TPTP Problem File: COM154^1.p
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%------------------------------------------------------------------------------
% File : COM154^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Computing Theory
% Problem : Abstract completeness 131
% Version : [Bla16] axioms : Especial.
% English :
% Refs : [BPT14] Blanchette et al. (2014), Abstract Completeness
% : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% Source : [Bla16]
% Names : abstract_completeness__131.p [Bla16]
% Status : Theorem
% Rating : 1.00 v7.5.0, 0.33 v7.2.0, 0.50 v7.1.0
% Syntax : Number of formulae : 373 ( 149 unt; 71 typ; 0 def)
% Number of atoms : 663 ( 257 equ; 0 cnn)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 2912 ( 108 ~; 9 |; 26 &;2479 @)
% ( 0 <=>; 290 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 6 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 181 ( 181 >; 0 *; 0 +; 0 <<)
% Number of symbols : 72 ( 69 usr; 8 con; 0-4 aty)
% Number of variables : 823 ( 53 ^; 688 !; 22 ?; 823 :)
% ( 60 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:53:38.952
%------------------------------------------------------------------------------
%----Could-be-implicit typings (6)
thf(ty_t_Stream_Ostream,type,
stream: $tType > $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 ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
thf(ty_tf_rule,type,
rule: $tType ).
%----Explicit typings (65)
thf(sy_cl_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oone,type,
one:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ominus,type,
minus:
!>[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_Num_Oneg__numeral,type,
neg_numeral:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[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_Transcendental_Oln,type,
ln:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Ozero__neq__one,type,
zero_neq_one:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Countable_Ocountable,type,
countable:
!>[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_Rings_Ozero__less__one,type,
zero_less_one:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Odense__order,type,
dense_order:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocomm__monoid__diff,type,
comm_monoid_diff:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Olinordered__semidom,type,
linordered_semidom:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
cancel1352612707id_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
cancel146912293up_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
linord1659791738miring:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni770627133id_add:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_Abstract__Completeness__Mirabelle__wdxnrclvrt_ORuleSystem__Defs_Ofenum,type,
abstra1774373515_fenum:
!>[Rule: $tType] : ( ( stream @ Rule ) > ( stream @ Rule ) ) ).
thf(sy_c_Countable__Set_Ocountable,type,
countable_countable:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Oone__class_Oone,type,
one_one:
!>[A: $tType] : A ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_List_Oappend,type,
append:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist_Omap,type,
map:
!>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( list @ A ) > ( list @ Aa ) ) ).
thf(sy_c_List_Omap__tailrec,type,
map_tailrec:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( list @ A ) > ( list @ B ) ) ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Onat_Ocase__nat,type,
case_nat:
!>[A: $tType] : ( A > ( nat > A ) > nat > A ) ).
thf(sy_c_Nat_Onat_Opred,type,
pred: nat > nat ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc,type,
neg_numeral_dbl_inc:
!>[A: $tType] : ( A > 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_OBex,type,
bex:
!>[A: $tType] : ( ( set @ A ) > ( A > $o ) > $o ) ).
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_Stream_Oflat,type,
flat:
!>[A: $tType] : ( ( stream @ ( list @ A ) ) > ( stream @ A ) ) ).
thf(sy_c_Stream_Oshift,type,
shift:
!>[A: $tType] : ( ( list @ A ) > ( stream @ A ) > ( stream @ A ) ) ).
thf(sy_c_Stream_Ositerate,type,
siterate:
!>[A: $tType] : ( ( A > A ) > A > ( stream @ A ) ) ).
thf(sy_c_Stream_Osmerge,type,
smerge:
!>[A: $tType] : ( ( stream @ ( stream @ A ) ) > ( stream @ A ) ) ).
thf(sy_c_Stream_Osnth,type,
snth:
!>[A: $tType] : ( ( stream @ A ) > nat > A ) ).
thf(sy_c_Stream_Ostake,type,
stake:
!>[A: $tType] : ( nat > ( stream @ A ) > ( list @ A ) ) ).
thf(sy_c_Stream_Ostream_Osmap,type,
smap:
!>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( stream @ A ) > ( stream @ Aa ) ) ).
thf(sy_c_Stream_Ostream_Osset,type,
sset:
!>[A: $tType] : ( ( stream @ A ) > ( set @ A ) ) ).
thf(sy_c_Transcendental_Oln__class_Oln,type,
ln_ln:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Opowr,type,
powr:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_m____,type,
m: nat ).
thf(sy_v_n____,type,
n: nat ).
thf(sy_v_na____,type,
na: nat ).
thf(sy_v_r____,type,
r: rule ).
thf(sy_v_rsa____,type,
rsa: list @ rule ).
thf(sy_v_rules,type,
rules: stream @ rule ).
%----Relevant facts (252)
thf(fact_0__092_060open_0620_A_060_An_____092_060close_062,axiom,
ord_less @ nat @ ( zero_zero @ nat ) @ n ).
% \<open>0 < n__\<close>
thf(fact_1_r,axiom,
( r
= ( snth @ rule @ rules @ m ) ) ).
% r
thf(fact_2_local_Oalw,axiom,
ord_less @ nat @ ( zero_zero @ nat ) @ na ).
% local.alw
thf(fact_3__092_060open_062r_A_092_060in_062_AR_092_060close_062,axiom,
member @ rule @ r @ ( sset @ rule @ rules ) ).
% \<open>r \<in> R\<close>
thf(fact_4__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062m_O_Ar_A_061_Arules_A_B_B_Am_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [M: nat] :
( r
!= ( snth @ rule @ rules @ M ) ) ).
% \<open>\<And>thesis. (\<And>m. r = rules !! m \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_5_sset__fenum,axiom,
( ( sset @ rule @ ( abstra1774373515_fenum @ rule @ rules ) )
= ( sset @ rule @ rules ) ) ).
% sset_fenum
thf(fact_6_fenum__def,axiom,
( ( abstra1774373515_fenum @ rule @ rules )
= ( flat @ rule
@ ( smap @ nat @ ( list @ rule )
@ ^ [N: nat] : ( stake @ rule @ N @ rules )
@ ( siterate @ nat @ suc @ ( one_one @ nat ) ) ) ) ) ).
% fenum_def
thf(fact_7_shift__left__inj,axiom,
! [A: $tType,Xs: list @ A,S1: stream @ A,S2: stream @ A] :
( ( ( shift @ A @ Xs @ S1 )
= ( shift @ A @ Xs @ S2 ) )
= ( S1 = S2 ) ) ).
% shift_left_inj
thf(fact_8_NE__R,axiom,
( ( sset @ rule @ rules )
!= ( bot_bot @ ( set @ rule ) ) ) ).
% NE_R
thf(fact_9_nat_Oinject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
= ( X2 = Y2 ) ) ).
% nat.inject
thf(fact_10_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_11_smap__siterate,axiom,
! [A: $tType,F: A > A,X: A] :
( ( smap @ A @ A @ F @ ( siterate @ A @ F @ X ) )
= ( siterate @ A @ F @ ( F @ X ) ) ) ).
% smap_siterate
thf(fact_12_stream_Omap__cong,axiom,
! [B: $tType,A: $tType,X: stream @ A,Ya: stream @ A,F: A > B,G: A > B] :
( ( X = Ya )
=> ( ! [Z: A] :
( ( member @ A @ Z @ ( sset @ A @ Ya ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( smap @ A @ B @ F @ X )
= ( smap @ A @ B @ G @ Ya ) ) ) ) ).
% stream.map_cong
thf(fact_13_stream_Omap__cong0,axiom,
! [B: $tType,A: $tType,X: stream @ A,F: A > B,G: A > B] :
( ! [Z: A] :
( ( member @ A @ Z @ ( sset @ A @ X ) )
=> ( ( F @ Z )
= ( G @ Z ) ) )
=> ( ( smap @ A @ B @ F @ X )
= ( smap @ A @ B @ G @ X ) ) ) ).
% stream.map_cong0
thf(fact_14_stream_Oinj__map__strong,axiom,
! [B: $tType,A: $tType,X: stream @ A,Xa: stream @ A,F: A > B,Fa: A > B] :
( ! [Z: A,Za: A] :
( ( member @ A @ Z @ ( sset @ A @ X ) )
=> ( ( member @ A @ Za @ ( sset @ A @ Xa ) )
=> ( ( ( F @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( smap @ A @ B @ F @ X )
= ( smap @ A @ B @ Fa @ Xa ) )
=> ( X = Xa ) ) ) ).
