @ A,Ys2: list @ A] :
( ~ ( ord_less @ A @ X3 @ Y2 )
=> ( ~ ( ord_less @ A @ Y2 @ X3 )
=> ( ( ord_lexordp_eq @ A @ Xs2 @ Ys2 )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ A @ Y2 @ Ys2 ) ) ) ) ) )
=> ( P @ X1 @ X2 ) ) ) ) ) ) ).
% lexordp_eq.inducts
thf(fact_82_sublists_Osimps_I1_J,axiom,
! [A: $tType] :
( ( sublists @ A @ ( nil @ A ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% sublists.simps(1)
thf(fact_83_product__lists_Osimps_I1_J,axiom,
! [A: $tType] :
( ( product_lists @ A @ ( nil @ ( list @ A ) ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% product_lists.simps(1)
thf(fact_84_merge__list_Oelims,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: list @ ( list @ A ),Xa: list @ ( list @ A ),Y: list @ A] :
( ( ( merge_list @ A @ X @ Xa )
= Y )
=> ( ( ( X
= ( nil @ ( list @ A ) ) )
=> ( ( Xa
= ( nil @ ( list @ A ) ) )
=> ( Y
!= ( nil @ A ) ) ) )
=> ( ( ( X
= ( nil @ ( list @ A ) ) )
=> ! [L: list @ A] :
( ( Xa
= ( cons @ ( list @ A ) @ L @ ( nil @ ( list @ A ) ) ) )
=> ( Y != L ) ) )
=> ( ! [La: list @ A,Acc2: list @ ( list @ A )] :
( ( X
= ( cons @ ( list @ A ) @ La @ Acc2 ) )
=> ( ( Xa
= ( nil @ ( list @ A ) ) )
=> ( Y
!= ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La @ Acc2 ) ) ) ) )
=> ( ! [La: list @ A,Acc2: list @ ( list @ A )] :
( ( X
= ( cons @ ( list @ A ) @ La @ Acc2 ) )
=> ! [L: list @ A] :
( ( Xa
= ( cons @ ( list @ A ) @ L @ ( nil @ ( list @ A ) ) ) )
=> ( Y
!= ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L @ ( cons @ ( list @ A ) @ La @ Acc2 ) ) ) ) ) )
=> ~ ! [L12: list @ A,L22: list @ A,Ls: list @ ( list @ A )] :
( ( Xa
= ( cons @ ( list @ A ) @ L12 @ ( cons @ ( list @ A ) @ L22 @ Ls ) ) )
=> ( Y
!= ( merge_list @ A @ ( cons @ ( list @ A ) @ ( merge @ A @ L12 @ L22 ) @ X ) @ Ls ) ) ) ) ) ) ) ) ) ).
% merge_list.elims
thf(fact_85_combinatorial__product_Osimps_I1_J,axiom,
! [A: $tType] :
( ( list_c659805718roduct @ A @ ( nil @ ( list @ A ) ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% combinatorial_product.simps(1)
thf(fact_86_map__tailrec__rev_Oelims,axiom,
! [A: $tType,B: $tType,X: A > B,Xa: list @ A,Xb: list @ B,Y: list @ B] :
( ( ( map_tailrec_rev @ A @ B @ X @ Xa @ Xb )
= Y )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( Y != Xb ) )
=> ~ ! [A3: A,As: list @ A] :
( ( Xa
= ( cons @ A @ A3 @ As ) )
=> ( Y
!= ( map_tailrec_rev @ A @ B @ X @ As @ ( cons @ B @ ( X @ A3 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_87_list__collect__set__map__simps_I2_J,axiom,
! [A: $tType,B: $tType,C: $tType,F2: B > ( set @ A ),X: C > B,A4: C] :
( ( list_collect_set @ B @ A @ F2 @ ( map @ C @ B @ X @ ( cons @ C @ A4 @ ( nil @ C ) ) ) )
= ( F2 @ ( X @ A4 ) ) ) ).
% list_collect_set_map_simps(2)
thf(fact_88_bind__simps_I1_J,axiom,
! [B: $tType,A: $tType,F2: B > ( list @ A )] :
( ( bind @ B @ A @ ( nil @ B ) @ F2 )
= ( nil @ A ) ) ).
% bind_simps(1)
thf(fact_89_splice_Osimps_I2_J,axiom,
! [A: $tType,V2: A,Va2: list @ A] :
( ( splice @ A @ ( cons @ A @ V2 @ Va2 ) @ ( nil @ A ) )
= ( cons @ A @ V2 @ Va2 ) ) ).
% splice.simps(2)
thf(fact_90_Nil__is__map__conv,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B] :
( ( ( nil @ A )
= ( map @ B @ A @ F2 @ Xs ) )
= ( Xs
= ( nil @ B ) ) ) ).
% Nil_is_map_conv
thf(fact_91_map__is__Nil__conv,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B] :
( ( ( map @ B @ A @ F2 @ Xs )
= ( nil @ A ) )
= ( Xs
= ( nil @ B ) ) ) ).
% map_is_Nil_conv
thf(fact_92_list_Omap__disc__iff,axiom,
! [B: $tType,A: $tType,F2: A > B,A4: list @ A] :
( ( ( map @ A @ B @ F2 @ A4 )
= ( nil @ B ) )
= ( A4
= ( nil @ A ) ) ) ).
% list.map_disc_iff
thf(fact_93_splice__Nil2,axiom,
! [A: $tType,Xs: list @ A] :
( ( splice @ A @ Xs @ ( nil @ A ) )
= Xs ) ).
