TPTP Problem File: ITP197^2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP197^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Type problem prob_386__3259180_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 : Type/prob_386__3259180_1 [Des21]
% Status : Theorem
% Rating : 0.00 v7.5.0
% Syntax : Number of formulae : 359 ( 148 unt; 60 typ; 0 def)
% Number of atoms : 713 ( 333 equ; 0 cnn)
% Maximal formula atoms : 11 ( 2 avg)
% Number of connectives : 3178 ( 107 ~; 22 |; 57 &;2646 @)
% ( 0 <=>; 346 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 7 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 261 ( 261 >; 0 *; 0 +; 0 <<)
% Number of symbols : 60 ( 57 usr; 4 con; 0-5 aty)
% Number of variables : 1051 ( 73 ^; 905 !; 31 ?;1051 :)
% ( 42 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:19:19.244
%------------------------------------------------------------------------------
% Could-be-implicit typings (5)
thf(ty_t_Type__Mirabelle__sfxnzmapfi_Otype__scheme,type,
type_M1610475166scheme: $tType ).
thf(ty_t_Type__Mirabelle__sfxnzmapfi_Otyp,type,
type_Mirabelle_typ: $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 (55)
thf(sy_cl_Lattices_Obounded__lattice,type,
bounded_lattice:
!>[A: $tType] : $o ).
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Obot,type,
bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ono__bot,type,
no_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ono__top,type,
no_top:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder__bot,type,
order_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Owellorder,type,
wellorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Odense__order,type,
dense_order:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Osemilattice__sup,type,
semilattice_sup:
!>[A: $tType] : $o ).
thf(sy_cl_Type__Mirabelle__sfxnzmapfi_Oof__nat,type,
type_M201404320of_nat:
!>[A: $tType] : $o ).
thf(sy_cl_Type__Mirabelle__sfxnzmapfi_Otyp__of,type,
type_M1535146424typ_of:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Obounded__semilattice__sup__bot,type,
bounde1808546759up_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Type__Mirabelle__sfxnzmapfi_Otype__struct,type,
type_M1470978233struct:
!>[A: $tType] : $o ).
thf(sy_c_HOL_OThe,type,
the:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Lattices_Osemilattice__neutr,type,
semilattice_neutr:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Lattices_Osup__class_Osup,type,
sup_sup:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_List_Oinsert,type,
insert:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist_OCons,type,
cons:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist_ONil,type,
nil:
!>[A: $tType] : ( list @ A ) ).
thf(sy_c_List_Oord__class_Olexordp__eq,type,
ord_lexordp_eq:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
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_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_Oinsert,type,
insert2:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Set_Ois__empty,type,
is_empty:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Set_Ois__singleton,type,
is_singleton:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Set_Othe__elem,type,
the_elem:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Type__Mirabelle__sfxnzmapfi_Ocod,type,
type_Mirabelle_cod: ( nat > type_Mirabelle_typ ) > ( set @ nat ) ).
thf(sy_c_Type__Mirabelle__sfxnzmapfi_Odom,type,
type_Mirabelle_dom: ( nat > type_Mirabelle_typ ) > ( set @ nat ) ).
thf(sy_c_Type__Mirabelle__sfxnzmapfi_Oid__subst,type,
type_M17369686_subst: nat > type_Mirabelle_typ ).
thf(sy_c_Type__Mirabelle__sfxnzmapfi_Omk__scheme,type,
type_M540877363scheme: type_Mirabelle_typ > type_M1610475166scheme ).
thf(sy_c_Type__Mirabelle__sfxnzmapfi_Onew__tv,type,
type_M1046766514new_tv:
!>[A: $tType] : ( nat > A > $o ) ).
thf(sy_c_Type__Mirabelle__sfxnzmapfi_Oof__nat__class_Oof__nat,type,
type_M167561660of_nat:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_Type__Mirabelle__sfxnzmapfi_Otyp_OFun,type,
type_Mirabelle_Fun: type_Mirabelle_typ > type_Mirabelle_typ > type_Mirabelle_typ ).
thf(sy_c_Type__Mirabelle__sfxnzmapfi_Otyp_OTVar,type,
type_Mirabelle_TVar: nat > type_Mirabelle_typ ).
thf(sy_c_Type__Mirabelle__sfxnzmapfi_Otyp_Ocase__typ,type,
type_M1695911984se_typ:
!>[A: $tType] : ( ( nat > A ) > ( type_Mirabelle_typ > type_Mirabelle_typ > A ) > type_Mirabelle_typ > A ) ).
thf(sy_c_Type__Mirabelle__sfxnzmapfi_Otyp__of__class_Otyp__of,type,
type_M604437996typ_of:
!>[A: $tType] : ( A > type_Mirabelle_typ ) ).
thf(sy_c_Type__Mirabelle__sfxnzmapfi_Otype__scheme_OBVar,type,
type_M1264524676e_BVar: nat > type_M1610475166scheme ).
thf(sy_c_Type__Mirabelle__sfxnzmapfi_Otype__scheme_OFVar,type,
type_M91906176e_FVar: nat > type_M1610475166scheme ).
thf(sy_c_Type__Mirabelle__sfxnzmapfi_Otype__scheme_OSFun,type,
type_M1518733067e_SFun: type_M1610475166scheme > type_M1610475166scheme > type_M1610475166scheme ).
thf(sy_c_Type__Mirabelle__sfxnzmapfi_Otype__scheme_Ocase__type__scheme,type,
type_M697345518scheme:
!>[A: $tType] : ( ( nat > A ) > ( nat > A ) > ( type_M1610475166scheme > type_M1610475166scheme > A ) > type_M1610475166scheme > A ) ).
thf(sy_c_Type__Mirabelle__sfxnzmapfi_Otype__scheme_Orec__type__scheme,type,
type_M1705333808scheme:
!>[A: $tType] : ( ( nat > A ) > ( nat > A ) > ( type_M1610475166scheme > type_M1610475166scheme > A > A > A ) > type_M1610475166scheme > A ) ).
thf(sy_c_Type__Mirabelle__sfxnzmapfi_Otype__struct__class_Oapp__subst,type,
type_M353778473_subst:
!>[A: $tType] : ( ( nat > type_Mirabelle_typ ) > A > A ) ).
thf(sy_c_Type__Mirabelle__sfxnzmapfi_Otype__struct__class_Obound__tv,type,
type_M779184887und_tv:
!>[A: $tType] : ( A > ( set @ nat ) ) ).
thf(sy_c_Type__Mirabelle__sfxnzmapfi_Otype__struct__class_Ofree__tv,type,
type_M950795831ree_tv:
!>[A: $tType] : ( A > ( set @ nat ) ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_S1,type,
s1: nat > type_Mirabelle_typ ).
thf(sy_v_S2,type,
s2: nat > type_Mirabelle_typ ).
thf(sy_v_n,type,
n: nat ).
thf(sy_v_x,type,
x: nat ).
% Relevant facts (255)
thf(fact_0_type__scheme_Oinject_I1_J,axiom,
! [X1: nat,Y1: nat] :
( ( ( type_M91906176e_FVar @ X1 )
= ( type_M91906176e_FVar @ Y1 ) )
= ( X1 = Y1 ) ) ).
% type_scheme.inject(1)
thf(fact_1_eq__free__eq__subst__type__scheme,axiom,
! [Sch: type_M1610475166scheme,S1: nat > type_Mirabelle_typ,S2: nat > type_Mirabelle_typ] :
( ! [N: nat] :
( ( member @ nat @ N @ ( type_M950795831ree_tv @ type_M1610475166scheme @ Sch ) )
=> ( ( S1 @ N )
= ( S2 @ N ) ) )
=> ( ( type_M353778473_subst @ type_M1610475166scheme @ S1 @ Sch )
= ( type_M353778473_subst @ type_M1610475166scheme @ S2 @ Sch ) ) ) ).
% eq_free_eq_subst_type_scheme
thf(fact_2_scheme__substitutions__only__on__free__variables,axiom,
! [Sch: type_M1610475166scheme,S: nat > type_Mirabelle_typ,S3: nat > type_Mirabelle_typ] :
( ! [X: nat] :
( ( member @ nat @ X @ ( type_M950795831ree_tv @ type_M1610475166scheme @ Sch ) )
=> ( ( S @ X )
= ( S3 @ X ) ) )
=> ( ( type_M353778473_subst @ type_M1610475166scheme @ S @ Sch )
= ( type_M353778473_subst @ type_M1610475166scheme @ S3 @ Sch ) ) ) ).
% scheme_substitutions_only_on_free_variables
thf(fact_3_app__subst__type__scheme_Osimps_I1_J,axiom,
! [S: nat > type_Mirabelle_typ,N2: nat] :
( ( type_M353778473_subst @ type_M1610475166scheme @ S @ ( type_M91906176e_FVar @ N2 ) )
= ( type_M540877363scheme @ ( S @ N2 ) ) ) ).
% app_subst_type_scheme.simps(1)
thf(fact_4_type__scheme_Osimps_I13_J,axiom,
! [A: $tType,F1: nat > A,F2: nat > A,F3: type_M1610475166scheme > type_M1610475166scheme > A > A > A,X1: nat] :
( ( type_M1705333808scheme @ A @ F1 @ F2 @ F3 @ ( type_M91906176e_FVar @ X1 ) )
= ( F1 @ X1 ) ) ).
% type_scheme.simps(13)
thf(fact_5_type__scheme_Osimps_I10_J,axiom,
! [A: $tType,F1: nat > A,F2: nat > A,F3: type_M1610475166scheme > type_M1610475166scheme > A,X1: nat] :
( ( type_M697345518scheme @ A @ F1 @ F2 @ F3 @ ( type_M91906176e_FVar @ X1 ) )
= ( F1 @ X1 ) ) ).
% type_scheme.simps(10)
thf(fact_6_app__subst__type__scheme_Osimps_I2_J,axiom,
! [S: nat > type_Mirabelle_typ,N2: nat] :
( ( type_M353778473_subst @ type_M1610475166scheme @ S @ ( type_M1264524676e_BVar @ N2 ) )
= ( type_M1264524676e_BVar @ N2 ) ) ).
% app_subst_type_scheme.simps(2)
thf(fact_7_app__subst__type__scheme_Osimps_I3_J,axiom,
! [S: nat > type_Mirabelle_typ,Sch1: type_M1610475166scheme,Sch2: type_M1610475166scheme] :
( ( type_M353778473_subst @ type_M1610475166scheme @ S @ ( type_M1518733067e_SFun @ Sch1 @ Sch2 ) )
= ( type_M1518733067e_SFun @ ( type_M353778473_subst @ type_M1610475166scheme @ S @ Sch1 ) @ ( type_M353778473_subst @ type_M1610475166scheme @ S @ Sch2 ) ) ) ).
% app_subst_type_scheme.simps(3)
thf(fact_8_type__scheme_Odistinct_I1_J,axiom,
! [X1: nat,X2: nat] :
( ( type_M91906176e_FVar @ X1 )
!= ( type_M1264524676e_BVar @ X2 ) ) ).
% type_scheme.distinct(1)
thf(fact_9_type__scheme_Odistinct_I3_J,axiom,
! [X1: nat,X31: type_M1610475166scheme,X32: type_M1610475166scheme] :
( ( type_M91906176e_FVar @ X1 )
!= ( type_M1518733067e_SFun @ X31 @ X32 ) ) ).
% type_scheme.distinct(3)
thf(fact_10_app__subst__Cons,axiom,
! [A: $tType] :
( ( type_M1470978233struct @ A )
=> ! [S: nat > type_Mirabelle_typ,X3: A,L: list @ A] :
( ( type_M353778473_subst @ ( list @ A ) @ S @ ( cons @ A @ X3 @ L ) )
= ( cons @ A @ ( type_M353778473_subst @ A @ S @ X3 ) @ ( type_M353778473_subst @ ( list @ A ) @ S @ L ) ) ) ) ).