% stream.inj_map_strong
thf(fact_15_countable__R,axiom,
countable_countable @ rule @ ( sset @ rule @ rules ) ).
% countable_R
thf(fact_16_RuleSystem__Defs_Ofenum__def,axiom,
! [Rule: $tType] :
( ( abstra1774373515_fenum @ Rule )
= ( ^ [Rules: stream @ Rule] :
( flat @ Rule
@ ( smap @ nat @ ( list @ Rule )
@ ^ [N: nat] : ( stake @ Rule @ N @ Rules )
@ ( siterate @ nat @ suc @ ( one_one @ nat ) ) ) ) ) ) ).
% RuleSystem_Defs.fenum_def
thf(fact_17_neq0__conv,axiom,
! [N2: nat] :
( ( N2
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ).
% neq0_conv
thf(fact_18_Suc__less__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
= ( ord_less @ nat @ M2 @ N2 ) ) ).
% Suc_less_eq
thf(fact_19_Suc__mono,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
=> ( ord_less @ nat @ ( suc @ M2 ) @ ( suc @ N2 ) ) ) ).
% Suc_mono
thf(fact_20_lessI,axiom,
! [N2: nat] : ( ord_less @ nat @ N2 @ ( suc @ N2 ) ) ).
% lessI
thf(fact_21_snth__smap,axiom,
! [A: $tType,B: $tType,F: B > A,S: stream @ B,N2: nat] :
( ( snth @ A @ ( smap @ B @ A @ F @ S ) @ N2 )
= ( F @ ( snth @ B @ S @ N2 ) ) ) ).
% snth_smap
thf(fact_22_zero__less__Suc,axiom,
! [N2: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) ).
% zero_less_Suc
thf(fact_23_less__Suc0,axiom,
! [N2: nat] :
( ( ord_less @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) )
= ( N2
= ( zero_zero @ nat ) ) ) ).
% less_Suc0
thf(fact_24_less__one,axiom,
! [N2: nat] :
( ( ord_less @ nat @ N2 @ ( one_one @ nat ) )
= ( N2
= ( zero_zero @ nat ) ) ) ).
% less_one
thf(fact_25_gr0I,axiom,
! [N2: nat] :
( ( N2
!= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ).
% gr0I
thf(fact_26_not__gr0,axiom,
! [N2: nat] :
( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) )
= ( N2
= ( zero_zero @ nat ) ) ) ).
% not_gr0
thf(fact_27_not__less0,axiom,
! [N2: nat] :
~ ( ord_less @ nat @ N2 @ ( zero_zero @ nat ) ) ).
% not_less0
thf(fact_28_less__zeroE,axiom,
! [N2: nat] :
~ ( ord_less @ nat @ N2 @ ( zero_zero @ nat ) ) ).
% less_zeroE
thf(fact_29_nat__neq__iff,axiom,
! [M2: nat,N2: nat] :
( ( M2 != N2 )
= ( ( ord_less @ nat @ M2 @ N2 )
| ( ord_less @ nat @ N2 @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_30_less__not__refl,axiom,
! [N2: nat] :
~ ( ord_less @ nat @ N2 @ N2 ) ).
% less_not_refl
thf(fact_31_less__not__refl2,axiom,
! [N2: nat,M2: nat] :
( ( ord_less @ nat @ N2 @ M2 )
=> ( M2 != N2 ) ) ).
% less_not_refl2
thf(fact_32_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less @ nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_33_measure__induct,axiom,
! [A: $tType,F: A > nat,P: A > $o,A2: A] :
( ! [X3: A] :
( ! [Y: A] :
( ( ord_less @ nat @ ( F @ Y ) @ ( F @ X3 ) )
=> ( P @ Y ) )
=> ( P @ X3 ) )
=> ( P @ A2 ) ) ).
% measure_induct
thf(fact_34_gr__implies__not0,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
=> ( N2
!= ( zero_zero @ nat ) ) ) ).
% gr_implies_not0
thf(fact_35_less__irrefl__nat,axiom,
! [N2: nat] :
~ ( ord_less @ nat @ N2 @ N2 ) ).
% less_irrefl_nat
thf(fact_36_nat__less__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less @ nat @ M3 @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N2 ) ) ).
% nat_less_induct
thf(fact_37_infinite__descent,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less @ nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) )
=> ( P @ N2 ) ) ).
% infinite_descent
thf(fact_38_infinite__descent0,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less @ nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N2 ) ) ) ).
% infinite_descent0
thf(fact_39_linorder__neqE__nat,axiom,
! [X: nat,Y3: nat] :
( ( X != Y3 )
=> ( ~ ( ord_less @ nat @ X @ Y3 )
=> ( ord_less @ nat @ Y3 @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_40_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less @ nat @ N2 @ ( zero_zero @ nat ) ) ).
% less_nat_zero_code
thf(fact_41_measure__induct__rule,axiom,
! [A: $tType,F: A > nat,P: A > $o,A2: A] :
( ! [X3: A] :
( ! [Y: A] :
( ( ord_less @ nat @ ( F @ Y ) @ ( F @ X3 ) )
=> ( P @ Y ) )
=> ( P @ X3 ) )
=> ( P @ A2 ) ) ).
% measure_induct_rule
thf(fact_42_infinite__descent__measure,axiom,
! [A: $tType,P: A > $o,V: A > nat,X: A] :
( ! [X3: A] :
( ~ ( P @ X3 )
=> ? [Y: A] :
( ( ord_less @ nat @ ( V @ Y ) @ ( V @ X3 ) )
& ~ ( P @ Y ) ) )
=> ( P @ X ) ) ).
% infinite_descent_measure
thf(fact_43_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_44_Collect__mem__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( collect @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_45_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_46_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X3: A] :
( ( F @ X3 )
= ( G @ X3 ) )
=> ( F = G ) ) ).
% ext
thf(fact_47_infinite__descent0__measure,axiom,
! [A: $tType,V: A > nat,P: A > $o,X: A] :
( ! [X3: A] :
( ( ( V @ X3 )
= ( zero_zero @ nat ) )
=> ( P @ X3 ) )
=> ( ! [X3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V @ X3 ) )
=> ( ~ ( P @ X3 )
=> ? [Y: A] :
( ( ord_less @ nat @ ( V @ Y ) @ ( V @ X3 ) )
& ~ ( P @ Y ) ) ) )
=> ( P @ X ) ) ) ).
% infinite_descent0_measure
thf(fact_48_One__nat__def,axiom,
( ( one_one @ nat )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% One_nat_def
thf(fact_49_lift__Suc__mono__less__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [F: nat > A,N2: nat,M2: nat] :
( ! [N3: nat] : ( ord_less @ A @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less @ A @ ( F @ N2 ) @ ( F @ M2 ) )
= ( ord_less @ nat @ N2 @ M2 ) ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_50_lift__Suc__mono__less,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [F: nat > A,N2: nat,N4: nat] :
( ! [N3: nat] : ( ord_less @ A @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less @ nat @ N2 @ N4 )
=> ( ord_less @ A @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_51_less__Suc__eq__0__disj,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ M2 @ ( suc @ N2 ) )
= ( ( M2
= ( zero_zero @ nat ) )
| ? [J: nat] :
( ( M2
= ( suc @ J ) )
& ( ord_less @ nat @ J @ N2 ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_52_gr0__implies__Suc,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ? [M: nat] :
( N2
= ( suc @ M ) ) ) ).
% gr0_implies_Suc
thf(fact_53_gr0__conv__Suc,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
= ( ? [M4: nat] :
( N2
= ( suc @ M4 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_54_RuleSystem__Defs_Ocountable__R,axiom,
! [Rule: $tType,Rules2: stream @ Rule] : ( countable_countable @ Rule @ ( sset @ Rule @ Rules2 ) ) ).
% RuleSystem_Defs.countable_R
thf(fact_55_RuleSystem__Defs_ONE__R,axiom,
! [Rule: $tType,Rules2: stream @ Rule] :
( ( sset @ Rule @ Rules2 )
!= ( bot_bot @ ( set @ Rule ) ) ) ).
