TPTP Problem File: ITP196^2.p
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%------------------------------------------------------------------------------
% File : ITP196^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Sturm_Theorem problem prob_153__5878286_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : Sturm_Theorem/prob_153__5878286_1 [Des21]
% Status : Theorem
% Rating : 0.00 v8.1.0, 0.25 v7.5.0
% Syntax : Number of formulae : 358 ( 154 unt; 72 typ; 0 def)
% Number of atoms : 681 ( 416 equ; 0 cnn)
% Maximal formula atoms : 11 ( 2 avg)
% Number of connectives : 5742 ( 147 ~; 27 |; 61 &;5187 @)
% ( 0 <=>; 320 =>; 0 <=; 0 <~>)
% Maximal formula depth : 34 ( 8 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 306 ( 306 >; 0 *; 0 +; 0 <<)
% Number of symbols : 71 ( 70 usr; 0 con; 1-5 aty)
% Number of variables : 1284 ( 13 ^;1145 !; 49 ?;1284 :)
% ( 77 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:25:56.019
%------------------------------------------------------------------------------
% Could-be-implicit typings (7)
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_Polynomial_Opoly,type,
poly: $tType > $tType ).
thf(ty_t_Option_Ooption,type,
option: $tType > $tType ).
thf(ty_t_Real_Oreal,type,
real: $tType ).
thf(ty_t_List_Olist,type,
list: $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
% Explicit typings (65)
thf(sy_cl_Groups_Oab__group__add,type,
ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__semiring__0,type,
comm_semiring_0:
!>[A: $tType] : $o ).
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Nat_Osize,type,
size:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__semiring__1,type,
comm_semiring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__add,type,
comm_monoid_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__no__zero__divisors,type,
semiri1193490041visors:
!>[A: $tType] : $o ).
thf(sy_c_BNF__Greatest__Fixpoint_OShift,type,
bNF_Greatest_Shift:
!>[A: $tType] : ( ( set @ ( list @ A ) ) > A > ( set @ ( list @ A ) ) ) ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc,type,
bNF_Greatest_Succ:
!>[A: $tType] : ( ( set @ ( list @ A ) ) > ( list @ A ) > ( set @ A ) ) ).
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_Groups_Otimes__class_Otimes,type,
times_times:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_List_Oappend,type,
append:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oarg__min__list,type,
arg_min_list:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( list @ A ) > A ) ).
thf(sy_c_List_Obind,type,
bind:
!>[A: $tType,B: $tType] : ( ( list @ A ) > ( A > ( list @ B ) ) > ( list @ B ) ) ).
thf(sy_c_List_Obutlast,type,
butlast:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oconcat,type,
concat:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ A ) ) ).
thf(sy_c_List_Oenumerate,type,
enumerate:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ ( product_prod @ nat @ A ) ) ) ).
thf(sy_c_List_Ofoldr,type,
foldr:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > ( list @ A ) > B > B ) ).
thf(sy_c_List_Ogen__length,type,
gen_length:
!>[A: $tType] : ( nat > ( list @ A ) > nat ) ).
thf(sy_c_List_Oinsert,type,
insert:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olast,type,
last:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_List_Olenlex,type,
lenlex:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olex,type,
lex:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olexord,type,
lexord:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olist_OCons,type,
cons:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist_ONil,type,
nil:
!>[A: $tType] : ( list @ A ) ).
thf(sy_c_List_Olist_Omap,type,
map:
!>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( list @ A ) > ( list @ Aa ) ) ).
thf(sy_c_List_Olist_Orec__list,type,
rec_list:
!>[C: $tType,A: $tType] : ( C > ( A > ( list @ A ) > C > C ) > ( list @ A ) > C ) ).
thf(sy_c_List_Olist__ex,type,
list_ex:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Olist__ex1,type,
list_ex1:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Olistrel,type,
listrel:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) ) ).
thf(sy_c_List_Olistrel1,type,
listrel1:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olistrelp,type,
listrelp:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( list @ A ) > ( list @ B ) > $o ) ).
thf(sy_c_List_Omap__filter,type,
map_filter:
!>[A: $tType,B: $tType] : ( ( A > ( option @ 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_Omap__tailrec__rev__rel,type,
map_tailrec_rev_rel:
!>[A: $tType,B: $tType] : ( ( product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) > ( product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) > $o ) ).
thf(sy_c_List_Omaps,type,
maps:
!>[A: $tType,B: $tType] : ( ( A > ( list @ B ) ) > ( list @ A ) > ( list @ B ) ) ).
thf(sy_c_List_Omember,type,
member:
!>[A: $tType] : ( ( list @ A ) > A > $o ) ).
thf(sy_c_List_Onull,type,
null:
!>[A: $tType] : ( ( list @ A ) > $o ) ).
thf(sy_c_List_Oord_Olexordp__eq,type,
lexordp_eq:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Oord__class_Olexordp__eq,type,
ord_lexordp_eq:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Oproduct,type,
product:
!>[A: $tType,B: $tType] : ( ( list @ A ) > ( list @ B ) > ( list @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_List_Oproduct__lists,type,
product_lists:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Orotate,type,
rotate:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Orotate1,type,
rotate1:
!>[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_Osubseqs,type,
subseqs:
!>[A: $tType] : ( ( list @ A ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Ozip,type,
zip:
!>[A: $tType,B: $tType] : ( ( list @ A ) > ( list @ B ) > ( list @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Multiset_Olinorder__class_Opart,type,
linorder_part:
!>[B: $tType,A: $tType] : ( ( B > A ) > A > ( list @ B ) > ( product_prod @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) ) ) ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
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_Polynomial_Oplus__coeffs,type,
plus_coeffs:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
thf(sy_c_Relation_Oirrefl,type,
irrefl:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Sturm__Theorem__Mirabelle__xzqolxckln_Oquasi__sturm__seq,type,
sturm_2060350619rm_seq: ( list @ ( poly @ real ) ) > $o ).
thf(sy_c_Sturm__Theorem__Mirabelle__xzqolxckln_Osturm__seq,type,
sturm_801371416rm_seq: ( list @ ( poly @ real ) ) > ( poly @ real ) > $o ).
thf(sy_c_Sturm__Theorem__Mirabelle__xzqolxckln_Osturm__seq__axioms,type,
sturm_1585582267axioms: ( list @ ( poly @ real ) ) > ( poly @ real ) > $o ).
thf(sy_c_Wellfounded_Oaccp,type,
accp:
!>[A: $tType] : ( ( A > A > $o ) > A > $o ) ).
thf(sy_c_member,type,
member2:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
% Relevant facts (254)
thf(fact_0_quasi__sturm__seq_Ops__not__Nil,axiom,
! [Ps: list @ ( poly @ real )] :
( ( sturm_2060350619rm_seq @ Ps )
=> ( Ps
!= ( nil @ ( poly @ real ) ) ) ) ).
% quasi_sturm_seq.ps_not_Nil
thf(fact_1_sturm__seq_Oquasi__sturm__seq,axiom,
! [Ps: list @ ( poly @ real ),P: poly @ real] :
( ( sturm_801371416rm_seq @ Ps @ P )
=> ( sturm_2060350619rm_seq @ Ps ) ) ).
% sturm_seq.quasi_sturm_seq
thf(fact_2_sturm__seq__def,axiom,
( sturm_801371416rm_seq
= ( ^ [Ps2: list @ ( poly @ real ),P2: poly @ real] :
( ( sturm_2060350619rm_seq @ Ps2 )
& ( sturm_1585582267axioms @ Ps2 @ P2 ) ) ) ) ).
% sturm_seq_def
thf(fact_3_sturm__seq_Ointro,axiom,
! [Ps: list @ ( poly @ real ),P: poly @ real] :
( ( sturm_2060350619rm_seq @ Ps )
=> ( ( sturm_1585582267axioms @ Ps @ P )
=> ( sturm_801371416rm_seq @ Ps @ P ) ) ) ).
% sturm_seq.intro
thf(fact_4_list__ex1__simps_I1_J,axiom,
! [A: $tType,P3: A > $o] :
~ ( list_ex1 @ A @ P3 @ ( nil @ A ) ) ).
% list_ex1_simps(1)
thf(fact_5_bind__simps_I1_J,axiom,
! [B: $tType,A: $tType,F: B > ( list @ A )] :
( ( bind @ B @ A @ ( nil @ B ) @ F )
= ( nil @ A ) ) ).
% bind_simps(1)
thf(fact_6_member__rec_I2_J,axiom,
! [A: $tType,Y: A] :
~ ( member @ A @ ( nil @ A ) @ Y ) ).
% member_rec(2)
thf(fact_7_gen__length__code_I1_J,axiom,
! [A: $tType,N: nat] :
( ( gen_length @ A @ N @ ( nil @ A ) )
= N ) ).
% gen_length_code(1)
thf(fact_8_maps__simps_I2_J,axiom,
! [B: $tType,A: $tType,F: B > ( list @ A )] :
( ( maps @ B @ A @ F @ ( nil @ B ) )
= ( nil @ A ) ) ).
% maps_simps(2)
thf(fact_9_map__filter__simps_I2_J,axiom,
! [B: $tType,A: $tType,F: B > ( option @ A )] :
( ( map_filter @ B @ A @ F @ ( nil @ B ) )
= ( nil @ A ) ) ).
% map_filter_simps(2)
thf(fact_10_null__rec_I2_J,axiom,
! [B: $tType] : ( null @ B @ ( nil @ B ) ) ).
% null_rec(2)
thf(fact_11_eq__Nil__null,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
= ( nil @ A ) )
= ( null @ A @ Xs ) ) ).
% eq_Nil_null
thf(fact_12_sturm__seq_Oaxioms_I2_J,axiom,
! [Ps: list @ ( poly @ real ),P: poly @ real] :
( ( sturm_801371416rm_seq @ Ps @ P )
=> ( sturm_1585582267axioms @ Ps @ P ) ) ).
% sturm_seq.axioms(2)
thf(fact_13_list__ex__simps_I2_J,axiom,
! [A: $tType,P3: A > $o] :
~ ( list_ex @ A @ P3 @ ( nil @ A ) ) ).
% list_ex_simps(2)
thf(fact_14_rotate1__is__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( rotate1 @ A @ Xs )
= ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% rotate1_is_Nil_conv
thf(fact_15_split__Nil__iff,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( splice @ A @ Xs @ Ys )
= ( nil @ A ) )
= ( ( Xs
= ( nil @ A ) )
& ( Ys
= ( nil @ A ) ) ) ) ).
% split_Nil_iff
thf(fact_16_splice__Nil2,axiom,
! [A: $tType,Xs: list @ A] :
( ( splice @ A @ Xs @ ( nil @ A ) )
= Xs ) ).
% splice_Nil2
thf(fact_17_enumerate__simps_I1_J,axiom,
! [A: $tType,N: nat] :
( ( enumerate @ A @ N @ ( nil @ A ) )
= ( nil @ ( product_prod @ nat @ A ) ) ) ).
% enumerate_simps(1)
thf(fact_18_product_Osimps_I1_J,axiom,
! [B: $tType,A: $tType,Uu: list @ B] :
( ( product @ A @ B @ ( nil @ A ) @ Uu )
= ( nil @ ( product_prod @ A @ B ) ) ) ).
% product.simps(1)
thf(fact_19_map__tailrec__rev_Osimps_I1_J,axiom,
! [A: $tType,B: $tType,F: A > B,Bs: list @ B] :
( ( map_tailrec_rev @ A @ B @ F @ ( nil @ A ) @ Bs )
= Bs ) ).