% app_subst_Cons
thf(fact_11_free__tv__mk__scheme,axiom,
! [T: type_Mirabelle_typ] :
( ( type_M950795831ree_tv @ type_M1610475166scheme @ ( type_M540877363scheme @ T ) )
= ( type_M950795831ree_tv @ type_Mirabelle_typ @ T ) ) ).
% free_tv_mk_scheme
thf(fact_12_free__tv__type__scheme_Osimps_I1_J,axiom,
! [M: nat] :
( ( type_M950795831ree_tv @ type_M1610475166scheme @ ( type_M91906176e_FVar @ M ) )
= ( insert2 @ nat @ M @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% free_tv_type_scheme.simps(1)
thf(fact_13_new__if__subst__type__scheme,axiom,
! [N2: nat,Sch: type_M1610475166scheme,S: nat > type_Mirabelle_typ,S3: nat > type_Mirabelle_typ] :
( ( type_M1046766514new_tv @ type_M1610475166scheme @ N2 @ Sch )
=> ( ( type_M353778473_subst @ type_M1610475166scheme
@ ^ [K: nat] : ( if @ type_Mirabelle_typ @ ( ord_less @ nat @ K @ N2 ) @ ( S @ K ) @ ( S3 @ K ) )
@ Sch )
= ( type_M353778473_subst @ type_M1610475166scheme @ S @ Sch ) ) ) ).
% new_if_subst_type_scheme
thf(fact_14_typ_Ocase__distrib,axiom,
! [A: $tType,B: $tType,H: A > B,F1: nat > A,F2: type_Mirabelle_typ > type_Mirabelle_typ > A,Typ: type_Mirabelle_typ] :
( ( H @ ( type_M1695911984se_typ @ A @ F1 @ F2 @ Typ ) )
= ( type_M1695911984se_typ @ B
@ ^ [X4: nat] : ( H @ ( F1 @ X4 ) )
@ ^ [X12: type_Mirabelle_typ,X22: type_Mirabelle_typ] : ( H @ ( F2 @ X12 @ X22 ) )
@ Typ ) ) ).
% typ.case_distrib
thf(fact_15_type__scheme_Oinject_I3_J,axiom,
! [X31: type_M1610475166scheme,X32: type_M1610475166scheme,Y31: type_M1610475166scheme,Y32: type_M1610475166scheme] :
( ( ( type_M1518733067e_SFun @ X31 @ X32 )
= ( type_M1518733067e_SFun @ Y31 @ Y32 ) )
= ( ( X31 = Y31 )
& ( X32 = Y32 ) ) ) ).
% type_scheme.inject(3)
thf(fact_16_type__scheme_Oinject_I2_J,axiom,
! [X2: nat,Y2: nat] :
( ( ( type_M1264524676e_BVar @ X2 )
= ( type_M1264524676e_BVar @ Y2 ) )
= ( X2 = Y2 ) ) ).
% type_scheme.inject(2)
thf(fact_17_new__tv__Cons,axiom,
! [A: $tType] :
( ( type_M1470978233struct @ A )
=> ! [N2: nat,X3: A,L: list @ A] :
( ( type_M1046766514new_tv @ ( list @ A ) @ N2 @ ( cons @ A @ X3 @ L ) )
= ( ( type_M1046766514new_tv @ A @ N2 @ X3 )
& ( type_M1046766514new_tv @ ( list @ A ) @ N2 @ L ) ) ) ) ).
% new_tv_Cons
thf(fact_18_subst__mk__scheme,axiom,
! [S: nat > type_Mirabelle_typ,T: type_Mirabelle_typ] :
( ( type_M353778473_subst @ type_M1610475166scheme @ S @ ( type_M540877363scheme @ T ) )
= ( type_M540877363scheme @ ( type_M353778473_subst @ type_Mirabelle_typ @ S @ T ) ) ) ).
% subst_mk_scheme
thf(fact_19_new__tv__Fun2,axiom,
! [N2: nat,T1: type_M1610475166scheme,T2: type_M1610475166scheme] :
( ( type_M1046766514new_tv @ type_M1610475166scheme @ N2 @ ( type_M1518733067e_SFun @ T1 @ T2 ) )
= ( ( type_M1046766514new_tv @ type_M1610475166scheme @ N2 @ T1 )
& ( type_M1046766514new_tv @ type_M1610475166scheme @ N2 @ T2 ) ) ) ).
% new_tv_Fun2
thf(fact_20_new__tv__BVar,axiom,
! [N2: nat,M: nat] : ( type_M1046766514new_tv @ type_M1610475166scheme @ N2 @ ( type_M1264524676e_BVar @ M ) ) ).
% new_tv_BVar
thf(fact_21_new__tv__FVar,axiom,
! [N2: nat,M: nat] :
( ( type_M1046766514new_tv @ type_M1610475166scheme @ N2 @ ( type_M91906176e_FVar @ M ) )
= ( ord_less @ nat @ M @ N2 ) ) ).
% new_tv_FVar
thf(fact_22_type__scheme_Osimps_I15_J,axiom,
! [A: $tType,F1: nat > A,F2: nat > A,F3: type_M1610475166scheme > type_M1610475166scheme > A > A > A,X31: type_M1610475166scheme,X32: type_M1610475166scheme] :
( ( type_M1705333808scheme @ A @ F1 @ F2 @ F3 @ ( type_M1518733067e_SFun @ X31 @ X32 ) )
= ( F3 @ X31 @ X32 @ ( type_M1705333808scheme @ A @ F1 @ F2 @ F3 @ X31 ) @ ( type_M1705333808scheme @ A @ F1 @ F2 @ F3 @ X32 ) ) ) ).
% type_scheme.simps(15)
thf(fact_23_type__scheme_Osimps_I14_J,axiom,
! [A: $tType,F1: nat > A,F2: nat > A,F3: type_M1610475166scheme > type_M1610475166scheme > A > A > A,X2: nat] :
( ( type_M1705333808scheme @ A @ F1 @ F2 @ F3 @ ( type_M1264524676e_BVar @ X2 ) )
= ( F2 @ X2 ) ) ).
% type_scheme.simps(14)
thf(fact_24_type__scheme_Osimps_I12_J,axiom,
! [A: $tType,F1: nat > A,F2: nat > A,F3: type_M1610475166scheme > type_M1610475166scheme > A,X31: type_M1610475166scheme,X32: type_M1610475166scheme] :
( ( type_M697345518scheme @ A @ F1 @ F2 @ F3 @ ( type_M1518733067e_SFun @ X31 @ X32 ) )
= ( F3 @ X31 @ X32 ) ) ).
% type_scheme.simps(12)
thf(fact_25_type__scheme_Osimps_I11_J,axiom,
! [A: $tType,F1: nat > A,F2: nat > A,F3: type_M1610475166scheme > type_M1610475166scheme > A,X2: nat] :
( ( type_M697345518scheme @ A @ F1 @ F2 @ F3 @ ( type_M1264524676e_BVar @ X2 ) )
= ( F2 @ X2 ) ) ).
% type_scheme.simps(11)
thf(fact_26_type__scheme_Odistinct_I5_J,axiom,
! [X2: nat,X31: type_M1610475166scheme,X32: type_M1610475166scheme] :
( ( type_M1264524676e_BVar @ X2 )
!= ( type_M1518733067e_SFun @ X31 @ X32 ) ) ).
% type_scheme.distinct(5)
thf(fact_27_mk__scheme__Fun,axiom,
! [T: type_Mirabelle_typ,Sch1: type_M1610475166scheme,Sch2: type_M1610475166scheme] :
( ( ( type_M540877363scheme @ T )
= ( type_M1518733067e_SFun @ Sch1 @ Sch2 ) )
=> ? [T12: type_Mirabelle_typ,T22: type_Mirabelle_typ] :
( ( Sch1
= ( type_M540877363scheme @ T12 ) )
& ( Sch2
= ( type_M540877363scheme @ T22 ) ) ) ) ).
% mk_scheme_Fun
thf(fact_28_eq__free__eq__subst__te,axiom,
! [T: type_Mirabelle_typ,S1: nat > type_Mirabelle_typ,S2: nat > type_Mirabelle_typ] :
( ! [N: nat] :
( ( member @ nat @ N @ ( type_M950795831ree_tv @ type_Mirabelle_typ @ T ) )
=> ( ( S1 @ N )
= ( S2 @ N ) ) )
=> ( ( type_M353778473_subst @ type_Mirabelle_typ @ S1 @ T )
= ( type_M353778473_subst @ type_Mirabelle_typ @ S2 @ T ) ) ) ).
% eq_free_eq_subst_te
thf(fact_29_eq__subst__te__eq__free,axiom,
! [S1: nat > type_Mirabelle_typ,T: type_Mirabelle_typ,S2: nat > type_Mirabelle_typ,N2: nat] :
( ( ( type_M353778473_subst @ type_Mirabelle_typ @ S1 @ T )
= ( type_M353778473_subst @ type_Mirabelle_typ @ S2 @ T ) )
=> ( ( member @ nat @ N2 @ ( type_M950795831ree_tv @ type_Mirabelle_typ @ T ) )
=> ( ( S1 @ N2 )
= ( S2 @ N2 ) ) ) ) ).
% eq_subst_te_eq_free
thf(fact_30_typ__substitutions__only__on__free__variables,axiom,
! [T: type_Mirabelle_typ,S: nat > type_Mirabelle_typ,S3: nat > type_Mirabelle_typ] :
( ! [X: nat] :
( ( member @ nat @ X @ ( type_M950795831ree_tv @ type_Mirabelle_typ @ T ) )
=> ( ( S @ X )
= ( S3 @ X ) ) )
=> ( ( type_M353778473_subst @ type_Mirabelle_typ @ S @ T )
= ( type_M353778473_subst @ type_Mirabelle_typ @ S3 @ T ) ) ) ).
% typ_substitutions_only_on_free_variables
thf(fact_31_mk__scheme__injective,axiom,
! [T: type_Mirabelle_typ,T3: type_Mirabelle_typ] :
( ( ( type_M540877363scheme @ T )
= ( type_M540877363scheme @ T3 ) )
=> ( T = T3 ) ) ).
% mk_scheme_injective
thf(fact_32_free__tv__type__scheme_Osimps_I2_J,axiom,
! [M: nat] :
( ( type_M950795831ree_tv @ type_M1610475166scheme @ ( type_M1264524676e_BVar @ M ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% free_tv_type_scheme.simps(2)
thf(fact_33_type__scheme_Oexhaust,axiom,
! [Y: type_M1610475166scheme] :
( ! [X13: nat] :
( Y
!= ( type_M91906176e_FVar @ X13 ) )
=> ( ! [X23: nat] :
( Y
!= ( type_M1264524676e_BVar @ X23 ) )
=> ~ ! [X312: type_M1610475166scheme,X322: type_M1610475166scheme] :
( Y
!= ( type_M1518733067e_SFun @ X312 @ X322 ) ) ) ) ).
% type_scheme.exhaust
thf(fact_34_type__scheme_Oinduct,axiom,
! [P: type_M1610475166scheme > $o,Type_scheme: type_M1610475166scheme] :
( ! [X: nat] : ( P @ ( type_M91906176e_FVar @ X ) )
=> ( ! [X: nat] : ( P @ ( type_M1264524676e_BVar @ X ) )
=> ( ! [X1a: type_M1610475166scheme,X2a: type_M1610475166scheme] :
( ( P @ X1a )
=> ( ( P @ X2a )
=> ( P @ ( type_M1518733067e_SFun @ X1a @ X2a ) ) ) )
=> ( P @ Type_scheme ) ) ) ) ).