% RuleSystem_Defs.NE_R
thf(fact_56_not__less__less__Suc__eq,axiom,
! [N2: nat,M2: nat] :
( ~ ( ord_less @ nat @ N2 @ M2 )
=> ( ( ord_less @ nat @ N2 @ ( suc @ M2 ) )
= ( N2 = M2 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_57_strict__inc__induct,axiom,
! [I: nat,J2: nat,P: nat > $o] :
( ( ord_less @ nat @ I @ J2 )
=> ( ! [I2: nat] :
( ( J2
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ J2 )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_58_less__Suc__induct,axiom,
! [I: nat,J2: nat,P: nat > nat > $o] :
( ( ord_less @ nat @ I @ J2 )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J3: nat,K: nat] :
( ( ord_less @ nat @ I2 @ J3 )
=> ( ( ord_less @ nat @ J3 @ K )
=> ( ( P @ I2 @ J3 )
=> ( ( P @ J3 @ K )
=> ( P @ I2 @ K ) ) ) ) )
=> ( P @ I @ J2 ) ) ) ) ).
% less_Suc_induct
thf(fact_59_less__trans__Suc,axiom,
! [I: nat,J2: nat,K2: nat] :
( ( ord_less @ nat @ I @ J2 )
=> ( ( ord_less @ nat @ J2 @ K2 )
=> ( ord_less @ nat @ ( suc @ I ) @ K2 ) ) ) ).
% less_trans_Suc
thf(fact_60_Suc__less__SucD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
=> ( ord_less @ nat @ M2 @ N2 ) ) ).
% Suc_less_SucD
thf(fact_61_less__antisym,axiom,
! [N2: nat,M2: nat] :
( ~ ( ord_less @ nat @ N2 @ M2 )
=> ( ( ord_less @ nat @ N2 @ ( suc @ M2 ) )
=> ( M2 = N2 ) ) ) ).
% less_antisym
thf(fact_62_Suc__less__eq2,axiom,
! [N2: nat,M2: nat] :
( ( ord_less @ nat @ ( suc @ N2 ) @ M2 )
= ( ? [M5: nat] :
( ( M2
= ( suc @ M5 ) )
& ( ord_less @ nat @ N2 @ M5 ) ) ) ) ).
% Suc_less_eq2
thf(fact_63_not__less__eq,axiom,
! [M2: nat,N2: nat] :
( ( ~ ( ord_less @ nat @ M2 @ N2 ) )
= ( ord_less @ nat @ N2 @ ( suc @ M2 ) ) ) ).
% not_less_eq
thf(fact_64_less__Suc__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ M2 @ ( suc @ N2 ) )
= ( ( ord_less @ nat @ M2 @ N2 )
| ( M2 = N2 ) ) ) ).
% less_Suc_eq
thf(fact_65_less__SucI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
=> ( ord_less @ nat @ M2 @ ( suc @ N2 ) ) ) ).
% less_SucI
thf(fact_66_less__SucE,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ M2 @ ( suc @ N2 ) )
=> ( ~ ( ord_less @ nat @ M2 @ N2 )
=> ( M2 = N2 ) ) ) ).
% less_SucE
thf(fact_67_Suc__lessI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
=> ( ( ( suc @ M2 )
!= N2 )
=> ( ord_less @ nat @ ( suc @ M2 ) @ N2 ) ) ) ).
% Suc_lessI
thf(fact_68_Suc__lessE,axiom,
! [I: nat,K2: nat] :
( ( ord_less @ nat @ ( suc @ I ) @ K2 )
=> ~ ! [J3: nat] :
( ( ord_less @ nat @ I @ J3 )
=> ( K2
!= ( suc @ J3 ) ) ) ) ).
% Suc_lessE
thf(fact_69_Suc__lessD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ ( suc @ M2 ) @ N2 )
=> ( ord_less @ nat @ M2 @ N2 ) ) ).
% Suc_lessD
thf(fact_70_lessE,axiom,
! [I: nat,K2: nat] :
( ( ord_less @ nat @ I @ K2 )
=> ( ( K2
!= ( suc @ I ) )
=> ~ ! [J3: nat] :
( ( ord_less @ nat @ I @ J3 )
=> ( K2
!= ( suc @ J3 ) ) ) ) ) ).
% lessE
thf(fact_71_not0__implies__Suc,axiom,
! [N2: nat] :
( ( N2
!= ( zero_zero @ nat ) )
=> ? [M: nat] :
( N2
= ( suc @ M ) ) ) ).
% not0_implies_Suc
thf(fact_72_old_Onat_Oinducts,axiom,
! [P: nat > $o,Nat: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [Nat3: nat] :
( ( P @ Nat3 )
=> ( P @ ( suc @ Nat3 ) ) )
=> ( P @ Nat ) ) ) ).
% old.nat.inducts
thf(fact_73_old_Onat_Oexhaust,axiom,
! [Y3: nat] :
( ( Y3
!= ( zero_zero @ nat ) )
=> ~ ! [Nat3: nat] :
( Y3
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_74_Zero__not__Suc,axiom,
! [M2: nat] :
( ( zero_zero @ nat )
!= ( suc @ M2 ) ) ).
% Zero_not_Suc
thf(fact_75_Zero__neq__Suc,axiom,
! [M2: nat] :
( ( zero_zero @ nat )
!= ( suc @ M2 ) ) ).
% Zero_neq_Suc
thf(fact_76_Suc__neq__Zero,axiom,
! [M2: nat] :
( ( suc @ M2 )
!= ( zero_zero @ nat ) ) ).
% Suc_neq_Zero
thf(fact_77_zero__induct,axiom,
! [P: nat > $o,K2: nat] :
( ( P @ K2 )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ ( zero_zero @ nat ) ) ) ) ).
% zero_induct
thf(fact_78_diff__induct,axiom,
! [P: nat > nat > $o,M2: nat,N2: nat] :
( ! [X3: nat] : ( P @ X3 @ ( zero_zero @ nat ) )
=> ( ! [Y4: nat] : ( P @ ( zero_zero @ nat ) @ ( suc @ Y4 ) )
=> ( ! [X3: nat,Y4: nat] :
( ( P @ X3 @ Y4 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y4 ) ) )
=> ( P @ M2 @ N2 ) ) ) ) ).
% diff_induct
thf(fact_79_nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N2 ) ) ) ).
% nat_induct
thf(fact_80_nat_OdiscI,axiom,
! [Nat: nat,X2: nat] :
( ( Nat
= ( suc @ X2 ) )
=> ( Nat
!= ( zero_zero @ nat ) ) ) ).
% nat.discI
thf(fact_81_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( ( zero_zero @ nat )
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_82_old_Onat_Odistinct_I2_J,axiom,
! [Nat4: nat] :
( ( suc @ Nat4 )
!= ( zero_zero @ nat ) ) ).
% old.nat.distinct(2)
thf(fact_83_nat_Odistinct_I1_J,axiom,
! [X2: nat] :
( ( zero_zero @ nat )
!= ( suc @ X2 ) ) ).
% nat.distinct(1)
thf(fact_84_snth__sset,axiom,
! [A: $tType,S: stream @ A,N2: nat] : ( member @ A @ ( snth @ A @ S @ N2 ) @ ( sset @ A @ S ) ) ).
% snth_sset
thf(fact_85_smap__alt,axiom,
! [A: $tType,B: $tType,F: B > A,S: stream @ B,S3: stream @ A] :
( ( ( smap @ B @ A @ F @ S )
= S3 )
= ( ! [N: nat] :
( ( F @ ( snth @ B @ S @ N ) )
= ( snth @ A @ S3 @ N ) ) ) ) ).
% smap_alt
thf(fact_86_n__not__Suc__n,axiom,
! [N2: nat] :
( N2
!= ( suc @ N2 ) ) ).
% n_not_Suc_n
thf(fact_87_Suc__inject,axiom,
! [X: nat,Y3: nat] :
( ( ( suc @ X )
= ( suc @ Y3 ) )
=> ( X = Y3 ) ) ).
% Suc_inject
thf(fact_88_stream_Omap__ident,axiom,
! [A: $tType,T: stream @ A] :
( ( smap @ A @ A
@ ^ [X4: A] : X4
@ T )
= T ) ).
% stream.map_ident
thf(fact_89_stream__smap__nats,axiom,
! [A: $tType,S: stream @ A] :
( S
= ( smap @ nat @ A @ ( snth @ A @ S ) @ ( siterate @ nat @ suc @ ( zero_zero @ nat ) ) ) ) ).