% map_tailrec_rev.simps(1)
thf(fact_20_list_Osimps_I6_J,axiom,
! [A: $tType,C: $tType,F1: C,F2: A > ( list @ A ) > C > C] :
( ( rec_list @ C @ A @ F1 @ F2 @ ( nil @ A ) )
= F1 ) ).
% list.simps(6)
thf(fact_21_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_22_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_23_listrelp_ONil,axiom,
! [A: $tType,B: $tType,R: A > B > $o] : ( listrelp @ A @ B @ R @ ( nil @ A ) @ ( nil @ B ) ) ).
% listrelp.Nil
thf(fact_24_ord_Olexordp__eq_Ocong,axiom,
! [A: $tType] :
( ( lexordp_eq @ A )
= ( lexordp_eq @ A ) ) ).
% ord.lexordp_eq.cong
thf(fact_25_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_26_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_27_splice_Osimps_I1_J,axiom,
! [A: $tType,Ys: list @ A] :
( ( splice @ A @ ( nil @ A ) @ Ys )
= Ys ) ).
% splice.simps(1)
thf(fact_28_rotate1_Osimps_I1_J,axiom,
! [A: $tType] :
( ( rotate1 @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% rotate1.simps(1)
thf(fact_29_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_30_zip__Nil,axiom,
! [B: $tType,A: $tType,Ys: list @ B] :
( ( zip @ A @ B @ ( nil @ A ) @ Ys )
= ( nil @ ( product_prod @ A @ B ) ) ) ).
% zip_Nil
thf(fact_31_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 ) )
=> ~ ! [A2: A,As: list @ A] :
( ( Xa
= ( cons @ A @ A2 @ As ) )
=> ( Y
!= ( map_tailrec_rev @ A @ B @ X @ As @ ( cons @ B @ ( X @ A2 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_32_listrelp_Oinducts,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X1: list @ A,X2: list @ B,P3: ( list @ A ) > ( list @ B ) > $o] :
( ( listrelp @ A @ B @ R @ X1 @ X2 )
=> ( ( P3 @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X3: A,Y2: B,Xs2: list @ A,Ys2: list @ B] :
( ( R @ X3 @ Y2 )
=> ( ( listrelp @ A @ B @ R @ Xs2 @ Ys2 )
=> ( ( P3 @ Xs2 @ Ys2 )
=> ( P3 @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ B @ Y2 @ Ys2 ) ) ) ) )
=> ( P3 @ X1 @ X2 ) ) ) ) ).
% listrelp.inducts
thf(fact_33_listrelp_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( listrelp @ A @ B )
= ( ^ [R2: A > B > $o,A1: list @ A,A22: list @ B] :
( ( ( A1
= ( nil @ A ) )
& ( A22
= ( nil @ B ) ) )
| ? [X4: A,Y3: B,Xs3: list @ A,Ys3: list @ B] :
( ( A1
= ( cons @ A @ X4 @ Xs3 ) )
& ( A22
= ( cons @ B @ Y3 @ Ys3 ) )
& ( R2 @ X4 @ Y3 )
& ( listrelp @ A @ B @ R2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% listrelp.simps
thf(fact_34_listrelp_Ocases,axiom,
! [A: $tType,B: $tType,R: A > B > $o,A12: list @ A,A23: list @ B] :
( ( listrelp @ A @ B @ R @ A12 @ A23 )
=> ( ( ( A12
= ( nil @ A ) )
=> ( A23
!= ( nil @ B ) ) )
=> ~ ! [X3: A,Y2: B,Xs2: list @ A] :
( ( A12
= ( cons @ A @ X3 @ Xs2 ) )
=> ! [Ys2: list @ B] :
( ( A23
= ( cons @ B @ Y2 @ Ys2 ) )
=> ( ( R @ X3 @ Y2 )
=> ~ ( listrelp @ A @ B @ R @ Xs2 @ Ys2 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_35_splice_Oelims,axiom,
! [A: $tType,X: list @ A,Xa: list @ A,Y: list @ A] :
( ( ( splice @ A @ X @ Xa )
= Y )
=> ( ( ( X
= ( nil @ A ) )
=> ( Y != Xa ) )
=> ~ ! [X3: A,Xs2: list @ A] :
( ( X
= ( cons @ A @ X3 @ Xs2 ) )
=> ( Y
!= ( cons @ A @ X3 @ ( splice @ A @ Xa @ Xs2 ) ) ) ) ) ) ).
% splice.elims
thf(fact_36_ord_Olexordp__eq_Oinducts,axiom,
! [A: $tType,Less: A > A > $o,X1: list @ A,X2: list @ A,P3: ( list @ A ) > ( list @ A ) > $o] :
( ( lexordp_eq @ A @ Less @ X1 @ X2 )
=> ( ! [X_1: list @ A] : ( P3 @ ( nil @ A ) @ X_1 )
=> ( ! [X3: A,Y2: A,Xs2: list @ A,Ys2: list @ A] :
( ( Less @ X3 @ Y2 )
=> ( P3 @ ( 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 )
=> ( ( P3 @ Xs2 @ Ys2 )
=> ( P3 @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ A @ Y2 @ Ys2 ) ) ) ) ) )
=> ( P3 @ X1 @ X2 ) ) ) ) ) ).
% ord.lexordp_eq.inducts
thf(fact_37_ord_Olexordp__eq_Osimps,axiom,
! [A: $tType] :
( ( lexordp_eq @ A )
= ( ^ [Less2: A > A > $o,A1: list @ A,A22: list @ A] :
( ? [Ys3: list @ A] :
( ( A1
= ( nil @ A ) )
& ( A22 = Ys3 ) )
| ? [X4: A,Y3: A,Xs3: list @ A,Ys3: list @ A] :
( ( A1
= ( cons @ A @ X4 @ Xs3 ) )
& ( A22
= ( cons @ A @ Y3 @ Ys3 ) )
& ( Less2 @ X4 @ Y3 ) )
| ? [X4: A,Y3: A,Xs3: list @ A,Ys3: list @ A] :
( ( A1
= ( 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_38_ord_Olexordp__eq_Ocases,axiom,
! [A: $tType,Less: A > A > $o,A12: list @ A,A23: list @ A] :
( ( lexordp_eq @ A @ Less @ A12 @ A23 )
=> ( ( A12
!= ( nil @ A ) )
=> ( ! [X3: A] :
( ? [Xs2: list @ A] :
( A12
= ( cons @ A @ X3 @ Xs2 ) )
=> ! [Y2: A] :
( ? [Ys2: list @ A] :
( A23
= ( cons @ A @ Y2 @ Ys2 ) )
=> ~ ( Less @ X3 @ Y2 ) ) )
=> ~ ! [X3: A,Y2: A,Xs2: list @ A] :
( ( A12
= ( cons @ A @ X3 @ Xs2 ) )
=> ! [Ys2: list @ A] :
( ( A23
= ( cons @ A @ Y2 @ Ys2 ) )
=> ( ~ ( Less @ X3 @ Y2 )
=> ( ~ ( Less @ Y2 @ X3 )
=> ~ ( lexordp_eq @ A @ Less @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ).
% ord.lexordp_eq.cases
thf(fact_39_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_40_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_41_list__ex__simps_I1_J,axiom,
! [A: $tType,P3: A > $o,X: A,Xs: list @ A] :
( ( list_ex @ A @ P3 @ ( cons @ A @ X @ Xs ) )
= ( ( P3 @ X )
| ( list_ex @ A @ P3 @ Xs ) ) ) ).
% list_ex_simps(1)
thf(fact_42_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_43_not__Cons__self2,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( cons @ A @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_44_mem__Collect__eq,axiom,
! [A: $tType,A3: A,P3: A > $o] :
( ( member2 @ A @ A3 @ ( collect @ A @ P3 ) )
= ( P3 @ A3 ) ) ).
% mem_Collect_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( collect @ A
@ ^ [X4: A] : ( member2 @ A @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_46_Collect__cong,axiom,
! [A: $tType,P3: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P3 @ X3 )
= ( Q @ X3 ) )
=> ( ( collect @ A @ P3 )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_47_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X3: A] :
( ( F @ X3 )
= ( G @ X3 ) )
=> ( F = G ) ) ).
% ext
thf(fact_48_strict__sorted_Oinduct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P3: ( list @ A ) > $o,A0: list @ A] :
( ( P3 @ ( nil @ A ) )
=> ( ! [X3: A,Ys2: list @ A] :
( ( P3 @ Ys2 )
=> ( P3 @ ( cons @ A @ X3 @ Ys2 ) ) )
=> ( P3 @ A0 ) ) ) ) ).
% strict_sorted.induct
thf(fact_49_strict__sorted_Ocases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: list @ A] :
( ( X
!= ( nil @ A ) )
=> ~ ! [X3: A,Ys2: list @ A] :
( X
!= ( cons @ A @ X3 @ Ys2 ) ) ) ) ).
% strict_sorted.cases
thf(fact_50_map__tailrec__rev_Oinduct,axiom,
! [A: $tType,B: $tType,P3: ( A > B ) > ( list @ A ) > ( list @ B ) > $o,A0: A > B,A12: list @ A,A23: list @ B] :
( ! [F3: A > B,X_1: list @ B] : ( P3 @ F3 @ ( nil @ A ) @ X_1 )
=> ( ! [F3: A > B,A2: A,As: list @ A,Bs2: list @ B] :
( ( P3 @ F3 @ As @ ( cons @ B @ ( F3 @ A2 ) @ Bs2 ) )
=> ( P3 @ F3 @ ( cons @ A @ A2 @ As ) @ Bs2 ) )
=> ( P3 @ A0 @ A12 @ A23 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_51_list__nonempty__induct,axiom,
! [A: $tType,Xs: list @ A,P3: ( list @ A ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X3: A] : ( P3 @ ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ( ! [X3: A,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( P3 @ Xs2 )
=> ( P3 @ ( cons @ A @ X3 @ Xs2 ) ) ) )
=> ( P3 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_52_successively_Oinduct,axiom,
! [A: $tType,P3: ( A > A > $o ) > ( list @ A ) > $o,A0: A > A > $o,A12: list @ A] :
( ! [P4: A > A > $o] : ( P3 @ P4 @ ( nil @ A ) )
=> ( ! [P4: A > A > $o,X3: A] : ( P3 @ P4 @ ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ( ! [P4: A > A > $o,X3: A,Y2: A,Xs2: list @ A] :
( ( P3 @ P4 @ ( cons @ A @ Y2 @ Xs2 ) )
=> ( P3 @ P4 @ ( cons @ A @ X3 @ ( cons @ A @ Y2 @ Xs2 ) ) ) )
=> ( P3 @ A0 @ A12 ) ) ) ) ).
% successively.induct
thf(fact_53_arg__min__list_Oinduct,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [P3: ( A > B ) > ( list @ A ) > $o,A0: A > B,A12: list @ A] :
( ! [F3: A > B,X3: A] : ( P3 @ F3 @ ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ( ! [F3: A > B,X3: A,Y2: A,Zs: list @ A] :
( ( P3 @ F3 @ ( cons @ A @ Y2 @ Zs ) )
=> ( P3 @ F3 @ ( cons @ A @ X3 @ ( cons @ A @ Y2 @ Zs ) ) ) )
=> ( ! [A2: A > B] : ( P3 @ A2 @ ( nil @ A ) )
=> ( P3 @ A0 @ A12 ) ) ) ) ) ).