% splice_Nil2
thf(fact_94_map__eq__Cons__conv,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B,Y: A,Ys: list @ A] :
( ( ( map @ B @ A @ F2 @ Xs )
= ( cons @ A @ Y @ Ys ) )
= ( ? [Z: B,Zs2: list @ B] :
( ( Xs
= ( cons @ B @ Z @ Zs2 ) )
& ( ( F2 @ Z )
= Y )
& ( ( map @ B @ A @ F2 @ Zs2 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_95_Cons__eq__map__conv,axiom,
! [A: $tType,B: $tType,X: A,Xs: list @ A,F2: B > A,Ys: list @ B] :
( ( ( cons @ A @ X @ Xs )
= ( map @ B @ A @ F2 @ Ys ) )
= ( ? [Z: B,Zs2: list @ B] :
( ( Ys
= ( cons @ B @ Z @ Zs2 ) )
& ( X
= ( F2 @ Z ) )
& ( Xs
= ( map @ B @ A @ F2 @ Zs2 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_96_map__eq__Cons__D,axiom,
! [B: $tType,A: $tType,F2: B > A,Xs: list @ B,Y: A,Ys: list @ A] :
( ( ( map @ B @ A @ F2 @ Xs )
= ( cons @ A @ Y @ Ys ) )
=> ? [Z2: B,Zs3: list @ B] :
( ( Xs
= ( cons @ B @ Z2 @ Zs3 ) )
& ( ( F2 @ Z2 )
= Y )
& ( ( map @ B @ A @ F2 @ Zs3 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_97_Cons__eq__map__D,axiom,
! [A: $tType,B: $tType,X: A,Xs: list @ A,F2: B > A,Ys: list @ B] :
( ( ( cons @ A @ X @ Xs )
= ( map @ B @ A @ F2 @ Ys ) )
=> ? [Z2: B,Zs3: list @ B] :
( ( Ys
= ( cons @ B @ Z2 @ Zs3 ) )
& ( X
= ( F2 @ Z2 ) )
& ( Xs
= ( map @ B @ A @ F2 @ Zs3 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_98_map__eq__consE,axiom,
! [B: $tType,A: $tType,F2: B > A,Ls2: list @ B,Fa: A,Fl: list @ A] :
( ( ( map @ B @ A @ F2 @ Ls2 )
= ( cons @ A @ Fa @ Fl ) )
=> ~ ! [A3: B,L: list @ B] :
( ( Ls2
= ( cons @ B @ A3 @ L ) )
=> ( ( ( F2 @ A3 )
= Fa )
=> ( ( map @ B @ A @ F2 @ L )
!= Fl ) ) ) ) ).
% map_eq_consE
thf(fact_99_list_Osimps_I9_J,axiom,
! [B: $tType,A: $tType,F2: A > B,X21: A,X22: list @ A] :
( ( map @ A @ B @ F2 @ ( cons @ A @ X21 @ X22 ) )
= ( cons @ B @ ( F2 @ X21 ) @ ( map @ A @ B @ F2 @ X22 ) ) ) ).
% list.simps(9)
thf(fact_100_map__consI_I1_J,axiom,
! [A: $tType,B: $tType,W: list @ A,F2: B > A,Ww: list @ B,A4: B] :
( ( W
= ( map @ B @ A @ F2 @ Ww ) )
=> ( ( cons @ A @ ( F2 @ A4 ) @ W )
= ( map @ B @ A @ F2 @ ( cons @ B @ A4 @ Ww ) ) ) ) ).
% map_consI(1)
thf(fact_101_list_Osimps_I8_J,axiom,
! [A: $tType,B: $tType,F2: A > B] :
( ( map @ A @ B @ F2 @ ( nil @ A ) )
= ( nil @ B ) ) ).
% list.simps(8)
thf(fact_102_splice_Osimps_I3_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: A,Ys: list @ A] :
( ( splice @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) )
= ( cons @ A @ X @ ( cons @ A @ Y @ ( splice @ A @ Xs @ Ys ) ) ) ) ).
% splice.simps(3)
thf(fact_103_splice_Osimps_I1_J,axiom,
! [A: $tType,Ys: list @ A] :
( ( splice @ A @ ( nil @ A ) @ Ys )
= Ys ) ).
% splice.simps(1)
thf(fact_104_map__tailrec__rev_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,F2: A > B,A4: A,As2: list @ A,Bs2: list @ B] :
( ( map_tailrec_rev @ A @ B @ F2 @ ( cons @ A @ A4 @ As2 ) @ Bs2 )
= ( map_tailrec_rev @ A @ B @ F2 @ As2 @ ( cons @ B @ ( F2 @ A4 ) @ Bs2 ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_105_map__tailrec__rev_Osimps_I1_J,axiom,
! [A: $tType,B: $tType,F2: A > B,Bs2: list @ B] :
( ( map_tailrec_rev @ A @ B @ F2 @ ( nil @ A ) @ Bs2 )
= Bs2 ) ).
% map_tailrec_rev.simps(1)
thf(fact_106_merge__list_Osimps_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ( ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( nil @ ( list @ A ) ) )
= ( nil @ A ) ) ) ).
% merge_list.simps(1)
thf(fact_107_merge__list_Osimps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [L3: list @ A] :
( ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L3 @ ( nil @ ( list @ A ) ) ) )
= L3 ) ) ).
% merge_list.simps(2)
thf(fact_108_merge__list_Osimps_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [La2: list @ A,Acc22: list @ ( list @ A )] :
( ( merge_list @ A @ ( cons @ ( list @ A ) @ La2 @ Acc22 ) @ ( nil @ ( list @ A ) ) )
= ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ La2 @ Acc22 ) ) ) ) ).
% merge_list.simps(3)
thf(fact_109_merge__list_Osimps_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [La2: list @ A,Acc22: list @ ( list @ A ),L3: list @ A] :
( ( merge_list @ A @ ( cons @ ( list @ A ) @ La2 @ Acc22 ) @ ( cons @ ( list @ A ) @ L3 @ ( nil @ ( list @ A ) ) ) )
= ( merge_list @ A @ ( nil @ ( list @ A ) ) @ ( cons @ ( list @ A ) @ L3 @ ( cons @ ( list @ A ) @ La2 @ Acc22 ) ) ) ) ) ).
% merge_list.simps(4)
thf(fact_110_merge__list_Osimps_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Acc22: list @ ( list @ A ),L1: list @ A,L2: list @ A,Ls2: list @ ( list @ A )] :
( ( merge_list @ A @ Acc22 @ ( cons @ ( list @ A ) @ L1 @ ( cons @ ( list @ A ) @ L2 @ Ls2 ) ) )
= ( merge_list @ A @ ( cons @ ( list @ A ) @ ( merge @ A @ L1 @ L2 ) @ Acc22 ) @ Ls2 ) ) ) ).