% stream_smap_nats
thf(fact_90_RuleSystem__Defs_Osset__fenum,axiom,
! [Rule: $tType,Rules2: stream @ Rule] :
( ( sset @ Rule @ ( abstra1774373515_fenum @ Rule @ Rules2 ) )
= ( sset @ Rule @ Rules2 ) ) ).
% RuleSystem_Defs.sset_fenum
thf(fact_91_countable__empty,axiom,
! [A: $tType] : ( countable_countable @ A @ ( bot_bot @ ( set @ A ) ) ) ).
% countable_empty
thf(fact_92_bex__empty,axiom,
! [A: $tType,P: A > $o] :
~ ? [X5: A] :
( ( member @ A @ X5 @ ( bot_bot @ ( set @ A ) ) )
& ( P @ X5 ) ) ).
% bex_empty
thf(fact_93_not__gr__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [N2: A] :
( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N2 ) )
= ( N2
= ( zero_zero @ A ) ) ) ) ).
% not_gr_zero
thf(fact_94_less__numeral__extra_I2_J,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% less_numeral_extra(2)
thf(fact_95_less__numeral__extra_I1_J,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% less_numeral_extra(1)
thf(fact_96_zero__less__one,axiom,
! [A: $tType] :
( ( zero_less_one @ A @ ( type2 @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% zero_less_one
thf(fact_97_not__one__less__zero,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A @ ( type2 @ A ) )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% not_one_less_zero
thf(fact_98_countableI__type,axiom,
! [A: $tType] :
( ( countable @ A @ ( type2 @ A ) )
=> ! [A3: set @ A] : ( countable_countable @ A @ A3 ) ) ).
% countableI_type
thf(fact_99_empty__iff,axiom,
! [A: $tType,C: A] :
~ ( member @ A @ C @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_100_all__not__in__conv,axiom,
! [A: $tType,A3: set @ A] :
( ( ! [X4: A] :
~ ( member @ A @ X4 @ A3 ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_101_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_102_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_103_not__psubset__empty,axiom,
! [A: $tType,A3: set @ A] :
~ ( ord_less @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% not_psubset_empty
thf(fact_104_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( ( zero_zero @ A )
= X )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_105_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( ( linordered_idom @ A @ ( type2 @ A ) )
=> ! [X: A,Y3: A] :
( ( X != Y3 )
=> ( ~ ( ord_less @ A @ X @ Y3 )
=> ( ord_less @ A @ Y3 @ X ) ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_106_one__reorient,axiom,
! [A: $tType] :
( ( one @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( ( one_one @ A )
= X )
= ( X
= ( one_one @ A ) ) ) ) ).
% one_reorient
thf(fact_107_ex__in__conv,axiom,
! [A: $tType,A3: set @ A] :
( ( ? [X4: A] : ( member @ A @ X4 @ A3 ) )
= ( A3
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_108_equals0I,axiom,
! [A: $tType,A3: set @ A] :
( ! [Y4: A] :
~ ( member @ A @ Y4 @ A3 )
=> ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_109_equals0D,axiom,
! [A: $tType,A3: set @ A,A2: A] :
( ( A3
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A2 @ A3 ) ) ).
% equals0D
thf(fact_110_emptyE,axiom,
! [A: $tType,A2: A] :
~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_111_Bex__def,axiom,
! [A: $tType] :
( ( bex @ A )
= ( ^ [A4: set @ A,P2: A > $o] :
? [X4: A] :
( ( member @ A @ X4 @ A4 )
& ( P2 @ X4 ) ) ) ) ).
% Bex_def
thf(fact_112_empty__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A
@ ^ [X4: A] : $false ) ) ).
% empty_def
thf(fact_113_countable__Collect,axiom,
! [A: $tType,A3: set @ A,Phi: A > $o] :
( ( countable_countable @ A @ A3 )
=> ( countable_countable @ A
@ ( collect @ A
@ ^ [A5: A] :
( ( member @ A @ A5 @ A3 )
& ( Phi @ A5 ) ) ) ) ) ).
% countable_Collect
thf(fact_114_zero__less__iff__neq__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [N2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ N2 )
= ( N2
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_iff_neq_zero
thf(fact_115_gr__implies__not__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [M2: A,N2: A] :
( ( ord_less @ A @ M2 @ N2 )
=> ( N2
!= ( zero_zero @ A ) ) ) ) ).
% gr_implies_not_zero
thf(fact_116_not__less__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [N2: A] :
~ ( ord_less @ A @ N2 @ ( zero_zero @ A ) ) ) ).
% not_less_zero
thf(fact_117_gr__zeroI,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A @ ( type2 @ A ) )
=> ! [N2: A] :
( ( N2
!= ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ N2 ) ) ) ).
% gr_zeroI
thf(fact_118_less__numeral__extra_I3_J,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% less_numeral_extra(3)
thf(fact_119_zero__neq__one,axiom,
! [A: $tType] :
( ( zero_neq_one @ A @ ( type2 @ A ) )
=> ( ( zero_zero @ A )
!= ( one_one @ A ) ) ) ).
% zero_neq_one
thf(fact_120_less__numeral__extra_I4_J,axiom,
! [A: $tType] :
( ( linordered_semidom @ A @ ( type2 @ A ) )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ).
% less_numeral_extra(4)
thf(fact_121_bot__apply,axiom,
! [C2: $tType,D: $tType] :
( ( bot @ C2 @ ( type2 @ C2 ) )
=> ( ( bot_bot @ ( D > C2 ) )
= ( ^ [X4: D] : ( bot_bot @ C2 ) ) ) ) ).
% bot_apply
thf(fact_122_snth__sset__smerge,axiom,
! [A: $tType,Ss: stream @ ( stream @ A ),N2: nat,M2: nat] : ( member @ A @ ( snth @ A @ ( snth @ ( stream @ A ) @ Ss @ N2 ) @ M2 ) @ ( sset @ A @ ( smerge @ A @ Ss ) ) ) ).
% snth_sset_smerge
thf(fact_123_list__decode_Ocases,axiom,
! [X: nat] :
( ( X
!= ( zero_zero @ nat ) )
=> ~ ! [N3: nat] :
( X
!= ( suc @ N3 ) ) ) ).
% list_decode.cases
thf(fact_124_dependent__nat__choice,axiom,
! [A: $tType,P: nat > A > $o,Q: nat > A > A > $o] :
( ? [X1: A] : ( P @ ( zero_zero @ nat ) @ X1 )
=> ( ! [X3: A,N3: nat] :
( ( P @ N3 @ X3 )
=> ? [Y: A] :
( ( P @ ( suc @ N3 ) @ Y )
& ( Q @ N3 @ X3 @ Y ) ) )
=> ? [F2: nat > A] :
! [N5: nat] :
( ( P @ N5 @ ( F2 @ N5 ) )
& ( Q @ N5 @ ( F2 @ N5 ) @ ( F2 @ ( suc @ N5 ) ) ) ) ) ) ).
% dependent_nat_choice
thf(fact_125_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ ( zero_zero @ nat ) )
=> ( ? [X1: nat] : ( P @ X1 )
=> ? [N3: nat] :
( ~ ( P @ N3 )
& ( P @ ( suc @ N3 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_126_psubsetD,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,C: A] :
( ( ord_less @ ( set @ A ) @ A3 @ B2 )
=> ( ( member @ A @ C @ A3 )
=> ( member @ A @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_127_less__set__def,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A4: set @ A,B3: set @ A] :
( ord_less @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ A4 )
@ ^ [X4: A] : ( member @ A @ X4 @ B3 ) ) ) ) ).
% less_set_def
thf(fact_128_psubset__trans,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,C3: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ B2 )
=> ( ( ord_less @ ( set @ A ) @ B2 @ C3 )
=> ( ord_less @ ( set @ A ) @ A3 @ C3 ) ) ) ).
% psubset_trans
thf(fact_129_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_130_bot__nat__def,axiom,
( ( bot_bot @ nat )
= ( zero_zero @ nat ) ) ).