% type_scheme.induct
thf(fact_35_new__tv__def,axiom,
! [A: $tType] :
( ( type_M1470978233struct @ A )
=> ( ( type_M1046766514new_tv @ A )
= ( ^ [N3: nat,Ts: A] :
! [M2: nat] :
( ( member @ nat @ M2 @ ( type_M950795831ree_tv @ A @ Ts ) )
=> ( ord_less @ nat @ M2 @ N3 ) ) ) ) ) ).
% new_tv_def
thf(fact_36_type__scheme_Ocase__distrib,axiom,
! [A: $tType,B: $tType,H: A > B,F1: nat > A,F2: nat > A,F3: type_M1610475166scheme > type_M1610475166scheme > A,Type_scheme: type_M1610475166scheme] :
( ( H @ ( type_M697345518scheme @ A @ F1 @ F2 @ F3 @ Type_scheme ) )
= ( type_M697345518scheme @ B
@ ^ [X4: nat] : ( H @ ( F1 @ X4 ) )
@ ^ [X4: nat] : ( H @ ( F2 @ X4 ) )
@ ^ [X12: type_M1610475166scheme,X22: type_M1610475166scheme] : ( H @ ( F3 @ X12 @ X22 ) )
@ Type_scheme ) ) ).
% type_scheme.case_distrib
thf(fact_37_singleton__conv,axiom,
! [A: $tType,A2: A] :
( ( collect @ A
@ ^ [X4: A] : X4 = A2 )
= ( insert2 @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singleton_conv
thf(fact_38_singleton__conv2,axiom,
! [A: $tType,A2: A] :
( ( collect @ A
@ ( ^ [Y3: A,Z: A] : Y3 = Z
@ A2 ) )
= ( insert2 @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singleton_conv2
thf(fact_39_singletonI,axiom,
! [A: $tType,A2: A] : ( member @ A @ A2 @ ( insert2 @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singletonI
thf(fact_40_insertCI,axiom,
! [A: $tType,A2: A,B2: set @ A,B3: A] :
( ( ~ ( member @ A @ A2 @ B2 )
=> ( A2 = B3 ) )
=> ( member @ A @ A2 @ ( insert2 @ A @ B3 @ B2 ) ) ) ).
% insertCI
thf(fact_41_insert__iff,axiom,
! [A: $tType,A2: A,B3: A,A3: set @ A] :
( ( member @ A @ A2 @ ( insert2 @ A @ B3 @ A3 ) )
= ( ( A2 = B3 )
| ( member @ A @ A2 @ A3 ) ) ) ).
% insert_iff
thf(fact_42_insert__absorb2,axiom,
! [A: $tType,X3: A,A3: set @ A] :
( ( insert2 @ A @ X3 @ ( insert2 @ A @ X3 @ A3 ) )
= ( insert2 @ A @ X3 @ A3 ) ) ).
% insert_absorb2
thf(fact_43_list_Oinject,axiom,
! [A: $tType,X21: A,X222: list @ A,Y21: A,Y22: list @ A] :
( ( ( cons @ A @ X21 @ X222 )
= ( cons @ A @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X222 = Y22 ) ) ) ).
% list.inject
thf(fact_44_empty__iff,axiom,
! [A: $tType,C: A] :
~ ( member @ A @ C @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_45_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_46_Collect__mem__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( collect @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_47_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X: A] :
( ( P @ X )
= ( Q @ X ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_48_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X: A] :
( ( F @ X )
= ( G @ X ) )
=> ( F = G ) ) ).
% ext
thf(fact_49_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_50_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_51_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_52_new__if__subst__type__scheme__list,axiom,
! [N2: nat,A3: list @ type_M1610475166scheme,S: nat > type_Mirabelle_typ,S3: nat > type_Mirabelle_typ] :
( ( type_M1046766514new_tv @ ( list @ type_M1610475166scheme ) @ N2 @ A3 )
=> ( ( type_M353778473_subst @ ( list @ type_M1610475166scheme )
@ ^ [K: nat] : ( if @ type_Mirabelle_typ @ ( ord_less @ nat @ K @ N2 ) @ ( S @ K ) @ ( S3 @ K ) )
@ A3 )
= ( type_M353778473_subst @ ( list @ type_M1610475166scheme ) @ S @ A3 ) ) ) ).
% new_if_subst_type_scheme_list
thf(fact_53_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_54_equals0I,axiom,
! [A: $tType,A3: set @ A] :
( ! [Y4: A] :
~ ( member @ A @ Y4 @ A3 )
=> ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_55_equals0D,axiom,
! [A: $tType,A3: set @ A,A2: A] :
( ( A3
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A2 @ A3 ) ) ).
% equals0D
thf(fact_56_emptyE,axiom,
! [A: $tType,A2: A] :
~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_57_not__Cons__self2,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( cons @ A @ X3 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_58_mk__disjoint__insert,axiom,
! [A: $tType,A2: A,A3: set @ A] :
( ( member @ A @ A2 @ A3 )
=> ? [B4: set @ A] :
( ( A3
= ( insert2 @ A @ A2 @ B4 ) )
& ~ ( member @ A @ A2 @ B4 ) ) ) ).
% mk_disjoint_insert
thf(fact_59_insert__commute,axiom,
! [A: $tType,X3: A,Y: A,A3: set @ A] :
( ( insert2 @ A @ X3 @ ( insert2 @ A @ Y @ A3 ) )
= ( insert2 @ A @ Y @ ( insert2 @ A @ X3 @ A3 ) ) ) ).
% insert_commute
thf(fact_60_insert__eq__iff,axiom,
! [A: $tType,A2: A,A3: set @ A,B3: A,B2: set @ A] :
( ~ ( member @ A @ A2 @ A3 )
=> ( ~ ( member @ A @ B3 @ B2 )
=> ( ( ( insert2 @ A @ A2 @ A3 )
= ( insert2 @ A @ B3 @ B2 ) )
= ( ( ( A2 = B3 )
=> ( A3 = B2 ) )
& ( ( A2 != B3 )
=> ? [C2: set @ A] :
( ( A3
= ( insert2 @ A @ B3 @ C2 ) )
& ~ ( member @ A @ B3 @ C2 )
& ( B2
= ( insert2 @ A @ A2 @ C2 ) )
& ~ ( member @ A @ A2 @ C2 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_61_insert__absorb,axiom,
! [A: $tType,A2: A,A3: set @ A] :
( ( member @ A @ A2 @ A3 )
=> ( ( insert2 @ A @ A2 @ A3 )
= A3 ) ) ).
% insert_absorb
thf(fact_62_insert__ident,axiom,
! [A: $tType,X3: A,A3: set @ A,B2: set @ A] :
( ~ ( member @ A @ X3 @ A3 )
=> ( ~ ( member @ A @ X3 @ B2 )
=> ( ( ( insert2 @ A @ X3 @ A3 )
= ( insert2 @ A @ X3 @ B2 ) )
= ( A3 = B2 ) ) ) ) ).
% insert_ident
thf(fact_63_Set_Oset__insert,axiom,
! [A: $tType,X3: A,A3: set @ A] :
( ( member @ A @ X3 @ A3 )
=> ~ ! [B4: set @ A] :
( ( A3
= ( insert2 @ A @ X3 @ B4 ) )
=> ( member @ A @ X3 @ B4 ) ) ) ).
% Set.set_insert
thf(fact_64_insertI2,axiom,
! [A: $tType,A2: A,B2: set @ A,B3: A] :
( ( member @ A @ A2 @ B2 )
=> ( member @ A @ A2 @ ( insert2 @ A @ B3 @ B2 ) ) ) ).
% insertI2
thf(fact_65_insertI1,axiom,
! [A: $tType,A2: A,B2: set @ A] : ( member @ A @ A2 @ ( insert2 @ A @ A2 @ B2 ) ) ).
% insertI1
thf(fact_66_insertE,axiom,
! [A: $tType,A2: A,B3: A,A3: set @ A] :
( ( member @ A @ A2 @ ( insert2 @ A @ B3 @ A3 ) )
=> ( ( A2 != B3 )
=> ( member @ A @ A2 @ A3 ) ) ) ).
% insertE
thf(fact_67_empty__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A
@ ^ [X4: A] : $false ) ) ).
% empty_def
thf(fact_68_insert__Collect,axiom,
! [A: $tType,A2: A,P: A > $o] :
( ( insert2 @ A @ A2 @ ( collect @ A @ P ) )
= ( collect @ A
@ ^ [U: A] :
( ( U != A2 )
=> ( P @ U ) ) ) ) ).
% insert_Collect
thf(fact_69_insert__compr,axiom,
! [A: $tType] :
( ( insert2 @ A )
= ( ^ [A4: A,B5: set @ A] :
( collect @ A
@ ^ [X4: A] :
( ( X4 = A4 )
| ( member @ A @ X4 @ B5 ) ) ) ) ) ).
% insert_compr
thf(fact_70_singleton__inject,axiom,
! [A: $tType,A2: A,B3: A] :
( ( ( insert2 @ A @ A2 @ ( bot_bot @ ( set @ A ) ) )
= ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( A2 = B3 ) ) ).
% singleton_inject
thf(fact_71_insert__not__empty,axiom,
! [A: $tType,A2: A,A3: set @ A] :
( ( insert2 @ A @ A2 @ A3 )
!= ( bot_bot @ ( set @ A ) ) ) ).
% insert_not_empty
thf(fact_72_doubleton__eq__iff,axiom,
! [A: $tType,A2: A,B3: A,C: A,D: A] :
( ( ( insert2 @ A @ A2 @ ( insert2 @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert2 @ A @ C @ ( insert2 @ A @ D @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ( ( A2 = C )
& ( B3 = D ) )
| ( ( A2 = D )
& ( B3 = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_73_singleton__iff,axiom,
! [A: $tType,B3: A,A2: A] :
( ( member @ A @ B3 @ ( insert2 @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( B3 = A2 ) ) ).
% singleton_iff
thf(fact_74_singletonD,axiom,
! [A: $tType,B3: A,A2: A] :
( ( member @ A @ B3 @ ( insert2 @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( B3 = A2 ) ) ).
% singletonD
thf(fact_75_Collect__conv__if2,axiom,
! [A: $tType,P: A > $o,A2: A] :
( ( ( P @ A2 )
=> ( ( collect @ A
@ ^ [X4: A] :
( ( A2 = X4 )
& ( P @ X4 ) ) )
= ( insert2 @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect @ A
@ ^ [X4: A] :
( ( A2 = X4 )
& ( P @ X4 ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_conv_if2
thf(fact_76_Collect__conv__if,axiom,
! [A: $tType,P: A > $o,A2: A] :
( ( ( P @ A2 )
=> ( ( collect @ A
@ ^ [X4: A] :
( ( X4 = A2 )
& ( P @ X4 ) ) )
= ( insert2 @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( P @ A2 )
=> ( ( collect @ A
@ ^ [X4: A] :
( ( X4 = A2 )
& ( P @ X4 ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_conv_if
thf(fact_77_the__elem__eq,axiom,
! [A: $tType,X3: A] :
( ( the_elem @ A @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= X3 ) ).
% the_elem_eq
thf(fact_78_bot__apply,axiom,
! [C3: $tType,D2: $tType] :
( ( bot @ C3 )
=> ( ( bot_bot @ ( D2 > C3 ) )
= ( ^ [X4: D2] : ( bot_bot @ C3 ) ) ) ) ).