% merge_list.simps(5)
thf(fact_111_splice_Oelims,axiom,
! [A: $tType,X: list @ A,Xa: list @ A,Y: list @ A] :
( ( ( splice @ A @ X @ Xa )
= Y )
=> ( ( ( X
= ( nil @ A ) )
=> ( Y != Xa ) )
=> ( ! [V: A,Va: list @ A] :
( ( X
= ( cons @ A @ V @ Va ) )
=> ( ( Xa
= ( nil @ A ) )
=> ( Y
!= ( cons @ A @ V @ Va ) ) ) )
=> ~ ! [X3: A,Xs2: list @ A] :
( ( X
= ( cons @ A @ X3 @ Xs2 ) )
=> ! [Y2: A,Ys2: list @ A] :
( ( Xa
= ( cons @ A @ Y2 @ Ys2 ) )
=> ( Y
!= ( cons @ A @ X3 @ ( cons @ A @ Y2 @ ( splice @ A @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ) ).
% splice.elims
thf(fact_112_map__eq__map__tailrec,axiom,
! [B: $tType,A: $tType] :
( ( map @ A @ B )
= ( map_tailrec @ A @ B ) ) ).
% map_eq_map_tailrec
thf(fact_113_list__collect__set__map__simps_I1_J,axiom,
! [C: $tType,B: $tType,A: $tType,F2: B > ( set @ A ),X: C > B] :
( ( list_collect_set @ B @ A @ F2 @ ( map @ C @ B @ X @ ( nil @ C ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% list_collect_set_map_simps(1)
thf(fact_114_list__collect__set__map__simps_I3_J,axiom,
! [A: $tType,B: $tType,C: $tType,F2: B > ( set @ A ),X: C > B,A4: C,L3: list @ C] :
( ( list_collect_set @ B @ A @ F2 @ ( map @ C @ B @ X @ ( cons @ C @ A4 @ L3 ) ) )
= ( sup_sup @ ( set @ A ) @ ( F2 @ ( X @ A4 ) ) @ ( list_collect_set @ B @ A @ F2 @ ( map @ C @ B @ X @ L3 ) ) ) ) ).
% list_collect_set_map_simps(3)
thf(fact_115_listrelp_Oinducts,axiom,
! [A: $tType,B: $tType,R4: A > B > $o,X1: list @ A,X2: list @ B,P: ( list @ A ) > ( list @ B ) > $o] :
( ( listrelp @ A @ B @ R4 @ X1 @ X2 )
=> ( ( P @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X3: A,Y2: B,Xs2: list @ A,Ys2: list @ B] :
( ( R4 @ X3 @ Y2 )
=> ( ( listrelp @ A @ B @ R4 @ Xs2 @ Ys2 )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ B @ Y2 @ Ys2 ) ) ) ) )
=> ( P @ X1 @ X2 ) ) ) ) ).
% listrelp.inducts
thf(fact_116_listrelp_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( listrelp @ A @ B )
= ( ^ [R5: A > B > $o,A12: list @ A,A22: list @ B] :
( ( ( A12
= ( nil @ A ) )
& ( A22
= ( nil @ B ) ) )
| ? [X4: A,Y3: B,Xs3: list @ A,Ys3: list @ B] :
( ( A12
= ( cons @ A @ X4 @ Xs3 ) )
& ( A22
= ( cons @ B @ Y3 @ Ys3 ) )
& ( R5 @ X4 @ Y3 )
& ( listrelp @ A @ B @ R5 @ Xs3 @ Ys3 ) ) ) ) ) ).
% listrelp.simps
thf(fact_117_listrelp_Ocases,axiom,
! [A: $tType,B: $tType,R4: A > B > $o,A1: list @ A,A2: list @ B] :
( ( listrelp @ A @ B @ R4 @ A1 @ A2 )
=> ( ( ( A1
= ( nil @ A ) )
=> ( A2
!= ( nil @ B ) ) )
=> ~ ! [X3: A,Y2: B,Xs2: list @ A] :
( ( A1
= ( cons @ A @ X3 @ Xs2 ) )
=> ! [Ys2: list @ B] :
( ( A2
= ( cons @ B @ Y2 @ Ys2 ) )
=> ( ( R4 @ X3 @ Y2 )
=> ~ ( listrelp @ A @ B @ R4 @ Xs2 @ Ys2 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_118_list__collect__set__simps_I1_J,axiom,
! [B: $tType,A: $tType,F2: B > ( set @ A )] :
( ( list_collect_set @ B @ A @ F2 @ ( nil @ B ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% list_collect_set_simps(1)
thf(fact_119_list__collect__set__simps_I3_J,axiom,
! [A: $tType,B: $tType,F2: B > ( set @ A ),A4: B,L3: list @ B] :
( ( list_collect_set @ B @ A @ F2 @ ( cons @ B @ A4 @ L3 ) )
= ( sup_sup @ ( set @ A ) @ ( F2 @ A4 ) @ ( list_collect_set @ B @ A @ F2 @ L3 ) ) ) ).
% list_collect_set_simps(3)
thf(fact_120_set__notEmptyE,axiom,
! [A: $tType,S: set @ A] :
( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [X3: A] :
~ ( member @ A @ X3 @ S ) ) ).
% set_notEmptyE
thf(fact_121_memb__imp__not__empty,axiom,
! [A: $tType,X: A,S: set @ A] :
( ( member @ A @ X @ S )
=> ( S
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% memb_imp_not_empty
thf(fact_122_listrelp_OCons,axiom,
! [A: $tType,B: $tType,R4: A > B > $o,X: A,Y: B,Xs: list @ A,Ys: list @ B] :
( ( R4 @ X @ Y )
=> ( ( listrelp @ A @ B @ R4 @ Xs @ Ys )
=> ( listrelp @ A @ B @ R4 @ ( cons @ A @ X @ Xs ) @ ( cons @ B @ Y @ Ys ) ) ) ) ).
% listrelp.Cons
thf(fact_123_listrelp_ONil,axiom,
! [A: $tType,B: $tType,R4: A > B > $o] : ( listrelp @ A @ B @ R4 @ ( nil @ A ) @ ( nil @ B ) ) ).
% listrelp.Nil
thf(fact_124_Un__empty,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( ( sup_sup @ ( set @ A ) @ A5 @ B4 )
= ( bot_bot @ ( set @ A ) ) )
= ( ( A5
= ( bot_bot @ ( set @ A ) ) )
& ( B4
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Un_empty
thf(fact_125_sup__bot_Oright__neutral,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A @ ( type2 @ A ) )
=> ! [A4: A] :
( ( sup_sup @ A @ A4 @ ( bot_bot @ A ) )
= A4 ) ) ).