% bot_nat_def
thf(fact_131_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B > A,B4: B,C: B] :
( ( A2
= ( F @ B4 ) )
=> ( ( ord_less @ B @ B4 @ C )
=> ( ! [X3: B,Y4: B] :
( ( ord_less @ B @ X3 @ Y4 )
=> ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_132_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B4: A,F: A > B,C: B] :
( ( ord_less @ A @ A2 @ B4 )
=> ( ( ( F @ B4 )
= C )
=> ( ! [X3: A,Y4: A] :
( ( ord_less @ A @ X3 @ Y4 )
=> ( ord_less @ B @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_133_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B > A,B4: B,C: B] :
( ( ord_less @ A @ A2 @ ( F @ B4 ) )
=> ( ( ord_less @ B @ B4 @ C )
=> ( ! [X3: B,Y4: B] :
( ( ord_less @ B @ X3 @ Y4 )
=> ( ord_less @ A @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_134_order__less__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 @ ( type2 @ C2 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B4: A,F: A > C2,C: C2] :
( ( ord_less @ A @ A2 @ B4 )
=> ( ( ord_less @ C2 @ ( F @ B4 ) @ C )
=> ( ! [X3: A,Y4: A] :
( ( ord_less @ A @ X3 @ Y4 )
=> ( ord_less @ C2 @ ( F @ X3 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_less_subst2
thf(fact_135_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A @ ( type2 @ A ) )
=> ! [X: A] :
? [Y4: A] : ( ord_less @ A @ Y4 @ X ) ) ).
% lt_ex
thf(fact_136_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A @ ( type2 @ A ) )
=> ! [X: A] :
? [X12: A] : ( ord_less @ A @ X @ X12 ) ) ).
% gt_ex
thf(fact_137_neqE,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y3: A] :
( ( X != Y3 )
=> ( ~ ( ord_less @ A @ X @ Y3 )
=> ( ord_less @ A @ Y3 @ X ) ) ) ) ).
% neqE
thf(fact_138_neq__iff,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y3: A] :
( ( X != Y3 )
= ( ( ord_less @ A @ X @ Y3 )
| ( ord_less @ A @ Y3 @ X ) ) ) ) ).
% neq_iff
thf(fact_139_order_Oasym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B4: A] :
( ( ord_less @ A @ A2 @ B4 )
=> ~ ( ord_less @ A @ B4 @ A2 ) ) ) ).
% order.asym
thf(fact_140_dense,axiom,
! [A: $tType] :
( ( dense_order @ A @ ( type2 @ A ) )
=> ! [X: A,Y3: A] :
( ( ord_less @ A @ X @ Y3 )
=> ? [Z: A] :
( ( ord_less @ A @ X @ Z )
& ( ord_less @ A @ Z @ Y3 ) ) ) ) ).
% dense
thf(fact_141_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y3: A] :
( ( ord_less @ A @ X @ Y3 )
=> ( X != Y3 ) ) ) ).
% less_imp_neq
thf(fact_142_less__asym,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y3: A] :
( ( ord_less @ A @ X @ Y3 )
=> ~ ( ord_less @ A @ Y3 @ X ) ) ) ).
% less_asym
thf(fact_143_less__asym_H,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [A2: A,B4: A] :
( ( ord_less @ A @ A2 @ B4 )
=> ~ ( ord_less @ A @ B4 @ A2 ) ) ) ).
% less_asym'
thf(fact_144_less__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y3: A,Z2: A] :
( ( ord_less @ A @ X @ Y3 )
=> ( ( ord_less @ A @ Y3 @ Z2 )
=> ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% less_trans
thf(fact_145_less__linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y3: A] :
( ( ord_less @ A @ X @ Y3 )
| ( X = Y3 )
| ( ord_less @ A @ Y3 @ X ) ) ) ).
% less_linear
thf(fact_146_less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A] :
~ ( ord_less @ A @ X @ X ) ) ).
% less_irrefl
thf(fact_147_ord__eq__less__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B4: A,C: A] :
( ( A2 = B4 )
=> ( ( ord_less @ A @ B4 @ C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% ord_eq_less_trans
thf(fact_148_ord__less__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B4: A,C: A] :
( ( ord_less @ A @ A2 @ B4 )
=> ( ( B4 = C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% ord_less_eq_trans
thf(fact_149_dual__order_Oasym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B4: A,A2: A] :
( ( ord_less @ A @ B4 @ A2 )
=> ~ ( ord_less @ A @ A2 @ B4 ) ) ) ).
% dual_order.asym
thf(fact_150_less__imp__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y3: A] :
( ( ord_less @ A @ X @ Y3 )
=> ( X != Y3 ) ) ) ).
% less_imp_not_eq
thf(fact_151_less__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y3: A] :
( ( ord_less @ A @ X @ Y3 )
=> ~ ( ord_less @ A @ Y3 @ X ) ) ) ).
% less_not_sym
thf(fact_152_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,A2: A] :
( ! [X3: A] :
( ! [Y: A] :
( ( ord_less @ A @ Y @ X3 )
=> ( P @ Y ) )
=> ( P @ X3 ) )
=> ( P @ A2 ) ) ) ).
% less_induct
thf(fact_153_antisym__conv3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y3: A,X: A] :
( ~ ( ord_less @ A @ Y3 @ X )
=> ( ( ~ ( ord_less @ A @ X @ Y3 ) )
= ( X = Y3 ) ) ) ) ).
% antisym_conv3
thf(fact_154_less__imp__not__eq2,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y3: A] :
( ( ord_less @ A @ X @ Y3 )
=> ( Y3 != X ) ) ) ).
% less_imp_not_eq2
thf(fact_155_less__imp__triv,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y3: A,P: $o] :
( ( ord_less @ A @ X @ Y3 )
=> ( ( ord_less @ A @ Y3 @ X )
=> P ) ) ) ).
% less_imp_triv
thf(fact_156_linorder__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y3: A] :
( ~ ( ord_less @ A @ X @ Y3 )
=> ( ( X != Y3 )
=> ( ord_less @ A @ Y3 @ X ) ) ) ) ).
% linorder_cases
thf(fact_157_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ A2 ) ) ).
% dual_order.irrefl
thf(fact_158_order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B4: A,C: A] :
( ( ord_less @ A @ A2 @ B4 )
=> ( ( ord_less @ A @ B4 @ C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% order.strict_trans
thf(fact_159_less__imp__not__less,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y3: A] :
( ( ord_less @ A @ X @ Y3 )
=> ~ ( ord_less @ A @ Y3 @ X ) ) ) ).
% less_imp_not_less
thf(fact_160_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B4: A,A2: A,C: A] :
( ( ord_less @ A @ B4 @ A2 )
=> ( ( ord_less @ A @ C @ B4 )
=> ( ord_less @ A @ C @ A2 ) ) ) ) ).
% dual_order.strict_trans
thf(fact_161_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y3: A] :
( ( ~ ( ord_less @ A @ X @ Y3 ) )
= ( ( ord_less @ A @ Y3 @ X )
| ( X = Y3 ) ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_162_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B4: A] :
( ( ord_less @ A @ A2 @ B4 )
=> ( A2 != B4 ) ) ) ).
% order.strict_implies_not_eq
thf(fact_163_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B4: A,A2: A] :
( ( ord_less @ A @ B4 @ A2 )
=> ( A2 != B4 ) ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_164_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_165_bot_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ ( bot_bot @ A ) ) ) ).
% bot.extremum_strict
thf(fact_166_bot_Onot__eq__extremum,axiom,
! [A: $tType] :
( ( order_bot @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( A2
!= ( bot_bot @ A ) )
= ( ord_less @ A @ ( bot_bot @ A ) @ A2 ) ) ) ).
% bot.not_eq_extremum
thf(fact_167_dbl__inc__simps_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A @ ( type2 @ A ) )
=> ( ( neg_numeral_dbl_inc @ A @ ( zero_zero @ A ) )
= ( one_one @ A ) ) ) ).
% dbl_inc_simps(2)
thf(fact_168_smerge__def,axiom,
! [A: $tType] :
( ( smerge @ A )
= ( ^ [Ss2: stream @ ( stream @ A )] :
( flat @ A
@ ( smap @ nat @ ( list @ A )
@ ^ [N: nat] :
( append @ A
@ ( map @ ( stream @ A ) @ A
@ ^ [S4: stream @ A] : ( snth @ A @ S4 @ N )
@ ( stake @ ( stream @ A ) @ ( suc @ N ) @ Ss2 ) )
@ ( stake @ A @ N @ ( snth @ ( stream @ A ) @ Ss2 @ N ) ) )
@ ( siterate @ nat @ suc @ ( zero_zero @ nat ) ) ) ) ) ) ).