% arg_min_list.induct
thf(fact_54_remdups__adj_Oinduct,axiom,
! [A: $tType,P3: ( list @ A ) > $o,A0: list @ A] :
( ( P3 @ ( nil @ A ) )
=> ( ! [X3: A] : ( P3 @ ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ( ! [X3: A,Y2: A,Xs2: list @ A] :
( ( ( X3 = Y2 )
=> ( P3 @ ( cons @ A @ X3 @ Xs2 ) ) )
=> ( ( ( X3 != Y2 )
=> ( P3 @ ( cons @ A @ Y2 @ Xs2 ) ) )
=> ( P3 @ ( cons @ A @ X3 @ ( cons @ A @ Y2 @ Xs2 ) ) ) ) )
=> ( P3 @ A0 ) ) ) ) ).
% remdups_adj.induct
thf(fact_55_sorted__wrt_Oinduct,axiom,
! [A: $tType,P3: ( A > A > $o ) > ( list @ A ) > $o,A0: A > A > $o,A12: list @ A] :
( ! [P4: A > A > $o] : ( P3 @ P4 @ ( nil @ A ) )
=> ( ! [P4: A > A > $o,X3: A,Ys2: list @ A] :
( ( P3 @ P4 @ Ys2 )
=> ( P3 @ P4 @ ( cons @ A @ X3 @ Ys2 ) ) )
=> ( P3 @ A0 @ A12 ) ) ) ).
% sorted_wrt.induct
thf(fact_56_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_57_shuffles_Oinduct,axiom,
! [A: $tType,P3: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A12: list @ A] :
( ! [X_1: list @ A] : ( P3 @ ( nil @ A ) @ X_1 )
=> ( ! [Xs2: list @ A] : ( P3 @ Xs2 @ ( nil @ A ) )
=> ( ! [X3: A,Xs2: list @ A,Y2: A,Ys2: list @ A] :
( ( P3 @ Xs2 @ ( cons @ A @ Y2 @ Ys2 ) )
=> ( ( P3 @ ( cons @ A @ X3 @ Xs2 ) @ Ys2 )
=> ( P3 @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ A @ Y2 @ Ys2 ) ) ) )
=> ( P3 @ A0 @ A12 ) ) ) ) ).
% shuffles.induct
thf(fact_58_min__list_Oinduct,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [P3: ( list @ A ) > $o,A0: list @ A] :
( ! [X3: A,Xs2: list @ A] :
( ! [X212: A,X222: list @ A] :
( ( Xs2
= ( cons @ A @ X212 @ X222 ) )
=> ( P3 @ Xs2 ) )
=> ( P3 @ ( cons @ A @ X3 @ Xs2 ) ) )
=> ( ( P3 @ ( nil @ A ) )
=> ( P3 @ A0 ) ) ) ) ).
% min_list.induct
thf(fact_59_min__list_Ocases,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: list @ A] :
( ! [X3: A,Xs2: list @ A] :
( X
!= ( cons @ A @ X3 @ Xs2 ) )
=> ( X
= ( nil @ A ) ) ) ) ).
% min_list.cases
thf(fact_60_induct__list012,axiom,
! [A: $tType,P3: ( list @ A ) > $o,Xs: list @ A] :
( ( P3 @ ( nil @ A ) )
=> ( ! [X3: A] : ( P3 @ ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ( ! [X3: A,Y2: A,Zs: list @ A] :
( ( P3 @ Zs )
=> ( ( P3 @ ( cons @ A @ Y2 @ Zs ) )
=> ( P3 @ ( cons @ A @ X3 @ ( cons @ A @ Y2 @ Zs ) ) ) ) )
=> ( P3 @ Xs ) ) ) ) ).
% induct_list012
thf(fact_61_splice_Oinduct,axiom,
! [A: $tType,P3: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A12: list @ A] :
( ! [X_1: list @ A] : ( P3 @ ( nil @ A ) @ X_1 )
=> ( ! [X3: A,Xs2: list @ A,Ys2: list @ A] :
( ( P3 @ Ys2 @ Xs2 )
=> ( P3 @ ( cons @ A @ X3 @ Xs2 ) @ Ys2 ) )
=> ( P3 @ A0 @ A12 ) ) ) ).
% splice.induct
thf(fact_62_list__induct2_H,axiom,
! [A: $tType,B: $tType,P3: ( list @ A ) > ( list @ B ) > $o,Xs: list @ A,Ys: list @ B] :
( ( P3 @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X3: A,Xs2: list @ A] : ( P3 @ ( cons @ A @ X3 @ Xs2 ) @ ( nil @ B ) )
=> ( ! [Y2: B,Ys2: list @ B] : ( P3 @ ( nil @ A ) @ ( cons @ B @ Y2 @ Ys2 ) )
=> ( ! [X3: A,Xs2: list @ A,Y2: B,Ys2: list @ B] :
( ( P3 @ Xs2 @ Ys2 )
=> ( P3 @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ B @ Y2 @ Ys2 ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_63_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_64_list_Oinducts,axiom,
! [A: $tType,P3: ( list @ A ) > $o,List: list @ A] :
( ( P3 @ ( nil @ A ) )
=> ( ! [X12: A,X23: list @ A] :
( ( P3 @ X23 )
=> ( P3 @ ( cons @ A @ X12 @ X23 ) ) )
=> ( P3 @ List ) ) ) ).
% list.inducts
thf(fact_65_list_Oexhaust,axiom,
! [A: $tType,Y: list @ A] :
( ( Y
!= ( nil @ A ) )
=> ~ ! [X213: A,X223: list @ A] :
( Y
!= ( cons @ A @ X213 @ X223 ) ) ) ).
% list.exhaust
thf(fact_66_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_67_list_Odistinct_I1_J,axiom,
! [A: $tType,X21: A,X22: list @ A] :
( ( nil @ A )
!= ( cons @ A @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_68_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_69_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_70_splice_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A] :
( ( splice @ A @ ( cons @ A @ X @ Xs ) @ Ys )
= ( cons @ A @ X @ ( splice @ A @ Ys @ Xs ) ) ) ).
% splice.simps(2)
thf(fact_71_list_Osimps_I7_J,axiom,
! [C: $tType,A: $tType,F1: C,F2: A > ( list @ A ) > C > C,X21: A,X22: list @ A] :
( ( rec_list @ C @ A @ F1 @ F2 @ ( cons @ A @ X21 @ X22 ) )
= ( F2 @ X21 @ X22 @ ( rec_list @ C @ A @ F1 @ F2 @ X22 ) ) ) ).
% list.simps(7)
thf(fact_72_listrelp_OCons,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X: A,Y: B,Xs: list @ A,Ys: list @ B] :
( ( R @ X @ Y )
=> ( ( listrelp @ A @ B @ R @ Xs @ Ys )
=> ( listrelp @ A @ B @ R @ ( cons @ A @ X @ Xs ) @ ( cons @ B @ Y @ Ys ) ) ) ) ).
% listrelp.Cons
thf(fact_73_sturm__seq_Ops__first__two,axiom,
! [Ps: list @ ( poly @ real ),P: poly @ real] :
( ( sturm_801371416rm_seq @ Ps @ P )
=> ~ ! [Q2: poly @ real,Ps3: list @ ( poly @ real )] :
( Ps
!= ( cons @ ( poly @ real ) @ P @ ( cons @ ( poly @ real ) @ Q2 @ Ps3 ) ) ) ) ).
% sturm_seq.ps_first_two
thf(fact_74_map__tailrec__rev_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,F: A > B,A3: A,As2: list @ A,Bs: list @ B] :
( ( map_tailrec_rev @ A @ B @ F @ ( cons @ A @ A3 @ As2 ) @ Bs )
= ( map_tailrec_rev @ A @ B @ F @ As2 @ ( cons @ B @ ( F @ A3 ) @ Bs ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_75_null__rec_I1_J,axiom,
! [A: $tType,X: A,Xs: list @ A] :
~ ( null @ A @ ( cons @ A @ X @ Xs ) ) ).
% null_rec(1)
thf(fact_76_member__rec_I1_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: A] :
( ( member @ A @ ( cons @ A @ X @ Xs ) @ Y )
= ( ( X = Y )
| ( member @ A @ Xs @ Y ) ) ) ).
% member_rec(1)
thf(fact_77_zip__eq__Nil__iff,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] :
( ( ( zip @ A @ B @ Xs @ Ys )
= ( nil @ ( product_prod @ A @ B ) ) )
= ( ( Xs
= ( nil @ A ) )
| ( Ys
= ( nil @ B ) ) ) ) ).
% zip_eq_Nil_iff
thf(fact_78_zip_Osimps_I1_J,axiom,
! [B: $tType,A: $tType,Xs: list @ A] :
( ( zip @ A @ B @ Xs @ ( nil @ B ) )
= ( nil @ ( product_prod @ A @ B ) ) ) ).
% zip.simps(1)
thf(fact_79_minus__poly__rev__list_Oinduct,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [P3: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A12: list @ A] :
( ! [X3: A,Xs2: list @ A,Y2: A,Ys2: list @ A] :
( ( P3 @ Xs2 @ Ys2 )
=> ( P3 @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ A @ Y2 @ Ys2 ) ) )
=> ( ! [Xs2: list @ A] : ( P3 @ Xs2 @ ( nil @ A ) )
=> ( ! [Y2: A,Ys2: list @ A] : ( P3 @ ( nil @ A ) @ ( cons @ A @ Y2 @ Ys2 ) )
=> ( P3 @ A0 @ A12 ) ) ) ) ) ).
% minus_poly_rev_list.induct
thf(fact_80_plus__coeffs_Oinduct,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [P3: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A12: list @ A] :
( ! [Xs2: list @ A] : ( P3 @ Xs2 @ ( nil @ A ) )
=> ( ! [V: A,Va: list @ A] : ( P3 @ ( nil @ A ) @ ( cons @ A @ V @ Va ) )
=> ( ! [X3: A,Xs2: list @ A,Y2: A,Ys2: list @ A] :
( ( P3 @ Xs2 @ Ys2 )
=> ( P3 @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ A @ Y2 @ Ys2 ) ) )
=> ( P3 @ A0 @ A12 ) ) ) ) ) ).
% plus_coeffs.induct
thf(fact_81_insert__Nil,axiom,
! [A: $tType,X: A] :
( ( insert @ A @ X @ ( nil @ A ) )
= ( cons @ A @ X @ ( nil @ A ) ) ) ).
% insert_Nil
thf(fact_82_arg__min__list_Osimps_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [F: A > B,X: A] :
( ( arg_min_list @ A @ B @ F @ ( cons @ A @ X @ ( nil @ A ) ) )
= X ) ) ).
% arg_min_list.simps(1)
thf(fact_83_lexordp__eq__simps_I3_J,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Xs: list @ A] :
~ ( ord_lexordp_eq @ A @ ( cons @ A @ X @ Xs ) @ ( nil @ A ) ) ) ).
% lexordp_eq_simps(3)
thf(fact_84_rotate1_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( rotate1 @ A @ ( cons @ A @ X @ Xs ) )
= ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ).
% rotate1.simps(2)
thf(fact_85_bind__simps_I2_J,axiom,
! [A: $tType,B: $tType,X: B,Xs: list @ B,F: B > ( list @ A )] :
( ( bind @ B @ A @ ( cons @ B @ X @ Xs ) @ F )
= ( append @ A @ ( F @ X ) @ ( bind @ B @ A @ Xs @ F ) ) ) ).
% bind_simps(2)
thf(fact_86_gen__length__code_I2_J,axiom,
! [B: $tType,N: nat,X: B,Xs: list @ B] :
( ( gen_length @ B @ N @ ( cons @ B @ X @ Xs ) )
= ( gen_length @ B @ ( suc @ N ) @ Xs ) ) ).