% bot_apply
thf(fact_79_bound__tv__type__scheme_Osimps_I2_J,axiom,
! [M: nat] :
( ( type_M779184887und_tv @ type_M1610475166scheme @ ( type_M1264524676e_BVar @ M ) )
= ( insert2 @ nat @ M @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% bound_tv_type_scheme.simps(2)
thf(fact_80_is__singletonI,axiom,
! [A: $tType,X3: A] : ( is_singleton @ A @ ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% is_singletonI
thf(fact_81_bot_Onot__eq__extremum,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A2: A] :
( ( A2
!= ( bot_bot @ A ) )
= ( ord_less @ A @ ( bot_bot @ A ) @ A2 ) ) ) ).
% bot.not_eq_extremum
thf(fact_82_bot_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ ( bot_bot @ A ) ) ) ).
% bot.extremum_strict
thf(fact_83_free__tv__TVar,axiom,
! [M: nat] :
( ( type_M950795831ree_tv @ type_Mirabelle_typ @ ( type_Mirabelle_TVar @ M ) )
= ( insert2 @ nat @ M @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% free_tv_TVar
thf(fact_84_bound__tv__type__scheme_Osimps_I1_J,axiom,
! [M: nat] :
( ( type_M779184887und_tv @ type_M1610475166scheme @ ( type_M91906176e_FVar @ M ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% bound_tv_type_scheme.simps(1)
thf(fact_85_typ_Oinject_I1_J,axiom,
! [X1: nat,Y1: nat] :
( ( ( type_Mirabelle_TVar @ X1 )
= ( type_Mirabelle_TVar @ Y1 ) )
= ( X1 = Y1 ) ) ).
% typ.inject(1)
thf(fact_86_new__tv__TVar,axiom,
! [N2: nat,M: nat] :
( ( type_M1046766514new_tv @ type_Mirabelle_typ @ N2 @ ( type_Mirabelle_TVar @ M ) )
= ( ord_less @ nat @ M @ N2 ) ) ).
% new_tv_TVar
thf(fact_87_new__tv__TVar__subst,axiom,
! [N2: nat] : ( type_M1046766514new_tv @ ( nat > type_Mirabelle_typ ) @ N2 @ type_Mirabelle_TVar ) ).
% new_tv_TVar_subst
thf(fact_88_eq__free__eq__subst__scheme__list,axiom,
! [A3: list @ type_M1610475166scheme,S1: nat > type_Mirabelle_typ,S2: nat > type_Mirabelle_typ] :
( ! [N: nat] :
( ( member @ nat @ N @ ( type_M950795831ree_tv @ ( list @ type_M1610475166scheme ) @ A3 ) )
=> ( ( S1 @ N )
= ( S2 @ N ) ) )
=> ( ( type_M353778473_subst @ ( list @ type_M1610475166scheme ) @ S1 @ A3 )
= ( type_M353778473_subst @ ( list @ type_M1610475166scheme ) @ S2 @ A3 ) ) ) ).
% eq_free_eq_subst_scheme_list
thf(fact_89_scheme__list__substitutions__only__on__free__variables,axiom,
! [A3: list @ type_M1610475166scheme,S: nat > type_Mirabelle_typ,S3: nat > type_Mirabelle_typ] :
( ! [X: nat] :
( ( member @ nat @ X @ ( type_M950795831ree_tv @ ( list @ type_M1610475166scheme ) @ A3 ) )
=> ( ( S @ X )
= ( S3 @ X ) ) )
=> ( ( type_M353778473_subst @ ( list @ type_M1610475166scheme ) @ S @ A3 )
= ( type_M353778473_subst @ ( list @ type_M1610475166scheme ) @ S3 @ A3 ) ) ) ).
% scheme_list_substitutions_only_on_free_variables
thf(fact_90_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_91_not__psubset__empty,axiom,
! [A: $tType,A3: set @ A] :
~ ( ord_less @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) ) ).
% not_psubset_empty
thf(fact_92_app__subst__TVar,axiom,
! [S: nat > type_Mirabelle_typ,N2: nat] :
( ( type_M353778473_subst @ type_Mirabelle_typ @ S @ ( type_Mirabelle_TVar @ N2 ) )
= ( S @ N2 ) ) ).
% app_subst_TVar
thf(fact_93_is__singleton__the__elem,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A5: set @ A] :
( A5
= ( insert2 @ A @ ( the_elem @ A @ A5 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_the_elem
thf(fact_94_typ_Osimps_I5_J,axiom,
! [A: $tType,F1: nat > A,F2: type_Mirabelle_typ > type_Mirabelle_typ > A,X1: nat] :
( ( type_M1695911984se_typ @ A @ F1 @ F2 @ ( type_Mirabelle_TVar @ X1 ) )
= ( F1 @ X1 ) ) ).
% typ.simps(5)
thf(fact_95_is__singletonI_H,axiom,
! [A: $tType,A3: set @ A] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X: A,Y4: A] :
( ( member @ A @ X @ A3 )
=> ( ( member @ A @ Y4 @ A3 )
=> ( X = Y4 ) ) )
=> ( is_singleton @ A @ A3 ) ) ) ).
% is_singletonI'
thf(fact_96_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,F: B > A,B3: B,C: B] :
( ( A2
= ( F @ B3 ) )
=> ( ( ord_less @ B @ B3 @ C )
=> ( ! [X: B,Y4: B] :
( ( ord_less @ B @ X @ Y4 )
=> ( ord_less @ A @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_97_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,B3: A,F: A > B,C: B] :
( ( ord_less @ A @ A2 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X: A,Y4: A] :
( ( ord_less @ A @ X @ Y4 )
=> ( ord_less @ B @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_98_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F: B > A,B3: B,C: B] :
( ( ord_less @ A @ A2 @ ( F @ B3 ) )
=> ( ( ord_less @ B @ B3 @ C )
=> ( ! [X: B,Y4: B] :
( ( ord_less @ B @ X @ Y4 )
=> ( ord_less @ A @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_99_order__less__subst2,axiom,
! [A: $tType,C3: $tType] :
( ( ( order @ C3 )
& ( order @ A ) )
=> ! [A2: A,B3: A,F: A > C3,C: C3] :
( ( ord_less @ A @ A2 @ B3 )
=> ( ( ord_less @ C3 @ ( F @ B3 ) @ C )
=> ( ! [X: A,Y4: A] :
( ( ord_less @ A @ X @ Y4 )
=> ( ord_less @ C3 @ ( F @ X ) @ ( F @ Y4 ) ) )
=> ( ord_less @ C3 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_less_subst2
thf(fact_100_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [X3: A] :
? [Y4: A] : ( ord_less @ A @ Y4 @ X3 ) ) ).
% lt_ex
thf(fact_101_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [X3: A] :
? [X_1: A] : ( ord_less @ A @ X3 @ X_1 ) ) ).
% gt_ex
thf(fact_102_neqE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A] :
( ( X3 != Y )
=> ( ~ ( ord_less @ A @ X3 @ Y )
=> ( ord_less @ A @ Y @ X3 ) ) ) ) ).
% neqE
thf(fact_103_neq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A] :
( ( X3 != Y )
= ( ( ord_less @ A @ X3 @ Y )
| ( ord_less @ A @ Y @ X3 ) ) ) ) ).
% neq_iff
thf(fact_104_order_Oasym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B3: A] :
( ( ord_less @ A @ A2 @ B3 )
=> ~ ( ord_less @ A @ B3 @ A2 ) ) ) ).
% order.asym
thf(fact_105_dense,axiom,
! [A: $tType] :
( ( dense_order @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ? [Z2: A] :
( ( ord_less @ A @ X3 @ Z2 )
& ( ord_less @ A @ Z2 @ Y ) ) ) ) ).
% dense
thf(fact_106_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( X3 != Y ) ) ) ).
% less_imp_neq
thf(fact_107_less__asym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ~ ( ord_less @ A @ Y @ X3 ) ) ) ).
% less_asym
thf(fact_108_less__asym_H,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B3: A] :
( ( ord_less @ A @ A2 @ B3 )
=> ~ ( ord_less @ A @ B3 @ A2 ) ) ) ).
% less_asym'
thf(fact_109_less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y: A,Z3: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ( ord_less @ A @ Y @ Z3 )
=> ( ord_less @ A @ X3 @ Z3 ) ) ) ) ).
% less_trans
thf(fact_110_less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
| ( X3 = Y )
| ( ord_less @ A @ Y @ X3 ) ) ) ).
% less_linear
thf(fact_111_less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A] :
~ ( ord_less @ A @ X3 @ X3 ) ) ).
% less_irrefl
thf(fact_112_ord__eq__less__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B3: A,C: A] :
( ( A2 = B3 )
=> ( ( ord_less @ A @ B3 @ C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% ord_eq_less_trans
thf(fact_113_ord__less__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B3: A,C: A] :
( ( ord_less @ A @ A2 @ B3 )
=> ( ( B3 = C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% ord_less_eq_trans
thf(fact_114_dual__order_Oasym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A2: A] :
( ( ord_less @ A @ B3 @ A2 )
=> ~ ( ord_less @ A @ A2 @ B3 ) ) ) ).
% dual_order.asym
thf(fact_115_less__imp__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( X3 != Y ) ) ) ).
% less_imp_not_eq
thf(fact_116_less__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ~ ( ord_less @ A @ Y @ X3 ) ) ) ).
% less_not_sym
thf(fact_117_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,A2: A] :
( ! [X: A] :
( ! [Y5: A] :
( ( ord_less @ A @ Y5 @ X )
=> ( P @ Y5 ) )
=> ( P @ X ) )
=> ( P @ A2 ) ) ) ).
% less_induct
thf(fact_118_antisym__conv3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X3: A] :
( ~ ( ord_less @ A @ Y @ X3 )
=> ( ( ~ ( ord_less @ A @ X3 @ Y ) )
= ( X3 = Y ) ) ) ) ).
% antisym_conv3
thf(fact_119_less__imp__not__eq2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ( Y != X3 ) ) ) ).
% less_imp_not_eq2
thf(fact_120_less__imp__triv,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y: A,P: $o] :
( ( ord_less @ A @ X3 @ Y )
=> ( ( ord_less @ A @ Y @ X3 )
=> P ) ) ) ).
% less_imp_triv
thf(fact_121_linorder__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A] :
( ~ ( ord_less @ A @ X3 @ Y )
=> ( ( X3 != Y )
=> ( ord_less @ A @ Y @ X3 ) ) ) ) ).
% linorder_cases
thf(fact_122_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ A2 ) ) ).
% dual_order.irrefl
thf(fact_123_order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B3: A,C: A] :
( ( ord_less @ A @ A2 @ B3 )
=> ( ( ord_less @ A @ B3 @ C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% order.strict_trans
thf(fact_124_less__imp__not__less,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X3: A,Y: A] :
( ( ord_less @ A @ X3 @ Y )
=> ~ ( ord_less @ A @ Y @ X3 ) ) ) ).
% less_imp_not_less
thf(fact_125_exists__least__iff,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ( ( ^ [P2: A > $o] :
? [X5: A] : ( P2 @ X5 ) )
= ( ^ [P3: A > $o] :
? [N3: A] :
( ( P3 @ N3 )
& ! [M2: A] :
( ( ord_less @ A @ M2 @ N3 )
=> ~ ( P3 @ M2 ) ) ) ) ) ) ).
% exists_least_iff
thf(fact_126_linorder__less__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A2: A,B3: A] :
( ! [A6: A,B6: A] :
( ( ord_less @ A @ A6 @ B6 )
=> ( P @ A6 @ B6 ) )
=> ( ! [A6: A] : ( P @ A6 @ A6 )
=> ( ! [A6: A,B6: A] :
( ( P @ B6 @ A6 )
=> ( P @ A6 @ B6 ) )
=> ( P @ A2 @ B3 ) ) ) ) ) ).