% smerge_def
thf(fact_169_stream__smap__fromN,axiom,
! [A: $tType,S: stream @ A,N2: nat] :
( S
= ( smap @ nat @ A
@ ^ [J: nat] : ( snth @ A @ S @ ( minus_minus @ nat @ J @ N2 ) )
@ ( siterate @ nat @ suc @ N2 ) ) ) ).
% stream_smap_fromN
thf(fact_170_Suc__diff__1,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( suc @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) )
= N2 ) ) ).
% Suc_diff_1
thf(fact_171_diff__self,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ A2 )
= ( zero_zero @ A ) ) ) ).
% diff_self
thf(fact_172_diff__0__right,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% diff_0_right
thf(fact_173_zero__diff,axiom,
! [A: $tType] :
( ( comm_monoid_diff @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( minus_minus @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% zero_diff
thf(fact_174_diff__zero,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% diff_zero
thf(fact_175_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ A2 )
= ( zero_zero @ A ) ) ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_176_diff__0__eq__0,axiom,
! [N2: nat] :
( ( minus_minus @ nat @ ( zero_zero @ nat ) @ N2 )
= ( zero_zero @ nat ) ) ).
% diff_0_eq_0
thf(fact_177_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus @ nat @ M2 @ M2 )
= ( zero_zero @ nat ) ) ).
% diff_self_eq_0
thf(fact_178_Suc__diff__diff,axiom,
! [M2: nat,N2: nat,K2: nat] :
( ( minus_minus @ nat @ ( minus_minus @ nat @ ( suc @ M2 ) @ N2 ) @ ( suc @ K2 ) )
= ( minus_minus @ nat @ ( minus_minus @ nat @ M2 @ N2 ) @ K2 ) ) ).
% Suc_diff_diff
thf(fact_179_diff__Suc__Suc,axiom,
! [M2: nat,N2: nat] :
( ( minus_minus @ nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
= ( minus_minus @ nat @ M2 @ N2 ) ) ).
% diff_Suc_Suc
thf(fact_180_shift__append,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,S: stream @ A] :
( ( shift @ A @ ( append @ A @ Xs @ Ys ) @ S )
= ( shift @ A @ Xs @ ( shift @ A @ Ys @ S ) ) ) ).
% shift_append
thf(fact_181_diff__gt__0__iff__gt,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B4 ) )
= ( ord_less @ A @ B4 @ A2 ) ) ) ).
% diff_gt_0_iff_gt
thf(fact_182_diff__numeral__special_I9_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A @ ( type2 @ A ) )
=> ( ( minus_minus @ A @ ( one_one @ A ) @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% diff_numeral_special(9)
thf(fact_183_zero__less__diff,axiom,
! [N2: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N2 @ M2 ) )
= ( ord_less @ nat @ M2 @ N2 ) ) ).
% zero_less_diff
thf(fact_184_diff__Suc__1,axiom,
! [N2: nat] :
( ( minus_minus @ nat @ ( suc @ N2 ) @ ( one_one @ nat ) )
= N2 ) ).
% diff_Suc_1
thf(fact_185_smap__shift,axiom,
! [A: $tType,B: $tType,F: B > A,Xs: list @ B,S: stream @ B] :
( ( smap @ B @ A @ F @ ( shift @ B @ Xs @ S ) )
= ( shift @ A @ ( map @ B @ A @ F @ Xs ) @ ( smap @ B @ A @ F @ S ) ) ) ).
% smap_shift
thf(fact_186_stake__smap,axiom,
! [A: $tType,B: $tType,N2: nat,F: B > A,S: stream @ B] :
( ( stake @ A @ N2 @ ( smap @ B @ A @ F @ S ) )
= ( map @ B @ A @ F @ ( stake @ B @ N2 @ S ) ) ) ).
% stake_smap
thf(fact_187_Suc__pred,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( suc @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) )
= N2 ) ) ).
% Suc_pred
thf(fact_188_diff__strict__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B4: A,C: A] :
( ( ord_less @ A @ A2 @ B4 )
=> ( ord_less @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B4 @ C ) ) ) ) ).
% diff_strict_right_mono
thf(fact_189_diff__strict__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [B4: A,A2: A,C: A] :
( ( ord_less @ A @ B4 @ A2 )
=> ( ord_less @ A @ ( minus_minus @ A @ C @ A2 ) @ ( minus_minus @ A @ C @ B4 ) ) ) ) ).
% diff_strict_left_mono
thf(fact_190_diff__eq__diff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B4: A,C: A,D2: A] :
( ( ( minus_minus @ A @ A2 @ B4 )
= ( minus_minus @ A @ C @ D2 ) )
=> ( ( ord_less @ A @ A2 @ B4 )
= ( ord_less @ A @ C @ D2 ) ) ) ) ).
% diff_eq_diff_less
thf(fact_191_diff__strict__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B4: A,D2: A,C: A] :
( ( ord_less @ A @ A2 @ B4 )
=> ( ( ord_less @ A @ D2 @ C )
=> ( ord_less @ A @ ( minus_minus @ A @ A2 @ C ) @ ( minus_minus @ A @ B4 @ D2 ) ) ) ) ) ).
% diff_strict_mono
thf(fact_192_diff__less__mono2,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less @ nat @ M2 @ N2 )
=> ( ( ord_less @ nat @ M2 @ L )
=> ( ord_less @ nat @ ( minus_minus @ nat @ L @ N2 ) @ ( minus_minus @ nat @ L @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_193_less__imp__diff__less,axiom,
! [J2: nat,K2: nat,N2: nat] :
( ( ord_less @ nat @ J2 @ K2 )
=> ( ord_less @ nat @ ( minus_minus @ nat @ J2 @ N2 ) @ K2 ) ) ).
% less_imp_diff_less
thf(fact_194_diff__commute,axiom,
! [I: nat,J2: nat,K2: nat] :
( ( minus_minus @ nat @ ( minus_minus @ nat @ I @ J2 ) @ K2 )
= ( minus_minus @ nat @ ( minus_minus @ nat @ I @ K2 ) @ J2 ) ) ).
% diff_commute
thf(fact_195_zero__induct__lemma,axiom,
! [P: nat > $o,K2: nat,I: nat] :
( ( P @ K2 )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ ( minus_minus @ nat @ K2 @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_196_diffs0__imp__equal,axiom,
! [M2: nat,N2: nat] :
( ( ( minus_minus @ nat @ M2 @ N2 )
= ( zero_zero @ nat ) )
=> ( ( ( minus_minus @ nat @ N2 @ M2 )
= ( zero_zero @ nat ) )
=> ( M2 = N2 ) ) ) ).
% diffs0_imp_equal
thf(fact_197_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus @ nat @ M2 @ ( zero_zero @ nat ) )
= M2 ) ).
% minus_nat.diff_0
thf(fact_198_eq__iff__diff__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ( ( ^ [Y5: A,Z3: A] : Y5 = Z3 )
= ( ^ [A5: A,B5: A] :
( ( minus_minus @ A @ A5 @ B5 )
= ( zero_zero @ A ) ) ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_199_diff__eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A @ ( type2 @ A ) )
=> ! [A2: A,B4: A,C: A,D2: A] :
( ( ( minus_minus @ A @ A2 @ B4 )
= ( minus_minus @ A @ C @ D2 ) )
=> ( ( A2 = B4 )
= ( C = D2 ) ) ) ) ).
% diff_eq_diff_eq
thf(fact_200_diff__right__commute,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A @ ( type2 @ A ) )
=> ! [A2: A,C: A,B4: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C ) @ B4 )
= ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B4 ) @ C ) ) ) ).
% diff_right_commute
thf(fact_201_less__iff__diff__less__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [A5: A,B5: A] : ( ord_less @ A @ ( minus_minus @ A @ A5 @ B5 ) @ ( zero_zero @ A ) ) ) ) ) ).
% less_iff_diff_less_0
thf(fact_202_diff__less,axiom,
! [N2: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ord_less @ nat @ ( minus_minus @ nat @ M2 @ N2 ) @ M2 ) ) ) ).