% gen_length_code(2)
thf(fact_87_append_Oassoc,axiom,
! [A: $tType,A3: list @ A,B2: list @ A,C2: list @ A] :
( ( append @ A @ ( append @ A @ A3 @ B2 ) @ C2 )
= ( append @ A @ A3 @ ( append @ A @ B2 @ C2 ) ) ) ).
% append.assoc
thf(fact_88_append__assoc,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs2: list @ A] :
( ( append @ A @ ( append @ A @ Xs @ Ys ) @ Zs2 )
= ( append @ A @ Xs @ ( append @ A @ Ys @ Zs2 ) ) ) ).
% append_assoc
thf(fact_89_append__same__eq,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A,Zs2: list @ A] :
( ( ( append @ A @ Ys @ Xs )
= ( append @ A @ Zs2 @ Xs ) )
= ( Ys = Zs2 ) ) ).
% append_same_eq
thf(fact_90_same__append__eq,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs2: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Xs @ Zs2 ) )
= ( Ys = Zs2 ) ) ).
% same_append_eq
thf(fact_91_append__Nil2,axiom,
! [A: $tType,Xs: list @ A] :
( ( append @ A @ Xs @ ( nil @ A ) )
= Xs ) ).
% append_Nil2
thf(fact_92_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_93_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_94_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_95_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_96_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_97_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_98_append_Oright__neutral,axiom,
! [A: $tType,A3: list @ A] :
( ( append @ A @ A3 @ ( nil @ A ) )
= A3 ) ).
% append.right_neutral
thf(fact_99_lexordp__eq__simps_I2_J,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Xs: list @ A] :
( ( ord_lexordp_eq @ A @ Xs @ ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ) ).
% lexordp_eq_simps(2)
thf(fact_100_lexordp__eq__simps_I1_J,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Ys: list @ A] : ( ord_lexordp_eq @ A @ ( nil @ A ) @ Ys ) ) ).
% lexordp_eq_simps(1)
thf(fact_101_list__ex__append,axiom,
! [A: $tType,P3: A > $o,Xs: list @ A,Ys: list @ A] :
( ( list_ex @ A @ P3 @ ( append @ A @ Xs @ Ys ) )
= ( ( list_ex @ A @ P3 @ Xs )
| ( list_ex @ A @ P3 @ Ys ) ) ) ).
% list_ex_append
thf(fact_102_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_103_append__eq__appendI,axiom,
! [A: $tType,Xs: list @ A,Xs1: list @ A,Zs2: list @ A,Ys: list @ A,Us: list @ A] :
( ( ( append @ A @ Xs @ Xs1 )
= Zs2 )
=> ( ( Ys
= ( append @ A @ Xs1 @ Us ) )
=> ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Zs2 @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_104_append__eq__append__conv2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs2: list @ A,Ts: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Zs2 @ Ts ) )
= ( ? [Us2: list @ A] :
( ( ( Xs
= ( append @ A @ Zs2 @ Us2 ) )
& ( ( append @ A @ Us2 @ Ys )
= Ts ) )
| ( ( ( append @ A @ Xs @ Us2 )
= Zs2 )
& ( Ys
= ( append @ A @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_105_lexordp__eq__pref,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [U: list @ A,V2: list @ A] : ( ord_lexordp_eq @ A @ U @ ( append @ A @ U @ V2 ) ) ) ).
% lexordp_eq_pref
thf(fact_106_lexordp__eq__refl,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Xs: list @ A] : ( ord_lexordp_eq @ A @ Xs @ Xs ) ) ).
% lexordp_eq_refl
thf(fact_107_lexordp__eq__trans,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys: list @ A,Zs2: list @ A] :
( ( ord_lexordp_eq @ A @ Xs @ Ys )
=> ( ( ord_lexordp_eq @ A @ Ys @ Zs2 )
=> ( ord_lexordp_eq @ A @ Xs @ Zs2 ) ) ) ) ).
% lexordp_eq_trans
thf(fact_108_lexordp__eq__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys: list @ A] :
( ( ord_lexordp_eq @ A @ Xs @ Ys )
| ( ord_lexordp_eq @ A @ Ys @ Xs ) ) ) ).
% lexordp_eq_linear
thf(fact_109_lexordp__eq__antisym,axiom,
! [A: $tType] :
( ( linorder @ 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_110_Cons__eq__appendI,axiom,
! [A: $tType,X: A,Xs1: list @ A,Ys: list @ A,Xs: list @ A,Zs2: list @ A] :
( ( ( cons @ A @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append @ A @ Xs1 @ Zs2 ) )
=> ( ( cons @ A @ X @ Xs )
= ( append @ A @ Ys @ Zs2 ) ) ) ) ).
% Cons_eq_appendI
thf(fact_111_append__Cons,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A] :
( ( append @ A @ ( cons @ A @ X @ Xs ) @ Ys )
= ( cons @ A @ X @ ( append @ A @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_112_eq__Nil__appendI,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs = Ys )
=> ( Xs
= ( append @ A @ ( nil @ A ) @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_113_append__Nil,axiom,
! [A: $tType,Ys: list @ A] :
( ( append @ A @ ( nil @ A ) @ Ys )
= Ys ) ).
% append_Nil
thf(fact_114_append_Oleft__neutral,axiom,
! [A: $tType,A3: list @ A] :
( ( append @ A @ ( nil @ A ) @ A3 )
= A3 ) ).
% append.left_neutral
thf(fact_115_ord_Olexordp__eq__pref,axiom,
! [A: $tType,Less: A > A > $o,U: list @ A,V2: list @ A] : ( lexordp_eq @ A @ Less @ U @ ( append @ A @ U @ V2 ) ) ).
% ord.lexordp_eq_pref
thf(fact_116_lexordp__eq_ONil,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Ys: list @ A] : ( ord_lexordp_eq @ A @ ( nil @ A ) @ Ys ) ) ).
% lexordp_eq.Nil
thf(fact_117_rev__induct,axiom,
! [A: $tType,P3: ( list @ A ) > $o,Xs: list @ A] :
( ( P3 @ ( nil @ A ) )
=> ( ! [X3: A,Xs2: list @ A] :
( ( P3 @ Xs2 )
=> ( P3 @ ( append @ A @ Xs2 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) )
=> ( P3 @ Xs ) ) ) ).
% rev_induct
thf(fact_118_rev__exhaust,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ~ ! [Ys2: list @ A,Y2: A] :
( Xs
!= ( append @ A @ Ys2 @ ( cons @ A @ Y2 @ ( nil @ A ) ) ) ) ) ).
% rev_exhaust
thf(fact_119_Cons__eq__append__conv,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A,Zs2: list @ A] :
( ( ( cons @ A @ X @ Xs )
= ( append @ A @ Ys @ Zs2 ) )
= ( ( ( Ys
= ( nil @ A ) )
& ( ( cons @ A @ X @ Xs )
= Zs2 ) )
| ? [Ys4: list @ A] :
( ( ( cons @ A @ X @ Ys4 )
= Ys )
& ( Xs
= ( append @ A @ Ys4 @ Zs2 ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_120_append__eq__Cons__conv,axiom,
! [A: $tType,Ys: list @ A,Zs2: list @ A,X: A,Xs: list @ A] :
( ( ( append @ A @ Ys @ Zs2 )
= ( cons @ A @ X @ Xs ) )
= ( ( ( Ys
= ( nil @ A ) )
& ( Zs2
= ( cons @ A @ X @ Xs ) ) )
| ? [Ys4: list @ A] :
( ( Ys
= ( cons @ A @ X @ Ys4 ) )
& ( ( append @ A @ Ys4 @ Zs2 )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_121_rev__nonempty__induct,axiom,
! [A: $tType,Xs: list @ A,P3: ( list @ A ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X3: A] : ( P3 @ ( cons @ A @ X3 @ ( nil @ A ) ) )
=> ( ! [X3: A,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( P3 @ Xs2 )
=> ( P3 @ ( append @ A @ Xs2 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) ) ) )
=> ( P3 @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_122_maps__simps_I1_J,axiom,
! [A: $tType,B: $tType,F: B > ( list @ A ),X: B,Xs: list @ B] :
( ( maps @ B @ A @ F @ ( cons @ B @ X @ Xs ) )
= ( append @ A @ ( F @ X ) @ ( maps @ B @ A @ F @ Xs ) ) ) ).
% maps_simps(1)
thf(fact_123_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_124_nat_Oinject,axiom,
! [X2: nat,Y23: nat] :
( ( ( suc @ X2 )
= ( suc @ Y23 ) )
= ( X2 = Y23 ) ) ).
% nat.inject
thf(fact_125_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_126_subseqs_Osimps_I1_J,axiom,
! [A: $tType] :
( ( subseqs @ A @ ( nil @ A ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% subseqs.simps(1)
thf(fact_127_concat__eq__append__conv,axiom,
! [A: $tType,Xss2: list @ ( list @ A ),Ys: list @ A,Zs2: list @ A] :
( ( ( concat @ A @ Xss2 )
= ( append @ A @ Ys @ Zs2 ) )
= ( ( ( Xss2
= ( nil @ ( list @ A ) ) )
=> ( ( Ys
= ( nil @ A ) )
& ( Zs2
= ( nil @ A ) ) ) )
& ( ( Xss2
!= ( nil @ ( list @ A ) ) )
=> ? [Xss1: list @ ( list @ A ),Xs3: list @ A,Xs4: list @ A,Xss22: list @ ( list @ A )] :
( ( Xss2
= ( append @ ( list @ A ) @ Xss1 @ ( cons @ ( list @ A ) @ ( append @ A @ Xs3 @ Xs4 ) @ Xss22 ) ) )
& ( Ys
= ( append @ A @ ( concat @ A @ Xss1 ) @ Xs3 ) )
& ( Zs2
= ( append @ A @ Xs4 @ ( concat @ A @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_128_concat__append,axiom,
! [A: $tType,Xs: list @ ( list @ A ),Ys: list @ ( list @ A )] :
( ( concat @ A @ ( append @ ( list @ A ) @ Xs @ Ys ) )
= ( append @ A @ ( concat @ A @ Xs ) @ ( concat @ A @ Ys ) ) ) ).
% concat_append
thf(fact_129_concat_Osimps_I1_J,axiom,
! [A: $tType] :
( ( concat @ A @ ( nil @ ( list @ A ) ) )
= ( nil @ A ) ) ).
% concat.simps(1)
thf(fact_130_concat_Osimps_I2_J,axiom,
! [A: $tType,X: list @ A,Xs: list @ ( list @ A )] :
( ( concat @ A @ ( cons @ ( list @ A ) @ X @ Xs ) )
= ( append @ A @ X @ ( concat @ A @ Xs ) ) ) ).
% concat.simps(2)
thf(fact_131_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_132_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_133_concat__eq__appendD,axiom,
! [A: $tType,Xss2: list @ ( list @ A ),Ys: list @ A,Zs2: list @ A] :
( ( ( concat @ A @ Xss2 )
= ( append @ A @ Ys @ Zs2 ) )
=> ( ( Xss2
!= ( nil @ ( list @ A ) ) )
=> ? [Xss12: list @ ( list @ A ),Xs2: list @ A,Xs5: list @ A,Xss23: list @ ( list @ A )] :
( ( Xss2
= ( append @ ( list @ A ) @ Xss12 @ ( cons @ ( list @ A ) @ ( append @ A @ Xs2 @ Xs5 ) @ Xss23 ) ) )
& ( Ys
= ( append @ A @ ( concat @ A @ Xss12 ) @ Xs2 ) )
& ( Zs2
= ( append @ A @ Xs5 @ ( concat @ A @ Xss23 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_134_butlast__snoc,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( butlast @ A @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_135_enumerate__simps_I2_J,axiom,
! [B: $tType,N: nat,X: B,Xs: list @ B] :
( ( enumerate @ B @ N @ ( cons @ B @ X @ Xs ) )
= ( cons @ ( product_prod @ nat @ B ) @ ( product_Pair @ nat @ B @ N @ X ) @ ( enumerate @ B @ ( suc @ N ) @ Xs ) ) ) ).