% sup_bot.right_neutral
thf(fact_126_sup__bot_Oleft__neutral,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A @ ( type2 @ A ) )
=> ! [A4: A] :
( ( sup_sup @ A @ ( bot_bot @ A ) @ A4 )
= A4 ) ) ).
% sup_bot.left_neutral
thf(fact_127_sup__eq__bot__iff,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ( sup_sup @ A @ X @ Y )
= ( bot_bot @ A ) )
= ( ( X
= ( bot_bot @ A ) )
& ( Y
= ( bot_bot @ A ) ) ) ) ) ).
% sup_eq_bot_iff
thf(fact_128_empty__iff,axiom,
! [A: $tType,C2: A] :
~ ( member @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_129_all__not__in__conv,axiom,
! [A: $tType,A5: set @ A] :
( ( ! [X4: A] :
~ ( member @ A @ X4 @ A5 ) )
= ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_130_Collect__empty__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X4: A] :
~ ( P @ X4 ) ) ) ).
% Collect_empty_eq
thf(fact_131_empty__Collect__eq,axiom,
! [A: $tType,P: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P ) )
= ( ! [X4: A] :
~ ( P @ X4 ) ) ) ).
% empty_Collect_eq
thf(fact_132_sup__apply,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B @ ( type2 @ B ) )
=> ( ( sup_sup @ ( A > B ) )
= ( ^ [F3: A > B,G2: A > B,X4: A] : ( sup_sup @ B @ ( F3 @ X4 ) @ ( G2 @ X4 ) ) ) ) ) ).
% sup_apply
thf(fact_133_sup_Oidem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A4: A] :
( ( sup_sup @ A @ A4 @ A4 )
= A4 ) ) ).
% sup.idem
thf(fact_134_sup__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( sup_sup @ A @ X @ X )
= X ) ) ).
% sup_idem
thf(fact_135_sup_Oleft__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A4: A,B3: A] :
( ( sup_sup @ A @ A4 @ ( sup_sup @ A @ A4 @ B3 ) )
= ( sup_sup @ A @ A4 @ B3 ) ) ) ).
% sup.left_idem
thf(fact_136_sup__left__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
= ( sup_sup @ A @ X @ Y ) ) ) ).
% sup_left_idem
thf(fact_137_sup_Oright__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A4: A,B3: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A4 @ B3 ) @ B3 )
= ( sup_sup @ A @ A4 @ B3 ) ) ) ).
% sup.right_idem
thf(fact_138_UnCI,axiom,
! [A: $tType,C2: A,B4: set @ A,A5: set @ A] :
( ( ~ ( member @ A @ C2 @ B4 )
=> ( member @ A @ C2 @ A5 ) )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) ) ) ).
% UnCI
thf(fact_139_Un__iff,axiom,
! [A: $tType,C2: A,A5: set @ A,B4: set @ A] :
( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
= ( ( member @ A @ C2 @ A5 )
| ( member @ A @ C2 @ B4 ) ) ) ).
% Un_iff
thf(fact_140_bot__eq__sup__iff,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ( bot_bot @ A )
= ( sup_sup @ A @ X @ Y ) )
= ( ( X
= ( bot_bot @ A ) )
& ( Y
= ( bot_bot @ A ) ) ) ) ) ).
% bot_eq_sup_iff
thf(fact_141_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_142_emptyE,axiom,
! [A: $tType,A4: A] :
~ ( member @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_143_equals0D,axiom,
! [A: $tType,A5: set @ A,A4: A] :
( ( A5
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A4 @ A5 ) ) ).
% equals0D
thf(fact_144_equals0I,axiom,
! [A: $tType,A5: set @ A] :
( ! [Y2: A] :
~ ( member @ A @ Y2 @ A5 )
=> ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_145_ex__in__conv,axiom,
! [A: $tType,A5: set @ A] :
( ( ? [X4: A] : ( member @ A @ X4 @ A5 ) )
= ( A5
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_146_not__psubset__empty,axiom,
! [A: $tType,A5: set @ A] :
~ ( ord_less @ ( set @ A ) @ A5 @ ( bot_bot @ ( set @ A ) ) ) ).
% not_psubset_empty
thf(fact_147_inf__sup__aci_I8_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
= ( sup_sup @ A @ X @ Y ) ) ) ).
% inf_sup_aci(8)
thf(fact_148_inf__sup__aci_I7_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z3: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z3 ) )
= ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z3 ) ) ) ) ).
% inf_sup_aci(7)
thf(fact_149_inf__sup__aci_I6_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z3: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z3 )
= ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z3 ) ) ) ) ).
% inf_sup_aci(6)
thf(fact_150_inf__sup__aci_I5_J,axiom,
! [A: $tType] :
( ( lattice @ A @ ( type2 @ A ) )
=> ( ( sup_sup @ A )
= ( ^ [X4: A,Y3: A] : ( sup_sup @ A @ Y3 @ X4 ) ) ) ) ).
% inf_sup_aci(5)
thf(fact_151_sup__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B @ ( type2 @ B ) )
=> ( ( sup_sup @ ( A > B ) )
= ( ^ [F3: A > B,G2: A > B,X4: A] : ( sup_sup @ B @ ( F3 @ X4 ) @ ( G2 @ X4 ) ) ) ) ) ).
% sup_fun_def
thf(fact_152_sup_Oassoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [A4: A,B3: A,C2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A4 @ B3 ) @ C2 )
= ( sup_sup @ A @ A4 @ ( sup_sup @ A @ B3 @ C2 ) ) ) ) ).
% sup.assoc
thf(fact_153_sup__assoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z3: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z3 )
= ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z3 ) ) ) ) ).
% sup_assoc
thf(fact_154_sup_Ocommute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( sup_sup @ A )
= ( ^ [A6: A,B5: A] : ( sup_sup @ A @ B5 @ A6 ) ) ) ) ).
% sup.commute
thf(fact_155_sup__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( sup_sup @ A )
= ( ^ [X4: A,Y3: A] : ( sup_sup @ A @ Y3 @ X4 ) ) ) ) ).