% linorder_less_wlog
thf(fact_127_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A2: A,C: A] :
( ( ord_less @ A @ B3 @ A2 )
=> ( ( ord_less @ A @ C @ B3 )
=> ( ord_less @ A @ C @ A2 ) ) ) ) ).
% dual_order.strict_trans
thf(fact_128_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: A,Y: A] :
( ( ~ ( ord_less @ A @ X3 @ Y ) )
= ( ( ord_less @ A @ Y @ X3 )
| ( X3 = Y ) ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_129_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B3: A] :
( ( ord_less @ A @ A2 @ B3 )
=> ( A2 != B3 ) ) ) ).
% order.strict_implies_not_eq
thf(fact_130_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B3: A,A2: A] :
( ( ord_less @ A @ B3 @ A2 )
=> ( A2 != B3 ) ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_131_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X4: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_132_mk__scheme_Osimps_I1_J,axiom,
! [N2: nat] :
( ( type_M540877363scheme @ ( type_Mirabelle_TVar @ N2 ) )
= ( type_M91906176e_FVar @ N2 ) ) ).
% mk_scheme.simps(1)
thf(fact_133_is__singletonE,axiom,
! [A: $tType,A3: set @ A] :
( ( is_singleton @ A @ A3 )
=> ~ ! [X: A] :
( A3
!= ( insert2 @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% is_singletonE
thf(fact_134_is__singleton__def,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A5: set @ A] :
? [X4: A] :
( A5
= ( insert2 @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_def
thf(fact_135_free__tv__id__subst,axiom,
( ( type_M950795831ree_tv @ ( nat > type_Mirabelle_typ ) @ type_M17369686_subst )
= ( bot_bot @ ( set @ nat ) ) ) ).
% free_tv_id_subst
thf(fact_136_Set_Ois__empty__def,axiom,
! [A: $tType] :
( ( is_empty @ A )
= ( ^ [A5: set @ A] :
( A5
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Set.is_empty_def
thf(fact_137_the__elem__def,axiom,
! [A: $tType] :
( ( the_elem @ A )
= ( ^ [X6: set @ A] :
( the @ A
@ ^ [X4: A] :
( X6
= ( insert2 @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% the_elem_def
thf(fact_138_mk__scheme_Osimps_I2_J,axiom,
! [T1: type_Mirabelle_typ,T2: type_Mirabelle_typ] :
( ( type_M540877363scheme @ ( type_Mirabelle_Fun @ T1 @ T2 ) )
= ( type_M1518733067e_SFun @ ( type_M540877363scheme @ T1 ) @ ( type_M540877363scheme @ T2 ) ) ) ).
% mk_scheme.simps(2)
thf(fact_139_Type__Mirabelle__sfxnzmapfi_Odom__def,axiom,
( type_Mirabelle_dom
= ( ^ [S4: nat > type_Mirabelle_typ] :
( collect @ nat
@ ^ [N3: nat] :
( ( S4 @ N3 )
!= ( type_Mirabelle_TVar @ N3 ) ) ) ) ) ).
% Type_Mirabelle_sfxnzmapfi.dom_def
thf(fact_140_id__subst__def,axiom,
type_M17369686_subst = type_Mirabelle_TVar ).
% id_subst_def
thf(fact_141_typ_Oinject_I2_J,axiom,
! [X21: type_Mirabelle_typ,X222: type_Mirabelle_typ,Y21: type_Mirabelle_typ,Y22: type_Mirabelle_typ] :
( ( ( type_Mirabelle_Fun @ X21 @ X222 )
= ( type_Mirabelle_Fun @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X222 = Y22 ) ) ) ).
% typ.inject(2)
thf(fact_142_new__tv__Fun,axiom,
! [N2: nat,T1: type_Mirabelle_typ,T2: type_Mirabelle_typ] :
( ( type_M1046766514new_tv @ type_Mirabelle_typ @ N2 @ ( type_Mirabelle_Fun @ T1 @ T2 ) )
= ( ( type_M1046766514new_tv @ type_Mirabelle_typ @ N2 @ T1 )
& ( type_M1046766514new_tv @ type_Mirabelle_typ @ N2 @ T2 ) ) ) ).
% new_tv_Fun
thf(fact_143_dom__id__subst,axiom,
( ( type_Mirabelle_dom @ type_M17369686_subst )
= ( bot_bot @ ( set @ nat ) ) ) ).
% dom_id_subst
thf(fact_144_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_145_less__set__def,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A5: set @ A,B5: set @ A] :
( ord_less @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ A5 )
@ ^ [X4: A] : ( member @ A @ X4 @ B5 ) ) ) ) ).
% less_set_def
thf(fact_146_psubset__trans,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,C4: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ B2 )
=> ( ( ord_less @ ( set @ A ) @ B2 @ C4 )
=> ( ord_less @ ( set @ A ) @ A3 @ C4 ) ) ) ).
% psubset_trans
thf(fact_147_new__tv__id__subst,axiom,
! [N2: nat] : ( type_M1046766514new_tv @ ( nat > type_Mirabelle_typ ) @ N2 @ type_M17369686_subst ) ).
% new_tv_id_subst
thf(fact_148_app__subst__Fun,axiom,
! [S: nat > type_Mirabelle_typ,T1: type_Mirabelle_typ,T2: type_Mirabelle_typ] :
( ( type_M353778473_subst @ type_Mirabelle_typ @ S @ ( type_Mirabelle_Fun @ T1 @ T2 ) )
= ( type_Mirabelle_Fun @ ( type_M353778473_subst @ type_Mirabelle_typ @ S @ T1 ) @ ( type_M353778473_subst @ type_Mirabelle_typ @ S @ T2 ) ) ) ).
% app_subst_Fun
thf(fact_149_typ_Oexhaust,axiom,
! [Y: type_Mirabelle_typ] :
( ! [X13: nat] :
( Y
!= ( type_Mirabelle_TVar @ X13 ) )
=> ~ ! [X212: type_Mirabelle_typ,X223: type_Mirabelle_typ] :
( Y
!= ( type_Mirabelle_Fun @ X212 @ X223 ) ) ) ).
% typ.exhaust
thf(fact_150_typ_Oinduct,axiom,
! [P: type_Mirabelle_typ > $o,Typ: type_Mirabelle_typ] :
( ! [X: nat] : ( P @ ( type_Mirabelle_TVar @ X ) )
=> ( ! [X1a: type_Mirabelle_typ,X23: type_Mirabelle_typ] :
( ( P @ X1a )
=> ( ( P @ X23 )
=> ( P @ ( type_Mirabelle_Fun @ X1a @ X23 ) ) ) )
=> ( P @ Typ ) ) ) ).
% typ.induct
thf(fact_151_typ_Odistinct_I1_J,axiom,
! [X1: nat,X21: type_Mirabelle_typ,X222: type_Mirabelle_typ] :
( ( type_Mirabelle_TVar @ X1 )
!= ( type_Mirabelle_Fun @ X21 @ X222 ) ) ).
% typ.distinct(1)
thf(fact_152_typ_Osimps_I6_J,axiom,
! [A: $tType,F1: nat > A,F2: type_Mirabelle_typ > type_Mirabelle_typ > A,X21: type_Mirabelle_typ,X222: type_Mirabelle_typ] :
( ( type_M1695911984se_typ @ A @ F1 @ F2 @ ( type_Mirabelle_Fun @ X21 @ X222 ) )
= ( F2 @ X21 @ X222 ) ) ).
% typ.simps(6)
thf(fact_153_the__sym__eq__trivial,axiom,
! [A: $tType,X3: A] :
( ( the @ A
@ ( ^ [Y3: A,Z: A] : Y3 = Z
@ X3 ) )
= X3 ) ).
% the_sym_eq_trivial
thf(fact_154_the__eq__trivial,axiom,
! [A: $tType,A2: A] :
( ( the @ A
@ ^ [X4: A] : X4 = A2 )
= A2 ) ).
% the_eq_trivial
thf(fact_155_the__equality,axiom,
! [A: $tType,P: A > $o,A2: A] :
( ( P @ A2 )
=> ( ! [X: A] :
( ( P @ X )
=> ( X = A2 ) )
=> ( ( the @ A @ P )
= A2 ) ) ) ).
% the_equality
thf(fact_156_cod__id__subst,axiom,
( ( type_Mirabelle_cod @ type_M17369686_subst )
= ( bot_bot @ ( set @ nat ) ) ) ).
% cod_id_subst
thf(fact_157_theI,axiom,
! [A: $tType,P: A > $o,A2: A] :
( ( P @ A2 )
=> ( ! [X: A] :
( ( P @ X )
=> ( X = A2 ) )
=> ( P @ ( the @ A @ P ) ) ) ) ).
% theI
thf(fact_158_theI_H,axiom,
! [A: $tType,P: A > $o] :
( ? [X7: A] :
( ( P @ X7 )
& ! [Y4: A] :
( ( P @ Y4 )
=> ( Y4 = X7 ) ) )
=> ( P @ ( the @ A @ P ) ) ) ).
% theI'
thf(fact_159_theI2,axiom,
! [A: $tType,P: A > $o,A2: A,Q: A > $o] :
( ( P @ A2 )
=> ( ! [X: A] :
( ( P @ X )
=> ( X = A2 ) )
=> ( ! [X: A] :
( ( P @ X )
=> ( Q @ X ) )
=> ( Q @ ( the @ A @ P ) ) ) ) ) ).
% theI2
thf(fact_160_If__def,axiom,
! [A: $tType] :
( ( if @ A )
= ( ^ [P3: $o,X4: A,Y6: A] :
( the @ A
@ ^ [Z4: A] :
( ( P3
=> ( Z4 = X4 ) )
& ( ~ P3
=> ( Z4 = Y6 ) ) ) ) ) ) ).
% If_def
thf(fact_161_the1I2,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ? [X7: A] :
( ( P @ X7 )
& ! [Y4: A] :
( ( P @ Y4 )
=> ( Y4 = X7 ) ) )
=> ( ! [X: A] :
( ( P @ X )
=> ( Q @ X ) )
=> ( Q @ ( the @ A @ P ) ) ) ) ).
% the1I2
thf(fact_162_the1__equality,axiom,
! [A: $tType,P: A > $o,A2: A] :
( ? [X7: A] :
( ( P @ X7 )
& ! [Y4: A] :
( ( P @ Y4 )
=> ( Y4 = X7 ) ) )
=> ( ( P @ A2 )
=> ( ( the @ A @ P )
= A2 ) ) ) ).
% the1_equality
thf(fact_163_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_164_bot__empty__eq,axiom,
! [A: $tType] :
( ( bot_bot @ ( A > $o ) )
= ( ^ [X4: A] : ( member @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% bot_empty_eq
thf(fact_165_free__tv__subst,axiom,
( ( type_M950795831ree_tv @ ( nat > type_Mirabelle_typ ) )
= ( ^ [S4: nat > type_Mirabelle_typ] : ( sup_sup @ ( set @ nat ) @ ( type_Mirabelle_dom @ S4 ) @ ( type_Mirabelle_cod @ S4 ) ) ) ) ).
% free_tv_subst
thf(fact_166_UnCI,axiom,
! [A: $tType,C: A,B2: set @ A,A3: set @ A] :
( ( ~ ( member @ A @ C @ B2 )
=> ( member @ A @ C @ A3 ) )
=> ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) ) ) ).
% UnCI
thf(fact_167_Un__iff,axiom,
! [A: $tType,C: A,A3: set @ A,B2: set @ A] :
( ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) )
= ( ( member @ A @ C @ A3 )
| ( member @ A @ C @ B2 ) ) ) ).