% enumerate_simps(2)
thf(fact_136_last__snoc,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( last @ A @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) )
= X ) ).
% last_snoc
thf(fact_137_plus__coeffs_Osimps_I2_J,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [V2: A,Va2: list @ A] :
( ( plus_coeffs @ A @ ( nil @ A ) @ ( cons @ A @ V2 @ Va2 ) )
= ( cons @ A @ V2 @ Va2 ) ) ) ).
% plus_coeffs.simps(2)
thf(fact_138_last__appendL,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A] :
( ( Ys
= ( nil @ A ) )
=> ( ( last @ A @ ( append @ A @ Xs @ Ys ) )
= ( last @ A @ Xs ) ) ) ).
% last_appendL
thf(fact_139_last__appendR,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A] :
( ( Ys
!= ( nil @ A ) )
=> ( ( last @ A @ ( append @ A @ Xs @ Ys ) )
= ( last @ A @ Ys ) ) ) ).
% last_appendR
thf(fact_140_zip__Cons__Cons,axiom,
! [A: $tType,B: $tType,X: A,Xs: list @ A,Y: B,Ys: list @ B] :
( ( zip @ A @ B @ ( cons @ A @ X @ Xs ) @ ( cons @ B @ Y @ Ys ) )
= ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) ).
% zip_Cons_Cons
thf(fact_141_append__butlast__last__id,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( append @ A @ ( butlast @ A @ Xs ) @ ( cons @ A @ ( last @ A @ Xs ) @ ( nil @ A ) ) )
= Xs ) ) ).
% append_butlast_last_id
thf(fact_142_butlast_Osimps_I1_J,axiom,
! [A: $tType] :
( ( butlast @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% butlast.simps(1)
thf(fact_143_snoc__eq__iff__butlast,axiom,
! [A: $tType,Xs: list @ A,X: A,Ys: list @ A] :
( ( ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) )
= Ys )
= ( ( Ys
!= ( nil @ A ) )
& ( ( butlast @ A @ Ys )
= Xs )
& ( ( last @ A @ Ys )
= X ) ) ) ).
% snoc_eq_iff_butlast
thf(fact_144_last_Osimps,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( ( Xs
= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X @ Xs ) )
= X ) )
& ( ( Xs
!= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X @ Xs ) )
= ( last @ A @ Xs ) ) ) ) ).
% last.simps
thf(fact_145_last__ConsL,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( Xs
= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X @ Xs ) )
= X ) ) ).
% last_ConsL
thf(fact_146_last__ConsR,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( last @ A @ ( cons @ A @ X @ Xs ) )
= ( last @ A @ Xs ) ) ) ).
% last_ConsR
thf(fact_147_zip__eq__ConsE,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,Xy: product_prod @ A @ B,Xys: list @ ( product_prod @ A @ B )] :
( ( ( zip @ A @ B @ Xs @ Ys )
= ( cons @ ( product_prod @ A @ B ) @ Xy @ Xys ) )
=> ~ ! [X3: A,Xs5: list @ A] :
( ( Xs
= ( cons @ A @ X3 @ Xs5 ) )
=> ! [Y2: B,Ys5: list @ B] :
( ( Ys
= ( cons @ B @ Y2 @ Ys5 ) )
=> ( ( Xy
= ( product_Pair @ A @ B @ X3 @ Y2 ) )
=> ( Xys
!= ( zip @ A @ B @ Xs5 @ Ys5 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_148_plus__coeffs_Osimps_I1_J,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [Xs: list @ A] :
( ( plus_coeffs @ A @ Xs @ ( nil @ A ) )
= Xs ) ) ).
% plus_coeffs.simps(1)
thf(fact_149_last__append,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A] :
( ( ( Ys
= ( nil @ A ) )
=> ( ( last @ A @ ( append @ A @ Xs @ Ys ) )
= ( last @ A @ Xs ) ) )
& ( ( Ys
!= ( nil @ A ) )
=> ( ( last @ A @ ( append @ A @ Xs @ Ys ) )
= ( last @ A @ Ys ) ) ) ) ).
% last_append
thf(fact_150_longest__common__suffix,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
? [Ss: list @ A,Xs5: list @ A,Ys5: list @ A] :
( ( Xs
= ( append @ A @ Xs5 @ Ss ) )
& ( Ys
= ( append @ A @ Ys5 @ Ss ) )
& ( ( Xs5
= ( nil @ A ) )
| ( Ys5
= ( nil @ A ) )
| ( ( last @ A @ Xs5 )
!= ( last @ A @ Ys5 ) ) ) ) ).
% longest_common_suffix
thf(fact_151_butlast_Osimps_I2_J,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( ( Xs
= ( nil @ A ) )
=> ( ( butlast @ A @ ( cons @ A @ X @ Xs ) )
= ( nil @ A ) ) )
& ( ( Xs
!= ( nil @ A ) )
=> ( ( butlast @ A @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X @ ( butlast @ A @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_152_butlast__append,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A] :
( ( ( Ys
= ( nil @ A ) )
=> ( ( butlast @ A @ ( append @ A @ Xs @ Ys ) )
= ( butlast @ A @ Xs ) ) )
& ( ( Ys
!= ( nil @ A ) )
=> ( ( butlast @ A @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ Xs @ ( butlast @ A @ Ys ) ) ) ) ) ).
% butlast_append
thf(fact_153_snoc__listrel1__snoc__iff,axiom,
! [A: $tType,Xs: list @ A,X: A,Ys: list @ A,Y: A,R: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) @ ( append @ A @ Ys @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) @ ( listrel1 @ A @ R ) )
= ( ( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R ) )
& ( X = Y ) )
| ( ( Xs = Ys )
& ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_154_last__zip,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] :
( ( Xs
!= ( nil @ A ) )
=> ( ( Ys
!= ( nil @ B ) )
=> ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( last @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) )
= ( product_Pair @ A @ B @ ( last @ A @ Xs ) @ ( last @ B @ Ys ) ) ) ) ) ) ).
% last_zip
thf(fact_155_concat__conv__foldr,axiom,
! [A: $tType] :
( ( concat @ A )
= ( ^ [Xss3: list @ ( list @ A )] : ( foldr @ ( list @ A ) @ ( list @ A ) @ ( append @ A ) @ Xss3 @ ( nil @ A ) ) ) ) ).
% concat_conv_foldr
thf(fact_156_rotate__Suc,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( rotate @ A @ ( suc @ N ) @ Xs )
= ( rotate1 @ A @ ( rotate @ A @ N @ Xs ) ) ) ).
% rotate_Suc
thf(fact_157_append__eq__append__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Us: list @ A,Vs: list @ A] :
( ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) )
| ( ( size_size @ ( list @ A ) @ Us )
= ( size_size @ ( list @ A ) @ Vs ) ) )
=> ( ( ( append @ A @ Xs @ Us )
= ( append @ A @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_158_rotate__is__Nil__conv,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ( rotate @ A @ N @ Xs )
= ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% rotate_is_Nil_conv
thf(fact_159_length__rotate,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( rotate @ A @ N @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_rotate
thf(fact_160_foldr__append,axiom,
! [B: $tType,A: $tType,F: B > A > A,Xs: list @ B,Ys: list @ B,A3: A] :
( ( foldr @ B @ A @ F @ ( append @ B @ Xs @ Ys ) @ A3 )
= ( foldr @ B @ A @ F @ Xs @ ( foldr @ B @ A @ F @ Ys @ A3 ) ) ) ).
% foldr_append
thf(fact_161_length__enumerate,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( size_size @ ( list @ ( product_prod @ nat @ A ) ) @ ( enumerate @ A @ N @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_enumerate
thf(fact_162_length__rotate1,axiom,
! [A: $tType,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( rotate1 @ A @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_rotate1
thf(fact_163_zip__append,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Us: list @ B,Ys: list @ A,Vs: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Us ) )
=> ( ( zip @ A @ B @ ( append @ A @ Xs @ Ys ) @ ( append @ B @ Us @ Vs ) )
= ( append @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Us ) @ ( zip @ A @ B @ Ys @ Vs ) ) ) ) ).
% zip_append
thf(fact_164_Cons__listrel1__Cons,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: A,Ys: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) ) @ ( listrel1 @ A @ R ) )
= ( ( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
& ( Xs = Ys ) )
| ( ( X = Y )
& ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_165_append__listrel1I,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R: set @ ( product_prod @ A @ A ),Us: list @ A,Vs: list @ A] :
( ( ( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R ) )
& ( Us = Vs ) )
| ( ( Xs = Ys )
& ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Us @ Vs ) @ ( listrel1 @ A @ R ) ) ) )
=> ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Us ) @ ( append @ A @ Ys @ Vs ) ) @ ( listrel1 @ A @ R ) ) ) ).
% append_listrel1I
thf(fact_166_listrel1__eq__len,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R ) )
=> ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) ) ) ).
% listrel1_eq_len
thf(fact_167_map__tailrec__rev_Ocases,axiom,
! [A: $tType,B: $tType,X: product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) )] :
( ! [F3: A > B,Bs2: list @ B] :
( X
!= ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F3 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ Bs2 ) ) )
=> ~ ! [F3: A > B,A2: A,As: list @ A,Bs2: list @ B] :
( X
!= ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F3 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A2 @ As ) @ Bs2 ) ) ) ) ).
% map_tailrec_rev.cases
thf(fact_168_not__listrel1__Nil,axiom,
! [A: $tType,Xs: list @ A,R: set @ ( product_prod @ A @ A )] :
~ ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ ( nil @ A ) ) @ ( listrel1 @ A @ R ) ) ).
% not_listrel1_Nil
thf(fact_169_not__Nil__listrel1,axiom,
! [A: $tType,Xs: list @ A,R: set @ ( product_prod @ A @ A )] :
~ ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Xs ) @ ( listrel1 @ A @ R ) ) ).
% not_Nil_listrel1
thf(fact_170_listrel1I2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R: set @ ( product_prod @ A @ A ),X: A] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R ) )
=> ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ X @ Ys ) ) @ ( listrel1 @ A @ R ) ) ) ).
% listrel1I2
thf(fact_171_neq__if__length__neq,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
!= ( size_size @ ( list @ A ) @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_172_Ex__list__of__length,axiom,
! [A: $tType,N: nat] :
? [Xs2: list @ A] :
( ( size_size @ ( list @ A ) @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_173_size__neq__size__imp__neq,axiom,
! [A: $tType] :
( ( size @ A )
=> ! [X: A,Y: A] :
( ( ( size_size @ A @ X )
!= ( size_size @ A @ Y ) )
=> ( X != Y ) ) ) ).
% size_neq_size_imp_neq
thf(fact_174_rotate__append,axiom,
! [A: $tType,L: list @ A,Q3: list @ A] :
( ( rotate @ A @ ( size_size @ ( list @ A ) @ L ) @ ( append @ A @ L @ Q3 ) )
= ( append @ A @ Q3 @ L ) ) ).
% rotate_append
thf(fact_175_rotate1__rotate__swap,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( rotate1 @ A @ ( rotate @ A @ N @ Xs ) )
= ( rotate @ A @ N @ ( rotate1 @ A @ Xs ) ) ) ).