% sup_commute
thf(fact_156_sup_Oleft__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B3: A,A4: A,C2: A] :
( ( sup_sup @ A @ B3 @ ( sup_sup @ A @ A4 @ C2 ) )
= ( sup_sup @ A @ A4 @ ( sup_sup @ A @ B3 @ C2 ) ) ) ) ).
% sup.left_commute
thf(fact_157_sup__left__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z3: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z3 ) )
= ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z3 ) ) ) ) ).
% sup_left_commute
thf(fact_158_UnE,axiom,
! [A: $tType,C2: A,A5: set @ A,B4: set @ A] :
( ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
=> ( ~ ( member @ A @ C2 @ A5 )
=> ( member @ A @ C2 @ B4 ) ) ) ).
% UnE
thf(fact_159_UnI1,axiom,
! [A: $tType,C2: A,A5: set @ A,B4: set @ A] :
( ( member @ A @ C2 @ A5 )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) ) ) ).
% UnI1
thf(fact_160_UnI2,axiom,
! [A: $tType,C2: A,B4: set @ A,A5: set @ A] :
( ( member @ A @ C2 @ B4 )
=> ( member @ A @ C2 @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) ) ) ).
% UnI2
thf(fact_161_bex__Un,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,P: A > $o] :
( ( ? [X4: A] :
( ( member @ A @ X4 @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
& ( P @ X4 ) ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ A5 )
& ( P @ X4 ) )
| ? [X4: A] :
( ( member @ A @ X4 @ B4 )
& ( P @ X4 ) ) ) ) ).
% bex_Un
thf(fact_162_ball__Un,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,P: A > $o] :
( ( ! [X4: A] :
( ( member @ A @ X4 @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
=> ( P @ X4 ) ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( P @ X4 ) )
& ! [X4: A] :
( ( member @ A @ X4 @ B4 )
=> ( P @ X4 ) ) ) ) ).
% ball_Un
thf(fact_163_Un__assoc,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C3: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) @ C3 )
= ( sup_sup @ ( set @ A ) @ A5 @ ( sup_sup @ ( set @ A ) @ B4 @ C3 ) ) ) ).
% Un_assoc
thf(fact_164_Un__absorb,axiom,
! [A: $tType,A5: set @ A] :
( ( sup_sup @ ( set @ A ) @ A5 @ A5 )
= A5 ) ).
% Un_absorb
thf(fact_165_Un__commute,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A7: set @ A,B6: set @ A] : ( sup_sup @ ( set @ A ) @ B6 @ A7 ) ) ) ).
% Un_commute
thf(fact_166_Un__left__absorb,axiom,
! [A: $tType,A5: set @ A,B4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A5 @ ( sup_sup @ ( set @ A ) @ A5 @ B4 ) )
= ( sup_sup @ ( set @ A ) @ A5 @ B4 ) ) ).
% Un_left_absorb
thf(fact_167_Un__left__commute,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A5 @ ( sup_sup @ ( set @ A ) @ B4 @ C3 ) )
= ( sup_sup @ ( set @ A ) @ B4 @ ( sup_sup @ ( set @ A ) @ A5 @ C3 ) ) ) ).
% Un_left_commute
thf(fact_168_sup_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [C2: A,B3: A,A4: A] :
( ( ord_less @ A @ C2 @ B3 )
=> ( ord_less @ A @ C2 @ ( sup_sup @ A @ A4 @ B3 ) ) ) ) ).
% sup.strict_coboundedI2
thf(fact_169_sup_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [C2: A,A4: A,B3: A] :
( ( ord_less @ A @ C2 @ A4 )
=> ( ord_less @ A @ C2 @ ( sup_sup @ A @ A4 @ B3 ) ) ) ) ).
% sup.strict_coboundedI1
thf(fact_170_sup_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [B5: A,A6: A] :
( ( A6
= ( sup_sup @ A @ A6 @ B5 ) )
& ( A6 != B5 ) ) ) ) ) ).
% sup.strict_order_iff
thf(fact_171_sup_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [B3: A,C2: A,A4: A] :
( ( ord_less @ A @ ( sup_sup @ A @ B3 @ C2 ) @ A4 )
=> ~ ( ( ord_less @ A @ B3 @ A4 )
=> ~ ( ord_less @ A @ C2 @ A4 ) ) ) ) ).
% sup.strict_boundedE
thf(fact_172_less__supI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,B3: A,A4: A] :
( ( ord_less @ A @ X @ B3 )
=> ( ord_less @ A @ X @ ( sup_sup @ A @ A4 @ B3 ) ) ) ) ).
% less_supI2
thf(fact_173_less__supI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A @ ( type2 @ A ) )
=> ! [X: A,A4: A,B3: A] :
( ( ord_less @ A @ X @ A4 )
=> ( ord_less @ A @ X @ ( sup_sup @ A @ A4 @ B3 ) ) ) ) ).
% less_supI1
thf(fact_174_sup__bot__left,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( sup_sup @ A @ ( bot_bot @ A ) @ X )
= X ) ) ).
% sup_bot_left
thf(fact_175_sup__bot__right,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( sup_sup @ A @ X @ ( bot_bot @ A ) )
= X ) ) ).
% sup_bot_right
thf(fact_176_Un__empty__left,axiom,
! [A: $tType,B4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B4 )
= B4 ) ).
% Un_empty_left
thf(fact_177_Un__empty__right,axiom,
! [A: $tType,A5: set @ A] :
( ( sup_sup @ ( set @ A ) @ A5 @ ( bot_bot @ ( set @ A ) ) )
= A5 ) ).
% Un_empty_right
thf(fact_178_bot__apply,axiom,
! [C: $tType,D: $tType] :
( ( bot @ C @ ( type2 @ C ) )
=> ( ( bot_bot @ ( D > C ) )
= ( ^ [X4: D] : ( bot_bot @ C ) ) ) ) ).
% bot_apply
thf(fact_179_bot_Onot__eq__extremum,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A4: A] :
( ( A4
!= ( bot_bot @ A ) )
= ( ord_less @ A @ ( bot_bot @ A ) @ A4 ) ) ) ).
% bot.not_eq_extremum
thf(fact_180_bot_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A4: A] :
~ ( ord_less @ A @ A4 @ ( bot_bot @ A ) ) ) ).