% Un_iff
thf(fact_168_Un__empty,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( ( sup_sup @ ( set @ A ) @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ( A3
= ( bot_bot @ ( set @ A ) ) )
& ( B2
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Un_empty
thf(fact_169_Un__insert__left,axiom,
! [A: $tType,A2: A,B2: set @ A,C4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( insert2 @ A @ A2 @ B2 ) @ C4 )
= ( insert2 @ A @ A2 @ ( sup_sup @ ( set @ A ) @ B2 @ C4 ) ) ) ).
% Un_insert_left
thf(fact_170_Un__insert__right,axiom,
! [A: $tType,A3: set @ A,A2: A,B2: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( insert2 @ A @ A2 @ B2 ) )
= ( insert2 @ A @ A2 @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) ) ) ).
% Un_insert_right
thf(fact_171_weaken__asm__Un,axiom,
! [A: $tType,A3: set @ A,P: A > $o,Q: $o,B2: set @ A] :
( ( ! [X7: A] :
( ( member @ A @ X7 @ A3 )
=> ( P @ X7 ) )
=> Q )
=> ( ! [X: A] :
( ( member @ A @ X @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) )
=> ( P @ X ) )
=> Q ) ) ).
% weaken_asm_Un
thf(fact_172_UnE,axiom,
! [A: $tType,C: A,A3: set @ A,B2: set @ A] :
( ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) )
=> ( ~ ( member @ A @ C @ A3 )
=> ( member @ A @ C @ B2 ) ) ) ).
% UnE
thf(fact_173_UnI1,axiom,
! [A: $tType,C: A,A3: set @ A,B2: set @ A] :
( ( member @ A @ C @ A3 )
=> ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) ) ) ).
% UnI1
thf(fact_174_UnI2,axiom,
! [A: $tType,C: A,B2: set @ A,A3: set @ A] :
( ( member @ A @ C @ B2 )
=> ( member @ A @ C @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) ) ) ).
% UnI2
thf(fact_175_Un__def,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A5: set @ A,B5: set @ A] :
( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A5 )
| ( member @ A @ X4 @ B5 ) ) ) ) ) ).
% Un_def
thf(fact_176_bex__Un,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,P: A > $o] :
( ( ? [X4: A] :
( ( member @ A @ X4 @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) )
& ( P @ X4 ) ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ A3 )
& ( P @ X4 ) )
| ? [X4: A] :
( ( member @ A @ X4 @ B2 )
& ( P @ X4 ) ) ) ) ).
% bex_Un
thf(fact_177_ball__Un,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,P: A > $o] :
( ( ! [X4: A] :
( ( member @ A @ X4 @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) )
=> ( P @ X4 ) ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A3 )
=> ( P @ X4 ) )
& ! [X4: A] :
( ( member @ A @ X4 @ B2 )
=> ( P @ X4 ) ) ) ) ).
% ball_Un
thf(fact_178_Un__assoc,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,C4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) @ C4 )
= ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B2 @ C4 ) ) ) ).
% Un_assoc
thf(fact_179_Un__absorb,axiom,
! [A: $tType,A3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ A3 )
= A3 ) ).
% Un_absorb
thf(fact_180_Un__commute,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A5: set @ A,B5: set @ A] : ( sup_sup @ ( set @ A ) @ B5 @ A5 ) ) ) ).
% Un_commute
thf(fact_181_Un__left__absorb,axiom,
! [A: $tType,A3: set @ A,B2: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ A3 @ B2 ) )
= ( sup_sup @ ( set @ A ) @ A3 @ B2 ) ) ).
% Un_left_absorb
thf(fact_182_Collect__disj__eq,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( collect @ A
@ ^ [X4: A] :
( ( P @ X4 )
| ( Q @ X4 ) ) )
= ( sup_sup @ ( set @ A ) @ ( collect @ A @ P ) @ ( collect @ A @ Q ) ) ) ).
% Collect_disj_eq
thf(fact_183_Un__left__commute,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,C4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B2 @ C4 ) )
= ( sup_sup @ ( set @ A ) @ B2 @ ( sup_sup @ ( set @ A ) @ A3 @ C4 ) ) ) ).
% Un_left_commute
thf(fact_184_insert__def,axiom,
! [A: $tType] :
( ( insert2 @ A )
= ( ^ [A4: A] :
( sup_sup @ ( set @ A )
@ ( collect @ A
@ ^ [X4: A] : X4 = A4 ) ) ) ) ).
% insert_def
thf(fact_185_Un__empty__left,axiom,
! [A: $tType,B2: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B2 )
= B2 ) ).
% Un_empty_left
thf(fact_186_Un__empty__right,axiom,
! [A: $tType,A3: set @ A] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
= A3 ) ).
% Un_empty_right
thf(fact_187_singleton__Un__iff,axiom,
! [A: $tType,X3: A,A3: set @ A,B2: set @ A] :
( ( ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) )
= ( sup_sup @ ( set @ A ) @ A3 @ B2 ) )
= ( ( ( A3
= ( bot_bot @ ( set @ A ) ) )
& ( B2
= ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) )
| ( ( A3
= ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
& ( B2
= ( bot_bot @ ( set @ A ) ) ) )
| ( ( A3
= ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
& ( B2
= ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_188_Un__singleton__iff,axiom,
! [A: $tType,A3: set @ A,B2: set @ A,X3: A] :
( ( ( sup_sup @ ( set @ A ) @ A3 @ B2 )
= ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( ( A3
= ( bot_bot @ ( set @ A ) ) )
& ( B2
= ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) )
| ( ( A3
= ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
& ( B2
= ( bot_bot @ ( set @ A ) ) ) )
| ( ( A3
= ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) )
& ( B2
= ( insert2 @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_189_insert__is__Un,axiom,
! [A: $tType] :
( ( insert2 @ A )
= ( ^ [A4: A] : ( sup_sup @ ( set @ A ) @ ( insert2 @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% insert_is_Un
thf(fact_190_bound__tv__list_Osimps_I2_J,axiom,
! [A: $tType] :
( ( type_M1470978233struct @ A )
=> ! [X3: A,L: list @ A] :
( ( type_M779184887und_tv @ ( list @ A ) @ ( cons @ A @ X3 @ L ) )
= ( sup_sup @ ( set @ nat ) @ ( type_M779184887und_tv @ A @ X3 ) @ ( type_M779184887und_tv @ ( list @ A ) @ L ) ) ) ) ).
% bound_tv_list.simps(2)
thf(fact_191_free__tv__Fun,axiom,
! [T1: type_Mirabelle_typ,T2: type_Mirabelle_typ] :
( ( type_M950795831ree_tv @ type_Mirabelle_typ @ ( type_Mirabelle_Fun @ T1 @ T2 ) )
= ( sup_sup @ ( set @ nat ) @ ( type_M950795831ree_tv @ type_Mirabelle_typ @ T1 ) @ ( type_M950795831ree_tv @ type_Mirabelle_typ @ T2 ) ) ) ).
% free_tv_Fun
thf(fact_192_free__tv__type__scheme_Osimps_I3_J,axiom,
! [S1: type_M1610475166scheme,S2: type_M1610475166scheme] :
( ( type_M950795831ree_tv @ type_M1610475166scheme @ ( type_M1518733067e_SFun @ S1 @ S2 ) )
= ( sup_sup @ ( set @ nat ) @ ( type_M950795831ree_tv @ type_M1610475166scheme @ S1 ) @ ( type_M950795831ree_tv @ type_M1610475166scheme @ S2 ) ) ) ).
% free_tv_type_scheme.simps(3)
thf(fact_193_bound__tv__type__scheme_Osimps_I3_J,axiom,
! [S1: type_M1610475166scheme,S2: type_M1610475166scheme] :
( ( type_M779184887und_tv @ type_M1610475166scheme @ ( type_M1518733067e_SFun @ S1 @ S2 ) )
= ( sup_sup @ ( set @ nat ) @ ( type_M779184887und_tv @ type_M1610475166scheme @ S1 ) @ ( type_M779184887und_tv @ type_M1610475166scheme @ S2 ) ) ) ).
% bound_tv_type_scheme.simps(3)
thf(fact_194_free__tv__list_Osimps_I2_J,axiom,
! [A: $tType] :
( ( type_M1470978233struct @ A )
=> ! [X3: A,L: list @ A] :
( ( type_M950795831ree_tv @ ( list @ A ) @ ( cons @ A @ X3 @ L ) )
= ( sup_sup @ ( set @ nat ) @ ( type_M950795831ree_tv @ A @ X3 ) @ ( type_M950795831ree_tv @ ( list @ A ) @ L ) ) ) ) ).
% free_tv_list.simps(2)
thf(fact_195_free__tv__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ( type_M201404320of_nat @ A )
& ( type_M1535146424typ_of @ B ) )
=> ( ( type_M950795831ree_tv @ ( A > B ) )
= ( ^ [F4: A > B] :
( sup_sup @ ( set @ nat )
@ ( type_Mirabelle_dom
@ ^ [N3: nat] : ( type_M604437996typ_of @ B @ ( F4 @ ( type_M167561660of_nat @ A @ N3 ) ) ) )
@ ( type_Mirabelle_cod
@ ^ [N3: nat] : ( type_M604437996typ_of @ B @ ( F4 @ ( type_M167561660of_nat @ A @ N3 ) ) ) ) ) ) ) ) ).
% free_tv_fun_def
thf(fact_196_sup__bot__left,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ! [X3: A] :
( ( sup_sup @ A @ ( bot_bot @ A ) @ X3 )
= X3 ) ) ).
% sup_bot_left
thf(fact_197_sup__bot__right,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ! [X3: A] :
( ( sup_sup @ A @ X3 @ ( bot_bot @ A ) )
= X3 ) ) ).
% sup_bot_right
thf(fact_198_sup__bot_Oright__neutral,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ! [A2: A] :
( ( sup_sup @ A @ A2 @ ( bot_bot @ A ) )
= A2 ) ) ).
% sup_bot.right_neutral
thf(fact_199_sup__bot_Oneutr__eq__iff,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ! [A2: A,B3: A] :
( ( ( bot_bot @ A )
= ( sup_sup @ A @ A2 @ B3 ) )
= ( ( A2
= ( bot_bot @ A ) )
& ( B3
= ( bot_bot @ A ) ) ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_200_sup__bot_Oleft__neutral,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ! [A2: A] :
( ( sup_sup @ A @ ( bot_bot @ A ) @ A2 )
= A2 ) ) ).
% sup_bot.left_neutral
thf(fact_201_sup__bot_Oeq__neutr__iff,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ! [A2: A,B3: A] :
( ( ( sup_sup @ A @ A2 @ B3 )
= ( bot_bot @ A ) )
= ( ( A2
= ( bot_bot @ A ) )
& ( B3
= ( bot_bot @ A ) ) ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_202_sup__eq__bot__iff,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ! [X3: A,Y: A] :
( ( ( sup_sup @ A @ X3 @ Y )
= ( bot_bot @ A ) )
= ( ( X3
= ( bot_bot @ A ) )
& ( Y
= ( bot_bot @ A ) ) ) ) ) ).
% sup_eq_bot_iff
thf(fact_203_bot__eq__sup__iff,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ! [X3: A,Y: A] :
( ( ( bot_bot @ A )
= ( sup_sup @ A @ X3 @ Y ) )
= ( ( X3
= ( bot_bot @ A ) )
& ( Y
= ( bot_bot @ A ) ) ) ) ) ).
% bot_eq_sup_iff
thf(fact_204_sup__Un__eq,axiom,
! [A: $tType,R: set @ A,S: set @ A] :
( ( sup_sup @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ R )
@ ^ [X4: A] : ( member @ A @ X4 @ S ) )
= ( ^ [X4: A] : ( member @ A @ X4 @ ( sup_sup @ ( set @ A ) @ R @ S ) ) ) ) ).