% rotate1_rotate_swap
thf(fact_176_arg__min__list_Ocases,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [X: product_prod @ ( A > B ) @ ( list @ A )] :
( ! [F3: A > B,X3: A] :
( X
!= ( product_Pair @ ( A > B ) @ ( list @ A ) @ F3 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) )
=> ( ! [F3: A > B,X3: A,Y2: A,Zs: list @ A] :
( X
!= ( product_Pair @ ( A > B ) @ ( list @ A ) @ F3 @ ( cons @ A @ X3 @ ( cons @ A @ Y2 @ Zs ) ) ) )
=> ~ ! [A2: A > B] :
( X
!= ( product_Pair @ ( A > B ) @ ( list @ A ) @ A2 @ ( nil @ A ) ) ) ) ) ) ).
% arg_min_list.cases
thf(fact_177_successively_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > A > $o ) @ ( list @ A )] :
( ! [P4: A > A > $o] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P4 @ ( nil @ A ) ) )
=> ( ! [P4: A > A > $o,X3: A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P4 @ ( cons @ A @ X3 @ ( nil @ A ) ) ) )
=> ~ ! [P4: A > A > $o,X3: A,Y2: A,Xs2: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P4 @ ( cons @ A @ X3 @ ( cons @ A @ Y2 @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_178_sorted__wrt_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > A > $o ) @ ( list @ A )] :
( ! [P4: A > A > $o] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P4 @ ( nil @ A ) ) )
=> ~ ! [P4: A > A > $o,X3: A,Ys2: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P4 @ ( cons @ A @ X3 @ Ys2 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_179_Suc__length__conv,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ( suc @ N )
= ( size_size @ ( list @ A ) @ Xs ) )
= ( ? [Y3: A,Ys3: list @ A] :
( ( Xs
= ( cons @ A @ Y3 @ Ys3 ) )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_180_length__Suc__conv,axiom,
! [A: $tType,Xs: list @ A,N: nat] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( suc @ N ) )
= ( ? [Y3: A,Ys3: list @ A] :
( ( Xs
= ( cons @ A @ Y3 @ Ys3 ) )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_181_list__induct4,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,Xs: list @ A,Ys: list @ B,Zs2: list @ C,Ws: list @ D,P3: ( list @ A ) > ( list @ B ) > ( list @ C ) > ( list @ D ) > $o] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys )
= ( size_size @ ( list @ C ) @ Zs2 ) )
=> ( ( ( size_size @ ( list @ C ) @ Zs2 )
= ( size_size @ ( list @ D ) @ Ws ) )
=> ( ( P3 @ ( nil @ A ) @ ( nil @ B ) @ ( nil @ C ) @ ( nil @ D ) )
=> ( ! [X3: A,Xs2: list @ A,Y2: B,Ys2: list @ B,Z: C,Zs: list @ C,W: D,Ws2: list @ D] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys2 )
= ( size_size @ ( list @ C ) @ Zs ) )
=> ( ( ( size_size @ ( list @ C ) @ Zs )
= ( size_size @ ( list @ D ) @ Ws2 ) )
=> ( ( P3 @ Xs2 @ Ys2 @ Zs @ Ws2 )
=> ( P3 @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ B @ Y2 @ Ys2 ) @ ( cons @ C @ Z @ Zs ) @ ( cons @ D @ W @ Ws2 ) ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs2 @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_182_list__induct3,axiom,
! [B: $tType,A: $tType,C: $tType,Xs: list @ A,Ys: list @ B,Zs2: list @ C,P3: ( list @ A ) > ( list @ B ) > ( list @ C ) > $o] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys )
= ( size_size @ ( list @ C ) @ Zs2 ) )
=> ( ( P3 @ ( nil @ A ) @ ( nil @ B ) @ ( nil @ C ) )
=> ( ! [X3: A,Xs2: list @ A,Y2: B,Ys2: list @ B,Z: C,Zs: list @ C] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys2 )
= ( size_size @ ( list @ C ) @ Zs ) )
=> ( ( P3 @ Xs2 @ Ys2 @ Zs )
=> ( P3 @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ B @ Y2 @ Ys2 ) @ ( cons @ C @ Z @ Zs ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs2 ) ) ) ) ) ).
% list_induct3
thf(fact_183_list__induct2,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,P3: ( list @ A ) > ( list @ B ) > $o] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( P3 @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X3: A,Xs2: list @ A,Y2: B,Ys2: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( P3 @ Xs2 @ Ys2 )
=> ( P3 @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ B @ Y2 @ Ys2 ) ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_184_pderiv__coeffs__code_Ocases,axiom,
! [A: $tType] :
( ( ( comm_semiring_1 @ A )
& ( semiri1193490041visors @ A ) )
=> ! [X: product_prod @ A @ ( list @ A )] :
( ! [F3: A,X3: A,Xs2: list @ A] :
( X
!= ( product_Pair @ A @ ( list @ A ) @ F3 @ ( cons @ A @ X3 @ Xs2 ) ) )
=> ~ ! [F3: A] :
( X
!= ( product_Pair @ A @ ( list @ A ) @ F3 @ ( nil @ A ) ) ) ) ) ).
% pderiv_coeffs_code.cases
thf(fact_185_shuffles_Ocases,axiom,
! [A: $tType,X: product_prod @ ( list @ A ) @ ( list @ A )] :
( ! [Ys2: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys2 ) )
=> ( ! [Xs2: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ ( nil @ A ) ) )
=> ~ ! [X3: A,Xs2: list @ A,Y2: A,Ys2: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ A @ Y2 @ Ys2 ) ) ) ) ) ).
% shuffles.cases
thf(fact_186_plus__coeffs_Ocases,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [X: product_prod @ ( list @ A ) @ ( list @ A )] :
( ! [Xs2: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ ( nil @ A ) ) )
=> ( ! [V: A,Va: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( cons @ A @ V @ Va ) ) )
=> ~ ! [X3: A,Xs2: list @ A,Y2: A,Ys2: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ A @ Y2 @ Ys2 ) ) ) ) ) ) ).
% plus_coeffs.cases
thf(fact_187_minus__poly__rev__list_Ocases,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [X: product_prod @ ( list @ A ) @ ( list @ A )] :
( ! [X3: A,Xs2: list @ A,Y2: A,Ys2: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ A @ Y2 @ Ys2 ) ) )
=> ( ! [Xs2: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ ( nil @ A ) ) )
=> ~ ! [Y2: A,Ys2: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( cons @ A @ Y2 @ Ys2 ) ) ) ) ) ) ).
% minus_poly_rev_list.cases
thf(fact_188_Cons__listrel1E2,axiom,
! [A: $tType,Xs: list @ A,Y: A,Ys: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ ( cons @ A @ Y @ Ys ) ) @ ( listrel1 @ A @ R ) )
=> ( ! [X3: A] :
( ( Xs
= ( cons @ A @ X3 @ Ys ) )
=> ~ ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y ) @ R ) )
=> ~ ! [Zs: list @ A] :
( ( Xs
= ( cons @ A @ Y @ Zs ) )
=> ~ ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Zs @ Ys ) @ ( listrel1 @ A @ R ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_189_Cons__listrel1E1,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ Ys ) @ ( listrel1 @ A @ R ) )
=> ( ! [Y2: A] :
( ( Ys
= ( cons @ A @ Y2 @ Xs ) )
=> ~ ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ R ) )
=> ~ ! [Zs: list @ A] :
( ( Ys
= ( cons @ A @ X @ Zs ) )
=> ~ ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Zs ) @ ( listrel1 @ A @ R ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_190_listrel1I1,axiom,
! [A: $tType,X: A,Y: A,R: set @ ( product_prod @ A @ A ),Xs: list @ A] :
( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
=> ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Xs ) ) @ ( listrel1 @ A @ R ) ) ) ).
% listrel1I1
thf(fact_191_same__length__different,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs != Ys )
=> ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) )
=> ? [Pre: list @ A,X3: A,Xs5: list @ A,Y2: A,Ys5: list @ A] :
( ( X3 != Y2 )
& ( Xs
= ( append @ A @ Pre @ ( append @ A @ ( cons @ A @ X3 @ ( nil @ A ) ) @ Xs5 ) ) )
& ( Ys
= ( append @ A @ Pre @ ( append @ A @ ( cons @ A @ Y2 @ ( nil @ A ) ) @ Ys5 ) ) ) ) ) ) ).
% same_length_different
thf(fact_192_listrel1E,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R ) )
=> ~ ! [X3: A,Y2: A] :
( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y2 ) @ R )
=> ! [Us3: list @ A,Vs2: list @ A] :
( ( Xs
= ( append @ A @ Us3 @ ( cons @ A @ X3 @ Vs2 ) ) )
=> ( Ys
!= ( append @ A @ Us3 @ ( cons @ A @ Y2 @ Vs2 ) ) ) ) ) ) ).
% listrel1E
thf(fact_193_listrel1I,axiom,
! [A: $tType,X: A,Y: A,R: set @ ( product_prod @ A @ A ),Xs: list @ A,Us: list @ A,Vs: list @ A,Ys: list @ A] :
( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
=> ( ( Xs
= ( append @ A @ Us @ ( cons @ A @ X @ Vs ) ) )
=> ( ( Ys
= ( append @ A @ Us @ ( cons @ A @ Y @ Vs ) ) )
=> ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R ) ) ) ) ) ).
% listrel1I
thf(fact_194_length__append__singleton,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( size_size @ ( list @ A ) @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) )
= ( suc @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% length_append_singleton
thf(fact_195_subset__eq__mset__impl_Ocases,axiom,
! [A: $tType,X: product_prod @ ( list @ A ) @ ( list @ A )] :
( ! [Ys2: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys2 ) )
=> ~ ! [X3: A,Xs2: list @ A,Ys2: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X3 @ Xs2 ) @ Ys2 ) ) ) ).
% subset_eq_mset_impl.cases
thf(fact_196_length__Cons,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( cons @ A @ X @ Xs ) )
= ( suc @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% length_Cons
thf(fact_197_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_198_Cons__in__lex,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: A,Ys: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) ) @ ( lex @ A @ R ) )
= ( ( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
& ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) ) )
| ( ( X = Y )
& ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lex @ A @ R ) ) ) ) ) ).
% Cons_in_lex
thf(fact_199_Nil2__notin__lex,axiom,
! [A: $tType,Xs: list @ A,R: set @ ( product_prod @ A @ A )] :
~ ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ ( nil @ A ) ) @ ( lex @ A @ R ) ) ).
% Nil2_notin_lex
thf(fact_200_Nil__notin__lex,axiom,
! [A: $tType,Ys: list @ A,R: set @ ( product_prod @ A @ A )] :
~ ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys ) @ ( lex @ A @ R ) ) ).
% Nil_notin_lex
thf(fact_201_lex__append__leftI,axiom,
! [A: $tType,Ys: list @ A,Zs2: list @ A,R: set @ ( product_prod @ A @ A ),Xs: list @ A] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs2 ) @ ( lex @ A @ R ) )
=> ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys ) @ ( append @ A @ Xs @ Zs2 ) ) @ ( lex @ A @ R ) ) ) ).
% lex_append_leftI
thf(fact_202_lex__append__left__iff,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),Xs: list @ A,Ys: list @ A,Zs2: list @ A] :
( ! [X3: A] :
~ ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ R )
=> ( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys ) @ ( append @ A @ Xs @ Zs2 ) ) @ ( lex @ A @ R ) )
= ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs2 ) @ ( lex @ A @ R ) ) ) ) ).