% bot.extremum_strict
thf(fact_181_Set_Ois__empty__def,axiom,
! [A: $tType] :
( ( is_empty @ A )
= ( ^ [A7: set @ A] :
( A7
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Set.is_empty_def
thf(fact_182_psubset__trans,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C3: set @ A] :
( ( ord_less @ ( set @ A ) @ A5 @ B4 )
=> ( ( ord_less @ ( set @ A ) @ B4 @ C3 )
=> ( ord_less @ ( set @ A ) @ A5 @ C3 ) ) ) ).
% psubset_trans
thf(fact_183_psubsetD,axiom,
! [A: $tType,A5: set @ A,B4: set @ A,C2: A] :
( ( ord_less @ ( set @ A ) @ A5 @ B4 )
=> ( ( member @ A @ C2 @ A5 )
=> ( member @ A @ C2 @ B4 ) ) ) ).
% psubsetD
thf(fact_184_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A4: A] :
( ( ord_less @ A @ B3 @ A4 )
=> ( A4 != B3 ) ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_185_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( A4 != B3 ) ) ) ).
% order.strict_implies_not_eq
thf(fact_186_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_187_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A4: A,C2: A] :
( ( ord_less @ A @ B3 @ A4 )
=> ( ( ord_less @ A @ C2 @ B3 )
=> ( ord_less @ A @ C2 @ A4 ) ) ) ) ).
% dual_order.strict_trans
thf(fact_188_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_189_order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less @ A @ B3 @ C2 )
=> ( ord_less @ A @ A4 @ C2 ) ) ) ) ).
% order.strict_trans
thf(fact_190_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A] :
~ ( ord_less @ A @ A4 @ A4 ) ) ).
% dual_order.irrefl
thf(fact_191_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_192_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_193_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_194_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_195_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,A4: A] :
( ! [X3: A] :
( ! [Y4: A] :
( ( ord_less @ A @ Y4 @ X3 )
=> ( P @ Y4 ) )
=> ( P @ X3 ) )
=> ( P @ A4 ) ) ) ).
% less_induct
thf(fact_196_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_197_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_198_dual__order_Oasym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B3: A,A4: A] :
( ( ord_less @ A @ B3 @ A4 )
=> ~ ( ord_less @ A @ A4 @ B3 ) ) ) ).
% dual_order.asym
thf(fact_199_ord__less__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A4: A,B3: A,C2: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( B3 = C2 )
=> ( ord_less @ A @ A4 @ C2 ) ) ) ) ).
% ord_less_eq_trans
thf(fact_200_ord__eq__less__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A4: A,B3: A,C2: A] :
( ( A4 = B3 )
=> ( ( ord_less @ A @ B3 @ C2 )
=> ( ord_less @ A @ A4 @ C2 ) ) ) ) ).
% ord_eq_less_trans
thf(fact_201_less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A] :
~ ( ord_less @ A @ X @ X ) ) ).
% less_irrefl
thf(fact_202_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_203_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_204_less__asym_H,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ~ ( ord_less @ A @ B3 @ A4 ) ) ) ).
% less_asym'
thf(fact_205_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_206_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_207_dense,axiom,
! [A: $tType] :
( ( dense_order @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ? [Z2: A] :
( ( ord_less @ A @ X @ Z2 )
& ( ord_less @ A @ Z2 @ Y ) ) ) ) ).
% dense
thf(fact_208_order_Oasym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ~ ( ord_less @ A @ B3 @ A4 ) ) ) ).
% order.asym
thf(fact_209_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_210_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_211_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A @ ( type2 @ A ) )
=> ! [X: A] :
? [X12: A] : ( ord_less @ A @ X @ X12 ) ) ).
% gt_ex
thf(fact_212_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A @ ( type2 @ A ) )
=> ! [X: A] :
? [Y2: A] : ( ord_less @ A @ Y2 @ X ) ) ).
% lt_ex
thf(fact_213_order__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C @ ( type2 @ C ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A4: A,B3: A,F2: A > C,C2: C] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ord_less @ C @ ( F2 @ B3 ) @ C2 )
=> ( ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( ord_less @ C @ ( F2 @ X3 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less @ C @ ( F2 @ A4 ) @ C2 ) ) ) ) ) ).
% order_less_subst2
thf(fact_214_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A4: A,F2: B > A,B3: B,C2: B] :
( ( ord_less @ A @ A4 @ ( F2 @ B3 ) )
=> ( ( ord_less @ B @ B3 @ C2 )
=> ( ! [X3: B,Y2: B] :
( ( ord_less @ B @ X3 @ Y2 )
=> ( ord_less @ A @ ( F2 @ X3 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less @ A @ A4 @ ( F2 @ C2 ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_215_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A4: A,B3: A,F2: A > B,C2: B] :
( ( ord_less @ A @ A4 @ B3 )
=> ( ( ( F2 @ B3 )
= C2 )
=> ( ! [X3: A,Y2: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( ord_less @ B @ ( F2 @ X3 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less @ B @ ( F2 @ A4 ) @ C2 ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_216_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A4: A,F2: B > A,B3: B,C2: B] :
( ( A4
= ( F2 @ B3 ) )
=> ( ( ord_less @ B @ B3 @ C2 )
=> ( ! [X3: B,Y2: B] :
( ( ord_less @ B @ X3 @ Y2 )
=> ( ord_less @ A @ ( F2 @ X3 ) @ ( F2 @ Y2 ) ) )
=> ( ord_less @ A @ A4 @ ( F2 @ C2 ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_217_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B @ ( type2 @ B ) )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X4: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_218_bot__empty__eq,axiom,
! [A: $tType] :
( ( bot_bot @ ( A > $o ) )
= ( ^ [X4: A] : ( member @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% bot_empty_eq
thf(fact_219_Collect__empty__eq__bot,axiom,
! [A: $tType,P: A > $o] :
( ( ( collect @ A @ P )
= ( bot_bot @ ( set @ A ) ) )
= ( P
= ( bot_bot @ ( A > $o ) ) ) ) ).
% Collect_empty_eq_bot
thf(fact_220_map__tailrec__def,axiom,
! [B: $tType,A: $tType] :
( ( map_tailrec @ A @ B )
= ( ^ [F3: A > B,As3: list @ A] : ( rev @ B @ ( map_tailrec_rev @ A @ B @ F3 @ As3 @ ( nil @ B ) ) ) ) ) ).
% map_tailrec_def
thf(fact_221_rev__is__rev__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( rev @ A @ Xs )
= ( rev @ A @ Ys ) )
= ( Xs = Ys ) ) ).