% sup_Un_eq
thf(fact_205_sup__set__def,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A5: set @ A,B5: set @ A] :
( collect @ A
@ ( sup_sup @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ A5 )
@ ^ [X4: A] : ( member @ A @ X4 @ B5 ) ) ) ) ) ).
% sup_set_def
thf(fact_206_less__supI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X3: A,A2: A,B3: A] :
( ( ord_less @ A @ X3 @ A2 )
=> ( ord_less @ A @ X3 @ ( sup_sup @ A @ A2 @ B3 ) ) ) ) ).
% less_supI1
thf(fact_207_less__supI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X3: A,B3: A,A2: A] :
( ( ord_less @ A @ X3 @ B3 )
=> ( ord_less @ A @ X3 @ ( sup_sup @ A @ A2 @ B3 ) ) ) ) ).
% less_supI2
thf(fact_208_sup_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B3: A,C: A,A2: A] :
( ( ord_less @ A @ ( sup_sup @ A @ B3 @ C ) @ A2 )
=> ~ ( ( ord_less @ A @ B3 @ A2 )
=> ~ ( ord_less @ A @ C @ A2 ) ) ) ) ).
% sup.strict_boundedE
thf(fact_209_sup_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less @ A )
= ( ^ [B7: A,A4: A] :
( ( A4
= ( sup_sup @ A @ A4 @ B7 ) )
& ( A4 != B7 ) ) ) ) ) ).
% sup.strict_order_iff
thf(fact_210_sup_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C: A,A2: A,B3: A] :
( ( ord_less @ A @ C @ A2 )
=> ( ord_less @ A @ C @ ( sup_sup @ A @ A2 @ B3 ) ) ) ) ).
% sup.strict_coboundedI1
thf(fact_211_sup_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C: A,B3: A,A2: A] :
( ( ord_less @ A @ C @ B3 )
=> ( ord_less @ A @ C @ ( sup_sup @ A @ A2 @ B3 ) ) ) ) ).
% sup.strict_coboundedI2
thf(fact_212_Nitpick_OThe__psimp,axiom,
! [A: $tType,P: A > $o,X3: A] :
( ( P
= ( ^ [Y3: A,Z: A] : Y3 = Z
@ X3 ) )
=> ( ( the @ A @ P )
= X3 ) ) ).
% Nitpick.The_psimp
thf(fact_213_theI__unique,axiom,
! [A: $tType,P: A > $o,X3: A] :
( ? [X7: A] :
( ( P @ X7 )
& ! [Y4: A] :
( ( P @ Y4 )
=> ( Y4 = X7 ) ) )
=> ( ( P @ X3 )
= ( X3
= ( the @ A @ P ) ) ) ) ).
% theI_unique
thf(fact_214_sup__bot_Osemilattice__neutr__axioms,axiom,
! [A: $tType] :
( ( bounde1808546759up_bot @ A )
=> ( semilattice_neutr @ A @ ( sup_sup @ A ) @ ( bot_bot @ A ) ) ) ).
% sup_bot.semilattice_neutr_axioms
thf(fact_215_lexordp__eq__simps_I4_J,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: A,Xs: list @ A,Y: A,Ys: list @ A] :
( ( ord_lexordp_eq @ A @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys ) )
= ( ( ord_less @ A @ X3 @ Y )
| ( ~ ( ord_less @ A @ Y @ X3 )
& ( ord_lexordp_eq @ A @ Xs @ Ys ) ) ) ) ) ).
% lexordp_eq_simps(4)
thf(fact_216_lexordp__eq_OCons__eq,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: A,Y: A,Xs: list @ A,Ys: list @ A] :
( ~ ( ord_less @ A @ X3 @ Y )
=> ( ~ ( ord_less @ A @ Y @ X3 )
=> ( ( ord_lexordp_eq @ A @ Xs @ Ys )
=> ( ord_lexordp_eq @ A @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) ) ) ) ).
% lexordp_eq.Cons_eq
thf(fact_217_lexordp__eq_OCons,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: A,Y: A,Xs: list @ A,Ys: list @ A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ord_lexordp_eq @ A @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) ) ).
% lexordp_eq.Cons
thf(fact_218_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_219_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_220_lexordp__eq__trans,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ord_lexordp_eq @ A @ Xs @ Ys )
=> ( ( ord_lexordp_eq @ A @ Ys @ Zs )
=> ( ord_lexordp_eq @ A @ Xs @ Zs ) ) ) ) ).
% lexordp_eq_trans
thf(fact_221_lexordp__eq__refl,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Xs: list @ A] : ( ord_lexordp_eq @ A @ Xs @ Xs ) ) ).
% lexordp_eq_refl
thf(fact_222_lexordp__eq_Ocases,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A1: list @ A,A22: list @ A] :
( ( ord_lexordp_eq @ A @ A1 @ A22 )
=> ( ( A1
!= ( nil @ A ) )
=> ( ! [X: A] :
( ? [Xs2: list @ A] :
( A1
= ( cons @ A @ X @ Xs2 ) )
=> ! [Y4: A] :
( ? [Ys2: list @ A] :
( A22
= ( cons @ A @ Y4 @ Ys2 ) )
=> ~ ( ord_less @ A @ X @ Y4 ) ) )
=> ~ ! [X: A,Y4: A,Xs2: list @ A] :
( ( A1
= ( cons @ A @ X @ Xs2 ) )
=> ! [Ys2: list @ A] :
( ( A22
= ( cons @ A @ Y4 @ Ys2 ) )
=> ( ~ ( ord_less @ A @ X @ Y4 )
=> ( ~ ( ord_less @ A @ Y4 @ X )
=> ~ ( ord_lexordp_eq @ A @ Xs2 @ Ys2 ) ) ) ) ) ) ) ) ) ).
% lexordp_eq.cases
thf(fact_223_lexordp__eq_Osimps,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_lexordp_eq @ A )
= ( ^ [A12: list @ A,A23: list @ A] :
( ? [Ys3: list @ A] :
( ( A12
= ( nil @ A ) )
& ( A23 = Ys3 ) )
| ? [X4: A,Y6: A,Xs3: list @ A,Ys3: list @ A] :
( ( A12
= ( cons @ A @ X4 @ Xs3 ) )
& ( A23
= ( cons @ A @ Y6 @ Ys3 ) )
& ( ord_less @ A @ X4 @ Y6 ) )
| ? [X4: A,Y6: A,Xs3: list @ A,Ys3: list @ A] :
( ( A12
= ( cons @ A @ X4 @ Xs3 ) )
& ( A23
= ( cons @ A @ Y6 @ Ys3 ) )
& ~ ( ord_less @ A @ X4 @ Y6 )
& ~ ( ord_less @ A @ Y6 @ X4 )
& ( ord_lexordp_eq @ A @ Xs3 @ Ys3 ) ) ) ) ) ) ).
% lexordp_eq.simps
thf(fact_224_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_225_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_226_app__subst__Nil,axiom,
! [A: $tType] :
( ( type_M1470978233struct @ A )
=> ! [S: nat > type_Mirabelle_typ] :
( ( type_M353778473_subst @ ( list @ A ) @ S @ ( nil @ A ) )
= ( nil @ A ) ) ) ).
% app_subst_Nil
thf(fact_227_lexordp__eq__simps_I3_J,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: A,Xs: list @ A] :
~ ( ord_lexordp_eq @ A @ ( cons @ A @ X3 @ Xs ) @ ( nil @ A ) ) ) ).
% lexordp_eq_simps(3)
thf(fact_228_lexordp__eq_ONil,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Ys: list @ A] : ( ord_lexordp_eq @ A @ ( nil @ A ) @ Ys ) ) ).
% lexordp_eq.Nil
thf(fact_229_new__tv__Nil,axiom,
! [A: $tType] :
( ( type_M1470978233struct @ A )
=> ! [N2: nat] : ( type_M1046766514new_tv @ ( list @ A ) @ N2 @ ( nil @ A ) ) ) ).
% new_tv_Nil
thf(fact_230_transpose_Ocases,axiom,
! [A: $tType,X3: list @ ( list @ A )] :
( ( X3
!= ( nil @ ( list @ A ) ) )
=> ( ! [Xss: list @ ( list @ A )] :
( X3
!= ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
=> ~ ! [X: A,Xs2: list @ A,Xss: list @ ( list @ A )] :
( X3
!= ( cons @ ( list @ A ) @ ( cons @ A @ X @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_231_list_Odistinct_I1_J,axiom,
! [A: $tType,X21: A,X222: list @ A] :
( ( nil @ A )
!= ( cons @ A @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_232_list_OdiscI,axiom,
! [A: $tType,List: list @ A,X21: A,X222: list @ A] :
( ( List
= ( cons @ A @ X21 @ X222 ) )
=> ( List
!= ( nil @ A ) ) ) ).
% list.discI
thf(fact_233_list_Oexhaust,axiom,
! [A: $tType,Y: list @ A] :
( ( Y
!= ( nil @ A ) )
=> ~ ! [X212: A,X223: list @ A] :
( Y
!= ( cons @ A @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_234_list_Oinducts,axiom,
! [A: $tType,P: ( list @ A ) > $o,List: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X13: A,X23: list @ A] :
( ( P @ X23 )
=> ( P @ ( cons @ A @ X13 @ X23 ) ) )
=> ( P @ List ) ) ) ).
% list.inducts
thf(fact_235_neq__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
= ( ? [Y6: A,Ys3: list @ A] :
( Xs
= ( cons @ A @ Y6 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_236_list__induct2_H,axiom,
! [A: $tType,B: $tType,P: ( list @ A ) > ( list @ B ) > $o,Xs: list @ A,Ys: list @ B] :
( ( P @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X: A,Xs2: list @ A] : ( P @ ( cons @ A @ X @ Xs2 ) @ ( nil @ B ) )
=> ( ! [Y4: B,Ys2: list @ B] : ( P @ ( nil @ A ) @ ( cons @ B @ Y4 @ Ys2 ) )
=> ( ! [X: A,Xs2: list @ A,Y4: B,Ys2: list @ B] :
( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ B @ Y4 @ Ys2 ) ) )
=> ( P @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_237_splice_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
( ! [X_1: list @ A] : ( P @ ( nil @ A ) @ X_1 )
=> ( ! [X: A,Xs2: list @ A,Ys2: list @ A] :
( ( P @ Ys2 @ Xs2 )
=> ( P @ ( cons @ A @ X @ Xs2 ) @ Ys2 ) )
=> ( P @ A0 @ A1 ) ) ) ).
% splice.induct
thf(fact_238_induct__list012,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X: A] : ( P @ ( cons @ A @ X @ ( nil @ A ) ) )
=> ( ! [X: A,Y4: A,Zs2: list @ A] :
( ( P @ Zs2 )
=> ( ( P @ ( cons @ A @ Y4 @ Zs2 ) )
=> ( P @ ( cons @ A @ X @ ( cons @ A @ Y4 @ Zs2 ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% induct_list012
thf(fact_239_min__list_Ocases,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: list @ A] :
( ! [X: A,Xs2: list @ A] :
( X3
!= ( cons @ A @ X @ Xs2 ) )
=> ( X3
= ( nil @ A ) ) ) ) ).
% min_list.cases
thf(fact_240_min__list_Oinduct,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [P: ( list @ A ) > $o,A0: list @ A] :
( ! [X: A,Xs2: list @ A] :
( ! [X213: A,X224: list @ A] :
( ( Xs2
= ( cons @ A @ X213 @ X224 ) )
=> ( P @ Xs2 ) )
=> ( P @ ( cons @ A @ X @ Xs2 ) ) )
=> ( ( P @ ( nil @ A ) )
=> ( P @ A0 ) ) ) ) ).