% lex_append_left_iff
thf(fact_203_lex__append__leftD,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),Xs: list @ A,Ys: list @ A,Zs2: list @ A] :
( ! [X3: A] :
~ ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ R )
=> ( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys ) @ ( append @ A @ Xs @ Zs2 ) ) @ ( lex @ A @ R ) )
=> ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs2 ) @ ( lex @ A @ R ) ) ) ) ).
% lex_append_leftD
thf(fact_204_lex__append__rightI,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R: set @ ( product_prod @ A @ A ),Vs: list @ A,Us: list @ A] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lex @ A @ R ) )
=> ( ( ( size_size @ ( list @ A ) @ Vs )
= ( size_size @ ( list @ A ) @ Us ) )
=> ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Us ) @ ( append @ A @ Ys @ Vs ) ) @ ( lex @ A @ R ) ) ) ) ).
% lex_append_rightI
thf(fact_205_SuccD,axiom,
! [A: $tType,K2: A,Kl2: set @ ( list @ A ),Kl3: list @ A] :
( ( member2 @ A @ K2 @ ( bNF_Greatest_Succ @ A @ Kl2 @ Kl3 ) )
=> ( member2 @ ( list @ A ) @ ( append @ A @ Kl3 @ ( cons @ A @ K2 @ ( nil @ A ) ) ) @ Kl2 ) ) ).
% SuccD
thf(fact_206_SuccI,axiom,
! [A: $tType,Kl3: list @ A,K2: A,Kl2: set @ ( list @ A )] :
( ( member2 @ ( list @ A ) @ ( append @ A @ Kl3 @ ( cons @ A @ K2 @ ( nil @ A ) ) ) @ Kl2 )
=> ( member2 @ A @ K2 @ ( bNF_Greatest_Succ @ A @ Kl2 @ Kl3 ) ) ) ).
% SuccI
thf(fact_207_empty__Shift,axiom,
! [A: $tType,Kl2: set @ ( list @ A ),K2: A] :
( ( member2 @ ( list @ A ) @ ( nil @ A ) @ Kl2 )
=> ( ( member2 @ A @ K2 @ ( bNF_Greatest_Succ @ A @ Kl2 @ ( nil @ A ) ) )
=> ( member2 @ ( list @ A ) @ ( nil @ A ) @ ( bNF_Greatest_Shift @ A @ Kl2 @ K2 ) ) ) ) ).
% empty_Shift
thf(fact_208_Succ__Shift,axiom,
! [A: $tType,Kl2: set @ ( list @ A ),K2: A,Kl3: list @ A] :
( ( bNF_Greatest_Succ @ A @ ( bNF_Greatest_Shift @ A @ Kl2 @ K2 ) @ Kl3 )
= ( bNF_Greatest_Succ @ A @ Kl2 @ ( cons @ A @ K2 @ Kl3 ) ) ) ).
% Succ_Shift
thf(fact_209_ShiftD,axiom,
! [A: $tType,Kl3: list @ A,Kl2: set @ ( list @ A ),K2: A] :
( ( member2 @ ( list @ A ) @ Kl3 @ ( bNF_Greatest_Shift @ A @ Kl2 @ K2 ) )
=> ( member2 @ ( list @ A ) @ ( cons @ A @ K2 @ Kl3 ) @ Kl2 ) ) ).
% ShiftD
thf(fact_210_length__product,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] :
( ( size_size @ ( list @ ( product_prod @ A @ B ) ) @ ( product @ A @ B @ Xs @ Ys ) )
= ( times_times @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) ) ) ).
% length_product
thf(fact_211_part__code_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F: B > A,Pivot: A] :
( ( linorder_part @ B @ A @ F @ Pivot @ ( nil @ B ) )
= ( product_Pair @ ( list @ B ) @ ( product_prod @ ( list @ B ) @ ( list @ B ) ) @ ( nil @ B ) @ ( product_Pair @ ( list @ B ) @ ( list @ B ) @ ( nil @ B ) @ ( nil @ B ) ) ) ) ) ).
% part_code(1)
thf(fact_212_Suc__mult__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ( times_times @ nat @ ( suc @ K2 ) @ M )
= ( times_times @ nat @ ( suc @ K2 ) @ N ) )
= ( M = N ) ) ).
% Suc_mult_cancel1
thf(fact_213_lexord__append__left__rightI,axiom,
! [A: $tType,A3: A,B2: A,R: set @ ( product_prod @ A @ A ),U: list @ A,X: list @ A,Y: list @ A] :
( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
=> ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ U @ ( cons @ A @ A3 @ X ) ) @ ( append @ A @ U @ ( cons @ A @ B2 @ Y ) ) ) @ ( lexord @ A @ R ) ) ) ).
% lexord_append_left_rightI
thf(fact_214_listrel_Oinducts,axiom,
! [A: $tType,B: $tType,X1: list @ A,X2: list @ B,R: set @ ( product_prod @ A @ B ),P3: ( list @ A ) > ( list @ B ) > $o] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ X1 @ X2 ) @ ( listrel @ A @ B @ R ) )
=> ( ( P3 @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X3: A,Y2: B,Xs2: list @ A,Ys2: list @ B] :
( ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y2 ) @ R )
=> ( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs2 @ Ys2 ) @ ( listrel @ A @ B @ R ) )
=> ( ( P3 @ Xs2 @ Ys2 )
=> ( P3 @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ B @ Y2 @ Ys2 ) ) ) ) )
=> ( P3 @ X1 @ X2 ) ) ) ) ).
% listrel.inducts
thf(fact_215_lexord__cons__cons,axiom,
! [A: $tType,A3: A,X: list @ A,B2: A,Y: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ A3 @ X ) @ ( cons @ A @ B2 @ Y ) ) @ ( lexord @ A @ R ) )
= ( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R )
| ( ( A3 = B2 )
& ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lexord @ A @ R ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_216_lexord__Nil__left,axiom,
! [A: $tType,Y: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Y ) @ ( lexord @ A @ R ) )
= ( ? [A5: A,X4: list @ A] :
( Y
= ( cons @ A @ A5 @ X4 ) ) ) ) ).
% lexord_Nil_left
thf(fact_217_lexord__append__leftI,axiom,
! [A: $tType,U: list @ A,V2: list @ A,R: set @ ( product_prod @ A @ A ),X: list @ A] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ U @ V2 ) @ ( lexord @ A @ R ) )
=> ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ X @ U ) @ ( append @ A @ X @ V2 ) ) @ ( lexord @ A @ R ) ) ) ).
% lexord_append_leftI
thf(fact_218_lexord__Nil__right,axiom,
! [A: $tType,X: list @ A,R: set @ ( product_prod @ A @ A )] :
~ ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ ( nil @ A ) ) @ ( lexord @ A @ R ) ) ).
% lexord_Nil_right
thf(fact_219_listrel__eq__len,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,R: set @ ( product_prod @ A @ B )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys ) @ ( listrel @ A @ B @ R ) )
=> ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) ) ) ).
% listrel_eq_len
thf(fact_220_listrel__Nil2,axiom,
! [B: $tType,A: $tType,Xs: list @ A,R: set @ ( product_prod @ A @ B )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ ( nil @ B ) ) @ ( listrel @ A @ B @ R ) )
=> ( Xs
= ( nil @ A ) ) ) ).
% listrel_Nil2
thf(fact_221_listrel__Nil1,axiom,
! [A: $tType,B: $tType,Xs: list @ B,R: set @ ( product_prod @ A @ B )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ Xs ) @ ( listrel @ A @ B @ R ) )
=> ( Xs
= ( nil @ B ) ) ) ).
% listrel_Nil1
thf(fact_222_listrel_ONil,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B )] : ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( nil @ B ) ) @ ( listrel @ A @ B @ R ) ) ).
% listrel.Nil
thf(fact_223_lexord__linear,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),X: list @ A,Y: list @ A] :
( ! [A2: A,B3: A] :
( ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B3 ) @ R )
| ( A2 = B3 )
| ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A2 ) @ R ) )
=> ( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lexord @ A @ R ) )
| ( X = Y )
| ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Y @ X ) @ ( lexord @ A @ R ) ) ) ) ).
% lexord_linear
thf(fact_224_lexord__irreflexive,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),Xs: list @ A] :
( ! [X3: A] :
~ ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ R )
=> ~ ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Xs ) @ ( lexord @ A @ R ) ) ) ).
% lexord_irreflexive
thf(fact_225_listrel__Cons2,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Y: B,Ys: list @ B,R: set @ ( product_prod @ A @ B )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ ( cons @ B @ Y @ Ys ) ) @ ( listrel @ A @ B @ R ) )
=> ~ ! [X3: A,Xs2: list @ A] :
( ( Xs
= ( cons @ A @ X3 @ Xs2 ) )
=> ( ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y ) @ R )
=> ~ ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs2 @ Ys ) @ ( listrel @ A @ B @ R ) ) ) ) ) ).
% listrel_Cons2
thf(fact_226_listrel__Cons1,axiom,
! [B: $tType,A: $tType,Y: A,Ys: list @ A,Xs: list @ B,R: set @ ( product_prod @ A @ B )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ Y @ Ys ) @ Xs ) @ ( listrel @ A @ B @ R ) )
=> ~ ! [Y2: B,Ys2: list @ B] :
( ( Xs
= ( cons @ B @ Y2 @ Ys2 ) )
=> ( ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y @ Y2 ) @ R )
=> ~ ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Ys @ Ys2 ) @ ( listrel @ A @ B @ R ) ) ) ) ) ).
% listrel_Cons1
thf(fact_227_listrel_OCons,axiom,
! [B: $tType,A: $tType,X: A,Y: B,R: set @ ( product_prod @ A @ B ),Xs: list @ A,Ys: list @ B] :
( ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ R )
=> ( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys ) @ ( listrel @ A @ B @ R ) )
=> ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ X @ Xs ) @ ( cons @ B @ Y @ Ys ) ) @ ( listrel @ A @ B @ R ) ) ) ) ).
% listrel.Cons
thf(fact_228_lexord__append__leftD,axiom,
! [A: $tType,X: list @ A,U: list @ A,V2: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ X @ U ) @ ( append @ A @ X @ V2 ) ) @ ( lexord @ A @ R ) )
=> ( ! [A2: A] :
~ ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ R )
=> ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ U @ V2 ) @ ( lexord @ A @ R ) ) ) ) ).
% lexord_append_leftD
thf(fact_229_lexord__append__rightI,axiom,
! [A: $tType,Y: list @ A,X: list @ A,R: set @ ( product_prod @ A @ A )] :
( ? [B4: A,Z2: list @ A] :
( Y
= ( cons @ A @ B4 @ Z2 ) )
=> ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ ( append @ A @ X @ Y ) ) @ ( lexord @ A @ R ) ) ) ).
% lexord_append_rightI
thf(fact_230_lexord__sufE,axiom,
! [A: $tType,Xs: list @ A,Zs2: list @ A,Ys: list @ A,Qs: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Zs2 ) @ ( append @ A @ Ys @ Qs ) ) @ ( lexord @ A @ R ) )
=> ( ( Xs != Ys )
=> ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) )
=> ( ( ( size_size @ ( list @ A ) @ Zs2 )
= ( size_size @ ( list @ A ) @ Qs ) )
=> ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lexord @ A @ R ) ) ) ) ) ) ).
% lexord_sufE
thf(fact_231_lexord__lex,axiom,
! [A: $tType,X: list @ A,Y: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lex @ A @ R ) )
= ( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lexord @ A @ R ) )
& ( ( size_size @ ( list @ A ) @ X )
= ( size_size @ ( list @ A ) @ Y ) ) ) ) ).