% rev_is_rev_conv
thf(fact_222_rev__rev__ident,axiom,
! [A: $tType,Xs: list @ A] :
( ( rev @ A @ ( rev @ A @ Xs ) )
= Xs ) ).
% rev_rev_ident
thf(fact_223_rev__is__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( rev @ A @ Xs )
= ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% rev_is_Nil_conv
thf(fact_224_Nil__is__rev__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( nil @ A )
= ( rev @ A @ Xs ) )
= ( Xs
= ( nil @ A ) ) ) ).
% Nil_is_rev_conv
thf(fact_225_singleton__rev__conv,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( ( cons @ A @ X @ ( nil @ A ) )
= ( rev @ A @ Xs ) )
= ( Xs
= ( cons @ A @ X @ ( nil @ A ) ) ) ) ).
% singleton_rev_conv
thf(fact_226_rev__singleton__conv,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( ( rev @ A @ Xs )
= ( cons @ A @ X @ ( nil @ A ) ) )
= ( Xs
= ( cons @ A @ X @ ( nil @ A ) ) ) ) ).
% rev_singleton_conv
thf(fact_227_rev_Osimps_I1_J,axiom,
! [A: $tType] :
( ( rev @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% rev.simps(1)
thf(fact_228_rev__map,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B] :
( ( rev @ A @ ( map @ B @ A @ F2 @ Xs ) )
= ( map @ B @ A @ F2 @ ( rev @ B @ Xs ) ) ) ).
% rev_map
thf(fact_229_rev__swap,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( rev @ A @ Xs )
= Ys )
= ( Xs
= ( rev @ A @ Ys ) ) ) ).
% rev_swap
thf(fact_230_list__collect__set__map__simps_I4_J,axiom,
! [A: $tType,B: $tType,C: $tType,F2: B > ( set @ A ),X: C > B,L3: list @ C,L4: list @ C] :
( ( list_collect_set @ B @ A @ F2 @ ( map @ C @ B @ X @ ( append @ C @ L3 @ L4 ) ) )
= ( sup_sup @ ( set @ A ) @ ( list_collect_set @ B @ A @ F2 @ ( map @ C @ B @ X @ L3 ) ) @ ( list_collect_set @ B @ A @ F2 @ ( map @ C @ B @ X @ L4 ) ) ) ) ).
% list_collect_set_map_simps(4)
thf(fact_231_sorted__list__of__set__empty,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ( ( linord467138063of_set @ A @ ( bot_bot @ ( set @ A ) ) )
= ( nil @ A ) ) ) ).
% sorted_list_of_set_empty
thf(fact_232_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_233_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_234_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_235_append__Nil2,axiom,
! [A: $tType,Xs: list @ A] :
( ( append @ A @ Xs @ ( nil @ A ) )
= Xs ) ).
% append_Nil2
thf(fact_236_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_237_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_238_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_239_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_240_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_241_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_242_map__append,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B,Ys: list @ B] :
( ( map @ B @ A @ F2 @ ( append @ B @ Xs @ Ys ) )
= ( append @ A @ ( map @ B @ A @ F2 @ Xs ) @ ( map @ B @ A @ F2 @ Ys ) ) ) ).
% map_append
thf(fact_243_rev__append,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( rev @ A @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( rev @ A @ Ys ) @ ( rev @ A @ Xs ) ) ) ).
% rev_append
thf(fact_244_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_245_list__ee__eq__leel_I1_J,axiom,
! [A: $tType,E1: A,E2: A,L1: list @ A,E12: A,E22: A,L2: list @ A] :
( ( ( cons @ A @ E1 @ ( cons @ A @ E2 @ ( nil @ A ) ) )
= ( append @ A @ L1 @ ( cons @ A @ E12 @ ( cons @ A @ E22 @ L2 ) ) ) )
= ( ( L1
= ( nil @ A ) )
& ( E1 = E12 )
& ( E2 = E22 )
& ( L2
= ( nil @ A ) ) ) ) ).
% list_ee_eq_leel(1)
thf(fact_246_list__ee__eq__leel_I2_J,axiom,
! [A: $tType,L1: list @ A,E12: A,E22: A,L2: list @ A,E1: A,E2: A] :
( ( ( append @ A @ L1 @ ( cons @ A @ E12 @ ( cons @ A @ E22 @ L2 ) ) )
= ( cons @ A @ E1 @ ( cons @ A @ E2 @ ( nil @ A ) ) ) )
= ( ( L1
= ( nil @ A ) )
& ( E1 = E12 )
& ( E2 = E22 )
& ( L2
= ( nil @ A ) ) ) ) ).
% list_ee_eq_leel(2)
thf(fact_247_list__se__match_I1_J,axiom,
! [A: $tType,L1: list @ A,L2: list @ A,A4: A] :
( ( L1
!= ( nil @ A ) )
=> ( ( ( append @ A @ L1 @ L2 )
= ( cons @ A @ A4 @ ( nil @ A ) ) )
= ( ( L1
= ( cons @ A @ A4 @ ( nil @ A ) ) )
& ( L2
= ( nil @ A ) ) ) ) ) ).
% list_se_match(1)
thf(fact_248_list__se__match_I2_J,axiom,
! [A: $tType,L2: list @ A,L1: list @ A,A4: A] :
( ( L2
!= ( nil @ A ) )
=> ( ( ( append @ A @ L1 @ L2 )
= ( cons @ A @ A4 @ ( nil @ A ) ) )
= ( ( L1
= ( nil @ A ) )
& ( L2
= ( cons @ A @ A4 @ ( nil @ A ) ) ) ) ) ) ).
% list_se_match(2)
thf(fact_249_list__se__match_I3_J,axiom,
! [A: $tType,L1: list @ A,A4: A,L2: list @ A] :
( ( L1
!= ( nil @ A ) )
=> ( ( ( cons @ A @ A4 @ ( nil @ A ) )
= ( append @ A @ L1 @ L2 ) )
= ( ( L1
= ( cons @ A @ A4 @ ( nil @ A ) ) )
& ( L2
= ( nil @ A ) ) ) ) ) ).