% min_list.induct
thf(fact_241_shuffles_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
( ! [X_1: list @ A] : ( P @ ( nil @ A ) @ X_1 )
=> ( ! [Xs2: list @ A] : ( P @ Xs2 @ ( nil @ A ) )
=> ( ! [X: A,Xs2: list @ A,Y4: A,Ys2: list @ A] :
( ( P @ Xs2 @ ( cons @ A @ Y4 @ Ys2 ) )
=> ( ( P @ ( cons @ A @ X @ Xs2 ) @ Ys2 )
=> ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y4 @ Ys2 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% shuffles.induct
thf(fact_242_remdups__adj_Ocases,axiom,
! [A: $tType,X3: list @ A] :
( ( X3
!= ( nil @ A ) )
=> ( ! [X: A] :
( X3
!= ( cons @ A @ X @ ( nil @ A ) ) )
=> ~ ! [X: A,Y4: A,Xs2: list @ A] :
( X3
!= ( cons @ A @ X @ ( cons @ A @ Y4 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_243_sorted__wrt_Oinduct,axiom,
! [A: $tType,P: ( A > A > $o ) > ( list @ A ) > $o,A0: A > A > $o,A1: list @ A] :
( ! [P4: A > A > $o] : ( P @ P4 @ ( nil @ A ) )
=> ( ! [P4: A > A > $o,X: A,Ys2: list @ A] :
( ( P @ P4 @ Ys2 )
=> ( P @ P4 @ ( cons @ A @ X @ Ys2 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% sorted_wrt.induct
thf(fact_244_remdups__adj_Oinduct,axiom,
! [A: $tType,P: ( list @ A ) > $o,A0: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X: A] : ( P @ ( cons @ A @ X @ ( nil @ A ) ) )
=> ( ! [X: A,Y4: A,Xs2: list @ A] :
( ( ( X = Y4 )
=> ( P @ ( cons @ A @ X @ Xs2 ) ) )
=> ( ( ( X != Y4 )
=> ( P @ ( cons @ A @ Y4 @ Xs2 ) ) )
=> ( P @ ( cons @ A @ X @ ( cons @ A @ Y4 @ Xs2 ) ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% remdups_adj.induct
thf(fact_245_arg__min__list_Oinduct,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [P: ( A > B ) > ( list @ A ) > $o,A0: A > B,A1: list @ A] :
( ! [F5: A > B,X: A] : ( P @ F5 @ ( cons @ A @ X @ ( nil @ A ) ) )
=> ( ! [F5: A > B,X: A,Y4: A,Zs2: list @ A] :
( ( P @ F5 @ ( cons @ A @ Y4 @ Zs2 ) )
=> ( P @ F5 @ ( cons @ A @ X @ ( cons @ A @ Y4 @ Zs2 ) ) ) )
=> ( ! [A6: A > B] : ( P @ A6 @ ( nil @ A ) )
=> ( P @ A0 @ A1 ) ) ) ) ) ).
% arg_min_list.induct
thf(fact_246_successively_Oinduct,axiom,
! [A: $tType,P: ( A > A > $o ) > ( list @ A ) > $o,A0: A > A > $o,A1: list @ A] :
( ! [P4: A > A > $o] : ( P @ P4 @ ( nil @ A ) )
=> ( ! [P4: A > A > $o,X: A] : ( P @ P4 @ ( cons @ A @ X @ ( nil @ A ) ) )
=> ( ! [P4: A > A > $o,X: A,Y4: A,Xs2: list @ A] :
( ( P @ P4 @ ( cons @ A @ Y4 @ Xs2 ) )
=> ( P @ P4 @ ( cons @ A @ X @ ( cons @ A @ Y4 @ Xs2 ) ) ) )
=> ( P @ A0 @ A1 ) ) ) ) ).
% successively.induct
thf(fact_247_list__nonempty__induct,axiom,
! [A: $tType,Xs: list @ A,P: ( list @ A ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X: A] : ( P @ ( cons @ A @ X @ ( nil @ A ) ) )
=> ( ! [X: A,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( P @ Xs2 )
=> ( P @ ( cons @ A @ X @ Xs2 ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_248_map__tailrec__rev_Oinduct,axiom,
! [A: $tType,B: $tType,P: ( A > B ) > ( list @ A ) > ( list @ B ) > $o,A0: A > B,A1: list @ A,A22: list @ B] :
( ! [F5: A > B,X_1: list @ B] : ( P @ F5 @ ( nil @ A ) @ X_1 )
=> ( ! [F5: A > B,A6: A,As: list @ A,Bs: list @ B] :
( ( P @ F5 @ As @ ( cons @ B @ ( F5 @ A6 ) @ Bs ) )
=> ( P @ F5 @ ( cons @ A @ A6 @ As ) @ Bs ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_249_strict__sorted_Ocases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: list @ A] :
( ( X3
!= ( nil @ A ) )
=> ~ ! [X: A,Ys2: list @ A] :
( X3
!= ( cons @ A @ X @ Ys2 ) ) ) ) ).
% strict_sorted.cases
thf(fact_250_strict__sorted_Oinduct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: ( list @ A ) > $o,A0: list @ A] :
( ( P @ ( nil @ A ) )
=> ( ! [X: A,Ys2: list @ A] :
( ( P @ Ys2 )
=> ( P @ ( cons @ A @ X @ Ys2 ) ) )
=> ( P @ A0 ) ) ) ) ).
% strict_sorted.induct
thf(fact_251_free__tv__list_Osimps_I1_J,axiom,
! [A: $tType] :
( ( type_M1470978233struct @ A )
=> ( ( type_M950795831ree_tv @ ( list @ A ) @ ( nil @ A ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ).
% free_tv_list.simps(1)
thf(fact_252_bound__tv__list_Osimps_I1_J,axiom,
! [A: $tType] :
( ( type_M1470978233struct @ A )
=> ( ( type_M779184887und_tv @ ( list @ A ) @ ( nil @ A ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ).
% bound_tv_list.simps(1)
thf(fact_253_lexordp__eq_Oinducts,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X1: list @ A,X2: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
( ( ord_lexordp_eq @ A @ X1 @ X2 )
=> ( ! [X_1: list @ A] : ( P @ ( nil @ A ) @ X_1 )
=> ( ! [X: A,Y4: A,Xs2: list @ A,Ys2: list @ A] :
( ( ord_less @ A @ X @ Y4 )
=> ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y4 @ Ys2 ) ) )
=> ( ! [X: A,Y4: A,Xs2: list @ A,Ys2: list @ A] :
( ~ ( ord_less @ A @ X @ Y4 )
=> ( ~ ( ord_less @ A @ Y4 @ X )
=> ( ( ord_lexordp_eq @ A @ Xs2 @ Ys2 )
=> ( ( P @ Xs2 @ Ys2 )
=> ( P @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y4 @ Ys2 ) ) ) ) ) )
=> ( P @ X1 @ X2 ) ) ) ) ) ) ).
% lexordp_eq.inducts
thf(fact_254_insert__Nil,axiom,
! [A: $tType,X3: A] :
( ( insert @ A @ X3 @ ( nil @ A ) )
= ( cons @ A @ X3 @ ( nil @ A ) ) ) ).
% insert_Nil
% Type constructors (40)
thf(tcon_HOL_Obool___Lattices_Obounded__lattice,axiom,
bounded_lattice @ $o ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice_1,axiom,
! [A7: $tType] : ( bounded_lattice @ ( set @ A7 ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice_2,axiom,
! [A7: $tType,A8: $tType] :
( ( bounded_lattice @ A8 )
=> ( bounded_lattice @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Type__Mirabelle__sfxnzmapfi_Otype__struct,axiom,
! [A7: $tType,A8: $tType] :
( ( ( type_M201404320of_nat @ A7 )
& ( type_M1535146424typ_of @ A8 ) )
=> ( type_M1470978233struct @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
! [A7: $tType,A8: $tType] :
( ( bounded_lattice @ A8 )
=> ( bounde1808546759up_bot @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A7: $tType,A8: $tType] :
( ( semilattice_sup @ A8 )
=> ( semilattice_sup @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__bot,axiom,
! [A7: $tType,A8: $tType] :
( ( order_bot @ A8 )
=> ( order_bot @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A7: $tType,A8: $tType] :
( ( preorder @ A8 )
=> ( preorder @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A7: $tType,A8: $tType] :
( ( order @ A8 )
=> ( order @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A7: $tType,A8: $tType] :
( ( ord @ A8 )
=> ( ord @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Obot,axiom,
! [A7: $tType,A8: $tType] :
( ( bot @ A8 )
=> ( bot @ ( A7 > A8 ) ) ) ).
thf(tcon_Nat_Onat___Type__Mirabelle__sfxnzmapfi_Oof__nat,axiom,
type_M201404320of_nat @ nat ).
thf(tcon_Nat_Onat___Lattices_Osemilattice__sup_3,axiom,
semilattice_sup @ nat ).
thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
wellorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder__bot_4,axiom,
order_bot @ nat ).
thf(tcon_Nat_Onat___Orderings_Opreorder_5,axiom,
preorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
linorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Ono__top,axiom,
no_top @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder_6,axiom,
order @ nat ).
thf(tcon_Nat_Onat___Orderings_Oord_7,axiom,
ord @ nat ).
thf(tcon_Nat_Onat___Orderings_Obot_8,axiom,
bot @ nat ).
thf(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_9,axiom,
! [A7: $tType] : ( bounde1808546759up_bot @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__sup_10,axiom,
! [A7: $tType] : ( semilattice_sup @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_11,axiom,
! [A7: $tType] : ( order_bot @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_12,axiom,
! [A7: $tType] : ( preorder @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_13,axiom,
! [A7: $tType] : ( order @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_14,axiom,
! [A7: $tType] : ( ord @ ( set @ A7 ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_15,axiom,
! [A7: $tType] : ( bot @ ( set @ A7 ) ) ).
thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_16,axiom,
bounde1808546759up_bot @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_17,axiom,
semilattice_sup @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_18,axiom,
order_bot @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_19,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder_20,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_21,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_22,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Orderings_Obot_23,axiom,
bot @ $o ).
thf(tcon_List_Olist___Type__Mirabelle__sfxnzmapfi_Otype__struct_24,axiom,
! [A7: $tType] :
( ( type_M1470978233struct @ A7 )
=> ( type_M1470978233struct @ ( list @ A7 ) ) ) ).
thf(tcon_Type__Mirabelle__sfxnzmapfi_Otyp___Type__Mirabelle__sfxnzmapfi_Otype__struct_25,axiom,
type_M1470978233struct @ type_Mirabelle_typ ).
thf(tcon_Type__Mirabelle__sfxnzmapfi_Otyp___Type__Mirabelle__sfxnzmapfi_Otyp__of,axiom,
type_M1535146424typ_of @ type_Mirabelle_typ ).
thf(tcon_Type__Mirabelle__sfxnzmapfi_Otype__scheme___Type__Mirabelle__sfxnzmapfi_Otype__struct_26,axiom,
type_M1470978233struct @ type_M1610475166scheme ).
% 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,X3: A,Y: A] :
( ( if @ A @ $false @ X3 @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X3: A,Y: A] :
( ( if @ A @ $true @ X3 @ Y )
= X3 ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( ( type_M353778473_subst @ type_M1610475166scheme @ s1 @ ( type_M91906176e_FVar @ x ) )
!= ( type_M353778473_subst @ type_M1610475166scheme @ s2 @ ( type_M91906176e_FVar @ x ) ) )
| ~ ( member @ nat @ n @ ( type_M950795831ree_tv @ type_M1610475166scheme @ ( type_M91906176e_FVar @ x ) ) )
| ( ( s1 @ n )
= ( s2 @ n ) ) ) ).
%------------------------------------------------------------------------------