% lexord_lex
thf(fact_232_listrel_Ocases,axiom,
! [B: $tType,A: $tType,A12: list @ A,A23: list @ B,R: set @ ( product_prod @ A @ B )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ A12 @ A23 ) @ ( listrel @ A @ B @ R ) )
=> ( ( ( A12
= ( nil @ A ) )
=> ( A23
!= ( nil @ B ) ) )
=> ~ ! [X3: A,Y2: B,Xs2: list @ A] :
( ( A12
= ( cons @ A @ X3 @ Xs2 ) )
=> ! [Ys2: list @ B] :
( ( A23
= ( cons @ B @ Y2 @ Ys2 ) )
=> ( ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y2 ) @ R )
=> ~ ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs2 @ Ys2 ) @ ( listrel @ A @ B @ R ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_233_listrel_Osimps,axiom,
! [B: $tType,A: $tType,A12: list @ A,A23: list @ B,R: set @ ( product_prod @ A @ B )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ A12 @ A23 ) @ ( listrel @ A @ B @ R ) )
= ( ( ( A12
= ( nil @ A ) )
& ( A23
= ( nil @ B ) ) )
| ? [X4: A,Y3: B,Xs3: list @ A,Ys3: list @ B] :
( ( A12
= ( cons @ A @ X4 @ Xs3 ) )
& ( A23
= ( cons @ B @ Y3 @ Ys3 ) )
& ( member2 @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) @ R )
& ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs3 @ Ys3 ) @ ( listrel @ A @ B @ R ) ) ) ) ) ).
% listrel.simps
thf(fact_234_lexord__same__pref__if__irrefl,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),Xs: list @ A,Ys: list @ A,Zs2: list @ A] :
( ( irrefl @ A @ R )
=> ( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys ) @ ( append @ A @ Xs @ Zs2 ) ) @ ( lexord @ A @ R ) )
= ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs2 ) @ ( lexord @ A @ R ) ) ) ) ).
% lexord_same_pref_if_irrefl
thf(fact_235_map__tailrec__rev_Opelims,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 )
=> ( ( accp @ ( product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( map_tailrec_rev_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xa @ Xb ) ) )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( ( Y = Xb )
=> ~ ( accp @ ( product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( map_tailrec_rev_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ Xb ) ) ) ) )
=> ~ ! [A2: A,As: list @ A] :
( ( Xa
= ( cons @ A @ A2 @ As ) )
=> ( ( Y
= ( map_tailrec_rev @ A @ B @ X @ As @ ( cons @ B @ ( X @ A2 ) @ Xb ) ) )
=> ~ ( accp @ ( product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( map_tailrec_rev_rel @ A @ B ) @ ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ X @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A2 @ As ) @ Xb ) ) ) ) ) ) ) ) ).
% map_tailrec_rev.pelims
thf(fact_236_lexord__irrefl,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A )] :
( ( irrefl @ A @ R3 )
=> ( irrefl @ ( list @ A ) @ ( lexord @ A @ R3 ) ) ) ).
% lexord_irrefl
thf(fact_237_irrefl__lex,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( irrefl @ A @ R )
=> ( irrefl @ ( list @ A ) @ ( lex @ A @ R ) ) ) ).
% irrefl_lex
thf(fact_238_lexl__not__refl,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),X: list @ A] :
( ( irrefl @ A @ R )
=> ~ ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ X ) @ ( lex @ A @ R ) ) ) ).
% lexl_not_refl
thf(fact_239_lenlex__append2,axiom,
! [A: $tType,R3: set @ ( product_prod @ A @ A ),Us: list @ A,Xs: list @ A,Ys: list @ A] :
( ( irrefl @ A @ R3 )
=> ( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Us @ Xs ) @ ( append @ A @ Us @ Ys ) ) @ ( lenlex @ A @ R3 ) )
= ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lenlex @ A @ R3 ) ) ) ) ).
% lenlex_append2
thf(fact_240_lenlex__append1,axiom,
! [A: $tType,Us: list @ A,Xs: list @ A,R3: set @ ( product_prod @ A @ A ),Vs: list @ A,Ys: list @ A] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Us @ Xs ) @ ( lenlex @ A @ R3 ) )
=> ( ( ( size_size @ ( list @ A ) @ Vs )
= ( size_size @ ( list @ A ) @ Ys ) )
=> ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Us @ Vs ) @ ( append @ A @ Xs @ Ys ) ) @ ( lenlex @ A @ R3 ) ) ) ) ).
% lenlex_append1
thf(fact_241_Nil__lenlex__iff1,axiom,
! [A: $tType,Ns: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ns ) @ ( lenlex @ A @ R ) )
= ( Ns
!= ( nil @ A ) ) ) ).
% Nil_lenlex_iff1
thf(fact_242_lenlex__irreflexive,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),Xs: list @ A] :
( ! [X3: A] :
~ ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ X3 ) @ R )
=> ~ ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Xs ) @ ( lenlex @ A @ R ) ) ) ).
% lenlex_irreflexive
thf(fact_243_Nil__lenlex__iff2,axiom,
! [A: $tType,Ns: list @ A,R: set @ ( product_prod @ A @ A )] :
~ ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ns @ ( nil @ A ) ) @ ( lenlex @ A @ R ) ) ).
% Nil_lenlex_iff2
thf(fact_244_Cons__lenlex__iff,axiom,
! [A: $tType,M: A,Ms: list @ A,N: A,Ns: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ M @ Ms ) @ ( cons @ A @ N @ Ns ) ) @ ( lenlex @ A @ R ) )
= ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ms ) @ ( size_size @ ( list @ A ) @ Ns ) )
| ( ( ( size_size @ ( list @ A ) @ Ms )
= ( size_size @ ( list @ A ) @ Ns ) )
& ( member2 @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ M @ N ) @ R ) )
| ( ( M = N )
& ( member2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ms @ Ns ) @ ( lenlex @ A @ R ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_245_product_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,X: A,Xs: list @ A,Ys: list @ B] :
( ( product @ A @ B @ ( cons @ A @ X @ Xs ) @ Ys )
= ( append @ ( product_prod @ A @ B ) @ ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X ) @ Ys ) @ ( product @ A @ B @ Xs @ Ys ) ) ) ).
% product.simps(2)
thf(fact_246_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less @ nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_247_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_248_lessI,axiom,
! [N: nat] : ( ord_less @ nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_249_list_Omap__disc__iff,axiom,
! [B: $tType,A: $tType,F: A > B,A3: list @ A] :
( ( ( map @ A @ B @ F @ A3 )
= ( nil @ B ) )
= ( A3
= ( nil @ A ) ) ) ).
% list.map_disc_iff
thf(fact_250_map__is__Nil__conv,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: list @ B] :
( ( ( map @ B @ A @ F @ Xs )
= ( nil @ A ) )
= ( Xs
= ( nil @ B ) ) ) ).
% map_is_Nil_conv
thf(fact_251_Nil__is__map__conv,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: list @ B] :
( ( ( nil @ A )
= ( map @ B @ A @ F @ Xs ) )
= ( Xs
= ( nil @ B ) ) ) ).
% Nil_is_map_conv
thf(fact_252_length__map,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: list @ B] :
( ( size_size @ ( list @ A ) @ ( map @ B @ A @ F @ Xs ) )
= ( size_size @ ( list @ B ) @ Xs ) ) ).
% length_map
thf(fact_253_map__append,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: list @ B,Ys: list @ B] :
( ( map @ B @ A @ F @ ( append @ B @ Xs @ Ys ) )
= ( append @ A @ ( map @ B @ A @ F @ Xs ) @ ( map @ B @ A @ F @ Ys ) ) ) ).
% map_append
% Type constructors (31)
thf(tcon_Polynomial_Opoly___Rings_Ocomm__semiring__0,axiom,
! [A6: $tType] :
( ( comm_semiring_0 @ A6 )
=> ( comm_semiring_0 @ ( poly @ A6 ) ) ) ).
thf(tcon_Polynomial_Opoly___Rings_Olinordered__idom,axiom,
! [A6: $tType] :
( ( linordered_idom @ A6 )
=> ( linordered_idom @ ( poly @ A6 ) ) ) ).
thf(tcon_Polynomial_Opoly___Groups_Oab__group__add,axiom,
! [A6: $tType] :
( ( ab_group_add @ A6 )
=> ( ab_group_add @ ( poly @ A6 ) ) ) ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring__0_1,axiom,
comm_semiring_0 @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__idom_2,axiom,
linordered_idom @ real ).
thf(tcon_Real_Oreal___Groups_Oab__group__add_3,axiom,
ab_group_add @ real ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__0_4,axiom,
comm_semiring_0 @ nat ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A6: $tType,A7: $tType] :
( ( ord @ A7 )
=> ( ord @ ( A6 > A7 ) ) ) ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors,axiom,
semiri1193490041visors @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1 @ nat ).
thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
linorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Oord_5,axiom,
ord @ nat ).
thf(tcon_Nat_Onat___Nat_Osize,axiom,
size @ nat ).
thf(tcon_Set_Oset___Orderings_Oord_6,axiom,
! [A6: $tType] : ( ord @ ( set @ A6 ) ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder_7,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_8,axiom,
ord @ $o ).
thf(tcon_List_Olist___Nat_Osize_9,axiom,
! [A6: $tType] : ( size @ ( list @ A6 ) ) ).
thf(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors_10,axiom,
semiri1193490041visors @ real ).
thf(tcon_Real_Oreal___Groups_Ocomm__monoid__add_11,axiom,
comm_monoid_add @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring__1_12,axiom,
comm_semiring_1 @ real ).
thf(tcon_Real_Oreal___Orderings_Olinorder_13,axiom,
linorder @ real ).
thf(tcon_Real_Oreal___Groups_Ogroup__add,axiom,
group_add @ real ).
thf(tcon_Real_Oreal___Orderings_Oord_14,axiom,
ord @ real ).
thf(tcon_Polynomial_Opoly___Rings_Osemiring__no__zero__divisors_15,axiom,
! [A6: $tType] :
( ( ( comm_semiring_0 @ A6 )
& ( semiri1193490041visors @ A6 ) )
=> ( semiri1193490041visors @ ( poly @ A6 ) ) ) ).
thf(tcon_Polynomial_Opoly___Groups_Ocomm__monoid__add_16,axiom,
! [A6: $tType] :
( ( comm_monoid_add @ A6 )
=> ( comm_monoid_add @ ( poly @ A6 ) ) ) ).
thf(tcon_Polynomial_Opoly___Rings_Ocomm__semiring__1_17,axiom,
! [A6: $tType] :
( ( comm_semiring_1 @ A6 )
=> ( comm_semiring_1 @ ( poly @ A6 ) ) ) ).
thf(tcon_Polynomial_Opoly___Orderings_Olinorder_18,axiom,
! [A6: $tType] :
( ( linordered_idom @ A6 )
=> ( linorder @ ( poly @ A6 ) ) ) ).
thf(tcon_Polynomial_Opoly___Groups_Ogroup__add_19,axiom,
! [A6: $tType] :
( ( ab_group_add @ A6 )
=> ( group_add @ ( poly @ A6 ) ) ) ).
thf(tcon_Polynomial_Opoly___Orderings_Oord_20,axiom,
! [A6: $tType] :
( ( linordered_idom @ A6 )
=> ( ord @ ( poly @ A6 ) ) ) ).
thf(tcon_Product__Type_Oprod___Nat_Osize_21,axiom,
! [A6: $tType,A7: $tType] : ( size @ ( product_prod @ A6 @ A7 ) ) ).
% Conjectures (1)
thf(conj_0,conjecture,
~ ( sturm_2060350619rm_seq @ ( nil @ ( poly @ real ) ) ) ).
%------------------------------------------------------------------------------