% diff_less
thf(fact_203_Suc__diff__Suc,axiom,
! [N2: nat,M2: nat] :
( ( ord_less @ nat @ N2 @ M2 )
=> ( ( suc @ ( minus_minus @ nat @ M2 @ ( suc @ N2 ) ) )
= ( minus_minus @ nat @ M2 @ N2 ) ) ) ).
% Suc_diff_Suc
thf(fact_204_diff__less__Suc,axiom,
! [M2: nat,N2: nat] : ( ord_less @ nat @ ( minus_minus @ nat @ M2 @ N2 ) @ ( suc @ M2 ) ) ).
% diff_less_Suc
thf(fact_205_diff__Suc__eq__diff__pred,axiom,
! [M2: nat,N2: nat] :
( ( minus_minus @ nat @ M2 @ ( suc @ N2 ) )
= ( minus_minus @ nat @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) @ N2 ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_206_diff__Suc__less,axiom,
! [N2: nat,I: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ord_less @ nat @ ( minus_minus @ nat @ N2 @ ( suc @ I ) ) @ N2 ) ) ).
% diff_Suc_less
thf(fact_207_Suc__diff__eq__diff__pred,axiom,
! [N2: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( minus_minus @ nat @ ( suc @ M2 ) @ N2 )
= ( minus_minus @ nat @ M2 @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_208_Suc__pred_H,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( N2
= ( suc @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ).
% Suc_pred'
thf(fact_209_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
thf(fact_210_Suc__if__eq,axiom,
! [A: $tType,F: nat > A,H: nat > A,G: A,N2: nat] :
( ! [N3: nat] :
( ( F @ ( suc @ N3 ) )
= ( H @ N3 ) )
=> ( ( ( F @ ( zero_zero @ nat ) )
= G )
=> ( ( ( N2
= ( zero_zero @ nat ) )
=> ( ( F @ N2 )
= G ) )
& ( ( N2
!= ( zero_zero @ nat ) )
=> ( ( F @ N2 )
= ( H @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% Suc_if_eq
thf(fact_211_map__ident,axiom,
! [A: $tType] :
( ( map @ A @ A
@ ^ [X4: A] : X4 )
= ( ^ [Xs2: list @ A] : Xs2 ) ) ).
% map_ident
thf(fact_212_Diff__empty,axiom,
! [A: $tType,A3: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
= A3 ) ).
% Diff_empty
thf(fact_213_empty__Diff,axiom,
! [A: $tType,A3: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 )
= ( bot_bot @ ( set @ A ) ) ) ).
% empty_Diff
thf(fact_214_Diff__cancel,axiom,
! [A: $tType,A3: set @ A] :
( ( minus_minus @ ( set @ A ) @ A3 @ A3 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_cancel
thf(fact_215_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_216_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_217_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_218_countable__Diff,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( countable_countable @ A @ A3 )
=> ( countable_countable @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) ) ) ).
% countable_Diff
thf(fact_219_psubset__imp__ex__mem,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ B2 )
=> ? [B6: A] : ( member @ A @ B6 @ ( minus_minus @ ( set @ A ) @ B2 @ A3 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_220_uncountable__minus__countable,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ~ ( countable_countable @ A @ A3 )
=> ( ( countable_countable @ A @ B2 )
=> ~ ( countable_countable @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) ) ) ) ).
% uncountable_minus_countable
thf(fact_221_append__eq__appendI,axiom,
! [A: $tType,Xs: list @ A,Xs1: list @ A,Zs: list @ A,Ys: list @ A,Us: list @ A] :
( ( ( append @ A @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append @ A @ Xs1 @ Us ) )
=> ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_222_append__eq__append__conv2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A,Ts: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Zs @ Ts ) )
= ( ? [Us2: list @ A] :
( ( ( Xs
= ( append @ A @ Zs @ Us2 ) )
& ( ( append @ A @ Us2 @ Ys )
= Ts ) )
| ( ( ( append @ A @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append @ A @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_223_list_Omap__ident,axiom,
! [A: $tType,T: list @ A] :
( ( map @ A @ A
@ ^ [X4: A] : X4
@ T )
= T ) ).
% list.map_ident
thf(fact_224_minus__apply,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B @ ( type2 @ B ) )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A4: A > B,B3: A > B,X4: A] : ( minus_minus @ B @ ( A4 @ X4 ) @ ( B3 @ X4 ) ) ) ) ) ).
% minus_apply
thf(fact_225_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_226_DiffI,axiom,
! [A: $tType,C: A,A3: set @ A,B2: set @ A] :
( ( member @ A @ C @ A3 )
=> ( ~ ( member @ A @ C @ B2 )
=> ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) ) ) ) ).
% DiffI
thf(fact_227_Diff__iff,axiom,
! [A: $tType,C: A,A3: set @ A,B2: set @ A] :
( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) )
= ( ( member @ A @ C @ A3 )
& ~ ( member @ A @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_228_Diff__idemp,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) @ B2 )
= ( minus_minus @ ( set @ A ) @ A3 @ B2 ) ) ).
% Diff_idemp
thf(fact_229_DiffE,axiom,
! [A: $tType,C: A,A3: set @ A,B2: set @ A] :
( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) )
=> ~ ( ( member @ A @ C @ A3 )
=> ( member @ A @ C @ B2 ) ) ) ).
% DiffE
thf(fact_230_DiffD1,axiom,
! [A: $tType,C: A,A3: set @ A,B2: set @ A] :
( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) )
=> ( member @ A @ C @ A3 ) ) ).
% DiffD1
thf(fact_231_DiffD2,axiom,
! [A: $tType,C: A,A3: set @ A,B2: set @ A] :
( ( member @ A @ C @ ( minus_minus @ ( set @ A ) @ A3 @ B2 ) )
=> ~ ( member @ A @ C @ B2 ) ) ).
% DiffD2
thf(fact_232_set__diff__eq,axiom,
! [A: $tType] :
( ( minus_minus @ ( set @ A ) )
= ( ^ [A4: set @ A,B3: set @ A] :
( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A4 )
& ~ ( member @ A @ X4 @ B3 ) ) ) ) ) ).
% set_diff_eq
thf(fact_233_minus__set__def,axiom,
! [A: $tType] :
( ( minus_minus @ ( set @ A ) )
= ( ^ [A4: set @ A,B3: set @ A] :
( collect @ A
@ ( minus_minus @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ A4 )
@ ^ [X4: A] : ( member @ A @ X4 @ B3 ) ) ) ) ) ).
% minus_set_def
thf(fact_234_fun__diff__def,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B @ ( type2 @ B ) )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A4: A > B,B3: A > B,X4: A] : ( minus_minus @ B @ ( A4 @ X4 ) @ ( B3 @ X4 ) ) ) ) ) ).
% fun_diff_def
thf(fact_235_bot__empty__eq,axiom,
! [A: $tType] :
( ( bot_bot @ ( A > $o ) )
= ( ^ [X4: A] : ( member @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% bot_empty_eq
thf(fact_236_Set_Ois__empty__def,axiom,
! [A: $tType] :
( ( is_empty @ A )
= ( ^ [A4: set @ A] :
( A4
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Set.is_empty_def
thf(fact_237_ln__one,axiom,
! [A: $tType] :
( ( ln @ A @ ( type2 @ A ) )
=> ( ( ln_ln @ A @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% ln_one
thf(fact_238_map__eq__map__tailrec,axiom,
! [B: $tType,A: $tType] :
( ( map @ A @ B )
= ( map_tailrec @ A @ B ) ) ).
% map_eq_map_tailrec
thf(fact_239_powr__zero__eq__one,axiom,
! [A: $tType] :
( ( ln @ A @ ( type2 @ A ) )
=> ! [X: A] :
( ( ( X
= ( zero_zero @ A ) )
=> ( ( powr @ A @ X @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) )
& ( ( X
!= ( zero_zero @ A ) )
=> ( ( powr @ A @ X @ ( zero_zero @ A ) )
= ( one_one @ A ) ) ) ) ) ).
% powr_zero_eq_one
thf(fact_240_powr__one__eq__one,axiom,
! [A: $tType] :
( ( ln @ A @ ( type2 @ A ) )
=> ! [A2: A] :
( ( powr @ A @ ( one_one @ A ) @ A2 )
= ( one_one @ A ) ) ) ).