% list_se_match(3)
thf(fact_250_list__se__match_I4_J,axiom,
! [A: $tType,L2: list @ A,A4: A,L1: list @ A] :
( ( L2
!= ( nil @ A ) )
=> ( ( ( cons @ A @ A4 @ ( nil @ A ) )
= ( append @ A @ L1 @ L2 ) )
= ( ( L1
= ( nil @ A ) )
& ( L2
= ( cons @ A @ A4 @ ( nil @ A ) ) ) ) ) ) ).
% list_se_match(4)
thf(fact_251_list__e__eq__lel_I1_J,axiom,
! [A: $tType,E3: A,L1: list @ A,E4: A,L2: list @ A] :
( ( ( cons @ A @ E3 @ ( nil @ A ) )
= ( append @ A @ L1 @ ( cons @ A @ E4 @ L2 ) ) )
= ( ( L1
= ( nil @ A ) )
& ( E4 = E3 )
& ( L2
= ( nil @ A ) ) ) ) ).
% list_e_eq_lel(1)
thf(fact_252_list__e__eq__lel_I2_J,axiom,
! [A: $tType,L1: list @ A,E4: A,L2: list @ A,E3: A] :
( ( ( append @ A @ L1 @ ( cons @ A @ E4 @ L2 ) )
= ( cons @ A @ E3 @ ( nil @ A ) ) )
= ( ( L1
= ( nil @ A ) )
& ( E4 = E3 )
& ( L2
= ( nil @ A ) ) ) ) ).
% list_e_eq_lel(2)
thf(fact_253_list__collect__set__simps_I4_J,axiom,
! [A: $tType,B: $tType,F2: B > ( set @ A ),L3: list @ B,L4: list @ B] :
( ( list_collect_set @ B @ A @ F2 @ ( append @ B @ L3 @ L4 ) )
= ( sup_sup @ ( set @ A ) @ ( list_collect_set @ B @ A @ F2 @ L3 ) @ ( list_collect_set @ B @ A @ F2 @ L4 ) ) ) ).
% list_collect_set_simps(4)
thf(fact_254_bind__simps_I2_J,axiom,
! [A: $tType,B: $tType,X: B,Xs: list @ B,F2: B > ( list @ A )] :
( ( bind @ B @ A @ ( cons @ B @ X @ Xs ) @ F2 )
= ( append @ A @ ( F2 @ X ) @ ( bind @ B @ A @ Xs @ F2 ) ) ) ).
% bind_simps(2)
%----Subclasses (4)
thf(subcl_Orderings_Olinorder___HOL_Otype,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ( type @ A @ ( type2 @ A ) ) ) ).
thf(subcl_Orderings_Olinorder___Orderings_Oord,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ( ord @ A @ ( type2 @ A ) ) ) ).
thf(subcl_Orderings_Olinorder___Orderings_Oorder,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ( order @ A @ ( type2 @ A ) ) ) ).
thf(subcl_Orderings_Olinorder___Orderings_Opreorder,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ( preorder @ A @ ( type2 @ A ) ) ) ).
%----Type constructors (31)
thf(tcon_HOL_Obool___Lattices_Obounded__lattice,axiom,
bounded_lattice @ $o @ ( type2 @ $o ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice_1,axiom,
! [A8: $tType] : ( bounded_lattice @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice_2,axiom,
! [A8: $tType,A9: $tType] :
( ( bounded_lattice @ A9 @ ( type2 @ A9 ) )
=> ( bounded_lattice @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
! [A8: $tType,A9: $tType] :
( ( bounded_lattice @ A9 @ ( type2 @ A9 ) )
=> ( bounde1808546759up_bot @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice__bot,axiom,
! [A8: $tType,A9: $tType] :
( ( bounded_lattice @ A9 @ ( type2 @ A9 ) )
=> ( bounded_lattice_bot @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A8: $tType,A9: $tType] :
( ( semilattice_sup @ A9 @ ( type2 @ A9 ) )
=> ( semilattice_sup @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder__bot,axiom,
! [A8: $tType,A9: $tType] :
( ( order_bot @ A9 @ ( type2 @ A9 ) )
=> ( order_bot @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A8: $tType,A9: $tType] :
( ( preorder @ A9 @ ( type2 @ A9 ) )
=> ( preorder @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Lattices_Olattice,axiom,
! [A8: $tType,A9: $tType] :
( ( lattice @ A9 @ ( type2 @ A9 ) )
=> ( lattice @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A8: $tType,A9: $tType] :
( ( order @ A9 @ ( type2 @ A9 ) )
=> ( order @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A8: $tType,A9: $tType] :
( ( ord @ A9 @ ( type2 @ A9 ) )
=> ( ord @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_fun___Orderings_Obot,axiom,
! [A8: $tType,A9: $tType] :
( ( bot @ A9 @ ( type2 @ A9 ) )
=> ( bot @ ( A8 > A9 ) @ ( type2 @ ( A8 > A9 ) ) ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_3,axiom,
! [A8: $tType] : ( bounde1808546759up_bot @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice__bot_4,axiom,
! [A8: $tType] : ( bounded_lattice_bot @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__sup_5,axiom,
! [A8: $tType] : ( semilattice_sup @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_6,axiom,
! [A8: $tType] : ( order_bot @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_7,axiom,
! [A8: $tType] : ( preorder @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Olattice_8,axiom,
! [A8: $tType] : ( lattice @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_9,axiom,
! [A8: $tType] : ( order @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_10,axiom,
! [A8: $tType] : ( ord @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_11,axiom,
! [A8: $tType] : ( bot @ ( set @ A8 ) @ ( type2 @ ( set @ A8 ) ) ) ).
thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_12,axiom,
bounde1808546759up_bot @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_13,axiom,
bounded_lattice_bot @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_14,axiom,
semilattice_sup @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_15,axiom,
order_bot @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_16,axiom,
preorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Lattices_Olattice_17,axiom,
lattice @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_18,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_19,axiom,
ord @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Obot_20,axiom,
bot @ $o @ ( type2 @ $o ) ).
%----Free types (1)
thf(tfree_0,hypothesis,
linorder @ a @ ( type2 @ a ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
( ( ( cons @ a @ x1 @ l1 )
= l2a )
= ( ( sorted1061247458sorted @ a @ ( cons @ a @ x1 @ l1 ) @ l2a )
& ( sorted1061247458sorted @ a @ l2a @ ( cons @ a @ x1 @ l1 ) ) ) ) ).
%------------------------------------------------------------------------------