% powr_one_eq_one
thf(fact_241_powr__eq__0__iff,axiom,
! [A: $tType] :
( ( ln @ A @ ( type2 @ A ) )
=> ! [W: A,Z2: A] :
( ( ( powr @ A @ W @ Z2 )
= ( zero_zero @ A ) )
= ( W
= ( zero_zero @ A ) ) ) ) ).
% powr_eq_0_iff
thf(fact_242_powr__0,axiom,
! [A: $tType] :
( ( ln @ A @ ( type2 @ A ) )
=> ! [Z2: A] :
( ( powr @ A @ ( zero_zero @ A ) @ Z2 )
= ( zero_zero @ A ) ) ) ).
% powr_0
thf(fact_243_diff__Suc,axiom,
! [M2: nat,N2: nat] :
( ( minus_minus @ nat @ M2 @ ( suc @ N2 ) )
= ( case_nat @ nat @ ( zero_zero @ nat )
@ ^ [K3: nat] : K3
@ ( minus_minus @ nat @ M2 @ N2 ) ) ) ).
% diff_Suc
thf(fact_244_fun__cong__unused__0,axiom,
! [A: $tType,B: $tType,C2: $tType] :
( ( zero @ B @ ( type2 @ B ) )
=> ! [F: ( A > B ) > C2,G: C2] :
( ( F
= ( ^ [X4: A > B] : G ) )
=> ( ( F
@ ^ [X4: A] : ( zero_zero @ B ) )
= G ) ) ) ).
% fun_cong_unused_0
thf(fact_245_nat_Ocase__distrib,axiom,
! [B: $tType,A: $tType,H: A > B,F1: A,F22: nat > A,Nat: nat] :
( ( H @ ( case_nat @ A @ F1 @ F22 @ Nat ) )
= ( case_nat @ B @ ( H @ F1 )
@ ^ [X4: nat] : ( H @ ( F22 @ X4 ) )
@ Nat ) ) ).
% nat.case_distrib
thf(fact_246_old_Onat_Osimps_I5_J,axiom,
! [A: $tType,F1: A,F22: nat > A,X2: nat] :
( ( case_nat @ A @ F1 @ F22 @ ( suc @ X2 ) )
= ( F22 @ X2 ) ) ).
% old.nat.simps(5)
thf(fact_247_old_Onat_Osimps_I4_J,axiom,
! [A: $tType,F1: A,F22: nat > A] :
( ( case_nat @ A @ F1 @ F22 @ ( zero_zero @ nat ) )
= F1 ) ).
% old.nat.simps(4)
thf(fact_248_Nitpick_Ocase__nat__unfold,axiom,
! [A: $tType] :
( ( case_nat @ A )
= ( ^ [X4: A,F3: nat > A,N: nat] :
( if @ A
@ ( N
= ( zero_zero @ nat ) )
@ X4
@ ( F3 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ).
% Nitpick.case_nat_unfold
thf(fact_249_nat_Osplit__sels_I1_J,axiom,
! [A: $tType,P: A > $o,F1: A,F22: nat > A,Nat: nat] :
( ( P @ ( case_nat @ A @ F1 @ F22 @ Nat ) )
= ( ( ( Nat
= ( zero_zero @ nat ) )
=> ( P @ F1 ) )
& ( ( Nat
= ( suc @ ( pred @ Nat ) ) )
=> ( P @ ( F22 @ ( pred @ Nat ) ) ) ) ) ) ).
% nat.split_sels(1)
thf(fact_250_nat_Odisc__eq__case_I2_J,axiom,
! [Nat: nat] :
( ( Nat
!= ( zero_zero @ nat ) )
= ( case_nat @ $o @ $false
@ ^ [Uu: nat] : $true
@ Nat ) ) ).
% nat.disc_eq_case(2)
thf(fact_251_nat_Odisc__eq__case_I1_J,axiom,
! [Nat: nat] :
( ( Nat
= ( zero_zero @ nat ) )
= ( case_nat @ $o @ $true
@ ^ [Uu: nat] : $false
@ Nat ) ) ).
% nat.disc_eq_case(1)
%----Type constructors (46)
thf(tcon_HOL_Obool___Finite__Set_Ofinite,axiom,
finite_finite @ $o @ ( type2 @ $o ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_1,axiom,
! [A6: $tType] :
( ( finite_finite @ A6 @ ( type2 @ A6 ) )
=> ( finite_finite @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ) ).
thf(tcon_fun___Finite__Set_Ofinite_2,axiom,
! [A6: $tType,A7: $tType] :
( ( ( finite_finite @ A6 @ ( type2 @ A6 ) )
& ( finite_finite @ A7 @ ( type2 @ A7 ) ) )
=> ( finite_finite @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder__bot,axiom,
! [A6: $tType,A7: $tType] :
( ( order_bot @ A7 @ ( type2 @ A7 ) )
=> ( order_bot @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Countable_Ocountable,axiom,
! [A6: $tType,A7: $tType] :
( ( ( finite_finite @ A6 @ ( type2 @ A6 ) )
& ( countable @ A7 @ ( type2 @ A7 ) ) )
=> ( countable @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A6: $tType,A7: $tType] :
( ( preorder @ A7 @ ( type2 @ A7 ) )
=> ( preorder @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A6: $tType,A7: $tType] :
( ( order @ A7 @ ( type2 @ A7 ) )
=> ( order @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A6: $tType,A7: $tType] :
( ( ord @ A7 @ ( type2 @ A7 ) )
=> ( ord @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Orderings_Obot,axiom,
! [A6: $tType,A7: $tType] :
( ( bot @ A7 @ ( type2 @ A7 ) )
=> ( bot @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_fun___Groups_Ominus,axiom,
! [A6: $tType,A7: $tType] :
( ( minus @ A7 @ ( type2 @ A7 ) )
=> ( minus @ ( A6 > A7 ) @ ( type2 @ ( A6 > A7 ) ) ) ) ).
thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
canoni770627133id_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring,axiom,
linord1659791738miring @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add,axiom,
cancel1352612707id_add @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Olinordered__semidom,axiom,
linordered_semidom @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__diff,axiom,
comm_monoid_diff @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Ozero__less__one,axiom,
zero_less_one @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
wellorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oorder__bot_3,axiom,
order_bot @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Countable_Ocountable_4,axiom,
countable @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Rings_Ozero__neq__one,axiom,
zero_neq_one @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Opreorder_5,axiom,
preorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
linorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Ono__top,axiom,
no_top @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oorder_6,axiom,
order @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oord_7,axiom,
ord @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Obot_8,axiom,
bot @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ominus_9,axiom,
minus @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Ozero,axiom,
zero @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Groups_Oone,axiom,
one @ nat @ ( type2 @ nat ) ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_10,axiom,
! [A6: $tType] : ( order_bot @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Countable_Ocountable_11,axiom,
! [A6: $tType] :
( ( finite_finite @ A6 @ ( type2 @ A6 ) )
=> ( countable @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_12,axiom,
! [A6: $tType] : ( preorder @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_13,axiom,
! [A6: $tType] : ( order @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_14,axiom,
! [A6: $tType] : ( ord @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_15,axiom,
! [A6: $tType] : ( bot @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_Set_Oset___Groups_Ominus_16,axiom,
! [A6: $tType] : ( minus @ ( set @ A6 ) @ ( type2 @ ( set @ A6 ) ) ) ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_17,axiom,
order_bot @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Countable_Ocountable_18,axiom,
countable @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_19,axiom,
preorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder_20,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_21,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_22,axiom,
ord @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Obot_23,axiom,
bot @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Groups_Ominus_24,axiom,
minus @ $o @ ( type2 @ $o ) ).
thf(tcon_List_Olist___Countable_Ocountable_25,axiom,
! [A6: $tType] :
( ( countable @ A6 @ ( type2 @ A6 ) )
=> ( countable @ ( list @ A6 ) @ ( type2 @ ( list @ A6 ) ) ) ) ).
%----Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X: A,Y3: A] :
( ( if @ A @ $false @ X @ Y3 )
= Y3 ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X: A,Y3: A] :
( ( if @ A @ $true @ X @ Y3 )
= X ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
( member @ rule @ r
@ ( sset @ rule
@ ( shift @ rule @ rsa
@ ( flat @ rule
@ ( smap @ nat @ ( list @ rule )
@ ^ [N: nat] : ( stake @ rule @ N @ rules )
@ ( siterate @ nat @ suc @ na ) ) ) ) ) ) ).
%------------------------------------------------------------------------------